LMFDB - Embedded newform 3344.2.a.p.1.3

Error: no document with id 251452352 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3344,2,Mod(1,3344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [3,0,-1,0,5,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.16425$ 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)

$f(q)$	$=$	$q+2.16425 q^{3} +4.16425 q^{5} +0.683969 q^{7} +1.68397 q^{9} +1.00000 q^{11} +2.68397 q^{13} +9.01247 q^{15} +3.36794 q^{17} +1.00000 q^{19} +1.48028 q^{21} +1.36794 q^{23} +12.3410 q^{25} -2.84822 q^{27} +2.79631 q^{29} -5.01247 q^{31} +2.16425 q^{33} +2.84822 q^{35} -11.6964 q^{37} +5.80877 q^{39} -7.01247 q^{41} -7.86068 q^{43} +7.01247 q^{45} -2.96056 q^{47} -6.53219 q^{49} +7.28905 q^{51} -10.3285 q^{53} +4.16425 q^{55} +2.16425 q^{57} +9.92112 q^{59} +2.00000 q^{61} +1.15178 q^{63} +11.1767 q^{65} +0.683969 q^{67} +2.96056 q^{69} -3.53219 q^{71} +0.407381 q^{73} +26.7089 q^{75} +0.683969 q^{77} +7.28905 q^{79} -11.2162 q^{81} +12.1892 q^{83} +14.0249 q^{85} +6.05191 q^{87} -16.6819 q^{89} +1.83575 q^{91} -10.8482 q^{93} +4.16425 q^{95} -11.6964 q^{97} +1.68397 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 5 q^{13} + 9 q^{15} + 4 q^{17} + 3 q^{19} - 2 q^{23} + 4 q^{25} + 2 q^{27} + 7 q^{29} + 3 q^{31} - q^{33} - 2 q^{35} - 14 q^{37} - 2 q^{39} - 3 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q - q^3 + 5 * q^5 - q^7 + 2 * q^9 + 3 * q^11 + 5 * q^13 + 9 * q^15 + 4 * q^17 + 3 * q^19 - 2 * q^23 + 4 * q^25 + 2 * q^27 + 7 * q^29 + 3 * q^31 - q^33 - 2 * q^35 - 14 * q^37 - 2 * q^39 - 3 * q^41 + 5 * q^43 + 3 * q^45 - 6 * q^49 - 2 * q^51 - 16 * q^53 + 5 * q^55 - q^57 + 12 * q^59 + 6 * q^61 + 14 * q^63 + 8 * q^65 - q^67 + 3 * q^71 + 4 * q^73 + 41 * q^75 - q^77 - 2 * q^79 - 17 * q^81 - 7 * q^83 + 6 * q^85 + 9 * q^87 + 16 * q^89 + 13 * q^91 - 22 * q^93 + 5 * q^95 - 14 * q^97 + 2 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.16425	1.24953	0.624765	−	0.780813i	$-0.285195\pi$
$3$	2.16425	1.24953	0.624765	+	0.780813i	$0.285195\pi$
$4$	0	0
$5$	4.16425	1.86231	0.931154	−	0.364626i	$-0.118803\pi$
$5$	4.16425	1.86231	0.931154	+	0.364626i	$0.118803\pi$
$6$	0	0
$7$	0.683969	0.258516	0.129258	−	0.991611i	$-0.458740\pi$
$7$	0.683969	0.258516	0.129258	+	0.991611i	$0.458740\pi$
$8$	0	0
$9$	1.68397	0.561323
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.68397	0.744399	0.372200	−	0.928153i	$-0.378604\pi$
$13$	2.68397	0.744399	0.372200	+	0.928153i	$0.378604\pi$
$14$	0	0
$15$	9.01247	2.32701
$16$	0	0
$17$	3.36794	0.816845	0.408423	−	0.912793i	$-0.366079\pi$
$17$	3.36794	0.816845	0.408423	+	0.912793i	$0.366079\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.48028	0.323023
$22$	0	0
$23$	1.36794	0.285235	0.142617	−	0.989778i	$-0.454448\pi$
$23$	1.36794	0.285235	0.142617	+	0.989778i	$0.454448\pi$
$24$	0	0
$25$	12.3410	2.46819
$26$	0	0
$27$	−2.84822	−0.548140
$28$	0	0
$29$	2.79631	0.519262	0.259631	−	0.965708i	$-0.416399\pi$
$29$	2.79631	0.519262	0.259631	+	0.965708i	$0.416399\pi$
$30$	0	0
$31$	−5.01247	−0.900265	−0.450133	−	0.892962i	$-0.648623\pi$
$31$	−5.01247	−0.900265	−0.450133	+	0.892962i	$0.648623\pi$
$32$	0	0
$33$	2.16425	0.376747
$34$	0	0
$35$	2.84822	0.481437
$36$	0	0
$37$	−11.6964	−1.92288	−0.961441	−	0.275011i	$-0.911318\pi$
$37$	−11.6964	−1.92288	−0.961441	+	0.275011i	$0.911318\pi$
$38$	0	0
$39$	5.80877	0.930148
$40$	0	0
$41$	−7.01247	−1.09516	−0.547582	−	0.836752i	$-0.684451\pi$
$41$	−7.01247	−1.09516	−0.547582	+	0.836752i	$0.684451\pi$
$42$	0	0
$43$	−7.86068	−1.19874	−0.599371	−	0.800471i	$-0.704583\pi$
$43$	−7.86068	−1.19874	−0.599371	+	0.800471i	$0.704583\pi$
$44$	0	0
$45$	7.01247	1.04536
$46$	0	0
$47$	−2.96056	−0.431842	−0.215921	−	0.976411i	$-0.569275\pi$
$47$	−2.96056	−0.431842	−0.215921	+	0.976411i	$0.569275\pi$
$48$	0	0
$49$	−6.53219	−0.933169
$50$	0	0
$51$	7.28905	1.02067
$52$	0	0
$53$	−10.3285	−1.41873	−0.709364	−	0.704842i	$-0.751018\pi$
$53$	−10.3285	−1.41873	−0.709364	+	0.704842i	$0.751018\pi$
$54$	0	0
$55$	4.16425	0.561507
$56$	0	0
$57$	2.16425	0.286662
$58$	0	0
$59$	9.92112	1.29162	0.645810	−	0.763499i	$-0.276520\pi$
$59$	9.92112	1.29162	0.645810	+	0.763499i	$0.276520\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	1.15178	0.145111
$64$	0	0
$65$	11.1767	1.38630
$66$	0	0
$67$	0.683969	0.0835601	0.0417801	−	0.999127i	$-0.486697\pi$
$67$	0.683969	0.0835601	0.0417801	+	0.999127i	$0.486697\pi$
$68$	0	0
$69$	2.96056	0.356409
$70$	0	0
$71$	−3.53219	−0.419193	−0.209597	−	0.977788i	$-0.567215\pi$
$71$	−3.53219	−0.419193	−0.209597	+	0.977788i	$0.567215\pi$
$72$	0	0
$73$	0.407381	0.0476803	0.0238402	−	0.999716i	$-0.492411\pi$
$73$	0.407381	0.0476803	0.0238402	+	0.999716i	$0.492411\pi$
$74$	0	0
$75$	26.7089	3.08408
$76$	0	0
$77$	0.683969	0.0779455
$78$	0	0
$79$	7.28905	0.820083	0.410041	−	0.912067i	$-0.365514\pi$
$79$	7.28905	0.820083	0.410041	+	0.912067i	$0.365514\pi$
$80$	0	0
$81$	−11.2162	−1.24624
$82$	0	0
$83$	12.1892	1.33794	0.668968	−	0.743291i	$-0.266736\pi$
$83$	12.1892	1.33794	0.668968	+	0.743291i	$0.266736\pi$
$84$	0	0
$85$	14.0249	1.52122
$86$	0	0
$87$	6.05191	0.648833
$88$	0	0
$89$	−16.6819	−1.76828	−0.884140	−	0.467222i	$-0.845255\pi$
$89$	−16.6819	−1.76828	−0.884140	+	0.467222i	$0.845255\pi$
$90$	0	0
$91$	1.83575	0.192439
$92$	0	0
$93$	−10.8482	−1.12491
$94$	0	0
$95$	4.16425	0.427243
$96$	0	0
$97$	−11.6964	−1.18759	−0.593796	−	0.804615i	$-0.702372\pi$
$97$	−11.6964	−1.18759	−0.593796	+	0.804615i	$0.702372\pi$
$98$	0	0
$99$	1.68397	0.169245
$100$	0	0
$101$	3.59262	0.357479	0.178739	−	0.983896i	$-0.442798\pi$
$101$	3.59262	0.357479	0.178739	+	0.983896i	$0.442798\pi$
$102$	0	0
$103$	4.46781	0.440227	0.220113	−	0.975474i	$-0.429357\pi$
$103$	4.46781	0.440227	0.220113	+	0.975474i	$0.429357\pi$
$104$	0	0
$105$	6.16425	0.601569
$106$	0	0
$107$	−9.69643	−0.937390	−0.468695	−	0.883360i	$-0.655276\pi$
$107$	−9.69643	−0.937390	−0.468695	+	0.883360i	$0.655276\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−25.3140	−2.40270
$112$	0	0
$113$	10.4323	0.981389	0.490695	−	0.871332i	$-0.336743\pi$
$113$	10.4323	0.981389	0.490695	+	0.871332i	$0.336743\pi$
$114$	0	0
$115$	5.69643	0.531195
$116$	0	0
$117$	4.51972	0.417848
$118$	0	0
$119$	2.30357	0.211168
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−15.1767	−1.36844
$124$	0	0
$125$	30.5696	2.73423
$126$	0	0
$127$	−15.2891	−1.35668	−0.678342	−	0.734746i	$-0.737301\pi$
$127$	−15.2891	−1.35668	−0.678342	+	0.734746i	$0.737301\pi$
$128$	0	0
$129$	−17.0125	−1.49786
$130$	0	0
$131$	−3.64453	−0.318424	−0.159212	−	0.987244i	$-0.550895\pi$
$131$	−3.64453	−0.318424	−0.159212	+	0.987244i	$0.550895\pi$
$132$	0	0
$133$	0.683969	0.0593076
$134$	0	0
$135$	−11.8607	−1.02080
$136$	0	0
$137$	14.6840	1.25454	0.627268	−	0.778803i	$-0.284173\pi$
$137$	14.6840	1.25454	0.627268	+	0.778803i	$0.284173\pi$
$138$	0	0
$139$	18.5966	1.57734	0.788670	−	0.614817i	$-0.210770\pi$
$139$	18.5966	1.57734	0.788670	+	0.614817i	$0.210770\pi$
$140$	0	0
$141$	−6.40738	−0.539599
$142$	0	0
$143$	2.68397	0.224445
$144$	0	0
$145$	11.6445	0.967025
$146$	0	0
$147$	−14.1373	−1.16602
$148$	0	0
$149$	4.30357	0.352562	0.176281	−	0.984340i	$-0.443593\pi$
$149$	4.30357	0.352562	0.176281	+	0.984340i	$0.443593\pi$
$150$	0	0
$151$	21.6964	1.76563	0.882815	−	0.469720i	$-0.155645\pi$
$151$	21.6964	1.76563	0.882815	+	0.469720i	$0.155645\pi$
$152$	0	0
$153$	5.67150	0.458514
$154$	0	0
$155$	−20.8731	−1.67657
$156$	0	0
$157$	−10.9001	−0.869925	−0.434962	−	0.900449i	$-0.643238\pi$
$157$	−10.9001	−0.869925	−0.434962	+	0.900449i	$0.643238\pi$
$158$	0	0
$159$	−22.3534	−1.77274
$160$	0	0
$161$	0.935628	0.0737378
$162$	0	0
$163$	−19.7214	−1.54470	−0.772348	−	0.635199i	$-0.780918\pi$
$163$	−19.7214	−1.54470	−0.772348	+	0.635199i	$0.780918\pi$
$164$	0	0
$165$	9.01247	0.701619
$166$	0	0
$167$	−7.77532	−0.601672	−0.300836	−	0.953676i	$-0.597266\pi$
$167$	−7.77532	−0.601672	−0.300836	+	0.953676i	$0.597266\pi$
$168$	0	0
$169$	−5.79631	−0.445870
$170$	0	0
$171$	1.68397	0.128776
$172$	0	0
$173$	15.7818	1.19987	0.599934	−	0.800050i	$-0.295194\pi$
$173$	15.7818	1.19987	0.599934	+	0.800050i	$0.295194\pi$
$174$	0	0
$175$	8.44084	0.638067
$176$	0	0
$177$	21.4718	1.61392
$178$	0	0
$179$	7.97302	0.595932	0.297966	−	0.954577i	$-0.403692\pi$
$179$	7.97302	0.595932	0.297966	+	0.954577i	$0.403692\pi$
$180$	0	0
$181$	−2.55318	−0.189776	−0.0948881	−	0.995488i	$-0.530249\pi$
$181$	−2.55318	−0.189776	−0.0948881	+	0.995488i	$0.530249\pi$
$182$	0	0
$183$	4.32850	0.319972
$184$	0	0
$185$	−48.7069	−3.58100
$186$	0	0
$187$	3.36794	0.246288
$188$	0	0
$189$	−1.94809	−0.141703
$190$	0	0
$191$	−8.22468	−0.595117	−0.297559	−	0.954704i	$-0.596172\pi$
$191$	−8.22468	−0.595117	−0.297559	+	0.954704i	$0.596172\pi$
$192$	0	0
$193$	−24.2141	−1.74297	−0.871485	−	0.490422i	$-0.836842\pi$
$193$	−24.2141	−1.74297	−0.871485	+	0.490422i	$0.836842\pi$
$194$	0	0
$195$	24.1892	1.73222
$196$	0	0
$197$	26.6570	1.89923	0.949616	−	0.313416i	$-0.101473\pi$
$197$	26.6570	1.89923	0.949616	+	0.313416i	$0.101473\pi$
$198$	0	0
$199$	7.06437	0.500780	0.250390	−	0.968145i	$-0.419441\pi$
$199$	7.06437	0.500780	0.250390	+	0.968145i	$0.419441\pi$
$200$	0	0
$201$	1.48028	0.104411
$202$	0	0
$203$	1.91259	0.134237
$204$	0	0
$205$	−29.2016	−2.03953
$206$	0	0
$207$	2.30357	0.160109
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	11.0644	0.761703	0.380851	−	0.924636i	$-0.375631\pi$
$211$	11.0644	0.761703	0.380851	+	0.924636i	$0.375631\pi$
$212$	0	0
$213$	−7.64453	−0.523794
$214$	0	0
$215$	−32.7338	−2.23243
$216$	0	0
$217$	−3.42837	−0.232733
$218$	0	0
$219$	0.881673	0.0595779
$220$	0	0
$221$	9.03944	0.608059
$222$	0	0
$223$	14.7359	0.986787	0.493394	−	0.869806i	$-0.335756\pi$
$223$	14.7359	0.986787	0.493394	+	0.869806i	$0.335756\pi$
$224$	0	0
$225$	20.7818	1.38545
$226$	0	0
$227$	−21.8002	−1.44693	−0.723467	−	0.690359i	$-0.757452\pi$
$227$	−21.8002	−1.44693	−0.723467	+	0.690359i	$0.757452\pi$
$228$	0	0
$229$	17.7483	1.17284	0.586422	−	0.810006i	$-0.300536\pi$
$229$	17.7483	1.17284	0.586422	+	0.810006i	$0.300536\pi$
$230$	0	0
$231$	1.48028	0.0973952
$232$	0	0
$233$	−20.3534	−1.33340	−0.666699	−	0.745327i	$-0.732293\pi$
$233$	−20.3534	−1.33340	−0.666699	+	0.745327i	$0.732293\pi$
$234$	0	0
$235$	−12.3285	−0.804222
$236$	0	0
$237$	15.7753	1.02472
$238$	0	0
$239$	−3.75687	−0.243012	−0.121506	−	0.992591i	$-0.538772\pi$
$239$	−3.75687	−0.243012	−0.121506	+	0.992591i	$0.538772\pi$
$240$	0	0
$241$	−9.14974	−0.589386	−0.294693	−	0.955592i	$-0.595217\pi$
$241$	−9.14974	−0.589386	−0.294693	+	0.955592i	$0.595217\pi$
$242$	0	0
$243$	−15.7299	−1.00907
$244$	0	0
$245$	−27.2016	−1.73785
$246$	0	0
$247$	2.68397	0.170777
$248$	0	0
$249$	26.3804	1.67179
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	1.36794	0.0860015
$254$	0	0
$255$	30.3534	1.90081
$256$	0	0
$257$	23.2102	1.44781	0.723905	−	0.689899i	$-0.242345\pi$
$257$	23.2102	1.44781	0.723905	+	0.689899i	$0.242345\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	4.70890	0.291474
$262$	0	0
$263$	22.2141	1.36978	0.684890	−	0.728646i	$-0.259850\pi$
$263$	22.2141	1.36978	0.684890	+	0.728646i	$0.259850\pi$
$264$	0	0
$265$	−43.0104	−2.64211
$266$	0	0
$267$	−36.1038	−2.20952
$268$	0	0
$269$	−18.9855	−1.15757	−0.578783	−	0.815482i	$-0.696472\pi$
$269$	−18.9855	−1.15757	−0.578783	+	0.815482i	$0.696472\pi$
$270$	0	0
$271$	−20.7338	−1.25949	−0.629745	−	0.776802i	$-0.716841\pi$
$271$	−20.7338	−1.25949	−0.629745	+	0.776802i	$0.716841\pi$
$272$	0	0
$273$	3.97302	0.240458
$274$	0	0
$275$	12.3410	0.744188
$276$	0	0
$277$	21.8422	1.31237	0.656186	−	0.754599i	$-0.272169\pi$
$277$	21.8422	1.31237	0.656186	+	0.754599i	$0.272169\pi$
$278$	0	0
$279$	−8.44084	−0.505340
$280$	0	0
$281$	0.164248	0.00979821	0.00489911	−	0.999988i	$-0.498441\pi$
$281$	0.164248	0.00979821	0.00489911	+	0.999988i	$0.498441\pi$
$282$	0	0
$283$	−4.30152	−0.255699	−0.127849	−	0.991794i	$-0.540807\pi$
$283$	−4.30152	−0.255699	−0.127849	+	0.991794i	$0.540807\pi$
$284$	0	0
$285$	9.01247	0.533852
$286$	0	0
$287$	−4.79631	−0.283117
$288$	0	0
$289$	−5.65699	−0.332764
$290$	0	0
$291$	−25.3140	−1.48393
$292$	0	0
$293$	32.5511	1.90166	0.950829	−	0.309717i	$-0.100234\pi$
$293$	32.5511	1.90166	0.950829	+	0.309717i	$0.100234\pi$
$294$	0	0
$295$	41.3140	2.40539
$296$	0	0
$297$	−2.84822	−0.165270
$298$	0	0
$299$	3.67150	0.212329
$300$	0	0
$301$	−5.37646	−0.309894
$302$	0	0
$303$	7.77532	0.446680
$304$	0	0
$305$	8.32850	0.476888
$306$	0	0
$307$	−22.2496	−1.26985	−0.634926	−	0.772573i	$-0.718970\pi$
$307$	−22.2496	−1.26985	−0.634926	+	0.772573i	$0.718970\pi$
$308$	0	0
$309$	9.66946	0.550076
$310$	0	0
$311$	20.9855	1.18998	0.594989	−	0.803734i	$-0.297156\pi$
$311$	20.9855	1.18998	0.594989	+	0.803734i	$0.297156\pi$
$312$	0	0
$313$	−11.4533	−0.647379	−0.323689	−	0.946163i	$-0.604923\pi$
$313$	−11.4533	−0.647379	−0.323689	+	0.946163i	$0.604923\pi$
$314$	0	0
$315$	4.79631	0.270241
$316$	0	0
$317$	10.7109	0.601587	0.300793	−	0.953689i	$-0.402749\pi$
$317$	10.7109	0.601587	0.300793	+	0.953689i	$0.402749\pi$
$318$	0	0
$319$	2.79631	0.156563
$320$	0	0
$321$	−20.9855	−1.17130
$322$	0	0
$323$	3.36794	0.187397
$324$	0	0
$325$	33.1228	1.83732
$326$	0	0
$327$	12.9855	0.718099
$328$	0	0
$329$	−2.02493	−0.111638
$330$	0	0
$331$	6.77138	0.372189	0.186094	−	0.982532i	$-0.440417\pi$
$331$	6.77138	0.372189	0.186094	+	0.982532i	$0.440417\pi$
$332$	0	0
$333$	−19.6964	−1.07936
$334$	0	0
$335$	2.84822	0.155615
$336$	0	0
$337$	21.0374	1.14598	0.572990	−	0.819562i	$-0.305783\pi$
$337$	21.0374	1.14598	0.572990	+	0.819562i	$0.305783\pi$
$338$	0	0
$339$	22.5781	1.22627
$340$	0	0
$341$	−5.01247	−0.271440
$342$	0	0
$343$	−9.25560	−0.499755
$344$	0	0
$345$	12.3285	0.663744
$346$	0	0
$347$	29.9710	1.60893	0.804463	−	0.594003i	$-0.202453\pi$
$347$	29.9710	1.60893	0.804463	+	0.594003i	$0.202453\pi$
$348$	0	0
$349$	−22.6570	−1.21280	−0.606400	−	0.795159i	$-0.707387\pi$
$349$	−22.6570	−1.21280	−0.606400	+	0.795159i	$0.707387\pi$
$350$	0	0
$351$	−7.64453	−0.408035
$352$	0	0
$353$	−1.18524	−0.0630839	−0.0315419	−	0.999502i	$-0.510042\pi$
$353$	−1.18524	−0.0630839	−0.0315419	+	0.999502i	$0.510042\pi$
$354$	0	0
$355$	−14.7089	−0.780667
$356$	0	0
$357$	4.98549	0.263860
$358$	0	0
$359$	−19.3659	−1.02209	−0.511046	−	0.859553i	$-0.670742\pi$
$359$	−19.3659	−1.02209	−0.511046	+	0.859553i	$0.670742\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.16425	0.113594
$364$	0	0
$365$	1.69643	0.0887954
$366$	0	0
$367$	18.3036	0.955438	0.477719	−	0.878513i	$-0.341464\pi$
$367$	18.3036	0.955438	0.477719	+	0.878513i	$0.341464\pi$
$368$	0	0
$369$	−11.8088	−0.614740
$370$	0	0
$371$	−7.06437	−0.366764
$372$	0	0
$373$	6.46781	0.334891	0.167445	−	0.985881i	$-0.446448\pi$
$373$	6.46781	0.334891	0.167445	+	0.985881i	$0.446448\pi$
$374$	0	0
$375$	66.1602	3.41650
$376$	0	0
$377$	7.50521	0.386538
$378$	0	0
$379$	−31.9191	−1.63957	−0.819786	−	0.572670i	$-0.805908\pi$
$379$	−31.9191	−1.63957	−0.819786	+	0.572670i	$0.805908\pi$
$380$	0	0
$381$	−33.0893	−1.69522
$382$	0	0
$383$	−5.66946	−0.289696	−0.144848	−	0.989454i	$-0.546269\pi$
$383$	−5.66946	−0.289696	−0.144848	+	0.989454i	$0.546269\pi$
$384$	0	0
$385$	2.84822	0.145159
$386$	0	0
$387$	−13.2371	−0.672882
$388$	0	0
$389$	20.1642	1.02237	0.511184	−	0.859471i	$-0.329207\pi$
$389$	20.1642	1.02237	0.511184	+	0.859471i	$0.329207\pi$
$390$	0	0
$391$	4.60713	0.232993
$392$	0	0
$393$	−7.88766	−0.397880
$394$	0	0
$395$	30.3534	1.52725
$396$	0	0
$397$	−15.8272	−0.794346	−0.397173	−	0.917744i	$-0.630009\pi$
$397$	−15.8272	−0.794346	−0.397173	+	0.917744i	$0.630009\pi$
$398$	0	0
$399$	1.48028	0.0741066
$400$	0	0
$401$	9.06437	0.452653	0.226327	−	0.974051i	$-0.427328\pi$
$401$	9.06437	0.452653	0.226327	+	0.974051i	$0.427328\pi$
$402$	0	0
$403$	−13.4533	−0.670157
$404$	0	0
$405$	−46.7069	−2.32088
$406$	0	0
$407$	−11.6964	−0.579771
$408$	0	0
$409$	2.90012	0.143402	0.0717010	−	0.997426i	$-0.477157\pi$
$409$	2.90012	0.143402	0.0717010	+	0.997426i	$0.477157\pi$
$410$	0	0
$411$	31.7797	1.56758
$412$	0	0
$413$	6.78574	0.333904
$414$	0	0
$415$	50.7588	2.49165
$416$	0	0
$417$	40.2476	1.97093
$418$	0	0
$419$	−19.7753	−0.966088	−0.483044	−	0.875596i	$-0.660469\pi$
$419$	−19.7753	−0.966088	−0.483044	+	0.875596i	$0.660469\pi$
$420$	0	0
$421$	14.9855	0.730348	0.365174	−	0.930939i	$-0.381009\pi$
$421$	14.9855	0.730348	0.365174	+	0.930939i	$0.381009\pi$
$422$	0	0
$423$	−4.98549	−0.242403
$424$	0	0
$425$	41.5636	2.01613
$426$	0	0
$427$	1.36794	0.0661992
$428$	0	0
$429$	5.80877	0.280450
$430$	0	0
$431$	−39.4427	−1.89989	−0.949945	−	0.312418i	$-0.898861\pi$
$431$	−39.4427	−1.89989	−0.949945	+	0.312418i	$0.898861\pi$
$432$	0	0
$433$	−1.23919	−0.0595518	−0.0297759	−	0.999557i	$-0.509479\pi$
$433$	−1.23919	−0.0595518	−0.0297759	+	0.999557i	$0.509479\pi$
$434$	0	0
$435$	25.2016	1.20833
$436$	0	0
$437$	1.36794	0.0654374
$438$	0	0
$439$	11.1852	0.533842	0.266921	−	0.963718i	$-0.413994\pi$
$439$	11.1852	0.533842	0.266921	+	0.963718i	$0.413994\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	0	0
$443$	−23.8422	−1.13278	−0.566389	−	0.824138i	$-0.691660\pi$
$443$	−23.8422	−1.13278	−0.566389	+	0.824138i	$0.691660\pi$
$444$	0	0
$445$	−69.4677	−3.29308
$446$	0	0
$447$	9.31398	0.440536
$448$	0	0
$449$	20.9066	0.986644	0.493322	−	0.869847i	$-0.335783\pi$
$449$	20.9066	0.986644	0.493322	+	0.869847i	$0.335783\pi$
$450$	0	0
$451$	−7.01247	−0.330204
$452$	0	0
$453$	46.9565	2.20621
$454$	0	0
$455$	7.64453	0.358381
$456$	0	0
$457$	−4.18270	−0.195658	−0.0978292	−	0.995203i	$-0.531190\pi$
$457$	−4.18270	−0.195658	−0.0978292	+	0.995203i	$0.531190\pi$
$458$	0	0
$459$	−9.59262	−0.447745
$460$	0	0
$461$	−31.5387	−1.46890	−0.734451	−	0.678662i	$-0.762560\pi$
$461$	−31.5387	−1.46890	−0.734451	+	0.678662i	$0.762560\pi$
$462$	0	0
$463$	4.60713	0.214112	0.107056	−	0.994253i	$-0.465858\pi$
$463$	4.60713	0.214112	0.107056	+	0.994253i	$0.465858\pi$
$464$	0	0
$465$	−45.1747	−2.09492
$466$	0	0
$467$	3.94605	0.182601	0.0913006	−	0.995823i	$-0.470898\pi$
$467$	3.94605	0.182601	0.0913006	+	0.995823i	$0.470898\pi$
$468$	0	0
$469$	0.467814	0.0216016
$470$	0	0
$471$	−23.5906	−1.08700
$472$	0	0
$473$	−7.86068	−0.361435
$474$	0	0
$475$	12.3410	0.566242
$476$	0	0
$477$	−17.3929	−0.796365
$478$	0	0
$479$	16.6840	0.762310	0.381155	−	0.924511i	$-0.375526\pi$
$479$	16.6840	0.762310	0.381155	+	0.924511i	$0.375526\pi$
$480$	0	0
$481$	−31.3929	−1.43139
$482$	0	0
$483$	2.02493	0.0921375
$484$	0	0
$485$	−48.7069	−2.21166
$486$	0	0
$487$	14.4388	0.654284	0.327142	−	0.944975i	$-0.393914\pi$
$487$	14.4388	0.654284	0.327142	+	0.944975i	$0.393914\pi$
$488$	0	0
$489$	−42.6819	−1.93014
$490$	0	0
$491$	−6.38040	−0.287944	−0.143972	−	0.989582i	$-0.545987\pi$
$491$	−6.38040	−0.287944	−0.143972	+	0.989582i	$0.545987\pi$
$492$	0	0
$493$	9.41780	0.424156
$494$	0	0
$495$	7.01247	0.315187
$496$	0	0
$497$	−2.41591	−0.108368
$498$	0	0
$499$	31.8252	1.42469	0.712345	−	0.701829i	$-0.247633\pi$
$499$	31.8252	1.42469	0.712345	+	0.701829i	$0.247633\pi$
$500$	0	0
$501$	−16.8277	−0.751807
$502$	0	0
$503$	−24.5716	−1.09559	−0.547797	−	0.836611i	$-0.684534\pi$
$503$	−24.5716	−1.09559	−0.547797	+	0.836611i	$0.684534\pi$
$504$	0	0
$505$	14.9606	0.665736
$506$	0	0
$507$	−12.5447	−0.557128
$508$	0	0
$509$	−5.56769	−0.246783	−0.123392	−	0.992358i	$-0.539377\pi$
$509$	−5.56769	−0.246783	−0.123392	+	0.992358i	$0.539377\pi$
$510$	0	0
$511$	0.278636	0.0123261
$512$	0	0
$513$	−2.84822	−0.125752
$514$	0	0
$515$	18.6051	0.819838
$516$	0	0
$517$	−2.96056	−0.130205
$518$	0	0
$519$	34.1557	1.49927
$520$	0	0
$521$	−10.8148	−0.473803	−0.236902	−	0.971534i	$-0.576132\pi$
$521$	−10.8148	−0.473803	−0.236902	+	0.971534i	$0.576132\pi$
$522$	0	0
$523$	18.4074	0.804899	0.402449	−	0.915442i	$-0.368159\pi$
$523$	18.4074	0.804899	0.402449	+	0.915442i	$0.368159\pi$
$524$	0	0
$525$	18.2681	0.797284
$526$	0	0
$527$	−16.8817	−0.735377
$528$	0	0
$529$	−21.1287	−0.918641
$530$	0	0
$531$	16.7069	0.725016
$532$	0	0
$533$	−18.8212	−0.815238
$534$	0	0
$535$	−40.3784	−1.74571
$536$	0	0
$537$	17.2556	0.744634
$538$	0	0
$539$	−6.53219	−0.281361
$540$	0	0
$541$	33.0644	1.42155	0.710774	−	0.703420i	$-0.248345\pi$
$541$	33.0644	1.42155	0.710774	+	0.703420i	$0.248345\pi$
$542$	0	0
$543$	−5.52571	−0.237131
$544$	0	0
$545$	24.9855	1.07026
$546$	0	0
$547$	−34.1287	−1.45924	−0.729620	−	0.683853i	$-0.760303\pi$
$547$	−34.1287	−1.45924	−0.729620	+	0.683853i	$0.760303\pi$
$548$	0	0
$549$	3.36794	0.143740
$550$	0	0
$551$	2.79631	0.119127
$552$	0	0
$553$	4.98549	0.212004
$554$	0	0
$555$	−105.414	−4.47456
$556$	0	0
$557$	20.5112	0.869087	0.434544	−	0.900651i	$-0.356910\pi$
$557$	20.5112	0.869087	0.434544	+	0.900651i	$0.356910\pi$
$558$	0	0
$559$	−21.0978	−0.892343
$560$	0	0
$561$	7.28905	0.307744
$562$	0	0
$563$	27.0104	1.13835	0.569177	−	0.822215i	$-0.307262\pi$
$563$	27.0104	1.13835	0.569177	+	0.822215i	$0.307262\pi$
$564$	0	0
$565$	43.4427	1.82765
$566$	0	0
$567$	−7.67150	−0.322173
$568$	0	0
$569$	7.56564	0.317168	0.158584	−	0.987345i	$-0.449307\pi$
$569$	7.56564	0.317168	0.158584	+	0.987345i	$0.449307\pi$
$570$	0	0
$571$	−42.2226	−1.76696	−0.883481	−	0.468467i	$-0.844807\pi$
$571$	−42.2226	−1.76696	−0.883481	+	0.468467i	$0.844807\pi$
$572$	0	0
$573$	−17.8002	−0.743616
$574$	0	0
$575$	16.8817	0.704014
$576$	0	0
$577$	−25.9191	−1.07902	−0.539512	−	0.841978i	$-0.681391\pi$
$577$	−25.9191	−1.07902	−0.539512	+	0.841978i	$0.681391\pi$
$578$	0	0
$579$	−52.4053	−2.17789
$580$	0	0
$581$	8.33702	0.345878
$582$	0	0
$583$	−10.3285	−0.427763
$584$	0	0
$585$	18.8212	0.778162
$586$	0	0
$587$	−20.6570	−0.852605	−0.426303	−	0.904581i	$-0.640184\pi$
$587$	−20.6570	−0.852605	−0.426303	+	0.904581i	$0.640184\pi$
$588$	0	0
$589$	−5.01247	−0.206535
$590$	0	0
$591$	57.6923	2.37315
$592$	0	0
$593$	−28.1287	−1.15511	−0.577555	−	0.816352i	$-0.695993\pi$
$593$	−28.1287	−1.15511	−0.577555	+	0.816352i	$0.695993\pi$
$594$	0	0
$595$	9.59262	0.393259
$596$	0	0
$597$	15.2891	0.625739
$598$	0	0
$599$	26.3179	1.07532	0.537661	−	0.843161i	$-0.319308\pi$
$599$	26.3179	1.07532	0.537661	+	0.843161i	$0.319308\pi$
$600$	0	0
$601$	38.4053	1.56659	0.783293	−	0.621653i	$-0.213538\pi$
$601$	38.4053	1.56659	0.783293	+	0.621653i	$0.213538\pi$
$602$	0	0
$603$	1.15178	0.0469042
$604$	0	0
$605$	4.16425	0.169301
$606$	0	0
$607$	33.2600	1.34998	0.674991	−	0.737826i	$-0.264147\pi$
$607$	33.2600	1.34998	0.674991	+	0.737826i	$0.264147\pi$
$608$	0	0
$609$	4.13932	0.167734
$610$	0	0
$611$	−7.94605	−0.321463
$612$	0	0
$613$	17.5506	0.708864	0.354432	−	0.935082i	$-0.384674\pi$
$613$	17.5506	0.708864	0.354432	+	0.935082i	$0.384674\pi$
$614$	0	0
$615$	−63.1996	−2.54845
$616$	0	0
$617$	14.6840	0.591154	0.295577	−	0.955319i	$-0.404488\pi$
$617$	14.6840	0.591154	0.295577	+	0.955319i	$0.404488\pi$
$618$	0	0
$619$	−11.1183	−0.446883	−0.223442	−	0.974717i	$-0.571729\pi$
$619$	−11.1183	−0.446883	−0.223442	+	0.974717i	$0.571729\pi$
$620$	0	0
$621$	−3.89619	−0.156349
$622$	0	0
$623$	−11.4099	−0.457129
$624$	0	0
$625$	65.5945	2.62378
$626$	0	0
$627$	2.16425	0.0864317
$628$	0	0
$629$	−39.3929	−1.57070
$630$	0	0
$631$	46.0748	1.83421	0.917104	−	0.398648i	$-0.130520\pi$
$631$	46.0748	1.83421	0.917104	+	0.398648i	$0.130520\pi$
$632$	0	0
$633$	23.9460	0.951770
$634$	0	0
$635$	−63.6674	−2.52656
$636$	0	0
$637$	−17.5322	−0.694651
$638$	0	0
$639$	−5.94809	−0.235303
$640$	0	0
$641$	−11.3181	−0.447037	−0.223519	−	0.974700i	$-0.571754\pi$
$641$	−11.3181	−0.447037	−0.223519	+	0.974700i	$0.571754\pi$
$642$	0	0
$643$	30.6280	1.20785	0.603925	−	0.797041i	$-0.293603\pi$
$643$	30.6280	1.20785	0.603925	+	0.797041i	$0.293603\pi$
$644$	0	0
$645$	−70.8441	−2.78948
$646$	0	0
$647$	31.2720	1.22943	0.614715	−	0.788750i	$-0.289271\pi$
$647$	31.2720	1.22943	0.614715	+	0.788750i	$0.289271\pi$
$648$	0	0
$649$	9.92112	0.389438
$650$	0	0
$651$	−7.41985	−0.290807
$652$	0	0
$653$	−2.09389	−0.0819402	−0.0409701	−	0.999160i	$-0.513045\pi$
$653$	−2.09389	−0.0819402	−0.0409701	+	0.999160i	$0.513045\pi$
$654$	0	0
$655$	−15.1767	−0.593003
$656$	0	0
$657$	0.686016	0.0267641
$658$	0	0
$659$	−24.7608	−0.964544	−0.482272	−	0.876022i	$-0.660188\pi$
$659$	−24.7608	−0.964544	−0.482272	+	0.876022i	$0.660188\pi$
$660$	0	0
$661$	36.4033	1.41592	0.707962	−	0.706251i	$-0.249615\pi$
$661$	36.4033	1.41592	0.707962	+	0.706251i	$0.249615\pi$
$662$	0	0
$663$	19.5636	0.759787
$664$	0	0
$665$	2.84822	0.110449
$666$	0	0
$667$	3.82518	0.148112
$668$	0	0
$669$	31.8921	1.23302
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−16.1728	−0.623415	−0.311707	−	0.950178i	$-0.600901\pi$
$673$	−16.1728	−0.623415	−0.311707	+	0.950178i	$0.600901\pi$
$674$	0	0
$675$	−35.1497	−1.35291
$676$	0	0
$677$	5.32456	0.204639	0.102320	−	0.994752i	$-0.467374\pi$
$677$	5.32456	0.204639	0.102320	+	0.994752i	$0.467374\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−47.1811	−1.80799
$682$	0	0
$683$	−49.1562	−1.88091	−0.940455	−	0.339918i	$-0.889601\pi$
$683$	−49.1562	−1.88091	−0.940455	+	0.339918i	$0.889601\pi$
$684$	0	0
$685$	61.1477	2.33633
$686$	0	0
$687$	38.4118	1.46550
$688$	0	0
$689$	−27.7214	−1.05610
$690$	0	0
$691$	6.51120	0.247698	0.123849	−	0.992301i	$-0.460476\pi$
$691$	6.51120	0.247698	0.123849	+	0.992301i	$0.460476\pi$
$692$	0	0
$693$	1.15178	0.0437526
$694$	0	0
$695$	77.4407	2.93749
$696$	0	0
$697$	−23.6175	−0.894578
$698$	0	0
$699$	−44.0499	−1.66612
$700$	0	0
$701$	−5.77532	−0.218131	−0.109065	−	0.994035i	$-0.534786\pi$
$701$	−5.77532	−0.218131	−0.109065	+	0.994035i	$0.534786\pi$
$702$	0	0
$703$	−11.6964	−0.441139
$704$	0	0
$705$	−26.6819	−1.00490
$706$	0	0
$707$	2.45724	0.0924140
$708$	0	0
$709$	−35.6155	−1.33757	−0.668784	−	0.743457i	$-0.733185\pi$
$709$	−35.6155	−1.33757	−0.668784	+	0.743457i	$0.733185\pi$
$710$	0	0
$711$	12.2745	0.460331
$712$	0	0
$713$	−6.85674	−0.256787
$714$	0	0
$715$	11.1767	0.417985
$716$	0	0
$717$	−8.13079	−0.303650
$718$	0	0
$719$	8.93563	0.333243	0.166621	−	0.986021i	$-0.446714\pi$
$719$	8.93563	0.333243	0.166621	+	0.986021i	$0.446714\pi$
$720$	0	0
$721$	3.05585	0.113806
$722$	0	0
$723$	−19.8023	−0.736455
$724$	0	0
$725$	34.5091	1.28164
$726$	0	0
$727$	1.42189	0.0527351	0.0263675	−	0.999652i	$-0.491606\pi$
$727$	1.42189	0.0527351	0.0263675	+	0.999652i	$0.491606\pi$
$728$	0	0
$729$	−0.394916	−0.0146265
$730$	0	0
$731$	−26.4743	−0.979187
$732$	0	0
$733$	−46.7567	−1.72700	−0.863499	−	0.504350i	$-0.831732\pi$
$733$	−46.7567	−1.72700	−0.863499	+	0.504350i	$0.831732\pi$
$734$	0	0
$735$	−58.8711	−2.17149
$736$	0	0
$737$	0.683969	0.0251943
$738$	0	0
$739$	34.4886	1.26869	0.634343	−	0.773052i	$-0.281271\pi$
$739$	34.4886	1.26869	0.634343	+	0.773052i	$0.281271\pi$
$740$	0	0
$741$	5.80877	0.213391
$742$	0	0
$743$	10.9606	0.402104	0.201052	−	0.979581i	$-0.435564\pi$
$743$	10.9606	0.402104	0.201052	+	0.979581i	$0.435564\pi$
$744$	0	0
$745$	17.9211	0.656579
$746$	0	0
$747$	20.5262	0.751014
$748$	0	0
$749$	−6.63206	−0.242330
$750$	0	0
$751$	−13.3140	−0.485834	−0.242917	−	0.970047i	$-0.578104\pi$
$751$	−13.3140	−0.485834	−0.242917	+	0.970047i	$0.578104\pi$
$752$	0	0
$753$	25.9710	0.946435
$754$	0	0
$755$	90.3493	3.28815
$756$	0	0
$757$	−8.54260	−0.310486	−0.155243	−	0.987876i	$-0.549616\pi$
$757$	−8.54260	−0.310486	−0.155243	+	0.987876i	$0.549616\pi$
$758$	0	0
$759$	2.96056	0.107461
$760$	0	0
$761$	32.7318	1.18653	0.593263	−	0.805009i	$-0.297839\pi$
$761$	32.7318	1.18653	0.593263	+	0.805009i	$0.297839\pi$
$762$	0	0
$763$	4.10381	0.148568
$764$	0	0
$765$	23.6175	0.853894
$766$	0	0
$767$	26.6280	0.961480
$768$	0	0
$769$	50.5491	1.82285	0.911423	−	0.411470i	$-0.134985\pi$
$769$	50.5491	1.82285	0.911423	+	0.411470i	$0.134985\pi$
$770$	0	0
$771$	50.2326	1.80908
$772$	0	0
$773$	−40.7359	−1.46517	−0.732584	−	0.680677i	$-0.761686\pi$
$773$	−40.7359	−1.46517	−0.732584	+	0.680677i	$0.761686\pi$
$774$	0	0
$775$	−61.8586	−2.22203
$776$	0	0
$777$	−17.3140	−0.621136
$778$	0	0
$779$	−7.01247	−0.251248
$780$	0	0
$781$	−3.53219	−0.126392
$782$	0	0
$783$	−7.96450	−0.284628
$784$	0	0
$785$	−45.3908	−1.62007
$786$	0	0
$787$	−8.53613	−0.304280	−0.152140	−	0.988359i	$-0.548616\pi$
$787$	−8.53613	−0.304280	−0.152140	+	0.988359i	$0.548616\pi$
$788$	0	0
$789$	48.0768	1.71158
$790$	0	0
$791$	7.13538	0.253705
$792$	0	0
$793$	5.36794	0.190621
$794$	0	0
$795$	−93.0852	−3.30139
$796$	0	0
$797$	−16.0249	−0.567632	−0.283816	−	0.958879i	$-0.591601\pi$
$797$	−16.0249	−0.567632	−0.283816	+	0.958879i	$0.591601\pi$
$798$	0	0
$799$	−9.97098	−0.352748
$800$	0	0
$801$	−28.0918	−0.992576
$802$	0	0
$803$	0.407381	0.0143762
$804$	0	0
$805$	3.89619	0.137322
$806$	0	0
$807$	−41.0893	−1.44641
$808$	0	0
$809$	−20.1458	−0.708288	−0.354144	−	0.935191i	$-0.615228\pi$
$809$	−20.1458	−0.708288	−0.354144	+	0.935191i	$0.615228\pi$
$810$	0	0
$811$	−8.27454	−0.290558	−0.145279	−	0.989391i	$-0.546408\pi$
$811$	−8.27454	−0.290558	−0.145279	+	0.989391i	$0.546408\pi$
$812$	0	0
$813$	−44.8731	−1.57377
$814$	0	0
$815$	−82.1247	−2.87670
$816$	0	0
$817$	−7.86068	−0.275010
$818$	0	0
$819$	3.09135	0.108021
$820$	0	0
$821$	−21.5137	−0.750835	−0.375417	−	0.926856i	$-0.622501\pi$
$821$	−21.5137	−0.750835	−0.375417	+	0.926856i	$0.622501\pi$
$822$	0	0
$823$	5.26412	0.183496	0.0917479	−	0.995782i	$-0.470755\pi$
$823$	5.26412	0.183496	0.0917479	+	0.995782i	$0.470755\pi$
$824$	0	0
$825$	26.7089	0.929885
$826$	0	0
$827$	−16.7069	−0.580954	−0.290477	−	0.956882i	$-0.593814\pi$
$827$	−16.7069	−0.580954	−0.290477	+	0.956882i	$0.593814\pi$
$828$	0	0
$829$	−5.55064	−0.192782	−0.0963908	−	0.995344i	$-0.530730\pi$
$829$	−5.55064	−0.192782	−0.0963908	+	0.995344i	$0.530730\pi$
$830$	0	0
$831$	47.2720	1.63985
$832$	0	0
$833$	−22.0000	−0.762255
$834$	0	0
$835$	−32.3784	−1.12050
$836$	0	0
$837$	14.2766	0.493471
$838$	0	0
$839$	−15.3744	−0.530784	−0.265392	−	0.964141i	$-0.585501\pi$
$839$	−15.3744	−0.530784	−0.265392	+	0.964141i	$0.585501\pi$
$840$	0	0
$841$	−21.1807	−0.730367
$842$	0	0
$843$	0.355473	0.0122431
$844$	0	0
$845$	−24.1373	−0.830347
$846$	0	0
$847$	0.683969	0.0235015
$848$	0	0
$849$	−9.30955	−0.319503
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	24.8028	0.849231	0.424616	−	0.905374i	$-0.360409\pi$
$853$	24.8028	0.849231	0.424616	+	0.905374i	$0.360409\pi$
$854$	0	0
$855$	7.01247	0.239821
$856$	0	0
$857$	−45.6445	−1.55919	−0.779594	−	0.626286i	$-0.784574\pi$
$857$	−45.6445	−1.55919	−0.779594	+	0.626286i	$0.784574\pi$
$858$	0	0
$859$	−54.8356	−1.87097	−0.935483	−	0.353371i	$-0.885035\pi$
$859$	−54.8356	−1.87097	−0.935483	+	0.353371i	$0.885035\pi$
$860$	0	0
$861$	−10.3804	−0.353763
$862$	0	0
$863$	−45.8771	−1.56167	−0.780837	−	0.624735i	$-0.785207\pi$
$863$	−45.8771	−1.56167	−0.780837	+	0.624735i	$0.785207\pi$
$864$	0	0
$865$	65.7193	2.23452
$866$	0	0
$867$	−12.2431	−0.415799
$868$	0	0
$869$	7.28905	0.247264
$870$	0	0
$871$	1.83575	0.0622021
$872$	0	0
$873$	−19.6964	−0.666623
$874$	0	0
$875$	20.9087	0.706841
$876$	0	0
$877$	−15.6609	−0.528832	−0.264416	−	0.964409i	$-0.585179\pi$
$877$	−15.6609	−0.528832	−0.264416	+	0.964409i	$0.585179\pi$
$878$	0	0
$879$	70.4487	2.37618
$880$	0	0
$881$	−3.39492	−0.114378	−0.0571888	−	0.998363i	$-0.518214\pi$
$881$	−3.39492	−0.114378	−0.0571888	+	0.998363i	$0.518214\pi$
$882$	0	0
$883$	33.6425	1.13216	0.566080	−	0.824350i	$-0.308459\pi$
$883$	33.6425	1.13216	0.566080	+	0.824350i	$0.308459\pi$
$884$	0	0
$885$	89.4137	3.00561
$886$	0	0
$887$	22.1287	0.743011	0.371505	−	0.928431i	$-0.378842\pi$
$887$	22.1287	0.743011	0.371505	+	0.928431i	$0.378842\pi$
$888$	0	0
$889$	−10.4572	−0.350725
$890$	0	0
$891$	−11.2162	−0.375755
$892$	0	0
$893$	−2.96056	−0.0990713
$894$	0	0
$895$	33.2016	1.10981
$896$	0	0
$897$	7.94605	0.265311
$898$	0	0
$899$	−14.0164	−0.467473
$900$	0	0
$901$	−34.7857	−1.15888
$902$	0	0
$903$	−11.6360	−0.387222
$904$	0	0
$905$	−10.6321	−0.353422
$906$	0	0
$907$	21.7633	0.722640	0.361320	−	0.932442i	$-0.382326\pi$
$907$	21.7633	0.722640	0.361320	+	0.932442i	$0.382326\pi$
$908$	0	0
$909$	6.04986	0.200661
$910$	0	0
$911$	−46.8356	−1.55173	−0.775866	−	0.630897i	$-0.782687\pi$
$911$	−46.8356	−1.55173	−0.775866	+	0.630897i	$0.782687\pi$
$912$	0	0
$913$	12.1892	0.403403
$914$	0	0
$915$	18.0249	0.595886
$916$	0	0
$917$	−2.49274	−0.0823177
$918$	0	0
$919$	42.0563	1.38731	0.693655	−	0.720307i	$-0.255999\pi$
$919$	42.0563	1.38731	0.693655	+	0.720307i	$0.255999\pi$
$920$	0	0
$921$	−48.1537	−1.58672
$922$	0	0
$923$	−9.48028	−0.312047
$924$	0	0
$925$	−144.345	−4.74604
$926$	0	0
$927$	7.52366	0.247109
$928$	0	0
$929$	36.1018	1.18446	0.592230	−	0.805769i	$-0.298248\pi$
$929$	36.1018	1.18446	0.592230	+	0.805769i	$0.298248\pi$
$930$	0	0
$931$	−6.53219	−0.214084
$932$	0	0
$933$	45.4178	1.48691
$934$	0	0
$935$	14.0249	0.458664
$936$	0	0
$937$	47.3638	1.54731	0.773655	−	0.633607i	$-0.218427\pi$
$937$	47.3638	1.54731	0.773655	+	0.633607i	$0.218427\pi$
$938$	0	0
$939$	−24.7878	−0.808919
$940$	0	0
$941$	−9.34301	−0.304573	−0.152287	−	0.988336i	$-0.548664\pi$
$941$	−9.34301	−0.304573	−0.152287	+	0.988336i	$0.548664\pi$
$942$	0	0
$943$	−9.59262	−0.312379
$944$	0	0
$945$	−8.11234	−0.263894
$946$	0	0
$947$	−22.2496	−0.723015	−0.361508	−	0.932369i	$-0.617738\pi$
$947$	−22.2496	−0.723015	−0.361508	+	0.932369i	$0.617738\pi$
$948$	0	0
$949$	1.09340	0.0354932
$950$	0	0
$951$	23.1811	0.751700
$952$	0	0
$953$	−31.9710	−1.03564	−0.517821	−	0.855489i	$-0.673257\pi$
$953$	−31.9710	−1.03564	−0.517821	+	0.855489i	$0.673257\pi$
$954$	0	0
$955$	−34.2496	−1.10829
$956$	0	0
$957$	6.05191	0.195630
$958$	0	0
$959$	10.0434	0.324318
$960$	0	0
$961$	−5.87519	−0.189522
$962$	0	0
$963$	−16.3285	−0.526178
$964$	0	0
$965$	−100.834	−3.24595
$966$	0	0
$967$	−5.97098	−0.192014	−0.0960068	−	0.995381i	$-0.530607\pi$
$967$	−5.97098	−0.192014	−0.0960068	+	0.995381i	$0.530607\pi$
$968$	0	0
$969$	7.28905	0.234158
$970$	0	0
$971$	19.0999	0.612944	0.306472	−	0.951880i	$-0.400851\pi$
$971$	19.0999	0.612944	0.306472	+	0.951880i	$0.400851\pi$
$972$	0	0
$973$	12.7195	0.407768
$974$	0	0
$975$	71.6859	2.29578
$976$	0	0
$977$	−10.2745	−0.328712	−0.164356	−	0.986401i	$-0.552555\pi$
$977$	−10.2745	−0.328712	−0.164356	+	0.986401i	$0.552555\pi$
$978$	0	0
$979$	−16.6819	−0.533157
$980$	0	0
$981$	10.1038	0.322590
$982$	0	0
$983$	−33.0084	−1.05280	−0.526402	−	0.850236i	$-0.676459\pi$
$983$	−33.0084	−1.05280	−0.526402	+	0.850236i	$0.676459\pi$
$984$	0	0
$985$	111.006	3.53696
$986$	0	0
$987$	−4.38245	−0.139495
$988$	0	0
$989$	−10.7529	−0.341923
$990$	0	0
$991$	−1.56564	−0.0497343	−0.0248671	−	0.999691i	$-0.507916\pi$
$991$	−1.56564	−0.0497343	−0.0248671	+	0.999691i	$0.507916\pi$
$992$	0	0
$993$	14.6549	0.465061
$994$	0	0
$995$	29.4178	0.932607
$996$	0	0
$997$	46.5361	1.47381	0.736907	−	0.675994i	$-0.236286\pi$
$997$	46.5361	1.47381	0.736907	+	0.675994i	$0.236286\pi$
$998$	0	0
$999$	33.3140	1.05401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.p.1.3		3
4.3	odd	2		418.2.a.h.1.1	✓	3
12.11	even	2		3762.2.a.bd.1.1		3
44.43	even	2		4598.2.a.bm.1.1		3
76.75	even	2		7942.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.h.1.1	✓	3	4.3	odd	2
3344.2.a.p.1.3		3	1.1	even	1	trivial
3762.2.a.bd.1.1		3	12.11	even	2
4598.2.a.bm.1.1		3	44.43	even	2
7942.2.a.bc.1.3		3	76.75	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.16425 q^{3} +4.16425 q^{5} +0.683969 q^{7} +1.68397 q^{9} +1.00000 q^{11} +2.68397 q^{13} +9.01247 q^{15} +3.36794 q^{17} +1.00000 q^{19} +1.48028 q^{21} +1.36794 q^{23} +12.3410 q^{25} -2.84822 q^{27} +2.79631 q^{29} -5.01247 q^{31} +2.16425 q^{33} +2.84822 q^{35} -11.6964 q^{37} +5.80877 q^{39} -7.01247 q^{41} -7.86068 q^{43} +7.01247 q^{45} -2.96056 q^{47} -6.53219 q^{49} +7.28905 q^{51} -10.3285 q^{53} +4.16425 q^{55} +2.16425 q^{57} +9.92112 q^{59} +2.00000 q^{61} +1.15178 q^{63} +11.1767 q^{65} +0.683969 q^{67} +2.96056 q^{69} -3.53219 q^{71} +0.407381 q^{73} +26.7089 q^{75} +0.683969 q^{77} +7.28905 q^{79} -11.2162 q^{81} +12.1892 q^{83} +14.0249 q^{85} +6.05191 q^{87} -16.6819 q^{89} +1.83575 q^{91} -10.8482 q^{93} +4.16425 q^{95} -11.6964 q^{97} +1.68397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 5 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 5 q^{13} + 9 q^{15} + 4 q^{17} + 3 q^{19} - 2 q^{23} + 4 q^{25} + 2 q^{27} + 7 q^{29} + 3 q^{31} - q^{33} - 2 q^{35} - 14 q^{37} - 2 q^{39} - 3 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q - q^3 + 5 * q^5 - q^7 + 2 * q^9 + 3 * q^11 + 5 * q^13 + 9 * q^15 + 4 * q^17 + 3 * q^19 - 2 * q^23 + 4 * q^25 + 2 * q^27 + 7 * q^29 + 3 * q^31 - q^33 - 2 * q^35 - 14 * q^37 - 2 * q^39 - 3 * q^41 + 5 * q^43 + 3 * q^45 - 6 * q^49 - 2 * q^51 - 16 * q^53 + 5 * q^55 - q^57 + 12 * q^59 + 6 * q^61 + 14 * q^63 + 8 * q^65 - q^67 + 3 * q^71 + 4 * q^73 + 41 * q^75 - q^77 - 2 * q^79 - 17 * q^81 - 7 * q^83 + 6 * q^85 + 9 * q^87 + 16 * q^89 + 13 * q^91 - 22 * q^93 + 5 * q^95 - 14 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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