LMFDB - Embedded newform 3330.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.46410 q^{11} +1.46410 q^{13} -1.26795 q^{14} +1.00000 q^{16} +1.46410 q^{17} -4.19615 q^{19} -1.00000 q^{20} -1.46410 q^{22} +8.00000 q^{23} +1.00000 q^{25} +1.46410 q^{26} -1.26795 q^{28} +8.92820 q^{29} -2.73205 q^{31} +1.00000 q^{32} +1.46410 q^{34} +1.26795 q^{35} +1.00000 q^{37} -4.19615 q^{38} -1.00000 q^{40} +2.00000 q^{41} -6.92820 q^{43} -1.46410 q^{44} +8.00000 q^{46} +1.26795 q^{47} -5.39230 q^{49} +1.00000 q^{50} +1.46410 q^{52} +6.00000 q^{53} +1.46410 q^{55} -1.26795 q^{56} +8.92820 q^{58} -0.196152 q^{59} +8.92820 q^{61} -2.73205 q^{62} +1.00000 q^{64} -1.46410 q^{65} +13.6603 q^{67} +1.46410 q^{68} +1.26795 q^{70} +10.9282 q^{71} +12.9282 q^{73} +1.00000 q^{74} -4.19615 q^{76} +1.85641 q^{77} +5.26795 q^{79} -1.00000 q^{80} +2.00000 q^{82} +5.26795 q^{83} -1.46410 q^{85} -6.92820 q^{86} -1.46410 q^{88} +2.00000 q^{89} -1.85641 q^{91} +8.00000 q^{92} +1.26795 q^{94} +4.19615 q^{95} -2.00000 q^{97} -5.39230 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 2 q^{25} - 4 q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} + 2 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 4 q^{55} - 6 q^{56} + 4 q^{58} + 10 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 4 q^{68} + 6 q^{70} + 8 q^{71} + 12 q^{73} + 2 q^{74} + 2 q^{76} - 24 q^{77} + 14 q^{79} - 2 q^{80} + 4 q^{82} + 14 q^{83} + 4 q^{85} + 4 q^{88} + 4 q^{89} + 24 q^{91} + 16 q^{92} + 6 q^{94} - 2 q^{95} - 4 q^{97} + 10 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 - 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^19 - 2 * q^20 + 4 * q^22 + 16 * q^23 + 2 * q^25 - 4 * q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^35 + 2 * q^37 + 2 * q^38 - 2 * q^40 + 4 * q^41 + 4 * q^44 + 16 * q^46 + 6 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 + 12 * q^53 - 4 * q^55 - 6 * q^56 + 4 * q^58 + 10 * q^59 + 4 * q^61 - 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 4 * q^68 + 6 * q^70 + 8 * q^71 + 12 * q^73 + 2 * q^74 + 2 * q^76 - 24 * q^77 + 14 * q^79 - 2 * q^80 + 4 * q^82 + 14 * q^83 + 4 * q^85 + 4 * q^88 + 4 * q^89 + 24 * q^91 + 16 * q^92 + 6 * q^94 - 2 * q^95 - 4 * q^97 + 10 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	−1.26795	−0.338874
$15$	0	0
$16$	1.00000	0.250000
$17$	1.46410	0.355097	0.177548	−	0.984112i	$-0.443183\pi$
$17$	1.46410	0.355097	0.177548	+	0.984112i	$0.443183\pi$
$18$	0	0
$19$	−4.19615	−0.962663	−0.481332	−	0.876539i	$-0.659847\pi$
$19$	−4.19615	−0.962663	−0.481332	+	0.876539i	$0.659847\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.46410	−0.312148
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.46410	0.287134
$27$	0	0
$28$	−1.26795	−0.239620
$29$	8.92820	1.65793	0.828963	−	0.559304i	$-0.188931\pi$
$29$	8.92820	1.65793	0.828963	+	0.559304i	$0.188931\pi$
$30$	0	0
$31$	−2.73205	−0.490691	−0.245345	−	0.969436i	$-0.578901\pi$
$31$	−2.73205	−0.490691	−0.245345	+	0.969436i	$0.578901\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.46410	0.251091
$35$	1.26795	0.214323
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−4.19615	−0.680706
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	−1.46410	−0.220722
$45$	0	0
$46$	8.00000	1.17954
$47$	1.26795	0.184949	0.0924747	−	0.995715i	$-0.470522\pi$
$47$	1.26795	0.184949	0.0924747	+	0.995715i	$0.470522\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	1.00000	0.141421
$51$	0	0
$52$	1.46410	0.203034
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	1.46410	0.197419
$56$	−1.26795	−0.169437
$57$	0	0
$58$	8.92820	1.17233
$59$	−0.196152	−0.0255369	−0.0127684	−	0.999918i	$-0.504064\pi$
$59$	−0.196152	−0.0255369	−0.0127684	+	0.999918i	$0.504064\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	−2.73205	−0.346971
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.46410	−0.181599
$66$	0	0
$67$	13.6603	1.66887	0.834433	−	0.551110i	$-0.185795\pi$
$67$	13.6603	1.66887	0.834433	+	0.551110i	$0.185795\pi$
$68$	1.46410	0.177548
$69$	0	0
$70$	1.26795	0.151549
$71$	10.9282	1.29694	0.648470	−	0.761241i	$-0.275409\pi$
$71$	10.9282	1.29694	0.648470	+	0.761241i	$0.275409\pi$
$72$	0	0
$73$	12.9282	1.51313	0.756566	−	0.653917i	$-0.226876\pi$
$73$	12.9282	1.51313	0.756566	+	0.653917i	$0.226876\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−4.19615	−0.481332
$77$	1.85641	0.211557
$78$	0	0
$79$	5.26795	0.592691	0.296345	−	0.955081i	$-0.404232\pi$
$79$	5.26795	0.592691	0.296345	+	0.955081i	$0.404232\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	2.00000	0.220863
$83$	5.26795	0.578233	0.289116	−	0.957294i	$-0.406639\pi$
$83$	5.26795	0.578233	0.289116	+	0.957294i	$0.406639\pi$
$84$	0	0
$85$	−1.46410	−0.158804
$86$	−6.92820	−0.747087
$87$	0	0
$88$	−1.46410	−0.156074
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−1.85641	−0.194604
$92$	8.00000	0.834058
$93$	0	0
$94$	1.26795	0.130779
$95$	4.19615	0.430516
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−5.39230	−0.544705
$99$	0	0
$100$	1.00000	0.100000
$101$	2.53590	0.252331	0.126166	−	0.992009i	$-0.459733\pi$
$101$	2.53590	0.252331	0.126166	+	0.992009i	$0.459733\pi$
$102$	0	0
$103$	−13.4641	−1.32666	−0.663329	−	0.748328i	$-0.730857\pi$
$103$	−13.4641	−1.32666	−0.663329	+	0.748328i	$0.730857\pi$
$104$	1.46410	0.143567
$105$	0	0
$106$	6.00000	0.582772
$107$	−6.73205	−0.650812	−0.325406	−	0.945574i	$-0.605501\pi$
$107$	−6.73205	−0.650812	−0.325406	+	0.945574i	$0.605501\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	1.46410	0.139597
$111$	0	0
$112$	−1.26795	−0.119810
$113$	17.4641	1.64288	0.821442	−	0.570292i	$-0.193170\pi$
$113$	17.4641	1.64288	0.821442	+	0.570292i	$0.193170\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	8.92820	0.828963
$117$	0	0
$118$	−0.196152	−0.0180573
$119$	−1.85641	−0.170177
$120$	0	0
$121$	−8.85641	−0.805128
$122$	8.92820	0.808322
$123$	0	0
$124$	−2.73205	−0.245345
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.6603	−1.21215	−0.606076	−	0.795407i	$-0.707257\pi$
$127$	−13.6603	−1.21215	−0.606076	+	0.795407i	$0.707257\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.46410	−0.128410
$131$	−12.5885	−1.09986	−0.549929	−	0.835211i	$-0.685345\pi$
$131$	−12.5885	−1.09986	−0.549929	+	0.835211i	$0.685345\pi$
$132$	0	0
$133$	5.32051	0.461347
$134$	13.6603	1.18007
$135$	0	0
$136$	1.46410	0.125546
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	1.26795	0.107161
$141$	0	0
$142$	10.9282	0.917074
$143$	−2.14359	−0.179256
$144$	0	0
$145$	−8.92820	−0.741447
$146$	12.9282	1.06995
$147$	0	0
$148$	1.00000	0.0821995
$149$	16.3923	1.34291	0.671455	−	0.741045i	$-0.265670\pi$
$149$	16.3923	1.34291	0.671455	+	0.741045i	$0.265670\pi$
$150$	0	0
$151$	8.39230	0.682956	0.341478	−	0.939890i	$-0.389073\pi$
$151$	8.39230	0.682956	0.341478	+	0.939890i	$0.389073\pi$
$152$	−4.19615	−0.340353
$153$	0	0
$154$	1.85641	0.149593
$155$	2.73205	0.219444
$156$	0	0
$157$	16.9282	1.35102	0.675509	−	0.737352i	$-0.263924\pi$
$157$	16.9282	1.35102	0.675509	+	0.737352i	$0.263924\pi$
$158$	5.26795	0.419096
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−10.1436	−0.799427
$162$	0	0
$163$	23.3205	1.82660	0.913302	−	0.407284i	$-0.133524\pi$
$163$	23.3205	1.82660	0.913302	+	0.407284i	$0.133524\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	5.26795	0.408872
$167$	−5.46410	−0.422825	−0.211412	−	0.977397i	$-0.567806\pi$
$167$	−5.46410	−0.422825	−0.211412	+	0.977397i	$0.567806\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	−1.46410	−0.112291
$171$	0	0
$172$	−6.92820	−0.528271
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	−1.26795	−0.0958479
$176$	−1.46410	−0.110361
$177$	0	0
$178$	2.00000	0.149906
$179$	17.6603	1.31999	0.659995	−	0.751270i	$-0.270559\pi$
$179$	17.6603	1.31999	0.659995	+	0.751270i	$0.270559\pi$
$180$	0	0
$181$	−1.46410	−0.108826	−0.0544129	−	0.998519i	$-0.517329\pi$
$181$	−1.46410	−0.108826	−0.0544129	+	0.998519i	$0.517329\pi$
$182$	−1.85641	−0.137606
$183$	0	0
$184$	8.00000	0.589768
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−2.14359	−0.156755
$188$	1.26795	0.0924747
$189$	0	0
$190$	4.19615	0.304421
$191$	5.26795	0.381175	0.190588	−	0.981670i	$-0.438961\pi$
$191$	5.26795	0.381175	0.190588	+	0.981670i	$0.438961\pi$
$192$	0	0
$193$	−11.8564	−0.853443	−0.426721	−	0.904383i	$-0.640332\pi$
$193$	−11.8564	−0.853443	−0.426721	+	0.904383i	$0.640332\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−5.39230	−0.385165
$197$	−18.7846	−1.33835	−0.669174	−	0.743106i	$-0.733352\pi$
$197$	−18.7846	−1.33835	−0.669174	+	0.743106i	$0.733352\pi$
$198$	0	0
$199$	26.0526	1.84682	0.923408	−	0.383819i	$-0.125391\pi$
$199$	26.0526	1.84682	0.923408	+	0.383819i	$0.125391\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	2.53590	0.178425
$203$	−11.3205	−0.794544
$204$	0	0
$205$	−2.00000	−0.139686
$206$	−13.4641	−0.938088
$207$	0	0
$208$	1.46410	0.101517
$209$	6.14359	0.424961
$210$	0	0
$211$	9.85641	0.678543	0.339272	−	0.940688i	$-0.389819\pi$
$211$	9.85641	0.678543	0.339272	+	0.940688i	$0.389819\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−6.73205	−0.460194
$215$	6.92820	0.472500
$216$	0	0
$217$	3.46410	0.235159
$218$	2.00000	0.135457
$219$	0	0
$220$	1.46410	0.0987097
$221$	2.14359	0.144194
$222$	0	0
$223$	−22.0526	−1.47675	−0.738374	−	0.674391i	$-0.764406\pi$
$223$	−22.0526	−1.47675	−0.738374	+	0.674391i	$0.764406\pi$
$224$	−1.26795	−0.0847184
$225$	0	0
$226$	17.4641	1.16169
$227$	−3.60770	−0.239451	−0.119726	−	0.992807i	$-0.538201\pi$
$227$	−3.60770	−0.239451	−0.119726	+	0.992807i	$0.538201\pi$
$228$	0	0
$229$	15.8564	1.04782	0.523910	−	0.851773i	$-0.324473\pi$
$229$	15.8564	1.04782	0.523910	+	0.851773i	$0.324473\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	8.92820	0.586165
$233$	−15.0718	−0.987386	−0.493693	−	0.869636i	$-0.664353\pi$
$233$	−15.0718	−0.987386	−0.493693	+	0.869636i	$0.664353\pi$
$234$	0	0
$235$	−1.26795	−0.0827119
$236$	−0.196152	−0.0127684
$237$	0	0
$238$	−1.85641	−0.120333
$239$	17.2679	1.11697	0.558485	−	0.829514i	$-0.311383\pi$
$239$	17.2679	1.11697	0.558485	+	0.829514i	$0.311383\pi$
$240$	0	0
$241$	−8.92820	−0.575116	−0.287558	−	0.957763i	$-0.592843\pi$
$241$	−8.92820	−0.575116	−0.287558	+	0.957763i	$0.592843\pi$
$242$	−8.85641	−0.569311
$243$	0	0
$244$	8.92820	0.571570
$245$	5.39230	0.344502
$246$	0	0
$247$	−6.14359	−0.390907
$248$	−2.73205	−0.173485
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−22.7321	−1.43483	−0.717417	−	0.696644i	$-0.754676\pi$
$251$	−22.7321	−1.43483	−0.717417	+	0.696644i	$0.754676\pi$
$252$	0	0
$253$	−11.7128	−0.736378
$254$	−13.6603	−0.857121
$255$	0	0
$256$	1.00000	0.0625000
$257$	25.4641	1.58841	0.794204	−	0.607652i	$-0.207888\pi$
$257$	25.4641	1.58841	0.794204	+	0.607652i	$0.207888\pi$
$258$	0	0
$259$	−1.26795	−0.0787865
$260$	−1.46410	−0.0907997
$261$	0	0
$262$	−12.5885	−0.777717
$263$	−30.0526	−1.85312	−0.926560	−	0.376147i	$-0.877249\pi$
$263$	−30.0526	−1.85312	−0.926560	+	0.376147i	$0.877249\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	5.32051	0.326221
$267$	0	0
$268$	13.6603	0.834433
$269$	0.392305	0.0239192	0.0119596	−	0.999928i	$-0.496193\pi$
$269$	0.392305	0.0239192	0.0119596	+	0.999928i	$0.496193\pi$
$270$	0	0
$271$	16.7846	1.01959	0.509796	−	0.860295i	$-0.329721\pi$
$271$	16.7846	1.01959	0.509796	+	0.860295i	$0.329721\pi$
$272$	1.46410	0.0887742
$273$	0	0
$274$	−2.00000	−0.120824
$275$	−1.46410	−0.0882886
$276$	0	0
$277$	−26.2487	−1.57713	−0.788566	−	0.614950i	$-0.789176\pi$
$277$	−26.2487	−1.57713	−0.788566	+	0.614950i	$0.789176\pi$
$278$	−6.92820	−0.415526
$279$	0	0
$280$	1.26795	0.0757745
$281$	−4.92820	−0.293992	−0.146996	−	0.989137i	$-0.546960\pi$
$281$	−4.92820	−0.293992	−0.146996	+	0.989137i	$0.546960\pi$
$282$	0	0
$283$	−4.39230	−0.261095	−0.130548	−	0.991442i	$-0.541674\pi$
$283$	−4.39230	−0.261095	−0.130548	+	0.991442i	$0.541674\pi$
$284$	10.9282	0.648470
$285$	0	0
$286$	−2.14359	−0.126753
$287$	−2.53590	−0.149689
$288$	0	0
$289$	−14.8564	−0.873906
$290$	−8.92820	−0.524282
$291$	0	0
$292$	12.9282	0.756566
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	0.196152	0.0114204
$296$	1.00000	0.0581238
$297$	0	0
$298$	16.3923	0.949581
$299$	11.7128	0.677369
$300$	0	0
$301$	8.78461	0.506336
$302$	8.39230	0.482923
$303$	0	0
$304$	−4.19615	−0.240666
$305$	−8.92820	−0.511227
$306$	0	0
$307$	−12.5885	−0.718461	−0.359231	−	0.933249i	$-0.616961\pi$
$307$	−12.5885	−0.718461	−0.359231	+	0.933249i	$0.616961\pi$
$308$	1.85641	0.105779
$309$	0	0
$310$	2.73205	0.155170
$311$	−27.1244	−1.53808	−0.769041	−	0.639200i	$-0.779266\pi$
$311$	−27.1244	−1.53808	−0.769041	+	0.639200i	$0.779266\pi$
$312$	0	0
$313$	3.85641	0.217977	0.108988	−	0.994043i	$-0.465239\pi$
$313$	3.85641	0.217977	0.108988	+	0.994043i	$0.465239\pi$
$314$	16.9282	0.955314
$315$	0	0
$316$	5.26795	0.296345
$317$	31.8564	1.78923	0.894617	−	0.446834i	$-0.147448\pi$
$317$	31.8564	1.78923	0.894617	+	0.446834i	$0.147448\pi$
$318$	0	0
$319$	−13.0718	−0.731880
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−10.1436	−0.565280
$323$	−6.14359	−0.341839
$324$	0	0
$325$	1.46410	0.0812137
$326$	23.3205	1.29160
$327$	0	0
$328$	2.00000	0.110432
$329$	−1.60770	−0.0886351
$330$	0	0
$331$	−8.87564	−0.487850	−0.243925	−	0.969794i	$-0.578435\pi$
$331$	−8.87564	−0.487850	−0.243925	+	0.969794i	$0.578435\pi$
$332$	5.26795	0.289116
$333$	0	0
$334$	−5.46410	−0.298982
$335$	−13.6603	−0.746339
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−10.8564	−0.590511
$339$	0	0
$340$	−1.46410	−0.0794021
$341$	4.00000	0.216612
$342$	0	0
$343$	15.7128	0.848412
$344$	−6.92820	−0.373544
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−17.0718	−0.916462	−0.458231	−	0.888833i	$-0.651517\pi$
$347$	−17.0718	−0.916462	−0.458231	+	0.888833i	$0.651517\pi$
$348$	0	0
$349$	19.3205	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$349$	19.3205	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$350$	−1.26795	−0.0677747
$351$	0	0
$352$	−1.46410	−0.0780369
$353$	−15.8564	−0.843951	−0.421976	−	0.906607i	$-0.638663\pi$
$353$	−15.8564	−0.843951	−0.421976	+	0.906607i	$0.638663\pi$
$354$	0	0
$355$	−10.9282	−0.580009
$356$	2.00000	0.106000
$357$	0	0
$358$	17.6603	0.933373
$359$	−8.39230	−0.442929	−0.221464	−	0.975168i	$-0.571084\pi$
$359$	−8.39230	−0.442929	−0.221464	+	0.975168i	$0.571084\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	−1.46410	−0.0769515
$363$	0	0
$364$	−1.85641	−0.0973021
$365$	−12.9282	−0.676693
$366$	0	0
$367$	−21.6603	−1.13066	−0.565328	−	0.824866i	$-0.691250\pi$
$367$	−21.6603	−1.13066	−0.565328	+	0.824866i	$0.691250\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	−7.60770	−0.394972
$372$	0	0
$373$	30.7846	1.59397	0.796983	−	0.604001i	$-0.206428\pi$
$373$	30.7846	1.59397	0.796983	+	0.604001i	$0.206428\pi$
$374$	−2.14359	−0.110843
$375$	0	0
$376$	1.26795	0.0653895
$377$	13.0718	0.673232
$378$	0	0
$379$	11.6077	0.596247	0.298124	−	0.954527i	$-0.403639\pi$
$379$	11.6077	0.596247	0.298124	+	0.954527i	$0.403639\pi$
$380$	4.19615	0.215258
$381$	0	0
$382$	5.26795	0.269532
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	−1.85641	−0.0946112
$386$	−11.8564	−0.603475
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−15.8564	−0.803952	−0.401976	−	0.915650i	$-0.631676\pi$
$389$	−15.8564	−0.803952	−0.401976	+	0.915650i	$0.631676\pi$
$390$	0	0
$391$	11.7128	0.592342
$392$	−5.39230	−0.272353
$393$	0	0
$394$	−18.7846	−0.946355
$395$	−5.26795	−0.265059
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	26.0526	1.30590
$399$	0	0
$400$	1.00000	0.0500000
$401$	19.0718	0.952400	0.476200	−	0.879337i	$-0.342014\pi$
$401$	19.0718	0.952400	0.476200	+	0.879337i	$0.342014\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	2.53590	0.126166
$405$	0	0
$406$	−11.3205	−0.561827
$407$	−1.46410	−0.0725728
$408$	0	0
$409$	16.9282	0.837046	0.418523	−	0.908206i	$-0.362548\pi$
$409$	16.9282	0.837046	0.418523	+	0.908206i	$0.362548\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−13.4641	−0.663329
$413$	0.248711	0.0122383
$414$	0	0
$415$	−5.26795	−0.258593
$416$	1.46410	0.0717835
$417$	0	0
$418$	6.14359	0.300493
$419$	34.2487	1.67316	0.836580	−	0.547846i	$-0.184552\pi$
$419$	34.2487	1.67316	0.836580	+	0.547846i	$0.184552\pi$
$420$	0	0
$421$	−27.8564	−1.35764	−0.678819	−	0.734306i	$-0.737508\pi$
$421$	−27.8564	−1.35764	−0.678819	+	0.734306i	$0.737508\pi$
$422$	9.85641	0.479802
$423$	0	0
$424$	6.00000	0.291386
$425$	1.46410	0.0710194
$426$	0	0
$427$	−11.3205	−0.547838
$428$	−6.73205	−0.325406
$429$	0	0
$430$	6.92820	0.334108
$431$	−8.19615	−0.394795	−0.197397	−	0.980324i	$-0.563249\pi$
$431$	−8.19615	−0.394795	−0.197397	+	0.980324i	$0.563249\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	3.46410	0.166282
$435$	0	0
$436$	2.00000	0.0957826
$437$	−33.5692	−1.60583
$438$	0	0
$439$	−31.5167	−1.50421	−0.752104	−	0.659044i	$-0.770961\pi$
$439$	−31.5167	−1.50421	−0.752104	+	0.659044i	$0.770961\pi$
$440$	1.46410	0.0697983
$441$	0	0
$442$	2.14359	0.101960
$443$	39.1244	1.85885	0.929427	−	0.369006i	$-0.120302\pi$
$443$	39.1244	1.85885	0.929427	+	0.369006i	$0.120302\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	−22.0526	−1.04422
$447$	0	0
$448$	−1.26795	−0.0599050
$449$	−33.7128	−1.59101	−0.795503	−	0.605950i	$-0.792793\pi$
$449$	−33.7128	−1.59101	−0.795503	+	0.605950i	$0.792793\pi$
$450$	0	0
$451$	−2.92820	−0.137884
$452$	17.4641	0.821442
$453$	0	0
$454$	−3.60770	−0.169318
$455$	1.85641	0.0870297
$456$	0	0
$457$	−4.14359	−0.193829	−0.0969146	−	0.995293i	$-0.530897\pi$
$457$	−4.14359	−0.193829	−0.0969146	+	0.995293i	$0.530897\pi$
$458$	15.8564	0.740921
$459$	0	0
$460$	−8.00000	−0.373002
$461$	26.7846	1.24748	0.623742	−	0.781630i	$-0.285612\pi$
$461$	26.7846	1.24748	0.623742	+	0.781630i	$0.285612\pi$
$462$	0	0
$463$	5.07180	0.235706	0.117853	−	0.993031i	$-0.462399\pi$
$463$	5.07180	0.235706	0.117853	+	0.993031i	$0.462399\pi$
$464$	8.92820	0.414481
$465$	0	0
$466$	−15.0718	−0.698188
$467$	−37.1769	−1.72034	−0.860171	−	0.510005i	$-0.829643\pi$
$467$	−37.1769	−1.72034	−0.860171	+	0.510005i	$0.829643\pi$
$468$	0	0
$469$	−17.3205	−0.799787
$470$	−1.26795	−0.0584861
$471$	0	0
$472$	−0.196152	−0.00902865
$473$	10.1436	0.466403
$474$	0	0
$475$	−4.19615	−0.192533
$476$	−1.85641	−0.0850883
$477$	0	0
$478$	17.2679	0.789818
$479$	34.0526	1.55590	0.777951	−	0.628325i	$-0.216259\pi$
$479$	34.0526	1.55590	0.777951	+	0.628325i	$0.216259\pi$
$480$	0	0
$481$	1.46410	0.0667573
$482$	−8.92820	−0.406669
$483$	0	0
$484$	−8.85641	−0.402564
$485$	2.00000	0.0908153
$486$	0	0
$487$	−16.3923	−0.742806	−0.371403	−	0.928472i	$-0.621123\pi$
$487$	−16.3923	−0.742806	−0.371403	+	0.928472i	$0.621123\pi$
$488$	8.92820	0.404161
$489$	0	0
$490$	5.39230	0.243600
$491$	1.07180	0.0483695	0.0241848	−	0.999708i	$-0.492301\pi$
$491$	1.07180	0.0483695	0.0241848	+	0.999708i	$0.492301\pi$
$492$	0	0
$493$	13.0718	0.588724
$494$	−6.14359	−0.276413
$495$	0	0
$496$	−2.73205	−0.122673
$497$	−13.8564	−0.621545
$498$	0	0
$499$	40.5885	1.81699	0.908494	−	0.417897i	$-0.137233\pi$
$499$	40.5885	1.81699	0.908494	+	0.417897i	$0.137233\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−22.7321	−1.01458
$503$	5.07180	0.226140	0.113070	−	0.993587i	$-0.463932\pi$
$503$	5.07180	0.226140	0.113070	+	0.993587i	$0.463932\pi$
$504$	0	0
$505$	−2.53590	−0.112846
$506$	−11.7128	−0.520698
$507$	0	0
$508$	−13.6603	−0.606076
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−16.3923	−0.725153
$512$	1.00000	0.0441942
$513$	0	0
$514$	25.4641	1.12317
$515$	13.4641	0.593299
$516$	0	0
$517$	−1.85641	−0.0816447
$518$	−1.26795	−0.0557105
$519$	0	0
$520$	−1.46410	−0.0642051
$521$	−33.4641	−1.46609	−0.733044	−	0.680181i	$-0.761901\pi$
$521$	−33.4641	−1.46609	−0.733044	+	0.680181i	$0.761901\pi$
$522$	0	0
$523$	−4.78461	−0.209216	−0.104608	−	0.994514i	$-0.533359\pi$
$523$	−4.78461	−0.209216	−0.104608	+	0.994514i	$0.533359\pi$
$524$	−12.5885	−0.549929
$525$	0	0
$526$	−30.0526	−1.31035
$527$	−4.00000	−0.174243
$528$	0	0
$529$	41.0000	1.78261
$530$	−6.00000	−0.260623
$531$	0	0
$532$	5.32051	0.230673
$533$	2.92820	0.126835
$534$	0	0
$535$	6.73205	0.291052
$536$	13.6603	0.590033
$537$	0	0
$538$	0.392305	0.0169135
$539$	7.89488	0.340057
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	16.7846	0.720961
$543$	0	0
$544$	1.46410	0.0627728
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−2.00000	−0.0854358
$549$	0	0
$550$	−1.46410	−0.0624295
$551$	−37.4641	−1.59602
$552$	0	0
$553$	−6.67949	−0.284041
$554$	−26.2487	−1.11520
$555$	0	0
$556$	−6.92820	−0.293821
$557$	32.1051	1.36034	0.680169	−	0.733056i	$-0.261906\pi$
$557$	32.1051	1.36034	0.680169	+	0.733056i	$0.261906\pi$
$558$	0	0
$559$	−10.1436	−0.429028
$560$	1.26795	0.0535806
$561$	0	0
$562$	−4.92820	−0.207884
$563$	−17.0718	−0.719490	−0.359745	−	0.933051i	$-0.617136\pi$
$563$	−17.0718	−0.719490	−0.359745	+	0.933051i	$0.617136\pi$
$564$	0	0
$565$	−17.4641	−0.734720
$566$	−4.39230	−0.184622
$567$	0	0
$568$	10.9282	0.458537
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−2.14359	−0.0896281
$573$	0	0
$574$	−2.53590	−0.105846
$575$	8.00000	0.333623
$576$	0	0
$577$	−3.60770	−0.150190	−0.0750952	−	0.997176i	$-0.523926\pi$
$577$	−3.60770	−0.150190	−0.0750952	+	0.997176i	$0.523926\pi$
$578$	−14.8564	−0.617945
$579$	0	0
$580$	−8.92820	−0.370723
$581$	−6.67949	−0.277112
$582$	0	0
$583$	−8.78461	−0.363821
$584$	12.9282	0.534973
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−3.21539	−0.132713	−0.0663567	−	0.997796i	$-0.521138\pi$
$587$	−3.21539	−0.132713	−0.0663567	+	0.997796i	$0.521138\pi$
$588$	0	0
$589$	11.4641	0.472370
$590$	0.196152	0.00807547
$591$	0	0
$592$	1.00000	0.0410997
$593$	−6.78461	−0.278611	−0.139305	−	0.990249i	$-0.544487\pi$
$593$	−6.78461	−0.278611	−0.139305	+	0.990249i	$0.544487\pi$
$594$	0	0
$595$	1.85641	0.0761052
$596$	16.3923	0.671455
$597$	0	0
$598$	11.7128	0.478973
$599$	−2.53590	−0.103614	−0.0518070	−	0.998657i	$-0.516498\pi$
$599$	−2.53590	−0.103614	−0.0518070	+	0.998657i	$0.516498\pi$
$600$	0	0
$601$	48.3923	1.97396	0.986982	−	0.160833i	$-0.0514180\pi$
$601$	48.3923	1.97396	0.986982	+	0.160833i	$0.0514180\pi$
$602$	8.78461	0.358034
$603$	0	0
$604$	8.39230	0.341478
$605$	8.85641	0.360064
$606$	0	0
$607$	−40.7846	−1.65540	−0.827698	−	0.561174i	$-0.810350\pi$
$607$	−40.7846	−1.65540	−0.827698	+	0.561174i	$0.810350\pi$
$608$	−4.19615	−0.170176
$609$	0	0
$610$	−8.92820	−0.361492
$611$	1.85641	0.0751022
$612$	0	0
$613$	3.07180	0.124069	0.0620344	−	0.998074i	$-0.480241\pi$
$613$	3.07180	0.124069	0.0620344	+	0.998074i	$0.480241\pi$
$614$	−12.5885	−0.508029
$615$	0	0
$616$	1.85641	0.0747967
$617$	−12.9282	−0.520470	−0.260235	−	0.965545i	$-0.583800\pi$
$617$	−12.9282	−0.520470	−0.260235	+	0.965545i	$0.583800\pi$
$618$	0	0
$619$	9.85641	0.396162	0.198081	−	0.980186i	$-0.436529\pi$
$619$	9.85641	0.396162	0.198081	+	0.980186i	$0.436529\pi$
$620$	2.73205	0.109722
$621$	0	0
$622$	−27.1244	−1.08759
$623$	−2.53590	−0.101599
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.85641	0.154133
$627$	0	0
$628$	16.9282	0.675509
$629$	1.46410	0.0583776
$630$	0	0
$631$	−20.9808	−0.835231	−0.417615	−	0.908624i	$-0.637134\pi$
$631$	−20.9808	−0.835231	−0.417615	+	0.908624i	$0.637134\pi$
$632$	5.26795	0.209548
$633$	0	0
$634$	31.8564	1.26518
$635$	13.6603	0.542091
$636$	0	0
$637$	−7.89488	−0.312807
$638$	−13.0718	−0.517517
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	13.4641	0.531800	0.265900	−	0.964001i	$-0.414331\pi$
$641$	13.4641	0.531800	0.265900	+	0.964001i	$0.414331\pi$
$642$	0	0
$643$	41.4641	1.63518	0.817592	−	0.575798i	$-0.195308\pi$
$643$	41.4641	1.63518	0.817592	+	0.575798i	$0.195308\pi$
$644$	−10.1436	−0.399714
$645$	0	0
$646$	−6.14359	−0.241716
$647$	−19.7128	−0.774991	−0.387495	−	0.921872i	$-0.626660\pi$
$647$	−19.7128	−0.774991	−0.387495	+	0.921872i	$0.626660\pi$
$648$	0	0
$649$	0.287187	0.0112731
$650$	1.46410	0.0574268
$651$	0	0
$652$	23.3205	0.913302
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	12.5885	0.491872
$656$	2.00000	0.0780869
$657$	0	0
$658$	−1.60770	−0.0626745
$659$	−6.92820	−0.269884	−0.134942	−	0.990853i	$-0.543085\pi$
$659$	−6.92820	−0.269884	−0.134942	+	0.990853i	$0.543085\pi$
$660$	0	0
$661$	−44.9282	−1.74750	−0.873752	−	0.486371i	$-0.838320\pi$
$661$	−44.9282	−1.74750	−0.873752	+	0.486371i	$0.838320\pi$
$662$	−8.87564	−0.344962
$663$	0	0
$664$	5.26795	0.204436
$665$	−5.32051	−0.206320
$666$	0	0
$667$	71.4256	2.76561
$668$	−5.46410	−0.211412
$669$	0	0
$670$	−13.6603	−0.527742
$671$	−13.0718	−0.504631
$672$	0	0
$673$	−19.0718	−0.735164	−0.367582	−	0.929991i	$-0.619814\pi$
$673$	−19.0718	−0.735164	−0.367582	+	0.929991i	$0.619814\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−10.8564	−0.417554
$677$	31.8564	1.22434	0.612171	−	0.790726i	$-0.290297\pi$
$677$	31.8564	1.22434	0.612171	+	0.790726i	$0.290297\pi$
$678$	0	0
$679$	2.53590	0.0973188
$680$	−1.46410	−0.0561457
$681$	0	0
$682$	4.00000	0.153168
$683$	−36.7846	−1.40752	−0.703762	−	0.710436i	$-0.748498\pi$
$683$	−36.7846	−1.40752	−0.703762	+	0.710436i	$0.748498\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	15.7128	0.599918
$687$	0	0
$688$	−6.92820	−0.264135
$689$	8.78461	0.334667
$690$	0	0
$691$	20.3923	0.775760	0.387880	−	0.921710i	$-0.373208\pi$
$691$	20.3923	0.775760	0.387880	+	0.921710i	$0.373208\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	−17.0718	−0.648037
$695$	6.92820	0.262802
$696$	0	0
$697$	2.92820	0.110914
$698$	19.3205	0.731292
$699$	0	0
$700$	−1.26795	−0.0479240
$701$	−15.8564	−0.598888	−0.299444	−	0.954114i	$-0.596801\pi$
$701$	−15.8564	−0.598888	−0.299444	+	0.954114i	$0.596801\pi$
$702$	0	0
$703$	−4.19615	−0.158261
$704$	−1.46410	−0.0551804
$705$	0	0
$706$	−15.8564	−0.596764
$707$	−3.21539	−0.120927
$708$	0	0
$709$	16.1436	0.606285	0.303143	−	0.952945i	$-0.401964\pi$
$709$	16.1436	0.606285	0.303143	+	0.952945i	$0.401964\pi$
$710$	−10.9282	−0.410128
$711$	0	0
$712$	2.00000	0.0749532
$713$	−21.8564	−0.818529
$714$	0	0
$715$	2.14359	0.0801659
$716$	17.6603	0.659995
$717$	0	0
$718$	−8.39230	−0.313198
$719$	−8.39230	−0.312980	−0.156490	−	0.987680i	$-0.550018\pi$
$719$	−8.39230	−0.312980	−0.156490	+	0.987680i	$0.550018\pi$
$720$	0	0
$721$	17.0718	0.635787
$722$	−1.39230	−0.0518162
$723$	0	0
$724$	−1.46410	−0.0544129
$725$	8.92820	0.331585
$726$	0	0
$727$	−32.7846	−1.21591	−0.607957	−	0.793970i	$-0.708011\pi$
$727$	−32.7846	−1.21591	−0.607957	+	0.793970i	$0.708011\pi$
$728$	−1.85641	−0.0688030
$729$	0	0
$730$	−12.9282	−0.478494
$731$	−10.1436	−0.375174
$732$	0	0
$733$	0.143594	0.00530375	0.00265187	−	0.999996i	$-0.499156\pi$
$733$	0.143594	0.00530375	0.00265187	+	0.999996i	$0.499156\pi$
$734$	−21.6603	−0.799495
$735$	0	0
$736$	8.00000	0.294884
$737$	−20.0000	−0.736709
$738$	0	0
$739$	−6.92820	−0.254858	−0.127429	−	0.991848i	$-0.540673\pi$
$739$	−6.92820	−0.254858	−0.127429	+	0.991848i	$0.540673\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−7.60770	−0.279287
$743$	7.12436	0.261367	0.130684	−	0.991424i	$-0.458283\pi$
$743$	7.12436	0.261367	0.130684	+	0.991424i	$0.458283\pi$
$744$	0	0
$745$	−16.3923	−0.600568
$746$	30.7846	1.12710
$747$	0	0
$748$	−2.14359	−0.0783775
$749$	8.53590	0.311895
$750$	0	0
$751$	0.392305	0.0143154	0.00715770	−	0.999974i	$-0.497722\pi$
$751$	0.392305	0.0143154	0.00715770	+	0.999974i	$0.497722\pi$
$752$	1.26795	0.0462373
$753$	0	0
$754$	13.0718	0.476047
$755$	−8.39230	−0.305427
$756$	0	0
$757$	−53.7128	−1.95223	−0.976113	−	0.217265i	$-0.930286\pi$
$757$	−53.7128	−1.95223	−0.976113	+	0.217265i	$0.930286\pi$
$758$	11.6077	0.421610
$759$	0	0
$760$	4.19615	0.152210
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	−2.53590	−0.0918057
$764$	5.26795	0.190588
$765$	0	0
$766$	8.00000	0.289052
$767$	−0.287187	−0.0103697
$768$	0	0
$769$	20.9282	0.754690	0.377345	−	0.926073i	$-0.376837\pi$
$769$	20.9282	0.754690	0.377345	+	0.926073i	$0.376837\pi$
$770$	−1.85641	−0.0669002
$771$	0	0
$772$	−11.8564	−0.426721
$773$	38.7846	1.39499	0.697493	−	0.716592i	$-0.254299\pi$
$773$	38.7846	1.39499	0.697493	+	0.716592i	$0.254299\pi$
$774$	0	0
$775$	−2.73205	−0.0981382
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−15.8564	−0.568480
$779$	−8.39230	−0.300686
$780$	0	0
$781$	−16.0000	−0.572525
$782$	11.7128	0.418849
$783$	0	0
$784$	−5.39230	−0.192582
$785$	−16.9282	−0.604193
$786$	0	0
$787$	11.8038	0.420762	0.210381	−	0.977620i	$-0.432530\pi$
$787$	11.8038	0.420762	0.210381	+	0.977620i	$0.432530\pi$
$788$	−18.7846	−0.669174
$789$	0	0
$790$	−5.26795	−0.187425
$791$	−22.1436	−0.787336
$792$	0	0
$793$	13.0718	0.464193
$794$	−22.0000	−0.780751
$795$	0	0
$796$	26.0526	0.923408
$797$	−17.4641	−0.618610	−0.309305	−	0.950963i	$-0.600096\pi$
$797$	−17.4641	−0.618610	−0.309305	+	0.950963i	$0.600096\pi$
$798$	0	0
$799$	1.85641	0.0656749
$800$	1.00000	0.0353553
$801$	0	0
$802$	19.0718	0.673449
$803$	−18.9282	−0.667962
$804$	0	0
$805$	10.1436	0.357515
$806$	−4.00000	−0.140894
$807$	0	0
$808$	2.53590	0.0892126
$809$	39.5692	1.39118	0.695590	−	0.718439i	$-0.255143\pi$
$809$	39.5692	1.39118	0.695590	+	0.718439i	$0.255143\pi$
$810$	0	0
$811$	−14.1436	−0.496649	−0.248324	−	0.968677i	$-0.579880\pi$
$811$	−14.1436	−0.496649	−0.248324	+	0.968677i	$0.579880\pi$
$812$	−11.3205	−0.397272
$813$	0	0
$814$	−1.46410	−0.0513167
$815$	−23.3205	−0.816882
$816$	0	0
$817$	29.0718	1.01709
$818$	16.9282	0.591881
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−10.4449	−0.364085	−0.182043	−	0.983291i	$-0.558271\pi$
$823$	−10.4449	−0.364085	−0.182043	+	0.983291i	$0.558271\pi$
$824$	−13.4641	−0.469044
$825$	0	0
$826$	0.248711	0.00865377
$827$	−4.39230	−0.152735	−0.0763677	−	0.997080i	$-0.524332\pi$
$827$	−4.39230	−0.152735	−0.0763677	+	0.997080i	$0.524332\pi$
$828$	0	0
$829$	31.5692	1.09644	0.548222	−	0.836333i	$-0.315305\pi$
$829$	31.5692	1.09644	0.548222	+	0.836333i	$0.315305\pi$
$830$	−5.26795	−0.182853
$831$	0	0
$832$	1.46410	0.0507586
$833$	−7.89488	−0.273541
$834$	0	0
$835$	5.46410	0.189093
$836$	6.14359	0.212481
$837$	0	0
$838$	34.2487	1.18310
$839$	−32.7846	−1.13185	−0.565925	−	0.824457i	$-0.691481\pi$
$839$	−32.7846	−1.13185	−0.565925	+	0.824457i	$0.691481\pi$
$840$	0	0
$841$	50.7128	1.74872
$842$	−27.8564	−0.959995
$843$	0	0
$844$	9.85641	0.339272
$845$	10.8564	0.373472
$846$	0	0
$847$	11.2295	0.385849
$848$	6.00000	0.206041
$849$	0	0
$850$	1.46410	0.0502183
$851$	8.00000	0.274236
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	−11.3205	−0.387380
$855$	0	0
$856$	−6.73205	−0.230097
$857$	4.14359	0.141542	0.0707712	−	0.997493i	$-0.477454\pi$
$857$	4.14359	0.141542	0.0707712	+	0.997493i	$0.477454\pi$
$858$	0	0
$859$	−14.4449	−0.492852	−0.246426	−	0.969162i	$-0.579256\pi$
$859$	−14.4449	−0.492852	−0.246426	+	0.969162i	$0.579256\pi$
$860$	6.92820	0.236250
$861$	0	0
$862$	−8.19615	−0.279162
$863$	−38.4449	−1.30868	−0.654339	−	0.756201i	$-0.727053\pi$
$863$	−38.4449	−1.30868	−0.654339	+	0.756201i	$0.727053\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	−34.0000	−1.15537
$867$	0	0
$868$	3.46410	0.117579
$869$	−7.71281	−0.261639
$870$	0	0
$871$	20.0000	0.677674
$872$	2.00000	0.0677285
$873$	0	0
$874$	−33.5692	−1.13550
$875$	1.26795	0.0428645
$876$	0	0
$877$	−50.7846	−1.71487	−0.857437	−	0.514589i	$-0.827945\pi$
$877$	−50.7846	−1.71487	−0.857437	+	0.514589i	$0.827945\pi$
$878$	−31.5167	−1.06364
$879$	0	0
$880$	1.46410	0.0493549
$881$	−47.3205	−1.59427	−0.797134	−	0.603802i	$-0.793652\pi$
$881$	−47.3205	−1.59427	−0.797134	+	0.603802i	$0.793652\pi$
$882$	0	0
$883$	−34.2487	−1.15256	−0.576280	−	0.817252i	$-0.695496\pi$
$883$	−34.2487	−1.15256	−0.576280	+	0.817252i	$0.695496\pi$
$884$	2.14359	0.0720969
$885$	0	0
$886$	39.1244	1.31441
$887$	−20.1962	−0.678120	−0.339060	−	0.940765i	$-0.610109\pi$
$887$	−20.1962	−0.678120	−0.339060	+	0.940765i	$0.610109\pi$
$888$	0	0
$889$	17.3205	0.580911
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	−22.0526	−0.738374
$893$	−5.32051	−0.178044
$894$	0	0
$895$	−17.6603	−0.590317
$896$	−1.26795	−0.0423592
$897$	0	0
$898$	−33.7128	−1.12501
$899$	−24.3923	−0.813529
$900$	0	0
$901$	8.78461	0.292658
$902$	−2.92820	−0.0974985
$903$	0	0
$904$	17.4641	0.580847
$905$	1.46410	0.0486684
$906$	0	0
$907$	5.75129	0.190968	0.0954842	−	0.995431i	$-0.469560\pi$
$907$	5.75129	0.190968	0.0954842	+	0.995431i	$0.469560\pi$
$908$	−3.60770	−0.119726
$909$	0	0
$910$	1.85641	0.0615393
$911$	27.9090	0.924665	0.462333	−	0.886707i	$-0.347013\pi$
$911$	27.9090	0.924665	0.462333	+	0.886707i	$0.347013\pi$
$912$	0	0
$913$	−7.71281	−0.255257
$914$	−4.14359	−0.137058
$915$	0	0
$916$	15.8564	0.523910
$917$	15.9615	0.527096
$918$	0	0
$919$	−13.2679	−0.437669	−0.218835	−	0.975762i	$-0.570226\pi$
$919$	−13.2679	−0.437669	−0.218835	+	0.975762i	$0.570226\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	26.7846	0.882104
$923$	16.0000	0.526646
$924$	0	0
$925$	1.00000	0.0328798
$926$	5.07180	0.166670
$927$	0	0
$928$	8.92820	0.293083
$929$	15.8564	0.520232	0.260116	−	0.965577i	$-0.416239\pi$
$929$	15.8564	0.520232	0.260116	+	0.965577i	$0.416239\pi$
$930$	0	0
$931$	22.6269	0.741568
$932$	−15.0718	−0.493693
$933$	0	0
$934$	−37.1769	−1.21647
$935$	2.14359	0.0701030
$936$	0	0
$937$	3.85641	0.125983	0.0629917	−	0.998014i	$-0.479936\pi$
$937$	3.85641	0.125983	0.0629917	+	0.998014i	$0.479936\pi$
$938$	−17.3205	−0.565535
$939$	0	0
$940$	−1.26795	−0.0413559
$941$	−28.3923	−0.925563	−0.462781	−	0.886472i	$-0.653148\pi$
$941$	−28.3923	−0.925563	−0.462781	+	0.886472i	$0.653148\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	−0.196152	−0.00638422
$945$	0	0
$946$	10.1436	0.329797
$947$	45.1769	1.46805	0.734026	−	0.679121i	$-0.237639\pi$
$947$	45.1769	1.46805	0.734026	+	0.679121i	$0.237639\pi$
$948$	0	0
$949$	18.9282	0.614435
$950$	−4.19615	−0.136141
$951$	0	0
$952$	−1.85641	−0.0601665
$953$	11.8564	0.384067	0.192033	−	0.981388i	$-0.438492\pi$
$953$	11.8564	0.384067	0.192033	+	0.981388i	$0.438492\pi$
$954$	0	0
$955$	−5.26795	−0.170467
$956$	17.2679	0.558485
$957$	0	0
$958$	34.0526	1.10019
$959$	2.53590	0.0818884
$960$	0	0
$961$	−23.5359	−0.759223
$962$	1.46410	0.0472045
$963$	0	0
$964$	−8.92820	−0.287558
$965$	11.8564	0.381671
$966$	0	0
$967$	−23.6077	−0.759172	−0.379586	−	0.925156i	$-0.623934\pi$
$967$	−23.6077	−0.759172	−0.379586	+	0.925156i	$0.623934\pi$
$968$	−8.85641	−0.284656
$969$	0	0
$970$	2.00000	0.0642161
$971$	−14.9282	−0.479069	−0.239534	−	0.970888i	$-0.576995\pi$
$971$	−14.9282	−0.479069	−0.239534	+	0.970888i	$0.576995\pi$
$972$	0	0
$973$	8.78461	0.281622
$974$	−16.3923	−0.525243
$975$	0	0
$976$	8.92820	0.285785
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	0	0
$979$	−2.92820	−0.0935858
$980$	5.39230	0.172251
$981$	0	0
$982$	1.07180	0.0342024
$983$	38.4449	1.22620	0.613100	−	0.790005i	$-0.289922\pi$
$983$	38.4449	1.22620	0.613100	+	0.790005i	$0.289922\pi$
$984$	0	0
$985$	18.7846	0.598527
$986$	13.0718	0.416291
$987$	0	0
$988$	−6.14359	−0.195454
$989$	−55.4256	−1.76243
$990$	0	0
$991$	5.94744	0.188927	0.0944633	−	0.995528i	$-0.469886\pi$
$991$	5.94744	0.188927	0.0944633	+	0.995528i	$0.469886\pi$
$992$	−2.73205	−0.0867427
$993$	0	0
$994$	−13.8564	−0.439499
$995$	−26.0526	−0.825922
$996$	0	0
$997$	−4.14359	−0.131229	−0.0656145	−	0.997845i	$-0.520901\pi$
$997$	−4.14359	−0.131229	−0.0656145	+	0.997845i	$0.520901\pi$
$998$	40.5885	1.28481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bd.1.2		2
3.2	odd	2		370.2.a.e.1.1	✓	2
12.11	even	2		2960.2.a.q.1.2		2
15.2	even	4		1850.2.b.l.149.2		4
15.8	even	4		1850.2.b.l.149.3		4
15.14	odd	2		1850.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.e.1.1	✓	2	3.2	odd	2
1850.2.a.x.1.2		2	15.14	odd	2
1850.2.b.l.149.2		4	15.2	even	4
1850.2.b.l.149.3		4	15.8	even	4
2960.2.a.q.1.2		2	12.11	even	2
3330.2.a.bd.1.2		2	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.46410 q^{11} +1.46410 q^{13} -1.26795 q^{14} +1.00000 q^{16} +1.46410 q^{17} -4.19615 q^{19} -1.00000 q^{20} -1.46410 q^{22} +8.00000 q^{23} +1.00000 q^{25} +1.46410 q^{26} -1.26795 q^{28} +8.92820 q^{29} -2.73205 q^{31} +1.00000 q^{32} +1.46410 q^{34} +1.26795 q^{35} +1.00000 q^{37} -4.19615 q^{38} -1.00000 q^{40} +2.00000 q^{41} -6.92820 q^{43} -1.46410 q^{44} +8.00000 q^{46} +1.26795 q^{47} -5.39230 q^{49} +1.00000 q^{50} +1.46410 q^{52} +6.00000 q^{53} +1.46410 q^{55} -1.26795 q^{56} +8.92820 q^{58} -0.196152 q^{59} +8.92820 q^{61} -2.73205 q^{62} +1.00000 q^{64} -1.46410 q^{65} +13.6603 q^{67} +1.46410 q^{68} +1.26795 q^{70} +10.9282 q^{71} +12.9282 q^{73} +1.00000 q^{74} -4.19615 q^{76} +1.85641 q^{77} +5.26795 q^{79} -1.00000 q^{80} +2.00000 q^{82} +5.26795 q^{83} -1.46410 q^{85} -6.92820 q^{86} -1.46410 q^{88} +2.00000 q^{89} -1.85641 q^{91} +8.00000 q^{92} +1.26795 q^{94} +4.19615 q^{95} -2.00000 q^{97} -5.39230 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 2 q^{25} - 4 q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} + 2 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 4 q^{55} - 6 q^{56} + 4 q^{58} + 10 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 4 q^{68} + 6 q^{70} + 8 q^{71} + 12 q^{73} + 2 q^{74} + 2 q^{76} - 24 q^{77} + 14 q^{79} - 2 q^{80} + 4 q^{82} + 14 q^{83} + 4 q^{85} + 4 q^{88} + 4 q^{89} + 24 q^{91} + 16 q^{92} + 6 q^{94} - 2 q^{95} - 4 q^{97} + 10 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 - 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^19 - 2 * q^20 + 4 * q^22 + 16 * q^23 + 2 * q^25 - 4 * q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^35 + 2 * q^37 + 2 * q^38 - 2 * q^40 + 4 * q^41 + 4 * q^44 + 16 * q^46 + 6 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 + 12 * q^53 - 4 * q^55 - 6 * q^56 + 4 * q^58 + 10 * q^59 + 4 * q^61 - 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 4 * q^68 + 6 * q^70 + 8 * q^71 + 12 * q^73 + 2 * q^74 + 2 * q^76 - 24 * q^77 + 14 * q^79 - 2 * q^80 + 4 * q^82 + 14 * q^83 + 4 * q^85 + 4 * q^88 + 4 * q^89 + 24 * q^91 + 16 * q^92 + 6 * q^94 - 2 * q^95 - 4 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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