LMFDB - Embedded newform 441.4.c.a.440.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.c (of order $2$, degree $1$, 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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} + \cdots + 810000 $ x^16 - 48x^14 + 1647x^12 - 27620x^10 + 336765x^8 - 1200006x^6 + 3242464x^4 - 1762200*x^2 + 810000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{8}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.3
Root			$3.91663 + 2.26127i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.4.c.a.440.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.52254i q^{2} -12.4534 q^{4} -1.26570 q^{5} +20.1405i q^{8} +O(q^{10})$	$q-4.52254i q^{2} -12.4534 q^{4} -1.26570 q^{5} +20.1405i q^{8} +5.72419i q^{10} -41.5978i q^{11} -85.7355i q^{13} -8.54055 q^{16} +77.7859 q^{17} -48.6865i q^{19} +15.7623 q^{20} -188.128 q^{22} +90.9487i q^{23} -123.398 q^{25} -387.742 q^{26} +151.196i q^{29} +88.1799i q^{31} +199.749i q^{32} -351.790i q^{34} +90.5828 q^{37} -220.187 q^{38} -25.4919i q^{40} -383.530 q^{41} -227.894 q^{43} +518.033i q^{44} +411.319 q^{46} +139.106 q^{47} +558.072i q^{50} +1067.70i q^{52} +334.573i q^{53} +52.6505i q^{55} +683.789 q^{58} -880.100 q^{59} -13.1571i q^{61} +398.797 q^{62} +835.050 q^{64} +108.516i q^{65} -442.424 q^{67} -968.696 q^{68} -341.552i q^{71} -921.703i q^{73} -409.664i q^{74} +606.311i q^{76} -413.129 q^{79} +10.8098 q^{80} +1734.53i q^{82} +954.307 q^{83} -98.4538 q^{85} +1030.66i q^{86} +837.803 q^{88} -29.6981 q^{89} -1132.62i q^{92} -629.111i q^{94} +61.6227i q^{95} -1199.63i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 64 q^{4}+O(q^{10}) $ 16 * q - 64 * q^4	$ 16 q - 64 q^{4} + 376 q^{16} + 528 q^{22} + 40 q^{25} + 2392 q^{37} + 328 q^{43} + 2784 q^{46} + 6744 q^{58} + 5432 q^{64} - 616 q^{67} + 4352 q^{79} - 4608 q^{85} - 1416 q^{88}+O(q^{100}) $ 16 * q - 64 * q^4 + 376 * q^16 + 528 * q^22 + 40 * q^25 + 2392 * q^37 + 328 * q^43 + 2784 * q^46 + 6744 * q^58 + 5432 * q^64 - 616 * q^67 + 4352 * q^79 - 4608 * q^85 - 1416 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 4.52254i	− 1.59896i	−0.600693	−	0.799480i	$-0.705109\pi$
$2$	− 4.52254i	− 1.59896i	0.600693	−	0.799480i	$-0.294891\pi$
$3$	0	0
$3$	0	0
$4$	−12.4534	−1.55667
$5$	−1.26570	−0.113208	−0.0566040	−	0.998397i	$-0.518027\pi$
$5$	−1.26570	−0.113208	−0.0566040	+	0.998397i	$0.518027\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	20.1405i	0.890094i
$9$	0	0
$10$	5.72419i	0.181015i
$11$	− 41.5978i	− 1.14020i	−0.821575	−	0.570101i	$-0.806904\pi$
$11$	− 41.5978i	− 1.14020i	0.821575	−	0.570101i	$-0.193096\pi$
$12$	0	0
$13$	− 85.7355i	− 1.82914i	−0.404433	−	0.914568i	$-0.632531\pi$
$13$	− 85.7355i	− 1.82914i	0.404433	−	0.914568i	$-0.367469\pi$
$14$	0	0
$15$	0	0
$16$	−8.54055	−0.133446
$17$	77.7859	1.10976	0.554878	−	0.831932i	$-0.312765\pi$
$17$	77.7859	1.10976	0.554878	+	0.831932i	$0.312765\pi$
$18$	0	0
$19$	− 48.6865i	− 0.587866i	−0.955826	−	0.293933i	$-0.905036\pi$
$19$	− 48.6865i	− 0.587866i	0.955826	−	0.293933i	$-0.0949642\pi$
$20$	15.7623	0.176227
$21$	0	0
$22$	−188.128	−1.82314
$23$	90.9487i	0.824527i	0.911065	+	0.412263i	$0.135262\pi$
$23$	90.9487i	0.824527i	−0.911065	+	0.412263i	$0.864738\pi$
$24$	0	0
$25$	−123.398	−0.987184
$26$	−387.742	−2.92471
$27$	0	0
$28$	0	0
$29$	151.196i	0.968151i	0.875026	+	0.484075i	$0.160844\pi$
$29$	151.196i	0.968151i	−0.875026	+	0.484075i	$0.839156\pi$
$30$	0	0
$31$	88.1799i	0.510890i	0.966824	+	0.255445i	$0.0822219\pi$
$31$	88.1799i	0.510890i	−0.966824	+	0.255445i	$0.917778\pi$
$32$	199.749i	1.10347i
$33$	0	0
$34$	− 351.790i	− 1.77445i
$35$	0	0
$36$	0	0
$37$	90.5828	0.402479	0.201239	−	0.979542i	$-0.435503\pi$
$37$	90.5828	0.402479	0.201239	+	0.979542i	$0.435503\pi$
$38$	−220.187	−0.939974
$39$	0	0
$40$	− 25.4919i	− 0.100766i
$41$	−383.530	−1.46091	−0.730455	−	0.682961i	$-0.760692\pi$
$41$	−383.530	−1.46091	−0.730455	+	0.682961i	$0.760692\pi$
$42$	0	0
$43$	−227.894	−0.808222	−0.404111	−	0.914710i	$-0.632419\pi$
$43$	−227.894	−0.808222	−0.404111	+	0.914710i	$0.632419\pi$
$44$	518.033i	1.77492i
$45$	0	0
$46$	411.319	1.31838
$47$	139.106	0.431716	0.215858	−	0.976425i	$-0.430745\pi$
$47$	139.106	0.431716	0.215858	+	0.976425i	$0.430745\pi$
$48$	0	0
$49$	0	0
$50$	558.072i	1.57847i
$51$	0	0
$52$	1067.70i	2.84736i
$53$	334.573i	0.867116i	0.901126	+	0.433558i	$0.142742\pi$
$53$	334.573i	0.867116i	−0.901126	+	0.433558i	$0.857258\pi$
$54$	0	0
$55$	52.6505i	0.129080i
$56$	0	0
$57$	0	0
$58$	683.789	1.54803
$59$	−880.100	−1.94202	−0.971010	−	0.239037i	$-0.923168\pi$
$59$	−880.100	−1.94202	−0.971010	+	0.239037i	$0.923168\pi$
$60$	0	0
$61$	− 13.1571i	− 0.0276162i	−0.999905	−	0.0138081i	$-0.995605\pi$
$61$	− 13.1571i	− 0.0276162i	0.999905	−	0.0138081i	$-0.00439540\pi$
$62$	398.797	0.816892
$63$	0	0
$64$	835.050	1.63096
$65$	108.516i	0.207073i
$66$	0	0
$67$	−442.424	−0.806728	−0.403364	−	0.915040i	$-0.632159\pi$
$67$	−442.424	−0.806728	−0.403364	+	0.915040i	$0.632159\pi$
$68$	−968.696	−1.72752
$69$	0	0
$70$	0	0
$71$	− 341.552i	− 0.570912i	−0.958392	−	0.285456i	$-0.907855\pi$
$71$	− 341.552i	− 0.570912i	0.958392	−	0.285456i	$-0.0921450\pi$
$72$	0	0
$73$	− 921.703i	− 1.47777i	−0.673832	−	0.738885i	$-0.735353\pi$
$73$	− 921.703i	− 1.47777i	0.673832	−	0.738885i	$-0.264647\pi$
$74$	− 409.664i	− 0.643547i
$75$	0	0
$76$	606.311i	0.915114i
$77$	0	0
$78$	0	0
$79$	−413.129	−0.588362	−0.294181	−	0.955750i	$-0.595047\pi$
$79$	−413.129	−0.588362	−0.294181	+	0.955750i	$0.595047\pi$
$80$	10.8098	0.0151072
$81$	0	0
$82$	1734.53i	2.33594i
$83$	954.307	1.26203	0.631017	−	0.775769i	$-0.282638\pi$
$83$	954.307	1.26203	0.631017	+	0.775769i	$0.282638\pi$
$84$	0	0
$85$	−98.4538	−0.125633
$86$	1030.66i	1.29231i
$87$	0	0
$88$	837.803	1.01489
$89$	−29.6981	−0.0353707	−0.0176853	−	0.999844i	$-0.505630\pi$
$89$	−29.6981	−0.0353707	−0.0176853	+	0.999844i	$0.505630\pi$
$90$	0	0
$91$	0	0
$92$	− 1132.62i	− 1.28352i
$93$	0	0
$94$	− 629.111i	− 0.690297i
$95$	61.6227i	0.0665511i
$96$	0	0
$97$	− 1199.63i	− 1.25572i	−0.778328	−	0.627858i	$-0.783932\pi$
$97$	− 1199.63i	− 1.25572i	0.778328	−	0.627858i	$-0.216068\pi$
$98$	0	0
$99$	0	0
$100$	1536.72	1.53672
$101$	654.844	0.645143	0.322571	−	0.946545i	$-0.395453\pi$
$101$	654.844	0.645143	0.322571	+	0.946545i	$0.395453\pi$
$102$	0	0
$103$	1369.49i	1.31009i	0.755589	+	0.655047i	$0.227351\pi$
$103$	1369.49i	1.31009i	−0.755589	+	0.655047i	$0.772649\pi$
$104$	1726.76	1.62810
$105$	0	0
$106$	1513.12	1.38648
$107$	− 428.753i	− 0.387375i	−0.981063	−	0.193688i	$-0.937955\pi$
$107$	− 428.753i	− 0.387375i	0.981063	−	0.193688i	$-0.0620448\pi$
$108$	0	0
$109$	−668.521	−0.587456	−0.293728	−	0.955889i	$-0.594896\pi$
$109$	−668.521	−0.587456	−0.293728	+	0.955889i	$0.594896\pi$
$110$	238.114	0.206393
$111$	0	0
$112$	0	0
$113$	− 914.837i	− 0.761598i	−0.924658	−	0.380799i	$-0.875649\pi$
$113$	− 914.837i	− 0.761598i	0.924658	−	0.380799i	$-0.124351\pi$
$114$	0	0
$115$	− 115.114i	− 0.0933429i
$116$	− 1882.90i	− 1.50709i
$117$	0	0
$118$	3980.29i	3.10521i
$119$	0	0
$120$	0	0
$121$	−399.379	−0.300060
$122$	−59.5034	−0.0441572
$123$	0	0
$124$	− 1098.14i	− 0.795287i
$125$	314.398	0.224965
$126$	0	0
$127$	1260.95	0.881034	0.440517	−	0.897744i	$-0.354795\pi$
$127$	1260.95	0.881034	0.440517	+	0.897744i	$0.354795\pi$
$128$	− 2178.55i	− 1.50436i
$129$	0	0
$130$	490.767	0.331101
$131$	1367.20	0.911853	0.455926	−	0.890017i	$-0.349308\pi$
$131$	1367.20	0.911853	0.455926	+	0.890017i	$0.349308\pi$
$132$	0	0
$133$	0	0
$134$	2000.88i	1.28992i
$135$	0	0
$136$	1566.65i	0.987787i
$137$	1101.16i	0.686703i	0.939207	+	0.343351i	$0.111562\pi$
$137$	1101.16i	0.686703i	−0.939207	+	0.343351i	$0.888438\pi$
$138$	0	0
$139$	− 2306.56i	− 1.40748i	−0.710458	−	0.703739i	$-0.751512\pi$
$139$	− 2306.56i	− 1.40748i	0.710458	−	0.703739i	$-0.248488\pi$
$140$	0	0
$141$	0	0
$142$	−1544.68	−0.912865
$143$	−3566.41	−2.08558
$144$	0	0
$145$	− 191.369i	− 0.109602i
$146$	−4168.44	−2.36289
$147$	0	0
$148$	−1128.06	−0.626527
$149$	1755.80i	0.965376i	0.875793	+	0.482688i	$0.160339\pi$
$149$	1755.80i	0.965376i	−0.875793	+	0.482688i	$0.839661\pi$
$150$	0	0
$151$	−524.981	−0.282930	−0.141465	−	0.989943i	$-0.545181\pi$
$151$	−524.981	−0.282930	−0.141465	+	0.989943i	$0.545181\pi$
$152$	980.573	0.523256
$153$	0	0
$154$	0	0
$155$	− 111.610i	− 0.0578368i
$156$	0	0
$157$	− 1318.02i	− 0.669995i	−0.942219	−	0.334998i	$-0.891264\pi$
$157$	− 1318.02i	− 0.669995i	0.942219	−	0.334998i	$-0.108736\pi$
$158$	1868.39i	0.940767i
$159$	0	0
$160$	− 252.823i	− 0.124921i
$161$	0	0
$162$	0	0
$163$	−447.832	−0.215196	−0.107598	−	0.994195i	$-0.534316\pi$
$163$	−447.832	−0.215196	−0.107598	+	0.994195i	$0.534316\pi$
$164$	4776.24	2.27416
$165$	0	0
$166$	− 4315.89i	− 2.01794i
$167$	−811.124	−0.375848	−0.187924	−	0.982184i	$-0.560176\pi$
$167$	−811.124	−0.375848	−0.187924	+	0.982184i	$0.560176\pi$
$168$	0	0
$169$	−5153.58	−2.34574
$170$	445.261i	0.200882i
$171$	0	0
$172$	2838.05	1.25814
$173$	2242.47	0.985503	0.492751	−	0.870170i	$-0.335991\pi$
$173$	2242.47	0.985503	0.492751	+	0.870170i	$0.335991\pi$
$174$	0	0
$175$	0	0
$176$	355.269i	0.152156i
$177$	0	0
$178$	134.311i	0.0565563i
$179$	2923.43i	1.22071i	0.792127	+	0.610357i	$0.208974\pi$
$179$	2923.43i	1.22071i	−0.792127	+	0.610357i	$0.791026\pi$
$180$	0	0
$181$	282.859i	0.116159i	0.998312	+	0.0580794i	$0.0184977\pi$
$181$	282.859i	0.116159i	−0.998312	+	0.0580794i	$0.981502\pi$
$182$	0	0
$183$	0	0
$184$	−1831.76	−0.733907
$185$	−114.651	−0.0455638
$186$	0	0
$187$	− 3235.72i	− 1.26534i
$188$	−1732.34	−0.672040
$189$	0	0
$190$	278.691	0.106412
$191$	− 4617.22i	− 1.74916i	−0.484878	−	0.874582i	$-0.661136\pi$
$191$	− 4617.22i	− 1.74916i	0.484878	−	0.874582i	$-0.338864\pi$
$192$	0	0
$193$	−4155.45	−1.54982	−0.774912	−	0.632070i	$-0.782206\pi$
$193$	−4155.45	−1.54982	−0.774912	+	0.632070i	$0.782206\pi$
$194$	−5425.39	−2.00784
$195$	0	0
$196$	0	0
$197$	− 1626.36i	− 0.588190i	−0.955776	−	0.294095i	$-0.904982\pi$
$197$	− 1626.36i	− 0.588190i	0.955776	−	0.294095i	$-0.0950183\pi$
$198$	0	0
$199$	174.199i	0.0620535i	0.999519	+	0.0310267i	$0.00987771\pi$
$199$	174.199i	0.0620535i	−0.999519	+	0.0310267i	$0.990122\pi$
$200$	− 2485.30i	− 0.878687i
$201$	0	0
$202$	− 2961.56i	− 1.03156i
$203$	0	0
$204$	0	0
$205$	485.435	0.165387
$206$	6193.56	2.09479
$207$	0	0
$208$	732.229i	0.244091i
$209$	−2025.25	−0.670286
$210$	0	0
$211$	−2942.35	−0.959999	−0.479999	−	0.877269i	$-0.659363\pi$
$211$	−2942.35	−0.959999	−0.479999	+	0.877269i	$0.659363\pi$
$212$	− 4166.56i	− 1.34981i
$213$	0	0
$214$	−1939.05	−0.619397
$215$	288.446	0.0914972
$216$	0	0
$217$	0	0
$218$	3023.42i	0.939319i
$219$	0	0
$220$	− 655.676i	− 0.200935i
$221$	− 6669.01i	− 2.02989i
$222$	0	0
$223$	− 3374.75i	− 1.01341i	−0.862120	−	0.506704i	$-0.830864\pi$
$223$	− 3374.75i	− 1.01341i	0.862120	−	0.506704i	$-0.169136\pi$
$224$	0	0
$225$	0	0
$226$	−4137.39	−1.21776
$227$	3030.85	0.886188	0.443094	−	0.896475i	$-0.353881\pi$
$227$	3030.85	0.886188	0.443094	+	0.896475i	$0.353881\pi$
$228$	0	0
$229$	− 1108.55i	− 0.319890i	−0.987126	−	0.159945i	$-0.948868\pi$
$229$	− 1108.55i	− 0.319890i	0.987126	−	0.159945i	$-0.0511317\pi$
$230$	−520.608	−0.149252
$231$	0	0
$232$	−3045.17	−0.861746
$233$	3099.87i	0.871585i	0.900047	+	0.435793i	$0.143532\pi$
$233$	3099.87i	0.871585i	−0.900047	+	0.435793i	$0.856468\pi$
$234$	0	0
$235$	−176.067	−0.0488737
$236$	10960.2	3.02309
$237$	0	0
$238$	0	0
$239$	1735.25i	0.469640i	0.972039	+	0.234820i	$0.0754501\pi$
$239$	1735.25i	0.469640i	−0.972039	+	0.234820i	$0.924550\pi$
$240$	0	0
$241$	− 1200.04i	− 0.320752i	−0.987056	−	0.160376i	$-0.948729\pi$
$241$	− 1200.04i	− 0.320752i	0.987056	−	0.160376i	$-0.0512707\pi$
$242$	1806.21i	0.479783i
$243$	0	0
$244$	163.850i	0.0429894i
$245$	0	0
$246$	0	0
$247$	−4174.16	−1.07529
$248$	−1775.99	−0.454740
$249$	0	0
$250$	− 1421.88i	− 0.359710i
$251$	3712.56	0.933603	0.466802	−	0.884362i	$-0.345406\pi$
$251$	3712.56	0.933603	0.466802	+	0.884362i	$0.345406\pi$
$252$	0	0
$253$	3783.27	0.940126
$254$	− 5702.70i	− 1.40874i
$255$	0	0
$256$	−3172.19	−0.774460
$257$	−779.148	−0.189113	−0.0945563	−	0.995520i	$-0.530143\pi$
$257$	−779.148	−0.189113	−0.0945563	+	0.995520i	$0.530143\pi$
$258$	0	0
$259$	0	0
$260$	− 1351.39i	− 0.322344i
$261$	0	0
$262$	− 6183.21i	− 1.45802i
$263$	− 1965.38i	− 0.460801i	−0.973096	−	0.230400i	$-0.925996\pi$
$263$	− 1965.38i	− 0.460801i	0.973096	−	0.230400i	$-0.0740035\pi$
$264$	0	0
$265$	− 423.470i	− 0.0981644i
$266$	0	0
$267$	0	0
$268$	5509.68	1.25581
$269$	−2472.82	−0.560486	−0.280243	−	0.959929i	$-0.590415\pi$
$269$	−2472.82	−0.560486	−0.280243	+	0.959929i	$0.590415\pi$
$270$	0	0
$271$	− 4729.42i	− 1.06012i	−0.847961	−	0.530058i	$-0.822170\pi$
$271$	− 4729.42i	− 1.06012i	0.847961	−	0.530058i	$-0.177830\pi$
$272$	−664.334	−0.148093
$273$	0	0
$274$	4980.03	1.09801
$275$	5133.09i	1.12559i
$276$	0	0
$277$	1173.16	0.254470	0.127235	−	0.991873i	$-0.459390\pi$
$277$	1173.16	0.254470	0.127235	+	0.991873i	$0.459390\pi$
$278$	−10431.5	−2.25050
$279$	0	0
$280$	0	0
$281$	− 8195.18i	− 1.73980i	−0.493229	−	0.869899i	$-0.664184\pi$
$281$	− 8195.18i	− 1.73980i	0.493229	−	0.869899i	$-0.335816\pi$
$282$	0	0
$283$	2589.34i	0.543889i	0.962313	+	0.271944i	$0.0876666\pi$
$283$	2589.34i	0.543889i	−0.962313	+	0.271944i	$0.912333\pi$
$284$	4253.47i	0.888722i
$285$	0	0
$286$	16129.2i	3.33476i
$287$	0	0
$288$	0	0
$289$	1137.64	0.231557
$290$	−865.474	−0.175250
$291$	0	0
$292$	11478.3i	2.30040i
$293$	8871.16	1.76880	0.884400	−	0.466729i	$-0.154568\pi$
$293$	8871.16	1.76880	0.884400	+	0.466729i	$0.154568\pi$
$294$	0	0
$295$	1113.94	0.219852
$296$	1824.39i	0.358244i
$297$	0	0
$298$	7940.69	1.54360
$299$	7797.53	1.50817
$300$	0	0
$301$	0	0
$302$	2374.25i	0.452393i
$303$	0	0
$304$	415.810i	0.0784484i
$305$	16.6529i	0.00312638i
$306$	0	0
$307$	2707.52i	0.503344i	0.967813	+	0.251672i	$0.0809804\pi$
$307$	2707.52i	0.503344i	−0.967813	+	0.251672i	$0.919020\pi$
$308$	0	0
$309$	0	0
$310$	−504.759	−0.0924786
$311$	−2161.09	−0.394033	−0.197017	−	0.980400i	$-0.563125\pi$
$311$	−2161.09	−0.394033	−0.197017	+	0.980400i	$0.563125\pi$
$312$	0	0
$313$	8429.61i	1.52227i	0.648595	+	0.761133i	$0.275357\pi$
$313$	8429.61i	1.52227i	−0.648595	+	0.761133i	$0.724643\pi$
$314$	−5960.79	−1.07130
$315$	0	0
$316$	5144.84	0.915886
$317$	− 9595.64i	− 1.70014i	−0.526669	−	0.850071i	$-0.676559\pi$
$317$	− 9595.64i	− 1.70014i	0.526669	−	0.850071i	$-0.323441\pi$
$318$	0	0
$319$	6289.42	1.10389
$320$	−1056.93	−0.184637
$321$	0	0
$322$	0	0
$323$	− 3787.12i	− 0.652387i
$324$	0	0
$325$	10579.6i	1.80569i
$326$	2025.34i	0.344089i
$327$	0	0
$328$	− 7724.50i	− 1.30035i
$329$	0	0
$330$	0	0
$331$	8543.91	1.41878	0.709390	−	0.704817i	$-0.248971\pi$
$331$	8543.91	1.41878	0.709390	+	0.704817i	$0.248971\pi$
$332$	−11884.3	−1.96457
$333$	0	0
$334$	3668.34i	0.600966i
$335$	559.978	0.0913280
$336$	0	0
$337$	598.875	0.0968036	0.0484018	−	0.998828i	$-0.484587\pi$
$337$	598.875	0.0968036	0.0484018	+	0.998828i	$0.484587\pi$
$338$	23307.3i	3.75074i
$339$	0	0
$340$	1226.08	0.195569
$341$	3668.09	0.582517
$342$	0	0
$343$	0	0
$344$	− 4589.91i	− 0.719394i
$345$	0	0
$346$	− 10141.7i	− 1.57578i
$347$	7100.97i	1.09856i	0.835638	+	0.549280i	$0.185098\pi$
$347$	7100.97i	1.09856i	−0.835638	+	0.549280i	$0.814902\pi$
$348$	0	0
$349$	3620.71i	0.555336i	0.960677	+	0.277668i	$0.0895616\pi$
$349$	3620.71i	0.555336i	−0.960677	+	0.277668i	$0.910438\pi$
$350$	0	0
$351$	0	0
$352$	8309.14	1.25818
$353$	2329.77	0.351278	0.175639	−	0.984455i	$-0.443801\pi$
$353$	2329.77	0.351278	0.175639	+	0.984455i	$0.443801\pi$
$354$	0	0
$355$	432.303i	0.0646317i
$356$	369.841	0.0550605
$357$	0	0
$358$	13221.3	1.95187
$359$	− 1757.95i	− 0.258443i	−0.991616	−	0.129222i	$-0.958752\pi$
$359$	− 1757.95i	− 0.258443i	0.991616	−	0.129222i	$-0.0412478\pi$
$360$	0	0
$361$	4488.62	0.654414
$362$	1279.24	0.185733
$363$	0	0
$364$	0	0
$365$	1166.60i	0.167295i
$366$	0	0
$367$	1684.58i	0.239603i	0.992798	+	0.119802i	$0.0382259\pi$
$367$	1684.58i	0.239603i	−0.992798	+	0.119802i	$0.961774\pi$
$368$	− 776.752i	− 0.110030i
$369$	0	0
$370$	518.513i	0.0728547i
$371$	0	0
$372$	0	0
$373$	−297.292	−0.0412686	−0.0206343	−	0.999787i	$-0.506569\pi$
$373$	−297.292	−0.0412686	−0.0206343	+	0.999787i	$0.506569\pi$
$374$	−14633.7	−2.02323
$375$	0	0
$376$	2801.66i	0.384268i
$377$	12962.9	1.77088
$378$	0	0
$379$	−7402.78	−1.00331	−0.501656	−	0.865067i	$-0.667276\pi$
$379$	−7402.78	−1.00331	−0.501656	+	0.865067i	$0.667276\pi$
$380$	− 767.410i	− 0.103598i
$381$	0	0
$382$	−20881.6	−2.79684
$383$	−12527.5	−1.67135	−0.835676	−	0.549223i	$-0.814924\pi$
$383$	−12527.5	−1.67135	−0.835676	+	0.549223i	$0.814924\pi$
$384$	0	0
$385$	0	0
$386$	18793.2i	2.47810i
$387$	0	0
$388$	14939.5i	1.95474i
$389$	− 7972.22i	− 1.03909i	−0.854442	−	0.519547i	$-0.826101\pi$
$389$	− 7972.22i	− 1.03909i	0.854442	−	0.519547i	$-0.173899\pi$
$390$	0	0
$391$	7074.52i	0.915023i
$392$	0	0
$393$	0	0
$394$	−7355.29	−0.940493
$395$	522.898	0.0666072
$396$	0	0
$397$	− 12508.3i	− 1.58130i	−0.612271	−	0.790648i	$-0.709744\pi$
$397$	− 12508.3i	− 1.58130i	0.612271	−	0.790648i	$-0.290256\pi$
$398$	787.822	0.0992210
$399$	0	0
$400$	1053.89	0.131736
$401$	− 8017.72i	− 0.998469i	−0.866467	−	0.499234i	$-0.833615\pi$
$401$	− 8017.72i	− 0.998469i	0.866467	−	0.499234i	$-0.166385\pi$
$402$	0	0
$403$	7560.15	0.934487
$404$	−8155.01	−1.00427
$405$	0	0
$406$	0	0
$407$	− 3768.05i	− 0.458907i
$408$	0	0
$409$	− 8736.51i	− 1.05622i	−0.849177	−	0.528109i	$-0.822901\pi$
$409$	− 8736.51i	− 1.05622i	0.849177	−	0.528109i	$-0.177099\pi$
$410$	− 2195.40i	− 0.264446i
$411$	0	0
$412$	− 17054.7i	− 2.03938i
$413$	0	0
$414$	0	0
$415$	−1207.87	−0.142872
$416$	17125.6	2.01840
$417$	0	0
$418$	9159.29i	1.07176i
$419$	−3926.67	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$419$	−3926.67	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$420$	0	0
$421$	1443.44	0.167100	0.0835499	−	0.996504i	$-0.473374\pi$
$421$	1443.44	0.167100	0.0835499	+	0.996504i	$0.473374\pi$
$422$	13306.9i	1.53500i
$423$	0	0
$424$	−6738.48	−0.771815
$425$	−9598.62	−1.09553
$426$	0	0
$427$	0	0
$428$	5339.42i	0.603016i
$429$	0	0
$430$	− 1304.51i	− 0.146300i
$431$	− 14803.8i	− 1.65447i	−0.561857	−	0.827234i	$-0.689913\pi$
$431$	− 14803.8i	− 1.65447i	0.561857	−	0.827234i	$-0.310087\pi$
$432$	0	0
$433$	− 15872.1i	− 1.76158i	−0.473508	−	0.880790i	$-0.657012\pi$
$433$	− 15872.1i	− 1.76158i	0.473508	−	0.880790i	$-0.342988\pi$
$434$	0	0
$435$	0	0
$436$	8325.34	0.914476
$437$	4427.97	0.484711
$438$	0	0
$439$	3032.91i	0.329733i	0.986316	+	0.164867i	$0.0527193\pi$
$439$	3032.91i	0.329733i	−0.986316	+	0.164867i	$0.947281\pi$
$440$	−1060.41	−0.114893
$441$	0	0
$442$	−30160.9	−3.24572
$443$	12848.2i	1.37796i	0.724782	+	0.688978i	$0.241940\pi$
$443$	12848.2i	1.37796i	−0.724782	+	0.688978i	$0.758060\pi$
$444$	0	0
$445$	37.5890	0.00400424
$446$	−15262.4	−1.62040
$447$	0	0
$448$	0	0
$449$	− 107.668i	− 0.0113166i	−0.999984	−	0.00565831i	$-0.998199\pi$
$449$	− 107.668i	− 0.0113166i	0.999984	−	0.00565831i	$-0.00180111\pi$
$450$	0	0
$451$	15954.0i	1.66573i
$452$	11392.8i	1.18556i
$453$	0	0
$454$	− 13707.1i	− 1.41698i
$455$	0	0
$456$	0	0
$457$	−9777.06	−1.00077	−0.500385	−	0.865803i	$-0.666808\pi$
$457$	−9777.06	−1.00077	−0.500385	+	0.865803i	$0.666808\pi$
$458$	−5013.45	−0.511492
$459$	0	0
$460$	1433.56i	0.145304i
$461$	638.874	0.0645452	0.0322726	−	0.999479i	$-0.489726\pi$
$461$	638.874	0.0645452	0.0322726	+	0.999479i	$0.489726\pi$
$462$	0	0
$463$	−5602.26	−0.562331	−0.281165	−	0.959659i	$-0.590721\pi$
$463$	−5602.26	−0.562331	−0.281165	+	0.959659i	$0.590721\pi$
$464$	− 1291.30i	− 0.129196i
$465$	0	0
$466$	14019.3	1.39363
$467$	5518.88	0.546859	0.273430	−	0.961892i	$-0.411842\pi$
$467$	5518.88	0.546859	0.273430	+	0.961892i	$0.411842\pi$
$468$	0	0
$469$	0	0
$470$	796.268i	0.0781470i
$471$	0	0
$472$	− 17725.7i	− 1.72858i
$473$	9479.91i	0.921536i
$474$	0	0
$475$	6007.82i	0.580332i
$476$	0	0
$477$	0	0
$478$	7847.74	0.750935
$479$	5468.79	0.521661	0.260830	−	0.965385i	$-0.416004\pi$
$479$	5468.79	0.521661	0.260830	+	0.965385i	$0.416004\pi$
$480$	0	0
$481$	− 7766.16i	− 0.736188i
$482$	−5427.22	−0.512869
$483$	0	0
$484$	4973.62	0.467094
$485$	1518.38i	0.142157i
$486$	0	0
$487$	−11733.4	−1.09177	−0.545886	−	0.837860i	$-0.683807\pi$
$487$	−11733.4	−1.09177	−0.545886	+	0.837860i	$0.683807\pi$
$488$	264.990	0.0245811
$489$	0	0
$490$	0	0
$491$	− 3514.92i	− 0.323068i	−0.986867	−	0.161534i	$-0.948356\pi$
$491$	− 3514.92i	− 0.323068i	0.986867	−	0.161534i	$-0.0516441\pi$
$492$	0	0
$493$	11760.9i	1.07441i
$494$	18877.8i	1.71934i
$495$	0	0
$496$	− 753.106i	− 0.0681763i
$497$	0	0
$498$	0	0
$499$	9888.98	0.887158	0.443579	−	0.896235i	$-0.353709\pi$
$499$	9888.98	0.887158	0.443579	+	0.896235i	$0.353709\pi$
$500$	−3915.32	−0.350196
$501$	0	0
$502$	− 16790.2i	− 1.49279i
$503$	−10172.2	−0.901698	−0.450849	−	0.892600i	$-0.648879\pi$
$503$	−10172.2	−0.901698	−0.450849	+	0.892600i	$0.648879\pi$
$504$	0	0
$505$	−828.838	−0.0730353
$506$	− 17110.0i	− 1.50322i
$507$	0	0
$508$	−15703.1	−1.37148
$509$	4299.12	0.374372	0.187186	−	0.982325i	$-0.440063\pi$
$509$	4299.12	0.374372	0.187186	+	0.982325i	$0.440063\pi$
$510$	0	0
$511$	0	0
$512$	− 3082.06i	− 0.266033i
$513$	0	0
$514$	3523.73i	0.302383i
$515$	− 1733.36i	− 0.148313i
$516$	0	0
$517$	− 5786.50i	− 0.492243i
$518$	0	0
$519$	0	0
$520$	−2185.57	−0.184314
$521$	10993.0	0.924396	0.462198	−	0.886777i	$-0.347061\pi$
$521$	10993.0	0.924396	0.462198	+	0.886777i	$0.347061\pi$
$522$	0	0
$523$	− 8529.54i	− 0.713137i	−0.934269	−	0.356569i	$-0.883947\pi$
$523$	− 8529.54i	− 0.713137i	0.934269	−	0.356569i	$-0.116053\pi$
$524$	−17026.2	−1.41946
$525$	0	0
$526$	−8888.51	−0.736801
$527$	6859.15i	0.566963i
$528$	0	0
$529$	3895.34	0.320156
$530$	−1915.16	−0.156961
$531$	0	0
$532$	0	0
$533$	32882.2i	2.67220i
$534$	0	0
$535$	542.674i	0.0438539i
$536$	− 8910.67i	− 0.718064i
$537$	0	0
$538$	11183.4i	0.896195i
$539$	0	0
$540$	0	0
$541$	−8705.86	−0.691856	−0.345928	−	0.938261i	$-0.612436\pi$
$541$	−8705.86	−0.691856	−0.345928	+	0.938261i	$0.612436\pi$
$542$	−21389.0	−1.69508
$543$	0	0
$544$	15537.7i	1.22458i
$545$	846.150	0.0665047
$546$	0	0
$547$	17183.8	1.34319	0.671596	−	0.740917i	$-0.265609\pi$
$547$	17183.8	1.34319	0.671596	+	0.740917i	$0.265609\pi$
$548$	− 13713.1i	− 1.06897i
$549$	0	0
$550$	23214.6	1.79977
$551$	7361.20	0.569143
$552$	0	0
$553$	0	0
$554$	− 5305.66i	− 0.406888i
$555$	0	0
$556$	28724.4i	2.19098i
$557$	− 10380.5i	− 0.789652i	−0.918756	−	0.394826i	$-0.870805\pi$
$557$	− 10380.5i	− 0.789652i	0.918756	−	0.394826i	$-0.129195\pi$
$558$	0	0
$559$	19538.6i	1.47835i
$560$	0	0
$561$	0	0
$562$	−37063.0	−2.78187
$563$	−18496.4	−1.38460	−0.692302	−	0.721608i	$-0.743403\pi$
$563$	−18496.4	−1.38460	−0.692302	+	0.721608i	$0.743403\pi$
$564$	0	0
$565$	1157.91i	0.0862189i
$566$	11710.4	0.869656
$567$	0	0
$568$	6879.04	0.508166
$569$	4033.88i	0.297204i	0.988897	+	0.148602i	$0.0474774\pi$
$569$	4033.88i	0.297204i	−0.988897	+	0.148602i	$0.952523\pi$
$570$	0	0
$571$	12860.0	0.942513	0.471257	−	0.881996i	$-0.343801\pi$
$571$	12860.0	0.942513	0.471257	+	0.881996i	$0.343801\pi$
$572$	44413.8	3.24657
$573$	0	0
$574$	0	0
$575$	− 11222.9i	− 0.813959i
$576$	0	0
$577$	− 20402.7i	− 1.47205i	−0.676953	−	0.736026i	$-0.736700\pi$
$577$	− 20402.7i	− 1.47205i	0.676953	−	0.736026i	$-0.263300\pi$
$578$	− 5145.02i	− 0.370250i
$579$	0	0
$580$	2383.19i	0.170615i
$581$	0	0
$582$	0	0
$583$	13917.5	0.988687
$584$	18563.6	1.31535
$585$	0	0
$586$	− 40120.2i	− 2.82824i
$587$	−16279.7	−1.14470	−0.572348	−	0.820011i	$-0.693967\pi$
$587$	−16279.7	−1.14470	−0.572348	+	0.820011i	$0.693967\pi$
$588$	0	0
$589$	4293.17	0.300335
$590$	− 5037.86i	− 0.351535i
$591$	0	0
$592$	−773.627	−0.0537093
$593$	−2684.31	−0.185888	−0.0929440	−	0.995671i	$-0.529628\pi$
$593$	−2684.31	−0.185888	−0.0929440	+	0.995671i	$0.529628\pi$
$594$	0	0
$595$	0	0
$596$	− 21865.7i	− 1.50277i
$597$	0	0
$598$	− 35264.7i	− 2.41150i
$599$	14115.8i	0.962864i	0.876483	+	0.481432i	$0.159883\pi$
$599$	14115.8i	0.962864i	−0.876483	+	0.481432i	$0.840117\pi$
$600$	0	0
$601$	11096.1i	0.753109i	0.926394	+	0.376555i	$0.122891\pi$
$601$	11096.1i	0.753109i	−0.926394	+	0.376555i	$0.877109\pi$
$602$	0	0
$603$	0	0
$604$	6537.79	0.440428
$605$	505.495	0.0339691
$606$	0	0
$607$	11076.7i	0.740674i	0.928897	+	0.370337i	$0.120758\pi$
$607$	11076.7i	0.740674i	−0.928897	+	0.370337i	$0.879242\pi$
$608$	9725.10	0.648692
$609$	0	0
$610$	75.3136	0.00499895
$611$	− 11926.3i	− 0.789667i
$612$	0	0
$613$	−7602.68	−0.500929	−0.250464	−	0.968126i	$-0.580583\pi$
$613$	−7602.68	−0.500929	−0.250464	+	0.968126i	$0.580583\pi$
$614$	12244.9	0.804827
$615$	0	0
$616$	0	0
$617$	− 11325.9i	− 0.738998i	−0.929231	−	0.369499i	$-0.879529\pi$
$617$	− 11325.9i	− 0.738998i	0.929231	−	0.369499i	$-0.120471\pi$
$618$	0	0
$619$	− 19162.4i	− 1.24427i	−0.782910	−	0.622136i	$-0.786265\pi$
$619$	− 19162.4i	− 1.24427i	0.782910	−	0.622136i	$-0.213735\pi$
$620$	1389.92i	0.0900328i
$621$	0	0
$622$	9773.64i	0.630044i
$623$	0	0
$624$	0	0
$625$	15026.8	0.961716
$626$	38123.2	2.43404
$627$	0	0
$628$	16413.8i	1.04296i
$629$	7046.06	0.446653
$630$	0	0
$631$	10140.7	0.639768	0.319884	−	0.947457i	$-0.396356\pi$
$631$	10140.7	0.639768	0.319884	+	0.947457i	$0.396356\pi$
$632$	− 8320.63i	− 0.523698i
$633$	0	0
$634$	−43396.7	−2.71846
$635$	−1595.99	−0.0997400
$636$	0	0
$637$	0	0
$638$	− 28444.2i	− 1.76507i
$639$	0	0
$640$	2757.40i	0.170306i
$641$	− 3407.13i	− 0.209943i	−0.994475	−	0.104972i	$-0.966525\pi$
$641$	− 3407.13i	− 0.209943i	0.994475	−	0.104972i	$-0.0334751\pi$
$642$	0	0
$643$	659.110i	0.0404242i	0.999796	+	0.0202121i	$0.00643415\pi$
$643$	659.110i	0.0404242i	−0.999796	+	0.0202121i	$0.993566\pi$
$644$	0	0
$645$	0	0
$646$	−17127.4	−1.04314
$647$	−6606.94	−0.401462	−0.200731	−	0.979646i	$-0.564332\pi$
$647$	−6606.94	−0.401462	−0.200731	+	0.979646i	$0.564332\pi$
$648$	0	0
$649$	36610.2i	2.21429i
$650$	47846.6	2.88723
$651$	0	0
$652$	5577.01	0.334989
$653$	− 25885.1i	− 1.55124i	−0.631198	−	0.775622i	$-0.717436\pi$
$653$	− 25885.1i	− 1.55124i	0.631198	−	0.775622i	$-0.282564\pi$
$654$	0	0
$655$	−1730.47	−0.103229
$656$	3275.56	0.194953
$657$	0	0
$658$	0	0
$659$	7468.86i	0.441495i	0.975331	+	0.220748i	$0.0708497\pi$
$659$	7468.86i	0.441495i	−0.975331	+	0.220748i	$0.929150\pi$
$660$	0	0
$661$	− 6353.11i	− 0.373839i	−0.982375	−	0.186919i	$-0.940150\pi$
$661$	− 6353.11i	− 0.373839i	0.982375	−	0.186919i	$-0.0598503\pi$
$662$	− 38640.2i	− 2.26857i
$663$	0	0
$664$	19220.3i	1.12333i
$665$	0	0
$666$	0	0
$667$	−13751.1	−0.798266
$668$	10101.2	0.585072
$669$	0	0
$670$	− 2532.52i	− 0.146030i
$671$	−547.306	−0.0314881
$672$	0	0
$673$	−20238.2	−1.15918	−0.579589	−	0.814909i	$-0.696787\pi$
$673$	−20238.2	−1.15918	−0.579589	+	0.814909i	$0.696787\pi$
$674$	− 2708.44i	− 0.154785i
$675$	0	0
$676$	64179.5	3.65154
$677$	−11316.7	−0.642449	−0.321224	−	0.947003i	$-0.604094\pi$
$677$	−11316.7	−0.642449	−0.321224	+	0.947003i	$0.604094\pi$
$678$	0	0
$679$	0	0
$680$	− 1982.91i	− 0.111825i
$681$	0	0
$682$	− 16589.1i	− 0.931422i
$683$	17169.7i	0.961906i	0.876746	+	0.480953i	$0.159709\pi$
$683$	17169.7i	0.961906i	−0.876746	+	0.480953i	$0.840291\pi$
$684$	0	0
$685$	− 1393.74i	− 0.0777402i
$686$	0	0
$687$	0	0
$688$	1946.34	0.107854
$689$	28684.8	1.58607
$690$	0	0
$691$	8358.40i	0.460157i	0.973172	+	0.230079i	$0.0738983\pi$
$691$	8358.40i	0.460157i	−0.973172	+	0.230079i	$0.926102\pi$
$692$	−27926.3	−1.53410
$693$	0	0
$694$	32114.4	1.75655
$695$	2919.41i	0.159338i
$696$	0	0
$697$	−29833.2	−1.62125
$698$	16374.8	0.887960
$699$	0	0
$700$	0	0
$701$	− 19235.8i	− 1.03641i	−0.855256	−	0.518206i	$-0.826600\pi$
$701$	− 19235.8i	− 1.03641i	0.855256	−	0.518206i	$-0.173400\pi$
$702$	0	0
$703$	− 4410.16i	− 0.236604i
$704$	− 34736.3i	− 1.85962i
$705$	0	0
$706$	− 10536.5i	− 0.561680i
$707$	0	0
$708$	0	0
$709$	−10320.3	−0.546669	−0.273334	−	0.961919i	$-0.588127\pi$
$709$	−10320.3	−0.546669	−0.273334	+	0.961919i	$0.588127\pi$
$710$	1955.11	0.103344
$711$	0	0
$712$	− 598.135i	− 0.0314833i
$713$	−8019.85	−0.421242
$714$	0	0
$715$	4514.02	0.236105
$716$	− 36406.6i	− 1.90025i
$717$	0	0
$718$	−7950.40	−0.413240
$719$	23359.5	1.21163	0.605815	−	0.795605i	$-0.292847\pi$
$719$	23359.5	1.21163	0.605815	+	0.795605i	$0.292847\pi$
$720$	0	0
$721$	0	0
$722$	− 20300.0i	− 1.04638i
$723$	0	0
$724$	− 3522.55i	− 0.180821i
$725$	− 18657.3i	− 0.955743i
$726$	0	0
$727$	− 22260.4i	− 1.13561i	−0.823162	−	0.567807i	$-0.807792\pi$
$727$	− 22260.4i	− 1.13561i	0.823162	−	0.567807i	$-0.192208\pi$
$728$	0	0
$729$	0	0
$730$	5276.00	0.267498
$731$	−17727.0	−0.896929
$732$	0	0
$733$	− 10447.5i	− 0.526451i	−0.964734	−	0.263226i	$-0.915214\pi$
$733$	− 10447.5i	− 0.526451i	0.964734	−	0.263226i	$-0.0847864\pi$
$734$	7618.58	0.383116
$735$	0	0
$736$	−18166.9	−0.909840
$737$	18403.9i	0.919832i
$738$	0	0
$739$	13190.9	0.656609	0.328304	−	0.944572i	$-0.393523\pi$
$739$	13190.9	0.656609	0.328304	+	0.944572i	$0.393523\pi$
$740$	1427.79	0.0709278
$741$	0	0
$742$	0	0
$743$	35382.2i	1.74703i	0.486795	+	0.873517i	$0.338166\pi$
$743$	35382.2i	1.74703i	−0.486795	+	0.873517i	$0.661834\pi$
$744$	0	0
$745$	− 2222.32i	− 0.109288i
$746$	1344.52i	0.0659869i
$747$	0	0
$748$	40295.6i	1.96973i
$749$	0	0
$750$	0	0
$751$	−29384.6	−1.42778	−0.713888	−	0.700259i	$-0.753068\pi$
$751$	−29384.6	−1.42778	−0.713888	+	0.700259i	$0.753068\pi$
$752$	−1188.04	−0.0576109
$753$	0	0
$754$	− 58625.1i	− 2.83156i
$755$	664.470	0.0320299
$756$	0	0
$757$	11329.1	0.543939	0.271969	−	0.962306i	$-0.412325\pi$
$757$	11329.1	0.543939	0.271969	+	0.962306i	$0.412325\pi$
$758$	33479.3i	1.60425i
$759$	0	0
$760$	−1241.11	−0.0592367
$761$	−25393.0	−1.20958	−0.604792	−	0.796383i	$-0.706744\pi$
$761$	−25393.0	−1.20958	−0.604792	+	0.796383i	$0.706744\pi$
$762$	0	0
$763$	0	0
$764$	57499.9i	2.72287i
$765$	0	0
$766$	56656.3i	2.67242i
$767$	75455.8i	3.55222i
$768$	0	0
$769$	− 18120.8i	− 0.849744i	−0.905253	−	0.424872i	$-0.860319\pi$
$769$	− 18120.8i	− 0.849744i	0.905253	−	0.424872i	$-0.139681\pi$
$770$	0	0
$771$	0	0
$772$	51749.4	2.41256
$773$	−17346.1	−0.807110	−0.403555	−	0.914955i	$-0.632226\pi$
$773$	−17346.1	−0.807110	−0.403555	+	0.914955i	$0.632226\pi$
$774$	0	0
$775$	− 10881.2i	− 0.504342i
$776$	24161.3	1.11771
$777$	0	0
$778$	−36054.7	−1.66147
$779$	18672.7i	0.858819i
$780$	0	0
$781$	−14207.8	−0.650955
$782$	31994.8	1.46308
$783$	0	0
$784$	0	0
$785$	1668.22i	0.0758488i
$786$	0	0
$787$	2080.22i	0.0942209i	0.998890	+	0.0471104i	$0.0150013\pi$
$787$	2080.22i	0.0942209i	−0.998890	+	0.0471104i	$0.984999\pi$
$788$	20253.7i	0.915619i
$789$	0	0
$790$	− 2364.83i	− 0.106502i
$791$	0	0
$792$	0	0
$793$	−1128.03	−0.0505138
$794$	−56569.4	−2.52843
$795$	0	0
$796$	− 2169.37i	− 0.0965969i
$797$	−39737.3	−1.76608	−0.883041	−	0.469297i	$-0.844507\pi$
$797$	−39737.3	−1.76608	−0.883041	+	0.469297i	$0.844507\pi$
$798$	0	0
$799$	10820.5	0.479099
$800$	− 24648.7i	− 1.08933i
$801$	0	0
$802$	−36260.5	−1.59651
$803$	−38340.8	−1.68495
$804$	0	0
$805$	0	0
$806$	− 34191.1i	− 1.49421i
$807$	0	0
$808$	13188.9i	0.574238i
$809$	− 11216.0i	− 0.487435i	−0.969846	−	0.243717i	$-0.921633\pi$
$809$	− 11216.0i	− 0.487435i	0.969846	−	0.243717i	$-0.0783669\pi$
$810$	0	0
$811$	14792.0i	0.640465i	0.947339	+	0.320232i	$0.103761\pi$
$811$	14792.0i	0.640465i	−0.947339	+	0.320232i	$0.896239\pi$
$812$	0	0
$813$	0	0
$814$	−17041.1	−0.733774
$815$	566.822	0.0243618
$816$	0	0
$817$	11095.4i	0.475126i
$818$	−39511.2	−1.68885
$819$	0	0
$820$	−6045.30	−0.257453
$821$	33062.1i	1.40545i	0.711460	+	0.702726i	$0.248034\pi$
$821$	33062.1i	1.40545i	−0.711460	+	0.702726i	$0.751966\pi$
$822$	0	0
$823$	20090.7	0.850933	0.425467	−	0.904974i	$-0.360110\pi$
$823$	20090.7	0.850933	0.425467	+	0.904974i	$0.360110\pi$
$824$	−27582.2	−1.16611
$825$	0	0
$826$	0	0
$827$	36882.5i	1.55082i	0.631458	+	0.775410i	$0.282457\pi$
$827$	36882.5i	1.55082i	−0.631458	+	0.775410i	$0.717543\pi$
$828$	0	0
$829$	− 37755.2i	− 1.58178i	−0.611961	−	0.790888i	$-0.709619\pi$
$829$	− 37755.2i	− 1.58178i	0.611961	−	0.790888i	$-0.290381\pi$
$830$	5462.64i	0.228447i
$831$	0	0
$832$	− 71593.5i	− 2.98324i
$833$	0	0
$834$	0	0
$835$	1026.64	0.0425490
$836$	25221.2	1.04341
$837$	0	0
$838$	17758.5i	0.732049i
$839$	−44250.4	−1.82085	−0.910426	−	0.413672i	$-0.864246\pi$
$839$	−44250.4	−1.82085	−0.910426	+	0.413672i	$0.864246\pi$
$840$	0	0
$841$	1528.81	0.0626844
$842$	− 6528.02i	− 0.267186i
$843$	0	0
$844$	36642.2	1.49440
$845$	6522.91	0.265556
$846$	0	0
$847$	0	0
$848$	− 2857.44i	− 0.115713i
$849$	0	0
$850$	43410.1i	1.75171i
$851$	8238.39i	0.331855i
$852$	0	0
$853$	− 14952.2i	− 0.600179i	−0.953911	−	0.300089i	$-0.902983\pi$
$853$	− 14952.2i	− 0.600179i	0.953911	−	0.300089i	$-0.0970165\pi$
$854$	0	0
$855$	0	0
$856$	8635.32	0.344801
$857$	2449.53	0.0976363	0.0488181	−	0.998808i	$-0.484455\pi$
$857$	2449.53	0.0976363	0.0488181	+	0.998808i	$0.484455\pi$
$858$	0	0
$859$	6641.80i	0.263813i	0.991262	+	0.131906i	$0.0421098\pi$
$859$	6641.80i	0.263813i	−0.991262	+	0.131906i	$0.957890\pi$
$860$	−3592.13	−0.142431
$861$	0	0
$862$	−66951.0	−2.64543
$863$	− 18866.9i	− 0.744191i	−0.928194	−	0.372096i	$-0.878639\pi$
$863$	− 18866.9i	− 0.744191i	0.928194	−	0.372096i	$-0.121361\pi$
$864$	0	0
$865$	−2838.30	−0.111567
$866$	−71782.2	−2.81669
$867$	0	0
$868$	0	0
$869$	17185.3i	0.670851i
$870$	0	0
$871$	37931.5i	1.47561i
$872$	− 13464.4i	− 0.522892i
$873$	0	0
$874$	− 20025.7i	− 0.775033i
$875$	0	0
$876$	0	0
$877$	11979.7	0.461260	0.230630	−	0.973042i	$-0.425921\pi$
$877$	11979.7	0.461260	0.230630	+	0.973042i	$0.425921\pi$
$878$	13716.5	0.527230
$879$	0	0
$880$	− 449.664i	− 0.0172252i
$881$	34504.0	1.31949	0.659743	−	0.751491i	$-0.270665\pi$
$881$	34504.0	1.31949	0.659743	+	0.751491i	$0.270665\pi$
$882$	0	0
$883$	−8148.85	−0.310567	−0.155283	−	0.987870i	$-0.549629\pi$
$883$	−8148.85	−0.310567	−0.155283	+	0.987870i	$0.549629\pi$
$884$	83051.7i	3.15988i
$885$	0	0
$886$	58106.3	2.20330
$887$	34955.2	1.32320	0.661602	−	0.749856i	$-0.269877\pi$
$887$	34955.2	1.32320	0.661602	+	0.749856i	$0.269877\pi$
$888$	0	0
$889$	0	0
$890$	− 169.998i	− 0.00640262i
$891$	0	0
$892$	42027.0i	1.57754i
$893$	− 6772.57i	− 0.253791i
$894$	0	0
$895$	− 3700.20i	− 0.138194i
$896$	0	0
$897$	0	0
$898$	−486.932	−0.0180948
$899$	−13332.4	−0.494618
$900$	0	0
$901$	26025.1i	0.962287i
$902$	72152.7	2.66344
$903$	0	0
$904$	18425.3	0.677894
$905$	− 358.016i	− 0.0131501i
$906$	0	0
$907$	−24989.0	−0.914824	−0.457412	−	0.889255i	$-0.651224\pi$
$907$	−24989.0	−0.914824	−0.457412	+	0.889255i	$0.651224\pi$
$908$	−37744.3	−1.37950
$909$	0	0
$910$	0	0
$911$	41609.1i	1.51325i	0.653849	+	0.756625i	$0.273153\pi$
$911$	41609.1i	1.51325i	−0.653849	+	0.756625i	$0.726847\pi$
$912$	0	0
$913$	− 39697.1i	− 1.43897i
$914$	44217.1i	1.60019i
$915$	0	0
$916$	13805.1i	0.497964i
$917$	0	0
$918$	0	0
$919$	50992.0	1.83033	0.915164	−	0.403082i	$-0.132061\pi$
$919$	50992.0	1.83033	0.915164	+	0.403082i	$0.132061\pi$
$920$	2318.46	0.0830840
$921$	0	0
$922$	− 2889.33i	− 0.103205i
$923$	−29283.1	−1.04428
$924$	0	0
$925$	−11177.7	−0.397321
$926$	25336.5i	0.899144i
$927$	0	0
$928$	−30201.3	−1.06832
$929$	−47493.0	−1.67728	−0.838642	−	0.544683i	$-0.816650\pi$
$929$	−47493.0	−1.67728	−0.838642	+	0.544683i	$0.816650\pi$
$930$	0	0
$931$	0	0
$932$	− 38603.9i	− 1.35677i
$933$	0	0
$934$	− 24959.4i	− 0.874406i
$935$	4095.46i	0.143247i
$936$	0	0
$937$	6811.98i	0.237500i	0.992924	+	0.118750i	$0.0378887\pi$
$937$	6811.98i	0.237500i	−0.992924	+	0.118750i	$0.962111\pi$
$938$	0	0
$939$	0	0
$940$	2192.62	0.0760802
$941$	32542.0	1.12735	0.563677	−	0.825995i	$-0.309386\pi$
$941$	32542.0	1.12735	0.563677	+	0.825995i	$0.309386\pi$
$942$	0	0
$943$	− 34881.5i	− 1.20456i
$944$	7516.54	0.259155
$945$	0	0
$946$	42873.3	1.47350
$947$	− 11685.4i	− 0.400976i	−0.979696	−	0.200488i	$-0.935747\pi$
$947$	− 11685.4i	− 0.400976i	0.979696	−	0.200488i	$-0.0642528\pi$
$948$	0	0
$949$	−79022.7	−2.70304
$950$	27170.6	0.927927
$951$	0	0
$952$	0	0
$953$	− 46457.5i	− 1.57912i	−0.613671	−	0.789562i	$-0.710308\pi$
$953$	− 46457.5i	− 1.57912i	0.613671	−	0.789562i	$-0.289692\pi$
$954$	0	0
$955$	5844.03i	0.198019i
$956$	− 21609.7i	− 0.731075i
$957$	0	0
$958$	− 24732.8i	− 0.834114i
$959$	0	0
$960$	0	0
$961$	22015.3	0.738992
$962$	−35122.8	−1.17714
$963$	0	0
$964$	14944.5i	0.499305i
$965$	5259.57	0.175452
$966$	0	0
$967$	−27949.1	−0.929455	−0.464728	−	0.885454i	$-0.653848\pi$
$967$	−27949.1	−0.929455	−0.464728	+	0.885454i	$0.653848\pi$
$968$	− 8043.71i	− 0.267081i
$969$	0	0
$970$	6866.94	0.227303
$971$	−3219.69	−0.106411	−0.0532053	−	0.998584i	$-0.516944\pi$
$971$	−3219.69	−0.106411	−0.0532053	+	0.998584i	$0.516944\pi$
$972$	0	0
$973$	0	0
$974$	53064.9i	1.74570i
$975$	0	0
$976$	112.369i	0.00368528i
$977$	− 38403.8i	− 1.25757i	−0.777579	−	0.628785i	$-0.783553\pi$
$977$	− 38403.8i	− 1.25757i	0.777579	−	0.628785i	$-0.216447\pi$
$978$	0	0
$979$	1235.38i	0.0403297i
$980$	0	0
$981$	0	0
$982$	−15896.4	−0.516572
$983$	−27418.2	−0.889627	−0.444814	−	0.895623i	$-0.646730\pi$
$983$	−27418.2	−0.889627	−0.444814	+	0.895623i	$0.646730\pi$
$984$	0	0
$985$	2058.49i	0.0665878i
$986$	53189.1	1.71794
$987$	0	0
$988$	51982.4	1.67387
$989$	− 20726.7i	− 0.666401i
$990$	0	0
$991$	43715.9	1.40129	0.700646	−	0.713509i	$-0.252895\pi$
$991$	43715.9	1.40129	0.700646	+	0.713509i	$0.252895\pi$
$992$	−17613.9	−0.563751
$993$	0	0
$994$	0	0
$995$	− 220.484i	− 0.00702495i
$996$	0	0
$997$	− 25323.5i	− 0.804415i	−0.915548	−	0.402208i	$-0.868243\pi$
$997$	− 25323.5i	− 0.804415i	0.915548	−	0.402208i	$-0.131757\pi$
$998$	− 44723.3i	− 1.41853i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.a.440.3		16
3.2	odd	2	inner	441.4.c.a.440.14		16
7.2	even	3		441.4.p.c.80.2		16
7.3	odd	6		441.4.p.c.215.7		16
7.4	even	3		63.4.p.a.26.7	yes	16
7.5	odd	6		63.4.p.a.17.2	✓	16
7.6	odd	2	inner	441.4.c.a.440.4		16
21.2	odd	6		441.4.p.c.80.7		16
21.5	even	6		63.4.p.a.17.7	yes	16
21.11	odd	6		63.4.p.a.26.2	yes	16
21.17	even	6		441.4.p.c.215.2		16
21.20	even	2	inner	441.4.c.a.440.13		16
28.11	odd	6		1008.4.bt.a.593.5		16
28.19	even	6		1008.4.bt.a.17.4		16
84.11	even	6		1008.4.bt.a.593.4		16
84.47	odd	6		1008.4.bt.a.17.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.p.a.17.2	✓	16	7.5	odd	6
63.4.p.a.17.7	yes	16	21.5	even	6
63.4.p.a.26.2	yes	16	21.11	odd	6
63.4.p.a.26.7	yes	16	7.4	even	3
441.4.c.a.440.3		16	1.1	even	1	trivial
441.4.c.a.440.4		16	7.6	odd	2	inner
441.4.c.a.440.13		16	21.20	even	2	inner
441.4.c.a.440.14		16	3.2	odd	2	inner
441.4.p.c.80.2		16	7.2	even	3
441.4.p.c.80.7		16	21.2	odd	6
441.4.p.c.215.2		16	21.17	even	6
441.4.p.c.215.7		16	7.3	odd	6
1008.4.bt.a.17.4		16	28.19	even	6
1008.4.bt.a.17.5		16	84.47	odd	6
1008.4.bt.a.593.4		16	84.11	even	6
1008.4.bt.a.593.5		16	28.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.52254i q^{2} -12.4534 q^{4} -1.26570 q^{5} +20.1405i q^{8} +O(q^{10})\)	\(q-4.52254i q^{2} -12.4534 q^{4} -1.26570 q^{5} +20.1405i q^{8} +5.72419i q^{10} -41.5978i q^{11} -85.7355i q^{13} -8.54055 q^{16} +77.7859 q^{17} -48.6865i q^{19} +15.7623 q^{20} -188.128 q^{22} +90.9487i q^{23} -123.398 q^{25} -387.742 q^{26} +151.196i q^{29} +88.1799i q^{31} +199.749i q^{32} -351.790i q^{34} +90.5828 q^{37} -220.187 q^{38} -25.4919i q^{40} -383.530 q^{41} -227.894 q^{43} +518.033i q^{44} +411.319 q^{46} +139.106 q^{47} +558.072i q^{50} +1067.70i q^{52} +334.573i q^{53} +52.6505i q^{55} +683.789 q^{58} -880.100 q^{59} -13.1571i q^{61} +398.797 q^{62} +835.050 q^{64} +108.516i q^{65} -442.424 q^{67} -968.696 q^{68} -341.552i q^{71} -921.703i q^{73} -409.664i q^{74} +606.311i q^{76} -413.129 q^{79} +10.8098 q^{80} +1734.53i q^{82} +954.307 q^{83} -98.4538 q^{85} +1030.66i q^{86} +837.803 q^{88} -29.6981 q^{89} -1132.62i q^{92} -629.111i q^{94} +61.6227i q^{95} -1199.63i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 64 q^{4}+O(q^{10}) \) 16 * q - 64 * q^4	\( 16 q - 64 q^{4} + 376 q^{16} + 528 q^{22} + 40 q^{25} + 2392 q^{37} + 328 q^{43} + 2784 q^{46} + 6744 q^{58} + 5432 q^{64} - 616 q^{67} + 4352 q^{79} - 4608 q^{85} - 1416 q^{88}+O(q^{100}) \) 16 * q - 64 * q^4 + 376 * q^16 + 528 * q^22 + 40 * q^25 + 2392 * q^37 + 328 * q^43 + 2784 * q^46 + 6744 * q^58 + 5432 * q^64 - 616 * q^67 + 4352 * q^79 - 4608 * q^85 - 1416 * q^88

Introduction

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