LMFDB - Embedded newform 1764.2.t.c.521.6

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			521.6
Root			$0.130526 + 0.991445i$ of defining polynomial
Character	$\chi$	$=$	1764.521
Dual form			1764.2.t.c.1097.6

$q$-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 $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

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

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	$-0.111921\pi$
$5$	0.382683	+	0.662827i	0.171141	+	0.296425i	−0.767678	+	0.640836i	$0.778588\pi$
$6$		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	$-0.235842\pi$
$11$	1.73205	+	1.00000i	0.522233	+	0.301511i	−0.215615	+	0.976478i	$0.569176\pi$
$12$		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
$17$	2.77164	−	4.80062i	0.672221	−	1.16432i	−0.305052	−	0.952336i	$-0.598674\pi$
$17$	2.77164	−	4.80062i	0.672221	−	1.16432i	0.977273	−	0.211985i	$-0.0679929\pi$
$18$		0			0
$19$	3.20041	−	1.84776i	0.734225	−	0.423905i	−0.0857408	−	0.996317i	$-0.527326\pi$
$19$	3.20041	−	1.84776i	0.734225	−	0.423905i	0.819966	+	0.572412i	$0.193992\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.74666	+	1.58579i	−0.572719	+	0.330659i	−0.758234	−	0.651982i	$-0.773938\pi$
$23$	−2.74666	+	1.58579i	−0.572719	+	0.330659i	0.185516	+	0.982641i	$0.440604\pi$
$24$		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	−0.128071	−	0.991765i	$-0.540879\pi$
$31$	−5.85172	−	3.37849i	−1.05100	−	0.606795i	−0.922929	+	0.384970i	$0.874212\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.121320	+	0.210133i	0.0199449	+	0.0345457i	0.875826	−	0.482628i	$-0.160318\pi$
$37$	0.121320	+	0.210133i	0.0199449	+	0.0345457i	−0.855881	+	0.517173i	$0.826984\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.74444			0.428610			0.214305	−	0.976767i	$-0.431251\pi$
$41$	2.74444			0.428610			0.214305	+	0.976767i	$0.431251\pi$
$42$		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	0.857859	+	0.513886i	$0.171794\pi$
$47$	5.99162	+	10.3778i	0.873967	+	1.51376i	0.0161088	+	0.999870i	$0.494872\pi$
$48$		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	$-0.0154019\pi$
$53$	10.6024	+	6.12132i	1.45636	+	0.840828i	0.457527	+	0.889196i	$0.348735\pi$
$54$		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	−0.00683301	−	0.999977i	$-0.502175\pi$
$59$	6.62567	−	11.4760i	0.862589	−	1.49405i	0.869422	−	0.494071i	$-0.164492\pi$
$60$		0			0
$61$	3.08669	−	1.78210i	0.395210	−	0.228175i	−0.289205	−	0.957267i	$-0.593391\pi$
$61$	3.08669	−	1.78210i	0.395210	−	0.228175i	0.684415	+	0.729093i	$0.260058\pi$
$62$		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	−0.695896	−	0.718143i	$-0.744992\pi$
$67$	2.24264	−	3.88437i	0.273982	−	0.474551i	0.969878	+	0.243592i	$0.0783257\pi$
$68$		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	−0.240470	−	0.970657i	$-0.577301\pi$
$73$	−10.2641	−	5.92596i	−1.20132	−	0.693581i	−0.960848	+	0.277075i	$0.910635\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.65685	+	9.79796i	0.636446	+	1.10236i	0.986207	+	0.165518i	$0.0529295\pi$
$79$	5.65685	+	9.79796i	0.636446	+	1.10236i	−0.349761	+	0.936839i	$0.613737\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.32957			−0.475232			−0.237616	−	0.971359i	$-0.576366\pi$
$83$	−4.32957			−0.475232			−0.237616	+	0.971359i	$0.576366\pi$
$84$		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	$-0.138591\pi$
$89$	0.831025	+	1.43938i	0.0880885	+	0.152574i	−0.818615	+	0.574343i	$0.805258\pi$
$90$		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	$-0.794494\pi$
$97$		−	11.8519i		−	1.20338i	0.798730	−	0.601690i	$-0.205506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.92596	−	10.2641i	0.589655	−	1.02131i	−0.404622	−	0.914484i	$-0.632597\pi$
$101$	5.92596	−	10.2641i	0.589655	−	1.02131i	0.994277	−	0.106829i	$-0.0340697\pi$
$102$		0			0
$103$	7.72648	−	4.46088i	0.761313	−	0.439544i	−0.0684542	−	0.997654i	$-0.521807\pi$
$103$	7.72648	−	4.46088i	0.761313	−	0.439544i	0.829767	+	0.558110i	$0.188473\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.21076	+	3.58579i	−0.600417	+	0.346651i	−0.769206	−	0.639001i	$-0.779348\pi$
$107$	−6.21076	+	3.58579i	−0.600417	+	0.346651i	0.168788	+	0.985652i	$0.446015\pi$
$108$		0			0
$109$	5.53553	−	9.58783i	0.530208	−	0.918347i	−0.469171	−	0.883107i	$-0.655447\pi$
$109$	5.53553	−	9.58783i	0.530208	−	0.918347i	0.999379	−	0.0352398i	$-0.0112195\pi$
$110$		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	0.952346	+	0.305018i	$0.0986626\pi$
$131$	8.47343	+	14.6764i	0.740327	+	1.28228i	−0.212020	+	0.977265i	$0.568004\pi$
$132$		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	$-0.180262\pi$
$137$	9.37769	+	5.41421i	0.801190	+	0.462567i	−0.0426968	+	0.999088i	$0.513595\pi$
$138$		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
$143$	0.317025	−	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	$-0.573130\pi$
$149$	6.12372	−	3.53553i	0.501675	−	0.289642i	0.729405	+	0.684082i	$0.239797\pi$
$150$		0			0
$151$	−5.07107	+	8.78335i	−0.412678	+	0.714779i	−0.995182	−	0.0980492i	$-0.968740\pi$
$151$	−5.07107	+	8.78335i	−0.412678	+	0.714779i	0.582504	+	0.812828i	$0.302073\pi$
$152$		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	0.271094	−	0.962553i	$-0.412615\pi$
$157$	−5.34972	−	3.08866i	−0.426954	−	0.246502i	−0.698048	+	0.716051i	$0.745948\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.41421	−	2.44949i	−0.110770	−	0.191859i	0.805311	−	0.592852i	$-0.201998\pi$
$163$	−1.41421	−	2.44949i	−0.110770	−	0.191859i	−0.916081	+	0.400994i	$0.868665\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.42742			−0.187839			−0.0939196	−	0.995580i	$-0.529940\pi$
$167$	−2.42742			−0.187839			−0.0939196	+	0.995580i	$0.529940\pi$
$168$		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	−0.963166	−	0.268908i	$-0.913337\pi$
$173$	−9.39731	−	16.2766i	−0.714464	−	1.23749i	0.248702	−	0.968580i	$-0.419996\pi$
$174$		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	$-0.446496\pi$
$179$	−8.06591	−	4.65685i	−0.602874	−	0.348070i	−0.770171	+	0.637837i	$0.779829\pi$
$180$		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
$185$	−0.0928546	+	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	$-0.882134\pi$
$191$	−14.9941	+	8.65685i	−1.08494	+	0.626388i	−0.152712	+	0.988271i	$0.548801\pi$
$192$		0			0
$193$	4.82843	−	8.36308i	0.347558	−	0.601988i	−0.638257	−	0.769823i	$-0.720344\pi$
$193$	4.82843	−	8.36308i	0.347558	−	0.601988i	0.985815	+	0.167835i	$0.0536777\pi$
$194$		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	0.709409	−	0.704797i	$-0.248962\pi$
$199$	6.40083	+	3.69552i	0.453742	+	0.261968i	−0.255667	+	0.966765i	$0.582295\pi$
$200$		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	$-0.739532\pi$
$223$		−	21.8017i		−	1.45995i	0.683474	−	0.729975i	$-0.260468\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.37849	+	5.85172i	−0.224238	+	0.388392i	−0.956091	−	0.293071i	$-0.905323\pi$
$227$	−3.37849	+	5.85172i	−0.224238	+	0.388392i	0.731852	+	0.681463i	$0.238656\pi$
$228$		0			0
$229$	−14.0136	+	8.09075i	−0.926044	+	0.534651i	−0.885558	−	0.464529i	$-0.846224\pi$
$229$	−14.0136	+	8.09075i	−0.926044	+	0.534651i	−0.0404854	+	0.999180i	$0.512890\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.1390	−	7.58579i	0.860762	−	0.496961i	−0.00350513	−	0.999994i	$-0.501116\pi$
$233$	13.1390	−	7.58579i	0.860762	−	0.496961i	0.864268	+	0.503032i	$0.167782\pi$
$234$		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	0.996754	−	0.0805055i	$-0.0256535\pi$
$241$	22.1283	+	12.7758i	1.42541	+	0.822962i	0.428657	+	0.903467i	$0.358987\pi$
$242$		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	$-0.797666\pi$
$251$	−25.4972			−1.60937			−0.804685	+	0.593702i	$0.797666\pi$
$252$		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	−0.984958	−	0.172792i	$-0.944721\pi$
$257$	−10.2940	−	17.8297i	−0.642122	−	1.11219i	0.342837	−	0.939395i	$-0.388612\pi$
$258$		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	$-0.540579\pi$
$263$	−17.0233	−	9.82843i	−1.04970	−	0.606047i	−0.922566	+	0.385838i	$0.873912\pi$
$264$		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	0.452939	+	0.891542i	$0.350376\pi$
$269$	−8.94897	+	15.5001i	−0.545628	+	0.945056i	−0.998567	+	0.0535141i	$0.982958\pi$
$270$		0			0
$271$	4.75351	−	2.74444i	0.288755	−	0.166713i	−0.348625	−	0.937262i	$-0.613351\pi$
$271$	4.75351	−	2.74444i	0.288755	−	0.166713i	0.637380	+	0.770549i	$0.280018\pi$
$272$		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	−0.992404	−	0.123021i	$-0.960742\pi$
$277$	−6.48528	+	11.2328i	−0.389663	+	0.674916i	0.602741	+	0.797937i	$0.294075\pi$
$278$		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	0.391651	−	0.920114i	$-0.371904\pi$
$283$	−3.52207	−	2.03347i	−0.209365	−	0.120877i	−0.601016	+	0.799237i	$0.705237\pi$
$284$		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	$-0.340514\pi$
$293$	16.4441			0.960676			0.480338	+	0.877083i	$0.340514\pi$
$294$		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	$0.281942\pi$
$307$			27.1367i			1.54877i	−0.632712	+	0.774387i	$0.718058\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.3617	+	26.6073i	−0.871084	+	1.50876i	−0.0102070	+	0.999948i	$0.503249\pi$
$311$	−15.3617	+	26.6073i	−0.871084	+	1.50876i	−0.860877	+	0.508814i	$0.830084\pi$
$312$		0			0
$313$	−25.0071	+	14.4379i	−1.41348	+	0.816076i	−0.995715	−	0.0924774i	$-0.970521\pi$
$313$	−25.0071	+	14.4379i	−1.41348	+	0.816076i	−0.417770	+	0.908553i	$0.637188\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.6463	−	7.87868i	0.766451	−	0.442511i	−0.0651561	−	0.997875i	$-0.520755\pi$
$317$	13.6463	−	7.87868i	0.766451	−	0.442511i	0.831607	+	0.555364i	$0.187421\pi$
$318$		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	0.947509	+	0.319730i	$0.103592\pi$
$331$	13.6569	+	23.6544i	0.750649	+	1.30016i	−0.196860	+	0.980432i	$0.563074\pi$
$332$		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	$-0.628419\pi$
$347$	−25.8067	−	14.8995i	−1.38538	−	0.799847i	−0.992790	+	0.119869i	$0.961753\pi$
$348$		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
$353$	−10.5726	+	18.3122i	−0.562720	+	0.974660i	0.434537	+	0.900654i	$0.356912\pi$
$353$	−10.5726	+	18.3122i	−0.562720	+	0.974660i	−0.997258	+	0.0740064i	$0.976421\pi$
$354$		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	$-0.838288\pi$
$359$	−16.8493	+	9.72792i	−0.889270	+	0.513420i	−0.0155661	+	0.999879i	$0.504955\pi$
$360$		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	0.984071	+	0.177778i	$0.0568909\pi$
$367$	31.2276	+	18.0292i	1.63007	+	0.941119i	0.645996	+	0.763341i	$0.276442\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−15.3137	−	26.5241i	−0.792914	−	1.37337i	−0.924155	−	0.382017i	$-0.875230\pi$
$373$	−15.3137	−	26.5241i	−0.792914	−	1.37337i	0.131242	−	0.991350i	$-0.458104\pi$
$374$		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	$-0.289939\pi$
$383$	−7.39104	−	12.8017i	−0.377664	−	0.654134i	−0.990722	+	0.135904i	$0.956606\pi$
$384$		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	$-0.342789\pi$
$389$	−1.01461	−	0.585786i	−0.0514429	−	0.0297006i	−0.525501	+	0.850793i	$0.676122\pi$
$390$		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	−0.772664	−	0.634815i	$-0.781076\pi$
$397$	−12.1388	+	7.00835i	−0.609230	+	0.351739i	0.163434	+	0.986554i	$0.447743\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.77589	−	2.75736i	0.238496	−	0.137696i	−0.375989	−	0.926624i	$-0.622697\pi$
$401$	4.77589	−	2.75736i	0.238496	−	0.137696i	0.614485	+	0.788928i	$0.289364\pi$
$402$		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	0.475463	−	0.879736i	$-0.342281\pi$
$409$	−0.984485	−	0.568393i	−0.0486796	−	0.0281052i	−0.524142	+	0.851631i	$0.675614\pi$
$410$		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	$-0.736154\pi$
$419$	−27.6620			−1.35138			−0.675688	+	0.737187i	$0.736154\pi$
$420$		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	$-0.629741\pi$
$431$	−28.8505	−	16.6569i	−1.38968	−	0.802332i	−0.993279	+	0.115745i	$0.963075\pi$
$432$		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
$437$	−5.86030	+	10.1503i	−0.280336	+	0.485557i
$438$		0			0
$439$	12.0251	−	6.94269i	0.573927	−	0.331357i	−0.184789	−	0.982778i	$-0.559160\pi$
$439$	12.0251	−	6.94269i	0.573927	−	0.331357i	0.758716	+	0.651421i	$0.225827\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	25.3864	−	14.6569i	1.20615	−	0.696368i	0.244230	−	0.969717i	$-0.421465\pi$
$443$	25.3864	−	14.6569i	1.20615	−	0.696368i	0.961915	+	0.273349i	$0.0881314\pi$
$444$		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	0.999857	+	0.0168929i	$0.00537742\pi$
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	−0.485299	+	0.874348i	$0.661289\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.17186			−0.240877			−0.120439	−	0.992721i	$-0.538430\pi$
$461$	−5.17186			−0.240877			−0.120439	+	0.992721i	$0.538430\pi$
$462$		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	$-0.120611\pi$
$467$	3.11586	+	5.39683i	0.144185	+	0.249735i	−0.784884	+	0.619643i	$0.787277\pi$
$468$		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	0.466213	+	0.884673i	$0.345618\pi$
$479$	−11.6662	+	20.2065i	−0.533043	+	0.923257i	−0.999255	+	0.0385845i	$0.987715\pi$
$480$		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	−0.998294	−	0.0583797i	$-0.981407\pi$
$487$	−9.89949	+	17.1464i	−0.448589	+	0.776979i	0.549706	+	0.835359i	$0.314740\pi$
$488$		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	−0.994858	−	0.101282i	$-0.967706\pi$
$499$	−13.0711	−	22.6398i	−0.585141	−	1.01349i	0.409716	−	0.912213i	$-0.365628\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	29.5641			1.31820			0.659100	−	0.752055i	$-0.270937\pi$
$503$	29.5641			1.31820			0.659100	+	0.752055i	$0.270937\pi$
$504$		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	$-0.259209\pi$
$509$	−6.46716	−	11.2014i	−0.286652	−	0.496495i	−0.973008	+	0.230770i	$0.925876\pi$
$510$		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	$-0.819150\pi$
$521$	1.01673	−	1.76104i	0.0445439	−	0.0771523i	0.887438	+	0.460928i	$0.152483\pi$
$522$		0			0
$523$	−15.7746	+	9.10748i	−0.689776	+	0.398242i	−0.803528	−	0.595267i	$-0.797046\pi$
$523$	−15.7746	+	9.10748i	−0.689776	+	0.398242i	0.113752	+	0.993509i	$0.463713\pi$
$524$		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	0.762308	−	0.647214i	$-0.224066\pi$
$541$	−4.17157	−	7.22538i	−0.179350	−	0.310643i	−0.941658	+	0.336571i	$0.890733\pi$
$542$		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	$-0.448646\pi$
$557$	−14.4868	−	8.36396i	−0.613826	−	0.354392i	−0.774461	+	0.632621i	$0.781979\pi$
$558$		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	$-0.679406\pi$
$563$	11.0322	−	19.1083i	0.464950	−	0.805317i	0.999199	+	0.0400098i	$0.0127389\pi$
$564$		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	$-0.663429\pi$
$569$	0.420266	−	0.242641i	0.0176185	−	0.0101720i	0.508783	+	0.860895i	$0.330095\pi$
$570$		0			0
$571$	1.31371	−	2.27541i	0.0549770	−	0.0952229i	−0.837227	−	0.546855i	$-0.815825\pi$
$571$	1.31371	−	2.27541i	0.0549770	−	0.0952229i	0.892204	+	0.451632i	$0.149158\pi$
$572$		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	0.0968824	−	0.995296i	$-0.469113\pi$
$577$	−17.2140	−	9.93850i	−0.716628	−	0.413745i	−0.813510	+	0.581551i	$0.802446\pi$
$578$		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	$-0.696374\pi$
$587$	−28.0334			−1.15706			−0.578531	+	0.815660i	$0.696374\pi$
$588$		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	−0.987153	−	0.159777i	$-0.948922\pi$
$593$	−15.3889	−	26.6544i	−0.631947	−	1.09457i	0.355206	−	0.934788i	$-0.384411\pi$
$594$		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	$-0.272396\pi$
$599$	8.06591	+	4.65685i	0.329564	+	0.190274i	−0.326083	+	0.945341i	$0.605729\pi$
$600$		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
$605$	2.67878	−	4.63979i	0.108908	−	0.188634i
$606$		0			0
$607$	10.9269	−	6.30864i	0.443509	−	0.256060i	−0.261576	−	0.965183i	$-0.584242\pi$
$607$	10.9269	−	6.30864i	0.443509	−	0.256060i	0.705085	+	0.709123i	$0.250909\pi$
$608$		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	0.103372	−	0.994643i	$-0.467037\pi$
$613$	22.6066	−	39.1558i	0.913072	−	1.58149i	0.809700	−	0.586844i	$-0.199630\pi$
$614$		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	0.468463	−	0.883483i	$-0.344808\pi$
$619$	−1.55310	−	0.896683i	−0.0624244	−	0.0360407i	−0.530887	+	0.847442i	$0.678141\pi$
$620$		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	$-0.486641\pi$
$641$	−20.3134	−	11.7279i	−0.802329	−	0.463225i	−0.844285	+	0.535894i	$0.819975\pi$
$642$		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
$647$	−3.19278	+	5.53006i	−0.125521	+	0.217409i	−0.921937	−	0.387341i	$-0.873394\pi$
$647$	−3.19278	+	5.53006i	−0.125521	+	0.217409i	0.796415	+	0.604750i	$0.206727\pi$
$648$		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	$-0.630102\pi$
$653$	5.07306	−	2.92893i	0.198524	−	0.114618i	0.595967	+	0.803009i	$0.296769\pi$
$654$		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	0.0514050	−	0.998678i	$-0.483630\pi$
$661$	−20.2536	−	11.6934i	−0.787773	−	0.454821i	−0.839178	+	0.543857i	$0.816963\pi$
$662$		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	0.999029	+	0.0440470i	$0.0140251\pi$
$677$	13.9895	+	24.2305i	0.537661	+	0.931255i	−0.461369	+	0.887208i	$0.652642\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	40.6777	+	23.4853i	1.55649	+	0.898639i	0.997589	+	0.0694045i	$0.0221099\pi$
$683$	40.6777	+	23.4853i	1.55649	+	0.898639i	0.558900	+	0.829235i	$0.311223\pi$
$684$		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	−0.0148906	−	0.999889i	$-0.504740\pi$
$691$	22.1754	−	12.8030i	0.843594	−	0.487049i	0.858484	+	0.512840i	$0.171407\pi$
$692$		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	0.995516	+	0.0945890i	$0.0301537\pi$
$709$	15.4350	+	26.7343i	0.579675	+	1.00403i	−0.415842	+	0.909437i	$0.636513\pi$
$710$		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	$-0.0473367\pi$
$719$	9.81845	+	17.0061i	0.366167	+	0.634219i	−0.622796	+	0.782384i	$0.714003\pi$
$720$		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	$-0.522329\pi$
$727$		−	53.7933i		−	1.99508i	0.0700903	−	0.997541i	$-0.477671\pi$
$728$		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	−0.812341	−	0.583183i	$-0.801807\pi$
$733$	−19.3162	+	11.1522i	−0.713460	+	0.411916i	0.0988808	+	0.995099i	$0.468474\pi$
$734$		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	−0.888474	−	0.458927i	$-0.848234\pi$
$739$	−1.27208	+	2.20330i	−0.0467941	+	0.0810498i	0.841680	+	0.539977i	$0.181567\pi$
$740$		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	0.980085	+	0.198578i	$0.0636322\pi$
$751$	18.1421	+	31.4231i	0.662016	+	1.14665i	−0.318069	+	0.948067i	$0.603034\pi$
$752$		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	0.687570	+	0.726118i	$0.258678\pi$
$761$	26.8310	+	46.4726i	0.972622	+	1.68463i	0.285052	+	0.958512i	$0.407989\pi$
$762$		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	$0.0828866\pi$
$769$			14.2793i			0.514926i	−0.966288	+	0.257463i	$0.917113\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16.7883	−	29.0783i	0.603835	−	1.04587i	−0.388400	−	0.921491i	$-0.626972\pi$
$773$	16.7883	−	29.0783i	0.603835	−	1.04587i	0.992234	−	0.124381i	$-0.0396947\pi$
$774$		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	−0.997448	−	0.0714009i	$-0.977253\pi$
$787$	−43.7076	−	25.2346i	−1.55801	−	0.899515i	−0.560559	−	0.828115i	$-0.689414\pi$
$788$		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	$-0.654687\pi$
$797$	−26.3714			−0.934122			−0.467061	+	0.884225i	$0.654687\pi$
$798$		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	$-0.211694\pi$
$809$	18.3712	+	10.6066i	0.645896	+	0.372908i	−0.140986	+	0.990012i	$0.545027\pi$
$810$		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
$815$	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	−0.544990	+	0.838443i	$0.683466\pi$
$821$	−44.2288	+	25.5355i	−1.54360	+	0.891196i	−0.998608	+	0.0527535i	$0.983200\pi$
$822$		0			0
$823$	20.1421	−	34.8872i	0.702111	−	1.21609i	−0.265613	−	0.964080i	$-0.585574\pi$
$823$	20.1421	−	34.8872i	0.702111	−	1.21609i	0.967724	−	0.252012i	$-0.0810922\pi$
$824$		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	−0.137371	−	0.990520i	$-0.543865\pi$
$829$	−30.6314	−	17.6850i	−1.06387	−	0.614226i	−0.926501	+	0.376293i	$0.877198\pi$
$830$		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	$-0.307828\pi$
$839$	32.8882			1.13543			0.567714	+	0.823226i	$0.307828\pi$
$840$		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	$-0.562763\pi$
$853$		−	57.2805i		−	1.96125i	0.195899	−	0.980624i	$-0.437237\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.8114	−	34.3143i	0.676743	−	1.17215i	−0.299213	−	0.954186i	$-0.596724\pi$
$857$	19.8114	−	34.3143i	0.676743	−	1.17215i	0.975956	−	0.217967i	$-0.0699424\pi$
$858$		0			0
$859$	11.9309	−	6.88830i	0.407077	−	0.235026i	−0.282456	−	0.959280i	$-0.591149\pi$
$859$	11.9309	−	6.88830i	0.407077	−	0.235026i	0.689533	+	0.724254i	$0.257816\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	21.7482	−	12.5563i	0.740319	−	0.427423i	−0.0818666	−	0.996643i	$-0.526088\pi$
$863$	21.7482	−	12.5563i	0.740319	−	0.427423i	0.822185	+	0.569220i	$0.192755\pi$
$864$		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	0.858077	−	0.513521i	$-0.171659\pi$
$877$	−0.464466	−	0.804479i	−0.0156839	−	0.0271653i	−0.873761	+	0.486356i	$0.838326\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−22.1187			−0.745198			−0.372599	−	0.927992i	$-0.621533\pi$
$881$	−22.1187			−0.745198			−0.372599	+	0.927992i	$0.621533\pi$
$882$		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	$-0.0582409\pi$
$887$	9.94977	+	17.2335i	0.334081	+	0.578644i	−0.649227	+	0.760595i	$0.724908\pi$
$888$		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	−0.459612	−	0.888120i	$-0.652012\pi$
$907$	16.2426	−	28.1331i	0.539328	−	0.934144i	0.998940	−	0.0460239i	$-0.0146550\pi$
$908$		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	0.965137	−	0.261744i	$-0.0842976\pi$
$919$	7.75736	+	13.4361i	0.255892	+	0.443217i	−0.709246	+	0.704961i	$0.750964\pi$
$920$		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	$-0.0474305\pi$
$929$	11.1522	+	19.3162i	0.365892	+	0.633744i	−0.623027	+	0.782201i	$0.714097\pi$
$930$		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	$0.0869773\pi$
$937$			16.5210i			0.539719i	−0.962900	+	0.269860i	$0.913023\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.18651	+	2.05510i	−0.0386792	+	0.0669943i	−0.884717	−	0.466129i	$-0.845648\pi$
$941$	−1.18651	+	2.05510i	−0.0386792	+	0.0669943i	0.846038	+	0.533123i	$0.178982\pi$
$942$		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	$-0.712484\pi$
$947$	−7.64564	+	4.41421i	−0.248450	+	0.143443i	0.370604	+	0.928791i	$0.379151\pi$
$948$		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	$-0.0339702\pi$
$971$	12.6173	+	21.8538i	0.404908	+	0.701321i	−0.589403	+	0.807839i	$0.700637\pi$
$972$		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	$-0.335804\pi$
$977$	−0.420266	−	0.242641i	−0.0134455	−	0.00776276i	−0.506708	+	0.862118i	$0.669138\pi$
$978$		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	$-0.797625\pi$
$983$	3.50981	−	6.07917i	0.111946	−	0.193895i	0.916555	+	0.399910i	$0.130958\pi$
$984$		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	−0.777058	−	0.629429i	$-0.783289\pi$
$991$	4.92893	−	8.53716i	0.156573	−	0.271192i	0.933631	+	0.358237i	$0.116622\pi$
$992$		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	0.696922	−	0.717147i	$-0.254553\pi$
$997$	13.3979	+	7.73526i	0.424314	+	0.244978i	−0.272607	+	0.962125i	$0.587886\pi$
$998$		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.521.6		16
3.2	odd	2	inner	1764.2.t.c.521.3		16
7.2	even	3	inner	1764.2.t.c.1097.5		16
7.3	odd	6		1764.2.f.b.881.5	yes	8
7.4	even	3		1764.2.f.b.881.3	✓	8
7.5	odd	6	inner	1764.2.t.c.1097.3		16
7.6	odd	2	inner	1764.2.t.c.521.4		16
21.2	odd	6	inner	1764.2.t.c.1097.4		16
21.5	even	6	inner	1764.2.t.c.1097.6		16
21.11	odd	6		1764.2.f.b.881.6	yes	8
21.17	even	6		1764.2.f.b.881.4	yes	8
21.20	even	2	inner	1764.2.t.c.521.5		16
28.3	even	6		7056.2.k.e.881.6		8
28.11	odd	6		7056.2.k.e.881.4		8
84.11	even	6		7056.2.k.e.881.5		8
84.59	odd	6		7056.2.k.e.881.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.f.b.881.3	✓	8	7.4	even	3
1764.2.f.b.881.4	yes	8	21.17	even	6
1764.2.f.b.881.5	yes	8	7.3	odd	6
1764.2.f.b.881.6	yes	8	21.11	odd	6
1764.2.t.c.521.3		16	3.2	odd	2	inner
1764.2.t.c.521.4		16	7.6	odd	2	inner
1764.2.t.c.521.5		16	21.20	even	2	inner
1764.2.t.c.521.6		16	1.1	even	1	trivial
1764.2.t.c.1097.3		16	7.5	odd	6	inner
1764.2.t.c.1097.4		16	21.2	odd	6	inner
1764.2.t.c.1097.5		16	7.2	even	3	inner
1764.2.t.c.1097.6		16	21.5	even	6	inner
7056.2.k.e.881.3		8	84.59	odd	6
7056.2.k.e.881.4		8	28.11	odd	6
7056.2.k.e.881.5		8	84.11	even	6
7056.2.k.e.881.6		8	28.3	even	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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