LMFDB - Embedded newform 495.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$1$
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	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.07863 q^{2} +33.2643 q^{4} +25.0000 q^{5} -39.3760 q^{7} +10.2141 q^{8} +O(q^{10})$	\(q+8.07863 q^{2} +33.2643 q^{4} +25.0000 q^{5} -39.3760 q^{7} +10.2141 q^{8} +201.966 q^{10} -121.000 q^{11} -220.765 q^{13} -318.105 q^{14} -981.943 q^{16} +200.343 q^{17} +350.345 q^{19} +831.608 q^{20} -977.515 q^{22} -1385.09 q^{23} +625.000 q^{25} -1783.48 q^{26} -1309.82 q^{28} -5506.64 q^{29} -2450.86 q^{31} -8259.61 q^{32} +1618.50 q^{34} -984.401 q^{35} -4060.46 q^{37} +2830.31 q^{38} +255.354 q^{40} -527.283 q^{41} -12078.9 q^{43} -4024.99 q^{44} -11189.6 q^{46} -563.023 q^{47} -15256.5 q^{49} +5049.15 q^{50} -7343.59 q^{52} +37203.0 q^{53} -3025.00 q^{55} -402.192 q^{56} -44486.2 q^{58} -2157.64 q^{59} -39938.0 q^{61} -19799.6 q^{62} -35304.2 q^{64} -5519.11 q^{65} -38473.2 q^{67} +6664.29 q^{68} -7952.62 q^{70} +13725.3 q^{71} -39736.3 q^{73} -32803.0 q^{74} +11654.0 q^{76} +4764.50 q^{77} +35672.9 q^{79} -24548.6 q^{80} -4259.73 q^{82} +79999.7 q^{83} +5008.58 q^{85} -97580.6 q^{86} -1235.91 q^{88} +37783.0 q^{89} +8692.83 q^{91} -46074.1 q^{92} -4548.46 q^{94} +8758.63 q^{95} -7616.35 q^{97} -123252. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8	$ 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 + 50 * q^10 - 363 * q^11 + 450 * q^13 + 1504 * q^14 - 1360 * q^16 + 334 * q^17 - 4036 * q^19 + 700 * q^20 - 242 * q^22 + 7060 * q^23 + 1875 * q^25 - 2932 * q^26 - 8320 * q^28 - 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3644 * q^34 - 5800 * q^35 + 2250 * q^37 + 12632 * q^38 - 600 * q^40 - 10654 * q^41 - 35528 * q^43 - 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 7667 * q^49 + 1250 * q^50 - 14520 * q^52 + 12826 * q^53 - 9075 * q^55 - 17088 * q^56 - 17196 * q^58 + 81876 * q^59 - 62298 * q^61 - 109184 * q^62 - 72256 * q^64 + 11250 * q^65 - 46148 * q^67 + 35832 * q^68 + 37600 * q^70 + 64724 * q^71 + 810 * q^73 + 44796 * q^74 + 44656 * q^76 + 28072 * q^77 + 43876 * q^79 - 34000 * q^80 + 56060 * q^82 + 101024 * q^83 + 8350 * q^85 - 24128 * q^86 + 2904 * q^88 - 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 74552 * q^94 - 100900 * q^95 - 319746 * q^97 - 431134 * 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 $

$2$	8.07863	1.42811	0.714057	−	0.700087i	$-0.246856\pi$
$2$	8.07863	1.42811	0.714057	+	0.700087i	$0.246856\pi$
$3$	0	0
$3$	0	0
$4$	33.2643	1.03951
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−39.3760	−0.303730	−0.151865	−	0.988401i	$-0.548528\pi$
$7$	−39.3760	−0.303730	−0.151865	+	0.988401i	$0.548528\pi$
$8$	10.2141	0.0564257
$9$	0	0
$10$	201.966	0.638672
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−220.765	−0.362302	−0.181151	−	0.983455i	$-0.557982\pi$
$13$	−220.765	−0.362302	−0.181151	+	0.983455i	$0.557982\pi$
$14$	−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	$0.473209\pi$
$18$	0	0
$19$	350.345	0.222645	0.111322	−	0.993784i	$-0.464491\pi$
$19$	350.345	0.222645	0.111322	+	0.993784i	$0.464491\pi$
$20$	831.608	0.464883
$21$	0	0
$22$	−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	$0.588009\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1783.48	−0.517409
$27$	0	0
$28$	−1309.82	−0.315730
$29$	−5506.64	−1.21588	−0.607942	−	0.793982i	$-0.708005\pi$
$29$	−5506.64	−1.21588	−0.607942	+	0.793982i	$0.708005\pi$
$30$	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
$33$	0	0
$34$	1618.50	0.240113
$35$	−984.401	−0.135832
$36$	0	0
$37$	−4060.46	−0.487609	−0.243804	−	0.969824i	$-0.578395\pi$
$37$	−4060.46	−0.487609	−0.243804	+	0.969824i	$0.578395\pi$
$38$	2830.31	0.317962
$39$	0	0
$40$	255.354	0.0252343
$41$	−527.283	−0.0489874	−0.0244937	−	0.999700i	$-0.507797\pi$
$41$	−527.283	−0.0489874	−0.0244937	+	0.999700i	$0.507797\pi$
$42$	0	0
$43$	−12078.9	−0.996218	−0.498109	−	0.867114i	$-0.665972\pi$
$43$	−12078.9	−0.996218	−0.498109	+	0.867114i	$0.665972\pi$
$44$	−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	$0.505917\pi$
$48$	0	0
$49$	−15256.5	−0.907748
$50$	5049.15	0.285623
$51$	0	0
$52$	−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	$0.136375\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−402.192	−0.0171381
$57$	0	0
$58$	−44486.2	−1.73642
$59$	−2157.64	−0.0806955	−0.0403477	−	0.999186i	$-0.512847\pi$
$59$	−2157.64	−0.0806955	−0.0403477	+	0.999186i	$0.512847\pi$
$60$	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
$63$	0	0
$64$	−35304.2	−1.07740
$65$	−5519.11	−0.162026
$66$	0	0
$67$	−38473.2	−1.04706	−0.523529	−	0.852008i	$-0.675385\pi$
$67$	−38473.2	−1.04706	−0.523529	+	0.852008i	$0.675385\pi$
$68$	6664.29	0.174776
$69$	0	0
$70$	−7952.62	−0.193984
$71$	13725.3	0.323129	0.161564	−	0.986862i	$-0.448346\pi$
$71$	13725.3	0.323129	0.161564	+	0.986862i	$0.448346\pi$
$72$	0	0
$73$	−39736.3	−0.872731	−0.436366	−	0.899769i	$-0.643735\pi$
$73$	−39736.3	−0.872731	−0.436366	+	0.899769i	$0.643735\pi$
$74$	−32803.0	−0.696361
$75$	0	0
$76$	11654.0	0.231441
$77$	4764.50	0.0915779
$78$	0	0
$79$	35672.9	0.643088	0.321544	−	0.946895i	$-0.395798\pi$
$79$	35672.9	0.643088	0.321544	+	0.946895i	$0.395798\pi$
$80$	−24548.6	−0.428846
$81$	0	0
$82$	−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	$0.280040\pi$
$84$	0	0
$85$	5008.58	0.0751913
$86$	−97580.6	−1.42271
$87$	0	0
$88$	−1235.91	−0.0170130
$89$	37783.0	0.505616	0.252808	−	0.967516i	$-0.418646\pi$
$89$	37783.0	0.505616	0.252808	+	0.967516i	$0.418646\pi$
$90$	0	0
$91$	8692.83	0.110042
$92$	−46074.1	−0.567528
$93$	0	0
$94$	−4548.46	−0.0530939
$95$	8758.63	0.0995697
$96$	0	0
$97$	−7616.35	−0.0821897	−0.0410948	−	0.999155i	$-0.513085\pi$
$97$	−7616.35	−0.0821897	−0.0410948	+	0.999155i	$0.513085\pi$
$98$	−123252.	−1.29637
$99$	0	0
$100$	20790.2	0.207902
$101$	−125963.	−1.22868	−0.614341	−	0.789041i	$-0.710578\pi$
$101$	−125963.	−1.22868	−0.614341	+	0.789041i	$0.710578\pi$
$102$	0	0
$103$	−165220.	−1.53451	−0.767257	−	0.641340i	$-0.778379\pi$
$103$	−165220.	−1.53451	−0.767257	+	0.641340i	$0.778379\pi$
$104$	−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	$0.482316\pi$
$108$	0	0
$109$	37970.9	0.306115	0.153057	−	0.988217i	$-0.451088\pi$
$109$	37970.9	0.306115	0.153057	+	0.988217i	$0.451088\pi$
$110$	−24437.9	−0.192567
$111$	0	0
$112$	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	$0.398396\pi$
$114$	0	0
$115$	−34627.3	−0.244159
$116$	−183175.	−1.26392
$117$	0	0
$118$	−17430.8	−0.115242
$119$	−7888.73	−0.0510669
$120$	0	0
$121$	14641.0	0.0909091
$122$	−322645.	−1.96257
$123$	0	0
$124$	−81526.1	−0.476148
$125$	15625.0	0.0894427
$126$	0	0
$127$	−141237.	−0.777033	−0.388517	−	0.921442i	$-0.627012\pi$
$127$	−141237.	−0.777033	−0.388517	+	0.921442i	$0.627012\pi$
$128$	−20902.2	−0.112763
$129$	0	0
$130$	−44586.9	−0.231392
$131$	−210469.	−1.07155	−0.535773	−	0.844362i	$-0.679980\pi$
$131$	−210469.	−1.07155	−0.535773	+	0.844362i	$0.679980\pi$
$132$	0	0
$133$	−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	$0.514179\pi$
$138$	0	0
$139$	132498.	0.581663	0.290831	−	0.956774i	$-0.406068\pi$
$139$	132498.	0.581663	0.290831	+	0.956774i	$0.406068\pi$
$140$	−32745.5	−0.141199
$141$	0	0
$142$	110882.	0.461465
$143$	26712.5	0.109238
$144$	0	0
$145$	−137666.	−0.543760
$146$	−321015.	−1.24636
$147$	0	0
$148$	−135069.	−0.506874
$149$	153316.	0.565748	0.282874	−	0.959157i	$-0.408712\pi$
$149$	153316.	0.565748	0.282874	+	0.959157i	$0.408712\pi$
$150$	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
$153$	0	0
$154$	38490.7	0.130784
$155$	−61271.4	−0.204846
$156$	0	0
$157$	−180692.	−0.585047	−0.292523	−	0.956258i	$-0.594495\pi$
$157$	−180692.	−0.585047	−0.292523	+	0.956258i	$0.594495\pi$
$158$	288188.	0.918403
$159$	0	0
$160$	−206490.	−0.637675
$161$	54539.4	0.165823
$162$	0	0
$163$	−176045.	−0.518985	−0.259493	−	0.965745i	$-0.583555\pi$
$163$	−176045.	−0.518985	−0.259493	+	0.965745i	$0.583555\pi$
$164$	−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	$0.633642\pi$
$168$	0	0
$169$	−322556.	−0.868737
$170$	40462.5	0.107382
$171$	0	0
$172$	−401795.	−1.03558
$173$	20784.9	0.0527999	0.0263999	−	0.999651i	$-0.491596\pi$
$173$	20784.9	0.0527999	0.0263999	+	0.999651i	$0.491596\pi$
$174$	0	0
$175$	−24610.0	−0.0607459
$176$	118815.	0.289128
$177$	0	0
$178$	305235.	0.722078
$179$	229326.	0.534958	0.267479	−	0.963564i	$-0.413809\pi$
$179$	229326.	0.534958	0.267479	+	0.963564i	$0.413809\pi$
$180$	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
$183$	0	0
$184$	−14147.5	−0.0308060
$185$	−101512.	−0.218065
$186$	0	0
$187$	−24241.5	−0.0506939
$188$	−18728.6	−0.0386465
$189$	0	0
$190$	70757.8	0.142197
$191$	496794.	0.985356	0.492678	−	0.870212i	$-0.336018\pi$
$191$	496794.	0.985356	0.492678	+	0.870212i	$0.336018\pi$
$192$	0	0
$193$	271362.	0.524391	0.262196	−	0.965015i	$-0.415553\pi$
$193$	271362.	0.524391	0.262196	+	0.965015i	$0.415553\pi$
$194$	−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	$0.443911\pi$
$198$	0	0
$199$	602637.	1.07876	0.539378	−	0.842064i	$-0.318660\pi$
$199$	602637.	1.07876	0.539378	+	0.842064i	$0.318660\pi$
$200$	6383.84	0.0112851
$201$	0	0
$202$	−1.01761e6	−1.75470
$203$	216830.	0.369300
$204$	0	0
$205$	−13182.1	−0.0219078
$206$	−1.33476e6	−2.19146
$207$	0	0
$208$	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
$213$	0	0
$214$	106252.	0.158600
$215$	−301971.	−0.445522
$216$	0	0
$217$	96505.0	0.139123
$218$	306753.	0.437167
$219$	0	0
$220$	−100625.	−0.140168
$221$	−44228.7	−0.0609149
$222$	0	0
$223$	959605.	1.29220	0.646101	−	0.763252i	$-0.276399\pi$
$223$	959605.	1.29220	0.646101	+	0.763252i	$0.276399\pi$
$224$	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	$0.440208\pi$
$228$	0	0
$229$	111161.	0.140076	0.0700379	−	0.997544i	$-0.477688\pi$
$229$	111161.	0.140076	0.0700379	+	0.997544i	$0.477688\pi$
$230$	−279741.	−0.348688
$231$	0	0
$232$	−56245.6	−0.0686071
$233$	453216.	0.546909	0.273455	−	0.961885i	$-0.411834\pi$
$233$	453216.	0.546909	0.273455	+	0.961885i	$0.411834\pi$
$234$	0	0
$235$	−14075.6	−0.0166263
$236$	−71772.5	−0.0838838
$237$	0	0
$238$	−63730.1	−0.0729294
$239$	−1.26387e6	−1.43123	−0.715614	−	0.698496i	$-0.753853\pi$
$239$	−1.26387e6	−1.43123	−0.715614	+	0.698496i	$0.753853\pi$
$240$	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
$243$	0	0
$244$	−1.32851e6	−1.42853
$245$	−381413.	−0.405957
$246$	0	0
$247$	−77343.8	−0.0806646
$248$	−25033.4	−0.0258458
$249$	0	0
$250$	126229.	0.127734
$251$	1.43370e6	1.43639	0.718197	−	0.695840i	$-0.244968\pi$
$251$	1.43370e6	1.43639	0.718197	+	0.695840i	$0.244968\pi$
$252$	0	0
$253$	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	$0.351260\pi$
$258$	0	0
$259$	159885.	0.148101
$260$	−183590.	−0.168428
$261$	0	0
$262$	−1.70030e6	−1.53029
$263$	−322280.	−0.287306	−0.143653	−	0.989628i	$-0.545885\pi$
$263$	−322280.	−0.287306	−0.143653	+	0.989628i	$0.545885\pi$
$264$	0	0
$265$	930075.	0.813586
$266$	−111446.	−0.0965744
$267$	0	0
$268$	−1.27978e6	−1.08843
$269$	−808737.	−0.681438	−0.340719	−	0.940165i	$-0.610671\pi$
$269$	−808737.	−0.681438	−0.340719	+	0.940165i	$0.610671\pi$
$270$	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
$273$	0	0
$274$	−158059.	−0.127187
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	236933.	0.185535	0.0927675	−	0.995688i	$-0.470429\pi$
$277$	236933.	0.185535	0.0927675	+	0.995688i	$0.470429\pi$
$278$	1.07040e6	0.830681
$279$	0	0
$280$	−10054.8	−0.00766441
$281$	450779.	0.340563	0.170282	−	0.985395i	$-0.445532\pi$
$281$	450779.	0.340563	0.170282	+	0.985395i	$0.445532\pi$
$282$	0	0
$283$	683413.	0.507245	0.253622	−	0.967303i	$-0.418378\pi$
$283$	683413.	0.507245	0.253622	+	0.967303i	$0.418378\pi$
$284$	456563.	0.335896
$285$	0	0
$286$	215801.	0.156005
$287$	20762.3	0.0148789
$288$	0	0
$289$	−1.37972e6	−0.971731
$290$	−1.11215e6	−0.776551
$291$	0	0
$292$	−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	$0.288717\pi$
$294$	0	0
$295$	−53941.0	−0.0360881
$296$	−41474.1	−0.0275136
$297$	0	0
$298$	1.23859e6	0.807953
$299$	305779.	0.197801
$300$	0	0
$301$	475617.	0.302581
$302$	2.00472e6	1.26484
$303$	0	0
$304$	−344019.	−0.213500
$305$	−998450.	−0.614578
$306$	0	0
$307$	−3.00860e6	−1.82187	−0.910937	−	0.412546i	$-0.864639\pi$
$307$	−3.00860e6	−1.82187	−0.910937	+	0.412546i	$0.864639\pi$
$308$	158488.	0.0951962
$309$	0	0
$310$	−494989.	−0.292544
$311$	2.04083e6	1.19648	0.598241	−	0.801316i	$-0.295866\pi$
$311$	2.04083e6	1.19648	0.598241	+	0.801316i	$0.295866\pi$
$312$	0	0
$313$	−1.73940e6	−1.00355	−0.501774	−	0.864999i	$-0.667319\pi$
$313$	−1.73940e6	−1.00355	−0.501774	+	0.864999i	$0.667319\pi$
$314$	−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	$0.509493\pi$
$318$	0	0
$319$	666304.	0.366603
$320$	−882605.	−0.481827
$321$	0	0
$322$	440604.	0.236815
$323$	70189.3	0.0374338
$324$	0	0
$325$	−137978.	−0.0724604
$326$	−1.42220e6	−0.741170
$327$	0	0
$328$	−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
$333$	0	0
$334$	−2.37365e6	−1.16426
$335$	−961829.	−0.468259
$336$	0	0
$337$	1.22550e6	0.587814	0.293907	−	0.955834i	$-0.405044\pi$
$337$	1.22550e6	0.587814	0.293907	+	0.955834i	$0.405044\pi$
$338$	−2.60581e6	−1.24066
$339$	0	0
$340$	166607.	0.0781621
$341$	296553.	0.138107
$342$	0	0
$343$	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	$0.825265\pi$
$348$	0	0
$349$	456968.	0.200827	0.100413	−	0.994946i	$-0.467983\pi$
$349$	456968.	0.200827	0.100413	+	0.994946i	$0.467983\pi$
$350$	−198815.	−0.0867521
$351$	0	0
$352$	999413.	0.429920
$353$	−4.32830e6	−1.84876	−0.924380	−	0.381474i	$-0.875417\pi$
$353$	−4.32830e6	−1.84876	−0.924380	+	0.381474i	$0.875417\pi$
$354$	0	0
$355$	343132.	0.144508
$356$	1.25683e6	0.525594
$357$	0	0
$358$	1.85264e6	0.763982
$359$	845380.	0.346191	0.173095	−	0.984905i	$-0.444623\pi$
$359$	845380.	0.346191	0.173095	+	0.984905i	$0.444623\pi$
$360$	0	0
$361$	−2.35336e6	−0.950429
$362$	−733599.	−0.294230
$363$	0	0
$364$	289161.	0.114390
$365$	−993408.	−0.390297
$366$	0	0
$367$	2.96016e6	1.14723	0.573615	−	0.819125i	$-0.305540\pi$
$367$	2.96016e6	1.14723	0.573615	+	0.819125i	$0.305540\pi$
$368$	1.36008e6	0.523534
$369$	0	0
$370$	−820075.	−0.311422
$371$	−1.46491e6	−0.552555
$372$	0	0
$373$	1.02299e6	0.380713	0.190357	−	0.981715i	$-0.439036\pi$
$373$	1.02299e6	0.380713	0.190357	+	0.981715i	$0.439036\pi$
$374$	−195839.	−0.0723968
$375$	0	0
$376$	−5750.80	−0.00209777
$377$	1.21567e6	0.440517
$378$	0	0
$379$	−4.19402e6	−1.49980	−0.749899	−	0.661552i	$-0.769898\pi$
$379$	−4.19402e6	−1.49980	−0.749899	+	0.661552i	$0.769898\pi$
$380$	291350.	0.103504
$381$	0	0
$382$	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	$0.407628\pi$
$384$	0	0
$385$	119113.	0.0409549
$386$	2.19223e6	0.748891
$387$	0	0
$388$	−253353.	−0.0854371
$389$	−769820.	−0.257938	−0.128969	−	0.991649i	$-0.541167\pi$
$389$	−769820.	−0.257938	−0.128969	+	0.991649i	$0.541167\pi$
$390$	0	0
$391$	−277493.	−0.0917933
$392$	−155832.	−0.0512203
$393$	0	0
$394$	1.54279e6	0.500688
$395$	891822.	0.287598
$396$	0	0
$397$	4.86631e6	1.54961	0.774806	−	0.632199i	$-0.217847\pi$
$397$	4.86631e6	1.54961	0.774806	+	0.632199i	$0.217847\pi$
$398$	4.86848e6	1.54059
$399$	0	0
$400$	−613714.	−0.191786
$401$	−717064.	−0.222688	−0.111344	−	0.993782i	$-0.535516\pi$
$401$	−717064.	−0.222688	−0.111344	+	0.993782i	$0.535516\pi$
$402$	0	0
$403$	541062.	0.165953
$404$	−4.19007e6	−1.27723
$405$	0	0
$406$	1.75169e6	0.527402
$407$	491316.	0.147019
$408$	0	0
$409$	−3.80091e6	−1.12352	−0.561758	−	0.827302i	$-0.689875\pi$
$409$	−3.80091e6	−1.12352	−0.561758	+	0.827302i	$0.689875\pi$
$410$	−106493.	−0.0312869
$411$	0	0
$412$	−5.49595e6	−1.59514
$413$	84959.4	0.0245096
$414$	0	0
$415$	1.99999e6	0.570044
$416$	1.82343e6	0.516601
$417$	0	0
$418$	−342468.	−0.0958691
$419$	1.55888e6	0.433787	0.216894	−	0.976195i	$-0.430408\pi$
$419$	1.55888e6	0.433787	0.216894	+	0.976195i	$0.430408\pi$
$420$	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
$423$	0	0
$424$	379997.	0.102651
$425$	125215.	0.0336266
$426$	0	0
$427$	1.57260e6	0.417397
$428$	437502.	0.115444
$429$	0	0
$430$	−2.43952e6	−0.636257
$431$	5.25900e6	1.36367	0.681836	−	0.731505i	$-0.261182\pi$
$431$	5.25900e6	1.36367	0.681836	+	0.731505i	$0.261182\pi$
$432$	0	0
$433$	−694549.	−0.178026	−0.0890130	−	0.996030i	$-0.528371\pi$
$433$	−694549.	−0.178026	−0.0890130	+	0.996030i	$0.528371\pi$
$434$	779629.	0.198684
$435$	0	0
$436$	1.26308e6	0.318210
$437$	−485260.	−0.121554
$438$	0	0
$439$	−7.80969e6	−1.93407	−0.967036	−	0.254640i	$-0.918043\pi$
$439$	−7.80969e6	−1.93407	−0.967036	+	0.254640i	$0.918043\pi$
$440$	−30897.8	−0.00760844
$441$	0	0
$442$	−357307.	−0.0869934
$443$	5.43876e6	1.31671	0.658356	−	0.752707i	$-0.271252\pi$
$443$	5.43876e6	1.31671	0.658356	+	0.752707i	$0.271252\pi$
$444$	0	0
$445$	944575.	0.226119
$446$	7.75229e6	1.84541
$447$	0	0
$448$	1.39014e6	0.327238
$449$	−5.46277e6	−1.27878	−0.639392	−	0.768881i	$-0.720814\pi$
$449$	−5.46277e6	−1.27878	−0.639392	+	0.768881i	$0.720814\pi$
$450$	0	0
$451$	63801.3	0.0147703
$452$	2.83378e6	0.652409
$453$	0	0
$454$	2.34243e6	0.533368
$455$	217321.	0.0492122
$456$	0	0
$457$	3.07144e6	0.687941	0.343970	−	0.938980i	$-0.388228\pi$
$457$	3.07144e6	0.687941	0.343970	+	0.938980i	$0.388228\pi$
$458$	898028.	0.200044
$459$	0	0
$460$	−1.15185e6	−0.253806
$461$	−3.85641e6	−0.845143	−0.422572	−	0.906329i	$-0.638873\pi$
$461$	−3.85641e6	−0.845143	−0.422572	+	0.906329i	$0.638873\pi$
$462$	0	0
$463$	3.20594e6	0.695029	0.347514	−	0.937675i	$-0.387026\pi$
$463$	3.20594e6	0.695029	0.347514	+	0.937675i	$0.387026\pi$
$464$	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	$0.416669\pi$
$468$	0	0
$469$	1.51492e6	0.318023
$470$	−113711.	−0.0237443
$471$	0	0
$472$	−22038.5	−0.00455330
$473$	1.46154e6	0.300371
$474$	0	0
$475$	218966.	0.0445289
$476$	−262413.	−0.0530846
$477$	0	0
$478$	−1.02104e7	−2.04396
$479$	−4.69186e6	−0.934342	−0.467171	−	0.884167i	$-0.654727\pi$
$479$	−4.69186e6	−0.934342	−0.467171	+	0.884167i	$0.654727\pi$
$480$	0	0
$481$	896406.	0.176662
$482$	1.08536e6	0.212793
$483$	0	0
$484$	487023.	0.0945010
$485$	−190409.	−0.0367564
$486$	0	0
$487$	5.73305e6	1.09538	0.547688	−	0.836683i	$-0.315508\pi$
$487$	5.73305e6	1.09538	0.547688	+	0.836683i	$0.315508\pi$
$488$	−407932.	−0.0775423
$489$	0	0
$490$	−3.08130e6	−0.579754
$491$	693872.	0.129890	0.0649449	−	0.997889i	$-0.479313\pi$
$491$	693872.	0.129890	0.0649449	+	0.997889i	$0.479313\pi$
$492$	0	0
$493$	−1.10322e6	−0.204430
$494$	−624832.	−0.115198
$495$	0	0
$496$	2.40660e6	0.439238
$497$	−540448.	−0.0981438
$498$	0	0
$499$	1.23153e6	0.221407	0.110704	−	0.993853i	$-0.464690\pi$
$499$	1.23153e6	0.221407	0.110704	+	0.993853i	$0.464690\pi$
$500$	519755.	0.0929767
$501$	0	0
$502$	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	$0.265847\pi$
$504$	0	0
$505$	−3.14907e6	−0.549483
$506$	1.35395e6	0.235085
$507$	0	0
$508$	−4.69816e6	−0.807734
$509$	1.55496e6	0.266026	0.133013	−	0.991114i	$-0.457535\pi$
$509$	1.55496e6	0.266026	0.133013	+	0.991114i	$0.457535\pi$
$510$	0	0
$511$	1.56466e6	0.265074
$512$	8.43141e6	1.42143
$513$	0	0
$514$	7.70649e6	1.28662
$515$	−4.13051e6	−0.686255
$516$	0	0
$517$	68125.8	0.0112095
$518$	1.29165e6	0.211505
$519$	0	0
$520$	−56373.0	−0.00914245
$521$	3.26766e6	0.527403	0.263701	−	0.964604i	$-0.415057\pi$
$521$	3.26766e6	0.527403	0.263701	+	0.964604i	$0.415057\pi$
$522$	0	0
$523$	−8.35303e6	−1.33533	−0.667667	−	0.744460i	$-0.732707\pi$
$523$	−8.35303e6	−1.33533	−0.667667	+	0.744460i	$0.732707\pi$
$524$	−7.00112e6	−1.11388
$525$	0	0
$526$	−2.60358e6	−0.410305
$527$	−491012.	−0.0770133
$528$	0	0
$529$	−4.51787e6	−0.701931
$530$	7.51374e6	1.16189
$531$	0	0
$532$	−458888.	−0.0702956
$533$	116405.	0.0177482
$534$	0	0
$535$	328807.	0.0496657
$536$	−392970.	−0.0590810
$537$	0	0
$538$	−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
$543$	0	0
$544$	−1.65476e6	−0.239738
$545$	949272.	0.136899
$546$	0	0
$547$	−3.84639e6	−0.549648	−0.274824	−	0.961495i	$-0.588620\pi$
$547$	−3.84639e6	−0.549648	−0.274824	+	0.961495i	$0.588620\pi$
$548$	−650819.	−0.0925782
$549$	0	0
$550$	−610947.	−0.0861185
$551$	−1.92923e6	−0.270710
$552$	0	0
$553$	−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	$0.868350\pi$
$558$	0	0
$559$	2.66658e6	0.360932
$560$	966625.	0.130253
$561$	0	0
$562$	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	$0.532043\pi$
$564$	0	0
$565$	2.12974e6	0.280677
$566$	5.52105e6	0.724403
$567$	0	0
$568$	140192.	0.0182328
$569$	−1.39221e7	−1.80271	−0.901353	−	0.433085i	$-0.857425\pi$
$569$	−1.39221e7	−1.80271	−0.901353	+	0.433085i	$0.857425\pi$
$570$	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
$573$	0	0
$574$	167731.	0.0212488
$575$	−865681.	−0.109191
$576$	0	0
$577$	3.96778e6	0.496144	0.248072	−	0.968742i	$-0.420203\pi$
$577$	3.96778e6	0.496144	0.248072	+	0.968742i	$0.420203\pi$
$578$	−1.11463e7	−1.38774
$579$	0	0
$580$	−4.57937e6	−0.565244
$581$	−3.15007e6	−0.387151
$582$	0	0
$583$	−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	$0.570073\pi$
$588$	0	0
$589$	−858645.	−0.101982
$590$	−435770.	−0.0515380
$591$	0	0
$592$	3.98714e6	0.467582
$593$	486759.	0.0568431	0.0284215	−	0.999596i	$-0.490952\pi$
$593$	486759.	0.0568431	0.0284215	+	0.999596i	$0.490952\pi$
$594$	0	0
$595$	−197218.	−0.0228378
$596$	5.09997e6	0.588101
$597$	0	0
$598$	2.47027e6	0.282483
$599$	−1.37413e7	−1.56481	−0.782404	−	0.622771i	$-0.786007\pi$
$599$	−1.37413e7	−1.56481	−0.782404	+	0.622771i	$0.786007\pi$
$600$	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
$603$	0	0
$604$	8.25457e6	0.920666
$605$	366025.	0.0406558
$606$	0	0
$607$	−3.41681e6	−0.376399	−0.188200	−	0.982131i	$-0.560265\pi$
$607$	−3.41681e6	−0.376399	−0.188200	+	0.982131i	$0.560265\pi$
$608$	−2.89371e6	−0.317465
$609$	0	0
$610$	−8.06611e6	−0.877687
$611$	124296.	0.0134695
$612$	0	0
$613$	−1.32402e7	−1.42313	−0.711564	−	0.702621i	$-0.752013\pi$
$613$	−1.32402e7	−1.42313	−0.711564	+	0.702621i	$0.752013\pi$
$614$	−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	$0.864087\pi$
$618$	0	0
$619$	1.60142e7	1.67988	0.839940	−	0.542679i	$-0.182590\pi$
$619$	1.60142e7	1.67988	0.839940	+	0.542679i	$0.182590\pi$
$620$	−2.03815e6	−0.212940
$621$	0	0
$622$	1.64871e7	1.70871
$623$	−1.48774e6	−0.153571
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.40520e7	−1.43318
$627$	0	0
$628$	−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
$633$	0	0
$634$	−862021.	−0.0851716
$635$	−3.53093e6	−0.347500
$636$	0	0
$637$	3.36810e6	0.328879
$638$	5.38282e6	0.523550
$639$	0	0
$640$	−522556.	−0.0504293
$641$	−8.77648e6	−0.843675	−0.421838	−	0.906671i	$-0.638615\pi$
$641$	−8.77648e6	−0.843675	−0.421838	+	0.906671i	$0.638615\pi$
$642$	0	0
$643$	−1.51011e7	−1.44040	−0.720198	−	0.693768i	$-0.755949\pi$
$643$	−1.51011e7	−1.44040	−0.720198	+	0.693768i	$0.755949\pi$
$644$	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	$0.765893\pi$
$648$	0	0
$649$	261075.	0.0243306
$650$	−1.11467e6	−0.103482
$651$	0	0
$652$	−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	$0.709427\pi$
$654$	0	0
$655$	−5.26173e6	−0.479210
$656$	517762.	0.0469754
$657$	0	0
$658$	179100.	0.0161262
$659$	882213.	0.0791334	0.0395667	−	0.999217i	$-0.487402\pi$
$659$	882213.	0.0791334	0.0395667	+	0.999217i	$0.487402\pi$
$660$	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
$663$	0	0
$664$	817128.	0.0719234
$665$	−344880.	−0.0302422
$666$	0	0
$667$	7.62720e6	0.663820
$668$	−9.77367e6	−0.847454
$669$	0	0
$670$	−7.77027e6	−0.668727
$671$	4.83250e6	0.414348
$672$	0	0
$673$	1.29320e6	0.110060	0.0550300	−	0.998485i	$-0.482475\pi$
$673$	1.29320e6	0.110060	0.0550300	+	0.998485i	$0.482475\pi$
$674$	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	$0.714442\pi$
$678$	0	0
$679$	299902.	0.0249634
$680$	51158.4	0.00424272
$681$	0	0
$682$	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	$0.615148\pi$
$684$	0	0
$685$	−489127.	−0.0398286
$686$	1.01996e7	0.827506
$687$	0	0
$688$	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
$693$	0	0
$694$	−3.09157e7	−2.43658
$695$	3.31244e6	0.260127
$696$	0	0
$697$	−105638.	−0.00823639
$698$	3.69167e6	0.286804
$699$	0	0
$700$	−818636.	−0.0631460
$701$	2.13485e7	1.64086	0.820432	−	0.571744i	$-0.193733\pi$
$701$	2.13485e7	1.64086	0.820432	+	0.571744i	$0.193733\pi$
$702$	0	0
$703$	−1.42256e6	−0.108563
$704$	4.27181e6	0.324848
$705$	0	0
$706$	−3.49667e7	−2.64024
$707$	4.95992e6	0.373187
$708$	0	0
$709$	6.90604e6	0.515957	0.257979	−	0.966151i	$-0.416944\pi$
$709$	6.90604e6	0.515957	0.257979	+	0.966151i	$0.416944\pi$
$710$	2.77204e6	0.206373
$711$	0	0
$712$	385921.	0.0285298
$713$	3.39466e6	0.250076
$714$	0	0
$715$	667813.	0.0488528
$716$	7.62837e6	0.556095
$717$	0	0
$718$	6.82951e6	0.494400
$719$	1.71863e7	1.23983	0.619913	−	0.784670i	$-0.287168\pi$
$719$	1.71863e7	1.23983	0.619913	+	0.784670i	$0.287168\pi$
$720$	0	0
$721$	6.50573e6	0.466077
$722$	−1.90119e7	−1.35732
$723$	0	0
$724$	−3.02065e6	−0.214167
$725$	−3.44165e6	−0.243177
$726$	0	0
$727$	−2.43516e7	−1.70880	−0.854401	−	0.519615i	$-0.826075\pi$
$727$	−2.43516e7	−1.70880	−0.854401	+	0.519615i	$0.826075\pi$
$728$	88789.8	0.00620919
$729$	0	0
$730$	−8.02538e6	−0.557389
$731$	−2.41992e6	−0.167497
$732$	0	0
$733$	−1.93102e7	−1.32747	−0.663737	−	0.747966i	$-0.731031\pi$
$733$	−1.93102e7	−1.32747	−0.663737	+	0.747966i	$0.731031\pi$
$734$	2.39141e7	1.63838
$735$	0	0
$736$	1.14403e7	0.778472
$737$	4.65525e6	0.315700
$738$	0	0
$739$	−9.23336e6	−0.621940	−0.310970	−	0.950420i	$-0.600654\pi$
$739$	−9.23336e6	−0.621940	−0.310970	+	0.950420i	$0.600654\pi$
$740$	−3.37672e6	−0.226681
$741$	0	0
$742$	−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	$0.585895\pi$
$744$	0	0
$745$	3.83291e6	0.253010
$746$	8.26434e6	0.543702
$747$	0	0
$748$	−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
$753$	0	0
$754$	9.82096e6	0.629109
$755$	6.20377e6	0.396085
$756$	0	0
$757$	−1.40172e7	−0.889040	−0.444520	−	0.895769i	$-0.646626\pi$
$757$	−1.40172e7	−0.889040	−0.444520	+	0.895769i	$0.646626\pi$
$758$	−3.38820e7	−2.14188
$759$	0	0
$760$	89461.9	0.00561829
$761$	1.98659e7	1.24350	0.621750	−	0.783215i	$-0.286422\pi$
$761$	1.98659e7	1.24350	0.621750	+	0.783215i	$0.286422\pi$
$762$	0	0
$763$	−1.49514e6	−0.0929761
$764$	1.65255e7	1.02429
$765$	0	0
$766$	1.32721e7	0.817275
$767$	476331.	0.0292361
$768$	0	0
$769$	−2.09738e7	−1.27897	−0.639486	−	0.768803i	$-0.720853\pi$
$769$	−2.09738e7	−1.27897	−0.639486	+	0.768803i	$0.720853\pi$
$770$	962267.	0.0584883
$771$	0	0
$772$	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	$0.574349\pi$
$774$	0	0
$775$	−1.53178e6	−0.0916101
$776$	−77794.4	−0.00463761
$777$	0	0
$778$	−6.21909e6	−0.368365
$779$	−184731.	−0.0109068
$780$	0	0
$781$	−1.66076e6	−0.0974270
$782$	−2.24177e6	−0.131091
$783$	0	0
$784$	1.49810e7	0.870466
$785$	−4.51731e6	−0.261641
$786$	0	0
$787$	−1.39975e6	−0.0805589	−0.0402794	−	0.999188i	$-0.512825\pi$
$787$	−1.39975e6	−0.0805589	−0.0402794	+	0.999188i	$0.512825\pi$
$788$	6.35256e6	0.364446
$789$	0	0
$790$	7.20470e6	0.410722
$791$	−3.35443e6	−0.190624
$792$	0	0
$793$	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	$0.611347\pi$
$798$	0	0
$799$	−112798.	−0.00625078
$800$	−5.16225e6	−0.285177
$801$	0	0
$802$	−5.79290e6	−0.318024
$803$	4.80810e6	0.263138
$804$	0	0
$805$	1.36348e6	0.0741584
$806$	4.37104e6	0.236999
$807$	0	0
$808$	−1.28660e6	−0.0693292
$809$	1.19607e7	0.642515	0.321258	−	0.946992i	$-0.395894\pi$
$809$	1.19607e7	0.642515	0.321258	+	0.946992i	$0.395894\pi$
$810$	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
$813$	0	0
$814$	3.96916e6	0.209961
$815$	−4.40113e6	−0.232097
$816$	0	0
$817$	−4.23177e6	−0.221803
$818$	−3.07062e7	−1.60451
$819$	0	0
$820$	−438493.	−0.0227734
$821$	−2.77684e7	−1.43778	−0.718890	−	0.695124i	$-0.755349\pi$
$821$	−2.77684e7	−1.43778	−0.718890	+	0.695124i	$0.755349\pi$
$822$	0	0
$823$	−2.63447e7	−1.35580	−0.677898	−	0.735156i	$-0.737109\pi$
$823$	−2.63447e7	−1.35580	−0.677898	+	0.735156i	$0.737109\pi$
$824$	−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	$0.179359\pi$
$828$	0	0
$829$	2.84581e7	1.43820	0.719101	−	0.694905i	$-0.244554\pi$
$829$	2.84581e7	1.43820	0.719101	+	0.694905i	$0.244554\pi$
$830$	1.61572e7	0.814087
$831$	0	0
$832$	7.79391e6	0.390344
$833$	−3.05654e6	−0.152622
$834$	0	0
$835$	−7.34545e6	−0.364588
$836$	−1.41013e6	−0.0697822
$837$	0	0
$838$	1.25936e7	0.619498
$839$	9.47931e6	0.464913	0.232457	−	0.972607i	$-0.425324\pi$
$839$	9.47931e6	0.464913	0.232457	+	0.972607i	$0.425324\pi$
$840$	0	0
$841$	9.81196e6	0.478372
$842$	3.34101e7	1.62404
$843$	0	0
$844$	3.05425e7	1.47587
$845$	−8.06390e6	−0.388511
$846$	0	0
$847$	−576505.	−0.0276118
$848$	−3.65312e7	−1.74451
$849$	0	0
$850$	1.01156e6	0.0480226
$851$	5.62411e6	0.266213
$852$	0	0
$853$	2.13145e7	1.00300	0.501501	−	0.865157i	$-0.332781\pi$
$853$	2.13145e7	1.00300	0.501501	+	0.865157i	$0.332781\pi$
$854$	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	$0.671851\pi$
$858$	0	0
$859$	−7.07193e6	−0.327006	−0.163503	−	0.986543i	$-0.552279\pi$
$859$	−7.07193e6	−0.327006	−0.163503	+	0.986543i	$0.552279\pi$
$860$	−1.00449e7	−0.463125
$861$	0	0
$862$	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	$0.460246\pi$
$864$	0	0
$865$	519623.	0.0236128
$866$	−5.61101e6	−0.254241
$867$	0	0
$868$	3.21017e6	0.144620
$869$	−4.31642e6	−0.193898
$870$	0	0
$871$	8.49351e6	0.379351
$872$	387840.	0.0172727
$873$	0	0
$874$	−3.92023e6	−0.173593
$875$	−615251.	−0.0271664
$876$	0	0
$877$	−2.02823e7	−0.890467	−0.445234	−	0.895414i	$-0.646879\pi$
$877$	−2.02823e7	−0.890467	−0.445234	+	0.895414i	$0.646879\pi$
$878$	−6.30917e7	−2.76208
$879$	0	0
$880$	2.97038e6	0.129302
$881$	6.58178e6	0.285696	0.142848	−	0.989745i	$-0.454374\pi$
$881$	6.58178e6	0.285696	0.142848	+	0.989745i	$0.454374\pi$
$882$	0	0
$883$	−1.63860e7	−0.707247	−0.353623	−	0.935388i	$-0.615051\pi$
$883$	−1.63860e7	−0.707247	−0.353623	+	0.935388i	$0.615051\pi$
$884$	−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	$0.480091\pi$
$888$	0	0
$889$	5.56136e6	0.236008
$890$	7.63087e6	0.322923
$891$	0	0
$892$	3.19206e7	1.34326
$893$	−197252.	−0.00827739
$894$	0	0
$895$	5.73314e6	0.239241
$896$	823047.	0.0342495
$897$	0	0
$898$	−4.41317e7	−1.82625
$899$	1.34960e7	0.556936
$900$	0	0
$901$	7.45337e6	0.305873
$902$	515427.	0.0210936
$903$	0	0
$904$	870140.	0.0354134
$905$	−2.27018e6	−0.0921381
$906$	0	0
$907$	−7.88236e6	−0.318154	−0.159077	−	0.987266i	$-0.550852\pi$
$907$	−7.88236e6	−0.318154	−0.159077	+	0.987266i	$0.550852\pi$
$908$	9.64512e6	0.388234
$909$	0	0
$910$	1.75566e6	0.0702807
$911$	2.16713e7	0.865146	0.432573	−	0.901599i	$-0.357606\pi$
$911$	2.16713e7	0.865146	0.432573	+	0.901599i	$0.357606\pi$
$912$	0	0
$913$	−9.67996e6	−0.384323
$914$	2.48130e7	0.982458
$915$	0	0
$916$	3.69769e6	0.145610
$917$	8.28745e6	0.325460
$918$	0	0
$919$	4.32037e7	1.68746	0.843728	−	0.536771i	$-0.180356\pi$
$919$	4.32037e7	1.68746	0.843728	+	0.536771i	$0.180356\pi$
$920$	−353688.	−0.0137769
$921$	0	0
$922$	−3.11545e7	−1.20696
$923$	−3.03006e6	−0.117070
$924$	0	0
$925$	−2.53779e6	−0.0975217
$926$	2.58996e7	0.992580
$927$	0	0
$928$	4.54827e7	1.73371
$929$	−3.70288e7	−1.40767	−0.703834	−	0.710364i	$-0.748530\pi$
$929$	−3.70288e7	−1.40767	−0.703834	+	0.710364i	$0.748530\pi$
$930$	0	0
$931$	−5.34505e6	−0.202105
$932$	1.50759e7	0.568518
$933$	0	0
$934$	1.97080e7	0.739224
$935$	−606038.	−0.0226710
$936$	0	0
$937$	1.22598e7	0.456180	0.228090	−	0.973640i	$-0.426752\pi$
$937$	1.22598e7	0.456180	0.228090	+	0.973640i	$0.426752\pi$
$938$	1.22385e7	0.454173
$939$	0	0
$940$	−468215.	−0.0172833
$941$	1.25646e7	0.462565	0.231283	−	0.972887i	$-0.425708\pi$
$941$	1.25646e7	0.462565	0.231283	+	0.972887i	$0.425708\pi$
$942$	0	0
$943$	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	$0.694116\pi$
$948$	0	0
$949$	8.77237e6	0.316192
$950$	1.76894e6	0.0635924
$951$	0	0
$952$	−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	$0.330614\pi$
$954$	0	0
$955$	1.24199e7	0.440665
$956$	−4.20419e7	−1.48778
$957$	0	0
$958$	−3.79038e7	−1.33435
$959$	770395.	0.0270500
$960$	0	0
$961$	−2.26225e7	−0.790190
$962$	7.24174e6	0.252293
$963$	0	0
$964$	4.46905e6	0.154890
$965$	6.78405e6	0.234515
$966$	0	0
$967$	−4.73439e7	−1.62816	−0.814081	−	0.580751i	$-0.802759\pi$
$967$	−4.73439e7	−1.62816	−0.814081	+	0.580751i	$0.802759\pi$
$968$	149545.	0.00512961
$969$	0	0
$970$	−1.53824e6	−0.0524923
$971$	4.33248e7	1.47465	0.737324	−	0.675539i	$-0.236089\pi$
$971$	4.33248e7	1.47465	0.737324	+	0.675539i	$0.236089\pi$
$972$	0	0
$973$	−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	$0.215866\pi$
$978$	0	0
$979$	−4.57174e6	−0.152449
$980$	−1.26875e7	−0.421997
$981$	0	0
$982$	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	$0.445923\pi$
$984$	0	0
$985$	4.77430e6	0.156790
$986$	−8.91250e6	−0.291949
$987$	0	0
$988$	−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
$993$	0	0
$994$	−4.36608e6	−0.140161
$995$	1.50659e7	0.482434
$996$	0	0
$997$	1.08535e7	0.345804	0.172902	−	0.984939i	$-0.444686\pi$
$997$	1.08535e7	0.345804	0.172902	+	0.984939i	$0.444686\pi$
$998$	9.94904e6	0.316195
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.b.1.1	✓	3	3.2	odd	2
495.6.a.d.1.3		3	1.1	even	1	trivial
825.6.a.i.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+8.07863 q^{2} +33.2643 q^{4} +25.0000 q^{5} -39.3760 q^{7} +10.2141 q^{8} +O(q^{10})\)	\(q+8.07863 q^{2} +33.2643 q^{4} +25.0000 q^{5} -39.3760 q^{7} +10.2141 q^{8} +201.966 q^{10} -121.000 q^{11} -220.765 q^{13} -318.105 q^{14} -981.943 q^{16} +200.343 q^{17} +350.345 q^{19} +831.608 q^{20} -977.515 q^{22} -1385.09 q^{23} +625.000 q^{25} -1783.48 q^{26} -1309.82 q^{28} -5506.64 q^{29} -2450.86 q^{31} -8259.61 q^{32} +1618.50 q^{34} -984.401 q^{35} -4060.46 q^{37} +2830.31 q^{38} +255.354 q^{40} -527.283 q^{41} -12078.9 q^{43} -4024.99 q^{44} -11189.6 q^{46} -563.023 q^{47} -15256.5 q^{49} +5049.15 q^{50} -7343.59 q^{52} +37203.0 q^{53} -3025.00 q^{55} -402.192 q^{56} -44486.2 q^{58} -2157.64 q^{59} -39938.0 q^{61} -19799.6 q^{62} -35304.2 q^{64} -5519.11 q^{65} -38473.2 q^{67} +6664.29 q^{68} -7952.62 q^{70} +13725.3 q^{71} -39736.3 q^{73} -32803.0 q^{74} +11654.0 q^{76} +4764.50 q^{77} +35672.9 q^{79} -24548.6 q^{80} -4259.73 q^{82} +79999.7 q^{83} +5008.58 q^{85} -97580.6 q^{86} -1235.91 q^{88} +37783.0 q^{89} +8692.83 q^{91} -46074.1 q^{92} -4548.46 q^{94} +8758.63 q^{95} -7616.35 q^{97} -123252. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8	\( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 28 * q^4 + 75 * q^5 - 232 * q^7 - 24 * q^8 + 50 * q^10 - 363 * q^11 + 450 * q^13 + 1504 * q^14 - 1360 * q^16 + 334 * q^17 - 4036 * q^19 + 700 * q^20 - 242 * q^22 + 7060 * q^23 + 1875 * q^25 - 2932 * q^26 - 8320 * q^28 - 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3644 * q^34 - 5800 * q^35 + 2250 * q^37 + 12632 * q^38 - 600 * q^40 - 10654 * q^41 - 35528 * q^43 - 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 7667 * q^49 + 1250 * q^50 - 14520 * q^52 + 12826 * q^53 - 9075 * q^55 - 17088 * q^56 - 17196 * q^58 + 81876 * q^59 - 62298 * q^61 - 109184 * q^62 - 72256 * q^64 + 11250 * q^65 - 46148 * q^67 + 35832 * q^68 + 37600 * q^70 + 64724 * q^71 + 810 * q^73 + 44796 * q^74 + 44656 * q^76 + 28072 * q^77 + 43876 * q^79 - 34000 * q^80 + 56060 * q^82 + 101024 * q^83 + 8350 * q^85 - 24128 * q^86 + 2904 * q^88 - 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 74552 * q^94 - 100900 * q^95 - 319746 * q^97 - 431134 * q^98

Introduction

L-functions

Modular forms

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Database

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