LMFDB - Embedded newform 450.6.c.h.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
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			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.6.c.h.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} +172.000i q^{7} +64.0000i q^{8} +O(q^{10})$	$q-4.00000i q^{2} -16.0000 q^{4} +172.000i q^{7} +64.0000i q^{8} -132.000 q^{11} -946.000i q^{13} +688.000 q^{14} +256.000 q^{16} -222.000i q^{17} -500.000 q^{19} +528.000i q^{22} -3564.00i q^{23} -3784.00 q^{26} -2752.00i q^{28} +2190.00 q^{29} +2312.00 q^{31} -1024.00i q^{32} -888.000 q^{34} +11242.0i q^{37} +2000.00i q^{38} -1242.00 q^{41} +20624.0i q^{43} +2112.00 q^{44} -14256.0 q^{46} +6588.00i q^{47} -12777.0 q^{49} +15136.0i q^{52} +21066.0i q^{53} -11008.0 q^{56} -8760.00i q^{58} +7980.00 q^{59} +16622.0 q^{61} -9248.00i q^{62} -4096.00 q^{64} -1808.00i q^{67} +3552.00i q^{68} +24528.0 q^{71} +20474.0i q^{73} +44968.0 q^{74} +8000.00 q^{76} -22704.0i q^{77} +46240.0 q^{79} +4968.00i q^{82} +51576.0i q^{83} +82496.0 q^{86} -8448.00i q^{88} -110310. q^{89} +162712. q^{91} +57024.0i q^{92} +26352.0 q^{94} +78382.0i q^{97} +51108.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} - 264 q^{11} + 1376 q^{14} + 512 q^{16} - 1000 q^{19} - 7568 q^{26} + 4380 q^{29} + 4624 q^{31} - 1776 q^{34} - 2484 q^{41} + 4224 q^{44} - 28512 q^{46} - 25554 q^{49} - 22016 q^{56} + 15960 q^{59} + 33244 q^{61} - 8192 q^{64} + 49056 q^{71} + 89936 q^{74} + 16000 q^{76} + 92480 q^{79} + 164992 q^{86} - 220620 q^{89} + 325424 q^{91} + 52704 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 - 264 * q^11 + 1376 * q^14 + 512 * q^16 - 1000 * q^19 - 7568 * q^26 + 4380 * q^29 + 4624 * q^31 - 1776 * q^34 - 2484 * q^41 + 4224 * q^44 - 28512 * q^46 - 25554 * q^49 - 22016 * q^56 + 15960 * q^59 + 33244 * q^61 - 8192 * q^64 + 49056 * q^71 + 89936 * q^74 + 16000 * q^76 + 92480 * q^79 + 164992 * q^86 - 220620 * q^89 + 325424 * q^91 + 52704 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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
$5$	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	$0.769127\pi$
$8$	64.0000i	0.353553i
$9$	0	0
$10$	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	$-0.717118\pi$
$13$	− 946.000i	− 1.55250i	0.630423	−	0.776252i	$-0.282882\pi$
$14$	688.000	0.938142
$15$	0	0
$16$	256.000	0.250000
$17$	− 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	$-0.550787\pi$
$19$	−500.000	−0.317750	−0.158875	+	0.987299i	$0.550787\pi$
$20$	0	0
$21$	0	0
$22$	528.000i	0.232583i
$23$	− 3564.00i	− 1.40481i	−0.711777	−	0.702406i	$-0.752109\pi$
$23$	− 3564.00i	− 1.40481i	0.711777	−	0.702406i	$-0.247891\pi$
$24$	0	0
$25$	0	0
$26$	−3784.00	−1.09779
$27$	0	0
$28$	− 2752.00i	− 0.663366i
$29$	2190.00	0.483559	0.241779	−	0.970331i	$-0.422269\pi$
$29$	2190.00	0.483559	0.241779	+	0.970331i	$0.422269\pi$
$30$	0	0
$31$	2312.00	0.432099	0.216050	−	0.976382i	$-0.430683\pi$
$31$	2312.00	0.432099	0.216050	+	0.976382i	$0.430683\pi$
$32$	− 1024.00i	− 0.176777i
$33$	0	0
$34$	−888.000	−0.131739
$35$	0	0
$36$	0	0
$37$	11242.0i	1.35002i	0.737810	+	0.675009i	$0.235860\pi$
$37$	11242.0i	1.35002i	−0.737810	+	0.675009i	$0.764140\pi$
$38$	2000.00i	0.224683i
$39$	0	0
$40$	0	0
$41$	−1242.00	−0.115388	−0.0576942	−	0.998334i	$-0.518375\pi$
$41$	−1242.00	−0.115388	−0.0576942	+	0.998334i	$0.518375\pi$
$42$	0	0
$43$	20624.0i	1.70099i	0.525983	+	0.850495i	$0.323697\pi$
$43$	20624.0i	1.70099i	−0.525983	+	0.850495i	$0.676303\pi$
$44$	2112.00	0.164461
$45$	0	0
$46$	−14256.0	−0.993352
$47$	6588.00i	0.435020i	0.976058	+	0.217510i	$0.0697934\pi$
$47$	6588.00i	0.435020i	−0.976058	+	0.217510i	$0.930207\pi$
$48$	0	0
$49$	−12777.0	−0.760219
$50$	0	0
$51$	0	0
$52$	15136.0i	0.776252i
$53$	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
$56$	−11008.0	−0.469071
$57$	0	0
$58$	− 8760.00i	− 0.341928i
$59$	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	$0.407682\pi$
$62$	− 9248.00i	− 0.305540i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	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	$-0.00783205\pi$
$68$	3552.00i	0.0931538i
$69$	0	0
$70$	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	$0.0721846\pi$
$73$	20474.0i	0.449672i	−0.974397	+	0.224836i	$0.927815\pi$
$74$	44968.0	0.954606
$75$	0	0
$76$	8000.00	0.158875
$77$	− 22704.0i	− 0.436391i
$78$	0	0
$79$	46240.0	0.833585	0.416793	−	0.909002i	$-0.363154\pi$
$79$	46240.0	0.833585	0.416793	+	0.909002i	$0.363154\pi$
$80$	0	0
$81$	0	0
$82$	4968.00i	0.0815919i
$83$	51576.0i	0.821774i	0.911686	+	0.410887i	$0.134781\pi$
$83$	51576.0i	0.821774i	−0.911686	+	0.410887i	$0.865219\pi$
$84$	0	0
$85$	0	0
$86$	82496.0	1.20278
$87$	0	0
$88$	− 8448.00i	− 0.116291i
$89$	−110310.	−1.47618	−0.738091	−	0.674701i	$-0.764272\pi$
$89$	−110310.	−1.47618	−0.738091	+	0.674701i	$0.764272\pi$
$90$	0	0
$91$	162712.	2.05976
$92$	57024.0i	0.702406i
$93$	0	0
$94$	26352.0	0.307605
$95$	0	0
$96$	0	0
$97$	78382.0i	0.845838i	0.906168	+	0.422919i	$0.138994\pi$
$97$	78382.0i	0.845838i	−0.906168	+	0.422919i	$0.861006\pi$
$98$	51108.0i	0.537556i
$99$	0	0
$100$	0	0
$101$	−141942.	−1.38455	−0.692273	−	0.721636i	$-0.743391\pi$
$101$	−141942.	−1.38455	−0.692273	+	0.721636i	$0.743391\pi$
$102$	0	0
$103$	− 436.000i	− 0.00404943i	−0.999998	−	0.00202471i	$-0.999356\pi$
$103$	− 436.000i	− 0.00404943i	0.999998	−	0.00202471i	$-0.000644487\pi$
$104$	60544.0	0.548893
$105$	0	0
$106$	84264.0	0.728413
$107$	232968.i	1.96715i	0.180508	+	0.983574i	$0.442226\pi$
$107$	232968.i	1.96715i	−0.180508	+	0.983574i	$0.557774\pi$
$108$	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
$111$	0	0
$112$	44032.0i	0.331683i
$113$	− 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
$116$	−35040.0	−0.241779
$117$	0	0
$118$	− 31920.0i	− 0.211037i
$119$	38184.0	0.247180
$120$	0	0
$121$	−143627.	−0.891811
$122$	− 66488.0i	− 0.404430i
$123$	0	0
$124$	−36992.0	−0.216050
$125$	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	$0.890641\pi$
$128$	16384.0i	0.0883883i
$129$	0	0
$130$	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
$134$	−7232.00	−0.0347934
$135$	0	0
$136$	14208.0	0.0658697
$137$	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	$-0.360338\pi$
$139$	193540.	0.849638	0.424819	+	0.905278i	$0.360338\pi$
$140$	0	0
$141$	0	0
$142$	− 98112.0i	− 0.408321i
$143$	124872.i	0.510652i
$144$	0	0
$145$	0	0
$146$	81896.0	0.317966
$147$	0	0
$148$	− 179872.i	− 0.675009i
$149$	140550.	0.518639	0.259320	−	0.965792i	$-0.416502\pi$
$149$	140550.	0.518639	0.259320	+	0.965792i	$0.416502\pi$
$150$	0	0
$151$	433952.	1.54881	0.774407	−	0.632688i	$-0.218048\pi$
$151$	433952.	1.54881	0.774407	+	0.632688i	$0.218048\pi$
$152$	− 32000.0i	− 0.112342i
$153$	0	0
$154$	−90816.0	−0.308575
$155$	0	0
$156$	0	0
$157$	555922.i	1.79997i	0.435923	+	0.899984i	$0.356422\pi$
$157$	555922.i	1.79997i	−0.435923	+	0.899984i	$0.643578\pi$
$158$	− 184960.i	− 0.589434i
$159$	0	0
$160$	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	$-0.0313062\pi$
$164$	19872.0	0.0576942
$165$	0	0
$166$	206304.	0.581082
$167$	− 205692.i	− 0.570724i	−0.958420	−	0.285362i	$-0.907886\pi$
$167$	− 205692.i	− 0.570724i	0.958420	−	0.285362i	$-0.0921138\pi$
$168$	0	0
$169$	−523623.	−1.41027
$170$	0	0
$171$	0	0
$172$	− 329984.i	− 0.850495i
$173$	− 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
$176$	−33792.0	−0.0822304
$177$	0	0
$178$	441240.i	1.04382i
$179$	−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	$0.196092\pi$
$182$	− 650848.i	− 1.45647i
$183$	0	0
$184$	228096.	0.496676
$185$	0	0
$186$	0	0
$187$	29304.0i	0.0612806i
$188$	− 105408.i	− 0.217510i
$189$	0	0
$190$	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	$-0.806389\pi$
$193$	− 591406.i	− 1.14286i	0.820651	−	0.571429i	$-0.193611\pi$
$194$	313528.	0.598098
$195$	0	0
$196$	204432.	0.380109
$197$	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	$-0.544923\pi$
$199$	−157160.	−0.281326	−0.140663	+	0.990058i	$0.544923\pi$
$200$	0	0
$201$	0	0
$202$	567768.i	0.979022i
$203$	376680.i	0.641553i
$204$	0	0
$205$	0	0
$206$	−1744.00	−0.00286338
$207$	0	0
$208$	− 242176.i	− 0.388126i
$209$	66000.0	0.104515
$210$	0	0
$211$	253052.	0.391294	0.195647	−	0.980674i	$-0.437319\pi$
$211$	253052.	0.391294	0.195647	+	0.980674i	$0.437319\pi$
$212$	− 337056.i	− 0.515065i
$213$	0	0
$214$	931872.	1.39098
$215$	0	0
$216$	0	0
$217$	397664.i	0.573280i
$218$	− 699400.i	− 0.996746i
$219$	0	0
$220$	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	$0.742878\pi$
$224$	176128.	0.234535
$225$	0	0
$226$	−731976.	−0.953292
$227$	− 626832.i	− 0.807396i	−0.914892	−	0.403698i	$-0.867725\pi$
$227$	− 626832.i	− 0.807396i	0.914892	−	0.403698i	$-0.132275\pi$
$228$	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
$231$	0	0
$232$	140160.i	0.170964i
$233$	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
$236$	−127680.	−0.149225
$237$	0	0
$238$	− 152736.i	− 0.174783i
$239$	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	$0.716197\pi$
$242$	574508.i	0.630605i
$243$	0	0
$244$	−265952.	−0.285975
$245$	0	0
$246$	0	0
$247$	473000.i	0.493309i
$248$	147968.i	0.152770i
$249$	0	0
$250$	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
$254$	489808.	0.476367
$255$	0	0
$256$	65536.0	0.0625000
$257$	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
$261$	0	0
$262$	− 967632.i	− 0.870877i
$263$	805476.i	0.718064i	0.933325	+	0.359032i	$0.116893\pi$
$263$	805476.i	0.718064i	−0.933325	+	0.359032i	$0.883107\pi$
$264$	0	0
$265$	0	0
$266$	−344000.	−0.298095
$267$	0	0
$268$	28928.0i	0.0246026i
$269$	−858690.	−0.723529	−0.361764	−	0.932270i	$-0.617825\pi$
$269$	−858690.	−0.723529	−0.361764	+	0.932270i	$0.617825\pi$
$270$	0	0
$271$	−383608.	−0.317296	−0.158648	−	0.987335i	$-0.550713\pi$
$271$	−383608.	−0.317296	−0.158648	+	0.987335i	$0.550713\pi$
$272$	− 56832.0i	− 0.0465769i
$273$	0	0
$274$	1.10839e6	0.891902
$275$	0	0
$276$	0	0
$277$	− 2.01076e6i	− 1.57456i	−0.616593	−	0.787282i	$-0.711488\pi$
$277$	− 2.01076e6i	− 1.57456i	0.616593	−	0.787282i	$-0.288512\pi$
$278$	− 774160.i	− 0.600785i
$279$	0	0
$280$	0	0
$281$	−202602.	−0.153066	−0.0765329	−	0.997067i	$-0.524385\pi$
$281$	−202602.	−0.153066	−0.0765329	+	0.997067i	$0.524385\pi$
$282$	0	0
$283$	− 221536.i	− 0.164429i	−0.996615	−	0.0822145i	$-0.973801\pi$
$283$	− 221536.i	− 0.164429i	0.996615	−	0.0822145i	$-0.0261992\pi$
$284$	−392448.	−0.288726
$285$	0	0
$286$	499488.	0.361085
$287$	− 213624.i	− 0.153089i
$288$	0	0
$289$	1.37057e6	0.965289
$290$	0	0
$291$	0	0
$292$	− 327584.i	− 0.224836i
$293$	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
$296$	−719488.	−0.477303
$297$	0	0
$298$	− 562200.i	− 0.366733i
$299$	−3.37154e6	−2.18098
$300$	0	0
$301$	−3.54733e6	−2.25676
$302$	− 1.73581e6i	− 1.09518i
$303$	0	0
$304$	−128000.	−0.0794376
$305$	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	$-0.143927\pi$
$308$	363264.i	0.218195i
$309$	0	0
$310$	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	$-0.904270\pi$
$313$	− 1.02689e6i	− 0.592463i	0.955116	−	0.296232i	$-0.0957300\pi$
$314$	2.22369e6	1.27277
$315$	0	0
$316$	−739840.	−0.416793
$317$	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
$321$	0	0
$322$	− 2.45203e6i	− 1.31791i
$323$	111000.i	0.0591993i
$324$	0	0
$325$	0	0
$326$	−266464.	−0.138866
$327$	0	0
$328$	− 79488.0i	− 0.0407959i
$329$	−1.13314e6	−0.577155
$330$	0	0
$331$	1.99413e6	1.00042	0.500212	−	0.865903i	$-0.333255\pi$
$331$	1.99413e6	1.00042	0.500212	+	0.865903i	$0.333255\pi$
$332$	− 825216.i	− 0.410887i
$333$	0	0
$334$	−822768.	−0.403563
$335$	0	0
$336$	0	0
$337$	987022.i	0.473426i	0.971580	+	0.236713i	$0.0760701\pi$
$337$	987022.i	0.473426i	−0.971580	+	0.236713i	$0.923930\pi$
$338$	2.09449e6i	0.997211i
$339$	0	0
$340$	0	0
$341$	−305184.	−0.142127
$342$	0	0
$343$	693160.i	0.318125i
$344$	−1.31994e6	−0.601391
$345$	0	0
$346$	−1.73542e6	−0.779316
$347$	2.20601e6i	0.983520i	0.870731	+	0.491760i	$0.163646\pi$
$347$	2.20601e6i	0.983520i	−0.870731	+	0.491760i	$0.836354\pi$
$348$	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
$351$	0	0
$352$	135168.i	0.0581456i
$353$	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
$356$	1.76496e6	0.738091
$357$	0	0
$358$	1.95672e6i	0.806903i
$359$	−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
$362$	− 2.87785e6i	− 1.15424i
$363$	0	0
$364$	−2.60339e6	−1.02988
$365$	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	$0.840685\pi$
$368$	− 912384.i	− 0.351203i
$369$	0	0
$370$	0	0
$371$	−3.62335e6	−1.36671
$372$	0	0
$373$	2.74525e6i	1.02167i	0.859679	+	0.510835i	$0.170664\pi$
$373$	2.74525e6i	1.02167i	−0.859679	+	0.510835i	$0.829336\pi$
$374$	117216.	0.0433319
$375$	0	0
$376$	−421632.	−0.153803
$377$	− 2.07174e6i	− 0.750727i
$378$	0	0
$379$	1.18906e6	0.425212	0.212606	−	0.977138i	$-0.431805\pi$
$379$	1.18906e6	0.425212	0.212606	+	0.977138i	$0.431805\pi$
$380$	0	0
$381$	0	0
$382$	− 743712.i	− 0.260763i
$383$	− 3.25760e6i	− 1.13475i	−0.823458	−	0.567377i	$-0.807958\pi$
$383$	− 3.25760e6i	− 1.13475i	0.823458	−	0.567377i	$-0.192042\pi$
$384$	0	0
$385$	0	0
$386$	−2.36562e6	−0.808123
$387$	0	0
$388$	− 1.25411e6i	− 0.422919i
$389$	1.98351e6	0.664600	0.332300	−	0.943174i	$-0.392175\pi$
$389$	1.98351e6	0.664600	0.332300	+	0.943174i	$0.392175\pi$
$390$	0	0
$391$	−791208.	−0.261727
$392$	− 817728.i	− 0.268778i
$393$	0	0
$394$	1.79791e6	0.583483
$395$	0	0
$396$	0	0
$397$	− 4.97416e6i	− 1.58396i	−0.610549	−	0.791978i	$-0.709051\pi$
$397$	− 4.97416e6i	− 1.58396i	0.610549	−	0.791978i	$-0.290949\pi$
$398$	628640.i	0.198927i
$399$	0	0
$400$	0	0
$401$	1.34264e6	0.416963	0.208482	−	0.978026i	$-0.433148\pi$
$401$	1.34264e6	0.416963	0.208482	+	0.978026i	$0.433148\pi$
$402$	0	0
$403$	− 2.18715e6i	− 0.670836i
$404$	2.27107e6	0.692273
$405$	0	0
$406$	1.50672e6	0.453646
$407$	− 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
$411$	0	0
$412$	6976.00i	0.00202471i
$413$	1.37256e6i	0.395964i
$414$	0	0
$415$	0	0
$416$	−968704.	−0.274447
$417$	0	0
$418$	− 264000.i	− 0.0739032i
$419$	−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	$0.570326\pi$
$422$	− 1.01221e6i	− 0.276687i
$423$	0	0
$424$	−1.34822e6	−0.364206
$425$	0	0
$426$	0	0
$427$	2.85898e6i	0.758826i
$428$	− 3.72749e6i	− 0.983574i
$429$	0	0
$430$	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	$0.158351\pi$
$433$	3.72355e6i	0.954416i	−0.878790	+	0.477208i	$0.841649\pi$
$434$	1.59066e6	0.405370
$435$	0	0
$436$	−2.79760e6	−0.704806
$437$	1.78200e6i	0.446379i
$438$	0	0
$439$	2.58340e6	0.639780	0.319890	−	0.947455i	$-0.396354\pi$
$439$	2.58340e6	0.639780	0.319890	+	0.947455i	$0.396354\pi$
$440$	0	0
$441$	0	0
$442$	840048.i	0.204526i
$443$	− 7.56206e6i	− 1.83076i	−0.402593	−	0.915379i	$-0.631891\pi$
$443$	− 7.56206e6i	− 1.83076i	0.402593	−	0.915379i	$-0.368109\pi$
$444$	0	0
$445$	0	0
$446$	4.29378e6	1.02212
$447$	0	0
$448$	− 704512.i	− 0.165842i
$449$	4.30773e6	1.00840	0.504200	−	0.863587i	$-0.331788\pi$
$449$	4.30773e6	1.00840	0.504200	+	0.863587i	$0.331788\pi$
$450$	0	0
$451$	163944.	0.0379537
$452$	2.92790e6i	0.674079i
$453$	0	0
$454$	−2.50733e6	−0.570915
$455$	0	0
$456$	0	0
$457$	2.24354e6i	0.502509i	0.967921	+	0.251254i	$0.0808431\pi$
$457$	2.24354e6i	0.502509i	−0.967921	+	0.251254i	$0.919157\pi$
$458$	− 466600.i	− 0.103940i
$459$	0	0
$460$	0	0
$461$	−1.65670e6	−0.363071	−0.181536	−	0.983384i	$-0.558107\pi$
$461$	−1.65670e6	−0.363071	−0.181536	+	0.983384i	$0.558107\pi$
$462$	0	0
$463$	− 2.89160e6i	− 0.626881i	−0.949608	−	0.313441i	$-0.898518\pi$
$463$	− 2.89160e6i	− 0.626881i	0.949608	−	0.313441i	$-0.101482\pi$
$464$	560640.	0.120890
$465$	0	0
$466$	2.97218e6	0.634032
$467$	− 6.52699e6i	− 1.38491i	−0.721462	−	0.692454i	$-0.756530\pi$
$467$	− 6.52699e6i	− 1.38491i	0.721462	−	0.692454i	$-0.243470\pi$
$468$	0	0
$469$	310976.	0.0652822
$470$	0	0
$471$	0	0
$472$	510720.i	0.105518i
$473$	− 2.72237e6i	− 0.559492i
$474$	0	0
$475$	0	0
$476$	−610944.	−0.123590
$477$	0	0
$478$	− 3.91488e6i	− 0.783698i
$479$	−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
$482$	4.53119e6i	0.888372i
$483$	0	0
$484$	2.29803e6	0.445905
$485$	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	$-0.0922666\pi$
$488$	1.06381e6i	0.202215i
$489$	0	0
$490$	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
$494$	1.89200e6	0.348822
$495$	0	0
$496$	591872.	0.108025
$497$	4.21882e6i	0.766125i
$498$	0	0
$499$	−9.20546e6	−1.65499	−0.827493	−	0.561477i	$-0.810233\pi$
$499$	−9.20546e6	−1.65499	−0.827493	+	0.561477i	$0.810233\pi$
$500$	0	0
$501$	0	0
$502$	3.62261e6i	0.641597i
$503$	3.35956e6i	0.592055i	0.955179	+	0.296027i	$0.0956620\pi$
$503$	3.35956e6i	0.592055i	−0.955179	+	0.296027i	$0.904338\pi$
$504$	0	0
$505$	0	0
$506$	1.88179e6	0.326735
$507$	0	0
$508$	− 1.95923e6i	− 0.336842i
$509$	−2.53701e6	−0.434038	−0.217019	−	0.976167i	$-0.569633\pi$
$509$	−2.53701e6	−0.434038	−0.217019	+	0.976167i	$0.569633\pi$
$510$	0	0
$511$	−3.52153e6	−0.596594
$512$	− 262144.i	− 0.0441942i
$513$	0	0
$514$	7.75975e6	1.29551
$515$	0	0
$516$	0	0
$517$	− 869616.i	− 0.143087i
$518$	7.73450e6i	1.26651i
$519$	0	0
$520$	0	0
$521$	9.31580e6	1.50358	0.751789	−	0.659404i	$-0.229191\pi$
$521$	9.31580e6	1.50358	0.751789	+	0.659404i	$0.229191\pi$
$522$	0	0
$523$	− 5.02802e6i	− 0.803790i	−0.915686	−	0.401895i	$-0.868352\pi$
$523$	− 5.02802e6i	− 0.803790i	0.915686	−	0.401895i	$-0.131648\pi$
$524$	−3.87053e6	−0.615803
$525$	0	0
$526$	3.22190e6	0.507748
$527$	− 513264.i	− 0.0805034i
$528$	0	0
$529$	−6.26575e6	−0.973496
$530$	0	0
$531$	0	0
$532$	1.37600e6i	0.210785i
$533$	1.17493e6i	0.179141i
$534$	0	0
$535$	0	0
$536$	115712.	0.0173967
$537$	0	0
$538$	3.43476e6i	0.511612i
$539$	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	$0.496862\pi$
$542$	1.53443e6i	0.224362i
$543$	0	0
$544$	−227328.	−0.0329348
$545$	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	$-0.0137787\pi$
$548$	− 4.43357e6i	− 0.630670i
$549$	0	0
$550$	0	0
$551$	−1.09500e6	−0.153651
$552$	0	0
$553$	7.95328e6i	1.10594i
$554$	−8.04303e6	−1.11339
$555$	0	0
$556$	−3.09664e6	−0.424819
$557$	− 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
$561$	0	0
$562$	810408.i	0.108234i
$563$	1.03029e7i	1.36990i	0.728588	+	0.684952i	$0.240177\pi$
$563$	1.03029e7i	1.36990i	−0.728588	+	0.684952i	$0.759823\pi$
$564$	0	0
$565$	0	0
$566$	−886144.	−0.116269
$567$	0	0
$568$	1.56979e6i	0.204160i
$569$	1.04769e6	0.135660	0.0678300	−	0.997697i	$-0.478392\pi$
$569$	1.04769e6	0.135660	0.0678300	+	0.997697i	$0.478392\pi$
$570$	0	0
$571$	1.40765e7	1.80677	0.903385	−	0.428830i	$-0.141074\pi$
$571$	1.40765e7	1.80677	0.903385	+	0.428830i	$0.141074\pi$
$572$	− 1.99795e6i	− 0.255326i
$573$	0	0
$574$	−854496.	−0.108251
$575$	0	0
$576$	0	0
$577$	− 1.62682e6i	− 0.203423i	−0.994814	−	0.101711i	$-0.967568\pi$
$577$	− 1.62682e6i	− 0.203423i	0.994814	−	0.101711i	$-0.0324318\pi$
$578$	− 5.48229e6i	− 0.682563i
$579$	0	0
$580$	0	0
$581$	−8.87107e6	−1.09027
$582$	0	0
$583$	− 2.78071e6i	− 0.338832i
$584$	−1.31034e6	−0.158983
$585$	0	0
$586$	1.29002e6	0.155186
$587$	6.96089e6i	0.833814i	0.908949	+	0.416907i	$0.136886\pi$
$587$	6.96089e6i	0.833814i	−0.908949	+	0.416907i	$0.863114\pi$
$588$	0	0
$589$	−1.15600e6	−0.137300
$590$	0	0
$591$	0	0
$592$	2.87795e6i	0.337504i
$593$	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
$596$	−2.24880e6	−0.259320
$597$	0	0
$598$	1.34862e7i	1.54218i
$599$	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	$0.522236\pi$
$602$	1.41893e7i	1.59577i
$603$	0	0
$604$	−6.94323e6	−0.774407
$605$	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	$0.759478\pi$
$608$	512000.i	0.0561709i
$609$	0	0
$610$	0	0
$611$	6.23225e6	0.675370
$612$	0	0
$613$	− 8.73491e6i	− 0.938873i	−0.882966	−	0.469437i	$-0.844457\pi$
$613$	− 8.73491e6i	− 0.938873i	0.882966	−	0.469437i	$-0.155543\pi$
$614$	−5.77203e6	−0.617885
$615$	0	0
$616$	1.45306e6	0.154287
$617$	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	$-0.220647\pi$
$619$	1.46658e7	1.53843	0.769216	+	0.638988i	$0.220647\pi$
$620$	0	0
$621$	0	0
$622$	685248.i	0.0710186i
$623$	− 1.89733e7i	− 1.95850i
$624$	0	0
$625$	0	0
$626$	−4.10754e6	−0.418935
$627$	0	0
$628$	− 8.89475e6i	− 0.899984i
$629$	2.49572e6	0.251519
$630$	0	0
$631$	−196288.	−0.0196255	−0.00981274	−	0.999952i	$-0.503124\pi$
$631$	−196288.	−0.0196255	−0.00981274	+	0.999952i	$0.503124\pi$
$632$	2.95936e6i	0.294717i
$633$	0	0
$634$	3.01183e6	0.297583
$635$	0	0
$636$	0	0
$637$	1.20870e7i	1.18024i
$638$	1.15632e6i	0.112467i
$639$	0	0
$640$	0	0
$641$	1.11596e7	1.07276	0.536381	−	0.843976i	$-0.319791\pi$
$641$	1.11596e7	1.07276	0.536381	+	0.843976i	$0.319791\pi$
$642$	0	0
$643$	− 2.25158e6i	− 0.214763i	−0.994218	−	0.107381i	$-0.965753\pi$
$643$	− 2.25158e6i	− 0.214763i	0.994218	−	0.107381i	$-0.0342466\pi$
$644$	−9.80813e6	−0.931905
$645$	0	0
$646$	444000.	0.0418602
$647$	8.05319e6i	0.756323i	0.925740	+	0.378161i	$0.123444\pi$
$647$	8.05319e6i	0.756323i	−0.925740	+	0.378161i	$0.876556\pi$
$648$	0	0
$649$	−1.05336e6	−0.0981669
$650$	0	0
$651$	0	0
$652$	1.06586e6i	0.0981928i
$653$	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
$656$	−317952.	−0.0288471
$657$	0	0
$658$	4.53254e6i	0.408110i
$659$	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	$0.524037\pi$
$662$	− 7.97653e6i	− 0.707406i
$663$	0	0
$664$	−3.30086e6	−0.290541
$665$	0	0
$666$	0	0
$667$	− 7.80516e6i	− 0.679309i
$668$	3.29107e6i	0.285362i
$669$	0	0
$670$	0	0
$671$	−2.19410e6	−0.188127
$672$	0	0
$673$	− 8.91605e6i	− 0.758813i	−0.925230	−	0.379406i	$-0.876128\pi$
$673$	− 8.91605e6i	− 0.758813i	0.925230	−	0.379406i	$-0.123872\pi$
$674$	3.94809e6	0.334763
$675$	0	0
$676$	8.37797e6	0.705134
$677$	− 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
$681$	0	0
$682$	1.22074e6i	0.100499i
$683$	5.33314e6i	0.437452i	0.975786	+	0.218726i	$0.0701902\pi$
$683$	5.33314e6i	0.437452i	−0.975786	+	0.218726i	$0.929810\pi$
$684$	0	0
$685$	0	0
$686$	2.77264e6	0.224949
$687$	0	0
$688$	5.27974e6i	0.425248i
$689$	1.99284e7	1.59928
$690$	0	0
$691$	698252.	0.0556310	0.0278155	−	0.999613i	$-0.491145\pi$
$691$	698252.	0.0556310	0.0278155	+	0.999613i	$0.491145\pi$
$692$	6.94166e6i	0.551059i
$693$	0	0
$694$	8.82403e6	0.695454
$695$	0	0
$696$	0	0
$697$	275724.i	0.0214977i
$698$	1.09675e7i	0.852056i
$699$	0	0
$700$	0	0
$701$	−1.79880e7	−1.38257	−0.691285	−	0.722582i	$-0.742955\pi$
$701$	−1.79880e7	−1.38257	−0.691285	+	0.722582i	$0.742955\pi$
$702$	0	0
$703$	− 5.62100e6i	− 0.428968i
$704$	540672.	0.0411152
$705$	0	0
$706$	9.55826e6	0.721718
$707$	− 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
$711$	0	0
$712$	− 7.05984e6i	− 0.521909i
$713$	− 8.23997e6i	− 0.607018i
$714$	0	0
$715$	0	0
$716$	7.82688e6	0.570566
$717$	0	0
$718$	1.11792e6i	0.0809282i
$719$	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
$722$	8.90440e6i	0.635714i
$723$	0	0
$724$	−1.15114e7	−0.816172
$725$	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	$0.912142\pi$
$728$	1.04136e7i	0.728234i
$729$	0	0
$730$	0	0
$731$	4.57853e6	0.316907
$732$	0	0
$733$	2.42083e7i	1.66420i	0.554627	+	0.832099i	$0.312861\pi$
$733$	2.42083e7i	1.66420i	−0.554627	+	0.832099i	$0.687139\pi$
$734$	9.90549e6	0.678634
$735$	0	0
$736$	−3.64954e6	−0.248338
$737$	238656.i	0.0161847i
$738$	0	0
$739$	−1.26850e7	−0.854434	−0.427217	−	0.904149i	$-0.640506\pi$
$739$	−1.26850e7	−0.854434	−0.427217	+	0.904149i	$0.640506\pi$
$740$	0	0
$741$	0	0
$742$	1.44934e7i	0.966409i
$743$	− 1.97632e7i	− 1.31337i	−0.754166	−	0.656684i	$-0.771959\pi$
$743$	− 1.97632e7i	− 1.31337i	0.754166	−	0.656684i	$-0.228041\pi$
$744$	0	0
$745$	0	0
$746$	1.09810e7	0.722429
$747$	0	0
$748$	− 468864.i	− 0.0306403i
$749$	−4.00705e7	−2.60988
$750$	0	0
$751$	−9.01761e6	−0.583434	−0.291717	−	0.956505i	$-0.594226\pi$
$751$	−9.01761e6	−0.583434	−0.291717	+	0.956505i	$0.594226\pi$
$752$	1.68653e6i	0.108755i
$753$	0	0
$754$	−8.28696e6	−0.530844
$755$	0	0
$756$	0	0
$757$	1.12556e6i	0.0713887i	0.999363	+	0.0356944i	$0.0113643\pi$
$757$	1.12556e6i	0.0713887i	−0.999363	+	0.0356944i	$0.988636\pi$
$758$	− 4.75624e6i	− 0.300670i
$759$	0	0
$760$	0	0
$761$	−2.25747e7	−1.41306	−0.706529	−	0.707684i	$-0.749740\pi$
$761$	−2.25747e7	−1.41306	−0.706529	+	0.707684i	$0.749740\pi$
$762$	0	0
$763$	3.00742e7i	1.87018i
$764$	−2.97485e6	−0.184387
$765$	0	0
$766$	−1.30304e7	−0.802392
$767$	− 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
$771$	0	0
$772$	9.46250e6i	0.571429i
$773$	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
$776$	−5.01645e6	−0.299049
$777$	0	0
$778$	− 7.93404e6i	− 0.469943i
$779$	621000.	0.0366647
$780$	0	0
$781$	−3.23770e6	−0.189937
$782$	3.16483e6i	0.185069i
$783$	0	0
$784$	−3.27091e6	−0.190055
$785$	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	$-0.213260\pi$
$788$	− 7.19165e6i	− 0.412584i
$789$	0	0
$790$	0	0
$791$	3.14750e7	1.78864
$792$	0	0
$793$	− 1.57244e7i	− 0.887956i
$794$	−1.98966e7	−1.12003
$795$	0	0
$796$	2.51456e6	0.140663
$797$	− 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
$801$	0	0
$802$	− 5.37055e6i	− 0.294838i
$803$	− 2.70257e6i	− 0.147907i
$804$	0	0
$805$	0	0
$806$	−8.74861e6	−0.474353
$807$	0	0
$808$	− 9.08429e6i	− 0.489511i
$809$	4.24929e6	0.228268	0.114134	−	0.993465i	$-0.463591\pi$
$809$	4.24929e6	0.228268	0.114134	+	0.993465i	$0.463591\pi$
$810$	0	0
$811$	3.42333e6	0.182767	0.0913833	−	0.995816i	$-0.470871\pi$
$811$	3.42333e6	0.182767	0.0913833	+	0.995816i	$0.470871\pi$
$812$	− 6.02688e6i	− 0.320776i
$813$	0	0
$814$	−5.93578e6	−0.313990
$815$	0	0
$816$	0	0
$817$	− 1.03120e7i	− 0.540490i
$818$	− 4.37692e6i	− 0.228710i
$819$	0	0
$820$	0	0
$821$	−3.10571e7	−1.60806	−0.804030	−	0.594588i	$-0.797315\pi$
$821$	−3.10571e7	−1.60806	−0.804030	+	0.594588i	$0.797315\pi$
$822$	0	0
$823$	− 3.11904e7i	− 1.60517i	−0.596538	−	0.802584i	$-0.703458\pi$
$823$	− 3.11904e7i	− 1.60517i	0.596538	−	0.802584i	$-0.296542\pi$
$824$	27904.0	0.00143169
$825$	0	0
$826$	5.49024e6	0.279989
$827$	− 8.28487e6i	− 0.421233i	−0.977569	−	0.210616i	$-0.932453\pi$
$827$	− 8.28487e6i	− 0.421233i	0.977569	−	0.210616i	$-0.0675471\pi$
$828$	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
$831$	0	0
$832$	3.87482e6i	0.194063i
$833$	2.83649e6i	0.141635i
$834$	0	0
$835$	0	0
$836$	−1.05600e6	−0.0522575
$837$	0	0
$838$	3.81624e6i	0.187727i
$839$	−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
$842$	6.37559e6i	0.309913i
$843$	0	0
$844$	−4.04883e6	−0.195647
$845$	0	0
$846$	0	0
$847$	− 2.47038e7i	− 1.18319i
$848$	5.39290e6i	0.257533i
$849$	0	0
$850$	0	0
$851$	4.00665e7	1.89652
$852$	0	0
$853$	6.28597e6i	0.295801i	0.989002	+	0.147901i	$0.0472516\pi$
$853$	6.28597e6i	0.295801i	−0.989002	+	0.147901i	$0.952748\pi$
$854$	1.14359e7	0.536571
$855$	0	0
$856$	−1.49100e7	−0.695492
$857$	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	$-0.607667\pi$
$859$	−1.43526e7	−0.663664	−0.331832	+	0.943338i	$0.607667\pi$
$860$	0	0
$861$	0	0
$862$	− 1.05866e7i	− 0.485275i
$863$	− 1.33278e7i	− 0.609158i	−0.952487	−	0.304579i	$-0.901484\pi$
$863$	− 1.33278e7i	− 0.609158i	0.952487	−	0.304579i	$-0.0985158\pi$
$864$	0	0
$865$	0	0
$866$	1.48942e7	0.674874
$867$	0	0
$868$	− 6.36262e6i	− 0.286640i
$869$	−6.10368e6	−0.274184
$870$	0	0
$871$	−1.71037e6	−0.0763913
$872$	1.11904e7i	0.498373i
$873$	0	0
$874$	7.12800e6	0.315638
$875$	0	0
$876$	0	0
$877$	− 3.24846e7i	− 1.42620i	−0.701065	−	0.713098i	$-0.747292\pi$
$877$	− 3.24846e7i	− 1.42620i	0.701065	−	0.713098i	$-0.252708\pi$
$878$	− 1.03336e7i	− 0.452392i
$879$	0	0
$880$	0	0
$881$	−1.54600e7	−0.671073	−0.335537	−	0.942027i	$-0.608918\pi$
$881$	−1.54600e7	−0.671073	−0.335537	+	0.942027i	$0.608918\pi$
$882$	0	0
$883$	− 1.69478e6i	− 0.0731494i	−0.999331	−	0.0365747i	$-0.988355\pi$
$883$	− 1.69478e6i	− 0.0731494i	0.999331	−	0.0365747i	$-0.0116447\pi$
$884$	3.36019e6	0.144622
$885$	0	0
$886$	−3.02483e7	−1.29454
$887$	− 2.87257e6i	− 0.122592i	−0.998120	−	0.0612960i	$-0.980477\pi$
$887$	− 2.87257e6i	− 0.122592i	0.998120	−	0.0612960i	$-0.0195233\pi$
$888$	0	0
$889$	−2.10617e7	−0.893799
$890$	0	0
$891$	0	0
$892$	− 1.71751e7i	− 0.722749i
$893$	− 3.29400e6i	− 0.138228i
$894$	0	0
$895$	0	0
$896$	−2.81805e6	−0.117268
$897$	0	0
$898$	− 1.72309e7i	− 0.713046i
$899$	5.06328e6	0.208945
$900$	0	0
$901$	4.67665e6	0.191921
$902$	− 655776.i	− 0.0268373i
$903$	0	0
$904$	1.17116e7	0.476646
$905$	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	$-0.294118\pi$
$908$	1.00293e7i	0.403698i
$909$	0	0
$910$	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
$914$	8.97417e6	0.355327
$915$	0	0
$916$	−1.86640e6	−0.0734964
$917$	4.16082e7i	1.63401i
$918$	0	0
$919$	−8.51348e6	−0.332520	−0.166260	−	0.986082i	$-0.553169\pi$
$919$	−8.51348e6	−0.332520	−0.166260	+	0.986082i	$0.553169\pi$
$920$	0	0
$921$	0	0
$922$	6.62681e6i	0.256730i
$923$	− 2.32035e7i	− 0.896497i
$924$	0	0
$925$	0	0
$926$	−1.15664e7	−0.443272
$927$	0	0
$928$	− 2.24256e6i	− 0.0854819i
$929$	−7.54587e6	−0.286860	−0.143430	−	0.989660i	$-0.545813\pi$
$929$	−7.54587e6	−0.286860	−0.143430	+	0.989660i	$0.545813\pi$
$930$	0	0
$931$	6.38850e6	0.241560
$932$	− 1.18887e7i	− 0.448328i
$933$	0	0
$934$	−2.61080e7	−0.979278
$935$	0	0
$936$	0	0
$937$	1.84500e7i	0.686512i	0.939242	+	0.343256i	$0.111530\pi$
$937$	1.84500e7i	0.686512i	−0.939242	+	0.343256i	$0.888470\pi$
$938$	− 1.24390e6i	− 0.0461615i
$939$	0	0
$940$	0	0
$941$	−6.75046e6	−0.248519	−0.124259	−	0.992250i	$-0.539656\pi$
$941$	−6.75046e6	−0.248519	−0.124259	+	0.992250i	$0.539656\pi$
$942$	0	0
$943$	4.42649e6i	0.162099i
$944$	2.04288e6	0.0746127
$945$	0	0
$946$	−1.08895e7	−0.395621
$947$	6.45677e6i	0.233959i	0.993134	+	0.116980i	$0.0373212\pi$
$947$	6.45677e6i	0.233959i	−0.993134	+	0.116980i	$0.962679\pi$
$948$	0	0
$949$	1.93684e7	0.698117
$950$	0	0
$951$	0	0
$952$	2.44378e6i	0.0873915i
$953$	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
$956$	−1.56595e7	−0.554158
$957$	0	0
$958$	2.38493e7i	0.839579i
$959$	−4.76609e7	−1.67346
$960$	0	0
$961$	−2.32838e7	−0.813290
$962$	− 4.25397e7i	− 1.48203i
$963$	0	0
$964$	1.81248e7	0.628174
$965$	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	$0.799200\pi$
$968$	− 9.19213e6i	− 0.315303i
$969$	0	0
$970$	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
$974$	−1.19676e7	−0.404214
$975$	0	0
$976$	4.25523e6	0.142988
$977$	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
$981$	0	0
$982$	− 4.81675e6i	− 0.159395i
$983$	− 4.59362e7i	− 1.51625i	−0.652108	−	0.758126i	$-0.726115\pi$
$983$	− 4.59362e7i	− 1.51625i	0.652108	−	0.758126i	$-0.273885\pi$
$984$	0	0
$985$	0	0
$986$	−1.94472e6	−0.0637037
$987$	0	0
$988$	− 7.56800e6i	− 0.246654i
$989$	7.35039e7	2.38957
$990$	0	0
$991$	−4.50298e7	−1.45652	−0.728260	−	0.685301i	$-0.759671\pi$
$991$	−4.50298e7	−1.45652	−0.728260	+	0.685301i	$0.759671\pi$
$992$	− 2.36749e6i	− 0.0763851i
$993$	0	0
$994$	1.68753e7	0.541732
$995$	0	0
$996$	0	0
$997$	2.37364e7i	0.756271i	0.925750	+	0.378136i	$0.123435\pi$
$997$	2.37364e7i	0.756271i	−0.925750	+	0.378136i	$0.876565\pi$
$998$	3.68218e7i	1.17025i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.b.1.1	✓	1	15.2	even	4
50.6.a.d.1.1		1	15.8	even	4
50.6.b.a.49.1		2	15.14	odd	2
50.6.b.a.49.2		2	3.2	odd	2
80.6.a.a.1.1		1	60.47	odd	4
90.6.a.d.1.1		1	5.2	odd	4
320.6.a.b.1.1		1	120.77	even	4
320.6.a.o.1.1		1	120.107	odd	4
400.6.a.n.1.1		1	60.23	odd	4
400.6.c.b.49.1		2	12.11	even	2
400.6.c.b.49.2		2	60.59	even	2
450.6.a.l.1.1		1	5.3	odd	4
450.6.c.h.199.1		2	1.1	even	1	trivial
450.6.c.h.199.2		2	5.4	even	2	inner
490.6.a.a.1.1		1	105.62	odd	4
720.6.a.j.1.1		1	20.7	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} +172.000i q^{7} +64.0000i q^{8} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} +172.000i q^{7} +64.0000i q^{8} -132.000 q^{11} -946.000i q^{13} +688.000 q^{14} +256.000 q^{16} -222.000i q^{17} -500.000 q^{19} +528.000i q^{22} -3564.00i q^{23} -3784.00 q^{26} -2752.00i q^{28} +2190.00 q^{29} +2312.00 q^{31} -1024.00i q^{32} -888.000 q^{34} +11242.0i q^{37} +2000.00i q^{38} -1242.00 q^{41} +20624.0i q^{43} +2112.00 q^{44} -14256.0 q^{46} +6588.00i q^{47} -12777.0 q^{49} +15136.0i q^{52} +21066.0i q^{53} -11008.0 q^{56} -8760.00i q^{58} +7980.00 q^{59} +16622.0 q^{61} -9248.00i q^{62} -4096.00 q^{64} -1808.00i q^{67} +3552.00i q^{68} +24528.0 q^{71} +20474.0i q^{73} +44968.0 q^{74} +8000.00 q^{76} -22704.0i q^{77} +46240.0 q^{79} +4968.00i q^{82} +51576.0i q^{83} +82496.0 q^{86} -8448.00i q^{88} -110310. q^{89} +162712. q^{91} +57024.0i q^{92} +26352.0 q^{94} +78382.0i q^{97} +51108.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} - 264 q^{11} + 1376 q^{14} + 512 q^{16} - 1000 q^{19} - 7568 q^{26} + 4380 q^{29} + 4624 q^{31} - 1776 q^{34} - 2484 q^{41} + 4224 q^{44} - 28512 q^{46} - 25554 q^{49} - 22016 q^{56} + 15960 q^{59} + 33244 q^{61} - 8192 q^{64} + 49056 q^{71} + 89936 q^{74} + 16000 q^{76} + 92480 q^{79} + 164992 q^{86} - 220620 q^{89} + 325424 q^{91} + 52704 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 - 264 * q^11 + 1376 * q^14 + 512 * q^16 - 1000 * q^19 - 7568 * q^26 + 4380 * q^29 + 4624 * q^31 - 1776 * q^34 - 2484 * q^41 + 4224 * q^44 - 28512 * q^46 - 25554 * q^49 - 22016 * q^56 + 15960 * q^59 + 33244 * q^61 - 8192 * q^64 + 49056 * q^71 + 89936 * q^74 + 16000 * q^76 + 92480 * q^79 + 164992 * q^86 - 220620 * q^89 + 325424 * q^91 + 52704 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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