LMFDB - Embedded newform 75.16.a.j.1.1

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			1.1
Root			$335.765$ 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)

$f(q)$	$=$	$q-296.765 q^{2} -2187.00 q^{3} +55301.4 q^{4} +649025. q^{6} -491831. q^{7} -6.68712e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$	\(q-296.765 q^{2} -2187.00 q^{3} +55301.4 q^{4} +649025. q^{6} -491831. q^{7} -6.68712e6 q^{8} +4.78297e6 q^{9} +9.58763e6 q^{11} -1.20944e8 q^{12} -1.59068e8 q^{13} +1.45958e8 q^{14} +1.72386e8 q^{16} -2.24882e8 q^{17} -1.41942e9 q^{18} +4.77850e9 q^{19} +1.07564e9 q^{21} -2.84527e9 q^{22} +3.94661e8 q^{23} +1.46247e10 q^{24} +4.72057e10 q^{26} -1.04604e10 q^{27} -2.71990e10 q^{28} +3.60539e10 q^{29} +2.24094e11 q^{31} +1.67965e11 q^{32} -2.09682e10 q^{33} +6.67372e10 q^{34} +2.64505e11 q^{36} +6.50682e11 q^{37} -1.41809e12 q^{38} +3.47881e11 q^{39} +8.51327e11 q^{41} -3.19211e11 q^{42} -3.43006e12 q^{43} +5.30209e11 q^{44} -1.17122e11 q^{46} -4.35199e12 q^{47} -3.77009e11 q^{48} -4.50566e12 q^{49} +4.91818e11 q^{51} -8.79666e12 q^{52} +1.02172e13 q^{53} +3.10427e12 q^{54} +3.28893e12 q^{56} -1.04506e13 q^{57} -1.06995e13 q^{58} -7.42071e12 q^{59} -1.50870e13 q^{61} -6.65032e13 q^{62} -2.35241e12 q^{63} -5.54950e13 q^{64} +6.22261e12 q^{66} -7.83383e13 q^{67} -1.24363e13 q^{68} -8.63124e11 q^{69} +5.60869e13 q^{71} -3.19843e13 q^{72} -6.54633e13 q^{73} -1.93100e14 q^{74} +2.64258e14 q^{76} -4.71550e12 q^{77} -1.03239e14 q^{78} +1.32994e13 q^{79} +2.28768e13 q^{81} -2.52644e14 q^{82} +1.92445e14 q^{83} +5.94841e13 q^{84} +1.01792e15 q^{86} -7.88499e13 q^{87} -6.41137e13 q^{88} -3.50728e14 q^{89} +7.82344e13 q^{91} +2.18253e13 q^{92} -4.90094e14 q^{93} +1.29152e15 q^{94} -3.67340e14 q^{96} -8.03059e14 q^{97} +1.33712e15 q^{98} +4.58574e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−296.765	−1.63941	−0.819705	−	0.572786i	$-0.805863\pi$
$2$	−296.765	−1.63941	−0.819705	+	0.572786i	$0.805863\pi$
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	55301.4	1.68766
$5$	0	0
$5$	0	0
$6$	649025.	0.946514
$7$	−491831.	−0.225726	−0.112863	−	0.993611i	$-0.536002\pi$
$7$	−491831.	−0.225726	−0.112863	+	0.993611i	$0.536002\pi$
$8$	−6.68712e6	−1.12736
$9$	4.78297e6	0.333333
$10$	0	0
$11$	9.58763e6	0.148343	0.0741714	−	0.997246i	$-0.476369\pi$
$11$	9.58763e6	0.148343	0.0741714	+	0.997246i	$0.476369\pi$
$12$	−1.20944e8	−0.974374
$13$	−1.59068e8	−0.703083	−0.351541	−	0.936172i	$-0.614342\pi$
$13$	−1.59068e8	−0.703083	−0.351541	+	0.936172i	$0.614342\pi$
$14$	1.45958e8	0.370057
$15$	0	0
$16$	1.72386e8	0.160547
$17$	−2.24882e8	−0.132919	−0.0664597	−	0.997789i	$-0.521170\pi$
$17$	−2.24882e8	−0.132919	−0.0664597	+	0.997789i	$0.521170\pi$
$18$	−1.41942e9	−0.546470
$19$	4.77850e9	1.22642	0.613211	−	0.789919i	$-0.289878\pi$
$19$	4.77850e9	1.22642	0.613211	+	0.789919i	$0.289878\pi$
$20$	0	0
$21$	1.07564e9	0.130323
$22$	−2.84527e9	−0.243195
$23$	3.94661e8	0.0241694	0.0120847	−	0.999927i	$-0.496153\pi$
$23$	3.94661e8	0.0241694	0.0120847	+	0.999927i	$0.496153\pi$
$24$	1.46247e10	0.650884
$25$	0	0
$26$	4.72057e10	1.15264
$27$	−1.04604e10	−0.192450
$28$	−2.71990e10	−0.380949
$29$	3.60539e10	0.388121	0.194061	−	0.980990i	$-0.437834\pi$
$29$	3.60539e10	0.388121	0.194061	+	0.980990i	$0.437834\pi$
$30$	0	0
$31$	2.24094e11	1.46291	0.731455	−	0.681890i	$-0.238842\pi$
$31$	2.24094e11	1.46291	0.731455	+	0.681890i	$0.238842\pi$
$32$	1.67965e11	0.864161
$33$	−2.09682e10	−0.0856457
$34$	6.67372e10	0.217910
$35$	0	0
$36$	2.64505e11	0.562555
$37$	6.50682e11	1.12682	0.563412	−	0.826176i	$-0.309488\pi$
$37$	6.50682e11	1.12682	0.563412	+	0.826176i	$0.309488\pi$
$38$	−1.41809e12	−2.01061
$39$	3.47881e11	0.405925
$40$	0	0
$41$	8.51327e11	0.682681	0.341340	−	0.939940i	$-0.389119\pi$
$41$	8.51327e11	0.682681	0.341340	+	0.939940i	$0.389119\pi$
$42$	−3.19211e11	−0.213652
$43$	−3.43006e12	−1.92437	−0.962184	−	0.272401i	$-0.912182\pi$
$43$	−3.43006e12	−1.92437	−0.962184	+	0.272401i	$0.912182\pi$
$44$	5.30209e11	0.250353
$45$	0	0
$46$	−1.17122e11	−0.0396236
$47$	−4.35199e12	−1.25301	−0.626505	−	0.779418i	$-0.715515\pi$
$47$	−4.35199e12	−1.25301	−0.626505	+	0.779418i	$0.715515\pi$
$48$	−3.77009e11	−0.0926919
$49$	−4.50566e12	−0.949048
$50$	0	0
$51$	4.91818e11	0.0767411
$52$	−8.79666e12	−1.18657
$53$	1.02172e13	1.19471	0.597353	−	0.801979i	$-0.296219\pi$
$53$	1.02172e13	1.19471	0.597353	+	0.801979i	$0.296219\pi$
$54$	3.10427e12	0.315505
$55$	0	0
$56$	3.28893e12	0.254475
$57$	−1.04506e13	−0.708075
$58$	−1.06995e13	−0.636290
$59$	−7.42071e12	−0.388200	−0.194100	−	0.980982i	$-0.562179\pi$
$59$	−7.42071e12	−0.388200	−0.194100	+	0.980982i	$0.562179\pi$
$60$	0	0
$61$	−1.50870e13	−0.614650	−0.307325	−	0.951605i	$-0.599434\pi$
$61$	−1.50870e13	−0.614650	−0.307325	+	0.951605i	$0.599434\pi$
$62$	−6.65032e13	−2.39831
$63$	−2.35241e12	−0.0752419
$64$	−5.54950e13	−1.57726
$65$	0	0
$66$	6.22261e12	0.140408
$67$	−7.83383e13	−1.57911	−0.789556	−	0.613678i	$-0.789689\pi$
$67$	−7.83383e13	−1.57911	−0.789556	+	0.613678i	$0.789689\pi$
$68$	−1.24363e13	−0.224324
$69$	−8.63124e11	−0.0139542
$70$	0	0
$71$	5.60869e13	0.731853	0.365927	−	0.930644i	$-0.380752\pi$
$71$	5.60869e13	0.731853	0.365927	+	0.930644i	$0.380752\pi$
$72$	−3.19843e13	−0.375788
$73$	−6.54633e13	−0.693548	−0.346774	−	0.937949i	$-0.612723\pi$
$73$	−6.54633e13	−0.693548	−0.346774	+	0.937949i	$0.612723\pi$
$74$	−1.93100e14	−1.84733
$75$	0	0
$76$	2.64258e14	2.06979
$77$	−4.71550e12	−0.0334848
$78$	−1.03239e14	−0.665478
$79$	1.32994e13	0.0779166	0.0389583	−	0.999241i	$-0.487596\pi$
$79$	1.32994e13	0.0779166	0.0389583	+	0.999241i	$0.487596\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	−2.52644e14	−1.11919
$83$	1.92445e14	0.778431	0.389216	−	0.921147i	$-0.372746\pi$
$83$	1.92445e14	0.778431	0.389216	+	0.921147i	$0.372746\pi$
$84$	5.94841e13	0.219941
$85$	0	0
$86$	1.01792e15	3.15483
$87$	−7.88499e13	−0.224082
$88$	−6.41137e13	−0.167236
$89$	−3.50728e14	−0.840515	−0.420257	−	0.907405i	$-0.638060\pi$
$89$	−3.50728e14	−0.840515	−0.420257	+	0.907405i	$0.638060\pi$
$90$	0	0
$91$	7.82344e13	0.158704
$92$	2.18253e13	0.0407898
$93$	−4.90094e14	−0.844612
$94$	1.29152e15	2.05420
$95$	0	0
$96$	−3.67340e14	−0.498924
$97$	−8.03059e14	−1.00916	−0.504579	−	0.863365i	$-0.668352\pi$
$97$	−8.03059e14	−1.00916	−0.504579	+	0.863365i	$0.668352\pi$
$98$	1.33712e15	1.55588
$99$	4.58574e13	0.0494476
$100$	0	0
$101$	−8.44880e14	−0.784124	−0.392062	−	0.919939i	$-0.628238\pi$
$101$	−8.44880e14	−0.784124	−0.392062	+	0.919939i	$0.628238\pi$
$102$	−1.45954e14	−0.125810
$103$	−5.43350e13	−0.0435312	−0.0217656	−	0.999763i	$-0.506929\pi$
$103$	−5.43350e13	−0.0435312	−0.0217656	+	0.999763i	$0.506929\pi$
$104$	1.06370e15	0.792630
$105$	0	0
$106$	−3.03209e15	−1.95861
$107$	9.15623e14	0.551237	0.275618	−	0.961267i	$-0.411117\pi$
$107$	9.15623e14	0.551237	0.275618	+	0.961267i	$0.411117\pi$
$108$	−5.78472e14	−0.324791
$109$	−7.26399e13	−0.0380607	−0.0190303	−	0.999819i	$-0.506058\pi$
$109$	−7.26399e13	−0.0380607	−0.0190303	+	0.999819i	$0.506058\pi$
$110$	0	0
$111$	−1.42304e15	−0.650572
$112$	−8.47849e13	−0.0362396
$113$	4.95171e15	1.98001	0.990004	−	0.141038i	$-0.0450438\pi$
$113$	4.95171e15	1.98001	0.990004	+	0.141038i	$0.0450438\pi$
$114$	3.10137e15	1.16082
$115$	0	0
$116$	1.99383e15	0.655019
$117$	−7.60815e14	−0.234361
$118$	2.20221e15	0.636419
$119$	1.10604e14	0.0300033
$120$	0	0
$121$	−4.08533e15	−0.977994
$122$	4.47728e15	1.00766
$123$	−1.86185e15	−0.394146
$124$	1.23927e16	2.46890
$125$	0	0
$126$	6.98114e14	0.123352
$127$	3.94438e15	0.656827	0.328413	−	0.944534i	$-0.393486\pi$
$127$	3.94438e15	0.656827	0.328413	+	0.944534i	$0.393486\pi$
$128$	1.09651e16	1.72162
$129$	7.50154e15	1.11103
$130$	0	0
$131$	8.40310e15	1.10893	0.554466	−	0.832207i	$-0.312923\pi$
$131$	8.40310e15	1.10893	0.554466	+	0.832207i	$0.312923\pi$
$132$	−1.15957e15	−0.144541
$133$	−2.35022e15	−0.276835
$134$	2.32480e16	2.58881
$135$	0	0
$136$	1.50382e15	0.149849
$137$	3.70993e15	0.349914	0.174957	−	0.984576i	$-0.444021\pi$
$137$	3.70993e15	0.349914	0.174957	+	0.984576i	$0.444021\pi$
$138$	2.56145e14	0.0228767
$139$	2.21046e15	0.187013	0.0935065	−	0.995619i	$-0.470192\pi$
$139$	2.21046e15	0.187013	0.0935065	+	0.995619i	$0.470192\pi$
$140$	0	0
$141$	9.51781e15	0.723425
$142$	−1.66446e16	−1.19981
$143$	−1.52508e15	−0.104297
$144$	8.24518e14	0.0535157
$145$	0	0
$146$	1.94272e16	1.13701
$147$	9.85389e15	0.547933
$148$	3.59836e16	1.90170
$149$	−1.20759e16	−0.606767	−0.303383	−	0.952869i	$-0.598116\pi$
$149$	−1.20759e16	−0.606767	−0.303383	+	0.952869i	$0.598116\pi$
$150$	0	0
$151$	1.42120e16	0.646143	0.323071	−	0.946375i	$-0.395285\pi$
$151$	1.42120e16	0.646143	0.323071	+	0.946375i	$0.395285\pi$
$152$	−3.19544e16	−1.38262
$153$	−1.07561e15	−0.0443065
$154$	1.39939e15	0.0548952
$155$	0	0
$156$	1.92383e16	0.685065
$157$	−4.12099e16	−1.39880	−0.699398	−	0.714733i	$-0.746548\pi$
$157$	−4.12099e16	−1.39880	−0.699398	+	0.714733i	$0.746548\pi$
$158$	−3.94680e15	−0.127737
$159$	−2.23449e16	−0.689764
$160$	0	0
$161$	−1.94107e14	−0.00545565
$162$	−6.78903e15	−0.182157
$163$	4.60181e16	1.17902	0.589510	−	0.807761i	$-0.299321\pi$
$163$	4.60181e16	1.17902	0.589510	+	0.807761i	$0.299321\pi$
$164$	4.70796e16	1.15214
$165$	0	0
$166$	−5.71108e16	−1.27617
$167$	−7.31192e16	−1.56192	−0.780959	−	0.624583i	$-0.785269\pi$
$167$	−7.31192e16	−1.56192	−0.780959	+	0.624583i	$0.785269\pi$
$168$	−7.19290e15	−0.146921
$169$	−2.58834e16	−0.505675
$170$	0	0
$171$	2.28554e16	0.408807
$172$	−1.89687e17	−3.24769
$173$	−9.37469e16	−1.53678	−0.768389	−	0.639983i	$-0.778941\pi$
$173$	−9.37469e16	−1.53678	−0.768389	+	0.639983i	$0.778941\pi$
$174$	2.33999e16	0.367362
$175$	0	0
$176$	1.65278e15	0.0238160
$177$	1.62291e16	0.224127
$178$	1.04084e17	1.37795
$179$	4.31918e16	0.548282	0.274141	−	0.961690i	$-0.411607\pi$
$179$	4.31918e16	0.548282	0.274141	+	0.961690i	$0.411607\pi$
$180$	0	0
$181$	3.16042e16	0.369110	0.184555	−	0.982822i	$-0.440916\pi$
$181$	3.16042e16	0.369110	0.184555	+	0.982822i	$0.440916\pi$
$182$	−2.32172e16	−0.260181
$183$	3.29952e16	0.354868
$184$	−2.63915e15	−0.0272477
$185$	0	0
$186$	1.45443e17	1.38466
$187$	−2.15609e15	−0.0197176
$188$	−2.40671e17	−2.11466
$189$	5.14473e15	0.0434409
$190$	0	0
$191$	1.21656e17	0.949258	0.474629	−	0.880186i	$-0.342582\pi$
$191$	1.21656e17	0.949258	0.474629	+	0.880186i	$0.342582\pi$
$192$	1.21368e17	0.910633
$193$	8.47640e16	0.611690	0.305845	−	0.952081i	$-0.401061\pi$
$193$	8.47640e16	0.611690	0.305845	+	0.952081i	$0.401061\pi$
$194$	2.38320e17	1.65442
$195$	0	0
$196$	−2.49169e17	−1.60167
$197$	1.85893e17	1.15018	0.575092	−	0.818089i	$-0.304966\pi$
$197$	1.85893e17	1.15018	0.575092	+	0.818089i	$0.304966\pi$
$198$	−1.36089e16	−0.0810648
$199$	−2.64557e17	−1.51747	−0.758736	−	0.651399i	$-0.774183\pi$
$199$	−2.64557e17	−1.51747	−0.758736	+	0.651399i	$0.774183\pi$
$200$	0	0
$201$	1.71326e17	0.911701
$202$	2.50731e17	1.28550
$203$	−1.77324e16	−0.0876089
$204$	2.71982e16	0.129513
$205$	0	0
$206$	1.61247e16	0.0713654
$207$	1.88765e15	0.00805647
$208$	−2.74211e16	−0.112878
$209$	4.58145e16	0.181931
$210$	0	0
$211$	−4.48707e17	−1.65899	−0.829496	−	0.558513i	$-0.811372\pi$
$211$	−4.48707e17	−1.65899	−0.829496	+	0.558513i	$0.811372\pi$
$212$	5.65023e17	2.01626
$213$	−1.22662e17	−0.422536
$214$	−2.71725e17	−0.903703
$215$	0	0
$216$	6.99496e16	0.216961
$217$	−1.10216e17	−0.330216
$218$	2.15570e16	0.0623971
$219$	1.43168e17	0.400420
$220$	0	0
$221$	3.57715e16	0.0934534
$222$	4.22309e17	1.06655
$223$	7.26819e17	1.77476	0.887380	−	0.461039i	$-0.152523\pi$
$223$	7.26819e17	1.77476	0.887380	+	0.461039i	$0.152523\pi$
$224$	−8.26106e16	−0.195063
$225$	0	0
$226$	−1.46949e18	−3.24605
$227$	3.10929e17	0.664458	0.332229	−	0.943199i	$-0.392199\pi$
$227$	3.10929e17	0.664458	0.332229	+	0.943199i	$0.392199\pi$
$228$	−5.77932e17	−1.19499
$229$	−6.87948e16	−0.137654	−0.0688271	−	0.997629i	$-0.521926\pi$
$229$	−6.87948e16	−0.137654	−0.0688271	+	0.997629i	$0.521926\pi$
$230$	0	0
$231$	1.03128e16	0.0193324
$232$	−2.41097e17	−0.437554
$233$	9.29812e17	1.63390	0.816951	−	0.576707i	$-0.195663\pi$
$233$	9.29812e17	1.63390	0.816951	+	0.576707i	$0.195663\pi$
$234$	2.25783e17	0.384214
$235$	0	0
$236$	−4.10376e17	−0.655151
$237$	−2.90859e16	−0.0449852
$238$	−3.28235e16	−0.0491878
$239$	−1.03843e18	−1.50796	−0.753982	−	0.656895i	$-0.771870\pi$
$239$	−1.03843e18	−1.50796	−0.753982	+	0.656895i	$0.771870\pi$
$240$	0	0
$241$	−1.09585e17	−0.149493	−0.0747467	−	0.997203i	$-0.523815\pi$
$241$	−1.09585e17	−0.149493	−0.0747467	+	0.997203i	$0.523815\pi$
$242$	1.21238e18	1.60333
$243$	−5.00315e16	−0.0641500
$244$	−8.34330e17	−1.03732
$245$	0	0
$246$	5.52533e17	0.646167
$247$	−7.60105e17	−0.862276
$248$	−1.49854e18	−1.64923
$249$	−4.20877e17	−0.449428
$250$	0	0
$251$	1.57273e18	1.58161	0.790807	−	0.612065i	$-0.209661\pi$
$251$	1.57273e18	1.58161	0.790807	+	0.612065i	$0.209661\pi$
$252$	−1.30092e17	−0.126983
$253$	3.78387e15	0.00358535
$254$	−1.17055e18	−1.07681
$255$	0	0
$256$	−1.43559e18	−1.24517
$257$	8.82468e17	0.743362	0.371681	−	0.928360i	$-0.378781\pi$
$257$	8.82468e17	0.743362	0.371681	+	0.928360i	$0.378781\pi$
$258$	−2.22619e18	−1.82144
$259$	−3.20026e17	−0.254353
$260$	0	0
$261$	1.72445e17	0.129374
$262$	−2.49374e18	−1.81799
$263$	−7.11055e16	−0.0503773	−0.0251887	−	0.999683i	$-0.508019\pi$
$263$	−7.11055e16	−0.0503773	−0.0251887	+	0.999683i	$0.508019\pi$
$264$	1.40217e17	0.0965539
$265$	0	0
$266$	6.97462e17	0.453846
$267$	7.67042e17	0.485271
$268$	−4.33222e18	−2.66501
$269$	−1.04614e18	−0.625820	−0.312910	−	0.949783i	$-0.601304\pi$
$269$	−1.04614e18	−0.625820	−0.312910	+	0.949783i	$0.601304\pi$
$270$	0	0
$271$	1.23567e18	0.699251	0.349626	−	0.936890i	$-0.386309\pi$
$271$	1.23567e18	0.699251	0.349626	+	0.936890i	$0.386309\pi$
$272$	−3.87666e16	−0.0213398
$273$	−1.71099e17	−0.0916277
$274$	−1.10098e18	−0.573652
$275$	0	0
$276$	−4.77320e16	−0.0235500
$277$	8.25031e17	0.396161	0.198081	−	0.980186i	$-0.436529\pi$
$277$	8.25031e17	0.396161	0.198081	+	0.980186i	$0.436529\pi$
$278$	−6.55987e17	−0.306591
$279$	1.07183e18	0.487637
$280$	0	0
$281$	3.50140e18	1.50988	0.754942	−	0.655791i	$-0.227665\pi$
$281$	3.50140e18	1.50988	0.754942	+	0.655791i	$0.227665\pi$
$282$	−2.82455e18	−1.18599
$283$	1.44557e18	0.591073	0.295536	−	0.955332i	$-0.404502\pi$
$283$	1.44557e18	0.591073	0.295536	+	0.955332i	$0.404502\pi$
$284$	3.10168e18	1.23512
$285$	0	0
$286$	4.52591e17	0.170986
$287$	−4.18710e17	−0.154099
$288$	8.03373e17	0.288054
$289$	−2.81185e18	−0.982332
$290$	0	0
$291$	1.75629e18	0.582638
$292$	−3.62021e18	−1.17048
$293$	4.06656e18	1.28151	0.640753	−	0.767747i	$-0.278622\pi$
$293$	4.06656e18	1.28151	0.640753	+	0.767747i	$0.278622\pi$
$294$	−2.92429e18	−0.898287
$295$	0	0
$296$	−4.35119e18	−1.27034
$297$	−1.00290e17	−0.0285486
$298$	3.58370e18	0.994740
$299$	−6.27778e16	−0.0169931
$300$	0	0
$301$	1.68701e18	0.434379
$302$	−4.21763e18	−1.05929
$303$	1.84775e18	0.452714
$304$	8.23748e17	0.196898
$305$	0	0
$306$	3.19202e17	0.0726365
$307$	4.80489e18	1.06695	0.533477	−	0.845815i	$-0.320885\pi$
$307$	4.80489e18	1.06695	0.533477	+	0.845815i	$0.320885\pi$
$308$	−2.60774e17	−0.0565110
$309$	1.18831e17	0.0251327
$310$	0	0
$311$	−9.00080e18	−1.81375	−0.906877	−	0.421395i	$-0.861541\pi$
$311$	−9.00080e18	−1.81375	−0.906877	+	0.421395i	$0.861541\pi$
$312$	−2.32632e18	−0.457625
$313$	4.80923e18	0.923619	0.461810	−	0.886979i	$-0.347200\pi$
$313$	4.80923e18	0.923619	0.461810	+	0.886979i	$0.347200\pi$
$314$	1.22297e19	2.29320
$315$	0	0
$316$	7.35477e17	0.131497
$317$	4.86508e18	0.849466	0.424733	−	0.905319i	$-0.360368\pi$
$317$	4.86508e18	0.849466	0.424733	+	0.905319i	$0.360368\pi$
$318$	6.63119e18	1.13081
$319$	3.45672e17	0.0575750
$320$	0	0
$321$	−2.00247e18	−0.318257
$322$	5.76041e16	0.00894405
$323$	−1.07460e18	−0.163015
$324$	1.26512e18	0.187518
$325$	0	0
$326$	−1.36565e19	−1.93290
$327$	1.58863e17	0.0219743
$328$	−5.69293e18	−0.769630
$329$	2.14045e18	0.282836
$330$	0	0
$331$	1.38852e19	1.75325	0.876623	−	0.481177i	$-0.159791\pi$
$331$	1.38852e19	1.75325	0.876623	+	0.481177i	$0.159791\pi$
$332$	1.06425e19	1.31373
$333$	3.11219e18	0.375608
$334$	2.16992e19	2.56062
$335$	0	0
$336$	1.85425e17	0.0209229
$337$	8.95745e18	0.988462	0.494231	−	0.869330i	$-0.335450\pi$
$337$	8.95745e18	0.988462	0.494231	+	0.869330i	$0.335450\pi$
$338$	7.68128e18	0.829008
$339$	−1.08294e19	−1.14316
$340$	0	0
$341$	2.14853e18	0.217012
$342$	−6.78269e18	−0.670203
$343$	4.55103e18	0.439950
$344$	2.29372e19	2.16946
$345$	0	0
$346$	2.78208e19	2.51941
$347$	2.96933e18	0.263140	0.131570	−	0.991307i	$-0.457998\pi$
$347$	2.96933e18	0.263140	0.131570	+	0.991307i	$0.457998\pi$
$348$	−4.36051e18	−0.378175
$349$	1.29369e19	1.09810	0.549049	−	0.835790i	$-0.314990\pi$
$349$	1.29369e19	1.09810	0.549049	+	0.835790i	$0.314990\pi$
$350$	0	0
$351$	1.66390e18	0.135308
$352$	1.61039e18	0.128192
$353$	7.17137e18	0.558845	0.279423	−	0.960168i	$-0.409857\pi$
$353$	7.17137e18	0.558845	0.279423	+	0.960168i	$0.409857\pi$
$354$	−4.81622e18	−0.367436
$355$	0	0
$356$	−1.93957e19	−1.41851
$357$	−2.41891e17	−0.0173224
$358$	−1.28178e19	−0.898858
$359$	1.05335e19	0.723377	0.361688	−	0.932299i	$-0.382200\pi$
$359$	1.05335e19	0.723377	0.361688	+	0.932299i	$0.382200\pi$
$360$	0	0
$361$	7.65296e18	0.504110
$362$	−9.37903e18	−0.605123
$363$	8.93461e18	0.564645
$364$	4.32647e18	0.267839
$365$	0	0
$366$	−9.79181e18	−0.581775
$367$	−1.95381e18	−0.113733	−0.0568664	−	0.998382i	$-0.518111\pi$
$367$	−1.95381e18	−0.113733	−0.0568664	+	0.998382i	$0.518111\pi$
$368$	6.80342e16	0.00388033
$369$	4.07187e18	0.227560
$370$	0	0
$371$	−5.02512e18	−0.269676
$372$	−2.71029e19	−1.42542
$373$	−9.56758e18	−0.493158	−0.246579	−	0.969123i	$-0.579306\pi$
$373$	−9.56758e18	−0.493158	−0.246579	+	0.969123i	$0.579306\pi$
$374$	6.39852e17	0.0323253
$375$	0	0
$376$	2.91023e19	1.41260
$377$	−5.73501e18	−0.272881
$378$	−1.52678e18	−0.0712175
$379$	−1.62097e19	−0.741277	−0.370638	−	0.928777i	$-0.620861\pi$
$379$	−1.62097e19	−0.741277	−0.370638	+	0.928777i	$0.620861\pi$
$380$	0	0
$381$	−8.62636e18	−0.379219
$382$	−3.61033e19	−1.55622
$383$	1.19452e19	0.504896	0.252448	−	0.967611i	$-0.418764\pi$
$383$	1.19452e19	0.504896	0.252448	+	0.967611i	$0.418764\pi$
$384$	−2.39806e19	−0.993976
$385$	0	0
$386$	−2.51550e19	−1.00281
$387$	−1.64059e19	−0.641456
$388$	−4.44103e19	−1.70312
$389$	−3.79025e19	−1.42576	−0.712879	−	0.701287i	$-0.752609\pi$
$389$	−3.79025e19	−1.42576	−0.712879	+	0.701287i	$0.752609\pi$
$390$	0	0
$391$	−8.87524e16	−0.00321258
$392$	3.01299e19	1.06992
$393$	−1.83776e19	−0.640242
$394$	−5.51666e19	−1.88562
$395$	0	0
$396$	2.53598e18	0.0834509
$397$	−3.80000e18	−0.122703	−0.0613515	−	0.998116i	$-0.519541\pi$
$397$	−3.80000e18	−0.122703	−0.0613515	+	0.998116i	$0.519541\pi$
$398$	7.85111e19	2.48776
$399$	5.13992e18	0.159831
$400$	0	0
$401$	2.02057e19	0.605190	0.302595	−	0.953119i	$-0.402147\pi$
$401$	2.02057e19	0.605190	0.302595	+	0.953119i	$0.402147\pi$
$402$	−5.08435e19	−1.49465
$403$	−3.56461e19	−1.02855
$404$	−4.67231e19	−1.32334
$405$	0	0
$406$	5.26237e18	0.143627
$407$	6.23850e18	0.167156
$408$	−3.28885e18	−0.0865152
$409$	3.96107e18	0.102303	0.0511515	−	0.998691i	$-0.483711\pi$
$409$	3.96107e18	0.102303	0.0511515	+	0.998691i	$0.483711\pi$
$410$	0	0
$411$	−8.11361e18	−0.202023
$412$	−3.00480e18	−0.0734660
$413$	3.64974e18	0.0876266
$414$	−5.60189e17	−0.0132079
$415$	0	0
$416$	−2.67178e19	−0.607577
$417$	−4.83428e18	−0.107972
$418$	−1.35961e19	−0.298259
$419$	2.62439e18	0.0565487	0.0282744	−	0.999600i	$-0.490999\pi$
$419$	2.62439e18	0.0565487	0.0282744	+	0.999600i	$0.490999\pi$
$420$	0	0
$421$	2.88629e19	0.600100	0.300050	−	0.953923i	$-0.402997\pi$
$421$	2.88629e19	0.600100	0.300050	+	0.953923i	$0.402997\pi$
$422$	1.33160e20	2.71977
$423$	−2.08154e19	−0.417670
$424$	−6.83233e19	−1.34687
$425$	0	0
$426$	3.64018e19	0.692709
$427$	7.42024e18	0.138742
$428$	5.06352e19	0.930303
$429$	3.33535e18	0.0602160
$430$	0	0
$431$	9.78746e19	1.70644	0.853219	−	0.521553i	$-0.174647\pi$
$431$	9.78746e19	1.70644	0.853219	+	0.521553i	$0.174647\pi$
$432$	−1.80322e18	−0.0308973
$433$	−7.17550e19	−1.20835	−0.604176	−	0.796851i	$-0.706497\pi$
$433$	−7.17550e19	−1.20835	−0.604176	+	0.796851i	$0.706497\pi$
$434$	3.27084e19	0.541360
$435$	0	0
$436$	−4.01709e18	−0.0642337
$437$	1.88589e18	0.0296419
$438$	−4.24873e19	−0.656453
$439$	6.58988e19	1.00091	0.500453	−	0.865764i	$-0.333167\pi$
$439$	6.58988e19	1.00091	0.500453	+	0.865764i	$0.333167\pi$
$440$	0	0
$441$	−2.15504e19	−0.316349
$442$	−1.06157e19	−0.153208
$443$	−1.30271e20	−1.84850	−0.924248	−	0.381792i	$-0.875307\pi$
$443$	−1.30271e20	−1.84850	−0.924248	+	0.381792i	$0.875307\pi$
$444$	−7.86962e19	−1.09795
$445$	0	0
$446$	−2.15694e20	−2.90956
$447$	2.64100e19	0.350317
$448$	2.72942e19	0.356028
$449$	1.17141e20	1.50266	0.751332	−	0.659924i	$-0.229412\pi$
$449$	1.17141e20	1.50266	0.751332	+	0.659924i	$0.229412\pi$
$450$	0	0
$451$	8.16222e18	0.101271
$452$	2.73837e20	3.34159
$453$	−3.10817e19	−0.373051
$454$	−9.22728e19	−1.08932
$455$	0	0
$456$	6.98843e19	0.798258
$457$	−4.96450e18	−0.0557833	−0.0278916	−	0.999611i	$-0.508879\pi$
$457$	−4.96450e18	−0.0557833	−0.0278916	+	0.999611i	$0.508879\pi$
$458$	2.04159e19	0.225672
$459$	2.35235e18	0.0255804
$460$	0	0
$461$	−9.75796e19	−1.02707	−0.513537	−	0.858067i	$-0.671665\pi$
$461$	−9.75796e19	−1.02707	−0.513537	+	0.858067i	$0.671665\pi$
$462$	−3.06048e18	−0.0316938
$463$	2.61466e19	0.266415	0.133207	−	0.991088i	$-0.457472\pi$
$463$	2.61466e19	0.266415	0.133207	+	0.991088i	$0.457472\pi$
$464$	6.21520e18	0.0623118
$465$	0	0
$466$	−2.75936e20	−2.67863
$467$	1.08959e20	1.04085	0.520425	−	0.853908i	$-0.325774\pi$
$467$	1.08959e20	1.04085	0.520425	+	0.853908i	$0.325774\pi$
$468$	−4.20741e19	−0.395523
$469$	3.85292e19	0.356446
$470$	0	0
$471$	9.01261e19	0.807595
$472$	4.96232e19	0.437642
$473$	−3.28861e19	−0.285466
$474$	8.63166e18	0.0737491
$475$	0	0
$476$	6.11657e18	0.0506356
$477$	4.88683e19	0.398235
$478$	3.08168e20	2.47217
$479$	3.61383e19	0.285398	0.142699	−	0.989766i	$-0.454422\pi$
$479$	3.61383e19	0.285398	0.142699	+	0.989766i	$0.454422\pi$
$480$	0	0
$481$	−1.03502e20	−0.792250
$482$	3.25209e19	0.245081
$483$	4.24512e17	0.00314982
$484$	−2.25924e20	−1.65053
$485$	0	0
$486$	1.48476e19	0.105168
$487$	−1.59588e20	−1.11310	−0.556550	−	0.830814i	$-0.687875\pi$
$487$	−1.59588e20	−1.11310	−0.556550	+	0.830814i	$0.687875\pi$
$488$	1.00888e20	0.692934
$489$	−1.00641e20	−0.680708
$490$	0	0
$491$	−2.32260e20	−1.52357	−0.761787	−	0.647828i	$-0.775678\pi$
$491$	−2.32260e20	−1.52357	−0.761787	+	0.647828i	$0.775678\pi$
$492$	−1.02963e20	−0.665186
$493$	−8.10789e18	−0.0515889
$494$	2.25572e20	1.41362
$495$	0	0
$496$	3.86307e19	0.234866
$497$	−2.75853e19	−0.165198
$498$	1.24901e20	0.736796
$499$	9.25050e19	0.537540	0.268770	−	0.963204i	$-0.413383\pi$
$499$	9.25050e19	0.537540	0.268770	+	0.963204i	$0.413383\pi$
$500$	0	0
$501$	1.59912e20	0.901773
$502$	−4.66730e20	−2.59291
$503$	−3.59183e20	−1.96587	−0.982937	−	0.183944i	$-0.941114\pi$
$503$	−3.59183e20	−1.96587	−0.982937	+	0.183944i	$0.941114\pi$
$504$	1.57309e19	0.0848250
$505$	0	0
$506$	−1.12292e18	−0.00587787
$507$	5.66070e19	0.291951
$508$	2.18130e20	1.10850
$509$	1.97454e20	0.988739	0.494369	−	0.869252i	$-0.335399\pi$
$509$	1.97454e20	0.988739	0.494369	+	0.869252i	$0.335399\pi$
$510$	0	0
$511$	3.21969e19	0.156552
$512$	6.67288e19	0.319734
$513$	−4.99848e19	−0.236025
$514$	−2.61885e20	−1.21868
$515$	0	0
$516$	4.14845e20	1.87505
$517$	−4.17253e19	−0.185875
$518$	9.49724e19	0.416989
$519$	2.05024e20	0.887259
$520$	0	0
$521$	4.47194e20	1.88024	0.940120	−	0.340845i	$-0.110713\pi$
$521$	4.47194e20	1.88024	0.940120	+	0.340845i	$0.110713\pi$
$522$	−5.11756e19	−0.212097
$523$	−1.64265e20	−0.671092	−0.335546	−	0.942024i	$-0.608921\pi$
$523$	−1.64265e20	−0.671092	−0.335546	+	0.942024i	$0.608921\pi$
$524$	4.64703e20	1.87150
$525$	0	0
$526$	2.11016e19	0.0825890
$527$	−5.03948e19	−0.194449
$528$	−3.61462e18	−0.0137502
$529$	−2.66479e20	−0.999416
$530$	0	0
$531$	−3.54930e19	−0.129400
$532$	−1.29970e20	−0.467204
$533$	−1.35419e20	−0.479981
$534$	−2.27631e20	−0.795559
$535$	0	0
$536$	5.23857e20	1.78023
$537$	−9.44605e19	−0.316551
$538$	3.10459e20	1.02598
$539$	−4.31986e19	−0.140784
$540$	0	0
$541$	1.97992e20	0.627577	0.313789	−	0.949493i	$-0.398402\pi$
$541$	1.97992e20	0.627577	0.313789	+	0.949493i	$0.398402\pi$
$542$	−3.66704e20	−1.14636
$543$	−6.91185e19	−0.213106
$544$	−3.77725e19	−0.114864
$545$	0	0
$546$	5.07761e19	0.150215
$547$	−3.02968e19	−0.0884082	−0.0442041	−	0.999023i	$-0.514075\pi$
$547$	−3.02968e19	−0.0884082	−0.0442041	+	0.999023i	$0.514075\pi$
$548$	2.05164e20	0.590537
$549$	−7.21604e19	−0.204883
$550$	0	0
$551$	1.72284e20	0.476000
$552$	5.77182e18	0.0157315
$553$	−6.54108e18	−0.0175878
$554$	−2.44840e20	−0.649471
$555$	0	0
$556$	1.22242e20	0.315615
$557$	3.10588e20	0.791172	0.395586	−	0.918429i	$-0.370542\pi$
$557$	3.10588e20	0.791172	0.395586	+	0.918429i	$0.370542\pi$
$558$	−3.18083e20	−0.799436
$559$	5.45611e20	1.35299
$560$	0	0
$561$	4.71537e18	0.0113840
$562$	−1.03909e21	−2.47532
$563$	2.93130e20	0.689044	0.344522	−	0.938778i	$-0.388041\pi$
$563$	2.93130e20	0.689044	0.344522	+	0.938778i	$0.388041\pi$
$564$	5.26348e20	1.22090
$565$	0	0
$566$	−4.28994e20	−0.969010
$567$	−1.12515e19	−0.0250806
$568$	−3.75060e20	−0.825065
$569$	−1.42639e20	−0.309668	−0.154834	−	0.987941i	$-0.549484\pi$
$569$	−1.42639e20	−0.309668	−0.154834	+	0.987941i	$0.549484\pi$
$570$	0	0
$571$	−3.48495e19	−0.0736929	−0.0368464	−	0.999321i	$-0.511731\pi$
$571$	−3.48495e19	−0.0736929	−0.0368464	+	0.999321i	$0.511731\pi$
$572$	−8.43391e19	−0.176019
$573$	−2.66062e20	−0.548055
$574$	1.24258e20	0.252631
$575$	0	0
$576$	−2.65431e20	−0.525754
$577$	2.16579e20	0.423445	0.211723	−	0.977330i	$-0.432093\pi$
$577$	2.16579e20	0.423445	0.211723	+	0.977330i	$0.432093\pi$
$578$	8.34459e20	1.61045
$579$	−1.85379e20	−0.353159
$580$	0	0
$581$	−9.46504e19	−0.175712
$582$	−5.21205e20	−0.955182
$583$	9.79583e19	0.177226
$584$	4.37761e20	0.781881
$585$	0	0
$586$	−1.20681e21	−2.10091
$587$	2.82916e20	0.486263	0.243131	−	0.969993i	$-0.421825\pi$
$587$	2.82916e20	0.486263	0.243131	+	0.969993i	$0.421825\pi$
$588$	5.44934e20	0.924727
$589$	1.07083e21	1.79414
$590$	0	0
$591$	−4.06549e20	−0.664059
$592$	1.12169e20	0.180908
$593$	−1.88434e20	−0.300089	−0.150044	−	0.988679i	$-0.547942\pi$
$593$	−1.88434e20	−0.300089	−0.150044	+	0.988679i	$0.547942\pi$
$594$	2.97626e19	0.0468028
$595$	0	0
$596$	−6.67813e20	−1.02402
$597$	5.78585e20	0.876113
$598$	1.86303e19	0.0278586
$599$	−3.60525e20	−0.532396	−0.266198	−	0.963918i	$-0.585767\pi$
$599$	−3.60525e20	−0.532396	−0.266198	+	0.963918i	$0.585767\pi$
$600$	0	0
$601$	7.45425e20	1.07361	0.536804	−	0.843707i	$-0.319632\pi$
$601$	7.45425e20	1.07361	0.536804	+	0.843707i	$0.319632\pi$
$602$	−5.00645e20	−0.712125
$603$	−3.74690e20	−0.526371
$604$	7.85944e20	1.09047
$605$	0	0
$606$	−5.48348e20	−0.742184
$607$	6.45660e20	0.863155	0.431577	−	0.902076i	$-0.357957\pi$
$607$	6.45660e20	0.863155	0.431577	+	0.902076i	$0.357957\pi$
$608$	8.02623e20	1.05983
$609$	3.87809e19	0.0505810
$610$	0	0
$611$	6.92261e20	0.880969
$612$	−5.94825e19	−0.0747745
$613$	7.08854e20	0.880244	0.440122	−	0.897938i	$-0.354935\pi$
$613$	7.08854e20	0.880244	0.440122	+	0.897938i	$0.354935\pi$
$614$	−1.42592e21	−1.74917
$615$	0	0
$616$	3.15331e19	0.0377495
$617$	9.70317e20	1.14756	0.573779	−	0.819010i	$-0.305477\pi$
$617$	9.70317e20	1.14756	0.573779	+	0.819010i	$0.305477\pi$
$618$	−3.52647e19	−0.0412028
$619$	2.10913e20	0.243458	0.121729	−	0.992563i	$-0.461156\pi$
$619$	2.10913e20	0.243458	0.121729	+	0.992563i	$0.461156\pi$
$620$	0	0
$621$	−4.12830e18	−0.00465140
$622$	2.67112e21	2.97349
$623$	1.72499e20	0.189726
$624$	5.99698e19	0.0651701
$625$	0	0
$626$	−1.42721e21	−1.51419
$627$	−1.00196e20	−0.105038
$628$	−2.27897e21	−2.36070
$629$	−1.46327e20	−0.149777
$630$	0	0
$631$	8.94290e20	0.893837	0.446918	−	0.894575i	$-0.352521\pi$
$631$	8.94290e20	0.893837	0.446918	+	0.894575i	$0.352521\pi$
$632$	−8.89349e19	−0.0878403
$633$	9.81322e20	0.957819
$634$	−1.44379e21	−1.39262
$635$	0	0
$636$	−1.23571e21	−1.16409
$637$	7.16705e20	0.667259
$638$	−1.02583e20	−0.0943890
$639$	2.68262e20	0.243951
$640$	0	0
$641$	1.36720e21	1.21450	0.607249	−	0.794512i	$-0.292273\pi$
$641$	1.36720e21	1.21450	0.607249	+	0.794512i	$0.292273\pi$
$642$	5.94262e20	0.521753
$643$	−1.31579e21	−1.14184	−0.570919	−	0.821006i	$-0.693413\pi$
$643$	−1.31579e21	−1.14184	−0.570919	+	0.821006i	$0.693413\pi$
$644$	−1.07344e19	−0.00920731
$645$	0	0
$646$	3.18904e20	0.267249
$647$	1.46944e21	1.21722	0.608610	−	0.793470i	$-0.291727\pi$
$647$	1.46944e21	1.21722	0.608610	+	0.793470i	$0.291727\pi$
$648$	−1.52980e20	−0.125263
$649$	−7.11470e19	−0.0575866
$650$	0	0
$651$	2.41043e20	0.190650
$652$	2.54486e21	1.98979
$653$	7.39005e19	0.0571214	0.0285607	−	0.999592i	$-0.490908\pi$
$653$	7.39005e19	0.0571214	0.0285607	+	0.999592i	$0.490908\pi$
$654$	−4.71451e19	−0.0360250
$655$	0	0
$656$	1.46757e20	0.109602
$657$	−3.13109e20	−0.231183
$658$	−6.35209e20	−0.463685
$659$	−8.64680e19	−0.0624044	−0.0312022	−	0.999513i	$-0.509934\pi$
$659$	−8.64680e19	−0.0624044	−0.0312022	+	0.999513i	$0.509934\pi$
$660$	0	0
$661$	2.58364e21	1.82273	0.911363	−	0.411603i	$-0.135031\pi$
$661$	2.58364e21	1.82273	0.911363	+	0.411603i	$0.135031\pi$
$662$	−4.12065e21	−2.87429
$663$	−7.82323e19	−0.0539554
$664$	−1.28690e21	−0.877576
$665$	0	0
$666$	−9.23589e20	−0.615775
$667$	1.42291e19	0.00938066
$668$	−4.04359e21	−2.63599
$669$	−1.58955e21	−1.02466
$670$	0	0
$671$	−1.44648e20	−0.0911789
$672$	1.80669e20	0.112620
$673$	4.17014e20	0.257062	0.128531	−	0.991705i	$-0.458974\pi$
$673$	4.17014e20	0.257062	0.128531	+	0.991705i	$0.458974\pi$
$674$	−2.65826e21	−1.62049
$675$	0	0
$676$	−1.43139e21	−0.853409
$677$	−2.27809e21	−1.34325	−0.671624	−	0.740892i	$-0.734403\pi$
$677$	−2.27809e21	−1.34325	−0.671624	+	0.740892i	$0.734403\pi$
$678$	3.21378e21	1.87411
$679$	3.94969e20	0.227793
$680$	0	0
$681$	−6.80002e20	−0.383625
$682$	−6.37609e20	−0.355772
$683$	1.19082e20	0.0657188	0.0328594	−	0.999460i	$-0.489539\pi$
$683$	1.19082e20	0.0657188	0.0328594	+	0.999460i	$0.489539\pi$
$684$	1.26394e21	0.689929
$685$	0	0
$686$	−1.35058e21	−0.721259
$687$	1.50454e20	0.0794747
$688$	−5.91295e20	−0.308952
$689$	−1.62522e21	−0.839977
$690$	0	0
$691$	3.08789e21	1.56162	0.780811	−	0.624767i	$-0.214806\pi$
$691$	3.08789e21	1.56162	0.780811	+	0.624767i	$0.214806\pi$
$692$	−5.18433e21	−2.59356
$693$	−2.25541e19	−0.0111616
$694$	−8.81192e20	−0.431394
$695$	0	0
$696$	5.27279e20	0.252622
$697$	−1.91449e20	−0.0907416
$698$	−3.83923e21	−1.80023
$699$	−2.03350e21	−0.943334
$700$	0	0
$701$	−2.74976e21	−1.24856	−0.624281	−	0.781200i	$-0.714608\pi$
$701$	−2.74976e21	−1.24856	−0.624281	+	0.781200i	$0.714608\pi$
$702$	−4.93788e20	−0.221826
$703$	3.10929e21	1.38196
$704$	−5.32065e20	−0.233975
$705$	0	0
$706$	−2.12821e21	−0.916177
$707$	4.15539e20	0.176997
$708$	8.97491e20	0.378252
$709$	−1.11379e21	−0.464471	−0.232235	−	0.972660i	$-0.574604\pi$
$709$	−1.11379e21	−0.464471	−0.232235	+	0.972660i	$0.574604\pi$
$710$	0	0
$711$	6.36108e19	0.0259722
$712$	2.34536e21	0.947566
$713$	8.84413e19	0.0353577
$714$	7.17849e19	0.0283986
$715$	0	0
$716$	2.38857e21	0.925315
$717$	2.27104e21	0.870624
$718$	−3.12597e21	−1.18591
$719$	−2.71734e21	−1.02018	−0.510091	−	0.860121i	$-0.670388\pi$
$719$	−2.71734e21	−1.02018	−0.510091	+	0.860121i	$0.670388\pi$
$720$	0	0
$721$	2.67236e19	0.00982610
$722$	−2.27113e21	−0.826443
$723$	2.39662e20	0.0863100
$724$	1.74776e21	0.622934
$725$	0	0
$726$	−2.65148e21	−0.925685
$727$	−4.83249e21	−1.66979	−0.834897	−	0.550406i	$-0.814473\pi$
$727$	−4.83249e21	−1.66979	−0.834897	+	0.550406i	$0.814473\pi$
$728$	−5.23163e20	−0.178917
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	7.71360e20	0.255786
$732$	1.82468e21	0.598899
$733$	2.93534e21	0.953627	0.476814	−	0.879004i	$-0.341792\pi$
$733$	2.93534e21	0.953627	0.476814	+	0.879004i	$0.341792\pi$
$734$	5.79821e20	0.186455
$735$	0	0
$736$	6.62894e19	0.0208863
$737$	−7.51079e20	−0.234250
$738$	−1.20839e21	−0.373065
$739$	4.76474e21	1.45615	0.728075	−	0.685497i	$-0.240415\pi$
$739$	4.76474e21	1.45615	0.728075	+	0.685497i	$0.240415\pi$
$740$	0	0
$741$	1.66235e21	0.497835
$742$	1.49128e21	0.442109
$743$	−5.33765e21	−1.56651	−0.783257	−	0.621698i	$-0.786443\pi$
$743$	−5.33765e21	−1.56651	−0.783257	+	0.621698i	$0.786443\pi$
$744$	3.27732e21	0.952185
$745$	0	0
$746$	2.83932e21	0.808488
$747$	9.20457e20	0.259477
$748$	−1.19235e20	−0.0332768
$749$	−4.50332e20	−0.124428
$750$	0	0
$751$	−7.12705e20	−0.193024	−0.0965118	−	0.995332i	$-0.530769\pi$
$751$	−7.12705e20	−0.193024	−0.0965118	+	0.995332i	$0.530769\pi$
$752$	−7.50224e20	−0.201167
$753$	−3.43956e21	−0.913146
$754$	1.70195e21	0.447365
$755$	0	0
$756$	2.84511e20	0.0733137
$757$	4.47746e21	1.14238	0.571192	−	0.820816i	$-0.306481\pi$
$757$	4.47746e21	1.14238	0.571192	+	0.820816i	$0.306481\pi$
$758$	4.81047e21	1.21526
$759$	−8.27532e18	−0.00207001
$760$	0	0
$761$	6.49938e21	1.59400	0.796998	−	0.603983i	$-0.206420\pi$
$761$	6.49938e21	1.59400	0.796998	+	0.603983i	$0.206420\pi$
$762$	2.56000e21	0.621695
$763$	3.57266e19	0.00859127
$764$	6.72776e21	1.60203
$765$	0	0
$766$	−3.54491e21	−0.827731
$767$	1.18039e21	0.272937
$768$	3.13963e21	0.718902
$769$	4.47834e21	1.01548	0.507738	−	0.861511i	$-0.330482\pi$
$769$	4.47834e21	1.01548	0.507738	+	0.861511i	$0.330482\pi$
$770$	0	0
$771$	−1.92996e21	−0.429180
$772$	4.68757e21	1.03233
$773$	4.94744e21	1.07903	0.539517	−	0.841975i	$-0.318607\pi$
$773$	4.94744e21	1.07903	0.539517	+	0.841975i	$0.318607\pi$
$774$	4.86868e21	1.05161
$775$	0	0
$776$	5.37015e21	1.13769
$777$	6.99897e20	0.146851
$778$	1.12481e22	2.33740
$779$	4.06807e21	0.837255
$780$	0	0
$781$	5.37741e20	0.108565
$782$	2.63386e19	0.00526674
$783$	−3.77137e20	−0.0746940
$784$	−7.76714e20	−0.152367
$785$	0	0
$786$	5.45382e21	1.04962
$787$	−1.84302e21	−0.351333	−0.175667	−	0.984450i	$-0.556208\pi$
$787$	−1.84302e21	−0.351333	−0.175667	+	0.984450i	$0.556208\pi$
$788$	1.02802e22	1.94112
$789$	1.55508e20	0.0290853
$790$	0	0
$791$	−2.43541e21	−0.446939
$792$	−3.06654e20	−0.0557454
$793$	2.39985e21	0.432150
$794$	1.12771e21	0.201161
$795$	0	0
$796$	−1.46303e22	−2.56098
$797$	7.83424e21	1.35850	0.679250	−	0.733907i	$-0.262305\pi$
$797$	7.83424e21	1.35850	0.679250	+	0.733907i	$0.262305\pi$
$798$	−1.52535e21	−0.262028
$799$	9.78687e20	0.166549
$800$	0	0
$801$	−1.67752e21	−0.280172
$802$	−5.99635e21	−0.992154
$803$	−6.27639e20	−0.102883
$804$	9.47456e21	1.53865
$805$	0	0
$806$	1.05785e22	1.68621
$807$	2.28792e21	0.361317
$808$	5.64982e21	0.883994
$809$	−9.09440e21	−1.40981	−0.704904	−	0.709303i	$-0.749010\pi$
$809$	−9.09440e21	−1.40981	−0.704904	+	0.709303i	$0.749010\pi$
$810$	0	0
$811$	−1.21679e22	−1.85166	−0.925829	−	0.377942i	$-0.876632\pi$
$811$	−1.21679e22	−1.85166	−0.925829	+	0.377942i	$0.876632\pi$
$812$	−9.80629e20	−0.147855
$813$	−2.70241e21	−0.403713
$814$	−1.85137e21	−0.274037
$815$	0	0
$816$	8.47826e19	0.0123206
$817$	−1.63905e22	−2.36009
$818$	−1.17551e21	−0.167716
$819$	3.74193e20	0.0529013
$820$	0	0
$821$	−5.93765e20	−0.0824216	−0.0412108	−	0.999150i	$-0.513122\pi$
$821$	−5.93765e20	−0.0824216	−0.0412108	+	0.999150i	$0.513122\pi$
$822$	2.40783e21	0.331198
$823$	−8.26954e21	−1.12715	−0.563576	−	0.826064i	$-0.690575\pi$
$823$	−8.26954e21	−1.12715	−0.563576	+	0.826064i	$0.690575\pi$
$824$	3.63345e20	0.0490755
$825$	0	0
$826$	−1.08311e21	−0.143656
$827$	−9.64614e21	−1.26783	−0.633917	−	0.773401i	$-0.718554\pi$
$827$	−9.64614e21	−1.26783	−0.633917	+	0.773401i	$0.718554\pi$
$828$	1.04390e20	0.0135966
$829$	1.16507e22	1.50381	0.751903	−	0.659273i	$-0.229136\pi$
$829$	1.16507e22	1.50381	0.751903	+	0.659273i	$0.229136\pi$
$830$	0	0
$831$	−1.80434e21	−0.228724
$832$	8.82745e21	1.10895
$833$	1.01324e21	0.126147
$834$	1.43464e21	0.177010
$835$	0	0
$836$	2.53361e21	0.307038
$837$	−2.34410e21	−0.281537
$838$	−7.78826e20	−0.0927066
$839$	−7.74734e19	−0.00913983	−0.00456991	−	0.999990i	$-0.501455\pi$
$839$	−7.74734e19	−0.00913983	−0.00456991	+	0.999990i	$0.501455\pi$
$840$	0	0
$841$	−7.32930e21	−0.849362
$842$	−8.56549e21	−0.983810
$843$	−7.65755e21	−0.871732
$844$	−2.48141e22	−2.79982
$845$	0	0
$846$	6.17729e21	0.684732
$847$	2.00929e21	0.220758
$848$	1.76130e21	0.191807
$849$	−3.16146e21	−0.341256
$850$	0	0
$851$	2.56799e20	0.0272347
$852$	−6.78338e21	−0.713098
$853$	9.23639e21	0.962464	0.481232	−	0.876593i	$-0.340189\pi$
$853$	9.23639e21	0.962464	0.481232	+	0.876593i	$0.340189\pi$
$854$	−2.20207e21	−0.227455
$855$	0	0
$856$	−6.12288e21	−0.621444
$857$	1.36084e22	1.36915	0.684577	−	0.728941i	$-0.259987\pi$
$857$	1.36084e22	1.36915	0.684577	+	0.728941i	$0.259987\pi$
$858$	−9.89816e20	−0.0987188
$859$	7.31883e21	0.723590	0.361795	−	0.932258i	$-0.382164\pi$
$859$	7.31883e21	0.723590	0.361795	+	0.932258i	$0.382164\pi$
$860$	0	0
$861$	9.15718e20	0.0889689
$862$	−2.90458e22	−2.79755
$863$	−1.31910e22	−1.25949	−0.629747	−	0.776800i	$-0.716842\pi$
$863$	−1.31910e22	−1.25949	−0.629747	+	0.776800i	$0.716842\pi$
$864$	−1.75698e21	−0.166308
$865$	0	0
$866$	2.12944e22	1.98098
$867$	6.14952e21	0.567150
$868$	−6.09512e21	−0.557294
$869$	1.27510e20	0.0115584
$870$	0	0
$871$	1.24611e22	1.11025
$872$	4.85752e20	0.0429082
$873$	−3.84101e21	−0.336386
$874$	−5.59666e20	−0.0485952
$875$	0	0
$876$	7.91741e21	0.675775
$877$	−1.77821e22	−1.50482	−0.752411	−	0.658694i	$-0.771109\pi$
$877$	−1.77821e22	−1.50482	−0.752411	+	0.658694i	$0.771109\pi$
$878$	−1.95564e22	−1.64090
$879$	−8.89357e21	−0.739877
$880$	0	0
$881$	−1.25488e22	−1.02632	−0.513159	−	0.858294i	$-0.671525\pi$
$881$	−1.25488e22	−1.02632	−0.513159	+	0.858294i	$0.671525\pi$
$882$	6.39542e21	0.518626
$883$	1.87696e22	1.50921	0.754605	−	0.656179i	$-0.227828\pi$
$883$	1.87696e22	1.50921	0.754605	+	0.656179i	$0.227828\pi$
$884$	1.97821e21	0.157718
$885$	0	0
$886$	3.86598e22	3.03044
$887$	−4.57191e21	−0.355362	−0.177681	−	0.984088i	$-0.556860\pi$
$887$	−4.57191e21	−0.355362	−0.177681	+	0.984088i	$0.556860\pi$
$888$	9.51605e21	0.733431
$889$	−1.93997e21	−0.148263
$890$	0	0
$891$	2.19334e20	0.0164825
$892$	4.01941e22	2.99520
$893$	−2.07960e22	−1.53672
$894$	−7.83755e21	−0.574313
$895$	0	0
$896$	−5.39296e21	−0.388613
$897$	1.37295e20	0.00981097
$898$	−3.47634e22	−2.46348
$899$	8.07947e21	0.567787
$900$	0	0
$901$	−2.29766e21	−0.158800
$902$	−2.42226e21	−0.166024
$903$	−3.68949e21	−0.250789
$904$	−3.31127e22	−2.23219
$905$	0	0
$906$	9.22395e21	0.611583
$907$	−2.14038e22	−1.40746	−0.703729	−	0.710469i	$-0.748483\pi$
$907$	−2.14038e22	−1.40746	−0.703729	+	0.710469i	$0.748483\pi$
$908$	1.71948e22	1.12138
$909$	−4.04104e21	−0.261375
$910$	0	0
$911$	−2.41994e22	−1.53963	−0.769815	−	0.638267i	$-0.779652\pi$
$911$	−2.41994e22	−1.53963	−0.769815	+	0.638267i	$0.779652\pi$
$912$	−1.80154e21	−0.113679
$913$	1.84509e21	0.115475
$914$	1.47329e21	0.0914516
$915$	0	0
$916$	−3.80445e21	−0.232314
$917$	−4.13291e21	−0.250314
$918$	−6.98095e20	−0.0419367
$919$	1.54883e22	0.922864	0.461432	−	0.887175i	$-0.347336\pi$
$919$	1.54883e22	0.922864	0.461432	+	0.887175i	$0.347336\pi$
$920$	0	0
$921$	−1.05083e22	−0.616006
$922$	2.89582e22	1.68380
$923$	−8.92161e21	−0.514553
$924$	5.70312e20	0.0326267
$925$	0	0
$926$	−7.75940e21	−0.436763
$927$	−2.59883e20	−0.0145104
$928$	6.05581e21	0.335400
$929$	−1.94192e22	−1.06687	−0.533436	−	0.845840i	$-0.679100\pi$
$929$	−1.94192e22	−1.06687	−0.533436	+	0.845840i	$0.679100\pi$
$930$	0	0
$931$	−2.15303e22	−1.16393
$932$	5.14199e22	2.75748
$933$	1.96848e22	1.04717
$934$	−3.23353e22	−1.70638
$935$	0	0
$936$	5.08766e21	0.264210
$937$	1.92188e22	0.990101	0.495051	−	0.868864i	$-0.335149\pi$
$937$	1.92188e22	0.990101	0.495051	+	0.868864i	$0.335149\pi$
$938$	−1.14341e22	−0.584361
$939$	−1.05178e22	−0.533252
$940$	0	0
$941$	−3.06847e22	−1.53109	−0.765544	−	0.643383i	$-0.777530\pi$
$941$	−3.06847e22	−1.53109	−0.765544	+	0.643383i	$0.777530\pi$
$942$	−2.67463e22	−1.32398
$943$	3.35986e20	0.0165000
$944$	−1.27923e21	−0.0623244
$945$	0	0
$946$	9.75945e21	0.467996
$947$	1.84073e21	0.0875721	0.0437861	−	0.999041i	$-0.486058\pi$
$947$	1.84073e21	0.0875721	0.0437861	+	0.999041i	$0.486058\pi$
$948$	−1.60849e21	−0.0759199
$949$	1.04131e22	0.487622
$950$	0	0
$951$	−1.06399e22	−0.490439
$952$	−7.39624e20	−0.0338247
$953$	2.95858e22	1.34242	0.671208	−	0.741269i	$-0.265776\pi$
$953$	2.95858e22	1.34242	0.671208	+	0.741269i	$0.265776\pi$
$954$	−1.45024e22	−0.652871
$955$	0	0
$956$	−5.74264e22	−2.54494
$957$	−7.55984e20	−0.0332409
$958$	−1.07246e22	−0.467885
$959$	−1.82466e21	−0.0789845
$960$	0	0
$961$	2.67529e22	1.14011
$962$	3.07159e22	1.29882
$963$	4.37939e21	0.183746
$964$	−6.06018e21	−0.252295
$965$	0	0
$966$	−1.25980e20	−0.00516385
$967$	−5.01721e20	−0.0204063	−0.0102031	−	0.999948i	$-0.503248\pi$
$967$	−5.01721e20	−0.0204063	−0.0102031	+	0.999948i	$0.503248\pi$
$968$	2.73191e22	1.10256
$969$	2.35015e21	0.0941169
$970$	0	0
$971$	−9.47242e21	−0.373523	−0.186761	−	0.982405i	$-0.559799\pi$
$971$	−9.47242e21	−0.373523	−0.186761	+	0.982405i	$0.559799\pi$
$972$	−2.76681e21	−0.108264
$973$	−1.08717e21	−0.0422136
$974$	4.73602e22	1.82483
$975$	0	0
$976$	−2.60078e21	−0.0986803
$977$	−3.43557e22	−1.29357	−0.646785	−	0.762672i	$-0.723887\pi$
$977$	−3.43557e22	−1.29357	−0.646785	+	0.762672i	$0.723887\pi$
$978$	2.98669e22	1.11596
$979$	−3.36265e21	−0.124684
$980$	0	0
$981$	−3.47434e20	−0.0126869
$982$	6.89266e22	2.49776
$983$	3.76919e22	1.35549	0.677746	−	0.735296i	$-0.262957\pi$
$983$	3.76919e22	1.35549	0.677746	+	0.735296i	$0.262957\pi$
$984$	1.24504e22	0.444346
$985$	0	0
$986$	2.40614e21	0.0845753
$987$	−4.68116e21	−0.163296
$988$	−4.20348e22	−1.45523
$989$	−1.35371e21	−0.0465108
$990$	0	0
$991$	−1.30482e22	−0.441567	−0.220784	−	0.975323i	$-0.570861\pi$
$991$	−1.30482e22	−0.441567	−0.220784	+	0.975323i	$0.570861\pi$
$992$	3.76400e22	1.26419
$993$	−3.03670e22	−1.01224
$994$	8.18635e21	0.270827
$995$	0	0
$996$	−2.32751e22	−0.758483
$997$	−2.05725e22	−0.665386	−0.332693	−	0.943035i	$-0.607957\pi$
$997$	−2.05725e22	−0.665386	−0.332693	+	0.943035i	$0.607957\pi$
$998$	−2.74522e22	−0.881249
$999$	−6.80637e21	−0.216857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.16.a.j.1.1	yes	6
5.2	odd	4		75.16.b.h.49.2		12
5.3	odd	4		75.16.b.h.49.11		12
5.4	even	2		75.16.a.i.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.16.a.i.1.6	✓	6	5.4	even	2
75.16.a.j.1.1	yes	6	1.1	even	1	trivial
75.16.b.h.49.2		12	5.2	odd	4
75.16.b.h.49.11		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)

\(f(q)\)	\(=\)	\(q-296.765 q^{2} -2187.00 q^{3} +55301.4 q^{4} +649025. q^{6} -491831. q^{7} -6.68712e6 q^{8} +4.78297e6 q^{9} +O(q^{10})\)	\(q-296.765 q^{2} -2187.00 q^{3} +55301.4 q^{4} +649025. q^{6} -491831. q^{7} -6.68712e6 q^{8} +4.78297e6 q^{9} +9.58763e6 q^{11} -1.20944e8 q^{12} -1.59068e8 q^{13} +1.45958e8 q^{14} +1.72386e8 q^{16} -2.24882e8 q^{17} -1.41942e9 q^{18} +4.77850e9 q^{19} +1.07564e9 q^{21} -2.84527e9 q^{22} +3.94661e8 q^{23} +1.46247e10 q^{24} +4.72057e10 q^{26} -1.04604e10 q^{27} -2.71990e10 q^{28} +3.60539e10 q^{29} +2.24094e11 q^{31} +1.67965e11 q^{32} -2.09682e10 q^{33} +6.67372e10 q^{34} +2.64505e11 q^{36} +6.50682e11 q^{37} -1.41809e12 q^{38} +3.47881e11 q^{39} +8.51327e11 q^{41} -3.19211e11 q^{42} -3.43006e12 q^{43} +5.30209e11 q^{44} -1.17122e11 q^{46} -4.35199e12 q^{47} -3.77009e11 q^{48} -4.50566e12 q^{49} +4.91818e11 q^{51} -8.79666e12 q^{52} +1.02172e13 q^{53} +3.10427e12 q^{54} +3.28893e12 q^{56} -1.04506e13 q^{57} -1.06995e13 q^{58} -7.42071e12 q^{59} -1.50870e13 q^{61} -6.65032e13 q^{62} -2.35241e12 q^{63} -5.54950e13 q^{64} +6.22261e12 q^{66} -7.83383e13 q^{67} -1.24363e13 q^{68} -8.63124e11 q^{69} +5.60869e13 q^{71} -3.19843e13 q^{72} -6.54633e13 q^{73} -1.93100e14 q^{74} +2.64258e14 q^{76} -4.71550e12 q^{77} -1.03239e14 q^{78} +1.32994e13 q^{79} +2.28768e13 q^{81} -2.52644e14 q^{82} +1.92445e14 q^{83} +5.94841e13 q^{84} +1.01792e15 q^{86} -7.88499e13 q^{87} -6.41137e13 q^{88} -3.50728e14 q^{89} +7.82344e13 q^{91} +2.18253e13 q^{92} -4.90094e14 q^{93} +1.29152e15 q^{94} -3.67340e14 q^{96} -8.03059e14 q^{97} +1.33712e15 q^{98} +4.58574e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9}+O(q^{10}) \) 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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