LMFDB - Embedded newform 196.8.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,8,Mod(1,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("196.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	196.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:	$61.2274649949$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 341x^{3} - 912x^{2} + 1764x + 4860 $ x^5 - x^4 - 341x^3 - 912x^2 + 1764*x + 4860
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.29744$ of defining polynomial
Character	$\chi$	$=$	196.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-58.7379 q^{3} +418.382 q^{5} +1263.14 q^{9} +256.583 q^{11} -8020.94 q^{13} -24574.9 q^{15} -25641.5 q^{17} +55127.7 q^{19} -37896.2 q^{23} +96918.1 q^{25} +54265.4 q^{27} +125755. q^{29} -160389. q^{31} -15071.1 q^{33} +141005. q^{37} +471134. q^{39} -674106. q^{41} +755814. q^{43} +528476. q^{45} +205501. q^{47} +1.50613e6 q^{51} +365777. q^{53} +107349. q^{55} -3.23809e6 q^{57} -1.92491e6 q^{59} -2.71920e6 q^{61} -3.35581e6 q^{65} -1.56543e6 q^{67} +2.22595e6 q^{69} -2.29364e6 q^{71} -1.17273e6 q^{73} -5.69277e6 q^{75} +720162. q^{79} -5.94993e6 q^{81} +2.39050e6 q^{83} -1.07279e7 q^{85} -7.38656e6 q^{87} -1.24140e7 q^{89} +9.42093e6 q^{93} +2.30644e7 q^{95} -1.48005e7 q^{97} +324101. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 27 q^{3} - 249 q^{5} + 5702 q^{9} - 6399 q^{11} - 13494 q^{13} + 9647 q^{15} - 3609 q^{17} + 12403 q^{19} + 13959 q^{23} + 162364 q^{25} + 80775 q^{27} + 13074 q^{29} + 20181 q^{31} - 264239 q^{33}+ \cdots - 59638906 q^{99}+O(q^{100}) $ 5 * q - 27 * q^3 - 249 * q^5 + 5702 * q^9 - 6399 * q^11 - 13494 * q^13 + 9647 * q^15 - 3609 * q^17 + 12403 * q^19 + 13959 * q^23 + 162364 * q^25 + 80775 * q^27 + 13074 * q^29 + 20181 * q^31 - 264239 * q^33 + 54763 * q^37 - 4750 * q^39 - 912234 * q^41 - 969212 * q^43 - 1936430 * q^45 - 1217253 * q^47 + 1353871 * q^51 + 2349159 * q^53 + 1561747 * q^55 - 4911597 * q^57 - 2188611 * q^59 - 6107445 * q^61 + 3022014 * q^65 + 3171581 * q^67 - 5627969 * q^69 - 9469488 * q^71 - 8034853 * q^73 - 29173588 * q^75 + 7589559 * q^79 + 9846485 * q^81 - 6726204 * q^83 - 24358835 * q^85 - 30033142 * q^87 - 29864757 * q^89 + 20217029 * q^93 + 25812009 * q^95 + 142790 * q^97 - 59638906 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−58.7379	−1.25601	−0.628007	−	0.778208i	$-0.716129\pi$
$3$	−58.7379	−1.25601	−0.628007	+	0.778208i	$0.716129\pi$
$4$	0	0
$5$	418.382	1.49685	0.748424	−	0.663221i	$-0.230811\pi$
$5$	418.382	1.49685	0.748424	+	0.663221i	$0.230811\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1263.14	0.577570
$10$	0	0
$11$	256.583	0.0581236	0.0290618	−	0.999578i	$-0.490748\pi$
$11$	256.583	0.0581236	0.0290618	+	0.999578i	$0.490748\pi$
$12$	0	0
$13$	−8020.94	−1.01257	−0.506283	−	0.862367i	$-0.668981\pi$
$13$	−8020.94	−1.01257	−0.506283	+	0.862367i	$0.668981\pi$
$14$	0	0
$15$	−24574.9	−1.88006
$16$	0	0
$17$	−25641.5	−1.26582	−0.632911	−	0.774225i	$-0.718140\pi$
$17$	−25641.5	−1.26582	−0.632911	+	0.774225i	$0.718140\pi$
$18$	0	0
$19$	55127.7	1.84388	0.921939	−	0.387336i	$-0.126605\pi$
$19$	55127.7	1.84388	0.921939	+	0.387336i	$0.126605\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−37896.2	−0.649454	−0.324727	−	0.945808i	$-0.605272\pi$
$23$	−37896.2	−0.649454	−0.324727	+	0.945808i	$0.605272\pi$
$24$	0	0
$25$	96918.1	1.24055
$26$	0	0
$27$	54265.4	0.530578
$28$	0	0
$29$	125755.	0.957482	0.478741	−	0.877956i	$-0.341093\pi$
$29$	125755.	0.957482	0.478741	+	0.877956i	$0.341093\pi$
$30$	0	0
$31$	−160389.	−0.966962	−0.483481	−	0.875355i	$-0.660628\pi$
$31$	−160389.	−0.966962	−0.483481	+	0.875355i	$0.660628\pi$
$32$	0	0
$33$	−15071.1	−0.0730041
$34$	0	0
$35$	0	0
$36$	0	0
$37$	141005.	0.457644	0.228822	−	0.973468i	$-0.426513\pi$
$37$	141005.	0.457644	0.228822	+	0.973468i	$0.426513\pi$
$38$	0	0
$39$	471134.	1.27180
$40$	0	0
$41$	−674106.	−1.52751	−0.763756	−	0.645505i	$-0.776647\pi$
$41$	−674106.	−1.52751	−0.763756	+	0.645505i	$0.776647\pi$
$42$	0	0
$43$	755814.	1.44969	0.724845	−	0.688912i	$-0.241911\pi$
$43$	755814.	1.44969	0.724845	+	0.688912i	$0.241911\pi$
$44$	0	0
$45$	528476.	0.864533
$46$	0	0
$47$	205501.	0.288717	0.144358	−	0.989525i	$-0.453888\pi$
$47$	205501.	0.288717	0.144358	+	0.989525i	$0.453888\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.50613e6	1.58989
$52$	0	0
$53$	365777.	0.337482	0.168741	−	0.985660i	$-0.446030\pi$
$53$	365777.	0.337482	0.168741	+	0.985660i	$0.446030\pi$
$54$	0	0
$55$	107349.	0.0870022
$56$	0	0
$57$	−3.23809e6	−2.31593
$58$	0	0
$59$	−1.92491e6	−1.22019	−0.610097	−	0.792327i	$-0.708870\pi$
$59$	−1.92491e6	−1.22019	−0.610097	+	0.792327i	$0.708870\pi$
$60$	0	0
$61$	−2.71920e6	−1.53386	−0.766931	−	0.641729i	$-0.778217\pi$
$61$	−2.71920e6	−1.53386	−0.766931	+	0.641729i	$0.778217\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.35581e6	−1.51566
$66$	0	0
$67$	−1.56543e6	−0.635875	−0.317938	−	0.948112i	$-0.602990\pi$
$67$	−1.56543e6	−0.635875	−0.317938	+	0.948112i	$0.602990\pi$
$68$	0	0
$69$	2.22595e6	0.815723
$70$	0	0
$71$	−2.29364e6	−0.760538	−0.380269	−	0.924876i	$-0.624169\pi$
$71$	−2.29364e6	−0.760538	−0.380269	+	0.924876i	$0.624169\pi$
$72$	0	0
$73$	−1.17273e6	−0.352832	−0.176416	−	0.984316i	$-0.556450\pi$
$73$	−1.17273e6	−0.352832	−0.176416	+	0.984316i	$0.556450\pi$
$74$	0	0
$75$	−5.69277e6	−1.55815
$76$	0	0
$77$	0	0
$78$	0	0
$79$	720162.	0.164337	0.0821685	−	0.996618i	$-0.473815\pi$
$79$	720162.	0.164337	0.0821685	+	0.996618i	$0.473815\pi$
$80$	0	0
$81$	−5.94993e6	−1.24398
$82$	0	0
$83$	2.39050e6	0.458898	0.229449	−	0.973321i	$-0.426308\pi$
$83$	2.39050e6	0.458898	0.229449	+	0.973321i	$0.426308\pi$
$84$	0	0
$85$	−1.07279e7	−1.89474
$86$	0	0
$87$	−7.38656e6	−1.20261
$88$	0	0
$89$	−1.24140e7	−1.86658	−0.933288	−	0.359128i	$-0.883074\pi$
$89$	−1.24140e7	−1.86658	−0.933288	+	0.359128i	$0.883074\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.42093e6	1.21452
$94$	0	0
$95$	2.30644e7	2.76000
$96$	0	0
$97$	−1.48005e7	−1.64655	−0.823276	−	0.567641i	$-0.807856\pi$
$97$	−1.48005e7	−1.64655	−0.823276	+	0.567641i	$0.807856\pi$
$98$	0	0
$99$	324101.	0.0335704
$100$	0	0
$101$	6.42121e6	0.620144	0.310072	−	0.950713i	$-0.399647\pi$
$101$	6.42121e6	0.620144	0.310072	+	0.950713i	$0.399647\pi$
$102$	0	0
$103$	3.46452e6	0.312401	0.156201	−	0.987725i	$-0.450075\pi$
$103$	3.46452e6	0.312401	0.156201	+	0.987725i	$0.450075\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.78860e7	1.41146	0.705732	−	0.708479i	$-0.250618\pi$
$107$	1.78860e7	1.41146	0.705732	+	0.708479i	$0.250618\pi$
$108$	0	0
$109$	−4.56937e6	−0.337959	−0.168979	−	0.985620i	$-0.554047\pi$
$109$	−4.56937e6	−0.337959	−0.168979	+	0.985620i	$0.554047\pi$
$110$	0	0
$111$	−8.28234e6	−0.574808
$112$	0	0
$113$	5.46079e6	0.356025	0.178013	−	0.984028i	$-0.443033\pi$
$113$	5.46079e6	0.356025	0.178013	+	0.984028i	$0.443033\pi$
$114$	0	0
$115$	−1.58551e7	−0.972133
$116$	0	0
$117$	−1.01316e7	−0.584828
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.94213e7	−0.996622
$122$	0	0
$123$	3.95956e7	1.91858
$124$	0	0
$125$	7.86270e6	0.360070
$126$	0	0
$127$	−1.65990e7	−0.719066	−0.359533	−	0.933132i	$-0.617064\pi$
$127$	−1.65990e7	−0.719066	−0.359533	+	0.933132i	$0.617064\pi$
$128$	0	0
$129$	−4.43950e7	−1.82083
$130$	0	0
$131$	−4.00970e7	−1.55834	−0.779169	−	0.626813i	$-0.784359\pi$
$131$	−4.00970e7	−1.55834	−0.779169	+	0.626813i	$0.784359\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.27036e7	0.794195
$136$	0	0
$137$	890583.	0.0295905	0.0147953	−	0.999891i	$-0.495290\pi$
$137$	890583.	0.0295905	0.0147953	+	0.999891i	$0.495290\pi$
$138$	0	0
$139$	1.42244e7	0.449243	0.224622	−	0.974446i	$-0.427885\pi$
$139$	1.42244e7	0.449243	0.224622	+	0.974446i	$0.427885\pi$
$140$	0	0
$141$	−1.20707e7	−0.362632
$142$	0	0
$143$	−2.05803e6	−0.0588541
$144$	0	0
$145$	5.26134e7	1.43320
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.98118e7	−1.23362	−0.616809	−	0.787113i	$-0.711575\pi$
$149$	−4.98118e7	−1.23362	−0.616809	+	0.787113i	$0.711575\pi$
$150$	0	0
$151$	−5.20497e7	−1.23026	−0.615132	−	0.788424i	$-0.710897\pi$
$151$	−5.20497e7	−1.23026	−0.615132	+	0.788424i	$0.710897\pi$
$152$	0	0
$153$	−3.23890e7	−0.731100
$154$	0	0
$155$	−6.71039e7	−1.44739
$156$	0	0
$157$	2.67138e7	0.550917	0.275459	−	0.961313i	$-0.411170\pi$
$157$	2.67138e7	0.550917	0.275459	+	0.961313i	$0.411170\pi$
$158$	0	0
$159$	−2.14850e7	−0.423882
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.71508e7	0.310189	0.155095	−	0.987900i	$-0.450432\pi$
$163$	1.71508e7	0.310189	0.155095	+	0.987900i	$0.450432\pi$
$164$	0	0
$165$	−6.30548e6	−0.109276
$166$	0	0
$167$	7.00218e7	1.16339	0.581695	−	0.813407i	$-0.302390\pi$
$167$	7.00218e7	1.16339	0.581695	+	0.813407i	$0.302390\pi$
$168$	0	0
$169$	1.58701e6	0.0252916
$170$	0	0
$171$	6.96342e7	1.06497
$172$	0	0
$173$	−5.17012e7	−0.759170	−0.379585	−	0.925157i	$-0.623933\pi$
$173$	−5.17012e7	−0.759170	−0.379585	+	0.925157i	$0.623933\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.13065e8	1.53258
$178$	0	0
$179$	−3.32281e7	−0.433032	−0.216516	−	0.976279i	$-0.569469\pi$
$179$	−3.32281e7	−0.433032	−0.216516	+	0.976279i	$0.569469\pi$
$180$	0	0
$181$	−1.23418e8	−1.54705	−0.773523	−	0.633769i	$-0.781507\pi$
$181$	−1.23418e8	−1.54705	−0.773523	+	0.633769i	$0.781507\pi$
$182$	0	0
$183$	1.59720e8	1.92655
$184$	0	0
$185$	5.89939e7	0.685024
$186$	0	0
$187$	−6.57917e6	−0.0735742
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.24575e7	−0.856275	−0.428137	−	0.903714i	$-0.640830\pi$
$191$	−8.24575e7	−0.856275	−0.428137	+	0.903714i	$0.640830\pi$
$192$	0	0
$193$	−1.05850e8	−1.05984	−0.529922	−	0.848047i	$-0.677779\pi$
$193$	−1.05850e8	−1.05984	−0.529922	+	0.848047i	$0.677779\pi$
$194$	0	0
$195$	1.97114e8	1.90369
$196$	0	0
$197$	1.23667e8	1.15245	0.576224	−	0.817292i	$-0.304525\pi$
$197$	1.23667e8	1.15245	0.576224	+	0.817292i	$0.304525\pi$
$198$	0	0
$199$	3.92583e7	0.353139	0.176569	−	0.984288i	$-0.443500\pi$
$199$	3.92583e7	0.353139	0.176569	+	0.984288i	$0.443500\pi$
$200$	0	0
$201$	9.19501e7	0.798668
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.82033e8	−2.28645
$206$	0	0
$207$	−4.78684e7	−0.375105
$208$	0	0
$209$	1.41448e7	0.107173
$210$	0	0
$211$	−1.64534e7	−0.120578	−0.0602890	−	0.998181i	$-0.519202\pi$
$211$	−1.64534e7	−0.120578	−0.0602890	+	0.998181i	$0.519202\pi$
$212$	0	0
$213$	1.34724e8	0.955246
$214$	0	0
$215$	3.16219e8	2.16997
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.88838e7	0.443162
$220$	0	0
$221$	2.05669e8	1.28173
$222$	0	0
$223$	−9.12765e7	−0.551178	−0.275589	−	0.961276i	$-0.588873\pi$
$223$	−9.12765e7	−0.551178	−0.275589	+	0.961276i	$0.588873\pi$
$224$	0	0
$225$	1.22422e8	0.716505
$226$	0	0
$227$	−3.30696e8	−1.87646	−0.938230	−	0.346014i	$-0.887535\pi$
$227$	−3.30696e8	−1.87646	−0.938230	+	0.346014i	$0.887535\pi$
$228$	0	0
$229$	2.76156e8	1.51960	0.759802	−	0.650154i	$-0.225296\pi$
$229$	2.76156e8	1.51960	0.759802	+	0.650154i	$0.225296\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.08365e8	1.59705	0.798526	−	0.601961i	$-0.205614\pi$
$233$	3.08365e8	1.59705	0.798526	+	0.601961i	$0.205614\pi$
$234$	0	0
$235$	8.59779e7	0.432165
$236$	0	0
$237$	−4.23008e7	−0.206409
$238$	0	0
$239$	−7.05871e7	−0.334451	−0.167225	−	0.985919i	$-0.553481\pi$
$239$	−7.05871e7	−0.334451	−0.167225	+	0.985919i	$0.553481\pi$
$240$	0	0
$241$	7.97343e6	0.0366932	0.0183466	−	0.999832i	$-0.494160\pi$
$241$	7.97343e6	0.0366932	0.0183466	+	0.999832i	$0.494160\pi$
$242$	0	0
$243$	2.30808e8	1.03188
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.42176e8	−1.86705
$248$	0	0
$249$	−1.40413e8	−0.576382
$250$	0	0
$251$	−1.09068e8	−0.435351	−0.217675	−	0.976021i	$-0.569847\pi$
$251$	−1.09068e8	−0.435351	−0.217675	+	0.976021i	$0.569847\pi$
$252$	0	0
$253$	−9.72351e6	−0.0377486
$254$	0	0
$255$	6.30137e8	2.37982
$256$	0	0
$257$	6.23013e7	0.228945	0.114473	−	0.993426i	$-0.463482\pi$
$257$	6.23013e7	0.228945	0.114473	+	0.993426i	$0.463482\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.58846e8	0.553013
$262$	0	0
$263$	−8.95838e7	−0.303658	−0.151829	−	0.988407i	$-0.548516\pi$
$263$	−8.95838e7	−0.303658	−0.151829	+	0.988407i	$0.548516\pi$
$264$	0	0
$265$	1.53034e8	0.505159
$266$	0	0
$267$	7.29171e8	2.34445
$268$	0	0
$269$	−2.77561e8	−0.869411	−0.434706	−	0.900573i	$-0.643148\pi$
$269$	−2.77561e8	−0.869411	−0.434706	+	0.900573i	$0.643148\pi$
$270$	0	0
$271$	1.90781e6	0.00582295	0.00291147	−	0.999996i	$-0.499073\pi$
$271$	1.90781e6	0.00582295	0.00291147	+	0.999996i	$0.499073\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.48675e7	0.0721054
$276$	0	0
$277$	−3.81539e8	−1.07860	−0.539299	−	0.842114i	$-0.681311\pi$
$277$	−3.81539e8	−1.07860	−0.539299	+	0.842114i	$0.681311\pi$
$278$	0	0
$279$	−2.02595e8	−0.558488
$280$	0	0
$281$	1.58621e8	0.426470	0.213235	−	0.977001i	$-0.431600\pi$
$281$	1.58621e8	0.426470	0.213235	+	0.977001i	$0.431600\pi$
$282$	0	0
$283$	2.43169e8	0.637759	0.318879	−	0.947795i	$-0.396694\pi$
$283$	2.43169e8	0.637759	0.318879	+	0.947795i	$0.396694\pi$
$284$	0	0
$285$	−1.35476e9	−3.46660
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.47149e8	0.602305
$290$	0	0
$291$	8.69351e8	2.06809
$292$	0	0
$293$	5.34536e8	1.24148	0.620740	−	0.784016i	$-0.286832\pi$
$293$	5.34536e8	1.24148	0.620740	+	0.784016i	$0.286832\pi$
$294$	0	0
$295$	−8.05348e8	−1.82644
$296$	0	0
$297$	1.39235e7	0.0308391
$298$	0	0
$299$	3.03963e8	0.657616
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.77169e8	−0.778909
$304$	0	0
$305$	−1.13766e9	−2.29596
$306$	0	0
$307$	6.57215e8	1.29635	0.648176	−	0.761490i	$-0.275532\pi$
$307$	6.57215e8	1.29635	0.648176	+	0.761490i	$0.275532\pi$
$308$	0	0
$309$	−2.03499e8	−0.392380
$310$	0	0
$311$	−5.54408e7	−0.104513	−0.0522563	−	0.998634i	$-0.516641\pi$
$311$	−5.54408e7	−0.104513	−0.0522563	+	0.998634i	$0.516641\pi$
$312$	0	0
$313$	−4.51349e8	−0.831968	−0.415984	−	0.909372i	$-0.636563\pi$
$313$	−4.51349e8	−0.831968	−0.415984	+	0.909372i	$0.636563\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.13880e7	−0.108237	−0.0541186	−	0.998535i	$-0.517235\pi$
$317$	−6.13880e7	−0.108237	−0.0541186	+	0.998535i	$0.517235\pi$
$318$	0	0
$319$	3.22664e7	0.0556523
$320$	0	0
$321$	−1.05059e9	−1.77282
$322$	0	0
$323$	−1.41356e9	−2.33402
$324$	0	0
$325$	−7.77375e8	−1.25614
$326$	0	0
$327$	2.68395e8	0.424481
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.07760e8	0.618026	0.309013	−	0.951058i	$-0.400001\pi$
$331$	4.07760e8	0.618026	0.309013	+	0.951058i	$0.400001\pi$
$332$	0	0
$333$	1.78110e8	0.264322
$334$	0	0
$335$	−6.54947e8	−0.951808
$336$	0	0
$337$	−2.63519e8	−0.375066	−0.187533	−	0.982258i	$-0.560049\pi$
$337$	−2.63519e8	−0.375066	−0.187533	+	0.982258i	$0.560049\pi$
$338$	0	0
$339$	−3.20755e8	−0.447172
$340$	0	0
$341$	−4.11531e7	−0.0562033
$342$	0	0
$343$	0	0
$344$	0	0
$345$	9.31294e8	1.22101
$346$	0	0
$347$	4.07599e8	0.523696	0.261848	−	0.965109i	$-0.415668\pi$
$347$	4.07599e8	0.523696	0.261848	+	0.965109i	$0.415668\pi$
$348$	0	0
$349$	−5.06614e7	−0.0637952	−0.0318976	−	0.999491i	$-0.510155\pi$
$349$	−5.06614e7	−0.0637952	−0.0318976	+	0.999491i	$0.510155\pi$
$350$	0	0
$351$	−4.35259e8	−0.537246
$352$	0	0
$353$	−8.52749e8	−1.03183	−0.515917	−	0.856639i	$-0.672549\pi$
$353$	−8.52749e8	−1.03183	−0.515917	+	0.856639i	$0.672549\pi$
$354$	0	0
$355$	−9.59616e8	−1.13841
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.50311e9	−1.71459	−0.857293	−	0.514828i	$-0.827856\pi$
$359$	−1.50311e9	−1.71459	−0.857293	+	0.514828i	$0.827856\pi$
$360$	0	0
$361$	2.14519e9	2.39988
$362$	0	0
$363$	1.14077e9	1.25177
$364$	0	0
$365$	−4.90649e8	−0.528136
$366$	0	0
$367$	−1.09915e9	−1.16071	−0.580355	−	0.814363i	$-0.697086\pi$
$367$	−1.09915e9	−1.16071	−0.580355	+	0.814363i	$0.697086\pi$
$368$	0	0
$369$	−8.51493e8	−0.882244
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.16058e9	1.15796	0.578981	−	0.815341i	$-0.303451\pi$
$373$	1.16058e9	1.15796	0.578981	+	0.815341i	$0.303451\pi$
$374$	0	0
$375$	−4.61839e8	−0.452252
$376$	0	0
$377$	−1.00867e9	−0.969515
$378$	0	0
$379$	1.63326e9	1.54106	0.770528	−	0.637406i	$-0.219993\pi$
$379$	1.63326e9	1.54106	0.770528	+	0.637406i	$0.219993\pi$
$380$	0	0
$381$	9.74990e8	0.903156
$382$	0	0
$383$	−1.22920e8	−0.111796	−0.0558981	−	0.998436i	$-0.517802\pi$
$383$	−1.22920e8	−0.111796	−0.0558981	+	0.998436i	$0.517802\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.54702e8	0.837297
$388$	0	0
$389$	5.58108e8	0.480723	0.240361	−	0.970684i	$-0.422734\pi$
$389$	5.58108e8	0.480723	0.240361	+	0.970684i	$0.422734\pi$
$390$	0	0
$391$	9.71717e8	0.822093
$392$	0	0
$393$	2.35521e9	1.95729
$394$	0	0
$395$	3.01302e8	0.245987
$396$	0	0
$397$	8.92169e8	0.715617	0.357808	−	0.933795i	$-0.383524\pi$
$397$	8.92169e8	0.715617	0.357808	+	0.933795i	$0.383524\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.71115e8	−0.519746	−0.259873	−	0.965643i	$-0.583681\pi$
$401$	−6.71115e8	−0.519746	−0.259873	+	0.965643i	$0.583681\pi$
$402$	0	0
$403$	1.28647e9	0.979113
$404$	0	0
$405$	−2.48934e9	−1.86205
$406$	0	0
$407$	3.61794e7	0.0266000
$408$	0	0
$409$	4.03330e8	0.291494	0.145747	−	0.989322i	$-0.453442\pi$
$409$	4.03330e8	0.291494	0.145747	+	0.989322i	$0.453442\pi$
$410$	0	0
$411$	−5.23110e7	−0.0371661
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.00014e9	0.686900
$416$	0	0
$417$	−8.35510e8	−0.564255
$418$	0	0
$419$	4.45684e8	0.295990	0.147995	−	0.988988i	$-0.452718\pi$
$419$	4.45684e8	0.295990	0.147995	+	0.988988i	$0.452718\pi$
$420$	0	0
$421$	−2.97060e9	−1.94025	−0.970123	−	0.242614i	$-0.921995\pi$
$421$	−2.97060e9	−1.94025	−0.970123	+	0.242614i	$0.921995\pi$
$422$	0	0
$423$	2.59578e8	0.166754
$424$	0	0
$425$	−2.48513e9	−1.57032
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.20885e8	0.0739215
$430$	0	0
$431$	1.13683e9	0.683950	0.341975	−	0.939709i	$-0.388904\pi$
$431$	1.13683e9	0.683950	0.341975	+	0.939709i	$0.388904\pi$
$432$	0	0
$433$	−1.17793e9	−0.697287	−0.348644	−	0.937255i	$-0.613358\pi$
$433$	−1.17793e9	−0.697287	−0.348644	+	0.937255i	$0.613358\pi$
$434$	0	0
$435$	−3.09040e9	−1.80012
$436$	0	0
$437$	−2.08913e9	−1.19751
$438$	0	0
$439$	1.14708e9	0.647093	0.323546	−	0.946212i	$-0.395125\pi$
$439$	1.14708e9	0.647093	0.323546	+	0.946212i	$0.395125\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.38444e8	−0.403557	−0.201778	−	0.979431i	$-0.564672\pi$
$443$	−7.38444e8	−0.403557	−0.201778	+	0.979431i	$0.564672\pi$
$444$	0	0
$445$	−5.19378e9	−2.79398
$446$	0	0
$447$	2.92584e9	1.54944
$448$	0	0
$449$	1.95056e9	1.01695	0.508473	−	0.861078i	$-0.330210\pi$
$449$	1.95056e9	1.01695	0.508473	+	0.861078i	$0.330210\pi$
$450$	0	0
$451$	−1.72964e8	−0.0887845
$452$	0	0
$453$	3.05729e9	1.54523
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.89090e6	−0.00484762	−0.00242381	−	0.999997i	$-0.500772\pi$
$457$	−9.89090e6	−0.00484762	−0.00242381	+	0.999997i	$0.500772\pi$
$458$	0	0
$459$	−1.39145e9	−0.671618
$460$	0	0
$461$	−8.01965e8	−0.381243	−0.190622	−	0.981664i	$-0.561050\pi$
$461$	−8.01965e8	−0.381243	−0.190622	+	0.981664i	$0.561050\pi$
$462$	0	0
$463$	−3.05301e8	−0.142953	−0.0714767	−	0.997442i	$-0.522771\pi$
$463$	−3.05301e8	−0.142953	−0.0714767	+	0.997442i	$0.522771\pi$
$464$	0	0
$465$	3.94154e9	1.81795
$466$	0	0
$467$	5.36242e8	0.243642	0.121821	−	0.992552i	$-0.461127\pi$
$467$	5.36242e8	0.243642	0.121821	+	0.992552i	$0.461127\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.56911e9	−0.691959
$472$	0	0
$473$	1.93929e8	0.0842613
$474$	0	0
$475$	5.34287e9	2.28743
$476$	0	0
$477$	4.62029e8	0.194919
$478$	0	0
$479$	1.98624e9	0.825768	0.412884	−	0.910784i	$-0.364521\pi$
$479$	1.98624e9	0.825768	0.412884	+	0.910784i	$0.364521\pi$
$480$	0	0
$481$	−1.13099e9	−0.463396
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.19226e9	−2.46464
$486$	0	0
$487$	8.28257e8	0.324948	0.162474	−	0.986713i	$-0.448053\pi$
$487$	8.28257e8	0.324948	0.162474	+	0.986713i	$0.448053\pi$
$488$	0	0
$489$	−1.00740e9	−0.389602
$490$	0	0
$491$	1.29504e9	0.493739	0.246869	−	0.969049i	$-0.420598\pi$
$491$	1.29504e9	0.493739	0.246869	+	0.969049i	$0.420598\pi$
$492$	0	0
$493$	−3.22454e9	−1.21200
$494$	0	0
$495$	1.35598e8	0.0502498
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.46144e7	−0.0268826	−0.0134413	−	0.999910i	$-0.504279\pi$
$499$	−7.46144e7	−0.0268826	−0.0134413	+	0.999910i	$0.504279\pi$
$500$	0	0
$501$	−4.11294e9	−1.46123
$502$	0	0
$503$	−2.88380e8	−0.101036	−0.0505181	−	0.998723i	$-0.516087\pi$
$503$	−2.88380e8	−0.101036	−0.0505181	+	0.998723i	$0.516087\pi$
$504$	0	0
$505$	2.68652e9	0.928260
$506$	0	0
$507$	−9.32178e7	−0.0317666
$508$	0	0
$509$	9.65027e8	0.324360	0.162180	−	0.986761i	$-0.448147\pi$
$509$	9.65027e8	0.324360	0.162180	+	0.986761i	$0.448147\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.99152e9	0.978321
$514$	0	0
$515$	1.44949e9	0.467617
$516$	0	0
$517$	5.27280e7	0.0167813
$518$	0	0
$519$	3.03682e9	0.953528
$520$	0	0
$521$	5.10390e9	1.58114	0.790570	−	0.612372i	$-0.209784\pi$
$521$	5.10390e9	1.58114	0.790570	+	0.612372i	$0.209784\pi$
$522$	0	0
$523$	3.37030e9	1.03018	0.515090	−	0.857136i	$-0.327758\pi$
$523$	3.37030e9	1.03018	0.515090	+	0.857136i	$0.327758\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.11262e9	1.22400
$528$	0	0
$529$	−1.96870e9	−0.578210
$530$	0	0
$531$	−2.43144e9	−0.704747
$532$	0	0
$533$	5.40696e9	1.54671
$534$	0	0
$535$	7.48317e9	2.11275
$536$	0	0
$537$	1.95175e9	0.543894
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.34596e9	0.908512	0.454256	−	0.890871i	$-0.349905\pi$
$541$	3.34596e9	0.908512	0.454256	+	0.890871i	$0.349905\pi$
$542$	0	0
$543$	7.24931e9	1.94311
$544$	0	0
$545$	−1.91174e9	−0.505872
$546$	0	0
$547$	4.98263e9	1.30168	0.650838	−	0.759216i	$-0.274418\pi$
$547$	4.98263e9	1.30168	0.650838	+	0.759216i	$0.274418\pi$
$548$	0	0
$549$	−3.43474e9	−0.885912
$550$	0	0
$551$	6.93256e9	1.76548
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.46518e9	−0.860399
$556$	0	0
$557$	4.05452e8	0.0994137	0.0497069	−	0.998764i	$-0.484171\pi$
$557$	4.05452e8	0.0994137	0.0497069	+	0.998764i	$0.484171\pi$
$558$	0	0
$559$	−6.06234e9	−1.46791
$560$	0	0
$561$	3.86447e8	0.0924101
$562$	0	0
$563$	−5.29340e9	−1.25013	−0.625065	−	0.780573i	$-0.714928\pi$
$563$	−5.29340e9	−1.25013	−0.625065	+	0.780573i	$0.714928\pi$
$564$	0	0
$565$	2.28469e9	0.532915
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.70120e9	−0.387136	−0.193568	−	0.981087i	$-0.562006\pi$
$569$	−1.70120e9	−0.387136	−0.193568	+	0.981087i	$0.562006\pi$
$570$	0	0
$571$	3.48432e9	0.783234	0.391617	−	0.920128i	$-0.371916\pi$
$571$	3.48432e9	0.783234	0.391617	+	0.920128i	$0.371916\pi$
$572$	0	0
$573$	4.84338e9	1.07549
$574$	0	0
$575$	−3.67283e9	−0.805681
$576$	0	0
$577$	6.22782e9	1.34965	0.674825	−	0.737978i	$-0.264219\pi$
$577$	6.22782e9	1.34965	0.674825	+	0.737978i	$0.264219\pi$
$578$	0	0
$579$	6.21743e9	1.33118
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.38519e7	0.0196157
$584$	0	0
$585$	−4.23888e9	−0.875398
$586$	0	0
$587$	−1.01119e9	−0.206348	−0.103174	−	0.994663i	$-0.532900\pi$
$587$	−1.01119e9	−0.206348	−0.103174	+	0.994663i	$0.532900\pi$
$588$	0	0
$589$	−8.84189e9	−1.78296
$590$	0	0
$591$	−7.26394e9	−1.44749
$592$	0	0
$593$	−7.91897e9	−1.55947	−0.779736	−	0.626109i	$-0.784647\pi$
$593$	−7.91897e9	−1.55947	−0.779736	+	0.626109i	$0.784647\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.30595e9	−0.443547
$598$	0	0
$599$	−2.34123e9	−0.445093	−0.222546	−	0.974922i	$-0.571437\pi$
$599$	−2.34123e9	−0.445093	−0.222546	+	0.974922i	$0.571437\pi$
$600$	0	0
$601$	1.13771e8	0.0213781	0.0106891	−	0.999943i	$-0.496598\pi$
$601$	1.13771e8	0.0213781	0.0106891	+	0.999943i	$0.496598\pi$
$602$	0	0
$603$	−1.97736e9	−0.367262
$604$	0	0
$605$	−8.12553e9	−1.49179
$606$	0	0
$607$	−4.03060e9	−0.731492	−0.365746	−	0.930715i	$-0.619186\pi$
$607$	−4.03060e9	−0.731492	−0.365746	+	0.930715i	$0.619186\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.64831e9	−0.292345
$612$	0	0
$613$	1.34356e9	0.235584	0.117792	−	0.993038i	$-0.462418\pi$
$613$	1.34356e9	0.235584	0.117792	+	0.993038i	$0.462418\pi$
$614$	0	0
$615$	1.65661e10	2.87181
$616$	0	0
$617$	6.24917e9	1.07109	0.535543	−	0.844508i	$-0.320107\pi$
$617$	6.24917e9	1.07109	0.535543	+	0.844508i	$0.320107\pi$
$618$	0	0
$619$	7.69026e9	1.30324	0.651619	−	0.758546i	$-0.274090\pi$
$619$	7.69026e9	1.30324	0.651619	+	0.758546i	$0.274090\pi$
$620$	0	0
$621$	−2.05645e9	−0.344586
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.28212e9	−0.701583
$626$	0	0
$627$	−8.30836e8	−0.134611
$628$	0	0
$629$	−3.61558e9	−0.579296
$630$	0	0
$631$	6.83436e9	1.08292	0.541459	−	0.840727i	$-0.317872\pi$
$631$	6.83436e9	1.08292	0.541459	+	0.840727i	$0.317872\pi$
$632$	0	0
$633$	9.66441e8	0.151448
$634$	0	0
$635$	−6.94471e9	−1.07633
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.89720e9	−0.439264
$640$	0	0
$641$	4.58177e9	0.687116	0.343558	−	0.939131i	$-0.388368\pi$
$641$	4.58177e9	0.687116	0.343558	+	0.939131i	$0.388368\pi$
$642$	0	0
$643$	−6.98580e9	−1.03628	−0.518140	−	0.855296i	$-0.673376\pi$
$643$	−6.98580e9	−1.03628	−0.518140	+	0.855296i	$0.673376\pi$
$644$	0	0
$645$	−1.85740e10	−2.72551
$646$	0	0
$647$	−9.79379e9	−1.42163	−0.710814	−	0.703380i	$-0.751673\pi$
$647$	−9.79379e9	−1.42163	−0.710814	+	0.703380i	$0.751673\pi$
$648$	0	0
$649$	−4.93899e8	−0.0709221
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.22131e10	1.71645	0.858223	−	0.513276i	$-0.171568\pi$
$653$	1.22131e10	1.71645	0.858223	+	0.513276i	$0.171568\pi$
$654$	0	0
$655$	−1.67758e10	−2.33260
$656$	0	0
$657$	−1.48133e9	−0.203785
$658$	0	0
$659$	−6.39022e9	−0.869795	−0.434898	−	0.900480i	$-0.643215\pi$
$659$	−6.39022e9	−0.869795	−0.434898	+	0.900480i	$0.643215\pi$
$660$	0	0
$661$	1.93942e9	0.261196	0.130598	−	0.991435i	$-0.458310\pi$
$661$	1.93942e9	0.261196	0.130598	+	0.991435i	$0.458310\pi$
$662$	0	0
$663$	−1.20806e10	−1.60987
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.76562e9	−0.621841
$668$	0	0
$669$	5.36140e9	0.692287
$670$	0	0
$671$	−6.97699e8	−0.0891536
$672$	0	0
$673$	−9.25555e8	−0.117044	−0.0585221	−	0.998286i	$-0.518639\pi$
$673$	−9.25555e8	−0.117044	−0.0585221	+	0.998286i	$0.518639\pi$
$674$	0	0
$675$	5.25930e9	0.658210
$676$	0	0
$677$	−5.49367e9	−0.680460	−0.340230	−	0.940342i	$-0.610505\pi$
$677$	−5.49367e9	−0.680460	−0.340230	+	0.940342i	$0.610505\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.94244e10	2.35686
$682$	0	0
$683$	−4.14007e9	−0.497205	−0.248603	−	0.968606i	$-0.579971\pi$
$683$	−4.14007e9	−0.497205	−0.248603	+	0.968606i	$0.579971\pi$
$684$	0	0
$685$	3.72603e8	0.0442925
$686$	0	0
$687$	−1.62208e10	−1.90864
$688$	0	0
$689$	−2.93387e9	−0.341723
$690$	0	0
$691$	−1.04694e10	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$691$	−1.04694e10	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.95122e9	0.672448
$696$	0	0
$697$	1.72851e10	1.93356
$698$	0	0
$699$	−1.81127e10	−2.00592
$700$	0	0
$701$	1.45142e10	1.59141	0.795703	−	0.605686i	$-0.207101\pi$
$701$	1.45142e10	1.59141	0.795703	+	0.605686i	$0.207101\pi$
$702$	0	0
$703$	7.77327e9	0.843840
$704$	0	0
$705$	−5.05016e9	−0.542805
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.12754e10	−1.18815	−0.594074	−	0.804410i	$-0.702481\pi$
$709$	−1.12754e10	−1.18815	−0.594074	+	0.804410i	$0.702481\pi$
$710$	0	0
$711$	9.09669e8	0.0949160
$712$	0	0
$713$	6.07814e9	0.627997
$714$	0	0
$715$	−8.61043e8	−0.0880955
$716$	0	0
$717$	4.14614e9	0.420075
$718$	0	0
$719$	5.68142e9	0.570041	0.285020	−	0.958521i	$-0.408000\pi$
$719$	5.68142e9	0.570041	0.285020	+	0.958521i	$0.408000\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.68343e8	−0.0460871
$724$	0	0
$725$	1.21879e10	1.18781
$726$	0	0
$727$	1.98501e9	0.191599	0.0957993	−	0.995401i	$-0.469459\pi$
$727$	1.98501e9	0.191599	0.0957993	+	0.995401i	$0.469459\pi$
$728$	0	0
$729$	−5.44705e8	−0.0520733
$730$	0	0
$731$	−1.93802e10	−1.83505
$732$	0	0
$733$	5.03969e9	0.472651	0.236325	−	0.971674i	$-0.424057\pi$
$733$	5.03969e9	0.472651	0.236325	+	0.971674i	$0.424057\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.01662e8	−0.0369594
$738$	0	0
$739$	−1.10178e10	−1.00425	−0.502124	−	0.864796i	$-0.667448\pi$
$739$	−1.10178e10	−1.00425	−0.502124	+	0.864796i	$0.667448\pi$
$740$	0	0
$741$	2.59725e10	2.34504
$742$	0	0
$743$	4.80549e9	0.429810	0.214905	−	0.976635i	$-0.431056\pi$
$743$	4.80549e9	0.429810	0.214905	+	0.976635i	$0.431056\pi$
$744$	0	0
$745$	−2.08403e10	−1.84654
$746$	0	0
$747$	3.01955e9	0.265045
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.65858e10	1.42888	0.714440	−	0.699697i	$-0.246682\pi$
$751$	1.65858e10	1.42888	0.714440	+	0.699697i	$0.246682\pi$
$752$	0	0
$753$	6.40643e9	0.546806
$754$	0	0
$755$	−2.17766e10	−1.84152
$756$	0	0
$757$	−1.01328e10	−0.848973	−0.424487	−	0.905434i	$-0.639545\pi$
$757$	−1.01328e10	−0.848973	−0.424487	+	0.905434i	$0.639545\pi$
$758$	0	0
$759$	5.71139e8	0.0474128
$760$	0	0
$761$	1.63036e9	0.134103	0.0670514	−	0.997750i	$-0.478641\pi$
$761$	1.63036e9	0.134103	0.0670514	+	0.997750i	$0.478641\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.35509e10	−1.09435
$766$	0	0
$767$	1.54396e10	1.23553
$768$	0	0
$769$	−3.58444e9	−0.284236	−0.142118	−	0.989850i	$-0.545391\pi$
$769$	−3.58444e9	−0.284236	−0.142118	+	0.989850i	$0.545391\pi$
$770$	0	0
$771$	−3.65945e9	−0.287558
$772$	0	0
$773$	−1.00689e10	−0.784069	−0.392034	−	0.919951i	$-0.628229\pi$
$773$	−1.00689e10	−0.784069	−0.392034	+	0.919951i	$0.628229\pi$
$774$	0	0
$775$	−1.55446e10	−1.19957
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.71619e10	−2.81654
$780$	0	0
$781$	−5.88508e8	−0.0442052
$782$	0	0
$783$	6.82411e9	0.508019
$784$	0	0
$785$	1.11765e10	0.824639
$786$	0	0
$787$	1.73869e10	1.27149	0.635743	−	0.771901i	$-0.280694\pi$
$787$	1.73869e10	1.27149	0.635743	+	0.771901i	$0.280694\pi$
$788$	0	0
$789$	5.26197e9	0.381398
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.18105e10	1.55314
$794$	0	0
$795$	−8.98891e9	−0.634486
$796$	0	0
$797$	−2.03291e10	−1.42238	−0.711189	−	0.703001i	$-0.751843\pi$
$797$	−2.03291e10	−1.42238	−0.711189	+	0.703001i	$0.751843\pi$
$798$	0	0
$799$	−5.26936e9	−0.365464
$800$	0	0
$801$	−1.56806e10	−1.07808
$802$	0	0
$803$	−3.00902e8	−0.0205079
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.63033e10	1.09199
$808$	0	0
$809$	−5.56009e9	−0.369200	−0.184600	−	0.982814i	$-0.559099\pi$
$809$	−5.56009e9	−0.369200	−0.184600	+	0.982814i	$0.559099\pi$
$810$	0	0
$811$	1.89473e10	1.24731	0.623655	−	0.781700i	$-0.285647\pi$
$811$	1.89473e10	1.24731	0.623655	+	0.781700i	$0.285647\pi$
$812$	0	0
$813$	−1.12061e8	−0.00731370
$814$	0	0
$815$	7.17556e9	0.464306
$816$	0	0
$817$	4.16663e10	2.67305
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.58667e10	−1.00066	−0.500329	−	0.865835i	$-0.666788\pi$
$821$	−1.58667e10	−1.00066	−0.500329	+	0.865835i	$0.666788\pi$
$822$	0	0
$823$	1.70234e10	1.06450	0.532251	−	0.846586i	$-0.321346\pi$
$823$	1.70234e10	1.06450	0.532251	+	0.846586i	$0.321346\pi$
$824$	0	0
$825$	−1.46067e9	−0.0905653
$826$	0	0
$827$	3.25344e9	0.200020	0.100010	−	0.994986i	$-0.468112\pi$
$827$	3.25344e9	0.200020	0.100010	+	0.994986i	$0.468112\pi$
$828$	0	0
$829$	2.58845e10	1.57797	0.788985	−	0.614412i	$-0.210607\pi$
$829$	2.58845e10	1.57797	0.788985	+	0.614412i	$0.210607\pi$
$830$	0	0
$831$	2.24108e10	1.35473
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.92958e10	1.74142
$836$	0	0
$837$	−8.70358e9	−0.513049
$838$	0	0
$839$	−2.19214e10	−1.28145	−0.640724	−	0.767771i	$-0.721366\pi$
$839$	−2.19214e10	−1.28145	−0.640724	+	0.767771i	$0.721366\pi$
$840$	0	0
$841$	−1.43567e9	−0.0832278
$842$	0	0
$843$	−9.31706e9	−0.535652
$844$	0	0
$845$	6.63977e8	0.0378577
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.42833e10	−0.801034
$850$	0	0
$851$	−5.34355e9	−0.297219
$852$	0	0
$853$	3.22899e10	1.78133	0.890666	−	0.454657i	$-0.150238\pi$
$853$	3.22899e10	1.78133	0.890666	+	0.454657i	$0.150238\pi$
$854$	0	0
$855$	2.91337e10	1.59409
$856$	0	0
$857$	2.54343e10	1.38034	0.690172	−	0.723646i	$-0.257535\pi$
$857$	2.54343e10	1.38034	0.690172	+	0.723646i	$0.257535\pi$
$858$	0	0
$859$	−1.99925e10	−1.07619	−0.538096	−	0.842883i	$-0.680856\pi$
$859$	−1.99925e10	−1.07619	−0.538096	+	0.842883i	$0.680856\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.38285e9	0.126200	0.0630998	−	0.998007i	$-0.479901\pi$
$863$	2.38285e9	0.126200	0.0630998	+	0.998007i	$0.479901\pi$
$864$	0	0
$865$	−2.16308e10	−1.13636
$866$	0	0
$867$	−1.45170e10	−0.756503
$868$	0	0
$869$	1.84781e8	0.00955186
$870$	0	0
$871$	1.25562e10	0.643866
$872$	0	0
$873$	−1.86952e10	−0.950999
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.02965e9	−0.101607	−0.0508033	−	0.998709i	$-0.516178\pi$
$877$	−2.02965e9	−0.101607	−0.0508033	+	0.998709i	$0.516178\pi$
$878$	0	0
$879$	−3.13975e10	−1.55932
$880$	0	0
$881$	2.02197e10	0.996231	0.498116	−	0.867111i	$-0.334026\pi$
$881$	2.02197e10	0.996231	0.498116	+	0.867111i	$0.334026\pi$
$882$	0	0
$883$	2.38857e10	1.16755	0.583775	−	0.811916i	$-0.301575\pi$
$883$	2.38857e10	1.16755	0.583775	+	0.811916i	$0.301575\pi$
$884$	0	0
$885$	4.73045e10	2.29404
$886$	0	0
$887$	−3.47065e10	−1.66985	−0.834926	−	0.550363i	$-0.814489\pi$
$887$	−3.47065e10	−1.66985	−0.834926	+	0.550363i	$0.814489\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.52665e9	−0.0723048
$892$	0	0
$893$	1.13288e10	0.532358
$894$	0	0
$895$	−1.39020e10	−0.648183
$896$	0	0
$897$	−1.78542e10	−0.825974
$898$	0	0
$899$	−2.01697e10	−0.925849
$900$	0	0
$901$	−9.37907e9	−0.427192
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.16358e10	−2.31569
$906$	0	0
$907$	−1.35367e10	−0.602405	−0.301203	−	0.953560i	$-0.597388\pi$
$907$	−1.35367e10	−0.602405	−0.301203	+	0.953560i	$0.597388\pi$
$908$	0	0
$909$	8.11092e9	0.358176
$910$	0	0
$911$	1.62909e10	0.713889	0.356945	−	0.934126i	$-0.383819\pi$
$911$	1.62909e10	0.713889	0.356945	+	0.934126i	$0.383819\pi$
$912$	0	0
$913$	6.13362e8	0.0266728
$914$	0	0
$915$	6.68239e10	2.88375
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.09325e10	0.464641	0.232320	−	0.972639i	$-0.425368\pi$
$919$	1.09325e10	0.464641	0.232320	+	0.972639i	$0.425368\pi$
$920$	0	0
$921$	−3.86035e10	−1.62824
$922$	0	0
$923$	1.83972e10	0.770096
$924$	0	0
$925$	1.36659e10	0.567732
$926$	0	0
$927$	4.37619e9	0.180433
$928$	0	0
$929$	2.90569e10	1.18903	0.594516	−	0.804084i	$-0.297344\pi$
$929$	2.90569e10	1.18903	0.594516	+	0.804084i	$0.297344\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.25648e9	0.131269
$934$	0	0
$935$	−2.75260e9	−0.110129
$936$	0	0
$937$	3.95737e10	1.57151	0.785757	−	0.618535i	$-0.212273\pi$
$937$	3.95737e10	1.57151	0.785757	+	0.618535i	$0.212273\pi$
$938$	0	0
$939$	2.65113e10	1.04496
$940$	0	0
$941$	−2.76957e10	−1.08355	−0.541774	−	0.840524i	$-0.682247\pi$
$941$	−2.76957e10	−1.08355	−0.541774	+	0.840524i	$0.682247\pi$
$942$	0	0
$943$	2.55461e10	0.992049
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.55874e10	−1.36167	−0.680834	−	0.732438i	$-0.738383\pi$
$947$	−3.55874e10	−1.36167	−0.680834	+	0.732438i	$0.738383\pi$
$948$	0	0
$949$	9.40641e9	0.357266
$950$	0	0
$951$	3.60581e9	0.135947
$952$	0	0
$953$	−2.97100e10	−1.11193	−0.555964	−	0.831206i	$-0.687651\pi$
$953$	−2.97100e10	−1.11193	−0.555964	+	0.831206i	$0.687651\pi$
$954$	0	0
$955$	−3.44987e10	−1.28171
$956$	0	0
$957$	−1.89526e9	−0.0699001
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.78790e9	−0.0649849
$962$	0	0
$963$	2.25926e10	0.815218
$964$	0	0
$965$	−4.42858e10	−1.58642
$966$	0	0
$967$	−2.76167e10	−0.982152	−0.491076	−	0.871117i	$-0.663396\pi$
$967$	−2.76167e10	−0.982152	−0.491076	+	0.871117i	$0.663396\pi$
$968$	0	0
$969$	8.30294e10	2.93156
$970$	0	0
$971$	−3.89828e10	−1.36649	−0.683245	−	0.730189i	$-0.739432\pi$
$971$	−3.89828e10	−1.36649	−0.683245	+	0.730189i	$0.739432\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.56614e10	1.57773
$976$	0	0
$977$	3.37230e10	1.15690	0.578450	−	0.815718i	$-0.303658\pi$
$977$	3.37230e10	1.15690	0.578450	+	0.815718i	$0.303658\pi$
$978$	0	0
$979$	−3.18521e9	−0.108492
$980$	0	0
$981$	−5.77177e9	−0.195195
$982$	0	0
$983$	−2.62407e10	−0.881126	−0.440563	−	0.897722i	$-0.645221\pi$
$983$	−2.62407e10	−0.881126	−0.440563	+	0.897722i	$0.645221\pi$
$984$	0	0
$985$	5.17399e10	1.72504
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.86425e10	−0.941507
$990$	0	0
$991$	−8.79302e9	−0.286999	−0.143500	−	0.989650i	$-0.545836\pi$
$991$	−8.79302e9	−0.286999	−0.143500	+	0.989650i	$0.545836\pi$
$992$	0	0
$993$	−2.39510e10	−0.776249
$994$	0	0
$995$	1.64249e10	0.528595
$996$	0	0
$997$	−3.94096e10	−1.25941	−0.629707	−	0.776833i	$-0.716825\pi$
$997$	−3.94096e10	−1.25941	−0.629707	+	0.776833i	$0.716825\pi$
$998$	0	0
$999$	7.65168e9	0.242816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.8.a.d.1.2		5
7.2	even	3		28.8.e.a.25.4	yes	10
7.3	odd	6		196.8.e.f.177.2		10
7.4	even	3		28.8.e.a.9.4	✓	10
7.5	odd	6		196.8.e.f.165.2		10
7.6	odd	2		196.8.a.e.1.4		5
21.2	odd	6		252.8.k.c.109.5		10
21.11	odd	6		252.8.k.c.37.5		10
28.11	odd	6		112.8.i.d.65.2		10
28.23	odd	6		112.8.i.d.81.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.e.a.9.4	✓	10	7.4	even	3
28.8.e.a.25.4	yes	10	7.2	even	3
112.8.i.d.65.2		10	28.11	odd	6
112.8.i.d.81.2		10	28.23	odd	6
196.8.a.d.1.2		5	1.1	even	1	trivial
196.8.a.e.1.4		5	7.6	odd	2
196.8.e.f.165.2		10	7.5	odd	6
196.8.e.f.177.2		10	7.3	odd	6
252.8.k.c.37.5		10	21.11	odd	6
252.8.k.c.109.5		10	21.2	odd	6

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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)

\(f(q)\)	\(=\)	\(q-58.7379 q^{3} +418.382 q^{5} +1263.14 q^{9} +256.583 q^{11} -8020.94 q^{13} -24574.9 q^{15} -25641.5 q^{17} +55127.7 q^{19} -37896.2 q^{23} +96918.1 q^{25} +54265.4 q^{27} +125755. q^{29} -160389. q^{31} -15071.1 q^{33} +141005. q^{37} +471134. q^{39} -674106. q^{41} +755814. q^{43} +528476. q^{45} +205501. q^{47} +1.50613e6 q^{51} +365777. q^{53} +107349. q^{55} -3.23809e6 q^{57} -1.92491e6 q^{59} -2.71920e6 q^{61} -3.35581e6 q^{65} -1.56543e6 q^{67} +2.22595e6 q^{69} -2.29364e6 q^{71} -1.17273e6 q^{73} -5.69277e6 q^{75} +720162. q^{79} -5.94993e6 q^{81} +2.39050e6 q^{83} -1.07279e7 q^{85} -7.38656e6 q^{87} -1.24140e7 q^{89} +9.42093e6 q^{93} +2.30644e7 q^{95} -1.48005e7 q^{97} +324101. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 27 q^{3} - 249 q^{5} + 5702 q^{9} - 6399 q^{11} - 13494 q^{13} + 9647 q^{15} - 3609 q^{17} + 12403 q^{19} + 13959 q^{23} + 162364 q^{25} + 80775 q^{27} + 13074 q^{29} + 20181 q^{31} - 264239 q^{33}+ \cdots - 59638906 q^{99}+O(q^{100}) \) 5 * q - 27 * q^3 - 249 * q^5 + 5702 * q^9 - 6399 * q^11 - 13494 * q^13 + 9647 * q^15 - 3609 * q^17 + 12403 * q^19 + 13959 * q^23 + 162364 * q^25 + 80775 * q^27 + 13074 * q^29 + 20181 * q^31 - 264239 * q^33 + 54763 * q^37 - 4750 * q^39 - 912234 * q^41 - 969212 * q^43 - 1936430 * q^45 - 1217253 * q^47 + 1353871 * q^51 + 2349159 * q^53 + 1561747 * q^55 - 4911597 * q^57 - 2188611 * q^59 - 6107445 * q^61 + 3022014 * q^65 + 3171581 * q^67 - 5627969 * q^69 - 9469488 * q^71 - 8034853 * q^73 - 29173588 * q^75 + 7589559 * q^79 + 9846485 * q^81 - 6726204 * q^83 - 24358835 * q^85 - 30033142 * q^87 - 29864757 * q^89 + 20217029 * q^93 + 25812009 * q^95 + 142790 * q^97 - 59638906 * q^99

Introduction

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