LMFDB - Embedded newform 234.6.b.b.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(181,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("234.181");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$234 = 2 \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	234.b (of order $2$ , degree $1$ , 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:	$37.5298138362$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	234.181
Dual form			234.6.b.b.181.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} -51.0000i q^{5} +105.000i q^{7} -64.0000i q^{8} +204.000 q^{10} -120.000i q^{11} +(-598.000 + 117.000i) q^{13} -420.000 q^{14} +256.000 q^{16} +1101.00 q^{17} -1170.00i q^{19} +816.000i q^{20} +480.000 q^{22} -1050.00 q^{23} +524.000 q^{25} +(-468.000 - 2392.00i) q^{26} -1680.00i q^{28} +4104.00 q^{29} +9624.00i q^{31} +1024.00i q^{32} +4404.00i q^{34} +5355.00 q^{35} +8709.00i q^{37} +4680.00 q^{38} -3264.00 q^{40} +9480.00i q^{41} +9995.00 q^{43} +1920.00i q^{44} -4200.00i q^{46} +2943.00i q^{47} +5782.00 q^{49} +2096.00i q^{50} +(9568.00 - 1872.00i) q^{52} +750.000 q^{53} -6120.00 q^{55} +6720.00 q^{56} +16416.0i q^{58} +40938.0i q^{59} -57920.0 q^{61} -38496.0 q^{62} -4096.00 q^{64} +(5967.00 + 30498.0i) q^{65} +22812.0i q^{67} -17616.0 q^{68} +21420.0i q^{70} -63741.0i q^{71} +58866.0i q^{73} -34836.0 q^{74} +18720.0i q^{76} +12600.0 q^{77} +63202.0 q^{79} -13056.0i q^{80} -37920.0 q^{82} -55458.0i q^{83} -56151.0i q^{85} +39980.0i q^{86} -7680.00 q^{88} +104778. i q^{89} +(-12285.0 - 62790.0i) q^{91} +16800.0 q^{92} -11772.0 q^{94} -59670.0 q^{95} +160452. i q^{97} +23128.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 32 q^{4} + 408 q^{10} - 1196 q^{13} - 840 q^{14} + 512 q^{16} + 2202 q^{17} + 960 q^{22} - 2100 q^{23} + 1048 q^{25} - 936 q^{26} + 8208 q^{29} + 10710 q^{35} + 9360 q^{38} - 6528 q^{40} + 19990 q^{43}+ \cdots - 119340 q^{95}+O(q^{100})$ 2 * q - 32 * q^4 + 408 * q^10 - 1196 * q^13 - 840 * q^14 + 512 * q^16 + 2202 * q^17 + 960 * q^22 - 2100 * q^23 + 1048 * q^25 - 936 * q^26 + 8208 * q^29 + 10710 * q^35 + 9360 * q^38 - 6528 * q^40 + 19990 * q^43 + 11564 * q^49 + 19136 * q^52 + 1500 * q^53 - 12240 * q^55 + 13440 * q^56 - 115840 * q^61 - 76992 * q^62 - 8192 * q^64 + 11934 * q^65 - 35232 * q^68 - 69672 * q^74 + 25200 * q^77 + 126404 * q^79 - 75840 * q^82 - 15360 * q^88 - 24570 * q^91 + 33600 * q^92 - 23544 * q^94 - 119340 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$ .

$n$	$145$	$209$
$\chi(n)$	$-1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$			4.00000i			0.707107i
$2$			4.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−16.0000			−0.500000
$5$		−	51.0000i		−	0.912316i	−0.889899	−	0.456158i	$-0.849225\pi$
$5$		−	51.0000i		−	0.912316i	0.889899	−	0.456158i	$-0.150775\pi$
$6$		0			0
$7$			105.000i			0.809924i	0.914334	+	0.404962i	$0.132715\pi$
$7$			105.000i			0.809924i	−0.914334	+	0.404962i	$0.867285\pi$
$8$		−	64.0000i		−	0.353553i
$9$		0			0
$10$	204.000			0.645105
$11$		−	120.000i		−	0.299020i	−0.988760	−	0.149510i	$-0.952230\pi$
$11$		−	120.000i		−	0.299020i	0.988760	−	0.149510i	$-0.0477695\pi$
$12$		0			0
$13$	−598.000	+	117.000i	−0.981393	+	0.192012i
$13$	−598.000	+	117.000i	−0.981393	+	0.192012i
$14$	−420.000			−0.572703
$15$		0			0
$16$	256.000			0.250000
$17$	1101.00			0.923985			0.461993	−	0.886884i	$-0.347135\pi$
$17$	1101.00			0.923985			0.461993	+	0.886884i	$0.347135\pi$
$18$		0			0
$19$		−	1170.00i		−	0.743536i	−0.928326	−	0.371768i	$-0.878752\pi$
$19$		−	1170.00i		−	0.743536i	0.928326	−	0.371768i	$-0.121248\pi$
$20$			816.000i			0.456158i
$21$		0			0
$22$	480.000			0.211439
$23$	−1050.00			−0.413875			−0.206938	−	0.978354i	$-0.566350\pi$
$23$	−1050.00			−0.413875			−0.206938	+	0.978354i	$0.566350\pi$
$24$		0			0
$25$	524.000			0.167680
$26$	−468.000	−	2392.00i	−0.135773	−	0.693949i
$27$		0			0
$28$		−	1680.00i		−	0.404962i
$29$	4104.00			0.906176			0.453088	−	0.891466i	$-0.350322\pi$
$29$	4104.00			0.906176			0.453088	+	0.891466i	$0.350322\pi$
$30$		0			0
$31$			9624.00i			1.79867i	0.437261	+	0.899335i	$0.355949\pi$
$31$			9624.00i			1.79867i	−0.437261	+	0.899335i	$0.644051\pi$
$32$			1024.00i			0.176777i
$33$		0			0
$34$			4404.00i			0.653356i
$35$	5355.00			0.738906
$36$		0			0
$37$			8709.00i			1.04584i	0.852383	+	0.522918i	$0.175157\pi$
$37$			8709.00i			1.04584i	−0.852383	+	0.522918i	$0.824843\pi$
$38$	4680.00			0.525759
$39$		0			0
$40$	−3264.00			−0.322552
$41$			9480.00i			0.880742i	0.897816	+	0.440371i	$0.145153\pi$
$41$			9480.00i			0.880742i	−0.897816	+	0.440371i	$0.854847\pi$
$42$		0			0
$43$	9995.00			0.824350			0.412175	−	0.911105i	$-0.364769\pi$
$43$	9995.00			0.824350			0.412175	+	0.911105i	$0.364769\pi$
$44$			1920.00i			0.149510i
$45$		0			0
$46$		−	4200.00i		−	0.292654i
$47$			2943.00i			0.194333i	0.995268	+	0.0971663i	$0.0309779\pi$
$47$			2943.00i			0.194333i	−0.995268	+	0.0971663i	$0.969022\pi$
$48$		0			0
$49$	5782.00			0.344023
$50$			2096.00i			0.118568i
$51$		0			0
$52$	9568.00	−	1872.00i	0.490696	−	0.0960058i
$53$	750.000			0.0366751			0.0183376	−	0.999832i	$-0.494163\pi$
$53$	750.000			0.0366751			0.0183376	+	0.999832i	$0.494163\pi$
$54$		0			0
$55$	−6120.00			−0.272800
$56$	6720.00			0.286351
$57$		0			0
$58$			16416.0i			0.640763i
$59$			40938.0i			1.53108i	0.643391	+	0.765538i	$0.277527\pi$
$59$			40938.0i			1.53108i	−0.643391	+	0.765538i	$0.722473\pi$
$60$		0			0
$61$	−57920.0			−1.99298			−0.996492	−	0.0836839i	$-0.973331\pi$
$61$	−57920.0			−1.99298			−0.996492	+	0.0836839i	$0.973331\pi$
$62$	−38496.0			−1.27185
$63$		0			0
$64$	−4096.00			−0.125000
$65$	5967.00	+	30498.0i	0.175175	+	0.895340i
$66$		0			0
$67$			22812.0i			0.620835i	0.950600	+	0.310418i	$0.100469\pi$
$67$			22812.0i			0.620835i	−0.950600	+	0.310418i	$0.899531\pi$
$68$	−17616.0			−0.461993
$69$		0			0
$70$			21420.0i			0.522486i
$71$		−	63741.0i		−	1.50063i	−0.661082	−	0.750314i	$-0.729902\pi$
$71$		−	63741.0i		−	1.50063i	0.661082	−	0.750314i	$-0.270098\pi$
$72$		0			0
$73$			58866.0i			1.29288i	0.762966	+	0.646439i	$0.223742\pi$
$73$			58866.0i			1.29288i	−0.762966	+	0.646439i	$0.776258\pi$
$74$	−34836.0			−0.739518
$75$		0			0
$76$			18720.0i			0.371768i
$77$	12600.0			0.242183
$78$		0			0
$79$	63202.0			1.13937			0.569683	−	0.821865i	$-0.307066\pi$
$79$	63202.0			1.13937			0.569683	+	0.821865i	$0.307066\pi$
$80$		−	13056.0i		−	0.228079i
$81$		0			0
$82$	−37920.0			−0.622779
$83$		−	55458.0i		−	0.883627i	−0.897107	−	0.441813i	$-0.854335\pi$
$83$		−	55458.0i		−	0.883627i	0.897107	−	0.441813i	$-0.145665\pi$
$84$		0			0
$85$		−	56151.0i		−	0.842966i
$86$			39980.0i			0.582903i
$87$		0			0
$88$	−7680.00			−0.105719
$89$			104778.i			1.40215i	0.713087	+	0.701076i	$0.247297\pi$
$89$			104778.i			1.40215i	−0.713087	+	0.701076i	$0.752703\pi$
$90$		0			0
$91$	−12285.0	−	62790.0i	−0.155515	−	0.794853i
$92$	16800.0			0.206938
$93$		0			0
$94$	−11772.0			−0.137414
$95$	−59670.0			−0.678339
$96$		0			0
$97$			160452.i			1.73147i	0.500500	+	0.865737i	$0.333150\pi$
$97$			160452.i			1.73147i	−0.500500	+	0.865737i	$0.666850\pi$
$98$			23128.0i			0.243261i
$99$		0			0
$100$	−8384.00			−0.0838400
$101$	113124.			1.10345			0.551723	−	0.834027i	$-0.313970\pi$
$101$	113124.			1.10345			0.551723	+	0.834027i	$0.313970\pi$
$102$		0			0
$103$	25046.0			0.232619			0.116310	−	0.993213i	$-0.462894\pi$
$103$	25046.0			0.232619			0.116310	+	0.993213i	$0.462894\pi$
$104$	7488.00	+	38272.0i	0.0678864	+	0.346975i
$105$		0			0
$106$			3000.00i			0.0259332i
$107$	−24924.0			−0.210455			−0.105227	−	0.994448i	$-0.533557\pi$
$107$	−24924.0			−0.210455			−0.105227	+	0.994448i	$0.533557\pi$
$108$		0			0
$109$			144831.i			1.16760i	0.811896	+	0.583802i	$0.198435\pi$
$109$			144831.i			1.16760i	−0.811896	+	0.583802i	$0.801565\pi$
$110$		−	24480.0i		−	0.192899i
$111$		0			0
$112$			26880.0i			0.202481i
$113$	−100266.			−0.738682			−0.369341	−	0.929294i	$-0.620417\pi$
$113$	−100266.			−0.738682			−0.369341	+	0.929294i	$0.620417\pi$
$114$		0			0
$115$			53550.0i			0.377585i
$116$	−65664.0			−0.453088
$117$		0			0
$118$	−163752.			−1.08263
$119$			115605.i			0.748358i
$120$		0			0
$121$	146651.			0.910587
$122$		−	231680.i		−	1.40925i
$123$		0			0
$124$		−	153984.i		−	0.899335i
$125$		−	186099.i		−	1.06529i
$126$		0			0
$127$	−202754.			−1.11548			−0.557738	−	0.830017i	$-0.688331\pi$
$127$	−202754.			−1.11548			−0.557738	+	0.830017i	$0.688331\pi$
$128$		−	16384.0i		−	0.0883883i
$129$		0			0
$130$	−121992.	+	23868.0i	−0.633101	+	0.123868i
$131$	303855.			1.54699			0.773496	−	0.633801i	$-0.218506\pi$
$131$	303855.			1.54699			0.773496	+	0.633801i	$0.218506\pi$
$132$		0			0
$133$	122850.			0.602207
$134$	−91248.0			−0.438997
$135$		0			0
$136$		−	70464.0i		−	0.326678i
$137$			63738.0i			0.290133i	0.989422	+	0.145066i	$0.0463396\pi$
$137$			63738.0i			0.290133i	−0.989422	+	0.145066i	$0.953660\pi$
$138$		0			0
$139$	13841.0			0.0607618			0.0303809	−	0.999538i	$-0.490328\pi$
$139$	13841.0			0.0607618			0.0303809	+	0.999538i	$0.490328\pi$
$140$	−85680.0			−0.369453
$141$		0			0
$142$	254964.			1.06110
$143$	14040.0	+	71760.0i	0.0574152	+	0.293456i
$144$		0			0
$145$		−	209304.i		−	0.826718i
$146$	−235464.			−0.914202
$147$		0			0
$148$		−	139344.i		−	0.522918i
$149$			276426.i			1.02003i	0.860165	+	0.510015i	$0.170360\pi$
$149$			276426.i			1.02003i	−0.860165	+	0.510015i	$0.829640\pi$
$150$		0			0
$151$		−	321333.i		−	1.14687i	−0.819252	−	0.573433i	$-0.805611\pi$
$151$		−	321333.i		−	1.14687i	0.819252	−	0.573433i	$-0.194389\pi$
$152$	−74880.0			−0.262880
$153$		0			0
$154$			50400.0i			0.171249i
$155$	490824.			1.64095
$156$		0			0
$157$	339506.			1.09925			0.549627	−	0.835410i	$-0.314770\pi$
$157$	339506.			1.09925			0.549627	+	0.835410i	$0.314770\pi$
$158$			252808.i			0.805653i
$159$		0			0
$160$	52224.0			0.161276
$161$		−	110250.i		−	0.335208i
$162$		0			0
$163$		−	395718.i		−	1.16659i	−0.812262	−	0.583293i	$-0.801764\pi$
$163$		−	395718.i		−	1.16659i	0.812262	−	0.583293i	$-0.198236\pi$
$164$		−	151680.i		−	0.440371i
$165$		0			0
$166$	221832.			0.624819
$167$		−	426708.i		−	1.18397i	−0.805950	−	0.591984i	$-0.798345\pi$
$167$		−	426708.i		−	1.18397i	0.805950	−	0.591984i	$-0.201655\pi$
$168$		0			0
$169$	343915.	−	139932.i	0.926263	−	0.376878i
$170$	224604.			0.596067
$171$		0			0
$172$	−159920.			−0.412175
$173$	−16026.0			−0.0407108			−0.0203554	−	0.999793i	$-0.506480\pi$
$173$	−16026.0			−0.0407108			−0.0203554	+	0.999793i	$0.506480\pi$
$174$		0			0
$175$			55020.0i			0.135808i
$176$		−	30720.0i		−	0.0747549i
$177$		0			0
$178$	−419112.			−0.991471
$179$	690045.			1.60970			0.804850	−	0.593479i	$-0.202246\pi$
$179$	690045.			1.60970			0.804850	+	0.593479i	$0.202246\pi$
$180$		0			0
$181$	−96478.0			−0.218893			−0.109446	−	0.993993i	$-0.534908\pi$
$181$	−96478.0			−0.218893			−0.109446	+	0.993993i	$0.534908\pi$
$182$	251160.	−	49140.0i	0.562046	−	0.109966i
$183$		0			0
$184$			67200.0i			0.146327i
$185$	444159.			0.954134
$186$		0			0
$187$		−	132120.i		−	0.276290i
$188$		−	47088.0i		−	0.0971663i
$189$		0			0
$190$		−	238680.i		−	0.479658i
$191$	−708180.			−1.40462			−0.702312	−	0.711869i	$-0.747849\pi$
$191$	−708180.			−1.40462			−0.702312	+	0.711869i	$0.747849\pi$
$192$		0			0
$193$		−	347862.i		−	0.672224i	−0.941822	−	0.336112i	$-0.890888\pi$
$193$		−	347862.i		−	0.672224i	0.941822	−	0.336112i	$-0.109112\pi$
$194$	−641808.			−1.22434
$195$		0			0
$196$	−92512.0			−0.172012
$197$		−	899589.i		−	1.65150i	−0.564036	−	0.825750i	$-0.690752\pi$
$197$		−	899589.i		−	1.65150i	0.564036	−	0.825750i	$-0.309248\pi$
$198$		0			0
$199$	143116.			0.256186			0.128093	−	0.991762i	$-0.459114\pi$
$199$	143116.			0.256186			0.128093	+	0.991762i	$0.459114\pi$
$200$		−	33536.0i		−	0.0592838i
$201$		0			0
$202$			452496.i			0.780255i
$203$			430920.i			0.733933i
$204$		0			0
$205$	483480.			0.803515
$206$			100184.i			0.164487i
$207$		0			0
$208$	−153088.	+	29952.0i	−0.245348	+	0.0480029i
$209$	−140400.			−0.222332
$210$		0			0
$211$	−339731.			−0.525326			−0.262663	−	0.964888i	$-0.584601\pi$
$211$	−339731.			−0.525326			−0.262663	+	0.964888i	$0.584601\pi$
$212$	−12000.0			−0.0183376
$213$		0			0
$214$		−	99696.0i		−	0.148814i
$215$		−	509745.i		−	0.752068i
$216$		0			0
$217$	−1.01052e6			−1.45679
$218$	−579324.			−0.825620
$219$		0			0
$220$	97920.0			0.136400
$221$	−658398.	+	128817.i	−0.906792	+	0.177416i
$222$		0			0
$223$			623757.i			0.839950i	0.907536	+	0.419975i	$0.137961\pi$
$223$			623757.i			0.839950i	−0.907536	+	0.419975i	$0.862039\pi$
$224$	−107520.			−0.143176
$225$		0			0
$226$		−	401064.i		−	0.522327i
$227$			177612.i			0.228775i	0.993436	+	0.114387i	$0.0364905\pi$
$227$			177612.i			0.228775i	−0.993436	+	0.114387i	$0.963510\pi$
$228$		0			0
$229$		−	1.18705e6i		−	1.49582i	−0.663799	−	0.747911i	$-0.731057\pi$
$229$		−	1.18705e6i		−	1.49582i	0.663799	−	0.747911i	$-0.268943\pi$
$230$	−214200.			−0.266993
$231$		0			0
$232$		−	262656.i		−	0.320381i
$233$	112317.			0.135536			0.0677682	−	0.997701i	$-0.478412\pi$
$233$	112317.			0.135536			0.0677682	+	0.997701i	$0.478412\pi$
$234$		0			0
$235$	150093.			0.177293
$236$		−	655008.i		−	0.765538i
$237$		0			0
$238$	−462420.			−0.529169
$239$			1.19805e6i			1.35669i	0.734743	+	0.678346i	$0.237303\pi$
$239$			1.19805e6i			1.35669i	−0.734743	+	0.678346i	$0.762697\pi$
$240$		0			0
$241$			1.16629e6i			1.29349i	0.762707	+	0.646744i	$0.223870\pi$
$241$			1.16629e6i			1.29349i	−0.762707	+	0.646744i	$0.776130\pi$
$242$			586604.i			0.643882i
$243$		0			0
$244$	926720.			0.996492
$245$		−	294882.i		−	0.313858i
$246$		0			0
$247$	136890.	+	699660.i	0.142767	+	0.729701i
$248$	615936.			0.635926
$249$		0			0
$250$	744396.			0.753276
$251$	−648996.			−0.650216			−0.325108	−	0.945677i	$-0.605401\pi$
$251$	−648996.			−0.650216			−0.325108	+	0.945677i	$0.605401\pi$
$252$		0			0
$253$			126000.i			0.123757i
$254$		−	811016.i		−	0.788760i
$255$		0			0
$256$	65536.0			0.0625000
$257$	945885.			0.893317			0.446658	−	0.894705i	$-0.352614\pi$
$257$	945885.			0.893317			0.446658	+	0.894705i	$0.352614\pi$
$258$		0			0
$259$	−914445.			−0.847048
$260$	−95472.0	−	487968.i	−0.0875876	−	0.447670i
$261$		0			0
$262$			1.21542e6i			1.09389i
$263$	−1.01222e6			−0.902375			−0.451188	−	0.892429i	$-0.649000\pi$
$263$	−1.01222e6			−0.902375			−0.451188	+	0.892429i	$0.649000\pi$
$264$		0			0
$265$		−	38250.0i		−	0.0334593i
$266$			491400.i			0.425825i
$267$		0			0
$268$		−	364992.i		−	0.310418i
$269$	1.01772e6			0.857527			0.428763	−	0.903417i	$-0.358949\pi$
$269$	1.01772e6			0.857527			0.428763	+	0.903417i	$0.358949\pi$
$270$		0			0
$271$			463461.i			0.383345i	0.981459	+	0.191673i	$0.0613912\pi$
$271$			463461.i			0.383345i	−0.981459	+	0.191673i	$0.938609\pi$
$272$	281856.			0.230996
$273$		0			0
$274$	−254952.			−0.205155
$275$		−	62880.0i		−	0.0501396i
$276$		0			0
$277$	332528.			0.260393			0.130196	−	0.991488i	$-0.458439\pi$
$277$	332528.			0.260393			0.130196	+	0.991488i	$0.458439\pi$
$278$			55364.0i			0.0429651i
$279$		0			0
$280$		−	342720.i		−	0.261243i
$281$			49122.0i			0.0371116i	0.999828	+	0.0185558i	$0.00590684\pi$
$281$			49122.0i			0.0371116i	−0.999828	+	0.0185558i	$0.994093\pi$
$282$		0			0
$283$	−1.55848e6			−1.15674			−0.578371	−	0.815774i	$-0.696311\pi$
$283$	−1.55848e6			−1.15674			−0.578371	+	0.815774i	$0.696311\pi$
$284$			1.01986e6i			0.750314i
$285$		0			0
$286$	−287040.	+	56160.0i	−0.207504	+	0.0405987i
$287$	−995400.			−0.713334
$288$		0			0
$289$	−207656.			−0.146251
$290$	837216.			0.584578
$291$		0			0
$292$		−	941856.i		−	0.646439i
$293$			218463.i			0.148665i	0.997234	+	0.0743325i	$0.0236826\pi$
$293$			218463.i			0.148665i	−0.997234	+	0.0743325i	$0.976317\pi$
$294$		0			0
$295$	2.08784e6			1.39682
$296$	557376.			0.369759
$297$		0			0
$298$	−1.10570e6			−0.721271
$299$	627900.	−	122850.i	0.406174	−	0.0794689i
$300$		0			0
$301$			1.04948e6i			0.667661i
$302$	1.28533e6			0.810957
$303$		0			0
$304$		−	299520.i		−	0.185884i
$305$			2.95392e6i			1.81823i
$306$		0			0
$307$			321102.i			0.194445i	0.995263	+	0.0972226i	$0.0309959\pi$
$307$			321102.i			0.194445i	−0.995263	+	0.0972226i	$0.969004\pi$
$308$	−201600.			−0.121092
$309$		0			0
$310$			1.96330e6i			1.16033i
$311$	−3.33725e6			−1.95654			−0.978269	−	0.207340i	$-0.933519\pi$
$311$	−3.33725e6			−1.95654			−0.978269	+	0.207340i	$0.933519\pi$
$312$		0			0
$313$	1.16568e6			0.672538			0.336269	−	0.941766i	$-0.390835\pi$
$313$	1.16568e6			0.672538			0.336269	+	0.941766i	$0.390835\pi$
$314$			1.35802e6i			0.777290i
$315$		0			0
$316$	−1.01123e6			−0.569683
$317$		−	73518.0i		−	0.0410909i	−0.999789	−	0.0205454i	$-0.993460\pi$
$317$		−	73518.0i		−	0.0410909i	0.999789	−	0.0205454i	$-0.00654028\pi$
$318$		0			0
$319$		−	492480.i		−	0.270964i
$320$			208896.i			0.114039i
$321$		0			0
$322$	441000.			0.237028
$323$		−	1.28817e6i		−	0.687016i
$324$		0			0
$325$	−313352.	+	61308.0i	−0.164560	+	0.0321965i
$326$	1.58287e6			0.824901
$327$		0			0
$328$	606720.			0.311389
$329$	−309015.			−0.157395
$330$		0			0
$331$			632682.i			0.317406i	0.987326	+	0.158703i	$0.0507313\pi$
$331$			632682.i			0.317406i	−0.987326	+	0.158703i	$0.949269\pi$
$332$			887328.i			0.441813i
$333$		0			0
$334$	1.70683e6			0.837191
$335$	1.16341e6			0.566398
$336$		0			0
$337$	−326843.			−0.156771			−0.0783853	−	0.996923i	$-0.524976\pi$
$337$	−326843.			−0.156771			−0.0783853	+	0.996923i	$0.524976\pi$
$338$	559728.	+	1.37566e6i	0.266493	+	0.654967i
$339$		0			0
$340$			898416.i			0.421483i
$341$	1.15488e6			0.537837
$342$		0			0
$343$			2.37184e6i			1.08856i
$344$		−	639680.i		−	0.291452i
$345$		0			0
$346$		−	64104.0i		−	0.0287869i
$347$	−2.96275e6			−1.32090			−0.660452	−	0.750868i	$-0.729635\pi$
$347$	−2.96275e6			−1.32090			−0.660452	+	0.750868i	$0.729635\pi$
$348$		0			0
$349$			866325.i			0.380730i	0.981713	+	0.190365i	$0.0609672\pi$
$349$			866325.i			0.380730i	−0.981713	+	0.190365i	$0.939033\pi$
$350$	−220080.			−0.0960308
$351$		0			0
$352$	122880.			0.0528597
$353$		−	1.66291e6i		−	0.710282i	−0.934813	−	0.355141i	$-0.884433\pi$
$353$		−	1.66291e6i		−	0.710282i	0.934813	−	0.355141i	$-0.115567\pi$
$354$		0			0
$355$	−3.25079e6			−1.36905
$356$		−	1.67645e6i		−	0.701076i
$357$		0			0
$358$			2.76018e6i			1.13823i
$359$		−	625536.i		−	0.256163i	−0.991764	−	0.128081i	$-0.959118\pi$
$359$		−	625536.i		−	0.256163i	0.991764	−	0.128081i	$-0.0408819\pi$
$360$		0			0
$361$	1.10720e6			0.447155
$362$		−	385912.i		−	0.154781i
$363$		0			0
$364$	196560.	+	1.00464e6i	0.0777574	+	0.397427i
$365$	3.00217e6			1.17951
$366$		0			0
$367$	1.08327e6			0.419829			0.209914	−	0.977720i	$-0.432681\pi$
$367$	1.08327e6			0.419829			0.209914	+	0.977720i	$0.432681\pi$
$368$	−268800.			−0.103469
$369$		0			0
$370$			1.77664e6i			0.674674i
$371$			78750.0i			0.0297041i
$372$		0			0
$373$	−1.78896e6			−0.665775			−0.332888	−	0.942967i	$-0.608023\pi$
$373$	−1.78896e6			−0.665775			−0.332888	+	0.942967i	$0.608023\pi$
$374$	528480.			0.195366
$375$		0			0
$376$	188352.			0.0687069
$377$	−2.45419e6	+	480168.i	−0.889314	+	0.173996i
$378$		0			0
$379$			868614.i			0.310620i	0.987866	+	0.155310i	$0.0496376\pi$
$379$			868614.i			0.310620i	−0.987866	+	0.155310i	$0.950362\pi$
$380$	954720.			0.339170
$381$		0			0
$382$		−	2.83272e6i		−	0.993220i
$383$			1.07972e6i			0.376108i	0.982159	+	0.188054i	$0.0602179\pi$
$383$			1.07972e6i			0.376108i	−0.982159	+	0.188054i	$0.939782\pi$
$384$		0			0
$385$		−	642600.i		−	0.220947i
$386$	1.39145e6			0.475334
$387$		0			0
$388$		−	2.56723e6i		−	0.865737i
$389$	−1.28822e6			−0.431634			−0.215817	−	0.976434i	$-0.569241\pi$
$389$	−1.28822e6			−0.431634			−0.215817	+	0.976434i	$0.569241\pi$
$390$		0			0
$391$	−1.15605e6			−0.382415
$392$		−	370048.i		−	0.121631i
$393$		0			0
$394$	3.59836e6			1.16779
$395$		−	3.22330e6i		−	1.03946i
$396$		0			0
$397$		−	5.46909e6i		−	1.74156i	−0.491672	−	0.870781i	$-0.663614\pi$
$397$		−	5.46909e6i		−	1.74156i	0.491672	−	0.870781i	$-0.336386\pi$
$398$			572464.i			0.181151i
$399$		0			0
$400$	134144.			0.0419200
$401$			1.58612e6i			0.492577i	0.969196	+	0.246289i	$0.0792112\pi$
$401$			1.58612e6i			0.492577i	−0.969196	+	0.246289i	$0.920789\pi$
$402$		0			0
$403$	−1.12601e6	−	5.75515e6i	−0.345365	−	1.76520i
$404$	−1.80998e6			−0.551723
$405$		0			0
$406$	−1.72368e6			−0.518969
$407$	1.04508e6			0.312726
$408$		0			0
$409$			6.44192e6i			1.90418i	0.305825	+	0.952088i	$0.401068\pi$
$409$			6.44192e6i			1.90418i	−0.305825	+	0.952088i	$0.598932\pi$
$410$			1.93392e6i			0.568171i
$411$		0			0
$412$	−400736.			−0.116310
$413$	−4.29849e6			−1.24005
$414$		0			0
$415$	−2.82836e6			−0.806147
$416$	−119808.	−	612352.i	−0.0339432	−	0.173487i
$417$		0			0
$418$		−	561600.i		−	0.157212i
$419$	4.30545e6			1.19807			0.599037	−	0.800721i	$-0.295550\pi$
$419$	4.30545e6			1.19807			0.599037	+	0.800721i	$0.295550\pi$
$420$		0			0
$421$			1.51346e6i			0.416164i	0.978111	+	0.208082i	$0.0667221\pi$
$421$			1.51346e6i			0.416164i	−0.978111	+	0.208082i	$0.933278\pi$
$422$		−	1.35892e6i		−	0.371462i
$423$		0			0
$424$		−	48000.0i		−	0.0129666i
$425$	576924.			0.154934
$426$		0			0
$427$		−	6.08160e6i		−	1.61417i
$428$	398784.			0.105227
$429$		0			0
$430$	2.03898e6			0.531792
$431$			1.43116e6i			0.371105i	0.982634	+	0.185552i	$0.0594074\pi$
$431$			1.43116e6i			0.371105i	−0.982634	+	0.185552i	$0.940593\pi$
$432$		0			0
$433$	429613.			0.110118			0.0550589	−	0.998483i	$-0.482465\pi$
$433$	429613.			0.110118			0.0550589	+	0.998483i	$0.482465\pi$
$434$		−	4.04208e6i		−	1.03010i
$435$		0			0
$436$		−	2.31730e6i		−	0.583802i
$437$			1.22850e6i			0.307731i
$438$		0			0
$439$	552038.			0.136712			0.0683562	−	0.997661i	$-0.478225\pi$
$439$	552038.			0.136712			0.0683562	+	0.997661i	$0.478225\pi$
$440$			391680.i			0.0964494i
$441$		0			0
$442$	−515268.	−	2.63359e6i	−0.125452	−	0.641199i
$443$	−2.15255e6			−0.521128			−0.260564	−	0.965457i	$-0.583908\pi$
$443$	−2.15255e6			−0.521128			−0.260564	+	0.965457i	$0.583908\pi$
$444$		0			0
$445$	5.34368e6			1.27921
$446$	−2.49503e6			−0.593934
$447$		0			0
$448$		−	430080.i		−	0.101240i
$449$		−	1.40429e6i		−	0.328731i	−0.986400	−	0.164365i	$-0.947442\pi$
$449$		−	1.40429e6i		−	0.328731i	0.986400	−	0.164365i	$-0.0525576\pi$
$450$		0			0
$451$	1.13760e6			0.263359
$452$	1.60426e6			0.369341
$453$		0			0
$454$	−710448.			−0.161768
$455$	−3.20229e6	+	626535.i	−0.725157	+	0.141879i
$456$		0			0
$457$		−	1.32818e6i		−	0.297485i	−0.988876	−	0.148743i	$-0.952477\pi$
$457$		−	1.32818e6i		−	0.297485i	0.988876	−	0.148743i	$-0.0475226\pi$
$458$	4.74820e6			1.05771
$459$		0			0
$460$		−	856800.i		−	0.188793i
$461$		−	5.89070e6i		−	1.29096i	−0.763775	−	0.645482i	$-0.776656\pi$
$461$		−	5.89070e6i		−	1.29096i	0.763775	−	0.645482i	$-0.223344\pi$
$462$		0			0
$463$			2.37139e6i			0.514104i	0.966398	+	0.257052i	$0.0827511\pi$
$463$			2.37139e6i			0.514104i	−0.966398	+	0.257052i	$0.917249\pi$
$464$	1.05062e6			0.226544
$465$		0			0
$466$			449268.i			0.0958387i
$467$	−7.17827e6			−1.52310			−0.761548	−	0.648108i	$-0.775560\pi$
$467$	−7.17827e6			−1.52310			−0.761548	+	0.648108i	$0.775560\pi$
$468$		0			0
$469$	−2.39526e6			−0.502829
$470$			600372.i			0.125365i
$471$		0			0
$472$	2.62003e6			0.541317
$473$		−	1.19940e6i		−	0.246497i
$474$		0			0
$475$		−	613080.i		−	0.124676i
$476$		−	1.84968e6i		−	0.374179i
$477$		0			0
$478$	−4.79221e6			−0.959326
$479$		−	7.25193e6i		−	1.44416i	−0.691810	−	0.722079i	$-0.743186\pi$
$479$		−	7.25193e6i		−	1.44416i	0.691810	−	0.722079i	$-0.256814\pi$
$480$		0			0
$481$	−1.01895e6	−	5.20798e6i	−0.200813	−	1.02638i
$482$	−4.66514e6			−0.914634
$483$		0			0
$484$	−2.34642e6			−0.455294
$485$	8.18305e6			1.57965
$486$		0			0
$487$		−	2.53364e6i		−	0.484087i	−0.970265	−	0.242043i	$-0.922182\pi$
$487$		−	2.53364e6i		−	0.484087i	0.970265	−	0.242043i	$-0.0778176\pi$
$488$			3.70688e6i			0.704626i
$489$		0			0
$490$	1.17953e6			0.221931
$491$	−8.46186e6			−1.58403			−0.792013	−	0.610504i	$-0.790967\pi$
$491$	−8.46186e6			−1.58403			−0.792013	+	0.610504i	$0.790967\pi$
$492$		0			0
$493$	4.51850e6			0.837293
$494$	−2.79864e6	+	547560.i	−0.515976	+	0.100952i
$495$		0			0
$496$			2.46374e6i			0.449667i
$497$	6.69280e6			1.21539
$498$		0			0
$499$		−	1.95383e6i		−	0.351265i	−0.984456	−	0.175633i	$-0.943803\pi$
$499$		−	1.95383e6i		−	0.351265i	0.984456	−	0.175633i	$-0.0561971\pi$
$500$			2.97758e6i			0.532646i
$501$		0			0
$502$		−	2.59598e6i		−	0.459772i
$503$	−119778.			−0.0211085			−0.0105542	−	0.999944i	$-0.503360\pi$
$503$	−119778.			−0.0211085			−0.0105542	+	0.999944i	$0.503360\pi$
$504$		0			0
$505$		−	5.76932e6i		−	1.00669i
$506$	−504000.			−0.0875093
$507$		0			0
$508$	3.24406e6			0.557738
$509$			1.03653e7i			1.77332i	0.462420	+	0.886661i	$0.346981\pi$
$509$			1.03653e7i			1.77332i	−0.462420	+	0.886661i	$0.653019\pi$
$510$		0			0
$511$	−6.18093e6			−1.04713
$512$			262144.i			0.0441942i
$513$		0			0
$514$			3.78354e6i			0.631670i
$515$		−	1.27735e6i		−	0.212222i
$516$		0			0
$517$	353160.			0.0581092
$518$		−	3.65778e6i		−	0.598954i
$519$		0			0
$520$	1.95187e6	−	381888.i	0.316550	−	0.0619338i
$521$	1.04899e7			1.69307			0.846537	−	0.532330i	$-0.178684\pi$
$521$	1.04899e7			1.69307			0.846537	+	0.532330i	$0.178684\pi$
$522$		0			0
$523$	4.42662e6			0.707649			0.353824	−	0.935312i	$-0.384881\pi$
$523$	4.42662e6			0.707649			0.353824	+	0.935312i	$0.384881\pi$
$524$	−4.86168e6			−0.773496
$525$		0			0
$526$		−	4.04890e6i		−	0.638076i
$527$			1.05960e7i			1.66194i
$528$		0			0
$529$	−5.33384e6			−0.828707
$530$	153000.			0.0236593
$531$		0			0
$532$	−1.96560e6			−0.301104
$533$	−1.10916e6	−	5.66904e6i	−0.169113	−	0.864354i
$534$		0			0
$535$			1.27112e6i			0.192001i
$536$	1.45997e6			0.219498
$537$		0			0
$538$			4.07088e6i			0.606363i
$539$		−	693840.i		−	0.102870i
$540$		0			0
$541$			2.26377e6i			0.332536i	0.986081	+	0.166268i	$0.0531717\pi$
$541$			2.26377e6i			0.332536i	−0.986081	+	0.166268i	$0.946828\pi$
$542$	−1.85384e6			−0.271066
$543$		0			0
$544$			1.12742e6i			0.163339i
$545$	7.38638e6			1.06522
$546$		0			0
$547$	7.21090e6			1.03044			0.515218	−	0.857059i	$-0.327711\pi$
$547$	7.21090e6			1.03044			0.515218	+	0.857059i	$0.327711\pi$
$548$		−	1.01981e6i		−	0.145066i
$549$		0			0
$550$	251520.			0.0354540
$551$		−	4.80168e6i		−	0.673774i
$552$		0			0
$553$			6.63621e6i			0.922799i
$554$			1.33011e6i			0.184125i
$555$		0			0
$556$	−221456.			−0.0303809
$557$		−	273507.i		−	0.0373534i	−0.999826	−	0.0186767i	$-0.994055\pi$
$557$		−	273507.i		−	0.0373534i	0.999826	−	0.0186767i	$-0.00594533\pi$
$558$		0			0
$559$	−5.97701e6	+	1.16942e6i	−0.809011	+	0.158285i
$560$	1.37088e6			0.184727
$561$		0			0
$562$	−196488.			−0.0262419
$563$	959349.			0.127557			0.0637787	−	0.997964i	$-0.479685\pi$
$563$	959349.			0.127557			0.0637787	+	0.997964i	$0.479685\pi$
$564$		0			0
$565$			5.11357e6i			0.673911i
$566$		−	6.23394e6i		−	0.817940i
$567$		0			0
$568$	−4.07942e6			−0.530552
$569$	1.19403e7			1.54609			0.773044	−	0.634352i	$-0.218733\pi$
$569$	1.19403e7			1.54609			0.773044	+	0.634352i	$0.218733\pi$
$570$		0			0
$571$	7.20205e6			0.924413			0.462206	−	0.886772i	$-0.347058\pi$
$571$	7.20205e6			0.924413			0.462206	+	0.886772i	$0.347058\pi$
$572$	−224640.	−	1.14816e6i	−0.0287076	−	0.146728i
$573$		0			0
$574$		−	3.98160e6i		−	0.504403i
$575$	−550200.			−0.0693986
$576$		0			0
$577$		−	1.66990e6i		−	0.208810i	−0.994535	−	0.104405i	$-0.966706\pi$
$577$		−	1.66990e6i		−	0.208810i	0.994535	−	0.104405i	$-0.0332938\pi$
$578$		−	830624.i		−	0.103415i
$579$		0			0
$580$			3.34886e6i			0.413359i
$581$	5.82309e6			0.715671
$582$		0			0
$583$		−	90000.0i		−	0.0109666i
$584$	3.76742e6			0.457101
$585$		0			0
$586$	−873852.			−0.105122
$587$		−	8.29913e6i		−	0.994117i	−0.867717	−	0.497059i	$-0.834413\pi$
$587$		−	8.29913e6i		−	0.994117i	0.867717	−	0.497059i	$-0.165587\pi$
$588$		0			0
$589$	1.12601e7			1.33738
$590$			8.35135e6i			0.987704i
$591$		0			0
$592$			2.22950e6i			0.261459i
$593$			4.48969e6i			0.524300i	0.965027	+	0.262150i	$0.0844315\pi$
$593$			4.48969e6i			0.524300i	−0.965027	+	0.262150i	$0.915568\pi$
$594$		0			0
$595$	5.89586e6			0.682738
$596$		−	4.42282e6i		−	0.510015i
$597$		0			0
$598$	491400.	+	2.51160e6i	0.0561930	+	0.287209i
$599$	−1.38261e6			−0.157446			−0.0787232	−	0.996897i	$-0.525084\pi$
$599$	−1.38261e6			−0.157446			−0.0787232	+	0.996897i	$0.525084\pi$
$600$		0			0
$601$	1.04021e7			1.17472			0.587359	−	0.809327i	$-0.300168\pi$
$601$	1.04021e7			1.17472			0.587359	+	0.809327i	$0.300168\pi$
$602$	−4.19790e6			−0.472107
$603$		0			0
$604$			5.14133e6i			0.573433i
$605$		−	7.47920e6i		−	0.830743i
$606$		0			0
$607$	−4.78668e6			−0.527306			−0.263653	−	0.964618i	$-0.584927\pi$
$607$	−4.78668e6			−0.527306			−0.263653	+	0.964618i	$0.584927\pi$
$608$	1.19808e6			0.131440
$609$		0			0
$610$	−1.18157e7			−1.28568
$611$	−344331.	−	1.75991e6i	−0.0373141	−	0.190717i
$612$		0			0
$613$			1.04783e7i			1.12627i	0.826366	+	0.563134i	$0.190404\pi$
$613$			1.04783e7i			1.12627i	−0.826366	+	0.563134i	$0.809596\pi$
$614$	−1.28441e6			−0.137493
$615$		0			0
$616$		−	806400.i		−	0.0856246i
$617$			1.79106e7i			1.89407i	0.321128	+	0.947036i	$0.395938\pi$
$617$			1.79106e7i			1.89407i	−0.321128	+	0.947036i	$0.604062\pi$
$618$		0			0
$619$		−	4.43222e6i		−	0.464938i	−0.972604	−	0.232469i	$-0.925320\pi$
$619$		−	4.43222e6i		−	0.464938i	0.972604	−	0.232469i	$-0.0746804\pi$
$620$	−7.85318e6			−0.820477
$621$		0			0
$622$		−	1.33490e7i		−	1.38348i
$623$	−1.10017e7			−1.13564
$624$		0			0
$625$	−7.85355e6			−0.804203
$626$			4.66270e6i			0.475556i
$627$		0			0
$628$	−5.43210e6			−0.549627
$629$			9.58861e6i			0.966338i
$630$		0			0
$631$			1.43291e7i			1.43267i	0.697756	+	0.716335i	$0.254182\pi$
$631$			1.43291e7i			1.43267i	−0.697756	+	0.716335i	$0.745818\pi$
$632$		−	4.04493e6i		−	0.402827i
$633$		0			0
$634$	294072.			0.0290556
$635$			1.03405e7i			1.01767i
$636$		0			0
$637$	−3.45764e6	+	676494.i	−0.337622	+	0.0660565i
$638$	1.96992e6			0.191601
$639$		0			0
$640$	−835584.			−0.0806381
$641$	−6.65869e6			−0.640094			−0.320047	−	0.947402i	$-0.603699\pi$
$641$	−6.65869e6			−0.640094			−0.320047	+	0.947402i	$0.603699\pi$
$642$		0			0
$643$			1.55224e7i			1.48058i	0.672286	+	0.740291i	$0.265312\pi$
$643$			1.55224e7i			1.48058i	−0.672286	+	0.740291i	$0.734688\pi$
$644$			1.76400e6i			0.167604i
$645$		0			0
$646$	5.15268e6			0.485794
$647$	2.44454e6			0.229581			0.114791	−	0.993390i	$-0.463380\pi$
$647$	2.44454e6			0.229581			0.114791	+	0.993390i	$0.463380\pi$
$648$		0			0
$649$	4.91256e6			0.457821
$650$	−245232.	−	1.25341e6i	−0.0227664	−	0.116361i
$651$		0			0
$652$			6.33149e6i			0.583293i
$653$	−1.16500e7			−1.06916			−0.534580	−	0.845118i	$-0.679530\pi$
$653$	−1.16500e7			−1.06916			−0.534580	+	0.845118i	$0.679530\pi$
$654$		0			0
$655$		−	1.54966e7i		−	1.41135i
$656$			2.42688e6i			0.220185i
$657$		0			0
$658$		−	1.23606e6i		−	0.111295i
$659$	−1.33185e7			−1.19465			−0.597326	−	0.801999i	$-0.703770\pi$
$659$	−1.33185e7			−1.19465			−0.597326	+	0.801999i	$0.703770\pi$
$660$		0			0
$661$		−	1.35722e7i		−	1.20822i	−0.796900	−	0.604112i	$-0.793528\pi$
$661$		−	1.35722e7i		−	1.20822i	0.796900	−	0.604112i	$-0.206472\pi$
$662$	−2.53073e6			−0.224440
$663$		0			0
$664$	−3.54931e6			−0.312409
$665$		−	6.26535e6i		−	0.549403i
$666$		0			0
$667$	−4.30920e6			−0.375044
$668$			6.82733e6i			0.591984i
$669$		0			0
$670$			4.65365e6i			0.400504i
$671$			6.95040e6i			0.595941i
$672$		0			0
$673$	−1.58674e7			−1.35042			−0.675209	−	0.737626i	$-0.735947\pi$
$673$	−1.58674e7			−1.35042			−0.675209	+	0.737626i	$0.735947\pi$
$674$		−	1.30737e6i		−	0.110854i
$675$		0			0
$676$	−5.50264e6	+	2.23891e6i	−0.463132	+	0.188439i
$677$	2.24264e7			1.88056			0.940281	−	0.340398i	$-0.110562\pi$
$677$	2.24264e7			1.88056			0.940281	+	0.340398i	$0.110562\pi$
$678$		0			0
$679$	−1.68475e7			−1.40236
$680$	−3.59366e6			−0.298034
$681$		0			0
$682$			4.61952e6i			0.380308i
$683$		−	8.11034e6i		−	0.665254i	−0.943059	−	0.332627i	$-0.892065\pi$
$683$		−	8.11034e6i		−	0.665254i	0.943059	−	0.332627i	$-0.107935\pi$
$684$		0			0
$685$	3.25064e6			0.264693
$686$	−9.48738e6			−0.769726
$687$		0			0
$688$	2.55872e6			0.206088
$689$	−448500.	+	87750.0i	−0.0359927	+	0.00704205i
$690$		0			0
$691$			2.00020e7i			1.59359i	0.604246	+	0.796797i	$0.293474\pi$
$691$			2.00020e7i			1.59359i	−0.604246	+	0.796797i	$0.706526\pi$
$692$	256416.			0.0203554
$693$		0			0
$694$		−	1.18510e7i		−	0.934020i
$695$		−	705891.i		−	0.0554339i
$696$		0			0
$697$			1.04375e7i			0.813793i
$698$	−3.46530e6			−0.269217
$699$		0			0
$700$		−	880320.i		−	0.0679040i
$701$	2.22272e6			0.170840			0.0854200	−	0.996345i	$-0.472777\pi$
$701$	2.22272e6			0.170840			0.0854200	+	0.996345i	$0.472777\pi$
$702$		0			0
$703$	1.01895e7			0.777617
$704$			491520.i			0.0373774i
$705$		0			0
$706$	6.65162e6			0.502245
$707$			1.18780e7i			0.893708i
$708$		0			0
$709$			2.03634e7i			1.52137i	0.649122	+	0.760684i	$0.275136\pi$
$709$			2.03634e7i			1.52137i	−0.649122	+	0.760684i	$0.724864\pi$
$710$		−	1.30032e7i		−	0.968062i
$711$		0			0
$712$	6.70579e6			0.495736
$713$		−	1.01052e7i		−	0.744425i
$714$		0			0
$715$	3.65976e6	−	716040.i	0.267724	−	0.0523808i
$716$	−1.10407e7			−0.804850
$717$		0			0
$718$	2.50214e6			0.181135
$719$	1.98255e7			1.43022			0.715108	−	0.699014i	$-0.246377\pi$
$719$	1.98255e7			1.43022			0.715108	+	0.699014i	$0.246377\pi$
$720$		0			0
$721$			2.62983e6i			0.188404i
$722$			4.42880e6i			0.316186i
$723$		0			0
$724$	1.54365e6			0.109446
$725$	2.15050e6			0.151948
$726$		0			0
$727$	−9.24667e6			−0.648857			−0.324429	−	0.945910i	$-0.605172\pi$
$727$	−9.24667e6			−0.648857			−0.324429	+	0.945910i	$0.605172\pi$
$728$	−4.01856e6	+	786240.i	−0.281023	+	0.0549828i
$729$		0			0
$730$			1.20087e7i			0.834041i
$731$	1.10045e7			0.761687
$732$		0			0
$733$		−	1.48114e7i		−	1.01821i	−0.860704	−	0.509105i	$-0.829976\pi$
$733$		−	1.48114e7i		−	1.01821i	0.860704	−	0.509105i	$-0.170024\pi$
$734$			4.33309e6i			0.296864i
$735$		0			0
$736$		−	1.07520e6i		−	0.0731635i
$737$	2.73744e6			0.185642
$738$		0			0
$739$		−	5.67210e6i		−	0.382061i	−0.981584	−	0.191031i	$-0.938817\pi$
$739$		−	5.67210e6i		−	0.382061i	0.981584	−	0.191031i	$-0.0611830\pi$
$740$	−7.10654e6			−0.477067
$741$		0			0
$742$	−315000.			−0.0210039
$743$			2.75704e7i			1.83219i	0.400960	+	0.916095i	$0.368677\pi$
$743$			2.75704e7i			1.83219i	−0.400960	+	0.916095i	$0.631323\pi$
$744$		0			0
$745$	1.40977e7			0.930590
$746$		−	7.15582e6i		−	0.470774i
$747$		0			0
$748$			2.11392e6i			0.138145i
$749$		−	2.61702e6i		−	0.170452i
$750$		0			0
$751$	−4.09636e6			−0.265032			−0.132516	−	0.991181i	$-0.542306\pi$
$751$	−4.09636e6			−0.265032			−0.132516	+	0.991181i	$0.542306\pi$
$752$			753408.i			0.0485831i
$753$		0			0
$754$	−1.92067e6	−	9.81677e6i	−0.123034	−	0.628840i
$755$	−1.63880e7			−1.04630
$756$		0			0
$757$	1.09396e7			0.693844			0.346922	−	0.937894i	$-0.387227\pi$
$757$	1.09396e7			0.693844			0.346922	+	0.937894i	$0.387227\pi$
$758$	−3.47446e6			−0.219641
$759$		0			0
$760$			3.81888e6i			0.239829i
$761$		−	1.36940e6i		−	0.0857172i	−0.999081	−	0.0428586i	$-0.986354\pi$
$761$		−	1.36940e6i		−	0.0857172i	0.999081	−	0.0428586i	$-0.0136465\pi$
$762$		0			0
$763$	−1.52073e7			−0.945670
$764$	1.13309e7			0.702312
$765$		0			0
$766$	−4.31886e6			−0.265948
$767$	−4.78975e6	−	2.44809e7i	−0.293984	−	1.50259i
$768$		0			0
$769$		−	1.08375e7i		−	0.660867i	−0.943829	−	0.330433i	$-0.892805\pi$
$769$		−	1.08375e7i		−	0.660867i	0.943829	−	0.330433i	$-0.107195\pi$
$770$	2.57040e6			0.156233
$771$		0			0
$772$			5.56579e6i			0.336112i
$773$			2.05445e7i			1.23665i	0.785922	+	0.618325i	$0.212188\pi$
$773$			2.05445e7i			1.23665i	−0.785922	+	0.618325i	$0.787812\pi$
$774$		0			0
$775$			5.04298e6i			0.301601i
$776$	1.02689e7			0.612168
$777$		0			0
$778$		−	5.15287e6i		−	0.305211i
$779$	1.10916e7			0.654863
$780$		0			0
$781$	−7.64892e6			−0.448717
$782$		−	4.62420e6i		−	0.270408i
$783$		0			0
$784$	1.48019e6			0.0860058
$785$		−	1.73148e7i		−	1.00287i
$786$		0			0
$787$		−	1.34637e7i		−	0.774869i	−0.921897	−	0.387435i	$-0.873361\pi$
$787$		−	1.34637e7i		−	0.774869i	0.921897	−	0.387435i	$-0.126639\pi$
$788$			1.43934e7i			0.825750i
$789$		0			0
$790$	1.28932e7			0.735010
$791$		−	1.05279e7i		−	0.598276i
$792$		0			0
$793$	3.46362e7	−	6.77664e6i	1.95590	−	0.382676i
$794$	2.18764e7			1.23147
$795$		0			0
$796$	−2.28986e6			−0.128093
$797$	−2.02451e7			−1.12895			−0.564475	−	0.825450i	$-0.690921\pi$
$797$	−2.02451e7			−1.12895			−0.564475	+	0.825450i	$0.690921\pi$
$798$		0			0
$799$			3.24024e6i			0.179560i
$800$			536576.i			0.0296419i
$801$		0			0
$802$	−6.34447e6			−0.348305
$803$	7.06392e6			0.386596
$804$		0			0
$805$	−5.62275e6			−0.305815
$806$	2.30206e7	−	4.50403e6i	1.24819	−	0.244210i
$807$		0			0
$808$		−	7.23994e6i		−	0.390127i
$809$	−2.48958e7			−1.33738			−0.668689	−	0.743542i	$-0.733144\pi$
$809$	−2.48958e7			−1.33738			−0.668689	+	0.743542i	$0.733144\pi$
$810$		0			0
$811$		−	2.53328e7i		−	1.35248i	−0.736681	−	0.676241i	$-0.763608\pi$
$811$		−	2.53328e7i		−	1.35248i	0.736681	−	0.676241i	$-0.236392\pi$
$812$		−	6.89472e6i		−	0.366967i
$813$		0			0
$814$			4.18032e6i			0.221130i
$815$	−2.01816e7			−1.06429
$816$		0			0
$817$		−	1.16942e7i		−	0.612934i
$818$	−2.57677e7			−1.34646
$819$		0			0
$820$	−7.73568e6			−0.401757
$821$			1.25922e7i			0.651993i	0.945371	+	0.325996i	$0.105700\pi$
$821$			1.25922e7i			0.651993i	−0.945371	+	0.325996i	$0.894300\pi$
$822$		0			0
$823$	−3.32776e7			−1.71259			−0.856294	−	0.516490i	$-0.827239\pi$
$823$	−3.32776e7			−1.71259			−0.856294	+	0.516490i	$0.827239\pi$
$824$		−	1.60294e6i		−	0.0822433i
$825$		0			0
$826$		−	1.71940e7i		−	0.876851i
$827$		−	2.98630e7i		−	1.51834i	−0.650891	−	0.759171i	$-0.725604\pi$
$827$		−	2.98630e7i		−	1.51834i	0.650891	−	0.759171i	$-0.274396\pi$
$828$		0			0
$829$	3.04980e7			1.54129			0.770647	−	0.637262i	$-0.219933\pi$
$829$	3.04980e7			1.54129			0.770647	+	0.637262i	$0.219933\pi$
$830$		−	1.13134e7i		−	0.570032i
$831$		0			0
$832$	2.44941e6	−	479232.i	0.122674	−	0.0240015i
$833$	6.36598e6			0.317872
$834$		0			0
$835$	−2.17621e7			−1.08015
$836$	2.24640e6			0.111166
$837$		0			0
$838$			1.72218e7i			0.847167i
$839$			1.46249e7i			0.717278i	0.933476	+	0.358639i	$0.116759\pi$
$839$			1.46249e7i			0.717278i	−0.933476	+	0.358639i	$0.883241\pi$
$840$		0			0
$841$	−3.66833e6			−0.178846
$842$	−6.05382e6			−0.294272
$843$		0			0
$844$	5.43570e6			0.262663
$845$	−7.13653e6	−	1.75397e7i	−0.343831	−	0.845044i
$846$		0			0
$847$			1.53984e7i			0.737506i
$848$	192000.			0.00916878
$849$		0			0
$850$			2.30770e6i			0.109555i
$851$		−	9.14445e6i		−	0.432846i
$852$		0			0
$853$		−	1.54032e7i		−	0.724832i	−0.932016	−	0.362416i	$-0.881952\pi$
$853$		−	1.54032e7i		−	0.724832i	0.932016	−	0.362416i	$-0.118048\pi$
$854$	2.43264e7			1.14139
$855$		0			0
$856$			1.59514e6i			0.0744069i
$857$	2.59910e7			1.20884			0.604422	−	0.796664i	$-0.293404\pi$
$857$	2.59910e7			1.20884			0.604422	+	0.796664i	$0.293404\pi$
$858$		0			0
$859$	1.26690e7			0.585815			0.292908	−	0.956141i	$-0.405377\pi$
$859$	1.26690e7			0.585815			0.292908	+	0.956141i	$0.405377\pi$
$860$			8.15592e6i			0.376034i
$861$		0			0
$862$	−5.72466e6			−0.262411
$863$		−	3.93618e7i		−	1.79907i	−0.436849	−	0.899535i	$-0.643906\pi$
$863$		−	3.93618e7i		−	1.79907i	0.436849	−	0.899535i	$-0.356094\pi$
$864$		0			0
$865$			817326.i			0.0371411i
$866$			1.71845e6i			0.0778651i
$867$		0			0
$868$	1.61683e7			0.728393
$869$		−	7.58424e6i		−	0.340693i
$870$		0			0
$871$	−2.66900e6	−	1.36416e7i	−0.119208	−	0.609283i
$872$	9.26918e6			0.412810
$873$		0			0
$874$	−4.91400e6			−0.217599
$875$	1.95404e7			0.862806
$876$		0			0
$877$			2.93636e7i			1.28917i	0.764532	+	0.644585i	$0.222970\pi$
$877$			2.93636e7i			1.28917i	−0.764532	+	0.644585i	$0.777030\pi$
$878$			2.20815e6i			0.0966702i
$879$		0			0
$880$	−1.56672e6			−0.0682001
$881$	2.47421e7			1.07398			0.536990	−	0.843589i	$-0.319561\pi$
$881$	2.47421e7			1.07398			0.536990	+	0.843589i	$0.319561\pi$
$882$		0			0
$883$	1.56178e7			0.674092			0.337046	−	0.941488i	$-0.390572\pi$
$883$	1.56178e7			0.674092			0.337046	+	0.941488i	$0.390572\pi$
$884$	1.05344e7	−	2.06107e6i	0.453396	−	0.0887079i
$885$		0			0
$886$		−	8.61020e6i		−	0.368493i
$887$	−1.41193e6			−0.0602566			−0.0301283	−	0.999546i	$-0.509592\pi$
$887$	−1.41193e6			−0.0602566			−0.0301283	+	0.999546i	$0.509592\pi$
$888$		0			0
$889$		−	2.12892e7i		−	0.903450i
$890$			2.13747e7i			0.904535i
$891$		0			0
$892$		−	9.98011e6i		−	0.419975i
$893$	3.44331e6			0.144493
$894$		0			0
$895$		−	3.51923e7i		−	1.46855i
$896$	1.72032e6			0.0715878
$897$		0			0
$898$	5.61715e6			0.232448
$899$			3.94969e7i			1.62991i
$900$		0			0
$901$	825750.			0.0338873
$902$			4.55040e6i			0.186223i
$903$		0			0
$904$			6.41702e6i			0.261164i
$905$			4.92038e6i			0.199700i
$906$		0			0
$907$	−1.48543e7			−0.599563			−0.299781	−	0.954008i	$-0.596914\pi$
$907$	−1.48543e7			−0.599563			−0.299781	+	0.954008i	$0.596914\pi$
$908$		−	2.84179e6i		−	0.114387i
$909$		0			0
$910$	−2.50614e6	−	1.28092e7i	−0.100323	−	0.512764i
$911$	−4.29118e7			−1.71309			−0.856547	−	0.516069i	$-0.827395\pi$
$911$	−4.29118e7			−1.71309			−0.856547	+	0.516069i	$0.827395\pi$
$912$		0			0
$913$	−6.65496e6			−0.264222
$914$	5.31271e6			0.210354
$915$		0			0
$916$			1.89928e7i			0.747911i
$917$			3.19048e7i			1.25295i
$918$		0			0
$919$	4.34706e7			1.69788			0.848939	−	0.528491i	$-0.177242\pi$
$919$	4.34706e7			1.69788			0.848939	+	0.528491i	$0.177242\pi$
$920$	3.42720e6			0.133497
$921$		0			0
$922$	2.35628e7			0.912850
$923$	7.45770e6	+	3.81171e7i	0.288138	+	1.47271i
$924$		0			0
$925$			4.56352e6i			0.175366i
$926$	−9.48557e6			−0.363526
$927$		0			0
$928$			4.20250e6i			0.160191i
$929$		−	2.97375e7i		−	1.13049i	−0.824925	−	0.565243i	$-0.808783\pi$
$929$		−	2.97375e7i		−	1.13049i	0.824925	−	0.565243i	$-0.191217\pi$
$930$		0			0
$931$		−	6.76494e6i		−	0.255794i
$932$	−1.79707e6			−0.0677682
$933$		0			0
$934$		−	2.87131e7i		−	1.07699i
$935$	−6.73812e6			−0.252063
$936$		0			0
$937$	1.46550e7			0.545303			0.272651	−	0.962113i	$-0.412099\pi$
$937$	1.46550e7			0.545303			0.272651	+	0.962113i	$0.412099\pi$
$938$		−	9.58104e6i		−	0.355554i
$939$		0			0
$940$	−2.40149e6			−0.0886463
$941$		−	8.80233e6i		−	0.324059i	−0.986786	−	0.162029i	$-0.948196\pi$
$941$		−	8.80233e6i		−	0.324059i	0.986786	−	0.162029i	$-0.0518039\pi$
$942$		0			0
$943$		−	9.95400e6i		−	0.364518i
$944$			1.04801e7i			0.382769i
$945$		0			0
$946$	4.79760e6			0.174300
$947$		−	9.00847e6i		−	0.326420i	−0.986591	−	0.163210i	$-0.947815\pi$
$947$		−	9.00847e6i		−	0.326420i	0.986591	−	0.163210i	$-0.0521847\pi$
$948$		0			0
$949$	−6.88732e6	−	3.52019e7i	−0.248247	−	1.26882i
$950$	2.45232e6			0.0881593
$951$		0			0
$952$	7.39872e6			0.264584
$953$	−1.31122e7			−0.467675			−0.233837	−	0.972276i	$-0.575128\pi$
$953$	−1.31122e7			−0.467675			−0.233837	+	0.972276i	$0.575128\pi$
$954$		0			0
$955$			3.61172e7i			1.28146i
$956$		−	1.91688e7i		−	0.678346i
$957$		0			0
$958$	2.90077e7			1.02117
$959$	−6.69249e6			−0.234986
$960$		0			0
$961$	−6.39922e7			−2.23521
$962$	2.08319e7	−	4.07581e6i	0.725758	−	0.141996i
$963$		0			0
$964$		−	1.86606e7i		−	0.646744i
$965$	−1.77410e7			−0.613280
$966$		0			0
$967$		−	1.53210e6i		−	0.0526892i	−0.999653	−	0.0263446i	$-0.991613\pi$
$967$		−	1.53210e6i		−	0.0526892i	0.999653	−	0.0263446i	$-0.00838672\pi$
$968$		−	9.38566e6i		−	0.321941i
$969$		0			0
$970$			3.27322e7i			1.11698i
$971$	−2.37514e7			−0.808426			−0.404213	−	0.914665i	$-0.632455\pi$
$971$	−2.37514e7			−0.808426			−0.404213	+	0.914665i	$0.632455\pi$
$972$		0			0
$973$			1.45330e6i			0.0492124i
$974$	1.01346e7			0.342301
$975$		0			0
$976$	−1.48275e7			−0.498246
$977$		−	1.97751e7i		−	0.662798i	−0.943491	−	0.331399i	$-0.892479\pi$
$977$		−	1.97751e7i		−	0.662798i	0.943491	−	0.331399i	$-0.107521\pi$
$978$		0			0
$979$	1.25734e7			0.419271
$980$			4.71811e6i			0.156929i
$981$		0			0
$982$		−	3.38475e7i		−	1.12008i
$983$		−	1.57006e7i		−	0.518241i	−0.965845	−	0.259121i	$-0.916567\pi$
$983$		−	1.57006e7i		−	0.518241i	0.965845	−	0.259121i	$-0.0834327\pi$
$984$		0			0
$985$	−4.58790e7			−1.50669
$986$			1.80740e7i			0.592055i
$987$		0			0
$988$	−2.19024e6	−	1.11946e7i	−0.0713837	−	0.364850i
$989$	−1.04948e7			−0.341178
$990$		0			0
$991$	1.65835e7			0.536406			0.268203	−	0.963362i	$-0.413570\pi$
$991$	1.65835e7			0.536406			0.268203	+	0.963362i	$0.413570\pi$
$992$	−9.85498e6			−0.317963
$993$		0			0
$994$			2.67712e7i			0.859414i
$995$		−	7.29892e6i		−	0.233723i
$996$		0			0
$997$	−2.08432e7			−0.664091			−0.332045	−	0.943263i	$-0.607739\pi$
$997$	−2.08432e7			−0.664091			−0.332045	+	0.943263i	$0.607739\pi$
$998$	7.81531e6			0.248382
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.6.b.b.181.2		2
3.2	odd	2		26.6.b.a.25.1	✓	2
12.11	even	2		208.6.f.b.129.2		2
13.12	even	2	inner	234.6.b.b.181.1		2
39.5	even	4		338.6.a.a.1.1		1
39.8	even	4		338.6.a.c.1.1		1
39.38	odd	2		26.6.b.a.25.2	yes	2
156.155	even	2		208.6.f.b.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.6.b.a.25.1	✓	2	3.2	odd	2
26.6.b.a.25.2	yes	2	39.38	odd	2
208.6.f.b.129.1		2	156.155	even	2
208.6.f.b.129.2		2	12.11	even	2
234.6.b.b.181.1		2	13.12	even	2	inner
234.6.b.b.181.2		2	1.1	even	1	trivial
338.6.a.a.1.1		1	39.5	even	4
338.6.a.c.1.1		1	39.8	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	234.6.b.b.181.2

Level	$234$
Weight	$6$
Character	234.181
Analytic conductor	$37.530$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$