LMFDB - Embedded newform 50.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,6,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	50.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:	$8.01919099065$
Analytic rank:	$0$
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	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -24.0000 q^{3} +16.0000 q^{4} -96.0000 q^{6} +172.000 q^{7} +64.0000 q^{8} +333.000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -24.0000 q^{3} +16.0000 q^{4} -96.0000 q^{6} +172.000 q^{7} +64.0000 q^{8} +333.000 q^{9} +132.000 q^{11} -384.000 q^{12} +946.000 q^{13} +688.000 q^{14} +256.000 q^{16} +222.000 q^{17} +1332.00 q^{18} +500.000 q^{19} -4128.00 q^{21} +528.000 q^{22} -3564.00 q^{23} -1536.00 q^{24} +3784.00 q^{26} -2160.00 q^{27} +2752.00 q^{28} +2190.00 q^{29} +2312.00 q^{31} +1024.00 q^{32} -3168.00 q^{33} +888.000 q^{34} +5328.00 q^{36} +11242.0 q^{37} +2000.00 q^{38} -22704.0 q^{39} +1242.00 q^{41} -16512.0 q^{42} -20624.0 q^{43} +2112.00 q^{44} -14256.0 q^{46} -6588.00 q^{47} -6144.00 q^{48} +12777.0 q^{49} -5328.00 q^{51} +15136.0 q^{52} +21066.0 q^{53} -8640.00 q^{54} +11008.0 q^{56} -12000.0 q^{57} +8760.00 q^{58} +7980.00 q^{59} +16622.0 q^{61} +9248.00 q^{62} +57276.0 q^{63} +4096.00 q^{64} -12672.0 q^{66} -1808.00 q^{67} +3552.00 q^{68} +85536.0 q^{69} -24528.0 q^{71} +21312.0 q^{72} -20474.0 q^{73} +44968.0 q^{74} +8000.00 q^{76} +22704.0 q^{77} -90816.0 q^{78} -46240.0 q^{79} -29079.0 q^{81} +4968.00 q^{82} +51576.0 q^{83} -66048.0 q^{84} -82496.0 q^{86} -52560.0 q^{87} +8448.00 q^{88} -110310. q^{89} +162712. q^{91} -57024.0 q^{92} -55488.0 q^{93} -26352.0 q^{94} -24576.0 q^{96} +78382.0 q^{97} +51108.0 q^{98} +43956.0 q^{99} +O(q^{100})\)

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
$3$	−24.0000	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−24.0000	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−96.0000	−1.08866
$7$	172.000	1.32673	0.663366	−	0.748295i	$-0.269127\pi$
$7$	172.000	1.32673	0.663366	+	0.748295i	$0.269127\pi$
$8$	64.0000	0.353553
$9$	333.000	1.37037
$10$	0	0
$11$	132.000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	132.000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	−384.000	−0.769800
$13$	946.000	1.55250	0.776252	−	0.630423i	$-0.217118\pi$
$13$	946.000	1.55250	0.776252	+	0.630423i	$0.217118\pi$
$14$	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	$0.470305\pi$
$18$	1332.00	0.968998
$19$	500.000	0.317750	0.158875	−	0.987299i	$-0.449213\pi$
$19$	500.000	0.317750	0.158875	+	0.987299i	$0.449213\pi$
$20$	0	0
$21$	−4128.00	−2.04264
$22$	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	$0.747891\pi$
$24$	−1536.00	−0.544331
$25$	0	0
$26$	3784.00	1.09779
$27$	−2160.00	−0.570222
$28$	2752.00	0.663366
$29$	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	$0.430683\pi$
$32$	1024.00	0.176777
$33$	−3168.00	−0.506408
$34$	888.000	0.131739
$35$	0	0
$36$	5328.00	0.685185
$37$	11242.0	1.35002	0.675009	−	0.737810i	$-0.264140\pi$
$37$	11242.0	1.35002	0.675009	+	0.737810i	$0.264140\pi$
$38$	2000.00	0.224683
$39$	−22704.0	−2.39024
$40$	0	0
$41$	1242.00	0.115388	0.0576942	−	0.998334i	$-0.481625\pi$
$41$	1242.00	0.115388	0.0576942	+	0.998334i	$0.481625\pi$
$42$	−16512.0	−1.44436
$43$	−20624.0	−1.70099	−0.850495	−	0.525983i	$-0.823697\pi$
$43$	−20624.0	−1.70099	−0.850495	+	0.525983i	$0.823697\pi$
$44$	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	$0.569793\pi$
$48$	−6144.00	−0.384900
$49$	12777.0	0.760219
$50$	0	0
$51$	−5328.00	−0.286839
$52$	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	$0.327768\pi$
$54$	−8640.00	−0.403208
$55$	0	0
$56$	11008.0	0.469071
$57$	−12000.0	−0.489209
$58$	8760.00	0.341928
$59$	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	$0.407682\pi$
$62$	9248.00	0.305540
$63$	57276.0	1.81811
$64$	4096.00	0.125000
$65$	0	0
$66$	−12672.0	−0.358084
$67$	−1808.00	−0.0492052	−0.0246026	−	0.999697i	$-0.507832\pi$
$67$	−1808.00	−0.0492052	−0.0246026	+	0.999697i	$0.507832\pi$
$68$	3552.00	0.0931538
$69$	85536.0	2.16285
$70$	0	0
$71$	−24528.0	−0.577452	−0.288726	−	0.957412i	$-0.593232\pi$
$71$	−24528.0	−0.577452	−0.288726	+	0.957412i	$0.593232\pi$
$72$	21312.0	0.484499
$73$	−20474.0	−0.449672	−0.224836	−	0.974397i	$-0.572185\pi$
$73$	−20474.0	−0.449672	−0.224836	+	0.974397i	$0.572185\pi$
$74$	44968.0	0.954606
$75$	0	0
$76$	8000.00	0.158875
$77$	22704.0	0.436391
$78$	−90816.0	−1.69015
$79$	−46240.0	−0.833585	−0.416793	−	0.909002i	$-0.636846\pi$
$79$	−46240.0	−0.833585	−0.416793	+	0.909002i	$0.636846\pi$
$80$	0	0
$81$	−29079.0	−0.492455
$82$	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	$0.365219\pi$
$84$	−66048.0	−1.02132
$85$	0	0
$86$	−82496.0	−1.20278
$87$	−52560.0	−0.744487
$88$	8448.00	0.116291
$89$	−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
$92$	−57024.0	−0.702406
$93$	−55488.0	−0.665260
$94$	−26352.0	−0.307605
$95$	0	0
$96$	−24576.0	−0.272166
$97$	78382.0	0.845838	0.422919	−	0.906168i	$-0.361006\pi$
$97$	78382.0	0.845838	0.422919	+	0.906168i	$0.361006\pi$
$98$	51108.0	0.537556
$99$	43956.0	0.450744
$100$	0	0
$101$	141942.	1.38455	0.692273	−	0.721636i	$-0.256609\pi$
$101$	141942.	1.38455	0.692273	+	0.721636i	$0.256609\pi$
$102$	−21312.0	−0.202826
$103$	436.000	0.00404943	0.00202471	−	0.999998i	$-0.499356\pi$
$103$	436.000	0.00404943	0.00202471	+	0.999998i	$0.499356\pi$
$104$	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	$0.942226\pi$
$108$	−34560.0	−0.285111
$109$	−174850.	−1.40961	−0.704806	−	0.709400i	$-0.748966\pi$
$109$	−174850.	−1.40961	−0.704806	+	0.709400i	$0.748966\pi$
$110$	0	0
$111$	−269808.	−2.07849
$112$	44032.0	0.331683
$113$	−182994.	−1.34816	−0.674079	−	0.738659i	$-0.735459\pi$
$113$	−182994.	−1.34816	−0.674079	+	0.738659i	$0.735459\pi$
$114$	−48000.0	−0.345923
$115$	0	0
$116$	35040.0	0.241779
$117$	315018.	2.12751
$118$	31920.0	0.211037
$119$	38184.0	0.247180
$120$	0	0
$121$	−143627.	−0.891811
$122$	66488.0	0.404430
$123$	−29808.0	−0.177652
$124$	36992.0	0.216050
$125$	0	0
$126$	229104.	1.28560
$127$	122452.	0.673685	0.336842	−	0.941561i	$-0.390641\pi$
$127$	122452.	0.673685	0.336842	+	0.941561i	$0.390641\pi$
$128$	16384.0	0.0883883
$129$	494976.	2.61885
$130$	0	0
$131$	−241908.	−1.23161	−0.615803	−	0.787900i	$-0.711168\pi$
$131$	−241908.	−1.23161	−0.615803	+	0.787900i	$0.711168\pi$
$132$	−50688.0	−0.253204
$133$	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	$0.717220\pi$
$138$	342144.	1.52937
$139$	−193540.	−0.849638	−0.424819	−	0.905278i	$-0.639662\pi$
$139$	−193540.	−0.849638	−0.424819	+	0.905278i	$0.639662\pi$
$140$	0	0
$141$	158112.	0.669757
$142$	−98112.0	−0.408321
$143$	124872.	0.510652
$144$	85248.0	0.342593
$145$	0	0
$146$	−81896.0	−0.317966
$147$	−306648.	−1.17043
$148$	179872.	0.675009
$149$	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	$0.218048\pi$
$152$	32000.0	0.112342
$153$	73926.0	0.255310
$154$	90816.0	0.308575
$155$	0	0
$156$	−363264.	−1.19512
$157$	555922.	1.79997	0.899984	−	0.435923i	$-0.143578\pi$
$157$	555922.	1.79997	0.899984	+	0.435923i	$0.143578\pi$
$158$	−184960.	−0.589434
$159$	−505584.	−1.58599
$160$	0	0
$161$	−613008.	−1.86381
$162$	−116316.	−0.348219
$163$	66616.0	0.196386	0.0981928	−	0.995167i	$-0.468694\pi$
$163$	66616.0	0.196386	0.0981928	+	0.995167i	$0.468694\pi$
$164$	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	$0.407886\pi$
$168$	−264192.	−0.722182
$169$	523623.	1.41027
$170$	0	0
$171$	166500.	0.435436
$172$	−329984.	−0.850495
$173$	−433854.	−1.10212	−0.551059	−	0.834466i	$-0.685776\pi$
$173$	−433854.	−1.10212	−0.551059	+	0.834466i	$0.685776\pi$
$174$	−210240.	−0.526432
$175$	0	0
$176$	33792.0	0.0822304
$177$	−191520.	−0.459495
$178$	−441240.	−1.04382
$179$	−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.	1.45647
$183$	−398928.	−0.880576
$184$	−228096.	−0.496676
$185$	0	0
$186$	−221952.	−0.470410
$187$	29304.0	0.0612806
$188$	−105408.	−0.217510
$189$	−371520.	−0.756533
$190$	0	0
$191$	−185928.	−0.368775	−0.184387	−	0.982854i	$-0.559030\pi$
$191$	−185928.	−0.368775	−0.184387	+	0.982854i	$0.559030\pi$
$192$	−98304.0	−0.192450
$193$	591406.	1.14286	0.571429	−	0.820651i	$-0.306389\pi$
$193$	591406.	1.14286	0.571429	+	0.820651i	$0.306389\pi$
$194$	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	$0.635374\pi$
$198$	175824.	0.318724
$199$	157160.	0.281326	0.140663	−	0.990058i	$-0.455077\pi$
$199$	157160.	0.281326	0.140663	+	0.990058i	$0.455077\pi$
$200$	0	0
$201$	43392.0	0.0757564
$202$	567768.	0.979022
$203$	376680.	0.641553
$204$	−85248.0	−0.143420
$205$	0	0
$206$	1744.00	0.00286338
$207$	−1.18681e6	−1.92511
$208$	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
$213$	588672.	0.889046
$214$	−931872.	−1.39098
$215$	0	0
$216$	−138240.	−0.201604
$217$	397664.	0.573280
$218$	−699400.	−0.996746
$219$	491376.	0.692315
$220$	0	0
$221$	210012.	0.289243
$222$	−1.07923e6	−1.46971
$223$	−1.07344e6	−1.44550	−0.722749	−	0.691111i	$-0.757122\pi$
$223$	−1.07344e6	−1.44550	−0.722749	+	0.691111i	$0.757122\pi$
$224$	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	$0.367725\pi$
$228$	−192000.	−0.244604
$229$	−116650.	−0.146993	−0.0734964	−	0.997295i	$-0.523416\pi$
$229$	−116650.	−0.146993	−0.0734964	+	0.997295i	$0.523416\pi$
$230$	0	0
$231$	−544896.	−0.671868
$232$	140160.	0.170964
$233$	743046.	0.896656	0.448328	−	0.893869i	$-0.352020\pi$
$233$	743046.	0.896656	0.448328	+	0.893869i	$0.352020\pi$
$234$	1.26007e6	1.50437
$235$	0	0
$236$	127680.	0.149225
$237$	1.10976e6	1.28339
$238$	152736.	0.174783
$239$	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.	−0.630605
$243$	1.22278e6	1.32841
$244$	265952.	0.285975
$245$	0	0
$246$	−119232.	−0.125619
$247$	473000.	0.493309
$248$	147968.	0.152770
$249$	−1.23782e6	−1.26520
$250$	0	0
$251$	905652.	0.907355	0.453677	−	0.891166i	$-0.350112\pi$
$251$	905652.	0.907355	0.453677	+	0.891166i	$0.350112\pi$
$252$	916416.	0.909057
$253$	−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	$0.868650\pi$
$258$	1.97990e6	1.85180
$259$	1.93362e6	1.79111
$260$	0	0
$261$	729270.	0.662654
$262$	−967632.	−0.870877
$263$	805476.	0.718064	0.359032	−	0.933325i	$-0.383107\pi$
$263$	805476.	0.718064	0.359032	+	0.933325i	$0.383107\pi$
$264$	−202752.	−0.179042
$265$	0	0
$266$	344000.	0.298095
$267$	2.64744e6	2.27273
$268$	−28928.0	−0.0246026
$269$	−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	$0.550713\pi$
$272$	56832.0	0.0465769
$273$	−3.90509e6	−3.17120
$274$	−1.10839e6	−0.891902
$275$	0	0
$276$	1.36858e6	1.08142
$277$	−2.01076e6	−1.57456	−0.787282	−	0.616593i	$-0.788512\pi$
$277$	−2.01076e6	−1.57456	−0.787282	+	0.616593i	$0.788512\pi$
$278$	−774160.	−0.600785
$279$	769896.	0.592136
$280$	0	0
$281$	202602.	0.153066	0.0765329	−	0.997067i	$-0.475615\pi$
$281$	202602.	0.153066	0.0765329	+	0.997067i	$0.475615\pi$
$282$	632448.	0.473589
$283$	221536.	0.164429	0.0822145	−	0.996615i	$-0.473801\pi$
$283$	221536.	0.164429	0.0822145	+	0.996615i	$0.473801\pi$
$284$	−392448.	−0.288726
$285$	0	0
$286$	499488.	0.361085
$287$	213624.	0.153089
$288$	340992.	0.242250
$289$	−1.37057e6	−0.965289
$290$	0	0
$291$	−1.88117e6	−1.30225
$292$	−327584.	−0.224836
$293$	322506.	0.219467	0.109733	−	0.993961i	$-0.465000\pi$
$293$	322506.	0.219467	0.109733	+	0.993961i	$0.465000\pi$
$294$	−1.22659e6	−0.827622
$295$	0	0
$296$	719488.	0.477303
$297$	−285120.	−0.187558
$298$	562200.	0.366733
$299$	−3.37154e6	−2.18098
$300$	0	0
$301$	−3.54733e6	−2.25676
$302$	1.73581e6	1.09518
$303$	−3.40661e6	−2.13165
$304$	128000.	0.0794376
$305$	0	0
$306$	295704.	0.180532
$307$	−1.44301e6	−0.873822	−0.436911	−	0.899505i	$-0.643927\pi$
$307$	−1.44301e6	−0.873822	−0.436911	+	0.899505i	$0.643927\pi$
$308$	363264.	0.218195
$309$	−10464.0	−0.00623450
$310$	0	0
$311$	171312.	0.100435	0.0502177	−	0.998738i	$-0.484008\pi$
$311$	171312.	0.100435	0.0502177	+	0.998738i	$0.484008\pi$
$312$	−1.45306e6	−0.845076
$313$	1.02689e6	0.592463	0.296232	−	0.955116i	$-0.404270\pi$
$313$	1.02689e6	0.592463	0.296232	+	0.955116i	$0.404270\pi$
$314$	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	$0.567484\pi$
$318$	−2.02234e6	−1.12146
$319$	289080.	0.159053
$320$	0	0
$321$	5.59123e6	3.02862
$322$	−2.45203e6	−1.31791
$323$	111000.	0.0591993
$324$	−465264.	−0.246228
$325$	0	0
$326$	266464.	0.138866
$327$	4.19640e6	2.17024
$328$	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
$333$	3.74359e6	1.85002
$334$	822768.	0.403563
$335$	0	0
$336$	−1.05677e6	−0.510660
$337$	987022.	0.473426	0.236713	−	0.971580i	$-0.423930\pi$
$337$	987022.	0.473426	0.236713	+	0.971580i	$0.423930\pi$
$338$	2.09449e6	0.997211
$339$	4.39186e6	2.07562
$340$	0	0
$341$	305184.	0.142127
$342$	666000.	0.307899
$343$	−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	$0.663646\pi$
$348$	−840960.	−0.372244
$349$	2.74187e6	1.20499	0.602495	−	0.798123i	$-0.294173\pi$
$349$	2.74187e6	1.20499	0.602495	+	0.798123i	$0.294173\pi$
$350$	0	0
$351$	−2.04336e6	−0.885273
$352$	135168.	0.0581456
$353$	2.38957e6	1.02066	0.510331	−	0.859978i	$-0.329523\pi$
$353$	2.38957e6	1.02066	0.510331	+	0.859978i	$0.329523\pi$
$354$	−766080.	−0.324912
$355$	0	0
$356$	−1.76496e6	−0.738091
$357$	−916416.	−0.380559
$358$	−1.95672e6	−0.806903
$359$	−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
$362$	2.87785e6	1.15424
$363$	3.44705e6	1.37303
$364$	2.60339e6	1.02988
$365$	0	0
$366$	−1.59571e6	−0.622661
$367$	2.47637e6	0.959734	0.479867	−	0.877341i	$-0.340685\pi$
$367$	2.47637e6	0.959734	0.479867	+	0.877341i	$0.340685\pi$
$368$	−912384.	−0.351203
$369$	413586.	0.158125
$370$	0	0
$371$	3.62335e6	1.36671
$372$	−887808.	−0.332630
$373$	−2.74525e6	−1.02167	−0.510835	−	0.859679i	$-0.670664\pi$
$373$	−2.74525e6	−1.02167	−0.510835	+	0.859679i	$0.670664\pi$
$374$	117216.	0.0433319
$375$	0	0
$376$	−421632.	−0.153803
$377$	2.07174e6	0.750727
$378$	−1.48608e6	−0.534949
$379$	−1.18906e6	−0.425212	−0.212606	−	0.977138i	$-0.568195\pi$
$379$	−1.18906e6	−0.425212	−0.212606	+	0.977138i	$0.568195\pi$
$380$	0	0
$381$	−2.93885e6	−1.03721
$382$	−743712.	−0.260763
$383$	−3.25760e6	−1.13475	−0.567377	−	0.823458i	$-0.692042\pi$
$383$	−3.25760e6	−1.13475	−0.567377	+	0.823458i	$0.692042\pi$
$384$	−393216.	−0.136083
$385$	0	0
$386$	2.36562e6	0.808123
$387$	−6.86779e6	−2.33099
$388$	1.25411e6	0.422919
$389$	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.	0.268778
$393$	5.80579e6	1.89618
$394$	−1.79791e6	−0.583483
$395$	0	0
$396$	703296.	0.225372
$397$	−4.97416e6	−1.58396	−0.791978	−	0.610549i	$-0.790949\pi$
$397$	−4.97416e6	−1.58396	−0.791978	+	0.610549i	$0.790949\pi$
$398$	628640.	0.198927
$399$	−2.06400e6	−0.649049
$400$	0	0
$401$	−1.34264e6	−0.416963	−0.208482	−	0.978026i	$-0.566852\pi$
$401$	−1.34264e6	−0.416963	−0.208482	+	0.978026i	$0.566852\pi$
$402$	173568.	0.0535679
$403$	2.18715e6	0.670836
$404$	2.27107e6	0.692273
$405$	0	0
$406$	1.50672e6	0.453646
$407$	1.48394e6	0.444050
$408$	−340992.	−0.101413
$409$	−1.09423e6	−0.323445	−0.161722	−	0.986836i	$-0.551705\pi$
$409$	−1.09423e6	−0.323445	−0.161722	+	0.986836i	$0.551705\pi$
$410$	0	0
$411$	6.65035e6	1.94196
$412$	6976.00	0.00202471
$413$	1.37256e6	0.395964
$414$	−4.74725e6	−1.36126
$415$	0	0
$416$	968704.	0.274447
$417$	4.64496e6	1.30810
$418$	264000.	0.0739032
$419$	−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	$0.570326\pi$
$422$	1.01221e6	0.276687
$423$	−2.19380e6	−0.596138
$424$	1.34822e6	0.364206
$425$	0	0
$426$	2.35469e6	0.628651
$427$	2.85898e6	0.758826
$428$	−3.72749e6	−0.983574
$429$	−2.99693e6	−0.786200
$430$	0	0
$431$	−2.64665e6	−0.686283	−0.343141	−	0.939284i	$-0.611491\pi$
$431$	−2.64665e6	−0.686283	−0.343141	+	0.939284i	$0.611491\pi$
$432$	−552960.	−0.142556
$433$	−3.72355e6	−0.954416	−0.477208	−	0.878790i	$-0.658351\pi$
$433$	−3.72355e6	−0.954416	−0.477208	+	0.878790i	$0.658351\pi$
$434$	1.59066e6	0.405370
$435$	0	0
$436$	−2.79760e6	−0.704806
$437$	−1.78200e6	−0.446379
$438$	1.96550e6	0.489541
$439$	−2.58340e6	−0.639780	−0.319890	−	0.947455i	$-0.603646\pi$
$439$	−2.58340e6	−0.639780	−0.319890	+	0.947455i	$0.603646\pi$
$440$	0	0
$441$	4.25474e6	1.04178
$442$	840048.	0.204526
$443$	−7.56206e6	−1.83076	−0.915379	−	0.402593i	$-0.868109\pi$
$443$	−7.56206e6	−1.83076	−0.915379	+	0.402593i	$0.868109\pi$
$444$	−4.31693e6	−1.03924
$445$	0	0
$446$	−4.29378e6	−1.02212
$447$	−3.37320e6	−0.798497
$448$	704512.	0.165842
$449$	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
$452$	−2.92790e6	−0.674079
$453$	−1.04148e7	−2.38456
$454$	2.50733e6	0.570915
$455$	0	0
$456$	−768000.	−0.172961
$457$	2.24354e6	0.502509	0.251254	−	0.967921i	$-0.419157\pi$
$457$	2.24354e6	0.502509	0.251254	+	0.967921i	$0.419157\pi$
$458$	−466600.	−0.103940
$459$	−479520.	−0.106237
$460$	0	0
$461$	1.65670e6	0.363071	0.181536	−	0.983384i	$-0.441893\pi$
$461$	1.65670e6	0.363071	0.181536	+	0.983384i	$0.441893\pi$
$462$	−2.17958e6	−0.475082
$463$	2.89160e6	0.626881	0.313441	−	0.949608i	$-0.398518\pi$
$463$	2.89160e6	0.626881	0.313441	+	0.949608i	$0.398518\pi$
$464$	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	$0.256530\pi$
$468$	5.04029e6	1.06375
$469$	−310976.	−0.0652822
$470$	0	0
$471$	−1.33421e7	−2.77123
$472$	510720.	0.105518
$473$	−2.72237e6	−0.559492
$474$	4.43904e6	0.907493
$475$	0	0
$476$	610944.	0.123590
$477$	7.01498e6	1.41166
$478$	3.91488e6	0.783698
$479$	−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
$482$	−4.53119e6	−0.888372
$483$	1.47122e7	2.86952
$484$	−2.29803e6	−0.445905
$485$	0	0
$486$	4.89110e6	0.939326
$487$	−2.99191e6	−0.571644	−0.285822	−	0.958283i	$-0.592267\pi$
$487$	−2.99191e6	−0.571644	−0.285822	+	0.958283i	$0.592267\pi$
$488$	1.06381e6	0.202215
$489$	−1.59878e6	−0.302355
$490$	0	0
$491$	−1.20419e6	−0.225419	−0.112710	−	0.993628i	$-0.535953\pi$
$491$	−1.20419e6	−0.225419	−0.112710	+	0.993628i	$0.535953\pi$
$492$	−476928.	−0.0888260
$493$	486180.	0.0900907
$494$	1.89200e6	0.348822
$495$	0	0
$496$	591872.	0.108025
$497$	−4.21882e6	−0.766125
$498$	−4.95130e6	−0.894634
$499$	9.20546e6	1.65499	0.827493	−	0.561477i	$-0.189767\pi$
$499$	9.20546e6	1.65499	0.827493	+	0.561477i	$0.189767\pi$
$500$	0	0
$501$	−4.93661e6	−0.878687
$502$	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	$0.404338\pi$
$504$	3.66566e6	0.642801
$505$	0	0
$506$	−1.88179e6	−0.326735
$507$	−1.25670e7	−2.17125
$508$	1.95923e6	0.336842
$509$	−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.	0.0441942
$513$	−1.08000e6	−0.181188
$514$	−7.75975e6	−1.29551
$515$	0	0
$516$	7.91962e6	1.30942
$517$	−869616.	−0.143087
$518$	7.73450e6	1.26651
$519$	1.04125e7	1.69682
$520$	0	0
$521$	−9.31580e6	−1.50358	−0.751789	−	0.659404i	$-0.770809\pi$
$521$	−9.31580e6	−1.50358	−0.751789	+	0.659404i	$0.770809\pi$
$522$	2.91708e6	0.468567
$523$	5.02802e6	0.803790	0.401895	−	0.915686i	$-0.368352\pi$
$523$	5.02802e6	0.803790	0.401895	+	0.915686i	$0.368352\pi$
$524$	−3.87053e6	−0.615803
$525$	0	0
$526$	3.22190e6	0.507748
$527$	513264.	0.0805034
$528$	−811008.	−0.126602
$529$	6.26575e6	0.973496
$530$	0	0
$531$	2.65734e6	0.408988
$532$	1.37600e6	0.210785
$533$	1.17493e6	0.179141
$534$	1.05898e7	1.60706
$535$	0	0
$536$	−115712.	−0.0173967
$537$	1.17403e7	1.75689
$538$	−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
$543$	−1.72671e7	−2.51316
$544$	227328.	0.0329348
$545$	0	0
$546$	−1.56204e7	−2.24238
$547$	−605648.	−0.0865470	−0.0432735	−	0.999063i	$-0.513779\pi$
$547$	−605648.	−0.0865470	−0.0432735	+	0.999063i	$0.513779\pi$
$548$	−4.43357e6	−0.630670
$549$	5.53513e6	0.783784
$550$	0	0
$551$	1.09500e6	0.153651
$552$	5.47430e6	0.764683
$553$	−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	$0.339815\pi$
$558$	3.07958e6	0.418703
$559$	−1.95103e7	−2.64079
$560$	0	0
$561$	−703296.	−0.0943476
$562$	810408.	0.108234
$563$	1.03029e7	1.36990	0.684952	−	0.728588i	$-0.259823\pi$
$563$	1.03029e7	1.36990	0.684952	+	0.728588i	$0.259823\pi$
$564$	2.52979e6	0.334878
$565$	0	0
$566$	886144.	0.116269
$567$	−5.00159e6	−0.653357
$568$	−1.56979e6	−0.204160
$569$	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	$0.141074\pi$
$572$	1.99795e6	0.255326
$573$	4.46227e6	0.567766
$574$	854496.	0.108251
$575$	0	0
$576$	1.36397e6	0.171296
$577$	−1.62682e6	−0.203423	−0.101711	−	0.994814i	$-0.532432\pi$
$577$	−1.62682e6	−0.203423	−0.101711	+	0.994814i	$0.532432\pi$
$578$	−5.48229e6	−0.682563
$579$	−1.41937e7	−1.75955
$580$	0	0
$581$	8.87107e6	1.09027
$582$	−7.52467e6	−0.920831
$583$	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	$0.636886\pi$
$588$	−4.90637e6	−0.585217
$589$	1.15600e6	0.137300
$590$	0	0
$591$	1.07875e7	1.27043
$592$	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	$0.269058\pi$
$594$	−1.14048e6	−0.132624
$595$	0	0
$596$	2.24880e6	0.259320
$597$	−3.77184e6	−0.433129
$598$	−1.34862e7	−1.54218
$599$	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	$0.522236\pi$
$602$	−1.41893e7	−1.59577
$603$	−602064.	−0.0674294
$604$	6.94323e6	0.774407
$605$	0	0
$606$	−1.36264e7	−1.50730
$607$	1.24498e7	1.37149	0.685743	−	0.727844i	$-0.259478\pi$
$607$	1.24498e7	1.37149	0.685743	+	0.727844i	$0.259478\pi$
$608$	512000.	0.0561709
$609$	−9.04032e6	−0.987735
$610$	0	0
$611$	−6.23225e6	−0.675370
$612$	1.18282e6	0.127655
$613$	8.73491e6	0.938873	0.469437	−	0.882966i	$-0.344457\pi$
$613$	8.73491e6	0.938873	0.469437	+	0.882966i	$0.344457\pi$
$614$	−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	$0.730957\pi$
$618$	−41856.0	−0.00440846
$619$	−1.46658e7	−1.53843	−0.769216	−	0.638988i	$-0.779353\pi$
$619$	−1.46658e7	−1.53843	−0.769216	+	0.638988i	$0.779353\pi$
$620$	0	0
$621$	7.69824e6	0.801055
$622$	685248.	0.0710186
$623$	−1.89733e7	−1.95850
$624$	−5.81222e6	−0.597559
$625$	0	0
$626$	4.10754e6	0.418935
$627$	−1.58400e6	−0.160911
$628$	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
$633$	−6.07325e6	−0.602437
$634$	−3.01183e6	−0.297583
$635$	0	0
$636$	−8.08934e6	−0.792995
$637$	1.20870e7	1.18024
$638$	1.15632e6	0.112467
$639$	−8.16782e6	−0.791324
$640$	0	0
$641$	−1.11596e7	−1.07276	−0.536381	−	0.843976i	$-0.680209\pi$
$641$	−1.11596e7	−1.07276	−0.536381	+	0.843976i	$0.680209\pi$
$642$	2.23649e7	2.14156
$643$	2.25158e6	0.214763	0.107381	−	0.994218i	$-0.465753\pi$
$643$	2.25158e6	0.214763	0.107381	+	0.994218i	$0.465753\pi$
$644$	−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	$0.623444\pi$
$648$	−1.86106e6	−0.174109
$649$	1.05336e6	0.0981669
$650$	0	0
$651$	−9.54394e6	−0.882623
$652$	1.06586e6	0.0981928
$653$	416466.	0.0382205	0.0191103	−	0.999817i	$-0.493917\pi$
$653$	416466.	0.0382205	0.0191103	+	0.999817i	$0.493917\pi$
$654$	1.67856e7	1.53459
$655$	0	0
$656$	317952.	0.0288471
$657$	−6.81784e6	−0.616217
$658$	−4.53254e6	−0.408110
$659$	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	$0.524037\pi$
$662$	7.97653e6	0.707406
$663$	−5.04029e6	−0.445319
$664$	3.30086e6	0.290541
$665$	0	0
$666$	1.49743e7	1.30816
$667$	−7.80516e6	−0.679309
$668$	3.29107e6	0.285362
$669$	2.57627e7	2.22549
$670$	0	0
$671$	2.19410e6	0.188127
$672$	−4.22707e6	−0.361091
$673$	8.91605e6	0.758813	0.379406	−	0.925230i	$-0.376128\pi$
$673$	8.91605e6	0.758813	0.379406	+	0.925230i	$0.376128\pi$
$674$	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	$0.295518\pi$
$678$	1.75674e7	1.46769
$679$	1.34817e7	1.12220
$680$	0	0
$681$	−1.50440e7	−1.24307
$682$	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	$0.429810\pi$
$684$	2.66400e6	0.217718
$685$	0	0
$686$	−2.77264e6	−0.224949
$687$	2.79960e6	0.226310
$688$	−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
$693$	7.56043e6	0.598017
$694$	−8.82403e6	−0.695454
$695$	0	0
$696$	−3.36384e6	−0.263216
$697$	275724.	0.0214977
$698$	1.09675e7	0.852056
$699$	−1.78331e7	−1.38049
$700$	0	0
$701$	1.79880e7	1.38257	0.691285	−	0.722582i	$-0.257045\pi$
$701$	1.79880e7	1.38257	0.691285	+	0.722582i	$0.257045\pi$
$702$	−8.17344e6	−0.625982
$703$	5.62100e6	0.428968
$704$	540672.	0.0411152
$705$	0	0
$706$	9.55826e6	0.721718
$707$	2.44140e7	1.83692
$708$	−3.06432e6	−0.229748
$709$	−1.39464e7	−1.04195	−0.520975	−	0.853572i	$-0.674432\pi$
$709$	−1.39464e7	−1.04195	−0.520975	+	0.853572i	$0.674432\pi$
$710$	0	0
$711$	−1.53979e7	−1.14232
$712$	−7.05984e6	−0.521909
$713$	−8.23997e6	−0.607018
$714$	−3.66566e6	−0.269096
$715$	0	0
$716$	−7.82688e6	−0.570566
$717$	−2.34893e7	−1.70636
$718$	−1.11792e6	−0.0809282
$719$	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
$722$	−8.90440e6	−0.635714
$723$	2.71872e7	1.93427
$724$	1.15114e7	0.816172
$725$	0	0
$726$	1.37882e7	0.970880
$727$	7.76729e6	0.545047	0.272523	−	0.962149i	$-0.412142\pi$
$727$	7.76729e6	0.545047	0.272523	+	0.962149i	$0.412142\pi$
$728$	1.04136e7	0.728234
$729$	−2.22804e7	−1.55276
$730$	0	0
$731$	−4.57853e6	−0.316907
$732$	−6.38285e6	−0.440288
$733$	−2.42083e7	−1.66420	−0.832099	−	0.554627i	$-0.812861\pi$
$733$	−2.42083e7	−1.66420	−0.832099	+	0.554627i	$0.812861\pi$
$734$	9.90549e6	0.678634
$735$	0	0
$736$	−3.64954e6	−0.248338
$737$	−238656.	−0.0161847
$738$	1.65434e6	0.111811
$739$	1.26850e7	0.854434	0.427217	−	0.904149i	$-0.359494\pi$
$739$	1.26850e7	0.854434	0.427217	+	0.904149i	$0.359494\pi$
$740$	0	0
$741$	−1.13520e7	−0.759498
$742$	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	$0.728041\pi$
$744$	−3.55123e6	−0.235205
$745$	0	0
$746$	−1.09810e7	−0.722429
$747$	1.71748e7	1.12613
$748$	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
$753$	−2.17356e7	−1.39696
$754$	8.28696e6	0.530844
$755$	0	0
$756$	−5.94432e6	−0.378266
$757$	1.12556e6	0.0713887	0.0356944	−	0.999363i	$-0.488636\pi$
$757$	1.12556e6	0.0713887	0.0356944	+	0.999363i	$0.488636\pi$
$758$	−4.75624e6	−0.300670
$759$	1.12908e7	0.711407
$760$	0	0
$761$	2.25747e7	1.41306	0.706529	−	0.707684i	$-0.250260\pi$
$761$	2.25747e7	1.41306	0.706529	+	0.707684i	$0.250260\pi$
$762$	−1.17554e7	−0.733415
$763$	−3.00742e7	−1.87018
$764$	−2.97485e6	−0.184387
$765$	0	0
$766$	−1.30304e7	−0.802392
$767$	7.54908e6	0.463346
$768$	−1.57286e6	−0.0962250
$769$	−632350.	−0.0385604	−0.0192802	−	0.999814i	$-0.506137\pi$
$769$	−632350.	−0.0385604	−0.0192802	+	0.999814i	$0.506137\pi$
$770$	0	0
$771$	4.65585e7	2.82074
$772$	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	$0.376329\pi$
$774$	−2.74712e7	−1.64826
$775$	0	0
$776$	5.01645e6	0.299049
$777$	−4.64070e7	−2.75760
$778$	7.93404e6	0.469943
$779$	621000.	0.0366647
$780$	0	0
$781$	−3.23770e6	−0.189937
$782$	−3.16483e6	−0.185069
$783$	−4.73040e6	−0.275736
$784$	3.27091e6	0.190055
$785$	0	0
$786$	2.32232e7	1.34080
$787$	−2.15792e7	−1.24194	−0.620968	−	0.783836i	$-0.713260\pi$
$787$	−2.15792e7	−1.24194	−0.620968	+	0.783836i	$0.713260\pi$
$788$	−7.19165e6	−0.412584
$789$	−1.93314e7	−1.10553
$790$	0	0
$791$	−3.14750e7	−1.78864
$792$	2.81318e6	0.159362
$793$	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	$0.168158\pi$
$798$	−8.25600e6	−0.458947
$799$	−1.46254e6	−0.0810475
$800$	0	0
$801$	−3.67332e7	−2.02292
$802$	−5.37055e6	−0.294838
$803$	−2.70257e6	−0.147907
$804$	694272.	0.0378782
$805$	0	0
$806$	8.74861e6	0.474353
$807$	2.06086e7	1.11395
$808$	9.08429e6	0.489511
$809$	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	$0.470871\pi$
$812$	6.02688e6	0.320776
$813$	9.20659e6	0.488509
$814$	5.93578e6	0.313990
$815$	0	0
$816$	−1.36397e6	−0.0717098
$817$	−1.03120e7	−0.540490
$818$	−4.37692e6	−0.228710
$819$	5.41831e7	2.82263
$820$	0	0
$821$	3.10571e7	1.60806	0.804030	−	0.594588i	$-0.202685\pi$
$821$	3.10571e7	1.60806	0.804030	+	0.594588i	$0.202685\pi$
$822$	2.66014e7	1.37317
$823$	3.11904e7	1.60517	0.802584	−	0.596538i	$-0.203458\pi$
$823$	3.11904e7	1.60517	0.802584	+	0.596538i	$0.203458\pi$
$824$	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	$0.432453\pi$
$828$	−1.89890e7	−0.962556
$829$	−1.81688e7	−0.918208	−0.459104	−	0.888383i	$-0.651829\pi$
$829$	−1.81688e7	−0.918208	−0.459104	+	0.888383i	$0.651829\pi$
$830$	0	0
$831$	4.82582e7	2.42420
$832$	3.87482e6	0.194063
$833$	2.83649e6	0.141635
$834$	1.85798e7	0.924968
$835$	0	0
$836$	1.05600e6	0.0522575
$837$	−4.99392e6	−0.246393
$838$	−3.81624e6	−0.187727
$839$	−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
$842$	−6.37559e6	−0.309913
$843$	−4.86245e6	−0.235660
$844$	4.04883e6	0.195647
$845$	0	0
$846$	−8.77522e6	−0.421533
$847$	−2.47038e7	−1.18319
$848$	5.39290e6	0.257533
$849$	−5.31686e6	−0.253155
$850$	0	0
$851$	−4.00665e7	−1.89652
$852$	9.41875e6	0.444523
$853$	−6.28597e6	−0.295801	−0.147901	−	0.989002i	$-0.547252\pi$
$853$	−6.28597e6	−0.295801	−0.147901	+	0.989002i	$0.547252\pi$
$854$	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	$0.616625\pi$
$858$	−1.19877e7	−0.555927
$859$	1.43526e7	0.663664	0.331832	−	0.943338i	$-0.392333\pi$
$859$	1.43526e7	0.663664	0.331832	+	0.943338i	$0.392333\pi$
$860$	0	0
$861$	−5.12698e6	−0.235697
$862$	−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	$0.598516\pi$
$864$	−2.21184e6	−0.100802
$865$	0	0
$866$	−1.48942e7	−0.674874
$867$	3.28938e7	1.48616
$868$	6.36262e6	0.286640
$869$	−6.10368e6	−0.274184
$870$	0	0
$871$	−1.71037e6	−0.0763913
$872$	−1.11904e7	−0.498373
$873$	2.61012e7	1.15911
$874$	−7.12800e6	−0.315638
$875$	0	0
$876$	7.86202e6	0.346157
$877$	−3.24846e7	−1.42620	−0.713098	−	0.701065i	$-0.752708\pi$
$877$	−3.24846e7	−1.42620	−0.713098	+	0.701065i	$0.752708\pi$
$878$	−1.03336e7	−0.452392
$879$	−7.74014e6	−0.337891
$880$	0	0
$881$	1.54600e7	0.671073	0.335537	−	0.942027i	$-0.391082\pi$
$881$	1.54600e7	0.671073	0.335537	+	0.942027i	$0.391082\pi$
$882$	1.70190e7	0.736651
$883$	1.69478e6	0.0731494	0.0365747	−	0.999331i	$-0.488355\pi$
$883$	1.69478e6	0.0731494	0.0365747	+	0.999331i	$0.488355\pi$
$884$	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	$0.480477\pi$
$888$	−1.72677e7	−0.734856
$889$	2.10617e7	0.893799
$890$	0	0
$891$	−3.83843e6	−0.161979
$892$	−1.71751e7	−0.722749
$893$	−3.29400e6	−0.138228
$894$	−1.34928e7	−0.564623
$895$	0	0
$896$	2.81805e6	0.117268
$897$	8.09171e7	3.35783
$898$	1.72309e7	0.713046
$899$	5.06328e6	0.208945
$900$	0	0
$901$	4.67665e6	0.191921
$902$	655776.	0.0268373
$903$	8.51359e7	3.47451
$904$	−1.17116e7	−0.476646
$905$	0	0
$906$	−4.16594e7	−1.68614
$907$	−3.95422e7	−1.59603	−0.798017	−	0.602635i	$-0.794118\pi$
$907$	−3.95422e7	−1.59603	−0.798017	+	0.602635i	$0.794118\pi$
$908$	1.00293e7	0.403698
$909$	4.72667e7	1.89734
$910$	0	0
$911$	1.13178e7	0.451819	0.225909	−	0.974148i	$-0.427465\pi$
$911$	1.13178e7	0.451819	0.225909	+	0.974148i	$0.427465\pi$
$912$	−3.07200e6	−0.122302
$913$	6.80803e6	0.270299
$914$	8.97417e6	0.355327
$915$	0	0
$916$	−1.86640e6	−0.0734964
$917$	−4.16082e7	−1.63401
$918$	−1.91808e6	−0.0751208
$919$	8.51348e6	0.332520	0.166260	−	0.986082i	$-0.446831\pi$
$919$	8.51348e6	0.332520	0.166260	+	0.986082i	$0.446831\pi$
$920$	0	0
$921$	3.46322e7	1.34534
$922$	6.62681e6	0.256730
$923$	−2.32035e7	−0.896497
$924$	−8.71834e6	−0.335934
$925$	0	0
$926$	1.15664e7	0.443272
$927$	145188.	0.00554921
$928$	2.24256e6	0.0854819
$929$	−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
$932$	1.18887e7	0.448328
$933$	−4.11149e6	−0.154630
$934$	2.61080e7	0.979278
$935$	0	0
$936$	2.01612e7	0.752187
$937$	1.84500e7	0.686512	0.343256	−	0.939242i	$-0.388470\pi$
$937$	1.84500e7	0.686512	0.343256	+	0.939242i	$0.388470\pi$
$938$	−1.24390e6	−0.0461615
$939$	−2.46453e7	−0.912157
$940$	0	0
$941$	6.75046e6	0.248519	0.124259	−	0.992250i	$-0.460344\pi$
$941$	6.75046e6	0.248519	0.124259	+	0.992250i	$0.460344\pi$
$942$	−5.33685e7	−1.95956
$943$	−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	$0.537321\pi$
$948$	1.77562e7	0.641694
$949$	−1.93684e7	−0.698117
$950$	0	0
$951$	1.80710e7	0.647934
$952$	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	$0.249884\pi$
$954$	2.80599e7	0.998195
$955$	0	0
$956$	1.56595e7	0.554158
$957$	−6.93792e6	−0.244878
$958$	−2.38493e7	−0.839579
$959$	−4.76609e7	−1.67346
$960$	0	0
$961$	−2.32838e7	−0.813290
$962$	4.25397e7	1.48203
$963$	−7.75783e7	−2.69572
$964$	−1.81248e7	−0.628174
$965$	0	0
$966$	5.88488e7	2.02906
$967$	3.43015e7	1.17963	0.589816	−	0.807538i	$-0.299200\pi$
$967$	3.43015e7	1.17963	0.589816	+	0.807538i	$0.299200\pi$
$968$	−9.19213e6	−0.315303
$969$	−2.66400e6	−0.0911433
$970$	0	0
$971$	−5.77115e6	−0.196433	−0.0982164	−	0.995165i	$-0.531314\pi$
$971$	−5.77115e6	−0.196433	−0.0982164	+	0.995165i	$0.531314\pi$
$972$	1.95644e7	0.664204
$973$	−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	$0.537897\pi$
$978$	−6.39514e6	−0.213798
$979$	−1.45609e7	−0.485548
$980$	0	0
$981$	−5.82250e7	−1.93169
$982$	−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	$0.773885\pi$
$984$	−1.90771e6	−0.0628095
$985$	0	0
$986$	1.94472e6	0.0637037
$987$	2.71953e7	0.888588
$988$	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
$993$	−4.78592e7	−1.54025
$994$	−1.68753e7	−0.541732
$995$	0	0
$996$	−1.98052e7	−0.632602
$997$	2.37364e7	0.756271	0.378136	−	0.925750i	$-0.376565\pi$
$997$	2.37364e7	0.756271	0.378136	+	0.925750i	$0.376565\pi$
$998$	3.68218e7	1.17025
$999$	−2.42827e7	−0.769810

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.b.1.1	✓	1	5.4	even	2
50.6.a.d.1.1		1	1.1	even	1	trivial
50.6.b.a.49.1		2	5.3	odd	4
50.6.b.a.49.2		2	5.2	odd	4
80.6.a.a.1.1		1	20.19	odd	2
90.6.a.d.1.1		1	15.14	odd	2
320.6.a.b.1.1		1	40.29	even	2
320.6.a.o.1.1		1	40.19	odd	2
400.6.a.n.1.1		1	4.3	odd	2
400.6.c.b.49.1		2	20.7	even	4
400.6.c.b.49.2		2	20.3	even	4
450.6.a.l.1.1		1	3.2	odd	2
450.6.c.h.199.1		2	15.2	even	4
450.6.c.h.199.2		2	15.8	even	4
490.6.a.a.1.1		1	35.34	odd	2
720.6.a.j.1.1		1	60.59	even	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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