LMFDB - Embedded newform 575.6.a.l.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,6,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, 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("575.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	575.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:	$92.2206963925$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.13410 q^{2} -27.5349 q^{3} -22.1774 q^{4} -86.2973 q^{6} -98.9863 q^{7} -169.798 q^{8} +515.172 q^{9} +O(q^{10})$	\(q+3.13410 q^{2} -27.5349 q^{3} -22.1774 q^{4} -86.2973 q^{6} -98.9863 q^{7} -169.798 q^{8} +515.172 q^{9} -172.976 q^{11} +610.653 q^{12} -499.623 q^{13} -310.233 q^{14} +177.514 q^{16} -185.294 q^{17} +1614.60 q^{18} -457.866 q^{19} +2725.58 q^{21} -542.125 q^{22} +529.000 q^{23} +4675.36 q^{24} -1565.87 q^{26} -7494.24 q^{27} +2195.26 q^{28} +4657.20 q^{29} +9047.26 q^{31} +5989.87 q^{32} +4762.89 q^{33} -580.731 q^{34} -11425.2 q^{36} -5379.45 q^{37} -1435.00 q^{38} +13757.1 q^{39} +957.099 q^{41} +8542.25 q^{42} +19714.5 q^{43} +3836.16 q^{44} +1657.94 q^{46} +22583.7 q^{47} -4887.83 q^{48} -7008.71 q^{49} +5102.06 q^{51} +11080.3 q^{52} -15608.8 q^{53} -23487.7 q^{54} +16807.6 q^{56} +12607.3 q^{57} +14596.1 q^{58} +506.470 q^{59} +31327.9 q^{61} +28355.1 q^{62} -50995.0 q^{63} +13092.4 q^{64} +14927.4 q^{66} -43689.5 q^{67} +4109.34 q^{68} -14566.0 q^{69} -18754.4 q^{71} -87475.0 q^{72} -39706.2 q^{73} -16859.8 q^{74} +10154.3 q^{76} +17122.3 q^{77} +43116.1 q^{78} -21075.0 q^{79} +81166.6 q^{81} +2999.65 q^{82} -59205.2 q^{83} -60446.3 q^{84} +61787.3 q^{86} -128236. q^{87} +29370.9 q^{88} -74791.3 q^{89} +49455.8 q^{91} -11731.8 q^{92} -249116. q^{93} +70779.5 q^{94} -164931. q^{96} +46828.3 q^{97} -21966.0 q^{98} -89112.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 8 q^{2} - 36 q^{3} + 416 q^{4} - 72 q^{6} - 297 q^{7} - 384 q^{8} + 2135 q^{9}+O(q^{10}) $ 27 * q - 8 * q^2 - 36 * q^3 + 416 * q^4 - 72 * q^6 - 297 * q^7 - 384 * q^8 + 2135 * q^9	\( 27 q - 8 q^{2} - 36 q^{3} + 416 q^{4} - 72 q^{6} - 297 q^{7} - 384 q^{8} + 2135 q^{9} - 190 q^{11} - 4518 q^{12} - 2704 q^{13} + 1994 q^{14} + 8316 q^{16} - 5821 q^{17} - 3240 q^{18} + 1220 q^{19} + 2384 q^{21} - 11826 q^{22} + 14283 q^{23} + 8970 q^{24} - 7572 q^{26} - 15294 q^{27} - 12736 q^{28} - 11479 q^{29} - 16985 q^{31} - 27436 q^{32} - 21686 q^{33} - 13060 q^{34} + 35780 q^{36} - 34585 q^{37} - 16760 q^{38} - 6708 q^{39} - 16921 q^{41} - 40528 q^{42} - 48832 q^{43} - 16954 q^{44} - 4232 q^{46} - 21712 q^{47} - 153792 q^{48} + 76886 q^{49} + 48288 q^{51} - 114550 q^{52} - 47803 q^{53} + 45250 q^{54} + 100898 q^{56} - 92832 q^{57} + 96246 q^{58} - 55053 q^{59} + 66462 q^{61} - 56374 q^{62} - 205431 q^{63} + 131054 q^{64} - 61644 q^{66} - 78037 q^{67} - 190622 q^{68} - 19044 q^{69} - 155407 q^{71} - 124230 q^{72} - 387138 q^{73} - 110330 q^{74} + 21110 q^{76} - 81124 q^{77} - 227044 q^{78} + 194674 q^{79} + 182211 q^{81} - 192838 q^{82} - 206961 q^{83} + 17918 q^{84} + 55990 q^{86} - 97856 q^{87} - 616608 q^{88} + 176894 q^{89} - 113316 q^{91} + 220064 q^{92} - 325304 q^{93} - 144902 q^{94} + 147208 q^{96} - 375272 q^{97} - 683362 q^{98} + 7044 q^{99}+O(q^{100}) \) 27 * q - 8 * q^2 - 36 * q^3 + 416 * q^4 - 72 * q^6 - 297 * q^7 - 384 * q^8 + 2135 * q^9 - 190 * q^11 - 4518 * q^12 - 2704 * q^13 + 1994 * q^14 + 8316 * q^16 - 5821 * q^17 - 3240 * q^18 + 1220 * q^19 + 2384 * q^21 - 11826 * q^22 + 14283 * q^23 + 8970 * q^24 - 7572 * q^26 - 15294 * q^27 - 12736 * q^28 - 11479 * q^29 - 16985 * q^31 - 27436 * q^32 - 21686 * q^33 - 13060 * q^34 + 35780 * q^36 - 34585 * q^37 - 16760 * q^38 - 6708 * q^39 - 16921 * q^41 - 40528 * q^42 - 48832 * q^43 - 16954 * q^44 - 4232 * q^46 - 21712 * q^47 - 153792 * q^48 + 76886 * q^49 + 48288 * q^51 - 114550 * q^52 - 47803 * q^53 + 45250 * q^54 + 100898 * q^56 - 92832 * q^57 + 96246 * q^58 - 55053 * q^59 + 66462 * q^61 - 56374 * q^62 - 205431 * q^63 + 131054 * q^64 - 61644 * q^66 - 78037 * q^67 - 190622 * q^68 - 19044 * q^69 - 155407 * q^71 - 124230 * q^72 - 387138 * q^73 - 110330 * q^74 + 21110 * q^76 - 81124 * q^77 - 227044 * q^78 + 194674 * q^79 + 182211 * q^81 - 192838 * q^82 - 206961 * q^83 + 17918 * q^84 + 55990 * q^86 - 97856 * q^87 - 616608 * q^88 + 176894 * q^89 - 113316 * q^91 + 220064 * q^92 - 325304 * q^93 - 144902 * q^94 + 147208 * q^96 - 375272 * q^97 - 683362 * q^98 + 7044 * 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 $

$2$	3.13410	0.554036	0.277018	−	0.960865i	$-0.410654\pi$
$2$	3.13410	0.554036	0.277018	+	0.960865i	$0.410654\pi$
$3$	−27.5349	−1.76637	−0.883183	−	0.469028i	$-0.844604\pi$
$3$	−27.5349	−1.76637	−0.883183	+	0.469028i	$0.844604\pi$
$4$	−22.1774	−0.693044
$5$	0	0
$5$	0	0
$6$	−86.2973	−0.978631
$7$	−98.9863	−0.763537	−0.381768	−	0.924258i	$-0.624685\pi$
$7$	−98.9863	−0.763537	−0.381768	+	0.924258i	$0.624685\pi$
$8$	−169.798	−0.938008
$9$	515.172	2.12005
$10$	0	0
$11$	−172.976	−0.431027	−0.215514	−	0.976501i	$-0.569143\pi$
$11$	−172.976	−0.431027	−0.215514	+	0.976501i	$0.569143\pi$
$12$	610.653	1.22417
$13$	−499.623	−0.819944	−0.409972	−	0.912098i	$-0.634462\pi$
$13$	−499.623	−0.819944	−0.409972	+	0.912098i	$0.634462\pi$
$14$	−310.233	−0.423027
$15$	0	0
$16$	177.514	0.173353
$17$	−185.294	−0.155503	−0.0777516	−	0.996973i	$-0.524774\pi$
$17$	−185.294	−0.155503	−0.0777516	+	0.996973i	$0.524774\pi$
$18$	1614.60	1.17458
$19$	−457.866	−0.290974	−0.145487	−	0.989360i	$-0.546475\pi$
$19$	−457.866	−0.290974	−0.145487	+	0.989360i	$0.546475\pi$
$20$	0	0
$21$	2725.58	1.34869
$22$	−542.125	−0.238805
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	4675.36	1.65687
$25$	0	0
$26$	−1565.87	−0.454279
$27$	−7494.24	−1.97842
$28$	2195.26	0.529164
$29$	4657.20	1.02832	0.514161	−	0.857693i	$-0.328103\pi$
$29$	4657.20	1.02832	0.514161	+	0.857693i	$0.328103\pi$
$30$	0	0
$31$	9047.26	1.69088	0.845440	−	0.534070i	$-0.179338\pi$
$31$	9047.26	1.69088	0.845440	+	0.534070i	$0.179338\pi$
$32$	5989.87	1.03405
$33$	4762.89	0.761352
$34$	−580.731	−0.0861545
$35$	0	0
$36$	−11425.2	−1.46929
$37$	−5379.45	−0.646002	−0.323001	−	0.946399i	$-0.604692\pi$
$37$	−5379.45	−0.646002	−0.323001	+	0.946399i	$0.604692\pi$
$38$	−1435.00	−0.161210
$39$	13757.1	1.44832
$40$	0	0
$41$	957.099	0.0889195	0.0444598	−	0.999011i	$-0.485843\pi$
$41$	957.099	0.0889195	0.0444598	+	0.999011i	$0.485843\pi$
$42$	8542.25	0.747221
$43$	19714.5	1.62598	0.812989	−	0.582279i	$-0.197839\pi$
$43$	19714.5	1.62598	0.812989	+	0.582279i	$0.197839\pi$
$44$	3836.16	0.298721
$45$	0	0
$46$	1657.94	0.115525
$47$	22583.7	1.49125	0.745623	−	0.666368i	$-0.232152\pi$
$47$	22583.7	1.49125	0.745623	+	0.666368i	$0.232152\pi$
$48$	−4887.83	−0.306206
$49$	−7008.71	−0.417012
$50$	0	0
$51$	5102.06	0.274676
$52$	11080.3	0.568257
$53$	−15608.8	−0.763271	−0.381635	−	0.924313i	$-0.624639\pi$
$53$	−15608.8	−0.763271	−0.381635	+	0.924313i	$0.624639\pi$
$54$	−23487.7	−1.09612
$55$	0	0
$56$	16807.6	0.716203
$57$	12607.3	0.513967
$58$	14596.1	0.569728
$59$	506.470	0.0189419	0.00947096	−	0.999955i	$-0.496985\pi$
$59$	506.470	0.0189419	0.00947096	+	0.999955i	$0.496985\pi$
$60$	0	0
$61$	31327.9	1.07797	0.538985	−	0.842316i	$-0.318808\pi$
$61$	31327.9	1.07797	0.538985	+	0.842316i	$0.318808\pi$
$62$	28355.1	0.936809
$63$	−50995.0	−1.61874
$64$	13092.4	0.399549
$65$	0	0
$66$	14927.4	0.421817
$67$	−43689.5	−1.18902	−0.594511	−	0.804087i	$-0.702654\pi$
$67$	−43689.5	−1.18902	−0.594511	+	0.804087i	$0.702654\pi$
$68$	4109.34	0.107771
$69$	−14566.0	−0.368313
$70$	0	0
$71$	−18754.4	−0.441526	−0.220763	−	0.975327i	$-0.570855\pi$
$71$	−18754.4	−0.441526	−0.220763	+	0.975327i	$0.570855\pi$
$72$	−87475.0	−1.98862
$73$	−39706.2	−0.872070	−0.436035	−	0.899930i	$-0.643618\pi$
$73$	−39706.2	−0.872070	−0.436035	+	0.899930i	$0.643618\pi$
$74$	−16859.8	−0.357909
$75$	0	0
$76$	10154.3	0.201658
$77$	17122.3	0.329105
$78$	43116.1	0.802423
$79$	−21075.0	−0.379927	−0.189964	−	0.981791i	$-0.560837\pi$
$79$	−21075.0	−0.379927	−0.189964	+	0.981791i	$0.560837\pi$
$80$	0	0
$81$	81166.6	1.37456
$82$	2999.65	0.0492646
$83$	−59205.2	−0.943333	−0.471666	−	0.881777i	$-0.656347\pi$
$83$	−59205.2	−0.943333	−0.471666	+	0.881777i	$0.656347\pi$
$84$	−60446.3	−0.934698
$85$	0	0
$86$	61787.3	0.900851
$87$	−128236.	−1.81639
$88$	29370.9	0.404307
$89$	−74791.3	−1.00087	−0.500433	−	0.865775i	$-0.666826\pi$
$89$	−74791.3	−1.00087	−0.500433	+	0.865775i	$0.666826\pi$
$90$	0	0
$91$	49455.8	0.626057
$92$	−11731.8	−0.144510
$93$	−249116.	−2.98672
$94$	70779.5	0.826205
$95$	0	0
$96$	−164931.	−1.82651
$97$	46828.3	0.505335	0.252668	−	0.967553i	$-0.418692\pi$
$97$	46828.3	0.505335	0.252668	+	0.967553i	$0.418692\pi$
$98$	−21966.0	−0.231040
$99$	−89112.5	−0.913799
$100$	0	0
$101$	124353.	1.21297	0.606487	−	0.795094i	$-0.292578\pi$
$101$	124353.	1.21297	0.606487	+	0.795094i	$0.292578\pi$
$102$	15990.4	0.152180
$103$	−165230.	−1.53460	−0.767299	−	0.641290i	$-0.778400\pi$
$103$	−165230.	−1.53460	−0.767299	+	0.641290i	$0.778400\pi$
$104$	84834.8	0.769114
$105$	0	0
$106$	−48919.5	−0.422880
$107$	136352.	1.15133	0.575667	−	0.817684i	$-0.304743\pi$
$107$	136352.	1.15133	0.575667	+	0.817684i	$0.304743\pi$
$108$	166203.	1.37113
$109$	223278.	1.80003	0.900016	−	0.435858i	$-0.143555\pi$
$109$	223278.	1.80003	0.900016	+	0.435858i	$0.143555\pi$
$110$	0	0
$111$	148123.	1.14108
$112$	−17571.4	−0.132362
$113$	106206.	0.782443	0.391222	−	0.920297i	$-0.372053\pi$
$113$	106206.	0.782443	0.391222	+	0.920297i	$0.372053\pi$
$114$	39512.6	0.284756
$115$	0	0
$116$	−103284.	−0.712673
$117$	−257392.	−1.73832
$118$	1587.33	0.0104945
$119$	18341.6	0.118732
$120$	0	0
$121$	−131130.	−0.814216
$122$	98184.8	0.597234
$123$	−26353.6	−0.157064
$124$	−200645.	−1.17185
$125$	0	0
$126$	−159824.	−0.896839
$127$	295410.	1.62524	0.812618	−	0.582797i	$-0.198042\pi$
$127$	295410.	1.62524	0.812618	+	0.582797i	$0.198042\pi$
$128$	−150643.	−0.812687
$129$	−542838.	−2.87207
$130$	0	0
$131$	−75570.6	−0.384746	−0.192373	−	0.981322i	$-0.561618\pi$
$131$	−75570.6	−0.384746	−0.192373	+	0.981322i	$0.561618\pi$
$132$	−105628.	−0.527650
$133$	45322.5	0.222170
$134$	−136927.	−0.658762
$135$	0	0
$136$	31462.5	0.145863
$137$	−400299.	−1.82215	−0.911073	−	0.412245i	$-0.864745\pi$
$137$	−400299.	−1.82215	−0.911073	+	0.412245i	$0.864745\pi$
$138$	−45651.3	−0.204059
$139$	66371.9	0.291371	0.145686	−	0.989331i	$-0.453461\pi$
$139$	66371.9	0.291371	0.145686	+	0.989331i	$0.453461\pi$
$140$	0	0
$141$	−621839.	−2.63409
$142$	−58778.1	−0.244622
$143$	86422.9	0.353418
$144$	91450.2	0.367518
$145$	0	0
$146$	−124443.	−0.483158
$147$	192984.	0.736595
$148$	119302.	0.447708
$149$	181608.	0.670147	0.335073	−	0.942192i	$-0.391239\pi$
$149$	181608.	0.670147	0.335073	+	0.942192i	$0.391239\pi$
$150$	0	0
$151$	−179415.	−0.640350	−0.320175	−	0.947358i	$-0.603742\pi$
$151$	−179415.	−0.640350	−0.320175	+	0.947358i	$0.603742\pi$
$152$	77744.5	0.272936
$153$	−95458.4	−0.329675
$154$	53663.0	0.182336
$155$	0	0
$156$	−305096.	−1.00375
$157$	−574225.	−1.85923	−0.929615	−	0.368531i	$-0.879861\pi$
$157$	−574225.	−1.85923	−0.929615	+	0.368531i	$0.879861\pi$
$158$	−66051.3	−0.210494
$159$	429786.	1.34822
$160$	0	0
$161$	−52363.8	−0.159208
$162$	254384.	0.761558
$163$	157294.	0.463705	0.231853	−	0.972751i	$-0.425521\pi$
$163$	157294.	0.463705	0.231853	+	0.972751i	$0.425521\pi$
$164$	−21226.0	−0.0616251
$165$	0	0
$166$	−185555.	−0.522640
$167$	150221.	0.416811	0.208406	−	0.978042i	$-0.433173\pi$
$167$	150221.	0.416811	0.208406	+	0.978042i	$0.433173\pi$
$168$	−462797.	−1.26508
$169$	−121670.	−0.327692
$170$	0	0
$171$	−235880.	−0.616880
$172$	−437217.	−1.12687
$173$	−408467.	−1.03763	−0.518813	−	0.854887i	$-0.673626\pi$
$173$	−408467.	−1.03763	−0.518813	+	0.854887i	$0.673626\pi$
$174$	−401903.	−1.00635
$175$	0	0
$176$	−30705.7	−0.0747200
$177$	−13945.6	−0.0334584
$178$	−234404.	−0.554517
$179$	83182.2	0.194043	0.0970214	−	0.995282i	$-0.469068\pi$
$179$	83182.2	0.194043	0.0970214	+	0.995282i	$0.469068\pi$
$180$	0	0
$181$	383086.	0.869160	0.434580	−	0.900633i	$-0.356897\pi$
$181$	383086.	0.869160	0.434580	+	0.900633i	$0.356897\pi$
$182$	155000.	0.346859
$183$	−862611.	−1.90409
$184$	−89822.9	−0.195588
$185$	0	0
$186$	−780754.	−1.65475
$187$	32051.5	0.0670261
$188$	−500847.	−1.03350
$189$	741827.	1.51060
$190$	0	0
$191$	906396.	1.79777	0.898886	−	0.438182i	$-0.144377\pi$
$191$	906396.	1.79777	0.898886	+	0.438182i	$0.144377\pi$
$192$	−360499.	−0.705750
$193$	−609358.	−1.17755	−0.588775	−	0.808297i	$-0.700390\pi$
$193$	−609358.	−1.17755	−0.588775	+	0.808297i	$0.700390\pi$
$194$	146765.	0.279974
$195$	0	0
$196$	155435.	0.289007
$197$	−690338.	−1.26735	−0.633675	−	0.773600i	$-0.718454\pi$
$197$	−690338.	−1.26735	−0.633675	+	0.773600i	$0.718454\pi$
$198$	−279288.	−0.506278
$199$	−118685.	−0.212452	−0.106226	−	0.994342i	$-0.533877\pi$
$199$	−118685.	−0.212452	−0.106226	+	0.994342i	$0.533877\pi$
$200$	0	0
$201$	1.20299e6	2.10025
$202$	389734.	0.672031
$203$	−460999.	−0.785162
$204$	−113150.	−0.190362
$205$	0	0
$206$	−517846.	−0.850223
$207$	272526.	0.442061
$208$	−88690.0	−0.142140
$209$	79199.9	0.125418
$210$	0	0
$211$	1.18355e6	1.83012	0.915062	−	0.403314i	$-0.132142\pi$
$211$	1.18355e6	1.83012	0.915062	+	0.403314i	$0.132142\pi$
$212$	346162.	0.528980
$213$	516400.	0.779897
$214$	427340.	0.637881
$215$	0	0
$216$	1.27250e6	1.85577
$217$	−895555.	−1.29105
$218$	699777.	0.997283
$219$	1.09331e6	1.54039
$220$	0	0
$221$	92577.3	0.127504
$222$	464232.	0.632198
$223$	1.37756e6	1.85501	0.927507	−	0.373806i	$-0.121948\pi$
$223$	1.37756e6	1.85501	0.927507	+	0.373806i	$0.121948\pi$
$224$	−592915.	−0.789537
$225$	0	0
$226$	332860.	0.433502
$227$	1.14830e6	1.47908	0.739539	−	0.673114i	$-0.235043\pi$
$227$	1.14830e6	1.47908	0.739539	+	0.673114i	$0.235043\pi$
$228$	−279597.	−0.356202
$229$	171814.	0.216506	0.108253	−	0.994123i	$-0.465474\pi$
$229$	171814.	0.216506	0.108253	+	0.994123i	$0.465474\pi$
$230$	0	0
$231$	−471461.	−0.581320
$232$	−790780.	−0.964575
$233$	651670.	0.786390	0.393195	−	0.919455i	$-0.371370\pi$
$233$	651670.	0.786390	0.393195	+	0.919455i	$0.371370\pi$
$234$	−806693.	−0.963094
$235$	0	0
$236$	−11232.2	−0.0131276
$237$	580300.	0.671091
$238$	57484.4	0.0657821
$239$	−1.53723e6	−1.74078	−0.870389	−	0.492365i	$-0.836133\pi$
$239$	−1.53723e6	−1.74078	−0.870389	+	0.492365i	$0.836133\pi$
$240$	0	0
$241$	913293.	1.01290	0.506451	−	0.862269i	$-0.330957\pi$
$241$	913293.	1.01290	0.506451	+	0.862269i	$0.330957\pi$
$242$	−410976.	−0.451105
$243$	−413814.	−0.449562
$244$	−694771.	−0.747080
$245$	0	0
$246$	−82595.0	−0.0870194
$247$	228760.	0.238583
$248$	−1.53620e6	−1.58606
$249$	1.63021e6	1.66627
$250$	0	0
$251$	432806.	0.433620	0.216810	−	0.976214i	$-0.430435\pi$
$251$	432806.	0.433620	0.216810	+	0.976214i	$0.430435\pi$
$252$	1.13094e6	1.12186
$253$	−91504.4	−0.0898754
$254$	925846.	0.900440
$255$	0	0
$256$	−891087.	−0.849807
$257$	−1.43836e6	−1.35842	−0.679210	−	0.733944i	$-0.737677\pi$
$257$	−1.43836e6	−1.35842	−0.679210	+	0.733944i	$0.737677\pi$
$258$	−1.70131e6	−1.59123
$259$	532492.	0.493246
$260$	0	0
$261$	2.39926e6	2.18010
$262$	−236846.	−0.213163
$263$	−172554.	−0.153828	−0.0769142	−	0.997038i	$-0.524507\pi$
$263$	−172554.	−0.153828	−0.0769142	+	0.997038i	$0.524507\pi$
$264$	−808727.	−0.714154
$265$	0	0
$266$	142045.	0.123090
$267$	2.05937e6	1.76790
$268$	968920.	0.824045
$269$	−243007.	−0.204757	−0.102378	−	0.994746i	$-0.532645\pi$
$269$	−243007.	−0.204757	−0.102378	+	0.994746i	$0.532645\pi$
$270$	0	0
$271$	−2.16968e6	−1.79462	−0.897310	−	0.441400i	$-0.854482\pi$
$271$	−2.16968e6	−1.79462	−0.897310	+	0.441400i	$0.854482\pi$
$272$	−32892.3	−0.0269570
$273$	−1.36176e6	−1.10585
$274$	−1.25458e6	−1.00954
$275$	0	0
$276$	323035.	0.255257
$277$	−858654.	−0.672386	−0.336193	−	0.941793i	$-0.609140\pi$
$277$	−858654.	−0.672386	−0.336193	+	0.941793i	$0.609140\pi$
$278$	208016.	0.161430
$279$	4.66090e6	3.58475
$280$	0	0
$281$	2.29812e6	1.73623	0.868115	−	0.496363i	$-0.165331\pi$
$281$	2.29812e6	1.73623	0.868115	+	0.496363i	$0.165331\pi$
$282$	−1.94891e6	−1.45938
$283$	−382115.	−0.283614	−0.141807	−	0.989894i	$-0.545291\pi$
$283$	−382115.	−0.283614	−0.141807	+	0.989894i	$0.545291\pi$
$284$	415923.	0.305997
$285$	0	0
$286$	270858.	0.195807
$287$	−94739.6	−0.0678933
$288$	3.08581e6	2.19224
$289$	−1.38552e6	−0.975819
$290$	0	0
$291$	−1.28942e6	−0.892607
$292$	880580.	0.604382
$293$	4228.48	0.00287750	0.00143875	−	0.999999i	$-0.499542\pi$
$293$	4228.48	0.00287750	0.00143875	+	0.999999i	$0.499542\pi$
$294$	604833.	0.408100
$295$	0	0
$296$	913418.	0.605955
$297$	1.29633e6	0.852752
$298$	569179.	0.371286
$299$	−264301.	−0.170970
$300$	0	0
$301$	−1.95147e6	−1.24149
$302$	−562306.	−0.354777
$303$	−3.42404e6	−2.14255
$304$	−81277.6	−0.0504414
$305$	0	0
$306$	−299176.	−0.182652
$307$	454799.	0.275406	0.137703	−	0.990474i	$-0.456028\pi$
$307$	454799.	0.275406	0.137703	+	0.990474i	$0.456028\pi$
$308$	−379728.	−0.228084
$309$	4.54958e6	2.71066
$310$	0	0
$311$	−2.90140e6	−1.70101	−0.850506	−	0.525966i	$-0.823704\pi$
$311$	−2.90140e6	−1.70101	−0.850506	+	0.525966i	$0.823704\pi$
$312$	−2.33592e6	−1.35854
$313$	2.58180e6	1.48957	0.744785	−	0.667304i	$-0.232552\pi$
$313$	2.58180e6	1.48957	0.744785	+	0.667304i	$0.232552\pi$
$314$	−1.79968e6	−1.03008
$315$	0	0
$316$	467389.	0.263306
$317$	−2.07352e6	−1.15894	−0.579468	−	0.814995i	$-0.696740\pi$
$317$	−2.07352e6	−1.15894	−0.579468	+	0.814995i	$0.696740\pi$
$318$	1.34699e6	0.746961
$319$	−805584.	−0.443235
$320$	0	0
$321$	−3.75443e6	−2.03368
$322$	−164113.	−0.0882073
$323$	84839.9	0.0452474
$324$	−1.80006e6	−0.952632
$325$	0	0
$326$	492974.	0.256909
$327$	−6.14795e6	−3.17951
$328$	−162513.	−0.0834072
$329$	−2.23547e6	−1.13862
$330$	0	0
$331$	3.30721e6	1.65917	0.829587	−	0.558377i	$-0.188576\pi$
$331$	3.30721e6	1.65917	0.829587	+	0.558377i	$0.188576\pi$
$332$	1.31302e6	0.653771
$333$	−2.77134e6	−1.36956
$334$	470808.	0.230929
$335$	0	0
$336$	483828.	0.233799
$337$	1.83038e6	0.877945	0.438973	−	0.898500i	$-0.355342\pi$
$337$	1.83038e6	0.877945	0.438973	+	0.898500i	$0.355342\pi$
$338$	−381325.	−0.181553
$339$	−2.92437e6	−1.38208
$340$	0	0
$341$	−1.56496e6	−0.728816
$342$	−739272.	−0.341774
$343$	2.35743e6	1.08194
$344$	−3.34747e6	−1.52518
$345$	0	0
$346$	−1.28018e6	−0.574883
$347$	1.81015e6	0.807030	0.403515	−	0.914973i	$-0.367788\pi$
$347$	1.81015e6	0.807030	0.403515	+	0.914973i	$0.367788\pi$
$348$	2.84393e6	1.25884
$349$	1.63329e6	0.717793	0.358896	−	0.933377i	$-0.383153\pi$
$349$	1.63329e6	0.717793	0.358896	+	0.933377i	$0.383153\pi$
$350$	0	0
$351$	3.74430e6	1.62219
$352$	−1.03610e6	−0.445704
$353$	833243.	0.355906	0.177953	−	0.984039i	$-0.443053\pi$
$353$	833243.	0.355906	0.177953	+	0.984039i	$0.443053\pi$
$354$	−43707.0	−0.0185371
$355$	0	0
$356$	1.65868e6	0.693644
$357$	−505034.	−0.209725
$358$	260701.	0.107507
$359$	−636036.	−0.260463	−0.130231	−	0.991484i	$-0.541572\pi$
$359$	−636036.	−0.260463	−0.130231	+	0.991484i	$0.541572\pi$
$360$	0	0
$361$	−2.26646e6	−0.915334
$362$	1.20063e6	0.481546
$363$	3.61066e6	1.43820
$364$	−1.09680e6	−0.433885
$365$	0	0
$366$	−2.70351e6	−1.05493
$367$	−372220.	−0.144256	−0.0721282	−	0.997395i	$-0.522979\pi$
$367$	−372220.	−0.144256	−0.0721282	+	0.997395i	$0.522979\pi$
$368$	93904.8	0.0361467
$369$	493071.	0.188514
$370$	0	0
$371$	1.54505e6	0.582785
$372$	5.52474e6	2.06992
$373$	18520.2	0.00689245	0.00344622	−	0.999994i	$-0.498903\pi$
$373$	18520.2	0.00689245	0.00344622	+	0.999994i	$0.498903\pi$
$374$	100453.	0.0371349
$375$	0	0
$376$	−3.83465e6	−1.39880
$377$	−2.32684e6	−0.843167
$378$	2.32496e6	0.836925
$379$	4.32941e6	1.54821	0.774106	−	0.633056i	$-0.218200\pi$
$379$	4.32941e6	1.54821	0.774106	+	0.633056i	$0.218200\pi$
$380$	0	0
$381$	−8.13410e6	−2.87076
$382$	2.84074e6	0.996031
$383$	−3.16493e6	−1.10247	−0.551235	−	0.834350i	$-0.685843\pi$
$383$	−3.16493e6	−1.10247	−0.551235	+	0.834350i	$0.685843\pi$
$384$	4.14794e6	1.43550
$385$	0	0
$386$	−1.90979e6	−0.652406
$387$	1.01564e7	3.44716
$388$	−1.03853e6	−0.350219
$389$	−2.47737e6	−0.830074	−0.415037	−	0.909804i	$-0.636231\pi$
$389$	−2.47737e6	−0.830074	−0.415037	+	0.909804i	$0.636231\pi$
$390$	0	0
$391$	−98020.6	−0.0324247
$392$	1.19006e6	0.391160
$393$	2.08083e6	0.679603
$394$	−2.16359e6	−0.702158
$395$	0	0
$396$	1.97628e6	0.633303
$397$	−1.98642e6	−0.632550	−0.316275	−	0.948667i	$-0.602432\pi$
$397$	−1.98642e6	−0.632550	−0.316275	+	0.948667i	$0.602432\pi$
$398$	−371970.	−0.117706
$399$	−1.24795e6	−0.392433
$400$	0	0
$401$	−3.05157e6	−0.947681	−0.473840	−	0.880611i	$-0.657133\pi$
$401$	−3.05157e6	−0.947681	−0.473840	+	0.880611i	$0.657133\pi$
$402$	3.77029e6	1.16361
$403$	−4.52022e6	−1.38643
$404$	−2.75782e6	−0.840643
$405$	0	0
$406$	−1.44482e6	−0.435008
$407$	930517.	0.278444
$408$	−866318.	−0.257648
$409$	−1.54349e6	−0.456243	−0.228122	−	0.973633i	$-0.573258\pi$
$409$	−1.54349e6	−0.456243	−0.228122	+	0.973633i	$0.573258\pi$
$410$	0	0
$411$	1.10222e7	3.21858
$412$	3.66436e6	1.06354
$413$	−50133.6	−0.0144628
$414$	854125.	0.244918
$415$	0	0
$416$	−2.99268e6	−0.847865
$417$	−1.82754e6	−0.514669
$418$	248221.	0.0694860
$419$	782997.	0.217884	0.108942	−	0.994048i	$-0.465254\pi$
$419$	782997.	0.217884	0.108942	+	0.994048i	$0.465254\pi$
$420$	0	0
$421$	−1.89001e6	−0.519707	−0.259854	−	0.965648i	$-0.583674\pi$
$421$	−1.89001e6	−0.519707	−0.259854	+	0.965648i	$0.583674\pi$
$422$	3.70937e6	1.01395
$423$	1.16345e7	3.16152
$424$	2.65033e6	0.715954
$425$	0	0
$426$	1.61845e6	0.432091
$427$	−3.10103e6	−0.823069
$428$	−3.02393e6	−0.797925
$429$	−2.37965e6	−0.624266
$430$	0	0
$431$	1.47121e6	0.381489	0.190745	−	0.981640i	$-0.438910\pi$
$431$	1.47121e6	0.381489	0.190745	+	0.981640i	$0.438910\pi$
$432$	−1.33033e6	−0.342966
$433$	−1638.32	−0.000419931 0	−0.000209966	−	1.00000i	$-0.500067\pi$
$433$	−1638.32	−0.000419931 0	−0.000209966		1.00000i	$0.500067\pi$
$434$	−2.80676e6	−0.715289
$435$	0	0
$436$	−4.95173e6	−1.24750
$437$	−242211.	−0.0606723
$438$	3.42654e6	0.853435
$439$	−6.97689e6	−1.72783	−0.863914	−	0.503640i	$-0.831994\pi$
$439$	−6.97689e6	−1.72783	−0.863914	+	0.503640i	$0.831994\pi$
$440$	0	0
$441$	−3.61069e6	−0.884085
$442$	290147.	0.0706418
$443$	3.49744e6	0.846723	0.423361	−	0.905961i	$-0.360850\pi$
$443$	3.49744e6	0.846723	0.423361	+	0.905961i	$0.360850\pi$
$444$	−3.28498e6	−0.790816
$445$	0	0
$446$	4.31740e6	1.02774
$447$	−5.00057e6	−1.18372
$448$	−1.29597e6	−0.305070
$449$	−4.00137e6	−0.936684	−0.468342	−	0.883547i	$-0.655149\pi$
$449$	−4.00137e6	−0.936684	−0.468342	+	0.883547i	$0.655149\pi$
$450$	0	0
$451$	−165555.	−0.0383267
$452$	−2.35537e6	−0.542267
$453$	4.94019e6	1.13109
$454$	3.59889e6	0.819463
$455$	0	0
$456$	−2.14069e6	−0.482105
$457$	4.37124e6	0.979072	0.489536	−	0.871983i	$-0.337166\pi$
$457$	4.37124e6	0.979072	0.489536	+	0.871983i	$0.337166\pi$
$458$	538482.	0.119952
$459$	1.38864e6	0.307651
$460$	0	0
$461$	1.70022e6	0.372610	0.186305	−	0.982492i	$-0.440349\pi$
$461$	1.70022e6	0.372610	0.186305	+	0.982492i	$0.440349\pi$
$462$	−1.47761e6	−0.322073
$463$	5.77097e6	1.25111	0.625557	−	0.780179i	$-0.284872\pi$
$463$	5.77097e6	1.25111	0.625557	+	0.780179i	$0.284872\pi$
$464$	826717.	0.178263
$465$	0	0
$466$	2.04240e6	0.435689
$467$	−3.25780e6	−0.691246	−0.345623	−	0.938373i	$-0.612332\pi$
$467$	−3.25780e6	−0.691246	−0.345623	+	0.938373i	$0.612332\pi$
$468$	5.70828e6	1.20473
$469$	4.32466e6	0.907863
$470$	0	0
$471$	1.58113e7	3.28408
$472$	−85997.4	−0.0177677
$473$	−3.41014e6	−0.700841
$474$	1.81872e6	0.371809
$475$	0	0
$476$	−406769.	−0.0822868
$477$	−8.04120e6	−1.61817
$478$	−4.81783e6	−0.964454
$479$	−5.35349e6	−1.06610	−0.533051	−	0.846083i	$-0.678955\pi$
$479$	−5.35349e6	−1.06610	−0.533051	+	0.846083i	$0.678955\pi$
$480$	0	0
$481$	2.68770e6	0.529685
$482$	2.86236e6	0.561185
$483$	1.44183e6	0.281220
$484$	2.90813e6	0.564287
$485$	0	0
$486$	−1.29694e6	−0.249074
$487$	−3.94606e6	−0.753948	−0.376974	−	0.926224i	$-0.623035\pi$
$487$	−3.94606e6	−0.753948	−0.376974	+	0.926224i	$0.623035\pi$
$488$	−5.31940e6	−1.01114
$489$	−4.33107e6	−0.819073
$490$	0	0
$491$	−4.46612e6	−0.836038	−0.418019	−	0.908438i	$-0.637276\pi$
$491$	−4.46612e6	−0.836038	−0.418019	+	0.908438i	$0.637276\pi$
$492$	584455.	0.108853
$493$	−862951.	−0.159908
$494$	716959.	0.132183
$495$	0	0
$496$	1.60601e6	0.293120
$497$	1.85642e6	0.337121
$498$	5.10925e6	0.923175
$499$	421495.	0.0757777	0.0378889	−	0.999282i	$-0.487937\pi$
$499$	421495.	0.0757777	0.0378889	+	0.999282i	$0.487937\pi$
$500$	0	0
$501$	−4.13633e6	−0.736242
$502$	1.35646e6	0.240241
$503$	165744.	0.0292090	0.0146045	−	0.999893i	$-0.495351\pi$
$503$	165744.	0.0292090	0.0146045	+	0.999893i	$0.495351\pi$
$504$	8.65882e6	1.51839
$505$	0	0
$506$	−286784.	−0.0497942
$507$	3.35017e6	0.578824
$508$	−6.55143e6	−1.12636
$509$	−2.54136e6	−0.434782	−0.217391	−	0.976085i	$-0.569755\pi$
$509$	−2.54136e6	−0.434782	−0.217391	+	0.976085i	$0.569755\pi$
$510$	0	0
$511$	3.93037e6	0.665857
$512$	2.02781e6	0.341863
$513$	3.43136e6	0.575669
$514$	−4.50796e6	−0.752614
$515$	0	0
$516$	1.20387e7	1.99047
$517$	−3.90643e6	−0.642768
$518$	1.66889e6	0.273276
$519$	1.12471e7	1.83283
$520$	0	0
$521$	−1.00488e7	−1.62189	−0.810944	−	0.585124i	$-0.801046\pi$
$521$	−1.00488e7	−1.62189	−0.810944	+	0.585124i	$0.801046\pi$
$522$	7.51952e6	1.20785
$523$	5.25311e6	0.839774	0.419887	−	0.907576i	$-0.362070\pi$
$523$	5.25311e6	0.839774	0.419887	+	0.907576i	$0.362070\pi$
$524$	1.67596e6	0.266646
$525$	0	0
$526$	−540803.	−0.0852265
$527$	−1.67641e6	−0.262937
$528$	845479.	0.131983
$529$	279841.	0.0434783
$530$	0	0
$531$	260919.	0.0401578
$532$	−1.00513e6	−0.153973
$533$	−478189.	−0.0729090
$534$	6.45429e6	0.979479
$535$	0	0
$536$	7.41837e6	1.11531
$537$	−2.29042e6	−0.342751
$538$	−761609.	−0.113443
$539$	1.21234e6	0.179743
$540$	0	0
$541$	4.45549e6	0.654489	0.327244	−	0.944940i	$-0.393880\pi$
$541$	4.45549e6	0.654489	0.327244	+	0.944940i	$0.393880\pi$
$542$	−6.80000e6	−0.994285
$543$	−1.05482e7	−1.53526
$544$	−1.10989e6	−0.160798
$545$	0	0
$546$	−4.26791e6	−0.612679
$547$	2.77840e6	0.397033	0.198516	−	0.980098i	$-0.436388\pi$
$547$	2.77840e6	0.397033	0.198516	+	0.980098i	$0.436388\pi$
$548$	8.87759e6	1.26283
$549$	1.61393e7	2.28535
$550$	0	0
$551$	−2.13237e6	−0.299215
$552$	2.47327e6	0.345480
$553$	2.08614e6	0.290089
$554$	−2.69111e6	−0.372527
$555$	0	0
$556$	−1.47196e6	−0.201933
$557$	1.26464e7	1.72715	0.863575	−	0.504220i	$-0.168220\pi$
$557$	1.26464e7	1.72715	0.863575	+	0.504220i	$0.168220\pi$
$558$	1.46077e7	1.98608
$559$	−9.84982e6	−1.33321
$560$	0	0
$561$	−882535.	−0.118393
$562$	7.20255e6	0.961935
$563$	3.99645e6	0.531377	0.265689	−	0.964059i	$-0.414401\pi$
$563$	3.99645e6	0.531377	0.265689	+	0.964059i	$0.414401\pi$
$564$	1.37908e7	1.82554
$565$	0	0
$566$	−1.19759e6	−0.157132
$567$	−8.03438e6	−1.04953
$568$	3.18444e6	0.414155
$569$	−1.09077e6	−0.141238	−0.0706191	−	0.997503i	$-0.522497\pi$
$569$	−1.09077e6	−0.141238	−0.0706191	+	0.997503i	$0.522497\pi$
$570$	0	0
$571$	226191.	0.0290325	0.0145162	−	0.999895i	$-0.495379\pi$
$571$	226191.	0.0290325	0.0145162	+	0.999895i	$0.495379\pi$
$572$	−1.91664e6	−0.244934
$573$	−2.49576e7	−3.17553
$574$	−296924.	−0.0376154
$575$	0	0
$576$	6.74485e6	0.847064
$577$	−3.28573e6	−0.410858	−0.205429	−	0.978672i	$-0.565859\pi$
$577$	−3.28573e6	−0.410858	−0.205429	+	0.978672i	$0.565859\pi$
$578$	−4.34237e6	−0.540639
$579$	1.67786e7	2.07999
$580$	0	0
$581$	5.86051e6	0.720269
$582$	−4.04116e6	−0.494537
$583$	2.69994e6	0.328991
$584$	6.74202e6	0.818008
$585$	0	0
$586$	13252.5	0.00159424
$587$	7.43016e6	0.890027	0.445013	−	0.895524i	$-0.353199\pi$
$587$	7.43016e6	0.890027	0.445013	+	0.895524i	$0.353199\pi$
$588$	−4.27989e6	−0.510493
$589$	−4.14244e6	−0.492003
$590$	0	0
$591$	1.90084e7	2.23860
$592$	−954928.	−0.111987
$593$	8.97683e6	1.04830	0.524151	−	0.851626i	$-0.324383\pi$
$593$	8.97683e6	1.04830	0.524151	+	0.851626i	$0.324383\pi$
$594$	4.06282e6	0.472456
$595$	0	0
$596$	−4.02760e6	−0.464441
$597$	3.26797e6	0.375269
$598$	−828345.	−0.0947237
$599$	−1.40377e7	−1.59856	−0.799280	−	0.600959i	$-0.794785\pi$
$599$	−1.40377e7	−1.59856	−0.799280	+	0.600959i	$0.794785\pi$
$600$	0	0
$601$	9.15574e6	1.03397	0.516984	−	0.855995i	$-0.327055\pi$
$601$	9.15574e6	1.03397	0.516984	+	0.855995i	$0.327055\pi$
$602$	−6.11610e6	−0.687833
$603$	−2.25076e7	−2.52079
$604$	3.97897e6	0.443790
$605$	0	0
$606$	−1.07313e7	−1.18705
$607$	−7.53570e6	−0.830141	−0.415070	−	0.909789i	$-0.636243\pi$
$607$	−7.53570e6	−0.830141	−0.415070	+	0.909789i	$0.636243\pi$
$608$	−2.74256e6	−0.300882
$609$	1.26936e7	1.38688
$610$	0	0
$611$	−1.12833e7	−1.22274
$612$	2.11702e6	0.228479
$613$	9.98554e6	1.07330	0.536649	−	0.843805i	$-0.319690\pi$
$613$	9.98554e6	1.07330	0.536649	+	0.843805i	$0.319690\pi$
$614$	1.42539e6	0.152585
$615$	0	0
$616$	−2.90732e6	−0.308703
$617$	−6.82945e6	−0.722226	−0.361113	−	0.932522i	$-0.617603\pi$
$617$	−6.82945e6	−0.722226	−0.361113	+	0.932522i	$0.617603\pi$
$618$	1.42589e7	1.50181
$619$	−9.08935e6	−0.953468	−0.476734	−	0.879047i	$-0.658180\pi$
$619$	−9.08935e6	−0.953468	−0.476734	+	0.879047i	$0.658180\pi$
$620$	0	0
$621$	−3.96445e6	−0.412529
$622$	−9.09330e6	−0.942422
$623$	7.40332e6	0.764199
$624$	2.44207e6	0.251071
$625$	0	0
$626$	8.09161e6	0.825276
$627$	−2.18076e6	−0.221534
$628$	1.27348e7	1.28853
$629$	996781.	0.100455
$630$	0	0
$631$	5.93201e6	0.593100	0.296550	−	0.955017i	$-0.404164\pi$
$631$	5.93201e6	0.593100	0.296550	+	0.955017i	$0.404164\pi$
$632$	3.57849e6	0.356375
$633$	−3.25890e7	−3.23267
$634$	−6.49862e6	−0.642093
$635$	0	0
$636$	−9.53154e6	−0.934373
$637$	3.50172e6	0.341926
$638$	−2.52478e6	−0.245568
$639$	−9.66173e6	−0.936058
$640$	0	0
$641$	−1.20892e7	−1.16212	−0.581062	−	0.813859i	$-0.697363\pi$
$641$	−1.20892e7	−1.16212	−0.581062	+	0.813859i	$0.697363\pi$
$642$	−1.17668e7	−1.12673
$643$	−1.09118e7	−1.04081	−0.520404	−	0.853920i	$-0.674219\pi$
$643$	−1.09118e7	−1.04081	−0.520404	+	0.853920i	$0.674219\pi$
$644$	1.16129e6	0.110338
$645$	0	0
$646$	265897.	0.0250687
$647$	8.06391e6	0.757330	0.378665	−	0.925534i	$-0.376383\pi$
$647$	8.06391e6	0.757330	0.378665	+	0.925534i	$0.376383\pi$
$648$	−1.37819e7	−1.28935
$649$	−87607.3	−0.00816448
$650$	0	0
$651$	2.46590e7	2.28047
$652$	−3.48836e6	−0.321368
$653$	−897296.	−0.0823479	−0.0411740	−	0.999152i	$-0.513110\pi$
$653$	−897296.	−0.0823479	−0.0411740	+	0.999152i	$0.513110\pi$
$654$	−1.92683e7	−1.76157
$655$	0	0
$656$	169898.	0.0154145
$657$	−2.04555e7	−1.84883
$658$	−7.00620e6	−0.630838
$659$	1.32258e6	0.118634	0.0593168	−	0.998239i	$-0.481108\pi$
$659$	1.32258e6	0.118634	0.0593168	+	0.998239i	$0.481108\pi$
$660$	0	0
$661$	−1.38914e7	−1.23663	−0.618317	−	0.785929i	$-0.712185\pi$
$661$	−1.38914e7	−1.23663	−0.618317	+	0.785929i	$0.712185\pi$
$662$	1.03651e7	0.919243
$663$	−2.54911e6	−0.225219
$664$	1.00529e7	0.884853
$665$	0	0
$666$	−8.68568e6	−0.758784
$667$	2.46366e6	0.214420
$668$	−3.33151e6	−0.288869
$669$	−3.79309e7	−3.27663
$670$	0	0
$671$	−5.41898e6	−0.464634
$672$	1.63259e7	1.39461
$673$	1.38217e7	1.17632	0.588159	−	0.808745i	$-0.299853\pi$
$673$	1.38217e7	1.17632	0.588159	+	0.808745i	$0.299853\pi$
$674$	5.73661e6	0.486414
$675$	0	0
$676$	2.69832e6	0.227105
$677$	−4.26668e6	−0.357782	−0.178891	−	0.983869i	$-0.557251\pi$
$677$	−4.26668e6	−0.357782	−0.178891	+	0.983869i	$0.557251\pi$
$678$	−9.16529e6	−0.765723
$679$	−4.63536e6	−0.385842
$680$	0	0
$681$	−3.16184e7	−2.61259
$682$	−4.90475e6	−0.403790
$683$	1.34750e6	0.110529	0.0552646	−	0.998472i	$-0.482400\pi$
$683$	1.34750e6	0.110529	0.0552646	+	0.998472i	$0.482400\pi$
$684$	5.23120e6	0.427525
$685$	0	0
$686$	7.38843e6	0.599434
$687$	−4.73088e6	−0.382429
$688$	3.49960e6	0.281869
$689$	7.79850e6	0.625839
$690$	0	0
$691$	−1.29881e7	−1.03478	−0.517392	−	0.855748i	$-0.673097\pi$
$691$	−1.29881e7	−1.03478	−0.517392	+	0.855748i	$0.673097\pi$
$692$	9.05873e6	0.719121
$693$	8.82092e6	0.697719
$694$	5.67318e6	0.447124
$695$	0	0
$696$	2.17741e7	1.70379
$697$	−177345.	−0.0138273
$698$	5.11889e6	0.397683
$699$	−1.79437e7	−1.38905
$700$	0	0
$701$	−1.78237e7	−1.36994	−0.684970	−	0.728571i	$-0.740185\pi$
$701$	−1.78237e7	−1.36994	−0.684970	+	0.728571i	$0.740185\pi$
$702$	1.17350e7	0.898754
$703$	2.46307e6	0.187970
$704$	−2.26468e6	−0.172216
$705$	0	0
$706$	2.61147e6	0.197185
$707$	−1.23092e7	−0.926150
$708$	309278.	0.0231881
$709$	−1.38491e7	−1.03468	−0.517341	−	0.855779i	$-0.673078\pi$
$709$	−1.38491e7	−1.03468	−0.517341	+	0.855779i	$0.673078\pi$
$710$	0	0
$711$	−1.08573e7	−0.805465
$712$	1.26994e7	0.938821
$713$	4.78600e6	0.352573
$714$	−1.58283e6	−0.116195
$715$	0	0
$716$	−1.84476e6	−0.134480
$717$	4.23274e7	3.07485
$718$	−1.99340e6	−0.144306
$719$	−6.39208e6	−0.461126	−0.230563	−	0.973057i	$-0.574057\pi$
$719$	−6.39208e6	−0.461126	−0.230563	+	0.973057i	$0.574057\pi$
$720$	0	0
$721$	1.63555e7	1.17172
$722$	−7.10331e6	−0.507128
$723$	−2.51475e7	−1.78916
$724$	−8.49585e6	−0.602366
$725$	0	0
$726$	1.13162e7	0.796817
$727$	−9.15468e6	−0.642403	−0.321201	−	0.947011i	$-0.604087\pi$
$727$	−9.15468e6	−0.642403	−0.321201	+	0.947011i	$0.604087\pi$
$728$	−8.39748e6	−0.587247
$729$	−8.32912e6	−0.580471
$730$	0	0
$731$	−3.65298e6	−0.252845
$732$	1.91305e7	1.31962
$733$	1.29905e7	0.893028	0.446514	−	0.894777i	$-0.352665\pi$
$733$	1.29905e7	0.893028	0.446514	+	0.894777i	$0.352665\pi$
$734$	−1.16658e6	−0.0799232
$735$	0	0
$736$	3.16864e6	0.215615
$737$	7.55725e6	0.512501
$738$	1.54533e6	0.104444
$739$	−3.98014e6	−0.268094	−0.134047	−	0.990975i	$-0.542797\pi$
$739$	−3.98014e6	−0.268094	−0.134047	+	0.990975i	$0.542797\pi$
$740$	0	0
$741$	−6.29890e6	−0.421424
$742$	4.84236e6	0.322884
$743$	1.96581e6	0.130638	0.0653192	−	0.997864i	$-0.479193\pi$
$743$	1.96581e6	0.130638	0.0653192	+	0.997864i	$0.479193\pi$
$744$	4.22992e7	2.80156
$745$	0	0
$746$	58044.2	0.00381867
$747$	−3.05009e7	−1.99991
$748$	−710819.	−0.0464520
$749$	−1.34970e7	−0.879086
$750$	0	0
$751$	3.00617e6	0.194497	0.0972487	−	0.995260i	$-0.468996\pi$
$751$	3.00617e6	0.194497	0.0972487	+	0.995260i	$0.468996\pi$
$752$	4.00891e6	0.258513
$753$	−1.19173e7	−0.765931
$754$	−7.29256e6	−0.467145
$755$	0	0
$756$	−1.64518e7	−1.04691
$757$	−8.54036e6	−0.541672	−0.270836	−	0.962626i	$-0.587300\pi$
$757$	−8.54036e6	−0.541672	−0.270836	+	0.962626i	$0.587300\pi$
$758$	1.35688e7	0.857766
$759$	2.51957e6	0.158753
$760$	0	0
$761$	−4.00401e6	−0.250630	−0.125315	−	0.992117i	$-0.539994\pi$
$761$	−4.00401e6	−0.250630	−0.125315	+	0.992117i	$0.539994\pi$
$762$	−2.54931e7	−1.59051
$763$	−2.21015e7	−1.37439
$764$	−2.01015e7	−1.24594
$765$	0	0
$766$	−9.91921e6	−0.610809
$767$	−253044.	−0.0155313
$768$	2.45360e7	1.50107
$769$	−3.19689e6	−0.194945	−0.0974725	−	0.995238i	$-0.531076\pi$
$769$	−3.19689e6	−0.194945	−0.0974725	+	0.995238i	$0.531076\pi$
$770$	0	0
$771$	3.96051e7	2.39947
$772$	1.35140e7	0.816094
$773$	−1.56246e7	−0.940503	−0.470251	−	0.882532i	$-0.655837\pi$
$773$	−1.56246e7	−0.940503	−0.470251	+	0.882532i	$0.655837\pi$
$774$	3.18311e7	1.90985
$775$	0	0
$776$	−7.95134e6	−0.474008
$777$	−1.46621e7	−0.871254
$778$	−7.76433e6	−0.459891
$779$	−438223.	−0.0258733
$780$	0	0
$781$	3.24406e6	0.190310
$782$	−307207.	−0.0179644
$783$	−3.49021e7	−2.03445
$784$	−1.24414e6	−0.0722904
$785$	0	0
$786$	6.52154e6	0.376525
$787$	−2.27061e7	−1.30679	−0.653395	−	0.757018i	$-0.726656\pi$
$787$	−2.27061e7	−1.30679	−0.653395	+	0.757018i	$0.726656\pi$
$788$	1.53099e7	0.878329
$789$	4.75127e6	0.271717
$790$	0	0
$791$	−1.05129e7	−0.597424
$792$	1.51311e7	0.857151
$793$	−1.56521e7	−0.883874
$794$	−6.22565e6	−0.350456
$795$	0	0
$796$	2.63212e6	0.147239
$797$	−2.22676e7	−1.24173	−0.620866	−	0.783916i	$-0.713219\pi$
$797$	−2.22676e7	−1.24173	−0.620866	+	0.783916i	$0.713219\pi$
$798$	−3.91121e6	−0.217422
$799$	−4.18462e6	−0.231894
$800$	0	0
$801$	−3.85304e7	−2.12189
$802$	−9.56393e6	−0.525050
$803$	6.86823e6	0.375886
$804$	−2.66791e7	−1.45557
$805$	0	0
$806$	−1.41668e7	−0.768131
$807$	6.69118e6	0.361676
$808$	−2.11148e7	−1.13778
$809$	3.12578e7	1.67914	0.839570	−	0.543251i	$-0.182807\pi$
$809$	3.12578e7	1.67914	0.839570	+	0.543251i	$0.182807\pi$
$810$	0	0
$811$	−5.17398e6	−0.276231	−0.138116	−	0.990416i	$-0.544105\pi$
$811$	−5.17398e6	−0.276231	−0.138116	+	0.990416i	$0.544105\pi$
$812$	1.02237e7	0.544152
$813$	5.97420e7	3.16996
$814$	2.91634e6	0.154268
$815$	0	0
$816$	905687.	0.0476160
$817$	−9.02660e6	−0.473118
$818$	−4.83747e6	−0.252775
$819$	2.54783e7	1.32727
$820$	0	0
$821$	−2.51567e7	−1.30255	−0.651277	−	0.758840i	$-0.725766\pi$
$821$	−2.51567e7	−1.30255	−0.651277	+	0.758840i	$0.725766\pi$
$822$	3.45447e7	1.78321
$823$	−2.17605e7	−1.11988	−0.559938	−	0.828535i	$-0.689175\pi$
$823$	−2.17605e7	−1.11988	−0.559938	+	0.828535i	$0.689175\pi$
$824$	2.80556e7	1.43946
$825$	0	0
$826$	−157124.	−0.00801294
$827$	2.25721e7	1.14765	0.573823	−	0.818979i	$-0.305460\pi$
$827$	2.25721e7	1.14765	0.573823	+	0.818979i	$0.305460\pi$
$828$	−6.04392e6	−0.306368
$829$	1.17900e6	0.0595836	0.0297918	−	0.999556i	$-0.490516\pi$
$829$	1.17900e6	0.0595836	0.0297918	+	0.999556i	$0.490516\pi$
$830$	0	0
$831$	2.36430e7	1.18768
$832$	−6.54127e6	−0.327608
$833$	1.29867e6	0.0648467
$834$	−5.72771e6	−0.285145
$835$	0	0
$836$	−1.75645e6	−0.0869200
$837$	−6.78024e7	−3.34527
$838$	2.45399e6	0.120716
$839$	5.57632e6	0.273491	0.136746	−	0.990606i	$-0.456336\pi$
$839$	5.57632e6	0.273491	0.136746	+	0.990606i	$0.456336\pi$
$840$	0	0
$841$	1.17832e6	0.0574477
$842$	−5.92349e6	−0.287937
$843$	−6.32786e7	−3.06682
$844$	−2.62481e7	−1.26836
$845$	0	0
$846$	3.64636e7	1.75160
$847$	1.29801e7	0.621684
$848$	−2.77077e6	−0.132316
$849$	1.05215e7	0.500966
$850$	0	0
$851$	−2.84573e6	−0.134701
$852$	−1.14524e7	−0.540503
$853$	−2.78507e7	−1.31058	−0.655289	−	0.755378i	$-0.727453\pi$
$853$	−2.78507e7	−1.31058	−0.655289	+	0.755378i	$0.727453\pi$
$854$	−9.71895e6	−0.456010
$855$	0	0
$856$	−2.31522e7	−1.07996
$857$	−7.89202e6	−0.367059	−0.183530	−	0.983014i	$-0.558752\pi$
$857$	−7.89202e6	−0.367059	−0.183530	+	0.983014i	$0.558752\pi$
$858$	−7.45806e6	−0.345866
$859$	1.81190e7	0.837823	0.418911	−	0.908027i	$-0.362412\pi$
$859$	1.81190e7	0.837823	0.418911	+	0.908027i	$0.362412\pi$
$860$	0	0
$861$	2.60865e6	0.119924
$862$	4.61093e6	0.211359
$863$	272254.	0.0124437	0.00622183	−	0.999981i	$-0.498020\pi$
$863$	272254.	0.0124437	0.00622183	+	0.999981i	$0.498020\pi$
$864$	−4.48895e7	−2.04579
$865$	0	0
$866$	−5134.65	−0.000232657 0
$867$	3.81503e7	1.72365
$868$	1.98611e7	0.894754
$869$	3.64548e6	0.163759
$870$	0	0
$871$	2.18283e7	0.974932
$872$	−3.79121e7	−1.68844
$873$	2.41247e7	1.07134
$874$	−759115.	−0.0336147
$875$	0	0
$876$	−2.42467e7	−1.06756
$877$	−8.51694e6	−0.373925	−0.186962	−	0.982367i	$-0.559864\pi$
$877$	−8.51694e6	−0.373925	−0.186962	+	0.982367i	$0.559864\pi$
$878$	−2.18663e7	−0.957279
$879$	−116431.	−0.00508272
$880$	0	0
$881$	−475215.	−0.0206277	−0.0103138	−	0.999947i	$-0.503283\pi$
$881$	−475215.	−0.0206277	−0.0103138	+	0.999947i	$0.503283\pi$
$882$	−1.13163e7	−0.489815
$883$	−2.40143e7	−1.03650	−0.518249	−	0.855230i	$-0.673416\pi$
$883$	−2.40143e7	−1.03650	−0.518249	+	0.855230i	$0.673416\pi$
$884$	−2.05312e6	−0.0883658
$885$	0	0
$886$	1.09613e7	0.469115
$887$	−840273.	−0.0358601	−0.0179300	−	0.999839i	$-0.505708\pi$
$887$	−840273.	−0.0358601	−0.0179300	+	0.999839i	$0.505708\pi$
$888$	−2.51509e7	−1.07034
$889$	−2.92416e7	−1.24093
$890$	0	0
$891$	−1.40399e7	−0.592474
$892$	−3.05506e7	−1.28561
$893$	−1.03403e7	−0.433914
$894$	−1.56723e7	−0.655826
$895$	0	0
$896$	1.49116e7	0.620517
$897$	7.27750e6	0.301996
$898$	−1.25407e7	−0.518957
$899$	4.21349e7	1.73877
$900$	0	0
$901$	2.89221e6	0.118691
$902$	−518867.	−0.0212344
$903$	5.37335e7	2.19293
$904$	−1.80335e7	−0.733938
$905$	0	0
$906$	1.54831e7	0.626666
$907$	−2.64919e7	−1.06929	−0.534643	−	0.845078i	$-0.679554\pi$
$907$	−2.64919e7	−1.06929	−0.534643	+	0.845078i	$0.679554\pi$
$908$	−2.54663e7	−1.02507
$909$	6.40630e7	2.57156
$910$	0	0
$911$	−1.82537e7	−0.728709	−0.364355	−	0.931260i	$-0.618710\pi$
$911$	−1.82537e7	−0.728709	−0.364355	+	0.931260i	$0.618710\pi$
$912$	2.23797e6	0.0890979
$913$	1.02411e7	0.406602
$914$	1.36999e7	0.542441
$915$	0	0
$916$	−3.81038e6	−0.150048
$917$	7.48045e6	0.293768
$918$	4.35214e6	0.170450
$919$	3.48435e7	1.36092	0.680461	−	0.732784i	$-0.261780\pi$
$919$	3.48435e7	1.36092	0.680461	+	0.732784i	$0.261780\pi$
$920$	0	0
$921$	−1.25228e7	−0.486468
$922$	5.32868e6	0.206439
$923$	9.37011e6	0.362027
$924$	1.04558e7	0.402880
$925$	0	0
$926$	1.80868e7	0.693162
$927$	−8.51217e7	−3.25342
$928$	2.78960e7	1.06334
$929$	−4.36870e6	−0.166078	−0.0830391	−	0.996546i	$-0.526463\pi$
$929$	−4.36870e6	−0.166078	−0.0830391	+	0.996546i	$0.526463\pi$
$930$	0	0
$931$	3.20905e6	0.121340
$932$	−1.44523e7	−0.545003
$933$	7.98899e7	3.00461
$934$	−1.02103e7	−0.382975
$935$	0	0
$936$	4.37045e7	1.63056
$937$	1.90567e7	0.709087	0.354543	−	0.935040i	$-0.384636\pi$
$937$	1.90567e7	0.709087	0.354543	+	0.935040i	$0.384636\pi$
$938$	1.35539e7	0.502989
$939$	−7.10895e7	−2.63113
$940$	0	0
$941$	−6.86363e6	−0.252685	−0.126343	−	0.991987i	$-0.540324\pi$
$941$	−6.86363e6	−0.252685	−0.126343	+	0.991987i	$0.540324\pi$
$942$	4.95541e7	1.81950
$943$	506305.	0.0185410
$944$	89905.5	0.00328364
$945$	0	0
$946$	−1.06877e7	−0.388291
$947$	2.04942e7	0.742602	0.371301	−	0.928513i	$-0.378912\pi$
$947$	2.04942e7	0.742602	0.371301	+	0.928513i	$0.378912\pi$
$948$	−1.28695e7	−0.465095
$949$	1.98381e7	0.715048
$950$	0	0
$951$	5.70942e7	2.04711
$952$	−3.11436e6	−0.111372
$953$	2.36728e6	0.0844341	0.0422170	−	0.999108i	$-0.486558\pi$
$953$	2.36728e6	0.0844341	0.0422170	+	0.999108i	$0.486558\pi$
$954$	−2.52019e7	−0.896526
$955$	0	0
$956$	3.40917e7	1.20644
$957$	2.21817e7	0.782916
$958$	−1.67784e7	−0.590659
$959$	3.96241e7	1.39128
$960$	0	0
$961$	5.32238e7	1.85908
$962$	8.42353e6	0.293465
$963$	7.02446e7	2.44089
$964$	−2.02545e7	−0.701986
$965$	0	0
$966$	4.51885e6	0.155806
$967$	193505.	0.00665465	0.00332732	−	0.999994i	$-0.498941\pi$
$967$	193505.	0.00665465	0.00332732	+	0.999994i	$0.498941\pi$
$968$	2.22656e7	0.763740
$969$	−2.33606e6	−0.0799236
$970$	0	0
$971$	1.18283e7	0.402602	0.201301	−	0.979529i	$-0.435483\pi$
$971$	1.18283e7	0.402602	0.201301	+	0.979529i	$0.435483\pi$
$972$	9.17733e6	0.311566
$973$	−6.56990e6	−0.222473
$974$	−1.23674e7	−0.417715
$975$	0	0
$976$	5.56113e6	0.186870
$977$	1.27220e7	0.426400	0.213200	−	0.977009i	$-0.431611\pi$
$977$	1.27220e7	0.426400	0.213200	+	0.977009i	$0.431611\pi$
$978$	−1.35740e7	−0.453796
$979$	1.29371e7	0.431401
$980$	0	0
$981$	1.15027e8	3.81616
$982$	−1.39973e7	−0.463196
$983$	2.62540e7	0.866588	0.433294	−	0.901253i	$-0.357351\pi$
$983$	2.62540e7	0.866588	0.433294	+	0.901253i	$0.357351\pi$
$984$	4.47478e6	0.147328
$985$	0	0
$986$	−2.70458e6	−0.0885946
$987$	6.15536e7	2.01122
$988$	−5.07331e6	−0.165348
$989$	1.04290e7	0.339040
$990$	0	0
$991$	5.49258e7	1.77661	0.888306	−	0.459253i	$-0.151883\pi$
$991$	5.49258e7	1.77661	0.888306	+	0.459253i	$0.151883\pi$
$992$	5.41919e7	1.74846
$993$	−9.10638e7	−2.93071
$994$	5.81823e6	0.186778
$995$	0	0
$996$	−3.61539e7	−1.15480
$997$	3.45837e7	1.10188	0.550939	−	0.834545i	$-0.314270\pi$
$997$	3.45837e7	1.10188	0.550939	+	0.834545i	$0.314270\pi$
$998$	1.32101e6	0.0419836
$999$	4.03149e7	1.27806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.6.a.l.1.19		27
5.2	odd	4		115.6.b.a.24.36	yes	54
5.3	odd	4		115.6.b.a.24.19	✓	54
5.4	even	2		575.6.a.m.1.9		27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.6.b.a.24.19	✓	54	5.3	odd	4
115.6.b.a.24.36	yes	54	5.2	odd	4
575.6.a.l.1.19		27	1.1	even	1	trivial
575.6.a.m.1.9		27	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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)

\(f(q)\)	\(=\)	\(q+3.13410 q^{2} -27.5349 q^{3} -22.1774 q^{4} -86.2973 q^{6} -98.9863 q^{7} -169.798 q^{8} +515.172 q^{9} +O(q^{10})\)	\(q+3.13410 q^{2} -27.5349 q^{3} -22.1774 q^{4} -86.2973 q^{6} -98.9863 q^{7} -169.798 q^{8} +515.172 q^{9} -172.976 q^{11} +610.653 q^{12} -499.623 q^{13} -310.233 q^{14} +177.514 q^{16} -185.294 q^{17} +1614.60 q^{18} -457.866 q^{19} +2725.58 q^{21} -542.125 q^{22} +529.000 q^{23} +4675.36 q^{24} -1565.87 q^{26} -7494.24 q^{27} +2195.26 q^{28} +4657.20 q^{29} +9047.26 q^{31} +5989.87 q^{32} +4762.89 q^{33} -580.731 q^{34} -11425.2 q^{36} -5379.45 q^{37} -1435.00 q^{38} +13757.1 q^{39} +957.099 q^{41} +8542.25 q^{42} +19714.5 q^{43} +3836.16 q^{44} +1657.94 q^{46} +22583.7 q^{47} -4887.83 q^{48} -7008.71 q^{49} +5102.06 q^{51} +11080.3 q^{52} -15608.8 q^{53} -23487.7 q^{54} +16807.6 q^{56} +12607.3 q^{57} +14596.1 q^{58} +506.470 q^{59} +31327.9 q^{61} +28355.1 q^{62} -50995.0 q^{63} +13092.4 q^{64} +14927.4 q^{66} -43689.5 q^{67} +4109.34 q^{68} -14566.0 q^{69} -18754.4 q^{71} -87475.0 q^{72} -39706.2 q^{73} -16859.8 q^{74} +10154.3 q^{76} +17122.3 q^{77} +43116.1 q^{78} -21075.0 q^{79} +81166.6 q^{81} +2999.65 q^{82} -59205.2 q^{83} -60446.3 q^{84} +61787.3 q^{86} -128236. q^{87} +29370.9 q^{88} -74791.3 q^{89} +49455.8 q^{91} -11731.8 q^{92} -249116. q^{93} +70779.5 q^{94} -164931. q^{96} +46828.3 q^{97} -21966.0 q^{98} -89112.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 8 q^{2} - 36 q^{3} + 416 q^{4} - 72 q^{6} - 297 q^{7} - 384 q^{8} + 2135 q^{9}+O(q^{10}) \) 27 * q - 8 * q^2 - 36 * q^3 + 416 * q^4 - 72 * q^6 - 297 * q^7 - 384 * q^8 + 2135 * q^9	\( 27 q - 8 q^{2} - 36 q^{3} + 416 q^{4} - 72 q^{6} - 297 q^{7} - 384 q^{8} + 2135 q^{9} - 190 q^{11} - 4518 q^{12} - 2704 q^{13} + 1994 q^{14} + 8316 q^{16} - 5821 q^{17} - 3240 q^{18} + 1220 q^{19} + 2384 q^{21} - 11826 q^{22} + 14283 q^{23} + 8970 q^{24} - 7572 q^{26} - 15294 q^{27} - 12736 q^{28} - 11479 q^{29} - 16985 q^{31} - 27436 q^{32} - 21686 q^{33} - 13060 q^{34} + 35780 q^{36} - 34585 q^{37} - 16760 q^{38} - 6708 q^{39} - 16921 q^{41} - 40528 q^{42} - 48832 q^{43} - 16954 q^{44} - 4232 q^{46} - 21712 q^{47} - 153792 q^{48} + 76886 q^{49} + 48288 q^{51} - 114550 q^{52} - 47803 q^{53} + 45250 q^{54} + 100898 q^{56} - 92832 q^{57} + 96246 q^{58} - 55053 q^{59} + 66462 q^{61} - 56374 q^{62} - 205431 q^{63} + 131054 q^{64} - 61644 q^{66} - 78037 q^{67} - 190622 q^{68} - 19044 q^{69} - 155407 q^{71} - 124230 q^{72} - 387138 q^{73} - 110330 q^{74} + 21110 q^{76} - 81124 q^{77} - 227044 q^{78} + 194674 q^{79} + 182211 q^{81} - 192838 q^{82} - 206961 q^{83} + 17918 q^{84} + 55990 q^{86} - 97856 q^{87} - 616608 q^{88} + 176894 q^{89} - 113316 q^{91} + 220064 q^{92} - 325304 q^{93} - 144902 q^{94} + 147208 q^{96} - 375272 q^{97} - 683362 q^{98} + 7044 q^{99}+O(q^{100}) \) 27 * q - 8 * q^2 - 36 * q^3 + 416 * q^4 - 72 * q^6 - 297 * q^7 - 384 * q^8 + 2135 * q^9 - 190 * q^11 - 4518 * q^12 - 2704 * q^13 + 1994 * q^14 + 8316 * q^16 - 5821 * q^17 - 3240 * q^18 + 1220 * q^19 + 2384 * q^21 - 11826 * q^22 + 14283 * q^23 + 8970 * q^24 - 7572 * q^26 - 15294 * q^27 - 12736 * q^28 - 11479 * q^29 - 16985 * q^31 - 27436 * q^32 - 21686 * q^33 - 13060 * q^34 + 35780 * q^36 - 34585 * q^37 - 16760 * q^38 - 6708 * q^39 - 16921 * q^41 - 40528 * q^42 - 48832 * q^43 - 16954 * q^44 - 4232 * q^46 - 21712 * q^47 - 153792 * q^48 + 76886 * q^49 + 48288 * q^51 - 114550 * q^52 - 47803 * q^53 + 45250 * q^54 + 100898 * q^56 - 92832 * q^57 + 96246 * q^58 - 55053 * q^59 + 66462 * q^61 - 56374 * q^62 - 205431 * q^63 + 131054 * q^64 - 61644 * q^66 - 78037 * q^67 - 190622 * q^68 - 19044 * q^69 - 155407 * q^71 - 124230 * q^72 - 387138 * q^73 - 110330 * q^74 + 21110 * q^76 - 81124 * q^77 - 227044 * q^78 + 194674 * q^79 + 182211 * q^81 - 192838 * q^82 - 206961 * q^83 + 17918 * q^84 + 55990 * q^86 - 97856 * q^87 - 616608 * q^88 + 176894 * q^89 - 113316 * q^91 + 220064 * q^92 - 325304 * q^93 - 144902 * q^94 + 147208 * q^96 - 375272 * q^97 - 683362 * q^98 + 7044 * q^99

Introduction

L-functions

Modular forms

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Database

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