LMFDB - Embedded newform 3344.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 7x + 1 $ x^4 - x^3 - 6x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.128950$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.128950 q^{3} +1.64261 q^{5} +1.12895 q^{7} -2.98337 q^{9} +O(q^{10})$	$q-0.128950 q^{3} +1.64261 q^{5} +1.12895 q^{7} -2.98337 q^{9} -1.00000 q^{11} -6.41416 q^{13} -0.211815 q^{15} +4.45309 q^{17} -1.00000 q^{19} -0.145578 q^{21} +3.59867 q^{23} -2.30184 q^{25} +0.771557 q^{27} -1.46972 q^{29} +7.75493 q^{31} +0.128950 q^{33} +1.85442 q^{35} -1.14558 q^{37} +0.827108 q^{39} -5.06624 q^{41} +0.301842 q^{43} -4.90051 q^{45} -8.57043 q^{47} -5.72547 q^{49} -0.574227 q^{51} -9.28521 q^{53} -1.64261 q^{55} +0.128950 q^{57} -2.85442 q^{59} -12.4198 q^{61} -3.36808 q^{63} -10.5360 q^{65} +0.0460859 q^{67} -0.464049 q^{69} -14.5782 q^{71} +8.82833 q^{73} +0.296823 q^{75} -1.12895 q^{77} +9.48040 q^{79} +8.85062 q^{81} -6.97957 q^{83} +7.31467 q^{85} +0.189520 q^{87} +1.79385 q^{89} -7.24127 q^{91} -1.00000 q^{93} -1.64261 q^{95} -7.82711 q^{97} +2.98337 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) $ 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9	$ 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41} - 5 q^{43} - 7 q^{45} + 4 q^{47} - 13 q^{49} - 4 q^{51} - 18 q^{53} + 3 q^{55} - q^{57} - 6 q^{61} + 6 q^{63} - 24 q^{65} - q^{67} - 20 q^{69} + q^{71} - 6 q^{73} + q^{75} - 3 q^{77} + 4 q^{79} - 16 q^{81} + 25 q^{83} - 4 q^{85} + 9 q^{87} - 14 q^{89} - 13 q^{91} - 4 q^{93} + 3 q^{95} - 36 q^{97} - q^{99}+O(q^{100}) $ 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.128950	−0.0744495	−0.0372247	−	0.999307i	$-0.511852\pi$
$3$	−0.128950	−0.0744495	−0.0372247	+	0.999307i	$0.511852\pi$
$4$	0	0
$5$	1.64261	0.734596	0.367298	−	0.930103i	$-0.380283\pi$
$5$	1.64261	0.734596	0.367298	+	0.930103i	$0.380283\pi$
$6$	0	0
$7$	1.12895	0.426703	0.213352	−	0.976975i	$-0.431562\pi$
$7$	1.12895	0.426703	0.213352	+	0.976975i	$0.431562\pi$
$8$	0	0
$9$	−2.98337	−0.994457
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.41416	−1.77897	−0.889485	−	0.456965i	$-0.848936\pi$
$13$	−6.41416	−1.77897	−0.889485	+	0.456965i	$0.848936\pi$
$14$	0	0
$15$	−0.211815	−0.0546903
$16$	0	0
$17$	4.45309	1.08003	0.540016	−	0.841655i	$-0.318418\pi$
$17$	4.45309	1.08003	0.540016	+	0.841655i	$0.318418\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.145578	−0.0317678
$22$	0	0
$23$	3.59867	0.750374	0.375187	−	0.926949i	$-0.377579\pi$
$23$	3.59867	0.750374	0.375187	+	0.926949i	$0.377579\pi$
$24$	0	0
$25$	−2.30184	−0.460368
$26$	0	0
$27$	0.771557	0.148486
$28$	0	0
$29$	−1.46972	−0.272919	−0.136460	−	0.990646i	$-0.543572\pi$
$29$	−1.46972	−0.272919	−0.136460	+	0.990646i	$0.543572\pi$
$30$	0	0
$31$	7.75493	1.39283	0.696413	−	0.717641i	$-0.254778\pi$
$31$	7.75493	1.39283	0.696413	+	0.717641i	$0.254778\pi$
$32$	0	0
$33$	0.128950	0.0224474
$34$	0	0
$35$	1.85442	0.313454
$36$	0	0
$37$	−1.14558	−0.188332	−0.0941660	−	0.995557i	$-0.530018\pi$
$37$	−1.14558	−0.188332	−0.0941660	+	0.995557i	$0.530018\pi$
$38$	0	0
$39$	0.827108	0.132443
$40$	0	0
$41$	−5.06624	−0.791213	−0.395607	−	0.918420i	$-0.629466\pi$
$41$	−5.06624	−0.791213	−0.395607	+	0.918420i	$0.629466\pi$
$42$	0	0
$43$	0.301842	0.0460305	0.0230153	−	0.999735i	$-0.492673\pi$
$43$	0.301842	0.0460305	0.0230153	+	0.999735i	$0.492673\pi$
$44$	0	0
$45$	−4.90051	−0.730525
$46$	0	0
$47$	−8.57043	−1.25013	−0.625063	−	0.780575i	$-0.714927\pi$
$47$	−8.57043	−1.25013	−0.625063	+	0.780575i	$0.714927\pi$
$48$	0	0
$49$	−5.72547	−0.817924
$50$	0	0
$51$	−0.574227	−0.0804078
$52$	0	0
$53$	−9.28521	−1.27542	−0.637711	−	0.770276i	$-0.720119\pi$
$53$	−9.28521	−1.27542	−0.637711	+	0.770276i	$0.720119\pi$
$54$	0	0
$55$	−1.64261	−0.221489
$56$	0	0
$57$	0.128950	0.0170799
$58$	0	0
$59$	−2.85442	−0.371614	−0.185807	−	0.982586i	$-0.559490\pi$
$59$	−2.85442	−0.371614	−0.185807	+	0.982586i	$0.559490\pi$
$60$	0	0
$61$	−12.4198	−1.59020	−0.795098	−	0.606481i	$-0.792581\pi$
$61$	−12.4198	−1.59020	−0.795098	+	0.606481i	$0.792581\pi$
$62$	0	0
$63$	−3.36808	−0.424338
$64$	0	0
$65$	−10.5360	−1.30682
$66$	0	0
$67$	0.0460859	0.00563029	0.00281515	−	0.999996i	$-0.499104\pi$
$67$	0.0460859	0.00563029	0.00281515	+	0.999996i	$0.499104\pi$
$68$	0	0
$69$	−0.464049	−0.0558649
$70$	0	0
$71$	−14.5782	−1.73012	−0.865059	−	0.501670i	$-0.832719\pi$
$71$	−14.5782	−1.73012	−0.865059	+	0.501670i	$0.832719\pi$
$72$	0	0
$73$	8.82833	1.03328	0.516639	−	0.856203i	$-0.327183\pi$
$73$	8.82833	1.03328	0.516639	+	0.856203i	$0.327183\pi$
$74$	0	0
$75$	0.296823	0.0342742
$76$	0	0
$77$	−1.12895	−0.128656
$78$	0	0
$79$	9.48040	1.06663	0.533314	−	0.845917i	$-0.320946\pi$
$79$	9.48040	1.06663	0.533314	+	0.845917i	$0.320946\pi$
$80$	0	0
$81$	8.85062	0.983403
$82$	0	0
$83$	−6.97957	−0.766108	−0.383054	−	0.923726i	$-0.625128\pi$
$83$	−6.97957	−0.766108	−0.383054	+	0.923726i	$0.625128\pi$
$84$	0	0
$85$	7.31467	0.793388
$86$	0	0
$87$	0.189520	0.0203187
$88$	0	0
$89$	1.79385	0.190148	0.0950740	−	0.995470i	$-0.469691\pi$
$89$	1.79385	0.190148	0.0950740	+	0.995470i	$0.469691\pi$
$90$	0	0
$91$	−7.24127	−0.759092
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−1.64261	−0.168528
$96$	0	0
$97$	−7.82711	−0.794722	−0.397361	−	0.917662i	$-0.630074\pi$
$97$	−7.82711	−0.794722	−0.397361	+	0.917662i	$0.630074\pi$
$98$	0	0
$99$	2.98337	0.299840
$100$	0	0
$101$	0.228443	0.0227309	0.0113655	−	0.999935i	$-0.496382\pi$
$101$	0.228443	0.0227309	0.0113655	+	0.999935i	$0.496382\pi$
$102$	0	0
$103$	14.8390	1.46213	0.731066	−	0.682307i	$-0.239023\pi$
$103$	14.8390	1.46213	0.731066	+	0.682307i	$0.239023\pi$
$104$	0	0
$105$	−0.239128	−0.0233365
$106$	0	0
$107$	−3.87886	−0.374984	−0.187492	−	0.982266i	$-0.560036\pi$
$107$	−3.87886	−0.374984	−0.187492	+	0.982266i	$0.560036\pi$
$108$	0	0
$109$	−1.48420	−0.142160	−0.0710802	−	0.997471i	$-0.522645\pi$
$109$	−1.48420	−0.142160	−0.0710802	+	0.997471i	$0.522645\pi$
$110$	0	0
$111$	0.147723	0.0140212
$112$	0	0
$113$	−13.5360	−1.27336	−0.636678	−	0.771130i	$-0.719692\pi$
$113$	−13.5360	−1.27336	−0.636678	+	0.771130i	$0.719692\pi$
$114$	0	0
$115$	5.91119	0.551222
$116$	0	0
$117$	19.1358	1.76911
$118$	0	0
$119$	5.02731	0.460853
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.653292	0.0589054
$124$	0	0
$125$	−11.9941	−1.07278
$126$	0	0
$127$	−8.67773	−0.770024	−0.385012	−	0.922912i	$-0.625803\pi$
$127$	−8.67773	−0.770024	−0.385012	+	0.922912i	$0.625803\pi$
$128$	0	0
$129$	−0.0389226	−0.00342695
$130$	0	0
$131$	0.720452	0.0629462	0.0314731	−	0.999505i	$-0.489980\pi$
$131$	0.720452	0.0629462	0.0314731	+	0.999505i	$0.489980\pi$
$132$	0	0
$133$	−1.12895	−0.0978924
$134$	0	0
$135$	1.26737	0.109077
$136$	0	0
$137$	−16.8996	−1.44383	−0.721914	−	0.691982i	$-0.756738\pi$
$137$	−16.8996	−1.44383	−0.721914	+	0.691982i	$0.756738\pi$
$138$	0	0
$139$	10.5227	0.892523	0.446261	−	0.894903i	$-0.352755\pi$
$139$	10.5227	0.892523	0.446261	+	0.894903i	$0.352755\pi$
$140$	0	0
$141$	1.10516	0.0930712
$142$	0	0
$143$	6.41416	0.536379
$144$	0	0
$145$	−2.41416	−0.200485
$146$	0	0
$147$	0.738301	0.0608940
$148$	0	0
$149$	20.3382	1.66617	0.833085	−	0.553146i	$-0.186573\pi$
$149$	20.3382	1.66617	0.833085	+	0.553146i	$0.186573\pi$
$150$	0	0
$151$	−10.0589	−0.818583	−0.409291	−	0.912404i	$-0.634224\pi$
$151$	−10.0589	−0.818583	−0.409291	+	0.912404i	$0.634224\pi$
$152$	0	0
$153$	−13.2852	−1.07405
$154$	0	0
$155$	12.7383	1.02316
$156$	0	0
$157$	−19.0646	−1.52152	−0.760760	−	0.649034i	$-0.775173\pi$
$157$	−19.0646	−1.52152	−0.760760	+	0.649034i	$0.775173\pi$
$158$	0	0
$159$	1.19733	0.0949545
$160$	0	0
$161$	4.06271	0.320187
$162$	0	0
$163$	8.13247	0.636984	0.318492	−	0.947925i	$-0.396824\pi$
$163$	8.13247	0.636984	0.318492	+	0.947925i	$0.396824\pi$
$164$	0	0
$165$	0.211815	0.0164897
$166$	0	0
$167$	−6.71099	−0.519312	−0.259656	−	0.965701i	$-0.583609\pi$
$167$	−6.71099	−0.519312	−0.259656	+	0.965701i	$0.583609\pi$
$168$	0	0
$169$	28.1415	2.16473
$170$	0	0
$171$	2.98337	0.228144
$172$	0	0
$173$	−3.15719	−0.240037	−0.120018	−	0.992772i	$-0.538295\pi$
$173$	−3.15719	−0.240037	−0.120018	+	0.992772i	$0.538295\pi$
$174$	0	0
$175$	−2.59867	−0.196441
$176$	0	0
$177$	0.368078	0.0276665
$178$	0	0
$179$	3.49703	0.261380	0.130690	−	0.991423i	$-0.458281\pi$
$179$	3.49703	0.261380	0.130690	+	0.991423i	$0.458281\pi$
$180$	0	0
$181$	−8.01850	−0.596010	−0.298005	−	0.954564i	$-0.596321\pi$
$181$	−8.01850	−0.596010	−0.298005	+	0.954564i	$0.596321\pi$
$182$	0	0
$183$	1.60154	0.118389
$184$	0	0
$185$	−1.88174	−0.138348
$186$	0	0
$187$	−4.45309	−0.325642
$188$	0	0
$189$	0.871050	0.0633595
$190$	0	0
$191$	13.9558	1.00980	0.504902	−	0.863176i	$-0.331528\pi$
$191$	13.9558	1.00980	0.504902	+	0.863176i	$0.331528\pi$
$192$	0	0
$193$	1.75707	0.126477	0.0632385	−	0.997998i	$-0.479857\pi$
$193$	1.75707	0.126477	0.0632385	+	0.997998i	$0.479857\pi$
$194$	0	0
$195$	1.35861	0.0972923
$196$	0	0
$197$	7.40256	0.527410	0.263705	−	0.964603i	$-0.415055\pi$
$197$	7.40256	0.527410	0.263705	+	0.964603i	$0.415055\pi$
$198$	0	0
$199$	−26.7029	−1.89292	−0.946459	−	0.322823i	$-0.895368\pi$
$199$	−26.7029	−1.89292	−0.946459	+	0.322823i	$0.895368\pi$
$200$	0	0
$201$	−0.00594279	−0.000419172 0
$202$	0	0
$203$	−1.65924	−0.116455
$204$	0	0
$205$	−8.32184	−0.581222
$206$	0	0
$207$	−10.7362	−0.746214
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−13.4847	−0.928324	−0.464162	−	0.885750i	$-0.653645\pi$
$211$	−13.4847	−0.928324	−0.464162	+	0.885750i	$0.653645\pi$
$212$	0	0
$213$	1.87987	0.128806
$214$	0	0
$215$	0.495808	0.0338138
$216$	0	0
$217$	8.75493	0.594323
$218$	0	0
$219$	−1.13842	−0.0769270
$220$	0	0
$221$	−28.5628	−1.92134
$222$	0	0
$223$	8.04746	0.538898	0.269449	−	0.963015i	$-0.413158\pi$
$223$	8.04746	0.538898	0.269449	+	0.963015i	$0.413158\pi$
$224$	0	0
$225$	6.86725	0.457817
$226$	0	0
$227$	0.924299	0.0613479	0.0306739	−	0.999529i	$-0.490235\pi$
$227$	0.924299	0.0613479	0.0306739	+	0.999529i	$0.490235\pi$
$228$	0	0
$229$	24.3072	1.60627	0.803134	−	0.595799i	$-0.203165\pi$
$229$	24.3072	1.60627	0.803134	+	0.595799i	$0.203165\pi$
$230$	0	0
$231$	0.145578	0.00957836
$232$	0	0
$233$	0.652072	0.0427187	0.0213593	−	0.999772i	$-0.493201\pi$
$233$	0.652072	0.0427187	0.0213593	+	0.999772i	$0.493201\pi$
$234$	0	0
$235$	−14.0778	−0.918337
$236$	0	0
$237$	−1.22250	−0.0794099
$238$	0	0
$239$	−6.19074	−0.400446	−0.200223	−	0.979750i	$-0.564167\pi$
$239$	−6.19074	−0.400446	−0.200223	+	0.979750i	$0.564167\pi$
$240$	0	0
$241$	−20.7844	−1.33884	−0.669420	−	0.742884i	$-0.733457\pi$
$241$	−20.7844	−1.33884	−0.669420	+	0.742884i	$0.733457\pi$
$242$	0	0
$243$	−3.45596	−0.221700
$244$	0	0
$245$	−9.40470	−0.600844
$246$	0	0
$247$	6.41416	0.408123
$248$	0	0
$249$	0.900018	0.0570363
$250$	0	0
$251$	−10.2469	−0.646781	−0.323391	−	0.946266i	$-0.604823\pi$
$251$	−10.2469	−0.646781	−0.323391	+	0.946266i	$0.604823\pi$
$252$	0	0
$253$	−3.59867	−0.226246
$254$	0	0
$255$	−0.943229	−0.0590673
$256$	0	0
$257$	14.3125	0.892791	0.446395	−	0.894836i	$-0.352707\pi$
$257$	14.3125	0.892791	0.446395	+	0.894836i	$0.352707\pi$
$258$	0	0
$259$	−1.29330	−0.0803618
$260$	0	0
$261$	4.38471	0.271407
$262$	0	0
$263$	19.4070	1.19669	0.598343	−	0.801240i	$-0.295826\pi$
$263$	19.4070	1.19669	0.598343	+	0.801240i	$0.295826\pi$
$264$	0	0
$265$	−15.2520	−0.936920
$266$	0	0
$267$	−0.231318	−0.0141564
$268$	0	0
$269$	−8.14251	−0.496458	−0.248229	−	0.968701i	$-0.579848\pi$
$269$	−8.14251	−0.496458	−0.248229	+	0.968701i	$0.579848\pi$
$270$	0	0
$271$	−10.1268	−0.615160	−0.307580	−	0.951522i	$-0.599519\pi$
$271$	−10.1268	−0.615160	−0.307580	+	0.951522i	$0.599519\pi$
$272$	0	0
$273$	0.933764	0.0565140
$274$	0	0
$275$	2.30184	0.138806
$276$	0	0
$277$	−8.85155	−0.531838	−0.265919	−	0.963995i	$-0.585675\pi$
$277$	−8.85155	−0.531838	−0.265919	+	0.963995i	$0.585675\pi$
$278$	0	0
$279$	−23.1358	−1.38511
$280$	0	0
$281$	1.83063	0.109206	0.0546031	−	0.998508i	$-0.482611\pi$
$281$	1.83063	0.109206	0.0546031	+	0.998508i	$0.482611\pi$
$282$	0	0
$283$	−24.7267	−1.46985	−0.734925	−	0.678149i	$-0.762782\pi$
$283$	−24.7267	−1.46985	−0.734925	+	0.678149i	$0.762782\pi$
$284$	0	0
$285$	0.211815	0.0125468
$286$	0	0
$287$	−5.71953	−0.337613
$288$	0	0
$289$	2.82998	0.166470
$290$	0	0
$291$	1.00931	0.0591667
$292$	0	0
$293$	10.2887	0.601074	0.300537	−	0.953770i	$-0.402834\pi$
$293$	10.2887	0.601074	0.300537	+	0.953770i	$0.402834\pi$
$294$	0	0
$295$	−4.68869	−0.272986
$296$	0	0
$297$	−0.771557	−0.0447703
$298$	0	0
$299$	−23.0824	−1.33489
$300$	0	0
$301$	0.340765	0.0196414
$302$	0	0
$303$	−0.0294577	−0.00169230
$304$	0	0
$305$	−20.4009	−1.16815
$306$	0	0
$307$	−15.5393	−0.886876	−0.443438	−	0.896305i	$-0.646241\pi$
$307$	−15.5393	−0.886876	−0.443438	+	0.896305i	$0.646241\pi$
$308$	0	0
$309$	−1.91349	−0.108855
$310$	0	0
$311$	12.2579	0.695082	0.347541	−	0.937665i	$-0.387017\pi$
$311$	12.2579	0.695082	0.347541	+	0.937665i	$0.387017\pi$
$312$	0	0
$313$	−6.75780	−0.381974	−0.190987	−	0.981593i	$-0.561169\pi$
$313$	−6.75780	−0.381974	−0.190987	+	0.981593i	$0.561169\pi$
$314$	0	0
$315$	−5.53243	−0.311717
$316$	0	0
$317$	−8.12960	−0.456604	−0.228302	−	0.973590i	$-0.573317\pi$
$317$	−8.12960	−0.456604	−0.228302	+	0.973590i	$0.573317\pi$
$318$	0	0
$319$	1.46972	0.0822882
$320$	0	0
$321$	0.500180	0.0279173
$322$	0	0
$323$	−4.45309	−0.247776
$324$	0	0
$325$	14.7644	0.818981
$326$	0	0
$327$	0.191388	0.0105838
$328$	0	0
$329$	−9.67559	−0.533432
$330$	0	0
$331$	−27.0452	−1.48654	−0.743268	−	0.668993i	$-0.766725\pi$
$331$	−27.0452	−1.48654	−0.743268	+	0.668993i	$0.766725\pi$
$332$	0	0
$333$	3.41769	0.187288
$334$	0	0
$335$	0.0757010	0.00413599
$336$	0	0
$337$	11.1290	0.606233	0.303116	−	0.952954i	$-0.401973\pi$
$337$	11.1290	0.606233	0.303116	+	0.952954i	$0.401973\pi$
$338$	0	0
$339$	1.74546	0.0948006
$340$	0	0
$341$	−7.75493	−0.419953
$342$	0	0
$343$	−14.3664	−0.775714
$344$	0	0
$345$	−0.762250	−0.0410381
$346$	0	0
$347$	12.2541	0.657835	0.328917	−	0.944359i	$-0.393316\pi$
$347$	12.2541	0.657835	0.328917	+	0.944359i	$0.393316\pi$
$348$	0	0
$349$	23.2268	1.24330	0.621651	−	0.783295i	$-0.286462\pi$
$349$	23.2268	1.24330	0.621651	+	0.783295i	$0.286462\pi$
$350$	0	0
$351$	−4.94890	−0.264152
$352$	0	0
$353$	35.9582	1.91386	0.956932	−	0.290314i	$-0.0937596\pi$
$353$	35.9582	1.91386	0.956932	+	0.290314i	$0.0937596\pi$
$354$	0	0
$355$	−23.9463	−1.27094
$356$	0	0
$357$	−0.648273	−0.0343103
$358$	0	0
$359$	8.52147	0.449746	0.224873	−	0.974388i	$-0.427803\pi$
$359$	8.52147	0.449746	0.224873	+	0.974388i	$0.427803\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.128950	−0.00676813
$364$	0	0
$365$	14.5015	0.759042
$366$	0	0
$367$	30.0736	1.56983	0.784915	−	0.619604i	$-0.212707\pi$
$367$	30.0736	1.56983	0.784915	+	0.619604i	$0.212707\pi$
$368$	0	0
$369$	15.1145	0.786828
$370$	0	0
$371$	−10.4825	−0.544227
$372$	0	0
$373$	18.6300	0.964624	0.482312	−	0.875999i	$-0.339797\pi$
$373$	18.6300	0.964624	0.482312	+	0.875999i	$0.339797\pi$
$374$	0	0
$375$	1.54664	0.0798680
$376$	0	0
$377$	9.42699	0.485515
$378$	0	0
$379$	−19.9350	−1.02399	−0.511996	−	0.858988i	$-0.671094\pi$
$379$	−19.9350	−1.02399	−0.511996	+	0.858988i	$0.671094\pi$
$380$	0	0
$381$	1.11900	0.0573279
$382$	0	0
$383$	21.0145	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$383$	21.0145	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$384$	0	0
$385$	−1.85442	−0.0945101
$386$	0	0
$387$	−0.900508	−0.0457754
$388$	0	0
$389$	22.6077	1.14626	0.573128	−	0.819466i	$-0.305730\pi$
$389$	22.6077	1.14626	0.573128	+	0.819466i	$0.305730\pi$
$390$	0	0
$391$	16.0252	0.810428
$392$	0	0
$393$	−0.0929025	−0.00468631
$394$	0	0
$395$	15.5726	0.783541
$396$	0	0
$397$	1.76969	0.0888182	0.0444091	−	0.999013i	$-0.485859\pi$
$397$	1.76969	0.0888182	0.0444091	+	0.999013i	$0.485859\pi$
$398$	0	0
$399$	0.145578	0.00728804
$400$	0	0
$401$	−9.75643	−0.487213	−0.243606	−	0.969874i	$-0.578330\pi$
$401$	−9.75643	−0.487213	−0.243606	+	0.969874i	$0.578330\pi$
$402$	0	0
$403$	−49.7414	−2.47780
$404$	0	0
$405$	14.5381	0.722404
$406$	0	0
$407$	1.14558	0.0567842
$408$	0	0
$409$	24.0359	1.18850	0.594248	−	0.804282i	$-0.297450\pi$
$409$	24.0359	1.18850	0.594248	+	0.804282i	$0.297450\pi$
$410$	0	0
$411$	2.17921	0.107492
$412$	0	0
$413$	−3.22250	−0.158569
$414$	0	0
$415$	−11.4647	−0.562780
$416$	0	0
$417$	−1.35690	−0.0664478
$418$	0	0
$419$	11.6567	0.569465	0.284732	−	0.958607i	$-0.408095\pi$
$419$	11.6567	0.569465	0.284732	+	0.958607i	$0.408095\pi$
$420$	0	0
$421$	−23.5099	−1.14580	−0.572900	−	0.819625i	$-0.694182\pi$
$421$	−23.5099	−1.14580	−0.572900	+	0.819625i	$0.694182\pi$
$422$	0	0
$423$	25.5688	1.24320
$424$	0	0
$425$	−10.2503	−0.497213
$426$	0	0
$427$	−14.0214	−0.678542
$428$	0	0
$429$	−0.827108	−0.0399332
$430$	0	0
$431$	0.170017	0.00818943	0.00409472	−	0.999992i	$-0.498697\pi$
$431$	0.170017	0.00818943	0.00409472	+	0.999992i	$0.498697\pi$
$432$	0	0
$433$	−33.1450	−1.59285	−0.796424	−	0.604738i	$-0.793278\pi$
$433$	−33.1450	−1.59285	−0.796424	+	0.604738i	$0.793278\pi$
$434$	0	0
$435$	0.311307	0.0149260
$436$	0	0
$437$	−3.59867	−0.172148
$438$	0	0
$439$	39.9511	1.90676	0.953380	−	0.301772i	$-0.0975782\pi$
$439$	39.9511	1.90676	0.953380	+	0.301772i	$0.0975782\pi$
$440$	0	0
$441$	17.0812	0.813391
$442$	0	0
$443$	19.0639	0.905755	0.452877	−	0.891573i	$-0.350398\pi$
$443$	19.0639	0.905755	0.452877	+	0.891573i	$0.350398\pi$
$444$	0	0
$445$	2.94659	0.139682
$446$	0	0
$447$	−2.62261	−0.124045
$448$	0	0
$449$	31.6306	1.49274	0.746369	−	0.665532i	$-0.231795\pi$
$449$	31.6306	1.49274	0.746369	+	0.665532i	$0.231795\pi$
$450$	0	0
$451$	5.06624	0.238560
$452$	0	0
$453$	1.29710	0.0609431
$454$	0	0
$455$	−11.8946	−0.557626
$456$	0	0
$457$	−29.8229	−1.39506	−0.697528	−	0.716558i	$-0.745717\pi$
$457$	−29.8229	−1.39506	−0.697528	+	0.716558i	$0.745717\pi$
$458$	0	0
$459$	3.43581	0.160370
$460$	0	0
$461$	7.01972	0.326941	0.163470	−	0.986548i	$-0.447731\pi$
$461$	7.01972	0.326941	0.163470	+	0.986548i	$0.447731\pi$
$462$	0	0
$463$	5.81198	0.270105	0.135053	−	0.990838i	$-0.456880\pi$
$463$	5.81198	0.270105	0.135053	+	0.990838i	$0.456880\pi$
$464$	0	0
$465$	−1.64261	−0.0761741
$466$	0	0
$467$	−33.1745	−1.53513	−0.767566	−	0.640970i	$-0.778532\pi$
$467$	−33.1745	−1.53513	−0.767566	+	0.640970i	$0.778532\pi$
$468$	0	0
$469$	0.0520287	0.00240246
$470$	0	0
$471$	2.45838	0.113276
$472$	0	0
$473$	−0.301842	−0.0138787
$474$	0	0
$475$	2.30184	0.105616
$476$	0	0
$477$	27.7012	1.26835
$478$	0	0
$479$	23.8694	1.09062	0.545310	−	0.838234i	$-0.316412\pi$
$479$	23.8694	1.09062	0.545310	+	0.838234i	$0.316412\pi$
$480$	0	0
$481$	7.34793	0.335037
$482$	0	0
$483$	−0.523888	−0.0238377
$484$	0	0
$485$	−12.8569	−0.583800
$486$	0	0
$487$	14.9530	0.677585	0.338792	−	0.940861i	$-0.389982\pi$
$487$	14.9530	0.677585	0.338792	+	0.940861i	$0.389982\pi$
$488$	0	0
$489$	−1.04868	−0.0474231
$490$	0	0
$491$	6.23411	0.281341	0.140671	−	0.990056i	$-0.455074\pi$
$491$	6.23411	0.281341	0.140671	+	0.990056i	$0.455074\pi$
$492$	0	0
$493$	−6.54477	−0.294762
$494$	0	0
$495$	4.90051	0.220261
$496$	0	0
$497$	−16.4581	−0.738247
$498$	0	0
$499$	3.09052	0.138351	0.0691753	−	0.997605i	$-0.477963\pi$
$499$	3.09052	0.138351	0.0691753	+	0.997605i	$0.477963\pi$
$500$	0	0
$501$	0.865383	0.0386625
$502$	0	0
$503$	29.5129	1.31592	0.657959	−	0.753054i	$-0.271420\pi$
$503$	29.5129	1.31592	0.657959	+	0.753054i	$0.271420\pi$
$504$	0	0
$505$	0.375242	0.0166980
$506$	0	0
$507$	−3.62885	−0.161163
$508$	0	0
$509$	−38.5657	−1.70939	−0.854697	−	0.519126i	$-0.826257\pi$
$509$	−38.5657	−1.70939	−0.854697	+	0.519126i	$0.826257\pi$
$510$	0	0
$511$	9.96674	0.440903
$512$	0	0
$513$	−0.771557	−0.0340651
$514$	0	0
$515$	24.3747	1.07408
$516$	0	0
$517$	8.57043	0.376927
$518$	0	0
$519$	0.407120	0.0178706
$520$	0	0
$521$	−0.567553	−0.0248650	−0.0124325	−	0.999923i	$-0.503957\pi$
$521$	−0.567553	−0.0248650	−0.0124325	+	0.999923i	$0.503957\pi$
$522$	0	0
$523$	−34.2603	−1.49810	−0.749050	−	0.662513i	$-0.769490\pi$
$523$	−34.2603	−1.49810	−0.749050	+	0.662513i	$0.769490\pi$
$524$	0	0
$525$	0.335099	0.0146249
$526$	0	0
$527$	34.5334	1.50430
$528$	0	0
$529$	−10.0496	−0.436939
$530$	0	0
$531$	8.51580	0.369554
$532$	0	0
$533$	32.4957	1.40754
$534$	0	0
$535$	−6.37144	−0.275462
$536$	0	0
$537$	−0.450943	−0.0194596
$538$	0	0
$539$	5.72547	0.246614
$540$	0	0
$541$	12.8981	0.554532	0.277266	−	0.960793i	$-0.410572\pi$
$541$	12.8981	0.554532	0.277266	+	0.960793i	$0.410572\pi$
$542$	0	0
$543$	1.03399	0.0443726
$544$	0	0
$545$	−2.43796	−0.104431
$546$	0	0
$547$	−9.94028	−0.425016	−0.212508	−	0.977159i	$-0.568163\pi$
$547$	−9.94028	−0.425016	−0.212508	+	0.977159i	$0.568163\pi$
$548$	0	0
$549$	37.0530	1.58138
$550$	0	0
$551$	1.46972	0.0626120
$552$	0	0
$553$	10.7029	0.455134
$554$	0	0
$555$	0.242650	0.0102999
$556$	0	0
$557$	10.4393	0.442325	0.221163	−	0.975237i	$-0.429015\pi$
$557$	10.4393	0.442325	0.221163	+	0.975237i	$0.429015\pi$
$558$	0	0
$559$	−1.93607	−0.0818868
$560$	0	0
$561$	0.574227	0.0242439
$562$	0	0
$563$	−43.7350	−1.84321	−0.921605	−	0.388129i	$-0.873122\pi$
$563$	−43.7350	−1.84321	−0.921605	+	0.388129i	$0.873122\pi$
$564$	0	0
$565$	−22.2342	−0.935402
$566$	0	0
$567$	9.99191	0.419621
$568$	0	0
$569$	10.1041	0.423585	0.211792	−	0.977315i	$-0.432070\pi$
$569$	10.1041	0.423585	0.211792	+	0.977315i	$0.432070\pi$
$570$	0	0
$571$	−19.3939	−0.811609	−0.405805	−	0.913960i	$-0.633009\pi$
$571$	−19.3939	−0.811609	−0.405805	+	0.913960i	$0.633009\pi$
$572$	0	0
$573$	−1.79960	−0.0751794
$574$	0	0
$575$	−8.28356	−0.345448
$576$	0	0
$577$	21.1142	0.878995	0.439498	−	0.898244i	$-0.355156\pi$
$577$	21.1142	0.878995	0.439498	+	0.898244i	$0.355156\pi$
$578$	0	0
$579$	−0.226575	−0.00941614
$580$	0	0
$581$	−7.87959	−0.326901
$582$	0	0
$583$	9.28521	0.384554
$584$	0	0
$585$	31.4327	1.29958
$586$	0	0
$587$	−27.0719	−1.11738	−0.558688	−	0.829378i	$-0.688695\pi$
$587$	−27.0719	−1.11738	−0.558688	+	0.829378i	$0.688695\pi$
$588$	0	0
$589$	−7.75493	−0.319536
$590$	0	0
$591$	−0.954561	−0.0392654
$592$	0	0
$593$	−4.59235	−0.188585	−0.0942926	−	0.995545i	$-0.530059\pi$
$593$	−4.59235	−0.188585	−0.0942926	+	0.995545i	$0.530059\pi$
$594$	0	0
$595$	8.25790	0.338541
$596$	0	0
$597$	3.44335	0.140927
$598$	0	0
$599$	−33.6460	−1.37474	−0.687369	−	0.726309i	$-0.741234\pi$
$599$	−33.6460	−1.37474	−0.687369	+	0.726309i	$0.741234\pi$
$600$	0	0
$601$	4.17948	0.170485	0.0852423	−	0.996360i	$-0.472834\pi$
$601$	4.17948	0.170485	0.0852423	+	0.996360i	$0.472834\pi$
$602$	0	0
$603$	−0.137491	−0.00559908
$604$	0	0
$605$	1.64261	0.0667815
$606$	0	0
$607$	29.3635	1.19183	0.595915	−	0.803048i	$-0.296790\pi$
$607$	29.3635	1.19183	0.595915	+	0.803048i	$0.296790\pi$
$608$	0	0
$609$	0.213959	0.00867005
$610$	0	0
$611$	54.9721	2.22393
$612$	0	0
$613$	3.11569	0.125841	0.0629207	−	0.998019i	$-0.479958\pi$
$613$	3.11569	0.125841	0.0629207	+	0.998019i	$0.479958\pi$
$614$	0	0
$615$	1.07310	0.0432717
$616$	0	0
$617$	0.218978	0.00881572	0.00440786	−	0.999990i	$-0.498597\pi$
$617$	0.218978	0.00881572	0.00440786	+	0.999990i	$0.498597\pi$
$618$	0	0
$619$	−2.41267	−0.0969733	−0.0484866	−	0.998824i	$-0.515440\pi$
$619$	−2.41267	−0.0969733	−0.0484866	+	0.998824i	$0.515440\pi$
$620$	0	0
$621$	2.77658	0.111420
$622$	0	0
$623$	2.02517	0.0811367
$624$	0	0
$625$	−8.19231	−0.327692
$626$	0	0
$627$	−0.128950	−0.00514978
$628$	0	0
$629$	−5.10136	−0.203405
$630$	0	0
$631$	−35.5246	−1.41421	−0.707105	−	0.707108i	$-0.749999\pi$
$631$	−35.5246	−1.41421	−0.707105	+	0.707108i	$0.749999\pi$
$632$	0	0
$633$	1.73885	0.0691132
$634$	0	0
$635$	−14.2541	−0.565657
$636$	0	0
$637$	36.7241	1.45506
$638$	0	0
$639$	43.4923	1.72053
$640$	0	0
$641$	−2.03038	−0.0801952	−0.0400976	−	0.999196i	$-0.512767\pi$
$641$	−2.03038	−0.0801952	−0.0400976	+	0.999196i	$0.512767\pi$
$642$	0	0
$643$	−30.2091	−1.19133	−0.595665	−	0.803233i	$-0.703111\pi$
$643$	−30.2091	−1.19133	−0.595665	+	0.803233i	$0.703111\pi$
$644$	0	0
$645$	−0.0639346	−0.00251742
$646$	0	0
$647$	41.3099	1.62406	0.812031	−	0.583615i	$-0.198362\pi$
$647$	41.3099	1.62406	0.812031	+	0.583615i	$0.198362\pi$
$648$	0	0
$649$	2.85442	0.112046
$650$	0	0
$651$	−1.12895	−0.0442470
$652$	0	0
$653$	−21.9990	−0.860887	−0.430444	−	0.902617i	$-0.641643\pi$
$653$	−21.9990	−0.860887	−0.430444	+	0.902617i	$0.641643\pi$
$654$	0	0
$655$	1.18342	0.0462400
$656$	0	0
$657$	−26.3382	−1.02755
$658$	0	0
$659$	25.9710	1.01169	0.505844	−	0.862625i	$-0.331181\pi$
$659$	25.9710	1.01169	0.505844	+	0.862625i	$0.331181\pi$
$660$	0	0
$661$	−23.4291	−0.911288	−0.455644	−	0.890162i	$-0.650591\pi$
$661$	−23.4291	−0.911288	−0.455644	+	0.890162i	$0.650591\pi$
$662$	0	0
$663$	3.68318	0.143043
$664$	0	0
$665$	−1.85442	−0.0719114
$666$	0	0
$667$	−5.28901	−0.204791
$668$	0	0
$669$	−1.03772	−0.0401207
$670$	0	0
$671$	12.4198	0.479462
$672$	0	0
$673$	11.4666	0.442007	0.221003	−	0.975273i	$-0.429067\pi$
$673$	11.4666	0.442007	0.221003	+	0.975273i	$0.429067\pi$
$674$	0	0
$675$	−1.77600	−0.0683584
$676$	0	0
$677$	18.1129	0.696135	0.348068	−	0.937469i	$-0.386838\pi$
$677$	18.1129	0.696135	0.348068	+	0.937469i	$0.386838\pi$
$678$	0	0
$679$	−8.83642	−0.339111
$680$	0	0
$681$	−0.119189	−0.00456732
$682$	0	0
$683$	19.7189	0.754522	0.377261	−	0.926107i	$-0.376866\pi$
$683$	19.7189	0.754522	0.377261	+	0.926107i	$0.376866\pi$
$684$	0	0
$685$	−27.7594	−1.06063
$686$	0	0
$687$	−3.13442	−0.119586
$688$	0	0
$689$	59.5569	2.26894
$690$	0	0
$691$	40.9755	1.55878	0.779391	−	0.626538i	$-0.215529\pi$
$691$	40.9755	1.55878	0.779391	+	0.626538i	$0.215529\pi$
$692$	0	0
$693$	3.36808	0.127943
$694$	0	0
$695$	17.2846	0.655644
$696$	0	0
$697$	−22.5604	−0.854536
$698$	0	0
$699$	−0.0840848	−0.00318038
$700$	0	0
$701$	23.9781	0.905639	0.452820	−	0.891602i	$-0.350418\pi$
$701$	23.9781	0.905639	0.452820	+	0.891602i	$0.350418\pi$
$702$	0	0
$703$	1.14558	0.0432063
$704$	0	0
$705$	1.81534	0.0683697
$706$	0	0
$707$	0.257900	0.00969935
$708$	0	0
$709$	30.8255	1.15767	0.578837	−	0.815443i	$-0.303507\pi$
$709$	30.8255	1.15767	0.578837	+	0.815443i	$0.303507\pi$
$710$	0	0
$711$	−28.2836	−1.06072
$712$	0	0
$713$	27.9074	1.04514
$714$	0	0
$715$	10.5360	0.394022
$716$	0	0
$717$	0.798298	0.0298130
$718$	0	0
$719$	−37.6357	−1.40357	−0.701787	−	0.712387i	$-0.747614\pi$
$719$	−37.6357	−1.40357	−0.701787	+	0.712387i	$0.747614\pi$
$720$	0	0
$721$	16.7525	0.623896
$722$	0	0
$723$	2.68015	0.0996759
$724$	0	0
$725$	3.38305	0.125643
$726$	0	0
$727$	20.4270	0.757595	0.378798	−	0.925480i	$-0.376338\pi$
$727$	20.4270	0.757595	0.378798	+	0.925480i	$0.376338\pi$
$728$	0	0
$729$	−26.1062	−0.966897
$730$	0	0
$731$	1.34413	0.0497144
$732$	0	0
$733$	−5.72397	−0.211420	−0.105710	−	0.994397i	$-0.533711\pi$
$733$	−5.72397	−0.211420	−0.105710	+	0.994397i	$0.533711\pi$
$734$	0	0
$735$	1.21274	0.0447325
$736$	0	0
$737$	−0.0460859	−0.00169760
$738$	0	0
$739$	−5.53623	−0.203653	−0.101827	−	0.994802i	$-0.532469\pi$
$739$	−5.53623	−0.203653	−0.101827	+	0.994802i	$0.532469\pi$
$740$	0	0
$741$	−0.827108	−0.0303846
$742$	0	0
$743$	37.6794	1.38232	0.691161	−	0.722701i	$-0.257099\pi$
$743$	37.6794	1.38232	0.691161	+	0.722701i	$0.257099\pi$
$744$	0	0
$745$	33.4076	1.22396
$746$	0	0
$747$	20.8227	0.761861
$748$	0	0
$749$	−4.37904	−0.160007
$750$	0	0
$751$	52.8939	1.93012	0.965062	−	0.262021i	$-0.0843891\pi$
$751$	52.8939	1.93012	0.965062	+	0.262021i	$0.0843891\pi$
$752$	0	0
$753$	1.32135	0.0481525
$754$	0	0
$755$	−16.5228	−0.601328
$756$	0	0
$757$	−43.6222	−1.58548	−0.792738	−	0.609563i	$-0.791345\pi$
$757$	−43.6222	−1.58548	−0.792738	+	0.609563i	$0.791345\pi$
$758$	0	0
$759$	0.464049	0.0168439
$760$	0	0
$761$	−38.5928	−1.39899	−0.699494	−	0.714639i	$-0.746591\pi$
$761$	−38.5928	−1.39899	−0.699494	+	0.714639i	$0.746591\pi$
$762$	0	0
$763$	−1.67559	−0.0606603
$764$	0	0
$765$	−21.8224	−0.788990
$766$	0	0
$767$	18.3087	0.661090
$768$	0	0
$769$	−40.7832	−1.47068	−0.735340	−	0.677699i	$-0.762977\pi$
$769$	−40.7832	−1.47068	−0.735340	+	0.677699i	$0.762977\pi$
$770$	0	0
$771$	−1.84560	−0.0664678
$772$	0	0
$773$	−40.0773	−1.44148	−0.720740	−	0.693205i	$-0.756198\pi$
$773$	−40.0773	−1.44148	−0.720740	+	0.693205i	$0.756198\pi$
$774$	0	0
$775$	−17.8506	−0.641213
$776$	0	0
$777$	0.166771	0.00598289
$778$	0	0
$779$	5.06624	0.181517
$780$	0	0
$781$	14.5782	0.521650
$782$	0	0
$783$	−1.13397	−0.0405248
$784$	0	0
$785$	−31.3156	−1.11770
$786$	0	0
$787$	35.8077	1.27641	0.638204	−	0.769867i	$-0.279678\pi$
$787$	35.8077	1.27641	0.638204	+	0.769867i	$0.279678\pi$
$788$	0	0
$789$	−2.50254	−0.0890927
$790$	0	0
$791$	−15.2814	−0.543345
$792$	0	0
$793$	79.6628	2.82891
$794$	0	0
$795$	1.96674	0.0697532
$796$	0	0
$797$	−52.1278	−1.84646	−0.923230	−	0.384248i	$-0.874461\pi$
$797$	−52.1278	−1.84646	−0.923230	+	0.384248i	$0.874461\pi$
$798$	0	0
$799$	−38.1649	−1.35018
$800$	0	0
$801$	−5.35173	−0.189094
$802$	0	0
$803$	−8.82833	−0.311545
$804$	0	0
$805$	6.67344	0.235208
$806$	0	0
$807$	1.04998	0.0369610
$808$	0	0
$809$	41.1046	1.44516	0.722580	−	0.691287i	$-0.242956\pi$
$809$	41.1046	1.44516	0.722580	+	0.691287i	$0.242956\pi$
$810$	0	0
$811$	17.2595	0.606063	0.303031	−	0.952981i	$-0.402001\pi$
$811$	17.2595	0.606063	0.303031	+	0.952981i	$0.402001\pi$
$812$	0	0
$813$	1.30585	0.0457983
$814$	0	0
$815$	13.3585	0.467926
$816$	0	0
$817$	−0.301842	−0.0105601
$818$	0	0
$819$	21.6034	0.754884
$820$	0	0
$821$	12.5296	0.437287	0.218644	−	0.975805i	$-0.429837\pi$
$821$	12.5296	0.437287	0.218644	+	0.975805i	$0.429837\pi$
$822$	0	0
$823$	54.2469	1.89093	0.945465	−	0.325725i	$-0.105608\pi$
$823$	54.2469	1.89093	0.945465	+	0.325725i	$0.105608\pi$
$824$	0	0
$825$	−0.296823	−0.0103341
$826$	0	0
$827$	46.2868	1.60955	0.804775	−	0.593580i	$-0.202286\pi$
$827$	46.2868	1.60955	0.804775	+	0.593580i	$0.202286\pi$
$828$	0	0
$829$	8.74332	0.303668	0.151834	−	0.988406i	$-0.451482\pi$
$829$	8.74332	0.303668	0.151834	+	0.988406i	$0.451482\pi$
$830$	0	0
$831$	1.14141	0.0395950
$832$	0	0
$833$	−25.4960	−0.883385
$834$	0	0
$835$	−11.0235	−0.381484
$836$	0	0
$837$	5.98337	0.206816
$838$	0	0
$839$	25.9376	0.895464	0.447732	−	0.894168i	$-0.352232\pi$
$839$	25.9376	0.895464	0.447732	+	0.894168i	$0.352232\pi$
$840$	0	0
$841$	−26.8399	−0.925515
$842$	0	0
$843$	−0.236060	−0.00813035
$844$	0	0
$845$	46.2254	1.59020
$846$	0	0
$847$	1.12895	0.0387912
$848$	0	0
$849$	3.18851	0.109429
$850$	0	0
$851$	−4.12255	−0.141319
$852$	0	0
$853$	33.0762	1.13251	0.566254	−	0.824231i	$-0.308392\pi$
$853$	33.0762	1.13251	0.566254	+	0.824231i	$0.308392\pi$
$854$	0	0
$855$	4.90051	0.167594
$856$	0	0
$857$	−43.1583	−1.47426	−0.737130	−	0.675750i	$-0.763820\pi$
$857$	−43.1583	−1.47426	−0.737130	+	0.675750i	$0.763820\pi$
$858$	0	0
$859$	−47.6814	−1.62687	−0.813434	−	0.581657i	$-0.802405\pi$
$859$	−47.6814	−1.62687	−0.813434	+	0.581657i	$0.802405\pi$
$860$	0	0
$861$	0.737535	0.0251351
$862$	0	0
$863$	−14.9865	−0.510148	−0.255074	−	0.966922i	$-0.582100\pi$
$863$	−14.9865	−0.510148	−0.255074	+	0.966922i	$0.582100\pi$
$864$	0	0
$865$	−5.18602	−0.176330
$866$	0	0
$867$	−0.364927	−0.0123936
$868$	0	0
$869$	−9.48040	−0.321601
$870$	0	0
$871$	−0.295603	−0.0100161
$872$	0	0
$873$	23.3512	0.790317
$874$	0	0
$875$	−13.5407	−0.457759
$876$	0	0
$877$	8.73680	0.295021	0.147510	−	0.989060i	$-0.452874\pi$
$877$	8.73680	0.295021	0.147510	+	0.989060i	$0.452874\pi$
$878$	0	0
$879$	−1.32674	−0.0447497
$880$	0	0
$881$	−44.9728	−1.51517	−0.757587	−	0.652735i	$-0.773622\pi$
$881$	−44.9728	−1.51517	−0.757587	+	0.652735i	$0.773622\pi$
$882$	0	0
$883$	12.0660	0.406052	0.203026	−	0.979173i	$-0.434922\pi$
$883$	12.0660	0.406052	0.203026	+	0.979173i	$0.434922\pi$
$884$	0	0
$885$	0.604608	0.0203237
$886$	0	0
$887$	−1.92345	−0.0645831	−0.0322916	−	0.999478i	$-0.510281\pi$
$887$	−1.92345	−0.0645831	−0.0322916	+	0.999478i	$0.510281\pi$
$888$	0	0
$889$	−9.79673	−0.328572
$890$	0	0
$891$	−8.85062	−0.296507
$892$	0	0
$893$	8.57043	0.286798
$894$	0	0
$895$	5.74424	0.192009
$896$	0	0
$897$	2.97649	0.0993819
$898$	0	0
$899$	−11.3975	−0.380129
$900$	0	0
$901$	−41.3479	−1.37750
$902$	0	0
$903$	−0.0439417	−0.00146229
$904$	0	0
$905$	−13.1712	−0.437827
$906$	0	0
$907$	−12.3904	−0.411416	−0.205708	−	0.978613i	$-0.565950\pi$
$907$	−12.3904	−0.411416	−0.205708	+	0.978613i	$0.565950\pi$
$908$	0	0
$909$	−0.681530	−0.0226049
$910$	0	0
$911$	−14.3785	−0.476381	−0.238190	−	0.971219i	$-0.576554\pi$
$911$	−14.3785	−0.476381	−0.238190	+	0.971219i	$0.576554\pi$
$912$	0	0
$913$	6.97957	0.230990
$914$	0	0
$915$	2.63070	0.0869683
$916$	0	0
$917$	0.813355	0.0268593
$918$	0	0
$919$	−31.2256	−1.03004	−0.515019	−	0.857179i	$-0.672215\pi$
$919$	−31.2256	−1.03004	−0.515019	+	0.857179i	$0.672215\pi$
$920$	0	0
$921$	2.00380	0.0660274
$922$	0	0
$923$	93.5072	3.07783
$924$	0	0
$925$	2.63694	0.0867021
$926$	0	0
$927$	−44.2703	−1.45403
$928$	0	0
$929$	15.3156	0.502487	0.251244	−	0.967924i	$-0.419160\pi$
$929$	15.3156	0.502487	0.251244	+	0.967924i	$0.419160\pi$
$930$	0	0
$931$	5.72547	0.187645
$932$	0	0
$933$	−1.58066	−0.0517485
$934$	0	0
$935$	−7.31467	−0.239215
$936$	0	0
$937$	42.6588	1.39360	0.696801	−	0.717264i	$-0.254606\pi$
$937$	42.6588	1.39360	0.696801	+	0.717264i	$0.254606\pi$
$938$	0	0
$939$	0.871420	0.0284377
$940$	0	0
$941$	16.0798	0.524186	0.262093	−	0.965043i	$-0.415587\pi$
$941$	16.0798	0.524186	0.262093	+	0.965043i	$0.415587\pi$
$942$	0	0
$943$	−18.2317	−0.593705
$944$	0	0
$945$	1.43079	0.0465437
$946$	0	0
$947$	29.1399	0.946921	0.473460	−	0.880815i	$-0.343005\pi$
$947$	29.1399	0.946921	0.473460	+	0.880815i	$0.343005\pi$
$948$	0	0
$949$	−56.6263	−1.83817
$950$	0	0
$951$	1.04831	0.0339939
$952$	0	0
$953$	−32.2868	−1.04587	−0.522936	−	0.852372i	$-0.675163\pi$
$953$	−32.2868	−1.04587	−0.522936	+	0.852372i	$0.675163\pi$
$954$	0	0
$955$	22.9239	0.741799
$956$	0	0
$957$	−0.189520	−0.00612632
$958$	0	0
$959$	−19.0788	−0.616086
$960$	0	0
$961$	29.1389	0.939965
$962$	0	0
$963$	11.5721	0.372905
$964$	0	0
$965$	2.88618	0.0929095
$966$	0	0
$967$	11.8637	0.381512	0.190756	−	0.981638i	$-0.438906\pi$
$967$	11.8637	0.381512	0.190756	+	0.981638i	$0.438906\pi$
$968$	0	0
$969$	0.574227	0.0184468
$970$	0	0
$971$	40.3560	1.29509	0.647543	−	0.762029i	$-0.275796\pi$
$971$	40.3560	1.29509	0.647543	+	0.762029i	$0.275796\pi$
$972$	0	0
$973$	11.8796	0.380842
$974$	0	0
$975$	−1.90387	−0.0609727
$976$	0	0
$977$	−43.8338	−1.40237	−0.701184	−	0.712980i	$-0.747345\pi$
$977$	−43.8338	−1.40237	−0.701184	+	0.712980i	$0.747345\pi$
$978$	0	0
$979$	−1.79385	−0.0573317
$980$	0	0
$981$	4.42792	0.141373
$982$	0	0
$983$	−30.7890	−0.982016	−0.491008	−	0.871155i	$-0.663371\pi$
$983$	−30.7890	−0.982016	−0.491008	+	0.871155i	$0.663371\pi$
$984$	0	0
$985$	12.1595	0.387434
$986$	0	0
$987$	1.24767	0.0397138
$988$	0	0
$989$	1.08623	0.0345401
$990$	0	0
$991$	11.7045	0.371805	0.185902	−	0.982568i	$-0.440479\pi$
$991$	11.7045	0.371805	0.185902	+	0.982568i	$0.440479\pi$
$992$	0	0
$993$	3.48748	0.110672
$994$	0	0
$995$	−43.8624	−1.39053
$996$	0	0
$997$	58.6744	1.85824	0.929119	−	0.369781i	$-0.120567\pi$
$997$	58.6744	1.85824	0.929119	+	0.369781i	$0.120567\pi$
$998$	0	0
$999$	−0.883879	−0.0279647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.s.1.2		4
4.3	odd	2		1672.2.a.f.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.f.1.3	✓	4	4.3	odd	2
3344.2.a.s.1.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-0.128950 q^{3} +1.64261 q^{5} +1.12895 q^{7} -2.98337 q^{9} +O(q^{10})\)	\(q-0.128950 q^{3} +1.64261 q^{5} +1.12895 q^{7} -2.98337 q^{9} -1.00000 q^{11} -6.41416 q^{13} -0.211815 q^{15} +4.45309 q^{17} -1.00000 q^{19} -0.145578 q^{21} +3.59867 q^{23} -2.30184 q^{25} +0.771557 q^{27} -1.46972 q^{29} +7.75493 q^{31} +0.128950 q^{33} +1.85442 q^{35} -1.14558 q^{37} +0.827108 q^{39} -5.06624 q^{41} +0.301842 q^{43} -4.90051 q^{45} -8.57043 q^{47} -5.72547 q^{49} -0.574227 q^{51} -9.28521 q^{53} -1.64261 q^{55} +0.128950 q^{57} -2.85442 q^{59} -12.4198 q^{61} -3.36808 q^{63} -10.5360 q^{65} +0.0460859 q^{67} -0.464049 q^{69} -14.5782 q^{71} +8.82833 q^{73} +0.296823 q^{75} -1.12895 q^{77} +9.48040 q^{79} +8.85062 q^{81} -6.97957 q^{83} +7.31467 q^{85} +0.189520 q^{87} +1.79385 q^{89} -7.24127 q^{91} -1.00000 q^{93} -1.64261 q^{95} -7.82711 q^{97} +2.98337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) \) 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9	\( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41} - 5 q^{43} - 7 q^{45} + 4 q^{47} - 13 q^{49} - 4 q^{51} - 18 q^{53} + 3 q^{55} - q^{57} - 6 q^{61} + 6 q^{63} - 24 q^{65} - q^{67} - 20 q^{69} + q^{71} - 6 q^{73} + q^{75} - 3 q^{77} + 4 q^{79} - 16 q^{81} + 25 q^{83} - 4 q^{85} + 9 q^{87} - 14 q^{89} - 13 q^{91} - 4 q^{93} + 3 q^{95} - 36 q^{97} - q^{99}+O(q^{100}) \) 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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