LMFDB - Embedded newform 400.6.c.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.0000i q^{3} +172.000i q^{7} -333.000 q^{9} +O(q^{10})$	$q+24.0000i q^{3} +172.000i q^{7} -333.000 q^{9} -132.000 q^{11} +946.000i q^{13} -222.000i q^{17} +500.000 q^{19} -4128.00 q^{21} +3564.00i q^{23} -2160.00i q^{27} -2190.00 q^{29} -2312.00 q^{31} -3168.00i q^{33} -11242.0i q^{37} -22704.0 q^{39} +1242.00 q^{41} +20624.0i q^{43} -6588.00i q^{47} -12777.0 q^{49} +5328.00 q^{51} +21066.0i q^{53} +12000.0i q^{57} +7980.00 q^{59} +16622.0 q^{61} -57276.0i q^{63} -1808.00i q^{67} -85536.0 q^{69} +24528.0 q^{71} -20474.0i q^{73} -22704.0i q^{77} -46240.0 q^{79} -29079.0 q^{81} -51576.0i q^{83} -52560.0i q^{87} +110310. q^{89} -162712. q^{91} -55488.0i q^{93} -78382.0i q^{97} +43956.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 666 q^{9}+O(q^{10}) $ 2 * q - 666 * q^9	$ 2 q - 666 q^{9} - 264 q^{11} + 1000 q^{19} - 8256 q^{21} - 4380 q^{29} - 4624 q^{31} - 45408 q^{39} + 2484 q^{41} - 25554 q^{49} + 10656 q^{51} + 15960 q^{59} + 33244 q^{61} - 171072 q^{69} + 49056 q^{71} - 92480 q^{79} - 58158 q^{81} + 220620 q^{89} - 325424 q^{91} + 87912 q^{99}+O(q^{100}) $ 2 * q - 666 * q^9 - 264 * q^11 + 1000 * q^19 - 8256 * q^21 - 4380 * q^29 - 4624 * q^31 - 45408 * q^39 + 2484 * q^41 - 25554 * q^49 + 10656 * q^51 + 15960 * q^59 + 33244 * q^61 - 171072 * q^69 + 49056 * q^71 - 92480 * q^79 - 58158 * q^81 + 220620 * q^89 - 325424 * q^91 + 87912 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	24.0000i	1.53960i	0.638285	+	0.769800i	$0.279644\pi$
$3$	24.0000i	1.53960i	−0.638285	+	0.769800i	$0.720356\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	172.000i	1.32673i	0.748295	+	0.663366i	$0.230873\pi$
$7$	172.000i	1.32673i	−0.748295	+	0.663366i	$0.769127\pi$
$8$	0	0
$9$	−333.000	−1.37037
$10$	0	0
$11$	−132.000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−132.000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	946.000i	1.55250i	0.630423	+	0.776252i	$0.282882\pi$
$13$	946.000i	1.55250i	−0.630423	+	0.776252i	$0.717118\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 222.000i	− 0.186308i	−0.995652	−	0.0931538i	$-0.970305\pi$
$17$	− 222.000i	− 0.186308i	0.995652	−	0.0931538i	$-0.0296948\pi$
$18$	0	0
$19$	500.000	0.317750	0.158875	−	0.987299i	$-0.449213\pi$
$19$	500.000	0.317750	0.158875	+	0.987299i	$0.449213\pi$
$20$	0	0
$21$	−4128.00	−2.04264
$22$	0	0
$23$	3564.00i	1.40481i	0.711777	+	0.702406i	$0.247891\pi$
$23$	3564.00i	1.40481i	−0.711777	+	0.702406i	$0.752109\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2160.00i	− 0.570222i
$28$	0	0
$29$	−2190.00	−0.483559	−0.241779	−	0.970331i	$-0.577731\pi$
$29$	−2190.00	−0.483559	−0.241779	+	0.970331i	$0.577731\pi$
$30$	0	0
$31$	−2312.00	−0.432099	−0.216050	−	0.976382i	$-0.569317\pi$
$31$	−2312.00	−0.432099	−0.216050	+	0.976382i	$0.569317\pi$
$32$	0	0
$33$	− 3168.00i	− 0.506408i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 11242.0i	− 1.35002i	−0.737810	−	0.675009i	$-0.764140\pi$
$37$	− 11242.0i	− 1.35002i	0.737810	−	0.675009i	$-0.235860\pi$
$38$	0	0
$39$	−22704.0	−2.39024
$40$	0	0
$41$	1242.00	0.115388	0.0576942	−	0.998334i	$-0.481625\pi$
$41$	1242.00	0.115388	0.0576942	+	0.998334i	$0.481625\pi$
$42$	0	0
$43$	20624.0i	1.70099i	0.525983	+	0.850495i	$0.323697\pi$
$43$	20624.0i	1.70099i	−0.525983	+	0.850495i	$0.676303\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6588.00i	− 0.435020i	−0.976058	−	0.217510i	$-0.930207\pi$
$47$	− 6588.00i	− 0.435020i	0.976058	−	0.217510i	$-0.0697934\pi$
$48$	0	0
$49$	−12777.0	−0.760219
$50$	0	0
$51$	5328.00	0.286839
$52$	0	0
$53$	21066.0i	1.03013i	0.857151	+	0.515065i	$0.172232\pi$
$53$	21066.0i	1.03013i	−0.857151	+	0.515065i	$0.827768\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12000.0i	0.489209i
$58$	0	0
$59$	7980.00	0.298451	0.149225	−	0.988803i	$-0.452322\pi$
$59$	7980.00	0.298451	0.149225	+	0.988803i	$0.452322\pi$
$60$	0	0
$61$	16622.0	0.571951	0.285975	−	0.958237i	$-0.407682\pi$
$61$	16622.0	0.571951	0.285975	+	0.958237i	$0.407682\pi$
$62$	0	0
$63$	− 57276.0i	− 1.81811i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1808.00i	− 0.0492052i	−0.999697	−	0.0246026i	$-0.992168\pi$
$67$	− 1808.00i	− 0.0492052i	0.999697	−	0.0246026i	$-0.00783205\pi$
$68$	0	0
$69$	−85536.0	−2.16285
$70$	0	0
$71$	24528.0	0.577452	0.288726	−	0.957412i	$-0.406768\pi$
$71$	24528.0	0.577452	0.288726	+	0.957412i	$0.406768\pi$
$72$	0	0
$73$	− 20474.0i	− 0.449672i	−0.974397	−	0.224836i	$-0.927815\pi$
$73$	− 20474.0i	− 0.449672i	0.974397	−	0.224836i	$-0.0721846\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 22704.0i	− 0.436391i
$78$	0	0
$79$	−46240.0	−0.833585	−0.416793	−	0.909002i	$-0.636846\pi$
$79$	−46240.0	−0.833585	−0.416793	+	0.909002i	$0.636846\pi$
$80$	0	0
$81$	−29079.0	−0.492455
$82$	0	0
$83$	− 51576.0i	− 0.821774i	−0.911686	−	0.410887i	$-0.865219\pi$
$83$	− 51576.0i	− 0.821774i	0.911686	−	0.410887i	$-0.134781\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 52560.0i	− 0.744487i
$88$	0	0
$89$	110310.	1.47618	0.738091	−	0.674701i	$-0.235728\pi$
$89$	110310.	1.47618	0.738091	+	0.674701i	$0.235728\pi$
$90$	0	0
$91$	−162712.	−2.05976
$92$	0	0
$93$	− 55488.0i	− 0.665260i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 78382.0i	− 0.845838i	−0.906168	−	0.422919i	$-0.861006\pi$
$97$	− 78382.0i	− 0.845838i	0.906168	−	0.422919i	$-0.138994\pi$
$98$	0	0
$99$	43956.0	0.450744
$100$	0	0
$101$	141942.	1.38455	0.692273	−	0.721636i	$-0.256609\pi$
$101$	141942.	1.38455	0.692273	+	0.721636i	$0.256609\pi$
$102$	0	0
$103$	− 436.000i	− 0.00404943i	−0.999998	−	0.00202471i	$-0.999356\pi$
$103$	− 436.000i	− 0.00404943i	0.999998	−	0.00202471i	$-0.000644487\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 232968.i	− 1.96715i	−0.180508	−	0.983574i	$-0.557774\pi$
$107$	− 232968.i	− 1.96715i	0.180508	−	0.983574i	$-0.442226\pi$
$108$	0	0
$109$	174850.	1.40961	0.704806	−	0.709400i	$-0.251034\pi$
$109$	174850.	1.40961	0.704806	+	0.709400i	$0.251034\pi$
$110$	0	0
$111$	269808.	2.07849
$112$	0	0
$113$	− 182994.i	− 1.34816i	−0.738659	−	0.674079i	$-0.764541\pi$
$113$	− 182994.i	− 1.34816i	0.738659	−	0.674079i	$-0.235459\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 315018.i	− 2.12751i
$118$	0	0
$119$	38184.0	0.247180
$120$	0	0
$121$	−143627.	−0.891811
$122$	0	0
$123$	29808.0i	0.177652i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	122452.i	0.673685i	0.941561	+	0.336842i	$0.109359\pi$
$127$	122452.i	0.673685i	−0.941561	+	0.336842i	$0.890641\pi$
$128$	0	0
$129$	−494976.	−2.61885
$130$	0	0
$131$	241908.	1.23161	0.615803	−	0.787900i	$-0.288832\pi$
$131$	241908.	1.23161	0.615803	+	0.787900i	$0.288832\pi$
$132$	0	0
$133$	86000.0i	0.421570i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	277098.i	1.26134i	0.776051	+	0.630670i	$0.217220\pi$
$137$	277098.i	1.26134i	−0.776051	+	0.630670i	$0.782780\pi$
$138$	0	0
$139$	−193540.	−0.849638	−0.424819	−	0.905278i	$-0.639662\pi$
$139$	−193540.	−0.849638	−0.424819	+	0.905278i	$0.639662\pi$
$140$	0	0
$141$	158112.	0.669757
$142$	0	0
$143$	− 124872.i	− 0.510652i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 306648.i	− 1.17043i
$148$	0	0
$149$	−140550.	−0.518639	−0.259320	−	0.965792i	$-0.583498\pi$
$149$	−140550.	−0.518639	−0.259320	+	0.965792i	$0.583498\pi$
$150$	0	0
$151$	−433952.	−1.54881	−0.774407	−	0.632688i	$-0.781952\pi$
$151$	−433952.	−1.54881	−0.774407	+	0.632688i	$0.781952\pi$
$152$	0	0
$153$	73926.0i	0.255310i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 555922.i	− 1.79997i	−0.435923	−	0.899984i	$-0.643578\pi$
$157$	− 555922.i	− 1.79997i	0.435923	−	0.899984i	$-0.356422\pi$
$158$	0	0
$159$	−505584.	−1.58599
$160$	0	0
$161$	−613008.	−1.86381
$162$	0	0
$163$	− 66616.0i	− 0.196386i	−0.995167	−	0.0981928i	$-0.968694\pi$
$163$	− 66616.0i	− 0.196386i	0.995167	−	0.0981928i	$-0.0313062\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	205692.i	0.570724i	0.958420	+	0.285362i	$0.0921138\pi$
$167$	205692.i	0.570724i	−0.958420	+	0.285362i	$0.907886\pi$
$168$	0	0
$169$	−523623.	−1.41027
$170$	0	0
$171$	−166500.	−0.435436
$172$	0	0
$173$	− 433854.i	− 1.10212i	−0.834466	−	0.551059i	$-0.814224\pi$
$173$	− 433854.i	− 1.10212i	0.834466	−	0.551059i	$-0.185776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	191520.i	0.459495i
$178$	0	0
$179$	−489180.	−1.14113	−0.570566	−	0.821252i	$-0.693276\pi$
$179$	−489180.	−1.14113	−0.570566	+	0.821252i	$0.693276\pi$
$180$	0	0
$181$	719462.	1.63234	0.816172	−	0.577810i	$-0.196092\pi$
$181$	719462.	1.63234	0.816172	+	0.577810i	$0.196092\pi$
$182$	0	0
$183$	398928.i	0.880576i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	29304.0i	0.0612806i
$188$	0	0
$189$	371520.	0.756533
$190$	0	0
$191$	185928.	0.368775	0.184387	−	0.982854i	$-0.440970\pi$
$191$	185928.	0.368775	0.184387	+	0.982854i	$0.440970\pi$
$192$	0	0
$193$	591406.i	1.14286i	0.820651	+	0.571429i	$0.193611\pi$
$193$	591406.i	1.14286i	−0.820651	+	0.571429i	$0.806389\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	449478.i	0.825169i	0.910919	+	0.412584i	$0.135374\pi$
$197$	449478.i	0.825169i	−0.910919	+	0.412584i	$0.864626\pi$
$198$	0	0
$199$	157160.	0.281326	0.140663	−	0.990058i	$-0.455077\pi$
$199$	157160.	0.281326	0.140663	+	0.990058i	$0.455077\pi$
$200$	0	0
$201$	43392.0	0.0757564
$202$	0	0
$203$	− 376680.i	− 0.641553i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.18681e6i	− 1.92511i
$208$	0	0
$209$	−66000.0	−0.104515
$210$	0	0
$211$	−253052.	−0.391294	−0.195647	−	0.980674i	$-0.562681\pi$
$211$	−253052.	−0.391294	−0.195647	+	0.980674i	$0.562681\pi$
$212$	0	0
$213$	588672.i	0.889046i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 397664.i	− 0.573280i
$218$	0	0
$219$	491376.	0.692315
$220$	0	0
$221$	210012.	0.289243
$222$	0	0
$223$	1.07344e6i	1.44550i	0.691111	+	0.722749i	$0.257122\pi$
$223$	1.07344e6i	1.44550i	−0.691111	+	0.722749i	$0.742878\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	626832.i	0.807396i	0.914892	+	0.403698i	$0.132275\pi$
$227$	626832.i	0.807396i	−0.914892	+	0.403698i	$0.867725\pi$
$228$	0	0
$229$	116650.	0.146993	0.0734964	−	0.997295i	$-0.476584\pi$
$229$	116650.	0.146993	0.0734964	+	0.997295i	$0.476584\pi$
$230$	0	0
$231$	544896.	0.671868
$232$	0	0
$233$	743046.i	0.896656i	0.893869	+	0.448328i	$0.147980\pi$
$233$	743046.i	0.896656i	−0.893869	+	0.448328i	$0.852020\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.10976e6i	− 1.28339i
$238$	0	0
$239$	978720.	1.10832	0.554158	−	0.832411i	$-0.313040\pi$
$239$	978720.	1.10832	0.554158	+	0.832411i	$0.313040\pi$
$240$	0	0
$241$	−1.13280e6	−1.25635	−0.628174	−	0.778073i	$-0.716197\pi$
$241$	−1.13280e6	−1.25635	−0.628174	+	0.778073i	$0.716197\pi$
$242$	0	0
$243$	− 1.22278e6i	− 1.32841i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	473000.i	0.493309i
$248$	0	0
$249$	1.23782e6	1.26520
$250$	0	0
$251$	−905652.	−0.907355	−0.453677	−	0.891166i	$-0.649888\pi$
$251$	−905652.	−0.907355	−0.453677	+	0.891166i	$0.649888\pi$
$252$	0	0
$253$	− 470448.i	− 0.462073i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.93994e6i	1.83212i	0.401036	+	0.916062i	$0.368650\pi$
$257$	1.93994e6i	1.83212i	−0.401036	+	0.916062i	$0.631350\pi$
$258$	0	0
$259$	1.93362e6	1.79111
$260$	0	0
$261$	729270.	0.662654
$262$	0	0
$263$	− 805476.i	− 0.718064i	−0.933325	−	0.359032i	$-0.883107\pi$
$263$	− 805476.i	− 0.718064i	0.933325	−	0.359032i	$-0.116893\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.64744e6i	2.27273i
$268$	0	0
$269$	858690.	0.723529	0.361764	−	0.932270i	$-0.382175\pi$
$269$	858690.	0.723529	0.361764	+	0.932270i	$0.382175\pi$
$270$	0	0
$271$	383608.	0.317296	0.158648	−	0.987335i	$-0.449287\pi$
$271$	383608.	0.317296	0.158648	+	0.987335i	$0.449287\pi$
$272$	0	0
$273$	− 3.90509e6i	− 3.17120i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.01076e6i	1.57456i	0.616593	+	0.787282i	$0.288512\pi$
$277$	2.01076e6i	1.57456i	−0.616593	+	0.787282i	$0.711488\pi$
$278$	0	0
$279$	769896.	0.592136
$280$	0	0
$281$	202602.	0.153066	0.0765329	−	0.997067i	$-0.475615\pi$
$281$	202602.	0.153066	0.0765329	+	0.997067i	$0.475615\pi$
$282$	0	0
$283$	− 221536.i	− 0.164429i	−0.996615	−	0.0822145i	$-0.973801\pi$
$283$	− 221536.i	− 0.164429i	0.996615	−	0.0822145i	$-0.0261992\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	213624.i	0.153089i
$288$	0	0
$289$	1.37057e6	0.965289
$290$	0	0
$291$	1.88117e6	1.30225
$292$	0	0
$293$	322506.i	0.219467i	0.993961	+	0.109733i	$0.0349997\pi$
$293$	322506.i	0.219467i	−0.993961	+	0.109733i	$0.965000\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	285120.i	0.187558i
$298$	0	0
$299$	−3.37154e6	−2.18098
$300$	0	0
$301$	−3.54733e6	−2.25676
$302$	0	0
$303$	3.40661e6i	2.13165i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.44301e6i	− 0.873822i	−0.899505	−	0.436911i	$-0.856073\pi$
$307$	− 1.44301e6i	− 0.873822i	0.899505	−	0.436911i	$-0.143927\pi$
$308$	0	0
$309$	10464.0	0.00623450
$310$	0	0
$311$	−171312.	−0.100435	−0.0502177	−	0.998738i	$-0.515992\pi$
$311$	−171312.	−0.100435	−0.0502177	+	0.998738i	$0.515992\pi$
$312$	0	0
$313$	1.02689e6i	0.592463i	0.955116	+	0.296232i	$0.0957300\pi$
$313$	1.02689e6i	0.592463i	−0.955116	+	0.296232i	$0.904270\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	752958.i	0.420845i	0.977610	+	0.210423i	$0.0674840\pi$
$317$	752958.i	0.420845i	−0.977610	+	0.210423i	$0.932516\pi$
$318$	0	0
$319$	289080.	0.159053
$320$	0	0
$321$	5.59123e6	3.02862
$322$	0	0
$323$	− 111000.i	− 0.0591993i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.19640e6i	2.17024i
$328$	0	0
$329$	1.13314e6	0.577155
$330$	0	0
$331$	−1.99413e6	−1.00042	−0.500212	−	0.865903i	$-0.666745\pi$
$331$	−1.99413e6	−1.00042	−0.500212	+	0.865903i	$0.666745\pi$
$332$	0	0
$333$	3.74359e6i	1.85002i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 987022.i	− 0.473426i	−0.971580	−	0.236713i	$-0.923930\pi$
$337$	− 987022.i	− 0.473426i	0.971580	−	0.236713i	$-0.0760701\pi$
$338$	0	0
$339$	4.39186e6	2.07562
$340$	0	0
$341$	305184.	0.142127
$342$	0	0
$343$	693160.i	0.318125i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.20601e6i	− 0.983520i	−0.870731	−	0.491760i	$-0.836354\pi$
$347$	− 2.20601e6i	− 0.983520i	0.870731	−	0.491760i	$-0.163646\pi$
$348$	0	0
$349$	−2.74187e6	−1.20499	−0.602495	−	0.798123i	$-0.705827\pi$
$349$	−2.74187e6	−1.20499	−0.602495	+	0.798123i	$0.705827\pi$
$350$	0	0
$351$	2.04336e6	0.885273
$352$	0	0
$353$	2.38957e6i	1.02066i	0.859978	+	0.510331i	$0.170477\pi$
$353$	2.38957e6i	1.02066i	−0.859978	+	0.510331i	$0.829523\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	916416.i	0.380559i
$358$	0	0
$359$	−279480.	−0.114450	−0.0572248	−	0.998361i	$-0.518225\pi$
$359$	−279480.	−0.114450	−0.0572248	+	0.998361i	$0.518225\pi$
$360$	0	0
$361$	−2.22610e6	−0.899035
$362$	0	0
$363$	− 3.44705e6i	− 1.37303i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.47637e6i	0.959734i	0.877341	+	0.479867i	$0.159315\pi$
$367$	2.47637e6i	0.959734i	−0.877341	+	0.479867i	$0.840685\pi$
$368$	0	0
$369$	−413586.	−0.158125
$370$	0	0
$371$	−3.62335e6	−1.36671
$372$	0	0
$373$	− 2.74525e6i	− 1.02167i	−0.859679	−	0.510835i	$-0.829336\pi$
$373$	− 2.74525e6i	− 1.02167i	0.859679	−	0.510835i	$-0.170664\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.07174e6i	− 0.750727i
$378$	0	0
$379$	−1.18906e6	−0.425212	−0.212606	−	0.977138i	$-0.568195\pi$
$379$	−1.18906e6	−0.425212	−0.212606	+	0.977138i	$0.568195\pi$
$380$	0	0
$381$	−2.93885e6	−1.03721
$382$	0	0
$383$	3.25760e6i	1.13475i	0.823458	+	0.567377i	$0.192042\pi$
$383$	3.25760e6i	1.13475i	−0.823458	+	0.567377i	$0.807958\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6.86779e6i	− 2.33099i
$388$	0	0
$389$	−1.98351e6	−0.664600	−0.332300	−	0.943174i	$-0.607825\pi$
$389$	−1.98351e6	−0.664600	−0.332300	+	0.943174i	$0.607825\pi$
$390$	0	0
$391$	791208.	0.261727
$392$	0	0
$393$	5.80579e6i	1.89618i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.97416e6i	1.58396i	0.610549	+	0.791978i	$0.290949\pi$
$397$	4.97416e6i	1.58396i	−0.610549	+	0.791978i	$0.709051\pi$
$398$	0	0
$399$	−2.06400e6	−0.649049
$400$	0	0
$401$	−1.34264e6	−0.416963	−0.208482	−	0.978026i	$-0.566852\pi$
$401$	−1.34264e6	−0.416963	−0.208482	+	0.978026i	$0.566852\pi$
$402$	0	0
$403$	− 2.18715e6i	− 0.670836i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.48394e6i	0.444050i
$408$	0	0
$409$	1.09423e6	0.323445	0.161722	−	0.986836i	$-0.448295\pi$
$409$	1.09423e6	0.323445	0.161722	+	0.986836i	$0.448295\pi$
$410$	0	0
$411$	−6.65035e6	−1.94196
$412$	0	0
$413$	1.37256e6i	0.395964i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.64496e6i	− 1.30810i
$418$	0	0
$419$	−954060.	−0.265485	−0.132743	−	0.991151i	$-0.542378\pi$
$419$	−954060.	−0.265485	−0.132743	+	0.991151i	$0.542378\pi$
$420$	0	0
$421$	−1.59390e6	−0.438284	−0.219142	−	0.975693i	$-0.570326\pi$
$421$	−1.59390e6	−0.438284	−0.219142	+	0.975693i	$0.570326\pi$
$422$	0	0
$423$	2.19380e6i	0.596138i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.85898e6i	0.758826i
$428$	0	0
$429$	2.99693e6	0.786200
$430$	0	0
$431$	2.64665e6	0.686283	0.343141	−	0.939284i	$-0.388509\pi$
$431$	2.64665e6	0.686283	0.343141	+	0.939284i	$0.388509\pi$
$432$	0	0
$433$	− 3.72355e6i	− 0.954416i	−0.878790	−	0.477208i	$-0.841649\pi$
$433$	− 3.72355e6i	− 0.954416i	0.878790	−	0.477208i	$-0.158351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.78200e6i	0.446379i
$438$	0	0
$439$	−2.58340e6	−0.639780	−0.319890	−	0.947455i	$-0.603646\pi$
$439$	−2.58340e6	−0.639780	−0.319890	+	0.947455i	$0.603646\pi$
$440$	0	0
$441$	4.25474e6	1.04178
$442$	0	0
$443$	7.56206e6i	1.83076i	0.402593	+	0.915379i	$0.368109\pi$
$443$	7.56206e6i	1.83076i	−0.402593	+	0.915379i	$0.631891\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.37320e6i	− 0.798497i
$448$	0	0
$449$	−4.30773e6	−1.00840	−0.504200	−	0.863587i	$-0.668212\pi$
$449$	−4.30773e6	−1.00840	−0.504200	+	0.863587i	$0.668212\pi$
$450$	0	0
$451$	−163944.	−0.0379537
$452$	0	0
$453$	− 1.04148e7i	− 2.38456i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.24354e6i	− 0.502509i	−0.967921	−	0.251254i	$-0.919157\pi$
$457$	− 2.24354e6i	− 0.502509i	0.967921	−	0.251254i	$-0.0808431\pi$
$458$	0	0
$459$	−479520.	−0.106237
$460$	0	0
$461$	1.65670e6	0.363071	0.181536	−	0.983384i	$-0.441893\pi$
$461$	1.65670e6	0.363071	0.181536	+	0.983384i	$0.441893\pi$
$462$	0	0
$463$	− 2.89160e6i	− 0.626881i	−0.949608	−	0.313441i	$-0.898518\pi$
$463$	− 2.89160e6i	− 0.626881i	0.949608	−	0.313441i	$-0.101482\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.52699e6i	1.38491i	0.721462	+	0.692454i	$0.243470\pi$
$467$	6.52699e6i	1.38491i	−0.721462	+	0.692454i	$0.756530\pi$
$468$	0	0
$469$	310976.	0.0652822
$470$	0	0
$471$	1.33421e7	2.77123
$472$	0	0
$473$	− 2.72237e6i	− 0.559492i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 7.01498e6i	− 1.41166i
$478$	0	0
$479$	−5.96232e6	−1.18734	−0.593672	−	0.804707i	$-0.702322\pi$
$479$	−5.96232e6	−1.18734	−0.593672	+	0.804707i	$0.702322\pi$
$480$	0	0
$481$	1.06349e7	2.09591
$482$	0	0
$483$	− 1.47122e7i	− 2.86952i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.99191e6i	− 0.571644i	−0.958283	−	0.285822i	$-0.907733\pi$
$487$	− 2.99191e6i	− 0.571644i	0.958283	−	0.285822i	$-0.0922666\pi$
$488$	0	0
$489$	1.59878e6	0.302355
$490$	0	0
$491$	1.20419e6	0.225419	0.112710	−	0.993628i	$-0.464047\pi$
$491$	1.20419e6	0.225419	0.112710	+	0.993628i	$0.464047\pi$
$492$	0	0
$493$	486180.i	0.0900907i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.21882e6i	0.766125i
$498$	0	0
$499$	9.20546e6	1.65499	0.827493	−	0.561477i	$-0.189767\pi$
$499$	9.20546e6	1.65499	0.827493	+	0.561477i	$0.189767\pi$
$500$	0	0
$501$	−4.93661e6	−0.878687
$502$	0	0
$503$	− 3.35956e6i	− 0.592055i	−0.955179	−	0.296027i	$-0.904338\pi$
$503$	− 3.35956e6i	− 0.592055i	0.955179	−	0.296027i	$-0.0956620\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.25670e7i	− 2.17125i
$508$	0	0
$509$	2.53701e6	0.434038	0.217019	−	0.976167i	$-0.430367\pi$
$509$	2.53701e6	0.434038	0.217019	+	0.976167i	$0.430367\pi$
$510$	0	0
$511$	3.52153e6	0.596594
$512$	0	0
$513$	− 1.08000e6i	− 0.181188i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	869616.i	0.143087i
$518$	0	0
$519$	1.04125e7	1.69682
$520$	0	0
$521$	−9.31580e6	−1.50358	−0.751789	−	0.659404i	$-0.770809\pi$
$521$	−9.31580e6	−1.50358	−0.751789	+	0.659404i	$0.770809\pi$
$522$	0	0
$523$	− 5.02802e6i	− 0.803790i	−0.915686	−	0.401895i	$-0.868352\pi$
$523$	− 5.02802e6i	− 0.803790i	0.915686	−	0.401895i	$-0.131648\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	513264.i	0.0805034i
$528$	0	0
$529$	−6.26575e6	−0.973496
$530$	0	0
$531$	−2.65734e6	−0.408988
$532$	0	0
$533$	1.17493e6i	0.179141i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.17403e7i	− 1.75689i
$538$	0	0
$539$	1.68656e6	0.250052
$540$	0	0
$541$	134222.	0.0197165	0.00985827	−	0.999951i	$-0.496862\pi$
$541$	134222.	0.0197165	0.00985827	+	0.999951i	$0.496862\pi$
$542$	0	0
$543$	1.72671e7i	2.51316i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 605648.i	− 0.0865470i	−0.999063	−	0.0432735i	$-0.986221\pi$
$547$	− 605648.i	− 0.0865470i	0.999063	−	0.0432735i	$-0.0137787\pi$
$548$	0	0
$549$	−5.53513e6	−0.783784
$550$	0	0
$551$	−1.09500e6	−0.153651
$552$	0	0
$553$	− 7.95328e6i	− 1.10594i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.06240e6i	− 0.964527i	−0.876026	−	0.482264i	$-0.839815\pi$
$557$	− 7.06240e6i	− 0.964527i	0.876026	−	0.482264i	$-0.160185\pi$
$558$	0	0
$559$	−1.95103e7	−2.64079
$560$	0	0
$561$	−703296.	−0.0943476
$562$	0	0
$563$	− 1.03029e7i	− 1.36990i	−0.728588	−	0.684952i	$-0.759823\pi$
$563$	− 1.03029e7i	− 1.36990i	0.728588	−	0.684952i	$-0.240177\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 5.00159e6i	− 0.653357i
$568$	0	0
$569$	−1.04769e6	−0.135660	−0.0678300	−	0.997697i	$-0.521608\pi$
$569$	−1.04769e6	−0.135660	−0.0678300	+	0.997697i	$0.521608\pi$
$570$	0	0
$571$	−1.40765e7	−1.80677	−0.903385	−	0.428830i	$-0.858926\pi$
$571$	−1.40765e7	−1.80677	−0.903385	+	0.428830i	$0.858926\pi$
$572$	0	0
$573$	4.46227e6i	0.567766i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.62682e6i	0.203423i	0.994814	+	0.101711i	$0.0324318\pi$
$577$	1.62682e6i	0.203423i	−0.994814	+	0.101711i	$0.967568\pi$
$578$	0	0
$579$	−1.41937e7	−1.75955
$580$	0	0
$581$	8.87107e6	1.09027
$582$	0	0
$583$	− 2.78071e6i	− 0.338832i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.96089e6i	− 0.833814i	−0.908949	−	0.416907i	$-0.863114\pi$
$587$	− 6.96089e6i	− 0.833814i	0.908949	−	0.416907i	$-0.136886\pi$
$588$	0	0
$589$	−1.15600e6	−0.137300
$590$	0	0
$591$	−1.07875e7	−1.27043
$592$	0	0
$593$	1.13639e7i	1.32706i	0.748150	+	0.663529i	$0.230942\pi$
$593$	1.13639e7i	1.32706i	−0.748150	+	0.663529i	$0.769058\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.77184e6i	0.433129i
$598$	0	0
$599$	1.48688e7	1.69321	0.846603	−	0.532224i	$-0.178644\pi$
$599$	1.48688e7	1.69321	0.846603	+	0.532224i	$0.178644\pi$
$600$	0	0
$601$	−1.23612e6	−0.139596	−0.0697981	−	0.997561i	$-0.522236\pi$
$601$	−1.23612e6	−0.139596	−0.0697981	+	0.997561i	$0.522236\pi$
$602$	0	0
$603$	602064.i	0.0674294i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.24498e7i	1.37149i	0.727844	+	0.685743i	$0.240522\pi$
$607$	1.24498e7i	1.37149i	−0.727844	+	0.685743i	$0.759478\pi$
$608$	0	0
$609$	9.04032e6	0.987735
$610$	0	0
$611$	6.23225e6	0.675370
$612$	0	0
$613$	8.73491e6i	0.938873i	0.882966	+	0.469437i	$0.155543\pi$
$613$	8.73491e6i	0.938873i	−0.882966	+	0.469437i	$0.844457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.25495e7i	1.32713i	0.748119	+	0.663565i	$0.230957\pi$
$617$	1.25495e7i	1.32713i	−0.748119	+	0.663565i	$0.769043\pi$
$618$	0	0
$619$	−1.46658e7	−1.53843	−0.769216	−	0.638988i	$-0.779353\pi$
$619$	−1.46658e7	−1.53843	−0.769216	+	0.638988i	$0.779353\pi$
$620$	0	0
$621$	7.69824e6	0.801055
$622$	0	0
$623$	1.89733e7i	1.95850i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.58400e6i	− 0.160911i
$628$	0	0
$629$	−2.49572e6	−0.251519
$630$	0	0
$631$	196288.	0.0196255	0.00981274	−	0.999952i	$-0.496876\pi$
$631$	196288.	0.0196255	0.00981274	+	0.999952i	$0.496876\pi$
$632$	0	0
$633$	− 6.07325e6i	− 0.602437i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.20870e7i	− 1.18024i
$638$	0	0
$639$	−8.16782e6	−0.791324
$640$	0	0
$641$	−1.11596e7	−1.07276	−0.536381	−	0.843976i	$-0.680209\pi$
$641$	−1.11596e7	−1.07276	−0.536381	+	0.843976i	$0.680209\pi$
$642$	0	0
$643$	− 2.25158e6i	− 0.214763i	−0.994218	−	0.107381i	$-0.965753\pi$
$643$	− 2.25158e6i	− 0.214763i	0.994218	−	0.107381i	$-0.0342466\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.05319e6i	− 0.756323i	−0.925740	−	0.378161i	$-0.876556\pi$
$647$	− 8.05319e6i	− 0.756323i	0.925740	−	0.378161i	$-0.123444\pi$
$648$	0	0
$649$	−1.05336e6	−0.0981669
$650$	0	0
$651$	9.54394e6	0.882623
$652$	0	0
$653$	416466.i	0.0382205i	0.999817	+	0.0191103i	$0.00608336\pi$
$653$	416466.i	0.0382205i	−0.999817	+	0.0191103i	$0.993917\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.81784e6i	0.616217i
$658$	0	0
$659$	1.31721e7	1.18152	0.590761	−	0.806847i	$-0.298828\pi$
$659$	1.31721e7	1.18152	0.590761	+	0.806847i	$0.298828\pi$
$660$	0	0
$661$	−1.69494e6	−0.150886	−0.0754432	−	0.997150i	$-0.524037\pi$
$661$	−1.69494e6	−0.150886	−0.0754432	+	0.997150i	$0.524037\pi$
$662$	0	0
$663$	5.04029e6i	0.445319i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.80516e6i	− 0.679309i
$668$	0	0
$669$	−2.57627e7	−2.22549
$670$	0	0
$671$	−2.19410e6	−0.188127
$672$	0	0
$673$	8.91605e6i	0.758813i	0.925230	+	0.379406i	$0.123872\pi$
$673$	8.91605e6i	0.758813i	−0.925230	+	0.379406i	$0.876128\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.42894e7i	− 1.19824i	−0.800661	−	0.599118i	$-0.795518\pi$
$677$	− 1.42894e7i	− 1.19824i	0.800661	−	0.599118i	$-0.204482\pi$
$678$	0	0
$679$	1.34817e7	1.12220
$680$	0	0
$681$	−1.50440e7	−1.24307
$682$	0	0
$683$	− 5.33314e6i	− 0.437452i	−0.975786	−	0.218726i	$-0.929810\pi$
$683$	− 5.33314e6i	− 0.437452i	0.975786	−	0.218726i	$-0.0701902\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.79960e6i	0.226310i
$688$	0	0
$689$	−1.99284e7	−1.59928
$690$	0	0
$691$	−698252.	−0.0556310	−0.0278155	−	0.999613i	$-0.508855\pi$
$691$	−698252.	−0.0556310	−0.0278155	+	0.999613i	$0.508855\pi$
$692$	0	0
$693$	7.56043e6i	0.598017i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 275724.i	− 0.0214977i
$698$	0	0
$699$	−1.78331e7	−1.38049
$700$	0	0
$701$	1.79880e7	1.38257	0.691285	−	0.722582i	$-0.257045\pi$
$701$	1.79880e7	1.38257	0.691285	+	0.722582i	$0.257045\pi$
$702$	0	0
$703$	− 5.62100e6i	− 0.428968i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.44140e7i	1.83692i
$708$	0	0
$709$	1.39464e7	1.04195	0.520975	−	0.853572i	$-0.325568\pi$
$709$	1.39464e7	1.04195	0.520975	+	0.853572i	$0.325568\pi$
$710$	0	0
$711$	1.53979e7	1.14232
$712$	0	0
$713$	− 8.23997e6i	− 0.607018i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.34893e7i	1.70636i
$718$	0	0
$719$	6.22272e6	0.448909	0.224454	−	0.974485i	$-0.427940\pi$
$719$	6.22272e6	0.448909	0.224454	+	0.974485i	$0.427940\pi$
$720$	0	0
$721$	74992.0	0.00537250
$722$	0	0
$723$	− 2.71872e7i	− 1.93427i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.76729e6i	0.545047i	0.962149	+	0.272523i	$0.0878582\pi$
$727$	7.76729e6i	0.545047i	−0.962149	+	0.272523i	$0.912142\pi$
$728$	0	0
$729$	2.22804e7	1.55276
$730$	0	0
$731$	4.57853e6	0.316907
$732$	0	0
$733$	− 2.42083e7i	− 1.66420i	−0.554627	−	0.832099i	$-0.687139\pi$
$733$	− 2.42083e7i	− 1.66420i	0.554627	−	0.832099i	$-0.312861\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	238656.i	0.0161847i
$738$	0	0
$739$	1.26850e7	0.854434	0.427217	−	0.904149i	$-0.359494\pi$
$739$	1.26850e7	0.854434	0.427217	+	0.904149i	$0.359494\pi$
$740$	0	0
$741$	−1.13520e7	−0.759498
$742$	0	0
$743$	1.97632e7i	1.31337i	0.754166	+	0.656684i	$0.228041\pi$
$743$	1.97632e7i	1.31337i	−0.754166	+	0.656684i	$0.771959\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.71748e7i	1.12613i
$748$	0	0
$749$	4.00705e7	2.60988
$750$	0	0
$751$	9.01761e6	0.583434	0.291717	−	0.956505i	$-0.405774\pi$
$751$	9.01761e6	0.583434	0.291717	+	0.956505i	$0.405774\pi$
$752$	0	0
$753$	− 2.17356e7i	− 1.39696i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.12556e6i	− 0.0713887i	−0.999363	−	0.0356944i	$-0.988636\pi$
$757$	− 1.12556e6i	− 0.0713887i	0.999363	−	0.0356944i	$-0.0113643\pi$
$758$	0	0
$759$	1.12908e7	0.711407
$760$	0	0
$761$	2.25747e7	1.41306	0.706529	−	0.707684i	$-0.250260\pi$
$761$	2.25747e7	1.41306	0.706529	+	0.707684i	$0.250260\pi$
$762$	0	0
$763$	3.00742e7i	1.87018i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.54908e6i	0.463346i
$768$	0	0
$769$	632350.	0.0385604	0.0192802	−	0.999814i	$-0.493863\pi$
$769$	632350.	0.0385604	0.0192802	+	0.999814i	$0.493863\pi$
$770$	0	0
$771$	−4.65585e7	−2.82074
$772$	0	0
$773$	1.25867e7i	0.757643i	0.925470	+	0.378822i	$0.123671\pi$
$773$	1.25867e7i	0.757643i	−0.925470	+	0.378822i	$0.876329\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.64070e7i	2.75760i
$778$	0	0
$779$	621000.	0.0366647
$780$	0	0
$781$	−3.23770e6	−0.189937
$782$	0	0
$783$	4.73040e6i	0.275736i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.15792e7i	− 1.24194i	−0.783836	−	0.620968i	$-0.786740\pi$
$787$	− 2.15792e7i	− 1.24194i	0.783836	−	0.620968i	$-0.213260\pi$
$788$	0	0
$789$	1.93314e7	1.10553
$790$	0	0
$791$	3.14750e7	1.78864
$792$	0	0
$793$	1.57244e7i	0.887956i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.09760e7i	− 1.72735i	−0.504052	−	0.863673i	$-0.668158\pi$
$797$	− 3.09760e7i	− 1.72735i	0.504052	−	0.863673i	$-0.331842\pi$
$798$	0	0
$799$	−1.46254e6	−0.0810475
$800$	0	0
$801$	−3.67332e7	−2.02292
$802$	0	0
$803$	2.70257e6i	0.147907i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.06086e7i	1.11395i
$808$	0	0
$809$	−4.24929e6	−0.228268	−0.114134	−	0.993465i	$-0.536409\pi$
$809$	−4.24929e6	−0.228268	−0.114134	+	0.993465i	$0.536409\pi$
$810$	0	0
$811$	−3.42333e6	−0.182767	−0.0913833	−	0.995816i	$-0.529129\pi$
$811$	−3.42333e6	−0.182767	−0.0913833	+	0.995816i	$0.529129\pi$
$812$	0	0
$813$	9.20659e6i	0.488509i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.03120e7i	0.540490i
$818$	0	0
$819$	5.41831e7	2.82263
$820$	0	0
$821$	3.10571e7	1.60806	0.804030	−	0.594588i	$-0.202685\pi$
$821$	3.10571e7	1.60806	0.804030	+	0.594588i	$0.202685\pi$
$822$	0	0
$823$	− 3.11904e7i	− 1.60517i	−0.596538	−	0.802584i	$-0.703458\pi$
$823$	− 3.11904e7i	− 1.60517i	0.596538	−	0.802584i	$-0.296542\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.28487e6i	0.421233i	0.977569	+	0.210616i	$0.0675471\pi$
$827$	8.28487e6i	0.421233i	−0.977569	+	0.210616i	$0.932453\pi$
$828$	0	0
$829$	1.81688e7	0.918208	0.459104	−	0.888383i	$-0.348171\pi$
$829$	1.81688e7	0.918208	0.459104	+	0.888383i	$0.348171\pi$
$830$	0	0
$831$	−4.82582e7	−2.42420
$832$	0	0
$833$	2.83649e6i	0.141635i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.99392e6i	0.246393i
$838$	0	0
$839$	−1.02743e7	−0.503902	−0.251951	−	0.967740i	$-0.581072\pi$
$839$	−1.02743e7	−0.503902	−0.251951	+	0.967740i	$0.581072\pi$
$840$	0	0
$841$	−1.57150e7	−0.766171
$842$	0	0
$843$	4.86245e6i	0.235660i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.47038e7i	− 1.18319i
$848$	0	0
$849$	5.31686e6	0.253155
$850$	0	0
$851$	4.00665e7	1.89652
$852$	0	0
$853$	− 6.28597e6i	− 0.295801i	−0.989002	−	0.147901i	$-0.952748\pi$
$853$	− 6.28597e6i	− 0.295801i	0.989002	−	0.147901i	$-0.0472516\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.54050e7i	0.716490i	0.933628	+	0.358245i	$0.116625\pi$
$857$	1.54050e7i	0.716490i	−0.933628	+	0.358245i	$0.883375\pi$
$858$	0	0
$859$	1.43526e7	0.663664	0.331832	−	0.943338i	$-0.392333\pi$
$859$	1.43526e7	0.663664	0.331832	+	0.943338i	$0.392333\pi$
$860$	0	0
$861$	−5.12698e6	−0.235697
$862$	0	0
$863$	1.33278e7i	0.609158i	0.952487	+	0.304579i	$0.0985158\pi$
$863$	1.33278e7i	0.609158i	−0.952487	+	0.304579i	$0.901484\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.28938e7i	1.48616i
$868$	0	0
$869$	6.10368e6	0.274184
$870$	0	0
$871$	1.71037e6	0.0763913
$872$	0	0
$873$	2.61012e7i	1.15911i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.24846e7i	1.42620i	0.701065	+	0.713098i	$0.252708\pi$
$877$	3.24846e7i	1.42620i	−0.701065	+	0.713098i	$0.747292\pi$
$878$	0	0
$879$	−7.74014e6	−0.337891
$880$	0	0
$881$	1.54600e7	0.671073	0.335537	−	0.942027i	$-0.391082\pi$
$881$	1.54600e7	0.671073	0.335537	+	0.942027i	$0.391082\pi$
$882$	0	0
$883$	− 1.69478e6i	− 0.0731494i	−0.999331	−	0.0365747i	$-0.988355\pi$
$883$	− 1.69478e6i	− 0.0731494i	0.999331	−	0.0365747i	$-0.0116447\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.87257e6i	0.122592i	0.998120	+	0.0612960i	$0.0195233\pi$
$887$	2.87257e6i	0.122592i	−0.998120	+	0.0612960i	$0.980477\pi$
$888$	0	0
$889$	−2.10617e7	−0.893799
$890$	0	0
$891$	3.83843e6	0.161979
$892$	0	0
$893$	− 3.29400e6i	− 0.138228i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.09171e7i	− 3.35783i
$898$	0	0
$899$	5.06328e6	0.208945
$900$	0	0
$901$	4.67665e6	0.191921
$902$	0	0
$903$	− 8.51359e7i	− 3.47451i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.95422e7i	− 1.59603i	−0.602635	−	0.798017i	$-0.705882\pi$
$907$	− 3.95422e7i	− 1.59603i	0.602635	−	0.798017i	$-0.294118\pi$
$908$	0	0
$909$	−4.72667e7	−1.89734
$910$	0	0
$911$	−1.13178e7	−0.451819	−0.225909	−	0.974148i	$-0.572535\pi$
$911$	−1.13178e7	−0.451819	−0.225909	+	0.974148i	$0.572535\pi$
$912$	0	0
$913$	6.80803e6i	0.270299i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.16082e7i	1.63401i
$918$	0	0
$919$	8.51348e6	0.332520	0.166260	−	0.986082i	$-0.446831\pi$
$919$	8.51348e6	0.332520	0.166260	+	0.986082i	$0.446831\pi$
$920$	0	0
$921$	3.46322e7	1.34534
$922$	0	0
$923$	2.32035e7i	0.896497i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	145188.i	0.00554921i
$928$	0	0
$929$	7.54587e6	0.286860	0.143430	−	0.989660i	$-0.454187\pi$
$929$	7.54587e6	0.286860	0.143430	+	0.989660i	$0.454187\pi$
$930$	0	0
$931$	−6.38850e6	−0.241560
$932$	0	0
$933$	− 4.11149e6i	− 0.154630i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.84500e7i	− 0.686512i	−0.939242	−	0.343256i	$-0.888470\pi$
$937$	− 1.84500e7i	− 0.686512i	0.939242	−	0.343256i	$-0.111530\pi$
$938$	0	0
$939$	−2.46453e7	−0.912157
$940$	0	0
$941$	6.75046e6	0.248519	0.124259	−	0.992250i	$-0.460344\pi$
$941$	6.75046e6	0.248519	0.124259	+	0.992250i	$0.460344\pi$
$942$	0	0
$943$	4.42649e6i	0.162099i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.45677e6i	− 0.233959i	−0.993134	−	0.116980i	$-0.962679\pi$
$947$	− 6.45677e6i	− 0.233959i	0.993134	−	0.116980i	$-0.0373212\pi$
$948$	0	0
$949$	1.93684e7	0.698117
$950$	0	0
$951$	−1.80710e7	−0.647934
$952$	0	0
$953$	3.96648e7i	1.41473i	0.706849	+	0.707364i	$0.250116\pi$
$953$	3.96648e7i	1.41473i	−0.706849	+	0.707364i	$0.749884\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.93792e6i	0.244878i
$958$	0	0
$959$	−4.76609e7	−1.67346
$960$	0	0
$961$	−2.32838e7	−0.813290
$962$	0	0
$963$	7.75783e7i	2.69572i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.43015e7i	1.17963i	0.807538	+	0.589816i	$0.200800\pi$
$967$	3.43015e7i	1.17963i	−0.807538	+	0.589816i	$0.799200\pi$
$968$	0	0
$969$	2.66400e6	0.0911433
$970$	0	0
$971$	5.77115e6	0.196433	0.0982164	−	0.995165i	$-0.468686\pi$
$971$	5.77115e6	0.196433	0.0982164	+	0.995165i	$0.468686\pi$
$972$	0	0
$973$	− 3.32889e7i	− 1.12724i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.08746e6i	0.237549i	0.992921	+	0.118775i	$0.0378966\pi$
$977$	7.08746e6i	0.237549i	−0.992921	+	0.118775i	$0.962103\pi$
$978$	0	0
$979$	−1.45609e7	−0.485548
$980$	0	0
$981$	−5.82250e7	−1.93169
$982$	0	0
$983$	4.59362e7i	1.51625i	0.652108	+	0.758126i	$0.273885\pi$
$983$	4.59362e7i	1.51625i	−0.652108	+	0.758126i	$0.726115\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.71953e7i	0.888588i
$988$	0	0
$989$	−7.35039e7	−2.38957
$990$	0	0
$991$	4.50298e7	1.45652	0.728260	−	0.685301i	$-0.240329\pi$
$991$	4.50298e7	1.45652	0.728260	+	0.685301i	$0.240329\pi$
$992$	0	0
$993$	− 4.78592e7i	− 1.54025i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.37364e7i	− 0.756271i	−0.925750	−	0.378136i	$-0.876565\pi$
$997$	− 2.37364e7i	− 0.756271i	0.925750	−	0.378136i	$-0.123435\pi$
$998$	0	0
$999$	−2.42827e7	−0.769810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.b.49.2		2
4.3	odd	2		50.6.b.a.49.1		2
5.2	odd	4		400.6.a.n.1.1		1
5.3	odd	4		80.6.a.a.1.1		1
5.4	even	2	inner	400.6.c.b.49.1		2
12.11	even	2		450.6.c.h.199.2		2
15.8	even	4		720.6.a.j.1.1		1
20.3	even	4		10.6.a.b.1.1	✓	1
20.7	even	4		50.6.a.d.1.1		1
20.19	odd	2		50.6.b.a.49.2		2
40.3	even	4		320.6.a.b.1.1		1
40.13	odd	4		320.6.a.o.1.1		1
60.23	odd	4		90.6.a.d.1.1		1
60.47	odd	4		450.6.a.l.1.1		1
60.59	even	2		450.6.c.h.199.1		2
140.83	odd	4		490.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.b.1.1	✓	1	20.3	even	4
50.6.a.d.1.1		1	20.7	even	4
50.6.b.a.49.1		2	4.3	odd	2
50.6.b.a.49.2		2	20.19	odd	2
80.6.a.a.1.1		1	5.3	odd	4
90.6.a.d.1.1		1	60.23	odd	4
320.6.a.b.1.1		1	40.3	even	4
320.6.a.o.1.1		1	40.13	odd	4
400.6.a.n.1.1		1	5.2	odd	4
400.6.c.b.49.1		2	5.4	even	2	inner
400.6.c.b.49.2		2	1.1	even	1	trivial
450.6.a.l.1.1		1	60.47	odd	4
450.6.c.h.199.1		2	60.59	even	2
450.6.c.h.199.2		2	12.11	even	2
490.6.a.a.1.1		1	140.83	odd	4
720.6.a.j.1.1		1	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+24.0000i q^{3} +172.000i q^{7} -333.000 q^{9} +O(q^{10})\)	\(q+24.0000i q^{3} +172.000i q^{7} -333.000 q^{9} -132.000 q^{11} +946.000i q^{13} -222.000i q^{17} +500.000 q^{19} -4128.00 q^{21} +3564.00i q^{23} -2160.00i q^{27} -2190.00 q^{29} -2312.00 q^{31} -3168.00i q^{33} -11242.0i q^{37} -22704.0 q^{39} +1242.00 q^{41} +20624.0i q^{43} -6588.00i q^{47} -12777.0 q^{49} +5328.00 q^{51} +21066.0i q^{53} +12000.0i q^{57} +7980.00 q^{59} +16622.0 q^{61} -57276.0i q^{63} -1808.00i q^{67} -85536.0 q^{69} +24528.0 q^{71} -20474.0i q^{73} -22704.0i q^{77} -46240.0 q^{79} -29079.0 q^{81} -51576.0i q^{83} -52560.0i q^{87} +110310. q^{89} -162712. q^{91} -55488.0i q^{93} -78382.0i q^{97} +43956.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 666 q^{9}+O(q^{10}) \) 2 * q - 666 * q^9	\( 2 q - 666 q^{9} - 264 q^{11} + 1000 q^{19} - 8256 q^{21} - 4380 q^{29} - 4624 q^{31} - 45408 q^{39} + 2484 q^{41} - 25554 q^{49} + 10656 q^{51} + 15960 q^{59} + 33244 q^{61} - 171072 q^{69} + 49056 q^{71} - 92480 q^{79} - 58158 q^{81} + 220620 q^{89} - 325424 q^{91} + 87912 q^{99}+O(q^{100}) \) 2 * q - 666 * q^9 - 264 * q^11 + 1000 * q^19 - 8256 * q^21 - 4380 * q^29 - 4624 * q^31 - 45408 * q^39 + 2484 * q^41 - 25554 * q^49 + 10656 * q^51 + 15960 * q^59 + 33244 * q^61 - 171072 * q^69 + 49056 * q^71 - 92480 * q^79 - 58158 * q^81 + 220620 * q^89 - 325424 * q^91 + 87912 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more