LMFDB - Embedded newform 531.8.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,8,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.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:	$165.876448532$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 $ x^16 - 6x^15 - 1493x^14 + 8791x^13 + 890490x^12 - 5107725x^11 - 269092298x^10 + 1488374176x^9 + 42885295136x^8 - 226132003872x^7 - 3353576629440x^6 + 16796366777600x^5 + 99470801612800x^4 - 494039551757568x^3 - 493048066650624x^2 + 3193975642099712*x - 2385018853548032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-17.9442$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.9442 q^{2} +193.993 q^{4} -1.22320 q^{5} -719.259 q^{7} -1184.20 q^{8} +O(q^{10})$	\(q-17.9442 q^{2} +193.993 q^{4} -1.22320 q^{5} -719.259 q^{7} -1184.20 q^{8} +21.9494 q^{10} -1110.85 q^{11} +8283.40 q^{13} +12906.5 q^{14} -3581.74 q^{16} -4834.97 q^{17} -31620.9 q^{19} -237.293 q^{20} +19933.3 q^{22} +43588.0 q^{23} -78123.5 q^{25} -148639. q^{26} -139531. q^{28} +232502. q^{29} -85314.0 q^{31} +215848. q^{32} +86759.6 q^{34} +879.800 q^{35} -52796.8 q^{37} +567411. q^{38} +1448.51 q^{40} +589172. q^{41} +367000. q^{43} -215497. q^{44} -782150. q^{46} -739433. q^{47} -306210. q^{49} +1.40186e6 q^{50} +1.60692e6 q^{52} -1.53905e6 q^{53} +1358.79 q^{55} +851743. q^{56} -4.17205e6 q^{58} -205379. q^{59} +325249. q^{61} +1.53089e6 q^{62} -3.41476e6 q^{64} -10132.3 q^{65} -1.72338e6 q^{67} -937953. q^{68} -15787.3 q^{70} +4.29199e6 q^{71} -1.55847e6 q^{73} +947394. q^{74} -6.13424e6 q^{76} +798988. q^{77} -3.39722e6 q^{79} +4381.20 q^{80} -1.05722e7 q^{82} +7.04945e6 q^{83} +5914.16 q^{85} -6.58552e6 q^{86} +1.31546e6 q^{88} -1.14007e7 q^{89} -5.95791e6 q^{91} +8.45577e6 q^{92} +1.32685e7 q^{94} +38678.8 q^{95} +1.36312e7 q^{97} +5.49468e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8}+O(q^{10}) $ 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8	\( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8} - 3479 q^{10} - 898 q^{11} - 8172 q^{13} + 13315 q^{14} + 3138 q^{16} + 44985 q^{17} - 40137 q^{19} - 130657 q^{20} + 109394 q^{22} + 2833 q^{23} + 285746 q^{25} + 129420 q^{26} + 112890 q^{28} - 144375 q^{29} - 141759 q^{31} + 36224 q^{32} - 341332 q^{34} + 78859 q^{35} - 297971 q^{37} - 329075 q^{38} - 203048 q^{40} - 659077 q^{41} - 1431608 q^{43} - 254916 q^{44} + 873113 q^{46} + 1574073 q^{47} + 1893545 q^{49} - 302533 q^{50} - 4972548 q^{52} - 587736 q^{53} - 4624036 q^{55} + 5798506 q^{56} - 6991380 q^{58} - 3286064 q^{59} - 6117131 q^{61} + 11570258 q^{62} - 19063011 q^{64} + 5335514 q^{65} - 16518710 q^{67} + 17284669 q^{68} - 39189486 q^{70} + 10882582 q^{71} - 21097441 q^{73} + 16717030 q^{74} - 40864952 q^{76} + 3404601 q^{77} - 3784458 q^{79} + 27466195 q^{80} - 24990117 q^{82} + 1951425 q^{83} - 23238675 q^{85} + 35910572 q^{86} - 27843055 q^{88} - 10499443 q^{89} + 699217 q^{91} + 20062766 q^{92} - 59358988 q^{94} + 29236333 q^{95} - 25158976 q^{97} - 2120460 q^{98}+O(q^{100}) \) 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8 - 3479 * q^10 - 898 * q^11 - 8172 * q^13 + 13315 * q^14 + 3138 * q^16 + 44985 * q^17 - 40137 * q^19 - 130657 * q^20 + 109394 * q^22 + 2833 * q^23 + 285746 * q^25 + 129420 * q^26 + 112890 * q^28 - 144375 * q^29 - 141759 * q^31 + 36224 * q^32 - 341332 * q^34 + 78859 * q^35 - 297971 * q^37 - 329075 * q^38 - 203048 * q^40 - 659077 * q^41 - 1431608 * q^43 - 254916 * q^44 + 873113 * q^46 + 1574073 * q^47 + 1893545 * q^49 - 302533 * q^50 - 4972548 * q^52 - 587736 * q^53 - 4624036 * q^55 + 5798506 * q^56 - 6991380 * q^58 - 3286064 * q^59 - 6117131 * q^61 + 11570258 * q^62 - 19063011 * q^64 + 5335514 * q^65 - 16518710 * q^67 + 17284669 * q^68 - 39189486 * q^70 + 10882582 * q^71 - 21097441 * q^73 + 16717030 * q^74 - 40864952 * q^76 + 3404601 * q^77 - 3784458 * q^79 + 27466195 * q^80 - 24990117 * q^82 + 1951425 * q^83 - 23238675 * q^85 + 35910572 * q^86 - 27843055 * q^88 - 10499443 * q^89 + 699217 * q^91 + 20062766 * q^92 - 59358988 * q^94 + 29236333 * q^95 - 25158976 * q^97 - 2120460 * q^98

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 $

$2$	−17.9442	−1.58606	−0.793028	−	0.609185i	$-0.791497\pi$
$2$	−17.9442	−1.58606	−0.793028	+	0.609185i	$0.791497\pi$
$3$	0	0
$3$	0	0
$4$	193.993	1.51557
$5$	−1.22320	−0.00437627	−0.00218813	−	0.999998i	$-0.500697\pi$
$5$	−1.22320	−0.00437627	−0.00218813	+	0.999998i	$0.500697\pi$
$6$	0	0
$7$	−719.259	−0.792578	−0.396289	−	0.918126i	$-0.629702\pi$
$7$	−719.259	−0.792578	−0.396289	+	0.918126i	$0.629702\pi$
$8$	−1184.20	−0.817727
$9$	0	0
$10$	21.9494	0.00694101
$11$	−1110.85	−0.251640	−0.125820	−	0.992053i	$-0.540156\pi$
$11$	−1110.85	−0.251640	−0.125820	+	0.992053i	$0.540156\pi$
$12$	0	0
$13$	8283.40	1.04570	0.522850	−	0.852425i	$-0.324869\pi$
$13$	8283.40	1.04570	0.522850	+	0.852425i	$0.324869\pi$
$14$	12906.5	1.25707
$15$	0	0
$16$	−3581.74	−0.218612
$17$	−4834.97	−0.238684	−0.119342	−	0.992853i	$-0.538078\pi$
$17$	−4834.97	−0.238684	−0.119342	+	0.992853i	$0.538078\pi$
$18$	0	0
$19$	−31620.9	−1.05764	−0.528818	−	0.848735i	$-0.677365\pi$
$19$	−31620.9	−1.05764	−0.528818	+	0.848735i	$0.677365\pi$
$20$	−237.293	−0.00663255
$21$	0	0
$22$	19933.3	0.399116
$23$	43588.0	0.746997	0.373499	−	0.927631i	$-0.378158\pi$
$23$	43588.0	0.746997	0.373499	+	0.927631i	$0.378158\pi$
$24$	0	0
$25$	−78123.5	−0.999981
$26$	−148639.	−1.65854
$27$	0	0
$28$	−139531.	−1.20121
$29$	232502.	1.77025	0.885123	−	0.465358i	$-0.154074\pi$
$29$	232502.	1.77025	0.885123	+	0.465358i	$0.154074\pi$
$30$	0	0
$31$	−85314.0	−0.514345	−0.257172	−	0.966366i	$-0.582791\pi$
$31$	−85314.0	−0.514345	−0.257172	+	0.966366i	$0.582791\pi$
$32$	215848.	1.16446
$33$	0	0
$34$	86759.6	0.378566
$35$	879.800	0.00346853
$36$	0	0
$37$	−52796.8	−0.171357	−0.0856784	−	0.996323i	$-0.527306\pi$
$37$	−52796.8	−0.171357	−0.0856784	+	0.996323i	$0.527306\pi$
$38$	567411.	1.67747
$39$	0	0
$40$	1448.51	0.00357859
$41$	589172.	1.33505	0.667527	−	0.744585i	$-0.267353\pi$
$41$	589172.	1.33505	0.667527	+	0.744585i	$0.267353\pi$
$42$	0	0
$43$	367000.	0.703926	0.351963	−	0.936014i	$-0.385514\pi$
$43$	367000.	0.703926	0.351963	+	0.936014i	$0.385514\pi$
$44$	−215497.	−0.381379
$45$	0	0
$46$	−782150.	−1.18478
$47$	−739433.	−1.03886	−0.519429	−	0.854514i	$-0.673855\pi$
$47$	−739433.	−1.03886	−0.519429	+	0.854514i	$0.673855\pi$
$48$	0	0
$49$	−306210.	−0.371820
$50$	1.40186e6	1.58603
$51$	0	0
$52$	1.60692e6	1.58483
$53$	−1.53905e6	−1.42000	−0.710000	−	0.704201i	$-0.751305\pi$
$53$	−1.53905e6	−1.42000	−0.710000	+	0.704201i	$0.751305\pi$
$54$	0	0
$55$	1358.79	0.00110125
$56$	851743.	0.648113
$57$	0	0
$58$	−4.17205e6	−2.80771
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	325249.	0.183469	0.0917343	−	0.995784i	$-0.470759\pi$
$61$	325249.	0.183469	0.0917343	+	0.995784i	$0.470759\pi$
$62$	1.53089e6	0.815779
$63$	0	0
$64$	−3.41476e6	−1.62828
$65$	−10132.3	−0.00457626
$66$	0	0
$67$	−1.72338e6	−0.700035	−0.350018	−	0.936743i	$-0.613824\pi$
$67$	−1.72338e6	−0.700035	−0.350018	+	0.936743i	$0.613824\pi$
$68$	−937953.	−0.361743
$69$	0	0
$70$	−15787.3	−0.00550129
$71$	4.29199e6	1.42316	0.711582	−	0.702603i	$-0.247979\pi$
$71$	4.29199e6	1.42316	0.711582	+	0.702603i	$0.247979\pi$
$72$	0	0
$73$	−1.55847e6	−0.468888	−0.234444	−	0.972130i	$-0.575327\pi$
$73$	−1.55847e6	−0.468888	−0.234444	+	0.972130i	$0.575327\pi$
$74$	947394.	0.271781
$75$	0	0
$76$	−6.13424e6	−1.60292
$77$	798988.	0.199445
$78$	0	0
$79$	−3.39722e6	−0.775226	−0.387613	−	0.921822i	$-0.626700\pi$
$79$	−3.39722e6	−0.775226	−0.387613	+	0.921822i	$0.626700\pi$
$80$	4381.20	0.000956704 0
$81$	0	0
$82$	−1.05722e7	−2.11747
$83$	7.04945e6	1.35326	0.676631	−	0.736322i	$-0.263439\pi$
$83$	7.04945e6	1.35326	0.676631	+	0.736322i	$0.263439\pi$
$84$	0	0
$85$	5914.16	0.00104454
$86$	−6.58552e6	−1.11647
$87$	0	0
$88$	1.31546e6	0.205773
$89$	−1.14007e7	−1.71422	−0.857109	−	0.515136i	$-0.827741\pi$
$89$	−1.14007e7	−1.71422	−0.857109	+	0.515136i	$0.827741\pi$
$90$	0	0
$91$	−5.95791e6	−0.828798
$92$	8.45577e6	1.13213
$93$	0	0
$94$	1.32685e7	1.64769
$95$	38678.8	0.00462850
$96$	0	0
$97$	1.36312e7	1.51647	0.758236	−	0.651980i	$-0.226062\pi$
$97$	1.36312e7	1.51647	0.758236	+	0.651980i	$0.226062\pi$
$98$	5.49468e6	0.589727
$99$	0	0
$100$	−1.51554e7	−1.51554
$101$	1.39105e7	1.34344	0.671719	−	0.740806i	$-0.265556\pi$
$101$	1.39105e7	1.34344	0.671719	+	0.740806i	$0.265556\pi$
$102$	0	0
$103$	5.48771e6	0.494835	0.247418	−	0.968909i	$-0.420418\pi$
$103$	5.48771e6	0.494835	0.247418	+	0.968909i	$0.420418\pi$
$104$	−9.80916e6	−0.855097
$105$	0	0
$106$	2.76171e7	2.25220
$107$	−735470.	−0.0580393	−0.0290196	−	0.999579i	$-0.509239\pi$
$107$	−735470.	−0.0580393	−0.0290196	+	0.999579i	$0.509239\pi$
$108$	0	0
$109$	2.33048e7	1.72366	0.861831	−	0.507196i	$-0.169318\pi$
$109$	2.33048e7	1.72366	0.861831	+	0.507196i	$0.169318\pi$
$110$	−24382.5	−0.00174664
$111$	0	0
$112$	2.57620e6	0.173267
$113$	−1.31124e7	−0.854885	−0.427442	−	0.904043i	$-0.640585\pi$
$113$	−1.31124e7	−0.854885	−0.427442	+	0.904043i	$0.640585\pi$
$114$	0	0
$115$	−53317.0	−0.00326906
$116$	4.51038e7	2.68294
$117$	0	0
$118$	3.68536e6	0.206487
$119$	3.47760e6	0.189176
$120$	0	0
$121$	−1.82532e7	−0.936677
$122$	−5.83632e6	−0.290991
$123$	0	0
$124$	−1.65503e7	−0.779527
$125$	191124.	0.00875245
$126$	0	0
$127$	2.57648e7	1.11613	0.558063	−	0.829799i	$-0.311545\pi$
$127$	2.57648e7	1.11613	0.558063	+	0.829799i	$0.311545\pi$
$128$	3.36464e7	1.41809
$129$	0	0
$130$	181815.	0.00725820
$131$	4.04125e7	1.57060	0.785302	−	0.619113i	$-0.212508\pi$
$131$	4.04125e7	1.57060	0.785302	+	0.619113i	$0.212508\pi$
$132$	0	0
$133$	2.27436e7	0.838259
$134$	3.09247e7	1.11029
$135$	0	0
$136$	5.72555e6	0.195178
$137$	−3.40595e7	−1.13166	−0.565830	−	0.824522i	$-0.691444\pi$
$137$	−3.40595e7	−1.13166	−0.565830	+	0.824522i	$0.691444\pi$
$138$	0	0
$139$	3.98464e6	0.125845	0.0629227	−	0.998018i	$-0.479958\pi$
$139$	3.98464e6	0.125845	0.0629227	+	0.998018i	$0.479958\pi$
$140$	170675.	0.00525682
$141$	0	0
$142$	−7.70163e7	−2.25722
$143$	−9.20160e6	−0.263140
$144$	0	0
$145$	−284397.	−0.00774707
$146$	2.79655e7	0.743682
$147$	0	0
$148$	−1.02422e7	−0.259704
$149$	−5.83891e6	−0.144604	−0.0723020	−	0.997383i	$-0.523035\pi$
$149$	−5.83891e6	−0.144604	−0.0723020	+	0.997383i	$0.523035\pi$
$150$	0	0
$151$	5.13811e7	1.21446	0.607231	−	0.794525i	$-0.292280\pi$
$151$	5.13811e7	1.21446	0.607231	+	0.794525i	$0.292280\pi$
$152$	3.74453e7	0.864858
$153$	0	0
$154$	−1.43372e7	−0.316330
$155$	104356.	0.00225091
$156$	0	0
$157$	−1.14443e7	−0.236016	−0.118008	−	0.993013i	$-0.537651\pi$
$157$	−1.14443e7	−0.236016	−0.118008	+	0.993013i	$0.537651\pi$
$158$	6.09602e7	1.22955
$159$	0	0
$160$	−264027.	−0.00509598
$161$	−3.13510e7	−0.592054
$162$	0	0
$163$	1.25340e7	0.226690	0.113345	−	0.993556i	$-0.463844\pi$
$163$	1.25340e7	0.226690	0.113345	+	0.993556i	$0.463844\pi$
$164$	1.14296e8	2.02337
$165$	0	0
$166$	−1.26497e8	−2.14635
$167$	3.77486e7	0.627182	0.313591	−	0.949558i	$-0.398468\pi$
$167$	3.77486e7	0.627182	0.313591	+	0.949558i	$0.398468\pi$
$168$	0	0
$169$	5.86614e6	0.0934865
$170$	−106125.	−0.00165671
$171$	0	0
$172$	7.11956e7	1.06685
$173$	2.06219e7	0.302808	0.151404	−	0.988472i	$-0.451621\pi$
$173$	2.06219e7	0.302808	0.151404	+	0.988472i	$0.451621\pi$
$174$	0	0
$175$	5.61910e7	0.792563
$176$	3.97877e6	0.0550116
$177$	0	0
$178$	2.04576e8	2.71884
$179$	1.01937e8	1.32845	0.664227	−	0.747531i	$-0.268761\pi$
$179$	1.01937e8	1.32845	0.664227	+	0.747531i	$0.268761\pi$
$180$	0	0
$181$	9.14678e6	0.114655	0.0573275	−	0.998355i	$-0.481742\pi$
$181$	9.14678e6	0.114655	0.0573275	+	0.998355i	$0.481742\pi$
$182$	1.06910e8	1.31452
$183$	0	0
$184$	−5.16167e7	−0.610840
$185$	64581.2	0.000749903 0
$186$	0	0
$187$	5.37092e6	0.0600625
$188$	−1.43445e8	−1.57447
$189$	0	0
$190$	−694059.	−0.00734106
$191$	−1.28136e8	−1.33062	−0.665311	−	0.746566i	$-0.731701\pi$
$191$	−1.28136e8	−1.33062	−0.665311	+	0.746566i	$0.731701\pi$
$192$	0	0
$193$	2.97509e7	0.297885	0.148943	−	0.988846i	$-0.452413\pi$
$193$	2.97509e7	0.297885	0.148943	+	0.988846i	$0.452413\pi$
$194$	−2.44601e8	−2.40521
$195$	0	0
$196$	−5.94027e7	−0.563521
$197$	1.48736e7	0.138607	0.0693036	−	0.997596i	$-0.477922\pi$
$197$	1.48736e7	0.138607	0.0693036	+	0.997596i	$0.477922\pi$
$198$	0	0
$199$	−1.07194e8	−0.964242	−0.482121	−	0.876105i	$-0.660133\pi$
$199$	−1.07194e8	−0.964242	−0.482121	+	0.876105i	$0.660133\pi$
$200$	9.25135e7	0.817712
$201$	0	0
$202$	−2.49612e8	−2.13077
$203$	−1.67229e8	−1.40306
$204$	0	0
$205$	−720678.	−0.00584256
$206$	−9.84724e7	−0.784836
$207$	0	0
$208$	−2.96690e7	−0.228602
$209$	3.51260e7	0.266144
$210$	0	0
$211$	8.62669e7	0.632201	0.316101	−	0.948726i	$-0.397626\pi$
$211$	8.62669e7	0.632201	0.316101	+	0.948726i	$0.397626\pi$
$212$	−2.98566e8	−2.15211
$213$	0	0
$214$	1.31974e7	0.0920535
$215$	−448917.	−0.00308057
$216$	0	0
$217$	6.13628e7	0.407658
$218$	−4.18185e8	−2.73382
$219$	0	0
$220$	263597.	0.00166902
$221$	−4.00500e7	−0.249591
$222$	0	0
$223$	1.22690e8	0.740871	0.370435	−	0.928858i	$-0.379208\pi$
$223$	1.22690e8	0.740871	0.370435	+	0.928858i	$0.379208\pi$
$224$	−1.55251e8	−0.922924
$225$	0	0
$226$	2.35291e8	1.35590
$227$	−1.12105e8	−0.636115	−0.318057	−	0.948071i	$-0.603030\pi$
$227$	−1.12105e8	−0.636115	−0.318057	+	0.948071i	$0.603030\pi$
$228$	0	0
$229$	1.28324e8	0.706131	0.353066	−	0.935599i	$-0.385139\pi$
$229$	1.28324e8	0.706131	0.353066	+	0.935599i	$0.385139\pi$
$230$	956729.	0.00518491
$231$	0	0
$232$	−2.75328e8	−1.44758
$233$	3.60112e7	0.186505	0.0932527	−	0.995642i	$-0.470274\pi$
$233$	3.60112e7	0.186505	0.0932527	+	0.995642i	$0.470274\pi$
$234$	0	0
$235$	904477.	0.00454632
$236$	−3.98422e7	−0.197311
$237$	0	0
$238$	−6.24026e7	−0.300043
$239$	2.30387e8	1.09160	0.545801	−	0.837915i	$-0.316225\pi$
$239$	2.30387e8	1.09160	0.545801	+	0.837915i	$0.316225\pi$
$240$	0	0
$241$	−3.09598e8	−1.42475	−0.712376	−	0.701798i	$-0.752381\pi$
$241$	−3.09598e8	−1.42475	−0.712376	+	0.701798i	$0.752381\pi$
$242$	3.27538e8	1.48562
$243$	0	0
$244$	6.30961e7	0.278060
$245$	374557.	0.00162718
$246$	0	0
$247$	−2.61928e8	−1.10597
$248$	1.01028e8	0.420594
$249$	0	0
$250$	−3.42956e6	−0.0138819
$251$	−1.81483e8	−0.724399	−0.362200	−	0.932101i	$-0.617974\pi$
$251$	−1.81483e8	−0.724399	−0.362200	+	0.932101i	$0.617974\pi$
$252$	0	0
$253$	−4.84196e7	−0.187975
$254$	−4.62327e8	−1.77024
$255$	0	0
$256$	−1.66668e8	−0.620887
$257$	−4.15191e8	−1.52574	−0.762872	−	0.646549i	$-0.776212\pi$
$257$	−4.15191e8	−1.52574	−0.762872	+	0.646549i	$0.776212\pi$
$258$	0	0
$259$	3.79745e7	0.135814
$260$	−1.96560e6	−0.00693566
$261$	0	0
$262$	−7.25169e8	−2.49106
$263$	3.81009e8	1.29149	0.645744	−	0.763554i	$-0.276547\pi$
$263$	3.81009e8	1.29149	0.645744	+	0.763554i	$0.276547\pi$
$264$	0	0
$265$	1.88258e6	0.00621430
$266$	−4.08115e8	−1.32953
$267$	0	0
$268$	−3.34325e8	−1.06095
$269$	9.53490e7	0.298664	0.149332	−	0.988787i	$-0.452288\pi$
$269$	9.53490e7	0.298664	0.149332	+	0.988787i	$0.452288\pi$
$270$	0	0
$271$	−4.17979e8	−1.27574	−0.637870	−	0.770144i	$-0.720184\pi$
$271$	−4.17979e8	−1.27574	−0.637870	+	0.770144i	$0.720184\pi$
$272$	1.73176e7	0.0521791
$273$	0	0
$274$	6.11169e8	1.79488
$275$	8.67834e7	0.251636
$276$	0	0
$277$	−3.26940e8	−0.924249	−0.462125	−	0.886815i	$-0.652913\pi$
$277$	−3.26940e8	−0.924249	−0.462125	+	0.886815i	$0.652913\pi$
$278$	−7.15011e7	−0.199598
$279$	0	0
$280$	−1.04186e6	−0.00283631
$281$	−9.47550e7	−0.254759	−0.127380	−	0.991854i	$-0.540657\pi$
$281$	−9.47550e7	−0.254759	−0.127380	+	0.991854i	$0.540657\pi$
$282$	0	0
$283$	−4.46938e8	−1.17218	−0.586091	−	0.810246i	$-0.699334\pi$
$283$	−4.46938e8	−1.17218	−0.586091	+	0.810246i	$0.699334\pi$
$284$	8.32618e8	2.15691
$285$	0	0
$286$	1.65115e8	0.417355
$287$	−4.23767e8	−1.05813
$288$	0	0
$289$	−3.86962e8	−0.943030
$290$	5.10327e6	0.0122873
$291$	0	0
$292$	−3.02333e8	−0.710634
$293$	−4.04818e8	−0.940207	−0.470103	−	0.882611i	$-0.655783\pi$
$293$	−4.04818e8	−0.940207	−0.470103	+	0.882611i	$0.655783\pi$
$294$	0	0
$295$	251220.	0.000569742 0
$296$	6.25217e7	0.140123
$297$	0	0
$298$	1.04774e8	0.229350
$299$	3.61056e8	0.781134
$300$	0	0
$301$	−2.63968e8	−0.557916
$302$	−9.21991e8	−1.92620
$303$	0	0
$304$	1.13258e8	0.231212
$305$	−397846.	−0.000802908 0
$306$	0	0
$307$	−7.95438e8	−1.56900	−0.784498	−	0.620131i	$-0.787079\pi$
$307$	−7.95438e8	−1.56900	−0.784498	+	0.620131i	$0.787079\pi$
$308$	1.54998e8	0.302273
$309$	0	0
$310$	−1.87259e6	−0.00357007
$311$	−3.49530e8	−0.658906	−0.329453	−	0.944172i	$-0.606864\pi$
$311$	−3.49530e8	−0.658906	−0.329453	+	0.944172i	$0.606864\pi$
$312$	0	0
$313$	9.82107e7	0.181031	0.0905157	−	0.995895i	$-0.471148\pi$
$313$	9.82107e7	0.181031	0.0905157	+	0.995895i	$0.471148\pi$
$314$	2.05359e8	0.374334
$315$	0	0
$316$	−6.59037e8	−1.17491
$317$	−9.11476e8	−1.60708	−0.803541	−	0.595250i	$-0.797053\pi$
$317$	−9.11476e8	−1.60708	−0.803541	+	0.595250i	$0.797053\pi$
$318$	0	0
$319$	−2.58274e8	−0.445465
$320$	4.17695e6	0.00712580
$321$	0	0
$322$	5.62568e8	0.939030
$323$	1.52886e8	0.252441
$324$	0	0
$325$	−6.47128e8	−1.04568
$326$	−2.24912e8	−0.359542
$327$	0	0
$328$	−6.97695e8	−1.09171
$329$	5.31844e8	0.823376
$330$	0	0
$331$	8.69089e8	1.31724	0.658622	−	0.752474i	$-0.271140\pi$
$331$	8.69089e8	1.31724	0.658622	+	0.752474i	$0.271140\pi$
$332$	1.36755e9	2.05097
$333$	0	0
$334$	−6.77368e8	−0.994745
$335$	2.10805e6	0.00306354
$336$	0	0
$337$	3.71807e7	0.0529192	0.0264596	−	0.999650i	$-0.491577\pi$
$337$	3.71807e7	0.0529192	0.0264596	+	0.999650i	$0.491577\pi$
$338$	−1.05263e8	−0.148275
$339$	0	0
$340$	1.14731e6	0.00158308
$341$	9.47709e7	0.129430
$342$	0	0
$343$	8.12585e8	1.08727
$344$	−4.34600e8	−0.575619
$345$	0	0
$346$	−3.70043e8	−0.480270
$347$	5.17499e8	0.664900	0.332450	−	0.943121i	$-0.392125\pi$
$347$	5.17499e8	0.664900	0.332450	+	0.943121i	$0.392125\pi$
$348$	0	0
$349$	−7.74296e8	−0.975030	−0.487515	−	0.873115i	$-0.662097\pi$
$349$	−7.74296e8	−0.975030	−0.487515	+	0.873115i	$0.662097\pi$
$350$	−1.00830e9	−1.25705
$351$	0	0
$352$	−2.39775e8	−0.293025
$353$	6.61876e8	0.800876	0.400438	−	0.916324i	$-0.368858\pi$
$353$	6.61876e8	0.800876	0.400438	+	0.916324i	$0.368858\pi$
$354$	0	0
$355$	−5.24998e6	−0.00622815
$356$	−2.21166e9	−2.59802
$357$	0	0
$358$	−1.82918e9	−2.10700
$359$	−6.26557e8	−0.714711	−0.357355	−	0.933968i	$-0.616321\pi$
$359$	−6.26557e8	−0.714711	−0.357355	+	0.933968i	$0.616321\pi$
$360$	0	0
$361$	1.06008e8	0.118595
$362$	−1.64131e8	−0.181849
$363$	0	0
$364$	−1.15579e9	−1.25610
$365$	1.90633e6	0.00205198
$366$	0	0
$367$	8.99146e7	0.0949509	0.0474755	−	0.998872i	$-0.484882\pi$
$367$	8.99146e7	0.0949509	0.0474755	+	0.998872i	$0.484882\pi$
$368$	−1.56121e8	−0.163303
$369$	0	0
$370$	−1.15886e6	−0.00118939
$371$	1.10698e9	1.12546
$372$	0	0
$373$	1.93821e8	0.193383	0.0966917	−	0.995314i	$-0.469174\pi$
$373$	1.93821e8	0.193383	0.0966917	+	0.995314i	$0.469174\pi$
$374$	−9.63768e7	−0.0952625
$375$	0	0
$376$	8.75633e8	0.849503
$377$	1.92591e9	1.85114
$378$	0	0
$379$	−2.28105e8	−0.215227	−0.107614	−	0.994193i	$-0.534321\pi$
$379$	−2.28105e8	−0.215227	−0.107614	+	0.994193i	$0.534321\pi$
$380$	7.50343e6	0.00701483
$381$	0	0
$382$	2.29930e9	2.11044
$383$	1.16094e8	0.105588	0.0527938	−	0.998605i	$-0.483187\pi$
$383$	1.16094e8	0.105588	0.0527938	+	0.998605i	$0.483187\pi$
$384$	0	0
$385$	−977325.	−0.000872824 0
$386$	−5.33855e8	−0.472463
$387$	0	0
$388$	2.64437e9	2.29832
$389$	−6.84222e8	−0.589351	−0.294675	−	0.955597i	$-0.595211\pi$
$389$	−6.84222e8	−0.589351	−0.294675	+	0.955597i	$0.595211\pi$
$390$	0	0
$391$	−2.10747e8	−0.178296
$392$	3.62612e8	0.304047
$393$	0	0
$394$	−2.66895e8	−0.219839
$395$	4.15549e6	0.00339260
$396$	0	0
$397$	−1.73331e9	−1.39031	−0.695153	−	0.718862i	$-0.744663\pi$
$397$	−1.73331e9	−1.39031	−0.695153	+	0.718862i	$0.744663\pi$
$398$	1.92351e9	1.52934
$399$	0	0
$400$	2.79818e8	0.218608
$401$	−1.39266e9	−1.07855	−0.539274	−	0.842130i	$-0.681301\pi$
$401$	−1.39266e9	−1.07855	−0.539274	+	0.842130i	$0.681301\pi$
$402$	0	0
$403$	−7.06690e8	−0.537850
$404$	2.69854e9	2.03608
$405$	0	0
$406$	3.00079e9	2.22533
$407$	5.86492e7	0.0431203
$408$	0	0
$409$	−1.48753e9	−1.07506	−0.537530	−	0.843244i	$-0.680643\pi$
$409$	−1.48753e9	−1.07506	−0.537530	+	0.843244i	$0.680643\pi$
$410$	1.29320e7	0.00926662
$411$	0	0
$412$	1.06458e9	0.749959
$413$	1.47721e8	0.103185
$414$	0	0
$415$	−8.62292e6	−0.00592224
$416$	1.78796e9	1.21767
$417$	0	0
$418$	−6.30307e8	−0.422119
$419$	−9.73899e8	−0.646792	−0.323396	−	0.946264i	$-0.604825\pi$
$419$	−9.73899e8	−0.646792	−0.323396	+	0.946264i	$0.604825\pi$
$420$	0	0
$421$	2.01358e9	1.31517	0.657584	−	0.753381i	$-0.271578\pi$
$421$	2.01358e9	1.31517	0.657584	+	0.753381i	$0.271578\pi$
$422$	−1.54799e9	−1.00271
$423$	0	0
$424$	1.82254e9	1.16117
$425$	3.77725e8	0.238679
$426$	0	0
$427$	−2.33938e8	−0.145413
$428$	−1.42676e8	−0.0879627
$429$	0	0
$430$	8.05544e6	0.00488596
$431$	−2.94920e8	−0.177433	−0.0887164	−	0.996057i	$-0.528276\pi$
$431$	−2.94920e8	−0.177433	−0.0887164	+	0.996057i	$0.528276\pi$
$432$	0	0
$433$	−1.42354e9	−0.842676	−0.421338	−	0.906904i	$-0.638439\pi$
$433$	−1.42354e9	−0.842676	−0.421338	+	0.906904i	$0.638439\pi$
$434$	−1.10111e9	−0.646569
$435$	0	0
$436$	4.52097e9	2.61233
$437$	−1.37829e9	−0.790052
$438$	0	0
$439$	−1.55418e9	−0.876750	−0.438375	−	0.898792i	$-0.644446\pi$
$439$	−1.55418e9	−0.876750	−0.438375	+	0.898792i	$0.644446\pi$
$440$	−1.60908e6	−0.000900519 0
$441$	0	0
$442$	7.18664e8	0.395866
$443$	6.41286e8	0.350460	0.175230	−	0.984528i	$-0.443933\pi$
$443$	6.41286e8	0.350460	0.175230	+	0.984528i	$0.443933\pi$
$444$	0	0
$445$	1.39454e7	0.00750187
$446$	−2.20157e9	−1.17506
$447$	0	0
$448$	2.45609e9	1.29054
$449$	−2.96725e9	−1.54701	−0.773503	−	0.633793i	$-0.781497\pi$
$449$	−2.96725e9	−1.54701	−0.773503	+	0.633793i	$0.781497\pi$
$450$	0	0
$451$	−6.54481e8	−0.335954
$452$	−2.54372e9	−1.29564
$453$	0	0
$454$	2.01164e9	1.00891
$455$	7.28774e6	0.00362704
$456$	0	0
$457$	−3.31780e9	−1.62609	−0.813043	−	0.582203i	$-0.802191\pi$
$457$	−3.31780e9	−1.62609	−0.813043	+	0.582203i	$0.802191\pi$
$458$	−2.30268e9	−1.11996
$459$	0	0
$460$	−1.03431e7	−0.00495450
$461$	−9.48650e8	−0.450975	−0.225488	−	0.974246i	$-0.572398\pi$
$461$	−9.48650e8	−0.450975	−0.225488	+	0.974246i	$0.572398\pi$
$462$	0	0
$463$	1.33385e9	0.624561	0.312281	−	0.949990i	$-0.398907\pi$
$463$	1.33385e9	0.624561	0.312281	+	0.949990i	$0.398907\pi$
$464$	−8.32761e8	−0.386997
$465$	0	0
$466$	−6.46191e8	−0.295808
$467$	3.76180e9	1.70918	0.854588	−	0.519306i	$-0.173810\pi$
$467$	3.76180e9	1.70918	0.854588	+	0.519306i	$0.173810\pi$
$468$	0	0
$469$	1.23956e9	0.554833
$470$	−1.62301e7	−0.00721072
$471$	0	0
$472$	2.43209e8	0.106459
$473$	−4.07682e8	−0.177136
$474$	0	0
$475$	2.47033e9	1.05762
$476$	6.74631e8	0.286709
$477$	0	0
$478$	−4.13410e9	−1.73134
$479$	2.93522e9	1.22030	0.610151	−	0.792285i	$-0.291109\pi$
$479$	2.93522e9	1.22030	0.610151	+	0.792285i	$0.291109\pi$
$480$	0	0
$481$	−4.37337e8	−0.179188
$482$	5.55549e9	2.25973
$483$	0	0
$484$	−3.54100e9	−1.41960
$485$	−1.66738e7	−0.00663649
$486$	0	0
$487$	−1.25935e9	−0.494077	−0.247038	−	0.969006i	$-0.579457\pi$
$487$	−1.25935e9	−0.494077	−0.247038	+	0.969006i	$0.579457\pi$
$488$	−3.85158e8	−0.150027
$489$	0	0
$490$	−6.72112e6	−0.00258081
$491$	−4.20973e8	−0.160498	−0.0802489	−	0.996775i	$-0.525572\pi$
$491$	−4.20973e8	−0.160498	−0.0802489	+	0.996775i	$0.525572\pi$
$492$	0	0
$493$	−1.12414e9	−0.422529
$494$	4.70009e9	1.75413
$495$	0	0
$496$	3.05572e8	0.112442
$497$	−3.08705e9	−1.12797
$498$	0	0
$499$	−4.86604e9	−1.75317	−0.876584	−	0.481248i	$-0.840184\pi$
$499$	−4.86604e9	−1.75317	−0.876584	+	0.481248i	$0.840184\pi$
$500$	3.70767e7	0.0132650
$501$	0	0
$502$	3.25656e9	1.14894
$503$	3.22489e9	1.12987	0.564933	−	0.825137i	$-0.308902\pi$
$503$	3.22489e9	1.12987	0.564933	+	0.825137i	$0.308902\pi$
$504$	0	0
$505$	−1.70154e7	−0.00587925
$506$	8.68850e8	0.298138
$507$	0	0
$508$	4.99819e9	1.69157
$509$	−1.08431e8	−0.0364452	−0.0182226	−	0.999834i	$-0.505801\pi$
$509$	−1.08431e8	−0.0364452	−0.0182226	+	0.999834i	$0.505801\pi$
$510$	0	0
$511$	1.12094e9	0.371630
$512$	−1.31602e9	−0.433329
$513$	0	0
$514$	7.45026e9	2.41992
$515$	−6.71259e6	−0.00216553
$516$	0	0
$517$	8.21398e8	0.261419
$518$	−6.81422e8	−0.215408
$519$	0	0
$520$	1.19986e7	0.00374213
$521$	−2.88309e8	−0.0893152	−0.0446576	−	0.999002i	$-0.514220\pi$
$521$	−2.88309e8	−0.0893152	−0.0446576	+	0.999002i	$0.514220\pi$
$522$	0	0
$523$	8.83228e8	0.269971	0.134985	−	0.990848i	$-0.456901\pi$
$523$	8.83228e8	0.269971	0.134985	+	0.990848i	$0.456901\pi$
$524$	7.83976e9	2.38036
$525$	0	0
$526$	−6.83689e9	−2.04837
$527$	4.12491e8	0.122766
$528$	0	0
$529$	−1.50492e9	−0.441995
$530$	−3.37813e7	−0.00985623
$531$	0	0
$532$	4.41211e9	1.27044
$533$	4.88035e9	1.39607
$534$	0	0
$535$	899630.	0.000253995 0
$536$	2.04082e9	0.572438
$537$	0	0
$538$	−1.71096e9	−0.473698
$539$	3.40153e8	0.0935650
$540$	0	0
$541$	−3.92838e9	−1.06665	−0.533326	−	0.845910i	$-0.679058\pi$
$541$	−3.92838e9	−1.06665	−0.533326	+	0.845910i	$0.679058\pi$
$542$	7.50029e9	2.02340
$543$	0	0
$544$	−1.04362e9	−0.277937
$545$	−2.85065e7	−0.00754321
$546$	0	0
$547$	1.10099e9	0.287624	0.143812	−	0.989605i	$-0.454064\pi$
$547$	1.10099e9	0.287624	0.143812	+	0.989605i	$0.454064\pi$
$548$	−6.60731e9	−1.71511
$549$	0	0
$550$	−1.55726e9	−0.399108
$551$	−7.35191e9	−1.87228
$552$	0	0
$553$	2.44348e9	0.614427
$554$	5.86667e9	1.46591
$555$	0	0
$556$	7.72994e8	0.190728
$557$	5.42968e9	1.33132	0.665658	−	0.746257i	$-0.268151\pi$
$557$	5.42968e9	1.33132	0.665658	+	0.746257i	$0.268151\pi$
$558$	0	0
$559$	3.04001e9	0.736095
$560$	−3.15121e6	−0.000758263 0
$561$	0	0
$562$	1.70030e9	0.404063
$563$	7.14917e8	0.168840	0.0844201	−	0.996430i	$-0.473096\pi$
$563$	7.14917e8	0.168840	0.0844201	+	0.996430i	$0.473096\pi$
$564$	0	0
$565$	1.60391e7	0.00374121
$566$	8.01993e9	1.85914
$567$	0	0
$568$	−5.08256e9	−1.16376
$569$	1.67905e9	0.382094	0.191047	−	0.981581i	$-0.438812\pi$
$569$	1.67905e9	0.382094	0.191047	+	0.981581i	$0.438812\pi$
$570$	0	0
$571$	−9.65128e8	−0.216950	−0.108475	−	0.994099i	$-0.534597\pi$
$571$	−9.65128e8	−0.216950	−0.108475	+	0.994099i	$0.534597\pi$
$572$	−1.78505e9	−0.398808
$573$	0	0
$574$	7.60416e9	1.67826
$575$	−3.40524e9	−0.746983
$576$	0	0
$577$	2.50966e9	0.543875	0.271938	−	0.962315i	$-0.412336\pi$
$577$	2.50966e9	0.543875	0.271938	+	0.962315i	$0.412336\pi$
$578$	6.94371e9	1.49570
$579$	0	0
$580$	−5.51712e7	−0.0117412
$581$	−5.07038e9	−1.07257
$582$	0	0
$583$	1.70966e9	0.357330
$584$	1.84554e9	0.383422
$585$	0	0
$586$	7.26413e9	1.49122
$587$	−3.24638e9	−0.662470	−0.331235	−	0.943548i	$-0.607465\pi$
$587$	−3.24638e9	−0.662470	−0.331235	+	0.943548i	$0.607465\pi$
$588$	0	0
$589$	2.69770e9	0.543990
$590$	−4.50794e6	−0.000903642 0
$591$	0	0
$592$	1.89104e8	0.0374606
$593$	−7.03607e9	−1.38560	−0.692801	−	0.721129i	$-0.743624\pi$
$593$	−7.03607e9	−1.38560	−0.692801	+	0.721129i	$0.743624\pi$
$594$	0	0
$595$	−4.25381e6	−0.000827883 0
$596$	−1.13271e9	−0.219158
$597$	0	0
$598$	−6.47886e9	−1.23892
$599$	−9.00254e9	−1.71148	−0.855738	−	0.517409i	$-0.826897\pi$
$599$	−9.00254e9	−1.71148	−0.855738	+	0.517409i	$0.826897\pi$
$600$	0	0
$601$	5.40983e9	1.01654	0.508268	−	0.861199i	$-0.330286\pi$
$601$	5.40983e9	1.01654	0.508268	+	0.861199i	$0.330286\pi$
$602$	4.73669e9	0.884886
$603$	0	0
$604$	9.96758e9	1.84061
$605$	2.23274e7	0.00409915
$606$	0	0
$607$	−6.20066e9	−1.12532	−0.562662	−	0.826687i	$-0.690223\pi$
$607$	−6.20066e9	−1.12532	−0.562662	+	0.826687i	$0.690223\pi$
$608$	−6.82531e9	−1.23157
$609$	0	0
$610$	7.13902e6	0.00127346
$611$	−6.12502e9	−1.08633
$612$	0	0
$613$	5.56027e9	0.974955	0.487477	−	0.873136i	$-0.337917\pi$
$613$	5.56027e9	0.974955	0.487477	+	0.873136i	$0.337917\pi$
$614$	1.42735e10	2.48851
$615$	0	0
$616$	−9.46158e8	−0.163091
$617$	−8.63571e9	−1.48013	−0.740065	−	0.672535i	$-0.765205\pi$
$617$	−8.63571e9	−1.48013	−0.740065	+	0.672535i	$0.765205\pi$
$618$	0	0
$619$	4.62517e9	0.783809	0.391904	−	0.920006i	$-0.371816\pi$
$619$	4.62517e9	0.783809	0.391904	+	0.920006i	$0.371816\pi$
$620$	2.02445e7	0.00341142
$621$	0	0
$622$	6.27203e9	1.04506
$623$	8.20004e9	1.35865
$624$	0	0
$625$	6.10316e9	0.999943
$626$	−1.76231e9	−0.287126
$627$	0	0
$628$	−2.22012e9	−0.357699
$629$	2.55271e8	0.0409001
$630$	0	0
$631$	−4.78738e9	−0.758568	−0.379284	−	0.925280i	$-0.623830\pi$
$631$	−4.78738e9	−0.758568	−0.379284	+	0.925280i	$0.623830\pi$
$632$	4.02297e9	0.633923
$633$	0	0
$634$	1.63557e10	2.54892
$635$	−3.15156e7	−0.00488447
$636$	0	0
$637$	−2.53646e9	−0.388812
$638$	4.63452e9	0.706533
$639$	0	0
$640$	−4.11564e7	−0.00620594
$641$	−1.18477e10	−1.77676	−0.888382	−	0.459104i	$-0.848170\pi$
$641$	−1.18477e10	−1.77676	−0.888382	+	0.459104i	$0.848170\pi$
$642$	0	0
$643$	−1.06317e10	−1.57712	−0.788562	−	0.614956i	$-0.789174\pi$
$643$	−1.06317e10	−1.57712	−0.788562	+	0.614956i	$0.789174\pi$
$644$	−6.08189e9	−0.897300
$645$	0	0
$646$	−2.74342e9	−0.400385
$647$	9.21966e9	1.33829	0.669145	−	0.743132i	$-0.266661\pi$
$647$	9.21966e9	1.33829	0.669145	+	0.743132i	$0.266661\pi$
$648$	0	0
$649$	2.28145e8	0.0327608
$650$	1.16122e10	1.65851
$651$	0	0
$652$	2.43150e9	0.343565
$653$	1.41112e10	1.98321	0.991605	−	0.129302i	$-0.0412738\pi$
$653$	1.41112e10	1.98321	0.991605	+	0.129302i	$0.0412738\pi$
$654$	0	0
$655$	−4.94328e7	−0.00687338
$656$	−2.11026e9	−0.291859
$657$	0	0
$658$	−9.54349e9	−1.30592
$659$	−6.80419e9	−0.926141	−0.463071	−	0.886321i	$-0.653252\pi$
$659$	−6.80419e9	−0.926141	−0.463071	+	0.886321i	$0.653252\pi$
$660$	0	0
$661$	2.76563e8	0.0372468	0.0186234	−	0.999827i	$-0.494072\pi$
$661$	2.76563e8	0.0372468	0.0186234	+	0.999827i	$0.494072\pi$
$662$	−1.55951e10	−2.08922
$663$	0	0
$664$	−8.34793e9	−1.10660
$665$	−2.78201e7	−0.00366845
$666$	0	0
$667$	1.01343e10	1.32237
$668$	7.32298e9	0.950540
$669$	0	0
$670$	−3.78272e7	−0.00485895
$671$	−3.61302e8	−0.0461681
$672$	0	0
$673$	−7.86611e9	−0.994734	−0.497367	−	0.867540i	$-0.665700\pi$
$673$	−7.86611e9	−0.994734	−0.497367	+	0.867540i	$0.665700\pi$
$674$	−6.67178e8	−0.0839328
$675$	0	0
$676$	1.13799e9	0.141686
$677$	−6.11786e8	−0.0757773	−0.0378886	−	0.999282i	$-0.512063\pi$
$677$	−6.11786e8	−0.0757773	−0.0378886	+	0.999282i	$0.512063\pi$
$678$	0	0
$679$	−9.80439e9	−1.20192
$680$	−7.00352e6	−0.000854152 0
$681$	0	0
$682$	−1.70059e9	−0.205283
$683$	1.81400e9	0.217854	0.108927	−	0.994050i	$-0.465259\pi$
$683$	1.81400e9	0.217854	0.108927	+	0.994050i	$0.465259\pi$
$684$	0	0
$685$	4.16617e7	0.00495245
$686$	−1.45812e10	−1.72448
$687$	0	0
$688$	−1.31450e9	−0.153887
$689$	−1.27486e10	−1.48489
$690$	0	0
$691$	8.17172e9	0.942194	0.471097	−	0.882081i	$-0.343858\pi$
$691$	8.17172e9	0.942194	0.471097	+	0.882081i	$0.343858\pi$
$692$	4.00051e9	0.458927
$693$	0	0
$694$	−9.28610e9	−1.05457
$695$	−4.87403e6	−0.000550734 0
$696$	0	0
$697$	−2.84863e9	−0.318656
$698$	1.38941e10	1.54645
$699$	0	0
$700$	1.09007e10	1.20119
$701$	−8.54219e9	−0.936605	−0.468302	−	0.883568i	$-0.655134\pi$
$701$	−8.54219e9	−0.936605	−0.468302	+	0.883568i	$0.655134\pi$
$702$	0	0
$703$	1.66948e9	0.181233
$704$	3.79328e9	0.409742
$705$	0	0
$706$	−1.18768e10	−1.27023
$707$	−1.00052e10	−1.06478
$708$	0	0
$709$	−8.20988e9	−0.865117	−0.432559	−	0.901606i	$-0.642389\pi$
$709$	−8.20988e9	−0.865117	−0.432559	+	0.901606i	$0.642389\pi$
$710$	9.42066e7	0.00987819
$711$	0	0
$712$	1.35006e10	1.40176
$713$	−3.71866e9	−0.384214
$714$	0	0
$715$	1.12554e7	0.00115157
$716$	1.97751e10	2.01337
$717$	0	0
$718$	1.12431e10	1.13357
$719$	1.72415e9	0.172991	0.0864957	−	0.996252i	$-0.472433\pi$
$719$	1.72415e9	0.172991	0.0864957	+	0.996252i	$0.472433\pi$
$720$	0	0
$721$	−3.94708e9	−0.392196
$722$	−1.90223e9	−0.188098
$723$	0	0
$724$	1.77441e9	0.173768
$725$	−1.81639e10	−1.77021
$726$	0	0
$727$	−1.17427e9	−0.113343	−0.0566717	−	0.998393i	$-0.518049\pi$
$727$	−1.17427e9	−0.113343	−0.0566717	+	0.998393i	$0.518049\pi$
$728$	7.05532e9	0.677731
$729$	0	0
$730$	−3.42075e7	−0.00325455
$731$	−1.77444e9	−0.168016
$732$	0	0
$733$	1.78077e10	1.67011	0.835053	−	0.550170i	$-0.185437\pi$
$733$	1.78077e10	1.67011	0.835053	+	0.550170i	$0.185437\pi$
$734$	−1.61344e9	−0.150597
$735$	0	0
$736$	9.40839e9	0.869847
$737$	1.91442e9	0.176157
$738$	0	0
$739$	−1.69312e10	−1.54324	−0.771620	−	0.636084i	$-0.780553\pi$
$739$	−1.69312e10	−1.54324	−0.771620	+	0.636084i	$0.780553\pi$
$740$	1.25283e7	0.00113653
$741$	0	0
$742$	−1.98638e10	−1.78504
$743$	−5.43777e9	−0.486362	−0.243181	−	0.969981i	$-0.578191\pi$
$743$	−5.43777e9	−0.486362	−0.243181	+	0.969981i	$0.578191\pi$
$744$	0	0
$745$	7.14219e6	0.000632826 0
$746$	−3.47795e9	−0.306717
$747$	0	0
$748$	1.04192e9	0.0910291
$749$	5.28993e8	0.0460006
$750$	0	0
$751$	−1.10319e10	−0.950406	−0.475203	−	0.879876i	$-0.657625\pi$
$751$	−1.10319e10	−0.950406	−0.475203	+	0.879876i	$0.657625\pi$
$752$	2.64845e9	0.227107
$753$	0	0
$754$	−3.45588e10	−2.93602
$755$	−6.28496e7	−0.00531481
$756$	0	0
$757$	9.41776e9	0.789064	0.394532	−	0.918882i	$-0.370907\pi$
$757$	9.41776e9	0.789064	0.394532	+	0.918882i	$0.370907\pi$
$758$	4.09315e9	0.341363
$759$	0	0
$760$	−4.58033e7	−0.00378485
$761$	−1.61763e10	−1.33056	−0.665278	−	0.746596i	$-0.731687\pi$
$761$	−1.61763e10	−1.33056	−0.665278	+	0.746596i	$0.731687\pi$
$762$	0	0
$763$	−1.67622e10	−1.36614
$764$	−2.48575e10	−2.01665
$765$	0	0
$766$	−2.08321e9	−0.167468
$767$	−1.70124e9	−0.136138
$768$	0	0
$769$	−1.61743e10	−1.28257	−0.641287	−	0.767301i	$-0.721599\pi$
$769$	−1.61743e10	−1.28257	−0.641287	+	0.767301i	$0.721599\pi$
$770$	1.75373e7	0.00138435
$771$	0	0
$772$	5.77147e9	0.451467
$773$	1.43501e9	0.111745	0.0558723	−	0.998438i	$-0.482206\pi$
$773$	1.43501e9	0.111745	0.0558723	+	0.998438i	$0.482206\pi$
$774$	0	0
$775$	6.66503e9	0.514335
$776$	−1.61421e10	−1.24006
$777$	0	0
$778$	1.22778e10	0.934743
$779$	−1.86302e10	−1.41200
$780$	0	0
$781$	−4.76775e9	−0.358125
$782$	3.78167e9	0.282788
$783$	0	0
$784$	1.09676e9	0.0812843
$785$	1.39987e7	0.00103287
$786$	0	0
$787$	−1.73282e10	−1.26719	−0.633597	−	0.773664i	$-0.718422\pi$
$787$	−1.73282e10	−1.26719	−0.633597	+	0.773664i	$0.718422\pi$
$788$	2.88539e9	0.210069
$789$	0	0
$790$	−7.45668e7	−0.00538085
$791$	9.43121e9	0.677563
$792$	0	0
$793$	2.69417e9	0.191853
$794$	3.11029e10	2.20510
$795$	0	0
$796$	−2.07950e10	−1.46138
$797$	1.59506e10	1.11602	0.558012	−	0.829833i	$-0.311564\pi$
$797$	1.59506e10	1.11602	0.558012	+	0.829833i	$0.311564\pi$
$798$	0	0
$799$	3.57514e9	0.247959
$800$	−1.68628e10	−1.16444
$801$	0	0
$802$	2.49901e10	1.71064
$803$	1.73123e9	0.117991
$804$	0	0
$805$	3.83487e7	0.00259099
$806$	1.26810e10	0.853060
$807$	0	0
$808$	−1.64727e10	−1.09857
$809$	−1.99696e10	−1.32602	−0.663010	−	0.748611i	$-0.730721\pi$
$809$	−1.99696e10	−1.32602	−0.663010	+	0.748611i	$0.730721\pi$
$810$	0	0
$811$	1.94704e10	1.28175	0.640874	−	0.767646i	$-0.278572\pi$
$811$	1.94704e10	1.28175	0.640874	+	0.767646i	$0.278572\pi$
$812$	−3.24413e10	−2.12644
$813$	0	0
$814$	−1.05241e9	−0.0683912
$815$	−1.53316e7	−0.000992055 0
$816$	0	0
$817$	−1.16049e10	−0.744498
$818$	2.66924e10	1.70511
$819$	0	0
$820$	−1.39807e8	−0.00885482
$821$	1.65074e10	1.04107	0.520534	−	0.853841i	$-0.325733\pi$
$821$	1.65074e10	1.04107	0.520534	+	0.853841i	$0.325733\pi$
$822$	0	0
$823$	1.23795e10	0.774113	0.387057	−	0.922056i	$-0.373492\pi$
$823$	1.23795e10	0.774113	0.387057	+	0.922056i	$0.373492\pi$
$824$	−6.49852e9	−0.404640
$825$	0	0
$826$	−2.65072e9	−0.163657
$827$	−7.73227e9	−0.475376	−0.237688	−	0.971342i	$-0.576390\pi$
$827$	−7.73227e9	−0.475376	−0.237688	+	0.971342i	$0.576390\pi$
$828$	0	0
$829$	8.83980e9	0.538892	0.269446	−	0.963016i	$-0.413159\pi$
$829$	8.83980e9	0.538892	0.269446	+	0.963016i	$0.413159\pi$
$830$	1.54731e8	0.00939300
$831$	0	0
$832$	−2.82858e10	−1.70269
$833$	1.48052e9	0.0887474
$834$	0	0
$835$	−4.61743e7	−0.00274472
$836$	6.81421e9	0.403361
$837$	0	0
$838$	1.74758e10	1.02585
$839$	1.45899e10	0.852877	0.426439	−	0.904516i	$-0.359768\pi$
$839$	1.45899e10	0.852877	0.426439	+	0.904516i	$0.359768\pi$
$840$	0	0
$841$	3.68072e10	2.13377
$842$	−3.61320e10	−2.08593
$843$	0	0
$844$	1.67352e10	0.958147
$845$	−7.17549e6	−0.000409122 0
$846$	0	0
$847$	1.31288e10	0.742390
$848$	5.51249e9	0.310429
$849$	0	0
$850$	−6.77796e9	−0.378559
$851$	−2.30130e9	−0.128003
$852$	0	0
$853$	−1.83684e10	−1.01333	−0.506665	−	0.862143i	$-0.669122\pi$
$853$	−1.83684e10	−1.01333	−0.506665	+	0.862143i	$0.669122\pi$
$854$	4.19783e9	0.230633
$855$	0	0
$856$	8.70941e8	0.0474603
$857$	−1.71154e10	−0.928871	−0.464436	−	0.885607i	$-0.653743\pi$
$857$	−1.71154e10	−0.928871	−0.464436	+	0.885607i	$0.653743\pi$
$858$	0	0
$859$	−1.64262e10	−0.884220	−0.442110	−	0.896961i	$-0.645770\pi$
$859$	−1.64262e10	−0.884220	−0.442110	+	0.896961i	$0.645770\pi$
$860$	−8.70868e7	−0.00466883
$861$	0	0
$862$	5.29210e9	0.281418
$863$	1.57145e10	0.832265	0.416132	−	0.909304i	$-0.363385\pi$
$863$	1.57145e10	0.832265	0.416132	+	0.909304i	$0.363385\pi$
$864$	0	0
$865$	−2.52248e7	−0.00132517
$866$	2.55442e10	1.33653
$867$	0	0
$868$	1.19040e10	0.617836
$869$	3.77379e9	0.195078
$870$	0	0
$871$	−1.42755e10	−0.732026
$872$	−2.75974e10	−1.40949
$873$	0	0
$874$	2.47323e10	1.25307
$875$	−1.37467e8	−0.00693700
$876$	0	0
$877$	4.28124e9	0.214324	0.107162	−	0.994242i	$-0.465824\pi$
$877$	4.28124e9	0.214324	0.107162	+	0.994242i	$0.465824\pi$
$878$	2.78885e10	1.39057
$879$	0	0
$880$	−4.86685e6	−0.000240746 0
$881$	−1.71858e10	−0.846748	−0.423374	−	0.905955i	$-0.639154\pi$
$881$	−1.71858e10	−0.846748	−0.423374	+	0.905955i	$0.639154\pi$
$882$	0	0
$883$	−1.87937e10	−0.918648	−0.459324	−	0.888269i	$-0.651908\pi$
$883$	−1.87937e10	−0.918648	−0.459324	+	0.888269i	$0.651908\pi$
$884$	−7.76943e9	−0.378274
$885$	0	0
$886$	−1.15073e10	−0.555849
$887$	−1.22073e10	−0.587339	−0.293669	−	0.955907i	$-0.594876\pi$
$887$	−1.22073e10	−0.587339	−0.293669	+	0.955907i	$0.594876\pi$
$888$	0	0
$889$	−1.85315e10	−0.884617
$890$	−2.50238e8	−0.0118984
$891$	0	0
$892$	2.38011e10	1.12284
$893$	2.33815e10	1.09873
$894$	0	0
$895$	−1.24690e8	−0.00581368
$896$	−2.42005e10	−1.12395
$897$	0	0
$898$	5.32448e10	2.45364
$899$	−1.98357e10	−0.910516
$900$	0	0
$901$	7.44129e9	0.338931
$902$	1.17441e10	0.532841
$903$	0	0
$904$	1.55276e10	0.699063
$905$	−1.11884e7	−0.000501761 0
$906$	0	0
$907$	1.54047e9	0.0685534	0.0342767	−	0.999412i	$-0.489087\pi$
$907$	1.54047e9	0.0685534	0.0342767	+	0.999412i	$0.489087\pi$
$908$	−2.17477e10	−0.964078
$909$	0	0
$910$	−1.30772e8	−0.00575269
$911$	−3.19695e10	−1.40094	−0.700472	−	0.713680i	$-0.747027\pi$
$911$	−3.19695e10	−1.40094	−0.700472	+	0.713680i	$0.747027\pi$
$912$	0	0
$913$	−7.83088e9	−0.340536
$914$	5.95352e10	2.57906
$915$	0	0
$916$	2.48941e10	1.07019
$917$	−2.90671e10	−1.24483
$918$	0	0
$919$	2.59602e10	1.10333	0.551663	−	0.834067i	$-0.313993\pi$
$919$	2.59602e10	1.10333	0.551663	+	0.834067i	$0.313993\pi$
$920$	6.31377e7	0.00267320
$921$	0	0
$922$	1.70227e10	0.715272
$923$	3.55523e10	1.48820
$924$	0	0
$925$	4.12467e9	0.171354
$926$	−2.39349e10	−0.990589
$927$	0	0
$928$	5.01851e10	2.06138
$929$	6.20074e9	0.253740	0.126870	−	0.991919i	$-0.459507\pi$
$929$	6.20074e9	0.253740	0.126870	+	0.991919i	$0.459507\pi$
$930$	0	0
$931$	9.68263e9	0.393251
$932$	6.98593e9	0.282663
$933$	0	0
$934$	−6.75024e10	−2.71085
$935$	−6.56974e6	−0.000262850 0
$936$	0	0
$937$	−2.50957e10	−0.996575	−0.498288	−	0.867012i	$-0.666038\pi$
$937$	−2.50957e10	−0.996575	−0.498288	+	0.867012i	$0.666038\pi$
$938$	−2.22428e10	−0.879995
$939$	0	0
$940$	1.75463e8	0.00689028
$941$	8.17848e9	0.319970	0.159985	−	0.987119i	$-0.448855\pi$
$941$	8.17848e9	0.319970	0.159985	+	0.987119i	$0.448855\pi$
$942$	0	0
$943$	2.56808e10	0.997282
$944$	7.35614e8	0.0284608
$945$	0	0
$946$	7.31552e9	0.280948
$947$	−1.81622e10	−0.694935	−0.347468	−	0.937692i	$-0.612958\pi$
$947$	−1.81622e10	−0.694935	−0.347468	+	0.937692i	$0.612958\pi$
$948$	0	0
$949$	−1.29094e10	−0.490315
$950$	−4.43281e10	−1.67744
$951$	0	0
$952$	−4.11815e9	−0.154694
$953$	4.42866e10	1.65748	0.828738	−	0.559637i	$-0.189059\pi$
$953$	4.42866e10	1.65748	0.828738	+	0.559637i	$0.189059\pi$
$954$	0	0
$955$	1.56737e8	0.00582316
$956$	4.46935e10	1.65440
$957$	0	0
$958$	−5.26702e10	−1.93547
$959$	2.44976e10	0.896929
$960$	0	0
$961$	−2.02341e10	−0.735449
$962$	7.84764e9	0.284202
$963$	0	0
$964$	−6.00600e10	−2.15931
$965$	−3.63914e7	−0.00130363
$966$	0	0
$967$	3.34893e10	1.19101	0.595503	−	0.803353i	$-0.296953\pi$
$967$	3.34893e10	1.19101	0.595503	+	0.803353i	$0.296953\pi$
$968$	2.16153e10	0.765946
$969$	0	0
$970$	2.99197e8	0.0105258
$971$	2.53801e10	0.889663	0.444831	−	0.895614i	$-0.353264\pi$
$971$	2.53801e10	0.889663	0.444831	+	0.895614i	$0.353264\pi$
$972$	0	0
$973$	−2.86599e9	−0.0997423
$974$	2.25980e10	0.783633
$975$	0	0
$976$	−1.16496e9	−0.0401084
$977$	1.10528e10	0.379177	0.189589	−	0.981864i	$-0.439285\pi$
$977$	1.10528e10	0.379177	0.189589	+	0.981864i	$0.439285\pi$
$978$	0	0
$979$	1.26644e10	0.431366
$980$	7.26616e7	0.00246612
$981$	0	0
$982$	7.55401e9	0.254558
$983$	3.71529e9	0.124754	0.0623771	−	0.998053i	$-0.480132\pi$
$983$	3.71529e9	0.124754	0.0623771	+	0.998053i	$0.480132\pi$
$984$	0	0
$985$	−1.81935e7	−0.000606582 0
$986$	2.01718e10	0.670154
$987$	0	0
$988$	−5.08123e10	−1.67618
$989$	1.59968e10	0.525831
$990$	0	0
$991$	−3.64285e10	−1.18901	−0.594503	−	0.804094i	$-0.702651\pi$
$991$	−3.64285e10	−1.18901	−0.594503	+	0.804094i	$0.702651\pi$
$992$	−1.84149e10	−0.598933
$993$	0	0
$994$	5.53946e10	1.78902
$995$	1.31121e8	0.00421978
$996$	0	0
$997$	4.94528e10	1.58037	0.790183	−	0.612871i	$-0.209985\pi$
$997$	4.94528e10	1.58037	0.790183	+	0.612871i	$0.209985\pi$
$998$	8.73170e10	2.78062
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.b.1.3		16
3.2	odd	2		177.8.a.a.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.14	✓	16	3.2	odd	2
531.8.a.b.1.3		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-17.9442 q^{2} +193.993 q^{4} -1.22320 q^{5} -719.259 q^{7} -1184.20 q^{8} +O(q^{10})\)	\(q-17.9442 q^{2} +193.993 q^{4} -1.22320 q^{5} -719.259 q^{7} -1184.20 q^{8} +21.9494 q^{10} -1110.85 q^{11} +8283.40 q^{13} +12906.5 q^{14} -3581.74 q^{16} -4834.97 q^{17} -31620.9 q^{19} -237.293 q^{20} +19933.3 q^{22} +43588.0 q^{23} -78123.5 q^{25} -148639. q^{26} -139531. q^{28} +232502. q^{29} -85314.0 q^{31} +215848. q^{32} +86759.6 q^{34} +879.800 q^{35} -52796.8 q^{37} +567411. q^{38} +1448.51 q^{40} +589172. q^{41} +367000. q^{43} -215497. q^{44} -782150. q^{46} -739433. q^{47} -306210. q^{49} +1.40186e6 q^{50} +1.60692e6 q^{52} -1.53905e6 q^{53} +1358.79 q^{55} +851743. q^{56} -4.17205e6 q^{58} -205379. q^{59} +325249. q^{61} +1.53089e6 q^{62} -3.41476e6 q^{64} -10132.3 q^{65} -1.72338e6 q^{67} -937953. q^{68} -15787.3 q^{70} +4.29199e6 q^{71} -1.55847e6 q^{73} +947394. q^{74} -6.13424e6 q^{76} +798988. q^{77} -3.39722e6 q^{79} +4381.20 q^{80} -1.05722e7 q^{82} +7.04945e6 q^{83} +5914.16 q^{85} -6.58552e6 q^{86} +1.31546e6 q^{88} -1.14007e7 q^{89} -5.95791e6 q^{91} +8.45577e6 q^{92} +1.32685e7 q^{94} +38678.8 q^{95} +1.36312e7 q^{97} +5.49468e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8}+O(q^{10}) \) 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8	\( 16 q + 6 q^{2} + 974 q^{4} + 68 q^{5} - 2343 q^{7} - 819 q^{8} - 3479 q^{10} - 898 q^{11} - 8172 q^{13} + 13315 q^{14} + 3138 q^{16} + 44985 q^{17} - 40137 q^{19} - 130657 q^{20} + 109394 q^{22} + 2833 q^{23} + 285746 q^{25} + 129420 q^{26} + 112890 q^{28} - 144375 q^{29} - 141759 q^{31} + 36224 q^{32} - 341332 q^{34} + 78859 q^{35} - 297971 q^{37} - 329075 q^{38} - 203048 q^{40} - 659077 q^{41} - 1431608 q^{43} - 254916 q^{44} + 873113 q^{46} + 1574073 q^{47} + 1893545 q^{49} - 302533 q^{50} - 4972548 q^{52} - 587736 q^{53} - 4624036 q^{55} + 5798506 q^{56} - 6991380 q^{58} - 3286064 q^{59} - 6117131 q^{61} + 11570258 q^{62} - 19063011 q^{64} + 5335514 q^{65} - 16518710 q^{67} + 17284669 q^{68} - 39189486 q^{70} + 10882582 q^{71} - 21097441 q^{73} + 16717030 q^{74} - 40864952 q^{76} + 3404601 q^{77} - 3784458 q^{79} + 27466195 q^{80} - 24990117 q^{82} + 1951425 q^{83} - 23238675 q^{85} + 35910572 q^{86} - 27843055 q^{88} - 10499443 q^{89} + 699217 q^{91} + 20062766 q^{92} - 59358988 q^{94} + 29236333 q^{95} - 25158976 q^{97} - 2120460 q^{98}+O(q^{100}) \) 16 * q + 6 * q^2 + 974 * q^4 + 68 * q^5 - 2343 * q^7 - 819 * q^8 - 3479 * q^10 - 898 * q^11 - 8172 * q^13 + 13315 * q^14 + 3138 * q^16 + 44985 * q^17 - 40137 * q^19 - 130657 * q^20 + 109394 * q^22 + 2833 * q^23 + 285746 * q^25 + 129420 * q^26 + 112890 * q^28 - 144375 * q^29 - 141759 * q^31 + 36224 * q^32 - 341332 * q^34 + 78859 * q^35 - 297971 * q^37 - 329075 * q^38 - 203048 * q^40 - 659077 * q^41 - 1431608 * q^43 - 254916 * q^44 + 873113 * q^46 + 1574073 * q^47 + 1893545 * q^49 - 302533 * q^50 - 4972548 * q^52 - 587736 * q^53 - 4624036 * q^55 + 5798506 * q^56 - 6991380 * q^58 - 3286064 * q^59 - 6117131 * q^61 + 11570258 * q^62 - 19063011 * q^64 + 5335514 * q^65 - 16518710 * q^67 + 17284669 * q^68 - 39189486 * q^70 + 10882582 * q^71 - 21097441 * q^73 + 16717030 * q^74 - 40864952 * q^76 + 3404601 * q^77 - 3784458 * q^79 + 27466195 * q^80 - 24990117 * q^82 + 1951425 * q^83 - 23238675 * q^85 + 35910572 * q^86 - 27843055 * q^88 - 10499443 * q^89 + 699217 * q^91 + 20062766 * q^92 - 59358988 * q^94 + 29236333 * q^95 - 25158976 * q^97 - 2120460 * q^98

Introduction

L-functions

Modular forms

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