LMFDB - Embedded newform 1320.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1320,2,Mod(1,1320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1320.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:	$10.5402530668$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.82843 q^{13} +1.00000 q^{15} +4.82843 q^{17} -2.82843 q^{21} -5.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} +5.65685 q^{31} +1.00000 q^{33} -2.82843 q^{35} -2.00000 q^{37} +4.82843 q^{39} +11.6569 q^{41} +12.4853 q^{43} +1.00000 q^{45} +5.65685 q^{47} +1.00000 q^{49} +4.82843 q^{51} +11.6569 q^{53} +1.00000 q^{55} -9.65685 q^{59} -7.65685 q^{61} -2.82843 q^{63} +4.82843 q^{65} -9.65685 q^{67} -5.65685 q^{69} +5.65685 q^{71} +3.17157 q^{73} +1.00000 q^{75} -2.82843 q^{77} -4.00000 q^{79} +1.00000 q^{81} +5.17157 q^{83} +4.82843 q^{85} -3.65685 q^{87} +2.00000 q^{89} -13.6569 q^{91} +5.65685 q^{93} -11.6569 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{33} - 4 q^{37} + 4 q^{39} + 12 q^{41} + 8 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} + 12 q^{53} + 2 q^{55} - 8 q^{59} - 4 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 16 q^{91} - 12 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^15 + 4 * q^17 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^33 - 4 * q^37 + 4 * q^39 + 12 * q^41 + 8 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 + 12 * q^53 + 2 * q^55 - 8 * q^59 - 4 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 - 16 * q^91 - 12 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	11.6569	1.82049	0.910247	−	0.414065i	$-0.135891\pi$
$41$	11.6569	1.82049	0.910247	+	0.414065i	$0.135891\pi$
$42$	0	0
$43$	12.4853	1.90399	0.951994	−	0.306117i	$-0.0990300\pi$
$43$	12.4853	1.90399	0.951994	+	0.306117i	$0.0990300\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.82843	0.676115
$52$	0	0
$53$	11.6569	1.60119	0.800596	−	0.599204i	$-0.204516\pi$
$53$	11.6569	1.60119	0.800596	+	0.599204i	$0.204516\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.65685	−1.25722	−0.628608	−	0.777723i	$-0.716375\pi$
$59$	−9.65685	−1.25722	−0.628608	+	0.777723i	$0.716375\pi$
$60$	0	0
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	4.82843	0.598893
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	3.17157	0.371205	0.185602	−	0.982625i	$-0.440576\pi$
$73$	3.17157	0.371205	0.185602	+	0.982625i	$0.440576\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.17157	0.567654	0.283827	−	0.958876i	$-0.408396\pi$
$83$	5.17157	0.567654	0.283827	+	0.958876i	$0.408396\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	0	0
$87$	−3.65685	−0.392056
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−13.6569	−1.43163
$92$	0	0
$93$	5.65685	0.586588
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−9.31371	−0.926749	−0.463374	−	0.886163i	$-0.653361\pi$
$101$	−9.31371	−0.926749	−0.463374	+	0.886163i	$0.653361\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−2.82843	−0.276026
$106$	0	0
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	0	0
$117$	4.82843	0.446388
$118$	0	0
$119$	−13.6569	−1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	11.6569	1.05106
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.1421	1.25491	0.627456	−	0.778652i	$-0.284096\pi$
$127$	14.1421	1.25491	0.627456	+	0.778652i	$0.284096\pi$
$128$	0	0
$129$	12.4853	1.09927
$130$	0	0
$131$	−9.65685	−0.843723	−0.421862	−	0.906660i	$-0.638623\pi$
$131$	−9.65685	−0.843723	−0.421862	+	0.906660i	$0.638623\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	5.65685	0.479808	0.239904	−	0.970797i	$-0.422884\pi$
$139$	5.65685	0.479808	0.239904	+	0.970797i	$0.422884\pi$
$140$	0	0
$141$	5.65685	0.476393
$142$	0	0
$143$	4.82843	0.403773
$144$	0	0
$145$	−3.65685	−0.303685
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	21.3137	1.74609	0.873044	−	0.487642i	$-0.162143\pi$
$149$	21.3137	1.74609	0.873044	+	0.487642i	$0.162143\pi$
$150$	0	0
$151$	−23.3137	−1.89724	−0.948621	−	0.316414i	$-0.897521\pi$
$151$	−23.3137	−1.89724	−0.948621	+	0.316414i	$0.897521\pi$
$152$	0	0
$153$	4.82843	0.390355
$154$	0	0
$155$	5.65685	0.454369
$156$	0	0
$157$	−23.6569	−1.88802	−0.944011	−	0.329913i	$-0.892981\pi$
$157$	−23.6569	−1.88802	−0.944011	+	0.329913i	$0.892981\pi$
$158$	0	0
$159$	11.6569	0.924449
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	−15.3137	−1.19946	−0.599731	−	0.800202i	$-0.704726\pi$
$163$	−15.3137	−1.19946	−0.599731	+	0.800202i	$0.704726\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−1.17157	−0.0906590	−0.0453295	−	0.998972i	$-0.514434\pi$
$167$	−1.17157	−0.0906590	−0.0453295	+	0.998972i	$0.514434\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.7990	−1.04912	−0.524559	−	0.851374i	$-0.675770\pi$
$173$	−13.7990	−1.04912	−0.524559	+	0.851374i	$0.675770\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	−9.65685	−0.725854
$178$	0	0
$179$	7.31371	0.546652	0.273326	−	0.961921i	$-0.411876\pi$
$179$	7.31371	0.546652	0.273326	+	0.961921i	$0.411876\pi$
$180$	0	0
$181$	9.31371	0.692283	0.346141	−	0.938182i	$-0.387492\pi$
$181$	9.31371	0.692283	0.346141	+	0.938182i	$0.387492\pi$
$182$	0	0
$183$	−7.65685	−0.566011
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	4.82843	0.353090
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	3.17157	0.228295	0.114147	−	0.993464i	$-0.463586\pi$
$193$	3.17157	0.228295	0.114147	+	0.993464i	$0.463586\pi$
$194$	0	0
$195$	4.82843	0.345771
$196$	0	0
$197$	12.1421	0.865091	0.432546	−	0.901612i	$-0.357615\pi$
$197$	12.1421	0.865091	0.432546	+	0.901612i	$0.357615\pi$
$198$	0	0
$199$	−19.3137	−1.36911	−0.684556	−	0.728960i	$-0.740004\pi$
$199$	−19.3137	−1.36911	−0.684556	+	0.728960i	$0.740004\pi$
$200$	0	0
$201$	−9.65685	−0.681142
$202$	0	0
$203$	10.3431	0.725947
$204$	0	0
$205$	11.6569	0.814150
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.68629	−0.322618	−0.161309	−	0.986904i	$-0.551572\pi$
$211$	−4.68629	−0.322618	−0.161309	+	0.986904i	$0.551572\pi$
$212$	0	0
$213$	5.65685	0.387601
$214$	0	0
$215$	12.4853	0.851489
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	3.17157	0.214315
$220$	0	0
$221$	23.3137	1.56825
$222$	0	0
$223$	−24.9706	−1.67215	−0.836076	−	0.548613i	$-0.815156\pi$
$223$	−24.9706	−1.67215	−0.836076	+	0.548613i	$0.815156\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−8.48528	−0.563188	−0.281594	−	0.959534i	$-0.590863\pi$
$227$	−8.48528	−0.563188	−0.281594	+	0.959534i	$0.590863\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−2.82843	−0.186097
$232$	0	0
$233$	18.4853	1.21101	0.605506	−	0.795841i	$-0.292971\pi$
$233$	18.4853	1.21101	0.605506	+	0.795841i	$0.292971\pi$
$234$	0	0
$235$	5.65685	0.369012
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−10.3431	−0.669042	−0.334521	−	0.942388i	$-0.608575\pi$
$239$	−10.3431	−0.669042	−0.334521	+	0.942388i	$0.608575\pi$
$240$	0	0
$241$	7.65685	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$241$	7.65685	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	5.17157	0.327735
$250$	0	0
$251$	−17.6569	−1.11449	−0.557245	−	0.830348i	$-0.688142\pi$
$251$	−17.6569	−1.11449	−0.557245	+	0.830348i	$0.688142\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	0	0
$255$	4.82843	0.302368
$256$	0	0
$257$	−22.9706	−1.43286	−0.716432	−	0.697657i	$-0.754226\pi$
$257$	−22.9706	−1.43286	−0.716432	+	0.697657i	$0.754226\pi$
$258$	0	0
$259$	5.65685	0.351500
$260$	0	0
$261$	−3.65685	−0.226354
$262$	0	0
$263$	12.4853	0.769875	0.384938	−	0.922943i	$-0.374223\pi$
$263$	12.4853	0.769875	0.384938	+	0.922943i	$0.374223\pi$
$264$	0	0
$265$	11.6569	0.716075
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	−24.6274	−1.50156	−0.750780	−	0.660552i	$-0.770322\pi$
$269$	−24.6274	−1.50156	−0.750780	+	0.660552i	$0.770322\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	−13.6569	−0.826550
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−19.1716	−1.15191	−0.575954	−	0.817482i	$-0.695369\pi$
$277$	−19.1716	−1.15191	−0.575954	+	0.817482i	$0.695369\pi$
$278$	0	0
$279$	5.65685	0.338667
$280$	0	0
$281$	−26.9706	−1.60893	−0.804464	−	0.594001i	$-0.797548\pi$
$281$	−26.9706	−1.60893	−0.804464	+	0.594001i	$0.797548\pi$
$282$	0	0
$283$	23.7990	1.41470	0.707352	−	0.706862i	$-0.249890\pi$
$283$	23.7990	1.41470	0.707352	+	0.706862i	$0.249890\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−32.9706	−1.94619
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	−11.6569	−0.683337
$292$	0	0
$293$	27.1716	1.58738	0.793690	−	0.608322i	$-0.208157\pi$
$293$	27.1716	1.58738	0.793690	+	0.608322i	$0.208157\pi$
$294$	0	0
$295$	−9.65685	−0.562244
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−27.3137	−1.57959
$300$	0	0
$301$	−35.3137	−2.03545
$302$	0	0
$303$	−9.31371	−0.535059
$304$	0	0
$305$	−7.65685	−0.438430
$306$	0	0
$307$	5.85786	0.334326	0.167163	−	0.985929i	$-0.446539\pi$
$307$	5.85786	0.334326	0.167163	+	0.985929i	$0.446539\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	8.97056	0.508674	0.254337	−	0.967116i	$-0.418143\pi$
$311$	8.97056	0.508674	0.254337	+	0.967116i	$0.418143\pi$
$312$	0	0
$313$	5.31371	0.300349	0.150174	−	0.988660i	$-0.452017\pi$
$313$	5.31371	0.300349	0.150174	+	0.988660i	$0.452017\pi$
$314$	0	0
$315$	−2.82843	−0.159364
$316$	0	0
$317$	20.6274	1.15855	0.579276	−	0.815132i	$-0.303336\pi$
$317$	20.6274	1.15855	0.579276	+	0.815132i	$0.303336\pi$
$318$	0	0
$319$	−3.65685	−0.204745
$320$	0	0
$321$	8.48528	0.473602
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.82843	0.267833
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	20.9706	1.15265	0.576323	−	0.817222i	$-0.304487\pi$
$331$	20.9706	1.15265	0.576323	+	0.817222i	$0.304487\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−9.65685	−0.527610
$336$	0	0
$337$	11.1716	0.608554	0.304277	−	0.952584i	$-0.401585\pi$
$337$	11.1716	0.608554	0.304277	+	0.952584i	$0.401585\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	0	0
$341$	5.65685	0.306336
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	−5.65685	−0.304555
$346$	0	0
$347$	1.85786	0.0997354	0.0498677	−	0.998756i	$-0.484120\pi$
$347$	1.85786	0.0997354	0.0498677	+	0.998756i	$0.484120\pi$
$348$	0	0
$349$	3.65685	0.195747	0.0978735	−	0.995199i	$-0.468796\pi$
$349$	3.65685	0.195747	0.0978735	+	0.995199i	$0.468796\pi$
$350$	0	0
$351$	4.82843	0.257722
$352$	0	0
$353$	−0.343146	−0.0182638	−0.00913190	−	0.999958i	$-0.502907\pi$
$353$	−0.343146	−0.0182638	−0.00913190	+	0.999958i	$0.502907\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	−13.6569	−0.722797
$358$	0	0
$359$	5.65685	0.298557	0.149279	−	0.988795i	$-0.452305\pi$
$359$	5.65685	0.298557	0.149279	+	0.988795i	$0.452305\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	3.17157	0.166008
$366$	0	0
$367$	8.97056	0.468260	0.234130	−	0.972205i	$-0.424776\pi$
$367$	8.97056	0.468260	0.234130	+	0.972205i	$0.424776\pi$
$368$	0	0
$369$	11.6569	0.606832
$370$	0	0
$371$	−32.9706	−1.71175
$372$	0	0
$373$	−31.4558	−1.62872	−0.814361	−	0.580359i	$-0.802912\pi$
$373$	−31.4558	−1.62872	−0.814361	+	0.580359i	$0.802912\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−17.6569	−0.909374
$378$	0	0
$379$	8.68629	0.446185	0.223092	−	0.974797i	$-0.428385\pi$
$379$	8.68629	0.446185	0.223092	+	0.974797i	$0.428385\pi$
$380$	0	0
$381$	14.1421	0.724524
$382$	0	0
$383$	22.6274	1.15621	0.578103	−	0.815963i	$-0.303793\pi$
$383$	22.6274	1.15621	0.578103	+	0.815963i	$0.303793\pi$
$384$	0	0
$385$	−2.82843	−0.144150
$386$	0	0
$387$	12.4853	0.634663
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−27.3137	−1.38131
$392$	0	0
$393$	−9.65685	−0.487124
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−1.02944	−0.0516660	−0.0258330	−	0.999666i	$-0.508224\pi$
$397$	−1.02944	−0.0516660	−0.0258330	+	0.999666i	$0.508224\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	27.3137	1.36059
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−36.6274	−1.81111	−0.905555	−	0.424230i	$-0.860545\pi$
$409$	−36.6274	−1.81111	−0.905555	+	0.424230i	$0.860545\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	27.3137	1.34402
$414$	0	0
$415$	5.17157	0.253863
$416$	0	0
$417$	5.65685	0.277017
$418$	0	0
$419$	18.6274	0.910009	0.455004	−	0.890489i	$-0.349638\pi$
$419$	18.6274	0.910009	0.455004	+	0.890489i	$0.349638\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	5.65685	0.275046
$424$	0	0
$425$	4.82843	0.234213
$426$	0	0
$427$	21.6569	1.04805
$428$	0	0
$429$	4.82843	0.233119
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	4.34315	0.208718	0.104359	−	0.994540i	$-0.466721\pi$
$433$	4.34315	0.208718	0.104359	+	0.994540i	$0.466721\pi$
$434$	0	0
$435$	−3.65685	−0.175333
$436$	0	0
$437$	0	0
$438$	0	0
$439$	14.3431	0.684561	0.342280	−	0.939598i	$-0.388801\pi$
$439$	14.3431	0.684561	0.342280	+	0.939598i	$0.388801\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	21.3137	1.00810
$448$	0	0
$449$	−33.3137	−1.57217	−0.786086	−	0.618118i	$-0.787896\pi$
$449$	−33.3137	−1.57217	−0.786086	+	0.618118i	$0.787896\pi$
$450$	0	0
$451$	11.6569	0.548900
$452$	0	0
$453$	−23.3137	−1.09537
$454$	0	0
$455$	−13.6569	−0.640243
$456$	0	0
$457$	−37.7990	−1.76816	−0.884081	−	0.467334i	$-0.845215\pi$
$457$	−37.7990	−1.76816	−0.884081	+	0.467334i	$0.845215\pi$
$458$	0	0
$459$	4.82843	0.225372
$460$	0	0
$461$	13.3137	0.620081	0.310041	−	0.950723i	$-0.399657\pi$
$461$	13.3137	0.620081	0.310041	+	0.950723i	$0.399657\pi$
$462$	0	0
$463$	−18.3431	−0.852478	−0.426239	−	0.904611i	$-0.640162\pi$
$463$	−18.3431	−0.852478	−0.426239	+	0.904611i	$0.640162\pi$
$464$	0	0
$465$	5.65685	0.262330
$466$	0	0
$467$	36.9706	1.71079	0.855397	−	0.517973i	$-0.173313\pi$
$467$	36.9706	1.71079	0.855397	+	0.517973i	$0.173313\pi$
$468$	0	0
$469$	27.3137	1.26123
$470$	0	0
$471$	−23.6569	−1.09005
$472$	0	0
$473$	12.4853	0.574074
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.6569	0.533731
$478$	0	0
$479$	−5.65685	−0.258468	−0.129234	−	0.991614i	$-0.541252\pi$
$479$	−5.65685	−0.258468	−0.129234	+	0.991614i	$0.541252\pi$
$480$	0	0
$481$	−9.65685	−0.440315
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	−11.6569	−0.529310
$486$	0	0
$487$	29.6569	1.34388	0.671940	−	0.740605i	$-0.265461\pi$
$487$	29.6569	1.34388	0.671940	+	0.740605i	$0.265461\pi$
$488$	0	0
$489$	−15.3137	−0.692510
$490$	0	0
$491$	−23.3137	−1.05213	−0.526066	−	0.850443i	$-0.676334\pi$
$491$	−23.3137	−1.05213	−0.526066	+	0.850443i	$0.676334\pi$
$492$	0	0
$493$	−17.6569	−0.795225
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−29.9411	−1.34035	−0.670174	−	0.742204i	$-0.733781\pi$
$499$	−29.9411	−1.34035	−0.670174	+	0.742204i	$0.733781\pi$
$500$	0	0
$501$	−1.17157	−0.0523420
$502$	0	0
$503$	−23.7990	−1.06114	−0.530572	−	0.847640i	$-0.678023\pi$
$503$	−23.7990	−1.06114	−0.530572	+	0.847640i	$0.678023\pi$
$504$	0	0
$505$	−9.31371	−0.414455
$506$	0	0
$507$	10.3137	0.458048
$508$	0	0
$509$	9.31371	0.412823	0.206411	−	0.978465i	$-0.433821\pi$
$509$	9.31371	0.412823	0.206411	+	0.978465i	$0.433821\pi$
$510$	0	0
$511$	−8.97056	−0.396834
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	−13.7990	−0.605708
$520$	0	0
$521$	27.9411	1.22412	0.612061	−	0.790810i	$-0.290340\pi$
$521$	27.9411	1.22412	0.612061	+	0.790810i	$0.290340\pi$
$522$	0	0
$523$	−7.79899	−0.341026	−0.170513	−	0.985355i	$-0.554542\pi$
$523$	−7.79899	−0.341026	−0.170513	+	0.985355i	$0.554542\pi$
$524$	0	0
$525$	−2.82843	−0.123443
$526$	0	0
$527$	27.3137	1.18980
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	−9.65685	−0.419072
$532$	0	0
$533$	56.2843	2.43794
$534$	0	0
$535$	8.48528	0.366851
$536$	0	0
$537$	7.31371	0.315610
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−6.68629	−0.287466	−0.143733	−	0.989616i	$-0.545911\pi$
$541$	−6.68629	−0.287466	−0.143733	+	0.989616i	$0.545911\pi$
$542$	0	0
$543$	9.31371	0.399689
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−15.7990	−0.675516	−0.337758	−	0.941233i	$-0.609669\pi$
$547$	−15.7990	−0.675516	−0.337758	+	0.941233i	$0.609669\pi$
$548$	0	0
$549$	−7.65685	−0.326787
$550$	0	0
$551$	0	0
$552$	0	0
$553$	11.3137	0.481108
$554$	0	0
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	5.51472	0.233666	0.116833	−	0.993152i	$-0.462726\pi$
$557$	5.51472	0.233666	0.116833	+	0.993152i	$0.462726\pi$
$558$	0	0
$559$	60.2843	2.54975
$560$	0	0
$561$	4.82843	0.203856
$562$	0	0
$563$	15.5147	0.653867	0.326934	−	0.945047i	$-0.393985\pi$
$563$	15.5147	0.653867	0.326934	+	0.945047i	$0.393985\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−14.6863	−0.615681	−0.307841	−	0.951438i	$-0.599606\pi$
$569$	−14.6863	−0.615681	−0.307841	+	0.951438i	$0.599606\pi$
$570$	0	0
$571$	18.3431	0.767637	0.383818	−	0.923409i	$-0.374609\pi$
$571$	18.3431	0.767637	0.383818	+	0.923409i	$0.374609\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	−5.65685	−0.235907
$576$	0	0
$577$	26.9706	1.12280	0.561400	−	0.827545i	$-0.310263\pi$
$577$	26.9706	1.12280	0.561400	+	0.827545i	$0.310263\pi$
$578$	0	0
$579$	3.17157	0.131806
$580$	0	0
$581$	−14.6274	−0.606848
$582$	0	0
$583$	11.6569	0.482778
$584$	0	0
$585$	4.82843	0.199631
$586$	0	0
$587$	−9.65685	−0.398581	−0.199291	−	0.979940i	$-0.563864\pi$
$587$	−9.65685	−0.398581	−0.199291	+	0.979940i	$0.563864\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.1421	0.499461
$592$	0	0
$593$	−41.7990	−1.71648	−0.858239	−	0.513250i	$-0.828442\pi$
$593$	−41.7990	−1.71648	−0.858239	+	0.513250i	$0.828442\pi$
$594$	0	0
$595$	−13.6569	−0.559876
$596$	0	0
$597$	−19.3137	−0.790457
$598$	0	0
$599$	12.6863	0.518348	0.259174	−	0.965831i	$-0.416550\pi$
$599$	12.6863	0.518348	0.259174	+	0.965831i	$0.416550\pi$
$600$	0	0
$601$	28.3431	1.15614	0.578071	−	0.815987i	$-0.303806\pi$
$601$	28.3431	1.15614	0.578071	+	0.815987i	$0.303806\pi$
$602$	0	0
$603$	−9.65685	−0.393258
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−47.1127	−1.91225	−0.956123	−	0.292966i	$-0.905358\pi$
$607$	−47.1127	−1.91225	−0.956123	+	0.292966i	$0.905358\pi$
$608$	0	0
$609$	10.3431	0.419125
$610$	0	0
$611$	27.3137	1.10499
$612$	0	0
$613$	7.17157	0.289657	0.144829	−	0.989457i	$-0.453737\pi$
$613$	7.17157	0.289657	0.144829	+	0.989457i	$0.453737\pi$
$614$	0	0
$615$	11.6569	0.470050
$616$	0	0
$617$	−9.31371	−0.374956	−0.187478	−	0.982269i	$-0.560031\pi$
$617$	−9.31371	−0.374956	−0.187478	+	0.982269i	$0.560031\pi$
$618$	0	0
$619$	14.3431	0.576500	0.288250	−	0.957555i	$-0.406927\pi$
$619$	14.3431	0.576500	0.288250	+	0.957555i	$0.406927\pi$
$620$	0	0
$621$	−5.65685	−0.227002
$622$	0	0
$623$	−5.65685	−0.226637
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.65685	−0.385044
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	−4.68629	−0.186263
$634$	0	0
$635$	14.1421	0.561214
$636$	0	0
$637$	4.82843	0.191309
$638$	0	0
$639$	5.65685	0.223782
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	23.3137	0.919403	0.459701	−	0.888074i	$-0.347956\pi$
$643$	23.3137	0.919403	0.459701	+	0.888074i	$0.347956\pi$
$644$	0	0
$645$	12.4853	0.491607
$646$	0	0
$647$	−30.6274	−1.20409	−0.602044	−	0.798463i	$-0.705647\pi$
$647$	−30.6274	−1.20409	−0.602044	+	0.798463i	$0.705647\pi$
$648$	0	0
$649$	−9.65685	−0.379065
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	−18.9706	−0.742375	−0.371188	−	0.928558i	$-0.621049\pi$
$653$	−18.9706	−0.742375	−0.371188	+	0.928558i	$0.621049\pi$
$654$	0	0
$655$	−9.65685	−0.377325
$656$	0	0
$657$	3.17157	0.123735
$658$	0	0
$659$	−0.686292	−0.0267341	−0.0133671	−	0.999911i	$-0.504255\pi$
$659$	−0.686292	−0.0267341	−0.0133671	+	0.999911i	$0.504255\pi$
$660$	0	0
$661$	−27.9411	−1.08678	−0.543392	−	0.839479i	$-0.682860\pi$
$661$	−27.9411	−1.08678	−0.543392	+	0.839479i	$0.682860\pi$
$662$	0	0
$663$	23.3137	0.905429
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.6863	0.800976
$668$	0	0
$669$	−24.9706	−0.965418
$670$	0	0
$671$	−7.65685	−0.295590
$672$	0	0
$673$	41.7990	1.61123	0.805616	−	0.592438i	$-0.201834\pi$
$673$	41.7990	1.61123	0.805616	+	0.592438i	$0.201834\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−8.14214	−0.312928	−0.156464	−	0.987684i	$-0.550009\pi$
$677$	−8.14214	−0.312928	−0.156464	+	0.987684i	$0.550009\pi$
$678$	0	0
$679$	32.9706	1.26529
$680$	0	0
$681$	−8.48528	−0.325157
$682$	0	0
$683$	25.6569	0.981732	0.490866	−	0.871235i	$-0.336680\pi$
$683$	25.6569	0.981732	0.490866	+	0.871235i	$0.336680\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	56.2843	2.14426
$690$	0	0
$691$	23.3137	0.886895	0.443448	−	0.896300i	$-0.353755\pi$
$691$	23.3137	0.886895	0.443448	+	0.896300i	$0.353755\pi$
$692$	0	0
$693$	−2.82843	−0.107443
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	56.2843	2.13192
$698$	0	0
$699$	18.4853	0.699178
$700$	0	0
$701$	42.9706	1.62298	0.811488	−	0.584369i	$-0.198658\pi$
$701$	42.9706	1.62298	0.811488	+	0.584369i	$0.198658\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	5.65685	0.213049
$706$	0	0
$707$	26.3431	0.990736
$708$	0	0
$709$	2.68629	0.100886	0.0504429	−	0.998727i	$-0.483937\pi$
$709$	2.68629	0.100886	0.0504429	+	0.998727i	$0.483937\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	4.82843	0.180573
$716$	0	0
$717$	−10.3431	−0.386272
$718$	0	0
$719$	−12.6863	−0.473119	−0.236559	−	0.971617i	$-0.576020\pi$
$719$	−12.6863	−0.473119	−0.236559	+	0.971617i	$0.576020\pi$
$720$	0	0
$721$	22.6274	0.842689
$722$	0	0
$723$	7.65685	0.284761
$724$	0	0
$725$	−3.65685	−0.135812
$726$	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
$729$	1.00000	0.0370370
$730$	0	0
$731$	60.2843	2.22969
$732$	0	0
$733$	34.4853	1.27374	0.636871	−	0.770970i	$-0.280228\pi$
$733$	34.4853	1.27374	0.636871	+	0.770970i	$0.280228\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−9.65685	−0.355715
$738$	0	0
$739$	5.65685	0.208091	0.104045	−	0.994573i	$-0.466821\pi$
$739$	5.65685	0.208091	0.104045	+	0.994573i	$0.466821\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.17157	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$743$	9.17157	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$744$	0	0
$745$	21.3137	0.780874
$746$	0	0
$747$	5.17157	0.189218
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	22.6274	0.825686	0.412843	−	0.910802i	$-0.364536\pi$
$751$	22.6274	0.825686	0.412843	+	0.910802i	$0.364536\pi$
$752$	0	0
$753$	−17.6569	−0.643452
$754$	0	0
$755$	−23.3137	−0.848473
$756$	0	0
$757$	−42.9706	−1.56179	−0.780896	−	0.624661i	$-0.785237\pi$
$757$	−42.9706	−1.56179	−0.780896	+	0.624661i	$0.785237\pi$
$758$	0	0
$759$	−5.65685	−0.205331
$760$	0	0
$761$	44.6274	1.61774	0.808871	−	0.587986i	$-0.200079\pi$
$761$	44.6274	1.61774	0.808871	+	0.587986i	$0.200079\pi$
$762$	0	0
$763$	5.65685	0.204792
$764$	0	0
$765$	4.82843	0.174572
$766$	0	0
$767$	−46.6274	−1.68362
$768$	0	0
$769$	42.9706	1.54956	0.774779	−	0.632232i	$-0.217861\pi$
$769$	42.9706	1.54956	0.774779	+	0.632232i	$0.217861\pi$
$770$	0	0
$771$	−22.9706	−0.827265
$772$	0	0
$773$	16.3431	0.587822	0.293911	−	0.955833i	$-0.405043\pi$
$773$	16.3431	0.587822	0.293911	+	0.955833i	$0.405043\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	5.65685	0.202939
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.65685	0.202418
$782$	0	0
$783$	−3.65685	−0.130685
$784$	0	0
$785$	−23.6569	−0.844349
$786$	0	0
$787$	54.4264	1.94009	0.970046	−	0.242922i	$-0.0781058\pi$
$787$	54.4264	1.94009	0.970046	+	0.242922i	$0.0781058\pi$
$788$	0	0
$789$	12.4853	0.444488
$790$	0	0
$791$	−28.2843	−1.00567
$792$	0	0
$793$	−36.9706	−1.31286
$794$	0	0
$795$	11.6569	0.413426
$796$	0	0
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	0	0
$799$	27.3137	0.966290
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	3.17157	0.111922
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	−24.6274	−0.866926
$808$	0	0
$809$	−23.6569	−0.831731	−0.415865	−	0.909426i	$-0.636521\pi$
$809$	−23.6569	−0.831731	−0.415865	+	0.909426i	$0.636521\pi$
$810$	0	0
$811$	37.6569	1.32231	0.661155	−	0.750249i	$-0.270066\pi$
$811$	37.6569	1.32231	0.661155	+	0.750249i	$0.270066\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	0	0
$815$	−15.3137	−0.536416
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−13.6569	−0.477209
$820$	0	0
$821$	20.3431	0.709981	0.354990	−	0.934870i	$-0.384484\pi$
$821$	20.3431	0.709981	0.354990	+	0.934870i	$0.384484\pi$
$822$	0	0
$823$	14.6274	0.509880	0.254940	−	0.966957i	$-0.417944\pi$
$823$	14.6274	0.509880	0.254940	+	0.966957i	$0.417944\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	15.1127	0.525520	0.262760	−	0.964861i	$-0.415367\pi$
$827$	15.1127	0.525520	0.262760	+	0.964861i	$0.415367\pi$
$828$	0	0
$829$	−11.9411	−0.414732	−0.207366	−	0.978263i	$-0.566489\pi$
$829$	−11.9411	−0.414732	−0.207366	+	0.978263i	$0.566489\pi$
$830$	0	0
$831$	−19.1716	−0.665054
$832$	0	0
$833$	4.82843	0.167295
$834$	0	0
$835$	−1.17157	−0.0405440
$836$	0	0
$837$	5.65685	0.195529
$838$	0	0
$839$	−8.97056	−0.309698	−0.154849	−	0.987938i	$-0.549489\pi$
$839$	−8.97056	−0.309698	−0.154849	+	0.987938i	$0.549489\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	−26.9706	−0.928916
$844$	0	0
$845$	10.3137	0.354802
$846$	0	0
$847$	−2.82843	−0.0971859
$848$	0	0
$849$	23.7990	0.816779
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	11.8579	0.406006	0.203003	−	0.979178i	$-0.434930\pi$
$853$	11.8579	0.406006	0.203003	+	0.979178i	$0.434930\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.4853	−1.58791	−0.793953	−	0.607979i	$-0.791981\pi$
$857$	−46.4853	−1.58791	−0.793953	+	0.607979i	$0.791981\pi$
$858$	0	0
$859$	−17.6569	−0.602444	−0.301222	−	0.953554i	$-0.597395\pi$
$859$	−17.6569	−0.602444	−0.301222	+	0.953554i	$0.597395\pi$
$860$	0	0
$861$	−32.9706	−1.12363
$862$	0	0
$863$	−11.3137	−0.385123	−0.192562	−	0.981285i	$-0.561680\pi$
$863$	−11.3137	−0.385123	−0.192562	+	0.981285i	$0.561680\pi$
$864$	0	0
$865$	−13.7990	−0.469180
$866$	0	0
$867$	6.31371	0.214425
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−46.6274	−1.57991
$872$	0	0
$873$	−11.6569	−0.394525
$874$	0	0
$875$	−2.82843	−0.0956183
$876$	0	0
$877$	5.79899	0.195818	0.0979090	−	0.995195i	$-0.468785\pi$
$877$	5.79899	0.195818	0.0979090	+	0.995195i	$0.468785\pi$
$878$	0	0
$879$	27.1716	0.916474
$880$	0	0
$881$	3.37258	0.113625	0.0568126	−	0.998385i	$-0.481906\pi$
$881$	3.37258	0.113625	0.0568126	+	0.998385i	$0.481906\pi$
$882$	0	0
$883$	−41.2548	−1.38834	−0.694168	−	0.719813i	$-0.744227\pi$
$883$	−41.2548	−1.38834	−0.694168	+	0.719813i	$0.744227\pi$
$884$	0	0
$885$	−9.65685	−0.324612
$886$	0	0
$887$	45.4558	1.52626	0.763129	−	0.646246i	$-0.223662\pi$
$887$	45.4558	1.52626	0.763129	+	0.646246i	$0.223662\pi$
$888$	0	0
$889$	−40.0000	−1.34156
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	7.31371	0.244470
$896$	0	0
$897$	−27.3137	−0.911978
$898$	0	0
$899$	−20.6863	−0.689926
$900$	0	0
$901$	56.2843	1.87510
$902$	0	0
$903$	−35.3137	−1.17517
$904$	0	0
$905$	9.31371	0.309598
$906$	0	0
$907$	−9.65685	−0.320651	−0.160325	−	0.987064i	$-0.551254\pi$
$907$	−9.65685	−0.320651	−0.160325	+	0.987064i	$0.551254\pi$
$908$	0	0
$909$	−9.31371	−0.308916
$910$	0	0
$911$	−12.6863	−0.420316	−0.210158	−	0.977667i	$-0.567398\pi$
$911$	−12.6863	−0.420316	−0.210158	+	0.977667i	$0.567398\pi$
$912$	0	0
$913$	5.17157	0.171154
$914$	0	0
$915$	−7.65685	−0.253128
$916$	0	0
$917$	27.3137	0.901978
$918$	0	0
$919$	−6.34315	−0.209241	−0.104621	−	0.994512i	$-0.533363\pi$
$919$	−6.34315	−0.209241	−0.104621	+	0.994512i	$0.533363\pi$
$920$	0	0
$921$	5.85786	0.193023
$922$	0	0
$923$	27.3137	0.899042
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	16.6274	0.545528	0.272764	−	0.962081i	$-0.412062\pi$
$929$	16.6274	0.545528	0.272764	+	0.962081i	$0.412062\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.97056	0.293683
$934$	0	0
$935$	4.82843	0.157906
$936$	0	0
$937$	−25.1127	−0.820396	−0.410198	−	0.911996i	$-0.634540\pi$
$937$	−25.1127	−0.820396	−0.410198	+	0.911996i	$0.634540\pi$
$938$	0	0
$939$	5.31371	0.173406
$940$	0	0
$941$	−17.3137	−0.564411	−0.282205	−	0.959354i	$-0.591066\pi$
$941$	−17.3137	−0.564411	−0.282205	+	0.959354i	$0.591066\pi$
$942$	0	0
$943$	−65.9411	−2.14734
$944$	0	0
$945$	−2.82843	−0.0920087
$946$	0	0
$947$	12.9706	0.421487	0.210743	−	0.977541i	$-0.432412\pi$
$947$	12.9706	0.421487	0.210743	+	0.977541i	$0.432412\pi$
$948$	0	0
$949$	15.3137	0.497104
$950$	0	0
$951$	20.6274	0.668890
$952$	0	0
$953$	−39.4558	−1.27810	−0.639050	−	0.769165i	$-0.720672\pi$
$953$	−39.4558	−1.27810	−0.639050	+	0.769165i	$0.720672\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	0	0
$957$	−3.65685	−0.118209
$958$	0	0
$959$	−5.65685	−0.182669
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	8.48528	0.273434
$964$	0	0
$965$	3.17157	0.102097
$966$	0	0
$967$	38.1421	1.22657	0.613284	−	0.789862i	$-0.289848\pi$
$967$	38.1421	1.22657	0.613284	+	0.789862i	$0.289848\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.9411	−1.21759	−0.608794	−	0.793328i	$-0.708347\pi$
$971$	−37.9411	−1.21759	−0.608794	+	0.793328i	$0.708347\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	4.82843	0.154633
$976$	0	0
$977$	−17.3137	−0.553915	−0.276957	−	0.960882i	$-0.589326\pi$
$977$	−17.3137	−0.553915	−0.276957	+	0.960882i	$0.589326\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	9.94113	0.317073	0.158536	−	0.987353i	$-0.449323\pi$
$983$	9.94113	0.317073	0.158536	+	0.987353i	$0.449323\pi$
$984$	0	0
$985$	12.1421	0.386881
$986$	0	0
$987$	−16.0000	−0.509286
$988$	0	0
$989$	−70.6274	−2.24582
$990$	0	0
$991$	21.6569	0.687953	0.343976	−	0.938978i	$-0.388226\pi$
$991$	21.6569	0.687953	0.343976	+	0.938978i	$0.388226\pi$
$992$	0	0
$993$	20.9706	0.665481
$994$	0	0
$995$	−19.3137	−0.612286
$996$	0	0
$997$	−31.4558	−0.996217	−0.498108	−	0.867115i	$-0.665972\pi$
$997$	−31.4558	−0.996217	−0.498108	+	0.867115i	$0.665972\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1320.2.a.q.1.1	✓	2
3.2	odd	2		3960.2.a.x.1.1		2
4.3	odd	2		2640.2.a.z.1.2		2
5.2	odd	4		6600.2.d.bb.1849.1		4
5.3	odd	4		6600.2.d.bb.1849.4		4
5.4	even	2		6600.2.a.bh.1.2		2
12.11	even	2		7920.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.q.1.1	✓	2	1.1	even	1	trivial
2640.2.a.z.1.2		2	4.3	odd	2
3960.2.a.x.1.1		2	3.2	odd	2
6600.2.a.bh.1.2		2	5.4	even	2
6600.2.d.bb.1849.1		4	5.2	odd	4
6600.2.d.bb.1849.4		4	5.3	odd	4
7920.2.a.bt.1.2		2	12.11	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1320.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.82843 q^{13} +1.00000 q^{15} +4.82843 q^{17} -2.82843 q^{21} -5.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} +5.65685 q^{31} +1.00000 q^{33} -2.82843 q^{35} -2.00000 q^{37} +4.82843 q^{39} +11.6569 q^{41} +12.4853 q^{43} +1.00000 q^{45} +5.65685 q^{47} +1.00000 q^{49} +4.82843 q^{51} +11.6569 q^{53} +1.00000 q^{55} -9.65685 q^{59} -7.65685 q^{61} -2.82843 q^{63} +4.82843 q^{65} -9.65685 q^{67} -5.65685 q^{69} +5.65685 q^{71} +3.17157 q^{73} +1.00000 q^{75} -2.82843 q^{77} -4.00000 q^{79} +1.00000 q^{81} +5.17157 q^{83} +4.82843 q^{85} -3.65685 q^{87} +2.00000 q^{89} -13.6569 q^{91} +5.65685 q^{93} -11.6569 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{33} - 4 q^{37} + 4 q^{39} + 12 q^{41} + 8 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} + 12 q^{53} + 2 q^{55} - 8 q^{59} - 4 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 16 q^{91} - 12 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^15 + 4 * q^17 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^33 - 4 * q^37 + 4 * q^39 + 12 * q^41 + 8 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 + 12 * q^53 + 2 * q^55 - 8 * q^59 - 4 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 - 16 * q^91 - 12 * q^97 + 2 * q^99

Introduction

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