LMFDB - Embedded newform 4400.2.b.bc.4049.1

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:	$6$
Coefficient field:	6.0.44836416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 1 $ x^6 + 12x^4 + 36x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-2.36147i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.bc.4049.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36147i q^{3} -0.576535i q^{7} -8.29947 q^{9} +O(q^{10})$	$q-3.36147i q^{3} -0.576535i q^{7} -8.29947 q^{9} +1.00000 q^{11} +3.72294i q^{13} +6.51454i q^{17} +1.00000 q^{19} -1.93800 q^{21} -4.36147i q^{23} +17.8140i q^{27} -2.42347 q^{29} -2.15307 q^{31} -3.36147i q^{33} -3.21507i q^{37} +12.5145 q^{39} +5.93800 q^{41} +4.93800i q^{43} +12.6609i q^{47} +6.66761 q^{49} +21.8984 q^{51} +8.02241i q^{53} -3.36147i q^{57} -0.660941 q^{59} -10.4525 q^{61} +4.78493i q^{63} -5.87601i q^{67} -14.6609 q^{69} +15.2308 q^{71} -12.5765i q^{73} -0.576535i q^{77} +12.0844 q^{79} +34.9828 q^{81} +7.45254i q^{83} +8.14640i q^{87} -11.4525 q^{89} +2.14640 q^{91} +7.23748i q^{93} +14.6543i q^{97} -8.29947 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{9}+O(q^{10}) $ 6 * q - 12 * q^9	$ 6 q - 12 q^{9} + 6 q^{11} + 6 q^{19} + 12 q^{21} - 24 q^{29} + 6 q^{31} + 42 q^{39} + 12 q^{41} - 12 q^{49} + 18 q^{51} + 48 q^{59} - 6 q^{61} - 36 q^{69} + 30 q^{71} + 30 q^{79} + 54 q^{81} - 12 q^{89} - 6 q^{91} - 12 q^{99}+O(q^{100}) $ 6 * q - 12 * q^9 + 6 * q^11 + 6 * q^19 + 12 * q^21 - 24 * q^29 + 6 * q^31 + 42 * q^39 + 12 * q^41 - 12 * q^49 + 18 * q^51 + 48 * q^59 - 6 * q^61 - 36 * q^69 + 30 * q^71 + 30 * q^79 + 54 * q^81 - 12 * q^89 - 6 * q^91 - 12 * 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
$3$	− 3.36147i	− 1.94074i	−0.241614	−	0.970372i	$-0.577677\pi$
$3$	− 3.36147i	− 1.94074i	0.241614	−	0.970372i	$-0.422323\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.576535i	− 0.217910i	−0.994047	−	0.108955i	$-0.965250\pi$
$7$	− 0.576535i	− 0.217910i	0.994047	−	0.108955i	$-0.0347504\pi$
$8$	0	0
$9$	−8.29947	−2.76649
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.72294i	1.03256i	0.856421	+	0.516279i	$0.172683\pi$
$13$	3.72294i	1.03256i	−0.856421	+	0.516279i	$0.827317\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.51454i	1.58001i	0.613102	+	0.790004i	$0.289921\pi$
$17$	6.51454i	1.58001i	−0.613102	+	0.790004i	$0.710079\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−1.93800	−0.422907
$22$	0	0
$23$	− 4.36147i	− 0.909429i	−0.890637	−	0.454715i	$-0.849741\pi$
$23$	− 4.36147i	− 0.909429i	0.890637	−	0.454715i	$-0.150259\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	17.8140i	3.42831i
$28$	0	0
$29$	−2.42347	−0.450026	−0.225013	−	0.974356i	$-0.572242\pi$
$29$	−2.42347	−0.450026	−0.225013	+	0.974356i	$0.572242\pi$
$30$	0	0
$31$	−2.15307	−0.386703	−0.193351	−	0.981130i	$-0.561936\pi$
$31$	−2.15307	−0.386703	−0.193351	+	0.981130i	$0.561936\pi$
$32$	0	0
$33$	− 3.36147i	− 0.585157i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.21507i	− 0.528554i	−0.964447	−	0.264277i	$-0.914867\pi$
$37$	− 3.21507i	− 0.528554i	0.964447	−	0.264277i	$-0.0851332\pi$
$38$	0	0
$39$	12.5145	2.00393
$40$	0	0
$41$	5.93800	0.927360	0.463680	−	0.886003i	$-0.346529\pi$
$41$	5.93800	0.927360	0.463680	+	0.886003i	$0.346529\pi$
$42$	0	0
$43$	4.93800i	0.753038i	0.926409	+	0.376519i	$0.122879\pi$
$43$	4.93800i	0.753038i	−0.926409	+	0.376519i	$0.877121\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.6609i	1.84679i	0.383853	+	0.923394i	$0.374597\pi$
$47$	12.6609i	1.84679i	−0.383853	+	0.923394i	$0.625403\pi$
$48$	0	0
$49$	6.66761	0.952515
$50$	0	0
$51$	21.8984	3.06639
$52$	0	0
$53$	8.02241i	1.10196i	0.834518	+	0.550981i	$0.185746\pi$
$53$	8.02241i	1.10196i	−0.834518	+	0.550981i	$0.814254\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.36147i	− 0.445237i
$58$	0	0
$59$	−0.660941	−0.0860472	−0.0430236	−	0.999074i	$-0.513699\pi$
$59$	−0.660941	−0.0860472	−0.0430236	+	0.999074i	$0.513699\pi$
$60$	0	0
$61$	−10.4525	−1.33831	−0.669155	−	0.743122i	$-0.733344\pi$
$61$	−10.4525	−1.33831	−0.669155	+	0.743122i	$0.733344\pi$
$62$	0	0
$63$	4.78493i	0.602845i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.87601i	− 0.717869i	−0.933363	−	0.358934i	$-0.883140\pi$
$67$	− 5.87601i	− 0.717869i	0.933363	−	0.358934i	$-0.116860\pi$
$68$	0	0
$69$	−14.6609	−1.76497
$70$	0	0
$71$	15.2308	1.80756	0.903782	−	0.427993i	$-0.140779\pi$
$71$	15.2308	1.80756	0.903782	+	0.427993i	$0.140779\pi$
$72$	0	0
$73$	− 12.5765i	− 1.47197i	−0.676997	−	0.735986i	$-0.736719\pi$
$73$	− 12.5765i	− 1.47197i	0.676997	−	0.735986i	$-0.263281\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.576535i	− 0.0657022i
$78$	0	0
$79$	12.0844	1.35960	0.679801	−	0.733397i	$-0.262066\pi$
$79$	12.0844	1.35960	0.679801	+	0.733397i	$0.262066\pi$
$80$	0	0
$81$	34.9828	3.88698
$82$	0	0
$83$	7.45254i	0.818023i	0.912529	+	0.409011i	$0.134126\pi$
$83$	7.45254i	0.818023i	−0.912529	+	0.409011i	$0.865874\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.14640i	0.873386i
$88$	0	0
$89$	−11.4525	−1.21397	−0.606983	−	0.794714i	$-0.707621\pi$
$89$	−11.4525	−1.21397	−0.606983	+	0.794714i	$0.707621\pi$
$90$	0	0
$91$	2.14640	0.225004
$92$	0	0
$93$	7.23748i	0.750491i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.6543i	1.48792i	0.668226	+	0.743958i	$0.267054\pi$
$97$	14.6543i	1.48792i	−0.668226	+	0.743958i	$0.732946\pi$
$98$	0	0
$99$	−8.29947	−0.834128
$100$	0	0
$101$	17.5145	1.74276	0.871381	−	0.490607i	$-0.163225\pi$
$101$	17.5145	1.74276	0.871381	+	0.490607i	$0.163225\pi$
$102$	0	0
$103$	13.4392i	1.32420i	0.749413	+	0.662102i	$0.230336\pi$
$103$	13.4392i	1.32420i	−0.749413	+	0.662102i	$0.769664\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.4459i	− 1.39654i	−0.715837	−	0.698268i	$-0.753955\pi$
$107$	− 14.4459i	− 1.39654i	0.715837	−	0.698268i	$-0.246045\pi$
$108$	0	0
$109$	3.99333	0.382492	0.191246	−	0.981542i	$-0.438747\pi$
$109$	3.99333	0.382492	0.191246	+	0.981542i	$0.438747\pi$
$110$	0	0
$111$	−10.8073	−1.02579
$112$	0	0
$113$	− 5.21507i	− 0.490592i	−0.969448	−	0.245296i	$-0.921115\pi$
$113$	− 5.21507i	− 0.490592i	0.969448	−	0.245296i	$-0.0788852\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 30.8984i	− 2.85656i
$118$	0	0
$119$	3.75586	0.344299
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 19.9604i	− 1.79977i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.3615i	1.18564i	0.805335	+	0.592819i	$0.201985\pi$
$127$	13.3615i	1.18564i	−0.805335	+	0.592819i	$0.798015\pi$
$128$	0	0
$129$	16.5989	1.46146
$130$	0	0
$131$	−14.6543	−1.28035	−0.640175	−	0.768229i	$-0.721138\pi$
$131$	−14.6543	−1.28035	−0.640175	+	0.768229i	$0.721138\pi$
$132$	0	0
$133$	− 0.576535i	− 0.0499919i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.29947i	− 0.196457i	−0.995164	−	0.0982286i	$-0.968682\pi$
$137$	− 2.29947i	− 0.196457i	0.995164	−	0.0982286i	$-0.0313176\pi$
$138$	0	0
$139$	1.43013	0.121302	0.0606511	−	0.998159i	$-0.480682\pi$
$139$	1.43013	0.121302	0.0606511	+	0.998159i	$0.480682\pi$
$140$	0	0
$141$	42.5594	3.58414
$142$	0	0
$143$	3.72294i	0.311328i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 22.4130i	− 1.84859i
$148$	0	0
$149$	9.87601	0.809074	0.404537	−	0.914522i	$-0.367433\pi$
$149$	9.87601	0.809074	0.404537	+	0.914522i	$0.367433\pi$
$150$	0	0
$151$	7.21507	0.587154	0.293577	−	0.955935i	$-0.405154\pi$
$151$	7.21507	0.587154	0.293577	+	0.955935i	$0.405154\pi$
$152$	0	0
$153$	− 54.0672i	− 4.37108i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.1979i	1.85139i	0.378267	+	0.925697i	$0.376520\pi$
$157$	23.1979i	1.85139i	−0.378267	+	0.925697i	$0.623480\pi$
$158$	0	0
$159$	26.9671	2.13863
$160$	0	0
$161$	−2.51454	−0.198173
$162$	0	0
$163$	− 6.88267i	− 0.539093i	−0.962987	−	0.269546i	$-0.913126\pi$
$163$	− 6.88267i	− 0.539093i	0.962987	−	0.269546i	$-0.0868737\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 17.6900i	− 1.36889i	−0.729062	−	0.684447i	$-0.760044\pi$
$167$	− 17.6900i	− 1.36889i	0.729062	−	0.684447i	$-0.239956\pi$
$168$	0	0
$169$	−0.860264	−0.0661741
$170$	0	0
$171$	−8.29947	−0.634677
$172$	0	0
$173$	− 4.44588i	− 0.338014i	−0.985615	−	0.169007i	$-0.945944\pi$
$173$	− 4.44588i	− 0.338014i	0.985615	−	0.169007i	$-0.0540560\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.22173i	0.166996i
$178$	0	0
$179$	4.70053	0.351334	0.175667	−	0.984450i	$-0.443792\pi$
$179$	4.70053	0.351334	0.175667	+	0.984450i	$0.443792\pi$
$180$	0	0
$181$	−17.8984	−1.33038	−0.665189	−	0.746675i	$-0.731649\pi$
$181$	−17.8984	−1.33038	−0.665189	+	0.746675i	$0.731649\pi$
$182$	0	0
$183$	35.1359i	2.59732i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.51454i	0.476390i
$188$	0	0
$189$	10.2704	0.747061
$190$	0	0
$191$	−7.99333	−0.578377	−0.289189	−	0.957272i	$-0.593385\pi$
$191$	−7.99333	−0.578377	−0.289189	+	0.957272i	$0.593385\pi$
$192$	0	0
$193$	11.7520i	0.845928i	0.906146	+	0.422964i	$0.139010\pi$
$193$	11.7520i	0.845928i	−0.906146	+	0.422964i	$0.860990\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.2666i	1.65767i	0.559491	+	0.828837i	$0.310997\pi$
$197$	23.2666i	1.65767i	−0.559491	+	0.828837i	$0.689003\pi$
$198$	0	0
$199$	7.86934	0.557843	0.278921	−	0.960314i	$-0.410023\pi$
$199$	7.86934	0.557843	0.278921	+	0.960314i	$0.410023\pi$
$200$	0	0
$201$	−19.7520	−1.39320
$202$	0	0
$203$	1.39721i	0.0980651i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	36.1979i	2.51593i
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−2.23081	−0.153575	−0.0767876	−	0.997047i	$-0.524466\pi$
$211$	−2.23081	−0.153575	−0.0767876	+	0.997047i	$0.524466\pi$
$212$	0	0
$213$	− 51.1979i	− 3.50802i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.24132i	0.0842662i
$218$	0	0
$219$	−42.2756	−2.85672
$220$	0	0
$221$	−24.2532	−1.63145
$222$	0	0
$223$	− 4.66094i	− 0.312120i	−0.987748	−	0.156060i	$-0.950121\pi$
$223$	− 4.66094i	− 0.312120i	0.987748	−	0.156060i	$-0.0498793\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.15974i	0.0769744i	0.999259	+	0.0384872i	$0.0122539\pi$
$227$	1.15974i	0.0769744i	−0.999259	+	0.0384872i	$0.987746\pi$
$228$	0	0
$229$	−10.0844	−0.666396	−0.333198	−	0.942857i	$-0.608128\pi$
$229$	−10.0844	−0.666396	−0.333198	+	0.942857i	$0.608128\pi$
$230$	0	0
$231$	−1.93800	−0.127511
$232$	0	0
$233$	− 9.29947i	− 0.609229i	−0.952476	−	0.304614i	$-0.901472\pi$
$233$	− 9.29947i	− 0.609229i	0.952476	−	0.304614i	$-0.0985276\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 40.6214i	− 2.63864i
$238$	0	0
$239$	−5.13066	−0.331875	−0.165937	−	0.986136i	$-0.553065\pi$
$239$	−5.13066	−0.331875	−0.165937	+	0.986136i	$0.553065\pi$
$240$	0	0
$241$	−23.1135	−1.48887	−0.744435	−	0.667695i	$-0.767281\pi$
$241$	−23.1135	−1.48887	−0.744435	+	0.667695i	$0.767281\pi$
$242$	0	0
$243$	− 64.1516i	− 4.11533i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.72294i	0.236885i
$248$	0	0
$249$	25.0515	1.58757
$250$	0	0
$251$	−1.06866	−0.0674534	−0.0337267	−	0.999431i	$-0.510738\pi$
$251$	−1.06866	−0.0674534	−0.0337267	+	0.999431i	$0.510738\pi$
$252$	0	0
$253$	− 4.36147i	− 0.274203i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.5079i	0.780220i	0.920768	+	0.390110i	$0.127563\pi$
$257$	12.5079i	0.780220i	−0.920768	+	0.390110i	$0.872437\pi$
$258$	0	0
$259$	−1.85360	−0.115177
$260$	0	0
$261$	20.1135	1.24499
$262$	0	0
$263$	− 5.64520i	− 0.348098i	−0.984737	−	0.174049i	$-0.944315\pi$
$263$	− 5.64520i	− 0.348098i	0.984737	−	0.174049i	$-0.0556851\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	38.4974i	2.35600i
$268$	0	0
$269$	−10.3772	−0.632710	−0.316355	−	0.948641i	$-0.602459\pi$
$269$	−10.3772	−0.632710	−0.316355	+	0.948641i	$0.602459\pi$
$270$	0	0
$271$	−18.1201	−1.10072	−0.550360	−	0.834927i	$-0.685510\pi$
$271$	−18.1201	−1.10072	−0.550360	+	0.834927i	$0.685510\pi$
$272$	0	0
$273$	− 7.21507i	− 0.436676i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.9537i	− 0.658147i	−0.944304	−	0.329073i	$-0.893264\pi$
$277$	− 10.9537i	− 0.658147i	0.944304	−	0.329073i	$-0.106736\pi$
$278$	0	0
$279$	17.8693	1.06981
$280$	0	0
$281$	−2.77160	−0.165340	−0.0826699	−	0.996577i	$-0.526345\pi$
$281$	−2.77160	−0.165340	−0.0826699	+	0.996577i	$0.526345\pi$
$282$	0	0
$283$	6.80734i	0.404655i	0.979318	+	0.202327i	$0.0648505\pi$
$283$	6.80734i	0.404655i	−0.979318	+	0.202327i	$0.935150\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.42347i	− 0.202081i
$288$	0	0
$289$	−25.4392	−1.49642
$290$	0	0
$291$	49.2599	2.88767
$292$	0	0
$293$	18.6123i	1.08734i	0.839299	+	0.543670i	$0.182966\pi$
$293$	18.6123i	1.08734i	−0.839299	+	0.543670i	$0.817034\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	17.8140i	1.03367i
$298$	0	0
$299$	16.2375	0.939037
$300$	0	0
$301$	2.84693	0.164094
$302$	0	0
$303$	− 58.8746i	− 3.38226i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 25.8231i	− 1.47380i	−0.676001	−	0.736901i	$-0.736289\pi$
$307$	− 25.8231i	− 1.47380i	0.676001	−	0.736901i	$-0.263711\pi$
$308$	0	0
$309$	45.1755	2.56994
$310$	0	0
$311$	−18.8760	−1.07036	−0.535180	−	0.844738i	$-0.679756\pi$
$311$	−18.8760	−1.07036	−0.535180	+	0.844738i	$0.679756\pi$
$312$	0	0
$313$	15.6147i	0.882594i	0.897361	+	0.441297i	$0.145482\pi$
$313$	15.6147i	0.882594i	−0.897361	+	0.441297i	$0.854518\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.1001i	1.52210i	0.648696	+	0.761048i	$0.275315\pi$
$317$	27.1001i	1.52210i	−0.648696	+	0.761048i	$0.724685\pi$
$318$	0	0
$319$	−2.42347	−0.135688
$320$	0	0
$321$	−48.5594	−2.71032
$322$	0	0
$323$	6.51454i	0.362479i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 13.4235i	− 0.742319i
$328$	0	0
$329$	7.29947	0.402433
$330$	0	0
$331$	9.93800	0.546242	0.273121	−	0.961980i	$-0.411944\pi$
$331$	9.93800	0.546242	0.273121	+	0.961980i	$0.411944\pi$
$332$	0	0
$333$	26.6834i	1.46224i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.30614i	0.234570i	0.993098	+	0.117285i	$0.0374192\pi$
$337$	4.30614i	0.234570i	−0.993098	+	0.117285i	$0.962581\pi$
$338$	0	0
$339$	−17.5303	−0.952114
$340$	0	0
$341$	−2.15307	−0.116595
$342$	0	0
$343$	− 7.87985i	− 0.425472i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.5765i	− 1.31934i	−0.751556	−	0.659669i	$-0.770697\pi$
$347$	− 24.5765i	− 1.31934i	0.751556	−	0.659669i	$-0.229303\pi$
$348$	0	0
$349$	13.1240	0.702511	0.351256	−	0.936280i	$-0.385755\pi$
$349$	13.1240	0.702511	0.351256	+	0.936280i	$0.385755\pi$
$350$	0	0
$351$	−66.3204	−3.53992
$352$	0	0
$353$	28.4459i	1.51402i	0.653403	+	0.757011i	$0.273341\pi$
$353$	28.4459i	1.51402i	−0.653403	+	0.757011i	$0.726659\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.6252i	− 0.668196i
$358$	0	0
$359$	35.6280	1.88038	0.940188	−	0.340657i	$-0.110650\pi$
$359$	35.6280	1.88038	0.940188	+	0.340657i	$0.110650\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 3.36147i	− 0.176431i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.0844i	0.683000i	0.939882	+	0.341500i	$0.110935\pi$
$367$	13.0844i	0.683000i	−0.939882	+	0.341500i	$0.889065\pi$
$368$	0	0
$369$	−49.2823	−2.56553
$370$	0	0
$371$	4.62520	0.240128
$372$	0	0
$373$	− 31.1226i	− 1.61147i	−0.592280	−	0.805733i	$-0.701772\pi$
$373$	− 31.1226i	− 1.61147i	0.592280	−	0.805733i	$-0.298228\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.02241i	− 0.464678i
$378$	0	0
$379$	10.9537	0.562656	0.281328	−	0.959612i	$-0.409225\pi$
$379$	10.9537	0.562656	0.281328	+	0.959612i	$0.409225\pi$
$380$	0	0
$381$	44.9142	2.30102
$382$	0	0
$383$	27.8298i	1.42203i	0.703175	+	0.711017i	$0.251765\pi$
$383$	27.8298i	1.42203i	−0.703175	+	0.711017i	$0.748235\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 40.9828i	− 2.08327i
$388$	0	0
$389$	21.4459	1.08735	0.543675	−	0.839296i	$-0.317033\pi$
$389$	21.4459	1.08735	0.543675	+	0.839296i	$0.317033\pi$
$390$	0	0
$391$	28.4130	1.43690
$392$	0	0
$393$	49.2599i	2.48483i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12.2666i	− 0.615641i	−0.951445	−	0.307820i	$-0.900400\pi$
$397$	− 12.2666i	− 0.615641i	0.951445	−	0.307820i	$-0.0995996\pi$
$398$	0	0
$399$	−1.93800	−0.0970215
$400$	0	0
$401$	16.9380	0.845844	0.422922	−	0.906166i	$-0.361005\pi$
$401$	16.9380	0.845844	0.422922	+	0.906166i	$0.361005\pi$
$402$	0	0
$403$	− 8.01574i	− 0.399293i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.21507i	− 0.159365i
$408$	0	0
$409$	39.6767	1.96189	0.980943	−	0.194296i	$-0.0622423\pi$
$409$	39.6767	1.96189	0.980943	+	0.194296i	$0.0622423\pi$
$410$	0	0
$411$	−7.72960	−0.381273
$412$	0	0
$413$	0.381055i	0.0187505i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.80734i	− 0.235417i
$418$	0	0
$419$	40.0276	1.95548	0.977739	−	0.209824i	$-0.0672891\pi$
$419$	40.0276	1.95548	0.977739	+	0.209824i	$0.0672891\pi$
$420$	0	0
$421$	−6.13307	−0.298908	−0.149454	−	0.988769i	$-0.547752\pi$
$421$	−6.13307	−0.298908	−0.149454	+	0.988769i	$0.547752\pi$
$422$	0	0
$423$	− 105.079i	− 5.10912i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.02625i	0.291631i
$428$	0	0
$429$	12.5145	0.604208
$430$	0	0
$431$	6.67427	0.321488	0.160744	−	0.986996i	$-0.448611\pi$
$431$	6.67427	0.321488	0.160744	+	0.986996i	$0.448611\pi$
$432$	0	0
$433$	6.62802i	0.318522i	0.987236	+	0.159261i	$0.0509112\pi$
$433$	6.62802i	0.318522i	−0.987236	+	0.159261i	$0.949089\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.36147i	− 0.208637i
$438$	0	0
$439$	13.4063	0.639847	0.319924	−	0.947443i	$-0.396343\pi$
$439$	13.4063	0.639847	0.319924	+	0.947443i	$0.396343\pi$
$440$	0	0
$441$	−55.3376	−2.63513
$442$	0	0
$443$	− 39.9208i	− 1.89670i	−0.317232	−	0.948348i	$-0.602753\pi$
$443$	− 39.9208i	− 1.89670i	0.317232	−	0.948348i	$-0.397247\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 33.1979i	− 1.57021i
$448$	0	0
$449$	−21.8207	−1.02978	−0.514891	−	0.857256i	$-0.672168\pi$
$449$	−21.8207	−1.02978	−0.514891	+	0.857256i	$0.672168\pi$
$450$	0	0
$451$	5.93800	0.279610
$452$	0	0
$453$	− 24.2532i	− 1.13952i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.00667i	0.327758i	0.986480	+	0.163879i	$0.0524007\pi$
$457$	7.00667i	0.327758i	−0.986480	+	0.163879i	$0.947599\pi$
$458$	0	0
$459$	−116.050	−5.41675
$460$	0	0
$461$	13.0911	0.609712	0.304856	−	0.952398i	$-0.401392\pi$
$461$	13.0911	0.609712	0.304856	+	0.952398i	$0.401392\pi$
$462$	0	0
$463$	29.7811i	1.38404i	0.721876	+	0.692022i	$0.243280\pi$
$463$	29.7811i	1.38404i	−0.721876	+	0.692022i	$0.756720\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20.9671i	− 0.970241i	−0.874447	−	0.485120i	$-0.838776\pi$
$467$	− 20.9671i	− 0.970241i	0.874447	−	0.485120i	$-0.161224\pi$
$468$	0	0
$469$	−3.38772	−0.156430
$470$	0	0
$471$	77.9790	3.59308
$472$	0	0
$473$	4.93800i	0.227050i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 66.5818i	− 3.04857i
$478$	0	0
$479$	19.2466	0.879397	0.439699	−	0.898145i	$-0.355085\pi$
$479$	19.2466	0.879397	0.439699	+	0.898145i	$0.355085\pi$
$480$	0	0
$481$	11.9695	0.545762
$482$	0	0
$483$	8.45254i	0.384604i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.2771i	0.646955i	0.946236	+	0.323478i	$0.104852\pi$
$487$	14.2771i	0.646955i	−0.946236	+	0.323478i	$0.895148\pi$
$488$	0	0
$489$	−23.1359	−1.04624
$490$	0	0
$491$	5.18215	0.233867	0.116933	−	0.993140i	$-0.462694\pi$
$491$	5.18215	0.233867	0.116933	+	0.993140i	$0.462694\pi$
$492$	0	0
$493$	− 15.7878i	− 0.711045i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.78109i	− 0.393886i
$498$	0	0
$499$	−0.315215	−0.0141110	−0.00705549	−	0.999975i	$-0.502246\pi$
$499$	−0.315215	−0.0141110	−0.00705549	+	0.999975i	$0.502246\pi$
$500$	0	0
$501$	−59.4644	−2.65668
$502$	0	0
$503$	27.6147i	1.23128i	0.788028	+	0.615639i	$0.211102\pi$
$503$	27.6147i	1.23128i	−0.788028	+	0.615639i	$0.788898\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.89175i	0.128427i
$508$	0	0
$509$	9.71627	0.430666	0.215333	−	0.976541i	$-0.430916\pi$
$509$	9.71627	0.430666	0.215333	+	0.976541i	$0.430916\pi$
$510$	0	0
$511$	−7.25081	−0.320757
$512$	0	0
$513$	17.8140i	0.786508i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.6609i	0.556828i
$518$	0	0
$519$	−14.9447	−0.655998
$520$	0	0
$521$	−16.5527	−0.725187	−0.362593	−	0.931947i	$-0.618109\pi$
$521$	−16.5527	−0.725187	−0.362593	+	0.931947i	$0.618109\pi$
$522$	0	0
$523$	3.30998i	0.144735i	0.997378	+	0.0723677i	$0.0230555\pi$
$523$	3.30998i	0.144735i	−0.997378	+	0.0723677i	$0.976944\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 14.0263i	− 0.610993i
$528$	0	0
$529$	3.97759	0.172939
$530$	0	0
$531$	5.48546	0.238049
$532$	0	0
$533$	22.1068i	0.957552i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.8007i	− 0.681850i
$538$	0	0
$539$	6.66761	0.287194
$540$	0	0
$541$	3.09774	0.133182	0.0665911	−	0.997780i	$-0.478788\pi$
$541$	3.09774	0.133182	0.0665911	+	0.997780i	$0.478788\pi$
$542$	0	0
$543$	60.1650i	2.58193i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.22414i	0.0950975i	0.998869	+	0.0475487i	$0.0151409\pi$
$547$	2.22414i	0.0950975i	−0.998869	+	0.0475487i	$0.984859\pi$
$548$	0	0
$549$	86.7506	3.70242
$550$	0	0
$551$	−2.42347	−0.103243
$552$	0	0
$553$	− 6.96708i	− 0.296270i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.7968i	1.22016i	0.792339	+	0.610080i	$0.208863\pi$
$557$	28.7968i	1.22016i	−0.792339	+	0.610080i	$0.791137\pi$
$558$	0	0
$559$	−18.3839	−0.777555
$560$	0	0
$561$	21.8984	0.924552
$562$	0	0
$563$	− 25.3128i	− 1.06681i	−0.845861	−	0.533404i	$-0.820913\pi$
$563$	− 25.3128i	− 1.06681i	0.845861	−	0.533404i	$-0.179087\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 20.1688i	− 0.847011i
$568$	0	0
$569$	19.4235	0.814274	0.407137	−	0.913367i	$-0.366527\pi$
$569$	19.4235	0.814274	0.407137	+	0.913367i	$0.366527\pi$
$570$	0	0
$571$	−39.4354	−1.65032	−0.825159	−	0.564900i	$-0.808915\pi$
$571$	−39.4354	−1.65032	−0.825159	+	0.564900i	$0.808915\pi$
$572$	0	0
$573$	26.8693i	1.12248i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.6543i	1.65083i	0.564527	+	0.825415i	$0.309059\pi$
$577$	39.6543i	1.65083i	−0.564527	+	0.825415i	$0.690941\pi$
$578$	0	0
$579$	39.5040	1.64173
$580$	0	0
$581$	4.29665	0.178255
$582$	0	0
$583$	8.02241i	0.332254i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.49880i	0.309508i	0.987953	+	0.154754i	$0.0494586\pi$
$587$	7.49880i	0.309508i	−0.987953	+	0.154754i	$0.950541\pi$
$588$	0	0
$589$	−2.15307	−0.0887157
$590$	0	0
$591$	78.2098	3.21712
$592$	0	0
$593$	− 18.0935i	− 0.743010i	−0.928431	−	0.371505i	$-0.878842\pi$
$593$	− 18.0935i	− 0.743010i	0.928431	−	0.371505i	$-0.121158\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 26.4525i	− 1.08263i
$598$	0	0
$599$	−11.8364	−0.483623	−0.241812	−	0.970323i	$-0.577742\pi$
$599$	−11.8364	−0.483623	−0.241812	+	0.970323i	$0.577742\pi$
$600$	0	0
$601$	3.05149	0.124473	0.0622364	−	0.998061i	$-0.480177\pi$
$601$	3.05149	0.124473	0.0622364	+	0.998061i	$0.480177\pi$
$602$	0	0
$603$	48.7678i	1.98598i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	45.5040i	1.84695i	0.383657	+	0.923476i	$0.374665\pi$
$607$	45.5040i	1.84695i	−0.383657	+	0.923476i	$0.625335\pi$
$608$	0	0
$609$	4.69668	0.190319
$610$	0	0
$611$	−47.1359	−1.90691
$612$	0	0
$613$	23.3615i	0.943561i	0.881716	+	0.471780i	$0.156388\pi$
$613$	23.3615i	0.943561i	−0.881716	+	0.471780i	$0.843612\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0329i	0.484427i	0.970223	+	0.242214i	$0.0778735\pi$
$617$	12.0329i	0.484427i	−0.970223	+	0.242214i	$0.922127\pi$
$618$	0	0
$619$	7.67668	0.308552	0.154276	−	0.988028i	$-0.450696\pi$
$619$	7.67668	0.308552	0.154276	+	0.988028i	$0.450696\pi$
$620$	0	0
$621$	77.6953	3.11780
$622$	0	0
$623$	6.60279i	0.264535i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.36147i	− 0.134244i
$628$	0	0
$629$	20.9447	0.835119
$630$	0	0
$631$	−9.28614	−0.369675	−0.184838	−	0.982769i	$-0.559176\pi$
$631$	−9.28614	−0.369675	−0.184838	+	0.982769i	$0.559176\pi$
$632$	0	0
$633$	7.49880i	0.298050i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	24.8231i	0.983527i
$638$	0	0
$639$	−126.408	−5.00061
$640$	0	0
$641$	11.2795	0.445512	0.222756	−	0.974874i	$-0.428495\pi$
$641$	11.2795	0.445512	0.222756	+	0.974874i	$0.428495\pi$
$642$	0	0
$643$	2.76776i	0.109150i	0.998510	+	0.0545748i	$0.0173803\pi$
$643$	2.76776i	0.109150i	−0.998510	+	0.0545748i	$0.982620\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.3996i	0.566108i	0.959104	+	0.283054i	$0.0913475\pi$
$647$	14.3996i	0.566108i	−0.959104	+	0.283054i	$0.908653\pi$
$648$	0	0
$649$	−0.660941	−0.0259442
$650$	0	0
$651$	4.17266	0.163539
$652$	0	0
$653$	− 15.9471i	− 0.624057i	−0.950073	−	0.312029i	$-0.898991\pi$
$653$	− 15.9471i	− 0.624057i	0.950073	−	0.312029i	$-0.101009\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	104.379i	4.07220i
$658$	0	0
$659$	0.560792	0.0218453	0.0109227	−	0.999940i	$-0.496523\pi$
$659$	0.560792	0.0218453	0.0109227	+	0.999940i	$0.496523\pi$
$660$	0	0
$661$	−36.6609	−1.42595	−0.712973	−	0.701192i	$-0.752652\pi$
$661$	−36.6609	−1.42595	−0.712973	+	0.701192i	$0.752652\pi$
$662$	0	0
$663$	81.5264i	3.16622i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.5699i	0.409267i
$668$	0	0
$669$	−15.6676	−0.605745
$670$	0	0
$671$	−10.4525	−0.403516
$672$	0	0
$673$	− 20.3681i	− 0.785134i	−0.919723	−	0.392567i	$-0.871587\pi$
$673$	− 20.3681i	− 0.785134i	0.919723	−	0.392567i	$-0.128413\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.23081i	0.354769i	0.984142	+	0.177384i	$0.0567636\pi$
$677$	9.23081i	0.354769i	−0.984142	+	0.177384i	$0.943236\pi$
$678$	0	0
$679$	8.44870	0.324231
$680$	0	0
$681$	3.89842	0.149388
$682$	0	0
$683$	− 31.7034i	− 1.21310i	−0.795047	−	0.606548i	$-0.792554\pi$
$683$	− 31.7034i	− 1.21310i	0.795047	−	0.606548i	$-0.207446\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	33.8984i	1.29331i
$688$	0	0
$689$	−29.8669	−1.13784
$690$	0	0
$691$	22.6080	0.860050	0.430025	−	0.902817i	$-0.358505\pi$
$691$	22.6080	0.860050	0.430025	+	0.902817i	$0.358505\pi$
$692$	0	0
$693$	4.78493i	0.181765i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	38.6834i	1.46524i
$698$	0	0
$699$	−31.2599	−1.18236
$700$	0	0
$701$	36.1092	1.36383	0.681913	−	0.731433i	$-0.261148\pi$
$701$	36.1092	1.36383	0.681913	+	0.731433i	$0.261148\pi$
$702$	0	0
$703$	− 3.21507i	− 0.121259i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.0977i	− 0.379765i
$708$	0	0
$709$	−18.0582	−0.678188	−0.339094	−	0.940752i	$-0.610121\pi$
$709$	−18.0582	−0.678188	−0.339094	+	0.940752i	$0.610121\pi$
$710$	0	0
$711$	−100.294	−3.76133
$712$	0	0
$713$	9.39055i	0.351679i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.2466i	0.644084i
$718$	0	0
$719$	23.6319	0.881320	0.440660	−	0.897674i	$-0.354744\pi$
$719$	23.6319	0.881320	0.440660	+	0.897674i	$0.354744\pi$
$720$	0	0
$721$	7.74817	0.288557
$722$	0	0
$723$	77.6953i	2.88952i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 40.0382i	− 1.48493i	−0.669883	−	0.742466i	$-0.733656\pi$
$727$	− 40.0382i	− 1.48493i	0.669883	−	0.742466i	$-0.266344\pi$
$728$	0	0
$729$	−110.695	−4.09982
$730$	0	0
$731$	−32.1688	−1.18981
$732$	0	0
$733$	− 15.4883i	− 0.572073i	−0.958219	−	0.286036i	$-0.907662\pi$
$733$	− 15.4883i	− 0.572073i	0.958219	−	0.286036i	$-0.0923378\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.87601i	− 0.216445i
$738$	0	0
$739$	−46.3867	−1.70636	−0.853181	−	0.521615i	$-0.825330\pi$
$739$	−46.3867	−1.70636	−0.853181	+	0.521615i	$0.825330\pi$
$740$	0	0
$741$	12.5145	0.459733
$742$	0	0
$743$	− 18.7453i	− 0.687700i	−0.939025	−	0.343850i	$-0.888269\pi$
$743$	− 18.7453i	− 0.687700i	0.939025	−	0.343850i	$-0.111731\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 61.8522i	− 2.26305i
$748$	0	0
$749$	−8.32855	−0.304319
$750$	0	0
$751$	18.6504	0.680564	0.340282	−	0.940323i	$-0.389477\pi$
$751$	18.6504	0.680564	0.340282	+	0.940323i	$0.389477\pi$
$752$	0	0
$753$	3.59228i	0.130910i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 28.4921i	− 1.03556i	−0.855513	−	0.517782i	$-0.826758\pi$
$757$	− 28.4921i	− 1.03556i	0.855513	−	0.517782i	$-0.173242\pi$
$758$	0	0
$759$	−14.6609	−0.532158
$760$	0	0
$761$	−14.8918	−0.539826	−0.269913	−	0.962885i	$-0.586995\pi$
$761$	−14.8918	−0.539826	−0.269913	+	0.962885i	$0.586995\pi$
$762$	0	0
$763$	− 2.30230i	− 0.0833487i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.46064i	− 0.0888486i
$768$	0	0
$769$	−26.2175	−0.945426	−0.472713	−	0.881216i	$-0.656725\pi$
$769$	−26.2175	−0.945426	−0.472713	+	0.881216i	$0.656725\pi$
$770$	0	0
$771$	42.0448	1.51421
$772$	0	0
$773$	37.7744i	1.35865i	0.733837	+	0.679326i	$0.237728\pi$
$773$	37.7744i	1.35865i	−0.733837	+	0.679326i	$0.762272\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.23081i	0.223529i
$778$	0	0
$779$	5.93800	0.212751
$780$	0	0
$781$	15.2308	0.545001
$782$	0	0
$783$	− 43.1716i	− 1.54283i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32.2441i	1.14938i	0.818371	+	0.574690i	$0.194877\pi$
$787$	32.2441i	1.14938i	−0.818371	+	0.574690i	$0.805123\pi$
$788$	0	0
$789$	−18.9762	−0.675569
$790$	0	0
$791$	−3.00667	−0.106905
$792$	0	0
$793$	− 38.9142i	− 1.38188i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.77827i	0.204677i	0.994750	+	0.102338i	$0.0326324\pi$
$797$	5.77827i	0.204677i	−0.994750	+	0.102338i	$0.967368\pi$
$798$	0	0
$799$	−82.4802	−2.91794
$800$	0	0
$801$	95.0501	3.35843
$802$	0	0
$803$	− 12.5765i	− 0.443816i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	34.8827i	1.22793i
$808$	0	0
$809$	33.9656	1.19417	0.597084	−	0.802179i	$-0.296326\pi$
$809$	33.9656	1.19417	0.597084	+	0.802179i	$0.296326\pi$
$810$	0	0
$811$	0.412955	0.0145008	0.00725041	−	0.999974i	$-0.497692\pi$
$811$	0.412955	0.0145008	0.00725041	+	0.999974i	$0.497692\pi$
$812$	0	0
$813$	60.9103i	2.13622i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.93800i	0.172759i
$818$	0	0
$819$	−17.8140	−0.622472
$820$	0	0
$821$	6.17025	0.215343	0.107672	−	0.994187i	$-0.465661\pi$
$821$	6.17025	0.215343	0.107672	+	0.994187i	$0.465661\pi$
$822$	0	0
$823$	34.3061i	1.19584i	0.801557	+	0.597918i	$0.204005\pi$
$823$	34.3061i	1.19584i	−0.801557	+	0.597918i	$0.795995\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.6108i	1.06444i	0.846605	+	0.532222i	$0.178643\pi$
$827$	30.6108i	1.06444i	−0.846605	+	0.532222i	$0.821357\pi$
$828$	0	0
$829$	−3.44064	−0.119498	−0.0597492	−	0.998213i	$-0.519030\pi$
$829$	−3.44064	−0.119498	−0.0597492	+	0.998213i	$0.519030\pi$
$830$	0	0
$831$	−36.8207	−1.27730
$832$	0	0
$833$	43.4364i	1.50498i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 38.3548i	− 1.32574i
$838$	0	0
$839$	−11.7611	−0.406038	−0.203019	−	0.979175i	$-0.565075\pi$
$839$	−11.7611	−0.406038	−0.203019	+	0.979175i	$0.565075\pi$
$840$	0	0
$841$	−23.1268	−0.797476
$842$	0	0
$843$	9.31665i	0.320882i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.576535i	− 0.0198100i
$848$	0	0
$849$	22.8827	0.785331
$850$	0	0
$851$	−14.0224	−0.480682
$852$	0	0
$853$	18.5279i	0.634382i	0.948362	+	0.317191i	$0.102740\pi$
$853$	18.5279i	0.634382i	−0.948362	+	0.317191i	$0.897260\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 45.7348i	− 1.56227i	−0.624361	−	0.781136i	$-0.714640\pi$
$857$	− 45.7348i	− 1.56227i	0.624361	−	0.781136i	$-0.285360\pi$
$858$	0	0
$859$	13.0606	0.445621	0.222810	−	0.974862i	$-0.428477\pi$
$859$	13.0606	0.445621	0.222810	+	0.974862i	$0.428477\pi$
$860$	0	0
$861$	−11.5079	−0.392187
$862$	0	0
$863$	26.7520i	0.910649i	0.890326	+	0.455325i	$0.150477\pi$
$863$	26.7520i	0.910649i	−0.890326	+	0.455325i	$0.849523\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	85.5131i	2.90418i
$868$	0	0
$869$	12.0844	0.409935
$870$	0	0
$871$	21.8760	0.741240
$872$	0	0
$873$	− 121.623i	− 4.11631i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35.3023i	1.19207i	0.802957	+	0.596037i	$0.203259\pi$
$877$	35.3023i	1.19207i	−0.802957	+	0.596037i	$0.796741\pi$
$878$	0	0
$879$	62.5646	2.11025
$880$	0	0
$881$	31.7282	1.06895	0.534475	−	0.845185i	$-0.320510\pi$
$881$	31.7282	1.06895	0.534475	+	0.845185i	$0.320510\pi$
$882$	0	0
$883$	29.9380i	1.00749i	0.863851	+	0.503747i	$0.168046\pi$
$883$	29.9380i	1.00749i	−0.863851	+	0.503747i	$0.831954\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.66094i	− 0.223653i	−0.993728	−	0.111826i	$-0.964330\pi$
$887$	− 6.66094i	− 0.223653i	0.993728	−	0.111826i	$-0.0356700\pi$
$888$	0	0
$889$	7.70335	0.258362
$890$	0	0
$891$	34.9828	1.17197
$892$	0	0
$893$	12.6609i	0.423682i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 54.5818i	− 1.82243i
$898$	0	0
$899$	5.21789	0.174026
$900$	0	0
$901$	−52.2623	−1.74111
$902$	0	0
$903$	− 9.56987i	− 0.318465i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	45.0157i	1.49472i	0.664418	+	0.747362i	$0.268680\pi$
$907$	45.0157i	1.49472i	−0.664418	+	0.747362i	$0.731320\pi$
$908$	0	0
$909$	−145.361	−4.82133
$910$	0	0
$911$	25.0739	0.830735	0.415368	−	0.909654i	$-0.363653\pi$
$911$	25.0739	0.830735	0.415368	+	0.909654i	$0.363653\pi$
$912$	0	0
$913$	7.45254i	0.246643i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.44870i	0.279001i
$918$	0	0
$919$	22.9184	0.756009	0.378004	−	0.925804i	$-0.376610\pi$
$919$	22.9184	0.756009	0.378004	+	0.925804i	$0.376610\pi$
$920$	0	0
$921$	−86.8035	−2.86027
$922$	0	0
$923$	56.7034i	1.86641i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 111.538i	− 3.66340i
$928$	0	0
$929$	4.45921	0.146302	0.0731509	−	0.997321i	$-0.476695\pi$
$929$	4.45921	0.146302	0.0731509	+	0.997321i	$0.476695\pi$
$930$	0	0
$931$	6.66761	0.218522
$932$	0	0
$933$	63.4511i	2.07730i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.15548i	− 0.135754i	−0.997694	−	0.0678768i	$-0.978378\pi$
$937$	− 4.15548i	− 0.135754i	0.997694	−	0.0678768i	$-0.0216225\pi$
$938$	0	0
$939$	52.4883	1.71289
$940$	0	0
$941$	33.5079	1.09233	0.546163	−	0.837679i	$-0.316088\pi$
$941$	33.5079	1.09233	0.546163	+	0.837679i	$0.316088\pi$
$942$	0	0
$943$	− 25.8984i	− 0.843368i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 19.0329i	− 0.618487i	−0.950983	−	0.309243i	$-0.899924\pi$
$947$	− 19.0329i	− 0.618487i	0.950983	−	0.309243i	$-0.100076\pi$
$948$	0	0
$949$	46.8217	1.51990
$950$	0	0
$951$	91.0963	2.95400
$952$	0	0
$953$	35.1979i	1.14017i	0.821585	+	0.570086i	$0.193090\pi$
$953$	35.1979i	1.14017i	−0.821585	+	0.570086i	$0.806910\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.14640i	0.263336i
$958$	0	0
$959$	−1.32573	−0.0428099
$960$	0	0
$961$	−26.3643	−0.850461
$962$	0	0
$963$	119.893i	3.86350i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	27.4855i	0.883873i	0.897046	+	0.441936i	$0.145708\pi$
$967$	27.4855i	0.883873i	−0.897046	+	0.441936i	$0.854292\pi$
$968$	0	0
$969$	21.8984	0.703479
$970$	0	0
$971$	8.92749	0.286497	0.143248	−	0.989687i	$-0.454245\pi$
$971$	8.92749	0.286497	0.143248	+	0.989687i	$0.454245\pi$
$972$	0	0
$973$	− 0.824521i	− 0.0264329i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.7191i	0.790834i	0.918502	+	0.395417i	$0.129400\pi$
$977$	24.7191i	0.790834i	−0.918502	+	0.395417i	$0.870600\pi$
$978$	0	0
$979$	−11.4525	−0.366025
$980$	0	0
$981$	−33.1426	−1.05816
$982$	0	0
$983$	− 12.2017i	− 0.389175i	−0.980885	−	0.194587i	$-0.937663\pi$
$983$	− 12.2017i	− 0.389175i	0.980885	−	0.194587i	$-0.0623368\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.5369i	− 0.781020i
$988$	0	0
$989$	21.5369	0.684835
$990$	0	0
$991$	−25.9737	−0.825083	−0.412542	−	0.910939i	$-0.635359\pi$
$991$	−25.9737	−0.825083	−0.412542	+	0.910939i	$0.635359\pi$
$992$	0	0
$993$	− 33.4063i	− 1.06012i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 62.2980i	− 1.97300i	−0.163765	−	0.986499i	$-0.552364\pi$
$997$	− 62.2980i	− 1.97300i	0.163765	−	0.986499i	$-0.447636\pi$
$998$	0	0
$999$	57.2732	1.81204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.bc.4049.1		6
4.3	odd	2		2200.2.b.l.1849.6		6
5.2	odd	4		4400.2.a.bx.1.1		3
5.3	odd	4		4400.2.a.ca.1.3		3
5.4	even	2	inner	4400.2.b.bc.4049.6		6
20.3	even	4		2200.2.a.t.1.1	✓	3
20.7	even	4		2200.2.a.w.1.3	yes	3
20.19	odd	2		2200.2.b.l.1849.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.t.1.1	✓	3	20.3	even	4
2200.2.a.w.1.3	yes	3	20.7	even	4
2200.2.b.l.1849.1		6	20.19	odd	2
2200.2.b.l.1849.6		6	4.3	odd	2
4400.2.a.bx.1.1		3	5.2	odd	4
4400.2.a.ca.1.3		3	5.3	odd	4
4400.2.b.bc.4049.1		6	1.1	even	1	trivial
4400.2.b.bc.4049.6		6	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)

\(f(q)\)	\(=\)	\(q-3.36147i q^{3} -0.576535i q^{7} -8.29947 q^{9} +O(q^{10})\)	\(q-3.36147i q^{3} -0.576535i q^{7} -8.29947 q^{9} +1.00000 q^{11} +3.72294i q^{13} +6.51454i q^{17} +1.00000 q^{19} -1.93800 q^{21} -4.36147i q^{23} +17.8140i q^{27} -2.42347 q^{29} -2.15307 q^{31} -3.36147i q^{33} -3.21507i q^{37} +12.5145 q^{39} +5.93800 q^{41} +4.93800i q^{43} +12.6609i q^{47} +6.66761 q^{49} +21.8984 q^{51} +8.02241i q^{53} -3.36147i q^{57} -0.660941 q^{59} -10.4525 q^{61} +4.78493i q^{63} -5.87601i q^{67} -14.6609 q^{69} +15.2308 q^{71} -12.5765i q^{73} -0.576535i q^{77} +12.0844 q^{79} +34.9828 q^{81} +7.45254i q^{83} +8.14640i q^{87} -11.4525 q^{89} +2.14640 q^{91} +7.23748i q^{93} +14.6543i q^{97} -8.29947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{9}+O(q^{10}) \) 6 * q - 12 * q^9	\( 6 q - 12 q^{9} + 6 q^{11} + 6 q^{19} + 12 q^{21} - 24 q^{29} + 6 q^{31} + 42 q^{39} + 12 q^{41} - 12 q^{49} + 18 q^{51} + 48 q^{59} - 6 q^{61} - 36 q^{69} + 30 q^{71} + 30 q^{79} + 54 q^{81} - 12 q^{89} - 6 q^{91} - 12 q^{99}+O(q^{100}) \) 6 * q - 12 * q^9 + 6 * q^11 + 6 * q^19 + 12 * q^21 - 24 * q^29 + 6 * q^31 + 42 * q^39 + 12 * q^41 - 12 * q^49 + 18 * q^51 + 48 * q^59 - 6 * q^61 - 36 * q^69 + 30 * q^71 + 30 * q^79 + 54 * q^81 - 12 * q^89 - 6 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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