LMFDB - Embedded newform 538.6.a.c.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,6,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, 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("538.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	538.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:	$86.2864950594$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +2.68265 q^{3} +16.0000 q^{4} +25.8421 q^{5} -10.7306 q^{6} +96.3372 q^{7} -64.0000 q^{8} -235.803 q^{9} -103.369 q^{10} -93.7407 q^{11} +42.9224 q^{12} -667.002 q^{13} -385.349 q^{14} +69.3253 q^{15} +256.000 q^{16} +1784.88 q^{17} +943.214 q^{18} +2442.53 q^{19} +413.474 q^{20} +258.439 q^{21} +374.963 q^{22} +700.737 q^{23} -171.689 q^{24} -2457.18 q^{25} +2668.01 q^{26} -1284.46 q^{27} +1541.40 q^{28} -5936.54 q^{29} -277.301 q^{30} -8370.35 q^{31} -1024.00 q^{32} -251.473 q^{33} -7139.51 q^{34} +2489.56 q^{35} -3772.85 q^{36} -2065.32 q^{37} -9770.11 q^{38} -1789.33 q^{39} -1653.90 q^{40} -12488.4 q^{41} -1033.75 q^{42} +14743.9 q^{43} -1499.85 q^{44} -6093.66 q^{45} -2802.95 q^{46} +10629.1 q^{47} +686.758 q^{48} -7526.14 q^{49} +9828.74 q^{50} +4788.20 q^{51} -10672.0 q^{52} +7099.35 q^{53} +5137.84 q^{54} -2422.46 q^{55} -6165.58 q^{56} +6552.44 q^{57} +23746.2 q^{58} +44165.1 q^{59} +1109.21 q^{60} +12326.7 q^{61} +33481.4 q^{62} -22716.6 q^{63} +4096.00 q^{64} -17236.8 q^{65} +1005.89 q^{66} -29753.4 q^{67} +28558.0 q^{68} +1879.83 q^{69} -9958.23 q^{70} -42415.5 q^{71} +15091.4 q^{72} -11634.1 q^{73} +8261.28 q^{74} -6591.76 q^{75} +39080.4 q^{76} -9030.72 q^{77} +7157.32 q^{78} +3427.83 q^{79} +6615.59 q^{80} +53854.5 q^{81} +49953.7 q^{82} -62674.7 q^{83} +4135.02 q^{84} +46125.0 q^{85} -58975.4 q^{86} -15925.7 q^{87} +5999.40 q^{88} -145342. q^{89} +24374.6 q^{90} -64257.1 q^{91} +11211.8 q^{92} -22454.7 q^{93} -42516.6 q^{94} +63120.1 q^{95} -2747.03 q^{96} -34590.9 q^{97} +30104.6 q^{98} +22104.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 120 q^{2} - 30 q^{3} + 480 q^{4} - 136 q^{5} + 120 q^{6} - 123 q^{7} - 1920 q^{8} + 2670 q^{9} + 544 q^{10} - 1058 q^{11} - 480 q^{12} - 371 q^{13} + 492 q^{14} - 1364 q^{15} + 7680 q^{16} - 1918 q^{17}+ \cdots - 78063 q^{99}+O(q^{100}) $ 30 * q - 120 * q^2 - 30 * q^3 + 480 * q^4 - 136 * q^5 + 120 * q^6 - 123 * q^7 - 1920 * q^8 + 2670 * q^9 + 544 * q^10 - 1058 * q^11 - 480 * q^12 - 371 * q^13 + 492 * q^14 - 1364 * q^15 + 7680 * q^16 - 1918 * q^17 - 10680 * q^18 - 3315 * q^19 - 2176 * q^20 - 1217 * q^21 + 4232 * q^22 - 9534 * q^23 + 1920 * q^24 + 22130 * q^25 + 1484 * q^26 - 12156 * q^27 - 1968 * q^28 - 8009 * q^29 + 5456 * q^30 - 3543 * q^31 - 30720 * q^32 - 3411 * q^33 + 7672 * q^34 - 11177 * q^35 + 42720 * q^36 - 6517 * q^37 + 13260 * q^38 - 2453 * q^39 + 8704 * q^40 - 5951 * q^41 + 4868 * q^42 - 18768 * q^43 - 16928 * q^44 - 22619 * q^45 + 38136 * q^46 - 41067 * q^47 - 7680 * q^48 + 107797 * q^49 - 88520 * q^50 - 4187 * q^51 - 5936 * q^52 - 64055 * q^53 + 48624 * q^54 + 31973 * q^55 + 7872 * q^56 - 25442 * q^57 + 32036 * q^58 - 66832 * q^59 - 21824 * q^60 + 46777 * q^61 + 14172 * q^62 - 115994 * q^63 + 122880 * q^64 - 68783 * q^65 + 13644 * q^66 - 107218 * q^67 - 30688 * q^68 + 228589 * q^69 + 44708 * q^70 + 4733 * q^71 - 170880 * q^72 + 150385 * q^73 + 26068 * q^74 + 133325 * q^75 - 53040 * q^76 + 292020 * q^77 + 9812 * q^78 + 31827 * q^79 - 34816 * q^80 + 425214 * q^81 + 23804 * q^82 + 107301 * q^83 - 19472 * q^84 + 338625 * q^85 + 75072 * q^86 + 288484 * q^87 + 67712 * q^88 - 99690 * q^89 + 90476 * q^90 - 109140 * q^91 - 152544 * q^92 - 114545 * q^93 + 164268 * q^94 - 155668 * q^95 + 30720 * q^96 - 79959 * q^97 - 431188 * q^98 - 78063 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	2.68265	0.172092	0.0860460	−	0.996291i	$-0.472577\pi$
$3$	2.68265	0.172092	0.0860460	+	0.996291i	$0.472577\pi$
$4$	16.0000	0.500000
$5$	25.8421	0.462278	0.231139	−	0.972921i	$-0.425755\pi$
$5$	25.8421	0.462278	0.231139	+	0.972921i	$0.425755\pi$
$6$	−10.7306	−0.121687
$7$	96.3372	0.743103	0.371551	−	0.928412i	$-0.378826\pi$
$7$	96.3372	0.743103	0.371551	+	0.928412i	$0.378826\pi$
$8$	−64.0000	−0.353553
$9$	−235.803	−0.970384
$10$	−103.369	−0.326880
$11$	−93.7407	−0.233586	−0.116793	−	0.993156i	$-0.537261\pi$
$11$	−93.7407	−0.233586	−0.116793	+	0.993156i	$0.537261\pi$
$12$	42.9224	0.0860460
$13$	−667.002	−1.09463	−0.547317	−	0.836926i	$-0.684351\pi$
$13$	−667.002	−1.09463	−0.547317	+	0.836926i	$0.684351\pi$
$14$	−385.349	−0.525453
$15$	69.3253	0.0795543
$16$	256.000	0.250000
$17$	1784.88	1.49791	0.748956	−	0.662620i	$-0.230556\pi$
$17$	1784.88	1.49791	0.748956	+	0.662620i	$0.230556\pi$
$18$	943.214	0.686165
$19$	2442.53	1.55223	0.776114	−	0.630593i	$-0.217188\pi$
$19$	2442.53	1.55223	0.776114	+	0.630593i	$0.217188\pi$
$20$	413.474	0.231139
$21$	258.439	0.127882
$22$	374.963	0.165170
$23$	700.737	0.276207	0.138104	−	0.990418i	$-0.455899\pi$
$23$	700.737	0.276207	0.138104	+	0.990418i	$0.455899\pi$
$24$	−171.689	−0.0608437
$25$	−2457.18	−0.786299
$26$	2668.01	0.774023
$27$	−1284.46	−0.339087
$28$	1541.40	0.371551
$29$	−5936.54	−1.31081	−0.655403	−	0.755279i	$-0.727501\pi$
$29$	−5936.54	−1.31081	−0.655403	+	0.755279i	$0.727501\pi$
$30$	−277.301	−0.0562534
$31$	−8370.35	−1.56437	−0.782185	−	0.623046i	$-0.785895\pi$
$31$	−8370.35	−1.56437	−0.782185	+	0.623046i	$0.785895\pi$
$32$	−1024.00	−0.176777
$33$	−251.473	−0.0401982
$34$	−7139.51	−1.05918
$35$	2489.56	0.343520
$36$	−3772.85	−0.485192
$37$	−2065.32	−0.248018	−0.124009	−	0.992281i	$-0.539575\pi$
$37$	−2065.32	−0.248018	−0.124009	+	0.992281i	$0.539575\pi$
$38$	−9770.11	−1.09759
$39$	−1789.33	−0.188378
$40$	−1653.90	−0.163440
$41$	−12488.4	−1.16024	−0.580120	−	0.814531i	$-0.696994\pi$
$41$	−12488.4	−1.16024	−0.580120	+	0.814531i	$0.696994\pi$
$42$	−1033.75	−0.0904262
$43$	14743.9	1.21602	0.608009	−	0.793930i	$-0.291969\pi$
$43$	14743.9	1.21602	0.608009	+	0.793930i	$0.291969\pi$
$44$	−1499.85	−0.116793
$45$	−6093.66	−0.448587
$46$	−2802.95	−0.195308
$47$	10629.1	0.701865	0.350933	−	0.936401i	$-0.385865\pi$
$47$	10629.1	0.701865	0.350933	+	0.936401i	$0.385865\pi$
$48$	686.758	0.0430230
$49$	−7526.14	−0.447798
$50$	9828.74	0.555997
$51$	4788.20	0.257778
$52$	−10672.0	−0.547317
$53$	7099.35	0.347160	0.173580	−	0.984820i	$-0.444467\pi$
$53$	7099.35	0.347160	0.173580	+	0.984820i	$0.444467\pi$
$54$	5137.84	0.239771
$55$	−2422.46	−0.107982
$56$	−6165.58	−0.262727
$57$	6552.44	0.267126
$58$	23746.2	0.926880
$59$	44165.1	1.65177	0.825884	−	0.563839i	$-0.190676\pi$
$59$	44165.1	1.65177	0.825884	+	0.563839i	$0.190676\pi$
$60$	1109.21	0.0397772
$61$	12326.7	0.424151	0.212076	−	0.977253i	$-0.431978\pi$
$61$	12326.7	0.424151	0.212076	+	0.977253i	$0.431978\pi$
$62$	33481.4	1.10618
$63$	−22716.6	−0.721095
$64$	4096.00	0.125000
$65$	−17236.8	−0.506025
$66$	1005.89	0.0284244
$67$	−29753.4	−0.809747	−0.404873	−	0.914373i	$-0.632684\pi$
$67$	−29753.4	−0.809747	−0.404873	+	0.914373i	$0.632684\pi$
$68$	28558.0	0.748956
$69$	1879.83	0.0475331
$70$	−9958.23	−0.242905
$71$	−42415.5	−0.998569	−0.499285	−	0.866438i	$-0.666404\pi$
$71$	−42415.5	−0.998569	−0.499285	+	0.866438i	$0.666404\pi$
$72$	15091.4	0.343083
$73$	−11634.1	−0.255520	−0.127760	−	0.991805i	$-0.540779\pi$
$73$	−11634.1	−0.255520	−0.127760	+	0.991805i	$0.540779\pi$
$74$	8261.28	0.175375
$75$	−6591.76	−0.135316
$76$	39080.4	0.776114
$77$	−9030.72	−0.173578
$78$	7157.32	0.133203
$79$	3427.83	0.0617948	0.0308974	−	0.999523i	$-0.490163\pi$
$79$	3427.83	0.0617948	0.0308974	+	0.999523i	$0.490163\pi$
$80$	6615.59	0.115570
$81$	53854.5	0.912030
$82$	49953.7	0.820413
$83$	−62674.7	−0.998613	−0.499307	−	0.866425i	$-0.666412\pi$
$83$	−62674.7	−0.998613	−0.499307	+	0.866425i	$0.666412\pi$
$84$	4135.02	0.0639410
$85$	46125.0	0.692451
$86$	−58975.4	−0.859854
$87$	−15925.7	−0.225579
$88$	5999.40	0.0825851
$89$	−145342.	−1.94498	−0.972492	−	0.232934i	$-0.925167\pi$
$89$	−145342.	−1.94498	−0.972492	+	0.232934i	$0.925167\pi$
$90$	24374.6	0.317199
$91$	−64257.1	−0.813425
$92$	11211.8	0.138104
$93$	−22454.7	−0.269215
$94$	−42516.6	−0.496294
$95$	63120.1	0.717561
$96$	−2747.03	−0.0304218
$97$	−34590.9	−0.373278	−0.186639	−	0.982429i	$-0.559759\pi$
$97$	−34590.9	−0.373278	−0.186639	+	0.982429i	$0.559759\pi$
$98$	30104.6	0.316641
$99$	22104.4	0.226668
$100$	−39314.9	−0.393149
$101$	64859.4	0.632659	0.316329	−	0.948649i	$-0.397550\pi$
$101$	64859.4	0.632659	0.316329	+	0.948649i	$0.397550\pi$
$102$	−19152.8	−0.182277
$103$	140594.	1.30580	0.652898	−	0.757446i	$-0.273553\pi$
$103$	140594.	1.30580	0.652898	+	0.757446i	$0.273553\pi$
$104$	42688.1	0.387011
$105$	6678.61	0.0591170
$106$	−28397.4	−0.245479
$107$	219999.	1.85764	0.928821	−	0.370528i	$-0.120823\pi$
$107$	219999.	1.85764	0.928821	+	0.370528i	$0.120823\pi$
$108$	−20551.4	−0.169544
$109$	−115630.	−0.932191	−0.466096	−	0.884734i	$-0.654340\pi$
$109$	−115630.	−0.932191	−0.466096	+	0.884734i	$0.654340\pi$
$110$	9689.84	0.0763545
$111$	−5540.52	−0.0426819
$112$	24662.3	0.185776
$113$	−57789.7	−0.425749	−0.212875	−	0.977080i	$-0.568283\pi$
$113$	−57789.7	−0.425749	−0.212875	+	0.977080i	$0.568283\pi$
$114$	−26209.7	−0.188886
$115$	18108.5	0.127685
$116$	−94984.7	−0.655403
$117$	157281.	1.06222
$118$	−176660.	−1.16798
$119$	171950.	1.11310
$120$	−4436.82	−0.0281267
$121$	−152264.	−0.945438
$122$	−49306.6	−0.299920
$123$	−33502.0	−0.199668
$124$	−133926.	−0.782185
$125$	−144256.	−0.825767
$126$	90866.6	0.509891
$127$	−98669.2	−0.542841	−0.271420	−	0.962461i	$-0.587493\pi$
$127$	−98669.2	−0.542841	−0.271420	+	0.962461i	$0.587493\pi$
$128$	−16384.0	−0.0883883
$129$	39552.6	0.209267
$130$	68947.0	0.357814
$131$	−140087.	−0.713213	−0.356606	−	0.934255i	$-0.616066\pi$
$131$	−140087.	−0.713213	−0.356606	+	0.934255i	$0.616066\pi$
$132$	−4023.57	−0.0200991
$133$	235306.	1.15346
$134$	119014.	0.572577
$135$	−33193.2	−0.156753
$136$	−114232.	−0.529592
$137$	−408746.	−1.86060	−0.930298	−	0.366804i	$-0.880452\pi$
$137$	−408746.	−1.86060	−0.930298	+	0.366804i	$0.880452\pi$
$138$	−7519.32	−0.0336110
$139$	−117122.	−0.514164	−0.257082	−	0.966390i	$-0.582761\pi$
$139$	−117122.	−0.514164	−0.257082	+	0.966390i	$0.582761\pi$
$140$	39832.9	0.171760
$141$	28514.3	0.120785
$142$	169662.	0.706095
$143$	62525.2	0.255691
$144$	−60365.7	−0.242596
$145$	−153413.	−0.605957
$146$	46536.3	0.180680
$147$	−20190.0	−0.0770625
$148$	−33045.1	−0.124009
$149$	−486496.	−1.79520	−0.897602	−	0.440807i	$-0.854692\pi$
$149$	−486496.	−1.79520	−0.897602	+	0.440807i	$0.854692\pi$
$150$	26367.0	0.0956826
$151$	−220928.	−0.788513	−0.394257	−	0.919000i	$-0.628998\pi$
$151$	−220928.	−0.788513	−0.394257	+	0.919000i	$0.628998\pi$
$152$	−156322.	−0.548795
$153$	−420880.	−1.45355
$154$	36122.9	0.122738
$155$	−216308.	−0.723174
$156$	−28629.3	−0.0941888
$157$	280919.	0.909562	0.454781	−	0.890603i	$-0.349718\pi$
$157$	280919.	0.909562	0.454781	+	0.890603i	$0.349718\pi$
$158$	−13711.3	−0.0436955
$159$	19045.1	0.0597433
$160$	−26462.3	−0.0817200
$161$	67507.0	0.205251
$162$	−215418.	−0.644903
$163$	514693.	1.51733	0.758663	−	0.651483i	$-0.225853\pi$
$163$	514693.	1.51733	0.758663	+	0.651483i	$0.225853\pi$
$164$	−199815.	−0.580120
$165$	−6498.60	−0.0185828
$166$	250699.	0.706126
$167$	−465392.	−1.29130	−0.645651	−	0.763633i	$-0.723414\pi$
$167$	−465392.	−1.29130	−0.645651	+	0.763633i	$0.723414\pi$
$168$	−16540.1	−0.0452131
$169$	73598.7	0.198223
$170$	−184500.	−0.489637
$171$	−575956.	−1.50626
$172$	235902.	0.608009
$173$	471105.	1.19675	0.598373	−	0.801218i	$-0.295814\pi$
$173$	471105.	1.19675	0.598373	+	0.801218i	$0.295814\pi$
$174$	63702.6	0.159509
$175$	−236718.	−0.584301
$176$	−23997.6	−0.0583965
$177$	118479.	0.284256
$178$	581368.	1.37531
$179$	−123175.	−0.287336	−0.143668	−	0.989626i	$-0.545890\pi$
$179$	−123175.	−0.287336	−0.143668	+	0.989626i	$0.545890\pi$
$180$	−97498.6	−0.224294
$181$	−104028.	−0.236023	−0.118011	−	0.993012i	$-0.537652\pi$
$181$	−104028.	−0.236023	−0.118011	+	0.993012i	$0.537652\pi$
$182$	257028.	0.575179
$183$	33068.1	0.0729930
$184$	−44847.2	−0.0976541
$185$	−53372.3	−0.114653
$186$	89818.8	0.190364
$187$	−167316.	−0.349891
$188$	170066.	0.350933
$189$	−123741.	−0.251977
$190$	−252480.	−0.507392
$191$	569053.	1.12868	0.564338	−	0.825544i	$-0.309132\pi$
$191$	569053.	1.12868	0.564338	+	0.825544i	$0.309132\pi$
$192$	10988.1	0.0215115
$193$	−548209.	−1.05938	−0.529692	−	0.848190i	$-0.677692\pi$
$193$	−548209.	−1.05938	−0.529692	+	0.848190i	$0.677692\pi$
$194$	138363.	0.263947
$195$	−46240.1	−0.0870828
$196$	−120418.	−0.223899
$197$	−690801.	−1.26820	−0.634100	−	0.773251i	$-0.718629\pi$
$197$	−690801.	−1.26820	−0.634100	+	0.773251i	$0.718629\pi$
$198$	−88417.5	−0.160279
$199$	−112880.	−0.202061	−0.101031	−	0.994883i	$-0.532214\pi$
$199$	−112880.	−0.202061	−0.101031	+	0.994883i	$0.532214\pi$
$200$	157260.	0.277999
$201$	−79817.8	−0.139351
$202$	−259438.	−0.447357
$203$	−571910.	−0.974064
$204$	76611.1	0.128889
$205$	−322727.	−0.536353
$206$	−562378.	−0.923337
$207$	−165236.	−0.268027
$208$	−170753.	−0.273658
$209$	−228964.	−0.362578
$210$	−26714.4	−0.0418021
$211$	−248207.	−0.383802	−0.191901	−	0.981414i	$-0.561465\pi$
$211$	−248207.	−0.383802	−0.191901	+	0.981414i	$0.561465\pi$
$212$	113590.	0.173580
$213$	−113786.	−0.171846
$214$	−879998.	−1.31355
$215$	381013.	0.562138
$216$	82205.5	0.119885
$217$	−806376.	−1.16249
$218$	462521.	0.659159
$219$	−31210.1	−0.0439729
$220$	−38759.4	−0.0539908
$221$	−1.19052e6	−1.63966
$222$	22162.1	0.0301806
$223$	−58638.9	−0.0789629	−0.0394815	−	0.999220i	$-0.512571\pi$
$223$	−58638.9	−0.0789629	−0.0394815	+	0.999220i	$0.512571\pi$
$224$	−98649.3	−0.131363
$225$	579412.	0.763012
$226$	231159.	0.301050
$227$	−1.23401e6	−1.58947	−0.794736	−	0.606955i	$-0.792391\pi$
$227$	−1.23401e6	−1.58947	−0.794736	+	0.606955i	$0.792391\pi$
$228$	104839.	0.133563
$229$	30449.5	0.0383700	0.0191850	−	0.999816i	$-0.493893\pi$
$229$	30449.5	0.0383700	0.0191850	+	0.999816i	$0.493893\pi$
$230$	−72434.1	−0.0902867
$231$	−24226.2	−0.0298714
$232$	379939.	0.463440
$233$	−453583.	−0.547353	−0.273676	−	0.961822i	$-0.588240\pi$
$233$	−453583.	−0.547353	−0.273676	+	0.961822i	$0.588240\pi$
$234$	−629125.	−0.751100
$235$	274680.	0.324457
$236$	706642.	0.825884
$237$	9195.67	0.0106344
$238$	−687800.	−0.787082
$239$	194029.	0.219721	0.109861	−	0.993947i	$-0.464960\pi$
$239$	194029.	0.219721	0.109861	+	0.993947i	$0.464960\pi$
$240$	17747.3	0.0198886
$241$	231818.	0.257101	0.128551	−	0.991703i	$-0.458968\pi$
$241$	231818.	0.257101	0.128551	+	0.991703i	$0.458968\pi$
$242$	609055.	0.668525
$243$	456597.	0.496040
$244$	197227.	0.212076
$245$	−194492.	−0.207007
$246$	134008.	0.141186
$247$	−1.62917e6	−1.69912
$248$	535703.	0.553088
$249$	−168134.	−0.171853
$250$	577022.	0.583905
$251$	−990077.	−0.991938	−0.495969	−	0.868340i	$-0.665187\pi$
$251$	−990077.	−0.991938	−0.495969	+	0.868340i	$0.665187\pi$
$252$	−363466.	−0.360548
$253$	−65687.6	−0.0645181
$254$	394677.	0.383846
$255$	123737.	0.119165
$256$	65536.0	0.0625000
$257$	948299.	0.895597	0.447798	−	0.894135i	$-0.352208\pi$
$257$	948299.	0.895597	0.447798	+	0.894135i	$0.352208\pi$
$258$	−158210.	−0.147974
$259$	−198967.	−0.184303
$260$	−275788.	−0.253013
$261$	1.39986e6	1.27199
$262$	560347.	0.504318
$263$	372596.	0.332162	0.166081	−	0.986112i	$-0.446889\pi$
$263$	372596.	0.332162	0.166081	+	0.986112i	$0.446889\pi$
$264$	16094.3	0.0142122
$265$	183462.	0.160484
$266$	−941225.	−0.815622
$267$	−389901.	−0.334716
$268$	−476054.	−0.404873
$269$	−72361.0	−0.0609711
$269$	−72361.0	−0.0609711
$270$	132773.	0.110841
$271$	1.75731e6	1.45354	0.726768	−	0.686883i	$-0.241022\pi$
$271$	1.75731e6	1.45354	0.726768	+	0.686883i	$0.241022\pi$
$272$	456929.	0.374478
$273$	−172379.	−0.139984
$274$	1.63498e6	1.31564
$275$	230338.	0.183668
$276$	30077.3	0.0237665
$277$	−1.31062e6	−1.02630	−0.513152	−	0.858298i	$-0.671522\pi$
$277$	−1.31062e6	−1.02630	−0.513152	+	0.858298i	$0.671522\pi$
$278$	468488.	0.363569
$279$	1.97376e6	1.51804
$280$	−159332.	−0.121453
$281$	−1.76221e6	−1.33135	−0.665674	−	0.746243i	$-0.731856\pi$
$281$	−1.76221e6	−1.33135	−0.665674	+	0.746243i	$0.731856\pi$
$282$	−114057.	−0.0854081
$283$	735849.	0.546163	0.273082	−	0.961991i	$-0.411957\pi$
$283$	735849.	0.546163	0.273082	+	0.961991i	$0.411957\pi$
$284$	−678647.	−0.499285
$285$	169329.	0.123486
$286$	−250101.	−0.180801
$287$	−1.20310e6	−0.862177
$288$	241463.	0.171541
$289$	1.76593e6	1.24374
$290$	613652.	0.428476
$291$	−92795.1	−0.0642381
$292$	−186145.	−0.127760
$293$	−2.08349e6	−1.41782	−0.708912	−	0.705297i	$-0.750814\pi$
$293$	−2.08349e6	−1.41782	−0.708912	+	0.705297i	$0.750814\pi$
$294$	80760.0	0.0544914
$295$	1.14132e6	0.763577
$296$	132180.	0.0876876
$297$	120406.	0.0792060
$298$	1.94599e6	1.26940
$299$	−467393.	−0.302346
$300$	−105468.	−0.0676578
$301$	1.42038e6	0.903626
$302$	883713.	0.557563
$303$	173995.	0.108875
$304$	625287.	0.388057
$305$	318547.	0.196076
$306$	1.68352e6	1.02781
$307$	−849605.	−0.514483	−0.257242	−	0.966347i	$-0.582814\pi$
$307$	−849605.	−0.514483	−0.257242	+	0.966347i	$0.582814\pi$
$308$	−144491.	−0.0867891
$309$	377165.	0.224717
$310$	865231.	0.511361
$311$	−2.36993e6	−1.38942	−0.694712	−	0.719288i	$-0.744468\pi$
$311$	−2.36993e6	−1.38942	−0.694712	+	0.719288i	$0.744468\pi$
$312$	114517.	0.0666015
$313$	−2.09181e6	−1.20687	−0.603436	−	0.797411i	$-0.706202\pi$
$313$	−2.09181e6	−1.20687	−0.603436	+	0.797411i	$0.706202\pi$
$314$	−1.12368e6	−0.643157
$315$	−587046.	−0.333347
$316$	54845.3	0.0308974
$317$	2.11077e6	1.17976	0.589880	−	0.807491i	$-0.299175\pi$
$317$	2.11077e6	1.17976	0.589880	+	0.807491i	$0.299175\pi$
$318$	−76180.2	−0.0422449
$319$	556496.	0.306186
$320$	105849.	0.0577848
$321$	590181.	0.319685
$322$	−270028.	−0.145134
$323$	4.35961e6	2.32510
$324$	861672.	0.456015
$325$	1.63895e6	0.860709
$326$	−2.05877e6	−1.07291
$327$	−310195.	−0.160423
$328$	799258.	0.410207
$329$	1.02398e6	0.521558
$330$	25994.4	0.0131400
$331$	−2.53054e6	−1.26953	−0.634765	−	0.772705i	$-0.718903\pi$
$331$	−2.53054e6	−1.26953	−0.634765	+	0.772705i	$0.718903\pi$
$332$	−1.00280e6	−0.499307
$333$	487009.	0.240673
$334$	1.86157e6	0.913088
$335$	−768891.	−0.374328
$336$	66160.3	0.0319705
$337$	−1.57223e6	−0.754120	−0.377060	−	0.926189i	$-0.623065\pi$
$337$	−1.57223e6	−0.754120	−0.377060	+	0.926189i	$0.623065\pi$
$338$	−294395.	−0.140165
$339$	−155029.	−0.0732680
$340$	738000.	0.346226
$341$	784643.	0.365415
$342$	2.30382e6	1.06508
$343$	−2.34419e6	−1.07586
$344$	−943607.	−0.429927
$345$	48578.8	0.0219735
$346$	−1.88442e6	−0.846227
$347$	−4.05633e6	−1.80846	−0.904232	−	0.427042i	$-0.859556\pi$
$347$	−4.05633e6	−1.80846	−0.904232	+	0.427042i	$0.859556\pi$
$348$	−254810.	−0.112790
$349$	−2.13094e6	−0.936502	−0.468251	−	0.883596i	$-0.655116\pi$
$349$	−2.13094e6	−0.936502	−0.468251	+	0.883596i	$0.655116\pi$
$350$	946873.	0.413163
$351$	856738.	0.371176
$352$	95990.5	0.0412925
$353$	1.74063e6	0.743480	0.371740	−	0.928337i	$-0.378761\pi$
$353$	1.74063e6	0.743480	0.371740	+	0.928337i	$0.378761\pi$
$354$	−473918.	−0.200999
$355$	−1.09611e6	−0.461617
$356$	−2.32547e6	−0.972492
$357$	461281.	0.191556
$358$	492700.	0.203177
$359$	767082.	0.314127	0.157064	−	0.987588i	$-0.449797\pi$
$359$	767082.	0.314127	0.157064	+	0.987588i	$0.449797\pi$
$360$	389994.	0.158600
$361$	3.48984e6	1.40941
$362$	416112.	0.166893
$363$	−408470.	−0.162702
$364$	−1.02811e6	−0.406713
$365$	−300649.	−0.118121
$366$	−132272.	−0.0516139
$367$	2.26150e6	0.876459	0.438230	−	0.898863i	$-0.355606\pi$
$367$	2.26150e6	0.876459	0.438230	+	0.898863i	$0.355606\pi$
$368$	179389.	0.0690519
$369$	2.94481e6	1.12588
$370$	213489.	0.0810721
$371$	683932.	0.257975
$372$	−359275.	−0.134608
$373$	5.24435e6	1.95173	0.975865	−	0.218375i	$-0.0700755\pi$
$373$	5.24435e6	1.95173	0.975865	+	0.218375i	$0.0700755\pi$
$374$	669263.	0.247410
$375$	−386987.	−0.142108
$376$	−680266.	−0.248147
$377$	3.95969e6	1.43485
$378$	494965.	0.178174
$379$	815203.	0.291520	0.145760	−	0.989320i	$-0.453437\pi$
$379$	815203.	0.291520	0.145760	+	0.989320i	$0.453437\pi$
$380$	1.00992e6	0.358780
$381$	−264695.	−0.0934185
$382$	−2.27621e6	−0.798094
$383$	91973.3	0.0320379	0.0160190	−	0.999872i	$-0.494901\pi$
$383$	91973.3	0.0320379	0.0160190	+	0.999872i	$0.494901\pi$
$384$	−43952.5	−0.0152109
$385$	−233373.	−0.0802414
$386$	2.19284e6	0.749097
$387$	−3.47665e6	−1.18000
$388$	−553454.	−0.186639
$389$	−5.39926e6	−1.80909	−0.904545	−	0.426378i	$-0.859789\pi$
$389$	−5.39926e6	−1.80909	−0.904545	+	0.426378i	$0.859789\pi$
$390$	184961.	0.0615769
$391$	1.25073e6	0.413734
$392$	481673.	0.158321
$393$	−375804.	−0.122738
$394$	2.76321e6	0.896753
$395$	88582.5	0.0285664
$396$	353670.	0.113334
$397$	2.54729e6	0.811152	0.405576	−	0.914061i	$-0.367071\pi$
$397$	2.54729e6	0.811152	0.405576	+	0.914061i	$0.367071\pi$
$398$	451519.	0.142879
$399$	631243.	0.198502
$400$	−629039.	−0.196575
$401$	−1.52464e6	−0.473486	−0.236743	−	0.971572i	$-0.576080\pi$
$401$	−1.52464e6	−0.473486	−0.236743	+	0.971572i	$0.576080\pi$
$402$	319271.	0.0985360
$403$	5.58304e6	1.71241
$404$	1.03775e6	0.316329
$405$	1.39171e6	0.421612
$406$	2.28764e6	0.688767
$407$	193605.	0.0579335
$408$	−306445.	−0.0911384
$409$	−3.78578e6	−1.11904	−0.559522	−	0.828815i	$-0.689015\pi$
$409$	−3.78578e6	−1.11904	−0.559522	+	0.828815i	$0.689015\pi$
$410$	1.29091e6	0.379259
$411$	−1.09652e6	−0.320194
$412$	2.24951e6	0.652898
$413$	4.25474e6	1.22743
$414$	660944.	0.189524
$415$	−1.61965e6	−0.461637
$416$	683010.	0.193506
$417$	−314197.	−0.0884834
$418$	915857.	0.256382
$419$	−2.27184e6	−0.632181	−0.316091	−	0.948729i	$-0.602370\pi$
$419$	−2.27184e6	−0.632181	−0.316091	+	0.948729i	$0.602370\pi$
$420$	106858.	0.0295585
$421$	−422161.	−0.116084	−0.0580420	−	0.998314i	$-0.518486\pi$
$421$	−422161.	−0.116084	−0.0580420	+	0.998314i	$0.518486\pi$
$422$	992826.	0.271389
$423$	−2.50639e6	−0.681079
$424$	−454359.	−0.122739
$425$	−4.38577e6	−1.17781
$426$	455143.	0.121513
$427$	1.18752e6	0.315188
$428$	3.51999e6	0.928821
$429$	167733.	0.0440023
$430$	−1.52405e6	−0.397492
$431$	1.83504e6	0.475832	0.237916	−	0.971286i	$-0.423536\pi$
$431$	1.83504e6	0.475832	0.237916	+	0.971286i	$0.423536\pi$
$432$	−328822.	−0.0847718
$433$	1.53555e6	0.393591	0.196795	−	0.980445i	$-0.436947\pi$
$433$	1.53555e6	0.393591	0.196795	+	0.980445i	$0.436947\pi$
$434$	3.22550e6	0.822003
$435$	−411553.	−0.104280
$436$	−1.85008e6	−0.466096
$437$	1.71157e6	0.428737
$438$	124840.	0.0310935
$439$	−4.38781e6	−1.08664	−0.543321	−	0.839525i	$-0.682833\pi$
$439$	−4.38781e6	−1.08664	−0.543321	+	0.839525i	$0.682833\pi$
$440$	155037.	0.0381773
$441$	1.77469e6	0.434536
$442$	4.76207e6	1.15942
$443$	7.94980e6	1.92463	0.962314	−	0.271941i	$-0.0876655\pi$
$443$	7.94980e6	1.92463	0.962314	+	0.271941i	$0.0876655\pi$
$444$	−88648.4	−0.0213409
$445$	−3.75595e6	−0.899124
$446$	234555.	0.0558352
$447$	−1.30510e6	−0.308940
$448$	394597.	0.0928878
$449$	5.67095e6	1.32752	0.663758	−	0.747948i	$-0.268961\pi$
$449$	5.67095e6	1.32752	0.663758	+	0.747948i	$0.268961\pi$
$450$	−2.31765e6	−0.539531
$451$	1.17067e6	0.271015
$452$	−924635.	−0.212875
$453$	−592673.	−0.135697
$454$	4.93603e6	1.12393
$455$	−1.66054e6	−0.376029
$456$	−419356.	−0.0944432
$457$	4.59081e6	1.02825	0.514125	−	0.857715i	$-0.328117\pi$
$457$	4.59081e6	1.02825	0.514125	+	0.857715i	$0.328117\pi$
$458$	−121798.	−0.0271317
$459$	−2.29260e6	−0.507923
$460$	289737.	0.0638423
$461$	1.51804e6	0.332683	0.166341	−	0.986068i	$-0.446805\pi$
$461$	1.51804e6	0.332683	0.166341	+	0.986068i	$0.446805\pi$
$462$	96904.9	0.0211223
$463$	3.78434e6	0.820422	0.410211	−	0.911991i	$-0.365455\pi$
$463$	3.78434e6	0.820422	0.410211	+	0.911991i	$0.365455\pi$
$464$	−1.51976e6	−0.327702
$465$	−580277.	−0.124452
$466$	1.81433e6	0.387037
$467$	−5.87911e6	−1.24744	−0.623720	−	0.781648i	$-0.714379\pi$
$467$	−5.87911e6	−1.24744	−0.623720	+	0.781648i	$0.714379\pi$
$468$	2.51650e6	0.531108
$469$	−2.86636e6	−0.601725
$470$	−1.09872e6	−0.229426
$471$	753607.	0.156528
$472$	−2.82657e6	−0.583989
$473$	−1.38210e6	−0.284044
$474$	−36782.7	−0.00751965
$475$	−6.00174e6	−1.22051
$476$	2.75120e6	0.556551
$477$	−1.67405e6	−0.336878
$478$	−776116.	−0.155366
$479$	4.27765e6	0.851856	0.425928	−	0.904757i	$-0.359948\pi$
$479$	4.27765e6	0.851856	0.425928	+	0.904757i	$0.359948\pi$
$480$	−70989.1	−0.0140633
$481$	1.37757e6	0.271489
$482$	−927271.	−0.181798
$483$	181098.	0.0353220
$484$	−2.43622e6	−0.472719
$485$	−893902.	−0.172558
$486$	−1.82639e6	−0.350753
$487$	4.92226e6	0.940464	0.470232	−	0.882543i	$-0.344170\pi$
$487$	4.92226e6	0.940464	0.470232	+	0.882543i	$0.344170\pi$
$488$	−788906.	−0.149960
$489$	1.38074e6	0.261120
$490$	777966.	0.146376
$491$	4.46862e6	0.836508	0.418254	−	0.908330i	$-0.362642\pi$
$491$	4.46862e6	0.836508	0.418254	+	0.908330i	$0.362642\pi$
$492$	−536032.	−0.0998339
$493$	−1.05960e7	−1.96347
$494$	6.51668e6	1.20146
$495$	571224.	0.104784
$496$	−2.14281e6	−0.391092
$497$	−4.08619e6	−0.742040
$498$	672537.	0.121519
$499$	2.31892e6	0.416902	0.208451	−	0.978033i	$-0.433158\pi$
$499$	2.31892e6	0.416902	0.208451	+	0.978033i	$0.433158\pi$
$500$	−2.30809e6	−0.412883
$501$	−1.24848e6	−0.222223
$502$	3.96031e6	0.701406
$503$	2.98267e6	0.525636	0.262818	−	0.964845i	$-0.415348\pi$
$503$	2.98267e6	0.525636	0.262818	+	0.964845i	$0.415348\pi$
$504$	1.45386e6	0.254946
$505$	1.67611e6	0.292464
$506$	262750.	0.0456212
$507$	197439.	0.0341125
$508$	−1.57871e6	−0.271420
$509$	−3.25087e6	−0.556167	−0.278083	−	0.960557i	$-0.589699\pi$
$509$	−3.25087e6	−0.556167	−0.278083	+	0.960557i	$0.589699\pi$
$510$	−494949.	−0.0842626
$511$	−1.12079e6	−0.189877
$512$	−262144.	−0.0441942
$513$	−3.13733e6	−0.526340
$514$	−3.79320e6	−0.633283
$515$	3.63326e6	0.603641
$516$	632841.	0.104633
$517$	−996384.	−0.163946
$518$	795868.	0.130322
$519$	1.26381e6	0.205950
$520$	1.10315e6	0.178907
$521$	−1.80103e6	−0.290688	−0.145344	−	0.989381i	$-0.546429\pi$
$521$	−1.80103e6	−0.290688	−0.145344	+	0.989381i	$0.546429\pi$
$522$	−5.59943e6	−0.899430
$523$	6.68223e6	1.06824	0.534118	−	0.845410i	$-0.320644\pi$
$523$	6.68223e6	1.06824	0.534118	+	0.845410i	$0.320644\pi$
$524$	−2.24139e6	−0.356606
$525$	−635032.	−0.100553
$526$	−1.49039e6	−0.234874
$527$	−1.49401e7	−2.34329
$528$	−64377.2	−0.0100496
$529$	−5.94531e6	−0.923709
$530$	−733850.	−0.113479
$531$	−1.04143e7	−1.60285
$532$	3.76490e6	0.576732
$533$	8.32980e6	1.27004
$534$	1.55961e6	0.236680
$535$	5.68525e6	0.858747
$536$	1.90422e6	0.286289
$537$	−330435.	−0.0494482
$538$	289444.	0.0431131
$539$	705506.	0.104599
$540$	−531091.	−0.0783763
$541$	2.12886e6	0.312718	0.156359	−	0.987700i	$-0.450024\pi$
$541$	2.12886e6	0.312718	0.156359	+	0.987700i	$0.450024\pi$
$542$	−7.02925e6	−1.02780
$543$	−279071.	−0.0406176
$544$	−1.82771e6	−0.264796
$545$	−2.98813e6	−0.430931
$546$	689517.	0.0989836
$547$	4.78809e6	0.684217	0.342109	−	0.939660i	$-0.388859\pi$
$547$	4.78809e6	0.684217	0.342109	+	0.939660i	$0.388859\pi$
$548$	−6.53994e6	−0.930298
$549$	−2.90667e6	−0.411590
$550$	−921353.	−0.129873
$551$	−1.45002e7	−2.03467
$552$	−120309.	−0.0168055
$553$	330228.	0.0459199
$554$	5.24247e6	0.725707
$555$	−143179.	−0.0197309
$556$	−1.87395e6	−0.257082
$557$	1.01290e7	1.38334	0.691669	−	0.722214i	$-0.256876\pi$
$557$	1.01290e7	1.38334	0.691669	+	0.722214i	$0.256876\pi$
$558$	−7.89503e6	−1.07342
$559$	−9.83418e6	−1.33109
$560$	637327.	0.0858800
$561$	−448849.	−0.0602134
$562$	7.04884e6	0.941405
$563$	−1.12196e6	−0.149179	−0.0745895	−	0.997214i	$-0.523765\pi$
$563$	−1.12196e6	−0.149179	−0.0745895	+	0.997214i	$0.523765\pi$
$564$	456228.	0.0603927
$565$	−1.49341e6	−0.196815
$566$	−2.94339e6	−0.386196
$567$	5.18819e6	0.677732
$568$	2.71459e6	0.353048
$569$	4.94630e6	0.640471	0.320235	−	0.947338i	$-0.396238\pi$
$569$	4.94630e6	0.640471	0.320235	+	0.947338i	$0.396238\pi$
$570$	−677316.	−0.0873181
$571$	7.72683e6	0.991771	0.495885	−	0.868388i	$-0.334844\pi$
$571$	7.72683e6	0.991771	0.495885	+	0.868388i	$0.334844\pi$
$572$	1.00040e6	0.127845
$573$	1.52657e6	0.194236
$574$	4.81239e6	0.609651
$575$	−1.72184e6	−0.217182
$576$	−965851.	−0.121298
$577$	344575.	0.0430868	0.0215434	−	0.999768i	$-0.493142\pi$
$577$	344575.	0.0430868	0.0215434	+	0.999768i	$0.493142\pi$
$578$	−7.06372e6	−0.879455
$579$	−1.47065e6	−0.182311
$580$	−2.45461e6	−0.302979
$581$	−6.03791e6	−0.742072
$582$	371180.	0.0454232
$583$	−665498.	−0.0810915
$584$	744580.	0.0903398
$585$	4.06448e6	0.491039
$586$	8.33396e6	1.00255
$587$	−668046.	−0.0800223	−0.0400112	−	0.999199i	$-0.512739\pi$
$587$	−668046.	−0.0800223	−0.0400112	+	0.999199i	$0.512739\pi$
$588$	−323040.	−0.0385312
$589$	−2.04448e7	−2.42826
$590$	−4.56528e6	−0.539930
$591$	−1.85318e6	−0.218247
$592$	−528722.	−0.0620045
$593$	1.38808e7	1.62098	0.810488	−	0.585756i	$-0.199202\pi$
$593$	1.38808e7	1.62098	0.810488	+	0.585756i	$0.199202\pi$
$594$	−481625.	−0.0560071
$595$	4.44356e6	0.514563
$596$	−7.78394e6	−0.897602
$597$	−302816.	−0.0347731
$598$	1.86957e6	0.213791
$599$	8.94895e6	1.01907	0.509536	−	0.860449i	$-0.329817\pi$
$599$	8.94895e6	1.01907	0.509536	+	0.860449i	$0.329817\pi$
$600$	421873.	0.0478413
$601$	1.44289e7	1.62947	0.814736	−	0.579832i	$-0.196882\pi$
$601$	1.44289e7	1.62947	0.814736	+	0.579832i	$0.196882\pi$
$602$	−5.68153e6	−0.638960
$603$	7.01595e6	0.785766
$604$	−3.53485e6	−0.394257
$605$	−3.93482e6	−0.437055
$606$	−695980.	−0.0769866
$607$	−5.32611e6	−0.586730	−0.293365	−	0.956001i	$-0.594775\pi$
$607$	−5.32611e6	−0.586730	−0.293365	+	0.956001i	$0.594775\pi$
$608$	−2.50115e6	−0.274398
$609$	−1.53423e6	−0.167629
$610$	−1.27419e6	−0.138647
$611$	−7.08966e6	−0.768285
$612$	−6.73408e6	−0.726775
$613$	859906.	0.0924272	0.0462136	−	0.998932i	$-0.485285\pi$
$613$	859906.	0.0924272	0.0462136	+	0.998932i	$0.485285\pi$
$614$	3.39842e6	0.363795
$615$	−865763.	−0.0923021
$616$	577966.	0.0613692
$617$	−1.20389e7	−1.27313	−0.636567	−	0.771221i	$-0.719646\pi$
$617$	−1.20389e7	−1.27313	−0.636567	+	0.771221i	$0.719646\pi$
$618$	−1.50866e6	−0.158899
$619$	−4.91989e6	−0.516094	−0.258047	−	0.966132i	$-0.583079\pi$
$619$	−4.91989e6	−0.516094	−0.258047	+	0.966132i	$0.583079\pi$
$620$	−3.46092e6	−0.361587
$621$	−900069.	−0.0936584
$622$	9.47972e6	0.982470
$623$	−1.40018e7	−1.44532
$624$	−458069.	−0.0470944
$625$	3.95083e6	0.404565
$626$	8.36724e6	0.853388
$627$	−614230.	−0.0623968
$628$	4.49471e6	0.454781
$629$	−3.68634e6	−0.371509
$630$	2.34819e6	0.235712
$631$	1.53560e7	1.53534	0.767670	−	0.640846i	$-0.221416\pi$
$631$	1.53560e7	1.53534	0.767670	+	0.640846i	$0.221416\pi$
$632$	−219381.	−0.0218478
$633$	−665851.	−0.0660492
$634$	−8.44310e6	−0.834216
$635$	−2.54982e6	−0.250943
$636$	304721.	0.0298717
$637$	5.01995e6	0.490175
$638$	−2.22598e6	−0.216506
$639$	1.00017e7	0.968996
$640$	−423397.	−0.0408600
$641$	9.59300e6	0.922167	0.461083	−	0.887357i	$-0.347461\pi$
$641$	9.59300e6	0.922167	0.461083	+	0.887357i	$0.347461\pi$
$642$	−2.36072e6	−0.226052
$643$	−1.68680e6	−0.160893	−0.0804465	−	0.996759i	$-0.525635\pi$
$643$	−1.68680e6	−0.160893	−0.0804465	+	0.996759i	$0.525635\pi$
$644$	1.08011e6	0.102625
$645$	1.02212e6	0.0967394
$646$	−1.74384e7	−1.64409
$647$	−1.68323e6	−0.158082	−0.0790409	−	0.996871i	$-0.525186\pi$
$647$	−1.68323e6	−0.158082	−0.0790409	+	0.996871i	$0.525186\pi$
$648$	−3.44669e6	−0.322451
$649$	−4.14007e6	−0.385830
$650$	−6.55579e6	−0.608613
$651$	−2.16322e6	−0.200055
$652$	8.23508e6	0.758663
$653$	−4.81296e6	−0.441702	−0.220851	−	0.975308i	$-0.570883\pi$
$653$	−4.81296e6	−0.441702	−0.220851	+	0.975308i	$0.570883\pi$
$654$	1.24078e6	0.113436
$655$	−3.62014e6	−0.329703
$656$	−3.19703e6	−0.290060
$657$	2.74335e6	0.247952
$658$	−4.09593e6	−0.368797
$659$	−8.55674e6	−0.767529	−0.383764	−	0.923431i	$-0.625372\pi$
$659$	−8.55674e6	−0.767529	−0.383764	+	0.923431i	$0.625372\pi$
$660$	−103978.	−0.00929138
$661$	−1.00927e7	−0.898473	−0.449237	−	0.893413i	$-0.648304\pi$
$661$	−1.00927e7	−0.898473	−0.449237	+	0.893413i	$0.648304\pi$
$662$	1.01222e7	0.897694
$663$	−3.19374e6	−0.282173
$664$	4.01118e6	0.353063
$665$	6.08081e6	0.533221
$666$	−1.94804e6	−0.170181
$667$	−4.15995e6	−0.362055
$668$	−7.44627e6	−0.645651
$669$	−157307.	−0.0135889
$670$	3.07556e6	0.264690
$671$	−1.15551e6	−0.0990758
$672$	−264641.	−0.0226066
$673$	1.40486e7	1.19563	0.597814	−	0.801635i	$-0.296036\pi$
$673$	1.40486e7	1.19563	0.597814	+	0.801635i	$0.296036\pi$
$674$	6.28891e6	0.533243
$675$	3.15616e6	0.266624
$676$	1.17758e6	0.0991113
$677$	9.40935e6	0.789020	0.394510	−	0.918892i	$-0.370914\pi$
$677$	9.40935e6	0.789020	0.394510	+	0.918892i	$0.370914\pi$
$678$	620117.	0.0518083
$679$	−3.33239e6	−0.277384
$680$	−2.95200e6	−0.244819
$681$	−3.31041e6	−0.273535
$682$	−3.13857e6	−0.258387
$683$	−1.09826e7	−0.900850	−0.450425	−	0.892814i	$-0.648728\pi$
$683$	−1.09826e7	−0.900850	−0.450425	+	0.892814i	$0.648728\pi$
$684$	−9.21530e6	−0.753128
$685$	−1.05629e7	−0.860113
$686$	9.37675e6	0.760750
$687$	81685.2	0.00660316
$688$	3.77443e6	0.304004
$689$	−4.73528e6	−0.380012
$690$	−194315.	−0.0155376
$691$	1.57836e7	1.25750	0.628752	−	0.777606i	$-0.283566\pi$
$691$	1.57836e7	1.25750	0.628752	+	0.777606i	$0.283566\pi$
$692$	7.53767e6	0.598373
$693$	2.12947e6	0.168438
$694$	1.62253e7	1.27878
$695$	−3.02668e6	−0.237687
$696$	1.01924e6	0.0797543
$697$	−2.22903e7	−1.73794
$698$	8.52378e6	0.662207
$699$	−1.21680e6	−0.0941950
$700$	−3.78749e6	−0.292150
$701$	1.19739e7	0.920325	0.460162	−	0.887835i	$-0.347791\pi$
$701$	1.19739e7	0.920325	0.460162	+	0.887835i	$0.347791\pi$
$702$	−3.42695e6	−0.262461
$703$	−5.04460e6	−0.384980
$704$	−383962.	−0.0291982
$705$	736869.	0.0558364
$706$	−6.96252e6	−0.525720
$707$	6.24837e6	0.470131
$708$	1.89567e6	0.142128
$709$	9.50451e6	0.710091	0.355046	−	0.934849i	$-0.384465\pi$
$709$	9.50451e6	0.710091	0.355046	+	0.934849i	$0.384465\pi$
$710$	4.38442e6	0.326412
$711$	−808295.	−0.0599647
$712$	9.30189e6	0.687656
$713$	−5.86541e6	−0.432091
$714$	−1.84513e6	−0.135450
$715$	1.61579e6	0.118200
$716$	−1.97080e6	−0.143668
$717$	520511.	0.0378122
$718$	−3.06833e6	−0.222121
$719$	−1.90912e7	−1.37724	−0.688622	−	0.725120i	$-0.741784\pi$
$719$	−1.90912e7	−1.37724	−0.688622	+	0.725120i	$0.741784\pi$
$720$	−1.55998e6	−0.112147
$721$	1.35445e7	0.970340
$722$	−1.39593e7	−0.996603
$723$	621885.	0.0442450
$724$	−1.66445e6	−0.118011
$725$	1.45872e7	1.03069
$726$	1.63388e6	0.115048
$727$	1.03393e7	0.725527	0.362764	−	0.931881i	$-0.381833\pi$
$727$	1.03393e7	0.725527	0.362764	+	0.931881i	$0.381833\pi$
$728$	4.11245e6	0.287589
$729$	−1.18617e7	−0.826666
$730$	1.20260e6	0.0835243
$731$	2.63160e7	1.82149
$732$	529089.	0.0364965
$733$	−9.49287e6	−0.652586	−0.326293	−	0.945269i	$-0.605800\pi$
$733$	−9.49287e6	−0.652586	−0.326293	+	0.945269i	$0.605800\pi$
$734$	−9.04601e6	−0.619750
$735$	−521752.	−0.0356243
$736$	−717554.	−0.0488270
$737$	2.78910e6	0.189145
$738$	−1.17792e7	−0.796116
$739$	8.62625e6	0.581046	0.290523	−	0.956868i	$-0.406171\pi$
$739$	8.62625e6	0.581046	0.290523	+	0.956868i	$0.406171\pi$
$740$	−853956.	−0.0573266
$741$	−4.37049e6	−0.292405
$742$	−2.73573e6	−0.182416
$743$	−1.15175e7	−0.765395	−0.382698	−	0.923874i	$-0.625005\pi$
$743$	−1.15175e7	−0.765395	−0.382698	+	0.923874i	$0.625005\pi$
$744$	1.43710e6	0.0951820
$745$	−1.25721e7	−0.829884
$746$	−2.09774e7	−1.38008
$747$	1.47789e7	0.969038
$748$	−2.67705e6	−0.174945
$749$	2.11941e7	1.38042
$750$	1.54795e6	0.100485
$751$	−4.89021e6	−0.316393	−0.158197	−	0.987408i	$-0.550568\pi$
$751$	−4.89021e6	−0.316393	−0.158197	+	0.987408i	$0.550568\pi$
$752$	2.72106e6	0.175466
$753$	−2.65603e6	−0.170705
$754$	−1.58387e7	−1.01459
$755$	−5.70926e6	−0.364512
$756$	−1.97986e6	−0.125988
$757$	1.39630e7	0.885606	0.442803	−	0.896619i	$-0.353984\pi$
$757$	1.39630e7	0.885606	0.442803	+	0.896619i	$0.353984\pi$
$758$	−3.26081e6	−0.206136
$759$	−176217.	−0.0111030
$760$	−4.03969e6	−0.253696
$761$	−2.10536e7	−1.31785	−0.658925	−	0.752209i	$-0.728988\pi$
$761$	−2.10536e7	−1.31785	−0.658925	+	0.752209i	$0.728988\pi$
$762$	1.05878e6	0.0660568
$763$	−1.11395e7	−0.692714
$764$	9.10485e6	0.564338
$765$	−1.08764e7	−0.671944
$766$	−367893.	−0.0226543
$767$	−2.94582e7	−1.80808
$768$	175810.	0.0107557
$769$	3.62465e6	0.221029	0.110515	−	0.993874i	$-0.464750\pi$
$769$	3.62465e6	0.221029	0.110515	+	0.993874i	$0.464750\pi$
$770$	933492.	0.0567393
$771$	2.54395e6	0.154125
$772$	−8.77135e6	−0.529692
$773$	−6.05290e6	−0.364347	−0.182173	−	0.983266i	$-0.558313\pi$
$773$	−6.05290e6	−0.364347	−0.182173	+	0.983266i	$0.558313\pi$
$774$	1.39066e7	0.834389
$775$	2.05675e7	1.23006
$776$	2.21382e6	0.131974
$777$	−533759.	−0.0317170
$778$	2.15970e7	1.27922
$779$	−3.05033e7	−1.80096
$780$	−739842.	−0.0435414
$781$	3.97605e6	0.233252
$782$	−5.00292e6	−0.292554
$783$	7.62526e6	0.444478
$784$	−1.92669e6	−0.111950
$785$	7.25955e6	0.420471
$786$	1.50321e6	0.0867890
$787$	1.08966e7	0.627126	0.313563	−	0.949567i	$-0.398477\pi$
$787$	1.08966e7	0.627126	0.313563	+	0.949567i	$0.398477\pi$
$788$	−1.10528e7	−0.634100
$789$	999545.	0.0571623
$790$	−354330.	−0.0201995
$791$	−5.56729e6	−0.316376
$792$	−1.41468e6	−0.0801393
$793$	−8.22191e6	−0.464290
$794$	−1.01892e7	−0.573571
$795$	492165.	0.0276180
$796$	−1.80607e6	−0.101031
$797$	−1.77449e7	−0.989528	−0.494764	−	0.869027i	$-0.664745\pi$
$797$	−1.77449e7	−0.989528	−0.494764	+	0.869027i	$0.664745\pi$
$798$	−2.52497e6	−0.140362
$799$	1.89717e7	1.05133
$800$	2.51616e6	0.138999
$801$	3.42721e7	1.88738
$802$	6.09857e6	0.334805
$803$	1.09059e6	0.0596858
$804$	−1.27709e6	−0.0696754
$805$	1.74453e6	0.0948828
$806$	−2.23322e7	−1.21086
$807$	−194119.	−0.0104926
$808$	−4.15100e6	−0.223679
$809$	−2.11929e7	−1.13846	−0.569232	−	0.822177i	$-0.692759\pi$
$809$	−2.11929e7	−1.13846	−0.569232	+	0.822177i	$0.692759\pi$
$810$	−5.56686e6	−0.298124
$811$	8.97945e6	0.479399	0.239700	−	0.970847i	$-0.422951\pi$
$811$	8.97945e6	0.479399	0.239700	+	0.970847i	$0.422951\pi$
$812$	−9.15056e6	−0.487032
$813$	4.71425e6	0.250142
$814$	−774418.	−0.0409651
$815$	1.33008e7	0.701427
$816$	1.22578e6	0.0644446
$817$	3.60122e7	1.88754
$818$	1.51431e7	0.791284
$819$	1.51520e7	0.789335
$820$	−5.16364e6	−0.268177
$821$	−3.01371e7	−1.56043	−0.780213	−	0.625514i	$-0.784889\pi$
$821$	−3.01371e7	−1.56043	−0.780213	+	0.625514i	$0.784889\pi$
$822$	4.38609e6	0.226411
$823$	−5.76171e6	−0.296518	−0.148259	−	0.988949i	$-0.547367\pi$
$823$	−5.76171e6	−0.296518	−0.148259	+	0.988949i	$0.547367\pi$
$824$	−8.99805e6	−0.461668
$825$	617916.	0.0316078
$826$	−1.70190e7	−0.867927
$827$	−2.12463e7	−1.08024	−0.540119	−	0.841589i	$-0.681621\pi$
$827$	−2.12463e7	−1.08024	−0.540119	+	0.841589i	$0.681621\pi$
$828$	−2.64378e6	−0.134014
$829$	2.78936e7	1.40967	0.704837	−	0.709370i	$-0.251020\pi$
$829$	2.78936e7	1.40967	0.704837	+	0.709370i	$0.251020\pi$
$830$	6.47859e6	0.326427
$831$	−3.51592e6	−0.176619
$832$	−2.73204e6	−0.136829
$833$	−1.34332e7	−0.670762
$834$	1.25679e6	0.0625672
$835$	−1.20267e7	−0.596940
$836$	−3.66343e6	−0.181289
$837$	1.07514e7	0.530458
$838$	9.08734e6	0.447020
$839$	1.88890e7	0.926410	0.463205	−	0.886251i	$-0.346699\pi$
$839$	1.88890e7	0.926410	0.463205	+	0.886251i	$0.346699\pi$
$840$	−427431.	−0.0209010
$841$	1.47314e7	0.718214
$842$	1.68864e6	0.0820838
$843$	−4.72738e6	−0.229114
$844$	−3.97130e6	−0.191901
$845$	1.90195e6	0.0916340
$846$	1.00256e7	0.481596
$847$	−1.46687e7	−0.702557
$848$	1.81743e6	0.0867899
$849$	1.97402e6	0.0939903
$850$	1.75431e7	0.832835
$851$	−1.44725e6	−0.0685044
$852$	−1.82057e6	−0.0859229
$853$	−3.31375e7	−1.55936	−0.779681	−	0.626177i	$-0.784619\pi$
$853$	−3.31375e7	−1.55936	−0.779681	+	0.626177i	$0.784619\pi$
$854$	−4.75006e6	−0.222872
$855$	−1.48839e7	−0.696310
$856$	−1.40800e7	−0.656776
$857$	2.62281e7	1.21987	0.609936	−	0.792451i	$-0.291195\pi$
$857$	2.62281e7	1.21987	0.609936	+	0.792451i	$0.291195\pi$
$858$	−670933.	−0.0311143
$859$	3.75341e7	1.73557	0.867787	−	0.496937i	$-0.165542\pi$
$859$	3.75341e7	1.73557	0.867787	+	0.496937i	$0.165542\pi$
$860$	6.09620e6	0.281069
$861$	−3.22749e6	−0.148374
$862$	−7.34018e6	−0.336464
$863$	−2.27040e7	−1.03771	−0.518854	−	0.854863i	$-0.673641\pi$
$863$	−2.27040e7	−1.03771	−0.518854	+	0.854863i	$0.673641\pi$
$864$	1.31529e6	0.0599427
$865$	1.21743e7	0.553230
$866$	−6.14221e6	−0.278311
$867$	4.73737e6	0.214037
$868$	−1.29020e7	−0.581244
$869$	−321327.	−0.0144344
$870$	1.64621e6	0.0737373
$871$	1.98456e7	0.886376
$872$	7.40033e6	0.329579
$873$	8.15664e6	0.362223
$874$	−6.84627e6	−0.303163
$875$	−1.38972e7	−0.613630
$876$	−499361.	−0.0219864
$877$	2.62522e7	1.15257	0.576284	−	0.817250i	$-0.304502\pi$
$877$	2.62522e7	1.15257	0.576284	+	0.817250i	$0.304502\pi$
$878$	1.75512e7	0.768372
$879$	−5.58927e6	−0.243996
$880$	−620150.	−0.0269954
$881$	3.20496e7	1.39118	0.695590	−	0.718439i	$-0.255143\pi$
$881$	3.20496e7	1.39118	0.695590	+	0.718439i	$0.255143\pi$
$882$	−7.09876e6	−0.307264
$883$	1.56098e7	0.673745	0.336873	−	0.941550i	$-0.390631\pi$
$883$	1.56098e7	0.673745	0.336873	+	0.941550i	$0.390631\pi$
$884$	−1.90483e7	−0.819832
$885$	3.06176e6	0.131405
$886$	−3.17992e7	−1.36092
$887$	−3.70919e7	−1.58296	−0.791480	−	0.611195i	$-0.790689\pi$
$887$	−3.70919e7	−1.58296	−0.791480	+	0.611195i	$0.790689\pi$
$888$	354594.	0.0150903
$889$	−9.50551e6	−0.403386
$890$	1.50238e7	0.635777
$891$	−5.04836e6	−0.213037
$892$	−938222.	−0.0394815
$893$	2.59620e7	1.08945
$894$	5.22039e6	0.218454
$895$	−3.18310e6	−0.132829
$896$	−1.57839e6	−0.0656816
$897$	−1.25385e6	−0.0520313
$898$	−2.26838e7	−0.938695
$899$	4.96910e7	2.05059
$900$	9.27060e6	0.381506
$901$	1.26715e7	0.520014
$902$	−4.68269e6	−0.191637
$903$	3.81038e6	0.155507
$904$	3.69854e6	0.150525
$905$	−2.68831e6	−0.109108
$906$	2.37069e6	0.0959521
$907$	3.26587e7	1.31820	0.659098	−	0.752057i	$-0.270938\pi$
$907$	3.26587e7	1.31820	0.659098	+	0.752057i	$0.270938\pi$
$908$	−1.97441e7	−0.794736
$909$	−1.52941e7	−0.613922
$910$	6.64216e6	0.265892
$911$	−2.64624e7	−1.05641	−0.528206	−	0.849117i	$-0.677135\pi$
$911$	−2.64624e7	−1.05641	−0.528206	+	0.849117i	$0.677135\pi$
$912$	1.67742e6	0.0667814
$913$	5.87517e6	0.233262
$914$	−1.83632e7	−0.727083
$915$	854550.	0.0337431
$916$	487192.	0.0191850
$917$	−1.34956e7	−0.529990
$918$	9.17042e6	0.359155
$919$	1.98496e7	0.775287	0.387643	−	0.921809i	$-0.373289\pi$
$919$	1.98496e7	0.775287	0.387643	+	0.921809i	$0.373289\pi$
$920$	−1.15895e6	−0.0451433
$921$	−2.27919e6	−0.0885384
$922$	−6.07215e6	−0.235242
$923$	2.82912e7	1.09307
$924$	−387620.	−0.0149357
$925$	5.07487e6	0.195016
$926$	−1.51373e7	−0.580126
$927$	−3.31527e7	−1.26712
$928$	6.07902e6	0.231720
$929$	3.80714e7	1.44730	0.723652	−	0.690165i	$-0.242462\pi$
$929$	3.80714e7	1.44730	0.723652	+	0.690165i	$0.242462\pi$
$930$	2.32111e6	0.0880011
$931$	−1.83828e7	−0.695085
$932$	−7.25733e6	−0.273676
$933$	−6.35769e6	−0.239108
$934$	2.35164e7	0.882073
$935$	−4.32379e6	−0.161747
$936$	−1.00660e7	−0.375550
$937$	−3.54747e7	−1.31999	−0.659994	−	0.751271i	$-0.729441\pi$
$937$	−3.54747e7	−1.31999	−0.659994	+	0.751271i	$0.729441\pi$
$938$	1.14654e7	0.425484
$939$	−5.61159e6	−0.207693
$940$	4.39488e6	0.162228
$941$	−1.47629e7	−0.543497	−0.271749	−	0.962368i	$-0.587602\pi$
$941$	−1.47629e7	−0.543497	−0.271749	+	0.962368i	$0.587602\pi$
$942$	−3.01443e6	−0.110682
$943$	−8.75109e6	−0.320467
$944$	1.13063e7	0.412942
$945$	−3.19774e6	−0.116483
$946$	5.52840e6	0.200850
$947$	1.95909e7	0.709871	0.354935	−	0.934891i	$-0.384503\pi$
$947$	1.95909e7	0.709871	0.354935	+	0.934891i	$0.384503\pi$
$948$	147131.	0.00531719
$949$	7.75994e6	0.279700
$950$	2.40070e7	0.863034
$951$	5.66246e6	0.203027
$952$	−1.10048e7	−0.393541
$953$	1.99480e7	0.711488	0.355744	−	0.934583i	$-0.384227\pi$
$953$	1.99480e7	0.711488	0.355744	+	0.934583i	$0.384227\pi$
$954$	6.69621e6	0.238209
$955$	1.47055e7	0.521762
$956$	3.10446e6	0.109861
$957$	1.49288e6	0.0526921
$958$	−1.71106e7	−0.602353
$959$	−3.93774e7	−1.38261
$960$	283957.	0.00994429
$961$	4.14336e7	1.44725
$962$	−5.51029e6	−0.191971
$963$	−5.18766e7	−1.80263
$964$	3.70908e6	0.128551
$965$	−1.41669e7	−0.489730
$966$	−724390.	−0.0249764
$967$	5.29988e7	1.82263	0.911317	−	0.411705i	$-0.135067\pi$
$967$	5.29988e7	1.82263	0.911317	+	0.411705i	$0.135067\pi$
$968$	9.74488e6	0.334263
$969$	1.16953e7	0.400131
$970$	3.57561e6	0.122017
$971$	−4.33148e7	−1.47431	−0.737153	−	0.675725i	$-0.763830\pi$
$971$	−4.33148e7	−1.47431	−0.737153	+	0.675725i	$0.763830\pi$
$972$	7.30554e6	0.248020
$973$	−1.12832e7	−0.382077
$974$	−1.96890e7	−0.665008
$975$	4.39672e6	0.148121
$976$	3.15563e6	0.106038
$977$	3.51591e7	1.17842	0.589212	−	0.807978i	$-0.299438\pi$
$977$	3.51591e7	1.17842	0.589212	+	0.807978i	$0.299438\pi$
$978$	−5.52296e6	−0.184639
$979$	1.36245e7	0.454321
$980$	−3.11187e6	−0.103504
$981$	2.72660e7	0.904584
$982$	−1.78745e7	−0.591500
$983$	−4.19275e7	−1.38393	−0.691967	−	0.721929i	$-0.743256\pi$
$983$	−4.19275e7	−1.38393	−0.691967	+	0.721929i	$0.743256\pi$
$984$	2.14413e6	0.0705932
$985$	−1.78518e7	−0.586261
$986$	4.23840e7	1.38838
$987$	2.74698e6	0.0897559
$988$	−2.60667e7	−0.849560
$989$	1.03316e7	0.335873
$990$	−2.28490e6	−0.0740932
$991$	−2.45776e7	−0.794980	−0.397490	−	0.917607i	$-0.630119\pi$
$991$	−2.45776e7	−0.794980	−0.397490	+	0.917607i	$0.630119\pi$
$992$	8.57124e6	0.276544
$993$	−6.78854e6	−0.218476
$994$	1.63447e7	0.524701
$995$	−2.91705e6	−0.0934085
$996$	−2.69015e6	−0.0859266
$997$	−9.59067e6	−0.305570	−0.152785	−	0.988259i	$-0.548824\pi$
$997$	−9.59067e6	−0.305570	−0.152785	+	0.988259i	$0.548824\pi$
$998$	−9.27567e6	−0.294794
$999$	2.65282e6	0.0840997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.6.a.c.1.17	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.6.a.c.1.17	✓	30	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:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +2.68265 q^{3} +16.0000 q^{4} +25.8421 q^{5} -10.7306 q^{6} +96.3372 q^{7} -64.0000 q^{8} -235.803 q^{9} -103.369 q^{10} -93.7407 q^{11} +42.9224 q^{12} -667.002 q^{13} -385.349 q^{14} +69.3253 q^{15} +256.000 q^{16} +1784.88 q^{17} +943.214 q^{18} +2442.53 q^{19} +413.474 q^{20} +258.439 q^{21} +374.963 q^{22} +700.737 q^{23} -171.689 q^{24} -2457.18 q^{25} +2668.01 q^{26} -1284.46 q^{27} +1541.40 q^{28} -5936.54 q^{29} -277.301 q^{30} -8370.35 q^{31} -1024.00 q^{32} -251.473 q^{33} -7139.51 q^{34} +2489.56 q^{35} -3772.85 q^{36} -2065.32 q^{37} -9770.11 q^{38} -1789.33 q^{39} -1653.90 q^{40} -12488.4 q^{41} -1033.75 q^{42} +14743.9 q^{43} -1499.85 q^{44} -6093.66 q^{45} -2802.95 q^{46} +10629.1 q^{47} +686.758 q^{48} -7526.14 q^{49} +9828.74 q^{50} +4788.20 q^{51} -10672.0 q^{52} +7099.35 q^{53} +5137.84 q^{54} -2422.46 q^{55} -6165.58 q^{56} +6552.44 q^{57} +23746.2 q^{58} +44165.1 q^{59} +1109.21 q^{60} +12326.7 q^{61} +33481.4 q^{62} -22716.6 q^{63} +4096.00 q^{64} -17236.8 q^{65} +1005.89 q^{66} -29753.4 q^{67} +28558.0 q^{68} +1879.83 q^{69} -9958.23 q^{70} -42415.5 q^{71} +15091.4 q^{72} -11634.1 q^{73} +8261.28 q^{74} -6591.76 q^{75} +39080.4 q^{76} -9030.72 q^{77} +7157.32 q^{78} +3427.83 q^{79} +6615.59 q^{80} +53854.5 q^{81} +49953.7 q^{82} -62674.7 q^{83} +4135.02 q^{84} +46125.0 q^{85} -58975.4 q^{86} -15925.7 q^{87} +5999.40 q^{88} -145342. q^{89} +24374.6 q^{90} -64257.1 q^{91} +11211.8 q^{92} -22454.7 q^{93} -42516.6 q^{94} +63120.1 q^{95} -2747.03 q^{96} -34590.9 q^{97} +30104.6 q^{98} +22104.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 120 q^{2} - 30 q^{3} + 480 q^{4} - 136 q^{5} + 120 q^{6} - 123 q^{7} - 1920 q^{8} + 2670 q^{9} + 544 q^{10} - 1058 q^{11} - 480 q^{12} - 371 q^{13} + 492 q^{14} - 1364 q^{15} + 7680 q^{16} - 1918 q^{17}+ \cdots - 78063 q^{99}+O(q^{100}) \) 30 * q - 120 * q^2 - 30 * q^3 + 480 * q^4 - 136 * q^5 + 120 * q^6 - 123 * q^7 - 1920 * q^8 + 2670 * q^9 + 544 * q^10 - 1058 * q^11 - 480 * q^12 - 371 * q^13 + 492 * q^14 - 1364 * q^15 + 7680 * q^16 - 1918 * q^17 - 10680 * q^18 - 3315 * q^19 - 2176 * q^20 - 1217 * q^21 + 4232 * q^22 - 9534 * q^23 + 1920 * q^24 + 22130 * q^25 + 1484 * q^26 - 12156 * q^27 - 1968 * q^28 - 8009 * q^29 + 5456 * q^30 - 3543 * q^31 - 30720 * q^32 - 3411 * q^33 + 7672 * q^34 - 11177 * q^35 + 42720 * q^36 - 6517 * q^37 + 13260 * q^38 - 2453 * q^39 + 8704 * q^40 - 5951 * q^41 + 4868 * q^42 - 18768 * q^43 - 16928 * q^44 - 22619 * q^45 + 38136 * q^46 - 41067 * q^47 - 7680 * q^48 + 107797 * q^49 - 88520 * q^50 - 4187 * q^51 - 5936 * q^52 - 64055 * q^53 + 48624 * q^54 + 31973 * q^55 + 7872 * q^56 - 25442 * q^57 + 32036 * q^58 - 66832 * q^59 - 21824 * q^60 + 46777 * q^61 + 14172 * q^62 - 115994 * q^63 + 122880 * q^64 - 68783 * q^65 + 13644 * q^66 - 107218 * q^67 - 30688 * q^68 + 228589 * q^69 + 44708 * q^70 + 4733 * q^71 - 170880 * q^72 + 150385 * q^73 + 26068 * q^74 + 133325 * q^75 - 53040 * q^76 + 292020 * q^77 + 9812 * q^78 + 31827 * q^79 - 34816 * q^80 + 425214 * q^81 + 23804 * q^82 + 107301 * q^83 - 19472 * q^84 + 338625 * q^85 + 75072 * q^86 + 288484 * q^87 + 67712 * q^88 - 99690 * q^89 + 90476 * q^90 - 109140 * q^91 - 152544 * q^92 - 114545 * q^93 + 164268 * q^94 - 155668 * q^95 + 30720 * q^96 - 79959 * q^97 - 431188 * q^98 - 78063 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more