LMFDB - Embedded newform 4896.2.f.c.2449.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4896,2,Mod(2449,4896)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$39.0947568296$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1649659456.5
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:	$\Z[a_1, \ldots, a_{11}]$
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			2449.8
Root			$1.40014 + 0.199044i$ of defining polynomial
Character	$\chi$	$=$	4896.2449
Dual form			4896.2.f.c.2449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.30391i q^{5} -3.45790 q^{7} +O(q^{10})$	$q+3.30391i q^{5} -3.45790 q^{7} +3.09665i q^{11} +2.50773i q^{13} +1.00000 q^{17} +4.43508i q^{19} +4.14265 q^{23} -5.91579 q^{25} +7.15862i q^{29} +6.22515 q^{31} -11.4246i q^{35} +6.74411i q^{37} +6.99830 q^{41} +2.22951i q^{43} -3.31525 q^{47} +4.95705 q^{49} +6.19330i q^{53} -10.2310 q^{55} +8.83732i q^{59} -14.3707i q^{61} -8.28530 q^{65} -6.44195i q^{67} -5.66069 q^{71} -15.4486 q^{73} -10.7079i q^{77} +8.14265 q^{79} +13.8528i q^{83} +3.30391i q^{85} -7.32655 q^{89} -8.67146i q^{91} -14.6531 q^{95} +8.36780 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} + 8 q^{17} + 12 q^{23} - 8 q^{25} + 12 q^{31} - 28 q^{47} - 8 q^{49} - 44 q^{55} - 24 q^{65} + 8 q^{73} + 44 q^{79} - 8 q^{89} - 16 q^{95} + 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 + 8 * q^17 + 12 * q^23 - 8 * q^25 + 12 * q^31 - 28 * q^47 - 8 * q^49 - 44 * q^55 - 24 * q^65 + 8 * q^73 + 44 * q^79 - 8 * q^89 - 16 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	3.30391i	1.47755i	0.673951	+	0.738776i	$0.264596\pi$
$5$	3.30391i	1.47755i	−0.673951	+	0.738776i	$0.735404\pi$
$6$	0	0
$7$	−3.45790	−1.30696	−0.653481	−	0.756943i	$-0.726692\pi$
$7$	−3.45790	−1.30696	−0.653481	+	0.756943i	$0.726692\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.09665i	0.933675i	0.884343	+	0.466838i	$0.154607\pi$
$11$	3.09665i	0.933675i	−0.884343	+	0.466838i	$0.845393\pi$
$12$	0	0
$13$	2.50773i	0.695519i	0.937584	+	0.347759i	$0.113057\pi$
$13$	2.50773i	0.695519i	−0.937584	+	0.347759i	$0.886943\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.43508i	1.01748i	0.860921	+	0.508739i	$0.169888\pi$
$19$	4.43508i	1.01748i	−0.860921	+	0.508739i	$0.830112\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.14265	0.863802	0.431901	−	0.901921i	$-0.357843\pi$
$23$	4.14265	0.863802	0.431901	+	0.901921i	$0.357843\pi$
$24$	0	0
$25$	−5.91579	−1.18316
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.15862i	1.32932i	0.747145	+	0.664661i	$0.231424\pi$
$29$	7.15862i	1.32932i	−0.747145	+	0.664661i	$0.768576\pi$
$30$	0	0
$31$	6.22515	1.11807	0.559035	−	0.829144i	$-0.311172\pi$
$31$	6.22515	1.11807	0.559035	+	0.829144i	$0.311172\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 11.4246i	− 1.93110i
$36$	0	0
$37$	6.74411i	1.10872i	0.832276	+	0.554362i	$0.187038\pi$
$37$	6.74411i	1.10872i	−0.832276	+	0.554362i	$0.812962\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.99830	1.09295	0.546475	−	0.837475i	$-0.315969\pi$
$41$	6.99830	1.09295	0.546475	+	0.837475i	$0.315969\pi$
$42$	0	0
$43$	2.22951i	0.339998i	0.985444	+	0.169999i	$0.0543764\pi$
$43$	2.22951i	0.339998i	−0.985444	+	0.169999i	$0.945624\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.31525	−0.483579	−0.241789	−	0.970329i	$-0.577734\pi$
$47$	−3.31525	−0.483579	−0.241789	+	0.970329i	$0.577734\pi$
$48$	0	0
$49$	4.95705	0.708149
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.19330i	0.850715i	0.905025	+	0.425358i	$0.139852\pi$
$53$	6.19330i	0.850715i	−0.905025	+	0.425358i	$0.860148\pi$
$54$	0	0
$55$	−10.2310	−1.37955
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.83732i	1.15052i	0.817970	+	0.575261i	$0.195099\pi$
$59$	8.83732i	1.15052i	−0.817970	+	0.575261i	$0.804901\pi$
$60$	0	0
$61$	− 14.3707i	− 1.83999i	−0.391936	−	0.919993i	$-0.628194\pi$
$61$	− 14.3707i	− 1.83999i	0.391936	−	0.919993i	$-0.371806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.28530	−1.02766
$66$	0	0
$67$	− 6.44195i	− 0.787009i	−0.919323	−	0.393505i	$-0.871262\pi$
$67$	− 6.44195i	− 0.787009i	0.919323	−	0.393505i	$-0.128738\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.66069	−0.671800	−0.335900	−	0.941898i	$-0.609040\pi$
$71$	−5.66069	−0.671800	−0.335900	+	0.941898i	$0.609040\pi$
$72$	0	0
$73$	−15.4486	−1.80812	−0.904061	−	0.427403i	$-0.859429\pi$
$73$	−15.4486	−1.80812	−0.904061	+	0.427403i	$0.859429\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.7079i	− 1.22028i
$78$	0	0
$79$	8.14265	0.916120	0.458060	−	0.888921i	$-0.348545\pi$
$79$	8.14265	0.916120	0.458060	+	0.888921i	$0.348545\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.8528i	1.52054i	0.649607	+	0.760270i	$0.274933\pi$
$83$	13.8528i	1.52054i	−0.649607	+	0.760270i	$0.725067\pi$
$84$	0	0
$85$	3.30391i	0.358359i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.32655	−0.776613	−0.388306	−	0.921530i	$-0.626940\pi$
$89$	−7.32655	−0.776613	−0.388306	+	0.921530i	$0.626940\pi$
$90$	0	0
$91$	− 8.67146i	− 0.909016i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−14.6531	−1.50338
$96$	0	0
$97$	8.36780	0.849622	0.424811	−	0.905282i	$-0.360341\pi$
$97$	8.36780	0.849622	0.424811	+	0.905282i	$0.360341\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 7.95480i	− 0.791532i	−0.918351	−	0.395766i	$-0.870479\pi$
$101$	− 7.95480i	− 0.791532i	0.918351	−	0.395766i	$-0.129521\pi$
$102$	0	0
$103$	7.91409	0.779798	0.389899	−	0.920858i	$-0.372510\pi$
$103$	7.91409	0.779798	0.389899	+	0.920858i	$0.372510\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.3605i	− 1.29161i	−0.763504	−	0.645803i	$-0.776523\pi$
$107$	− 13.3605i	− 1.29161i	0.763504	−	0.645803i	$-0.223477\pi$
$108$	0	0
$109$	2.12606i	0.203640i	0.994803	+	0.101820i	$0.0324665\pi$
$109$	2.12606i	0.203640i	−0.994803	+	0.101820i	$0.967533\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.28530	−0.214983	−0.107491	−	0.994206i	$-0.534282\pi$
$113$	−2.28530	−0.214983	−0.107491	+	0.994206i	$0.534282\pi$
$114$	0	0
$115$	13.6869i	1.27631i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.45790	−0.316985
$120$	0	0
$121$	1.41076	0.128251
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.02569i	− 0.270626i
$126$	0	0
$127$	−2.36780	−0.210109	−0.105054	−	0.994466i	$-0.533502\pi$
$127$	−2.36780	−0.210109	−0.105054	+	0.994466i	$0.533502\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.73893i	0.239301i	0.992816	+	0.119651i	$0.0381774\pi$
$131$	2.73893i	0.239301i	−0.992816	+	0.119651i	$0.961823\pi$
$132$	0	0
$133$	− 15.3361i	− 1.32981i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−22.7044	−1.93977	−0.969885	−	0.243564i	$-0.921683\pi$
$137$	−22.7044	−1.93977	−0.969885	+	0.243564i	$0.921683\pi$
$138$	0	0
$139$	0.502570i	0.0426275i	0.999773	+	0.0213137i	$0.00678488\pi$
$139$	0.502570i	0.0426275i	−0.999773	+	0.0213137i	$0.993215\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.76556	−0.649388
$144$	0	0
$145$	−23.6514	−1.96414
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.21213i	− 0.590840i	−0.955367	−	0.295420i	$-0.904540\pi$
$149$	− 7.21213i	− 0.590840i	0.955367	−	0.295420i	$-0.0954596\pi$
$150$	0	0
$151$	−3.16671	−0.257704	−0.128852	−	0.991664i	$-0.541129\pi$
$151$	−3.16671	−0.257704	−0.128852	+	0.991664i	$0.541129\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.5673i	1.65201i
$156$	0	0
$157$	− 3.16761i	− 0.252803i	−0.991979	−	0.126401i	$-0.959657\pi$
$157$	− 3.16761i	− 0.252803i	0.991979	−	0.126401i	$-0.0403427\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−14.3248	−1.12896
$162$	0	0
$163$	− 1.66331i	− 0.130281i	−0.997876	−	0.0651404i	$-0.979250\pi$
$163$	− 1.66331i	− 0.130281i	0.997876	−	0.0651404i	$-0.0207495\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.57648	−0.663668	−0.331834	−	0.943338i	$-0.607667\pi$
$167$	−8.57648	−0.663668	−0.331834	+	0.943338i	$0.607667\pi$
$168$	0	0
$169$	6.71130	0.516254
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.48175i	− 0.340741i	−0.985380	−	0.170371i	$-0.945504\pi$
$173$	− 4.48175i	− 0.340741i	0.985380	−	0.170371i	$-0.0544965\pi$
$174$	0	0
$175$	20.4562	1.54634
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3.60606i	− 0.269530i	−0.990878	−	0.134765i	$-0.956972\pi$
$179$	− 3.60606i	− 0.269530i	0.990878	−	0.134765i	$-0.0430279\pi$
$180$	0	0
$181$	− 17.5384i	− 1.30362i	−0.758384	−	0.651808i	$-0.774011\pi$
$181$	− 17.5384i	− 1.30362i	0.758384	−	0.651808i	$-0.225989\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−22.2819	−1.63820
$186$	0	0
$187$	3.09665i	0.226449i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.76386	0.634130	0.317065	−	0.948404i	$-0.397303\pi$
$191$	8.76386	0.634130	0.317065	+	0.948404i	$0.397303\pi$
$192$	0	0
$193$	−18.4847	−1.33056	−0.665278	−	0.746595i	$-0.731687\pi$
$193$	−18.4847	−1.33056	−0.665278	+	0.746595i	$0.731687\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.87744i	− 0.276256i	−0.990414	−	0.138128i	$-0.955891\pi$
$197$	− 3.87744i	− 0.276256i	0.990414	−	0.138128i	$-0.0441085\pi$
$198$	0	0
$199$	−13.1409	−0.931537	−0.465769	−	0.884907i	$-0.654222\pi$
$199$	−13.1409	−0.931537	−0.465769	+	0.884907i	$0.654222\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 24.7538i	− 1.73737i
$204$	0	0
$205$	23.1217i	1.61489i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.7339	−0.949994
$210$	0	0
$211$	− 13.5024i	− 0.929543i	−0.885431	−	0.464771i	$-0.846137\pi$
$211$	− 13.5024i	− 0.929543i	0.885431	−	0.464771i	$-0.153863\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.36610	−0.502364
$216$	0	0
$217$	−21.5259	−1.46128
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.50773i	0.168688i
$222$	0	0
$223$	7.85147	0.525773	0.262887	−	0.964827i	$-0.415326\pi$
$223$	7.85147	0.525773	0.262887	+	0.964827i	$0.415326\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.1858i	− 1.40615i	−0.711115	−	0.703076i	$-0.751809\pi$
$227$	− 21.1858i	− 1.40615i	0.711115	−	0.703076i	$-0.248191\pi$
$228$	0	0
$229$	− 13.6745i	− 0.903633i	−0.892111	−	0.451817i	$-0.850776\pi$
$229$	− 13.6745i	− 0.903633i	0.892111	−	0.451817i	$-0.149224\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.1994	1.19228	0.596141	−	0.802880i	$-0.296700\pi$
$233$	18.1994	1.19228	0.596141	+	0.802880i	$0.296700\pi$
$234$	0	0
$235$	− 10.9533i	− 0.714512i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.93227	0.189673	0.0948364	−	0.995493i	$-0.469767\pi$
$239$	2.93227	0.189673	0.0948364	+	0.995493i	$0.469767\pi$
$240$	0	0
$241$	27.3180	1.75971	0.879853	−	0.475247i	$-0.157641\pi$
$241$	27.3180	1.75971	0.879853	+	0.475247i	$0.157641\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	16.3776i	1.04633i
$246$	0	0
$247$	−11.1220	−0.707675
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.8286i	1.18845i	0.804300	+	0.594224i	$0.202541\pi$
$251$	18.8286i	1.18845i	−0.804300	+	0.594224i	$0.797459\pi$
$252$	0	0
$253$	12.8283i	0.806510i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.58924	−0.473404	−0.236702	−	0.971582i	$-0.576066\pi$
$257$	−7.58924	−0.473404	−0.236702	+	0.971582i	$0.576066\pi$
$258$	0	0
$259$	− 23.3204i	− 1.44906i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.0310	1.85179	0.925895	−	0.377782i	$-0.123313\pi$
$263$	30.0310	1.85179	0.925895	+	0.377782i	$0.123313\pi$
$264$	0	0
$265$	−20.4621	−1.25698
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3.24478i	− 0.197837i	−0.995096	−	0.0989187i	$-0.968462\pi$
$269$	− 3.24478i	− 0.197837i	0.995096	−	0.0989187i	$-0.0315384\pi$
$270$	0	0
$271$	11.6453	0.707400	0.353700	−	0.935359i	$-0.384923\pi$
$271$	11.6453	0.707400	0.353700	+	0.935359i	$0.384923\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 18.3191i	− 1.10469i
$276$	0	0
$277$	16.9340i	1.01747i	0.860924	+	0.508734i	$0.169886\pi$
$277$	16.9340i	1.01747i	−0.860924	+	0.508734i	$0.830114\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.38127	−0.440330	−0.220165	−	0.975463i	$-0.570660\pi$
$281$	−7.38127	−0.440330	−0.220165	+	0.975463i	$0.570660\pi$
$282$	0	0
$283$	19.3209i	1.14851i	0.818678	+	0.574253i	$0.194707\pi$
$283$	19.3209i	1.14851i	−0.818678	+	0.574253i	$0.805293\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.1994	−1.42844
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.71966i	0.334146i	0.985945	+	0.167073i	$0.0534316\pi$
$293$	5.71966i	0.334146i	−0.985945	+	0.167073i	$0.946568\pi$
$294$	0	0
$295$	−29.1977	−1.69995
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.3886i	0.600790i
$300$	0	0
$301$	− 7.70943i	− 0.444364i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	47.4796	2.71867
$306$	0	0
$307$	− 0.222648i	− 0.0127072i	−0.999980	−	0.00635359i	$-0.997978\pi$
$307$	− 0.222648i	− 0.0127072i	0.999980	−	0.00635359i	$-0.00202242\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.88560	0.220332	0.110166	−	0.993913i	$-0.464862\pi$
$311$	3.88560	0.220332	0.110166	+	0.993913i	$0.464862\pi$
$312$	0	0
$313$	−20.4022	−1.15320	−0.576600	−	0.817027i	$-0.695621\pi$
$313$	−20.4022	−1.15320	−0.576600	+	0.817027i	$0.695621\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.32960i	− 0.355506i	−0.984075	−	0.177753i	$-0.943117\pi$
$317$	− 6.32960i	− 0.355506i	0.984075	−	0.177753i	$-0.0568827\pi$
$318$	0	0
$319$	−22.1677	−1.24115
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.43508i	0.246775i
$324$	0	0
$325$	− 14.8352i	− 0.822909i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.4638	0.632019
$330$	0	0
$331$	20.8922i	1.14834i	0.818736	+	0.574170i	$0.194675\pi$
$331$	20.8922i	1.14834i	−0.818736	+	0.574170i	$0.805325\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	21.2836	1.16285
$336$	0	0
$337$	8.82988	0.480994	0.240497	−	0.970650i	$-0.422690\pi$
$337$	8.82988	0.480994	0.240497	+	0.970650i	$0.422690\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.2771i	1.04391i
$342$	0	0
$343$	7.06433	0.381438
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.62301i	0.140811i	0.997518	+	0.0704053i	$0.0224292\pi$
$347$	2.62301i	0.140811i	−0.997518	+	0.0704053i	$0.977571\pi$
$348$	0	0
$349$	− 13.3575i	− 0.715013i	−0.933911	−	0.357506i	$-0.883627\pi$
$349$	− 13.3575i	− 0.715013i	0.933911	−	0.357506i	$-0.116373\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.510209	−0.0271557	−0.0135779	−	0.999908i	$-0.504322\pi$
$353$	−0.510209	−0.0271557	−0.0135779	+	0.999908i	$0.504322\pi$
$354$	0	0
$355$	− 18.7024i	− 0.992619i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.861534	−0.0454700	−0.0227350	−	0.999742i	$-0.507237\pi$
$359$	−0.861534	−0.0454700	−0.0227350	+	0.999742i	$0.507237\pi$
$360$	0	0
$361$	−0.669976	−0.0352619
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 51.0407i	− 2.67159i
$366$	0	0
$367$	−3.07492	−0.160509	−0.0802547	−	0.996774i	$-0.525573\pi$
$367$	−3.07492	−0.160509	−0.0802547	+	0.996774i	$0.525573\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 21.4158i	− 1.11185i
$372$	0	0
$373$	8.12750i	0.420826i	0.977613	+	0.210413i	$0.0674809\pi$
$373$	8.12750i	0.420826i	−0.977613	+	0.210413i	$0.932519\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−17.9519	−0.924568
$378$	0	0
$379$	− 9.01502i	− 0.463070i	−0.972827	−	0.231535i	$-0.925625\pi$
$379$	− 9.01502i	− 0.463070i	0.972827	−	0.231535i	$-0.0743748\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.79721	−0.0918331	−0.0459165	−	0.998945i	$-0.514621\pi$
$383$	−1.79721	−0.0918331	−0.0459165	+	0.998945i	$0.514621\pi$
$384$	0	0
$385$	35.3779	1.80302
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 18.7490i	− 0.950614i	−0.879820	−	0.475307i	$-0.842337\pi$
$389$	− 18.7490i	− 0.950614i	0.879820	−	0.475307i	$-0.157663\pi$
$390$	0	0
$391$	4.14265	0.209503
$392$	0	0
$393$	0	0
$394$	0	0
$395$	26.9025i	1.35361i
$396$	0	0
$397$	23.9280i	1.20091i	0.799658	+	0.600456i	$0.205014\pi$
$397$	23.9280i	1.20091i	−0.799658	+	0.600456i	$0.794986\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.58236	−0.328708	−0.164354	−	0.986401i	$-0.552554\pi$
$401$	−6.58236	−0.328708	−0.164354	+	0.986401i	$0.552554\pi$
$402$	0	0
$403$	15.6110i	0.777639i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.8841	−1.03519
$408$	0	0
$409$	29.7717	1.47212	0.736058	−	0.676919i	$-0.236685\pi$
$409$	29.7717	1.47212	0.736058	+	0.676919i	$0.236685\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 30.5585i	− 1.50369i
$414$	0	0
$415$	−45.7683	−2.24668
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 24.2375i	− 1.18408i	−0.805909	−	0.592040i	$-0.798323\pi$
$419$	− 24.2375i	− 1.18408i	0.805909	−	0.592040i	$-0.201677\pi$
$420$	0	0
$421$	19.8507i	0.967462i	0.875217	+	0.483731i	$0.160719\pi$
$421$	19.8507i	0.967462i	−0.875217	+	0.483731i	$0.839281\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.91579	−0.286958
$426$	0	0
$427$	49.6925i	2.40479i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.84218	−0.0887346	−0.0443673	−	0.999015i	$-0.514127\pi$
$431$	−1.84218	−0.0887346	−0.0443673	+	0.999015i	$0.514127\pi$
$432$	0	0
$433$	8.03955	0.386356	0.193178	−	0.981164i	$-0.438120\pi$
$433$	8.03955	0.386356	0.193178	+	0.981164i	$0.438120\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.3730i	0.878900i
$438$	0	0
$439$	−14.4184	−0.688153	−0.344077	−	0.938942i	$-0.611808\pi$
$439$	−14.4184	−0.688153	−0.344077	+	0.938942i	$0.611808\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.7835i	1.46257i	0.682072	+	0.731285i	$0.261079\pi$
$443$	30.7835i	1.46257i	−0.682072	+	0.731285i	$0.738921\pi$
$444$	0	0
$445$	− 24.2062i	− 1.14749i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.6305	−0.596070	−0.298035	−	0.954555i	$-0.596331\pi$
$449$	−12.6305	−0.596070	−0.298035	+	0.954555i	$0.596331\pi$
$450$	0	0
$451$	21.6713i	1.02046i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	28.6497	1.34312
$456$	0	0
$457$	20.3592	0.952364	0.476182	−	0.879347i	$-0.342020\pi$
$457$	20.3592	0.952364	0.476182	+	0.879347i	$0.342020\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	33.5671i	1.56338i	0.623669	+	0.781689i	$0.285641\pi$
$461$	33.5671i	1.56338i	−0.623669	+	0.781689i	$0.714359\pi$
$462$	0	0
$463$	21.8316	1.01460	0.507300	−	0.861770i	$-0.330644\pi$
$463$	21.8316	1.01460	0.507300	+	0.861770i	$0.330644\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 23.2705i	− 1.07683i	−0.842680	−	0.538415i	$-0.819023\pi$
$467$	− 23.2705i	− 1.07683i	0.842680	−	0.538415i	$-0.180977\pi$
$468$	0	0
$469$	22.2756i	1.02859i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.90402	−0.317447
$474$	0	0
$475$	− 26.2370i	− 1.20384i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−43.5363	−1.98923	−0.994613	−	0.103662i	$-0.966944\pi$
$479$	−43.5363	−1.98923	−0.994613	+	0.103662i	$0.966944\pi$
$480$	0	0
$481$	−16.9124	−0.771139
$482$	0	0
$483$	0	0
$484$	0	0
$485$	27.6464i	1.25536i
$486$	0	0
$487$	−20.2912	−0.919481	−0.459741	−	0.888053i	$-0.652058\pi$
$487$	−20.2912	−0.919481	−0.459741	+	0.888053i	$0.652058\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 39.6117i	− 1.78765i	−0.448416	−	0.893825i	$-0.648012\pi$
$491$	− 39.6117i	− 1.78765i	0.448416	−	0.893825i	$-0.351988\pi$
$492$	0	0
$493$	7.15862i	0.322408i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19.5741	0.878017
$498$	0	0
$499$	− 9.69556i	− 0.434033i	−0.976168	−	0.217016i	$-0.930367\pi$
$499$	− 9.69556i	− 0.434033i	0.976168	−	0.217016i	$-0.0696325\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−39.2861	−1.75168	−0.875840	−	0.482602i	$-0.839692\pi$
$503$	−39.2861	−1.75168	−0.875840	+	0.482602i	$0.839692\pi$
$504$	0	0
$505$	26.2819	1.16953
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.64374i	0.338803i	0.985547	+	0.169401i	$0.0541834\pi$
$509$	7.64374i	0.338803i	−0.985547	+	0.169401i	$0.945817\pi$
$510$	0	0
$511$	53.4197	2.36315
$512$	0	0
$513$	0	0
$514$	0	0
$515$	26.1474i	1.15219i
$516$	0	0
$517$	− 10.2662i	− 0.451505i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.66487	−0.204372	−0.102186	−	0.994765i	$-0.532584\pi$
$521$	−4.66487	−0.204372	−0.102186	+	0.994765i	$0.532584\pi$
$522$	0	0
$523$	− 34.7381i	− 1.51899i	−0.650511	−	0.759496i	$-0.725445\pi$
$523$	− 34.7381i	− 1.51899i	0.650511	−	0.759496i	$-0.274555\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.22515	0.271172
$528$	0	0
$529$	−5.83846	−0.253846
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.5498i	0.760168i
$534$	0	0
$535$	44.1417	1.90841
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15.3502i	0.661181i
$540$	0	0
$541$	24.8504i	1.06840i	0.845358	+	0.534200i	$0.179387\pi$
$541$	24.8504i	1.06840i	−0.845358	+	0.534200i	$0.820613\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.02431	−0.300888
$546$	0	0
$547$	7.32619i	0.313245i	0.987659	+	0.156623i	$0.0500606\pi$
$547$	7.32619i	0.313245i	−0.987659	+	0.156623i	$0.949939\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−31.7491	−1.35256
$552$	0	0
$553$	−28.1564	−1.19733
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.3646i	1.54082i	0.637551	+	0.770408i	$0.279947\pi$
$557$	36.3646i	1.54082i	−0.637551	+	0.770408i	$0.720053\pi$
$558$	0	0
$559$	−5.59101	−0.236475
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.9579i	0.883270i	0.897195	+	0.441635i	$0.145601\pi$
$563$	20.9579i	0.883270i	−0.897195	+	0.441635i	$0.854399\pi$
$564$	0	0
$565$	− 7.55041i	− 0.317648i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.3214	0.726150	0.363075	−	0.931760i	$-0.381727\pi$
$569$	17.3214	0.726150	0.363075	+	0.931760i	$0.381727\pi$
$570$	0	0
$571$	41.5633i	1.73937i	0.493606	+	0.869686i	$0.335678\pi$
$571$	41.5633i	1.73937i	−0.493606	+	0.869686i	$0.664322\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.5070	−1.02201
$576$	0	0
$577$	−3.88631	−0.161789	−0.0808946	−	0.996723i	$-0.525778\pi$
$577$	−3.88631	−0.161789	−0.0808946	+	0.996723i	$0.525778\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 47.9015i	− 1.98729i
$582$	0	0
$583$	−19.1785	−0.794292
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.7386i	0.608327i	0.952620	+	0.304163i	$0.0983769\pi$
$587$	14.7386i	0.608327i	−0.952620	+	0.304163i	$0.901623\pi$
$588$	0	0
$589$	27.6091i	1.13761i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	40.2819	1.65418	0.827090	−	0.562070i	$-0.189995\pi$
$593$	40.2819	1.65418	0.827090	+	0.562070i	$0.189995\pi$
$594$	0	0
$595$	− 11.4246i	− 0.468361i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15.4982	0.633238	0.316619	−	0.948553i	$-0.397452\pi$
$599$	15.4982	0.633238	0.316619	+	0.948553i	$0.397452\pi$
$600$	0	0
$601$	31.9519	1.30334	0.651672	−	0.758501i	$-0.274068\pi$
$601$	31.9519	1.30334	0.651672	+	0.758501i	$0.274068\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.66101i	0.189497i
$606$	0	0
$607$	3.04667	0.123661	0.0618303	−	0.998087i	$-0.480306\pi$
$607$	3.04667	0.123661	0.0618303	+	0.998087i	$0.480306\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 8.31374i	− 0.336338i
$612$	0	0
$613$	− 34.0750i	− 1.37628i	−0.725580	−	0.688138i	$-0.758428\pi$
$613$	− 34.0750i	− 1.37628i	0.725580	−	0.688138i	$-0.241572\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.1395	−1.61595	−0.807977	−	0.589213i	$-0.799438\pi$
$617$	−40.1395	−1.61595	−0.807977	+	0.589213i	$0.799438\pi$
$618$	0	0
$619$	9.21372i	0.370331i	0.982707	+	0.185165i	$0.0592821\pi$
$619$	9.21372i	0.370331i	−0.982707	+	0.185165i	$0.940718\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	25.3344	1.01500
$624$	0	0
$625$	−19.5824	−0.783295
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.74411i	0.268905i
$630$	0	0
$631$	−23.0526	−0.917708	−0.458854	−	0.888512i	$-0.651740\pi$
$631$	−23.0526	−0.917708	−0.458854	+	0.888512i	$0.651740\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.82300i	− 0.310446i
$636$	0	0
$637$	12.4309i	0.492531i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11.6322	−0.459444	−0.229722	−	0.973256i	$-0.573782\pi$
$641$	−11.6322	−0.459444	−0.229722	+	0.973256i	$0.573782\pi$
$642$	0	0
$643$	21.1290i	0.833247i	0.909079	+	0.416624i	$0.136787\pi$
$643$	21.1290i	0.833247i	−0.909079	+	0.416624i	$0.863213\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.5854	−1.16312	−0.581560	−	0.813503i	$-0.697558\pi$
$647$	−29.5854	−1.16312	−0.581560	+	0.813503i	$0.697558\pi$
$648$	0	0
$649$	−27.3661	−1.07421
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.85861i	0.111866i	0.998435	+	0.0559330i	$0.0178133\pi$
$653$	2.85861i	0.111866i	−0.998435	+	0.0559330i	$0.982187\pi$
$654$	0	0
$655$	−9.04915	−0.353580
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.654259i	0.0254863i	0.999919	+	0.0127431i	$0.00405638\pi$
$659$	0.654259i	0.0254863i	−0.999919	+	0.0127431i	$0.995944\pi$
$660$	0	0
$661$	16.3412i	0.635599i	0.948158	+	0.317800i	$0.102944\pi$
$661$	16.3412i	0.635599i	−0.948158	+	0.317800i	$0.897056\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	50.6689	1.96486
$666$	0	0
$667$	29.6556i	1.14827i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.5012	1.71795
$672$	0	0
$673$	18.7921	0.724382	0.362191	−	0.932104i	$-0.382029\pi$
$673$	18.7921	0.724382	0.362191	+	0.932104i	$0.382029\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.5706i	0.790593i	0.918554	+	0.395296i	$0.129358\pi$
$677$	20.5706i	0.790593i	−0.918554	+	0.395296i	$0.870642\pi$
$678$	0	0
$679$	−28.9350	−1.11042
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 40.4925i	− 1.54940i	−0.632327	−	0.774701i	$-0.717900\pi$
$683$	− 40.4925i	− 1.54940i	0.632327	−	0.774701i	$-0.282100\pi$
$684$	0	0
$685$	− 75.0133i	− 2.86611i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−15.5311	−0.591688
$690$	0	0
$691$	− 9.29651i	− 0.353656i	−0.984242	−	0.176828i	$-0.943416\pi$
$691$	− 9.29651i	− 0.353656i	0.984242	−	0.176828i	$-0.0565836\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.66044	−0.0629843
$696$	0	0
$697$	6.99830	0.265079
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 46.6793i	− 1.76305i	−0.472134	−	0.881527i	$-0.656516\pi$
$701$	− 46.6793i	− 1.76305i	0.472134	−	0.881527i	$-0.343484\pi$
$702$	0	0
$703$	−29.9107	−1.12810
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.5069i	1.03450i
$708$	0	0
$709$	39.5821i	1.48654i	0.668993	+	0.743269i	$0.266726\pi$
$709$	39.5821i	1.48654i	−0.668993	+	0.743269i	$0.733274\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	25.7886	0.965792
$714$	0	0
$715$	− 25.6567i	− 0.959505i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	38.3669	1.43084	0.715422	−	0.698693i	$-0.246235\pi$
$719$	38.3669	1.43084	0.715422	+	0.698693i	$0.246235\pi$
$720$	0	0
$721$	−27.3661	−1.01917
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 42.3489i	− 1.57280i
$726$	0	0
$727$	17.3344	0.642899	0.321450	−	0.946927i	$-0.395830\pi$
$727$	17.3344	0.642899	0.321450	+	0.946927i	$0.395830\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.22951i	0.0824615i
$732$	0	0
$733$	45.3211i	1.67397i	0.547224	+	0.836986i	$0.315685\pi$
$733$	45.3211i	1.67397i	−0.547224	+	0.836986i	$0.684315\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.9485	0.734811
$738$	0	0
$739$	− 20.9602i	− 0.771035i	−0.922701	−	0.385517i	$-0.874023\pi$
$739$	− 20.9602i	− 0.771035i	0.922701	−	0.385517i	$-0.125977\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.7354	−0.760707	−0.380353	−	0.924841i	$-0.624198\pi$
$743$	−20.7354	−0.760707	−0.380353	+	0.924841i	$0.624198\pi$
$744$	0	0
$745$	23.8282	0.872997
$746$	0	0
$747$	0	0
$748$	0	0
$749$	46.1991i	1.68808i
$750$	0	0
$751$	17.6725	0.644877	0.322439	−	0.946590i	$-0.395497\pi$
$751$	17.6725	0.644877	0.322439	+	0.946590i	$0.395497\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 10.4625i	− 0.380770i
$756$	0	0
$757$	22.0131i	0.800081i	0.916498	+	0.400040i	$0.131004\pi$
$757$	22.0131i	0.800081i	−0.916498	+	0.400040i	$0.868996\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.9024	−0.685211	−0.342606	−	0.939479i	$-0.611309\pi$
$761$	−18.9024	−0.685211	−0.342606	+	0.939479i	$0.611309\pi$
$762$	0	0
$763$	− 7.35170i	− 0.266149i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.1616	−0.800209
$768$	0	0
$769$	44.8954	1.61897	0.809486	−	0.587140i	$-0.199746\pi$
$769$	44.8954	1.61897	0.809486	+	0.587140i	$0.199746\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.2196i	0.511445i	0.966750	+	0.255722i	$0.0823133\pi$
$773$	14.2196i	0.511445i	−0.966750	+	0.255722i	$0.917687\pi$
$774$	0	0
$775$	−36.8267	−1.32285
$776$	0	0
$777$	0	0
$778$	0	0
$779$	31.0380i	1.11205i
$780$	0	0
$781$	− 17.5292i	− 0.627243i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.4655	0.373529
$786$	0	0
$787$	− 37.9008i	− 1.35102i	−0.737353	−	0.675508i	$-0.763924\pi$
$787$	− 37.9008i	− 1.35102i	0.737353	−	0.675508i	$-0.236076\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.90232	0.280974
$792$	0	0
$793$	36.0379	1.27974
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.8276i	0.879440i	0.898135	+	0.439720i	$0.144922\pi$
$797$	24.8276i	0.879440i	−0.898135	+	0.439720i	$0.855078\pi$
$798$	0	0
$799$	−3.31525	−0.117285
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 47.8389i	− 1.68820i
$804$	0	0
$805$	− 47.3279i	− 1.66809i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.87801	−0.241818	−0.120909	−	0.992664i	$-0.538581\pi$
$809$	−6.87801	−0.241818	−0.120909	+	0.992664i	$0.538581\pi$
$810$	0	0
$811$	− 5.35901i	− 0.188180i	−0.995564	−	0.0940901i	$-0.970006\pi$
$811$	− 5.35901i	− 0.188180i	0.995564	−	0.0940901i	$-0.0299942\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.49543	0.192497
$816$	0	0
$817$	−9.88808	−0.345940
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.17151i	0.0757861i	0.999282	+	0.0378931i	$0.0120646\pi$
$821$	2.17151i	0.0757861i	−0.999282	+	0.0378931i	$0.987935\pi$
$822$	0	0
$823$	−38.9210	−1.35670	−0.678350	−	0.734739i	$-0.737305\pi$
$823$	−38.9210	−1.35670	−0.678350	+	0.734739i	$0.737305\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 22.5203i	− 0.783109i	−0.920155	−	0.391554i	$-0.871938\pi$
$827$	− 22.5203i	− 0.783109i	0.920155	−	0.391554i	$-0.128062\pi$
$828$	0	0
$829$	11.4269i	0.396873i	0.980114	+	0.198436i	$0.0635863\pi$
$829$	11.4269i	0.396873i	−0.980114	+	0.198436i	$0.936414\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.95705	0.171751
$834$	0	0
$835$	− 28.3359i	− 0.980604i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.1591	−0.972161	−0.486081	−	0.873914i	$-0.661574\pi$
$839$	−28.1591	−0.972161	−0.486081	+	0.873914i	$0.661574\pi$
$840$	0	0
$841$	−22.2458	−0.767097
$842$	0	0
$843$	0	0
$844$	0	0
$845$	22.1735i	0.762792i
$846$	0	0
$847$	−4.87826	−0.167619
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27.9385i	0.957718i
$852$	0	0
$853$	− 0.516607i	− 0.0176883i	−0.999961	−	0.00884415i	$-0.997185\pi$
$853$	− 0.516607i	− 0.0176883i	0.999961	−	0.00884415i	$-0.00281522\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.72307	0.263815	0.131907	−	0.991262i	$-0.457890\pi$
$857$	7.72307	0.263815	0.131907	+	0.991262i	$0.457890\pi$
$858$	0	0
$859$	− 36.3045i	− 1.23869i	−0.785118	−	0.619347i	$-0.787398\pi$
$859$	− 36.3045i	− 1.23869i	0.785118	−	0.619347i	$-0.212602\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.9299	1.35923	0.679615	−	0.733569i	$-0.262147\pi$
$863$	39.9299	1.35923	0.679615	+	0.733569i	$0.262147\pi$
$864$	0	0
$865$	14.8073	0.503462
$866$	0	0
$867$	0	0
$868$	0	0
$869$	25.2149i	0.855358i
$870$	0	0
$871$	16.1547	0.547380
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.4625i	0.353698i
$876$	0	0
$877$	3.73210i	0.126024i	0.998013	+	0.0630121i	$0.0200707\pi$
$877$	3.73210i	0.126024i	−0.998013	+	0.0630121i	$0.979929\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.3644	0.888239	0.444120	−	0.895968i	$-0.353517\pi$
$881$	26.3644	0.888239	0.444120	+	0.895968i	$0.353517\pi$
$882$	0	0
$883$	− 19.9236i	− 0.670483i	−0.942132	−	0.335241i	$-0.891182\pi$
$883$	− 19.9236i	− 0.670483i	0.942132	−	0.335241i	$-0.108818\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.2662	0.982663	0.491331	−	0.870973i	$-0.336510\pi$
$887$	29.2662	0.982663	0.491331	+	0.870973i	$0.336510\pi$
$888$	0	0
$889$	8.18762	0.274604
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 14.7034i	− 0.492031i
$894$	0	0
$895$	11.9141	0.398244
$896$	0	0
$897$	0	0
$898$	0	0
$899$	44.5635i	1.48628i
$900$	0	0
$901$	6.19330i	0.206329i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	57.9451	1.92616
$906$	0	0
$907$	− 54.0895i	− 1.79601i	−0.439983	−	0.898006i	$-0.645015\pi$
$907$	− 54.0895i	− 1.79601i	0.439983	−	0.898006i	$-0.354985\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.8223	1.71695	0.858475	−	0.512856i	$-0.171413\pi$
$911$	51.8223	1.71695	0.858475	+	0.512856i	$0.171413\pi$
$912$	0	0
$913$	−42.8972	−1.41969
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.47092i	− 0.312757i
$918$	0	0
$919$	19.5888	0.646174	0.323087	−	0.946369i	$-0.395279\pi$
$919$	19.5888	0.646174	0.323087	+	0.946369i	$0.395279\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 14.1955i	− 0.467249i
$924$	0	0
$925$	− 39.8967i	− 1.31180i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.9090	0.423530	0.211765	−	0.977321i	$-0.432079\pi$
$929$	12.9090	0.423530	0.211765	+	0.977321i	$0.432079\pi$
$930$	0	0
$931$	21.9849i	0.720526i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.2310	−0.334591
$936$	0	0
$937$	4.57400	0.149426	0.0747130	−	0.997205i	$-0.476196\pi$
$937$	4.57400	0.149426	0.0747130	+	0.997205i	$0.476196\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.2967i	0.889846i	0.895569	+	0.444923i	$0.146769\pi$
$941$	27.2967i	0.889846i	−0.895569	+	0.444923i	$0.853231\pi$
$942$	0	0
$943$	28.9915	0.944093
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.0284i	0.520854i	0.965494	+	0.260427i	$0.0838634\pi$
$947$	16.0284i	0.520854i	−0.965494	+	0.260427i	$0.916137\pi$
$948$	0	0
$949$	− 38.7409i	− 1.25758i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.0588	1.10327	0.551636	−	0.834085i	$-0.314004\pi$
$953$	34.0588	1.10327	0.551636	+	0.834085i	$0.314004\pi$
$954$	0	0
$955$	28.9549i	0.936960i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	78.5095	2.53520
$960$	0	0
$961$	7.75255	0.250082
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 61.0717i	− 1.96597i
$966$	0	0
$967$	22.8242	0.733978	0.366989	−	0.930225i	$-0.380389\pi$
$967$	22.8242	0.733978	0.366989	+	0.930225i	$0.380389\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 5.49360i	− 0.176298i	−0.996107	−	0.0881490i	$-0.971905\pi$
$971$	− 5.49360i	− 0.176298i	0.996107	−	0.0881490i	$-0.0280951\pi$
$972$	0	0
$973$	− 1.73784i	− 0.0557125i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.2254	0.775039	0.387520	−	0.921861i	$-0.373332\pi$
$977$	24.2254	0.775039	0.387520	+	0.921861i	$0.373332\pi$
$978$	0	0
$979$	− 22.6878i	− 0.725104i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.00418	−0.159609	−0.0798043	−	0.996811i	$-0.525430\pi$
$983$	−5.00418	−0.159609	−0.0798043	+	0.996811i	$0.525430\pi$
$984$	0	0
$985$	12.8107	0.408182
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.23609i	0.293691i
$990$	0	0
$991$	43.0455	1.36739	0.683693	−	0.729770i	$-0.260373\pi$
$991$	43.0455	1.36739	0.683693	+	0.729770i	$0.260373\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 43.4164i	− 1.37639i
$996$	0	0
$997$	− 3.28681i	− 0.104094i	−0.998645	−	0.0520471i	$-0.983425\pi$
$997$	− 3.28681i	− 0.104094i	0.998645	−	0.0520471i	$-0.0165746\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.f.c.2449.8		8
3.2	odd	2		544.2.c.a.273.3		8
4.3	odd	2		1224.2.f.d.613.7		8
8.3	odd	2		1224.2.f.d.613.8		8
8.5	even	2	inner	4896.2.f.c.2449.1		8
12.11	even	2		136.2.c.a.69.2	yes	8
24.5	odd	2		544.2.c.a.273.6		8
24.11	even	2		136.2.c.a.69.1	✓	8
48.5	odd	4		4352.2.a.be.1.3		8
48.11	even	4		4352.2.a.bc.1.6		8
48.29	odd	4		4352.2.a.be.1.6		8
48.35	even	4		4352.2.a.bc.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.c.a.69.1	✓	8	24.11	even	2
136.2.c.a.69.2	yes	8	12.11	even	2
544.2.c.a.273.3		8	3.2	odd	2
544.2.c.a.273.6		8	24.5	odd	2
1224.2.f.d.613.7		8	4.3	odd	2
1224.2.f.d.613.8		8	8.3	odd	2
4352.2.a.bc.1.3		8	48.35	even	4
4352.2.a.bc.1.6		8	48.11	even	4
4352.2.a.be.1.3		8	48.5	odd	4
4352.2.a.be.1.6		8	48.29	odd	4
4896.2.f.c.2449.1		8	8.5	even	2	inner
4896.2.f.c.2449.8		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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+3.30391i q^{5} -3.45790 q^{7} +O(q^{10})\)	\(q+3.30391i q^{5} -3.45790 q^{7} +3.09665i q^{11} +2.50773i q^{13} +1.00000 q^{17} +4.43508i q^{19} +4.14265 q^{23} -5.91579 q^{25} +7.15862i q^{29} +6.22515 q^{31} -11.4246i q^{35} +6.74411i q^{37} +6.99830 q^{41} +2.22951i q^{43} -3.31525 q^{47} +4.95705 q^{49} +6.19330i q^{53} -10.2310 q^{55} +8.83732i q^{59} -14.3707i q^{61} -8.28530 q^{65} -6.44195i q^{67} -5.66069 q^{71} -15.4486 q^{73} -10.7079i q^{77} +8.14265 q^{79} +13.8528i q^{83} +3.30391i q^{85} -7.32655 q^{89} -8.67146i q^{91} -14.6531 q^{95} +8.36780 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} + 8 q^{17} + 12 q^{23} - 8 q^{25} + 12 q^{31} - 28 q^{47} - 8 q^{49} - 44 q^{55} - 24 q^{65} + 8 q^{73} + 44 q^{79} - 8 q^{89} - 16 q^{95} + 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 + 8 * q^17 + 12 * q^23 - 8 * q^25 + 12 * q^31 - 28 * q^47 - 8 * q^49 - 44 * q^55 - 24 * q^65 + 8 * q^73 + 44 * q^79 - 8 * q^89 - 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

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Database

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