LMFDB - Embedded newform 90.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,6,Mod(1,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("90.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

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:	yes
Analytic conductor:	$14.4345437832$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	90.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} -25.0000 q^{5} -172.000 q^{7} +64.0000 q^{8} +O(q^{10})$

\(q+4.00000 q^{2} +16.0000 q^{4} -25.0000 q^{5} -172.000 q^{7} +64.0000 q^{8} -100.000 q^{10} -132.000 q^{11} -946.000 q^{13} -688.000 q^{14} +256.000 q^{16} +222.000 q^{17} +500.000 q^{19} -400.000 q^{20} -528.000 q^{22} -3564.00 q^{23} +625.000 q^{25} -3784.00 q^{26} -2752.00 q^{28} -2190.00 q^{29} +2312.00 q^{31} +1024.00 q^{32} +888.000 q^{34} +4300.00 q^{35} -11242.0 q^{37} +2000.00 q^{38} -1600.00 q^{40} -1242.00 q^{41} +20624.0 q^{43} -2112.00 q^{44} -14256.0 q^{46} -6588.00 q^{47} +12777.0 q^{49} +2500.00 q^{50} -15136.0 q^{52} +21066.0 q^{53} +3300.00 q^{55} -11008.0 q^{56} -8760.00 q^{58} -7980.00 q^{59} +16622.0 q^{61} +9248.00 q^{62} +4096.00 q^{64} +23650.0 q^{65} +1808.00 q^{67} +3552.00 q^{68} +17200.0 q^{70} +24528.0 q^{71} +20474.0 q^{73} -44968.0 q^{74} +8000.00 q^{76} +22704.0 q^{77} -46240.0 q^{79} -6400.00 q^{80} -4968.00 q^{82} +51576.0 q^{83} -5550.00 q^{85} +82496.0 q^{86} -8448.00 q^{88} +110310. q^{89} +162712. q^{91} -57024.0 q^{92} -26352.0 q^{94} -12500.0 q^{95} -78382.0 q^{97} +51108.0 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−172.000	−1.32673	−0.663366	−	0.748295i	$-0.730873\pi$
$7$	−172.000	−1.32673	−0.663366	+	0.748295i	$0.730873\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	−100.000	−0.316228
$11$	−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
$13$	−946.000	−1.55250	−0.776252	−	0.630423i	$-0.782882\pi$
$13$	−946.000	−1.55250	−0.776252	+	0.630423i	$0.782882\pi$
$14$	−688.000	−0.938142
$15$	0	0
$16$	256.000	0.250000
$17$	222.000	0.186308	0.0931538	−	0.995652i	$-0.470305\pi$
$17$	222.000	0.186308	0.0931538	+	0.995652i	$0.470305\pi$
$18$	0	0
$19$	500.000	0.317750	0.158875	−	0.987299i	$-0.449213\pi$
$19$	500.000	0.317750	0.158875	+	0.987299i	$0.449213\pi$
$20$	−400.000	−0.223607
$21$	0	0
$22$	−528.000	−0.232583
$23$	−3564.00	−1.40481	−0.702406	−	0.711777i	$-0.747891\pi$
$23$	−3564.00	−1.40481	−0.702406	+	0.711777i	$0.747891\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−3784.00	−1.09779
$27$	0	0
$28$	−2752.00	−0.663366
$29$	−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	$-0.430683\pi$
$31$	2312.00	0.432099	0.216050	+	0.976382i	$0.430683\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	888.000	0.131739
$35$	4300.00	0.593333
$36$	0	0
$37$	−11242.0	−1.35002	−0.675009	−	0.737810i	$-0.735860\pi$
$37$	−11242.0	−1.35002	−0.675009	+	0.737810i	$0.735860\pi$
$38$	2000.00	0.224683
$39$	0	0
$40$	−1600.00	−0.158114
$41$	−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
$43$	20624.0	1.70099	0.850495	−	0.525983i	$-0.176303\pi$
$43$	20624.0	1.70099	0.850495	+	0.525983i	$0.176303\pi$
$44$	−2112.00	−0.164461
$45$	0	0
$46$	−14256.0	−0.993352
$47$	−6588.00	−0.435020	−0.217510	−	0.976058i	$-0.569793\pi$
$47$	−6588.00	−0.435020	−0.217510	+	0.976058i	$0.569793\pi$
$48$	0	0
$49$	12777.0	0.760219
$50$	2500.00	0.141421
$51$	0	0
$52$	−15136.0	−0.776252
$53$	21066.0	1.03013	0.515065	−	0.857151i	$-0.327768\pi$
$53$	21066.0	1.03013	0.515065	+	0.857151i	$0.327768\pi$
$54$	0	0
$55$	3300.00	0.147098
$56$	−11008.0	−0.469071
$57$	0	0
$58$	−8760.00	−0.341928
$59$	−7980.00	−0.298451	−0.149225	−	0.988803i	$-0.547678\pi$
$59$	−7980.00	−0.298451	−0.149225	+	0.988803i	$0.547678\pi$
$60$	0	0
$61$	16622.0	0.571951	0.285975	−	0.958237i	$-0.407682\pi$
$61$	16622.0	0.571951	0.285975	+	0.958237i	$0.407682\pi$
$62$	9248.00	0.305540
$63$	0	0
$64$	4096.00	0.125000
$65$	23650.0	0.694301
$66$	0	0
$67$	1808.00	0.0492052	0.0246026	−	0.999697i	$-0.492168\pi$
$67$	1808.00	0.0492052	0.0246026	+	0.999697i	$0.492168\pi$
$68$	3552.00	0.0931538
$69$	0	0
$70$	17200.0	0.419550
$71$	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
$73$	20474.0	0.449672	0.224836	−	0.974397i	$-0.427815\pi$
$73$	20474.0	0.449672	0.224836	+	0.974397i	$0.427815\pi$
$74$	−44968.0	−0.954606
$75$	0	0
$76$	8000.00	0.158875
$77$	22704.0	0.436391
$78$	0	0
$79$	−46240.0	−0.833585	−0.416793	−	0.909002i	$-0.636846\pi$
$79$	−46240.0	−0.833585	−0.416793	+	0.909002i	$0.636846\pi$
$80$	−6400.00	−0.111803
$81$	0	0
$82$	−4968.00	−0.0815919
$83$	51576.0	0.821774	0.410887	−	0.911686i	$-0.365219\pi$
$83$	51576.0	0.821774	0.410887	+	0.911686i	$0.365219\pi$
$84$	0	0
$85$	−5550.00	−0.0833193
$86$	82496.0	1.20278
$87$	0	0
$88$	−8448.00	−0.116291
$89$	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
$92$	−57024.0	−0.702406
$93$	0	0
$94$	−26352.0	−0.307605
$95$	−12500.0	−0.142102
$96$	0	0
$97$	−78382.0	−0.845838	−0.422919	−	0.906168i	$-0.638994\pi$
$97$	−78382.0	−0.845838	−0.422919	+	0.906168i	$0.638994\pi$
$98$	51108.0	0.537556
$99$	0	0
$100$	10000.0	0.100000
$101$	−141942.	−1.38455	−0.692273	−	0.721636i	$-0.743391\pi$
$101$	−141942.	−1.38455	−0.692273	+	0.721636i	$0.743391\pi$
$102$	0	0
$103$	−436.000	−0.00404943	−0.00202471	−	0.999998i	$-0.500644\pi$
$103$	−436.000	−0.00404943	−0.00202471	+	0.999998i	$0.500644\pi$
$104$	−60544.0	−0.548893
$105$	0	0
$106$	84264.0	0.728413
$107$	−232968.	−1.96715	−0.983574	−	0.180508i	$-0.942226\pi$
$107$	−232968.	−1.96715	−0.983574	+	0.180508i	$0.942226\pi$
$108$	0	0
$109$	−174850.	−1.40961	−0.704806	−	0.709400i	$-0.748966\pi$
$109$	−174850.	−1.40961	−0.704806	+	0.709400i	$0.748966\pi$
$110$	13200.0	0.104014
$111$	0	0
$112$	−44032.0	−0.331683
$113$	−182994.	−1.34816	−0.674079	−	0.738659i	$-0.735459\pi$
$113$	−182994.	−1.34816	−0.674079	+	0.738659i	$0.735459\pi$
$114$	0	0
$115$	89100.0	0.628251
$116$	−35040.0	−0.241779
$117$	0	0
$118$	−31920.0	−0.211037
$119$	−38184.0	−0.247180
$120$	0	0
$121$	−143627.	−0.891811
$122$	66488.0	0.404430
$123$	0	0
$124$	36992.0	0.216050
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−122452.	−0.673685	−0.336842	−	0.941561i	$-0.609359\pi$
$127$	−122452.	−0.673685	−0.336842	+	0.941561i	$0.609359\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	94600.0	0.490945
$131$	241908.	1.23161	0.615803	−	0.787900i	$-0.288832\pi$
$131$	241908.	1.23161	0.615803	+	0.787900i	$0.288832\pi$
$132$	0	0
$133$	−86000.0	−0.421570
$134$	7232.00	0.0347934
$135$	0	0
$136$	14208.0	0.0658697
$137$	−277098.	−1.26134	−0.630670	−	0.776051i	$-0.717220\pi$
$137$	−277098.	−1.26134	−0.630670	+	0.776051i	$0.717220\pi$
$138$	0	0
$139$	−193540.	−0.849638	−0.424819	−	0.905278i	$-0.639662\pi$
$139$	−193540.	−0.849638	−0.424819	+	0.905278i	$0.639662\pi$
$140$	68800.0	0.296666
$141$	0	0
$142$	98112.0	0.408321
$143$	124872.	0.510652
$144$	0	0
$145$	54750.0	0.216254
$146$	81896.0	0.317966
$147$	0	0
$148$	−179872.	−0.675009
$149$	−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	$-0.218048\pi$
$151$	433952.	1.54881	0.774407	+	0.632688i	$0.218048\pi$
$152$	32000.0	0.112342
$153$	0	0
$154$	90816.0	0.308575
$155$	−57800.0	−0.193241
$156$	0	0
$157$	−555922.	−1.79997	−0.899984	−	0.435923i	$-0.856422\pi$
$157$	−555922.	−1.79997	−0.899984	+	0.435923i	$0.856422\pi$
$158$	−184960.	−0.589434
$159$	0	0
$160$	−25600.0	−0.0790569
$161$	613008.	1.86381
$162$	0	0
$163$	−66616.0	−0.196386	−0.0981928	−	0.995167i	$-0.531306\pi$
$163$	−66616.0	−0.196386	−0.0981928	+	0.995167i	$0.531306\pi$
$164$	−19872.0	−0.0576942
$165$	0	0
$166$	206304.	0.581082
$167$	205692.	0.570724	0.285362	−	0.958420i	$-0.407886\pi$
$167$	205692.	0.570724	0.285362	+	0.958420i	$0.407886\pi$
$168$	0	0
$169$	523623.	1.41027
$170$	−22200.0	−0.0589156
$171$	0	0
$172$	329984.	0.850495
$173$	−433854.	−1.10212	−0.551059	−	0.834466i	$-0.685776\pi$
$173$	−433854.	−1.10212	−0.551059	+	0.834466i	$0.685776\pi$
$174$	0	0
$175$	−107500.	−0.265346
$176$	−33792.0	−0.0822304
$177$	0	0
$178$	441240.	1.04382
$179$	489180.	1.14113	0.570566	−	0.821252i	$-0.306724\pi$
$179$	489180.	1.14113	0.570566	+	0.821252i	$0.306724\pi$
$180$	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.	1.45647
$183$	0	0
$184$	−228096.	−0.496676
$185$	281050.	0.603746
$186$	0	0
$187$	−29304.0	−0.0612806
$188$	−105408.	−0.217510
$189$	0	0
$190$	−50000.0	−0.100481
$191$	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.	−1.14286	−0.571429	−	0.820651i	$-0.693611\pi$
$193$	−591406.	−1.14286	−0.571429	+	0.820651i	$0.693611\pi$
$194$	−313528.	−0.598098
$195$	0	0
$196$	204432.	0.380109
$197$	−449478.	−0.825169	−0.412584	−	0.910919i	$-0.635374\pi$
$197$	−449478.	−0.825169	−0.412584	+	0.910919i	$0.635374\pi$
$198$	0	0
$199$	157160.	0.281326	0.140663	−	0.990058i	$-0.455077\pi$
$199$	157160.	0.281326	0.140663	+	0.990058i	$0.455077\pi$
$200$	40000.0	0.0707107
$201$	0	0
$202$	−567768.	−0.979022
$203$	376680.	0.641553
$204$	0	0
$205$	31050.0	0.0516032
$206$	−1744.00	−0.00286338
$207$	0	0
$208$	−242176.	−0.388126
$209$	−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.	0.515065
$213$	0	0
$214$	−931872.	−1.39098
$215$	−515600.	−0.760706
$216$	0	0
$217$	−397664.	−0.573280
$218$	−699400.	−0.996746
$219$	0	0
$220$	52800.0	0.0735491
$221$	−210012.	−0.289243
$222$	0	0
$223$	1.07344e6	1.44550	0.722749	−	0.691111i	$-0.242878\pi$
$223$	1.07344e6	1.44550	0.722749	+	0.691111i	$0.242878\pi$
$224$	−176128.	−0.234535
$225$	0	0
$226$	−731976.	−0.953292
$227$	626832.	0.807396	0.403698	−	0.914892i	$-0.367725\pi$
$227$	626832.	0.807396	0.403698	+	0.914892i	$0.367725\pi$
$228$	0	0
$229$	−116650.	−0.146993	−0.0734964	−	0.997295i	$-0.523416\pi$
$229$	−116650.	−0.146993	−0.0734964	+	0.997295i	$0.523416\pi$
$230$	356400.	0.444240
$231$	0	0
$232$	−140160.	−0.170964
$233$	743046.	0.896656	0.448328	−	0.893869i	$-0.352020\pi$
$233$	743046.	0.896656	0.448328	+	0.893869i	$0.352020\pi$
$234$	0	0
$235$	164700.	0.194547
$236$	−127680.	−0.149225
$237$	0	0
$238$	−152736.	−0.174783
$239$	−978720.	−1.10832	−0.554158	−	0.832411i	$-0.686960\pi$
$239$	−978720.	−1.10832	−0.554158	+	0.832411i	$0.686960\pi$
$240$	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.	−0.630605
$243$	0	0
$244$	265952.	0.285975
$245$	−319425.	−0.339980
$246$	0	0
$247$	−473000.	−0.493309
$248$	147968.	0.152770
$249$	0	0
$250$	−62500.0	−0.0632456
$251$	−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.	0.462073
$254$	−489808.	−0.476367
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.93994e6	−1.83212	−0.916062	−	0.401036i	$-0.868650\pi$
$257$	−1.93994e6	−1.83212	−0.916062	+	0.401036i	$0.868650\pi$
$258$	0	0
$259$	1.93362e6	1.79111
$260$	378400.	0.347150
$261$	0	0
$262$	967632.	0.870877
$263$	805476.	0.718064	0.359032	−	0.933325i	$-0.383107\pi$
$263$	805476.	0.718064	0.359032	+	0.933325i	$0.383107\pi$
$264$	0	0
$265$	−526650.	−0.460689
$266$	−344000.	−0.298095
$267$	0	0
$268$	28928.0	0.0246026
$269$	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	$-0.550713\pi$
$271$	−383608.	−0.317296	−0.158648	+	0.987335i	$0.550713\pi$
$272$	56832.0	0.0465769
$273$	0	0
$274$	−1.10839e6	−0.891902
$275$	−82500.0	−0.0657843
$276$	0	0
$277$	2.01076e6	1.57456	0.787282	−	0.616593i	$-0.211488\pi$
$277$	2.01076e6	1.57456	0.787282	+	0.616593i	$0.211488\pi$
$278$	−774160.	−0.600785
$279$	0	0
$280$	275200.	0.209775
$281$	−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.	−0.164429	−0.0822145	−	0.996615i	$-0.526199\pi$
$283$	−221536.	−0.164429	−0.0822145	+	0.996615i	$0.526199\pi$
$284$	392448.	0.288726
$285$	0	0
$286$	499488.	0.361085
$287$	213624.	0.153089
$288$	0	0
$289$	−1.37057e6	−0.965289
$290$	219000.	0.152915
$291$	0	0
$292$	327584.	0.224836
$293$	322506.	0.219467	0.109733	−	0.993961i	$-0.465000\pi$
$293$	322506.	0.219467	0.109733	+	0.993961i	$0.465000\pi$
$294$	0	0
$295$	199500.	0.133471
$296$	−719488.	−0.477303
$297$	0	0
$298$	−562200.	−0.366733
$299$	3.37154e6	2.18098
$300$	0	0
$301$	−3.54733e6	−2.25676
$302$	1.73581e6	1.09518
$303$	0	0
$304$	128000.	0.0794376
$305$	−415550.	−0.255784
$306$	0	0
$307$	1.44301e6	0.873822	0.436911	−	0.899505i	$-0.356073\pi$
$307$	1.44301e6	0.873822	0.436911	+	0.899505i	$0.356073\pi$
$308$	363264.	0.218195
$309$	0	0
$310$	−231200.	−0.136642
$311$	−171312.	−0.100435	−0.0502177	−	0.998738i	$-0.515992\pi$
$311$	−171312.	−0.100435	−0.0502177	+	0.998738i	$0.515992\pi$
$312$	0	0
$313$	−1.02689e6	−0.592463	−0.296232	−	0.955116i	$-0.595730\pi$
$313$	−1.02689e6	−0.592463	−0.296232	+	0.955116i	$0.595730\pi$
$314$	−2.22369e6	−1.27277
$315$	0	0
$316$	−739840.	−0.416793
$317$	−752958.	−0.420845	−0.210423	−	0.977610i	$-0.567484\pi$
$317$	−752958.	−0.420845	−0.210423	+	0.977610i	$0.567484\pi$
$318$	0	0
$319$	289080.	0.159053
$320$	−102400.	−0.0559017
$321$	0	0
$322$	2.45203e6	1.31791
$323$	111000.	0.0591993
$324$	0	0
$325$	−591250.	−0.310501
$326$	−266464.	−0.138866
$327$	0	0
$328$	−79488.0	−0.0407959
$329$	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.	0.410887
$333$	0	0
$334$	822768.	0.403563
$335$	−45200.0	−0.0220053
$336$	0	0
$337$	−987022.	−0.473426	−0.236713	−	0.971580i	$-0.576070\pi$
$337$	−987022.	−0.473426	−0.236713	+	0.971580i	$0.576070\pi$
$338$	2.09449e6	0.997211
$339$	0	0
$340$	−88800.0	−0.0416597
$341$	−305184.	−0.142127
$342$	0	0
$343$	693160.	0.318125
$344$	1.31994e6	0.601391
$345$	0	0
$346$	−1.73542e6	−0.779316
$347$	−2.20601e6	−0.983520	−0.491760	−	0.870731i	$-0.663646\pi$
$347$	−2.20601e6	−0.983520	−0.491760	+	0.870731i	$0.663646\pi$
$348$	0	0
$349$	2.74187e6	1.20499	0.602495	−	0.798123i	$-0.294173\pi$
$349$	2.74187e6	1.20499	0.602495	+	0.798123i	$0.294173\pi$
$350$	−430000.	−0.187628
$351$	0	0
$352$	−135168.	−0.0581456
$353$	2.38957e6	1.02066	0.510331	−	0.859978i	$-0.329523\pi$
$353$	2.38957e6	1.02066	0.510331	+	0.859978i	$0.329523\pi$
$354$	0	0
$355$	−613200.	−0.258245
$356$	1.76496e6	0.738091
$357$	0	0
$358$	1.95672e6	0.806903
$359$	279480.	0.114450	0.0572248	−	0.998361i	$-0.481775\pi$
$359$	279480.	0.114450	0.0572248	+	0.998361i	$0.481775\pi$
$360$	0	0
$361$	−2.22610e6	−0.899035
$362$	2.87785e6	1.15424
$363$	0	0
$364$	2.60339e6	1.02988
$365$	−511850.	−0.201099
$366$	0	0
$367$	−2.47637e6	−0.959734	−0.479867	−	0.877341i	$-0.659315\pi$
$367$	−2.47637e6	−0.959734	−0.479867	+	0.877341i	$0.659315\pi$
$368$	−912384.	−0.351203
$369$	0	0
$370$	1.12420e6	0.426913
$371$	−3.62335e6	−1.36671
$372$	0	0
$373$	2.74525e6	1.02167	0.510835	−	0.859679i	$-0.329336\pi$
$373$	2.74525e6	1.02167	0.510835	+	0.859679i	$0.329336\pi$
$374$	−117216.	−0.0433319
$375$	0	0
$376$	−421632.	−0.153803
$377$	2.07174e6	0.750727
$378$	0	0
$379$	−1.18906e6	−0.425212	−0.212606	−	0.977138i	$-0.568195\pi$
$379$	−1.18906e6	−0.425212	−0.212606	+	0.977138i	$0.568195\pi$
$380$	−200000.	−0.0710511
$381$	0	0
$382$	743712.	0.260763
$383$	−3.25760e6	−1.13475	−0.567377	−	0.823458i	$-0.692042\pi$
$383$	−3.25760e6	−1.13475	−0.567377	+	0.823458i	$0.692042\pi$
$384$	0	0
$385$	−567600.	−0.195160
$386$	−2.36562e6	−0.808123
$387$	0	0
$388$	−1.25411e6	−0.422919
$389$	−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
$392$	817728.	0.268778
$393$	0	0
$394$	−1.79791e6	−0.583483
$395$	1.15600e6	0.372791
$396$	0	0
$397$	4.97416e6	1.58396	0.791978	−	0.610549i	$-0.209051\pi$
$397$	4.97416e6	1.58396	0.791978	+	0.610549i	$0.209051\pi$
$398$	628640.	0.198927
$399$	0	0
$400$	160000.	0.0500000
$401$	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
$403$	−2.18715e6	−0.670836
$404$	−2.27107e6	−0.692273
$405$	0	0
$406$	1.50672e6	0.453646
$407$	1.48394e6	0.444050
$408$	0	0
$409$	−1.09423e6	−0.323445	−0.161722	−	0.986836i	$-0.551705\pi$
$409$	−1.09423e6	−0.323445	−0.161722	+	0.986836i	$0.551705\pi$
$410$	124200.	0.0364890
$411$	0	0
$412$	−6976.00	−0.00202471
$413$	1.37256e6	0.395964
$414$	0	0
$415$	−1.28940e6	−0.367509
$416$	−968704.	−0.274447
$417$	0	0
$418$	−264000.	−0.0739032
$419$	954060.	0.265485	0.132743	−	0.991151i	$-0.457622\pi$
$419$	954060.	0.265485	0.132743	+	0.991151i	$0.457622\pi$
$420$	0	0
$421$	−1.59390e6	−0.438284	−0.219142	−	0.975693i	$-0.570326\pi$
$421$	−1.59390e6	−0.438284	−0.219142	+	0.975693i	$0.570326\pi$
$422$	1.01221e6	0.276687
$423$	0	0
$424$	1.34822e6	0.364206
$425$	138750.	0.0372615
$426$	0	0
$427$	−2.85898e6	−0.758826
$428$	−3.72749e6	−0.983574
$429$	0	0
$430$	−2.06240e6	−0.537900
$431$	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
$433$	3.72355e6	0.954416	0.477208	−	0.878790i	$-0.341649\pi$
$433$	3.72355e6	0.954416	0.477208	+	0.878790i	$0.341649\pi$
$434$	−1.59066e6	−0.405370
$435$	0	0
$436$	−2.79760e6	−0.704806
$437$	−1.78200e6	−0.446379
$438$	0	0
$439$	−2.58340e6	−0.639780	−0.319890	−	0.947455i	$-0.603646\pi$
$439$	−2.58340e6	−0.639780	−0.319890	+	0.947455i	$0.603646\pi$
$440$	211200.	0.0520071
$441$	0	0
$442$	−840048.	−0.204526
$443$	−7.56206e6	−1.83076	−0.915379	−	0.402593i	$-0.868109\pi$
$443$	−7.56206e6	−1.83076	−0.915379	+	0.402593i	$0.868109\pi$
$444$	0	0
$445$	−2.75775e6	−0.660169
$446$	4.29378e6	1.02212
$447$	0	0
$448$	−704512.	−0.165842
$449$	−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
$452$	−2.92790e6	−0.674079
$453$	0	0
$454$	2.50733e6	0.570915
$455$	−4.06780e6	−0.921152
$456$	0	0
$457$	−2.24354e6	−0.502509	−0.251254	−	0.967921i	$-0.580843\pi$
$457$	−2.24354e6	−0.502509	−0.251254	+	0.967921i	$0.580843\pi$
$458$	−466600.	−0.103940
$459$	0	0
$460$	1.42560e6	0.314125
$461$	−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
$463$	−2.89160e6	−0.626881	−0.313441	−	0.949608i	$-0.601482\pi$
$463$	−2.89160e6	−0.626881	−0.313441	+	0.949608i	$0.601482\pi$
$464$	−560640.	−0.120890
$465$	0	0
$466$	2.97218e6	0.634032
$467$	6.52699e6	1.38491	0.692454	−	0.721462i	$-0.256530\pi$
$467$	6.52699e6	1.38491	0.692454	+	0.721462i	$0.256530\pi$
$468$	0	0
$469$	−310976.	−0.0652822
$470$	658800.	0.137565
$471$	0	0
$472$	−510720.	−0.105518
$473$	−2.72237e6	−0.559492
$474$	0	0
$475$	312500.	0.0635501
$476$	−610944.	−0.123590
$477$	0	0
$478$	−3.91488e6	−0.783698
$479$	5.96232e6	1.18734	0.593672	−	0.804707i	$-0.297678\pi$
$479$	5.96232e6	1.18734	0.593672	+	0.804707i	$0.297678\pi$
$480$	0	0
$481$	1.06349e7	2.09591
$482$	−4.53119e6	−0.888372
$483$	0	0
$484$	−2.29803e6	−0.445905
$485$	1.95955e6	0.378270
$486$	0	0
$487$	2.99191e6	0.571644	0.285822	−	0.958283i	$-0.407733\pi$
$487$	2.99191e6	0.571644	0.285822	+	0.958283i	$0.407733\pi$
$488$	1.06381e6	0.202215
$489$	0	0
$490$	−1.27770e6	−0.240402
$491$	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.	−0.0900907
$494$	−1.89200e6	−0.348822
$495$	0	0
$496$	591872.	0.108025
$497$	−4.21882e6	−0.766125
$498$	0	0
$499$	9.20546e6	1.65499	0.827493	−	0.561477i	$-0.189767\pi$
$499$	9.20546e6	1.65499	0.827493	+	0.561477i	$0.189767\pi$
$500$	−250000.	−0.0447214
$501$	0	0
$502$	−3.62261e6	−0.641597
$503$	3.35956e6	0.592055	0.296027	−	0.955179i	$-0.404338\pi$
$503$	3.35956e6	0.592055	0.296027	+	0.955179i	$0.404338\pi$
$504$	0	0
$505$	3.54855e6	0.619188
$506$	1.88179e6	0.326735
$507$	0	0
$508$	−1.95923e6	−0.336842
$509$	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
$512$	262144.	0.0441942
$513$	0	0
$514$	−7.75975e6	−1.29551
$515$	10900.0	0.00181096
$516$	0	0
$517$	869616.	0.143087
$518$	7.73450e6	1.26651
$519$	0	0
$520$	1.51360e6	0.245472
$521$	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
$523$	−5.02802e6	−0.803790	−0.401895	−	0.915686i	$-0.631648\pi$
$523$	−5.02802e6	−0.803790	−0.401895	+	0.915686i	$0.631648\pi$
$524$	3.87053e6	0.615803
$525$	0	0
$526$	3.22190e6	0.507748
$527$	513264.	0.0805034
$528$	0	0
$529$	6.26575e6	0.973496
$530$	−2.10660e6	−0.325756
$531$	0	0
$532$	−1.37600e6	−0.210785
$533$	1.17493e6	0.179141
$534$	0	0
$535$	5.82420e6	0.879735
$536$	115712.	0.0173967
$537$	0	0
$538$	3.43476e6	0.511612
$539$	−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	$0.496862\pi$
$542$	−1.53443e6	−0.224362
$543$	0	0
$544$	227328.	0.0329348
$545$	4.37125e6	0.630397
$546$	0	0
$547$	605648.	0.0865470	0.0432735	−	0.999063i	$-0.486221\pi$
$547$	605648.	0.0865470	0.0432735	+	0.999063i	$0.486221\pi$
$548$	−4.43357e6	−0.630670
$549$	0	0
$550$	−330000.	−0.0465165
$551$	−1.09500e6	−0.153651
$552$	0	0
$553$	7.95328e6	1.10594
$554$	8.04303e6	1.11339
$555$	0	0
$556$	−3.09664e6	−0.424819
$557$	7.06240e6	0.964527	0.482264	−	0.876026i	$-0.339815\pi$
$557$	7.06240e6	0.964527	0.482264	+	0.876026i	$0.339815\pi$
$558$	0	0
$559$	−1.95103e7	−2.64079
$560$	1.10080e6	0.148333
$561$	0	0
$562$	−810408.	−0.108234
$563$	1.03029e7	1.36990	0.684952	−	0.728588i	$-0.259823\pi$
$563$	1.03029e7	1.36990	0.684952	+	0.728588i	$0.259823\pi$
$564$	0	0
$565$	4.57485e6	0.602915
$566$	−886144.	−0.116269
$567$	0	0
$568$	1.56979e6	0.204160
$569$	−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	$-0.141074\pi$
$571$	1.40765e7	1.80677	0.903385	+	0.428830i	$0.141074\pi$
$572$	1.99795e6	0.255326
$573$	0	0
$574$	854496.	0.108251
$575$	−2.22750e6	−0.280962
$576$	0	0
$577$	1.62682e6	0.203423	0.101711	−	0.994814i	$-0.467568\pi$
$577$	1.62682e6	0.203423	0.101711	+	0.994814i	$0.467568\pi$
$578$	−5.48229e6	−0.682563
$579$	0	0
$580$	876000.	0.108127
$581$	−8.87107e6	−1.09027
$582$	0	0
$583$	−2.78071e6	−0.338832
$584$	1.31034e6	0.158983
$585$	0	0
$586$	1.29002e6	0.155186
$587$	−6.96089e6	−0.833814	−0.416907	−	0.908949i	$-0.636886\pi$
$587$	−6.96089e6	−0.833814	−0.416907	+	0.908949i	$0.636886\pi$
$588$	0	0
$589$	1.15600e6	0.137300
$590$	798000.	0.0943785
$591$	0	0
$592$	−2.87795e6	−0.337504
$593$	1.13639e7	1.32706	0.663529	−	0.748150i	$-0.269058\pi$
$593$	1.13639e7	1.32706	0.663529	+	0.748150i	$0.269058\pi$
$594$	0	0
$595$	954600.	0.110542
$596$	−2.24880e6	−0.259320
$597$	0	0
$598$	1.34862e7	1.54218
$599$	−1.48688e7	−1.69321	−0.846603	−	0.532224i	$-0.821356\pi$
$599$	−1.48688e7	−1.69321	−0.846603	+	0.532224i	$0.821356\pi$
$600$	0	0
$601$	−1.23612e6	−0.139596	−0.0697981	−	0.997561i	$-0.522236\pi$
$601$	−1.23612e6	−0.139596	−0.0697981	+	0.997561i	$0.522236\pi$
$602$	−1.41893e7	−1.59577
$603$	0	0
$604$	6.94323e6	0.774407
$605$	3.59067e6	0.398830
$606$	0	0
$607$	−1.24498e7	−1.37149	−0.685743	−	0.727844i	$-0.740522\pi$
$607$	−1.24498e7	−1.37149	−0.685743	+	0.727844i	$0.740522\pi$
$608$	512000.	0.0561709
$609$	0	0
$610$	−1.66220e6	−0.180867
$611$	6.23225e6	0.675370
$612$	0	0
$613$	−8.73491e6	−0.938873	−0.469437	−	0.882966i	$-0.655543\pi$
$613$	−8.73491e6	−0.938873	−0.469437	+	0.882966i	$0.655543\pi$
$614$	5.77203e6	0.617885
$615$	0	0
$616$	1.45306e6	0.154287
$617$	−1.25495e7	−1.32713	−0.663565	−	0.748119i	$-0.730957\pi$
$617$	−1.25495e7	−1.32713	−0.663565	+	0.748119i	$0.730957\pi$
$618$	0	0
$619$	−1.46658e7	−1.53843	−0.769216	−	0.638988i	$-0.779353\pi$
$619$	−1.46658e7	−1.53843	−0.769216	+	0.638988i	$0.779353\pi$
$620$	−924800.	−0.0966203
$621$	0	0
$622$	−685248.	−0.0710186
$623$	−1.89733e7	−1.95850
$624$	0	0
$625$	390625.	0.0400000
$626$	−4.10754e6	−0.418935
$627$	0	0
$628$	−8.89475e6	−0.899984
$629$	−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	$0.503124\pi$
$632$	−2.95936e6	−0.294717
$633$	0	0
$634$	−3.01183e6	−0.297583
$635$	3.06130e6	0.301281
$636$	0	0
$637$	−1.20870e7	−1.18024
$638$	1.15632e6	0.112467
$639$	0	0
$640$	−409600.	−0.0395285
$641$	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
$643$	−2.25158e6	−0.214763	−0.107381	−	0.994218i	$-0.534247\pi$
$643$	−2.25158e6	−0.214763	−0.107381	+	0.994218i	$0.534247\pi$
$644$	9.80813e6	0.931905
$645$	0	0
$646$	444000.	0.0418602
$647$	−8.05319e6	−0.756323	−0.378161	−	0.925740i	$-0.623444\pi$
$647$	−8.05319e6	−0.756323	−0.378161	+	0.925740i	$0.623444\pi$
$648$	0	0
$649$	1.05336e6	0.0981669
$650$	−2.36500e6	−0.219557
$651$	0	0
$652$	−1.06586e6	−0.0981928
$653$	416466.	0.0382205	0.0191103	−	0.999817i	$-0.493917\pi$
$653$	416466.	0.0382205	0.0191103	+	0.999817i	$0.493917\pi$
$654$	0	0
$655$	−6.04770e6	−0.550791
$656$	−317952.	−0.0288471
$657$	0	0
$658$	4.53254e6	0.408110
$659$	−1.31721e7	−1.18152	−0.590761	−	0.806847i	$-0.701172\pi$
$659$	−1.31721e7	−1.18152	−0.590761	+	0.806847i	$0.701172\pi$
$660$	0	0
$661$	−1.69494e6	−0.150886	−0.0754432	−	0.997150i	$-0.524037\pi$
$661$	−1.69494e6	−0.150886	−0.0754432	+	0.997150i	$0.524037\pi$
$662$	7.97653e6	0.707406
$663$	0	0
$664$	3.30086e6	0.290541
$665$	2.15000e6	0.188532
$666$	0	0
$667$	7.80516e6	0.679309
$668$	3.29107e6	0.285362
$669$	0	0
$670$	−180800.	−0.0155601
$671$	−2.19410e6	−0.188127
$672$	0	0
$673$	−8.91605e6	−0.758813	−0.379406	−	0.925230i	$-0.623872\pi$
$673$	−8.91605e6	−0.758813	−0.379406	+	0.925230i	$0.623872\pi$
$674$	−3.94809e6	−0.334763
$675$	0	0
$676$	8.37797e6	0.705134
$677$	1.42894e7	1.19824	0.599118	−	0.800661i	$-0.295518\pi$
$677$	1.42894e7	1.19824	0.599118	+	0.800661i	$0.295518\pi$
$678$	0	0
$679$	1.34817e7	1.12220
$680$	−355200.	−0.0294578
$681$	0	0
$682$	−1.22074e6	−0.100499
$683$	5.33314e6	0.437452	0.218726	−	0.975786i	$-0.429810\pi$
$683$	5.33314e6	0.437452	0.218726	+	0.975786i	$0.429810\pi$
$684$	0	0
$685$	6.92745e6	0.564088
$686$	2.77264e6	0.224949
$687$	0	0
$688$	5.27974e6	0.425248
$689$	−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	$0.491145\pi$
$692$	−6.94166e6	−0.551059
$693$	0	0
$694$	−8.82403e6	−0.695454
$695$	4.83850e6	0.379969
$696$	0	0
$697$	−275724.	−0.0214977
$698$	1.09675e7	0.852056
$699$	0	0
$700$	−1.72000e6	−0.132673
$701$	−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
$703$	−5.62100e6	−0.428968
$704$	−540672.	−0.0411152
$705$	0	0
$706$	9.55826e6	0.721718
$707$	2.44140e7	1.83692
$708$	0	0
$709$	−1.39464e7	−1.04195	−0.520975	−	0.853572i	$-0.674432\pi$
$709$	−1.39464e7	−1.04195	−0.520975	+	0.853572i	$0.674432\pi$
$710$	−2.45280e6	−0.182607
$711$	0	0
$712$	7.05984e6	0.521909
$713$	−8.23997e6	−0.607018
$714$	0	0
$715$	−3.12180e6	−0.228370
$716$	7.82688e6	0.570566
$717$	0	0
$718$	1.11792e6	0.0809282
$719$	−6.22272e6	−0.448909	−0.224454	−	0.974485i	$-0.572060\pi$
$719$	−6.22272e6	−0.448909	−0.224454	+	0.974485i	$0.572060\pi$
$720$	0	0
$721$	74992.0	0.00537250
$722$	−8.90440e6	−0.635714
$723$	0	0
$724$	1.15114e7	0.816172
$725$	−1.36875e6	−0.0967117
$726$	0	0
$727$	−7.76729e6	−0.545047	−0.272523	−	0.962149i	$-0.587858\pi$
$727$	−7.76729e6	−0.545047	−0.272523	+	0.962149i	$0.587858\pi$
$728$	1.04136e7	0.728234
$729$	0	0
$730$	−2.04740e6	−0.142199
$731$	4.57853e6	0.316907
$732$	0	0
$733$	2.42083e7	1.66420	0.832099	−	0.554627i	$-0.187139\pi$
$733$	2.42083e7	1.66420	0.832099	+	0.554627i	$0.187139\pi$
$734$	−9.90549e6	−0.678634
$735$	0	0
$736$	−3.64954e6	−0.248338
$737$	−238656.	−0.0161847
$738$	0	0
$739$	1.26850e7	0.854434	0.427217	−	0.904149i	$-0.359494\pi$
$739$	1.26850e7	0.854434	0.427217	+	0.904149i	$0.359494\pi$
$740$	4.49680e6	0.301873
$741$	0	0
$742$	−1.44934e7	−0.966409
$743$	−1.97632e7	−1.31337	−0.656684	−	0.754166i	$-0.728041\pi$
$743$	−1.97632e7	−1.31337	−0.656684	+	0.754166i	$0.728041\pi$
$744$	0	0
$745$	3.51375e6	0.231942
$746$	1.09810e7	0.722429
$747$	0	0
$748$	−468864.	−0.0306403
$749$	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	$0.594226\pi$
$752$	−1.68653e6	−0.108755
$753$	0	0
$754$	8.28696e6	0.530844
$755$	−1.08488e7	−0.692651
$756$	0	0
$757$	−1.12556e6	−0.0713887	−0.0356944	−	0.999363i	$-0.511364\pi$
$757$	−1.12556e6	−0.0713887	−0.0356944	+	0.999363i	$0.511364\pi$
$758$	−4.75624e6	−0.300670
$759$	0	0
$760$	−800000.	−0.0502407
$761$	−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
$763$	3.00742e7	1.87018
$764$	2.97485e6	0.184387
$765$	0	0
$766$	−1.30304e7	−0.802392
$767$	7.54908e6	0.463346
$768$	0	0
$769$	−632350.	−0.0385604	−0.0192802	−	0.999814i	$-0.506137\pi$
$769$	−632350.	−0.0385604	−0.0192802	+	0.999814i	$0.506137\pi$
$770$	−2.27040e6	−0.137999
$771$	0	0
$772$	−9.46250e6	−0.571429
$773$	1.25867e7	0.757643	0.378822	−	0.925470i	$-0.376329\pi$
$773$	1.25867e7	0.757643	0.378822	+	0.925470i	$0.376329\pi$
$774$	0	0
$775$	1.44500e6	0.0864199
$776$	−5.01645e6	−0.299049
$777$	0	0
$778$	−7.93404e6	−0.469943
$779$	−621000.	−0.0366647
$780$	0	0
$781$	−3.23770e6	−0.189937
$782$	−3.16483e6	−0.185069
$783$	0	0
$784$	3.27091e6	0.190055
$785$	1.38980e7	0.804970
$786$	0	0
$787$	2.15792e7	1.24194	0.620968	−	0.783836i	$-0.286740\pi$
$787$	2.15792e7	1.24194	0.620968	+	0.783836i	$0.286740\pi$
$788$	−7.19165e6	−0.412584
$789$	0	0
$790$	4.62400e6	0.263603
$791$	3.14750e7	1.78864
$792$	0	0
$793$	−1.57244e7	−0.887956
$794$	1.98966e7	1.12003
$795$	0	0
$796$	2.51456e6	0.140663
$797$	3.09760e7	1.72735	0.863673	−	0.504052i	$-0.168158\pi$
$797$	3.09760e7	1.72735	0.863673	+	0.504052i	$0.168158\pi$
$798$	0	0
$799$	−1.46254e6	−0.0810475
$800$	640000.	0.0353553
$801$	0	0
$802$	5.37055e6	0.294838
$803$	−2.70257e6	−0.147907
$804$	0	0
$805$	−1.53252e7	−0.833521
$806$	−8.74861e6	−0.474353
$807$	0	0
$808$	−9.08429e6	−0.489511
$809$	−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	$-0.470871\pi$
$811$	3.42333e6	0.182767	0.0913833	+	0.995816i	$0.470871\pi$
$812$	6.02688e6	0.320776
$813$	0	0
$814$	5.93578e6	0.313990
$815$	1.66540e6	0.0878263
$816$	0	0
$817$	1.03120e7	0.540490
$818$	−4.37692e6	−0.228710
$819$	0	0
$820$	496800.	0.0258016
$821$	−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
$823$	−3.11904e7	−1.60517	−0.802584	−	0.596538i	$-0.796542\pi$
$823$	−3.11904e7	−1.60517	−0.802584	+	0.596538i	$0.796542\pi$
$824$	−27904.0	−0.00143169
$825$	0	0
$826$	5.49024e6	0.279989
$827$	8.28487e6	0.421233	0.210616	−	0.977569i	$-0.432453\pi$
$827$	8.28487e6	0.421233	0.210616	+	0.977569i	$0.432453\pi$
$828$	0	0
$829$	−1.81688e7	−0.918208	−0.459104	−	0.888383i	$-0.651829\pi$
$829$	−1.81688e7	−0.918208	−0.459104	+	0.888383i	$0.651829\pi$
$830$	−5.15760e6	−0.259868
$831$	0	0
$832$	−3.87482e6	−0.194063
$833$	2.83649e6	0.141635
$834$	0	0
$835$	−5.14230e6	−0.255236
$836$	−1.05600e6	−0.0522575
$837$	0	0
$838$	3.81624e6	0.187727
$839$	1.02743e7	0.503902	0.251951	−	0.967740i	$-0.418928\pi$
$839$	1.02743e7	0.503902	0.251951	+	0.967740i	$0.418928\pi$
$840$	0	0
$841$	−1.57150e7	−0.766171
$842$	−6.37559e6	−0.309913
$843$	0	0
$844$	4.04883e6	0.195647
$845$	−1.30906e7	−0.630691
$846$	0	0
$847$	2.47038e7	1.18319
$848$	5.39290e6	0.257533
$849$	0	0
$850$	555000.	0.0263479
$851$	4.00665e7	1.89652
$852$	0	0
$853$	6.28597e6	0.295801	0.147901	−	0.989002i	$-0.452748\pi$
$853$	6.28597e6	0.295801	0.147901	+	0.989002i	$0.452748\pi$
$854$	−1.14359e7	−0.536571
$855$	0	0
$856$	−1.49100e7	−0.695492
$857$	−1.54050e7	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$857$	−1.54050e7	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$858$	0	0
$859$	1.43526e7	0.663664	0.331832	−	0.943338i	$-0.392333\pi$
$859$	1.43526e7	0.663664	0.331832	+	0.943338i	$0.392333\pi$
$860$	−8.24960e6	−0.380353
$861$	0	0
$862$	1.05866e7	0.485275
$863$	−1.33278e7	−0.609158	−0.304579	−	0.952487i	$-0.598516\pi$
$863$	−1.33278e7	−0.609158	−0.304579	+	0.952487i	$0.598516\pi$
$864$	0	0
$865$	1.08464e7	0.492882
$866$	1.48942e7	0.674874
$867$	0	0
$868$	−6.36262e6	−0.286640
$869$	6.10368e6	0.274184
$870$	0	0
$871$	−1.71037e6	−0.0763913
$872$	−1.11904e7	−0.498373
$873$	0	0
$874$	−7.12800e6	−0.315638
$875$	2.68750e6	0.118667
$876$	0	0
$877$	3.24846e7	1.42620	0.713098	−	0.701065i	$-0.247292\pi$
$877$	3.24846e7	1.42620	0.713098	+	0.701065i	$0.247292\pi$
$878$	−1.03336e7	−0.452392
$879$	0	0
$880$	844800.	0.0367745
$881$	−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
$883$	−1.69478e6	−0.0731494	−0.0365747	−	0.999331i	$-0.511645\pi$
$883$	−1.69478e6	−0.0731494	−0.0365747	+	0.999331i	$0.511645\pi$
$884$	−3.36019e6	−0.144622
$885$	0	0
$886$	−3.02483e7	−1.29454
$887$	2.87257e6	0.122592	0.0612960	−	0.998120i	$-0.480477\pi$
$887$	2.87257e6	0.122592	0.0612960	+	0.998120i	$0.480477\pi$
$888$	0	0
$889$	2.10617e7	0.893799
$890$	−1.10310e7	−0.466810
$891$	0	0
$892$	1.71751e7	0.722749
$893$	−3.29400e6	−0.138228
$894$	0	0
$895$	−1.22295e7	−0.510330
$896$	−2.81805e6	−0.117268
$897$	0	0
$898$	−1.72309e7	−0.713046
$899$	−5.06328e6	−0.208945
$900$	0	0
$901$	4.67665e6	0.191921
$902$	655776.	0.0268373
$903$	0	0
$904$	−1.17116e7	−0.476646
$905$	−1.79866e7	−0.730006
$906$	0	0
$907$	3.95422e7	1.59603	0.798017	−	0.602635i	$-0.205882\pi$
$907$	3.95422e7	1.59603	0.798017	+	0.602635i	$0.205882\pi$
$908$	1.00293e7	0.403698
$909$	0	0
$910$	−1.62712e7	−0.651353
$911$	−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
$913$	−6.80803e6	−0.270299
$914$	−8.97417e6	−0.355327
$915$	0	0
$916$	−1.86640e6	−0.0734964
$917$	−4.16082e7	−1.63401
$918$	0	0
$919$	8.51348e6	0.332520	0.166260	−	0.986082i	$-0.446831\pi$
$919$	8.51348e6	0.332520	0.166260	+	0.986082i	$0.446831\pi$
$920$	5.70240e6	0.222120
$921$	0	0
$922$	−6.62681e6	−0.256730
$923$	−2.32035e7	−0.896497
$924$	0	0
$925$	−7.02625e6	−0.270003
$926$	−1.15664e7	−0.443272
$927$	0	0
$928$	−2.24256e6	−0.0854819
$929$	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
$932$	1.18887e7	0.448328
$933$	0	0
$934$	2.61080e7	0.979278
$935$	732600.	0.0274055
$936$	0	0
$937$	−1.84500e7	−0.686512	−0.343256	−	0.939242i	$-0.611530\pi$
$937$	−1.84500e7	−0.686512	−0.343256	+	0.939242i	$0.611530\pi$
$938$	−1.24390e6	−0.0461615
$939$	0	0
$940$	2.63520e6	0.0972734
$941$	−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
$943$	4.42649e6	0.162099
$944$	−2.04288e6	−0.0746127
$945$	0	0
$946$	−1.08895e7	−0.395621
$947$	−6.45677e6	−0.233959	−0.116980	−	0.993134i	$-0.537321\pi$
$947$	−6.45677e6	−0.233959	−0.116980	+	0.993134i	$0.537321\pi$
$948$	0	0
$949$	−1.93684e7	−0.698117
$950$	1.25000e6	0.0449367
$951$	0	0
$952$	−2.44378e6	−0.0873915
$953$	3.96648e7	1.41473	0.707364	−	0.706849i	$-0.249884\pi$
$953$	3.96648e7	1.41473	0.707364	+	0.706849i	$0.249884\pi$
$954$	0	0
$955$	−4.64820e6	−0.164921
$956$	−1.56595e7	−0.554158
$957$	0	0
$958$	2.38493e7	0.839579
$959$	4.76609e7	1.67346
$960$	0	0
$961$	−2.32838e7	−0.813290
$962$	4.25397e7	1.48203
$963$	0	0
$964$	−1.81248e7	−0.628174
$965$	1.47851e7	0.511102
$966$	0	0
$967$	−3.43015e7	−1.17963	−0.589816	−	0.807538i	$-0.700800\pi$
$967$	−3.43015e7	−1.17963	−0.589816	+	0.807538i	$0.700800\pi$
$968$	−9.19213e6	−0.315303
$969$	0	0
$970$	7.83820e6	0.267477
$971$	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
$973$	3.32889e7	1.12724
$974$	1.19676e7	0.404214
$975$	0	0
$976$	4.25523e6	0.142988
$977$	−7.08746e6	−0.237549	−0.118775	−	0.992921i	$-0.537897\pi$
$977$	−7.08746e6	−0.237549	−0.118775	+	0.992921i	$0.537897\pi$
$978$	0	0
$979$	−1.45609e7	−0.485548
$980$	−5.11080e6	−0.169990
$981$	0	0
$982$	4.81675e6	0.159395
$983$	−4.59362e7	−1.51625	−0.758126	−	0.652108i	$-0.773885\pi$
$983$	−4.59362e7	−1.51625	−0.758126	+	0.652108i	$0.773885\pi$
$984$	0	0
$985$	1.12370e7	0.369027
$986$	−1.94472e6	−0.0637037
$987$	0	0
$988$	−7.56800e6	−0.246654
$989$	−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	$0.759671\pi$
$992$	2.36749e6	0.0763851
$993$	0	0
$994$	−1.68753e7	−0.541732
$995$	−3.92900e6	−0.125813
$996$	0	0
$997$	−2.37364e7	−0.756271	−0.378136	−	0.925750i	$-0.623435\pi$
$997$	−2.37364e7	−0.756271	−0.378136	+	0.925750i	$0.623435\pi$
$998$	3.68218e7	1.17025
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.b.1.1	✓	1	3.2	odd	2
50.6.a.d.1.1		1	15.14	odd	2
50.6.b.a.49.1		2	15.2	even	4
50.6.b.a.49.2		2	15.8	even	4
80.6.a.a.1.1		1	12.11	even	2
90.6.a.d.1.1		1	1.1	even	1	trivial
320.6.a.b.1.1		1	24.5	odd	2
320.6.a.o.1.1		1	24.11	even	2
400.6.a.n.1.1		1	60.59	even	2
400.6.c.b.49.1		2	60.23	odd	4
400.6.c.b.49.2		2	60.47	odd	4
450.6.a.l.1.1		1	5.4	even	2
450.6.c.h.199.1		2	5.3	odd	4
450.6.c.h.199.2		2	5.2	odd	4
490.6.a.a.1.1		1	21.20	even	2
720.6.a.j.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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