LMFDB - Embedded newform 4680.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +2.70156 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +2.70156 q^{7} -0.701562 q^{11} -1.00000 q^{13} +0.701562 q^{17} -2.00000 q^{19} -6.70156 q^{23} +1.00000 q^{25} -9.40312 q^{29} -9.40312 q^{31} +2.70156 q^{35} -6.70156 q^{37} -10.7016 q^{41} +4.00000 q^{43} +1.40312 q^{47} +0.298438 q^{49} -10.7016 q^{53} -0.701562 q^{55} +14.8062 q^{59} +2.70156 q^{61} -1.00000 q^{65} -4.00000 q^{67} -15.5078 q^{71} +13.4031 q^{73} -1.89531 q^{77} +4.70156 q^{79} -4.00000 q^{83} +0.701562 q^{85} -8.10469 q^{89} -2.70156 q^{91} -2.00000 q^{95} +18.1047 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - q^7	$ 2 q + 2 q^{5} - q^{7} + 5 q^{11} - 2 q^{13} - 5 q^{17} - 4 q^{19} - 7 q^{23} + 2 q^{25} - 6 q^{29} - 6 q^{31} - q^{35} - 7 q^{37} - 15 q^{41} + 8 q^{43} - 10 q^{47} + 7 q^{49} - 15 q^{53} + 5 q^{55} + 4 q^{59} - q^{61} - 2 q^{65} - 8 q^{67} + q^{71} + 14 q^{73} - 23 q^{77} + 3 q^{79} - 8 q^{83} - 5 q^{85} + 3 q^{89} + q^{91} - 4 q^{95} + 17 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - q^7 + 5 * q^11 - 2 * q^13 - 5 * q^17 - 4 * q^19 - 7 * q^23 + 2 * q^25 - 6 * q^29 - 6 * q^31 - q^35 - 7 * q^37 - 15 * q^41 + 8 * q^43 - 10 * q^47 + 7 * q^49 - 15 * q^53 + 5 * q^55 + 4 * q^59 - q^61 - 2 * q^65 - 8 * q^67 + q^71 + 14 * q^73 - 23 * q^77 + 3 * q^79 - 8 * q^83 - 5 * q^85 + 3 * q^89 + q^91 - 4 * q^95 + 17 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.70156	1.02109	0.510547	−	0.859850i	$-0.329443\pi$
$7$	2.70156	1.02109	0.510547	+	0.859850i	$0.329443\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.701562	−0.211529	−0.105764	−	0.994391i	$-0.533729\pi$
$11$	−0.701562	−0.211529	−0.105764	+	0.994391i	$0.533729\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.701562	0.170154	0.0850769	−	0.996374i	$-0.472886\pi$
$17$	0.701562	0.170154	0.0850769	+	0.996374i	$0.472886\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.70156	−1.39737	−0.698686	−	0.715428i	$-0.746232\pi$
$23$	−6.70156	−1.39737	−0.698686	+	0.715428i	$0.746232\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.40312	−1.74612	−0.873058	−	0.487616i	$-0.837867\pi$
$29$	−9.40312	−1.74612	−0.873058	+	0.487616i	$0.837867\pi$
$30$	0	0
$31$	−9.40312	−1.68885	−0.844425	−	0.535673i	$-0.820058\pi$
$31$	−9.40312	−1.68885	−0.844425	+	0.535673i	$0.820058\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.70156	0.456647
$36$	0	0
$37$	−6.70156	−1.10173	−0.550865	−	0.834594i	$-0.685702\pi$
$37$	−6.70156	−1.10173	−0.550865	+	0.834594i	$0.685702\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.7016	−1.67130	−0.835652	−	0.549260i	$-0.814910\pi$
$41$	−10.7016	−1.67130	−0.835652	+	0.549260i	$0.814910\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.40312	0.204667	0.102333	−	0.994750i	$-0.467369\pi$
$47$	1.40312	0.204667	0.102333	+	0.994750i	$0.467369\pi$
$48$	0	0
$49$	0.298438	0.0426340
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.7016	−1.46997	−0.734986	−	0.678082i	$-0.762811\pi$
$53$	−10.7016	−1.46997	−0.734986	+	0.678082i	$0.762811\pi$
$54$	0	0
$55$	−0.701562	−0.0945986
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.8062	1.92761	0.963805	−	0.266609i	$-0.0859033\pi$
$59$	14.8062	1.92761	0.963805	+	0.266609i	$0.0859033\pi$
$60$	0	0
$61$	2.70156	0.345900	0.172950	−	0.984931i	$-0.444670\pi$
$61$	2.70156	0.345900	0.172950	+	0.984931i	$0.444670\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.5078	−1.84044	−0.920219	−	0.391403i	$-0.871990\pi$
$71$	−15.5078	−1.84044	−0.920219	+	0.391403i	$0.871990\pi$
$72$	0	0
$73$	13.4031	1.56872	0.784359	−	0.620308i	$-0.212992\pi$
$73$	13.4031	1.56872	0.784359	+	0.620308i	$0.212992\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.89531	−0.215991
$78$	0	0
$79$	4.70156	0.528967	0.264484	−	0.964390i	$-0.414799\pi$
$79$	4.70156	0.528967	0.264484	+	0.964390i	$0.414799\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0.701562	0.0760951
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.10469	−0.859095	−0.429548	−	0.903044i	$-0.641327\pi$
$89$	−8.10469	−0.859095	−0.429548	+	0.903044i	$0.641327\pi$
$90$	0	0
$91$	−2.70156	−0.283201
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	18.1047	1.83825	0.919126	−	0.393963i	$-0.128896\pi$
$97$	18.1047	1.83825	0.919126	+	0.393963i	$0.128896\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.80625	0.677247	0.338624	−	0.940922i	$-0.390039\pi$
$101$	6.80625	0.677247	0.338624	+	0.940922i	$0.390039\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.1047	−1.75025	−0.875123	−	0.483900i	$-0.839220\pi$
$107$	−18.1047	−1.75025	−0.875123	+	0.483900i	$0.839220\pi$
$108$	0	0
$109$	10.8062	1.03505	0.517525	−	0.855668i	$-0.326853\pi$
$109$	10.8062	1.03505	0.517525	+	0.855668i	$0.326853\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.59688	−0.620582	−0.310291	−	0.950642i	$-0.600426\pi$
$113$	−6.59688	−0.620582	−0.310291	+	0.950642i	$0.600426\pi$
$114$	0	0
$115$	−6.70156	−0.624924
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.89531	0.173743
$120$	0	0
$121$	−10.5078	−0.955256
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.4031	−1.18933	−0.594667	−	0.803972i	$-0.702716\pi$
$127$	−13.4031	−1.18933	−0.594667	+	0.803972i	$0.702716\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.8062	1.11889	0.559444	−	0.828868i	$-0.311015\pi$
$131$	12.8062	1.11889	0.559444	+	0.828868i	$0.311015\pi$
$132$	0	0
$133$	−5.40312	−0.468510
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−4.70156	−0.398781	−0.199391	−	0.979920i	$-0.563896\pi$
$139$	−4.70156	−0.398781	−0.199391	+	0.979920i	$0.563896\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.701562	0.0586676
$144$	0	0
$145$	−9.40312	−0.780887
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.29844	−0.106372	−0.0531861	−	0.998585i	$-0.516938\pi$
$149$	−1.29844	−0.106372	−0.0531861	+	0.998585i	$0.516938\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.40312	−0.755277
$156$	0	0
$157$	8.80625	0.702815	0.351408	−	0.936223i	$-0.385703\pi$
$157$	8.80625	0.702815	0.351408	+	0.936223i	$0.385703\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−18.1047	−1.42685
$162$	0	0
$163$	11.2984	0.884962	0.442481	−	0.896778i	$-0.354098\pi$
$163$	11.2984	0.884962	0.442481	+	0.896778i	$0.354098\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.8062	1.76480	0.882400	−	0.470500i	$-0.155926\pi$
$167$	22.8062	1.76480	0.882400	+	0.470500i	$0.155926\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	2.70156	0.204219
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.0000	1.64436	0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000	1.64436	0.822179	+	0.569230i	$0.192758\pi$
$180$	0	0
$181$	−1.29844	−0.0965121	−0.0482561	−	0.998835i	$-0.515366\pi$
$181$	−1.29844	−0.0965121	−0.0482561	+	0.998835i	$0.515366\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.70156	−0.492709
$186$	0	0
$187$	−0.492189	−0.0359925
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−22.1047	−1.59113	−0.795565	−	0.605868i	$-0.792826\pi$
$193$	−22.1047	−1.59113	−0.795565	+	0.605868i	$0.792826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.2094	1.01238	0.506188	−	0.862423i	$-0.331054\pi$
$197$	14.2094	1.01238	0.506188	+	0.862423i	$0.331054\pi$
$198$	0	0
$199$	2.80625	0.198930	0.0994648	−	0.995041i	$-0.468287\pi$
$199$	2.80625	0.198930	0.0994648	+	0.995041i	$0.468287\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−25.4031	−1.78295
$204$	0	0
$205$	−10.7016	−0.747430
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.40312	0.0970561
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−25.4031	−1.72448
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.701562	−0.0471922
$222$	0	0
$223$	3.40312	0.227890	0.113945	−	0.993487i	$-0.463651\pi$
$223$	3.40312	0.227890	0.113945	+	0.993487i	$0.463651\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.5969	0.703339	0.351670	−	0.936124i	$-0.385614\pi$
$227$	10.5969	0.703339	0.351670	+	0.936124i	$0.385614\pi$
$228$	0	0
$229$	−25.4031	−1.67869	−0.839343	−	0.543602i	$-0.817060\pi$
$229$	−25.4031	−1.67869	−0.839343	+	0.543602i	$0.817060\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.1047	0.661980	0.330990	−	0.943634i	$-0.392617\pi$
$233$	10.1047	0.661980	0.330990	+	0.943634i	$0.392617\pi$
$234$	0	0
$235$	1.40312	0.0915297
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.10469	0.394879	0.197440	−	0.980315i	$-0.436737\pi$
$239$	6.10469	0.394879	0.197440	+	0.980315i	$0.436737\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.298438	0.0190665
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.2094	1.14937	0.574683	−	0.818376i	$-0.305125\pi$
$251$	18.2094	1.14937	0.574683	+	0.818376i	$0.305125\pi$
$252$	0	0
$253$	4.70156	0.295585
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.2094	−1.26063	−0.630313	−	0.776341i	$-0.717073\pi$
$257$	−20.2094	−1.26063	−0.630313	+	0.776341i	$0.717073\pi$
$258$	0	0
$259$	−18.1047	−1.12497
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.59688	−0.283456	−0.141728	−	0.989906i	$-0.545266\pi$
$263$	−4.59688	−0.283456	−0.141728	+	0.989906i	$0.545266\pi$
$264$	0	0
$265$	−10.7016	−0.657392
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.8062	0.902753	0.451376	−	0.892334i	$-0.350933\pi$
$269$	14.8062	0.902753	0.451376	+	0.892334i	$0.350933\pi$
$270$	0	0
$271$	−6.59688	−0.400732	−0.200366	−	0.979721i	$-0.564213\pi$
$271$	−6.59688	−0.400732	−0.200366	+	0.979721i	$0.564213\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.701562	−0.0423058
$276$	0	0
$277$	8.80625	0.529116	0.264558	−	0.964370i	$-0.414774\pi$
$277$	8.80625	0.529116	0.264558	+	0.964370i	$0.414774\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.9109	−1.70656
$288$	0	0
$289$	−16.5078	−0.971048
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.4031	−0.899860	−0.449930	−	0.893064i	$-0.648551\pi$
$293$	−15.4031	−0.899860	−0.449930	+	0.893064i	$0.648551\pi$
$294$	0	0
$295$	14.8062	0.862053
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.70156	0.387561
$300$	0	0
$301$	10.8062	0.622862
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.70156	0.154691
$306$	0	0
$307$	−13.8953	−0.793047	−0.396524	−	0.918024i	$-0.629784\pi$
$307$	−13.8953	−0.793047	−0.396524	+	0.918024i	$0.629784\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	16.8062	0.949945	0.474973	−	0.880001i	$-0.342458\pi$
$313$	16.8062	0.949945	0.474973	+	0.880001i	$0.342458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.2094	−0.798078	−0.399039	−	0.916934i	$-0.630656\pi$
$317$	−14.2094	−0.798078	−0.399039	+	0.916934i	$0.630656\pi$
$318$	0	0
$319$	6.59688	0.369354
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.40312	−0.0780719
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.79063	0.208984
$330$	0	0
$331$	−14.2094	−0.781018	−0.390509	−	0.920599i	$-0.627701\pi$
$331$	−14.2094	−0.781018	−0.390509	+	0.920599i	$0.627701\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−6.20937	−0.338246	−0.169123	−	0.985595i	$-0.554094\pi$
$337$	−6.20937	−0.338246	−0.169123	+	0.985595i	$0.554094\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.59688	0.357241
$342$	0	0
$343$	−18.1047	−0.977561
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.10469	−0.327717	−0.163858	−	0.986484i	$-0.552394\pi$
$347$	−6.10469	−0.327717	−0.163858	+	0.986484i	$0.552394\pi$
$348$	0	0
$349$	−34.8062	−1.86314	−0.931568	−	0.363567i	$-0.881559\pi$
$349$	−34.8062	−1.86314	−0.931568	+	0.363567i	$0.881559\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.8062	−0.681608	−0.340804	−	0.940134i	$-0.610699\pi$
$353$	−12.8062	−0.681608	−0.340804	+	0.940134i	$0.610699\pi$
$354$	0	0
$355$	−15.5078	−0.823069
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.6125	1.14066	0.570332	−	0.821414i	$-0.306815\pi$
$359$	21.6125	1.14066	0.570332	+	0.821414i	$0.306815\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.4031	0.701552
$366$	0	0
$367$	1.19375	0.0623133	0.0311567	−	0.999515i	$-0.490081\pi$
$367$	1.19375	0.0623133	0.0311567	+	0.999515i	$0.490081\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−28.9109	−1.50098
$372$	0	0
$373$	−8.59688	−0.445129	−0.222565	−	0.974918i	$-0.571443\pi$
$373$	−8.59688	−0.445129	−0.222565	+	0.974918i	$0.571443\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.40312	0.484286
$378$	0	0
$379$	15.6125	0.801960	0.400980	−	0.916087i	$-0.368670\pi$
$379$	15.6125	0.801960	0.400980	+	0.916087i	$0.368670\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.40312	0.276087	0.138043	−	0.990426i	$-0.455919\pi$
$383$	5.40312	0.276087	0.138043	+	0.990426i	$0.455919\pi$
$384$	0	0
$385$	−1.89531	−0.0965941
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.6125	−1.29861	−0.649303	−	0.760530i	$-0.724939\pi$
$389$	−25.6125	−1.29861	−0.649303	+	0.760530i	$0.724939\pi$
$390$	0	0
$391$	−4.70156	−0.237768
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.70156	0.236561
$396$	0	0
$397$	−22.7016	−1.13936	−0.569679	−	0.821867i	$-0.692933\pi$
$397$	−22.7016	−1.13936	−0.569679	+	0.821867i	$0.692933\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.806248	−0.0402621	−0.0201311	−	0.999797i	$-0.506408\pi$
$401$	−0.806248	−0.0402621	−0.0201311	+	0.999797i	$0.506408\pi$
$402$	0	0
$403$	9.40312	0.468403
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.70156	0.233048
$408$	0	0
$409$	−4.80625	−0.237654	−0.118827	−	0.992915i	$-0.537913\pi$
$409$	−4.80625	−0.237654	−0.118827	+	0.992915i	$0.537913\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	40.0000	1.96827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.6125	−1.34896	−0.674479	−	0.738294i	$-0.735632\pi$
$419$	−27.6125	−1.34896	−0.674479	+	0.738294i	$0.735632\pi$
$420$	0	0
$421$	−25.6125	−1.24828	−0.624138	−	0.781314i	$-0.714550\pi$
$421$	−25.6125	−1.24828	−0.624138	+	0.781314i	$0.714550\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.701562	0.0340308
$426$	0	0
$427$	7.29844	0.353196
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	12.5969	0.605367	0.302684	−	0.953091i	$-0.402117\pi$
$433$	12.5969	0.605367	0.302684	+	0.953091i	$0.402117\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.4031	0.641158
$438$	0	0
$439$	−1.89531	−0.0904584	−0.0452292	−	0.998977i	$-0.514402\pi$
$439$	−1.89531	−0.0904584	−0.0452292	+	0.998977i	$0.514402\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.3141	1.25022	0.625109	−	0.780537i	$-0.285054\pi$
$443$	26.3141	1.25022	0.625109	+	0.780537i	$0.285054\pi$
$444$	0	0
$445$	−8.10469	−0.384199
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.89531	−0.183831	−0.0919156	−	0.995767i	$-0.529299\pi$
$449$	−3.89531	−0.183831	−0.0919156	+	0.995767i	$0.529299\pi$
$450$	0	0
$451$	7.50781	0.353529
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.70156	−0.126651
$456$	0	0
$457$	7.29844	0.341407	0.170703	−	0.985322i	$-0.445396\pi$
$457$	7.29844	0.341407	0.170703	+	0.985322i	$0.445396\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19.8953	−0.926617	−0.463309	−	0.886197i	$-0.653338\pi$
$461$	−19.8953	−0.926617	−0.463309	+	0.886197i	$0.653338\pi$
$462$	0	0
$463$	−4.10469	−0.190761	−0.0953805	−	0.995441i	$-0.530407\pi$
$463$	−4.10469	−0.190761	−0.0953805	+	0.995441i	$0.530407\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.1047	−0.837785	−0.418892	−	0.908036i	$-0.637582\pi$
$467$	−18.1047	−0.837785	−0.418892	+	0.908036i	$0.637582\pi$
$468$	0	0
$469$	−10.8062	−0.498986
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.80625	−0.129031
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.7016	1.31141	0.655704	−	0.755018i	$-0.272372\pi$
$479$	28.7016	1.31141	0.655704	+	0.755018i	$0.272372\pi$
$480$	0	0
$481$	6.70156	0.305565
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.1047	0.822091
$486$	0	0
$487$	−36.3141	−1.64555	−0.822774	−	0.568369i	$-0.807574\pi$
$487$	−36.3141	−1.64555	−0.822774	+	0.568369i	$0.807574\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.20937	0.280225	0.140113	−	0.990136i	$-0.455254\pi$
$491$	6.20937	0.280225	0.140113	+	0.990136i	$0.455254\pi$
$492$	0	0
$493$	−6.59688	−0.297108
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−41.8953	−1.87926
$498$	0	0
$499$	15.1938	0.680166	0.340083	−	0.940395i	$-0.389545\pi$
$499$	15.1938	0.680166	0.340083	+	0.940395i	$0.389545\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.4031	−1.40020	−0.700098	−	0.714047i	$-0.746860\pi$
$503$	−31.4031	−1.40020	−0.700098	+	0.714047i	$0.746860\pi$
$504$	0	0
$505$	6.80625	0.302874
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.3141	1.43230	0.716148	−	0.697949i	$-0.245904\pi$
$509$	32.3141	1.43230	0.716148	+	0.697949i	$0.245904\pi$
$510$	0	0
$511$	36.2094	1.60181
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	−0.984379	−0.0432929
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.59688	−0.287364
$528$	0	0
$529$	21.9109	0.952649
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.7016	0.463536
$534$	0	0
$535$	−18.1047	−0.782734
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.209373	−0.00901832
$540$	0	0
$541$	40.2094	1.72874	0.864368	−	0.502860i	$-0.167719\pi$
$541$	40.2094	1.72874	0.864368	+	0.502860i	$0.167719\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.8062	0.462889
$546$	0	0
$547$	−25.6125	−1.09511	−0.547556	−	0.836769i	$-0.684442\pi$
$547$	−25.6125	−1.09511	−0.547556	+	0.836769i	$0.684442\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	18.8062	0.801173
$552$	0	0
$553$	12.7016	0.540125
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.1047	−0.931601	−0.465801	−	0.884890i	$-0.654234\pi$
$563$	−22.1047	−0.931601	−0.465801	+	0.884890i	$0.654234\pi$
$564$	0	0
$565$	−6.59688	−0.277533
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.4031	1.14880	0.574399	−	0.818575i	$-0.305236\pi$
$569$	27.4031	1.14880	0.574399	+	0.818575i	$0.305236\pi$
$570$	0	0
$571$	26.3141	1.10121	0.550605	−	0.834766i	$-0.314397\pi$
$571$	26.3141	1.10121	0.550605	+	0.834766i	$0.314397\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.70156	−0.279474
$576$	0	0
$577$	−31.5078	−1.31169	−0.655844	−	0.754897i	$-0.727687\pi$
$577$	−31.5078	−1.31169	−0.655844	+	0.754897i	$0.727687\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.8062	−0.448319
$582$	0	0
$583$	7.50781	0.310942
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.2094	0.999228	0.499614	−	0.866248i	$-0.333475\pi$
$587$	24.2094	0.999228	0.499614	+	0.866248i	$0.333475\pi$
$588$	0	0
$589$	18.8062	0.774898
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.40312	0.139750	0.0698748	−	0.997556i	$-0.477740\pi$
$593$	3.40312	0.139750	0.0698748	+	0.997556i	$0.477740\pi$
$594$	0	0
$595$	1.89531	0.0777003
$596$	0	0
$597$	0	0
$598$	0	0
$599$	42.8062	1.74902	0.874508	−	0.485011i	$-0.161184\pi$
$599$	42.8062	1.74902	0.874508	+	0.485011i	$0.161184\pi$
$600$	0	0
$601$	9.29844	0.379291	0.189646	−	0.981853i	$-0.439266\pi$
$601$	9.29844	0.379291	0.189646	+	0.981853i	$0.439266\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.5078	−0.427203
$606$	0	0
$607$	−41.6125	−1.68900	−0.844500	−	0.535556i	$-0.820102\pi$
$607$	−41.6125	−1.68900	−0.844500	+	0.535556i	$0.820102\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.40312	−0.0567643
$612$	0	0
$613$	−45.7172	−1.84650	−0.923250	−	0.384200i	$-0.874477\pi$
$613$	−45.7172	−1.84650	−0.923250	+	0.384200i	$0.874477\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	39.6125	1.59216	0.796080	−	0.605191i	$-0.206903\pi$
$619$	39.6125	1.59216	0.796080	+	0.605191i	$0.206903\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21.8953	−0.877217
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.70156	−0.187464
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.4031	−0.531887
$636$	0	0
$637$	−0.298438	−0.0118245
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.2094	0.403246	0.201623	−	0.979463i	$-0.435378\pi$
$641$	10.2094	0.403246	0.201623	+	0.979463i	$0.435378\pi$
$642$	0	0
$643$	31.2984	1.23429	0.617145	−	0.786849i	$-0.288289\pi$
$643$	31.2984	1.23429	0.617145	+	0.786849i	$0.288289\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.1047	−0.790397	−0.395198	−	0.918596i	$-0.629324\pi$
$647$	−20.1047	−0.790397	−0.395198	+	0.918596i	$0.629324\pi$
$648$	0	0
$649$	−10.3875	−0.407745
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.6125	−0.610964	−0.305482	−	0.952198i	$-0.598818\pi$
$653$	−15.6125	−0.610964	−0.305482	+	0.952198i	$0.598818\pi$
$654$	0	0
$655$	12.8062	0.500382
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.4031	1.06747	0.533737	−	0.845650i	$-0.320787\pi$
$659$	27.4031	1.06747	0.533737	+	0.845650i	$0.320787\pi$
$660$	0	0
$661$	−12.0000	−0.466746	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000	−0.466746	−0.233373	+	0.972387i	$0.574976\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.40312	−0.209524
$666$	0	0
$667$	63.0156	2.43997
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.89531	−0.0731678
$672$	0	0
$673$	−3.19375	−0.123110	−0.0615550	−	0.998104i	$-0.519606\pi$
$673$	−3.19375	−0.123110	−0.0615550	+	0.998104i	$0.519606\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.10469	0.157756	0.0788780	−	0.996884i	$-0.474866\pi$
$677$	4.10469	0.157756	0.0788780	+	0.996884i	$0.474866\pi$
$678$	0	0
$679$	48.9109	1.87703
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.7016	0.407697
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.70156	−0.178340
$696$	0	0
$697$	−7.50781	−0.284379
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25.6125	0.967371	0.483685	−	0.875242i	$-0.339298\pi$
$701$	25.6125	0.967371	0.483685	+	0.875242i	$0.339298\pi$
$702$	0	0
$703$	13.4031	0.505508
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.3875	0.691533
$708$	0	0
$709$	42.8062	1.60762	0.803811	−	0.594884i	$-0.202802\pi$
$709$	42.8062	1.60762	0.803811	+	0.594884i	$0.202802\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	63.0156	2.35995
$714$	0	0
$715$	0.701562	0.0262369
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.19375	−0.342869	−0.171435	−	0.985196i	$-0.554840\pi$
$719$	−9.19375	−0.342869	−0.171435	+	0.985196i	$0.554840\pi$
$720$	0	0
$721$	32.4187	1.20734
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.40312	−0.349223
$726$	0	0
$727$	−9.40312	−0.348743	−0.174371	−	0.984680i	$-0.555789\pi$
$727$	−9.40312	−0.348743	−0.174371	+	0.984680i	$0.555789\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.80625	0.103793
$732$	0	0
$733$	41.7172	1.54086	0.770430	−	0.637525i	$-0.220042\pi$
$733$	41.7172	1.54086	0.770430	+	0.637525i	$0.220042\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.80625	0.103369
$738$	0	0
$739$	−8.80625	−0.323943	−0.161972	−	0.986795i	$-0.551785\pi$
$739$	−8.80625	−0.323943	−0.161972	+	0.986795i	$0.551785\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.5969	0.388762	0.194381	−	0.980926i	$-0.437730\pi$
$743$	10.5969	0.388762	0.194381	+	0.980926i	$0.437730\pi$
$744$	0	0
$745$	−1.29844	−0.0475711
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−48.9109	−1.78717
$750$	0	0
$751$	−42.3141	−1.54406	−0.772031	−	0.635585i	$-0.780759\pi$
$751$	−42.3141	−1.54406	−0.772031	+	0.635585i	$0.780759\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−47.4031	−1.72290	−0.861448	−	0.507846i	$-0.830442\pi$
$757$	−47.4031	−1.72290	−0.861448	+	0.507846i	$0.830442\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	29.1938	1.05688
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.8062	−0.534623
$768$	0	0
$769$	51.4031	1.85364	0.926822	−	0.375501i	$-0.122529\pi$
$769$	51.4031	1.85364	0.926822	+	0.375501i	$0.122529\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.5969	1.17243	0.586214	−	0.810156i	$-0.300618\pi$
$773$	32.5969	1.17243	0.586214	+	0.810156i	$0.300618\pi$
$774$	0	0
$775$	−9.40312	−0.337770
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21.4031	0.766847
$780$	0	0
$781$	10.8797	0.389306
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.80625	0.314308
$786$	0	0
$787$	5.19375	0.185137	0.0925686	−	0.995706i	$-0.470492\pi$
$787$	5.19375	0.185137	0.0925686	+	0.995706i	$0.470492\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17.8219	−0.633673
$792$	0	0
$793$	−2.70156	−0.0959353
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.7016	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$797$	14.7016	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$798$	0	0
$799$	0.984379	0.0348248
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.40312	−0.331829
$804$	0	0
$805$	−18.1047	−0.638106
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.806248	0.0283462	0.0141731	−	0.999900i	$-0.495488\pi$
$809$	0.806248	0.0283462	0.0141731	+	0.999900i	$0.495488\pi$
$810$	0	0
$811$	−14.2094	−0.498959	−0.249479	−	0.968380i	$-0.580259\pi$
$811$	−14.2094	−0.498959	−0.249479	+	0.968380i	$0.580259\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.2984	0.395767
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.49219	0.0869780	0.0434890	−	0.999054i	$-0.486153\pi$
$821$	2.49219	0.0869780	0.0434890	+	0.999054i	$0.486153\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.5969	−1.20305	−0.601526	−	0.798854i	$-0.705440\pi$
$827$	−34.5969	−1.20305	−0.601526	+	0.798854i	$0.705440\pi$
$828$	0	0
$829$	−47.6125	−1.65365	−0.826825	−	0.562459i	$-0.809855\pi$
$829$	−47.6125	−1.65365	−0.826825	+	0.562459i	$0.809855\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.209373	0.00725433
$834$	0	0
$835$	22.8062	0.789243
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.1203	−1.55773	−0.778863	−	0.627194i	$-0.784203\pi$
$839$	−45.1203	−1.55773	−0.778863	+	0.627194i	$0.784203\pi$
$840$	0	0
$841$	59.4187	2.04892
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−28.3875	−0.975406
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.9109	1.53953
$852$	0	0
$853$	−21.7172	−0.743582	−0.371791	−	0.928316i	$-0.621256\pi$
$853$	−21.7172	−0.743582	−0.371791	+	0.928316i	$0.621256\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.7016	−0.433877	−0.216939	−	0.976185i	$-0.569607\pi$
$857$	−12.7016	−0.433877	−0.216939	+	0.976185i	$0.569607\pi$
$858$	0	0
$859$	31.2984	1.06789	0.533944	−	0.845520i	$-0.320709\pi$
$859$	31.2984	1.06789	0.533944	+	0.845520i	$0.320709\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.4031	−0.456248	−0.228124	−	0.973632i	$-0.573259\pi$
$863$	−13.4031	−0.456248	−0.228124	+	0.973632i	$0.573259\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.29844	−0.111892
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.70156	0.0913295
$876$	0	0
$877$	−3.61250	−0.121985	−0.0609927	−	0.998138i	$-0.519427\pi$
$877$	−3.61250	−0.121985	−0.0609927	+	0.998138i	$0.519427\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.8062	0.431453	0.215727	−	0.976454i	$-0.430788\pi$
$881$	12.8062	0.431453	0.215727	+	0.976454i	$0.430788\pi$
$882$	0	0
$883$	−17.6125	−0.592708	−0.296354	−	0.955078i	$-0.595771\pi$
$883$	−17.6125	−0.592708	−0.296354	+	0.955078i	$0.595771\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−18.4922	−0.620907	−0.310453	−	0.950589i	$-0.600481\pi$
$887$	−18.4922	−0.620907	−0.310453	+	0.950589i	$0.600481\pi$
$888$	0	0
$889$	−36.2094	−1.21442
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.80625	−0.0939075
$894$	0	0
$895$	22.0000	0.735379
$896$	0	0
$897$	0	0
$898$	0	0
$899$	88.4187	2.94893
$900$	0	0
$901$	−7.50781	−0.250121
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.29844	−0.0431615
$906$	0	0
$907$	−16.2094	−0.538223	−0.269112	−	0.963109i	$-0.586730\pi$
$907$	−16.2094	−0.538223	−0.269112	+	0.963109i	$0.586730\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.80625	−0.225501	−0.112751	−	0.993623i	$-0.535966\pi$
$911$	−6.80625	−0.225501	−0.112751	+	0.993623i	$0.535966\pi$
$912$	0	0
$913$	2.80625	0.0928733
$914$	0	0
$915$	0	0
$916$	0	0
$917$	34.5969	1.14249
$918$	0	0
$919$	7.08907	0.233847	0.116923	−	0.993141i	$-0.462697\pi$
$919$	7.08907	0.233847	0.116923	+	0.993141i	$0.462697\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.5078	0.510446
$924$	0	0
$925$	−6.70156	−0.220346
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.10469	−0.265906	−0.132953	−	0.991122i	$-0.542446\pi$
$929$	−8.10469	−0.265906	−0.132953	+	0.991122i	$0.542446\pi$
$930$	0	0
$931$	−0.596876	−0.0195618
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.492189	−0.0160963
$936$	0	0
$937$	52.8062	1.72510	0.862552	−	0.505968i	$-0.168864\pi$
$937$	52.8062	1.72510	0.862552	+	0.505968i	$0.168864\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	54.7016	1.78322	0.891610	−	0.452804i	$-0.149576\pi$
$941$	54.7016	1.78322	0.891610	+	0.452804i	$0.149576\pi$
$942$	0	0
$943$	71.7172	2.33543
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.61250	0.312364	0.156182	−	0.987728i	$-0.450081\pi$
$947$	9.61250	0.312364	0.156182	+	0.987728i	$0.450081\pi$
$948$	0	0
$949$	−13.4031	−0.435084
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.91093	−0.159081	−0.0795404	−	0.996832i	$-0.525345\pi$
$953$	−4.91093	−0.159081	−0.0795404	+	0.996832i	$0.525345\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−48.6281	−1.57028
$960$	0	0
$961$	57.4187	1.85222
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22.1047	−0.711575
$966$	0	0
$967$	39.8219	1.28058	0.640292	−	0.768131i	$-0.278813\pi$
$967$	39.8219	1.28058	0.640292	+	0.768131i	$0.278813\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−57.0156	−1.82972	−0.914859	−	0.403773i	$-0.867699\pi$
$971$	−57.0156	−1.82972	−0.914859	+	0.403773i	$0.867699\pi$
$972$	0	0
$973$	−12.7016	−0.407193
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26.2094	−0.838512	−0.419256	−	0.907868i	$-0.637709\pi$
$977$	−26.2094	−0.838512	−0.419256	+	0.907868i	$0.637709\pi$
$978$	0	0
$979$	5.68594	0.181723
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.0000	0.637901	0.318950	−	0.947771i	$-0.396670\pi$
$983$	20.0000	0.637901	0.318950	+	0.947771i	$0.396670\pi$
$984$	0	0
$985$	14.2094	0.452748
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.8062	−0.852389
$990$	0	0
$991$	27.7172	0.880465	0.440233	−	0.897884i	$-0.354896\pi$
$991$	27.7172	0.880465	0.440233	+	0.897884i	$0.354896\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.80625	0.0889641
$996$	0	0
$997$	−51.4031	−1.62795	−0.813977	−	0.580898i	$-0.802702\pi$
$997$	−51.4031	−1.62795	−0.813977	+	0.580898i	$0.802702\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bb.1.2		2
3.2	odd	2		1560.2.a.m.1.2	✓	2
4.3	odd	2		9360.2.a.ct.1.1		2
12.11	even	2		3120.2.a.bf.1.1		2
15.14	odd	2		7800.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.m.1.2	✓	2	3.2	odd	2
3120.2.a.bf.1.1		2	12.11	even	2
4680.2.a.bb.1.2		2	1.1	even	1	trivial
7800.2.a.be.1.1		2	15.14	odd	2
9360.2.a.ct.1.1		2	4.3	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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.70156 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +2.70156 q^{7} -0.701562 q^{11} -1.00000 q^{13} +0.701562 q^{17} -2.00000 q^{19} -6.70156 q^{23} +1.00000 q^{25} -9.40312 q^{29} -9.40312 q^{31} +2.70156 q^{35} -6.70156 q^{37} -10.7016 q^{41} +4.00000 q^{43} +1.40312 q^{47} +0.298438 q^{49} -10.7016 q^{53} -0.701562 q^{55} +14.8062 q^{59} +2.70156 q^{61} -1.00000 q^{65} -4.00000 q^{67} -15.5078 q^{71} +13.4031 q^{73} -1.89531 q^{77} +4.70156 q^{79} -4.00000 q^{83} +0.701562 q^{85} -8.10469 q^{89} -2.70156 q^{91} -2.00000 q^{95} +18.1047 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - q^7	\( 2 q + 2 q^{5} - q^{7} + 5 q^{11} - 2 q^{13} - 5 q^{17} - 4 q^{19} - 7 q^{23} + 2 q^{25} - 6 q^{29} - 6 q^{31} - q^{35} - 7 q^{37} - 15 q^{41} + 8 q^{43} - 10 q^{47} + 7 q^{49} - 15 q^{53} + 5 q^{55} + 4 q^{59} - q^{61} - 2 q^{65} - 8 q^{67} + q^{71} + 14 q^{73} - 23 q^{77} + 3 q^{79} - 8 q^{83} - 5 q^{85} + 3 q^{89} + q^{91} - 4 q^{95} + 17 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - q^7 + 5 * q^11 - 2 * q^13 - 5 * q^17 - 4 * q^19 - 7 * q^23 + 2 * q^25 - 6 * q^29 - 6 * q^31 - q^35 - 7 * q^37 - 15 * q^41 + 8 * q^43 - 10 * q^47 + 7 * q^49 - 15 * q^53 + 5 * q^55 + 4 * q^59 - q^61 - 2 * q^65 - 8 * q^67 + q^71 + 14 * q^73 - 23 * q^77 + 3 * q^79 - 8 * q^83 - 5 * q^85 + 3 * q^89 + q^91 - 4 * q^95 + 17 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more