LMFDB - Embedded newform 9016.2.a.bs.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9016,2,Mod(1,9016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9016 = 2^{3} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9016.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:	$71.9931224624$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 28x^{12} + 293x^{10} - 1394x^{8} + 2848x^{6} - 1722x^{4} + 332x^{2} - 8 $ x^14 - 28x^12 + 293x^10 - 1394x^8 + 2848x^6 - 1722x^4 + 332x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.667834$ of defining polynomial
Character	$\chi$	$=$	9016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.667834 q^{3} +2.41880 q^{5} -2.55400 q^{9} +O(q^{10})$	$q-0.667834 q^{3} +2.41880 q^{5} -2.55400 q^{9} +3.80534 q^{11} -3.16518 q^{13} -1.61535 q^{15} -3.18546 q^{17} +4.09969 q^{19} -1.00000 q^{23} +0.850581 q^{25} +3.70915 q^{27} -8.76815 q^{29} -3.04765 q^{31} -2.54134 q^{33} +9.87048 q^{37} +2.11381 q^{39} +8.57374 q^{41} -4.03084 q^{43} -6.17760 q^{45} -4.48043 q^{47} +2.12736 q^{51} -3.38986 q^{53} +9.20435 q^{55} -2.73791 q^{57} -8.92748 q^{59} -3.70025 q^{61} -7.65592 q^{65} +3.69426 q^{67} +0.667834 q^{69} +0.602036 q^{71} +4.66237 q^{73} -0.568046 q^{75} -8.09598 q^{79} +5.18490 q^{81} -2.78084 q^{83} -7.70497 q^{85} +5.85567 q^{87} -15.1459 q^{89} +2.03532 q^{93} +9.91632 q^{95} -0.777467 q^{97} -9.71883 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{9}+O(q^{10}) $ 14 * q + 14 * q^9	$ 14 q + 14 q^{9} - 8 q^{11} - 28 q^{15} - 14 q^{23} + 10 q^{25} + 4 q^{29} + 12 q^{37} - 24 q^{39} - 44 q^{43} - 28 q^{51} - 4 q^{53} - 8 q^{57} - 44 q^{65} - 28 q^{67} - 28 q^{71} - 36 q^{79} + 18 q^{81} + 8 q^{85} + 44 q^{93} - 8 q^{95} - 60 q^{99}+O(q^{100}) $ 14 * q + 14 * q^9 - 8 * q^11 - 28 * q^15 - 14 * q^23 + 10 * q^25 + 4 * q^29 + 12 * q^37 - 24 * q^39 - 44 * q^43 - 28 * q^51 - 4 * q^53 - 8 * q^57 - 44 * q^65 - 28 * q^67 - 28 * q^71 - 36 * q^79 + 18 * q^81 + 8 * q^85 + 44 * q^93 - 8 * q^95 - 60 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.667834	−0.385574	−0.192787	−	0.981241i	$-0.561753\pi$
$3$	−0.667834	−0.385574	−0.192787	+	0.981241i	$0.561753\pi$
$4$	0	0
$5$	2.41880	1.08172	0.540860	−	0.841113i	$-0.318099\pi$
$5$	2.41880	1.08172	0.540860	+	0.841113i	$0.318099\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.55400	−0.851333
$10$	0	0
$11$	3.80534	1.14735	0.573677	−	0.819082i	$-0.305517\pi$
$11$	3.80534	1.14735	0.573677	+	0.819082i	$0.305517\pi$
$12$	0	0
$13$	−3.16518	−0.877862	−0.438931	−	0.898521i	$-0.644643\pi$
$13$	−3.16518	−0.877862	−0.438931	+	0.898521i	$0.644643\pi$
$14$	0	0
$15$	−1.61535	−0.417083
$16$	0	0
$17$	−3.18546	−0.772587	−0.386293	−	0.922376i	$-0.626245\pi$
$17$	−3.18546	−0.772587	−0.386293	+	0.922376i	$0.626245\pi$
$18$	0	0
$19$	4.09969	0.940534	0.470267	−	0.882524i	$-0.344158\pi$
$19$	4.09969	0.940534	0.470267	+	0.882524i	$0.344158\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.850581	0.170116
$26$	0	0
$27$	3.70915	0.713826
$28$	0	0
$29$	−8.76815	−1.62820	−0.814102	−	0.580722i	$-0.802770\pi$
$29$	−8.76815	−1.62820	−0.814102	+	0.580722i	$0.802770\pi$
$30$	0	0
$31$	−3.04765	−0.547374	−0.273687	−	0.961819i	$-0.588243\pi$
$31$	−3.04765	−0.547374	−0.273687	+	0.961819i	$0.588243\pi$
$32$	0	0
$33$	−2.54134	−0.442390
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.87048	1.62270	0.811349	−	0.584563i	$-0.198734\pi$
$37$	9.87048	1.62270	0.811349	+	0.584563i	$0.198734\pi$
$38$	0	0
$39$	2.11381	0.338481
$40$	0	0
$41$	8.57374	1.33899	0.669497	−	0.742815i	$-0.266510\pi$
$41$	8.57374	1.33899	0.669497	+	0.742815i	$0.266510\pi$
$42$	0	0
$43$	−4.03084	−0.614697	−0.307348	−	0.951597i	$-0.599442\pi$
$43$	−4.03084	−0.614697	−0.307348	+	0.951597i	$0.599442\pi$
$44$	0	0
$45$	−6.17760	−0.920903
$46$	0	0
$47$	−4.48043	−0.653538	−0.326769	−	0.945104i	$-0.605960\pi$
$47$	−4.48043	−0.653538	−0.326769	+	0.945104i	$0.605960\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.12736	0.297889
$52$	0	0
$53$	−3.38986	−0.465633	−0.232816	−	0.972521i	$-0.574794\pi$
$53$	−3.38986	−0.465633	−0.232816	+	0.972521i	$0.574794\pi$
$54$	0	0
$55$	9.20435	1.24111
$56$	0	0
$57$	−2.73791	−0.362645
$58$	0	0
$59$	−8.92748	−1.16226	−0.581130	−	0.813811i	$-0.697389\pi$
$59$	−8.92748	−1.16226	−0.581130	+	0.813811i	$0.697389\pi$
$60$	0	0
$61$	−3.70025	−0.473768	−0.236884	−	0.971538i	$-0.576126\pi$
$61$	−3.70025	−0.473768	−0.236884	+	0.971538i	$0.576126\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.65592	−0.949600
$66$	0	0
$67$	3.69426	0.451326	0.225663	−	0.974205i	$-0.427545\pi$
$67$	3.69426	0.451326	0.225663	+	0.974205i	$0.427545\pi$
$68$	0	0
$69$	0.667834	0.0803977
$70$	0	0
$71$	0.602036	0.0714486	0.0357243	−	0.999362i	$-0.488626\pi$
$71$	0.602036	0.0714486	0.0357243	+	0.999362i	$0.488626\pi$
$72$	0	0
$73$	4.66237	0.545689	0.272844	−	0.962058i	$-0.412036\pi$
$73$	4.66237	0.545689	0.272844	+	0.962058i	$0.412036\pi$
$74$	0	0
$75$	−0.568046	−0.0655924
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.09598	−0.910869	−0.455434	−	0.890269i	$-0.650516\pi$
$79$	−8.09598	−0.910869	−0.455434	+	0.890269i	$0.650516\pi$
$80$	0	0
$81$	5.18490	0.576100
$82$	0	0
$83$	−2.78084	−0.305236	−0.152618	−	0.988285i	$-0.548770\pi$
$83$	−2.78084	−0.305236	−0.152618	+	0.988285i	$0.548770\pi$
$84$	0	0
$85$	−7.70497	−0.835722
$86$	0	0
$87$	5.85567	0.627793
$88$	0	0
$89$	−15.1459	−1.60546	−0.802729	−	0.596343i	$-0.796620\pi$
$89$	−15.1459	−1.60546	−0.802729	+	0.596343i	$0.796620\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.03532	0.211053
$94$	0	0
$95$	9.91632	1.01739
$96$	0	0
$97$	−0.777467	−0.0789398	−0.0394699	−	0.999221i	$-0.512567\pi$
$97$	−0.777467	−0.0789398	−0.0394699	+	0.999221i	$0.512567\pi$
$98$	0	0
$99$	−9.71883	−0.976780
$100$	0	0
$101$	19.3602	1.92641	0.963204	−	0.268772i	$-0.0866178\pi$
$101$	19.3602	1.92641	0.963204	+	0.268772i	$0.0866178\pi$
$102$	0	0
$103$	2.83463	0.279304	0.139652	−	0.990201i	$-0.455402\pi$
$103$	2.83463	0.279304	0.139652	+	0.990201i	$0.455402\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.3203	−1.28772	−0.643862	−	0.765141i	$-0.722669\pi$
$107$	−13.3203	−1.28772	−0.643862	+	0.765141i	$0.722669\pi$
$108$	0	0
$109$	−7.56484	−0.724580	−0.362290	−	0.932065i	$-0.618005\pi$
$109$	−7.56484	−0.724580	−0.362290	+	0.932065i	$0.618005\pi$
$110$	0	0
$111$	−6.59184	−0.625670
$112$	0	0
$113$	−9.59371	−0.902500	−0.451250	−	0.892398i	$-0.649022\pi$
$113$	−9.59371	−0.902500	−0.451250	+	0.892398i	$0.649022\pi$
$114$	0	0
$115$	−2.41880	−0.225554
$116$	0	0
$117$	8.08386	0.747353
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.48062	0.316420
$122$	0	0
$123$	−5.72583	−0.516281
$124$	0	0
$125$	−10.0366	−0.897701
$126$	0	0
$127$	13.4828	1.19641	0.598203	−	0.801344i	$-0.295881\pi$
$127$	13.4828	1.19641	0.598203	+	0.801344i	$0.295881\pi$
$128$	0	0
$129$	2.69193	0.237011
$130$	0	0
$131$	−5.04677	−0.440939	−0.220469	−	0.975394i	$-0.570759\pi$
$131$	−5.04677	−0.440939	−0.220469	+	0.975394i	$0.570759\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.97168	0.772159
$136$	0	0
$137$	−1.43510	−0.122609	−0.0613044	−	0.998119i	$-0.519526\pi$
$137$	−1.43510	−0.122609	−0.0613044	+	0.998119i	$0.519526\pi$
$138$	0	0
$139$	−17.0132	−1.44304	−0.721522	−	0.692392i	$-0.756557\pi$
$139$	−17.0132	−1.44304	−0.721522	+	0.692392i	$0.756557\pi$
$140$	0	0
$141$	2.99219	0.251987
$142$	0	0
$143$	−12.0446	−1.00722
$144$	0	0
$145$	−21.2084	−1.76126
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.90332	−0.155926	−0.0779631	−	0.996956i	$-0.524842\pi$
$149$	−1.90332	−0.155926	−0.0779631	+	0.996956i	$0.524842\pi$
$150$	0	0
$151$	−3.36694	−0.273998	−0.136999	−	0.990571i	$-0.543746\pi$
$151$	−3.36694	−0.273998	−0.136999	+	0.990571i	$0.543746\pi$
$152$	0	0
$153$	8.13565	0.657728
$154$	0	0
$155$	−7.37165	−0.592105
$156$	0	0
$157$	7.18304	0.573269	0.286635	−	0.958040i	$-0.407463\pi$
$157$	7.18304	0.573269	0.286635	+	0.958040i	$0.407463\pi$
$158$	0	0
$159$	2.26386	0.179536
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.5712	−1.84624	−0.923119	−	0.384513i	$-0.874369\pi$
$163$	−23.5712	−1.84624	−0.923119	+	0.384513i	$0.874369\pi$
$164$	0	0
$165$	−6.14698	−0.478541
$166$	0	0
$167$	3.84320	0.297395	0.148698	−	0.988883i	$-0.452492\pi$
$167$	3.84320	0.297395	0.148698	+	0.988883i	$0.452492\pi$
$168$	0	0
$169$	−2.98165	−0.229358
$170$	0	0
$171$	−10.4706	−0.800707
$172$	0	0
$173$	8.28797	0.630123	0.315061	−	0.949071i	$-0.397975\pi$
$173$	8.28797	0.630123	0.315061	+	0.949071i	$0.397975\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.96208	0.448137
$178$	0	0
$179$	11.7670	0.879509	0.439755	−	0.898118i	$-0.355065\pi$
$179$	11.7670	0.879509	0.439755	+	0.898118i	$0.355065\pi$
$180$	0	0
$181$	3.18188	0.236507	0.118254	−	0.992983i	$-0.462270\pi$
$181$	3.18188	0.236507	0.118254	+	0.992983i	$0.462270\pi$
$182$	0	0
$183$	2.47115	0.182673
$184$	0	0
$185$	23.8747	1.75530
$186$	0	0
$187$	−12.1217	−0.886430
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.09836	−0.0794749	−0.0397374	−	0.999210i	$-0.512652\pi$
$191$	−1.09836	−0.0794749	−0.0397374	+	0.999210i	$0.512652\pi$
$192$	0	0
$193$	11.6317	0.837270	0.418635	−	0.908155i	$-0.362509\pi$
$193$	11.6317	0.837270	0.418635	+	0.908155i	$0.362509\pi$
$194$	0	0
$195$	5.11288	0.366141
$196$	0	0
$197$	−19.9102	−1.41854	−0.709271	−	0.704936i	$-0.750976\pi$
$197$	−19.9102	−1.41854	−0.709271	+	0.704936i	$0.750976\pi$
$198$	0	0
$199$	19.5924	1.38887	0.694434	−	0.719556i	$-0.255655\pi$
$199$	19.5924	1.38887	0.694434	+	0.719556i	$0.255655\pi$
$200$	0	0
$201$	−2.46715	−0.174019
$202$	0	0
$203$	0	0
$204$	0	0
$205$	20.7381	1.44841
$206$	0	0
$207$	2.55400	0.177515
$208$	0	0
$209$	15.6007	1.07912
$210$	0	0
$211$	−5.50815	−0.379197	−0.189598	−	0.981862i	$-0.560719\pi$
$211$	−5.50815	−0.379197	−0.189598	+	0.981862i	$0.560719\pi$
$212$	0	0
$213$	−0.402060	−0.0275487
$214$	0	0
$215$	−9.74978	−0.664929
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.11369	−0.210403
$220$	0	0
$221$	10.0825	0.678225
$222$	0	0
$223$	−22.9710	−1.53826	−0.769128	−	0.639095i	$-0.779309\pi$
$223$	−22.9710	−1.53826	−0.769128	+	0.639095i	$0.779309\pi$
$224$	0	0
$225$	−2.17238	−0.144825
$226$	0	0
$227$	−17.9665	−1.19248	−0.596241	−	0.802806i	$-0.703340\pi$
$227$	−17.9665	−1.19248	−0.596241	+	0.802806i	$0.703340\pi$
$228$	0	0
$229$	−8.54384	−0.564593	−0.282296	−	0.959327i	$-0.591096\pi$
$229$	−8.54384	−0.564593	−0.282296	+	0.959327i	$0.591096\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.8068	−1.62515	−0.812574	−	0.582858i	$-0.801934\pi$
$233$	−24.8068	−1.62515	−0.812574	+	0.582858i	$0.801934\pi$
$234$	0	0
$235$	−10.8373	−0.706945
$236$	0	0
$237$	5.40677	0.351207
$238$	0	0
$239$	−28.5179	−1.84467	−0.922335	−	0.386391i	$-0.873722\pi$
$239$	−28.5179	−1.84467	−0.922335	+	0.386391i	$0.873722\pi$
$240$	0	0
$241$	18.2794	1.17748	0.588740	−	0.808323i	$-0.299624\pi$
$241$	18.2794	1.17748	0.588740	+	0.808323i	$0.299624\pi$
$242$	0	0
$243$	−14.5901	−0.935955
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.9762	−0.825659
$248$	0	0
$249$	1.85714	0.117691
$250$	0	0
$251$	9.65201	0.609229	0.304615	−	0.952476i	$-0.401472\pi$
$251$	9.65201	0.609229	0.304615	+	0.952476i	$0.401472\pi$
$252$	0	0
$253$	−3.80534	−0.239240
$254$	0	0
$255$	5.14564	0.322233
$256$	0	0
$257$	22.5047	1.40380	0.701901	−	0.712274i	$-0.252335\pi$
$257$	22.5047	1.40380	0.701901	+	0.712274i	$0.252335\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	22.3938	1.38614
$262$	0	0
$263$	1.46152	0.0901213	0.0450606	−	0.998984i	$-0.485652\pi$
$263$	1.46152	0.0901213	0.0450606	+	0.998984i	$0.485652\pi$
$264$	0	0
$265$	−8.19938	−0.503684
$266$	0	0
$267$	10.1149	0.619023
$268$	0	0
$269$	−8.22604	−0.501551	−0.250775	−	0.968045i	$-0.580686\pi$
$269$	−8.22604	−0.501551	−0.250775	+	0.968045i	$0.580686\pi$
$270$	0	0
$271$	−22.1234	−1.34390	−0.671951	−	0.740596i	$-0.734543\pi$
$271$	−22.1234	−1.34390	−0.671951	+	0.740596i	$0.734543\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.23675	0.195183
$276$	0	0
$277$	10.6518	0.640006	0.320003	−	0.947417i	$-0.396316\pi$
$277$	10.6518	0.640006	0.320003	+	0.947417i	$0.396316\pi$
$278$	0	0
$279$	7.78369	0.465997
$280$	0	0
$281$	22.7930	1.35972	0.679858	−	0.733343i	$-0.262041\pi$
$281$	22.7930	1.35972	0.679858	+	0.733343i	$0.262041\pi$
$282$	0	0
$283$	14.9240	0.887138	0.443569	−	0.896240i	$-0.353712\pi$
$283$	14.9240	0.887138	0.443569	+	0.896240i	$0.353712\pi$
$284$	0	0
$285$	−6.62246	−0.392280
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.85287	−0.403110
$290$	0	0
$291$	0.519219	0.0304371
$292$	0	0
$293$	11.8325	0.691262	0.345631	−	0.938371i	$-0.387665\pi$
$293$	11.8325	0.691262	0.345631	+	0.938371i	$0.387665\pi$
$294$	0	0
$295$	−21.5938	−1.25724
$296$	0	0
$297$	14.1146	0.819011
$298$	0	0
$299$	3.16518	0.183047
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−12.9294	−0.742773
$304$	0	0
$305$	−8.95014	−0.512484
$306$	0	0
$307$	11.9854	0.684041	0.342021	−	0.939692i	$-0.388889\pi$
$307$	11.9854	0.684041	0.342021	+	0.939692i	$0.388889\pi$
$308$	0	0
$309$	−1.89306	−0.107693
$310$	0	0
$311$	14.3107	0.811484	0.405742	−	0.913988i	$-0.367013\pi$
$311$	14.3107	0.811484	0.405742	+	0.913988i	$0.367013\pi$
$312$	0	0
$313$	19.8945	1.12450	0.562251	−	0.826966i	$-0.309935\pi$
$313$	19.8945	1.12450	0.562251	+	0.826966i	$0.309935\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.0652	−1.18314	−0.591569	−	0.806255i	$-0.701491\pi$
$317$	−21.0652	−1.18314	−0.591569	+	0.806255i	$0.701491\pi$
$318$	0	0
$319$	−33.3658	−1.86813
$320$	0	0
$321$	8.89577	0.496513
$322$	0	0
$323$	−13.0594	−0.726644
$324$	0	0
$325$	−2.69224	−0.149339
$326$	0	0
$327$	5.05206	0.279379
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.5544	−1.40460	−0.702298	−	0.711883i	$-0.747843\pi$
$331$	−25.5544	−1.40460	−0.702298	+	0.711883i	$0.747843\pi$
$332$	0	0
$333$	−25.2092	−1.38146
$334$	0	0
$335$	8.93567	0.488208
$336$	0	0
$337$	−6.63068	−0.361196	−0.180598	−	0.983557i	$-0.557803\pi$
$337$	−6.63068	−0.361196	−0.180598	+	0.983557i	$0.557803\pi$
$338$	0	0
$339$	6.40700	0.347981
$340$	0	0
$341$	−11.5973	−0.628031
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.61535	0.0869678
$346$	0	0
$347$	−34.0429	−1.82752	−0.913759	−	0.406256i	$-0.866834\pi$
$347$	−34.0429	−1.82752	−0.913759	+	0.406256i	$0.866834\pi$
$348$	0	0
$349$	−2.02935	−0.108629	−0.0543143	−	0.998524i	$-0.517297\pi$
$349$	−2.02935	−0.108629	−0.0543143	+	0.998524i	$0.517297\pi$
$350$	0	0
$351$	−11.7401	−0.626641
$352$	0	0
$353$	27.7487	1.47691	0.738457	−	0.674301i	$-0.235555\pi$
$353$	27.7487	1.47691	0.738457	+	0.674301i	$0.235555\pi$
$354$	0	0
$355$	1.45620	0.0772873
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0883	1.06022	0.530111	−	0.847929i	$-0.322150\pi$
$359$	20.0883	1.06022	0.530111	+	0.847929i	$0.322150\pi$
$360$	0	0
$361$	−2.19253	−0.115396
$362$	0	0
$363$	−2.32448	−0.122003
$364$	0	0
$365$	11.2773	0.590282
$366$	0	0
$367$	−9.20680	−0.480591	−0.240295	−	0.970700i	$-0.577244\pi$
$367$	−9.20680	−0.480591	−0.240295	+	0.970700i	$0.577244\pi$
$368$	0	0
$369$	−21.8973	−1.13993
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5.26635	−0.272681	−0.136341	−	0.990662i	$-0.543534\pi$
$373$	−5.26635	−0.272681	−0.136341	+	0.990662i	$0.543534\pi$
$374$	0	0
$375$	6.70278	0.346130
$376$	0	0
$377$	27.7527	1.42934
$378$	0	0
$379$	−38.3694	−1.97090	−0.985452	−	0.169953i	$-0.945638\pi$
$379$	−38.3694	−1.97090	−0.985452	+	0.169953i	$0.945638\pi$
$380$	0	0
$381$	−9.00428	−0.461303
$382$	0	0
$383$	11.2557	0.575138	0.287569	−	0.957760i	$-0.407153\pi$
$383$	11.2557	0.575138	0.287569	+	0.957760i	$0.407153\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.2947	0.523311
$388$	0	0
$389$	24.4404	1.23918	0.619590	−	0.784926i	$-0.287299\pi$
$389$	24.4404	1.23918	0.619590	+	0.784926i	$0.287299\pi$
$390$	0	0
$391$	3.18546	0.161095
$392$	0	0
$393$	3.37041	0.170014
$394$	0	0
$395$	−19.5825	−0.985304
$396$	0	0
$397$	−17.6316	−0.884903	−0.442451	−	0.896792i	$-0.645891\pi$
$397$	−17.6316	−0.884903	−0.442451	+	0.896792i	$0.645891\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.1758	−0.707908	−0.353954	−	0.935263i	$-0.615163\pi$
$401$	−14.1758	−0.707908	−0.353954	+	0.935263i	$0.615163\pi$
$402$	0	0
$403$	9.64635	0.480519
$404$	0	0
$405$	12.5412	0.623178
$406$	0	0
$407$	37.5606	1.86181
$408$	0	0
$409$	−11.4581	−0.566565	−0.283282	−	0.959037i	$-0.591423\pi$
$409$	−11.4581	−0.566565	−0.283282	+	0.959037i	$0.591423\pi$
$410$	0	0
$411$	0.958407	0.0472748
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.72628	−0.330180
$416$	0	0
$417$	11.3620	0.556400
$418$	0	0
$419$	17.2023	0.840386	0.420193	−	0.907435i	$-0.361962\pi$
$419$	17.2023	0.840386	0.420193	+	0.907435i	$0.361962\pi$
$420$	0	0
$421$	16.5858	0.808342	0.404171	−	0.914683i	$-0.367560\pi$
$421$	16.5858	0.808342	0.404171	+	0.914683i	$0.367560\pi$
$422$	0	0
$423$	11.4430	0.556379
$424$	0	0
$425$	−2.70949	−0.131429
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.04378	0.388357
$430$	0	0
$431$	−13.6461	−0.657309	−0.328655	−	0.944450i	$-0.606595\pi$
$431$	−13.6461	−0.657309	−0.328655	+	0.944450i	$0.606595\pi$
$432$	0	0
$433$	19.5599	0.939989	0.469995	−	0.882669i	$-0.344256\pi$
$433$	19.5599	0.939989	0.469995	+	0.882669i	$0.344256\pi$
$434$	0	0
$435$	14.1637	0.679096
$436$	0	0
$437$	−4.09969	−0.196115
$438$	0	0
$439$	10.6336	0.507515	0.253758	−	0.967268i	$-0.418333\pi$
$439$	10.6336	0.507515	0.253758	+	0.967268i	$0.418333\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0640	−1.14332	−0.571658	−	0.820492i	$-0.693700\pi$
$443$	−24.0640	−1.14332	−0.571658	+	0.820492i	$0.693700\pi$
$444$	0	0
$445$	−36.6348	−1.73666
$446$	0	0
$447$	1.27110	0.0601211
$448$	0	0
$449$	−7.88113	−0.371933	−0.185967	−	0.982556i	$-0.559542\pi$
$449$	−7.88113	−0.371933	−0.185967	+	0.982556i	$0.559542\pi$
$450$	0	0
$451$	32.6260	1.53630
$452$	0	0
$453$	2.24856	0.105646
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.838026	0.0392012	0.0196006	−	0.999808i	$-0.493761\pi$
$457$	0.838026	0.0392012	0.0196006	+	0.999808i	$0.493761\pi$
$458$	0	0
$459$	−11.8153	−0.551492
$460$	0	0
$461$	−19.5359	−0.909879	−0.454940	−	0.890522i	$-0.650339\pi$
$461$	−19.5359	−0.909879	−0.454940	+	0.890522i	$0.650339\pi$
$462$	0	0
$463$	1.74802	0.0812376	0.0406188	−	0.999175i	$-0.487067\pi$
$463$	1.74802	0.0812376	0.0406188	+	0.999175i	$0.487067\pi$
$464$	0	0
$465$	4.92303	0.228300
$466$	0	0
$467$	5.49335	0.254202	0.127101	−	0.991890i	$-0.459433\pi$
$467$	5.49335	0.254202	0.127101	+	0.991890i	$0.459433\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.79708	−0.221038
$472$	0	0
$473$	−15.3387	−0.705275
$474$	0	0
$475$	3.48712	0.160000
$476$	0	0
$477$	8.65769	0.396409
$478$	0	0
$479$	−32.3885	−1.47987	−0.739935	−	0.672679i	$-0.765144\pi$
$479$	−32.3885	−1.47987	−0.739935	+	0.672679i	$0.765144\pi$
$480$	0	0
$481$	−31.2418	−1.42450
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.88053	−0.0853907
$486$	0	0
$487$	−3.67109	−0.166353	−0.0831764	−	0.996535i	$-0.526506\pi$
$487$	−3.67109	−0.166353	−0.0831764	+	0.996535i	$0.526506\pi$
$488$	0	0
$489$	15.7416	0.711862
$490$	0	0
$491$	−2.27194	−0.102531	−0.0512656	−	0.998685i	$-0.516325\pi$
$491$	−2.27194	−0.102531	−0.0512656	+	0.998685i	$0.516325\pi$
$492$	0	0
$493$	27.9305	1.25793
$494$	0	0
$495$	−23.5079	−1.05660
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−40.9296	−1.83226	−0.916129	−	0.400883i	$-0.868703\pi$
$499$	−40.9296	−1.83226	−0.916129	+	0.400883i	$0.868703\pi$
$500$	0	0
$501$	−2.56662	−0.114668
$502$	0	0
$503$	36.7633	1.63919	0.819596	−	0.572941i	$-0.194198\pi$
$503$	36.7633	1.63919	0.819596	+	0.572941i	$0.194198\pi$
$504$	0	0
$505$	46.8283	2.08383
$506$	0	0
$507$	1.99125	0.0884345
$508$	0	0
$509$	−12.8986	−0.571720	−0.285860	−	0.958271i	$-0.592279\pi$
$509$	−12.8986	−0.571720	−0.285860	+	0.958271i	$0.592279\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	15.2064	0.671377
$514$	0	0
$515$	6.85640	0.302129
$516$	0	0
$517$	−17.0496	−0.749840
$518$	0	0
$519$	−5.53499	−0.242959
$520$	0	0
$521$	5.80956	0.254521	0.127261	−	0.991869i	$-0.459382\pi$
$521$	5.80956	0.254521	0.127261	+	0.991869i	$0.459382\pi$
$522$	0	0
$523$	37.7033	1.64865	0.824325	−	0.566116i	$-0.191555\pi$
$523$	37.7033	1.64865	0.824325	+	0.566116i	$0.191555\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.70815	0.422894
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	22.8008	0.989469
$532$	0	0
$533$	−27.1374	−1.17545
$534$	0	0
$535$	−32.2192	−1.39296
$536$	0	0
$537$	−7.85842	−0.339116
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.41443	0.146798	0.0733989	−	0.997303i	$-0.476615\pi$
$541$	3.41443	0.146798	0.0733989	+	0.997303i	$0.476615\pi$
$542$	0	0
$543$	−2.12497	−0.0911910
$544$	0	0
$545$	−18.2978	−0.783792
$546$	0	0
$547$	−8.03875	−0.343712	−0.171856	−	0.985122i	$-0.554976\pi$
$547$	−8.03875	−0.343712	−0.171856	+	0.985122i	$0.554976\pi$
$548$	0	0
$549$	9.45042	0.403334
$550$	0	0
$551$	−35.9467	−1.53138
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−15.9443	−0.676799
$556$	0	0
$557$	44.8119	1.89874	0.949370	−	0.314161i	$-0.101723\pi$
$557$	44.8119	1.89874	0.949370	+	0.314161i	$0.101723\pi$
$558$	0	0
$559$	12.7583	0.539619
$560$	0	0
$561$	8.09531	0.341784
$562$	0	0
$563$	−11.0541	−0.465874	−0.232937	−	0.972492i	$-0.574834\pi$
$563$	−11.0541	−0.465874	−0.232937	+	0.972492i	$0.574834\pi$
$564$	0	0
$565$	−23.2052	−0.976252
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.2272	1.01566	0.507830	−	0.861458i	$-0.330448\pi$
$569$	24.2272	1.01566	0.507830	+	0.861458i	$0.330448\pi$
$570$	0	0
$571$	−7.95085	−0.332733	−0.166366	−	0.986064i	$-0.553203\pi$
$571$	−7.95085	−0.332733	−0.166366	+	0.986064i	$0.553203\pi$
$572$	0	0
$573$	0.733525	0.0306434
$574$	0	0
$575$	−0.850581	−0.0354717
$576$	0	0
$577$	−0.0894489	−0.00372381	−0.00186190	−	0.999998i	$-0.500593\pi$
$577$	−0.0894489	−0.00372381	−0.00186190	+	0.999998i	$0.500593\pi$
$578$	0	0
$579$	−7.76806	−0.322830
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.8996	−0.534246
$584$	0	0
$585$	19.5532	0.808426
$586$	0	0
$587$	38.6258	1.59426	0.797128	−	0.603810i	$-0.206351\pi$
$587$	38.6258	1.59426	0.797128	+	0.603810i	$0.206351\pi$
$588$	0	0
$589$	−12.4944	−0.514824
$590$	0	0
$591$	13.2967	0.546953
$592$	0	0
$593$	−36.0964	−1.48230	−0.741151	−	0.671338i	$-0.765720\pi$
$593$	−36.0964	−1.48230	−0.741151	+	0.671338i	$0.765720\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.0845	−0.535512
$598$	0	0
$599$	−35.5061	−1.45074	−0.725370	−	0.688359i	$-0.758331\pi$
$599$	−35.5061	−1.45074	−0.725370	+	0.688359i	$0.758331\pi$
$600$	0	0
$601$	−25.7458	−1.05019	−0.525096	−	0.851043i	$-0.675971\pi$
$601$	−25.7458	−1.05019	−0.525096	+	0.851043i	$0.675971\pi$
$602$	0	0
$603$	−9.43513	−0.384228
$604$	0	0
$605$	8.41892	0.342278
$606$	0	0
$607$	−29.1721	−1.18406	−0.592030	−	0.805916i	$-0.701673\pi$
$607$	−29.1721	−1.18406	−0.592030	+	0.805916i	$0.701673\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.1814	0.573717
$612$	0	0
$613$	44.4079	1.79362	0.896808	−	0.442419i	$-0.145880\pi$
$613$	44.4079	1.79362	0.896808	+	0.442419i	$0.145880\pi$
$614$	0	0
$615$	−13.8496	−0.558471
$616$	0	0
$617$	−12.2812	−0.494424	−0.247212	−	0.968961i	$-0.579514\pi$
$617$	−12.2812	−0.494424	−0.247212	+	0.968961i	$0.579514\pi$
$618$	0	0
$619$	−33.4711	−1.34532	−0.672659	−	0.739953i	$-0.734848\pi$
$619$	−33.4711	−1.34532	−0.672659	+	0.739953i	$0.734848\pi$
$620$	0	0
$621$	−3.70915	−0.148843
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.5294	−1.14118
$626$	0	0
$627$	−10.4187	−0.416082
$628$	0	0
$629$	−31.4420	−1.25367
$630$	0	0
$631$	6.08577	0.242271	0.121135	−	0.992636i	$-0.461347\pi$
$631$	6.08577	0.242271	0.121135	+	0.992636i	$0.461347\pi$
$632$	0	0
$633$	3.67853	0.146208
$634$	0	0
$635$	32.6122	1.29418
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.53760	−0.0608265
$640$	0	0
$641$	3.46711	0.136943	0.0684713	−	0.997653i	$-0.478188\pi$
$641$	3.46711	0.136943	0.0684713	+	0.997653i	$0.478188\pi$
$642$	0	0
$643$	27.5145	1.08507	0.542534	−	0.840034i	$-0.317465\pi$
$643$	27.5145	1.08507	0.542534	+	0.840034i	$0.317465\pi$
$644$	0	0
$645$	6.51123	0.256379
$646$	0	0
$647$	−21.2376	−0.834936	−0.417468	−	0.908692i	$-0.637082\pi$
$647$	−21.2376	−0.834936	−0.417468	+	0.908692i	$0.637082\pi$
$648$	0	0
$649$	−33.9721	−1.33352
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.8504	1.55947	0.779734	−	0.626111i	$-0.215354\pi$
$653$	39.8504	1.55947	0.779734	+	0.626111i	$0.215354\pi$
$654$	0	0
$655$	−12.2071	−0.476972
$656$	0	0
$657$	−11.9077	−0.464563
$658$	0	0
$659$	6.32398	0.246347	0.123174	−	0.992385i	$-0.460693\pi$
$659$	6.32398	0.246347	0.123174	+	0.992385i	$0.460693\pi$
$660$	0	0
$661$	−24.6355	−0.958210	−0.479105	−	0.877758i	$-0.659039\pi$
$661$	−24.6355	−0.958210	−0.479105	+	0.877758i	$0.659039\pi$
$662$	0	0
$663$	−6.73346	−0.261506
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.76815	0.339504
$668$	0	0
$669$	15.3408	0.593111
$670$	0	0
$671$	−14.0807	−0.543579
$672$	0	0
$673$	−30.0545	−1.15851	−0.579257	−	0.815145i	$-0.696657\pi$
$673$	−30.0545	−1.15851	−0.579257	+	0.815145i	$0.696657\pi$
$674$	0	0
$675$	3.15493	0.121433
$676$	0	0
$677$	−6.93079	−0.266372	−0.133186	−	0.991091i	$-0.542521\pi$
$677$	−6.93079	−0.266372	−0.133186	+	0.991091i	$0.542521\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11.9987	0.459790
$682$	0	0
$683$	−44.8183	−1.71492	−0.857462	−	0.514547i	$-0.827960\pi$
$683$	−44.8183	−1.71492	−0.857462	+	0.514547i	$0.827960\pi$
$684$	0	0
$685$	−3.47121	−0.132628
$686$	0	0
$687$	5.70587	0.217692
$688$	0	0
$689$	10.7295	0.408762
$690$	0	0
$691$	−48.6081	−1.84914	−0.924570	−	0.381012i	$-0.875576\pi$
$691$	−48.6081	−1.84914	−0.924570	+	0.381012i	$0.875576\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−41.1516	−1.56097
$696$	0	0
$697$	−27.3113	−1.03449
$698$	0	0
$699$	16.5668	0.626615
$700$	0	0
$701$	−20.4935	−0.774028	−0.387014	−	0.922074i	$-0.626494\pi$
$701$	−20.4935	−0.774028	−0.387014	+	0.922074i	$0.626494\pi$
$702$	0	0
$703$	40.4659	1.52620
$704$	0	0
$705$	7.23749	0.272580
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−28.4566	−1.06871	−0.534355	−	0.845260i	$-0.679445\pi$
$709$	−28.4566	−1.06871	−0.534355	+	0.845260i	$0.679445\pi$
$710$	0	0
$711$	20.6771	0.775452
$712$	0	0
$713$	3.04765	0.114135
$714$	0	0
$715$	−29.1334	−1.08953
$716$	0	0
$717$	19.0452	0.711257
$718$	0	0
$719$	7.81891	0.291596	0.145798	−	0.989314i	$-0.453425\pi$
$719$	7.81891	0.291596	0.145798	+	0.989314i	$0.453425\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−12.2076	−0.454005
$724$	0	0
$725$	−7.45802	−0.276984
$726$	0	0
$727$	−5.81431	−0.215641	−0.107820	−	0.994170i	$-0.534387\pi$
$727$	−5.81431	−0.215641	−0.107820	+	0.994170i	$0.534387\pi$
$728$	0	0
$729$	−5.81094	−0.215220
$730$	0	0
$731$	12.8401	0.474907
$732$	0	0
$733$	25.4875	0.941404	0.470702	−	0.882292i	$-0.344001\pi$
$733$	25.4875	0.941404	0.470702	+	0.882292i	$0.344001\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.0579	0.517830
$738$	0	0
$739$	34.7915	1.27983	0.639913	−	0.768448i	$-0.278971\pi$
$739$	34.7915	1.27983	0.639913	+	0.768448i	$0.278971\pi$
$740$	0	0
$741$	8.66598	0.318353
$742$	0	0
$743$	−45.1812	−1.65754	−0.828770	−	0.559590i	$-0.810959\pi$
$743$	−45.1812	−1.65754	−0.828770	+	0.559590i	$0.810959\pi$
$744$	0	0
$745$	−4.60375	−0.168668
$746$	0	0
$747$	7.10225	0.259858
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.6009	−0.386831	−0.193415	−	0.981117i	$-0.561957\pi$
$751$	−10.6009	−0.386831	−0.193415	+	0.981117i	$0.561957\pi$
$752$	0	0
$753$	−6.44594	−0.234903
$754$	0	0
$755$	−8.14395	−0.296389
$756$	0	0
$757$	−27.9275	−1.01504	−0.507522	−	0.861639i	$-0.669438\pi$
$757$	−27.9275	−1.01504	−0.507522	+	0.861639i	$0.669438\pi$
$758$	0	0
$759$	2.54134	0.0922446
$760$	0	0
$761$	−4.11857	−0.149298	−0.0746490	−	0.997210i	$-0.523784\pi$
$761$	−4.11857	−0.149298	−0.0746490	+	0.997210i	$0.523784\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	19.6785	0.711477
$766$	0	0
$767$	28.2571	1.02030
$768$	0	0
$769$	−35.0398	−1.26357	−0.631784	−	0.775145i	$-0.717677\pi$
$769$	−35.0398	−1.26357	−0.631784	+	0.775145i	$0.717677\pi$
$770$	0	0
$771$	−15.0294	−0.541270
$772$	0	0
$773$	2.18705	0.0786628	0.0393314	−	0.999226i	$-0.487477\pi$
$773$	2.18705	0.0786628	0.0393314	+	0.999226i	$0.487477\pi$
$774$	0	0
$775$	−2.59227	−0.0931171
$776$	0	0
$777$	0	0
$778$	0	0
$779$	35.1497	1.25937
$780$	0	0
$781$	2.29095	0.0819768
$782$	0	0
$783$	−32.5224	−1.16225
$784$	0	0
$785$	17.3743	0.620116
$786$	0	0
$787$	−12.6369	−0.450457	−0.225228	−	0.974306i	$-0.572313\pi$
$787$	−12.6369	−0.450457	−0.225228	+	0.974306i	$0.572313\pi$
$788$	0	0
$789$	−0.976054	−0.0347484
$790$	0	0
$791$	0	0
$792$	0	0
$793$	11.7119	0.415903
$794$	0	0
$795$	5.47583	0.194208
$796$	0	0
$797$	−46.1650	−1.63525	−0.817625	−	0.575752i	$-0.804709\pi$
$797$	−46.1650	−1.63525	−0.817625	+	0.575752i	$0.804709\pi$
$798$	0	0
$799$	14.2722	0.504915
$800$	0	0
$801$	38.6825	1.36678
$802$	0	0
$803$	17.7419	0.626098
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.49363	0.193385
$808$	0	0
$809$	6.05184	0.212771	0.106386	−	0.994325i	$-0.466072\pi$
$809$	6.05184	0.212771	0.106386	+	0.994325i	$0.466072\pi$
$810$	0	0
$811$	−19.7586	−0.693820	−0.346910	−	0.937898i	$-0.612769\pi$
$811$	−19.7586	−0.693820	−0.346910	+	0.937898i	$0.612769\pi$
$812$	0	0
$813$	14.7748	0.518174
$814$	0	0
$815$	−57.0140	−1.99711
$816$	0	0
$817$	−16.5252	−0.578143
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.74090	−0.0607577	−0.0303788	−	0.999538i	$-0.509671\pi$
$821$	−1.74090	−0.0607577	−0.0303788	+	0.999538i	$0.509671\pi$
$822$	0	0
$823$	14.2921	0.498190	0.249095	−	0.968479i	$-0.419867\pi$
$823$	14.2921	0.498190	0.249095	+	0.968479i	$0.419867\pi$
$824$	0	0
$825$	−2.16161	−0.0752576
$826$	0	0
$827$	−2.79872	−0.0973210	−0.0486605	−	0.998815i	$-0.515495\pi$
$827$	−2.79872	−0.0973210	−0.0486605	+	0.998815i	$0.515495\pi$
$828$	0	0
$829$	40.1973	1.39611	0.698054	−	0.716045i	$-0.254049\pi$
$829$	40.1973	1.39611	0.698054	+	0.716045i	$0.254049\pi$
$830$	0	0
$831$	−7.11365	−0.246770
$832$	0	0
$833$	0	0
$834$	0	0
$835$	9.29591	0.321698
$836$	0	0
$837$	−11.3042	−0.390730
$838$	0	0
$839$	30.8463	1.06493	0.532466	−	0.846452i	$-0.321265\pi$
$839$	30.8463	1.06493	0.532466	+	0.846452i	$0.321265\pi$
$840$	0	0
$841$	47.8804	1.65105
$842$	0	0
$843$	−15.2219	−0.524272
$844$	0	0
$845$	−7.21202	−0.248101
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.96673	−0.342057
$850$	0	0
$851$	−9.87048	−0.338356
$852$	0	0
$853$	−5.44619	−0.186474	−0.0932369	−	0.995644i	$-0.529721\pi$
$853$	−5.44619	−0.186474	−0.0932369	+	0.995644i	$0.529721\pi$
$854$	0	0
$855$	−25.3263	−0.866140
$856$	0	0
$857$	6.39602	0.218484	0.109242	−	0.994015i	$-0.465158\pi$
$857$	6.39602	0.218484	0.109242	+	0.994015i	$0.465158\pi$
$858$	0	0
$859$	13.3323	0.454893	0.227446	−	0.973791i	$-0.426962\pi$
$859$	13.3323	0.454893	0.227446	+	0.973791i	$0.426962\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.5028	1.82126	0.910629	−	0.413225i	$-0.135598\pi$
$863$	53.5028	1.82126	0.910629	+	0.413225i	$0.135598\pi$
$864$	0	0
$865$	20.0469	0.681616
$866$	0	0
$867$	4.57658	0.155429
$868$	0	0
$869$	−30.8080	−1.04509
$870$	0	0
$871$	−11.6930	−0.396202
$872$	0	0
$873$	1.98565	0.0672040
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.9838	0.438433	0.219217	−	0.975676i	$-0.429650\pi$
$877$	12.9838	0.438433	0.219217	+	0.975676i	$0.429650\pi$
$878$	0	0
$879$	−7.90214	−0.266533
$880$	0	0
$881$	32.8084	1.10534	0.552671	−	0.833400i	$-0.313609\pi$
$881$	32.8084	1.10534	0.552671	+	0.833400i	$0.313609\pi$
$882$	0	0
$883$	32.3677	1.08926	0.544629	−	0.838677i	$-0.316670\pi$
$883$	32.3677	1.08926	0.544629	+	0.838677i	$0.316670\pi$
$884$	0	0
$885$	14.4211	0.484758
$886$	0	0
$887$	56.9719	1.91293	0.956465	−	0.291846i	$-0.0942694\pi$
$887$	56.9719	1.91293	0.956465	+	0.291846i	$0.0942694\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	19.7303	0.660990
$892$	0	0
$893$	−18.3684	−0.614675
$894$	0	0
$895$	28.4621	0.951382
$896$	0	0
$897$	−2.11381	−0.0705781
$898$	0	0
$899$	26.7222	0.891236
$900$	0	0
$901$	10.7982	0.359742
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.69632	0.255834
$906$	0	0
$907$	54.7760	1.81881	0.909404	−	0.415914i	$-0.136538\pi$
$907$	54.7760	1.81881	0.909404	+	0.415914i	$0.136538\pi$
$908$	0	0
$909$	−49.4458	−1.64001
$910$	0	0
$911$	−41.9935	−1.39131	−0.695653	−	0.718378i	$-0.744885\pi$
$911$	−41.9935	−1.39131	−0.695653	+	0.718378i	$0.744885\pi$
$912$	0	0
$913$	−10.5820	−0.350214
$914$	0	0
$915$	5.97721	0.197600
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.25436	−0.305273	−0.152637	−	0.988282i	$-0.548776\pi$
$919$	−9.25436	−0.305273	−0.152637	+	0.988282i	$0.548776\pi$
$920$	0	0
$921$	−8.00423	−0.263749
$922$	0	0
$923$	−1.90555	−0.0627220
$924$	0	0
$925$	8.39564	0.276047
$926$	0	0
$927$	−7.23964	−0.237781
$928$	0	0
$929$	−36.5897	−1.20047	−0.600235	−	0.799824i	$-0.704926\pi$
$929$	−36.5897	−1.20047	−0.600235	+	0.799824i	$0.704926\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−9.55715	−0.312887
$934$	0	0
$935$	−29.3201	−0.958868
$936$	0	0
$937$	−28.9748	−0.946564	−0.473282	−	0.880911i	$-0.656931\pi$
$937$	−28.9748	−0.946564	−0.473282	+	0.880911i	$0.656931\pi$
$938$	0	0
$939$	−13.2862	−0.433579
$940$	0	0
$941$	−41.8693	−1.36490	−0.682450	−	0.730933i	$-0.739085\pi$
$941$	−41.8693	−1.36490	−0.682450	+	0.730933i	$0.739085\pi$
$942$	0	0
$943$	−8.57374	−0.279199
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.6540	0.606174	0.303087	−	0.952963i	$-0.401983\pi$
$947$	18.6540	0.606174	0.303087	+	0.952963i	$0.401983\pi$
$948$	0	0
$949$	−14.7572	−0.479039
$950$	0	0
$951$	14.0680	0.456187
$952$	0	0
$953$	−44.7516	−1.44965	−0.724823	−	0.688935i	$-0.758078\pi$
$953$	−44.7516	−1.44965	−0.724823	+	0.688935i	$0.758078\pi$
$954$	0	0
$955$	−2.65672	−0.0859695
$956$	0	0
$957$	22.2828	0.720301
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21.7118	−0.700382
$962$	0	0
$963$	34.0201	1.09628
$964$	0	0
$965$	28.1348	0.905691
$966$	0	0
$967$	6.90180	0.221947	0.110973	−	0.993823i	$-0.464603\pi$
$967$	6.90180	0.221947	0.110973	+	0.993823i	$0.464603\pi$
$968$	0	0
$969$	8.72150	0.280175
$970$	0	0
$971$	21.9016	0.702854	0.351427	−	0.936215i	$-0.385696\pi$
$971$	21.9016	0.702854	0.351427	+	0.936215i	$0.385696\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.79797	0.0575811
$976$	0	0
$977$	−35.3190	−1.12996	−0.564978	−	0.825106i	$-0.691115\pi$
$977$	−35.3190	−1.12996	−0.564978	+	0.825106i	$0.691115\pi$
$978$	0	0
$979$	−57.6352	−1.84203
$980$	0	0
$981$	19.3206	0.616859
$982$	0	0
$983$	−13.7037	−0.437080	−0.218540	−	0.975828i	$-0.570129\pi$
$983$	−13.7037	−0.437080	−0.218540	+	0.975828i	$0.570129\pi$
$984$	0	0
$985$	−48.1587	−1.53446
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.03084	0.128173
$990$	0	0
$991$	−8.04739	−0.255634	−0.127817	−	0.991798i	$-0.540797\pi$
$991$	−8.04739	−0.255634	−0.127817	+	0.991798i	$0.540797\pi$
$992$	0	0
$993$	17.0661	0.541576
$994$	0	0
$995$	47.3901	1.50237
$996$	0	0
$997$	−39.7641	−1.25934	−0.629671	−	0.776862i	$-0.716810\pi$
$997$	−39.7641	−1.25934	−0.629671	+	0.776862i	$0.716810\pi$
$998$	0	0
$999$	36.6111	1.15832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bs.1.5	✓	14
7.6	odd	2	inner	9016.2.a.bs.1.10	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9016.2.a.bs.1.5	✓	14	1.1	even	1	trivial
9016.2.a.bs.1.10	yes	14	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q-0.667834 q^{3} +2.41880 q^{5} -2.55400 q^{9} +O(q^{10})\)	\(q-0.667834 q^{3} +2.41880 q^{5} -2.55400 q^{9} +3.80534 q^{11} -3.16518 q^{13} -1.61535 q^{15} -3.18546 q^{17} +4.09969 q^{19} -1.00000 q^{23} +0.850581 q^{25} +3.70915 q^{27} -8.76815 q^{29} -3.04765 q^{31} -2.54134 q^{33} +9.87048 q^{37} +2.11381 q^{39} +8.57374 q^{41} -4.03084 q^{43} -6.17760 q^{45} -4.48043 q^{47} +2.12736 q^{51} -3.38986 q^{53} +9.20435 q^{55} -2.73791 q^{57} -8.92748 q^{59} -3.70025 q^{61} -7.65592 q^{65} +3.69426 q^{67} +0.667834 q^{69} +0.602036 q^{71} +4.66237 q^{73} -0.568046 q^{75} -8.09598 q^{79} +5.18490 q^{81} -2.78084 q^{83} -7.70497 q^{85} +5.85567 q^{87} -15.1459 q^{89} +2.03532 q^{93} +9.91632 q^{95} -0.777467 q^{97} -9.71883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{9}+O(q^{10}) \) 14 * q + 14 * q^9	\( 14 q + 14 q^{9} - 8 q^{11} - 28 q^{15} - 14 q^{23} + 10 q^{25} + 4 q^{29} + 12 q^{37} - 24 q^{39} - 44 q^{43} - 28 q^{51} - 4 q^{53} - 8 q^{57} - 44 q^{65} - 28 q^{67} - 28 q^{71} - 36 q^{79} + 18 q^{81} + 8 q^{85} + 44 q^{93} - 8 q^{95} - 60 q^{99}+O(q^{100}) \) 14 * q + 14 * q^9 - 8 * q^11 - 28 * q^15 - 14 * q^23 + 10 * q^25 + 4 * q^29 + 12 * q^37 - 24 * q^39 - 44 * q^43 - 28 * q^51 - 4 * q^53 - 8 * q^57 - 44 * q^65 - 28 * q^67 - 28 * q^71 - 36 * q^79 + 18 * q^81 + 8 * q^85 + 44 * q^93 - 8 * q^95 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more