LMFDB - Embedded newform 975.6.a.u.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,6,Mod(1,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$975 = 3 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,2,-99,224,0,-18,-55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$156.374224318$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800$ x^11 - 2x^10 - 286x^9 + 442x^8 + 28715x^7 - 29138x^6 - 1208172x^5 + 509768x^4 + 20144528x^3 - 3093984x^2 - 98068800x + 55036800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{5}\cdot 3\cdot 5^{3}\cdot 11$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$9.28847$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.28847 q^{2} -9.00000 q^{3} +54.2758 q^{4} -83.5963 q^{6} +85.6782 q^{7} +206.908 q^{8} +81.0000 q^{9} +119.113 q^{11} -488.482 q^{12} +169.000 q^{13} +795.820 q^{14} +185.034 q^{16} -763.350 q^{17} +752.366 q^{18} -2433.97 q^{19} -771.104 q^{21} +1106.38 q^{22} -4050.77 q^{23} -1862.17 q^{24} +1569.75 q^{26} -729.000 q^{27} +4650.25 q^{28} -5525.35 q^{29} +9537.22 q^{31} -4902.37 q^{32} -1072.02 q^{33} -7090.35 q^{34} +4396.34 q^{36} +5704.17 q^{37} -22607.9 q^{38} -1521.00 q^{39} -10517.8 q^{41} -7162.38 q^{42} +6708.08 q^{43} +6464.96 q^{44} -37625.5 q^{46} -577.173 q^{47} -1665.31 q^{48} -9466.24 q^{49} +6870.15 q^{51} +9172.60 q^{52} +12025.3 q^{53} -6771.30 q^{54} +17727.5 q^{56} +21905.7 q^{57} -51322.0 q^{58} -39563.2 q^{59} +49752.0 q^{61} +88586.3 q^{62} +6939.94 q^{63} -51456.6 q^{64} -9957.42 q^{66} -20394.4 q^{67} -41431.4 q^{68} +36456.9 q^{69} +29124.8 q^{71} +16759.5 q^{72} -57924.8 q^{73} +52983.0 q^{74} -132106. q^{76} +10205.4 q^{77} -14127.8 q^{78} +88938.2 q^{79} +6561.00 q^{81} -97694.5 q^{82} -68552.2 q^{83} -41852.3 q^{84} +62307.8 q^{86} +49728.1 q^{87} +24645.5 q^{88} -130659. q^{89} +14479.6 q^{91} -219859. q^{92} -85835.0 q^{93} -5361.06 q^{94} +44121.3 q^{96} +84079.6 q^{97} -87927.0 q^{98} +9648.17 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100})$ 11 * q + 2 * q^2 - 99 * q^3 + 224 * q^4 - 18 * q^6 - 55 * q^7 + 270 * q^8 + 891 * q^9 - 125 * q^11 - 2016 * q^12 + 1859 * q^13 - 1311 * q^14 + 5756 * q^16 - 4507 * q^17 + 162 * q^18 + 142 * q^19 + 495 * q^21 - 5975 * q^22 - 4672 * q^23 - 2430 * q^24 + 338 * q^26 - 8019 * q^27 - 4149 * q^28 + 5197 * q^29 - 277 * q^31 + 2272 * q^32 + 1125 * q^33 - 21225 * q^34 + 18144 * q^36 - 1696 * q^37 - 27485 * q^38 - 16731 * q^39 + 2040 * q^41 + 11799 * q^42 - 31436 * q^43 - 39883 * q^44 + 67530 * q^46 + 15499 * q^47 - 51804 * q^48 + 29212 * q^49 + 40563 * q^51 + 37856 * q^52 - 119577 * q^53 - 1458 * q^54 + 1677 * q^56 - 1278 * q^57 - 91154 * q^58 + 17115 * q^59 + 26273 * q^61 - 45211 * q^62 - 4455 * q^63 + 137480 * q^64 + 53775 * q^66 + 26313 * q^67 - 191297 * q^68 + 42048 * q^69 - 105756 * q^71 + 21870 * q^72 - 103230 * q^73 - 6501 * q^74 + 124495 * q^76 - 104415 * q^77 - 3042 * q^78 + 210732 * q^79 + 72171 * q^81 - 60283 * q^82 - 144247 * q^83 + 37341 * q^84 - 42260 * q^86 - 46773 * q^87 - 415403 * q^88 + 117088 * q^89 - 9295 * q^91 - 70700 * q^92 + 2493 * q^93 + 383414 * q^94 - 20448 * q^96 - 112934 * q^97 - 159305 * q^98 - 10125 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	9.28847	1.64199	0.820993	−	0.570938i	$-0.193420\pi$
$2$	9.28847	1.64199	0.820993	+	0.570938i	$0.193420\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	54.2758	1.69612
$5$	0	0
$5$	0	0
$6$	−83.5963	−0.948001
$7$	85.6782	0.660884	0.330442	−	0.943826i	$-0.392802\pi$
$7$	85.6782	0.660884	0.330442	+	0.943826i	$0.392802\pi$
$8$	206.908	1.14302
$9$	81.0000	0.333333
$10$	0	0
$11$	119.113	0.296810	0.148405	−	0.988927i	$-0.452586\pi$
$11$	119.113	0.296810	0.148405	+	0.988927i	$0.452586\pi$
$12$	−488.482	−0.979254
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	795.820	1.08516
$15$	0	0
$16$	185.034	0.180697
$17$	−763.350	−0.640621	−0.320311	−	0.947313i	$-0.603787\pi$
$17$	−763.350	−0.640621	−0.320311	+	0.947313i	$0.603787\pi$
$18$	752.366	0.547329
$19$	−2433.97	−1.54679	−0.773395	−	0.633925i	$-0.781443\pi$
$19$	−2433.97	−1.54679	−0.773395	+	0.633925i	$0.781443\pi$
$20$	0	0
$21$	−771.104	−0.381562
$22$	1106.38	0.487358
$23$	−4050.77	−1.59668	−0.798340	−	0.602207i	$-0.794288\pi$
$23$	−4050.77	−1.59668	−0.798340	+	0.602207i	$0.794288\pi$
$24$	−1862.17	−0.659920
$25$	0	0
$26$	1569.75	0.455405
$27$	−729.000	−0.192450
$28$	4650.25	1.12094
$29$	−5525.35	−1.22001	−0.610007	−	0.792396i	$-0.708833\pi$
$29$	−5525.35	−1.22001	−0.610007	+	0.792396i	$0.708833\pi$
$30$	0	0
$31$	9537.22	1.78245	0.891226	−	0.453560i	$-0.149846\pi$
$31$	9537.22	1.78245	0.891226	+	0.453560i	$0.149846\pi$
$32$	−4902.37	−0.846313
$33$	−1072.02	−0.171363
$34$	−7090.35	−1.05189
$35$	0	0
$36$	4396.34	0.565373
$37$	5704.17	0.684996	0.342498	−	0.939519i	$-0.388727\pi$
$37$	5704.17	0.684996	0.342498	+	0.939519i	$0.388727\pi$
$38$	−22607.9	−2.53981
$39$	−1521.00	−0.160128
$40$	0	0
$41$	−10517.8	−0.977161	−0.488581	−	0.872519i	$-0.662485\pi$
$41$	−10517.8	−0.977161	−0.488581	+	0.872519i	$0.662485\pi$
$42$	−7162.38	−0.626519
$43$	6708.08	0.553257	0.276629	−	0.960977i	$-0.410783\pi$
$43$	6708.08	0.553257	0.276629	+	0.960977i	$0.410783\pi$
$44$	6464.96	0.503425
$45$	0	0
$46$	−37625.5	−2.62173
$47$	−577.173	−0.0381120	−0.0190560	−	0.999818i	$-0.506066\pi$
$47$	−577.173	−0.0381120	−0.0190560	+	0.999818i	$0.506066\pi$
$48$	−1665.31	−0.104326
$49$	−9466.24	−0.563232
$50$	0	0
$51$	6870.15	0.369863
$52$	9172.60	0.470418
$53$	12025.3	0.588037	0.294019	−	0.955800i	$-0.405007\pi$
$53$	12025.3	0.588037	0.294019	+	0.955800i	$0.405007\pi$
$54$	−6771.30	−0.316000
$55$	0	0
$56$	17727.5	0.755401
$57$	21905.7	0.893039
$58$	−51322.0	−2.00324
$59$	−39563.2	−1.47966	−0.739830	−	0.672794i	$-0.765094\pi$
$59$	−39563.2	−1.47966	−0.739830	+	0.672794i	$0.765094\pi$
$60$	0	0
$61$	49752.0	1.71193	0.855964	−	0.517035i	$-0.172964\pi$
$61$	49752.0	1.71193	0.855964	+	0.517035i	$0.172964\pi$
$62$	88586.3	2.92676
$63$	6939.94	0.220295
$64$	−51456.6	−1.57033
$65$	0	0
$66$	−9957.42	−0.281376
$67$	−20394.4	−0.555041	−0.277520	−	0.960720i	$-0.589513\pi$
$67$	−20394.4	−0.555041	−0.277520	+	0.960720i	$0.589513\pi$
$68$	−41431.4	−1.08657
$69$	36456.9	0.921844
$70$	0	0
$71$	29124.8	0.685672	0.342836	−	0.939395i	$-0.388612\pi$
$71$	29124.8	0.685672	0.342836	+	0.939395i	$0.388612\pi$
$72$	16759.5	0.381005
$73$	−57924.8	−1.27221	−0.636103	−	0.771604i	$-0.719455\pi$
$73$	−57924.8	−1.27221	−0.636103	+	0.771604i	$0.719455\pi$
$74$	52983.0	1.12475
$75$	0	0
$76$	−132106.	−2.62354
$77$	10205.4	0.196157
$78$	−14127.8	−0.262928
$79$	88938.2	1.60332	0.801661	−	0.597779i	$-0.203950\pi$
$79$	88938.2	1.60332	0.801661	+	0.597779i	$0.203950\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−97694.5	−1.60448
$83$	−68552.2	−1.09226	−0.546130	−	0.837701i	$-0.683899\pi$
$83$	−68552.2	−1.09226	−0.546130	+	0.837701i	$0.683899\pi$
$84$	−41852.3	−0.647173
$85$	0	0
$86$	62307.8	0.908440
$87$	49728.1	0.704375
$88$	24645.5	0.339258
$89$	−130659.	−1.74850	−0.874249	−	0.485478i	$-0.838646\pi$
$89$	−130659.	−1.74850	−0.874249	+	0.485478i	$0.838646\pi$
$90$	0	0
$91$	14479.6	0.183296
$92$	−219859.	−2.70816
$93$	−85835.0	−1.02910
$94$	−5361.06	−0.0625793
$95$	0	0
$96$	44121.3	0.488619
$97$	84079.6	0.907322	0.453661	−	0.891174i	$-0.350118\pi$
$97$	84079.6	0.907322	0.453661	+	0.891174i	$0.350118\pi$
$98$	−87927.0	−0.924819
$99$	9648.17	0.0989366
$100$	0	0
$101$	13354.3	0.130262	0.0651308	−	0.997877i	$-0.479254\pi$
$101$	13354.3	0.130262	0.0651308	+	0.997877i	$0.479254\pi$
$102$	63813.2	0.607309
$103$	−139847.	−1.29885	−0.649426	−	0.760425i	$-0.724991\pi$
$103$	−139847.	−1.29885	−0.649426	+	0.760425i	$0.724991\pi$
$104$	34967.4	0.317015
$105$	0	0
$106$	111696.	0.965549
$107$	−42834.2	−0.361686	−0.180843	−	0.983512i	$-0.557883\pi$
$107$	−42834.2	−0.361686	−0.180843	+	0.983512i	$0.557883\pi$
$108$	−39567.0	−0.326418
$109$	−92100.9	−0.742502	−0.371251	−	0.928532i	$-0.621071\pi$
$109$	−92100.9	−0.742502	−0.371251	+	0.928532i	$0.621071\pi$
$110$	0	0
$111$	−51337.5	−0.395483
$112$	15853.4	0.119420
$113$	80661.8	0.594254	0.297127	−	0.954838i	$-0.403972\pi$
$113$	80661.8	0.594254	0.297127	+	0.954838i	$0.403972\pi$
$114$	203471.	1.46636
$115$	0	0
$116$	−299892.	−2.06929
$117$	13689.0	0.0924500
$118$	−367482.	−2.42958
$119$	−65402.4	−0.423376
$120$	0	0
$121$	−146863.	−0.911904
$122$	462120.	2.81096
$123$	94660.4	0.564164
$124$	517640.	3.02325
$125$	0	0
$126$	64461.4	0.361721
$127$	−209632.	−1.15332	−0.576658	−	0.816986i	$-0.695644\pi$
$127$	−209632.	−1.15332	−0.576658	+	0.816986i	$0.695644\pi$
$128$	−321078.	−1.73215
$129$	−60372.7	−0.319423
$130$	0	0
$131$	−236797.	−1.20559	−0.602793	−	0.797897i	$-0.705946\pi$
$131$	−236797.	−1.20559	−0.602793	+	0.797897i	$0.705946\pi$
$132$	−58184.7	−0.290652
$133$	−208538.	−1.02225
$134$	−189433.	−0.911369
$135$	0	0
$136$	−157943.	−0.732240
$137$	−113905.	−0.518490	−0.259245	−	0.965812i	$-0.583474\pi$
$137$	−113905.	−0.518490	−0.259245	+	0.965812i	$0.583474\pi$
$138$	338629.	1.51365
$139$	166120.	0.729266	0.364633	−	0.931151i	$-0.381195\pi$
$139$	166120.	0.729266	0.364633	+	0.931151i	$0.381195\pi$
$140$	0	0
$141$	5194.56	0.0220040
$142$	270525.	1.12586
$143$	20130.1	0.0823203
$144$	14987.8	0.0602324
$145$	0	0
$146$	−538033.	−2.08894
$147$	85196.2	0.325182
$148$	309598.	1.16183
$149$	−319768.	−1.17997	−0.589983	−	0.807416i	$-0.700866\pi$
$149$	−319768.	−1.17997	−0.589983	+	0.807416i	$0.700866\pi$
$150$	0	0
$151$	−449813.	−1.60542	−0.802712	−	0.596367i	$-0.796610\pi$
$151$	−449813.	−1.60542	−0.802712	+	0.596367i	$0.796610\pi$
$152$	−503607.	−1.76800
$153$	−61831.3	−0.213540
$154$	94792.7	0.322087
$155$	0	0
$156$	−82553.4	−0.271596
$157$	7895.85	0.0255652	0.0127826	−	0.999918i	$-0.495931\pi$
$157$	7895.85	0.0255652	0.0127826	+	0.999918i	$0.495931\pi$
$158$	826101.	2.63263
$159$	−108227.	−0.339503
$160$	0	0
$161$	−347063.	−1.05522
$162$	60941.7	0.182443
$163$	−235832.	−0.695240	−0.347620	−	0.937636i	$-0.613010\pi$
$163$	−235832.	−0.695240	−0.347620	+	0.937636i	$0.613010\pi$
$164$	−570863.	−1.65738
$165$	0	0
$166$	−636745.	−1.79348
$167$	−177897.	−0.493604	−0.246802	−	0.969066i	$-0.579380\pi$
$167$	−177897.	−0.493604	−0.246802	+	0.969066i	$0.579380\pi$
$168$	−159547.	−0.436131
$169$	28561.0	0.0769231
$170$	0	0
$171$	−197152.	−0.515596
$172$	364086.	0.938389
$173$	129563.	0.329129	0.164564	−	0.986366i	$-0.447378\pi$
$173$	129563.	0.329129	0.164564	+	0.986366i	$0.447378\pi$
$174$	461898.	1.15657
$175$	0	0
$176$	22040.0	0.0536327
$177$	356069.	0.854282
$178$	−1.21363e6	−2.87101
$179$	90173.3	0.210351	0.105176	−	0.994454i	$-0.466459\pi$
$179$	90173.3	0.210351	0.105176	+	0.994454i	$0.466459\pi$
$180$	0	0
$181$	235482.	0.534271	0.267135	−	0.963659i	$-0.413923\pi$
$181$	235482.	0.534271	0.267135	+	0.963659i	$0.413923\pi$
$182$	134494.	0.300970
$183$	−447768.	−0.988382
$184$	−838136.	−1.82503
$185$	0	0
$186$	−797276.	−1.68977
$187$	−90925.1	−0.190143
$188$	−31326.5	−0.0646424
$189$	−62459.4	−0.127187
$190$	0	0
$191$	208853.	0.414245	0.207122	−	0.978315i	$-0.433590\pi$
$191$	208853.	0.414245	0.207122	+	0.978315i	$0.433590\pi$
$192$	463109.	0.906631
$193$	930559.	1.79825	0.899126	−	0.437690i	$-0.144203\pi$
$193$	930559.	1.79825	0.899126	+	0.437690i	$0.144203\pi$
$194$	780971.	1.48981
$195$	0	0
$196$	−513788.	−0.955308
$197$	−343197.	−0.630054	−0.315027	−	0.949083i	$-0.602014\pi$
$197$	−343197.	−0.630054	−0.315027	+	0.949083i	$0.602014\pi$
$198$	89616.8	0.162453
$199$	589911.	1.05598	0.527988	−	0.849252i	$-0.322947\pi$
$199$	589911.	1.05598	0.527988	+	0.849252i	$0.322947\pi$
$200$	0	0
$201$	183550.	0.320453
$202$	124041.	0.213888
$203$	−473402.	−0.806287
$204$	372882.	0.627331
$205$	0	0
$206$	−1.29896e6	−2.13270
$207$	−328112.	−0.532227
$208$	31270.7	0.0501164
$209$	−289918.	−0.459102
$210$	0	0
$211$	235802.	0.364621	0.182311	−	0.983241i	$-0.441642\pi$
$211$	235802.	0.364621	0.182311	+	0.983241i	$0.441642\pi$
$212$	652680.	0.997380
$213$	−262123.	−0.395873
$214$	−397865.	−0.593883
$215$	0	0
$216$	−150836.	−0.219973
$217$	817132.	1.17799
$218$	−855477.	−1.21918
$219$	521323.	0.734508
$220$	0	0
$221$	−129006.	−0.177676
$222$	−476847.	−0.649377
$223$	380770.	0.512744	0.256372	−	0.966578i	$-0.417473\pi$
$223$	380770.	0.512744	0.256372	+	0.966578i	$0.417473\pi$
$224$	−420026.	−0.559315
$225$	0	0
$226$	749225.	0.975756
$227$	970558.	1.25014	0.625068	−	0.780571i	$-0.285071\pi$
$227$	970558.	1.25014	0.625068	+	0.780571i	$0.285071\pi$
$228$	1.18895e6	1.51470
$229$	−735622.	−0.926970	−0.463485	−	0.886105i	$-0.653401\pi$
$229$	−735622.	−0.926970	−0.463485	+	0.886105i	$0.653401\pi$
$230$	0	0
$231$	−91848.7	−0.113251
$232$	−1.14324e6	−1.39449
$233$	1.20744e6	1.45705	0.728526	−	0.685018i	$-0.240206\pi$
$233$	1.20744e6	1.45705	0.728526	+	0.685018i	$0.240206\pi$
$234$	127150.	0.151802
$235$	0	0
$236$	−2.14733e6	−2.50968
$237$	−800444.	−0.925679
$238$	−607489.	−0.695178
$239$	350894.	0.397357	0.198678	−	0.980065i	$-0.436335\pi$
$239$	350894.	0.397357	0.198678	+	0.980065i	$0.436335\pi$
$240$	0	0
$241$	1.32445e6	1.46890	0.734449	−	0.678664i	$-0.237441\pi$
$241$	1.32445e6	1.46890	0.734449	+	0.678664i	$0.237441\pi$
$242$	−1.36413e6	−1.49733
$243$	−59049.0	−0.0641500
$244$	2.70033e6	2.90363
$245$	0	0
$246$	879251.	0.926350
$247$	−411341.	−0.429002
$248$	1.97333e6	2.03737
$249$	616969.	0.630616
$250$	0	0
$251$	336335.	0.336967	0.168483	−	0.985704i	$-0.446113\pi$
$251$	336335.	0.336967	0.168483	+	0.985704i	$0.446113\pi$
$252$	376670.	0.373646
$253$	−482500.	−0.473911
$254$	−1.94716e6	−1.89373
$255$	0	0
$256$	−1.33571e6	−1.27383
$257$	−1.35530e6	−1.27997	−0.639987	−	0.768386i	$-0.721060\pi$
$257$	−1.35530e6	−1.27997	−0.639987	+	0.768386i	$0.721060\pi$
$258$	−560770.	−0.524488
$259$	488723.	0.452703
$260$	0	0
$261$	−447553.	−0.406671
$262$	−2.19948e6	−1.97956
$263$	−649642.	−0.579141	−0.289571	−	0.957157i	$-0.593513\pi$
$263$	−649642.	−0.579141	−0.289571	+	0.957157i	$0.593513\pi$
$264$	−221809.	−0.195871
$265$	0	0
$266$	−1.93700e6	−1.67852
$267$	1.17593e6	1.00950
$268$	−1.10692e6	−0.941415
$269$	941767.	0.793529	0.396765	−	0.917920i	$-0.370133\pi$
$269$	941767.	0.793529	0.396765	+	0.917920i	$0.370133\pi$
$270$	0	0
$271$	244104.	0.201907	0.100954	−	0.994891i	$-0.467811\pi$
$271$	244104.	0.201907	0.100954	+	0.994891i	$0.467811\pi$
$272$	−141246.	−0.115758
$273$	−130317.	−0.105826
$274$	−1.05800e6	−0.851353
$275$	0	0
$276$	1.97873e6	1.56356
$277$	581955.	0.455711	0.227856	−	0.973695i	$-0.426829\pi$
$277$	581955.	0.455711	0.227856	+	0.973695i	$0.426829\pi$
$278$	1.54300e6	1.19744
$279$	772515.	0.594150
$280$	0	0
$281$	1.56647e6	1.18347	0.591733	−	0.806134i	$-0.298444\pi$
$281$	1.56647e6	1.18347	0.591733	+	0.806134i	$0.298444\pi$
$282$	48249.5	0.0361302
$283$	166122.	0.123300	0.0616498	−	0.998098i	$-0.480364\pi$
$283$	166122.	0.123300	0.0616498	+	0.998098i	$0.480364\pi$
$284$	1.58077e6	1.16298
$285$	0	0
$286$	186978.	0.135169
$287$	−901148.	−0.645790
$288$	−397092.	−0.282104
$289$	−837154.	−0.589605
$290$	0	0
$291$	−756717.	−0.523843
$292$	−3.14391e6	−2.15781
$293$	−1.96523e6	−1.33735	−0.668674	−	0.743555i	$-0.733138\pi$
$293$	−1.96523e6	−1.33735	−0.668674	+	0.743555i	$0.733138\pi$
$294$	791343.	0.533945
$295$	0	0
$296$	1.18024e6	0.782961
$297$	−86833.6	−0.0571211
$298$	−2.97016e6	−1.93749
$299$	−684580.	−0.442839
$300$	0	0
$301$	574736.	0.365639
$302$	−4.17808e6	−2.63608
$303$	−120188.	−0.0752065
$304$	−450367.	−0.279501
$305$	0	0
$306$	−574319.	−0.350630
$307$	2.09330e6	1.26761	0.633805	−	0.773493i	$-0.281492\pi$
$307$	2.09330e6	1.26761	0.633805	+	0.773493i	$0.281492\pi$
$308$	553907.	0.332705
$309$	1.25862e6	0.749892
$310$	0	0
$311$	−1.26784e6	−0.743299	−0.371650	−	0.928373i	$-0.621208\pi$
$311$	−1.26784e6	−0.743299	−0.371650	+	0.928373i	$0.621208\pi$
$312$	−314707.	−0.183029
$313$	−2.90120e6	−1.67385	−0.836925	−	0.547317i	$-0.815649\pi$
$313$	−2.90120e6	−1.67385	−0.836925	+	0.547317i	$0.815649\pi$
$314$	73340.4	0.0419777
$315$	0	0
$316$	4.82719e6	2.71942
$317$	−3.09468e6	−1.72969	−0.864844	−	0.502040i	$-0.832583\pi$
$317$	−3.09468e6	−1.72969	−0.864844	+	0.502040i	$0.832583\pi$
$318$	−1.00527e6	−0.557460
$319$	−658142.	−0.362112
$320$	0	0
$321$	385508.	0.208819
$322$	−3.22368e6	−1.73266
$323$	1.85797e6	0.990906
$324$	356103.	0.188458
$325$	0	0
$326$	−2.19052e6	−1.14157
$327$	828908.	0.428684
$328$	−2.17622e6	−1.11691
$329$	−49451.2	−0.0251876
$330$	0	0
$331$	2.33082e6	1.16934	0.584668	−	0.811273i	$-0.301225\pi$
$331$	2.33082e6	1.16934	0.584668	+	0.811273i	$0.301225\pi$
$332$	−3.72072e6	−1.85260
$333$	462038.	0.228332
$334$	−1.65240e6	−0.810490
$335$	0	0
$336$	−142680.	−0.0689471
$337$	−1.31758e6	−0.631977	−0.315988	−	0.948763i	$-0.602336\pi$
$337$	−1.31758e6	−0.631977	−0.315988	+	0.948763i	$0.602336\pi$
$338$	265288.	0.126307
$339$	−725956.	−0.343093
$340$	0	0
$341$	1.13601e6	0.529049
$342$	−1.83124e6	−0.846602
$343$	−2.25104e6	−1.03312
$344$	1.38795e6	0.632381
$345$	0	0
$346$	1.20344e6	0.540425
$347$	1.67075e6	0.744883	0.372442	−	0.928056i	$-0.378521\pi$
$347$	1.67075e6	0.744883	0.372442	+	0.928056i	$0.378521\pi$
$348$	2.69903e6	1.19470
$349$	−3.19784e6	−1.40538	−0.702688	−	0.711498i	$-0.748017\pi$
$349$	−3.19784e6	−1.40538	−0.702688	+	0.711498i	$0.748017\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	−583937.	−0.251194
$353$	170177.	0.0726883	0.0363442	−	0.999339i	$-0.488429\pi$
$353$	170177.	0.0726883	0.0363442	+	0.999339i	$0.488429\pi$
$354$	3.30734e6	1.40272
$355$	0	0
$356$	−7.09163e6	−2.96566
$357$	588622.	0.244436
$358$	837573.	0.345394
$359$	647400.	0.265116	0.132558	−	0.991175i	$-0.457681\pi$
$359$	647400.	0.265116	0.132558	+	0.991175i	$0.457681\pi$
$360$	0	0
$361$	3.44811e6	1.39256
$362$	2.18727e6	0.877265
$363$	1.32177e6	0.526488
$364$	785892.	0.310892
$365$	0	0
$366$	−4.15908e6	−1.62291
$367$	−2.63522e6	−1.02130	−0.510648	−	0.859790i	$-0.670594\pi$
$367$	−2.63522e6	−1.02130	−0.510648	+	0.859790i	$0.670594\pi$
$368$	−749530.	−0.288516
$369$	−851944.	−0.325720
$370$	0	0
$371$	1.03030e6	0.388624
$372$	−4.65876e6	−1.74547
$373$	−3.87103e6	−1.44064	−0.720318	−	0.693644i	$-0.756004\pi$
$373$	−3.87103e6	−1.44064	−0.720318	+	0.693644i	$0.756004\pi$
$374$	−844555.	−0.312212
$375$	0	0
$376$	−119422.	−0.0435626
$377$	−933784.	−0.338371
$378$	−580153.	−0.208840
$379$	2.54174e6	0.908936	0.454468	−	0.890763i	$-0.349829\pi$
$379$	2.54174e6	0.908936	0.454468	+	0.890763i	$0.349829\pi$
$380$	0	0
$381$	1.88669e6	0.665868
$382$	1.93992e6	0.680184
$383$	3.54092e6	1.23344	0.616721	−	0.787182i	$-0.288461\pi$
$383$	3.54092e6	1.23344	0.616721	+	0.787182i	$0.288461\pi$
$384$	2.88970e6	1.00006
$385$	0	0
$386$	8.64347e6	2.95270
$387$	543354.	0.184419
$388$	4.56349e6	1.53892
$389$	1.95942e6	0.656527	0.328263	−	0.944586i	$-0.393537\pi$
$389$	1.95942e6	0.656527	0.328263	+	0.944586i	$0.393537\pi$
$390$	0	0
$391$	3.09215e6	1.02287
$392$	−1.95864e6	−0.643783
$393$	2.13117e6	0.696046
$394$	−3.18778e6	−1.03454
$395$	0	0
$396$	523662.	0.167808
$397$	−5.10515e6	−1.62567	−0.812835	−	0.582494i	$-0.802077\pi$
$397$	−5.10515e6	−1.62567	−0.812835	+	0.582494i	$0.802077\pi$
$398$	5.47937e6	1.73390
$399$	1.87684e6	0.590195
$400$	0	0
$401$	−596902.	−0.185371	−0.0926855	−	0.995695i	$-0.529545\pi$
$401$	−596902.	−0.185371	−0.0926855	+	0.995695i	$0.529545\pi$
$402$	1.70490e6	0.526179
$403$	1.61179e6	0.494363
$404$	724812.	0.220939
$405$	0	0
$406$	−4.39718e6	−1.32391
$407$	679442.	0.203314
$408$	1.42149e6	0.422759
$409$	2.88580e6	0.853018	0.426509	−	0.904483i	$-0.359743\pi$
$409$	2.88580e6	0.853018	0.426509	+	0.904483i	$0.359743\pi$
$410$	0	0
$411$	1.02514e6	0.299350
$412$	−7.59029e6	−2.20301
$413$	−3.38971e6	−0.977884
$414$	−3.04766e6	−0.873909
$415$	0	0
$416$	−828500.	−0.234725
$417$	−1.49508e6	−0.421042
$418$	−2.69290e6	−0.753840
$419$	5.48041e6	1.52503	0.762515	−	0.646971i	$-0.223965\pi$
$419$	5.48041e6	1.52503	0.762515	+	0.646971i	$0.223965\pi$
$420$	0	0
$421$	4.48786e6	1.23405	0.617026	−	0.786943i	$-0.288337\pi$
$421$	4.48786e6	1.23405	0.617026	+	0.786943i	$0.288337\pi$
$422$	2.19024e6	0.598703
$423$	−46751.0	−0.0127040
$424$	2.48812e6	0.672135
$425$	0	0
$426$	−2.43472e6	−0.650018
$427$	4.26266e6	1.13139
$428$	−2.32486e6	−0.613462
$429$	−181171.	−0.0475276
$430$	0	0
$431$	−139450.	−0.0361597	−0.0180798	−	0.999837i	$-0.505755\pi$
$431$	−139450.	−0.0361597	−0.0180798	+	0.999837i	$0.505755\pi$
$432$	−134890.	−0.0347752
$433$	2.04709e6	0.524708	0.262354	−	0.964972i	$-0.415501\pi$
$433$	2.04709e6	0.524708	0.262354	+	0.964972i	$0.415501\pi$
$434$	7.58991e6	1.93425
$435$	0	0
$436$	−4.99885e6	−1.25937
$437$	9.85945e6	2.46973
$438$	4.84230e6	1.20605
$439$	5.07121e6	1.25589	0.627943	−	0.778259i	$-0.283897\pi$
$439$	5.07121e6	1.25589	0.627943	+	0.778259i	$0.283897\pi$
$440$	0	0
$441$	−766766.	−0.187744
$442$	−1.19827e6	−0.291742
$443$	−2.31892e6	−0.561406	−0.280703	−	0.959795i	$-0.590568\pi$
$443$	−2.31892e6	−0.561406	−0.280703	+	0.959795i	$0.590568\pi$
$444$	−2.78638e6	−0.670785
$445$	0	0
$446$	3.53678e6	0.841919
$447$	2.87791e6	0.681254
$448$	−4.40871e6	−1.03781
$449$	7.95151e6	1.86137	0.930687	−	0.365816i	$-0.119210\pi$
$449$	7.95151e6	1.86137	0.930687	+	0.365816i	$0.119210\pi$
$450$	0	0
$451$	−1.25281e6	−0.290031
$452$	4.37798e6	1.00792
$453$	4.04832e6	0.926892
$454$	9.01501e6	2.05270
$455$	0	0
$456$	4.53247e6	1.02076
$457$	528506.	0.118375	0.0591874	−	0.998247i	$-0.481149\pi$
$457$	528506.	0.118375	0.0591874	+	0.998247i	$0.481149\pi$
$458$	−6.83280e6	−1.52207
$459$	556482.	0.123288
$460$	0	0
$461$	−3.79727e6	−0.832183	−0.416091	−	0.909323i	$-0.636600\pi$
$461$	−3.79727e6	−0.832183	−0.416091	+	0.909323i	$0.636600\pi$
$462$	−853134.	−0.185957
$463$	−8.19835e6	−1.77735	−0.888677	−	0.458533i	$-0.848375\pi$
$463$	−8.19835e6	−1.77735	−0.888677	+	0.458533i	$0.848375\pi$
$464$	−1.02238e6	−0.220453
$465$	0	0
$466$	1.12153e7	2.39246
$467$	2.15445e6	0.457135	0.228568	−	0.973528i	$-0.426596\pi$
$467$	2.15445e6	0.457135	0.228568	+	0.973528i	$0.426596\pi$
$468$	742981.	0.156806
$469$	−1.74736e6	−0.366818
$470$	0	0
$471$	−71062.6	−0.0147601
$472$	−8.18595e6	−1.69127
$473$	799021.	0.164212
$474$	−7.43491e6	−1.51995
$475$	0	0
$476$	−3.54977e6	−0.718096
$477$	974046.	0.196012
$478$	3.25927e6	0.652455
$479$	−2.04730e6	−0.407702	−0.203851	−	0.979002i	$-0.565346\pi$
$479$	−2.04730e6	−0.407702	−0.203851	+	0.979002i	$0.565346\pi$
$480$	0	0
$481$	964004.	0.189984
$482$	1.23021e7	2.41191
$483$	3.12356e6	0.609232
$484$	−7.97110e6	−1.54670
$485$	0	0
$486$	−548475.	−0.105333
$487$	−2.21510e6	−0.423225	−0.211612	−	0.977354i	$-0.567871\pi$
$487$	−2.21510e6	−0.423225	−0.211612	+	0.977354i	$0.567871\pi$
$488$	1.02941e7	1.95676
$489$	2.12249e6	0.401397
$490$	0	0
$491$	−6.52362e6	−1.22119	−0.610597	−	0.791941i	$-0.709070\pi$
$491$	−6.52362e6	−1.22119	−0.610597	+	0.791941i	$0.709070\pi$
$492$	5.13777e6	0.956889
$493$	4.21777e6	0.781566
$494$	−3.82073e6	−0.704415
$495$	0	0
$496$	1.76471e6	0.322084
$497$	2.49536e6	0.453150
$498$	5.73071e6	1.03546
$499$	−3.53790e6	−0.636055	−0.318027	−	0.948082i	$-0.603020\pi$
$499$	−3.53790e6	−0.636055	−0.318027	+	0.948082i	$0.603020\pi$
$500$	0	0
$501$	1.60108e6	0.284982
$502$	3.12403e6	0.553295
$503$	3.76357e6	0.663254	0.331627	−	0.943411i	$-0.392402\pi$
$503$	3.76357e6	0.663254	0.331627	+	0.943411i	$0.392402\pi$
$504$	1.43593e6	0.251800
$505$	0	0
$506$	−4.48169e6	−0.778154
$507$	−257049.	−0.0444116
$508$	−1.13779e7	−1.95616
$509$	8.60164e6	1.47159	0.735795	−	0.677204i	$-0.236809\pi$
$509$	8.60164e6	1.47159	0.735795	+	0.677204i	$0.236809\pi$
$510$	0	0
$511$	−4.96289e6	−0.840781
$512$	−2.13222e6	−0.359466
$513$	1.77436e6	0.297680
$514$	−1.25886e7	−2.10170
$515$	0	0
$516$	−3.27677e6	−0.541779
$517$	−68749.0	−0.0113120
$518$	4.53949e6	0.743332
$519$	−1.16607e6	−0.190023
$520$	0	0
$521$	−3.20683e6	−0.517585	−0.258792	−	0.965933i	$-0.583325\pi$
$521$	−3.20683e6	−0.517585	−0.258792	+	0.965933i	$0.583325\pi$
$522$	−4.15709e6	−0.667748
$523$	−4.37109e6	−0.698772	−0.349386	−	0.936979i	$-0.613610\pi$
$523$	−4.37109e6	−0.698772	−0.349386	+	0.936979i	$0.613610\pi$
$524$	−1.28523e7	−2.04482
$525$	0	0
$526$	−6.03418e6	−0.950942
$527$	−7.28024e6	−1.14188
$528$	−198360.	−0.0309649
$529$	9.97239e6	1.54939
$530$	0	0
$531$	−3.20462e6	−0.493220
$532$	−1.13186e7	−1.73385
$533$	−1.77751e6	−0.271016
$534$	1.09226e7	1.65758
$535$	0	0
$536$	−4.21977e6	−0.634420
$537$	−811560.	−0.121446
$538$	8.74758e6	1.30296
$539$	−1.12755e6	−0.167173
$540$	0	0
$541$	8.19236e6	1.20342	0.601708	−	0.798716i	$-0.294487\pi$
$541$	8.19236e6	1.20342	0.601708	+	0.798716i	$0.294487\pi$
$542$	2.26735e6	0.331529
$543$	−2.11934e6	−0.308461
$544$	3.74222e6	0.542166
$545$	0	0
$546$	−1.21044e6	−0.173765
$547$	−5.17040e6	−0.738849	−0.369425	−	0.929261i	$-0.620445\pi$
$547$	−5.17040e6	−0.738849	−0.369425	+	0.929261i	$0.620445\pi$
$548$	−6.18227e6	−0.879420
$549$	4.02991e6	0.570643
$550$	0	0
$551$	1.34485e7	1.88710
$552$	7.54322e6	1.05368
$553$	7.62007e6	1.05961
$554$	5.40547e6	0.748272
$555$	0	0
$556$	9.01631e6	1.23692
$557$	1.30342e7	1.78011	0.890055	−	0.455853i	$-0.150666\pi$
$557$	1.30342e7	1.78011	0.890055	+	0.455853i	$0.150666\pi$
$558$	7.17549e6	0.975587
$559$	1.13367e6	0.153446
$560$	0	0
$561$	818326.	0.109779
$562$	1.45501e7	1.94324
$563$	−6.23908e6	−0.829563	−0.414781	−	0.909921i	$-0.636142\pi$
$563$	−6.23908e6	−0.829563	−0.414781	+	0.909921i	$0.636142\pi$
$564$	281939.	0.0373213
$565$	0	0
$566$	1.54302e6	0.202456
$567$	562135.	0.0734316
$568$	6.02614e6	0.783734
$569$	9.96520e6	1.29034	0.645172	−	0.764038i	$-0.276786\pi$
$569$	9.96520e6	1.29034	0.645172	+	0.764038i	$0.276786\pi$
$570$	0	0
$571$	−5.09148e6	−0.653513	−0.326756	−	0.945109i	$-0.605956\pi$
$571$	−5.09148e6	−0.653513	−0.326756	+	0.945109i	$0.605956\pi$
$572$	1.09258e6	0.139625
$573$	−1.87968e6	−0.239164
$574$	−8.37029e6	−1.06038
$575$	0	0
$576$	−4.16799e6	−0.523444
$577$	3.40798e6	0.426145	0.213073	−	0.977036i	$-0.431653\pi$
$577$	3.40798e6	0.426145	0.213073	+	0.977036i	$0.431653\pi$
$578$	−7.77589e6	−0.968123
$579$	−8.37503e6	−1.03822
$580$	0	0
$581$	−5.87343e6	−0.721857
$582$	−7.02874e6	−0.860142
$583$	1.43237e6	0.174535
$584$	−1.19851e7	−1.45415
$585$	0	0
$586$	−1.82540e7	−2.19591
$587$	−4.58327e6	−0.549010	−0.274505	−	0.961586i	$-0.588514\pi$
$587$	−4.58327e6	−0.549010	−0.274505	+	0.961586i	$0.588514\pi$
$588$	4.62409e6	0.551547
$589$	−2.32133e7	−2.75708
$590$	0	0
$591$	3.08877e6	0.363762
$592$	1.05546e6	0.123777
$593$	−1.55994e7	−1.82168	−0.910838	−	0.412765i	$-0.864563\pi$
$593$	−1.55994e7	−1.82168	−0.910838	+	0.412765i	$0.864563\pi$
$594$	−806551.	−0.0937920
$595$	0	0
$596$	−1.73557e7	−2.00136
$597$	−5.30920e6	−0.609668
$598$	−6.35870e6	−0.727136
$599$	−223331.	−0.0254321	−0.0127161	−	0.999919i	$-0.504048\pi$
$599$	−223331.	−0.0254321	−0.0127161	+	0.999919i	$0.504048\pi$
$600$	0	0
$601$	3.14524e6	0.355195	0.177598	−	0.984103i	$-0.443167\pi$
$601$	3.14524e6	0.355195	0.177598	+	0.984103i	$0.443167\pi$
$602$	5.33842e6	0.600374
$603$	−1.65195e6	−0.185014
$604$	−2.44140e7	−2.72299
$605$	0	0
$606$	−1.11637e6	−0.123488
$607$	−774656.	−0.0853370	−0.0426685	−	0.999089i	$-0.513586\pi$
$607$	−774656.	−0.0853370	−0.0426685	+	0.999089i	$0.513586\pi$
$608$	1.19322e7	1.30907
$609$	4.26062e6	0.465510
$610$	0	0
$611$	−97542.3	−0.0105704
$612$	−3.35594e6	−0.362190
$613$	−4.51473e6	−0.485267	−0.242633	−	0.970118i	$-0.578011\pi$
$613$	−4.51473e6	−0.485267	−0.242633	+	0.970118i	$0.578011\pi$
$614$	1.94436e7	2.08140
$615$	0	0
$616$	2.11158e6	0.224210
$617$	1.89227e6	0.200111	0.100056	−	0.994982i	$-0.468098\pi$
$617$	1.89227e6	0.200111	0.100056	+	0.994982i	$0.468098\pi$
$618$	1.16907e7	1.23131
$619$	1.13652e7	1.19220	0.596099	−	0.802911i	$-0.296716\pi$
$619$	1.13652e7	1.19220	0.596099	+	0.802911i	$0.296716\pi$
$620$	0	0
$621$	2.95301e6	0.307281
$622$	−1.17763e7	−1.22049
$623$	−1.11947e7	−1.15555
$624$	−281437.	−0.0289347
$625$	0	0
$626$	−2.69477e7	−2.74844
$627$	2.60926e6	0.265063
$628$	428553.	0.0433616
$629$	−4.35427e6	−0.438823
$630$	0	0
$631$	9.66441e6	0.966278	0.483139	−	0.875544i	$-0.339497\pi$
$631$	9.66441e6	0.966278	0.483139	+	0.875544i	$0.339497\pi$
$632$	1.84020e7	1.83262
$633$	−2.12222e6	−0.210514
$634$	−2.87449e7	−2.84012
$635$	0	0
$636$	−5.87412e6	−0.575838
$637$	−1.59979e6	−0.156212
$638$	−6.11314e6	−0.594583
$639$	2.35911e6	0.228557
$640$	0	0
$641$	1.42384e7	1.36873	0.684363	−	0.729141i	$-0.260080\pi$
$641$	1.42384e7	1.36873	0.684363	+	0.729141i	$0.260080\pi$
$642$	3.58078e6	0.342879
$643$	−1.28164e7	−1.22248	−0.611238	−	0.791447i	$-0.709328\pi$
$643$	−1.28164e7	−1.22248	−0.611238	+	0.791447i	$0.709328\pi$
$644$	−1.88371e7	−1.78978
$645$	0	0
$646$	1.72577e7	1.62705
$647$	−1.07646e7	−1.01097	−0.505485	−	0.862836i	$-0.668686\pi$
$647$	−1.07646e7	−1.01097	−0.505485	+	0.862836i	$0.668686\pi$
$648$	1.35752e6	0.127002
$649$	−4.71251e6	−0.439178
$650$	0	0
$651$	−7.35419e6	−0.680115
$652$	−1.28000e7	−1.17921
$653$	2.77756e6	0.254906	0.127453	−	0.991845i	$-0.459320\pi$
$653$	2.77756e6	0.254906	0.127453	+	0.991845i	$0.459320\pi$
$654$	7.69930e6	0.703893
$655$	0	0
$656$	−1.94615e6	−0.176570
$657$	−4.69191e6	−0.424069
$658$	−459326.	−0.0413577
$659$	40832.4	0.00366262	0.00183131	−	0.999998i	$-0.499417\pi$
$659$	40832.4	0.00366262	0.00183131	+	0.999998i	$0.499417\pi$
$660$	0	0
$661$	−3.77383e6	−0.335953	−0.167977	−	0.985791i	$-0.553723\pi$
$661$	−3.77383e6	−0.335953	−0.167977	+	0.985791i	$0.553723\pi$
$662$	2.16498e7	1.92003
$663$	1.16105e6	0.102581
$664$	−1.41840e7	−1.24847
$665$	0	0
$666$	4.29162e6	0.374918
$667$	2.23819e7	1.94797
$668$	−9.65552e6	−0.837210
$669$	−3.42693e6	−0.296033
$670$	0	0
$671$	5.92612e6	0.508117
$672$	3.78024e6	0.322921
$673$	−1.78358e7	−1.51794	−0.758970	−	0.651126i	$-0.774297\pi$
$673$	−1.78358e7	−1.51794	−0.758970	+	0.651126i	$0.774297\pi$
$674$	−1.22383e7	−1.03770
$675$	0	0
$676$	1.55017e6	0.130471
$677$	1.34680e7	1.12936	0.564681	−	0.825310i	$-0.308999\pi$
$677$	1.34680e7	1.12936	0.564681	+	0.825310i	$0.308999\pi$
$678$	−6.74303e6	−0.563353
$679$	7.20379e6	0.599635
$680$	0	0
$681$	−8.73503e6	−0.721766
$682$	1.05518e7	0.868691
$683$	1.13143e7	0.928059	0.464030	−	0.885820i	$-0.346403\pi$
$683$	1.13143e7	0.928059	0.464030	+	0.885820i	$0.346403\pi$
$684$	−1.07005e7	−0.874512
$685$	0	0
$686$	−2.09088e7	−1.69636
$687$	6.62059e6	0.535186
$688$	1.24122e6	0.0999720
$689$	2.03227e6	0.163092
$690$	0	0
$691$	−1.61896e7	−1.28986	−0.644929	−	0.764242i	$-0.723113\pi$
$691$	−1.61896e7	−1.28986	−0.644929	+	0.764242i	$0.723113\pi$
$692$	7.03213e6	0.558241
$693$	826638.	0.0653857
$694$	1.55187e7	1.22309
$695$	0	0
$696$	1.02891e7	0.805111
$697$	8.02878e6	0.625990
$698$	−2.97030e7	−2.30761
$699$	−1.08669e7	−0.841229
$700$	0	0
$701$	5.30840e6	0.408008	0.204004	−	0.978970i	$-0.434604\pi$
$701$	5.30840e6	0.408008	0.204004	+	0.978970i	$0.434604\pi$
$702$	−1.14435e6	−0.0876427
$703$	−1.38838e7	−1.05954
$704$	−6.12916e6	−0.466090
$705$	0	0
$706$	1.58069e6	0.119353
$707$	1.14417e6	0.0860878
$708$	1.93259e7	1.44896
$709$	4.99200e6	0.372957	0.186478	−	0.982459i	$-0.440293\pi$
$709$	4.99200e6	0.372957	0.186478	+	0.982459i	$0.440293\pi$
$710$	0	0
$711$	7.20400e6	0.534441
$712$	−2.70344e7	−1.99856
$713$	−3.86331e7	−2.84601
$714$	5.46740e6	0.401361
$715$	0	0
$716$	4.89423e6	0.356781
$717$	−3.15804e6	−0.229414
$718$	6.01336e6	0.435317
$719$	−7.83351e6	−0.565111	−0.282556	−	0.959251i	$-0.591182\pi$
$719$	−7.83351e6	−0.565111	−0.282556	+	0.959251i	$0.591182\pi$
$720$	0	0
$721$	−1.19818e7	−0.858390
$722$	3.20277e7	2.28656
$723$	−1.19200e7	−0.848069
$724$	1.27810e7	0.906186
$725$	0	0
$726$	1.22772e7	0.864486
$727$	2.20732e7	1.54892	0.774460	−	0.632622i	$-0.218021\pi$
$727$	2.20732e7	1.54892	0.774460	+	0.632622i	$0.218021\pi$
$728$	2.99595e6	0.209510
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.12061e6	−0.354428
$732$	−2.43029e7	−1.67641
$733$	1.89500e7	1.30272	0.651358	−	0.758770i	$-0.274199\pi$
$733$	1.89500e7	1.30272	0.651358	+	0.758770i	$0.274199\pi$
$734$	−2.44772e7	−1.67695
$735$	0	0
$736$	1.98584e7	1.35129
$737$	−2.42925e6	−0.164742
$738$	−7.91326e6	−0.534828
$739$	−9.68780e6	−0.652550	−0.326275	−	0.945275i	$-0.605794\pi$
$739$	−9.68780e6	−0.652550	−0.326275	+	0.945275i	$0.605794\pi$
$740$	0	0
$741$	3.70207e6	0.247685
$742$	9.56994e6	0.638116
$743$	−1.93530e6	−0.128610	−0.0643051	−	0.997930i	$-0.520483\pi$
$743$	−1.93530e6	−0.128610	−0.0643051	+	0.997930i	$0.520483\pi$
$744$	−1.77599e7	−1.17628
$745$	0	0
$746$	−3.59560e7	−2.36551
$747$	−5.55273e6	−0.364087
$748$	−4.93503e6	−0.322504
$749$	−3.66996e6	−0.239032
$750$	0	0
$751$	1.66142e7	1.07493	0.537464	−	0.843287i	$-0.319383\pi$
$751$	1.66142e7	1.07493	0.537464	+	0.843287i	$0.319383\pi$
$752$	−106797.	−0.00688673
$753$	−3.02701e6	−0.194548
$754$	−8.67342e6	−0.555600
$755$	0	0
$756$	−3.39003e6	−0.215724
$757$	1.89893e7	1.20439	0.602196	−	0.798348i	$-0.294292\pi$
$757$	1.89893e7	1.20439	0.602196	+	0.798348i	$0.294292\pi$
$758$	2.36089e7	1.49246
$759$	4.34250e6	0.273612
$760$	0	0
$761$	−2.11475e7	−1.32373	−0.661863	−	0.749625i	$-0.730234\pi$
$761$	−2.11475e7	−1.32373	−0.661863	+	0.749625i	$0.730234\pi$
$762$	1.75245e7	1.09335
$763$	−7.89104e6	−0.490708
$764$	1.13356e7	0.702608
$765$	0	0
$766$	3.28897e7	2.02530
$767$	−6.68619e6	−0.410384
$768$	1.20214e7	0.735447
$769$	−2.47106e7	−1.50684	−0.753420	−	0.657540i	$-0.771597\pi$
$769$	−2.47106e7	−1.50684	−0.753420	+	0.657540i	$0.771597\pi$
$770$	0	0
$771$	1.21977e7	0.738993
$772$	5.05068e7	3.05005
$773$	2.84076e7	1.70996	0.854980	−	0.518660i	$-0.173569\pi$
$773$	2.84076e7	1.70996	0.854980	+	0.518660i	$0.173569\pi$
$774$	5.04693e6	0.302813
$775$	0	0
$776$	1.73967e7	1.03708
$777$	−4.39851e6	−0.261368
$778$	1.82000e7	1.07801
$779$	2.56001e7	1.51146
$780$	0	0
$781$	3.46915e6	0.203514
$782$	2.87214e7	1.67953
$783$	4.02798e6	0.234792
$784$	−1.75158e6	−0.101774
$785$	0	0
$786$	1.97954e7	1.14290
$787$	1.57312e6	0.0905365	0.0452683	−	0.998975i	$-0.485586\pi$
$787$	1.57312e6	0.0905365	0.0452683	+	0.998975i	$0.485586\pi$
$788$	−1.86273e7	−1.06865
$789$	5.84678e6	0.334367
$790$	0	0
$791$	6.91096e6	0.392733
$792$	1.99628e6	0.113086
$793$	8.40808e6	0.474804
$794$	−4.74191e7	−2.66933
$795$	0	0
$796$	3.20179e7	1.79106
$797$	−3.10824e7	−1.73328	−0.866641	−	0.498932i	$-0.833726\pi$
$797$	−3.10824e7	−1.73328	−0.866641	+	0.498932i	$0.833726\pi$
$798$	1.74330e7	0.969093
$799$	440585.	0.0244153
$800$	0	0
$801$	−1.05834e7	−0.582833
$802$	−5.54431e6	−0.304377
$803$	−6.89961e6	−0.377603
$804$	9.96232e6	0.543526
$805$	0	0
$806$	1.49711e7	0.811737
$807$	−8.47591e6	−0.458144
$808$	2.76310e6	0.148891
$809$	2.30163e7	1.23642	0.618208	−	0.786014i	$-0.287859\pi$
$809$	2.30163e7	1.23642	0.618208	+	0.786014i	$0.287859\pi$
$810$	0	0
$811$	3.15218e7	1.68290	0.841451	−	0.540334i	$-0.181702\pi$
$811$	3.15218e7	1.68290	0.841451	+	0.540334i	$0.181702\pi$
$812$	−2.56942e7	−1.36756
$813$	−2.19694e6	−0.116571
$814$	6.31098e6	0.333838
$815$	0	0
$816$	1.27121e6	0.0668332
$817$	−1.63273e7	−0.855772
$818$	2.68047e7	1.40064
$819$	1.17285e6	0.0610988
$820$	0	0
$821$	2.55863e7	1.32480	0.662398	−	0.749152i	$-0.269539\pi$
$821$	2.55863e7	1.32480	0.662398	+	0.749152i	$0.269539\pi$
$822$	9.52201e6	0.491529
$823$	−2.24169e7	−1.15365	−0.576826	−	0.816867i	$-0.695709\pi$
$823$	−2.24169e7	−1.15365	−0.576826	+	0.816867i	$0.695709\pi$
$824$	−2.89354e7	−1.48461
$825$	0	0
$826$	−3.14852e7	−1.60567
$827$	3.61037e7	1.83564	0.917822	−	0.396992i	$-0.129946\pi$
$827$	3.61037e7	1.83564	0.917822	+	0.396992i	$0.129946\pi$
$828$	−1.78085e7	−0.902719
$829$	−173851.	−0.00878599	−0.00439300	−	0.999990i	$-0.501398\pi$
$829$	−173851.	−0.00878599	−0.00439300	+	0.999990i	$0.501398\pi$
$830$	0	0
$831$	−5.23759e6	−0.263105
$832$	−8.69617e6	−0.435531
$833$	7.22605e6	0.360818
$834$	−1.38870e7	−0.691345
$835$	0	0
$836$	−1.57355e7	−0.778692
$837$	−6.95264e6	−0.343033
$838$	5.09047e7	2.50408
$839$	−1.92270e7	−0.942987	−0.471493	−	0.881870i	$-0.656285\pi$
$839$	−1.92270e7	−0.942987	−0.471493	+	0.881870i	$0.656285\pi$
$840$	0	0
$841$	1.00183e7	0.488432
$842$	4.16853e7	2.02630
$843$	−1.40982e7	−0.683275
$844$	1.27984e7	0.618441
$845$	0	0
$846$	−434246.	−0.0208598
$847$	−1.25830e7	−0.602663
$848$	2.22508e6	0.106257
$849$	−1.49510e6	−0.0711870
$850$	0	0
$851$	−2.31063e7	−1.09372
$852$	−1.42269e7	−0.671447
$853$	−1.35017e7	−0.635352	−0.317676	−	0.948199i	$-0.602902\pi$
$853$	−1.35017e7	−0.635352	−0.317676	+	0.948199i	$0.602902\pi$
$854$	3.95936e7	1.85772
$855$	0	0
$856$	−8.86274e6	−0.413412
$857$	−1.86409e7	−0.866992	−0.433496	−	0.901156i	$-0.642720\pi$
$857$	−1.86409e7	−0.866992	−0.433496	+	0.901156i	$0.642720\pi$
$858$	−1.68280e6	−0.0780397
$859$	1.54673e7	0.715208	0.357604	−	0.933873i	$-0.383594\pi$
$859$	1.54673e7	0.715208	0.357604	+	0.933873i	$0.383594\pi$
$860$	0	0
$861$	8.11034e6	0.372847
$862$	−1.29528e6	−0.0593737
$863$	−7.44388e6	−0.340230	−0.170115	−	0.985424i	$-0.554414\pi$
$863$	−7.44388e6	−0.340230	−0.170115	+	0.985424i	$0.554414\pi$
$864$	3.57383e6	0.162873
$865$	0	0
$866$	1.90144e7	0.861564
$867$	7.53439e6	0.340408
$868$	4.43505e7	1.99802
$869$	1.05937e7	0.475882
$870$	0	0
$871$	−3.44666e6	−0.153941
$872$	−1.90564e7	−0.848691
$873$	6.81045e6	0.302441
$874$	9.15793e7	4.05526
$875$	0	0
$876$	2.82952e7	1.24581
$877$	1.50507e7	0.660783	0.330391	−	0.943844i	$-0.392819\pi$
$877$	1.50507e7	0.660783	0.330391	+	0.943844i	$0.392819\pi$
$878$	4.71038e7	2.06215
$879$	1.76871e7	0.772119
$880$	0	0
$881$	4.50487e7	1.95543	0.977716	−	0.209932i	$-0.0673244\pi$
$881$	4.50487e7	1.95543	0.977716	+	0.209932i	$0.0673244\pi$
$882$	−7.12208e6	−0.308273
$883$	−1.45280e7	−0.627053	−0.313527	−	0.949579i	$-0.601510\pi$
$883$	−1.45280e7	−0.627053	−0.313527	+	0.949579i	$0.601510\pi$
$884$	−7.00190e6	−0.301360
$885$	0	0
$886$	−2.15393e7	−0.921820
$887$	−2.50271e7	−1.06807	−0.534037	−	0.845461i	$-0.679326\pi$
$887$	−2.50271e7	−1.06807	−0.534037	+	0.845461i	$0.679326\pi$
$888$	−1.06221e7	−0.452043
$889$	−1.79609e7	−0.762209
$890$	0	0
$891$	781502.	0.0329789
$892$	2.06666e7	0.869675
$893$	1.40482e6	0.0589512
$894$	2.67314e7	1.11861
$895$	0	0
$896$	−2.75094e7	−1.14475
$897$	6.16122e6	0.255673
$898$	7.38574e7	3.05635
$899$	−5.26965e7	−2.17461
$900$	0	0
$901$	−9.17948e6	−0.376709
$902$	−1.16367e7	−0.476227
$903$	−5.17263e6	−0.211102
$904$	1.66896e7	0.679241
$905$	0	0
$906$	3.76027e7	1.52194
$907$	1.25157e7	0.505168	0.252584	−	0.967575i	$-0.418720\pi$
$907$	1.25157e7	0.505168	0.252584	+	0.967575i	$0.418720\pi$
$908$	5.26778e7	2.12038
$909$	1.08169e6	0.0434205
$910$	0	0
$911$	3.94640e6	0.157545	0.0787727	−	0.996893i	$-0.474900\pi$
$911$	3.94640e6	0.157545	0.0787727	+	0.996893i	$0.474900\pi$
$912$	4.05330e6	0.161370
$913$	−8.16547e6	−0.324194
$914$	4.90901e6	0.194370
$915$	0	0
$916$	−3.99264e7	−1.57225
$917$	−2.02884e7	−0.796753
$918$	5.16887e6	0.202436
$919$	9.49362e6	0.370803	0.185401	−	0.982663i	$-0.440641\pi$
$919$	9.49362e6	0.370803	0.185401	+	0.982663i	$0.440641\pi$
$920$	0	0
$921$	−1.88397e7	−0.731855
$922$	−3.52708e7	−1.36643
$923$	4.92209e6	0.190171
$924$	−4.98516e6	−0.192087
$925$	0	0
$926$	−7.61502e7	−2.91839
$927$	−1.13276e7	−0.432951
$928$	2.70873e7	1.03251
$929$	−3.50533e7	−1.33257	−0.666284	−	0.745698i	$-0.732116\pi$
$929$	−3.50533e7	−1.33257	−0.666284	+	0.745698i	$0.732116\pi$
$930$	0	0
$931$	2.30405e7	0.871201
$932$	6.55346e7	2.47133
$933$	1.14106e7	0.429144
$934$	2.00116e7	0.750610
$935$	0	0
$936$	2.83236e6	0.105672
$937$	2.44398e6	0.0909388	0.0454694	−	0.998966i	$-0.485522\pi$
$937$	2.44398e6	0.0909388	0.0454694	+	0.998966i	$0.485522\pi$
$938$	−1.62303e7	−0.602310
$939$	2.61108e7	0.966398
$940$	0	0
$941$	−5.35649e7	−1.97200	−0.985998	−	0.166759i	$-0.946670\pi$
$941$	−5.35649e7	−1.97200	−0.985998	+	0.166759i	$0.946670\pi$
$942$	−660063.	−0.0242359
$943$	4.26053e7	1.56021
$944$	−7.32054e6	−0.267370
$945$	0	0
$946$	7.42169e6	0.269634
$947$	4.21959e7	1.52896	0.764478	−	0.644650i	$-0.222997\pi$
$947$	4.21959e7	1.52896	0.764478	+	0.644650i	$0.222997\pi$
$948$	−4.34447e7	−1.57006
$949$	−9.78929e6	−0.352846
$950$	0	0
$951$	2.78521e7	0.998636
$952$	−1.35323e7	−0.483926
$953$	1.76524e7	0.629608	0.314804	−	0.949157i	$-0.398061\pi$
$953$	1.76524e7	0.629608	0.314804	+	0.949157i	$0.398061\pi$
$954$	9.04740e6	0.321850
$955$	0	0
$956$	1.90450e7	0.673964
$957$	5.92328e6	0.209065
$958$	−1.90163e7	−0.669441
$959$	−9.75916e6	−0.342662
$960$	0	0
$961$	6.23295e7	2.17713
$962$	8.95413e6	0.311951
$963$	−3.46957e6	−0.120562
$964$	7.18853e7	2.49142
$965$	0	0
$966$	2.90132e7	1.00035
$967$	3.24710e6	0.111668	0.0558340	−	0.998440i	$-0.482218\pi$
$967$	3.24710e6	0.111668	0.0558340	+	0.998440i	$0.482218\pi$
$968$	−3.03871e7	−1.04232
$969$	−1.67217e7	−0.572100
$970$	0	0
$971$	−989341.	−0.0336743	−0.0168371	−	0.999858i	$-0.505360\pi$
$971$	−989341.	−0.0336743	−0.0168371	+	0.999858i	$0.505360\pi$
$972$	−3.20493e6	−0.108806
$973$	1.42329e7	0.481960
$974$	−2.05749e7	−0.694929
$975$	0	0
$976$	9.20580e6	0.309341
$977$	−1.01621e7	−0.340603	−0.170301	−	0.985392i	$-0.554474\pi$
$977$	−1.01621e7	−0.340603	−0.170301	+	0.985392i	$0.554474\pi$
$978$	1.97147e7	0.659088
$979$	−1.55633e7	−0.518972
$980$	0	0
$981$	−7.46018e6	−0.247501
$982$	−6.05944e7	−2.00518
$983$	−4.05016e7	−1.33687	−0.668433	−	0.743772i	$-0.733035\pi$
$983$	−4.05016e7	−1.33687	−0.668433	+	0.743772i	$0.733035\pi$
$984$	1.95860e7	0.644848
$985$	0	0
$986$	3.91767e7	1.28332
$987$	445061.	0.0145421
$988$	−2.23258e7	−0.727638
$989$	−2.71729e7	−0.883375
$990$	0	0
$991$	−2.37017e7	−0.766648	−0.383324	−	0.923614i	$-0.625221\pi$
$991$	−2.37017e7	−0.766648	−0.383324	+	0.923614i	$0.625221\pi$
$992$	−4.67550e7	−1.50851
$993$	−2.09774e7	−0.675117
$994$	2.31781e7	0.744066
$995$	0	0
$996$	3.34865e7	1.06960
$997$	3.98801e7	1.27063	0.635315	−	0.772253i	$-0.280870\pi$
$997$	3.98801e7	1.27063	0.635315	+	0.772253i	$0.280870\pi$
$998$	−3.28617e7	−1.04439
$999$	−4.15834e6	−0.131828

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.6.a.u.1.10	yes	11
5.4	even	2		975.6.a.r.1.2	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.6.a.r.1.2	✓	11	5.4	even	2
975.6.a.u.1.10	yes	11	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	975.6.a.u.1.10

Level	$975$
Weight	$6$
Character	975.1
Self dual	yes
Analytic conductor	$156.374$
Analytic rank	$1$
Dimension	$11$
CM	no
Inner twists	$1$