LMFDB - Embedded newform 3447.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3447,2,Mod(1,3447)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3447, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3447.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3447 = 3^{2} \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3447.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:	$27.5244335767$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 383)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	3447.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.58110 q^{2} +4.66207 q^{4} +4.25360 q^{5} -1.55327 q^{7} -6.87105 q^{8} -10.9790 q^{10} +1.50526 q^{11} -0.166872 q^{13} +4.00915 q^{14} +8.41073 q^{16} +7.68231 q^{17} +7.82416 q^{19} +19.8306 q^{20} -3.88523 q^{22} -1.82098 q^{23} +13.0931 q^{25} +0.430713 q^{26} -7.24147 q^{28} -2.12508 q^{29} -5.38423 q^{31} -7.96681 q^{32} -19.8288 q^{34} -6.60701 q^{35} +3.94801 q^{37} -20.1949 q^{38} -29.2267 q^{40} +0.332492 q^{41} +4.68328 q^{43} +7.01764 q^{44} +4.70013 q^{46} +1.24434 q^{47} -4.58734 q^{49} -33.7947 q^{50} -0.777969 q^{52} +9.61215 q^{53} +6.40280 q^{55} +10.6726 q^{56} +5.48504 q^{58} -15.2225 q^{59} -4.86732 q^{61} +13.8972 q^{62} +3.74166 q^{64} -0.709808 q^{65} +9.31251 q^{67} +35.8154 q^{68} +17.0533 q^{70} -7.05460 q^{71} +8.88768 q^{73} -10.1902 q^{74} +36.4767 q^{76} -2.33809 q^{77} -0.129066 q^{79} +35.7759 q^{80} -0.858194 q^{82} -3.08459 q^{83} +32.6775 q^{85} -12.0880 q^{86} -10.3428 q^{88} +9.24888 q^{89} +0.259198 q^{91} -8.48953 q^{92} -3.21177 q^{94} +33.2808 q^{95} -10.1487 q^{97} +11.8404 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) $ 24 * q - 5 * q^2 + 29 * q^4 - 3 * q^5 + 17 * q^7 - 15 * q^8 + q^10 + 28 * q^13 + 8 * q^14 + 35 * q^16 - 16 * q^17 + 13 * q^19 + 4 * q^20 + 12 * q^22 - 7 * q^23 + 67 * q^25 + 14 * q^26 + 39 * q^28 + 2 * q^29 - 20 * q^32 - 18 * q^34 + 10 * q^35 + 35 * q^37 + 5 * q^38 - 36 * q^40 + 34 * q^43 + 36 * q^44 - 17 * q^46 + 3 * q^47 + 49 * q^49 + 20 * q^50 + 29 * q^52 + 9 * q^53 + 10 * q^55 + 47 * q^56 + 12 * q^58 + 6 * q^59 + 3 * q^61 + 58 * q^62 + 15 * q^64 - 8 * q^65 + 42 * q^67 + 23 * q^68 - 59 * q^70 + 13 * q^71 + 83 * q^73 + 45 * q^74 + 4 * q^76 + 21 * q^77 - 3 * q^79 + 93 * q^80 - 13 * q^82 + 23 * q^83 + 52 * q^85 + 43 * q^86 - 25 * q^88 + 38 * q^89 - 17 * q^91 + 61 * q^92 - 71 * q^94 + 66 * q^95 + 63 * q^97 + 29 * q^98

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 $

$2$	−2.58110	−1.82511	−0.912556	−	0.408952i	$-0.865894\pi$
$2$	−2.58110	−1.82511	−0.912556	+	0.408952i	$0.865894\pi$
$3$	0	0
$3$	0	0
$4$	4.66207	2.33103
$5$	4.25360	1.90227	0.951134	−	0.308777i	$-0.0999198\pi$
$5$	4.25360	1.90227	0.951134	+	0.308777i	$0.0999198\pi$
$6$	0	0
$7$	−1.55327	−0.587083	−0.293541	−	0.955946i	$-0.594834\pi$
$7$	−1.55327	−0.587083	−0.293541	+	0.955946i	$0.594834\pi$
$8$	−6.87105	−2.42928
$9$	0	0
$10$	−10.9790	−3.47185
$11$	1.50526	0.453854	0.226927	−	0.973912i	$-0.427132\pi$
$11$	1.50526	0.453854	0.226927	+	0.973912i	$0.427132\pi$
$12$	0	0
$13$	−0.166872	−0.0462820	−0.0231410	−	0.999732i	$-0.507367\pi$
$13$	−0.166872	−0.0462820	−0.0231410	+	0.999732i	$0.507367\pi$
$14$	4.00915	1.07149
$15$	0	0
$16$	8.41073	2.10268
$17$	7.68231	1.86323	0.931617	−	0.363442i	$-0.118399\pi$
$17$	7.68231	1.86323	0.931617	+	0.363442i	$0.118399\pi$
$18$	0	0
$19$	7.82416	1.79498	0.897492	−	0.441030i	$-0.145387\pi$
$19$	7.82416	1.79498	0.897492	+	0.441030i	$0.145387\pi$
$20$	19.8306	4.43425
$21$	0	0
$22$	−3.88523	−0.828335
$23$	−1.82098	−0.379701	−0.189850	−	0.981813i	$-0.560800\pi$
$23$	−1.82098	−0.379701	−0.189850	+	0.981813i	$0.560800\pi$
$24$	0	0
$25$	13.0931	2.61863
$26$	0.430713	0.0844698
$27$	0	0
$28$	−7.24147	−1.36851
$29$	−2.12508	−0.394618	−0.197309	−	0.980341i	$-0.563220\pi$
$29$	−2.12508	−0.394618	−0.197309	+	0.980341i	$0.563220\pi$
$30$	0	0
$31$	−5.38423	−0.967037	−0.483518	−	0.875334i	$-0.660641\pi$
$31$	−5.38423	−0.967037	−0.483518	+	0.875334i	$0.660641\pi$
$32$	−7.96681	−1.40835
$33$	0	0
$34$	−19.8288	−3.40061
$35$	−6.60701	−1.11679
$36$	0	0
$37$	3.94801	0.649049	0.324525	−	0.945877i	$-0.394796\pi$
$37$	3.94801	0.649049	0.324525	+	0.945877i	$0.394796\pi$
$38$	−20.1949	−3.27605
$39$	0	0
$40$	−29.2267	−4.62115
$41$	0.332492	0.0519265	0.0259632	−	0.999663i	$-0.491735\pi$
$41$	0.332492	0.0519265	0.0259632	+	0.999663i	$0.491735\pi$
$42$	0	0
$43$	4.68328	0.714194	0.357097	−	0.934067i	$-0.383767\pi$
$43$	4.68328	0.714194	0.357097	+	0.934067i	$0.383767\pi$
$44$	7.01764	1.05795
$45$	0	0
$46$	4.70013	0.692996
$47$	1.24434	0.181506	0.0907531	−	0.995873i	$-0.471073\pi$
$47$	1.24434	0.181506	0.0907531	+	0.995873i	$0.471073\pi$
$48$	0	0
$49$	−4.58734	−0.655334
$50$	−33.7947	−4.77929
$51$	0	0
$52$	−0.777969	−0.107885
$53$	9.61215	1.32033	0.660165	−	0.751120i	$-0.270486\pi$
$53$	9.61215	1.32033	0.660165	+	0.751120i	$0.270486\pi$
$54$	0	0
$55$	6.40280	0.863353
$56$	10.6726	1.42619
$57$	0	0
$58$	5.48504	0.720222
$59$	−15.2225	−1.98181	−0.990903	−	0.134580i	$-0.957032\pi$
$59$	−15.2225	−1.98181	−0.990903	+	0.134580i	$0.957032\pi$
$60$	0	0
$61$	−4.86732	−0.623197	−0.311599	−	0.950214i	$-0.600864\pi$
$61$	−4.86732	−0.623197	−0.311599	+	0.950214i	$0.600864\pi$
$62$	13.8972	1.76495
$63$	0	0
$64$	3.74166	0.467708
$65$	−0.709808	−0.0880408
$66$	0	0
$67$	9.31251	1.13770	0.568852	−	0.822440i	$-0.307388\pi$
$67$	9.31251	1.13770	0.568852	+	0.822440i	$0.307388\pi$
$68$	35.8154	4.34326
$69$	0	0
$70$	17.0533	2.03826
$71$	−7.05460	−0.837227	−0.418614	−	0.908164i	$-0.637484\pi$
$71$	−7.05460	−0.837227	−0.418614	+	0.908164i	$0.637484\pi$
$72$	0	0
$73$	8.88768	1.04022	0.520112	−	0.854098i	$-0.325890\pi$
$73$	8.88768	1.04022	0.520112	+	0.854098i	$0.325890\pi$
$74$	−10.1902	−1.18459
$75$	0	0
$76$	36.4767	4.18417
$77$	−2.33809	−0.266450
$78$	0	0
$79$	−0.129066	−0.0145211	−0.00726053	−	0.999974i	$-0.502311\pi$
$79$	−0.129066	−0.0145211	−0.00726053	+	0.999974i	$0.502311\pi$
$80$	35.7759	3.99987
$81$	0	0
$82$	−0.858194	−0.0947716
$83$	−3.08459	−0.338577	−0.169289	−	0.985567i	$-0.554147\pi$
$83$	−3.08459	−0.338577	−0.169289	+	0.985567i	$0.554147\pi$
$84$	0	0
$85$	32.6775	3.54437
$86$	−12.0880	−1.30348
$87$	0	0
$88$	−10.3428	−1.10254
$89$	9.24888	0.980379	0.490189	−	0.871616i	$-0.336928\pi$
$89$	9.24888	0.980379	0.490189	+	0.871616i	$0.336928\pi$
$90$	0	0
$91$	0.259198	0.0271714
$92$	−8.48953	−0.885095
$93$	0	0
$94$	−3.21177	−0.331269
$95$	33.2808	3.41454
$96$	0	0
$97$	−10.1487	−1.03044	−0.515222	−	0.857057i	$-0.672291\pi$
$97$	−10.1487	−1.03044	−0.515222	+	0.857057i	$0.672291\pi$
$98$	11.8404	1.19606
$99$	0	0
$100$	61.0411	6.10411
$101$	6.98437	0.694971	0.347485	−	0.937685i	$-0.387036\pi$
$101$	6.98437	0.694971	0.347485	+	0.937685i	$0.387036\pi$
$102$	0	0
$103$	10.6733	1.05167	0.525837	−	0.850586i	$-0.323752\pi$
$103$	10.6733	1.05167	0.525837	+	0.850586i	$0.323752\pi$
$104$	1.14659	0.112432
$105$	0	0
$106$	−24.8099	−2.40975
$107$	−6.46852	−0.625336	−0.312668	−	0.949862i	$-0.601223\pi$
$107$	−6.46852	−0.625336	−0.312668	+	0.949862i	$0.601223\pi$
$108$	0	0
$109$	−5.24266	−0.502156	−0.251078	−	0.967967i	$-0.580785\pi$
$109$	−5.24266	−0.502156	−0.251078	+	0.967967i	$0.580785\pi$
$110$	−16.5262	−1.57572
$111$	0	0
$112$	−13.0642	−1.23445
$113$	−4.72006	−0.444026	−0.222013	−	0.975044i	$-0.571263\pi$
$113$	−4.72006	−0.444026	−0.222013	+	0.975044i	$0.571263\pi$
$114$	0	0
$115$	−7.74573	−0.722293
$116$	−9.90727	−0.919867
$117$	0	0
$118$	39.2909	3.61702
$119$	−11.9327	−1.09387
$120$	0	0
$121$	−8.73418	−0.794016
$122$	12.5630	1.13740
$123$	0	0
$124$	−25.1017	−2.25420
$125$	34.4250	3.07906
$126$	0	0
$127$	−10.1154	−0.897599	−0.448800	−	0.893632i	$-0.648148\pi$
$127$	−10.1154	−0.897599	−0.448800	+	0.893632i	$0.648148\pi$
$128$	6.27603	0.554728
$129$	0	0
$130$	1.83208	0.160684
$131$	3.14047	0.274385	0.137192	−	0.990544i	$-0.456192\pi$
$131$	3.14047	0.274385	0.137192	+	0.990544i	$0.456192\pi$
$132$	0	0
$133$	−12.1531	−1.05380
$134$	−24.0365	−2.07644
$135$	0	0
$136$	−52.7856	−4.52632
$137$	−8.77278	−0.749509	−0.374755	−	0.927124i	$-0.622273\pi$
$137$	−8.77278	−0.749509	−0.374755	+	0.927124i	$0.622273\pi$
$138$	0	0
$139$	−13.3382	−1.13133	−0.565667	−	0.824634i	$-0.691381\pi$
$139$	−13.3382	−1.13133	−0.565667	+	0.824634i	$0.691381\pi$
$140$	−30.8023	−2.60327
$141$	0	0
$142$	18.2086	1.52803
$143$	−0.251187	−0.0210053
$144$	0	0
$145$	−9.03925	−0.750669
$146$	−22.9400	−1.89852
$147$	0	0
$148$	18.4059	1.51296
$149$	5.16087	0.422795	0.211397	−	0.977400i	$-0.432199\pi$
$149$	5.16087	0.422795	0.211397	+	0.977400i	$0.432199\pi$
$150$	0	0
$151$	7.15492	0.582259	0.291130	−	0.956684i	$-0.405969\pi$
$151$	7.15492	0.582259	0.291130	+	0.956684i	$0.405969\pi$
$152$	−53.7602	−4.36053
$153$	0	0
$154$	6.03484	0.486301
$155$	−22.9024	−1.83956
$156$	0	0
$157$	18.9675	1.51377	0.756887	−	0.653546i	$-0.226719\pi$
$157$	18.9675	1.51377	0.756887	+	0.653546i	$0.226719\pi$
$158$	0.333132	0.0265025
$159$	0	0
$160$	−33.8877	−2.67905
$161$	2.82848	0.222916
$162$	0	0
$163$	−2.08285	−0.163142	−0.0815708	−	0.996668i	$-0.525994\pi$
$163$	−2.08285	−0.163142	−0.0815708	+	0.996668i	$0.525994\pi$
$164$	1.55010	0.121042
$165$	0	0
$166$	7.96162	0.617941
$167$	−8.65764	−0.669948	−0.334974	−	0.942227i	$-0.608728\pi$
$167$	−8.65764	−0.669948	−0.334974	+	0.942227i	$0.608728\pi$
$168$	0	0
$169$	−12.9722	−0.997858
$170$	−84.3438	−6.46887
$171$	0	0
$172$	21.8338	1.66481
$173$	0.0384860	0.00292603	0.00146302	−	0.999999i	$-0.499534\pi$
$173$	0.0384860	0.00292603	0.00146302	+	0.999999i	$0.499534\pi$
$174$	0	0
$175$	−20.3372	−1.53735
$176$	12.6604	0.954312
$177$	0	0
$178$	−23.8723	−1.78930
$179$	11.9385	0.892327	0.446163	−	0.894951i	$-0.352790\pi$
$179$	11.9385	0.892327	0.446163	+	0.894951i	$0.352790\pi$
$180$	0	0
$181$	−23.4257	−1.74122	−0.870611	−	0.491972i	$-0.836276\pi$
$181$	−23.4257	−1.74122	−0.870611	+	0.491972i	$0.836276\pi$
$182$	−0.669016	−0.0495908
$183$	0	0
$184$	12.5121	0.922401
$185$	16.7933	1.23467
$186$	0	0
$187$	11.5639	0.845636
$188$	5.80121	0.423097
$189$	0	0
$190$	−85.9011	−6.23192
$191$	−13.4857	−0.975794	−0.487897	−	0.872901i	$-0.662236\pi$
$191$	−13.4857	−0.975794	−0.487897	+	0.872901i	$0.662236\pi$
$192$	0	0
$193$	25.9852	1.87045	0.935226	−	0.354051i	$-0.115196\pi$
$193$	25.9852	1.87045	0.935226	+	0.354051i	$0.115196\pi$
$194$	26.1948	1.88068
$195$	0	0
$196$	−21.3865	−1.52761
$197$	11.3574	0.809183	0.404592	−	0.914497i	$-0.367414\pi$
$197$	11.3574	0.809183	0.404592	+	0.914497i	$0.367414\pi$
$198$	0	0
$199$	−16.7821	−1.18965	−0.594827	−	0.803853i	$-0.702780\pi$
$199$	−16.7821	−1.18965	−0.594827	+	0.803853i	$0.702780\pi$
$200$	−89.9636	−6.36139
$201$	0	0
$202$	−18.0273	−1.26840
$203$	3.30084	0.231673
$204$	0	0
$205$	1.41429	0.0987781
$206$	−27.5489	−1.91942
$207$	0	0
$208$	−1.40352	−0.0973164
$209$	11.7774	0.814661
$210$	0	0
$211$	7.33286	0.504815	0.252407	−	0.967621i	$-0.418778\pi$
$211$	7.33286	0.504815	0.252407	+	0.967621i	$0.418778\pi$
$212$	44.8125	3.07774
$213$	0	0
$214$	16.6959	1.14131
$215$	19.9208	1.35859
$216$	0	0
$217$	8.36319	0.567731
$218$	13.5318	0.916491
$219$	0	0
$220$	29.8503	2.01250
$221$	−1.28196	−0.0862342
$222$	0	0
$223$	3.77173	0.252574	0.126287	−	0.991994i	$-0.459694\pi$
$223$	3.77173	0.252574	0.126287	+	0.991994i	$0.459694\pi$
$224$	12.3746	0.826816
$225$	0	0
$226$	12.1829	0.810397
$227$	−2.68713	−0.178351	−0.0891757	−	0.996016i	$-0.528423\pi$
$227$	−2.68713	−0.178351	−0.0891757	+	0.996016i	$0.528423\pi$
$228$	0	0
$229$	−5.12320	−0.338550	−0.169275	−	0.985569i	$-0.554143\pi$
$229$	−5.12320	−0.338550	−0.169275	+	0.985569i	$0.554143\pi$
$230$	19.9925	1.31827
$231$	0	0
$232$	14.6016	0.958639
$233$	16.4945	1.08059	0.540295	−	0.841476i	$-0.318313\pi$
$233$	16.4945	1.08059	0.540295	+	0.841476i	$0.318313\pi$
$234$	0	0
$235$	5.29294	0.345273
$236$	−70.9685	−4.61965
$237$	0	0
$238$	30.7996	1.99644
$239$	6.38220	0.412830	0.206415	−	0.978465i	$-0.433820\pi$
$239$	6.38220	0.412830	0.206415	+	0.978465i	$0.433820\pi$
$240$	0	0
$241$	4.11739	0.265225	0.132612	−	0.991168i	$-0.457663\pi$
$241$	4.11739	0.265225	0.132612	+	0.991168i	$0.457663\pi$
$242$	22.5438	1.44917
$243$	0	0
$244$	−22.6918	−1.45269
$245$	−19.5127	−1.24662
$246$	0	0
$247$	−1.30563	−0.0830755
$248$	36.9954	2.34921
$249$	0	0
$250$	−88.8542	−5.61963
$251$	19.6004	1.23717	0.618583	−	0.785719i	$-0.287707\pi$
$251$	19.6004	1.23717	0.618583	+	0.785719i	$0.287707\pi$
$252$	0	0
$253$	−2.74106	−0.172329
$254$	26.1089	1.63822
$255$	0	0
$256$	−23.6824	−1.48015
$257$	−19.4944	−1.21603	−0.608014	−	0.793926i	$-0.708034\pi$
$257$	−19.4944	−1.21603	−0.608014	+	0.793926i	$0.708034\pi$
$258$	0	0
$259$	−6.13235	−0.381045
$260$	−3.30917	−0.205226
$261$	0	0
$262$	−8.10587	−0.500782
$263$	−8.00691	−0.493727	−0.246864	−	0.969050i	$-0.579400\pi$
$263$	−8.00691	−0.493727	−0.246864	+	0.969050i	$0.579400\pi$
$264$	0	0
$265$	40.8863	2.51162
$266$	31.3682	1.92331
$267$	0	0
$268$	43.4155	2.65203
$269$	−6.64635	−0.405235	−0.202617	−	0.979258i	$-0.564945\pi$
$269$	−6.64635	−0.405235	−0.202617	+	0.979258i	$0.564945\pi$
$270$	0	0
$271$	16.7286	1.01619	0.508095	−	0.861301i	$-0.330350\pi$
$271$	16.7286	1.01619	0.508095	+	0.861301i	$0.330350\pi$
$272$	64.6138	3.91779
$273$	0	0
$274$	22.6434	1.36794
$275$	19.7086	1.18847
$276$	0	0
$277$	13.7869	0.828373	0.414186	−	0.910192i	$-0.364066\pi$
$277$	13.7869	0.828373	0.414186	+	0.910192i	$0.364066\pi$
$278$	34.4273	2.06481
$279$	0	0
$280$	45.3971	2.71300
$281$	−24.5323	−1.46347	−0.731737	−	0.681587i	$-0.761290\pi$
$281$	−24.5323	−1.46347	−0.731737	+	0.681587i	$0.761290\pi$
$282$	0	0
$283$	−22.6649	−1.34729	−0.673644	−	0.739056i	$-0.735272\pi$
$283$	−22.6649	−1.34729	−0.673644	+	0.739056i	$0.735272\pi$
$284$	−32.8890	−1.95160
$285$	0	0
$286$	0.648338	0.0383370
$287$	−0.516451	−0.0304851
$288$	0	0
$289$	42.0179	2.47164
$290$	23.3312	1.37006
$291$	0	0
$292$	41.4349	2.42480
$293$	−2.60088	−0.151945	−0.0759725	−	0.997110i	$-0.524206\pi$
$293$	−2.60088	−0.151945	−0.0759725	+	0.997110i	$0.524206\pi$
$294$	0	0
$295$	−64.7506	−3.76993
$296$	−27.1270	−1.57673
$297$	0	0
$298$	−13.3207	−0.771647
$299$	0.303871	0.0175733
$300$	0	0
$301$	−7.27443	−0.419291
$302$	−18.4676	−1.06269
$303$	0	0
$304$	65.8069	3.77428
$305$	−20.7037	−1.18549
$306$	0	0
$307$	−18.3612	−1.04793	−0.523964	−	0.851741i	$-0.675547\pi$
$307$	−18.3612	−1.04793	−0.523964	+	0.851741i	$0.675547\pi$
$308$	−10.9003	−0.621104
$309$	0	0
$310$	59.1133	3.35741
$311$	−9.92640	−0.562874	−0.281437	−	0.959580i	$-0.590811\pi$
$311$	−9.92640	−0.562874	−0.281437	+	0.959580i	$0.590811\pi$
$312$	0	0
$313$	−2.26752	−0.128168	−0.0640838	−	0.997945i	$-0.520412\pi$
$313$	−2.26752	−0.128168	−0.0640838	+	0.997945i	$0.520412\pi$
$314$	−48.9571	−2.76281
$315$	0	0
$316$	−0.601714	−0.0338491
$317$	−23.9947	−1.34767	−0.673837	−	0.738880i	$-0.735355\pi$
$317$	−23.9947	−1.34767	−0.673837	+	0.738880i	$0.735355\pi$
$318$	0	0
$319$	−3.19881	−0.179099
$320$	15.9155	0.889706
$321$	0	0
$322$	−7.30059	−0.406846
$323$	60.1076	3.34447
$324$	0	0
$325$	−2.18488	−0.121195
$326$	5.37604	0.297752
$327$	0	0
$328$	−2.28457	−0.126144
$329$	−1.93281	−0.106559
$330$	0	0
$331$	−2.07464	−0.114032	−0.0570162	−	0.998373i	$-0.518159\pi$
$331$	−2.07464	−0.114032	−0.0570162	+	0.998373i	$0.518159\pi$
$332$	−14.3805	−0.789235
$333$	0	0
$334$	22.3462	1.22273
$335$	39.6117	2.16422
$336$	0	0
$337$	1.88017	0.102419	0.0512097	−	0.998688i	$-0.483692\pi$
$337$	1.88017	0.102419	0.0512097	+	0.998688i	$0.483692\pi$
$338$	33.4824	1.82120
$339$	0	0
$340$	152.345	8.26205
$341$	−8.10470	−0.438894
$342$	0	0
$343$	17.9983	0.971818
$344$	−32.1791	−1.73498
$345$	0	0
$346$	−0.0993360	−0.00534034
$347$	28.8389	1.54815	0.774076	−	0.633093i	$-0.218215\pi$
$347$	28.8389	1.54815	0.774076	+	0.633093i	$0.218215\pi$
$348$	0	0
$349$	0.147924	0.00791819	0.00395910	−	0.999992i	$-0.498740\pi$
$349$	0.147924	0.00791819	0.00395910	+	0.999992i	$0.498740\pi$
$350$	52.4924	2.80584
$351$	0	0
$352$	−11.9922	−0.639184
$353$	2.49447	0.132767	0.0663836	−	0.997794i	$-0.478854\pi$
$353$	2.49447	0.132767	0.0663836	+	0.997794i	$0.478854\pi$
$354$	0	0
$355$	−30.0075	−1.59263
$356$	43.1189	2.28530
$357$	0	0
$358$	−30.8145	−1.62860
$359$	−13.3856	−0.706467	−0.353234	−	0.935535i	$-0.614918\pi$
$359$	−13.3856	−0.706467	−0.353234	+	0.935535i	$0.614918\pi$
$360$	0	0
$361$	42.2174	2.22197
$362$	60.4641	3.17792
$363$	0	0
$364$	1.20840	0.0633374
$365$	37.8046	1.97879
$366$	0	0
$367$	21.8109	1.13852	0.569260	−	0.822158i	$-0.307230\pi$
$367$	21.8109	1.13852	0.569260	+	0.822158i	$0.307230\pi$
$368$	−15.3158	−0.798390
$369$	0	0
$370$	−43.3451	−2.25340
$371$	−14.9303	−0.775143
$372$	0	0
$373$	−13.9236	−0.720935	−0.360467	−	0.932772i	$-0.617383\pi$
$373$	−13.9236	−0.720935	−0.360467	+	0.932772i	$0.617383\pi$
$374$	−29.8476	−1.54338
$375$	0	0
$376$	−8.54995	−0.440930
$377$	0.354617	0.0182637
$378$	0	0
$379$	11.6065	0.596187	0.298094	−	0.954537i	$-0.403649\pi$
$379$	11.6065	0.596187	0.298094	+	0.954537i	$0.403649\pi$
$380$	155.158	7.95941
$381$	0	0
$382$	34.8080	1.78093
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	0	0
$385$	−9.94530	−0.506859
$386$	−67.0702	−3.41378
$387$	0	0
$388$	−47.3139	−2.40200
$389$	11.0407	0.559786	0.279893	−	0.960031i	$-0.409701\pi$
$389$	11.0407	0.559786	0.279893	+	0.960031i	$0.409701\pi$
$390$	0	0
$391$	−13.9893	−0.707471
$392$	31.5199	1.59199
$393$	0	0
$394$	−29.3146	−1.47685
$395$	−0.548995	−0.0276229
$396$	0	0
$397$	−11.4621	−0.575265	−0.287633	−	0.957741i	$-0.592868\pi$
$397$	−11.4621	−0.575265	−0.287633	+	0.957741i	$0.592868\pi$
$398$	43.3164	2.17125
$399$	0	0
$400$	110.123	5.50614
$401$	17.0940	0.853635	0.426817	−	0.904338i	$-0.359635\pi$
$401$	17.0940	0.853635	0.426817	+	0.904338i	$0.359635\pi$
$402$	0	0
$403$	0.898479	0.0447564
$404$	32.5616	1.62000
$405$	0	0
$406$	−8.51978	−0.422830
$407$	5.94280	0.294574
$408$	0	0
$409$	15.8540	0.783929	0.391965	−	0.919980i	$-0.371796\pi$
$409$	15.8540	0.783929	0.391965	+	0.919980i	$0.371796\pi$
$410$	−3.65041	−0.180281
$411$	0	0
$412$	49.7597	2.45149
$413$	23.6448	1.16348
$414$	0	0
$415$	−13.1206	−0.644065
$416$	1.32944	0.0651811
$417$	0	0
$418$	−30.3987	−1.48685
$419$	−13.9445	−0.681234	−0.340617	−	0.940202i	$-0.610636\pi$
$419$	−13.9445	−0.681234	−0.340617	+	0.940202i	$0.610636\pi$
$420$	0	0
$421$	−22.0016	−1.07229	−0.536147	−	0.844125i	$-0.680121\pi$
$421$	−22.0016	−1.07229	−0.536147	+	0.844125i	$0.680121\pi$
$422$	−18.9268	−0.921343
$423$	0	0
$424$	−66.0456	−3.20746
$425$	100.585	4.87911
$426$	0	0
$427$	7.56029	0.365868
$428$	−30.1567	−1.45768
$429$	0	0
$430$	−51.4176	−2.47958
$431$	33.5644	1.61674	0.808370	−	0.588675i	$-0.200350\pi$
$431$	33.5644	1.61674	0.808370	+	0.588675i	$0.200350\pi$
$432$	0	0
$433$	−24.2287	−1.16436	−0.582178	−	0.813061i	$-0.697799\pi$
$433$	−24.2287	−1.16436	−0.582178	+	0.813061i	$0.697799\pi$
$434$	−21.5862	−1.03617
$435$	0	0
$436$	−24.4416	−1.17054
$437$	−14.2476	−0.681557
$438$	0	0
$439$	9.16240	0.437297	0.218649	−	0.975804i	$-0.429835\pi$
$439$	9.16240	0.437297	0.218649	+	0.975804i	$0.429835\pi$
$440$	−43.9940	−2.09733
$441$	0	0
$442$	3.30887	0.157387
$443$	−1.91989	−0.0912170	−0.0456085	−	0.998959i	$-0.514523\pi$
$443$	−1.91989	−0.0912170	−0.0456085	+	0.998959i	$0.514523\pi$
$444$	0	0
$445$	39.3410	1.86494
$446$	−9.73521	−0.460976
$447$	0	0
$448$	−5.81183	−0.274583
$449$	17.4866	0.825243	0.412621	−	0.910903i	$-0.364613\pi$
$449$	17.4866	0.825243	0.412621	+	0.910903i	$0.364613\pi$
$450$	0	0
$451$	0.500488	0.0235671
$452$	−22.0052	−1.03504
$453$	0	0
$454$	6.93576	0.325511
$455$	1.10253	0.0516872
$456$	0	0
$457$	24.5140	1.14672	0.573359	−	0.819304i	$-0.305640\pi$
$457$	24.5140	1.14672	0.573359	+	0.819304i	$0.305640\pi$
$458$	13.2235	0.617892
$459$	0	0
$460$	−36.1111	−1.68369
$461$	−11.9651	−0.557268	−0.278634	−	0.960397i	$-0.589882\pi$
$461$	−11.9651	−0.557268	−0.278634	+	0.960397i	$0.589882\pi$
$462$	0	0
$463$	−12.5200	−0.581852	−0.290926	−	0.956745i	$-0.593963\pi$
$463$	−12.5200	−0.581852	−0.290926	+	0.956745i	$0.593963\pi$
$464$	−17.8735	−0.829756
$465$	0	0
$466$	−42.5739	−1.97220
$467$	−33.4951	−1.54997	−0.774985	−	0.631980i	$-0.782242\pi$
$467$	−33.4951	−1.54997	−0.774985	+	0.631980i	$0.782242\pi$
$468$	0	0
$469$	−14.4649	−0.667926
$470$	−13.6616	−0.630163
$471$	0	0
$472$	104.595	4.81437
$473$	7.04958	0.324140
$474$	0	0
$475$	102.443	4.70039
$476$	−55.6312	−2.54985
$477$	0	0
$478$	−16.4731	−0.753461
$479$	−26.2231	−1.19816	−0.599082	−	0.800687i	$-0.704468\pi$
$479$	−26.2231	−1.19816	−0.599082	+	0.800687i	$0.704468\pi$
$480$	0	0
$481$	−0.658813	−0.0300393
$482$	−10.6274	−0.484065
$483$	0	0
$484$	−40.7193	−1.85088
$485$	−43.1685	−1.96018
$486$	0	0
$487$	−32.4390	−1.46995	−0.734976	−	0.678093i	$-0.762807\pi$
$487$	−32.4390	−1.46995	−0.734976	+	0.678093i	$0.762807\pi$
$488$	33.4437	1.51392
$489$	0	0
$490$	50.3642	2.27522
$491$	37.1110	1.67479	0.837397	−	0.546595i	$-0.184076\pi$
$491$	37.1110	1.67479	0.837397	+	0.546595i	$0.184076\pi$
$492$	0	0
$493$	−16.3255	−0.735265
$494$	3.36997	0.151622
$495$	0	0
$496$	−45.2853	−2.03337
$497$	10.9577	0.491521
$498$	0	0
$499$	5.79833	0.259569	0.129784	−	0.991542i	$-0.458571\pi$
$499$	5.79833	0.259569	0.129784	+	0.991542i	$0.458571\pi$
$500$	160.491	7.17740
$501$	0	0
$502$	−50.5906	−2.25797
$503$	−10.9238	−0.487067	−0.243533	−	0.969893i	$-0.578307\pi$
$503$	−10.9238	−0.487067	−0.243533	+	0.969893i	$0.578307\pi$
$504$	0	0
$505$	29.7087	1.32202
$506$	7.07494	0.314519
$507$	0	0
$508$	−47.1588	−2.09233
$509$	7.86855	0.348767	0.174384	−	0.984678i	$-0.444207\pi$
$509$	7.86855	0.348767	0.174384	+	0.984678i	$0.444207\pi$
$510$	0	0
$511$	−13.8050	−0.610697
$512$	48.5745	2.14671
$513$	0	0
$514$	50.3170	2.21939
$515$	45.4001	2.00057
$516$	0	0
$517$	1.87307	0.0823773
$518$	15.8282	0.695451
$519$	0	0
$520$	4.87713	0.213876
$521$	14.6913	0.643637	0.321818	−	0.946801i	$-0.395706\pi$
$521$	14.6913	0.643637	0.321818	+	0.946801i	$0.395706\pi$
$522$	0	0
$523$	8.37124	0.366049	0.183024	−	0.983108i	$-0.441411\pi$
$523$	8.37124	0.366049	0.183024	+	0.983108i	$0.441411\pi$
$524$	14.6411	0.639600
$525$	0	0
$526$	20.6666	0.901107
$527$	−41.3633	−1.80182
$528$	0	0
$529$	−19.6840	−0.855827
$530$	−105.531	−4.58400
$531$	0	0
$532$	−56.6584	−2.45645
$533$	−0.0554836	−0.00240326
$534$	0	0
$535$	−27.5145	−1.18956
$536$	−63.9868	−2.76381
$537$	0	0
$538$	17.1549	0.739599
$539$	−6.90516	−0.297426
$540$	0	0
$541$	45.2764	1.94658	0.973292	−	0.229573i	$-0.0737328\pi$
$541$	45.2764	1.94658	0.973292	+	0.229573i	$0.0737328\pi$
$542$	−43.1782	−1.85466
$543$	0	0
$544$	−61.2035	−2.62408
$545$	−22.3002	−0.955236
$546$	0	0
$547$	33.4144	1.42870	0.714348	−	0.699790i	$-0.246723\pi$
$547$	33.4144	1.42870	0.714348	+	0.699790i	$0.246723\pi$
$548$	−40.8993	−1.74713
$549$	0	0
$550$	−50.8699	−2.16910
$551$	−16.6270	−0.708333
$552$	0	0
$553$	0.200475	0.00852506
$554$	−35.5853	−1.51187
$555$	0	0
$556$	−62.1838	−2.63718
$557$	19.5238	0.827252	0.413626	−	0.910447i	$-0.364262\pi$
$557$	19.5238	0.827252	0.413626	+	0.910447i	$0.364262\pi$
$558$	0	0
$559$	−0.781510	−0.0330543
$560$	−55.5698	−2.34825
$561$	0	0
$562$	63.3202	2.67100
$563$	−6.70777	−0.282699	−0.141349	−	0.989960i	$-0.545144\pi$
$563$	−6.70777	−0.282699	−0.141349	+	0.989960i	$0.545144\pi$
$564$	0	0
$565$	−20.0773	−0.844657
$566$	58.5003	2.45895
$567$	0	0
$568$	48.4725	2.03386
$569$	−1.49454	−0.0626543	−0.0313271	−	0.999509i	$-0.509973\pi$
$569$	−1.49454	−0.0626543	−0.0313271	+	0.999509i	$0.509973\pi$
$570$	0	0
$571$	30.9339	1.29454	0.647271	−	0.762260i	$-0.275910\pi$
$571$	30.9339	1.29454	0.647271	+	0.762260i	$0.275910\pi$
$572$	−1.17105	−0.0489640
$573$	0	0
$574$	1.33301	0.0556388
$575$	−23.8423	−0.994294
$576$	0	0
$577$	27.8334	1.15872	0.579359	−	0.815072i	$-0.303303\pi$
$577$	27.8334	1.15872	0.579359	+	0.815072i	$0.303303\pi$
$578$	−108.452	−4.51102
$579$	0	0
$580$	−42.1416	−1.74983
$581$	4.79121	0.198773
$582$	0	0
$583$	14.4688	0.599238
$584$	−61.0677	−2.52700
$585$	0	0
$586$	6.71313	0.277317
$587$	16.6001	0.685159	0.342580	−	0.939489i	$-0.388699\pi$
$587$	16.6001	0.685159	0.342580	+	0.939489i	$0.388699\pi$
$588$	0	0
$589$	−42.1271	−1.73582
$590$	167.128	6.88054
$591$	0	0
$592$	33.2057	1.36474
$593$	8.39100	0.344577	0.172288	−	0.985047i	$-0.444884\pi$
$593$	8.39100	0.344577	0.172288	+	0.985047i	$0.444884\pi$
$594$	0	0
$595$	−50.7571	−2.08084
$596$	24.0603	0.985548
$597$	0	0
$598$	−0.784321	−0.0320733
$599$	9.91342	0.405051	0.202526	−	0.979277i	$-0.435085\pi$
$599$	9.91342	0.405051	0.202526	+	0.979277i	$0.435085\pi$
$600$	0	0
$601$	9.98252	0.407195	0.203598	−	0.979055i	$-0.434737\pi$
$601$	9.98252	0.407195	0.203598	+	0.979055i	$0.434737\pi$
$602$	18.7760	0.765253
$603$	0	0
$604$	33.3567	1.35727
$605$	−37.1517	−1.51043
$606$	0	0
$607$	−13.6457	−0.553860	−0.276930	−	0.960890i	$-0.589317\pi$
$607$	−13.6457	−0.553860	−0.276930	+	0.960890i	$0.589317\pi$
$608$	−62.3336	−2.52796
$609$	0	0
$610$	53.4382	2.16365
$611$	−0.207646	−0.00840047
$612$	0	0
$613$	−4.77723	−0.192951	−0.0964753	−	0.995335i	$-0.530757\pi$
$613$	−4.77723	−0.192951	−0.0964753	+	0.995335i	$0.530757\pi$
$614$	47.3920	1.91258
$615$	0	0
$616$	16.0651	0.647283
$617$	−41.7811	−1.68204	−0.841021	−	0.541002i	$-0.818045\pi$
$617$	−41.7811	−1.68204	−0.841021	+	0.541002i	$0.818045\pi$
$618$	0	0
$619$	2.02804	0.0815138	0.0407569	−	0.999169i	$-0.487023\pi$
$619$	2.02804	0.0815138	0.0407569	+	0.999169i	$0.487023\pi$
$620$	−106.772	−4.28809
$621$	0	0
$622$	25.6210	1.02731
$623$	−14.3660	−0.575563
$624$	0	0
$625$	80.9644	3.23858
$626$	5.85268	0.233920
$627$	0	0
$628$	88.4279	3.52866
$629$	30.3298	1.20933
$630$	0	0
$631$	−34.8608	−1.38779	−0.693893	−	0.720079i	$-0.744106\pi$
$631$	−34.8608	−1.38779	−0.693893	+	0.720079i	$0.744106\pi$
$632$	0.886819	0.0352758
$633$	0	0
$634$	61.9326	2.45966
$635$	−43.0270	−1.70747
$636$	0	0
$637$	0.765499	0.0303302
$638$	8.25644	0.326876
$639$	0	0
$640$	26.6957	1.05524
$641$	−13.2080	−0.521683	−0.260842	−	0.965382i	$-0.584000\pi$
$641$	−13.2080	−0.521683	−0.260842	+	0.965382i	$0.584000\pi$
$642$	0	0
$643$	−25.7071	−1.01379	−0.506894	−	0.862008i	$-0.669207\pi$
$643$	−25.7071	−1.01379	−0.506894	+	0.862008i	$0.669207\pi$
$644$	13.1866	0.519624
$645$	0	0
$646$	−155.144	−6.10404
$647$	−5.44019	−0.213876	−0.106938	−	0.994266i	$-0.534105\pi$
$647$	−5.44019	−0.213876	−0.106938	+	0.994266i	$0.534105\pi$
$648$	0	0
$649$	−22.9139	−0.899451
$650$	5.63939	0.221195
$651$	0	0
$652$	−9.71039	−0.380288
$653$	16.6654	0.652167	0.326084	−	0.945341i	$-0.394271\pi$
$653$	16.6654	0.652167	0.326084	+	0.945341i	$0.394271\pi$
$654$	0	0
$655$	13.3583	0.521953
$656$	2.79650	0.109185
$657$	0	0
$658$	4.98876	0.194482
$659$	−8.50057	−0.331135	−0.165568	−	0.986198i	$-0.552946\pi$
$659$	−8.50057	−0.331135	−0.165568	+	0.986198i	$0.552946\pi$
$660$	0	0
$661$	−7.81643	−0.304024	−0.152012	−	0.988379i	$-0.548575\pi$
$661$	−7.81643	−0.304024	−0.152012	+	0.988379i	$0.548575\pi$
$662$	5.35484	0.208122
$663$	0	0
$664$	21.1944	0.822500
$665$	−51.6943	−2.00462
$666$	0	0
$667$	3.86973	0.149837
$668$	−40.3625	−1.56167
$669$	0	0
$670$	−102.242	−3.94994
$671$	−7.32661	−0.282841
$672$	0	0
$673$	0.334170	0.0128813	0.00644065	−	0.999979i	$-0.497950\pi$
$673$	0.334170	0.0128813	0.00644065	+	0.999979i	$0.497950\pi$
$674$	−4.85290	−0.186927
$675$	0	0
$676$	−60.4770	−2.32604
$677$	−9.17504	−0.352625	−0.176313	−	0.984334i	$-0.556417\pi$
$677$	−9.17504	−0.352625	−0.176313	+	0.984334i	$0.556417\pi$
$678$	0	0
$679$	15.7637	0.604956
$680$	−224.529	−8.61028
$681$	0	0
$682$	20.9190	0.801030
$683$	20.6041	0.788392	0.394196	−	0.919026i	$-0.371023\pi$
$683$	20.6041	0.788392	0.394196	+	0.919026i	$0.371023\pi$
$684$	0	0
$685$	−37.3159	−1.42577
$686$	−46.4554	−1.77368
$687$	0	0
$688$	39.3899	1.50172
$689$	−1.60400	−0.0611076
$690$	0	0
$691$	−31.4224	−1.19537	−0.597683	−	0.801733i	$-0.703912\pi$
$691$	−31.4224	−1.19537	−0.597683	+	0.801733i	$0.703912\pi$
$692$	0.179424	0.00682068
$693$	0	0
$694$	−74.4360	−2.82555
$695$	−56.7356	−2.15210
$696$	0	0
$697$	2.55430	0.0967512
$698$	−0.381807	−0.0144516
$699$	0	0
$700$	−94.8135	−3.58361
$701$	17.3859	0.656655	0.328328	−	0.944564i	$-0.393515\pi$
$701$	17.3859	0.656655	0.328328	+	0.944564i	$0.393515\pi$
$702$	0	0
$703$	30.8899	1.16503
$704$	5.63219	0.212271
$705$	0	0
$706$	−6.43847	−0.242315
$707$	−10.8486	−0.408005
$708$	0	0
$709$	38.2186	1.43533	0.717665	−	0.696389i	$-0.245211\pi$
$709$	38.2186	1.43533	0.717665	+	0.696389i	$0.245211\pi$
$710$	77.4522	2.90673
$711$	0	0
$712$	−63.5495	−2.38162
$713$	9.80459	0.367185
$714$	0	0
$715$	−1.06845	−0.0399577
$716$	55.6582	2.08004
$717$	0	0
$718$	34.5497	1.28938
$719$	19.7424	0.736268	0.368134	−	0.929773i	$-0.379997\pi$
$719$	19.7424	0.736268	0.368134	+	0.929773i	$0.379997\pi$
$720$	0	0
$721$	−16.5786	−0.617419
$722$	−108.967	−4.05534
$723$	0	0
$724$	−109.212	−4.05885
$725$	−27.8240	−1.03336
$726$	0	0
$727$	−22.2866	−0.826564	−0.413282	−	0.910603i	$-0.635618\pi$
$727$	−22.2866	−0.826564	−0.413282	+	0.910603i	$0.635618\pi$
$728$	−1.78097	−0.0660070
$729$	0	0
$730$	−97.5775	−3.61150
$731$	35.9784	1.33071
$732$	0	0
$733$	26.6840	0.985596	0.492798	−	0.870144i	$-0.335974\pi$
$733$	26.6840	0.985596	0.492798	+	0.870144i	$0.335974\pi$
$734$	−56.2960	−2.07792
$735$	0	0
$736$	14.5074	0.534750
$737$	14.0178	0.516352
$738$	0	0
$739$	26.9464	0.991240	0.495620	−	0.868540i	$-0.334941\pi$
$739$	26.9464	0.991240	0.495620	+	0.868540i	$0.334941\pi$
$740$	78.2914	2.87805
$741$	0	0
$742$	38.5366	1.41472
$743$	−10.8798	−0.399140	−0.199570	−	0.979884i	$-0.563954\pi$
$743$	−10.8798	−0.399140	−0.199570	+	0.979884i	$0.563954\pi$
$744$	0	0
$745$	21.9523	0.804269
$746$	35.9381	1.31579
$747$	0	0
$748$	53.9117	1.97121
$749$	10.0474	0.367124
$750$	0	0
$751$	46.3601	1.69170	0.845852	−	0.533417i	$-0.179092\pi$
$751$	46.3601	1.69170	0.845852	+	0.533417i	$0.179092\pi$
$752$	10.4658	0.381650
$753$	0	0
$754$	−0.915301	−0.0333333
$755$	30.4342	1.10761
$756$	0	0
$757$	41.1991	1.49741	0.748704	−	0.662905i	$-0.230676\pi$
$757$	41.1991	1.49741	0.748704	+	0.662905i	$0.230676\pi$
$758$	−29.9576	−1.08811
$759$	0	0
$760$	−228.675	−8.29490
$761$	9.29105	0.336800	0.168400	−	0.985719i	$-0.446140\pi$
$761$	9.29105	0.336800	0.168400	+	0.985719i	$0.446140\pi$
$762$	0	0
$763$	8.14330	0.294807
$764$	−62.8714	−2.27461
$765$	0	0
$766$	2.58110	0.0932589
$767$	2.54022	0.0917219
$768$	0	0
$769$	−26.4255	−0.952928	−0.476464	−	0.879194i	$-0.658082\pi$
$769$	−26.4255	−0.952928	−0.476464	+	0.879194i	$0.658082\pi$
$770$	25.6698	0.925075
$771$	0	0
$772$	121.145	4.36009
$773$	−28.5566	−1.02711	−0.513554	−	0.858057i	$-0.671671\pi$
$773$	−28.5566	−1.02711	−0.513554	+	0.858057i	$0.671671\pi$
$774$	0	0
$775$	−70.4965	−2.53231
$776$	69.7323	2.50324
$777$	0	0
$778$	−28.4971	−1.02167
$779$	2.60147	0.0932072
$780$	0	0
$781$	−10.6190	−0.379979
$782$	36.1078	1.29121
$783$	0	0
$784$	−38.5829	−1.37796
$785$	80.6804	2.87961
$786$	0	0
$787$	31.1708	1.11112	0.555559	−	0.831477i	$-0.312504\pi$
$787$	31.1708	1.11112	0.555559	+	0.831477i	$0.312504\pi$
$788$	52.9491	1.88623
$789$	0	0
$790$	1.41701	0.0504150
$791$	7.33155	0.260680
$792$	0	0
$793$	0.812221	0.0288428
$794$	29.5848	1.04992
$795$	0	0
$796$	−78.2395	−2.77313
$797$	−23.3338	−0.826524	−0.413262	−	0.910612i	$-0.635611\pi$
$797$	−23.3338	−0.826524	−0.413262	+	0.910612i	$0.635611\pi$
$798$	0	0
$799$	9.55943	0.338188
$800$	−104.311	−3.68793
$801$	0	0
$802$	−44.1214	−1.55798
$803$	13.3783	0.472110
$804$	0	0
$805$	12.0312	0.424046
$806$	−2.31906	−0.0816855
$807$	0	0
$808$	−47.9900	−1.68828
$809$	−29.6986	−1.04415	−0.522074	−	0.852900i	$-0.674841\pi$
$809$	−29.6986	−1.04415	−0.522074	+	0.852900i	$0.674841\pi$
$810$	0	0
$811$	−26.5649	−0.932820	−0.466410	−	0.884569i	$-0.654453\pi$
$811$	−26.5649	−0.932820	−0.466410	+	0.884569i	$0.654453\pi$
$812$	15.3887	0.540038
$813$	0	0
$814$	−15.3390	−0.537630
$815$	−8.85962	−0.310339
$816$	0	0
$817$	36.6427	1.28197
$818$	−40.9207	−1.43076
$819$	0	0
$820$	6.59350	0.230255
$821$	35.1382	1.22633	0.613165	−	0.789955i	$-0.289896\pi$
$821$	35.1382	1.22633	0.613165	+	0.789955i	$0.289896\pi$
$822$	0	0
$823$	−3.41873	−0.119169	−0.0595847	−	0.998223i	$-0.518978\pi$
$823$	−3.41873	−0.119169	−0.0595847	+	0.998223i	$0.518978\pi$
$824$	−73.3370	−2.55481
$825$	0	0
$826$	−61.0295	−2.12349
$827$	23.0916	0.802974	0.401487	−	0.915865i	$-0.368494\pi$
$827$	23.0916	0.802974	0.401487	+	0.915865i	$0.368494\pi$
$828$	0	0
$829$	49.9112	1.73349	0.866743	−	0.498754i	$-0.166209\pi$
$829$	49.9112	1.73349	0.866743	+	0.498754i	$0.166209\pi$
$830$	33.8655	1.17549
$831$	0	0
$832$	−0.624379	−0.0216465
$833$	−35.2413	−1.22104
$834$	0	0
$835$	−36.8262	−1.27442
$836$	54.9071	1.89900
$837$	0	0
$838$	35.9922	1.24333
$839$	5.90244	0.203775	0.101887	−	0.994796i	$-0.467512\pi$
$839$	5.90244	0.203775	0.101887	+	0.994796i	$0.467512\pi$
$840$	0	0
$841$	−24.4840	−0.844277
$842$	56.7884	1.95706
$843$	0	0
$844$	34.1863	1.17674
$845$	−55.1784	−1.89819
$846$	0	0
$847$	13.5666	0.466153
$848$	80.8453	2.77624
$849$	0	0
$850$	−259.621	−8.90492
$851$	−7.18926	−0.246444
$852$	0	0
$853$	−25.5062	−0.873315	−0.436657	−	0.899628i	$-0.643838\pi$
$853$	−25.5062	−0.873315	−0.436657	+	0.899628i	$0.643838\pi$
$854$	−19.5139	−0.667750
$855$	0	0
$856$	44.4456	1.51912
$857$	29.6954	1.01437	0.507187	−	0.861836i	$-0.330685\pi$
$857$	29.6954	1.01437	0.507187	+	0.861836i	$0.330685\pi$
$858$	0	0
$859$	45.1715	1.54123	0.770616	−	0.637300i	$-0.219949\pi$
$859$	45.1715	1.54123	0.770616	+	0.637300i	$0.219949\pi$
$860$	92.8722	3.16692
$861$	0	0
$862$	−86.6330	−2.95073
$863$	−9.41843	−0.320607	−0.160303	−	0.987068i	$-0.551247\pi$
$863$	−9.41843	−0.320607	−0.160303	+	0.987068i	$0.551247\pi$
$864$	0	0
$865$	0.163704	0.00556610
$866$	62.5366	2.12508
$867$	0	0
$868$	38.9898	1.32340
$869$	−0.194278	−0.00659044
$870$	0	0
$871$	−1.55400	−0.0526552
$872$	36.0226	1.21988
$873$	0	0
$874$	36.7746	1.24392
$875$	−53.4714	−1.80766
$876$	0	0
$877$	−50.5960	−1.70851	−0.854253	−	0.519857i	$-0.825985\pi$
$877$	−50.5960	−1.70851	−0.854253	+	0.519857i	$0.825985\pi$
$878$	−23.6490	−0.798117
$879$	0	0
$880$	53.8522	1.81536
$881$	2.72131	0.0916833	0.0458416	−	0.998949i	$-0.485403\pi$
$881$	2.72131	0.0916833	0.0458416	+	0.998949i	$0.485403\pi$
$882$	0	0
$883$	−38.9747	−1.31160	−0.655801	−	0.754934i	$-0.727669\pi$
$883$	−38.9747	−1.31160	−0.655801	+	0.754934i	$0.727669\pi$
$884$	−5.97660	−0.201015
$885$	0	0
$886$	4.95544	0.166481
$887$	−43.2030	−1.45061	−0.725307	−	0.688426i	$-0.758302\pi$
$887$	−43.2030	−1.45061	−0.725307	+	0.688426i	$0.758302\pi$
$888$	0	0
$889$	15.7120	0.526965
$890$	−101.543	−3.40373
$891$	0	0
$892$	17.5841	0.588758
$893$	9.73593	0.325801
$894$	0	0
$895$	50.7817	1.69745
$896$	−9.74840	−0.325671
$897$	0	0
$898$	−45.1346	−1.50616
$899$	11.4419	0.381610
$900$	0	0
$901$	73.8435	2.46008
$902$	−1.29181	−0.0430125
$903$	0	0
$904$	32.4318	1.07867
$905$	−99.6438	−3.31227
$906$	0	0
$907$	−6.38275	−0.211936	−0.105968	−	0.994370i	$-0.533794\pi$
$907$	−6.38275	−0.211936	−0.105968	+	0.994370i	$0.533794\pi$
$908$	−12.5276	−0.415743
$909$	0	0
$910$	−2.84573	−0.0943350
$911$	−48.1594	−1.59559	−0.797797	−	0.602927i	$-0.794001\pi$
$911$	−48.1594	−1.59559	−0.797797	+	0.602927i	$0.794001\pi$
$912$	0	0
$913$	−4.64312	−0.153665
$914$	−63.2731	−2.09289
$915$	0	0
$916$	−23.8847	−0.789172
$917$	−4.87802	−0.161086
$918$	0	0
$919$	−32.2668	−1.06439	−0.532193	−	0.846623i	$-0.678632\pi$
$919$	−32.2668	−1.06439	−0.532193	+	0.846623i	$0.678632\pi$
$920$	53.2213	1.75466
$921$	0	0
$922$	30.8830	1.01708
$923$	1.17722	0.0387486
$924$	0	0
$925$	51.6918	1.69962
$926$	32.3153	1.06195
$927$	0	0
$928$	16.9301	0.555759
$929$	46.6399	1.53020	0.765102	−	0.643909i	$-0.222688\pi$
$929$	46.6399	1.53020	0.765102	+	0.643909i	$0.222688\pi$
$930$	0	0
$931$	−35.8921	−1.17631
$932$	76.8984	2.51889
$933$	0	0
$934$	86.4542	2.82887
$935$	49.1883	1.60863
$936$	0	0
$937$	−8.11948	−0.265252	−0.132626	−	0.991166i	$-0.542341\pi$
$937$	−8.11948	−0.265252	−0.132626	+	0.991166i	$0.542341\pi$
$938$	37.3353	1.21904
$939$	0	0
$940$	24.6760	0.804844
$941$	36.2879	1.18295	0.591476	−	0.806322i	$-0.298545\pi$
$941$	36.2879	1.18295	0.591476	+	0.806322i	$0.298545\pi$
$942$	0	0
$943$	−0.605461	−0.0197165
$944$	−128.033	−4.16711
$945$	0	0
$946$	−18.1957	−0.591592
$947$	27.6412	0.898217	0.449109	−	0.893477i	$-0.351742\pi$
$947$	27.6412	0.898217	0.449109	+	0.893477i	$0.351742\pi$
$948$	0	0
$949$	−1.48311	−0.0481437
$950$	−264.415	−8.57874
$951$	0	0
$952$	81.9905	2.65733
$953$	−9.40326	−0.304602	−0.152301	−	0.988334i	$-0.548668\pi$
$953$	−9.40326	−0.304602	−0.152301	+	0.988334i	$0.548668\pi$
$954$	0	0
$955$	−57.3630	−1.85622
$956$	29.7542	0.962320
$957$	0	0
$958$	67.6844	2.18678
$959$	13.6265	0.440024
$960$	0	0
$961$	−2.01003	−0.0648396
$962$	1.70046	0.0548251
$963$	0	0
$964$	19.1956	0.618248
$965$	110.531	3.55810
$966$	0	0
$967$	11.7730	0.378595	0.189297	−	0.981920i	$-0.439379\pi$
$967$	11.7730	0.378595	0.189297	+	0.981920i	$0.439379\pi$
$968$	60.0130	1.92889
$969$	0	0
$970$	111.422	3.57755
$971$	18.4018	0.590542	0.295271	−	0.955414i	$-0.404590\pi$
$971$	18.4018	0.590542	0.295271	+	0.955414i	$0.404590\pi$
$972$	0	0
$973$	20.7179	0.664187
$974$	83.7283	2.68283
$975$	0	0
$976$	−40.9378	−1.31039
$977$	−23.1625	−0.741035	−0.370517	−	0.928826i	$-0.620820\pi$
$977$	−23.1625	−0.741035	−0.370517	+	0.928826i	$0.620820\pi$
$978$	0	0
$979$	13.9220	0.444949
$980$	−90.9696	−2.90592
$981$	0	0
$982$	−95.7870	−3.05669
$983$	−33.3799	−1.06465	−0.532327	−	0.846539i	$-0.678682\pi$
$983$	−33.3799	−1.06465	−0.532327	+	0.846539i	$0.678682\pi$
$984$	0	0
$985$	48.3100	1.53928
$986$	42.1378	1.34194
$987$	0	0
$988$	−6.08695	−0.193652
$989$	−8.52817	−0.271180
$990$	0	0
$991$	−13.1016	−0.416185	−0.208093	−	0.978109i	$-0.566726\pi$
$991$	−13.1016	−0.416185	−0.208093	+	0.978109i	$0.566726\pi$
$992$	42.8952	1.36192
$993$	0	0
$994$	−28.2830	−0.897082
$995$	−71.3846	−2.26304
$996$	0	0
$997$	31.4260	0.995272	0.497636	−	0.867386i	$-0.334202\pi$
$997$	31.4260	0.995272	0.497636	+	0.867386i	$0.334202\pi$
$998$	−14.9660	−0.473742
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3447.2.a.j.1.2		24
3.2	odd	2		383.2.a.c.1.23	✓	24
12.11	even	2		6128.2.a.p.1.4		24
15.14	odd	2		9575.2.a.e.1.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
383.2.a.c.1.23	✓	24	3.2	odd	2
3447.2.a.j.1.2		24	1.1	even	1	trivial
6128.2.a.p.1.4		24	12.11	even	2
9575.2.a.e.1.2		24	15.14	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3447 = 3^{2} \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3447.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58110 q^{2} +4.66207 q^{4} +4.25360 q^{5} -1.55327 q^{7} -6.87105 q^{8} -10.9790 q^{10} +1.50526 q^{11} -0.166872 q^{13} +4.00915 q^{14} +8.41073 q^{16} +7.68231 q^{17} +7.82416 q^{19} +19.8306 q^{20} -3.88523 q^{22} -1.82098 q^{23} +13.0931 q^{25} +0.430713 q^{26} -7.24147 q^{28} -2.12508 q^{29} -5.38423 q^{31} -7.96681 q^{32} -19.8288 q^{34} -6.60701 q^{35} +3.94801 q^{37} -20.1949 q^{38} -29.2267 q^{40} +0.332492 q^{41} +4.68328 q^{43} +7.01764 q^{44} +4.70013 q^{46} +1.24434 q^{47} -4.58734 q^{49} -33.7947 q^{50} -0.777969 q^{52} +9.61215 q^{53} +6.40280 q^{55} +10.6726 q^{56} +5.48504 q^{58} -15.2225 q^{59} -4.86732 q^{61} +13.8972 q^{62} +3.74166 q^{64} -0.709808 q^{65} +9.31251 q^{67} +35.8154 q^{68} +17.0533 q^{70} -7.05460 q^{71} +8.88768 q^{73} -10.1902 q^{74} +36.4767 q^{76} -2.33809 q^{77} -0.129066 q^{79} +35.7759 q^{80} -0.858194 q^{82} -3.08459 q^{83} +32.6775 q^{85} -12.0880 q^{86} -10.3428 q^{88} +9.24888 q^{89} +0.259198 q^{91} -8.48953 q^{92} -3.21177 q^{94} +33.2808 q^{95} -10.1487 q^{97} +11.8404 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) \) 24 * q - 5 * q^2 + 29 * q^4 - 3 * q^5 + 17 * q^7 - 15 * q^8 + q^10 + 28 * q^13 + 8 * q^14 + 35 * q^16 - 16 * q^17 + 13 * q^19 + 4 * q^20 + 12 * q^22 - 7 * q^23 + 67 * q^25 + 14 * q^26 + 39 * q^28 + 2 * q^29 - 20 * q^32 - 18 * q^34 + 10 * q^35 + 35 * q^37 + 5 * q^38 - 36 * q^40 + 34 * q^43 + 36 * q^44 - 17 * q^46 + 3 * q^47 + 49 * q^49 + 20 * q^50 + 29 * q^52 + 9 * q^53 + 10 * q^55 + 47 * q^56 + 12 * q^58 + 6 * q^59 + 3 * q^61 + 58 * q^62 + 15 * q^64 - 8 * q^65 + 42 * q^67 + 23 * q^68 - 59 * q^70 + 13 * q^71 + 83 * q^73 + 45 * q^74 + 4 * q^76 + 21 * q^77 - 3 * q^79 + 93 * q^80 - 13 * q^82 + 23 * q^83 + 52 * q^85 + 43 * q^86 - 25 * q^88 + 38 * q^89 - 17 * q^91 + 61 * q^92 - 71 * q^94 + 66 * q^95 + 63 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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