LMFDB - Embedded newform 1075.6.a.i.1.5

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			1.5
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)

$f(q)$	$=$	$q-9.45595 q^{2} +20.2754 q^{3} +57.4150 q^{4} -191.723 q^{6} +137.133 q^{7} -240.323 q^{8} +168.090 q^{9} +O(q^{10})$	\(q-9.45595 q^{2} +20.2754 q^{3} +57.4150 q^{4} -191.723 q^{6} +137.133 q^{7} -240.323 q^{8} +168.090 q^{9} -338.952 q^{11} +1164.11 q^{12} +651.032 q^{13} -1296.72 q^{14} +435.205 q^{16} +1314.25 q^{17} -1589.45 q^{18} -3120.69 q^{19} +2780.42 q^{21} +3205.11 q^{22} +8.80880 q^{23} -4872.64 q^{24} -6156.13 q^{26} -1518.82 q^{27} +7873.50 q^{28} +2177.56 q^{29} +9577.68 q^{31} +3575.07 q^{32} -6872.37 q^{33} -12427.5 q^{34} +9650.91 q^{36} -13752.5 q^{37} +29509.1 q^{38} +13199.9 q^{39} -1311.40 q^{41} -26291.6 q^{42} -1849.00 q^{43} -19460.9 q^{44} -83.2956 q^{46} -6661.76 q^{47} +8823.93 q^{48} +1998.50 q^{49} +26646.9 q^{51} +37379.0 q^{52} +34093.9 q^{53} +14361.9 q^{54} -32956.3 q^{56} -63273.1 q^{57} -20590.9 q^{58} -29444.2 q^{59} +22157.6 q^{61} -90566.1 q^{62} +23050.8 q^{63} -47732.3 q^{64} +64984.8 q^{66} +61568.0 q^{67} +75457.7 q^{68} +178.602 q^{69} +38408.1 q^{71} -40396.0 q^{72} -11799.5 q^{73} +130043. q^{74} -179175. q^{76} -46481.5 q^{77} -124818. q^{78} -10887.0 q^{79} -71640.6 q^{81} +12400.5 q^{82} -39381.8 q^{83} +159638. q^{84} +17484.1 q^{86} +44150.8 q^{87} +81458.0 q^{88} +108231. q^{89} +89278.1 q^{91} +505.758 q^{92} +194191. q^{93} +62993.2 q^{94} +72485.9 q^{96} -20691.1 q^{97} -18897.7 q^{98} -56974.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.45595	−1.67159	−0.835796	−	0.549040i	$-0.814993\pi$
$2$	−9.45595	−1.67159	−0.835796	+	0.549040i	$0.814993\pi$
$3$	20.2754	1.30067	0.650333	−	0.759650i	$-0.274630\pi$
$3$	20.2754	1.30067	0.650333	+	0.759650i	$0.274630\pi$
$4$	57.4150	1.79422
$5$	0	0
$5$	0	0
$6$	−191.723	−2.17418
$7$	137.133	1.05778	0.528892	−	0.848689i	$-0.322608\pi$
$7$	137.133	1.05778	0.528892	+	0.848689i	$0.322608\pi$
$8$	−240.323	−1.32761
$9$	168.090	0.691730
$10$	0	0
$11$	−338.952	−0.844609	−0.422305	−	0.906454i	$-0.638779\pi$
$11$	−338.952	−0.844609	−0.422305	+	0.906454i	$0.638779\pi$
$12$	1164.11	2.33368
$13$	651.032	1.06843	0.534213	−	0.845350i	$-0.320608\pi$
$13$	651.032	1.06843	0.534213	+	0.845350i	$0.320608\pi$
$14$	−1296.72	−1.76818
$15$	0	0
$16$	435.205	0.425005
$17$	1314.25	1.10295	0.551475	−	0.834192i	$-0.314065\pi$
$17$	1314.25	1.10295	0.551475	+	0.834192i	$0.314065\pi$
$18$	−1589.45	−1.15629
$19$	−3120.69	−1.98320	−0.991600	−	0.129340i	$-0.958714\pi$
$19$	−3120.69	−1.98320	−0.991600	+	0.129340i	$0.958714\pi$
$20$	0	0
$21$	2780.42	1.37582
$22$	3205.11	1.41184
$23$	8.80880	0.00347214	0.00173607	−	0.999998i	$-0.499447\pi$
$23$	8.80880	0.00347214	0.00173607	+	0.999998i	$0.499447\pi$
$24$	−4872.64	−1.72678
$25$	0	0
$26$	−6156.13	−1.78597
$27$	−1518.82	−0.400956
$28$	7873.50	1.89790
$29$	2177.56	0.480812	0.240406	−	0.970672i	$-0.422719\pi$
$29$	2177.56	0.480812	0.240406	+	0.970672i	$0.422719\pi$
$30$	0	0
$31$	9577.68	1.79001	0.895006	−	0.446053i	$-0.147171\pi$
$31$	9577.68	1.79001	0.895006	+	0.446053i	$0.147171\pi$
$32$	3575.07	0.617177
$33$	−6872.37	−1.09855
$34$	−12427.5	−1.84368
$35$	0	0
$36$	9650.91	1.24111
$37$	−13752.5	−1.65150	−0.825750	−	0.564037i	$-0.809248\pi$
$37$	−13752.5	−1.65150	−0.825750	+	0.564037i	$0.809248\pi$
$38$	29509.1	3.31510
$39$	13199.9	1.38966
$40$	0	0
$41$	−1311.40	−0.121836	−0.0609178	−	0.998143i	$-0.519403\pi$
$41$	−1311.40	−0.121836	−0.0609178	+	0.998143i	$0.519403\pi$
$42$	−26291.6	−2.29982
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−19460.9	−1.51541
$45$	0	0
$46$	−83.2956	−0.00580400
$47$	−6661.76	−0.439890	−0.219945	−	0.975512i	$-0.570588\pi$
$47$	−6661.76	−0.439890	−0.219945	+	0.975512i	$0.570588\pi$
$48$	8823.93	0.552789
$49$	1998.50	0.118909
$50$	0	0
$51$	26646.9	1.43457
$52$	37379.0	1.91699
$53$	34093.9	1.66720	0.833600	−	0.552369i	$-0.186276\pi$
$53$	34093.9	1.66720	0.833600	+	0.552369i	$0.186276\pi$
$54$	14361.9	0.670236
$55$	0	0
$56$	−32956.3	−1.40433
$57$	−63273.1	−2.57948
$58$	−20590.9	−0.803721
$59$	−29444.2	−1.10121	−0.550604	−	0.834767i	$-0.685603\pi$
$59$	−29444.2	−1.10121	−0.550604	+	0.834767i	$0.685603\pi$
$60$	0	0
$61$	22157.6	0.762427	0.381214	−	0.924487i	$-0.375506\pi$
$61$	22157.6	0.762427	0.381214	+	0.924487i	$0.375506\pi$
$62$	−90566.1	−2.99217
$63$	23050.8	0.731701
$64$	−47732.3	−1.45667
$65$	0	0
$66$	64984.8	1.83633
$67$	61568.0	1.67559	0.837796	−	0.545984i	$-0.183844\pi$
$67$	61568.0	1.67559	0.837796	+	0.545984i	$0.183844\pi$
$68$	75457.7	1.97893
$69$	178.602	0.00451609
$70$	0	0
$71$	38408.1	0.904227	0.452113	−	0.891960i	$-0.350670\pi$
$71$	38408.1	0.904227	0.452113	+	0.891960i	$0.350670\pi$
$72$	−40396.0	−0.918348
$73$	−11799.5	−0.259153	−0.129577	−	0.991569i	$-0.541362\pi$
$73$	−11799.5	−0.259153	−0.129577	+	0.991569i	$0.541362\pi$
$74$	130043.	2.76063
$75$	0	0
$76$	−179175.	−3.55830
$77$	−46481.5	−0.893415
$78$	−124818.	−2.32295
$79$	−10887.0	−0.196264	−0.0981318	−	0.995173i	$-0.531287\pi$
$79$	−10887.0	−0.196264	−0.0981318	+	0.995173i	$0.531287\pi$
$80$	0	0
$81$	−71640.6	−1.21324
$82$	12400.5	0.203659
$83$	−39381.8	−0.627480	−0.313740	−	0.949509i	$-0.601582\pi$
$83$	−39381.8	−0.627480	−0.313740	+	0.949509i	$0.601582\pi$
$84$	159638.	2.46853
$85$	0	0
$86$	17484.1	0.254915
$87$	44150.8	0.625375
$88$	81458.0	1.12131
$89$	108231.	1.44836	0.724180	−	0.689611i	$-0.242219\pi$
$89$	108231.	1.44836	0.724180	+	0.689611i	$0.242219\pi$
$90$	0	0
$91$	89278.1	1.13016
$92$	505.758	0.00622978
$93$	194191.	2.32821
$94$	62993.2	0.735317
$95$	0	0
$96$	72485.9	0.802741
$97$	−20691.1	−0.223282	−0.111641	−	0.993749i	$-0.535611\pi$
$97$	−20691.1	−0.223282	−0.111641	+	0.993749i	$0.535611\pi$
$98$	−18897.7	−0.198767
$99$	−56974.5	−0.584241
$100$	0	0
$101$	111591.	1.08850	0.544249	−	0.838924i	$-0.316815\pi$
$101$	111591.	1.08850	0.544249	+	0.838924i	$0.316815\pi$
$102$	−251972.	−2.39801
$103$	183497.	1.70426	0.852128	−	0.523334i	$-0.175312\pi$
$103$	183497.	1.70426	0.852128	+	0.523334i	$0.175312\pi$
$104$	−156458.	−1.41845
$105$	0	0
$106$	−322391.	−2.78688
$107$	208844.	1.76345	0.881725	−	0.471764i	$-0.156383\pi$
$107$	208844.	1.76345	0.881725	+	0.471764i	$0.156383\pi$
$108$	−87203.2	−0.719404
$109$	12414.4	0.100083	0.0500415	−	0.998747i	$-0.484065\pi$
$109$	12414.4	0.100083	0.0500415	+	0.998747i	$0.484065\pi$
$110$	0	0
$111$	−278838.	−2.14805
$112$	59681.0	0.449563
$113$	108995.	0.802989	0.401494	−	0.915861i	$-0.368491\pi$
$113$	108995.	0.802989	0.401494	+	0.915861i	$0.368491\pi$
$114$	598308.	4.31184
$115$	0	0
$116$	125025.	0.862682
$117$	109432.	0.739062
$118$	278423.	1.84077
$119$	180227.	1.16668
$120$	0	0
$121$	−46162.9	−0.286635
$122$	−209521.	−1.27447
$123$	−26589.0	−0.158467
$124$	549903.	3.21168
$125$	0	0
$126$	−217967.	−1.22311
$127$	−19705.7	−0.108413	−0.0542067	−	0.998530i	$-0.517263\pi$
$127$	−19705.7	−0.108413	−0.0542067	+	0.998530i	$0.517263\pi$
$128$	336952.	1.81779
$129$	−37489.1	−0.198350
$130$	0	0
$131$	223041.	1.13555	0.567776	−	0.823183i	$-0.307804\pi$
$131$	223041.	1.13555	0.567776	+	0.823183i	$0.307804\pi$
$132$	−394577.	−1.97105
$133$	−427950.	−2.09780
$134$	−582184.	−2.80091
$135$	0	0
$136$	−315845.	−1.46429
$137$	37278.8	0.169691	0.0848457	−	0.996394i	$-0.472960\pi$
$137$	37278.8	0.169691	0.0848457	+	0.996394i	$0.472960\pi$
$138$	−1688.85	−0.00754906
$139$	377640.	1.65783	0.828917	−	0.559371i	$-0.188957\pi$
$139$	377640.	1.65783	0.828917	+	0.559371i	$0.188957\pi$
$140$	0	0
$141$	−135070.	−0.572150
$142$	−363186.	−1.51150
$143$	−220668.	−0.902402
$144$	73153.7	0.293988
$145$	0	0
$146$	111576.	0.433199
$147$	40520.3	0.154660
$148$	−789602.	−2.96315
$149$	390055.	1.43933	0.719665	−	0.694321i	$-0.244295\pi$
$149$	390055.	1.43933	0.719665	+	0.694321i	$0.244295\pi$
$150$	0	0
$151$	153419.	0.547567	0.273783	−	0.961791i	$-0.411725\pi$
$151$	153419.	0.547567	0.273783	+	0.961791i	$0.411725\pi$
$152$	749975.	2.63292
$153$	220913.	0.762943
$154$	439527.	1.49343
$155$	0	0
$156$	757874.	2.49336
$157$	−22983.1	−0.0744147	−0.0372074	−	0.999308i	$-0.511846\pi$
$157$	−22983.1	−0.0744147	−0.0372074	+	0.999308i	$0.511846\pi$
$158$	102947.	0.328073
$159$	691267.	2.16847
$160$	0	0
$161$	1207.98	0.00367278
$162$	677430.	2.02804
$163$	−393509.	−1.16007	−0.580037	−	0.814590i	$-0.696962\pi$
$163$	−393509.	−1.16007	−0.580037	+	0.814590i	$0.696962\pi$
$164$	−75293.8	−0.218600
$165$	0	0
$166$	372392.	1.04889
$167$	−492293.	−1.36594	−0.682971	−	0.730445i	$-0.739313\pi$
$167$	−492293.	−1.36594	−0.682971	+	0.730445i	$0.739313\pi$
$168$	−668201.	−1.82656
$169$	52550.3	0.141533
$170$	0	0
$171$	−524558.	−1.37184
$172$	−106160.	−0.273616
$173$	−651868.	−1.65594	−0.827970	−	0.560773i	$-0.810504\pi$
$173$	−651868.	−1.65594	−0.827970	+	0.560773i	$0.810504\pi$
$174$	−417488.	−1.04537
$175$	0	0
$176$	−147513.	−0.358963
$177$	−596991.	−1.43230
$178$	−1.02343e6	−2.42107
$179$	126092.	0.294140	0.147070	−	0.989126i	$-0.453016\pi$
$179$	126092.	0.294140	0.147070	+	0.989126i	$0.453016\pi$
$180$	0	0
$181$	−501981.	−1.13891	−0.569457	−	0.822021i	$-0.692846\pi$
$181$	−501981.	−1.13891	−0.569457	+	0.822021i	$0.692846\pi$
$182$	−844210.	−1.88917
$183$	449254.	0.991663
$184$	−2116.96	−0.00460965
$185$	0	0
$186$	−1.83626e6	−3.89181
$187$	−445467.	−0.931562
$188$	−382485.	−0.789259
$189$	−208281.	−0.424126
$190$	0	0
$191$	104202.	0.206677	0.103339	−	0.994646i	$-0.467047\pi$
$191$	104202.	0.206677	0.103339	+	0.994646i	$0.467047\pi$
$192$	−967789.	−1.89464
$193$	−4364.34	−0.00843385	−0.00421692	−	0.999991i	$-0.501342\pi$
$193$	−4364.34	−0.00843385	−0.00421692	+	0.999991i	$0.501342\pi$
$194$	195654.	0.373237
$195$	0	0
$196$	114744.	0.213348
$197$	167124.	0.306813	0.153406	−	0.988163i	$-0.450976\pi$
$197$	167124.	0.306813	0.153406	+	0.988163i	$0.450976\pi$
$198$	538748.	0.976613
$199$	−771304.	−1.38068	−0.690340	−	0.723485i	$-0.742539\pi$
$199$	−771304.	−1.38068	−0.690340	+	0.723485i	$0.742539\pi$
$200$	0	0
$201$	1.24831e6	2.17938
$202$	−1.05520e6	−1.81952
$203$	298616.	0.508595
$204$	1.52993e6	2.57393
$205$	0	0
$206$	−1.73513e6	−2.84882
$207$	1480.67	0.00240178
$208$	283332.	0.454086
$209$	1.05776e6	1.67503
$210$	0	0
$211$	−672514.	−1.03991	−0.519954	−	0.854194i	$-0.674051\pi$
$211$	−672514.	−1.03991	−0.519954	+	0.854194i	$0.674051\pi$
$212$	1.95750e6	2.99132
$213$	778739.	1.17610
$214$	−1.97482e6	−2.94777
$215$	0	0
$216$	365008.	0.532314
$217$	1.31342e6	1.89345
$218$	−117390.	−0.167298
$219$	−239239.	−0.337072
$220$	0	0
$221$	855619.	1.17842
$222$	2.63668e6	3.59066
$223$	−163098.	−0.219627	−0.109814	−	0.993952i	$-0.535025\pi$
$223$	−163098.	−0.219627	−0.109814	+	0.993952i	$0.535025\pi$
$224$	490261.	0.652840
$225$	0	0
$226$	−1.03065e6	−1.34227
$227$	−85164.2	−0.109696	−0.0548482	−	0.998495i	$-0.517467\pi$
$227$	−85164.2	−0.109696	−0.0548482	+	0.998495i	$0.517467\pi$
$228$	−3.63283e6	−4.62815
$229$	−492569.	−0.620696	−0.310348	−	0.950623i	$-0.600445\pi$
$229$	−492569.	−0.620696	−0.310348	+	0.950623i	$0.600445\pi$
$230$	0	0
$231$	−942429.	−1.16203
$232$	−523318.	−0.638331
$233$	1.49484e6	1.80386	0.901932	−	0.431878i	$-0.142149\pi$
$233$	1.49484e6	1.80386	0.901932	+	0.431878i	$0.142149\pi$
$234$	−1.03479e6	−1.23541
$235$	0	0
$236$	−1.69054e6	−1.97581
$237$	−220738.	−0.255273
$238$	−1.70422e6	−1.95022
$239$	−899223.	−1.01829	−0.509146	−	0.860680i	$-0.670039\pi$
$239$	−899223.	−1.01829	−0.509146	+	0.860680i	$0.670039\pi$
$240$	0	0
$241$	1.53467e6	1.70205	0.851023	−	0.525129i	$-0.175983\pi$
$241$	1.53467e6	1.70205	0.851023	+	0.525129i	$0.175983\pi$
$242$	436514.	0.479137
$243$	−1.08347e6	−1.17706
$244$	1.27218e6	1.36796
$245$	0	0
$246$	251424.	0.264893
$247$	−2.03167e6	−2.11890
$248$	−2.30174e6	−2.37644
$249$	−798480.	−0.816142
$250$	0	0
$251$	469024.	0.469906	0.234953	−	0.972007i	$-0.424506\pi$
$251$	469024.	0.469906	0.234953	+	0.972007i	$0.424506\pi$
$252$	1.32346e6	1.31283
$253$	−2985.76	−0.00293260
$254$	186336.	0.181223
$255$	0	0
$256$	−1.65877e6	−1.58192
$257$	−706623.	−0.667352	−0.333676	−	0.942688i	$-0.608289\pi$
$257$	−706623.	−0.667352	−0.333676	+	0.942688i	$0.608289\pi$
$258$	354496.	0.331560
$259$	−1.88593e6	−1.74693
$260$	0	0
$261$	366027.	0.332592
$262$	−2.10907e6	−1.89818
$263$	2.19751e6	1.95903	0.979517	−	0.201364i	$-0.0645373\pi$
$263$	2.19751e6	1.95903	0.979517	+	0.201364i	$0.0645373\pi$
$264$	1.65159e6	1.45845
$265$	0	0
$266$	4.04667e6	3.50666
$267$	2.19442e6	1.88383
$268$	3.53493e6	3.00638
$269$	334676.	0.281996	0.140998	−	0.990010i	$-0.454969\pi$
$269$	334676.	0.281996	0.140998	+	0.990010i	$0.454969\pi$
$270$	0	0
$271$	−946818.	−0.783147	−0.391573	−	0.920147i	$-0.628069\pi$
$271$	−946818.	−0.783147	−0.391573	+	0.920147i	$0.628069\pi$
$272$	571968.	0.468759
$273$	1.81015e6	1.46997
$274$	−352506.	−0.283655
$275$	0	0
$276$	10254.4	0.00810286
$277$	−1.77489e6	−1.38987	−0.694933	−	0.719074i	$-0.744566\pi$
$277$	−1.77489e6	−1.38987	−0.694933	+	0.719074i	$0.744566\pi$
$278$	−3.57095e6	−2.77122
$279$	1.60992e6	1.23820
$280$	0	0
$281$	−703497.	−0.531491	−0.265746	−	0.964043i	$-0.585618\pi$
$281$	−703497.	−0.531491	−0.265746	+	0.964043i	$0.585618\pi$
$282$	1.27721e6	0.956401
$283$	1.36829e6	1.01558	0.507788	−	0.861482i	$-0.330463\pi$
$283$	1.36829e6	1.01558	0.507788	+	0.861482i	$0.330463\pi$
$284$	2.20520e6	1.62238
$285$	0	0
$286$	2.08663e6	1.50845
$287$	−179836.	−0.128876
$288$	600935.	0.426920
$289$	307396.	0.216498
$290$	0	0
$291$	−419520.	−0.290415
$292$	−677469.	−0.464978
$293$	1.79442e6	1.22111	0.610557	−	0.791972i	$-0.290946\pi$
$293$	1.79442e6	1.22111	0.610557	+	0.791972i	$0.290946\pi$
$294$	−383158.	−0.258529
$295$	0	0
$296$	3.30506e6	2.19255
$297$	514807.	0.338652
$298$	−3.68834e6	−2.40597
$299$	5734.82	0.00370972
$300$	0	0
$301$	−253559.	−0.161311
$302$	−1.45072e6	−0.915308
$303$	2.26256e6	1.41577
$304$	−1.35814e6	−0.842869
$305$	0	0
$306$	−2.08894e6	−1.27533
$307$	352848.	0.213669	0.106834	−	0.994277i	$-0.465929\pi$
$307$	352848.	0.213669	0.106834	+	0.994277i	$0.465929\pi$
$308$	−2.66874e6	−1.60298
$309$	3.72046e6	2.21667
$310$	0	0
$311$	−449548.	−0.263557	−0.131779	−	0.991279i	$-0.542069\pi$
$311$	−449548.	−0.263557	−0.131779	+	0.991279i	$0.542069\pi$
$312$	−3.17225e6	−1.84493
$313$	2.39703e6	1.38297	0.691486	−	0.722390i	$-0.256957\pi$
$313$	2.39703e6	1.38297	0.691486	+	0.722390i	$0.256957\pi$
$314$	217327.	0.124391
$315$	0	0
$316$	−625077.	−0.352140
$317$	−817400.	−0.456864	−0.228432	−	0.973560i	$-0.573360\pi$
$317$	−817400.	−0.456864	−0.228432	+	0.973560i	$0.573360\pi$
$318$	−6.53659e6	−3.62479
$319$	−738087.	−0.406098
$320$	0	0
$321$	4.23439e6	2.29366
$322$	−11422.6	−0.00613939
$323$	−4.10137e6	−2.18737
$324$	−4.11325e6	−2.17682
$325$	0	0
$326$	3.72100e6	1.93917
$327$	251707.	0.130175
$328$	315159.	0.161750
$329$	−913548.	−0.465309
$330$	0	0
$331$	598983.	0.300500	0.150250	−	0.988648i	$-0.451992\pi$
$331$	598983.	0.300500	0.150250	+	0.988648i	$0.451992\pi$
$332$	−2.26111e6	−1.12584
$333$	−2.31167e6	−1.14239
$334$	4.65510e6	2.28330
$335$	0	0
$336$	1.21005e6	0.584731
$337$	1.81953e6	0.872737	0.436368	−	0.899768i	$-0.356264\pi$
$337$	1.81953e6	0.872737	0.436368	+	0.899768i	$0.356264\pi$
$338$	−496913.	−0.236586
$339$	2.20991e6	1.04442
$340$	0	0
$341$	−3.24637e6	−1.51186
$342$	4.96019e6	2.29315
$343$	−2.03074e6	−0.932005
$344$	444358.	0.202459
$345$	0	0
$346$	6.16403e6	2.76806
$347$	1.44735e6	0.645282	0.322641	−	0.946521i	$-0.395429\pi$
$347$	1.44735e6	0.645282	0.322641	+	0.946521i	$0.395429\pi$
$348$	2.53492e6	1.12206
$349$	2.58244e6	1.13492	0.567462	−	0.823399i	$-0.307925\pi$
$349$	2.58244e6	1.13492	0.567462	+	0.823399i	$0.307925\pi$
$350$	0	0
$351$	−988802.	−0.428392
$352$	−1.21178e6	−0.521273
$353$	−84106.4	−0.0359246	−0.0179623	−	0.999839i	$-0.505718\pi$
$353$	−84106.4	−0.0359246	−0.0179623	+	0.999839i	$0.505718\pi$
$354$	5.64512e6	2.39423
$355$	0	0
$356$	6.21408e6	2.59867
$357$	3.65417e6	1.51746
$358$	−1.19232e6	−0.491682
$359$	−640667.	−0.262359	−0.131180	−	0.991359i	$-0.541876\pi$
$359$	−640667.	−0.262359	−0.131180	+	0.991359i	$0.541876\pi$
$360$	0	0
$361$	7.26261e6	2.93308
$362$	4.74671e6	1.90380
$363$	−935969.	−0.372816
$364$	5.12591e6	2.02776
$365$	0	0
$366$	−4.24812e6	−1.65766
$367$	−2.41020e6	−0.934089	−0.467045	−	0.884234i	$-0.654681\pi$
$367$	−2.41020e6	−0.934089	−0.467045	+	0.884234i	$0.654681\pi$
$368$	3833.63	0.00147568
$369$	−220433.	−0.0842773
$370$	0	0
$371$	4.67541e6	1.76354
$372$	1.11495e7	4.17732
$373$	1.62915e6	0.606304	0.303152	−	0.952942i	$-0.401961\pi$
$373$	1.62915e6	0.606304	0.303152	+	0.952942i	$0.401961\pi$
$374$	4.21232e6	1.55719
$375$	0	0
$376$	1.60098e6	0.584003
$377$	1.41766e6	0.513712
$378$	1.96949e6	0.708965
$379$	−3.55332e6	−1.27068	−0.635341	−	0.772232i	$-0.719140\pi$
$379$	−3.55332e6	−1.27068	−0.635341	+	0.772232i	$0.719140\pi$
$380$	0	0
$381$	−399541.	−0.141010
$382$	−985328.	−0.345480
$383$	2.15175e6	0.749541	0.374770	−	0.927118i	$-0.377722\pi$
$383$	2.15175e6	0.749541	0.374770	+	0.927118i	$0.377722\pi$
$384$	6.83182e6	2.36433
$385$	0	0
$386$	41269.0	0.0140979
$387$	−310799.	−0.105488
$388$	−1.18798e6	−0.400617
$389$	2.70507e6	0.906368	0.453184	−	0.891417i	$-0.350288\pi$
$389$	2.70507e6	0.906368	0.453184	+	0.891417i	$0.350288\pi$
$390$	0	0
$391$	11577.0	0.00382960
$392$	−480285.	−0.157864
$393$	4.52225e6	1.47697
$394$	−1.58032e6	−0.512866
$395$	0	0
$396$	−3.27119e6	−1.04826
$397$	−391351.	−0.124621	−0.0623103	−	0.998057i	$-0.519847\pi$
$397$	−391351.	−0.124621	−0.0623103	+	0.998057i	$0.519847\pi$
$398$	7.29341e6	2.30793
$399$	−8.67684e6	−2.72853
$400$	0	0
$401$	1.98245e6	0.615660	0.307830	−	0.951441i	$-0.400397\pi$
$401$	1.98245e6	0.615660	0.307830	+	0.951441i	$0.400397\pi$
$402$	−1.18040e7	−3.64304
$403$	6.23538e6	1.91250
$404$	6.40703e6	1.95300
$405$	0	0
$406$	−2.82369e6	−0.850164
$407$	4.66144e6	1.39487
$408$	−6.40387e6	−1.90455
$409$	3.79823e6	1.12272	0.561362	−	0.827570i	$-0.310278\pi$
$409$	3.79823e6	1.12272	0.561362	+	0.827570i	$0.310278\pi$
$410$	0	0
$411$	755840.	0.220712
$412$	1.05355e7	3.05781
$413$	−4.03777e6	−1.16484
$414$	−14001.2	−0.00401480
$415$	0	0
$416$	2.32749e6	0.659408
$417$	7.65679e6	2.15629
$418$	−1.00022e7	−2.79997
$419$	707118.	0.196769	0.0983845	−	0.995148i	$-0.468633\pi$
$419$	707118.	0.196769	0.0983845	+	0.995148i	$0.468633\pi$
$420$	0	0
$421$	5.72052e6	1.57301	0.786503	−	0.617587i	$-0.211890\pi$
$421$	5.72052e6	1.57301	0.786503	+	0.617587i	$0.211890\pi$
$422$	6.35926e6	1.73830
$423$	−1.11978e6	−0.304285
$424$	−8.19357e6	−2.21339
$425$	0	0
$426$	−7.36372e6	−1.96595
$427$	3.03854e6	0.806484
$428$	1.19908e7	3.16402
$429$	−4.47413e6	−1.17372
$430$	0	0
$431$	−3.65226e6	−0.947040	−0.473520	−	0.880783i	$-0.657017\pi$
$431$	−3.65226e6	−0.947040	−0.473520	+	0.880783i	$0.657017\pi$
$432$	−660998.	−0.170408
$433$	−1.19470e6	−0.306224	−0.153112	−	0.988209i	$-0.548930\pi$
$433$	−1.19470e6	−0.306224	−0.153112	+	0.988209i	$0.548930\pi$
$434$	−1.24196e7	−3.16507
$435$	0	0
$436$	712775.	0.179571
$437$	−27489.5	−0.00688595
$438$	2.26223e6	0.563446
$439$	−3.87138e6	−0.958747	−0.479373	−	0.877611i	$-0.659136\pi$
$439$	−3.87138e6	−0.958747	−0.479373	+	0.877611i	$0.659136\pi$
$440$	0	0
$441$	335928.	0.0822526
$442$	−8.09070e6	−1.96984
$443$	2.72249e6	0.659107	0.329554	−	0.944137i	$-0.393102\pi$
$443$	2.72249e6	0.659107	0.329554	+	0.944137i	$0.393102\pi$
$444$	−1.60095e7	−3.85407
$445$	0	0
$446$	1.54225e6	0.367127
$447$	7.90851e6	1.87209
$448$	−6.54567e6	−1.54085
$449$	2.42630e6	0.567975	0.283987	−	0.958828i	$-0.408343\pi$
$449$	2.42630e6	0.567975	0.283987	+	0.958828i	$0.408343\pi$
$450$	0	0
$451$	444499.	0.102903
$452$	6.25794e6	1.44074
$453$	3.11063e6	0.712201
$454$	805308.	0.183368
$455$	0	0
$456$	1.52060e7	3.42455
$457$	−1.13756e6	−0.254790	−0.127395	−	0.991852i	$-0.540662\pi$
$457$	−1.13756e6	−0.254790	−0.127395	+	0.991852i	$0.540662\pi$
$458$	4.65771e6	1.03755
$459$	−1.99611e6	−0.442235
$460$	0	0
$461$	−1.03645e6	−0.227142	−0.113571	−	0.993530i	$-0.536229\pi$
$461$	−1.03645e6	−0.227142	−0.113571	+	0.993530i	$0.536229\pi$
$462$	8.91156e6	1.94245
$463$	−2.71947e6	−0.589564	−0.294782	−	0.955564i	$-0.595247\pi$
$463$	−2.71947e6	−0.589564	−0.294782	+	0.955564i	$0.595247\pi$
$464$	947684.	0.204347
$465$	0	0
$466$	−1.41351e7	−3.01532
$467$	317120.	0.0672871	0.0336435	−	0.999434i	$-0.489289\pi$
$467$	317120.	0.0672871	0.0336435	+	0.999434i	$0.489289\pi$
$468$	6.28306e6	1.32604
$469$	8.44301e6	1.77242
$470$	0	0
$471$	−465990.	−0.0967886
$472$	7.07612e6	1.46198
$473$	626721.	0.128802
$474$	2.08728e6	0.426713
$475$	0	0
$476$	1.03478e7	2.09329
$477$	5.73086e6	1.15325
$478$	8.50301e6	1.70217
$479$	2.05803e6	0.409838	0.204919	−	0.978779i	$-0.434307\pi$
$479$	2.05803e6	0.409838	0.204919	+	0.978779i	$0.434307\pi$
$480$	0	0
$481$	−8.95335e6	−1.76450
$482$	−1.45117e7	−2.84513
$483$	24492.2	0.00477705
$484$	−2.65044e6	−0.514286
$485$	0	0
$486$	1.02452e7	1.96757
$487$	−4.36350e6	−0.833706	−0.416853	−	0.908974i	$-0.636867\pi$
$487$	−4.36350e6	−0.833706	−0.416853	+	0.908974i	$0.636867\pi$
$488$	−5.32499e6	−1.01221
$489$	−7.97854e6	−1.50887
$490$	0	0
$491$	1.02643e6	0.192144	0.0960719	−	0.995374i	$-0.469372\pi$
$491$	1.02643e6	0.192144	0.0960719	+	0.995374i	$0.469372\pi$
$492$	−1.52661e6	−0.284325
$493$	2.86186e6	0.530311
$494$	1.92114e7	3.54194
$495$	0	0
$496$	4.16825e6	0.760764
$497$	5.26703e6	0.956477
$498$	7.55039e6	1.36426
$499$	7.09662e6	1.27585	0.637926	−	0.770098i	$-0.279793\pi$
$499$	7.09662e6	1.27585	0.637926	+	0.770098i	$0.279793\pi$
$500$	0	0
$501$	−9.98142e6	−1.77663
$502$	−4.43507e6	−0.785491
$503$	5.90960e6	1.04145	0.520725	−	0.853725i	$-0.325662\pi$
$503$	5.90960e6	1.04145	0.520725	+	0.853725i	$0.325662\pi$
$504$	−5.53963e6	−0.971415
$505$	0	0
$506$	28233.2	0.00490212
$507$	1.06548e6	0.184087
$508$	−1.13141e6	−0.194518
$509$	9.84216e6	1.68382	0.841910	−	0.539617i	$-0.181431\pi$
$509$	9.84216e6	1.68382	0.841910	+	0.539617i	$0.181431\pi$
$510$	0	0
$511$	−1.61810e6	−0.274128
$512$	4.90276e6	0.826544
$513$	4.73977e6	0.795177
$514$	6.68179e6	1.11554
$515$	0	0
$516$	−2.15244e6	−0.355883
$517$	2.25801e6	0.371535
$518$	1.78332e7	2.92016
$519$	−1.32169e7	−2.15382
$520$	0	0
$521$	−6.90885e6	−1.11509	−0.557547	−	0.830145i	$-0.688257\pi$
$521$	−6.90885e6	−1.11509	−0.557547	+	0.830145i	$0.688257\pi$
$522$	−3.46113e6	−0.555958
$523$	−5.78387e6	−0.924623	−0.462311	−	0.886718i	$-0.652980\pi$
$523$	−5.78387e6	−0.924623	−0.462311	+	0.886718i	$0.652980\pi$
$524$	1.28059e7	2.03743
$525$	0	0
$526$	−2.07796e7	−3.27470
$527$	1.25875e7	1.97429
$528$	−2.99089e6	−0.466890
$529$	−6.43627e6	−0.999988
$530$	0	0
$531$	−4.94928e6	−0.761738
$532$	−2.45708e7	−3.76391
$533$	−853761.	−0.130172
$534$	−2.07503e7	−3.14900
$535$	0	0
$536$	−1.47962e7	−2.22453
$537$	2.55655e6	0.382577
$538$	−3.16468e6	−0.471383
$539$	−677394.	−0.100431
$540$	0	0
$541$	−2.94760e6	−0.432987	−0.216493	−	0.976284i	$-0.569462\pi$
$541$	−2.94760e6	−0.432987	−0.216493	+	0.976284i	$0.569462\pi$
$542$	8.95307e6	1.30910
$543$	−1.01778e7	−1.48134
$544$	4.69854e6	0.680715
$545$	0	0
$546$	−1.71167e7	−2.45718
$547$	−1.05903e7	−1.51336	−0.756678	−	0.653787i	$-0.773179\pi$
$547$	−1.05903e7	−1.51336	−0.756678	+	0.653787i	$0.773179\pi$
$548$	2.14036e6	0.304464
$549$	3.72448e6	0.527394
$550$	0	0
$551$	−6.79549e6	−0.953546
$552$	−42922.2	−0.00599562
$553$	−1.49297e6	−0.207605
$554$	1.67833e7	2.32329
$555$	0	0
$556$	2.16822e7	2.97452
$557$	−6.30012e6	−0.860420	−0.430210	−	0.902729i	$-0.641561\pi$
$557$	−6.30012e6	−0.860420	−0.430210	+	0.902729i	$0.641561\pi$
$558$	−1.52233e7	−2.06977
$559$	−1.20376e6	−0.162933
$560$	0	0
$561$	−9.03201e6	−1.21165
$562$	6.65223e6	0.888437
$563$	−3.79492e6	−0.504582	−0.252291	−	0.967651i	$-0.581184\pi$
$563$	−3.79492e6	−0.504582	−0.252291	+	0.967651i	$0.581184\pi$
$564$	−7.75502e6	−1.02656
$565$	0	0
$566$	−1.29385e7	−1.69763
$567$	−9.82430e6	−1.28335
$568$	−9.23037e6	−1.20046
$569$	3.15439e6	0.408446	0.204223	−	0.978924i	$-0.434533\pi$
$569$	3.15439e6	0.408446	0.204223	+	0.978924i	$0.434533\pi$
$570$	0	0
$571$	1.09791e7	1.40921	0.704606	−	0.709599i	$-0.251124\pi$
$571$	1.09791e7	1.40921	0.704606	+	0.709599i	$0.251124\pi$
$572$	−1.26697e7	−1.61911
$573$	2.11273e6	0.268818
$574$	1.70052e6	0.215428
$575$	0	0
$576$	−8.02333e6	−1.00762
$577$	2.01076e6	0.251432	0.125716	−	0.992066i	$-0.459877\pi$
$577$	2.01076e6	0.251432	0.125716	+	0.992066i	$0.459877\pi$
$578$	−2.90672e6	−0.361896
$579$	−88488.7	−0.0109696
$580$	0	0
$581$	−5.40055e6	−0.663739
$582$	3.96696e6	0.485456
$583$	−1.15562e7	−1.40813
$584$	2.83570e6	0.344055
$585$	0	0
$586$	−1.69680e7	−2.04120
$587$	5.31174e6	0.636270	0.318135	−	0.948045i	$-0.396944\pi$
$587$	5.31174e6	0.636270	0.318135	+	0.948045i	$0.396944\pi$
$588$	2.32647e6	0.277495
$589$	−2.98890e7	−3.54995
$590$	0	0
$591$	3.38850e6	0.399061
$592$	−5.98517e6	−0.701895
$593$	−1.41135e7	−1.64815	−0.824077	−	0.566477i	$-0.808306\pi$
$593$	−1.41135e7	−1.64815	−0.824077	+	0.566477i	$0.808306\pi$
$594$	−4.86799e6	−0.566087
$595$	0	0
$596$	2.23950e7	2.58248
$597$	−1.56385e7	−1.79580
$598$	−54228.2	−0.00620115
$599$	−4.97732e6	−0.566799	−0.283399	−	0.959002i	$-0.591462\pi$
$599$	−4.97732e6	−0.566799	−0.283399	+	0.959002i	$0.591462\pi$
$600$	0	0
$601$	1.50383e7	1.69829	0.849144	−	0.528162i	$-0.177119\pi$
$601$	1.50383e7	1.69829	0.849144	+	0.528162i	$0.177119\pi$
$602$	2.39764e6	0.269646
$603$	1.03490e7	1.15906
$604$	8.80856e6	0.982455
$605$	0	0
$606$	−2.13946e7	−2.36659
$607$	2.10206e6	0.231566	0.115783	−	0.993275i	$-0.463062\pi$
$607$	2.10206e6	0.231566	0.115783	+	0.993275i	$0.463062\pi$
$608$	−1.11567e7	−1.22399
$609$	6.05454e6	0.661512
$610$	0	0
$611$	−4.33702e6	−0.469990
$612$	1.26837e7	1.36889
$613$	1.49954e7	1.61179	0.805894	−	0.592059i	$-0.201685\pi$
$613$	1.49954e7	1.61179	0.805894	+	0.592059i	$0.201685\pi$
$614$	−3.33651e6	−0.357167
$615$	0	0
$616$	1.11706e7	1.18611
$617$	−8.48082e6	−0.896860	−0.448430	−	0.893818i	$-0.648017\pi$
$617$	−8.48082e6	−0.896860	−0.448430	+	0.893818i	$0.648017\pi$
$618$	−3.51805e7	−3.70536
$619$	1.51512e7	1.58936	0.794679	−	0.607030i	$-0.207639\pi$
$619$	1.51512e7	1.58936	0.794679	+	0.607030i	$0.207639\pi$
$620$	0	0
$621$	−13379.0	−0.00139218
$622$	4.25090e6	0.440560
$623$	1.48420e7	1.53205
$624$	5.74467e6	0.590613
$625$	0	0
$626$	−2.26662e7	−2.31176
$627$	2.14465e7	2.17865
$628$	−1.31957e6	−0.133516
$629$	−1.80743e7	−1.82152
$630$	0	0
$631$	−3.96111e6	−0.396044	−0.198022	−	0.980198i	$-0.563452\pi$
$631$	−3.96111e6	−0.396044	−0.198022	+	0.980198i	$0.563452\pi$
$632$	2.61640e6	0.260562
$633$	−1.36355e7	−1.35257
$634$	7.72930e6	0.763689
$635$	0	0
$636$	3.96891e7	3.89071
$637$	1.30109e6	0.127045
$638$	6.97932e6	0.678830
$639$	6.45604e6	0.625481
$640$	0	0
$641$	−1.61653e7	−1.55395	−0.776976	−	0.629531i	$-0.783247\pi$
$641$	−1.61653e7	−1.55395	−0.776976	+	0.629531i	$0.783247\pi$
$642$	−4.00402e7	−3.83406
$643$	1.40394e7	1.33912	0.669561	−	0.742757i	$-0.266482\pi$
$643$	1.40394e7	1.33912	0.669561	+	0.742757i	$0.266482\pi$
$644$	69356.1	0.00658977
$645$	0	0
$646$	3.87823e7	3.65639
$647$	1.58913e7	1.49244	0.746221	−	0.665698i	$-0.231866\pi$
$647$	1.58913e7	1.49244	0.746221	+	0.665698i	$0.231866\pi$
$648$	1.72169e7	1.61071
$649$	9.98015e6	0.930090
$650$	0	0
$651$	2.66300e7	2.46274
$652$	−2.25933e7	−2.08143
$653$	4.77396e6	0.438123	0.219061	−	0.975711i	$-0.429701\pi$
$653$	4.77396e6	0.438123	0.219061	+	0.975711i	$0.429701\pi$
$654$	−2.38013e6	−0.217599
$655$	0	0
$656$	−570725.	−0.0517807
$657$	−1.98338e6	−0.179264
$658$	8.63846e6	0.777807
$659$	1.81644e7	1.62933	0.814663	−	0.579934i	$-0.196922\pi$
$659$	1.81644e7	1.62933	0.814663	+	0.579934i	$0.196922\pi$
$660$	0	0
$661$	−7.16582e6	−0.637915	−0.318957	−	0.947769i	$-0.603333\pi$
$661$	−7.16582e6	−0.637915	−0.318957	+	0.947769i	$0.603333\pi$
$662$	−5.66395e6	−0.502313
$663$	1.73480e7	1.53273
$664$	9.46436e6	0.833050
$665$	0	0
$666$	2.18590e7	1.90961
$667$	19181.7	0.00166945
$668$	−2.82650e7	−2.45080
$669$	−3.30687e6	−0.285662
$670$	0	0
$671$	−7.51036e6	−0.643953
$672$	9.94021e6	0.849127
$673$	−9.45709e6	−0.804859	−0.402429	−	0.915451i	$-0.631834\pi$
$673$	−9.45709e6	−0.804859	−0.402429	+	0.915451i	$0.631834\pi$
$674$	−1.72053e7	−1.45886
$675$	0	0
$676$	3.01718e6	0.253942
$677$	8.88930e6	0.745412	0.372706	−	0.927950i	$-0.378430\pi$
$677$	8.88930e6	0.745412	0.372706	+	0.927950i	$0.378430\pi$
$678$	−2.08968e7	−1.74584
$679$	−2.83744e6	−0.236185
$680$	0	0
$681$	−1.72673e6	−0.142678
$682$	3.06975e7	2.52722
$683$	−7.16994e6	−0.588117	−0.294059	−	0.955787i	$-0.595006\pi$
$683$	−7.16994e6	−0.588117	−0.294059	+	0.955787i	$0.595006\pi$
$684$	−3.01175e7	−2.46138
$685$	0	0
$686$	1.92025e7	1.55793
$687$	−9.98702e6	−0.807317
$688$	−804693.	−0.0648126
$689$	2.21963e7	1.78128
$690$	0	0
$691$	−4.88452e6	−0.389159	−0.194579	−	0.980887i	$-0.562334\pi$
$691$	−4.88452e6	−0.389159	−0.194579	+	0.980887i	$0.562334\pi$
$692$	−3.74270e7	−2.97112
$693$	−7.81309e6	−0.618002
$694$	−1.36861e7	−1.07865
$695$	0	0
$696$	−1.06105e7	−0.830255
$697$	−1.72350e6	−0.134378
$698$	−2.44194e7	−1.89713
$699$	3.03083e7	2.34622
$700$	0	0
$701$	−2.05830e6	−0.158203	−0.0791014	−	0.996867i	$-0.525205\pi$
$701$	−2.05830e6	−0.158203	−0.0791014	+	0.996867i	$0.525205\pi$
$702$	9.35006e6	0.716097
$703$	4.29174e7	3.27526
$704$	1.61789e7	1.23032
$705$	0	0
$706$	795306.	0.0600513
$707$	1.53029e7	1.15140
$708$	−3.42763e7	−2.56987
$709$	1.10915e7	0.828656	0.414328	−	0.910128i	$-0.364017\pi$
$709$	1.10915e7	0.828656	0.414328	+	0.910128i	$0.364017\pi$
$710$	0	0
$711$	−1.83000e6	−0.135761
$712$	−2.60104e7	−1.92286
$713$	84367.9	0.00621518
$714$	−3.45537e7	−2.53658
$715$	0	0
$716$	7.23956e6	0.527752
$717$	−1.82321e7	−1.32446
$718$	6.05811e6	0.438557
$719$	4.87775e6	0.351882	0.175941	−	0.984401i	$-0.443703\pi$
$719$	4.87775e6	0.351882	0.175941	+	0.984401i	$0.443703\pi$
$720$	0	0
$721$	2.51635e7	1.80274
$722$	−6.86749e7	−4.90292
$723$	3.11159e7	2.21379
$724$	−2.88212e7	−2.04346
$725$	0	0
$726$	8.85047e6	0.623196
$727$	6.80698e6	0.477659	0.238830	−	0.971061i	$-0.423236\pi$
$727$	6.80698e6	0.477659	0.238830	+	0.971061i	$0.423236\pi$
$728$	−2.14556e7	−1.50042
$729$	−4.55899e6	−0.317724
$730$	0	0
$731$	−2.43005e6	−0.168198
$732$	2.57939e7	1.77926
$733$	8.44989e6	0.580887	0.290443	−	0.956892i	$-0.406197\pi$
$733$	8.44989e6	0.580887	0.290443	+	0.956892i	$0.406197\pi$
$734$	2.27908e7	1.56142
$735$	0	0
$736$	31492.1	0.00214293
$737$	−2.08686e7	−1.41522
$738$	2.08440e6	0.140877
$739$	−2.15089e7	−1.44879	−0.724397	−	0.689383i	$-0.757882\pi$
$739$	−2.15089e7	−1.44879	−0.724397	+	0.689383i	$0.757882\pi$
$740$	0	0
$741$	−4.11929e7	−2.75598
$742$	−4.42104e7	−2.94792
$743$	1.40133e7	0.931256	0.465628	−	0.884981i	$-0.345829\pi$
$743$	1.40133e7	0.931256	0.465628	+	0.884981i	$0.345829\pi$
$744$	−4.66686e7	−3.09095
$745$	0	0
$746$	−1.54052e7	−1.01349
$747$	−6.61969e6	−0.434047
$748$	−2.55765e7	−1.67143
$749$	2.86395e7	1.86535
$750$	0	0
$751$	1.39591e7	0.903145	0.451572	−	0.892234i	$-0.350863\pi$
$751$	1.39591e7	0.903145	0.451572	+	0.892234i	$0.350863\pi$
$752$	−2.89923e6	−0.186955
$753$	9.50964e6	0.611190
$754$	−1.34053e7	−0.858716
$755$	0	0
$756$	−1.19584e7	−0.760975
$757$	6.93655e6	0.439950	0.219975	−	0.975505i	$-0.429402\pi$
$757$	6.93655e6	0.439950	0.219975	+	0.975505i	$0.429402\pi$
$758$	3.36000e7	2.12406
$759$	−60537.3	−0.00381433
$760$	0	0
$761$	2.15452e7	1.34862	0.674309	−	0.738450i	$-0.264442\pi$
$761$	2.15452e7	1.34862	0.674309	+	0.738450i	$0.264442\pi$
$762$	3.77804e6	0.235711
$763$	1.70243e6	0.105866
$764$	5.98276e6	0.370824
$765$	0	0
$766$	−2.03469e7	−1.25293
$767$	−1.91691e7	−1.17656
$768$	−3.36321e7	−2.05755
$769$	−171548.	−0.0104609	−0.00523047	−	0.999986i	$-0.501665\pi$
$769$	−171548.	−0.0104609	−0.00523047	+	0.999986i	$0.501665\pi$
$770$	0	0
$771$	−1.43270e7	−0.868001
$772$	−250579.	−0.0151322
$773$	−1.04798e6	−0.0630816	−0.0315408	−	0.999502i	$-0.510041\pi$
$773$	−1.04798e6	−0.0630816	−0.0315408	+	0.999502i	$0.510041\pi$
$774$	2.93890e6	0.176333
$775$	0	0
$776$	4.97255e6	0.296432
$777$	−3.82379e7	−2.27217
$778$	−2.55790e7	−1.51508
$779$	4.09246e6	0.241624
$780$	0	0
$781$	−1.30185e7	−0.763718
$782$	−109471.	−0.00640152
$783$	−3.30732e6	−0.192785
$784$	869755.	0.0505367
$785$	0	0
$786$	−4.27621e7	−2.46890
$787$	1.63532e6	0.0941167	0.0470584	−	0.998892i	$-0.485015\pi$
$787$	1.63532e6	0.0941167	0.0470584	+	0.998892i	$0.485015\pi$
$788$	9.59544e6	0.550490
$789$	4.45553e7	2.54805
$790$	0	0
$791$	1.49468e7	0.849389
$792$	1.36923e7	0.775645
$793$	1.44253e7	0.814597
$794$	3.70059e6	0.208315
$795$	0	0
$796$	−4.42844e7	−2.47724
$797$	1.44879e7	0.807903	0.403952	−	0.914780i	$-0.367636\pi$
$797$	1.44879e7	0.807903	0.403952	+	0.914780i	$0.367636\pi$
$798$	8.20478e7	4.56100
$799$	−8.75521e6	−0.485177
$800$	0	0
$801$	1.81926e7	1.00187
$802$	−1.87459e7	−1.02913
$803$	3.99946e6	0.218883
$804$	7.16720e7	3.91029
$805$	0	0
$806$	−5.89615e7	−3.19691
$807$	6.78567e6	0.366783
$808$	−2.68180e7	−1.44510
$809$	−3.54370e7	−1.90364	−0.951821	−	0.306655i	$-0.900790\pi$
$809$	−3.54370e7	−1.90364	−0.951821	+	0.306655i	$0.900790\pi$
$810$	0	0
$811$	−1.04660e7	−0.558765	−0.279382	−	0.960180i	$-0.590130\pi$
$811$	−1.04660e7	−0.558765	−0.279382	+	0.960180i	$0.590130\pi$
$812$	1.71450e7	0.912532
$813$	−1.91971e7	−1.01861
$814$	−4.40784e7	−2.33166
$815$	0	0
$816$	1.15969e7	0.609698
$817$	5.77016e6	0.302435
$818$	−3.59159e7	−1.87674
$819$	1.50068e7	0.781768
$820$	0	0
$821$	1.41169e7	0.730940	0.365470	−	0.930823i	$-0.380908\pi$
$821$	1.41169e7	0.730940	0.365470	+	0.930823i	$0.380908\pi$
$822$	−7.14719e6	−0.368940
$823$	3.04760e6	0.156841	0.0784203	−	0.996920i	$-0.475012\pi$
$823$	3.04760e6	0.156841	0.0784203	+	0.996920i	$0.475012\pi$
$824$	−4.40985e7	−2.26259
$825$	0	0
$826$	3.81810e7	1.94714
$827$	3.31116e6	0.168351	0.0841755	−	0.996451i	$-0.473174\pi$
$827$	3.31116e6	0.168351	0.0841755	+	0.996451i	$0.473174\pi$
$828$	85013.0	0.00430933
$829$	2.25117e7	1.13768	0.568842	−	0.822447i	$-0.307392\pi$
$829$	2.25117e7	1.13768	0.568842	+	0.822447i	$0.307392\pi$
$830$	0	0
$831$	−3.59866e7	−1.80775
$832$	−3.10752e7	−1.55635
$833$	2.62653e6	0.131150
$834$	−7.24023e7	−3.60443
$835$	0	0
$836$	6.07315e7	3.00537
$837$	−1.45468e7	−0.717717
$838$	−6.68647e6	−0.328917
$839$	−1.82302e7	−0.894101	−0.447050	−	0.894509i	$-0.647526\pi$
$839$	−1.82302e7	−0.894101	−0.447050	+	0.894509i	$0.647526\pi$
$840$	0	0
$841$	−1.57694e7	−0.768820
$842$	−5.40930e7	−2.62942
$843$	−1.42636e7	−0.691292
$844$	−3.86124e7	−1.86582
$845$	0	0
$846$	1.05886e7	0.508640
$847$	−6.33046e6	−0.303198
$848$	1.48378e7	0.708567
$849$	2.77426e7	1.32092
$850$	0	0
$851$	−121143.	−0.00573424
$852$	4.47113e7	2.11018
$853$	2.77877e7	1.30761	0.653807	−	0.756662i	$-0.273171\pi$
$853$	2.77877e7	1.30761	0.653807	+	0.756662i	$0.273171\pi$
$854$	−2.87323e7	−1.34811
$855$	0	0
$856$	−5.01901e7	−2.34118
$857$	−4.19965e7	−1.95326	−0.976632	−	0.214919i	$-0.931051\pi$
$857$	−4.19965e7	−1.95326	−0.976632	+	0.214919i	$0.931051\pi$
$858$	4.23072e7	1.96199
$859$	5.23333e6	0.241989	0.120994	−	0.992653i	$-0.461392\pi$
$859$	5.23333e6	0.241989	0.120994	+	0.992653i	$0.461392\pi$
$860$	0	0
$861$	−3.64623e6	−0.167624
$862$	3.45356e7	1.58306
$863$	2.65684e6	0.121433	0.0607167	−	0.998155i	$-0.480661\pi$
$863$	2.65684e6	0.121433	0.0607167	+	0.998155i	$0.480661\pi$
$864$	−5.42989e6	−0.247461
$865$	0	0
$866$	1.12970e7	0.511882
$867$	6.23257e6	0.281591
$868$	7.54099e7	3.39726
$869$	3.69016e6	0.165766
$870$	0	0
$871$	4.00828e7	1.79024
$872$	−2.98348e6	−0.132871
$873$	−3.47797e6	−0.154451
$874$	259940.	0.0115105
$875$	0	0
$876$	−1.37359e7	−0.604781
$877$	−4.41009e7	−1.93619	−0.968096	−	0.250579i	$-0.919379\pi$
$877$	−4.41009e7	−1.93619	−0.968096	+	0.250579i	$0.919379\pi$
$878$	3.66075e7	1.60263
$879$	3.63826e7	1.58826
$880$	0	0
$881$	2.66333e7	1.15607	0.578036	−	0.816011i	$-0.303819\pi$
$881$	2.66333e7	1.15607	0.578036	+	0.816011i	$0.303819\pi$
$882$	−3.17652e6	−0.137493
$883$	9.47190e6	0.408823	0.204412	−	0.978885i	$-0.434472\pi$
$883$	9.47190e6	0.408823	0.204412	+	0.978885i	$0.434472\pi$
$884$	4.91254e7	2.11434
$885$	0	0
$886$	−2.57437e7	−1.10176
$887$	2.03039e7	0.866505	0.433253	−	0.901273i	$-0.357366\pi$
$887$	2.03039e7	0.866505	0.433253	+	0.901273i	$0.357366\pi$
$888$	6.70112e7	2.85177
$889$	−2.70231e6	−0.114678
$890$	0	0
$891$	2.42827e7	1.02471
$892$	−9.36428e6	−0.394060
$893$	2.07893e7	0.872390
$894$	−7.47825e7	−3.12937
$895$	0	0
$896$	4.62072e7	1.92283
$897$	116276.	0.00482511
$898$	−2.29430e7	−0.949422
$899$	2.08560e7	0.860659
$900$	0	0
$901$	4.48080e7	1.83884
$902$	−4.20317e6	−0.172013
$903$	−5.14100e6	−0.209811
$904$	−2.61940e7	−1.06606
$905$	0	0
$906$	−2.94139e7	−1.19051
$907$	−1.74867e6	−0.0705811	−0.0352906	−	0.999377i	$-0.511236\pi$
$907$	−1.74867e6	−0.0705811	−0.0352906	+	0.999377i	$0.511236\pi$
$908$	−4.88970e6	−0.196819
$909$	1.87574e7	0.752946
$910$	0	0
$911$	5.23623e6	0.209037	0.104518	−	0.994523i	$-0.466670\pi$
$911$	5.23623e6	0.209037	0.104518	+	0.994523i	$0.466670\pi$
$912$	−2.75368e7	−1.09629
$913$	1.33485e7	0.529976
$914$	1.07567e7	0.425904
$915$	0	0
$916$	−2.82809e7	−1.11366
$917$	3.05864e7	1.20117
$918$	1.88751e7	0.739236
$919$	−2.81194e7	−1.09829	−0.549145	−	0.835727i	$-0.685047\pi$
$919$	−2.81194e7	−1.09829	−0.549145	+	0.835727i	$0.685047\pi$
$920$	0	0
$921$	7.15412e6	0.277912
$922$	9.80066e6	0.379689
$923$	2.50049e7	0.966099
$924$	−5.41096e7	−2.08494
$925$	0	0
$926$	2.57151e7	0.985511
$927$	3.08440e7	1.17888
$928$	7.78493e6	0.296746
$929$	−1.80587e6	−0.0686511	−0.0343255	−	0.999411i	$-0.510928\pi$
$929$	−1.80587e6	−0.0686511	−0.0343255	+	0.999411i	$0.510928\pi$
$930$	0	0
$931$	−6.23669e6	−0.235820
$932$	8.58261e7	3.23653
$933$	−9.11475e6	−0.342800
$934$	−2.99867e6	−0.112477
$935$	0	0
$936$	−2.62991e7	−0.981187
$937$	1.09417e7	0.407134	0.203567	−	0.979061i	$-0.434746\pi$
$937$	1.09417e7	0.407134	0.203567	+	0.979061i	$0.434746\pi$
$938$	−7.98367e7	−2.96275
$939$	4.86007e7	1.79878
$940$	0	0
$941$	−2.67877e7	−0.986193	−0.493097	−	0.869975i	$-0.664135\pi$
$941$	−2.67877e7	−0.986193	−0.493097	+	0.869975i	$0.664135\pi$
$942$	4.40638e6	0.161791
$943$	−11551.8	−0.000423030 0
$944$	−1.28142e7	−0.468018
$945$	0	0
$946$	−5.92625e6	−0.215304
$947$	2.47956e7	0.898461	0.449230	−	0.893416i	$-0.351698\pi$
$947$	2.47956e7	0.898461	0.449230	+	0.893416i	$0.351698\pi$
$948$	−1.26737e7	−0.458016
$949$	−7.68186e6	−0.276886
$950$	0	0
$951$	−1.65731e7	−0.594227
$952$	−4.33128e7	−1.54890
$953$	−3.81550e7	−1.36088	−0.680439	−	0.732805i	$-0.738211\pi$
$953$	−3.81550e7	−1.36088	−0.680439	+	0.732805i	$0.738211\pi$
$954$	−5.41907e7	−1.92777
$955$	0	0
$956$	−5.16289e7	−1.82704
$957$	−1.49650e7	−0.528198
$958$	−1.94606e7	−0.685083
$959$	5.11215e6	0.179497
$960$	0	0
$961$	6.31028e7	2.20415
$962$	8.46624e7	2.94953
$963$	3.51047e7	1.21983
$964$	8.81129e7	3.05384
$965$	0	0
$966$	−231597.	−0.00798529
$967$	−250672.	−0.00862064	−0.00431032	−	0.999991i	$-0.501372\pi$
$967$	−250672.	−0.00862064	−0.00431032	+	0.999991i	$0.501372\pi$
$968$	1.10940e7	0.380540
$969$	−8.31567e7	−2.84504
$970$	0	0
$971$	−8.88137e6	−0.302296	−0.151148	−	0.988511i	$-0.548297\pi$
$971$	−8.88137e6	−0.302296	−0.151148	+	0.988511i	$0.548297\pi$
$972$	−6.22072e7	−2.11191
$973$	5.17870e7	1.75363
$974$	4.12611e7	1.39362
$975$	0	0
$976$	9.64310e6	0.324035
$977$	2.71638e7	0.910445	0.455222	−	0.890378i	$-0.349560\pi$
$977$	2.71638e7	0.910445	0.455222	+	0.890378i	$0.349560\pi$
$978$	7.54447e7	2.52221
$979$	−3.66850e7	−1.22330
$980$	0	0
$981$	2.08675e6	0.0692304
$982$	−9.70589e6	−0.321186
$983$	−5.70815e7	−1.88413	−0.942066	−	0.335427i	$-0.891119\pi$
$983$	−5.70815e7	−1.88413	−0.942066	+	0.335427i	$0.891119\pi$
$984$	6.38996e6	0.210383
$985$	0	0
$986$	−2.70616e7	−0.886464
$987$	−1.85225e7	−0.605211
$988$	−1.16648e8	−3.80178
$989$	−16287.5	−0.000529497 0
$990$	0	0
$991$	−3.47776e7	−1.12490	−0.562452	−	0.826830i	$-0.690142\pi$
$991$	−3.47776e7	−1.12490	−0.562452	+	0.826830i	$0.690142\pi$
$992$	3.42409e7	1.10475
$993$	1.21446e7	0.390850
$994$	−4.98048e7	−1.59884
$995$	0	0
$996$	−4.58447e7	−1.46434
$997$	−5.36931e7	−1.71073	−0.855364	−	0.518028i	$-0.826666\pi$
$997$	−5.36931e7	−1.71073	−0.855364	+	0.518028i	$0.826666\pi$
$998$	−6.71053e7	−2.13270
$999$	2.08876e7	0.662179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.i.1.5	✓	37
5.4	even	2		1075.6.a.j.1.33	yes	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1075.6.a.i.1.5	✓	37	1.1	even	1	trivial
1075.6.a.j.1.33	yes	37	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-9.45595 q^{2} +20.2754 q^{3} +57.4150 q^{4} -191.723 q^{6} +137.133 q^{7} -240.323 q^{8} +168.090 q^{9} +O(q^{10})\)	\(q-9.45595 q^{2} +20.2754 q^{3} +57.4150 q^{4} -191.723 q^{6} +137.133 q^{7} -240.323 q^{8} +168.090 q^{9} -338.952 q^{11} +1164.11 q^{12} +651.032 q^{13} -1296.72 q^{14} +435.205 q^{16} +1314.25 q^{17} -1589.45 q^{18} -3120.69 q^{19} +2780.42 q^{21} +3205.11 q^{22} +8.80880 q^{23} -4872.64 q^{24} -6156.13 q^{26} -1518.82 q^{27} +7873.50 q^{28} +2177.56 q^{29} +9577.68 q^{31} +3575.07 q^{32} -6872.37 q^{33} -12427.5 q^{34} +9650.91 q^{36} -13752.5 q^{37} +29509.1 q^{38} +13199.9 q^{39} -1311.40 q^{41} -26291.6 q^{42} -1849.00 q^{43} -19460.9 q^{44} -83.2956 q^{46} -6661.76 q^{47} +8823.93 q^{48} +1998.50 q^{49} +26646.9 q^{51} +37379.0 q^{52} +34093.9 q^{53} +14361.9 q^{54} -32956.3 q^{56} -63273.1 q^{57} -20590.9 q^{58} -29444.2 q^{59} +22157.6 q^{61} -90566.1 q^{62} +23050.8 q^{63} -47732.3 q^{64} +64984.8 q^{66} +61568.0 q^{67} +75457.7 q^{68} +178.602 q^{69} +38408.1 q^{71} -40396.0 q^{72} -11799.5 q^{73} +130043. q^{74} -179175. q^{76} -46481.5 q^{77} -124818. q^{78} -10887.0 q^{79} -71640.6 q^{81} +12400.5 q^{82} -39381.8 q^{83} +159638. q^{84} +17484.1 q^{86} +44150.8 q^{87} +81458.0 q^{88} +108231. q^{89} +89278.1 q^{91} +505.758 q^{92} +194191. q^{93} +62993.2 q^{94} +72485.9 q^{96} -20691.1 q^{97} -18897.7 q^{98} -56974.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 630 q^{4} + 291 q^{6} - 213 q^{8} + 3535 q^{9}+O(q^{10}) \) 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

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Database

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