LMFDB - Embedded newform 75.16.a.j.1.6

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			1.6
Root			$-307.210$ 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)

$f(q)$	$=$	$q+346.210 q^{2} -2187.00 q^{3} +87093.0 q^{4} -757160. q^{6} -4.02969e6 q^{7} +1.88078e7 q^{8} +4.78297e6 q^{9} +O(q^{10})$	\(q+346.210 q^{2} -2187.00 q^{3} +87093.0 q^{4} -757160. q^{6} -4.02969e6 q^{7} +1.88078e7 q^{8} +4.78297e6 q^{9} -2.10466e7 q^{11} -1.90472e8 q^{12} -1.69289e6 q^{13} -1.39512e9 q^{14} +3.65759e9 q^{16} +3.24177e9 q^{17} +1.65591e9 q^{18} -1.81453e9 q^{19} +8.81294e9 q^{21} -7.28653e9 q^{22} +1.20057e10 q^{23} -4.11328e10 q^{24} -5.86095e8 q^{26} -1.04604e10 q^{27} -3.50958e11 q^{28} +1.00382e11 q^{29} +1.92165e11 q^{31} +6.49997e11 q^{32} +4.60289e10 q^{33} +1.12233e12 q^{34} +4.16563e11 q^{36} +1.21165e10 q^{37} -6.28206e11 q^{38} +3.70235e9 q^{39} +1.74434e12 q^{41} +3.05112e12 q^{42} +2.51589e12 q^{43} -1.83301e12 q^{44} +4.15648e12 q^{46} -3.60557e12 q^{47} -7.99915e12 q^{48} +1.14909e13 q^{49} -7.08975e12 q^{51} -1.47439e11 q^{52} -4.83559e12 q^{53} -3.62147e12 q^{54} -7.57898e13 q^{56} +3.96837e12 q^{57} +3.47532e13 q^{58} -1.68448e13 q^{59} -2.26927e13 q^{61} +6.65295e13 q^{62} -1.92739e13 q^{63} +1.05183e14 q^{64} +1.59356e13 q^{66} +2.44503e13 q^{67} +2.82336e14 q^{68} -2.62564e13 q^{69} +9.17615e12 q^{71} +8.99573e13 q^{72} -1.13065e14 q^{73} +4.19484e12 q^{74} -1.58033e14 q^{76} +8.48113e13 q^{77} +1.28179e12 q^{78} +2.09329e14 q^{79} +2.28768e13 q^{81} +6.03906e14 q^{82} -1.53111e14 q^{83} +7.67546e14 q^{84} +8.71025e14 q^{86} -2.19535e14 q^{87} -3.95841e14 q^{88} +3.76237e14 q^{89} +6.82183e12 q^{91} +1.04561e15 q^{92} -4.20266e14 q^{93} -1.24828e15 q^{94} -1.42154e15 q^{96} +6.17001e14 q^{97} +3.97825e15 q^{98} -1.00665e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	346.210	1.91256	0.956278	−	0.292460i	$-0.0944739\pi$
$2$	346.210	1.91256	0.956278	+	0.292460i	$0.0944739\pi$
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	87093.0	2.65787
$5$	0	0
$5$	0	0
$6$	−757160.	−1.10421
$7$	−4.02969e6	−1.84942	−0.924712	−	0.380667i	$-0.875695\pi$
$7$	−4.02969e6	−1.84942	−0.924712	+	0.380667i	$0.875695\pi$
$8$	1.88078e7	3.17077
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−2.10466e7	−0.325639	−0.162820	−	0.986656i	$-0.552059\pi$
$11$	−2.10466e7	−0.325639	−0.162820	+	0.986656i	$0.552059\pi$
$12$	−1.90472e8	−1.53452
$13$	−1.69289e6	−0.00748262	−0.00374131	−	0.999993i	$-0.501191\pi$
$13$	−1.69289e6	−0.00748262	−0.00374131	+	0.999993i	$0.501191\pi$
$14$	−1.39512e9	−3.53713
$15$	0	0
$16$	3.65759e9	3.40640
$17$	3.24177e9	1.91609	0.958044	−	0.286621i	$-0.0925319\pi$
$17$	3.24177e9	1.91609	0.958044	+	0.286621i	$0.0925319\pi$
$18$	1.65591e9	0.637518
$19$	−1.81453e9	−0.465705	−0.232853	−	0.972512i	$-0.574806\pi$
$19$	−1.81453e9	−0.465705	−0.232853	+	0.972512i	$0.574806\pi$
$20$	0	0
$21$	8.81294e9	1.06777
$22$	−7.28653e9	−0.622803
$23$	1.20057e10	0.735238	0.367619	−	0.929976i	$-0.380173\pi$
$23$	1.20057e10	0.735238	0.367619	+	0.929976i	$0.380173\pi$
$24$	−4.11328e10	−1.83064
$25$	0	0
$26$	−5.86095e8	−0.0143109
$27$	−1.04604e10	−0.192450
$28$	−3.50958e11	−4.91553
$29$	1.00382e11	1.08061	0.540307	−	0.841468i	$-0.318308\pi$
$29$	1.00382e11	1.08061	0.540307	+	0.841468i	$0.318308\pi$
$30$	0	0
$31$	1.92165e11	1.25448	0.627238	−	0.778827i	$-0.284185\pi$
$31$	1.92165e11	1.25448	0.627238	+	0.778827i	$0.284185\pi$
$32$	6.49997e11	3.34416
$33$	4.60289e10	0.188008
$34$	1.12233e12	3.66463
$35$	0	0
$36$	4.16563e11	0.885956
$37$	1.21165e10	0.0209828	0.0104914	−	0.999945i	$-0.496660\pi$
$37$	1.21165e10	0.0209828	0.0104914	+	0.999945i	$0.496660\pi$
$38$	−6.28206e11	−0.890687
$39$	3.70235e9	0.00432009
$40$	0	0
$41$	1.74434e12	1.39879	0.699394	−	0.714737i	$-0.253453\pi$
$41$	1.74434e12	1.39879	0.699394	+	0.714737i	$0.253453\pi$
$42$	3.05112e12	2.04216
$43$	2.51589e12	1.41149	0.705745	−	0.708466i	$-0.250612\pi$
$43$	2.51589e12	1.41149	0.705745	+	0.708466i	$0.250612\pi$
$44$	−1.83301e12	−0.865506
$45$	0	0
$46$	4.15648e12	1.40618
$47$	−3.60557e12	−1.03810	−0.519051	−	0.854743i	$-0.673715\pi$
$47$	−3.60557e12	−1.03810	−0.519051	+	0.854743i	$0.673715\pi$
$48$	−7.99915e12	−1.96668
$49$	1.14909e13	2.42037
$50$	0	0
$51$	−7.08975e12	−1.10625
$52$	−1.47439e11	−0.0198878
$53$	−4.83559e12	−0.565432	−0.282716	−	0.959204i	$-0.591235\pi$
$53$	−4.83559e12	−0.565432	−0.282716	+	0.959204i	$0.591235\pi$
$54$	−3.62147e12	−0.368071
$55$	0	0
$56$	−7.57898e13	−5.86409
$57$	3.96837e12	0.268875
$58$	3.47532e13	2.06674
$59$	−1.68448e13	−0.881205	−0.440602	−	0.897702i	$-0.645235\pi$
$59$	−1.68448e13	−0.881205	−0.440602	+	0.897702i	$0.645235\pi$
$60$	0	0
$61$	−2.26927e13	−0.924511	−0.462256	−	0.886747i	$-0.652960\pi$
$61$	−2.26927e13	−0.924511	−0.462256	+	0.886747i	$0.652960\pi$
$62$	6.65295e13	2.39926
$63$	−1.92739e13	−0.616475
$64$	1.05183e14	2.98949
$65$	0	0
$66$	1.59356e13	0.359575
$67$	2.44503e13	0.492860	0.246430	−	0.969161i	$-0.420742\pi$
$67$	2.44503e13	0.492860	0.246430	+	0.969161i	$0.420742\pi$
$68$	2.82336e14	5.09271
$69$	−2.62564e13	−0.424490
$70$	0	0
$71$	9.17615e12	0.119735	0.0598677	−	0.998206i	$-0.480932\pi$
$71$	9.17615e12	0.119735	0.0598677	+	0.998206i	$0.480932\pi$
$72$	8.99573e13	1.05692
$73$	−1.13065e14	−1.19787	−0.598933	−	0.800799i	$-0.704408\pi$
$73$	−1.13065e14	−1.19787	−0.598933	+	0.800799i	$0.704408\pi$
$74$	4.19484e12	0.0401308
$75$	0	0
$76$	−1.58033e14	−1.23778
$77$	8.48113e13	0.602245
$78$	1.28179e12	0.00826242
$79$	2.09329e14	1.22638	0.613190	−	0.789935i	$-0.289886\pi$
$79$	2.09329e14	1.22638	0.613190	+	0.789935i	$0.289886\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	6.03906e14	2.67526
$83$	−1.53111e14	−0.619329	−0.309664	−	0.950846i	$-0.600217\pi$
$83$	−1.53111e14	−0.619329	−0.309664	+	0.950846i	$0.600217\pi$
$84$	7.67546e14	2.83798
$85$	0	0
$86$	8.71025e14	2.69955
$87$	−2.19535e14	−0.623893
$88$	−3.95841e14	−1.03253
$89$	3.76237e14	0.901646	0.450823	−	0.892613i	$-0.351131\pi$
$89$	3.76237e14	0.901646	0.450823	+	0.892613i	$0.351131\pi$
$90$	0	0
$91$	6.82183e12	0.0138385
$92$	1.04561e15	1.95417
$93$	−4.20266e14	−0.724272
$94$	−1.24828e15	−1.98543
$95$	0	0
$96$	−1.42154e15	−1.93075
$97$	6.17001e14	0.775351	0.387675	−	0.921796i	$-0.373278\pi$
$97$	6.17001e14	0.775351	0.387675	+	0.921796i	$0.373278\pi$
$98$	3.97825e15	4.62909
$99$	−1.00665e14	−0.108546
$100$	0	0
$101$	6.73958e14	0.625493	0.312746	−	0.949837i	$-0.398751\pi$
$101$	6.73958e14	0.625493	0.312746	+	0.949837i	$0.398751\pi$
$102$	−2.45454e15	−2.11577
$103$	−3.00478e14	−0.240731	−0.120366	−	0.992730i	$-0.538407\pi$
$103$	−3.00478e14	−0.240731	−0.120366	+	0.992730i	$0.538407\pi$
$104$	−3.18396e13	−0.0237256
$105$	0	0
$106$	−1.67413e15	−1.08142
$107$	−1.29643e15	−0.780499	−0.390249	−	0.920709i	$-0.627611\pi$
$107$	−1.29643e15	−0.780499	−0.390249	+	0.920709i	$0.627611\pi$
$108$	−9.11024e14	−0.511507
$109$	3.47880e15	1.82277	0.911384	−	0.411558i	$-0.135015\pi$
$109$	3.47880e15	1.82277	0.911384	+	0.411558i	$0.135015\pi$
$110$	0	0
$111$	−2.64988e13	−0.0121144
$112$	−1.47390e16	−6.29987
$113$	−4.00096e14	−0.159984	−0.0799919	−	0.996796i	$-0.525489\pi$
$113$	−4.00096e14	−0.159984	−0.0799919	+	0.996796i	$0.525489\pi$
$114$	1.37389e15	0.514239
$115$	0	0
$116$	8.74257e15	2.87213
$117$	−8.09704e12	−0.00249421
$118$	−5.83185e15	−1.68535
$119$	−1.30633e16	−3.54366
$120$	0	0
$121$	−3.73429e15	−0.893959
$122$	−7.85642e15	−1.76818
$123$	−3.81487e15	−0.807590
$124$	1.67363e16	3.33423
$125$	0	0
$126$	−6.67281e15	−1.17904
$127$	6.37216e15	1.06111	0.530553	−	0.847652i	$-0.321984\pi$
$127$	6.37216e15	1.06111	0.530553	+	0.847652i	$0.321984\pi$
$128$	1.51163e16	2.37341
$129$	−5.50225e15	−0.814924
$130$	0	0
$131$	9.76047e15	1.28806	0.644030	−	0.765000i	$-0.277261\pi$
$131$	9.76047e15	1.28806	0.644030	+	0.765000i	$0.277261\pi$
$132$	4.00880e15	0.499700
$133$	7.31198e15	0.861287
$134$	8.46494e15	0.942623
$135$	0	0
$136$	6.09707e16	6.07547
$137$	6.25809e15	0.590252	0.295126	−	0.955458i	$-0.404638\pi$
$137$	6.25809e15	0.590252	0.295126	+	0.955458i	$0.404638\pi$
$138$	−9.09023e15	−0.811861
$139$	1.85802e15	0.157195	0.0785976	−	0.996906i	$-0.474956\pi$
$139$	1.85802e15	0.157195	0.0785976	+	0.996906i	$0.474956\pi$
$140$	0	0
$141$	7.88539e15	0.599349
$142$	3.17687e15	0.229001
$143$	3.56296e13	0.00243663
$144$	1.74941e16	1.13547
$145$	0	0
$146$	−3.91443e16	−2.29099
$147$	−2.51305e16	−1.39740
$148$	1.05526e15	0.0557696
$149$	1.69137e16	0.849849	0.424924	−	0.905229i	$-0.360301\pi$
$149$	1.69137e16	0.849849	0.424924	+	0.905229i	$0.360301\pi$
$150$	0	0
$151$	−1.02023e16	−0.463842	−0.231921	−	0.972735i	$-0.574501\pi$
$151$	−1.02023e16	−0.463842	−0.231921	+	0.972735i	$0.574501\pi$
$152$	−3.41273e16	−1.47664
$153$	1.55053e16	0.638696
$154$	2.93625e16	1.15183
$155$	0	0
$156$	3.22449e14	0.0114822
$157$	6.78067e15	0.230158	0.115079	−	0.993356i	$-0.463288\pi$
$157$	6.78067e15	0.230158	0.115079	+	0.993356i	$0.463288\pi$
$158$	7.24715e16	2.34552
$159$	1.05754e16	0.326452
$160$	0	0
$161$	−4.83792e16	−1.35977
$162$	7.92016e15	0.212506
$163$	−2.90902e16	−0.745316	−0.372658	−	0.927969i	$-0.621554\pi$
$163$	−2.90902e16	−0.745316	−0.372658	+	0.927969i	$0.621554\pi$
$164$	1.51920e17	3.71779
$165$	0	0
$166$	−5.30086e16	−1.18450
$167$	−5.40971e16	−1.15558	−0.577791	−	0.816185i	$-0.696085\pi$
$167$	−5.40971e16	−1.15558	−0.577791	+	0.816185i	$0.696085\pi$
$168$	1.65752e17	3.38563
$169$	−5.11830e16	−0.999944
$170$	0	0
$171$	−8.67882e15	−0.155235
$172$	2.19116e17	3.75156
$173$	−3.89515e16	−0.638525	−0.319263	−	0.947666i	$-0.603435\pi$
$173$	−3.89515e16	−0.638525	−0.319263	+	0.947666i	$0.603435\pi$
$174$	−7.60053e16	−1.19323
$175$	0	0
$176$	−7.69798e16	−1.10926
$177$	3.68397e16	0.508764
$178$	1.30257e17	1.72445
$179$	−4.49482e16	−0.570577	−0.285289	−	0.958442i	$-0.592089\pi$
$179$	−4.49482e16	−0.570577	−0.285289	+	0.958442i	$0.592089\pi$
$180$	0	0
$181$	3.28485e16	0.383642	0.191821	−	0.981430i	$-0.438561\pi$
$181$	3.28485e16	0.383642	0.191821	+	0.981430i	$0.438561\pi$
$182$	2.36178e15	0.0264670
$183$	4.96289e16	0.533767
$184$	2.25801e17	2.33127
$185$	0	0
$186$	−1.45500e17	−1.38521
$187$	−6.82282e16	−0.623953
$188$	−3.14020e17	−2.75914
$189$	4.21520e16	0.355922
$190$	0	0
$191$	−1.32114e17	−1.03086	−0.515428	−	0.856933i	$-0.672367\pi$
$191$	−1.32114e17	−1.03086	−0.515428	+	0.856933i	$0.672367\pi$
$192$	−2.30036e17	−1.72598
$193$	−2.31533e17	−1.67084	−0.835418	−	0.549615i	$-0.814774\pi$
$193$	−2.31533e17	−1.67084	−0.835418	+	0.549615i	$0.814774\pi$
$194$	2.13612e17	1.48290
$195$	0	0
$196$	1.00077e18	6.43303
$197$	7.76170e16	0.480242	0.240121	−	0.970743i	$-0.422813\pi$
$197$	7.76170e16	0.480242	0.240121	+	0.970743i	$0.422813\pi$
$198$	−3.48513e16	−0.207601
$199$	7.08319e16	0.406285	0.203142	−	0.979149i	$-0.434885\pi$
$199$	7.08319e16	0.406285	0.203142	+	0.979149i	$0.434885\pi$
$200$	0	0
$201$	−5.34729e16	−0.284553
$202$	2.33331e17	1.19629
$203$	−4.04509e17	−1.99852
$204$	−6.17468e17	−2.94028
$205$	0	0
$206$	−1.04028e17	−0.460412
$207$	5.74228e16	0.245079
$208$	−6.19190e15	−0.0254888
$209$	3.81896e16	0.151652
$210$	0	0
$211$	2.89451e17	1.07018	0.535090	−	0.844795i	$-0.320278\pi$
$211$	2.89451e17	1.07018	0.535090	+	0.844795i	$0.320278\pi$
$212$	−4.21146e17	−1.50284
$213$	−2.00682e16	−0.0691293
$214$	−4.48838e17	−1.49275
$215$	0	0
$216$	−1.96737e17	−0.610214
$217$	−7.74368e17	−2.32006
$218$	1.20440e18	3.48614
$219$	2.47274e17	0.691588
$220$	0	0
$221$	−5.48796e15	−0.0143374
$222$	−9.17412e15	−0.0231695
$223$	5.73361e15	0.0140004	0.00700022	−	0.999975i	$-0.497772\pi$
$223$	5.73361e15	0.0140004	0.00700022	+	0.999975i	$0.497772\pi$
$224$	−2.61929e18	−6.18476
$225$	0	0
$226$	−1.38517e17	−0.305978
$227$	−4.88606e16	−0.104415	−0.0522077	−	0.998636i	$-0.516626\pi$
$227$	−4.88606e16	−0.104415	−0.0522077	+	0.998636i	$0.516626\pi$
$228$	3.45617e17	0.714635
$229$	9.48405e17	1.89770	0.948850	−	0.315726i	$-0.102248\pi$
$229$	9.48405e17	1.89770	0.948850	+	0.315726i	$0.102248\pi$
$230$	0	0
$231$	−1.85482e17	−0.347706
$232$	1.88797e18	3.42638
$233$	−4.81044e16	−0.0845308	−0.0422654	−	0.999106i	$-0.513458\pi$
$233$	−4.81044e16	−0.0845308	−0.0422654	+	0.999106i	$0.513458\pi$
$234$	−2.80327e15	−0.00477031
$235$	0	0
$236$	−1.46707e18	−2.34213
$237$	−4.57802e17	−0.708051
$238$	−4.52265e18	−6.77745
$239$	−4.55145e17	−0.660945	−0.330473	−	0.943816i	$-0.607208\pi$
$239$	−4.55145e17	−0.660945	−0.330473	+	0.943816i	$0.607208\pi$
$240$	0	0
$241$	1.57004e17	0.214181	0.107091	−	0.994249i	$-0.465846\pi$
$241$	1.57004e17	0.214181	0.107091	+	0.994249i	$0.465846\pi$
$242$	−1.29285e18	−1.70975
$243$	−5.00315e16	−0.0641500
$244$	−1.97637e18	−2.45723
$245$	0	0
$246$	−1.32074e18	−1.54456
$247$	3.07179e15	0.00348470
$248$	3.61422e18	3.97765
$249$	3.34854e17	0.357570
$250$	0	0
$251$	−1.73508e17	−0.174489	−0.0872443	−	0.996187i	$-0.527806\pi$
$251$	−1.73508e17	−0.174489	−0.0872443	+	0.996187i	$0.527806\pi$
$252$	−1.67862e18	−1.63851
$253$	−2.52679e17	−0.239422
$254$	2.20610e18	2.02943
$255$	0	0
$256$	1.78678e18	1.54978
$257$	1.55990e18	1.31401	0.657005	−	0.753887i	$-0.271823\pi$
$257$	1.55990e18	1.31401	0.657005	+	0.753887i	$0.271823\pi$
$258$	−1.90493e18	−1.55859
$259$	−4.88257e16	−0.0388061
$260$	0	0
$261$	4.80124e17	0.360205
$262$	3.37917e18	2.46349
$263$	−1.30855e18	−0.927090	−0.463545	−	0.886073i	$-0.653423\pi$
$263$	−1.30855e18	−0.927090	−0.463545	+	0.886073i	$0.653423\pi$
$264$	8.65704e17	0.596129
$265$	0	0
$266$	2.53148e18	1.64726
$267$	−8.22830e17	−0.520566
$268$	2.12945e18	1.30996
$269$	−8.25713e17	−0.493954	−0.246977	−	0.969021i	$-0.579437\pi$
$269$	−8.25713e17	−0.493954	−0.246977	+	0.969021i	$0.579437\pi$
$270$	0	0
$271$	−8.90468e17	−0.503905	−0.251952	−	0.967740i	$-0.581073\pi$
$271$	−8.90468e17	−0.503905	−0.251952	+	0.967740i	$0.581073\pi$
$272$	1.18571e19	6.52696
$273$	−1.49193e16	−0.00798969
$274$	2.16661e18	1.12889
$275$	0	0
$276$	−2.28675e18	−1.12824
$277$	2.73024e18	1.31100	0.655500	−	0.755195i	$-0.272458\pi$
$277$	2.73024e18	1.31100	0.655500	+	0.755195i	$0.272458\pi$
$278$	6.43264e17	0.300645
$279$	9.19121e17	0.418159
$280$	0	0
$281$	−3.13968e18	−1.35390	−0.676951	−	0.736028i	$-0.736699\pi$
$281$	−3.13968e18	−1.35390	−0.676951	+	0.736028i	$0.736699\pi$
$282$	2.73000e18	1.14629
$283$	−2.69231e18	−1.10085	−0.550423	−	0.834886i	$-0.685533\pi$
$283$	−2.69231e18	−1.10085	−0.550423	+	0.834886i	$0.685533\pi$
$284$	7.99178e17	0.318241
$285$	0	0
$286$	1.23353e16	0.00466020
$287$	−7.02914e18	−2.58695
$288$	3.10892e18	1.11472
$289$	7.64666e18	2.67139
$290$	0	0
$291$	−1.34938e18	−0.447649
$292$	−9.84721e18	−3.18377
$293$	−5.52432e18	−1.74089	−0.870445	−	0.492266i	$-0.836169\pi$
$293$	−5.52432e18	−1.74089	−0.870445	+	0.492266i	$0.836169\pi$
$294$	−8.70042e18	−2.67261
$295$	0	0
$296$	2.27885e17	0.0665316
$297$	2.20155e17	0.0626693
$298$	5.85569e18	1.62538
$299$	−2.03243e16	−0.00550151
$300$	0	0
$301$	−1.01383e19	−2.61045
$302$	−3.53213e18	−0.887124
$303$	−1.47395e18	−0.361128
$304$	−6.63679e18	−1.58638
$305$	0	0
$306$	5.36808e18	1.22154
$307$	−3.75582e18	−0.834002	−0.417001	−	0.908906i	$-0.636919\pi$
$307$	−3.75582e18	−0.834002	−0.417001	+	0.908906i	$0.636919\pi$
$308$	7.38647e18	1.60069
$309$	6.57145e17	0.138986
$310$	0	0
$311$	−1.12161e18	−0.226015	−0.113008	−	0.993594i	$-0.536048\pi$
$311$	−1.12161e18	−0.226015	−0.113008	+	0.993594i	$0.536048\pi$
$312$	6.96332e16	0.0136980
$313$	2.35629e18	0.452528	0.226264	−	0.974066i	$-0.427349\pi$
$313$	2.35629e18	0.452528	0.226264	+	0.974066i	$0.427349\pi$
$314$	2.34753e18	0.440189
$315$	0	0
$316$	1.82311e19	3.25956
$317$	5.24394e18	0.915615	0.457808	−	0.889051i	$-0.348635\pi$
$317$	5.24394e18	0.915615	0.457808	+	0.889051i	$0.348635\pi$
$318$	3.66131e18	0.624358
$319$	−2.11270e18	−0.351891
$320$	0	0
$321$	2.83530e18	0.450621
$322$	−1.67493e19	−2.60063
$323$	−5.88228e18	−0.892333
$324$	1.99241e18	0.295319
$325$	0	0
$326$	−1.00713e19	−1.42546
$327$	−7.60814e18	−1.05238
$328$	3.28072e19	4.43523
$329$	1.45293e19	1.91989
$330$	0	0
$331$	7.42149e18	0.937090	0.468545	−	0.883440i	$-0.344778\pi$
$331$	7.42149e18	0.937090	0.468545	+	0.883440i	$0.344778\pi$
$332$	−1.33349e19	−1.64609
$333$	5.79528e16	0.00699427
$334$	−1.87289e19	−2.21011
$335$	0	0
$336$	3.22341e19	3.63723
$337$	8.80142e18	0.971244	0.485622	−	0.874169i	$-0.338593\pi$
$337$	8.80142e18	0.971244	0.485622	+	0.874169i	$0.338593\pi$
$338$	−1.77201e19	−1.91245
$339$	8.75010e17	0.0923667
$340$	0	0
$341$	−4.04443e18	−0.408507
$342$	−3.00469e18	−0.296896
$343$	−2.71734e19	−2.62687
$344$	4.73184e19	4.47551
$345$	0	0
$346$	−1.34854e19	−1.22122
$347$	−9.68316e17	−0.0858116	−0.0429058	−	0.999079i	$-0.513662\pi$
$347$	−9.68316e17	−0.0858116	−0.0429058	+	0.999079i	$0.513662\pi$
$348$	−1.91200e19	−1.65823
$349$	−1.37936e19	−1.17081	−0.585404	−	0.810742i	$-0.699064\pi$
$349$	−1.37936e19	−1.17081	−0.585404	+	0.810742i	$0.699064\pi$
$350$	0	0
$351$	1.77082e16	0.00144003
$352$	−1.36802e19	−1.08899
$353$	−1.73051e19	−1.34854	−0.674271	−	0.738484i	$-0.735542\pi$
$353$	−1.73051e19	−1.34854	−0.674271	+	0.738484i	$0.735542\pi$
$354$	1.27542e19	0.973039
$355$	0	0
$356$	3.27676e19	2.39646
$357$	2.85695e19	2.04593
$358$	−1.55615e19	−1.09126
$359$	−9.37891e18	−0.644086	−0.322043	−	0.946725i	$-0.604370\pi$
$359$	−9.37891e18	−0.644086	−0.322043	+	0.946725i	$0.604370\pi$
$360$	0	0
$361$	−1.18886e19	−0.783118
$362$	1.13725e19	0.733737
$363$	8.16689e18	0.516128
$364$	5.94134e17	0.0367810
$365$	0	0
$366$	1.71820e19	1.02086
$367$	−9.10392e18	−0.529947	−0.264974	−	0.964256i	$-0.585363\pi$
$367$	−9.10392e18	−0.529947	−0.264974	+	0.964256i	$0.585363\pi$
$368$	4.39119e19	2.50451
$369$	8.34311e18	0.466262
$370$	0	0
$371$	1.94859e19	1.04572
$372$	−3.66022e19	−1.92502
$373$	2.40440e19	1.23934	0.619671	−	0.784862i	$-0.287266\pi$
$373$	2.40440e19	1.23934	0.619671	+	0.784862i	$0.287266\pi$
$374$	−2.36213e19	−1.19335
$375$	0	0
$376$	−6.78130e19	−3.29158
$377$	−1.69936e17	−0.00808583
$378$	1.45934e19	0.680720
$379$	2.10822e19	0.964098	0.482049	−	0.876144i	$-0.339893\pi$
$379$	2.10822e19	0.964098	0.482049	+	0.876144i	$0.339893\pi$
$380$	0	0
$381$	−1.39359e19	−0.612630
$382$	−4.57390e19	−1.97157
$383$	2.51496e19	1.06302	0.531510	−	0.847052i	$-0.321625\pi$
$383$	2.51496e19	1.06302	0.531510	+	0.847052i	$0.321625\pi$
$384$	−3.30594e19	−1.37029
$385$	0	0
$386$	−8.01591e19	−3.19557
$387$	1.20334e19	0.470497
$388$	5.37365e19	2.06078
$389$	1.13782e19	0.428009	0.214005	−	0.976833i	$-0.431349\pi$
$389$	1.13782e19	0.428009	0.214005	+	0.976833i	$0.431349\pi$
$390$	0	0
$391$	3.89197e19	1.40878
$392$	2.16118e20	7.67443
$393$	−2.13462e19	−0.743662
$394$	2.68718e19	0.918490
$395$	0	0
$396$	−8.76724e18	−0.288502
$397$	−3.70948e19	−1.19780	−0.598900	−	0.800824i	$-0.704395\pi$
$397$	−3.70948e19	−1.19780	−0.598900	+	0.800824i	$0.704395\pi$
$398$	2.45227e19	0.777042
$399$	−1.59913e19	−0.497264
$400$	0	0
$401$	4.03228e19	1.20772	0.603862	−	0.797089i	$-0.293628\pi$
$401$	4.03228e19	1.20772	0.603862	+	0.797089i	$0.293628\pi$
$402$	−1.85128e19	−0.544223
$403$	−3.25315e17	−0.00938677
$404$	5.86970e19	1.66248
$405$	0	0
$406$	−1.40045e20	−3.82227
$407$	−2.55011e17	−0.00683283
$408$	−1.33343e20	−3.50767
$409$	−9.98931e18	−0.257995	−0.128997	−	0.991645i	$-0.541176\pi$
$409$	−9.98931e18	−0.257995	−0.128997	+	0.991645i	$0.541176\pi$
$410$	0	0
$411$	−1.36864e19	−0.340782
$412$	−2.61695e19	−0.639833
$413$	6.78795e19	1.62972
$414$	1.98803e19	0.468728
$415$	0	0
$416$	−1.10037e18	−0.0250231
$417$	−4.06349e18	−0.0907567
$418$	1.32216e19	0.290043
$419$	1.41273e19	0.304407	0.152204	−	0.988349i	$-0.451363\pi$
$419$	1.41273e19	0.304407	0.152204	+	0.988349i	$0.451363\pi$
$420$	0	0
$421$	6.55854e18	0.136361	0.0681807	−	0.997673i	$-0.478281\pi$
$421$	6.55854e18	0.136361	0.0681807	+	0.997673i	$0.478281\pi$
$422$	1.00211e20	2.04678
$423$	−1.72453e19	−0.346034
$424$	−9.09470e19	−1.79285
$425$	0	0
$426$	−6.94781e18	−0.132214
$427$	9.14446e19	1.70981
$428$	−1.12910e20	−2.07446
$429$	−7.79219e16	−0.00140679
$430$	0	0
$431$	2.53750e18	0.0442412	0.0221206	−	0.999755i	$-0.492958\pi$
$431$	2.53750e18	0.0442412	0.0221206	+	0.999755i	$0.492958\pi$
$432$	−3.82597e19	−0.655561
$433$	−4.32001e19	−0.727488	−0.363744	−	0.931499i	$-0.618502\pi$
$433$	−4.32001e19	−0.727488	−0.363744	+	0.931499i	$0.618502\pi$
$434$	−2.68093e20	−4.43724
$435$	0	0
$436$	3.02980e20	4.84468
$437$	−2.17846e19	−0.342405
$438$	8.56086e19	1.32270
$439$	−3.85404e19	−0.585372	−0.292686	−	0.956209i	$-0.594549\pi$
$439$	−3.85404e19	−0.585372	−0.292686	+	0.956209i	$0.594549\pi$
$440$	0	0
$441$	5.49604e19	0.806790
$442$	−1.89999e18	−0.0274210
$443$	−1.96889e19	−0.279379	−0.139690	−	0.990195i	$-0.544610\pi$
$443$	−1.96889e19	−0.279379	−0.139690	+	0.990195i	$0.544610\pi$
$444$	−2.30786e18	−0.0321986
$445$	0	0
$446$	1.98503e18	0.0267766
$447$	−3.69903e19	−0.490661
$448$	−4.23856e20	−5.52883
$449$	−1.02900e18	−0.0131998	−0.00659990	−	0.999978i	$-0.502101\pi$
$449$	−1.02900e18	−0.0131998	−0.00659990	+	0.999978i	$0.502101\pi$
$450$	0	0
$451$	−3.67124e19	−0.455500
$452$	−3.48456e19	−0.425216
$453$	2.23124e19	0.267800
$454$	−1.69160e19	−0.199700
$455$	0	0
$456$	7.46365e19	0.852540
$457$	1.16817e20	1.31260	0.656302	−	0.754498i	$-0.272120\pi$
$457$	1.16817e20	1.31260	0.656302	+	0.754498i	$0.272120\pi$
$458$	3.28347e20	3.62946
$459$	−3.39101e19	−0.368751
$460$	0	0
$461$	1.64177e20	1.72804	0.864021	−	0.503456i	$-0.167938\pi$
$461$	1.64177e20	1.72804	0.864021	+	0.503456i	$0.167938\pi$
$462$	−6.42157e19	−0.665008
$463$	−1.45978e19	−0.148741	−0.0743706	−	0.997231i	$-0.523695\pi$
$463$	−1.45978e19	−0.148741	−0.0743706	+	0.997231i	$0.523695\pi$
$464$	3.67156e20	3.68100
$465$	0	0
$466$	−1.66542e19	−0.161670
$467$	−1.88194e20	−1.79775	−0.898875	−	0.438206i	$-0.855614\pi$
$467$	−1.88194e20	−1.79775	−0.898875	+	0.438206i	$0.855614\pi$
$468$	−7.05196e17	−0.00662927
$469$	−9.85273e19	−0.911508
$470$	0	0
$471$	−1.48293e19	−0.132882
$472$	−3.16815e20	−2.79409
$473$	−5.29509e19	−0.459637
$474$	−1.58495e20	−1.35419
$475$	0	0
$476$	−1.13773e21	−9.41858
$477$	−2.31285e19	−0.188477
$478$	−1.57576e20	−1.26409
$479$	1.98397e19	0.156682	0.0783408	−	0.996927i	$-0.475038\pi$
$479$	1.98397e19	0.156682	0.0783408	+	0.996927i	$0.475038\pi$
$480$	0	0
$481$	−2.05119e16	−0.000157006 0
$482$	5.43562e19	0.409634
$483$	1.05805e20	0.785062
$484$	−3.25231e20	−2.37603
$485$	0	0
$486$	−1.73214e19	−0.122690
$487$	1.86146e20	1.29833	0.649166	−	0.760647i	$-0.275118\pi$
$487$	1.86146e20	1.29833	0.649166	+	0.760647i	$0.275118\pi$
$488$	−4.26800e20	−2.93141
$489$	6.36204e19	0.430308
$490$	0	0
$491$	2.44745e20	1.60547	0.802735	−	0.596335i	$-0.203377\pi$
$491$	2.44745e20	1.60547	0.802735	+	0.596335i	$0.203377\pi$
$492$	−3.32248e20	−2.14647
$493$	3.25416e20	2.07055
$494$	1.06348e18	0.00666468
$495$	0	0
$496$	7.02862e20	4.27324
$497$	−3.69771e19	−0.221442
$498$	1.15930e20	0.683872
$499$	1.13728e20	0.660868	0.330434	−	0.943829i	$-0.392805\pi$
$499$	1.13728e20	0.660868	0.330434	+	0.943829i	$0.392805\pi$
$500$	0	0
$501$	1.18310e20	0.667175
$502$	−6.00702e19	−0.333719
$503$	6.44974e19	0.353006	0.176503	−	0.984300i	$-0.443521\pi$
$503$	6.44974e19	0.353006	0.176503	+	0.984300i	$0.443521\pi$
$504$	−3.62500e20	−1.95470
$505$	0	0
$506$	−8.74798e19	−0.457909
$507$	1.11937e20	0.577318
$508$	5.54971e20	2.82028
$509$	−1.54167e20	−0.771982	−0.385991	−	0.922503i	$-0.626140\pi$
$509$	−1.54167e20	−0.771982	−0.385991	+	0.922503i	$0.626140\pi$
$510$	0	0
$511$	4.55619e20	2.21536
$512$	1.23267e20	0.590638
$513$	1.89806e19	0.0896251
$514$	5.40052e20	2.51312
$515$	0	0
$516$	−4.79208e20	−2.16596
$517$	7.58850e19	0.338047
$518$	−1.69039e19	−0.0742189
$519$	8.51869e19	0.368653
$520$	0	0
$521$	−3.79150e19	−0.159415	−0.0797073	−	0.996818i	$-0.525399\pi$
$521$	−3.79150e19	−0.159415	−0.0797073	+	0.996818i	$0.525399\pi$
$522$	1.66224e20	0.688912
$523$	2.70228e17	0.00110399	0.000551997	−	1.00000i	$-0.499824\pi$
$523$	2.70228e17	0.00110399	0.000551997		1.00000i	$0.499824\pi$
$524$	8.50069e20	3.42349
$525$	0	0
$526$	−4.53032e20	−1.77311
$527$	6.22957e20	2.40369
$528$	1.68355e20	0.640429
$529$	−1.22499e20	−0.459425
$530$	0	0
$531$	−8.05684e19	−0.293735
$532$	6.36823e20	2.28919
$533$	−2.95297e18	−0.0104666
$534$	−2.84871e20	−0.995611
$535$	0	0
$536$	4.59858e20	1.56274
$537$	9.83017e19	0.329423
$538$	−2.85870e20	−0.944715
$539$	−2.41843e20	−0.788168
$540$	0	0
$541$	5.43054e20	1.72133	0.860663	−	0.509175i	$-0.170049\pi$
$541$	5.43054e20	1.72133	0.860663	+	0.509175i	$0.170049\pi$
$542$	−3.08288e20	−0.963746
$543$	−7.18397e19	−0.221496
$544$	2.10714e21	6.40770
$545$	0	0
$546$	−5.16522e18	−0.0152807
$547$	2.42415e20	0.707383	0.353691	−	0.935362i	$-0.384926\pi$
$547$	2.42415e20	0.707383	0.353691	+	0.935362i	$0.384926\pi$
$548$	5.45036e20	1.56881
$549$	−1.08538e20	−0.308170
$550$	0	0
$551$	−1.82146e20	−0.503248
$552$	−4.93827e20	−1.34596
$553$	−8.43530e20	−2.26810
$554$	9.45235e20	2.50736
$555$	0	0
$556$	1.61821e20	0.417804
$557$	4.93071e20	1.25602	0.628008	−	0.778207i	$-0.283870\pi$
$557$	4.93071e20	1.25602	0.628008	+	0.778207i	$0.283870\pi$
$558$	3.18209e20	0.799752
$559$	−4.25912e18	−0.0105616
$560$	0	0
$561$	1.49215e20	0.360240
$562$	−1.08699e21	−2.58941
$563$	−1.49364e20	−0.351102	−0.175551	−	0.984470i	$-0.556171\pi$
$563$	−1.49364e20	−0.351102	−0.175551	+	0.984470i	$0.556171\pi$
$564$	6.86762e20	1.59299
$565$	0	0
$566$	−9.32102e20	−2.10543
$567$	−9.21864e19	−0.205492
$568$	1.72584e20	0.379653
$569$	−2.91076e19	−0.0631924	−0.0315962	−	0.999501i	$-0.510059\pi$
$569$	−2.91076e19	−0.0631924	−0.0315962	+	0.999501i	$0.510059\pi$
$570$	0	0
$571$	−4.58400e20	−0.969334	−0.484667	−	0.874699i	$-0.661059\pi$
$571$	−4.58400e20	−0.969334	−0.484667	+	0.874699i	$0.661059\pi$
$572$	3.10309e18	0.00647625
$573$	2.88933e20	0.595165
$574$	−2.43356e21	−4.94769
$575$	0	0
$576$	5.03088e20	0.996496
$577$	−4.22911e20	−0.826856	−0.413428	−	0.910537i	$-0.635669\pi$
$577$	−4.22911e20	−0.826856	−0.413428	+	0.910537i	$0.635669\pi$
$578$	2.64735e21	5.10919
$579$	5.06364e20	0.964658
$580$	0	0
$581$	6.16991e20	1.14540
$582$	−4.67169e20	−0.856153
$583$	1.01773e20	0.184127
$584$	−2.12652e21	−3.79815
$585$	0	0
$586$	−1.91257e21	−3.32955
$587$	3.03547e20	0.521724	0.260862	−	0.965376i	$-0.415993\pi$
$587$	3.03547e20	0.521724	0.260862	+	0.965376i	$0.415993\pi$
$588$	−2.18869e21	−3.71411
$589$	−3.48689e20	−0.584217
$590$	0	0
$591$	−1.69748e20	−0.277268
$592$	4.43171e19	0.0714758
$593$	9.20475e19	0.146589	0.0732946	−	0.997310i	$-0.476649\pi$
$593$	9.20475e19	0.146589	0.0732946	+	0.997310i	$0.476649\pi$
$594$	7.62197e19	0.119858
$595$	0	0
$596$	1.47307e21	2.25879
$597$	−1.54909e20	−0.234569
$598$	−7.03647e18	−0.0105219
$599$	9.43301e20	1.39299	0.696496	−	0.717560i	$-0.254741\pi$
$599$	9.43301e20	1.39299	0.696496	+	0.717560i	$0.254741\pi$
$600$	0	0
$601$	3.55593e20	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$601$	3.55593e20	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$602$	−3.50996e21	−4.99262
$603$	1.16945e20	0.164287
$604$	−8.88548e20	−1.23283
$605$	0	0
$606$	−5.10294e20	−0.690678
$607$	1.25191e20	0.167362	0.0836811	−	0.996493i	$-0.473332\pi$
$607$	1.25191e20	0.167362	0.0836811	+	0.996493i	$0.473332\pi$
$608$	−1.17944e21	−1.55739
$609$	8.84660e20	1.15384
$610$	0	0
$611$	6.10384e18	0.00776773
$612$	1.35040e21	1.69757
$613$	−7.76949e20	−0.964804	−0.482402	−	0.875950i	$-0.660236\pi$
$613$	−7.76949e20	−0.964804	−0.482402	+	0.875950i	$0.660236\pi$
$614$	−1.30030e21	−1.59507
$615$	0	0
$616$	1.59512e21	1.90958
$617$	−1.10310e21	−1.30460	−0.652301	−	0.757960i	$-0.726196\pi$
$617$	−1.10310e21	−1.30460	−0.652301	+	0.757960i	$0.726196\pi$
$618$	2.27510e20	0.265819
$619$	8.61567e20	0.994510	0.497255	−	0.867604i	$-0.334341\pi$
$619$	8.61567e20	0.994510	0.497255	+	0.867604i	$0.334341\pi$
$620$	0	0
$621$	−1.25584e20	−0.141497
$622$	−3.88311e20	−0.432267
$623$	−1.51612e21	−1.66753
$624$	1.35417e19	0.0147159
$625$	0	0
$626$	8.15769e20	0.865485
$627$	−8.35207e19	−0.0875563
$628$	5.90549e20	0.611729
$629$	3.92789e19	0.0402049
$630$	0	0
$631$	−1.84357e21	−1.84263	−0.921317	−	0.388813i	$-0.872885\pi$
$631$	−1.84357e21	−1.84263	−0.921317	+	0.388813i	$0.872885\pi$
$632$	3.93702e21	3.88857
$633$	−6.33030e20	−0.617869
$634$	1.81550e21	1.75117
$635$	0	0
$636$	9.21046e20	0.867667
$637$	−1.94528e19	−0.0181107
$638$	−7.31437e20	−0.673010
$639$	4.38892e19	0.0399118
$640$	0	0
$641$	−2.43079e20	−0.215929	−0.107965	−	0.994155i	$-0.534433\pi$
$641$	−2.43079e20	−0.215929	−0.107965	+	0.994155i	$0.534433\pi$
$642$	9.81608e20	0.861838
$643$	−9.73782e20	−0.845044	−0.422522	−	0.906353i	$-0.638855\pi$
$643$	−9.73782e20	−0.845044	−0.422522	+	0.906353i	$0.638855\pi$
$644$	−4.21349e21	−3.61408
$645$	0	0
$646$	−2.03650e21	−1.70664
$647$	4.58676e20	0.379948	0.189974	−	0.981789i	$-0.439160\pi$
$647$	4.58676e20	0.379948	0.189974	+	0.981789i	$0.439160\pi$
$648$	4.30263e20	0.352307
$649$	3.54527e20	0.286955
$650$	0	0
$651$	1.69354e21	1.33949
$652$	−2.53356e21	−1.98095
$653$	6.64062e20	0.513286	0.256643	−	0.966506i	$-0.417384\pi$
$653$	6.64062e20	0.513286	0.256643	+	0.966506i	$0.417384\pi$
$654$	−2.63401e21	−2.01273
$655$	0	0
$656$	6.38007e21	4.76482
$657$	−5.40788e20	−0.399289
$658$	5.03020e21	3.67190
$659$	2.17788e21	1.57179	0.785893	−	0.618362i	$-0.212204\pi$
$659$	2.17788e21	1.57179	0.785893	+	0.618362i	$0.212204\pi$
$660$	0	0
$661$	1.23778e21	0.873240	0.436620	−	0.899646i	$-0.356176\pi$
$661$	1.23778e21	0.873240	0.436620	+	0.899646i	$0.356176\pi$
$662$	2.56939e21	1.79224
$663$	1.20022e19	0.00827768
$664$	−2.87969e21	−1.96375
$665$	0	0
$666$	2.00638e19	0.0133769
$667$	1.20515e21	0.794510
$668$	−4.71148e21	−3.07138
$669$	−1.25394e19	−0.00808316
$670$	0	0
$671$	4.77604e20	0.301057
$672$	5.72838e21	3.57078
$673$	−2.70828e21	−1.66948	−0.834740	−	0.550644i	$-0.814382\pi$
$673$	−2.70828e21	−1.66948	−0.834740	+	0.550644i	$0.814382\pi$
$674$	3.04714e21	1.85756
$675$	0	0
$676$	−4.45769e21	−2.65772
$677$	1.04477e20	0.0616033	0.0308016	−	0.999526i	$-0.490194\pi$
$677$	1.04477e20	0.0616033	0.0308016	+	0.999526i	$0.490194\pi$
$678$	3.02937e20	0.176656
$679$	−2.48633e21	−1.43395
$680$	0	0
$681$	1.06858e20	0.0602843
$682$	−1.40022e21	−0.781292
$683$	−3.08055e21	−1.70009	−0.850046	−	0.526709i	$-0.823426\pi$
$683$	−3.08055e21	−1.70009	−0.850046	+	0.526709i	$0.823426\pi$
$684$	−7.55865e20	−0.412595
$685$	0	0
$686$	−9.40770e21	−5.02403
$687$	−2.07416e21	−1.09564
$688$	9.20209e21	4.80810
$689$	8.18612e18	0.00423091
$690$	0	0
$691$	−2.41164e21	−1.21963	−0.609813	−	0.792546i	$-0.708755\pi$
$691$	−2.41164e21	−1.21963	−0.609813	+	0.792546i	$0.708755\pi$
$692$	−3.39240e21	−1.69712
$693$	4.05650e20	0.200748
$694$	−3.35240e20	−0.164120
$695$	0	0
$696$	−4.12899e21	−1.97822
$697$	5.65474e21	2.68020
$698$	−4.77546e21	−2.23923
$699$	1.05204e20	0.0488039
$700$	0	0
$701$	−7.42818e20	−0.337285	−0.168643	−	0.985677i	$-0.553938\pi$
$701$	−7.42818e20	−0.337285	−0.168643	+	0.985677i	$0.553938\pi$
$702$	6.13076e18	0.00275414
$703$	−2.19857e19	−0.00977181
$704$	−2.21375e21	−0.973494
$705$	0	0
$706$	−5.99120e21	−2.57916
$707$	−2.71584e21	−1.15680
$708$	3.20848e21	1.35223
$709$	1.51598e20	0.0632190	0.0316095	−	0.999500i	$-0.489937\pi$
$709$	1.51598e20	0.0632190	0.0316095	+	0.999500i	$0.489937\pi$
$710$	0	0
$711$	1.00121e21	0.408794
$712$	7.07620e21	2.85891
$713$	2.30708e21	0.922339
$714$	9.89104e21	3.91296
$715$	0	0
$716$	−3.91467e21	−1.51652
$717$	9.95402e20	0.381597
$718$	−3.24707e21	−1.23185
$719$	4.93790e21	1.85386	0.926928	−	0.375240i	$-0.122440\pi$
$719$	4.93790e21	1.85386	0.926928	+	0.375240i	$0.122440\pi$
$720$	0	0
$721$	1.21083e21	0.445215
$722$	−4.11595e21	−1.49776
$723$	−3.43367e20	−0.123658
$724$	2.86088e21	1.01967
$725$	0	0
$726$	2.82746e21	0.987123
$727$	−4.57793e21	−1.58183	−0.790917	−	0.611924i	$-0.790396\pi$
$727$	−4.57793e21	−1.58183	−0.790917	+	0.611924i	$0.790396\pi$
$728$	1.28304e20	0.0438788
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	8.15594e21	2.70454
$732$	4.32233e21	1.41868
$733$	2.18646e21	0.710332	0.355166	−	0.934803i	$-0.384424\pi$
$733$	2.18646e21	0.710332	0.355166	+	0.934803i	$0.384424\pi$
$734$	−3.15186e21	−1.01355
$735$	0	0
$736$	7.80366e21	2.45875
$737$	−5.14596e20	−0.160495
$738$	2.88847e21	0.891753
$739$	−3.45456e21	−1.05575	−0.527873	−	0.849323i	$-0.677010\pi$
$739$	−3.45456e21	−1.05575	−0.527873	+	0.849323i	$0.677010\pi$
$740$	0	0
$741$	−6.71801e18	−0.00201189
$742$	6.74622e21	2.00000
$743$	3.02510e21	0.887817	0.443909	−	0.896072i	$-0.353591\pi$
$743$	3.02510e21	0.887817	0.443909	+	0.896072i	$0.353591\pi$
$744$	−7.90429e21	−2.29650
$745$	0	0
$746$	8.32427e21	2.37031
$747$	−7.32326e20	−0.206443
$748$	−5.94220e21	−1.65839
$749$	5.22423e21	1.44347
$750$	0	0
$751$	1.73179e21	0.469025	0.234512	−	0.972113i	$-0.424651\pi$
$751$	1.73179e21	0.469025	0.234512	+	0.972113i	$0.424651\pi$
$752$	−1.31877e22	−3.53619
$753$	3.79463e20	0.100741
$754$	−5.88334e19	−0.0154646
$755$	0	0
$756$	3.67115e21	0.945994
$757$	4.91616e21	1.25432	0.627158	−	0.778892i	$-0.284218\pi$
$757$	4.91616e21	1.25432	0.627158	+	0.778892i	$0.284218\pi$
$758$	7.29885e21	1.84389
$759$	5.52608e20	0.138231
$760$	0	0
$761$	−7.80081e21	−1.91318	−0.956588	−	0.291444i	$-0.905864\pi$
$761$	−7.80081e21	−1.91318	−0.956588	+	0.291444i	$0.905864\pi$
$762$	−4.82475e21	−1.17169
$763$	−1.40185e22	−3.37107
$764$	−1.15062e22	−2.73988
$765$	0	0
$766$	8.70705e21	2.03308
$767$	2.85165e19	0.00659372
$768$	−3.90768e21	−0.894767
$769$	−3.82301e21	−0.866878	−0.433439	−	0.901183i	$-0.642700\pi$
$769$	−3.82301e21	−0.866878	−0.433439	+	0.901183i	$0.642700\pi$
$770$	0	0
$771$	−3.41150e21	−0.758644
$772$	−2.01649e22	−4.44086
$773$	−3.63029e21	−0.791762	−0.395881	−	0.918302i	$-0.629561\pi$
$773$	−3.63029e21	−0.791762	−0.395881	+	0.918302i	$0.629561\pi$
$774$	4.16608e21	0.899851
$775$	0	0
$776$	1.16045e22	2.45845
$777$	1.06782e20	0.0224047
$778$	3.93926e21	0.818591
$779$	−3.16515e21	−0.651423
$780$	0	0
$781$	−1.93127e20	−0.0389906
$782$	1.34744e22	2.69437
$783$	−1.05003e21	−0.207964
$784$	4.20289e22	8.24474
$785$	0	0
$786$	−7.39024e21	−1.42229
$787$	−5.65960e21	−1.07889	−0.539443	−	0.842022i	$-0.681365\pi$
$787$	−5.65960e21	−1.07889	−0.539443	+	0.842022i	$0.681365\pi$
$788$	6.75990e21	1.27642
$789$	2.86180e21	0.535256
$790$	0	0
$791$	1.61226e21	0.295878
$792$	−1.89330e21	−0.344175
$793$	3.84162e19	0.00691777
$794$	−1.28426e22	−2.29086
$795$	0	0
$796$	6.16896e21	1.07985
$797$	−3.25052e20	−0.0563657	−0.0281829	−	0.999603i	$-0.508972\pi$
$797$	−3.25052e20	−0.0563657	−0.0281829	+	0.999603i	$0.508972\pi$
$798$	−5.53634e21	−0.951046
$799$	−1.16884e22	−1.98910
$800$	0	0
$801$	1.79953e21	0.300549
$802$	1.39601e22	2.30984
$803$	2.37964e21	0.390072
$804$	−4.65712e21	−0.756304
$805$	0	0
$806$	−1.12627e20	−0.0179527
$807$	1.80583e21	0.285185
$808$	1.26757e22	1.98329
$809$	1.50217e21	0.232866	0.116433	−	0.993199i	$-0.462854\pi$
$809$	1.50217e21	0.232866	0.116433	+	0.993199i	$0.462854\pi$
$810$	0	0
$811$	−6.98302e21	−1.06264	−0.531321	−	0.847171i	$-0.678304\pi$
$811$	−6.98302e21	−1.06264	−0.531321	+	0.847171i	$0.678304\pi$
$812$	−3.52299e22	−5.31179
$813$	1.94745e21	0.290930
$814$	−8.82871e19	−0.0130682
$815$	0	0
$816$	−2.59314e22	−3.76834
$817$	−4.56515e21	−0.657339
$818$	−3.45840e21	−0.493429
$819$	3.26286e19	0.00461285
$820$	0	0
$821$	−2.50543e21	−0.347783	−0.173892	−	0.984765i	$-0.555634\pi$
$821$	−2.50543e21	−0.347783	−0.173892	+	0.984765i	$0.555634\pi$
$822$	−4.73838e21	−0.651765
$823$	1.81605e21	0.247531	0.123765	−	0.992312i	$-0.460503\pi$
$823$	1.81605e21	0.247531	0.123765	+	0.992312i	$0.460503\pi$
$824$	−5.65134e21	−0.763303
$825$	0	0
$826$	2.35005e22	3.11693
$827$	3.59221e21	0.472139	0.236070	−	0.971736i	$-0.424141\pi$
$827$	3.59221e21	0.472139	0.236070	+	0.971736i	$0.424141\pi$
$828$	5.00113e21	0.651389
$829$	−9.33600e21	−1.20504	−0.602521	−	0.798103i	$-0.705837\pi$
$829$	−9.33600e21	−1.20504	−0.602521	+	0.798103i	$0.705837\pi$
$830$	0	0
$831$	−5.97104e21	−0.756906
$832$	−1.78064e20	−0.0223692
$833$	3.72508e22	4.63765
$834$	−1.40682e21	−0.173577
$835$	0	0
$836$	3.32605e21	0.403071
$837$	−2.01012e21	−0.241424
$838$	4.89102e21	0.582196
$839$	−2.67886e21	−0.316035	−0.158017	−	0.987436i	$-0.550510\pi$
$839$	−2.67886e21	−0.316035	−0.158017	+	0.987436i	$0.550510\pi$
$840$	0	0
$841$	1.44736e21	0.167728
$842$	2.27063e21	0.260799
$843$	6.86647e21	0.781676
$844$	2.52092e22	2.84440
$845$	0	0
$846$	−5.97050e21	−0.661810
$847$	1.50480e22	1.65331
$848$	−1.76866e22	−1.92608
$849$	5.88808e21	0.635573
$850$	0	0
$851$	1.45467e20	0.0154274
$852$	−1.74780e21	−0.183737
$853$	−5.29415e21	−0.551669	−0.275835	−	0.961205i	$-0.588954\pi$
$853$	−5.29415e21	−0.551669	−0.275835	+	0.961205i	$0.588954\pi$
$854$	3.16590e22	3.27011
$855$	0	0
$856$	−2.43831e22	−2.47478
$857$	4.61042e21	0.463857	0.231929	−	0.972733i	$-0.425496\pi$
$857$	4.61042e21	0.463857	0.231929	+	0.972733i	$0.425496\pi$
$858$	−2.69773e19	−0.00269057
$859$	−8.98854e20	−0.0888670	−0.0444335	−	0.999012i	$-0.514148\pi$
$859$	−8.98854e20	−0.0888670	−0.0444335	+	0.999012i	$0.514148\pi$
$860$	0	0
$861$	1.53727e22	1.49358
$862$	8.78508e20	0.0846138
$863$	−9.64772e21	−0.921179	−0.460589	−	0.887613i	$-0.652362\pi$
$863$	−9.64772e21	−0.921179	−0.460589	+	0.887613i	$0.652362\pi$
$864$	−6.79920e21	−0.643583
$865$	0	0
$866$	−1.49563e22	−1.39136
$867$	−1.67232e22	−1.54233
$868$	−6.74420e22	−6.16641
$869$	−4.40565e21	−0.399358
$870$	0	0
$871$	−4.13917e19	−0.00368789
$872$	6.54288e22	5.77957
$873$	2.95110e21	0.258450
$874$	−7.54205e21	−0.654868
$875$	0	0
$876$	2.15358e22	1.83815
$877$	1.74308e22	1.47509	0.737547	−	0.675296i	$-0.235984\pi$
$877$	1.74308e22	1.47509	0.737547	+	0.675296i	$0.235984\pi$
$878$	−1.33430e22	−1.11956
$879$	1.20817e22	1.00510
$880$	0	0
$881$	1.02622e22	0.839310	0.419655	−	0.907684i	$-0.362151\pi$
$881$	1.02622e22	0.839310	0.419655	+	0.907684i	$0.362151\pi$
$882$	1.90278e22	1.54303
$883$	1.86767e22	1.50174	0.750870	−	0.660451i	$-0.229635\pi$
$883$	1.86767e22	1.50174	0.750870	+	0.660451i	$0.229635\pi$
$884$	−4.77963e20	−0.0381068
$885$	0	0
$886$	−6.81650e21	−0.534328
$887$	−7.56786e21	−0.588228	−0.294114	−	0.955770i	$-0.595025\pi$
$887$	−7.56786e21	−0.588228	−0.294114	+	0.955770i	$0.595025\pi$
$888$	−4.98384e20	−0.0384120
$889$	−2.56779e22	−1.96244
$890$	0	0
$891$	−4.81479e20	−0.0361821
$892$	4.99358e20	0.0372113
$893$	6.54241e21	0.483450
$894$	−1.28064e22	−0.938415
$895$	0	0
$896$	−6.09142e22	−4.38944
$897$	4.44493e19	0.00317630
$898$	−3.56249e20	−0.0252453
$899$	1.92900e22	1.35561
$900$	0	0
$901$	−1.56759e22	−1.08342
$902$	−1.27102e22	−0.871169
$903$	2.21724e22	1.50714
$904$	−7.52494e21	−0.507271
$905$	0	0
$906$	7.72477e21	0.512182
$907$	2.84042e22	1.86779	0.933895	−	0.357547i	$-0.116387\pi$
$907$	2.84042e22	1.86779	0.933895	+	0.357547i	$0.116387\pi$
$908$	−4.25542e21	−0.277522
$909$	3.22352e21	0.208498
$910$	0	0
$911$	1.78901e22	1.13821	0.569107	−	0.822263i	$-0.307289\pi$
$911$	1.78901e22	1.13821	0.569107	+	0.822263i	$0.307289\pi$
$912$	1.45147e22	0.915895
$913$	3.22247e21	0.201678
$914$	4.04431e22	2.51043
$915$	0	0
$916$	8.25995e22	5.04384
$917$	−3.93317e22	−2.38217
$918$	−1.17400e22	−0.705257
$919$	4.47380e21	0.266569	0.133285	−	0.991078i	$-0.457448\pi$
$919$	4.47380e21	0.266569	0.133285	+	0.991078i	$0.457448\pi$
$920$	0	0
$921$	8.21398e21	0.481511
$922$	5.68395e22	3.30498
$923$	−1.55342e19	−0.000895935 0
$924$	−1.61542e22	−0.924158
$925$	0	0
$926$	−5.05391e21	−0.284476
$927$	−1.43718e21	−0.0802438
$928$	6.52480e22	3.61374
$929$	−3.53797e22	−1.94373	−0.971865	−	0.235537i	$-0.924315\pi$
$929$	−3.53797e22	−1.94373	−0.971865	+	0.235537i	$0.924315\pi$
$930$	0	0
$931$	−2.08505e22	−1.12718
$932$	−4.18955e21	−0.224672
$933$	2.45296e21	0.130490
$934$	−6.51545e22	−3.43830
$935$	0	0
$936$	−1.52288e20	−0.00790854
$937$	1.22204e22	0.629561	0.314780	−	0.949165i	$-0.398069\pi$
$937$	1.22204e22	0.629561	0.314780	+	0.949165i	$0.398069\pi$
$938$	−3.41111e22	−1.74331
$939$	−5.15320e21	−0.261267
$940$	0	0
$941$	−1.46595e22	−0.731468	−0.365734	−	0.930719i	$-0.619182\pi$
$941$	−1.46595e22	−0.731468	−0.365734	+	0.930719i	$0.619182\pi$
$942$	−5.13405e21	−0.254143
$943$	2.09420e22	1.02844
$944$	−6.16115e22	−3.00173
$945$	0	0
$946$	−1.83321e22	−0.879081
$947$	2.05124e22	0.975869	0.487934	−	0.872880i	$-0.337751\pi$
$947$	2.05124e22	0.975869	0.487934	+	0.872880i	$0.337751\pi$
$948$	−3.98713e22	−1.88191
$949$	1.91407e20	0.00896318
$950$	0	0
$951$	−1.14685e22	−0.528631
$952$	−2.45693e23	−11.2361
$953$	−1.77677e22	−0.806185	−0.403092	−	0.915159i	$-0.632065\pi$
$953$	−1.77677e22	−0.806185	−0.403092	+	0.915159i	$0.632065\pi$
$954$	−8.00729e21	−0.360473
$955$	0	0
$956$	−3.96400e22	−1.75671
$957$	4.62047e21	0.203164
$958$	6.86868e21	0.299662
$959$	−2.52182e22	−1.09163
$960$	0	0
$961$	1.34623e22	0.573712
$962$	−7.10141e18	−0.000300283 0
$963$	−6.20080e21	−0.260166
$964$	1.36739e22	0.569266
$965$	0	0
$966$	3.66308e22	1.50148
$967$	−3.69698e22	−1.50366	−0.751828	−	0.659359i	$-0.770828\pi$
$967$	−3.69698e22	−1.50366	−0.751828	+	0.659359i	$0.770828\pi$
$968$	−7.02339e22	−2.83453
$969$	1.28645e22	0.515189
$970$	0	0
$971$	−3.25843e22	−1.28488	−0.642442	−	0.766334i	$-0.722079\pi$
$971$	−3.25843e22	−1.28488	−0.642442	+	0.766334i	$0.722079\pi$
$972$	−4.35740e21	−0.170502
$973$	−7.48725e21	−0.290721
$974$	6.44454e22	2.48313
$975$	0	0
$976$	−8.30005e22	−3.14925
$977$	4.41262e21	0.166145	0.0830724	−	0.996544i	$-0.473527\pi$
$977$	4.41262e21	0.166145	0.0830724	+	0.996544i	$0.473527\pi$
$978$	2.20260e22	0.822989
$979$	−7.91850e21	−0.293611
$980$	0	0
$981$	1.66390e22	0.607589
$982$	8.47330e22	3.07055
$983$	3.23436e22	1.16315	0.581577	−	0.813491i	$-0.302436\pi$
$983$	3.23436e22	1.16315	0.581577	+	0.813491i	$0.302436\pi$
$984$	−7.17494e22	−2.56068
$985$	0	0
$986$	1.12662e23	3.96005
$987$	−3.17757e22	−1.10845
$988$	2.67532e20	0.00926187
$989$	3.02050e22	1.03778
$990$	0	0
$991$	−2.38336e22	−0.806560	−0.403280	−	0.915077i	$-0.632130\pi$
$991$	−2.38336e22	−0.806560	−0.403280	+	0.915077i	$0.632130\pi$
$992$	1.24907e23	4.19517
$993$	−1.62308e22	−0.541029
$994$	−1.28018e22	−0.423520
$995$	0	0
$996$	2.91635e22	0.950373
$997$	−5.57936e22	−1.80456	−0.902279	−	0.431152i	$-0.858107\pi$
$997$	−5.57936e22	−1.80456	−0.902279	+	0.431152i	$0.858107\pi$
$998$	3.93739e22	1.26395
$999$	−1.26743e20	−0.00403814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.16.a.j.1.6	yes	6
5.2	odd	4		75.16.b.h.49.12		12
5.3	odd	4		75.16.b.h.49.1		12
5.4	even	2		75.16.a.i.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.16.a.i.1.1	✓	6	5.4	even	2
75.16.a.j.1.6	yes	6	1.1	even	1	trivial
75.16.b.h.49.1		12	5.3	odd	4
75.16.b.h.49.12		12	5.2	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)

\(f(q)\)	\(=\)	\(q+346.210 q^{2} -2187.00 q^{3} +87093.0 q^{4} -757160. q^{6} -4.02969e6 q^{7} +1.88078e7 q^{8} +4.78297e6 q^{9} +O(q^{10})\)	\(q+346.210 q^{2} -2187.00 q^{3} +87093.0 q^{4} -757160. q^{6} -4.02969e6 q^{7} +1.88078e7 q^{8} +4.78297e6 q^{9} -2.10466e7 q^{11} -1.90472e8 q^{12} -1.69289e6 q^{13} -1.39512e9 q^{14} +3.65759e9 q^{16} +3.24177e9 q^{17} +1.65591e9 q^{18} -1.81453e9 q^{19} +8.81294e9 q^{21} -7.28653e9 q^{22} +1.20057e10 q^{23} -4.11328e10 q^{24} -5.86095e8 q^{26} -1.04604e10 q^{27} -3.50958e11 q^{28} +1.00382e11 q^{29} +1.92165e11 q^{31} +6.49997e11 q^{32} +4.60289e10 q^{33} +1.12233e12 q^{34} +4.16563e11 q^{36} +1.21165e10 q^{37} -6.28206e11 q^{38} +3.70235e9 q^{39} +1.74434e12 q^{41} +3.05112e12 q^{42} +2.51589e12 q^{43} -1.83301e12 q^{44} +4.15648e12 q^{46} -3.60557e12 q^{47} -7.99915e12 q^{48} +1.14909e13 q^{49} -7.08975e12 q^{51} -1.47439e11 q^{52} -4.83559e12 q^{53} -3.62147e12 q^{54} -7.57898e13 q^{56} +3.96837e12 q^{57} +3.47532e13 q^{58} -1.68448e13 q^{59} -2.26927e13 q^{61} +6.65295e13 q^{62} -1.92739e13 q^{63} +1.05183e14 q^{64} +1.59356e13 q^{66} +2.44503e13 q^{67} +2.82336e14 q^{68} -2.62564e13 q^{69} +9.17615e12 q^{71} +8.99573e13 q^{72} -1.13065e14 q^{73} +4.19484e12 q^{74} -1.58033e14 q^{76} +8.48113e13 q^{77} +1.28179e12 q^{78} +2.09329e14 q^{79} +2.28768e13 q^{81} +6.03906e14 q^{82} -1.53111e14 q^{83} +7.67546e14 q^{84} +8.71025e14 q^{86} -2.19535e14 q^{87} -3.95841e14 q^{88} +3.76237e14 q^{89} +6.82183e12 q^{91} +1.04561e15 q^{92} -4.20266e14 q^{93} -1.24828e15 q^{94} -1.42154e15 q^{96} +6.17001e14 q^{97} +3.97825e15 q^{98} -1.00665e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9}+O(q^{10}) \) 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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