LMFDB - Embedded newform 4304.2.a.l.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 314 x^{12} - 283 x^{11} - 1803 x^{10} + 1435 x^{9} + \cdots + 172 $ x^16 - x^15 - 28x^14 + 27x^13 + 314x^12 - 283x^11 - 1803x^10 + 1435x^9 + 5637x^8 - 3547x^7 - 9470x^6 + 3701x^5 + 7860x^4 - 1001x^3 - 2363x^2 - 43x + 172
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 269)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$0.282877$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.37910 q^{3} +1.97313 q^{5} -2.44119 q^{7} -1.09807 q^{9} -3.43591 q^{11} +4.69311 q^{13} +2.72116 q^{15} +1.90195 q^{17} +1.23155 q^{19} -3.36666 q^{21} -6.50000 q^{23} -1.10675 q^{25} -5.65167 q^{27} -5.54253 q^{29} -4.88221 q^{31} -4.73848 q^{33} -4.81680 q^{35} -10.2236 q^{37} +6.47229 q^{39} -5.60633 q^{41} +2.85084 q^{43} -2.16664 q^{45} +1.80999 q^{47} -1.04058 q^{49} +2.62299 q^{51} +2.04207 q^{53} -6.77951 q^{55} +1.69844 q^{57} +1.02290 q^{59} +12.6403 q^{61} +2.68060 q^{63} +9.26012 q^{65} -1.65225 q^{67} -8.96418 q^{69} +4.59265 q^{71} +5.17859 q^{73} -1.52632 q^{75} +8.38772 q^{77} -13.5494 q^{79} -4.50003 q^{81} +13.9606 q^{83} +3.75280 q^{85} -7.64373 q^{87} -9.64059 q^{89} -11.4568 q^{91} -6.73308 q^{93} +2.43001 q^{95} -16.2488 q^{97} +3.77287 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{3} - q^{5} - 11 q^{7} + 21 q^{9} - 16 q^{11} - q^{13} + 5 q^{15} - 2 q^{17} - 35 q^{19} - q^{23} + 13 q^{25} - 11 q^{27} + 2 q^{29} - 13 q^{31} - 8 q^{33} - 9 q^{35} + 4 q^{37} - 11 q^{39}+ \cdots - 90 q^{99}+O(q^{100}) $ 16 * q - 5 * q^3 - q^5 - 11 * q^7 + 21 * q^9 - 16 * q^11 - q^13 + 5 * q^15 - 2 * q^17 - 35 * q^19 - q^23 + 13 * q^25 - 11 * q^27 + 2 * q^29 - 13 * q^31 - 8 * q^33 - 9 * q^35 + 4 * q^37 - 11 * q^39 - 2 * q^41 - 51 * q^43 - 35 * q^45 + 9 * q^47 + 27 * q^49 - 40 * q^51 - 29 * q^53 + 8 * q^55 - 16 * q^57 + 13 * q^59 + 14 * q^61 + 14 * q^63 - 29 * q^65 - 25 * q^67 - 12 * q^69 + 9 * q^71 + 4 * q^73 - 22 * q^75 - 10 * q^77 - 51 * q^79 + 2 * q^83 + 4 * q^85 - q^87 - 35 * q^89 - 65 * q^91 - 36 * q^93 - 18 * q^95 - 14 * q^97 - 90 * q^99

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
$3$	1.37910	0.796227	0.398113	−	0.917336i	$-0.369665\pi$
$3$	1.37910	0.796227	0.398113	+	0.917336i	$0.369665\pi$
$4$	0	0
$5$	1.97313	0.882412	0.441206	−	0.897406i	$-0.354551\pi$
$5$	1.97313	0.882412	0.441206	+	0.897406i	$0.354551\pi$
$6$	0	0
$7$	−2.44119	−0.922684	−0.461342	−	0.887222i	$-0.652632\pi$
$7$	−2.44119	−0.922684	−0.461342	+	0.887222i	$0.652632\pi$
$8$	0	0
$9$	−1.09807	−0.366023
$10$	0	0
$11$	−3.43591	−1.03597	−0.517983	−	0.855391i	$-0.673317\pi$
$11$	−3.43591	−1.03597	−0.517983	+	0.855391i	$0.673317\pi$
$12$	0	0
$13$	4.69311	1.30163	0.650817	−	0.759235i	$-0.274427\pi$
$13$	4.69311	1.30163	0.650817	+	0.759235i	$0.274427\pi$
$14$	0	0
$15$	2.72116	0.702600
$16$	0	0
$17$	1.90195	0.461291	0.230645	−	0.973038i	$-0.425916\pi$
$17$	1.90195	0.461291	0.230645	+	0.973038i	$0.425916\pi$
$18$	0	0
$19$	1.23155	0.282537	0.141269	−	0.989971i	$-0.454882\pi$
$19$	1.23155	0.282537	0.141269	+	0.989971i	$0.454882\pi$
$20$	0	0
$21$	−3.36666	−0.734666
$22$	0	0
$23$	−6.50000	−1.35534	−0.677672	−	0.735365i	$-0.737011\pi$
$23$	−6.50000	−1.35534	−0.677672	+	0.735365i	$0.737011\pi$
$24$	0	0
$25$	−1.10675	−0.221349
$26$	0	0
$27$	−5.65167	−1.08766
$28$	0	0
$29$	−5.54253	−1.02922	−0.514611	−	0.857424i	$-0.672064\pi$
$29$	−5.54253	−1.02922	−0.514611	+	0.857424i	$0.672064\pi$
$30$	0	0
$31$	−4.88221	−0.876871	−0.438436	−	0.898763i	$-0.644467\pi$
$31$	−4.88221	−0.876871	−0.438436	+	0.898763i	$0.644467\pi$
$32$	0	0
$33$	−4.73848	−0.824864
$34$	0	0
$35$	−4.81680	−0.814187
$36$	0	0
$37$	−10.2236	−1.68075	−0.840377	−	0.542003i	$-0.817666\pi$
$37$	−10.2236	−1.68075	−0.840377	+	0.542003i	$0.817666\pi$
$38$	0	0
$39$	6.47229	1.03640
$40$	0	0
$41$	−5.60633	−0.875561	−0.437780	−	0.899082i	$-0.644235\pi$
$41$	−5.60633	−0.875561	−0.437780	+	0.899082i	$0.644235\pi$
$42$	0	0
$43$	2.85084	0.434749	0.217374	−	0.976088i	$-0.430251\pi$
$43$	2.85084	0.434749	0.217374	+	0.976088i	$0.430251\pi$
$44$	0	0
$45$	−2.16664	−0.322983
$46$	0	0
$47$	1.80999	0.264014	0.132007	−	0.991249i	$-0.457858\pi$
$47$	1.80999	0.264014	0.132007	+	0.991249i	$0.457858\pi$
$48$	0	0
$49$	−1.04058	−0.148654
$50$	0	0
$51$	2.62299	0.367292
$52$	0	0
$53$	2.04207	0.280500	0.140250	−	0.990116i	$-0.455209\pi$
$53$	2.04207	0.280500	0.140250	+	0.990116i	$0.455209\pi$
$54$	0	0
$55$	−6.77951	−0.914149
$56$	0	0
$57$	1.69844	0.224964
$58$	0	0
$59$	1.02290	0.133170	0.0665850	−	0.997781i	$-0.478790\pi$
$59$	1.02290	0.133170	0.0665850	+	0.997781i	$0.478790\pi$
$60$	0	0
$61$	12.6403	1.61843	0.809213	−	0.587515i	$-0.199894\pi$
$61$	12.6403	1.61843	0.809213	+	0.587515i	$0.199894\pi$
$62$	0	0
$63$	2.68060	0.337724
$64$	0	0
$65$	9.26012	1.14858
$66$	0	0
$67$	−1.65225	−0.201854	−0.100927	−	0.994894i	$-0.532181\pi$
$67$	−1.65225	−0.201854	−0.100927	+	0.994894i	$0.532181\pi$
$68$	0	0
$69$	−8.96418	−1.07916
$70$	0	0
$71$	4.59265	0.545047	0.272523	−	0.962149i	$-0.412142\pi$
$71$	4.59265	0.545047	0.272523	+	0.962149i	$0.412142\pi$
$72$	0	0
$73$	5.17859	0.606108	0.303054	−	0.952973i	$-0.401994\pi$
$73$	5.17859	0.606108	0.303054	+	0.952973i	$0.401994\pi$
$74$	0	0
$75$	−1.52632	−0.176244
$76$	0	0
$77$	8.38772	0.955870
$78$	0	0
$79$	−13.5494	−1.52443	−0.762213	−	0.647326i	$-0.775887\pi$
$79$	−13.5494	−1.52443	−0.762213	+	0.647326i	$0.775887\pi$
$80$	0	0
$81$	−4.50003	−0.500004
$82$	0	0
$83$	13.9606	1.53237	0.766187	−	0.642618i	$-0.222152\pi$
$83$	13.9606	1.53237	0.766187	+	0.642618i	$0.222152\pi$
$84$	0	0
$85$	3.75280	0.407049
$86$	0	0
$87$	−7.64373	−0.819494
$88$	0	0
$89$	−9.64059	−1.02190	−0.510950	−	0.859610i	$-0.670706\pi$
$89$	−9.64059	−1.02190	−0.510950	+	0.859610i	$0.670706\pi$
$90$	0	0
$91$	−11.4568	−1.20100
$92$	0	0
$93$	−6.73308	−0.698188
$94$	0	0
$95$	2.43001	0.249314
$96$	0	0
$97$	−16.2488	−1.64982	−0.824908	−	0.565267i	$-0.808773\pi$
$97$	−16.2488	−1.64982	−0.824908	+	0.565267i	$0.808773\pi$
$98$	0	0
$99$	3.77287	0.379188
$100$	0	0
$101$	15.4748	1.53980	0.769900	−	0.638165i	$-0.220306\pi$
$101$	15.4748	1.53980	0.769900	+	0.638165i	$0.220306\pi$
$102$	0	0
$103$	−11.6936	−1.15220	−0.576101	−	0.817379i	$-0.695426\pi$
$103$	−11.6936	−1.15220	−0.576101	+	0.817379i	$0.695426\pi$
$104$	0	0
$105$	−6.64287	−0.648278
$106$	0	0
$107$	−0.0867365	−0.00838514	−0.00419257	−	0.999991i	$-0.501335\pi$
$107$	−0.0867365	−0.00838514	−0.00419257	+	0.999991i	$0.501335\pi$
$108$	0	0
$109$	3.18524	0.305091	0.152545	−	0.988296i	$-0.451253\pi$
$109$	3.18524	0.305091	0.152545	+	0.988296i	$0.451253\pi$
$110$	0	0
$111$	−14.0994	−1.33826
$112$	0	0
$113$	−2.15866	−0.203070	−0.101535	−	0.994832i	$-0.532375\pi$
$113$	−2.15866	−0.203070	−0.101535	+	0.994832i	$0.532375\pi$
$114$	0	0
$115$	−12.8254	−1.19597
$116$	0	0
$117$	−5.15336	−0.476428
$118$	0	0
$119$	−4.64303	−0.425626
$120$	0	0
$121$	0.805487	0.0732261
$122$	0	0
$123$	−7.73171	−0.697145
$124$	0	0
$125$	−12.0494	−1.07773
$126$	0	0
$127$	−3.13665	−0.278333	−0.139166	−	0.990269i	$-0.544442\pi$
$127$	−3.13665	−0.278333	−0.139166	+	0.990269i	$0.544442\pi$
$128$	0	0
$129$	3.93160	0.346158
$130$	0	0
$131$	19.1069	1.66938	0.834689	−	0.550721i	$-0.185647\pi$
$131$	19.1069	1.66938	0.834689	+	0.550721i	$0.185647\pi$
$132$	0	0
$133$	−3.00645	−0.260693
$134$	0	0
$135$	−11.1515	−0.959768
$136$	0	0
$137$	−7.77105	−0.663925	−0.331963	−	0.943293i	$-0.607711\pi$
$137$	−7.77105	−0.663925	−0.331963	+	0.943293i	$0.607711\pi$
$138$	0	0
$139$	−7.50884	−0.636891	−0.318446	−	0.947941i	$-0.603161\pi$
$139$	−7.50884	−0.636891	−0.318446	+	0.947941i	$0.603161\pi$
$140$	0	0
$141$	2.49616	0.210215
$142$	0	0
$143$	−16.1251	−1.34845
$144$	0	0
$145$	−10.9361	−0.908198
$146$	0	0
$147$	−1.43507	−0.118362
$148$	0	0
$149$	−21.5548	−1.76584	−0.882921	−	0.469522i	$-0.844426\pi$
$149$	−21.5548	−1.76584	−0.882921	+	0.469522i	$0.844426\pi$
$150$	0	0
$151$	8.16873	0.664762	0.332381	−	0.943145i	$-0.392148\pi$
$151$	8.16873	0.664762	0.332381	+	0.943145i	$0.392148\pi$
$152$	0	0
$153$	−2.08848	−0.168843
$154$	0	0
$155$	−9.63325	−0.773761
$156$	0	0
$157$	19.5185	1.55774	0.778872	−	0.627183i	$-0.215792\pi$
$157$	19.5185	1.55774	0.778872	+	0.627183i	$0.215792\pi$
$158$	0	0
$159$	2.81623	0.223342
$160$	0	0
$161$	15.8677	1.25055
$162$	0	0
$163$	−5.85613	−0.458687	−0.229344	−	0.973345i	$-0.573658\pi$
$163$	−5.85613	−0.458687	−0.229344	+	0.973345i	$0.573658\pi$
$164$	0	0
$165$	−9.34965	−0.727870
$166$	0	0
$167$	16.1245	1.24775	0.623876	−	0.781523i	$-0.285557\pi$
$167$	16.1245	1.24775	0.623876	+	0.781523i	$0.285557\pi$
$168$	0	0
$169$	9.02526	0.694251
$170$	0	0
$171$	−1.35233	−0.103415
$172$	0	0
$173$	−13.5232	−1.02815	−0.514074	−	0.857746i	$-0.671864\pi$
$173$	−13.5232	−1.02815	−0.514074	+	0.857746i	$0.671864\pi$
$174$	0	0
$175$	2.70178	0.204236
$176$	0	0
$177$	1.41068	0.106034
$178$	0	0
$179$	−22.1712	−1.65715	−0.828577	−	0.559875i	$-0.810849\pi$
$179$	−22.1712	−1.65715	−0.828577	+	0.559875i	$0.810849\pi$
$180$	0	0
$181$	3.61117	0.268416	0.134208	−	0.990953i	$-0.457151\pi$
$181$	3.61117	0.268416	0.134208	+	0.990953i	$0.457151\pi$
$182$	0	0
$183$	17.4323	1.28863
$184$	0	0
$185$	−20.1726	−1.48312
$186$	0	0
$187$	−6.53494	−0.477882
$188$	0	0
$189$	13.7968	1.00357
$190$	0	0
$191$	7.41632	0.536626	0.268313	−	0.963332i	$-0.413534\pi$
$191$	7.41632	0.536626	0.268313	+	0.963332i	$0.413534\pi$
$192$	0	0
$193$	−23.0099	−1.65629	−0.828145	−	0.560514i	$-0.810604\pi$
$193$	−23.0099	−1.65629	−0.828145	+	0.560514i	$0.810604\pi$
$194$	0	0
$195$	12.7707	0.914528
$196$	0	0
$197$	0.810204	0.0577247	0.0288623	−	0.999583i	$-0.490812\pi$
$197$	0.810204	0.0577247	0.0288623	+	0.999583i	$0.490812\pi$
$198$	0	0
$199$	−16.0676	−1.13900	−0.569500	−	0.821991i	$-0.692863\pi$
$199$	−16.0676	−1.13900	−0.569500	+	0.821991i	$0.692863\pi$
$200$	0	0
$201$	−2.27863	−0.160722
$202$	0	0
$203$	13.5304	0.949647
$204$	0	0
$205$	−11.0620	−0.772605
$206$	0	0
$207$	7.13745	0.496087
$208$	0	0
$209$	−4.23150	−0.292699
$210$	0	0
$211$	9.98098	0.687119	0.343559	−	0.939131i	$-0.388367\pi$
$211$	9.98098	0.687119	0.343559	+	0.939131i	$0.388367\pi$
$212$	0	0
$213$	6.33374	0.433981
$214$	0	0
$215$	5.62508	0.383627
$216$	0	0
$217$	11.9184	0.809075
$218$	0	0
$219$	7.14182	0.482599
$220$	0	0
$221$	8.92606	0.600432
$222$	0	0
$223$	19.6088	1.31310	0.656550	−	0.754283i	$-0.272015\pi$
$223$	19.6088	1.31310	0.656550	+	0.754283i	$0.272015\pi$
$224$	0	0
$225$	1.21529	0.0810191
$226$	0	0
$227$	11.4505	0.759998	0.379999	−	0.924987i	$-0.375924\pi$
$227$	11.4505	0.759998	0.379999	+	0.924987i	$0.375924\pi$
$228$	0	0
$229$	22.6433	1.49631	0.748156	−	0.663523i	$-0.230939\pi$
$229$	22.6433	1.49631	0.748156	+	0.663523i	$0.230939\pi$
$230$	0	0
$231$	11.5675	0.761089
$232$	0	0
$233$	−15.8879	−1.04085	−0.520427	−	0.853906i	$-0.674227\pi$
$233$	−15.8879	−1.04085	−0.520427	+	0.853906i	$0.674227\pi$
$234$	0	0
$235$	3.57134	0.232969
$236$	0	0
$237$	−18.6860	−1.21379
$238$	0	0
$239$	−7.35894	−0.476010	−0.238005	−	0.971264i	$-0.576493\pi$
$239$	−7.35894	−0.476010	−0.238005	+	0.971264i	$0.576493\pi$
$240$	0	0
$241$	−13.1161	−0.844882	−0.422441	−	0.906390i	$-0.638827\pi$
$241$	−13.1161	−0.844882	−0.422441	+	0.906390i	$0.638827\pi$
$242$	0	0
$243$	10.7490	0.689548
$244$	0	0
$245$	−2.05320	−0.131174
$246$	0	0
$247$	5.77980	0.367760
$248$	0	0
$249$	19.2531	1.22012
$250$	0	0
$251$	0.274032	0.0172967	0.00864837	−	0.999963i	$-0.497247\pi$
$251$	0.274032	0.0172967	0.00864837	+	0.999963i	$0.497247\pi$
$252$	0	0
$253$	22.3334	1.40409
$254$	0	0
$255$	5.17551	0.324103
$256$	0	0
$257$	−5.65777	−0.352922	−0.176461	−	0.984308i	$-0.556465\pi$
$257$	−5.65777	−0.352922	−0.176461	+	0.984308i	$0.556465\pi$
$258$	0	0
$259$	24.9578	1.55080
$260$	0	0
$261$	6.08609	0.376719
$262$	0	0
$263$	−18.5754	−1.14541	−0.572703	−	0.819763i	$-0.694105\pi$
$263$	−18.5754	−1.14541	−0.572703	+	0.819763i	$0.694105\pi$
$264$	0	0
$265$	4.02928	0.247517
$266$	0	0
$267$	−13.2954	−0.813665
$268$	0	0
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	5.19643	0.315661	0.157830	−	0.987466i	$-0.449550\pi$
$271$	5.19643	0.315661	0.157830	+	0.987466i	$0.449550\pi$
$272$	0	0
$273$	−15.8001	−0.956266
$274$	0	0
$275$	3.80269	0.229311
$276$	0	0
$277$	6.13328	0.368513	0.184257	−	0.982878i	$-0.441012\pi$
$277$	6.13328	0.368513	0.184257	+	0.982878i	$0.441012\pi$
$278$	0	0
$279$	5.36101	0.320955
$280$	0	0
$281$	19.4792	1.16203	0.581017	−	0.813891i	$-0.302655\pi$
$281$	19.4792	1.16203	0.581017	+	0.813891i	$0.302655\pi$
$282$	0	0
$283$	−1.43510	−0.0853076	−0.0426538	−	0.999090i	$-0.513581\pi$
$283$	−1.43510	−0.0853076	−0.0426538	+	0.999090i	$0.513581\pi$
$284$	0	0
$285$	3.35124	0.198511
$286$	0	0
$287$	13.6861	0.807866
$288$	0	0
$289$	−13.3826	−0.787211
$290$	0	0
$291$	−22.4088	−1.31363
$292$	0	0
$293$	−29.8473	−1.74370	−0.871848	−	0.489776i	$-0.837078\pi$
$293$	−29.8473	−1.74370	−0.871848	+	0.489776i	$0.837078\pi$
$294$	0	0
$295$	2.01832	0.117511
$296$	0	0
$297$	19.4186	1.12678
$298$	0	0
$299$	−30.5052	−1.76416
$300$	0	0
$301$	−6.95944	−0.401136
$302$	0	0
$303$	21.3414	1.22603
$304$	0	0
$305$	24.9410	1.42812
$306$	0	0
$307$	14.6428	0.835710	0.417855	−	0.908514i	$-0.362782\pi$
$307$	14.6428	0.835710	0.417855	+	0.908514i	$0.362782\pi$
$308$	0	0
$309$	−16.1267	−0.917413
$310$	0	0
$311$	14.9926	0.850155	0.425077	−	0.905157i	$-0.360247\pi$
$311$	14.9926	0.850155	0.425077	+	0.905157i	$0.360247\pi$
$312$	0	0
$313$	−14.3883	−0.813272	−0.406636	−	0.913590i	$-0.633298\pi$
$313$	−14.3883	−0.813272	−0.406636	+	0.913590i	$0.633298\pi$
$314$	0	0
$315$	5.28918	0.298012
$316$	0	0
$317$	18.4565	1.03662	0.518309	−	0.855193i	$-0.326562\pi$
$317$	18.4565	1.03662	0.518309	+	0.855193i	$0.326562\pi$
$318$	0	0
$319$	19.0436	1.06624
$320$	0	0
$321$	−0.119619	−0.00667647
$322$	0	0
$323$	2.34235	0.130332
$324$	0	0
$325$	−5.19408	−0.288116
$326$	0	0
$327$	4.39278	0.242922
$328$	0	0
$329$	−4.41853	−0.243601
$330$	0	0
$331$	−29.1435	−1.60187	−0.800935	−	0.598752i	$-0.795664\pi$
$331$	−29.1435	−1.60187	−0.800935	+	0.598752i	$0.795664\pi$
$332$	0	0
$333$	11.2263	0.615195
$334$	0	0
$335$	−3.26011	−0.178119
$336$	0	0
$337$	−13.3401	−0.726682	−0.363341	−	0.931656i	$-0.618364\pi$
$337$	−13.3401	−0.726682	−0.363341	+	0.931656i	$0.618364\pi$
$338$	0	0
$339$	−2.97702	−0.161689
$340$	0	0
$341$	16.7748	0.908409
$342$	0	0
$343$	19.6286	1.05984
$344$	0	0
$345$	−17.6875	−0.952264
$346$	0	0
$347$	−15.4841	−0.831232	−0.415616	−	0.909540i	$-0.636434\pi$
$347$	−15.4841	−0.831232	−0.415616	+	0.909540i	$0.636434\pi$
$348$	0	0
$349$	4.95774	0.265382	0.132691	−	0.991157i	$-0.457638\pi$
$349$	4.95774	0.265382	0.132691	+	0.991157i	$0.457638\pi$
$350$	0	0
$351$	−26.5239	−1.41574
$352$	0	0
$353$	−19.7221	−1.04970	−0.524851	−	0.851194i	$-0.675879\pi$
$353$	−19.7221	−1.04970	−0.524851	+	0.851194i	$0.675879\pi$
$354$	0	0
$355$	9.06190	0.480956
$356$	0	0
$357$	−6.40322	−0.338895
$358$	0	0
$359$	−10.0142	−0.528527	−0.264263	−	0.964451i	$-0.585129\pi$
$359$	−10.0142	−0.528527	−0.264263	+	0.964451i	$0.585129\pi$
$360$	0	0
$361$	−17.4833	−0.920173
$362$	0	0
$363$	1.11085	0.0583046
$364$	0	0
$365$	10.2180	0.534837
$366$	0	0
$367$	1.28284	0.0669637	0.0334819	−	0.999439i	$-0.489340\pi$
$367$	1.28284	0.0669637	0.0334819	+	0.999439i	$0.489340\pi$
$368$	0	0
$369$	6.15614	0.320476
$370$	0	0
$371$	−4.98509	−0.258813
$372$	0	0
$373$	33.4573	1.73235	0.866176	−	0.499739i	$-0.166571\pi$
$373$	33.4573	1.73235	0.866176	+	0.499739i	$0.166571\pi$
$374$	0	0
$375$	−16.6174	−0.858120
$376$	0	0
$377$	−26.0117	−1.33967
$378$	0	0
$379$	−32.2690	−1.65755	−0.828775	−	0.559583i	$-0.810961\pi$
$379$	−32.2690	−1.65755	−0.828775	+	0.559583i	$0.810961\pi$
$380$	0	0
$381$	−4.32577	−0.221616
$382$	0	0
$383$	13.8572	0.708071	0.354036	−	0.935232i	$-0.384809\pi$
$383$	13.8572	0.708071	0.354036	+	0.935232i	$0.384809\pi$
$384$	0	0
$385$	16.5501	0.843471
$386$	0	0
$387$	−3.13042	−0.159128
$388$	0	0
$389$	29.3155	1.48636	0.743178	−	0.669094i	$-0.233317\pi$
$389$	29.3155	1.48636	0.743178	+	0.669094i	$0.233317\pi$
$390$	0	0
$391$	−12.3627	−0.625208
$392$	0	0
$393$	26.3504	1.32920
$394$	0	0
$395$	−26.7348	−1.34517
$396$	0	0
$397$	−4.33222	−0.217428	−0.108714	−	0.994073i	$-0.534673\pi$
$397$	−4.33222	−0.217428	−0.108714	+	0.994073i	$0.534673\pi$
$398$	0	0
$399$	−4.14621	−0.207570
$400$	0	0
$401$	11.9708	0.597794	0.298897	−	0.954285i	$-0.403381\pi$
$401$	11.9708	0.597794	0.298897	+	0.954285i	$0.403381\pi$
$402$	0	0
$403$	−22.9127	−1.14137
$404$	0	0
$405$	−8.87916	−0.441209
$406$	0	0
$407$	35.1275	1.74120
$408$	0	0
$409$	10.5969	0.523985	0.261992	−	0.965070i	$-0.415620\pi$
$409$	10.5969	0.523985	0.261992	+	0.965070i	$0.415620\pi$
$410$	0	0
$411$	−10.7171	−0.528635
$412$	0	0
$413$	−2.49709	−0.122874
$414$	0	0
$415$	27.5461	1.35218
$416$	0	0
$417$	−10.3555	−0.507110
$418$	0	0
$419$	−0.320387	−0.0156519	−0.00782595	−	0.999969i	$-0.502491\pi$
$419$	−0.320387	−0.0156519	−0.00782595	+	0.999969i	$0.502491\pi$
$420$	0	0
$421$	24.0501	1.17213	0.586064	−	0.810265i	$-0.300677\pi$
$421$	24.0501	1.17213	0.586064	+	0.810265i	$0.300677\pi$
$422$	0	0
$423$	−1.98749	−0.0966352
$424$	0	0
$425$	−2.10498	−0.102106
$426$	0	0
$427$	−30.8574	−1.49330
$428$	0	0
$429$	−22.2382	−1.07367
$430$	0	0
$431$	22.7364	1.09518	0.547588	−	0.836748i	$-0.315546\pi$
$431$	22.7364	1.09518	0.547588	+	0.836748i	$0.315546\pi$
$432$	0	0
$433$	−26.8292	−1.28933	−0.644663	−	0.764467i	$-0.723002\pi$
$433$	−26.8292	−1.28933	−0.644663	+	0.764467i	$0.723002\pi$
$434$	0	0
$435$	−15.0821	−0.723131
$436$	0	0
$437$	−8.00508	−0.382935
$438$	0	0
$439$	−26.3083	−1.25562	−0.627812	−	0.778365i	$-0.716049\pi$
$439$	−26.3083	−1.25562	−0.627812	+	0.778365i	$0.716049\pi$
$440$	0	0
$441$	1.14263	0.0544109
$442$	0	0
$443$	−6.31703	−0.300131	−0.150066	−	0.988676i	$-0.547949\pi$
$443$	−6.31703	−0.300131	−0.150066	+	0.988676i	$0.547949\pi$
$444$	0	0
$445$	−19.0222	−0.901737
$446$	0	0
$447$	−29.7264	−1.40601
$448$	0	0
$449$	23.1873	1.09428	0.547138	−	0.837042i	$-0.315717\pi$
$449$	23.1873	1.09428	0.547138	+	0.837042i	$0.315717\pi$
$450$	0	0
$451$	19.2628	0.907052
$452$	0	0
$453$	11.2655	0.529301
$454$	0	0
$455$	−22.6057	−1.05977
$456$	0	0
$457$	28.8658	1.35029	0.675144	−	0.737686i	$-0.264082\pi$
$457$	28.8658	1.35029	0.675144	+	0.737686i	$0.264082\pi$
$458$	0	0
$459$	−10.7492	−0.501729
$460$	0	0
$461$	−9.40857	−0.438201	−0.219100	−	0.975702i	$-0.570312\pi$
$461$	−9.40857	−0.438201	−0.219100	+	0.975702i	$0.570312\pi$
$462$	0	0
$463$	−38.4998	−1.78924	−0.894618	−	0.446831i	$-0.852553\pi$
$463$	−38.4998	−1.78924	−0.894618	+	0.446831i	$0.852553\pi$
$464$	0	0
$465$	−13.2853	−0.616089
$466$	0	0
$467$	−29.2898	−1.35537	−0.677686	−	0.735352i	$-0.737017\pi$
$467$	−29.2898	−1.35537	−0.677686	+	0.735352i	$0.737017\pi$
$468$	0	0
$469$	4.03346	0.186248
$470$	0	0
$471$	26.9180	1.24032
$472$	0	0
$473$	−9.79522	−0.450385
$474$	0	0
$475$	−1.36302	−0.0625394
$476$	0	0
$477$	−2.24234	−0.102670
$478$	0	0
$479$	14.2259	0.650000	0.325000	−	0.945714i	$-0.394636\pi$
$479$	14.2259	0.650000	0.325000	+	0.945714i	$0.394636\pi$
$480$	0	0
$481$	−47.9806	−2.18773
$482$	0	0
$483$	21.8833	0.995724
$484$	0	0
$485$	−32.0610	−1.45582
$486$	0	0
$487$	34.8431	1.57889	0.789446	−	0.613820i	$-0.210368\pi$
$487$	34.8431	1.57889	0.789446	+	0.613820i	$0.210368\pi$
$488$	0	0
$489$	−8.07622	−0.365219
$490$	0	0
$491$	29.5777	1.33482	0.667411	−	0.744689i	$-0.267402\pi$
$491$	29.5777	1.33482	0.667411	+	0.744689i	$0.267402\pi$
$492$	0	0
$493$	−10.5416	−0.474771
$494$	0	0
$495$	7.44438	0.334600
$496$	0	0
$497$	−11.2115	−0.502906
$498$	0	0
$499$	22.3770	1.00173	0.500866	−	0.865525i	$-0.333015\pi$
$499$	22.3770	1.00173	0.500866	+	0.865525i	$0.333015\pi$
$500$	0	0
$501$	22.2374	0.993494
$502$	0	0
$503$	−1.82300	−0.0812836	−0.0406418	−	0.999174i	$-0.512940\pi$
$503$	−1.82300	−0.0812836	−0.0406418	+	0.999174i	$0.512940\pi$
$504$	0	0
$505$	30.5338	1.35874
$506$	0	0
$507$	12.4468	0.552781
$508$	0	0
$509$	−2.76277	−0.122458	−0.0612288	−	0.998124i	$-0.519502\pi$
$509$	−2.76277	−0.122458	−0.0612288	+	0.998124i	$0.519502\pi$
$510$	0	0
$511$	−12.6419	−0.559246
$512$	0	0
$513$	−6.96032	−0.307305
$514$	0	0
$515$	−23.0730	−1.01672
$516$	0	0
$517$	−6.21895	−0.273509
$518$	0	0
$519$	−18.6499	−0.818639
$520$	0	0
$521$	14.7532	0.646352	0.323176	−	0.946339i	$-0.395249\pi$
$521$	14.7532	0.646352	0.323176	+	0.946339i	$0.395249\pi$
$522$	0	0
$523$	−3.00571	−0.131431	−0.0657154	−	0.997838i	$-0.520933\pi$
$523$	−3.00571	−0.131431	−0.0657154	+	0.997838i	$0.520933\pi$
$524$	0	0
$525$	3.72604	0.162618
$526$	0	0
$527$	−9.28573	−0.404493
$528$	0	0
$529$	19.2500	0.836956
$530$	0	0
$531$	−1.12321	−0.0487434
$532$	0	0
$533$	−26.3111	−1.13966
$534$	0	0
$535$	−0.171143	−0.00739914
$536$	0	0
$537$	−30.5764	−1.31947
$538$	0	0
$539$	3.57534	0.154001
$540$	0	0
$541$	22.5866	0.971076	0.485538	−	0.874216i	$-0.338624\pi$
$541$	22.5866	0.971076	0.485538	+	0.874216i	$0.338624\pi$
$542$	0	0
$543$	4.98018	0.213720
$544$	0	0
$545$	6.28491	0.269216
$546$	0	0
$547$	−6.83246	−0.292135	−0.146067	−	0.989275i	$-0.546662\pi$
$547$	−6.83246	−0.292135	−0.146067	+	0.989275i	$0.546662\pi$
$548$	0	0
$549$	−13.8799	−0.592382
$550$	0	0
$551$	−6.82591	−0.290793
$552$	0	0
$553$	33.0767	1.40656
$554$	0	0
$555$	−27.8201	−1.18090
$556$	0	0
$557$	22.0266	0.933298	0.466649	−	0.884443i	$-0.345461\pi$
$557$	22.0266	0.933298	0.466649	+	0.884443i	$0.345461\pi$
$558$	0	0
$559$	13.3793	0.565883
$560$	0	0
$561$	−9.01236	−0.380502
$562$	0	0
$563$	36.1440	1.52329	0.761645	−	0.647995i	$-0.224392\pi$
$563$	36.1440	1.52329	0.761645	+	0.647995i	$0.224392\pi$
$564$	0	0
$565$	−4.25932	−0.179191
$566$	0	0
$567$	10.9854	0.461345
$568$	0	0
$569$	−4.48775	−0.188136	−0.0940682	−	0.995566i	$-0.529987\pi$
$569$	−4.48775	−0.188136	−0.0940682	+	0.995566i	$0.529987\pi$
$570$	0	0
$571$	15.3385	0.641896	0.320948	−	0.947097i	$-0.395999\pi$
$571$	15.3385	0.641896	0.320948	+	0.947097i	$0.395999\pi$
$572$	0	0
$573$	10.2279	0.427276
$574$	0	0
$575$	7.19386	0.300005
$576$	0	0
$577$	25.5418	1.06332	0.531659	−	0.846958i	$-0.321569\pi$
$577$	25.5418	1.06332	0.531659	+	0.846958i	$0.321569\pi$
$578$	0	0
$579$	−31.7331	−1.31878
$580$	0	0
$581$	−34.0805	−1.41390
$582$	0	0
$583$	−7.01638	−0.290589
$584$	0	0
$585$	−10.1683	−0.420406
$586$	0	0
$587$	−6.30137	−0.260085	−0.130043	−	0.991508i	$-0.541511\pi$
$587$	−6.30137	−0.260085	−0.130043	+	0.991508i	$0.541511\pi$
$588$	0	0
$589$	−6.01269	−0.247749
$590$	0	0
$591$	1.11736	0.0459619
$592$	0	0
$593$	−11.9031	−0.488801	−0.244400	−	0.969674i	$-0.578591\pi$
$593$	−11.9031	−0.488801	−0.244400	+	0.969674i	$0.578591\pi$
$594$	0	0
$595$	−9.16131	−0.375577
$596$	0	0
$597$	−22.1589	−0.906902
$598$	0	0
$599$	44.5084	1.81856	0.909282	−	0.416180i	$-0.136632\pi$
$599$	44.5084	1.81856	0.909282	+	0.416180i	$0.136632\pi$
$600$	0	0
$601$	18.3848	0.749930	0.374965	−	0.927039i	$-0.377655\pi$
$601$	18.3848	0.749930	0.374965	+	0.927039i	$0.377655\pi$
$602$	0	0
$603$	1.81429	0.0738834
$604$	0	0
$605$	1.58933	0.0646156
$606$	0	0
$607$	21.1238	0.857389	0.428694	−	0.903450i	$-0.358974\pi$
$607$	21.1238	0.857389	0.428694	+	0.903450i	$0.358974\pi$
$608$	0	0
$609$	18.6598	0.756134
$610$	0	0
$611$	8.49446	0.343649
$612$	0	0
$613$	2.99053	0.120786	0.0603931	−	0.998175i	$-0.480765\pi$
$613$	2.99053	0.120786	0.0603931	+	0.998175i	$0.480765\pi$
$614$	0	0
$615$	−15.2557	−0.615169
$616$	0	0
$617$	−10.4767	−0.421775	−0.210888	−	0.977510i	$-0.567635\pi$
$617$	−10.4767	−0.421775	−0.210888	+	0.977510i	$0.567635\pi$
$618$	0	0
$619$	9.34147	0.375465	0.187733	−	0.982220i	$-0.439886\pi$
$619$	9.34147	0.375465	0.187733	+	0.982220i	$0.439886\pi$
$620$	0	0
$621$	36.7358	1.47416
$622$	0	0
$623$	23.5345	0.942892
$624$	0	0
$625$	−18.2414	−0.729655
$626$	0	0
$627$	−5.83568	−0.233055
$628$	0	0
$629$	−19.4448	−0.775316
$630$	0	0
$631$	21.8463	0.869688	0.434844	−	0.900506i	$-0.356804\pi$
$631$	21.8463	0.869688	0.434844	+	0.900506i	$0.356804\pi$
$632$	0	0
$633$	13.7648	0.547102
$634$	0	0
$635$	−6.18902	−0.245604
$636$	0	0
$637$	−4.88355	−0.193493
$638$	0	0
$639$	−5.04305	−0.199500
$640$	0	0
$641$	39.9747	1.57891	0.789453	−	0.613811i	$-0.210364\pi$
$641$	39.9747	1.57891	0.789453	+	0.613811i	$0.210364\pi$
$642$	0	0
$643$	15.1256	0.596495	0.298247	−	0.954489i	$-0.403598\pi$
$643$	15.1256	0.596495	0.298247	+	0.954489i	$0.403598\pi$
$644$	0	0
$645$	7.75757	0.305454
$646$	0	0
$647$	−43.0209	−1.69133	−0.845664	−	0.533716i	$-0.820795\pi$
$647$	−43.0209	−1.69133	−0.845664	+	0.533716i	$0.820795\pi$
$648$	0	0
$649$	−3.51459	−0.137960
$650$	0	0
$651$	16.4368	0.644207
$652$	0	0
$653$	33.2486	1.30112	0.650559	−	0.759455i	$-0.274534\pi$
$653$	33.2486	1.30112	0.650559	+	0.759455i	$0.274534\pi$
$654$	0	0
$655$	37.7005	1.47308
$656$	0	0
$657$	−5.68645	−0.221850
$658$	0	0
$659$	−42.2827	−1.64710	−0.823549	−	0.567245i	$-0.808009\pi$
$659$	−42.2827	−1.64710	−0.823549	+	0.567245i	$0.808009\pi$
$660$	0	0
$661$	−16.5817	−0.644955	−0.322477	−	0.946577i	$-0.604516\pi$
$661$	−16.5817	−0.644955	−0.322477	+	0.946577i	$0.604516\pi$
$662$	0	0
$663$	12.3100	0.478080
$664$	0	0
$665$	−5.93213	−0.230038
$666$	0	0
$667$	36.0264	1.39495
$668$	0	0
$669$	27.0425	1.04552
$670$	0	0
$671$	−43.4310	−1.67663
$672$	0	0
$673$	12.5066	0.482096	0.241048	−	0.970513i	$-0.422509\pi$
$673$	12.5066	0.482096	0.241048	+	0.970513i	$0.422509\pi$
$674$	0	0
$675$	6.25497	0.240754
$676$	0	0
$677$	41.3779	1.59028	0.795141	−	0.606424i	$-0.207397\pi$
$677$	41.3779	1.59028	0.795141	+	0.606424i	$0.207397\pi$
$678$	0	0
$679$	39.6664	1.52226
$680$	0	0
$681$	15.7915	0.605130
$682$	0	0
$683$	12.2490	0.468694	0.234347	−	0.972153i	$-0.424705\pi$
$683$	12.2490	0.468694	0.234347	+	0.972153i	$0.424705\pi$
$684$	0	0
$685$	−15.3333	−0.585856
$686$	0	0
$687$	31.2275	1.19140
$688$	0	0
$689$	9.58367	0.365109
$690$	0	0
$691$	29.4843	1.12164	0.560819	−	0.827939i	$-0.310486\pi$
$691$	29.4843	1.12164	0.560819	+	0.827939i	$0.310486\pi$
$692$	0	0
$693$	−9.21030	−0.349871
$694$	0	0
$695$	−14.8159	−0.562000
$696$	0	0
$697$	−10.6630	−0.403888
$698$	0	0
$699$	−21.9111	−0.828755
$700$	0	0
$701$	−16.4542	−0.621465	−0.310733	−	0.950497i	$-0.600574\pi$
$701$	−16.4542	−0.621465	−0.310733	+	0.950497i	$0.600574\pi$
$702$	0	0
$703$	−12.5909	−0.474875
$704$	0	0
$705$	4.92526	0.185496
$706$	0	0
$707$	−37.7770	−1.42075
$708$	0	0
$709$	−46.2347	−1.73638	−0.868190	−	0.496232i	$-0.834717\pi$
$709$	−46.2347	−1.73638	−0.868190	+	0.496232i	$0.834717\pi$
$710$	0	0
$711$	14.8782	0.557976
$712$	0	0
$713$	31.7344	1.18846
$714$	0	0
$715$	−31.8170	−1.18989
$716$	0	0
$717$	−10.1487	−0.379012
$718$	0	0
$719$	23.8436	0.889218	0.444609	−	0.895725i	$-0.353343\pi$
$719$	23.8436	0.889218	0.444609	+	0.895725i	$0.353343\pi$
$720$	0	0
$721$	28.5462	1.06312
$722$	0	0
$723$	−18.0885	−0.672717
$724$	0	0
$725$	6.13418	0.227818
$726$	0	0
$727$	−38.1659	−1.41549	−0.707747	−	0.706466i	$-0.750288\pi$
$727$	−38.1659	−1.41549	−0.707747	+	0.706466i	$0.750288\pi$
$728$	0	0
$729$	28.3241	1.04904
$730$	0	0
$731$	5.42215	0.200546
$732$	0	0
$733$	−50.9077	−1.88032	−0.940159	−	0.340737i	$-0.889323\pi$
$733$	−50.9077	−1.88032	−0.940159	+	0.340737i	$0.889323\pi$
$734$	0	0
$735$	−2.83158	−0.104444
$736$	0	0
$737$	5.67698	0.209114
$738$	0	0
$739$	−48.7104	−1.79184	−0.895920	−	0.444216i	$-0.853482\pi$
$739$	−48.7104	−1.79184	−0.895920	+	0.444216i	$0.853482\pi$
$740$	0	0
$741$	7.97095	0.292820
$742$	0	0
$743$	−0.675629	−0.0247864	−0.0123932	−	0.999923i	$-0.503945\pi$
$743$	−0.675629	−0.0247864	−0.0123932	+	0.999923i	$0.503945\pi$
$744$	0	0
$745$	−42.5306	−1.55820
$746$	0	0
$747$	−15.3297	−0.560885
$748$	0	0
$749$	0.211741	0.00773683
$750$	0	0
$751$	−20.4428	−0.745967	−0.372983	−	0.927838i	$-0.621665\pi$
$751$	−20.4428	−0.745967	−0.372983	+	0.927838i	$0.621665\pi$
$752$	0	0
$753$	0.377919	0.0137721
$754$	0	0
$755$	16.1180	0.586594
$756$	0	0
$757$	−21.5003	−0.781441	−0.390720	−	0.920509i	$-0.627774\pi$
$757$	−21.5003	−0.781441	−0.390720	+	0.920509i	$0.627774\pi$
$758$	0	0
$759$	30.8001	1.11797
$760$	0	0
$761$	29.7165	1.07722	0.538612	−	0.842554i	$-0.318949\pi$
$761$	29.7165	1.07722	0.538612	+	0.842554i	$0.318949\pi$
$762$	0	0
$763$	−7.77579	−0.281503
$764$	0	0
$765$	−4.12084	−0.148989
$766$	0	0
$767$	4.80057	0.173339
$768$	0	0
$769$	1.14904	0.0414355	0.0207178	−	0.999785i	$-0.493405\pi$
$769$	1.14904	0.0414355	0.0207178	+	0.999785i	$0.493405\pi$
$770$	0	0
$771$	−7.80266	−0.281006
$772$	0	0
$773$	3.89004	0.139915	0.0699576	−	0.997550i	$-0.477714\pi$
$773$	3.89004	0.139915	0.0699576	+	0.997550i	$0.477714\pi$
$774$	0	0
$775$	5.40337	0.194095
$776$	0	0
$777$	34.4195	1.23479
$778$	0	0
$779$	−6.90448	−0.247379
$780$	0	0
$781$	−15.7799	−0.564650
$782$	0	0
$783$	31.3245	1.11945
$784$	0	0
$785$	38.5126	1.37457
$786$	0	0
$787$	−19.9603	−0.711508	−0.355754	−	0.934580i	$-0.615776\pi$
$787$	−19.9603	−0.711508	−0.355754	+	0.934580i	$0.615776\pi$
$788$	0	0
$789$	−25.6174	−0.912003
$790$	0	0
$791$	5.26970	0.187369
$792$	0	0
$793$	59.3223	2.10660
$794$	0	0
$795$	5.55680	0.197079
$796$	0	0
$797$	−42.9686	−1.52203	−0.761014	−	0.648736i	$-0.775298\pi$
$797$	−42.9686	−1.52203	−0.761014	+	0.648736i	$0.775298\pi$
$798$	0	0
$799$	3.44251	0.121787
$800$	0	0
$801$	10.5860	0.374040
$802$	0	0
$803$	−17.7932	−0.627908
$804$	0	0
$805$	31.3092	1.10350
$806$	0	0
$807$	1.37910	0.0485468
$808$	0	0
$809$	28.0766	0.987119	0.493560	−	0.869712i	$-0.335695\pi$
$809$	28.0766	0.987119	0.493560	+	0.869712i	$0.335695\pi$
$810$	0	0
$811$	1.23855	0.0434915	0.0217458	−	0.999764i	$-0.493078\pi$
$811$	1.23855	0.0434915	0.0217458	+	0.999764i	$0.493078\pi$
$812$	0	0
$813$	7.16642	0.251337
$814$	0	0
$815$	−11.5549	−0.404751
$816$	0	0
$817$	3.51095	0.122833
$818$	0	0
$819$	12.5803	0.439593
$820$	0	0
$821$	9.91227	0.345941	0.172970	−	0.984927i	$-0.444664\pi$
$821$	9.91227	0.345941	0.172970	+	0.984927i	$0.444664\pi$
$822$	0	0
$823$	−4.82339	−0.168133	−0.0840663	−	0.996460i	$-0.526791\pi$
$823$	−4.82339	−0.168133	−0.0840663	+	0.996460i	$0.526791\pi$
$824$	0	0
$825$	5.24430	0.182583
$826$	0	0
$827$	−24.6328	−0.856566	−0.428283	−	0.903645i	$-0.640881\pi$
$827$	−24.6328	−0.856566	−0.428283	+	0.903645i	$0.640881\pi$
$828$	0	0
$829$	14.6300	0.508122	0.254061	−	0.967188i	$-0.418234\pi$
$829$	14.6300	0.508122	0.254061	+	0.967188i	$0.418234\pi$
$830$	0	0
$831$	8.45844	0.293420
$832$	0	0
$833$	−1.97913	−0.0685728
$834$	0	0
$835$	31.8158	1.10103
$836$	0	0
$837$	27.5926	0.953741
$838$	0	0
$839$	42.5008	1.46729	0.733645	−	0.679533i	$-0.237818\pi$
$839$	42.5008	1.46729	0.733645	+	0.679533i	$0.237818\pi$
$840$	0	0
$841$	1.71964	0.0592979
$842$	0	0
$843$	26.8639	0.925243
$844$	0	0
$845$	17.8080	0.612615
$846$	0	0
$847$	−1.96635	−0.0675646
$848$	0	0
$849$	−1.97915	−0.0679242
$850$	0	0
$851$	66.4535	2.27800
$852$	0	0
$853$	27.8087	0.952153	0.476077	−	0.879404i	$-0.342058\pi$
$853$	27.8087	0.952153	0.476077	+	0.879404i	$0.342058\pi$
$854$	0	0
$855$	−2.66832	−0.0912548
$856$	0	0
$857$	−25.9123	−0.885149	−0.442574	−	0.896732i	$-0.645935\pi$
$857$	−25.9123	−0.885149	−0.442574	+	0.896732i	$0.645935\pi$
$858$	0	0
$859$	−53.1905	−1.81484	−0.907418	−	0.420229i	$-0.861950\pi$
$859$	−53.1905	−1.81484	−0.907418	+	0.420229i	$0.861950\pi$
$860$	0	0
$861$	18.8746	0.643244
$862$	0	0
$863$	27.3604	0.931360	0.465680	−	0.884953i	$-0.345810\pi$
$863$	27.3604	0.931360	0.465680	+	0.884953i	$0.345810\pi$
$864$	0	0
$865$	−26.6830	−0.907251
$866$	0	0
$867$	−18.4560	−0.626798
$868$	0	0
$869$	46.5545	1.57925
$870$	0	0
$871$	−7.75419	−0.262741
$872$	0	0
$873$	17.8423	0.603871
$874$	0	0
$875$	29.4150	0.994407
$876$	0	0
$877$	−58.1170	−1.96247	−0.981235	−	0.192813i	$-0.938239\pi$
$877$	−58.1170	−1.96247	−0.981235	+	0.192813i	$0.938239\pi$
$878$	0	0
$879$	−41.1625	−1.38838
$880$	0	0
$881$	−46.7912	−1.57644	−0.788218	−	0.615396i	$-0.788996\pi$
$881$	−46.7912	−1.57644	−0.788218	+	0.615396i	$0.788996\pi$
$882$	0	0
$883$	−25.8997	−0.871593	−0.435797	−	0.900045i	$-0.643533\pi$
$883$	−25.8997	−0.871593	−0.435797	+	0.900045i	$0.643533\pi$
$884$	0	0
$885$	2.78347	0.0935653
$886$	0	0
$887$	−47.8720	−1.60738	−0.803692	−	0.595046i	$-0.797134\pi$
$887$	−47.8720	−1.60738	−0.803692	+	0.595046i	$0.797134\pi$
$888$	0	0
$889$	7.65716	0.256813
$890$	0	0
$891$	15.4617	0.517987
$892$	0	0
$893$	2.22909	0.0745937
$894$	0	0
$895$	−43.7468	−1.46229
$896$	0	0
$897$	−42.0699	−1.40467
$898$	0	0
$899$	27.0598	0.902495
$900$	0	0
$901$	3.88392	0.129392
$902$	0	0
$903$	−9.59780	−0.319395
$904$	0	0
$905$	7.12532	0.236854
$906$	0	0
$907$	55.0189	1.82687	0.913437	−	0.406980i	$-0.133418\pi$
$907$	55.0189	1.82687	0.913437	+	0.406980i	$0.133418\pi$
$908$	0	0
$909$	−16.9924	−0.563603
$910$	0	0
$911$	46.9311	1.55490	0.777448	−	0.628947i	$-0.216514\pi$
$911$	46.9311	1.55490	0.777448	+	0.628947i	$0.216514\pi$
$912$	0	0
$913$	−47.9674	−1.58749
$914$	0	0
$915$	34.3963	1.13711
$916$	0	0
$917$	−46.6436	−1.54031
$918$	0	0
$919$	10.0327	0.330948	0.165474	−	0.986214i	$-0.447085\pi$
$919$	10.0327	0.330948	0.165474	+	0.986214i	$0.447085\pi$
$920$	0	0
$921$	20.1940	0.665415
$922$	0	0
$923$	21.5538	0.709452
$924$	0	0
$925$	11.3150	0.372034
$926$	0	0
$927$	12.8404	0.421733
$928$	0	0
$929$	−24.5291	−0.804773	−0.402387	−	0.915470i	$-0.631819\pi$
$929$	−24.5291	−0.804773	−0.402387	+	0.915470i	$0.631819\pi$
$930$	0	0
$931$	−1.28153	−0.0420003
$932$	0	0
$933$	20.6764	0.676916
$934$	0	0
$935$	−12.8943	−0.421689
$936$	0	0
$937$	8.89765	0.290673	0.145337	−	0.989382i	$-0.453573\pi$
$937$	8.89765	0.290673	0.145337	+	0.989382i	$0.453573\pi$
$938$	0	0
$939$	−19.8429	−0.647549
$940$	0	0
$941$	−36.1992	−1.18006	−0.590030	−	0.807381i	$-0.700884\pi$
$941$	−36.1992	−1.18006	−0.590030	+	0.807381i	$0.700884\pi$
$942$	0	0
$943$	36.4411	1.18669
$944$	0	0
$945$	27.2229	0.885562
$946$	0	0
$947$	−15.3584	−0.499081	−0.249541	−	0.968364i	$-0.580280\pi$
$947$	−15.3584	−0.499081	−0.249541	+	0.968364i	$0.580280\pi$
$948$	0	0
$949$	24.3037	0.788931
$950$	0	0
$951$	25.4534	0.825383
$952$	0	0
$953$	−5.78646	−0.187442	−0.0937209	−	0.995599i	$-0.529876\pi$
$953$	−5.78646	−0.187442	−0.0937209	+	0.995599i	$0.529876\pi$
$954$	0	0
$955$	14.6334	0.473525
$956$	0	0
$957$	26.2632	0.848968
$958$	0	0
$959$	18.9706	0.612593
$960$	0	0
$961$	−7.16401	−0.231097
$962$	0	0
$963$	0.0952428	0.00306916
$964$	0	0
$965$	−45.4016	−1.46153
$966$	0	0
$967$	27.8199	0.894629	0.447315	−	0.894377i	$-0.352380\pi$
$967$	27.8199	0.894629	0.447315	+	0.894377i	$0.352380\pi$
$968$	0	0
$969$	3.23035	0.103774
$970$	0	0
$971$	26.2776	0.843290	0.421645	−	0.906761i	$-0.361453\pi$
$971$	26.2776	0.843290	0.421645	+	0.906761i	$0.361453\pi$
$972$	0	0
$973$	18.3305	0.587649
$974$	0	0
$975$	−7.16319	−0.229406
$976$	0	0
$977$	−38.8447	−1.24275	−0.621377	−	0.783512i	$-0.713426\pi$
$977$	−38.8447	−1.24275	−0.621377	+	0.783512i	$0.713426\pi$
$978$	0	0
$979$	33.1242	1.05865
$980$	0	0
$981$	−3.49762	−0.111670
$982$	0	0
$983$	−15.9632	−0.509147	−0.254574	−	0.967053i	$-0.581935\pi$
$983$	−15.9632	−0.509147	−0.254574	+	0.967053i	$0.581935\pi$
$984$	0	0
$985$	1.59864	0.0509369
$986$	0	0
$987$	−6.09361	−0.193962
$988$	0	0
$989$	−18.5304	−0.589234
$990$	0	0
$991$	−14.6111	−0.464138	−0.232069	−	0.972699i	$-0.574550\pi$
$991$	−14.6111	−0.464138	−0.232069	+	0.972699i	$0.574550\pi$
$992$	0	0
$993$	−40.1919	−1.27545
$994$	0	0
$995$	−31.7035	−1.00507
$996$	0	0
$997$	44.1438	1.39805	0.699024	−	0.715098i	$-0.253618\pi$
$997$	44.1438	1.39805	0.699024	+	0.715098i	$0.253618\pi$
$998$	0	0
$999$	57.7805	1.82809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.l.1.12		16
4.3	odd	2		269.2.a.c.1.9	✓	16
12.11	even	2		2421.2.a.i.1.8		16
20.19	odd	2		6725.2.a.i.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
269.2.a.c.1.9	✓	16	4.3	odd	2
2421.2.a.i.1.8		16	12.11	even	2
4304.2.a.l.1.12		16	1.1	even	1	trivial
6725.2.a.i.1.8		16	20.19	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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37910 q^{3} +1.97313 q^{5} -2.44119 q^{7} -1.09807 q^{9} -3.43591 q^{11} +4.69311 q^{13} +2.72116 q^{15} +1.90195 q^{17} +1.23155 q^{19} -3.36666 q^{21} -6.50000 q^{23} -1.10675 q^{25} -5.65167 q^{27} -5.54253 q^{29} -4.88221 q^{31} -4.73848 q^{33} -4.81680 q^{35} -10.2236 q^{37} +6.47229 q^{39} -5.60633 q^{41} +2.85084 q^{43} -2.16664 q^{45} +1.80999 q^{47} -1.04058 q^{49} +2.62299 q^{51} +2.04207 q^{53} -6.77951 q^{55} +1.69844 q^{57} +1.02290 q^{59} +12.6403 q^{61} +2.68060 q^{63} +9.26012 q^{65} -1.65225 q^{67} -8.96418 q^{69} +4.59265 q^{71} +5.17859 q^{73} -1.52632 q^{75} +8.38772 q^{77} -13.5494 q^{79} -4.50003 q^{81} +13.9606 q^{83} +3.75280 q^{85} -7.64373 q^{87} -9.64059 q^{89} -11.4568 q^{91} -6.73308 q^{93} +2.43001 q^{95} -16.2488 q^{97} +3.77287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{3} - q^{5} - 11 q^{7} + 21 q^{9} - 16 q^{11} - q^{13} + 5 q^{15} - 2 q^{17} - 35 q^{19} - q^{23} + 13 q^{25} - 11 q^{27} + 2 q^{29} - 13 q^{31} - 8 q^{33} - 9 q^{35} + 4 q^{37} - 11 q^{39}+ \cdots - 90 q^{99}+O(q^{100}) \) 16 * q - 5 * q^3 - q^5 - 11 * q^7 + 21 * q^9 - 16 * q^11 - q^13 + 5 * q^15 - 2 * q^17 - 35 * q^19 - q^23 + 13 * q^25 - 11 * q^27 + 2 * q^29 - 13 * q^31 - 8 * q^33 - 9 * q^35 + 4 * q^37 - 11 * q^39 - 2 * q^41 - 51 * q^43 - 35 * q^45 + 9 * q^47 + 27 * q^49 - 40 * q^51 - 29 * q^53 + 8 * q^55 - 16 * q^57 + 13 * q^59 + 14 * q^61 + 14 * q^63 - 29 * q^65 - 25 * q^67 - 12 * q^69 + 9 * q^71 + 4 * q^73 - 22 * q^75 - 10 * q^77 - 51 * q^79 + 2 * q^83 + 4 * q^85 - q^87 - 35 * q^89 - 65 * q^91 - 36 * q^93 - 18 * q^95 - 14 * q^97 - 90 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more