LMFDB - Embedded newform 3751.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3751,2,Mod(1,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3751 = 11^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3751.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,1,2,17,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.9518857982$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 23 x^{13} + 21 x^{12} + 204 x^{11} - 160 x^{10} - 880 x^{9} + 535 x^{8} + 1918 x^{7} + \cdots + 4 $ x^15 - x^14 - 23x^13 + 21x^12 + 204x^11 - 160x^10 - 880x^9 + 535x^8 + 1918x^7 - 751x^6 - 1980x^5 + 313x^4 + 845x^3 + 37x^2 - 72*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 23 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.74615$ of defining polynomial
Character	$\chi$	$=$	3751.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.74615 q^{2} -1.55132 q^{3} +1.04905 q^{4} +1.03434 q^{5} +2.70884 q^{6} +5.16265 q^{7} +1.66050 q^{8} -0.593409 q^{9} -1.80611 q^{10} -1.62742 q^{12} -1.03844 q^{13} -9.01478 q^{14} -1.60459 q^{15} -4.99759 q^{16} -2.01238 q^{17} +1.03618 q^{18} -1.68345 q^{19} +1.08508 q^{20} -8.00892 q^{21} +2.13060 q^{23} -2.57596 q^{24} -3.93014 q^{25} +1.81328 q^{26} +5.57452 q^{27} +5.41589 q^{28} +4.45313 q^{29} +2.80186 q^{30} -1.00000 q^{31} +5.40557 q^{32} +3.51393 q^{34} +5.33993 q^{35} -0.622518 q^{36} -2.24305 q^{37} +2.93957 q^{38} +1.61095 q^{39} +1.71752 q^{40} -9.06471 q^{41} +13.9848 q^{42} +10.0419 q^{43} -0.613786 q^{45} -3.72036 q^{46} +9.18087 q^{47} +7.75286 q^{48} +19.6529 q^{49} +6.86263 q^{50} +3.12184 q^{51} -1.08938 q^{52} +1.38926 q^{53} -9.73398 q^{54} +8.57257 q^{56} +2.61158 q^{57} -7.77585 q^{58} +0.187412 q^{59} -1.68330 q^{60} -0.401179 q^{61} +1.74615 q^{62} -3.06356 q^{63} +0.556232 q^{64} -1.07410 q^{65} +4.41292 q^{67} -2.11109 q^{68} -3.30524 q^{69} -9.32434 q^{70} -4.17895 q^{71} -0.985356 q^{72} -16.3576 q^{73} +3.91671 q^{74} +6.09691 q^{75} -1.76603 q^{76} -2.81297 q^{78} +0.702478 q^{79} -5.16921 q^{80} -6.86764 q^{81} +15.8284 q^{82} +12.3800 q^{83} -8.40178 q^{84} -2.08148 q^{85} -17.5346 q^{86} -6.90822 q^{87} +9.32302 q^{89} +1.07177 q^{90} -5.36110 q^{91} +2.23511 q^{92} +1.55132 q^{93} -16.0312 q^{94} -1.74126 q^{95} -8.38576 q^{96} +17.0539 q^{97} -34.3171 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + 2 q^{3} + 17 q^{4} + 8 q^{5} - 2 q^{6} + 3 q^{8} + 25 q^{9} - 15 q^{10} + 11 q^{12} - 4 q^{13} + 9 q^{14} + 15 q^{15} + 29 q^{16} - 2 q^{17} + 4 q^{18} + 5 q^{19} + 17 q^{20} - 15 q^{21}+ \cdots - 31 q^{98}+O(q^{100}) $ 15 * q + q^2 + 2 * q^3 + 17 * q^4 + 8 * q^5 - 2 * q^6 + 3 * q^8 + 25 * q^9 - 15 * q^10 + 11 * q^12 - 4 * q^13 + 9 * q^14 + 15 * q^15 + 29 * q^16 - 2 * q^17 + 4 * q^18 + 5 * q^19 + 17 * q^20 - 15 * q^21 + 17 * q^23 - 18 * q^24 + 27 * q^25 + 22 * q^26 + 17 * q^27 + 35 * q^28 + 7 * q^29 + 27 * q^30 - 15 * q^31 - 23 * q^32 + 15 * q^34 + 9 * q^35 - 3 * q^36 + 17 * q^37 + 4 * q^38 + 18 * q^39 - 51 * q^40 - 7 * q^41 - 17 * q^42 - 12 * q^43 + 18 * q^45 + 20 * q^46 - 8 * q^47 + 21 * q^48 + 37 * q^49 + 16 * q^50 - 52 * q^51 + 35 * q^52 + 26 * q^53 - 48 * q^54 + 8 * q^56 + 44 * q^57 - 55 * q^58 + 11 * q^59 + 108 * q^60 + 17 * q^61 - q^62 + 22 * q^63 - 3 * q^64 - 12 * q^65 - 31 * q^67 + 30 * q^68 - 16 * q^69 - 8 * q^70 + 35 * q^71 + 59 * q^72 - q^73 + 5 * q^74 + 61 * q^75 - 12 * q^76 + q^78 + 14 * q^79 + 27 * q^80 + 83 * q^81 + 39 * q^82 + 26 * q^83 - 46 * q^84 + 16 * q^85 + 51 * q^86 + 43 * q^87 + 73 * q^89 - 45 * q^90 - 23 * q^91 + 19 * q^92 - 2 * q^93 + 46 * q^94 - 59 * q^95 + 53 * q^96 + 34 * q^97 - 31 * 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 $

$2$	−1.74615	−1.23472	−0.617359	−	0.786682i	$-0.711797\pi$
$2$	−1.74615	−1.23472	−0.617359	+	0.786682i	$0.711797\pi$
$3$	−1.55132	−0.895654	−0.447827	−	0.894120i	$-0.647802\pi$
$3$	−1.55132	−0.895654	−0.447827	+	0.894120i	$0.647802\pi$
$4$	1.04905	0.524527
$5$	1.03434	0.462570	0.231285	−	0.972886i	$-0.425707\pi$
$5$	1.03434	0.462570	0.231285	+	0.972886i	$0.425707\pi$
$6$	2.70884	1.10588
$7$	5.16265	1.95130	0.975649	−	0.219338i	$-0.0703898\pi$
$7$	5.16265	1.95130	0.975649	+	0.219338i	$0.0703898\pi$
$8$	1.66050	0.587075
$9$	−0.593409	−0.197803
$10$	−1.80611	−0.571144
$11$	0	0
$11$	0	0
$12$	−1.62742	−0.469795
$13$	−1.03844	−0.288011	−0.144006	−	0.989577i	$-0.545998\pi$
$13$	−1.03844	−0.288011	−0.144006	+	0.989577i	$0.545998\pi$
$14$	−9.01478	−2.40930
$15$	−1.60459	−0.414303
$16$	−4.99759	−1.24940
$17$	−2.01238	−0.488074	−0.244037	−	0.969766i	$-0.578472\pi$
$17$	−2.01238	−0.488074	−0.244037	+	0.969766i	$0.578472\pi$
$18$	1.03618	0.244231
$19$	−1.68345	−0.386211	−0.193105	−	0.981178i	$-0.561856\pi$
$19$	−1.68345	−0.386211	−0.193105	+	0.981178i	$0.561856\pi$
$20$	1.08508	0.242631
$21$	−8.00892	−1.74769
$22$	0	0
$23$	2.13060	0.444261	0.222130	−	0.975017i	$-0.428699\pi$
$23$	2.13060	0.444261	0.222130	+	0.975017i	$0.428699\pi$
$24$	−2.57596	−0.525816
$25$	−3.93014	−0.786029
$26$	1.81328	0.355612
$27$	5.57452	1.07282
$28$	5.41589	1.02351
$29$	4.45313	0.826925	0.413463	−	0.910521i	$-0.364319\pi$
$29$	4.45313	0.826925	0.413463	+	0.910521i	$0.364319\pi$
$30$	2.80186	0.511547
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.40557	0.955579
$33$	0	0
$34$	3.51393	0.602633
$35$	5.33993	0.902613
$36$	−0.622518	−0.103753
$37$	−2.24305	−0.368755	−0.184378	−	0.982855i	$-0.559027\pi$
$37$	−2.24305	−0.368755	−0.184378	+	0.982855i	$0.559027\pi$
$38$	2.93957	0.476861
$39$	1.61095	0.257959
$40$	1.71752	0.271564
$41$	−9.06471	−1.41567	−0.707835	−	0.706378i	$-0.750328\pi$
$41$	−9.06471	−1.41567	−0.707835	+	0.706378i	$0.750328\pi$
$42$	13.9848	2.15790
$43$	10.0419	1.53137	0.765685	−	0.643215i	$-0.222400\pi$
$43$	10.0419	1.53137	0.765685	+	0.643215i	$0.222400\pi$
$44$	0	0
$45$	−0.613786	−0.0914978
$46$	−3.72036	−0.548537
$47$	9.18087	1.33917	0.669583	−	0.742737i	$-0.266473\pi$
$47$	9.18087	1.33917	0.669583	+	0.742737i	$0.266473\pi$
$48$	7.75286	1.11903
$49$	19.6529	2.80756
$50$	6.86263	0.970523
$51$	3.12184	0.437146
$52$	−1.08938	−0.151070
$53$	1.38926	0.190830	0.0954148	−	0.995438i	$-0.469582\pi$
$53$	1.38926	0.190830	0.0954148	+	0.995438i	$0.469582\pi$
$54$	−9.73398	−1.32463
$55$	0	0
$56$	8.57257	1.14556
$57$	2.61158	0.345912
$58$	−7.77585	−1.02102
$59$	0.187412	0.0243990	0.0121995	−	0.999926i	$-0.496117\pi$
$59$	0.187412	0.0243990	0.0121995	+	0.999926i	$0.496117\pi$
$60$	−1.68330	−0.217313
$61$	−0.401179	−0.0513657	−0.0256828	−	0.999670i	$-0.508176\pi$
$61$	−0.401179	−0.0513657	−0.0256828	+	0.999670i	$0.508176\pi$
$62$	1.74615	0.221762
$63$	−3.06356	−0.385973
$64$	0.556232	0.0695290
$65$	−1.07410	−0.133225
$66$	0	0
$67$	4.41292	0.539124	0.269562	−	0.962983i	$-0.413121\pi$
$67$	4.41292	0.539124	0.269562	+	0.962983i	$0.413121\pi$
$68$	−2.11109	−0.256008
$69$	−3.30524	−0.397904
$70$	−9.32434	−1.11447
$71$	−4.17895	−0.495951	−0.247975	−	0.968766i	$-0.579765\pi$
$71$	−4.17895	−0.495951	−0.247975	+	0.968766i	$0.579765\pi$
$72$	−0.985356	−0.116125
$73$	−16.3576	−1.91451	−0.957256	−	0.289241i	$-0.906597\pi$
$73$	−16.3576	−1.91451	−0.957256	+	0.289241i	$0.906597\pi$
$74$	3.91671	0.455308
$75$	6.09691	0.704010
$76$	−1.76603	−0.202578
$77$	0	0
$78$	−2.81297	−0.318506
$79$	0.702478	0.0790350	0.0395175	−	0.999219i	$-0.487418\pi$
$79$	0.702478	0.0790350	0.0395175	+	0.999219i	$0.487418\pi$
$80$	−5.16921	−0.577935
$81$	−6.86764	−0.763071
$82$	15.8284	1.74795
$83$	12.3800	1.35888	0.679438	−	0.733733i	$-0.262224\pi$
$83$	12.3800	1.35888	0.679438	+	0.733733i	$0.262224\pi$
$84$	−8.40178	−0.916709
$85$	−2.08148	−0.225769
$86$	−17.5346	−1.89081
$87$	−6.90822	−0.740639
$88$	0	0
$89$	9.32302	0.988239	0.494119	−	0.869394i	$-0.335491\pi$
$89$	9.32302	0.988239	0.494119	+	0.869394i	$0.335491\pi$
$90$	1.07177	0.112974
$91$	−5.36110	−0.561996
$92$	2.23511	0.233027
$93$	1.55132	0.160864
$94$	−16.0312	−1.65349
$95$	−1.74126	−0.178650
$96$	−8.38576	−0.855868
$97$	17.0539	1.73156	0.865781	−	0.500423i	$-0.166822\pi$
$97$	17.0539	1.73156	0.865781	+	0.500423i	$0.166822\pi$
$98$	−34.3171	−3.46655
$99$	0	0
$100$	−4.12293	−0.412293
$101$	−11.8808	−1.18218	−0.591090	−	0.806606i	$-0.701302\pi$
$101$	−11.8808	−1.18218	−0.591090	+	0.806606i	$0.701302\pi$
$102$	−5.45122	−0.539751
$103$	3.87876	0.382185	0.191093	−	0.981572i	$-0.438797\pi$
$103$	3.87876	0.382185	0.191093	+	0.981572i	$0.438797\pi$
$104$	−1.72433	−0.169084
$105$	−8.28393	−0.808429
$106$	−2.42586	−0.235620
$107$	8.02539	0.775843	0.387922	−	0.921692i	$-0.373193\pi$
$107$	8.02539	0.775843	0.387922	+	0.921692i	$0.373193\pi$
$108$	5.84797	0.562721
$109$	−5.83751	−0.559132	−0.279566	−	0.960126i	$-0.590191\pi$
$109$	−5.83751	−0.559132	−0.279566	+	0.960126i	$0.590191\pi$
$110$	0	0
$111$	3.47969	0.330277
$112$	−25.8008	−2.43795
$113$	6.54085	0.615311	0.307656	−	0.951498i	$-0.400456\pi$
$113$	6.54085	0.615311	0.307656	+	0.951498i	$0.400456\pi$
$114$	−4.56021	−0.427103
$115$	2.20376	0.205502
$116$	4.67157	0.433744
$117$	0.616220	0.0569695
$118$	−0.327250	−0.0301258
$119$	−10.3892	−0.952378
$120$	−2.66442	−0.243227
$121$	0	0
$122$	0.700520	0.0634221
$123$	14.0623	1.26795
$124$	−1.04905	−0.0942078
$125$	−9.23679	−0.826164
$126$	5.34945	0.476567
$127$	−1.78015	−0.157963	−0.0789815	−	0.996876i	$-0.525167\pi$
$127$	−1.78015	−0.157963	−0.0789815	+	0.996876i	$0.525167\pi$
$128$	−11.7824	−1.04143
$129$	−15.5781	−1.37158
$130$	1.87554	0.164496
$131$	−6.46564	−0.564906	−0.282453	−	0.959281i	$-0.591148\pi$
$131$	−6.46564	−0.564906	−0.282453	+	0.959281i	$0.591148\pi$
$132$	0	0
$133$	−8.69108	−0.753613
$134$	−7.70564	−0.665666
$135$	5.76595	0.496254
$136$	−3.34156	−0.286536
$137$	12.8339	1.09647	0.548235	−	0.836324i	$-0.315300\pi$
$137$	12.8339	1.09647	0.548235	+	0.836324i	$0.315300\pi$
$138$	5.77146	0.491299
$139$	−19.6791	−1.66916	−0.834579	−	0.550889i	$-0.814289\pi$
$139$	−19.6791	−1.66916	−0.834579	+	0.550889i	$0.814289\pi$
$140$	5.60187	0.473444
$141$	−14.2425	−1.19943
$142$	7.29710	0.612359
$143$	0	0
$144$	2.96562	0.247135
$145$	4.60604	0.382511
$146$	28.5629	2.36388
$147$	−30.4880	−2.51461
$148$	−2.35308	−0.193422
$149$	5.19130	0.425288	0.212644	−	0.977130i	$-0.431793\pi$
$149$	5.19130	0.425288	0.212644	+	0.977130i	$0.431793\pi$
$150$	−10.6461	−0.869253
$151$	5.97827	0.486505	0.243253	−	0.969963i	$-0.421786\pi$
$151$	5.97827	0.486505	0.243253	+	0.969963i	$0.421786\pi$
$152$	−2.79537	−0.226735
$153$	1.19417	0.0965425
$154$	0	0
$155$	−1.03434	−0.0830801
$156$	1.68997	0.135306
$157$	−19.7761	−1.57831	−0.789154	−	0.614195i	$-0.789481\pi$
$157$	−19.7761	−1.57831	−0.789154	+	0.614195i	$0.789481\pi$
$158$	−1.22664	−0.0975858
$159$	−2.15519	−0.170917
$160$	5.59119	0.442022
$161$	10.9995	0.866885
$162$	11.9920	0.942177
$163$	8.19256	0.641690	0.320845	−	0.947132i	$-0.396033\pi$
$163$	8.19256	0.641690	0.320845	+	0.947132i	$0.396033\pi$
$164$	−9.50936	−0.742557
$165$	0	0
$166$	−21.6173	−1.67783
$167$	3.87450	0.299818	0.149909	−	0.988700i	$-0.452102\pi$
$167$	3.87450	0.299818	0.149909	+	0.988700i	$0.452102\pi$
$168$	−13.2988	−1.02602
$169$	−11.9216	−0.917050
$170$	3.63459	0.278760
$171$	0.998977	0.0763937
$172$	10.5345	0.803245
$173$	10.6239	0.807724	0.403862	−	0.914820i	$-0.367668\pi$
$173$	10.6239	0.807724	0.403862	+	0.914820i	$0.367668\pi$
$174$	12.0628	0.914480
$175$	−20.2899	−1.53378
$176$	0	0
$177$	−0.290736	−0.0218531
$178$	−16.2794	−1.22020
$179$	−5.47863	−0.409492	−0.204746	−	0.978815i	$-0.565637\pi$
$179$	−5.47863	−0.409492	−0.204746	+	0.978815i	$0.565637\pi$
$180$	−0.643894	−0.0479931
$181$	−13.3156	−0.989738	−0.494869	−	0.868967i	$-0.664784\pi$
$181$	−13.3156	−0.989738	−0.494869	+	0.868967i	$0.664784\pi$
$182$	9.36130	0.693906
$183$	0.622356	0.0460059
$184$	3.53786	0.260814
$185$	−2.32007	−0.170575
$186$	−2.70884	−0.198622
$187$	0	0
$188$	9.63122	0.702429
$189$	28.7793	2.09339
$190$	3.04051	0.220582
$191$	24.0961	1.74354	0.871768	−	0.489919i	$-0.162974\pi$
$191$	24.0961	1.74354	0.871768	+	0.489919i	$0.162974\pi$
$192$	−0.862893	−0.0622739
$193$	−0.942745	−0.0678602	−0.0339301	−	0.999424i	$-0.510802\pi$
$193$	−0.942745	−0.0678602	−0.0339301	+	0.999424i	$0.510802\pi$
$194$	−29.7788	−2.13799
$195$	1.66627	0.119324
$196$	20.6170	1.47264
$197$	21.0166	1.49737	0.748684	−	0.662927i	$-0.230686\pi$
$197$	21.0166	1.49737	0.748684	+	0.662927i	$0.230686\pi$
$198$	0	0
$199$	10.9245	0.774416	0.387208	−	0.921992i	$-0.373440\pi$
$199$	10.9245	0.774416	0.387208	+	0.921992i	$0.373440\pi$
$200$	−6.52600	−0.461458
$201$	−6.84585	−0.482869
$202$	20.7456	1.45966
$203$	22.9899	1.61358
$204$	3.27498	0.229295
$205$	−9.37598	−0.654847
$206$	−6.77290	−0.471891
$207$	−1.26432	−0.0878762
$208$	5.18970	0.359841
$209$	0	0
$210$	14.4650	0.998181
$211$	15.8599	1.09184	0.545920	−	0.837837i	$-0.316180\pi$
$211$	15.8599	1.09184	0.545920	+	0.837837i	$0.316180\pi$
$212$	1.45741	0.100095
$213$	6.48289	0.444200
$214$	−14.0136	−0.957947
$215$	10.3867	0.708367
$216$	9.25649	0.629825
$217$	−5.16265	−0.350463
$218$	10.1932	0.690370
$219$	25.3759	1.71474
$220$	0	0
$221$	2.08974	0.140571
$222$	−6.07607	−0.407799
$223$	−25.6392	−1.71693	−0.858464	−	0.512873i	$-0.828581\pi$
$223$	−25.6392	−1.71693	−0.858464	+	0.512873i	$0.828581\pi$
$224$	27.9071	1.86462
$225$	2.33218	0.155479
$226$	−11.4213	−0.759735
$227$	11.9399	0.792477	0.396238	−	0.918148i	$-0.370315\pi$
$227$	11.9399	0.792477	0.396238	+	0.918148i	$0.370315\pi$
$228$	2.73968	0.181440
$229$	21.3753	1.41252	0.706260	−	0.707953i	$-0.250381\pi$
$229$	21.3753	1.41252	0.706260	+	0.707953i	$0.250381\pi$
$230$	−3.84811	−0.253737
$231$	0	0
$232$	7.39442	0.485467
$233$	5.61403	0.367788	0.183894	−	0.982946i	$-0.441130\pi$
$233$	5.61403	0.367788	0.183894	+	0.982946i	$0.441130\pi$
$234$	−1.07601	−0.0703412
$235$	9.49613	0.619459
$236$	0.196605	0.0127979
$237$	−1.08977	−0.0707880
$238$	18.1412	1.17592
$239$	−1.12943	−0.0730569	−0.0365284	−	0.999333i	$-0.511630\pi$
$239$	−1.12943	−0.0730569	−0.0365284	+	0.999333i	$0.511630\pi$
$240$	8.01909	0.517630
$241$	28.0781	1.80867	0.904334	−	0.426826i	$-0.140368\pi$
$241$	28.0781	1.80867	0.904334	+	0.426826i	$0.140368\pi$
$242$	0	0
$243$	−6.06968	−0.389370
$244$	−0.420858	−0.0269427
$245$	20.3278	1.29870
$246$	−24.5549	−1.56556
$247$	1.74817	0.111233
$248$	−1.66050	−0.105442
$249$	−19.2053	−1.21708
$250$	16.1289	1.02008
$251$	20.8684	1.31720	0.658600	−	0.752493i	$-0.271149\pi$
$251$	20.8684	1.31720	0.658600	+	0.752493i	$0.271149\pi$
$252$	−3.21384	−0.202453
$253$	0	0
$254$	3.10842	0.195040
$255$	3.22905	0.202211
$256$	19.4614	1.21634
$257$	16.8106	1.04862	0.524309	−	0.851528i	$-0.324324\pi$
$257$	16.8106	1.04862	0.524309	+	0.851528i	$0.324324\pi$
$258$	27.2018	1.69351
$259$	−11.5801	−0.719551
$260$	−1.12679	−0.0698803
$261$	−2.64253	−0.163568
$262$	11.2900	0.697499
$263$	24.5626	1.51460	0.757298	−	0.653070i	$-0.226519\pi$
$263$	24.5626	1.51460	0.757298	+	0.653070i	$0.226519\pi$
$264$	0	0
$265$	1.43697	0.0882721
$266$	15.1760	0.930498
$267$	−14.4630	−0.885120
$268$	4.62939	0.282785
$269$	6.60736	0.402858	0.201429	−	0.979503i	$-0.435441\pi$
$269$	6.60736	0.402858	0.201429	+	0.979503i	$0.435441\pi$
$270$	−10.0682	−0.612733
$271$	−23.9482	−1.45475	−0.727375	−	0.686241i	$-0.759260\pi$
$271$	−23.9482	−1.45475	−0.727375	+	0.686241i	$0.759260\pi$
$272$	10.0571	0.609799
$273$	8.31677	0.503354
$274$	−22.4099	−1.35383
$275$	0	0
$276$	−3.46737	−0.208711
$277$	−22.5790	−1.35664	−0.678319	−	0.734767i	$-0.737291\pi$
$277$	−22.5790	−1.35664	−0.678319	+	0.734767i	$0.737291\pi$
$278$	34.3627	2.06094
$279$	0.593409	0.0355265
$280$	8.86695	0.529901
$281$	−16.6500	−0.993253	−0.496627	−	0.867964i	$-0.665428\pi$
$281$	−16.6500	−0.993253	−0.496627	+	0.867964i	$0.665428\pi$
$282$	24.8695	1.48096
$283$	14.8002	0.879782	0.439891	−	0.898051i	$-0.355017\pi$
$283$	14.8002	0.879782	0.439891	+	0.898051i	$0.355017\pi$
$284$	−4.38395	−0.260139
$285$	2.70125	0.160008
$286$	0	0
$287$	−46.7979	−2.76239
$288$	−3.20771	−0.189016
$289$	−12.9503	−0.761784
$290$	−8.04286	−0.472293
$291$	−26.4561	−1.55088
$292$	−17.1600	−1.00421
$293$	−19.6163	−1.14600	−0.573000	−	0.819556i	$-0.694220\pi$
$293$	−19.6163	−1.14600	−0.573000	+	0.819556i	$0.694220\pi$
$294$	53.2367	3.10483
$295$	0.193848	0.0112862
$296$	−3.72458	−0.216487
$297$	0	0
$298$	−9.06480	−0.525110
$299$	−2.21250	−0.127952
$300$	6.39598	0.369272
$301$	51.8426	2.98816
$302$	−10.4390	−0.600696
$303$	18.4309	1.05883
$304$	8.41322	0.482531
$305$	−0.414955	−0.0237602
$306$	−2.08520	−0.119203
$307$	−0.877288	−0.0500695	−0.0250347	−	0.999687i	$-0.507970\pi$
$307$	−0.877288	−0.0500695	−0.0250347	+	0.999687i	$0.507970\pi$
$308$	0	0
$309$	−6.01719	−0.342306
$310$	1.80611	0.102580
$311$	−17.5848	−0.997143	−0.498571	−	0.866849i	$-0.666142\pi$
$311$	−17.5848	−0.997143	−0.498571	+	0.866849i	$0.666142\pi$
$312$	2.67498	0.151441
$313$	28.6327	1.61842	0.809209	−	0.587521i	$-0.199896\pi$
$313$	28.6327	1.61842	0.809209	+	0.587521i	$0.199896\pi$
$314$	34.5322	1.94876
$315$	−3.16876	−0.178540
$316$	0.736937	0.0414560
$317$	−27.5796	−1.54902	−0.774512	−	0.632560i	$-0.782004\pi$
$317$	−27.5796	−1.54902	−0.774512	+	0.632560i	$0.782004\pi$
$318$	3.76329	0.211035
$319$	0	0
$320$	0.575332	0.0321620
$321$	−12.4499	−0.694888
$322$	−19.2069	−1.07036
$323$	3.38775	0.188500
$324$	−7.20452	−0.400251
$325$	4.08122	0.226385
$326$	−14.3055	−0.792306
$327$	9.05584	0.500789
$328$	−15.0519	−0.831104
$329$	47.3976	2.61311
$330$	0	0
$331$	3.18557	0.175095	0.0875473	−	0.996160i	$-0.472097\pi$
$331$	3.18557	0.175095	0.0875473	+	0.996160i	$0.472097\pi$
$332$	12.9872	0.712767
$333$	1.33105	0.0729409
$334$	−6.76548	−0.370190
$335$	4.56445	0.249383
$336$	40.0253	2.18356
$337$	4.68171	0.255029	0.127514	−	0.991837i	$-0.459300\pi$
$337$	4.68171	0.255029	0.127514	+	0.991837i	$0.459300\pi$
$338$	20.8170	1.13230
$339$	−10.1469	−0.551106
$340$	−2.18359	−0.118422
$341$	0	0
$342$	−1.74437	−0.0943246
$343$	65.3227	3.52709
$344$	16.6745	0.899030
$345$	−3.41874	−0.184059
$346$	−18.5510	−0.997310
$347$	−21.3724	−1.14733	−0.573665	−	0.819090i	$-0.694479\pi$
$347$	−21.3724	−1.14733	−0.573665	+	0.819090i	$0.694479\pi$
$348$	−7.24710	−0.388485
$349$	−2.84481	−0.152279	−0.0761397	−	0.997097i	$-0.524259\pi$
$349$	−2.84481	−0.152279	−0.0761397	+	0.997097i	$0.524259\pi$
$350$	35.4294	1.89378
$351$	−5.78881	−0.308984
$352$	0	0
$353$	18.5284	0.986167	0.493083	−	0.869982i	$-0.335870\pi$
$353$	18.5284	0.986167	0.493083	+	0.869982i	$0.335870\pi$
$354$	0.507670	0.0269824
$355$	−4.32246	−0.229412
$356$	9.78035	0.518358
$357$	16.1170	0.853001
$358$	9.56653	0.505607
$359$	18.0400	0.952116	0.476058	−	0.879414i	$-0.342065\pi$
$359$	18.0400	0.952116	0.476058	+	0.879414i	$0.342065\pi$
$360$	−1.01919	−0.0537161
$361$	−16.1660	−0.850841
$362$	23.2510	1.22205
$363$	0	0
$364$	−5.62408	−0.294782
$365$	−16.9193	−0.885597
$366$	−1.08673	−0.0568043
$367$	−3.14948	−0.164401	−0.0822007	−	0.996616i	$-0.526195\pi$
$367$	−3.14948	−0.164401	−0.0822007	+	0.996616i	$0.526195\pi$
$368$	−10.6479	−0.555059
$369$	5.37908	0.280024
$370$	4.05121	0.210612
$371$	7.17226	0.372365
$372$	1.62742	0.0843776
$373$	−15.8465	−0.820502	−0.410251	−	0.911973i	$-0.634559\pi$
$373$	−15.8465	−0.820502	−0.410251	+	0.911973i	$0.634559\pi$
$374$	0	0
$375$	14.3292	0.739957
$376$	15.2448	0.786192
$377$	−4.62430	−0.238164
$378$	−50.2531	−2.58474
$379$	5.14644	0.264355	0.132178	−	0.991226i	$-0.457803\pi$
$379$	5.14644	0.264355	0.132178	+	0.991226i	$0.457803\pi$
$380$	−1.82668	−0.0937066
$381$	2.76159	0.141480
$382$	−42.0756	−2.15277
$383$	−7.75309	−0.396165	−0.198082	−	0.980185i	$-0.563471\pi$
$383$	−7.75309	−0.396165	−0.198082	+	0.980185i	$0.563471\pi$
$384$	18.2783	0.932759
$385$	0	0
$386$	1.64618	0.0837882
$387$	−5.95894	−0.302910
$388$	17.8905	0.908251
$389$	37.1510	1.88363	0.941815	−	0.336132i	$-0.109119\pi$
$389$	37.1510	1.88363	0.941815	+	0.336132i	$0.109119\pi$
$390$	−2.90956	−0.147331
$391$	−4.28758	−0.216832
$392$	32.6337	1.64825
$393$	10.0303	0.505960
$394$	−36.6981	−1.84883
$395$	0.726601	0.0365592
$396$	0	0
$397$	16.5590	0.831071	0.415536	−	0.909577i	$-0.363594\pi$
$397$	16.5590	0.831071	0.415536	+	0.909577i	$0.363594\pi$
$398$	−19.0758	−0.956185
$399$	13.4826	0.674976
$400$	19.6413	0.982063
$401$	−3.90261	−0.194887	−0.0974436	−	0.995241i	$-0.531067\pi$
$401$	−3.90261	−0.194887	−0.0974436	+	0.995241i	$0.531067\pi$
$402$	11.9539	0.596206
$403$	1.03844	0.0517284
$404$	−12.4636	−0.620085
$405$	−7.10346	−0.352974
$406$	−40.1440	−1.99231
$407$	0	0
$408$	5.18382	0.256637
$409$	−21.9827	−1.08697	−0.543487	−	0.839418i	$-0.682896\pi$
$409$	−21.9827	−1.08697	−0.543487	+	0.839418i	$0.682896\pi$
$410$	16.3719	0.808551
$411$	−19.9094	−0.982059
$412$	4.06902	0.200466
$413$	0.967543	0.0476097
$414$	2.20769	0.108502
$415$	12.8051	0.628576
$416$	−5.61336	−0.275217
$417$	30.5285	1.49499
$418$	0	0
$419$	4.57515	0.223511	0.111755	−	0.993736i	$-0.464353\pi$
$419$	4.57515	0.223511	0.111755	+	0.993736i	$0.464353\pi$
$420$	−8.69029	−0.424043
$421$	35.5483	1.73252	0.866260	−	0.499594i	$-0.166517\pi$
$421$	35.5483	1.73252	0.866260	+	0.499594i	$0.166517\pi$
$422$	−27.6938	−1.34811
$423$	−5.44801	−0.264891
$424$	2.30686	0.112031
$425$	7.90894	0.383640
$426$	−11.3201	−0.548462
$427$	−2.07115	−0.100230
$428$	8.41906	0.406951
$429$	0	0
$430$	−18.1368	−0.874633
$431$	36.1631	1.74192	0.870958	−	0.491357i	$-0.163499\pi$
$431$	36.1631	1.74192	0.870958	+	0.491357i	$0.163499\pi$
$432$	−27.8592	−1.34038
$433$	20.8699	1.00294	0.501471	−	0.865175i	$-0.332793\pi$
$433$	20.8699	1.00294	0.501471	+	0.865175i	$0.332793\pi$
$434$	9.01478	0.432723
$435$	−7.14544	−0.342598
$436$	−6.12386	−0.293280
$437$	−3.58677	−0.171578
$438$	−44.3102	−2.11722
$439$	−20.8831	−0.996697	−0.498349	−	0.866977i	$-0.666060\pi$
$439$	−20.8831	−0.996697	−0.498349	+	0.866977i	$0.666060\pi$
$440$	0	0
$441$	−11.6622	−0.555345
$442$	−3.64900	−0.173565
$443$	18.9234	0.899076	0.449538	−	0.893261i	$-0.351589\pi$
$443$	18.9234	0.899076	0.449538	+	0.893261i	$0.351589\pi$
$444$	3.65038	0.173239
$445$	9.64317	0.457130
$446$	44.7700	2.11992
$447$	−8.05336	−0.380911
$448$	2.87163	0.135672
$449$	−28.4778	−1.34395	−0.671975	−	0.740574i	$-0.734554\pi$
$449$	−28.4778	−1.34395	−0.671975	+	0.740574i	$0.734554\pi$
$450$	−4.07235	−0.191972
$451$	0	0
$452$	6.86170	0.322747
$453$	−9.27421	−0.435740
$454$	−20.8488	−0.978485
$455$	−5.54519	−0.259963
$456$	4.33652	0.203076
$457$	−10.9957	−0.514357	−0.257178	−	0.966364i	$-0.582793\pi$
$457$	−10.9957	−0.514357	−0.257178	+	0.966364i	$0.582793\pi$
$458$	−37.3246	−1.74406
$459$	−11.2181	−0.523614
$460$	2.31186	0.107791
$461$	15.0337	0.700190	0.350095	−	0.936714i	$-0.386149\pi$
$461$	15.0337	0.700190	0.350095	+	0.936714i	$0.386149\pi$
$462$	0	0
$463$	26.3870	1.22631	0.613155	−	0.789963i	$-0.289900\pi$
$463$	26.3870	1.22631	0.613155	+	0.789963i	$0.289900\pi$
$464$	−22.2549	−1.03316
$465$	1.60459	0.0744111
$466$	−9.80297	−0.454114
$467$	−10.4252	−0.482420	−0.241210	−	0.970473i	$-0.577544\pi$
$467$	−10.4252	−0.482420	−0.241210	+	0.970473i	$0.577544\pi$
$468$	0.646447	0.0298820
$469$	22.7824	1.05199
$470$	−16.5817	−0.764857
$471$	30.6791	1.41362
$472$	0.311198	0.0143240
$473$	0	0
$474$	1.90290	0.0874032
$475$	6.61622	0.303573
$476$	−10.8988	−0.499548
$477$	−0.824400	−0.0377467
$478$	1.97216	0.0902046
$479$	−13.7900	−0.630081	−0.315040	−	0.949078i	$-0.602018\pi$
$479$	−13.7900	−0.630081	−0.315040	+	0.949078i	$0.602018\pi$
$480$	−8.67372	−0.395899
$481$	2.32927	0.106206
$482$	−49.0286	−2.23319
$483$	−17.0638	−0.776430
$484$	0	0
$485$	17.6395	0.800970
$486$	10.5986	0.480762
$487$	0.906414	0.0410736	0.0205368	−	0.999789i	$-0.493462\pi$
$487$	0.906414	0.0410736	0.0205368	+	0.999789i	$0.493462\pi$
$488$	−0.666157	−0.0301555
$489$	−12.7093	−0.574733
$490$	−35.4955	−1.60352
$491$	15.8451	0.715081	0.357540	−	0.933898i	$-0.383615\pi$
$491$	15.8451	0.715081	0.357540	+	0.933898i	$0.383615\pi$
$492$	14.7521	0.665074
$493$	−8.96139	−0.403601
$494$	−3.05257	−0.137341
$495$	0	0
$496$	4.99759	0.224399
$497$	−21.5745	−0.967747
$498$	33.5353	1.50275
$499$	23.9009	1.06995	0.534976	−	0.844867i	$-0.320321\pi$
$499$	23.9009	1.06995	0.534976	+	0.844867i	$0.320321\pi$
$500$	−9.68989	−0.433345
$501$	−6.01059	−0.268533
$502$	−36.4394	−1.62637
$503$	−2.26525	−0.101002	−0.0505012	−	0.998724i	$-0.516082\pi$
$503$	−2.26525	−0.101002	−0.0505012	+	0.998724i	$0.516082\pi$
$504$	−5.08704	−0.226595
$505$	−12.2887	−0.546842
$506$	0	0
$507$	18.4943	0.821359
$508$	−1.86748	−0.0828558
$509$	11.7099	0.519032	0.259516	−	0.965739i	$-0.416437\pi$
$509$	11.7099	0.519032	0.259516	+	0.965739i	$0.416437\pi$
$510$	−5.63841	−0.249673
$511$	−84.4486	−3.73578
$512$	−10.4178	−0.460408
$513$	−9.38446	−0.414334
$514$	−29.3540	−1.29475
$515$	4.01195	0.176788
$516$	−16.3423	−0.719430
$517$	0	0
$518$	20.2206	0.888442
$519$	−16.4811	−0.723441
$520$	−1.78354	−0.0782134
$521$	24.1991	1.06018	0.530091	−	0.847941i	$-0.322158\pi$
$521$	24.1991	1.06018	0.530091	+	0.847941i	$0.322158\pi$
$522$	4.61426	0.201961
$523$	−38.7527	−1.69454	−0.847269	−	0.531164i	$-0.821755\pi$
$523$	−38.7527	−1.69454	−0.847269	+	0.531164i	$0.821755\pi$
$524$	−6.78280	−0.296308
$525$	31.4762	1.37373
$526$	−42.8901	−1.87010
$527$	2.01238	0.0876607
$528$	0	0
$529$	−18.4605	−0.802632
$530$	−2.50916	−0.108991
$531$	−0.111212	−0.00482619
$532$	−9.11741	−0.395290
$533$	9.41315	0.407729
$534$	25.2546	1.09287
$535$	8.30097	0.358882
$536$	7.32765	0.316506
$537$	8.49910	0.366763
$538$	−11.5375	−0.497416
$539$	0	0
$540$	6.04879	0.260298
$541$	−4.67360	−0.200934	−0.100467	−	0.994940i	$-0.532034\pi$
$541$	−4.67360	−0.200934	−0.100467	+	0.994940i	$0.532034\pi$
$542$	41.8172	1.79620
$543$	20.6567	0.886464
$544$	−10.8781	−0.466393
$545$	−6.03796	−0.258638
$546$	−14.5224	−0.621500
$547$	−22.1290	−0.946166	−0.473083	−	0.881018i	$-0.656859\pi$
$547$	−22.1290	−0.946166	−0.473083	+	0.881018i	$0.656859\pi$
$548$	13.4634	0.575128
$549$	0.238063	0.0101603
$550$	0	0
$551$	−7.49664	−0.319368
$552$	−5.48835	−0.233600
$553$	3.62665	0.154221
$554$	39.4263	1.67506
$555$	3.59918	0.152776
$556$	−20.6444	−0.875518
$557$	−42.6618	−1.80764	−0.903818	−	0.427917i	$-0.859248\pi$
$557$	−42.6618	−1.80764	−0.903818	+	0.427917i	$0.859248\pi$
$558$	−1.03618	−0.0438652
$559$	−10.4279	−0.441052
$560$	−26.6868	−1.12772
$561$	0	0
$562$	29.0734	1.22639
$563$	28.1479	1.18629	0.593145	−	0.805095i	$-0.297886\pi$
$563$	28.1479	1.18629	0.593145	+	0.805095i	$0.297886\pi$
$564$	−14.9411	−0.629133
$565$	6.76545	0.284625
$566$	−25.8435	−1.08628
$567$	−35.4552	−1.48898
$568$	−6.93915	−0.291160
$569$	30.5045	1.27881	0.639407	−	0.768868i	$-0.279180\pi$
$569$	30.5045	1.27881	0.639407	+	0.768868i	$0.279180\pi$
$570$	−4.71680	−0.197565
$571$	−0.740624	−0.0309942	−0.0154971	−	0.999880i	$-0.504933\pi$
$571$	−0.740624	−0.0309942	−0.0154971	+	0.999880i	$0.504933\pi$
$572$	0	0
$573$	−37.3808	−1.56161
$574$	81.7164	3.41077
$575$	−8.37356	−0.349202
$576$	−0.330073	−0.0137530
$577$	15.5564	0.647621	0.323810	−	0.946122i	$-0.395036\pi$
$577$	15.5564	0.647621	0.323810	+	0.946122i	$0.395036\pi$
$578$	22.6133	0.940588
$579$	1.46250	0.0607793
$580$	4.83199	0.200637
$581$	63.9133	2.65157
$582$	46.1964	1.91490
$583$	0	0
$584$	−27.1618	−1.12396
$585$	0.637380	0.0263524
$586$	34.2532	1.41499
$587$	−35.0950	−1.44853	−0.724263	−	0.689524i	$-0.757820\pi$
$587$	−35.0950	−1.44853	−0.724263	+	0.689524i	$0.757820\pi$
$588$	−31.9835	−1.31898
$589$	1.68345	0.0693655
$590$	−0.338488	−0.0139353
$591$	−32.6034	−1.34112
$592$	11.2099	0.460722
$593$	30.5749	1.25556	0.627779	−	0.778391i	$-0.283964\pi$
$593$	30.5749	1.25556	0.627779	+	0.778391i	$0.283964\pi$
$594$	0	0
$595$	−10.7460	−0.440542
$596$	5.44595	0.223075
$597$	−16.9474	−0.693609
$598$	3.86336	0.157985
$599$	7.07483	0.289070	0.144535	−	0.989500i	$-0.453831\pi$
$599$	7.07483	0.289070	0.144535	+	0.989500i	$0.453831\pi$
$600$	10.1239	0.413307
$601$	5.02076	0.204801	0.102401	−	0.994743i	$-0.467348\pi$
$601$	5.02076	0.204801	0.102401	+	0.994743i	$0.467348\pi$
$602$	−90.5252	−3.68953
$603$	−2.61867	−0.106640
$604$	6.27153	0.255185
$605$	0	0
$606$	−32.1831	−1.30735
$607$	−17.0642	−0.692613	−0.346306	−	0.938122i	$-0.612564\pi$
$607$	−17.0642	−0.692613	−0.346306	+	0.938122i	$0.612564\pi$
$608$	−9.10003	−0.369055
$609$	−35.6647	−1.44521
$610$	0.724575	0.0293372
$611$	−9.53377	−0.385695
$612$	1.25274	0.0506391
$613$	35.8658	1.44861	0.724304	−	0.689481i	$-0.242161\pi$
$613$	35.8658	1.44861	0.724304	+	0.689481i	$0.242161\pi$
$614$	1.53188	0.0618216
$615$	14.5451	0.586517
$616$	0	0
$617$	46.5443	1.87380	0.936901	−	0.349594i	$-0.113681\pi$
$617$	46.5443	1.87380	0.936901	+	0.349594i	$0.113681\pi$
$618$	10.5069	0.422651
$619$	8.97981	0.360929	0.180465	−	0.983581i	$-0.442240\pi$
$619$	8.97981	0.360929	0.180465	+	0.983581i	$0.442240\pi$
$620$	−1.08508	−0.0435777
$621$	11.8771	0.476611
$622$	30.7058	1.23119
$623$	48.1315	1.92835
$624$	−8.05088	−0.322293
$625$	10.0967	0.403870
$626$	−49.9972	−1.99829
$627$	0	0
$628$	−20.7462	−0.827865
$629$	4.51387	0.179980
$630$	5.53315	0.220446
$631$	15.4902	0.616654	0.308327	−	0.951280i	$-0.400231\pi$
$631$	15.4902	0.616654	0.308327	+	0.951280i	$0.400231\pi$
$632$	1.16646	0.0463995
$633$	−24.6037	−0.977911
$634$	48.1582	1.91261
$635$	−1.84128	−0.0730690
$636$	−2.26090	−0.0896507
$637$	−20.4084	−0.808610
$638$	0	0
$639$	2.47983	0.0981006
$640$	−12.1870	−0.481733
$641$	35.6917	1.40974	0.704868	−	0.709338i	$-0.251006\pi$
$641$	35.6917	1.40974	0.704868	+	0.709338i	$0.251006\pi$
$642$	21.7395	0.857990
$643$	8.98201	0.354216	0.177108	−	0.984191i	$-0.443326\pi$
$643$	8.98201	0.354216	0.177108	+	0.984191i	$0.443326\pi$
$644$	11.5391	0.454704
$645$	−16.1131	−0.634452
$646$	−5.91554	−0.232744
$647$	−33.3542	−1.31129	−0.655645	−	0.755070i	$-0.727603\pi$
$647$	−33.3542	−1.31129	−0.655645	+	0.755070i	$0.727603\pi$
$648$	−11.4037	−0.447980
$649$	0	0
$650$	−7.12643	−0.279522
$651$	8.00892	0.313894
$652$	8.59443	0.336584
$653$	−27.9202	−1.09260	−0.546301	−	0.837589i	$-0.683965\pi$
$653$	−27.9202	−1.09260	−0.546301	+	0.837589i	$0.683965\pi$
$654$	−15.8129	−0.618333
$655$	−6.68766	−0.261309
$656$	45.3017	1.76874
$657$	9.70675	0.378696
$658$	−82.7635	−3.22646
$659$	23.8914	0.930676	0.465338	−	0.885133i	$-0.345933\pi$
$659$	23.8914	0.930676	0.465338	+	0.885133i	$0.345933\pi$
$660$	0	0
$661$	−25.1167	−0.976928	−0.488464	−	0.872584i	$-0.662443\pi$
$661$	−25.1167	−0.976928	−0.488464	+	0.872584i	$0.662443\pi$
$662$	−5.56249	−0.216192
$663$	−3.24185	−0.125903
$664$	20.5569	0.797762
$665$	−8.98953	−0.348599
$666$	−2.32421	−0.0900614
$667$	9.48784	0.367371
$668$	4.06456	0.157262
$669$	39.7746	1.53777
$670$	−7.97024	−0.307917
$671$	0	0
$672$	−43.2927	−1.67005
$673$	−36.0397	−1.38923	−0.694614	−	0.719382i	$-0.744425\pi$
$673$	−36.0397	−1.38923	−0.694614	+	0.719382i	$0.744425\pi$
$674$	−8.17498	−0.314888
$675$	−21.9087	−0.843265
$676$	−12.5064	−0.481017
$677$	36.1242	1.38837	0.694183	−	0.719799i	$-0.255766\pi$
$677$	36.1242	1.38837	0.694183	+	0.719799i	$0.255766\pi$
$678$	17.7181	0.680460
$679$	88.0434	3.37879
$680$	−3.45630	−0.132543
$681$	−18.5225	−0.709785
$682$	0	0
$683$	−38.9893	−1.49188	−0.745942	−	0.666010i	$-0.768001\pi$
$683$	−38.9893	−1.49188	−0.745942	+	0.666010i	$0.768001\pi$
$684$	1.04798	0.0400705
$685$	13.2746	0.507195
$686$	−114.063	−4.35496
$687$	−33.1599	−1.26513
$688$	−50.1852	−1.91329
$689$	−1.44266	−0.0549611
$690$	5.96964	0.227260
$691$	21.0565	0.801027	0.400514	−	0.916291i	$-0.368832\pi$
$691$	21.0565	0.801027	0.400514	+	0.916291i	$0.368832\pi$
$692$	11.1451	0.423673
$693$	0	0
$694$	37.3195	1.41663
$695$	−20.3548	−0.772103
$696$	−11.4711	−0.434811
$697$	18.2416	0.690952
$698$	4.96748	0.188022
$699$	−8.70916	−0.329411
$700$	−21.2852	−0.804506
$701$	−42.7951	−1.61635	−0.808174	−	0.588943i	$-0.799544\pi$
$701$	−42.7951	−1.61635	−0.808174	+	0.588943i	$0.799544\pi$
$702$	10.1081	0.381507
$703$	3.77607	0.142417
$704$	0	0
$705$	−14.7315	−0.554821
$706$	−32.3534	−1.21764
$707$	−61.3362	−2.30679
$708$	−0.304998	−0.0114625
$709$	14.8226	0.556673	0.278336	−	0.960484i	$-0.410217\pi$
$709$	14.8226	0.556673	0.278336	+	0.960484i	$0.410217\pi$
$710$	7.54767	0.283259
$711$	−0.416857	−0.0156334
$712$	15.4809	0.580170
$713$	−2.13060	−0.0797916
$714$	−28.1427	−1.05322
$715$	0	0
$716$	−5.74738	−0.214790
$717$	1.75211	0.0654337
$718$	−31.5007	−1.17559
$719$	12.1671	0.453757	0.226879	−	0.973923i	$-0.427148\pi$
$719$	12.1671	0.453757	0.226879	+	0.973923i	$0.427148\pi$
$720$	3.06745	0.114317
$721$	20.0247	0.745757
$722$	28.2283	1.05055
$723$	−43.5580	−1.61994
$724$	−13.9687	−0.519144
$725$	−17.5014	−0.649987
$726$	0	0
$727$	2.61267	0.0968986	0.0484493	−	0.998826i	$-0.484572\pi$
$727$	2.61267	0.0968986	0.0484493	+	0.998826i	$0.484572\pi$
$728$	−8.90210	−0.329934
$729$	30.0189	1.11181
$730$	29.5437	1.09346
$731$	−20.2081	−0.747422
$732$	0.652885	0.0241313
$733$	32.9162	1.21579	0.607894	−	0.794018i	$-0.292014\pi$
$733$	32.9162	1.21579	0.607894	+	0.794018i	$0.292014\pi$
$734$	5.49947	0.202989
$735$	−31.5349	−1.16318
$736$	11.5171	0.424526
$737$	0	0
$738$	−9.39271	−0.345750
$739$	21.4619	0.789489	0.394744	−	0.918791i	$-0.370833\pi$
$739$	21.4619	0.789489	0.394744	+	0.918791i	$0.370833\pi$
$740$	−2.43388	−0.0894713
$741$	−2.71196	−0.0996264
$742$	−12.5239	−0.459766
$743$	43.7010	1.60324	0.801618	−	0.597837i	$-0.203973\pi$
$743$	43.7010	1.60324	0.801618	+	0.597837i	$0.203973\pi$
$744$	2.57596	0.0944394
$745$	5.36956	0.196725
$746$	27.6705	1.01309
$747$	−7.34638	−0.268790
$748$	0	0
$749$	41.4322	1.51390
$750$	−25.0210	−0.913638
$751$	−6.57083	−0.239773	−0.119887	−	0.992788i	$-0.538253\pi$
$751$	−6.57083	−0.239773	−0.119887	+	0.992788i	$0.538253\pi$
$752$	−45.8822	−1.67315
$753$	−32.3735	−1.17976
$754$	8.07475	0.294065
$755$	6.18356	0.225043
$756$	30.1910	1.09804
$757$	−5.01442	−0.182252	−0.0911261	−	0.995839i	$-0.529047\pi$
$757$	−5.01442	−0.182252	−0.0911261	+	0.995839i	$0.529047\pi$
$758$	−8.98648	−0.326404
$759$	0	0
$760$	−2.89136	−0.104881
$761$	18.8130	0.681969	0.340984	−	0.940069i	$-0.389240\pi$
$761$	18.8130	0.681969	0.340984	+	0.940069i	$0.389240\pi$
$762$	−4.82215	−0.174688
$763$	−30.1370	−1.09103
$764$	25.2781	0.914531
$765$	1.23517	0.0446577
$766$	13.5381	0.489151
$767$	−0.194616	−0.00702718
$768$	−30.1909	−1.08942
$769$	−22.4564	−0.809798	−0.404899	−	0.914362i	$-0.632693\pi$
$769$	−22.4564	−0.809798	−0.404899	+	0.914362i	$0.632693\pi$
$770$	0	0
$771$	−26.0787	−0.939200
$772$	−0.988989	−0.0355945
$773$	−25.9568	−0.933601	−0.466801	−	0.884363i	$-0.654593\pi$
$773$	−25.9568	−0.933601	−0.466801	+	0.884363i	$0.654593\pi$
$774$	10.4052	0.374008
$775$	3.93014	0.141175
$776$	28.3180	1.01656
$777$	17.9644	0.644469
$778$	−64.8713	−2.32575
$779$	15.2600	0.546747
$780$	1.74801	0.0625886
$781$	0	0
$782$	7.48677	0.267726
$783$	24.8241	0.887140
$784$	−98.2174	−3.50776
$785$	−20.4552	−0.730079
$786$	−17.5144	−0.624718
$787$	−9.12898	−0.325413	−0.162706	−	0.986675i	$-0.552022\pi$
$787$	−9.12898	−0.325413	−0.162706	+	0.986675i	$0.552022\pi$
$788$	22.0475	0.785409
$789$	−38.1044	−1.35655
$790$	−1.26876	−0.0451403
$791$	33.7681	1.20066
$792$	0	0
$793$	0.416600	0.0147939
$794$	−28.9145	−1.02614
$795$	−2.22919	−0.0790613
$796$	11.4604	0.406202
$797$	4.76088	0.168639	0.0843195	−	0.996439i	$-0.473128\pi$
$797$	4.76088	0.168639	0.0843195	+	0.996439i	$0.473128\pi$
$798$	−23.5428	−0.833405
$799$	−18.4754	−0.653613
$800$	−21.2447	−0.751112
$801$	−5.53237	−0.195477
$802$	6.81456	0.240631
$803$	0	0
$804$	−7.18166	−0.253278
$805$	11.3773	0.400995
$806$	−1.81328	−0.0638699
$807$	−10.2501	−0.360821
$808$	−19.7280	−0.694029
$809$	27.2254	0.957195	0.478598	−	0.878034i	$-0.341145\pi$
$809$	27.2254	0.957195	0.478598	+	0.878034i	$0.341145\pi$
$810$	12.4037	0.435823
$811$	20.0077	0.702566	0.351283	−	0.936269i	$-0.385745\pi$
$811$	20.0077	0.702566	0.351283	+	0.936269i	$0.385745\pi$
$812$	24.1177	0.846364
$813$	37.1513	1.30295
$814$	0	0
$815$	8.47388	0.296827
$816$	−15.6017	−0.546169
$817$	−16.9050	−0.591432
$818$	38.3852	1.34211
$819$	3.18133	0.111164
$820$	−9.83590	−0.343485
$821$	30.5671	1.06680	0.533400	−	0.845863i	$-0.320914\pi$
$821$	30.5671	1.06680	0.533400	+	0.845863i	$0.320914\pi$
$822$	34.7649	1.21257
$823$	44.2392	1.54208	0.771041	−	0.636786i	$-0.219737\pi$
$823$	44.2392	1.54208	0.771041	+	0.636786i	$0.219737\pi$
$824$	6.44067	0.224371
$825$	0	0
$826$	−1.68948	−0.0587845
$827$	−10.4041	−0.361787	−0.180893	−	0.983503i	$-0.557899\pi$
$827$	−10.4041	−0.361787	−0.180893	+	0.983503i	$0.557899\pi$
$828$	−1.32634	−0.0460934
$829$	10.3646	0.359978	0.179989	−	0.983669i	$-0.442394\pi$
$829$	10.3646	0.359978	0.179989	+	0.983669i	$0.442394\pi$
$830$	−22.3596	−0.776114
$831$	35.0272	1.21508
$832$	−0.577613	−0.0200251
$833$	−39.5492	−1.37030
$834$	−53.3075	−1.84589
$835$	4.00755	0.138687
$836$	0	0
$837$	−5.57452	−0.192684
$838$	−7.98891	−0.275972
$839$	13.2843	0.458624	0.229312	−	0.973353i	$-0.426352\pi$
$839$	13.2843	0.458624	0.229312	+	0.973353i	$0.426352\pi$
$840$	−13.7555	−0.474609
$841$	−9.16964	−0.316195
$842$	−62.0729	−2.13917
$843$	25.8294	0.889612
$844$	16.6379	0.572699
$845$	−12.3310	−0.424200
$846$	9.51307	0.327066
$847$	0	0
$848$	−6.94296	−0.238422
$849$	−22.9599	−0.787981
$850$	−13.8102	−0.473687
$851$	−4.77904	−0.163824
$852$	6.80090	0.232995
$853$	0.915849	0.0313581	0.0156790	−	0.999877i	$-0.495009\pi$
$853$	0.915849	0.0313581	0.0156790	+	0.999877i	$0.495009\pi$
$854$	3.61654	0.123755
$855$	1.03328	0.0353375
$856$	13.3261	0.455478
$857$	−28.9505	−0.988929	−0.494464	−	0.869198i	$-0.664636\pi$
$857$	−28.9505	−0.988929	−0.494464	+	0.869198i	$0.664636\pi$
$858$	0	0
$859$	−2.78855	−0.0951439	−0.0475720	−	0.998868i	$-0.515148\pi$
$859$	−2.78855	−0.0951439	−0.0475720	+	0.998868i	$0.515148\pi$
$860$	10.8962	0.371557
$861$	72.5985	2.47415
$862$	−63.1464	−2.15077
$863$	18.4520	0.628114	0.314057	−	0.949404i	$-0.398312\pi$
$863$	18.4520	0.628114	0.314057	+	0.949404i	$0.398312\pi$
$864$	30.1335	1.02516
$865$	10.9888	0.373629
$866$	−36.4420	−1.23835
$867$	20.0901	0.682295
$868$	−5.41589	−0.183827
$869$	0	0
$870$	12.4770	0.423011
$871$	−4.58255	−0.155274
$872$	−9.69318	−0.328252
$873$	−10.1200	−0.342508
$874$	6.26305	0.211851
$875$	−47.6863	−1.61209
$876$	26.6206	0.899428
$877$	−21.6929	−0.732518	−0.366259	−	0.930513i	$-0.619362\pi$
$877$	−21.6929	−0.732518	−0.366259	+	0.930513i	$0.619362\pi$
$878$	36.4651	1.23064
$879$	30.4312	1.02642
$880$	0	0
$881$	38.2546	1.28883	0.644415	−	0.764676i	$-0.277101\pi$
$881$	38.2546	1.28883	0.644415	+	0.764676i	$0.277101\pi$
$882$	20.3641	0.685694
$883$	−34.1002	−1.14756	−0.573781	−	0.819009i	$-0.694524\pi$
$883$	−34.1002	−1.14756	−0.573781	+	0.819009i	$0.694524\pi$
$884$	2.19224	0.0737331
$885$	−0.300720	−0.0101086
$886$	−33.0431	−1.11010
$887$	15.2390	0.511675	0.255837	−	0.966720i	$-0.417649\pi$
$887$	15.2390	0.511675	0.255837	+	0.966720i	$0.417649\pi$
$888$	5.77802	0.193898
$889$	−9.19030	−0.308233
$890$	−16.8385	−0.564426
$891$	0	0
$892$	−26.8969	−0.900575
$893$	−15.4556	−0.517201
$894$	14.0624	0.470317
$895$	−5.66676	−0.189419
$896$	−60.8284	−2.03213
$897$	3.43229	0.114601
$898$	49.7266	1.65940
$899$	−4.45313	−0.148520
$900$	2.44658	0.0815528
$901$	−2.79572	−0.0931389
$902$	0	0
$903$	−80.4245	−2.67636
$904$	10.8611	0.361234
$905$	−13.7728	−0.457824
$906$	16.1942	0.538016
$907$	33.3375	1.10695	0.553477	−	0.832864i	$-0.313301\pi$
$907$	33.3375	1.10695	0.553477	+	0.832864i	$0.313301\pi$
$908$	12.5256	0.415675
$909$	7.05016	0.233839
$910$	9.68276	0.320980
$911$	2.03204	0.0673244	0.0336622	−	0.999433i	$-0.489283\pi$
$911$	2.03204	0.0673244	0.0336622	+	0.999433i	$0.489283\pi$
$912$	−13.0516	−0.432181
$913$	0	0
$914$	19.2002	0.635085
$915$	0.643727	0.0212810
$916$	22.4238	0.740904
$917$	−33.3798	−1.10230
$918$	19.5885	0.646516
$919$	−40.6193	−1.33991	−0.669953	−	0.742404i	$-0.733686\pi$
$919$	−40.6193	−1.33991	−0.669953	+	0.742404i	$0.733686\pi$
$920$	3.65935	0.120645
$921$	1.36095	0.0448449
$922$	−26.2512	−0.864537
$923$	4.33959	0.142839
$924$	0	0
$925$	8.81551	0.289852
$926$	−46.0758	−1.51415
$927$	−2.30169	−0.0755974
$928$	24.0717	0.790192
$929$	52.8569	1.73418	0.867090	−	0.498151i	$-0.165988\pi$
$929$	52.8569	1.73418	0.867090	+	0.498151i	$0.165988\pi$
$930$	−2.80186	−0.0918766
$931$	−33.0848	−1.08431
$932$	5.88942	0.192914
$933$	27.2796	0.893096
$934$	18.2040	0.595652
$935$	0	0
$936$	1.02323	0.0334454
$937$	−29.2702	−0.956214	−0.478107	−	0.878301i	$-0.658677\pi$
$937$	−29.2702	−0.956214	−0.478107	+	0.878301i	$0.658677\pi$
$938$	−39.7815	−1.29891
$939$	−44.4185	−1.44954
$940$	9.96194	0.324923
$941$	−48.7226	−1.58831	−0.794156	−	0.607714i	$-0.792087\pi$
$941$	−48.7226	−1.58831	−0.794156	+	0.607714i	$0.792087\pi$
$942$	−53.5704	−1.74542
$943$	−19.3133	−0.628927
$944$	−0.936610	−0.0304841
$945$	29.7676	0.968339
$946$	0	0
$947$	−21.0241	−0.683190	−0.341595	−	0.939847i	$-0.610967\pi$
$947$	−21.0241	−0.683190	−0.341595	+	0.939847i	$0.610967\pi$
$948$	−1.14322	−0.0371302
$949$	16.9864	0.551401
$950$	−11.5529	−0.374827
$951$	42.7847	1.38739
$952$	−17.2513	−0.559117
$953$	36.8870	1.19489	0.597444	−	0.801911i	$-0.296183\pi$
$953$	36.8870	1.19489	0.597444	+	0.801911i	$0.296183\pi$
$954$	1.43953	0.0466065
$955$	24.9236	0.806508
$956$	−1.18483	−0.0383203
$957$	0	0
$958$	24.0794	0.777971
$959$	66.2567	2.13954
$960$	−0.892523	−0.0288061
$961$	1.00000	0.0322581
$962$	−4.06727	−0.131134
$963$	−4.76234	−0.153464
$964$	29.4554	0.948694
$965$	−0.975117	−0.0313901
$966$	29.7960	0.958671
$967$	34.8209	1.11977	0.559883	−	0.828572i	$-0.310846\pi$
$967$	34.8209	1.11977	0.559883	+	0.828572i	$0.310846\pi$
$968$	0	0
$969$	−5.25548	−0.168830
$970$	−30.8013	−0.988971
$971$	−24.3021	−0.779893	−0.389946	−	0.920838i	$-0.627506\pi$
$971$	−24.3021	−0.779893	−0.389946	+	0.920838i	$0.627506\pi$
$972$	−6.36741	−0.204235
$973$	−101.596	−3.25702
$974$	−1.58274	−0.0507142
$975$	−6.33127	−0.202763
$976$	2.00493	0.0641762
$977$	−26.0018	−0.831871	−0.415936	−	0.909394i	$-0.636546\pi$
$977$	−26.0018	−0.831871	−0.415936	+	0.909394i	$0.636546\pi$
$978$	22.1923	0.709633
$979$	0	0
$980$	21.3249	0.681200
$981$	3.46403	0.110598
$982$	−27.6680	−0.882923
$983$	37.2627	1.18850	0.594248	−	0.804282i	$-0.297450\pi$
$983$	37.2627	1.18850	0.594248	+	0.804282i	$0.297450\pi$
$984$	23.3504	0.744382
$985$	21.7382	0.692638
$986$	15.6480	0.498333
$987$	−73.5288	−2.34045
$988$	1.83392	0.0583447
$989$	21.3952	0.680328
$990$	0	0
$991$	36.0204	1.14423	0.572113	−	0.820175i	$-0.306124\pi$
$991$	36.0204	1.14423	0.572113	+	0.820175i	$0.306124\pi$
$992$	−5.40557	−0.171627
$993$	−4.94183	−0.156824
$994$	37.6724	1.19489
$995$	11.2996	0.358222
$996$	−20.1473	−0.638393
$997$	−29.1766	−0.924031	−0.462016	−	0.886872i	$-0.652874\pi$
$997$	−29.1766	−0.924031	−0.462016	+	0.886872i	$0.652874\pi$
$998$	−41.7347	−1.32109
$999$	−12.5039	−0.395607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3751.2.a.m.1.4	yes	15
11.10	odd	2		3751.2.a.l.1.12	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3751.2.a.l.1.12	✓	15	11.10	odd	2
3751.2.a.m.1.4	yes	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3751 = 11^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3751.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74615 q^{2} -1.55132 q^{3} +1.04905 q^{4} +1.03434 q^{5} +2.70884 q^{6} +5.16265 q^{7} +1.66050 q^{8} -0.593409 q^{9} -1.80611 q^{10} -1.62742 q^{12} -1.03844 q^{13} -9.01478 q^{14} -1.60459 q^{15} -4.99759 q^{16} -2.01238 q^{17} +1.03618 q^{18} -1.68345 q^{19} +1.08508 q^{20} -8.00892 q^{21} +2.13060 q^{23} -2.57596 q^{24} -3.93014 q^{25} +1.81328 q^{26} +5.57452 q^{27} +5.41589 q^{28} +4.45313 q^{29} +2.80186 q^{30} -1.00000 q^{31} +5.40557 q^{32} +3.51393 q^{34} +5.33993 q^{35} -0.622518 q^{36} -2.24305 q^{37} +2.93957 q^{38} +1.61095 q^{39} +1.71752 q^{40} -9.06471 q^{41} +13.9848 q^{42} +10.0419 q^{43} -0.613786 q^{45} -3.72036 q^{46} +9.18087 q^{47} +7.75286 q^{48} +19.6529 q^{49} +6.86263 q^{50} +3.12184 q^{51} -1.08938 q^{52} +1.38926 q^{53} -9.73398 q^{54} +8.57257 q^{56} +2.61158 q^{57} -7.77585 q^{58} +0.187412 q^{59} -1.68330 q^{60} -0.401179 q^{61} +1.74615 q^{62} -3.06356 q^{63} +0.556232 q^{64} -1.07410 q^{65} +4.41292 q^{67} -2.11109 q^{68} -3.30524 q^{69} -9.32434 q^{70} -4.17895 q^{71} -0.985356 q^{72} -16.3576 q^{73} +3.91671 q^{74} +6.09691 q^{75} -1.76603 q^{76} -2.81297 q^{78} +0.702478 q^{79} -5.16921 q^{80} -6.86764 q^{81} +15.8284 q^{82} +12.3800 q^{83} -8.40178 q^{84} -2.08148 q^{85} -17.5346 q^{86} -6.90822 q^{87} +9.32302 q^{89} +1.07177 q^{90} -5.36110 q^{91} +2.23511 q^{92} +1.55132 q^{93} -16.0312 q^{94} -1.74126 q^{95} -8.38576 q^{96} +17.0539 q^{97} -34.3171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 2 q^{3} + 17 q^{4} + 8 q^{5} - 2 q^{6} + 3 q^{8} + 25 q^{9} - 15 q^{10} + 11 q^{12} - 4 q^{13} + 9 q^{14} + 15 q^{15} + 29 q^{16} - 2 q^{17} + 4 q^{18} + 5 q^{19} + 17 q^{20} - 15 q^{21}+ \cdots - 31 q^{98}+O(q^{100}) \) 15 * q + q^2 + 2 * q^3 + 17 * q^4 + 8 * q^5 - 2 * q^6 + 3 * q^8 + 25 * q^9 - 15 * q^10 + 11 * q^12 - 4 * q^13 + 9 * q^14 + 15 * q^15 + 29 * q^16 - 2 * q^17 + 4 * q^18 + 5 * q^19 + 17 * q^20 - 15 * q^21 + 17 * q^23 - 18 * q^24 + 27 * q^25 + 22 * q^26 + 17 * q^27 + 35 * q^28 + 7 * q^29 + 27 * q^30 - 15 * q^31 - 23 * q^32 + 15 * q^34 + 9 * q^35 - 3 * q^36 + 17 * q^37 + 4 * q^38 + 18 * q^39 - 51 * q^40 - 7 * q^41 - 17 * q^42 - 12 * q^43 + 18 * q^45 + 20 * q^46 - 8 * q^47 + 21 * q^48 + 37 * q^49 + 16 * q^50 - 52 * q^51 + 35 * q^52 + 26 * q^53 - 48 * q^54 + 8 * q^56 + 44 * q^57 - 55 * q^58 + 11 * q^59 + 108 * q^60 + 17 * q^61 - q^62 + 22 * q^63 - 3 * q^64 - 12 * q^65 - 31 * q^67 + 30 * q^68 - 16 * q^69 - 8 * q^70 + 35 * q^71 + 59 * q^72 - q^73 + 5 * q^74 + 61 * q^75 - 12 * q^76 + q^78 + 14 * q^79 + 27 * q^80 + 83 * q^81 + 39 * q^82 + 26 * q^83 - 46 * q^84 + 16 * q^85 + 51 * q^86 + 43 * q^87 + 73 * q^89 - 45 * q^90 - 23 * q^91 + 19 * q^92 - 2 * q^93 + 46 * q^94 - 59 * q^95 + 53 * q^96 + 34 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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