LMFDB - Embedded newform 1352.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,2,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.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:	$10.7957743533$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.27743$ of defining polynomial
Character	$\chi$	$=$	1352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.27743 q^{3} +1.12182 q^{5} +2.61023 q^{7} +2.18667 q^{9} +O(q^{10})$	$q+2.27743 q^{3} +1.12182 q^{5} +2.61023 q^{7} +2.18667 q^{9} +2.61023 q^{11} +2.55485 q^{15} -5.55485 q^{17} +7.94462 q^{19} +5.94462 q^{21} -2.27743 q^{23} -3.74153 q^{25} -1.85229 q^{27} +7.79849 q^{29} -8.26259 q^{31} +5.94462 q^{33} +2.92820 q^{35} +3.60233 q^{37} -5.73205 q^{41} +4.03380 q^{43} +2.45305 q^{45} +2.66562 q^{47} -0.186674 q^{49} -12.6508 q^{51} -9.54002 q^{53} +2.92820 q^{55} +18.0933 q^{57} -3.83070 q^{59} +7.68457 q^{61} +5.70773 q^{63} +1.76311 q^{67} -5.18667 q^{69} +1.01642 q^{71} +0.323330 q^{73} -8.52106 q^{75} +6.81333 q^{77} -2.66877 q^{79} -10.7785 q^{81} +2.77953 q^{83} -6.23152 q^{85} +17.7605 q^{87} -11.5416 q^{89} -18.8174 q^{93} +8.91240 q^{95} +15.6524 q^{97} +5.70773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 6 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} - 12 q^{15} + 16 q^{19} + 8 q^{21} + 2 q^{23} + 10 q^{25} - 14 q^{27} + 8 q^{29} + 4 q^{31} + 8 q^{33} - 16 q^{35} + 12 q^{37} - 16 q^{41} + 6 q^{43} + 28 q^{45} + 20 q^{47} + 2 q^{49} - 34 q^{51} + 10 q^{53} - 16 q^{55} + 16 q^{57} + 4 q^{59} + 4 q^{61} + 8 q^{63} + 8 q^{67} - 18 q^{69} + 16 q^{71} + 24 q^{73} - 22 q^{75} + 30 q^{77} + 8 q^{79} + 20 q^{81} + 24 q^{83} + 20 q^{85} + 10 q^{87} + 16 q^{89} - 16 q^{93} + 16 q^{95} + 32 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 - 12 * q^15 + 16 * q^19 + 8 * q^21 + 2 * q^23 + 10 * q^25 - 14 * q^27 + 8 * q^29 + 4 * q^31 + 8 * q^33 - 16 * q^35 + 12 * q^37 - 16 * q^41 + 6 * q^43 + 28 * q^45 + 20 * q^47 + 2 * q^49 - 34 * q^51 + 10 * q^53 - 16 * q^55 + 16 * q^57 + 4 * q^59 + 4 * q^61 + 8 * q^63 + 8 * q^67 - 18 * q^69 + 16 * q^71 + 24 * q^73 - 22 * q^75 + 30 * q^77 + 8 * q^79 + 20 * q^81 + 24 * q^83 + 20 * q^85 + 10 * q^87 + 16 * q^89 - 16 * q^93 + 16 * q^95 + 32 * q^97 + 8 * 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
$3$	2.27743	1.31487	0.657437	−	0.753510i	$-0.271641\pi$
$3$	2.27743	1.31487	0.657437	+	0.753510i	$0.271641\pi$
$4$	0	0
$5$	1.12182	0.501691	0.250846	−	0.968027i	$-0.419291\pi$
$5$	1.12182	0.501691	0.250846	+	0.968027i	$0.419291\pi$
$6$	0	0
$7$	2.61023	0.986576	0.493288	−	0.869866i	$-0.335795\pi$
$7$	2.61023	0.986576	0.493288	+	0.869866i	$0.335795\pi$
$8$	0	0
$9$	2.18667	0.728891
$10$	0	0
$11$	2.61023	0.787015	0.393508	−	0.919321i	$-0.371261\pi$
$11$	2.61023	0.787015	0.393508	+	0.919321i	$0.371261\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.55485	0.659661
$16$	0	0
$17$	−5.55485	−1.34725	−0.673625	−	0.739073i	$-0.735264\pi$
$17$	−5.55485	−1.34725	−0.673625	+	0.739073i	$0.735264\pi$
$18$	0	0
$19$	7.94462	1.82262	0.911310	−	0.411720i	$-0.135072\pi$
$19$	7.94462	1.82262	0.911310	+	0.411720i	$0.135072\pi$
$20$	0	0
$21$	5.94462	1.29722
$22$	0	0
$23$	−2.27743	−0.474876	−0.237438	−	0.971403i	$-0.576308\pi$
$23$	−2.27743	−0.474876	−0.237438	+	0.971403i	$0.576308\pi$
$24$	0	0
$25$	−3.74153	−0.748306
$26$	0	0
$27$	−1.85229	−0.356473
$28$	0	0
$29$	7.79849	1.44814	0.724071	−	0.689725i	$-0.242269\pi$
$29$	7.79849	1.44814	0.724071	+	0.689725i	$0.242269\pi$
$30$	0	0
$31$	−8.26259	−1.48400	−0.742002	−	0.670397i	$-0.766124\pi$
$31$	−8.26259	−1.48400	−0.742002	+	0.670397i	$0.766124\pi$
$32$	0	0
$33$	5.94462	1.03483
$34$	0	0
$35$	2.92820	0.494957
$36$	0	0
$37$	3.60233	0.592220	0.296110	−	0.955154i	$-0.404310\pi$
$37$	3.60233	0.592220	0.296110	+	0.955154i	$0.404310\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.73205	−0.895196	−0.447598	−	0.894235i	$-0.647720\pi$
$41$	−5.73205	−0.895196	−0.447598	+	0.894235i	$0.647720\pi$
$42$	0	0
$43$	4.03380	0.615148	0.307574	−	0.951524i	$-0.400483\pi$
$43$	4.03380	0.615148	0.307574	+	0.951524i	$0.400483\pi$
$44$	0	0
$45$	2.45305	0.365679
$46$	0	0
$47$	2.66562	0.388820	0.194410	−	0.980920i	$-0.437721\pi$
$47$	2.66562	0.388820	0.194410	+	0.980920i	$0.437721\pi$
$48$	0	0
$49$	−0.186674	−0.0266678
$50$	0	0
$51$	−12.6508	−1.77146
$52$	0	0
$53$	−9.54002	−1.31042	−0.655211	−	0.755446i	$-0.727420\pi$
$53$	−9.54002	−1.31042	−0.655211	+	0.755446i	$0.727420\pi$
$54$	0	0
$55$	2.92820	0.394839
$56$	0	0
$57$	18.0933	2.39652
$58$	0	0
$59$	−3.83070	−0.498715	−0.249358	−	0.968411i	$-0.580219\pi$
$59$	−3.83070	−0.498715	−0.249358	+	0.968411i	$0.580219\pi$
$60$	0	0
$61$	7.68457	0.983909	0.491954	−	0.870621i	$-0.336283\pi$
$61$	7.68457	0.983909	0.491954	+	0.870621i	$0.336283\pi$
$62$	0	0
$63$	5.70773	0.719107
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.76311	0.215399	0.107699	−	0.994184i	$-0.465652\pi$
$67$	1.76311	0.215399	0.107699	+	0.994184i	$0.465652\pi$
$68$	0	0
$69$	−5.18667	−0.624402
$70$	0	0
$71$	1.01642	0.120626	0.0603132	−	0.998180i	$-0.480790\pi$
$71$	1.01642	0.120626	0.0603132	+	0.998180i	$0.480790\pi$
$72$	0	0
$73$	0.323330	0.0378429	0.0189214	−	0.999821i	$-0.493977\pi$
$73$	0.323330	0.0378429	0.0189214	+	0.999821i	$0.493977\pi$
$74$	0	0
$75$	−8.52106	−0.983927
$76$	0	0
$77$	6.81333	0.776451
$78$	0	0
$79$	−2.66877	−0.300260	−0.150130	−	0.988666i	$-0.547969\pi$
$79$	−2.66877	−0.300260	−0.150130	+	0.988666i	$0.547969\pi$
$80$	0	0
$81$	−10.7785	−1.19761
$82$	0	0
$83$	2.77953	0.305093	0.152547	−	0.988296i	$-0.451253\pi$
$83$	2.77953	0.305093	0.152547	+	0.988296i	$0.451253\pi$
$84$	0	0
$85$	−6.23152	−0.675904
$86$	0	0
$87$	17.7605	1.90412
$88$	0	0
$89$	−11.5416	−1.22341	−0.611703	−	0.791087i	$-0.709515\pi$
$89$	−11.5416	−1.22341	−0.611703	+	0.791087i	$0.709515\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18.8174	−1.95128
$94$	0	0
$95$	8.91240	0.914393
$96$	0	0
$97$	15.6524	1.58926	0.794628	−	0.607097i	$-0.207666\pi$
$97$	15.6524	1.58926	0.794628	+	0.607097i	$0.207666\pi$
$98$	0	0
$99$	5.70773	0.573649
$100$	0	0
$101$	16.3533	1.62722	0.813609	−	0.581412i	$-0.197500\pi$
$101$	16.3533	1.62722	0.813609	+	0.581412i	$0.197500\pi$
$102$	0	0
$103$	−2.24363	−0.221072	−0.110536	−	0.993872i	$-0.535257\pi$
$103$	−2.24363	−0.221072	−0.110536	+	0.993872i	$0.535257\pi$
$104$	0	0
$105$	6.66877	0.650805
$106$	0	0
$107$	−10.8323	−1.04720	−0.523598	−	0.851965i	$-0.675411\pi$
$107$	−10.8323	−1.04720	−0.523598	+	0.851965i	$0.675411\pi$
$108$	0	0
$109$	2.92820	0.280471	0.140236	−	0.990118i	$-0.455214\pi$
$109$	2.92820	0.280471	0.140236	+	0.990118i	$0.455214\pi$
$110$	0	0
$111$	8.20405	0.778694
$112$	0	0
$113$	10.0421	0.944683	0.472342	−	0.881416i	$-0.343409\pi$
$113$	10.0421	0.944683	0.472342	+	0.881416i	$0.343409\pi$
$114$	0	0
$115$	−2.55485	−0.238241
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−14.4995	−1.32916
$120$	0	0
$121$	−4.18667	−0.380607
$122$	0	0
$123$	−13.0543	−1.17707
$124$	0	0
$125$	−9.80639	−0.877110
$126$	0	0
$127$	0.391342	0.0347260	0.0173630	−	0.999849i	$-0.494473\pi$
$127$	0.391342	0.0347260	0.0173630	+	0.999849i	$0.494473\pi$
$128$	0	0
$129$	9.18667	0.808842
$130$	0	0
$131$	−19.3533	−1.69091	−0.845455	−	0.534047i	$-0.820670\pi$
$131$	−19.3533	−1.69091	−0.845455	+	0.534047i	$0.820670\pi$
$132$	0	0
$133$	20.7373	1.79815
$134$	0	0
$135$	−2.07793	−0.178840
$136$	0	0
$137$	−3.97568	−0.339666	−0.169833	−	0.985473i	$-0.554323\pi$
$137$	−3.97568	−0.339666	−0.169833	+	0.985473i	$0.554323\pi$
$138$	0	0
$139$	−18.9620	−1.60834	−0.804168	−	0.594402i	$-0.797389\pi$
$139$	−18.9620	−1.60834	−0.804168	+	0.594402i	$0.797389\pi$
$140$	0	0
$141$	6.07074	0.511249
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.74847	0.726521
$146$	0	0
$147$	−0.425137	−0.0350647
$148$	0	0
$149$	−1.46946	−0.120383	−0.0601915	−	0.998187i	$-0.519171\pi$
$149$	−1.46946	−0.120383	−0.0601915	+	0.998187i	$0.519171\pi$
$150$	0	0
$151$	−5.59382	−0.455218	−0.227609	−	0.973753i	$-0.573091\pi$
$151$	−5.59382	−0.455218	−0.227609	+	0.973753i	$0.573091\pi$
$152$	0	0
$153$	−12.1467	−0.981999
$154$	0	0
$155$	−9.26910	−0.744512
$156$	0	0
$157$	0.430307	0.0343422	0.0171711	−	0.999853i	$-0.494534\pi$
$157$	0.430307	0.0343422	0.0171711	+	0.999853i	$0.494534\pi$
$158$	0	0
$159$	−21.7267	−1.72304
$160$	0	0
$161$	−5.94462	−0.468502
$162$	0	0
$163$	3.68203	0.288399	0.144199	−	0.989549i	$-0.453939\pi$
$163$	3.68203	0.288399	0.144199	+	0.989549i	$0.453939\pi$
$164$	0	0
$165$	6.66877	0.519163
$166$	0	0
$167$	16.4666	1.27423	0.637113	−	0.770770i	$-0.280128\pi$
$167$	16.4666	1.27423	0.637113	+	0.770770i	$0.280128\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	17.3723	1.32849
$172$	0	0
$173$	23.7794	1.80792	0.903959	−	0.427620i	$-0.140648\pi$
$173$	23.7794	1.80792	0.903959	+	0.427620i	$0.140648\pi$
$174$	0	0
$175$	−9.76627	−0.738261
$176$	0	0
$177$	−8.72415	−0.655747
$178$	0	0
$179$	5.07591	0.379392	0.189696	−	0.981843i	$-0.439250\pi$
$179$	5.07591	0.379392	0.189696	+	0.981843i	$0.439250\pi$
$180$	0	0
$181$	9.16667	0.681353	0.340676	−	0.940181i	$-0.389344\pi$
$181$	9.16667	0.681353	0.340676	+	0.940181i	$0.389344\pi$
$182$	0	0
$183$	17.5011	1.29371
$184$	0	0
$185$	4.04116	0.297112
$186$	0	0
$187$	−14.4995	−1.06031
$188$	0	0
$189$	−4.83491	−0.351688
$190$	0	0
$191$	−16.9841	−1.22893	−0.614463	−	0.788945i	$-0.710627\pi$
$191$	−16.9841	−1.22893	−0.614463	+	0.788945i	$0.710627\pi$
$192$	0	0
$193$	−20.4874	−1.47471	−0.737356	−	0.675504i	$-0.763926\pi$
$193$	−20.4874	−1.47471	−0.737356	+	0.675504i	$0.763926\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.5416	1.10729	0.553646	−	0.832752i	$-0.313236\pi$
$197$	15.5416	1.10729	0.553646	+	0.832752i	$0.313236\pi$
$198$	0	0
$199$	−19.3575	−1.37221	−0.686107	−	0.727501i	$-0.740682\pi$
$199$	−19.3575	−1.37221	−0.686107	+	0.727501i	$0.740682\pi$
$200$	0	0
$201$	4.01536	0.283222
$202$	0	0
$203$	20.3559	1.42870
$204$	0	0
$205$	−6.43031	−0.449112
$206$	0	0
$207$	−4.97999	−0.346133
$208$	0	0
$209$	20.7373	1.43443
$210$	0	0
$211$	19.8744	1.36821	0.684105	−	0.729383i	$-0.260193\pi$
$211$	19.8744	1.36821	0.684105	+	0.729383i	$0.260193\pi$
$212$	0	0
$213$	2.31481	0.158608
$214$	0	0
$215$	4.52518	0.308614
$216$	0	0
$217$	−21.5673	−1.46408
$218$	0	0
$219$	0.736360	0.0497586
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.2758	1.15688	0.578438	−	0.815726i	$-0.303662\pi$
$223$	17.2758	1.15688	0.578438	+	0.815726i	$0.303662\pi$
$224$	0	0
$225$	−8.18151	−0.545434
$226$	0	0
$227$	−8.05538	−0.534654	−0.267327	−	0.963606i	$-0.586140\pi$
$227$	−8.05538	−0.534654	−0.267327	+	0.963606i	$0.586140\pi$
$228$	0	0
$229$	10.9282	0.722156	0.361078	−	0.932536i	$-0.382409\pi$
$229$	10.9282	0.722156	0.361078	+	0.932536i	$0.382409\pi$
$230$	0	0
$231$	15.5169	1.02093
$232$	0	0
$233$	−4.66877	−0.305861	−0.152931	−	0.988237i	$-0.548871\pi$
$233$	−4.66877	−0.305861	−0.152931	+	0.988237i	$0.548871\pi$
$234$	0	0
$235$	2.99033	0.195068
$236$	0	0
$237$	−6.07793	−0.394804
$238$	0	0
$239$	−17.4862	−1.13109	−0.565545	−	0.824718i	$-0.691334\pi$
$239$	−17.4862	−1.13109	−0.565545	+	0.824718i	$0.691334\pi$
$240$	0	0
$241$	14.1086	0.908812	0.454406	−	0.890795i	$-0.349852\pi$
$241$	14.1086	0.908812	0.454406	+	0.890795i	$0.349852\pi$
$242$	0	0
$243$	−18.9903	−1.21823
$244$	0	0
$245$	−0.209414	−0.0133790
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.33018	0.401159
$250$	0	0
$251$	−18.5728	−1.17231	−0.586154	−	0.810200i	$-0.699359\pi$
$251$	−18.5728	−1.17231	−0.586154	+	0.810200i	$0.699359\pi$
$252$	0	0
$253$	−5.94462	−0.373735
$254$	0	0
$255$	−14.1918	−0.888728
$256$	0	0
$257$	−4.33123	−0.270175	−0.135087	−	0.990834i	$-0.543132\pi$
$257$	−4.33123	−0.270175	−0.135087	+	0.990834i	$0.543132\pi$
$258$	0	0
$259$	9.40294	0.584270
$260$	0	0
$261$	17.0528	1.05554
$262$	0	0
$263$	−4.22564	−0.260564	−0.130282	−	0.991477i	$-0.541588\pi$
$263$	−4.22564	−0.260564	−0.130282	+	0.991477i	$0.541588\pi$
$264$	0	0
$265$	−10.7021	−0.657427
$266$	0	0
$267$	−26.2851	−1.60862
$268$	0	0
$269$	−8.22879	−0.501718	−0.250859	−	0.968024i	$-0.580713\pi$
$269$	−8.22879	−0.501718	−0.250859	+	0.968024i	$0.580713\pi$
$270$	0	0
$271$	7.71994	0.468953	0.234477	−	0.972122i	$-0.424662\pi$
$271$	7.71994	0.468953	0.234477	+	0.972122i	$0.424662\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.76627	−0.588928
$276$	0	0
$277$	−9.50307	−0.570984	−0.285492	−	0.958381i	$-0.592157\pi$
$277$	−9.50307	−0.570984	−0.285492	+	0.958381i	$0.592157\pi$
$278$	0	0
$279$	−18.0676	−1.08168
$280$	0	0
$281$	−23.0300	−1.37386	−0.686928	−	0.726726i	$-0.741041\pi$
$281$	−23.0300	−1.37386	−0.686928	+	0.726726i	$0.741041\pi$
$282$	0	0
$283$	−13.3713	−0.794843	−0.397422	−	0.917636i	$-0.630095\pi$
$283$	−13.3713	−0.794843	−0.397422	+	0.917636i	$0.630095\pi$
$284$	0	0
$285$	20.2973	1.20231
$286$	0	0
$287$	−14.9620	−0.883179
$288$	0	0
$289$	13.8564	0.815083
$290$	0	0
$291$	35.6471	2.08967
$292$	0	0
$293$	−17.3966	−1.01632	−0.508161	−	0.861262i	$-0.669674\pi$
$293$	−17.3966	−1.01632	−0.508161	+	0.861262i	$0.669674\pi$
$294$	0	0
$295$	−4.29735	−0.250201
$296$	0	0
$297$	−4.83491	−0.280550
$298$	0	0
$299$	0	0
$300$	0	0
$301$	10.5292	0.606890
$302$	0	0
$303$	37.2435	2.13959
$304$	0	0
$305$	8.62068	0.493618
$306$	0	0
$307$	−4.00315	−0.228472	−0.114236	−	0.993454i	$-0.536442\pi$
$307$	−4.00315	−0.228472	−0.114236	+	0.993454i	$0.536442\pi$
$308$	0	0
$309$	−5.10971	−0.290681
$310$	0	0
$311$	28.4630	1.61399	0.806996	−	0.590557i	$-0.201092\pi$
$311$	28.4630	1.61399	0.806996	+	0.590557i	$0.201092\pi$
$312$	0	0
$313$	−3.55906	−0.201170	−0.100585	−	0.994928i	$-0.532071\pi$
$313$	−3.55906	−0.201170	−0.100585	+	0.994928i	$0.532071\pi$
$314$	0	0
$315$	6.40303	0.360770
$316$	0	0
$317$	−1.38125	−0.0775787	−0.0387894	−	0.999247i	$-0.512350\pi$
$317$	−1.38125	−0.0775787	−0.0387894	+	0.999247i	$0.512350\pi$
$318$	0	0
$319$	20.3559	1.13971
$320$	0	0
$321$	−24.6697	−1.37693
$322$	0	0
$323$	−44.1312	−2.45553
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.66877	0.368784
$328$	0	0
$329$	6.95788	0.383600
$330$	0	0
$331$	9.42768	0.518192	0.259096	−	0.965852i	$-0.416575\pi$
$331$	9.42768	0.518192	0.259096	+	0.965852i	$0.416575\pi$
$332$	0	0
$333$	7.87713	0.431664
$334$	0	0
$335$	1.97789	0.108064
$336$	0	0
$337$	−1.21215	−0.0660298	−0.0330149	−	0.999455i	$-0.510511\pi$
$337$	−1.21215	−0.0660298	−0.0330149	+	0.999455i	$0.510511\pi$
$338$	0	0
$339$	22.8702	1.24214
$340$	0	0
$341$	−21.5673	−1.16793
$342$	0	0
$343$	−18.7589	−1.01289
$344$	0	0
$345$	−5.81849	−0.313257
$346$	0	0
$347$	−14.2774	−0.766452	−0.383226	−	0.923655i	$-0.625187\pi$
$347$	−14.2774	−0.766452	−0.383226	+	0.923655i	$0.625187\pi$
$348$	0	0
$349$	23.4245	1.25389	0.626943	−	0.779065i	$-0.284306\pi$
$349$	23.4245	1.25389	0.626943	+	0.779065i	$0.284306\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.11086	−0.431698	−0.215849	−	0.976427i	$-0.569252\pi$
$353$	−8.11086	−0.431698	−0.215849	+	0.976427i	$0.569252\pi$
$354$	0	0
$355$	1.14023	0.0605172
$356$	0	0
$357$	−33.0215	−1.74768
$358$	0	0
$359$	−26.3302	−1.38965	−0.694827	−	0.719177i	$-0.744519\pi$
$359$	−26.3302	−1.38965	−0.694827	+	0.719177i	$0.744519\pi$
$360$	0	0
$361$	44.1170	2.32195
$362$	0	0
$363$	−9.53485	−0.500450
$364$	0	0
$365$	0.362716	0.0189854
$366$	0	0
$367$	−10.0856	−0.526463	−0.263232	−	0.964733i	$-0.584788\pi$
$367$	−10.0856	−0.526463	−0.263232	+	0.964733i	$0.584788\pi$
$368$	0	0
$369$	−12.5341	−0.652501
$370$	0	0
$371$	−24.9017	−1.29283
$372$	0	0
$373$	0.610849	0.0316286	0.0158143	−	0.999875i	$-0.494966\pi$
$373$	0.610849	0.0316286	0.0158143	+	0.999875i	$0.494966\pi$
$374$	0	0
$375$	−22.3333	−1.15329
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.60603	−0.185229	−0.0926146	−	0.995702i	$-0.529522\pi$
$379$	−3.60603	−0.185229	−0.0926146	+	0.995702i	$0.529522\pi$
$380$	0	0
$381$	0.891254	0.0456603
$382$	0	0
$383$	−11.9794	−0.612118	−0.306059	−	0.952013i	$-0.599010\pi$
$383$	−11.9794	−0.612118	−0.306059	+	0.952013i	$0.599010\pi$
$384$	0	0
$385$	7.64330	0.389539
$386$	0	0
$387$	8.82060	0.448376
$388$	0	0
$389$	−10.1245	−0.513335	−0.256667	−	0.966500i	$-0.582625\pi$
$389$	−10.1245	−0.513335	−0.256667	+	0.966500i	$0.582625\pi$
$390$	0	0
$391$	12.6508	0.639777
$392$	0	0
$393$	−44.0758	−2.22333
$394$	0	0
$395$	−2.99387	−0.150638
$396$	0	0
$397$	11.5353	0.578939	0.289470	−	0.957187i	$-0.406521\pi$
$397$	11.5353	0.578939	0.289470	+	0.957187i	$0.406521\pi$
$398$	0	0
$399$	47.2277	2.36434
$400$	0	0
$401$	5.08539	0.253952	0.126976	−	0.991906i	$-0.459473\pi$
$401$	5.08539	0.253952	0.126976	+	0.991906i	$0.459473\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−12.0915	−0.600830
$406$	0	0
$407$	9.40294	0.466086
$408$	0	0
$409$	−38.8310	−1.92007	−0.960036	−	0.279878i	$-0.909706\pi$
$409$	−38.8310	−1.92007	−0.960036	+	0.279878i	$0.909706\pi$
$410$	0	0
$411$	−9.05433	−0.446617
$412$	0	0
$413$	−9.99904	−0.492020
$414$	0	0
$415$	3.11812	0.153063
$416$	0	0
$417$	−43.1846	−2.11476
$418$	0	0
$419$	−18.7350	−0.915265	−0.457633	−	0.889141i	$-0.651303\pi$
$419$	−18.7350	−0.915265	−0.457633	+	0.889141i	$0.651303\pi$
$420$	0	0
$421$	7.50063	0.365558	0.182779	−	0.983154i	$-0.441491\pi$
$421$	7.50063	0.365558	0.182779	+	0.983154i	$0.441491\pi$
$422$	0	0
$423$	5.82883	0.283408
$424$	0	0
$425$	20.7836	1.00815
$426$	0	0
$427$	20.0585	0.970701
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.8697	−0.523574	−0.261787	−	0.965126i	$-0.584312\pi$
$431$	−10.8697	−0.523574	−0.261787	+	0.965126i	$0.584312\pi$
$432$	0	0
$433$	0.156034	0.00749850	0.00374925	−	0.999993i	$-0.498807\pi$
$433$	0.156034	0.00749850	0.00374925	+	0.999993i	$0.498807\pi$
$434$	0	0
$435$	19.9240	0.955283
$436$	0	0
$437$	−18.0933	−0.865520
$438$	0	0
$439$	−7.70870	−0.367916	−0.183958	−	0.982934i	$-0.558891\pi$
$439$	−7.70870	−0.367916	−0.183958	+	0.982934i	$0.558891\pi$
$440$	0	0
$441$	−0.408196	−0.0194379
$442$	0	0
$443$	−5.59697	−0.265920	−0.132960	−	0.991121i	$-0.542448\pi$
$443$	−5.59697	−0.265920	−0.132960	+	0.991121i	$0.542448\pi$
$444$	0	0
$445$	−12.9475	−0.613772
$446$	0	0
$447$	−3.34659	−0.158288
$448$	0	0
$449$	−18.9057	−0.892213	−0.446107	−	0.894980i	$-0.647190\pi$
$449$	−18.9057	−0.892213	−0.446107	+	0.894980i	$0.647190\pi$
$450$	0	0
$451$	−14.9620	−0.704533
$452$	0	0
$453$	−12.7395	−0.598555
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.5274	0.492450	0.246225	−	0.969213i	$-0.420810\pi$
$457$	10.5274	0.492450	0.246225	+	0.969213i	$0.420810\pi$
$458$	0	0
$459$	10.2892	0.480259
$460$	0	0
$461$	1.68026	0.0782576	0.0391288	−	0.999234i	$-0.487542\pi$
$461$	1.68026	0.0782576	0.0391288	+	0.999234i	$0.487542\pi$
$462$	0	0
$463$	18.7795	0.872759	0.436379	−	0.899763i	$-0.356261\pi$
$463$	18.7795	0.872759	0.436379	+	0.899763i	$0.356261\pi$
$464$	0	0
$465$	−21.1097	−0.978939
$466$	0	0
$467$	16.6604	0.770949	0.385475	−	0.922718i	$-0.374038\pi$
$467$	16.6604	0.770949	0.385475	+	0.922718i	$0.374038\pi$
$468$	0	0
$469$	4.60214	0.212507
$470$	0	0
$471$	0.979992	0.0451556
$472$	0	0
$473$	10.5292	0.484131
$474$	0	0
$475$	−29.7250	−1.36388
$476$	0	0
$477$	−20.8609	−0.955155
$478$	0	0
$479$	12.3211	0.562966	0.281483	−	0.959566i	$-0.409174\pi$
$479$	12.3211	0.562966	0.281483	+	0.959566i	$0.409174\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−13.5384	−0.616020
$484$	0	0
$485$	17.5591	0.797316
$486$	0	0
$487$	37.2062	1.68597	0.842986	−	0.537936i	$-0.180796\pi$
$487$	37.2062	1.68597	0.842986	+	0.537936i	$0.180796\pi$
$488$	0	0
$489$	8.38556	0.379208
$490$	0	0
$491$	−14.7342	−0.664944	−0.332472	−	0.943113i	$-0.607883\pi$
$491$	−14.7342	−0.664944	−0.332472	+	0.943113i	$0.607883\pi$
$492$	0	0
$493$	−43.3195	−1.95101
$494$	0	0
$495$	6.40303	0.287795
$496$	0	0
$497$	2.65309	0.119007
$498$	0	0
$499$	19.2287	0.860795	0.430397	−	0.902639i	$-0.358373\pi$
$499$	19.2287	0.860795	0.430397	+	0.902639i	$0.358373\pi$
$500$	0	0
$501$	37.5016	1.67545
$502$	0	0
$503$	34.8081	1.55201	0.776007	−	0.630724i	$-0.217242\pi$
$503$	34.8081	1.55201	0.776007	+	0.630724i	$0.217242\pi$
$504$	0	0
$505$	18.3454	0.816361
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.22352	−0.231529	−0.115764	−	0.993277i	$-0.536932\pi$
$509$	−5.22352	−0.231529	−0.115764	+	0.993277i	$0.536932\pi$
$510$	0	0
$511$	0.843966	0.0373349
$512$	0	0
$513$	−14.7157	−0.649716
$514$	0	0
$515$	−2.51694	−0.110910
$516$	0	0
$517$	6.95788	0.306007
$518$	0	0
$519$	54.1559	2.37718
$520$	0	0
$521$	−7.92917	−0.347383	−0.173692	−	0.984800i	$-0.555570\pi$
$521$	−7.92917	−0.347383	−0.173692	+	0.984800i	$0.555570\pi$
$522$	0	0
$523$	31.9365	1.39649	0.698243	−	0.715860i	$-0.253965\pi$
$523$	31.9365	1.39649	0.698243	+	0.715860i	$0.253965\pi$
$524$	0	0
$525$	−22.2420	−0.970719
$526$	0	0
$527$	45.8975	1.99933
$528$	0	0
$529$	−17.8133	−0.774492
$530$	0	0
$531$	−8.37650	−0.363509
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−12.1518	−0.525369
$536$	0	0
$537$	11.5600	0.498852
$538$	0	0
$539$	−0.487264	−0.0209880
$540$	0	0
$541$	17.1534	0.737483	0.368742	−	0.929532i	$-0.379789\pi$
$541$	17.1534	0.737483	0.368742	+	0.929532i	$0.379789\pi$
$542$	0	0
$543$	20.8764	0.895893
$544$	0	0
$545$	3.28491	0.140710
$546$	0	0
$547$	15.4885	0.662241	0.331121	−	0.943588i	$-0.392573\pi$
$547$	15.4885	0.662241	0.331121	+	0.943588i	$0.392573\pi$
$548$	0	0
$549$	16.8037	0.717163
$550$	0	0
$551$	61.9560	2.63941
$552$	0	0
$553$	−6.96612	−0.296229
$554$	0	0
$555$	9.20344	0.390664
$556$	0	0
$557$	−31.7510	−1.34533	−0.672667	−	0.739946i	$-0.734851\pi$
$557$	−31.7510	−1.34533	−0.672667	+	0.739946i	$0.734851\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−33.0215	−1.39417
$562$	0	0
$563$	−34.1477	−1.43915	−0.719577	−	0.694412i	$-0.755664\pi$
$563$	−34.1477	−1.43915	−0.719577	+	0.694412i	$0.755664\pi$
$564$	0	0
$565$	11.2654	0.473939
$566$	0	0
$567$	−28.1344	−1.18153
$568$	0	0
$569$	2.27974	0.0955715	0.0477858	−	0.998858i	$-0.484784\pi$
$569$	2.27974	0.0955715	0.0477858	+	0.998858i	$0.484784\pi$
$570$	0	0
$571$	−24.5631	−1.02793	−0.513967	−	0.857810i	$-0.671824\pi$
$571$	−24.5631	−1.02793	−0.513967	+	0.857810i	$0.671824\pi$
$572$	0	0
$573$	−38.6801	−1.61588
$574$	0	0
$575$	8.52106	0.355353
$576$	0	0
$577$	−10.6670	−0.444073	−0.222037	−	0.975038i	$-0.571270\pi$
$577$	−10.6670	−0.444073	−0.222037	+	0.975038i	$0.571270\pi$
$578$	0	0
$579$	−46.6585	−1.93906
$580$	0	0
$581$	7.25523	0.300998
$582$	0	0
$583$	−24.9017	−1.03132
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.01221	0.0830528	0.0415264	−	0.999137i	$-0.486778\pi$
$587$	2.01221	0.0830528	0.0415264	+	0.999137i	$0.486778\pi$
$588$	0	0
$589$	−65.6431	−2.70478
$590$	0	0
$591$	35.3948	1.45595
$592$	0	0
$593$	38.9058	1.59767	0.798834	−	0.601552i	$-0.205451\pi$
$593$	38.9058	1.59767	0.798834	+	0.601552i	$0.205451\pi$
$594$	0	0
$595$	−16.2657	−0.666830
$596$	0	0
$597$	−44.0852	−1.80429
$598$	0	0
$599$	14.2815	0.583528	0.291764	−	0.956490i	$-0.405758\pi$
$599$	14.2815	0.583528	0.291764	+	0.956490i	$0.405758\pi$
$600$	0	0
$601$	34.1179	1.39170	0.695850	−	0.718187i	$-0.255028\pi$
$601$	34.1179	1.39170	0.695850	+	0.718187i	$0.255028\pi$
$602$	0	0
$603$	3.85536	0.157002
$604$	0	0
$605$	−4.69668	−0.190947
$606$	0	0
$607$	6.56654	0.266528	0.133264	−	0.991081i	$-0.457454\pi$
$607$	6.56654	0.266528	0.133264	+	0.991081i	$0.457454\pi$
$608$	0	0
$609$	46.3590	1.87856
$610$	0	0
$611$	0	0
$612$	0	0
$613$	46.2983	1.86997	0.934985	−	0.354687i	$-0.115412\pi$
$613$	46.2983	1.86997	0.934985	+	0.354687i	$0.115412\pi$
$614$	0	0
$615$	−14.6446	−0.590525
$616$	0	0
$617$	19.7479	0.795019	0.397509	−	0.917598i	$-0.369875\pi$
$617$	19.7479	0.795019	0.397509	+	0.917598i	$0.369875\pi$
$618$	0	0
$619$	6.07074	0.244004	0.122002	−	0.992530i	$-0.461069\pi$
$619$	6.07074	0.244004	0.122002	+	0.992530i	$0.461069\pi$
$620$	0	0
$621$	4.21845	0.169281
$622$	0	0
$623$	−30.1263	−1.20698
$624$	0	0
$625$	7.70668	0.308267
$626$	0	0
$627$	47.2277	1.88609
$628$	0	0
$629$	−20.0104	−0.797869
$630$	0	0
$631$	24.2072	0.963674	0.481837	−	0.876261i	$-0.339970\pi$
$631$	24.2072	0.963674	0.481837	+	0.876261i	$0.339970\pi$
$632$	0	0
$633$	45.2625	1.79902
$634$	0	0
$635$	0.439014	0.0174217
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.22257	0.0879236
$640$	0	0
$641$	33.7743	1.33400	0.667002	−	0.745056i	$-0.267577\pi$
$641$	33.7743	1.33400	0.667002	+	0.745056i	$0.267577\pi$
$642$	0	0
$643$	32.8052	1.29371	0.646856	−	0.762612i	$-0.276083\pi$
$643$	32.8052	1.29371	0.646856	+	0.762612i	$0.276083\pi$
$644$	0	0
$645$	10.3058	0.405789
$646$	0	0
$647$	38.0199	1.49472	0.747359	−	0.664421i	$-0.231322\pi$
$647$	38.0199	1.49472	0.747359	+	0.664421i	$0.231322\pi$
$648$	0	0
$649$	−9.99904	−0.392497
$650$	0	0
$651$	−49.1179	−1.92508
$652$	0	0
$653$	−20.0010	−0.782698	−0.391349	−	0.920242i	$-0.627991\pi$
$653$	−20.0010	−0.782698	−0.391349	+	0.920242i	$0.627991\pi$
$654$	0	0
$655$	−21.7109	−0.848315
$656$	0	0
$657$	0.707017	0.0275833
$658$	0	0
$659$	11.1918	0.435969	0.217984	−	0.975952i	$-0.430052\pi$
$659$	11.1918	0.435969	0.217984	+	0.975952i	$0.430052\pi$
$660$	0	0
$661$	26.0812	1.01444	0.507220	−	0.861816i	$-0.330673\pi$
$661$	26.0812	1.01444	0.507220	+	0.861816i	$0.330673\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.2635	0.902118
$666$	0	0
$667$	−17.7605	−0.687689
$668$	0	0
$669$	39.3445	1.52115
$670$	0	0
$671$	20.0585	0.774351
$672$	0	0
$673$	−45.3333	−1.74747	−0.873736	−	0.486401i	$-0.838309\pi$
$673$	−45.3333	−1.74747	−0.873736	+	0.486401i	$0.838309\pi$
$674$	0	0
$675$	6.93040	0.266751
$676$	0	0
$677$	23.6942	0.910644	0.455322	−	0.890327i	$-0.349524\pi$
$677$	23.6942	0.910644	0.455322	+	0.890327i	$0.349524\pi$
$678$	0	0
$679$	40.8563	1.56792
$680$	0	0
$681$	−18.3455	−0.703003
$682$	0	0
$683$	42.8972	1.64142	0.820709	−	0.571347i	$-0.193579\pi$
$683$	42.8972	1.64142	0.820709	+	0.571347i	$0.193579\pi$
$684$	0	0
$685$	−4.45998	−0.170407
$686$	0	0
$687$	24.8882	0.949544
$688$	0	0
$689$	0	0
$690$	0	0
$691$	48.1280	1.83088	0.915439	−	0.402458i	$-0.131844\pi$
$691$	48.1280	1.83088	0.915439	+	0.402458i	$0.131844\pi$
$692$	0	0
$693$	14.8985	0.565948
$694$	0	0
$695$	−21.2719	−0.806888
$696$	0	0
$697$	31.8407	1.20605
$698$	0	0
$699$	−10.6328	−0.402169
$700$	0	0
$701$	21.6433	0.817456	0.408728	−	0.912656i	$-0.365972\pi$
$701$	21.6433	0.817456	0.408728	+	0.912656i	$0.365972\pi$
$702$	0	0
$703$	28.6192	1.07939
$704$	0	0
$705$	6.81026	0.256489
$706$	0	0
$707$	42.6861	1.60537
$708$	0	0
$709$	12.1761	0.457285	0.228642	−	0.973510i	$-0.426571\pi$
$709$	12.1761	0.457285	0.228642	+	0.973510i	$0.426571\pi$
$710$	0	0
$711$	−5.83573	−0.218857
$712$	0	0
$713$	18.8174	0.704719
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−39.8236	−1.48724
$718$	0	0
$719$	−17.3353	−0.646499	−0.323250	−	0.946314i	$-0.604775\pi$
$719$	−17.3353	−0.646499	−0.323250	+	0.946314i	$0.604775\pi$
$720$	0	0
$721$	−5.85641	−0.218104
$722$	0	0
$723$	32.1312	1.19497
$724$	0	0
$725$	−29.1783	−1.08365
$726$	0	0
$727$	21.9600	0.814451	0.407225	−	0.913328i	$-0.366496\pi$
$727$	21.9600	0.814451	0.407225	+	0.913328i	$0.366496\pi$
$728$	0	0
$729$	−10.9137	−0.404210
$730$	0	0
$731$	−22.4071	−0.828758
$732$	0	0
$733$	−4.24733	−0.156879	−0.0784393	−	0.996919i	$-0.524994\pi$
$733$	−4.24733	−0.156879	−0.0784393	+	0.996919i	$0.524994\pi$
$734$	0	0
$735$	−0.476926	−0.0175917
$736$	0	0
$737$	4.60214	0.169522
$738$	0	0
$739$	13.9878	0.514549	0.257275	−	0.966338i	$-0.417176\pi$
$739$	13.9878	0.514549	0.257275	+	0.966338i	$0.417176\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−48.2830	−1.77133	−0.885666	−	0.464322i	$-0.846298\pi$
$743$	−48.2830	−1.77133	−0.885666	+	0.464322i	$0.846298\pi$
$744$	0	0
$745$	−1.64847	−0.0603952
$746$	0	0
$747$	6.07793	0.222380
$748$	0	0
$749$	−28.2748	−1.03314
$750$	0	0
$751$	−15.8060	−0.576768	−0.288384	−	0.957515i	$-0.593118\pi$
$751$	−15.8060	−0.576768	−0.288384	+	0.957515i	$0.593118\pi$
$752$	0	0
$753$	−42.2983	−1.54144
$754$	0	0
$755$	−6.27524	−0.228379
$756$	0	0
$757$	10.6021	0.385341	0.192671	−	0.981263i	$-0.438285\pi$
$757$	10.6021	0.385341	0.192671	+	0.981263i	$0.438285\pi$
$758$	0	0
$759$	−13.5384	−0.491414
$760$	0	0
$761$	35.6768	1.29328	0.646641	−	0.762794i	$-0.276173\pi$
$761$	35.6768	1.29328	0.646641	+	0.762794i	$0.276173\pi$
$762$	0	0
$763$	7.64330	0.276706
$764$	0	0
$765$	−13.6263	−0.492660
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−3.42452	−0.123491	−0.0617457	−	0.998092i	$-0.519667\pi$
$769$	−3.42452	−0.123491	−0.0617457	+	0.998092i	$0.519667\pi$
$770$	0	0
$771$	−9.86406	−0.355246
$772$	0	0
$773$	29.8339	1.07305	0.536525	−	0.843885i	$-0.319737\pi$
$773$	29.8339	1.07305	0.536525	+	0.843885i	$0.319737\pi$
$774$	0	0
$775$	30.9147	1.11049
$776$	0	0
$777$	21.4145	0.768241
$778$	0	0
$779$	−45.5390	−1.63160
$780$	0	0
$781$	2.65309	0.0949349
$782$	0	0
$783$	−14.4451	−0.516224
$784$	0	0
$785$	0.482725	0.0172292
$786$	0	0
$787$	28.9057	1.03038	0.515188	−	0.857077i	$-0.327722\pi$
$787$	28.9057	1.03038	0.515188	+	0.857077i	$0.327722\pi$
$788$	0	0
$789$	−9.62358	−0.342609
$790$	0	0
$791$	26.2123	0.932002
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−24.3733	−0.864434
$796$	0	0
$797$	0.450315	0.0159510	0.00797548	−	0.999968i	$-0.497461\pi$
$797$	0.450315	0.0159510	0.00797548	+	0.999968i	$0.497461\pi$
$798$	0	0
$799$	−14.8071	−0.523838
$800$	0	0
$801$	−25.2377	−0.891730
$802$	0	0
$803$	0.843966	0.0297829
$804$	0	0
$805$	−6.66877	−0.235043
$806$	0	0
$807$	−18.7405	−0.659696
$808$	0	0
$809$	32.5210	1.14338	0.571688	−	0.820471i	$-0.306289\pi$
$809$	32.5210	1.14338	0.571688	+	0.820471i	$0.306289\pi$
$810$	0	0
$811$	0.00315434	0.000110764 0	5.53819e−5	−	1.00000i	$-0.499982\pi$
$811$	0.00315434	0.000110764 0	5.53819e−5		1.00000i	$0.499982\pi$
$812$	0	0
$813$	17.5816	0.616614
$814$	0	0
$815$	4.13056	0.144687
$816$	0	0
$817$	32.0470	1.12118
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.2544	1.50959	0.754795	−	0.655961i	$-0.227737\pi$
$821$	43.2544	1.50959	0.754795	+	0.655961i	$0.227737\pi$
$822$	0	0
$823$	40.7868	1.42174	0.710869	−	0.703324i	$-0.248302\pi$
$823$	40.7868	1.42174	0.710869	+	0.703324i	$0.248302\pi$
$824$	0	0
$825$	−22.2420	−0.774366
$826$	0	0
$827$	15.0453	0.523175	0.261588	−	0.965180i	$-0.415754\pi$
$827$	15.0453	0.523175	0.261588	+	0.965180i	$0.415754\pi$
$828$	0	0
$829$	−34.0392	−1.18223	−0.591115	−	0.806588i	$-0.701312\pi$
$829$	−34.0392	−1.18223	−0.591115	+	0.806588i	$0.701312\pi$
$830$	0	0
$831$	−21.6425	−0.750771
$832$	0	0
$833$	1.03695	0.0359282
$834$	0	0
$835$	18.4725	0.639269
$836$	0	0
$837$	15.3047	0.529008
$838$	0	0
$839$	−44.2104	−1.52631	−0.763156	−	0.646215i	$-0.776351\pi$
$839$	−44.2104	−1.52631	−0.763156	+	0.646215i	$0.776351\pi$
$840$	0	0
$841$	31.8164	1.09712
$842$	0	0
$843$	−52.4492	−1.80645
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.9282	−0.375498
$848$	0	0
$849$	−30.4522	−1.04512
$850$	0	0
$851$	−8.20405	−0.281231
$852$	0	0
$853$	−47.0540	−1.61110	−0.805550	−	0.592528i	$-0.798130\pi$
$853$	−47.0540	−1.61110	−0.805550	+	0.592528i	$0.798130\pi$
$854$	0	0
$855$	19.4885	0.666493
$856$	0	0
$857$	7.26478	0.248160	0.124080	−	0.992272i	$-0.460402\pi$
$857$	7.26478	0.248160	0.124080	+	0.992272i	$0.460402\pi$
$858$	0	0
$859$	−18.3788	−0.627077	−0.313538	−	0.949575i	$-0.601514\pi$
$859$	−18.3788	−0.627077	−0.313538	+	0.949575i	$0.601514\pi$
$860$	0	0
$861$	−34.0749	−1.16127
$862$	0	0
$863$	−15.0390	−0.511932	−0.255966	−	0.966686i	$-0.582394\pi$
$863$	−15.0390	−0.511932	−0.255966	+	0.966686i	$0.582394\pi$
$864$	0	0
$865$	26.6762	0.907017
$866$	0	0
$867$	31.5570	1.07173
$868$	0	0
$869$	−6.96612	−0.236309
$870$	0	0
$871$	0	0
$872$	0	0
$873$	34.2266	1.15839
$874$	0	0
$875$	−25.5970	−0.865336
$876$	0	0
$877$	26.2469	0.886294	0.443147	−	0.896449i	$-0.353862\pi$
$877$	26.2469	0.886294	0.443147	+	0.896449i	$0.353862\pi$
$878$	0	0
$879$	−39.6195	−1.33633
$880$	0	0
$881$	14.4071	0.485386	0.242693	−	0.970103i	$-0.421969\pi$
$881$	14.4071	0.485386	0.242693	+	0.970103i	$0.421969\pi$
$882$	0	0
$883$	36.9503	1.24348	0.621739	−	0.783225i	$-0.286427\pi$
$883$	36.9503	1.24348	0.621739	+	0.783225i	$0.286427\pi$
$884$	0	0
$885$	−9.78689	−0.328983
$886$	0	0
$887$	−47.8205	−1.60566	−0.802828	−	0.596211i	$-0.796672\pi$
$887$	−47.8205	−1.60566	−0.802828	+	0.596211i	$0.796672\pi$
$888$	0	0
$889$	1.02150	0.0342599
$890$	0	0
$891$	−28.1344	−0.942536
$892$	0	0
$893$	21.1773	0.708671
$894$	0	0
$895$	5.69424	0.190337
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−64.4357	−2.14905
$900$	0	0
$901$	52.9934	1.76547
$902$	0	0
$903$	23.9794	0.797984
$904$	0	0
$905$	10.2833	0.341829
$906$	0	0
$907$	21.4705	0.712917	0.356459	−	0.934311i	$-0.383984\pi$
$907$	21.4705	0.712917	0.356459	+	0.934311i	$0.383984\pi$
$908$	0	0
$909$	35.7594	1.18607
$910$	0	0
$911$	−28.2594	−0.936277	−0.468138	−	0.883655i	$-0.655075\pi$
$911$	−28.2594	−0.936277	−0.468138	+	0.883655i	$0.655075\pi$
$912$	0	0
$913$	7.25523	0.240113
$914$	0	0
$915$	19.6330	0.649046
$916$	0	0
$917$	−50.5168	−1.66821
$918$	0	0
$919$	45.4463	1.49914	0.749568	−	0.661928i	$-0.230261\pi$
$919$	45.4463	1.49914	0.749568	+	0.661928i	$0.230261\pi$
$920$	0	0
$921$	−9.11689	−0.300412
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−13.4782	−0.443162
$926$	0	0
$927$	−4.90609	−0.161137
$928$	0	0
$929$	17.4449	0.572347	0.286174	−	0.958178i	$-0.407617\pi$
$929$	17.4449	0.572347	0.286174	+	0.958178i	$0.407617\pi$
$930$	0	0
$931$	−1.48306	−0.0486052
$932$	0	0
$933$	64.8225	2.12219
$934$	0	0
$935$	−16.2657	−0.531947
$936$	0	0
$937$	44.7794	1.46288	0.731440	−	0.681906i	$-0.238848\pi$
$937$	44.7794	1.46288	0.731440	+	0.681906i	$0.238848\pi$
$938$	0	0
$939$	−8.10550	−0.264513
$940$	0	0
$941$	−0.525176	−0.0171202	−0.00856012	−	0.999963i	$-0.502725\pi$
$941$	−0.525176	−0.0171202	−0.00856012	+	0.999963i	$0.502725\pi$
$942$	0	0
$943$	13.0543	0.425107
$944$	0	0
$945$	−5.42388	−0.176439
$946$	0	0
$947$	−24.3346	−0.790769	−0.395384	−	0.918516i	$-0.629389\pi$
$947$	−24.3346	−0.790769	−0.395384	+	0.918516i	$0.629389\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−3.14570	−0.102006
$952$	0	0
$953$	53.0513	1.71850	0.859250	−	0.511555i	$-0.170930\pi$
$953$	53.0513	1.71850	0.859250	+	0.511555i	$0.170930\pi$
$954$	0	0
$955$	−19.0530	−0.616542
$956$	0	0
$957$	46.3590	1.49857
$958$	0	0
$959$	−10.3775	−0.335106
$960$	0	0
$961$	37.2704	1.20227
$962$	0	0
$963$	−23.6867	−0.763292
$964$	0	0
$965$	−22.9831	−0.739851
$966$	0	0
$967$	26.5793	0.854732	0.427366	−	0.904079i	$-0.359442\pi$
$967$	26.5793	0.854732	0.427366	+	0.904079i	$0.359442\pi$
$968$	0	0
$969$	−100.506	−3.22871
$970$	0	0
$971$	−55.0999	−1.76824	−0.884121	−	0.467258i	$-0.845242\pi$
$971$	−55.0999	−1.76824	−0.884121	+	0.467258i	$0.845242\pi$
$972$	0	0
$973$	−49.4953	−1.58675
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.1600	−1.09287	−0.546437	−	0.837500i	$-0.684016\pi$
$977$	−34.1600	−1.09287	−0.546437	+	0.837500i	$0.684016\pi$
$978$	0	0
$979$	−30.1263	−0.962840
$980$	0	0
$981$	6.40303	0.204433
$982$	0	0
$983$	13.8912	0.443059	0.221530	−	0.975154i	$-0.428895\pi$
$983$	13.8912	0.443059	0.221530	+	0.975154i	$0.428895\pi$
$984$	0	0
$985$	17.4348	0.555519
$986$	0	0
$987$	15.8461	0.504386
$988$	0	0
$989$	−9.18667	−0.292119
$990$	0	0
$991$	−17.2416	−0.547698	−0.273849	−	0.961773i	$-0.588297\pi$
$991$	−17.2416	−0.547698	−0.273849	+	0.961773i	$0.588297\pi$
$992$	0	0
$993$	21.4708	0.681357
$994$	0	0
$995$	−21.7155	−0.688428
$996$	0	0
$997$	−1.45674	−0.0461354	−0.0230677	−	0.999734i	$-0.507343\pi$
$997$	−1.45674	−0.0461354	−0.0230677	+	0.999734i	$0.507343\pi$
$998$	0	0
$999$	−6.67257	−0.211111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.a.l.1.4		4
4.3	odd	2		2704.2.a.be.1.1		4
13.2	odd	12		1352.2.o.f.1161.1		8
13.3	even	3		1352.2.i.l.529.1		8
13.4	even	6		1352.2.i.k.1329.1		8
13.5	odd	4		1352.2.f.f.337.7		8
13.6	odd	12		104.2.o.a.49.1	yes	8
13.7	odd	12		1352.2.o.f.361.1		8
13.8	odd	4		1352.2.f.f.337.8		8
13.9	even	3		1352.2.i.l.1329.1		8
13.10	even	6		1352.2.i.k.529.1		8
13.11	odd	12		104.2.o.a.17.1	✓	8
13.12	even	2		1352.2.a.k.1.4		4
39.11	even	12		936.2.bi.b.433.3		8
39.32	even	12		936.2.bi.b.361.2		8
52.11	even	12		208.2.w.c.17.4		8
52.19	even	12		208.2.w.c.49.4		8
52.31	even	4		2704.2.f.q.337.1		8
52.47	even	4		2704.2.f.q.337.2		8
52.51	odd	2		2704.2.a.bd.1.1		4
104.11	even	12		832.2.w.i.641.1		8
104.19	even	12		832.2.w.i.257.1		8
104.37	odd	12		832.2.w.g.641.4		8
104.45	odd	12		832.2.w.g.257.4		8
156.11	odd	12		1872.2.by.n.433.3		8
156.71	odd	12		1872.2.by.n.1297.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.o.a.17.1	✓	8	13.11	odd	12
104.2.o.a.49.1	yes	8	13.6	odd	12
208.2.w.c.17.4		8	52.11	even	12
208.2.w.c.49.4		8	52.19	even	12
832.2.w.g.257.4		8	104.45	odd	12
832.2.w.g.641.4		8	104.37	odd	12
832.2.w.i.257.1		8	104.19	even	12
832.2.w.i.641.1		8	104.11	even	12
936.2.bi.b.361.2		8	39.32	even	12
936.2.bi.b.433.3		8	39.11	even	12
1352.2.a.k.1.4		4	13.12	even	2
1352.2.a.l.1.4		4	1.1	even	1	trivial
1352.2.f.f.337.7		8	13.5	odd	4
1352.2.f.f.337.8		8	13.8	odd	4
1352.2.i.k.529.1		8	13.10	even	6
1352.2.i.k.1329.1		8	13.4	even	6
1352.2.i.l.529.1		8	13.3	even	3
1352.2.i.l.1329.1		8	13.9	even	3
1352.2.o.f.361.1		8	13.7	odd	12
1352.2.o.f.1161.1		8	13.2	odd	12
1872.2.by.n.433.3		8	156.11	odd	12
1872.2.by.n.1297.2		8	156.71	odd	12
2704.2.a.bd.1.1		4	52.51	odd	2
2704.2.a.be.1.1		4	4.3	odd	2
2704.2.f.q.337.1		8	52.31	even	4
2704.2.f.q.337.2		8	52.47	even	4

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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q+2.27743 q^{3} +1.12182 q^{5} +2.61023 q^{7} +2.18667 q^{9} +O(q^{10})\)	\(q+2.27743 q^{3} +1.12182 q^{5} +2.61023 q^{7} +2.18667 q^{9} +2.61023 q^{11} +2.55485 q^{15} -5.55485 q^{17} +7.94462 q^{19} +5.94462 q^{21} -2.27743 q^{23} -3.74153 q^{25} -1.85229 q^{27} +7.79849 q^{29} -8.26259 q^{31} +5.94462 q^{33} +2.92820 q^{35} +3.60233 q^{37} -5.73205 q^{41} +4.03380 q^{43} +2.45305 q^{45} +2.66562 q^{47} -0.186674 q^{49} -12.6508 q^{51} -9.54002 q^{53} +2.92820 q^{55} +18.0933 q^{57} -3.83070 q^{59} +7.68457 q^{61} +5.70773 q^{63} +1.76311 q^{67} -5.18667 q^{69} +1.01642 q^{71} +0.323330 q^{73} -8.52106 q^{75} +6.81333 q^{77} -2.66877 q^{79} -10.7785 q^{81} +2.77953 q^{83} -6.23152 q^{85} +17.7605 q^{87} -11.5416 q^{89} -18.8174 q^{93} +8.91240 q^{95} +15.6524 q^{97} +5.70773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 6 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} - 12 q^{15} + 16 q^{19} + 8 q^{21} + 2 q^{23} + 10 q^{25} - 14 q^{27} + 8 q^{29} + 4 q^{31} + 8 q^{33} - 16 q^{35} + 12 q^{37} - 16 q^{41} + 6 q^{43} + 28 q^{45} + 20 q^{47} + 2 q^{49} - 34 q^{51} + 10 q^{53} - 16 q^{55} + 16 q^{57} + 4 q^{59} + 4 q^{61} + 8 q^{63} + 8 q^{67} - 18 q^{69} + 16 q^{71} + 24 q^{73} - 22 q^{75} + 30 q^{77} + 8 q^{79} + 20 q^{81} + 24 q^{83} + 20 q^{85} + 10 q^{87} + 16 q^{89} - 16 q^{93} + 16 q^{95} + 32 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 - 12 * q^15 + 16 * q^19 + 8 * q^21 + 2 * q^23 + 10 * q^25 - 14 * q^27 + 8 * q^29 + 4 * q^31 + 8 * q^33 - 16 * q^35 + 12 * q^37 - 16 * q^41 + 6 * q^43 + 28 * q^45 + 20 * q^47 + 2 * q^49 - 34 * q^51 + 10 * q^53 - 16 * q^55 + 16 * q^57 + 4 * q^59 + 4 * q^61 + 8 * q^63 + 8 * q^67 - 18 * q^69 + 16 * q^71 + 24 * q^73 - 22 * q^75 + 30 * q^77 + 8 * q^79 + 20 * q^81 + 24 * q^83 + 20 * q^85 + 10 * q^87 + 16 * q^89 - 16 * q^93 + 16 * q^95 + 32 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more