LMFDB - Embedded newform 5472.2.g.b.2737.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(2737,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5472.2737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.g (of order $2$, degree $1$, 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:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2737.7
Root			$1.12629 - 0.855255i$ of defining polynomial
Character	$\chi$	$=$	5472.2737
Dual form			5472.2.g.b.2737.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.51356i q^{5} -0.580162 q^{7} +O(q^{10})$	$q-1.51356i q^{5} -0.580162 q^{7} -1.31799i q^{11} +3.89230i q^{13} +1.20142 q^{17} -1.00000i q^{19} +5.85527 q^{23} +2.70913 q^{25} -1.29188i q^{29} -2.96413 q^{31} +0.878110i q^{35} -1.18418i q^{37} +9.04577 q^{41} -8.38816i q^{43} -12.8560 q^{47} -6.66341 q^{49} +3.07183i q^{53} -1.99485 q^{55} +0.258163i q^{59} +14.7200i q^{61} +5.89123 q^{65} +9.54884i q^{67} -6.93697 q^{71} +15.2934 q^{73} +0.764646i q^{77} +13.1332 q^{79} -3.70615i q^{83} -1.81843i q^{85} +8.36653 q^{89} -2.25816i q^{91} -1.51356 q^{95} +17.0442 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7}+O(q^{10}) $ 16 * q + 8 * q^7	$ 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) $ 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	− 1.51356i	− 0.676885i	−0.940987	−	0.338442i	$-0.890100\pi$
$5$	− 1.51356i	− 0.676885i	0.940987	−	0.338442i	$-0.109900\pi$
$6$	0	0
$7$	−0.580162	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$7$	−0.580162	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.31799i	− 0.397388i	−0.980062	−	0.198694i	$-0.936330\pi$
$11$	− 1.31799i	− 0.397388i	0.980062	−	0.198694i	$-0.0636700\pi$
$12$	0	0
$13$	3.89230i	1.07953i	0.841816	+	0.539765i	$0.181487\pi$
$13$	3.89230i	1.07953i	−0.841816	+	0.539765i	$0.818513\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.20142	0.291388	0.145694	−	0.989330i	$-0.453459\pi$
$17$	1.20142	0.291388	0.145694	+	0.989330i	$0.453459\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.85527	1.22091	0.610454	−	0.792052i	$-0.290987\pi$
$23$	5.85527	1.22091	0.610454	+	0.792052i	$0.290987\pi$
$24$	0	0
$25$	2.70913	0.541827
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.29188i	− 0.239896i	−0.992780	−	0.119948i	$-0.961727\pi$
$29$	− 1.29188i	− 0.239896i	0.992780	−	0.119948i	$-0.0382727\pi$
$30$	0	0
$31$	−2.96413	−0.532374	−0.266187	−	0.963921i	$-0.585764\pi$
$31$	−2.96413	−0.532374	−0.266187	+	0.963921i	$0.585764\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.878110i	0.148428i
$36$	0	0
$37$	− 1.18418i	− 0.194677i	−0.995251	−	0.0973387i	$-0.968967\pi$
$37$	− 1.18418i	− 0.194677i	0.995251	−	0.0973387i	$-0.0310330\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.04577	1.41271	0.706356	−	0.707856i	$-0.250338\pi$
$41$	9.04577	1.41271	0.706356	+	0.707856i	$0.250338\pi$
$42$	0	0
$43$	− 8.38816i	− 1.27918i	−0.768715	−	0.639592i	$-0.779104\pi$
$43$	− 8.38816i	− 1.27918i	0.768715	−	0.639592i	$-0.220896\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.8560	−1.87524	−0.937620	−	0.347661i	$-0.886976\pi$
$47$	−12.8560	−1.87524	−0.937620	+	0.347661i	$0.886976\pi$
$48$	0	0
$49$	−6.66341	−0.951916
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.07183i	0.421949i	0.977492	+	0.210974i	$0.0676636\pi$
$53$	3.07183i	0.421949i	−0.977492	+	0.210974i	$0.932336\pi$
$54$	0	0
$55$	−1.99485	−0.268986
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.258163i	0.0336100i	0.999859	+	0.0168050i	$0.00534945\pi$
$59$	0.258163i	0.0336100i	−0.999859	+	0.0168050i	$0.994651\pi$
$60$	0	0
$61$	14.7200i	1.88470i	0.334623	+	0.942352i	$0.391391\pi$
$61$	14.7200i	1.88470i	−0.334623	+	0.942352i	$0.608609\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.89123	0.730717
$66$	0	0
$67$	9.54884i	1.16658i	0.812265	+	0.583288i	$0.198234\pi$
$67$	9.54884i	1.16658i	−0.812265	+	0.583288i	$0.801766\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.93697	−0.823267	−0.411634	−	0.911349i	$-0.635042\pi$
$71$	−6.93697	−0.823267	−0.411634	+	0.911349i	$0.635042\pi$
$72$	0	0
$73$	15.2934	1.78996	0.894978	−	0.446110i	$-0.147191\pi$
$73$	15.2934	1.78996	0.894978	+	0.446110i	$0.147191\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.764646i	0.0871395i
$78$	0	0
$79$	13.1332	1.47760	0.738798	−	0.673927i	$-0.235394\pi$
$79$	13.1332	1.47760	0.738798	+	0.673927i	$0.235394\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.70615i	− 0.406803i	−0.979095	−	0.203402i	$-0.934800\pi$
$83$	− 3.70615i	− 0.406803i	0.979095	−	0.203402i	$-0.0651997\pi$
$84$	0	0
$85$	− 1.81843i	− 0.197236i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.36653	0.886850	0.443425	−	0.896311i	$-0.353763\pi$
$89$	8.36653	0.886850	0.443425	+	0.896311i	$0.353763\pi$
$90$	0	0
$91$	− 2.25816i	− 0.236720i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.51356	−0.155288
$96$	0	0
$97$	17.0442	1.73057	0.865286	−	0.501278i	$-0.167137\pi$
$97$	17.0442	1.73057	0.865286	+	0.501278i	$0.167137\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.97721i	0.196739i	0.995150	+	0.0983697i	$0.0313628\pi$
$101$	1.97721i	0.196739i	−0.995150	+	0.0983697i	$0.968637\pi$
$102$	0	0
$103$	2.30853	0.227466	0.113733	−	0.993511i	$-0.463719\pi$
$103$	2.30853	0.227466	0.113733	+	0.993511i	$0.463719\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.0396i	− 1.45393i	−0.686675	−	0.726965i	$-0.740930\pi$
$107$	− 15.0396i	− 1.45393i	0.686675	−	0.726965i	$-0.259070\pi$
$108$	0	0
$109$	− 13.4002i	− 1.28351i	−0.766911	−	0.641753i	$-0.778207\pi$
$109$	− 13.4002i	− 1.28351i	0.766911	−	0.641753i	$-0.221793\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.89717	−0.460687	−0.230343	−	0.973109i	$-0.573985\pi$
$113$	−4.89717	−0.460687	−0.230343	+	0.973109i	$0.573985\pi$
$114$	0	0
$115$	− 8.86230i	− 0.826414i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.697020	−0.0638957
$120$	0	0
$121$	9.26291	0.842083
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.6682i	− 1.04364i
$126$	0	0
$127$	2.61625	0.232155	0.116078	−	0.993240i	$-0.462968\pi$
$127$	2.61625	0.232155	0.116078	+	0.993240i	$0.462968\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.64498i	0.231093i	0.993302	+	0.115547i	$0.0368620\pi$
$131$	2.64498i	0.231093i	−0.993302	+	0.115547i	$0.963138\pi$
$132$	0	0
$133$	0.580162i	0.0503064i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.88035	−0.416957	−0.208478	−	0.978027i	$-0.566851\pi$
$137$	−4.88035	−0.416957	−0.208478	+	0.978027i	$0.566851\pi$
$138$	0	0
$139$	− 10.3334i	− 0.876468i	−0.898861	−	0.438234i	$-0.855604\pi$
$139$	− 10.3334i	− 0.876468i	0.898861	−	0.438234i	$-0.144396\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.13000	0.428992
$144$	0	0
$145$	−1.95533	−0.162382
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.35324i	0.684324i	0.939641	+	0.342162i	$0.111159\pi$
$149$	8.35324i	0.684324i	−0.939641	+	0.342162i	$0.888841\pi$
$150$	0	0
$151$	5.84941	0.476019	0.238009	−	0.971263i	$-0.423505\pi$
$151$	5.84941	0.476019	0.238009	+	0.971263i	$0.423505\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.48639i	0.360356i
$156$	0	0
$157$	− 0.361645i	− 0.0288624i	−0.999896	−	0.0144312i	$-0.995406\pi$
$157$	− 0.361645i	− 0.0288624i	0.999896	−	0.0144312i	$-0.00459375\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.39700	−0.267721
$162$	0	0
$163$	− 0.146029i	− 0.0114379i	−0.999984	−	0.00571894i	$-0.998180\pi$
$163$	− 0.146029i	− 0.0114379i	0.999984	−	0.00571894i	$-0.00182041\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.406603	0.0314639	0.0157319	−	0.999876i	$-0.494992\pi$
$167$	0.406603	0.0314639	0.0157319	+	0.999876i	$0.494992\pi$
$168$	0	0
$169$	−2.14999	−0.165384
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 12.6466i	− 0.961507i	−0.876856	−	0.480753i	$-0.840363\pi$
$173$	− 12.6466i	− 0.961507i	0.876856	−	0.480753i	$-0.159637\pi$
$174$	0	0
$175$	−1.57174	−0.118812
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 5.55598i	− 0.415274i	−0.978206	−	0.207637i	$-0.933423\pi$
$179$	− 5.55598i	− 0.415274i	0.978206	−	0.207637i	$-0.0665772\pi$
$180$	0	0
$181$	− 9.13401i	− 0.678926i	−0.940619	−	0.339463i	$-0.889755\pi$
$181$	− 9.13401i	− 0.678926i	0.940619	−	0.339463i	$-0.110245\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.79232	−0.131774
$186$	0	0
$187$	− 1.58346i	− 0.115794i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.47532	0.396180	0.198090	−	0.980184i	$-0.436526\pi$
$191$	5.47532	0.396180	0.198090	+	0.980184i	$0.436526\pi$
$192$	0	0
$193$	16.9697	1.22151	0.610753	−	0.791821i	$-0.290867\pi$
$193$	16.9697	1.22151	0.610753	+	0.791821i	$0.290867\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.7727i	1.69374i	0.531803	+	0.846868i	$0.321515\pi$
$197$	23.7727i	1.69374i	−0.531803	+	0.846868i	$0.678485\pi$
$198$	0	0
$199$	17.5747	1.24584	0.622918	−	0.782287i	$-0.285947\pi$
$199$	17.5747	1.24584	0.622918	+	0.782287i	$0.285947\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.749498i	0.0526044i
$204$	0	0
$205$	− 13.6913i	− 0.956244i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.31799	−0.0911671
$210$	0	0
$211$	− 17.2804i	− 1.18963i	−0.803863	−	0.594815i	$-0.797225\pi$
$211$	− 17.2804i	− 1.18963i	0.803863	−	0.594815i	$-0.202775\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.6960	−0.865860
$216$	0	0
$217$	1.71968	0.116739
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.67630i	0.314562i
$222$	0	0
$223$	−11.6027	−0.776973	−0.388487	−	0.921454i	$-0.627002\pi$
$223$	−11.6027	−0.776973	−0.388487	+	0.921454i	$0.627002\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 27.4484i	− 1.82182i	−0.412609	−	0.910908i	$-0.635382\pi$
$227$	− 27.4484i	− 1.82182i	0.412609	−	0.910908i	$-0.364618\pi$
$228$	0	0
$229$	− 21.8680i	− 1.44508i	−0.691329	−	0.722540i	$-0.742975\pi$
$229$	− 21.8680i	− 1.44508i	0.691329	−	0.722540i	$-0.257025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.1975	1.45421	0.727103	−	0.686528i	$-0.240866\pi$
$233$	22.1975	1.45421	0.727103	+	0.686528i	$0.240866\pi$
$234$	0	0
$235$	19.4583i	1.26932i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.1350	1.36711	0.683555	−	0.729899i	$-0.260433\pi$
$239$	21.1350	1.36711	0.683555	+	0.729899i	$0.260433\pi$
$240$	0	0
$241$	−17.2437	−1.11076	−0.555382	−	0.831595i	$-0.687428\pi$
$241$	−17.2437	−1.11076	−0.555382	+	0.831595i	$0.687428\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10.0855i	0.644338i
$246$	0	0
$247$	3.89230	0.247661
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.08851i	− 0.384303i	−0.981365	−	0.192152i	$-0.938453\pi$
$251$	− 6.08851i	− 0.384303i	0.981365	−	0.192152i	$-0.0615465\pi$
$252$	0	0
$253$	− 7.71717i	− 0.485174i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.1741	−0.634643	−0.317322	−	0.948318i	$-0.602783\pi$
$257$	−10.1741	−0.634643	−0.317322	+	0.948318i	$0.602783\pi$
$258$	0	0
$259$	0.687014i	0.0426890i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.8414	−0.976822	−0.488411	−	0.872614i	$-0.662423\pi$
$263$	−15.8414	−0.976822	−0.488411	+	0.872614i	$0.662423\pi$
$264$	0	0
$265$	4.64940	0.285611
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 22.8533i	− 1.39339i	−0.717367	−	0.696695i	$-0.754653\pi$
$269$	− 22.8533i	− 1.39339i	0.717367	−	0.696695i	$-0.245347\pi$
$270$	0	0
$271$	−12.2830	−0.746142	−0.373071	−	0.927803i	$-0.621695\pi$
$271$	−12.2830	−0.746142	−0.373071	+	0.927803i	$0.621695\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.57060i	− 0.215316i
$276$	0	0
$277$	0.643776i	0.0386808i	0.999813	+	0.0193404i	$0.00615662\pi$
$277$	0.643776i	0.0386808i	−0.999813	+	0.0193404i	$0.993843\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.93027	0.532735	0.266368	−	0.963872i	$-0.414177\pi$
$281$	8.93027	0.532735	0.266368	+	0.963872i	$0.414177\pi$
$282$	0	0
$283$	− 14.7397i	− 0.876187i	−0.898929	−	0.438094i	$-0.855654\pi$
$283$	− 14.7397i	− 0.876187i	0.898929	−	0.438094i	$-0.144346\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.24801	−0.309780
$288$	0	0
$289$	−15.5566	−0.915093
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 23.9816i	− 1.40102i	−0.713643	−	0.700509i	$-0.752956\pi$
$293$	− 23.9816i	− 1.40102i	0.713643	−	0.700509i	$-0.247044\pi$
$294$	0	0
$295$	0.390746	0.0227501
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.7905i	1.31801i
$300$	0	0
$301$	4.86649i	0.280500i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.2796	1.27573
$306$	0	0
$307$	1.82132i	0.103948i	0.998648	+	0.0519740i	$0.0165513\pi$
$307$	1.82132i	0.103948i	−0.998648	+	0.0519740i	$0.983449\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.53302	−0.257044	−0.128522	−	0.991707i	$-0.541023\pi$
$311$	−4.53302	−0.257044	−0.128522	+	0.991707i	$0.541023\pi$
$312$	0	0
$313$	12.4149	0.701730	0.350865	−	0.936426i	$-0.385888\pi$
$313$	12.4149	0.701730	0.350865	+	0.936426i	$0.385888\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.0948i	− 1.29713i	−0.761157	−	0.648567i	$-0.775368\pi$
$317$	− 23.0948i	− 1.29713i	0.761157	−	0.648567i	$-0.224632\pi$
$318$	0	0
$319$	−1.70268	−0.0953317
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.20142i	− 0.0668490i
$324$	0	0
$325$	10.5448i	0.584918i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.45856	0.411204
$330$	0	0
$331$	26.3743i	1.44966i	0.688927	+	0.724831i	$0.258082\pi$
$331$	26.3743i	1.44966i	−0.688927	+	0.724831i	$0.741918\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.4527	0.789638
$336$	0	0
$337$	3.82102	0.208144	0.104072	−	0.994570i	$-0.466813\pi$
$337$	3.82102	0.208144	0.104072	+	0.994570i	$0.466813\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.90669i	0.211559i
$342$	0	0
$343$	7.92699	0.428017
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.8808i	1.60408i	0.597268	+	0.802042i	$0.296253\pi$
$347$	29.8808i	1.60408i	−0.597268	+	0.802042i	$0.703747\pi$
$348$	0	0
$349$	12.7016i	0.679902i	0.940443	+	0.339951i	$0.110411\pi$
$349$	12.7016i	0.679902i	−0.940443	+	0.339951i	$0.889589\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.8398	−1.21564	−0.607820	−	0.794074i	$-0.707956\pi$
$353$	−22.8398	−1.21564	−0.607820	+	0.794074i	$0.707956\pi$
$354$	0	0
$355$	10.4995i	0.557257i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.4003	1.86835	0.934177	−	0.356810i	$-0.116136\pi$
$359$	35.4003	1.86835	0.934177	+	0.356810i	$0.116136\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 23.1475i	− 1.21159i
$366$	0	0
$367$	17.2148	0.898608	0.449304	−	0.893379i	$-0.351672\pi$
$367$	17.2148	0.898608	0.449304	+	0.893379i	$0.351672\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.78216i	− 0.0925251i
$372$	0	0
$373$	− 20.3506i	− 1.05372i	−0.849953	−	0.526858i	$-0.823370\pi$
$373$	− 20.3506i	− 1.05372i	0.849953	−	0.526858i	$-0.176630\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.02837	0.258974
$378$	0	0
$379$	− 4.22765i	− 0.217160i	−0.994088	−	0.108580i	$-0.965370\pi$
$379$	− 4.22765i	− 0.217160i	0.994088	−	0.108580i	$-0.0346303\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.71281	0.496301	0.248151	−	0.968721i	$-0.420177\pi$
$383$	9.71281	0.496301	0.248151	+	0.968721i	$0.420177\pi$
$384$	0	0
$385$	1.15734	0.0589834
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.3291i	1.13213i	0.824361	+	0.566064i	$0.191534\pi$
$389$	22.3291i	1.13213i	−0.824361	+	0.566064i	$0.808466\pi$
$390$	0	0
$391$	7.03466	0.355758
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 19.8778i	− 1.00016i
$396$	0	0
$397$	− 23.9210i	− 1.20056i	−0.799789	−	0.600281i	$-0.795055\pi$
$397$	− 23.9210i	− 1.20056i	0.799789	−	0.600281i	$-0.204945\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.5237	0.825154	0.412577	−	0.910923i	$-0.364629\pi$
$401$	16.5237	0.825154	0.412577	+	0.910923i	$0.364629\pi$
$402$	0	0
$403$	− 11.5373i	− 0.574713i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.56073	−0.0773625
$408$	0	0
$409$	16.0228	0.792278	0.396139	−	0.918191i	$-0.370350\pi$
$409$	16.0228	0.792278	0.396139	+	0.918191i	$0.370350\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 0.149776i	− 0.00737002i
$414$	0	0
$415$	−5.60949	−0.275359
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.7496i	0.622858i	0.950269	+	0.311429i	$0.100808\pi$
$419$	12.7496i	0.622858i	−0.950269	+	0.311429i	$0.899192\pi$
$420$	0	0
$421$	1.89293i	0.0922560i	0.998936	+	0.0461280i	$0.0146882\pi$
$421$	1.89293i	0.0922560i	−0.998936	+	0.0461280i	$0.985312\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.25482	0.157882
$426$	0	0
$427$	− 8.53999i	− 0.413279i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	37.2166	1.79266	0.896331	−	0.443386i	$-0.146223\pi$
$431$	37.2166	1.79266	0.896331	+	0.443386i	$0.146223\pi$
$432$	0	0
$433$	−36.0314	−1.73156	−0.865780	−	0.500425i	$-0.833177\pi$
$433$	−36.0314	−1.73156	−0.865780	+	0.500425i	$0.833177\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.85527i	− 0.280095i
$438$	0	0
$439$	18.6188	0.888629	0.444315	−	0.895871i	$-0.353447\pi$
$439$	18.6188	0.888629	0.444315	+	0.895871i	$0.353447\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.26887i	0.107797i	0.998546	+	0.0538987i	$0.0171648\pi$
$443$	2.26887i	0.107797i	−0.998546	+	0.0538987i	$0.982835\pi$
$444$	0	0
$445$	− 12.6633i	− 0.600296i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.248909	−0.0117467	−0.00587337	−	0.999983i	$-0.501870\pi$
$449$	−0.248909	−0.0117467	−0.00587337	+	0.999983i	$0.501870\pi$
$450$	0	0
$451$	− 11.9222i	− 0.561395i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.41787	−0.160232
$456$	0	0
$457$	−6.98767	−0.326869	−0.163435	−	0.986554i	$-0.552257\pi$
$457$	−6.98767	−0.326869	−0.163435	+	0.986554i	$0.552257\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	34.1624i	1.59110i	0.605888	+	0.795550i	$0.292818\pi$
$461$	34.1624i	1.59110i	−0.605888	+	0.795550i	$0.707182\pi$
$462$	0	0
$463$	−34.5311	−1.60480	−0.802399	−	0.596788i	$-0.796444\pi$
$463$	−34.5311	−1.60480	−0.802399	+	0.596788i	$0.796444\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.6496i	1.83476i	0.398008	+	0.917382i	$0.369702\pi$
$467$	39.6496i	1.83476i	−0.398008	+	0.917382i	$0.630298\pi$
$468$	0	0
$469$	− 5.53987i	− 0.255808i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.0555	−0.508332
$474$	0	0
$475$	− 2.70913i	− 0.124304i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.8669	−1.13620	−0.568098	−	0.822961i	$-0.692321\pi$
$479$	−24.8669	−1.13620	−0.568098	+	0.822961i	$0.692321\pi$
$480$	0	0
$481$	4.60917	0.210160
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 25.7974i	− 1.17140i
$486$	0	0
$487$	39.7320	1.80043	0.900214	−	0.435448i	$-0.143410\pi$
$487$	39.7320	1.80043	0.900214	+	0.435448i	$0.143410\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 3.45561i	− 0.155949i	−0.996955	−	0.0779747i	$-0.975155\pi$
$491$	− 3.45561i	− 0.155949i	0.996955	−	0.0779747i	$-0.0248453\pi$
$492$	0	0
$493$	− 1.55209i	− 0.0699027i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.02457	0.180527
$498$	0	0
$499$	− 17.5837i	− 0.787156i	−0.919291	−	0.393578i	$-0.871237\pi$
$499$	− 17.5837i	− 0.787156i	0.919291	−	0.393578i	$-0.128763\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.72712	0.121596	0.0607981	−	0.998150i	$-0.480635\pi$
$503$	2.72712	0.121596	0.0607981	+	0.998150i	$0.480635\pi$
$504$	0	0
$505$	2.99262	0.133170
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.73123i	0.298356i	0.988810	+	0.149178i	$0.0476628\pi$
$509$	6.73123i	0.298356i	−0.988810	+	0.149178i	$0.952337\pi$
$510$	0	0
$511$	−8.87264	−0.392503
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.49410i	− 0.153968i
$516$	0	0
$517$	16.9441i	0.745198i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−38.2190	−1.67440	−0.837202	−	0.546894i	$-0.815810\pi$
$521$	−38.2190	−1.67440	−0.837202	+	0.546894i	$0.815810\pi$
$522$	0	0
$523$	5.37182i	0.234893i	0.993079	+	0.117447i	$0.0374709\pi$
$523$	5.37182i	0.234893i	−0.993079	+	0.117447i	$0.962529\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.56118	−0.155127
$528$	0	0
$529$	11.2842	0.490616
$530$	0	0
$531$	0	0
$532$	0	0
$533$	35.2089i	1.52506i
$534$	0	0
$535$	−22.7633	−0.984143
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.78229i	0.378280i
$540$	0	0
$541$	34.3497i	1.47681i	0.674359	+	0.738404i	$0.264420\pi$
$541$	34.3497i	1.47681i	−0.674359	+	0.738404i	$0.735580\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−20.2820	−0.868786
$546$	0	0
$547$	2.58358i	0.110466i	0.998473	+	0.0552330i	$0.0175902\pi$
$547$	2.58358i	0.110466i	−0.998473	+	0.0552330i	$0.982410\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.29188	−0.0550358
$552$	0	0
$553$	−7.61936	−0.324008
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.498941i	0.0211408i	0.999944	+	0.0105704i	$0.00336472\pi$
$557$	0.498941i	0.0211408i	−0.999944	+	0.0105704i	$0.996635\pi$
$558$	0	0
$559$	32.6492	1.38092
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.5553i	1.37204i	0.727581	+	0.686021i	$0.240644\pi$
$563$	32.5553i	1.37204i	−0.727581	+	0.686021i	$0.759356\pi$
$564$	0	0
$565$	7.41216i	0.311832i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.6834	−0.573636	−0.286818	−	0.957985i	$-0.592598\pi$
$569$	−13.6834	−0.573636	−0.286818	+	0.957985i	$0.592598\pi$
$570$	0	0
$571$	− 12.2520i	− 0.512732i	−0.966580	−	0.256366i	$-0.917475\pi$
$571$	− 12.2520i	− 0.512732i	0.966580	−	0.256366i	$-0.0825253\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.8627	0.661521
$576$	0	0
$577$	14.6672	0.610605	0.305302	−	0.952255i	$-0.401242\pi$
$577$	14.6672	0.610605	0.305302	+	0.952255i	$0.401242\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.15017i	0.0892040i
$582$	0	0
$583$	4.04864	0.167677
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.53206i	− 0.228333i	−0.993462	−	0.114166i	$-0.963580\pi$
$587$	− 5.53206i	− 0.228333i	0.993462	−	0.114166i	$-0.0364197\pi$
$588$	0	0
$589$	2.96413i	0.122135i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.5870	−0.886470	−0.443235	−	0.896405i	$-0.646169\pi$
$593$	−21.5870	−0.886470	−0.443235	+	0.896405i	$0.646169\pi$
$594$	0	0
$595$	1.05498i	0.0432500i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28.3513	−1.15840	−0.579202	−	0.815184i	$-0.696636\pi$
$599$	−28.3513	−1.15840	−0.579202	+	0.815184i	$0.696636\pi$
$600$	0	0
$601$	−21.9758	−0.896410	−0.448205	−	0.893931i	$-0.647936\pi$
$601$	−21.9758	−0.896410	−0.448205	+	0.893931i	$0.647936\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 14.0200i	− 0.569993i
$606$	0	0
$607$	7.89484	0.320441	0.160221	−	0.987081i	$-0.448779\pi$
$607$	7.89484	0.320441	0.160221	+	0.987081i	$0.448779\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 50.0394i	− 2.02438i
$612$	0	0
$613$	− 29.5174i	− 1.19220i	−0.802912	−	0.596098i	$-0.796717\pi$
$613$	− 29.5174i	− 1.19220i	0.802912	−	0.596098i	$-0.203283\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.08379	−0.124149	−0.0620744	−	0.998072i	$-0.519772\pi$
$617$	−3.08379	−0.124149	−0.0620744	+	0.998072i	$0.519772\pi$
$618$	0	0
$619$	− 10.9487i	− 0.440066i	−0.975492	−	0.220033i	$-0.929384\pi$
$619$	− 10.9487i	− 0.440066i	0.975492	−	0.220033i	$-0.0706164\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.85394	−0.194469
$624$	0	0
$625$	−4.11492	−0.164597
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.42270i	− 0.0567267i
$630$	0	0
$631$	20.7432	0.825773	0.412886	−	0.910783i	$-0.364521\pi$
$631$	20.7432	0.825773	0.412886	+	0.910783i	$0.364521\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 3.95986i	− 0.157142i
$636$	0	0
$637$	− 25.9360i	− 1.02762i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.5310	1.36389	0.681946	−	0.731403i	$-0.261134\pi$
$641$	34.5310	1.36389	0.681946	+	0.731403i	$0.261134\pi$
$642$	0	0
$643$	21.5300i	0.849062i	0.905413	+	0.424531i	$0.139561\pi$
$643$	21.5300i	0.849062i	−0.905413	+	0.424531i	$0.860439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.7527	0.894502	0.447251	−	0.894409i	$-0.352403\pi$
$647$	22.7527	0.894502	0.447251	+	0.894409i	$0.352403\pi$
$648$	0	0
$649$	0.340256	0.0133562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 35.6805i	− 1.39629i	−0.715958	−	0.698143i	$-0.754010\pi$
$653$	− 35.6805i	− 1.39629i	0.715958	−	0.698143i	$-0.245990\pi$
$654$	0	0
$655$	4.00334	0.156423
$656$	0	0
$657$	0	0
$658$	0	0
$659$	35.8589i	1.39686i	0.715676	+	0.698432i	$0.246119\pi$
$659$	35.8589i	1.39686i	−0.715676	+	0.698432i	$0.753881\pi$
$660$	0	0
$661$	25.4409i	0.989537i	0.869025	+	0.494769i	$0.164747\pi$
$661$	25.4409i	0.989537i	−0.869025	+	0.494769i	$0.835253\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.878110	0.0340516
$666$	0	0
$667$	− 7.56429i	− 0.292890i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.4008	0.748959
$672$	0	0
$673$	34.2142	1.31886	0.659431	−	0.751765i	$-0.270797\pi$
$673$	34.2142	1.31886	0.659431	+	0.751765i	$0.270797\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.9103i	1.45701i	0.685040	+	0.728505i	$0.259785\pi$
$677$	37.9103i	1.45701i	−0.685040	+	0.728505i	$0.740215\pi$
$678$	0	0
$679$	−9.88837	−0.379481
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 30.3542i	− 1.16147i	−0.814092	−	0.580736i	$-0.802765\pi$
$683$	− 30.3542i	− 1.16147i	0.814092	−	0.580736i	$-0.197235\pi$
$684$	0	0
$685$	7.38671i	0.282232i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11.9565	−0.455506
$690$	0	0
$691$	3.50694i	0.133410i	0.997773	+	0.0667052i	$0.0212487\pi$
$691$	3.50694i	0.133410i	−0.997773	+	0.0667052i	$0.978751\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.6402	−0.593268
$696$	0	0
$697$	10.8678	0.411647
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.2168i	0.499190i	0.968350	+	0.249595i	$0.0802975\pi$
$701$	13.2168i	0.499190i	−0.968350	+	0.249595i	$0.919703\pi$
$702$	0	0
$703$	−1.18418	−0.0446621
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.14710i	− 0.0431411i
$708$	0	0
$709$	23.2882i	0.874605i	0.899314	+	0.437302i	$0.144066\pi$
$709$	23.2882i	0.874605i	−0.899314	+	0.437302i	$0.855934\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−17.3558	−0.649979
$714$	0	0
$715$	− 7.76457i	− 0.290378i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.192843	0.00719184	0.00359592	−	0.999994i	$-0.498855\pi$
$719$	0.192843	0.00719184	0.00359592	+	0.999994i	$0.498855\pi$
$720$	0	0
$721$	−1.33932	−0.0498789
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.49987i	− 0.129982i
$726$	0	0
$727$	−14.5604	−0.540014	−0.270007	−	0.962858i	$-0.587026\pi$
$727$	−14.5604	−0.540014	−0.270007	+	0.962858i	$0.587026\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 10.0777i	− 0.372739i
$732$	0	0
$733$	− 29.0856i	− 1.07430i	−0.843487	−	0.537150i	$-0.819501\pi$
$733$	− 29.0856i	− 1.07430i	0.843487	−	0.537150i	$-0.180499\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.5853	0.463584
$738$	0	0
$739$	− 28.8090i	− 1.05975i	−0.848074	−	0.529877i	$-0.822238\pi$
$739$	− 28.8090i	− 1.05975i	0.848074	−	0.529877i	$-0.177762\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.2025	1.54826	0.774129	−	0.633027i	$-0.218188\pi$
$743$	42.2025	1.54826	0.774129	+	0.633027i	$0.218188\pi$
$744$	0	0
$745$	12.6431	0.463208
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.72538i	0.318818i
$750$	0	0
$751$	−2.64036	−0.0963483	−0.0481741	−	0.998839i	$-0.515340\pi$
$751$	−2.64036	−0.0963483	−0.0481741	+	0.998839i	$0.515340\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 8.85344i	− 0.322210i
$756$	0	0
$757$	4.51764i	0.164197i	0.996624	+	0.0820983i	$0.0261621\pi$
$757$	4.51764i	0.164197i	−0.996624	+	0.0820983i	$0.973838\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.5124	0.852323	0.426162	−	0.904647i	$-0.359866\pi$
$761$	23.5124	0.852323	0.426162	+	0.904647i	$0.359866\pi$
$762$	0	0
$763$	7.77428i	0.281448i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.00485	−0.0362830
$768$	0	0
$769$	−29.6727	−1.07002	−0.535012	−	0.844845i	$-0.679693\pi$
$769$	−29.6727	−1.07002	−0.535012	+	0.844845i	$0.679693\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.1647i	0.365598i	0.983150	+	0.182799i	$0.0585158\pi$
$773$	10.1647i	0.365598i	−0.983150	+	0.182799i	$0.941484\pi$
$774$	0	0
$775$	−8.03023	−0.288454
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 9.04577i	− 0.324098i
$780$	0	0
$781$	9.14285i	0.327157i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.547372	−0.0195365
$786$	0	0
$787$	− 18.8481i	− 0.671863i	−0.941886	−	0.335931i	$-0.890949\pi$
$787$	− 18.8481i	− 0.671863i	0.941886	−	0.335931i	$-0.109051\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.84115	0.101020
$792$	0	0
$793$	−57.2947	−2.03459
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.9098i	0.492709i	0.969180	+	0.246355i	$0.0792328\pi$
$797$	13.9098i	0.492709i	−0.969180	+	0.246355i	$0.920767\pi$
$798$	0	0
$799$	−15.4455	−0.546423
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 20.1565i	− 0.711308i
$804$	0	0
$805$	5.14157i	0.181217i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−19.7630	−0.694831	−0.347415	−	0.937711i	$-0.612941\pi$
$809$	−19.7630	−0.694831	−0.347415	+	0.937711i	$0.612941\pi$
$810$	0	0
$811$	30.1099i	1.05730i	0.848839	+	0.528651i	$0.177302\pi$
$811$	30.1099i	1.05730i	−0.848839	+	0.528651i	$0.822698\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.221024	−0.00774213
$816$	0	0
$817$	−8.38816	−0.293465
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.1788i	0.913645i	0.889558	+	0.456823i	$0.151013\pi$
$821$	26.1788i	0.913645i	−0.889558	+	0.456823i	$0.848987\pi$
$822$	0	0
$823$	−41.9817	−1.46339	−0.731696	−	0.681631i	$-0.761271\pi$
$823$	−41.9817	−1.46339	−0.731696	+	0.681631i	$0.761271\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.6394i	0.717703i	0.933395	+	0.358851i	$0.116832\pi$
$827$	20.6394i	0.717703i	−0.933395	+	0.358851i	$0.883168\pi$
$828$	0	0
$829$	36.6292i	1.27218i	0.771613	+	0.636092i	$0.219450\pi$
$829$	36.6292i	1.27218i	−0.771613	+	0.636092i	$0.780550\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.00558	−0.277377
$834$	0	0
$835$	− 0.615418i	− 0.0212974i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.71855	0.128379	0.0641893	−	0.997938i	$-0.479554\pi$
$839$	3.71855	0.128379	0.0641893	+	0.997938i	$0.479554\pi$
$840$	0	0
$841$	27.3311	0.942450
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.25414i	0.111946i
$846$	0	0
$847$	−5.37399	−0.184652
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 6.93367i	− 0.237683i
$852$	0	0
$853$	49.7965i	1.70500i	0.522726	+	0.852501i	$0.324915\pi$
$853$	49.7965i	1.70500i	−0.522726	+	0.852501i	$0.675085\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.6409	−1.21747	−0.608735	−	0.793374i	$-0.708323\pi$
$857$	−35.6409	−1.21747	−0.608735	+	0.793374i	$0.708323\pi$
$858$	0	0
$859$	43.5752i	1.48677i	0.668866	+	0.743383i	$0.266780\pi$
$859$	43.5752i	1.48677i	−0.668866	+	0.743383i	$0.733220\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43.6579	−1.48613	−0.743067	−	0.669217i	$-0.766630\pi$
$863$	−43.6579	−1.48613	−0.743067	+	0.669217i	$0.766630\pi$
$864$	0	0
$865$	−19.1415	−0.650829
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 17.3093i	− 0.587179i
$870$	0	0
$871$	−37.1669	−1.25935
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.76947i	0.228850i
$876$	0	0
$877$	− 5.23078i	− 0.176631i	−0.996093	−	0.0883154i	$-0.971852\pi$
$877$	− 5.23078i	− 0.176631i	0.996093	−	0.0883154i	$-0.0281483\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.0358	0.742405	0.371203	−	0.928552i	$-0.378946\pi$
$881$	22.0358	0.742405	0.371203	+	0.928552i	$0.378946\pi$
$882$	0	0
$883$	56.9658i	1.91705i	0.285008	+	0.958525i	$0.408004\pi$
$883$	56.9658i	1.91705i	−0.285008	+	0.958525i	$0.591996\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.28694	0.110365	0.0551823	−	0.998476i	$-0.482426\pi$
$887$	3.28694	0.110365	0.0551823	+	0.998476i	$0.482426\pi$
$888$	0	0
$889$	−1.51785	−0.0509071
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.8560i	0.430210i
$894$	0	0
$895$	−8.40932	−0.281092
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.82929i	0.127714i
$900$	0	0
$901$	3.69057i	0.122951i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.8249	−0.459555
$906$	0	0
$907$	− 0.0491571i	− 0.00163223i	−1.00000		0.000816117i	$-0.999740\pi$
$907$	− 0.0491571i	− 0.00163223i	1.00000		0.000816117i	$-0.000259778\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32.4990	1.07674	0.538370	−	0.842708i	$-0.319040\pi$
$911$	32.4990	1.07674	0.538370	+	0.842708i	$0.319040\pi$
$912$	0	0
$913$	−4.88466	−0.161659
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.53452i	− 0.0506742i
$918$	0	0
$919$	19.8145	0.653619	0.326810	−	0.945090i	$-0.394026\pi$
$919$	19.8145	0.653619	0.326810	+	0.945090i	$0.394026\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 27.0008i	− 0.888742i
$924$	0	0
$925$	− 3.20809i	− 0.105481i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43.9293	−1.44127	−0.720637	−	0.693313i	$-0.756150\pi$
$929$	−43.9293	−1.44127	−0.720637	+	0.693313i	$0.756150\pi$
$930$	0	0
$931$	6.66341i	0.218385i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.39666	−0.0783793
$936$	0	0
$937$	−23.9026	−0.780863	−0.390431	−	0.920632i	$-0.627674\pi$
$937$	−23.9026	−0.780863	−0.390431	+	0.920632i	$0.627674\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 20.4109i	− 0.665375i	−0.943037	−	0.332688i	$-0.892045\pi$
$941$	− 20.4109i	− 0.665375i	0.943037	−	0.332688i	$-0.107955\pi$
$942$	0	0
$943$	52.9654	1.72479
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.72706i	0.0886176i	0.999018	+	0.0443088i	$0.0141085\pi$
$947$	2.72706i	0.0886176i	−0.999018	+	0.0443088i	$0.985891\pi$
$948$	0	0
$949$	59.5265i	1.93231i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32.7679	1.06146	0.530729	−	0.847542i	$-0.321918\pi$
$953$	32.7679	1.06146	0.530729	+	0.847542i	$0.321918\pi$
$954$	0	0
$955$	− 8.28723i	− 0.268168i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.83139	0.0914305
$960$	0	0
$961$	−22.2139	−0.716578
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 25.6847i	− 0.826818i
$966$	0	0
$967$	−36.4163	−1.17107	−0.585535	−	0.810647i	$-0.699115\pi$
$967$	−36.4163	−1.17107	−0.585535	+	0.810647i	$0.699115\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.40329i	0.205491i	0.994708	+	0.102746i	$0.0327628\pi$
$971$	6.40329i	0.205491i	−0.994708	+	0.102746i	$0.967237\pi$
$972$	0	0
$973$	5.99505i	0.192192i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.6473	−0.372629	−0.186315	−	0.982490i	$-0.559654\pi$
$977$	−11.6473	−0.372629	−0.186315	+	0.982490i	$0.559654\pi$
$978$	0	0
$979$	− 11.0270i	− 0.352424i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−34.7061	−1.10695	−0.553477	−	0.832864i	$-0.686699\pi$
$983$	−34.7061	−1.10695	−0.553477	+	0.832864i	$0.686699\pi$
$984$	0	0
$985$	35.9815	1.14646
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 49.1150i	− 1.56176i
$990$	0	0
$991$	−3.30503	−0.104988	−0.0524939	−	0.998621i	$-0.516717\pi$
$991$	−3.30503	−0.104988	−0.0524939	+	0.998621i	$0.516717\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 26.6003i	− 0.843288i
$996$	0	0
$997$	15.9009i	0.503586i	0.967781	+	0.251793i	$0.0810203\pi$
$997$	15.9009i	0.503586i	−0.967781	+	0.251793i	$0.918980\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.g.b.2737.7		16
3.2	odd	2		608.2.c.b.305.12		16
4.3	odd	2		1368.2.g.b.685.6		16
8.3	odd	2		1368.2.g.b.685.5		16
8.5	even	2	inner	5472.2.g.b.2737.10		16
12.11	even	2		152.2.c.b.77.11	✓	16
24.5	odd	2		608.2.c.b.305.5		16
24.11	even	2		152.2.c.b.77.12	yes	16
48.5	odd	4		4864.2.a.bp.1.6		8
48.11	even	4		4864.2.a.bq.1.3		8
48.29	odd	4		4864.2.a.bn.1.3		8
48.35	even	4		4864.2.a.bo.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.11	✓	16	12.11	even	2
152.2.c.b.77.12	yes	16	24.11	even	2
608.2.c.b.305.5		16	24.5	odd	2
608.2.c.b.305.12		16	3.2	odd	2
1368.2.g.b.685.5		16	8.3	odd	2
1368.2.g.b.685.6		16	4.3	odd	2
4864.2.a.bn.1.3		8	48.29	odd	4
4864.2.a.bo.1.6		8	48.35	even	4
4864.2.a.bp.1.6		8	48.5	odd	4
4864.2.a.bq.1.3		8	48.11	even	4
5472.2.g.b.2737.7		16	1.1	even	1	trivial
5472.2.g.b.2737.10		16	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.51356i q^{5} -0.580162 q^{7} +O(q^{10})\)	\(q-1.51356i q^{5} -0.580162 q^{7} -1.31799i q^{11} +3.89230i q^{13} +1.20142 q^{17} -1.00000i q^{19} +5.85527 q^{23} +2.70913 q^{25} -1.29188i q^{29} -2.96413 q^{31} +0.878110i q^{35} -1.18418i q^{37} +9.04577 q^{41} -8.38816i q^{43} -12.8560 q^{47} -6.66341 q^{49} +3.07183i q^{53} -1.99485 q^{55} +0.258163i q^{59} +14.7200i q^{61} +5.89123 q^{65} +9.54884i q^{67} -6.93697 q^{71} +15.2934 q^{73} +0.764646i q^{77} +13.1332 q^{79} -3.70615i q^{83} -1.81843i q^{85} +8.36653 q^{89} -2.25816i q^{91} -1.51356 q^{95} +17.0442 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7}+O(q^{10}) \) 16 * q + 8 * q^7	\( 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) \) 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

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Database

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