LMFDB - Embedded newform 825.6.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.03932$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.07863 q^{2} -9.00000 q^{3} +33.2643 q^{4} -72.7077 q^{6} +39.3760 q^{7} +10.2141 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+8.07863 q^{2} -9.00000 q^{3} +33.2643 q^{4} -72.7077 q^{6} +39.3760 q^{7} +10.2141 q^{8} +81.0000 q^{9} +121.000 q^{11} -299.379 q^{12} +220.765 q^{13} +318.105 q^{14} -981.943 q^{16} +200.343 q^{17} +654.369 q^{18} +350.345 q^{19} -354.384 q^{21} +977.515 q^{22} -1385.09 q^{23} -91.9273 q^{24} +1783.48 q^{26} -729.000 q^{27} +1309.82 q^{28} +5506.64 q^{29} -2450.86 q^{31} -8259.61 q^{32} -1089.00 q^{33} +1618.50 q^{34} +2694.41 q^{36} +4060.46 q^{37} +2830.31 q^{38} -1986.88 q^{39} +527.283 q^{41} -2862.94 q^{42} +12078.9 q^{43} +4024.99 q^{44} -11189.6 q^{46} -563.023 q^{47} +8837.48 q^{48} -15256.5 q^{49} -1803.09 q^{51} +7343.59 q^{52} +37203.0 q^{53} -5889.32 q^{54} +402.192 q^{56} -3153.11 q^{57} +44486.2 q^{58} +2157.64 q^{59} -39938.0 q^{61} -19799.6 q^{62} +3189.46 q^{63} -35304.2 q^{64} -8797.63 q^{66} +38473.2 q^{67} +6664.29 q^{68} +12465.8 q^{69} -13725.3 q^{71} +827.345 q^{72} +39736.3 q^{73} +32803.0 q^{74} +11654.0 q^{76} +4764.50 q^{77} -16051.3 q^{78} +35672.9 q^{79} +6561.00 q^{81} +4259.73 q^{82} +79999.7 q^{83} -11788.4 q^{84} +97580.6 q^{86} -49559.8 q^{87} +1235.91 q^{88} -37783.0 q^{89} +8692.83 q^{91} -46074.1 q^{92} +22057.7 q^{93} -4548.46 q^{94} +74336.5 q^{96} +7616.35 q^{97} -123252. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 + 363 * q^11 - 252 * q^12 - 450 * q^13 - 1504 * q^14 - 1360 * q^16 + 334 * q^17 + 162 * q^18 - 4036 * q^19 - 2088 * q^21 + 242 * q^22 + 7060 * q^23 + 216 * q^24 + 2932 * q^26 - 2187 * q^27 + 8320 * q^28 + 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3267 * q^33 - 3644 * q^34 + 2268 * q^36 - 2250 * q^37 + 12632 * q^38 + 4050 * q^39 + 10654 * q^41 + 13536 * q^42 + 35528 * q^43 + 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 12240 * q^48 + 7667 * q^49 - 3006 * q^51 + 14520 * q^52 + 12826 * q^53 - 1458 * q^54 + 17088 * q^56 + 36324 * q^57 + 17196 * q^58 - 81876 * q^59 - 62298 * q^61 - 109184 * q^62 + 18792 * q^63 - 72256 * q^64 - 2178 * q^66 + 46148 * q^67 + 35832 * q^68 - 63540 * q^69 - 64724 * q^71 - 1944 * q^72 - 810 * q^73 - 44796 * q^74 + 44656 * q^76 + 28072 * q^77 - 26388 * q^78 + 43876 * q^79 + 19683 * q^81 - 56060 * q^82 + 101024 * q^83 - 74880 * q^84 + 24128 * q^86 - 36378 * q^87 - 2904 * q^88 + 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 5472 * q^93 + 74552 * q^94 - 27936 * q^96 + 319746 * q^97 - 431134 * q^98 + 29403 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.07863	1.42811	0.714057	−	0.700087i	$-0.246856\pi$
$2$	8.07863	1.42811	0.714057	+	0.700087i	$0.246856\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	33.2643	1.03951
$5$	0	0
$5$	0	0
$6$	−72.7077	−0.824522
$7$	39.3760	0.303730	0.151865	−	0.988401i	$-0.451472\pi$
$7$	39.3760	0.303730	0.151865	+	0.988401i	$0.451472\pi$
$8$	10.2141	0.0564257
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−299.379	−0.600162
$13$	220.765	0.362302	0.181151	−	0.983455i	$-0.442018\pi$
$13$	220.765	0.362302	0.181151	+	0.983455i	$0.442018\pi$
$14$	318.105	0.433761
$15$	0	0
$16$	−981.943	−0.958928
$17$	200.343	0.168133	0.0840664	−	0.996460i	$-0.473209\pi$
$17$	200.343	0.168133	0.0840664	+	0.996460i	$0.473209\pi$
$18$	654.369	0.476038
$19$	350.345	0.222645	0.111322	−	0.993784i	$-0.464491\pi$
$19$	350.345	0.222645	0.111322	+	0.993784i	$0.464491\pi$
$20$	0	0
$21$	−354.384	−0.175358
$22$	977.515	0.430593
$23$	−1385.09	−0.545957	−0.272978	−	0.962020i	$-0.588009\pi$
$23$	−1385.09	−0.545957	−0.272978	+	0.962020i	$0.588009\pi$
$24$	−91.9273	−0.0325774
$25$	0	0
$26$	1783.48	0.517409
$27$	−729.000	−0.192450
$28$	1309.82	0.315730
$29$	5506.64	1.21588	0.607942	−	0.793982i	$-0.291995\pi$
$29$	5506.64	1.21588	0.607942	+	0.793982i	$0.291995\pi$
$30$	0	0
$31$	−2450.86	−0.458051	−0.229025	−	0.973420i	$-0.573554\pi$
$31$	−2450.86	−0.458051	−0.229025	+	0.973420i	$0.573554\pi$
$32$	−8259.61	−1.42588
$33$	−1089.00	−0.174078
$34$	1618.50	0.240113
$35$	0	0
$36$	2694.41	0.346504
$37$	4060.46	0.487609	0.243804	−	0.969824i	$-0.421605\pi$
$37$	4060.46	0.487609	0.243804	+	0.969824i	$0.421605\pi$
$38$	2830.31	0.317962
$39$	−1986.88	−0.209175
$40$	0	0
$41$	527.283	0.0489874	0.0244937	−	0.999700i	$-0.492203\pi$
$41$	527.283	0.0489874	0.0244937	+	0.999700i	$0.492203\pi$
$42$	−2862.94	−0.250432
$43$	12078.9	0.996218	0.498109	−	0.867114i	$-0.334028\pi$
$43$	12078.9	0.996218	0.498109	+	0.867114i	$0.334028\pi$
$44$	4024.99	0.313424
$45$	0	0
$46$	−11189.6	−0.779689
$47$	−563.023	−0.0371776	−0.0185888	−	0.999827i	$-0.505917\pi$
$47$	−563.023	−0.0371776	−0.0185888	+	0.999827i	$0.505917\pi$
$48$	8837.48	0.553638
$49$	−15256.5	−0.907748
$50$	0	0
$51$	−1803.09	−0.0970715
$52$	7343.59	0.376617
$53$	37203.0	1.81923	0.909617	−	0.415449i	$-0.136375\pi$
$53$	37203.0	1.81923	0.909617	+	0.415449i	$0.136375\pi$
$54$	−5889.32	−0.274841
$55$	0	0
$56$	402.192	0.0171381
$57$	−3153.11	−0.128544
$58$	44486.2	1.73642
$59$	2157.64	0.0806955	0.0403477	−	0.999186i	$-0.487153\pi$
$59$	2157.64	0.0806955	0.0403477	+	0.999186i	$0.487153\pi$
$60$	0	0
$61$	−39938.0	−1.37424	−0.687119	−	0.726545i	$-0.741125\pi$
$61$	−39938.0	−1.37424	−0.687119	+	0.726545i	$0.741125\pi$
$62$	−19799.6	−0.654149
$63$	3189.46	0.101243
$64$	−35304.2	−1.07740
$65$	0	0
$66$	−8797.63	−0.248603
$67$	38473.2	1.04706	0.523529	−	0.852008i	$-0.324615\pi$
$67$	38473.2	1.04706	0.523529	+	0.852008i	$0.324615\pi$
$68$	6664.29	0.174776
$69$	12465.8	0.315208
$70$	0	0
$71$	−13725.3	−0.323129	−0.161564	−	0.986862i	$-0.551654\pi$
$71$	−13725.3	−0.323129	−0.161564	+	0.986862i	$0.551654\pi$
$72$	827.345	0.0188086
$73$	39736.3	0.872731	0.436366	−	0.899769i	$-0.356265\pi$
$73$	39736.3	0.872731	0.436366	+	0.899769i	$0.356265\pi$
$74$	32803.0	0.696361
$75$	0	0
$76$	11654.0	0.231441
$77$	4764.50	0.0915779
$78$	−16051.3	−0.298726
$79$	35672.9	0.643088	0.321544	−	0.946895i	$-0.395798\pi$
$79$	35672.9	0.643088	0.321544	+	0.946895i	$0.395798\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	4259.73	0.0699596
$83$	79999.7	1.27466	0.637328	−	0.770593i	$-0.280040\pi$
$83$	79999.7	1.27466	0.637328	+	0.770593i	$0.280040\pi$
$84$	−11788.4	−0.182287
$85$	0	0
$86$	97580.6	1.42271
$87$	−49559.8	−0.701991
$88$	1235.91	0.0170130
$89$	−37783.0	−0.505616	−0.252808	−	0.967516i	$-0.581354\pi$
$89$	−37783.0	−0.505616	−0.252808	+	0.967516i	$0.581354\pi$
$90$	0	0
$91$	8692.83	0.110042
$92$	−46074.1	−0.567528
$93$	22057.7	0.264456
$94$	−4548.46	−0.0530939
$95$	0	0
$96$	74336.5	0.823235
$97$	7616.35	0.0821897	0.0410948	−	0.999155i	$-0.486915\pi$
$97$	7616.35	0.0821897	0.0410948	+	0.999155i	$0.486915\pi$
$98$	−123252.	−1.29637
$99$	9801.00	0.100504
$100$	0	0
$101$	125963.	1.22868	0.614341	−	0.789041i	$-0.289422\pi$
$101$	125963.	1.22868	0.614341	+	0.789041i	$0.289422\pi$
$102$	−14566.5	−0.138629
$103$	165220.	1.53451	0.767257	−	0.641340i	$-0.221621\pi$
$103$	165220.	1.53451	0.767257	+	0.641340i	$0.221621\pi$
$104$	2254.92	0.0204431
$105$	0	0
$106$	300550.	2.59807
$107$	13152.3	0.111056	0.0555279	−	0.998457i	$-0.482316\pi$
$107$	13152.3	0.111056	0.0555279	+	0.998457i	$0.482316\pi$
$108$	−24249.7	−0.200054
$109$	37970.9	0.306115	0.153057	−	0.988217i	$-0.451088\pi$
$109$	37970.9	0.306115	0.153057	+	0.988217i	$0.451088\pi$
$110$	0	0
$111$	−36544.2	−0.281521
$112$	−38665.0	−0.291255
$113$	85189.7	0.627612	0.313806	−	0.949487i	$-0.398396\pi$
$113$	85189.7	0.627612	0.313806	+	0.949487i	$0.398396\pi$
$114$	−25472.8	−0.183575
$115$	0	0
$116$	183175.	1.26392
$117$	17881.9	0.120767
$118$	17430.8	0.115242
$119$	7888.73	0.0510669
$120$	0	0
$121$	14641.0	0.0909091
$122$	−322645.	−1.96257
$123$	−4745.55	−0.0282829
$124$	−81526.1	−0.476148
$125$	0	0
$126$	25766.5	0.144587
$127$	141237.	0.777033	0.388517	−	0.921442i	$-0.372988\pi$
$127$	141237.	0.777033	0.388517	+	0.921442i	$0.372988\pi$
$128$	−20902.2	−0.112763
$129$	−108710.	−0.575167
$130$	0	0
$131$	210469.	1.07155	0.535773	−	0.844362i	$-0.320020\pi$
$131$	210469.	1.07155	0.535773	+	0.844362i	$0.320020\pi$
$132$	−36224.9	−0.180956
$133$	13795.2	0.0676237
$134$	310811.	1.49532
$135$	0	0
$136$	2046.33	0.00948701
$137$	−19565.1	−0.0890594	−0.0445297	−	0.999008i	$-0.514179\pi$
$137$	−19565.1	−0.0890594	−0.0445297	+	0.999008i	$0.514179\pi$
$138$	100707.	0.450154
$139$	132498.	0.581663	0.290831	−	0.956774i	$-0.406068\pi$
$139$	132498.	0.581663	0.290831	+	0.956774i	$0.406068\pi$
$140$	0	0
$141$	5067.21	0.0214645
$142$	−110882.	−0.461465
$143$	26712.5	0.109238
$144$	−79537.3	−0.319643
$145$	0	0
$146$	321015.	1.24636
$147$	137309.	0.524089
$148$	135069.	0.506874
$149$	−153316.	−0.565748	−0.282874	−	0.959157i	$-0.591288\pi$
$149$	−153316.	−0.565748	−0.282874	+	0.959157i	$0.591288\pi$
$150$	0	0
$151$	248151.	0.885672	0.442836	−	0.896603i	$-0.353972\pi$
$151$	248151.	0.885672	0.442836	+	0.896603i	$0.353972\pi$
$152$	3578.47	0.0125629
$153$	16227.8	0.0560443
$154$	38490.7	0.130784
$155$	0	0
$156$	−66092.3	−0.217440
$157$	180692.	0.585047	0.292523	−	0.956258i	$-0.405505\pi$
$157$	180692.	0.585047	0.292523	+	0.956258i	$0.405505\pi$
$158$	288188.	0.918403
$159$	−334827.	−1.05033
$160$	0	0
$161$	−54539.4	−0.165823
$162$	53003.9	0.158679
$163$	176045.	0.518985	0.259493	−	0.965745i	$-0.416445\pi$
$163$	176045.	0.518985	0.259493	+	0.965745i	$0.416445\pi$
$164$	17539.7	0.0509229
$165$	0	0
$166$	646288.	1.82035
$167$	−293818.	−0.815244	−0.407622	−	0.913151i	$-0.633642\pi$
$167$	−293818.	−0.815244	−0.407622	+	0.913151i	$0.633642\pi$
$168$	−3619.73	−0.00989471
$169$	−322556.	−0.868737
$170$	0	0
$171$	28378.0	0.0742148
$172$	401795.	1.03558
$173$	20784.9	0.0527999	0.0263999	−	0.999651i	$-0.491596\pi$
$173$	20784.9	0.0527999	0.0263999	+	0.999651i	$0.491596\pi$
$174$	−400375.	−1.00252
$175$	0	0
$176$	−118815.	−0.289128
$177$	−19418.8	−0.0465896
$178$	−305235.	−0.722078
$179$	−229326.	−0.534958	−0.267479	−	0.963564i	$-0.586191\pi$
$179$	−229326.	−0.534958	−0.267479	+	0.963564i	$0.586191\pi$
$180$	0	0
$181$	−90807.3	−0.206027	−0.103014	−	0.994680i	$-0.532849\pi$
$181$	−90807.3	−0.206027	−0.103014	+	0.994680i	$0.532849\pi$
$182$	70226.2	0.157152
$183$	359442.	0.793416
$184$	−14147.5	−0.0308060
$185$	0	0
$186$	178196.	0.377673
$187$	24241.5	0.0506939
$188$	−18728.6	−0.0386465
$189$	−28705.1	−0.0584528
$190$	0	0
$191$	−496794.	−0.985356	−0.492678	−	0.870212i	$-0.663982\pi$
$191$	−496794.	−0.985356	−0.492678	+	0.870212i	$0.663982\pi$
$192$	317738.	0.622036
$193$	−271362.	−0.524391	−0.262196	−	0.965015i	$-0.584447\pi$
$193$	−271362.	−0.524391	−0.262196	+	0.965015i	$0.584447\pi$
$194$	61529.7	0.117376
$195$	0	0
$196$	−507498.	−0.943614
$197$	190972.	0.350594	0.175297	−	0.984516i	$-0.443911\pi$
$197$	190972.	0.350594	0.175297	+	0.984516i	$0.443911\pi$
$198$	79178.7	0.143531
$199$	602637.	1.07876	0.539378	−	0.842064i	$-0.318660\pi$
$199$	602637.	1.07876	0.539378	+	0.842064i	$0.318660\pi$
$200$	0	0
$201$	−346258.	−0.604519
$202$	1.01761e6	1.75470
$203$	216830.	0.369300
$204$	−59978.6	−0.100907
$205$	0	0
$206$	1.33476e6	2.19146
$207$	−112192.	−0.181986
$208$	−216778.	−0.347422
$209$	42391.8	0.0671299
$210$	0	0
$211$	918176.	1.41978	0.709888	−	0.704315i	$-0.248746\pi$
$211$	918176.	1.41978	0.709888	+	0.704315i	$0.248746\pi$
$212$	1.23753e6	1.89111
$213$	123528.	0.186559
$214$	106252.	0.158600
$215$	0	0
$216$	−7446.11	−0.0108591
$217$	−96505.0	−0.139123
$218$	306753.	0.437167
$219$	−357627.	−0.503872
$220$	0	0
$221$	44228.7	0.0609149
$222$	−295227.	−0.402044
$223$	−959605.	−1.29220	−0.646101	−	0.763252i	$-0.723601\pi$
$223$	−959605.	−1.29220	−0.646101	+	0.763252i	$0.723601\pi$
$224$	−325231.	−0.433083
$225$	0	0
$226$	688217.	0.896301
$227$	289954.	0.373477	0.186739	−	0.982410i	$-0.440208\pi$
$227$	289954.	0.373477	0.186739	+	0.982410i	$0.440208\pi$
$228$	−104886.	−0.133623
$229$	111161.	0.140076	0.0700379	−	0.997544i	$-0.477688\pi$
$229$	111161.	0.140076	0.0700379	+	0.997544i	$0.477688\pi$
$230$	0	0
$231$	−42880.5	−0.0528725
$232$	56245.6	0.0686071
$233$	453216.	0.546909	0.273455	−	0.961885i	$-0.411834\pi$
$233$	453216.	0.546909	0.273455	+	0.961885i	$0.411834\pi$
$234$	144462.	0.172470
$235$	0	0
$236$	71772.5	0.0838838
$237$	−321056.	−0.371287
$238$	63730.1	0.0729294
$239$	1.26387e6	1.43123	0.715614	−	0.698496i	$-0.246147\pi$
$239$	1.26387e6	1.43123	0.715614	+	0.698496i	$0.246147\pi$
$240$	0	0
$241$	134350.	0.149003	0.0745013	−	0.997221i	$-0.476264\pi$
$241$	134350.	0.149003	0.0745013	+	0.997221i	$0.476264\pi$
$242$	118279.	0.129829
$243$	−59049.0	−0.0641500
$244$	−1.32851e6	−1.42853
$245$	0	0
$246$	−38337.6	−0.0403912
$247$	77343.8	0.0806646
$248$	−25033.4	−0.0258458
$249$	−719997.	−0.735923
$250$	0	0
$251$	−1.43370e6	−1.43639	−0.718197	−	0.695840i	$-0.755032\pi$
$251$	−1.43370e6	−1.43639	−0.718197	+	0.695840i	$0.755032\pi$
$252$	106095.	0.105243
$253$	−167596.	−0.164612
$254$	1.14100e6	1.10969
$255$	0	0
$256$	960873.	0.916360
$257$	953935.	0.900920	0.450460	−	0.892797i	$-0.351260\pi$
$257$	953935.	0.900920	0.450460	+	0.892797i	$0.351260\pi$
$258$	−878226.	−0.821404
$259$	159885.	0.148101
$260$	0	0
$261$	446038.	0.405294
$262$	1.70030e6	1.53029
$263$	−322280.	−0.287306	−0.143653	−	0.989628i	$-0.545885\pi$
$263$	−322280.	−0.287306	−0.143653	+	0.989628i	$0.545885\pi$
$264$	−11123.2	−0.00982245
$265$	0	0
$266$	111446.	0.0965744
$267$	340047.	0.291918
$268$	1.27978e6	1.08843
$269$	808737.	0.681438	0.340719	−	0.940165i	$-0.389329\pi$
$269$	808737.	0.681438	0.340719	+	0.940165i	$0.389329\pi$
$270$	0	0
$271$	−909303.	−0.752117	−0.376058	−	0.926596i	$-0.622721\pi$
$271$	−909303.	−0.752117	−0.376058	+	0.926596i	$0.622721\pi$
$272$	−196726.	−0.161227
$273$	−78235.5	−0.0635327
$274$	−158059.	−0.127187
$275$	0	0
$276$	414667.	0.327662
$277$	−236933.	−0.185535	−0.0927675	−	0.995688i	$-0.529571\pi$
$277$	−236933.	−0.185535	−0.0927675	+	0.995688i	$0.529571\pi$
$278$	1.07040e6	0.830681
$279$	−198519.	−0.152684
$280$	0	0
$281$	−450779.	−0.340563	−0.170282	−	0.985395i	$-0.554468\pi$
$281$	−450779.	−0.340563	−0.170282	+	0.985395i	$0.554468\pi$
$282$	40936.1	0.0306538
$283$	−683413.	−0.507245	−0.253622	−	0.967303i	$-0.581622\pi$
$283$	−683413.	−0.507245	−0.253622	+	0.967303i	$0.581622\pi$
$284$	−456563.	−0.335896
$285$	0	0
$286$	215801.	0.156005
$287$	20762.3	0.0148789
$288$	−669028.	−0.475295
$289$	−1.37972e6	−0.971731
$290$	0	0
$291$	−68547.1	−0.0474522
$292$	1.32180e6	0.907213
$293$	1.81068e6	1.23218	0.616088	−	0.787678i	$-0.288717\pi$
$293$	1.81068e6	1.23218	0.616088	+	0.787678i	$0.288717\pi$
$294$	1.10927e6	0.748459
$295$	0	0
$296$	41474.1	0.0275136
$297$	−88209.0	−0.0580259
$298$	−1.23859e6	−0.807953
$299$	−305779.	−0.197801
$300$	0	0
$301$	475617.	0.302581
$302$	2.00472e6	1.26484
$303$	−1.13367e6	−0.709380
$304$	−344019.	−0.213500
$305$	0	0
$306$	131099.	0.0800376
$307$	3.00860e6	1.82187	0.910937	−	0.412546i	$-0.135361\pi$
$307$	3.00860e6	1.82187	0.910937	+	0.412546i	$0.135361\pi$
$308$	158488.	0.0951962
$309$	−1.48698e6	−0.885952
$310$	0	0
$311$	−2.04083e6	−1.19648	−0.598241	−	0.801316i	$-0.704134\pi$
$311$	−2.04083e6	−1.19648	−0.598241	+	0.801316i	$0.704134\pi$
$312$	−20294.3	−0.0118029
$313$	1.73940e6	1.00355	0.501774	−	0.864999i	$-0.332681\pi$
$313$	1.73940e6	1.00355	0.501774	+	0.864999i	$0.332681\pi$
$314$	1.45975e6	0.835513
$315$	0	0
$316$	1.18663e6	0.668497
$317$	−106704.	−0.0596392	−0.0298196	−	0.999555i	$-0.509493\pi$
$317$	−106704.	−0.0596392	−0.0298196	+	0.999555i	$0.509493\pi$
$318$	−2.70495e6	−1.50000
$319$	666304.	0.366603
$320$	0	0
$321$	−118370.	−0.0641181
$322$	−440604.	−0.236815
$323$	70189.3	0.0374338
$324$	218247.	0.115501
$325$	0	0
$326$	1.42220e6	0.741170
$327$	−341738.	−0.176736
$328$	5385.74	0.00276415
$329$	−22169.6	−0.0112919
$330$	0	0
$331$	1.54759e6	0.776402	0.388201	−	0.921575i	$-0.373097\pi$
$331$	1.54759e6	0.776402	0.388201	+	0.921575i	$0.373097\pi$
$332$	2.66114e6	1.32502
$333$	328898.	0.162536
$334$	−2.37365e6	−1.16426
$335$	0	0
$336$	347985.	0.168156
$337$	−1.22550e6	−0.587814	−0.293907	−	0.955834i	$-0.594956\pi$
$337$	−1.22550e6	−0.587814	−0.293907	+	0.955834i	$0.594956\pi$
$338$	−2.60581e6	−1.24066
$339$	−766708.	−0.362352
$340$	0	0
$341$	−296553.	−0.138107
$342$	229255.	0.105987
$343$	−1.26253e6	−0.579440
$344$	123375.	0.0562123
$345$	0	0
$346$	167914.	0.0754042
$347$	−3.82684e6	−1.70615	−0.853074	−	0.521789i	$-0.825265\pi$
$347$	−3.82684e6	−1.70615	−0.853074	+	0.521789i	$0.825265\pi$
$348$	−1.64857e6	−0.729727
$349$	456968.	0.200827	0.100413	−	0.994946i	$-0.467983\pi$
$349$	456968.	0.200827	0.100413	+	0.994946i	$0.467983\pi$
$350$	0	0
$351$	−160937.	−0.0697251
$352$	−999413.	−0.429920
$353$	−4.32830e6	−1.84876	−0.924380	−	0.381474i	$-0.875417\pi$
$353$	−4.32830e6	−1.84876	−0.924380	+	0.381474i	$0.875417\pi$
$354$	−156877.	−0.0665352
$355$	0	0
$356$	−1.25683e6	−0.525594
$357$	−70998.5	−0.0294835
$358$	−1.85264e6	−0.763982
$359$	−845380.	−0.346191	−0.173095	−	0.984905i	$-0.555377\pi$
$359$	−845380.	−0.346191	−0.173095	+	0.984905i	$0.555377\pi$
$360$	0	0
$361$	−2.35336e6	−0.950429
$362$	−733599.	−0.294230
$363$	−131769.	−0.0524864
$364$	289161.	0.114390
$365$	0	0
$366$	2.90380e6	1.13309
$367$	−2.96016e6	−1.14723	−0.573615	−	0.819125i	$-0.694460\pi$
$367$	−2.96016e6	−1.14723	−0.573615	+	0.819125i	$0.694460\pi$
$368$	1.36008e6	0.523534
$369$	42709.9	0.0163291
$370$	0	0
$371$	1.46491e6	0.552555
$372$	733735.	0.274904
$373$	−1.02299e6	−0.380713	−0.190357	−	0.981715i	$-0.560964\pi$
$373$	−1.02299e6	−0.380713	−0.190357	+	0.981715i	$0.560964\pi$
$374$	195839.	0.0723968
$375$	0	0
$376$	−5750.80	−0.00209777
$377$	1.21567e6	0.440517
$378$	−231898.	−0.0834772
$379$	−4.19402e6	−1.49980	−0.749899	−	0.661552i	$-0.769898\pi$
$379$	−4.19402e6	−1.49980	−0.749899	+	0.661552i	$0.769898\pi$
$380$	0	0
$381$	−1.27113e6	−0.448620
$382$	−4.01342e6	−1.40720
$383$	1.64287e6	0.572276	0.286138	−	0.958188i	$-0.407628\pi$
$383$	1.64287e6	0.572276	0.286138	+	0.958188i	$0.407628\pi$
$384$	188120.	0.0651039
$385$	0	0
$386$	−2.19223e6	−0.748891
$387$	978387.	0.332073
$388$	253353.	0.0854371
$389$	769820.	0.257938	0.128969	−	0.991649i	$-0.458833\pi$
$389$	769820.	0.257938	0.128969	+	0.991649i	$0.458833\pi$
$390$	0	0
$391$	−277493.	−0.0917933
$392$	−155832.	−0.0512203
$393$	−1.89422e6	−0.618657
$394$	1.54279e6	0.500688
$395$	0	0
$396$	326024.	0.104475
$397$	−4.86631e6	−1.54961	−0.774806	−	0.632199i	$-0.782153\pi$
$397$	−4.86631e6	−1.54961	−0.774806	+	0.632199i	$0.782153\pi$
$398$	4.86848e6	1.54059
$399$	−124157.	−0.0390426
$400$	0	0
$401$	717064.	0.222688	0.111344	−	0.993782i	$-0.464484\pi$
$401$	717064.	0.222688	0.111344	+	0.993782i	$0.464484\pi$
$402$	−2.79730e6	−0.863323
$403$	−541062.	−0.165953
$404$	4.19007e6	1.27723
$405$	0	0
$406$	1.75169e6	0.527402
$407$	491316.	0.147019
$408$	−18417.0	−0.00547733
$409$	−3.80091e6	−1.12352	−0.561758	−	0.827302i	$-0.689875\pi$
$409$	−3.80091e6	−1.12352	−0.561758	+	0.827302i	$0.689875\pi$
$410$	0	0
$411$	176086.	0.0514185
$412$	5.49595e6	1.59514
$413$	84959.4	0.0245096
$414$	−906361.	−0.259896
$415$	0	0
$416$	−1.82343e6	−0.516601
$417$	−1.19248e6	−0.335823
$418$	342468.	0.0958691
$419$	−1.55888e6	−0.433787	−0.216894	−	0.976195i	$-0.569592\pi$
$419$	−1.55888e6	−0.433787	−0.216894	+	0.976195i	$0.569592\pi$
$420$	0	0
$421$	4.13561e6	1.13719	0.568596	−	0.822617i	$-0.307487\pi$
$421$	4.13561e6	1.13719	0.568596	+	0.822617i	$0.307487\pi$
$422$	7.41761e6	2.02760
$423$	−45604.9	−0.0123925
$424$	379997.	0.102651
$425$	0	0
$426$	997935.	0.266427
$427$	−1.57260e6	−0.417397
$428$	437502.	0.115444
$429$	−240413.	−0.0630687
$430$	0	0
$431$	−5.25900e6	−1.36367	−0.681836	−	0.731505i	$-0.738818\pi$
$431$	−5.25900e6	−1.36367	−0.681836	+	0.731505i	$0.738818\pi$
$432$	715836.	0.184546
$433$	694549.	0.178026	0.0890130	−	0.996030i	$-0.471629\pi$
$433$	694549.	0.178026	0.0890130	+	0.996030i	$0.471629\pi$
$434$	−779629.	−0.198684
$435$	0	0
$436$	1.26308e6	0.318210
$437$	−485260.	−0.121554
$438$	−2.88914e6	−0.719586
$439$	−7.80969e6	−1.93407	−0.967036	−	0.254640i	$-0.918043\pi$
$439$	−7.80969e6	−1.93407	−0.967036	+	0.254640i	$0.918043\pi$
$440$	0	0
$441$	−1.23578e6	−0.302583
$442$	357307.	0.0869934
$443$	5.43876e6	1.31671	0.658356	−	0.752707i	$-0.271252\pi$
$443$	5.43876e6	1.31671	0.658356	+	0.752707i	$0.271252\pi$
$444$	−1.21562e6	−0.292644
$445$	0	0
$446$	−7.75229e6	−1.84541
$447$	1.37985e6	0.326635
$448$	−1.39014e6	−0.327238
$449$	5.46277e6	1.27878	0.639392	−	0.768881i	$-0.279186\pi$
$449$	5.46277e6	1.27878	0.639392	+	0.768881i	$0.279186\pi$
$450$	0	0
$451$	63801.3	0.0147703
$452$	2.83378e6	0.652409
$453$	−2.23336e6	−0.511343
$454$	2.34243e6	0.533368
$455$	0	0
$456$	−32206.3	−0.00725318
$457$	−3.07144e6	−0.687941	−0.343970	−	0.938980i	$-0.611772\pi$
$457$	−3.07144e6	−0.687941	−0.343970	+	0.938980i	$0.611772\pi$
$458$	898028.	0.200044
$459$	−146050.	−0.0323572
$460$	0	0
$461$	3.85641e6	0.845143	0.422572	−	0.906329i	$-0.361127\pi$
$461$	3.85641e6	0.845143	0.422572	+	0.906329i	$0.361127\pi$
$462$	−346416.	−0.0755080
$463$	−3.20594e6	−0.695029	−0.347514	−	0.937675i	$-0.612974\pi$
$463$	−3.20594e6	−0.695029	−0.347514	+	0.937675i	$0.612974\pi$
$464$	−5.40721e6	−1.16594
$465$	0	0
$466$	3.66136e6	0.781049
$467$	2.43952e6	0.517622	0.258811	−	0.965928i	$-0.416669\pi$
$467$	2.43952e6	0.517622	0.258811	+	0.965928i	$0.416669\pi$
$468$	594830.	0.125539
$469$	1.51492e6	0.318023
$470$	0	0
$471$	−1.62623e6	−0.337777
$472$	22038.5	0.00455330
$473$	1.46154e6	0.300371
$474$	−2.59369e6	−0.530240
$475$	0	0
$476$	262413.	0.0530846
$477$	3.01344e6	0.606411
$478$	1.02104e7	2.04396
$479$	4.69186e6	0.934342	0.467171	−	0.884167i	$-0.345273\pi$
$479$	4.69186e6	0.934342	0.467171	+	0.884167i	$0.345273\pi$
$480$	0	0
$481$	896406.	0.176662
$482$	1.08536e6	0.212793
$483$	490854.	0.0957381
$484$	487023.	0.0945010
$485$	0	0
$486$	−477035.	−0.0916136
$487$	−5.73305e6	−1.09538	−0.547688	−	0.836683i	$-0.684492\pi$
$487$	−5.73305e6	−1.09538	−0.547688	+	0.836683i	$0.684492\pi$
$488$	−407932.	−0.0775423
$489$	−1.58441e6	−0.299636
$490$	0	0
$491$	−693872.	−0.129890	−0.0649449	−	0.997889i	$-0.520687\pi$
$491$	−693872.	−0.129890	−0.0649449	+	0.997889i	$0.520687\pi$
$492$	−157858.	−0.0294004
$493$	1.10322e6	0.204430
$494$	624832.	0.115198
$495$	0	0
$496$	2.40660e6	0.439238
$497$	−540448.	−0.0981438
$498$	−5.81659e6	−1.05098
$499$	1.23153e6	0.221407	0.110704	−	0.993853i	$-0.464690\pi$
$499$	1.23153e6	0.221407	0.110704	+	0.993853i	$0.464690\pi$
$500$	0	0
$501$	2.64436e6	0.470681
$502$	−1.15823e7	−2.05133
$503$	7.61551e6	1.34208	0.671041	−	0.741420i	$-0.265847\pi$
$503$	7.61551e6	1.34208	0.671041	+	0.741420i	$0.265847\pi$
$504$	32577.6	0.00571272
$505$	0	0
$506$	−1.35395e6	−0.235085
$507$	2.90300e6	0.501566
$508$	4.69816e6	0.807734
$509$	−1.55496e6	−0.266026	−0.133013	−	0.991114i	$-0.542465\pi$
$509$	−1.55496e6	−0.266026	−0.133013	+	0.991114i	$0.542465\pi$
$510$	0	0
$511$	1.56466e6	0.265074
$512$	8.43141e6	1.42143
$513$	−255402.	−0.0428480
$514$	7.70649e6	1.28662
$515$	0	0
$516$	−3.61616e6	−0.597892
$517$	−68125.8	−0.0112095
$518$	1.29165e6	0.211505
$519$	−187064.	−0.0304840
$520$	0	0
$521$	−3.26766e6	−0.527403	−0.263701	−	0.964604i	$-0.584943\pi$
$521$	−3.26766e6	−0.527403	−0.263701	+	0.964604i	$0.584943\pi$
$522$	3.60338e6	0.578807
$523$	8.35303e6	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$523$	8.35303e6	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$524$	7.00112e6	1.11388
$525$	0	0
$526$	−2.60358e6	−0.410305
$527$	−491012.	−0.0770133
$528$	1.06934e6	0.166928
$529$	−4.51787e6	−0.701931
$530$	0	0
$531$	174769.	0.0268985
$532$	458888.	0.0702956
$533$	116405.	0.0177482
$534$	2.74711e6	0.416892
$535$	0	0
$536$	392970.	0.0590810
$537$	2.06393e6	0.308858
$538$	6.53349e6	0.973172
$539$	−1.84604e6	−0.273696
$540$	0	0
$541$	4.72814e6	0.694540	0.347270	−	0.937765i	$-0.387109\pi$
$541$	4.72814e6	0.694540	0.347270	+	0.937765i	$0.387109\pi$
$542$	−7.34593e6	−1.07411
$543$	817266.	0.118950
$544$	−1.65476e6	−0.239738
$545$	0	0
$546$	−632036.	−0.0907320
$547$	3.84639e6	0.549648	0.274824	−	0.961495i	$-0.411380\pi$
$547$	3.84639e6	0.549648	0.274824	+	0.961495i	$0.411380\pi$
$548$	−650819.	−0.0925782
$549$	−3.23498e6	−0.458079
$550$	0	0
$551$	1.92923e6	0.270710
$552$	127328.	0.0177859
$553$	1.40466e6	0.195325
$554$	−1.91409e6	−0.264965
$555$	0	0
$556$	4.40745e6	0.604644
$557$	−1.34095e7	−1.83137	−0.915684	−	0.401900i	$-0.868350\pi$
$557$	−1.34095e7	−1.83137	−0.915684	+	0.401900i	$0.868350\pi$
$558$	−1.60376e6	−0.218050
$559$	2.66658e6	0.360932
$560$	0	0
$561$	−218174.	−0.0292682
$562$	−3.64168e6	−0.486363
$563$	−1.51163e6	−0.200990	−0.100495	−	0.994938i	$-0.532043\pi$
$563$	−1.51163e6	−0.200990	−0.100495	+	0.994938i	$0.532043\pi$
$564$	168557.	0.0223126
$565$	0	0
$566$	−5.52105e6	−0.724403
$567$	258346.	0.0337477
$568$	−140192.	−0.0182328
$569$	1.39221e7	1.80271	0.901353	−	0.433085i	$-0.142575\pi$
$569$	1.39221e7	1.80271	0.901353	+	0.433085i	$0.142575\pi$
$570$	0	0
$571$	−3.44073e6	−0.441632	−0.220816	−	0.975316i	$-0.570872\pi$
$571$	−3.44073e6	−0.441632	−0.220816	+	0.975316i	$0.570872\pi$
$572$	888574.	0.113554
$573$	4.47115e6	0.568896
$574$	167731.	0.0212488
$575$	0	0
$576$	−2.85964e6	−0.359133
$577$	−3.96778e6	−0.496144	−0.248072	−	0.968742i	$-0.579797\pi$
$577$	−3.96778e6	−0.496144	−0.248072	+	0.968742i	$0.579797\pi$
$578$	−1.11463e7	−1.38774
$579$	2.44226e6	0.302758
$580$	0	0
$581$	3.15007e6	0.387151
$582$	−553767.	−0.0677672
$583$	4.50156e6	0.548519
$584$	405872.	0.0492445
$585$	0	0
$586$	1.46278e7	1.75969
$587$	−3.64595e6	−0.436733	−0.218366	−	0.975867i	$-0.570073\pi$
$587$	−3.64595e6	−0.436733	−0.218366	+	0.975867i	$0.570073\pi$
$588$	4.56748e6	0.544796
$589$	−858645.	−0.101982
$590$	0	0
$591$	−1.71875e6	−0.202416
$592$	−3.98714e6	−0.467582
$593$	486759.	0.0568431	0.0284215	−	0.999596i	$-0.490952\pi$
$593$	486759.	0.0568431	0.0284215	+	0.999596i	$0.490952\pi$
$594$	−712608.	−0.0828676
$595$	0	0
$596$	−5.09997e6	−0.588101
$597$	−5.42373e6	−0.622820
$598$	−2.47027e6	−0.282483
$599$	1.37413e7	1.56481	0.782404	−	0.622771i	$-0.213993\pi$
$599$	1.37413e7	1.56481	0.782404	+	0.622771i	$0.213993\pi$
$600$	0	0
$601$	7.84470e6	0.885911	0.442955	−	0.896544i	$-0.353930\pi$
$601$	7.84470e6	0.885911	0.442955	+	0.896544i	$0.353930\pi$
$602$	3.84234e6	0.432120
$603$	3.11633e6	0.349019
$604$	8.25457e6	0.920666
$605$	0	0
$606$	−9.15848e6	−1.01308
$607$	3.41681e6	0.376399	0.188200	−	0.982131i	$-0.439735\pi$
$607$	3.41681e6	0.376399	0.188200	+	0.982131i	$0.439735\pi$
$608$	−2.89371e6	−0.317465
$609$	−1.95147e6	−0.213215
$610$	0	0
$611$	−124296.	−0.0134695
$612$	539807.	0.0582586
$613$	1.32402e7	1.42313	0.711564	−	0.702621i	$-0.247987\pi$
$613$	1.32402e7	1.42313	0.711564	+	0.702621i	$0.247987\pi$
$614$	2.43054e7	2.60184
$615$	0	0
$616$	48665.3	0.00516735
$617$	−1.72143e7	−1.82044	−0.910220	−	0.414126i	$-0.864087\pi$
$617$	−1.72143e7	−1.82044	−0.910220	+	0.414126i	$0.864087\pi$
$618$	−1.20128e7	−1.26524
$619$	1.60142e7	1.67988	0.839940	−	0.542679i	$-0.182590\pi$
$619$	1.60142e7	1.67988	0.839940	+	0.542679i	$0.182590\pi$
$620$	0	0
$621$	1.00973e6	0.105069
$622$	−1.64871e7	−1.70871
$623$	−1.48774e6	−0.153571
$624$	1.95100e6	0.200584
$625$	0	0
$626$	1.40520e7	1.43318
$627$	−381526.	−0.0387574
$628$	6.01061e6	0.608162
$629$	813486.	0.0819830
$630$	0	0
$631$	−7.75049e6	−0.774918	−0.387459	−	0.921887i	$-0.626647\pi$
$631$	−7.75049e6	−0.774918	−0.387459	+	0.921887i	$0.626647\pi$
$632$	364368.	0.0362867
$633$	−8.26358e6	−0.819708
$634$	−862021.	−0.0851716
$635$	0	0
$636$	−1.11378e7	−1.09183
$637$	−3.36810e6	−0.328879
$638$	5.38282e6	0.523550
$639$	−1.11175e6	−0.107710
$640$	0	0
$641$	8.77648e6	0.843675	0.421838	−	0.906671i	$-0.361385\pi$
$641$	8.77648e6	0.843675	0.421838	+	0.906671i	$0.361385\pi$
$642$	−956272.	−0.0915680
$643$	1.51011e7	1.44040	0.720198	−	0.693768i	$-0.244051\pi$
$643$	1.51011e7	1.44040	0.720198	+	0.693768i	$0.244051\pi$
$644$	−1.81422e6	−0.172375
$645$	0	0
$646$	567034.	0.0534598
$647$	−1.57910e7	−1.48303	−0.741516	−	0.670936i	$-0.765893\pi$
$647$	−1.57910e7	−1.48303	−0.741516	+	0.670936i	$0.765893\pi$
$648$	67015.0	0.00626952
$649$	261075.	0.0243306
$650$	0	0
$651$	868545.	0.0803230
$652$	5.85603e6	0.539491
$653$	−1.33260e7	−1.22297	−0.611484	−	0.791256i	$-0.709427\pi$
$653$	−1.33260e7	−1.22297	−0.611484	+	0.791256i	$0.709427\pi$
$654$	−2.76078e6	−0.252399
$655$	0	0
$656$	−517762.	−0.0469754
$657$	3.21864e6	0.290910
$658$	−179100.	−0.0161262
$659$	−882213.	−0.0791334	−0.0395667	−	0.999217i	$-0.512598\pi$
$659$	−882213.	−0.0791334	−0.0395667	+	0.999217i	$0.512598\pi$
$660$	0	0
$661$	−5.59700e6	−0.498255	−0.249127	−	0.968471i	$-0.580144\pi$
$661$	−5.59700e6	−0.498255	−0.249127	+	0.968471i	$0.580144\pi$
$662$	1.25024e7	1.10879
$663$	−398058.	−0.0351692
$664$	817128.	0.0719234
$665$	0	0
$666$	2.65704e6	0.232120
$667$	−7.62720e6	−0.663820
$668$	−9.77367e6	−0.847454
$669$	8.63644e6	0.746053
$670$	0	0
$671$	−4.83250e6	−0.414348
$672$	2.92708e6	0.250041
$673$	−1.29320e6	−0.110060	−0.0550300	−	0.998485i	$-0.517525\pi$
$673$	−1.29320e6	−0.110060	−0.0550300	+	0.998485i	$0.517525\pi$
$674$	−9.90040e6	−0.839466
$675$	0	0
$676$	−1.07296e7	−0.903062
$677$	−1.48798e7	−1.24775	−0.623873	−	0.781526i	$-0.714442\pi$
$677$	−1.48798e7	−1.24775	−0.623873	+	0.781526i	$0.714442\pi$
$678$	−6.19395e6	−0.517480
$679$	299902.	0.0249634
$680$	0	0
$681$	−2.60958e6	−0.215627
$682$	−2.39575e6	−0.197233
$683$	−8.62925e6	−0.707817	−0.353909	−	0.935280i	$-0.615148\pi$
$683$	−8.62925e6	−0.707817	−0.353909	+	0.935280i	$0.615148\pi$
$684$	943974.	0.0771471
$685$	0	0
$686$	−1.01996e7	−0.827506
$687$	−1.00045e6	−0.0808729
$688$	−1.18607e7	−0.955302
$689$	8.21310e6	0.659112
$690$	0	0
$691$	−1.79734e7	−1.43198	−0.715988	−	0.698113i	$-0.754023\pi$
$691$	−1.79734e7	−1.43198	−0.715988	+	0.698113i	$0.754023\pi$
$692$	691396.	0.0548860
$693$	385925.	0.0305260
$694$	−3.09157e7	−2.43658
$695$	0	0
$696$	−506211.	−0.0396103
$697$	105638.	0.00823639
$698$	3.69167e6	0.286804
$699$	−4.07894e6	−0.315758
$700$	0	0
$701$	−2.13485e7	−1.64086	−0.820432	−	0.571744i	$-0.806267\pi$
$701$	−2.13485e7	−1.64086	−0.820432	+	0.571744i	$0.806267\pi$
$702$	−1.30015e6	−0.0995754
$703$	1.42256e6	0.108563
$704$	−4.27181e6	−0.324848
$705$	0	0
$706$	−3.49667e7	−2.64024
$707$	4.95992e6	0.373187
$708$	−645953.	−0.0484303
$709$	6.90604e6	0.515957	0.257979	−	0.966151i	$-0.416944\pi$
$709$	6.90604e6	0.515957	0.257979	+	0.966151i	$0.416944\pi$
$710$	0	0
$711$	2.88950e6	0.214363
$712$	−385921.	−0.0285298
$713$	3.39466e6	0.250076
$714$	−573571.	−0.0421058
$715$	0	0
$716$	−7.62837e6	−0.556095
$717$	−1.13749e7	−0.826320
$718$	−6.82951e6	−0.494400
$719$	−1.71863e7	−1.23983	−0.619913	−	0.784670i	$-0.712832\pi$
$719$	−1.71863e7	−1.23983	−0.619913	+	0.784670i	$0.712832\pi$
$720$	0	0
$721$	6.50573e6	0.466077
$722$	−1.90119e7	−1.35732
$723$	−1.20915e6	−0.0860266
$724$	−3.02065e6	−0.214167
$725$	0	0
$726$	−1.06451e6	−0.0749566
$727$	2.43516e7	1.70880	0.854401	−	0.519615i	$-0.173925\pi$
$727$	2.43516e7	1.70880	0.854401	+	0.519615i	$0.173925\pi$
$728$	88789.8	0.00620919
$729$	531441.	0.0370370
$730$	0	0
$731$	2.41992e6	0.167497
$732$	1.19566e7	0.824765
$733$	1.93102e7	1.32747	0.663737	−	0.747966i	$-0.268969\pi$
$733$	1.93102e7	1.32747	0.663737	+	0.747966i	$0.268969\pi$
$734$	−2.39141e7	−1.63838
$735$	0	0
$736$	1.14403e7	0.778472
$737$	4.65525e6	0.315700
$738$	345038.	0.0233199
$739$	−9.23336e6	−0.621940	−0.310970	−	0.950420i	$-0.600654\pi$
$739$	−9.23336e6	−0.621940	−0.310970	+	0.950420i	$0.600654\pi$
$740$	0	0
$741$	−696094.	−0.0465717
$742$	1.18345e7	0.789112
$743$	−8.02302e6	−0.533170	−0.266585	−	0.963811i	$-0.585895\pi$
$743$	−8.02302e6	−0.533170	−0.266585	+	0.963811i	$0.585895\pi$
$744$	225300.	0.0149221
$745$	0	0
$746$	−8.26434e6	−0.543702
$747$	6.47998e6	0.424885
$748$	806379.	0.0526969
$749$	517885.	0.0337309
$750$	0	0
$751$	1.19930e6	0.0775938	0.0387969	−	0.999247i	$-0.487647\pi$
$751$	1.19930e6	0.0775938	0.0387969	+	0.999247i	$0.487647\pi$
$752$	552856.	0.0356507
$753$	1.29033e7	0.829302
$754$	9.82096e6	0.629109
$755$	0	0
$756$	−954858.	−0.0607623
$757$	1.40172e7	0.889040	0.444520	−	0.895769i	$-0.353374\pi$
$757$	1.40172e7	0.889040	0.444520	+	0.895769i	$0.353374\pi$
$758$	−3.38820e7	−2.14188
$759$	1.50836e6	0.0950389
$760$	0	0
$761$	−1.98659e7	−1.24350	−0.621750	−	0.783215i	$-0.713578\pi$
$761$	−1.98659e7	−1.24350	−0.621750	+	0.783215i	$0.713578\pi$
$762$	−1.02690e7	−0.640681
$763$	1.49514e6	0.0929761
$764$	−1.65255e7	−1.02429
$765$	0	0
$766$	1.32721e7	0.817275
$767$	476331.	0.0292361
$768$	−8.64785e6	−0.529060
$769$	−2.09738e7	−1.27897	−0.639486	−	0.768803i	$-0.720853\pi$
$769$	−2.09738e7	−1.27897	−0.639486	+	0.768803i	$0.720853\pi$
$770$	0	0
$771$	−8.58542e6	−0.520146
$772$	−9.02668e6	−0.545111
$773$	−7.69041e6	−0.462914	−0.231457	−	0.972845i	$-0.574349\pi$
$773$	−7.69041e6	−0.462914	−0.231457	+	0.972845i	$0.574349\pi$
$774$	7.90403e6	0.474238
$775$	0	0
$776$	77794.4	0.00463761
$777$	−1.43896e6	−0.0855062
$778$	6.21909e6	0.368365
$779$	184731.	0.0109068
$780$	0	0
$781$	−1.66076e6	−0.0974270
$782$	−2.24177e6	−0.131091
$783$	−4.01434e6	−0.233997
$784$	1.49810e7	0.870466
$785$	0	0
$786$	−1.53027e7	−0.883513
$787$	1.39975e6	0.0805589	0.0402794	−	0.999188i	$-0.487175\pi$
$787$	1.39975e6	0.0805589	0.0402794	+	0.999188i	$0.487175\pi$
$788$	6.35256e6	0.364446
$789$	2.90052e6	0.165876
$790$	0	0
$791$	3.35443e6	0.190624
$792$	100109.	0.00567100
$793$	−8.81689e6	−0.497889
$794$	−3.93131e7	−2.21302
$795$	0	0
$796$	2.00463e7	1.12138
$797$	−1.22917e7	−0.685432	−0.342716	−	0.939439i	$-0.611347\pi$
$797$	−1.22917e7	−0.685432	−0.342716	+	0.939439i	$0.611347\pi$
$798$	−1.00302e6	−0.0557573
$799$	−112798.	−0.00625078
$800$	0	0
$801$	−3.06042e6	−0.168539
$802$	5.79290e6	0.318024
$803$	4.80810e6	0.263138
$804$	−1.15181e7	−0.628404
$805$	0	0
$806$	−4.37104e6	−0.236999
$807$	−7.27863e6	−0.393429
$808$	1.28660e6	0.0693292
$809$	−1.19607e7	−0.642515	−0.321258	−	0.946992i	$-0.604106\pi$
$809$	−1.19607e7	−0.642515	−0.321258	+	0.946992i	$0.604106\pi$
$810$	0	0
$811$	−1.15112e7	−0.614565	−0.307282	−	0.951618i	$-0.599420\pi$
$811$	−1.15112e7	−0.614565	−0.307282	+	0.951618i	$0.599420\pi$
$812$	7.21270e6	0.383891
$813$	8.18373e6	0.434235
$814$	3.96916e6	0.209961
$815$	0	0
$816$	1.77053e6	0.0930846
$817$	4.23177e6	0.221803
$818$	−3.07062e7	−1.60451
$819$	704120.	0.0366806
$820$	0	0
$821$	2.77684e7	1.43778	0.718890	−	0.695124i	$-0.244651\pi$
$821$	2.77684e7	1.43778	0.718890	+	0.695124i	$0.244651\pi$
$822$	1.42253e6	0.0734315
$823$	2.63447e7	1.35580	0.677898	−	0.735156i	$-0.262891\pi$
$823$	2.63447e7	1.35580	0.677898	+	0.735156i	$0.262891\pi$
$824$	1.68758e6	0.0865860
$825$	0	0
$826$	686356.	0.0350025
$827$	3.32551e7	1.69081	0.845406	−	0.534125i	$-0.179359\pi$
$827$	3.32551e7	1.69081	0.845406	+	0.534125i	$0.179359\pi$
$828$	−3.73200e6	−0.189176
$829$	2.84581e7	1.43820	0.719101	−	0.694905i	$-0.244554\pi$
$829$	2.84581e7	1.43820	0.719101	+	0.694905i	$0.244554\pi$
$830$	0	0
$831$	2.13239e6	0.107119
$832$	−7.79391e6	−0.390344
$833$	−3.05654e6	−0.152622
$834$	−9.63360e6	−0.479594
$835$	0	0
$836$	1.41013e6	0.0697822
$837$	1.78667e6	0.0881519
$838$	−1.25936e7	−0.619498
$839$	−9.47931e6	−0.464913	−0.232457	−	0.972607i	$-0.574676\pi$
$839$	−9.47931e6	−0.464913	−0.232457	+	0.972607i	$0.574676\pi$
$840$	0	0
$841$	9.81196e6	0.478372
$842$	3.34101e7	1.62404
$843$	4.05701e6	0.196624
$844$	3.05425e7	1.47587
$845$	0	0
$846$	−368425.	−0.0176980
$847$	576505.	0.0276118
$848$	−3.65312e7	−1.74451
$849$	6.15072e6	0.292858
$850$	0	0
$851$	−5.62411e6	−0.266213
$852$	4.10907e6	0.193930
$853$	−2.13145e7	−1.00300	−0.501501	−	0.865157i	$-0.667219\pi$
$853$	−2.13145e7	−1.00300	−0.501501	+	0.865157i	$0.667219\pi$
$854$	−1.27045e7	−0.596090
$855$	0	0
$856$	134339.	0.00626640
$857$	−2.21044e7	−1.02808	−0.514039	−	0.857767i	$-0.671851\pi$
$857$	−2.21044e7	−1.02808	−0.514039	+	0.857767i	$0.671851\pi$
$858$	−1.94221e6	−0.0900693
$859$	−7.07193e6	−0.327006	−0.163503	−	0.986543i	$-0.552279\pi$
$859$	−7.07193e6	−0.327006	−0.163503	+	0.986543i	$0.552279\pi$
$860$	0	0
$861$	−186861.	−0.00859035
$862$	−4.24855e7	−1.94748
$863$	5.45080e6	0.249134	0.124567	−	0.992211i	$-0.460246\pi$
$863$	5.45080e6	0.249134	0.124567	+	0.992211i	$0.460246\pi$
$864$	6.02125e6	0.274412
$865$	0	0
$866$	5.61101e6	0.254241
$867$	1.24175e7	0.561029
$868$	−3.21017e6	−0.144620
$869$	4.31642e6	0.193898
$870$	0	0
$871$	8.49351e6	0.379351
$872$	387840.	0.0172727
$873$	616924.	0.0273966
$874$	−3.92023e6	−0.173593
$875$	0	0
$876$	−1.18962e7	−0.523780
$877$	2.02823e7	0.890467	0.445234	−	0.895414i	$-0.353121\pi$
$877$	2.02823e7	0.890467	0.445234	+	0.895414i	$0.353121\pi$
$878$	−6.30917e7	−2.76208
$879$	−1.62961e7	−0.711397
$880$	0	0
$881$	−6.58178e6	−0.285696	−0.142848	−	0.989745i	$-0.545626\pi$
$881$	−6.58178e6	−0.285696	−0.142848	+	0.989745i	$0.545626\pi$
$882$	−9.98340e6	−0.432123
$883$	1.63860e7	0.707247	0.353623	−	0.935388i	$-0.384949\pi$
$883$	1.63860e7	0.707247	0.353623	+	0.935388i	$0.384949\pi$
$884$	1.47124e6	0.0633217
$885$	0	0
$886$	4.39378e7	1.88042
$887$	2.92929e6	0.125013	0.0625063	−	0.998045i	$-0.480091\pi$
$887$	2.92929e6	0.125013	0.0625063	+	0.998045i	$0.480091\pi$
$888$	−373267.	−0.0158850
$889$	5.56136e6	0.236008
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.19206e7	−1.34326
$893$	−197252.	−0.00827739
$894$	1.11473e7	0.466472
$895$	0	0
$896$	−823047.	−0.0342495
$897$	2.75201e6	0.114201
$898$	4.41317e7	1.82625
$899$	−1.34960e7	−0.556936
$900$	0	0
$901$	7.45337e6	0.305873
$902$	515427.	0.0210936
$903$	−4.28056e6	−0.174695
$904$	870140.	0.0354134
$905$	0	0
$906$	−1.80425e7	−0.730256
$907$	7.88236e6	0.318154	0.159077	−	0.987266i	$-0.449148\pi$
$907$	7.88236e6	0.318154	0.159077	+	0.987266i	$0.449148\pi$
$908$	9.64512e6	0.388234
$909$	1.02030e7	0.409561
$910$	0	0
$911$	−2.16713e7	−0.865146	−0.432573	−	0.901599i	$-0.642394\pi$
$911$	−2.16713e7	−0.865146	−0.432573	+	0.901599i	$0.642394\pi$
$912$	3.09617e6	0.123264
$913$	9.67996e6	0.384323
$914$	−2.48130e7	−0.982458
$915$	0	0
$916$	3.69769e6	0.145610
$917$	8.28745e6	0.325460
$918$	−1.17989e6	−0.0462097
$919$	4.32037e7	1.68746	0.843728	−	0.536771i	$-0.180356\pi$
$919$	4.32037e7	1.68746	0.843728	+	0.536771i	$0.180356\pi$
$920$	0	0
$921$	−2.70774e7	−1.05186
$922$	3.11545e7	1.20696
$923$	−3.03006e6	−0.117070
$924$	−1.42639e6	−0.0549616
$925$	0	0
$926$	−2.58996e7	−0.992580
$927$	1.33829e7	0.511504
$928$	−4.54827e7	−1.73371
$929$	3.70288e7	1.40767	0.703834	−	0.710364i	$-0.251470\pi$
$929$	3.70288e7	1.40767	0.703834	+	0.710364i	$0.251470\pi$
$930$	0	0
$931$	−5.34505e6	−0.202105
$932$	1.50759e7	0.568518
$933$	1.83675e7	0.690790
$934$	1.97080e7	0.739224
$935$	0	0
$936$	182648.	0.00681438
$937$	−1.22598e7	−0.456180	−0.228090	−	0.973640i	$-0.573248\pi$
$937$	−1.22598e7	−0.456180	−0.228090	+	0.973640i	$0.573248\pi$
$938$	1.22385e7	0.454173
$939$	−1.56546e7	−0.579399
$940$	0	0
$941$	−1.25646e7	−0.462565	−0.231283	−	0.972887i	$-0.574292\pi$
$941$	−1.25646e7	−0.462565	−0.231283	+	0.972887i	$0.574292\pi$
$942$	−1.31377e7	−0.482384
$943$	−730335.	−0.0267450
$944$	−2.11868e6	−0.0773812
$945$	0	0
$946$	1.18073e7	0.428964
$947$	−3.16122e7	−1.14546	−0.572730	−	0.819744i	$-0.694116\pi$
$947$	−3.16122e7	−1.14546	−0.572730	+	0.819744i	$0.694116\pi$
$948$	−1.06797e7	−0.385957
$949$	8.77237e6	0.316192
$950$	0	0
$951$	960334.	0.0344327
$952$	80576.6	0.00288148
$953$	2.84509e7	1.01476	0.507380	−	0.861722i	$-0.330614\pi$
$953$	2.84509e7	1.01476	0.507380	+	0.861722i	$0.330614\pi$
$954$	2.43445e7	0.866024
$955$	0	0
$956$	4.20419e7	1.48778
$957$	−5.99673e6	−0.211658
$958$	3.79038e7	1.33435
$959$	−770395.	−0.0270500
$960$	0	0
$961$	−2.26225e7	−0.790190
$962$	7.24174e6	0.252293
$963$	1.06533e6	0.0370186
$964$	4.46905e6	0.154890
$965$	0	0
$966$	3.96543e6	0.136725
$967$	4.73439e7	1.62816	0.814081	−	0.580751i	$-0.197241\pi$
$967$	4.73439e7	1.62816	0.814081	+	0.580751i	$0.197241\pi$
$968$	149545.	0.00512961
$969$	−631704.	−0.0216124
$970$	0	0
$971$	−4.33248e7	−1.47465	−0.737324	−	0.675539i	$-0.763911\pi$
$971$	−4.33248e7	−1.47465	−0.737324	+	0.675539i	$0.763911\pi$
$972$	−1.96423e6	−0.0666846
$973$	5.21723e6	0.176668
$974$	−4.63152e7	−1.56432
$975$	0	0
$976$	3.92168e7	1.31780
$977$	4.64678e7	1.55745	0.778727	−	0.627363i	$-0.215866\pi$
$977$	4.64678e7	1.55745	0.778727	+	0.627363i	$0.215866\pi$
$978$	−1.27998e7	−0.427915
$979$	−4.57174e6	−0.152449
$980$	0	0
$981$	3.07564e6	0.102038
$982$	−5.60553e6	−0.185498
$983$	1.02443e7	0.338141	0.169071	−	0.985604i	$-0.445923\pi$
$983$	1.02443e7	0.338141	0.169071	+	0.985604i	$0.445923\pi$
$984$	−48471.7	−0.00159588
$985$	0	0
$986$	8.91250e6	0.291949
$987$	199527.	0.00651940
$988$	2.57279e6	0.0838517
$989$	−1.67303e7	−0.543892
$990$	0	0
$991$	−6.52638e6	−0.211100	−0.105550	−	0.994414i	$-0.533660\pi$
$991$	−6.52638e6	−0.211100	−0.105550	+	0.994414i	$0.533660\pi$
$992$	2.02431e7	0.653127
$993$	−1.39283e7	−0.448256
$994$	−4.36608e6	−0.140161
$995$	0	0
$996$	−2.39502e7	−0.765000
$997$	−1.08535e7	−0.345804	−0.172902	−	0.984939i	$-0.555314\pi$
$997$	−1.08535e7	−0.345804	−0.172902	+	0.984939i	$0.555314\pi$
$998$	9.94904e6	0.316195
$999$	−2.96008e6	−0.0938403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.i.1.3		3
5.4	even	2		165.6.a.b.1.1	✓	3
15.14	odd	2		495.6.a.d.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.b.1.1	✓	3	5.4	even	2
495.6.a.d.1.3		3	15.14	odd	2
825.6.a.i.1.3		3	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+8.07863 q^{2} -9.00000 q^{3} +33.2643 q^{4} -72.7077 q^{6} +39.3760 q^{7} +10.2141 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+8.07863 q^{2} -9.00000 q^{3} +33.2643 q^{4} -72.7077 q^{6} +39.3760 q^{7} +10.2141 q^{8} +81.0000 q^{9} +121.000 q^{11} -299.379 q^{12} +220.765 q^{13} +318.105 q^{14} -981.943 q^{16} +200.343 q^{17} +654.369 q^{18} +350.345 q^{19} -354.384 q^{21} +977.515 q^{22} -1385.09 q^{23} -91.9273 q^{24} +1783.48 q^{26} -729.000 q^{27} +1309.82 q^{28} +5506.64 q^{29} -2450.86 q^{31} -8259.61 q^{32} -1089.00 q^{33} +1618.50 q^{34} +2694.41 q^{36} +4060.46 q^{37} +2830.31 q^{38} -1986.88 q^{39} +527.283 q^{41} -2862.94 q^{42} +12078.9 q^{43} +4024.99 q^{44} -11189.6 q^{46} -563.023 q^{47} +8837.48 q^{48} -15256.5 q^{49} -1803.09 q^{51} +7343.59 q^{52} +37203.0 q^{53} -5889.32 q^{54} +402.192 q^{56} -3153.11 q^{57} +44486.2 q^{58} +2157.64 q^{59} -39938.0 q^{61} -19799.6 q^{62} +3189.46 q^{63} -35304.2 q^{64} -8797.63 q^{66} +38473.2 q^{67} +6664.29 q^{68} +12465.8 q^{69} -13725.3 q^{71} +827.345 q^{72} +39736.3 q^{73} +32803.0 q^{74} +11654.0 q^{76} +4764.50 q^{77} -16051.3 q^{78} +35672.9 q^{79} +6561.00 q^{81} +4259.73 q^{82} +79999.7 q^{83} -11788.4 q^{84} +97580.6 q^{86} -49559.8 q^{87} +1235.91 q^{88} -37783.0 q^{89} +8692.83 q^{91} -46074.1 q^{92} +22057.7 q^{93} -4548.46 q^{94} +74336.5 q^{96} +7616.35 q^{97} -123252. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9	\( 3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 + 363 * q^11 - 252 * q^12 - 450 * q^13 - 1504 * q^14 - 1360 * q^16 + 334 * q^17 + 162 * q^18 - 4036 * q^19 - 2088 * q^21 + 242 * q^22 + 7060 * q^23 + 216 * q^24 + 2932 * q^26 - 2187 * q^27 + 8320 * q^28 + 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3267 * q^33 - 3644 * q^34 + 2268 * q^36 - 2250 * q^37 + 12632 * q^38 + 4050 * q^39 + 10654 * q^41 + 13536 * q^42 + 35528 * q^43 + 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 12240 * q^48 + 7667 * q^49 - 3006 * q^51 + 14520 * q^52 + 12826 * q^53 - 1458 * q^54 + 17088 * q^56 + 36324 * q^57 + 17196 * q^58 - 81876 * q^59 - 62298 * q^61 - 109184 * q^62 + 18792 * q^63 - 72256 * q^64 - 2178 * q^66 + 46148 * q^67 + 35832 * q^68 - 63540 * q^69 - 64724 * q^71 - 1944 * q^72 - 810 * q^73 - 44796 * q^74 + 44656 * q^76 + 28072 * q^77 - 26388 * q^78 + 43876 * q^79 + 19683 * q^81 - 56060 * q^82 + 101024 * q^83 - 74880 * q^84 + 24128 * q^86 - 36378 * q^87 - 2904 * q^88 + 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 5472 * q^93 + 74552 * q^94 - 27936 * q^96 + 319746 * q^97 - 431134 * q^98 + 29403 * q^99

Introduction

L-functions

Modular forms

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Database

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