LMFDB - Embedded newform 390.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,6,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("390.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	390.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:	$62.5496897271$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} +36.0000 q^{6} +103.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} +36.0000 q^{6} +103.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -100.000 q^{10} -643.000 q^{11} +144.000 q^{12} -169.000 q^{13} +412.000 q^{14} -225.000 q^{15} +256.000 q^{16} -1327.00 q^{17} +324.000 q^{18} -2414.00 q^{19} -400.000 q^{20} +927.000 q^{21} -2572.00 q^{22} -1745.00 q^{23} +576.000 q^{24} +625.000 q^{25} -676.000 q^{26} +729.000 q^{27} +1648.00 q^{28} +5974.00 q^{29} -900.000 q^{30} -8882.00 q^{31} +1024.00 q^{32} -5787.00 q^{33} -5308.00 q^{34} -2575.00 q^{35} +1296.00 q^{36} +8739.00 q^{37} -9656.00 q^{38} -1521.00 q^{39} -1600.00 q^{40} +11909.0 q^{41} +3708.00 q^{42} -13124.0 q^{43} -10288.0 q^{44} -2025.00 q^{45} -6980.00 q^{46} +6078.00 q^{47} +2304.00 q^{48} -6198.00 q^{49} +2500.00 q^{50} -11943.0 q^{51} -2704.00 q^{52} +16533.0 q^{53} +2916.00 q^{54} +16075.0 q^{55} +6592.00 q^{56} -21726.0 q^{57} +23896.0 q^{58} -960.000 q^{59} -3600.00 q^{60} -49139.0 q^{61} -35528.0 q^{62} +8343.00 q^{63} +4096.00 q^{64} +4225.00 q^{65} -23148.0 q^{66} -14804.0 q^{67} -21232.0 q^{68} -15705.0 q^{69} -10300.0 q^{70} -79359.0 q^{71} +5184.00 q^{72} -43638.0 q^{73} +34956.0 q^{74} +5625.00 q^{75} -38624.0 q^{76} -66229.0 q^{77} -6084.00 q^{78} -74063.0 q^{79} -6400.00 q^{80} +6561.00 q^{81} +47636.0 q^{82} +98148.0 q^{83} +14832.0 q^{84} +33175.0 q^{85} -52496.0 q^{86} +53766.0 q^{87} -41152.0 q^{88} +115951. q^{89} -8100.00 q^{90} -17407.0 q^{91} -27920.0 q^{92} -79938.0 q^{93} +24312.0 q^{94} +60350.0 q^{95} +9216.00 q^{96} +123433. q^{97} -24792.0 q^{98} -52083.0 q^{99} +O(q^{100})\)

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	36.0000	0.408248
$7$	103.000	0.794497	0.397248	−	0.917711i	$-0.369965\pi$
$7$	103.000	0.794497	0.397248	+	0.917711i	$0.369965\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−100.000	−0.316228
$11$	−643.000	−1.60225	−0.801123	−	0.598500i	$-0.795764\pi$
$11$	−643.000	−1.60225	−0.801123	+	0.598500i	$0.795764\pi$
$12$	144.000	0.288675
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	412.000	0.561794
$15$	−225.000	−0.258199
$16$	256.000	0.250000
$17$	−1327.00	−1.11365	−0.556825	−	0.830630i	$-0.687981\pi$
$17$	−1327.00	−1.11365	−0.556825	+	0.830630i	$0.687981\pi$
$18$	324.000	0.235702
$19$	−2414.00	−1.53410	−0.767049	−	0.641588i	$-0.778276\pi$
$19$	−2414.00	−1.53410	−0.767049	+	0.641588i	$0.778276\pi$
$20$	−400.000	−0.223607
$21$	927.000	0.458703
$22$	−2572.00	−1.13296
$23$	−1745.00	−0.687822	−0.343911	−	0.939002i	$-0.611752\pi$
$23$	−1745.00	−0.687822	−0.343911	+	0.939002i	$0.611752\pi$
$24$	576.000	0.204124
$25$	625.000	0.200000
$26$	−676.000	−0.196116
$27$	729.000	0.192450
$28$	1648.00	0.397248
$29$	5974.00	1.31908	0.659539	−	0.751671i	$-0.270752\pi$
$29$	5974.00	1.31908	0.659539	+	0.751671i	$0.270752\pi$
$30$	−900.000	−0.182574
$31$	−8882.00	−1.65999	−0.829997	−	0.557768i	$-0.811658\pi$
$31$	−8882.00	−1.65999	−0.829997	+	0.557768i	$0.811658\pi$
$32$	1024.00	0.176777
$33$	−5787.00	−0.925057
$34$	−5308.00	−0.787469
$35$	−2575.00	−0.355310
$36$	1296.00	0.166667
$37$	8739.00	1.04944	0.524720	−	0.851275i	$-0.324170\pi$
$37$	8739.00	1.04944	0.524720	+	0.851275i	$0.324170\pi$
$38$	−9656.00	−1.08477
$39$	−1521.00	−0.160128
$40$	−1600.00	−0.158114
$41$	11909.0	1.10641	0.553204	−	0.833046i	$-0.313405\pi$
$41$	11909.0	1.10641	0.553204	+	0.833046i	$0.313405\pi$
$42$	3708.00	0.324352
$43$	−13124.0	−1.08242	−0.541209	−	0.840888i	$-0.682033\pi$
$43$	−13124.0	−1.08242	−0.541209	+	0.840888i	$0.682033\pi$
$44$	−10288.0	−0.801123
$45$	−2025.00	−0.149071
$46$	−6980.00	−0.486363
$47$	6078.00	0.401343	0.200672	−	0.979659i	$-0.435688\pi$
$47$	6078.00	0.401343	0.200672	+	0.979659i	$0.435688\pi$
$48$	2304.00	0.144338
$49$	−6198.00	−0.368775
$50$	2500.00	0.141421
$51$	−11943.0	−0.642966
$52$	−2704.00	−0.138675
$53$	16533.0	0.808466	0.404233	−	0.914656i	$-0.367538\pi$
$53$	16533.0	0.808466	0.404233	+	0.914656i	$0.367538\pi$
$54$	2916.00	0.136083
$55$	16075.0	0.716546
$56$	6592.00	0.280897
$57$	−21726.0	−0.885712
$58$	23896.0	0.932728
$59$	−960.000	−0.0359039	−0.0179519	−	0.999839i	$-0.505715\pi$
$59$	−960.000	−0.0359039	−0.0179519	+	0.999839i	$0.505715\pi$
$60$	−3600.00	−0.129099
$61$	−49139.0	−1.69084	−0.845418	−	0.534104i	$-0.820649\pi$
$61$	−49139.0	−1.69084	−0.845418	+	0.534104i	$0.820649\pi$
$62$	−35528.0	−1.17379
$63$	8343.00	0.264832
$64$	4096.00	0.125000
$65$	4225.00	0.124035
$66$	−23148.0	−0.654114
$67$	−14804.0	−0.402895	−0.201448	−	0.979499i	$-0.564565\pi$
$67$	−14804.0	−0.402895	−0.201448	+	0.979499i	$0.564565\pi$
$68$	−21232.0	−0.556825
$69$	−15705.0	−0.397114
$70$	−10300.0	−0.251242
$71$	−79359.0	−1.86832	−0.934158	−	0.356860i	$-0.883847\pi$
$71$	−79359.0	−1.86832	−0.934158	+	0.356860i	$0.883847\pi$
$72$	5184.00	0.117851
$73$	−43638.0	−0.958424	−0.479212	−	0.877699i	$-0.659077\pi$
$73$	−43638.0	−0.958424	−0.479212	+	0.877699i	$0.659077\pi$
$74$	34956.0	0.742066
$75$	5625.00	0.115470
$76$	−38624.0	−0.767049
$77$	−66229.0	−1.27298
$78$	−6084.00	−0.113228
$79$	−74063.0	−1.33516	−0.667580	−	0.744538i	$-0.732670\pi$
$79$	−74063.0	−1.33516	−0.667580	+	0.744538i	$0.732670\pi$
$80$	−6400.00	−0.111803
$81$	6561.00	0.111111
$82$	47636.0	0.782349
$83$	98148.0	1.56382	0.781909	−	0.623393i	$-0.214246\pi$
$83$	98148.0	1.56382	0.781909	+	0.623393i	$0.214246\pi$
$84$	14832.0	0.229351
$85$	33175.0	0.498039
$86$	−52496.0	−0.765385
$87$	53766.0	0.761570
$88$	−41152.0	−0.566480
$89$	115951.	1.55167	0.775835	−	0.630936i	$-0.217329\pi$
$89$	115951.	1.55167	0.775835	+	0.630936i	$0.217329\pi$
$90$	−8100.00	−0.105409
$91$	−17407.0	−0.220354
$92$	−27920.0	−0.343911
$93$	−79938.0	−0.958398
$94$	24312.0	0.283793
$95$	60350.0	0.686070
$96$	9216.00	0.102062
$97$	123433.	1.33199	0.665997	−	0.745955i	$-0.268007\pi$
$97$	123433.	1.33199	0.665997	+	0.745955i	$0.268007\pi$
$98$	−24792.0	−0.260763
$99$	−52083.0	−0.534082
$100$	10000.0	0.100000
$101$	−88136.0	−0.859706	−0.429853	−	0.902899i	$-0.641435\pi$
$101$	−88136.0	−0.859706	−0.429853	+	0.902899i	$0.641435\pi$
$102$	−47772.0	−0.454646
$103$	−49436.0	−0.459145	−0.229573	−	0.973292i	$-0.573733\pi$
$103$	−49436.0	−0.459145	−0.229573	+	0.973292i	$0.573733\pi$
$104$	−10816.0	−0.0980581
$105$	−23175.0	−0.205138
$106$	66132.0	0.571672
$107$	5939.00	0.0501480	0.0250740	−	0.999686i	$-0.492018\pi$
$107$	5939.00	0.0501480	0.0250740	+	0.999686i	$0.492018\pi$
$108$	11664.0	0.0962250
$109$	−41828.0	−0.337210	−0.168605	−	0.985684i	$-0.553926\pi$
$109$	−41828.0	−0.337210	−0.168605	+	0.985684i	$0.553926\pi$
$110$	64300.0	0.506675
$111$	78651.0	0.605894
$112$	26368.0	0.198624
$113$	61354.0	0.452009	0.226004	−	0.974126i	$-0.427434\pi$
$113$	61354.0	0.452009	0.226004	+	0.974126i	$0.427434\pi$
$114$	−86904.0	−0.626293
$115$	43625.0	0.307603
$116$	95584.0	0.659539
$117$	−13689.0	−0.0924500
$118$	−3840.00	−0.0253879
$119$	−136681.	−0.884791
$120$	−14400.0	−0.0912871
$121$	252398.	1.56719
$122$	−196556.	−1.19560
$123$	107181.	0.638785
$124$	−142112.	−0.829997
$125$	−15625.0	−0.0894427
$126$	33372.0	0.187265
$127$	−226486.	−1.24604	−0.623020	−	0.782206i	$-0.714095\pi$
$127$	−226486.	−1.24604	−0.623020	+	0.782206i	$0.714095\pi$
$128$	16384.0	0.0883883
$129$	−118116.	−0.624934
$130$	16900.0	0.0877058
$131$	9814.00	0.0499652	0.0249826	−	0.999688i	$-0.492047\pi$
$131$	9814.00	0.0499652	0.0249826	+	0.999688i	$0.492047\pi$
$132$	−92592.0	−0.462529
$133$	−248642.	−1.21884
$134$	−59216.0	−0.284890
$135$	−18225.0	−0.0860663
$136$	−84928.0	−0.393735
$137$	12314.0	0.0560529	0.0280264	−	0.999607i	$-0.491078\pi$
$137$	12314.0	0.0560529	0.0280264	+	0.999607i	$0.491078\pi$
$138$	−62820.0	−0.280802
$139$	25139.0	0.110360	0.0551799	−	0.998476i	$-0.482427\pi$
$139$	25139.0	0.110360	0.0551799	+	0.998476i	$0.482427\pi$
$140$	−41200.0	−0.177655
$141$	54702.0	0.231716
$142$	−317436.	−1.32110
$143$	108667.	0.444383
$144$	20736.0	0.0833333
$145$	−149350.	−0.589909
$146$	−174552.	−0.677708
$147$	−55782.0	−0.212912
$148$	139824.	0.524720
$149$	−218453.	−0.806106	−0.403053	−	0.915177i	$-0.632051\pi$
$149$	−218453.	−0.806106	−0.403053	+	0.915177i	$0.632051\pi$
$150$	22500.0	0.0816497
$151$	266744.	0.952034	0.476017	−	0.879436i	$-0.342080\pi$
$151$	266744.	0.952034	0.476017	+	0.879436i	$0.342080\pi$
$152$	−154496.	−0.542386
$153$	−107487.	−0.371217
$154$	−264916.	−0.900132
$155$	222050.	0.742372
$156$	−24336.0	−0.0800641
$157$	236690.	0.766356	0.383178	−	0.923674i	$-0.374830\pi$
$157$	236690.	0.766356	0.383178	+	0.923674i	$0.374830\pi$
$158$	−296252.	−0.944101
$159$	148797.	0.466768
$160$	−25600.0	−0.0790569
$161$	−179735.	−0.546472
$162$	26244.0	0.0785674
$163$	−189537.	−0.558760	−0.279380	−	0.960181i	$-0.590129\pi$
$163$	−189537.	−0.558760	−0.279380	+	0.960181i	$0.590129\pi$
$164$	190544.	0.553204
$165$	144675.	0.413698
$166$	392592.	1.10579
$167$	595728.	1.65294	0.826470	−	0.562981i	$-0.190346\pi$
$167$	595728.	1.65294	0.826470	+	0.562981i	$0.190346\pi$
$168$	59328.0	0.162176
$169$	28561.0	0.0769231
$170$	132700.	0.352167
$171$	−195534.	−0.511366
$172$	−209984.	−0.541209
$173$	−132194.	−0.335812	−0.167906	−	0.985803i	$-0.553701\pi$
$173$	−132194.	−0.335812	−0.167906	+	0.985803i	$0.553701\pi$
$174$	215064.	0.538511
$175$	64375.0	0.158899
$176$	−164608.	−0.400562
$177$	−8640.00	−0.0207291
$178$	463804.	1.09720
$179$	−538898.	−1.25711	−0.628556	−	0.777764i	$-0.716354\pi$
$179$	−538898.	−1.25711	−0.628556	+	0.777764i	$0.716354\pi$
$180$	−32400.0	−0.0745356
$181$	2369.00	0.00537488	0.00268744	−	0.999996i	$-0.499145\pi$
$181$	2369.00	0.00537488	0.00268744	+	0.999996i	$0.499145\pi$
$182$	−69628.0	−0.155814
$183$	−442251.	−0.976205
$184$	−111680.	−0.243182
$185$	−218475.	−0.469324
$186$	−319752.	−0.677690
$187$	853261.	1.78434
$188$	97248.0	0.200672
$189$	75087.0	0.152901
$190$	241400.	0.485125
$191$	−974872.	−1.93359	−0.966795	−	0.255555i	$-0.917742\pi$
$191$	−974872.	−1.93359	−0.966795	+	0.255555i	$0.917742\pi$
$192$	36864.0	0.0721688
$193$	−184365.	−0.356275	−0.178137	−	0.984006i	$-0.557007\pi$
$193$	−184365.	−0.356275	−0.178137	+	0.984006i	$0.557007\pi$
$194$	493732.	0.941861
$195$	38025.0	0.0716115
$196$	−99168.0	−0.184387
$197$	−192820.	−0.353986	−0.176993	−	0.984212i	$-0.556637\pi$
$197$	−192820.	−0.353986	−0.176993	+	0.984212i	$0.556637\pi$
$198$	−208332.	−0.377653
$199$	564636.	1.01073	0.505366	−	0.862905i	$-0.331358\pi$
$199$	564636.	1.01073	0.505366	+	0.862905i	$0.331358\pi$
$200$	40000.0	0.0707107
$201$	−133236.	−0.232612
$202$	−352544.	−0.607904
$203$	615322.	1.04800
$204$	−191088.	−0.321483
$205$	−297725.	−0.494801
$206$	−197744.	−0.324665
$207$	−141345.	−0.229274
$208$	−43264.0	−0.0693375
$209$	1.55220e6	2.45800
$210$	−92700.0	−0.145055
$211$	55196.0	0.0853496	0.0426748	−	0.999089i	$-0.486412\pi$
$211$	55196.0	0.0853496	0.0426748	+	0.999089i	$0.486412\pi$
$212$	264528.	0.404233
$213$	−714231.	−1.07867
$214$	23756.0	0.0354600
$215$	328100.	0.484072
$216$	46656.0	0.0680414
$217$	−914846.	−1.31886
$218$	−167312.	−0.238444
$219$	−392742.	−0.553346
$220$	257200.	0.358273
$221$	224263.	0.308871
$222$	314604.	0.428432
$223$	1.40422e6	1.89092	0.945462	−	0.325731i	$-0.105611\pi$
$223$	1.40422e6	1.89092	0.945462	+	0.325731i	$0.105611\pi$
$224$	105472.	0.140449
$225$	50625.0	0.0666667
$226$	245416.	0.319618
$227$	388878.	0.500897	0.250449	−	0.968130i	$-0.419422\pi$
$227$	388878.	0.500897	0.250449	+	0.968130i	$0.419422\pi$
$228$	−347616.	−0.442856
$229$	−232862.	−0.293434	−0.146717	−	0.989179i	$-0.546871\pi$
$229$	−232862.	−0.293434	−0.146717	+	0.989179i	$0.546871\pi$
$230$	174500.	0.217508
$231$	−596061.	−0.734955
$232$	382336.	0.466364
$233$	577619.	0.697030	0.348515	−	0.937303i	$-0.386686\pi$
$233$	577619.	0.697030	0.348515	+	0.937303i	$0.386686\pi$
$234$	−54756.0	−0.0653720
$235$	−151950.	−0.179486
$236$	−15360.0	−0.0179519
$237$	−666567.	−0.770855
$238$	−546724.	−0.625642
$239$	−869287.	−0.984393	−0.492196	−	0.870484i	$-0.663806\pi$
$239$	−869287.	−0.984393	−0.492196	+	0.870484i	$0.663806\pi$
$240$	−57600.0	−0.0645497
$241$	−187678.	−0.208147	−0.104074	−	0.994570i	$-0.533188\pi$
$241$	−187678.	−0.208147	−0.104074	+	0.994570i	$0.533188\pi$
$242$	1.00959e6	1.10817
$243$	59049.0	0.0641500
$244$	−786224.	−0.845418
$245$	154950.	0.164921
$246$	428724.	0.451690
$247$	407966.	0.425482
$248$	−568448.	−0.586897
$249$	883332.	0.902871
$250$	−62500.0	−0.0632456
$251$	1.75261e6	1.75590	0.877951	−	0.478750i	$-0.158910\pi$
$251$	1.75261e6	1.75590	0.877951	+	0.478750i	$0.158910\pi$
$252$	133488.	0.132416
$253$	1.12204e6	1.10206
$254$	−905944.	−0.881083
$255$	298575.	0.287543
$256$	65536.0	0.0625000
$257$	−1.52690e6	−1.44204	−0.721022	−	0.692912i	$-0.756327\pi$
$257$	−1.52690e6	−1.44204	−0.721022	+	0.692912i	$0.756327\pi$
$258$	−472464.	−0.441895
$259$	900117.	0.833776
$260$	67600.0	0.0620174
$261$	483894.	0.439692
$262$	39256.0	0.0353307
$263$	1.87973e6	1.67574	0.837868	−	0.545873i	$-0.183802\pi$
$263$	1.87973e6	1.67574	0.837868	+	0.545873i	$0.183802\pi$
$264$	−370368.	−0.327057
$265$	−413325.	−0.361557
$266$	−994568.	−0.861847
$267$	1.04356e6	0.895857
$268$	−236864.	−0.201448
$269$	−25880.0	−0.0218064	−0.0109032	−	0.999941i	$-0.503471\pi$
$269$	−25880.0	−0.0218064	−0.0109032	+	0.999941i	$0.503471\pi$
$270$	−72900.0	−0.0608581
$271$	−1.59213e6	−1.31691	−0.658455	−	0.752620i	$-0.728790\pi$
$271$	−1.59213e6	−1.31691	−0.658455	+	0.752620i	$0.728790\pi$
$272$	−339712.	−0.278412
$273$	−156663.	−0.127221
$274$	49256.0	0.0396354
$275$	−401875.	−0.320449
$276$	−251280.	−0.198557
$277$	568282.	0.445005	0.222502	−	0.974932i	$-0.428578\pi$
$277$	568282.	0.445005	0.222502	+	0.974932i	$0.428578\pi$
$278$	100556.	0.0780362
$279$	−719442.	−0.553331
$280$	−164800.	−0.125621
$281$	522442.	0.394705	0.197352	−	0.980333i	$-0.436766\pi$
$281$	522442.	0.394705	0.197352	+	0.980333i	$0.436766\pi$
$282$	218808.	0.163848
$283$	2.33016e6	1.72950	0.864750	−	0.502203i	$-0.167477\pi$
$283$	2.33016e6	1.72950	0.864750	+	0.502203i	$0.167477\pi$
$284$	−1.26974e6	−0.934158
$285$	543150.	0.396103
$286$	434668.	0.314226
$287$	1.22663e6	0.879038
$288$	82944.0	0.0589256
$289$	341072.	0.240216
$290$	−597400.	−0.417129
$291$	1.11090e6	0.769027
$292$	−698208.	−0.479212
$293$	814884.	0.554532	0.277266	−	0.960793i	$-0.410572\pi$
$293$	814884.	0.554532	0.277266	+	0.960793i	$0.410572\pi$
$294$	−223128.	−0.150552
$295$	24000.0	0.0160567
$296$	559296.	0.371033
$297$	−468747.	−0.308352
$298$	−873812.	−0.570003
$299$	294905.	0.190767
$300$	90000.0	0.0577350
$301$	−1.35177e6	−0.859978
$302$	1.06698e6	0.673189
$303$	−793224.	−0.496351
$304$	−617984.	−0.383525
$305$	1.22848e6	0.756165
$306$	−429948.	−0.262490
$307$	−1.83378e6	−1.11045	−0.555227	−	0.831699i	$-0.687369\pi$
$307$	−1.83378e6	−1.11045	−0.555227	+	0.831699i	$0.687369\pi$
$308$	−1.05966e6	−0.636490
$309$	−444924.	−0.265088
$310$	888200.	0.524936
$311$	135880.	0.0796626	0.0398313	−	0.999206i	$-0.487318\pi$
$311$	135880.	0.0796626	0.0398313	+	0.999206i	$0.487318\pi$
$312$	−97344.0	−0.0566139
$313$	−1.24075e6	−0.715852	−0.357926	−	0.933750i	$-0.616516\pi$
$313$	−1.24075e6	−0.715852	−0.357926	+	0.933750i	$0.616516\pi$
$314$	946760.	0.541896
$315$	−208575.	−0.118437
$316$	−1.18501e6	−0.667580
$317$	2.57671e6	1.44018	0.720090	−	0.693880i	$-0.244100\pi$
$317$	2.57671e6	1.44018	0.720090	+	0.693880i	$0.244100\pi$
$318$	595188.	0.330055
$319$	−3.84128e6	−2.11349
$320$	−102400.	−0.0559017
$321$	53451.0	0.0289530
$322$	−718940.	−0.386414
$323$	3.20338e6	1.70845
$324$	104976.	0.0555556
$325$	−105625.	−0.0554700
$326$	−758148.	−0.395103
$327$	−376452.	−0.194688
$328$	762176.	0.391175
$329$	626034.	0.318866
$330$	578700.	0.292529
$331$	2.27467e6	1.14116	0.570582	−	0.821241i	$-0.306718\pi$
$331$	2.27467e6	1.14116	0.570582	+	0.821241i	$0.306718\pi$
$332$	1.57037e6	0.781909
$333$	707859.	0.349813
$334$	2.38291e6	1.16880
$335$	370100.	0.180180
$336$	237312.	0.114676
$337$	−1.15190e6	−0.552510	−0.276255	−	0.961084i	$-0.589093\pi$
$337$	−1.15190e6	−0.552510	−0.276255	+	0.961084i	$0.589093\pi$
$338$	114244.	0.0543928
$339$	552186.	0.260967
$340$	530800.	0.249020
$341$	5.71113e6	2.65972
$342$	−782136.	−0.361590
$343$	−2.36951e6	−1.08749
$344$	−839936.	−0.382693
$345$	392625.	0.177595
$346$	−528776.	−0.237455
$347$	−1.55650e6	−0.693944	−0.346972	−	0.937875i	$-0.612790\pi$
$347$	−1.55650e6	−0.693944	−0.346972	+	0.937875i	$0.612790\pi$
$348$	860256.	0.380785
$349$	−1.54784e6	−0.680239	−0.340119	−	0.940382i	$-0.610467\pi$
$349$	−1.54784e6	−0.680239	−0.340119	+	0.940382i	$0.610467\pi$
$350$	257500.	0.112359
$351$	−123201.	−0.0533761
$352$	−658432.	−0.283240
$353$	1.25273e6	0.535084	0.267542	−	0.963546i	$-0.413789\pi$
$353$	1.25273e6	0.535084	0.267542	+	0.963546i	$0.413789\pi$
$354$	−34560.0	−0.0146577
$355$	1.98397e6	0.835536
$356$	1.85522e6	0.775835
$357$	−1.23013e6	−0.510834
$358$	−2.15559e6	−0.888912
$359$	−3.55985e6	−1.45779	−0.728896	−	0.684625i	$-0.759966\pi$
$359$	−3.55985e6	−1.45779	−0.728896	+	0.684625i	$0.759966\pi$
$360$	−129600.	−0.0527046
$361$	3.35130e6	1.35346
$362$	9476.00	0.00380061
$363$	2.27158e6	0.904819
$364$	−278512.	−0.110177
$365$	1.09095e6	0.428620
$366$	−1.76900e6	−0.690281
$367$	−1.15780e6	−0.448713	−0.224356	−	0.974507i	$-0.572028\pi$
$367$	−1.15780e6	−0.448713	−0.224356	+	0.974507i	$0.572028\pi$
$368$	−446720.	−0.171955
$369$	964629.	0.368803
$370$	−873900.	−0.331862
$371$	1.70290e6	0.642324
$372$	−1.27901e6	−0.479199
$373$	2.78529e6	1.03657	0.518285	−	0.855208i	$-0.326571\pi$
$373$	2.78529e6	1.03657	0.518285	+	0.855208i	$0.326571\pi$
$374$	3.41304e6	1.26172
$375$	−140625.	−0.0516398
$376$	388992.	0.141896
$377$	−1.00961e6	−0.365846
$378$	300348.	0.108117
$379$	5.00808e6	1.79091	0.895454	−	0.445155i	$-0.146851\pi$
$379$	5.00808e6	1.79091	0.895454	+	0.445155i	$0.146851\pi$
$380$	965600.	0.343035
$381$	−2.03837e6	−0.719402
$382$	−3.89949e6	−1.36725
$383$	212546.	0.0740382	0.0370191	−	0.999315i	$-0.488214\pi$
$383$	212546.	0.0740382	0.0370191	+	0.999315i	$0.488214\pi$
$384$	147456.	0.0510310
$385$	1.65572e6	0.569294
$386$	−737460.	−0.251924
$387$	−1.06304e6	−0.360806
$388$	1.97493e6	0.665997
$389$	3.88700e6	1.30239	0.651194	−	0.758912i	$-0.274269\pi$
$389$	3.88700e6	1.30239	0.651194	+	0.758912i	$0.274269\pi$
$390$	152100.	0.0506370
$391$	2.31562e6	0.765992
$392$	−396672.	−0.130382
$393$	88326.0	0.0288474
$394$	−771280.	−0.250306
$395$	1.85157e6	0.597102
$396$	−833328.	−0.267041
$397$	1.64040e6	0.522365	0.261183	−	0.965289i	$-0.415888\pi$
$397$	1.64040e6	0.522365	0.261183	+	0.965289i	$0.415888\pi$
$398$	2.25854e6	0.714695
$399$	−2.23778e6	−0.703695
$400$	160000.	0.0500000
$401$	−150198.	−0.0466448	−0.0233224	−	0.999728i	$-0.507424\pi$
$401$	−150198.	−0.0466448	−0.0233224	+	0.999728i	$0.507424\pi$
$402$	−532944.	−0.164481
$403$	1.50106e6	0.460400
$404$	−1.41018e6	−0.429853
$405$	−164025.	−0.0496904
$406$	2.46129e6	0.741050
$407$	−5.61918e6	−1.68146
$408$	−764352.	−0.227323
$409$	−2.57857e6	−0.762202	−0.381101	−	0.924533i	$-0.624455\pi$
$409$	−2.57857e6	−0.762202	−0.381101	+	0.924533i	$0.624455\pi$
$410$	−1.19090e6	−0.349877
$411$	110826.	0.0323621
$412$	−790976.	−0.229573
$413$	−98880.0	−0.0285255
$414$	−565380.	−0.162121
$415$	−2.45370e6	−0.699361
$416$	−173056.	−0.0490290
$417$	226251.	0.0637163
$418$	6.20881e6	1.73807
$419$	−4.32385e6	−1.20319	−0.601597	−	0.798799i	$-0.705469\pi$
$419$	−4.32385e6	−1.20319	−0.601597	+	0.798799i	$0.705469\pi$
$420$	−370800.	−0.102569
$421$	−2.32436e6	−0.639142	−0.319571	−	0.947562i	$-0.603539\pi$
$421$	−2.32436e6	−0.639142	−0.319571	+	0.947562i	$0.603539\pi$
$422$	220784.	0.0603513
$423$	492318.	0.133781
$424$	1.05811e6	0.285836
$425$	−829375.	−0.222730
$426$	−2.85692e6	−0.762737
$427$	−5.06132e6	−1.34336
$428$	95024.0	0.0250740
$429$	978003.	0.256565
$430$	1.31240e6	0.342291
$431$	5.49690e6	1.42536	0.712680	−	0.701490i	$-0.247481\pi$
$431$	5.49690e6	1.42536	0.712680	+	0.701490i	$0.247481\pi$
$432$	186624.	0.0481125
$433$	−6.12771e6	−1.57065	−0.785324	−	0.619086i	$-0.787503\pi$
$433$	−6.12771e6	−1.57065	−0.785324	+	0.619086i	$0.787503\pi$
$434$	−3.65938e6	−0.932575
$435$	−1.34415e6	−0.340584
$436$	−669248.	−0.168605
$437$	4.21243e6	1.05519
$438$	−1.57097e6	−0.391275
$439$	−6.96121e6	−1.72395	−0.861973	−	0.506954i	$-0.830771\pi$
$439$	−6.96121e6	−1.72395	−0.861973	+	0.506954i	$0.830771\pi$
$440$	1.02880e6	0.253337
$441$	−502038.	−0.122925
$442$	897052.	0.218405
$443$	−4.29435e6	−1.03965	−0.519826	−	0.854272i	$-0.674003\pi$
$443$	−4.29435e6	−1.03965	−0.519826	+	0.854272i	$0.674003\pi$
$444$	1.25842e6	0.302947
$445$	−2.89878e6	−0.693928
$446$	5.61690e6	1.33709
$447$	−1.96608e6	−0.465406
$448$	421888.	0.0993121
$449$	3.82985e6	0.896531	0.448266	−	0.893900i	$-0.352042\pi$
$449$	3.82985e6	0.896531	0.448266	+	0.893900i	$0.352042\pi$
$450$	202500.	0.0471405
$451$	−7.65749e6	−1.77274
$452$	981664.	0.226004
$453$	2.40070e6	0.549657
$454$	1.55551e6	0.354188
$455$	435175.	0.0985452
$456$	−1.39046e6	−0.313147
$457$	−6.84847e6	−1.53392	−0.766960	−	0.641695i	$-0.778232\pi$
$457$	−6.84847e6	−1.53392	−0.766960	+	0.641695i	$0.778232\pi$
$458$	−931448.	−0.207489
$459$	−967383.	−0.214322
$460$	698000.	0.153802
$461$	3.84575e6	0.842807	0.421404	−	0.906873i	$-0.361538\pi$
$461$	3.84575e6	0.842807	0.421404	+	0.906873i	$0.361538\pi$
$462$	−2.38424e6	−0.519692
$463$	6.47496e6	1.40373	0.701867	−	0.712308i	$-0.252350\pi$
$463$	6.47496e6	1.40373	0.701867	+	0.712308i	$0.252350\pi$
$464$	1.52934e6	0.329769
$465$	1.99845e6	0.428609
$466$	2.31048e6	0.492875
$467$	−3.42249e6	−0.726190	−0.363095	−	0.931752i	$-0.618280\pi$
$467$	−3.42249e6	−0.726190	−0.363095	+	0.931752i	$0.618280\pi$
$468$	−219024.	−0.0462250
$469$	−1.52481e6	−0.320099
$470$	−607800.	−0.126916
$471$	2.13021e6	0.442456
$472$	−61440.0	−0.0126939
$473$	8.43873e6	1.73430
$474$	−2.66627e6	−0.545077
$475$	−1.50875e6	−0.306820
$476$	−2.18690e6	−0.442396
$477$	1.33917e6	0.269489
$478$	−3.47715e6	−0.696071
$479$	9.52529e6	1.89688	0.948439	−	0.316959i	$-0.102662\pi$
$479$	9.52529e6	1.89688	0.948439	+	0.316959i	$0.102662\pi$
$480$	−230400.	−0.0456435
$481$	−1.47689e6	−0.291062
$482$	−750712.	−0.147182
$483$	−1.61762e6	−0.315506
$484$	4.03837e6	0.783597
$485$	−3.08582e6	−0.595685
$486$	236196.	0.0453609
$487$	−3.54206e6	−0.676758	−0.338379	−	0.941010i	$-0.609879\pi$
$487$	−3.54206e6	−0.676758	−0.338379	+	0.941010i	$0.609879\pi$
$488$	−3.14490e6	−0.597801
$489$	−1.70583e6	−0.322600
$490$	619800.	0.116617
$491$	−3.01713e6	−0.564795	−0.282397	−	0.959298i	$-0.591130\pi$
$491$	−3.01713e6	−0.564795	−0.282397	+	0.959298i	$0.591130\pi$
$492$	1.71490e6	0.319393
$493$	−7.92750e6	−1.46899
$494$	1.63186e6	0.300861
$495$	1.30208e6	0.238849
$496$	−2.27379e6	−0.414999
$497$	−8.17398e6	−1.48437
$498$	3.53333e6	0.638426
$499$	−9.82741e6	−1.76680	−0.883401	−	0.468618i	$-0.844752\pi$
$499$	−9.82741e6	−1.76680	−0.883401	+	0.468618i	$0.844752\pi$
$500$	−250000.	−0.0447214
$501$	5.36155e6	0.954325
$502$	7.01043e6	1.24161
$503$	−5.37944e6	−0.948020	−0.474010	−	0.880520i	$-0.657194\pi$
$503$	−5.37944e6	−0.948020	−0.474010	+	0.880520i	$0.657194\pi$
$504$	533952.	0.0936323
$505$	2.20340e6	0.384472
$506$	4.48814e6	0.779274
$507$	257049.	0.0444116
$508$	−3.62378e6	−0.623020
$509$	−9.06666e6	−1.55115	−0.775574	−	0.631257i	$-0.782539\pi$
$509$	−9.06666e6	−1.55115	−0.775574	+	0.631257i	$0.782539\pi$
$510$	1.19430e6	0.203324
$511$	−4.49471e6	−0.761465
$512$	262144.	0.0441942
$513$	−1.75981e6	−0.295237
$514$	−6.10761e6	−1.01968
$515$	1.23590e6	0.205336
$516$	−1.88986e6	−0.312467
$517$	−3.90815e6	−0.643051
$518$	3.60047e6	0.589569
$519$	−1.18975e6	−0.193881
$520$	270400.	0.0438529
$521$	4.08543e6	0.659392	0.329696	−	0.944087i	$-0.393054\pi$
$521$	4.08543e6	0.659392	0.329696	+	0.944087i	$0.393054\pi$
$522$	1.93558e6	0.310909
$523$	3.69388e6	0.590512	0.295256	−	0.955418i	$-0.404595\pi$
$523$	3.69388e6	0.590512	0.295256	+	0.955418i	$0.404595\pi$
$524$	157024.	0.0249826
$525$	579375.	0.0917406
$526$	7.51891e6	1.18492
$527$	1.17864e7	1.84865
$528$	−1.48147e6	−0.231264
$529$	−3.39132e6	−0.526901
$530$	−1.65330e6	−0.255660
$531$	−77760.0	−0.0119680
$532$	−3.97827e6	−0.609418
$533$	−2.01262e6	−0.306863
$534$	4.17424e6	0.633467
$535$	−148475.	−0.0224269
$536$	−947456.	−0.142445
$537$	−4.85008e6	−0.725794
$538$	−103520.	−0.0154194
$539$	3.98531e6	0.590868
$540$	−291600.	−0.0430331
$541$	2.13682e6	0.313888	0.156944	−	0.987607i	$-0.449836\pi$
$541$	2.13682e6	0.313888	0.156944	+	0.987607i	$0.449836\pi$
$542$	−6.36854e6	−0.931197
$543$	21321.0	0.00310319
$544$	−1.35885e6	−0.196867
$545$	1.04570e6	0.150805
$546$	−626652.	−0.0899590
$547$	−8.61030e6	−1.23041	−0.615205	−	0.788367i	$-0.710927\pi$
$547$	−8.61030e6	−1.23041	−0.615205	+	0.788367i	$0.710927\pi$
$548$	197024.	0.0280264
$549$	−3.98026e6	−0.563612
$550$	−1.60750e6	−0.226592
$551$	−1.44212e7	−2.02359
$552$	−1.00512e6	−0.140401
$553$	−7.62849e6	−1.06078
$554$	2.27313e6	0.314666
$555$	−1.96628e6	−0.270964
$556$	402224.	0.0551799
$557$	−3.08025e6	−0.420676	−0.210338	−	0.977629i	$-0.567456\pi$
$557$	−3.08025e6	−0.420676	−0.210338	+	0.977629i	$0.567456\pi$
$558$	−2.87777e6	−0.391264
$559$	2.21796e6	0.300209
$560$	−659200.	−0.0888274
$561$	7.67935e6	1.03019
$562$	2.08977e6	0.279098
$563$	1.13496e7	1.50908	0.754538	−	0.656256i	$-0.227861\pi$
$563$	1.13496e7	1.50908	0.754538	+	0.656256i	$0.227861\pi$
$564$	875232.	0.115858
$565$	−1.53385e6	−0.202144
$566$	9.32066e6	1.22294
$567$	675783.	0.0882774
$568$	−5.07898e6	−0.660549
$569$	−1.23908e7	−1.60443	−0.802213	−	0.597038i	$-0.796344\pi$
$569$	−1.23908e7	−1.60443	−0.802213	+	0.597038i	$0.796344\pi$
$570$	2.17260e6	0.280087
$571$	2.15172e6	0.276183	0.138091	−	0.990419i	$-0.455903\pi$
$571$	2.15172e6	0.276183	0.138091	+	0.990419i	$0.455903\pi$
$572$	1.73867e6	0.222192
$573$	−8.77385e6	−1.11636
$574$	4.90651e6	0.621574
$575$	−1.09062e6	−0.137564
$576$	331776.	0.0416667
$577$	−2.18008e6	−0.272605	−0.136302	−	0.990667i	$-0.543522\pi$
$577$	−2.18008e6	−0.272605	−0.136302	+	0.990667i	$0.543522\pi$
$578$	1.36429e6	0.169858
$579$	−1.65928e6	−0.205695
$580$	−2.38960e6	−0.294955
$581$	1.01092e7	1.24245
$582$	4.44359e6	0.543784
$583$	−1.06307e7	−1.29536
$584$	−2.79283e6	−0.338854
$585$	342225.	0.0413449
$586$	3.25954e6	0.392113
$587$	−7.82323e6	−0.937110	−0.468555	−	0.883434i	$-0.655225\pi$
$587$	−7.82323e6	−0.937110	−0.468555	+	0.883434i	$0.655225\pi$
$588$	−892512.	−0.106456
$589$	2.14411e7	2.54659
$590$	96000.0	0.0113538
$591$	−1.73538e6	−0.204374
$592$	2.23718e6	0.262360
$593$	−1.38189e7	−1.61375	−0.806876	−	0.590721i	$-0.798844\pi$
$593$	−1.38189e7	−1.61375	−0.806876	+	0.590721i	$0.798844\pi$
$594$	−1.87499e6	−0.218038
$595$	3.41703e6	0.395691
$596$	−3.49525e6	−0.403053
$597$	5.08172e6	0.583546
$598$	1.17962e6	0.134893
$599$	−3.55571e6	−0.404910	−0.202455	−	0.979292i	$-0.564892\pi$
$599$	−3.55571e6	−0.404910	−0.202455	+	0.979292i	$0.564892\pi$
$600$	360000.	0.0408248
$601$	1.36329e7	1.53958	0.769792	−	0.638295i	$-0.220360\pi$
$601$	1.36329e7	1.53958	0.769792	+	0.638295i	$0.220360\pi$
$602$	−5.40709e6	−0.608096
$603$	−1.19912e6	−0.134298
$604$	4.26790e6	0.476017
$605$	−6.30995e6	−0.700870
$606$	−3.17290e6	−0.350973
$607$	−1.71241e7	−1.88641	−0.943205	−	0.332210i	$-0.892206\pi$
$607$	−1.71241e7	−1.88641	−0.943205	+	0.332210i	$0.892206\pi$
$608$	−2.47194e6	−0.271193
$609$	5.53790e6	0.605065
$610$	4.91390e6	0.534690
$611$	−1.02718e6	−0.111313
$612$	−1.71979e6	−0.185608
$613$	−1.47558e7	−1.58603	−0.793013	−	0.609204i	$-0.791489\pi$
$613$	−1.47558e7	−1.58603	−0.793013	+	0.609204i	$0.791489\pi$
$614$	−7.33511e6	−0.785210
$615$	−2.67953e6	−0.285674
$616$	−4.23866e6	−0.450066
$617$	−1.23039e7	−1.30116	−0.650580	−	0.759438i	$-0.725474\pi$
$617$	−1.23039e7	−1.30116	−0.650580	+	0.759438i	$0.725474\pi$
$618$	−1.77970e6	−0.187445
$619$	1.07302e7	1.12560	0.562799	−	0.826594i	$-0.309725\pi$
$619$	1.07302e7	1.12560	0.562799	+	0.826594i	$0.309725\pi$
$620$	3.55280e6	0.371186
$621$	−1.27210e6	−0.132371
$622$	543520.	0.0563300
$623$	1.19430e7	1.23280
$624$	−389376.	−0.0400320
$625$	390625.	0.0400000
$626$	−4.96300e6	−0.506184
$627$	1.39698e7	1.41913
$628$	3.78704e6	0.383178
$629$	−1.15967e7	−1.16871
$630$	−834300.	−0.0837473
$631$	1.52237e7	1.52212	0.761058	−	0.648684i	$-0.224680\pi$
$631$	1.52237e7	1.52212	0.761058	+	0.648684i	$0.224680\pi$
$632$	−4.74003e6	−0.472051
$633$	496764.	0.0492766
$634$	1.03068e7	1.01836
$635$	5.66215e6	0.557246
$636$	2.38075e6	0.233384
$637$	1.04746e6	0.102280
$638$	−1.53651e7	−1.49446
$639$	−6.42808e6	−0.622772
$640$	−409600.	−0.0395285
$641$	3.82087e6	0.367297	0.183649	−	0.982992i	$-0.441209\pi$
$641$	3.82087e6	0.367297	0.183649	+	0.982992i	$0.441209\pi$
$642$	213804.	0.0204728
$643$	−1.24847e7	−1.19083	−0.595417	−	0.803417i	$-0.703013\pi$
$643$	−1.24847e7	−1.19083	−0.595417	+	0.803417i	$0.703013\pi$
$644$	−2.87576e6	−0.273236
$645$	2.95290e6	0.279479
$646$	1.28135e7	1.20806
$647$	−8.74421e6	−0.821221	−0.410610	−	0.911811i	$-0.634684\pi$
$647$	−8.74421e6	−0.821221	−0.410610	+	0.911811i	$0.634684\pi$
$648$	419904.	0.0392837
$649$	617280.	0.0575268
$650$	−422500.	−0.0392232
$651$	−8.23361e6	−0.761444
$652$	−3.03259e6	−0.279380
$653$	5.25331e6	0.482115	0.241057	−	0.970511i	$-0.422506\pi$
$653$	5.25331e6	0.482115	0.241057	+	0.970511i	$0.422506\pi$
$654$	−1.50581e6	−0.137666
$655$	−245350.	−0.0223451
$656$	3.04870e6	0.276602
$657$	−3.53468e6	−0.319475
$658$	2.50414e6	0.225472
$659$	−1.35720e6	−0.121739	−0.0608696	−	0.998146i	$-0.519387\pi$
$659$	−1.35720e6	−0.121739	−0.0608696	+	0.998146i	$0.519387\pi$
$660$	2.31480e6	0.206849
$661$	1.30861e7	1.16495	0.582476	−	0.812848i	$-0.302084\pi$
$661$	1.30861e7	1.16495	0.582476	+	0.812848i	$0.302084\pi$
$662$	9.09867e6	0.806925
$663$	2.01837e6	0.178327
$664$	6.28147e6	0.552893
$665$	6.21605e6	0.545080
$666$	2.83144e6	0.247355
$667$	−1.04246e7	−0.907290
$668$	9.53165e6	0.826470
$669$	1.26380e7	1.09173
$670$	1.48040e6	0.127407
$671$	3.15964e7	2.70914
$672$	949248.	0.0810880
$673$	7.47801e6	0.636426	0.318213	−	0.948019i	$-0.396917\pi$
$673$	7.47801e6	0.636426	0.318213	+	0.948019i	$0.396917\pi$
$674$	−4.60760e6	−0.390683
$675$	455625.	0.0384900
$676$	456976.	0.0384615
$677$	3.34737e6	0.280693	0.140346	−	0.990102i	$-0.455178\pi$
$677$	3.34737e6	0.280693	0.140346	+	0.990102i	$0.455178\pi$
$678$	2.20874e6	0.184532
$679$	1.27136e7	1.05826
$680$	2.12320e6	0.176083
$681$	3.49990e6	0.289193
$682$	2.28445e7	1.88071
$683$	−1.30723e7	−1.07226	−0.536131	−	0.844134i	$-0.680115\pi$
$683$	−1.30723e7	−1.07226	−0.536131	+	0.844134i	$0.680115\pi$
$684$	−3.12854e6	−0.255683
$685$	−307850.	−0.0250676
$686$	−9.47806e6	−0.768970
$687$	−2.09576e6	−0.169414
$688$	−3.35974e6	−0.270605
$689$	−2.79408e6	−0.224228
$690$	1.57050e6	0.125578
$691$	−9.11168e6	−0.725944	−0.362972	−	0.931800i	$-0.618238\pi$
$691$	−9.11168e6	−0.725944	−0.362972	+	0.931800i	$0.618238\pi$
$692$	−2.11510e6	−0.167906
$693$	−5.36455e6	−0.424326
$694$	−6.22599e6	−0.490693
$695$	−628475.	−0.0493544
$696$	3.44102e6	0.269256
$697$	−1.58032e7	−1.23215
$698$	−6.19134e6	−0.481001
$699$	5.19857e6	0.402431
$700$	1.03000e6	0.0794497
$701$	−1.16057e7	−0.892024	−0.446012	−	0.895027i	$-0.647156\pi$
$701$	−1.16057e7	−0.892024	−0.446012	+	0.895027i	$0.647156\pi$
$702$	−492804.	−0.0377426
$703$	−2.10959e7	−1.60994
$704$	−2.63373e6	−0.200281
$705$	−1.36755e6	−0.103626
$706$	5.01094e6	0.378362
$707$	−9.07801e6	−0.683034
$708$	−138240.	−0.0103646
$709$	3.88752e6	0.290441	0.145220	−	0.989399i	$-0.453611\pi$
$709$	3.88752e6	0.290441	0.145220	+	0.989399i	$0.453611\pi$
$710$	7.93590e6	0.590813
$711$	−5.99910e6	−0.445054
$712$	7.42086e6	0.548598
$713$	1.54991e7	1.14178
$714$	−4.92052e6	−0.361214
$715$	−2.71668e6	−0.198734
$716$	−8.62237e6	−0.628556
$717$	−7.82358e6	−0.568339
$718$	−1.42394e7	−1.03081
$719$	−1.14559e7	−0.826435	−0.413217	−	0.910632i	$-0.635595\pi$
$719$	−1.14559e7	−0.826435	−0.413217	+	0.910632i	$0.635595\pi$
$720$	−518400.	−0.0372678
$721$	−5.09191e6	−0.364790
$722$	1.34052e7	0.957040
$723$	−1.68910e6	−0.120174
$724$	37904.0	0.00268744
$725$	3.73375e6	0.263815
$726$	9.08633e6	0.639804
$727$	−1.17269e7	−0.822901	−0.411451	−	0.911432i	$-0.634978\pi$
$727$	−1.17269e7	−0.822901	−0.411451	+	0.911432i	$0.634978\pi$
$728$	−1.11405e6	−0.0779068
$729$	531441.	0.0370370
$730$	4.36380e6	0.303080
$731$	1.74155e7	1.20543
$732$	−7.07602e6	−0.488103
$733$	1.14752e7	0.788863	0.394431	−	0.918925i	$-0.370942\pi$
$733$	1.14752e7	0.788863	0.394431	+	0.918925i	$0.370942\pi$
$734$	−4.63120e6	−0.317288
$735$	1.39455e6	0.0952173
$736$	−1.78688e6	−0.121591
$737$	9.51897e6	0.645537
$738$	3.85852e6	0.260783
$739$	1.63126e6	0.109878	0.0549391	−	0.998490i	$-0.482504\pi$
$739$	1.63126e6	0.109878	0.0549391	+	0.998490i	$0.482504\pi$
$740$	−3.49560e6	−0.234662
$741$	3.67169e6	0.245652
$742$	6.81160e6	0.454192
$743$	1.24501e7	0.827369	0.413684	−	0.910420i	$-0.364242\pi$
$743$	1.24501e7	0.827369	0.413684	+	0.910420i	$0.364242\pi$
$744$	−5.11603e6	−0.338845
$745$	5.46132e6	0.360502
$746$	1.11412e7	0.732966
$747$	7.94999e6	0.521273
$748$	1.36522e7	0.892171
$749$	611717.	0.0398424
$750$	−562500.	−0.0365148
$751$	−9.08418e6	−0.587741	−0.293871	−	0.955845i	$-0.594943\pi$
$751$	−9.08418e6	−0.587741	−0.293871	+	0.955845i	$0.594943\pi$
$752$	1.55597e6	0.100336
$753$	1.57735e7	1.01377
$754$	−4.03842e6	−0.258692
$755$	−6.66860e6	−0.425762
$756$	1.20139e6	0.0764505
$757$	2.83339e7	1.79708	0.898538	−	0.438896i	$-0.144630\pi$
$757$	2.83339e7	1.79708	0.898538	+	0.438896i	$0.144630\pi$
$758$	2.00323e7	1.26636
$759$	1.00983e7	0.636274
$760$	3.86240e6	0.242562
$761$	3.67620e6	0.230111	0.115055	−	0.993359i	$-0.463295\pi$
$761$	3.67620e6	0.230111	0.115055	+	0.993359i	$0.463295\pi$
$762$	−8.15350e6	−0.508694
$763$	−4.30828e6	−0.267913
$764$	−1.55980e7	−0.966795
$765$	2.68718e6	0.166013
$766$	850184.	0.0523529
$767$	162240.	0.00995794
$768$	589824.	0.0360844
$769$	5.27327e6	0.321562	0.160781	−	0.986990i	$-0.448599\pi$
$769$	5.27327e6	0.321562	0.160781	+	0.986990i	$0.448599\pi$
$770$	6.62290e6	0.402551
$771$	−1.37421e7	−0.832564
$772$	−2.94984e6	−0.178137
$773$	3.05161e7	1.83688	0.918440	−	0.395559i	$-0.129449\pi$
$773$	3.05161e7	1.83688	0.918440	+	0.395559i	$0.129449\pi$
$774$	−4.25218e6	−0.255128
$775$	−5.55125e6	−0.331999
$776$	7.89971e6	0.470931
$777$	8.10105e6	0.481381
$778$	1.55480e7	0.920927
$779$	−2.87483e7	−1.69734
$780$	608400.	0.0358057
$781$	5.10278e7	2.99350
$782$	9.26246e6	0.541638
$783$	4.35505e6	0.253857
$784$	−1.58669e6	−0.0921937
$785$	−5.91725e6	−0.342725
$786$	353304.	0.0203982
$787$	1.00521e7	0.578523	0.289261	−	0.957250i	$-0.406590\pi$
$787$	1.00521e7	0.578523	0.289261	+	0.957250i	$0.406590\pi$
$788$	−3.08512e6	−0.176993
$789$	1.69176e7	0.967487
$790$	7.40630e6	0.422215
$791$	6.31946e6	0.359119
$792$	−3.33331e6	−0.188827
$793$	8.30449e6	0.468954
$794$	6.56161e6	0.369368
$795$	−3.71992e6	−0.208745
$796$	9.03418e6	0.505366
$797$	−1.29949e7	−0.724650	−0.362325	−	0.932052i	$-0.618017\pi$
$797$	−1.29949e7	−0.724650	−0.362325	+	0.932052i	$0.618017\pi$
$798$	−8.95111e6	−0.497588
$799$	−8.06551e6	−0.446956
$800$	640000.	0.0353553
$801$	9.39203e6	0.517223
$802$	−600792.	−0.0329829
$803$	2.80592e7	1.53563
$804$	−2.13178e6	−0.116306
$805$	4.49337e6	0.244390
$806$	6.00423e6	0.325552
$807$	−232920.	−0.0125899
$808$	−5.64070e6	−0.303952
$809$	9.02984e6	0.485075	0.242537	−	0.970142i	$-0.422020\pi$
$809$	9.02984e6	0.485075	0.242537	+	0.970142i	$0.422020\pi$
$810$	−656100.	−0.0351364
$811$	1.50515e7	0.803576	0.401788	−	0.915733i	$-0.368389\pi$
$811$	1.50515e7	0.803576	0.401788	+	0.915733i	$0.368389\pi$
$812$	9.84515e6	0.524001
$813$	−1.43292e7	−0.760319
$814$	−2.24767e7	−1.18897
$815$	4.73842e6	0.249885
$816$	−3.05741e6	−0.160741
$817$	3.16813e7	1.66054
$818$	−1.03143e7	−0.538958
$819$	−1.40997e6	−0.0734513
$820$	−4.76360e6	−0.247401
$821$	−2.62190e7	−1.35756	−0.678778	−	0.734344i	$-0.737490\pi$
$821$	−2.62190e7	−1.35756	−0.678778	+	0.734344i	$0.737490\pi$
$822$	443304.	0.0228835
$823$	−3.76829e7	−1.93930	−0.969650	−	0.244497i	$-0.921377\pi$
$823$	−3.76829e7	−1.93930	−0.969650	+	0.244497i	$0.921377\pi$
$824$	−3.16390e6	−0.162332
$825$	−3.61688e6	−0.185011
$826$	−395520.	−0.0201706
$827$	−2.12321e7	−1.07952	−0.539758	−	0.841821i	$-0.681484\pi$
$827$	−2.12321e7	−1.07952	−0.539758	+	0.841821i	$0.681484\pi$
$828$	−2.26152e6	−0.114637
$829$	2.17674e7	1.10007	0.550036	−	0.835141i	$-0.314614\pi$
$829$	2.17674e7	1.10007	0.550036	+	0.835141i	$0.314614\pi$
$830$	−9.81480e6	−0.494523
$831$	5.11454e6	0.256924
$832$	−692224.	−0.0346688
$833$	8.22475e6	0.410686
$834$	905004.	0.0450542
$835$	−1.48932e7	−0.739217
$836$	2.48352e7	1.22900
$837$	−6.47498e6	−0.319466
$838$	−1.72954e7	−0.850787
$839$	5.00782e6	0.245609	0.122804	−	0.992431i	$-0.460811\pi$
$839$	5.00782e6	0.245609	0.122804	+	0.992431i	$0.460811\pi$
$840$	−1.48320e6	−0.0725273
$841$	1.51775e7	0.739965
$842$	−9.29742e6	−0.451942
$843$	4.70198e6	0.227883
$844$	883136.	0.0426748
$845$	−714025.	−0.0344010
$846$	1.96927e6	0.0945975
$847$	2.59970e7	1.24513
$848$	4.23245e6	0.202117
$849$	2.09715e7	0.998527
$850$	−3.31750e6	−0.157494
$851$	−1.52496e7	−0.721827
$852$	−1.14277e7	−0.539336
$853$	1.42963e7	0.672744	0.336372	−	0.941729i	$-0.390800\pi$
$853$	1.42963e7	0.672744	0.336372	+	0.941729i	$0.390800\pi$
$854$	−2.02453e7	−0.949902
$855$	4.88835e6	0.228690
$856$	380096.	0.0177300
$857$	3.53685e6	0.164500	0.0822498	−	0.996612i	$-0.473789\pi$
$857$	3.53685e6	0.164500	0.0822498	+	0.996612i	$0.473789\pi$
$858$	3.91201e6	0.181419
$859$	1.99184e7	0.921024	0.460512	−	0.887653i	$-0.347666\pi$
$859$	1.99184e7	0.921024	0.460512	+	0.887653i	$0.347666\pi$
$860$	5.24960e6	0.242036
$861$	1.10396e7	0.507513
$862$	2.19876e7	1.00788
$863$	−3.04763e7	−1.39295	−0.696476	−	0.717580i	$-0.745250\pi$
$863$	−3.04763e7	−1.39295	−0.696476	+	0.717580i	$0.745250\pi$
$864$	746496.	0.0340207
$865$	3.30485e6	0.150180
$866$	−2.45108e7	−1.11062
$867$	3.06965e6	0.138689
$868$	−1.46375e7	−0.659430
$869$	4.76225e7	2.13926
$870$	−5.37660e6	−0.240829
$871$	2.50188e6	0.111743
$872$	−2.67699e6	−0.119222
$873$	9.99807e6	0.443998
$874$	1.68497e7	0.746129
$875$	−1.60938e6	−0.0710619
$876$	−6.28387e6	−0.276673
$877$	−1.21491e7	−0.533389	−0.266694	−	0.963781i	$-0.585931\pi$
$877$	−1.21491e7	−0.533389	−0.266694	+	0.963781i	$0.585931\pi$
$878$	−2.78449e7	−1.21901
$879$	7.33396e6	0.320159
$880$	4.11520e6	0.179137
$881$	−2.72337e7	−1.18213	−0.591067	−	0.806623i	$-0.701293\pi$
$881$	−2.72337e7	−1.18213	−0.591067	+	0.806623i	$0.701293\pi$
$882$	−2.00815e6	−0.0869211
$883$	2.68280e7	1.15794	0.578971	−	0.815348i	$-0.303454\pi$
$883$	2.68280e7	1.15794	0.578971	+	0.815348i	$0.303454\pi$
$884$	3.58821e6	0.154435
$885$	216000.	0.00927034
$886$	−1.71774e7	−0.735145
$887$	2.10785e7	0.899560	0.449780	−	0.893139i	$-0.351502\pi$
$887$	2.10785e7	0.899560	0.449780	+	0.893139i	$0.351502\pi$
$888$	5.03366e6	0.214216
$889$	−2.33281e7	−0.989975
$890$	−1.15951e7	−0.490681
$891$	−4.21872e6	−0.178027
$892$	2.24676e7	0.945462
$893$	−1.46723e7	−0.615700
$894$	−7.86431e6	−0.329092
$895$	1.34724e7	0.562198
$896$	1.68755e6	0.0702243
$897$	2.65414e6	0.110140
$898$	1.53194e7	0.633943
$899$	−5.30611e7	−2.18966
$900$	810000.	0.0333333
$901$	−2.19393e7	−0.900348
$902$	−3.06299e7	−1.25352
$903$	−1.21659e7	−0.496508
$904$	3.92666e6	0.159809
$905$	−59225.0	−0.00240372
$906$	9.60278e6	0.388666
$907$	−2.55173e7	−1.02995	−0.514975	−	0.857205i	$-0.672199\pi$
$907$	−2.55173e7	−1.02995	−0.514975	+	0.857205i	$0.672199\pi$
$908$	6.22205e6	0.250449
$909$	−7.13902e6	−0.286569
$910$	1.74070e6	0.0696820
$911$	5.70826e6	0.227881	0.113940	−	0.993488i	$-0.463653\pi$
$911$	5.70826e6	0.227881	0.113940	+	0.993488i	$0.463653\pi$
$912$	−5.56186e6	−0.221428
$913$	−6.31092e7	−2.50562
$914$	−2.73939e7	−1.08465
$915$	1.10563e7	0.436572
$916$	−3.72579e6	−0.146717
$917$	1.01084e6	0.0396972
$918$	−3.86953e6	−0.151549
$919$	1.47381e7	0.575641	0.287820	−	0.957684i	$-0.407069\pi$
$919$	1.47381e7	0.575641	0.287820	+	0.957684i	$0.407069\pi$
$920$	2.79200e6	0.108754
$921$	−1.65040e7	−0.641121
$922$	1.53830e7	0.595955
$923$	1.34117e7	0.518178
$924$	−9.53698e6	−0.367478
$925$	5.46188e6	0.209888
$926$	2.58998e7	0.992589
$927$	−4.00432e6	−0.153048
$928$	6.11738e6	0.233182
$929$	−3.28710e6	−0.124961	−0.0624805	−	0.998046i	$-0.519901\pi$
$929$	−3.28710e6	−0.124961	−0.0624805	+	0.998046i	$0.519901\pi$
$930$	7.99380e6	0.303072
$931$	1.49620e7	0.565737
$932$	9.24190e6	0.348515
$933$	1.22292e6	0.0459932
$934$	−1.36900e7	−0.513494
$935$	−2.13315e7	−0.797982
$936$	−876096.	−0.0326860
$937$	−4.73883e6	−0.176328	−0.0881641	−	0.996106i	$-0.528100\pi$
$937$	−4.73883e6	−0.176328	−0.0881641	+	0.996106i	$0.528100\pi$
$938$	−6.09925e6	−0.226344
$939$	−1.11668e7	−0.413298
$940$	−2.43120e6	−0.0897431
$941$	−1.39635e7	−0.514069	−0.257035	−	0.966402i	$-0.582745\pi$
$941$	−1.39635e7	−0.514069	−0.257035	+	0.966402i	$0.582745\pi$
$942$	8.52084e6	0.312864
$943$	−2.07812e7	−0.761012
$944$	−245760.	−0.00897597
$945$	−1.87717e6	−0.0683794
$946$	3.37549e7	1.22634
$947$	−1.03912e7	−0.376522	−0.188261	−	0.982119i	$-0.560285\pi$
$947$	−1.03912e7	−0.376522	−0.188261	+	0.982119i	$0.560285\pi$
$948$	−1.06651e7	−0.385428
$949$	7.37482e6	0.265819
$950$	−6.03500e6	−0.216954
$951$	2.31904e7	0.831489
$952$	−8.74758e6	−0.312821
$953$	−5.32754e7	−1.90018	−0.950090	−	0.311976i	$-0.899009\pi$
$953$	−5.32754e7	−1.90018	−0.950090	+	0.311976i	$0.899009\pi$
$954$	5.35669e6	0.190557
$955$	2.43718e7	0.864727
$956$	−1.39086e7	−0.492196
$957$	−3.45715e7	−1.22022
$958$	3.81012e7	1.34130
$959$	1.26834e6	0.0445338
$960$	−921600.	−0.0322749
$961$	5.02608e7	1.75558
$962$	−5.90756e6	−0.205812
$963$	481059.	0.0167160
$964$	−3.00285e6	−0.104074
$965$	4.60912e6	0.159331
$966$	−6.47046e6	−0.223096
$967$	−2.90829e7	−1.00017	−0.500083	−	0.865978i	$-0.666697\pi$
$967$	−2.90829e7	−1.00017	−0.500083	+	0.865978i	$0.666697\pi$
$968$	1.61535e7	0.554086
$969$	2.88304e7	0.986373
$970$	−1.23433e7	−0.421213
$971$	2.23811e7	0.761786	0.380893	−	0.924619i	$-0.375617\pi$
$971$	2.23811e7	0.761786	0.380893	+	0.924619i	$0.375617\pi$
$972$	944784.	0.0320750
$973$	2.58932e6	0.0876805
$974$	−1.41682e7	−0.478540
$975$	−950625.	−0.0320256
$976$	−1.25796e7	−0.422709
$977$	−3.96823e6	−0.133003	−0.0665014	−	0.997786i	$-0.521184\pi$
$977$	−3.96823e6	−0.133003	−0.0665014	+	0.997786i	$0.521184\pi$
$978$	−6.82333e6	−0.228113
$979$	−7.45565e7	−2.48616
$980$	2.47920e6	0.0824606
$981$	−3.38807e6	−0.112403
$982$	−1.20685e7	−0.399370
$983$	−3.04408e7	−1.00478	−0.502391	−	0.864640i	$-0.667546\pi$
$983$	−3.04408e7	−1.00478	−0.502391	+	0.864640i	$0.667546\pi$
$984$	6.85958e6	0.225845
$985$	4.82050e6	0.158308
$986$	−3.17100e7	−1.03873
$987$	5.63431e6	0.184097
$988$	6.52746e6	0.212741
$989$	2.29014e7	0.744511
$990$	5.20830e6	0.168892
$991$	−3.95144e7	−1.27812	−0.639060	−	0.769157i	$-0.720676\pi$
$991$	−3.95144e7	−1.27812	−0.639060	+	0.769157i	$0.720676\pi$
$992$	−9.09517e6	−0.293448
$993$	2.04720e7	0.658851
$994$	−3.26959e7	−1.04961
$995$	−1.41159e7	−0.452013
$996$	1.41333e7	0.451435
$997$	−3.89138e7	−1.23984	−0.619920	−	0.784665i	$-0.712835\pi$
$997$	−3.89138e7	−1.23984	−0.619920	+	0.784665i	$0.712835\pi$
$998$	−3.93097e7	−1.24932
$999$	6.37073e6	0.201965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.6.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.6.a.e.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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