LMFDB - Embedded newform 1075.6.a.j.1.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.36705 q^{2} +9.50108 q^{3} +22.2734 q^{4} +69.9949 q^{6} -66.6986 q^{7} -71.6560 q^{8} -152.730 q^{9} +O(q^{10})$	\(q+7.36705 q^{2} +9.50108 q^{3} +22.2734 q^{4} +69.9949 q^{6} -66.6986 q^{7} -71.6560 q^{8} -152.730 q^{9} -25.5585 q^{11} +211.622 q^{12} -1089.67 q^{13} -491.372 q^{14} -1240.64 q^{16} +2011.78 q^{17} -1125.17 q^{18} +511.460 q^{19} -633.709 q^{21} -188.291 q^{22} +2820.90 q^{23} -680.809 q^{24} -8027.62 q^{26} -3759.86 q^{27} -1485.61 q^{28} +6601.56 q^{29} -614.980 q^{31} -6846.89 q^{32} -242.833 q^{33} +14820.9 q^{34} -3401.81 q^{36} +205.824 q^{37} +3767.95 q^{38} -10353.0 q^{39} +16153.8 q^{41} -4668.56 q^{42} +1849.00 q^{43} -569.276 q^{44} +20781.8 q^{46} +13797.2 q^{47} -11787.5 q^{48} -12358.3 q^{49} +19114.1 q^{51} -24270.6 q^{52} -19950.4 q^{53} -27699.1 q^{54} +4779.36 q^{56} +4859.42 q^{57} +48634.1 q^{58} +45706.9 q^{59} -8980.02 q^{61} -4530.59 q^{62} +10186.9 q^{63} -10740.8 q^{64} -1788.97 q^{66} +25253.4 q^{67} +44809.4 q^{68} +26801.6 q^{69} +31165.4 q^{71} +10944.0 q^{72} +57916.4 q^{73} +1516.31 q^{74} +11392.0 q^{76} +1704.72 q^{77} -76271.0 q^{78} +61188.6 q^{79} +1390.59 q^{81} +119006. q^{82} +44354.7 q^{83} -14114.9 q^{84} +13621.7 q^{86} +62722.0 q^{87} +1831.42 q^{88} -15149.0 q^{89} +72679.2 q^{91} +62831.3 q^{92} -5842.97 q^{93} +101645. q^{94} -65052.9 q^{96} -62923.3 q^{97} -91044.2 q^{98} +3903.54 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9}+O(q^{10}) $ 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9	\( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9} + 675 q^{11} - 4446 q^{12} + 1241 q^{13} + 2375 q^{14} + 10518 q^{16} + 1153 q^{17} - 6680 q^{18} + 4065 q^{19} + 9953 q^{21} + 9283 q^{22} - 360 q^{23} + 2265 q^{24} + 23695 q^{26} + 1323 q^{27} - 30375 q^{28} + 19290 q^{29} + 23291 q^{31} + 8166 q^{32} - 10388 q^{33} - 13153 q^{34} + 148705 q^{36} + 13501 q^{37} - 8127 q^{38} - 1327 q^{39} + 38345 q^{41} - 21835 q^{42} + 68413 q^{43} + 47768 q^{44} + 48755 q^{46} + 84859 q^{47} - 208720 q^{48} + 107255 q^{49} + 62027 q^{51} + 128320 q^{52} - 53559 q^{53} + 44158 q^{54} + 107538 q^{56} + 104239 q^{57} - 85186 q^{58} + 48186 q^{59} + 82364 q^{61} + 206506 q^{62} - 75269 q^{63} + 161467 q^{64} + 91969 q^{66} + 38168 q^{67} + 95991 q^{68} + 287103 q^{69} + 155302 q^{71} + 9979 q^{72} + 31927 q^{73} + 59946 q^{74} + 225407 q^{76} - 80007 q^{77} - 67815 q^{78} + 150174 q^{79} + 417489 q^{81} + 60603 q^{82} + 266568 q^{83} + 586273 q^{84} - 57554 q^{87} + 323054 q^{88} + 334356 q^{89} + 51747 q^{91} - 258529 q^{92} - 285287 q^{93} + 302744 q^{94} + 287282 q^{96} - 78640 q^{97} - 397117 q^{98} + 362152 q^{99}+O(q^{100}) \) 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9 + 675 * q^11 - 4446 * q^12 + 1241 * q^13 + 2375 * q^14 + 10518 * q^16 + 1153 * q^17 - 6680 * q^18 + 4065 * q^19 + 9953 * q^21 + 9283 * q^22 - 360 * q^23 + 2265 * q^24 + 23695 * q^26 + 1323 * q^27 - 30375 * q^28 + 19290 * q^29 + 23291 * q^31 + 8166 * q^32 - 10388 * q^33 - 13153 * q^34 + 148705 * q^36 + 13501 * q^37 - 8127 * q^38 - 1327 * q^39 + 38345 * q^41 - 21835 * q^42 + 68413 * q^43 + 47768 * q^44 + 48755 * q^46 + 84859 * q^47 - 208720 * q^48 + 107255 * q^49 + 62027 * q^51 + 128320 * q^52 - 53559 * q^53 + 44158 * q^54 + 107538 * q^56 + 104239 * q^57 - 85186 * q^58 + 48186 * q^59 + 82364 * q^61 + 206506 * q^62 - 75269 * q^63 + 161467 * q^64 + 91969 * q^66 + 38168 * q^67 + 95991 * q^68 + 287103 * q^69 + 155302 * q^71 + 9979 * q^72 + 31927 * q^73 + 59946 * q^74 + 225407 * q^76 - 80007 * q^77 - 67815 * q^78 + 150174 * q^79 + 417489 * q^81 + 60603 * q^82 + 266568 * q^83 + 586273 * q^84 - 57554 * q^87 + 323054 * q^88 + 334356 * q^89 + 51747 * q^91 - 258529 * q^92 - 285287 * q^93 + 302744 * q^94 + 287282 * q^96 - 78640 * q^97 - 397117 * q^98 + 362152 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	7.36705	1.30232	0.651162	−	0.758939i	$-0.274282\pi$
$2$	7.36705	1.30232	0.651162	+	0.758939i	$0.274282\pi$
$3$	9.50108	0.609494	0.304747	−	0.952433i	$-0.401428\pi$
$3$	9.50108	0.609494	0.304747	+	0.952433i	$0.401428\pi$
$4$	22.2734	0.696045
$5$	0	0
$5$	0	0
$6$	69.9949	0.793759
$7$	−66.6986	−0.514484	−0.257242	−	0.966347i	$-0.582814\pi$
$7$	−66.6986	−0.514484	−0.257242	+	0.966347i	$0.582814\pi$
$8$	−71.6560	−0.395847
$9$	−152.730	−0.628517
$10$	0	0
$11$	−25.5585	−0.0636875	−0.0318437	−	0.999493i	$-0.510138\pi$
$11$	−25.5585	−0.0636875	−0.0318437	+	0.999493i	$0.510138\pi$
$12$	211.622	0.424236
$13$	−1089.67	−1.78828	−0.894139	−	0.447790i	$-0.852211\pi$
$13$	−1089.67	−1.78828	−0.894139	+	0.447790i	$0.852211\pi$
$14$	−491.372	−0.670024
$15$	0	0
$16$	−1240.64	−1.21157
$17$	2011.78	1.68834	0.844169	−	0.536078i	$-0.180095\pi$
$17$	2011.78	1.68834	0.844169	+	0.536078i	$0.180095\pi$
$18$	−1125.17	−0.818532
$19$	511.460	0.325033	0.162517	−	0.986706i	$-0.448039\pi$
$19$	511.460	0.325033	0.162517	+	0.986706i	$0.448039\pi$
$20$	0	0
$21$	−633.709	−0.313575
$22$	−188.291	−0.0829416
$23$	2820.90	1.11191	0.555954	−	0.831213i	$-0.312353\pi$
$23$	2820.90	1.11191	0.555954	+	0.831213i	$0.312353\pi$
$24$	−680.809	−0.241267
$25$	0	0
$26$	−8027.62	−2.32891
$27$	−3759.86	−0.992572
$28$	−1485.61	−0.358104
$29$	6601.56	1.45765	0.728823	−	0.684702i	$-0.240068\pi$
$29$	6601.56	1.45765	0.728823	+	0.684702i	$0.240068\pi$
$30$	0	0
$31$	−614.980	−0.114936	−0.0574681	−	0.998347i	$-0.518303\pi$
$31$	−614.980	−0.114936	−0.0574681	+	0.998347i	$0.518303\pi$
$32$	−6846.89	−1.18200
$33$	−242.833	−0.0388171
$34$	14820.9	2.19876
$35$	0	0
$36$	−3401.81	−0.437476
$37$	205.824	0.0247167	0.0123584	−	0.999924i	$-0.496066\pi$
$37$	205.824	0.0247167	0.0123584	+	0.999924i	$0.496066\pi$
$38$	3767.95	0.423298
$39$	−10353.0	−1.08994
$40$	0	0
$41$	16153.8	1.50077	0.750387	−	0.660999i	$-0.229867\pi$
$41$	16153.8	1.50077	0.750387	+	0.660999i	$0.229867\pi$
$42$	−4668.56	−0.408376
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	−569.276	−0.0443294
$45$	0	0
$46$	20781.8	1.44806
$47$	13797.2	0.911061	0.455531	−	0.890220i	$-0.349450\pi$
$47$	13797.2	0.911061	0.455531	+	0.890220i	$0.349450\pi$
$48$	−11787.5	−0.738443
$49$	−12358.3	−0.735306
$50$	0	0
$51$	19114.1	1.02903
$52$	−24270.6	−1.24472
$53$	−19950.4	−0.975579	−0.487789	−	0.872961i	$-0.662197\pi$
$53$	−19950.4	−0.975579	−0.487789	+	0.872961i	$0.662197\pi$
$54$	−27699.1	−1.29265
$55$	0	0
$56$	4779.36	0.203657
$57$	4859.42	0.198106
$58$	48634.1	1.89833
$59$	45706.9	1.70943	0.854715	−	0.519097i	$-0.173732\pi$
$59$	45706.9	1.70943	0.854715	+	0.519097i	$0.173732\pi$
$60$	0	0
$61$	−8980.02	−0.308996	−0.154498	−	0.987993i	$-0.549376\pi$
$61$	−8980.02	−0.308996	−0.154498	+	0.987993i	$0.549376\pi$
$62$	−4530.59	−0.149684
$63$	10186.9	0.323362
$64$	−10740.8	−0.327784
$65$	0	0
$66$	−1788.97	−0.0505525
$67$	25253.4	0.687279	0.343640	−	0.939102i	$-0.388340\pi$
$67$	25253.4	0.687279	0.343640	+	0.939102i	$0.388340\pi$
$68$	44809.4	1.17516
$69$	26801.6	0.677702
$70$	0	0
$71$	31165.4	0.733714	0.366857	−	0.930277i	$-0.380434\pi$
$71$	31165.4	0.733714	0.366857	+	0.930277i	$0.380434\pi$
$72$	10944.0	0.248797
$73$	57916.4	1.27202	0.636010	−	0.771680i	$-0.280584\pi$
$73$	57916.4	1.27202	0.636010	+	0.771680i	$0.280584\pi$
$74$	1516.31	0.0321892
$75$	0	0
$76$	11392.0	0.226238
$77$	1704.72	0.0327662
$78$	−76271.0	−1.41946
$79$	61188.6	1.10307	0.551535	−	0.834152i	$-0.314042\pi$
$79$	61188.6	1.10307	0.551535	+	0.834152i	$0.314042\pi$
$80$	0	0
$81$	1390.59	0.0235498
$82$	119006.	1.95449
$83$	44354.7	0.706714	0.353357	−	0.935488i	$-0.385040\pi$
$83$	44354.7	0.706714	0.353357	+	0.935488i	$0.385040\pi$
$84$	−14114.9	−0.218262
$85$	0	0
$86$	13621.7	0.198602
$87$	62722.0	0.888427
$88$	1831.42	0.0252105
$89$	−15149.0	−0.202726	−0.101363	−	0.994849i	$-0.532320\pi$
$89$	−15149.0	−0.202726	−0.101363	+	0.994849i	$0.532320\pi$
$90$	0	0
$91$	72679.2	0.920040
$92$	62831.3	0.773938
$93$	−5842.97	−0.0700529
$94$	101645.	1.18650
$95$	0	0
$96$	−65052.9	−0.720424
$97$	−62923.3	−0.679020	−0.339510	−	0.940602i	$-0.610261\pi$
$97$	−62923.3	−0.679020	−0.339510	+	0.940602i	$0.610261\pi$
$98$	−91044.2	−0.957606
$99$	3903.54	0.0400286
$100$	0	0
$101$	−3693.31	−0.0360257	−0.0180129	−	0.999838i	$-0.505734\pi$
$101$	−3693.31	−0.0360257	−0.0180129	+	0.999838i	$0.505734\pi$
$102$	140815.	1.34013
$103$	−58495.8	−0.543290	−0.271645	−	0.962398i	$-0.587568\pi$
$103$	−58495.8	−0.543290	−0.271645	+	0.962398i	$0.587568\pi$
$104$	78081.1	0.707885
$105$	0	0
$106$	−146976.	−1.27052
$107$	−134320.	−1.13418	−0.567089	−	0.823657i	$-0.691930\pi$
$107$	−134320.	−1.13418	−0.567089	+	0.823657i	$0.691930\pi$
$108$	−83745.0	−0.690875
$109$	−16991.0	−0.136979	−0.0684895	−	0.997652i	$-0.521818\pi$
$109$	−16991.0	−0.136979	−0.0684895	+	0.997652i	$0.521818\pi$
$110$	0	0
$111$	1955.55	0.0150647
$112$	82749.2	0.623331
$113$	14536.5	0.107094	0.0535468	−	0.998565i	$-0.482947\pi$
$113$	14536.5	0.107094	0.0535468	+	0.998565i	$0.482947\pi$
$114$	35799.6	0.257998
$115$	0	0
$116$	147040.	1.01459
$117$	166424.	1.12396
$118$	336725.	2.22623
$119$	−134183.	−0.868622
$120$	0	0
$121$	−160398.	−0.995944
$122$	−66156.3	−0.402413
$123$	153479.	0.914713
$124$	−13697.7	−0.0800007
$125$	0	0
$126$	75047.1	0.421121
$127$	275301.	1.51460	0.757300	−	0.653067i	$-0.226518\pi$
$127$	275301.	1.51460	0.757300	+	0.653067i	$0.226518\pi$
$128$	139972.	0.755123
$129$	17567.5	0.0929470
$130$	0	0
$131$	−170834.	−0.869755	−0.434878	−	0.900490i	$-0.643208\pi$
$131$	−170834.	−0.869755	−0.434878	+	0.900490i	$0.643208\pi$
$132$	−5408.74	−0.0270185
$133$	−34113.7	−0.167224
$134$	186043.	0.895060
$135$	0	0
$136$	−144156.	−0.668324
$137$	−58147.5	−0.264685	−0.132343	−	0.991204i	$-0.542250\pi$
$137$	−58147.5	−0.264685	−0.132343	+	0.991204i	$0.542250\pi$
$138$	197449.	0.882586
$139$	−336533.	−1.47738	−0.738688	−	0.674047i	$-0.764554\pi$
$139$	−336533.	−1.47738	−0.738688	+	0.674047i	$0.764554\pi$
$140$	0	0
$141$	131089.	0.555287
$142$	229597.	0.955532
$143$	27850.2	0.113891
$144$	189483.	0.761490
$145$	0	0
$146$	426673.	1.65658
$147$	−117417.	−0.448165
$148$	4584.40	0.0172040
$149$	373838.	1.37949	0.689743	−	0.724054i	$-0.257723\pi$
$149$	373838.	1.37949	0.689743	+	0.724054i	$0.257723\pi$
$150$	0	0
$151$	−262321.	−0.936249	−0.468125	−	0.883662i	$-0.655070\pi$
$151$	−262321.	−0.936249	−0.468125	+	0.883662i	$0.655070\pi$
$152$	−36649.2	−0.128663
$153$	−307259.	−1.06115
$154$	12558.7	0.0426721
$155$	0	0
$156$	−230597.	−0.758651
$157$	474594.	1.53664	0.768321	−	0.640064i	$-0.221092\pi$
$157$	474594.	1.53664	0.768321	+	0.640064i	$0.221092\pi$
$158$	450780.	1.43655
$159$	−189550.	−0.594610
$160$	0	0
$161$	−188150.	−0.572059
$162$	10244.6	0.0306695
$163$	549379.	1.61958	0.809791	−	0.586718i	$-0.199580\pi$
$163$	549379.	1.61958	0.809791	+	0.586718i	$0.199580\pi$
$164$	359801.	1.04461
$165$	0	0
$166$	326763.	0.920371
$167$	−254954.	−0.707408	−0.353704	−	0.935357i	$-0.615078\pi$
$167$	−254954.	−0.707408	−0.353704	+	0.935357i	$0.615078\pi$
$168$	45409.0	0.124128
$169$	816078.	2.19794
$170$	0	0
$171$	−78115.0	−0.204289
$172$	41183.6	0.106146
$173$	−374038.	−0.950168	−0.475084	−	0.879940i	$-0.657582\pi$
$173$	−374038.	−0.950168	−0.475084	+	0.879940i	$0.657582\pi$
$174$	462076.	1.15702
$175$	0	0
$176$	31709.0	0.0771616
$177$	434264.	1.04189
$178$	−111604.	−0.264015
$179$	−838738.	−1.95656	−0.978281	−	0.207281i	$-0.933539\pi$
$179$	−838738.	−1.95656	−0.978281	+	0.207281i	$0.933539\pi$
$180$	0	0
$181$	−505861.	−1.14772	−0.573858	−	0.818955i	$-0.694554\pi$
$181$	−505861.	−1.14772	−0.573858	+	0.818955i	$0.694554\pi$
$182$	535431.	1.19819
$183$	−85319.9	−0.188331
$184$	−202135.	−0.440146
$185$	0	0
$186$	−43045.4	−0.0912315
$187$	−51418.2	−0.107526
$188$	307312.	0.634140
$189$	250777.	0.510662
$190$	0	0
$191$	−648003.	−1.28527	−0.642634	−	0.766173i	$-0.722158\pi$
$191$	−648003.	−1.28527	−0.642634	+	0.766173i	$0.722158\pi$
$192$	−102049.	−0.199782
$193$	386904.	0.747671	0.373835	−	0.927495i	$-0.378043\pi$
$193$	386904.	0.747671	0.373835	+	0.927495i	$0.378043\pi$
$194$	−463560.	−0.884303
$195$	0	0
$196$	−275262.	−0.511807
$197$	−665075.	−1.22097	−0.610485	−	0.792028i	$-0.709025\pi$
$197$	−665075.	−1.22097	−0.610485	+	0.792028i	$0.709025\pi$
$198$	28757.6	0.0521302
$199$	731671.	1.30973	0.654867	−	0.755744i	$-0.272725\pi$
$199$	731671.	1.30973	0.654867	+	0.755744i	$0.272725\pi$
$200$	0	0
$201$	239935.	0.418893
$202$	−27208.8	−0.0469171
$203$	−440315.	−0.749935
$204$	425737.	0.716253
$205$	0	0
$206$	−430941.	−0.707539
$207$	−430836.	−0.698853
$208$	1.35189e6	2.16662
$209$	−13072.2	−0.0207005
$210$	0	0
$211$	546573.	0.845167	0.422583	−	0.906324i	$-0.361123\pi$
$211$	546573.	0.845167	0.422583	+	0.906324i	$0.361123\pi$
$212$	−444365.	−0.679047
$213$	296105.	0.447194
$214$	−989541.	−1.47706
$215$	0	0
$216$	269416.	0.392907
$217$	41018.3	0.0591328
$218$	−125174.	−0.178391
$219$	550268.	0.775290
$220$	0	0
$221$	−2.19217e6	−3.01921
$222$	14406.6	0.0196191
$223$	−164730.	−0.221825	−0.110913	−	0.993830i	$-0.535377\pi$
$223$	−164730.	−0.221825	−0.110913	+	0.993830i	$0.535377\pi$
$224$	456678.	0.608122
$225$	0	0
$226$	107091.	0.139470
$227$	107134.	0.137995	0.0689976	−	0.997617i	$-0.478020\pi$
$227$	107134.	0.137995	0.0689976	+	0.997617i	$0.478020\pi$
$228$	108236.	0.137891
$229$	867612.	1.09329	0.546647	−	0.837363i	$-0.315904\pi$
$229$	867612.	1.09329	0.546647	+	0.837363i	$0.315904\pi$
$230$	0	0
$231$	16196.7	0.0199708
$232$	−473042.	−0.577005
$233$	17432.1	0.0210359	0.0105180	−	0.999945i	$-0.496652\pi$
$233$	17432.1	0.0210359	0.0105180	+	0.999945i	$0.496652\pi$
$234$	1.22606e6	1.46376
$235$	0	0
$236$	1.01805e6	1.18984
$237$	581358.	0.672315
$238$	−988535.	−1.13123
$239$	1.54852e6	1.75357	0.876786	−	0.480881i	$-0.159683\pi$
$239$	1.54852e6	1.75357	0.876786	+	0.480881i	$0.159683\pi$
$240$	0	0
$241$	1.04874e6	1.16312	0.581559	−	0.813504i	$-0.302443\pi$
$241$	1.04874e6	1.16312	0.581559	+	0.813504i	$0.302443\pi$
$242$	−1.18166e6	−1.29704
$243$	926857.	1.00693
$244$	−200016.	−0.215075
$245$	0	0
$246$	1.13068e6	1.19125
$247$	−557320.	−0.581249
$248$	44067.0	0.0454971
$249$	421417.	0.430738
$250$	0	0
$251$	129997.	0.130241	0.0651206	−	0.997877i	$-0.479257\pi$
$251$	129997.	0.130241	0.0651206	+	0.997877i	$0.479257\pi$
$252$	226896.	0.225074
$253$	−72098.1	−0.0708146
$254$	2.02816e6	1.97250
$255$	0	0
$256$	1.37489e6	1.31120
$257$	448726.	0.423788	0.211894	−	0.977293i	$-0.432037\pi$
$257$	448726.	0.423788	0.211894	+	0.977293i	$0.432037\pi$
$258$	129421.	0.121047
$259$	−13728.1	−0.0127164
$260$	0	0
$261$	−1.00825e6	−0.916154
$262$	−1.25855e6	−1.13270
$263$	1.21109e6	1.07966	0.539831	−	0.841773i	$-0.318488\pi$
$263$	1.21109e6	1.07966	0.539831	+	0.841773i	$0.318488\pi$
$264$	17400.5	0.0153657
$265$	0	0
$266$	−251317.	−0.217780
$267$	−143932.	−0.123561
$268$	562481.	0.478378
$269$	1.37462e6	1.15825	0.579126	−	0.815238i	$-0.303394\pi$
$269$	1.37462e6	1.15825	0.579126	+	0.815238i	$0.303394\pi$
$270$	0	0
$271$	−357043.	−0.295323	−0.147662	−	0.989038i	$-0.547175\pi$
$271$	−357043.	−0.295323	−0.147662	+	0.989038i	$0.547175\pi$
$272$	−2.49591e6	−2.04553
$273$	690531.	0.560759
$274$	−428375.	−0.344705
$275$	0	0
$276$	596965.	0.471711
$277$	1.66003e6	1.29992	0.649961	−	0.759968i	$-0.274785\pi$
$277$	1.66003e6	1.29992	0.649961	+	0.759968i	$0.274785\pi$
$278$	−2.47926e6	−1.92402
$279$	93925.6	0.0722393
$280$	0	0
$281$	−1.34331e6	−1.01487	−0.507434	−	0.861691i	$-0.669406\pi$
$281$	−1.34331e6	−1.01487	−0.507434	+	0.861691i	$0.669406\pi$
$282$	965736.	0.723162
$283$	−376294.	−0.279293	−0.139647	−	0.990201i	$-0.544597\pi$
$283$	−376294.	−0.279293	−0.139647	+	0.990201i	$0.544597\pi$
$284$	694161.	0.510698
$285$	0	0
$286$	205174.	0.148323
$287$	−1.07744e6	−0.772124
$288$	1.04572e6	0.742909
$289$	2.62742e6	1.85048
$290$	0	0
$291$	−597840.	−0.413859
$292$	1.29000e6	0.885384
$293$	−1.24274e6	−0.845687	−0.422844	−	0.906203i	$-0.638968\pi$
$293$	−1.24274e6	−0.845687	−0.422844	+	0.906203i	$0.638968\pi$
$294$	−865018.	−0.583656
$295$	0	0
$296$	−14748.5	−0.00978405
$297$	96096.4	0.0632144
$298$	2.75408e6	1.79654
$299$	−3.07384e6	−1.98840
$300$	0	0
$301$	−123326.	−0.0784581
$302$	−1.93254e6	−1.21930
$303$	−35090.4	−0.0219575
$304$	−634539.	−0.393799
$305$	0	0
$306$	−2.26359e6	−1.38196
$307$	2.43147e6	1.47239	0.736196	−	0.676768i	$-0.236620\pi$
$307$	2.43147e6	1.47239	0.736196	+	0.676768i	$0.236620\pi$
$308$	37969.9	0.0228067
$309$	−555773.	−0.331132
$310$	0	0
$311$	2.14742e6	1.25897	0.629486	−	0.777012i	$-0.283265\pi$
$311$	2.14742e6	1.25897	0.629486	+	0.777012i	$0.283265\pi$
$312$	741854.	0.431452
$313$	−1.99799e6	−1.15274	−0.576372	−	0.817188i	$-0.695532\pi$
$313$	−1.99799e6	−1.15274	−0.576372	+	0.817188i	$0.695532\pi$
$314$	3.49636e6	2.00121
$315$	0	0
$316$	1.36288e6	0.767787
$317$	2.37566e6	1.32781	0.663904	−	0.747818i	$-0.268898\pi$
$317$	2.37566e6	1.32781	0.663904	+	0.747818i	$0.268898\pi$
$318$	−1.39643e6	−0.774374
$319$	−168726.	−0.0928337
$320$	0	0
$321$	−1.27618e6	−0.691274
$322$	−1.38611e6	−0.745005
$323$	1.02895e6	0.548765
$324$	30973.3	0.0163917
$325$	0	0
$326$	4.04730e6	2.10922
$327$	−161433.	−0.0834879
$328$	−1.15752e6	−0.594077
$329$	−920257.	−0.468726
$330$	0	0
$331$	2.37745e6	1.19273	0.596364	−	0.802714i	$-0.296611\pi$
$331$	2.37745e6	1.19273	0.596364	+	0.802714i	$0.296611\pi$
$332$	987931.	0.491905
$333$	−31435.3	−0.0155349
$334$	−1.87826e6	−0.921274
$335$	0	0
$336$	786207.	0.379917
$337$	−3.62105e6	−1.73684	−0.868421	−	0.495828i	$-0.834865\pi$
$337$	−3.62105e6	−1.73684	−0.868421	+	0.495828i	$0.834865\pi$
$338$	6.01209e6	2.86242
$339$	138112.	0.0652729
$340$	0	0
$341$	15718.0	0.00731999
$342$	−575477.	−0.266050
$343$	1.94528e6	0.892787
$344$	−132492.	−0.0603661
$345$	0	0
$346$	−2.75556e6	−1.23743
$347$	−2.29886e6	−1.02492	−0.512458	−	0.858712i	$-0.671265\pi$
$347$	−2.29886e6	−1.02492	−0.512458	+	0.858712i	$0.671265\pi$
$348$	1.39703e6	0.618385
$349$	−1.45422e6	−0.639097	−0.319549	−	0.947570i	$-0.603531\pi$
$349$	−1.45422e6	−0.639097	−0.319549	+	0.947570i	$0.603531\pi$
$350$	0	0
$351$	4.09699e6	1.77499
$352$	174996.	0.0752788
$353$	3.44593e6	1.47187	0.735936	−	0.677051i	$-0.236742\pi$
$353$	3.44593e6	1.47187	0.735936	+	0.677051i	$0.236742\pi$
$354$	3.19925e6	1.35687
$355$	0	0
$356$	−337421.	−0.141107
$357$	−1.27489e6	−0.529420
$358$	−6.17903e6	−2.54808
$359$	−520229.	−0.213039	−0.106519	−	0.994311i	$-0.533971\pi$
$359$	−520229.	−0.213039	−0.106519	+	0.994311i	$0.533971\pi$
$360$	0	0
$361$	−2.21451e6	−0.894354
$362$	−3.72670e6	−1.49470
$363$	−1.52395e6	−0.607022
$364$	1.61882e6	0.640389
$365$	0	0
$366$	−628556.	−0.245268
$367$	−2.43493e6	−0.943672	−0.471836	−	0.881686i	$-0.656409\pi$
$367$	−2.43493e6	−0.943672	−0.471836	+	0.881686i	$0.656409\pi$
$368$	−3.49974e6	−1.34715
$369$	−2.46716e6	−0.943261
$370$	0	0
$371$	1.33067e6	0.501920
$372$	−130143.	−0.0487600
$373$	−762409.	−0.283737	−0.141869	−	0.989886i	$-0.545311\pi$
$373$	−762409.	−0.283737	−0.141869	+	0.989886i	$0.545311\pi$
$374$	−378801.	−0.140033
$375$	0	0
$376$	−988655.	−0.360641
$377$	−7.19350e6	−2.60667
$378$	1.84749e6	0.665047
$379$	4.70418e6	1.68223	0.841117	−	0.540854i	$-0.181899\pi$
$379$	4.70418e6	1.68223	0.841117	+	0.540854i	$0.181899\pi$
$380$	0	0
$381$	2.61565e6	0.923141
$382$	−4.77387e6	−1.67383
$383$	−3.43046e6	−1.19497	−0.597483	−	0.801881i	$-0.703833\pi$
$383$	−3.43046e6	−1.19497	−0.597483	+	0.801881i	$0.703833\pi$
$384$	1.32989e6	0.460243
$385$	0	0
$386$	2.85034e6	0.973709
$387$	−282397.	−0.0958479
$388$	−1.40152e6	−0.472629
$389$	2.44486e6	0.819181	0.409591	−	0.912269i	$-0.365672\pi$
$389$	2.44486e6	0.819181	0.409591	+	0.912269i	$0.365672\pi$
$390$	0	0
$391$	5.67505e6	1.87728
$392$	885546.	0.291069
$393$	−1.62311e6	−0.530111
$394$	−4.89964e6	−1.59010
$395$	0	0
$396$	86945.3	0.0278617
$397$	3.00200e6	0.955948	0.477974	−	0.878374i	$-0.341371\pi$
$397$	3.00200e6	0.955948	0.477974	+	0.878374i	$0.341371\pi$
$398$	5.39026e6	1.70570
$399$	−324117.	−0.101922
$400$	0	0
$401$	−6.39474e6	−1.98592	−0.992961	−	0.118441i	$-0.962210\pi$
$401$	−6.39474e6	−1.98592	−0.992961	+	0.118441i	$0.962210\pi$
$402$	1.76761e6	0.545534
$403$	670122.	0.205538
$404$	−82262.8	−0.0250755
$405$	0	0
$406$	−3.24383e6	−0.976658
$407$	−5260.55	−0.00157415
$408$	−1.36964e6	−0.407339
$409$	3.34173e6	0.987788	0.493894	−	0.869522i	$-0.335573\pi$
$409$	3.34173e6	0.987788	0.493894	+	0.869522i	$0.335573\pi$
$410$	0	0
$411$	−552464.	−0.161324
$412$	−1.30290e6	−0.378154
$413$	−3.04858e6	−0.879474
$414$	−3.17399e6	−0.910132
$415$	0	0
$416$	7.46083e6	2.11375
$417$	−3.19743e6	−0.900453
$418$	−96303.2	−0.0269588
$419$	1.93808e6	0.539309	0.269654	−	0.962957i	$-0.413091\pi$
$419$	1.93808e6	0.539309	0.269654	+	0.962957i	$0.413091\pi$
$420$	0	0
$421$	3.16052e6	0.869068	0.434534	−	0.900655i	$-0.356913\pi$
$421$	3.16052e6	0.869068	0.434534	+	0.900655i	$0.356913\pi$
$422$	4.02663e6	1.10068
$423$	−2.10725e6	−0.572617
$424$	1.42957e6	0.386180
$425$	0	0
$426$	2.18142e6	0.582392
$427$	598955.	0.158973
$428$	−2.99177e6	−0.789439
$429$	264607.	0.0694158
$430$	0	0
$431$	3.37134e6	0.874197	0.437098	−	0.899414i	$-0.356006\pi$
$431$	3.37134e6	0.874197	0.437098	+	0.899414i	$0.356006\pi$
$432$	4.66464e6	1.20257
$433$	−2.53422e6	−0.649567	−0.324783	−	0.945788i	$-0.605291\pi$
$433$	−2.53422e6	−0.649567	−0.324783	+	0.945788i	$0.605291\pi$
$434$	302184.	0.0770100
$435$	0	0
$436$	−378449.	−0.0953436
$437$	1.44278e6	0.361407
$438$	4.05385e6	1.00968
$439$	2.23440e6	0.553350	0.276675	−	0.960964i	$-0.410767\pi$
$439$	2.23440e6	0.553350	0.276675	+	0.960964i	$0.410767\pi$
$440$	0	0
$441$	1.88748e6	0.462152
$442$	−1.61498e7	−3.93199
$443$	−137672.	−0.0333300	−0.0166650	−	0.999861i	$-0.505305\pi$
$443$	−137672.	−0.0333300	−0.0166650	+	0.999861i	$0.505305\pi$
$444$	43556.7	0.0104857
$445$	0	0
$446$	−1.21358e6	−0.288888
$447$	3.55186e6	0.840789
$448$	716398.	0.168640
$449$	3.10610e6	0.727109	0.363555	−	0.931573i	$-0.381563\pi$
$449$	3.10610e6	0.727109	0.363555	+	0.931573i	$0.381563\pi$
$450$	0	0
$451$	−412867.	−0.0955805
$452$	323778.	0.0745420
$453$	−2.49234e6	−0.570639
$454$	789264.	0.179714
$455$	0	0
$456$	−348207.	−0.0784196
$457$	−3.87510e6	−0.867945	−0.433972	−	0.900926i	$-0.642888\pi$
$457$	−3.87510e6	−0.867945	−0.433972	+	0.900926i	$0.642888\pi$
$458$	6.39175e6	1.42382
$459$	−7.56402e6	−1.67580
$460$	0	0
$461$	−765644.	−0.167793	−0.0838966	−	0.996474i	$-0.526737\pi$
$461$	−765644.	−0.167793	−0.0838966	+	0.996474i	$0.526737\pi$
$462$	119322.	0.0260084
$463$	1.59705e6	0.346230	0.173115	−	0.984902i	$-0.444617\pi$
$463$	1.59705e6	0.346230	0.173115	+	0.984902i	$0.444617\pi$
$464$	−8.19019e6	−1.76603
$465$	0	0
$466$	128424.	0.0273955
$467$	−4.57319e6	−0.970347	−0.485173	−	0.874418i	$-0.661243\pi$
$467$	−4.57319e6	−0.970347	−0.485173	+	0.874418i	$0.661243\pi$
$468$	3.70684e6	0.782328
$469$	−1.68437e6	−0.353594
$470$	0	0
$471$	4.50915e6	0.936575
$472$	−3.27517e6	−0.676673
$473$	−47257.7	−0.00971225
$474$	4.28289e6	0.875571
$475$	0	0
$476$	−2.98872e6	−0.604600
$477$	3.04702e6	0.613168
$478$	1.14081e7	2.28372
$479$	7.76005e6	1.54535	0.772673	−	0.634804i	$-0.218919\pi$
$479$	7.76005e6	1.54535	0.772673	+	0.634804i	$0.218919\pi$
$480$	0	0
$481$	−224279.	−0.0442003
$482$	7.72610e6	1.51476
$483$	−1.78763e6	−0.348667
$484$	−3.57261e6	−0.693222
$485$	0	0
$486$	6.82821e6	1.31134
$487$	−7.17772e6	−1.37140	−0.685700	−	0.727884i	$-0.740504\pi$
$487$	−7.17772e6	−1.37140	−0.685700	+	0.727884i	$0.740504\pi$
$488$	643473.	0.122315
$489$	5.21969e6	0.987126
$490$	0	0
$491$	9.99422e6	1.87088	0.935439	−	0.353489i	$-0.115005\pi$
$491$	9.99422e6	1.87088	0.935439	+	0.353489i	$0.115005\pi$
$492$	3.41850e6	0.636682
$493$	1.32809e7	2.46100
$494$	−4.10581e6	−0.756974
$495$	0	0
$496$	762971.	0.139253
$497$	−2.07869e6	−0.377484
$498$	3.10460e6	0.560961
$499$	−5.28372e6	−0.949922	−0.474961	−	0.880007i	$-0.657538\pi$
$499$	−5.28372e6	−0.949922	−0.474961	+	0.880007i	$0.657538\pi$
$500$	0	0
$501$	−2.42233e6	−0.431161
$502$	957693.	0.169616
$503$	2.28211e6	0.402176	0.201088	−	0.979573i	$-0.435552\pi$
$503$	2.28211e6	0.402176	0.201088	+	0.979573i	$0.435552\pi$
$504$	−729949.	−0.128002
$505$	0	0
$506$	−531151.	−0.0922235
$507$	7.75362e6	1.33963
$508$	6.13190e6	1.05423
$509$	481383.	0.0823562	0.0411781	−	0.999152i	$-0.486889\pi$
$509$	481383.	0.0823562	0.0411781	+	0.999152i	$0.486889\pi$
$510$	0	0
$511$	−3.86294e6	−0.654434
$512$	5.64977e6	0.952480
$513$	−1.92302e6	−0.322619
$514$	3.30579e6	0.551909
$515$	0	0
$516$	391289.	0.0646953
$517$	−352637.	−0.0580232
$518$	−101136.	−0.0165608
$519$	−3.55376e6	−0.579122
$520$	0	0
$521$	−6.13066e6	−0.989493	−0.494746	−	0.869037i	$-0.664739\pi$
$521$	−6.13066e6	−0.989493	−0.494746	+	0.869037i	$0.664739\pi$
$522$	−7.42786e6	−1.19313
$523$	1.64507e6	0.262985	0.131492	−	0.991317i	$-0.458023\pi$
$523$	1.64507e6	0.262985	0.131492	+	0.991317i	$0.458023\pi$
$524$	−3.80507e6	−0.605389
$525$	0	0
$526$	8.92218e6	1.40607
$527$	−1.23721e6	−0.194051
$528$	301270.	0.0470295
$529$	1.52116e6	0.236339
$530$	0	0
$531$	−6.98079e6	−1.07441
$532$	−759829.	−0.116396
$533$	−1.76022e7	−2.68380
$534$	−1.06036e6	−0.160916
$535$	0	0
$536$	−1.80956e6	−0.272058
$537$	−7.96892e6	−1.19251
$538$	1.01269e7	1.50842
$539$	315860.	0.0468298
$540$	0	0
$541$	571944.	0.0840156	0.0420078	−	0.999117i	$-0.486625\pi$
$541$	571944.	0.0840156	0.0420078	+	0.999117i	$0.486625\pi$
$542$	−2.63036e6	−0.384606
$543$	−4.80622e6	−0.699527
$544$	−1.37745e7	−1.99562
$545$	0	0
$546$	5.08717e6	0.730289
$547$	7.03075e6	1.00469	0.502346	−	0.864667i	$-0.332470\pi$
$547$	7.03075e6	1.00469	0.502346	+	0.864667i	$0.332470\pi$
$548$	−1.29514e6	−0.184233
$549$	1.37152e6	0.194209
$550$	0	0
$551$	3.37643e6	0.473783
$552$	−1.92050e6	−0.268266
$553$	−4.08120e6	−0.567512
$554$	1.22295e7	1.69292
$555$	0	0
$556$	−7.49576e6	−1.02832
$557$	−7.45152e6	−1.01767	−0.508835	−	0.860864i	$-0.669924\pi$
$557$	−7.45152e6	−1.01767	−0.508835	+	0.860864i	$0.669924\pi$
$558$	691954.	0.0940788
$559$	−2.01479e6	−0.272710
$560$	0	0
$561$	−488529.	−0.0655364
$562$	−9.89622e6	−1.32169
$563$	−5.05721e6	−0.672420	−0.336210	−	0.941787i	$-0.609145\pi$
$563$	−5.05721e6	−0.672420	−0.336210	+	0.941787i	$0.609145\pi$
$564$	2.91980e6	0.386505
$565$	0	0
$566$	−2.77217e6	−0.363730
$567$	−92750.7	−0.0121160
$568$	−2.23319e6	−0.290439
$569$	3.49121e6	0.452059	0.226030	−	0.974120i	$-0.427425\pi$
$569$	3.49121e6	0.452059	0.226030	+	0.974120i	$0.427425\pi$
$570$	0	0
$571$	−1.10250e7	−1.41510	−0.707550	−	0.706663i	$-0.750200\pi$
$571$	−1.10250e7	−1.41510	−0.707550	+	0.706663i	$0.750200\pi$
$572$	620321.	0.0792732
$573$	−6.15673e6	−0.783363
$574$	−7.93753e6	−1.00555
$575$	0	0
$576$	1.64044e6	0.206018
$577$	−2.18881e6	−0.273696	−0.136848	−	0.990592i	$-0.543697\pi$
$577$	−2.18881e6	−0.273696	−0.136848	+	0.990592i	$0.543697\pi$
$578$	1.93563e7	2.40993
$579$	3.67601e6	0.455701
$580$	0	0
$581$	−2.95839e6	−0.363593
$582$	−4.40431e6	−0.538978
$583$	509903.	0.0621321
$584$	−4.15006e6	−0.503526
$585$	0	0
$586$	−9.15530e6	−1.10136
$587$	8.83072e6	1.05779	0.528897	−	0.848686i	$-0.322606\pi$
$587$	8.83072e6	1.05779	0.528897	+	0.848686i	$0.322606\pi$
$588$	−2.61528e6	−0.311943
$589$	−314537.	−0.0373580
$590$	0	0
$591$	−6.31892e6	−0.744174
$592$	−255354.	−0.0299459
$593$	4.81497e6	0.562286	0.281143	−	0.959666i	$-0.409286\pi$
$593$	4.81497e6	0.562286	0.281143	+	0.959666i	$0.409286\pi$
$594$	707947.	0.0823255
$595$	0	0
$596$	8.32665e6	0.960185
$597$	6.95166e6	0.798275
$598$	−2.26452e7	−2.58954
$599$	8.66559e6	0.986804	0.493402	−	0.869801i	$-0.335753\pi$
$599$	8.66559e6	0.986804	0.493402	+	0.869801i	$0.335753\pi$
$600$	0	0
$601$	9.36875e6	1.05802	0.529012	−	0.848614i	$-0.322563\pi$
$601$	9.36875e6	1.05802	0.529012	+	0.848614i	$0.322563\pi$
$602$	−908547.	−0.102178
$603$	−3.85694e6	−0.431967
$604$	−5.84280e6	−0.651672
$605$	0	0
$606$	−258513.	−0.0285957
$607$	6.30459e6	0.694520	0.347260	−	0.937769i	$-0.387112\pi$
$607$	6.30459e6	0.694520	0.347260	+	0.937769i	$0.387112\pi$
$608$	−3.50191e6	−0.384190
$609$	−4.18347e6	−0.457081
$610$	0	0
$611$	−1.50344e7	−1.62923
$612$	−6.84372e6	−0.738607
$613$	1.67434e7	1.79967	0.899834	−	0.436232i	$-0.143687\pi$
$613$	1.67434e7	1.79967	0.899834	+	0.436232i	$0.143687\pi$
$614$	1.79128e7	1.91753
$615$	0	0
$616$	−122153.	−0.0129704
$617$	1.56051e7	1.65027	0.825135	−	0.564936i	$-0.191099\pi$
$617$	1.56051e7	1.65027	0.825135	+	0.564936i	$0.191099\pi$
$618$	−4.09441e6	−0.431241
$619$	744665.	0.0781150	0.0390575	−	0.999237i	$-0.487564\pi$
$619$	744665.	0.0781150	0.0390575	+	0.999237i	$0.487564\pi$
$620$	0	0
$621$	−1.06062e7	−1.10365
$622$	1.58202e7	1.63959
$623$	1.01042e6	0.104299
$624$	1.28444e7	1.32054
$625$	0	0
$626$	−1.47193e7	−1.50124
$627$	−124200.	−0.0126169
$628$	1.05708e7	1.06957
$629$	414073.	0.0417302
$630$	0	0
$631$	−1.84430e7	−1.84399	−0.921995	−	0.387201i	$-0.873442\pi$
$631$	−1.84430e7	−1.84399	−0.921995	+	0.387201i	$0.873442\pi$
$632$	−4.38453e6	−0.436647
$633$	5.19304e6	0.515124
$634$	1.75016e7	1.72924
$635$	0	0
$636$	−4.22194e6	−0.413875
$637$	1.34664e7	1.31493
$638$	−1.24301e6	−0.120900
$639$	−4.75988e6	−0.461151
$640$	0	0
$641$	−2.21961e6	−0.213370	−0.106685	−	0.994293i	$-0.534024\pi$
$641$	−2.21961e6	−0.213370	−0.106685	+	0.994293i	$0.534024\pi$
$642$	−9.40171e6	−0.900263
$643$	3.14061e6	0.299562	0.149781	−	0.988719i	$-0.452143\pi$
$643$	3.14061e6	0.299562	0.149781	+	0.988719i	$0.452143\pi$
$644$	−4.19076e6	−0.398179
$645$	0	0
$646$	7.58031e6	0.714670
$647$	6.08633e6	0.571604	0.285802	−	0.958289i	$-0.407740\pi$
$647$	6.08633e6	0.571604	0.285802	+	0.958289i	$0.407740\pi$
$648$	−99644.4	−0.00932214
$649$	−1.16820e6	−0.108869
$650$	0	0
$651$	389718.	0.0360411
$652$	1.22366e7	1.12730
$653$	−492611.	−0.0452086	−0.0226043	−	0.999744i	$-0.507196\pi$
$653$	−492611.	−0.0452086	−0.0226043	+	0.999744i	$0.507196\pi$
$654$	−1.18929e6	−0.108728
$655$	0	0
$656$	−2.00411e7	−1.81829
$657$	−8.84554e6	−0.799486
$658$	−6.77958e6	−0.610433
$659$	−3.70503e6	−0.332336	−0.166168	−	0.986097i	$-0.553139\pi$
$659$	−3.70503e6	−0.332336	−0.166168	+	0.986097i	$0.553139\pi$
$660$	0	0
$661$	6.69686e6	0.596166	0.298083	−	0.954540i	$-0.403653\pi$
$661$	6.69686e6	0.596166	0.298083	+	0.954540i	$0.403653\pi$
$662$	1.75148e7	1.55332
$663$	−2.08280e7	−1.84019
$664$	−3.17828e6	−0.279751
$665$	0	0
$666$	−231586.	−0.0202314
$667$	1.86224e7	1.62077
$668$	−5.67870e6	−0.492388
$669$	−1.56511e6	−0.135201
$670$	0	0
$671$	229516.	0.0196792
$672$	4.33894e6	0.370647
$673$	−1.51059e7	−1.28561	−0.642804	−	0.766031i	$-0.722229\pi$
$673$	−1.51059e7	−1.28561	−0.642804	+	0.766031i	$0.722229\pi$
$674$	−2.66765e7	−2.26193
$675$	0	0
$676$	1.81769e7	1.52986
$677$	−4.94664e6	−0.414800	−0.207400	−	0.978256i	$-0.566500\pi$
$677$	−4.94664e6	−0.414800	−0.207400	+	0.978256i	$0.566500\pi$
$678$	1.01748e6	0.0850064
$679$	4.19690e6	0.349345
$680$	0	0
$681$	1.01789e6	0.0841073
$682$	115795.	0.00953299
$683$	−3.47608e6	−0.285127	−0.142563	−	0.989786i	$-0.545534\pi$
$683$	−3.47608e6	−0.285127	−0.142563	+	0.989786i	$0.545534\pi$
$684$	−1.73989e6	−0.142194
$685$	0	0
$686$	1.43310e7	1.16270
$687$	8.24325e6	0.666357
$688$	−2.29395e6	−0.184762
$689$	2.17393e7	1.74461
$690$	0	0
$691$	8.19327e6	0.652773	0.326386	−	0.945236i	$-0.394169\pi$
$691$	8.19327e6	0.652773	0.326386	+	0.945236i	$0.394169\pi$
$692$	−8.33112e6	−0.661360
$693$	−260361.	−0.0205941
$694$	−1.69358e7	−1.33477
$695$	0	0
$696$	−4.49441e6	−0.351681
$697$	3.24980e7	2.53381
$698$	−1.07133e7	−0.832311
$699$	165624.	0.0128213
$700$	0	0
$701$	9.64159e6	0.741060	0.370530	−	0.928820i	$-0.379176\pi$
$701$	9.64159e6	0.741060	0.370530	+	0.928820i	$0.379176\pi$
$702$	3.01827e7	2.31161
$703$	105270.	0.00803375
$704$	274520.	0.0208757
$705$	0	0
$706$	2.53864e7	1.91685
$707$	246339.	0.0185346
$708$	9.67257e6	0.725201
$709$	−2.43923e6	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$709$	−2.43923e6	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$710$	0	0
$711$	−9.34531e6	−0.693298
$712$	1.08552e6	0.0802486
$713$	−1.73480e6	−0.127798
$714$	−9.39215e6	−0.689476
$715$	0	0
$716$	−1.86816e7	−1.36186
$717$	1.47127e7	1.06879
$718$	−3.83255e6	−0.277445
$719$	5.60150e6	0.404094	0.202047	−	0.979376i	$-0.435241\pi$
$719$	5.60150e6	0.404094	0.202047	+	0.979376i	$0.435241\pi$
$720$	0	0
$721$	3.90159e6	0.279514
$722$	−1.63144e7	−1.16474
$723$	9.96413e6	0.708914
$724$	−1.12673e7	−0.798863
$725$	0	0
$726$	−1.12270e7	−0.790539
$727$	−156391.	−0.0109743	−0.00548713	−	0.999985i	$-0.501747\pi$
$727$	−156391.	−0.0109743	−0.00548713	+	0.999985i	$0.501747\pi$
$728$	−5.20790e6	−0.364195
$729$	8.46823e6	0.590165
$730$	0	0
$731$	3.71979e6	0.257469
$732$	−1.90037e6	−0.131087
$733$	−2.46182e7	−1.69237	−0.846187	−	0.532887i	$-0.821107\pi$
$733$	−2.46182e7	−1.69237	−0.846187	+	0.532887i	$0.821107\pi$
$734$	−1.79382e7	−1.22897
$735$	0	0
$736$	−1.93144e7	−1.31428
$737$	−645440.	−0.0437711
$738$	−1.81757e7	−1.22843
$739$	−1.96107e6	−0.132093	−0.0660467	−	0.997817i	$-0.521039\pi$
$739$	−1.96107e6	−0.132093	−0.0660467	+	0.997817i	$0.521039\pi$
$740$	0	0
$741$	−5.29514e6	−0.354268
$742$	9.80308e6	0.653661
$743$	−243464.	−0.0161794	−0.00808970	−	0.999967i	$-0.502575\pi$
$743$	−243464.	−0.0161794	−0.00808970	+	0.999967i	$0.502575\pi$
$744$	418684.	0.0277302
$745$	0	0
$746$	−5.61671e6	−0.369517
$747$	−6.77427e6	−0.444182
$748$	−1.14526e6	−0.0748429
$749$	8.95895e6	0.583516
$750$	0	0
$751$	1.50964e7	0.976727	0.488364	−	0.872640i	$-0.337594\pi$
$751$	1.50964e7	0.976727	0.488364	+	0.872640i	$0.337594\pi$
$752$	−1.71175e7	−1.10381
$753$	1.23511e6	0.0793812
$754$	−5.29949e7	−3.39473
$755$	0	0
$756$	5.58567e6	0.355444
$757$	−8.06369e6	−0.511440	−0.255720	−	0.966751i	$-0.582312\pi$
$757$	−8.06369e6	−0.511440	−0.255720	+	0.966751i	$0.582312\pi$
$758$	3.46560e7	2.19081
$759$	−685010.	−0.0431611
$760$	0	0
$761$	−1.68891e7	−1.05717	−0.528586	−	0.848880i	$-0.677277\pi$
$761$	−1.68891e7	−1.05717	−0.528586	+	0.848880i	$0.677277\pi$
$762$	1.92697e7	1.20223
$763$	1.13328e6	0.0704735
$764$	−1.44333e7	−0.894604
$765$	0	0
$766$	−2.52724e7	−1.55623
$767$	−4.98052e7	−3.05694
$768$	1.30629e7	0.799168
$769$	−8.53810e6	−0.520649	−0.260325	−	0.965521i	$-0.583830\pi$
$769$	−8.53810e6	−0.520649	−0.260325	+	0.965521i	$0.583830\pi$
$770$	0	0
$771$	4.26338e6	0.258296
$772$	8.61769e6	0.520413
$773$	−1.04982e7	−0.631928	−0.315964	−	0.948771i	$-0.602328\pi$
$773$	−1.04982e7	−0.631928	−0.315964	+	0.948771i	$0.602328\pi$
$774$	−2.08043e6	−0.124825
$775$	0	0
$776$	4.50884e6	0.268788
$777$	−130432.	−0.00775055
$778$	1.80114e7	1.06684
$779$	8.26202e6	0.487801
$780$	0	0
$781$	−796541.	−0.0467284
$782$	4.18084e7	2.44482
$783$	−2.48209e7	−1.44682
$784$	1.53322e7	0.890872
$785$	0	0
$786$	−1.19575e7	−0.690376
$787$	1.34594e7	0.774621	0.387310	−	0.921949i	$-0.373404\pi$
$787$	1.34594e7	0.774621	0.387310	+	0.921949i	$0.373404\pi$
$788$	−1.48135e7	−0.849850
$789$	1.15067e7	0.658048
$790$	0	0
$791$	−969564.	−0.0550979
$792$	−279712.	−0.0158452
$793$	9.78522e6	0.552571
$794$	2.21159e7	1.24495
$795$	0	0
$796$	1.62968e7	0.911634
$797$	3.06439e7	1.70883	0.854414	−	0.519593i	$-0.173916\pi$
$797$	3.06439e7	1.70883	0.854414	+	0.519593i	$0.173916\pi$
$798$	−2.38778e6	−0.132736
$799$	2.77571e7	1.53818
$800$	0	0
$801$	2.31371e6	0.127417
$802$	−4.71104e7	−2.58631
$803$	−1.48026e6	−0.0810118
$804$	5.34417e6	0.291568
$805$	0	0
$806$	4.93682e6	0.267676
$807$	1.30604e7	0.705948
$808$	264648.	0.0142607
$809$	2.81912e7	1.51441	0.757203	−	0.653180i	$-0.226565\pi$
$809$	2.81912e7	1.51441	0.757203	+	0.653180i	$0.226565\pi$
$810$	0	0
$811$	1.73407e7	0.925792	0.462896	−	0.886412i	$-0.346810\pi$
$811$	1.73407e7	0.925792	0.462896	+	0.886412i	$0.346810\pi$
$812$	−9.80734e6	−0.521989
$813$	−3.39230e6	−0.179998
$814$	−38754.7	−0.00205005
$815$	0	0
$816$	−2.37138e7	−1.24674
$817$	945689.	0.0495671
$818$	2.46187e7	1.28642
$819$	−1.11003e7	−0.578260
$820$	0	0
$821$	2.20533e7	1.14187	0.570933	−	0.820996i	$-0.306581\pi$
$821$	2.20533e7	1.14187	0.570933	+	0.820996i	$0.306581\pi$
$822$	−4.07003e6	−0.210096
$823$	−2.39967e7	−1.23496	−0.617478	−	0.786588i	$-0.711846\pi$
$823$	−2.39967e7	−1.23496	−0.617478	+	0.786588i	$0.711846\pi$
$824$	4.19157e6	0.215060
$825$	0	0
$826$	−2.24591e7	−1.14536
$827$	−8.59344e6	−0.436921	−0.218461	−	0.975846i	$-0.570104\pi$
$827$	−8.59344e6	−0.436921	−0.218461	+	0.975846i	$0.570104\pi$
$828$	−9.59619e6	−0.486433
$829$	−2.87141e7	−1.45114	−0.725569	−	0.688150i	$-0.758423\pi$
$829$	−2.87141e7	−1.45114	−0.725569	+	0.688150i	$0.758423\pi$
$830$	0	0
$831$	1.57721e7	0.792295
$832$	1.17039e7	0.586169
$833$	−2.48622e7	−1.24145
$834$	−2.35556e7	−1.17268
$835$	0	0
$836$	−291162.	−0.0144085
$837$	2.31224e6	0.114082
$838$	1.42780e7	0.702354
$839$	2.71354e7	1.33086	0.665429	−	0.746461i	$-0.268249\pi$
$839$	2.71354e7	1.33086	0.665429	+	0.746461i	$0.268249\pi$
$840$	0	0
$841$	2.30695e7	1.12473
$842$	2.32837e7	1.13181
$843$	−1.27629e7	−0.618556
$844$	1.21741e7	0.588274
$845$	0	0
$846$	−1.55242e7	−0.745732
$847$	1.06983e7	0.512397
$848$	2.47514e7	1.18198
$849$	−3.57519e6	−0.170228
$850$	0	0
$851$	580609.	0.0274827
$852$	6.59527e6	0.311268
$853$	−2.33450e7	−1.09855	−0.549277	−	0.835640i	$-0.685097\pi$
$853$	−2.33450e7	−1.09855	−0.549277	+	0.835640i	$0.685097\pi$
$854$	4.41253e6	0.207035
$855$	0	0
$856$	9.62483e6	0.448961
$857$	−3.05379e7	−1.42032	−0.710161	−	0.704039i	$-0.751378\pi$
$857$	−3.05379e7	−1.42032	−0.710161	+	0.704039i	$0.751378\pi$
$858$	1.94937e6	0.0904018
$859$	7.18900e6	0.332419	0.166209	−	0.986090i	$-0.446847\pi$
$859$	7.18900e6	0.332419	0.166209	+	0.986090i	$0.446847\pi$
$860$	0	0
$861$	−1.02368e7	−0.470605
$862$	2.48368e7	1.13849
$863$	575848.	0.0263197	0.0131598	−	0.999913i	$-0.495811\pi$
$863$	575848.	0.0263197	0.0131598	+	0.999913i	$0.495811\pi$
$864$	2.57433e7	1.17322
$865$	0	0
$866$	−1.86697e7	−0.845946
$867$	2.49633e7	1.12786
$868$	913619.	0.0411591
$869$	−1.56389e6	−0.0702517
$870$	0	0
$871$	−2.75178e7	−1.22905
$872$	1.21751e6	0.0542227
$873$	9.61025e6	0.426775
$874$	1.06290e7	0.470668
$875$	0	0
$876$	1.22564e7	0.539637
$877$	2.79779e6	0.122833	0.0614166	−	0.998112i	$-0.480438\pi$
$877$	2.79779e6	0.122833	0.0614166	+	0.998112i	$0.480438\pi$
$878$	1.64610e7	0.720640
$879$	−1.18073e7	−0.515442
$880$	0	0
$881$	−4.27129e7	−1.85404	−0.927021	−	0.375009i	$-0.877640\pi$
$881$	−4.27129e7	−1.85404	−0.927021	+	0.375009i	$0.877640\pi$
$882$	1.39051e7	0.601872
$883$	−5.59248e6	−0.241381	−0.120690	−	0.992690i	$-0.538511\pi$
$883$	−5.59248e6	−0.241381	−0.120690	+	0.992690i	$0.538511\pi$
$884$	−4.88272e7	−2.10151
$885$	0	0
$886$	−1.01424e6	−0.0434065
$887$	−2.16203e7	−0.922684	−0.461342	−	0.887222i	$-0.652632\pi$
$887$	−2.16203e7	−0.922684	−0.461342	+	0.887222i	$0.652632\pi$
$888$	−140127.	−0.00596332
$889$	−1.83622e7	−0.779238
$890$	0	0
$891$	−35541.5	−0.00149983
$892$	−3.66911e6	−0.154400
$893$	7.05673e6	0.296125
$894$	2.61667e7	1.09498
$895$	0	0
$896$	−9.33597e6	−0.388498
$897$	−2.92048e7	−1.21192
$898$	2.28828e7	0.946931
$899$	−4.05983e6	−0.167536
$900$	0	0
$901$	−4.01359e7	−1.64711
$902$	−3.04162e6	−0.124477
$903$	−1.17173e6	−0.0478197
$904$	−1.04163e6	−0.0423927
$905$	0	0
$906$	−1.83612e7	−0.743156
$907$	2.30659e7	0.931007	0.465504	−	0.885046i	$-0.345873\pi$
$907$	2.30659e7	0.931007	0.465504	+	0.885046i	$0.345873\pi$
$908$	2.38625e6	0.0960509
$909$	564078.	0.0226428
$910$	0	0
$911$	−2.84918e6	−0.113743	−0.0568714	−	0.998382i	$-0.518112\pi$
$911$	−2.84918e6	−0.113743	−0.0568714	+	0.998382i	$0.518112\pi$
$912$	−6.02881e6	−0.240018
$913$	−1.13364e6	−0.0450088
$914$	−2.85480e7	−1.13034
$915$	0	0
$916$	1.93247e7	0.760982
$917$	1.13944e7	0.447475
$918$	−5.57245e7	−2.18243
$919$	−3.20156e7	−1.25047	−0.625234	−	0.780438i	$-0.714996\pi$
$919$	−3.20156e7	−1.25047	−0.625234	+	0.780438i	$0.714996\pi$
$920$	0	0
$921$	2.31016e7	0.897415
$922$	−5.64054e6	−0.218521
$923$	−3.39599e7	−1.31208
$924$	360755.	0.0139006
$925$	0	0
$926$	1.17655e7	0.450904
$927$	8.93403e6	0.341467
$928$	−4.52002e7	−1.72294
$929$	−8.55559e6	−0.325245	−0.162623	−	0.986688i	$-0.551995\pi$
$929$	−8.55559e6	−0.325245	−0.162623	+	0.986688i	$0.551995\pi$
$930$	0	0
$931$	−6.32077e6	−0.238999
$932$	388274.	0.0146419
$933$	2.04028e7	0.767336
$934$	−3.36909e7	−1.26370
$935$	0	0
$936$	−1.19253e7	−0.444917
$937$	−4.60556e7	−1.71370	−0.856848	−	0.515570i	$-0.827580\pi$
$937$	−4.60556e7	−1.71370	−0.856848	+	0.515570i	$0.827580\pi$
$938$	−1.24088e7	−0.460494
$939$	−1.89831e7	−0.702590
$940$	0	0
$941$	1.16126e7	0.427518	0.213759	−	0.976886i	$-0.431429\pi$
$941$	1.16126e7	0.427518	0.213759	+	0.976886i	$0.431429\pi$
$942$	3.32192e7	1.21972
$943$	4.55684e7	1.66872
$944$	−5.67059e7	−2.07109
$945$	0	0
$946$	−348150.	−0.0126485
$947$	−2.29169e7	−0.830388	−0.415194	−	0.909733i	$-0.636286\pi$
$947$	−2.29169e7	−0.830388	−0.415194	+	0.909733i	$0.636286\pi$
$948$	1.29488e7	0.467962
$949$	−6.31095e7	−2.27473
$950$	0	0
$951$	2.25713e7	0.809292
$952$	9.61504e6	0.343842
$953$	−5.04899e7	−1.80083	−0.900415	−	0.435033i	$-0.856737\pi$
$953$	−5.04899e7	−1.80083	−0.900415	+	0.435033i	$0.856737\pi$
$954$	2.24475e7	0.798542
$955$	0	0
$956$	3.44910e7	1.22057
$957$	−1.60308e6	−0.0565816
$958$	5.71687e7	2.01254
$959$	3.87836e6	0.136176
$960$	0	0
$961$	−2.82510e7	−0.986790
$962$	−1.65227e6	−0.0575631
$963$	2.05146e7	0.712849
$964$	2.33590e7	0.809583
$965$	0	0
$966$	−1.31696e7	−0.454076
$967$	1.26378e6	0.0434617	0.0217308	−	0.999764i	$-0.493082\pi$
$967$	1.26378e6	0.0434617	0.0217308	+	0.999764i	$0.493082\pi$
$968$	1.14935e7	0.394242
$969$	9.77610e6	0.334469
$970$	0	0
$971$	−6.80423e6	−0.231596	−0.115798	−	0.993273i	$-0.536942\pi$
$971$	−6.80423e6	−0.231596	−0.115798	+	0.993273i	$0.536942\pi$
$972$	2.06443e7	0.700866
$973$	2.24463e7	0.760087
$974$	−5.28786e7	−1.78601
$975$	0	0
$976$	1.11410e7	0.374369
$977$	−5.75856e6	−0.193009	−0.0965045	−	0.995333i	$-0.530766\pi$
$977$	−5.75856e6	−0.193009	−0.0965045	+	0.995333i	$0.530766\pi$
$978$	3.84537e7	1.28556
$979$	387187.	0.0129111
$980$	0	0
$981$	2.59503e6	0.0860936
$982$	7.36280e7	2.43649
$983$	−5.32605e7	−1.75801	−0.879004	−	0.476814i	$-0.841792\pi$
$983$	−5.32605e7	−1.75801	−0.879004	+	0.476814i	$0.841792\pi$
$984$	−1.09977e7	−0.362087
$985$	0	0
$986$	9.78413e7	3.20501
$987$	−8.74343e6	−0.285686
$988$	−1.24134e7	−0.404576
$989$	5.21585e6	0.169564
$990$	0	0
$991$	474785.	0.0153572	0.00767862	−	0.999971i	$-0.497556\pi$
$991$	474785.	0.0153572	0.00767862	+	0.999971i	$0.497556\pi$
$992$	4.21070e6	0.135855
$993$	2.25883e7	0.726961
$994$	−1.53138e7	−0.491606
$995$	0	0
$996$	9.38641e6	0.299813
$997$	−3.28106e7	−1.04539	−0.522693	−	0.852521i	$-0.675072\pi$
$997$	−3.28106e7	−1.04539	−0.522693	+	0.852521i	$0.675072\pi$
$998$	−3.89254e7	−1.23711
$999$	−773867.	−0.0245331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.j.1.30	yes	37
5.4	even	2		1075.6.a.i.1.8	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1075.6.a.i.1.8	✓	37	5.4	even	2
1075.6.a.j.1.30	yes	37	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q+7.36705 q^{2} +9.50108 q^{3} +22.2734 q^{4} +69.9949 q^{6} -66.6986 q^{7} -71.6560 q^{8} -152.730 q^{9} +O(q^{10})\)	\(q+7.36705 q^{2} +9.50108 q^{3} +22.2734 q^{4} +69.9949 q^{6} -66.6986 q^{7} -71.6560 q^{8} -152.730 q^{9} -25.5585 q^{11} +211.622 q^{12} -1089.67 q^{13} -491.372 q^{14} -1240.64 q^{16} +2011.78 q^{17} -1125.17 q^{18} +511.460 q^{19} -633.709 q^{21} -188.291 q^{22} +2820.90 q^{23} -680.809 q^{24} -8027.62 q^{26} -3759.86 q^{27} -1485.61 q^{28} +6601.56 q^{29} -614.980 q^{31} -6846.89 q^{32} -242.833 q^{33} +14820.9 q^{34} -3401.81 q^{36} +205.824 q^{37} +3767.95 q^{38} -10353.0 q^{39} +16153.8 q^{41} -4668.56 q^{42} +1849.00 q^{43} -569.276 q^{44} +20781.8 q^{46} +13797.2 q^{47} -11787.5 q^{48} -12358.3 q^{49} +19114.1 q^{51} -24270.6 q^{52} -19950.4 q^{53} -27699.1 q^{54} +4779.36 q^{56} +4859.42 q^{57} +48634.1 q^{58} +45706.9 q^{59} -8980.02 q^{61} -4530.59 q^{62} +10186.9 q^{63} -10740.8 q^{64} -1788.97 q^{66} +25253.4 q^{67} +44809.4 q^{68} +26801.6 q^{69} +31165.4 q^{71} +10944.0 q^{72} +57916.4 q^{73} +1516.31 q^{74} +11392.0 q^{76} +1704.72 q^{77} -76271.0 q^{78} +61188.6 q^{79} +1390.59 q^{81} +119006. q^{82} +44354.7 q^{83} -14114.9 q^{84} +13621.7 q^{86} +62722.0 q^{87} +1831.42 q^{88} -15149.0 q^{89} +72679.2 q^{91} +62831.3 q^{92} -5842.97 q^{93} +101645. q^{94} -65052.9 q^{96} -62923.3 q^{97} -91044.2 q^{98} +3903.54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9}+O(q^{10}) \) 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9	\( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9} + 675 q^{11} - 4446 q^{12} + 1241 q^{13} + 2375 q^{14} + 10518 q^{16} + 1153 q^{17} - 6680 q^{18} + 4065 q^{19} + 9953 q^{21} + 9283 q^{22} - 360 q^{23} + 2265 q^{24} + 23695 q^{26} + 1323 q^{27} - 30375 q^{28} + 19290 q^{29} + 23291 q^{31} + 8166 q^{32} - 10388 q^{33} - 13153 q^{34} + 148705 q^{36} + 13501 q^{37} - 8127 q^{38} - 1327 q^{39} + 38345 q^{41} - 21835 q^{42} + 68413 q^{43} + 47768 q^{44} + 48755 q^{46} + 84859 q^{47} - 208720 q^{48} + 107255 q^{49} + 62027 q^{51} + 128320 q^{52} - 53559 q^{53} + 44158 q^{54} + 107538 q^{56} + 104239 q^{57} - 85186 q^{58} + 48186 q^{59} + 82364 q^{61} + 206506 q^{62} - 75269 q^{63} + 161467 q^{64} + 91969 q^{66} + 38168 q^{67} + 95991 q^{68} + 287103 q^{69} + 155302 q^{71} + 9979 q^{72} + 31927 q^{73} + 59946 q^{74} + 225407 q^{76} - 80007 q^{77} - 67815 q^{78} + 150174 q^{79} + 417489 q^{81} + 60603 q^{82} + 266568 q^{83} + 586273 q^{84} - 57554 q^{87} + 323054 q^{88} + 334356 q^{89} + 51747 q^{91} - 258529 q^{92} - 285287 q^{93} + 302744 q^{94} + 287282 q^{96} - 78640 q^{97} - 397117 q^{98} + 362152 q^{99}+O(q^{100}) \) 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9 + 675 * q^11 - 4446 * q^12 + 1241 * q^13 + 2375 * q^14 + 10518 * q^16 + 1153 * q^17 - 6680 * q^18 + 4065 * q^19 + 9953 * q^21 + 9283 * q^22 - 360 * q^23 + 2265 * q^24 + 23695 * q^26 + 1323 * q^27 - 30375 * q^28 + 19290 * q^29 + 23291 * q^31 + 8166 * q^32 - 10388 * q^33 - 13153 * q^34 + 148705 * q^36 + 13501 * q^37 - 8127 * q^38 - 1327 * q^39 + 38345 * q^41 - 21835 * q^42 + 68413 * q^43 + 47768 * q^44 + 48755 * q^46 + 84859 * q^47 - 208720 * q^48 + 107255 * q^49 + 62027 * q^51 + 128320 * q^52 - 53559 * q^53 + 44158 * q^54 + 107538 * q^56 + 104239 * q^57 - 85186 * q^58 + 48186 * q^59 + 82364 * q^61 + 206506 * q^62 - 75269 * q^63 + 161467 * q^64 + 91969 * q^66 + 38168 * q^67 + 95991 * q^68 + 287103 * q^69 + 155302 * q^71 + 9979 * q^72 + 31927 * q^73 + 59946 * q^74 + 225407 * q^76 - 80007 * q^77 - 67815 * q^78 + 150174 * q^79 + 417489 * q^81 + 60603 * q^82 + 266568 * q^83 + 586273 * q^84 - 57554 * q^87 + 323054 * q^88 + 334356 * q^89 + 51747 * q^91 - 258529 * q^92 - 285287 * q^93 + 302744 * q^94 + 287282 * q^96 - 78640 * q^97 - 397117 * q^98 + 362152 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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