LMFDB - Embedded newform 2592.2.d.j.1297.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(1297,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2592.1297");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.d (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:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.31435290000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{4} - 8x + 16 $ x^8 - x^7 - 2x^4 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.5
Root			$-0.295065 - 1.38309i$ of defining polynomial
Character	$\chi$	$=$	2592.1297
Dual form			2592.2.d.j.1297.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.696047i q^{5} +1.59013 q^{7} +O(q^{10})$	$q+0.696047i q^{5} +1.59013 q^{7} -2.73921i q^{11} +5.50539i q^{13} -5.65175 q^{17} +0.963328i q^{19} +6.57714 q^{23} +4.51552 q^{25} +3.29178i q^{29} -7.39688 q^{31} +1.10680i q^{35} +6.25538i q^{37} +1.86377 q^{41} +3.46223i q^{43} +7.71337 q^{47} -4.47149 q^{49} +2.54179i q^{53} +1.90662 q^{55} -5.33494i q^{59} +9.16503i q^{61} -3.83201 q^{65} +6.87947i q^{67} -3.68351 q^{71} +2.83201 q^{73} -4.35569i q^{77} +5.75740 q^{79} +6.63916i q^{83} -3.93388i q^{85} -2.98701 q^{89} +8.75427i q^{91} -0.670522 q^{95} +2.49675 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{7}+O(q^{10}) $ 8 * q + 6 * q^7	$ 8 q + 6 q^{7} - 14 q^{17} - 10 q^{23} - 2 q^{25} - 10 q^{31} + 8 q^{41} + 6 q^{47} - 18 q^{49} + 2 q^{55} + 14 q^{65} - 36 q^{71} - 22 q^{73} - 30 q^{79} + 32 q^{89} + 44 q^{95}+O(q^{100}) $ 8 * q + 6 * q^7 - 14 * q^17 - 10 * q^23 - 2 * q^25 - 10 * q^31 + 8 * q^41 + 6 * q^47 - 18 * q^49 + 2 * q^55 + 14 * q^65 - 36 * q^71 - 22 * q^73 - 30 * q^79 + 32 * q^89 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.696047i	0.311282i	0.987814	+	0.155641i	$0.0497442\pi$
$5$	0.696047i	0.311282i	−0.987814	+	0.155641i	$0.950256\pi$
$6$	0	0
$7$	1.59013	0.601012	0.300506	−	0.953780i	$-0.402844\pi$
$7$	1.59013	0.601012	0.300506	+	0.953780i	$0.402844\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.73921i	− 0.825902i	−0.910753	−	0.412951i	$-0.864498\pi$
$11$	− 2.73921i	− 0.825902i	0.910753	−	0.412951i	$-0.135502\pi$
$12$	0	0
$13$	5.50539i	1.52692i	0.645855	+	0.763460i	$0.276501\pi$
$13$	5.50539i	1.52692i	−0.645855	+	0.763460i	$0.723499\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.65175	−1.37075	−0.685375	−	0.728190i	$-0.740362\pi$
$17$	−5.65175	−1.37075	−0.685375	+	0.728190i	$0.740362\pi$
$18$	0	0
$19$	0.963328i	0.221003i	0.993876	+	0.110501i	$0.0352457\pi$
$19$	0.963328i	0.221003i	−0.993876	+	0.110501i	$0.964754\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.57714	1.37143	0.685714	−	0.727871i	$-0.259490\pi$
$23$	6.57714	1.37143	0.685714	+	0.727871i	$0.259490\pi$
$24$	0	0
$25$	4.51552	0.903104
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.29178i	0.611268i	0.952149	+	0.305634i	$0.0988684\pi$
$29$	3.29178i	0.611268i	−0.952149	+	0.305634i	$0.901132\pi$
$30$	0	0
$31$	−7.39688	−1.32852	−0.664259	−	0.747502i	$-0.731253\pi$
$31$	−7.39688	−1.32852	−0.664259	+	0.747502i	$0.731253\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.10680i	0.187084i
$36$	0	0
$37$	6.25538i	1.02838i	0.857677	+	0.514189i	$0.171907\pi$
$37$	6.25538i	1.02838i	−0.857677	+	0.514189i	$0.828093\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.86377	0.291072	0.145536	−	0.989353i	$-0.453509\pi$
$41$	1.86377	0.291072	0.145536	+	0.989353i	$0.453509\pi$
$42$	0	0
$43$	3.46223i	0.527985i	0.964525	+	0.263992i	$0.0850393\pi$
$43$	3.46223i	0.527985i	−0.964525	+	0.263992i	$0.914961\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.71337	1.12511	0.562555	−	0.826760i	$-0.309818\pi$
$47$	7.71337	1.12511	0.562555	+	0.826760i	$0.309818\pi$
$48$	0	0
$49$	−4.47149	−0.638784
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.54179i	0.349141i	0.984645	+	0.174571i	$0.0558537\pi$
$53$	2.54179i	0.349141i	−0.984645	+	0.174571i	$0.944146\pi$
$54$	0	0
$55$	1.90662	0.257088
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.33494i	− 0.694550i	−0.937763	−	0.347275i	$-0.887107\pi$
$59$	− 5.33494i	− 0.694550i	0.937763	−	0.347275i	$-0.112893\pi$
$60$	0	0
$61$	9.16503i	1.17346i	0.809782	+	0.586731i	$0.199585\pi$
$61$	9.16503i	1.17346i	−0.809782	+	0.586731i	$0.800415\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.83201	−0.475302
$66$	0	0
$67$	6.87947i	0.840461i	0.907417	+	0.420231i	$0.138051\pi$
$67$	6.87947i	0.840461i	−0.907417	+	0.420231i	$0.861949\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.68351	−0.437153	−0.218576	−	0.975820i	$-0.570141\pi$
$71$	−3.68351	−0.437153	−0.218576	+	0.975820i	$0.570141\pi$
$72$	0	0
$73$	2.83201	0.331461	0.165731	−	0.986171i	$-0.447002\pi$
$73$	2.83201	0.331461	0.165731	+	0.986171i	$0.447002\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.35569i	− 0.496377i
$78$	0	0
$79$	5.75740	0.647758	0.323879	−	0.946099i	$-0.395013\pi$
$79$	5.75740	0.647758	0.323879	+	0.946099i	$0.395013\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.63916i	0.728743i	0.931254	+	0.364371i	$0.118716\pi$
$83$	6.63916i	0.728743i	−0.931254	+	0.364371i	$0.881284\pi$
$84$	0	0
$85$	− 3.93388i	− 0.426689i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.98701	−0.316622	−0.158311	−	0.987389i	$-0.550605\pi$
$89$	−2.98701	−0.316622	−0.158311	+	0.987389i	$0.550605\pi$
$90$	0	0
$91$	8.75427i	0.917697i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.670522	−0.0687941
$96$	0	0
$97$	2.49675	0.253506	0.126753	−	0.991934i	$-0.459544\pi$
$97$	2.49675	0.253506	0.126753	+	0.991934i	$0.459544\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.49321i	− 0.944610i	−0.881435	−	0.472305i	$-0.843422\pi$
$101$	− 9.49321i	− 0.944610i	0.881435	−	0.472305i	$-0.156578\pi$
$102$	0	0
$103$	14.7444	1.45281	0.726405	−	0.687267i	$-0.241190\pi$
$103$	14.7444	1.45281	0.726405	+	0.687267i	$0.241190\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.83384i	− 0.757325i	−0.925535	−	0.378663i	$-0.876384\pi$
$107$	− 7.83384i	− 0.757325i	0.925535	−	0.378663i	$-0.123616\pi$
$108$	0	0
$109$	− 0.242400i	− 0.0232177i	−0.999933	−	0.0116089i	$-0.996305\pi$
$109$	− 0.242400i	− 0.0232177i	0.999933	−	0.0116089i	$-0.00369529\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.69578	−0.818030	−0.409015	−	0.912528i	$-0.634128\pi$
$113$	−8.69578	−0.818030	−0.409015	+	0.912528i	$0.634128\pi$
$114$	0	0
$115$	4.57799i	0.426900i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.98701	−0.823838
$120$	0	0
$121$	3.49675	0.317886
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.62325i	0.592401i
$126$	0	0
$127$	1.72754	0.153295	0.0766473	−	0.997058i	$-0.475578\pi$
$127$	1.72754	0.153295	0.0766473	+	0.997058i	$0.475578\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.63916i	0.580066i	0.957017	+	0.290033i	$0.0936663\pi$
$131$	6.63916i	0.580066i	−0.957017	+	0.290033i	$0.906334\pi$
$132$	0	0
$133$	1.53182i	0.132825i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.62649	−0.309832	−0.154916	−	0.987928i	$-0.549511\pi$
$137$	−3.62649	−0.309832	−0.154916	+	0.987928i	$0.549511\pi$
$138$	0	0
$139$	17.3111i	1.46831i	0.678981	+	0.734155i	$0.262422\pi$
$139$	17.3111i	1.46831i	−0.678981	+	0.734155i	$0.737578\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.0804	1.26109
$144$	0	0
$145$	−2.29123	−0.190276
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.6218i	1.77133i	0.464324	+	0.885665i	$0.346297\pi$
$149$	21.6218i	1.77133i	−0.464324	+	0.885665i	$0.653703\pi$
$150$	0	0
$151$	−12.7004	−1.03354	−0.516771	−	0.856124i	$-0.672866\pi$
$151$	−12.7004	−1.03354	−0.516771	+	0.856124i	$0.672866\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 5.14857i	− 0.413543i
$156$	0	0
$157$	17.4689i	1.39417i	0.716990	+	0.697083i	$0.245519\pi$
$157$	17.4689i	1.39417i	−0.716990	+	0.697083i	$0.754481\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10.4585	0.824245
$162$	0	0
$163$	8.56748i	0.671057i	0.942030	+	0.335528i	$0.108915\pi$
$163$	8.56748i	0.671057i	−0.942030	+	0.335528i	$0.891085\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.9506	−0.924769	−0.462384	−	0.886680i	$-0.653006\pi$
$167$	−11.9506	−0.924769	−0.462384	+	0.886680i	$0.653006\pi$
$168$	0	0
$169$	−17.3093	−1.33148
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.0450i	0.991791i	0.868382	+	0.495895i	$0.165160\pi$
$173$	13.0450i	0.991791i	−0.868382	+	0.495895i	$0.834840\pi$
$174$	0	0
$175$	7.18026	0.542777
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.31875i	0.248055i	0.992279	+	0.124028i	$0.0395811\pi$
$179$	3.31875i	0.248055i	−0.992279	+	0.124028i	$0.960419\pi$
$180$	0	0
$181$	14.9128i	1.10846i	0.832363	+	0.554231i	$0.186987\pi$
$181$	14.9128i	1.10846i	−0.832363	+	0.554231i	$0.813013\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.35403	−0.320115
$186$	0	0
$187$	15.4813i	1.13211i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.31767	0.529488	0.264744	−	0.964319i	$-0.414713\pi$
$191$	7.31767	0.529488	0.264744	+	0.964319i	$0.414713\pi$
$192$	0	0
$193$	20.4708	1.47352	0.736759	−	0.676156i	$-0.236355\pi$
$193$	20.4708	1.47352	0.736759	+	0.676156i	$0.236355\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.5437i	− 1.46368i	−0.681479	−	0.731838i	$-0.738663\pi$
$197$	− 20.5437i	− 1.46368i	0.681479	−	0.731838i	$-0.261337\pi$
$198$	0	0
$199$	−1.95597	−0.138655	−0.0693275	−	0.997594i	$-0.522085\pi$
$199$	−1.95597	−0.138655	−0.0693275	+	0.997594i	$0.522085\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.23435i	0.367379i
$204$	0	0
$205$	1.29727i	0.0906054i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.63876	0.182527
$210$	0	0
$211$	− 10.5191i	− 0.724165i	−0.932146	−	0.362082i	$-0.882066\pi$
$211$	− 10.5191i	− 0.724165i	0.932146	−	0.362082i	$-0.117934\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.40987	−0.164352
$216$	0	0
$217$	−11.7620	−0.798456
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 31.1151i	− 2.09303i
$222$	0	0
$223$	3.86259	0.258658	0.129329	−	0.991602i	$-0.458718\pi$
$223$	3.86259	0.258658	0.129329	+	0.991602i	$0.458718\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.0715i	− 1.06670i	−0.845894	−	0.533351i	$-0.820933\pi$
$227$	− 16.0715i	− 1.06670i	0.845894	−	0.533351i	$-0.179067\pi$
$228$	0	0
$229$	− 8.61775i	− 0.569477i	−0.958605	−	0.284739i	$-0.908093\pi$
$229$	− 8.61775i	− 0.569477i	0.958605	−	0.284739i	$-0.0919068\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.1535	1.58235	0.791176	−	0.611589i	$-0.209469\pi$
$233$	24.1535	1.58235	0.791176	+	0.611589i	$0.209469\pi$
$234$	0	0
$235$	5.36886i	0.350226i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.02985	−0.260670	−0.130335	−	0.991470i	$-0.541605\pi$
$239$	−4.02985	−0.260670	−0.130335	+	0.991470i	$0.541605\pi$
$240$	0	0
$241$	−5.63297	−0.362852	−0.181426	−	0.983405i	$-0.558071\pi$
$241$	−5.63297	−0.362852	−0.181426	+	0.983405i	$0.558071\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.11237i	− 0.198842i
$246$	0	0
$247$	−5.30350	−0.337453
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.8828i	0.876276i	0.898908	+	0.438138i	$0.144362\pi$
$251$	13.8828i	0.876276i	−0.898908	+	0.438138i	$0.855638\pi$
$252$	0	0
$253$	− 18.0161i	− 1.13267i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.8508	0.676853	0.338427	−	0.940993i	$-0.390105\pi$
$257$	10.8508	0.676853	0.338427	+	0.940993i	$0.390105\pi$
$258$	0	0
$259$	9.94686i	0.618068i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−23.2101	−1.43120	−0.715598	−	0.698512i	$-0.753846\pi$
$263$	−23.2101	−1.43120	−0.715598	+	0.698512i	$0.753846\pi$
$264$	0	0
$265$	−1.76920	−0.108681
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.01966i	− 0.245083i	−0.992463	−	0.122541i	$-0.960896\pi$
$269$	− 4.01966i	− 0.245083i	0.992463	−	0.122541i	$-0.0391044\pi$
$270$	0	0
$271$	6.75621	0.410411	0.205205	−	0.978719i	$-0.434214\pi$
$271$	6.75621	0.410411	0.205205	+	0.978719i	$0.434214\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 12.3689i	− 0.745875i
$276$	0	0
$277$	2.11997i	0.127377i	0.997970	+	0.0636885i	$0.0202864\pi$
$277$	2.11997i	0.127377i	−0.997970	+	0.0636885i	$0.979714\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.1160	1.55795	0.778976	−	0.627054i	$-0.215739\pi$
$281$	26.1160	1.55795	0.778976	+	0.627054i	$0.215739\pi$
$282$	0	0
$283$	− 19.0960i	− 1.13514i	−0.823326	−	0.567569i	$-0.807884\pi$
$283$	− 19.0960i	− 1.13514i	0.823326	−	0.567569i	$-0.192116\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.96364	0.174938
$288$	0	0
$289$	14.9423	0.878956
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.85568i	0.342092i	0.985263	+	0.171046i	$0.0547147\pi$
$293$	5.85568i	0.342092i	−0.985263	+	0.171046i	$0.945285\pi$
$294$	0	0
$295$	3.71337	0.216201
$296$	0	0
$297$	0	0
$298$	0	0
$299$	36.2097i	2.09406i
$300$	0	0
$301$	5.50539i	0.317325i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.37929	−0.365277
$306$	0	0
$307$	− 13.7071i	− 0.782305i	−0.920326	−	0.391152i	$-0.872077\pi$
$307$	− 13.7071i	− 0.782305i	0.920326	−	0.391152i	$-0.127923\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.1596	−1.08644	−0.543221	−	0.839590i	$-0.682795\pi$
$311$	−19.1596	−1.08644	−0.543221	+	0.839590i	$0.682795\pi$
$312$	0	0
$313$	25.2205	1.42555	0.712773	−	0.701395i	$-0.247439\pi$
$313$	25.2205	1.42555	0.712773	+	0.701395i	$0.247439\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.47026i	0.138744i	0.997591	+	0.0693719i	$0.0220995\pi$
$317$	2.47026i	0.138744i	−0.997591	+	0.0693719i	$0.977900\pi$
$318$	0	0
$319$	9.01686	0.504847
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.44449i	− 0.302939i
$324$	0	0
$325$	24.8597i	1.37897i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.2652	0.676205
$330$	0	0
$331$	− 28.2213i	− 1.55118i	−0.631235	−	0.775592i	$-0.717452\pi$
$331$	− 28.2213i	− 1.55118i	0.631235	−	0.775592i	$-0.282548\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.78843	−0.261620
$336$	0	0
$337$	−11.2113	−0.610718	−0.305359	−	0.952237i	$-0.598777\pi$
$337$	−11.2113	−0.610718	−0.305359	+	0.952237i	$0.598777\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.2616i	1.09723i
$342$	0	0
$343$	−18.2411	−0.984929
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.5887i	1.10526i	0.833428	+	0.552628i	$0.186375\pi$
$347$	20.5887i	1.10526i	−0.833428	+	0.552628i	$0.813625\pi$
$348$	0	0
$349$	3.39445i	0.181701i	0.995865	+	0.0908505i	$0.0289585\pi$
$349$	3.39445i	0.181701i	−0.995865	+	0.0908505i	$0.971041\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.00648	0.0535696	0.0267848	−	0.999641i	$-0.491473\pi$
$353$	1.00648	0.0535696	0.0267848	+	0.999641i	$0.491473\pi$
$354$	0	0
$355$	− 2.56390i	− 0.136078i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.4772	−1.66131	−0.830653	−	0.556791i	$-0.812032\pi$
$359$	−31.4772	−1.66131	−0.830653	+	0.556791i	$0.812032\pi$
$360$	0	0
$361$	18.0720	0.951158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.97121i	0.103178i
$366$	0	0
$367$	−17.3333	−0.904793	−0.452397	−	0.891817i	$-0.649431\pi$
$367$	−17.3333	−0.904793	−0.452397	+	0.891817i	$0.649431\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.04177i	0.209838i
$372$	0	0
$373$	− 13.0183i	− 0.674064i	−0.941493	−	0.337032i	$-0.890577\pi$
$373$	− 13.0183i	− 0.674064i	0.941493	−	0.337032i	$-0.109423\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.1225	−0.933357
$378$	0	0
$379$	− 22.8643i	− 1.17446i	−0.809421	−	0.587229i	$-0.800219\pi$
$379$	− 22.8643i	− 1.17446i	0.809421	−	0.587229i	$-0.199781\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.0234	1.53412	0.767061	−	0.641574i	$-0.221718\pi$
$383$	30.0234	1.53412	0.767061	+	0.641574i	$0.221718\pi$
$384$	0	0
$385$	3.03177	0.154513
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 38.0444i	− 1.92893i	−0.264218	−	0.964463i	$-0.585114\pi$
$389$	− 38.0444i	− 1.92893i	0.264218	−	0.964463i	$-0.414886\pi$
$390$	0	0
$391$	−37.1723	−1.87989
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.00742i	0.201635i
$396$	0	0
$397$	− 37.4510i	− 1.87961i	−0.341709	−	0.939806i	$-0.611006\pi$
$397$	− 37.4510i	− 1.87961i	0.341709	−	0.939806i	$-0.388994\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.70804	0.235108	0.117554	−	0.993066i	$-0.462495\pi$
$401$	4.70804	0.235108	0.117554	+	0.993066i	$0.462495\pi$
$402$	0	0
$403$	− 40.7227i	− 2.02854i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.1348	0.849339
$408$	0	0
$409$	−10.7275	−0.530443	−0.265221	−	0.964188i	$-0.585445\pi$
$409$	−10.7275	−0.530443	−0.265221	+	0.964188i	$0.585445\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8.48324i	− 0.417433i
$414$	0	0
$415$	−4.62117	−0.226844
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 4.12921i	− 0.201725i	−0.994900	−	0.100863i	$-0.967840\pi$
$419$	− 4.12921i	− 0.201725i	0.994900	−	0.100863i	$-0.0321602\pi$
$420$	0	0
$421$	− 15.8565i	− 0.772797i	−0.922332	−	0.386399i	$-0.873719\pi$
$421$	− 15.8565i	− 0.772797i	0.922332	−	0.386399i	$-0.126281\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−25.5206	−1.23793
$426$	0	0
$427$	14.5736i	0.705265i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.1853	−0.779619	−0.389810	−	0.920895i	$-0.627459\pi$
$431$	−16.1853	−0.779619	−0.389810	+	0.920895i	$0.627459\pi$
$432$	0	0
$433$	−32.8306	−1.57774	−0.788868	−	0.614563i	$-0.789332\pi$
$433$	−32.8306	−1.57774	−0.788868	+	0.614563i	$0.789332\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.33594i	0.303089i
$438$	0	0
$439$	21.8546	1.04307	0.521533	−	0.853231i	$-0.325360\pi$
$439$	21.8546	1.04307	0.521533	+	0.853231i	$0.325360\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 35.1606i	− 1.67053i	−0.549848	−	0.835265i	$-0.685314\pi$
$443$	− 35.1606i	− 1.67053i	0.549848	−	0.835265i	$-0.314686\pi$
$444$	0	0
$445$	− 2.07910i	− 0.0985587i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.21851	0.151891	0.0759453	−	0.997112i	$-0.475803\pi$
$449$	3.21851	0.151891	0.0759453	+	0.997112i	$0.475803\pi$
$450$	0	0
$451$	− 5.10526i	− 0.240397i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.09338	−0.285662
$456$	0	0
$457$	−8.11025	−0.379381	−0.189691	−	0.981844i	$-0.560749\pi$
$457$	−8.11025	−0.379381	−0.189691	+	0.981844i	$0.560749\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.9940i	0.977788i	0.872343	+	0.488894i	$0.162599\pi$
$461$	20.9940i	0.977788i	−0.872343	+	0.488894i	$0.837401\pi$
$462$	0	0
$463$	−8.90011	−0.413623	−0.206812	−	0.978381i	$-0.566309\pi$
$463$	−8.90011	−0.413623	−0.206812	+	0.978381i	$0.566309\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 26.0527i	− 1.20557i	−0.797902	−	0.602787i	$-0.794057\pi$
$467$	− 26.0527i	− 1.20557i	0.797902	−	0.602787i	$-0.205943\pi$
$468$	0	0
$469$	10.9392i	0.505128i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.48375	0.436063
$474$	0	0
$475$	4.34993i	0.199588i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17.4229	0.796071	0.398035	−	0.917370i	$-0.369692\pi$
$479$	17.4229	0.796071	0.398035	+	0.917370i	$0.369692\pi$
$480$	0	0
$481$	−34.4383	−1.57025
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.73785i	0.0789118i
$486$	0	0
$487$	29.7367	1.34750	0.673750	−	0.738959i	$-0.264682\pi$
$487$	29.7367	1.34750	0.673750	+	0.738959i	$0.264682\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 23.8268i	− 1.07529i	−0.843171	−	0.537645i	$-0.819314\pi$
$491$	− 23.8268i	− 1.07529i	0.843171	−	0.537645i	$-0.180686\pi$
$492$	0	0
$493$	− 18.6043i	− 0.837895i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.85726	−0.262734
$498$	0	0
$499$	19.4708i	0.871632i	0.900036	+	0.435816i	$0.143540\pi$
$499$	19.4708i	0.871632i	−0.900036	+	0.435816i	$0.856460\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.23494	−0.0550631	−0.0275316	−	0.999621i	$-0.508765\pi$
$503$	−1.23494	−0.0550631	−0.0275316	+	0.999621i	$0.508765\pi$
$504$	0	0
$505$	6.60772	0.294040
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.453647i	0.0201075i	0.999949	+	0.0100538i	$0.00320027\pi$
$509$	0.453647i	0.0201075i	−0.999949	+	0.0100538i	$0.996800\pi$
$510$	0	0
$511$	4.50325	0.199212
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.2628i	0.452233i
$516$	0	0
$517$	− 21.1285i	− 0.929231i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.0873	−1.27434	−0.637170	−	0.770724i	$-0.719895\pi$
$521$	−29.0873	−1.27434	−0.637170	+	0.770724i	$0.719895\pi$
$522$	0	0
$523$	− 2.95874i	− 0.129377i	−0.997906	−	0.0646883i	$-0.979395\pi$
$523$	− 2.95874i	− 0.129377i	0.997906	−	0.0646883i	$-0.0206053\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	41.8053	1.82107
$528$	0	0
$529$	20.2587	0.880815
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.2608i	0.444444i
$534$	0	0
$535$	5.45272	0.235741
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.2483i	0.527573i
$540$	0	0
$541$	− 14.9753i	− 0.643838i	−0.946767	−	0.321919i	$-0.895672\pi$
$541$	− 14.9753i	− 0.643838i	0.946767	−	0.321919i	$-0.104328\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.168722	0.00722724
$546$	0	0
$547$	18.2132i	0.778741i	0.921081	+	0.389370i	$0.127307\pi$
$547$	18.2132i	0.778741i	−0.921081	+	0.389370i	$0.872693\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.17106	−0.135092
$552$	0	0
$553$	9.15500	0.389310
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.7359i	1.51418i	0.653310	+	0.757090i	$0.273380\pi$
$557$	35.7359i	1.51418i	−0.653310	+	0.757090i	$0.726620\pi$
$558$	0	0
$559$	−19.0609	−0.806190
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.28675i	0.391390i	0.980665	+	0.195695i	$0.0626962\pi$
$563$	9.28675i	0.391390i	−0.980665	+	0.195695i	$0.937304\pi$
$564$	0	0
$565$	− 6.05267i	− 0.254638i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.3345	0.475168	0.237584	−	0.971367i	$-0.423644\pi$
$569$	11.3345	0.475168	0.237584	+	0.971367i	$0.423644\pi$
$570$	0	0
$571$	− 43.6296i	− 1.82584i	−0.408138	−	0.912920i	$-0.633822\pi$
$571$	− 43.6296i	− 1.82584i	0.408138	−	0.912920i	$-0.366178\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	29.6992	1.23854
$576$	0	0
$577$	6.98123	0.290632	0.145316	−	0.989385i	$-0.453580\pi$
$577$	6.98123	0.290632	0.145316	+	0.989385i	$0.453580\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.5571i	0.437983i
$582$	0	0
$583$	6.96248	0.288356
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.48213i	0.350095i	0.984560	+	0.175048i	$0.0560079\pi$
$587$	8.48213i	0.350095i	−0.984560	+	0.175048i	$0.943992\pi$
$588$	0	0
$589$	− 7.12562i	− 0.293606i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.40869	−0.386368	−0.193184	−	0.981163i	$-0.561881\pi$
$593$	−9.40869	−0.386368	−0.193184	+	0.981163i	$0.561881\pi$
$594$	0	0
$595$	− 6.25538i	− 0.256445i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.9247	−1.22269	−0.611344	−	0.791365i	$-0.709371\pi$
$599$	−29.9247	−1.22269	−0.611344	+	0.791365i	$0.709371\pi$
$600$	0	0
$601$	−3.63946	−0.148456	−0.0742282	−	0.997241i	$-0.523649\pi$
$601$	−3.63946	−0.148456	−0.0742282	+	0.997241i	$0.523649\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.43390i	0.0989520i
$606$	0	0
$607$	7.26716	0.294965	0.147482	−	0.989065i	$-0.452883\pi$
$607$	7.26716	0.294965	0.147482	+	0.989065i	$0.452883\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	42.4651i	1.71795i
$612$	0	0
$613$	− 32.6469i	− 1.31859i	−0.751882	−	0.659297i	$-0.770854\pi$
$613$	− 32.6469i	− 1.31859i	0.751882	−	0.659297i	$-0.229146\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.3502	−1.26211	−0.631056	−	0.775737i	$-0.717378\pi$
$617$	−31.3502	−1.26211	−0.631056	+	0.775737i	$0.717378\pi$
$618$	0	0
$619$	1.99289i	0.0801010i	0.999198	+	0.0400505i	$0.0127519\pi$
$619$	1.99289i	0.0801010i	−0.999198	+	0.0400505i	$0.987248\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.74973	−0.190294
$624$	0	0
$625$	17.9675	0.718700
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 35.3538i	− 1.40965i
$630$	0	0
$631$	15.4643	0.615623	0.307812	−	0.951447i	$-0.400403\pi$
$631$	15.4643	0.615623	0.307812	+	0.951447i	$0.400403\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.20245i	0.0477178i
$636$	0	0
$637$	− 24.6173i	− 0.975372i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.7275	0.976679	0.488340	−	0.872654i	$-0.337603\pi$
$641$	24.7275	0.976679	0.488340	+	0.872654i	$0.337603\pi$
$642$	0	0
$643$	− 46.2084i	− 1.82228i	−0.412096	−	0.911141i	$-0.635203\pi$
$643$	− 46.2084i	− 1.82228i	0.412096	−	0.911141i	$-0.364797\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.36971	0.250419	0.125210	−	0.992130i	$-0.460040\pi$
$647$	6.36971	0.250419	0.125210	+	0.992130i	$0.460040\pi$
$648$	0	0
$649$	−14.6135	−0.573630
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 36.8174i	− 1.44078i	−0.693571	−	0.720389i	$-0.743963\pi$
$653$	− 36.8174i	− 1.44078i	0.693571	−	0.720389i	$-0.256037\pi$
$654$	0	0
$655$	−4.62117	−0.180564
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.46589i	− 0.290830i	−0.989371	−	0.145415i	$-0.953548\pi$
$659$	− 7.46589i	− 0.290830i	0.989371	−	0.145415i	$-0.0464517\pi$
$660$	0	0
$661$	− 2.90965i	− 0.113172i	−0.998398	−	0.0565862i	$-0.981978\pi$
$661$	− 2.90965i	− 0.113172i	0.998398	−	0.0565862i	$-0.0180216\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.06622	−0.0413461
$666$	0	0
$667$	21.6505i	0.838310i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	25.1049	0.969165
$672$	0	0
$673$	−42.1054	−1.62304	−0.811522	−	0.584323i	$-0.801360\pi$
$673$	−42.1054	−1.62304	−0.811522	+	0.584323i	$0.801360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.0983i	1.46424i	0.681177	+	0.732119i	$0.261468\pi$
$677$	38.0983i	1.46424i	−0.681177	+	0.732119i	$0.738532\pi$
$678$	0	0
$679$	3.97015	0.152360
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 47.0728i	− 1.80119i	−0.434659	−	0.900595i	$-0.643131\pi$
$683$	− 47.0728i	− 1.80119i	0.434659	−	0.900595i	$-0.356869\pi$
$684$	0	0
$685$	− 2.52421i	− 0.0964450i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13.9935	−0.533111
$690$	0	0
$691$	− 3.90892i	− 0.148702i	−0.997232	−	0.0743512i	$-0.976311\pi$
$691$	− 3.90892i	− 0.148702i	0.997232	−	0.0743512i	$-0.0236886\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.0494	−0.457058
$696$	0	0
$697$	−10.5336	−0.398987
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 16.4480i	− 0.621231i	−0.950536	−	0.310615i	$-0.899465\pi$
$701$	− 16.4480i	− 0.621231i	0.950536	−	0.310615i	$-0.100535\pi$
$702$	0	0
$703$	−6.02598	−0.227274
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.0954i	− 0.567722i
$708$	0	0
$709$	− 7.90745i	− 0.296971i	−0.988915	−	0.148485i	$-0.952560\pi$
$709$	− 7.90745i	− 0.296971i	0.988915	−	0.148485i	$-0.0474398\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−48.6503	−1.82197
$714$	0	0
$715$	10.4967i	0.392553i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	37.0556	1.38194	0.690970	−	0.722884i	$-0.257184\pi$
$719$	37.0556	1.38194	0.690970	+	0.722884i	$0.257184\pi$
$720$	0	0
$721$	23.4455	0.873156
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14.8641i	0.552038i
$726$	0	0
$727$	5.66934	0.210264	0.105132	−	0.994458i	$-0.466473\pi$
$727$	5.66934	0.210264	0.105132	+	0.994458i	$0.466473\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 19.5676i	− 0.723735i
$732$	0	0
$733$	− 12.5335i	− 0.462937i	−0.972842	−	0.231469i	$-0.925647\pi$
$733$	− 12.5335i	− 0.462937i	0.972842	−	0.231469i	$-0.0743530\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.8443	0.694139
$738$	0	0
$739$	1.83358i	0.0674492i	0.999431	+	0.0337246i	$0.0107369\pi$
$739$	1.83358i	0.0674492i	−0.999431	+	0.0337246i	$0.989263\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.3177	−1.14893	−0.574467	−	0.818528i	$-0.694791\pi$
$743$	−31.3177	−1.14893	−0.574467	+	0.818528i	$0.694791\pi$
$744$	0	0
$745$	−15.0498	−0.551382
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 12.4568i	− 0.455162i
$750$	0	0
$751$	7.28932	0.265991	0.132996	−	0.991117i	$-0.457540\pi$
$751$	7.28932	0.265991	0.132996	+	0.991117i	$0.457540\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 8.84005i	− 0.321722i
$756$	0	0
$757$	− 12.8156i	− 0.465792i	−0.972502	−	0.232896i	$-0.925180\pi$
$757$	− 12.8156i	− 0.465792i	0.972502	−	0.232896i	$-0.0748202\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.1600	−0.912050	−0.456025	−	0.889967i	$-0.650727\pi$
$761$	−25.1600	−0.912050	−0.456025	+	0.889967i	$0.650727\pi$
$762$	0	0
$763$	− 0.385447i	− 0.0139541i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29.3709	1.06052
$768$	0	0
$769$	21.4636	0.773996	0.386998	−	0.922081i	$-0.373512\pi$
$769$	21.4636	0.773996	0.386998	+	0.922081i	$0.373512\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.2122i	0.726981i	0.931598	+	0.363491i	$0.118415\pi$
$773$	20.2122i	0.726981i	−0.931598	+	0.363491i	$0.881585\pi$
$774$	0	0
$775$	−33.4007	−1.19979
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.79542i	0.0643277i
$780$	0	0
$781$	10.0899i	0.361045i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.1591	−0.433978
$786$	0	0
$787$	20.2911i	0.723299i	0.932314	+	0.361649i	$0.117786\pi$
$787$	20.2911i	0.723299i	−0.932314	+	0.361649i	$0.882214\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.8274	−0.491646
$792$	0	0
$793$	−50.4570	−1.79178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.3429i	1.18107i	0.807013	+	0.590533i	$0.201083\pi$
$797$	33.3429i	1.18107i	−0.807013	+	0.590533i	$0.798917\pi$
$798$	0	0
$799$	−43.5940	−1.54224
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 7.75745i	− 0.273754i
$804$	0	0
$805$	7.27960i	0.256572i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.6920	1.07907	0.539536	−	0.841962i	$-0.318600\pi$
$809$	30.6920	1.07907	0.539536	+	0.841962i	$0.318600\pi$
$810$	0	0
$811$	− 49.5457i	− 1.73978i	−0.493241	−	0.869892i	$-0.664188\pi$
$811$	− 49.5457i	− 1.73978i	0.493241	−	0.869892i	$-0.335812\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.96337	−0.208888
$816$	0	0
$817$	−3.33526	−0.116686
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 38.0750i	− 1.32883i	−0.747365	−	0.664414i	$-0.768681\pi$
$821$	− 38.0750i	− 1.32883i	0.747365	−	0.664414i	$-0.231319\pi$
$822$	0	0
$823$	22.5252	0.785178	0.392589	−	0.919714i	$-0.371579\pi$
$823$	22.5252	0.785178	0.392589	+	0.919714i	$0.371579\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.5317i	1.16601i	0.812468	+	0.583006i	$0.198124\pi$
$827$	33.5317i	1.16601i	−0.812468	+	0.583006i	$0.801876\pi$
$828$	0	0
$829$	37.7559i	1.31132i	0.755058	+	0.655658i	$0.227609\pi$
$829$	37.7559i	1.31132i	−0.755058	+	0.655658i	$0.772391\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25.2717	0.875614
$834$	0	0
$835$	− 8.31821i	− 0.287863i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.8121	−0.683989	−0.341994	−	0.939702i	$-0.611102\pi$
$839$	−19.8121	−0.683989	−0.341994	+	0.939702i	$0.611102\pi$
$840$	0	0
$841$	18.1642	0.626352
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 12.0481i	− 0.414466i
$846$	0	0
$847$	5.56028	0.191053
$848$	0	0
$849$	0	0
$850$	0	0
$851$	41.1425i	1.41035i
$852$	0	0
$853$	− 6.87537i	− 0.235408i	−0.993049	−	0.117704i	$-0.962447\pi$
$853$	− 6.87537i	− 0.235408i	0.993049	−	0.117704i	$-0.0375534\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.74632	0.264609	0.132305	−	0.991209i	$-0.457762\pi$
$857$	7.74632	0.264609	0.132305	+	0.991209i	$0.457762\pi$
$858$	0	0
$859$	0.686576i	0.0234257i	0.999931	+	0.0117128i	$0.00372839\pi$
$859$	0.686576i	0.0234257i	−0.999931	+	0.0117128i	$0.996272\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.9194	1.46099	0.730496	−	0.682917i	$-0.239289\pi$
$863$	42.9194	1.46099	0.730496	+	0.682917i	$0.239289\pi$
$864$	0	0
$865$	−9.07991	−0.308726
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 15.7707i	− 0.534984i
$870$	0	0
$871$	−37.8742	−1.28332
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.5318i	0.356040i
$876$	0	0
$877$	− 17.0328i	− 0.575157i	−0.957757	−	0.287578i	$-0.907150\pi$
$877$	− 17.0328i	− 0.575157i	0.957757	−	0.287578i	$-0.0928501\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.90546	−0.266342	−0.133171	−	0.991093i	$-0.542516\pi$
$881$	−7.90546	−0.266342	−0.133171	+	0.991093i	$0.542516\pi$
$882$	0	0
$883$	− 7.53298i	− 0.253505i	−0.991934	−	0.126752i	$-0.959545\pi$
$883$	− 7.53298i	− 0.253505i	0.991934	−	0.126752i	$-0.0404554\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.0544	0.471900	0.235950	−	0.971765i	$-0.424180\pi$
$887$	14.0544	0.471900	0.235950	+	0.971765i	$0.424180\pi$
$888$	0	0
$889$	2.74702	0.0921320
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.43051i	0.248652i
$894$	0	0
$895$	−2.31001	−0.0772150
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 24.3489i	− 0.812081i
$900$	0	0
$901$	− 14.3655i	− 0.478585i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.3800	−0.345043
$906$	0	0
$907$	− 45.9522i	− 1.52582i	−0.646505	−	0.762910i	$-0.723770\pi$
$907$	− 45.9522i	− 1.52582i	0.646505	−	0.762910i	$-0.276230\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.5822	1.70899	0.854497	−	0.519456i	$-0.173865\pi$
$911$	51.5822	1.70899	0.854497	+	0.519456i	$0.173865\pi$
$912$	0	0
$913$	18.1860	0.601870
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.5571i	0.348627i
$918$	0	0
$919$	21.2048	0.699481	0.349741	−	0.936847i	$-0.386270\pi$
$919$	21.2048	0.699481	0.349741	+	0.936847i	$0.386270\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 20.2792i	− 0.667497i
$924$	0	0
$925$	28.2463i	0.928732i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.41033	−0.111889	−0.0559446	−	0.998434i	$-0.517817\pi$
$929$	−3.41033	−0.111889	−0.0559446	+	0.998434i	$0.517817\pi$
$930$	0	0
$931$	− 4.30751i	− 0.141173i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.7757	−0.352403
$936$	0	0
$937$	29.4448	0.961919	0.480959	−	0.876743i	$-0.340288\pi$
$937$	29.4448	0.961919	0.480959	+	0.876743i	$0.340288\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.8670i	1.52782i	0.645323	+	0.763910i	$0.276723\pi$
$941$	46.8670i	1.52782i	−0.645323	+	0.763910i	$0.723277\pi$
$942$	0	0
$943$	12.2583	0.399185
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.4679i	1.18505i	0.805553	+	0.592524i	$0.201869\pi$
$947$	36.4679i	1.18505i	−0.805553	+	0.592524i	$0.798131\pi$
$948$	0	0
$949$	15.5913i	0.506115i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.0590	1.23285	0.616426	−	0.787413i	$-0.288580\pi$
$953$	38.0590	1.23285	0.616426	+	0.787413i	$0.288580\pi$
$954$	0	0
$955$	5.09344i	0.164820i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.76659	−0.186213
$960$	0	0
$961$	23.7138	0.764962
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.2486i	0.458679i
$966$	0	0
$967$	22.9728	0.738756	0.369378	−	0.929279i	$-0.379571\pi$
$967$	22.9728	0.738756	0.369378	+	0.929279i	$0.379571\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 53.8829i	− 1.72919i	−0.502474	−	0.864593i	$-0.667577\pi$
$971$	− 53.8829i	− 1.72919i	0.502474	−	0.864593i	$-0.332423\pi$
$972$	0	0
$973$	27.5269i	0.882473i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38.2048	−1.22228	−0.611140	−	0.791523i	$-0.709289\pi$
$977$	−38.2048	−1.22228	−0.611140	+	0.791523i	$0.709289\pi$
$978$	0	0
$979$	8.18203i	0.261499i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.6614	−1.55206	−0.776028	−	0.630698i	$-0.782769\pi$
$983$	−48.6614	−1.55206	−0.776028	+	0.630698i	$0.782769\pi$
$984$	0	0
$985$	14.2994	0.455615
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.7715i	0.724093i
$990$	0	0
$991$	12.7822	0.406040	0.203020	−	0.979175i	$-0.434924\pi$
$991$	12.7822	0.406040	0.203020	+	0.979175i	$0.434924\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.36145i	− 0.0431608i
$996$	0	0
$997$	24.3827i	0.772209i	0.922455	+	0.386104i	$0.126180\pi$
$997$	24.3827i	0.772209i	−0.922455	+	0.386104i	$0.873820\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.d.j.1297.5		8
3.2	odd	2		2592.2.d.k.1297.4		8
4.3	odd	2		648.2.d.j.325.5		8
8.3	odd	2		648.2.d.j.325.6		8
8.5	even	2	inner	2592.2.d.j.1297.4		8
9.2	odd	6		864.2.r.b.145.5		16
9.4	even	3		288.2.r.b.241.7		16
9.5	odd	6		864.2.r.b.721.4		16
9.7	even	3		288.2.r.b.49.2		16
12.11	even	2		648.2.d.k.325.4		8
24.5	odd	2		2592.2.d.k.1297.5		8
24.11	even	2		648.2.d.k.325.3		8
36.7	odd	6		72.2.n.b.13.1	✓	16
36.11	even	6		216.2.n.b.37.8		16
36.23	even	6		216.2.n.b.181.2		16
36.31	odd	6		72.2.n.b.61.7	yes	16
72.5	odd	6		864.2.r.b.721.5		16
72.11	even	6		216.2.n.b.37.2		16
72.13	even	6		288.2.r.b.241.2		16
72.29	odd	6		864.2.r.b.145.4		16
72.43	odd	6		72.2.n.b.13.7	yes	16
72.59	even	6		216.2.n.b.181.8		16
72.61	even	6		288.2.r.b.49.7		16
72.67	odd	6		72.2.n.b.61.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.n.b.13.1	✓	16	36.7	odd	6
72.2.n.b.13.7	yes	16	72.43	odd	6
72.2.n.b.61.1	yes	16	72.67	odd	6
72.2.n.b.61.7	yes	16	36.31	odd	6
216.2.n.b.37.2		16	72.11	even	6
216.2.n.b.37.8		16	36.11	even	6
216.2.n.b.181.2		16	36.23	even	6
216.2.n.b.181.8		16	72.59	even	6
288.2.r.b.49.2		16	9.7	even	3
288.2.r.b.49.7		16	72.61	even	6
288.2.r.b.241.2		16	72.13	even	6
288.2.r.b.241.7		16	9.4	even	3
648.2.d.j.325.5		8	4.3	odd	2
648.2.d.j.325.6		8	8.3	odd	2
648.2.d.k.325.3		8	24.11	even	2
648.2.d.k.325.4		8	12.11	even	2
864.2.r.b.145.4		16	72.29	odd	6
864.2.r.b.145.5		16	9.2	odd	6
864.2.r.b.721.4		16	9.5	odd	6
864.2.r.b.721.5		16	72.5	odd	6
2592.2.d.j.1297.4		8	8.5	even	2	inner
2592.2.d.j.1297.5		8	1.1	even	1	trivial
2592.2.d.k.1297.4		8	3.2	odd	2
2592.2.d.k.1297.5		8	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.696047i q^{5} +1.59013 q^{7} +O(q^{10})\)	\(q+0.696047i q^{5} +1.59013 q^{7} -2.73921i q^{11} +5.50539i q^{13} -5.65175 q^{17} +0.963328i q^{19} +6.57714 q^{23} +4.51552 q^{25} +3.29178i q^{29} -7.39688 q^{31} +1.10680i q^{35} +6.25538i q^{37} +1.86377 q^{41} +3.46223i q^{43} +7.71337 q^{47} -4.47149 q^{49} +2.54179i q^{53} +1.90662 q^{55} -5.33494i q^{59} +9.16503i q^{61} -3.83201 q^{65} +6.87947i q^{67} -3.68351 q^{71} +2.83201 q^{73} -4.35569i q^{77} +5.75740 q^{79} +6.63916i q^{83} -3.93388i q^{85} -2.98701 q^{89} +8.75427i q^{91} -0.670522 q^{95} +2.49675 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{7}+O(q^{10}) \) 8 * q + 6 * q^7	\( 8 q + 6 q^{7} - 14 q^{17} - 10 q^{23} - 2 q^{25} - 10 q^{31} + 8 q^{41} + 6 q^{47} - 18 q^{49} + 2 q^{55} + 14 q^{65} - 36 q^{71} - 22 q^{73} - 30 q^{79} + 32 q^{89} + 44 q^{95}+O(q^{100}) \) 8 * q + 6 * q^7 - 14 * q^17 - 10 * q^23 - 2 * q^25 - 10 * q^31 + 8 * q^41 + 6 * q^47 - 18 * q^49 + 2 * q^55 + 14 * q^65 - 36 * q^71 - 22 * q^73 - 30 * q^79 + 32 * q^89 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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