LMFDB - Embedded newform 4400.2.b.ba.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:	$\Q(i, \sqrt{5})$
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			4049.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.ba.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +O(q^{10})$	$q-0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +1.00000 q^{11} -2.23607i q^{13} +1.85410i q^{17} +3.47214 q^{19} +1.00000 q^{21} +9.32624i q^{23} -3.47214i q^{27} +6.61803 q^{29} -0.236068 q^{31} -0.618034i q^{33} -6.23607i q^{37} -1.38197 q^{39} -9.94427 q^{41} -0.472136i q^{43} +8.70820i q^{47} +4.38197 q^{49} +1.14590 q^{51} +2.85410i q^{53} -2.14590i q^{57} -8.23607 q^{59} -4.09017 q^{61} +4.23607i q^{63} +5.76393 q^{69} +13.7082 q^{71} +4.09017i q^{73} +1.61803i q^{77} -3.85410 q^{79} +5.70820 q^{81} +7.85410i q^{83} -4.09017i q^{87} -7.56231 q^{89} +3.61803 q^{91} +0.145898i q^{93} -16.0902i q^{97} +2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 0.618034i	− 0.356822i	−0.983956	−	0.178411i	$-0.942904\pi$
$3$	− 0.618034i	− 0.356822i	0.983956	−	0.178411i	$-0.0570957\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.61803i	0.611559i	0.952102	+	0.305780i	$0.0989171\pi$
$7$	1.61803i	0.611559i	−0.952102	+	0.305780i	$0.901083\pi$
$8$	0	0
$9$	2.61803	0.872678
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 2.23607i	− 0.620174i	−0.950708	−	0.310087i	$-0.899642\pi$
$13$	− 2.23607i	− 0.620174i	0.950708	−	0.310087i	$-0.100358\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.85410i	0.449686i	0.974395	+	0.224843i	$0.0721869\pi$
$17$	1.85410i	0.449686i	−0.974395	+	0.224843i	$0.927813\pi$
$18$	0	0
$19$	3.47214	0.796563	0.398281	−	0.917263i	$-0.369607\pi$
$19$	3.47214	0.796563	0.398281	+	0.917263i	$0.369607\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	9.32624i	1.94466i	0.233622	+	0.972328i	$0.424942\pi$
$23$	9.32624i	1.94466i	−0.233622	+	0.972328i	$0.575058\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.47214i	− 0.668213i
$28$	0	0
$29$	6.61803	1.22894	0.614469	−	0.788941i	$-0.289370\pi$
$29$	6.61803	1.22894	0.614469	+	0.788941i	$0.289370\pi$
$30$	0	0
$31$	−0.236068	−0.0423991	−0.0211995	−	0.999775i	$-0.506749\pi$
$31$	−0.236068	−0.0423991	−0.0211995	+	0.999775i	$0.506749\pi$
$32$	0	0
$33$	− 0.618034i	− 0.107586i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.23607i	− 1.02520i	−0.858627	−	0.512602i	$-0.828682\pi$
$37$	− 6.23607i	− 1.02520i	0.858627	−	0.512602i	$-0.171318\pi$
$38$	0	0
$39$	−1.38197	−0.221292
$40$	0	0
$41$	−9.94427	−1.55303	−0.776517	−	0.630096i	$-0.783015\pi$
$41$	−9.94427	−1.55303	−0.776517	+	0.630096i	$0.783015\pi$
$42$	0	0
$43$	− 0.472136i	− 0.0720001i	−0.999352	−	0.0360000i	$-0.988538\pi$
$43$	− 0.472136i	− 0.0720001i	0.999352	−	0.0360000i	$-0.0114616\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.70820i	1.27022i	0.772421	+	0.635111i	$0.219046\pi$
$47$	8.70820i	1.27022i	−0.772421	+	0.635111i	$0.780954\pi$
$48$	0	0
$49$	4.38197	0.625995
$50$	0	0
$51$	1.14590	0.160458
$52$	0	0
$53$	2.85410i	0.392041i	0.980600	+	0.196021i	$0.0628019\pi$
$53$	2.85410i	0.392041i	−0.980600	+	0.196021i	$0.937198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.14590i	− 0.284231i
$58$	0	0
$59$	−8.23607	−1.07224	−0.536122	−	0.844140i	$-0.680111\pi$
$59$	−8.23607	−1.07224	−0.536122	+	0.844140i	$0.680111\pi$
$60$	0	0
$61$	−4.09017	−0.523693	−0.261846	−	0.965110i	$-0.584331\pi$
$61$	−4.09017	−0.523693	−0.261846	+	0.965110i	$0.584331\pi$
$62$	0	0
$63$	4.23607i	0.533694i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	5.76393	0.693896
$70$	0	0
$71$	13.7082	1.62686	0.813432	−	0.581660i	$-0.197596\pi$
$71$	13.7082	1.62686	0.813432	+	0.581660i	$0.197596\pi$
$72$	0	0
$73$	4.09017i	0.478718i	0.970931	+	0.239359i	$0.0769373\pi$
$73$	4.09017i	0.478718i	−0.970931	+	0.239359i	$0.923063\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.61803i	0.184392i
$78$	0	0
$79$	−3.85410	−0.433620	−0.216810	−	0.976214i	$-0.569565\pi$
$79$	−3.85410	−0.433620	−0.216810	+	0.976214i	$0.569565\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	7.85410i	0.862100i	0.902328	+	0.431050i	$0.141857\pi$
$83$	7.85410i	0.862100i	−0.902328	+	0.431050i	$0.858143\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.09017i	− 0.438512i
$88$	0	0
$89$	−7.56231	−0.801603	−0.400801	−	0.916165i	$-0.631268\pi$
$89$	−7.56231	−0.801603	−0.400801	+	0.916165i	$0.631268\pi$
$90$	0	0
$91$	3.61803	0.379273
$92$	0	0
$93$	0.145898i	0.0151289i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 16.0902i	− 1.63371i	−0.576844	−	0.816855i	$-0.695716\pi$
$97$	− 16.0902i	− 1.63371i	0.576844	−	0.816855i	$-0.304284\pi$
$98$	0	0
$99$	2.61803	0.263122
$100$	0	0
$101$	12.5623	1.25000	0.624998	−	0.780626i	$-0.285100\pi$
$101$	12.5623	1.25000	0.624998	+	0.780626i	$0.285100\pi$
$102$	0	0
$103$	10.1459i	0.999705i	0.866111	+	0.499853i	$0.166612\pi$
$103$	10.1459i	0.999705i	−0.866111	+	0.499853i	$0.833388\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.94427i	0.574654i	0.957832	+	0.287327i	$0.0927667\pi$
$107$	5.94427i	0.574654i	−0.957832	+	0.287327i	$0.907233\pi$
$108$	0	0
$109$	18.7984	1.80056	0.900279	−	0.435314i	$-0.143363\pi$
$109$	18.7984	1.80056	0.900279	+	0.435314i	$0.143363\pi$
$110$	0	0
$111$	−3.85410	−0.365815
$112$	0	0
$113$	− 4.23607i	− 0.398496i	−0.979949	−	0.199248i	$-0.936150\pi$
$113$	− 4.23607i	− 0.398496i	0.979949	−	0.199248i	$-0.0638499\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 5.85410i	− 0.541212i
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.14590i	0.554157i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.61803i	0.232313i	0.993231	+	0.116156i	$0.0370574\pi$
$127$	2.61803i	0.232313i	−0.993231	+	0.116156i	$0.962943\pi$
$128$	0	0
$129$	−0.291796	−0.0256912
$130$	0	0
$131$	3.90983	0.341603	0.170802	−	0.985305i	$-0.445364\pi$
$131$	3.90983	0.341603	0.170802	+	0.985305i	$0.445364\pi$
$132$	0	0
$133$	5.61803i	0.487145i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.61803i	− 0.736288i	−0.929769	−	0.368144i	$-0.879993\pi$
$137$	− 8.61803i	− 0.736288i	0.929769	−	0.368144i	$-0.120007\pi$
$138$	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
$141$	5.38197	0.453243
$142$	0	0
$143$	− 2.23607i	− 0.186989i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.70820i	− 0.223369i
$148$	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
$151$	−17.6525	−1.43654	−0.718269	−	0.695765i	$-0.755065\pi$
$151$	−17.6525	−1.43654	−0.718269	+	0.695765i	$0.755065\pi$
$152$	0	0
$153$	4.85410i	0.392431i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.41641i	0.751511i	0.926719	+	0.375756i	$0.122617\pi$
$157$	9.41641i	0.751511i	−0.926719	+	0.375756i	$0.877383\pi$
$158$	0	0
$159$	1.76393	0.139889
$160$	0	0
$161$	−15.0902	−1.18927
$162$	0	0
$163$	17.6180i	1.37995i	0.723833	+	0.689975i	$0.242379\pi$
$163$	17.6180i	1.37995i	−0.723833	+	0.689975i	$0.757621\pi$
$164$	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
$171$	9.09017	0.695143
$172$	0	0
$173$	20.8885i	1.58813i	0.607835	+	0.794063i	$0.292038\pi$
$173$	20.8885i	1.58813i	−0.607835	+	0.794063i	$0.707962\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.09017i	0.382601i
$178$	0	0
$179$	−1.90983	−0.142747	−0.0713737	−	0.997450i	$-0.522738\pi$
$179$	−1.90983	−0.142747	−0.0713737	+	0.997450i	$0.522738\pi$
$180$	0	0
$181$	1.90983	0.141957	0.0709783	−	0.997478i	$-0.477388\pi$
$181$	1.90983	0.141957	0.0709783	+	0.997478i	$0.477388\pi$
$182$	0	0
$183$	2.52786i	0.186865i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.85410i	0.135585i
$188$	0	0
$189$	5.61803	0.408652
$190$	0	0
$191$	12.6180	0.913009	0.456504	−	0.889721i	$-0.349101\pi$
$191$	12.6180	0.913009	0.456504	+	0.889721i	$0.349101\pi$
$192$	0	0
$193$	− 4.94427i	− 0.355896i	−0.984040	−	0.177948i	$-0.943054\pi$
$193$	− 4.94427i	− 0.355896i	0.984040	−	0.177948i	$-0.0569460\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.43769i	0.102431i	0.998688	+	0.0512157i	$0.0163096\pi$
$197$	1.43769i	0.102431i	−0.998688	+	0.0512157i	$0.983690\pi$
$198$	0	0
$199$	0.437694	0.0310273	0.0155137	−	0.999880i	$-0.495062\pi$
$199$	0.437694	0.0310273	0.0155137	+	0.999880i	$0.495062\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.7082i	0.751569i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	24.4164i	1.69706i
$208$	0	0
$209$	3.47214	0.240173
$210$	0	0
$211$	6.70820	0.461812	0.230906	−	0.972976i	$-0.425831\pi$
$211$	6.70820	0.461812	0.230906	+	0.972976i	$0.425831\pi$
$212$	0	0
$213$	− 8.47214i	− 0.580501i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.381966i	− 0.0259295i
$218$	0	0
$219$	2.52786	0.170817
$220$	0	0
$221$	4.14590	0.278883
$222$	0	0
$223$	17.1803i	1.15048i	0.817984	+	0.575240i	$0.195091\pi$
$223$	17.1803i	1.15048i	−0.817984	+	0.575240i	$0.804909\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.7984i	− 1.11495i	−0.830195	−	0.557474i	$-0.811771\pi$
$227$	− 16.7984i	− 1.11495i	0.830195	−	0.557474i	$-0.188229\pi$
$228$	0	0
$229$	−10.5066	−0.694294	−0.347147	−	0.937811i	$-0.612850\pi$
$229$	−10.5066	−0.694294	−0.347147	+	0.937811i	$0.612850\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	24.8541i	1.62825i	0.580692	+	0.814123i	$0.302782\pi$
$233$	24.8541i	1.62825i	−0.580692	+	0.814123i	$0.697218\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.38197i	0.154725i
$238$	0	0
$239$	9.90983	0.641014	0.320507	−	0.947246i	$-0.396147\pi$
$239$	9.90983	0.641014	0.320507	+	0.947246i	$0.396147\pi$
$240$	0	0
$241$	12.0344	0.775207	0.387603	−	0.921826i	$-0.373303\pi$
$241$	12.0344	0.775207	0.387603	+	0.921826i	$0.373303\pi$
$242$	0	0
$243$	− 13.9443i	− 0.894525i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 7.76393i	− 0.494007i
$248$	0	0
$249$	4.85410	0.307616
$250$	0	0
$251$	13.8541	0.874463	0.437232	−	0.899349i	$-0.355959\pi$
$251$	13.8541	0.874463	0.437232	+	0.899349i	$0.355959\pi$
$252$	0	0
$253$	9.32624i	0.586336i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.9443i	− 1.18171i	−0.806777	−	0.590856i	$-0.798790\pi$
$257$	− 18.9443i	− 1.18171i	0.806777	−	0.590856i	$-0.201210\pi$
$258$	0	0
$259$	10.0902	0.626973
$260$	0	0
$261$	17.3262	1.07247
$262$	0	0
$263$	− 14.1246i	− 0.870961i	−0.900198	−	0.435480i	$-0.856579\pi$
$263$	− 14.1246i	− 0.870961i	0.900198	−	0.435480i	$-0.143421\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.67376i	0.286030i
$268$	0	0
$269$	−0.798374	−0.0486777	−0.0243389	−	0.999704i	$-0.507748\pi$
$269$	−0.798374	−0.0486777	−0.0243389	+	0.999704i	$0.507748\pi$
$270$	0	0
$271$	8.41641	0.511260	0.255630	−	0.966775i	$-0.417717\pi$
$271$	8.41641	0.511260	0.255630	+	0.966775i	$0.417717\pi$
$272$	0	0
$273$	− 2.23607i	− 0.135333i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.94427i	− 0.116820i	−0.998293	−	0.0584100i	$-0.981397\pi$
$277$	− 1.94427i	− 0.116820i	0.998293	−	0.0584100i	$-0.0186031\pi$
$278$	0	0
$279$	−0.618034	−0.0370007
$280$	0	0
$281$	13.3607	0.797031	0.398516	−	0.917162i	$-0.369525\pi$
$281$	13.3607	0.797031	0.398516	+	0.917162i	$0.369525\pi$
$282$	0	0
$283$	− 19.8541i	− 1.18020i	−0.807329	−	0.590102i	$-0.799088\pi$
$283$	− 19.8541i	− 1.18020i	0.807329	−	0.590102i	$-0.200912\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 16.0902i	− 0.949773i
$288$	0	0
$289$	13.5623	0.797783
$290$	0	0
$291$	−9.94427	−0.582944
$292$	0	0
$293$	− 15.8885i	− 0.928219i	−0.885778	−	0.464109i	$-0.846374\pi$
$293$	− 15.8885i	− 0.928219i	0.885778	−	0.464109i	$-0.153626\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.47214i	− 0.201474i
$298$	0	0
$299$	20.8541	1.20602
$300$	0	0
$301$	0.763932	0.0440323
$302$	0	0
$303$	− 7.76393i	− 0.446026i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.85410i	0.105819i	0.998599	+	0.0529096i	$0.0168495\pi$
$307$	1.85410i	0.105819i	−0.998599	+	0.0529096i	$0.983150\pi$
$308$	0	0
$309$	6.27051	0.356717
$310$	0	0
$311$	24.8885	1.41130	0.705650	−	0.708561i	$-0.250655\pi$
$311$	24.8885	1.41130	0.705650	+	0.708561i	$0.250655\pi$
$312$	0	0
$313$	14.2918i	0.807820i	0.914799	+	0.403910i	$0.132349\pi$
$313$	14.2918i	0.807820i	−0.914799	+	0.403910i	$0.867651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.2705i	1.64400i	0.569491	+	0.821998i	$0.307140\pi$
$317$	29.2705i	1.64400i	−0.569491	+	0.821998i	$0.692860\pi$
$318$	0	0
$319$	6.61803	0.370539
$320$	0	0
$321$	3.67376	0.205049
$322$	0	0
$323$	6.43769i	0.358203i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 11.6180i	− 0.642479i
$328$	0	0
$329$	−14.0902	−0.776816
$330$	0	0
$331$	−1.58359	−0.0870421	−0.0435210	−	0.999053i	$-0.513858\pi$
$331$	−1.58359	−0.0870421	−0.0435210	+	0.999053i	$0.513858\pi$
$332$	0	0
$333$	− 16.3262i	− 0.894672i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 24.4721i	− 1.33308i	−0.745468	−	0.666541i	$-0.767774\pi$
$337$	− 24.4721i	− 1.33308i	0.745468	−	0.666541i	$-0.232226\pi$
$338$	0	0
$339$	−2.61803	−0.142192
$340$	0	0
$341$	−0.236068	−0.0127838
$342$	0	0
$343$	18.4164i	0.994393i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.2705i	− 0.873447i	−0.899596	−	0.436723i	$-0.856139\pi$
$347$	− 16.2705i	− 0.873447i	0.899596	−	0.436723i	$-0.143861\pi$
$348$	0	0
$349$	−35.4721	−1.89878	−0.949390	−	0.314100i	$-0.898297\pi$
$349$	−35.4721	−1.89878	−0.949390	+	0.314100i	$0.898297\pi$
$350$	0	0
$351$	−7.76393	−0.414408
$352$	0	0
$353$	9.58359i	0.510083i	0.966930	+	0.255042i	$0.0820892\pi$
$353$	9.58359i	0.510083i	−0.966930	+	0.255042i	$0.917911\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.85410i	0.0981295i
$358$	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
$361$	−6.94427	−0.365488
$362$	0	0
$363$	− 0.618034i	− 0.0324384i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.79837i	0.407072i	0.979067	+	0.203536i	$0.0652434\pi$
$367$	7.79837i	0.407072i	−0.979067	+	0.203536i	$0.934757\pi$
$368$	0	0
$369$	−26.0344	−1.35530
$370$	0	0
$371$	−4.61803	−0.239756
$372$	0	0
$373$	− 8.70820i	− 0.450894i	−0.974255	−	0.225447i	$-0.927616\pi$
$373$	− 8.70820i	− 0.450894i	0.974255	−	0.225447i	$-0.0723842\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.7984i	− 0.762155i
$378$	0	0
$379$	36.3050	1.86486	0.932430	−	0.361351i	$-0.117684\pi$
$379$	36.3050	1.86486	0.932430	+	0.361351i	$0.117684\pi$
$380$	0	0
$381$	1.61803	0.0828944
$382$	0	0
$383$	− 2.58359i	− 0.132015i	−0.997819	−	0.0660077i	$-0.978974\pi$
$383$	− 2.58359i	− 0.132015i	0.997819	−	0.0660077i	$-0.0210262\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.23607i	− 0.0628329i
$388$	0	0
$389$	26.4721	1.34219	0.671095	−	0.741371i	$-0.265824\pi$
$389$	26.4721	1.34219	0.671095	+	0.741371i	$0.265824\pi$
$390$	0	0
$391$	−17.2918	−0.874484
$392$	0	0
$393$	− 2.41641i	− 0.121892i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.1459i	1.01109i	0.862799	+	0.505547i	$0.168709\pi$
$397$	20.1459i	1.01109i	−0.862799	+	0.505547i	$0.831291\pi$
$398$	0	0
$399$	3.47214	0.173824
$400$	0	0
$401$	−18.3607	−0.916889	−0.458444	−	0.888723i	$-0.651593\pi$
$401$	−18.3607	−0.916889	−0.458444	+	0.888723i	$0.651593\pi$
$402$	0	0
$403$	0.527864i	0.0262948i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.23607i	− 0.309110i
$408$	0	0
$409$	−34.1246	−1.68735	−0.843677	−	0.536852i	$-0.819614\pi$
$409$	−34.1246	−1.68735	−0.843677	+	0.536852i	$0.819614\pi$
$410$	0	0
$411$	−5.32624	−0.262724
$412$	0	0
$413$	− 13.3262i	− 0.655741i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.32624i	0.211857i
$418$	0	0
$419$	34.8885	1.70442	0.852208	−	0.523202i	$-0.175263\pi$
$419$	34.8885	1.70442	0.852208	+	0.523202i	$0.175263\pi$
$420$	0	0
$421$	8.09017	0.394291	0.197145	−	0.980374i	$-0.436833\pi$
$421$	8.09017	0.394291	0.197145	+	0.980374i	$0.436833\pi$
$422$	0	0
$423$	22.7984i	1.10849i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.61803i	− 0.320269i
$428$	0	0
$429$	−1.38197	−0.0667219
$430$	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
$433$	38.7771i	1.86351i	0.363090	+	0.931754i	$0.381722\pi$
$433$	38.7771i	1.86351i	−0.363090	+	0.931754i	$0.618278\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	32.3820i	1.54904i
$438$	0	0
$439$	7.67376	0.366249	0.183124	−	0.983090i	$-0.441379\pi$
$439$	7.67376	0.366249	0.183124	+	0.983090i	$0.441379\pi$
$440$	0	0
$441$	11.4721	0.546292
$442$	0	0
$443$	− 27.7082i	− 1.31646i	−0.752818	−	0.658228i	$-0.771306\pi$
$443$	− 27.7082i	− 1.31646i	0.752818	−	0.658228i	$-0.228694\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 7.41641i	− 0.350784i
$448$	0	0
$449$	38.1591	1.80084	0.900419	−	0.435025i	$-0.143260\pi$
$449$	38.1591	1.80084	0.900419	+	0.435025i	$0.143260\pi$
$450$	0	0
$451$	−9.94427	−0.468257
$452$	0	0
$453$	10.9098i	0.512589i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 24.5623i	− 1.14898i	−0.818513	−	0.574488i	$-0.805201\pi$
$457$	− 24.5623i	− 1.14898i	0.818513	−	0.574488i	$-0.194799\pi$
$458$	0	0
$459$	6.43769	0.300486
$460$	0	0
$461$	22.1246	1.03045	0.515223	−	0.857056i	$-0.327709\pi$
$461$	22.1246	1.03045	0.515223	+	0.857056i	$0.327709\pi$
$462$	0	0
$463$	− 5.29180i	− 0.245931i	−0.992411	−	0.122965i	$-0.960760\pi$
$463$	− 5.29180i	− 0.245931i	0.992411	−	0.122965i	$-0.0392404\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.34752i	− 0.108630i	−0.998524	−	0.0543152i	$-0.982702\pi$
$467$	− 2.34752i	− 0.108630i	0.998524	−	0.0543152i	$-0.0172976\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.81966	0.268156
$472$	0	0
$473$	− 0.472136i	− 0.0217088i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.47214i	0.342126i
$478$	0	0
$479$	−36.5967	−1.67215	−0.836074	−	0.548617i	$-0.815155\pi$
$479$	−36.5967	−1.67215	−0.836074	+	0.548617i	$0.815155\pi$
$480$	0	0
$481$	−13.9443	−0.635804
$482$	0	0
$483$	9.32624i	0.424359i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7.18034i	− 0.325372i	−0.986678	−	0.162686i	$-0.947984\pi$
$487$	− 7.18034i	− 0.325372i	0.986678	−	0.162686i	$-0.0520158\pi$
$488$	0	0
$489$	10.8885	0.492397
$490$	0	0
$491$	−29.8328	−1.34634	−0.673168	−	0.739490i	$-0.735067\pi$
$491$	−29.8328	−1.34634	−0.673168	+	0.739490i	$0.735067\pi$
$492$	0	0
$493$	12.2705i	0.552636i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.1803i	0.994924i
$498$	0	0
$499$	−37.6869	−1.68710	−0.843549	−	0.537052i	$-0.819538\pi$
$499$	−37.6869	−1.68710	−0.843549	+	0.537052i	$0.819538\pi$
$500$	0	0
$501$	−5.56231	−0.248506
$502$	0	0
$503$	− 32.7639i	− 1.46087i	−0.682981	−	0.730436i	$-0.739317\pi$
$503$	− 32.7639i	− 1.46087i	0.682981	−	0.730436i	$-0.260683\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.94427i	− 0.219583i
$508$	0	0
$509$	−15.2148	−0.674383	−0.337192	−	0.941436i	$-0.609477\pi$
$509$	−15.2148	−0.674383	−0.337192	+	0.941436i	$0.609477\pi$
$510$	0	0
$511$	−6.61803	−0.292765
$512$	0	0
$513$	− 12.0557i	− 0.532273i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.70820i	0.382986i
$518$	0	0
$519$	12.9098	0.566679
$520$	0	0
$521$	−7.34752	−0.321901	−0.160950	−	0.986962i	$-0.551456\pi$
$521$	−7.34752	−0.321901	−0.160950	+	0.986962i	$0.551456\pi$
$522$	0	0
$523$	− 8.47214i	− 0.370461i	−0.982695	−	0.185230i	$-0.940697\pi$
$523$	− 8.47214i	− 0.370461i	0.982695	−	0.185230i	$-0.0593031\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 0.437694i	− 0.0190663i
$528$	0	0
$529$	−63.9787	−2.78168
$530$	0	0
$531$	−21.5623	−0.935724
$532$	0	0
$533$	22.2361i	0.963151i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.18034i	0.0509354i
$538$	0	0
$539$	4.38197	0.188745
$540$	0	0
$541$	−9.50658	−0.408720	−0.204360	−	0.978896i	$-0.565511\pi$
$541$	−9.50658	−0.408720	−0.204360	+	0.978896i	$0.565511\pi$
$542$	0	0
$543$	− 1.18034i	− 0.0506532i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 42.5623i	− 1.81983i	−0.414792	−	0.909916i	$-0.636146\pi$
$547$	− 42.5623i	− 1.81983i	0.414792	−	0.909916i	$-0.363854\pi$
$548$	0	0
$549$	−10.7082	−0.457015
$550$	0	0
$551$	22.9787	0.978926
$552$	0	0
$553$	− 6.23607i	− 0.265185i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.29180i	0.0547352i	0.999625	+	0.0273676i	$0.00871246\pi$
$557$	1.29180i	0.0547352i	−0.999625	+	0.0273676i	$0.991288\pi$
$558$	0	0
$559$	−1.05573	−0.0446525
$560$	0	0
$561$	1.14590	0.0483799
$562$	0	0
$563$	− 4.38197i	− 0.184678i	−0.995728	−	0.0923389i	$-0.970566\pi$
$563$	− 4.38197i	− 0.184678i	0.995728	−	0.0923389i	$-0.0294343\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.23607i	0.387878i
$568$	0	0
$569$	−44.0902	−1.84836	−0.924178	−	0.381962i	$-0.875249\pi$
$569$	−44.0902	−1.84836	−0.924178	+	0.381962i	$0.875249\pi$
$570$	0	0
$571$	14.4508	0.604749	0.302375	−	0.953189i	$-0.402221\pi$
$571$	14.4508	0.604749	0.302375	+	0.953189i	$0.402221\pi$
$572$	0	0
$573$	− 7.79837i	− 0.325782i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 7.20163i	− 0.299808i	−0.988701	−	0.149904i	$-0.952104\pi$
$577$	− 7.20163i	− 0.299808i	0.988701	−	0.149904i	$-0.0478964\pi$
$578$	0	0
$579$	−3.05573	−0.126992
$580$	0	0
$581$	−12.7082	−0.527225
$582$	0	0
$583$	2.85410i	0.118205i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.2705i	1.29067i	0.763899	+	0.645336i	$0.223283\pi$
$587$	31.2705i	1.29067i	−0.763899	+	0.645336i	$0.776717\pi$
$588$	0	0
$589$	−0.819660	−0.0337735
$590$	0	0
$591$	0.888544	0.0365498
$592$	0	0
$593$	28.8885i	1.18631i	0.805088	+	0.593155i	$0.202118\pi$
$593$	28.8885i	1.18631i	−0.805088	+	0.593155i	$0.797882\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 0.270510i	− 0.0110712i
$598$	0	0
$599$	14.9098	0.609199	0.304600	−	0.952480i	$-0.401477\pi$
$599$	14.9098	0.609199	0.304600	+	0.952480i	$0.401477\pi$
$600$	0	0
$601$	16.7426	0.682947	0.341473	−	0.939891i	$-0.389074\pi$
$601$	16.7426	0.682947	0.341473	+	0.939891i	$0.389074\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 26.9443i	− 1.09363i	−0.837252	−	0.546817i	$-0.815839\pi$
$607$	− 26.9443i	− 1.09363i	0.837252	−	0.546817i	$-0.184161\pi$
$608$	0	0
$609$	6.61803	0.268176
$610$	0	0
$611$	19.4721	0.787758
$612$	0	0
$613$	− 12.9098i	− 0.521423i	−0.965417	−	0.260712i	$-0.916043\pi$
$613$	− 12.9098i	− 0.521423i	0.965417	−	0.260712i	$-0.0839572\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.7639i	0.594374i	0.954819	+	0.297187i	$0.0960484\pi$
$617$	14.7639i	0.594374i	−0.954819	+	0.297187i	$0.903952\pi$
$618$	0	0
$619$	−34.7082	−1.39504	−0.697520	−	0.716565i	$-0.745713\pi$
$619$	−34.7082	−1.39504	−0.697520	+	0.716565i	$0.745713\pi$
$620$	0	0
$621$	32.3820	1.29944
$622$	0	0
$623$	− 12.2361i	− 0.490228i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.14590i	− 0.0856989i
$628$	0	0
$629$	11.5623	0.461019
$630$	0	0
$631$	−5.50658	−0.219213	−0.109607	−	0.993975i	$-0.534959\pi$
$631$	−5.50658	−0.219213	−0.109607	+	0.993975i	$0.534959\pi$
$632$	0	0
$633$	− 4.14590i	− 0.164785i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.79837i	− 0.388226i
$638$	0	0
$639$	35.8885	1.41973
$640$	0	0
$641$	−36.8885	−1.45701	−0.728505	−	0.685041i	$-0.759784\pi$
$641$	−36.8885	−1.45701	−0.728505	+	0.685041i	$0.759784\pi$
$642$	0	0
$643$	26.9443i	1.06258i	0.847191	+	0.531289i	$0.178292\pi$
$643$	26.9443i	1.06258i	−0.847191	+	0.531289i	$0.821708\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.94427i	0.155065i	0.996990	+	0.0775327i	$0.0247042\pi$
$647$	3.94427i	0.155065i	−0.996990	+	0.0775327i	$0.975296\pi$
$648$	0	0
$649$	−8.23607	−0.323294
$650$	0	0
$651$	−0.236068	−0.00925223
$652$	0	0
$653$	− 3.67376i	− 0.143765i	−0.997413	−	0.0718827i	$-0.977099\pi$
$653$	− 3.67376i	− 0.143765i	0.997413	−	0.0718827i	$-0.0229007\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.7082i	0.417767i
$658$	0	0
$659$	7.20163	0.280536	0.140268	−	0.990114i	$-0.455204\pi$
$659$	7.20163	0.280536	0.140268	+	0.990114i	$0.455204\pi$
$660$	0	0
$661$	22.5967	0.878912	0.439456	−	0.898264i	$-0.355171\pi$
$661$	22.5967	0.878912	0.439456	+	0.898264i	$0.355171\pi$
$662$	0	0
$663$	− 2.56231i	− 0.0995117i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	61.7214i	2.38986i
$668$	0	0
$669$	10.6180	0.410517
$670$	0	0
$671$	−4.09017	−0.157899
$672$	0	0
$673$	6.52786i	0.251631i	0.992054	+	0.125815i	$0.0401547\pi$
$673$	6.52786i	0.251631i	−0.992054	+	0.125815i	$0.959845\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.2361i	1.43110i	0.698562	+	0.715549i	$0.253823\pi$
$677$	37.2361i	1.43110i	−0.698562	+	0.715549i	$0.746177\pi$
$678$	0	0
$679$	26.0344	0.999110
$680$	0	0
$681$	−10.3820	−0.397838
$682$	0	0
$683$	− 40.2361i	− 1.53959i	−0.638291	−	0.769795i	$-0.720358\pi$
$683$	− 40.2361i	− 1.53959i	0.638291	−	0.769795i	$-0.279642\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.49342i	0.247740i
$688$	0	0
$689$	6.38197	0.243134
$690$	0	0
$691$	−25.7984	−0.981416	−0.490708	−	0.871324i	$-0.663262\pi$
$691$	−25.7984	−0.981416	−0.490708	+	0.871324i	$0.663262\pi$
$692$	0	0
$693$	4.23607i	0.160915i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.4377i	− 0.698377i
$698$	0	0
$699$	15.3607	0.580994
$700$	0	0
$701$	14.4721	0.546605	0.273303	−	0.961928i	$-0.411884\pi$
$701$	14.4721	0.546605	0.273303	+	0.961928i	$0.411884\pi$
$702$	0	0
$703$	− 21.6525i	− 0.816639i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.3262i	0.764447i
$708$	0	0
$709$	28.3607	1.06511	0.532554	−	0.846396i	$-0.321232\pi$
$709$	28.3607	1.06511	0.532554	+	0.846396i	$0.321232\pi$
$710$	0	0
$711$	−10.0902	−0.378411
$712$	0	0
$713$	− 2.20163i	− 0.0824515i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.12461i	− 0.228728i
$718$	0	0
$719$	−24.0557	−0.897127	−0.448564	−	0.893751i	$-0.648064\pi$
$719$	−24.0557	−0.897127	−0.448564	+	0.893751i	$0.648064\pi$
$720$	0	0
$721$	−16.4164	−0.611379
$722$	0	0
$723$	− 7.43769i	− 0.276611i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3.49342i	− 0.129564i	−0.997899	−	0.0647819i	$-0.979365\pi$
$727$	− 3.49342i	− 0.129564i	0.997899	−	0.0647819i	$-0.0206352\pi$
$728$	0	0
$729$	8.50658	0.315058
$730$	0	0
$731$	0.875388	0.0323774
$732$	0	0
$733$	− 36.5279i	− 1.34919i	−0.738189	−	0.674594i	$-0.764319\pi$
$733$	− 36.5279i	− 1.34919i	0.738189	−	0.674594i	$-0.235681\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	6.96556	0.256232	0.128116	−	0.991759i	$-0.459107\pi$
$739$	6.96556	0.256232	0.128116	+	0.991759i	$0.459107\pi$
$740$	0	0
$741$	−4.79837	−0.176273
$742$	0	0
$743$	− 41.0344i	− 1.50541i	−0.658359	−	0.752704i	$-0.728749\pi$
$743$	− 41.0344i	− 1.50541i	0.658359	−	0.752704i	$-0.271251\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.5623i	0.752335i
$748$	0	0
$749$	−9.61803	−0.351435
$750$	0	0
$751$	−35.1591	−1.28297	−0.641486	−	0.767135i	$-0.721682\pi$
$751$	−35.1591	−1.28297	−0.641486	+	0.767135i	$0.721682\pi$
$752$	0	0
$753$	− 8.56231i	− 0.312028i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.83282i	0.0666148i	0.999445	+	0.0333074i	$0.0106040\pi$
$757$	1.83282i	0.0666148i	−0.999445	+	0.0333074i	$0.989396\pi$
$758$	0	0
$759$	5.76393	0.209217
$760$	0	0
$761$	−2.11146	−0.0765402	−0.0382701	−	0.999267i	$-0.512185\pi$
$761$	−2.11146	−0.0765402	−0.0382701	+	0.999267i	$0.512185\pi$
$762$	0	0
$763$	30.4164i	1.10115i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.4164i	0.664978i
$768$	0	0
$769$	−12.0557	−0.434741	−0.217370	−	0.976089i	$-0.569748\pi$
$769$	−12.0557	−0.434741	−0.217370	+	0.976089i	$0.569748\pi$
$770$	0	0
$771$	−11.7082	−0.421661
$772$	0	0
$773$	− 44.0902i	− 1.58581i	−0.609343	−	0.792907i	$-0.708567\pi$
$773$	− 44.0902i	− 1.58581i	0.609343	−	0.792907i	$-0.291433\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.23607i	− 0.223718i
$778$	0	0
$779$	−34.5279	−1.23709
$780$	0	0
$781$	13.7082	0.490518
$782$	0	0
$783$	− 22.9787i	− 0.821192i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.4721i	0.836691i	0.908288	+	0.418346i	$0.137390\pi$
$787$	23.4721i	0.836691i	−0.908288	+	0.418346i	$0.862610\pi$
$788$	0	0
$789$	−8.72949	−0.310778
$790$	0	0
$791$	6.85410	0.243704
$792$	0	0
$793$	9.14590i	0.324780i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.7984i	1.02009i	0.860147	+	0.510045i	$0.170371\pi$
$797$	28.7984i	1.02009i	−0.860147	+	0.510045i	$0.829629\pi$
$798$	0	0
$799$	−16.1459	−0.571201
$800$	0	0
$801$	−19.7984	−0.699541
$802$	0	0
$803$	4.09017i	0.144339i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.493422i	0.0173693i
$808$	0	0
$809$	−15.4164	−0.542012	−0.271006	−	0.962578i	$-0.587356\pi$
$809$	−15.4164	−0.542012	−0.271006	+	0.962578i	$0.587356\pi$
$810$	0	0
$811$	−55.1803	−1.93764	−0.968822	−	0.247758i	$-0.920306\pi$
$811$	−55.1803	−1.93764	−0.968822	+	0.247758i	$0.920306\pi$
$812$	0	0
$813$	− 5.20163i	− 0.182429i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.63932i	− 0.0573526i
$818$	0	0
$819$	9.47214	0.330983
$820$	0	0
$821$	−36.3607	−1.26900	−0.634498	−	0.772924i	$-0.718793\pi$
$821$	−36.3607	−1.26900	−0.634498	+	0.772924i	$0.718793\pi$
$822$	0	0
$823$	30.4721i	1.06219i	0.847312	+	0.531096i	$0.178220\pi$
$823$	30.4721i	1.06219i	−0.847312	+	0.531096i	$0.821780\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 25.5967i	− 0.890086i	−0.895509	−	0.445043i	$-0.853188\pi$
$827$	− 25.5967i	− 0.890086i	0.895509	−	0.445043i	$-0.146812\pi$
$828$	0	0
$829$	31.3820	1.08994	0.544970	−	0.838455i	$-0.316541\pi$
$829$	31.3820	1.08994	0.544970	+	0.838455i	$0.316541\pi$
$830$	0	0
$831$	−1.20163	−0.0416839
$832$	0	0
$833$	8.12461i	0.281501i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.819660i	0.0283316i
$838$	0	0
$839$	−39.3262	−1.35769	−0.678846	−	0.734280i	$-0.737520\pi$
$839$	−39.3262	−1.35769	−0.678846	+	0.734280i	$0.737520\pi$
$840$	0	0
$841$	14.7984	0.510289
$842$	0	0
$843$	− 8.25735i	− 0.284398i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.61803i	0.0555963i
$848$	0	0
$849$	−12.2705	−0.421123
$850$	0	0
$851$	58.1591	1.99367
$852$	0	0
$853$	− 27.3820i	− 0.937541i	−0.883320	−	0.468770i	$-0.844697\pi$
$853$	− 27.3820i	− 0.937541i	0.883320	−	0.468770i	$-0.155303\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11.6525i	− 0.398041i	−0.979995	−	0.199020i	$-0.936224\pi$
$857$	− 11.6525i	− 0.398041i	0.979995	−	0.199020i	$-0.0637760\pi$
$858$	0	0
$859$	−41.4853	−1.41546	−0.707730	−	0.706483i	$-0.750281\pi$
$859$	−41.4853	−1.41546	−0.707730	+	0.706483i	$0.750281\pi$
$860$	0	0
$861$	−9.94427	−0.338900
$862$	0	0
$863$	− 7.94427i	− 0.270426i	−0.990817	−	0.135213i	$-0.956828\pi$
$863$	− 7.94427i	− 0.270426i	0.990817	−	0.135213i	$-0.0431719\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.38197i	− 0.284666i
$868$	0	0
$869$	−3.85410	−0.130741
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 42.1246i	− 1.42570i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.4721i	1.29911i	0.760314	+	0.649556i	$0.225045\pi$
$877$	38.4721i	1.29911i	−0.760314	+	0.649556i	$0.774955\pi$
$878$	0	0
$879$	−9.81966	−0.331209
$880$	0	0
$881$	−29.3262	−0.988026	−0.494013	−	0.869454i	$-0.664470\pi$
$881$	−29.3262	−0.988026	−0.494013	+	0.869454i	$0.664470\pi$
$882$	0	0
$883$	25.3607i	0.853455i	0.904380	+	0.426727i	$0.140334\pi$
$883$	25.3607i	0.853455i	−0.904380	+	0.426727i	$0.859666\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.7639i	0.797915i	0.916969	+	0.398957i	$0.130628\pi$
$887$	23.7639i	0.797915i	−0.916969	+	0.398957i	$0.869372\pi$
$888$	0	0
$889$	−4.23607	−0.142073
$890$	0	0
$891$	5.70820	0.191232
$892$	0	0
$893$	30.2361i	1.01181i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 12.8885i	− 0.430336i
$898$	0	0
$899$	−1.56231	−0.0521058
$900$	0	0
$901$	−5.29180	−0.176295
$902$	0	0
$903$	− 0.472136i	− 0.0157117i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 38.8328i	− 1.28942i	−0.764426	−	0.644711i	$-0.776978\pi$
$907$	− 38.8328i	− 1.28942i	0.764426	−	0.644711i	$-0.223022\pi$
$908$	0	0
$909$	32.8885	1.09084
$910$	0	0
$911$	−7.30495	−0.242024	−0.121012	−	0.992651i	$-0.538614\pi$
$911$	−7.30495	−0.242024	−0.121012	+	0.992651i	$0.538614\pi$
$912$	0	0
$913$	7.85410i	0.259933i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.32624i	0.208911i
$918$	0	0
$919$	−7.63932	−0.251998	−0.125999	−	0.992030i	$-0.540214\pi$
$919$	−7.63932	−0.251998	−0.125999	+	0.992030i	$0.540214\pi$
$920$	0	0
$921$	1.14590	0.0377586
$922$	0	0
$923$	− 30.6525i	− 1.00894i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	26.5623i	0.872421i
$928$	0	0
$929$	0.819660	0.0268922	0.0134461	−	0.999910i	$-0.495720\pi$
$929$	0.819660	0.0268922	0.0134461	+	0.999910i	$0.495720\pi$
$930$	0	0
$931$	15.2148	0.498644
$932$	0	0
$933$	− 15.3820i	− 0.503583i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 52.9443i	− 1.72961i	−0.502104	−	0.864807i	$-0.667441\pi$
$937$	− 52.9443i	− 1.72961i	0.502104	−	0.864807i	$-0.332559\pi$
$938$	0	0
$939$	8.83282	0.288248
$940$	0	0
$941$	−48.4164	−1.57833	−0.789165	−	0.614181i	$-0.789486\pi$
$941$	−48.4164	−1.57833	−0.789165	+	0.614181i	$0.789486\pi$
$942$	0	0
$943$	− 92.7426i	− 3.02012i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24.7082i	− 0.802909i	−0.915879	−	0.401454i	$-0.868505\pi$
$947$	− 24.7082i	− 0.802909i	0.915879	−	0.401454i	$-0.131495\pi$
$948$	0	0
$949$	9.14590	0.296888
$950$	0	0
$951$	18.0902	0.586614
$952$	0	0
$953$	− 28.4721i	− 0.922303i	−0.887321	−	0.461151i	$-0.847436\pi$
$953$	− 28.4721i	− 0.922303i	0.887321	−	0.461151i	$-0.152564\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4.09017i	− 0.132216i
$958$	0	0
$959$	13.9443	0.450284
$960$	0	0
$961$	−30.9443	−0.998202
$962$	0	0
$963$	15.5623i	0.501488i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 45.1591i	− 1.45222i	−0.687580	−	0.726109i	$-0.741327\pi$
$967$	− 45.1591i	− 1.45222i	0.687580	−	0.726109i	$-0.258673\pi$
$968$	0	0
$969$	3.97871	0.127815
$970$	0	0
$971$	2.85410	0.0915925	0.0457962	−	0.998951i	$-0.485418\pi$
$971$	2.85410	0.0915925	0.0457962	+	0.998951i	$0.485418\pi$
$972$	0	0
$973$	− 11.3262i	− 0.363103i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 38.5967i	− 1.23482i	−0.786642	−	0.617410i	$-0.788182\pi$
$977$	− 38.5967i	− 1.23482i	0.786642	−	0.617410i	$-0.211818\pi$
$978$	0	0
$979$	−7.56231	−0.241692
$980$	0	0
$981$	49.2148	1.57131
$982$	0	0
$983$	50.2492i	1.60270i	0.598195	+	0.801351i	$0.295885\pi$
$983$	50.2492i	1.60270i	−0.598195	+	0.801351i	$0.704115\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.70820i	0.277185i
$988$	0	0
$989$	4.40325	0.140015
$990$	0	0
$991$	−28.4377	−0.903353	−0.451677	−	0.892182i	$-0.649174\pi$
$991$	−28.4377	−0.903353	−0.451677	+	0.892182i	$0.649174\pi$
$992$	0	0
$993$	0.978714i	0.0310585i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.9787i	1.23447i	0.786780	+	0.617234i	$0.211747\pi$
$997$	38.9787i	1.23447i	−0.786780	+	0.617234i	$0.788253\pi$
$998$	0	0
$999$	−21.6525	−0.685054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.ba.4049.2		4
4.3	odd	2		2200.2.b.j.1849.3		4
5.2	odd	4		4400.2.a.bq.1.1		2
5.3	odd	4		4400.2.a.bk.1.2		2
5.4	even	2	inner	4400.2.b.ba.4049.3		4
20.3	even	4		2200.2.a.r.1.1	yes	2
20.7	even	4		2200.2.a.n.1.2	✓	2
20.19	odd	2		2200.2.b.j.1849.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.n.1.2	✓	2	20.7	even	4
2200.2.a.r.1.1	yes	2	20.3	even	4
2200.2.b.j.1849.2		4	20.19	odd	2
2200.2.b.j.1849.3		4	4.3	odd	2
4400.2.a.bk.1.2		2	5.3	odd	4
4400.2.a.bq.1.1		2	5.2	odd	4
4400.2.b.ba.4049.2		4	1.1	even	1	trivial
4400.2.b.ba.4049.3		4	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +O(q^{10})\)	\(q-0.618034i q^{3} +1.61803i q^{7} +2.61803 q^{9} +1.00000 q^{11} -2.23607i q^{13} +1.85410i q^{17} +3.47214 q^{19} +1.00000 q^{21} +9.32624i q^{23} -3.47214i q^{27} +6.61803 q^{29} -0.236068 q^{31} -0.618034i q^{33} -6.23607i q^{37} -1.38197 q^{39} -9.94427 q^{41} -0.472136i q^{43} +8.70820i q^{47} +4.38197 q^{49} +1.14590 q^{51} +2.85410i q^{53} -2.14590i q^{57} -8.23607 q^{59} -4.09017 q^{61} +4.23607i q^{63} +5.76393 q^{69} +13.7082 q^{71} +4.09017i q^{73} +1.61803i q^{77} -3.85410 q^{79} +5.70820 q^{81} +7.85410i q^{83} -4.09017i q^{87} -7.56231 q^{89} +3.61803 q^{91} +0.145898i q^{93} -16.0902i q^{97} +2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{9}+O(q^{10}) \) 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

Properties

Related objects

Downloads

Learn more