LMFDB - Embedded newform 4896.2.f.d.2449.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4896,2,Mod(2449,4896)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4896, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4896.2449"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4896 = 2^{5} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4896.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.0947568296$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4469724736.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 2x^{5} - 4x^{4} + 4x^{3} - 8x + 16 $ x^8 - x^7 + 2x^5 - 4x^4 + 4x^3 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.6
Root			$1.33209 - 0.474920i$ of defining polynomial
Character	$\chi$	$=$	4896.2449
Dual form			4896.2.f.d.2449.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58069i q^{5} +5.01127 q^{7} +2.65428i q^{11} -6.34060i q^{13} -1.00000 q^{17} +2.44834i q^{19} +2.83821 q^{23} +2.50141 q^{25} -1.58069i q^{29} -1.68293 q^{31} +7.92129i q^{35} +1.16881i q^{37} +3.50141 q^{41} +6.52280i q^{43} +8.82975 q^{47} +18.1129 q^{49} -7.37263i q^{53} -4.19561 q^{55} -10.7815i q^{59} +1.58069i q^{61} +10.0225 q^{65} -9.82097i q^{67} +2.51268 q^{71} -2.52114 q^{73} +13.3013i q^{77} -0.815662 q^{79} +8.22246i q^{83} -1.58069i q^{85} -1.45337 q^{89} -31.7745i q^{91} -3.87008 q^{95} -7.15528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{7} - 8 q^{17} - 16 q^{23} - 8 q^{25} - 24 q^{31} + 4 q^{47} + 8 q^{49} + 12 q^{55} + 24 q^{65} - 36 q^{71} + 8 q^{73} - 24 q^{79} - 8 q^{89} + 32 q^{95} - 8 q^{97}+O(q^{100}) $ 8 * q + 12 * q^7 - 8 * q^17 - 16 * q^23 - 8 * q^25 - 24 * q^31 + 4 * q^47 + 8 * q^49 + 12 * q^55 + 24 * q^65 - 36 * q^71 + 8 * q^73 - 24 * q^79 - 8 * q^89 + 32 * q^95 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4896\mathbb{Z}\right)^\times$.

$n$	$613$	$2143$	$3809$	$4321$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	1.58069i	0.706908i	0.935452	+	0.353454i	$0.114993\pi$
$5$	1.58069i	0.706908i	−0.935452	+	0.353454i	$0.885007\pi$
$6$	0	0
$7$	5.01127	1.89408	0.947042	−	0.321110i	$-0.104056\pi$
$7$	5.01127	1.89408	0.947042	+	0.321110i	$0.104056\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.65428i	0.800297i	0.916450	+	0.400148i	$0.131041\pi$
$11$	2.65428i	0.800297i	−0.916450	+	0.400148i	$0.868959\pi$
$12$	0	0
$13$	− 6.34060i	− 1.75857i	−0.476300	−	0.879283i	$-0.658022\pi$
$13$	− 6.34060i	− 1.75857i	0.476300	−	0.879283i	$-0.341978\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.44834i	0.561688i	0.959753	+	0.280844i	$0.0906144\pi$
$19$	2.44834i	0.561688i	−0.959753	+	0.280844i	$0.909386\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.83821	0.591808	0.295904	−	0.955218i	$-0.404379\pi$
$23$	2.83821	0.591808	0.295904	+	0.955218i	$0.404379\pi$
$24$	0	0
$25$	2.50141	0.500281
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.58069i	− 0.293528i	−0.989172	−	0.146764i	$-0.953114\pi$
$29$	− 1.58069i	− 0.293528i	0.989172	−	0.146764i	$-0.0468857\pi$
$30$	0	0
$31$	−1.68293	−0.302264	−0.151132	−	0.988514i	$-0.548292\pi$
$31$	−1.68293	−0.302264	−0.151132	+	0.988514i	$0.548292\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.92129i	1.33894i
$36$	0	0
$37$	1.16881i	0.192151i	0.995374	+	0.0960757i	$0.0306291\pi$
$37$	1.16881i	0.192151i	−0.995374	+	0.0960757i	$0.969371\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.50141	0.546828	0.273414	−	0.961897i	$-0.411847\pi$
$41$	3.50141	0.546828	0.273414	+	0.961897i	$0.411847\pi$
$42$	0	0
$43$	6.52280i	0.994718i	0.867545	+	0.497359i	$0.165697\pi$
$43$	6.52280i	0.994718i	−0.867545	+	0.497359i	$0.834303\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.82975	1.28795	0.643975	−	0.765046i	$-0.277284\pi$
$47$	8.82975	1.28795	0.643975	+	0.765046i	$0.277284\pi$
$48$	0	0
$49$	18.1129	2.58755
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.37263i	− 1.01271i	−0.862326	−	0.506354i	$-0.830993\pi$
$53$	− 7.37263i	− 1.01271i	0.862326	−	0.506354i	$-0.169007\pi$
$54$	0	0
$55$	−4.19561	−0.565736
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.7815i	− 1.40363i	−0.712357	−	0.701817i	$-0.752372\pi$
$59$	− 10.7815i	− 1.40363i	0.712357	−	0.701817i	$-0.247628\pi$
$60$	0	0
$61$	1.58069i	0.202387i	0.994867	+	0.101194i	$0.0322661\pi$
$61$	1.58069i	0.202387i	−0.994867	+	0.101194i	$0.967734\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.0225	1.24314
$66$	0	0
$67$	− 9.82097i	− 1.19982i	−0.800067	−	0.599911i	$-0.795203\pi$
$67$	− 9.82097i	− 1.19982i	0.800067	−	0.599911i	$-0.204797\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.51268	0.298200	0.149100	−	0.988822i	$-0.452362\pi$
$71$	2.51268	0.298200	0.149100	+	0.988822i	$0.452362\pi$
$72$	0	0
$73$	−2.52114	−0.295077	−0.147539	−	0.989056i	$-0.547135\pi$
$73$	−2.52114	−0.295077	−0.147539	+	0.989056i	$0.547135\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.3013i	1.51583i
$78$	0	0
$79$	−0.815662	−0.0917692	−0.0458846	−	0.998947i	$-0.514611\pi$
$79$	−0.815662	−0.0917692	−0.0458846	+	0.998947i	$0.514611\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.22246i	0.902532i	0.892390	+	0.451266i	$0.149027\pi$
$83$	8.22246i	0.902532i	−0.892390	+	0.451266i	$0.850973\pi$
$84$	0	0
$85$	− 1.58069i	− 0.171450i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.45337	−0.154057	−0.0770286	−	0.997029i	$-0.524543\pi$
$89$	−1.45337	−0.154057	−0.0770286	+	0.997029i	$0.524543\pi$
$90$	0	0
$91$	− 31.7745i	− 3.33087i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.87008	−0.397062
$96$	0	0
$97$	−7.15528	−0.726508	−0.363254	−	0.931690i	$-0.618334\pi$
$97$	−7.15528	−0.726508	−0.363254	+	0.931690i	$0.618334\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.67822i	0.863515i	0.901990	+	0.431758i	$0.142106\pi$
$101$	8.67822i	0.863515i	−0.901990	+	0.431758i	$0.857894\pi$
$102$	0	0
$103$	7.13273	0.702809	0.351404	−	0.936224i	$-0.385704\pi$
$103$	7.13273	0.702809	0.351404	+	0.936224i	$0.385704\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.26701i	− 0.509181i	−0.967049	−	0.254590i	$-0.918059\pi$
$107$	− 5.26701i	− 0.509181i	0.967049	−	0.254590i	$-0.0819407\pi$
$108$	0	0
$109$	8.35100i	0.799880i	0.916541	+	0.399940i	$0.130969\pi$
$109$	8.35100i	0.799880i	−0.916541	+	0.399940i	$0.869031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.02255	0.754698	0.377349	−	0.926071i	$-0.376836\pi$
$113$	8.02255	0.754698	0.377349	+	0.926071i	$0.376836\pi$
$114$	0	0
$115$	4.48634i	0.418354i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.01127	−0.459383
$120$	0	0
$121$	3.95478	0.359525
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.8574i	1.06056i
$126$	0	0
$127$	−13.5014	−1.19806	−0.599028	−	0.800728i	$-0.704446\pi$
$127$	−13.5014	−1.19806	−0.599028	+	0.800728i	$0.704446\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.90498i	0.515920i	0.966156	+	0.257960i	$0.0830503\pi$
$131$	5.90498i	0.515920i	−0.966156	+	0.257960i	$0.916950\pi$
$132$	0	0
$133$	12.2693i	1.06388i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.56636	−0.133824	−0.0669118	−	0.997759i	$-0.521315\pi$
$137$	−1.56636	−0.133824	−0.0669118	+	0.997759i	$0.521315\pi$
$138$	0	0
$139$	3.29225i	0.279245i	0.990205	+	0.139623i	$0.0445889\pi$
$139$	3.29225i	0.279245i	−0.990205	+	0.139623i	$0.955411\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.8297	1.40737
$144$	0	0
$145$	2.49859	0.207497
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.70203i	0.221359i	0.993856	+	0.110679i	$0.0353027\pi$
$149$	2.70203i	0.221359i	−0.993856	+	0.110679i	$0.964697\pi$
$150$	0	0
$151$	−2.88982	−0.235170	−0.117585	−	0.993063i	$-0.537515\pi$
$151$	−2.88982	−0.235170	−0.117585	+	0.993063i	$0.537515\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.66020i	− 0.213673i
$156$	0	0
$157$	1.01421i	0.0809428i	0.999181	+	0.0404714i	$0.0128860\pi$
$157$	1.01421i	0.0809428i	−0.999181	+	0.0404714i	$0.987114\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	14.2230	1.12093
$162$	0	0
$163$	15.5259i	1.21608i	0.793905	+	0.608042i	$0.208045\pi$
$163$	15.5259i	1.21608i	−0.793905	+	0.608042i	$0.791955\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.9011	−1.23046	−0.615232	−	0.788346i	$-0.710938\pi$
$167$	−15.9011	−1.23046	−0.615232	+	0.788346i	$0.710938\pi$
$168$	0	0
$169$	−27.2032	−2.09255
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 21.1752i	− 1.60992i	−0.593331	−	0.804959i	$-0.702187\pi$
$173$	− 21.1752i	− 1.60992i	0.593331	−	0.804959i	$-0.297813\pi$
$174$	0	0
$175$	12.5352	0.947574
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.5913i	1.24009i	0.784566	+	0.620045i	$0.212886\pi$
$179$	16.5913i	1.24009i	−0.784566	+	0.620045i	$0.787114\pi$
$180$	0	0
$181$	− 11.7859i	− 0.876042i	−0.898965	−	0.438021i	$-0.855679\pi$
$181$	− 11.7859i	− 0.876042i	0.898965	−	0.438021i	$-0.144321\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.84753	−0.135833
$186$	0	0
$187$	− 2.65428i	− 0.194100i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.96248	0.576145	0.288072	−	0.957609i	$-0.406986\pi$
$191$	7.96248	0.576145	0.288072	+	0.957609i	$0.406986\pi$
$192$	0	0
$193$	25.2229	1.81559	0.907793	−	0.419419i	$-0.137766\pi$
$193$	25.2229	1.81559	0.907793	+	0.419419i	$0.137766\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.8833i	− 0.917895i	−0.888463	−	0.458948i	$-0.848227\pi$
$197$	− 12.8833i	− 0.917895i	0.888463	−	0.458948i	$-0.151773\pi$
$198$	0	0
$199$	−0.339615	−0.0240747	−0.0120373	−	0.999928i	$-0.503832\pi$
$199$	−0.339615	−0.0240747	−0.0120373	+	0.999928i	$0.503832\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.92129i	− 0.555966i
$204$	0	0
$205$	5.53465i	0.386557i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.49859	−0.449517
$210$	0	0
$211$	6.36434i	0.438139i	0.975709	+	0.219069i	$0.0703021\pi$
$211$	6.36434i	0.438139i	−0.975709	+	0.219069i	$0.929698\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.3106	−0.703174
$216$	0	0
$217$	−8.43364	−0.572512
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.34060i	0.426515i
$222$	0	0
$223$	−13.0629	−0.874755	−0.437378	−	0.899278i	$-0.644093\pi$
$223$	−13.0629	−0.874755	−0.437378	+	0.899278i	$0.644093\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 20.3705i	− 1.35204i	−0.736885	−	0.676018i	$-0.763704\pi$
$227$	− 20.3705i	− 1.35204i	0.736885	−	0.676018i	$-0.236296\pi$
$228$	0	0
$229$	17.9245i	1.18448i	0.805761	+	0.592241i	$0.201757\pi$
$229$	17.9245i	1.18448i	−0.805761	+	0.592241i	$0.798243\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.8926	−1.04116	−0.520580	−	0.853813i	$-0.674284\pi$
$233$	−15.8926	−1.04116	−0.520580	+	0.853813i	$0.674284\pi$
$234$	0	0
$235$	13.9571i	0.910463i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.3537	−1.05783	−0.528916	−	0.848674i	$-0.677402\pi$
$239$	−16.3537	−1.05783	−0.528916	+	0.848674i	$0.677402\pi$
$240$	0	0
$241$	29.6820	1.91199	0.955994	−	0.293386i	$-0.0947820\pi$
$241$	29.6820	1.91199	0.955994	+	0.293386i	$0.0947820\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	28.6309i	1.82916i
$246$	0	0
$247$	15.5240	0.987765
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.7425i	1.49861i	0.662225	+	0.749305i	$0.269612\pi$
$251$	23.7425i	1.49861i	−0.662225	+	0.749305i	$0.730388\pi$
$252$	0	0
$253$	7.53342i	0.473622i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.9548	1.18237	0.591183	−	0.806537i	$-0.298661\pi$
$257$	18.9548	1.18237	0.591183	+	0.806537i	$0.298661\pi$
$258$	0	0
$259$	5.85723i	0.363951i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.54650	0.588662	0.294331	−	0.955703i	$-0.404903\pi$
$263$	9.54650	0.588662	0.294331	+	0.955703i	$0.404903\pi$
$264$	0	0
$265$	11.6539	0.715892
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.3467i	1.24056i	0.784379	+	0.620282i	$0.212982\pi$
$269$	20.3467i	1.24056i	−0.784379	+	0.620282i	$0.787018\pi$
$270$	0	0
$271$	−28.8748	−1.75402	−0.877011	−	0.480471i	$-0.840466\pi$
$271$	−28.8748	−1.75402	−0.877011	+	0.480471i	$0.840466\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.63944i	0.400373i
$276$	0	0
$277$	− 10.4151i	− 0.625780i	−0.949789	−	0.312890i	$-0.898703\pi$
$277$	− 10.4151i	− 0.625780i	0.949789	−	0.312890i	$-0.101297\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.2032	1.80177	0.900885	−	0.434057i	$-0.142918\pi$
$281$	30.2032	1.80177	0.900885	+	0.434057i	$0.142918\pi$
$282$	0	0
$283$	− 26.6458i	− 1.58393i	−0.610568	−	0.791964i	$-0.709059\pi$
$283$	− 26.6458i	− 1.58393i	0.610568	−	0.791964i	$-0.290941\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.5465	1.03574
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.52580i	0.205979i	0.994682	+	0.102990i	$0.0328408\pi$
$293$	3.52580i	0.205979i	−0.994682	+	0.102990i	$0.967159\pi$
$294$	0	0
$295$	17.0423	0.992240
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 17.9959i	− 1.04073i
$300$	0	0
$301$	32.6875i	1.88408i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.49859	−0.143069
$306$	0	0
$307$	− 1.34483i	− 0.0767534i	−0.999263	−	0.0383767i	$-0.987781\pi$
$307$	− 1.34483i	− 0.0767534i	0.999263	−	0.0383767i	$-0.0122187\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.5803	1.84746	0.923730	−	0.383044i	$-0.125124\pi$
$311$	32.5803	1.84746	0.923730	+	0.383044i	$0.125124\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 25.8814i	− 1.45364i	−0.686826	−	0.726822i	$-0.740996\pi$
$317$	− 25.8814i	− 1.45364i	0.686826	−	0.726822i	$-0.259004\pi$
$318$	0	0
$319$	4.19561	0.234909
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.44834i	− 0.136229i
$324$	0	0
$325$	− 15.8604i	− 0.879777i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	44.2483	2.43949
$330$	0	0
$331$	1.26171i	0.0693499i	0.999399	+	0.0346749i	$0.0110396\pi$
$331$	1.26171i	0.0693499i	−0.999399	+	0.0346749i	$0.988960\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	15.5240	0.848164
$336$	0	0
$337$	17.8120	0.970279	0.485140	−	0.874437i	$-0.338769\pi$
$337$	17.8120	0.970279	0.485140	+	0.874437i	$0.338769\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4.46698i	− 0.241901i
$342$	0	0
$343$	55.6896	3.00696
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.34210i	0.233096i	0.993185	+	0.116548i	$0.0371829\pi$
$347$	4.34210i	0.233096i	−0.993185	+	0.116548i	$0.962817\pi$
$348$	0	0
$349$	12.4908i	0.668615i	0.942464	+	0.334307i	$0.108502\pi$
$349$	12.4908i	0.668615i	−0.942464	+	0.334307i	$0.891498\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.2782	−0.547055	−0.273527	−	0.961864i	$-0.588190\pi$
$353$	−10.2782	−0.547055	−0.273527	+	0.961864i	$0.588190\pi$
$354$	0	0
$355$	3.97178i	0.210800i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.45826	0.393632	0.196816	−	0.980440i	$-0.436940\pi$
$359$	7.45826	0.393632	0.196816	+	0.980440i	$0.436940\pi$
$360$	0	0
$361$	13.0056	0.684506
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.98516i	− 0.208593i
$366$	0	0
$367$	6.92083	0.361264	0.180632	−	0.983551i	$-0.442186\pi$
$367$	6.92083	0.361264	0.180632	+	0.983551i	$0.442186\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 36.9463i	− 1.91815i
$372$	0	0
$373$	− 22.0882i	− 1.14369i	−0.820363	−	0.571843i	$-0.806229\pi$
$373$	− 22.0882i	− 1.14369i	0.820363	−	0.571843i	$-0.193771\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.0225	−0.516187
$378$	0	0
$379$	35.4430i	1.82058i	0.413968	+	0.910291i	$0.364143\pi$
$379$	35.4430i	1.82058i	−0.413968	+	0.910291i	$0.635857\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.5493	−1.45880	−0.729401	−	0.684087i	$-0.760201\pi$
$383$	−28.5493	−1.45880	−0.729401	+	0.684087i	$0.760201\pi$
$384$	0	0
$385$	−21.0254	−1.07155
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.0218i	0.964443i	0.876049	+	0.482222i	$0.160170\pi$
$389$	19.0218i	0.964443i	−0.876049	+	0.482222i	$0.839830\pi$
$390$	0	0
$391$	−2.83821	−0.143534
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.28931i	− 0.0648724i
$396$	0	0
$397$	6.52486i	0.327473i	0.986504	+	0.163737i	$0.0523547\pi$
$397$	6.52486i	0.327473i	−0.986504	+	0.163737i	$0.947645\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.41096	0.270210	0.135105	−	0.990831i	$-0.456863\pi$
$401$	5.41096	0.270210	0.135105	+	0.990831i	$0.456863\pi$
$402$	0	0
$403$	10.6708i	0.531550i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.10236	−0.153778
$408$	0	0
$409$	−28.4307	−1.40581	−0.702904	−	0.711285i	$-0.748114\pi$
$409$	−28.4307	−1.40581	−0.702904	+	0.711285i	$0.748114\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 54.0291i	− 2.65860i
$414$	0	0
$415$	−12.9972	−0.638007
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.4224i	0.704581i	0.935891	+	0.352290i	$0.114597\pi$
$419$	14.4224i	0.704581i	−0.935891	+	0.352290i	$0.885403\pi$
$420$	0	0
$421$	− 21.2645i	− 1.03637i	−0.855270	−	0.518183i	$-0.826609\pi$
$421$	− 21.2645i	− 1.03637i	0.855270	−	0.518183i	$-0.173391\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.50141	−0.121336
$426$	0	0
$427$	7.92129i	0.383338i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.4752	0.889917	0.444958	−	0.895551i	$-0.353218\pi$
$431$	18.4752	0.889917	0.444958	+	0.895551i	$0.353218\pi$
$432$	0	0
$433$	−27.1580	−1.30513	−0.652564	−	0.757733i	$-0.726307\pi$
$433$	−27.1580	−1.30513	−0.652564	+	0.757733i	$0.726307\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.94891i	0.332411i
$438$	0	0
$439$	−2.29647	−0.109605	−0.0548023	−	0.998497i	$-0.517453\pi$
$439$	−2.29647	−0.109605	−0.0548023	+	0.998497i	$0.517453\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 14.9912i	− 0.712254i	−0.934438	−	0.356127i	$-0.884097\pi$
$443$	− 14.9912i	− 0.712254i	0.934438	−	0.356127i	$-0.115903\pi$
$444$	0	0
$445$	− 2.29734i	− 0.108904i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.6567	0.974849	0.487425	−	0.873165i	$-0.337936\pi$
$449$	20.6567	0.974849	0.487425	+	0.873165i	$0.337936\pi$
$450$	0	0
$451$	9.29372i	0.437624i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	50.2257	2.35462
$456$	0	0
$457$	26.4206	1.23590	0.617952	−	0.786216i	$-0.287963\pi$
$457$	26.4206	1.23590	0.617952	+	0.786216i	$0.287963\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.2364i	1.82742i	0.406365	+	0.913711i	$0.366796\pi$
$461$	39.2364i	1.82742i	−0.406365	+	0.913711i	$0.633204\pi$
$462$	0	0
$463$	−7.30774	−0.339620	−0.169810	−	0.985477i	$-0.554315\pi$
$463$	−7.30774	−0.339620	−0.169810	+	0.985477i	$0.554315\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.3923i	1.12874i	0.825522	+	0.564370i	$0.190881\pi$
$467$	24.3923i	1.12874i	−0.825522	+	0.564370i	$0.809119\pi$
$468$	0	0
$469$	− 49.2156i	− 2.27256i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17.3134	−0.796069
$474$	0	0
$475$	6.12430i	0.281002i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.68012	−0.305223	−0.152611	−	0.988286i	$-0.548768\pi$
$479$	−6.68012	−0.305223	−0.152611	+	0.988286i	$0.548768\pi$
$480$	0	0
$481$	7.41096	0.337911
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 11.3103i	− 0.513575i
$486$	0	0
$487$	−17.2314	−0.780829	−0.390414	−	0.920639i	$-0.627668\pi$
$487$	−17.2314	−0.780829	−0.390414	+	0.920639i	$0.627668\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 8.73548i	− 0.394227i	−0.980381	−	0.197113i	$-0.936843\pi$
$491$	− 8.73548i	− 0.394227i	0.980381	−	0.197113i	$-0.0631567\pi$
$492$	0	0
$493$	1.58069i	0.0711909i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.5917	0.564816
$498$	0	0
$499$	5.23137i	0.234188i	0.993121	+	0.117094i	$0.0373579\pi$
$499$	5.23137i	0.234188i	−0.993121	+	0.117094i	$0.962642\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.39969	0.106997	0.0534984	−	0.998568i	$-0.482963\pi$
$503$	2.39969	0.106997	0.0534984	+	0.998568i	$0.482963\pi$
$504$	0	0
$505$	−13.7176	−0.610426
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 6.72283i	− 0.297984i	−0.988838	−	0.148992i	$-0.952397\pi$
$509$	− 6.72283i	− 0.297984i	0.988838	−	0.148992i	$-0.0476029\pi$
$510$	0	0
$511$	−12.6341	−0.558901
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.2747i	0.496821i
$516$	0	0
$517$	23.4367i	1.03074i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−23.2754	−1.01971	−0.509857	−	0.860259i	$-0.670302\pi$
$521$	−23.2754	−1.01971	−0.509857	+	0.860259i	$0.670302\pi$
$522$	0	0
$523$	− 3.46255i	− 0.151407i	−0.997130	−	0.0757034i	$-0.975880\pi$
$523$	− 3.46255i	− 0.151407i	0.997130	−	0.0757034i	$-0.0241202\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.68293	0.0733097
$528$	0	0
$529$	−14.9446	−0.649764
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 22.2010i	− 0.961632i
$534$	0	0
$535$	8.32553	0.359944
$536$	0	0
$537$	0	0
$538$	0	0
$539$	48.0767i	2.07081i
$540$	0	0
$541$	9.41268i	0.404683i	0.979315	+	0.202341i	$0.0648551\pi$
$541$	9.41268i	0.404683i	−0.979315	+	0.202341i	$0.935145\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−13.2004	−0.565442
$546$	0	0
$547$	− 8.92340i	− 0.381537i	−0.981635	−	0.190768i	$-0.938902\pi$
$547$	− 8.92340i	− 0.381537i	0.981635	−	0.190768i	$-0.0610980\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.87008	0.164871
$552$	0	0
$553$	−4.08751	−0.173818
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.36902i	− 0.354607i	−0.984156	−	0.177303i	$-0.943263\pi$
$557$	− 8.36902i	− 0.354607i	0.984156	−	0.177303i	$-0.0567374\pi$
$558$	0	0
$559$	41.3585	1.74928
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.1127i	0.510488i	0.966877	+	0.255244i	$0.0821557\pi$
$563$	12.1127i	0.510488i	−0.966877	+	0.255244i	$0.917844\pi$
$564$	0	0
$565$	12.6812i	0.533502i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.8542	−0.790411	−0.395206	−	0.918593i	$-0.629327\pi$
$569$	−18.8542	−0.790411	−0.395206	+	0.918593i	$0.629327\pi$
$570$	0	0
$571$	− 3.35775i	− 0.140517i	−0.997529	−	0.0702587i	$-0.977618\pi$
$571$	− 3.35775i	− 0.140517i	0.997529	−	0.0702587i	$-0.0223825\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.09951	0.296070
$576$	0	0
$577$	−24.7442	−1.03011	−0.515057	−	0.857156i	$-0.672229\pi$
$577$	−24.7442	−1.03011	−0.515057	+	0.857156i	$0.672229\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	41.2050i	1.70947i
$582$	0	0
$583$	19.5690	0.810467
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.2517i	1.04225i	0.853481	+	0.521124i	$0.174487\pi$
$587$	25.2517i	1.04225i	−0.853481	+	0.521124i	$0.825513\pi$
$588$	0	0
$589$	− 4.12039i	− 0.169778i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.1581	−0.909924	−0.454962	−	0.890511i	$-0.650347\pi$
$593$	−22.1581	−0.909924	−0.454962	+	0.890511i	$0.650347\pi$
$594$	0	0
$595$	− 7.92129i	− 0.324741i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−34.4007	−1.40558	−0.702788	−	0.711399i	$-0.748062\pi$
$599$	−34.4007	−1.40558	−0.702788	+	0.711399i	$0.748062\pi$
$600$	0	0
$601$	−25.2908	−1.03163	−0.515817	−	0.856699i	$-0.672512\pi$
$601$	−25.2908	−1.03163	−0.515817	+	0.856699i	$0.672512\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.25129i	0.254151i
$606$	0	0
$607$	−14.2041	−0.576526	−0.288263	−	0.957551i	$-0.593078\pi$
$607$	−14.2041	−0.576526	−0.288263	+	0.957551i	$0.593078\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 55.9859i	− 2.26495i
$612$	0	0
$613$	33.8370i	1.36666i	0.730108	+	0.683332i	$0.239470\pi$
$613$	33.8370i	1.36666i	−0.730108	+	0.683332i	$0.760530\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.3261	−1.34166	−0.670830	−	0.741611i	$-0.734062\pi$
$617$	−33.3261	−1.34166	−0.670830	+	0.741611i	$0.734062\pi$
$618$	0	0
$619$	− 17.9900i	− 0.723080i	−0.932357	−	0.361540i	$-0.882251\pi$
$619$	− 17.9900i	− 0.723080i	0.932357	−	0.361540i	$-0.117749\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.28325	−0.291797
$624$	0	0
$625$	−6.23595	−0.249438
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.16881i	− 0.0466035i
$630$	0	0
$631$	9.75904	0.388501	0.194251	−	0.980952i	$-0.437773\pi$
$631$	9.75904	0.388501	0.194251	+	0.980952i	$0.437773\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 21.3416i	− 0.846915i
$636$	0	0
$637$	− 114.846i	− 4.55038i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.1525	0.479994	0.239997	−	0.970774i	$-0.422854\pi$
$641$	12.1525	0.479994	0.239997	+	0.970774i	$0.422854\pi$
$642$	0	0
$643$	− 20.1310i	− 0.793890i	−0.917842	−	0.396945i	$-0.870070\pi$
$643$	− 20.1310i	− 0.793890i	0.917842	−	0.396945i	$-0.129930\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.21254	0.322868	0.161434	−	0.986883i	$-0.448388\pi$
$647$	8.21254	0.322868	0.161434	+	0.986883i	$0.448388\pi$
$648$	0	0
$649$	28.6172	1.12332
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 39.5293i	− 1.54690i	−0.633856	−	0.773451i	$-0.718529\pi$
$653$	− 39.5293i	− 1.54690i	0.633856	−	0.773451i	$-0.281471\pi$
$654$	0	0
$655$	−9.33396	−0.364708
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.5399i	0.605348i	0.953094	+	0.302674i	$0.0978793\pi$
$659$	15.5399i	0.605348i	−0.953094	+	0.302674i	$0.902121\pi$
$660$	0	0
$661$	46.1373i	1.79453i	0.441489	+	0.897267i	$0.354450\pi$
$661$	46.1373i	1.79453i	−0.441489	+	0.897267i	$0.645550\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−19.3940	−0.752068
$666$	0	0
$667$	− 4.48634i	− 0.173712i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.19561	−0.161970
$672$	0	0
$673$	−32.0095	−1.23388	−0.616938	−	0.787012i	$-0.711627\pi$
$673$	−32.0095	−1.23388	−0.616938	+	0.787012i	$0.711627\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 13.7551i	− 0.528650i	−0.964434	−	0.264325i	$-0.914851\pi$
$677$	− 13.7551i	− 0.528650i	0.964434	−	0.264325i	$-0.0851491\pi$
$678$	0	0
$679$	−35.8571	−1.37607
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 23.7998i	− 0.910674i	−0.890319	−	0.455337i	$-0.849519\pi$
$683$	− 23.7998i	− 0.910674i	0.890319	−	0.455337i	$-0.150481\pi$
$684$	0	0
$685$	− 2.47594i	− 0.0946010i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−46.7469	−1.78091
$690$	0	0
$691$	− 37.8178i	− 1.43866i	−0.694671	−	0.719328i	$-0.744450\pi$
$691$	− 37.8178i	− 1.43866i	0.694671	−	0.719328i	$-0.255550\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.20404	−0.197401
$696$	0	0
$697$	−3.50141	−0.132625
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.14594i	0.156590i	0.996930	+	0.0782950i	$0.0249476\pi$
$701$	4.14594i	0.156590i	−0.996930	+	0.0782950i	$0.975052\pi$
$702$	0	0
$703$	−2.86165	−0.107929
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43.4889i	1.63557i
$708$	0	0
$709$	− 47.2230i	− 1.77350i	−0.462252	−	0.886748i	$-0.652959\pi$
$709$	− 47.2230i	− 1.77350i	0.462252	−	0.886748i	$-0.347041\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.77652	−0.178882
$714$	0	0
$715$	26.6027i	0.994884i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.5647	−0.729642	−0.364821	−	0.931078i	$-0.618870\pi$
$719$	−19.5647	−0.729642	−0.364821	+	0.931078i	$0.618870\pi$
$720$	0	0
$721$	35.7441	1.33118
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.95396i	− 0.146846i
$726$	0	0
$727$	−2.57406	−0.0954668	−0.0477334	−	0.998860i	$-0.515200\pi$
$727$	−2.57406	−0.0954668	−0.0477334	+	0.998860i	$0.515200\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 6.52280i	− 0.241255i
$732$	0	0
$733$	8.78689i	0.324551i	0.986745	+	0.162276i	$0.0518833\pi$
$733$	8.78689i	0.324551i	−0.986745	+	0.162276i	$0.948117\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.0676	0.960214
$738$	0	0
$739$	21.3574i	0.785643i	0.919615	+	0.392822i	$0.128501\pi$
$739$	21.3574i	0.785643i	−0.919615	+	0.392822i	$0.871499\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.6886	−0.575557	−0.287779	−	0.957697i	$-0.592917\pi$
$743$	−15.6886	−0.575557	−0.287779	+	0.957697i	$0.592917\pi$
$744$	0	0
$745$	−4.27108	−0.156480
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 26.3944i	− 0.964431i
$750$	0	0
$751$	19.4799	0.710832	0.355416	−	0.934708i	$-0.384339\pi$
$751$	19.4799	0.710832	0.355416	+	0.934708i	$0.384339\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 4.56792i	− 0.166244i
$756$	0	0
$757$	− 18.0194i	− 0.654927i	−0.944864	−	0.327463i	$-0.893806\pi$
$757$	− 18.0194i	− 0.654927i	0.944864	−	0.327463i	$-0.106194\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.8671	1.15518	0.577592	−	0.816326i	$-0.303993\pi$
$761$	31.8671	1.15518	0.577592	+	0.816326i	$0.303993\pi$
$762$	0	0
$763$	41.8491i	1.51504i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−68.3613	−2.46838
$768$	0	0
$769$	−2.60475	−0.0939296	−0.0469648	−	0.998897i	$-0.514955\pi$
$769$	−2.60475	−0.0939296	−0.0469648	+	0.998897i	$0.514955\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 9.40703i	− 0.338347i	−0.985586	−	0.169174i	$-0.945890\pi$
$773$	− 9.40703i	− 0.338347i	0.985586	−	0.169174i	$-0.0541099\pi$
$774$	0	0
$775$	−4.20970	−0.151217
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.57264i	0.307147i
$780$	0	0
$781$	6.66936i	0.238649i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.60316	−0.0572191
$786$	0	0
$787$	− 3.38310i	− 0.120594i	−0.998180	−	0.0602972i	$-0.980795\pi$
$787$	− 3.38310i	− 0.120594i	0.998180	−	0.0602972i	$-0.0192048\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	40.2032	1.42946
$792$	0	0
$793$	10.0225	0.355911
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.3351i	0.720307i	0.932893	+	0.360153i	$0.117276\pi$
$797$	20.3351i	0.720307i	−0.932893	+	0.360153i	$0.882724\pi$
$798$	0	0
$799$	−8.82975	−0.312374
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 6.69183i	− 0.236149i
$804$	0	0
$805$	22.4823i	0.792397i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−44.1507	−1.55226	−0.776128	−	0.630576i	$-0.782819\pi$
$809$	−44.1507	−1.55226	−0.776128	+	0.630576i	$0.782819\pi$
$810$	0	0
$811$	− 7.73677i	− 0.271675i	−0.990731	−	0.135837i	$-0.956628\pi$
$811$	− 7.73677i	− 0.271675i	0.990731	−	0.135837i	$-0.0433725\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−24.5417	−0.859660
$816$	0	0
$817$	−15.9700	−0.558721
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 48.5660i	− 1.69496i	−0.530824	−	0.847482i	$-0.678118\pi$
$821$	− 48.5660i	− 1.69496i	0.530824	−	0.847482i	$-0.321882\pi$
$822$	0	0
$823$	37.3368	1.30148	0.650740	−	0.759301i	$-0.274459\pi$
$823$	37.3368	1.30148	0.650740	+	0.759301i	$0.274459\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.3380i	− 1.26360i	−0.775133	−	0.631798i	$-0.782317\pi$
$827$	− 36.3380i	− 1.26360i	0.775133	−	0.631798i	$-0.217683\pi$
$828$	0	0
$829$	10.6053i	0.368337i	0.982895	+	0.184169i	$0.0589593\pi$
$829$	10.6053i	0.368337i	−0.982895	+	0.184169i	$0.941041\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.1129	−0.627574
$834$	0	0
$835$	− 25.1348i	− 0.869824i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.2821	−1.49426	−0.747132	−	0.664676i	$-0.768570\pi$
$839$	−43.2821	−1.49426	−0.747132	+	0.664676i	$0.768570\pi$
$840$	0	0
$841$	26.5014	0.913842
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 42.9999i	− 1.47924i
$846$	0	0
$847$	19.8185	0.680971
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.31733i	0.113717i
$852$	0	0
$853$	− 39.7672i	− 1.36160i	−0.732467	−	0.680802i	$-0.761631\pi$
$853$	− 39.7672i	− 1.36160i	0.732467	−	0.680802i	$-0.238369\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.9672	0.511271	0.255636	−	0.966773i	$-0.417715\pi$
$857$	14.9672	0.511271	0.255636	+	0.966773i	$0.417715\pi$
$858$	0	0
$859$	− 47.5063i	− 1.62090i	−0.585811	−	0.810448i	$-0.699224\pi$
$859$	− 47.5063i	− 1.62090i	0.585811	−	0.810448i	$-0.300776\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20.9221	−0.712198	−0.356099	−	0.934448i	$-0.615893\pi$
$863$	−20.9221	−0.712198	−0.356099	+	0.934448i	$0.615893\pi$
$864$	0	0
$865$	33.4715	1.13806
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.16500i	− 0.0734426i
$870$	0	0
$871$	−62.2708	−2.10997
$872$	0	0
$873$	0	0
$874$	0	0
$875$	59.4208i	2.00879i
$876$	0	0
$877$	29.6926i	1.00265i	0.865260	+	0.501324i	$0.167153\pi$
$877$	29.6926i	1.00265i	−0.865260	+	0.501324i	$0.832847\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.1976	0.545710	0.272855	−	0.962055i	$-0.412032\pi$
$881$	16.1976	0.545710	0.272855	+	0.962055i	$0.412032\pi$
$882$	0	0
$883$	− 37.4853i	− 1.26148i	−0.775993	−	0.630741i	$-0.782751\pi$
$883$	− 37.4853i	− 1.26148i	0.775993	−	0.630741i	$-0.217249\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.05161	−0.270347	−0.135173	−	0.990822i	$-0.543159\pi$
$887$	−8.05161	−0.270347	−0.135173	+	0.990822i	$0.543159\pi$
$888$	0	0
$889$	−67.6592	−2.26922
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21.6182i	0.723427i
$894$	0	0
$895$	−26.2257	−0.876630
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.66020i	0.0887227i
$900$	0	0
$901$	7.37263i	0.245618i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.6300	0.619281
$906$	0	0
$907$	33.3609i	1.10773i	0.832606	+	0.553865i	$0.186848\pi$
$907$	33.3609i	1.10773i	−0.832606	+	0.553865i	$0.813152\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.7866	−0.854347	−0.427174	−	0.904170i	$-0.640491\pi$
$911$	−25.7866	−0.854347	−0.427174	+	0.904170i	$0.640491\pi$
$912$	0	0
$913$	−21.8247	−0.722293
$914$	0	0
$915$	0	0
$916$	0	0
$917$	29.5915i	0.977196i
$918$	0	0
$919$	25.8720	0.853440	0.426720	−	0.904384i	$-0.359669\pi$
$919$	25.8720	0.853440	0.426720	+	0.904384i	$0.359669\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 15.9319i	− 0.524404i
$924$	0	0
$925$	2.92367i	0.0961297i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.6849	−1.33483	−0.667413	−	0.744687i	$-0.732599\pi$
$929$	−40.6849	−1.33483	−0.667413	+	0.744687i	$0.732599\pi$
$930$	0	0
$931$	44.3465i	1.45340i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.19561	0.137211
$936$	0	0
$937$	−0.426795	−0.0139428	−0.00697140	−	0.999976i	$-0.502219\pi$
$937$	−0.426795	−0.0139428	−0.00697140	+	0.999976i	$0.502219\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.4487i	0.699208i	0.936898	+	0.349604i	$0.113684\pi$
$941$	21.4487i	0.699208i	−0.936898	+	0.349604i	$0.886316\pi$
$942$	0	0
$943$	9.93772	0.323617
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 45.9708i	− 1.49385i	−0.664908	−	0.746925i	$-0.731529\pi$
$947$	− 45.9708i	− 1.49385i	0.664908	−	0.746925i	$-0.268471\pi$
$948$	0	0
$949$	15.9856i	0.518913i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−28.3714	−0.919038	−0.459519	−	0.888168i	$-0.651978\pi$
$953$	−28.3714	−0.919038	−0.459519	+	0.888168i	$0.651978\pi$
$954$	0	0
$955$	12.5862i	0.407281i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.84948	−0.253473
$960$	0	0
$961$	−28.1677	−0.908637
$962$	0	0
$963$	0	0
$964$	0	0
$965$	39.8697i	1.28345i
$966$	0	0
$967$	29.4541	0.947180	0.473590	−	0.880745i	$-0.342958\pi$
$967$	29.4541	0.947180	0.473590	+	0.880745i	$0.342958\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.8816i	1.24777i	0.781516	+	0.623885i	$0.214446\pi$
$971$	38.8816i	1.24777i	−0.781516	+	0.623885i	$0.785554\pi$
$972$	0	0
$973$	16.4984i	0.528914i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−51.5465	−1.64912	−0.824559	−	0.565775i	$-0.808577\pi$
$977$	−51.5465	−1.64912	−0.824559	+	0.565775i	$0.808577\pi$
$978$	0	0
$979$	− 3.85766i	− 0.123291i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.7261	1.26707	0.633533	−	0.773716i	$-0.281604\pi$
$983$	39.7261	1.26707	0.633533	+	0.773716i	$0.281604\pi$
$984$	0	0
$985$	20.3645	0.648868
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.5131i	0.588682i
$990$	0	0
$991$	−8.01408	−0.254576	−0.127288	−	0.991866i	$-0.540627\pi$
$991$	−8.01408	−0.254576	−0.127288	+	0.991866i	$0.540627\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 0.536828i	− 0.0170186i
$996$	0	0
$997$	18.6589i	0.590934i	0.955353	+	0.295467i	$0.0954753\pi$
$997$	18.6589i	0.590934i	−0.955353	+	0.295467i	$0.904525\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.f.d.2449.6		8
3.2	odd	2		544.2.c.b.273.7		8
4.3	odd	2		1224.2.f.c.613.7		8
8.3	odd	2		1224.2.f.c.613.8		8
8.5	even	2	inner	4896.2.f.d.2449.3		8
12.11	even	2		136.2.c.b.69.2	yes	8
24.5	odd	2		544.2.c.b.273.2		8
24.11	even	2		136.2.c.b.69.1	✓	8
48.5	odd	4		4352.2.a.bb.1.7		8
48.11	even	4		4352.2.a.bf.1.2		8
48.29	odd	4		4352.2.a.bb.1.2		8
48.35	even	4		4352.2.a.bf.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.c.b.69.1	✓	8	24.11	even	2
136.2.c.b.69.2	yes	8	12.11	even	2
544.2.c.b.273.2		8	24.5	odd	2
544.2.c.b.273.7		8	3.2	odd	2
1224.2.f.c.613.7		8	4.3	odd	2
1224.2.f.c.613.8		8	8.3	odd	2
4352.2.a.bb.1.2		8	48.29	odd	4
4352.2.a.bb.1.7		8	48.5	odd	4
4352.2.a.bf.1.2		8	48.11	even	4
4352.2.a.bf.1.7		8	48.35	even	4
4896.2.f.d.2449.3		8	8.5	even	2	inner
4896.2.f.d.2449.6		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4896.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.58069i q^{5} +5.01127 q^{7} +2.65428i q^{11} -6.34060i q^{13} -1.00000 q^{17} +2.44834i q^{19} +2.83821 q^{23} +2.50141 q^{25} -1.58069i q^{29} -1.68293 q^{31} +7.92129i q^{35} +1.16881i q^{37} +3.50141 q^{41} +6.52280i q^{43} +8.82975 q^{47} +18.1129 q^{49} -7.37263i q^{53} -4.19561 q^{55} -10.7815i q^{59} +1.58069i q^{61} +10.0225 q^{65} -9.82097i q^{67} +2.51268 q^{71} -2.52114 q^{73} +13.3013i q^{77} -0.815662 q^{79} +8.22246i q^{83} -1.58069i q^{85} -1.45337 q^{89} -31.7745i q^{91} -3.87008 q^{95} -7.15528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{7} - 8 q^{17} - 16 q^{23} - 8 q^{25} - 24 q^{31} + 4 q^{47} + 8 q^{49} + 12 q^{55} + 24 q^{65} - 36 q^{71} + 8 q^{73} - 24 q^{79} - 8 q^{89} + 32 q^{95} - 8 q^{97}+O(q^{100}) \) 8 * q + 12 * q^7 - 8 * q^17 - 16 * q^23 - 8 * q^25 - 24 * q^31 + 4 * q^47 + 8 * q^49 + 12 * q^55 + 24 * q^65 - 36 * q^71 + 8 * q^73 - 24 * q^79 - 8 * q^89 + 32 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

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Database

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