LMFDB - Embedded newform 625.6.a.d.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,6,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, 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("625.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	625.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:	$100.239887383$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.6667 q^{2} +5.92758 q^{3} +81.7795 q^{4} +63.2280 q^{6} -120.405 q^{7} +530.986 q^{8} -207.864 q^{9} +O(q^{10})$	\(q+10.6667 q^{2} +5.92758 q^{3} +81.7795 q^{4} +63.2280 q^{6} -120.405 q^{7} +530.986 q^{8} -207.864 q^{9} -442.746 q^{11} +484.755 q^{12} -891.643 q^{13} -1284.33 q^{14} +3046.94 q^{16} +480.421 q^{17} -2217.23 q^{18} +117.717 q^{19} -713.708 q^{21} -4722.66 q^{22} -2134.63 q^{23} +3147.46 q^{24} -9510.93 q^{26} -2672.53 q^{27} -9846.63 q^{28} +640.905 q^{29} -1623.14 q^{31} +15509.5 q^{32} -2624.41 q^{33} +5124.53 q^{34} -16999.0 q^{36} +12094.0 q^{37} +1255.65 q^{38} -5285.28 q^{39} -12573.4 q^{41} -7612.94 q^{42} -11083.8 q^{43} -36207.5 q^{44} -22769.6 q^{46} -19979.2 q^{47} +18061.0 q^{48} -2309.73 q^{49} +2847.74 q^{51} -72918.1 q^{52} +36909.9 q^{53} -28507.2 q^{54} -63933.1 q^{56} +697.775 q^{57} +6836.37 q^{58} -16033.2 q^{59} +43151.3 q^{61} -17313.6 q^{62} +25027.8 q^{63} +67933.2 q^{64} -27993.9 q^{66} -18063.3 q^{67} +39288.6 q^{68} -12653.2 q^{69} -26156.6 q^{71} -110373. q^{72} +15093.0 q^{73} +129004. q^{74} +9626.81 q^{76} +53308.6 q^{77} -56376.8 q^{78} +43779.0 q^{79} +34669.3 q^{81} -134117. q^{82} -23599.3 q^{83} -58366.7 q^{84} -118229. q^{86} +3799.01 q^{87} -235092. q^{88} +48842.2 q^{89} +107358. q^{91} -174569. q^{92} -9621.28 q^{93} -213113. q^{94} +91933.5 q^{96} -13981.0 q^{97} -24637.4 q^{98} +92030.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 9 q^{2} + 2 q^{3} + 289 q^{4} - 111 q^{6} - 101 q^{7} + 540 q^{8} + 1166 q^{9}+O(q^{10}) $ 22 * q + 9 * q^2 + 2 * q^3 + 289 * q^4 - 111 * q^6 - 101 * q^7 + 540 * q^8 + 1166 * q^9	\( 22 q + 9 q^{2} + 2 q^{3} + 289 q^{4} - 111 q^{6} - 101 q^{7} + 540 q^{8} + 1166 q^{9} - 1046 q^{11} - 1271 q^{12} - 1453 q^{13} - 2222 q^{14} + 1317 q^{16} + 2174 q^{17} + 4272 q^{18} - 3765 q^{19} - 4811 q^{21} - 12 q^{22} - 1488 q^{23} - 9810 q^{24} - 8941 q^{26} - 13570 q^{27} - 5587 q^{28} - 9035 q^{29} - 10426 q^{31} + 15339 q^{32} + 6874 q^{33} - 16027 q^{34} - 7098 q^{36} + 24944 q^{37} - 47645 q^{38} - 28703 q^{39} - 36606 q^{41} - 34782 q^{42} + 30597 q^{43} - 47892 q^{44} - 30881 q^{46} - 81481 q^{47} - 20803 q^{48} - 12241 q^{49} - 60566 q^{51} - 79836 q^{52} + 75802 q^{53} - 65085 q^{54} - 83335 q^{56} + 146075 q^{57} - 30030 q^{58} - 87095 q^{59} - 71476 q^{61} + 131553 q^{62} - 111758 q^{63} - 30736 q^{64} - 64222 q^{66} - 85706 q^{67} + 61943 q^{68} - 93448 q^{69} - 125571 q^{71} + 318570 q^{72} + 157982 q^{73} - 89682 q^{74} - 134640 q^{76} - 236842 q^{77} + 145204 q^{78} - 173645 q^{79} - 156538 q^{81} - 176667 q^{82} - 166408 q^{83} - 213437 q^{84} - 182891 q^{86} - 175085 q^{87} - 612480 q^{88} - 280940 q^{89} - 182911 q^{91} - 534131 q^{92} - 190216 q^{93} - 376702 q^{94} - 484051 q^{96} + 636469 q^{97} + 819128 q^{98} - 340388 q^{99}+O(q^{100}) \) 22 * q + 9 * q^2 + 2 * q^3 + 289 * q^4 - 111 * q^6 - 101 * q^7 + 540 * q^8 + 1166 * q^9 - 1046 * q^11 - 1271 * q^12 - 1453 * q^13 - 2222 * q^14 + 1317 * q^16 + 2174 * q^17 + 4272 * q^18 - 3765 * q^19 - 4811 * q^21 - 12 * q^22 - 1488 * q^23 - 9810 * q^24 - 8941 * q^26 - 13570 * q^27 - 5587 * q^28 - 9035 * q^29 - 10426 * q^31 + 15339 * q^32 + 6874 * q^33 - 16027 * q^34 - 7098 * q^36 + 24944 * q^37 - 47645 * q^38 - 28703 * q^39 - 36606 * q^41 - 34782 * q^42 + 30597 * q^43 - 47892 * q^44 - 30881 * q^46 - 81481 * q^47 - 20803 * q^48 - 12241 * q^49 - 60566 * q^51 - 79836 * q^52 + 75802 * q^53 - 65085 * q^54 - 83335 * q^56 + 146075 * q^57 - 30030 * q^58 - 87095 * q^59 - 71476 * q^61 + 131553 * q^62 - 111758 * q^63 - 30736 * q^64 - 64222 * q^66 - 85706 * q^67 + 61943 * q^68 - 93448 * q^69 - 125571 * q^71 + 318570 * q^72 + 157982 * q^73 - 89682 * q^74 - 134640 * q^76 - 236842 * q^77 + 145204 * q^78 - 173645 * q^79 - 156538 * q^81 - 176667 * q^82 - 166408 * q^83 - 213437 * q^84 - 182891 * q^86 - 175085 * q^87 - 612480 * q^88 - 280940 * q^89 - 182911 * q^91 - 534131 * q^92 - 190216 * q^93 - 376702 * q^94 - 484051 * q^96 + 636469 * q^97 + 819128 * q^98 - 340388 * 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 $

$2$	10.6667	1.88563	0.942816	−	0.333313i	$-0.108167\pi$
$2$	10.6667	1.88563	0.942816	+	0.333313i	$0.108167\pi$
$3$	5.92758	0.380254	0.190127	−	0.981759i	$-0.439110\pi$
$3$	5.92758	0.380254	0.190127	+	0.981759i	$0.439110\pi$
$4$	81.7795	2.55561
$5$	0	0
$5$	0	0
$6$	63.2280	0.717020
$7$	−120.405	−0.928748	−0.464374	−	0.885639i	$-0.653721\pi$
$7$	−120.405	−0.928748	−0.464374	+	0.885639i	$0.653721\pi$
$8$	530.986	2.93331
$9$	−207.864	−0.855407
$10$	0	0
$11$	−442.746	−1.10325	−0.551624	−	0.834093i	$-0.685992\pi$
$11$	−442.746	−1.10325	−0.551624	+	0.834093i	$0.685992\pi$
$12$	484.755	0.971782
$13$	−891.643	−1.46330	−0.731648	−	0.681682i	$-0.761249\pi$
$13$	−891.643	−1.46330	−0.731648	+	0.681682i	$0.761249\pi$
$14$	−1284.33	−1.75128
$15$	0	0
$16$	3046.94	2.97553
$17$	480.421	0.403181	0.201590	−	0.979470i	$-0.435389\pi$
$17$	480.421	0.403181	0.201590	+	0.979470i	$0.435389\pi$
$18$	−2217.23	−1.61298
$19$	117.717	0.0748090	0.0374045	−	0.999300i	$-0.488091\pi$
$19$	117.717	0.0748090	0.0374045	+	0.999300i	$0.488091\pi$
$20$	0	0
$21$	−713.708	−0.353161
$22$	−4722.66	−2.08032
$23$	−2134.63	−0.841401	−0.420701	−	0.907200i	$-0.638216\pi$
$23$	−2134.63	−0.841401	−0.420701	+	0.907200i	$0.638216\pi$
$24$	3147.46	1.11540
$25$	0	0
$26$	−9510.93	−2.75924
$27$	−2672.53	−0.705527
$28$	−9846.63	−2.37352
$29$	640.905	0.141514	0.0707569	−	0.997494i	$-0.477459\pi$
$29$	640.905	0.141514	0.0707569	+	0.997494i	$0.477459\pi$
$30$	0	0
$31$	−1623.14	−0.303355	−0.151677	−	0.988430i	$-0.548468\pi$
$31$	−1623.14	−0.303355	−0.151677	+	0.988430i	$0.548468\pi$
$32$	15509.5	2.67745
$33$	−2624.41	−0.419515
$34$	5124.53	0.760251
$35$	0	0
$36$	−16999.0	−2.18609
$37$	12094.0	1.45233	0.726167	−	0.687518i	$-0.241300\pi$
$37$	12094.0	1.45233	0.726167	+	0.687518i	$0.241300\pi$
$38$	1255.65	0.141062
$39$	−5285.28	−0.556425
$40$	0	0
$41$	−12573.4	−1.16813	−0.584067	−	0.811706i	$-0.698539\pi$
$41$	−12573.4	−1.16813	−0.584067	+	0.811706i	$0.698539\pi$
$42$	−7612.94	−0.665931
$43$	−11083.8	−0.914154	−0.457077	−	0.889427i	$-0.651104\pi$
$43$	−11083.8	−0.914154	−0.457077	+	0.889427i	$0.651104\pi$
$44$	−36207.5	−2.81947
$45$	0	0
$46$	−22769.6	−1.58657
$47$	−19979.2	−1.31927	−0.659633	−	0.751588i	$-0.729288\pi$
$47$	−19979.2	−1.31927	−0.659633	+	0.751588i	$0.729288\pi$
$48$	18061.0	1.13146
$49$	−2309.73	−0.137427
$50$	0	0
$51$	2847.74	0.153311
$52$	−72918.1	−3.73962
$53$	36909.9	1.80490	0.902450	−	0.430794i	$-0.141767\pi$
$53$	36909.9	1.80490	0.902450	+	0.430794i	$0.141767\pi$
$54$	−28507.2	−1.33036
$55$	0	0
$56$	−63933.1	−2.72430
$57$	697.775	0.0284464
$58$	6836.37	0.266843
$59$	−16033.2	−0.599641	−0.299820	−	0.953996i	$-0.596927\pi$
$59$	−16033.2	−0.599641	−0.299820	+	0.953996i	$0.596927\pi$
$60$	0	0
$61$	43151.3	1.48480	0.742402	−	0.669955i	$-0.233687\pi$
$61$	43151.3	1.48480	0.742402	+	0.669955i	$0.233687\pi$
$62$	−17313.6	−0.572016
$63$	25027.8	0.794457
$64$	67933.2	2.07316
$65$	0	0
$66$	−27993.9	−0.791050
$67$	−18063.3	−0.491597	−0.245799	−	0.969321i	$-0.579050\pi$
$67$	−18063.3	−0.491597	−0.245799	+	0.969321i	$0.579050\pi$
$68$	39288.6	1.03037
$69$	−12653.2	−0.319946
$70$	0	0
$71$	−26156.6	−0.615794	−0.307897	−	0.951420i	$-0.599625\pi$
$71$	−26156.6	−0.615794	−0.307897	+	0.951420i	$0.599625\pi$
$72$	−110373.	−2.50917
$73$	15093.0	0.331488	0.165744	−	0.986169i	$-0.446997\pi$
$73$	15093.0	0.331488	0.165744	+	0.986169i	$0.446997\pi$
$74$	129004.	2.73857
$75$	0	0
$76$	9626.81	0.191183
$77$	53308.6	1.02464
$78$	−56376.8	−1.04921
$79$	43779.0	0.789220	0.394610	−	0.918849i	$-0.370880\pi$
$79$	43779.0	0.789220	0.394610	+	0.918849i	$0.370880\pi$
$80$	0	0
$81$	34669.3	0.587127
$82$	−134117.	−2.20267
$83$	−23599.3	−0.376013	−0.188007	−	0.982168i	$-0.560203\pi$
$83$	−23599.3	−0.376013	−0.188007	+	0.982168i	$0.560203\pi$
$84$	−58366.7	−0.902541
$85$	0	0
$86$	−118229.	−1.72376
$87$	3799.01	0.0538112
$88$	−235092.	−3.23616
$89$	48842.2	0.653612	0.326806	−	0.945091i	$-0.394028\pi$
$89$	48842.2	0.653612	0.326806	+	0.945091i	$0.394028\pi$
$90$	0	0
$91$	107358.	1.35903
$92$	−174569.	−2.15029
$93$	−9621.28	−0.115352
$94$	−213113.	−2.48765
$95$	0	0
$96$	91933.5	1.01811
$97$	−13981.0	−0.150873	−0.0754363	−	0.997151i	$-0.524035\pi$
$97$	−13981.0	−0.150873	−0.0754363	+	0.997151i	$0.524035\pi$
$98$	−24637.4	−0.259137
$99$	92030.8	0.943725
$100$	0	0
$101$	381.587	0.00372212	0.00186106	−	0.999998i	$-0.499408\pi$
$101$	381.587	0.00372212	0.00186106	+	0.999998i	$0.499408\pi$
$102$	30376.1	0.289089
$103$	110235.	1.02383	0.511913	−	0.859038i	$-0.328937\pi$
$103$	110235.	1.02383	0.511913	+	0.859038i	$0.328937\pi$
$104$	−473449.	−4.29230
$105$	0	0
$106$	393709.	3.40338
$107$	−124723.	−1.05314	−0.526572	−	0.850131i	$-0.676523\pi$
$107$	−124723.	−1.05314	−0.526572	+	0.850131i	$0.676523\pi$
$108$	−218558.	−1.80305
$109$	36391.5	0.293382	0.146691	−	0.989182i	$-0.453138\pi$
$109$	36391.5	0.293382	0.146691	+	0.989182i	$0.453138\pi$
$110$	0	0
$111$	71688.3	0.552256
$112$	−366866.	−2.76352
$113$	89912.1	0.662402	0.331201	−	0.943560i	$-0.392546\pi$
$113$	89912.1	0.662402	0.331201	+	0.943560i	$0.392546\pi$
$114$	7442.98	0.0536395
$115$	0	0
$116$	52412.9	0.361654
$117$	185340.	1.25171
$118$	−171023.	−1.13070
$119$	−57844.9	−0.374454
$120$	0	0
$121$	34972.9	0.217154
$122$	460284.	2.79979
$123$	−74529.7	−0.444188
$124$	−132739.	−0.775257
$125$	0	0
$126$	266965.	1.49805
$127$	−159416.	−0.877044	−0.438522	−	0.898720i	$-0.644498\pi$
$127$	−159416.	−0.877044	−0.438522	+	0.898720i	$0.644498\pi$
$128$	228324.	1.23176
$129$	−65700.4	−0.347611
$130$	0	0
$131$	−191343.	−0.974167	−0.487084	−	0.873355i	$-0.661939\pi$
$131$	−191343.	−0.974167	−0.487084	+	0.873355i	$0.661939\pi$
$132$	−214623.	−1.07212
$133$	−14173.6	−0.0694787
$134$	−192676.	−0.926972
$135$	0	0
$136$	255097.	1.18265
$137$	−279442.	−1.27201	−0.636005	−	0.771685i	$-0.719414\pi$
$137$	−279442.	−1.27201	−0.636005	+	0.771685i	$0.719414\pi$
$138$	−134968.	−0.603301
$139$	197654.	0.867699	0.433850	−	0.900985i	$-0.357155\pi$
$139$	197654.	0.867699	0.433850	+	0.900985i	$0.357155\pi$
$140$	0	0
$141$	−118428.	−0.501657
$142$	−279006.	−1.16116
$143$	394771.	1.61438
$144$	−633349.	−2.54529
$145$	0	0
$146$	160993.	0.625065
$147$	−13691.1	−0.0522572
$148$	989044.	3.71160
$149$	49298.1	0.181913	0.0909567	−	0.995855i	$-0.471008\pi$
$149$	49298.1	0.181913	0.0909567	+	0.995855i	$0.471008\pi$
$150$	0	0
$151$	173320.	0.618594	0.309297	−	0.950966i	$-0.399906\pi$
$151$	173320.	0.618594	0.309297	+	0.950966i	$0.399906\pi$
$152$	62505.8	0.219438
$153$	−99862.2	−0.344884
$154$	568630.	1.93209
$155$	0	0
$156$	−432228.	−1.42201
$157$	−41142.9	−0.133213	−0.0666064	−	0.997779i	$-0.521217\pi$
$157$	−41142.9	−0.133213	−0.0666064	+	0.997779i	$0.521217\pi$
$158$	466979.	1.48818
$159$	218786.	0.686321
$160$	0	0
$161$	257019.	0.781450
$162$	369808.	1.10711
$163$	226902.	0.668912	0.334456	−	0.942411i	$-0.391447\pi$
$163$	226902.	0.668912	0.334456	+	0.942411i	$0.391447\pi$
$164$	−1.02825e6	−2.98529
$165$	0	0
$166$	−251727.	−0.709023
$167$	106692.	0.296034	0.148017	−	0.988985i	$-0.452711\pi$
$167$	106692.	0.296034	0.148017	+	0.988985i	$0.452711\pi$
$168$	−378969.	−1.03593
$169$	423734.	1.14124
$170$	0	0
$171$	−24469.0	−0.0639921
$172$	−906432.	−2.33622
$173$	−115853.	−0.294302	−0.147151	−	0.989114i	$-0.547010\pi$
$173$	−115853.	−0.294302	−0.147151	+	0.989114i	$0.547010\pi$
$174$	40523.1	0.101468
$175$	0	0
$176$	−1.34902e6	−3.28275
$177$	−95038.3	−0.228016
$178$	520988.	1.23247
$179$	−593880.	−1.38537	−0.692686	−	0.721239i	$-0.743573\pi$
$179$	−593880.	−1.38537	−0.692686	+	0.721239i	$0.743573\pi$
$180$	0	0
$181$	−738669.	−1.67592	−0.837961	−	0.545731i	$-0.816252\pi$
$181$	−738669.	−1.67592	−0.837961	+	0.545731i	$0.816252\pi$
$182$	1.14516e6	2.56264
$183$	255783.	0.564603
$184$	−1.13346e6	−2.46809
$185$	0	0
$186$	−102628.	−0.217512
$187$	−212705.	−0.444808
$188$	−1.63389e6	−3.37153
$189$	321785.	0.655256
$190$	0	0
$191$	847947.	1.68184	0.840922	−	0.541157i	$-0.182013\pi$
$191$	847947.	1.68184	0.840922	+	0.541157i	$0.182013\pi$
$192$	402679.	0.788327
$193$	−274628.	−0.530703	−0.265352	−	0.964152i	$-0.585488\pi$
$193$	−274628.	−0.530703	−0.265352	+	0.964152i	$0.585488\pi$
$194$	−149132.	−0.284490
$195$	0	0
$196$	−188889.	−0.351210
$197$	−172037.	−0.315833	−0.157916	−	0.987452i	$-0.550478\pi$
$197$	−172037.	−0.315833	−0.157916	+	0.987452i	$0.550478\pi$
$198$	981670.	1.77952
$199$	−1.05743e6	−1.89287	−0.946433	−	0.322899i	$-0.895342\pi$
$199$	−1.05743e6	−1.89287	−0.946433	+	0.322899i	$0.895342\pi$
$200$	0	0
$201$	−107071.	−0.186932
$202$	4070.29	0.00701854
$203$	−77167.9	−0.131431
$204$	232886.	0.391804
$205$	0	0
$206$	1.17585e6	1.93056
$207$	443712.	0.719740
$208$	−2.71679e6	−4.35409
$209$	−52118.5	−0.0825328
$210$	0	0
$211$	−1.17241e6	−1.81290	−0.906449	−	0.422315i	$-0.861218\pi$
$211$	−1.17241e6	−1.81290	−0.906449	+	0.422315i	$0.861218\pi$
$212$	3.01847e6	4.61262
$213$	−155045.	−0.234158
$214$	−1.33039e6	−1.98584
$215$	0	0
$216$	−1.41908e6	−2.06953
$217$	195433.	0.281740
$218$	388179.	0.553211
$219$	89464.9	0.126050
$220$	0	0
$221$	−428364.	−0.589973
$222$	764681.	1.04135
$223$	26647.3	0.0358832	0.0179416	−	0.999839i	$-0.494289\pi$
$223$	26647.3	0.0358832	0.0179416	+	0.999839i	$0.494289\pi$
$224$	−1.86741e6	−2.48668
$225$	0	0
$226$	959070.	1.24905
$227$	−641494.	−0.826282	−0.413141	−	0.910667i	$-0.635568\pi$
$227$	−641494.	−0.826282	−0.413141	+	0.910667i	$0.635568\pi$
$228$	57063.7	0.0726980
$229$	790242.	0.995799	0.497899	−	0.867235i	$-0.334105\pi$
$229$	790242.	0.995799	0.497899	+	0.867235i	$0.334105\pi$
$230$	0	0
$231$	315991.	0.389623
$232$	340311.	0.415103
$233$	538442.	0.649754	0.324877	−	0.945756i	$-0.394677\pi$
$233$	538442.	0.649754	0.324877	+	0.945756i	$0.394677\pi$
$234$	1.97698e6	2.36027
$235$	0	0
$236$	−1.31119e6	−1.53245
$237$	259503.	0.300104
$238$	−617017.	−0.706082
$239$	420883.	0.476614	0.238307	−	0.971190i	$-0.423408\pi$
$239$	420883.	0.476614	0.238307	+	0.971190i	$0.423408\pi$
$240$	0	0
$241$	45463.9	0.0504225	0.0252113	−	0.999682i	$-0.491974\pi$
$241$	45463.9	0.0504225	0.0252113	+	0.999682i	$0.491974\pi$
$242$	373048.	0.409473
$243$	854930.	0.928784
$244$	3.52889e6	3.79458
$245$	0	0
$246$	−794990.	−0.837575
$247$	−104961.	−0.109468
$248$	−861863.	−0.889834
$249$	−139887.	−0.142981
$250$	0	0
$251$	682911.	0.684194	0.342097	−	0.939665i	$-0.388863\pi$
$251$	682911.	0.684194	0.342097	+	0.939665i	$0.388863\pi$
$252$	2.04676e6	2.03032
$253$	945099.	0.928273
$254$	−1.70045e6	−1.65378
$255$	0	0
$256$	261610.	0.249491
$257$	504722.	0.476672	0.238336	−	0.971183i	$-0.423398\pi$
$257$	504722.	0.476672	0.238336	+	0.971183i	$0.423398\pi$
$258$	−700809.	−0.655467
$259$	−1.45618e6	−1.34885
$260$	0	0
$261$	−133221.	−0.121052
$262$	−2.04100e6	−1.83692
$263$	1.45944e6	1.30105	0.650527	−	0.759483i	$-0.274548\pi$
$263$	1.45944e6	1.30105	0.650527	+	0.759483i	$0.274548\pi$
$264$	−1.39352e6	−1.23057
$265$	0	0
$266$	−151186.	−0.131011
$267$	289516.	0.248539
$268$	−1.47721e6	−1.25633
$269$	−969441.	−0.816847	−0.408423	−	0.912793i	$-0.633921\pi$
$269$	−969441.	−0.816847	−0.408423	+	0.912793i	$0.633921\pi$
$270$	0	0
$271$	−1.88150e6	−1.55626	−0.778130	−	0.628104i	$-0.783831\pi$
$271$	−1.88150e6	−1.55626	−0.778130	+	0.628104i	$0.783831\pi$
$272$	1.46382e6	1.19968
$273$	636372.	0.516779
$274$	−2.98074e6	−2.39854
$275$	0	0
$276$	−1.03477e6	−0.817658
$277$	1.19182e6	0.933280	0.466640	−	0.884447i	$-0.345464\pi$
$277$	1.19182e6	0.933280	0.466640	+	0.884447i	$0.345464\pi$
$278$	2.10833e6	1.63616
$279$	337392.	0.259492
$280$	0	0
$281$	−1.45386e6	−1.09839	−0.549197	−	0.835693i	$-0.685066\pi$
$281$	−1.45386e6	−1.09839	−0.549197	+	0.835693i	$0.685066\pi$
$282$	−1.26324e6	−0.945940
$283$	1.02913e6	0.763847	0.381923	−	0.924194i	$-0.375262\pi$
$283$	1.02913e6	0.763847	0.381923	+	0.924194i	$0.375262\pi$
$284$	−2.13907e6	−1.57373
$285$	0	0
$286$	4.21092e6	3.04412
$287$	1.51389e6	1.08490
$288$	−3.22385e6	−2.29031
$289$	−1.18905e6	−0.837445
$290$	0	0
$291$	−82873.8	−0.0573700
$292$	1.23430e6	0.847155
$293$	−787720.	−0.536047	−0.268023	−	0.963412i	$-0.586370\pi$
$293$	−787720.	−0.536047	−0.268023	+	0.963412i	$0.586370\pi$
$294$	−146040.	−0.0985379
$295$	0	0
$296$	6.42175e6	4.26014
$297$	1.18325e6	0.778370
$298$	525851.	0.343022
$299$	1.90333e6	1.23122
$300$	0	0
$301$	1.33455e6	0.849019
$302$	1.84876e6	1.16644
$303$	2261.89	0.00141535
$304$	358676.	0.222596
$305$	0	0
$306$	−1.06521e6	−0.650324
$307$	−1.55665e6	−0.942637	−0.471318	−	0.881963i	$-0.656222\pi$
$307$	−1.55665e6	−0.942637	−0.471318	+	0.881963i	$0.656222\pi$
$308$	4.35955e6	2.61858
$309$	653426.	0.389314
$310$	0	0
$311$	2.72339e6	1.59665	0.798323	−	0.602229i	$-0.205721\pi$
$311$	2.72339e6	1.59665	0.798323	+	0.602229i	$0.205721\pi$
$312$	−2.80641e6	−1.63217
$313$	144019.	0.0830919	0.0415460	−	0.999137i	$-0.486772\pi$
$313$	144019.	0.0830919	0.0415460	+	0.999137i	$0.486772\pi$
$314$	−438861.	−0.251190
$315$	0	0
$316$	3.58022e6	2.01694
$317$	−577983.	−0.323048	−0.161524	−	0.986869i	$-0.551641\pi$
$317$	−577983.	−0.323048	−0.161524	+	0.986869i	$0.551641\pi$
$318$	2.33374e6	1.29415
$319$	−283758.	−0.156125
$320$	0	0
$321$	−739306.	−0.400463
$322$	2.74156e6	1.47353
$323$	56553.6	0.0301616
$324$	2.83524e6	1.50047
$325$	0	0
$326$	2.42031e6	1.26132
$327$	215713.	0.111560
$328$	−6.67628e6	−3.42650
$329$	2.40558e6	1.22527
$330$	0	0
$331$	1.98342e6	0.995050	0.497525	−	0.867450i	$-0.334242\pi$
$331$	1.98342e6	0.995050	0.497525	+	0.867450i	$0.334242\pi$
$332$	−1.92994e6	−0.960944
$333$	−2.51391e6	−1.24234
$334$	1.13806e6	0.558211
$335$	0	0
$336$	−2.17463e6	−1.05084
$337$	−913445.	−0.438135	−0.219067	−	0.975710i	$-0.570301\pi$
$337$	−913445.	−0.438135	−0.219067	+	0.975710i	$0.570301\pi$
$338$	4.51986e6	2.15196
$339$	532961.	0.251881
$340$	0	0
$341$	718637.	0.334675
$342$	−261005.	−0.120666
$343$	2.30174e6	1.05638
$344$	−5.88536e6	−2.68150
$345$	0	0
$346$	−1.23578e6	−0.554946
$347$	−2.80370e6	−1.24999	−0.624997	−	0.780627i	$-0.714900\pi$
$347$	−2.80370e6	−1.24999	−0.624997	+	0.780627i	$0.714900\pi$
$348$	310682.	0.137520
$349$	1.92703e6	0.846887	0.423444	−	0.905922i	$-0.360821\pi$
$349$	1.92703e6	0.846887	0.423444	+	0.905922i	$0.360821\pi$
$350$	0	0
$351$	2.38294e6	1.03239
$352$	−6.86675e6	−2.95389
$353$	758405.	0.323940	0.161970	−	0.986796i	$-0.448215\pi$
$353$	758405.	0.323940	0.161970	+	0.986796i	$0.448215\pi$
$354$	−1.01375e6	−0.429955
$355$	0	0
$356$	3.99429e6	1.67038
$357$	−342880.	−0.142388
$358$	−6.33477e6	−2.61230
$359$	467609.	0.191490	0.0957452	−	0.995406i	$-0.469477\pi$
$359$	467609.	0.191490	0.0957452	+	0.995406i	$0.469477\pi$
$360$	0	0
$361$	−2.46224e6	−0.994404
$362$	−7.87920e6	−3.16017
$363$	207305.	0.0825739
$364$	8.77967e6	3.47316
$365$	0	0
$366$	2.72837e6	1.06463
$367$	−3.84424e6	−1.48986	−0.744930	−	0.667143i	$-0.767517\pi$
$367$	−3.84424e6	−1.48986	−0.744930	+	0.667143i	$0.767517\pi$
$368$	−6.50410e6	−2.50362
$369$	2.61355e6	0.999229
$370$	0	0
$371$	−4.44412e6	−1.67630
$372$	−786823.	−0.294795
$373$	3.10489e6	1.15551	0.577755	−	0.816210i	$-0.303929\pi$
$373$	3.10489e6	1.15551	0.577755	+	0.816210i	$0.303929\pi$
$374$	−2.26887e6	−0.838745
$375$	0	0
$376$	−1.06086e7	−3.86982
$377$	−571458.	−0.207077
$378$	3.43240e6	1.23557
$379$	−1.92276e6	−0.687586	−0.343793	−	0.939046i	$-0.611712\pi$
$379$	−1.92276e6	−0.687586	−0.343793	+	0.939046i	$0.611712\pi$
$380$	0	0
$381$	−944949.	−0.333500
$382$	9.04484e6	3.17134
$383$	3.69535e6	1.28724	0.643619	−	0.765346i	$-0.277432\pi$
$383$	3.69535e6	1.28724	0.643619	+	0.765346i	$0.277432\pi$
$384$	1.35341e6	0.468382
$385$	0	0
$386$	−2.92939e6	−1.00071
$387$	2.30393e6	0.781973
$388$	−1.14336e6	−0.385571
$389$	−3.66232e6	−1.22711	−0.613554	−	0.789653i	$-0.710261\pi$
$389$	−3.66232e6	−1.22711	−0.613554	+	0.789653i	$0.710261\pi$
$390$	0	0
$391$	−1.02552e6	−0.339237
$392$	−1.22644e6	−0.403116
$393$	−1.13420e6	−0.370431
$394$	−1.83508e6	−0.595545
$395$	0	0
$396$	7.52624e6	2.41179
$397$	599240.	0.190820	0.0954102	−	0.995438i	$-0.469584\pi$
$397$	599240.	0.190820	0.0954102	+	0.995438i	$0.469584\pi$
$398$	−1.12794e7	−3.56925
$399$	−84015.3	−0.0264196
$400$	0	0
$401$	950637.	0.295226	0.147613	−	0.989045i	$-0.452841\pi$
$401$	950637.	0.295226	0.147613	+	0.989045i	$0.452841\pi$
$402$	−1.14210e6	−0.352485
$403$	1.44726e6	0.443898
$404$	31206.0	0.00951228
$405$	0	0
$406$	−823130.	−0.247830
$407$	−5.35458e6	−1.60228
$408$	1.51211e6	0.449709
$409$	1.80612e6	0.533873	0.266937	−	0.963714i	$-0.413989\pi$
$409$	1.80612e6	0.533873	0.266937	+	0.963714i	$0.413989\pi$
$410$	0	0
$411$	−1.65642e6	−0.483688
$412$	9.01495e6	2.61650
$413$	1.93048e6	0.556915
$414$	4.73297e6	1.35717
$415$	0	0
$416$	−1.38289e7	−3.91791
$417$	1.17161e6	0.329946
$418$	−555935.	−0.155627
$419$	−5.78098e6	−1.60867	−0.804334	−	0.594177i	$-0.797478\pi$
$419$	−5.78098e6	−1.60867	−0.804334	+	0.594177i	$0.797478\pi$
$420$	0	0
$421$	−6.09006e6	−1.67462	−0.837311	−	0.546727i	$-0.815873\pi$
$421$	−6.09006e6	−1.67462	−0.837311	+	0.546727i	$0.815873\pi$
$422$	−1.25058e7	−3.41846
$423$	4.15294e6	1.12851
$424$	1.95986e7	5.29433
$425$	0	0
$426$	−1.65383e6	−0.441536
$427$	−5.19561e6	−1.37901
$428$	−1.01998e7	−2.69142
$429$	2.34004e6	0.613874
$430$	0	0
$431$	−6.90927e6	−1.79159	−0.895796	−	0.444465i	$-0.853394\pi$
$431$	−6.90927e6	−1.79159	−0.895796	+	0.444465i	$0.853394\pi$
$432$	−8.14305e6	−2.09932
$433$	3.78791e6	0.970911	0.485456	−	0.874261i	$-0.338654\pi$
$433$	3.78791e6	0.970911	0.485456	+	0.874261i	$0.338654\pi$
$434$	2.08464e6	0.531259
$435$	0	0
$436$	2.97608e6	0.749770
$437$	−251281.	−0.0629444
$438$	954300.	0.237684
$439$	2.35490e6	0.583192	0.291596	−	0.956542i	$-0.405814\pi$
$439$	2.35490e6	0.583192	0.291596	+	0.956542i	$0.405814\pi$
$440$	0	0
$441$	480110.	0.117556
$442$	−4.56925e6	−1.11247
$443$	5.17434e6	1.25270	0.626348	−	0.779544i	$-0.284549\pi$
$443$	5.17434e6	1.25270	0.626348	+	0.779544i	$0.284549\pi$
$444$	5.86263e6	1.41135
$445$	0	0
$446$	284240.	0.0676625
$447$	292219.	0.0691734
$448$	−8.17947e6	−1.92544
$449$	−713943.	−0.167128	−0.0835638	−	0.996502i	$-0.526630\pi$
$449$	−713943.	−0.167128	−0.0835638	+	0.996502i	$0.526630\pi$
$450$	0	0
$451$	5.56681e6	1.28874
$452$	7.35297e6	1.69284
$453$	1.02737e6	0.235223
$454$	−6.84266e6	−1.55806
$455$	0	0
$456$	370508.	0.0834422
$457$	3.40931e6	0.763617	0.381809	−	0.924241i	$-0.375301\pi$
$457$	3.40931e6	0.763617	0.381809	+	0.924241i	$0.375301\pi$
$458$	8.42932e6	1.87771
$459$	−1.28394e6	−0.284455
$460$	0	0
$461$	−1.02085e6	−0.223722	−0.111861	−	0.993724i	$-0.535681\pi$
$461$	−1.02085e6	−0.223722	−0.111861	+	0.993724i	$0.535681\pi$
$462$	3.37060e6	0.734687
$463$	−606404.	−0.131465	−0.0657324	−	0.997837i	$-0.520938\pi$
$463$	−606404.	−0.131465	−0.0657324	+	0.997837i	$0.520938\pi$
$464$	1.95280e6	0.421079
$465$	0	0
$466$	5.74342e6	1.22520
$467$	6.52897e6	1.38533	0.692664	−	0.721261i	$-0.256437\pi$
$467$	6.52897e6	1.38533	0.692664	+	0.721261i	$0.256437\pi$
$468$	1.51570e7	3.19889
$469$	2.17490e6	0.456570
$470$	0	0
$471$	−243878.	−0.0506548
$472$	−8.51342e6	−1.75893
$473$	4.90733e6	1.00854
$474$	2.76806e6	0.565886
$475$	0	0
$476$	−4.73053e6	−0.956957
$477$	−7.67223e6	−1.54392
$478$	4.48946e6	0.898719
$479$	−6.39974e6	−1.27445	−0.637226	−	0.770677i	$-0.719918\pi$
$479$	−6.39974e6	−1.27445	−0.637226	+	0.770677i	$0.719918\pi$
$480$	0	0
$481$	−1.07835e7	−2.12520
$482$	484952.	0.0950783
$483$	1.52350e6	0.297150
$484$	2.86007e6	0.554962
$485$	0	0
$486$	9.11932e6	1.75135
$487$	2.87162e6	0.548661	0.274331	−	0.961635i	$-0.411544\pi$
$487$	2.87162e6	0.548661	0.274331	+	0.961635i	$0.411544\pi$
$488$	2.29127e7	4.35539
$489$	1.34498e6	0.254357
$490$	0	0
$491$	−4.25927e6	−0.797319	−0.398659	−	0.917099i	$-0.630524\pi$
$491$	−4.25927e6	−0.797319	−0.398659	+	0.917099i	$0.630524\pi$
$492$	−6.09500e6	−1.13517
$493$	307904.	0.0570556
$494$	−1.11959e6	−0.206416
$495$	0	0
$496$	−4.94561e6	−0.902642
$497$	3.14937e6	0.571917
$498$	−1.49213e6	−0.269609
$499$	−819085.	−0.147258	−0.0736288	−	0.997286i	$-0.523458\pi$
$499$	−819085.	−0.147258	−0.0736288	+	0.997286i	$0.523458\pi$
$500$	0	0
$501$	632426.	0.112568
$502$	7.28444e6	1.29014
$503$	4.74111e6	0.835527	0.417763	−	0.908556i	$-0.362814\pi$
$503$	4.74111e6	0.835527	0.417763	+	0.908556i	$0.362814\pi$
$504$	1.32894e7	2.33039
$505$	0	0
$506$	1.00811e7	1.75038
$507$	2.51171e6	0.433961
$508$	−1.30369e7	−2.24138
$509$	−9.67261e6	−1.65481	−0.827407	−	0.561602i	$-0.810185\pi$
$509$	−9.67261e6	−1.65481	−0.827407	+	0.561602i	$0.810185\pi$
$510$	0	0
$511$	−1.81727e6	−0.307869
$512$	−4.51583e6	−0.761312
$513$	−314601.	−0.0527797
$514$	5.38374e6	0.898828
$515$	0	0
$516$	−5.37295e6	−0.888358
$517$	8.84569e6	1.45548
$518$	−1.55327e7	−2.54344
$519$	−686731.	−0.111910
$520$	0	0
$521$	1.94535e6	0.313980	0.156990	−	0.987600i	$-0.449821\pi$
$521$	1.94535e6	0.313980	0.156990	+	0.987600i	$0.449821\pi$
$522$	−1.42103e6	−0.228259
$523$	−6.10286e6	−0.975617	−0.487809	−	0.872951i	$-0.662204\pi$
$523$	−6.10286e6	−0.975617	−0.487809	+	0.872951i	$0.662204\pi$
$524$	−1.56479e7	−2.48959
$525$	0	0
$526$	1.55674e7	2.45331
$527$	−779790.	−0.122307
$528$	−7.99644e6	−1.24828
$529$	−1.87970e6	−0.292044
$530$	0	0
$531$	3.33273e6	0.512937
$532$	−1.15911e6	−0.177560
$533$	1.12110e7	1.70933
$534$	3.08820e6	0.468653
$535$	0	0
$536$	−9.59134e6	−1.44201
$537$	−3.52027e6	−0.526794
$538$	−1.03408e7	−1.54027
$539$	1.02263e6	0.151616
$540$	0	0
$541$	7.48331e6	1.09926	0.549630	−	0.835408i	$-0.314769\pi$
$541$	7.48331e6	1.09926	0.549630	+	0.835408i	$0.314769\pi$
$542$	−2.00695e7	−2.93453
$543$	−4.37852e6	−0.637276
$544$	7.45107e6	1.07950
$545$	0	0
$546$	6.78802e6	0.974455
$547$	1.01882e7	1.45590	0.727948	−	0.685632i	$-0.240474\pi$
$547$	1.01882e7	1.45590	0.727948	+	0.685632i	$0.240474\pi$
$548$	−2.28527e7	−3.25076
$549$	−8.96958e6	−1.27011
$550$	0	0
$551$	75445.1	0.0105865
$552$	−6.71866e6	−0.938502
$553$	−5.27119e6	−0.732986
$554$	1.27129e7	1.75982
$555$	0	0
$556$	1.61641e7	2.21750
$557$	3.28894e6	0.449178	0.224589	−	0.974454i	$-0.427896\pi$
$557$	3.28894e6	0.449178	0.224589	+	0.974454i	$0.427896\pi$
$558$	3.59887e6	0.489306
$559$	9.88283e6	1.33768
$560$	0	0
$561$	−1.26082e6	−0.169140
$562$	−1.55080e7	−2.07117
$563$	−7.09348e6	−0.943166	−0.471583	−	0.881822i	$-0.656317\pi$
$563$	−7.09348e6	−0.943166	−0.471583	+	0.881822i	$0.656317\pi$
$564$	−9.68499e6	−1.28204
$565$	0	0
$566$	1.09775e7	1.44033
$567$	−4.17434e6	−0.545293
$568$	−1.38888e7	−1.80631
$569$	−2.13368e6	−0.276279	−0.138139	−	0.990413i	$-0.544112\pi$
$569$	−2.13368e6	−0.276279	−0.138139	+	0.990413i	$0.544112\pi$
$570$	0	0
$571$	69756.6	0.00895355	0.00447677	−	0.999990i	$-0.498575\pi$
$571$	69756.6	0.00895355	0.00447677	+	0.999990i	$0.498575\pi$
$572$	3.22842e7	4.12572
$573$	5.02628e6	0.639528
$574$	1.61483e7	2.04573
$575$	0	0
$576$	−1.41209e7	−1.77339
$577$	1.02943e7	1.28723	0.643616	−	0.765349i	$-0.277433\pi$
$577$	1.02943e7	1.28723	0.643616	+	0.765349i	$0.277433\pi$
$578$	−1.26833e7	−1.57911
$579$	−1.62788e6	−0.201802
$580$	0	0
$581$	2.84146e6	0.349222
$582$	−883993.	−0.108179
$583$	−1.63417e7	−1.99125
$584$	8.01416e6	0.972358
$585$	0	0
$586$	−8.40241e6	−1.01079
$587$	5.84442e6	0.700077	0.350039	−	0.936735i	$-0.386168\pi$
$587$	5.84442e6	0.700077	0.350039	+	0.936735i	$0.386168\pi$
$588$	−1.11965e6	−0.133549
$589$	−191070.	−0.0226937
$590$	0	0
$591$	−1.01977e6	−0.120097
$592$	3.68498e7	4.32147
$593$	8.58492e6	1.00253	0.501267	−	0.865292i	$-0.332867\pi$
$593$	8.58492e6	1.00253	0.501267	+	0.865292i	$0.332867\pi$
$594$	1.26215e7	1.46772
$595$	0	0
$596$	4.03158e6	0.464900
$597$	−6.26802e6	−0.719771
$598$	2.03023e7	2.32163
$599$	−4.48945e6	−0.511242	−0.255621	−	0.966777i	$-0.582280\pi$
$599$	−4.48945e6	−0.511242	−0.255621	+	0.966777i	$0.582280\pi$
$600$	0	0
$601$	−3.96161e6	−0.447389	−0.223694	−	0.974659i	$-0.571812\pi$
$601$	−3.96161e6	−0.447389	−0.223694	+	0.974659i	$0.571812\pi$
$602$	1.42353e7	1.60094
$603$	3.75470e6	0.420516
$604$	1.41740e7	1.58088
$605$	0	0
$606$	24127.0	0.00266883
$607$	−1.42627e7	−1.57120	−0.785599	−	0.618736i	$-0.787645\pi$
$607$	−1.42627e7	−1.57120	−0.785599	+	0.618736i	$0.787645\pi$
$608$	1.82572e6	0.200297
$609$	−457419.	−0.0499771
$610$	0	0
$611$	1.78143e7	1.93048
$612$	−8.16668e6	−0.881388
$613$	1.78171e7	1.91508	0.957538	−	0.288307i	$-0.0930925\pi$
$613$	1.78171e7	1.91508	0.957538	+	0.288307i	$0.0930925\pi$
$614$	−1.66044e7	−1.77747
$615$	0	0
$616$	2.83061e7	3.00558
$617$	1.85246e7	1.95901	0.979504	−	0.201425i	$-0.0645572\pi$
$617$	1.85246e7	1.95901	0.979504	+	0.201425i	$0.0645572\pi$
$618$	6.96993e6	0.734103
$619$	1.04691e7	1.09820	0.549102	−	0.835755i	$-0.314970\pi$
$619$	1.04691e7	1.09820	0.549102	+	0.835755i	$0.314970\pi$
$620$	0	0
$621$	5.70487e6	0.593631
$622$	2.90497e7	3.01069
$623$	−5.88083e6	−0.607041
$624$	−1.61040e7	−1.65566
$625$	0	0
$626$	1.53621e6	0.156681
$627$	−308937.	−0.0313835
$628$	−3.36465e6	−0.340440
$629$	5.81023e6	0.585553
$630$	0	0
$631$	−60149.4	−0.00601392	−0.00300696	−	0.999995i	$-0.500957\pi$
$631$	−60149.4	−0.00601392	−0.00300696	+	0.999995i	$0.500957\pi$
$632$	2.32460e7	2.31502
$633$	−6.94956e6	−0.689363
$634$	−6.16520e6	−0.609150
$635$	0	0
$636$	1.78922e7	1.75397
$637$	2.05946e6	0.201096
$638$	−3.02677e6	−0.294394
$639$	5.43701e6	0.526754
$640$	0	0
$641$	1.40256e7	1.34827	0.674136	−	0.738608i	$-0.264516\pi$
$641$	1.40256e7	1.34827	0.674136	+	0.738608i	$0.264516\pi$
$642$	−7.88600e6	−0.755125
$643$	5.64945e6	0.538863	0.269432	−	0.963020i	$-0.413164\pi$
$643$	5.64945e6	0.538863	0.269432	+	0.963020i	$0.413164\pi$
$644$	2.10189e7	1.99708
$645$	0	0
$646$	603243.	0.0568736
$647$	−1.55564e7	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$647$	−1.55564e7	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$648$	1.84089e7	1.72222
$649$	7.09865e6	0.661552
$650$	0	0
$651$	1.15845e6	0.107133
$652$	1.85559e7	1.70948
$653$	2.45452e6	0.225260	0.112630	−	0.993637i	$-0.464072\pi$
$653$	2.45452e6	0.225260	0.112630	+	0.993637i	$0.464072\pi$
$654$	2.30096e6	0.210361
$655$	0	0
$656$	−3.83104e7	−3.47582
$657$	−3.13729e6	−0.283557
$658$	2.56597e7	2.31040
$659$	158118.	0.0141830	0.00709150	−	0.999975i	$-0.497743\pi$
$659$	158118.	0.0141830	0.00709150	+	0.999975i	$0.497743\pi$
$660$	0	0
$661$	−1.69262e7	−1.50680	−0.753398	−	0.657565i	$-0.771587\pi$
$661$	−1.69262e7	−1.50680	−0.753398	+	0.657565i	$0.771587\pi$
$662$	2.11567e7	1.87630
$663$	−2.53916e6	−0.224340
$664$	−1.25309e7	−1.10296
$665$	0	0
$666$	−2.68152e7	−2.34259
$667$	−1.36809e6	−0.119070
$668$	8.72523e6	0.756547
$669$	157954.	0.0136447
$670$	0	0
$671$	−1.91050e7	−1.63810
$672$	−1.10692e7	−0.945570
$673$	−6.52562e6	−0.555373	−0.277686	−	0.960672i	$-0.589568\pi$
$673$	−6.52562e6	−0.555373	−0.277686	+	0.960672i	$0.589568\pi$
$674$	−9.74349e6	−0.826161
$675$	0	0
$676$	3.46527e7	2.91656
$677$	−1.82770e7	−1.53261	−0.766306	−	0.642476i	$-0.777907\pi$
$677$	−1.82770e7	−1.53261	−0.766306	+	0.642476i	$0.777907\pi$
$678$	5.68496e6	0.474956
$679$	1.68338e6	0.140123
$680$	0	0
$681$	−3.80251e6	−0.314197
$682$	7.66552e6	0.631075
$683$	8.58042e6	0.703812	0.351906	−	0.936035i	$-0.385534\pi$
$683$	8.58042e6	0.703812	0.351906	+	0.936035i	$0.385534\pi$
$684$	−2.00106e6	−0.163539
$685$	0	0
$686$	2.45521e7	1.99195
$687$	4.68422e6	0.378657
$688$	−3.37719e7	−2.72009
$689$	−3.29104e7	−2.64111
$690$	0	0
$691$	−2.16287e7	−1.72320	−0.861601	−	0.507586i	$-0.830538\pi$
$691$	−2.16287e7	−1.72320	−0.861601	+	0.507586i	$0.830538\pi$
$692$	−9.47444e6	−0.752122
$693$	−1.10809e7	−0.876483
$694$	−2.99064e7	−2.35703
$695$	0	0
$696$	2.01722e6	0.157845
$697$	−6.04052e6	−0.470969
$698$	2.05552e7	1.59692
$699$	3.19166e6	0.247072
$700$	0	0
$701$	−2.16485e7	−1.66392	−0.831959	−	0.554837i	$-0.812781\pi$
$701$	−2.16485e7	−1.66392	−0.831959	+	0.554837i	$0.812781\pi$
$702$	2.54182e7	1.94672
$703$	1.42367e6	0.108648
$704$	−3.00771e7	−2.28720
$705$	0	0
$706$	8.08971e6	0.610831
$707$	−45944.8	−0.00345691
$708$	−7.77219e6	−0.582720
$709$	1.15365e7	0.861904	0.430952	−	0.902375i	$-0.358178\pi$
$709$	1.15365e7	0.861904	0.430952	+	0.902375i	$0.358178\pi$
$710$	0	0
$711$	−9.10006e6	−0.675104
$712$	2.59345e7	1.91725
$713$	3.46480e6	0.255243
$714$	−3.65742e6	−0.268491
$715$	0	0
$716$	−4.85672e7	−3.54047
$717$	2.49482e6	0.181235
$718$	4.98787e6	0.361081
$719$	1.16519e7	0.840570	0.420285	−	0.907392i	$-0.361930\pi$
$719$	1.16519e7	0.840570	0.420285	+	0.907392i	$0.361930\pi$
$720$	0	0
$721$	−1.32728e7	−0.950876
$722$	−2.62641e7	−1.87508
$723$	269491.	0.0191734
$724$	−6.04080e7	−4.28300
$725$	0	0
$726$	2.21127e6	0.155704
$727$	−8.27347e6	−0.580566	−0.290283	−	0.956941i	$-0.593749\pi$
$727$	−8.27347e6	−0.580566	−0.290283	+	0.956941i	$0.593749\pi$
$728$	5.70055e7	3.98647
$729$	−3.35697e6	−0.233953
$730$	0	0
$731$	−5.32492e6	−0.368570
$732$	2.09178e7	1.44290
$733$	−2.68922e7	−1.84870	−0.924350	−	0.381545i	$-0.875392\pi$
$733$	−2.68922e7	−1.84870	−0.924350	+	0.381545i	$0.875392\pi$
$734$	−4.10055e7	−2.80933
$735$	0	0
$736$	−3.31069e7	−2.25281
$737$	7.99744e6	0.542353
$738$	2.78781e7	1.88418
$739$	2.29150e7	1.54351	0.771753	−	0.635922i	$-0.219380\pi$
$739$	2.29150e7	1.54351	0.771753	+	0.635922i	$0.219380\pi$
$740$	0	0
$741$	−622166.	−0.0416256
$742$	−4.74043e7	−3.16088
$743$	7.03515e6	0.467521	0.233761	−	0.972294i	$-0.424897\pi$
$743$	7.03515e6	0.467521	0.233761	+	0.972294i	$0.424897\pi$
$744$	−5.10876e6	−0.338363
$745$	0	0
$746$	3.31191e7	2.17887
$747$	4.90543e6	0.321644
$748$	−1.73949e7	−1.13676
$749$	1.50172e7	0.978105
$750$	0	0
$751$	−2.11756e7	−1.37005	−0.685023	−	0.728522i	$-0.740208\pi$
$751$	−2.11756e7	−1.37005	−0.685023	+	0.728522i	$0.740208\pi$
$752$	−6.08754e7	−3.92552
$753$	4.04801e6	0.260168
$754$	−6.09560e6	−0.390470
$755$	0	0
$756$	2.63154e7	1.67458
$757$	6.10083e6	0.386945	0.193473	−	0.981106i	$-0.438025\pi$
$757$	6.10083e6	0.386945	0.193473	+	0.981106i	$0.438025\pi$
$758$	−2.05096e7	−1.29653
$759$	5.60215e6	0.352980
$760$	0	0
$761$	−4.42795e6	−0.277167	−0.138583	−	0.990351i	$-0.544255\pi$
$761$	−4.42795e6	−0.277167	−0.138583	+	0.990351i	$0.544255\pi$
$762$	−1.00795e7	−0.628858
$763$	−4.38170e6	−0.272478
$764$	6.93447e7	4.29814
$765$	0	0
$766$	3.94174e7	2.42726
$767$	1.42959e7	0.877453
$768$	1.55071e6	0.0948699
$769$	−3.30798e6	−0.201719	−0.100860	−	0.994901i	$-0.532159\pi$
$769$	−3.30798e6	−0.201719	−0.100860	+	0.994901i	$0.532159\pi$
$770$	0	0
$771$	2.99178e6	0.181257
$772$	−2.24590e7	−1.35627
$773$	−1.66539e7	−1.00246	−0.501232	−	0.865313i	$-0.667120\pi$
$773$	−1.66539e7	−1.00246	−0.501232	+	0.865313i	$0.667120\pi$
$774$	2.45754e7	1.47451
$775$	0	0
$776$	−7.42373e6	−0.442556
$777$	−8.63160e6	−0.512907
$778$	−3.90651e7	−2.31387
$779$	−1.48010e6	−0.0873869
$780$	0	0
$781$	1.15807e7	0.679373
$782$	−1.09390e7	−0.639676
$783$	−1.71284e6	−0.0998417
$784$	−7.03763e6	−0.408918
$785$	0	0
$786$	−1.20982e7	−0.698497
$787$	1.03913e7	0.598045	0.299022	−	0.954246i	$-0.403339\pi$
$787$	1.03913e7	0.598045	0.299022	+	0.954246i	$0.403339\pi$
$788$	−1.40691e7	−0.807146
$789$	8.65092e6	0.494732
$790$	0	0
$791$	−1.08258e7	−0.615205
$792$	4.88670e7	2.76824
$793$	−3.84755e7	−2.17271
$794$	6.39195e6	0.359817
$795$	0	0
$796$	−8.64763e7	−4.83743
$797$	1.68557e6	0.0939940	0.0469970	−	0.998895i	$-0.485035\pi$
$797$	1.68557e6	0.0939940	0.0469970	+	0.998895i	$0.485035\pi$
$798$	−896170.	−0.0498176
$799$	−9.59841e6	−0.531903
$800$	0	0
$801$	−1.01525e7	−0.559104
$802$	1.01402e7	0.556687
$803$	−6.68236e6	−0.365714
$804$	−8.75625e6	−0.477725
$805$	0	0
$806$	1.54375e7	0.837029
$807$	−5.74644e6	−0.310609
$808$	202617.	0.0109181
$809$	8.48627e6	0.455875	0.227937	−	0.973676i	$-0.426802\pi$
$809$	8.48627e6	0.455875	0.227937	+	0.973676i	$0.426802\pi$
$810$	0	0
$811$	−1.14182e7	−0.609601	−0.304801	−	0.952416i	$-0.598590\pi$
$811$	−1.14182e7	−0.609601	−0.304801	+	0.952416i	$0.598590\pi$
$812$	−6.31075e6	−0.335885
$813$	−1.11528e7	−0.591775
$814$	−5.71160e7	−3.02132
$815$	0	0
$816$	8.67689e6	0.456183
$817$	−1.30475e6	−0.0683869
$818$	1.92654e7	1.00669
$819$	−2.23158e7	−1.16253
$820$	0	0
$821$	3.66175e7	1.89597	0.947983	−	0.318320i	$-0.103119\pi$
$821$	3.66175e7	1.89597	0.947983	+	0.318320i	$0.103119\pi$
$822$	−1.76686e7	−0.912057
$823$	−1.27704e7	−0.657212	−0.328606	−	0.944467i	$-0.606579\pi$
$823$	−1.27704e7	−0.657212	−0.328606	+	0.944467i	$0.606579\pi$
$824$	5.85331e7	3.00319
$825$	0	0
$826$	2.05919e7	1.05014
$827$	3.02070e7	1.53583	0.767917	−	0.640549i	$-0.221293\pi$
$827$	3.02070e7	1.53583	0.767917	+	0.640549i	$0.221293\pi$
$828$	3.62866e7	1.83937
$829$	−3.87577e6	−0.195872	−0.0979359	−	0.995193i	$-0.531224\pi$
$829$	−3.87577e6	−0.195872	−0.0979359	+	0.995193i	$0.531224\pi$
$830$	0	0
$831$	7.06462e6	0.354884
$832$	−6.05721e7	−3.03364
$833$	−1.10965e6	−0.0554079
$834$	1.24973e7	0.622158
$835$	0	0
$836$	−4.26223e6	−0.210922
$837$	4.33789e6	0.214025
$838$	−6.16643e7	−3.03336
$839$	−3.49516e6	−0.171420	−0.0857102	−	0.996320i	$-0.527316\pi$
$839$	−3.49516e6	−0.171420	−0.0857102	+	0.996320i	$0.527316\pi$
$840$	0	0
$841$	−2.01004e7	−0.979974
$842$	−6.49612e7	−3.15772
$843$	−8.61789e6	−0.417669
$844$	−9.58791e7	−4.63306
$845$	0	0
$846$	4.42984e7	2.12795
$847$	−4.21090e6	−0.201682
$848$	1.12462e8	5.37054
$849$	6.10028e6	0.290456
$850$	0	0
$851$	−2.58163e7	−1.22200
$852$	−1.26795e7	−0.598417
$853$	−3.02315e7	−1.42261	−0.711307	−	0.702881i	$-0.751896\pi$
$853$	−3.02315e7	−1.42261	−0.711307	+	0.702881i	$0.751896\pi$
$854$	−5.54203e7	−2.60030
$855$	0	0
$856$	−6.62262e7	−3.08920
$857$	−4.90078e6	−0.227936	−0.113968	−	0.993484i	$-0.536356\pi$
$857$	−4.90078e6	−0.227936	−0.113968	+	0.993484i	$0.536356\pi$
$858$	2.49606e7	1.15754
$859$	−3.90589e7	−1.80608	−0.903040	−	0.429557i	$-0.858670\pi$
$859$	−3.90589e7	−1.80608	−0.903040	+	0.429557i	$0.858670\pi$
$860$	0	0
$861$	8.97372e6	0.412539
$862$	−7.36995e7	−3.37828
$863$	−3.66262e7	−1.67404	−0.837019	−	0.547173i	$-0.815704\pi$
$863$	−3.66262e7	−1.67404	−0.837019	+	0.547173i	$0.815704\pi$
$864$	−4.14495e7	−1.88901
$865$	0	0
$866$	4.04046e7	1.83078
$867$	−7.04820e6	−0.318442
$868$	1.59824e7	0.720018
$869$	−1.93830e7	−0.870704
$870$	0	0
$871$	1.61060e7	0.719353
$872$	1.93234e7	0.860580
$873$	2.90615e6	0.129057
$874$	−2.68036e6	−0.118690
$875$	0	0
$876$	7.31640e6	0.322134
$877$	−3.05781e7	−1.34249	−0.671246	−	0.741235i	$-0.734241\pi$
$877$	−3.05781e7	−1.34249	−0.671246	+	0.741235i	$0.734241\pi$
$878$	2.51191e7	1.09969
$879$	−4.66927e6	−0.203834
$880$	0	0
$881$	−1.07731e7	−0.467629	−0.233815	−	0.972281i	$-0.575121\pi$
$881$	−1.07731e7	−0.467629	−0.233815	+	0.972281i	$0.575121\pi$
$882$	5.12121e6	0.221667
$883$	2.56044e7	1.10513	0.552563	−	0.833471i	$-0.313650\pi$
$883$	2.56044e7	1.10513	0.552563	+	0.833471i	$0.313650\pi$
$884$	−3.50314e7	−1.50774
$885$	0	0
$886$	5.51934e7	2.36212
$887$	−1.57569e7	−0.672453	−0.336227	−	0.941781i	$-0.609151\pi$
$887$	−1.57569e7	−0.672453	−0.336227	+	0.941781i	$0.609151\pi$
$888$	3.80654e7	1.61994
$889$	1.91944e7	0.814553
$890$	0	0
$891$	−1.53497e7	−0.647746
$892$	2.17920e6	0.0917034
$893$	−2.35188e6	−0.0986930
$894$	3.11702e6	0.130436
$895$	0	0
$896$	−2.74912e7	−1.14399
$897$	1.12821e7	0.468177
$898$	−7.61545e6	−0.315141
$899$	−1.04028e6	−0.0429289
$900$	0	0
$901$	1.77323e7	0.727701
$902$	5.93798e7	2.43009
$903$	7.91063e6	0.322843
$904$	4.77420e7	1.94303
$905$	0	0
$906$	1.09587e7	0.443544
$907$	−1.31837e7	−0.532131	−0.266065	−	0.963955i	$-0.585724\pi$
$907$	−1.31837e7	−0.532131	−0.266065	+	0.963955i	$0.585724\pi$
$908$	−5.24611e7	−2.11165
$909$	−79318.1	−0.00318392
$910$	0	0
$911$	−1.28956e7	−0.514808	−0.257404	−	0.966304i	$-0.582867\pi$
$911$	−1.28956e7	−0.514808	−0.257404	+	0.966304i	$0.582867\pi$
$912$	2.12608e6	0.0846433
$913$	1.04485e7	0.414836
$914$	3.63662e7	1.43990
$915$	0	0
$916$	6.46256e7	2.54487
$917$	2.30385e7	0.904756
$918$	−1.36955e7	−0.536377
$919$	4.43910e7	1.73383	0.866914	−	0.498459i	$-0.166100\pi$
$919$	4.43910e7	1.73383	0.866914	+	0.498459i	$0.166100\pi$
$920$	0	0
$921$	−9.22715e6	−0.358442
$922$	−1.08891e7	−0.421858
$923$	2.33223e7	0.901089
$924$	2.58416e7	0.995725
$925$	0	0
$926$	−6.46836e6	−0.247894
$927$	−2.29138e7	−0.875787
$928$	9.94008e6	0.378896
$929$	2.76819e7	1.05234	0.526170	−	0.850379i	$-0.323628\pi$
$929$	2.76819e7	1.05234	0.526170	+	0.850379i	$0.323628\pi$
$930$	0	0
$931$	−271894.	−0.0102808
$932$	4.40335e7	1.66052
$933$	1.61431e7	0.607132
$934$	6.96429e7	2.61222
$935$	0	0
$936$	9.84130e7	3.67166
$937$	−7.81376e6	−0.290744	−0.145372	−	0.989377i	$-0.546438\pi$
$937$	−7.81376e6	−0.290744	−0.145372	+	0.989377i	$0.546438\pi$
$938$	2.31991e7	0.860923
$939$	853684.	0.0315961
$940$	0	0
$941$	−1.76275e7	−0.648958	−0.324479	−	0.945893i	$-0.605189\pi$
$941$	−1.76275e7	−0.648958	−0.324479	+	0.945893i	$0.605189\pi$
$942$	−2.60139e6	−0.0955163
$943$	2.68395e7	0.982869
$944$	−4.88524e7	−1.78425
$945$	0	0
$946$	5.23452e7	1.90173
$947$	4.10115e7	1.48604	0.743021	−	0.669268i	$-0.233392\pi$
$947$	4.10115e7	1.48604	0.743021	+	0.669268i	$0.233392\pi$
$948$	2.12221e7	0.766949
$949$	−1.34576e7	−0.485066
$950$	0	0
$951$	−3.42604e6	−0.122840
$952$	−3.07148e7	−1.09839
$953$	−2.62994e7	−0.938022	−0.469011	−	0.883192i	$-0.655389\pi$
$953$	−2.62994e7	−0.938022	−0.469011	+	0.883192i	$0.655389\pi$
$954$	−8.18378e7	−2.91127
$955$	0	0
$956$	3.44196e7	1.21804
$957$	−1.68200e6	−0.0593671
$958$	−6.82644e7	−2.40315
$959$	3.36461e7	1.18138
$960$	0	0
$961$	−2.59946e7	−0.907976
$962$	−1.15025e8	−4.00734
$963$	2.59254e7	0.900866
$964$	3.71802e6	0.128860
$965$	0	0
$966$	1.62508e7	0.560315
$967$	3.92036e6	0.134822	0.0674109	−	0.997725i	$-0.478526\pi$
$967$	3.92036e6	0.134822	0.0674109	+	0.997725i	$0.478526\pi$
$968$	1.85701e7	0.636981
$969$	335226.	0.0114691
$970$	0	0
$971$	−2.97706e6	−0.101331	−0.0506653	−	0.998716i	$-0.516134\pi$
$971$	−2.97706e6	−0.101331	−0.0506653	+	0.998716i	$0.516134\pi$
$972$	6.99157e7	2.37361
$973$	−2.37985e7	−0.805874
$974$	3.06308e7	1.03457
$975$	0	0
$976$	1.31479e8	4.41808
$977$	−1.41516e7	−0.474318	−0.237159	−	0.971471i	$-0.576216\pi$
$977$	−1.41516e7	−0.474318	−0.237159	+	0.971471i	$0.576216\pi$
$978$	1.43466e7	0.479624
$979$	−2.16247e7	−0.721096
$980$	0	0
$981$	−7.56447e6	−0.250961
$982$	−4.54326e7	−1.50345
$983$	−8.13535e6	−0.268530	−0.134265	−	0.990945i	$-0.542867\pi$
$983$	−8.13535e6	−0.268530	−0.134265	+	0.990945i	$0.542867\pi$
$984$	−3.95742e7	−1.30294
$985$	0	0
$986$	3.28434e6	0.107586
$987$	1.42593e7	0.465913
$988$	−8.58367e6	−0.279757
$989$	2.36599e7	0.769170
$990$	0	0
$991$	−7.88149e6	−0.254932	−0.127466	−	0.991843i	$-0.540684\pi$
$991$	−7.88149e6	−0.254932	−0.127466	+	0.991843i	$0.540684\pi$
$992$	−2.51740e7	−0.812218
$993$	1.17569e7	0.378372
$994$	3.35936e7	1.07843
$995$	0	0
$996$	−1.14399e7	−0.365403
$997$	−3.12716e7	−0.996352	−0.498176	−	0.867076i	$-0.665997\pi$
$997$	−3.12716e7	−0.996352	−0.498176	+	0.867076i	$0.665997\pi$
$998$	−8.73698e6	−0.277674
$999$	−3.23217e7	−1.02466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.6.a.d.1.22		22
5.4	even	2		625.6.a.c.1.1		22
25.6	even	5		25.6.d.a.11.1	✓	44
25.21	even	5		25.6.d.a.16.1	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.1	✓	44	25.6	even	5
25.6.d.a.16.1	yes	44	25.21	even	5
625.6.a.c.1.1		22	5.4	even	2
625.6.a.d.1.22		22	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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q+10.6667 q^{2} +5.92758 q^{3} +81.7795 q^{4} +63.2280 q^{6} -120.405 q^{7} +530.986 q^{8} -207.864 q^{9} +O(q^{10})\)	\(q+10.6667 q^{2} +5.92758 q^{3} +81.7795 q^{4} +63.2280 q^{6} -120.405 q^{7} +530.986 q^{8} -207.864 q^{9} -442.746 q^{11} +484.755 q^{12} -891.643 q^{13} -1284.33 q^{14} +3046.94 q^{16} +480.421 q^{17} -2217.23 q^{18} +117.717 q^{19} -713.708 q^{21} -4722.66 q^{22} -2134.63 q^{23} +3147.46 q^{24} -9510.93 q^{26} -2672.53 q^{27} -9846.63 q^{28} +640.905 q^{29} -1623.14 q^{31} +15509.5 q^{32} -2624.41 q^{33} +5124.53 q^{34} -16999.0 q^{36} +12094.0 q^{37} +1255.65 q^{38} -5285.28 q^{39} -12573.4 q^{41} -7612.94 q^{42} -11083.8 q^{43} -36207.5 q^{44} -22769.6 q^{46} -19979.2 q^{47} +18061.0 q^{48} -2309.73 q^{49} +2847.74 q^{51} -72918.1 q^{52} +36909.9 q^{53} -28507.2 q^{54} -63933.1 q^{56} +697.775 q^{57} +6836.37 q^{58} -16033.2 q^{59} +43151.3 q^{61} -17313.6 q^{62} +25027.8 q^{63} +67933.2 q^{64} -27993.9 q^{66} -18063.3 q^{67} +39288.6 q^{68} -12653.2 q^{69} -26156.6 q^{71} -110373. q^{72} +15093.0 q^{73} +129004. q^{74} +9626.81 q^{76} +53308.6 q^{77} -56376.8 q^{78} +43779.0 q^{79} +34669.3 q^{81} -134117. q^{82} -23599.3 q^{83} -58366.7 q^{84} -118229. q^{86} +3799.01 q^{87} -235092. q^{88} +48842.2 q^{89} +107358. q^{91} -174569. q^{92} -9621.28 q^{93} -213113. q^{94} +91933.5 q^{96} -13981.0 q^{97} -24637.4 q^{98} +92030.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 9 q^{2} + 2 q^{3} + 289 q^{4} - 111 q^{6} - 101 q^{7} + 540 q^{8} + 1166 q^{9}+O(q^{10}) \) 22 * q + 9 * q^2 + 2 * q^3 + 289 * q^4 - 111 * q^6 - 101 * q^7 + 540 * q^8 + 1166 * q^9	\( 22 q + 9 q^{2} + 2 q^{3} + 289 q^{4} - 111 q^{6} - 101 q^{7} + 540 q^{8} + 1166 q^{9} - 1046 q^{11} - 1271 q^{12} - 1453 q^{13} - 2222 q^{14} + 1317 q^{16} + 2174 q^{17} + 4272 q^{18} - 3765 q^{19} - 4811 q^{21} - 12 q^{22} - 1488 q^{23} - 9810 q^{24} - 8941 q^{26} - 13570 q^{27} - 5587 q^{28} - 9035 q^{29} - 10426 q^{31} + 15339 q^{32} + 6874 q^{33} - 16027 q^{34} - 7098 q^{36} + 24944 q^{37} - 47645 q^{38} - 28703 q^{39} - 36606 q^{41} - 34782 q^{42} + 30597 q^{43} - 47892 q^{44} - 30881 q^{46} - 81481 q^{47} - 20803 q^{48} - 12241 q^{49} - 60566 q^{51} - 79836 q^{52} + 75802 q^{53} - 65085 q^{54} - 83335 q^{56} + 146075 q^{57} - 30030 q^{58} - 87095 q^{59} - 71476 q^{61} + 131553 q^{62} - 111758 q^{63} - 30736 q^{64} - 64222 q^{66} - 85706 q^{67} + 61943 q^{68} - 93448 q^{69} - 125571 q^{71} + 318570 q^{72} + 157982 q^{73} - 89682 q^{74} - 134640 q^{76} - 236842 q^{77} + 145204 q^{78} - 173645 q^{79} - 156538 q^{81} - 176667 q^{82} - 166408 q^{83} - 213437 q^{84} - 182891 q^{86} - 175085 q^{87} - 612480 q^{88} - 280940 q^{89} - 182911 q^{91} - 534131 q^{92} - 190216 q^{93} - 376702 q^{94} - 484051 q^{96} + 636469 q^{97} + 819128 q^{98} - 340388 q^{99}+O(q^{100}) \) 22 * q + 9 * q^2 + 2 * q^3 + 289 * q^4 - 111 * q^6 - 101 * q^7 + 540 * q^8 + 1166 * q^9 - 1046 * q^11 - 1271 * q^12 - 1453 * q^13 - 2222 * q^14 + 1317 * q^16 + 2174 * q^17 + 4272 * q^18 - 3765 * q^19 - 4811 * q^21 - 12 * q^22 - 1488 * q^23 - 9810 * q^24 - 8941 * q^26 - 13570 * q^27 - 5587 * q^28 - 9035 * q^29 - 10426 * q^31 + 15339 * q^32 + 6874 * q^33 - 16027 * q^34 - 7098 * q^36 + 24944 * q^37 - 47645 * q^38 - 28703 * q^39 - 36606 * q^41 - 34782 * q^42 + 30597 * q^43 - 47892 * q^44 - 30881 * q^46 - 81481 * q^47 - 20803 * q^48 - 12241 * q^49 - 60566 * q^51 - 79836 * q^52 + 75802 * q^53 - 65085 * q^54 - 83335 * q^56 + 146075 * q^57 - 30030 * q^58 - 87095 * q^59 - 71476 * q^61 + 131553 * q^62 - 111758 * q^63 - 30736 * q^64 - 64222 * q^66 - 85706 * q^67 + 61943 * q^68 - 93448 * q^69 - 125571 * q^71 + 318570 * q^72 + 157982 * q^73 - 89682 * q^74 - 134640 * q^76 - 236842 * q^77 + 145204 * q^78 - 173645 * q^79 - 156538 * q^81 - 176667 * q^82 - 166408 * q^83 - 213437 * q^84 - 182891 * q^86 - 175085 * q^87 - 612480 * q^88 - 280940 * q^89 - 182911 * q^91 - 534131 * q^92 - 190216 * q^93 - 376702 * q^94 - 484051 * q^96 + 636469 * q^97 + 819128 * q^98 - 340388 * q^99

Introduction

L-functions

Modular forms

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Database

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