LMFDB - Embedded newform 405.8.a.h.1.2

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			1.2
Root			$-19.4093$ 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)

$f(q)$	$=$	$q-16.6772 q^{2} +150.129 q^{4} +125.000 q^{5} +1291.61 q^{7} -369.053 q^{8} +O(q^{10})$	\(q-16.6772 q^{2} +150.129 q^{4} +125.000 q^{5} +1291.61 q^{7} -369.053 q^{8} -2084.65 q^{10} +3093.07 q^{11} -6584.94 q^{13} -21540.5 q^{14} -13061.8 q^{16} -2968.89 q^{17} +16303.7 q^{19} +18766.1 q^{20} -51583.7 q^{22} +53674.6 q^{23} +15625.0 q^{25} +109818. q^{26} +193909. q^{28} -6505.59 q^{29} +27889.1 q^{31} +265072. q^{32} +49512.8 q^{34} +161451. q^{35} +187837. q^{37} -271901. q^{38} -46131.6 q^{40} +745267. q^{41} -740828. q^{43} +464360. q^{44} -895143. q^{46} +1.18652e6 q^{47} +844717. q^{49} -260581. q^{50} -988592. q^{52} -1.54113e6 q^{53} +386633. q^{55} -476673. q^{56} +108495. q^{58} +1.91516e6 q^{59} -273773. q^{61} -465112. q^{62} -2.74876e6 q^{64} -823118. q^{65} +722010. q^{67} -445717. q^{68} -2.69256e6 q^{70} +641799. q^{71} +6.29811e6 q^{73} -3.13259e6 q^{74} +2.44767e6 q^{76} +3.99504e6 q^{77} -2.49001e6 q^{79} -1.63272e6 q^{80} -1.24290e7 q^{82} -1.27949e6 q^{83} -371111. q^{85} +1.23549e7 q^{86} -1.14151e6 q^{88} -4.02273e6 q^{89} -8.50518e6 q^{91} +8.05813e6 q^{92} -1.97878e7 q^{94} +2.03797e6 q^{95} -1.04326e7 q^{97} -1.40875e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−16.6772	−1.47407	−0.737035	−	0.675854i	$-0.763775\pi$
$2$	−16.6772	−1.47407	−0.737035	+	0.675854i	$0.763775\pi$
$3$	0	0
$3$	0	0
$4$	150.129	1.17288
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	1291.61	1.42327	0.711637	−	0.702547i	$-0.247954\pi$
$7$	1291.61	1.42327	0.711637	+	0.702547i	$0.247954\pi$
$8$	−369.053	−0.254844
$9$	0	0
$10$	−2084.65	−0.659224
$11$	3093.07	0.700672	0.350336	−	0.936624i	$-0.386067\pi$
$11$	3093.07	0.700672	0.350336	+	0.936624i	$0.386067\pi$
$12$	0	0
$13$	−6584.94	−0.831285	−0.415643	−	0.909528i	$-0.636443\pi$
$13$	−6584.94	−0.831285	−0.415643	+	0.909528i	$0.636443\pi$
$14$	−21540.5	−2.09801
$15$	0	0
$16$	−13061.8	−0.797227
$17$	−2968.89	−0.146562	−0.0732812	−	0.997311i	$-0.523347\pi$
$17$	−2968.89	−0.146562	−0.0732812	+	0.997311i	$0.523347\pi$
$18$	0	0
$19$	16303.7	0.545318	0.272659	−	0.962111i	$-0.412097\pi$
$19$	16303.7	0.545318	0.272659	+	0.962111i	$0.412097\pi$
$20$	18766.1	0.524530
$21$	0	0
$22$	−51583.7	−1.03284
$23$	53674.6	0.919860	0.459930	−	0.887955i	$-0.347874\pi$
$23$	53674.6	0.919860	0.459930	+	0.887955i	$0.347874\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	109818.	1.22537
$27$	0	0
$28$	193909.	1.66934
$29$	−6505.59	−0.0495329	−0.0247665	−	0.999693i	$-0.507884\pi$
$29$	−6505.59	−0.0495329	−0.0247665	+	0.999693i	$0.507884\pi$
$30$	0	0
$31$	27889.1	0.168139	0.0840694	−	0.996460i	$-0.473208\pi$
$31$	27889.1	0.168139	0.0840694	+	0.996460i	$0.473208\pi$
$32$	265072.	1.43001
$33$	0	0
$34$	49512.8	0.216043
$35$	161451.	0.636508
$36$	0	0
$37$	187837.	0.609642	0.304821	−	0.952410i	$-0.401403\pi$
$37$	187837.	0.609642	0.304821	+	0.952410i	$0.401403\pi$
$38$	−271901.	−0.803837
$39$	0	0
$40$	−46131.6	−0.113970
$41$	745267.	1.68876	0.844381	−	0.535743i	$-0.179968\pi$
$41$	745267.	1.68876	0.844381	+	0.535743i	$0.179968\pi$
$42$	0	0
$43$	−740828.	−1.42095	−0.710474	−	0.703724i	$-0.751519\pi$
$43$	−740828.	−1.42095	−0.710474	+	0.703724i	$0.751519\pi$
$44$	464360.	0.821807
$45$	0	0
$46$	−895143.	−1.35594
$47$	1.18652e6	1.66699	0.833494	−	0.552529i	$-0.186337\pi$
$47$	1.18652e6	1.66699	0.833494	+	0.552529i	$0.186337\pi$
$48$	0	0
$49$	844717.	1.02571
$50$	−260581.	−0.294814
$51$	0	0
$52$	−988592.	−0.975002
$53$	−1.54113e6	−1.42192	−0.710958	−	0.703234i	$-0.751738\pi$
$53$	−1.54113e6	−1.42192	−0.710958	+	0.703234i	$0.751738\pi$
$54$	0	0
$55$	386633.	0.313350
$56$	−476673.	−0.362713
$57$	0	0
$58$	108495.	0.0730150
$59$	1.91516e6	1.21401	0.607005	−	0.794698i	$-0.292371\pi$
$59$	1.91516e6	1.21401	0.607005	+	0.794698i	$0.292371\pi$
$60$	0	0
$61$	−273773.	−0.154432	−0.0772158	−	0.997014i	$-0.524603\pi$
$61$	−273773.	−0.154432	−0.0772158	+	0.997014i	$0.524603\pi$
$62$	−465112.	−0.247849
$63$	0	0
$64$	−2.74876e6	−1.31071
$65$	−823118.	−0.371762
$66$	0	0
$67$	722010.	0.293279	0.146640	−	0.989190i	$-0.453154\pi$
$67$	722010.	0.293279	0.146640	+	0.989190i	$0.453154\pi$
$68$	−445717.	−0.171901
$69$	0	0
$70$	−2.69256e6	−0.938257
$71$	641799.	0.212811	0.106406	−	0.994323i	$-0.466066\pi$
$71$	641799.	0.212811	0.106406	+	0.994323i	$0.466066\pi$
$72$	0	0
$73$	6.29811e6	1.89487	0.947436	−	0.319945i	$-0.103664\pi$
$73$	6.29811e6	1.89487	0.947436	+	0.319945i	$0.103664\pi$
$74$	−3.13259e6	−0.898655
$75$	0	0
$76$	2.44767e6	0.639595
$77$	3.99504e6	0.997249
$78$	0	0
$79$	−2.49001e6	−0.568207	−0.284103	−	0.958794i	$-0.591696\pi$
$79$	−2.49001e6	−0.568207	−0.284103	+	0.958794i	$0.591696\pi$
$80$	−1.63272e6	−0.356531
$81$	0	0
$82$	−1.24290e7	−2.48935
$83$	−1.27949e6	−0.245619	−0.122809	−	0.992430i	$-0.539190\pi$
$83$	−1.27949e6	−0.245619	−0.122809	+	0.992430i	$0.539190\pi$
$84$	0	0
$85$	−371111.	−0.0655447
$86$	1.23549e7	2.09458
$87$	0	0
$88$	−1.14151e6	−0.178562
$89$	−4.02273e6	−0.604862	−0.302431	−	0.953171i	$-0.597798\pi$
$89$	−4.02273e6	−0.604862	−0.302431	+	0.953171i	$0.597798\pi$
$90$	0	0
$91$	−8.50518e6	−1.18315
$92$	8.05813e6	1.07889
$93$	0	0
$94$	−1.97878e7	−2.45726
$95$	2.03797e6	0.243874
$96$	0	0
$97$	−1.04326e7	−1.16062	−0.580312	−	0.814395i	$-0.697069\pi$
$97$	−1.04326e7	−1.16062	−0.580312	+	0.814395i	$0.697069\pi$
$98$	−1.40875e7	−1.51197
$99$	0	0
$100$	2.34577e6	0.234577
$101$	−2.85676e6	−0.275898	−0.137949	−	0.990439i	$-0.544051\pi$
$101$	−2.85676e6	−0.275898	−0.137949	+	0.990439i	$0.544051\pi$
$102$	0	0
$103$	−1.54437e7	−1.39258	−0.696289	−	0.717761i	$-0.745167\pi$
$103$	−1.54437e7	−1.39258	−0.696289	+	0.717761i	$0.745167\pi$
$104$	2.43019e6	0.211848
$105$	0	0
$106$	2.57018e7	2.09601
$107$	−1.71058e6	−0.134989	−0.0674947	−	0.997720i	$-0.521501\pi$
$107$	−1.71058e6	−0.134989	−0.0674947	+	0.997720i	$0.521501\pi$
$108$	0	0
$109$	−1.74985e7	−1.29422	−0.647109	−	0.762397i	$-0.724022\pi$
$109$	−1.74985e7	−1.29422	−0.647109	+	0.762397i	$0.724022\pi$
$110$	−6.44796e6	−0.461900
$111$	0	0
$112$	−1.68707e7	−1.13467
$113$	2.57301e7	1.67752	0.838758	−	0.544504i	$-0.183282\pi$
$113$	2.57301e7	1.67752	0.838758	+	0.544504i	$0.183282\pi$
$114$	0	0
$115$	6.70933e6	0.411374
$116$	−976680.	−0.0580964
$117$	0	0
$118$	−3.19395e7	−1.78954
$119$	−3.83465e6	−0.208599
$120$	0	0
$121$	−9.92011e6	−0.509059
$122$	4.56577e6	0.227643
$123$	0	0
$124$	4.18696e6	0.197207
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−595322.	−0.0257893	−0.0128946	−	0.999917i	$-0.504105\pi$
$127$	−595322.	−0.0257893	−0.0128946	+	0.999917i	$0.504105\pi$
$128$	1.19124e7	0.502070
$129$	0	0
$130$	1.37273e7	0.548004
$131$	3.39445e7	1.31923	0.659613	−	0.751605i	$-0.270720\pi$
$131$	3.39445e7	1.31923	0.659613	+	0.751605i	$0.270720\pi$
$132$	0	0
$133$	2.10581e7	0.776137
$134$	−1.20411e7	−0.432314
$135$	0	0
$136$	1.09568e6	0.0373505
$137$	1.60349e6	0.0532776	0.0266388	−	0.999645i	$-0.491520\pi$
$137$	1.60349e6	0.0532776	0.0266388	+	0.999645i	$0.491520\pi$
$138$	0	0
$139$	4.43945e7	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$139$	4.43945e7	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$140$	2.42386e7	0.746550
$141$	0	0
$142$	−1.07034e7	−0.313699
$143$	−2.03677e7	−0.582459
$144$	0	0
$145$	−813199.	−0.0221518
$146$	−1.05035e8	−2.79318
$147$	0	0
$148$	2.81998e7	0.715039
$149$	−2.10622e7	−0.521617	−0.260808	−	0.965391i	$-0.583989\pi$
$149$	−2.10622e7	−0.521617	−0.260808	+	0.965391i	$0.583989\pi$
$150$	0	0
$151$	−4.98801e7	−1.17898	−0.589492	−	0.807774i	$-0.700672\pi$
$151$	−4.98801e7	−1.17898	−0.589492	+	0.807774i	$0.700672\pi$
$152$	−6.01695e6	−0.138971
$153$	0	0
$154$	−6.66261e7	−1.47002
$155$	3.48613e6	0.0751940
$156$	0	0
$157$	−982230.	−0.0202565	−0.0101283	−	0.999949i	$-0.503224\pi$
$157$	−982230.	−0.0202565	−0.0101283	+	0.999949i	$0.503224\pi$
$158$	4.15264e7	0.837577
$159$	0	0
$160$	3.31341e7	0.639521
$161$	6.93268e7	1.30921
$162$	0	0
$163$	−5.85963e7	−1.05977	−0.529887	−	0.848068i	$-0.677766\pi$
$163$	−5.85963e7	−1.05977	−0.529887	+	0.848068i	$0.677766\pi$
$164$	1.11886e8	1.98072
$165$	0	0
$166$	2.13382e7	0.362060
$167$	9.88516e6	0.164239	0.0821195	−	0.996622i	$-0.473831\pi$
$167$	9.88516e6	0.164239	0.0821195	+	0.996622i	$0.473831\pi$
$168$	0	0
$169$	−1.93871e7	−0.308965
$170$	6.18910e6	0.0966176
$171$	0	0
$172$	−1.11220e8	−1.66661
$173$	1.14608e8	1.68289	0.841443	−	0.540346i	$-0.181707\pi$
$173$	1.14608e8	1.68289	0.841443	+	0.540346i	$0.181707\pi$
$174$	0	0
$175$	2.01814e7	0.284655
$176$	−4.04009e7	−0.558595
$177$	0	0
$178$	6.70879e7	0.891609
$179$	−9.66344e6	−0.125935	−0.0629674	−	0.998016i	$-0.520056\pi$
$179$	−9.66344e6	−0.125935	−0.0629674	+	0.998016i	$0.520056\pi$
$180$	0	0
$181$	6.10729e7	0.765551	0.382775	−	0.923841i	$-0.374968\pi$
$181$	6.10729e7	0.765551	0.382775	+	0.923841i	$0.374968\pi$
$182$	1.41843e8	1.74404
$183$	0	0
$184$	−1.98088e7	−0.234421
$185$	2.34796e7	0.272640
$186$	0	0
$187$	−9.18297e6	−0.102692
$188$	1.78131e8	1.95518
$189$	0	0
$190$	−3.39876e7	−0.359487
$191$	−7.64863e7	−0.794267	−0.397134	−	0.917761i	$-0.629995\pi$
$191$	−7.64863e7	−0.794267	−0.397134	+	0.917761i	$0.629995\pi$
$192$	0	0
$193$	1.60890e8	1.61093	0.805467	−	0.592641i	$-0.201915\pi$
$193$	1.60890e8	1.61093	0.805467	+	0.592641i	$0.201915\pi$
$194$	1.73986e8	1.71084
$195$	0	0
$196$	1.26817e8	1.20304
$197$	1.40302e8	1.30747	0.653736	−	0.756723i	$-0.273201\pi$
$197$	1.40302e8	1.30747	0.653736	+	0.756723i	$0.273201\pi$
$198$	0	0
$199$	5.55084e7	0.499313	0.249656	−	0.968334i	$-0.419682\pi$
$199$	5.55084e7	0.499313	0.249656	+	0.968334i	$0.419682\pi$
$200$	−5.76645e6	−0.0509687
$201$	0	0
$202$	4.76427e7	0.406693
$203$	−8.40270e6	−0.0704990
$204$	0	0
$205$	9.31584e7	0.755237
$206$	2.57557e8	2.05276
$207$	0	0
$208$	8.60109e7	0.662723
$209$	5.04286e7	0.382089
$210$	0	0
$211$	−4.13648e7	−0.303139	−0.151570	−	0.988447i	$-0.548433\pi$
$211$	−4.13648e7	−0.303139	−0.151570	+	0.988447i	$0.548433\pi$
$212$	−2.31369e8	−1.66774
$213$	0	0
$214$	2.85277e7	0.198984
$215$	−9.26036e7	−0.635467
$216$	0	0
$217$	3.60218e7	0.239308
$218$	2.91826e8	1.90777
$219$	0	0
$220$	5.80449e7	0.367523
$221$	1.95500e7	0.121835
$222$	0	0
$223$	2.07882e8	1.25530	0.627652	−	0.778494i	$-0.284016\pi$
$223$	2.07882e8	1.25530	0.627652	+	0.778494i	$0.284016\pi$
$224$	3.42371e8	2.03530
$225$	0	0
$226$	−4.29106e8	−2.47278
$227$	1.22142e7	0.0693068	0.0346534	−	0.999399i	$-0.488967\pi$
$227$	1.22142e7	0.0693068	0.0346534	+	0.999399i	$0.488967\pi$
$228$	0	0
$229$	2.14195e8	1.17865	0.589324	−	0.807897i	$-0.299394\pi$
$229$	2.14195e8	1.17865	0.589324	+	0.807897i	$0.299394\pi$
$230$	−1.11893e8	−0.606394
$231$	0	0
$232$	2.40091e6	0.0126232
$233$	−3.22775e8	−1.67169	−0.835843	−	0.548968i	$-0.815021\pi$
$233$	−3.22775e8	−1.67169	−0.835843	+	0.548968i	$0.815021\pi$
$234$	0	0
$235$	1.48315e8	0.745500
$236$	2.87521e8	1.42389
$237$	0	0
$238$	6.39513e7	0.307489
$239$	−3.35588e8	−1.59006	−0.795030	−	0.606570i	$-0.792545\pi$
$239$	−3.35588e8	−1.59006	−0.795030	+	0.606570i	$0.792545\pi$
$240$	0	0
$241$	3.14739e8	1.44841	0.724204	−	0.689585i	$-0.242207\pi$
$241$	3.14739e8	1.44841	0.724204	+	0.689585i	$0.242207\pi$
$242$	1.65440e8	0.750388
$243$	0	0
$244$	−4.11013e7	−0.181130
$245$	1.05590e8	0.458712
$246$	0	0
$247$	−1.07359e8	−0.453315
$248$	−1.02925e7	−0.0428491
$249$	0	0
$250$	−3.25727e7	−0.131845
$251$	1.17568e8	0.469281	0.234640	−	0.972082i	$-0.424609\pi$
$251$	1.17568e8	0.469281	0.234640	+	0.972082i	$0.424609\pi$
$252$	0	0
$253$	1.66019e8	0.644520
$254$	9.92831e6	0.0380152
$255$	0	0
$256$	1.53176e8	0.570625
$257$	9.62735e7	0.353786	0.176893	−	0.984230i	$-0.443395\pi$
$257$	9.62735e7	0.353786	0.176893	+	0.984230i	$0.443395\pi$
$258$	0	0
$259$	2.42612e8	0.867688
$260$	−1.23574e8	−0.436034
$261$	0	0
$262$	−5.66099e8	−1.94463
$263$	−3.22043e8	−1.09161	−0.545806	−	0.837912i	$-0.683776\pi$
$263$	−3.22043e8	−1.09161	−0.545806	+	0.837912i	$0.683776\pi$
$264$	0	0
$265$	−1.92641e8	−0.635900
$266$	−3.51190e8	−1.14408
$267$	0	0
$268$	1.08395e8	0.343983
$269$	2.27302e8	0.711985	0.355992	−	0.934489i	$-0.384143\pi$
$269$	2.27302e8	0.711985	0.355992	+	0.934489i	$0.384143\pi$
$270$	0	0
$271$	−5.09084e8	−1.55381	−0.776903	−	0.629620i	$-0.783211\pi$
$271$	−5.09084e8	−1.55381	−0.776903	+	0.629620i	$0.783211\pi$
$272$	3.87789e7	0.116844
$273$	0	0
$274$	−2.67418e7	−0.0785350
$275$	4.83292e7	0.140134
$276$	0	0
$277$	4.78802e8	1.35356	0.676779	−	0.736187i	$-0.263375\pi$
$277$	4.78802e8	1.35356	0.676779	+	0.736187i	$0.263375\pi$
$278$	−7.40375e8	−2.06678
$279$	0	0
$280$	−5.95841e7	−0.162210
$281$	−6.33584e8	−1.70346	−0.851730	−	0.523981i	$-0.824446\pi$
$281$	−6.33584e8	−1.70346	−0.851730	+	0.523981i	$0.824446\pi$
$282$	0	0
$283$	4.68789e8	1.22949	0.614744	−	0.788727i	$-0.289259\pi$
$283$	4.68789e8	1.22949	0.614744	+	0.788727i	$0.289259\pi$
$284$	9.63527e7	0.249603
$285$	0	0
$286$	3.39676e8	0.858585
$287$	9.62595e8	2.40357
$288$	0	0
$289$	−4.01524e8	−0.978519
$290$	1.35619e7	0.0326533
$291$	0	0
$292$	9.45530e8	2.22247
$293$	2.78503e8	0.646835	0.323417	−	0.946256i	$-0.395168\pi$
$293$	2.78503e8	0.646835	0.323417	+	0.946256i	$0.395168\pi$
$294$	0	0
$295$	2.39395e8	0.542922
$296$	−6.93218e7	−0.155363
$297$	0	0
$298$	3.51258e8	0.768900
$299$	−3.53444e8	−0.764666
$300$	0	0
$301$	−9.56862e8	−2.02240
$302$	8.31860e8	1.73791
$303$	0	0
$304$	−2.12956e8	−0.434742
$305$	−3.42216e7	−0.0690639
$306$	0	0
$307$	8.83603e8	1.74290	0.871450	−	0.490484i	$-0.163180\pi$
$307$	8.83603e8	1.74290	0.871450	+	0.490484i	$0.163180\pi$
$308$	5.99772e8	1.16966
$309$	0	0
$310$	−5.81390e7	−0.110841
$311$	−6.65857e8	−1.25522	−0.627610	−	0.778528i	$-0.715967\pi$
$311$	−6.65857e8	−1.25522	−0.627610	+	0.778528i	$0.715967\pi$
$312$	0	0
$313$	5.80232e6	0.0106954	0.00534769	−	0.999986i	$-0.498298\pi$
$313$	5.80232e6	0.0106954	0.00534769	+	0.999986i	$0.498298\pi$
$314$	1.63809e7	0.0298595
$315$	0	0
$316$	−3.73823e8	−0.666441
$317$	−2.07938e8	−0.366628	−0.183314	−	0.983054i	$-0.558682\pi$
$317$	−2.07938e8	−0.366628	−0.183314	+	0.983054i	$0.558682\pi$
$318$	0	0
$319$	−2.01222e7	−0.0347063
$320$	−3.43595e8	−0.586168
$321$	0	0
$322$	−1.15618e9	−1.92987
$323$	−4.84040e7	−0.0799232
$324$	0	0
$325$	−1.02890e8	−0.166257
$326$	9.77222e8	1.56218
$327$	0	0
$328$	−2.75043e8	−0.430370
$329$	1.53252e9	2.37258
$330$	0	0
$331$	−6.57347e8	−0.996315	−0.498158	−	0.867087i	$-0.665990\pi$
$331$	−6.57347e8	−0.996315	−0.498158	+	0.867087i	$0.665990\pi$
$332$	−1.92088e8	−0.288083
$333$	0	0
$334$	−1.64857e8	−0.242100
$335$	9.02512e7	0.131158
$336$	0	0
$337$	5.72992e6	0.00815537	0.00407768	−	0.999992i	$-0.498702\pi$
$337$	5.72992e6	0.00815537	0.00407768	+	0.999992i	$0.498702\pi$
$338$	3.23322e8	0.455436
$339$	0	0
$340$	−5.57146e7	−0.0768764
$341$	8.62627e7	0.117810
$342$	0	0
$343$	2.73483e7	0.0365932
$344$	2.73405e8	0.362119
$345$	0	0
$346$	−1.91135e9	−2.48069
$347$	9.07363e8	1.16581	0.582905	−	0.812540i	$-0.301916\pi$
$347$	9.07363e8	1.16581	0.582905	+	0.812540i	$0.301916\pi$
$348$	0	0
$349$	−4.63726e8	−0.583945	−0.291973	−	0.956427i	$-0.594312\pi$
$349$	−4.63726e8	−0.583945	−0.291973	+	0.956427i	$0.594312\pi$
$350$	−3.36570e8	−0.419601
$351$	0	0
$352$	8.19887e8	1.00197
$353$	7.47466e8	0.904440	0.452220	−	0.891906i	$-0.350632\pi$
$353$	7.47466e8	0.904440	0.452220	+	0.891906i	$0.350632\pi$
$354$	0	0
$355$	8.02249e7	0.0951721
$356$	−6.03930e8	−0.709433
$357$	0	0
$358$	1.61159e8	0.185637
$359$	−6.90906e8	−0.788113	−0.394057	−	0.919086i	$-0.628929\pi$
$359$	−6.90906e8	−0.788113	−0.394057	+	0.919086i	$0.628929\pi$
$360$	0	0
$361$	−6.28060e8	−0.702628
$362$	−1.01853e9	−1.12848
$363$	0	0
$364$	−1.27688e9	−1.38770
$365$	7.87263e8	0.847413
$366$	0	0
$367$	−1.87089e9	−1.97568	−0.987840	−	0.155474i	$-0.950310\pi$
$367$	−1.87089e9	−1.97568	−0.987840	+	0.155474i	$0.950310\pi$
$368$	−7.01086e8	−0.733337
$369$	0	0
$370$	−3.91574e8	−0.401891
$371$	−1.99054e9	−2.02378
$372$	0	0
$373$	5.98516e8	0.597166	0.298583	−	0.954384i	$-0.403486\pi$
$373$	5.98516e8	0.597166	0.298583	+	0.954384i	$0.403486\pi$
$374$	1.53146e8	0.151376
$375$	0	0
$376$	−4.37889e8	−0.424821
$377$	4.28390e7	0.0411760
$378$	0	0
$379$	1.58681e9	1.49723	0.748614	−	0.663007i	$-0.230720\pi$
$379$	1.58681e9	1.49723	0.748614	+	0.663007i	$0.230720\pi$
$380$	3.05959e8	0.286036
$381$	0	0
$382$	1.27558e9	1.17081
$383$	1.64017e9	1.49174	0.745868	−	0.666093i	$-0.232035\pi$
$383$	1.64017e9	1.49174	0.745868	+	0.666093i	$0.232035\pi$
$384$	0	0
$385$	4.99380e8	0.445983
$386$	−2.68319e9	−2.37463
$387$	0	0
$388$	−1.56624e9	−1.36128
$389$	−2.21576e8	−0.190853	−0.0954267	−	0.995436i	$-0.530422\pi$
$389$	−2.21576e8	−0.190853	−0.0954267	+	0.995436i	$0.530422\pi$
$390$	0	0
$391$	−1.59354e8	−0.134817
$392$	−3.11745e8	−0.261396
$393$	0	0
$394$	−2.33985e9	−1.92731
$395$	−3.11251e8	−0.254110
$396$	0	0
$397$	9.00275e8	0.722118	0.361059	−	0.932543i	$-0.382415\pi$
$397$	9.00275e8	0.722118	0.361059	+	0.932543i	$0.382415\pi$
$398$	−9.25725e8	−0.736022
$399$	0	0
$400$	−2.04090e8	−0.159445
$401$	3.92263e8	0.303789	0.151894	−	0.988397i	$-0.451463\pi$
$401$	3.92263e8	0.303789	0.151894	+	0.988397i	$0.451463\pi$
$402$	0	0
$403$	−1.83648e8	−0.139771
$404$	−4.28882e8	−0.323596
$405$	0	0
$406$	1.40134e8	0.103920
$407$	5.80992e8	0.427159
$408$	0	0
$409$	2.50090e9	1.80745	0.903723	−	0.428118i	$-0.140823\pi$
$409$	2.50090e9	1.80745	0.903723	+	0.428118i	$0.140823\pi$
$410$	−1.55362e9	−1.11327
$411$	0	0
$412$	−2.31854e9	−1.63333
$413$	2.47364e9	1.72787
$414$	0	0
$415$	−1.59936e8	−0.109844
$416$	−1.74549e9	−1.18875
$417$	0	0
$418$	−8.41008e8	−0.563226
$419$	−6.45030e8	−0.428382	−0.214191	−	0.976792i	$-0.568711\pi$
$419$	−6.45030e8	−0.428382	−0.214191	+	0.976792i	$0.568711\pi$
$420$	0	0
$421$	1.93703e9	1.26517	0.632585	−	0.774491i	$-0.281994\pi$
$421$	1.93703e9	1.26517	0.632585	+	0.774491i	$0.281994\pi$
$422$	6.89848e8	0.446848
$423$	0	0
$424$	5.68759e8	0.362366
$425$	−4.63889e7	−0.0293125
$426$	0	0
$427$	−3.53608e8	−0.219798
$428$	−2.56808e8	−0.158327
$429$	0	0
$430$	1.54437e9	0.936723
$431$	−2.58210e9	−1.55347	−0.776733	−	0.629830i	$-0.783125\pi$
$431$	−2.58210e9	−1.55347	−0.776733	+	0.629830i	$0.783125\pi$
$432$	0	0
$433$	−3.66536e8	−0.216975	−0.108487	−	0.994098i	$-0.534601\pi$
$433$	−3.66536e8	−0.216975	−0.108487	+	0.994098i	$0.534601\pi$
$434$	−6.00744e8	−0.352757
$435$	0	0
$436$	−2.62703e9	−1.51797
$437$	8.75098e8	0.501616
$438$	0	0
$439$	−5.05372e8	−0.285092	−0.142546	−	0.989788i	$-0.545529\pi$
$439$	−5.05372e8	−0.285092	−0.142546	+	0.989788i	$0.545529\pi$
$440$	−1.42688e8	−0.0798553
$441$	0	0
$442$	−3.26039e8	−0.179594
$443$	−1.89898e9	−1.03778	−0.518892	−	0.854840i	$-0.673655\pi$
$443$	−1.89898e9	−1.03778	−0.518892	+	0.854840i	$0.673655\pi$
$444$	0	0
$445$	−5.02842e8	−0.270502
$446$	−3.46689e9	−1.85041
$447$	0	0
$448$	−3.55033e9	−1.86550
$449$	−1.48564e9	−0.774553	−0.387276	−	0.921964i	$-0.626584\pi$
$449$	−1.48564e9	−0.774553	−0.387276	+	0.921964i	$0.626584\pi$
$450$	0	0
$451$	2.30516e9	1.18327
$452$	3.86284e9	1.96753
$453$	0	0
$454$	−2.03699e8	−0.102163
$455$	−1.06315e9	−0.529120
$456$	0	0
$457$	6.01055e8	0.294583	0.147291	−	0.989093i	$-0.452945\pi$
$457$	6.01055e8	0.294583	0.147291	+	0.989093i	$0.452945\pi$
$458$	−3.57217e9	−1.73741
$459$	0	0
$460$	1.00727e9	0.482494
$461$	−1.63116e9	−0.775430	−0.387715	−	0.921779i	$-0.626736\pi$
$461$	−1.63116e9	−0.775430	−0.387715	+	0.921779i	$0.626736\pi$
$462$	0	0
$463$	−1.97894e9	−0.926615	−0.463308	−	0.886197i	$-0.653338\pi$
$463$	−1.97894e9	−0.926615	−0.463308	+	0.886197i	$0.653338\pi$
$464$	8.49745e7	0.0394890
$465$	0	0
$466$	5.38299e9	2.46418
$467$	3.06374e9	1.39201	0.696006	−	0.718036i	$-0.254959\pi$
$467$	3.06374e9	1.39201	0.696006	+	0.718036i	$0.254959\pi$
$468$	0	0
$469$	9.32556e8	0.417417
$470$	−2.47348e9	−1.09892
$471$	0	0
$472$	−7.06794e8	−0.309383
$473$	−2.29143e9	−0.995618
$474$	0	0
$475$	2.54746e8	0.109064
$476$	−5.75693e8	−0.244662
$477$	0	0
$478$	5.59667e9	2.34386
$479$	−1.20009e9	−0.498928	−0.249464	−	0.968384i	$-0.580254\pi$
$479$	−1.20009e9	−0.498928	−0.249464	+	0.968384i	$0.580254\pi$
$480$	0	0
$481$	−1.23689e9	−0.506786
$482$	−5.24897e9	−2.13506
$483$	0	0
$484$	−1.48930e9	−0.597067
$485$	−1.30407e9	−0.519046
$486$	0	0
$487$	4.61897e9	1.81215	0.906074	−	0.423120i	$-0.139065\pi$
$487$	4.61897e9	1.81215	0.906074	+	0.423120i	$0.139065\pi$
$488$	1.01037e8	0.0393559
$489$	0	0
$490$	−1.76094e9	−0.676173
$491$	−6.05440e7	−0.0230827	−0.0115413	−	0.999933i	$-0.503674\pi$
$491$	−6.05440e7	−0.0230827	−0.0115413	+	0.999933i	$0.503674\pi$
$492$	0	0
$493$	1.93144e7	0.00725967
$494$	1.79045e9	0.668218
$495$	0	0
$496$	−3.64280e8	−0.134045
$497$	8.28955e8	0.302889
$498$	0	0
$499$	5.87078e8	0.211516	0.105758	−	0.994392i	$-0.466273\pi$
$499$	5.87078e8	0.211516	0.105758	+	0.994392i	$0.466273\pi$
$500$	2.93221e8	0.104906
$501$	0	0
$502$	−1.96071e9	−0.691753
$503$	4.36786e9	1.53031	0.765157	−	0.643844i	$-0.222661\pi$
$503$	4.36786e9	1.53031	0.765157	+	0.643844i	$0.222661\pi$
$504$	0	0
$505$	−3.57094e8	−0.123385
$506$	−2.76874e9	−0.950069
$507$	0	0
$508$	−8.93752e7	−0.0302478
$509$	5.14131e9	1.72807	0.864035	−	0.503431i	$-0.167929\pi$
$509$	5.14131e9	1.72807	0.864035	+	0.503431i	$0.167929\pi$
$510$	0	0
$511$	8.13471e9	2.69692
$512$	−4.07934e9	−1.34321
$513$	0	0
$514$	−1.60557e9	−0.521506
$515$	−1.93046e9	−0.622780
$516$	0	0
$517$	3.66998e9	1.16801
$518$	−4.04609e9	−1.27903
$519$	0	0
$520$	3.03774e8	0.0947412
$521$	2.42343e9	0.750754	0.375377	−	0.926872i	$-0.377513\pi$
$521$	2.42343e9	0.750754	0.375377	+	0.926872i	$0.377513\pi$
$522$	0	0
$523$	2.57764e9	0.787890	0.393945	−	0.919134i	$-0.371110\pi$
$523$	2.57764e9	0.787890	0.393945	+	0.919134i	$0.371110\pi$
$524$	5.09605e9	1.54730
$525$	0	0
$526$	5.37077e9	1.60911
$527$	−8.27995e7	−0.0246428
$528$	0	0
$529$	−5.23858e8	−0.153857
$530$	3.21272e9	0.937362
$531$	0	0
$532$	3.16144e9	0.910319
$533$	−4.90754e9	−1.40384
$534$	0	0
$535$	−2.13822e8	−0.0603691
$536$	−2.66460e8	−0.0747404
$537$	0	0
$538$	−3.79077e9	−1.04952
$539$	2.61277e9	0.718687
$540$	0	0
$541$	1.16765e9	0.317046	0.158523	−	0.987355i	$-0.449327\pi$
$541$	1.16765e9	0.317046	0.158523	+	0.987355i	$0.449327\pi$
$542$	8.49010e9	2.29042
$543$	0	0
$544$	−7.86971e8	−0.209586
$545$	−2.18731e9	−0.578792
$546$	0	0
$547$	−4.03057e9	−1.05296	−0.526478	−	0.850189i	$-0.676488\pi$
$547$	−4.03057e9	−1.05296	−0.526478	+	0.850189i	$0.676488\pi$
$548$	2.40731e8	0.0624885
$549$	0	0
$550$	−8.05995e8	−0.206568
$551$	−1.06066e8	−0.0270112
$552$	0	0
$553$	−3.21613e9	−0.808714
$554$	−7.98508e9	−1.99524
$555$	0	0
$556$	6.66490e9	1.64449
$557$	2.54101e9	0.623036	0.311518	−	0.950240i	$-0.399163\pi$
$557$	2.54101e9	0.623036	0.311518	+	0.950240i	$0.399163\pi$
$558$	0	0
$559$	4.87831e9	1.18121
$560$	−2.10884e9	−0.507441
$561$	0	0
$562$	1.05664e10	2.51102
$563$	−2.37886e9	−0.561810	−0.280905	−	0.959736i	$-0.590635\pi$
$563$	−2.37886e9	−0.561810	−0.280905	+	0.959736i	$0.590635\pi$
$564$	0	0
$565$	3.21626e9	0.750208
$566$	−7.81808e9	−1.81235
$567$	0	0
$568$	−2.36858e8	−0.0542336
$569$	7.82287e9	1.78022	0.890109	−	0.455747i	$-0.150628\pi$
$569$	7.82287e9	1.78022	0.890109	+	0.455747i	$0.150628\pi$
$570$	0	0
$571$	−5.07422e9	−1.14063	−0.570313	−	0.821428i	$-0.693178\pi$
$571$	−5.07422e9	−1.14063	−0.570313	+	0.821428i	$0.693178\pi$
$572$	−3.05778e9	−0.683156
$573$	0	0
$574$	−1.60534e10	−3.54304
$575$	8.38666e8	0.183972
$576$	0	0
$577$	−2.38484e9	−0.516826	−0.258413	−	0.966035i	$-0.583200\pi$
$577$	−2.38484e9	−0.516826	−0.258413	+	0.966035i	$0.583200\pi$
$578$	6.69630e9	1.44241
$579$	0	0
$580$	−1.22085e8	−0.0259815
$581$	−1.65260e9	−0.349583
$582$	0	0
$583$	−4.76682e9	−0.996297
$584$	−2.32434e9	−0.482896
$585$	0	0
$586$	−4.64465e9	−0.953480
$587$	5.41369e9	1.10474	0.552370	−	0.833599i	$-0.313724\pi$
$587$	5.41369e9	1.10474	0.552370	+	0.833599i	$0.313724\pi$
$588$	0	0
$589$	4.54696e8	0.0916891
$590$	−3.99243e9	−0.800305
$591$	0	0
$592$	−2.45348e9	−0.486023
$593$	5.48942e9	1.08102	0.540512	−	0.841336i	$-0.318231\pi$
$593$	5.48942e9	1.08102	0.540512	+	0.841336i	$0.318231\pi$
$594$	0	0
$595$	−4.79331e8	−0.0932882
$596$	−3.16205e9	−0.611796
$597$	0	0
$598$	5.89447e9	1.12717
$599$	3.97376e9	0.755453	0.377726	−	0.925917i	$-0.376706\pi$
$599$	3.97376e9	0.755453	0.377726	+	0.925917i	$0.376706\pi$
$600$	0	0
$601$	5.52792e9	1.03873	0.519363	−	0.854554i	$-0.326169\pi$
$601$	5.52792e9	1.03873	0.519363	+	0.854554i	$0.326169\pi$
$602$	1.59578e10	2.98116
$603$	0	0
$604$	−7.48845e9	−1.38281
$605$	−1.24001e9	−0.227658
$606$	0	0
$607$	1.06901e10	1.94009	0.970044	−	0.242928i	$-0.0781078\pi$
$607$	1.06901e10	1.94009	0.970044	+	0.242928i	$0.0781078\pi$
$608$	4.32168e9	0.779811
$609$	0	0
$610$	5.70721e8	0.101805
$611$	−7.81316e9	−1.38574
$612$	0	0
$613$	−8.06363e9	−1.41390	−0.706950	−	0.707263i	$-0.749929\pi$
$613$	−8.06363e9	−1.41390	−0.706950	+	0.707263i	$0.749929\pi$
$614$	−1.47360e10	−2.56916
$615$	0	0
$616$	−1.47438e9	−0.254143
$617$	3.13272e9	0.536937	0.268468	−	0.963288i	$-0.413482\pi$
$617$	3.13272e9	0.536937	0.268468	+	0.963288i	$0.413482\pi$
$618$	0	0
$619$	1.01732e10	1.72401	0.862007	−	0.506896i	$-0.169207\pi$
$619$	1.01732e10	1.72401	0.862007	+	0.506896i	$0.169207\pi$
$620$	5.23370e8	0.0881938
$621$	0	0
$622$	1.11046e10	1.85028
$623$	−5.19581e9	−0.860885
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	−9.67664e7	−0.0157657
$627$	0	0
$628$	−1.47461e8	−0.0237585
$629$	−5.57667e8	−0.0893506
$630$	0	0
$631$	7.25139e9	1.14900	0.574498	−	0.818506i	$-0.305197\pi$
$631$	7.25139e9	1.14900	0.574498	+	0.818506i	$0.305197\pi$
$632$	9.18946e8	0.144804
$633$	0	0
$634$	3.46782e9	0.540435
$635$	−7.44153e7	−0.0115333
$636$	0	0
$637$	−5.56241e9	−0.852658
$638$	3.35583e8	0.0511596
$639$	0	0
$640$	1.48905e9	0.224533
$641$	−1.13527e9	−0.170254	−0.0851269	−	0.996370i	$-0.527130\pi$
$641$	−1.13527e9	−0.170254	−0.0851269	+	0.996370i	$0.527130\pi$
$642$	0	0
$643$	−7.38263e9	−1.09515	−0.547574	−	0.836757i	$-0.684448\pi$
$643$	−7.38263e9	−1.09515	−0.547574	+	0.836757i	$0.684448\pi$
$644$	1.04080e10	1.53556
$645$	0	0
$646$	8.07244e8	0.117812
$647$	−8.97838e9	−1.30327	−0.651633	−	0.758534i	$-0.725916\pi$
$647$	−8.97838e9	−1.30327	−0.651633	+	0.758534i	$0.725916\pi$
$648$	0	0
$649$	5.92371e9	0.850623
$650$	1.71591e9	0.245075
$651$	0	0
$652$	−8.79701e9	−1.24299
$653$	1.06065e10	1.49066	0.745328	−	0.666698i	$-0.232293\pi$
$653$	1.06065e10	1.49066	0.745328	+	0.666698i	$0.232293\pi$
$654$	0	0
$655$	4.24306e9	0.589976
$656$	−9.73450e9	−1.34633
$657$	0	0
$658$	−2.55582e10	−3.49735
$659$	1.17260e10	1.59607	0.798033	−	0.602614i	$-0.205874\pi$
$659$	1.17260e10	1.59607	0.798033	+	0.602614i	$0.205874\pi$
$660$	0	0
$661$	−4.42990e9	−0.596607	−0.298304	−	0.954471i	$-0.596421\pi$
$661$	−4.42990e9	−0.596607	−0.298304	+	0.954471i	$0.596421\pi$
$662$	1.09627e10	1.46864
$663$	0	0
$664$	4.72198e8	0.0625944
$665$	2.63226e9	0.347099
$666$	0	0
$667$	−3.49185e8	−0.0455634
$668$	1.48405e9	0.192633
$669$	0	0
$670$	−1.50514e9	−0.193337
$671$	−8.46798e8	−0.108206
$672$	0	0
$673$	7.26995e9	0.919345	0.459673	−	0.888088i	$-0.347967\pi$
$673$	7.26995e9	0.919345	0.459673	+	0.888088i	$0.347967\pi$
$674$	−9.55590e7	−0.0120216
$675$	0	0
$676$	−2.91056e9	−0.362380
$677$	−4.65457e8	−0.0576527	−0.0288263	−	0.999584i	$-0.509177\pi$
$677$	−4.65457e8	−0.0576527	−0.0288263	+	0.999584i	$0.509177\pi$
$678$	0	0
$679$	−1.34749e10	−1.65189
$680$	1.36960e8	0.0167037
$681$	0	0
$682$	−1.43862e9	−0.173661
$683$	7.75325e9	0.931132	0.465566	−	0.885013i	$-0.345851\pi$
$683$	7.75325e9	0.931132	0.465566	+	0.885013i	$0.345851\pi$
$684$	0	0
$685$	2.00437e8	0.0238265
$686$	−4.56093e8	−0.0539410
$687$	0	0
$688$	9.67652e9	1.13282
$689$	1.01483e10	1.18202
$690$	0	0
$691$	6.26499e8	0.0722349	0.0361175	−	0.999348i	$-0.488501\pi$
$691$	6.26499e8	0.0722349	0.0361175	+	0.999348i	$0.488501\pi$
$692$	1.72061e10	1.97383
$693$	0	0
$694$	−1.51323e10	−1.71849
$695$	5.54931e9	0.627035
$696$	0	0
$697$	−2.21262e9	−0.247509
$698$	7.73365e9	0.860777
$699$	0	0
$700$	3.02982e9	0.333867
$701$	−5.09570e9	−0.558715	−0.279358	−	0.960187i	$-0.590121\pi$
$701$	−5.09570e9	−0.558715	−0.279358	+	0.960187i	$0.590121\pi$
$702$	0	0
$703$	3.06245e9	0.332449
$704$	−8.50211e9	−0.918380
$705$	0	0
$706$	−1.24656e10	−1.33321
$707$	−3.68982e9	−0.392678
$708$	0	0
$709$	−3.71303e9	−0.391262	−0.195631	−	0.980678i	$-0.562675\pi$
$709$	−3.71303e9	−0.391262	−0.195631	+	0.980678i	$0.562675\pi$
$710$	−1.33793e9	−0.140290
$711$	0	0
$712$	1.48460e9	0.154145
$713$	1.49694e9	0.154664
$714$	0	0
$715$	−2.54596e9	−0.260483
$716$	−1.45076e9	−0.147707
$717$	0	0
$718$	1.15224e10	1.16173
$719$	−1.42804e10	−1.43281	−0.716404	−	0.697685i	$-0.754213\pi$
$719$	−1.42804e10	−1.43281	−0.716404	+	0.697685i	$0.754213\pi$
$720$	0	0
$721$	−1.99472e10	−1.98202
$722$	1.04743e10	1.03572
$723$	0	0
$724$	9.16883e9	0.897902
$725$	−1.01650e8	−0.00990659
$726$	0	0
$727$	5.64805e9	0.545166	0.272583	−	0.962132i	$-0.412122\pi$
$727$	5.64805e9	0.545166	0.272583	+	0.962132i	$0.412122\pi$
$728$	3.13886e9	0.301518
$729$	0	0
$730$	−1.31294e10	−1.24915
$731$	2.19944e9	0.208258
$732$	0	0
$733$	−1.84012e10	−1.72577	−0.862885	−	0.505400i	$-0.831345\pi$
$733$	−1.84012e10	−1.72577	−0.862885	+	0.505400i	$0.831345\pi$
$734$	3.12012e10	2.91229
$735$	0	0
$736$	1.42277e10	1.31541
$737$	2.23322e9	0.205493
$738$	0	0
$739$	−5.77829e9	−0.526676	−0.263338	−	0.964704i	$-0.584823\pi$
$739$	−5.77829e9	−0.526676	−0.263338	+	0.964704i	$0.584823\pi$
$740$	3.52497e9	0.319775
$741$	0	0
$742$	3.31967e10	2.98319
$743$	1.66224e8	0.0148673	0.00743365	−	0.999972i	$-0.497634\pi$
$743$	1.66224e8	0.0148673	0.00743365	+	0.999972i	$0.497634\pi$
$744$	0	0
$745$	−2.63277e9	−0.233274
$746$	−9.98157e9	−0.880264
$747$	0	0
$748$	−1.37863e9	−0.120446
$749$	−2.20940e9	−0.192127
$750$	0	0
$751$	−1.03825e10	−0.894464	−0.447232	−	0.894418i	$-0.647590\pi$
$751$	−1.03825e10	−0.894464	−0.447232	+	0.894418i	$0.647590\pi$
$752$	−1.54980e10	−1.32897
$753$	0	0
$754$	−7.14434e8	−0.0606963
$755$	−6.23501e9	−0.527257
$756$	0	0
$757$	2.83264e9	0.237332	0.118666	−	0.992934i	$-0.462138\pi$
$757$	2.83264e9	0.237332	0.118666	+	0.992934i	$0.462138\pi$
$758$	−2.64636e10	−2.20702
$759$	0	0
$760$	−7.52119e8	−0.0621497
$761$	−7.36080e9	−0.605450	−0.302725	−	0.953078i	$-0.597896\pi$
$761$	−7.36080e9	−0.605450	−0.302725	+	0.953078i	$0.597896\pi$
$762$	0	0
$763$	−2.26012e10	−1.84203
$764$	−1.14828e10	−0.931583
$765$	0	0
$766$	−2.73534e10	−2.19893
$767$	−1.26112e10	−1.00919
$768$	0	0
$769$	−3.21130e9	−0.254647	−0.127323	−	0.991861i	$-0.540639\pi$
$769$	−3.21130e9	−0.254647	−0.127323	+	0.991861i	$0.540639\pi$
$770$	−8.32826e9	−0.657411
$771$	0	0
$772$	2.41542e10	1.88944
$773$	−1.57623e10	−1.22741	−0.613706	−	0.789535i	$-0.710322\pi$
$773$	−1.57623e10	−1.22741	−0.613706	+	0.789535i	$0.710322\pi$
$774$	0	0
$775$	4.35767e8	0.0336278
$776$	3.85018e9	0.295777
$777$	0	0
$778$	3.69528e9	0.281332
$779$	1.21506e10	0.920912
$780$	0	0
$781$	1.98513e9	0.149111
$782$	2.65758e9	0.198730
$783$	0	0
$784$	−1.10335e10	−0.817724
$785$	−1.22779e8	−0.00905898
$786$	0	0
$787$	1.08358e9	0.0792408	0.0396204	−	0.999215i	$-0.487385\pi$
$787$	1.08358e9	0.0792408	0.0396204	+	0.999215i	$0.487385\pi$
$788$	2.10634e10	1.53351
$789$	0	0
$790$	5.19080e9	0.374576
$791$	3.32333e10	2.38757
$792$	0	0
$793$	1.80278e9	0.128377
$794$	−1.50141e10	−1.06445
$795$	0	0
$796$	8.33343e9	0.585636
$797$	1.47647e10	1.03305	0.516525	−	0.856272i	$-0.327225\pi$
$797$	1.47647e10	1.03305	0.516525	+	0.856272i	$0.327225\pi$
$798$	0	0
$799$	−3.52265e9	−0.244318
$800$	4.14176e9	0.286002
$801$	0	0
$802$	−6.54184e9	−0.447806
$803$	1.94805e10	1.32768
$804$	0	0
$805$	8.66585e9	0.585498
$806$	3.06273e9	0.206033
$807$	0	0
$808$	1.05429e9	0.0703108
$809$	1.71421e10	1.13827	0.569133	−	0.822245i	$-0.307279\pi$
$809$	1.71421e10	1.13827	0.569133	+	0.822245i	$0.307279\pi$
$810$	0	0
$811$	−5.84966e9	−0.385086	−0.192543	−	0.981289i	$-0.561673\pi$
$811$	−5.84966e9	−0.385086	−0.192543	+	0.981289i	$0.561673\pi$
$812$	−1.26149e9	−0.0826871
$813$	0	0
$814$	−9.68932e9	−0.629663
$815$	−7.32454e9	−0.473946
$816$	0	0
$817$	−1.20783e10	−0.774868
$818$	−4.17081e10	−2.66430
$819$	0	0
$820$	1.39858e10	0.885806
$821$	1.36240e10	0.859218	0.429609	−	0.903015i	$-0.358651\pi$
$821$	1.36240e10	0.859218	0.429609	+	0.903015i	$0.358651\pi$
$822$	0	0
$823$	9.85740e9	0.616401	0.308200	−	0.951321i	$-0.400273\pi$
$823$	9.85740e9	0.616401	0.308200	+	0.951321i	$0.400273\pi$
$824$	5.69953e9	0.354890
$825$	0	0
$826$	−4.12534e10	−2.54700
$827$	2.59487e10	1.59531	0.797657	−	0.603111i	$-0.206072\pi$
$827$	2.59487e10	1.59531	0.797657	+	0.603111i	$0.206072\pi$
$828$	0	0
$829$	8.01238e9	0.488451	0.244225	−	0.969718i	$-0.421466\pi$
$829$	8.01238e9	0.488451	0.244225	+	0.969718i	$0.421466\pi$
$830$	2.66728e9	0.161918
$831$	0	0
$832$	1.81004e10	1.08958
$833$	−2.50787e9	−0.150331
$834$	0	0
$835$	1.23565e9	0.0734499
$836$	7.57080e9	0.448146
$837$	0	0
$838$	1.07573e10	0.631465
$839$	1.73655e10	1.01513	0.507563	−	0.861615i	$-0.330546\pi$
$839$	1.73655e10	1.01513	0.507563	+	0.861615i	$0.330546\pi$
$840$	0	0
$841$	−1.72076e10	−0.997546
$842$	−3.23043e10	−1.86495
$843$	0	0
$844$	−6.21006e9	−0.355547
$845$	−2.42338e9	−0.138173
$846$	0	0
$847$	−1.28129e10	−0.724530
$848$	2.01299e10	1.13359
$849$	0	0
$850$	7.73637e8	0.0432087
$851$	1.00821e10	0.560785
$852$	0	0
$853$	−1.69232e10	−0.933598	−0.466799	−	0.884363i	$-0.654593\pi$
$853$	−1.69232e10	−0.933598	−0.466799	+	0.884363i	$0.654593\pi$
$854$	5.89720e9	0.323998
$855$	0	0
$856$	6.31294e8	0.0344012
$857$	2.82758e10	1.53455	0.767276	−	0.641317i	$-0.221612\pi$
$857$	2.82758e10	1.53455	0.767276	+	0.641317i	$0.221612\pi$
$858$	0	0
$859$	1.71804e10	0.924818	0.462409	−	0.886667i	$-0.346985\pi$
$859$	1.71804e10	0.924818	0.462409	+	0.886667i	$0.346985\pi$
$860$	−1.39025e10	−0.745329
$861$	0	0
$862$	4.30621e10	2.28992
$863$	1.43610e10	0.760583	0.380292	−	0.924867i	$-0.375824\pi$
$863$	1.43610e10	0.760583	0.380292	+	0.924867i	$0.375824\pi$
$864$	0	0
$865$	1.43260e10	0.752610
$866$	6.11280e9	0.319836
$867$	0	0
$868$	5.40793e9	0.280680
$869$	−7.70177e9	−0.398127
$870$	0	0
$871$	−4.75439e9	−0.243799
$872$	6.45787e9	0.329823
$873$	0	0
$874$	−1.45942e10	−0.739418
$875$	2.52268e9	0.127302
$876$	0	0
$877$	−3.03867e10	−1.52119	−0.760597	−	0.649225i	$-0.775094\pi$
$877$	−3.03867e10	−1.52119	−0.760597	+	0.649225i	$0.775094\pi$
$878$	8.42820e9	0.420246
$879$	0	0
$880$	−5.05011e9	−0.249811
$881$	4.96158e9	0.244458	0.122229	−	0.992502i	$-0.460996\pi$
$881$	4.96158e9	0.244458	0.122229	+	0.992502i	$0.460996\pi$
$882$	0	0
$883$	2.52404e10	1.23377	0.616884	−	0.787054i	$-0.288395\pi$
$883$	2.52404e10	1.23377	0.616884	+	0.787054i	$0.288395\pi$
$884$	2.93502e9	0.142899
$885$	0	0
$886$	3.16697e10	1.52977
$887$	−1.98679e10	−0.955917	−0.477958	−	0.878382i	$-0.658623\pi$
$887$	−1.98679e10	−0.955917	−0.477958	+	0.878382i	$0.658623\pi$
$888$	0	0
$889$	−7.68925e8	−0.0367052
$890$	8.38599e9	0.398740
$891$	0	0
$892$	3.12091e10	1.47233
$893$	1.93447e10	0.909039
$894$	0	0
$895$	−1.20793e9	−0.0563198
$896$	1.53862e10	0.714584
$897$	0	0
$898$	2.47763e10	1.14175
$899$	−1.81435e8	−0.00832841
$900$	0	0
$901$	4.57545e9	0.208400
$902$	−3.84436e10	−1.74422
$903$	0	0
$904$	−9.49577e9	−0.427504
$905$	7.63412e9	0.342365
$906$	0	0
$907$	2.35625e10	1.04857	0.524283	−	0.851544i	$-0.324333\pi$
$907$	2.35625e10	1.04857	0.524283	+	0.851544i	$0.324333\pi$
$908$	1.83371e9	0.0812889
$909$	0	0
$910$	1.77303e10	0.779960
$911$	1.76541e10	0.773628	0.386814	−	0.922158i	$-0.373575\pi$
$911$	1.76541e10	0.773628	0.386814	+	0.922158i	$0.373575\pi$
$912$	0	0
$913$	−3.95753e9	−0.172098
$914$	−1.00239e10	−0.434236
$915$	0	0
$916$	3.21569e10	1.38242
$917$	4.38430e10	1.87762
$918$	0	0
$919$	−3.40865e10	−1.44870	−0.724349	−	0.689433i	$-0.757860\pi$
$919$	−3.40865e10	−1.44870	−0.724349	+	0.689433i	$0.757860\pi$
$920$	−2.47610e9	−0.104836
$921$	0	0
$922$	2.72032e10	1.14304
$923$	−4.22621e9	−0.176907
$924$	0	0
$925$	2.93495e9	0.121928
$926$	3.30032e10	1.36590
$927$	0	0
$928$	−1.72445e9	−0.0708327
$929$	−8.82983e9	−0.361324	−0.180662	−	0.983545i	$-0.557824\pi$
$929$	−8.82983e9	−0.361324	−0.180662	+	0.983545i	$0.557824\pi$
$930$	0	0
$931$	1.37721e10	0.559338
$932$	−4.84580e10	−1.96069
$933$	0	0
$934$	−5.10947e10	−2.05192
$935$	−1.14787e9	−0.0459254
$936$	0	0
$937$	3.16670e10	1.25753	0.628766	−	0.777595i	$-0.283560\pi$
$937$	3.16670e10	1.25753	0.628766	+	0.777595i	$0.283560\pi$
$938$	−1.55524e10	−0.615302
$939$	0	0
$940$	2.22664e10	0.874385
$941$	−1.37265e10	−0.537028	−0.268514	−	0.963276i	$-0.586533\pi$
$941$	−1.37265e10	−0.537028	−0.268514	+	0.963276i	$0.586533\pi$
$942$	0	0
$943$	4.00020e10	1.55342
$944$	−2.50153e10	−0.967841
$945$	0	0
$946$	3.82147e10	1.46761
$947$	2.34840e10	0.898559	0.449279	−	0.893391i	$-0.351681\pi$
$947$	2.34840e10	0.898559	0.449279	+	0.893391i	$0.351681\pi$
$948$	0	0
$949$	−4.14727e10	−1.57518
$950$	−4.24845e9	−0.160767
$951$	0	0
$952$	1.41519e9	0.0531600
$953$	−4.66674e10	−1.74658	−0.873290	−	0.487201i	$-0.838018\pi$
$953$	−4.66674e10	−1.74658	−0.873290	+	0.487201i	$0.838018\pi$
$954$	0	0
$955$	−9.56078e9	−0.355207
$956$	−5.03815e10	−1.86496
$957$	0	0
$958$	2.00141e10	0.735455
$959$	2.07109e9	0.0758287
$960$	0	0
$961$	−2.67348e10	−0.971729
$962$	2.06279e10	0.747039
$963$	0	0
$964$	4.72515e10	1.69882
$965$	2.01112e10	0.720431
$966$	0	0
$967$	−3.18326e10	−1.13209	−0.566043	−	0.824376i	$-0.691526\pi$
$967$	−3.18326e10	−1.13209	−0.566043	+	0.824376i	$0.691526\pi$
$968$	3.66105e9	0.129730
$969$	0	0
$970$	2.17483e10	0.765111
$971$	1.13336e10	0.397282	0.198641	−	0.980072i	$-0.436347\pi$
$971$	1.13336e10	0.397282	0.198641	+	0.980072i	$0.436347\pi$
$972$	0	0
$973$	5.73404e10	1.99556
$974$	−7.70315e10	−2.67123
$975$	0	0
$976$	3.57596e9	0.123117
$977$	−5.70282e10	−1.95640	−0.978202	−	0.207655i	$-0.933417\pi$
$977$	−5.70282e10	−1.95640	−0.978202	+	0.207655i	$0.933417\pi$
$978$	0	0
$979$	−1.24426e10	−0.423810
$980$	1.58521e10	0.538016
$981$	0	0
$982$	1.00971e9	0.0340255
$983$	5.41084e9	0.181688	0.0908442	−	0.995865i	$-0.471043\pi$
$983$	5.41084e9	0.181688	0.0908442	+	0.995865i	$0.471043\pi$
$984$	0	0
$985$	1.75378e10	0.584719
$986$	−3.22110e8	−0.0107013
$987$	0	0
$988$	−1.61178e10	−0.531686
$989$	−3.97637e10	−1.30707
$990$	0	0
$991$	2.51666e10	0.821423	0.410711	−	0.911765i	$-0.365280\pi$
$991$	2.51666e10	0.821423	0.410711	+	0.911765i	$0.365280\pi$
$992$	7.39262e9	0.240441
$993$	0	0
$994$	−1.38246e10	−0.446480
$995$	6.93855e9	0.223300
$996$	0	0
$997$	4.23132e10	1.35221	0.676103	−	0.736807i	$-0.263667\pi$
$997$	4.23132e10	1.35221	0.676103	+	0.736807i	$0.263667\pi$
$998$	−9.79081e9	−0.311790
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.h.1.2	yes	14
3.2	odd	2		405.8.a.g.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.8.a.g.1.13	✓	14	3.2	odd	2
405.8.a.h.1.2	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)

\(f(q)\)	\(=\)	\(q-16.6772 q^{2} +150.129 q^{4} +125.000 q^{5} +1291.61 q^{7} -369.053 q^{8} +O(q^{10})\)	\(q-16.6772 q^{2} +150.129 q^{4} +125.000 q^{5} +1291.61 q^{7} -369.053 q^{8} -2084.65 q^{10} +3093.07 q^{11} -6584.94 q^{13} -21540.5 q^{14} -13061.8 q^{16} -2968.89 q^{17} +16303.7 q^{19} +18766.1 q^{20} -51583.7 q^{22} +53674.6 q^{23} +15625.0 q^{25} +109818. q^{26} +193909. q^{28} -6505.59 q^{29} +27889.1 q^{31} +265072. q^{32} +49512.8 q^{34} +161451. q^{35} +187837. q^{37} -271901. q^{38} -46131.6 q^{40} +745267. q^{41} -740828. q^{43} +464360. q^{44} -895143. q^{46} +1.18652e6 q^{47} +844717. q^{49} -260581. q^{50} -988592. q^{52} -1.54113e6 q^{53} +386633. q^{55} -476673. q^{56} +108495. q^{58} +1.91516e6 q^{59} -273773. q^{61} -465112. q^{62} -2.74876e6 q^{64} -823118. q^{65} +722010. q^{67} -445717. q^{68} -2.69256e6 q^{70} +641799. q^{71} +6.29811e6 q^{73} -3.13259e6 q^{74} +2.44767e6 q^{76} +3.99504e6 q^{77} -2.49001e6 q^{79} -1.63272e6 q^{80} -1.24290e7 q^{82} -1.27949e6 q^{83} -371111. q^{85} +1.23549e7 q^{86} -1.14151e6 q^{88} -4.02273e6 q^{89} -8.50518e6 q^{91} +8.05813e6 q^{92} -1.97878e7 q^{94} +2.03797e6 q^{95} -1.04326e7 q^{97} -1.40875e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8}+O(q^{10}) \) 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

Varieties

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Groups

Database

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