LMFDB - Embedded newform 1764.2.f.b.881.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(881,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1764.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.f (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.6
Root			$-0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1764.881
Dual form			1764.2.f.b.881.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765367 q^{5} +O(q^{10})$	$q+0.765367 q^{5} +2.00000i q^{11} -0.317025i q^{13} +5.54328 q^{17} +3.69552i q^{19} +3.17157i q^{23} -4.41421 q^{25} -6.82843i q^{29} +6.75699i q^{31} -0.242641 q^{37} -2.74444 q^{41} +6.82843 q^{43} +11.9832 q^{47} +12.2426i q^{53} +1.53073i q^{55} +13.2513 q^{59} +3.56420i q^{61} -0.242641i q^{65} -4.48528 q^{67} -9.31371i q^{71} +11.8519i q^{73} -11.3137 q^{79} +4.32957 q^{83} +4.24264 q^{85} +1.66205 q^{89} +2.82843i q^{95} -11.8519i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{67}+O(q^{100}) $ 8 * q - 24 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.765367	0.342282	0.171141	−	0.985247i	$-0.445255\pi$
$5$	0.765367	0.342282	0.171141	+	0.985247i	$0.445255\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	− 0.317025i	− 0.0879270i	−0.999033	−	0.0439635i	$-0.986001\pi$
$13$	− 0.317025i	− 0.0879270i	0.999033	−	0.0439635i	$-0.0139985\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.54328	1.34444	0.672221	−	0.740350i	$-0.265340\pi$
$17$	5.54328	1.34444	0.672221	+	0.740350i	$0.265340\pi$
$18$	0	0
$19$	3.69552i	0.847810i	0.905707	+	0.423905i	$0.139341\pi$
$19$	3.69552i	0.847810i	−0.905707	+	0.423905i	$0.860659\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.17157i	0.661319i	0.943750	+	0.330659i	$0.107271\pi$
$23$	3.17157i	0.661319i	−0.943750	+	0.330659i	$0.892729\pi$
$24$	0	0
$25$	−4.41421	−0.882843
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.82843i	− 1.26801i	−0.773330	−	0.634004i	$-0.781410\pi$
$29$	− 6.82843i	− 1.26801i	0.773330	−	0.634004i	$-0.218590\pi$
$30$	0	0
$31$	6.75699i	1.21359i	0.794858	+	0.606795i	$0.207545\pi$
$31$	6.75699i	1.21359i	−0.794858	+	0.606795i	$0.792455\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.242641	−0.0398899	−0.0199449	−	0.999801i	$-0.506349\pi$
$37$	−0.242641	−0.0398899	−0.0199449	+	0.999801i	$0.506349\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.74444	−0.428610	−0.214305	−	0.976767i	$-0.568749\pi$
$41$	−2.74444	−0.428610	−0.214305	+	0.976767i	$0.568749\pi$
$42$	0	0
$43$	6.82843	1.04133	0.520663	−	0.853762i	$-0.325685\pi$
$43$	6.82843	1.04133	0.520663	+	0.853762i	$0.325685\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.9832	1.74793	0.873967	−	0.485985i	$-0.161539\pi$
$47$	11.9832	1.74793	0.873967	+	0.485985i	$0.161539\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.2426i	1.68166i	0.541302	+	0.840828i	$0.317931\pi$
$53$	12.2426i	1.68166i	−0.541302	+	0.840828i	$0.682069\pi$
$54$	0	0
$55$	1.53073i	0.206404i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.2513	1.72518	0.862589	−	0.505906i	$-0.168842\pi$
$59$	13.2513	1.72518	0.862589	+	0.505906i	$0.168842\pi$
$60$	0	0
$61$	3.56420i	0.456349i	0.973620	+	0.228175i	$0.0732757\pi$
$61$	3.56420i	0.456349i	−0.973620	+	0.228175i	$0.926724\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.242641i	− 0.0300959i
$66$	0	0
$67$	−4.48528	−0.547964	−0.273982	−	0.961735i	$-0.588341\pi$
$67$	−4.48528	−0.547964	−0.273982	+	0.961735i	$0.588341\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 9.31371i	− 1.10533i	−0.833402	−	0.552667i	$-0.813610\pi$
$71$	− 9.31371i	− 1.10533i	0.833402	−	0.552667i	$-0.186390\pi$
$72$	0	0
$73$	11.8519i	1.38716i	0.720378	+	0.693581i	$0.243968\pi$
$73$	11.8519i	1.38716i	−0.720378	+	0.693581i	$0.756032\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.32957	0.475232	0.237616	−	0.971359i	$-0.423634\pi$
$83$	4.32957	0.475232	0.237616	+	0.971359i	$0.423634\pi$
$84$	0	0
$85$	4.24264	0.460179
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.66205	0.176177	0.0880885	−	0.996113i	$-0.471924\pi$
$89$	1.66205	0.176177	0.0880885	+	0.996113i	$0.471924\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.82843i	0.290191i
$96$	0	0
$97$	− 11.8519i	− 1.20338i	−0.798730	−	0.601690i	$-0.794494\pi$
$97$	− 11.8519i	− 1.20338i	0.798730	−	0.601690i	$-0.205506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.8519	1.17931	0.589655	−	0.807655i	$-0.299264\pi$
$101$	11.8519	1.17931	0.589655	+	0.807655i	$0.299264\pi$
$102$	0	0
$103$	8.92177i	0.879088i	0.898221	+	0.439544i	$0.144860\pi$
$103$	8.92177i	0.879088i	−0.898221	+	0.439544i	$0.855140\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.17157i	0.693302i	0.937994	+	0.346651i	$0.112681\pi$
$107$	7.17157i	0.693302i	−0.937994	+	0.346651i	$0.887319\pi$
$108$	0	0
$109$	−11.0711	−1.06042	−0.530208	−	0.847868i	$-0.677886\pi$
$109$	−11.0711	−1.06042	−0.530208	+	0.847868i	$0.677886\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 10.5858i	− 0.995827i	−0.867227	−	0.497914i	$-0.834100\pi$
$113$	− 10.5858i	− 0.995827i	0.867227	−	0.497914i	$-0.165900\pi$
$114$	0	0
$115$	2.42742i	0.226358i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.20533	−0.644464
$126$	0	0
$127$	1.17157	0.103960	0.0519801	−	0.998648i	$-0.483447\pi$
$127$	1.17157	0.103960	0.0519801	+	0.998648i	$0.483447\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.9469	1.48065	0.740327	−	0.672247i	$-0.234671\pi$
$131$	16.9469	1.48065	0.740327	+	0.672247i	$0.234671\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.8284i	0.925135i	0.886584	+	0.462567i	$0.153072\pi$
$137$	10.8284i	0.925135i	−0.886584	+	0.462567i	$0.846928\pi$
$138$	0	0
$139$	15.6788i	1.32985i	0.746908	+	0.664927i	$0.231538\pi$
$139$	15.6788i	1.32985i	−0.746908	+	0.664927i	$0.768462\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.634051	0.0530220
$144$	0	0
$145$	− 5.22625i	− 0.434017i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.07107i	− 0.579284i	−0.957135	−	0.289642i	$-0.906464\pi$
$149$	− 7.07107i	− 0.579284i	0.957135	−	0.289642i	$-0.0935363\pi$
$150$	0	0
$151$	10.1421	0.825355	0.412678	−	0.910877i	$-0.364594\pi$
$151$	10.1421	0.825355	0.412678	+	0.910877i	$0.364594\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.17157i	0.415391i
$156$	0	0
$157$	6.17733i	0.493004i	0.969142	+	0.246502i	$0.0792813\pi$
$157$	6.17733i	0.493004i	−0.969142	+	0.246502i	$0.920719\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.82843	0.221540	0.110770	−	0.993846i	$-0.464668\pi$
$163$	2.82843	0.221540	0.110770	+	0.993846i	$0.464668\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.42742	0.187839	0.0939196	−	0.995580i	$-0.470060\pi$
$167$	2.42742	0.187839	0.0939196	+	0.995580i	$0.470060\pi$
$168$	0	0
$169$	12.8995	0.992269
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.7946	−1.42893	−0.714464	−	0.699672i	$-0.753329\pi$
$173$	−18.7946	−1.42893	−0.714464	+	0.699672i	$0.753329\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.31371i	− 0.696139i	−0.937469	−	0.348070i	$-0.886837\pi$
$179$	− 9.31371i	− 0.696139i	0.937469	−	0.348070i	$-0.113163\pi$
$180$	0	0
$181$	1.66205i	0.123539i	0.998090	+	0.0617696i	$0.0196744\pi$
$181$	1.66205i	0.123539i	−0.998090	+	0.0617696i	$0.980326\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.185709	−0.0136536
$186$	0	0
$187$	11.0866i	0.810729i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.3137i	1.25278i	0.779511	+	0.626388i	$0.215467\pi$
$191$	17.3137i	1.25278i	−0.779511	+	0.626388i	$0.784533\pi$
$192$	0	0
$193$	−9.65685	−0.695116	−0.347558	−	0.937659i	$-0.612989\pi$
$193$	−9.65685	−0.695116	−0.347558	+	0.937659i	$0.612989\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.07107i	0.218805i	0.993998	+	0.109402i	$0.0348937\pi$
$197$	3.07107i	0.218805i	−0.993998	+	0.109402i	$0.965106\pi$
$198$	0	0
$199$	− 7.39104i	− 0.523937i	−0.965076	−	0.261968i	$-0.915628\pi$
$199$	− 7.39104i	− 0.523937i	0.965076	−	0.261968i	$-0.0843716\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.10051	−0.146706
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.39104	−0.511249
$210$	0	0
$211$	−18.6274	−1.28236	−0.641182	−	0.767389i	$-0.721556\pi$
$211$	−18.6274	−1.28236	−0.641182	+	0.767389i	$0.721556\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.22625	0.356427
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.75736i	− 0.118213i
$222$	0	0
$223$	− 21.8017i	− 1.45995i	−0.683474	−	0.729975i	$-0.739532\pi$
$223$	− 21.8017i	− 1.45995i	0.683474	−	0.729975i	$-0.260468\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.75699	−0.448477	−0.224238	−	0.974534i	$-0.571989\pi$
$227$	−6.75699	−0.448477	−0.224238	+	0.974534i	$0.571989\pi$
$228$	0	0
$229$	− 16.1815i	− 1.06930i	−0.845073	−	0.534651i	$-0.820443\pi$
$229$	− 16.1815i	− 1.06930i	0.845073	−	0.534651i	$-0.179557\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.1716i	− 0.993923i	−0.867773	−	0.496961i	$-0.834449\pi$
$233$	− 15.1716i	− 0.993923i	0.867773	−	0.496961i	$-0.165551\pi$
$234$	0	0
$235$	9.17157	0.598287
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.82843i	0.571063i	0.958369	+	0.285532i	$0.0921702\pi$
$239$	8.82843i	0.571063i	−0.958369	+	0.285532i	$0.907830\pi$
$240$	0	0
$241$	− 25.5516i	− 1.64592i	−0.568097	−	0.822962i	$-0.692320\pi$
$241$	− 25.5516i	− 1.64592i	0.568097	−	0.822962i	$-0.307680\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.17157	0.0745454
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.4972	1.60937	0.804685	−	0.593702i	$-0.202334\pi$
$251$	25.4972	1.60937	0.804685	+	0.593702i	$0.202334\pi$
$252$	0	0
$253$	−6.34315	−0.398790
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.5880	−1.28424	−0.642122	−	0.766603i	$-0.721946\pi$
$257$	−20.5880	−1.28424	−0.642122	+	0.766603i	$0.721946\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 19.6569i	− 1.21209i	−0.795429	−	0.606047i	$-0.792754\pi$
$263$	− 19.6569i	− 1.21209i	0.795429	−	0.606047i	$-0.207246\pi$
$264$	0	0
$265$	9.37011i	0.575601i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.8979	−1.09126	−0.545628	−	0.838027i	$-0.683709\pi$
$269$	−17.8979	−1.09126	−0.545628	+	0.838027i	$0.683709\pi$
$270$	0	0
$271$	5.48888i	0.333426i	0.986005	+	0.166713i	$0.0533153\pi$
$271$	5.48888i	0.333426i	−0.986005	+	0.166713i	$0.946685\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 8.82843i	− 0.532374i
$276$	0	0
$277$	12.9706	0.779326	0.389663	−	0.920958i	$-0.372592\pi$
$277$	12.9706	0.779326	0.389663	+	0.920958i	$0.372592\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.4853i	0.625499i	0.949836	+	0.312750i	$0.101250\pi$
$281$	10.4853i	0.625499i	−0.949836	+	0.312750i	$0.898750\pi$
$282$	0	0
$283$	4.06694i	0.241754i	0.992667	+	0.120877i	$0.0385707\pi$
$283$	4.06694i	0.241754i	−0.992667	+	0.120877i	$0.961429\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.7279	0.807525
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.4441	−0.960676	−0.480338	−	0.877083i	$-0.659486\pi$
$293$	−16.4441	−0.960676	−0.480338	+	0.877083i	$0.659486\pi$
$294$	0	0
$295$	10.1421	0.590498
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.00547	0.0581478
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.72792i	0.156200i
$306$	0	0
$307$	27.1367i	1.54877i	0.632712	+	0.774387i	$0.281942\pi$
$307$	27.1367i	1.54877i	−0.632712	+	0.774387i	$0.718058\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.7235	−1.74217	−0.871084	−	0.491134i	$-0.836582\pi$
$311$	−30.7235	−1.74217	−0.871084	+	0.491134i	$0.836582\pi$
$312$	0	0
$313$	− 28.8757i	− 1.63215i	−0.577945	−	0.816076i	$-0.696145\pi$
$313$	− 28.8757i	− 1.63215i	0.577945	−	0.816076i	$-0.303855\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.7574i	− 0.885021i	−0.896763	−	0.442511i	$-0.854088\pi$
$317$	− 15.7574i	− 0.885021i	0.896763	−	0.442511i	$-0.145912\pi$
$318$	0	0
$319$	13.6569	0.764637
$320$	0	0
$321$	0	0
$322$	0	0
$323$	20.4853i	1.13983i
$324$	0	0
$325$	1.39942i	0.0776257i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−27.3137	−1.50130	−0.750649	−	0.660702i	$-0.770259\pi$
$331$	−27.3137	−1.50130	−0.750649	+	0.660702i	$0.770259\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.43289	−0.187559
$336$	0	0
$337$	−27.0711	−1.47466	−0.737328	−	0.675535i	$-0.763913\pi$
$337$	−27.0711	−1.47466	−0.737328	+	0.675535i	$0.763913\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.5140	−0.731823
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 29.7990i	− 1.59969i	−0.600204	−	0.799847i	$-0.704914\pi$
$347$	− 29.7990i	− 1.59969i	0.600204	−	0.799847i	$-0.295086\pi$
$348$	0	0
$349$	− 31.9372i	− 1.70956i	−0.518993	−	0.854779i	$-0.673693\pi$
$349$	− 31.9372i	− 1.70956i	0.518993	−	0.854779i	$-0.326307\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.1451	−1.12544	−0.562720	−	0.826647i	$-0.690245\pi$
$353$	−21.1451	−1.12544	−0.562720	+	0.826647i	$0.690245\pi$
$354$	0	0
$355$	− 7.12840i	− 0.378336i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.4558i	1.02684i	0.858137	+	0.513420i	$0.171622\pi$
$359$	19.4558i	1.02684i	−0.858137	+	0.513420i	$0.828378\pi$
$360$	0	0
$361$	5.34315	0.281218
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.07107i	0.474801i
$366$	0	0
$367$	− 36.0585i	− 1.88224i	−0.338075	−	0.941119i	$-0.609776\pi$
$367$	− 36.0585i	− 1.88224i	0.338075	−	0.941119i	$-0.390224\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	30.6274	1.58583	0.792914	−	0.609334i	$-0.208563\pi$
$373$	30.6274	1.58583	0.792914	+	0.609334i	$0.208563\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.16478	−0.111492
$378$	0	0
$379$	−36.2843	−1.86380	−0.931899	−	0.362718i	$-0.881849\pi$
$379$	−36.2843	−1.86380	−0.931899	+	0.362718i	$0.881849\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.7821	−0.755329	−0.377664	−	0.925943i	$-0.623273\pi$
$383$	−14.7821	−0.755329	−0.377664	+	0.925943i	$0.623273\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.17157i	− 0.0594011i	−0.999559	−	0.0297006i	$-0.990545\pi$
$389$	− 1.17157i	− 0.0594011i	0.999559	−	0.0297006i	$-0.00945537\pi$
$390$	0	0
$391$	17.5809i	0.889105i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.65914	−0.435688
$396$	0	0
$397$	− 14.0167i	− 0.703478i	−0.936098	−	0.351739i	$-0.885590\pi$
$397$	− 14.0167i	− 0.703478i	0.936098	−	0.351739i	$-0.114410\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 5.51472i	− 0.275392i	−0.990475	−	0.137696i	$-0.956030\pi$
$401$	− 5.51472i	− 0.275392i	0.990475	−	0.137696i	$-0.0439697\pi$
$402$	0	0
$403$	2.14214	0.106707
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 0.485281i	− 0.0240545i
$408$	0	0
$409$	1.13679i	0.0562104i	0.999605	+	0.0281052i	$0.00894734\pi$
$409$	1.13679i	0.0562104i	−0.999605	+	0.0281052i	$0.991053\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.31371	0.162664
$416$	0	0
$417$	0	0
$418$	0	0
$419$	27.6620	1.35138	0.675688	−	0.737187i	$-0.263846\pi$
$419$	27.6620	1.35138	0.675688	+	0.737187i	$0.263846\pi$
$420$	0	0
$421$	15.3137	0.746344	0.373172	−	0.927762i	$-0.378270\pi$
$421$	15.3137	0.746344	0.373172	+	0.927762i	$0.378270\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−24.4692	−1.18693
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 33.3137i	− 1.60466i	−0.596877	−	0.802332i	$-0.703592\pi$
$431$	− 33.3137i	− 1.60466i	0.596877	−	0.802332i	$-0.296408\pi$
$432$	0	0
$433$	− 0.502734i	− 0.0241599i	−0.999927	−	0.0120799i	$-0.996155\pi$
$433$	− 0.502734i	− 0.0241599i	0.999927	−	0.0120799i	$-0.00384526\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.7206	−0.560673
$438$	0	0
$439$	13.8854i	0.662713i	0.943506	+	0.331357i	$0.107506\pi$
$439$	13.8854i	0.662713i	−0.943506	+	0.331357i	$0.892494\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 29.3137i	− 1.39274i	−0.717685	−	0.696368i	$-0.754798\pi$
$443$	− 29.3137i	− 1.39274i	0.717685	−	0.696368i	$-0.245202\pi$
$444$	0	0
$445$	1.27208	0.0603023
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.5563i	0.545378i	0.962102	+	0.272689i	$0.0879130\pi$
$449$	11.5563i	0.545378i	−0.962102	+	0.272689i	$0.912087\pi$
$450$	0	0
$451$	− 5.48888i	− 0.258461i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.17186	0.240877	0.120439	−	0.992721i	$-0.461570\pi$
$461$	5.17186	0.240877	0.120439	+	0.992721i	$0.461570\pi$
$462$	0	0
$463$	−2.82843	−0.131448	−0.0657241	−	0.997838i	$-0.520936\pi$
$463$	−2.82843	−0.131448	−0.0657241	+	0.997838i	$0.520936\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.23172	0.288370	0.144185	−	0.989551i	$-0.453944\pi$
$467$	6.23172	0.288370	0.144185	+	0.989551i	$0.453944\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.6569i	0.627943i
$474$	0	0
$475$	− 16.3128i	− 0.748483i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.3324	−1.06609	−0.533043	−	0.846088i	$-0.678952\pi$
$479$	−23.3324	−1.06609	−0.533043	+	0.846088i	$0.678952\pi$
$480$	0	0
$481$	0.0769232i	0.00350740i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9.07107i	− 0.411896i
$486$	0	0
$487$	19.7990	0.897178	0.448589	−	0.893738i	$-0.351927\pi$
$487$	19.7990	0.897178	0.448589	+	0.893738i	$0.351927\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	34.4853i	1.55630i	0.628079	+	0.778149i	$0.283841\pi$
$491$	34.4853i	1.55630i	−0.628079	+	0.778149i	$0.716159\pi$
$492$	0	0
$493$	− 37.8519i	− 1.70476i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.1421	1.17028	0.585141	−	0.810931i	$-0.301039\pi$
$499$	26.1421	1.17028	0.585141	+	0.810931i	$0.301039\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.5641	−1.31820	−0.659100	−	0.752055i	$-0.729063\pi$
$503$	−29.5641	−1.31820	−0.659100	+	0.752055i	$0.729063\pi$
$504$	0	0
$505$	9.07107	0.403657
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.9343	−0.573303	−0.286652	−	0.958035i	$-0.592542\pi$
$509$	−12.9343	−0.573303	−0.286652	+	0.958035i	$0.592542\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.82843i	0.300896i
$516$	0	0
$517$	23.9665i	1.05404i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.03347	0.0890879	0.0445439	−	0.999007i	$-0.485817\pi$
$521$	2.03347	0.0890879	0.0445439	+	0.999007i	$0.485817\pi$
$522$	0	0
$523$	− 18.2150i	− 0.796485i	−0.917280	−	0.398242i	$-0.869620\pi$
$523$	− 18.2150i	− 0.796485i	0.917280	−	0.398242i	$-0.130380\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	37.4558i	1.63160i
$528$	0	0
$529$	12.9411	0.562658
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.870058i	0.0376864i
$534$	0	0
$535$	5.48888i	0.237305i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.34315	0.358700	0.179350	−	0.983785i	$-0.442601\pi$
$541$	8.34315	0.358700	0.179350	+	0.983785i	$0.442601\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.47343	−0.362962
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	25.2346	1.07503
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 16.7279i	− 0.708785i	−0.935097	−	0.354392i	$-0.884688\pi$
$557$	− 16.7279i	− 0.708785i	0.935097	−	0.354392i	$-0.115312\pi$
$558$	0	0
$559$	− 2.16478i	− 0.0915606i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.0643	0.929900	0.464950	−	0.885337i	$-0.346072\pi$
$563$	22.0643	0.929900	0.464950	+	0.885337i	$0.346072\pi$
$564$	0	0
$565$	− 8.10201i	− 0.340854i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 0.485281i	− 0.0203441i	−0.999948	−	0.0101720i	$-0.996762\pi$
$569$	− 0.485281i	− 0.0203441i	0.999948	−	0.0101720i	$-0.00323791\pi$
$570$	0	0
$571$	−2.62742	−0.109954	−0.0549770	−	0.998488i	$-0.517509\pi$
$571$	−2.62742	−0.109954	−0.0549770	+	0.998488i	$0.517509\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 14.0000i	− 0.583840i
$576$	0	0
$577$	19.8770i	0.827491i	0.910393	+	0.413745i	$0.135780\pi$
$577$	19.8770i	0.827491i	−0.910393	+	0.413745i	$0.864220\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−24.4853	−1.01408
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.0334	1.15706	0.578531	−	0.815660i	$-0.303626\pi$
$587$	28.0334	1.15706	0.578531	+	0.815660i	$0.303626\pi$
$588$	0	0
$589$	−24.9706	−1.02889
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.7779	−1.26389	−0.631947	−	0.775011i	$-0.717744\pi$
$593$	−30.7779	−1.26389	−0.631947	+	0.775011i	$0.717744\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.31371i	0.380548i	0.981731	+	0.190274i	$0.0609376\pi$
$599$	9.31371i	0.380548i	−0.981731	+	0.190274i	$0.939062\pi$
$600$	0	0
$601$	− 23.9121i	− 0.975394i	−0.873013	−	0.487697i	$-0.837837\pi$
$601$	− 23.9121i	− 0.975394i	0.873013	−	0.487697i	$-0.162163\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.35757	0.217816
$606$	0	0
$607$	12.6173i	0.512120i	0.966661	+	0.256060i	$0.0824245\pi$
$607$	12.6173i	0.512120i	−0.966661	+	0.256060i	$0.917576\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.79899i	− 0.153691i
$612$	0	0
$613$	−45.2132	−1.82614	−0.913072	−	0.407798i	$-0.866297\pi$
$613$	−45.2132	−1.82614	−0.913072	+	0.407798i	$0.866297\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 17.4558i	− 0.702746i	−0.936236	−	0.351373i	$-0.885715\pi$
$617$	− 17.4558i	− 0.702746i	0.936236	−	0.351373i	$-0.114285\pi$
$618$	0	0
$619$	1.79337i	0.0720815i	0.999350	+	0.0360407i	$0.0114746\pi$
$619$	1.79337i	0.0720815i	−0.999350	+	0.0360407i	$0.988525\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.5563	0.662254
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.34502	−0.0536296
$630$	0	0
$631$	−12.4853	−0.497031	−0.248516	−	0.968628i	$-0.579943\pi$
$631$	−12.4853	−0.497031	−0.248516	+	0.968628i	$0.579943\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.896683	0.0355838
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 23.4558i	− 0.926450i	−0.886241	−	0.463225i	$-0.846692\pi$
$641$	− 23.4558i	− 0.926450i	0.886241	−	0.463225i	$-0.153308\pi$
$642$	0	0
$643$	32.5168i	1.28234i	0.767400	+	0.641169i	$0.221550\pi$
$643$	32.5168i	1.28234i	−0.767400	+	0.641169i	$0.778450\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.38557	−0.251043	−0.125521	−	0.992091i	$-0.540060\pi$
$647$	−6.38557	−0.251043	−0.125521	+	0.992091i	$0.540060\pi$
$648$	0	0
$649$	26.5027i	1.04032i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 5.85786i	− 0.229236i	−0.993410	−	0.114618i	$-0.963436\pi$
$653$	− 5.85786i	− 0.229236i	0.993410	−	0.114618i	$-0.0365644\pi$
$654$	0	0
$655$	12.9706	0.506802
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.97056i	− 0.115717i	−0.998325	−	0.0578583i	$-0.981573\pi$
$659$	− 2.97056i	− 0.115717i	0.998325	−	0.0578583i	$-0.0184272\pi$
$660$	0	0
$661$	23.3868i	0.909642i	0.890583	+	0.454821i	$0.150297\pi$
$661$	23.3868i	0.909642i	−0.890583	+	0.454821i	$0.849703\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.6569	0.838557
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.12840	−0.275189
$672$	0	0
$673$	20.0416	0.772548	0.386274	−	0.922384i	$-0.373762\pi$
$673$	20.0416	0.772548	0.386274	+	0.922384i	$0.373762\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.9790	1.07532	0.537661	−	0.843161i	$-0.319308\pi$
$677$	27.9790	1.07532	0.537661	+	0.843161i	$0.319308\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	46.9706i	1.79728i	0.438688	+	0.898639i	$0.355443\pi$
$683$	46.9706i	1.79728i	−0.438688	+	0.898639i	$0.644557\pi$
$684$	0	0
$685$	8.28772i	0.316657i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.88123	0.147863
$690$	0	0
$691$	25.6060i	0.974098i	0.873375	+	0.487049i	$0.161927\pi$
$691$	25.6060i	0.974098i	−0.873375	+	0.487049i	$0.838073\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000i	0.455186i
$696$	0	0
$697$	−15.2132	−0.576241
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 15.4558i	− 0.583759i	−0.956455	−	0.291880i	$-0.905719\pi$
$701$	− 15.4558i	− 0.583759i	0.956455	−	0.291880i	$-0.0942807\pi$
$702$	0	0
$703$	− 0.896683i	− 0.0338190i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−30.8701	−1.15935	−0.579675	−	0.814848i	$-0.696820\pi$
$709$	−30.8701	−1.15935	−0.579675	+	0.814848i	$0.696820\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.4303	−0.802570
$714$	0	0
$715$	0.485281	0.0181485
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.6369	0.732333	0.366167	−	0.930549i	$-0.380670\pi$
$719$	19.6369	0.732333	0.366167	+	0.930549i	$0.380670\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	30.1421i	1.11945i
$726$	0	0
$727$	− 53.7933i	− 1.99508i	−0.0700903	−	0.997541i	$-0.522329\pi$
$727$	− 53.7933i	− 1.99508i	0.0700903	−	0.997541i	$-0.477671\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	37.8519	1.40000
$732$	0	0
$733$	− 22.3044i	− 0.823833i	−0.911222	−	0.411916i	$-0.864860\pi$
$733$	− 22.3044i	− 0.823833i	0.911222	−	0.411916i	$-0.135140\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.97056i	− 0.330435i
$738$	0	0
$739$	2.54416	0.0935883	0.0467941	−	0.998905i	$-0.485100\pi$
$739$	2.54416	0.0935883	0.0467941	+	0.998905i	$0.485100\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 43.6569i	− 1.60161i	−0.598922	−	0.800807i	$-0.704404\pi$
$743$	− 43.6569i	− 1.60161i	0.598922	−	0.800807i	$-0.295596\pi$
$744$	0	0
$745$	− 5.41196i	− 0.198279i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−36.2843	−1.32403	−0.662016	−	0.749490i	$-0.730299\pi$
$751$	−36.2843	−1.32403	−0.662016	+	0.749490i	$0.730299\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.76245	0.282505
$756$	0	0
$757$	18.1005	0.657874	0.328937	−	0.944352i	$-0.393310\pi$
$757$	18.1005	0.657874	0.328937	+	0.944352i	$0.393310\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	53.6619	1.94524	0.972622	−	0.232394i	$-0.0746557\pi$
$761$	53.6619	1.94524	0.972622	+	0.232394i	$0.0746557\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.20101i	− 0.151690i
$768$	0	0
$769$	14.2793i	0.514926i	0.966288	+	0.257463i	$0.0828866\pi$
$769$	14.2793i	0.514926i	−0.966288	+	0.257463i	$0.917113\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.5767	1.20767	0.603835	−	0.797110i	$-0.293639\pi$
$773$	33.5767	1.20767	0.603835	+	0.797110i	$0.293639\pi$
$774$	0	0
$775$	− 29.8268i	− 1.07141i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 10.1421i	− 0.363380i
$780$	0	0
$781$	18.6274	0.666541
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.72792i	0.168747i
$786$	0	0
$787$	50.4692i	1.79903i	0.436889	+	0.899515i	$0.356080\pi$
$787$	50.4692i	1.79903i	−0.436889	+	0.899515i	$0.643920\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.12994	0.0401254
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.3714	0.934122	0.467061	−	0.884225i	$-0.345313\pi$
$797$	26.3714	0.934122	0.467061	+	0.884225i	$0.345313\pi$
$798$	0	0
$799$	66.4264	2.35000
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.7038	−0.836490
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.2132i	0.745817i	0.927868	+	0.372908i	$0.121639\pi$
$809$	21.2132i	0.745817i	−0.927868	+	0.372908i	$0.878361\pi$
$810$	0	0
$811$	42.1814i	1.48119i	0.671951	+	0.740595i	$0.265456\pi$
$811$	42.1814i	1.48119i	−0.671951	+	0.740595i	$0.734544\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.16478	0.0758291
$816$	0	0
$817$	25.2346i	0.882846i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.0711i	1.78239i	0.453618	+	0.891196i	$0.350133\pi$
$821$	51.0711i	1.78239i	−0.453618	+	0.891196i	$0.649867\pi$
$822$	0	0
$823$	−40.2843	−1.40422	−0.702111	−	0.712068i	$-0.747759\pi$
$823$	−40.2843	−1.40422	−0.702111	+	0.712068i	$0.747759\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.4853i	0.642796i	0.946944	+	0.321398i	$0.104153\pi$
$827$	18.4853i	0.642796i	−0.946944	+	0.321398i	$0.895847\pi$
$828$	0	0
$829$	35.3701i	1.22845i	0.789130	+	0.614226i	$0.210532\pi$
$829$	35.3701i	1.22845i	−0.789130	+	0.614226i	$0.789468\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.85786	0.0642940
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.8882	−1.13543	−0.567714	−	0.823226i	$-0.692172\pi$
$839$	−32.8882	−1.13543	−0.567714	+	0.823226i	$0.692172\pi$
$840$	0	0
$841$	−17.6274	−0.607842
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.87285	0.339636
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 0.769553i	− 0.0263799i
$852$	0	0
$853$	− 57.2805i	− 1.96125i	−0.195899	−	0.980624i	$-0.562763\pi$
$853$	− 57.2805i	− 1.96125i	0.195899	−	0.980624i	$-0.437237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.6227	1.35349	0.676743	−	0.736219i	$-0.263391\pi$
$857$	39.6227	1.35349	0.676743	+	0.736219i	$0.263391\pi$
$858$	0	0
$859$	13.7766i	0.470052i	0.971989	+	0.235026i	$0.0755175\pi$
$859$	13.7766i	0.470052i	−0.971989	+	0.235026i	$0.924483\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 25.1127i	− 0.854846i	−0.904052	−	0.427423i	$-0.859421\pi$
$863$	− 25.1127i	− 0.854846i	0.904052	−	0.427423i	$-0.140579\pi$
$864$	0	0
$865$	−14.3848	−0.489097
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 22.6274i	− 0.767583i
$870$	0	0
$871$	1.42195i	0.0481809i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.928932	0.0313678	0.0156839	−	0.999877i	$-0.495007\pi$
$877$	0.928932	0.0313678	0.0156839	+	0.999877i	$0.495007\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.1187	0.745198	0.372599	−	0.927992i	$-0.378467\pi$
$881$	22.1187	0.745198	0.372599	+	0.927992i	$0.378467\pi$
$882$	0	0
$883$	−21.8579	−0.735576	−0.367788	−	0.929910i	$-0.619885\pi$
$883$	−21.8579	−0.735576	−0.367788	+	0.929910i	$0.619885\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.8995	0.668161	0.334081	−	0.942545i	$-0.391574\pi$
$887$	19.8995	0.668161	0.334081	+	0.942545i	$0.391574\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	44.2843i	1.48192i
$894$	0	0
$895$	− 7.12840i	− 0.238276i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	46.1396	1.53884
$900$	0	0
$901$	67.8644i	2.26089i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.27208i	0.0422853i
$906$	0	0
$907$	−32.4853	−1.07866	−0.539328	−	0.842096i	$-0.681322\pi$
$907$	−32.4853	−1.07866	−0.539328	+	0.842096i	$0.681322\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 42.2843i	− 1.40094i	−0.713682	−	0.700470i	$-0.752974\pi$
$911$	− 42.2843i	− 1.40094i	0.713682	−	0.700470i	$-0.247026\pi$
$912$	0	0
$913$	8.65914i	0.286576i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−15.5147	−0.511783	−0.255892	−	0.966705i	$-0.582369\pi$
$919$	−15.5147	−0.511783	−0.255892	+	0.966705i	$0.582369\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.95268	−0.0971887
$924$	0	0
$925$	1.07107	0.0352165
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22.3044	0.731784	0.365892	−	0.930657i	$-0.380764\pi$
$929$	22.3044	0.731784	0.365892	+	0.930657i	$0.380764\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.48528i	0.277498i
$936$	0	0
$937$	16.5210i	0.539719i	0.962900	+	0.269860i	$0.0869773\pi$
$937$	16.5210i	0.539719i	−0.962900	+	0.269860i	$0.913023\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.37302	−0.0773584	−0.0386792	−	0.999252i	$-0.512315\pi$
$941$	−2.37302	−0.0773584	−0.0386792	+	0.999252i	$0.512315\pi$
$942$	0	0
$943$	− 8.70420i	− 0.283448i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.82843i	0.286885i	0.989659	+	0.143443i	$0.0458173\pi$
$947$	8.82843i	0.286885i	−0.989659	+	0.143443i	$0.954183\pi$
$948$	0	0
$949$	3.75736	0.121969
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 58.8701i	− 1.90699i	−0.301412	−	0.953494i	$-0.597458\pi$
$953$	− 58.8701i	− 1.90699i	0.301412	−	0.953494i	$-0.402542\pi$
$954$	0	0
$955$	13.2513i	0.428803i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−14.6569	−0.472802
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.39104	−0.237926
$966$	0	0
$967$	15.5147	0.498920	0.249460	−	0.968385i	$-0.419747\pi$
$967$	15.5147	0.498920	0.249460	+	0.968385i	$0.419747\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.2346	0.809816	0.404908	−	0.914357i	$-0.367304\pi$
$971$	25.2346	0.809816	0.404908	+	0.914357i	$0.367304\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 0.485281i	− 0.0155255i	−0.999970	−	0.00776276i	$-0.997529\pi$
$977$	− 0.485281i	− 0.0155255i	0.999970	−	0.00776276i	$-0.00247099\pi$
$978$	0	0
$979$	3.32410i	0.106239i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.01962	0.223891	0.111946	−	0.993714i	$-0.464292\pi$
$983$	7.01962	0.223891	0.111946	+	0.993714i	$0.464292\pi$
$984$	0	0
$985$	2.35049i	0.0748930i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.6569i	0.688648i
$990$	0	0
$991$	−9.85786	−0.313145	−0.156573	−	0.987666i	$-0.550045\pi$
$991$	−9.85786	−0.313145	−0.156573	+	0.987666i	$0.550045\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 5.65685i	− 0.179334i
$996$	0	0
$997$	− 15.4705i	− 0.489956i	−0.969529	−	0.244978i	$-0.921219\pi$
$997$	− 15.4705i	− 0.489956i	0.969529	−	0.244978i	$-0.0787808\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.f.b.881.6	yes	8
3.2	odd	2	inner	1764.2.f.b.881.3	✓	8
4.3	odd	2		7056.2.k.e.881.5		8
7.2	even	3		1764.2.t.c.521.3		16
7.3	odd	6		1764.2.t.c.1097.6		16
7.4	even	3		1764.2.t.c.1097.4		16
7.5	odd	6		1764.2.t.c.521.5		16
7.6	odd	2	inner	1764.2.f.b.881.4	yes	8
12.11	even	2		7056.2.k.e.881.4		8
21.2	odd	6		1764.2.t.c.521.6		16
21.5	even	6		1764.2.t.c.521.4		16
21.11	odd	6		1764.2.t.c.1097.5		16
21.17	even	6		1764.2.t.c.1097.3		16
21.20	even	2	inner	1764.2.f.b.881.5	yes	8
28.27	even	2		7056.2.k.e.881.3		8
84.83	odd	2		7056.2.k.e.881.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.f.b.881.3	✓	8	3.2	odd	2	inner
1764.2.f.b.881.4	yes	8	7.6	odd	2	inner
1764.2.f.b.881.5	yes	8	21.20	even	2	inner
1764.2.f.b.881.6	yes	8	1.1	even	1	trivial
1764.2.t.c.521.3		16	7.2	even	3
1764.2.t.c.521.4		16	21.5	even	6
1764.2.t.c.521.5		16	7.5	odd	6
1764.2.t.c.521.6		16	21.2	odd	6
1764.2.t.c.1097.3		16	21.17	even	6
1764.2.t.c.1097.4		16	7.4	even	3
1764.2.t.c.1097.5		16	21.11	odd	6
1764.2.t.c.1097.6		16	7.3	odd	6
7056.2.k.e.881.3		8	28.27	even	2
7056.2.k.e.881.4		8	12.11	even	2
7056.2.k.e.881.5		8	4.3	odd	2
7056.2.k.e.881.6		8	84.83	odd	2

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.765367 q^{5} +O(q^{10})\)	\(q+0.765367 q^{5} +2.00000i q^{11} -0.317025i q^{13} +5.54328 q^{17} +3.69552i q^{19} +3.17157i q^{23} -4.41421 q^{25} -6.82843i q^{29} +6.75699i q^{31} -0.242641 q^{37} -2.74444 q^{41} +6.82843 q^{43} +11.9832 q^{47} +12.2426i q^{53} +1.53073i q^{55} +13.2513 q^{59} +3.56420i q^{61} -0.242641i q^{65} -4.48528 q^{67} -9.31371i q^{71} +11.8519i q^{73} -11.3137 q^{79} +4.32957 q^{83} +4.24264 q^{85} +1.66205 q^{89} +2.82843i q^{95} -11.8519i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{67}+O(q^{100}) \) 8 * q - 24 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^67

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