LMFDB - Embedded newform 4400.2.b.ba.4049.4

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:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.ba.4049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803i q^{3} -0.618034i q^{7} +0.381966 q^{9} +O(q^{10})$	$q+1.61803i q^{3} -0.618034i q^{7} +0.381966 q^{9} +1.00000 q^{11} +2.23607i q^{13} -4.85410i q^{17} -5.47214 q^{19} +1.00000 q^{21} -6.32624i q^{23} +5.47214i q^{27} +4.38197 q^{29} +4.23607 q^{31} +1.61803i q^{33} -1.76393i q^{37} -3.61803 q^{39} +7.94427 q^{41} +8.47214i q^{43} -4.70820i q^{47} +6.61803 q^{49} +7.85410 q^{51} -3.85410i q^{53} -8.85410i q^{57} -3.76393 q^{59} +7.09017 q^{61} -0.236068i q^{63} +10.2361 q^{69} +0.291796 q^{71} -7.09017i q^{73} -0.618034i q^{77} +2.85410 q^{79} -7.70820 q^{81} +1.14590i q^{83} +7.09017i q^{87} +12.5623 q^{89} +1.38197 q^{91} +6.85410i q^{93} -4.90983i q^{97} +0.381966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{9}+O(q^{10}) $ 4 * q + 6 * q^9	$ 4 q + 6 q^{9} + 4 q^{11} - 4 q^{19} + 4 q^{21} + 22 q^{29} + 8 q^{31} - 10 q^{39} - 4 q^{41} + 22 q^{49} + 18 q^{51} - 24 q^{59} + 6 q^{61} + 32 q^{69} + 28 q^{71} - 2 q^{79} - 4 q^{81} + 10 q^{89} + 10 q^{91} + 6 q^{99}+O(q^{100}) $ 4 * q + 6 * q^9 + 4 * q^11 - 4 * q^19 + 4 * q^21 + 22 * q^29 + 8 * q^31 - 10 * q^39 - 4 * q^41 + 22 * q^49 + 18 * q^51 - 24 * q^59 + 6 * q^61 + 32 * q^69 + 28 * q^71 - 2 * q^79 - 4 * q^81 + 10 * q^89 + 10 * 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
$3$	1.61803i	0.934172i	0.884212	+	0.467086i	$0.154696\pi$
$3$	1.61803i	0.934172i	−0.884212	+	0.467086i	$0.845304\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.618034i	− 0.233595i	−0.993156	−	0.116797i	$-0.962737\pi$
$7$	− 0.618034i	− 0.233595i	0.993156	−	0.116797i	$-0.0372628\pi$
$8$	0	0
$9$	0.381966	0.127322
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.23607i	0.620174i	0.950708	+	0.310087i	$0.100358\pi$
$13$	2.23607i	0.620174i	−0.950708	+	0.310087i	$0.899642\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.85410i	− 1.17729i	−0.808391	−	0.588646i	$-0.799661\pi$
$17$	− 4.85410i	− 1.17729i	0.808391	−	0.588646i	$-0.200339\pi$
$18$	0	0
$19$	−5.47214	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$19$	−5.47214	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	− 6.32624i	− 1.31911i	−0.751656	−	0.659556i	$-0.770744\pi$
$23$	− 6.32624i	− 1.31911i	0.751656	−	0.659556i	$-0.229256\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.47214i	1.05311i
$28$	0	0
$29$	4.38197	0.813711	0.406855	−	0.913493i	$-0.366625\pi$
$29$	4.38197	0.813711	0.406855	+	0.913493i	$0.366625\pi$
$30$	0	0
$31$	4.23607	0.760820	0.380410	−	0.924818i	$-0.375783\pi$
$31$	4.23607	0.760820	0.380410	+	0.924818i	$0.375783\pi$
$32$	0	0
$33$	1.61803i	0.281664i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.76393i	− 0.289989i	−0.989432	−	0.144994i	$-0.953684\pi$
$37$	− 1.76393i	− 0.289989i	0.989432	−	0.144994i	$-0.0463164\pi$
$38$	0	0
$39$	−3.61803	−0.579349
$40$	0	0
$41$	7.94427	1.24069	0.620343	−	0.784330i	$-0.286993\pi$
$41$	7.94427	1.24069	0.620343	+	0.784330i	$0.286993\pi$
$42$	0	0
$43$	8.47214i	1.29199i	0.763342	+	0.645994i	$0.223557\pi$
$43$	8.47214i	1.29199i	−0.763342	+	0.645994i	$0.776443\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.70820i	− 0.686762i	−0.939196	−	0.343381i	$-0.888428\pi$
$47$	− 4.70820i	− 0.686762i	0.939196	−	0.343381i	$-0.111572\pi$
$48$	0	0
$49$	6.61803	0.945433
$50$	0	0
$51$	7.85410	1.09979
$52$	0	0
$53$	− 3.85410i	− 0.529402i	−0.964331	−	0.264701i	$-0.914727\pi$
$53$	− 3.85410i	− 0.529402i	0.964331	−	0.264701i	$-0.0852732\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.85410i	− 1.17275i
$58$	0	0
$59$	−3.76393	−0.490022	−0.245011	−	0.969520i	$-0.578792\pi$
$59$	−3.76393	−0.490022	−0.245011	+	0.969520i	$0.578792\pi$
$60$	0	0
$61$	7.09017	0.907803	0.453902	−	0.891052i	$-0.350032\pi$
$61$	7.09017	0.907803	0.453902	+	0.891052i	$0.350032\pi$
$62$	0	0
$63$	− 0.236068i	− 0.0297418i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	10.2361	1.23228
$70$	0	0
$71$	0.291796	0.0346298	0.0173149	−	0.999850i	$-0.494488\pi$
$71$	0.291796	0.0346298	0.0173149	+	0.999850i	$0.494488\pi$
$72$	0	0
$73$	− 7.09017i	− 0.829842i	−0.909858	−	0.414921i	$-0.863809\pi$
$73$	− 7.09017i	− 0.829842i	0.909858	−	0.414921i	$-0.136191\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.618034i	− 0.0704315i
$78$	0	0
$79$	2.85410	0.321112	0.160556	−	0.987027i	$-0.448671\pi$
$79$	2.85410	0.321112	0.160556	+	0.987027i	$0.448671\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	1.14590i	0.125779i	0.998021	+	0.0628893i	$0.0200315\pi$
$83$	1.14590i	0.125779i	−0.998021	+	0.0628893i	$0.979968\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.09017i	0.760146i
$88$	0	0
$89$	12.5623	1.33160	0.665801	−	0.746129i	$-0.268090\pi$
$89$	12.5623	1.33160	0.665801	+	0.746129i	$0.268090\pi$
$90$	0	0
$91$	1.38197	0.144869
$92$	0	0
$93$	6.85410i	0.710737i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.90983i	− 0.498518i	−0.968437	−	0.249259i	$-0.919813\pi$
$97$	− 4.90983i	− 0.498518i	0.968437	−	0.249259i	$-0.0801870\pi$
$98$	0	0
$99$	0.381966	0.0383890
$100$	0	0
$101$	−7.56231	−0.752478	−0.376239	−	0.926523i	$-0.622783\pi$
$101$	−7.56231	−0.752478	−0.376239	+	0.926523i	$0.622783\pi$
$102$	0	0
$103$	16.8541i	1.66068i	0.557254	+	0.830342i	$0.311855\pi$
$103$	16.8541i	1.66068i	−0.557254	+	0.830342i	$0.688145\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.9443i	− 1.15470i	−0.816498	−	0.577348i	$-0.804088\pi$
$107$	− 11.9443i	− 1.15470i	0.816498	−	0.577348i	$-0.195912\pi$
$108$	0	0
$109$	−5.79837	−0.555383	−0.277692	−	0.960670i	$-0.589569\pi$
$109$	−5.79837	−0.555383	−0.277692	+	0.960670i	$0.589569\pi$
$110$	0	0
$111$	2.85410	0.270899
$112$	0	0
$113$	0.236068i	0.0222074i	0.999938	+	0.0111037i	$0.00353449\pi$
$113$	0.236068i	0.0222074i	−0.999938	+	0.0111037i	$0.996466\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.854102i	0.0789618i
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.8541i	1.15902i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.381966i	0.0338940i	0.999856	+	0.0169470i	$0.00539466\pi$
$127$	0.381966i	0.0338940i	−0.999856	+	0.0169470i	$0.994605\pi$
$128$	0	0
$129$	−13.7082	−1.20694
$130$	0	0
$131$	15.0902	1.31843	0.659217	−	0.751953i	$-0.270888\pi$
$131$	15.0902	1.31843	0.659217	+	0.751953i	$0.270888\pi$
$132$	0	0
$133$	3.38197i	0.293254i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.38197i	− 0.545248i	−0.962121	−	0.272624i	$-0.912108\pi$
$137$	− 6.38197i	− 0.545248i	0.962121	−	0.272624i	$-0.0878915\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	7.61803	0.641554
$142$	0	0
$143$	2.23607i	0.186989i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.7082i	0.883198i
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	13.6525	1.11102	0.555511	−	0.831509i	$-0.312523\pi$
$151$	13.6525	1.11102	0.555511	+	0.831509i	$0.312523\pi$
$152$	0	0
$153$	− 1.85410i	− 0.149895i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.4164i	− 1.38998i	−0.719019	−	0.694990i	$-0.755409\pi$
$157$	− 17.4164i	− 1.38998i	0.719019	−	0.694990i	$-0.244591\pi$
$158$	0	0
$159$	6.23607	0.494552
$160$	0	0
$161$	−3.90983	−0.308138
$162$	0	0
$163$	15.3820i	1.20481i	0.798191	+	0.602404i	$0.205790\pi$
$163$	15.3820i	1.20481i	−0.798191	+	0.602404i	$0.794210\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.00000i	− 0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$	− 9.00000i	− 0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$	0	0
$169$	8.00000	0.615385
$170$	0	0
$171$	−2.09017	−0.159839
$172$	0	0
$173$	− 14.8885i	− 1.13196i	−0.824421	−	0.565978i	$-0.808499\pi$
$173$	− 14.8885i	− 1.13196i	0.824421	−	0.565978i	$-0.191501\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.09017i	− 0.457765i
$178$	0	0
$179$	−13.0902	−0.978405	−0.489203	−	0.872170i	$-0.662712\pi$
$179$	−13.0902	−0.978405	−0.489203	+	0.872170i	$0.662712\pi$
$180$	0	0
$181$	13.0902	0.972985	0.486492	−	0.873685i	$-0.338276\pi$
$181$	13.0902	0.972985	0.486492	+	0.873685i	$0.338276\pi$
$182$	0	0
$183$	11.4721i	0.848045i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.85410i	− 0.354967i
$188$	0	0
$189$	3.38197	0.246002
$190$	0	0
$191$	10.3820	0.751213	0.375606	−	0.926779i	$-0.377434\pi$
$191$	10.3820	0.751213	0.375606	+	0.926779i	$0.377434\pi$
$192$	0	0
$193$	12.9443i	0.931749i	0.884851	+	0.465875i	$0.154260\pi$
$193$	12.9443i	0.931749i	−0.884851	+	0.465875i	$0.845740\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.5623i	1.53625i	0.640300	+	0.768125i	$0.278810\pi$
$197$	21.5623i	1.53625i	−0.640300	+	0.768125i	$0.721190\pi$
$198$	0	0
$199$	20.5623	1.45762	0.728812	−	0.684714i	$-0.240073\pi$
$199$	20.5623	1.45762	0.728812	+	0.684714i	$0.240073\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.70820i	− 0.190079i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.41641i	− 0.167952i
$208$	0	0
$209$	−5.47214	−0.378516
$210$	0	0
$211$	−6.70820	−0.461812	−0.230906	−	0.972976i	$-0.574169\pi$
$211$	−6.70820	−0.461812	−0.230906	+	0.972976i	$0.574169\pi$
$212$	0	0
$213$	0.472136i	0.0323502i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.61803i	− 0.177724i
$218$	0	0
$219$	11.4721	0.775215
$220$	0	0
$221$	10.8541	0.730126
$222$	0	0
$223$	− 5.18034i	− 0.346901i	−0.984843	−	0.173451i	$-0.944508\pi$
$223$	− 5.18034i	− 0.346901i	0.984843	−	0.173451i	$-0.0554917\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.79837i	0.517596i	0.965931	+	0.258798i	$0.0833264\pi$
$227$	7.79837i	0.517596i	−0.965931	+	0.258798i	$0.916674\pi$
$228$	0	0
$229$	27.5066	1.81769	0.908843	−	0.417139i	$-0.136967\pi$
$229$	27.5066	1.81769	0.908843	+	0.417139i	$0.136967\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	18.1459i	1.18878i	0.804178	+	0.594389i	$0.202606\pi$
$233$	18.1459i	1.18878i	−0.804178	+	0.594389i	$0.797394\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.61803i	0.299974i
$238$	0	0
$239$	21.0902	1.36421	0.682105	−	0.731254i	$-0.261065\pi$
$239$	21.0902	1.36421	0.682105	+	0.731254i	$0.261065\pi$
$240$	0	0
$241$	−17.0344	−1.09728	−0.548642	−	0.836057i	$-0.684855\pi$
$241$	−17.0344	−1.09728	−0.548642	+	0.836057i	$0.684855\pi$
$242$	0	0
$243$	3.94427i	0.253025i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 12.2361i	− 0.778562i
$248$	0	0
$249$	−1.85410	−0.117499
$250$	0	0
$251$	7.14590	0.451045	0.225523	−	0.974238i	$-0.427591\pi$
$251$	7.14590	0.451045	0.225523	+	0.974238i	$0.427591\pi$
$252$	0	0
$253$	− 6.32624i	− 0.397727i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.05573i	− 0.0658545i	−0.999458	−	0.0329273i	$-0.989517\pi$
$257$	− 1.05573i	− 0.0658545i	0.999458	−	0.0329273i	$-0.0104830\pi$
$258$	0	0
$259$	−1.09017	−0.0677399
$260$	0	0
$261$	1.67376	0.103603
$262$	0	0
$263$	26.1246i	1.61091i	0.592655	+	0.805456i	$0.298080\pi$
$263$	26.1246i	1.61091i	−0.592655	+	0.805456i	$0.701920\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	20.3262i	1.24395i
$268$	0	0
$269$	23.7984	1.45101	0.725506	−	0.688216i	$-0.241606\pi$
$269$	23.7984	1.45101	0.725506	+	0.688216i	$0.241606\pi$
$270$	0	0
$271$	−18.4164	−1.11872	−0.559359	−	0.828926i	$-0.688952\pi$
$271$	−18.4164	−1.11872	−0.559359	+	0.828926i	$0.688952\pi$
$272$	0	0
$273$	2.23607i	0.135333i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.9443i	0.957998i	0.877815	+	0.478999i	$0.159000\pi$
$277$	15.9443i	0.957998i	−0.877815	+	0.478999i	$0.841000\pi$
$278$	0	0
$279$	1.61803	0.0968692
$280$	0	0
$281$	−31.3607	−1.87082	−0.935411	−	0.353563i	$-0.884970\pi$
$281$	−31.3607	−1.87082	−0.935411	+	0.353563i	$0.884970\pi$
$282$	0	0
$283$	− 13.1459i	− 0.781443i	−0.920509	−	0.390721i	$-0.872226\pi$
$283$	− 13.1459i	− 0.781443i	0.920509	−	0.390721i	$-0.127774\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.90983i	− 0.289818i
$288$	0	0
$289$	−6.56231	−0.386018
$290$	0	0
$291$	7.94427	0.465701
$292$	0	0
$293$	19.8885i	1.16190i	0.813939	+	0.580951i	$0.197319\pi$
$293$	19.8885i	1.16190i	−0.813939	+	0.580951i	$0.802681\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.47214i	0.317526i
$298$	0	0
$299$	14.1459	0.818078
$300$	0	0
$301$	5.23607	0.301802
$302$	0	0
$303$	− 12.2361i	− 0.702944i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.85410i	− 0.277038i	−0.990360	−	0.138519i	$-0.955766\pi$
$307$	− 4.85410i	− 0.277038i	0.990360	−	0.138519i	$-0.0442342\pi$
$308$	0	0
$309$	−27.2705	−1.55137
$310$	0	0
$311$	−10.8885	−0.617433	−0.308716	−	0.951154i	$-0.599899\pi$
$311$	−10.8885	−0.617433	−0.308716	+	0.951154i	$0.599899\pi$
$312$	0	0
$313$	27.7082i	1.56616i	0.621921	+	0.783080i	$0.286353\pi$
$313$	27.7082i	1.56616i	−0.621921	+	0.783080i	$0.713647\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.27051i	− 0.239856i	−0.992783	−	0.119928i	$-0.961734\pi$
$317$	− 4.27051i	− 0.239856i	0.992783	−	0.119928i	$-0.0382663\pi$
$318$	0	0
$319$	4.38197	0.245343
$320$	0	0
$321$	19.3262	1.07869
$322$	0	0
$323$	26.5623i	1.47797i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9.38197i	− 0.518824i
$328$	0	0
$329$	−2.90983	−0.160424
$330$	0	0
$331$	−28.4164	−1.56191	−0.780954	−	0.624589i	$-0.785266\pi$
$331$	−28.4164	−1.56191	−0.780954	+	0.624589i	$0.785266\pi$
$332$	0	0
$333$	− 0.673762i	− 0.0369219i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 15.5279i	− 0.845857i	−0.906163	−	0.422928i	$-0.861002\pi$
$337$	− 15.5279i	− 0.845857i	0.906163	−	0.422928i	$-0.138998\pi$
$338$	0	0
$339$	−0.381966	−0.0207455
$340$	0	0
$341$	4.23607	0.229396
$342$	0	0
$343$	− 8.41641i	− 0.454443i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.2705i	0.927130i	0.886063	+	0.463565i	$0.153430\pi$
$347$	17.2705i	0.927130i	−0.886063	+	0.463565i	$0.846570\pi$
$348$	0	0
$349$	−26.5279	−1.42000	−0.710002	−	0.704200i	$-0.751306\pi$
$349$	−26.5279	−1.42000	−0.710002	+	0.704200i	$0.751306\pi$
$350$	0	0
$351$	−12.2361	−0.653113
$352$	0	0
$353$	36.4164i	1.93825i	0.246570	+	0.969125i	$0.420696\pi$
$353$	36.4164i	1.93825i	−0.246570	+	0.969125i	$0.579304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.85410i	− 0.256906i
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	10.9443	0.576014
$362$	0	0
$363$	1.61803i	0.0849248i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 16.7984i	− 0.876868i	−0.898763	−	0.438434i	$-0.855533\pi$
$367$	− 16.7984i	− 0.876868i	0.898763	−	0.438434i	$-0.144467\pi$
$368$	0	0
$369$	3.03444	0.157967
$370$	0	0
$371$	−2.38197	−0.123666
$372$	0	0
$373$	4.70820i	0.243782i	0.992544	+	0.121891i	$0.0388958\pi$
$373$	4.70820i	0.243782i	−0.992544	+	0.121891i	$0.961104\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.79837i	0.504642i
$378$	0	0
$379$	−26.3050	−1.35119	−0.675597	−	0.737271i	$-0.736114\pi$
$379$	−26.3050	−1.35119	−0.675597	+	0.737271i	$0.736114\pi$
$380$	0	0
$381$	−0.618034	−0.0316628
$382$	0	0
$383$	− 29.4164i	− 1.50311i	−0.659671	−	0.751554i	$-0.729305\pi$
$383$	− 29.4164i	− 1.50311i	0.659671	−	0.751554i	$-0.270695\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.23607i	0.164499i
$388$	0	0
$389$	17.5279	0.888698	0.444349	−	0.895854i	$-0.353435\pi$
$389$	17.5279	0.888698	0.444349	+	0.895854i	$0.353435\pi$
$390$	0	0
$391$	−30.7082	−1.55298
$392$	0	0
$393$	24.4164i	1.23164i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.8541i	1.34777i	0.738837	+	0.673884i	$0.235375\pi$
$397$	26.8541i	1.34777i	−0.738837	+	0.673884i	$0.764625\pi$
$398$	0	0
$399$	−5.47214	−0.273949
$400$	0	0
$401$	26.3607	1.31639	0.658195	−	0.752848i	$-0.271320\pi$
$401$	26.3607	1.31639	0.658195	+	0.752848i	$0.271320\pi$
$402$	0	0
$403$	9.47214i	0.471841i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.76393i	− 0.0874349i
$408$	0	0
$409$	6.12461	0.302843	0.151421	−	0.988469i	$-0.451615\pi$
$409$	6.12461	0.302843	0.151421	+	0.988469i	$0.451615\pi$
$410$	0	0
$411$	10.3262	0.509356
$412$	0	0
$413$	2.32624i	0.114467i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 11.3262i	− 0.554648i
$418$	0	0
$419$	−0.888544	−0.0434082	−0.0217041	−	0.999764i	$-0.506909\pi$
$419$	−0.888544	−0.0434082	−0.0217041	+	0.999764i	$0.506909\pi$
$420$	0	0
$421$	−3.09017	−0.150606	−0.0753028	−	0.997161i	$-0.523992\pi$
$421$	−3.09017	−0.150606	−0.0753028	+	0.997161i	$0.523992\pi$
$422$	0	0
$423$	− 1.79837i	− 0.0874399i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.38197i	− 0.212058i
$428$	0	0
$429$	−3.61803	−0.174680
$430$	0	0
$431$	1.00000	0.0481683	0.0240842	−	0.999710i	$-0.492333\pi$
$431$	1.00000	0.0481683	0.0240842	+	0.999710i	$0.492333\pi$
$432$	0	0
$433$	− 32.7771i	− 1.57517i	−0.616208	−	0.787583i	$-0.711332\pi$
$433$	− 32.7771i	− 1.57517i	0.616208	−	0.787583i	$-0.288668\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.6180i	1.65601i
$438$	0	0
$439$	23.3262	1.11330	0.556650	−	0.830747i	$-0.312086\pi$
$439$	23.3262	1.11330	0.556650	+	0.830747i	$0.312086\pi$
$440$	0	0
$441$	2.52786	0.120374
$442$	0	0
$443$	− 14.2918i	− 0.679024i	−0.940602	−	0.339512i	$-0.889738\pi$
$443$	− 14.2918i	− 0.679024i	0.940602	−	0.339512i	$-0.110262\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	19.4164i	0.918365i
$448$	0	0
$449$	−31.1591	−1.47049	−0.735243	−	0.677803i	$-0.762932\pi$
$449$	−31.1591	−1.47049	−0.735243	+	0.677803i	$0.762932\pi$
$450$	0	0
$451$	7.94427	0.374081
$452$	0	0
$453$	22.0902i	1.03789i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4.43769i	− 0.207587i	−0.994599	−	0.103793i	$-0.966902\pi$
$457$	− 4.43769i	− 0.207587i	0.994599	−	0.103793i	$-0.0330980\pi$
$458$	0	0
$459$	26.5623	1.23982
$460$	0	0
$461$	−18.1246	−0.844147	−0.422074	−	0.906562i	$-0.638698\pi$
$461$	−18.1246	−0.844147	−0.422074	+	0.906562i	$0.638698\pi$
$462$	0	0
$463$	− 18.7082i	− 0.869444i	−0.900565	−	0.434722i	$-0.856847\pi$
$463$	− 18.7082i	− 0.869444i	0.900565	−	0.434722i	$-0.143153\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 33.6525i	− 1.55725i	−0.627489	−	0.778625i	$-0.715917\pi$
$467$	− 33.6525i	− 1.55725i	0.627489	−	0.778625i	$-0.284083\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.1803	1.29848
$472$	0	0
$473$	8.47214i	0.389549i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.47214i	− 0.0674045i
$478$	0	0
$479$	12.5967	0.575560	0.287780	−	0.957697i	$-0.407083\pi$
$479$	12.5967	0.575560	0.287780	+	0.957697i	$0.407083\pi$
$480$	0	0
$481$	3.94427	0.179843
$482$	0	0
$483$	− 6.32624i	− 0.287854i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.1803i	0.687887i	0.938990	+	0.343943i	$0.111763\pi$
$487$	15.1803i	0.687887i	−0.938990	+	0.343943i	$0.888237\pi$
$488$	0	0
$489$	−24.8885	−1.12550
$490$	0	0
$491$	23.8328	1.07556	0.537780	−	0.843085i	$-0.319263\pi$
$491$	23.8328	1.07556	0.537780	+	0.843085i	$0.319263\pi$
$492$	0	0
$493$	− 21.2705i	− 0.957976i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.180340i	− 0.00808935i
$498$	0	0
$499$	22.6869	1.01561	0.507803	−	0.861473i	$-0.330458\pi$
$499$	22.6869	1.01561	0.507803	+	0.861473i	$0.330458\pi$
$500$	0	0
$501$	14.5623	0.650596
$502$	0	0
$503$	− 37.2361i	− 1.66027i	−0.557559	−	0.830137i	$-0.688262\pi$
$503$	− 37.2361i	− 1.66027i	0.557559	−	0.830137i	$-0.311738\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.9443i	0.574875i
$508$	0	0
$509$	36.2148	1.60519	0.802596	−	0.596523i	$-0.203452\pi$
$509$	36.2148	1.60519	0.802596	+	0.596523i	$0.203452\pi$
$510$	0	0
$511$	−4.38197	−0.193847
$512$	0	0
$513$	− 29.9443i	− 1.32207i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.70820i	− 0.207067i
$518$	0	0
$519$	24.0902	1.05744
$520$	0	0
$521$	−38.6525	−1.69340	−0.846698	−	0.532074i	$-0.821413\pi$
$521$	−38.6525	−1.69340	−0.846698	+	0.532074i	$0.821413\pi$
$522$	0	0
$523$	0.472136i	0.0206451i	0.999947	+	0.0103225i	$0.00328582\pi$
$523$	0.472136i	0.0206451i	−0.999947	+	0.0103225i	$0.996714\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.5623i	− 0.895708i
$528$	0	0
$529$	−17.0213	−0.740056
$530$	0	0
$531$	−1.43769	−0.0623906
$532$	0	0
$533$	17.7639i	0.769441i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 21.1803i	− 0.913999i
$538$	0	0
$539$	6.61803	0.285059
$540$	0	0
$541$	28.5066	1.22559	0.612797	−	0.790241i	$-0.290044\pi$
$541$	28.5066	1.22559	0.612797	+	0.790241i	$0.290044\pi$
$542$	0	0
$543$	21.1803i	0.908935i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 22.4377i	− 0.959367i	−0.877442	−	0.479683i	$-0.840752\pi$
$547$	− 22.4377i	− 0.959367i	0.877442	−	0.479683i	$-0.159248\pi$
$548$	0	0
$549$	2.70820	0.115583
$550$	0	0
$551$	−23.9787	−1.02153
$552$	0	0
$553$	− 1.76393i	− 0.0750100i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.7082i	0.623207i	0.950212	+	0.311603i	$0.100866\pi$
$557$	14.7082i	0.623207i	−0.950212	+	0.311603i	$0.899134\pi$
$558$	0	0
$559$	−18.9443	−0.801257
$560$	0	0
$561$	7.85410	0.331600
$562$	0	0
$563$	− 6.61803i	− 0.278917i	−0.990228	−	0.139458i	$-0.955464\pi$
$563$	− 6.61803i	− 0.278917i	0.990228	−	0.139458i	$-0.0445362\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.76393i	0.200066i
$568$	0	0
$569$	−32.9098	−1.37965	−0.689826	−	0.723975i	$-0.742313\pi$
$569$	−32.9098	−1.37965	−0.689826	+	0.723975i	$0.742313\pi$
$570$	0	0
$571$	−41.4508	−1.73466	−0.867332	−	0.497730i	$-0.834167\pi$
$571$	−41.4508	−1.73466	−0.867332	+	0.497730i	$0.834167\pi$
$572$	0	0
$573$	16.7984i	0.701762i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 31.7984i	− 1.32378i	−0.749599	−	0.661892i	$-0.769754\pi$
$577$	− 31.7984i	− 1.32378i	0.749599	−	0.661892i	$-0.230246\pi$
$578$	0	0
$579$	−20.9443	−0.870414
$580$	0	0
$581$	0.708204	0.0293812
$582$	0	0
$583$	− 3.85410i	− 0.159621i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.27051i	− 0.0937140i	−0.998902	−	0.0468570i	$-0.985079\pi$
$587$	− 2.27051i	− 0.0937140i	0.998902	−	0.0468570i	$-0.0149205\pi$
$588$	0	0
$589$	−23.1803	−0.955129
$590$	0	0
$591$	−34.8885	−1.43512
$592$	0	0
$593$	− 6.88854i	− 0.282879i	−0.989947	−	0.141439i	$-0.954827\pi$
$593$	− 6.88854i	− 0.282879i	0.989947	−	0.141439i	$-0.0451730\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.2705i	1.36167i
$598$	0	0
$599$	26.0902	1.06602	0.533008	−	0.846110i	$-0.321062\pi$
$599$	26.0902	1.06602	0.533008	+	0.846110i	$0.321062\pi$
$600$	0	0
$601$	−25.7426	−1.05006	−0.525032	−	0.851082i	$-0.675947\pi$
$601$	−25.7426	−1.05006	−0.525032	+	0.851082i	$0.675947\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9.05573i	− 0.367561i	−0.982967	−	0.183780i	$-0.941166\pi$
$607$	− 9.05573i	− 0.367561i	0.982967	−	0.183780i	$-0.0588335\pi$
$608$	0	0
$609$	4.38197	0.177566
$610$	0	0
$611$	10.5279	0.425912
$612$	0	0
$613$	− 24.0902i	− 0.972993i	−0.873683	−	0.486496i	$-0.838275\pi$
$613$	− 24.0902i	− 0.972993i	0.873683	−	0.486496i	$-0.161725\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.2361i	0.774415i	0.921993	+	0.387207i	$0.126560\pi$
$617$	19.2361i	0.774415i	−0.921993	+	0.387207i	$0.873440\pi$
$618$	0	0
$619$	−21.2918	−0.855790	−0.427895	−	0.903829i	$-0.640745\pi$
$619$	−21.2918	−0.855790	−0.427895	+	0.903829i	$0.640745\pi$
$620$	0	0
$621$	34.6180	1.38917
$622$	0	0
$623$	− 7.76393i	− 0.311055i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8.85410i	− 0.353599i
$628$	0	0
$629$	−8.56231	−0.341401
$630$	0	0
$631$	32.5066	1.29407	0.647033	−	0.762462i	$-0.276009\pi$
$631$	32.5066	1.29407	0.647033	+	0.762462i	$0.276009\pi$
$632$	0	0
$633$	− 10.8541i	− 0.431412i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.7984i	0.586333i
$638$	0	0
$639$	0.111456	0.00440914
$640$	0	0
$641$	−1.11146	−0.0438999	−0.0219499	−	0.999759i	$-0.506987\pi$
$641$	−1.11146	−0.0438999	−0.0219499	+	0.999759i	$0.506987\pi$
$642$	0	0
$643$	9.05573i	0.357123i	0.983929	+	0.178562i	$0.0571444\pi$
$643$	9.05573i	0.357123i	−0.983929	+	0.178562i	$0.942856\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 13.9443i	− 0.548206i	−0.961700	−	0.274103i	$-0.911619\pi$
$647$	− 13.9443i	− 0.548206i	0.961700	−	0.274103i	$-0.0883809\pi$
$648$	0	0
$649$	−3.76393	−0.147747
$650$	0	0
$651$	4.23607	0.166025
$652$	0	0
$653$	− 19.3262i	− 0.756294i	−0.925746	−	0.378147i	$-0.876561\pi$
$653$	− 19.3262i	− 0.756294i	0.925746	−	0.378147i	$-0.123439\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.70820i	− 0.105657i
$658$	0	0
$659$	31.7984	1.23869	0.619344	−	0.785119i	$-0.287398\pi$
$659$	31.7984	1.23869	0.619344	+	0.785119i	$0.287398\pi$
$660$	0	0
$661$	−26.5967	−1.03449	−0.517247	−	0.855836i	$-0.673043\pi$
$661$	−26.5967	−1.03449	−0.517247	+	0.855836i	$0.673043\pi$
$662$	0	0
$663$	17.5623i	0.682063i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 27.7214i	− 1.07338i
$668$	0	0
$669$	8.38197	0.324066
$670$	0	0
$671$	7.09017	0.273713
$672$	0	0
$673$	15.4721i	0.596407i	0.954502	+	0.298204i	$0.0963874\pi$
$673$	15.4721i	0.596407i	−0.954502	+	0.298204i	$0.903613\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.7639i	1.25922i	0.776911	+	0.629610i	$0.216785\pi$
$677$	32.7639i	1.25922i	−0.776911	+	0.629610i	$0.783215\pi$
$678$	0	0
$679$	−3.03444	−0.116451
$680$	0	0
$681$	−12.6180	−0.483524
$682$	0	0
$683$	− 35.7639i	− 1.36847i	−0.729262	−	0.684234i	$-0.760137\pi$
$683$	− 35.7639i	− 1.36847i	0.729262	−	0.684234i	$-0.239863\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	44.5066i	1.69803i
$688$	0	0
$689$	8.61803	0.328321
$690$	0	0
$691$	−1.20163	−0.0457120	−0.0228560	−	0.999739i	$-0.507276\pi$
$691$	−1.20163	−0.0457120	−0.0228560	+	0.999739i	$0.507276\pi$
$692$	0	0
$693$	− 0.236068i	− 0.00896748i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 38.5623i	− 1.46065i
$698$	0	0
$699$	−29.3607	−1.11052
$700$	0	0
$701$	5.52786	0.208785	0.104392	−	0.994536i	$-0.466710\pi$
$701$	5.52786	0.208785	0.104392	+	0.994536i	$0.466710\pi$
$702$	0	0
$703$	9.65248i	0.364050i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.67376i	0.175775i
$708$	0	0
$709$	−16.3607	−0.614438	−0.307219	−	0.951639i	$-0.599398\pi$
$709$	−16.3607	−0.614438	−0.307219	+	0.951639i	$0.599398\pi$
$710$	0	0
$711$	1.09017	0.0408846
$712$	0	0
$713$	− 26.7984i	− 1.00361i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	34.1246i	1.27441i
$718$	0	0
$719$	−41.9443	−1.56426	−0.782129	−	0.623117i	$-0.785866\pi$
$719$	−41.9443	−1.56426	−0.782129	+	0.623117i	$0.785866\pi$
$720$	0	0
$721$	10.4164	0.387927
$722$	0	0
$723$	− 27.5623i	− 1.02505i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 41.5066i	− 1.53939i	−0.638410	−	0.769697i	$-0.720407\pi$
$727$	− 41.5066i	− 1.53939i	0.638410	−	0.769697i	$-0.279593\pi$
$728$	0	0
$729$	−29.5066	−1.09284
$730$	0	0
$731$	41.1246	1.52105
$732$	0	0
$733$	− 45.4721i	− 1.67955i	−0.542933	−	0.839776i	$-0.682686\pi$
$733$	− 45.4721i	− 1.67955i	0.542933	−	0.839776i	$-0.317314\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	36.0344	1.32555	0.662774	−	0.748819i	$-0.269379\pi$
$739$	36.0344	1.32555	0.662774	+	0.748819i	$0.269379\pi$
$740$	0	0
$741$	19.7984	0.727311
$742$	0	0
$743$	− 11.9656i	− 0.438974i	−0.975615	−	0.219487i	$-0.929562\pi$
$743$	− 11.9656i	− 0.438974i	0.975615	−	0.219487i	$-0.0704383\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.437694i	0.0160144i
$748$	0	0
$749$	−7.38197	−0.269731
$750$	0	0
$751$	34.1591	1.24648	0.623241	−	0.782030i	$-0.285816\pi$
$751$	34.1591	1.24648	0.623241	+	0.782030i	$0.285816\pi$
$752$	0	0
$753$	11.5623i	0.421354i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 51.8328i	− 1.88390i	−0.335759	−	0.941948i	$-0.608993\pi$
$757$	− 51.8328i	− 1.88390i	0.335759	−	0.941948i	$-0.391007\pi$
$758$	0	0
$759$	10.2361	0.371546
$760$	0	0
$761$	−37.8885	−1.37346	−0.686729	−	0.726913i	$-0.740954\pi$
$761$	−37.8885	−1.37346	−0.686729	+	0.726913i	$0.740954\pi$
$762$	0	0
$763$	3.58359i	0.129735i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.41641i	− 0.303899i
$768$	0	0
$769$	−29.9443	−1.07982	−0.539909	−	0.841723i	$-0.681541\pi$
$769$	−29.9443	−1.07982	−0.539909	+	0.841723i	$0.681541\pi$
$770$	0	0
$771$	1.70820	0.0615195
$772$	0	0
$773$	− 32.9098i	− 1.18368i	−0.806054	−	0.591842i	$-0.798401\pi$
$773$	− 32.9098i	− 1.18368i	0.806054	−	0.591842i	$-0.201599\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.76393i	− 0.0632807i
$778$	0	0
$779$	−43.4721	−1.55755
$780$	0	0
$781$	0.291796	0.0104413
$782$	0	0
$783$	23.9787i	0.856929i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.5279i	0.517862i	0.965896	+	0.258931i	$0.0833703\pi$
$787$	14.5279i	0.517862i	−0.965896	+	0.258931i	$0.916630\pi$
$788$	0	0
$789$	−42.2705	−1.50487
$790$	0	0
$791$	0.145898	0.00518754
$792$	0	0
$793$	15.8541i	0.562996i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.20163i	0.148829i	0.997227	+	0.0744146i	$0.0237088\pi$
$797$	4.20163i	0.148829i	−0.997227	+	0.0744146i	$0.976291\pi$
$798$	0	0
$799$	−22.8541	−0.808520
$800$	0	0
$801$	4.79837	0.169542
$802$	0	0
$803$	− 7.09017i	− 0.250207i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38.5066i	1.35550i
$808$	0	0
$809$	11.4164	0.401380	0.200690	−	0.979655i	$-0.435682\pi$
$809$	11.4164	0.401380	0.200690	+	0.979655i	$0.435682\pi$
$810$	0	0
$811$	−32.8197	−1.15245	−0.576227	−	0.817290i	$-0.695476\pi$
$811$	−32.8197	−1.15245	−0.576227	+	0.817290i	$0.695476\pi$
$812$	0	0
$813$	− 29.7984i	− 1.04507i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 46.3607i	− 1.62195i
$818$	0	0
$819$	0.527864	0.0184451
$820$	0	0
$821$	8.36068	0.291790	0.145895	−	0.989300i	$-0.453394\pi$
$821$	8.36068	0.291790	0.145895	+	0.989300i	$0.453394\pi$
$822$	0	0
$823$	21.5279i	0.750414i	0.926941	+	0.375207i	$0.122428\pi$
$823$	21.5279i	0.750414i	−0.926941	+	0.375207i	$0.877572\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.5967i	0.820539i	0.911964	+	0.410270i	$0.134565\pi$
$827$	23.5967i	0.820539i	−0.911964	+	0.410270i	$0.865435\pi$
$828$	0	0
$829$	33.6180	1.16760	0.583801	−	0.811897i	$-0.301565\pi$
$829$	33.6180	1.16760	0.583801	+	0.811897i	$0.301565\pi$
$830$	0	0
$831$	−25.7984	−0.894936
$832$	0	0
$833$	− 32.1246i	− 1.11305i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	23.1803i	0.801230i
$838$	0	0
$839$	−23.6738	−0.817309	−0.408655	−	0.912689i	$-0.634002\pi$
$839$	−23.6738	−0.817309	−0.408655	+	0.912689i	$0.634002\pi$
$840$	0	0
$841$	−9.79837	−0.337875
$842$	0	0
$843$	− 50.7426i	− 1.74767i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.618034i	− 0.0212359i
$848$	0	0
$849$	21.2705	0.730002
$850$	0	0
$851$	−11.1591	−0.382527
$852$	0	0
$853$	− 29.6180i	− 1.01410i	−0.861916	−	0.507051i	$-0.830736\pi$
$853$	− 29.6180i	− 1.01410i	0.861916	−	0.507051i	$-0.169264\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.6525i	0.671316i	0.941984	+	0.335658i	$0.108959\pi$
$857$	19.6525i	0.671316i	−0.941984	+	0.335658i	$0.891041\pi$
$858$	0	0
$859$	43.4853	1.48370	0.741850	−	0.670566i	$-0.233949\pi$
$859$	43.4853	1.48370	0.741850	+	0.670566i	$0.233949\pi$
$860$	0	0
$861$	7.94427	0.270740
$862$	0	0
$863$	9.94427i	0.338507i	0.985573	+	0.169253i	$0.0541357\pi$
$863$	9.94427i	0.338507i	−0.985573	+	0.169253i	$0.945864\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 10.6180i	− 0.360607i
$868$	0	0
$869$	2.85410	0.0968188
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 1.87539i	− 0.0634723i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.5279i	0.997085i	0.866865	+	0.498543i	$0.166131\pi$
$877$	29.5279i	0.997085i	−0.866865	+	0.498543i	$0.833869\pi$
$878$	0	0
$879$	−32.1803	−1.08542
$880$	0	0
$881$	−13.6738	−0.460681	−0.230340	−	0.973110i	$-0.573984\pi$
$881$	−13.6738	−0.460681	−0.230340	+	0.973110i	$0.573984\pi$
$882$	0	0
$883$	− 19.3607i	− 0.651539i	−0.945449	−	0.325769i	$-0.894377\pi$
$883$	− 19.3607i	− 0.651539i	0.945449	−	0.325769i	$-0.105623\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.2361i	0.948074i	0.880505	+	0.474037i	$0.157204\pi$
$887$	28.2361i	0.948074i	−0.880505	+	0.474037i	$0.842796\pi$
$888$	0	0
$889$	0.236068	0.00791747
$890$	0	0
$891$	−7.70820	−0.258235
$892$	0	0
$893$	25.7639i	0.862157i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	22.8885i	0.764226i
$898$	0	0
$899$	18.5623	0.619088
$900$	0	0
$901$	−18.7082	−0.623261
$902$	0	0
$903$	8.47214i	0.281935i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.8328i	0.492516i	0.969204	+	0.246258i	$0.0792010\pi$
$907$	14.8328i	0.492516i	−0.969204	+	0.246258i	$0.920799\pi$
$908$	0	0
$909$	−2.88854	−0.0958070
$910$	0	0
$911$	55.3050	1.83233	0.916167	−	0.400796i	$-0.131266\pi$
$911$	55.3050	1.83233	0.916167	+	0.400796i	$0.131266\pi$
$912$	0	0
$913$	1.14590i	0.0379237i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.32624i	− 0.307980i
$918$	0	0
$919$	−52.3607	−1.72722	−0.863610	−	0.504161i	$-0.831802\pi$
$919$	−52.3607	−1.72722	−0.863610	+	0.504161i	$0.831802\pi$
$920$	0	0
$921$	7.85410	0.258801
$922$	0	0
$923$	0.652476i	0.0214765i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.43769i	0.211442i
$928$	0	0
$929$	23.1803	0.760522	0.380261	−	0.924879i	$-0.375834\pi$
$929$	23.1803	0.760522	0.380261	+	0.924879i	$0.375834\pi$
$930$	0	0
$931$	−36.2148	−1.18689
$932$	0	0
$933$	− 17.6180i	− 0.576789i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 35.0557i	− 1.14522i	−0.819828	−	0.572610i	$-0.805931\pi$
$937$	− 35.0557i	− 1.14522i	0.819828	−	0.572610i	$-0.194069\pi$
$938$	0	0
$939$	−44.8328	−1.46306
$940$	0	0
$941$	−21.5836	−0.703605	−0.351802	−	0.936074i	$-0.614431\pi$
$941$	−21.5836	−0.703605	−0.351802	+	0.936074i	$0.614431\pi$
$942$	0	0
$943$	− 50.2574i	− 1.63660i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 11.2918i	− 0.366934i	−0.983026	−	0.183467i	$-0.941268\pi$
$947$	− 11.2918i	− 0.366934i	0.983026	−	0.183467i	$-0.0587321\pi$
$948$	0	0
$949$	15.8541	0.514646
$950$	0	0
$951$	6.90983	0.224067
$952$	0	0
$953$	− 19.5279i	− 0.632570i	−0.948664	−	0.316285i	$-0.897564\pi$
$953$	− 19.5279i	− 0.632570i	0.948664	−	0.316285i	$-0.102436\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.09017i	0.229193i
$958$	0	0
$959$	−3.94427	−0.127367
$960$	0	0
$961$	−13.0557	−0.421153
$962$	0	0
$963$	− 4.56231i	− 0.147018i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.1591i	0.776903i	0.921469	+	0.388451i	$0.126990\pi$
$967$	24.1591i	0.776903i	−0.921469	+	0.388451i	$0.873010\pi$
$968$	0	0
$969$	−42.9787	−1.38068
$970$	0	0
$971$	−3.85410	−0.123684	−0.0618420	−	0.998086i	$-0.519697\pi$
$971$	−3.85410	−0.123684	−0.0618420	+	0.998086i	$0.519697\pi$
$972$	0	0
$973$	4.32624i	0.138693i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.5967i	0.339020i	0.985528	+	0.169510i	$0.0542185\pi$
$977$	10.5967i	0.339020i	−0.985528	+	0.169510i	$0.945781\pi$
$978$	0	0
$979$	12.5623	0.401493
$980$	0	0
$981$	−2.21478	−0.0707125
$982$	0	0
$983$	− 30.2492i	− 0.964800i	−0.875951	−	0.482400i	$-0.839765\pi$
$983$	− 30.2492i	− 0.964800i	0.875951	−	0.482400i	$-0.160235\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.70820i	− 0.149864i
$988$	0	0
$989$	53.5967	1.70428
$990$	0	0
$991$	−48.5623	−1.54263	−0.771316	−	0.636452i	$-0.780401\pi$
$991$	−48.5623	−1.54263	−0.771316	+	0.636452i	$0.780401\pi$
$992$	0	0
$993$	− 45.9787i	− 1.45909i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 7.97871i	− 0.252688i	−0.991986	−	0.126344i	$-0.959676\pi$
$997$	− 7.97871i	− 0.252688i	0.991986	−	0.126344i	$-0.0403244\pi$
$998$	0	0
$999$	9.65248	0.305391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.ba.4049.4		4
4.3	odd	2		2200.2.b.j.1849.1		4
5.2	odd	4		4400.2.a.bq.1.2		2
5.3	odd	4		4400.2.a.bk.1.1		2
5.4	even	2	inner	4400.2.b.ba.4049.1		4
20.3	even	4		2200.2.a.r.1.2	yes	2
20.7	even	4		2200.2.a.n.1.1	✓	2
20.19	odd	2		2200.2.b.j.1849.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.n.1.1	✓	2	20.7	even	4
2200.2.a.r.1.2	yes	2	20.3	even	4
2200.2.b.j.1849.1		4	4.3	odd	2
2200.2.b.j.1849.4		4	20.19	odd	2
4400.2.a.bk.1.1		2	5.3	odd	4
4400.2.a.bq.1.2		2	5.2	odd	4
4400.2.b.ba.4049.1		4	5.4	even	2	inner
4400.2.b.ba.4049.4		4	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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.61803i q^{3} -0.618034i q^{7} +0.381966 q^{9} +O(q^{10})\)	\(q+1.61803i q^{3} -0.618034i q^{7} +0.381966 q^{9} +1.00000 q^{11} +2.23607i q^{13} -4.85410i q^{17} -5.47214 q^{19} +1.00000 q^{21} -6.32624i q^{23} +5.47214i q^{27} +4.38197 q^{29} +4.23607 q^{31} +1.61803i q^{33} -1.76393i q^{37} -3.61803 q^{39} +7.94427 q^{41} +8.47214i q^{43} -4.70820i q^{47} +6.61803 q^{49} +7.85410 q^{51} -3.85410i q^{53} -8.85410i q^{57} -3.76393 q^{59} +7.09017 q^{61} -0.236068i q^{63} +10.2361 q^{69} +0.291796 q^{71} -7.09017i q^{73} -0.618034i q^{77} +2.85410 q^{79} -7.70820 q^{81} +1.14590i q^{83} +7.09017i q^{87} +12.5623 q^{89} +1.38197 q^{91} +6.85410i q^{93} -4.90983i q^{97} +0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{9}+O(q^{10}) \) 4 * q + 6 * q^9	\( 4 q + 6 q^{9} + 4 q^{11} - 4 q^{19} + 4 q^{21} + 22 q^{29} + 8 q^{31} - 10 q^{39} - 4 q^{41} + 22 q^{49} + 18 q^{51} - 24 q^{59} + 6 q^{61} + 32 q^{69} + 28 q^{71} - 2 q^{79} - 4 q^{81} + 10 q^{89} + 10 q^{91} + 6 q^{99}+O(q^{100}) \) 4 * q + 6 * q^9 + 4 * q^11 - 4 * q^19 + 4 * q^21 + 22 * q^29 + 8 * q^31 - 10 * q^39 - 4 * q^41 + 22 * q^49 + 18 * q^51 - 24 * q^59 + 6 * q^61 + 32 * q^69 + 28 * q^71 - 2 * q^79 - 4 * q^81 + 10 * q^89 + 10 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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