LMFDB - Embedded newform 4400.2.b.w.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (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:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.w.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155i q^{3} -0.438447i q^{7} +0.561553 q^{9} +O(q^{10})$	$q-1.56155i q^{3} -0.438447i q^{7} +0.561553 q^{9} +1.00000 q^{11} -7.12311i q^{13} +4.68466i q^{17} +5.56155 q^{19} -0.684658 q^{21} -7.12311i q^{23} -5.56155i q^{27} -4.43845 q^{29} +5.56155 q^{31} -1.56155i q^{33} +11.5616i q^{37} -11.1231 q^{39} +4.24621 q^{41} +5.12311i q^{43} -13.3693i q^{47} +6.80776 q^{49} +7.31534 q^{51} +2.68466i q^{53} -8.68466i q^{57} -7.12311 q^{59} -8.43845 q^{61} -0.246211i q^{63} -11.1231 q^{69} +8.68466 q^{71} +7.12311i q^{73} -0.438447i q^{77} +13.3693 q^{79} -7.00000 q^{81} -6.00000i q^{83} +6.93087i q^{87} +2.68466 q^{89} -3.12311 q^{91} -8.68466i q^{93} -13.1231i q^{97} +0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} + 4 q^{11} + 14 q^{19} + 22 q^{21} - 26 q^{29} + 14 q^{31} - 28 q^{39} - 16 q^{41} - 14 q^{49} + 54 q^{51} - 12 q^{59} - 42 q^{61} - 28 q^{69} + 10 q^{71} + 4 q^{79} - 28 q^{81} - 14 q^{89} + 4 q^{91} - 6 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 + 4 * q^11 + 14 * q^19 + 22 * q^21 - 26 * q^29 + 14 * q^31 - 28 * q^39 - 16 * q^41 - 14 * q^49 + 54 * q^51 - 12 * q^59 - 42 * q^61 - 28 * q^69 + 10 * q^71 + 4 * q^79 - 28 * q^81 - 14 * q^89 + 4 * q^91 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 1.56155i	− 0.901563i	−0.892634	−	0.450781i	$-0.851145\pi$
$3$	− 1.56155i	− 0.901563i	0.892634	−	0.450781i	$-0.148855\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.438447i	− 0.165717i	−0.996561	−	0.0828587i	$-0.973595\pi$
$7$	− 0.438447i	− 0.165717i	0.996561	−	0.0828587i	$-0.0264050\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 7.12311i	− 1.97559i	−0.155747	−	0.987797i	$-0.549778\pi$
$13$	− 7.12311i	− 1.97559i	0.155747	−	0.987797i	$-0.450222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.68466i	1.13620i	0.822961	+	0.568098i	$0.192321\pi$
$17$	4.68466i	1.13620i	−0.822961	+	0.568098i	$0.807679\pi$
$18$	0	0
$19$	5.56155	1.27591	0.637954	−	0.770075i	$-0.279781\pi$
$19$	5.56155	1.27591	0.637954	+	0.770075i	$0.279781\pi$
$20$	0	0
$21$	−0.684658	−0.149405
$22$	0	0
$23$	− 7.12311i	− 1.48527i	−0.669696	−	0.742635i	$-0.733576\pi$
$23$	− 7.12311i	− 1.48527i	0.669696	−	0.742635i	$-0.266424\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.56155i	− 1.07032i
$28$	0	0
$29$	−4.43845	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$29$	−4.43845	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	− 1.56155i	− 0.271831i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.5616i	1.90071i	0.311171	+	0.950354i	$0.399279\pi$
$37$	11.5616i	1.90071i	−0.311171	+	0.950354i	$0.600721\pi$
$38$	0	0
$39$	−11.1231	−1.78112
$40$	0	0
$41$	4.24621	0.663147	0.331573	−	0.943429i	$-0.392421\pi$
$41$	4.24621	0.663147	0.331573	+	0.943429i	$0.392421\pi$
$42$	0	0
$43$	5.12311i	0.781266i	0.920546	+	0.390633i	$0.127744\pi$
$43$	5.12311i	0.781266i	−0.920546	+	0.390633i	$0.872256\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13.3693i	− 1.95012i	−0.221952	−	0.975058i	$-0.571243\pi$
$47$	− 13.3693i	− 1.95012i	0.221952	−	0.975058i	$-0.428757\pi$
$48$	0	0
$49$	6.80776	0.972538
$50$	0	0
$51$	7.31534	1.02435
$52$	0	0
$53$	2.68466i	0.368766i	0.982854	+	0.184383i	$0.0590287\pi$
$53$	2.68466i	0.368766i	−0.982854	+	0.184383i	$0.940971\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.68466i	− 1.15031i
$58$	0	0
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	0	0
$61$	−8.43845	−1.08043	−0.540216	−	0.841526i	$-0.681658\pi$
$61$	−8.43845	−1.08043	−0.540216	+	0.841526i	$0.681658\pi$
$62$	0	0
$63$	− 0.246211i	− 0.0310197i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−11.1231	−1.33906
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	7.12311i	0.833696i	0.908976	+	0.416848i	$0.136865\pi$
$73$	7.12311i	0.833696i	−0.908976	+	0.416848i	$0.863135\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.438447i	− 0.0499657i
$78$	0	0
$79$	13.3693	1.50417	0.752083	−	0.659069i	$-0.229049\pi$
$79$	13.3693	1.50417	0.752083	+	0.659069i	$0.229049\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.93087i	0.743067i
$88$	0	0
$89$	2.68466	0.284573	0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466	0.284573	0.142287	+	0.989825i	$0.454555\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	− 8.68466i	− 0.900557i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.1231i	− 1.33245i	−0.745751	−	0.666225i	$-0.767909\pi$
$97$	− 13.1231i	− 1.33245i	0.745751	−	0.666225i	$-0.232091\pi$
$98$	0	0
$99$	0.561553	0.0564382
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.12311i	0.108575i	0.998525	+	0.0542874i	$0.0172887\pi$
$107$	1.12311i	0.108575i	−0.998525	+	0.0542874i	$0.982711\pi$
$108$	0	0
$109$	−12.2462	−1.17297	−0.586487	−	0.809959i	$-0.699490\pi$
$109$	−12.2462	−1.17297	−0.586487	+	0.809959i	$0.699490\pi$
$110$	0	0
$111$	18.0540	1.71361
$112$	0	0
$113$	− 9.12311i	− 0.858230i	−0.903250	−	0.429115i	$-0.858826\pi$
$113$	− 9.12311i	− 0.858230i	0.903250	−	0.429115i	$-0.141174\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	2.05398	0.188288
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 6.63068i	− 0.597869i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.1231i	− 1.16449i	−0.813014	−	0.582244i	$-0.802175\pi$
$127$	− 13.1231i	− 1.16449i	0.813014	−	0.582244i	$-0.197825\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−3.80776	−0.332686	−0.166343	−	0.986068i	$-0.553196\pi$
$131$	−3.80776	−0.332686	−0.166343	+	0.986068i	$0.553196\pi$
$132$	0	0
$133$	− 2.43845i	− 0.211440i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.12311i	0.0959534i	0.998848	+	0.0479767i	$0.0152773\pi$
$137$	1.12311i	0.0959534i	−0.998848	+	0.0479767i	$0.984723\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−20.8769	−1.75815
$142$	0	0
$143$	− 7.12311i	− 0.595664i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 10.6307i	− 0.876804i
$148$	0	0
$149$	10.6847	0.875321	0.437661	−	0.899140i	$-0.355807\pi$
$149$	10.6847	0.875321	0.437661	+	0.899140i	$0.355807\pi$
$150$	0	0
$151$	−15.1231	−1.23070	−0.615350	−	0.788254i	$-0.710985\pi$
$151$	−15.1231	−1.23070	−0.615350	+	0.788254i	$0.710985\pi$
$152$	0	0
$153$	2.63068i	0.212678i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.31534i	0.743445i	0.928344	+	0.371723i	$0.121233\pi$
$157$	9.31534i	0.743445i	−0.928344	+	0.371723i	$0.878767\pi$
$158$	0	0
$159$	4.19224	0.332466
$160$	0	0
$161$	−3.12311	−0.246135
$162$	0	0
$163$	0.192236i	0.0150571i	0.999972	+	0.00752854i	$0.00239643\pi$
$163$	0.192236i	0.0150571i	−0.999972	+	0.00752854i	$0.997604\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 10.6847i	− 0.826804i	−0.910549	−	0.413402i	$-0.864340\pi$
$167$	− 10.6847i	− 0.826804i	0.910549	−	0.413402i	$-0.135660\pi$
$168$	0	0
$169$	−37.7386	−2.90297
$170$	0	0
$171$	3.12311	0.238830
$172$	0	0
$173$	− 19.1231i	− 1.45390i	−0.686689	−	0.726951i	$-0.740937\pi$
$173$	− 19.1231i	− 1.45390i	0.686689	−	0.726951i	$-0.259063\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.1231i	0.836064i
$178$	0	0
$179$	−2.24621	−0.167890	−0.0839449	−	0.996470i	$-0.526752\pi$
$179$	−2.24621	−0.167890	−0.0839449	+	0.996470i	$0.526752\pi$
$180$	0	0
$181$	8.24621	0.612936	0.306468	−	0.951881i	$-0.400853\pi$
$181$	8.24621	0.612936	0.306468	+	0.951881i	$0.400853\pi$
$182$	0	0
$183$	13.1771i	0.974078i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.68466i	0.342576i
$188$	0	0
$189$	−2.43845	−0.177371
$190$	0	0
$191$	6.24621	0.451960	0.225980	−	0.974132i	$-0.427442\pi$
$191$	6.24621	0.451960	0.225980	+	0.974132i	$0.427442\pi$
$192$	0	0
$193$	25.5616i	1.83996i	0.391964	+	0.919980i	$0.371796\pi$
$193$	25.5616i	1.83996i	−0.391964	+	0.919980i	$0.628204\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.7386i	− 1.90505i	−0.304462	−	0.952524i	$-0.598477\pi$
$197$	− 26.7386i	− 1.90505i	0.304462	−	0.952524i	$-0.401523\pi$
$198$	0	0
$199$	−16.6847	−1.18274	−0.591372	−	0.806399i	$-0.701414\pi$
$199$	−16.6847	−1.18274	−0.591372	+	0.806399i	$0.701414\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.94602i	0.136584i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	5.56155	0.384701
$210$	0	0
$211$	−7.31534	−0.503609	−0.251804	−	0.967778i	$-0.581024\pi$
$211$	−7.31534	−0.503609	−0.251804	+	0.967778i	$0.581024\pi$
$212$	0	0
$213$	− 13.5616i	− 0.929222i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.43845i	− 0.165533i
$218$	0	0
$219$	11.1231	0.751630
$220$	0	0
$221$	33.3693	2.24466
$222$	0	0
$223$	16.8769i	1.13016i	0.825036	+	0.565080i	$0.191155\pi$
$223$	16.8769i	1.13016i	−0.825036	+	0.565080i	$0.808845\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.24621i	0.547320i	0.961826	+	0.273660i	$0.0882343\pi$
$227$	8.24621i	0.547320i	−0.961826	+	0.273660i	$0.911766\pi$
$228$	0	0
$229$	0.246211	0.0162701	0.00813505	−	0.999967i	$-0.497411\pi$
$229$	0.246211	0.0162701	0.00813505	+	0.999967i	$0.497411\pi$
$230$	0	0
$231$	−0.684658	−0.0450472
$232$	0	0
$233$	10.0540i	0.658658i	0.944215	+	0.329329i	$0.106822\pi$
$233$	10.0540i	0.658658i	−0.944215	+	0.329329i	$0.893178\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 20.8769i	− 1.35610i
$238$	0	0
$239$	19.6155	1.26882	0.634412	−	0.772995i	$-0.281242\pi$
$239$	19.6155	1.26882	0.634412	+	0.772995i	$0.281242\pi$
$240$	0	0
$241$	−14.4924	−0.933539	−0.466769	−	0.884379i	$-0.654582\pi$
$241$	−14.4924	−0.933539	−0.466769	+	0.884379i	$0.654582\pi$
$242$	0	0
$243$	− 5.75379i	− 0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 39.6155i	− 2.52068i
$248$	0	0
$249$	−9.36932	−0.593756
$250$	0	0
$251$	2.63068	0.166047	0.0830236	−	0.996548i	$-0.473542\pi$
$251$	2.63068	0.166047	0.0830236	+	0.996548i	$0.473542\pi$
$252$	0	0
$253$	− 7.12311i	− 0.447826i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 26.4924i	− 1.65255i	−0.563266	−	0.826276i	$-0.690455\pi$
$257$	− 26.4924i	− 1.65255i	0.563266	−	0.826276i	$-0.309545\pi$
$258$	0	0
$259$	5.06913	0.314980
$260$	0	0
$261$	−2.49242	−0.154277
$262$	0	0
$263$	8.05398i	0.496629i	0.968679	+	0.248315i	$0.0798767\pi$
$263$	8.05398i	0.496629i	−0.968679	+	0.248315i	$0.920123\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.19224i	− 0.256561i
$268$	0	0
$269$	−9.12311	−0.556246	−0.278123	−	0.960546i	$-0.589712\pi$
$269$	−9.12311	−0.556246	−0.278123	+	0.960546i	$0.589712\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	4.87689i	0.295163i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.75379i	− 0.345712i	−0.984947	−	0.172856i	$-0.944701\pi$
$277$	− 5.75379i	− 0.345712i	0.984947	−	0.172856i	$-0.0552995\pi$
$278$	0	0
$279$	3.12311	0.186975
$280$	0	0
$281$	−8.24621	−0.491928	−0.245964	−	0.969279i	$-0.579104\pi$
$281$	−8.24621	−0.491928	−0.245964	+	0.969279i	$0.579104\pi$
$282$	0	0
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.86174i	− 0.109895i
$288$	0	0
$289$	−4.94602	−0.290943
$290$	0	0
$291$	−20.4924	−1.20129
$292$	0	0
$293$	− 8.87689i	− 0.518594i	−0.965798	−	0.259297i	$-0.916509\pi$
$293$	− 8.87689i	− 0.518594i	0.965798	−	0.259297i	$-0.0834908\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.56155i	− 0.322714i
$298$	0	0
$299$	−50.7386	−2.93429
$300$	0	0
$301$	2.24621	0.129469
$302$	0	0
$303$	3.12311i	0.179418i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 18.0000i	− 1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$	− 18.0000i	− 1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.5616	0.769005	0.384503	−	0.923124i	$-0.374373\pi$
$311$	13.5616	0.769005	0.384503	+	0.923124i	$0.374373\pi$
$312$	0	0
$313$	24.7386i	1.39831i	0.714970	+	0.699155i	$0.246440\pi$
$313$	24.7386i	1.39831i	−0.714970	+	0.699155i	$0.753560\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.6847i	1.04943i	0.851276	+	0.524717i	$0.175829\pi$
$317$	18.6847i	1.04943i	−0.851276	+	0.524717i	$0.824171\pi$
$318$	0	0
$319$	−4.43845	−0.248505
$320$	0	0
$321$	1.75379	0.0978869
$322$	0	0
$323$	26.0540i	1.44968i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	19.1231i	1.05751i
$328$	0	0
$329$	−5.86174	−0.323168
$330$	0	0
$331$	−30.7386	−1.68955	−0.844774	−	0.535123i	$-0.820265\pi$
$331$	−30.7386	−1.68955	−0.844774	+	0.535123i	$0.820265\pi$
$332$	0	0
$333$	6.49242i	0.355783i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.8078i	0.861104i	0.902566	+	0.430552i	$0.141681\pi$
$337$	15.8078i	0.861104i	−0.902566	+	0.430552i	$0.858319\pi$
$338$	0	0
$339$	−14.2462	−0.773748
$340$	0	0
$341$	5.56155	0.301175
$342$	0	0
$343$	− 6.05398i	− 0.326884i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.0000i	0.751559i	0.926709	+	0.375780i	$0.122625\pi$
$347$	14.0000i	0.751559i	−0.926709	+	0.375780i	$0.877375\pi$
$348$	0	0
$349$	18.4924	0.989877	0.494938	−	0.868928i	$-0.335191\pi$
$349$	18.4924	0.989877	0.494938	+	0.868928i	$0.335191\pi$
$350$	0	0
$351$	−39.6155	−2.11452
$352$	0	0
$353$	− 10.0000i	− 0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$	− 10.0000i	− 0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.20739i	− 0.169753i
$358$	0	0
$359$	−30.7386	−1.62232	−0.811162	−	0.584822i	$-0.801164\pi$
$359$	−30.7386	−1.62232	−0.811162	+	0.584822i	$0.801164\pi$
$360$	0	0
$361$	11.9309	0.627941
$362$	0	0
$363$	− 1.56155i	− 0.0819603i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.4924i	0.860897i	0.902615	+	0.430449i	$0.141645\pi$
$367$	16.4924i	0.860897i	−0.902615	+	0.430449i	$0.858355\pi$
$368$	0	0
$369$	2.38447	0.124131
$370$	0	0
$371$	1.17708	0.0611110
$372$	0	0
$373$	1.36932i	0.0709005i	0.999371	+	0.0354503i	$0.0112865\pi$
$373$	1.36932i	0.0709005i	−0.999371	+	0.0354503i	$0.988713\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.6155i	1.62828i
$378$	0	0
$379$	−23.1231	−1.18775	−0.593877	−	0.804556i	$-0.702403\pi$
$379$	−23.1231	−1.18775	−0.593877	+	0.804556i	$0.702403\pi$
$380$	0	0
$381$	−20.4924	−1.04986
$382$	0	0
$383$	− 25.3693i	− 1.29631i	−0.761508	−	0.648156i	$-0.775541\pi$
$383$	− 25.3693i	− 1.29631i	0.761508	−	0.648156i	$-0.224459\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.87689i	0.146241i
$388$	0	0
$389$	−18.4924	−0.937603	−0.468802	−	0.883304i	$-0.655314\pi$
$389$	−18.4924	−0.937603	−0.468802	+	0.883304i	$0.655314\pi$
$390$	0	0
$391$	33.3693	1.68756
$392$	0	0
$393$	5.94602i	0.299937i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	−3.80776	−0.190627
$400$	0	0
$401$	10.1922	0.508976	0.254488	−	0.967076i	$-0.418093\pi$
$401$	10.1922	0.508976	0.254488	+	0.967076i	$0.418093\pi$
$402$	0	0
$403$	− 39.6155i	− 1.97339i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.5616i	0.573085i
$408$	0	0
$409$	7.36932	0.364389	0.182195	−	0.983262i	$-0.441680\pi$
$409$	7.36932	0.364389	0.182195	+	0.983262i	$0.441680\pi$
$410$	0	0
$411$	1.75379	0.0865080
$412$	0	0
$413$	3.12311i	0.153678i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.24621i	0.305878i
$418$	0	0
$419$	−32.4924	−1.58736	−0.793679	−	0.608336i	$-0.791837\pi$
$419$	−32.4924	−1.58736	−0.793679	+	0.608336i	$0.791837\pi$
$420$	0	0
$421$	−13.6155	−0.663580	−0.331790	−	0.943353i	$-0.607653\pi$
$421$	−13.6155	−0.663580	−0.331790	+	0.943353i	$0.607653\pi$
$422$	0	0
$423$	− 7.50758i	− 0.365031i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.69981i	0.179047i
$428$	0	0
$429$	−11.1231	−0.537029
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	13.1231i	0.630656i	0.948983	+	0.315328i	$0.102115\pi$
$433$	13.1231i	0.630656i	−0.948983	+	0.315328i	$0.897885\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 39.6155i	− 1.89507i
$438$	0	0
$439$	−6.24621	−0.298115	−0.149058	−	0.988828i	$-0.547624\pi$
$439$	−6.24621	−0.298115	−0.149058	+	0.988828i	$0.547624\pi$
$440$	0	0
$441$	3.82292	0.182044
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 16.6847i	− 0.789157i
$448$	0	0
$449$	16.2462	0.766706	0.383353	−	0.923602i	$-0.374769\pi$
$449$	16.2462	0.766706	0.383353	+	0.923602i	$0.374769\pi$
$450$	0	0
$451$	4.24621	0.199946
$452$	0	0
$453$	23.6155i	1.10955i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.68466i	− 0.406251i	−0.979153	−	0.203126i	$-0.934890\pi$
$457$	− 8.68466i	− 0.406251i	0.979153	−	0.203126i	$-0.0651100\pi$
$458$	0	0
$459$	26.0540	1.21610
$460$	0	0
$461$	16.0540	0.747708	0.373854	−	0.927488i	$-0.378036\pi$
$461$	16.0540	0.747708	0.373854	+	0.927488i	$0.378036\pi$
$462$	0	0
$463$	− 2.63068i	− 0.122258i	−0.998130	−	0.0611291i	$-0.980530\pi$
$463$	− 2.63068i	− 0.122258i	0.998130	−	0.0611291i	$-0.0194701\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.1922i	− 0.564189i	−0.959387	−	0.282095i	$-0.908971\pi$
$467$	− 12.1922i	− 0.564189i	0.959387	−	0.282095i	$-0.0910292\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.5464	0.670263
$472$	0	0
$473$	5.12311i	0.235561i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.50758i	0.0690272i
$478$	0	0
$479$	−2.24621	−0.102632	−0.0513160	−	0.998682i	$-0.516342\pi$
$479$	−2.24621	−0.102632	−0.0513160	+	0.998682i	$0.516342\pi$
$480$	0	0
$481$	82.3542	3.75503
$482$	0	0
$483$	4.87689i	0.221906i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.7538i	0.623244i	0.950206	+	0.311622i	$0.100872\pi$
$487$	13.7538i	0.623244i	−0.950206	+	0.311622i	$0.899128\pi$
$488$	0	0
$489$	0.300187	0.0135749
$490$	0	0
$491$	28.6847	1.29452	0.647260	−	0.762269i	$-0.275915\pi$
$491$	28.6847	1.29452	0.647260	+	0.762269i	$0.275915\pi$
$492$	0	0
$493$	− 20.7926i	− 0.936452i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.80776i	− 0.170802i
$498$	0	0
$499$	30.7386	1.37605	0.688025	−	0.725687i	$-0.258478\pi$
$499$	30.7386	1.37605	0.688025	+	0.725687i	$0.258478\pi$
$500$	0	0
$501$	−16.6847	−0.745416
$502$	0	0
$503$	5.12311i	0.228428i	0.993456	+	0.114214i	$0.0364350\pi$
$503$	5.12311i	0.228428i	−0.993456	+	0.114214i	$0.963565\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	58.9309i	2.61721i
$508$	0	0
$509$	41.6155	1.84458	0.922288	−	0.386504i	$-0.126317\pi$
$509$	41.6155	1.84458	0.922288	+	0.386504i	$0.126317\pi$
$510$	0	0
$511$	3.12311	0.138158
$512$	0	0
$513$	− 30.9309i	− 1.36563i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13.3693i	− 0.587982i
$518$	0	0
$519$	−29.8617	−1.31078
$520$	0	0
$521$	2.49242	0.109195	0.0545975	−	0.998508i	$-0.482612\pi$
$521$	2.49242	0.109195	0.0545975	+	0.998508i	$0.482612\pi$
$522$	0	0
$523$	33.1231i	1.44837i	0.689605	+	0.724186i	$0.257784\pi$
$523$	33.1231i	1.44837i	−0.689605	+	0.724186i	$0.742216\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.0540i	1.13493i
$528$	0	0
$529$	−27.7386	−1.20603
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 30.2462i	− 1.31011i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.50758i	0.151363i
$538$	0	0
$539$	6.80776	0.293231
$540$	0	0
$541$	20.9309	0.899888	0.449944	−	0.893057i	$-0.351444\pi$
$541$	20.9309	0.899888	0.449944	+	0.893057i	$0.351444\pi$
$542$	0	0
$543$	− 12.8769i	− 0.552600i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.3693i	1.51228i	0.654408	+	0.756141i	$0.272918\pi$
$547$	35.3693i	1.51228i	−0.654408	+	0.756141i	$0.727082\pi$
$548$	0	0
$549$	−4.73863	−0.202240
$550$	0	0
$551$	−24.6847	−1.05160
$552$	0	0
$553$	− 5.86174i	− 0.249267i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.3693i	1.75287i	0.481516	+	0.876437i	$0.340086\pi$
$557$	41.3693i	1.75287i	−0.481516	+	0.876437i	$0.659914\pi$
$558$	0	0
$559$	36.4924	1.54347
$560$	0	0
$561$	7.31534	0.308854
$562$	0	0
$563$	22.9848i	0.968696i	0.874876	+	0.484348i	$0.160943\pi$
$563$	22.9848i	0.968696i	−0.874876	+	0.484348i	$0.839057\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.06913i	0.128891i
$568$	0	0
$569$	37.6155	1.57692	0.788462	−	0.615083i	$-0.210877\pi$
$569$	37.6155	1.57692	0.788462	+	0.615083i	$0.210877\pi$
$570$	0	0
$571$	28.3002	1.18433	0.592163	−	0.805818i	$-0.298274\pi$
$571$	28.3002	1.18433	0.592163	+	0.805818i	$0.298274\pi$
$572$	0	0
$573$	− 9.75379i	− 0.407470i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 32.2462i	− 1.34243i	−0.741264	−	0.671214i	$-0.765773\pi$
$577$	− 32.2462i	− 1.34243i	0.741264	−	0.671214i	$-0.234227\pi$
$578$	0	0
$579$	39.9157	1.65884
$580$	0	0
$581$	−2.63068	−0.109139
$582$	0	0
$583$	2.68466i	0.111187i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 34.0540i	− 1.40556i	−0.711408	−	0.702779i	$-0.751942\pi$
$587$	− 34.0540i	− 1.40556i	0.711408	−	0.702779i	$-0.248058\pi$
$588$	0	0
$589$	30.9309	1.27448
$590$	0	0
$591$	−41.7538	−1.71752
$592$	0	0
$593$	19.6155i	0.805513i	0.915307	+	0.402757i	$0.131948\pi$
$593$	19.6155i	0.805513i	−0.915307	+	0.402757i	$0.868052\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	26.0540i	1.06632i
$598$	0	0
$599$	−1.06913	−0.0436835	−0.0218417	−	0.999761i	$-0.506953\pi$
$599$	−1.06913	−0.0436835	−0.0218417	+	0.999761i	$0.506953\pi$
$600$	0	0
$601$	38.9848	1.59022	0.795112	−	0.606462i	$-0.207412\pi$
$601$	38.9848	1.59022	0.795112	+	0.606462i	$0.207412\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 5.80776i	− 0.235730i	−0.993030	−	0.117865i	$-0.962395\pi$
$607$	− 5.80776i	− 0.235730i	0.993030	−	0.117865i	$-0.0376050\pi$
$608$	0	0
$609$	3.03882	0.123139
$610$	0	0
$611$	−95.2311	−3.85264
$612$	0	0
$613$	11.1231i	0.449258i	0.974444	+	0.224629i	$0.0721170\pi$
$613$	11.1231i	0.449258i	−0.974444	+	0.224629i	$0.927883\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.6155i	0.548141i	0.961710	+	0.274070i	$0.0883701\pi$
$617$	13.6155i	0.548141i	−0.961710	+	0.274070i	$0.911630\pi$
$618$	0	0
$619$	14.7386	0.592396	0.296198	−	0.955127i	$-0.404281\pi$
$619$	14.7386	0.592396	0.296198	+	0.955127i	$0.404281\pi$
$620$	0	0
$621$	−39.6155	−1.58972
$622$	0	0
$623$	− 1.17708i	− 0.0471588i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8.68466i	− 0.346832i
$628$	0	0
$629$	−54.1619	−2.15958
$630$	0	0
$631$	12.1922	0.485365	0.242683	−	0.970106i	$-0.421973\pi$
$631$	12.1922	0.485365	0.242683	+	0.970106i	$0.421973\pi$
$632$	0	0
$633$	11.4233i	0.454035i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 48.4924i	− 1.92134i
$638$	0	0
$639$	4.87689	0.192927
$640$	0	0
$641$	16.4384	0.649280	0.324640	−	0.945838i	$-0.394757\pi$
$641$	16.4384	0.649280	0.324640	+	0.945838i	$0.394757\pi$
$642$	0	0
$643$	− 28.3002i	− 1.11605i	−0.829824	−	0.558025i	$-0.811559\pi$
$643$	− 28.3002i	− 1.11605i	0.829824	−	0.558025i	$-0.188441\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 36.8769i	− 1.44978i	−0.688864	−	0.724890i	$-0.741890\pi$
$647$	− 36.8769i	− 1.44978i	0.688864	−	0.724890i	$-0.258110\pi$
$648$	0	0
$649$	−7.12311	−0.279606
$650$	0	0
$651$	−3.80776	−0.149238
$652$	0	0
$653$	− 24.4384i	− 0.956350i	−0.878264	−	0.478175i	$-0.841298\pi$
$653$	− 24.4384i	− 0.956350i	0.878264	−	0.478175i	$-0.158702\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.00000i	0.156055i
$658$	0	0
$659$	−5.06913	−0.197465	−0.0987326	−	0.995114i	$-0.531479\pi$
$659$	−5.06913	−0.197465	−0.0987326	+	0.995114i	$0.531479\pi$
$660$	0	0
$661$	−30.4924	−1.18602	−0.593009	−	0.805196i	$-0.702060\pi$
$661$	−30.4924	−1.18602	−0.593009	+	0.805196i	$0.702060\pi$
$662$	0	0
$663$	− 52.1080i	− 2.02371i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	31.6155i	1.22416i
$668$	0	0
$669$	26.3542	1.01891
$670$	0	0
$671$	−8.43845	−0.325763
$672$	0	0
$673$	13.0691i	0.503778i	0.967756	+	0.251889i	$0.0810518\pi$
$673$	13.0691i	0.503778i	−0.967756	+	0.251889i	$0.918948\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.7538i	1.14353i	0.820417	+	0.571765i	$0.193741\pi$
$677$	29.7538i	1.14353i	−0.820417	+	0.571765i	$0.806259\pi$
$678$	0	0
$679$	−5.75379	−0.220810
$680$	0	0
$681$	12.8769	0.493444
$682$	0	0
$683$	− 5.17708i	− 0.198095i	−0.995083	−	0.0990477i	$-0.968420\pi$
$683$	− 5.17708i	− 0.198095i	0.995083	−	0.0990477i	$-0.0315797\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 0.384472i	− 0.0146685i
$688$	0	0
$689$	19.1231	0.728532
$690$	0	0
$691$	35.6155	1.35488	0.677439	−	0.735579i	$-0.263090\pi$
$691$	35.6155	1.35488	0.677439	+	0.735579i	$0.263090\pi$
$692$	0	0
$693$	− 0.246211i	− 0.00935279i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.8920i	0.753465i
$698$	0	0
$699$	15.6998	0.593821
$700$	0	0
$701$	−16.4384	−0.620872	−0.310436	−	0.950594i	$-0.600475\pi$
$701$	−16.4384	−0.620872	−0.310436	+	0.950594i	$0.600475\pi$
$702$	0	0
$703$	64.3002i	2.42513i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.876894i	0.0329790i
$708$	0	0
$709$	15.7538	0.591646	0.295823	−	0.955243i	$-0.404406\pi$
$709$	15.7538	0.591646	0.295823	+	0.955243i	$0.404406\pi$
$710$	0	0
$711$	7.50758	0.281556
$712$	0	0
$713$	− 39.6155i	− 1.48361i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 30.6307i	− 1.14392i
$718$	0	0
$719$	−2.82292	−0.105277	−0.0526386	−	0.998614i	$-0.516763\pi$
$719$	−2.82292	−0.105277	−0.0526386	+	0.998614i	$0.516763\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.6307i	0.841644i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.1231i	0.709237i	0.935011	+	0.354618i	$0.115389\pi$
$727$	19.1231i	0.709237i	−0.935011	+	0.354618i	$0.884611\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	30.2462i	1.11717i	0.829448	+	0.558585i	$0.188655\pi$
$733$	30.2462i	1.11717i	−0.829448	+	0.558585i	$0.811345\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	5.75379	0.211657	0.105828	−	0.994384i	$-0.466251\pi$
$739$	5.75379	0.211657	0.105828	+	0.994384i	$0.466251\pi$
$740$	0	0
$741$	−61.8617	−2.27255
$742$	0	0
$743$	16.4384i	0.603068i	0.953455	+	0.301534i	$0.0974987\pi$
$743$	16.4384i	0.603068i	−0.953455	+	0.301534i	$0.902501\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.36932i	− 0.123277i
$748$	0	0
$749$	0.492423	0.0179927
$750$	0	0
$751$	51.8078	1.89049	0.945246	−	0.326358i	$-0.105822\pi$
$751$	51.8078	1.89049	0.945246	+	0.326358i	$0.105822\pi$
$752$	0	0
$753$	− 4.10795i	− 0.149702i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	0	0
$759$	−11.1231	−0.403743
$760$	0	0
$761$	24.2462	0.878924	0.439462	−	0.898261i	$-0.355169\pi$
$761$	24.2462	0.878924	0.439462	+	0.898261i	$0.355169\pi$
$762$	0	0
$763$	5.36932i	0.194382i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	50.7386i	1.83207i
$768$	0	0
$769$	−20.2462	−0.730097	−0.365049	−	0.930988i	$-0.618948\pi$
$769$	−20.2462	−0.730097	−0.365049	+	0.930988i	$0.618948\pi$
$770$	0	0
$771$	−41.3693	−1.48988
$772$	0	0
$773$	− 15.0691i	− 0.541999i	−0.962579	−	0.270999i	$-0.912646\pi$
$773$	− 15.0691i	− 0.541999i	0.962579	−	0.270999i	$-0.0873542\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.91571i	− 0.283975i
$778$	0	0
$779$	23.6155	0.846114
$780$	0	0
$781$	8.68466	0.310762
$782$	0	0
$783$	24.6847i	0.882158i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 39.8617i	− 1.42092i	−0.703739	−	0.710459i	$-0.748487\pi$
$787$	− 39.8617i	− 1.42092i	0.703739	−	0.710459i	$-0.251513\pi$
$788$	0	0
$789$	12.5767	0.447743
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	60.1080i	2.13450i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.75379i	0.132966i	0.997788	+	0.0664830i	$0.0211778\pi$
$797$	3.75379i	0.132966i	−0.997788	+	0.0664830i	$0.978822\pi$
$798$	0	0
$799$	62.6307	2.21571
$800$	0	0
$801$	1.50758	0.0532676
$802$	0	0
$803$	7.12311i	0.251369i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.2462i	0.501490i
$808$	0	0
$809$	−8.73863	−0.307234	−0.153617	−	0.988130i	$-0.549092\pi$
$809$	−8.73863	−0.307234	−0.153617	+	0.988130i	$0.549092\pi$
$810$	0	0
$811$	−26.9309	−0.945671	−0.472835	−	0.881151i	$-0.656769\pi$
$811$	−26.9309	−0.945671	−0.472835	+	0.881151i	$0.656769\pi$
$812$	0	0
$813$	− 6.24621i	− 0.219064i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.4924i	0.996824i
$818$	0	0
$819$	−1.75379	−0.0612823
$820$	0	0
$821$	−18.4924	−0.645390	−0.322695	−	0.946503i	$-0.604589\pi$
$821$	−18.4924	−0.645390	−0.322695	+	0.946503i	$0.604589\pi$
$822$	0	0
$823$	− 20.4924i	− 0.714321i	−0.934043	−	0.357160i	$-0.883745\pi$
$823$	− 20.4924i	− 0.714321i	0.934043	−	0.357160i	$-0.116255\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.8617i	0.968848i	0.874833	+	0.484424i	$0.160971\pi$
$827$	27.8617i	0.968848i	−0.874833	+	0.484424i	$0.839029\pi$
$828$	0	0
$829$	41.1231	1.42826	0.714132	−	0.700011i	$-0.246822\pi$
$829$	41.1231	1.42826	0.714132	+	0.700011i	$0.246822\pi$
$830$	0	0
$831$	−8.98485	−0.311681
$832$	0	0
$833$	31.8920i	1.10499i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 30.9309i	− 1.06913i
$838$	0	0
$839$	52.4924	1.81224	0.906120	−	0.423021i	$-0.139030\pi$
$839$	52.4924	1.81224	0.906120	+	0.423021i	$0.139030\pi$
$840$	0	0
$841$	−9.30019	−0.320696
$842$	0	0
$843$	12.8769i	0.443504i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.438447i	− 0.0150652i
$848$	0	0
$849$	−9.36932	−0.321554
$850$	0	0
$851$	82.3542	2.82306
$852$	0	0
$853$	28.8769i	0.988726i	0.869256	+	0.494363i	$0.164599\pi$
$853$	28.8769i	0.988726i	−0.869256	+	0.494363i	$0.835401\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.80776i	− 0.266708i	−0.991068	−	0.133354i	$-0.957425\pi$
$857$	− 7.80776i	− 0.266708i	0.991068	−	0.133354i	$-0.0425747\pi$
$858$	0	0
$859$	−33.8617	−1.15535	−0.577674	−	0.816268i	$-0.696039\pi$
$859$	−33.8617	−1.15535	−0.577674	+	0.816268i	$0.696039\pi$
$860$	0	0
$861$	−2.90720	−0.0990773
$862$	0	0
$863$	20.8769i	0.710658i	0.934741	+	0.355329i	$0.115631\pi$
$863$	20.8769i	0.710658i	−0.934741	+	0.355329i	$0.884369\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.72348i	0.262303i
$868$	0	0
$869$	13.3693	0.453523
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 7.36932i	− 0.249414i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.6155i	0.527299i	0.964619	+	0.263649i	$0.0849262\pi$
$877$	15.6155i	0.527299i	−0.964619	+	0.263649i	$0.915074\pi$
$878$	0	0
$879$	−13.8617	−0.467545
$880$	0	0
$881$	38.4924	1.29684	0.648421	−	0.761282i	$-0.275430\pi$
$881$	38.4924	1.29684	0.648421	+	0.761282i	$0.275430\pi$
$882$	0	0
$883$	42.5464i	1.43180i	0.698203	+	0.715900i	$0.253983\pi$
$883$	42.5464i	1.43180i	−0.698203	+	0.715900i	$0.746017\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.8769i	0.633824i	0.948455	+	0.316912i	$0.102646\pi$
$887$	18.8769i	0.633824i	−0.948455	+	0.316912i	$0.897354\pi$
$888$	0	0
$889$	−5.75379	−0.192976
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	− 74.3542i	− 2.48817i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	79.2311i	2.64545i
$898$	0	0
$899$	−24.6847	−0.823279
$900$	0	0
$901$	−12.5767	−0.418991
$902$	0	0
$903$	− 3.50758i	− 0.116725i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	31.3153i	1.03981i	0.854224	+	0.519904i	$0.174032\pi$
$907$	31.3153i	1.03981i	−0.854224	+	0.519904i	$0.825968\pi$
$908$	0	0
$909$	−1.12311	−0.0372511
$910$	0	0
$911$	−45.5616	−1.50952	−0.754761	−	0.656000i	$-0.772247\pi$
$911$	−45.5616	−1.50952	−0.754761	+	0.656000i	$0.772247\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.66950i	0.0551319i
$918$	0	0
$919$	47.2311	1.55801	0.779004	−	0.627018i	$-0.215725\pi$
$919$	47.2311	1.55801	0.779004	+	0.627018i	$0.215725\pi$
$920$	0	0
$921$	−28.1080	−0.926188
$922$	0	0
$923$	− 61.8617i	− 2.03620i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.5464	1.19905	0.599524	−	0.800357i	$-0.295357\pi$
$929$	36.5464	1.19905	0.599524	+	0.800357i	$0.295357\pi$
$930$	0	0
$931$	37.8617	1.24087
$932$	0	0
$933$	− 21.1771i	− 0.693307i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8.38447i	− 0.273909i	−0.990577	−	0.136954i	$-0.956269\pi$
$937$	− 8.38447i	− 0.273909i	0.990577	−	0.136954i	$-0.0437314\pi$
$938$	0	0
$939$	38.6307	1.26066
$940$	0	0
$941$	−28.0540	−0.914533	−0.457267	−	0.889330i	$-0.651172\pi$
$941$	−28.0540	−0.914533	−0.457267	+	0.889330i	$0.651172\pi$
$942$	0	0
$943$	− 30.2462i	− 0.984952i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 50.0540i	− 1.62654i	−0.581890	−	0.813268i	$-0.697686\pi$
$947$	− 50.0540i	− 1.62654i	0.581890	−	0.813268i	$-0.302314\pi$
$948$	0	0
$949$	50.7386	1.64705
$950$	0	0
$951$	29.1771	0.946132
$952$	0	0
$953$	2.43845i	0.0789891i	0.999220	+	0.0394945i	$0.0125748\pi$
$953$	2.43845i	0.0789891i	−0.999220	+	0.0394945i	$0.987425\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.93087i	0.224043i
$958$	0	0
$959$	0.492423	0.0159012
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	0.630683i	0.0203235i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.43845i	0.271362i	0.990753	+	0.135681i	$0.0433223\pi$
$967$	8.43845i	0.271362i	−0.990753	+	0.135681i	$0.956678\pi$
$968$	0	0
$969$	40.6847	1.30698
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	1.75379i	0.0562239i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.8617i	1.14732i	0.819094	+	0.573659i	$0.194477\pi$
$977$	35.8617i	1.14732i	−0.819094	+	0.573659i	$0.805523\pi$
$978$	0	0
$979$	2.68466	0.0858021
$980$	0	0
$981$	−6.87689	−0.219562
$982$	0	0
$983$	− 41.3693i	− 1.31948i	−0.751496	−	0.659738i	$-0.770667\pi$
$983$	− 41.3693i	− 1.31948i	0.751496	−	0.659738i	$-0.229333\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	9.15342i	0.291356i
$988$	0	0
$989$	36.4924	1.16039
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	48.0000i	1.52323i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.7386i	1.60691i	0.595366	+	0.803454i	$0.297007\pi$
$997$	50.7386i	1.60691i	−0.595366	+	0.803454i	$0.702993\pi$
$998$	0	0
$999$	64.3002	2.03437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.w.4049.2		4
4.3	odd	2		2200.2.b.f.1849.3		4
5.2	odd	4		4400.2.a.bt.1.1		2
5.3	odd	4		880.2.a.k.1.2		2
5.4	even	2	inner	4400.2.b.w.4049.3		4
15.8	even	4		7920.2.a.by.1.2		2
20.3	even	4		440.2.a.g.1.1	✓	2
20.7	even	4		2200.2.a.l.1.2		2
20.19	odd	2		2200.2.b.f.1849.2		4
40.3	even	4		3520.2.a.bm.1.2		2
40.13	odd	4		3520.2.a.br.1.1		2
55.43	even	4		9680.2.a.bm.1.2		2
60.23	odd	4		3960.2.a.bf.1.1		2
220.43	odd	4		4840.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.g.1.1	✓	2	20.3	even	4
880.2.a.k.1.2		2	5.3	odd	4
2200.2.a.l.1.2		2	20.7	even	4
2200.2.b.f.1849.2		4	20.19	odd	2
2200.2.b.f.1849.3		4	4.3	odd	2
3520.2.a.bm.1.2		2	40.3	even	4
3520.2.a.br.1.1		2	40.13	odd	4
3960.2.a.bf.1.1		2	60.23	odd	4
4400.2.a.bt.1.1		2	5.2	odd	4
4400.2.b.w.4049.2		4	1.1	even	1	trivial
4400.2.b.w.4049.3		4	5.4	even	2	inner
4840.2.a.m.1.1		2	220.43	odd	4
7920.2.a.by.1.2		2	15.8	even	4
9680.2.a.bm.1.2		2	55.43	even	4

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.56155i q^{3} -0.438447i q^{7} +0.561553 q^{9} +O(q^{10})\)	\(q-1.56155i q^{3} -0.438447i q^{7} +0.561553 q^{9} +1.00000 q^{11} -7.12311i q^{13} +4.68466i q^{17} +5.56155 q^{19} -0.684658 q^{21} -7.12311i q^{23} -5.56155i q^{27} -4.43845 q^{29} +5.56155 q^{31} -1.56155i q^{33} +11.5616i q^{37} -11.1231 q^{39} +4.24621 q^{41} +5.12311i q^{43} -13.3693i q^{47} +6.80776 q^{49} +7.31534 q^{51} +2.68466i q^{53} -8.68466i q^{57} -7.12311 q^{59} -8.43845 q^{61} -0.246211i q^{63} -11.1231 q^{69} +8.68466 q^{71} +7.12311i q^{73} -0.438447i q^{77} +13.3693 q^{79} -7.00000 q^{81} -6.00000i q^{83} +6.93087i q^{87} +2.68466 q^{89} -3.12311 q^{91} -8.68466i q^{93} -13.1231i q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} + 4 q^{11} + 14 q^{19} + 22 q^{21} - 26 q^{29} + 14 q^{31} - 28 q^{39} - 16 q^{41} - 14 q^{49} + 54 q^{51} - 12 q^{59} - 42 q^{61} - 28 q^{69} + 10 q^{71} + 4 q^{79} - 28 q^{81} - 14 q^{89} + 4 q^{91} - 6 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 + 4 * q^11 + 14 * q^19 + 22 * q^21 - 26 * q^29 + 14 * q^31 - 28 * q^39 - 16 * q^41 - 14 * q^49 + 54 * q^51 - 12 * q^59 - 42 * q^61 - 28 * q^69 + 10 * q^71 + 4 * q^79 - 28 * q^81 - 14 * q^89 + 4 * q^91 - 6 * q^99

Introduction

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