LMFDB - Embedded newform 1764.3.d.h.685.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(685,1764)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.d (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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.2
Root			$1.60021 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1764.685
Dual form			1764.3.d.h.685.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.37964i q^{5} +O(q^{10})$	$q-5.37964i q^{5} +8.59159 q^{11} -21.0158i q^{13} +5.48622i q^{17} -7.24098i q^{19} -28.0556 q^{23} -3.94054 q^{25} -40.3447 q^{29} -40.5101i q^{31} +66.6370 q^{37} -33.6357i q^{41} +0.932907 q^{43} +85.6544i q^{47} +44.5954 q^{53} -46.2197i q^{55} +63.6950i q^{59} +32.0084i q^{61} -113.058 q^{65} -47.7341 q^{67} -14.9676 q^{71} -140.298i q^{73} -122.307 q^{79} -33.1852i q^{83} +29.5139 q^{85} -36.1246i q^{89} -38.9539 q^{95} +16.2175i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{23} + 72 q^{25} - 80 q^{29} + 128 q^{37} - 112 q^{43} + 144 q^{53} - 240 q^{65} - 64 q^{67} - 224 q^{71} - 432 q^{79} - 96 q^{85} + 272 q^{95}+O(q^{100}) $ 8 * q + 16 * q^23 + 72 * q^25 - 80 * q^29 + 128 * q^37 - 112 * q^43 + 144 * q^53 - 240 * q^65 - 64 * q^67 - 224 * q^71 - 432 * q^79 - 96 * q^85 + 272 * q^95

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
$5$	− 5.37964i	− 1.07593i	−0.842968	−	0.537964i	$-0.819194\pi$
$5$	− 5.37964i	− 1.07593i	0.842968	−	0.537964i	$-0.180806\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8.59159	0.781054	0.390527	−	0.920592i	$-0.372293\pi$
$11$	8.59159	0.781054	0.390527	+	0.920592i	$0.372293\pi$
$12$	0	0
$13$	− 21.0158i	− 1.61660i	−0.588769	−	0.808301i	$-0.700387\pi$
$13$	− 21.0158i	− 1.61660i	0.588769	−	0.808301i	$-0.299613\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.48622i	0.322719i	0.986896	+	0.161359i	$0.0515878\pi$
$17$	5.48622i	0.322719i	−0.986896	+	0.161359i	$0.948412\pi$
$18$	0	0
$19$	− 7.24098i	− 0.381104i	−0.981677	−	0.190552i	$-0.938972\pi$
$19$	− 7.24098i	− 0.381104i	0.981677	−	0.190552i	$-0.0610278\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−28.0556	−1.21981	−0.609905	−	0.792474i	$-0.708792\pi$
$23$	−28.0556	−1.21981	−0.609905	+	0.792474i	$0.708792\pi$
$24$	0	0
$25$	−3.94054	−0.157621
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−40.3447	−1.39120	−0.695599	−	0.718431i	$-0.744861\pi$
$29$	−40.3447	−1.39120	−0.695599	+	0.718431i	$0.744861\pi$
$30$	0	0
$31$	− 40.5101i	− 1.30678i	−0.757023	−	0.653388i	$-0.773347\pi$
$31$	− 40.5101i	− 1.30678i	0.757023	−	0.653388i	$-0.226653\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	66.6370	1.80100	0.900500	−	0.434856i	$-0.143201\pi$
$37$	66.6370	1.80100	0.900500	+	0.434856i	$0.143201\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 33.6357i	− 0.820383i	−0.911999	−	0.410191i	$-0.865462\pi$
$41$	− 33.6357i	− 0.820383i	0.911999	−	0.410191i	$-0.134538\pi$
$42$	0	0
$43$	0.932907	0.0216955	0.0108478	−	0.999941i	$-0.496547\pi$
$43$	0.932907	0.0216955	0.0108478	+	0.999941i	$0.496547\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	85.6544i	1.82243i	0.411927	+	0.911217i	$0.364856\pi$
$47$	85.6544i	1.82243i	−0.411927	+	0.911217i	$0.635144\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	44.5954	0.841422	0.420711	−	0.907195i	$-0.361781\pi$
$53$	44.5954	0.841422	0.420711	+	0.907195i	$0.361781\pi$
$54$	0	0
$55$	− 46.2197i	− 0.840358i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	63.6950i	1.07958i	0.841801	+	0.539788i	$0.181496\pi$
$59$	63.6950i	1.07958i	−0.841801	+	0.539788i	$0.818504\pi$
$60$	0	0
$61$	32.0084i	0.524727i	0.964969	+	0.262364i	$0.0845020\pi$
$61$	32.0084i	0.524727i	−0.964969	+	0.262364i	$0.915498\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−113.058	−1.73935
$66$	0	0
$67$	−47.7341	−0.712450	−0.356225	−	0.934400i	$-0.615936\pi$
$67$	−47.7341	−0.712450	−0.356225	+	0.934400i	$0.615936\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.9676	−0.210811	−0.105405	−	0.994429i	$-0.533614\pi$
$71$	−14.9676	−0.210811	−0.105405	+	0.994429i	$0.533614\pi$
$72$	0	0
$73$	− 140.298i	− 1.92188i	−0.276750	−	0.960942i	$-0.589258\pi$
$73$	− 140.298i	− 1.92188i	0.276750	−	0.960942i	$-0.410742\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−122.307	−1.54820	−0.774098	−	0.633066i	$-0.781796\pi$
$79$	−122.307	−1.54820	−0.774098	+	0.633066i	$0.781796\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 33.1852i	− 0.399822i	−0.979814	−	0.199911i	$-0.935935\pi$
$83$	− 33.1852i	− 0.399822i	0.979814	−	0.199911i	$-0.0640652\pi$
$84$	0	0
$85$	29.5139	0.347222
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 36.1246i	− 0.405894i	−0.979190	−	0.202947i	$-0.934948\pi$
$89$	− 36.1246i	− 0.405894i	0.979190	−	0.202947i	$-0.0650519\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−38.9539	−0.410041
$96$	0	0
$97$	16.2175i	0.167191i	0.996500	+	0.0835956i	$0.0266404\pi$
$97$	16.2175i	0.167191i	−0.996500	+	0.0835956i	$0.973360\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 119.464i	− 1.18281i	−0.806373	−	0.591407i	$-0.798573\pi$
$101$	− 119.464i	− 1.18281i	0.806373	−	0.591407i	$-0.201427\pi$
$102$	0	0
$103$	− 8.93188i	− 0.0867173i	−0.999060	−	0.0433586i	$-0.986194\pi$
$103$	− 8.93188i	− 0.0867173i	0.999060	−	0.0433586i	$-0.0138058\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−173.856	−1.62482	−0.812410	−	0.583086i	$-0.801845\pi$
$107$	−173.856	−1.62482	−0.812410	+	0.583086i	$0.801845\pi$
$108$	0	0
$109$	−160.315	−1.47078	−0.735388	−	0.677646i	$-0.763000\pi$
$109$	−160.315	−1.47078	−0.735388	+	0.677646i	$0.763000\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	81.4420	0.720725	0.360363	−	0.932812i	$-0.382653\pi$
$113$	81.4420	0.720725	0.360363	+	0.932812i	$0.382653\pi$
$114$	0	0
$115$	150.929i	1.31243i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−47.1846	−0.389955
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 113.292i	− 0.906339i
$126$	0	0
$127$	117.172	0.922613	0.461307	−	0.887241i	$-0.347381\pi$
$127$	117.172	0.922613	0.461307	+	0.887241i	$0.347381\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	244.150i	1.86374i	0.362795	+	0.931869i	$0.381823\pi$
$131$	244.150i	1.86374i	−0.362795	+	0.931869i	$0.618177\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−245.412	−1.79133	−0.895663	−	0.444733i	$-0.853299\pi$
$137$	−245.412	−1.79133	−0.895663	+	0.444733i	$0.853299\pi$
$138$	0	0
$139$	− 17.1371i	− 0.123288i	−0.998098	−	0.0616441i	$-0.980366\pi$
$139$	− 17.1371i	− 0.123288i	0.998098	−	0.0616441i	$-0.0196344\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 180.559i	− 1.26265i
$144$	0	0
$145$	217.040i	1.49683i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−134.242	−0.900952	−0.450476	−	0.892789i	$-0.648746\pi$
$149$	−134.242	−0.900952	−0.450476	+	0.892789i	$0.648746\pi$
$150$	0	0
$151$	199.009	1.31794	0.658971	−	0.752168i	$-0.270992\pi$
$151$	199.009	1.31794	0.658971	+	0.752168i	$0.270992\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−217.930	−1.40600
$156$	0	0
$157$	77.0189i	0.490566i	0.969451	+	0.245283i	$0.0788810\pi$
$157$	77.0189i	0.490566i	−0.969451	+	0.245283i	$0.921119\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−230.379	−1.41337	−0.706684	−	0.707529i	$-0.749810\pi$
$163$	−230.379	−1.41337	−0.706684	+	0.707529i	$0.749810\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 229.231i	− 1.37264i	−0.727298	−	0.686321i	$-0.759224\pi$
$167$	− 229.231i	− 1.37264i	0.727298	−	0.686321i	$-0.240776\pi$
$168$	0	0
$169$	−272.665	−1.61340
$170$	0	0
$171$	0	0
$172$	0	0
$173$	112.473i	0.650133i	0.945691	+	0.325066i	$0.105387\pi$
$173$	112.473i	0.650133i	−0.945691	+	0.325066i	$0.894613\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−105.494	−0.589352	−0.294676	−	0.955597i	$-0.595212\pi$
$179$	−105.494	−0.589352	−0.294676	+	0.955597i	$0.595212\pi$
$180$	0	0
$181$	15.2683i	0.0843553i	0.999110	+	0.0421776i	$0.0134295\pi$
$181$	15.2683i	0.0843553i	−0.999110	+	0.0421776i	$0.986570\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 358.483i	− 1.93775i
$186$	0	0
$187$	47.1353i	0.252061i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	306.186	1.60307	0.801534	−	0.597950i	$-0.204018\pi$
$191$	306.186	1.60307	0.801534	+	0.597950i	$0.204018\pi$
$192$	0	0
$193$	−1.84100	−0.00953885	−0.00476943	−	0.999989i	$-0.501518\pi$
$193$	−1.84100	−0.00953885	−0.00476943	+	0.999989i	$0.501518\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	255.334	1.29611	0.648056	−	0.761593i	$-0.275582\pi$
$197$	255.334	1.29611	0.648056	+	0.761593i	$0.275582\pi$
$198$	0	0
$199$	− 36.6483i	− 0.184162i	−0.995752	−	0.0920812i	$-0.970648\pi$
$199$	− 36.6483i	− 0.184162i	0.995752	−	0.0920812i	$-0.0293519\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−180.948	−0.882673
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 62.2116i	− 0.297663i
$210$	0	0
$211$	−126.571	−0.599862	−0.299931	−	0.953961i	$-0.596964\pi$
$211$	−126.571	−0.599862	−0.299931	+	0.953961i	$0.596964\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 5.01871i	− 0.0233428i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	115.297	0.521708
$222$	0	0
$223$	− 212.193i	− 0.951536i	−0.879571	−	0.475768i	$-0.842170\pi$
$223$	− 212.193i	− 0.951536i	0.879571	−	0.475768i	$-0.157830\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	105.489i	0.464708i	0.972631	+	0.232354i	$0.0746428\pi$
$227$	105.489i	0.464708i	−0.972631	+	0.232354i	$0.925357\pi$
$228$	0	0
$229$	6.99086i	0.0305278i	0.999884	+	0.0152639i	$0.00485883\pi$
$229$	6.99086i	0.0305278i	−0.999884	+	0.0152639i	$0.995141\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−336.046	−1.44226	−0.721128	−	0.692801i	$-0.756376\pi$
$233$	−336.046	−1.44226	−0.721128	+	0.692801i	$0.756376\pi$
$234$	0	0
$235$	460.790	1.96081
$236$	0	0
$237$	0	0
$238$	0	0
$239$	232.382	0.972309	0.486155	−	0.873873i	$-0.338399\pi$
$239$	232.382	0.972309	0.486155	+	0.873873i	$0.338399\pi$
$240$	0	0
$241$	− 63.7346i	− 0.264459i	−0.991219	−	0.132229i	$-0.957786\pi$
$241$	− 63.7346i	− 0.264459i	0.991219	−	0.132229i	$-0.0422136\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−152.175	−0.616094
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 415.450i	− 1.65518i	−0.561334	−	0.827589i	$-0.689712\pi$
$251$	− 415.450i	− 1.65518i	0.561334	−	0.827589i	$-0.310288\pi$
$252$	0	0
$253$	−241.043	−0.952737
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 265.598i	− 1.03345i	−0.856150	−	0.516727i	$-0.827150\pi$
$257$	− 265.598i	− 1.03345i	0.856150	−	0.516727i	$-0.172850\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−156.373	−0.594575	−0.297288	−	0.954788i	$-0.596082\pi$
$263$	−156.373	−0.594575	−0.297288	+	0.954788i	$0.596082\pi$
$264$	0	0
$265$	− 239.907i	− 0.905310i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 332.861i	− 1.23740i	−0.785626	−	0.618701i	$-0.787659\pi$
$269$	− 332.861i	− 1.23740i	0.785626	−	0.618701i	$-0.212341\pi$
$270$	0	0
$271$	− 209.089i	− 0.771546i	−0.922594	−	0.385773i	$-0.873935\pi$
$271$	− 209.089i	− 0.771546i	0.922594	−	0.385773i	$-0.126065\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−33.8555	−0.123111
$276$	0	0
$277$	−263.231	−0.950293	−0.475146	−	0.879907i	$-0.657605\pi$
$277$	−263.231	−0.950293	−0.475146	+	0.879907i	$0.657605\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−391.519	−1.39331	−0.696653	−	0.717408i	$-0.745328\pi$
$281$	−391.519	−1.39331	−0.696653	+	0.717408i	$0.745328\pi$
$282$	0	0
$283$	109.394i	0.386550i	0.981145	+	0.193275i	$0.0619109\pi$
$283$	109.394i	0.386550i	−0.981145	+	0.193275i	$0.938089\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	258.901	0.895853
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 35.3685i	− 0.120712i	−0.998177	−	0.0603558i	$-0.980776\pi$
$293$	− 35.3685i	− 0.120712i	0.998177	−	0.0603558i	$-0.0192235\pi$
$294$	0	0
$295$	342.656	1.16155
$296$	0	0
$297$	0	0
$298$	0	0
$299$	589.613i	1.97195i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	172.194	0.564569
$306$	0	0
$307$	125.621i	0.409189i	0.978847	+	0.204594i	$0.0655875\pi$
$307$	125.621i	0.409189i	−0.978847	+	0.204594i	$0.934412\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	315.735i	1.01523i	0.861585	+	0.507613i	$0.169472\pi$
$311$	315.735i	1.01523i	−0.861585	+	0.507613i	$0.830528\pi$
$312$	0	0
$313$	− 51.6046i	− 0.164871i	−0.996596	−	0.0824355i	$-0.973730\pi$
$313$	− 51.6046i	− 0.164871i	0.996596	−	0.0824355i	$-0.0262698\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.99391	0.0157537	0.00787683	−	0.999969i	$-0.497493\pi$
$317$	4.99391	0.0157537	0.00787683	+	0.999969i	$0.497493\pi$
$318$	0	0
$319$	−346.625	−1.08660
$320$	0	0
$321$	0	0
$322$	0	0
$323$	39.7256	0.122990
$324$	0	0
$325$	82.8137i	0.254811i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	454.780	1.37396	0.686980	−	0.726677i	$-0.258936\pi$
$331$	454.780	1.37396	0.686980	+	0.726677i	$0.258936\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	256.792i	0.766545i
$336$	0	0
$337$	−183.824	−0.545471	−0.272736	−	0.962089i	$-0.587928\pi$
$337$	−183.824	−0.545471	−0.272736	+	0.962089i	$0.587928\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 348.046i	− 1.02066i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−144.776	−0.417223	−0.208611	−	0.977999i	$-0.566894\pi$
$347$	−144.776	−0.417223	−0.208611	+	0.977999i	$0.566894\pi$
$348$	0	0
$349$	187.069i	0.536015i	0.963417	+	0.268007i	$0.0863651\pi$
$349$	187.069i	0.536015i	−0.963417	+	0.268007i	$0.913635\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	222.601i	0.630597i	0.948993	+	0.315299i	$0.102105\pi$
$353$	222.601i	0.630597i	−0.948993	+	0.315299i	$0.897895\pi$
$354$	0	0
$355$	80.5201i	0.226817i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−62.9856	−0.175447	−0.0877237	−	0.996145i	$-0.527959\pi$
$359$	−62.9856	−0.175447	−0.0877237	+	0.996145i	$0.527959\pi$
$360$	0	0
$361$	308.568	0.854759
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−754.750	−2.06781
$366$	0	0
$367$	− 547.100i	− 1.49074i	−0.666653	−	0.745368i	$-0.732274\pi$
$367$	− 547.100i	− 1.49074i	0.666653	−	0.745368i	$-0.267726\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−32.4417	−0.0869750	−0.0434875	−	0.999054i	$-0.513847\pi$
$373$	−32.4417	−0.0869750	−0.0434875	+	0.999054i	$0.513847\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	847.878i	2.24901i
$378$	0	0
$379$	508.859	1.34263	0.671317	−	0.741170i	$-0.265729\pi$
$379$	508.859	1.34263	0.671317	+	0.741170i	$0.265729\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 26.2839i	− 0.0686263i	−0.999411	−	0.0343131i	$-0.989076\pi$
$383$	− 26.2839i	− 0.0686263i	0.999411	−	0.0343131i	$-0.0109244\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	360.128	0.925779	0.462890	−	0.886416i	$-0.346813\pi$
$389$	360.128	0.925779	0.462890	+	0.886416i	$0.346813\pi$
$390$	0	0
$391$	− 153.919i	− 0.393656i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	657.970i	1.66575i
$396$	0	0
$397$	570.992i	1.43827i	0.694872	+	0.719134i	$0.255461\pi$
$397$	570.992i	1.43827i	−0.694872	+	0.719134i	$0.744539\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−68.4014	−0.170577	−0.0852885	−	0.996356i	$-0.527181\pi$
$401$	−68.4014	−0.170577	−0.0852885	+	0.996356i	$0.527181\pi$
$402$	0	0
$403$	−851.353	−2.11254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	572.518	1.40668
$408$	0	0
$409$	− 15.3649i	− 0.0375671i	−0.999824	−	0.0187835i	$-0.994021\pi$
$409$	− 15.3649i	− 0.0375671i	0.999824	−	0.0187835i	$-0.00597934\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−178.524	−0.430179
$416$	0	0
$417$	0	0
$418$	0	0
$419$	366.079i	0.873696i	0.899535	+	0.436848i	$0.143905\pi$
$419$	366.079i	0.873696i	−0.899535	+	0.436848i	$0.856095\pi$
$420$	0	0
$421$	607.135	1.44213	0.721063	−	0.692870i	$-0.243654\pi$
$421$	607.135	1.44213	0.721063	+	0.692870i	$0.243654\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 21.6186i	− 0.0508674i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	586.353	1.36045	0.680224	−	0.733004i	$-0.261882\pi$
$431$	586.353	1.36045	0.680224	+	0.733004i	$0.261882\pi$
$432$	0	0
$433$	518.769i	1.19808i	0.800719	+	0.599040i	$0.204451\pi$
$433$	518.769i	1.19808i	−0.800719	+	0.599040i	$0.795549\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	203.150i	0.464875i
$438$	0	0
$439$	− 221.524i	− 0.504610i	−0.967648	−	0.252305i	$-0.918811\pi$
$439$	− 221.524i	− 0.504610i	0.967648	−	0.252305i	$-0.0811886\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	226.415	0.511095	0.255548	−	0.966796i	$-0.417744\pi$
$443$	226.415	0.511095	0.255548	+	0.966796i	$0.417744\pi$
$444$	0	0
$445$	−194.337	−0.436713
$446$	0	0
$447$	0	0
$448$	0	0
$449$	378.422	0.842810	0.421405	−	0.906873i	$-0.361537\pi$
$449$	378.422	0.842810	0.421405	+	0.906873i	$0.361537\pi$
$450$	0	0
$451$	− 288.984i	− 0.640763i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	438.153	0.958760	0.479380	−	0.877607i	$-0.340862\pi$
$457$	438.153	0.958760	0.479380	+	0.877607i	$0.340862\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 46.3981i	− 0.100647i	−0.998733	−	0.0503234i	$-0.983975\pi$
$461$	− 46.3981i	− 0.100647i	0.998733	−	0.0503234i	$-0.0160252\pi$
$462$	0	0
$463$	367.455	0.793639	0.396820	−	0.917897i	$-0.370114\pi$
$463$	367.455	0.793639	0.396820	+	0.917897i	$0.370114\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 529.465i	− 1.13376i	−0.823801	−	0.566879i	$-0.808151\pi$
$467$	− 529.465i	− 1.13376i	0.823801	−	0.566879i	$-0.191849\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.01516	0.0169454
$474$	0	0
$475$	28.5334i	0.0600702i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	289.761i	0.604928i	0.953161	+	0.302464i	$0.0978093\pi$
$479$	289.761i	0.604928i	−0.953161	+	0.302464i	$0.902191\pi$
$480$	0	0
$481$	− 1400.43i	− 2.91150i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	87.2445	0.179886
$486$	0	0
$487$	439.636	0.902744	0.451372	−	0.892336i	$-0.350935\pi$
$487$	439.636	0.902744	0.451372	+	0.892336i	$0.350935\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	320.561	0.652874	0.326437	−	0.945219i	$-0.394152\pi$
$491$	320.561	0.652874	0.326437	+	0.945219i	$0.394152\pi$
$492$	0	0
$493$	− 221.340i	− 0.448965i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	771.973	1.54704	0.773520	−	0.633772i	$-0.218494\pi$
$499$	771.973	1.54704	0.773520	+	0.633772i	$0.218494\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 101.632i	− 0.202052i	−0.994884	−	0.101026i	$-0.967787\pi$
$503$	− 101.632i	− 0.202052i	0.994884	−	0.101026i	$-0.0322126\pi$
$504$	0	0
$505$	−642.675	−1.27262
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 605.407i	− 1.18941i	−0.803946	−	0.594703i	$-0.797270\pi$
$509$	− 605.407i	− 1.18941i	0.803946	−	0.594703i	$-0.202730\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−48.0503	−0.0933016
$516$	0	0
$517$	735.907i	1.42342i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	383.293i	0.735688i	0.929888	+	0.367844i	$0.119904\pi$
$521$	383.293i	0.735688i	−0.929888	+	0.367844i	$0.880096\pi$
$522$	0	0
$523$	448.095i	0.856777i	0.903595	+	0.428389i	$0.140919\pi$
$523$	448.095i	0.856777i	−0.903595	+	0.428389i	$0.859081\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	222.247	0.421721
$528$	0	0
$529$	258.119	0.487937
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−706.882	−1.32623
$534$	0	0
$535$	935.282i	1.74819i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−271.258	−0.501400	−0.250700	−	0.968065i	$-0.580661\pi$
$541$	−271.258	−0.501400	−0.250700	+	0.968065i	$0.580661\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	862.435i	1.58245i
$546$	0	0
$547$	−590.544	−1.07961	−0.539803	−	0.841791i	$-0.681501\pi$
$547$	−590.544	−1.07961	−0.539803	+	0.841791i	$0.681501\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	292.135i	0.530191i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−188.295	−0.338051	−0.169026	−	0.985612i	$-0.554062\pi$
$557$	−188.295	−0.338051	−0.169026	+	0.985612i	$0.554062\pi$
$558$	0	0
$559$	− 19.6058i	− 0.0350730i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 204.157i	− 0.362623i	−0.983426	−	0.181312i	$-0.941966\pi$
$563$	− 204.157i	− 0.362623i	0.983426	−	0.181312i	$-0.0580342\pi$
$564$	0	0
$565$	− 438.129i	− 0.775449i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	660.913	1.16153	0.580767	−	0.814070i	$-0.302753\pi$
$569$	660.913	1.16153	0.580767	+	0.814070i	$0.302753\pi$
$570$	0	0
$571$	−533.978	−0.935162	−0.467581	−	0.883950i	$-0.654874\pi$
$571$	−533.978	−0.935162	−0.467581	+	0.883950i	$0.654874\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	110.554	0.192268
$576$	0	0
$577$	− 53.4337i	− 0.0926062i	−0.998927	−	0.0463031i	$-0.985256\pi$
$577$	− 53.4337i	− 0.0926062i	0.998927	−	0.0463031i	$-0.0147440\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	383.145	0.657196
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 413.063i	− 0.703684i	−0.936059	−	0.351842i	$-0.885555\pi$
$587$	− 413.063i	− 0.703684i	0.936059	−	0.351842i	$-0.114445\pi$
$588$	0	0
$589$	−293.333	−0.498018
$590$	0	0
$591$	0	0
$592$	0	0
$593$	122.452i	0.206496i	0.994656	+	0.103248i	$0.0329235\pi$
$593$	122.452i	0.206496i	−0.994656	+	0.103248i	$0.967076\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	645.136	1.07702	0.538511	−	0.842619i	$-0.318987\pi$
$599$	645.136	1.07702	0.538511	+	0.842619i	$0.318987\pi$
$600$	0	0
$601$	− 683.488i	− 1.13725i	−0.822596	−	0.568626i	$-0.807475\pi$
$601$	− 683.488i	− 1.13725i	0.822596	−	0.568626i	$-0.192525\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	253.836i	0.419564i
$606$	0	0
$607$	257.253i	0.423811i	0.977290	+	0.211905i	$0.0679669\pi$
$607$	257.253i	0.423811i	−0.977290	+	0.211905i	$0.932033\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1800.10	2.94615
$612$	0	0
$613$	885.713	1.44488	0.722441	−	0.691433i	$-0.243020\pi$
$613$	885.713	1.44488	0.722441	+	0.691433i	$0.243020\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.0724	−0.0746717	−0.0373358	−	0.999303i	$-0.511887\pi$
$617$	−46.0724	−0.0746717	−0.0373358	+	0.999303i	$0.511887\pi$
$618$	0	0
$619$	1096.33i	1.77112i	0.464522	+	0.885562i	$0.346226\pi$
$619$	1096.33i	1.77112i	−0.464522	+	0.885562i	$0.653774\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−707.986	−1.13278
$626$	0	0
$627$	0	0
$628$	0	0
$629$	365.585i	0.581217i
$630$	0	0
$631$	606.319	0.960886	0.480443	−	0.877026i	$-0.340476\pi$
$631$	606.319	0.960886	0.480443	+	0.877026i	$0.340476\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 630.343i	− 0.992666i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−939.184	−1.46519	−0.732593	−	0.680667i	$-0.761690\pi$
$641$	−939.184	−1.46519	−0.732593	+	0.680667i	$0.761690\pi$
$642$	0	0
$643$	− 992.960i	− 1.54426i	−0.635464	−	0.772131i	$-0.719191\pi$
$643$	− 992.960i	− 1.54426i	0.635464	−	0.772131i	$-0.280809\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 265.184i	− 0.409867i	−0.978776	−	0.204933i	$-0.934302\pi$
$647$	− 265.184i	− 0.409867i	0.978776	−	0.204933i	$-0.0656978\pi$
$648$	0	0
$649$	547.241i	0.843206i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1134.14	1.73682	0.868409	−	0.495849i	$-0.165143\pi$
$653$	1134.14	1.73682	0.868409	+	0.495849i	$0.165143\pi$
$654$	0	0
$655$	1313.44	2.00525
$656$	0	0
$657$	0	0
$658$	0	0
$659$	924.147	1.40235	0.701174	−	0.712990i	$-0.252660\pi$
$659$	924.147	1.40235	0.701174	+	0.712990i	$0.252660\pi$
$660$	0	0
$661$	234.367i	0.354564i	0.984160	+	0.177282i	$0.0567305\pi$
$661$	234.367i	0.354564i	−0.984160	+	0.177282i	$0.943269\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1131.90	1.69700
$668$	0	0
$669$	0	0
$670$	0	0
$671$	275.003i	0.409840i
$672$	0	0
$673$	−465.127	−0.691125	−0.345563	−	0.938396i	$-0.612312\pi$
$673$	−465.127	−0.691125	−0.345563	+	0.938396i	$0.612312\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1295.93i	− 1.91422i	−0.289724	−	0.957110i	$-0.593563\pi$
$677$	− 1295.93i	− 1.91422i	0.289724	−	0.957110i	$-0.406437\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.2612	−0.0238085	−0.0119042	−	0.999929i	$-0.503789\pi$
$683$	−16.2612	−0.0238085	−0.0119042	+	0.999929i	$0.503789\pi$
$684$	0	0
$685$	1320.23i	1.92734i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 937.209i	− 1.36025i
$690$	0	0
$691$	− 68.5364i	− 0.0991843i	−0.998770	−	0.0495922i	$-0.984208\pi$
$691$	− 68.5364i	− 0.0991843i	0.998770	−	0.0495922i	$-0.0157922\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−92.1912	−0.132649
$696$	0	0
$697$	184.533	0.264753
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−923.360	−1.31720	−0.658602	−	0.752491i	$-0.728852\pi$
$701$	−923.360	−1.31720	−0.658602	+	0.752491i	$0.728852\pi$
$702$	0	0
$703$	− 482.518i	− 0.686369i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−912.863	−1.28754	−0.643768	−	0.765221i	$-0.722630\pi$
$709$	−912.863	−1.28754	−0.643768	+	0.765221i	$0.722630\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1136.54i	1.59402i
$714$	0	0
$715$	−971.345	−1.35852
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 545.020i	− 0.758025i	−0.925391	−	0.379013i	$-0.876264\pi$
$719$	− 545.020i	− 0.758025i	0.925391	−	0.379013i	$-0.123736\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	158.980	0.219283
$726$	0	0
$727$	750.292i	1.03204i	0.856577	+	0.516019i	$0.172587\pi$
$727$	750.292i	1.03204i	−0.856577	+	0.516019i	$0.827413\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.11813i	0.00700155i
$732$	0	0
$733$	− 926.599i	− 1.26412i	−0.774920	−	0.632059i	$-0.782210\pi$
$733$	− 926.599i	− 1.26412i	0.774920	−	0.632059i	$-0.217790\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−410.112	−0.556461
$738$	0	0
$739$	488.147	0.660551	0.330275	−	0.943885i	$-0.392858\pi$
$739$	488.147	0.660551	0.330275	+	0.943885i	$0.392858\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1091.72	1.46935	0.734674	−	0.678421i	$-0.237335\pi$
$743$	1091.72	1.46935	0.734674	+	0.678421i	$0.237335\pi$
$744$	0	0
$745$	722.173i	0.969360i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−480.443	−0.639738	−0.319869	−	0.947462i	$-0.603639\pi$
$751$	−480.443	−0.639738	−0.319869	+	0.947462i	$0.603639\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1070.60i	− 1.41801i
$756$	0	0
$757$	941.400	1.24359	0.621796	−	0.783179i	$-0.286403\pi$
$757$	941.400	1.24359	0.621796	+	0.783179i	$0.286403\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1192.34i	1.56681i	0.621509	+	0.783407i	$0.286520\pi$
$761$	1192.34i	1.56681i	−0.621509	+	0.783407i	$0.713480\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1338.60	1.74524
$768$	0	0
$769$	− 908.294i	− 1.18114i	−0.806988	−	0.590568i	$-0.798904\pi$
$769$	− 908.294i	− 1.18114i	0.806988	−	0.590568i	$-0.201096\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1189.38i	− 1.53866i	−0.638854	−	0.769328i	$-0.720591\pi$
$773$	− 1189.38i	− 1.53866i	0.638854	−	0.769328i	$-0.279409\pi$
$774$	0	0
$775$	159.632i	0.205976i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−243.556	−0.312652
$780$	0	0
$781$	−128.595	−0.164654
$782$	0	0
$783$	0	0
$784$	0	0
$785$	414.334	0.527814
$786$	0	0
$787$	1057.12i	1.34323i	0.740902	+	0.671613i	$0.234398\pi$
$787$	1057.12i	1.34323i	−0.740902	+	0.671613i	$0.765602\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	672.682	0.848275
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1037.94i	− 1.30231i	−0.758945	−	0.651154i	$-0.774285\pi$
$797$	− 1037.94i	− 1.30231i	0.758945	−	0.651154i	$-0.225715\pi$
$798$	0	0
$799$	−469.919	−0.588133
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1205.38i	− 1.50109i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1087.36	−1.34408	−0.672041	−	0.740514i	$-0.734582\pi$
$809$	−1087.36	−1.34408	−0.672041	+	0.740514i	$0.734582\pi$
$810$	0	0
$811$	926.995i	1.14303i	0.820593	+	0.571514i	$0.193644\pi$
$811$	926.995i	1.14303i	−0.820593	+	0.571514i	$0.806356\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1239.36i	1.52068i
$816$	0	0
$817$	− 6.75517i	− 0.00826826i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	430.300	0.524117	0.262058	−	0.965052i	$-0.415599\pi$
$821$	430.300	0.524117	0.262058	+	0.965052i	$0.415599\pi$
$822$	0	0
$823$	−1044.74	−1.26942	−0.634712	−	0.772749i	$-0.718881\pi$
$823$	−1044.74	−1.26942	−0.634712	+	0.772749i	$0.718881\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	610.313	0.737984	0.368992	−	0.929433i	$-0.379703\pi$
$827$	610.313	0.737984	0.368992	+	0.929433i	$0.379703\pi$
$828$	0	0
$829$	− 1413.90i	− 1.70555i	−0.522279	−	0.852775i	$-0.674918\pi$
$829$	− 1413.90i	− 1.70555i	0.522279	−	0.852775i	$-0.325082\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1233.18	−1.47687
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 46.3303i	− 0.0552209i	−0.999619	−	0.0276105i	$-0.991210\pi$
$839$	− 46.3303i	− 0.0552209i	0.999619	−	0.0276105i	$-0.00878980\pi$
$840$	0	0
$841$	786.696	0.935430
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1466.84i	1.73591i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1869.54	−2.19688
$852$	0	0
$853$	773.111i	0.906343i	0.891423	+	0.453172i	$0.149708\pi$
$853$	773.111i	0.906343i	−0.891423	+	0.453172i	$0.850292\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 438.159i	− 0.511271i	−0.966773	−	0.255636i	$-0.917715\pi$
$857$	− 438.159i	− 0.511271i	0.966773	−	0.255636i	$-0.0822847\pi$
$858$	0	0
$859$	72.3419i	0.0842164i	0.999113	+	0.0421082i	$0.0134074\pi$
$859$	72.3419i	0.0842164i	−0.999113	+	0.0421082i	$0.986593\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−426.495	−0.494200	−0.247100	−	0.968990i	$-0.579478\pi$
$863$	−426.495	−0.494200	−0.247100	+	0.968990i	$0.579478\pi$
$864$	0	0
$865$	605.064	0.699496
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1050.82	−1.20922
$870$	0	0
$871$	1003.17i	1.15175i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	416.485	0.474898	0.237449	−	0.971400i	$-0.423689\pi$
$877$	416.485	0.474898	0.237449	+	0.971400i	$0.423689\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1249.01i	1.41771i	0.705353	+	0.708857i	$0.250789\pi$
$881$	1249.01i	1.41771i	−0.705353	+	0.708857i	$0.749211\pi$
$882$	0	0
$883$	81.5906	0.0924016	0.0462008	−	0.998932i	$-0.485289\pi$
$883$	81.5906	0.0924016	0.0462008	+	0.998932i	$0.485289\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	917.730i	1.03464i	0.855791	+	0.517322i	$0.173071\pi$
$887$	917.730i	1.03464i	−0.855791	+	0.517322i	$0.826929\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	620.222	0.694537
$894$	0	0
$895$	567.520i	0.634100i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1634.37i	1.81798i
$900$	0	0
$901$	244.660i	0.271543i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	82.1380	0.0907602
$906$	0	0
$907$	−1451.40	−1.60022	−0.800110	−	0.599853i	$-0.795226\pi$
$907$	−1451.40	−1.60022	−0.800110	+	0.599853i	$0.795226\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	562.064	0.616975	0.308487	−	0.951228i	$-0.400177\pi$
$911$	562.064	0.616975	0.308487	+	0.951228i	$0.400177\pi$
$912$	0	0
$913$	− 285.114i	− 0.312282i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	538.158	0.585591	0.292795	−	0.956175i	$-0.405414\pi$
$919$	538.158	0.585591	0.292795	+	0.956175i	$0.405414\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	314.556i	0.340797i
$924$	0	0
$925$	−262.586	−0.283876
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 827.080i	− 0.890291i	−0.895458	−	0.445145i	$-0.853152\pi$
$929$	− 827.080i	− 0.890291i	0.895458	−	0.445145i	$-0.146848\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	253.571	0.271199
$936$	0	0
$937$	− 1501.10i	− 1.60203i	−0.598644	−	0.801016i	$-0.704293\pi$
$937$	− 1501.10i	− 1.60203i	0.598644	−	0.801016i	$-0.295707\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	766.672i	0.814742i	0.913263	+	0.407371i	$0.133554\pi$
$941$	766.672i	0.814742i	−0.913263	+	0.407371i	$0.866446\pi$
$942$	0	0
$943$	943.671i	1.00071i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1362.23	−1.43847	−0.719237	−	0.694765i	$-0.755508\pi$
$947$	−1362.23	−1.43847	−0.719237	+	0.694765i	$0.755508\pi$
$948$	0	0
$949$	−2948.47	−3.10692
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1431.37	−1.50196	−0.750980	−	0.660325i	$-0.770419\pi$
$953$	−1431.37	−1.50196	−0.750980	+	0.660325i	$0.770419\pi$
$954$	0	0
$955$	− 1647.17i	− 1.72479i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−680.067	−0.707666
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.90391i	0.0102631i
$966$	0	0
$967$	−426.276	−0.440823	−0.220411	−	0.975407i	$-0.570740\pi$
$967$	−426.276	−0.440823	−0.220411	+	0.975407i	$0.570740\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1728.41i	1.78003i	0.455934	+	0.890013i	$0.349305\pi$
$971$	1728.41i	1.78003i	−0.455934	+	0.890013i	$0.650695\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	712.320	0.729089	0.364545	−	0.931186i	$-0.381225\pi$
$977$	712.320	0.729089	0.364545	+	0.931186i	$0.381225\pi$
$978$	0	0
$979$	− 310.368i	− 0.317025i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1042.56i	− 1.06059i	−0.847812	−	0.530297i	$-0.822080\pi$
$983$	− 1042.56i	− 1.06059i	0.847812	−	0.530297i	$-0.177920\pi$
$984$	0	0
$985$	− 1373.61i	− 1.39452i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.1733	−0.0264644
$990$	0	0
$991$	1397.41	1.41010	0.705052	−	0.709156i	$-0.250924\pi$
$991$	1397.41	1.41010	0.705052	+	0.709156i	$0.250924\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−197.155	−0.198146
$996$	0	0
$997$	− 717.962i	− 0.720122i	−0.932929	−	0.360061i	$-0.882756\pi$
$997$	− 717.962i	− 0.720122i	0.932929	−	0.360061i	$-0.117244\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.d.h.685.2		8
3.2	odd	2		588.3.d.c.97.4	✓	8
7.2	even	3		1764.3.z.l.325.1		8
7.3	odd	6		1764.3.z.l.901.1		8
7.4	even	3		1764.3.z.m.901.4		8
7.5	odd	6		1764.3.z.m.325.4		8
7.6	odd	2	inner	1764.3.d.h.685.7		8
12.11	even	2		2352.3.f.j.97.8		8
21.2	odd	6		588.3.m.f.325.4		8
21.5	even	6		588.3.m.e.325.1		8
21.11	odd	6		588.3.m.e.313.1		8
21.17	even	6		588.3.m.f.313.4		8
21.20	even	2		588.3.d.c.97.5	yes	8
84.83	odd	2		2352.3.f.j.97.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.4	✓	8	3.2	odd	2
588.3.d.c.97.5	yes	8	21.20	even	2
588.3.m.e.313.1		8	21.11	odd	6
588.3.m.e.325.1		8	21.5	even	6
588.3.m.f.313.4		8	21.17	even	6
588.3.m.f.325.4		8	21.2	odd	6
1764.3.d.h.685.2		8	1.1	even	1	trivial
1764.3.d.h.685.7		8	7.6	odd	2	inner
1764.3.z.l.325.1		8	7.2	even	3
1764.3.z.l.901.1		8	7.3	odd	6
1764.3.z.m.325.4		8	7.5	odd	6
1764.3.z.m.901.4		8	7.4	even	3
2352.3.f.j.97.1		8	84.83	odd	2
2352.3.f.j.97.8		8	12.11	even	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.37964i q^{5} +O(q^{10})\)	\(q-5.37964i q^{5} +8.59159 q^{11} -21.0158i q^{13} +5.48622i q^{17} -7.24098i q^{19} -28.0556 q^{23} -3.94054 q^{25} -40.3447 q^{29} -40.5101i q^{31} +66.6370 q^{37} -33.6357i q^{41} +0.932907 q^{43} +85.6544i q^{47} +44.5954 q^{53} -46.2197i q^{55} +63.6950i q^{59} +32.0084i q^{61} -113.058 q^{65} -47.7341 q^{67} -14.9676 q^{71} -140.298i q^{73} -122.307 q^{79} -33.1852i q^{83} +29.5139 q^{85} -36.1246i q^{89} -38.9539 q^{95} +16.2175i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{23} + 72 q^{25} - 80 q^{29} + 128 q^{37} - 112 q^{43} + 144 q^{53} - 240 q^{65} - 64 q^{67} - 224 q^{71} - 432 q^{79} - 96 q^{85} + 272 q^{95}+O(q^{100}) \) 8 * q + 16 * q^23 + 72 * q^25 - 80 * q^29 + 128 * q^37 - 112 * q^43 + 144 * q^53 - 240 * q^65 - 64 * q^67 - 224 * q^71 - 432 * q^79 - 96 * q^85 + 272 * q^95

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