LMFDB - Embedded newform 1088.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,2,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, 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("1088.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1088.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:	$8.68772373992$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +2.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +2.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} -1.41421 q^{11} +4.00000 q^{13} -2.82843 q^{15} -1.00000 q^{17} +2.82843 q^{19} +6.00000 q^{21} +4.24264 q^{23} -1.00000 q^{25} +5.65685 q^{27} +6.00000 q^{29} +7.07107 q^{31} +2.00000 q^{33} -8.48528 q^{35} +2.00000 q^{37} -5.65685 q^{39} -6.00000 q^{41} -8.48528 q^{43} -2.00000 q^{45} +11.3137 q^{47} +11.0000 q^{49} +1.41421 q^{51} +6.00000 q^{53} -2.82843 q^{55} -4.00000 q^{57} +8.48528 q^{59} -6.00000 q^{61} +4.24264 q^{63} +8.00000 q^{65} +5.65685 q^{67} -6.00000 q^{69} -7.07107 q^{71} +10.0000 q^{73} +1.41421 q^{75} +6.00000 q^{77} -12.7279 q^{79} -5.00000 q^{81} +14.1421 q^{83} -2.00000 q^{85} -8.48528 q^{87} -8.00000 q^{89} -16.9706 q^{91} -10.0000 q^{93} +5.65685 q^{95} +14.0000 q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^5 - 2 * q^9	$ 2 q + 4 q^{5} - 2 q^{9} + 8 q^{13} - 2 q^{17} + 12 q^{21} - 2 q^{25} + 12 q^{29} + 4 q^{33} + 4 q^{37} - 12 q^{41} - 4 q^{45} + 22 q^{49} + 12 q^{53} - 8 q^{57} - 12 q^{61} + 16 q^{65} - 12 q^{69} + 20 q^{73} + 12 q^{77} - 10 q^{81} - 4 q^{85} - 16 q^{89} - 20 q^{93} + 28 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^9 + 8 * q^13 - 2 * q^17 + 12 * q^21 - 2 * q^25 + 12 * q^29 + 4 * q^33 + 4 * q^37 - 12 * q^41 - 4 * q^45 + 22 * q^49 + 12 * q^53 - 8 * q^57 - 12 * q^61 + 16 * q^65 - 12 * q^69 + 20 * q^73 + 12 * q^77 - 10 * q^81 - 4 * q^85 - 16 * q^89 - 20 * q^93 + 28 * 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.82843	−0.730297
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	4.24264	0.884652	0.442326	−	0.896854i	$-0.354153\pi$
$23$	4.24264	0.884652	0.442326	+	0.896854i	$0.354153\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	7.07107	1.27000	0.635001	−	0.772512i	$-0.281000\pi$
$31$	7.07107	1.27000	0.635001	+	0.772512i	$0.281000\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−8.48528	−1.43427
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−5.65685	−0.905822
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	1.41421	0.198030
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	4.24264	0.534522
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−7.07107	−0.839181	−0.419591	−	0.907713i	$-0.637826\pi$
$71$	−7.07107	−0.839181	−0.419591	+	0.907713i	$0.637826\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−12.7279	−1.43200	−0.716002	−	0.698099i	$-0.754030\pi$
$79$	−12.7279	−1.43200	−0.716002	+	0.698099i	$0.754030\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−8.48528	−0.909718
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	−16.9706	−1.77900
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	1.41421	0.142134
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	16.9706	1.67216	0.836080	−	0.548608i	$-0.184842\pi$
$103$	16.9706	1.67216	0.836080	+	0.548608i	$0.184842\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	−7.07107	−0.683586	−0.341793	−	0.939775i	$-0.611034\pi$
$107$	−7.07107	−0.683586	−0.341793	+	0.939775i	$0.611034\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.82843	−0.268462
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	8.48528	0.791257
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	4.24264	0.388922
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	8.48528	0.765092
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−8.48528	−0.752947	−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528	−0.752947	−0.376473	+	0.926427i	$0.622863\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	−1.41421	−0.123560	−0.0617802	−	0.998090i	$-0.519678\pi$
$131$	−1.41421	−0.123560	−0.0617802	+	0.998090i	$0.519678\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	11.3137	0.973729
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−9.89949	−0.839664	−0.419832	−	0.907602i	$-0.637911\pi$
$139$	−9.89949	−0.839664	−0.419832	+	0.907602i	$0.637911\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	−15.5563	−1.28307
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	14.1421	1.13592
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−8.48528	−0.672927
$160$	0	0
$161$	−18.0000	−1.41860
$162$	0	0
$163$	15.5563	1.21847	0.609234	−	0.792991i	$-0.291477\pi$
$163$	15.5563	1.21847	0.609234	+	0.792991i	$0.291477\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−12.7279	−0.984916	−0.492458	−	0.870336i	$-0.663902\pi$
$167$	−12.7279	−0.984916	−0.492458	+	0.870336i	$0.663902\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	−26.0000	−1.97674	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−26.0000	−1.97674	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	4.24264	0.320713
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	2.82843	0.211407	0.105703	−	0.994398i	$-0.466291\pi$
$179$	2.82843	0.211407	0.105703	+	0.994398i	$0.466291\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	1.41421	0.103418
$188$	0	0
$189$	−24.0000	−1.74574
$190$	0	0
$191$	5.65685	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	−11.3137	−0.810191
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−9.89949	−0.701757	−0.350878	−	0.936421i	$-0.614117\pi$
$199$	−9.89949	−0.701757	−0.350878	+	0.936421i	$0.614117\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−25.4558	−1.78665
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	−4.24264	−0.294884
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−21.2132	−1.46038	−0.730189	−	0.683246i	$-0.760568\pi$
$211$	−21.2132	−1.46038	−0.730189	+	0.683246i	$0.760568\pi$
$212$	0	0
$213$	10.0000	0.685189
$214$	0	0
$215$	−16.9706	−1.15738
$216$	0	0
$217$	−30.0000	−2.03653
$218$	0	0
$219$	−14.1421	−0.955637
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	8.48528	0.568216	0.284108	−	0.958792i	$-0.408302\pi$
$223$	8.48528	0.568216	0.284108	+	0.958792i	$0.408302\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	9.89949	0.657053	0.328526	−	0.944495i	$-0.393448\pi$
$227$	9.89949	0.657053	0.328526	+	0.944495i	$0.393448\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	−8.48528	−0.558291
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	22.6274	1.47605
$236$	0	0
$237$	18.0000	1.16923
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	22.0000	1.40553
$246$	0	0
$247$	11.3137	0.719874
$248$	0	0
$249$	−20.0000	−1.26745
$250$	0	0
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	2.82843	0.177123
$256$	0	0
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	0	0
$259$	−8.48528	−0.527250
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−2.82843	−0.174408	−0.0872041	−	0.996190i	$-0.527793\pi$
$263$	−2.82843	−0.174408	−0.0872041	+	0.996190i	$0.527793\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	11.3137	0.692388
$268$	0	0
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−22.6274	−1.37452	−0.687259	−	0.726413i	$-0.741186\pi$
$271$	−22.6274	−1.37452	−0.687259	+	0.726413i	$0.741186\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	1.41421	0.0852803
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	−7.07107	−0.423334
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	25.4558	1.50261
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−19.7990	−1.16064
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	16.9706	0.988064
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	16.9706	0.981433
$300$	0	0
$301$	36.0000	2.07501
$302$	0	0
$303$	−16.9706	−0.974933
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	33.9411	1.93712	0.968561	−	0.248776i	$-0.0800281\pi$
$307$	33.9411	1.93712	0.968561	+	0.248776i	$0.0800281\pi$
$308$	0	0
$309$	−24.0000	−1.36531
$310$	0	0
$311$	15.5563	0.882120	0.441060	−	0.897478i	$-0.354603\pi$
$311$	15.5563	0.882120	0.441060	+	0.897478i	$0.354603\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	8.48528	0.478091
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−8.48528	−0.475085
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	−2.82843	−0.157378
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−14.1421	−0.782062
$328$	0	0
$329$	−48.0000	−2.64633
$330$	0	0
$331$	14.1421	0.777322	0.388661	−	0.921381i	$-0.372938\pi$
$331$	14.1421	0.777322	0.388661	+	0.921381i	$0.372938\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	−8.48528	−0.460857
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	9.89949	0.531433	0.265716	−	0.964051i	$-0.414392\pi$
$347$	9.89949	0.531433	0.265716	+	0.964051i	$0.414392\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	22.6274	1.20776
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−14.1421	−0.750587
$356$	0	0
$357$	−6.00000	−0.317554
$358$	0	0
$359$	31.1127	1.64207	0.821033	−	0.570881i	$-0.193398\pi$
$359$	31.1127	1.64207	0.821033	+	0.570881i	$0.193398\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	12.7279	0.668043
$364$	0	0
$365$	20.0000	1.04685
$366$	0	0
$367$	29.6985	1.55025	0.775124	−	0.631809i	$-0.217687\pi$
$367$	29.6985	1.55025	0.775124	+	0.631809i	$0.217687\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−25.4558	−1.32160
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	16.9706	0.876356
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	−24.0416	−1.23494	−0.617468	−	0.786596i	$-0.711841\pi$
$379$	−24.0416	−1.23494	−0.617468	+	0.786596i	$0.711841\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	−14.1421	−0.722629	−0.361315	−	0.932444i	$-0.617672\pi$
$383$	−14.1421	−0.722629	−0.361315	+	0.932444i	$0.617672\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	8.48528	0.431331
$388$	0	0
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	−4.24264	−0.214560
$392$	0	0
$393$	2.00000	0.100887
$394$	0	0
$395$	−25.4558	−1.28082
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	16.9706	0.849591
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	28.2843	1.40894
$404$	0	0
$405$	−10.0000	−0.496904
$406$	0	0
$407$	−2.82843	−0.140200
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−36.0000	−1.77144
$414$	0	0
$415$	28.2843	1.38842
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	−9.89949	−0.483622	−0.241811	−	0.970323i	$-0.577741\pi$
$419$	−9.89949	−0.483622	−0.241811	+	0.970323i	$0.577741\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	0	0
$423$	−11.3137	−0.550091
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	25.4558	1.23189
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	−35.3553	−1.70301	−0.851503	−	0.524349i	$-0.824309\pi$
$431$	−35.3553	−1.70301	−0.851503	+	0.524349i	$0.824309\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	−16.9706	−0.813676
$436$	0	0
$437$	12.0000	0.574038
$438$	0	0
$439$	15.5563	0.742464	0.371232	−	0.928540i	$-0.378935\pi$
$439$	15.5563	0.742464	0.371232	+	0.928540i	$0.378935\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	0	0
$443$	−28.2843	−1.34383	−0.671913	−	0.740630i	$-0.734527\pi$
$443$	−28.2843	−1.34383	−0.671913	+	0.740630i	$0.734527\pi$
$444$	0	0
$445$	−16.0000	−0.758473
$446$	0	0
$447$	−25.4558	−1.20402
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	8.48528	0.399556
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	−33.9411	−1.59118
$456$	0	0
$457$	36.0000	1.68401	0.842004	−	0.539471i	$-0.181376\pi$
$457$	36.0000	1.68401	0.842004	+	0.539471i	$0.181376\pi$
$458$	0	0
$459$	−5.65685	−0.264039
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−11.3137	−0.525793	−0.262896	−	0.964824i	$-0.584678\pi$
$463$	−11.3137	−0.525793	−0.262896	+	0.964824i	$0.584678\pi$
$464$	0	0
$465$	−20.0000	−0.927478
$466$	0	0
$467$	−31.1127	−1.43972	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	−31.1127	−1.43972	−0.719862	+	0.694117i	$0.755795\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	2.82843	0.130327
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	1.41421	0.0646171	0.0323085	−	0.999478i	$-0.489714\pi$
$479$	1.41421	0.0646171	0.0323085	+	0.999478i	$0.489714\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	25.4558	1.15828
$484$	0	0
$485$	28.0000	1.27141
$486$	0	0
$487$	−4.24264	−0.192252	−0.0961262	−	0.995369i	$-0.530645\pi$
$487$	−4.24264	−0.192252	−0.0961262	+	0.995369i	$0.530645\pi$
$488$	0	0
$489$	−22.0000	−0.994874
$490$	0	0
$491$	36.7696	1.65939	0.829693	−	0.558219i	$-0.188515\pi$
$491$	36.7696	1.65939	0.829693	+	0.558219i	$0.188515\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	30.0000	1.34568
$498$	0	0
$499$	−4.24264	−0.189927	−0.0949633	−	0.995481i	$-0.530273\pi$
$499$	−4.24264	−0.189927	−0.0949633	+	0.995481i	$0.530273\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	0	0
$503$	−32.5269	−1.45030	−0.725152	−	0.688589i	$-0.758230\pi$
$503$	−32.5269	−1.45030	−0.725152	+	0.688589i	$0.758230\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	−4.24264	−0.188422
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−42.4264	−1.87683
$512$	0	0
$513$	16.0000	0.706417
$514$	0	0
$515$	33.9411	1.49562
$516$	0	0
$517$	−16.0000	−0.703679
$518$	0	0
$519$	36.7696	1.61400
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−39.5980	−1.73150	−0.865749	−	0.500478i	$-0.833158\pi$
$523$	−39.5980	−1.73150	−0.865749	+	0.500478i	$0.833158\pi$
$524$	0	0
$525$	−6.00000	−0.261861
$526$	0	0
$527$	−7.07107	−0.308021
$528$	0	0
$529$	−5.00000	−0.217391
$530$	0	0
$531$	−8.48528	−0.368230
$532$	0	0
$533$	−24.0000	−1.03956
$534$	0	0
$535$	−14.1421	−0.611418
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	−15.5563	−0.670059
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	8.48528	0.364138
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	−18.3848	−0.786076	−0.393038	−	0.919522i	$-0.628576\pi$
$547$	−18.3848	−0.786076	−0.393038	+	0.919522i	$0.628576\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	16.9706	0.722970
$552$	0	0
$553$	54.0000	2.29631
$554$	0	0
$555$	−5.65685	−0.240120
$556$	0	0
$557$	28.0000	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$558$	0	0
$559$	−33.9411	−1.43556
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	19.7990	0.834428	0.417214	−	0.908808i	$-0.363007\pi$
$563$	19.7990	0.834428	0.417214	+	0.908808i	$0.363007\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	21.2132	0.890871
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	−43.8406	−1.83467	−0.917336	−	0.398113i	$-0.869665\pi$
$571$	−43.8406	−1.83467	−0.917336	+	0.398113i	$0.869665\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	−4.24264	−0.176930
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	0	0
$579$	31.1127	1.29300
$580$	0	0
$581$	−60.0000	−2.48922
$582$	0	0
$583$	−8.48528	−0.351424
$584$	0	0
$585$	−8.00000	−0.330759
$586$	0	0
$587$	25.4558	1.05068	0.525338	−	0.850894i	$-0.323939\pi$
$587$	25.4558	1.05068	0.525338	+	0.850894i	$0.323939\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	2.82843	0.116346
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	8.48528	0.347863
$596$	0	0
$597$	14.0000	0.572982
$598$	0	0
$599$	11.3137	0.462266	0.231133	−	0.972922i	$-0.425757\pi$
$599$	11.3137	0.462266	0.231133	+	0.972922i	$0.425757\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−5.65685	−0.230365
$604$	0	0
$605$	−18.0000	−0.731804
$606$	0	0
$607$	35.3553	1.43503	0.717514	−	0.696544i	$-0.245280\pi$
$607$	35.3553	1.43503	0.717514	+	0.696544i	$0.245280\pi$
$608$	0	0
$609$	36.0000	1.45879
$610$	0	0
$611$	45.2548	1.83081
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	16.9706	0.684319
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−12.7279	−0.511578	−0.255789	−	0.966733i	$-0.582335\pi$
$619$	−12.7279	−0.511578	−0.255789	+	0.966733i	$0.582335\pi$
$620$	0	0
$621$	24.0000	0.963087
$622$	0	0
$623$	33.9411	1.35982
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	5.65685	0.225913
$628$	0	0
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	2.82843	0.112598	0.0562990	−	0.998414i	$-0.482070\pi$
$631$	2.82843	0.112598	0.0562990	+	0.998414i	$0.482070\pi$
$632$	0	0
$633$	30.0000	1.19239
$634$	0	0
$635$	−16.9706	−0.673456
$636$	0	0
$637$	44.0000	1.74334
$638$	0	0
$639$	7.07107	0.279727
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−35.3553	−1.39428	−0.697139	−	0.716936i	$-0.745544\pi$
$643$	−35.3553	−1.39428	−0.697139	+	0.716936i	$0.745544\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	11.3137	0.444788	0.222394	−	0.974957i	$-0.428613\pi$
$647$	11.3137	0.444788	0.222394	+	0.974957i	$0.428613\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	42.4264	1.66282
$652$	0	0
$653$	38.0000	1.48705	0.743527	−	0.668705i	$-0.233151\pi$
$653$	38.0000	1.48705	0.743527	+	0.668705i	$0.233151\pi$
$654$	0	0
$655$	−2.82843	−0.110516
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−11.3137	−0.440720	−0.220360	−	0.975419i	$-0.570723\pi$
$659$	−11.3137	−0.440720	−0.220360	+	0.975419i	$0.570723\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	5.65685	0.219694
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	25.4558	0.985654
$668$	0	0
$669$	−12.0000	−0.463947
$670$	0	0
$671$	8.48528	0.327571
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−59.3970	−2.27945
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	4.24264	0.162340	0.0811701	−	0.996700i	$-0.474134\pi$
$683$	4.24264	0.162340	0.0811701	+	0.996700i	$0.474134\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.6274	−0.863290
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	15.5563	0.591791	0.295896	−	0.955220i	$-0.404382\pi$
$691$	15.5563	0.591791	0.295896	+	0.955220i	$0.404382\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−19.7990	−0.751018
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	14.1421	0.534905
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	5.65685	0.213352
$704$	0	0
$705$	−32.0000	−1.20519
$706$	0	0
$707$	−50.9117	−1.91473
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	12.7279	0.477334
$712$	0	0
$713$	30.0000	1.12351
$714$	0	0
$715$	−11.3137	−0.423109
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	−18.3848	−0.685636	−0.342818	−	0.939402i	$-0.611381\pi$
$719$	−18.3848	−0.685636	−0.342818	+	0.939402i	$0.611381\pi$
$720$	0	0
$721$	−72.0000	−2.68142
$722$	0	0
$723$	−14.1421	−0.525952
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	11.3137	0.419602	0.209801	−	0.977744i	$-0.432718\pi$
$727$	11.3137	0.419602	0.209801	+	0.977744i	$0.432718\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	8.48528	0.313839
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	−31.1127	−1.14761
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−8.48528	−0.312136	−0.156068	−	0.987746i	$-0.549882\pi$
$739$	−8.48528	−0.312136	−0.156068	+	0.987746i	$0.549882\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	0	0
$743$	7.07107	0.259412	0.129706	−	0.991552i	$-0.458597\pi$
$743$	7.07107	0.259412	0.129706	+	0.991552i	$0.458597\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	0	0
$747$	−14.1421	−0.517434
$748$	0	0
$749$	30.0000	1.09618
$750$	0	0
$751$	32.5269	1.18692	0.593462	−	0.804862i	$-0.297761\pi$
$751$	32.5269	1.18692	0.593462	+	0.804862i	$0.297761\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	28.2843	1.02937
$756$	0	0
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	0	0
$759$	8.48528	0.307996
$760$	0	0
$761$	−40.0000	−1.45000	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000	−1.45000	−0.724999	+	0.688749i	$0.758160\pi$
$762$	0	0
$763$	−42.4264	−1.53594
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	33.9411	1.22554
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	39.5980	1.42609
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	−7.07107	−0.254000
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	−16.9706	−0.608034
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	33.9411	1.21296
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	1.41421	0.0504113	0.0252056	−	0.999682i	$-0.491976\pi$
$787$	1.41421	0.0504113	0.0252056	+	0.999682i	$0.491976\pi$
$788$	0	0
$789$	4.00000	0.142404
$790$	0	0
$791$	−25.4558	−0.905106
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	−16.9706	−0.601884
$796$	0	0
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	0	0
$799$	−11.3137	−0.400250
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	−14.1421	−0.499065
$804$	0	0
$805$	−36.0000	−1.26883
$806$	0	0
$807$	−2.82843	−0.0995654
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−21.2132	−0.744896	−0.372448	−	0.928053i	$-0.621482\pi$
$811$	−21.2132	−0.744896	−0.372448	+	0.928053i	$0.621482\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	0	0
$815$	31.1127	1.08983
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	16.9706	0.592999
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−1.41421	−0.0492964	−0.0246482	−	0.999696i	$-0.507847\pi$
$823$	−1.41421	−0.0492964	−0.0246482	+	0.999696i	$0.507847\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−9.89949	−0.344239	−0.172120	−	0.985076i	$-0.555062\pi$
$827$	−9.89949	−0.344239	−0.172120	+	0.985076i	$0.555062\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−19.7990	−0.686819
$832$	0	0
$833$	−11.0000	−0.381127
$834$	0	0
$835$	−25.4558	−0.880936
$836$	0	0
$837$	40.0000	1.38260
$838$	0	0
$839$	−9.89949	−0.341769	−0.170884	−	0.985291i	$-0.554662\pi$
$839$	−9.89949	−0.341769	−0.170884	+	0.985291i	$0.554662\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	31.1127	1.07158
$844$	0	0
$845$	6.00000	0.206406
$846$	0	0
$847$	38.1838	1.31201
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	8.48528	0.290872
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	−5.65685	−0.193460
$856$	0	0
$857$	50.0000	1.70797	0.853984	−	0.520300i	$-0.174180\pi$
$857$	50.0000	1.70797	0.853984	+	0.520300i	$0.174180\pi$
$858$	0	0
$859$	8.48528	0.289514	0.144757	−	0.989467i	$-0.453760\pi$
$859$	8.48528	0.289514	0.144757	+	0.989467i	$0.453760\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	0	0
$863$	−22.6274	−0.770246	−0.385123	−	0.922865i	$-0.625841\pi$
$863$	−22.6274	−0.770246	−0.385123	+	0.922865i	$0.625841\pi$
$864$	0	0
$865$	−52.0000	−1.76805
$866$	0	0
$867$	−1.41421	−0.0480292
$868$	0	0
$869$	18.0000	0.610608
$870$	0	0
$871$	22.6274	0.766701
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	50.9117	1.72113
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	8.48528	0.286201
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	22.6274	0.761473	0.380737	−	0.924684i	$-0.375670\pi$
$883$	22.6274	0.761473	0.380737	+	0.924684i	$0.375670\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	−15.5563	−0.522331	−0.261166	−	0.965294i	$-0.584107\pi$
$887$	−15.5563	−0.522331	−0.261166	+	0.965294i	$0.584107\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	7.07107	0.236890
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	5.65685	0.189088
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	42.4264	1.41500
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−50.9117	−1.69423
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	43.8406	1.45570	0.727852	−	0.685734i	$-0.240519\pi$
$907$	43.8406	1.45570	0.727852	+	0.685734i	$0.240519\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	24.0416	0.796535	0.398267	−	0.917269i	$-0.369612\pi$
$911$	24.0416	0.796535	0.398267	+	0.917269i	$0.369612\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	16.9706	0.561029
$916$	0	0
$917$	6.00000	0.198137
$918$	0	0
$919$	−5.65685	−0.186602	−0.0933012	−	0.995638i	$-0.529742\pi$
$919$	−5.65685	−0.186602	−0.0933012	+	0.995638i	$0.529742\pi$
$920$	0	0
$921$	−48.0000	−1.58165
$922$	0	0
$923$	−28.2843	−0.930988
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−16.9706	−0.557386
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	31.1127	1.01968
$932$	0	0
$933$	−22.0000	−0.720248
$934$	0	0
$935$	2.82843	0.0924995
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	−25.4558	−0.830720
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	0	0
$943$	−25.4558	−0.828956
$944$	0	0
$945$	−48.0000	−1.56144
$946$	0	0
$947$	−9.89949	−0.321690	−0.160845	−	0.986980i	$-0.551422\pi$
$947$	−9.89949	−0.321690	−0.160845	+	0.986980i	$0.551422\pi$
$948$	0	0
$949$	40.0000	1.29845
$950$	0	0
$951$	25.4558	0.825462
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	7.07107	0.227862
$964$	0	0
$965$	−44.0000	−1.41641
$966$	0	0
$967$	−8.48528	−0.272868	−0.136434	−	0.990649i	$-0.543564\pi$
$967$	−8.48528	−0.272868	−0.136434	+	0.990649i	$0.543564\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−48.0833	−1.54307	−0.771533	−	0.636190i	$-0.780510\pi$
$971$	−48.0833	−1.54307	−0.771533	+	0.636190i	$0.780510\pi$
$972$	0	0
$973$	42.0000	1.34646
$974$	0	0
$975$	5.65685	0.181164
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	11.3137	0.361588
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−18.3848	−0.586383	−0.293192	−	0.956054i	$-0.594717\pi$
$983$	−18.3848	−0.586383	−0.293192	+	0.956054i	$0.594717\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	0	0
$987$	67.8823	2.16072
$988$	0	0
$989$	−36.0000	−1.14473
$990$	0	0
$991$	26.8701	0.853556	0.426778	−	0.904357i	$-0.359649\pi$
$991$	26.8701	0.853556	0.426778	+	0.904357i	$0.359649\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	−19.7990	−0.627670
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	0	0
$999$	11.3137	0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.2.a.q.1.1		2
3.2	odd	2		9792.2.a.cg.1.1		2
4.3	odd	2	inner	1088.2.a.q.1.2		2
8.3	odd	2		544.2.a.g.1.1	✓	2
8.5	even	2		544.2.a.g.1.2	yes	2
12.11	even	2		9792.2.a.cg.1.2		2
24.5	odd	2		4896.2.a.z.1.1		2
24.11	even	2		4896.2.a.z.1.2		2
136.67	odd	2		9248.2.a.p.1.2		2
136.101	even	2		9248.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.a.g.1.1	✓	2	8.3	odd	2
544.2.a.g.1.2	yes	2	8.5	even	2
1088.2.a.q.1.1		2	1.1	even	1	trivial
1088.2.a.q.1.2		2	4.3	odd	2	inner
4896.2.a.z.1.1		2	24.5	odd	2
4896.2.a.z.1.2		2	24.11	even	2
9248.2.a.p.1.1		2	136.101	even	2
9248.2.a.p.1.2		2	136.67	odd	2
9792.2.a.cg.1.1		2	3.2	odd	2
9792.2.a.cg.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +2.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +2.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} -1.41421 q^{11} +4.00000 q^{13} -2.82843 q^{15} -1.00000 q^{17} +2.82843 q^{19} +6.00000 q^{21} +4.24264 q^{23} -1.00000 q^{25} +5.65685 q^{27} +6.00000 q^{29} +7.07107 q^{31} +2.00000 q^{33} -8.48528 q^{35} +2.00000 q^{37} -5.65685 q^{39} -6.00000 q^{41} -8.48528 q^{43} -2.00000 q^{45} +11.3137 q^{47} +11.0000 q^{49} +1.41421 q^{51} +6.00000 q^{53} -2.82843 q^{55} -4.00000 q^{57} +8.48528 q^{59} -6.00000 q^{61} +4.24264 q^{63} +8.00000 q^{65} +5.65685 q^{67} -6.00000 q^{69} -7.07107 q^{71} +10.0000 q^{73} +1.41421 q^{75} +6.00000 q^{77} -12.7279 q^{79} -5.00000 q^{81} +14.1421 q^{83} -2.00000 q^{85} -8.48528 q^{87} -8.00000 q^{89} -16.9706 q^{91} -10.0000 q^{93} +5.65685 q^{95} +14.0000 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^5 - 2 * q^9	\( 2 q + 4 q^{5} - 2 q^{9} + 8 q^{13} - 2 q^{17} + 12 q^{21} - 2 q^{25} + 12 q^{29} + 4 q^{33} + 4 q^{37} - 12 q^{41} - 4 q^{45} + 22 q^{49} + 12 q^{53} - 8 q^{57} - 12 q^{61} + 16 q^{65} - 12 q^{69} + 20 q^{73} + 12 q^{77} - 10 q^{81} - 4 q^{85} - 16 q^{89} - 20 q^{93} + 28 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^9 + 8 * q^13 - 2 * q^17 + 12 * q^21 - 2 * q^25 + 12 * q^29 + 4 * q^33 + 4 * q^37 - 12 * q^41 - 4 * q^45 + 22 * q^49 + 12 * q^53 - 8 * q^57 - 12 * q^61 + 16 * q^65 - 12 * q^69 + 20 * q^73 + 12 * q^77 - 10 * q^81 - 4 * q^85 - 16 * q^89 - 20 * q^93 + 28 * q^97

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