LMFDB - Embedded newform 961.6.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [961,6,Mod(1,961)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(961, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("961.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	961.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:	$154.128850840$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 $ x^8 - x^7 - 199x^6 + 256x^5 + 12633x^4 - 18583x^3 - 260319x^2 + 410640x + 275908
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 5\cdot 13 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.21681$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.21681 q^{2} +24.0001 q^{3} -30.5194 q^{4} +80.9514 q^{5} -29.2035 q^{6} +175.330 q^{7} +76.0739 q^{8} +333.006 q^{9} -98.5022 q^{10} +506.435 q^{11} -732.469 q^{12} -75.1097 q^{13} -213.342 q^{14} +1942.85 q^{15} +884.053 q^{16} -2340.12 q^{17} -405.204 q^{18} +1946.64 q^{19} -2470.59 q^{20} +4207.94 q^{21} -616.233 q^{22} +829.073 q^{23} +1825.78 q^{24} +3428.14 q^{25} +91.3939 q^{26} +2160.17 q^{27} -5350.95 q^{28} +619.340 q^{29} -2364.07 q^{30} -3510.09 q^{32} +12154.5 q^{33} +2847.47 q^{34} +14193.2 q^{35} -10163.2 q^{36} +3487.14 q^{37} -2368.68 q^{38} -1802.64 q^{39} +6158.29 q^{40} +13137.9 q^{41} -5120.24 q^{42} +14336.7 q^{43} -15456.1 q^{44} +26957.4 q^{45} -1008.82 q^{46} -2946.60 q^{47} +21217.4 q^{48} +13933.5 q^{49} -4171.38 q^{50} -56163.1 q^{51} +2292.30 q^{52} -21954.4 q^{53} -2628.50 q^{54} +40996.7 q^{55} +13338.0 q^{56} +46719.6 q^{57} -753.616 q^{58} +28172.2 q^{59} -59294.5 q^{60} +12952.3 q^{61} +58385.9 q^{63} -24018.6 q^{64} -6080.24 q^{65} -14789.7 q^{66} -34804.0 q^{67} +71419.0 q^{68} +19897.9 q^{69} -17270.4 q^{70} +19907.8 q^{71} +25333.1 q^{72} -45077.6 q^{73} -4243.18 q^{74} +82275.7 q^{75} -59410.3 q^{76} +88793.1 q^{77} +2193.47 q^{78} -63368.0 q^{79} +71565.4 q^{80} -29076.3 q^{81} -15986.2 q^{82} +24250.2 q^{83} -128424. q^{84} -189436. q^{85} -17445.0 q^{86} +14864.2 q^{87} +38526.5 q^{88} -7020.86 q^{89} -32801.9 q^{90} -13169.0 q^{91} -25302.8 q^{92} +3585.44 q^{94} +157583. q^{95} -84242.5 q^{96} +23289.5 q^{97} -16954.3 q^{98} +168646. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 7 q^{2} + 2 q^{3} + 149 q^{4} + 128 q^{5} - 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} - 574 q^{11} + 46 q^{12} + 122 q^{13} - 309 q^{14} + 524 q^{15} + 833 q^{16} - 1932 q^{17} - 6845 q^{18}+ \cdots + 180150 q^{99}+O(q^{100}) $ 8 * q + 7 * q^2 + 2 * q^3 + 149 * q^4 + 128 * q^5 - 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 - 574 * q^11 + 46 * q^12 + 122 * q^13 - 309 * q^14 + 524 * q^15 + 833 * q^16 - 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 - 996 * q^21 + 9000 * q^22 + 4136 * q^23 + 17468 * q^24 + 3308 * q^25 - 1524 * q^26 + 2624 * q^27 - 21245 * q^28 - 5050 * q^29 + 30450 * q^30 + 21045 * q^32 + 27920 * q^33 + 24738 * q^34 + 25700 * q^35 + 21637 * q^36 - 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 + 9994 * q^42 - 21058 * q^43 - 13168 * q^44 - 1344 * q^45 - 6198 * q^46 + 5348 * q^47 + 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 + 60386 * q^52 - 4926 * q^53 + 24476 * q^54 + 56892 * q^55 - 87652 * q^56 - 22072 * q^57 + 25002 * q^58 + 16944 * q^59 + 138556 * q^60 + 73682 * q^61 - 138784 * q^63 - 1300 * q^64 - 63316 * q^65 - 128796 * q^66 - 134768 * q^67 + 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 - 70792 * q^73 - 307468 * q^74 + 302902 * q^75 - 65405 * q^76 - 107932 * q^77 + 36820 * q^78 + 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 - 21882 * q^83 - 299144 * q^84 + 130464 * q^85 - 480478 * q^86 - 440244 * q^87 + 71982 * q^88 - 164392 * q^89 + 397969 * q^90 + 60028 * q^91 - 31012 * q^92 - 170388 * q^94 - 7404 * q^95 + 331010 * q^96 - 131948 * q^97 + 288768 * q^98 + 180150 * 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 $

$2$	−1.21681	−0.215103	−0.107551	−	0.994200i	$-0.534301\pi$
$2$	−1.21681	−0.215103	−0.107551	+	0.994200i	$0.534301\pi$
$3$	24.0001	1.53961	0.769805	−	0.638280i	$-0.220354\pi$
$3$	24.0001	1.53961	0.769805	+	0.638280i	$0.220354\pi$
$4$	−30.5194	−0.953731
$5$	80.9514	1.44810	0.724052	−	0.689746i	$-0.242278\pi$
$5$	80.9514	1.44810	0.724052	+	0.689746i	$0.242278\pi$
$6$	−29.2035	−0.331174
$7$	175.330	1.35242	0.676208	−	0.736711i	$-0.263622\pi$
$7$	175.330	1.35242	0.676208	+	0.736711i	$0.263622\pi$
$8$	76.0739	0.420253
$9$	333.006	1.37040
$10$	−98.5022	−0.311491
$11$	506.435	1.26195	0.630975	−	0.775803i	$-0.282655\pi$
$11$	506.435	1.26195	0.630975	+	0.775803i	$0.282655\pi$
$12$	−732.469	−1.46837
$13$	−75.1097	−0.123264	−0.0616322	−	0.998099i	$-0.519631\pi$
$13$	−75.1097	−0.123264	−0.0616322	+	0.998099i	$0.519631\pi$
$14$	−213.342	−0.290909
$15$	1942.85	2.22951
$16$	884.053	0.863333
$17$	−2340.12	−1.96388	−0.981941	−	0.189187i	$-0.939415\pi$
$17$	−2340.12	−1.96388	−0.981941	+	0.189187i	$0.939415\pi$
$18$	−405.204	−0.294776
$19$	1946.64	1.23709	0.618546	−	0.785749i	$-0.287722\pi$
$19$	1946.64	1.23709	0.618546	+	0.785749i	$0.287722\pi$
$20$	−2470.59	−1.38110
$21$	4207.94	2.08219
$22$	−616.233	−0.271449
$23$	829.073	0.326793	0.163397	−	0.986560i	$-0.447755\pi$
$23$	829.073	0.326793	0.163397	+	0.986560i	$0.447755\pi$
$24$	1825.78	0.647026
$25$	3428.14	1.09700
$26$	91.3939	0.0265145
$27$	2160.17	0.570267
$28$	−5350.95	−1.28984
$29$	619.340	0.136752	0.0683760	−	0.997660i	$-0.478218\pi$
$29$	619.340	0.136752	0.0683760	+	0.997660i	$0.478218\pi$
$30$	−2364.07	−0.479575
$31$	0	0
$31$	0	0
$32$	−3510.09	−0.605958
$33$	12154.5	1.94291
$34$	2847.47	0.422437
$35$	14193.2	1.95844
$36$	−10163.2	−1.30699
$37$	3487.14	0.418760	0.209380	−	0.977834i	$-0.432855\pi$
$37$	3487.14	0.418760	0.209380	+	0.977834i	$0.432855\pi$
$38$	−2368.68	−0.266102
$39$	−1802.64	−0.189779
$40$	6158.29	0.608570
$41$	13137.9	1.22058	0.610288	−	0.792179i	$-0.291054\pi$
$41$	13137.9	1.22058	0.610288	+	0.792179i	$0.291054\pi$
$42$	−5120.24	−0.447885
$43$	14336.7	1.18244	0.591219	−	0.806511i	$-0.298647\pi$
$43$	14336.7	1.18244	0.591219	+	0.806511i	$0.298647\pi$
$44$	−15456.1	−1.20356
$45$	26957.4	1.98448
$46$	−1008.82	−0.0702942
$47$	−2946.60	−0.194571	−0.0972853	−	0.995257i	$-0.531016\pi$
$47$	−2946.60	−0.194571	−0.0972853	+	0.995257i	$0.531016\pi$
$48$	21217.4	1.32920
$49$	13933.5	0.829029
$50$	−4171.38	−0.235969
$51$	−56163.1	−3.02361
$52$	2292.30	0.117561
$53$	−21954.4	−1.07357	−0.536786	−	0.843718i	$-0.680362\pi$
$53$	−21954.4	−1.07357	−0.536786	+	0.843718i	$0.680362\pi$
$54$	−2628.50	−0.122666
$55$	40996.7	1.82744
$56$	13338.0	0.568357
$57$	46719.6	1.90464
$58$	−753.616	−0.0294158
$59$	28172.2	1.05364	0.526818	−	0.849978i	$-0.323385\pi$
$59$	28172.2	1.05364	0.526818	+	0.849978i	$0.323385\pi$
$60$	−59294.5	−2.12636
$61$	12952.3	0.445679	0.222839	−	0.974855i	$-0.428467\pi$
$61$	12952.3	0.445679	0.222839	+	0.974855i	$0.428467\pi$
$62$	0	0
$63$	58385.9	1.85335
$64$	−24018.6	−0.732990
$65$	−6080.24	−0.178500
$66$	−14789.7	−0.417926
$67$	−34804.0	−0.947201	−0.473601	−	0.880740i	$-0.657046\pi$
$67$	−34804.0	−0.947201	−0.473601	+	0.880740i	$0.657046\pi$
$68$	71419.0	1.87301
$69$	19897.9	0.503134
$70$	−17270.4	−0.421266
$71$	19907.8	0.468682	0.234341	−	0.972155i	$-0.424707\pi$
$71$	19907.8	0.468682	0.234341	+	0.972155i	$0.424707\pi$
$72$	25333.1	0.575914
$73$	−45077.6	−0.990042	−0.495021	−	0.868881i	$-0.664840\pi$
$73$	−45077.6	−0.990042	−0.495021	+	0.868881i	$0.664840\pi$
$74$	−4243.18	−0.0900765
$75$	82275.7	1.68896
$76$	−59410.3	−1.17985
$77$	88793.1	1.70668
$78$	2193.47	0.0408220
$79$	−63368.0	−1.14236	−0.571179	−	0.820825i	$-0.693514\pi$
$79$	−63368.0	−1.14236	−0.571179	+	0.820825i	$0.693514\pi$
$80$	71565.4	1.25020
$81$	−29076.3	−0.492409
$82$	−15986.2	−0.262550
$83$	24250.2	0.386385	0.193193	−	0.981161i	$-0.438116\pi$
$83$	24250.2	0.386385	0.193193	+	0.981161i	$0.438116\pi$
$84$	−128424.	−1.98585
$85$	−189436.	−2.84390
$86$	−17445.0	−0.254346
$87$	14864.2	0.210545
$88$	38526.5	0.530339
$89$	−7020.86	−0.0939540	−0.0469770	−	0.998896i	$-0.514959\pi$
$89$	−7020.86	−0.0939540	−0.0469770	+	0.998896i	$0.514959\pi$
$90$	−32801.9	−0.426867
$91$	−13169.0	−0.166705
$92$	−25302.8	−0.311673
$93$	0	0
$94$	3585.44	0.0418527
$95$	157583.	1.79144
$96$	−84242.5	−0.932939
$97$	23289.5	0.251322	0.125661	−	0.992073i	$-0.459895\pi$
$97$	23289.5	0.251322	0.125661	+	0.992073i	$0.459895\pi$
$98$	−16954.3	−0.178326
$99$	168646.	1.72937
$100$	−104625.	−1.04625
$101$	61812.6	0.602939	0.301469	−	0.953476i	$-0.402523\pi$
$101$	61812.6	0.602939	0.301469	+	0.953476i	$0.402523\pi$
$102$	68339.6	0.650387
$103$	−120090.	−1.11535	−0.557677	−	0.830058i	$-0.688307\pi$
$103$	−120090.	−1.11535	−0.557677	+	0.830058i	$0.688307\pi$
$104$	−5713.89	−0.0518022
$105$	340638.	3.01523
$106$	26714.2	0.230929
$107$	−40972.2	−0.345963	−0.172982	−	0.984925i	$-0.555340\pi$
$107$	−40972.2	−0.345963	−0.172982	+	0.984925i	$0.555340\pi$
$108$	−65927.0	−0.543881
$109$	−214237.	−1.72714	−0.863570	−	0.504230i	$-0.831777\pi$
$109$	−214237.	−1.72714	−0.863570	+	0.504230i	$0.831777\pi$
$110$	−49885.0	−0.393087
$111$	83691.9	0.644727
$112$	155001.	1.16759
$113$	−164200.	−1.20970	−0.604850	−	0.796340i	$-0.706767\pi$
$113$	−164200.	−1.20970	−0.604850	+	0.796340i	$0.706767\pi$
$114$	−56848.7	−0.409693
$115$	67114.7	0.473231
$116$	−18901.9	−0.130425
$117$	−25012.0	−0.168921
$118$	−34280.1	−0.226640
$119$	−410292.	−2.65599
$120$	147800.	0.936960
$121$	95425.8	0.592519
$122$	−15760.4	−0.0958668
$123$	315310.	1.87921
$124$	0	0
$125$	24539.4	0.140472
$126$	−71044.3	−0.398660
$127$	289909.	1.59497	0.797486	−	0.603338i	$-0.206163\pi$
$127$	289909.	1.59497	0.797486	+	0.603338i	$0.206163\pi$
$128$	141549.	0.763627
$129$	344083.	1.82049
$130$	7398.47	0.0383958
$131$	120430.	0.613134	0.306567	−	0.951849i	$-0.400820\pi$
$131$	120430.	0.613134	0.306567	+	0.951849i	$0.400820\pi$
$132$	−370948.	−1.85301
$133$	341304.	1.67306
$134$	42349.7	0.203746
$135$	174869.	0.825805
$136$	−178022.	−0.825327
$137$	−108636.	−0.494508	−0.247254	−	0.968951i	$-0.579528\pi$
$137$	−108636.	−0.494508	−0.247254	+	0.968951i	$0.579528\pi$
$138$	−24211.8	−0.108226
$139$	−143692.	−0.630808	−0.315404	−	0.948958i	$-0.602140\pi$
$139$	−143692.	−0.630808	−0.315404	+	0.948958i	$0.602140\pi$
$140$	−433167.	−1.86782
$141$	−70718.9	−0.299563
$142$	−24224.0	−0.100815
$143$	−38038.2	−0.155554
$144$	294395.	1.18311
$145$	50136.4	0.198031
$146$	54850.7	0.212961
$147$	334406.	1.27638
$148$	−106425.	−0.399385
$149$	70270.4	0.259303	0.129651	−	0.991560i	$-0.458614\pi$
$149$	70270.4	0.259303	0.129651	+	0.991560i	$0.458614\pi$
$150$	−100114.	−0.363300
$151$	245097.	0.874774	0.437387	−	0.899273i	$-0.355904\pi$
$151$	245097.	0.874774	0.437387	+	0.899273i	$0.355904\pi$
$152$	148089.	0.519891
$153$	−779274.	−2.69130
$154$	−108044.	−0.367112
$155$	0	0
$156$	55015.6	0.180998
$157$	287025.	0.929332	0.464666	−	0.885486i	$-0.346174\pi$
$157$	287025.	0.929332	0.464666	+	0.885486i	$0.346174\pi$
$158$	77106.6	0.245725
$159$	−526908.	−1.65288
$160$	−284147.	−0.877491
$161$	145361.	0.441960
$162$	35380.2	0.105919
$163$	461431.	1.36031	0.680155	−	0.733069i	$-0.261913\pi$
$163$	461431.	1.36031	0.680155	+	0.733069i	$0.261913\pi$
$164$	−400959.	−1.16410
$165$	983926.	2.81354
$166$	−29507.8	−0.0831126
$167$	233748.	0.648568	0.324284	−	0.945960i	$-0.394877\pi$
$167$	233748.	0.648568	0.324284	+	0.945960i	$0.394877\pi$
$168$	320114.	0.875048
$169$	−365652.	−0.984806
$170$	230507.	0.611732
$171$	648244.	1.69531
$172$	−437548.	−1.12773
$173$	−326801.	−0.830171	−0.415086	−	0.909782i	$-0.636248\pi$
$173$	−326801.	−0.830171	−0.415086	+	0.909782i	$0.636248\pi$
$174$	−18086.9	−0.0452888
$175$	601054.	1.48361
$176$	447716.	1.08948
$177$	676136.	1.62219
$178$	8543.02	0.0202098
$179$	−65628.1	−0.153094	−0.0765468	−	0.997066i	$-0.524389\pi$
$179$	−65628.1	−0.153094	−0.0765468	+	0.997066i	$0.524389\pi$
$180$	−822722.	−1.89266
$181$	576563.	1.30813	0.654064	−	0.756439i	$-0.273063\pi$
$181$	576563.	1.30813	0.654064	+	0.756439i	$0.273063\pi$
$182$	16024.1	0.0358587
$183$	310857.	0.686171
$184$	63070.8	0.137336
$185$	282289.	0.606408
$186$	0	0
$187$	−1.18512e6	−2.47832
$188$	89928.5	0.185568
$189$	378741.	0.771238
$190$	−191748.	−0.385343
$191$	675263.	1.33934	0.669668	−	0.742661i	$-0.266437\pi$
$191$	675263.	1.33934	0.669668	+	0.742661i	$0.266437\pi$
$192$	−576450.	−1.12852
$193$	−244938.	−0.473330	−0.236665	−	0.971591i	$-0.576054\pi$
$193$	−244938.	−0.473330	−0.236665	+	0.971591i	$0.576054\pi$
$194$	−28338.8	−0.0540601
$195$	−145927.	−0.274820
$196$	−425241.	−0.790670
$197$	−512084.	−0.940104	−0.470052	−	0.882639i	$-0.655765\pi$
$197$	−512084.	−0.940104	−0.470052	+	0.882639i	$0.655765\pi$
$198$	−205210.	−0.371993
$199$	−299086.	−0.535381	−0.267690	−	0.963505i	$-0.586260\pi$
$199$	−299086.	−0.535381	−0.267690	+	0.963505i	$0.586260\pi$
$200$	260792.	0.461019
$201$	−835301.	−1.45832
$202$	−75213.9	−0.129694
$203$	108589.	0.184946
$204$	1.71406e6	2.88371
$205$	1.06353e6	1.76752
$206$	146126.	0.239916
$207$	276087.	0.447837
$208$	−66401.0	−0.106418
$209$	985848.	1.56115
$210$	−414491.	−0.648585
$211$	−916231.	−1.41677	−0.708384	−	0.705827i	$-0.750576\pi$
$211$	−916231.	−1.41677	−0.708384	+	0.705827i	$0.750576\pi$
$212$	670034.	1.02390
$213$	477790.	0.721587
$214$	49855.2	0.0744177
$215$	1.16058e6	1.71229
$216$	164332.	0.239656
$217$	0	0
$218$	260684.	0.371513
$219$	−1.08187e6	−1.52428
$220$	−1.25119e6	−1.74288
$221$	175766.	0.242077
$222$	−101837.	−0.138683
$223$	−230064.	−0.309804	−0.154902	−	0.987930i	$-0.549506\pi$
$223$	−230064.	−0.309804	−0.154902	+	0.987930i	$0.549506\pi$
$224$	−615422.	−0.819508
$225$	1.14159e6	1.50333
$226$	199800.	0.260210
$227$	−1.19248e6	−1.53599	−0.767993	−	0.640458i	$-0.778744\pi$
$227$	−1.19248e6	−1.53599	−0.767993	+	0.640458i	$0.778744\pi$
$228$	−1.42585e6	−1.81651
$229$	−571545.	−0.720215	−0.360107	−	0.932911i	$-0.617260\pi$
$229$	−571545.	−0.720215	−0.360107	+	0.932911i	$0.617260\pi$
$230$	−81665.5	−0.101793
$231$	2.13105e6	2.62762
$232$	47115.6	0.0574705
$233$	602554.	0.727120	0.363560	−	0.931571i	$-0.381561\pi$
$233$	602554.	0.727120	0.363560	+	0.931571i	$0.381561\pi$
$234$	30434.8	0.0363354
$235$	−238532.	−0.281758
$236$	−859798.	−1.00488
$237$	−1.52084e6	−1.75879
$238$	499246.	0.571310
$239$	−1.05476e6	−1.19443	−0.597214	−	0.802082i	$-0.703726\pi$
$239$	−1.05476e6	−1.19443	−0.597214	+	0.802082i	$0.703726\pi$
$240$	1.71758e6	1.92481
$241$	575.787	0.000638586 0	0.000319293	−	1.00000i	$-0.499898\pi$
$241$	575.787	0.000638586 0	0.000319293		1.00000i	$0.499898\pi$
$242$	−116115.	−0.127453
$243$	−1.22275e6	−1.32838
$244$	−395296.	−0.425057
$245$	1.12794e6	1.20052
$246$	−383671.	−0.404224
$247$	−146212.	−0.152489
$248$	0	0
$249$	582009.	0.594883
$250$	−29859.6	−0.0302158
$251$	−383652.	−0.384373	−0.192187	−	0.981358i	$-0.561558\pi$
$251$	−383652.	−0.384373	−0.192187	+	0.981358i	$0.561558\pi$
$252$	−1.78190e6	−1.76759
$253$	419872.	0.412397
$254$	−352763.	−0.343083
$255$	−4.54649e6	−4.37850
$256$	596358.	0.568731
$257$	456359.	0.430996	0.215498	−	0.976504i	$-0.430862\pi$
$257$	456359.	0.430996	0.215498	+	0.976504i	$0.430862\pi$
$258$	−418682.	−0.391593
$259$	611400.	0.566338
$260$	185565.	0.170241
$261$	206244.	0.187405
$262$	−146540.	−0.131887
$263$	−598156.	−0.533243	−0.266622	−	0.963801i	$-0.585907\pi$
$263$	−598156.	−0.533243	−0.266622	+	0.963801i	$0.585907\pi$
$264$	924642.	0.816514
$265$	−1.77724e6	−1.55464
$266$	−415300.	−0.359880
$267$	−168502.	−0.144652
$268$	1.06220e6	0.903375
$269$	1.95486e6	1.64716	0.823578	−	0.567204i	$-0.191975\pi$
$269$	1.95486e6	1.64716	0.823578	+	0.567204i	$0.191975\pi$
$270$	−212781.	−0.177633
$271$	594232.	0.491510	0.245755	−	0.969332i	$-0.420964\pi$
$271$	594232.	0.491510	0.245755	+	0.969332i	$0.420964\pi$
$272$	−2.06879e6	−1.69548
$273$	−316057.	−0.256660
$274$	132189.	0.106370
$275$	1.73613e6	1.38436
$276$	−607271.	−0.479854
$277$	2.28581e6	1.78995	0.894975	−	0.446116i	$-0.147193\pi$
$277$	2.28581e6	1.78995	0.894975	+	0.446116i	$0.147193\pi$
$278$	174846.	0.135689
$279$	0	0
$280$	1.07973e6	0.823040
$281$	−112074.	−0.0846716	−0.0423358	−	0.999103i	$-0.513480\pi$
$281$	−112074.	−0.0846716	−0.0423358	+	0.999103i	$0.513480\pi$
$282$	86051.1	0.0644368
$283$	−1.46373e6	−1.08641	−0.543206	−	0.839599i	$-0.682790\pi$
$283$	−1.46373e6	−1.08641	−0.543206	+	0.839599i	$0.682790\pi$
$284$	−607575.	−0.446996
$285$	3.78202e6	2.75811
$286$	46285.1	0.0334600
$287$	2.30346e6	1.65073
$288$	−1.16888e6	−0.830404
$289$	4.05629e6	2.85683
$290$	−61006.3	−0.0425971
$291$	558951.	0.386938
$292$	1.37574e6	0.944234
$293$	158097.	0.107586	0.0537928	−	0.998552i	$-0.482869\pi$
$293$	158097.	0.107586	0.0537928	+	0.998552i	$0.482869\pi$
$294$	−406907.	−0.274553
$295$	2.28058e6	1.52577
$296$	265281.	0.175985
$297$	1.09399e6	0.719648
$298$	−85505.4	−0.0557767
$299$	−62271.4	−0.0402820
$300$	−2.51101e6	−1.61081
$301$	2.51365e6	1.59915
$302$	−298236.	−0.188166
$303$	1.48351e6	0.928290
$304$	1.72093e6	1.06802
$305$	1.04851e6	0.645389
$306$	948225.	0.578906
$307$	−1.36437e6	−0.826202	−0.413101	−	0.910685i	$-0.635554\pi$
$307$	−1.36437e6	−0.826202	−0.413101	+	0.910685i	$0.635554\pi$
$308$	−2.70991e6	−1.62772
$309$	−2.88217e6	−1.71721
$310$	0	0
$311$	2.05789e6	1.20649	0.603243	−	0.797558i	$-0.293875\pi$
$311$	2.05789e6	1.20649	0.603243	+	0.797558i	$0.293875\pi$
$312$	−137134.	−0.0797552
$313$	1.45467e6	0.839276	0.419638	−	0.907692i	$-0.362157\pi$
$313$	1.45467e6	0.839276	0.419638	+	0.907692i	$0.362157\pi$
$314$	−349254.	−0.199902
$315$	4.72642e6	2.68384
$316$	1.93395e6	1.08950
$317$	−1.61011e6	−0.899930	−0.449965	−	0.893046i	$-0.648563\pi$
$317$	−1.61011e6	−0.899930	−0.449965	+	0.893046i	$0.648563\pi$
$318$	641144.	0.355540
$319$	313656.	0.172574
$320$	−1.94434e6	−1.06145
$321$	−983339.	−0.532648
$322$	−176876.	−0.0950670
$323$	−4.55537e6	−2.42950
$324$	887390.	0.469626
$325$	−257486.	−0.135222
$326$	−561472.	−0.292606
$327$	−5.14171e6	−2.65912
$328$	999449.	0.512951
$329$	−516627.	−0.263140
$330$	−1.19725e6	−0.605200
$331$	2.70621e6	1.35766	0.678831	−	0.734295i	$-0.262487\pi$
$331$	2.70621e6	1.35766	0.678831	+	0.734295i	$0.262487\pi$
$332$	−740102.	−0.368508
$333$	1.16124e6	0.573868
$334$	−284425.	−0.139509
$335$	−2.81743e6	−1.37165
$336$	3.72004e6	1.79763
$337$	871084.	0.417816	0.208908	−	0.977935i	$-0.433009\pi$
$337$	871084.	0.417816	0.208908	+	0.977935i	$0.433009\pi$
$338$	444927.	0.211835
$339$	−3.94083e6	−1.86246
$340$	5.78147e6	2.71232
$341$	0	0
$342$	−788787.	−0.364665
$343$	−503812.	−0.231224
$344$	1.09065e6	0.496923
$345$	1.61076e6	0.728590
$346$	397653.	0.178572
$347$	−328245.	−0.146344	−0.0731718	−	0.997319i	$-0.523312\pi$
$347$	−328245.	−0.146344	−0.0731718	+	0.997319i	$0.523312\pi$
$348$	−453647.	−0.200803
$349$	−736227.	−0.323555	−0.161778	−	0.986827i	$-0.551723\pi$
$349$	−736227.	−0.323555	−0.161778	+	0.986827i	$0.551723\pi$
$350$	−731366.	−0.319128
$351$	−162250.	−0.0702936
$352$	−1.77763e6	−0.764690
$353$	4.26934e6	1.82357	0.911787	−	0.410663i	$-0.134703\pi$
$353$	4.26934e6	1.82357	0.911787	+	0.410663i	$0.134703\pi$
$354$	−822726.	−0.348937
$355$	1.61157e6	0.678700
$356$	214272.	0.0896068
$357$	−9.84706e6	−4.08918
$358$	79856.6	0.0329309
$359$	2.29560e6	0.940072	0.470036	−	0.882647i	$-0.344241\pi$
$359$	2.29560e6	0.940072	0.470036	+	0.882647i	$0.344241\pi$
$360$	2.05075e6	0.833982
$361$	1.31331e6	0.530395
$362$	−701565.	−0.281382
$363$	2.29023e6	0.912248
$364$	401909.	0.158991
$365$	−3.64910e6	−1.43368
$366$	−378252.	−0.147597
$367$	−4.38724e6	−1.70030	−0.850152	−	0.526538i	$-0.823490\pi$
$367$	−4.38724e6	−1.70030	−0.850152	+	0.526538i	$0.823490\pi$
$368$	732945.	0.282131
$369$	4.37499e6	1.67267
$370$	−343491.	−0.130440
$371$	−3.84925e6	−1.45192
$372$	0	0
$373$	−4.93978e6	−1.83838	−0.919190	−	0.393814i	$-0.871155\pi$
$373$	−4.93978e6	−1.83838	−0.919190	+	0.393814i	$0.871155\pi$
$374$	1.44206e6	0.533094
$375$	588948.	0.216271
$376$	−224160.	−0.0817689
$377$	−46518.4	−0.0168567
$378$	−460855.	−0.165895
$379$	4.01947e6	1.43738	0.718688	−	0.695333i	$-0.244743\pi$
$379$	4.01947e6	1.43738	0.718688	+	0.695333i	$0.244743\pi$
$380$	−4.80935e6	−1.70855
$381$	6.95786e6	2.45563
$382$	−821663.	−0.288095
$383$	−535084.	−0.186391	−0.0931956	−	0.995648i	$-0.529708\pi$
$383$	−535084.	−0.186391	−0.0931956	+	0.995648i	$0.529708\pi$
$384$	3.39719e6	1.17569
$385$	7.18793e6	2.47145
$386$	298042.	0.101815
$387$	4.77422e6	1.62041
$388$	−710781.	−0.239694
$389$	3.98622e6	1.33563	0.667817	−	0.744326i	$-0.267229\pi$
$389$	3.98622e6	1.33563	0.667817	+	0.744326i	$0.267229\pi$
$390$	177564.	0.0591145
$391$	−1.94013e6	−0.641784
$392$	1.05998e6	0.348402
$393$	2.89033e6	0.943987
$394$	623107.	0.202219
$395$	−5.12973e6	−1.65425
$396$	−5.14698e6	−1.64936
$397$	−674935.	−0.214924	−0.107462	−	0.994209i	$-0.534272\pi$
$397$	−674935.	−0.214924	−0.107462	+	0.994209i	$0.534272\pi$
$398$	363929.	0.115162
$399$	8.19134e6	2.57586
$400$	3.03066e6	0.947080
$401$	−5.19608e6	−1.61367	−0.806835	−	0.590776i	$-0.798822\pi$
$401$	−5.19608e6	−1.61367	−0.806835	+	0.590776i	$0.798822\pi$
$402$	1.01640e6	0.313689
$403$	0	0
$404$	−1.88648e6	−0.575041
$405$	−2.35377e6	−0.713059
$406$	−132131.	−0.0397823
$407$	1.76601e6	0.528455
$408$	−4.27255e6	−1.27068
$409$	−91627.7	−0.0270844	−0.0135422	−	0.999908i	$-0.504311\pi$
$409$	−91627.7	−0.0270844	−0.0135422	+	0.999908i	$0.504311\pi$
$410$	−1.29411e6	−0.380199
$411$	−2.60728e6	−0.761349
$412$	3.66507e6	1.06375
$413$	4.93942e6	1.42495
$414$	−335944.	−0.0963309
$415$	1.96309e6	0.559526
$416$	263642.	0.0746931
$417$	−3.44864e6	−0.971197
$418$	−1.19959e6	−0.335807
$419$	3.16751e6	0.881420	0.440710	−	0.897649i	$-0.354727\pi$
$419$	3.16751e6	0.881420	0.440710	+	0.897649i	$0.354727\pi$
$420$	−1.03961e7	−2.87572
$421$	3.85120e6	1.05899	0.529493	−	0.848314i	$-0.322382\pi$
$421$	3.85120e6	1.05899	0.529493	+	0.848314i	$0.322382\pi$
$422$	1.11487e6	0.304751
$423$	−981238.	−0.266639
$424$	−1.67016e6	−0.451172
$425$	−8.02224e6	−2.15439
$426$	−581378.	−0.155215
$427$	2.27092e6	0.602743
$428$	1.25045e6	0.329956
$429$	−912922.	−0.239492
$430$	−1.41220e6	−0.368319
$431$	−1.32060e6	−0.342435	−0.171218	−	0.985233i	$-0.554770\pi$
$431$	−1.32060e6	−0.342435	−0.171218	+	0.985233i	$0.554770\pi$
$432$	1.90970e6	0.492330
$433$	2.61887e6	0.671264	0.335632	−	0.941993i	$-0.391050\pi$
$433$	2.61887e6	0.671264	0.335632	+	0.941993i	$0.391050\pi$
$434$	0	0
$435$	1.20328e6	0.304891
$436$	6.53837e6	1.64723
$437$	1.61391e6	0.404273
$438$	1.31642e6	0.327877
$439$	2.64481e6	0.654988	0.327494	−	0.944853i	$-0.393796\pi$
$439$	2.64481e6	0.654988	0.327494	+	0.944853i	$0.393796\pi$
$440$	3.11878e6	0.767985
$441$	4.63994e6	1.13610
$442$	−213873.	−0.0520714
$443$	−3.24058e6	−0.784538	−0.392269	−	0.919851i	$-0.628310\pi$
$443$	−3.24058e6	−0.784538	−0.392269	+	0.919851i	$0.628310\pi$
$444$	−2.55423e6	−0.614896
$445$	−568349.	−0.136055
$446$	279944.	0.0666397
$447$	1.68650e6	0.399225
$448$	−4.21117e6	−0.991307
$449$	1.72501e6	0.403810	0.201905	−	0.979405i	$-0.435287\pi$
$449$	1.72501e6	0.403810	0.201905	+	0.979405i	$0.435287\pi$
$450$	−1.38910e6	−0.323371
$451$	6.65348e6	1.54031
$452$	5.01129e6	1.15373
$453$	5.88237e6	1.34681
$454$	1.45102e6	0.330395
$455$	−1.06605e6	−0.241406
$456$	3.55415e6	0.800430
$457$	3.14112e6	0.703548	0.351774	−	0.936085i	$-0.385578\pi$
$457$	3.14112e6	0.703548	0.351774	+	0.936085i	$0.385578\pi$
$458$	695460.	0.154920
$459$	−5.05505e6	−1.11994
$460$	−2.04830e6	−0.451335
$461$	5.12837e6	1.12390	0.561949	−	0.827172i	$-0.310052\pi$
$461$	5.12837e6	1.12390	0.561949	+	0.827172i	$0.310052\pi$
$462$	−2.59307e6	−0.565209
$463$	−1.40561e6	−0.304727	−0.152363	−	0.988325i	$-0.548688\pi$
$463$	−1.40561e6	−0.304727	−0.152363	+	0.988325i	$0.548688\pi$
$464$	547529.	0.118063
$465$	0	0
$466$	−733191.	−0.156406
$467$	3.10368e6	0.658543	0.329271	−	0.944235i	$-0.393197\pi$
$467$	3.10368e6	0.658543	0.329271	+	0.944235i	$0.393197\pi$
$468$	763351.	0.161105
$469$	−6.10217e6	−1.28101
$470$	290247.	0.0606070
$471$	6.88864e6	1.43081
$472$	2.14317e6	0.442794
$473$	7.26062e6	1.49218
$474$	1.85057e6	0.378320
$475$	6.67335e6	1.35709
$476$	1.25219e7	2.53309
$477$	−7.31095e6	−1.47122
$478$	1.28344e6	0.256925
$479$	−5.66714e6	−1.12856	−0.564281	−	0.825583i	$-0.690846\pi$
$479$	−5.66714e6	−1.12856	−0.564281	+	0.825583i	$0.690846\pi$
$480$	−6.81956e6	−1.35099
$481$	−261918.	−0.0516182
$482$	−700.621	−0.000137362 0
$483$	3.48869e6	0.680446
$484$	−2.91234e6	−0.565104
$485$	1.88532e6	0.363940
$486$	1.48785e6	0.285739
$487$	−4.64591e6	−0.887664	−0.443832	−	0.896110i	$-0.646381\pi$
$487$	−4.64591e6	−0.887664	−0.443832	+	0.896110i	$0.646381\pi$
$488$	985331.	0.187298
$489$	1.10744e7	2.09434
$490$	−1.37248e6	−0.258235
$491$	3.54033e6	0.662734	0.331367	−	0.943502i	$-0.392490\pi$
$491$	3.54033e6	0.662734	0.331367	+	0.943502i	$0.392490\pi$
$492$	−9.62308e6	−1.79226
$493$	−1.44933e6	−0.268565
$494$	177911.	0.0328009
$495$	1.36522e7	2.50431
$496$	0	0
$497$	3.49043e6	0.633852
$498$	−708192.	−0.127961
$499$	8.46318e6	1.52154	0.760768	−	0.649024i	$-0.224823\pi$
$499$	8.46318e6	1.52154	0.760768	+	0.649024i	$0.224823\pi$
$500$	−748927.	−0.133972
$501$	5.60997e6	0.998542
$502$	466830.	0.0826798
$503$	−4.85894e6	−0.856291	−0.428145	−	0.903710i	$-0.640833\pi$
$503$	−4.85894e6	−0.856291	−0.428145	+	0.903710i	$0.640833\pi$
$504$	4.44165e6	0.778875
$505$	5.00382e6	0.873118
$506$	−510902.	−0.0887078
$507$	−8.77569e6	−1.51622
$508$	−8.84786e6	−1.52117
$509$	609647.	0.104300	0.0521500	−	0.998639i	$-0.483393\pi$
$509$	609647.	0.104300	0.0521500	+	0.998639i	$0.483393\pi$
$510$	5.53219e6	0.941828
$511$	−7.90344e6	−1.33895
$512$	−5.25521e6	−0.885962
$513$	4.20507e6	0.705472
$514$	−555300.	−0.0927085
$515$	−9.72144e6	−1.61515
$516$	−1.05012e7	−1.73626
$517$	−1.49226e6	−0.245538
$518$	−743955.	−0.121821
$519$	−7.84326e6	−1.27814
$520$	−462548.	−0.0750150
$521$	5.65251e6	0.912319	0.456159	−	0.889898i	$-0.349225\pi$
$521$	5.65251e6	0.912319	0.456159	+	0.889898i	$0.349225\pi$
$522$	−250959.	−0.0403113
$523$	−9.55036e6	−1.52674	−0.763371	−	0.645960i	$-0.776457\pi$
$523$	−9.55036e6	−1.52674	−0.763371	+	0.645960i	$0.776457\pi$
$524$	−3.67544e6	−0.584765
$525$	1.44254e7	2.28417
$526$	727840.	0.114702
$527$	0	0
$528$	1.07452e7	1.67738
$529$	−5.74898e6	−0.893206
$530$	2.16255e6	0.334408
$531$	9.38152e6	1.44390
$532$	−1.04164e7	−1.59565
$533$	−986781.	−0.150454
$534$	205034.	0.0311151
$535$	−3.31676e6	−0.500991
$536$	−2.64768e6	−0.398064
$537$	−1.57508e6	−0.235704
$538$	−2.37868e6	−0.354308
$539$	7.05641e6	1.04619
$540$	−5.33688e6	−0.787596
$541$	6.53766e6	0.960349	0.480175	−	0.877173i	$-0.340573\pi$
$541$	6.53766e6	0.960349	0.480175	+	0.877173i	$0.340573\pi$
$542$	−723064.	−0.105725
$543$	1.38376e7	2.01401
$544$	8.21402e6	1.19003
$545$	−1.73428e7	−2.50108
$546$	384580.	0.0552083
$547$	2.74038e6	0.391600	0.195800	−	0.980644i	$-0.437270\pi$
$547$	2.74038e6	0.391600	0.195800	+	0.980644i	$0.437270\pi$
$548$	3.31551e6	0.471627
$549$	4.31320e6	0.610757
$550$	−2.11253e6	−0.297781
$551$	1.20563e6	0.169175
$552$	1.51371e6	0.211444
$553$	−1.11103e7	−1.54494
$554$	−2.78139e6	−0.385023
$555$	6.77498e6	0.933632
$556$	4.38541e6	0.601621
$557$	−1.20334e7	−1.64343	−0.821715	−	0.569898i	$-0.806983\pi$
$557$	−1.20334e7	−1.64343	−0.821715	+	0.569898i	$0.806983\pi$
$558$	0	0
$559$	−1.07683e6	−0.145753
$560$	1.25475e7	1.69078
$561$	−2.84430e7	−3.81565
$562$	136372.	0.0182131
$563$	−199268.	−0.0264951	−0.0132476	−	0.999912i	$-0.504217\pi$
$563$	−199268.	−0.0264951	−0.0132476	+	0.999912i	$0.504217\pi$
$564$	2.15830e6	0.285702
$565$	−1.32922e7	−1.75177
$566$	1.78107e6	0.233691
$567$	−5.09793e6	−0.665942
$568$	1.51447e6	0.196965
$569$	−7.06938e6	−0.915379	−0.457689	−	0.889112i	$-0.651323\pi$
$569$	−7.06938e6	−0.915379	−0.457689	+	0.889112i	$0.651323\pi$
$570$	−4.60199e6	−0.593278
$571$	7.84100e6	1.00642	0.503212	−	0.864163i	$-0.332151\pi$
$571$	7.84100e6	1.00642	0.503212	+	0.864163i	$0.332151\pi$
$572$	1.16090e6	0.148356
$573$	1.62064e7	2.06205
$574$	−2.80286e6	−0.355076
$575$	2.84218e6	0.358494
$576$	−7.99835e6	−1.00449
$577$	1.25739e7	1.57228	0.786138	−	0.618051i	$-0.212078\pi$
$577$	1.25739e7	1.57228	0.786138	+	0.618051i	$0.212078\pi$
$578$	−4.93572e6	−0.614513
$579$	−5.87856e6	−0.728743
$580$	−1.53013e6	−0.188868
$581$	4.25179e6	0.522554
$582$	−680135.	−0.0832315
$583$	−1.11185e7	−1.35480
$584$	−3.42923e6	−0.416068
$585$	−2.02476e6	−0.244615
$586$	−192373.	−0.0231420
$587$	−4.16970e6	−0.499471	−0.249735	−	0.968314i	$-0.580344\pi$
$587$	−4.16970e6	−0.499471	−0.249735	+	0.968314i	$0.580344\pi$
$588$	−1.02059e7	−1.21732
$589$	0	0
$590$	−2.77502e6	−0.328198
$591$	−1.22901e7	−1.44739
$592$	3.08282e6	0.361530
$593$	−7.45642e6	−0.870750	−0.435375	−	0.900249i	$-0.643384\pi$
$593$	−7.45642e6	−0.870750	−0.435375	+	0.900249i	$0.643384\pi$
$594$	−1.33117e6	−0.154798
$595$	−3.32137e7	−3.84614
$596$	−2.14461e6	−0.247305
$597$	−7.17810e6	−0.824277
$598$	75772.2	0.00866477
$599$	−1.39190e7	−1.58504	−0.792519	−	0.609848i	$-0.791231\pi$
$599$	−1.39190e7	−1.58504	−0.792519	+	0.609848i	$0.791231\pi$
$600$	6.25904e6	0.709790
$601$	−2.17085e6	−0.245157	−0.122578	−	0.992459i	$-0.539116\pi$
$601$	−2.17085e6	−0.245157	−0.122578	+	0.992459i	$0.539116\pi$
$602$	−3.05862e6	−0.343981
$603$	−1.15900e7	−1.29804
$604$	−7.48022e6	−0.834299
$605$	7.72486e6	0.858029
$606$	−1.80514e6	−0.199678
$607$	3.26024e6	0.359152	0.179576	−	0.983744i	$-0.442527\pi$
$607$	3.26024e6	0.359152	0.179576	+	0.983744i	$0.442527\pi$
$608$	−6.83288e6	−0.749626
$609$	2.60614e6	0.284744
$610$	−1.27583e6	−0.138825
$611$	221319.	0.0239836
$612$	2.37830e7	2.56677
$613$	3.65825e6	0.393207	0.196604	−	0.980483i	$-0.437009\pi$
$613$	3.65825e6	0.393207	0.196604	+	0.980483i	$0.437009\pi$
$614$	1.66017e6	0.177718
$615$	2.55248e7	2.72129
$616$	6.75484e6	0.717238
$617$	−1.34766e7	−1.42518	−0.712588	−	0.701582i	$-0.752477\pi$
$617$	−1.34766e7	−1.42518	−0.712588	+	0.701582i	$0.752477\pi$
$618$	3.50704e6	0.369377
$619$	−3.84410e6	−0.403244	−0.201622	−	0.979463i	$-0.564621\pi$
$619$	−3.84410e6	−0.403244	−0.201622	+	0.979463i	$0.564621\pi$
$620$	0	0
$621$	1.79094e6	0.186359
$622$	−2.50406e6	−0.259519
$623$	−1.23096e6	−0.127065
$624$	−1.59363e6	−0.163843
$625$	−8.72643e6	−0.893586
$626$	−1.77005e6	−0.180531
$627$	2.36605e7	2.40356
$628$	−8.75983e6	−0.886333
$629$	−8.16033e6	−0.822396
$630$	−5.75114e6	−0.577301
$631$	4.72147e6	0.472067	0.236033	−	0.971745i	$-0.424153\pi$
$631$	4.72147e6	0.472067	0.236033	+	0.971745i	$0.424153\pi$
$632$	−4.82066e6	−0.480080
$633$	−2.19897e7	−2.18127
$634$	1.95920e6	0.193577
$635$	2.34686e7	2.30968
$636$	1.60809e7	1.57640
$637$	−1.04654e6	−0.102190
$638$	−381658.	−0.0371212
$639$	6.62943e6	0.642280
$640$	1.14586e7	1.10581
$641$	6.19170e6	0.595202	0.297601	−	0.954690i	$-0.403813\pi$
$641$	6.19170e6	0.595202	0.297601	+	0.954690i	$0.403813\pi$
$642$	1.19653e6	0.114574
$643$	−1.34263e7	−1.28064	−0.640322	−	0.768106i	$-0.721199\pi$
$643$	−1.34263e7	−1.28064	−0.640322	+	0.768106i	$0.721199\pi$
$644$	−4.43633e6	−0.421511
$645$	2.78540e7	2.63626
$646$	5.54300e6	0.522593
$647$	−1.70903e7	−1.60505	−0.802526	−	0.596617i	$-0.796511\pi$
$647$	−1.70903e7	−1.60505	−0.802526	+	0.596617i	$0.796511\pi$
$648$	−2.21195e6	−0.206936
$649$	1.42674e7	1.32964
$650$	313311.	0.0290865
$651$	0	0
$652$	−1.40826e7	−1.29737
$653$	−5.78620e6	−0.531020	−0.265510	−	0.964108i	$-0.585540\pi$
$653$	−5.78620e6	−0.531020	−0.265510	+	0.964108i	$0.585540\pi$
$654$	6.25646e6	0.571984
$655$	9.74896e6	0.887882
$656$	1.16146e7	1.05376
$657$	−1.50111e7	−1.35675
$658$	628635.	0.0566022
$659$	1.67851e7	1.50560	0.752801	−	0.658249i	$-0.228702\pi$
$659$	1.67851e7	1.50560	0.752801	+	0.658249i	$0.228702\pi$
$660$	−3.00288e7	−2.68336
$661$	−1.50047e7	−1.33575	−0.667873	−	0.744275i	$-0.732795\pi$
$661$	−1.50047e7	−1.33575	−0.667873	+	0.744275i	$0.732795\pi$
$662$	−3.29293e6	−0.292037
$663$	4.21840e6	0.372704
$664$	1.84481e6	0.162380
$665$	2.76290e7	2.42277
$666$	−1.41300e6	−0.123441
$667$	513478.	0.0446897
$668$	−7.13383e6	−0.618560
$669$	−5.52157e6	−0.476977
$670$	3.42827e6	0.295045
$671$	6.55950e6	0.562425
$672$	−1.47702e7	−1.26172
$673$	1.49353e7	1.27109	0.635544	−	0.772065i	$-0.280776\pi$
$673$	1.49353e7	1.27109	0.635544	+	0.772065i	$0.280776\pi$
$674$	−1.05994e6	−0.0898735
$675$	7.40535e6	0.625585
$676$	1.11595e7	0.939240
$677$	9.81799e6	0.823286	0.411643	−	0.911345i	$-0.364955\pi$
$677$	9.81799e6	0.823286	0.411643	+	0.911345i	$0.364955\pi$
$678$	4.79522e6	0.400622
$679$	4.08334e6	0.339892
$680$	−1.44111e7	−1.19516
$681$	−2.86197e7	−2.36482
$682$	0	0
$683$	−2.06589e7	−1.69455	−0.847277	−	0.531152i	$-0.821759\pi$
$683$	−2.06589e7	−1.69455	−0.847277	+	0.531152i	$0.821759\pi$
$684$	−1.97840e7	−1.61687
$685$	−8.79426e6	−0.716099
$686$	613041.	0.0497370
$687$	−1.37172e7	−1.10885
$688$	1.26744e7	1.02084
$689$	1.64899e6	0.132333
$690$	−1.95998e6	−0.156722
$691$	−1.40490e7	−1.11931	−0.559657	−	0.828725i	$-0.689067\pi$
$691$	−1.40490e7	−1.11931	−0.559657	+	0.828725i	$0.689067\pi$
$692$	9.97375e6	0.791760
$693$	2.95687e7	2.33883
$694$	399410.	0.0314789
$695$	−1.16321e7	−0.913475
$696$	1.13078e6	0.0884821
$697$	−3.07441e7	−2.39707
$698$	895845.	0.0695976
$699$	1.44614e7	1.11948
$700$	−1.83438e7	−1.41496
$701$	−4.52609e6	−0.347879	−0.173939	−	0.984756i	$-0.555650\pi$
$701$	−4.52609e6	−0.347879	−0.173939	+	0.984756i	$0.555650\pi$
$702$	197426.	0.0151204
$703$	6.78822e6	0.518045
$704$	−1.21639e7	−0.924997
$705$	−5.72480e6	−0.433798
$706$	−5.19495e6	−0.392256
$707$	1.08376e7	0.815424
$708$	−2.06353e7	−1.54713
$709$	−1.35285e7	−1.01073	−0.505364	−	0.862906i	$-0.668642\pi$
$709$	−1.35285e7	−1.01073	−0.505364	+	0.862906i	$0.668642\pi$
$710$	−1.96096e6	−0.145990
$711$	−2.11020e7	−1.56549
$712$	−534104.	−0.0394844
$713$	0	0
$714$	1.19820e7	0.879594
$715$	−3.07925e6	−0.225258
$716$	2.00293e6	0.146010
$717$	−2.53144e7	−1.83895
$718$	−2.79330e6	−0.202212
$719$	−1.91936e7	−1.38463	−0.692316	−	0.721595i	$-0.743409\pi$
$719$	−1.91936e7	−1.38463	−0.692316	+	0.721595i	$0.743409\pi$
$720$	2.38317e7	1.71326
$721$	−2.10553e7	−1.50842
$722$	−1.59804e6	−0.114090
$723$	13819.0	0.000983173 0
$724$	−1.75963e7	−1.24760
$725$	2.12318e6	0.150018
$726$	−2.78677e6	−0.196227
$727$	−2.19694e7	−1.54164	−0.770820	−	0.637053i	$-0.780153\pi$
$727$	−2.19694e7	−1.54164	−0.770820	+	0.637053i	$0.780153\pi$
$728$	−1.00181e6	−0.0700582
$729$	−2.22808e7	−1.55278
$730$	4.44024e6	0.308390
$731$	−3.35496e7	−2.32217
$732$	−9.48715e6	−0.654422
$733$	1.81467e7	1.24749	0.623746	−	0.781627i	$-0.285610\pi$
$733$	1.81467e7	1.24749	0.623746	+	0.781627i	$0.285610\pi$
$734$	5.33842e6	0.365740
$735$	2.70706e7	1.84833
$736$	−2.91012e6	−0.198023
$737$	−1.76260e7	−1.19532
$738$	−5.32352e6	−0.359797
$739$	−2.14767e7	−1.44663	−0.723315	−	0.690519i	$-0.757382\pi$
$739$	−2.14767e7	−1.44663	−0.723315	+	0.690519i	$0.757382\pi$
$740$	−8.61530e6	−0.578350
$741$	−3.50910e6	−0.234774
$742$	4.68379e6	0.312311
$743$	1.10502e7	0.734342	0.367171	−	0.930153i	$-0.380326\pi$
$743$	1.10502e7	0.734342	0.367171	+	0.930153i	$0.380326\pi$
$744$	0	0
$745$	5.68849e6	0.375497
$746$	6.01075e6	0.395441
$747$	8.07549e6	0.529501
$748$	3.61691e7	2.36365
$749$	−7.18364e6	−0.467886
$750$	−716635.	−0.0465206
$751$	−8.12377e6	−0.525603	−0.262801	−	0.964850i	$-0.584646\pi$
$751$	−8.12377e6	−0.525603	−0.262801	+	0.964850i	$0.584646\pi$
$752$	−2.60495e6	−0.167979
$753$	−9.20770e6	−0.591785
$754$	56603.9	0.00362592
$755$	1.98410e7	1.26676
$756$	−1.15590e7	−0.735553
$757$	78813.2	0.00499873	0.00249936	−	0.999997i	$-0.499204\pi$
$757$	78813.2	0.00499873	0.00249936	+	0.999997i	$0.499204\pi$
$758$	−4.89091e6	−0.309184
$759$	1.00770e7	0.634930
$760$	1.19880e7	0.752857
$761$	2.73850e6	0.171416	0.0857078	−	0.996320i	$-0.472685\pi$
$761$	2.73850e6	0.171416	0.0857078	+	0.996320i	$0.472685\pi$
$762$	−8.46637e6	−0.528214
$763$	−3.75620e7	−2.33581
$764$	−2.06086e7	−1.27737
$765$	−6.30834e7	−3.89728
$766$	651094.	0.0400933
$767$	−2.11600e6	−0.129876
$768$	1.43127e7	0.875624
$769$	−1.92734e7	−1.17528	−0.587640	−	0.809122i	$-0.699943\pi$
$769$	−1.92734e7	−1.17528	−0.587640	+	0.809122i	$0.699943\pi$
$770$	−8.74632e6	−0.531616
$771$	1.09527e7	0.663566
$772$	7.47537e6	0.451429
$773$	1.14370e7	0.688433	0.344216	−	0.938890i	$-0.388145\pi$
$773$	1.14370e7	0.688433	0.344216	+	0.938890i	$0.388145\pi$
$774$	−5.80930e6	−0.348555
$775$	0	0
$776$	1.77172e6	0.105619
$777$	1.46737e7	0.871940
$778$	−4.85046e6	−0.287299
$779$	2.55747e7	1.50996
$780$	4.45359e6	0.262104
$781$	1.00820e7	0.591453
$782$	2.36076e6	0.138049
$783$	1.33788e6	0.0779852
$784$	1.23179e7	0.715728
$785$	2.32351e7	1.34577
$786$	−3.51697e6	−0.203054
$787$	345460.	0.0198820	0.00994102	−	0.999951i	$-0.496836\pi$
$787$	345460.	0.0198820	0.00994102	+	0.999951i	$0.496836\pi$
$788$	1.56285e7	0.896606
$789$	−1.43558e7	−0.820986
$790$	6.24189e6	0.355835
$791$	−2.87892e7	−1.63602
$792$	1.28296e7	0.726774
$793$	−972843.	−0.0549363
$794$	821265.	0.0462308
$795$	−4.26540e7	−2.39354
$796$	9.12791e6	0.510609
$797$	−1.10741e7	−0.617538	−0.308769	−	0.951137i	$-0.599917\pi$
$797$	−1.10741e7	−0.617538	−0.308769	+	0.951137i	$0.599917\pi$
$798$	−9.96727e6	−0.554075
$799$	6.89540e6	0.382114
$800$	−1.20331e7	−0.664739
$801$	−2.33799e6	−0.128754
$802$	6.32262e6	0.347105
$803$	−2.28289e7	−1.24938
$804$	2.54929e7	1.39084
$805$	1.17672e7	0.640005
$806$	0	0
$807$	4.69169e7	2.53598
$808$	4.70232e6	0.253387
$809$	−2.88079e7	−1.54754	−0.773768	−	0.633469i	$-0.781630\pi$
$809$	−2.88079e7	−1.54754	−0.773768	+	0.633469i	$0.781630\pi$
$810$	2.86407e6	0.153381
$811$	1.44089e7	0.769272	0.384636	−	0.923068i	$-0.374327\pi$
$811$	1.44089e7	0.769272	0.384636	+	0.923068i	$0.374327\pi$
$812$	−3.31406e6	−0.176388
$813$	1.42616e7	0.756734
$814$	−2.14889e6	−0.113672
$815$	3.73535e7	1.96987
$816$	−4.96512e7	−2.61038
$817$	2.79084e7	1.46278
$818$	111493.	0.00582592
$819$	−4.38535e6	−0.228452
$820$	−3.24582e7	−1.68574
$821$	1.03391e7	0.535335	0.267668	−	0.963511i	$-0.413747\pi$
$821$	1.03391e7	0.535335	0.267668	+	0.963511i	$0.413747\pi$
$822$	3.17256e6	0.163768
$823$	−2.20396e7	−1.13424	−0.567118	−	0.823637i	$-0.691942\pi$
$823$	−2.20396e7	−1.13424	−0.567118	+	0.823637i	$0.691942\pi$
$824$	−9.13570e6	−0.468731
$825$	4.16674e7	2.13138
$826$	−6.01031e6	−0.306512
$827$	−562181.	−0.0285833	−0.0142916	−	0.999898i	$-0.504549\pi$
$827$	−562181.	−0.0285833	−0.0142916	+	0.999898i	$0.504549\pi$
$828$	−8.42599e6	−0.427116
$829$	−2.09616e7	−1.05935	−0.529674	−	0.848201i	$-0.677686\pi$
$829$	−2.09616e7	−1.05935	−0.529674	+	0.848201i	$0.677686\pi$
$830$	−2.38870e6	−0.120356
$831$	5.48598e7	2.75582
$832$	1.80403e6	0.0903516
$833$	−3.26060e7	−1.62811
$834$	4.19632e6	0.208907
$835$	1.89222e7	0.939194
$836$	−3.00875e7	−1.48892
$837$	0	0
$838$	−3.85424e6	−0.189596
$839$	9.20970e6	0.451690	0.225845	−	0.974163i	$-0.427486\pi$
$839$	9.20970e6	0.451690	0.225845	+	0.974163i	$0.427486\pi$
$840$	2.59137e7	1.26716
$841$	−2.01276e7	−0.981299
$842$	−4.68616e6	−0.227791
$843$	−2.68978e6	−0.130361
$844$	2.79628e7	1.35122
$845$	−2.96000e7	−1.42610
$846$	1.19398e6	0.0573548
$847$	1.67310e7	0.801333
$848$	−1.94088e7	−0.926851
$849$	−3.51297e7	−1.67265
$850$	9.76151e6	0.463415
$851$	2.89110e6	0.136848
$852$	−1.45819e7	−0.688199
$853$	1.87872e6	0.0884074	0.0442037	−	0.999023i	$-0.485925\pi$
$853$	1.87872e6	0.0884074	0.0442037	+	0.999023i	$0.485925\pi$
$854$	−2.76327e6	−0.129652
$855$	5.24763e7	2.45498
$856$	−3.11692e6	−0.145392
$857$	3.75786e7	1.74779	0.873893	−	0.486119i	$-0.161588\pi$
$857$	3.75786e7	1.74779	0.873893	+	0.486119i	$0.161588\pi$
$858$	1.11085e6	0.0515154
$859$	−3.98087e7	−1.84075	−0.920375	−	0.391037i	$-0.872116\pi$
$859$	−3.98087e7	−1.84075	−0.920375	+	0.391037i	$0.872116\pi$
$860$	−3.54201e7	−1.63307
$861$	5.52833e7	2.54148
$862$	1.60691e6	0.0736588
$863$	8.80298e6	0.402349	0.201174	−	0.979555i	$-0.435524\pi$
$863$	8.80298e6	0.402349	0.201174	+	0.979555i	$0.435524\pi$
$864$	−7.58237e6	−0.345558
$865$	−2.64550e7	−1.20217
$866$	−3.18665e6	−0.144391
$867$	9.73516e7	4.39841
$868$	0	0
$869$	−3.20918e7	−1.44160
$870$	−1.46416e6	−0.0655828
$871$	2.61412e6	0.116756
$872$	−1.62978e7	−0.725836
$873$	7.75555e6	0.344411
$874$	−1.96381e6	−0.0869603
$875$	4.30248e6	0.189976
$876$	3.30180e7	1.45375
$877$	−5.16712e6	−0.226856	−0.113428	−	0.993546i	$-0.536183\pi$
$877$	−5.16712e6	−0.226856	−0.113428	+	0.993546i	$0.536183\pi$
$878$	−3.21822e6	−0.140890
$879$	3.79435e6	0.165640
$880$	3.62432e7	1.57769
$881$	1.77656e7	0.771151	0.385576	−	0.922676i	$-0.374003\pi$
$881$	1.77656e7	0.771151	0.385576	+	0.922676i	$0.374003\pi$
$882$	−5.64591e6	−0.244378
$883$	2.22016e7	0.958258	0.479129	−	0.877744i	$-0.340953\pi$
$883$	2.22016e7	0.958258	0.479129	+	0.877744i	$0.340953\pi$
$884$	−5.36426e6	−0.230876
$885$	5.47342e7	2.34910
$886$	3.94316e6	0.168756
$887$	5.04692e6	0.215386	0.107693	−	0.994184i	$-0.465654\pi$
$887$	5.04692e6	0.215386	0.107693	+	0.994184i	$0.465654\pi$
$888$	6.36677e6	0.270949
$889$	5.08297e7	2.15707
$890$	691570.	0.0292658
$891$	−1.47252e7	−0.621396
$892$	7.02142e6	0.295470
$893$	−5.73598e6	−0.240702
$894$	−2.05214e6	−0.0858743
$895$	−5.31269e6	−0.221695
$896$	2.48177e7	1.03274
$897$	−1.49452e6	−0.0620185
$898$	−2.09901e6	−0.0868607
$899$	0	0
$900$	−3.48407e7	−1.43377
$901$	5.13758e7	2.10837
$902$	−8.09599e6	−0.331325
$903$	6.03280e7	2.46206
$904$	−1.24914e7	−0.508380
$905$	4.66736e7	1.89431
$906$	−7.15770e6	−0.289703
$907$	1.61290e7	0.651013	0.325507	−	0.945540i	$-0.394465\pi$
$907$	1.61290e7	0.651013	0.325507	+	0.945540i	$0.394465\pi$
$908$	3.63938e7	1.46492
$909$	2.05840e7	0.826266
$910$	1.29717e6	0.0519271
$911$	753030.	0.0300619	0.0150309	−	0.999887i	$-0.495215\pi$
$911$	753030.	0.0300619	0.0150309	+	0.999887i	$0.495215\pi$
$912$	4.13026e7	1.64434
$913$	1.22812e7	0.487599
$914$	−3.82213e6	−0.151335
$915$	2.51643e7	0.993647
$916$	1.74432e7	0.686891
$917$	2.11149e7	0.829212
$918$	6.15101e6	0.240902
$919$	2.81472e7	1.09938	0.549688	−	0.835370i	$-0.314747\pi$
$919$	2.81472e7	1.09938	0.549688	+	0.835370i	$0.314747\pi$
$920$	5.10568e6	0.198877
$921$	−3.27451e7	−1.27203
$922$	−6.24022e6	−0.241754
$923$	−1.49527e6	−0.0577718
$924$	−6.50383e7	−2.50605
$925$	1.19544e7	0.459382
$926$	1.71035e6	0.0655476
$927$	−3.99907e7	−1.52848
$928$	−2.17394e6	−0.0828661
$929$	−1.47192e7	−0.559559	−0.279780	−	0.960064i	$-0.590261\pi$
$929$	−1.47192e7	−0.559559	−0.279780	+	0.960064i	$0.590261\pi$
$930$	0	0
$931$	2.71235e7	1.02558
$932$	−1.83896e7	−0.693477
$933$	4.93898e7	1.85752
$934$	−3.77657e6	−0.141654
$935$	−9.59371e7	−3.58887
$936$	−1.90276e6	−0.0709896
$937$	3.08692e6	0.114862	0.0574309	−	0.998349i	$-0.481709\pi$
$937$	3.08692e6	0.114862	0.0574309	+	0.998349i	$0.481709\pi$
$938$	7.42516e6	0.275549
$939$	3.49124e7	1.29216
$940$	7.27984e6	0.268722
$941$	−3.56078e7	−1.31091	−0.655453	−	0.755236i	$-0.727522\pi$
$941$	−3.56078e7	−1.31091	−0.655453	+	0.755236i	$0.727522\pi$
$942$	−8.38214e6	−0.307771
$943$	1.08922e7	0.398876
$944$	2.49057e7	0.909639
$945$	3.06597e7	1.11683
$946$	−8.83476e6	−0.320972
$947$	4.19014e7	1.51829	0.759144	−	0.650923i	$-0.225618\pi$
$947$	4.19014e7	1.51829	0.759144	+	0.650923i	$0.225618\pi$
$948$	4.64151e7	1.67741
$949$	3.38577e6	0.122037
$950$	−8.12017e6	−0.291915
$951$	−3.86430e7	−1.38554
$952$	−3.12125e7	−1.11619
$953$	1.51476e7	0.540271	0.270136	−	0.962822i	$-0.412931\pi$
$953$	1.51476e7	0.540271	0.270136	+	0.962822i	$0.412931\pi$
$954$	8.89600e6	0.316464
$955$	5.46635e7	1.93950
$956$	3.21907e7	1.13916
$957$	7.52778e6	0.265697
$958$	6.89581e6	0.242757
$959$	−1.90471e7	−0.668780
$960$	−4.66644e7	−1.63421
$961$	0	0
$962$	318704.	0.0111032
$963$	−1.36440e7	−0.474107
$964$	−17572.7	−0.000609039 0
$965$	−1.98281e7	−0.685430
$966$	−4.24505e6	−0.146366
$967$	−8.64612e6	−0.297341	−0.148671	−	0.988887i	$-0.547499\pi$
$967$	−8.64612e6	−0.297341	−0.148671	+	0.988887i	$0.547499\pi$
$968$	7.25942e6	0.249008
$969$	−1.09329e8	−3.74048
$970$	−2.29407e6	−0.0782846
$971$	3.08834e7	1.05118	0.525591	−	0.850738i	$-0.323844\pi$
$971$	3.08834e7	1.05118	0.525591	+	0.850738i	$0.323844\pi$
$972$	3.73177e7	1.26692
$973$	−2.51935e7	−0.853114
$974$	5.65317e6	0.190939
$975$	−6.17971e6	−0.208188
$976$	1.14505e7	0.384769
$977$	−2.81945e7	−0.944993	−0.472497	−	0.881333i	$-0.656647\pi$
$977$	−2.81945e7	−0.944993	−0.472497	+	0.881333i	$0.656647\pi$
$978$	−1.34754e7	−0.450500
$979$	−3.55561e6	−0.118565
$980$	−3.44239e7	−1.14497
$981$	−7.13422e7	−2.36687
$982$	−4.30789e6	−0.142556
$983$	4.06003e7	1.34013	0.670063	−	0.742304i	$-0.266267\pi$
$983$	4.06003e7	1.34013	0.670063	+	0.742304i	$0.266267\pi$
$984$	2.39869e7	0.789744
$985$	−4.14540e7	−1.36137
$986$	1.76355e6	0.0577691
$987$	−1.23991e7	−0.405133
$988$	4.46229e6	0.145434
$989$	1.18862e7	0.386413
$990$	−1.66120e7	−0.538685
$991$	1.11802e7	0.361631	0.180815	−	0.983517i	$-0.442126\pi$
$991$	1.11802e7	0.361631	0.180815	+	0.983517i	$0.442126\pi$
$992$	0	0
$993$	6.49494e7	2.09027
$994$	−4.24718e6	−0.136343
$995$	−2.42114e7	−0.775287
$996$	−1.77626e7	−0.567358
$997$	2.64679e7	0.843298	0.421649	−	0.906759i	$-0.361451\pi$
$997$	2.64679e7	0.843298	0.421649	+	0.906759i	$0.361451\pi$
$998$	−1.02980e7	−0.327287
$999$	7.53281e6	0.238805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.6.a.c.1.4		8
31.30	odd	2		31.6.a.b.1.4	✓	8
93.92	even	2		279.6.a.f.1.5		8
124.123	even	2		496.6.a.h.1.7		8
155.154	odd	2		775.6.a.b.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.6.a.b.1.4	✓	8	31.30	odd	2
279.6.a.f.1.5		8	93.92	even	2
496.6.a.h.1.7		8	124.123	even	2
775.6.a.b.1.5		8	155.154	odd	2
961.6.a.c.1.4		8	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:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21681 q^{2} +24.0001 q^{3} -30.5194 q^{4} +80.9514 q^{5} -29.2035 q^{6} +175.330 q^{7} +76.0739 q^{8} +333.006 q^{9} -98.5022 q^{10} +506.435 q^{11} -732.469 q^{12} -75.1097 q^{13} -213.342 q^{14} +1942.85 q^{15} +884.053 q^{16} -2340.12 q^{17} -405.204 q^{18} +1946.64 q^{19} -2470.59 q^{20} +4207.94 q^{21} -616.233 q^{22} +829.073 q^{23} +1825.78 q^{24} +3428.14 q^{25} +91.3939 q^{26} +2160.17 q^{27} -5350.95 q^{28} +619.340 q^{29} -2364.07 q^{30} -3510.09 q^{32} +12154.5 q^{33} +2847.47 q^{34} +14193.2 q^{35} -10163.2 q^{36} +3487.14 q^{37} -2368.68 q^{38} -1802.64 q^{39} +6158.29 q^{40} +13137.9 q^{41} -5120.24 q^{42} +14336.7 q^{43} -15456.1 q^{44} +26957.4 q^{45} -1008.82 q^{46} -2946.60 q^{47} +21217.4 q^{48} +13933.5 q^{49} -4171.38 q^{50} -56163.1 q^{51} +2292.30 q^{52} -21954.4 q^{53} -2628.50 q^{54} +40996.7 q^{55} +13338.0 q^{56} +46719.6 q^{57} -753.616 q^{58} +28172.2 q^{59} -59294.5 q^{60} +12952.3 q^{61} +58385.9 q^{63} -24018.6 q^{64} -6080.24 q^{65} -14789.7 q^{66} -34804.0 q^{67} +71419.0 q^{68} +19897.9 q^{69} -17270.4 q^{70} +19907.8 q^{71} +25333.1 q^{72} -45077.6 q^{73} -4243.18 q^{74} +82275.7 q^{75} -59410.3 q^{76} +88793.1 q^{77} +2193.47 q^{78} -63368.0 q^{79} +71565.4 q^{80} -29076.3 q^{81} -15986.2 q^{82} +24250.2 q^{83} -128424. q^{84} -189436. q^{85} -17445.0 q^{86} +14864.2 q^{87} +38526.5 q^{88} -7020.86 q^{89} -32801.9 q^{90} -13169.0 q^{91} -25302.8 q^{92} +3585.44 q^{94} +157583. q^{95} -84242.5 q^{96} +23289.5 q^{97} -16954.3 q^{98} +168646. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 7 q^{2} + 2 q^{3} + 149 q^{4} + 128 q^{5} - 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} - 574 q^{11} + 46 q^{12} + 122 q^{13} - 309 q^{14} + 524 q^{15} + 833 q^{16} - 1932 q^{17} - 6845 q^{18}+ \cdots + 180150 q^{99}+O(q^{100}) \) 8 * q + 7 * q^2 + 2 * q^3 + 149 * q^4 + 128 * q^5 - 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 - 574 * q^11 + 46 * q^12 + 122 * q^13 - 309 * q^14 + 524 * q^15 + 833 * q^16 - 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 - 996 * q^21 + 9000 * q^22 + 4136 * q^23 + 17468 * q^24 + 3308 * q^25 - 1524 * q^26 + 2624 * q^27 - 21245 * q^28 - 5050 * q^29 + 30450 * q^30 + 21045 * q^32 + 27920 * q^33 + 24738 * q^34 + 25700 * q^35 + 21637 * q^36 - 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 + 9994 * q^42 - 21058 * q^43 - 13168 * q^44 - 1344 * q^45 - 6198 * q^46 + 5348 * q^47 + 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 + 60386 * q^52 - 4926 * q^53 + 24476 * q^54 + 56892 * q^55 - 87652 * q^56 - 22072 * q^57 + 25002 * q^58 + 16944 * q^59 + 138556 * q^60 + 73682 * q^61 - 138784 * q^63 - 1300 * q^64 - 63316 * q^65 - 128796 * q^66 - 134768 * q^67 + 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 - 70792 * q^73 - 307468 * q^74 + 302902 * q^75 - 65405 * q^76 - 107932 * q^77 + 36820 * q^78 + 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 - 21882 * q^83 - 299144 * q^84 + 130464 * q^85 - 480478 * q^86 - 440244 * q^87 + 71982 * q^88 - 164392 * q^89 + 397969 * q^90 + 60028 * q^91 - 31012 * q^92 - 170388 * q^94 - 7404 * q^95 + 331010 * q^96 - 131948 * q^97 + 288768 * q^98 + 180150 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more