LMFDB - Embedded newform 75.16.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,16,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$107.020128825$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 140297x^{4} - 1279200x^{3} + 3920349703x^{2} - 70310137200x - 19672158033999 $ x^6 - 140297x^4 - 1279200x^3 + 3920349703x^2 - 70310137200x - 19672158033999
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4}\cdot 5^{7} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$117.249$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-78.2494 q^{2} -2187.00 q^{3} -26645.0 q^{4} +171131. q^{6} -3.46184e6 q^{7} +4.64903e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$	\(q-78.2494 q^{2} -2187.00 q^{3} -26645.0 q^{4} +171131. q^{6} -3.46184e6 q^{7} +4.64903e6 q^{8} +4.78297e6 q^{9} +1.01045e8 q^{11} +5.82727e7 q^{12} +2.20454e8 q^{13} +2.70887e8 q^{14} +5.09321e8 q^{16} -2.62046e9 q^{17} -3.74264e8 q^{18} -6.27209e9 q^{19} +7.57105e9 q^{21} -7.90671e9 q^{22} -1.11856e10 q^{23} -1.01674e10 q^{24} -1.72504e10 q^{26} -1.04604e10 q^{27} +9.22410e10 q^{28} +7.36780e10 q^{29} -2.48692e11 q^{31} -1.92194e11 q^{32} -2.20985e11 q^{33} +2.05050e11 q^{34} -1.27442e11 q^{36} +6.82738e11 q^{37} +4.90787e11 q^{38} -4.82133e11 q^{39} -1.24675e12 q^{41} -5.92430e11 q^{42} -8.68448e11 q^{43} -2.69235e12 q^{44} +8.75268e11 q^{46} -6.07787e12 q^{47} -1.11388e12 q^{48} +7.23681e12 q^{49} +5.73095e12 q^{51} -5.87400e12 q^{52} -9.46247e12 q^{53} +8.18516e11 q^{54} -1.60942e13 q^{56} +1.37171e13 q^{57} -5.76526e12 q^{58} -7.03327e11 q^{59} +3.58228e13 q^{61} +1.94600e13 q^{62} -1.65579e13 q^{63} -1.65039e12 q^{64} +1.72920e13 q^{66} -3.16496e13 q^{67} +6.98223e13 q^{68} +2.44630e13 q^{69} -4.79440e13 q^{71} +2.22362e13 q^{72} -2.59677e13 q^{73} -5.34239e13 q^{74} +1.67120e14 q^{76} -3.49802e14 q^{77} +3.77266e13 q^{78} -1.86784e14 q^{79} +2.28768e13 q^{81} +9.75575e13 q^{82} -1.24846e14 q^{83} -2.01731e14 q^{84} +6.79555e13 q^{86} -1.61134e14 q^{87} +4.69762e14 q^{88} +5.32647e14 q^{89} -7.63177e14 q^{91} +2.98041e14 q^{92} +5.43889e14 q^{93} +4.75590e14 q^{94} +4.20327e14 q^{96} -1.01011e14 q^{97} -5.66276e14 q^{98} +4.83295e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9}+O(q^{10}) $ 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9} + 107489124 q^{11} - 203635944 q^{12} - 109881686 q^{13} - 563984442 q^{14} + 3622829560 q^{16} + 3573042876 q^{17} + 1119214746 q^{18} - 1602340942 q^{19} + 5664815514 q^{21} + 4024661012 q^{22} - 6555818844 q^{23} - 30645092556 q^{24} - 25715894778 q^{26} - 62762119218 q^{27} - 270752117896 q^{28} + 126894468996 q^{29} + 151760841646 q^{31} + 385411085208 q^{32} - 235078714188 q^{33} + 1431919606684 q^{34} + 445351809528 q^{36} + 616109002068 q^{37} - 2822785016634 q^{38} + 240311247282 q^{39} + 1091281712616 q^{41} + 1233433974654 q^{42} - 2444971199030 q^{43} + 1413344578176 q^{44} - 5480862370044 q^{46} - 8369143269660 q^{47} - 7923128247720 q^{48} + 19523846053580 q^{49} - 7814244769812 q^{51} - 10261294060344 q^{52} - 16571417665824 q^{53} - 2447722649502 q^{54} - 75252275829540 q^{56} + 3504319640154 q^{57} - 3994751501708 q^{58} + 8796604455252 q^{59} - 6959665405750 q^{61} + 52277129313066 q^{62} - 12388951529118 q^{63} + 50304241850208 q^{64} - 8801933633244 q^{66} - 53487461742094 q^{67} + 307147088145312 q^{68} + 14337575811828 q^{69} + 104634162717912 q^{71} + 67020817419972 q^{72} + 177000981923236 q^{73} - 45005277967812 q^{74} + 76188538526328 q^{76} + 117850730172876 q^{77} + 56240661879486 q^{78} + 185514024366160 q^{79} + 137260754729766 q^{81} + 654376907588896 q^{82} + 435827733256908 q^{83} + 592134881838552 q^{84} + 15\!\cdots\!14 q^{86}+ \cdots + 514117147929156 q^{99}+O(q^{100}) \) 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9 + 107489124 * q^11 - 203635944 * q^12 - 109881686 * q^13 - 563984442 * q^14 + 3622829560 * q^16 + 3573042876 * q^17 + 1119214746 * q^18 - 1602340942 * q^19 + 5664815514 * q^21 + 4024661012 * q^22 - 6555818844 * q^23 - 30645092556 * q^24 - 25715894778 * q^26 - 62762119218 * q^27 - 270752117896 * q^28 + 126894468996 * q^29 + 151760841646 * q^31 + 385411085208 * q^32 - 235078714188 * q^33 + 1431919606684 * q^34 + 445351809528 * q^36 + 616109002068 * q^37 - 2822785016634 * q^38 + 240311247282 * q^39 + 1091281712616 * q^41 + 1233433974654 * q^42 - 2444971199030 * q^43 + 1413344578176 * q^44 - 5480862370044 * q^46 - 8369143269660 * q^47 - 7923128247720 * q^48 + 19523846053580 * q^49 - 7814244769812 * q^51 - 10261294060344 * q^52 - 16571417665824 * q^53 - 2447722649502 * q^54 - 75252275829540 * q^56 + 3504319640154 * q^57 - 3994751501708 * q^58 + 8796604455252 * q^59 - 6959665405750 * q^61 + 52277129313066 * q^62 - 12388951529118 * q^63 + 50304241850208 * q^64 - 8801933633244 * q^66 - 53487461742094 * q^67 + 307147088145312 * q^68 + 14337575811828 * q^69 + 104634162717912 * q^71 + 67020817419972 * q^72 + 177000981923236 * q^73 - 45005277967812 * q^74 + 76188538526328 * q^76 + 117850730172876 * q^77 + 56240661879486 * q^78 + 185514024366160 * q^79 + 137260754729766 * q^81 + 654376907588896 * q^82 + 435827733256908 * q^83 + 592134881838552 * q^84 + 1570057453210014 * q^86 - 277518203694252 * q^87 + 174719041441128 * q^88 + 431517021638688 * q^89 + 466827529337246 * q^91 + 50075827036128 * q^92 - 331900960679802 * q^93 + 1404749684955772 * q^94 - 842894043349896 * q^96 - 1013615616184854 * q^97 + 5117833640095236 * q^98 + 514117147929156 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−78.2494	−0.432271	−0.216135	−	0.976363i	$-0.569345\pi$
$2$	−78.2494	−0.432271	−0.216135	+	0.976363i	$0.569345\pi$
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	−26645.0	−0.813142
$5$	0	0
$5$	0	0
$6$	171131.	0.249572
$7$	−3.46184e6	−1.58881	−0.794406	−	0.607388i	$-0.792218\pi$
$7$	−3.46184e6	−1.58881	−0.794406	+	0.607388i	$0.792218\pi$
$8$	4.64903e6	0.783768
$9$	4.78297e6	0.333333
$10$	0	0
$11$	1.01045e8	1.56340	0.781699	−	0.623656i	$-0.214353\pi$
$11$	1.01045e8	1.56340	0.781699	+	0.623656i	$0.214353\pi$
$12$	5.82727e7	0.469468
$13$	2.20454e8	0.974412	0.487206	−	0.873287i	$-0.338016\pi$
$13$	2.20454e8	0.974412	0.487206	+	0.873287i	$0.338016\pi$
$14$	2.70887e8	0.686797
$15$	0	0
$16$	5.09321e8	0.474342
$17$	−2.62046e9	−1.54886	−0.774428	−	0.632662i	$-0.781962\pi$
$17$	−2.62046e9	−1.54886	−0.774428	+	0.632662i	$0.781962\pi$
$18$	−3.74264e8	−0.144090
$19$	−6.27209e9	−1.60976	−0.804878	−	0.593441i	$-0.797769\pi$
$19$	−6.27209e9	−1.60976	−0.804878	+	0.593441i	$0.797769\pi$
$20$	0	0
$21$	7.57105e9	0.917301
$22$	−7.90671e9	−0.675812
$23$	−1.11856e10	−0.685017	−0.342509	−	0.939515i	$-0.611277\pi$
$23$	−1.11856e10	−0.685017	−0.342509	+	0.939515i	$0.611277\pi$
$24$	−1.01674e10	−0.452509
$25$	0	0
$26$	−1.72504e10	−0.421210
$27$	−1.04604e10	−0.192450
$28$	9.22410e10	1.29193
$29$	7.36780e10	0.793146	0.396573	−	0.918003i	$-0.370199\pi$
$29$	7.36780e10	0.793146	0.396573	+	0.918003i	$0.370199\pi$
$30$	0	0
$31$	−2.48692e11	−1.62349	−0.811744	−	0.584013i	$-0.801482\pi$
$31$	−2.48692e11	−1.62349	−0.811744	+	0.584013i	$0.801482\pi$
$32$	−1.92194e11	−0.988812
$33$	−2.20985e11	−0.902629
$34$	2.05050e11	0.669525
$35$	0	0
$36$	−1.27442e11	−0.271047
$37$	6.82738e11	1.18234	0.591169	−	0.806548i	$-0.298667\pi$
$37$	6.82738e11	1.18234	0.591169	+	0.806548i	$0.298667\pi$
$38$	4.90787e11	0.695850
$39$	−4.82133e11	−0.562577
$40$	0	0
$41$	−1.24675e12	−0.999772	−0.499886	−	0.866091i	$-0.666625\pi$
$41$	−1.24675e12	−0.999772	−0.499886	+	0.866091i	$0.666625\pi$
$42$	−5.92430e11	−0.396522
$43$	−8.68448e11	−0.487226	−0.243613	−	0.969873i	$-0.578333\pi$
$43$	−8.68448e11	−0.487226	−0.243613	+	0.969873i	$0.578333\pi$
$44$	−2.69235e12	−1.27126
$45$	0	0
$46$	8.75268e11	0.296113
$47$	−6.07787e12	−1.74992	−0.874959	−	0.484198i	$-0.839112\pi$
$47$	−6.07787e12	−1.74992	−0.874959	+	0.484198i	$0.839112\pi$
$48$	−1.11388e12	−0.273861
$49$	7.23681e12	1.52432
$50$	0	0
$51$	5.73095e12	0.894232
$52$	−5.87400e12	−0.792336
$53$	−9.46247e12	−1.10646	−0.553229	−	0.833029i	$-0.686605\pi$
$53$	−9.46247e12	−1.10646	−0.553229	+	0.833029i	$0.686605\pi$
$54$	8.18516e11	0.0831906
$55$	0	0
$56$	−1.60942e13	−1.24526
$57$	1.37171e13	0.929393
$58$	−5.76526e12	−0.342854
$59$	−7.03327e11	−0.0367932	−0.0183966	−	0.999831i	$-0.505856\pi$
$59$	−7.03327e11	−0.0367932	−0.0183966	+	0.999831i	$0.505856\pi$
$60$	0	0
$61$	3.58228e13	1.45944	0.729718	−	0.683748i	$-0.239651\pi$
$61$	3.58228e13	1.45944	0.729718	+	0.683748i	$0.239651\pi$
$62$	1.94600e13	0.701787
$63$	−1.65579e13	−0.529604
$64$	−1.65039e12	−0.0469069
$65$	0	0
$66$	1.72920e13	0.390180
$67$	−3.16496e13	−0.637980	−0.318990	−	0.947758i	$-0.603344\pi$
$67$	−3.16496e13	−0.637980	−0.318990	+	0.947758i	$0.603344\pi$
$68$	6.98223e13	1.25944
$69$	2.44630e13	0.395495
$70$	0	0
$71$	−4.79440e13	−0.625600	−0.312800	−	0.949819i	$-0.601267\pi$
$71$	−4.79440e13	−0.625600	−0.312800	+	0.949819i	$0.601267\pi$
$72$	2.22362e13	0.261256
$73$	−2.59677e13	−0.275113	−0.137557	−	0.990494i	$-0.543925\pi$
$73$	−2.59677e13	−0.275113	−0.137557	+	0.990494i	$0.543925\pi$
$74$	−5.34239e13	−0.511090
$75$	0	0
$76$	1.67120e14	1.30896
$77$	−3.49802e14	−2.48395
$78$	3.77266e13	0.243186
$79$	−1.86784e14	−1.09430	−0.547150	−	0.837035i	$-0.684287\pi$
$79$	−1.86784e14	−1.09430	−0.547150	+	0.837035i	$0.684287\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	9.75575e13	0.432172
$83$	−1.24846e14	−0.504998	−0.252499	−	0.967597i	$-0.581252\pi$
$83$	−1.24846e14	−0.504998	−0.252499	+	0.967597i	$0.581252\pi$
$84$	−2.01731e14	−0.745896
$85$	0	0
$86$	6.79555e13	0.210613
$87$	−1.61134e14	−0.457923
$88$	4.69762e14	1.22534
$89$	5.32647e14	1.27648	0.638241	−	0.769837i	$-0.279662\pi$
$89$	5.32647e14	1.27648	0.638241	+	0.769837i	$0.279662\pi$
$90$	0	0
$91$	−7.63177e14	−1.54816
$92$	2.98041e14	0.557016
$93$	5.43889e14	0.937321
$94$	4.75590e14	0.756438
$95$	0	0
$96$	4.20327e14	0.570891
$97$	−1.01011e14	−0.126935	−0.0634676	−	0.997984i	$-0.520216\pi$
$97$	−1.01011e14	−0.126935	−0.0634676	+	0.997984i	$0.520216\pi$
$98$	−5.66276e14	−0.658920
$99$	4.83295e14	0.521133
$100$	0	0
$101$	1.31421e15	1.21970	0.609851	−	0.792516i	$-0.291229\pi$
$101$	1.31421e15	1.21970	0.609851	+	0.792516i	$0.291229\pi$
$102$	−4.48443e14	−0.386550
$103$	7.28966e14	0.584021	0.292010	−	0.956415i	$-0.405676\pi$
$103$	7.28966e14	0.584021	0.292010	+	0.956415i	$0.405676\pi$
$104$	1.02490e15	0.763714
$105$	0	0
$106$	7.40432e14	0.478290
$107$	1.54393e14	0.0929499	0.0464750	−	0.998919i	$-0.485201\pi$
$107$	1.54393e14	0.0929499	0.0464750	+	0.998919i	$0.485201\pi$
$108$	2.78716e14	0.156489
$109$	1.96659e14	0.103042	0.0515210	−	0.998672i	$-0.483593\pi$
$109$	1.96659e14	0.103042	0.0515210	+	0.998672i	$0.483593\pi$
$110$	0	0
$111$	−1.49315e15	−0.682623
$112$	−1.76319e15	−0.753639
$113$	−3.20161e15	−1.28021	−0.640104	−	0.768288i	$-0.721109\pi$
$113$	−3.20161e15	−1.28021	−0.640104	+	0.768288i	$0.721109\pi$
$114$	−1.07335e15	−0.401749
$115$	0	0
$116$	−1.96315e15	−0.644940
$117$	1.05442e15	0.324804
$118$	5.50349e13	0.0159046
$119$	9.07163e15	2.46084
$120$	0	0
$121$	6.03284e15	1.44421
$122$	−2.80311e15	−0.630872
$123$	2.72664e15	0.577219
$124$	6.62641e15	1.32013
$125$	0	0
$126$	1.29565e15	0.228932
$127$	−4.46130e15	−0.742905	−0.371452	−	0.928452i	$-0.621140\pi$
$127$	−4.46130e15	−0.742905	−0.371452	+	0.928452i	$0.621140\pi$
$128$	6.42694e15	1.00909
$129$	1.89929e15	0.281300
$130$	0	0
$131$	−3.55038e15	−0.468533	−0.234267	−	0.972172i	$-0.575269\pi$
$131$	−3.55038e15	−0.468533	−0.234267	+	0.972172i	$0.575269\pi$
$132$	5.88816e15	0.733965
$133$	2.17130e16	2.55760
$134$	2.47656e15	0.275780
$135$	0	0
$136$	−1.21826e16	−1.21394
$137$	−7.46806e15	−0.704374	−0.352187	−	0.935930i	$-0.614562\pi$
$137$	−7.46806e15	−0.704374	−0.352187	+	0.935930i	$0.614562\pi$
$138$	−1.91421e15	−0.170961
$139$	8.98826e15	0.760440	0.380220	−	0.924896i	$-0.375848\pi$
$139$	8.98826e15	0.760440	0.380220	+	0.924896i	$0.375848\pi$
$140$	0	0
$141$	1.32923e16	1.01032
$142$	3.75159e15	0.270428
$143$	2.22758e16	1.52339
$144$	2.43606e15	0.158114
$145$	0	0
$146$	2.03195e15	0.118923
$147$	−1.58269e16	−0.880067
$148$	−1.81916e16	−0.961408
$149$	−3.49266e15	−0.175493	−0.0877463	−	0.996143i	$-0.527966\pi$
$149$	−3.49266e15	−0.175493	−0.0877463	+	0.996143i	$0.527966\pi$
$150$	0	0
$151$	2.80364e16	1.27466	0.637332	−	0.770590i	$-0.280038\pi$
$151$	2.80364e16	1.27466	0.637332	+	0.770590i	$0.280038\pi$
$152$	−2.91591e16	−1.26168
$153$	−1.25336e16	−0.516285
$154$	2.73718e16	1.07374
$155$	0	0
$156$	1.28464e16	0.457455
$157$	−3.50997e16	−1.19140	−0.595698	−	0.803209i	$-0.703124\pi$
$157$	−3.50997e16	−1.19140	−0.595698	+	0.803209i	$0.703124\pi$
$158$	1.46157e16	0.473034
$159$	2.06944e16	0.638814
$160$	0	0
$161$	3.87229e16	1.08836
$162$	−1.79009e15	−0.0480301
$163$	3.95945e16	1.01444	0.507222	−	0.861816i	$-0.330673\pi$
$163$	3.95945e16	1.01444	0.507222	+	0.861816i	$0.330673\pi$
$164$	3.32197e16	0.812956
$165$	0	0
$166$	9.76914e15	0.218296
$167$	4.45679e15	0.0952026	0.0476013	−	0.998866i	$-0.484842\pi$
$167$	4.45679e15	0.0952026	0.0476013	+	0.998866i	$0.484842\pi$
$168$	3.51981e16	0.718951
$169$	−2.58594e15	−0.0505205
$170$	0	0
$171$	−2.99992e16	−0.536585
$172$	2.31398e16	0.396184
$173$	−3.69090e16	−0.605043	−0.302522	−	0.953143i	$-0.597828\pi$
$173$	−3.69090e16	−0.605043	−0.302522	+	0.953143i	$0.597828\pi$
$174$	1.26086e16	0.197947
$175$	0	0
$176$	5.14643e16	0.741585
$177$	1.53818e15	0.0212425
$178$	−4.16793e16	−0.551786
$179$	9.49228e16	1.20496	0.602480	−	0.798134i	$-0.294179\pi$
$179$	9.49228e16	1.20496	0.602480	+	0.798134i	$0.294179\pi$
$180$	0	0
$181$	−8.32066e16	−0.971781	−0.485890	−	0.874020i	$-0.661505\pi$
$181$	−8.32066e16	−0.971781	−0.485890	+	0.874020i	$0.661505\pi$
$182$	5.97182e16	0.669223
$183$	−7.83444e16	−0.842606
$184$	−5.20023e16	−0.536895
$185$	0	0
$186$	−4.25590e16	−0.405177
$187$	−2.64785e17	−2.42148
$188$	1.61945e17	1.42293
$189$	3.62121e16	0.305767
$190$	0	0
$191$	1.49268e17	1.16471	0.582354	−	0.812935i	$-0.302132\pi$
$191$	1.49268e17	1.16471	0.582354	+	0.812935i	$0.302132\pi$
$192$	3.60940e15	0.0270817
$193$	2.20053e16	0.158799	0.0793995	−	0.996843i	$-0.474700\pi$
$193$	2.20053e16	0.158799	0.0793995	+	0.996843i	$0.474700\pi$
$194$	7.90408e15	0.0548704
$195$	0	0
$196$	−1.92825e17	−1.23949
$197$	6.40931e16	0.396565	0.198283	−	0.980145i	$-0.436464\pi$
$197$	6.40931e16	0.396565	0.198283	+	0.980145i	$0.436464\pi$
$198$	−3.78175e16	−0.225271
$199$	8.47225e16	0.485960	0.242980	−	0.970031i	$-0.421875\pi$
$199$	8.47225e16	0.485960	0.242980	+	0.970031i	$0.421875\pi$
$200$	0	0
$201$	6.92176e16	0.368338
$202$	−1.02836e17	−0.527242
$203$	−2.55062e17	−1.26016
$204$	−1.52701e17	−0.727138
$205$	0	0
$206$	−5.70411e16	−0.252455
$207$	−5.35005e16	−0.228339
$208$	1.12282e17	0.462204
$209$	−6.33763e17	−2.51669
$210$	0	0
$211$	−6.22050e16	−0.229989	−0.114994	−	0.993366i	$-0.536685\pi$
$211$	−6.22050e16	−0.229989	−0.114994	+	0.993366i	$0.536685\pi$
$212$	2.52128e17	0.899708
$213$	1.04853e17	0.361190
$214$	−1.20811e16	−0.0401795
$215$	0	0
$216$	−4.86305e16	−0.150836
$217$	8.60933e17	2.57942
$218$	−1.53884e16	−0.0445421
$219$	5.67913e16	0.158837
$220$	0	0
$221$	−5.77691e17	−1.50922
$222$	1.16838e17	0.295078
$223$	2.79504e17	0.682498	0.341249	−	0.939973i	$-0.389150\pi$
$223$	2.79504e17	0.682498	0.341249	+	0.939973i	$0.389150\pi$
$224$	6.65344e17	1.57104
$225$	0	0
$226$	2.50524e17	0.553397
$227$	2.28476e17	0.488254	0.244127	−	0.969743i	$-0.421499\pi$
$227$	2.28476e17	0.488254	0.244127	+	0.969743i	$0.421499\pi$
$228$	−3.65491e17	−0.755728
$229$	−5.55617e17	−1.11176	−0.555878	−	0.831264i	$-0.687618\pi$
$229$	−5.55617e17	−1.11176	−0.555878	+	0.831264i	$0.687618\pi$
$230$	0	0
$231$	7.65017e17	1.43411
$232$	3.42532e17	0.621642
$233$	1.05598e17	0.185560	0.0927800	−	0.995687i	$-0.470425\pi$
$233$	1.05598e17	0.185560	0.0927800	+	0.995687i	$0.470425\pi$
$234$	−8.25081e16	−0.140403
$235$	0	0
$236$	1.87402e16	0.0299181
$237$	4.08496e17	0.631794
$238$	−7.09850e17	−1.06375
$239$	−8.29802e17	−1.20501	−0.602505	−	0.798115i	$-0.705831\pi$
$239$	−8.29802e17	−1.20501	−0.602505	+	0.798115i	$0.705831\pi$
$240$	0	0
$241$	−4.80037e17	−0.654857	−0.327429	−	0.944876i	$-0.606182\pi$
$241$	−4.80037e17	−0.654857	−0.327429	+	0.944876i	$0.606182\pi$
$242$	−4.72066e17	−0.624292
$243$	−5.00315e16	−0.0641500
$244$	−9.54499e17	−1.18673
$245$	0	0
$246$	−2.13358e17	−0.249515
$247$	−1.38271e18	−1.56857
$248$	−1.15618e18	−1.27244
$249$	2.73039e17	0.291561
$250$	0	0
$251$	−4.54180e17	−0.456747	−0.228373	−	0.973574i	$-0.573341\pi$
$251$	−4.54180e17	−0.456747	−0.228373	+	0.973574i	$0.573341\pi$
$252$	4.41186e17	0.430643
$253$	−1.13025e18	−1.07095
$254$	3.49094e17	0.321136
$255$	0	0
$256$	−4.48824e17	−0.389293
$257$	1.88871e18	1.59099	0.795495	−	0.605961i	$-0.207211\pi$
$257$	1.88871e18	1.59099	0.795495	+	0.605961i	$0.207211\pi$
$258$	−1.48619e17	−0.121598
$259$	−2.36353e18	−1.87851
$260$	0	0
$261$	3.52400e17	0.264382
$262$	2.77815e17	0.202533
$263$	−2.56182e17	−0.181502	−0.0907508	−	0.995874i	$-0.528927\pi$
$263$	−2.56182e17	−0.181502	−0.0907508	+	0.995874i	$0.528927\pi$
$264$	−1.02737e18	−0.707452
$265$	0	0
$266$	−1.69903e18	−1.10557
$267$	−1.16490e18	−0.736977
$268$	8.43304e17	0.518768
$269$	4.63238e17	0.277116	0.138558	−	0.990354i	$-0.455753\pi$
$269$	4.63238e17	0.277116	0.138558	+	0.990354i	$0.455753\pi$
$270$	0	0
$271$	2.90705e17	0.164507	0.0822533	−	0.996611i	$-0.473788\pi$
$271$	2.90705e17	0.164507	0.0822533	+	0.996611i	$0.473788\pi$
$272$	−1.33465e18	−0.734687
$273$	1.66907e18	0.893829
$274$	5.84371e17	0.304481
$275$	0	0
$276$	−6.51816e17	−0.321593
$277$	−6.52563e17	−0.313346	−0.156673	−	0.987651i	$-0.550077\pi$
$277$	−6.52563e17	−0.313346	−0.156673	+	0.987651i	$0.550077\pi$
$278$	−7.03326e17	−0.328716
$279$	−1.18949e18	−0.541163
$280$	0	0
$281$	1.37831e18	0.594358	0.297179	−	0.954822i	$-0.403954\pi$
$281$	1.37831e18	0.594358	0.297179	+	0.954822i	$0.403954\pi$
$282$	−1.04011e18	−0.436730
$283$	1.36759e18	0.559188	0.279594	−	0.960118i	$-0.409800\pi$
$283$	1.36759e18	0.559188	0.279594	+	0.960118i	$0.409800\pi$
$284$	1.27747e18	0.508701
$285$	0	0
$286$	−1.74307e18	−0.658519
$287$	4.31606e18	1.58845
$288$	−9.19256e17	−0.329604
$289$	4.00440e18	1.39895
$290$	0	0
$291$	2.20912e17	0.0732861
$292$	6.91909e17	0.223706
$293$	−4.05606e18	−1.27820	−0.639098	−	0.769126i	$-0.720692\pi$
$293$	−4.05606e18	−1.27820	−0.639098	+	0.769126i	$0.720692\pi$
$294$	1.23845e18	0.380427
$295$	0	0
$296$	3.17407e18	0.926679
$297$	−1.05697e18	−0.300876
$298$	2.73298e17	0.0758604
$299$	−2.46591e18	−0.667489
$300$	0	0
$301$	3.00643e18	0.774110
$302$	−2.19383e18	−0.551000
$303$	−2.87417e18	−0.704196
$304$	−3.19450e18	−0.763574
$305$	0	0
$306$	9.80745e17	0.223175
$307$	−5.38152e18	−1.19500	−0.597498	−	0.801870i	$-0.703839\pi$
$307$	−5.38152e18	−1.19500	−0.597498	+	0.801870i	$0.703839\pi$
$308$	9.32049e18	2.01980
$309$	−1.59425e18	−0.337184
$310$	0	0
$311$	1.64819e18	0.332128	0.166064	−	0.986115i	$-0.446894\pi$
$311$	1.64819e18	0.332128	0.166064	+	0.986115i	$0.446894\pi$
$312$	−2.24145e18	−0.440930
$313$	2.80252e18	0.538227	0.269114	−	0.963108i	$-0.413269\pi$
$313$	2.80252e18	0.538227	0.269114	+	0.963108i	$0.413269\pi$
$314$	2.74653e18	0.515006
$315$	0	0
$316$	4.97686e18	0.889820
$317$	6.48032e18	1.13149	0.565747	−	0.824579i	$-0.308588\pi$
$317$	6.48032e18	1.13149	0.565747	+	0.824579i	$0.308588\pi$
$318$	−1.61932e18	−0.276141
$319$	7.44479e18	1.24000
$320$	0	0
$321$	−3.37657e17	−0.0536647
$322$	−3.03004e18	−0.470467
$323$	1.64358e19	2.49328
$324$	−6.09553e17	−0.0903491
$325$	0	0
$326$	−3.09824e18	−0.438514
$327$	−4.30093e17	−0.0594913
$328$	−5.79619e18	−0.783590
$329$	2.10406e19	2.78029
$330$	0	0
$331$	3.37435e18	0.426069	0.213034	−	0.977045i	$-0.431665\pi$
$331$	3.37435e18	0.426069	0.213034	+	0.977045i	$0.431665\pi$
$332$	3.32653e18	0.410635
$333$	3.26552e18	0.394112
$334$	−3.48741e17	−0.0411533
$335$	0	0
$336$	3.85609e18	0.435114
$337$	7.53974e18	0.832017	0.416008	−	0.909361i	$-0.363429\pi$
$337$	7.53974e18	0.832017	0.416008	+	0.909361i	$0.363429\pi$
$338$	2.02348e17	0.0218385
$339$	7.00193e18	0.739129
$340$	0	0
$341$	−2.51291e19	−2.53816
$342$	2.34742e18	0.231950
$343$	−8.61739e18	−0.833047
$344$	−4.03744e18	−0.381872
$345$	0	0
$346$	2.88811e18	0.261543
$347$	−8.51134e18	−0.754270	−0.377135	−	0.926158i	$-0.623091\pi$
$347$	−8.51134e18	−0.754270	−0.377135	+	0.926158i	$0.623091\pi$
$348$	4.29342e18	0.372356
$349$	3.80730e18	0.323167	0.161583	−	0.986859i	$-0.448340\pi$
$349$	3.80730e18	0.323167	0.161583	+	0.986859i	$0.448340\pi$
$350$	0	0
$351$	−2.30603e18	−0.187526
$352$	−1.94202e19	−1.54591
$353$	2.08662e19	1.62605	0.813024	−	0.582230i	$-0.197820\pi$
$353$	2.08662e19	1.62605	0.813024	+	0.582230i	$0.197820\pi$
$354$	−1.20361e17	−0.00918253
$355$	0	0
$356$	−1.41924e19	−1.03796
$357$	−1.98397e19	−1.42077
$358$	−7.42765e18	−0.520869
$359$	−2.66034e18	−0.182696	−0.0913481	−	0.995819i	$-0.529118\pi$
$359$	−2.66034e18	−0.182696	−0.0913481	+	0.995819i	$0.529118\pi$
$360$	0	0
$361$	2.41579e19	1.59131
$362$	6.51086e18	0.420073
$363$	−1.31938e19	−0.833818
$364$	2.03349e19	1.25887
$365$	0	0
$366$	6.13040e18	0.364234
$367$	−2.52954e18	−0.147247	−0.0736235	−	0.997286i	$-0.523456\pi$
$367$	−2.52954e18	−0.147247	−0.0736235	+	0.997286i	$0.523456\pi$
$368$	−5.69707e18	−0.324932
$369$	−5.96317e18	−0.333257
$370$	0	0
$371$	3.27576e19	1.75795
$372$	−1.44920e19	−0.762175
$373$	2.35095e19	1.21179	0.605896	−	0.795544i	$-0.292815\pi$
$373$	2.35095e19	1.21179	0.605896	+	0.795544i	$0.292815\pi$
$374$	2.07192e19	1.04673
$375$	0	0
$376$	−2.82562e19	−1.37153
$377$	1.62426e19	0.772851
$378$	−2.83358e18	−0.132174
$379$	−3.41424e19	−1.56135	−0.780674	−	0.624938i	$-0.785124\pi$
$379$	−3.41424e19	−1.56135	−0.780674	+	0.624938i	$0.785124\pi$
$380$	0	0
$381$	9.75685e18	0.428916
$382$	−1.16801e19	−0.503469
$383$	−2.57649e19	−1.08902	−0.544512	−	0.838753i	$-0.683285\pi$
$383$	−2.57649e19	−1.08902	−0.544512	+	0.838753i	$0.683285\pi$
$384$	−1.40557e19	−0.582598
$385$	0	0
$386$	−1.72190e18	−0.0686442
$387$	−4.15376e18	−0.162409
$388$	2.69145e18	0.103216
$389$	3.29387e19	1.23904	0.619518	−	0.784982i	$-0.287328\pi$
$389$	3.29387e19	1.23904	0.619518	+	0.784982i	$0.287328\pi$
$390$	0	0
$391$	2.93115e19	1.06099
$392$	3.36442e19	1.19471
$393$	7.76468e18	0.270508
$394$	−5.01524e18	−0.171424
$395$	0	0
$396$	−1.28774e19	−0.423755
$397$	3.36948e19	1.08801	0.544006	−	0.839081i	$-0.316907\pi$
$397$	3.36948e19	1.08801	0.544006	+	0.839081i	$0.316907\pi$
$398$	−6.62949e18	−0.210067
$399$	−4.74863e19	−1.47663
$400$	0	0
$401$	−3.83761e19	−1.14942	−0.574709	−	0.818358i	$-0.694885\pi$
$401$	−3.83761e19	−1.14942	−0.574709	+	0.818358i	$0.694885\pi$
$402$	−5.41623e18	−0.159222
$403$	−5.48251e19	−1.58195
$404$	−3.50171e19	−0.991792
$405$	0	0
$406$	1.99584e19	0.544730
$407$	6.89873e19	1.84846
$408$	2.66434e19	0.700871
$409$	−2.73785e19	−0.707108	−0.353554	−	0.935414i	$-0.615027\pi$
$409$	−2.73785e19	−0.707108	−0.353554	+	0.935414i	$0.615027\pi$
$410$	0	0
$411$	1.63326e19	0.406671
$412$	−1.94233e19	−0.474892
$413$	2.43481e18	0.0584574
$414$	4.18638e18	0.0987043
$415$	0	0
$416$	−4.23698e19	−0.963511
$417$	−1.96573e19	−0.439040
$418$	4.95915e19	1.08789
$419$	7.87526e19	1.69691	0.848457	−	0.529265i	$-0.177532\pi$
$419$	7.87526e19	1.69691	0.848457	+	0.529265i	$0.177532\pi$
$420$	0	0
$421$	4.53752e19	0.943415	0.471708	−	0.881755i	$-0.343638\pi$
$421$	4.53752e19	0.943415	0.471708	+	0.881755i	$0.343638\pi$
$422$	4.86750e18	0.0994174
$423$	−2.90703e19	−0.583306
$424$	−4.39913e19	−0.867207
$425$	0	0
$426$	−8.20472e18	−0.156132
$427$	−1.24013e20	−2.31877
$428$	−4.11380e18	−0.0755815
$429$	−4.87171e19	−0.879532
$430$	0	0
$431$	−7.34166e19	−1.28001	−0.640007	−	0.768369i	$-0.721069\pi$
$431$	−7.34166e19	−1.28001	−0.640007	+	0.768369i	$0.721069\pi$
$432$	−5.32767e18	−0.0912871
$433$	−8.49578e19	−1.43069	−0.715343	−	0.698774i	$-0.753729\pi$
$433$	−8.49578e19	−1.43069	−0.715343	+	0.698774i	$0.753729\pi$
$434$	−6.73675e19	−1.11501
$435$	0	0
$436$	−5.23998e18	−0.0837878
$437$	7.01572e19	1.10271
$438$	−4.44388e18	−0.0686605
$439$	−1.19247e20	−1.81119	−0.905597	−	0.424139i	$-0.860577\pi$
$439$	−1.19247e20	−1.81119	−0.905597	+	0.424139i	$0.860577\pi$
$440$	0	0
$441$	3.46134e19	0.508107
$442$	4.52040e19	0.652394
$443$	3.85306e19	0.546736	0.273368	−	0.961910i	$-0.411862\pi$
$443$	3.85306e19	0.546736	0.273368	+	0.961910i	$0.411862\pi$
$444$	3.97850e19	0.555069
$445$	0	0
$446$	−2.18710e19	−0.295024
$447$	7.63844e18	0.101321
$448$	5.71340e18	0.0745262
$449$	−4.49146e19	−0.576156	−0.288078	−	0.957607i	$-0.593016\pi$
$449$	−4.49146e19	−0.576156	−0.288078	+	0.957607i	$0.593016\pi$
$450$	0	0
$451$	−1.25978e20	−1.56304
$452$	8.53071e19	1.04099
$453$	−6.13157e19	−0.735927
$454$	−1.78781e19	−0.211058
$455$	0	0
$456$	6.37710e19	0.728429
$457$	1.20034e20	1.34876	0.674379	−	0.738385i	$-0.264411\pi$
$457$	1.20034e20	1.34876	0.674379	+	0.738385i	$0.264411\pi$
$458$	4.34767e19	0.480580
$459$	2.74110e19	0.298077
$460$	0	0
$461$	−1.01623e20	−1.06963	−0.534817	−	0.844968i	$-0.679620\pi$
$461$	−1.01623e20	−1.06963	−0.534817	+	0.844968i	$0.679620\pi$
$462$	−5.98621e19	−0.619922
$463$	−8.36310e19	−0.852138	−0.426069	−	0.904691i	$-0.640102\pi$
$463$	−8.36310e19	−0.852138	−0.426069	+	0.904691i	$0.640102\pi$
$464$	3.75257e19	0.376222
$465$	0	0
$466$	−8.26294e18	−0.0802122
$467$	1.09319e20	1.04429	0.522143	−	0.852858i	$-0.325133\pi$
$467$	1.09319e20	1.04429	0.522143	+	0.852858i	$0.325133\pi$
$468$	−2.80952e19	−0.264112
$469$	1.09566e20	1.01363
$470$	0	0
$471$	7.67630e19	0.687853
$472$	−3.26979e18	−0.0288373
$473$	−8.77523e19	−0.761728
$474$	−3.19646e19	−0.273106
$475$	0	0
$476$	−2.41714e20	−2.00101
$477$	−4.52587e19	−0.368820
$478$	6.49315e19	0.520890
$479$	2.24501e20	1.77297	0.886485	−	0.462758i	$-0.153140\pi$
$479$	2.24501e20	1.77297	0.886485	+	0.462758i	$0.153140\pi$
$480$	0	0
$481$	1.50512e20	1.15208
$482$	3.75626e19	0.283076
$483$	−8.46870e19	−0.628367
$484$	−1.60745e20	−1.17435
$485$	0	0
$486$	3.91494e18	0.0277302
$487$	−1.09079e20	−0.760807	−0.380403	−	0.924821i	$-0.624215\pi$
$487$	−1.09079e20	−0.760807	−0.380403	+	0.924821i	$0.624215\pi$
$488$	1.66541e20	1.14386
$489$	−8.65931e19	−0.585689
$490$	0	0
$491$	2.91004e20	1.90892	0.954461	−	0.298337i	$-0.0964319\pi$
$491$	2.91004e20	1.90892	0.954461	+	0.298337i	$0.0964319\pi$
$492$	−7.26515e19	−0.469361
$493$	−1.93070e20	−1.22847
$494$	1.08196e20	0.678045
$495$	0	0
$496$	−1.26664e20	−0.770088
$497$	1.65975e20	0.993960
$498$	−2.13651e19	−0.126033
$499$	2.32620e20	1.35174	0.675870	−	0.737021i	$-0.263768\pi$
$499$	2.32620e20	1.35174	0.675870	+	0.737021i	$0.263768\pi$
$500$	0	0
$501$	−9.74700e18	−0.0549652
$502$	3.55393e19	0.197438
$503$	1.81171e20	0.991581	0.495791	−	0.868442i	$-0.334878\pi$
$503$	1.81171e20	0.991581	0.495791	+	0.868442i	$0.334878\pi$
$504$	−7.69782e19	−0.415087
$505$	0	0
$506$	8.84415e19	0.462942
$507$	5.65544e18	0.0291680
$508$	1.18871e20	0.604087
$509$	−2.78987e20	−1.39702	−0.698508	−	0.715602i	$-0.746152\pi$
$509$	−2.78987e20	−1.39702	−0.698508	+	0.715602i	$0.746152\pi$
$510$	0	0
$511$	8.98960e19	0.437103
$512$	−1.75478e20	−0.840809
$513$	6.56082e19	0.309798
$514$	−1.47791e20	−0.687738
$515$	0	0
$516$	−5.06068e19	−0.228737
$517$	−6.14138e20	−2.73582
$518$	1.84945e20	0.812026
$519$	8.07200e19	0.349322
$520$	0	0
$521$	3.99817e20	1.68104	0.840520	−	0.541781i	$-0.182250\pi$
$521$	3.99817e20	1.68104	0.840520	+	0.541781i	$0.182250\pi$
$522$	−2.75751e19	−0.114285
$523$	3.30222e20	1.34910	0.674549	−	0.738230i	$-0.264338\pi$
$523$	3.30222e20	1.34910	0.674549	+	0.738230i	$0.264338\pi$
$524$	9.46000e19	0.380984
$525$	0	0
$526$	2.00461e19	0.0784578
$527$	6.51688e20	2.51455
$528$	−1.12552e20	−0.428154
$529$	−1.41517e20	−0.530752
$530$	0	0
$531$	−3.36399e18	−0.0122644
$532$	−5.78543e20	−2.07969
$533$	−2.74851e20	−0.974190
$534$	9.11526e19	0.318574
$535$	0	0
$536$	−1.47140e20	−0.500028
$537$	−2.07596e20	−0.695684
$538$	−3.62481e19	−0.119789
$539$	7.31243e20	2.38312
$540$	0	0
$541$	−5.16696e19	−0.163778	−0.0818890	−	0.996641i	$-0.526095\pi$
$541$	−5.16696e19	−0.163778	−0.0818890	+	0.996641i	$0.526095\pi$
$542$	−2.27475e19	−0.0711114
$543$	1.81973e20	0.561058
$544$	5.03636e20	1.53153
$545$	0	0
$546$	−1.30604e20	−0.386376
$547$	2.67306e20	0.780018	0.390009	−	0.920811i	$-0.372472\pi$
$547$	2.67306e20	0.780018	0.390009	+	0.920811i	$0.372472\pi$
$548$	1.98987e20	0.572756
$549$	1.71339e20	0.486479
$550$	0	0
$551$	−4.62115e20	−1.27677
$552$	1.13729e20	0.309976
$553$	6.46617e20	1.73863
$554$	5.10626e19	0.135450
$555$	0	0
$556$	−2.39493e20	−0.618345
$557$	−4.07141e20	−1.03712	−0.518562	−	0.855040i	$-0.673533\pi$
$557$	−4.07141e20	−1.03712	−0.518562	+	0.855040i	$0.673533\pi$
$558$	9.30765e19	0.233929
$559$	−1.91453e20	−0.474759
$560$	0	0
$561$	5.79084e20	1.39804
$562$	−1.07852e20	−0.256924
$563$	−1.15359e20	−0.271168	−0.135584	−	0.990766i	$-0.543291\pi$
$563$	−1.15359e20	−0.271168	−0.135584	+	0.990766i	$0.543291\pi$
$564$	−3.54174e20	−0.821530
$565$	0	0
$566$	−1.07013e20	−0.241721
$567$	−7.91959e19	−0.176535
$568$	−2.22893e20	−0.490325
$569$	1.00468e20	0.218114	0.109057	−	0.994035i	$-0.465217\pi$
$569$	1.00468e20	0.218114	0.109057	+	0.994035i	$0.465217\pi$
$570$	0	0
$571$	7.72381e20	1.63328	0.816641	−	0.577147i	$-0.195834\pi$
$571$	7.72381e20	1.63328	0.816641	+	0.577147i	$0.195834\pi$
$572$	−5.93539e20	−1.23874
$573$	−3.26450e20	−0.672445
$574$	−3.37729e20	−0.686640
$575$	0	0
$576$	−7.89377e18	−0.0156356
$577$	7.07914e20	1.38408	0.692042	−	0.721858i	$-0.256711\pi$
$577$	7.07914e20	1.38408	0.692042	+	0.721858i	$0.256711\pi$
$578$	−3.13342e20	−0.604727
$579$	−4.81256e19	−0.0916827
$580$	0	0
$581$	4.32198e20	0.802346
$582$	−1.72862e19	−0.0316794
$583$	−9.56135e20	−1.72984
$584$	−1.20725e20	−0.215625
$585$	0	0
$586$	3.17384e20	0.552526
$587$	−4.51629e20	−0.776240	−0.388120	−	0.921609i	$-0.626875\pi$
$587$	−4.51629e20	−0.776240	−0.388120	+	0.921609i	$0.626875\pi$
$588$	4.21708e20	0.715620
$589$	1.55982e21	2.61342
$590$	0	0
$591$	−1.40172e20	−0.228957
$592$	3.47733e20	0.560832
$593$	2.48017e20	0.394977	0.197488	−	0.980305i	$-0.436722\pi$
$593$	2.48017e20	0.394977	0.197488	+	0.980305i	$0.436722\pi$
$594$	8.27070e19	0.130060
$595$	0	0
$596$	9.30620e19	0.142700
$597$	−1.85288e20	−0.280569
$598$	1.92956e20	0.288536
$599$	2.83458e20	0.418588	0.209294	−	0.977853i	$-0.432883\pi$
$599$	2.83458e20	0.418588	0.209294	+	0.977853i	$0.432883\pi$
$600$	0	0
$601$	−1.20634e20	−0.173744	−0.0868720	−	0.996219i	$-0.527687\pi$
$601$	−1.20634e20	−0.173744	−0.0868720	+	0.996219i	$0.527687\pi$
$602$	−2.35251e20	−0.334625
$603$	−1.51379e20	−0.212660
$604$	−7.47032e20	−1.03648
$605$	0	0
$606$	2.24902e20	0.304403
$607$	3.48366e20	0.465716	0.232858	−	0.972511i	$-0.425192\pi$
$607$	3.48366e20	0.465716	0.232858	+	0.972511i	$0.425192\pi$
$608$	1.20545e21	1.59175
$609$	5.57820e20	0.727553
$610$	0	0
$611$	−1.33989e21	−1.70514
$612$	3.33958e20	0.419813
$613$	−2.56913e20	−0.319031	−0.159516	−	0.987195i	$-0.550993\pi$
$613$	−2.56913e20	−0.319031	−0.159516	+	0.987195i	$0.550993\pi$
$614$	4.21100e20	0.516562
$615$	0	0
$616$	−1.62624e21	−1.94684
$617$	9.27973e20	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$617$	9.27973e20	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$618$	1.24749e20	0.145755
$619$	−4.42908e20	−0.511250	−0.255625	−	0.966776i	$-0.582281\pi$
$619$	−4.42908e20	−0.511250	−0.255625	+	0.966776i	$0.582281\pi$
$620$	0	0
$621$	1.17006e20	0.131832
$622$	−1.28970e20	−0.143569
$623$	−1.84394e21	−2.02809
$624$	−2.45560e20	−0.266854
$625$	0	0
$626$	−2.19295e20	−0.232660
$627$	1.38604e21	1.45301
$628$	9.35233e20	0.968774
$629$	−1.78909e21	−1.83127
$630$	0	0
$631$	1.56005e21	1.55926	0.779628	−	0.626243i	$-0.215408\pi$
$631$	1.56005e21	1.55926	0.779628	+	0.626243i	$0.215408\pi$
$632$	−8.68364e20	−0.857677
$633$	1.36042e20	0.132784
$634$	−5.07081e20	−0.489112
$635$	0	0
$636$	−5.51403e20	−0.519447
$637$	1.59538e21	1.48532
$638$	−5.82551e20	−0.536017
$639$	−2.29315e20	−0.208533
$640$	0	0
$641$	5.40363e20	0.480010	0.240005	−	0.970772i	$-0.422851\pi$
$641$	5.40363e20	0.480010	0.240005	+	0.970772i	$0.422851\pi$
$642$	2.64215e19	0.0231977
$643$	5.44502e20	0.472517	0.236258	−	0.971690i	$-0.424079\pi$
$643$	5.44502e20	0.472517	0.236258	+	0.971690i	$0.424079\pi$
$644$	−1.03177e21	−0.884993
$645$	0	0
$646$	−1.28609e21	−1.07777
$647$	9.19093e20	0.761338	0.380669	−	0.924711i	$-0.375694\pi$
$647$	9.19093e20	0.761338	0.380669	+	0.924711i	$0.375694\pi$
$648$	1.06355e20	0.0870854
$649$	−7.10677e19	−0.0575224
$650$	0	0
$651$	−1.88286e21	−1.48923
$652$	−1.05500e21	−0.824886
$653$	−6.96084e19	−0.0538038	−0.0269019	−	0.999638i	$-0.508564\pi$
$653$	−6.96084e19	−0.0538038	−0.0269019	+	0.999638i	$0.508564\pi$
$654$	3.36545e19	0.0257164
$655$	0	0
$656$	−6.34996e20	−0.474233
$657$	−1.24203e20	−0.0917044
$658$	−1.64642e21	−1.20184
$659$	−1.84357e21	−1.33052	−0.665258	−	0.746614i	$-0.731678\pi$
$659$	−1.84357e21	−1.33052	−0.665258	+	0.746614i	$0.731678\pi$
$660$	0	0
$661$	1.89730e21	1.33852	0.669260	−	0.743028i	$-0.266611\pi$
$661$	1.89730e21	1.33852	0.669260	+	0.743028i	$0.266611\pi$
$662$	−2.64041e20	−0.184177
$663$	1.26341e21	0.871351
$664$	−5.80414e20	−0.395801
$665$	0	0
$666$	−2.55525e20	−0.170363
$667$	−8.24134e20	−0.543318
$668$	−1.18751e20	−0.0774132
$669$	−6.11275e20	−0.394041
$670$	0	0
$671$	3.61971e21	2.28168
$672$	−1.45511e21	−0.907038
$673$	5.54468e20	0.341793	0.170897	−	0.985289i	$-0.445334\pi$
$673$	5.54468e20	0.341793	0.170897	+	0.985289i	$0.445334\pi$
$674$	−5.89980e20	−0.359657
$675$	0	0
$676$	6.89024e19	0.0410803
$677$	−1.26360e21	−0.745067	−0.372534	−	0.928019i	$-0.621511\pi$
$677$	−1.26360e21	−0.745067	−0.372534	+	0.928019i	$0.621511\pi$
$678$	−5.47897e20	−0.319504
$679$	3.49686e20	0.201676
$680$	0	0
$681$	−4.99676e20	−0.281894
$682$	1.96634e21	1.09717
$683$	−1.74862e21	−0.965030	−0.482515	−	0.875888i	$-0.660277\pi$
$683$	−1.74862e21	−0.965030	−0.482515	+	0.875888i	$0.660277\pi$
$684$	7.99329e20	0.436320
$685$	0	0
$686$	6.74305e20	0.360102
$687$	1.21514e21	0.641873
$688$	−4.42318e20	−0.231111
$689$	−2.08604e21	−1.07815
$690$	0	0
$691$	−3.33317e21	−1.68567	−0.842833	−	0.538175i	$-0.819114\pi$
$691$	−3.33317e21	−1.68567	−0.842833	+	0.538175i	$0.819114\pi$
$692$	9.83441e20	0.491986
$693$	−1.67309e21	−0.827982
$694$	6.66007e20	0.326049
$695$	0	0
$696$	−7.49116e20	−0.358905
$697$	3.26706e21	1.54850
$698$	−2.97919e20	−0.139695
$699$	−2.30942e20	−0.107133
$700$	0	0
$701$	3.70211e21	1.68099	0.840494	−	0.541821i	$-0.182265\pi$
$701$	3.70211e21	1.68099	0.840494	+	0.541821i	$0.182265\pi$
$702$	1.80445e20	0.0810619
$703$	−4.28219e21	−1.90327
$704$	−1.66764e20	−0.0733342
$705$	0	0
$706$	−1.63277e21	−0.702893
$707$	−4.54959e21	−1.93788
$708$	−4.09847e19	−0.0172732
$709$	5.25611e20	0.219189	0.109594	−	0.993976i	$-0.465045\pi$
$709$	5.25611e20	0.219189	0.109594	+	0.993976i	$0.465045\pi$
$710$	0	0
$711$	−8.93381e20	−0.364766
$712$	2.47629e21	1.00047
$713$	2.78177e21	1.11212
$714$	1.55244e21	0.614156
$715$	0	0
$716$	−2.52922e21	−0.979804
$717$	1.81478e21	0.695712
$718$	2.08170e20	0.0789743
$719$	−9.77221e20	−0.366882	−0.183441	−	0.983031i	$-0.558724\pi$
$719$	−9.77221e20	−0.366882	−0.183441	+	0.983031i	$0.558724\pi$
$720$	0	0
$721$	−2.52357e21	−0.927898
$722$	−1.89034e21	−0.687878
$723$	1.04984e21	0.378082
$724$	2.21704e21	0.790196
$725$	0	0
$726$	1.03241e21	0.360435
$727$	−1.02590e21	−0.354483	−0.177241	−	0.984167i	$-0.556717\pi$
$727$	−1.02590e21	−0.354483	−0.177241	+	0.984167i	$0.556717\pi$
$728$	−3.54804e21	−1.21340
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	2.27573e21	0.754642
$732$	2.08749e21	0.685158
$733$	1.30167e21	0.422885	0.211443	−	0.977390i	$-0.432184\pi$
$733$	1.30167e21	0.422885	0.211443	+	0.977390i	$0.432184\pi$
$734$	1.97935e20	0.0636505
$735$	0	0
$736$	2.14980e21	0.677353
$737$	−3.19803e21	−0.997416
$738$	4.66615e20	0.144057
$739$	−5.42053e19	−0.0165657	−0.00828283	−	0.999966i	$-0.502637\pi$
$739$	−5.42053e19	−0.0165657	−0.00828283	+	0.999966i	$0.502637\pi$
$740$	0	0
$741$	3.02398e21	0.905612
$742$	−2.56326e21	−0.759912
$743$	4.02854e20	0.118231	0.0591155	−	0.998251i	$-0.481172\pi$
$743$	4.02854e20	0.118231	0.0591155	+	0.998251i	$0.481172\pi$
$744$	2.52856e21	0.734643
$745$	0	0
$746$	−1.83961e21	−0.523822
$747$	−5.97135e20	−0.168333
$748$	7.05519e21	1.96901
$749$	−5.34484e20	−0.147680
$750$	0	0
$751$	5.38166e21	1.45753	0.728764	−	0.684765i	$-0.240095\pi$
$751$	5.38166e21	1.45753	0.728764	+	0.684765i	$0.240095\pi$
$752$	−3.09558e21	−0.830059
$753$	9.93292e20	0.263703
$754$	−1.27097e21	−0.334081
$755$	0	0
$756$	−9.64873e20	−0.248632
$757$	1.61105e20	0.0411047	0.0205523	−	0.999789i	$-0.493458\pi$
$757$	1.61105e20	0.0411047	0.0205523	+	0.999789i	$0.493458\pi$
$758$	2.67162e21	0.674925
$759$	2.47186e21	0.618316
$760$	0	0
$761$	−4.49139e21	−1.10153	−0.550764	−	0.834661i	$-0.685664\pi$
$761$	−4.49139e21	−1.10153	−0.550764	+	0.834661i	$0.685664\pi$
$762$	−7.63468e20	−0.185408
$763$	−6.80802e20	−0.163714
$764$	−3.97726e21	−0.947073
$765$	0	0
$766$	2.01609e21	0.470754
$767$	−1.55051e20	−0.0358517
$768$	9.81578e20	0.224758
$769$	−1.02900e20	−0.0233328	−0.0116664	−	0.999932i	$-0.503714\pi$
$769$	−1.02900e20	−0.0233328	−0.0116664	+	0.999932i	$0.503714\pi$
$770$	0	0
$771$	−4.13061e21	−0.918558
$772$	−5.86333e20	−0.129126
$773$	2.97240e21	0.648277	0.324139	−	0.946010i	$-0.394926\pi$
$773$	2.97240e21	0.648277	0.324139	+	0.946010i	$0.394926\pi$
$774$	3.25029e20	0.0702045
$775$	0	0
$776$	−4.69605e20	−0.0994878
$777$	5.16905e21	1.08456
$778$	−2.57743e21	−0.535599
$779$	7.81973e21	1.60939
$780$	0	0
$781$	−4.84450e21	−0.978061
$782$	−2.29361e21	−0.458636
$783$	−7.70698e20	−0.152641
$784$	3.68586e21	0.723049
$785$	0	0
$786$	−6.07582e20	−0.116933
$787$	−2.39304e21	−0.456183	−0.228091	−	0.973640i	$-0.573248\pi$
$787$	−2.39304e21	−0.456183	−0.228091	+	0.973640i	$0.573248\pi$
$788$	−1.70776e21	−0.322464
$789$	5.60270e20	0.104790
$790$	0	0
$791$	1.10835e22	2.03401
$792$	2.24685e21	0.408447
$793$	7.89727e21	1.42209
$794$	−2.63660e21	−0.470316
$795$	0	0
$796$	−2.25744e21	−0.395155
$797$	−6.03860e21	−1.04713	−0.523563	−	0.851987i	$-0.675398\pi$
$797$	−6.03860e21	−1.04713	−0.523563	+	0.851987i	$0.675398\pi$
$798$	3.71577e21	0.638304
$799$	1.59268e22	2.71037
$800$	0	0
$801$	2.54763e21	0.425494
$802$	3.00291e21	0.496860
$803$	−2.62390e21	−0.430112
$804$	−1.84431e21	−0.299511
$805$	0	0
$806$	4.29003e21	0.683829
$807$	−1.01310e21	−0.159993
$808$	6.10980e21	0.955965
$809$	1.13034e22	1.75224	0.876121	−	0.482091i	$-0.160123\pi$
$809$	1.13034e22	1.75224	0.876121	+	0.482091i	$0.160123\pi$
$810$	0	0
$811$	−3.27617e21	−0.498552	−0.249276	−	0.968433i	$-0.580193\pi$
$811$	−3.27617e21	−0.498552	−0.249276	+	0.968433i	$0.580193\pi$
$812$	6.79613e21	1.02469
$813$	−6.35773e20	−0.0949779
$814$	−5.39821e21	−0.799037
$815$	0	0
$816$	2.91889e21	0.424172
$817$	5.44698e21	0.784314
$818$	2.14235e21	0.305662
$819$	−3.65025e21	−0.516052
$820$	0	0
$821$	−2.03953e21	−0.283110	−0.141555	−	0.989930i	$-0.545210\pi$
$821$	−2.03953e21	−0.283110	−0.141555	+	0.989930i	$0.545210\pi$
$822$	−1.27802e21	−0.175792
$823$	1.00578e22	1.37090	0.685449	−	0.728121i	$-0.259606\pi$
$823$	1.00578e22	1.37090	0.685449	+	0.728121i	$0.259606\pi$
$824$	3.38899e21	0.457737
$825$	0	0
$826$	−1.90522e20	−0.0252694
$827$	−4.09485e20	−0.0538204	−0.0269102	−	0.999638i	$-0.508567\pi$
$827$	−4.09485e20	−0.0538204	−0.0269102	+	0.999638i	$0.508567\pi$
$828$	1.42552e21	0.185672
$829$	−1.46492e21	−0.189084	−0.0945419	−	0.995521i	$-0.530139\pi$
$829$	−1.46492e21	−0.189084	−0.0945419	+	0.995521i	$0.530139\pi$
$830$	0	0
$831$	1.42715e21	0.180910
$832$	−3.63835e20	−0.0457067
$833$	−1.89638e22	−2.36095
$834$	1.53817e21	0.189784
$835$	0	0
$836$	1.68866e22	2.04643
$837$	2.60141e21	0.312440
$838$	−6.16234e21	−0.733526
$839$	−8.03115e21	−0.947465	−0.473732	−	0.880669i	$-0.657094\pi$
$839$	−8.03115e21	−0.947465	−0.473732	+	0.880669i	$0.657094\pi$
$840$	0	0
$841$	−3.20074e21	−0.370920
$842$	−3.55058e21	−0.407811
$843$	−3.01436e21	−0.343153
$844$	1.65745e21	0.187014
$845$	0	0
$846$	2.27473e21	0.252146
$847$	−2.08848e22	−2.29458
$848$	−4.81943e21	−0.524840
$849$	−2.99092e21	−0.322847
$850$	0	0
$851$	−7.63685e21	−0.809921
$852$	−2.79382e21	−0.293699
$853$	3.56240e21	0.371214	0.185607	−	0.982624i	$-0.440575\pi$
$853$	3.56240e21	0.371214	0.185607	+	0.982624i	$0.440575\pi$
$854$	9.70393e21	1.00234
$855$	0	0
$856$	7.17778e20	0.0728512
$857$	−8.50561e21	−0.855755	−0.427878	−	0.903837i	$-0.640739\pi$
$857$	−8.50561e21	−0.855755	−0.427878	+	0.903837i	$0.640739\pi$
$858$	3.81208e21	0.380196
$859$	−1.72313e22	−1.70360	−0.851802	−	0.523863i	$-0.824490\pi$
$859$	−1.72313e22	−1.70360	−0.851802	+	0.523863i	$0.824490\pi$
$860$	0	0
$861$	−9.43922e21	−0.917091
$862$	5.74480e21	0.553313
$863$	−1.75686e21	−0.167748	−0.0838739	−	0.996476i	$-0.526729\pi$
$863$	−1.75686e21	−0.167748	−0.0838739	+	0.996476i	$0.526729\pi$
$864$	2.01041e21	0.190297
$865$	0	0
$866$	6.64790e21	0.618444
$867$	−8.75761e21	−0.807686
$868$	−2.29396e22	−2.09743
$869$	−1.88736e22	−1.71083
$870$	0	0
$871$	−6.97727e21	−0.621655
$872$	9.14273e20	0.0807611
$873$	−4.83134e20	−0.0423118
$874$	−5.48976e21	−0.476669
$875$	0	0
$876$	−1.51321e21	−0.129157
$877$	8.78202e21	0.743186	0.371593	−	0.928396i	$-0.378812\pi$
$877$	8.78202e21	0.743186	0.371593	+	0.928396i	$0.378812\pi$
$878$	9.33104e21	0.782927
$879$	8.87060e21	0.737966
$880$	0	0
$881$	−1.89714e22	−1.55160	−0.775800	−	0.630979i	$-0.782654\pi$
$881$	−1.89714e22	−1.55160	−0.775800	+	0.630979i	$0.782654\pi$
$882$	−2.70848e21	−0.219640
$883$	−2.57358e21	−0.206934	−0.103467	−	0.994633i	$-0.532994\pi$
$883$	−2.57358e21	−0.206934	−0.103467	+	0.994633i	$0.532994\pi$
$884$	1.53926e22	1.22721
$885$	0	0
$886$	−3.01499e21	−0.236338
$887$	2.10966e22	1.63978	0.819888	−	0.572524i	$-0.194036\pi$
$887$	2.10966e22	1.63978	0.819888	+	0.572524i	$0.194036\pi$
$888$	−6.94170e21	−0.535018
$889$	1.54443e22	1.18034
$890$	0	0
$891$	2.31159e21	0.173711
$892$	−7.44739e21	−0.554968
$893$	3.81209e22	2.81694
$894$	−5.97703e20	−0.0437980
$895$	0	0
$896$	−2.22491e22	−1.60325
$897$	5.39296e21	0.385375
$898$	3.51454e21	0.249056
$899$	−1.83231e22	−1.28766
$900$	0	0
$901$	2.47960e22	1.71374
$902$	9.85770e21	0.675657
$903$	−6.57506e21	−0.446932
$904$	−1.48844e22	−1.00339
$905$	0	0
$906$	4.79791e21	0.318120
$907$	−5.79065e20	−0.0380779	−0.0190389	−	0.999819i	$-0.506061\pi$
$907$	−5.79065e20	−0.0380779	−0.0190389	+	0.999819i	$0.506061\pi$
$908$	−6.08774e21	−0.397020
$909$	6.28582e21	0.406568
$910$	0	0
$911$	−5.69963e21	−0.362626	−0.181313	−	0.983425i	$-0.558035\pi$
$911$	−5.69963e21	−0.362626	−0.181313	+	0.983425i	$0.558035\pi$
$912$	6.98638e21	0.440850
$913$	−1.26151e22	−0.789513
$914$	−9.39261e21	−0.583029
$915$	0	0
$916$	1.48044e22	0.904016
$917$	1.22909e22	0.744411
$918$	−2.14489e21	−0.128850
$919$	−2.26024e22	−1.34675	−0.673376	−	0.739300i	$-0.735156\pi$
$919$	−2.26024e22	−1.34675	−0.673376	+	0.739300i	$0.735156\pi$
$920$	0	0
$921$	1.17694e22	0.689932
$922$	7.95194e21	0.462372
$923$	−1.05694e22	−0.609592
$924$	−2.03839e22	−1.16613
$925$	0	0
$926$	6.54407e21	0.368354
$927$	3.48662e21	0.194674
$928$	−1.41604e22	−0.784272
$929$	−1.77183e22	−0.973429	−0.486715	−	0.873561i	$-0.661805\pi$
$929$	−1.77183e22	−0.973429	−0.486715	+	0.873561i	$0.661805\pi$
$930$	0	0
$931$	−4.53899e22	−2.45378
$932$	−2.81365e21	−0.150887
$933$	−3.60460e21	−0.191754
$934$	−8.55415e21	−0.451414
$935$	0	0
$936$	4.90205e21	0.254571
$937$	−2.12163e22	−1.09300	−0.546502	−	0.837458i	$-0.684041\pi$
$937$	−2.12163e22	−1.09300	−0.546502	+	0.837458i	$0.684041\pi$
$938$	−8.57346e21	−0.438162
$939$	−6.12911e21	−0.310746
$940$	0	0
$941$	−9.21161e21	−0.459635	−0.229818	−	0.973234i	$-0.573813\pi$
$941$	−9.21161e21	−0.459635	−0.229818	+	0.973234i	$0.573813\pi$
$942$	−6.00666e21	−0.297339
$943$	1.39457e22	0.684861
$944$	−3.58219e20	−0.0174525
$945$	0	0
$946$	6.86656e21	0.329273
$947$	2.54165e22	1.20918	0.604590	−	0.796537i	$-0.293337\pi$
$947$	2.54165e22	1.20918	0.604590	+	0.796537i	$0.293337\pi$
$948$	−1.08844e22	−0.513738
$949$	−5.72467e21	−0.268074
$950$	0	0
$951$	−1.41725e22	−0.653268
$952$	4.21743e22	1.92873
$953$	2.06299e22	0.936051	0.468026	−	0.883715i	$-0.344965\pi$
$953$	2.06299e22	0.936051	0.468026	+	0.883715i	$0.344965\pi$
$954$	3.54146e21	0.159430
$955$	0	0
$956$	2.21101e22	0.979844
$957$	−1.62818e22	−0.715916
$958$	−1.75670e22	−0.766403
$959$	2.58533e22	1.11912
$960$	0	0
$961$	3.83824e22	1.63571
$962$	−1.17775e22	−0.498012
$963$	7.38456e20	0.0309833
$964$	1.27906e22	0.532492
$965$	0	0
$966$	6.62670e21	0.271625
$967$	2.07791e22	0.845137	0.422569	−	0.906331i	$-0.361129\pi$
$967$	2.07791e22	0.845137	0.422569	+	0.906331i	$0.361129\pi$
$968$	2.80469e22	1.13193
$969$	−3.59450e22	−1.43950
$970$	0	0
$971$	−3.25558e22	−1.28376	−0.641881	−	0.766804i	$-0.721846\pi$
$971$	−3.25558e22	−1.28376	−0.641881	+	0.766804i	$0.721846\pi$
$972$	1.33309e21	0.0521631
$973$	−3.11160e22	−1.20820
$974$	8.53537e21	0.328875
$975$	0	0
$976$	1.82453e22	0.692272
$977$	2.58774e22	0.974342	0.487171	−	0.873307i	$-0.338029\pi$
$977$	2.58774e22	0.974342	0.487171	+	0.873307i	$0.338029\pi$
$978$	6.77586e21	0.253176
$979$	5.38213e22	1.99565
$980$	0	0
$981$	9.40612e20	0.0343473
$982$	−2.27709e22	−0.825171
$983$	−3.80367e22	−1.36789	−0.683946	−	0.729532i	$-0.739738\pi$
$983$	−3.80367e22	−1.36789	−0.683946	+	0.729532i	$0.739738\pi$
$984$	1.26763e22	0.452406
$985$	0	0
$986$	1.51076e22	0.531031
$987$	−4.60159e22	−1.60520
$988$	3.68423e22	1.27547
$989$	9.71413e21	0.333758
$990$	0	0
$991$	−3.47193e22	−1.17495	−0.587474	−	0.809243i	$-0.699878\pi$
$991$	−3.47193e22	−1.17495	−0.587474	+	0.809243i	$0.699878\pi$
$992$	4.77970e22	1.60533
$993$	−7.37970e21	−0.245991
$994$	−1.29874e22	−0.429660
$995$	0	0
$996$	−7.27512e21	−0.237080
$997$	−4.76641e22	−1.54162	−0.770811	−	0.637063i	$-0.780149\pi$
$997$	−4.76641e22	−1.54162	−0.770811	+	0.637063i	$0.780149\pi$
$998$	−1.82024e22	−0.584318
$999$	−7.14169e21	−0.227541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.16.a.j.1.3	yes	6
5.2	odd	4		75.16.b.h.49.6		12
5.3	odd	4		75.16.b.h.49.7		12
5.4	even	2		75.16.a.i.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.16.a.i.1.4	✓	6	5.4	even	2
75.16.a.j.1.3	yes	6	1.1	even	1	trivial
75.16.b.h.49.6		12	5.2	odd	4
75.16.b.h.49.7		12	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-78.2494 q^{2} -2187.00 q^{3} -26645.0 q^{4} +171131. q^{6} -3.46184e6 q^{7} +4.64903e6 q^{8} +4.78297e6 q^{9} +O(q^{10})\)	\(q-78.2494 q^{2} -2187.00 q^{3} -26645.0 q^{4} +171131. q^{6} -3.46184e6 q^{7} +4.64903e6 q^{8} +4.78297e6 q^{9} +1.01045e8 q^{11} +5.82727e7 q^{12} +2.20454e8 q^{13} +2.70887e8 q^{14} +5.09321e8 q^{16} -2.62046e9 q^{17} -3.74264e8 q^{18} -6.27209e9 q^{19} +7.57105e9 q^{21} -7.90671e9 q^{22} -1.11856e10 q^{23} -1.01674e10 q^{24} -1.72504e10 q^{26} -1.04604e10 q^{27} +9.22410e10 q^{28} +7.36780e10 q^{29} -2.48692e11 q^{31} -1.92194e11 q^{32} -2.20985e11 q^{33} +2.05050e11 q^{34} -1.27442e11 q^{36} +6.82738e11 q^{37} +4.90787e11 q^{38} -4.82133e11 q^{39} -1.24675e12 q^{41} -5.92430e11 q^{42} -8.68448e11 q^{43} -2.69235e12 q^{44} +8.75268e11 q^{46} -6.07787e12 q^{47} -1.11388e12 q^{48} +7.23681e12 q^{49} +5.73095e12 q^{51} -5.87400e12 q^{52} -9.46247e12 q^{53} +8.18516e11 q^{54} -1.60942e13 q^{56} +1.37171e13 q^{57} -5.76526e12 q^{58} -7.03327e11 q^{59} +3.58228e13 q^{61} +1.94600e13 q^{62} -1.65579e13 q^{63} -1.65039e12 q^{64} +1.72920e13 q^{66} -3.16496e13 q^{67} +6.98223e13 q^{68} +2.44630e13 q^{69} -4.79440e13 q^{71} +2.22362e13 q^{72} -2.59677e13 q^{73} -5.34239e13 q^{74} +1.67120e14 q^{76} -3.49802e14 q^{77} +3.77266e13 q^{78} -1.86784e14 q^{79} +2.28768e13 q^{81} +9.75575e13 q^{82} -1.24846e14 q^{83} -2.01731e14 q^{84} +6.79555e13 q^{86} -1.61134e14 q^{87} +4.69762e14 q^{88} +5.32647e14 q^{89} -7.63177e14 q^{91} +2.98041e14 q^{92} +5.43889e14 q^{93} +4.75590e14 q^{94} +4.20327e14 q^{96} -1.01011e14 q^{97} -5.66276e14 q^{98} +4.83295e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9}+O(q^{10}) \) 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9} + 107489124 q^{11} - 203635944 q^{12} - 109881686 q^{13} - 563984442 q^{14} + 3622829560 q^{16} + 3573042876 q^{17} + 1119214746 q^{18} - 1602340942 q^{19} + 5664815514 q^{21} + 4024661012 q^{22} - 6555818844 q^{23} - 30645092556 q^{24} - 25715894778 q^{26} - 62762119218 q^{27} - 270752117896 q^{28} + 126894468996 q^{29} + 151760841646 q^{31} + 385411085208 q^{32} - 235078714188 q^{33} + 1431919606684 q^{34} + 445351809528 q^{36} + 616109002068 q^{37} - 2822785016634 q^{38} + 240311247282 q^{39} + 1091281712616 q^{41} + 1233433974654 q^{42} - 2444971199030 q^{43} + 1413344578176 q^{44} - 5480862370044 q^{46} - 8369143269660 q^{47} - 7923128247720 q^{48} + 19523846053580 q^{49} - 7814244769812 q^{51} - 10261294060344 q^{52} - 16571417665824 q^{53} - 2447722649502 q^{54} - 75252275829540 q^{56} + 3504319640154 q^{57} - 3994751501708 q^{58} + 8796604455252 q^{59} - 6959665405750 q^{61} + 52277129313066 q^{62} - 12388951529118 q^{63} + 50304241850208 q^{64} - 8801933633244 q^{66} - 53487461742094 q^{67} + 307147088145312 q^{68} + 14337575811828 q^{69} + 104634162717912 q^{71} + 67020817419972 q^{72} + 177000981923236 q^{73} - 45005277967812 q^{74} + 76188538526328 q^{76} + 117850730172876 q^{77} + 56240661879486 q^{78} + 185514024366160 q^{79} + 137260754729766 q^{81} + 654376907588896 q^{82} + 435827733256908 q^{83} + 592134881838552 q^{84} + 15\!\cdots\!14 q^{86}+ \cdots + 514117147929156 q^{99}+O(q^{100}) \) 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9 + 107489124 * q^11 - 203635944 * q^12 - 109881686 * q^13 - 563984442 * q^14 + 3622829560 * q^16 + 3573042876 * q^17 + 1119214746 * q^18 - 1602340942 * q^19 + 5664815514 * q^21 + 4024661012 * q^22 - 6555818844 * q^23 - 30645092556 * q^24 - 25715894778 * q^26 - 62762119218 * q^27 - 270752117896 * q^28 + 126894468996 * q^29 + 151760841646 * q^31 + 385411085208 * q^32 - 235078714188 * q^33 + 1431919606684 * q^34 + 445351809528 * q^36 + 616109002068 * q^37 - 2822785016634 * q^38 + 240311247282 * q^39 + 1091281712616 * q^41 + 1233433974654 * q^42 - 2444971199030 * q^43 + 1413344578176 * q^44 - 5480862370044 * q^46 - 8369143269660 * q^47 - 7923128247720 * q^48 + 19523846053580 * q^49 - 7814244769812 * q^51 - 10261294060344 * q^52 - 16571417665824 * q^53 - 2447722649502 * q^54 - 75252275829540 * q^56 + 3504319640154 * q^57 - 3994751501708 * q^58 + 8796604455252 * q^59 - 6959665405750 * q^61 + 52277129313066 * q^62 - 12388951529118 * q^63 + 50304241850208 * q^64 - 8801933633244 * q^66 - 53487461742094 * q^67 + 307147088145312 * q^68 + 14337575811828 * q^69 + 104634162717912 * q^71 + 67020817419972 * q^72 + 177000981923236 * q^73 - 45005277967812 * q^74 + 76188538526328 * q^76 + 117850730172876 * q^77 + 56240661879486 * q^78 + 185514024366160 * q^79 + 137260754729766 * q^81 + 654376907588896 * q^82 + 435827733256908 * q^83 + 592134881838552 * q^84 + 1570057453210014 * q^86 - 277518203694252 * q^87 + 174719041441128 * q^88 + 431517021638688 * q^89 + 466827529337246 * q^91 + 50075827036128 * q^92 - 331900960679802 * q^93 + 1404749684955772 * q^94 - 842894043349896 * q^96 - 1013615616184854 * q^97 + 5117833640095236 * q^98 + 514117147929156 * q^99

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