LMFDB - Embedded newform 368.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,4,Mod(1,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("368.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	368.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:	$21.7127028821$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.10469 q^{3} -1.40312 q^{5} +3.40312 q^{7} +55.8953 q^{9} +O(q^{10})$	$q-9.10469 q^{3} -1.40312 q^{5} +3.40312 q^{7} +55.8953 q^{9} -61.6125 q^{11} -77.9109 q^{13} +12.7750 q^{15} -14.8375 q^{17} +46.6281 q^{19} -30.9844 q^{21} +23.0000 q^{23} -123.031 q^{25} -263.083 q^{27} -200.602 q^{29} +20.9266 q^{31} +560.962 q^{33} -4.77501 q^{35} +233.109 q^{37} +709.355 q^{39} +309.733 q^{41} +399.287 q^{43} -78.4281 q^{45} +190.245 q^{47} -331.419 q^{49} +135.091 q^{51} +516.994 q^{53} +86.4500 q^{55} -424.534 q^{57} -498.450 q^{59} +906.731 q^{61} +190.219 q^{63} +109.319 q^{65} +33.6750 q^{67} -209.408 q^{69} +61.1578 q^{71} -79.3453 q^{73} +1120.16 q^{75} -209.675 q^{77} -533.916 q^{79} +886.113 q^{81} -766.878 q^{83} +20.8188 q^{85} +1826.41 q^{87} +57.9468 q^{89} -265.141 q^{91} -190.530 q^{93} -65.4250 q^{95} +1601.06 q^{97} -3443.85 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 10 q^{5} - 6 q^{7} + 131 q^{9}+O(q^{10}) $ 2 * q + q^3 + 10 * q^5 - 6 * q^7 + 131 * q^9	$ 2 q + q^{3} + 10 q^{5} - 6 q^{7} + 131 q^{9} - 72 q^{11} - 111 q^{13} + 128 q^{15} + 124 q^{17} - 22 q^{19} - 126 q^{21} + 46 q^{23} - 118 q^{25} + 223 q^{27} + 15 q^{29} - 67 q^{31} + 456 q^{33} - 112 q^{35} + 18 q^{37} + 375 q^{39} + 485 q^{41} + 440 q^{43} + 778 q^{45} - 215 q^{47} - 586 q^{49} + 1538 q^{51} + 240 q^{53} - 32 q^{55} - 1118 q^{57} - 792 q^{59} + 456 q^{61} - 516 q^{63} - 268 q^{65} - 240 q^{67} + 23 q^{69} + 705 q^{71} + 27 q^{73} + 1171 q^{75} - 112 q^{77} - 594 q^{79} + 3770 q^{81} - 394 q^{83} + 1604 q^{85} + 4005 q^{87} - 486 q^{89} + 46 q^{91} - 1079 q^{93} - 848 q^{95} + 2152 q^{97} - 4224 q^{99}+O(q^{100}) $ 2 * q + q^3 + 10 * q^5 - 6 * q^7 + 131 * q^9 - 72 * q^11 - 111 * q^13 + 128 * q^15 + 124 * q^17 - 22 * q^19 - 126 * q^21 + 46 * q^23 - 118 * q^25 + 223 * q^27 + 15 * q^29 - 67 * q^31 + 456 * q^33 - 112 * q^35 + 18 * q^37 + 375 * q^39 + 485 * q^41 + 440 * q^43 + 778 * q^45 - 215 * q^47 - 586 * q^49 + 1538 * q^51 + 240 * q^53 - 32 * q^55 - 1118 * q^57 - 792 * q^59 + 456 * q^61 - 516 * q^63 - 268 * q^65 - 240 * q^67 + 23 * q^69 + 705 * q^71 + 27 * q^73 + 1171 * q^75 - 112 * q^77 - 594 * q^79 + 3770 * q^81 - 394 * q^83 + 1604 * q^85 + 4005 * q^87 - 486 * q^89 + 46 * q^91 - 1079 * q^93 - 848 * q^95 + 2152 * q^97 - 4224 * 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
$3$	−9.10469	−1.75220	−0.876099	−	0.482132i	$-0.839863\pi$
$3$	−9.10469	−1.75220	−0.876099	+	0.482132i	$0.839863\pi$
$4$	0	0
$5$	−1.40312	−0.125499	−0.0627496	−	0.998029i	$-0.519987\pi$
$5$	−1.40312	−0.125499	−0.0627496	+	0.998029i	$0.519987\pi$
$6$	0	0
$7$	3.40312	0.183751	0.0918757	−	0.995770i	$-0.470714\pi$
$7$	3.40312	0.183751	0.0918757	+	0.995770i	$0.470714\pi$
$8$	0	0
$9$	55.8953	2.07020
$10$	0	0
$11$	−61.6125	−1.68881	−0.844403	−	0.535708i	$-0.820045\pi$
$11$	−61.6125	−1.68881	−0.844403	+	0.535708i	$0.820045\pi$
$12$	0	0
$13$	−77.9109	−1.66220	−0.831100	−	0.556123i	$-0.812288\pi$
$13$	−77.9109	−1.66220	−0.831100	+	0.556123i	$0.812288\pi$
$14$	0	0
$15$	12.7750	0.219899
$16$	0	0
$17$	−14.8375	−0.211684	−0.105842	−	0.994383i	$-0.533754\pi$
$17$	−14.8375	−0.211684	−0.105842	+	0.994383i	$0.533754\pi$
$18$	0	0
$19$	46.6281	0.563012	0.281506	−	0.959559i	$-0.409166\pi$
$19$	46.6281	0.563012	0.281506	+	0.959559i	$0.409166\pi$
$20$	0	0
$21$	−30.9844	−0.321969
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−123.031	−0.984250
$26$	0	0
$27$	−263.083	−1.87520
$28$	0	0
$29$	−200.602	−1.28451	−0.642255	−	0.766491i	$-0.722001\pi$
$29$	−200.602	−1.28451	−0.642255	+	0.766491i	$0.722001\pi$
$30$	0	0
$31$	20.9266	0.121243	0.0606213	−	0.998161i	$-0.480692\pi$
$31$	20.9266	0.121243	0.0606213	+	0.998161i	$0.480692\pi$
$32$	0	0
$33$	560.962	2.95912
$34$	0	0
$35$	−4.77501	−0.0230607
$36$	0	0
$37$	233.109	1.03576	0.517878	−	0.855455i	$-0.326722\pi$
$37$	233.109	1.03576	0.517878	+	0.855455i	$0.326722\pi$
$38$	0	0
$39$	709.355	2.91250
$40$	0	0
$41$	309.733	1.17981	0.589904	−	0.807473i	$-0.299166\pi$
$41$	309.733	1.17981	0.589904	+	0.807473i	$0.299166\pi$
$42$	0	0
$43$	399.287	1.41606	0.708032	−	0.706180i	$-0.249583\pi$
$43$	399.287	1.41606	0.708032	+	0.706180i	$0.249583\pi$
$44$	0	0
$45$	−78.4281	−0.259808
$46$	0	0
$47$	190.245	0.590428	0.295214	−	0.955431i	$-0.404609\pi$
$47$	190.245	0.590428	0.295214	+	0.955431i	$0.404609\pi$
$48$	0	0
$49$	−331.419	−0.966235
$50$	0	0
$51$	135.091	0.370911
$52$	0	0
$53$	516.994	1.33990	0.669949	−	0.742408i	$-0.266316\pi$
$53$	516.994	1.33990	0.669949	+	0.742408i	$0.266316\pi$
$54$	0	0
$55$	86.4500	0.211944
$56$	0	0
$57$	−424.534	−0.986508
$58$	0	0
$59$	−498.450	−1.09988	−0.549938	−	0.835206i	$-0.685349\pi$
$59$	−498.450	−1.09988	−0.549938	+	0.835206i	$0.685349\pi$
$60$	0	0
$61$	906.731	1.90320	0.951599	−	0.307344i	$-0.0994401\pi$
$61$	906.731	1.90320	0.951599	+	0.307344i	$0.0994401\pi$
$62$	0	0
$63$	190.219	0.380402
$64$	0	0
$65$	109.319	0.208605
$66$	0	0
$67$	33.6750	0.0614038	0.0307019	−	0.999529i	$-0.490226\pi$
$67$	33.6750	0.0614038	0.0307019	+	0.999529i	$0.490226\pi$
$68$	0	0
$69$	−209.408	−0.365358
$70$	0	0
$71$	61.1578	0.102227	0.0511134	−	0.998693i	$-0.483723\pi$
$71$	61.1578	0.102227	0.0511134	+	0.998693i	$0.483723\pi$
$72$	0	0
$73$	−79.3453	−0.127215	−0.0636073	−	0.997975i	$-0.520261\pi$
$73$	−79.3453	−0.127215	−0.0636073	+	0.997975i	$0.520261\pi$
$74$	0	0
$75$	1120.16	1.72460
$76$	0	0
$77$	−209.675	−0.310321
$78$	0	0
$79$	−533.916	−0.760382	−0.380191	−	0.924908i	$-0.624142\pi$
$79$	−533.916	−0.760382	−0.380191	+	0.924908i	$0.624142\pi$
$80$	0	0
$81$	886.113	1.21552
$82$	0	0
$83$	−766.878	−1.01417	−0.507083	−	0.861897i	$-0.669276\pi$
$83$	−766.878	−1.01417	−0.507083	+	0.861897i	$0.669276\pi$
$84$	0	0
$85$	20.8188	0.0265661
$86$	0	0
$87$	1826.41	2.25071
$88$	0	0
$89$	57.9468	0.0690152	0.0345076	−	0.999404i	$-0.489014\pi$
$89$	57.9468	0.0690152	0.0345076	+	0.999404i	$0.489014\pi$
$90$	0	0
$91$	−265.141	−0.305432
$92$	0	0
$93$	−190.530	−0.212441
$94$	0	0
$95$	−65.4250	−0.0706576
$96$	0	0
$97$	1601.06	1.67590	0.837952	−	0.545744i	$-0.183753\pi$
$97$	1601.06	1.67590	0.837952	+	0.545744i	$0.183753\pi$
$98$	0	0
$99$	−3443.85	−3.49616
$100$	0	0
$101$	367.425	0.361982	0.180991	−	0.983485i	$-0.442070\pi$
$101$	367.425	0.361982	0.180991	+	0.983485i	$0.442070\pi$
$102$	0	0
$103$	1051.19	1.00560	0.502800	−	0.864403i	$-0.332303\pi$
$103$	1051.19	1.00560	0.502800	+	0.864403i	$0.332303\pi$
$104$	0	0
$105$	43.4749	0.0404068
$106$	0	0
$107$	1104.18	0.997616	0.498808	−	0.866712i	$-0.333771\pi$
$107$	1104.18	0.997616	0.498808	+	0.866712i	$0.333771\pi$
$108$	0	0
$109$	−1264.69	−1.11133	−0.555667	−	0.831405i	$-0.687537\pi$
$109$	−1264.69	−1.11133	−0.555667	+	0.831405i	$0.687537\pi$
$110$	0	0
$111$	−2122.39	−1.81485
$112$	0	0
$113$	−2102.16	−1.75004	−0.875021	−	0.484085i	$-0.839153\pi$
$113$	−2102.16	−1.75004	−0.875021	+	0.484085i	$0.839153\pi$
$114$	0	0
$115$	−32.2719	−0.0261684
$116$	0	0
$117$	−4354.86	−3.44108
$118$	0	0
$119$	−50.4938	−0.0388972
$120$	0	0
$121$	2465.10	1.85207
$122$	0	0
$123$	−2820.02	−2.06726
$124$	0	0
$125$	348.019	0.249022
$126$	0	0
$127$	−349.639	−0.244295	−0.122147	−	0.992512i	$-0.538978\pi$
$127$	−349.639	−0.244295	−0.122147	+	0.992512i	$0.538978\pi$
$128$	0	0
$129$	−3635.39	−2.48122
$130$	0	0
$131$	412.864	0.275360	0.137680	−	0.990477i	$-0.456036\pi$
$131$	412.864	0.275360	0.137680	+	0.990477i	$0.456036\pi$
$132$	0	0
$133$	158.681	0.103454
$134$	0	0
$135$	369.138	0.235336
$136$	0	0
$137$	2357.69	1.47030	0.735150	−	0.677905i	$-0.237112\pi$
$137$	2357.69	1.47030	0.735150	+	0.677905i	$0.237112\pi$
$138$	0	0
$139$	−85.9515	−0.0524483	−0.0262241	−	0.999656i	$-0.508348\pi$
$139$	−85.9515	−0.0524483	−0.0262241	+	0.999656i	$0.508348\pi$
$140$	0	0
$141$	−1732.12	−1.03455
$142$	0	0
$143$	4800.29	2.80713
$144$	0	0
$145$	281.469	0.161205
$146$	0	0
$147$	3017.46	1.69304
$148$	0	0
$149$	−559.531	−0.307642	−0.153821	−	0.988099i	$-0.549158\pi$
$149$	−559.531	−0.307642	−0.153821	+	0.988099i	$0.549158\pi$
$150$	0	0
$151$	−210.467	−0.113428	−0.0567138	−	0.998390i	$-0.518062\pi$
$151$	−210.467	−0.113428	−0.0567138	+	0.998390i	$0.518062\pi$
$152$	0	0
$153$	−829.346	−0.438227
$154$	0	0
$155$	−29.3626	−0.0152159
$156$	0	0
$157$	−436.734	−0.222008	−0.111004	−	0.993820i	$-0.535407\pi$
$157$	−436.734	−0.222008	−0.111004	+	0.993820i	$0.535407\pi$
$158$	0	0
$159$	−4707.07	−2.34776
$160$	0	0
$161$	78.2719	0.0383148
$162$	0	0
$163$	1199.47	0.576377	0.288189	−	0.957574i	$-0.406947\pi$
$163$	1199.47	0.576377	0.288189	+	0.957574i	$0.406947\pi$
$164$	0	0
$165$	−787.100	−0.371368
$166$	0	0
$167$	−1388.58	−0.643423	−0.321711	−	0.946838i	$-0.604258\pi$
$167$	−1388.58	−0.643423	−0.321711	+	0.946838i	$0.604258\pi$
$168$	0	0
$169$	3873.11	1.76291
$170$	0	0
$171$	2606.29	1.16555
$172$	0	0
$173$	2726.37	1.19816	0.599082	−	0.800688i	$-0.295532\pi$
$173$	2726.37	1.19816	0.599082	+	0.800688i	$0.295532\pi$
$174$	0	0
$175$	−418.691	−0.180857
$176$	0	0
$177$	4538.23	1.92720
$178$	0	0
$179$	−1197.97	−0.500226	−0.250113	−	0.968217i	$-0.580468\pi$
$179$	−1197.97	−0.500226	−0.250113	+	0.968217i	$0.580468\pi$
$180$	0	0
$181$	486.384	0.199738	0.0998692	−	0.995001i	$-0.468158\pi$
$181$	486.384	0.199738	0.0998692	+	0.995001i	$0.468158\pi$
$182$	0	0
$183$	−8255.50	−3.33478
$184$	0	0
$185$	−327.081	−0.129986
$186$	0	0
$187$	914.175	0.357492
$188$	0	0
$189$	−895.303	−0.344570
$190$	0	0
$191$	2208.62	0.836704	0.418352	−	0.908285i	$-0.362608\pi$
$191$	2208.62	0.836704	0.418352	+	0.908285i	$0.362608\pi$
$192$	0	0
$193$	961.280	0.358520	0.179260	−	0.983802i	$-0.442630\pi$
$193$	961.280	0.358520	0.179260	+	0.983802i	$0.442630\pi$
$194$	0	0
$195$	−995.313	−0.365517
$196$	0	0
$197$	−5228.47	−1.89093	−0.945465	−	0.325725i	$-0.894392\pi$
$197$	−5228.47	−1.89093	−0.945465	+	0.325725i	$0.894392\pi$
$198$	0	0
$199$	−3656.52	−1.30253	−0.651266	−	0.758850i	$-0.725762\pi$
$199$	−3656.52	−1.30253	−0.651266	+	0.758850i	$0.725762\pi$
$200$	0	0
$201$	−306.600	−0.107592
$202$	0	0
$203$	−682.672	−0.236030
$204$	0	0
$205$	−434.594	−0.148065
$206$	0	0
$207$	1285.59	0.431666
$208$	0	0
$209$	−2872.87	−0.950818
$210$	0	0
$211$	2849.63	0.929747	0.464874	−	0.885377i	$-0.346100\pi$
$211$	2849.63	0.929747	0.464874	+	0.885377i	$0.346100\pi$
$212$	0	0
$213$	−556.823	−0.179122
$214$	0	0
$215$	−560.250	−0.177715
$216$	0	0
$217$	71.2157	0.0222785
$218$	0	0
$219$	722.414	0.222905
$220$	0	0
$221$	1156.00	0.351860
$222$	0	0
$223$	3924.56	1.17851	0.589256	−	0.807946i	$-0.299421\pi$
$223$	3924.56	1.17851	0.589256	+	0.807946i	$0.299421\pi$
$224$	0	0
$225$	−6876.87	−2.03759
$226$	0	0
$227$	2692.10	0.787142	0.393571	−	0.919294i	$-0.371240\pi$
$227$	2692.10	0.787142	0.393571	+	0.919294i	$0.371240\pi$
$228$	0	0
$229$	−2226.63	−0.642533	−0.321267	−	0.946989i	$-0.604109\pi$
$229$	−2226.63	−0.642533	−0.321267	+	0.946989i	$0.604109\pi$
$230$	0	0
$231$	1909.02	0.543743
$232$	0	0
$233$	−138.836	−0.0390363	−0.0195181	−	0.999810i	$-0.506213\pi$
$233$	−138.836	−0.0390363	−0.0195181	+	0.999810i	$0.506213\pi$
$234$	0	0
$235$	−266.938	−0.0740983
$236$	0	0
$237$	4861.13	1.33234
$238$	0	0
$239$	−2165.37	−0.586052	−0.293026	−	0.956105i	$-0.594662\pi$
$239$	−2165.37	−0.586052	−0.293026	+	0.956105i	$0.594662\pi$
$240$	0	0
$241$	4147.85	1.10866	0.554329	−	0.832297i	$-0.312975\pi$
$241$	4147.85	1.10866	0.554329	+	0.832297i	$0.312975\pi$
$242$	0	0
$243$	−964.543	−0.254631
$244$	0	0
$245$	465.022	0.121262
$246$	0	0
$247$	−3632.84	−0.935838
$248$	0	0
$249$	6982.18	1.77702
$250$	0	0
$251$	−1007.93	−0.253466	−0.126733	−	0.991937i	$-0.540449\pi$
$251$	−1007.93	−0.253466	−0.126733	+	0.991937i	$0.540449\pi$
$252$	0	0
$253$	−1417.09	−0.352140
$254$	0	0
$255$	−189.549	−0.0465491
$256$	0	0
$257$	−3690.68	−0.895791	−0.447895	−	0.894086i	$-0.647826\pi$
$257$	−3690.68	−0.895791	−0.447895	+	0.894086i	$0.647826\pi$
$258$	0	0
$259$	793.300	0.190321
$260$	0	0
$261$	−11212.7	−2.65919
$262$	0	0
$263$	−2881.53	−0.675601	−0.337800	−	0.941218i	$-0.609683\pi$
$263$	−2881.53	−0.675601	−0.337800	+	0.941218i	$0.609683\pi$
$264$	0	0
$265$	−725.406	−0.168156
$266$	0	0
$267$	−527.588	−0.120928
$268$	0	0
$269$	2417.30	0.547902	0.273951	−	0.961744i	$-0.411669\pi$
$269$	2417.30	0.547902	0.273951	+	0.961744i	$0.411669\pi$
$270$	0	0
$271$	−2756.05	−0.617779	−0.308889	−	0.951098i	$-0.599957\pi$
$271$	−2756.05	−0.617779	−0.308889	+	0.951098i	$0.599957\pi$
$272$	0	0
$273$	2414.02	0.535177
$274$	0	0
$275$	7580.26	1.66221
$276$	0	0
$277$	−3117.08	−0.676126	−0.338063	−	0.941123i	$-0.609772\pi$
$277$	−3117.08	−0.676126	−0.338063	+	0.941123i	$0.609772\pi$
$278$	0	0
$279$	1169.70	0.250996
$280$	0	0
$281$	2917.44	0.619360	0.309680	−	0.950841i	$-0.399778\pi$
$281$	2917.44	0.619360	0.309680	+	0.950841i	$0.399778\pi$
$282$	0	0
$283$	6795.66	1.42742	0.713710	−	0.700441i	$-0.247013\pi$
$283$	6795.66	1.42742	0.713710	+	0.700441i	$0.247013\pi$
$284$	0	0
$285$	595.674	0.123806
$286$	0	0
$287$	1054.06	0.216791
$288$	0	0
$289$	−4692.85	−0.955190
$290$	0	0
$291$	−14577.1	−2.93651
$292$	0	0
$293$	−805.147	−0.160536	−0.0802682	−	0.996773i	$-0.525578\pi$
$293$	−805.147	−0.160536	−0.0802682	+	0.996773i	$0.525578\pi$
$294$	0	0
$295$	699.387	0.138034
$296$	0	0
$297$	16209.2	3.16684
$298$	0	0
$299$	−1791.95	−0.346593
$300$	0	0
$301$	1358.82	0.260204
$302$	0	0
$303$	−3345.29	−0.634264
$304$	0	0
$305$	−1272.26	−0.238850
$306$	0	0
$307$	4728.10	0.878980	0.439490	−	0.898247i	$-0.355159\pi$
$307$	4728.10	0.878980	0.439490	+	0.898247i	$0.355159\pi$
$308$	0	0
$309$	−9570.76	−1.76201
$310$	0	0
$311$	−5833.88	−1.06369	−0.531847	−	0.846840i	$-0.678502\pi$
$311$	−5833.88	−1.06369	−0.531847	+	0.846840i	$0.678502\pi$
$312$	0	0
$313$	8024.33	1.44908	0.724540	−	0.689233i	$-0.242052\pi$
$313$	8024.33	1.44908	0.724540	+	0.689233i	$0.242052\pi$
$314$	0	0
$315$	−266.900	−0.0477401
$316$	0	0
$317$	−4527.26	−0.802134	−0.401067	−	0.916049i	$-0.631360\pi$
$317$	−4527.26	−0.802134	−0.401067	+	0.916049i	$0.631360\pi$
$318$	0	0
$319$	12359.6	2.16929
$320$	0	0
$321$	−10053.2	−1.74802
$322$	0	0
$323$	−691.844	−0.119180
$324$	0	0
$325$	9585.48	1.63602
$326$	0	0
$327$	11514.6	1.94728
$328$	0	0
$329$	647.428	0.108492
$330$	0	0
$331$	935.443	0.155337	0.0776686	−	0.996979i	$-0.475252\pi$
$331$	935.443	0.155337	0.0776686	+	0.996979i	$0.475252\pi$
$332$	0	0
$333$	13029.7	2.14422
$334$	0	0
$335$	−47.2502	−0.00770613
$336$	0	0
$337$	−1650.15	−0.266735	−0.133367	−	0.991067i	$-0.542579\pi$
$337$	−1650.15	−0.266735	−0.133367	+	0.991067i	$0.542579\pi$
$338$	0	0
$339$	19139.5	3.06642
$340$	0	0
$341$	−1289.34	−0.204755
$342$	0	0
$343$	−2295.13	−0.361299
$344$	0	0
$345$	293.825	0.0458522
$346$	0	0
$347$	−8343.39	−1.29077	−0.645384	−	0.763858i	$-0.723303\pi$
$347$	−8343.39	−1.29077	−0.645384	+	0.763858i	$0.723303\pi$
$348$	0	0
$349$	6870.72	1.05381	0.526907	−	0.849923i	$-0.323351\pi$
$349$	6870.72	1.05381	0.526907	+	0.849923i	$0.323351\pi$
$350$	0	0
$351$	20497.0	3.11695
$352$	0	0
$353$	3698.10	0.557592	0.278796	−	0.960350i	$-0.410065\pi$
$353$	3698.10	0.557592	0.278796	+	0.960350i	$0.410065\pi$
$354$	0	0
$355$	−85.8121	−0.0128294
$356$	0	0
$357$	459.730	0.0681555
$358$	0	0
$359$	1539.79	0.226371	0.113185	−	0.993574i	$-0.463895\pi$
$359$	1539.79	0.226371	0.113185	+	0.993574i	$0.463895\pi$
$360$	0	0
$361$	−4684.82	−0.683018
$362$	0	0
$363$	−22444.0	−3.24519
$364$	0	0
$365$	111.331	0.0159653
$366$	0	0
$367$	3516.58	0.500174	0.250087	−	0.968223i	$-0.419541\pi$
$367$	3516.58	0.500174	0.250087	+	0.968223i	$0.419541\pi$
$368$	0	0
$369$	17312.6	2.44244
$370$	0	0
$371$	1759.39	0.246208
$372$	0	0
$373$	190.852	0.0264932	0.0132466	−	0.999912i	$-0.495783\pi$
$373$	190.852	0.0264932	0.0132466	+	0.999912i	$0.495783\pi$
$374$	0	0
$375$	−3168.60	−0.436336
$376$	0	0
$377$	15629.1	2.13511
$378$	0	0
$379$	9899.32	1.34167	0.670836	−	0.741606i	$-0.265935\pi$
$379$	9899.32	1.34167	0.670836	+	0.741606i	$0.265935\pi$
$380$	0	0
$381$	3183.35	0.428053
$382$	0	0
$383$	−12121.1	−1.61713	−0.808565	−	0.588407i	$-0.799755\pi$
$383$	−12121.1	−1.61713	−0.808565	+	0.588407i	$0.799755\pi$
$384$	0	0
$385$	294.200	0.0389450
$386$	0	0
$387$	22318.3	2.93153
$388$	0	0
$389$	−2479.68	−0.323200	−0.161600	−	0.986856i	$-0.551665\pi$
$389$	−2479.68	−0.323200	−0.161600	+	0.986856i	$0.551665\pi$
$390$	0	0
$391$	−341.262	−0.0441391
$392$	0	0
$393$	−3759.00	−0.482484
$394$	0	0
$395$	749.150	0.0954274
$396$	0	0
$397$	4988.43	0.630635	0.315318	−	0.948986i	$-0.397889\pi$
$397$	4988.43	0.630635	0.315318	+	0.948986i	$0.397889\pi$
$398$	0	0
$399$	−1444.74	−0.181272
$400$	0	0
$401$	6200.67	0.772186	0.386093	−	0.922460i	$-0.373824\pi$
$401$	6200.67	0.772186	0.386093	+	0.922460i	$0.373824\pi$
$402$	0	0
$403$	−1630.41	−0.201529
$404$	0	0
$405$	−1243.33	−0.152547
$406$	0	0
$407$	−14362.4	−1.74919
$408$	0	0
$409$	−12275.7	−1.48410	−0.742049	−	0.670346i	$-0.766146\pi$
$409$	−12275.7	−1.48410	−0.742049	+	0.670346i	$0.766146\pi$
$410$	0	0
$411$	−21466.0	−2.57626
$412$	0	0
$413$	−1696.29	−0.202104
$414$	0	0
$415$	1076.03	0.127277
$416$	0	0
$417$	782.561	0.0918998
$418$	0	0
$419$	−7354.79	−0.857530	−0.428765	−	0.903416i	$-0.641051\pi$
$419$	−7354.79	−0.857530	−0.428765	+	0.903416i	$0.641051\pi$
$420$	0	0
$421$	−2676.45	−0.309839	−0.154919	−	0.987927i	$-0.549512\pi$
$421$	−2676.45	−0.309839	−0.154919	+	0.987927i	$0.549512\pi$
$422$	0	0
$423$	10633.8	1.22230
$424$	0	0
$425$	1825.47	0.208350
$426$	0	0
$427$	3085.72	0.349715
$428$	0	0
$429$	−43705.1	−4.91865
$430$	0	0
$431$	880.806	0.0984383	0.0492192	−	0.998788i	$-0.484327\pi$
$431$	880.806	0.0984383	0.0492192	+	0.998788i	$0.484327\pi$
$432$	0	0
$433$	728.134	0.0808127	0.0404063	−	0.999183i	$-0.487135\pi$
$433$	728.134	0.0808127	0.0404063	+	0.999183i	$0.487135\pi$
$434$	0	0
$435$	−2562.69	−0.282463
$436$	0	0
$437$	1072.45	0.117396
$438$	0	0
$439$	8143.35	0.885332	0.442666	−	0.896687i	$-0.354033\pi$
$439$	8143.35	0.885332	0.442666	+	0.896687i	$0.354033\pi$
$440$	0	0
$441$	−18524.8	−2.00030
$442$	0	0
$443$	−15146.3	−1.62443	−0.812213	−	0.583360i	$-0.801738\pi$
$443$	−15146.3	−1.62443	−0.812213	+	0.583360i	$0.801738\pi$
$444$	0	0
$445$	−81.3066	−0.00866136
$446$	0	0
$447$	5094.36	0.539049
$448$	0	0
$449$	−9991.34	−1.05016	−0.525079	−	0.851054i	$-0.675964\pi$
$449$	−9991.34	−1.05016	−0.525079	+	0.851054i	$0.675964\pi$
$450$	0	0
$451$	−19083.4	−1.99247
$452$	0	0
$453$	1916.24	0.198748
$454$	0	0
$455$	372.025	0.0383315
$456$	0	0
$457$	9744.14	0.997400	0.498700	−	0.866775i	$-0.333811\pi$
$457$	9744.14	0.997400	0.498700	+	0.866775i	$0.333811\pi$
$458$	0	0
$459$	3903.49	0.396948
$460$	0	0
$461$	13262.0	1.33985	0.669925	−	0.742429i	$-0.266326\pi$
$461$	13262.0	1.33985	0.669925	+	0.742429i	$0.266326\pi$
$462$	0	0
$463$	3249.19	0.326139	0.163070	−	0.986615i	$-0.447860\pi$
$463$	3249.19	0.326139	0.163070	+	0.986615i	$0.447860\pi$
$464$	0	0
$465$	267.337	0.0266612
$466$	0	0
$467$	15841.9	1.56975	0.784877	−	0.619652i	$-0.212726\pi$
$467$	15841.9	1.56975	0.784877	+	0.619652i	$0.212726\pi$
$468$	0	0
$469$	114.600	0.0112830
$470$	0	0
$471$	3976.33	0.389001
$472$	0	0
$473$	−24601.1	−2.39146
$474$	0	0
$475$	−5736.72	−0.554144
$476$	0	0
$477$	28897.5	2.77385
$478$	0	0
$479$	−10766.3	−1.02698	−0.513491	−	0.858095i	$-0.671648\pi$
$479$	−10766.3	−1.02698	−0.513491	+	0.858095i	$0.671648\pi$
$480$	0	0
$481$	−18161.8	−1.72163
$482$	0	0
$483$	−712.641	−0.0671351
$484$	0	0
$485$	−2246.48	−0.210325
$486$	0	0
$487$	18053.6	1.67985	0.839925	−	0.542703i	$-0.182599\pi$
$487$	18053.6	1.67985	0.839925	+	0.542703i	$0.182599\pi$
$488$	0	0
$489$	−10920.8	−1.00993
$490$	0	0
$491$	2531.32	0.232662	0.116331	−	0.993210i	$-0.462887\pi$
$491$	2531.32	0.232662	0.116331	+	0.993210i	$0.462887\pi$
$492$	0	0
$493$	2976.42	0.271909
$494$	0	0
$495$	4832.15	0.438766
$496$	0	0
$497$	208.128	0.0187843
$498$	0	0
$499$	−7554.37	−0.677716	−0.338858	−	0.940838i	$-0.610041\pi$
$499$	−7554.37	−0.677716	−0.338858	+	0.940838i	$0.610041\pi$
$500$	0	0
$501$	12642.6	1.12740
$502$	0	0
$503$	12465.0	1.10495	0.552474	−	0.833530i	$-0.313684\pi$
$503$	12465.0	1.10495	0.552474	+	0.833530i	$0.313684\pi$
$504$	0	0
$505$	−515.543	−0.0454284
$506$	0	0
$507$	−35263.5	−3.08897
$508$	0	0
$509$	2161.55	0.188230	0.0941149	−	0.995561i	$-0.469998\pi$
$509$	2161.55	0.188230	0.0941149	+	0.995561i	$0.469998\pi$
$510$	0	0
$511$	−270.022	−0.0233759
$512$	0	0
$513$	−12267.1	−1.05576
$514$	0	0
$515$	−1474.95	−0.126202
$516$	0	0
$517$	−11721.5	−0.997119
$518$	0	0
$519$	−24822.8	−2.09942
$520$	0	0
$521$	−2880.90	−0.242255	−0.121127	−	0.992637i	$-0.538651\pi$
$521$	−2880.90	−0.242255	−0.121127	+	0.992637i	$0.538651\pi$
$522$	0	0
$523$	16965.4	1.41844	0.709222	−	0.704985i	$-0.249046\pi$
$523$	16965.4	1.41844	0.709222	+	0.704985i	$0.249046\pi$
$524$	0	0
$525$	3812.05	0.316898
$526$	0	0
$527$	−310.498	−0.0256651
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−27861.0	−2.27696
$532$	0	0
$533$	−24131.6	−1.96108
$534$	0	0
$535$	−1549.30	−0.125200
$536$	0	0
$537$	10907.1	0.876495
$538$	0	0
$539$	20419.5	1.63178
$540$	0	0
$541$	−2210.68	−0.175683	−0.0878417	−	0.996134i	$-0.527997\pi$
$541$	−2210.68	−0.175683	−0.0878417	+	0.996134i	$0.527997\pi$
$542$	0	0
$543$	−4428.38	−0.349981
$544$	0	0
$545$	1774.52	0.139472
$546$	0	0
$547$	−12826.0	−1.00256	−0.501280	−	0.865285i	$-0.667137\pi$
$547$	−12826.0	−1.00256	−0.501280	+	0.865285i	$0.667137\pi$
$548$	0	0
$549$	50682.0	3.93999
$550$	0	0
$551$	−9353.67	−0.723194
$552$	0	0
$553$	−1816.98	−0.139721
$554$	0	0
$555$	2977.97	0.227762
$556$	0	0
$557$	−13291.7	−1.01111	−0.505554	−	0.862795i	$-0.668712\pi$
$557$	−13291.7	−1.01111	−0.505554	+	0.862795i	$0.668712\pi$
$558$	0	0
$559$	−31108.9	−2.35378
$560$	0	0
$561$	−8323.28	−0.626397
$562$	0	0
$563$	6781.89	0.507678	0.253839	−	0.967246i	$-0.418307\pi$
$563$	6781.89	0.507678	0.253839	+	0.967246i	$0.418307\pi$
$564$	0	0
$565$	2949.59	0.219629
$566$	0	0
$567$	3015.55	0.223353
$568$	0	0
$569$	−9189.69	−0.677068	−0.338534	−	0.940954i	$-0.609931\pi$
$569$	−9189.69	−0.677068	−0.338534	+	0.940954i	$0.609931\pi$
$570$	0	0
$571$	14790.2	1.08398	0.541988	−	0.840386i	$-0.317672\pi$
$571$	14790.2	1.08398	0.541988	+	0.840386i	$0.317672\pi$
$572$	0	0
$573$	−20108.8	−1.46607
$574$	0	0
$575$	−2829.72	−0.205230
$576$	0	0
$577$	1346.56	0.0971540	0.0485770	−	0.998819i	$-0.484531\pi$
$577$	1346.56	0.0971540	0.0485770	+	0.998819i	$0.484531\pi$
$578$	0	0
$579$	−8752.15	−0.628198
$580$	0	0
$581$	−2609.78	−0.186355
$582$	0	0
$583$	−31853.3	−2.26283
$584$	0	0
$585$	6110.40	0.431853
$586$	0	0
$587$	5862.71	0.412232	0.206116	−	0.978528i	$-0.433918\pi$
$587$	5862.71	0.412232	0.206116	+	0.978528i	$0.433918\pi$
$588$	0	0
$589$	975.766	0.0682610
$590$	0	0
$591$	47603.6	3.31328
$592$	0	0
$593$	−2663.02	−0.184413	−0.0922066	−	0.995740i	$-0.529392\pi$
$593$	−2663.02	−0.184413	−0.0922066	+	0.995740i	$0.529392\pi$
$594$	0	0
$595$	70.8491	0.00488156
$596$	0	0
$597$	33291.5	2.28229
$598$	0	0
$599$	16378.3	1.11720	0.558598	−	0.829439i	$-0.311340\pi$
$599$	16378.3	1.11720	0.558598	+	0.829439i	$0.311340\pi$
$600$	0	0
$601$	11584.6	0.786268	0.393134	−	0.919481i	$-0.371391\pi$
$601$	11584.6	0.786268	0.393134	+	0.919481i	$0.371391\pi$
$602$	0	0
$603$	1882.27	0.127118
$604$	0	0
$605$	−3458.84	−0.232433
$606$	0	0
$607$	20218.9	1.35200	0.675998	−	0.736904i	$-0.263713\pi$
$607$	20218.9	1.35200	0.675998	+	0.736904i	$0.263713\pi$
$608$	0	0
$609$	6215.51	0.413572
$610$	0	0
$611$	−14822.2	−0.981410
$612$	0	0
$613$	8024.50	0.528722	0.264361	−	0.964424i	$-0.414839\pi$
$613$	8024.50	0.528722	0.264361	+	0.964424i	$0.414839\pi$
$614$	0	0
$615$	3956.84	0.259439
$616$	0	0
$617$	8933.48	0.582899	0.291449	−	0.956586i	$-0.405863\pi$
$617$	8933.48	0.582899	0.291449	+	0.956586i	$0.405863\pi$
$618$	0	0
$619$	10355.7	0.672421	0.336211	−	0.941787i	$-0.390855\pi$
$619$	10355.7	0.672421	0.336211	+	0.941787i	$0.390855\pi$
$620$	0	0
$621$	−6050.90	−0.391005
$622$	0	0
$623$	197.200	0.0126816
$624$	0	0
$625$	14890.6	0.952998
$626$	0	0
$627$	26156.6	1.66602
$628$	0	0
$629$	−3458.76	−0.219252
$630$	0	0
$631$	−19288.8	−1.21692	−0.608459	−	0.793586i	$-0.708212\pi$
$631$	−19288.8	−1.21692	−0.608459	+	0.793586i	$0.708212\pi$
$632$	0	0
$633$	−25945.0	−1.62910
$634$	0	0
$635$	490.587	0.0306588
$636$	0	0
$637$	25821.1	1.60608
$638$	0	0
$639$	3418.44	0.211630
$640$	0	0
$641$	−12912.1	−0.795628	−0.397814	−	0.917466i	$-0.630231\pi$
$641$	−12912.1	−0.795628	−0.397814	+	0.917466i	$0.630231\pi$
$642$	0	0
$643$	30943.4	1.89781	0.948903	−	0.315568i	$-0.102195\pi$
$643$	30943.4	1.89781	0.948903	+	0.315568i	$0.102195\pi$
$644$	0	0
$645$	5100.90	0.311392
$646$	0	0
$647$	−15938.8	−0.968499	−0.484250	−	0.874930i	$-0.660907\pi$
$647$	−15938.8	−0.968499	−0.484250	+	0.874930i	$0.660907\pi$
$648$	0	0
$649$	30710.7	1.85748
$650$	0	0
$651$	−648.396	−0.0390363
$652$	0	0
$653$	3395.93	0.203512	0.101756	−	0.994809i	$-0.467554\pi$
$653$	3395.93	0.203512	0.101756	+	0.994809i	$0.467554\pi$
$654$	0	0
$655$	−579.300	−0.0345574
$656$	0	0
$657$	−4435.03	−0.263359
$658$	0	0
$659$	15030.2	0.888460	0.444230	−	0.895913i	$-0.353477\pi$
$659$	15030.2	0.888460	0.444230	+	0.895913i	$0.353477\pi$
$660$	0	0
$661$	−15082.3	−0.887496	−0.443748	−	0.896152i	$-0.646351\pi$
$661$	−15082.3	−0.887496	−0.443748	+	0.896152i	$0.646351\pi$
$662$	0	0
$663$	−10525.0	−0.616529
$664$	0	0
$665$	−222.650	−0.0129834
$666$	0	0
$667$	−4613.84	−0.267839
$668$	0	0
$669$	−35731.9	−2.06499
$670$	0	0
$671$	−55866.0	−3.21413
$672$	0	0
$673$	32456.3	1.85899	0.929493	−	0.368839i	$-0.120245\pi$
$673$	32456.3	1.85899	0.929493	+	0.368839i	$0.120245\pi$
$674$	0	0
$675$	32367.4	1.84566
$676$	0	0
$677$	−34072.6	−1.93429	−0.967145	−	0.254224i	$-0.918180\pi$
$677$	−34072.6	−1.93429	−0.967145	+	0.254224i	$0.918180\pi$
$678$	0	0
$679$	5448.59	0.307950
$680$	0	0
$681$	−24510.7	−1.37923
$682$	0	0
$683$	20945.9	1.17346	0.586729	−	0.809784i	$-0.300416\pi$
$683$	20945.9	1.17346	0.586729	+	0.809784i	$0.300416\pi$
$684$	0	0
$685$	−3308.13	−0.184522
$686$	0	0
$687$	20272.8	1.12585
$688$	0	0
$689$	−40279.5	−2.22718
$690$	0	0
$691$	34095.6	1.87707	0.938536	−	0.345182i	$-0.112183\pi$
$691$	34095.6	1.87707	0.938536	+	0.345182i	$0.112183\pi$
$692$	0	0
$693$	−11719.8	−0.642425
$694$	0	0
$695$	120.601	0.00658222
$696$	0	0
$697$	−4595.66	−0.249746
$698$	0	0
$699$	1264.06	0.0683993
$700$	0	0
$701$	26916.2	1.45023	0.725115	−	0.688628i	$-0.241786\pi$
$701$	26916.2	1.45023	0.725115	+	0.688628i	$0.241786\pi$
$702$	0	0
$703$	10869.5	0.583142
$704$	0	0
$705$	2430.38	0.129835
$706$	0	0
$707$	1250.39	0.0665147
$708$	0	0
$709$	−15391.4	−0.815282	−0.407641	−	0.913142i	$-0.633649\pi$
$709$	−15391.4	−0.815282	−0.407641	+	0.913142i	$0.633649\pi$
$710$	0	0
$711$	−29843.4	−1.57414
$712$	0	0
$713$	481.311	0.0252808
$714$	0	0
$715$	−6735.40	−0.352293
$716$	0	0
$717$	19715.0	1.02688
$718$	0	0
$719$	−16237.9	−0.842240	−0.421120	−	0.907005i	$-0.638363\pi$
$719$	−16237.9	−0.842240	−0.421120	+	0.907005i	$0.638363\pi$
$720$	0	0
$721$	3577.33	0.184781
$722$	0	0
$723$	−37764.9	−1.94259
$724$	0	0
$725$	24680.3	1.26428
$726$	0	0
$727$	−13661.2	−0.696925	−0.348462	−	0.937323i	$-0.613296\pi$
$727$	−13661.2	−0.696925	−0.348462	+	0.937323i	$0.613296\pi$
$728$	0	0
$729$	−15143.2	−0.769353
$730$	0	0
$731$	−5924.42	−0.299758
$732$	0	0
$733$	12222.7	0.615900	0.307950	−	0.951403i	$-0.400357\pi$
$733$	12222.7	0.615900	0.307950	+	0.951403i	$0.400357\pi$
$734$	0	0
$735$	−4233.88	−0.212475
$736$	0	0
$737$	−2074.80	−0.103699
$738$	0	0
$739$	−10840.1	−0.539594	−0.269797	−	0.962917i	$-0.586957\pi$
$739$	−10840.1	−0.539594	−0.269797	+	0.962917i	$0.586957\pi$
$740$	0	0
$741$	33075.9	1.63977
$742$	0	0
$743$	−11839.8	−0.584603	−0.292302	−	0.956326i	$-0.594421\pi$
$743$	−11839.8	−0.584603	−0.292302	+	0.956326i	$0.594421\pi$
$744$	0	0
$745$	785.092	0.0386088
$746$	0	0
$747$	−42864.9	−2.09952
$748$	0	0
$749$	3757.66	0.183313
$750$	0	0
$751$	−24428.6	−1.18697	−0.593484	−	0.804846i	$-0.702248\pi$
$751$	−24428.6	−1.18697	−0.593484	+	0.804846i	$0.702248\pi$
$752$	0	0
$753$	9176.87	0.444122
$754$	0	0
$755$	295.312	0.0142351
$756$	0	0
$757$	22284.6	1.06994	0.534972	−	0.844870i	$-0.320322\pi$
$757$	22284.6	1.06994	0.534972	+	0.844870i	$0.320322\pi$
$758$	0	0
$759$	12902.1	0.617020
$760$	0	0
$761$	−5150.71	−0.245352	−0.122676	−	0.992447i	$-0.539148\pi$
$761$	−5150.71	−0.245352	−0.122676	+	0.992447i	$0.539148\pi$
$762$	0	0
$763$	−4303.90	−0.204209
$764$	0	0
$765$	1163.68	0.0549971
$766$	0	0
$767$	38834.7	1.82821
$768$	0	0
$769$	20286.3	0.951289	0.475645	−	0.879638i	$-0.342215\pi$
$769$	20286.3	0.951289	0.475645	+	0.879638i	$0.342215\pi$
$770$	0	0
$771$	33602.5	1.56960
$772$	0	0
$773$	−5981.92	−0.278337	−0.139169	−	0.990269i	$-0.544443\pi$
$773$	−5981.92	−0.278337	−0.139169	+	0.990269i	$0.544443\pi$
$774$	0	0
$775$	−2574.62	−0.119333
$776$	0	0
$777$	−7222.75	−0.333481
$778$	0	0
$779$	14442.3	0.664246
$780$	0	0
$781$	−3768.09	−0.172641
$782$	0	0
$783$	52774.8	2.40871
$784$	0	0
$785$	612.793	0.0278618
$786$	0	0
$787$	1118.63	0.0506667	0.0253333	−	0.999679i	$-0.491935\pi$
$787$	1118.63	0.0506667	0.0253333	+	0.999679i	$0.491935\pi$
$788$	0	0
$789$	26235.5	1.18379
$790$	0	0
$791$	−7153.92	−0.321573
$792$	0	0
$793$	−70644.3	−3.16350
$794$	0	0
$795$	6604.60	0.294643
$796$	0	0
$797$	30098.0	1.33768	0.668838	−	0.743409i	$-0.266792\pi$
$797$	30098.0	1.33768	0.668838	+	0.743409i	$0.266792\pi$
$798$	0	0
$799$	−2822.76	−0.124984
$800$	0	0
$801$	3238.96	0.142875
$802$	0	0
$803$	4888.66	0.214841
$804$	0	0
$805$	−109.825	−0.00480848
$806$	0	0
$807$	−22008.8	−0.960033
$808$	0	0
$809$	730.186	0.0317330	0.0158665	−	0.999874i	$-0.494949\pi$
$809$	730.186	0.0317330	0.0158665	+	0.999874i	$0.494949\pi$
$810$	0	0
$811$	34542.4	1.49562	0.747810	−	0.663912i	$-0.231105\pi$
$811$	34542.4	1.49562	0.747810	+	0.663912i	$0.231105\pi$
$812$	0	0
$813$	25093.0	1.08247
$814$	0	0
$815$	−1683.00	−0.0723349
$816$	0	0
$817$	18618.0	0.797261
$818$	0	0
$819$	−14820.1	−0.632304
$820$	0	0
$821$	25416.3	1.08043	0.540217	−	0.841526i	$-0.318342\pi$
$821$	25416.3	1.08043	0.540217	+	0.841526i	$0.318342\pi$
$822$	0	0
$823$	36849.3	1.56073	0.780367	−	0.625322i	$-0.215032\pi$
$823$	36849.3	1.56073	0.780367	+	0.625322i	$0.215032\pi$
$824$	0	0
$825$	−69015.9	−2.91252
$826$	0	0
$827$	−8488.52	−0.356922	−0.178461	−	0.983947i	$-0.557112\pi$
$827$	−8488.52	−0.356922	−0.178461	+	0.983947i	$0.557112\pi$
$828$	0	0
$829$	−17502.1	−0.733260	−0.366630	−	0.930367i	$-0.619488\pi$
$829$	−17502.1	−0.733260	−0.366630	+	0.930367i	$0.619488\pi$
$830$	0	0
$831$	28380.0	1.18471
$832$	0	0
$833$	4917.42	0.204536
$834$	0	0
$835$	1948.35	0.0807491
$836$	0	0
$837$	−5505.42	−0.227354
$838$	0	0
$839$	6056.79	0.249229	0.124615	−	0.992205i	$-0.460231\pi$
$839$	6056.79	0.249229	0.124615	+	0.992205i	$0.460231\pi$
$840$	0	0
$841$	15852.0	0.649964
$842$	0	0
$843$	−26562.4	−1.08524
$844$	0	0
$845$	−5434.46	−0.221244
$846$	0	0
$847$	8389.04	0.340320
$848$	0	0
$849$	−61872.3	−2.50112
$850$	0	0
$851$	5361.52	0.215970
$852$	0	0
$853$	30466.7	1.22293	0.611465	−	0.791271i	$-0.290580\pi$
$853$	30466.7	1.22293	0.611465	+	0.791271i	$0.290580\pi$
$854$	0	0
$855$	−3656.95	−0.146275
$856$	0	0
$857$	42451.4	1.69208	0.846039	−	0.533121i	$-0.178981\pi$
$857$	42451.4	1.69208	0.846039	+	0.533121i	$0.178981\pi$
$858$	0	0
$859$	−23261.8	−0.923960	−0.461980	−	0.886890i	$-0.652861\pi$
$859$	−23261.8	−0.923960	−0.461980	+	0.886890i	$0.652861\pi$
$860$	0	0
$861$	−9596.88	−0.379861
$862$	0	0
$863$	−37167.9	−1.46606	−0.733031	−	0.680196i	$-0.761895\pi$
$863$	−37167.9	−1.46606	−0.733031	+	0.680196i	$0.761895\pi$
$864$	0	0
$865$	−3825.44	−0.150369
$866$	0	0
$867$	42726.9	1.67368
$868$	0	0
$869$	32895.9	1.28414
$870$	0	0
$871$	−2623.65	−0.102065
$872$	0	0
$873$	89491.5	3.46945
$874$	0	0
$875$	1184.35	0.0457581
$876$	0	0
$877$	−47613.3	−1.83328	−0.916641	−	0.399712i	$-0.869110\pi$
$877$	−47613.3	−1.83328	−0.916641	+	0.399712i	$0.869110\pi$
$878$	0	0
$879$	7330.61	0.281292
$880$	0	0
$881$	−8298.27	−0.317339	−0.158670	−	0.987332i	$-0.550720\pi$
$881$	−8298.27	−0.317339	−0.158670	+	0.987332i	$0.550720\pi$
$882$	0	0
$883$	24325.8	0.927099	0.463549	−	0.886071i	$-0.346576\pi$
$883$	24325.8	0.927099	0.463549	+	0.886071i	$0.346576\pi$
$884$	0	0
$885$	−6367.70	−0.241862
$886$	0	0
$887$	21964.2	0.831439	0.415720	−	0.909493i	$-0.363530\pi$
$887$	21964.2	0.831439	0.415720	+	0.909493i	$0.363530\pi$
$888$	0	0
$889$	−1189.87	−0.0448895
$890$	0	0
$891$	−54595.6	−2.05277
$892$	0	0
$893$	8870.78	0.332418
$894$	0	0
$895$	1680.90	0.0627780
$896$	0	0
$897$	16315.2	0.607299
$898$	0	0
$899$	−4197.90	−0.155737
$900$	0	0
$901$	−7670.89	−0.283634
$902$	0	0
$903$	−12371.7	−0.455929
$904$	0	0
$905$	−682.458	−0.0250670
$906$	0	0
$907$	4228.93	0.154817	0.0774086	−	0.996999i	$-0.475335\pi$
$907$	4228.93	0.154817	0.0774086	+	0.996999i	$0.475335\pi$
$908$	0	0
$909$	20537.3	0.749373
$910$	0	0
$911$	40118.0	1.45902	0.729510	−	0.683970i	$-0.239748\pi$
$911$	40118.0	1.45902	0.729510	+	0.683970i	$0.239748\pi$
$912$	0	0
$913$	47249.3	1.71273
$914$	0	0
$915$	11583.5	0.418512
$916$	0	0
$917$	1405.03	0.0505977
$918$	0	0
$919$	−38199.4	−1.37114	−0.685572	−	0.728005i	$-0.740448\pi$
$919$	−38199.4	−1.37114	−0.685572	+	0.728005i	$0.740448\pi$
$920$	0	0
$921$	−43047.9	−1.54015
$922$	0	0
$923$	−4764.87	−0.169921
$924$	0	0
$925$	−28679.7	−1.01944
$926$	0	0
$927$	58756.6	2.08179
$928$	0	0
$929$	13530.5	0.477848	0.238924	−	0.971038i	$-0.423205\pi$
$929$	13530.5	0.477848	0.238924	+	0.971038i	$0.423205\pi$
$930$	0	0
$931$	−15453.4	−0.544002
$932$	0	0
$933$	53115.6	1.86380
$934$	0	0
$935$	−1282.70	−0.0448650
$936$	0	0
$937$	34004.9	1.18558	0.592792	−	0.805356i	$-0.298026\pi$
$937$	34004.9	1.18558	0.592792	+	0.805356i	$0.298026\pi$
$938$	0	0
$939$	−73059.0	−2.53907
$940$	0	0
$941$	40271.5	1.39513	0.697563	−	0.716523i	$-0.254268\pi$
$941$	40271.5	1.39513	0.697563	+	0.716523i	$0.254268\pi$
$942$	0	0
$943$	7123.85	0.246007
$944$	0	0
$945$	1256.22	0.0432433
$946$	0	0
$947$	30897.3	1.06022	0.530109	−	0.847930i	$-0.322151\pi$
$947$	30897.3	1.06022	0.530109	+	0.847930i	$0.322151\pi$
$948$	0	0
$949$	6181.87	0.211456
$950$	0	0
$951$	41219.3	1.40550
$952$	0	0
$953$	−34614.8	−1.17658	−0.588291	−	0.808649i	$-0.700199\pi$
$953$	−34614.8	−1.17658	−0.588291	+	0.808649i	$0.700199\pi$
$954$	0	0
$955$	−3098.98	−0.105006
$956$	0	0
$957$	−112530.	−3.80102
$958$	0	0
$959$	8023.51	0.270170
$960$	0	0
$961$	−29353.1	−0.985300
$962$	0	0
$963$	61718.4	2.06526
$964$	0	0
$965$	−1348.79	−0.0449940
$966$	0	0
$967$	−13078.3	−0.434923	−0.217461	−	0.976069i	$-0.569778\pi$
$967$	−13078.3	−0.434923	−0.217461	+	0.976069i	$0.569778\pi$
$968$	0	0
$969$	6299.03	0.208827
$970$	0	0
$971$	−11408.6	−0.377055	−0.188527	−	0.982068i	$-0.560371\pi$
$971$	−11408.6	−0.377055	−0.188527	+	0.982068i	$0.560371\pi$
$972$	0	0
$973$	−292.504	−0.00963745
$974$	0	0
$975$	−87272.8	−2.86663
$976$	0	0
$977$	33058.5	1.08253	0.541267	−	0.840851i	$-0.317945\pi$
$977$	33058.5	1.08253	0.541267	+	0.840851i	$0.317945\pi$
$978$	0	0
$979$	−3570.25	−0.116553
$980$	0	0
$981$	−70690.3	−2.30068
$982$	0	0
$983$	−26380.8	−0.855967	−0.427984	−	0.903787i	$-0.640776\pi$
$983$	−26380.8	−0.855967	−0.427984	+	0.903787i	$0.640776\pi$
$984$	0	0
$985$	7336.19	0.237310
$986$	0	0
$987$	−5894.63	−0.190099
$988$	0	0
$989$	9183.61	0.295270
$990$	0	0
$991$	46295.9	1.48399	0.741997	−	0.670403i	$-0.233879\pi$
$991$	46295.9	1.48399	0.741997	+	0.670403i	$0.233879\pi$
$992$	0	0
$993$	−8516.91	−0.272181
$994$	0	0
$995$	5130.55	0.163467
$996$	0	0
$997$	−35436.0	−1.12565	−0.562824	−	0.826577i	$-0.690285\pi$
$997$	−35436.0	−1.12565	−0.562824	+	0.826577i	$0.690285\pi$
$998$	0	0
$999$	−61327.1	−1.94224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.4.a.g.1.1		2
4.3	odd	2		46.4.a.c.1.2	✓	2
8.3	odd	2		1472.4.a.m.1.1		2
8.5	even	2		1472.4.a.l.1.2		2
12.11	even	2		414.4.a.j.1.2		2
20.3	even	4		1150.4.b.i.599.4		4
20.7	even	4		1150.4.b.i.599.1		4
20.19	odd	2		1150.4.a.k.1.1		2
28.27	even	2		2254.4.a.d.1.1		2
92.91	even	2		1058.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.4.a.c.1.2	✓	2	4.3	odd	2
368.4.a.g.1.1		2	1.1	even	1	trivial
414.4.a.j.1.2		2	12.11	even	2
1058.4.a.f.1.2		2	92.91	even	2
1150.4.a.k.1.1		2	20.19	odd	2
1150.4.b.i.599.1		4	20.7	even	4
1150.4.b.i.599.4		4	20.3	even	4
1472.4.a.l.1.2		2	8.5	even	2
1472.4.a.m.1.1		2	8.3	odd	2
2254.4.a.d.1.1		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)

\(f(q)\)	\(=\)	\(q-9.10469 q^{3} -1.40312 q^{5} +3.40312 q^{7} +55.8953 q^{9} +O(q^{10})\)	\(q-9.10469 q^{3} -1.40312 q^{5} +3.40312 q^{7} +55.8953 q^{9} -61.6125 q^{11} -77.9109 q^{13} +12.7750 q^{15} -14.8375 q^{17} +46.6281 q^{19} -30.9844 q^{21} +23.0000 q^{23} -123.031 q^{25} -263.083 q^{27} -200.602 q^{29} +20.9266 q^{31} +560.962 q^{33} -4.77501 q^{35} +233.109 q^{37} +709.355 q^{39} +309.733 q^{41} +399.287 q^{43} -78.4281 q^{45} +190.245 q^{47} -331.419 q^{49} +135.091 q^{51} +516.994 q^{53} +86.4500 q^{55} -424.534 q^{57} -498.450 q^{59} +906.731 q^{61} +190.219 q^{63} +109.319 q^{65} +33.6750 q^{67} -209.408 q^{69} +61.1578 q^{71} -79.3453 q^{73} +1120.16 q^{75} -209.675 q^{77} -533.916 q^{79} +886.113 q^{81} -766.878 q^{83} +20.8188 q^{85} +1826.41 q^{87} +57.9468 q^{89} -265.141 q^{91} -190.530 q^{93} -65.4250 q^{95} +1601.06 q^{97} -3443.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 10 q^{5} - 6 q^{7} + 131 q^{9}+O(q^{10}) \) 2 * q + q^3 + 10 * q^5 - 6 * q^7 + 131 * q^9	\( 2 q + q^{3} + 10 q^{5} - 6 q^{7} + 131 q^{9} - 72 q^{11} - 111 q^{13} + 128 q^{15} + 124 q^{17} - 22 q^{19} - 126 q^{21} + 46 q^{23} - 118 q^{25} + 223 q^{27} + 15 q^{29} - 67 q^{31} + 456 q^{33} - 112 q^{35} + 18 q^{37} + 375 q^{39} + 485 q^{41} + 440 q^{43} + 778 q^{45} - 215 q^{47} - 586 q^{49} + 1538 q^{51} + 240 q^{53} - 32 q^{55} - 1118 q^{57} - 792 q^{59} + 456 q^{61} - 516 q^{63} - 268 q^{65} - 240 q^{67} + 23 q^{69} + 705 q^{71} + 27 q^{73} + 1171 q^{75} - 112 q^{77} - 594 q^{79} + 3770 q^{81} - 394 q^{83} + 1604 q^{85} + 4005 q^{87} - 486 q^{89} + 46 q^{91} - 1079 q^{93} - 848 q^{95} + 2152 q^{97} - 4224 q^{99}+O(q^{100}) \) 2 * q + q^3 + 10 * q^5 - 6 * q^7 + 131 * q^9 - 72 * q^11 - 111 * q^13 + 128 * q^15 + 124 * q^17 - 22 * q^19 - 126 * q^21 + 46 * q^23 - 118 * q^25 + 223 * q^27 + 15 * q^29 - 67 * q^31 + 456 * q^33 - 112 * q^35 + 18 * q^37 + 375 * q^39 + 485 * q^41 + 440 * q^43 + 778 * q^45 - 215 * q^47 - 586 * q^49 + 1538 * q^51 + 240 * q^53 - 32 * q^55 - 1118 * q^57 - 792 * q^59 + 456 * q^61 - 516 * q^63 - 268 * q^65 - 240 * q^67 + 23 * q^69 + 705 * q^71 + 27 * q^73 + 1171 * q^75 - 112 * q^77 - 594 * q^79 + 3770 * q^81 - 394 * q^83 + 1604 * q^85 + 4005 * q^87 - 486 * q^89 + 46 * q^91 - 1079 * q^93 - 848 * q^95 + 2152 * q^97 - 4224 * q^99

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