LMFDB - Embedded newform 75.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,6,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	75.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:	$12.0287864860$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} -16.0000 q^{4} +36.0000 q^{6} +225.000 q^{7} +192.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} -16.0000 q^{4} +36.0000 q^{6} +225.000 q^{7} +192.000 q^{8} +81.0000 q^{9} -434.000 q^{11} +144.000 q^{12} -613.000 q^{13} -900.000 q^{14} -256.000 q^{16} +878.000 q^{17} -324.000 q^{18} -731.000 q^{19} -2025.00 q^{21} +1736.00 q^{22} -2850.00 q^{23} -1728.00 q^{24} +2452.00 q^{26} -729.000 q^{27} -3600.00 q^{28} -7582.00 q^{29} +2175.00 q^{31} -5120.00 q^{32} +3906.00 q^{33} -3512.00 q^{34} -1296.00 q^{36} +9310.00 q^{37} +2924.00 q^{38} +5517.00 q^{39} -12040.0 q^{41} +8100.00 q^{42} +1121.00 q^{43} +6944.00 q^{44} +11400.0 q^{46} -29878.0 q^{47} +2304.00 q^{48} +33818.0 q^{49} -7902.00 q^{51} +9808.00 q^{52} -5740.00 q^{53} +2916.00 q^{54} +43200.0 q^{56} +6579.00 q^{57} +30328.0 q^{58} -5174.00 q^{59} -38717.0 q^{61} -8700.00 q^{62} +18225.0 q^{63} +28672.0 q^{64} -15624.0 q^{66} -31707.0 q^{67} -14048.0 q^{68} +25650.0 q^{69} +64472.0 q^{71} +15552.0 q^{72} +19790.0 q^{73} -37240.0 q^{74} +11696.0 q^{76} -97650.0 q^{77} -22068.0 q^{78} -105000. q^{79} +6561.00 q^{81} +48160.0 q^{82} +3318.00 q^{83} +32400.0 q^{84} -4484.00 q^{86} +68238.0 q^{87} -83328.0 q^{88} -65376.0 q^{89} -137925. q^{91} +45600.0 q^{92} -19575.0 q^{93} +119512. q^{94} +46080.0 q^{96} +89143.0 q^{97} -135272. q^{98} -35154.0 q^{99} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−4.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	225.000	1.73555	0.867776	−	0.496956i	$-0.165549\pi$
$7$	225.000	1.73555	0.867776	+	0.496956i	$0.165549\pi$
$8$	192.000	1.06066
$9$	81.0000	0.333333
$10$	0	0
$11$	−434.000	−1.08145	−0.540727	−	0.841198i	$-0.681851\pi$
$11$	−434.000	−1.08145	−0.540727	+	0.841198i	$0.681851\pi$
$12$	144.000	0.288675
$13$	−613.000	−1.00601	−0.503005	−	0.864284i	$-0.667772\pi$
$13$	−613.000	−1.00601	−0.503005	+	0.864284i	$0.667772\pi$
$14$	−900.000	−1.22722
$15$	0	0
$16$	−256.000	−0.250000
$17$	878.000	0.736838	0.368419	−	0.929660i	$-0.379899\pi$
$17$	878.000	0.736838	0.368419	+	0.929660i	$0.379899\pi$
$18$	−324.000	−0.235702
$19$	−731.000	−0.464551	−0.232275	−	0.972650i	$-0.574617\pi$
$19$	−731.000	−0.464551	−0.232275	+	0.972650i	$0.574617\pi$
$20$	0	0
$21$	−2025.00	−1.00202
$22$	1736.00	0.764703
$23$	−2850.00	−1.12338	−0.561688	−	0.827349i	$-0.689848\pi$
$23$	−2850.00	−1.12338	−0.561688	+	0.827349i	$0.689848\pi$
$24$	−1728.00	−0.612372
$25$	0	0
$26$	2452.00	0.711356
$27$	−729.000	−0.192450
$28$	−3600.00	−0.867776
$29$	−7582.00	−1.67413	−0.837064	−	0.547105i	$-0.815730\pi$
$29$	−7582.00	−1.67413	−0.837064	+	0.547105i	$0.815730\pi$
$30$	0	0
$31$	2175.00	0.406495	0.203247	−	0.979127i	$-0.434850\pi$
$31$	2175.00	0.406495	0.203247	+	0.979127i	$0.434850\pi$
$32$	−5120.00	−0.883883
$33$	3906.00	0.624378
$34$	−3512.00	−0.521023
$35$	0	0
$36$	−1296.00	−0.166667
$37$	9310.00	1.11801	0.559005	−	0.829165i	$-0.311183\pi$
$37$	9310.00	1.11801	0.559005	+	0.829165i	$0.311183\pi$
$38$	2924.00	0.328487
$39$	5517.00	0.580820
$40$	0	0
$41$	−12040.0	−1.11858	−0.559290	−	0.828972i	$-0.688926\pi$
$41$	−12040.0	−1.11858	−0.559290	+	0.828972i	$0.688926\pi$
$42$	8100.00	0.708536
$43$	1121.00	0.0924559	0.0462279	−	0.998931i	$-0.485280\pi$
$43$	1121.00	0.0924559	0.0462279	+	0.998931i	$0.485280\pi$
$44$	6944.00	0.540727
$45$	0	0
$46$	11400.0	0.794347
$47$	−29878.0	−1.97291	−0.986454	−	0.164038i	$-0.947548\pi$
$47$	−29878.0	−1.97291	−0.986454	+	0.164038i	$0.947548\pi$
$48$	2304.00	0.144338
$49$	33818.0	2.01214
$50$	0	0
$51$	−7902.00	−0.425414
$52$	9808.00	0.503005
$53$	−5740.00	−0.280687	−0.140343	−	0.990103i	$-0.544821\pi$
$53$	−5740.00	−0.280687	−0.140343	+	0.990103i	$0.544821\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	43200.0	1.84083
$57$	6579.00	0.268209
$58$	30328.0	1.18379
$59$	−5174.00	−0.193507	−0.0967534	−	0.995308i	$-0.530846\pi$
$59$	−5174.00	−0.193507	−0.0967534	+	0.995308i	$0.530846\pi$
$60$	0	0
$61$	−38717.0	−1.33222	−0.666112	−	0.745852i	$-0.732043\pi$
$61$	−38717.0	−1.33222	−0.666112	+	0.745852i	$0.732043\pi$
$62$	−8700.00	−0.287435
$63$	18225.0	0.578517
$64$	28672.0	0.875000
$65$	0	0
$66$	−15624.0	−0.441502
$67$	−31707.0	−0.862915	−0.431458	−	0.902133i	$-0.642001\pi$
$67$	−31707.0	−0.862915	−0.431458	+	0.902133i	$0.642001\pi$
$68$	−14048.0	−0.368419
$69$	25650.0	0.648582
$70$	0	0
$71$	64472.0	1.51784	0.758919	−	0.651185i	$-0.225728\pi$
$71$	64472.0	1.51784	0.758919	+	0.651185i	$0.225728\pi$
$72$	15552.0	0.353553
$73$	19790.0	0.434649	0.217324	−	0.976099i	$-0.430267\pi$
$73$	19790.0	0.434649	0.217324	+	0.976099i	$0.430267\pi$
$74$	−37240.0	−0.790552
$75$	0	0
$76$	11696.0	0.232275
$77$	−97650.0	−1.87692
$78$	−22068.0	−0.410702
$79$	−105000.	−1.89287	−0.946437	−	0.322889i	$-0.895346\pi$
$79$	−105000.	−1.89287	−0.946437	+	0.322889i	$0.895346\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	48160.0	0.790955
$83$	3318.00	0.0528666	0.0264333	−	0.999651i	$-0.491585\pi$
$83$	3318.00	0.0528666	0.0264333	+	0.999651i	$0.491585\pi$
$84$	32400.0	0.501010
$85$	0	0
$86$	−4484.00	−0.0653762
$87$	68238.0	0.966559
$88$	−83328.0	−1.14706
$89$	−65376.0	−0.874870	−0.437435	−	0.899250i	$-0.644113\pi$
$89$	−65376.0	−0.874870	−0.437435	+	0.899250i	$0.644113\pi$
$90$	0	0
$91$	−137925.	−1.74598
$92$	45600.0	0.561688
$93$	−19575.0	−0.234690
$94$	119512.	1.39506
$95$	0	0
$96$	46080.0	0.510310
$97$	89143.0	0.961962	0.480981	−	0.876731i	$-0.340281\pi$
$97$	89143.0	0.961962	0.480981	+	0.876731i	$0.340281\pi$
$98$	−135272.	−1.42280
$99$	−35154.0	−0.360485
$100$	0	0
$101$	−94644.0	−0.923187	−0.461593	−	0.887092i	$-0.652722\pi$
$101$	−94644.0	−0.923187	−0.461593	+	0.887092i	$0.652722\pi$
$102$	31608.0	0.300813
$103$	113956.	1.05839	0.529193	−	0.848501i	$-0.322495\pi$
$103$	113956.	1.05839	0.529193	+	0.848501i	$0.322495\pi$
$104$	−117696.	−1.06703
$105$	0	0
$106$	22960.0	0.198476
$107$	19212.0	0.162223	0.0811116	−	0.996705i	$-0.474153\pi$
$107$	19212.0	0.162223	0.0811116	+	0.996705i	$0.474153\pi$
$108$	11664.0	0.0962250
$109$	−152791.	−1.23178	−0.615888	−	0.787834i	$-0.711203\pi$
$109$	−152791.	−1.23178	−0.615888	+	0.787834i	$0.711203\pi$
$110$	0	0
$111$	−83790.0	−0.645483
$112$	−57600.0	−0.433888
$113$	−4012.00	−0.0295573	−0.0147787	−	0.999891i	$-0.504704\pi$
$113$	−4012.00	−0.0295573	−0.0147787	+	0.999891i	$0.504704\pi$
$114$	−26316.0	−0.189652
$115$	0	0
$116$	121312.	0.837064
$117$	−49653.0	−0.335337
$118$	20696.0	0.136830
$119$	197550.	1.27882
$120$	0	0
$121$	27305.0	0.169543
$122$	154868.	0.942024
$123$	108360.	0.645812
$124$	−34800.0	−0.203247
$125$	0	0
$126$	−72900.0	−0.409073
$127$	36368.0	0.200083	0.100041	−	0.994983i	$-0.468102\pi$
$127$	36368.0	0.200083	0.100041	+	0.994983i	$0.468102\pi$
$128$	49152.0	0.265165
$129$	−10089.0	−0.0533794
$130$	0	0
$131$	−43932.0	−0.223667	−0.111834	−	0.993727i	$-0.535672\pi$
$131$	−43932.0	−0.223667	−0.111834	+	0.993727i	$0.535672\pi$
$132$	−62496.0	−0.312189
$133$	−164475.	−0.806252
$134$	126828.	0.610173
$135$	0	0
$136$	168576.	0.781535
$137$	−185214.	−0.843087	−0.421544	−	0.906808i	$-0.638512\pi$
$137$	−185214.	−0.843087	−0.421544	+	0.906808i	$0.638512\pi$
$138$	−102600.	−0.458616
$139$	115916.	0.508869	0.254435	−	0.967090i	$-0.418111\pi$
$139$	115916.	0.508869	0.254435	+	0.967090i	$0.418111\pi$
$140$	0	0
$141$	268902.	1.13906
$142$	−257888.	−1.07327
$143$	266042.	1.08795
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	−79160.0	−0.307343
$147$	−304362.	−1.16171
$148$	−148960.	−0.559005
$149$	33122.0	0.122222	0.0611112	−	0.998131i	$-0.480536\pi$
$149$	33122.0	0.122222	0.0611112	+	0.998131i	$0.480536\pi$
$150$	0	0
$151$	431317.	1.53941	0.769705	−	0.638400i	$-0.220403\pi$
$151$	431317.	1.53941	0.769705	+	0.638400i	$0.220403\pi$
$152$	−140352.	−0.492731
$153$	71118.0	0.245613
$154$	390600.	1.32718
$155$	0	0
$156$	−88272.0	−0.290410
$157$	−40439.0	−0.130934	−0.0654668	−	0.997855i	$-0.520854\pi$
$157$	−40439.0	−0.130934	−0.0654668	+	0.997855i	$0.520854\pi$
$158$	420000.	1.33846
$159$	51660.0	0.162055
$160$	0	0
$161$	−641250.	−1.94968
$162$	−26244.0	−0.0785674
$163$	116299.	0.342852	0.171426	−	0.985197i	$-0.445163\pi$
$163$	116299.	0.342852	0.171426	+	0.985197i	$0.445163\pi$
$164$	192640.	0.559290
$165$	0	0
$166$	−13272.0	−0.0373823
$167$	632700.	1.75552	0.877762	−	0.479097i	$-0.159036\pi$
$167$	632700.	1.75552	0.877762	+	0.479097i	$0.159036\pi$
$168$	−388800.	−1.06280
$169$	4476.00	0.0120552
$170$	0	0
$171$	−59211.0	−0.154850
$172$	−17936.0	−0.0462279
$173$	−108222.	−0.274916	−0.137458	−	0.990508i	$-0.543893\pi$
$173$	−108222.	−0.274916	−0.137458	+	0.990508i	$0.543893\pi$
$174$	−272952.	−0.683460
$175$	0	0
$176$	111104.	0.270363
$177$	46566.0	0.111721
$178$	261504.	0.618626
$179$	496570.	1.15837	0.579186	−	0.815196i	$-0.303371\pi$
$179$	496570.	1.15837	0.579186	+	0.815196i	$0.303371\pi$
$180$	0	0
$181$	105365.	0.239056	0.119528	−	0.992831i	$-0.461862\pi$
$181$	105365.	0.239056	0.119528	+	0.992831i	$0.461862\pi$
$182$	551700.	1.23460
$183$	348453.	0.769160
$184$	−547200.	−1.19152
$185$	0	0
$186$	78300.0	0.165951
$187$	−381052.	−0.796857
$188$	478048.	0.986454
$189$	−164025.	−0.334007
$190$	0	0
$191$	427346.	0.847610	0.423805	−	0.905753i	$-0.360694\pi$
$191$	427346.	0.847610	0.423805	+	0.905753i	$0.360694\pi$
$192$	−258048.	−0.505181
$193$	−646427.	−1.24918	−0.624592	−	0.780951i	$-0.714735\pi$
$193$	−646427.	−1.24918	−0.624592	+	0.780951i	$0.714735\pi$
$194$	−356572.	−0.680210
$195$	0	0
$196$	−541088.	−1.00607
$197$	546954.	1.00412	0.502060	−	0.864833i	$-0.332576\pi$
$197$	546954.	1.00412	0.502060	+	0.864833i	$0.332576\pi$
$198$	140616.	0.254901
$199$	−581783.	−1.04143	−0.520713	−	0.853732i	$-0.674334\pi$
$199$	−581783.	−1.04143	−0.520713	+	0.853732i	$0.674334\pi$
$200$	0	0
$201$	285363.	0.498204
$202$	378576.	0.652792
$203$	−1.70595e6	−2.90554
$204$	126432.	0.212707
$205$	0	0
$206$	−455824.	−0.748392
$207$	−230850.	−0.374459
$208$	156928.	0.251502
$209$	317254.	0.502390
$210$	0	0
$211$	−441355.	−0.682467	−0.341234	−	0.939978i	$-0.610845\pi$
$211$	−441355.	−0.682467	−0.341234	+	0.939978i	$0.610845\pi$
$212$	91840.0	0.140343
$213$	−580248.	−0.876324
$214$	−76848.0	−0.114709
$215$	0	0
$216$	−139968.	−0.204124
$217$	489375.	0.705493
$218$	611164.	0.870997
$219$	−178110.	−0.250945
$220$	0	0
$221$	−538214.	−0.741266
$222$	335160.	0.456425
$223$	330457.	0.444993	0.222496	−	0.974934i	$-0.428579\pi$
$223$	330457.	0.444993	0.222496	+	0.974934i	$0.428579\pi$
$224$	−1.15200e6	−1.53402
$225$	0	0
$226$	16048.0	0.0209002
$227$	−1.12508e6	−1.44917	−0.724584	−	0.689186i	$-0.757968\pi$
$227$	−1.12508e6	−1.44917	−0.724584	+	0.689186i	$0.757968\pi$
$228$	−105264.	−0.134104
$229$	−977379.	−1.23161	−0.615807	−	0.787897i	$-0.711170\pi$
$229$	−977379.	−1.23161	−0.615807	+	0.787897i	$0.711170\pi$
$230$	0	0
$231$	878850.	1.08364
$232$	−1.45574e6	−1.77568
$233$	−334584.	−0.403753	−0.201876	−	0.979411i	$-0.564704\pi$
$233$	−334584.	−0.403753	−0.201876	+	0.979411i	$0.564704\pi$
$234$	198612.	0.237119
$235$	0	0
$236$	82784.0	0.0967534
$237$	945000.	1.09285
$238$	−790200.	−0.904263
$239$	218764.	0.247731	0.123866	−	0.992299i	$-0.460471\pi$
$239$	218764.	0.247731	0.123866	+	0.992299i	$0.460471\pi$
$240$	0	0
$241$	734041.	0.814100	0.407050	−	0.913406i	$-0.366557\pi$
$241$	734041.	0.814100	0.407050	+	0.913406i	$0.366557\pi$
$242$	−109220.	−0.119885
$243$	−59049.0	−0.0641500
$244$	619472.	0.666112
$245$	0	0
$246$	−433440.	−0.456658
$247$	448103.	0.467343
$248$	417600.	0.431153
$249$	−29862.0	−0.0305225
$250$	0	0
$251$	1.18381e6	1.18604	0.593019	−	0.805189i	$-0.297936\pi$
$251$	1.18381e6	1.18604	0.593019	+	0.805189i	$0.297936\pi$
$252$	−291600.	−0.289259
$253$	1.23690e6	1.21488
$254$	−145472.	−0.141480
$255$	0	0
$256$	−1.11411e6	−1.06250
$257$	333732.	0.315185	0.157592	−	0.987504i	$-0.449627\pi$
$257$	333732.	0.315185	0.157592	+	0.987504i	$0.449627\pi$
$258$	40356.0	0.0377449
$259$	2.09475e6	1.94036
$260$	0	0
$261$	−614142.	−0.558043
$262$	175728.	0.158157
$263$	731104.	0.651763	0.325882	−	0.945411i	$-0.394339\pi$
$263$	731104.	0.651763	0.325882	+	0.945411i	$0.394339\pi$
$264$	749952.	0.662253
$265$	0	0
$266$	657900.	0.570106
$267$	588384.	0.505106
$268$	507312.	0.431458
$269$	734254.	0.618679	0.309340	−	0.950952i	$-0.399892\pi$
$269$	734254.	0.618679	0.309340	+	0.950952i	$0.399892\pi$
$270$	0	0
$271$	335056.	0.277137	0.138568	−	0.990353i	$-0.455750\pi$
$271$	335056.	0.277137	0.138568	+	0.990353i	$0.455750\pi$
$272$	−224768.	−0.184210
$273$	1.24132e6	1.00804
$274$	740856.	0.596153
$275$	0	0
$276$	−410400.	−0.324291
$277$	1.58260e6	1.23929	0.619643	−	0.784884i	$-0.287277\pi$
$277$	1.58260e6	1.23929	0.619643	+	0.784884i	$0.287277\pi$
$278$	−463664.	−0.359825
$279$	176175.	0.135498
$280$	0	0
$281$	1.08713e6	0.821329	0.410665	−	0.911786i	$-0.365297\pi$
$281$	1.08713e6	0.821329	0.410665	+	0.911786i	$0.365297\pi$
$282$	−1.07561e6	−0.805436
$283$	607407.	0.450831	0.225415	−	0.974263i	$-0.427626\pi$
$283$	607407.	0.450831	0.225415	+	0.974263i	$0.427626\pi$
$284$	−1.03155e6	−0.758919
$285$	0	0
$286$	−1.06417e6	−0.769299
$287$	−2.70900e6	−1.94135
$288$	−414720.	−0.294628
$289$	−648973.	−0.457069
$290$	0	0
$291$	−802287.	−0.555389
$292$	−316640.	−0.217324
$293$	−1.23301e6	−0.839066	−0.419533	−	0.907740i	$-0.637806\pi$
$293$	−1.23301e6	−0.839066	−0.419533	+	0.907740i	$0.637806\pi$
$294$	1.21745e6	0.821452
$295$	0	0
$296$	1.78752e6	1.18583
$297$	316386.	0.208126
$298$	−132488.	−0.0864243
$299$	1.74705e6	1.13013
$300$	0	0
$301$	252225.	0.160462
$302$	−1.72527e6	−1.08853
$303$	851796.	0.533002
$304$	187136.	0.116138
$305$	0	0
$306$	−284472.	−0.173674
$307$	−562753.	−0.340778	−0.170389	−	0.985377i	$-0.554502\pi$
$307$	−562753.	−0.340778	−0.170389	+	0.985377i	$0.554502\pi$
$308$	1.56240e6	0.938459
$309$	−1.02560e6	−0.611060
$310$	0	0
$311$	−2.39645e6	−1.40497	−0.702487	−	0.711697i	$-0.747927\pi$
$311$	−2.39645e6	−1.40497	−0.702487	+	0.711697i	$0.747927\pi$
$312$	1.05926e6	0.616053
$313$	−1.52833e6	−0.881773	−0.440886	−	0.897563i	$-0.645336\pi$
$313$	−1.52833e6	−0.881773	−0.440886	+	0.897563i	$0.645336\pi$
$314$	161756.	0.0925841
$315$	0	0
$316$	1.68000e6	0.946437
$317$	−1.00033e6	−0.559106	−0.279553	−	0.960130i	$-0.590186\pi$
$317$	−1.00033e6	−0.559106	−0.279553	+	0.960130i	$0.590186\pi$
$318$	−206640.	−0.114590
$319$	3.29059e6	1.81049
$320$	0	0
$321$	−172908.	−0.0936596
$322$	2.56500e6	1.37863
$323$	−641818.	−0.342299
$324$	−104976.	−0.0555556
$325$	0	0
$326$	−465196.	−0.242433
$327$	1.37512e6	0.711166
$328$	−2.31168e6	−1.18643
$329$	−6.72255e6	−3.42408
$330$	0	0
$331$	2.29093e6	1.14932	0.574662	−	0.818391i	$-0.305134\pi$
$331$	2.29093e6	1.14932	0.574662	+	0.818391i	$0.305134\pi$
$332$	−53088.0	−0.0264333
$333$	754110.	0.372670
$334$	−2.53080e6	−1.24134
$335$	0	0
$336$	518400.	0.250505
$337$	−1.28904e6	−0.618290	−0.309145	−	0.951015i	$-0.600043\pi$
$337$	−1.28904e6	−0.618290	−0.309145	+	0.951015i	$0.600043\pi$
$338$	−17904.0	−0.00852429
$339$	36108.0	0.0170649
$340$	0	0
$341$	−943950.	−0.439605
$342$	236844.	0.109496
$343$	3.82748e6	1.75662
$344$	215232.	0.0980643
$345$	0	0
$346$	432888.	0.194395
$347$	−19102.0	−0.00851638	−0.00425819	−	0.999991i	$-0.501355\pi$
$347$	−19102.0	−0.00851638	−0.00425819	+	0.999991i	$0.501355\pi$
$348$	−1.09181e6	−0.483279
$349$	−3.24690e6	−1.42694	−0.713470	−	0.700686i	$-0.752877\pi$
$349$	−3.24690e6	−1.42694	−0.713470	+	0.700686i	$0.752877\pi$
$350$	0	0
$351$	446877.	0.193607
$352$	2.22208e6	0.955879
$353$	−4.28804e6	−1.83156	−0.915782	−	0.401677i	$-0.868428\pi$
$353$	−4.28804e6	−1.83156	−0.915782	+	0.401677i	$0.868428\pi$
$354$	−186264.	−0.0789989
$355$	0	0
$356$	1.04602e6	0.437435
$357$	−1.77795e6	−0.738327
$358$	−1.98628e6	−0.819092
$359$	88656.0	0.0363055	0.0181527	−	0.999835i	$-0.494221\pi$
$359$	88656.0	0.0363055	0.0181527	+	0.999835i	$0.494221\pi$
$360$	0	0
$361$	−1.94174e6	−0.784192
$362$	−421460.	−0.169038
$363$	−245745.	−0.0978854
$364$	2.20680e6	0.872990
$365$	0	0
$366$	−1.39381e6	−0.543878
$367$	2.04184e6	0.791328	0.395664	−	0.918395i	$-0.370515\pi$
$367$	2.04184e6	0.791328	0.395664	+	0.918395i	$0.370515\pi$
$368$	729600.	0.280844
$369$	−975240.	−0.372860
$370$	0	0
$371$	−1.29150e6	−0.487147
$372$	313200.	0.117345
$373$	806321.	0.300079	0.150040	−	0.988680i	$-0.452060\pi$
$373$	806321.	0.300079	0.150040	+	0.988680i	$0.452060\pi$
$374$	1.52421e6	0.563463
$375$	0	0
$376$	−5.73658e6	−2.09259
$377$	4.64777e6	1.68419
$378$	656100.	0.236179
$379$	−2.41314e6	−0.862950	−0.431475	−	0.902125i	$-0.642007\pi$
$379$	−2.41314e6	−0.862950	−0.431475	+	0.902125i	$0.642007\pi$
$380$	0	0
$381$	−327312.	−0.115518
$382$	−1.70938e6	−0.599351
$383$	1.45856e6	0.508076	0.254038	−	0.967194i	$-0.418241\pi$
$383$	1.45856e6	0.508076	0.254038	+	0.967194i	$0.418241\pi$
$384$	−442368.	−0.153093
$385$	0	0
$386$	2.58571e6	0.883306
$387$	90801.0	0.0308186
$388$	−1.42629e6	−0.480981
$389$	2.65618e6	0.889985	0.444993	−	0.895534i	$-0.353206\pi$
$389$	2.65618e6	0.889985	0.444993	+	0.895534i	$0.353206\pi$
$390$	0	0
$391$	−2.50230e6	−0.827747
$392$	6.49306e6	2.13419
$393$	395388.	0.129134
$394$	−2.18782e6	−0.710019
$395$	0	0
$396$	562464.	0.180242
$397$	−1.68664e6	−0.537090	−0.268545	−	0.963267i	$-0.586543\pi$
$397$	−1.68664e6	−0.537090	−0.268545	+	0.963267i	$0.586543\pi$
$398$	2.32713e6	0.736399
$399$	1.48028e6	0.465490
$400$	0	0
$401$	−2.12533e6	−0.660033	−0.330017	−	0.943975i	$-0.607054\pi$
$401$	−2.12533e6	−0.660033	−0.330017	+	0.943975i	$0.607054\pi$
$402$	−1.14145e6	−0.352284
$403$	−1.33328e6	−0.408938
$404$	1.51430e6	0.461593
$405$	0	0
$406$	6.82380e6	2.05452
$407$	−4.04054e6	−1.20908
$408$	−1.51718e6	−0.451219
$409$	1.97158e6	0.582782	0.291391	−	0.956604i	$-0.405882\pi$
$409$	1.97158e6	0.582782	0.291391	+	0.956604i	$0.405882\pi$
$410$	0	0
$411$	1.66693e6	0.486757
$412$	−1.82330e6	−0.529193
$413$	−1.16415e6	−0.335841
$414$	923400.	0.264782
$415$	0	0
$416$	3.13856e6	0.889195
$417$	−1.04324e6	−0.293796
$418$	−1.26902e6	−0.355244
$419$	1.75744e6	0.489042	0.244521	−	0.969644i	$-0.421369\pi$
$419$	1.75744e6	0.489042	0.244521	+	0.969644i	$0.421369\pi$
$420$	0	0
$421$	−2.83947e6	−0.780785	−0.390392	−	0.920649i	$-0.627661\pi$
$421$	−2.83947e6	−0.780785	−0.390392	+	0.920649i	$0.627661\pi$
$422$	1.76542e6	0.482577
$423$	−2.42012e6	−0.657636
$424$	−1.10208e6	−0.297713
$425$	0	0
$426$	2.32099e6	0.619655
$427$	−8.71132e6	−2.31214
$428$	−307392.	−0.0811116
$429$	−2.39438e6	−0.628130
$430$	0	0
$431$	−4.69335e6	−1.21700	−0.608499	−	0.793555i	$-0.708228\pi$
$431$	−4.69335e6	−1.21700	−0.608499	+	0.793555i	$0.708228\pi$
$432$	186624.	0.0481125
$433$	6.71343e6	1.72078	0.860389	−	0.509639i	$-0.170221\pi$
$433$	6.71343e6	1.72078	0.860389	+	0.509639i	$0.170221\pi$
$434$	−1.95750e6	−0.498859
$435$	0	0
$436$	2.44466e6	0.615888
$437$	2.08335e6	0.521866
$438$	712440.	0.177445
$439$	3.92022e6	0.970842	0.485421	−	0.874281i	$-0.338666\pi$
$439$	3.92022e6	0.970842	0.485421	+	0.874281i	$0.338666\pi$
$440$	0	0
$441$	2.73926e6	0.670713
$442$	2.15286e6	0.524154
$443$	6.93031e6	1.67781	0.838906	−	0.544276i	$-0.183195\pi$
$443$	6.93031e6	1.67781	0.838906	+	0.544276i	$0.183195\pi$
$444$	1.34064e6	0.322741
$445$	0	0
$446$	−1.32183e6	−0.314657
$447$	−298098.	−0.0705651
$448$	6.45120e6	1.51861
$449$	4.38605e6	1.02673	0.513367	−	0.858169i	$-0.328398\pi$
$449$	4.38605e6	1.02673	0.513367	+	0.858169i	$0.328398\pi$
$450$	0	0
$451$	5.22536e6	1.20969
$452$	64192.0	0.0147787
$453$	−3.88185e6	−0.888779
$454$	4.50032e6	1.02472
$455$	0	0
$456$	1.26317e6	0.284478
$457$	−2.41951e6	−0.541922	−0.270961	−	0.962590i	$-0.587341\pi$
$457$	−2.41951e6	−0.541922	−0.270961	+	0.962590i	$0.587341\pi$
$458$	3.90952e6	0.870882
$459$	−640062.	−0.141805
$460$	0	0
$461$	−366228.	−0.0802600	−0.0401300	−	0.999194i	$-0.512777\pi$
$461$	−366228.	−0.0802600	−0.0401300	+	0.999194i	$0.512777\pi$
$462$	−3.51540e6	−0.766249
$463$	−6.62534e6	−1.43634	−0.718168	−	0.695870i	$-0.755019\pi$
$463$	−6.62534e6	−1.43634	−0.718168	+	0.695870i	$0.755019\pi$
$464$	1.94099e6	0.418532
$465$	0	0
$466$	1.33834e6	0.285496
$467$	−3.38476e6	−0.718183	−0.359092	−	0.933302i	$-0.616913\pi$
$467$	−3.38476e6	−0.718183	−0.359092	+	0.933302i	$0.616913\pi$
$468$	794448.	0.167668
$469$	−7.13408e6	−1.49763
$470$	0	0
$471$	363951.	0.0755946
$472$	−993408.	−0.205245
$473$	−486514.	−0.0999868
$474$	−3.78000e6	−0.772762
$475$	0	0
$476$	−3.16080e6	−0.639410
$477$	−464940.	−0.0935623
$478$	−875056.	−0.175173
$479$	6.61905e6	1.31813	0.659063	−	0.752088i	$-0.270953\pi$
$479$	6.61905e6	1.31813	0.659063	+	0.752088i	$0.270953\pi$
$480$	0	0
$481$	−5.70703e6	−1.12473
$482$	−2.93616e6	−0.575656
$483$	5.77125e6	1.12565
$484$	−436880.	−0.0847713
$485$	0	0
$486$	236196.	0.0453609
$487$	287383.	0.0549084	0.0274542	−	0.999623i	$-0.491260\pi$
$487$	287383.	0.0549084	0.0274542	+	0.999623i	$0.491260\pi$
$488$	−7.43366e6	−1.41304
$489$	−1.04669e6	−0.197946
$490$	0	0
$491$	6.51337e6	1.21928	0.609638	−	0.792680i	$-0.291315\pi$
$491$	6.51337e6	1.21928	0.609638	+	0.792680i	$0.291315\pi$
$492$	−1.73376e6	−0.322906
$493$	−6.65700e6	−1.23356
$494$	−1.79241e6	−0.330461
$495$	0	0
$496$	−556800.	−0.101624
$497$	1.45062e7	2.63428
$498$	119448.	0.0215827
$499$	−1.17037e6	−0.210413	−0.105206	−	0.994450i	$-0.533550\pi$
$499$	−1.17037e6	−0.210413	−0.105206	+	0.994450i	$0.533550\pi$
$500$	0	0
$501$	−5.69430e6	−1.01355
$502$	−4.73525e6	−0.838655
$503$	1.40651e6	0.247870	0.123935	−	0.992290i	$-0.460449\pi$
$503$	1.40651e6	0.247870	0.123935	+	0.992290i	$0.460449\pi$
$504$	3.49920e6	0.613610
$505$	0	0
$506$	−4.94760e6	−0.859050
$507$	−40284.0	−0.00696006
$508$	−581888.	−0.100041
$509$	3.65645e6	0.625554	0.312777	−	0.949827i	$-0.398741\pi$
$509$	3.65645e6	0.625554	0.312777	+	0.949827i	$0.398741\pi$
$510$	0	0
$511$	4.45275e6	0.754356
$512$	2.88358e6	0.486136
$513$	532899.	0.0894029
$514$	−1.33493e6	−0.222869
$515$	0	0
$516$	161424.	0.0266897
$517$	1.29671e7	2.13361
$518$	−8.37900e6	−1.37204
$519$	973998.	0.158723
$520$	0	0
$521$	−2.70889e6	−0.437217	−0.218609	−	0.975813i	$-0.570152\pi$
$521$	−2.70889e6	−0.437217	−0.218609	+	0.975813i	$0.570152\pi$
$522$	2.45657e6	0.394596
$523$	1.14014e7	1.82265	0.911326	−	0.411686i	$-0.135060\pi$
$523$	1.14014e7	1.82265	0.911326	+	0.411686i	$0.135060\pi$
$524$	702912.	0.111834
$525$	0	0
$526$	−2.92442e6	−0.460866
$527$	1.90965e6	0.299521
$528$	−999936.	−0.156094
$529$	1.68616e6	0.261974
$530$	0	0
$531$	−419094.	−0.0645023
$532$	2.63160e6	0.403126
$533$	7.38052e6	1.12530
$534$	−2.35354e6	−0.357164
$535$	0	0
$536$	−6.08774e6	−0.915260
$537$	−4.46913e6	−0.668786
$538$	−2.93702e6	−0.437472
$539$	−1.46770e7	−2.17603
$540$	0	0
$541$	6.95050e6	1.02099	0.510497	−	0.859880i	$-0.329461\pi$
$541$	6.95050e6	1.02099	0.510497	+	0.859880i	$0.329461\pi$
$542$	−1.34022e6	−0.195965
$543$	−948285.	−0.138019
$544$	−4.49536e6	−0.651279
$545$	0	0
$546$	−4.96530e6	−0.712794
$547$	8.08532e6	1.15539	0.577695	−	0.816252i	$-0.303952\pi$
$547$	8.08532e6	1.15539	0.577695	+	0.816252i	$0.303952\pi$
$548$	2.96342e6	0.421544
$549$	−3.13608e6	−0.444075
$550$	0	0
$551$	5.54244e6	0.777718
$552$	4.92480e6	0.687925
$553$	−2.36250e7	−3.28518
$554$	−6.33040e6	−0.876307
$555$	0	0
$556$	−1.85466e6	−0.254435
$557$	−1.46025e7	−1.99429	−0.997146	−	0.0754991i	$-0.975945\pi$
$557$	−1.46025e7	−1.99429	−0.997146	+	0.0754991i	$0.975945\pi$
$558$	−704700.	−0.0958118
$559$	−687173.	−0.0930115
$560$	0	0
$561$	3.42947e6	0.460065
$562$	−4.34854e6	−0.580767
$563$	−3.54652e6	−0.471554	−0.235777	−	0.971807i	$-0.575764\pi$
$563$	−3.54652e6	−0.471554	−0.235777	+	0.971807i	$0.575764\pi$
$564$	−4.30243e6	−0.569530
$565$	0	0
$566$	−2.42963e6	−0.318786
$567$	1.47623e6	0.192839
$568$	1.23786e7	1.60991
$569$	−1.42681e7	−1.84750	−0.923750	−	0.382996i	$-0.874892\pi$
$569$	−1.42681e7	−1.84750	−0.923750	+	0.382996i	$0.874892\pi$
$570$	0	0
$571$	−5.48534e6	−0.704066	−0.352033	−	0.935988i	$-0.614509\pi$
$571$	−5.48534e6	−0.704066	−0.352033	+	0.935988i	$0.614509\pi$
$572$	−4.25667e6	−0.543976
$573$	−3.84611e6	−0.489368
$574$	1.08360e7	1.37274
$575$	0	0
$576$	2.32243e6	0.291667
$577$	2.13253e6	0.266659	0.133329	−	0.991072i	$-0.457433\pi$
$577$	2.13253e6	0.266659	0.133329	+	0.991072i	$0.457433\pi$
$578$	2.59589e6	0.323197
$579$	5.81784e6	0.721216
$580$	0	0
$581$	746550.	0.0917526
$582$	3.20915e6	0.392719
$583$	2.49116e6	0.303550
$584$	3.79968e6	0.461015
$585$	0	0
$586$	4.93202e6	0.593309
$587$	−2.39303e6	−0.286650	−0.143325	−	0.989676i	$-0.545779\pi$
$587$	−2.39303e6	−0.286650	−0.143325	+	0.989676i	$0.545779\pi$
$588$	4.86979e6	0.580854
$589$	−1.58992e6	−0.188838
$590$	0	0
$591$	−4.92259e6	−0.579728
$592$	−2.38336e6	−0.279502
$593$	−1.83478e6	−0.214264	−0.107132	−	0.994245i	$-0.534167\pi$
$593$	−1.83478e6	−0.214264	−0.107132	+	0.994245i	$0.534167\pi$
$594$	−1.26554e6	−0.147167
$595$	0	0
$596$	−529952.	−0.0611112
$597$	5.23605e6	0.601267
$598$	−6.98820e6	−0.799121
$599$	−126508.	−0.0144062	−0.00720312	−	0.999974i	$-0.502293\pi$
$599$	−126508.	−0.0144062	−0.00720312	+	0.999974i	$0.502293\pi$
$600$	0	0
$601$	1.03791e7	1.17212	0.586059	−	0.810268i	$-0.300679\pi$
$601$	1.03791e7	1.17212	0.586059	+	0.810268i	$0.300679\pi$
$602$	−1.00890e6	−0.113464
$603$	−2.56827e6	−0.287638
$604$	−6.90107e6	−0.769705
$605$	0	0
$606$	−3.40718e6	−0.376889
$607$	366608.	0.0403859	0.0201930	−	0.999796i	$-0.493572\pi$
$607$	366608.	0.0403859	0.0201930	+	0.999796i	$0.493572\pi$
$608$	3.74272e6	0.410609
$609$	1.53535e7	1.67751
$610$	0	0
$611$	1.83152e7	1.98476
$612$	−1.13789e6	−0.122806
$613$	7.49272e6	0.805357	0.402678	−	0.915342i	$-0.368079\pi$
$613$	7.49272e6	0.805357	0.402678	+	0.915342i	$0.368079\pi$
$614$	2.25101e6	0.240967
$615$	0	0
$616$	−1.87488e7	−1.99077
$617$	−2.23748e6	−0.236618	−0.118309	−	0.992977i	$-0.537747\pi$
$617$	−2.23748e6	−0.236618	−0.118309	+	0.992977i	$0.537747\pi$
$618$	4.10242e6	0.432084
$619$	6.61648e6	0.694065	0.347033	−	0.937853i	$-0.387189\pi$
$619$	6.61648e6	0.694065	0.347033	+	0.937853i	$0.387189\pi$
$620$	0	0
$621$	2.07765e6	0.216194
$622$	9.58582e6	0.993466
$623$	−1.47096e7	−1.51838
$624$	−1.41235e6	−0.145205
$625$	0	0
$626$	6.11332e6	0.623507
$627$	−2.85529e6	−0.290055
$628$	647024.	0.0654668
$629$	8.17418e6	0.823792
$630$	0	0
$631$	−6.16481e6	−0.616377	−0.308189	−	0.951325i	$-0.599723\pi$
$631$	−6.16481e6	−0.616377	−0.308189	+	0.951325i	$0.599723\pi$
$632$	−2.01600e7	−2.00770
$633$	3.97219e6	0.394023
$634$	4.00131e6	0.395348
$635$	0	0
$636$	−826560.	−0.0810273
$637$	−2.07304e7	−2.02423
$638$	−1.31624e7	−1.28021
$639$	5.22223e6	0.505946
$640$	0	0
$641$	4.79502e6	0.460941	0.230471	−	0.973079i	$-0.425973\pi$
$641$	4.79502e6	0.460941	0.230471	+	0.973079i	$0.425973\pi$
$642$	691632.	0.0662274
$643$	1.20124e6	0.114578	0.0572890	−	0.998358i	$-0.481754\pi$
$643$	1.20124e6	0.114578	0.0572890	+	0.998358i	$0.481754\pi$
$644$	1.02600e7	0.974839
$645$	0	0
$646$	2.56727e6	0.242042
$647$	1.01480e7	0.953058	0.476529	−	0.879159i	$-0.341895\pi$
$647$	1.01480e7	0.953058	0.476529	+	0.879159i	$0.341895\pi$
$648$	1.25971e6	0.117851
$649$	2.24552e6	0.209269
$650$	0	0
$651$	−4.40438e6	−0.407316
$652$	−1.86078e6	−0.171426
$653$	1.98614e7	1.82275	0.911375	−	0.411576i	$-0.135022\pi$
$653$	1.98614e7	1.82275	0.911375	+	0.411576i	$0.135022\pi$
$654$	−5.50048e6	−0.502870
$655$	0	0
$656$	3.08224e6	0.279645
$657$	1.60299e6	0.144883
$658$	2.68902e7	2.42119
$659$	−5.73308e6	−0.514250	−0.257125	−	0.966378i	$-0.582775\pi$
$659$	−5.73308e6	−0.514250	−0.257125	+	0.966378i	$0.582775\pi$
$660$	0	0
$661$	−7.92244e6	−0.705270	−0.352635	−	0.935761i	$-0.614714\pi$
$661$	−7.92244e6	−0.705270	−0.352635	+	0.935761i	$0.614714\pi$
$662$	−9.16373e6	−0.812694
$663$	4.84393e6	0.427970
$664$	637056.	0.0560735
$665$	0	0
$666$	−3.01644e6	−0.263517
$667$	2.16087e7	1.88068
$668$	−1.01232e7	−0.877762
$669$	−2.97411e6	−0.256917
$670$	0	0
$671$	1.68032e7	1.44074
$672$	1.03680e7	0.885670
$673$	7.75353e6	0.659875	0.329938	−	0.944003i	$-0.392972\pi$
$673$	7.75353e6	0.659875	0.329938	+	0.944003i	$0.392972\pi$
$674$	5.15616e6	0.437197
$675$	0	0
$676$	−71616.0	−0.00602758
$677$	3.77777e6	0.316785	0.158392	−	0.987376i	$-0.449369\pi$
$677$	3.77777e6	0.316785	0.158392	+	0.987376i	$0.449369\pi$
$678$	−144432.	−0.0120667
$679$	2.00572e7	1.66953
$680$	0	0
$681$	1.01257e7	0.836678
$682$	3.77580e6	0.310848
$683$	6.18290e6	0.507155	0.253578	−	0.967315i	$-0.418393\pi$
$683$	6.18290e6	0.507155	0.253578	+	0.967315i	$0.418393\pi$
$684$	947376.	0.0774252
$685$	0	0
$686$	−1.53099e7	−1.24212
$687$	8.79641e6	0.711072
$688$	−286976.	−0.0231140
$689$	3.51862e6	0.282374
$690$	0	0
$691$	−2.38475e7	−1.89997	−0.949986	−	0.312293i	$-0.898903\pi$
$691$	−2.38475e7	−1.89997	−0.949986	+	0.312293i	$0.898903\pi$
$692$	1.73155e6	0.137458
$693$	−7.90965e6	−0.625640
$694$	76408.0	0.00602199
$695$	0	0
$696$	1.31017e7	1.02519
$697$	−1.05711e7	−0.824212
$698$	1.29876e7	1.00900
$699$	3.01126e6	0.233107
$700$	0	0
$701$	−5.81240e6	−0.446746	−0.223373	−	0.974733i	$-0.571707\pi$
$701$	−5.81240e6	−0.446746	−0.223373	+	0.974733i	$0.571707\pi$
$702$	−1.78751e6	−0.136901
$703$	−6.80561e6	−0.519372
$704$	−1.24436e7	−0.946272
$705$	0	0
$706$	1.71522e7	1.29511
$707$	−2.12949e7	−1.60224
$708$	−745056.	−0.0558606
$709$	−7.53133e6	−0.562673	−0.281337	−	0.959609i	$-0.590778\pi$
$709$	−7.53133e6	−0.562673	−0.281337	+	0.959609i	$0.590778\pi$
$710$	0	0
$711$	−8.50500e6	−0.630958
$712$	−1.25522e7	−0.927939
$713$	−6.19875e6	−0.456647
$714$	7.11180e6	0.522076
$715$	0	0
$716$	−7.94512e6	−0.579186
$717$	−1.96888e6	−0.143028
$718$	−354624.	−0.0256718
$719$	−8.66228e6	−0.624899	−0.312450	−	0.949934i	$-0.601150\pi$
$719$	−8.66228e6	−0.624899	−0.312450	+	0.949934i	$0.601150\pi$
$720$	0	0
$721$	2.56401e7	1.83688
$722$	7.76695e6	0.554508
$723$	−6.60637e6	−0.470021
$724$	−1.68584e6	−0.119528
$725$	0	0
$726$	982980.	0.0692155
$727$	−2.94444e6	−0.206617	−0.103309	−	0.994649i	$-0.532943\pi$
$727$	−2.94444e6	−0.206617	−0.103309	+	0.994649i	$0.532943\pi$
$728$	−2.64816e7	−1.85189
$729$	531441.	0.0370370
$730$	0	0
$731$	984238.	0.0681250
$732$	−5.57525e6	−0.384580
$733$	−1.92363e7	−1.32239	−0.661197	−	0.750213i	$-0.729951\pi$
$733$	−1.92363e7	−1.32239	−0.661197	+	0.750213i	$0.729951\pi$
$734$	−8.16736e6	−0.559553
$735$	0	0
$736$	1.45920e7	0.992934
$737$	1.37608e7	0.933203
$738$	3.90096e6	0.263652
$739$	−2.63909e7	−1.77764	−0.888819	−	0.458258i	$-0.848474\pi$
$739$	−2.63909e7	−1.77764	−0.888819	+	0.458258i	$0.848474\pi$
$740$	0	0
$741$	−4.03293e6	−0.269820
$742$	5.16600e6	0.344465
$743$	11516.0	0.000765296 0	0.000382648	−	1.00000i	$-0.499878\pi$
$743$	11516.0	0.000765296 0	0.000382648		1.00000i	$0.499878\pi$
$744$	−3.75840e6	−0.248926
$745$	0	0
$746$	−3.22528e6	−0.212188
$747$	268758.	0.0176222
$748$	6.09683e6	0.398428
$749$	4.32270e6	0.281547
$750$	0	0
$751$	−2.44836e7	−1.58407	−0.792036	−	0.610474i	$-0.790979\pi$
$751$	−2.44836e7	−1.58407	−0.792036	+	0.610474i	$0.790979\pi$
$752$	7.64877e6	0.493227
$753$	−1.06543e7	−0.684759
$754$	−1.85911e7	−1.19090
$755$	0	0
$756$	2.62440e6	0.167003
$757$	1.01385e7	0.643031	0.321515	−	0.946904i	$-0.395808\pi$
$757$	1.01385e7	0.643031	0.321515	+	0.946904i	$0.395808\pi$
$758$	9.65258e6	0.610197
$759$	−1.11321e7	−0.701411
$760$	0	0
$761$	−1.34571e7	−0.842343	−0.421172	−	0.906981i	$-0.638381\pi$
$761$	−1.34571e7	−0.842343	−0.421172	+	0.906981i	$0.638381\pi$
$762$	1.30925e6	0.0816835
$763$	−3.43780e7	−2.13781
$764$	−6.83754e6	−0.423805
$765$	0	0
$766$	−5.83426e6	−0.359264
$767$	3.17166e6	0.194670
$768$	1.00270e7	0.613435
$769$	2.13573e7	1.30236	0.651181	−	0.758923i	$-0.274274\pi$
$769$	2.13573e7	1.30236	0.651181	+	0.758923i	$0.274274\pi$
$770$	0	0
$771$	−3.00359e6	−0.181972
$772$	1.03428e7	0.624592
$773$	2.55468e7	1.53776	0.768879	−	0.639394i	$-0.220815\pi$
$773$	2.55468e7	1.53776	0.768879	+	0.639394i	$0.220815\pi$
$774$	−363204.	−0.0217921
$775$	0	0
$776$	1.71155e7	1.02031
$777$	−1.88528e7	−1.12027
$778$	−1.06247e7	−0.629315
$779$	8.80124e6	0.519637
$780$	0	0
$781$	−2.79808e7	−1.64147
$782$	1.00092e7	0.585305
$783$	5.52728e6	0.322186
$784$	−8.65741e6	−0.503034
$785$	0	0
$786$	−1.58155e6	−0.0913118
$787$	−1.32668e7	−0.763535	−0.381768	−	0.924258i	$-0.624685\pi$
$787$	−1.32668e7	−0.763535	−0.381768	+	0.924258i	$0.624685\pi$
$788$	−8.75126e6	−0.502060
$789$	−6.57994e6	−0.376296
$790$	0	0
$791$	−902700.	−0.0512982
$792$	−6.74957e6	−0.382352
$793$	2.37335e7	1.34023
$794$	6.74657e6	0.379780
$795$	0	0
$796$	9.30853e6	0.520713
$797$	1.24919e7	0.696601	0.348300	−	0.937383i	$-0.386759\pi$
$797$	1.24919e7	0.696601	0.348300	+	0.937383i	$0.386759\pi$
$798$	−5.92110e6	−0.329151
$799$	−2.62329e7	−1.45371
$800$	0	0
$801$	−5.29546e6	−0.291623
$802$	8.50133e6	0.466714
$803$	−8.58886e6	−0.470053
$804$	−4.56581e6	−0.249102
$805$	0	0
$806$	5.33310e6	0.289163
$807$	−6.60829e6	−0.357195
$808$	−1.81716e7	−0.979188
$809$	−7.83514e6	−0.420897	−0.210448	−	0.977605i	$-0.567492\pi$
$809$	−7.83514e6	−0.420897	−0.210448	+	0.977605i	$0.567492\pi$
$810$	0	0
$811$	3.04751e7	1.62702	0.813510	−	0.581551i	$-0.197554\pi$
$811$	3.04751e7	1.62702	0.813510	+	0.581551i	$0.197554\pi$
$812$	2.72952e7	1.45277
$813$	−3.01550e6	−0.160005
$814$	1.61622e7	0.854945
$815$	0	0
$816$	2.02291e6	0.106353
$817$	−819451.	−0.0429505
$818$	−7.88632e6	−0.412089
$819$	−1.11719e7	−0.581994
$820$	0	0
$821$	−2.90330e7	−1.50326	−0.751629	−	0.659586i	$-0.770732\pi$
$821$	−2.90330e7	−1.50326	−0.751629	+	0.659586i	$0.770732\pi$
$822$	−6.66770e6	−0.344189
$823$	1.99935e7	1.02894	0.514468	−	0.857510i	$-0.327989\pi$
$823$	1.99935e7	1.02894	0.514468	+	0.857510i	$0.327989\pi$
$824$	2.18796e7	1.12259
$825$	0	0
$826$	4.65660e6	0.237476
$827$	196388.	0.00998507	0.00499254	−	0.999988i	$-0.498411\pi$
$827$	196388.	0.00998507	0.00499254	+	0.999988i	$0.498411\pi$
$828$	3.69360e6	0.187229
$829$	7.05093e6	0.356336	0.178168	−	0.984000i	$-0.442983\pi$
$829$	7.05093e6	0.356336	0.178168	+	0.984000i	$0.442983\pi$
$830$	0	0
$831$	−1.42434e7	−0.715502
$832$	−1.75759e7	−0.880258
$833$	2.96922e7	1.48262
$834$	4.17298e6	0.207745
$835$	0	0
$836$	−5.07606e6	−0.251195
$837$	−1.58558e6	−0.0782300
$838$	−7.02978e6	−0.345805
$839$	−3.26118e7	−1.59945	−0.799724	−	0.600368i	$-0.795021\pi$
$839$	−3.26118e7	−1.59945	−0.799724	+	0.600368i	$0.795021\pi$
$840$	0	0
$841$	3.69756e7	1.80271
$842$	1.13579e7	0.552098
$843$	−9.78421e6	−0.474195
$844$	7.06168e6	0.341234
$845$	0	0
$846$	9.68047e6	0.465019
$847$	6.14362e6	0.294250
$848$	1.46944e6	0.0701717
$849$	−5.46666e6	−0.260287
$850$	0	0
$851$	−2.65335e7	−1.25595
$852$	9.28397e6	0.438162
$853$	1.84687e7	0.869089	0.434545	−	0.900650i	$-0.356909\pi$
$853$	1.84687e7	0.869089	0.434545	+	0.900650i	$0.356909\pi$
$854$	3.48453e7	1.63493
$855$	0	0
$856$	3.68870e6	0.172064
$857$	3.14310e7	1.46186	0.730930	−	0.682452i	$-0.239087\pi$
$857$	3.14310e7	1.46186	0.730930	+	0.682452i	$0.239087\pi$
$858$	9.57751e6	0.444155
$859$	2.80767e7	1.29826	0.649132	−	0.760676i	$-0.275132\pi$
$859$	2.80767e7	1.29826	0.649132	+	0.760676i	$0.275132\pi$
$860$	0	0
$861$	2.43810e7	1.12084
$862$	1.87734e7	0.860547
$863$	2.11951e7	0.968744	0.484372	−	0.874862i	$-0.339048\pi$
$863$	2.11951e7	0.968744	0.484372	+	0.874862i	$0.339048\pi$
$864$	3.73248e6	0.170103
$865$	0	0
$866$	−2.68537e7	−1.21677
$867$	5.84076e6	0.263889
$868$	−7.83000e6	−0.352746
$869$	4.55700e7	2.04706
$870$	0	0
$871$	1.94364e7	0.868101
$872$	−2.93359e7	−1.30650
$873$	7.22058e6	0.320654
$874$	−8.33340e6	−0.369015
$875$	0	0
$876$	2.84976e6	0.125472
$877$	−3.70911e7	−1.62843	−0.814217	−	0.580561i	$-0.802833\pi$
$877$	−3.70911e7	−1.62843	−0.814217	+	0.580561i	$0.802833\pi$
$878$	−1.56809e7	−0.686489
$879$	1.10971e7	0.484435
$880$	0	0
$881$	−9.16386e6	−0.397776	−0.198888	−	0.980022i	$-0.563733\pi$
$881$	−9.16386e6	−0.397776	−0.198888	+	0.980022i	$0.563733\pi$
$882$	−1.09570e7	−0.474265
$883$	−1.36396e7	−0.588706	−0.294353	−	0.955697i	$-0.595104\pi$
$883$	−1.36396e7	−0.588706	−0.294353	+	0.955697i	$0.595104\pi$
$884$	8.61142e6	0.370633
$885$	0	0
$886$	−2.77212e7	−1.18639
$887$	−2.90401e7	−1.23934	−0.619669	−	0.784863i	$-0.712733\pi$
$887$	−2.90401e7	−1.23934	−0.619669	+	0.784863i	$0.712733\pi$
$888$	−1.60877e7	−0.684638
$889$	8.18280e6	0.347254
$890$	0	0
$891$	−2.84747e6	−0.120162
$892$	−5.28731e6	−0.222496
$893$	2.18408e7	0.916516
$894$	1.19239e6	0.0498971
$895$	0	0
$896$	1.10592e7	0.460207
$897$	−1.57234e7	−0.652479
$898$	−1.75442e7	−0.726011
$899$	−1.64908e7	−0.680525
$900$	0	0
$901$	−5.03972e6	−0.206821
$902$	−2.09014e7	−0.855382
$903$	−2.27002e6	−0.0926427
$904$	−770304.	−0.0313503
$905$	0	0
$906$	1.55274e7	0.628461
$907$	−2.12100e7	−0.856095	−0.428048	−	0.903756i	$-0.640798\pi$
$907$	−2.12100e7	−0.856095	−0.428048	+	0.903756i	$0.640798\pi$
$908$	1.80013e7	0.724584
$909$	−7.66616e6	−0.307729
$910$	0	0
$911$	1.95267e7	0.779531	0.389766	−	0.920914i	$-0.372556\pi$
$911$	1.95267e7	0.779531	0.389766	+	0.920914i	$0.372556\pi$
$912$	−1.68422e6	−0.0670522
$913$	−1.44001e6	−0.0571728
$914$	9.67804e6	0.383197
$915$	0	0
$916$	1.56381e7	0.615807
$917$	−9.88470e6	−0.388186
$918$	2.56025e6	0.100271
$919$	−2.85930e7	−1.11679	−0.558395	−	0.829575i	$-0.688583\pi$
$919$	−2.85930e7	−1.11679	−0.558395	+	0.829575i	$0.688583\pi$
$920$	0	0
$921$	5.06478e6	0.196748
$922$	1.46491e6	0.0567524
$923$	−3.95213e7	−1.52696
$924$	−1.40616e7	−0.541820
$925$	0	0
$926$	2.65014e7	1.01564
$927$	9.23044e6	0.352795
$928$	3.88198e7	1.47973
$929$	2.38769e7	0.907692	0.453846	−	0.891080i	$-0.350052\pi$
$929$	2.38769e7	0.907692	0.453846	+	0.891080i	$0.350052\pi$
$930$	0	0
$931$	−2.47210e7	−0.934741
$932$	5.35334e6	0.201876
$933$	2.15681e7	0.811162
$934$	1.35390e7	0.507832
$935$	0	0
$936$	−9.53338e6	−0.355678
$937$	−3.46672e7	−1.28994	−0.644970	−	0.764208i	$-0.723130\pi$
$937$	−3.46672e7	−1.28994	−0.644970	+	0.764208i	$0.723130\pi$
$938$	2.85363e7	1.05899
$939$	1.37550e7	0.509092
$940$	0	0
$941$	4.64188e7	1.70891	0.854455	−	0.519525i	$-0.173891\pi$
$941$	4.64188e7	1.70891	0.854455	+	0.519525i	$0.173891\pi$
$942$	−1.45580e6	−0.0534534
$943$	3.43140e7	1.25659
$944$	1.32454e6	0.0483767
$945$	0	0
$946$	1.94606e6	0.0707013
$947$	2.10829e7	0.763934	0.381967	−	0.924176i	$-0.375247\pi$
$947$	2.10829e7	0.763934	0.381967	+	0.924176i	$0.375247\pi$
$948$	−1.51200e7	−0.546425
$949$	−1.21313e7	−0.437261
$950$	0	0
$951$	9.00295e6	0.322800
$952$	3.79296e7	1.35639
$953$	−3.67438e7	−1.31055	−0.655273	−	0.755392i	$-0.727446\pi$
$953$	−3.67438e7	−1.31055	−0.655273	+	0.755392i	$0.727446\pi$
$954$	1.85976e6	0.0661585
$955$	0	0
$956$	−3.50022e6	−0.123866
$957$	−2.96153e7	−1.04529
$958$	−2.64762e7	−0.932056
$959$	−4.16732e7	−1.46322
$960$	0	0
$961$	−2.38985e7	−0.834762
$962$	2.28281e7	0.795303
$963$	1.55617e6	0.0540744
$964$	−1.17447e7	−0.407050
$965$	0	0
$966$	−2.30850e7	−0.795952
$967$	3.76234e7	1.29388	0.646938	−	0.762543i	$-0.276049\pi$
$967$	3.76234e7	1.29388	0.646938	+	0.762543i	$0.276049\pi$
$968$	5.24256e6	0.179827
$969$	5.77636e6	0.197626
$970$	0	0
$971$	−4.89205e7	−1.66511	−0.832555	−	0.553943i	$-0.813123\pi$
$971$	−4.89205e7	−1.66511	−0.832555	+	0.553943i	$0.813123\pi$
$972$	944784.	0.0320750
$973$	2.60811e7	0.883169
$974$	−1.14953e6	−0.0388261
$975$	0	0
$976$	9.91155e6	0.333056
$977$	2.84450e7	0.953386	0.476693	−	0.879070i	$-0.341835\pi$
$977$	2.84450e7	0.953386	0.476693	+	0.879070i	$0.341835\pi$
$978$	4.18676e6	0.139969
$979$	2.83732e7	0.946131
$980$	0	0
$981$	−1.23761e7	−0.410592
$982$	−2.60535e7	−0.862158
$983$	−4.51611e7	−1.49067	−0.745334	−	0.666692i	$-0.767710\pi$
$983$	−4.51611e7	−1.49067	−0.745334	+	0.666692i	$0.767710\pi$
$984$	2.08051e7	0.684987
$985$	0	0
$986$	2.66280e7	0.872260
$987$	6.05030e7	1.97690
$988$	−7.16965e6	−0.233671
$989$	−3.19485e6	−0.103863
$990$	0	0
$991$	−2.85517e7	−0.923524	−0.461762	−	0.887004i	$-0.652783\pi$
$991$	−2.85517e7	−0.923524	−0.461762	+	0.887004i	$0.652783\pi$
$992$	−1.11360e7	−0.359294
$993$	−2.06184e7	−0.663562
$994$	−5.80248e7	−1.86272
$995$	0	0
$996$	477792.	0.0152613
$997$	−3.26527e7	−1.04035	−0.520177	−	0.854058i	$-0.674134\pi$
$997$	−3.26527e7	−1.04035	−0.520177	+	0.854058i	$0.674134\pi$
$998$	4.68148e6	0.148784
$999$	−6.78699e6	−0.215161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.a.b.1.1	✓	1
3.2	odd	2		225.6.a.g.1.1		1
5.2	odd	4		75.6.b.c.49.1		2
5.3	odd	4		75.6.b.c.49.2		2
5.4	even	2		75.6.a.d.1.1	yes	1
15.2	even	4		225.6.b.c.199.2		2
15.8	even	4		225.6.b.c.199.1		2
15.14	odd	2		225.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.a.b.1.1	✓	1	1.1	even	1	trivial
75.6.a.d.1.1	yes	1	5.4	even	2
75.6.b.c.49.1		2	5.2	odd	4
75.6.b.c.49.2		2	5.3	odd	4
225.6.a.b.1.1		1	15.14	odd	2
225.6.a.g.1.1		1	3.2	odd	2
225.6.b.c.199.1		2	15.8	even	4
225.6.b.c.199.2		2	15.2	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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