LMFDB - Embedded newform 3330.2.a.bd.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.46410 q^{11} -5.46410 q^{13} -4.73205 q^{14} +1.00000 q^{16} -5.46410 q^{17} +6.19615 q^{19} -1.00000 q^{20} +5.46410 q^{22} +8.00000 q^{23} +1.00000 q^{25} -5.46410 q^{26} -4.73205 q^{28} -4.92820 q^{29} +0.732051 q^{31} +1.00000 q^{32} -5.46410 q^{34} +4.73205 q^{35} +1.00000 q^{37} +6.19615 q^{38} -1.00000 q^{40} +2.00000 q^{41} +6.92820 q^{43} +5.46410 q^{44} +8.00000 q^{46} +4.73205 q^{47} +15.3923 q^{49} +1.00000 q^{50} -5.46410 q^{52} +6.00000 q^{53} -5.46410 q^{55} -4.73205 q^{56} -4.92820 q^{58} +10.1962 q^{59} -4.92820 q^{61} +0.732051 q^{62} +1.00000 q^{64} +5.46410 q^{65} -3.66025 q^{67} -5.46410 q^{68} +4.73205 q^{70} -2.92820 q^{71} -0.928203 q^{73} +1.00000 q^{74} +6.19615 q^{76} -25.8564 q^{77} +8.73205 q^{79} -1.00000 q^{80} +2.00000 q^{82} +8.73205 q^{83} +5.46410 q^{85} +6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +25.8564 q^{91} +8.00000 q^{92} +4.73205 q^{94} -6.19615 q^{95} -2.00000 q^{97} +15.3923 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.73205	−1.78855	−0.894274	−	0.447521i	$-0.852307\pi$
$7$	−4.73205	−1.78855	−0.894274	+	0.447521i	$0.852307\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	−4.73205	−1.26469
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.46410	−1.32524	−0.662620	−	0.748956i	$-0.730555\pi$
$17$	−5.46410	−1.32524	−0.662620	+	0.748956i	$0.730555\pi$
$18$	0	0
$19$	6.19615	1.42149	0.710747	−	0.703447i	$-0.248357\pi$
$19$	6.19615	1.42149	0.710747	+	0.703447i	$0.248357\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	5.46410	1.16495
$23$	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
$26$	−5.46410	−1.07160
$27$	0	0
$28$	−4.73205	−0.894274
$29$	−4.92820	−0.915144	−0.457572	−	0.889172i	$-0.651281\pi$
$29$	−4.92820	−0.915144	−0.457572	+	0.889172i	$0.651281\pi$
$30$	0	0
$31$	0.732051	0.131480	0.0657401	−	0.997837i	$-0.479059\pi$
$31$	0.732051	0.131480	0.0657401	+	0.997837i	$0.479059\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.46410	−0.937086
$35$	4.73205	0.799863
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	6.19615	1.00515
$39$	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	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	5.46410	0.823744
$45$	0	0
$46$	8.00000	1.17954
$47$	4.73205	0.690241	0.345120	−	0.938558i	$-0.387838\pi$
$47$	4.73205	0.690241	0.345120	+	0.938558i	$0.387838\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.46410	−0.757735
$53$	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
$55$	−5.46410	−0.736779
$56$	−4.73205	−0.632347
$57$	0	0
$58$	−4.92820	−0.647105
$59$	10.1962	1.32743	0.663713	−	0.747987i	$-0.268980\pi$
$59$	10.1962	1.32743	0.663713	+	0.747987i	$0.268980\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0.732051	0.0929705
$63$	0	0
$64$	1.00000	0.125000
$65$	5.46410	0.677738
$66$	0	0
$67$	−3.66025	−0.447171	−0.223586	−	0.974684i	$-0.571776\pi$
$67$	−3.66025	−0.447171	−0.223586	+	0.974684i	$0.571776\pi$
$68$	−5.46410	−0.662620
$69$	0	0
$70$	4.73205	0.565588
$71$	−2.92820	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$71$	−2.92820	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$72$	0	0
$73$	−0.928203	−0.108638	−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203	−0.108638	−0.0543190	+	0.998524i	$0.517299\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	6.19615	0.710747
$77$	−25.8564	−2.94661
$78$	0	0
$79$	8.73205	0.982432	0.491216	−	0.871038i	$-0.336552\pi$
$79$	8.73205	0.982432	0.491216	+	0.871038i	$0.336552\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	2.00000	0.220863
$83$	8.73205	0.958467	0.479234	−	0.877687i	$-0.340915\pi$
$83$	8.73205	0.958467	0.479234	+	0.877687i	$0.340915\pi$
$84$	0	0
$85$	5.46410	0.592665
$86$	6.92820	0.747087
$87$	0	0
$88$	5.46410	0.582475
$89$	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
$91$	25.8564	2.71049
$92$	8.00000	0.834058
$93$	0	0
$94$	4.73205	0.488074
$95$	−6.19615	−0.635712
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	15.3923	1.55486
$99$	0	0
$100$	1.00000	0.100000
$101$	9.46410	0.941713	0.470857	−	0.882210i	$-0.343945\pi$
$101$	9.46410	0.941713	0.470857	+	0.882210i	$0.343945\pi$
$102$	0	0
$103$	−6.53590	−0.644001	−0.322001	−	0.946739i	$-0.604355\pi$
$103$	−6.53590	−0.644001	−0.322001	+	0.946739i	$0.604355\pi$
$104$	−5.46410	−0.535799
$105$	0	0
$106$	6.00000	0.582772
$107$	−3.26795	−0.315925	−0.157962	−	0.987445i	$-0.550492\pi$
$107$	−3.26795	−0.315925	−0.157962	+	0.987445i	$0.550492\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−5.46410	−0.520982
$111$	0	0
$112$	−4.73205	−0.447137
$113$	10.5359	0.991134	0.495567	−	0.868570i	$-0.334960\pi$
$113$	10.5359	0.991134	0.495567	+	0.868570i	$0.334960\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	−4.92820	−0.457572
$117$	0	0
$118$	10.1962	0.938632
$119$	25.8564	2.37025
$120$	0	0
$121$	18.8564	1.71422
$122$	−4.92820	−0.446179
$123$	0	0
$124$	0.732051	0.0657401
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.66025	0.324795	0.162398	−	0.986725i	$-0.448077\pi$
$127$	3.66025	0.324795	0.162398	+	0.986725i	$0.448077\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.46410	0.479233
$131$	18.5885	1.62408	0.812041	−	0.583601i	$-0.198357\pi$
$131$	18.5885	1.62408	0.812041	+	0.583601i	$0.198357\pi$
$132$	0	0
$133$	−29.3205	−2.54241
$134$	−3.66025	−0.316198
$135$	0	0
$136$	−5.46410	−0.468543
$137$	−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	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	4.73205	0.399931
$141$	0	0
$142$	−2.92820	−0.245729
$143$	−29.8564	−2.49672
$144$	0	0
$145$	4.92820	0.409265
$146$	−0.928203	−0.0768186
$147$	0	0
$148$	1.00000	0.0821995
$149$	−4.39230	−0.359832	−0.179916	−	0.983682i	$-0.557583\pi$
$149$	−4.39230	−0.359832	−0.179916	+	0.983682i	$0.557583\pi$
$150$	0	0
$151$	−12.3923	−1.00847	−0.504236	−	0.863566i	$-0.668226\pi$
$151$	−12.3923	−1.00847	−0.504236	+	0.863566i	$0.668226\pi$
$152$	6.19615	0.502574
$153$	0	0
$154$	−25.8564	−2.08357
$155$	−0.732051	−0.0587997
$156$	0	0
$157$	3.07180	0.245156	0.122578	−	0.992459i	$-0.460884\pi$
$157$	3.07180	0.245156	0.122578	+	0.992459i	$0.460884\pi$
$158$	8.73205	0.694685
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−37.8564	−2.98350
$162$	0	0
$163$	−11.3205	−0.886691	−0.443345	−	0.896351i	$-0.646208\pi$
$163$	−11.3205	−0.886691	−0.443345	+	0.896351i	$0.646208\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	8.73205	0.677739
$167$	1.46410	0.113296	0.0566478	−	0.998394i	$-0.481959\pi$
$167$	1.46410	0.113296	0.0566478	+	0.998394i	$0.481959\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	5.46410	0.419077
$171$	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
$175$	−4.73205	−0.357709
$176$	5.46410	0.411872
$177$	0	0
$178$	2.00000	0.149906
$179$	0.339746	0.0253938	0.0126969	−	0.999919i	$-0.495958\pi$
$179$	0.339746	0.0253938	0.0126969	+	0.999919i	$0.495958\pi$
$180$	0	0
$181$	5.46410	0.406143	0.203072	−	0.979164i	$-0.434908\pi$
$181$	5.46410	0.406143	0.203072	+	0.979164i	$0.434908\pi$
$182$	25.8564	1.91660
$183$	0	0
$184$	8.00000	0.589768
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−29.8564	−2.18332
$188$	4.73205	0.345120
$189$	0	0
$190$	−6.19615	−0.449516
$191$	8.73205	0.631829	0.315915	−	0.948788i	$-0.397689\pi$
$191$	8.73205	0.631829	0.315915	+	0.948788i	$0.397689\pi$
$192$	0	0
$193$	15.8564	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$193$	15.8564	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	15.3923	1.09945
$197$	22.7846	1.62334	0.811668	−	0.584119i	$-0.198560\pi$
$197$	22.7846	1.62334	0.811668	+	0.584119i	$0.198560\pi$
$198$	0	0
$199$	−12.0526	−0.854383	−0.427192	−	0.904161i	$-0.640497\pi$
$199$	−12.0526	−0.854383	−0.427192	+	0.904161i	$0.640497\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	9.46410	0.665892
$203$	23.3205	1.63678
$204$	0	0
$205$	−2.00000	−0.139686
$206$	−6.53590	−0.455378
$207$	0	0
$208$	−5.46410	−0.378867
$209$	33.8564	2.34190
$210$	0	0
$211$	−17.8564	−1.22929	−0.614643	−	0.788806i	$-0.710700\pi$
$211$	−17.8564	−1.22929	−0.614643	+	0.788806i	$0.710700\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−3.26795	−0.223392
$215$	−6.92820	−0.472500
$216$	0	0
$217$	−3.46410	−0.235159
$218$	2.00000	0.135457
$219$	0	0
$220$	−5.46410	−0.368390
$221$	29.8564	2.00836
$222$	0	0
$223$	16.0526	1.07496	0.537479	−	0.843277i	$-0.319377\pi$
$223$	16.0526	1.07496	0.537479	+	0.843277i	$0.319377\pi$
$224$	−4.73205	−0.316173
$225$	0	0
$226$	10.5359	0.700838
$227$	−24.3923	−1.61897	−0.809487	−	0.587138i	$-0.800255\pi$
$227$	−24.3923	−1.61897	−0.809487	+	0.587138i	$0.800255\pi$
$228$	0	0
$229$	−11.8564	−0.783493	−0.391747	−	0.920073i	$-0.628129\pi$
$229$	−11.8564	−0.783493	−0.391747	+	0.920073i	$0.628129\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−4.92820	−0.323552
$233$	−28.9282	−1.89515	−0.947575	−	0.319534i	$-0.896474\pi$
$233$	−28.9282	−1.89515	−0.947575	+	0.319534i	$0.896474\pi$
$234$	0	0
$235$	−4.73205	−0.308685
$236$	10.1962	0.663713
$237$	0	0
$238$	25.8564	1.67602
$239$	20.7321	1.34104	0.670522	−	0.741889i	$-0.266070\pi$
$239$	20.7321	1.34104	0.670522	+	0.741889i	$0.266070\pi$
$240$	0	0
$241$	4.92820	0.317453	0.158727	−	0.987323i	$-0.449261\pi$
$241$	4.92820	0.317453	0.158727	+	0.987323i	$0.449261\pi$
$242$	18.8564	1.21214
$243$	0	0
$244$	−4.92820	−0.315496
$245$	−15.3923	−0.983378
$246$	0	0
$247$	−33.8564	−2.15423
$248$	0.732051	0.0464853
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−19.2679	−1.21618	−0.608091	−	0.793867i	$-0.708064\pi$
$251$	−19.2679	−1.21618	−0.608091	+	0.793867i	$0.708064\pi$
$252$	0	0
$253$	43.7128	2.74820
$254$	3.66025	0.229665
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.5359	1.15624	0.578119	−	0.815953i	$-0.303787\pi$
$257$	18.5359	1.15624	0.578119	+	0.815953i	$0.303787\pi$
$258$	0	0
$259$	−4.73205	−0.294035
$260$	5.46410	0.338869
$261$	0	0
$262$	18.5885	1.14840
$263$	8.05256	0.496542	0.248271	−	0.968691i	$-0.420138\pi$
$263$	8.05256	0.496542	0.248271	+	0.968691i	$0.420138\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−29.3205	−1.79776
$267$	0	0
$268$	−3.66025	−0.223586
$269$	−20.3923	−1.24334	−0.621670	−	0.783279i	$-0.713546\pi$
$269$	−20.3923	−1.24334	−0.621670	+	0.783279i	$0.713546\pi$
$270$	0	0
$271$	−24.7846	−1.50556	−0.752779	−	0.658273i	$-0.771287\pi$
$271$	−24.7846	−1.50556	−0.752779	+	0.658273i	$0.771287\pi$
$272$	−5.46410	−0.331310
$273$	0	0
$274$	−2.00000	−0.120824
$275$	5.46410	0.329498
$276$	0	0
$277$	22.2487	1.33680	0.668398	−	0.743804i	$-0.266980\pi$
$277$	22.2487	1.33680	0.668398	+	0.743804i	$0.266980\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	4.73205	0.282794
$281$	8.92820	0.532612	0.266306	−	0.963889i	$-0.414197\pi$
$281$	8.92820	0.532612	0.266306	+	0.963889i	$0.414197\pi$
$282$	0	0
$283$	16.3923	0.974421	0.487211	−	0.873284i	$-0.338014\pi$
$283$	16.3923	0.974421	0.487211	+	0.873284i	$0.338014\pi$
$284$	−2.92820	−0.173757
$285$	0	0
$286$	−29.8564	−1.76545
$287$	−9.46410	−0.558648
$288$	0	0
$289$	12.8564	0.756259
$290$	4.92820	0.289394
$291$	0	0
$292$	−0.928203	−0.0543190
$293$	−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
$295$	−10.1962	−0.593643
$296$	1.00000	0.0581238
$297$	0	0
$298$	−4.39230	−0.254439
$299$	−43.7128	−2.52798
$300$	0	0
$301$	−32.7846	−1.88967
$302$	−12.3923	−0.713097
$303$	0	0
$304$	6.19615	0.355374
$305$	4.92820	0.282188
$306$	0	0
$307$	18.5885	1.06090	0.530450	−	0.847716i	$-0.322023\pi$
$307$	18.5885	1.06090	0.530450	+	0.847716i	$0.322023\pi$
$308$	−25.8564	−1.47331
$309$	0	0
$310$	−0.732051	−0.0415777
$311$	−2.87564	−0.163063	−0.0815314	−	0.996671i	$-0.525981\pi$
$311$	−2.87564	−0.163063	−0.0815314	+	0.996671i	$0.525981\pi$
$312$	0	0
$313$	−23.8564	−1.34844	−0.674222	−	0.738529i	$-0.735521\pi$
$313$	−23.8564	−1.34844	−0.674222	+	0.738529i	$0.735521\pi$
$314$	3.07180	0.173352
$315$	0	0
$316$	8.73205	0.491216
$317$	4.14359	0.232727	0.116364	−	0.993207i	$-0.462876\pi$
$317$	4.14359	0.232727	0.116364	+	0.993207i	$0.462876\pi$
$318$	0	0
$319$	−26.9282	−1.50769
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−37.8564	−2.10966
$323$	−33.8564	−1.88382
$324$	0	0
$325$	−5.46410	−0.303094
$326$	−11.3205	−0.626985
$327$	0	0
$328$	2.00000	0.110432
$329$	−22.3923	−1.23453
$330$	0	0
$331$	−33.1244	−1.82068	−0.910340	−	0.413862i	$-0.864180\pi$
$331$	−33.1244	−1.82068	−0.910340	+	0.413862i	$0.864180\pi$
$332$	8.73205	0.479234
$333$	0	0
$334$	1.46410	0.0801121
$335$	3.66025	0.199981
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	16.8564	0.916868
$339$	0	0
$340$	5.46410	0.296333
$341$	4.00000	0.216612
$342$	0	0
$343$	−39.7128	−2.14429
$344$	6.92820	0.373544
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−30.9282	−1.66031	−0.830156	−	0.557530i	$-0.811749\pi$
$347$	−30.9282	−1.66031	−0.830156	+	0.557530i	$0.811749\pi$
$348$	0	0
$349$	−15.3205	−0.820088	−0.410044	−	0.912066i	$-0.634487\pi$
$349$	−15.3205	−0.820088	−0.410044	+	0.912066i	$0.634487\pi$
$350$	−4.73205	−0.252939
$351$	0	0
$352$	5.46410	0.291238
$353$	11.8564	0.631053	0.315526	−	0.948917i	$-0.397819\pi$
$353$	11.8564	0.631053	0.315526	+	0.948917i	$0.397819\pi$
$354$	0	0
$355$	2.92820	0.155413
$356$	2.00000	0.106000
$357$	0	0
$358$	0.339746	0.0179561
$359$	12.3923	0.654041	0.327020	−	0.945017i	$-0.393955\pi$
$359$	12.3923	0.654041	0.327020	+	0.945017i	$0.393955\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	5.46410	0.287187
$363$	0	0
$364$	25.8564	1.35524
$365$	0.928203	0.0485844
$366$	0	0
$367$	−4.33975	−0.226533	−0.113266	−	0.993565i	$-0.536131\pi$
$367$	−4.33975	−0.226533	−0.113266	+	0.993565i	$0.536131\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	−28.3923	−1.47406
$372$	0	0
$373$	−10.7846	−0.558406	−0.279203	−	0.960232i	$-0.590070\pi$
$373$	−10.7846	−0.558406	−0.279203	+	0.960232i	$0.590070\pi$
$374$	−29.8564	−1.54384
$375$	0	0
$376$	4.73205	0.244037
$377$	26.9282	1.38687
$378$	0	0
$379$	32.3923	1.66388	0.831940	−	0.554865i	$-0.187230\pi$
$379$	32.3923	1.66388	0.831940	+	0.554865i	$0.187230\pi$
$380$	−6.19615	−0.317856
$381$	0	0
$382$	8.73205	0.446771
$383$	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
$385$	25.8564	1.31776
$386$	15.8564	0.807070
$387$	0	0
$388$	−2.00000	−0.101535
$389$	11.8564	0.601144	0.300572	−	0.953759i	$-0.402822\pi$
$389$	11.8564	0.601144	0.300572	+	0.953759i	$0.402822\pi$
$390$	0	0
$391$	−43.7128	−2.21065
$392$	15.3923	0.777429
$393$	0	0
$394$	22.7846	1.14787
$395$	−8.73205	−0.439357
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−12.0526	−0.604140
$399$	0	0
$400$	1.00000	0.0500000
$401$	32.9282	1.64436	0.822178	−	0.569230i	$-0.192759\pi$
$401$	32.9282	1.64436	0.822178	+	0.569230i	$0.192759\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	9.46410	0.470857
$405$	0	0
$406$	23.3205	1.15738
$407$	5.46410	0.270845
$408$	0	0
$409$	3.07180	0.151891	0.0759453	−	0.997112i	$-0.475803\pi$
$409$	3.07180	0.151891	0.0759453	+	0.997112i	$0.475803\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−6.53590	−0.322001
$413$	−48.2487	−2.37416
$414$	0	0
$415$	−8.73205	−0.428640
$416$	−5.46410	−0.267900
$417$	0	0
$418$	33.8564	1.65597
$419$	−14.2487	−0.696095	−0.348048	−	0.937477i	$-0.613155\pi$
$419$	−14.2487	−0.696095	−0.348048	+	0.937477i	$0.613155\pi$
$420$	0	0
$421$	−0.143594	−0.00699832	−0.00349916	−	0.999994i	$-0.501114\pi$
$421$	−0.143594	−0.00699832	−0.00349916	+	0.999994i	$0.501114\pi$
$422$	−17.8564	−0.869236
$423$	0	0
$424$	6.00000	0.291386
$425$	−5.46410	−0.265048
$426$	0	0
$427$	23.3205	1.12856
$428$	−3.26795	−0.157962
$429$	0	0
$430$	−6.92820	−0.334108
$431$	2.19615	0.105785	0.0528925	−	0.998600i	$-0.483156\pi$
$431$	2.19615	0.105785	0.0528925	+	0.998600i	$0.483156\pi$
$432$	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
$437$	49.5692	2.37122
$438$	0	0
$439$	13.5167	0.645115	0.322558	−	0.946550i	$-0.395457\pi$
$439$	13.5167	0.645115	0.322558	+	0.946550i	$0.395457\pi$
$440$	−5.46410	−0.260491
$441$	0	0
$442$	29.8564	1.42012
$443$	14.8756	0.706763	0.353382	−	0.935479i	$-0.385032\pi$
$443$	14.8756	0.706763	0.353382	+	0.935479i	$0.385032\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	16.0526	0.760111
$447$	0	0
$448$	−4.73205	−0.223568
$449$	21.7128	1.02469	0.512345	−	0.858779i	$-0.328777\pi$
$449$	21.7128	1.02469	0.512345	+	0.858779i	$0.328777\pi$
$450$	0	0
$451$	10.9282	0.514589
$452$	10.5359	0.495567
$453$	0	0
$454$	−24.3923	−1.14479
$455$	−25.8564	−1.21217
$456$	0	0
$457$	−31.8564	−1.49018	−0.745090	−	0.666964i	$-0.767593\pi$
$457$	−31.8564	−1.49018	−0.745090	+	0.666964i	$0.767593\pi$
$458$	−11.8564	−0.554013
$459$	0	0
$460$	−8.00000	−0.373002
$461$	−14.7846	−0.688588	−0.344294	−	0.938862i	$-0.611882\pi$
$461$	−14.7846	−0.688588	−0.344294	+	0.938862i	$0.611882\pi$
$462$	0	0
$463$	18.9282	0.879668	0.439834	−	0.898079i	$-0.355037\pi$
$463$	18.9282	0.879668	0.439834	+	0.898079i	$0.355037\pi$
$464$	−4.92820	−0.228786
$465$	0	0
$466$	−28.9282	−1.34007
$467$	25.1769	1.16505	0.582524	−	0.812813i	$-0.302065\pi$
$467$	25.1769	1.16505	0.582524	+	0.812813i	$0.302065\pi$
$468$	0	0
$469$	17.3205	0.799787
$470$	−4.73205	−0.218273
$471$	0	0
$472$	10.1962	0.469316
$473$	37.8564	1.74064
$474$	0	0
$475$	6.19615	0.284299
$476$	25.8564	1.18513
$477$	0	0
$478$	20.7321	0.948262
$479$	−4.05256	−0.185166	−0.0925831	−	0.995705i	$-0.529512\pi$
$479$	−4.05256	−0.185166	−0.0925831	+	0.995705i	$0.529512\pi$
$480$	0	0
$481$	−5.46410	−0.249142
$482$	4.92820	0.224474
$483$	0	0
$484$	18.8564	0.857109
$485$	2.00000	0.0908153
$486$	0	0
$487$	4.39230	0.199034	0.0995172	−	0.995036i	$-0.468270\pi$
$487$	4.39230	0.199034	0.0995172	+	0.995036i	$0.468270\pi$
$488$	−4.92820	−0.223089
$489$	0	0
$490$	−15.3923	−0.695353
$491$	14.9282	0.673700	0.336850	−	0.941558i	$-0.390638\pi$
$491$	14.9282	0.673700	0.336850	+	0.941558i	$0.390638\pi$
$492$	0	0
$493$	26.9282	1.21279
$494$	−33.8564	−1.52327
$495$	0	0
$496$	0.732051	0.0328701
$497$	13.8564	0.621545
$498$	0	0
$499$	9.41154	0.421319	0.210659	−	0.977560i	$-0.432439\pi$
$499$	9.41154	0.421319	0.210659	+	0.977560i	$0.432439\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−19.2679	−0.859971
$503$	18.9282	0.843967	0.421983	−	0.906604i	$-0.361334\pi$
$503$	18.9282	0.843967	0.421983	+	0.906604i	$0.361334\pi$
$504$	0	0
$505$	−9.46410	−0.421147
$506$	43.7128	1.94327
$507$	0	0
$508$	3.66025	0.162398
$509$	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
$511$	4.39230	0.194304
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.5359	0.817583
$515$	6.53590	0.288006
$516$	0	0
$517$	25.8564	1.13716
$518$	−4.73205	−0.207914
$519$	0	0
$520$	5.46410	0.239617
$521$	−26.5359	−1.16256	−0.581279	−	0.813704i	$-0.697448\pi$
$521$	−26.5359	−1.16256	−0.581279	+	0.813704i	$0.697448\pi$
$522$	0	0
$523$	36.7846	1.60848	0.804239	−	0.594306i	$-0.202573\pi$
$523$	36.7846	1.60848	0.804239	+	0.594306i	$0.202573\pi$
$524$	18.5885	0.812041
$525$	0	0
$526$	8.05256	0.351108
$527$	−4.00000	−0.174243
$528$	0	0
$529$	41.0000	1.78261
$530$	−6.00000	−0.260623
$531$	0	0
$532$	−29.3205	−1.27121
$533$	−10.9282	−0.473353
$534$	0	0
$535$	3.26795	0.141286
$536$	−3.66025	−0.158099
$537$	0	0
$538$	−20.3923	−0.879175
$539$	84.1051	3.62266
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−24.7846	−1.06459
$543$	0	0
$544$	−5.46410	−0.234271
$545$	−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
$550$	5.46410	0.232990
$551$	−30.5359	−1.30087
$552$	0	0
$553$	−41.3205	−1.75713
$554$	22.2487	0.945257
$555$	0	0
$556$	6.92820	0.293821
$557$	−44.1051	−1.86879	−0.934397	−	0.356234i	$-0.884061\pi$
$557$	−44.1051	−1.86879	−0.934397	+	0.356234i	$0.884061\pi$
$558$	0	0
$559$	−37.8564	−1.60116
$560$	4.73205	0.199966
$561$	0	0
$562$	8.92820	0.376614
$563$	−30.9282	−1.30347	−0.651734	−	0.758447i	$-0.725958\pi$
$563$	−30.9282	−1.30347	−0.651734	+	0.758447i	$0.725958\pi$
$564$	0	0
$565$	−10.5359	−0.443249
$566$	16.3923	0.689020
$567$	0	0
$568$	−2.92820	−0.122865
$569$	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	$0.580790\pi$
$572$	−29.8564	−1.24836
$573$	0	0
$574$	−9.46410	−0.395024
$575$	8.00000	0.333623
$576$	0	0
$577$	−24.3923	−1.01546	−0.507732	−	0.861515i	$-0.669516\pi$
$577$	−24.3923	−1.01546	−0.507732	+	0.861515i	$0.669516\pi$
$578$	12.8564	0.534756
$579$	0	0
$580$	4.92820	0.204633
$581$	−41.3205	−1.71426
$582$	0	0
$583$	32.7846	1.35780
$584$	−0.928203	−0.0384093
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−44.7846	−1.84846	−0.924229	−	0.381838i	$-0.875291\pi$
$587$	−44.7846	−1.84846	−0.924229	+	0.381838i	$0.875291\pi$
$588$	0	0
$589$	4.53590	0.186898
$590$	−10.1962	−0.419769
$591$	0	0
$592$	1.00000	0.0410997
$593$	34.7846	1.42843	0.714216	−	0.699925i	$-0.246783\pi$
$593$	34.7846	1.42843	0.714216	+	0.699925i	$0.246783\pi$
$594$	0	0
$595$	−25.8564	−1.06001
$596$	−4.39230	−0.179916
$597$	0	0
$598$	−43.7128	−1.78755
$599$	−9.46410	−0.386693	−0.193346	−	0.981131i	$-0.561934\pi$
$599$	−9.46410	−0.386693	−0.193346	+	0.981131i	$0.561934\pi$
$600$	0	0
$601$	27.6077	1.12614	0.563071	−	0.826409i	$-0.309620\pi$
$601$	27.6077	1.12614	0.563071	+	0.826409i	$0.309620\pi$
$602$	−32.7846	−1.33620
$603$	0	0
$604$	−12.3923	−0.504236
$605$	−18.8564	−0.766622
$606$	0	0
$607$	0.784610	0.0318463	0.0159232	−	0.999873i	$-0.494931\pi$
$607$	0.784610	0.0318463	0.0159232	+	0.999873i	$0.494931\pi$
$608$	6.19615	0.251287
$609$	0	0
$610$	4.92820	0.199537
$611$	−25.8564	−1.04604
$612$	0	0
$613$	16.9282	0.683724	0.341862	−	0.939750i	$-0.388943\pi$
$613$	16.9282	0.683724	0.341862	+	0.939750i	$0.388943\pi$
$614$	18.5885	0.750169
$615$	0	0
$616$	−25.8564	−1.04178
$617$	0.928203	0.0373681	0.0186840	−	0.999825i	$-0.494052\pi$
$617$	0.928203	0.0373681	0.0186840	+	0.999825i	$0.494052\pi$
$618$	0	0
$619$	−17.8564	−0.717710	−0.358855	−	0.933393i	$-0.616833\pi$
$619$	−17.8564	−0.717710	−0.358855	+	0.933393i	$0.616833\pi$
$620$	−0.732051	−0.0293999
$621$	0	0
$622$	−2.87564	−0.115303
$623$	−9.46410	−0.379171
$624$	0	0
$625$	1.00000	0.0400000
$626$	−23.8564	−0.953494
$627$	0	0
$628$	3.07180	0.122578
$629$	−5.46410	−0.217868
$630$	0	0
$631$	30.9808	1.23332	0.616662	−	0.787228i	$-0.288484\pi$
$631$	30.9808	1.23332	0.616662	+	0.787228i	$0.288484\pi$
$632$	8.73205	0.347342
$633$	0	0
$634$	4.14359	0.164563
$635$	−3.66025	−0.145253
$636$	0	0
$637$	−84.1051	−3.33237
$638$	−26.9282	−1.06610
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	6.53590	0.258152	0.129076	−	0.991635i	$-0.458799\pi$
$641$	6.53590	0.258152	0.129076	+	0.991635i	$0.458799\pi$
$642$	0	0
$643$	34.5359	1.36196	0.680981	−	0.732301i	$-0.261553\pi$
$643$	34.5359	1.36196	0.680981	+	0.732301i	$0.261553\pi$
$644$	−37.8564	−1.49175
$645$	0	0
$646$	−33.8564	−1.33206
$647$	35.7128	1.40402	0.702008	−	0.712169i	$-0.252287\pi$
$647$	35.7128	1.40402	0.702008	+	0.712169i	$0.252287\pi$
$648$	0	0
$649$	55.7128	2.18692
$650$	−5.46410	−0.214320
$651$	0	0
$652$	−11.3205	−0.443345
$653$	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
$655$	−18.5885	−0.726311
$656$	2.00000	0.0780869
$657$	0	0
$658$	−22.3923	−0.872943
$659$	6.92820	0.269884	0.134942	−	0.990853i	$-0.456915\pi$
$659$	6.92820	0.269884	0.134942	+	0.990853i	$0.456915\pi$
$660$	0	0
$661$	−31.0718	−1.20855	−0.604276	−	0.796775i	$-0.706538\pi$
$661$	−31.0718	−1.20855	−0.604276	+	0.796775i	$0.706538\pi$
$662$	−33.1244	−1.28741
$663$	0	0
$664$	8.73205	0.338869
$665$	29.3205	1.13700
$666$	0	0
$667$	−39.4256	−1.52657
$668$	1.46410	0.0566478
$669$	0	0
$670$	3.66025	0.141408
$671$	−26.9282	−1.03955
$672$	0	0
$673$	−32.9282	−1.26929	−0.634644	−	0.772804i	$-0.718853\pi$
$673$	−32.9282	−1.26929	−0.634644	+	0.772804i	$0.718853\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	16.8564	0.648323
$677$	4.14359	0.159251	0.0796256	−	0.996825i	$-0.474628\pi$
$677$	4.14359	0.159251	0.0796256	+	0.996825i	$0.474628\pi$
$678$	0	0
$679$	9.46410	0.363199
$680$	5.46410	0.209539
$681$	0	0
$682$	4.00000	0.153168
$683$	4.78461	0.183078	0.0915390	−	0.995801i	$-0.470821\pi$
$683$	4.78461	0.183078	0.0915390	+	0.995801i	$0.470821\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	−39.7128	−1.51624
$687$	0	0
$688$	6.92820	0.264135
$689$	−32.7846	−1.24899
$690$	0	0
$691$	−0.392305	−0.0149240	−0.00746199	−	0.999972i	$-0.502375\pi$
$691$	−0.392305	−0.0149240	−0.00746199	+	0.999972i	$0.502375\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	−30.9282	−1.17402
$695$	−6.92820	−0.262802
$696$	0	0
$697$	−10.9282	−0.413935
$698$	−15.3205	−0.579890
$699$	0	0
$700$	−4.73205	−0.178855
$701$	11.8564	0.447810	0.223905	−	0.974611i	$-0.428119\pi$
$701$	11.8564	0.447810	0.223905	+	0.974611i	$0.428119\pi$
$702$	0	0
$703$	6.19615	0.233692
$704$	5.46410	0.205936
$705$	0	0
$706$	11.8564	0.446222
$707$	−44.7846	−1.68430
$708$	0	0
$709$	43.8564	1.64706	0.823531	−	0.567271i	$-0.192001\pi$
$709$	43.8564	1.64706	0.823531	+	0.567271i	$0.192001\pi$
$710$	2.92820	0.109894
$711$	0	0
$712$	2.00000	0.0749532
$713$	5.85641	0.219324
$714$	0	0
$715$	29.8564	1.11657
$716$	0.339746	0.0126969
$717$	0	0
$718$	12.3923	0.462477
$719$	12.3923	0.462155	0.231077	−	0.972935i	$-0.425775\pi$
$719$	12.3923	0.462155	0.231077	+	0.972935i	$0.425775\pi$
$720$	0	0
$721$	30.9282	1.15183
$722$	19.3923	0.721707
$723$	0	0
$724$	5.46410	0.203072
$725$	−4.92820	−0.183029
$726$	0	0
$727$	8.78461	0.325803	0.162902	−	0.986642i	$-0.447915\pi$
$727$	8.78461	0.325803	0.162902	+	0.986642i	$0.447915\pi$
$728$	25.8564	0.958302
$729$	0	0
$730$	0.928203	0.0343543
$731$	−37.8564	−1.40017
$732$	0	0
$733$	27.8564	1.02890	0.514450	−	0.857520i	$-0.327996\pi$
$733$	27.8564	1.02890	0.514450	+	0.857520i	$0.327996\pi$
$734$	−4.33975	−0.160183
$735$	0	0
$736$	8.00000	0.294884
$737$	−20.0000	−0.736709
$738$	0	0
$739$	6.92820	0.254858	0.127429	−	0.991848i	$-0.459327\pi$
$739$	6.92820	0.254858	0.127429	+	0.991848i	$0.459327\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−28.3923	−1.04231
$743$	−17.1244	−0.628232	−0.314116	−	0.949385i	$-0.601708\pi$
$743$	−17.1244	−0.628232	−0.314116	+	0.949385i	$0.601708\pi$
$744$	0	0
$745$	4.39230	0.160922
$746$	−10.7846	−0.394853
$747$	0	0
$748$	−29.8564	−1.09166
$749$	15.4641	0.565046
$750$	0	0
$751$	−20.3923	−0.744126	−0.372063	−	0.928208i	$-0.621349\pi$
$751$	−20.3923	−0.744126	−0.372063	+	0.928208i	$0.621349\pi$
$752$	4.73205	0.172560
$753$	0	0
$754$	26.9282	0.980667
$755$	12.3923	0.451002
$756$	0	0
$757$	1.71281	0.0622532	0.0311266	−	0.999515i	$-0.490090\pi$
$757$	1.71281	0.0622532	0.0311266	+	0.999515i	$0.490090\pi$
$758$	32.3923	1.17654
$759$	0	0
$760$	−6.19615	−0.224758
$761$	−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
$763$	−9.46410	−0.342623
$764$	8.73205	0.315915
$765$	0	0
$766$	8.00000	0.289052
$767$	−55.7128	−2.01167
$768$	0	0
$769$	7.07180	0.255016	0.127508	−	0.991838i	$-0.459302\pi$
$769$	7.07180	0.255016	0.127508	+	0.991838i	$0.459302\pi$
$770$	25.8564	0.931800
$771$	0	0
$772$	15.8564	0.570685
$773$	−2.78461	−0.100155	−0.0500777	−	0.998745i	$-0.515947\pi$
$773$	−2.78461	−0.100155	−0.0500777	+	0.998745i	$0.515947\pi$
$774$	0	0
$775$	0.732051	0.0262960
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	11.8564	0.425073
$779$	12.3923	0.444000
$780$	0	0
$781$	−16.0000	−0.572525
$782$	−43.7128	−1.56317
$783$	0	0
$784$	15.3923	0.549725
$785$	−3.07180	−0.109637
$786$	0	0
$787$	22.1962	0.791207	0.395604	−	0.918421i	$-0.370535\pi$
$787$	22.1962	0.791207	0.395604	+	0.918421i	$0.370535\pi$
$788$	22.7846	0.811668
$789$	0	0
$790$	−8.73205	−0.310672
$791$	−49.8564	−1.77269
$792$	0	0
$793$	26.9282	0.956249
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−12.0526	−0.427192
$797$	−10.5359	−0.373201	−0.186600	−	0.982436i	$-0.559747\pi$
$797$	−10.5359	−0.373201	−0.186600	+	0.982436i	$0.559747\pi$
$798$	0	0
$799$	−25.8564	−0.914734
$800$	1.00000	0.0353553
$801$	0	0
$802$	32.9282	1.16274
$803$	−5.07180	−0.178980
$804$	0	0
$805$	37.8564	1.33426
$806$	−4.00000	−0.140894
$807$	0	0
$808$	9.46410	0.332946
$809$	−43.5692	−1.53181	−0.765906	−	0.642952i	$-0.777709\pi$
$809$	−43.5692	−1.53181	−0.765906	+	0.642952i	$0.777709\pi$
$810$	0	0
$811$	−41.8564	−1.46978	−0.734889	−	0.678188i	$-0.762766\pi$
$811$	−41.8564	−1.46978	−0.734889	+	0.678188i	$0.762766\pi$
$812$	23.3205	0.818389
$813$	0	0
$814$	5.46410	0.191517
$815$	11.3205	0.396540
$816$	0	0
$817$	42.9282	1.50187
$818$	3.07180	0.107403
$819$	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
$823$	48.4449	1.68868	0.844341	−	0.535806i	$-0.179992\pi$
$823$	48.4449	1.68868	0.844341	+	0.535806i	$0.179992\pi$
$824$	−6.53590	−0.227689
$825$	0	0
$826$	−48.2487	−1.67879
$827$	16.3923	0.570016	0.285008	−	0.958525i	$-0.408004\pi$
$827$	16.3923	0.570016	0.285008	+	0.958525i	$0.408004\pi$
$828$	0	0
$829$	−51.5692	−1.79107	−0.895537	−	0.444988i	$-0.853208\pi$
$829$	−51.5692	−1.79107	−0.895537	+	0.444988i	$0.853208\pi$
$830$	−8.73205	−0.303094
$831$	0	0
$832$	−5.46410	−0.189434
$833$	−84.1051	−2.91407
$834$	0	0
$835$	−1.46410	−0.0506673
$836$	33.8564	1.17095
$837$	0	0
$838$	−14.2487	−0.492214
$839$	8.78461	0.303278	0.151639	−	0.988436i	$-0.451545\pi$
$839$	8.78461	0.303278	0.151639	+	0.988436i	$0.451545\pi$
$840$	0	0
$841$	−4.71281	−0.162511
$842$	−0.143594	−0.00494856
$843$	0	0
$844$	−17.8564	−0.614643
$845$	−16.8564	−0.579878
$846$	0	0
$847$	−89.2295	−3.06596
$848$	6.00000	0.206041
$849$	0	0
$850$	−5.46410	−0.187417
$851$	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	$0.671685\pi$
$854$	23.3205	0.798011
$855$	0	0
$856$	−3.26795	−0.111696
$857$	31.8564	1.08819	0.544097	−	0.839022i	$-0.316872\pi$
$857$	31.8564	1.08819	0.544097	+	0.839022i	$0.316872\pi$
$858$	0	0
$859$	44.4449	1.51644	0.758220	−	0.651999i	$-0.226069\pi$
$859$	44.4449	1.51644	0.758220	+	0.651999i	$0.226069\pi$
$860$	−6.92820	−0.236250
$861$	0	0
$862$	2.19615	0.0748012
$863$	20.4449	0.695951	0.347976	−	0.937504i	$-0.386869\pi$
$863$	20.4449	0.695951	0.347976	+	0.937504i	$0.386869\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	−34.0000	−1.15537
$867$	0	0
$868$	−3.46410	−0.117579
$869$	47.7128	1.61855
$870$	0	0
$871$	20.0000	0.677674
$872$	2.00000	0.0677285
$873$	0	0
$874$	49.5692	1.67670
$875$	4.73205	0.159973
$876$	0	0
$877$	−9.21539	−0.311182	−0.155591	−	0.987822i	$-0.549728\pi$
$877$	−9.21539	−0.311182	−0.155591	+	0.987822i	$0.549728\pi$
$878$	13.5167	0.456165
$879$	0	0
$880$	−5.46410	−0.184195
$881$	−12.6795	−0.427183	−0.213591	−	0.976923i	$-0.568516\pi$
$881$	−12.6795	−0.427183	−0.213591	+	0.976923i	$0.568516\pi$
$882$	0	0
$883$	14.2487	0.479507	0.239754	−	0.970834i	$-0.422933\pi$
$883$	14.2487	0.479507	0.239754	+	0.970834i	$0.422933\pi$
$884$	29.8564	1.00418
$885$	0	0
$886$	14.8756	0.499757
$887$	−9.80385	−0.329181	−0.164590	−	0.986362i	$-0.552630\pi$
$887$	−9.80385	−0.329181	−0.164590	+	0.986362i	$0.552630\pi$
$888$	0	0
$889$	−17.3205	−0.580911
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	16.0526	0.537479
$893$	29.3205	0.981173
$894$	0	0
$895$	−0.339746	−0.0113565
$896$	−4.73205	−0.158087
$897$	0	0
$898$	21.7128	0.724566
$899$	−3.60770	−0.120323
$900$	0	0
$901$	−32.7846	−1.09221
$902$	10.9282	0.363869
$903$	0	0
$904$	10.5359	0.350419
$905$	−5.46410	−0.181633
$906$	0	0
$907$	54.2487	1.80130	0.900649	−	0.434546i	$-0.143091\pi$
$907$	54.2487	1.80130	0.900649	+	0.434546i	$0.143091\pi$
$908$	−24.3923	−0.809487
$909$	0	0
$910$	−25.8564	−0.857132
$911$	−37.9090	−1.25598	−0.627990	−	0.778221i	$-0.716122\pi$
$911$	−37.9090	−1.25598	−0.627990	+	0.778221i	$0.716122\pi$
$912$	0	0
$913$	47.7128	1.57906
$914$	−31.8564	−1.05372
$915$	0	0
$916$	−11.8564	−0.391747
$917$	−87.9615	−2.90475
$918$	0	0
$919$	−16.7321	−0.551939	−0.275970	−	0.961166i	$-0.588999\pi$
$919$	−16.7321	−0.551939	−0.275970	+	0.961166i	$0.588999\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	−14.7846	−0.486905
$923$	16.0000	0.526646
$924$	0	0
$925$	1.00000	0.0328798
$926$	18.9282	0.622019
$927$	0	0
$928$	−4.92820	−0.161776
$929$	−11.8564	−0.388996	−0.194498	−	0.980903i	$-0.562308\pi$
$929$	−11.8564	−0.388996	−0.194498	+	0.980903i	$0.562308\pi$
$930$	0	0
$931$	95.3731	3.12573
$932$	−28.9282	−0.947575
$933$	0	0
$934$	25.1769	0.823814
$935$	29.8564	0.976409
$936$	0	0
$937$	−23.8564	−0.779355	−0.389677	−	0.920951i	$-0.627413\pi$
$937$	−23.8564	−0.779355	−0.389677	+	0.920951i	$0.627413\pi$
$938$	17.3205	0.565535
$939$	0	0
$940$	−4.73205	−0.154342
$941$	−7.60770	−0.248004	−0.124002	−	0.992282i	$-0.539573\pi$
$941$	−7.60770	−0.248004	−0.124002	+	0.992282i	$0.539573\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	10.1962	0.331856
$945$	0	0
$946$	37.8564	1.23082
$947$	−17.1769	−0.558175	−0.279087	−	0.960266i	$-0.590032\pi$
$947$	−17.1769	−0.558175	−0.279087	+	0.960266i	$0.590032\pi$
$948$	0	0
$949$	5.07180	0.164637
$950$	6.19615	0.201030
$951$	0	0
$952$	25.8564	0.838011
$953$	−15.8564	−0.513639	−0.256820	−	0.966459i	$-0.582675\pi$
$953$	−15.8564	−0.513639	−0.256820	+	0.966459i	$0.582675\pi$
$954$	0	0
$955$	−8.73205	−0.282563
$956$	20.7321	0.670522
$957$	0	0
$958$	−4.05256	−0.130932
$959$	9.46410	0.305612
$960$	0	0
$961$	−30.4641	−0.982713
$962$	−5.46410	−0.176170
$963$	0	0
$964$	4.92820	0.158727
$965$	−15.8564	−0.510436
$966$	0	0
$967$	−44.3923	−1.42756	−0.713780	−	0.700370i	$-0.753018\pi$
$967$	−44.3923	−1.42756	−0.713780	+	0.700370i	$0.753018\pi$
$968$	18.8564	0.606068
$969$	0	0
$970$	2.00000	0.0642161
$971$	−1.07180	−0.0343956	−0.0171978	−	0.999852i	$-0.505474\pi$
$971$	−1.07180	−0.0343956	−0.0171978	+	0.999852i	$0.505474\pi$
$972$	0	0
$973$	−32.7846	−1.05103
$974$	4.39230	0.140739
$975$	0	0
$976$	−4.92820	−0.157748
$977$	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
$979$	10.9282	0.349267
$980$	−15.3923	−0.491689
$981$	0	0
$982$	14.9282	0.476378
$983$	−20.4449	−0.652090	−0.326045	−	0.945354i	$-0.605716\pi$
$983$	−20.4449	−0.652090	−0.326045	+	0.945354i	$0.605716\pi$
$984$	0	0
$985$	−22.7846	−0.725978
$986$	26.9282	0.857569
$987$	0	0
$988$	−33.8564	−1.07712
$989$	55.4256	1.76243
$990$	0	0
$991$	44.0526	1.39938	0.699688	−	0.714449i	$-0.253322\pi$
$991$	44.0526	1.39938	0.699688	+	0.714449i	$0.253322\pi$
$992$	0.732051	0.0232426
$993$	0	0
$994$	13.8564	0.439499
$995$	12.0526	0.382092
$996$	0	0
$997$	−31.8564	−1.00890	−0.504451	−	0.863440i	$-0.668305\pi$
$997$	−31.8564	−1.00890	−0.504451	+	0.863440i	$0.668305\pi$
$998$	9.41154	0.297917
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bd.1.1		2
3.2	odd	2		370.2.a.e.1.2	✓	2
12.11	even	2		2960.2.a.q.1.1		2
15.2	even	4		1850.2.b.l.149.1		4
15.8	even	4		1850.2.b.l.149.4		4
15.14	odd	2		1850.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.e.1.2	✓	2	3.2	odd	2
1850.2.a.x.1.1		2	15.14	odd	2
1850.2.b.l.149.1		4	15.2	even	4
1850.2.b.l.149.4		4	15.8	even	4
2960.2.a.q.1.1		2	12.11	even	2
3330.2.a.bd.1.1		2	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.73205 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.46410 q^{11} -5.46410 q^{13} -4.73205 q^{14} +1.00000 q^{16} -5.46410 q^{17} +6.19615 q^{19} -1.00000 q^{20} +5.46410 q^{22} +8.00000 q^{23} +1.00000 q^{25} -5.46410 q^{26} -4.73205 q^{28} -4.92820 q^{29} +0.732051 q^{31} +1.00000 q^{32} -5.46410 q^{34} +4.73205 q^{35} +1.00000 q^{37} +6.19615 q^{38} -1.00000 q^{40} +2.00000 q^{41} +6.92820 q^{43} +5.46410 q^{44} +8.00000 q^{46} +4.73205 q^{47} +15.3923 q^{49} +1.00000 q^{50} -5.46410 q^{52} +6.00000 q^{53} -5.46410 q^{55} -4.73205 q^{56} -4.92820 q^{58} +10.1962 q^{59} -4.92820 q^{61} +0.732051 q^{62} +1.00000 q^{64} +5.46410 q^{65} -3.66025 q^{67} -5.46410 q^{68} +4.73205 q^{70} -2.92820 q^{71} -0.928203 q^{73} +1.00000 q^{74} +6.19615 q^{76} -25.8564 q^{77} +8.73205 q^{79} -1.00000 q^{80} +2.00000 q^{82} +8.73205 q^{83} +5.46410 q^{85} +6.92820 q^{86} +5.46410 q^{88} +2.00000 q^{89} +25.8564 q^{91} +8.00000 q^{92} +4.73205 q^{94} -6.19615 q^{95} -2.00000 q^{97} +15.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8}+O(q^{10}) \) 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

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Database

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