LMFDB - Embedded newform 1352.2.a.k.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:	$1$
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	$-0.580709\pi$
$5$	−1.12182	−0.501691	−0.250846	+	0.968027i	$0.580709\pi$
$6$	0	0
$7$	−2.61023	−0.986576	−0.493288	−	0.869866i	$-0.664205\pi$
$7$	−2.61023	−0.986576	−0.493288	+	0.869866i	$0.664205\pi$
$8$	0	0
$9$	2.18667	0.728891
$10$	0	0
$11$	−2.61023	−0.787015	−0.393508	−	0.919321i	$-0.628739\pi$
$11$	−2.61023	−0.787015	−0.393508	+	0.919321i	$0.628739\pi$
$12$	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	$-0.864928\pi$
$19$	−7.94462	−1.82262	−0.911310	+	0.411720i	$0.864928\pi$
$20$	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	$-0.233876\pi$
$31$	8.26259	1.48400	0.742002	+	0.670397i	$0.233876\pi$
$32$	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	$-0.595690\pi$
$37$	−3.60233	−0.592220	−0.296110	+	0.955154i	$0.595690\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.73205	0.895196	0.447598	−	0.894235i	$-0.352280\pi$
$41$	5.73205	0.895196	0.447598	+	0.894235i	$0.352280\pi$
$42$	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	$-0.562279\pi$
$47$	−2.66562	−0.388820	−0.194410	+	0.980920i	$0.562279\pi$
$48$	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	$-0.419781\pi$
$59$	3.83070	0.498715	0.249358	+	0.968411i	$0.419781\pi$
$60$	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	$-0.534348\pi$
$67$	−1.76311	−0.215399	−0.107699	+	0.994184i	$0.534348\pi$
$68$	0	0
$69$	−5.18667	−0.624402
$70$	0	0
$71$	−1.01642	−0.120626	−0.0603132	−	0.998180i	$-0.519210\pi$
$71$	−1.01642	−0.120626	−0.0603132	+	0.998180i	$0.519210\pi$
$72$	0	0
$73$	−0.323330	−0.0378429	−0.0189214	−	0.999821i	$-0.506023\pi$
$73$	−0.323330	−0.0378429	−0.0189214	+	0.999821i	$0.506023\pi$
$74$	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	$-0.548747\pi$
$83$	−2.77953	−0.305093	−0.152547	+	0.988296i	$0.548747\pi$
$84$	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	$-0.290485\pi$
$89$	11.5416	1.22341	0.611703	+	0.791087i	$0.290485\pi$
$90$	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	$-0.792334\pi$
$97$	−15.6524	−1.58926	−0.794628	+	0.607097i	$0.792334\pi$
$98$	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	$-0.544786\pi$
$109$	−2.92820	−0.280471	−0.140236	+	0.990118i	$0.544786\pi$
$110$	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	$-0.445677\pi$
$137$	3.97568	0.339666	0.169833	+	0.985473i	$0.445677\pi$
$138$	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	$-0.480829\pi$
$149$	1.46946	0.120383	0.0601915	+	0.998187i	$0.480829\pi$
$150$	0	0
$151$	5.59382	0.455218	0.227609	−	0.973753i	$-0.426909\pi$
$151$	5.59382	0.455218	0.227609	+	0.973753i	$0.426909\pi$
$152$	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	$-0.546061\pi$
$163$	−3.68203	−0.288399	−0.144199	+	0.989549i	$0.546061\pi$
$164$	0	0
$165$	6.66877	0.519163
$166$	0	0
$167$	−16.4666	−1.27423	−0.637113	−	0.770770i	$-0.719872\pi$
$167$	−16.4666	−1.27423	−0.637113	+	0.770770i	$0.719872\pi$
$168$	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	$-0.236074\pi$
$193$	20.4874	1.47471	0.737356	+	0.675504i	$0.236074\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.5416	−1.10729	−0.553646	−	0.832752i	$-0.686764\pi$
$197$	−15.5416	−1.10729	−0.553646	+	0.832752i	$0.686764\pi$
$198$	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	$-0.696338\pi$
$223$	−17.2758	−1.15688	−0.578438	+	0.815726i	$0.696338\pi$
$224$	0	0
$225$	−8.18151	−0.545434
$226$	0	0
$227$	8.05538	0.534654	0.267327	−	0.963606i	$-0.413860\pi$
$227$	8.05538	0.534654	0.267327	+	0.963606i	$0.413860\pi$
$228$	0	0
$229$	−10.9282	−0.722156	−0.361078	−	0.932536i	$-0.617591\pi$
$229$	−10.9282	−0.722156	−0.361078	+	0.932536i	$0.617591\pi$
$230$	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	$-0.308666\pi$
$239$	17.4862	1.13109	0.565545	+	0.824718i	$0.308666\pi$
$240$	0	0
$241$	−14.1086	−0.908812	−0.454406	−	0.890795i	$-0.650148\pi$
$241$	−14.1086	−0.908812	−0.454406	+	0.890795i	$0.650148\pi$
$242$	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	$-0.575338\pi$
$271$	−7.71994	−0.468953	−0.234477	+	0.972122i	$0.575338\pi$
$272$	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	$-0.258959\pi$
$281$	23.0300	1.37386	0.686928	+	0.726726i	$0.258959\pi$
$282$	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	$-0.330326\pi$
$293$	17.3966	1.01632	0.508161	+	0.861262i	$0.330326\pi$
$294$	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	$-0.463558\pi$
$307$	4.00315	0.228472	0.114236	+	0.993454i	$0.463558\pi$
$308$	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	$-0.487650\pi$
$317$	1.38125	0.0775787	0.0387894	+	0.999247i	$0.487650\pi$
$318$	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	$-0.583425\pi$
$331$	−9.42768	−0.518192	−0.259096	+	0.965852i	$0.583425\pi$
$332$	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	$-0.715694\pi$
$349$	−23.4245	−1.25389	−0.626943	+	0.779065i	$0.715694\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.11086	0.431698	0.215849	−	0.976427i	$-0.430748\pi$
$353$	8.11086	0.431698	0.215849	+	0.976427i	$0.430748\pi$
$354$	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	$-0.255481\pi$
$359$	26.3302	1.38965	0.694827	+	0.719177i	$0.255481\pi$
$360$	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	$-0.470478\pi$
$379$	3.60603	0.185229	0.0926146	+	0.995702i	$0.470478\pi$
$380$	0	0
$381$	0.891254	0.0456603
$382$	0	0
$383$	11.9794	0.612118	0.306059	−	0.952013i	$-0.400990\pi$
$383$	11.9794	0.612118	0.306059	+	0.952013i	$0.400990\pi$
$384$	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	$-0.593479\pi$
$397$	−11.5353	−0.578939	−0.289470	+	0.957187i	$0.593479\pi$
$398$	0	0
$399$	47.2277	2.36434
$400$	0	0
$401$	−5.08539	−0.253952	−0.126976	−	0.991906i	$-0.540527\pi$
$401$	−5.08539	−0.253952	−0.126976	+	0.991906i	$0.540527\pi$
$402$	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	$-0.0902940\pi$
$409$	38.8310	1.92007	0.960036	+	0.279878i	$0.0902940\pi$
$410$	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	$-0.558509\pi$
$421$	−7.50063	−0.365558	−0.182779	+	0.983154i	$0.558509\pi$
$422$	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	$-0.415688\pi$
$431$	10.8697	0.523574	0.261787	+	0.965126i	$0.415688\pi$
$432$	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	$-0.352810\pi$
$449$	18.9057	0.892213	0.446107	+	0.894980i	$0.352810\pi$
$450$	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	$-0.579190\pi$
$457$	−10.5274	−0.492450	−0.246225	+	0.969213i	$0.579190\pi$
$458$	0	0
$459$	10.2892	0.480259
$460$	0	0
$461$	−1.68026	−0.0782576	−0.0391288	−	0.999234i	$-0.512458\pi$
$461$	−1.68026	−0.0782576	−0.0391288	+	0.999234i	$0.512458\pi$
$462$	0	0
$463$	−18.7795	−0.872759	−0.436379	−	0.899763i	$-0.643739\pi$
$463$	−18.7795	−0.872759	−0.436379	+	0.899763i	$0.643739\pi$
$464$	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	$-0.590826\pi$
$479$	−12.3211	−0.562966	−0.281483	+	0.959566i	$0.590826\pi$
$480$	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	$-0.819204\pi$
$487$	−37.2062	−1.68597	−0.842986	+	0.537936i	$0.819204\pi$
$488$	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	$-0.641627\pi$
$499$	−19.2287	−0.860795	−0.430397	+	0.902639i	$0.641627\pi$
$500$	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	$-0.463068\pi$
$509$	5.22352	0.231529	0.115764	+	0.993277i	$0.463068\pi$
$510$	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	$-0.620211\pi$
$541$	−17.1534	−0.737483	−0.368742	+	0.929532i	$0.620211\pi$
$542$	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	$-0.265149\pi$
$557$	31.7510	1.34533	0.672667	+	0.739946i	$0.265149\pi$
$558$	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	$-0.428730\pi$
$577$	10.6670	0.444073	0.222037	+	0.975038i	$0.428730\pi$
$578$	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	$-0.513222\pi$
$587$	−2.01221	−0.0830528	−0.0415264	+	0.999137i	$0.513222\pi$
$588$	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	$-0.794549\pi$
$593$	−38.9058	−1.59767	−0.798834	+	0.601552i	$0.794549\pi$
$594$	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	$-0.884588\pi$
$613$	−46.2983	−1.86997	−0.934985	+	0.354687i	$0.884588\pi$
$614$	0	0
$615$	−14.6446	−0.590525
$616$	0	0
$617$	−19.7479	−0.795019	−0.397509	−	0.917598i	$-0.630125\pi$
$617$	−19.7479	−0.795019	−0.397509	+	0.917598i	$0.630125\pi$
$618$	0	0
$619$	−6.07074	−0.244004	−0.122002	−	0.992530i	$-0.538931\pi$
$619$	−6.07074	−0.244004	−0.122002	+	0.992530i	$0.538931\pi$
$620$	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	$-0.660030\pi$
$631$	−24.2072	−0.963674	−0.481837	+	0.876261i	$0.660030\pi$
$632$	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	$-0.723917\pi$
$643$	−32.8052	−1.29371	−0.646856	+	0.762612i	$0.723917\pi$
$644$	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	$-0.669327\pi$
$661$	−26.0812	−1.01444	−0.507220	+	0.861816i	$0.669327\pi$
$662$	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	$-0.806421\pi$
$683$	−42.8972	−1.64142	−0.820709	+	0.571347i	$0.806421\pi$
$684$	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	$-0.868156\pi$
$691$	−48.1280	−1.83088	−0.915439	+	0.402458i	$0.868156\pi$
$692$	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	$-0.573429\pi$
$709$	−12.1761	−0.457285	−0.228642	+	0.973510i	$0.573429\pi$
$710$	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	$-0.475006\pi$
$733$	4.24733	0.156879	0.0784393	+	0.996919i	$0.475006\pi$
$734$	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	$-0.582824\pi$
$739$	−13.9878	−0.514549	−0.257275	+	0.966338i	$0.582824\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.2830	1.77133	0.885666	−	0.464322i	$-0.153702\pi$
$743$	48.2830	1.77133	0.885666	+	0.464322i	$0.153702\pi$
$744$	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	$-0.723827\pi$
$761$	−35.6768	−1.29328	−0.646641	+	0.762794i	$0.723827\pi$
$762$	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	$-0.480333\pi$
$769$	3.42452	0.123491	0.0617457	+	0.998092i	$0.480333\pi$
$770$	0	0
$771$	−9.86406	−0.355246
$772$	0	0
$773$	−29.8339	−1.07305	−0.536525	−	0.843885i	$-0.680263\pi$
$773$	−29.8339	−1.07305	−0.536525	+	0.843885i	$0.680263\pi$
$774$	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	$-0.672278\pi$
$787$	−28.9057	−1.03038	−0.515188	+	0.857077i	$0.672278\pi$
$788$	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	$-0.500018\pi$
$811$	−0.00315434	−0.000110764 0	−5.53819e−5		1.00000i	$0.500018\pi$
$812$	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	$-0.772263\pi$
$821$	−43.2544	−1.50959	−0.754795	+	0.655961i	$0.772263\pi$
$822$	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	$-0.584246\pi$
$827$	−15.0453	−0.523175	−0.261588	+	0.965180i	$0.584246\pi$
$828$	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	$-0.223649\pi$
$839$	44.2104	1.52631	0.763156	+	0.646215i	$0.223649\pi$
$840$	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	$-0.201870\pi$
$853$	47.0540	1.61110	0.805550	+	0.592528i	$0.201870\pi$
$854$	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	$-0.417606\pi$
$863$	15.0390	0.511932	0.255966	+	0.966686i	$0.417606\pi$
$864$	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	$-0.646138\pi$
$877$	−26.2469	−0.886294	−0.443147	+	0.896449i	$0.646138\pi$
$878$	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	$-0.592383\pi$
$929$	−17.4449	−0.572347	−0.286174	+	0.958178i	$0.592383\pi$
$930$	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	$-0.497275\pi$
$941$	0.525176	0.0171202	0.00856012	+	0.999963i	$0.497275\pi$
$942$	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	$-0.370611\pi$
$947$	24.3346	0.790769	0.395384	+	0.918516i	$0.370611\pi$
$948$	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	$-0.640558\pi$
$967$	−26.5793	−0.854732	−0.427366	+	0.904079i	$0.640558\pi$
$968$	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	$-0.315984\pi$
$977$	34.1600	1.09287	0.546437	+	0.837500i	$0.315984\pi$
$978$	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	$-0.571105\pi$
$983$	−13.8912	−0.443059	−0.221530	+	0.975154i	$0.571105\pi$
$984$	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.k.1.4		4
4.3	odd	2		2704.2.a.bd.1.1		4
13.2	odd	12		104.2.o.a.17.1	✓	8
13.3	even	3		1352.2.i.k.529.1		8
13.4	even	6		1352.2.i.l.1329.1		8
13.5	odd	4		1352.2.f.f.337.8		8
13.6	odd	12		1352.2.o.f.361.1		8
13.7	odd	12		104.2.o.a.49.1	yes	8
13.8	odd	4		1352.2.f.f.337.7		8
13.9	even	3		1352.2.i.k.1329.1		8
13.10	even	6		1352.2.i.l.529.1		8
13.11	odd	12		1352.2.o.f.1161.1		8
13.12	even	2		1352.2.a.l.1.4		4
39.2	even	12		936.2.bi.b.433.3		8
39.20	even	12		936.2.bi.b.361.2		8
52.7	even	12		208.2.w.c.49.4		8
52.15	even	12		208.2.w.c.17.4		8
52.31	even	4		2704.2.f.q.337.2		8
52.47	even	4		2704.2.f.q.337.1		8
52.51	odd	2		2704.2.a.be.1.1		4
104.59	even	12		832.2.w.i.257.1		8
104.67	even	12		832.2.w.i.641.1		8
104.85	odd	12		832.2.w.g.257.4		8
104.93	odd	12		832.2.w.g.641.4		8
156.59	odd	12		1872.2.by.n.1297.2		8
156.119	odd	12		1872.2.by.n.433.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.o.a.17.1	✓	8	13.2	odd	12
104.2.o.a.49.1	yes	8	13.7	odd	12
208.2.w.c.17.4		8	52.15	even	12
208.2.w.c.49.4		8	52.7	even	12
832.2.w.g.257.4		8	104.85	odd	12
832.2.w.g.641.4		8	104.93	odd	12
832.2.w.i.257.1		8	104.59	even	12
832.2.w.i.641.1		8	104.67	even	12
936.2.bi.b.361.2		8	39.20	even	12
936.2.bi.b.433.3		8	39.2	even	12
1352.2.a.k.1.4		4	1.1	even	1	trivial
1352.2.a.l.1.4		4	13.12	even	2
1352.2.f.f.337.7		8	13.8	odd	4
1352.2.f.f.337.8		8	13.5	odd	4
1352.2.i.k.529.1		8	13.3	even	3
1352.2.i.k.1329.1		8	13.9	even	3
1352.2.i.l.529.1		8	13.10	even	6
1352.2.i.l.1329.1		8	13.4	even	6
1352.2.o.f.361.1		8	13.6	odd	12
1352.2.o.f.1161.1		8	13.11	odd	12
1872.2.by.n.433.3		8	156.119	odd	12
1872.2.by.n.1297.2		8	156.59	odd	12
2704.2.a.bd.1.1		4	4.3	odd	2
2704.2.a.be.1.1		4	52.51	odd	2
2704.2.f.q.337.1		8	52.47	even	4
2704.2.f.q.337.2		8	52.31	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

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