LMFDB - Embedded newform 9680.2.a.cw.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.27433728.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 15x^{2} - 3 $ x^6 - 9x^4 + 15x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.480901$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60168 q^{3} +1.00000 q^{5} -0.480901 q^{7} -0.434624 q^{9} +O(q^{10})$	$q+1.60168 q^{3} +1.00000 q^{5} -0.480901 q^{7} -0.434624 q^{9} -4.79559 q^{13} +1.60168 q^{15} +2.50230 q^{17} +5.75739 q^{19} -0.770249 q^{21} -4.43462 q^{23} +1.00000 q^{25} -5.50117 q^{27} +9.01248 q^{29} -7.97209 q^{31} -0.480901 q^{35} +5.20336 q^{37} -7.68099 q^{39} -8.45124 q^{41} -8.53158 q^{43} -0.434624 q^{45} -9.60168 q^{47} -6.76873 q^{49} +4.00788 q^{51} +6.10051 q^{53} +9.22149 q^{57} -2.76873 q^{59} +4.02534 q^{61} +0.209011 q^{63} -4.79559 q^{65} -7.60168 q^{67} -7.10284 q^{69} +2.76873 q^{71} -5.00460 q^{73} +1.60168 q^{75} +3.62478 q^{79} -7.50723 q^{81} +1.71459 q^{83} +2.50230 q^{85} +14.4351 q^{87} -15.7408 q^{89} +2.30620 q^{91} -12.7687 q^{93} +5.75739 q^{95} -3.63798 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + 12 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + 12 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + 12 q^{9} - 6 q^{15} - 12 q^{23} + 6 q^{25} - 30 q^{27} + 12 q^{45} - 42 q^{47} - 24 q^{49} + 24 q^{53} - 30 q^{67} - 24 q^{69} - 6 q^{75} + 30 q^{81} - 30 q^{89} - 36 q^{91} - 60 q^{93} + 24 q^{97}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + 12 * q^9 - 6 * q^15 - 12 * q^23 + 6 * q^25 - 30 * q^27 + 12 * q^45 - 42 * q^47 - 24 * q^49 + 24 * q^53 - 30 * q^67 - 24 * q^69 - 6 * q^75 + 30 * q^81 - 30 * q^89 - 36 * q^91 - 60 * q^93 + 24 * 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
$3$	1.60168	0.924730	0.462365	−	0.886690i	$-0.347001\pi$
$3$	1.60168	0.924730	0.462365	+	0.886690i	$0.347001\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.480901	−0.181763	−0.0908817	−	0.995862i	$-0.528969\pi$
$7$	−0.480901	−0.181763	−0.0908817	+	0.995862i	$0.528969\pi$
$8$	0	0
$9$	−0.434624	−0.144875
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.79559	−1.33006	−0.665028	−	0.746818i	$-0.731581\pi$
$13$	−4.79559	−1.33006	−0.665028	+	0.746818i	$0.731581\pi$
$14$	0	0
$15$	1.60168	0.413552
$16$	0	0
$17$	2.50230	0.606897	0.303448	−	0.952848i	$-0.401862\pi$
$17$	2.50230	0.606897	0.303448	+	0.952848i	$0.401862\pi$
$18$	0	0
$19$	5.75739	1.32084	0.660418	−	0.750898i	$-0.270379\pi$
$19$	5.75739	1.32084	0.660418	+	0.750898i	$0.270379\pi$
$20$	0	0
$21$	−0.770249	−0.168082
$22$	0	0
$23$	−4.43462	−0.924683	−0.462342	−	0.886702i	$-0.652991\pi$
$23$	−4.43462	−0.924683	−0.462342	+	0.886702i	$0.652991\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.50117	−1.05870
$28$	0	0
$29$	9.01248	1.67358	0.836788	−	0.547527i	$-0.184431\pi$
$29$	9.01248	1.67358	0.836788	+	0.547527i	$0.184431\pi$
$30$	0	0
$31$	−7.97209	−1.43183	−0.715915	−	0.698187i	$-0.753990\pi$
$31$	−7.97209	−1.43183	−0.715915	+	0.698187i	$0.753990\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.480901	−0.0812871
$36$	0	0
$37$	5.20336	0.855427	0.427713	−	0.903914i	$-0.359319\pi$
$37$	5.20336	0.855427	0.427713	+	0.903914i	$0.359319\pi$
$38$	0	0
$39$	−7.68099	−1.22994
$40$	0	0
$41$	−8.45124	−1.31986	−0.659931	−	0.751326i	$-0.729415\pi$
$41$	−8.45124	−1.31986	−0.659931	+	0.751326i	$0.729415\pi$
$42$	0	0
$43$	−8.53158	−1.30105	−0.650527	−	0.759483i	$-0.725452\pi$
$43$	−8.53158	−1.30105	−0.650527	+	0.759483i	$0.725452\pi$
$44$	0	0
$45$	−0.434624	−0.0647899
$46$	0	0
$47$	−9.60168	−1.40055	−0.700274	−	0.713874i	$-0.746939\pi$
$47$	−9.60168	−1.40055	−0.700274	+	0.713874i	$0.746939\pi$
$48$	0	0
$49$	−6.76873	−0.966962
$50$	0	0
$51$	4.00788	0.561216
$52$	0	0
$53$	6.10051	0.837970	0.418985	−	0.907993i	$-0.362386\pi$
$53$	6.10051	0.837970	0.418985	+	0.907993i	$0.362386\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.22149	1.22142
$58$	0	0
$59$	−2.76873	−0.360459	−0.180229	−	0.983625i	$-0.557684\pi$
$59$	−2.76873	−0.360459	−0.180229	+	0.983625i	$0.557684\pi$
$60$	0	0
$61$	4.02534	0.515392	0.257696	−	0.966226i	$-0.417037\pi$
$61$	4.02534	0.515392	0.257696	+	0.966226i	$0.417037\pi$
$62$	0	0
$63$	0.209011	0.0263329
$64$	0	0
$65$	−4.79559	−0.594820
$66$	0	0
$67$	−7.60168	−0.928693	−0.464346	−	0.885654i	$-0.653711\pi$
$67$	−7.60168	−0.928693	−0.464346	+	0.885654i	$0.653711\pi$
$68$	0	0
$69$	−7.10284	−0.855082
$70$	0	0
$71$	2.76873	0.328588	0.164294	−	0.986411i	$-0.447465\pi$
$71$	2.76873	0.328588	0.164294	+	0.986411i	$0.447465\pi$
$72$	0	0
$73$	−5.00460	−0.585744	−0.292872	−	0.956152i	$-0.594611\pi$
$73$	−5.00460	−0.585744	−0.292872	+	0.956152i	$0.594611\pi$
$74$	0	0
$75$	1.60168	0.184946
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.62478	0.407819	0.203910	−	0.978990i	$-0.434635\pi$
$79$	3.62478	0.407819	0.203910	+	0.978990i	$0.434635\pi$
$80$	0	0
$81$	−7.50723	−0.834137
$82$	0	0
$83$	1.71459	0.188201	0.0941005	−	0.995563i	$-0.470002\pi$
$83$	1.71459	0.188201	0.0941005	+	0.995563i	$0.470002\pi$
$84$	0	0
$85$	2.50230	0.271413
$86$	0	0
$87$	14.4351	1.54761
$88$	0	0
$89$	−15.7408	−1.66852	−0.834262	−	0.551368i	$-0.814106\pi$
$89$	−15.7408	−1.66852	−0.834262	+	0.551368i	$0.814106\pi$
$90$	0	0
$91$	2.30620	0.241756
$92$	0	0
$93$	−12.7687	−1.32406
$94$	0	0
$95$	5.75739	0.590696
$96$	0	0
$97$	−3.63798	−0.369381	−0.184691	−	0.982797i	$-0.559128\pi$
$97$	−3.63798	−0.369381	−0.184691	+	0.982797i	$0.559128\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.69845	0.766025	0.383012	−	0.923743i	$-0.374887\pi$
$101$	7.69845	0.766025	0.383012	+	0.923743i	$0.374887\pi$
$102$	0	0
$103$	−9.10284	−0.896930	−0.448465	−	0.893800i	$-0.648029\pi$
$103$	−9.10284	−0.896930	−0.448465	+	0.893800i	$0.648029\pi$
$104$	0	0
$105$	−0.770249	−0.0751686
$106$	0	0
$107$	−16.7913	−1.62327	−0.811637	−	0.584163i	$-0.801423\pi$
$107$	−16.7913	−1.62327	−0.811637	+	0.584163i	$0.801423\pi$
$108$	0	0
$109$	17.8817	1.71276	0.856380	−	0.516346i	$-0.172708\pi$
$109$	17.8817	1.71276	0.856380	+	0.516346i	$0.172708\pi$
$110$	0	0
$111$	8.33411	0.791039
$112$	0	0
$113$	−0.462531	−0.0435113	−0.0217556	−	0.999763i	$-0.506926\pi$
$113$	−0.462531	−0.0435113	−0.0217556	+	0.999763i	$0.506926\pi$
$114$	0	0
$115$	−4.43462	−0.413531
$116$	0	0
$117$	2.08428	0.192692
$118$	0	0
$119$	−1.20336	−0.110312
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−13.5362	−1.22052
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.8295	1.40464	0.702319	−	0.711862i	$-0.252148\pi$
$127$	15.8295	1.40464	0.702319	+	0.711862i	$0.252148\pi$
$128$	0	0
$129$	−13.6649	−1.20312
$130$	0	0
$131$	9.60460	0.839158	0.419579	−	0.907719i	$-0.362178\pi$
$131$	9.60460	0.839158	0.419579	+	0.907719i	$0.362178\pi$
$132$	0	0
$133$	−2.76873	−0.240080
$134$	0	0
$135$	−5.50117	−0.473465
$136$	0	0
$137$	3.97209	0.339359	0.169679	−	0.985499i	$-0.445727\pi$
$137$	3.97209	0.339359	0.169679	+	0.985499i	$0.445727\pi$
$138$	0	0
$139$	9.22149	0.782157	0.391078	−	0.920357i	$-0.372102\pi$
$139$	9.22149	0.782157	0.391078	+	0.920357i	$0.372102\pi$
$140$	0	0
$141$	−15.3788	−1.29513
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.01248	0.748446
$146$	0	0
$147$	−10.8413	−0.894179
$148$	0	0
$149$	−7.65012	−0.626722	−0.313361	−	0.949634i	$-0.601455\pi$
$149$	−7.65012	−0.626722	−0.313361	+	0.949634i	$0.601455\pi$
$150$	0	0
$151$	−2.87198	−0.233719	−0.116859	−	0.993148i	$-0.537283\pi$
$151$	−2.87198	−0.233719	−0.116859	+	0.993148i	$0.537283\pi$
$152$	0	0
$153$	−1.08756	−0.0879240
$154$	0	0
$155$	−7.97209	−0.640334
$156$	0	0
$157$	17.4817	1.39519	0.697594	−	0.716493i	$-0.254254\pi$
$157$	17.4817	1.39519	0.697594	+	0.716493i	$0.254254\pi$
$158$	0	0
$159$	9.77107	0.774896
$160$	0	0
$161$	2.13261	0.168074
$162$	0	0
$163$	−13.1671	−1.03132	−0.515662	−	0.856792i	$-0.672454\pi$
$163$	−13.1671	−1.03132	−0.515662	+	0.856792i	$0.672454\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.8778	−1.22866	−0.614331	−	0.789049i	$-0.710574\pi$
$167$	−15.8778	−1.22866	−0.614331	+	0.789049i	$0.710574\pi$
$168$	0	0
$169$	9.99767	0.769051
$170$	0	0
$171$	−2.50230	−0.191356
$172$	0	0
$173$	−22.8206	−1.73501	−0.867507	−	0.497425i	$-0.834279\pi$
$173$	−22.8206	−1.73501	−0.867507	+	0.497425i	$0.834279\pi$
$174$	0	0
$175$	−0.480901	−0.0363527
$176$	0	0
$177$	−4.43462	−0.333327
$178$	0	0
$179$	0.462531	0.0345712	0.0172856	−	0.999851i	$-0.494498\pi$
$179$	0.462531	0.0345712	0.0172856	+	0.999851i	$0.494498\pi$
$180$	0	0
$181$	−3.12842	−0.232534	−0.116267	−	0.993218i	$-0.537093\pi$
$181$	−3.12842	−0.232534	−0.116267	+	0.993218i	$0.537093\pi$
$182$	0	0
$183$	6.44730	0.476598
$184$	0	0
$185$	5.20336	0.382559
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.64552	0.192433
$190$	0	0
$191$	5.43695	0.393404	0.196702	−	0.980463i	$-0.436977\pi$
$191$	5.43695	0.393404	0.196702	+	0.980463i	$0.436977\pi$
$192$	0	0
$193$	3.20675	0.230827	0.115414	−	0.993318i	$-0.463181\pi$
$193$	3.20675	0.230827	0.115414	+	0.993318i	$0.463181\pi$
$194$	0	0
$195$	−7.68099	−0.550047
$196$	0	0
$197$	−19.4048	−1.38253	−0.691267	−	0.722600i	$-0.742947\pi$
$197$	−19.4048	−1.38253	−0.691267	+	0.722600i	$0.742947\pi$
$198$	0	0
$199$	−20.3509	−1.44264	−0.721319	−	0.692603i	$-0.756464\pi$
$199$	−20.3509	−1.44264	−0.721319	+	0.692603i	$0.756464\pi$
$200$	0	0
$201$	−12.1755	−0.858790
$202$	0	0
$203$	−4.33411	−0.304195
$204$	0	0
$205$	−8.45124	−0.590260
$206$	0	0
$207$	1.92739	0.133963
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−6.87987	−0.473630	−0.236815	−	0.971555i	$-0.576103\pi$
$211$	−6.87987	−0.473630	−0.236815	+	0.971555i	$0.576103\pi$
$212$	0	0
$213$	4.43462	0.303855
$214$	0	0
$215$	−8.53158	−0.581849
$216$	0	0
$217$	3.83379	0.260254
$218$	0	0
$219$	−8.01576	−0.541655
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	1.03863	0.0695521	0.0347760	−	0.999395i	$-0.488928\pi$
$223$	1.03863	0.0695521	0.0347760	+	0.999395i	$0.488928\pi$
$224$	0	0
$225$	−0.434624	−0.0289749
$226$	0	0
$227$	−0.480901	−0.0319185	−0.0159593	−	0.999873i	$-0.505080\pi$
$227$	−0.480901	−0.0319185	−0.0159593	+	0.999873i	$0.505080\pi$
$228$	0	0
$229$	13.5398	0.894735	0.447368	−	0.894350i	$-0.352362\pi$
$229$	13.5398	0.894735	0.447368	+	0.894350i	$0.352362\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.1543	−1.38586	−0.692932	−	0.721003i	$-0.743681\pi$
$233$	−21.1543	−1.38586	−0.692932	+	0.721003i	$0.743681\pi$
$234$	0	0
$235$	−9.60168	−0.626344
$236$	0	0
$237$	5.80573	0.377123
$238$	0	0
$239$	−2.67639	−0.173122	−0.0865608	−	0.996247i	$-0.527588\pi$
$239$	−2.67639	−0.173122	−0.0865608	+	0.996247i	$0.527588\pi$
$240$	0	0
$241$	−4.56912	−0.294323	−0.147161	−	0.989112i	$-0.547014\pi$
$241$	−4.56912	−0.294323	−0.147161	+	0.989112i	$0.547014\pi$
$242$	0	0
$243$	4.47932	0.287349
$244$	0	0
$245$	−6.76873	−0.432439
$246$	0	0
$247$	−27.6101	−1.75679
$248$	0	0
$249$	2.74623	0.174035
$250$	0	0
$251$	11.4370	0.721894	0.360947	−	0.932586i	$-0.382453\pi$
$251$	11.4370	0.721894	0.360947	+	0.932586i	$0.382453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.00788	0.250983
$256$	0	0
$257$	−20.0447	−1.25035	−0.625177	−	0.780483i	$-0.714973\pi$
$257$	−20.0447	−1.25035	−0.625177	+	0.780483i	$0.714973\pi$
$258$	0	0
$259$	−2.50230	−0.155485
$260$	0	0
$261$	−3.91704	−0.242459
$262$	0	0
$263$	6.55852	0.404416	0.202208	−	0.979343i	$-0.435188\pi$
$263$	6.55852	0.404416	0.202208	+	0.979343i	$0.435188\pi$
$264$	0	0
$265$	6.10051	0.374752
$266$	0	0
$267$	−25.2118	−1.54293
$268$	0	0
$269$	1.89716	0.115672	0.0578358	−	0.998326i	$-0.481580\pi$
$269$	1.89716	0.115672	0.0578358	+	0.998326i	$0.481580\pi$
$270$	0	0
$271$	−11.3541	−0.689713	−0.344856	−	0.938655i	$-0.612072\pi$
$271$	−11.3541	−0.689713	−0.344856	+	0.938655i	$0.612072\pi$
$272$	0	0
$273$	3.69380	0.223559
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.8340	−1.37196	−0.685980	−	0.727620i	$-0.740626\pi$
$277$	−22.8340	−1.37196	−0.685980	+	0.727620i	$0.740626\pi$
$278$	0	0
$279$	3.46486	0.207436
$280$	0	0
$281$	22.6115	1.34889	0.674446	−	0.738324i	$-0.264383\pi$
$281$	22.6115	1.34889	0.674446	+	0.738324i	$0.264383\pi$
$282$	0	0
$283$	−27.5667	−1.63867	−0.819335	−	0.573316i	$-0.805657\pi$
$283$	−27.5667	−1.63867	−0.819335	+	0.573316i	$0.805657\pi$
$284$	0	0
$285$	9.22149	0.546234
$286$	0	0
$287$	4.06421	0.239903
$288$	0	0
$289$	−10.7385	−0.631676
$290$	0	0
$291$	−5.82688	−0.341578
$292$	0	0
$293$	21.8587	1.27700	0.638501	−	0.769621i	$-0.279555\pi$
$293$	21.8587	1.27700	0.638501	+	0.769621i	$0.279555\pi$
$294$	0	0
$295$	−2.76873	−0.161202
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.2666	1.22988
$300$	0	0
$301$	4.10284	0.236484
$302$	0	0
$303$	12.3305	0.708366
$304$	0	0
$305$	4.02534	0.230490
$306$	0	0
$307$	−9.80019	−0.559326	−0.279663	−	0.960098i	$-0.590223\pi$
$307$	−9.80019	−0.559326	−0.279663	+	0.960098i	$0.590223\pi$
$308$	0	0
$309$	−14.5798	−0.829418
$310$	0	0
$311$	−9.77107	−0.554066	−0.277033	−	0.960860i	$-0.589351\pi$
$311$	−9.77107	−0.554066	−0.277033	+	0.960860i	$0.589351\pi$
$312$	0	0
$313$	−0.897155	−0.0507102	−0.0253551	−	0.999679i	$-0.508072\pi$
$313$	−0.897155	−0.0507102	−0.0253551	+	0.999679i	$0.508072\pi$
$314$	0	0
$315$	0.209011	0.0117764
$316$	0	0
$317$	−2.66822	−0.149862	−0.0749311	−	0.997189i	$-0.523874\pi$
$317$	−2.66822	−0.149862	−0.0749311	+	0.997189i	$0.523874\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−26.8942	−1.50109
$322$	0	0
$323$	14.4067	0.801611
$324$	0	0
$325$	−4.79559	−0.266011
$326$	0	0
$327$	28.6408	1.58384
$328$	0	0
$329$	4.61746	0.254569
$330$	0	0
$331$	−18.9744	−1.04293	−0.521464	−	0.853273i	$-0.674614\pi$
$331$	−18.9744	−1.04293	−0.521464	+	0.853273i	$0.674614\pi$
$332$	0	0
$333$	−2.26150	−0.123930
$334$	0	0
$335$	−7.60168	−0.415324
$336$	0	0
$337$	4.79559	0.261232	0.130616	−	0.991433i	$-0.458304\pi$
$337$	4.79559	0.261232	0.130616	+	0.991433i	$0.458304\pi$
$338$	0	0
$339$	−0.740827	−0.0402362
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6.62140	0.357522
$344$	0	0
$345$	−7.10284	−0.382404
$346$	0	0
$347$	−0.258469	−0.0138754	−0.00693768	−	0.999976i	$-0.502208\pi$
$347$	−0.258469	−0.0138754	−0.00693768	+	0.999976i	$0.502208\pi$
$348$	0	0
$349$	−1.34491	−0.0719913	−0.0359956	−	0.999352i	$-0.511460\pi$
$349$	−1.34491	−0.0719913	−0.0359956	+	0.999352i	$0.511460\pi$
$350$	0	0
$351$	26.3813	1.40813
$352$	0	0
$353$	35.4537	1.88701	0.943506	−	0.331355i	$-0.107506\pi$
$353$	35.4537	1.88701	0.943506	+	0.331355i	$0.107506\pi$
$354$	0	0
$355$	2.76873	0.146949
$356$	0	0
$357$	−1.92739	−0.102008
$358$	0	0
$359$	−7.84167	−0.413867	−0.206934	−	0.978355i	$-0.566348\pi$
$359$	−7.84167	−0.413867	−0.206934	+	0.978355i	$0.566348\pi$
$360$	0	0
$361$	14.1475	0.744608
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.00460	−0.261953
$366$	0	0
$367$	−1.06654	−0.0556730	−0.0278365	−	0.999612i	$-0.508862\pi$
$367$	−1.06654	−0.0556730	−0.0278365	+	0.999612i	$0.508862\pi$
$368$	0	0
$369$	3.67311	0.191215
$370$	0	0
$371$	−2.93374	−0.152312
$372$	0	0
$373$	37.6388	1.94886	0.974431	−	0.224689i	$-0.0721366\pi$
$373$	37.6388	1.94886	0.974431	+	0.224689i	$0.0721366\pi$
$374$	0	0
$375$	1.60168	0.0827104
$376$	0	0
$377$	−43.2201	−2.22595
$378$	0	0
$379$	−35.1475	−1.80541	−0.902704	−	0.430262i	$-0.858421\pi$
$379$	−35.1475	−1.80541	−0.902704	+	0.430262i	$0.858421\pi$
$380$	0	0
$381$	25.3537	1.29891
$382$	0	0
$383$	−0.824549	−0.0421325	−0.0210662	−	0.999778i	$-0.506706\pi$
$383$	−0.824549	−0.0421325	−0.0210662	+	0.999778i	$0.506706\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.70803	0.188490
$388$	0	0
$389$	4.84367	0.245584	0.122792	−	0.992432i	$-0.460815\pi$
$389$	4.84367	0.245584	0.122792	+	0.992432i	$0.460815\pi$
$390$	0	0
$391$	−11.0968	−0.561187
$392$	0	0
$393$	15.3835	0.775994
$394$	0	0
$395$	3.62478	0.182382
$396$	0	0
$397$	−7.33178	−0.367971	−0.183986	−	0.982929i	$-0.558900\pi$
$397$	−7.33178	−0.367971	−0.183986	+	0.982929i	$0.558900\pi$
$398$	0	0
$399$	−4.43462	−0.222009
$400$	0	0
$401$	−31.1499	−1.55555	−0.777775	−	0.628542i	$-0.783652\pi$
$401$	−31.1499	−1.55555	−0.777775	+	0.628542i	$0.783652\pi$
$402$	0	0
$403$	38.2309	1.90442
$404$	0	0
$405$	−7.50723	−0.373037
$406$	0	0
$407$	0	0
$408$	0	0
$409$	7.23209	0.357604	0.178802	−	0.983885i	$-0.442778\pi$
$409$	7.23209	0.357604	0.178802	+	0.983885i	$0.442778\pi$
$410$	0	0
$411$	6.36202	0.313815
$412$	0	0
$413$	1.33149	0.0655182
$414$	0	0
$415$	1.71459	0.0841660
$416$	0	0
$417$	14.7699	0.723284
$418$	0	0
$419$	2.30620	0.112665	0.0563327	−	0.998412i	$-0.482059\pi$
$419$	2.30620	0.112665	0.0563327	+	0.998412i	$0.482059\pi$
$420$	0	0
$421$	6.63798	0.323515	0.161758	−	0.986831i	$-0.448284\pi$
$421$	6.63798	0.323515	0.161758	+	0.986831i	$0.448284\pi$
$422$	0	0
$423$	4.17312	0.202904
$424$	0	0
$425$	2.50230	0.121379
$426$	0	0
$427$	−1.93579	−0.0936794
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.8432	−1.58200	−0.791000	−	0.611816i	$-0.790439\pi$
$431$	−32.8432	−1.58200	−0.791000	+	0.611816i	$0.790439\pi$
$432$	0	0
$433$	2.36202	0.113511	0.0567557	−	0.998388i	$-0.481924\pi$
$433$	2.36202	0.113511	0.0567557	+	0.998388i	$0.481924\pi$
$434$	0	0
$435$	14.4351	0.692110
$436$	0	0
$437$	−25.5319	−1.22135
$438$	0	0
$439$	30.8847	1.47404	0.737022	−	0.675869i	$-0.236231\pi$
$439$	30.8847	1.47404	0.737022	+	0.675869i	$0.236231\pi$
$440$	0	0
$441$	2.94185	0.140088
$442$	0	0
$443$	13.7748	0.654460	0.327230	−	0.944945i	$-0.393885\pi$
$443$	13.7748	0.654460	0.327230	+	0.944945i	$0.393885\pi$
$444$	0	0
$445$	−15.7408	−0.746187
$446$	0	0
$447$	−12.2530	−0.579548
$448$	0	0
$449$	−38.9163	−1.83657	−0.918286	−	0.395917i	$-0.870427\pi$
$449$	−38.9163	−1.83657	−0.918286	+	0.395917i	$0.870427\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−4.60000	−0.216127
$454$	0	0
$455$	2.30620	0.108116
$456$	0	0
$457$	1.33149	0.0622843	0.0311422	−	0.999515i	$-0.490086\pi$
$457$	1.33149	0.0622843	0.0311422	+	0.999515i	$0.490086\pi$
$458$	0	0
$459$	−13.7656	−0.642522
$460$	0	0
$461$	5.73993	0.267335	0.133668	−	0.991026i	$-0.457325\pi$
$461$	5.73993	0.267335	0.133668	+	0.991026i	$0.457325\pi$
$462$	0	0
$463$	3.44535	0.160119	0.0800595	−	0.996790i	$-0.474489\pi$
$463$	3.44535	0.160119	0.0800595	+	0.996790i	$0.474489\pi$
$464$	0	0
$465$	−12.7687	−0.592136
$466$	0	0
$467$	27.2164	1.25943	0.629713	−	0.776828i	$-0.283173\pi$
$467$	27.2164	1.25943	0.629713	+	0.776828i	$0.283173\pi$
$468$	0	0
$469$	3.65565	0.168802
$470$	0	0
$471$	28.0000	1.29017
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.75739	0.264167
$476$	0	0
$477$	−2.65143	−0.121401
$478$	0	0
$479$	−13.9822	−0.638861	−0.319431	−	0.947610i	$-0.603492\pi$
$479$	−13.9822	−0.638861	−0.319431	+	0.947610i	$0.603492\pi$
$480$	0	0
$481$	−24.9532	−1.13777
$482$	0	0
$483$	3.41576	0.155423
$484$	0	0
$485$	−3.63798	−0.165192
$486$	0	0
$487$	7.45375	0.337761	0.168881	−	0.985636i	$-0.445985\pi$
$487$	7.45375	0.337761	0.168881	+	0.985636i	$0.445985\pi$
$488$	0	0
$489$	−21.0894	−0.953696
$490$	0	0
$491$	15.9756	0.720969	0.360484	−	0.932765i	$-0.382611\pi$
$491$	15.9756	0.720969	0.360484	+	0.932765i	$0.382611\pi$
$492$	0	0
$493$	22.5519	1.01569
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.33149	−0.0597253
$498$	0	0
$499$	24.3509	1.09010	0.545048	−	0.838405i	$-0.316511\pi$
$499$	24.3509	1.09010	0.545048	+	0.838405i	$0.316511\pi$
$500$	0	0
$501$	−25.4311	−1.13618
$502$	0	0
$503$	−27.5667	−1.22914	−0.614569	−	0.788863i	$-0.710670\pi$
$503$	−27.5667	−1.22914	−0.614569	+	0.788863i	$0.710670\pi$
$504$	0	0
$505$	7.69845	0.342577
$506$	0	0
$507$	16.0131	0.711165
$508$	0	0
$509$	17.6850	0.783874	0.391937	−	0.919992i	$-0.371805\pi$
$509$	17.6850	0.783874	0.391937	+	0.919992i	$0.371805\pi$
$510$	0	0
$511$	2.40672	0.106467
$512$	0	0
$513$	−31.6724	−1.39837
$514$	0	0
$515$	−9.10284	−0.401119
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−36.5512	−1.60442
$520$	0	0
$521$	−5.38993	−0.236137	−0.118068	−	0.993005i	$-0.537670\pi$
$521$	−5.38993	−0.236137	−0.118068	+	0.993005i	$0.537670\pi$
$522$	0	0
$523$	37.6737	1.64735	0.823677	−	0.567059i	$-0.191919\pi$
$523$	37.6737	1.64735	0.823677	+	0.567059i	$0.191919\pi$
$524$	0	0
$525$	−0.770249	−0.0336164
$526$	0	0
$527$	−19.9486	−0.868973
$528$	0	0
$529$	−3.33411	−0.144961
$530$	0	0
$531$	1.20336	0.0522213
$532$	0	0
$533$	40.5287	1.75549
$534$	0	0
$535$	−16.7913	−0.725950
$536$	0	0
$537$	0.740827	0.0319690
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−28.9651	−1.24531	−0.622653	−	0.782498i	$-0.713945\pi$
$541$	−28.9651	−1.24531	−0.622653	+	0.782498i	$0.713945\pi$
$542$	0	0
$543$	−5.01073	−0.215031
$544$	0	0
$545$	17.8817	0.765970
$546$	0	0
$547$	−19.7396	−0.844002	−0.422001	−	0.906595i	$-0.638672\pi$
$547$	−19.7396	−0.844002	−0.422001	+	0.906595i	$0.638672\pi$
$548$	0	0
$549$	−1.74951	−0.0746672
$550$	0	0
$551$	51.8884	2.21052
$552$	0	0
$553$	−1.74316	−0.0741266
$554$	0	0
$555$	8.33411	0.353763
$556$	0	0
$557$	−16.6801	−0.706757	−0.353378	−	0.935481i	$-0.614967\pi$
$557$	−16.6801	−0.706757	−0.353378	+	0.935481i	$0.614967\pi$
$558$	0	0
$559$	40.9139	1.73048
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.90680	−0.206797	−0.103399	−	0.994640i	$-0.532972\pi$
$563$	−4.90680	−0.206797	−0.103399	+	0.994640i	$0.532972\pi$
$564$	0	0
$565$	−0.462531	−0.0194588
$566$	0	0
$567$	3.61023	0.151616
$568$	0	0
$569$	22.4334	0.940457	0.470229	−	0.882545i	$-0.344171\pi$
$569$	22.4334	0.940457	0.470229	+	0.882545i	$0.344171\pi$
$570$	0	0
$571$	0.195590	0.00818520	0.00409260	−	0.999992i	$-0.498697\pi$
$571$	0.195590	0.00818520	0.00409260	+	0.999992i	$0.498697\pi$
$572$	0	0
$573$	8.70826	0.363793
$574$	0	0
$575$	−4.43462	−0.184937
$576$	0	0
$577$	−41.2481	−1.71718	−0.858590	−	0.512664i	$-0.828659\pi$
$577$	−41.2481	−1.71718	−0.858590	+	0.512664i	$0.828659\pi$
$578$	0	0
$579$	5.13619	0.213453
$580$	0	0
$581$	−0.824549	−0.0342081
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.08428	0.0861743
$586$	0	0
$587$	1.77480	0.0732538	0.0366269	−	0.999329i	$-0.488339\pi$
$587$	1.77480	0.0732538	0.0366269	+	0.999329i	$0.488339\pi$
$588$	0	0
$589$	−45.8984	−1.89121
$590$	0	0
$591$	−31.0802	−1.27847
$592$	0	0
$593$	29.0094	1.19127	0.595636	−	0.803254i	$-0.296900\pi$
$593$	29.0094	1.19127	0.595636	+	0.803254i	$0.296900\pi$
$594$	0	0
$595$	−1.20336	−0.0493329
$596$	0	0
$597$	−32.5956	−1.33405
$598$	0	0
$599$	−40.2672	−1.64527	−0.822636	−	0.568568i	$-0.807498\pi$
$599$	−40.2672	−1.64527	−0.822636	+	0.568568i	$0.807498\pi$
$600$	0	0
$601$	13.6957	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$601$	13.6957	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$602$	0	0
$603$	3.30387	0.134544
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.1543	0.858626	0.429313	−	0.903156i	$-0.358756\pi$
$607$	21.1543	0.858626	0.429313	+	0.903156i	$0.358756\pi$
$608$	0	0
$609$	−6.94185	−0.281298
$610$	0	0
$611$	46.0457	1.86281
$612$	0	0
$613$	23.2520	0.939139	0.469570	−	0.882895i	$-0.344409\pi$
$613$	23.2520	0.939139	0.469570	+	0.882895i	$0.344409\pi$
$614$	0	0
$615$	−13.5362	−0.545831
$616$	0	0
$617$	−13.1028	−0.527501	−0.263750	−	0.964591i	$-0.584960\pi$
$617$	−13.1028	−0.527501	−0.263750	+	0.964591i	$0.584960\pi$
$618$	0	0
$619$	−16.6682	−0.669952	−0.334976	−	0.942227i	$-0.608728\pi$
$619$	−16.6682	−0.669952	−0.334976	+	0.942227i	$0.608728\pi$
$620$	0	0
$621$	24.3956	0.978962
$622$	0	0
$623$	7.56978	0.303277
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.0204	0.519156
$630$	0	0
$631$	2.66356	0.106035	0.0530173	−	0.998594i	$-0.483116\pi$
$631$	2.66356	0.106035	0.0530173	+	0.998594i	$0.483116\pi$
$632$	0	0
$633$	−11.0193	−0.437979
$634$	0	0
$635$	15.8295	0.628173
$636$	0	0
$637$	32.4601	1.28611
$638$	0	0
$639$	−1.20336	−0.0476041
$640$	0	0
$641$	−0.462531	−0.0182689	−0.00913445	−	0.999958i	$-0.502908\pi$
$641$	−0.462531	−0.0182689	−0.00913445	+	0.999958i	$0.502908\pi$
$642$	0	0
$643$	33.4621	1.31962	0.659809	−	0.751433i	$-0.270637\pi$
$643$	33.4621	1.31962	0.659809	+	0.751433i	$0.270637\pi$
$644$	0	0
$645$	−13.6649	−0.538053
$646$	0	0
$647$	40.2588	1.58274	0.791368	−	0.611340i	$-0.209369\pi$
$647$	40.2588	1.58274	0.791368	+	0.611340i	$0.209369\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.14050	0.240665
$652$	0	0
$653$	15.3486	0.600636	0.300318	−	0.953839i	$-0.402907\pi$
$653$	15.3486	0.600636	0.300318	+	0.953839i	$0.402907\pi$
$654$	0	0
$655$	9.60460	0.375283
$656$	0	0
$657$	2.17512	0.0848595
$658$	0	0
$659$	36.6635	1.42821	0.714104	−	0.700039i	$-0.246834\pi$
$659$	36.6635	1.42821	0.714104	+	0.700039i	$0.246834\pi$
$660$	0	0
$661$	−8.03024	−0.312340	−0.156170	−	0.987730i	$-0.549915\pi$
$661$	−8.03024	−0.312340	−0.156170	+	0.987730i	$0.549915\pi$
$662$	0	0
$663$	−19.2201	−0.746449
$664$	0	0
$665$	−2.76873	−0.107367
$666$	0	0
$667$	−39.9670	−1.54753
$668$	0	0
$669$	1.66356	0.0643169
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11.7587	0.453265	0.226632	−	0.973980i	$-0.427228\pi$
$673$	11.7587	0.453265	0.226632	+	0.973980i	$0.427228\pi$
$674$	0	0
$675$	−5.50117	−0.211740
$676$	0	0
$677$	27.2948	1.04902	0.524512	−	0.851403i	$-0.324248\pi$
$677$	27.2948	1.04902	0.524512	+	0.851403i	$0.324248\pi$
$678$	0	0
$679$	1.74951	0.0671400
$680$	0	0
$681$	−0.770249	−0.0295160
$682$	0	0
$683$	−49.5738	−1.89689	−0.948444	−	0.316945i	$-0.897343\pi$
$683$	−49.5738	−1.89689	−0.948444	+	0.316945i	$0.897343\pi$
$684$	0	0
$685$	3.97209	0.151766
$686$	0	0
$687$	21.6864	0.827388
$688$	0	0
$689$	−29.2556	−1.11455
$690$	0	0
$691$	25.3253	0.963421	0.481710	−	0.876330i	$-0.340016\pi$
$691$	25.3253	0.963421	0.481710	+	0.876330i	$0.340016\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.22149	0.349791
$696$	0	0
$697$	−21.1475	−0.801020
$698$	0	0
$699$	−33.8824	−1.28155
$700$	0	0
$701$	18.2555	0.689500	0.344750	−	0.938695i	$-0.387964\pi$
$701$	18.2555	0.689500	0.344750	+	0.938695i	$0.387964\pi$
$702$	0	0
$703$	29.9578	1.12988
$704$	0	0
$705$	−15.3788	−0.579199
$706$	0	0
$707$	−3.70219	−0.139235
$708$	0	0
$709$	4.33178	0.162683	0.0813417	−	0.996686i	$-0.474079\pi$
$709$	4.33178	0.162683	0.0813417	+	0.996686i	$0.474079\pi$
$710$	0	0
$711$	−1.57541	−0.0590827
$712$	0	0
$713$	35.3532	1.32399
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.28673	−0.160091
$718$	0	0
$719$	24.1005	0.898797	0.449399	−	0.893331i	$-0.351638\pi$
$719$	24.1005	0.898797	0.449399	+	0.893331i	$0.351638\pi$
$720$	0	0
$721$	4.37757	0.163029
$722$	0	0
$723$	−7.31826	−0.272169
$724$	0	0
$725$	9.01248	0.334715
$726$	0	0
$727$	25.1066	0.931151	0.465576	−	0.885008i	$-0.345847\pi$
$727$	25.1066	0.931151	0.465576	+	0.885008i	$0.345847\pi$
$728$	0	0
$729$	29.6961	1.09986
$730$	0	0
$731$	−21.3486	−0.789605
$732$	0	0
$733$	17.4463	0.644393	0.322196	−	0.946673i	$-0.395579\pi$
$733$	17.4463	0.644393	0.322196	+	0.946673i	$0.395579\pi$
$734$	0	0
$735$	−10.8413	−0.399889
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−5.20019	−0.191292	−0.0956460	−	0.995415i	$-0.530492\pi$
$739$	−5.20019	−0.191292	−0.0956460	+	0.995415i	$0.530492\pi$
$740$	0	0
$741$	−44.2225	−1.62455
$742$	0	0
$743$	−37.1928	−1.36447	−0.682235	−	0.731133i	$-0.738992\pi$
$743$	−37.1928	−1.36447	−0.682235	+	0.731133i	$0.738992\pi$
$744$	0	0
$745$	−7.65012	−0.280279
$746$	0	0
$747$	−0.745203	−0.0272656
$748$	0	0
$749$	8.07494	0.295052
$750$	0	0
$751$	−41.4212	−1.51148	−0.755740	−	0.654872i	$-0.772723\pi$
$751$	−41.4212	−1.51148	−0.755740	+	0.654872i	$0.772723\pi$
$752$	0	0
$753$	18.3183	0.667557
$754$	0	0
$755$	−2.87198	−0.104522
$756$	0	0
$757$	3.91628	0.142340	0.0711698	−	0.997464i	$-0.477327\pi$
$757$	3.91628	0.142340	0.0711698	+	0.997464i	$0.477327\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39.1309	1.41849	0.709247	−	0.704960i	$-0.249035\pi$
$761$	39.1309	1.41849	0.709247	+	0.704960i	$0.249035\pi$
$762$	0	0
$763$	−8.59935	−0.311317
$764$	0	0
$765$	−1.08756	−0.0393208
$766$	0	0
$767$	13.2777	0.479430
$768$	0	0
$769$	−11.6755	−0.421028	−0.210514	−	0.977591i	$-0.567514\pi$
$769$	−11.6755	−0.421028	−0.210514	+	0.977591i	$0.567514\pi$
$770$	0	0
$771$	−32.1052	−1.15624
$772$	0	0
$773$	−7.76640	−0.279338	−0.139669	−	0.990198i	$-0.544604\pi$
$773$	−7.76640	−0.279338	−0.139669	+	0.990198i	$0.544604\pi$
$774$	0	0
$775$	−7.97209	−0.286366
$776$	0	0
$777$	−4.00788	−0.143782
$778$	0	0
$779$	−48.6571	−1.74332
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−49.5791	−1.77181
$784$	0	0
$785$	17.4817	0.623947
$786$	0	0
$787$	−1.42928	−0.0509484	−0.0254742	−	0.999675i	$-0.508110\pi$
$787$	−1.42928	−0.0509484	−0.0254742	+	0.999675i	$0.508110\pi$
$788$	0	0
$789$	10.5046	0.373975
$790$	0	0
$791$	0.222432	0.00790876
$792$	0	0
$793$	−19.3039	−0.685501
$794$	0	0
$795$	9.77107	0.346544
$796$	0	0
$797$	−36.3788	−1.28860	−0.644302	−	0.764771i	$-0.722852\pi$
$797$	−36.3788	−1.28860	−0.644302	+	0.764771i	$0.722852\pi$
$798$	0	0
$799$	−24.0263	−0.849989
$800$	0	0
$801$	6.84134	0.241727
$802$	0	0
$803$	0	0
$804$	0	0
$805$	2.13261	0.0751648
$806$	0	0
$807$	3.03863	0.106965
$808$	0	0
$809$	−27.8735	−0.979980	−0.489990	−	0.871728i	$-0.663000\pi$
$809$	−27.8735	−0.979980	−0.489990	+	0.871728i	$0.663000\pi$
$810$	0	0
$811$	39.1793	1.37577	0.687885	−	0.725820i	$-0.258539\pi$
$811$	39.1793	1.37577	0.687885	+	0.725820i	$0.258539\pi$
$812$	0	0
$813$	−18.1856	−0.637798
$814$	0	0
$815$	−13.1671	−0.461222
$816$	0	0
$817$	−49.1196	−1.71848
$818$	0	0
$819$	−1.00233	−0.0350243
$820$	0	0
$821$	13.7772	0.480827	0.240414	−	0.970671i	$-0.422717\pi$
$821$	13.7772	0.480827	0.240414	+	0.970671i	$0.422717\pi$
$822$	0	0
$823$	19.8800	0.692972	0.346486	−	0.938055i	$-0.387375\pi$
$823$	19.8800	0.692972	0.346486	+	0.938055i	$0.387375\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.3272	0.463431	0.231716	−	0.972784i	$-0.425566\pi$
$827$	13.3272	0.463431	0.231716	+	0.972784i	$0.425566\pi$
$828$	0	0
$829$	51.1899	1.77790	0.888950	−	0.458005i	$-0.151436\pi$
$829$	51.1899	1.77790	0.888950	+	0.458005i	$0.151436\pi$
$830$	0	0
$831$	−36.5727	−1.26869
$832$	0	0
$833$	−16.9374	−0.586846
$834$	0	0
$835$	−15.8778	−0.549474
$836$	0	0
$837$	43.8558	1.51588
$838$	0	0
$839$	16.8739	0.582552	0.291276	−	0.956639i	$-0.405920\pi$
$839$	16.8739	0.582552	0.291276	+	0.956639i	$0.405920\pi$
$840$	0	0
$841$	52.2248	1.80086
$842$	0	0
$843$	36.2164	1.24736
$844$	0	0
$845$	9.99767	0.343930
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−44.1530	−1.51533
$850$	0	0
$851$	−23.0749	−0.790999
$852$	0	0
$853$	−5.42262	−0.185667	−0.0928335	−	0.995682i	$-0.529592\pi$
$853$	−5.42262	−0.185667	−0.0928335	+	0.995682i	$0.529592\pi$
$854$	0	0
$855$	−2.50230	−0.0855768
$856$	0	0
$857$	−27.8386	−0.950947	−0.475474	−	0.879730i	$-0.657723\pi$
$857$	−27.8386	−0.950947	−0.475474	+	0.879730i	$0.657723\pi$
$858$	0	0
$859$	−48.1899	−1.64422	−0.822109	−	0.569330i	$-0.807203\pi$
$859$	−48.1899	−1.64422	−0.822109	+	0.569330i	$0.807203\pi$
$860$	0	0
$861$	6.50956	0.221845
$862$	0	0
$863$	−26.8609	−0.914354	−0.457177	−	0.889376i	$-0.651139\pi$
$863$	−26.8609	−0.914354	−0.457177	+	0.889376i	$0.651139\pi$
$864$	0	0
$865$	−22.8206	−0.775922
$866$	0	0
$867$	−17.1996	−0.584130
$868$	0	0
$869$	0	0
$870$	0	0
$871$	36.4545	1.23521
$872$	0	0
$873$	1.58115	0.0535140
$874$	0	0
$875$	−0.480901	−0.0162574
$876$	0	0
$877$	−39.8837	−1.34678	−0.673389	−	0.739289i	$-0.735162\pi$
$877$	−39.8837	−1.34678	−0.673389	+	0.739289i	$0.735162\pi$
$878$	0	0
$879$	35.0107	1.18088
$880$	0	0
$881$	−2.82222	−0.0950829	−0.0475415	−	0.998869i	$-0.515139\pi$
$881$	−2.82222	−0.0950829	−0.0475415	+	0.998869i	$0.515139\pi$
$882$	0	0
$883$	−50.5287	−1.70043	−0.850213	−	0.526439i	$-0.823527\pi$
$883$	−50.5287	−1.70043	−0.850213	+	0.526439i	$0.823527\pi$
$884$	0	0
$885$	−4.43462	−0.149068
$886$	0	0
$887$	−28.3195	−0.950875	−0.475437	−	0.879750i	$-0.657710\pi$
$887$	−28.3195	−0.950875	−0.475437	+	0.879750i	$0.657710\pi$
$888$	0	0
$889$	−7.61241	−0.255312
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−55.2806	−1.84990
$894$	0	0
$895$	0.462531	0.0154607
$896$	0	0
$897$	34.0623	1.13731
$898$	0	0
$899$	−71.8483	−2.39628
$900$	0	0
$901$	15.2653	0.508561
$902$	0	0
$903$	6.57144	0.218684
$904$	0	0
$905$	−3.12842	−0.103992
$906$	0	0
$907$	−51.0508	−1.69511	−0.847556	−	0.530705i	$-0.821927\pi$
$907$	−51.0508	−1.69511	−0.847556	+	0.530705i	$0.821927\pi$
$908$	0	0
$909$	−3.34593	−0.110978
$910$	0	0
$911$	25.1429	0.833021	0.416510	−	0.909131i	$-0.363253\pi$
$911$	25.1429	0.833021	0.416510	+	0.909131i	$0.363253\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6.44730	0.213141
$916$	0	0
$917$	−4.61886	−0.152528
$918$	0	0
$919$	−25.6151	−0.844965	−0.422482	−	0.906371i	$-0.638841\pi$
$919$	−25.6151	−0.844965	−0.422482	+	0.906371i	$0.638841\pi$
$920$	0	0
$921$	−15.6968	−0.517226
$922$	0	0
$923$	−13.2777	−0.437041
$924$	0	0
$925$	5.20336	0.171085
$926$	0	0
$927$	3.95631	0.129942
$928$	0	0
$929$	−8.60774	−0.282411	−0.141205	−	0.989980i	$-0.545098\pi$
$929$	−8.60774	−0.282411	−0.141205	+	0.989980i	$0.545098\pi$
$930$	0	0
$931$	−38.9702	−1.27720
$932$	0	0
$933$	−15.6501	−0.512362
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.96640	−0.194914	−0.0974569	−	0.995240i	$-0.531071\pi$
$937$	−5.96640	−0.194914	−0.0974569	+	0.995240i	$0.531071\pi$
$938$	0	0
$939$	−1.43695	−0.0468933
$940$	0	0
$941$	−22.8380	−0.744498	−0.372249	−	0.928133i	$-0.621413\pi$
$941$	−22.8380	−0.744498	−0.372249	+	0.928133i	$0.621413\pi$
$942$	0	0
$943$	37.4781	1.22045
$944$	0	0
$945$	2.64552	0.0860586
$946$	0	0
$947$	2.30620	0.0749415	0.0374708	−	0.999298i	$-0.488070\pi$
$947$	2.30620	0.0749415	0.0374708	+	0.999298i	$0.488070\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	−4.27363	−0.138582
$952$	0	0
$953$	28.6129	0.926861	0.463431	−	0.886133i	$-0.346618\pi$
$953$	28.6129	0.926861	0.463431	+	0.886133i	$0.346618\pi$
$954$	0	0
$955$	5.43695	0.175936
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.91018	−0.0616830
$960$	0	0
$961$	32.5543	1.05014
$962$	0	0
$963$	7.29789	0.235171
$964$	0	0
$965$	3.20675	0.103229
$966$	0	0
$967$	31.7422	1.02076	0.510380	−	0.859949i	$-0.329505\pi$
$967$	31.7422	1.02076	0.510380	+	0.859949i	$0.329505\pi$
$968$	0	0
$969$	23.0749	0.741274
$970$	0	0
$971$	37.0023	1.18746	0.593731	−	0.804664i	$-0.297654\pi$
$971$	37.0023	1.18746	0.593731	+	0.804664i	$0.297654\pi$
$972$	0	0
$973$	−4.43462	−0.142168
$974$	0	0
$975$	−7.68099	−0.245989
$976$	0	0
$977$	33.8716	1.08365	0.541824	−	0.840492i	$-0.317734\pi$
$977$	33.8716	1.08365	0.541824	+	0.840492i	$0.317734\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−7.77184	−0.248136
$982$	0	0
$983$	37.7516	1.20409	0.602044	−	0.798463i	$-0.294353\pi$
$983$	37.7516	1.20409	0.602044	+	0.798463i	$0.294353\pi$
$984$	0	0
$985$	−19.4048	−0.618288
$986$	0	0
$987$	7.39568	0.235407
$988$	0	0
$989$	37.8343	1.20306
$990$	0	0
$991$	44.9354	1.42742	0.713710	−	0.700441i	$-0.247013\pi$
$991$	44.9354	1.42742	0.713710	+	0.700441i	$0.247013\pi$
$992$	0	0
$993$	−30.3909	−0.964427
$994$	0	0
$995$	−20.3509	−0.645167
$996$	0	0
$997$	−0.396526	−0.0125581	−0.00627906	−	0.999980i	$-0.501999\pi$
$997$	−0.396526	−0.0125581	−0.00627906	+	0.999980i	$0.501999\pi$
$998$	0	0
$999$	−28.6245	−0.905640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cw.1.5		6
4.3	odd	2		605.2.a.m.1.3	✓	6
11.10	odd	2	inner	9680.2.a.cw.1.6		6
12.11	even	2		5445.2.a.bx.1.4		6
20.19	odd	2		3025.2.a.bg.1.4		6
44.3	odd	10		605.2.g.q.251.3		24
44.7	even	10		605.2.g.q.511.4		24
44.15	odd	10		605.2.g.q.511.3		24
44.19	even	10		605.2.g.q.251.4		24
44.27	odd	10		605.2.g.q.366.4		24
44.31	odd	10		605.2.g.q.81.4		24
44.35	even	10		605.2.g.q.81.3		24
44.39	even	10		605.2.g.q.366.3		24
44.43	even	2		605.2.a.m.1.4	yes	6
132.131	odd	2		5445.2.a.bx.1.3		6
220.219	even	2		3025.2.a.bg.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.m.1.3	✓	6	4.3	odd	2
605.2.a.m.1.4	yes	6	44.43	even	2
605.2.g.q.81.3		24	44.35	even	10
605.2.g.q.81.4		24	44.31	odd	10
605.2.g.q.251.3		24	44.3	odd	10
605.2.g.q.251.4		24	44.19	even	10
605.2.g.q.366.3		24	44.39	even	10
605.2.g.q.366.4		24	44.27	odd	10
605.2.g.q.511.3		24	44.15	odd	10
605.2.g.q.511.4		24	44.7	even	10
3025.2.a.bg.1.3		6	220.219	even	2
3025.2.a.bg.1.4		6	20.19	odd	2
5445.2.a.bx.1.3		6	132.131	odd	2
5445.2.a.bx.1.4		6	12.11	even	2
9680.2.a.cw.1.5		6	1.1	even	1	trivial
9680.2.a.cw.1.6		6	11.10	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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.60168 q^{3} +1.00000 q^{5} -0.480901 q^{7} -0.434624 q^{9} +O(q^{10})\)	\(q+1.60168 q^{3} +1.00000 q^{5} -0.480901 q^{7} -0.434624 q^{9} -4.79559 q^{13} +1.60168 q^{15} +2.50230 q^{17} +5.75739 q^{19} -0.770249 q^{21} -4.43462 q^{23} +1.00000 q^{25} -5.50117 q^{27} +9.01248 q^{29} -7.97209 q^{31} -0.480901 q^{35} +5.20336 q^{37} -7.68099 q^{39} -8.45124 q^{41} -8.53158 q^{43} -0.434624 q^{45} -9.60168 q^{47} -6.76873 q^{49} +4.00788 q^{51} +6.10051 q^{53} +9.22149 q^{57} -2.76873 q^{59} +4.02534 q^{61} +0.209011 q^{63} -4.79559 q^{65} -7.60168 q^{67} -7.10284 q^{69} +2.76873 q^{71} -5.00460 q^{73} +1.60168 q^{75} +3.62478 q^{79} -7.50723 q^{81} +1.71459 q^{83} +2.50230 q^{85} +14.4351 q^{87} -15.7408 q^{89} +2.30620 q^{91} -12.7687 q^{93} +5.75739 q^{95} -3.63798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + 12 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + 12 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + 12 q^{9} - 6 q^{15} - 12 q^{23} + 6 q^{25} - 30 q^{27} + 12 q^{45} - 42 q^{47} - 24 q^{49} + 24 q^{53} - 30 q^{67} - 24 q^{69} - 6 q^{75} + 30 q^{81} - 30 q^{89} - 36 q^{91} - 60 q^{93} + 24 q^{97}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + 12 * q^9 - 6 * q^15 - 12 * q^23 + 6 * q^25 - 30 * q^27 + 12 * q^45 - 42 * q^47 - 24 * q^49 + 24 * q^53 - 30 * q^67 - 24 * q^69 - 6 * q^75 + 30 * q^81 - 30 * q^89 - 36 * q^91 - 60 * q^93 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more