LMFDB - Embedded newform 405.8.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,8,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	405.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:	$126.515935321$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 1221 x^{12} + 3034 x^{11} + 559330 x^{10} - 1662468 x^{9} - 119658132 x^{8} + \cdots + 11075674978368 $ x^14 - 2x^13 - 1221x^12 + 3034x^11 + 559330x^10 - 1662468x^9 - 119658132x^8 + 388985472x^7 + 12117415968x^6 - 35096050656x^5 - 526695898608x^4 + 532226461536x^3 + 9146880114336x^2 + 18387947059776*x + 11075674978368
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{28}\cdot 13 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.07113$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.660923 q^{2} -127.563 q^{4} +125.000 q^{5} +32.2409 q^{7} -168.908 q^{8} +O(q^{10})$	\(q+0.660923 q^{2} -127.563 q^{4} +125.000 q^{5} +32.2409 q^{7} -168.908 q^{8} +82.6154 q^{10} -4638.68 q^{11} -15318.5 q^{13} +21.3088 q^{14} +16216.5 q^{16} -14884.6 q^{17} -23699.8 q^{19} -15945.4 q^{20} -3065.81 q^{22} +16633.5 q^{23} +15625.0 q^{25} -10124.3 q^{26} -4112.75 q^{28} -44127.9 q^{29} +96346.6 q^{31} +32338.0 q^{32} -9837.61 q^{34} +4030.12 q^{35} +173876. q^{37} -15663.7 q^{38} -21113.5 q^{40} -303592. q^{41} +277174. q^{43} +591725. q^{44} +10993.5 q^{46} -615618. q^{47} -822504. q^{49} +10326.9 q^{50} +1.95407e6 q^{52} -1.65626e6 q^{53} -579835. q^{55} -5445.74 q^{56} -29165.2 q^{58} -1.18178e6 q^{59} +1.28415e6 q^{61} +63677.7 q^{62} -2.05433e6 q^{64} -1.91481e6 q^{65} -1.36150e6 q^{67} +1.89873e6 q^{68} +2663.60 q^{70} -2.78262e6 q^{71} +945768. q^{73} +114919. q^{74} +3.02322e6 q^{76} -149555. q^{77} -6.56175e6 q^{79} +2.02706e6 q^{80} -200651. q^{82} +8.16030e6 q^{83} -1.86058e6 q^{85} +183191. q^{86} +783509. q^{88} +1.24031e7 q^{89} -493881. q^{91} -2.12182e6 q^{92} -406876. q^{94} -2.96247e6 q^{95} +7.93596e6 q^{97} -543612. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8}+O(q^{10}) $ 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8	\( 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8} + 2000 q^{10} + 10648 q^{11} + 17268 q^{13} + 9180 q^{14} + 34122 q^{16} + 3886 q^{17} + 22986 q^{19} + 89250 q^{20} - 49706 q^{22} + 155040 q^{23} + 218750 q^{25} + 7804 q^{26} + 256160 q^{28} + 195836 q^{29} + 92786 q^{31} + 157778 q^{32} + 787348 q^{34} + 192250 q^{35} + 876406 q^{37} + 329320 q^{38} - 15750 q^{40} + 795164 q^{41} + 730350 q^{43} + 2360876 q^{44} - 225654 q^{46} + 2687842 q^{47} + 1663586 q^{49} + 250000 q^{50} + 3875836 q^{52} + 2533750 q^{53} + 1331000 q^{55} + 2055276 q^{56} - 318934 q^{58} + 2283340 q^{59} + 2600400 q^{61} + 5702022 q^{62} + 474098 q^{64} + 2158500 q^{65} + 2422160 q^{67} + 1114364 q^{68} + 1147500 q^{70} + 6395324 q^{71} - 540774 q^{73} + 260516 q^{74} - 2417042 q^{76} - 1384890 q^{77} - 307384 q^{79} + 4265250 q^{80} - 14044738 q^{82} + 10991322 q^{83} + 485750 q^{85} + 2847712 q^{86} - 19226592 q^{88} + 2094000 q^{89} - 9496256 q^{91} + 39213378 q^{92} - 28132682 q^{94} + 2873250 q^{95} - 12859994 q^{97} + 28336478 q^{98}+O(q^{100}) \) 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8 + 2000 * q^10 + 10648 * q^11 + 17268 * q^13 + 9180 * q^14 + 34122 * q^16 + 3886 * q^17 + 22986 * q^19 + 89250 * q^20 - 49706 * q^22 + 155040 * q^23 + 218750 * q^25 + 7804 * q^26 + 256160 * q^28 + 195836 * q^29 + 92786 * q^31 + 157778 * q^32 + 787348 * q^34 + 192250 * q^35 + 876406 * q^37 + 329320 * q^38 - 15750 * q^40 + 795164 * q^41 + 730350 * q^43 + 2360876 * q^44 - 225654 * q^46 + 2687842 * q^47 + 1663586 * q^49 + 250000 * q^50 + 3875836 * q^52 + 2533750 * q^53 + 1331000 * q^55 + 2055276 * q^56 - 318934 * q^58 + 2283340 * q^59 + 2600400 * q^61 + 5702022 * q^62 + 474098 * q^64 + 2158500 * q^65 + 2422160 * q^67 + 1114364 * q^68 + 1147500 * q^70 + 6395324 * q^71 - 540774 * q^73 + 260516 * q^74 - 2417042 * q^76 - 1384890 * q^77 - 307384 * q^79 + 4265250 * q^80 - 14044738 * q^82 + 10991322 * q^83 + 485750 * q^85 + 2847712 * q^86 - 19226592 * q^88 + 2094000 * q^89 - 9496256 * q^91 + 39213378 * q^92 - 28132682 * q^94 + 2873250 * q^95 - 12859994 * q^97 + 28336478 * 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 $

$2$	0.660923	0.0584179	0.0292090	−	0.999573i	$-0.490701\pi$
$2$	0.660923	0.0584179	0.0292090	+	0.999573i	$0.490701\pi$
$3$	0	0
$3$	0	0
$4$	−127.563	−0.996587
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	32.2409	0.0355275	0.0177637	−	0.999842i	$-0.494345\pi$
$7$	32.2409	0.0355275	0.0177637	+	0.999842i	$0.494345\pi$
$8$	−168.908	−0.116636
$9$	0	0
$10$	82.6154	0.0261253
$11$	−4638.68	−1.05080	−0.525400	−	0.850855i	$-0.676084\pi$
$11$	−4638.68	−1.05080	−0.525400	+	0.850855i	$0.676084\pi$
$12$	0	0
$13$	−15318.5	−1.93381	−0.966904	−	0.255142i	$-0.917878\pi$
$13$	−15318.5	−1.93381	−0.966904	+	0.255142i	$0.917878\pi$
$14$	21.3088	0.00207544
$15$	0	0
$16$	16216.5	0.989774
$17$	−14884.6	−0.734797	−0.367398	−	0.930064i	$-0.619751\pi$
$17$	−14884.6	−0.734797	−0.367398	+	0.930064i	$0.619751\pi$
$18$	0	0
$19$	−23699.8	−0.792695	−0.396348	−	0.918101i	$-0.629722\pi$
$19$	−23699.8	−0.792695	−0.396348	+	0.918101i	$0.629722\pi$
$20$	−15945.4	−0.445687
$21$	0	0
$22$	−3065.81	−0.0613855
$23$	16633.5	0.285060	0.142530	−	0.989791i	$-0.454476\pi$
$23$	16633.5	0.285060	0.142530	+	0.989791i	$0.454476\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−10124.3	−0.112969
$27$	0	0
$28$	−4112.75	−0.0354062
$29$	−44127.9	−0.335986	−0.167993	−	0.985788i	$-0.553729\pi$
$29$	−44127.9	−0.335986	−0.167993	+	0.985788i	$0.553729\pi$
$30$	0	0
$31$	96346.6	0.580859	0.290429	−	0.956896i	$-0.406202\pi$
$31$	96346.6	0.580859	0.290429	+	0.956896i	$0.406202\pi$
$32$	32338.0	0.174457
$33$	0	0
$34$	−9837.61	−0.0429253
$35$	4030.12	0.0158884
$36$	0	0
$37$	173876.	0.564332	0.282166	−	0.959366i	$-0.408947\pi$
$37$	173876.	0.564332	0.282166	+	0.959366i	$0.408947\pi$
$38$	−15663.7	−0.0463076
$39$	0	0
$40$	−21113.5	−0.0521614
$41$	−303592.	−0.687935	−0.343967	−	0.938982i	$-0.611771\pi$
$41$	−303592.	−0.687935	−0.343967	+	0.938982i	$0.611771\pi$
$42$	0	0
$43$	277174.	0.531634	0.265817	−	0.964024i	$-0.414358\pi$
$43$	277174.	0.531634	0.265817	+	0.964024i	$0.414358\pi$
$44$	591725.	1.04721
$45$	0	0
$46$	10993.5	0.0166526
$47$	−615618.	−0.864906	−0.432453	−	0.901656i	$-0.642352\pi$
$47$	−615618.	−0.864906	−0.432453	+	0.901656i	$0.642352\pi$
$48$	0	0
$49$	−822504.	−0.998738
$50$	10326.9	0.0116836
$51$	0	0
$52$	1.95407e6	1.92721
$53$	−1.65626e6	−1.52814	−0.764069	−	0.645134i	$-0.776801\pi$
$53$	−1.65626e6	−1.52814	−0.764069	+	0.645134i	$0.776801\pi$
$54$	0	0
$55$	−579835.	−0.469932
$56$	−5445.74	−0.00414380
$57$	0	0
$58$	−29165.2	−0.0196276
$59$	−1.18178e6	−0.749123	−0.374561	−	0.927202i	$-0.622207\pi$
$59$	−1.18178e6	−0.749123	−0.374561	+	0.927202i	$0.622207\pi$
$60$	0	0
$61$	1.28415e6	0.724372	0.362186	−	0.932106i	$-0.382030\pi$
$61$	1.28415e6	0.724372	0.362186	+	0.932106i	$0.382030\pi$
$62$	63677.7	0.0339326
$63$	0	0
$64$	−2.05433e6	−0.979582
$65$	−1.91481e6	−0.864825
$66$	0	0
$67$	−1.36150e6	−0.553041	−0.276520	−	0.961008i	$-0.589181\pi$
$67$	−1.36150e6	−0.553041	−0.276520	+	0.961008i	$0.589181\pi$
$68$	1.89873e6	0.732289
$69$	0	0
$70$	2663.60	0.000928165 0
$71$	−2.78262e6	−0.922677	−0.461338	−	0.887224i	$-0.652631\pi$
$71$	−2.78262e6	−0.922677	−0.461338	+	0.887224i	$0.652631\pi$
$72$	0	0
$73$	945768.	0.284547	0.142274	−	0.989827i	$-0.454559\pi$
$73$	945768.	0.284547	0.142274	+	0.989827i	$0.454559\pi$
$74$	114919.	0.0329671
$75$	0	0
$76$	3.02322e6	0.789990
$77$	−149555.	−0.0373323
$78$	0	0
$79$	−6.56175e6	−1.49736	−0.748678	−	0.662934i	$-0.769311\pi$
$79$	−6.56175e6	−1.49736	−0.748678	+	0.662934i	$0.769311\pi$
$80$	2.02706e6	0.442640
$81$	0	0
$82$	−200651.	−0.0401877
$83$	8.16030e6	1.56651	0.783255	−	0.621701i	$-0.213558\pi$
$83$	8.16030e6	1.56651	0.783255	+	0.621701i	$0.213558\pi$
$84$	0	0
$85$	−1.86058e6	−0.328611
$86$	183191.	0.0310569
$87$	0	0
$88$	783509.	0.122562
$89$	1.24031e7	1.86495	0.932474	−	0.361237i	$-0.117646\pi$
$89$	1.24031e7	1.86495	0.932474	+	0.361237i	$0.117646\pi$
$90$	0	0
$91$	−493881.	−0.0687033
$92$	−2.12182e6	−0.284087
$93$	0	0
$94$	−406876.	−0.0505260
$95$	−2.96247e6	−0.354504
$96$	0	0
$97$	7.93596e6	0.882874	0.441437	−	0.897292i	$-0.354469\pi$
$97$	7.93596e6	0.882874	0.441437	+	0.897292i	$0.354469\pi$
$98$	−543612.	−0.0583442
$99$	0	0
$100$	−1.99317e6	−0.199317
$101$	1.13780e7	1.09886	0.549428	−	0.835541i	$-0.314846\pi$
$101$	1.13780e7	1.09886	0.549428	+	0.835541i	$0.314846\pi$
$102$	0	0
$103$	1.18241e7	1.06620	0.533100	−	0.846052i	$-0.321027\pi$
$103$	1.18241e7	1.06620	0.533100	+	0.846052i	$0.321027\pi$
$104$	2.58740e6	0.225552
$105$	0	0
$106$	−1.09466e6	−0.0892707
$107$	−6.92060e6	−0.546136	−0.273068	−	0.961995i	$-0.588038\pi$
$107$	−6.92060e6	−0.546136	−0.273068	+	0.961995i	$0.588038\pi$
$108$	0	0
$109$	−5.86549e6	−0.433822	−0.216911	−	0.976191i	$-0.569598\pi$
$109$	−5.86549e6	−0.433822	−0.216911	+	0.976191i	$0.569598\pi$
$110$	−383226.	−0.0274524
$111$	0	0
$112$	522833.	0.0351642
$113$	2.57566e6	0.167925	0.0839623	−	0.996469i	$-0.473242\pi$
$113$	2.57566e6	0.167925	0.0839623	+	0.996469i	$0.473242\pi$
$114$	0	0
$115$	2.07918e6	0.127483
$116$	5.62910e6	0.334839
$117$	0	0
$118$	−781063.	−0.0437622
$119$	−479895.	−0.0261055
$120$	0	0
$121$	2.03018e6	0.104180
$122$	848726.	0.0423163
$123$	0	0
$124$	−1.22903e7	−0.578876
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−2.44072e7	−1.05732	−0.528658	−	0.848835i	$-0.677305\pi$
$127$	−2.44072e7	−1.05732	−0.528658	+	0.848835i	$0.677305\pi$
$128$	−5.49702e6	−0.231682
$129$	0	0
$130$	−1.26554e6	−0.0505213
$131$	4.32334e7	1.68024	0.840118	−	0.542403i	$-0.182486\pi$
$131$	4.32334e7	1.68024	0.840118	+	0.542403i	$0.182486\pi$
$132$	0	0
$133$	−764102.	−0.0281625
$134$	−899850.	−0.0323075
$135$	0	0
$136$	2.51413e6	0.0857041
$137$	3.21729e7	1.06898	0.534488	−	0.845176i	$-0.320505\pi$
$137$	3.21729e7	1.06898	0.534488	+	0.845176i	$0.320505\pi$
$138$	0	0
$139$	4.57835e7	1.44596	0.722982	−	0.690867i	$-0.242771\pi$
$139$	4.57835e7	1.44596	0.722982	+	0.690867i	$0.242771\pi$
$140$	−514094.	−0.0158341
$141$	0	0
$142$	−1.83910e6	−0.0539008
$143$	7.10574e7	2.03204
$144$	0	0
$145$	−5.51599e6	−0.150257
$146$	625080.	0.0166227
$147$	0	0
$148$	−2.21802e7	−0.562406
$149$	4.15357e7	1.02865	0.514327	−	0.857594i	$-0.328042\pi$
$149$	4.15357e7	1.02865	0.514327	+	0.857594i	$0.328042\pi$
$150$	0	0
$151$	3.58469e7	0.847291	0.423645	−	0.905828i	$-0.360750\pi$
$151$	3.58469e7	0.847291	0.423645	+	0.905828i	$0.360750\pi$
$152$	4.00307e6	0.0924572
$153$	0	0
$154$	−98844.6	−0.00218087
$155$	1.20433e7	0.259768
$156$	0	0
$157$	2.55103e7	0.526099	0.263050	−	0.964782i	$-0.415272\pi$
$157$	2.55103e7	0.526099	0.263050	+	0.964782i	$0.415272\pi$
$158$	−4.33682e6	−0.0874724
$159$	0	0
$160$	4.04225e6	0.0780195
$161$	536279.	0.0101274
$162$	0	0
$163$	−8.59386e7	−1.55429	−0.777145	−	0.629322i	$-0.783333\pi$
$163$	−8.59386e7	−1.55429	−0.777145	+	0.629322i	$0.783333\pi$
$164$	3.87272e7	0.685587
$165$	0	0
$166$	5.39334e6	0.0915122
$167$	−7.20013e6	−0.119628	−0.0598140	−	0.998210i	$-0.519051\pi$
$167$	−7.20013e6	−0.119628	−0.0598140	+	0.998210i	$0.519051\pi$
$168$	0	0
$169$	1.71906e8	2.73961
$170$	−1.22970e6	−0.0191968
$171$	0	0
$172$	−3.53572e7	−0.529819
$173$	−7.62070e7	−1.11901	−0.559504	−	0.828828i	$-0.689008\pi$
$173$	−7.62070e7	−1.11901	−0.559504	+	0.828828i	$0.689008\pi$
$174$	0	0
$175$	503764.	0.00710549
$176$	−7.52229e7	−1.04005
$177$	0	0
$178$	8.19753e6	0.108946
$179$	−9.99349e7	−1.30236	−0.651181	−	0.758923i	$-0.725726\pi$
$179$	−9.99349e7	−1.30236	−0.651181	+	0.758923i	$0.725726\pi$
$180$	0	0
$181$	−9.04302e7	−1.13354	−0.566772	−	0.823875i	$-0.691808\pi$
$181$	−9.04302e7	−1.13354	−0.566772	+	0.823875i	$0.691808\pi$
$182$	−326417.	−0.00401350
$183$	0	0
$184$	−2.80952e6	−0.0332483
$185$	2.17345e7	0.252377
$186$	0	0
$187$	6.90451e7	0.772125
$188$	7.85302e7	0.861954
$189$	0	0
$190$	−1.95796e6	−0.0207094
$191$	6.57205e7	0.682471	0.341236	−	0.939978i	$-0.389155\pi$
$191$	6.57205e7	0.682471	0.341236	+	0.939978i	$0.389155\pi$
$192$	0	0
$193$	−1.08575e8	−1.08713	−0.543564	−	0.839368i	$-0.682925\pi$
$193$	−1.08575e8	−1.08713	−0.543564	+	0.839368i	$0.682925\pi$
$194$	5.24506e6	0.0515756
$195$	0	0
$196$	1.04921e8	0.995329
$197$	7.90877e7	0.737017	0.368508	−	0.929624i	$-0.379869\pi$
$197$	7.90877e7	0.737017	0.368508	+	0.929624i	$0.379869\pi$
$198$	0	0
$199$	−1.08106e8	−0.972440	−0.486220	−	0.873837i	$-0.661625\pi$
$199$	−1.08106e8	−0.972440	−0.486220	+	0.873837i	$0.661625\pi$
$200$	−2.63918e6	−0.0233273
$201$	0	0
$202$	7.51998e6	0.0641929
$203$	−1.42273e6	−0.0119367
$204$	0	0
$205$	−3.79490e7	−0.307654
$206$	7.81484e6	0.0622852
$207$	0	0
$208$	−2.48411e8	−1.91403
$209$	1.09936e8	0.832964
$210$	0	0
$211$	1.16697e8	0.855207	0.427604	−	0.903966i	$-0.359358\pi$
$211$	1.16697e8	0.855207	0.427604	+	0.903966i	$0.359358\pi$
$212$	2.11278e8	1.52292
$213$	0	0
$214$	−4.57398e6	−0.0319041
$215$	3.46467e7	0.237754
$216$	0	0
$217$	3.10630e6	0.0206364
$218$	−3.87664e6	−0.0253430
$219$	0	0
$220$	7.39656e7	0.468328
$221$	2.28010e8	1.42096
$222$	0	0
$223$	−4.30894e7	−0.260198	−0.130099	−	0.991501i	$-0.541529\pi$
$223$	−4.30894e7	−0.260198	−0.130099	+	0.991501i	$0.541529\pi$
$224$	1.04261e6	0.00619802
$225$	0	0
$226$	1.70232e6	0.00980981
$227$	2.80665e8	1.59257	0.796284	−	0.604922i	$-0.206796\pi$
$227$	2.80665e8	1.59257	0.796284	+	0.604922i	$0.206796\pi$
$228$	0	0
$229$	−9.06919e7	−0.499050	−0.249525	−	0.968368i	$-0.580275\pi$
$229$	−9.06919e7	−0.499050	−0.249525	+	0.968368i	$0.580275\pi$
$230$	1.37418e6	0.00744726
$231$	0	0
$232$	7.45355e6	0.0391882
$233$	2.55680e8	1.32419	0.662095	−	0.749419i	$-0.269667\pi$
$233$	2.55680e8	1.32419	0.662095	+	0.749419i	$0.269667\pi$
$234$	0	0
$235$	−7.69523e7	−0.386798
$236$	1.50751e8	0.746566
$237$	0	0
$238$	−317174.	−0.00152503
$239$	−5.17788e7	−0.245335	−0.122667	−	0.992448i	$-0.539145\pi$
$239$	−5.17788e7	−0.245335	−0.122667	+	0.992448i	$0.539145\pi$
$240$	0	0
$241$	1.07136e8	0.493035	0.246517	−	0.969138i	$-0.420714\pi$
$241$	1.07136e8	0.493035	0.246517	+	0.969138i	$0.420714\pi$
$242$	1.34179e6	0.00608600
$243$	0	0
$244$	−1.63810e8	−0.721900
$245$	−1.02813e8	−0.446649
$246$	0	0
$247$	3.63043e8	1.53292
$248$	−1.62737e7	−0.0677493
$249$	0	0
$250$	1.29087e6	0.00522506
$251$	−3.55731e8	−1.41992	−0.709959	−	0.704243i	$-0.751287\pi$
$251$	−3.55731e8	−1.41992	−0.709959	+	0.704243i	$0.751287\pi$
$252$	0	0
$253$	−7.71574e7	−0.299541
$254$	−1.61313e7	−0.0617662
$255$	0	0
$256$	2.59322e8	0.966048
$257$	6.54993e7	0.240697	0.120349	−	0.992732i	$-0.461599\pi$
$257$	6.54993e7	0.240697	0.120349	+	0.992732i	$0.461599\pi$
$258$	0	0
$259$	5.60593e6	0.0200493
$260$	2.44259e8	0.861873
$261$	0	0
$262$	2.85740e7	0.0981559
$263$	9.26946e7	0.314202	0.157101	−	0.987583i	$-0.449785\pi$
$263$	9.26946e7	0.314202	0.157101	+	0.987583i	$0.449785\pi$
$264$	0	0
$265$	−2.07032e8	−0.683404
$266$	−505013.	−0.00164519
$267$	0	0
$268$	1.73678e8	0.551153
$269$	5.78313e8	1.81146	0.905732	−	0.423850i	$-0.139322\pi$
$269$	5.78313e8	1.81146	0.905732	+	0.423850i	$0.139322\pi$
$270$	0	0
$271$	−1.52159e8	−0.464413	−0.232206	−	0.972667i	$-0.574594\pi$
$271$	−1.52159e8	−0.464413	−0.232206	+	0.972667i	$0.574594\pi$
$272$	−2.41376e8	−0.727283
$273$	0	0
$274$	2.12638e7	0.0624473
$275$	−7.24794e7	−0.210160
$276$	0	0
$277$	2.74170e8	0.775070	0.387535	−	0.921855i	$-0.373327\pi$
$277$	2.74170e8	0.775070	0.387535	+	0.921855i	$0.373327\pi$
$278$	3.02594e7	0.0844702
$279$	0	0
$280$	−680717.	−0.00185316
$281$	4.55579e8	1.22487	0.612437	−	0.790519i	$-0.290189\pi$
$281$	4.55579e8	1.22487	0.612437	+	0.790519i	$0.290189\pi$
$282$	0	0
$283$	2.48493e8	0.651720	0.325860	−	0.945418i	$-0.394346\pi$
$283$	2.48493e8	0.651720	0.325860	+	0.945418i	$0.394346\pi$
$284$	3.54960e8	0.919528
$285$	0	0
$286$	4.69635e7	0.118708
$287$	−9.78810e6	−0.0244406
$288$	0	0
$289$	−1.88786e8	−0.460073
$290$	−3.64565e6	−0.00877772
$291$	0	0
$292$	−1.20645e8	−0.283576
$293$	6.67978e7	0.155141	0.0775703	−	0.996987i	$-0.475284\pi$
$293$	6.67978e7	0.155141	0.0775703	+	0.996987i	$0.475284\pi$
$294$	0	0
$295$	−1.47722e8	−0.335018
$296$	−2.93690e7	−0.0658217
$297$	0	0
$298$	2.74519e7	0.0600919
$299$	−2.54799e8	−0.551250
$300$	0	0
$301$	8.93634e6	0.0188876
$302$	2.36921e7	0.0494970
$303$	0	0
$304$	−3.84326e8	−0.784589
$305$	1.60519e8	0.323949
$306$	0	0
$307$	−3.39294e8	−0.669255	−0.334628	−	0.942350i	$-0.608611\pi$
$307$	−3.39294e8	−0.669255	−0.334628	+	0.942350i	$0.608611\pi$
$308$	1.90778e7	0.0372049
$309$	0	0
$310$	7.95971e6	0.0151751
$311$	9.71850e8	1.83205	0.916027	−	0.401118i	$-0.131378\pi$
$311$	9.71850e8	1.83205	0.916027	+	0.401118i	$0.131378\pi$
$312$	0	0
$313$	−4.34322e8	−0.800583	−0.400292	−	0.916388i	$-0.631091\pi$
$313$	−4.34322e8	−0.800583	−0.400292	+	0.916388i	$0.631091\pi$
$314$	1.68604e7	0.0307336
$315$	0	0
$316$	8.37038e8	1.49225
$317$	4.20773e8	0.741892	0.370946	−	0.928654i	$-0.379033\pi$
$317$	4.20773e8	0.741892	0.370946	+	0.928654i	$0.379033\pi$
$318$	0	0
$319$	2.04695e8	0.353054
$320$	−2.56792e8	−0.438083
$321$	0	0
$322$	354439.	0.000591624 0
$323$	3.52762e8	0.582470
$324$	0	0
$325$	−2.39351e8	−0.386761
$326$	−5.67988e7	−0.0907983
$327$	0	0
$328$	5.12791e7	0.0802383
$329$	−1.98481e7	−0.0307279
$330$	0	0
$331$	4.76792e8	0.722656	0.361328	−	0.932439i	$-0.382324\pi$
$331$	4.76792e8	0.722656	0.361328	+	0.932439i	$0.382324\pi$
$332$	−1.04095e9	−1.56116
$333$	0	0
$334$	−4.75874e6	−0.00698842
$335$	−1.70188e8	−0.247327
$336$	0	0
$337$	−4.04773e8	−0.576112	−0.288056	−	0.957614i	$-0.593009\pi$
$337$	−4.04773e8	−0.576112	−0.288056	+	0.957614i	$0.593009\pi$
$338$	1.13617e8	0.160042
$339$	0	0
$340$	2.37342e8	0.327490
$341$	−4.46921e8	−0.610366
$342$	0	0
$343$	−5.30701e7	−0.0710101
$344$	−4.68168e7	−0.0620079
$345$	0	0
$346$	−5.03670e7	−0.0653701
$347$	−1.35373e9	−1.73932	−0.869661	−	0.493649i	$-0.835663\pi$
$347$	−1.35373e9	−1.73932	−0.869661	+	0.493649i	$0.835663\pi$
$348$	0	0
$349$	2.20962e8	0.278246	0.139123	−	0.990275i	$-0.455572\pi$
$349$	2.20962e8	0.278246	0.139123	+	0.990275i	$0.455572\pi$
$350$	332950.	0.000415088 0
$351$	0	0
$352$	−1.50006e8	−0.183319
$353$	−1.22401e9	−1.48106	−0.740529	−	0.672024i	$-0.765425\pi$
$353$	−1.22401e9	−1.48106	−0.740529	+	0.672024i	$0.765425\pi$
$354$	0	0
$355$	−3.47827e8	−0.412634
$356$	−1.58218e9	−1.85858
$357$	0	0
$358$	−6.60493e7	−0.0760813
$359$	−6.08625e8	−0.694255	−0.347128	−	0.937818i	$-0.612843\pi$
$359$	−6.08625e8	−0.694255	−0.347128	+	0.937818i	$0.612843\pi$
$360$	0	0
$361$	−3.32194e8	−0.371634
$362$	−5.97674e7	−0.0662193
$363$	0	0
$364$	6.30010e7	0.0684688
$365$	1.18221e8	0.127253
$366$	0	0
$367$	1.36615e9	1.44267	0.721337	−	0.692584i	$-0.243528\pi$
$367$	1.36615e9	1.44267	0.721337	+	0.692584i	$0.243528\pi$
$368$	2.69736e8	0.282144
$369$	0	0
$370$	1.43649e7	0.0147433
$371$	−5.33993e7	−0.0542909
$372$	0	0
$373$	−1.62737e9	−1.62369	−0.811847	−	0.583871i	$-0.801538\pi$
$373$	−1.62737e9	−1.62369	−0.811847	+	0.583871i	$0.801538\pi$
$374$	4.56335e7	0.0451059
$375$	0	0
$376$	1.03983e8	0.100880
$377$	6.75972e8	0.649731
$378$	0	0
$379$	1.47071e9	1.38768	0.693841	−	0.720129i	$-0.255917\pi$
$379$	1.47071e9	1.38768	0.693841	+	0.720129i	$0.255917\pi$
$380$	3.77902e8	0.353294
$381$	0	0
$382$	4.34362e7	0.0398685
$383$	−1.02178e9	−0.929313	−0.464657	−	0.885491i	$-0.653822\pi$
$383$	−1.02178e9	−0.929313	−0.464657	+	0.885491i	$0.653822\pi$
$384$	0	0
$385$	−1.86944e7	−0.0166955
$386$	−7.17600e7	−0.0635077
$387$	0	0
$388$	−1.01234e9	−0.879861
$389$	4.95486e8	0.426784	0.213392	−	0.976967i	$-0.431549\pi$
$389$	4.95486e8	0.426784	0.213392	+	0.976967i	$0.431549\pi$
$390$	0	0
$391$	−2.47583e8	−0.209461
$392$	1.38927e8	0.116489
$393$	0	0
$394$	5.22709e7	0.0430550
$395$	−8.20219e8	−0.669638
$396$	0	0
$397$	2.13468e9	1.71225	0.856125	−	0.516769i	$-0.172866\pi$
$397$	2.13468e9	1.71225	0.856125	+	0.516769i	$0.172866\pi$
$398$	−7.14495e7	−0.0568079
$399$	0	0
$400$	2.53382e8	0.197955
$401$	6.77994e7	0.0525074	0.0262537	−	0.999655i	$-0.491642\pi$
$401$	6.77994e7	0.0525074	0.0262537	+	0.999655i	$0.491642\pi$
$402$	0	0
$403$	−1.47588e9	−1.12327
$404$	−1.45141e9	−1.09511
$405$	0	0
$406$	−940313.	−0.000697318 0
$407$	−8.06557e8	−0.593000
$408$	0	0
$409$	−4.21780e8	−0.304828	−0.152414	−	0.988317i	$-0.548705\pi$
$409$	−4.21780e8	−0.304828	−0.152414	+	0.988317i	$0.548705\pi$
$410$	−2.50814e7	−0.0179725
$411$	0	0
$412$	−1.50832e9	−1.06256
$413$	−3.81015e7	−0.0266144
$414$	0	0
$415$	1.02004e9	0.700564
$416$	−4.95368e8	−0.337366
$417$	0	0
$418$	7.26590e7	0.0486600
$419$	4.18473e8	0.277919	0.138960	−	0.990298i	$-0.455624\pi$
$419$	4.18473e8	0.277919	0.138960	+	0.990298i	$0.455624\pi$
$420$	0	0
$421$	−3.00965e9	−1.96575	−0.982875	−	0.184276i	$-0.941006\pi$
$421$	−3.00965e9	−1.96575	−0.982875	+	0.184276i	$0.941006\pi$
$422$	7.71278e7	0.0499594
$423$	0	0
$424$	2.79755e8	0.178237
$425$	−2.32573e8	−0.146959
$426$	0	0
$427$	4.14022e7	0.0257351
$428$	8.82814e8	0.544272
$429$	0	0
$430$	2.28988e7	0.0138891
$431$	6.83210e8	0.411039	0.205520	−	0.978653i	$-0.434112\pi$
$431$	6.83210e8	0.411039	0.205520	+	0.978653i	$0.434112\pi$
$432$	0	0
$433$	2.34708e9	1.38938	0.694689	−	0.719310i	$-0.255542\pi$
$433$	2.34708e9	1.38938	0.694689	+	0.719310i	$0.255542\pi$
$434$	2.05303e6	0.00120554
$435$	0	0
$436$	7.48220e8	0.432341
$437$	−3.94209e8	−0.225965
$438$	0	0
$439$	−9.05198e8	−0.510643	−0.255322	−	0.966856i	$-0.582181\pi$
$439$	−9.05198e8	−0.510643	−0.255322	+	0.966856i	$0.582181\pi$
$440$	9.79386e7	0.0548112
$441$	0	0
$442$	1.50697e8	0.0830093
$443$	−3.08556e8	−0.168625	−0.0843124	−	0.996439i	$-0.526869\pi$
$443$	−3.08556e8	−0.168625	−0.0843124	+	0.996439i	$0.526869\pi$
$444$	0	0
$445$	1.55039e9	0.834030
$446$	−2.84788e7	−0.0152002
$447$	0	0
$448$	−6.62336e7	−0.0348021
$449$	−2.54487e9	−1.32680	−0.663398	−	0.748267i	$-0.730886\pi$
$449$	−2.54487e9	−1.32680	−0.663398	+	0.748267i	$0.730886\pi$
$450$	0	0
$451$	1.40827e9	0.722882
$452$	−3.28560e8	−0.167352
$453$	0	0
$454$	1.85498e8	0.0930346
$455$	−6.17351e7	−0.0307250
$456$	0	0
$457$	−1.66227e9	−0.814694	−0.407347	−	0.913273i	$-0.633546\pi$
$457$	−1.66227e9	−0.814694	−0.407347	+	0.913273i	$0.633546\pi$
$458$	−5.99404e7	−0.0291535
$459$	0	0
$460$	−2.65227e8	−0.127047
$461$	−2.25720e9	−1.07304	−0.536521	−	0.843887i	$-0.680262\pi$
$461$	−2.25720e9	−1.07304	−0.536521	+	0.843887i	$0.680262\pi$
$462$	0	0
$463$	−7.79690e8	−0.365081	−0.182540	−	0.983198i	$-0.558432\pi$
$463$	−7.79690e8	−0.365081	−0.182540	+	0.983198i	$0.558432\pi$
$464$	−7.15599e8	−0.332550
$465$	0	0
$466$	1.68985e8	0.0773565
$467$	2.10933e9	0.958376	0.479188	−	0.877712i	$-0.340931\pi$
$467$	2.10933e9	0.958376	0.479188	+	0.877712i	$0.340931\pi$
$468$	0	0
$469$	−4.38962e7	−0.0196481
$470$	−5.08595e7	−0.0225959
$471$	0	0
$472$	1.99611e8	0.0873750
$473$	−1.28572e9	−0.558640
$474$	0	0
$475$	−3.70309e8	−0.158539
$476$	6.12169e7	0.0260164
$477$	0	0
$478$	−3.42218e7	−0.0143319
$479$	−9.11828e7	−0.0379087	−0.0189543	−	0.999820i	$-0.506034\pi$
$479$	−9.11828e7	−0.0379087	−0.0189543	+	0.999820i	$0.506034\pi$
$480$	0	0
$481$	−2.66352e9	−1.09131
$482$	7.08090e7	0.0288021
$483$	0	0
$484$	−2.58976e8	−0.103825
$485$	9.91995e8	0.394833
$486$	0	0
$487$	3.29198e9	1.29153	0.645767	−	0.763535i	$-0.276538\pi$
$487$	3.29198e9	1.29153	0.645767	+	0.763535i	$0.276538\pi$
$488$	−2.16903e8	−0.0844882
$489$	0	0
$490$	−6.79515e7	−0.0260923
$491$	−5.22909e8	−0.199361	−0.0996807	−	0.995019i	$-0.531782\pi$
$491$	−5.22909e8	−0.199361	−0.0996807	+	0.995019i	$0.531782\pi$
$492$	0	0
$493$	6.56829e8	0.246881
$494$	2.39944e8	0.0895500
$495$	0	0
$496$	1.56240e9	0.574919
$497$	−8.97142e7	−0.0327804
$498$	0	0
$499$	−3.88556e9	−1.39992	−0.699958	−	0.714184i	$-0.746798\pi$
$499$	−3.88556e9	−1.39992	−0.699958	+	0.714184i	$0.746798\pi$
$500$	−2.49147e8	−0.0891375
$501$	0	0
$502$	−2.35111e8	−0.0829487
$503$	−5.41477e9	−1.89711	−0.948555	−	0.316614i	$-0.897454\pi$
$503$	−5.41477e9	−1.89711	−0.948555	+	0.316614i	$0.897454\pi$
$504$	0	0
$505$	1.42225e9	0.491424
$506$	−5.09951e7	−0.0174985
$507$	0	0
$508$	3.11346e9	1.05371
$509$	−3.71235e8	−0.124778	−0.0623888	−	0.998052i	$-0.519872\pi$
$509$	−3.71235e8	−0.124778	−0.0623888	+	0.998052i	$0.519872\pi$
$510$	0	0
$511$	3.04924e7	0.0101092
$512$	8.75010e8	0.288117
$513$	0	0
$514$	4.32900e7	0.0140610
$515$	1.47802e9	0.476819
$516$	0	0
$517$	2.85566e9	0.908843
$518$	3.70509e6	0.00117124
$519$	0	0
$520$	3.23425e8	0.100870
$521$	−4.81158e8	−0.149058	−0.0745290	−	0.997219i	$-0.523745\pi$
$521$	−4.81158e8	−0.149058	−0.0745290	+	0.997219i	$0.523745\pi$
$522$	0	0
$523$	5.56626e9	1.70140	0.850702	−	0.525649i	$-0.176177\pi$
$523$	5.56626e9	1.70140	0.850702	+	0.525649i	$0.176177\pi$
$524$	−5.51500e9	−1.67450
$525$	0	0
$526$	6.12640e7	0.0183550
$527$	−1.43409e9	−0.426813
$528$	0	0
$529$	−3.12815e9	−0.918741
$530$	−1.36833e8	−0.0399231
$531$	0	0
$532$	9.74713e7	0.0280663
$533$	4.65057e9	1.33033
$534$	0	0
$535$	−8.65075e8	−0.244239
$536$	2.29968e8	0.0645047
$537$	0	0
$538$	3.82220e8	0.105822
$539$	3.81533e9	1.04947
$540$	0	0
$541$	−3.29748e9	−0.895349	−0.447675	−	0.894197i	$-0.647748\pi$
$541$	−3.29748e9	−0.895349	−0.447675	+	0.894197i	$0.647748\pi$
$542$	−1.00565e8	−0.0271300
$543$	0	0
$544$	−4.81340e8	−0.128190
$545$	−7.33186e8	−0.194011
$546$	0	0
$547$	−1.18317e9	−0.309095	−0.154547	−	0.987985i	$-0.549392\pi$
$547$	−1.18317e9	−0.309095	−0.154547	+	0.987985i	$0.549392\pi$
$548$	−4.10407e9	−1.06533
$549$	0	0
$550$	−4.79033e7	−0.0122771
$551$	1.04582e9	0.266334
$552$	0	0
$553$	−2.11557e8	−0.0531973
$554$	1.81205e8	0.0452780
$555$	0	0
$556$	−5.84029e9	−1.44103
$557$	−1.30364e9	−0.319641	−0.159821	−	0.987146i	$-0.551092\pi$
$557$	−1.30364e9	−0.319641	−0.159821	+	0.987146i	$0.551092\pi$
$558$	0	0
$559$	−4.24587e9	−1.02808
$560$	6.53542e7	0.0157259
$561$	0	0
$562$	3.01103e8	0.0715546
$563$	6.67266e9	1.57587	0.787934	−	0.615760i	$-0.211151\pi$
$563$	6.67266e9	1.57587	0.787934	+	0.615760i	$0.211151\pi$
$564$	0	0
$565$	3.21958e8	0.0750982
$566$	1.64235e8	0.0380721
$567$	0	0
$568$	4.70006e8	0.107618
$569$	−4.06197e9	−0.924365	−0.462183	−	0.886785i	$-0.652934\pi$
$569$	−4.06197e9	−0.924365	−0.462183	+	0.886785i	$0.652934\pi$
$570$	0	0
$571$	−3.50410e9	−0.787680	−0.393840	−	0.919179i	$-0.628854\pi$
$571$	−3.50410e9	−0.787680	−0.393840	+	0.919179i	$0.628854\pi$
$572$	−9.06431e9	−2.02511
$573$	0	0
$574$	−6.46918e6	−0.00142777
$575$	2.59898e8	0.0570119
$576$	0	0
$577$	−3.24448e9	−0.703121	−0.351560	−	0.936165i	$-0.614349\pi$
$577$	−3.24448e9	−0.703121	−0.351560	+	0.936165i	$0.614349\pi$
$578$	−1.24773e8	−0.0268765
$579$	0	0
$580$	7.03638e8	0.149745
$581$	2.63096e8	0.0556541
$582$	0	0
$583$	7.68286e9	1.60577
$584$	−1.59747e8	−0.0331886
$585$	0	0
$586$	4.41482e7	0.00906299
$587$	3.08962e9	0.630480	0.315240	−	0.949012i	$-0.397915\pi$
$587$	3.08962e9	0.630480	0.315240	+	0.949012i	$0.397915\pi$
$588$	0	0
$589$	−2.28339e9	−0.460444
$590$	−9.76329e7	−0.0195710
$591$	0	0
$592$	2.81966e9	0.558561
$593$	−8.35319e9	−1.64498	−0.822490	−	0.568779i	$-0.807416\pi$
$593$	−8.35319e9	−1.64498	−0.822490	+	0.568779i	$0.807416\pi$
$594$	0	0
$595$	−5.99869e7	−0.0116747
$596$	−5.29843e9	−1.02514
$597$	0	0
$598$	−1.68403e8	−0.0322029
$599$	4.71205e9	0.895810	0.447905	−	0.894081i	$-0.352170\pi$
$599$	4.71205e9	0.895810	0.447905	+	0.894081i	$0.352170\pi$
$600$	0	0
$601$	7.78892e9	1.46358	0.731790	−	0.681530i	$-0.238685\pi$
$601$	7.78892e9	1.46358	0.731790	+	0.681530i	$0.238685\pi$
$602$	5.90623e6	0.00110337
$603$	0	0
$604$	−4.57275e9	−0.844399
$605$	2.53773e8	0.0465909
$606$	0	0
$607$	−9.70797e9	−1.76185	−0.880923	−	0.473259i	$-0.843077\pi$
$607$	−9.70797e9	−1.76185	−0.880923	+	0.473259i	$0.843077\pi$
$608$	−7.66403e8	−0.138291
$609$	0	0
$610$	1.06091e8	0.0189244
$611$	9.43032e9	1.67256
$612$	0	0
$613$	−6.07402e9	−1.06504	−0.532518	−	0.846419i	$-0.678754\pi$
$613$	−6.07402e9	−1.06504	−0.532518	+	0.846419i	$0.678754\pi$
$614$	−2.24247e8	−0.0390965
$615$	0	0
$616$	2.52610e7	0.00435430
$617$	−5.24192e9	−0.898447	−0.449224	−	0.893419i	$-0.648299\pi$
$617$	−5.24192e9	−0.898447	−0.449224	+	0.893419i	$0.648299\pi$
$618$	0	0
$619$	−1.88254e9	−0.319026	−0.159513	−	0.987196i	$-0.550992\pi$
$619$	−1.88254e9	−0.319026	−0.159513	+	0.987196i	$0.550992\pi$
$620$	−1.53628e9	−0.258881
$621$	0	0
$622$	6.42319e8	0.107025
$623$	3.99889e8	0.0662569
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−2.87053e8	−0.0467684
$627$	0	0
$628$	−3.25418e9	−0.524304
$629$	−2.58809e9	−0.414669
$630$	0	0
$631$	5.70580e9	0.904094	0.452047	−	0.891994i	$-0.350694\pi$
$631$	5.70580e9	0.904094	0.452047	+	0.891994i	$0.350694\pi$
$632$	1.10833e9	0.174646
$633$	0	0
$634$	2.78099e8	0.0433398
$635$	−3.05090e9	−0.472846
$636$	0	0
$637$	1.25995e10	1.93137
$638$	1.35288e8	0.0206247
$639$	0	0
$640$	−6.87128e8	−0.103611
$641$	4.12549e9	0.618688	0.309344	−	0.950950i	$-0.399890\pi$
$641$	4.12549e9	0.618688	0.309344	+	0.950950i	$0.399890\pi$
$642$	0	0
$643$	3.38755e9	0.502512	0.251256	−	0.967921i	$-0.419156\pi$
$643$	3.38755e9	0.502512	0.251256	+	0.967921i	$0.419156\pi$
$644$	−6.84094e7	−0.0100929
$645$	0	0
$646$	2.33149e8	0.0340267
$647$	1.00682e10	1.46145	0.730727	−	0.682669i	$-0.239181\pi$
$647$	1.00682e10	1.46145	0.730727	+	0.682669i	$0.239181\pi$
$648$	0	0
$649$	5.48188e9	0.787178
$650$	−1.58193e8	−0.0225938
$651$	0	0
$652$	1.09626e10	1.54899
$653$	8.81112e9	1.23833	0.619163	−	0.785263i	$-0.287472\pi$
$653$	8.81112e9	1.23833	0.619163	+	0.785263i	$0.287472\pi$
$654$	0	0
$655$	5.40418e9	0.751425
$656$	−4.92319e9	−0.680900
$657$	0	0
$658$	−1.31181e7	−0.00179506
$659$	7.70767e9	1.04912	0.524559	−	0.851374i	$-0.324230\pi$
$659$	7.70767e9	1.04912	0.524559	+	0.851374i	$0.324230\pi$
$660$	0	0
$661$	7.47802e9	1.00712	0.503560	−	0.863960i	$-0.332023\pi$
$661$	7.47802e9	1.00712	0.503560	+	0.863960i	$0.332023\pi$
$662$	3.15123e8	0.0422160
$663$	0	0
$664$	−1.37834e9	−0.182712
$665$	−9.55127e7	−0.0125946
$666$	0	0
$667$	−7.34001e8	−0.0957759
$668$	9.18472e8	0.119220
$669$	0	0
$670$	−1.12481e8	−0.0144483
$671$	−5.95677e9	−0.761170
$672$	0	0
$673$	−5.73807e9	−0.725626	−0.362813	−	0.931862i	$-0.618184\pi$
$673$	−5.73807e9	−0.725626	−0.362813	+	0.931862i	$0.618184\pi$
$674$	−2.67524e8	−0.0336552
$675$	0	0
$676$	−2.19289e10	−2.73026
$677$	−3.36926e9	−0.417324	−0.208662	−	0.977988i	$-0.566911\pi$
$677$	−3.36926e9	−0.417324	−0.208662	+	0.977988i	$0.566911\pi$
$678$	0	0
$679$	2.55863e8	0.0313663
$680$	3.14266e8	0.0383280
$681$	0	0
$682$	−2.95381e8	−0.0356563
$683$	1.25702e10	1.50963	0.754814	−	0.655939i	$-0.227727\pi$
$683$	1.25702e10	1.50963	0.754814	+	0.655939i	$0.227727\pi$
$684$	0	0
$685$	4.02161e9	0.478060
$686$	−3.50752e7	−0.00414826
$687$	0	0
$688$	4.49477e9	0.526197
$689$	2.53713e10	2.95513
$690$	0	0
$691$	−1.14942e10	−1.32527	−0.662637	−	0.748940i	$-0.730563\pi$
$691$	−1.14942e10	−1.32527	−0.662637	+	0.748940i	$0.730563\pi$
$692$	9.72120e9	1.11519
$693$	0	0
$694$	−8.94714e8	−0.101608
$695$	5.72294e9	0.646655
$696$	0	0
$697$	4.51887e9	0.505492
$698$	1.46039e8	0.0162545
$699$	0	0
$700$	−6.42618e7	−0.00708125
$701$	9.46829e9	1.03815	0.519073	−	0.854730i	$-0.326277\pi$
$701$	9.46829e9	1.03815	0.519073	+	0.854730i	$0.326277\pi$
$702$	0	0
$703$	−4.12083e9	−0.447343
$704$	9.52939e9	1.02934
$705$	0	0
$706$	−8.08974e8	−0.0865203
$707$	3.66837e8	0.0390396
$708$	0	0
$709$	−9.68980e7	−0.0102106	−0.00510532	−	0.999987i	$-0.501625\pi$
$709$	−9.68980e7	−0.0102106	−0.00510532	+	0.999987i	$0.501625\pi$
$710$	−2.29887e8	−0.0241052
$711$	0	0
$712$	−2.09499e9	−0.217521
$713$	1.60258e9	0.165579
$714$	0	0
$715$	8.88217e9	0.908758
$716$	1.27480e10	1.29792
$717$	0	0
$718$	−4.02254e8	−0.0405570
$719$	−1.07728e10	−1.08088	−0.540441	−	0.841382i	$-0.681742\pi$
$719$	−1.07728e10	−1.08088	−0.540441	+	0.841382i	$0.681742\pi$
$720$	0	0
$721$	3.81221e8	0.0378794
$722$	−2.19554e8	−0.0217101
$723$	0	0
$724$	1.15356e10	1.12968
$725$	−6.89499e8	−0.0671971
$726$	0	0
$727$	8.38306e9	0.809156	0.404578	−	0.914503i	$-0.367418\pi$
$727$	8.38306e9	0.809156	0.404578	+	0.914503i	$0.367418\pi$
$728$	8.34203e7	0.00801331
$729$	0	0
$730$	7.81350e7	0.00743388
$731$	−4.12563e9	−0.390643
$732$	0	0
$733$	−1.28256e10	−1.20285	−0.601426	−	0.798928i	$-0.705401\pi$
$733$	−1.28256e10	−1.20285	−0.601426	+	0.798928i	$0.705401\pi$
$734$	9.02923e8	0.0842780
$735$	0	0
$736$	5.37894e8	0.0497306
$737$	6.31558e9	0.581135
$738$	0	0
$739$	−1.41983e10	−1.29414	−0.647071	−	0.762430i	$-0.724006\pi$
$739$	−1.41983e10	−1.29414	−0.647071	+	0.762430i	$0.724006\pi$
$740$	−2.77253e9	−0.251516
$741$	0	0
$742$	−3.52929e7	−0.00317156
$743$	4.55042e9	0.406996	0.203498	−	0.979075i	$-0.434769\pi$
$743$	4.55042e9	0.406996	0.203498	+	0.979075i	$0.434769\pi$
$744$	0	0
$745$	5.19196e9	0.460028
$746$	−1.07556e9	−0.0948528
$747$	0	0
$748$	−8.80762e9	−0.769490
$749$	−2.23126e8	−0.0194028
$750$	0	0
$751$	−6.62273e9	−0.570554	−0.285277	−	0.958445i	$-0.592086\pi$
$751$	−6.62273e9	−0.570554	−0.285277	+	0.958445i	$0.592086\pi$
$752$	−9.98314e9	−0.856061
$753$	0	0
$754$	4.46765e8	0.0379560
$755$	4.48086e9	0.378920
$756$	0	0
$757$	1.26339e10	1.05853	0.529263	−	0.848458i	$-0.322468\pi$
$757$	1.26339e10	1.05853	0.529263	+	0.848458i	$0.322468\pi$
$758$	9.72026e8	0.0810654
$759$	0	0
$760$	5.00384e8	0.0413481
$761$	7.85870e9	0.646405	0.323202	−	0.946330i	$-0.395241\pi$
$761$	7.85870e9	0.646405	0.323202	+	0.946330i	$0.395241\pi$
$762$	0	0
$763$	−1.89109e8	−0.0154126
$764$	−8.38352e9	−0.680142
$765$	0	0
$766$	−6.75319e8	−0.0542885
$767$	1.81030e10	1.44866
$768$	0	0
$769$	−1.51143e10	−1.19852	−0.599262	−	0.800553i	$-0.704539\pi$
$769$	−1.51143e10	−1.19852	−0.599262	+	0.800553i	$0.704539\pi$
$770$	−1.23556e7	−0.000975316 0
$771$	0	0
$772$	1.38502e10	1.08342
$773$	1.17793e10	0.917257	0.458628	−	0.888628i	$-0.348341\pi$
$773$	1.17793e10	0.917257	0.458628	+	0.888628i	$0.348341\pi$
$774$	0	0
$775$	1.50542e9	0.116172
$776$	−1.34044e9	−0.102975
$777$	0	0
$778$	3.27478e8	0.0249318
$779$	7.19506e9	0.545323
$780$	0	0
$781$	1.29077e10	0.969548
$782$	−1.63634e8	−0.0122363
$783$	0	0
$784$	−1.33381e10	−0.988524
$785$	3.18879e9	0.235279
$786$	0	0
$787$	1.22032e10	0.892406	0.446203	−	0.894932i	$-0.352776\pi$
$787$	1.22032e10	0.892406	0.446203	+	0.894932i	$0.352776\pi$
$788$	−1.00887e10	−0.734502
$789$	0	0
$790$	−5.42102e8	−0.0391189
$791$	8.30417e7	0.00596594
$792$	0	0
$793$	−1.96712e10	−1.40080
$794$	1.41086e9	0.100026
$795$	0	0
$796$	1.37903e10	0.969121
$797$	−9.44357e9	−0.660742	−0.330371	−	0.943851i	$-0.607174\pi$
$797$	−9.44357e9	−0.660742	−0.330371	+	0.943851i	$0.607174\pi$
$798$	0	0
$799$	9.16326e9	0.635530
$800$	5.05281e8	0.0348914
$801$	0	0
$802$	4.48102e7	0.00306737
$803$	−4.38711e9	−0.299002
$804$	0	0
$805$	6.70348e7	0.00452913
$806$	−9.75444e8	−0.0656190
$807$	0	0
$808$	−1.92183e9	−0.128167
$809$	−5.05980e9	−0.335980	−0.167990	−	0.985789i	$-0.553728\pi$
$809$	−5.05980e9	−0.335980	−0.167990	+	0.985789i	$0.553728\pi$
$810$	0	0
$811$	−1.57961e9	−0.103986	−0.0519931	−	0.998647i	$-0.516557\pi$
$811$	−1.57961e9	−0.103986	−0.0519931	+	0.998647i	$0.516557\pi$
$812$	1.81487e8	0.0118960
$813$	0	0
$814$	−5.33072e8	−0.0346418
$815$	−1.07423e10	−0.695099
$816$	0	0
$817$	−6.56895e9	−0.421423
$818$	−2.78764e8	−0.0178074
$819$	0	0
$820$	4.84090e9	0.306604
$821$	2.90631e10	1.83291	0.916456	−	0.400136i	$-0.131037\pi$
$821$	2.90631e10	1.83291	0.916456	+	0.400136i	$0.131037\pi$
$822$	0	0
$823$	−2.81795e9	−0.176211	−0.0881056	−	0.996111i	$-0.528081\pi$
$823$	−2.81795e9	−0.176211	−0.0881056	+	0.996111i	$0.528081\pi$
$824$	−1.99719e9	−0.124358
$825$	0	0
$826$	−2.51822e7	−0.00155476
$827$	2.02876e10	1.24727	0.623635	−	0.781715i	$-0.285655\pi$
$827$	2.02876e10	1.24727	0.623635	+	0.781715i	$0.285655\pi$
$828$	0	0
$829$	−1.09989e10	−0.670516	−0.335258	−	0.942126i	$-0.608824\pi$
$829$	−1.09989e10	−0.670516	−0.335258	+	0.942126i	$0.608824\pi$
$830$	6.74167e8	0.0409255
$831$	0	0
$832$	3.14692e10	1.89432
$833$	1.22427e10	0.733870
$834$	0	0
$835$	−9.00017e8	−0.0534993
$836$	−1.40237e10	−0.830121
$837$	0	0
$838$	2.76579e8	0.0162355
$839$	−1.59263e10	−0.930997	−0.465499	−	0.885049i	$-0.654125\pi$
$839$	−1.59263e10	−0.930997	−0.465499	+	0.885049i	$0.654125\pi$
$840$	0	0
$841$	−1.53026e10	−0.887114
$842$	−1.98915e9	−0.114835
$843$	0	0
$844$	−1.48862e10	−0.852289
$845$	2.14883e10	1.22519
$846$	0	0
$847$	6.54549e7	0.00370127
$848$	−2.68586e10	−1.51251
$849$	0	0
$850$	−1.53713e8	−0.00858506
$851$	2.89217e9	0.160868
$852$	0	0
$853$	7.73776e9	0.426868	0.213434	−	0.976957i	$-0.431535\pi$
$853$	7.73776e9	0.426868	0.213434	+	0.976957i	$0.431535\pi$
$854$	2.73637e7	0.00150339
$855$	0	0
$856$	1.16894e9	0.0636993
$857$	2.43053e10	1.31907	0.659535	−	0.751674i	$-0.270753\pi$
$857$	2.43053e10	1.31907	0.659535	+	0.751674i	$0.270753\pi$
$858$	0	0
$859$	−2.86538e10	−1.54243	−0.771216	−	0.636573i	$-0.780351\pi$
$859$	−2.86538e10	−1.54243	−0.771216	+	0.636573i	$0.780351\pi$
$860$	−4.41964e9	−0.236942
$861$	0	0
$862$	4.51549e8	0.0240121
$863$	−1.25051e10	−0.662292	−0.331146	−	0.943580i	$-0.607435\pi$
$863$	−1.25051e10	−0.662292	−0.331146	+	0.943580i	$0.607435\pi$
$864$	0	0
$865$	−9.52587e9	−0.500436
$866$	1.55124e9	0.0811646
$867$	0	0
$868$	−3.96250e8	−0.0205660
$869$	3.04379e10	1.57342
$870$	0	0
$871$	2.08561e10	1.06947
$872$	9.90725e8	0.0505994
$873$	0	0
$874$	−2.60542e8	−0.0132004
$875$	6.29706e7	0.00317767
$876$	0	0
$877$	−1.78548e9	−0.0893834	−0.0446917	−	0.999001i	$-0.514231\pi$
$877$	−1.78548e9	−0.0893834	−0.0446917	+	0.999001i	$0.514231\pi$
$878$	−5.98266e8	−0.0298307
$879$	0	0
$880$	−9.40287e9	−0.465126
$881$	−2.37289e10	−1.16913	−0.584563	−	0.811348i	$-0.698734\pi$
$881$	−2.37289e10	−1.16913	−0.584563	+	0.811348i	$0.698734\pi$
$882$	0	0
$883$	−7.48664e8	−0.0365952	−0.0182976	−	0.999833i	$-0.505825\pi$
$883$	−7.48664e8	−0.0365952	−0.0182976	+	0.999833i	$0.505825\pi$
$884$	−2.90857e10	−1.41611
$885$	0	0
$886$	−2.03932e8	−0.00985071
$887$	−3.12317e10	−1.50267	−0.751334	−	0.659922i	$-0.770589\pi$
$887$	−3.12317e10	−1.50267	−0.751334	+	0.659922i	$0.770589\pi$
$888$	0	0
$889$	−7.86911e8	−0.0375638
$890$	1.02469e9	0.0487223
$891$	0	0
$892$	5.49662e9	0.259310
$893$	1.45900e10	0.685607
$894$	0	0
$895$	−1.24919e10	−0.582434
$896$	−1.77229e8	−0.00823108
$897$	0	0
$898$	−1.68197e9	−0.0775087
$899$	−4.25158e9	−0.195160
$900$	0	0
$901$	2.46528e10	1.12287
$902$	9.30757e8	0.0422293
$903$	0	0
$904$	−4.35049e8	−0.0195861
$905$	−1.13038e10	−0.506936
$906$	0	0
$907$	2.33083e10	1.03725	0.518626	−	0.855001i	$-0.326444\pi$
$907$	2.33083e10	1.03725	0.518626	+	0.855001i	$0.326444\pi$
$908$	−3.58026e10	−1.58713
$909$	0	0
$910$	−4.08022e7	−0.00179489
$911$	−1.47817e10	−0.647755	−0.323878	−	0.946099i	$-0.604987\pi$
$911$	−1.47817e10	−0.647755	−0.323878	+	0.946099i	$0.604987\pi$
$912$	0	0
$913$	−3.78530e10	−1.64609
$914$	−1.09863e9	−0.0475927
$915$	0	0
$916$	1.15689e10	0.497347
$917$	1.39389e9	0.0596946
$918$	0	0
$919$	−1.90815e10	−0.810977	−0.405489	−	0.914100i	$-0.632899\pi$
$919$	−1.90815e10	−0.810977	−0.405489	+	0.914100i	$0.632899\pi$
$920$	−3.51190e8	−0.0148691
$921$	0	0
$922$	−1.49184e9	−0.0626848
$923$	4.26254e10	1.78428
$924$	0	0
$925$	2.71682e9	0.112866
$926$	−5.15316e8	−0.0213272
$927$	0	0
$928$	−1.42701e9	−0.0586150
$929$	−3.90390e10	−1.59751	−0.798756	−	0.601655i	$-0.794508\pi$
$929$	−3.90390e10	−1.59751	−0.798756	+	0.601655i	$0.794508\pi$
$930$	0	0
$931$	1.94931e10	0.791695
$932$	−3.26153e10	−1.31967
$933$	0	0
$934$	1.39411e9	0.0559863
$935$	8.63064e9	0.345305
$936$	0	0
$937$	4.36982e10	1.73530	0.867651	−	0.497174i	$-0.165629\pi$
$937$	4.36982e10	1.73530	0.867651	+	0.497174i	$0.165629\pi$
$938$	−2.90120e7	−0.00114780
$939$	0	0
$940$	9.81628e9	0.385478
$941$	1.00925e10	0.394853	0.197427	−	0.980318i	$-0.436742\pi$
$941$	1.00925e10	0.394853	0.197427	+	0.980318i	$0.436742\pi$
$942$	0	0
$943$	−5.04980e9	−0.196102
$944$	−1.91642e10	−0.741462
$945$	0	0
$946$	−8.49762e8	−0.0326346
$947$	2.40570e10	0.920483	0.460242	−	0.887794i	$-0.347763\pi$
$947$	2.40570e10	0.920483	0.460242	+	0.887794i	$0.347763\pi$
$948$	0	0
$949$	−1.44877e10	−0.550260
$950$	−2.44746e8	−0.00926152
$951$	0	0
$952$	8.10579e7	0.00304485
$953$	4.15590e10	1.55539	0.777696	−	0.628641i	$-0.216388\pi$
$953$	4.15590e10	1.55539	0.777696	+	0.628641i	$0.216388\pi$
$954$	0	0
$955$	8.21507e9	0.305210
$956$	6.60507e9	0.244498
$957$	0	0
$958$	−6.02648e7	−0.00221455
$959$	1.03728e9	0.0379780
$960$	0	0
$961$	−1.82299e10	−0.662603
$962$	−1.76038e9	−0.0637520
$963$	0	0
$964$	−1.36667e10	−0.491352
$965$	−1.35719e10	−0.486178
$966$	0	0
$967$	2.29700e10	0.816900	0.408450	−	0.912781i	$-0.366069\pi$
$967$	2.29700e10	0.816900	0.408450	+	0.912781i	$0.366069\pi$
$968$	−3.42913e8	−0.0121512
$969$	0	0
$970$	6.55633e8	0.0230653
$971$	−3.27851e10	−1.14924	−0.574618	−	0.818421i	$-0.694850\pi$
$971$	−3.27851e10	−1.14924	−0.574618	+	0.818421i	$0.694850\pi$
$972$	0	0
$973$	1.47610e9	0.0513714
$974$	2.17574e9	0.0754487
$975$	0	0
$976$	2.08244e10	0.716965
$977$	−3.87148e10	−1.32815	−0.664074	−	0.747667i	$-0.731174\pi$
$977$	−3.87148e10	−1.32815	−0.664074	+	0.747667i	$0.731174\pi$
$978$	0	0
$979$	−5.75342e10	−1.95969
$980$	1.31151e10	0.445125
$981$	0	0
$982$	−3.45603e8	−0.0116463
$983$	−4.43335e10	−1.48866	−0.744329	−	0.667814i	$-0.767230\pi$
$983$	−4.43335e10	−1.48866	−0.744329	+	0.667814i	$0.767230\pi$
$984$	0	0
$985$	9.88597e9	0.329604
$986$	4.34114e8	0.0144223
$987$	0	0
$988$	−4.63110e10	−1.52769
$989$	4.61036e9	0.151547
$990$	0	0
$991$	2.58820e10	0.844772	0.422386	−	0.906416i	$-0.361193\pi$
$991$	2.58820e10	0.844772	0.422386	+	0.906416i	$0.361193\pi$
$992$	3.11566e9	0.101335
$993$	0	0
$994$	−5.92942e7	−0.00191496
$995$	−1.35132e10	−0.434888
$996$	0	0
$997$	1.78036e10	0.568949	0.284475	−	0.958684i	$-0.408181\pi$
$997$	1.78036e10	0.568949	0.284475	+	0.958684i	$0.408181\pi$
$998$	−2.56806e9	−0.0817802
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.h.1.7	yes	14
3.2	odd	2		405.8.a.g.1.8	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.8.a.g.1.8	✓	14	3.2	odd	2
405.8.a.h.1.7	yes	14	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+0.660923 q^{2} -127.563 q^{4} +125.000 q^{5} +32.2409 q^{7} -168.908 q^{8} +O(q^{10})\)	\(q+0.660923 q^{2} -127.563 q^{4} +125.000 q^{5} +32.2409 q^{7} -168.908 q^{8} +82.6154 q^{10} -4638.68 q^{11} -15318.5 q^{13} +21.3088 q^{14} +16216.5 q^{16} -14884.6 q^{17} -23699.8 q^{19} -15945.4 q^{20} -3065.81 q^{22} +16633.5 q^{23} +15625.0 q^{25} -10124.3 q^{26} -4112.75 q^{28} -44127.9 q^{29} +96346.6 q^{31} +32338.0 q^{32} -9837.61 q^{34} +4030.12 q^{35} +173876. q^{37} -15663.7 q^{38} -21113.5 q^{40} -303592. q^{41} +277174. q^{43} +591725. q^{44} +10993.5 q^{46} -615618. q^{47} -822504. q^{49} +10326.9 q^{50} +1.95407e6 q^{52} -1.65626e6 q^{53} -579835. q^{55} -5445.74 q^{56} -29165.2 q^{58} -1.18178e6 q^{59} +1.28415e6 q^{61} +63677.7 q^{62} -2.05433e6 q^{64} -1.91481e6 q^{65} -1.36150e6 q^{67} +1.89873e6 q^{68} +2663.60 q^{70} -2.78262e6 q^{71} +945768. q^{73} +114919. q^{74} +3.02322e6 q^{76} -149555. q^{77} -6.56175e6 q^{79} +2.02706e6 q^{80} -200651. q^{82} +8.16030e6 q^{83} -1.86058e6 q^{85} +183191. q^{86} +783509. q^{88} +1.24031e7 q^{89} -493881. q^{91} -2.12182e6 q^{92} -406876. q^{94} -2.96247e6 q^{95} +7.93596e6 q^{97} -543612. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8}+O(q^{10}) \) 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8	\( 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8} + 2000 q^{10} + 10648 q^{11} + 17268 q^{13} + 9180 q^{14} + 34122 q^{16} + 3886 q^{17} + 22986 q^{19} + 89250 q^{20} - 49706 q^{22} + 155040 q^{23} + 218750 q^{25} + 7804 q^{26} + 256160 q^{28} + 195836 q^{29} + 92786 q^{31} + 157778 q^{32} + 787348 q^{34} + 192250 q^{35} + 876406 q^{37} + 329320 q^{38} - 15750 q^{40} + 795164 q^{41} + 730350 q^{43} + 2360876 q^{44} - 225654 q^{46} + 2687842 q^{47} + 1663586 q^{49} + 250000 q^{50} + 3875836 q^{52} + 2533750 q^{53} + 1331000 q^{55} + 2055276 q^{56} - 318934 q^{58} + 2283340 q^{59} + 2600400 q^{61} + 5702022 q^{62} + 474098 q^{64} + 2158500 q^{65} + 2422160 q^{67} + 1114364 q^{68} + 1147500 q^{70} + 6395324 q^{71} - 540774 q^{73} + 260516 q^{74} - 2417042 q^{76} - 1384890 q^{77} - 307384 q^{79} + 4265250 q^{80} - 14044738 q^{82} + 10991322 q^{83} + 485750 q^{85} + 2847712 q^{86} - 19226592 q^{88} + 2094000 q^{89} - 9496256 q^{91} + 39213378 q^{92} - 28132682 q^{94} + 2873250 q^{95} - 12859994 q^{97} + 28336478 q^{98}+O(q^{100}) \) 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8 + 2000 * q^10 + 10648 * q^11 + 17268 * q^13 + 9180 * q^14 + 34122 * q^16 + 3886 * q^17 + 22986 * q^19 + 89250 * q^20 - 49706 * q^22 + 155040 * q^23 + 218750 * q^25 + 7804 * q^26 + 256160 * q^28 + 195836 * q^29 + 92786 * q^31 + 157778 * q^32 + 787348 * q^34 + 192250 * q^35 + 876406 * q^37 + 329320 * q^38 - 15750 * q^40 + 795164 * q^41 + 730350 * q^43 + 2360876 * q^44 - 225654 * q^46 + 2687842 * q^47 + 1663586 * q^49 + 250000 * q^50 + 3875836 * q^52 + 2533750 * q^53 + 1331000 * q^55 + 2055276 * q^56 - 318934 * q^58 + 2283340 * q^59 + 2600400 * q^61 + 5702022 * q^62 + 474098 * q^64 + 2158500 * q^65 + 2422160 * q^67 + 1114364 * q^68 + 1147500 * q^70 + 6395324 * q^71 - 540774 * q^73 + 260516 * q^74 - 2417042 * q^76 - 1384890 * q^77 - 307384 * q^79 + 4265250 * q^80 - 14044738 * q^82 + 10991322 * q^83 + 485750 * q^85 + 2847712 * q^86 - 19226592 * q^88 + 2094000 * q^89 - 9496256 * q^91 + 39213378 * q^92 - 28132682 * q^94 + 2873250 * q^95 - 12859994 * q^97 + 28336478 * q^98

Introduction

L-functions

Modular forms

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Database

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