LMFDB - Embedded newform 1764.2.t.c.1097.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(521,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("1764.521");

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.t (of order $6$, degree $2$, not 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:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1097.3
Root			$-0.793353 + 0.608761i$ of defining polynomial
Character	$\chi$	$=$	1764.1097
Dual form			1764.2.t.c.521.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.382683 + 0.662827i) q^{5} +O(q^{10})$	$q+(-0.382683 + 0.662827i) q^{5} +(-1.73205 + 1.00000i) q^{11} +0.317025i q^{13} +(-2.77164 - 4.80062i) q^{17} +(3.20041 + 1.84776i) q^{19} +(2.74666 + 1.58579i) q^{23} +(2.20711 + 3.82282i) q^{25} +6.82843i q^{29} +(-5.85172 + 3.37849i) q^{31} +(0.121320 - 0.210133i) q^{37} -2.74444 q^{41} +6.82843 q^{43} +(-5.99162 + 10.3778i) q^{47} +(-10.6024 + 6.12132i) q^{53} -1.53073i q^{55} +(-6.62567 - 11.4760i) q^{59} +(3.08669 + 1.78210i) q^{61} +(-0.210133 - 0.121320i) q^{65} +(2.24264 + 3.88437i) q^{67} +9.31371i q^{71} +(-10.2641 + 5.92596i) q^{73} +(5.65685 - 9.79796i) q^{79} +4.32957 q^{83} +4.24264 q^{85} +(-0.831025 + 1.43938i) q^{89} +(-2.44949 + 1.41421i) q^{95} +11.8519i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67}+O(q^{100}) $ 16 * q + 24 * q^25 - 32 * q^37 + 64 * q^43 - 32 * q^67

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−0.382683	+	0.662827i	−0.171141	+	0.296425i	−0.938819	−	0.344411i	$-0.888079\pi$
$5$	−0.382683	+	0.662827i	−0.171141	+	0.296425i	0.767678	+	0.640836i	$0.221412\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.73205	+	1.00000i	−0.522233	+	0.301511i	−0.737848	−	0.674967i	$-0.764158\pi$
$11$	−1.73205	+	1.00000i	−0.522233	+	0.301511i	0.215615	+	0.976478i	$0.430824\pi$
$12$		0			0
$13$			0.317025i			0.0879270i	0.999033	+	0.0439635i	$0.0139985\pi$
$13$			0.317025i			0.0879270i	−0.999033	+	0.0439635i	$0.986001\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.77164	−	4.80062i	−0.672221	−	1.16432i	−0.977273	−	0.211985i	$-0.932007\pi$
$17$	−2.77164	−	4.80062i	−0.672221	−	1.16432i	0.305052	−	0.952336i	$-0.401326\pi$
$18$		0			0
$19$	3.20041	+	1.84776i	0.734225	+	0.423905i	0.819966	−	0.572412i	$-0.193992\pi$
$19$	3.20041	+	1.84776i	0.734225	+	0.423905i	−0.0857408	+	0.996317i	$0.527326\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.74666	+	1.58579i	0.572719	+	0.330659i	0.758234	−	0.651982i	$-0.226062\pi$
$23$	2.74666	+	1.58579i	0.572719	+	0.330659i	−0.185516	+	0.982641i	$0.559396\pi$
$24$		0			0
$25$	2.20711	+	3.82282i	0.441421	+	0.764564i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.82843i			1.26801i	0.773330	+	0.634004i	$0.218590\pi$
$29$			6.82843i			1.26801i	−0.773330	+	0.634004i	$0.781410\pi$
$30$		0			0
$31$	−5.85172	+	3.37849i	−1.05100	+	0.606795i	−0.922929	−	0.384970i	$-0.874212\pi$
$31$	−5.85172	+	3.37849i	−1.05100	+	0.606795i	−0.128071	+	0.991765i	$0.540879\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.121320	−	0.210133i	0.0199449	−	0.0345457i	−0.855881	−	0.517173i	$-0.826984\pi$
$37$	0.121320	−	0.210133i	0.0199449	−	0.0345457i	0.875826	+	0.482628i	$0.160318\pi$
$38$		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
$47$	−5.99162	+	10.3778i	−0.873967	+	1.51376i	−0.0161088	+	0.999870i	$0.505128\pi$
$47$	−5.99162	+	10.3778i	−0.873967	+	1.51376i	−0.857859	+	0.513886i	$0.828206\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.6024	+	6.12132i	−1.45636	+	0.840828i	−0.998830	−	0.0483676i	$-0.984598\pi$
$53$	−10.6024	+	6.12132i	−1.45636	+	0.840828i	−0.457527	+	0.889196i	$0.651265\pi$
$54$		0			0
$55$		−	1.53073i		−	0.206404i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.62567	−	11.4760i	−0.862589	−	1.49405i	−0.869422	−	0.494071i	$-0.835508\pi$
$59$	−6.62567	−	11.4760i	−0.862589	−	1.49405i	0.00683301	−	0.999977i	$-0.497825\pi$
$60$		0			0
$61$	3.08669	+	1.78210i	0.395210	+	0.228175i	0.684415	−	0.729093i	$-0.260058\pi$
$61$	3.08669	+	1.78210i	0.395210	+	0.228175i	−0.289205	+	0.957267i	$0.593391\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.210133	−	0.121320i	−0.0260638	−	0.0150479i
$66$		0			0
$67$	2.24264	+	3.88437i	0.273982	+	0.474551i	0.969878	−	0.243592i	$-0.0783257\pi$
$67$	2.24264	+	3.88437i	0.273982	+	0.474551i	−0.695896	+	0.718143i	$0.744992\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.31371i			1.10533i	0.833402	+	0.552667i	$0.186390\pi$
$71$			9.31371i			1.10533i	−0.833402	+	0.552667i	$0.813610\pi$
$72$		0			0
$73$	−10.2641	+	5.92596i	−1.20132	+	0.693581i	−0.960848	−	0.277075i	$-0.910635\pi$
$73$	−10.2641	+	5.92596i	−1.20132	+	0.693581i	−0.240470	+	0.970657i	$0.577301\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.65685	−	9.79796i	0.636446	−	1.10236i	−0.349761	−	0.936839i	$-0.613737\pi$
$79$	5.65685	−	9.79796i	0.636446	−	1.10236i	0.986207	−	0.165518i	$-0.0529295\pi$
$80$		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
$89$	−0.831025	+	1.43938i	−0.0880885	+	0.152574i	−0.906703	−	0.421769i	$-0.861409\pi$
$89$	−0.831025	+	1.43938i	−0.0880885	+	0.152574i	0.818615	+	0.574343i	$0.194742\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.44949	+	1.41421i	−0.251312	+	0.145095i
$96$		0			0
$97$			11.8519i			1.20338i	0.798730	+	0.601690i	$0.205506\pi$
$97$			11.8519i			1.20338i	−0.798730	+	0.601690i	$0.794494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.92596	−	10.2641i	−0.589655	−	1.02131i	−0.994277	−	0.106829i	$-0.965930\pi$
$101$	−5.92596	−	10.2641i	−0.589655	−	1.02131i	0.404622	−	0.914484i	$-0.367403\pi$
$102$		0			0
$103$	7.72648	+	4.46088i	0.761313	+	0.439544i	0.829767	−	0.558110i	$-0.188473\pi$
$103$	7.72648	+	4.46088i	0.761313	+	0.439544i	−0.0684542	+	0.997654i	$0.521807\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.21076	+	3.58579i	0.600417	+	0.346651i	0.769206	−	0.639001i	$-0.220652\pi$
$107$	6.21076	+	3.58579i	0.600417	+	0.346651i	−0.168788	+	0.985652i	$0.553985\pi$
$108$		0			0
$109$	5.53553	+	9.58783i	0.530208	+	0.918347i	0.999379	+	0.0352398i	$0.0112195\pi$
$109$	5.53553	+	9.58783i	0.530208	+	0.918347i	−0.469171	+	0.883107i	$0.655447\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.5858i			0.995827i	0.867227	+	0.497914i	$0.165900\pi$
$113$			10.5858i			0.995827i	−0.867227	+	0.497914i	$0.834100\pi$
$114$		0			0
$115$	−2.10220	+	1.21371i	−0.196032	+	0.113179i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.50000	+	6.06218i	−0.318182	+	0.551107i
$122$		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
$131$	−8.47343	+	14.6764i	−0.740327	+	1.28228i	0.212020	+	0.977265i	$0.431996\pi$
$131$	−8.47343	+	14.6764i	−0.740327	+	1.28228i	−0.952346	+	0.305018i	$0.901337\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.37769	+	5.41421i	−0.801190	+	0.462567i	−0.843887	−	0.536521i	$-0.819738\pi$
$137$	−9.37769	+	5.41421i	−0.801190	+	0.462567i	0.0426968	+	0.999088i	$0.486405\pi$
$138$		0			0
$139$		−	15.6788i		−	1.32985i	−0.746908	−	0.664927i	$-0.768462\pi$
$139$		−	15.6788i		−	1.32985i	0.746908	−	0.664927i	$-0.231538\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.317025	−	0.549104i	−0.0265110	−	0.0459184i
$144$		0			0
$145$	−4.52607	−	2.61313i	−0.375869	−	0.217008i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.12372	−	3.53553i	−0.501675	−	0.289642i	0.227730	−	0.973724i	$-0.426870\pi$
$149$	−6.12372	−	3.53553i	−0.501675	−	0.289642i	−0.729405	+	0.684082i	$0.760203\pi$
$150$		0			0
$151$	−5.07107	−	8.78335i	−0.412678	−	0.714779i	0.582504	−	0.812828i	$-0.302073\pi$
$151$	−5.07107	−	8.78335i	−0.412678	−	0.714779i	−0.995182	+	0.0980492i	$0.968740\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.17157i		−	0.415391i
$156$		0			0
$157$	−5.34972	+	3.08866i	−0.426954	+	0.246502i	−0.698048	−	0.716051i	$-0.745948\pi$
$157$	−5.34972	+	3.08866i	−0.426954	+	0.246502i	0.271094	+	0.962553i	$0.412615\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.41421	+	2.44949i	−0.110770	+	0.191859i	−0.916081	−	0.400994i	$-0.868665\pi$
$163$	−1.41421	+	2.44949i	−0.110770	+	0.191859i	0.805311	+	0.592852i	$0.201998\pi$
$164$		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
$173$	9.39731	−	16.2766i	0.714464	−	1.23749i	−0.248702	−	0.968580i	$-0.580004\pi$
$173$	9.39731	−	16.2766i	0.714464	−	1.23749i	0.963166	−	0.268908i	$-0.0866628\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.06591	−	4.65685i	0.602874	−	0.348070i	−0.167297	−	0.985907i	$-0.553504\pi$
$179$	8.06591	−	4.65685i	0.602874	−	0.348070i	0.770171	+	0.637837i	$0.220171\pi$
$180$		0			0
$181$		−	1.66205i		−	0.123539i	−0.998090	−	0.0617696i	$-0.980326\pi$
$181$		−	1.66205i		−	0.123539i	0.998090	−	0.0617696i	$-0.0196744\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0928546	+	0.160829i	0.00682680	+	0.0118244i
$186$		0			0
$187$	9.60124	+	5.54328i	0.702112	+	0.405365i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.9941	+	8.65685i	1.08494	+	0.626388i	0.932224	−	0.361883i	$-0.117866\pi$
$191$	14.9941	+	8.65685i	1.08494	+	0.626388i	0.152712	+	0.988271i	$0.451199\pi$
$192$		0			0
$193$	4.82843	+	8.36308i	0.347558	+	0.601988i	0.985815	−	0.167835i	$-0.0536777\pi$
$193$	4.82843	+	8.36308i	0.347558	+	0.601988i	−0.638257	+	0.769823i	$0.720344\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.07107i		−	0.218805i	−0.993998	−	0.109402i	$-0.965106\pi$
$197$		−	3.07107i		−	0.218805i	0.993998	−	0.109402i	$-0.0348937\pi$
$198$		0			0
$199$	6.40083	−	3.69552i	0.453742	−	0.261968i	−0.255667	−	0.966765i	$-0.582295\pi$
$199$	6.40083	−	3.69552i	0.453742	−	0.261968i	0.709409	+	0.704797i	$0.248962\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.05025	−	1.81909i	0.0733528	−	0.127051i
$206$		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
$215$	−2.61313	+	4.52607i	−0.178214	+	0.308675i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.52192	−	0.878680i	0.102375	−	0.0591064i
$222$		0			0
$223$			21.8017i			1.45995i	0.683474	+	0.729975i	$0.260468\pi$
$223$			21.8017i			1.45995i	−0.683474	+	0.729975i	$0.739532\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.37849	+	5.85172i	0.224238	+	0.388392i	0.956091	−	0.293071i	$-0.0946772\pi$
$227$	3.37849	+	5.85172i	0.224238	+	0.388392i	−0.731852	+	0.681463i	$0.761344\pi$
$228$		0			0
$229$	−14.0136	−	8.09075i	−0.926044	−	0.534651i	−0.0404854	−	0.999180i	$-0.512890\pi$
$229$	−14.0136	−	8.09075i	−0.926044	−	0.534651i	−0.885558	+	0.464529i	$0.846224\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.1390	−	7.58579i	−0.860762	−	0.496961i	0.00350513	−	0.999994i	$-0.498884\pi$
$233$	−13.1390	−	7.58579i	−0.860762	−	0.496961i	−0.864268	+	0.503032i	$0.832218\pi$
$234$		0			0
$235$	−4.58579	−	7.94282i	−0.299144	−	0.518132i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.82843i		−	0.571063i	−0.958369	−	0.285532i	$-0.907830\pi$
$239$		−	8.82843i		−	0.571063i	0.958369	−	0.285532i	$-0.0921702\pi$
$240$		0			0
$241$	22.1283	−	12.7758i	1.42541	−	0.822962i	0.428657	−	0.903467i	$-0.358987\pi$
$241$	22.1283	−	12.7758i	1.42541	−	0.822962i	0.996754	+	0.0805055i	$0.0256535\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.585786	+	1.01461i	−0.0372727	+	0.0645582i
$248$		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
$257$	10.2940	−	17.8297i	0.642122	−	1.11219i	−0.342837	−	0.939395i	$-0.611388\pi$
$257$	10.2940	−	17.8297i	0.642122	−	1.11219i	0.984958	−	0.172792i	$-0.0552789\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	17.0233	−	9.82843i	1.04970	−	0.606047i	0.127137	−	0.991885i	$-0.459421\pi$
$263$	17.0233	−	9.82843i	1.04970	−	0.606047i	0.922566	+	0.385838i	$0.126088\pi$
$264$		0			0
$265$		−	9.37011i		−	0.575601i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.94897	+	15.5001i	0.545628	+	0.945056i	0.998567	+	0.0535141i	$0.0170422\pi$
$269$	8.94897	+	15.5001i	0.545628	+	0.945056i	−0.452939	+	0.891542i	$0.649624\pi$
$270$		0			0
$271$	4.75351	+	2.74444i	0.288755	+	0.166713i	0.637380	−	0.770549i	$-0.280018\pi$
$271$	4.75351	+	2.74444i	0.288755	+	0.166713i	−0.348625	+	0.937262i	$0.613351\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.64564	−	4.41421i	−0.461050	−	0.266187i
$276$		0			0
$277$	−6.48528	−	11.2328i	−0.389663	−	0.674916i	0.602741	−	0.797937i	$-0.294075\pi$
$277$	−6.48528	−	11.2328i	−0.389663	−	0.674916i	−0.992404	+	0.123021i	$0.960742\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.4853i		−	0.625499i	−0.949836	−	0.312750i	$-0.898750\pi$
$281$		−	10.4853i		−	0.625499i	0.949836	−	0.312750i	$-0.101250\pi$
$282$		0			0
$283$	−3.52207	+	2.03347i	−0.209365	+	0.120877i	−0.601016	−	0.799237i	$-0.705237\pi$
$283$	−3.52207	+	2.03347i	−0.209365	+	0.120877i	0.391651	+	0.920114i	$0.371904\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−6.86396	+	11.8887i	−0.403762	+	0.699337i
$290$		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
$299$	−0.502734	+	0.870762i	−0.0290739	+	0.0503574i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.36245	+	1.36396i	−0.135273	+	0.0781002i
$306$		0			0
$307$		−	27.1367i		−	1.54877i	−0.632712	−	0.774387i	$-0.718058\pi$
$307$		−	27.1367i		−	1.54877i	0.632712	−	0.774387i	$-0.281942\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.3617	+	26.6073i	0.871084	+	1.50876i	0.860877	+	0.508814i	$0.169916\pi$
$311$	15.3617	+	26.6073i	0.871084	+	1.50876i	0.0102070	+	0.999948i	$0.496751\pi$
$312$		0			0
$313$	−25.0071	−	14.4379i	−1.41348	−	0.816076i	−0.417770	−	0.908553i	$-0.637188\pi$
$313$	−25.0071	−	14.4379i	−1.41348	−	0.816076i	−0.995715	+	0.0924774i	$0.970521\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.6463	−	7.87868i	−0.766451	−	0.442511i	0.0651561	−	0.997875i	$-0.479245\pi$
$317$	−13.6463	−	7.87868i	−0.766451	−	0.442511i	−0.831607	+	0.555364i	$0.812579\pi$
$318$		0			0
$319$	−6.82843	−	11.8272i	−0.382319	−	0.662195i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	20.4853i		−	1.13983i
$324$		0			0
$325$	−1.21193	+	0.699709i	−0.0672258	+	0.0388129i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.6569	−	23.6544i	0.750649	−	1.30016i	−0.196860	−	0.980432i	$-0.563074\pi$
$331$	13.6569	−	23.6544i	0.750649	−	1.30016i	0.947509	−	0.319730i	$-0.103592\pi$
$332$		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
$341$	6.75699	−	11.7034i	0.365911	−	0.633777i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	25.8067	−	14.8995i	1.38538	−	0.799847i	0.392586	−	0.919715i	$-0.371581\pi$
$347$	25.8067	−	14.8995i	1.38538	−	0.799847i	0.992790	+	0.119869i	$0.0382474\pi$
$348$		0			0
$349$			31.9372i			1.70956i	0.518993	+	0.854779i	$0.326307\pi$
$349$			31.9372i			1.70956i	−0.518993	+	0.854779i	$0.673693\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.5726	+	18.3122i	0.562720	+	0.974660i	0.997258	+	0.0740064i	$0.0235785\pi$
$353$	10.5726	+	18.3122i	0.562720	+	0.974660i	−0.434537	+	0.900654i	$0.643088\pi$
$354$		0			0
$355$	−6.17338	−	3.56420i	−0.327649	−	0.189168i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.8493	+	9.72792i	0.889270	+	0.513420i	0.873704	−	0.486459i	$-0.161712\pi$
$359$	16.8493	+	9.72792i	0.889270	+	0.513420i	0.0155661	+	0.999879i	$0.495045\pi$
$360$		0			0
$361$	−2.67157	−	4.62730i	−0.140609	−	0.243542i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	9.07107i		−	0.474801i
$366$		0			0
$367$	31.2276	−	18.0292i	1.63007	−	0.941119i	0.645996	−	0.763341i	$-0.276442\pi$
$367$	31.2276	−	18.0292i	1.63007	−	0.941119i	0.984071	−	0.177778i	$-0.0568909\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−15.3137	+	26.5241i	−0.792914	+	1.37337i	0.131242	+	0.991350i	$0.458104\pi$
$373$	−15.3137	+	26.5241i	−0.792914	+	1.37337i	−0.924155	+	0.382017i	$0.875230\pi$
$374$		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
$383$	7.39104	−	12.8017i	0.377664	−	0.654134i	−0.613058	−	0.790038i	$-0.710061\pi$
$383$	7.39104	−	12.8017i	0.377664	−	0.654134i	0.990722	+	0.135904i	$0.0433940\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.01461	−	0.585786i	0.0514429	−	0.0297006i	−0.474058	−	0.880494i	$-0.657211\pi$
$389$	1.01461	−	0.585786i	0.0514429	−	0.0297006i	0.525501	+	0.850793i	$0.323878\pi$
$390$		0			0
$391$		−	17.5809i		−	0.889105i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.32957	+	7.49903i	0.217844	+	0.377317i
$396$		0			0
$397$	−12.1388	−	7.00835i	−0.609230	−	0.351739i	0.163434	−	0.986554i	$-0.447743\pi$
$397$	−12.1388	−	7.00835i	−0.609230	−	0.351739i	−0.772664	+	0.634815i	$0.781076\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.77589	−	2.75736i	−0.238496	−	0.137696i	0.375989	−	0.926624i	$-0.377303\pi$
$401$	−4.77589	−	2.75736i	−0.238496	−	0.137696i	−0.614485	+	0.788928i	$0.710636\pi$
$402$		0			0
$403$	−1.07107	−	1.85514i	−0.0533537	−	0.0924113i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.485281i			0.0240545i
$408$		0			0
$409$	−0.984485	+	0.568393i	−0.0486796	+	0.0281052i	−0.524142	−	0.851631i	$-0.675614\pi$
$409$	−0.984485	+	0.568393i	−0.0486796	+	0.0281052i	0.475463	+	0.879736i	$0.342281\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.65685	+	2.86976i	−0.0813318	+	0.140871i
$416$		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
$425$	12.2346	−	21.1910i	0.593465	−	1.02791i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	28.8505	−	16.6569i	1.38968	−	0.802332i	0.396402	−	0.918077i	$-0.370259\pi$
$431$	28.8505	−	16.6569i	1.38968	−	0.802332i	0.993279	+	0.115745i	$0.0369255\pi$
$432$		0			0
$433$			0.502734i			0.0241599i	0.999927	+	0.0120799i	$0.00384526\pi$
$433$			0.502734i			0.0241599i	−0.999927	+	0.0120799i	$0.996155\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.86030	+	10.1503i	0.280336	+	0.485557i
$438$		0			0
$439$	12.0251	+	6.94269i	0.573927	+	0.331357i	0.758716	−	0.651421i	$-0.225827\pi$
$439$	12.0251	+	6.94269i	0.573927	+	0.331357i	−0.184789	+	0.982778i	$0.559160\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−25.3864	−	14.6569i	−1.20615	−	0.696368i	−0.244230	−	0.969717i	$-0.578535\pi$
$443$	−25.3864	−	14.6569i	−1.20615	−	0.696368i	−0.961915	+	0.273349i	$0.911869\pi$
$444$		0			0
$445$	−0.636039	−	1.10165i	−0.0301511	−	0.0522233i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	11.5563i		−	0.545378i	−0.962102	−	0.272689i	$-0.912087\pi$
$449$		−	11.5563i		−	0.545378i	0.962102	−	0.272689i	$-0.0879130\pi$
$450$		0			0
$451$	4.75351	−	2.74444i	0.223834	−	0.129231i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	−0.485299	−	0.874348i	$-0.661289\pi$
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	0.999857	−	0.0168929i	$-0.00537742\pi$
$458$		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
$467$	−3.11586	+	5.39683i	−0.144185	+	0.249735i	−0.929069	−	0.369908i	$-0.879389\pi$
$467$	−3.11586	+	5.39683i	−0.144185	+	0.249735i	0.784884	+	0.619643i	$0.212723\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.8272	+	6.82843i	−0.543814	+	0.313971i
$474$		0			0
$475$			16.3128i			0.748483i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.6662	+	20.2065i	0.533043	+	0.923257i	0.999255	+	0.0385845i	$0.0122849\pi$
$479$	11.6662	+	20.2065i	0.533043	+	0.923257i	−0.466213	+	0.884673i	$0.654382\pi$
$480$		0			0
$481$	0.0666175	+	0.0384616i	0.00303750	+	0.00175370i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.85578	−	4.53553i	−0.356712	−	0.205948i
$486$		0			0
$487$	−9.89949	−	17.1464i	−0.448589	−	0.776979i	0.549706	−	0.835359i	$-0.314740\pi$
$487$	−9.89949	−	17.1464i	−0.448589	−	0.776979i	−0.998294	+	0.0583797i	$0.981407\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	34.4853i		−	1.55630i	−0.628079	−	0.778149i	$-0.716159\pi$
$491$		−	34.4853i		−	1.55630i	0.628079	−	0.778149i	$-0.283841\pi$
$492$		0			0
$493$	32.7807	−	18.9259i	1.47637	−	0.852381i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.0711	+	22.6398i	−0.585141	+	1.01349i	0.409716	+	0.912213i	$0.365628\pi$
$499$	−13.0711	+	22.6398i	−0.585141	+	1.01349i	−0.994858	+	0.101282i	$0.967706\pi$
$500$		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
$509$	6.46716	−	11.2014i	0.286652	−	0.496495i	−0.686357	−	0.727265i	$-0.740791\pi$
$509$	6.46716	−	11.2014i	0.286652	−	0.496495i	0.973008	+	0.230770i	$0.0741244\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.91359	+	3.41421i	−0.260584	+	0.150448i
$516$		0			0
$517$		−	23.9665i		−	1.05404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.01673	−	1.76104i	−0.0445439	−	0.0771523i	0.842894	−	0.538080i	$-0.180850\pi$
$521$	−1.01673	−	1.76104i	−0.0445439	−	0.0771523i	−0.887438	+	0.460928i	$0.847517\pi$
$522$		0			0
$523$	−15.7746	−	9.10748i	−0.689776	−	0.398242i	0.113752	−	0.993509i	$-0.463713\pi$
$523$	−15.7746	−	9.10748i	−0.689776	−	0.398242i	−0.803528	+	0.595267i	$0.797046\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	32.4377	+	18.7279i	1.41301	+	0.815801i
$528$		0			0
$529$	−6.47056	−	11.2073i	−0.281329	−	0.487276i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	0.870058i		−	0.0376864i
$534$		0			0
$535$	−4.75351	+	2.74444i	−0.205512	+	0.118653i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−4.17157	+	7.22538i	−0.179350	+	0.310643i	−0.941658	−	0.336571i	$-0.890733\pi$
$541$	−4.17157	+	7.22538i	−0.179350	+	0.310643i	0.762308	+	0.647214i	$0.224066\pi$
$542$		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
$551$	−12.6173	+	21.8538i	−0.537515	+	0.931003i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.4868	−	8.36396i	0.613826	−	0.354392i	−0.160636	−	0.987014i	$-0.551354\pi$
$557$	14.4868	−	8.36396i	0.613826	−	0.354392i	0.774461	+	0.632621i	$0.218021\pi$
$558$		0			0
$559$			2.16478i			0.0915606i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11.0322	−	19.1083i	−0.464950	−	0.805317i	0.534249	−	0.845327i	$-0.320594\pi$
$563$	−11.0322	−	19.1083i	−0.464950	−	0.805317i	−0.999199	+	0.0400098i	$0.987261\pi$
$564$		0			0
$565$	−7.01655	−	4.05101i	−0.295188	−	0.170427i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.420266	−	0.242641i	−0.0176185	−	0.0101720i	0.491165	−	0.871067i	$-0.336571\pi$
$569$	−0.420266	−	0.242641i	−0.0176185	−	0.0101720i	−0.508783	+	0.860895i	$0.669905\pi$
$570$		0			0
$571$	1.31371	+	2.27541i	0.0549770	+	0.0952229i	0.892204	−	0.451632i	$-0.149158\pi$
$571$	1.31371	+	2.27541i	0.0549770	+	0.0952229i	−0.837227	+	0.546855i	$0.815825\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			14.0000i			0.583840i
$576$		0			0
$577$	−17.2140	+	9.93850i	−0.716628	+	0.413745i	−0.813510	−	0.581551i	$-0.802446\pi$
$577$	−17.2140	+	9.93850i	−0.716628	+	0.413745i	0.0968824	+	0.995296i	$0.469113\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.2426	−	21.2049i	0.507038	−	0.878216i
$584$		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
$593$	15.3889	−	26.6544i	0.631947	−	1.09457i	−0.355206	−	0.934788i	$-0.615589\pi$
$593$	15.3889	−	26.6544i	0.631947	−	1.09457i	0.987153	−	0.159777i	$-0.0510775\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.06591	+	4.65685i	−0.329564	+	0.190274i	−0.655648	−	0.755067i	$-0.727604\pi$
$599$	−8.06591	+	4.65685i	−0.329564	+	0.190274i	0.326083	+	0.945341i	$0.394271\pi$
$600$		0			0
$601$			23.9121i			0.975394i	0.873013	+	0.487697i	$0.162163\pi$
$601$			23.9121i			0.975394i	−0.873013	+	0.487697i	$0.837837\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.67878	−	4.63979i	−0.108908	−	0.188634i
$606$		0			0
$607$	10.9269	+	6.30864i	0.443509	+	0.256060i	0.705085	−	0.709123i	$-0.250909\pi$
$607$	10.9269	+	6.30864i	0.443509	+	0.256060i	−0.261576	+	0.965183i	$0.584242\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.29002	−	1.89949i	−0.133100	−	0.0768453i
$612$		0			0
$613$	22.6066	+	39.1558i	0.913072	+	1.58149i	0.809700	+	0.586844i	$0.199630\pi$
$613$	22.6066	+	39.1558i	0.913072	+	1.58149i	0.103372	+	0.994643i	$0.467037\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			17.4558i			0.702746i	0.936236	+	0.351373i	$0.114285\pi$
$617$			17.4558i			0.702746i	−0.936236	+	0.351373i	$0.885715\pi$
$618$		0			0
$619$	−1.55310	+	0.896683i	−0.0624244	+	0.0360407i	−0.530887	−	0.847442i	$-0.678141\pi$
$619$	−1.55310	+	0.896683i	−0.0624244	+	0.0360407i	0.468463	+	0.883483i	$0.344808\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−8.27817	+	14.3382i	−0.331127	+	0.573529i
$626$		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
$635$	−0.448342	+	0.776550i	−0.0177919	+	0.0308165i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.3134	−	11.7279i	0.802329	−	0.463225i	−0.0419557	−	0.999119i	$-0.513359\pi$
$641$	20.3134	−	11.7279i	0.802329	−	0.463225i	0.844285	+	0.535894i	$0.180025\pi$
$642$		0			0
$643$		−	32.5168i		−	1.28234i	−0.767400	−	0.641169i	$-0.778450\pi$
$643$		−	32.5168i		−	1.28234i	0.767400	−	0.641169i	$-0.221550\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.19278	+	5.53006i	0.125521	+	0.217409i	0.921937	−	0.387341i	$-0.126606\pi$
$647$	3.19278	+	5.53006i	0.125521	+	0.217409i	−0.796415	+	0.604750i	$0.793273\pi$
$648$		0			0
$649$	22.9520	+	13.2513i	0.900944	+	0.520161i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.07306	−	2.92893i	−0.198524	−	0.114618i	0.397443	−	0.917627i	$-0.369898\pi$
$653$	−5.07306	−	2.92893i	−0.198524	−	0.114618i	−0.595967	+	0.803009i	$0.703231\pi$
$654$		0			0
$655$	−6.48528	−	11.2328i	−0.253401	−	0.438903i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			2.97056i			0.115717i	0.998325	+	0.0578583i	$0.0184272\pi$
$659$			2.97056i			0.115717i	−0.998325	+	0.0578583i	$0.981573\pi$
$660$		0			0
$661$	−20.2536	+	11.6934i	−0.787773	+	0.454821i	−0.839178	−	0.543857i	$-0.816963\pi$
$661$	−20.2536	+	11.6934i	−0.787773	+	0.454821i	0.0514050	+	0.998678i	$0.483630\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−10.8284	+	18.7554i	−0.419278	+	0.726211i
$668$		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
$677$	−13.9895	+	24.2305i	−0.537661	+	0.931255i	0.461369	+	0.887208i	$0.347358\pi$
$677$	−13.9895	+	24.2305i	−0.537661	+	0.931255i	−0.999029	+	0.0440470i	$0.985975\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−40.6777	+	23.4853i	−1.55649	+	0.898639i	−0.558900	+	0.829235i	$0.688777\pi$
$683$	−40.6777	+	23.4853i	−1.55649	+	0.898639i	−0.997589	+	0.0694045i	$0.977890\pi$
$684$		0			0
$685$		−	8.28772i		−	0.316657i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.94061	−	3.36124i	−0.0739315	−	0.128053i
$690$		0			0
$691$	22.1754	+	12.8030i	0.843594	+	0.487049i	0.858484	−	0.512840i	$-0.171407\pi$
$691$	22.1754	+	12.8030i	0.843594	+	0.487049i	−0.0148906	+	0.999889i	$0.504740\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.3923	+	6.00000i	0.394203	+	0.227593i
$696$		0			0
$697$	7.60660	+	13.1750i	0.288121	+	0.499039i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			15.4558i			0.583759i	0.956455	+	0.291880i	$0.0942807\pi$
$701$			15.4558i			0.583759i	−0.956455	+	0.291880i	$0.905719\pi$
$702$		0			0
$703$	0.776550	−	0.448342i	0.0292881	−	0.0169095i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	15.4350	−	26.7343i	0.579675	−	1.00403i	−0.415842	−	0.909437i	$-0.636513\pi$
$709$	15.4350	−	26.7343i	0.579675	−	1.00403i	0.995516	−	0.0945890i	$-0.0301537\pi$
$710$		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
$719$	−9.81845	+	17.0061i	−0.366167	+	0.634219i	−0.988963	−	0.148165i	$-0.952663\pi$
$719$	−9.81845	+	17.0061i	−0.366167	+	0.634219i	0.622796	+	0.782384i	$0.285997\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−26.1039	+	15.0711i	−0.969473	+	0.559725i
$726$		0			0
$727$			53.7933i			1.99508i	0.0700903	+	0.997541i	$0.477671\pi$
$727$			53.7933i			1.99508i	−0.0700903	+	0.997541i	$0.522329\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−18.9259	−	32.7807i	−0.700001	−	1.21244i
$732$		0			0
$733$	−19.3162	−	11.1522i	−0.713460	−	0.411916i	0.0988808	−	0.995099i	$-0.468474\pi$
$733$	−19.3162	−	11.1522i	−0.713460	−	0.411916i	−0.812341	+	0.583183i	$0.801807\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.76874	−	4.48528i	−0.286165	−	0.165217i
$738$		0			0
$739$	−1.27208	−	2.20330i	−0.0467941	−	0.0810498i	0.841680	−	0.539977i	$-0.181567\pi$
$739$	−1.27208	−	2.20330i	−0.0467941	−	0.0810498i	−0.888474	+	0.458927i	$0.848234\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			43.6569i			1.60161i	0.598922	+	0.800807i	$0.295596\pi$
$743$			43.6569i			1.60161i	−0.598922	+	0.800807i	$0.704404\pi$
$744$		0			0
$745$	4.68690	−	2.70598i	0.171715	−	0.0991395i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18.1421	−	31.4231i	0.662016	−	1.14665i	−0.318069	−	0.948067i	$-0.603034\pi$
$751$	18.1421	−	31.4231i	0.662016	−	1.14665i	0.980085	−	0.198578i	$-0.0636322\pi$
$752$		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
$761$	−26.8310	+	46.4726i	−0.972622	+	1.68463i	−0.285052	+	0.958512i	$0.592011\pi$
$761$	−26.8310	+	46.4726i	−0.972622	+	1.68463i	−0.687570	+	0.726118i	$0.741322\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.63818	−	2.10051i	0.131367	−	0.0758448i
$768$		0			0
$769$		−	14.2793i		−	0.514926i	−0.966288	−	0.257463i	$-0.917113\pi$
$769$		−	14.2793i		−	0.514926i	0.966288	−	0.257463i	$-0.0828866\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.7883	−	29.0783i	−0.603835	−	1.04587i	−0.992234	−	0.124381i	$-0.960305\pi$
$773$	−16.7883	−	29.0783i	−0.603835	−	1.04587i	0.388400	−	0.921491i	$-0.373028\pi$
$774$		0			0
$775$	−25.8307	−	14.9134i	−0.927868	−	0.535705i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.78335	−	5.07107i	−0.314696	−	0.181690i
$780$		0			0
$781$	−9.31371	−	16.1318i	−0.333271	−	0.577242i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	4.72792i		−	0.168747i
$786$		0			0
$787$	−43.7076	+	25.2346i	−1.55801	+	0.899515i	−0.560559	+	0.828115i	$0.689414\pi$
$787$	−43.7076	+	25.2346i	−1.55801	+	0.899515i	−0.997448	+	0.0714009i	$0.977253\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.564971	+	0.978559i	−0.0200627	+	0.0347496i
$794$		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
$803$	11.8519	−	20.5281i	0.418245	−	0.724422i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.3712	+	10.6066i	−0.645896	+	0.372908i	−0.786882	−	0.617103i	$-0.788306\pi$
$809$	−18.3712	+	10.6066i	−0.645896	+	0.372908i	0.140986	+	0.990012i	$0.454973\pi$
$810$		0			0
$811$		−	42.1814i		−	1.48119i	−0.671951	−	0.740595i	$-0.734544\pi$
$811$		−	42.1814i		−	1.48119i	0.671951	−	0.740595i	$-0.265456\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.08239	−	1.87476i	−0.0379145	−	0.0656699i
$816$		0			0
$817$	21.8538	+	12.6173i	0.764567	+	0.441423i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	44.2288	+	25.5355i	1.54360	+	0.891196i	0.998608	+	0.0527535i	$0.0167998\pi$
$821$	44.2288	+	25.5355i	1.54360	+	0.891196i	0.544990	+	0.838443i	$0.316534\pi$
$822$		0			0
$823$	20.1421	+	34.8872i	0.702111	+	1.21609i	0.967724	+	0.252012i	$0.0810922\pi$
$823$	20.1421	+	34.8872i	0.702111	+	1.21609i	−0.265613	+	0.964080i	$0.585574\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	18.4853i		−	0.642796i	−0.946944	−	0.321398i	$-0.895847\pi$
$827$		−	18.4853i		−	0.642796i	0.946944	−	0.321398i	$-0.104153\pi$
$828$		0			0
$829$	−30.6314	+	17.6850i	−1.06387	+	0.614226i	−0.926501	−	0.376293i	$-0.877198\pi$
$829$	−30.6314	+	17.6850i	−1.06387	+	0.614226i	−0.137371	+	0.990520i	$0.543865\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.928932	+	1.60896i	−0.0321470	+	0.0556803i
$836$		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
$845$	−4.93642	+	8.55014i	−0.169818	+	0.294134i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.666452	−	0.384776i	0.0228457	−	0.0131900i
$852$		0			0
$853$			57.2805i			1.96125i	0.195899	+	0.980624i	$0.437237\pi$
$853$			57.2805i			1.96125i	−0.195899	+	0.980624i	$0.562763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.8114	−	34.3143i	−0.676743	−	1.17215i	−0.975956	−	0.217967i	$-0.930058\pi$
$857$	−19.8114	−	34.3143i	−0.676743	−	1.17215i	0.299213	−	0.954186i	$-0.403276\pi$
$858$		0			0
$859$	11.9309	+	6.88830i	0.407077	+	0.235026i	0.689533	−	0.724254i	$-0.257816\pi$
$859$	11.9309	+	6.88830i	0.407077	+	0.235026i	−0.282456	+	0.959280i	$0.591149\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.7482	−	12.5563i	−0.740319	−	0.427423i	0.0818666	−	0.996643i	$-0.473912\pi$
$863$	−21.7482	−	12.5563i	−0.740319	−	0.427423i	−0.822185	+	0.569220i	$0.807245\pi$
$864$		0			0
$865$	7.19239	+	12.4576i	0.244549	+	0.423570i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			22.6274i			0.767583i
$870$		0			0
$871$	−1.23144	+	0.710974i	−0.0417259	+	0.0240904i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.464466	+	0.804479i	−0.0156839	+	0.0271653i	−0.873761	−	0.486356i	$-0.838326\pi$
$877$	−0.464466	+	0.804479i	−0.0156839	+	0.0271653i	0.858077	+	0.513521i	$0.171659\pi$
$878$		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
$887$	−9.94977	+	17.2335i	−0.334081	+	0.578644i	−0.983308	−	0.181950i	$-0.941759\pi$
$887$	−9.94977	+	17.2335i	−0.334081	+	0.578644i	0.649227	+	0.760595i	$0.275092\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−38.3513	+	22.1421i	−1.28338	+	0.740958i
$894$		0			0
$895$			7.12840i			0.238276i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−23.0698	−	39.9581i	−0.769421	−	1.33268i
$900$		0			0
$901$	58.7723	+	33.9322i	1.95799	+	1.13044i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.10165	+	0.636039i	0.0366201	+	0.0211427i
$906$		0			0
$907$	16.2426	+	28.1331i	0.539328	+	0.934144i	0.998940	+	0.0460239i	$0.0146550\pi$
$907$	16.2426	+	28.1331i	0.539328	+	0.934144i	−0.459612	+	0.888120i	$0.652012\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			42.2843i			1.40094i	0.713682	+	0.700470i	$0.247026\pi$
$911$			42.2843i			1.40094i	−0.713682	+	0.700470i	$0.752974\pi$
$912$		0			0
$913$	−7.49903	+	4.32957i	−0.248182	+	0.143288i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	7.75736	−	13.4361i	0.255892	−	0.443217i	−0.709246	−	0.704961i	$-0.750964\pi$
$919$	7.75736	−	13.4361i	0.255892	−	0.443217i	0.965137	+	0.261744i	$0.0842976\pi$
$920$		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
$929$	−11.1522	+	19.3162i	−0.365892	+	0.633744i	−0.988919	−	0.148457i	$-0.952569\pi$
$929$	−11.1522	+	19.3162i	−0.365892	+	0.633744i	0.623027	+	0.782201i	$0.285903\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7.34847	+	4.24264i	−0.240321	+	0.138749i
$936$		0			0
$937$		−	16.5210i		−	0.539719i	−0.962900	−	0.269860i	$-0.913023\pi$
$937$		−	16.5210i		−	0.539719i	0.962900	−	0.269860i	$-0.0869773\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.18651	+	2.05510i	0.0386792	+	0.0669943i	0.884717	−	0.466129i	$-0.154352\pi$
$941$	1.18651	+	2.05510i	0.0386792	+	0.0669943i	−0.846038	+	0.533123i	$0.821018\pi$
$942$		0			0
$943$	−7.53806	−	4.35210i	−0.245473	−	0.141724i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.64564	+	4.41421i	0.248450	+	0.143443i	0.619054	−	0.785348i	$-0.287516\pi$
$947$	7.64564	+	4.41421i	0.248450	+	0.143443i	−0.370604	+	0.928791i	$0.620849\pi$
$948$		0			0
$949$	−1.87868	−	3.25397i	−0.0609845	−	0.105628i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			58.8701i			1.90699i	0.301412	+	0.953494i	$0.402542\pi$
$953$			58.8701i			1.90699i	−0.301412	+	0.953494i	$0.597458\pi$
$954$		0			0
$955$	−11.4760	+	6.62567i	−0.371355	+	0.214402i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.32843	−	12.6932i	0.236401	−	0.409458i
$962$		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
$971$	−12.6173	+	21.8538i	−0.404908	+	0.701321i	−0.994311	−	0.106518i	$-0.966030\pi$
$971$	−12.6173	+	21.8538i	−0.404908	+	0.701321i	0.589403	+	0.807839i	$0.299363\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.420266	−	0.242641i	0.0134455	−	0.00776276i	−0.493262	−	0.869881i	$-0.664196\pi$
$977$	0.420266	−	0.242641i	0.0134455	−	0.00776276i	0.506708	+	0.862118i	$0.330862\pi$
$978$		0			0
$979$		−	3.32410i		−	0.106239i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.50981	−	6.07917i	−0.111946	−	0.193895i	0.804609	−	0.593805i	$-0.202375\pi$
$983$	−3.50981	−	6.07917i	−0.111946	−	0.193895i	−0.916555	+	0.399910i	$0.869042\pi$
$984$		0			0
$985$	2.03559	+	1.17525i	0.0648592	+	0.0374465i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.7554	+	10.8284i	0.596387	+	0.344324i
$990$		0			0
$991$	4.92893	+	8.53716i	0.156573	+	0.271192i	0.933631	−	0.358237i	$-0.116622\pi$
$991$	4.92893	+	8.53716i	0.156573	+	0.271192i	−0.777058	+	0.629429i	$0.783289\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			5.65685i			0.179334i
$996$		0			0
$997$	13.3979	−	7.73526i	0.424314	−	0.244978i	−0.272607	−	0.962125i	$-0.587886\pi$
$997$	13.3979	−	7.73526i	0.424314	−	0.244978i	0.696922	+	0.717147i	$0.254553\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.f.b.881.3	✓	8	7.5	odd	6
1764.2.f.b.881.4	yes	8	21.2	odd	6
1764.2.f.b.881.5	yes	8	7.2	even	3
1764.2.f.b.881.6	yes	8	21.5	even	6
1764.2.t.c.521.3		16	21.17	even	6	inner
1764.2.t.c.521.4		16	7.4	even	3	inner
1764.2.t.c.521.5		16	21.11	odd	6	inner
1764.2.t.c.521.6		16	7.3	odd	6	inner
1764.2.t.c.1097.3		16	1.1	even	1	trivial
1764.2.t.c.1097.4		16	21.20	even	2	inner
1764.2.t.c.1097.5		16	7.6	odd	2	inner
1764.2.t.c.1097.6		16	3.2	odd	2	inner
7056.2.k.e.881.3		8	84.23	even	6
7056.2.k.e.881.4		8	28.19	even	6
7056.2.k.e.881.5		8	84.47	odd	6
7056.2.k.e.881.6		8	28.23	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.382683 + 0.662827i) q^{5} +O(q^{10})\)	\(q+(-0.382683 + 0.662827i) q^{5} +(-1.73205 + 1.00000i) q^{11} +0.317025i q^{13} +(-2.77164 - 4.80062i) q^{17} +(3.20041 + 1.84776i) q^{19} +(2.74666 + 1.58579i) q^{23} +(2.20711 + 3.82282i) q^{25} +6.82843i q^{29} +(-5.85172 + 3.37849i) q^{31} +(0.121320 - 0.210133i) q^{37} -2.74444 q^{41} +6.82843 q^{43} +(-5.99162 + 10.3778i) q^{47} +(-10.6024 + 6.12132i) q^{53} -1.53073i q^{55} +(-6.62567 - 11.4760i) q^{59} +(3.08669 + 1.78210i) q^{61} +(-0.210133 - 0.121320i) q^{65} +(2.24264 + 3.88437i) q^{67} +9.31371i q^{71} +(-10.2641 + 5.92596i) q^{73} +(5.65685 - 9.79796i) q^{79} +4.32957 q^{83} +4.24264 q^{85} +(-0.831025 + 1.43938i) q^{89} +(-2.44949 + 1.41421i) q^{95} +11.8519i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67}+O(q^{100}) \) 16 * q + 24 * q^25 - 32 * q^37 + 64 * q^43 - 32 * q^67

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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