LMFDB - Embedded newform 1458.3.b.c.1457.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,3,Mod(1457,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1458.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1458.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$39.7276225437$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.6
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -2.89319i q^{5} -2.31381 q^{7} +2.82843i q^{8} -4.09159 q^{10} -0.623989i q^{11} -23.4295 q^{13} +3.27222i q^{14} +4.00000 q^{16} -3.13795i q^{17} -4.08237 q^{19} +5.78639i q^{20} -0.882453 q^{22} +45.3388i q^{23} +16.6294 q^{25} +33.1344i q^{26} +4.62762 q^{28} -19.9603i q^{29} +45.3593 q^{31} -5.65685i q^{32} -4.43774 q^{34} +6.69429i q^{35} -37.5461 q^{37} +5.77335i q^{38} +8.18319 q^{40} -46.4043i q^{41} +60.5559 q^{43} +1.24798i q^{44} +64.1187 q^{46} +3.10162i q^{47} -43.6463 q^{49} -23.5176i q^{50} +46.8591 q^{52} +79.9023i q^{53} -1.80532 q^{55} -6.54444i q^{56} -28.2281 q^{58} +65.3035i q^{59} +35.3203 q^{61} -64.1478i q^{62} -8.00000 q^{64} +67.7862i q^{65} -28.3848 q^{67} +6.27591i q^{68} +9.46716 q^{70} +81.6675i q^{71} +25.2095 q^{73} +53.0982i q^{74} +8.16474 q^{76} +1.44379i q^{77} +66.5457 q^{79} -11.5728i q^{80} -65.6256 q^{82} -49.9266i q^{83} -9.07871 q^{85} -85.6389i q^{86} +1.76491 q^{88} +128.325i q^{89} +54.2115 q^{91} -90.6776i q^{92} +4.38636 q^{94} +11.8111i q^{95} +140.945 q^{97} +61.7252i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) $ 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$.

$n$	$731$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 2.89319i	− 0.578639i	−0.957233	−	0.289319i	$-0.906571\pi$
$5$	− 2.89319i	− 0.578639i	0.957233	−	0.289319i	$-0.0934289\pi$
$6$	0	0
$7$	−2.31381	−0.330544	−0.165272	−	0.986248i	$-0.552850\pi$
$7$	−2.31381	−0.330544	−0.165272	+	0.986248i	$0.552850\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−4.09159	−0.409159
$11$	− 0.623989i	− 0.0567263i	−0.999598	−	0.0283631i	$-0.990971\pi$
$11$	− 0.623989i	− 0.0567263i	0.999598	−	0.0283631i	$-0.00902948\pi$
$12$	0	0
$13$	−23.4295	−1.80227	−0.901136	−	0.433536i	$-0.857266\pi$
$13$	−23.4295	−1.80227	−0.901136	+	0.433536i	$0.857266\pi$
$14$	3.27222i	0.233730i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 3.13795i	− 0.184586i	−0.995732	−	0.0922928i	$-0.970580\pi$
$17$	− 3.13795i	− 0.184586i	0.995732	−	0.0922928i	$-0.0294196\pi$
$18$	0	0
$19$	−4.08237	−0.214862	−0.107431	−	0.994213i	$-0.534262\pi$
$19$	−4.08237	−0.214862	−0.107431	+	0.994213i	$0.534262\pi$
$20$	5.78639i	0.289319i
$21$	0	0
$22$	−0.882453	−0.0401115
$23$	45.3388i	1.97125i	0.168942	+	0.985626i	$0.445965\pi$
$23$	45.3388i	1.97125i	−0.168942	+	0.985626i	$0.554035\pi$
$24$	0	0
$25$	16.6294	0.665177
$26$	33.1344i	1.27440i
$27$	0	0
$28$	4.62762	0.165272
$29$	− 19.9603i	− 0.688285i	−0.938917	−	0.344142i	$-0.888170\pi$
$29$	− 19.9603i	− 0.688285i	0.938917	−	0.344142i	$-0.111830\pi$
$30$	0	0
$31$	45.3593	1.46320	0.731602	−	0.681732i	$-0.238773\pi$
$31$	45.3593	1.46320	0.731602	+	0.681732i	$0.238773\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−4.43774	−0.130522
$35$	6.69429i	0.191266i
$36$	0	0
$37$	−37.5461	−1.01476	−0.507379	−	0.861723i	$-0.669386\pi$
$37$	−37.5461	−1.01476	−0.507379	+	0.861723i	$0.669386\pi$
$38$	5.77335i	0.151930i
$39$	0	0
$40$	8.18319	0.204580
$41$	− 46.4043i	− 1.13181i	−0.824470	−	0.565906i	$-0.808526\pi$
$41$	− 46.4043i	− 1.13181i	0.824470	−	0.565906i	$-0.191474\pi$
$42$	0	0
$43$	60.5559	1.40828	0.704138	−	0.710063i	$-0.251334\pi$
$43$	60.5559	1.40828	0.704138	+	0.710063i	$0.251334\pi$
$44$	1.24798i	0.0283631i
$45$	0	0
$46$	64.1187	1.39389
$47$	3.10162i	0.0659920i	0.999455	+	0.0329960i	$0.0105049\pi$
$47$	3.10162i	0.0659920i	−0.999455	+	0.0329960i	$0.989495\pi$
$48$	0	0
$49$	−43.6463	−0.890741
$50$	− 23.5176i	− 0.470351i
$51$	0	0
$52$	46.8591	0.901136
$53$	79.9023i	1.50759i	0.657109	+	0.753795i	$0.271779\pi$
$53$	79.9023i	1.50759i	−0.657109	+	0.753795i	$0.728221\pi$
$54$	0	0
$55$	−1.80532	−0.0328240
$56$	− 6.54444i	− 0.116865i
$57$	0	0
$58$	−28.2281	−0.486691
$59$	65.3035i	1.10684i	0.832903	+	0.553420i	$0.186677\pi$
$59$	65.3035i	1.10684i	−0.832903	+	0.553420i	$0.813323\pi$
$60$	0	0
$61$	35.3203	0.579022	0.289511	−	0.957175i	$-0.406507\pi$
$61$	35.3203	0.579022	0.289511	+	0.957175i	$0.406507\pi$
$62$	− 64.1478i	− 1.03464i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	67.7862i	1.04286i
$66$	0	0
$67$	−28.3848	−0.423653	−0.211827	−	0.977307i	$-0.567941\pi$
$67$	−28.3848	−0.423653	−0.211827	+	0.977307i	$0.567941\pi$
$68$	6.27591i	0.0922928i
$69$	0	0
$70$	9.46716	0.135245
$71$	81.6675i	1.15025i	0.818067	+	0.575123i	$0.195046\pi$
$71$	81.6675i	1.15025i	−0.818067	+	0.575123i	$0.804954\pi$
$72$	0	0
$73$	25.2095	0.345336	0.172668	−	0.984980i	$-0.444761\pi$
$73$	25.2095	0.345336	0.172668	+	0.984980i	$0.444761\pi$
$74$	53.0982i	0.717543i
$75$	0	0
$76$	8.16474	0.107431
$77$	1.44379i	0.0187505i
$78$	0	0
$79$	66.5457	0.842350	0.421175	−	0.906979i	$-0.361618\pi$
$79$	66.5457	0.842350	0.421175	+	0.906979i	$0.361618\pi$
$80$	− 11.5728i	− 0.144660i
$81$	0	0
$82$	−65.6256	−0.800312
$83$	− 49.9266i	− 0.601526i	−0.953699	−	0.300763i	$-0.902759\pi$
$83$	− 49.9266i	− 0.601526i	0.953699	−	0.300763i	$-0.0972413\pi$
$84$	0	0
$85$	−9.07871	−0.106808
$86$	− 85.6389i	− 0.995801i
$87$	0	0
$88$	1.76491	0.0200558
$89$	128.325i	1.44186i	0.693010	+	0.720928i	$0.256284\pi$
$89$	128.325i	1.44186i	−0.693010	+	0.720928i	$0.743716\pi$
$90$	0	0
$91$	54.2115	0.595731
$92$	− 90.6776i	− 0.985626i
$93$	0	0
$94$	4.38636	0.0466634
$95$	11.8111i	0.124327i
$96$	0	0
$97$	140.945	1.45304	0.726521	−	0.687145i	$-0.241136\pi$
$97$	140.945	1.45304	0.726521	+	0.687145i	$0.241136\pi$
$98$	61.7252i	0.629849i
$99$	0	0
$100$	−33.2589	−0.332589
$101$	− 39.7372i	− 0.393438i	−0.980460	−	0.196719i	$-0.936971\pi$
$101$	− 39.7372i	− 0.393438i	0.980460	−	0.196719i	$-0.0630286\pi$
$102$	0	0
$103$	46.5321	0.451768	0.225884	−	0.974154i	$-0.427473\pi$
$103$	46.5321	0.451768	0.225884	+	0.974154i	$0.427473\pi$
$104$	− 66.2688i	− 0.637200i
$105$	0	0
$106$	112.999	1.06603
$107$	− 107.937i	− 1.00876i	−0.863482	−	0.504379i	$-0.831721\pi$
$107$	− 107.937i	− 1.00876i	0.863482	−	0.504379i	$-0.168279\pi$
$108$	0	0
$109$	61.3938	0.563246	0.281623	−	0.959525i	$-0.409127\pi$
$109$	61.3938	0.563246	0.281623	+	0.959525i	$0.409127\pi$
$110$	2.55311i	0.0232101i
$111$	0	0
$112$	−9.25523	−0.0826360
$113$	− 40.9196i	− 0.362121i	−0.983472	−	0.181060i	$-0.942047\pi$
$113$	− 40.9196i	− 0.362121i	0.983472	−	0.181060i	$-0.0579529\pi$
$114$	0	0
$115$	131.174	1.14064
$116$	39.9205i	0.344142i
$117$	0	0
$118$	92.3531	0.782653
$119$	7.26063i	0.0610137i
$120$	0	0
$121$	120.611	0.996782
$122$	− 49.9505i	− 0.409430i
$123$	0	0
$124$	−90.7187	−0.731602
$125$	− 120.442i	− 0.963536i
$126$	0	0
$127$	−213.683	−1.68254	−0.841272	−	0.540613i	$-0.818192\pi$
$127$	−213.683	−1.68254	−0.841272	+	0.540613i	$0.818192\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	95.8642	0.737417
$131$	14.6591i	0.111902i	0.998434	+	0.0559509i	$0.0178190\pi$
$131$	14.6591i	0.111902i	−0.998434	+	0.0559509i	$0.982181\pi$
$132$	0	0
$133$	9.44583	0.0710213
$134$	40.1421i	0.299568i
$135$	0	0
$136$	8.87548	0.0652609
$137$	135.210i	0.986932i	0.869765	+	0.493466i	$0.164270\pi$
$137$	135.210i	0.986932i	−0.869765	+	0.493466i	$0.835730\pi$
$138$	0	0
$139$	74.6479	0.537035	0.268518	−	0.963275i	$-0.413466\pi$
$139$	74.6479	0.537035	0.268518	+	0.963275i	$0.413466\pi$
$140$	− 13.3886i	− 0.0956328i
$141$	0	0
$142$	115.495	0.813347
$143$	14.6198i	0.102236i
$144$	0	0
$145$	−57.7489	−0.398268
$146$	− 35.6516i	− 0.244189i
$147$	0	0
$148$	75.0922	0.507379
$149$	219.693i	1.47445i	0.675646	+	0.737226i	$0.263865\pi$
$149$	219.693i	1.47445i	−0.675646	+	0.737226i	$0.736135\pi$
$150$	0	0
$151$	121.176	0.802487	0.401244	−	0.915971i	$-0.368578\pi$
$151$	121.176	0.802487	0.401244	+	0.915971i	$0.368578\pi$
$152$	− 11.5467i	− 0.0759651i
$153$	0	0
$154$	2.04183	0.0132586
$155$	− 131.233i	− 0.846667i
$156$	0	0
$157$	69.6092	0.443370	0.221685	−	0.975118i	$-0.428844\pi$
$157$	69.6092	0.443370	0.221685	+	0.975118i	$0.428844\pi$
$158$	− 94.1098i	− 0.595632i
$159$	0	0
$160$	−16.3664	−0.102290
$161$	− 104.905i	− 0.651586i
$162$	0	0
$163$	67.8167	0.416054	0.208027	−	0.978123i	$-0.433296\pi$
$163$	67.8167	0.416054	0.208027	+	0.978123i	$0.433296\pi$
$164$	92.8086i	0.565906i
$165$	0	0
$166$	−70.6069	−0.425343
$167$	234.053i	1.40151i	0.713400	+	0.700757i	$0.247154\pi$
$167$	234.053i	1.40151i	−0.713400	+	0.700757i	$0.752846\pi$
$168$	0	0
$169$	379.944	2.24819
$170$	12.8392i	0.0755249i
$171$	0	0
$172$	−121.112	−0.704138
$173$	− 53.5839i	− 0.309734i	−0.987935	−	0.154867i	$-0.950505\pi$
$173$	− 53.5839i	− 0.309734i	0.987935	−	0.154867i	$-0.0494949\pi$
$174$	0	0
$175$	−38.4773	−0.219870
$176$	− 2.49596i	− 0.0141816i
$177$	0	0
$178$	181.479	1.01955
$179$	9.91919i	0.0554145i	0.999616	+	0.0277072i	$0.00882062\pi$
$179$	9.91919i	0.0554145i	−0.999616	+	0.0277072i	$0.991179\pi$
$180$	0	0
$181$	108.312	0.598407	0.299203	−	0.954189i	$-0.403279\pi$
$181$	108.312	0.598407	0.299203	+	0.954189i	$0.403279\pi$
$182$	− 76.6666i	− 0.421245i
$183$	0	0
$184$	−128.237	−0.696943
$185$	108.628i	0.587179i
$186$	0	0
$187$	−1.95805	−0.0104708
$188$	− 6.20324i	− 0.0329960i
$189$	0	0
$190$	16.7034	0.0879127
$191$	156.045i	0.816990i	0.912760	+	0.408495i	$0.133946\pi$
$191$	156.045i	0.816990i	−0.912760	+	0.408495i	$0.866054\pi$
$192$	0	0
$193$	−263.364	−1.36458	−0.682289	−	0.731083i	$-0.739015\pi$
$193$	−263.364	−1.36458	−0.682289	+	0.731083i	$0.739015\pi$
$194$	− 199.326i	− 1.02746i
$195$	0	0
$196$	87.2926	0.445370
$197$	− 144.239i	− 0.732178i	−0.930580	−	0.366089i	$-0.880696\pi$
$197$	− 144.239i	− 0.732178i	0.930580	−	0.366089i	$-0.119304\pi$
$198$	0	0
$199$	266.274	1.33806	0.669030	−	0.743235i	$-0.266710\pi$
$199$	266.274	1.33806	0.669030	+	0.743235i	$0.266710\pi$
$200$	47.0351i	0.235176i
$201$	0	0
$202$	−56.1969	−0.278202
$203$	46.1842i	0.227508i
$204$	0	0
$205$	−134.257	−0.654910
$206$	− 65.8064i	− 0.319448i
$207$	0	0
$208$	−93.7182	−0.450568
$209$	2.54735i	0.0121883i
$210$	0	0
$211$	−204.009	−0.966867	−0.483434	−	0.875381i	$-0.660611\pi$
$211$	−204.009	−0.966867	−0.483434	+	0.875381i	$0.660611\pi$
$212$	− 159.805i	− 0.753795i
$213$	0	0
$214$	−152.646	−0.713299
$215$	− 175.200i	− 0.814883i
$216$	0	0
$217$	−104.953	−0.483654
$218$	− 86.8240i	− 0.398275i
$219$	0	0
$220$	3.61064	0.0164120
$221$	73.5209i	0.332674i
$222$	0	0
$223$	−18.0316	−0.0808592	−0.0404296	−	0.999182i	$-0.512873\pi$
$223$	−18.0316	−0.0808592	−0.0404296	+	0.999182i	$0.512873\pi$
$224$	13.0889i	0.0584325i
$225$	0	0
$226$	−57.8691	−0.256058
$227$	− 107.573i	− 0.473888i	−0.971523	−	0.236944i	$-0.923854\pi$
$227$	− 107.573i	− 0.473888i	0.971523	−	0.236944i	$-0.0761458\pi$
$228$	0	0
$229$	−139.356	−0.608540	−0.304270	−	0.952586i	$-0.598412\pi$
$229$	−139.356	−0.608540	−0.304270	+	0.952586i	$0.598412\pi$
$230$	− 185.508i	− 0.806556i
$231$	0	0
$232$	56.4561	0.243345
$233$	− 65.2969i	− 0.280244i	−0.990134	−	0.140122i	$-0.955250\pi$
$233$	− 65.2969i	− 0.280244i	0.990134	−	0.140122i	$-0.0447495\pi$
$234$	0	0
$235$	8.97359	0.0381855
$236$	− 130.607i	− 0.553420i
$237$	0	0
$238$	10.2681	0.0431432
$239$	79.9321i	0.334444i	0.985919	+	0.167222i	$0.0534796\pi$
$239$	79.9321i	0.334444i	−0.985919	+	0.167222i	$0.946520\pi$
$240$	0	0
$241$	242.780	1.00739	0.503693	−	0.863883i	$-0.331974\pi$
$241$	242.780	1.00739	0.503693	+	0.863883i	$0.331974\pi$
$242$	− 170.569i	− 0.704831i
$243$	0	0
$244$	−70.6406	−0.289511
$245$	126.277i	0.515417i
$246$	0	0
$247$	95.6481	0.387239
$248$	128.296i	0.517321i
$249$	0	0
$250$	−170.331	−0.681323
$251$	− 127.798i	− 0.509157i	−0.967052	−	0.254579i	$-0.918063\pi$
$251$	− 127.798i	− 0.509157i	0.967052	−	0.254579i	$-0.0819368\pi$
$252$	0	0
$253$	28.2909	0.111822
$254$	302.193i	1.18974i
$255$	0	0
$256$	16.0000	0.0625000
$257$	309.939i	1.20599i	0.797746	+	0.602994i	$0.206026\pi$
$257$	309.939i	1.20599i	−0.797746	+	0.602994i	$0.793974\pi$
$258$	0	0
$259$	86.8744	0.335423
$260$	− 135.572i	− 0.521432i
$261$	0	0
$262$	20.7312	0.0791266
$263$	376.490i	1.43152i	0.698346	+	0.715760i	$0.253920\pi$
$263$	376.490i	1.43152i	−0.698346	+	0.715760i	$0.746080\pi$
$264$	0	0
$265$	231.173	0.872350
$266$	− 13.3584i	− 0.0502196i
$267$	0	0
$268$	56.7695	0.211827
$269$	265.016i	0.985188i	0.870259	+	0.492594i	$0.163951\pi$
$269$	265.016i	0.985188i	−0.870259	+	0.492594i	$0.836049\pi$
$270$	0	0
$271$	32.4939	0.119904	0.0599518	−	0.998201i	$-0.480905\pi$
$271$	32.4939	0.119904	0.0599518	+	0.998201i	$0.480905\pi$
$272$	− 12.5518i	− 0.0461464i
$273$	0	0
$274$	191.215	0.697866
$275$	− 10.3766i	− 0.0377330i
$276$	0	0
$277$	−467.389	−1.68733	−0.843663	−	0.536873i	$-0.819605\pi$
$277$	−467.389	−1.68733	−0.843663	+	0.536873i	$0.819605\pi$
$278$	− 105.568i	− 0.379741i
$279$	0	0
$280$	−18.9343	−0.0676226
$281$	− 120.373i	− 0.428374i	−0.976793	−	0.214187i	$-0.931290\pi$
$281$	− 120.373i	− 0.428374i	0.976793	−	0.214187i	$-0.0687102\pi$
$282$	0	0
$283$	320.113	1.13114	0.565570	−	0.824700i	$-0.308656\pi$
$283$	320.113	1.13114	0.565570	+	0.824700i	$0.308656\pi$
$284$	− 163.335i	− 0.575123i
$285$	0	0
$286$	20.6755	0.0722919
$287$	107.371i	0.374114i
$288$	0	0
$289$	279.153	0.965928
$290$	81.6692i	0.281618i
$291$	0	0
$292$	−50.4190	−0.172668
$293$	374.267i	1.27736i	0.769472	+	0.638680i	$0.220519\pi$
$293$	374.267i	1.27736i	−0.769472	+	0.638680i	$0.779481\pi$
$294$	0	0
$295$	188.936	0.640460
$296$	− 106.196i	− 0.358771i
$297$	0	0
$298$	310.693	1.04260
$299$	− 1062.27i	− 3.55273i
$300$	0	0
$301$	−140.115	−0.465497
$302$	− 171.368i	− 0.567444i
$303$	0	0
$304$	−16.3295	−0.0537154
$305$	− 102.188i	− 0.335044i
$306$	0	0
$307$	−328.889	−1.07130	−0.535650	−	0.844440i	$-0.679933\pi$
$307$	−328.889	−1.07130	−0.535650	+	0.844440i	$0.679933\pi$
$308$	− 2.88758i	− 0.00937526i
$309$	0	0
$310$	−185.592	−0.598684
$311$	− 117.218i	− 0.376906i	−0.982082	−	0.188453i	$-0.939653\pi$
$311$	− 117.218i	− 0.376906i	0.982082	−	0.188453i	$-0.0603473\pi$
$312$	0	0
$313$	−373.935	−1.19468	−0.597340	−	0.801988i	$-0.703776\pi$
$313$	−373.935	−1.19468	−0.597340	+	0.801988i	$0.703776\pi$
$314$	− 98.4422i	− 0.313510i
$315$	0	0
$316$	−133.091	−0.421175
$317$	186.406i	0.588031i	0.955801	+	0.294016i	$0.0949917\pi$
$317$	186.406i	0.588031i	−0.955801	+	0.294016i	$0.905008\pi$
$318$	0	0
$319$	−12.4550	−0.0390438
$320$	23.1455i	0.0723298i
$321$	0	0
$322$	−148.358	−0.460741
$323$	12.8103i	0.0396604i
$324$	0	0
$325$	−389.620	−1.19883
$326$	− 95.9073i	− 0.294194i
$327$	0	0
$328$	131.251	0.400156
$329$	− 7.17656i	− 0.0218132i
$330$	0	0
$331$	−465.115	−1.40518	−0.702590	−	0.711595i	$-0.747973\pi$
$331$	−465.115	−1.40518	−0.702590	+	0.711595i	$0.747973\pi$
$332$	99.8533i	0.300763i
$333$	0	0
$334$	331.001	0.991020
$335$	82.1226i	0.245142i
$336$	0	0
$337$	−445.216	−1.32112	−0.660558	−	0.750775i	$-0.729680\pi$
$337$	−445.216	−1.32112	−0.660558	+	0.750775i	$0.729680\pi$
$338$	− 537.321i	− 1.58971i
$339$	0	0
$340$	18.1574	0.0534042
$341$	− 28.3037i	− 0.0830021i
$342$	0	0
$343$	214.366	0.624973
$344$	171.278i	0.497901i
$345$	0	0
$346$	−75.7791	−0.219015
$347$	− 485.183i	− 1.39822i	−0.715013	−	0.699111i	$-0.753579\pi$
$347$	− 485.183i	− 1.39822i	0.715013	−	0.699111i	$-0.246421\pi$
$348$	0	0
$349$	237.937	0.681767	0.340883	−	0.940106i	$-0.389274\pi$
$349$	237.937	0.681767	0.340883	+	0.940106i	$0.389274\pi$
$350$	54.4152i	0.155472i
$351$	0	0
$352$	−3.52981	−0.0100279
$353$	508.399i	1.44022i	0.693858	+	0.720111i	$0.255909\pi$
$353$	508.399i	1.44022i	−0.693858	+	0.720111i	$0.744091\pi$
$354$	0	0
$355$	236.280	0.665577
$356$	− 256.651i	− 0.720928i
$357$	0	0
$358$	14.0279	0.0391839
$359$	− 507.938i	− 1.41487i	−0.706779	−	0.707435i	$-0.749852\pi$
$359$	− 507.938i	− 1.41487i	0.706779	−	0.707435i	$-0.250148\pi$
$360$	0	0
$361$	−344.334	−0.953834
$362$	− 153.176i	− 0.423137i
$363$	0	0
$364$	−108.423	−0.297865
$365$	− 72.9360i	− 0.199825i
$366$	0	0
$367$	91.5734	0.249519	0.124759	−	0.992187i	$-0.460184\pi$
$367$	91.5734	0.249519	0.124759	+	0.992187i	$0.460184\pi$
$368$	181.355i	0.492813i
$369$	0	0
$370$	153.623	0.415198
$371$	− 184.879i	− 0.498325i
$372$	0	0
$373$	−229.014	−0.613979	−0.306990	−	0.951713i	$-0.599322\pi$
$373$	−229.014	−0.613979	−0.306990	+	0.951713i	$0.599322\pi$
$374$	2.76910i	0.00740401i
$375$	0	0
$376$	−8.77271	−0.0233317
$377$	467.660i	1.24048i
$378$	0	0
$379$	−143.466	−0.378539	−0.189269	−	0.981925i	$-0.560612\pi$
$379$	−143.466	−0.378539	−0.189269	+	0.981925i	$0.560612\pi$
$380$	− 23.6222i	− 0.0621636i
$381$	0	0
$382$	220.681	0.577699
$383$	158.007i	0.412551i	0.978494	+	0.206275i	$0.0661343\pi$
$383$	158.007i	0.412551i	−0.978494	+	0.206275i	$0.933866\pi$
$384$	0	0
$385$	4.17716	0.0108498
$386$	372.452i	0.964902i
$387$	0	0
$388$	−281.890	−0.726521
$389$	219.357i	0.563899i	0.959429	+	0.281950i	$0.0909811\pi$
$389$	219.357i	0.563899i	−0.959429	+	0.281950i	$0.909019\pi$
$390$	0	0
$391$	142.271	0.363865
$392$	− 123.450i	− 0.314924i
$393$	0	0
$394$	−203.985	−0.517728
$395$	− 192.529i	− 0.487416i
$396$	0	0
$397$	79.9691	0.201433	0.100717	−	0.994915i	$-0.467886\pi$
$397$	79.9691	0.201433	0.100717	+	0.994915i	$0.467886\pi$
$398$	− 376.568i	− 0.946151i
$399$	0	0
$400$	66.5177	0.166294
$401$	598.931i	1.49359i	0.665052	+	0.746797i	$0.268409\pi$
$401$	598.931i	1.49359i	−0.665052	+	0.746797i	$0.731591\pi$
$402$	0	0
$403$	−1062.75	−2.63709
$404$	79.4744i	0.196719i
$405$	0	0
$406$	65.3143	0.160873
$407$	23.4283i	0.0575635i
$408$	0	0
$409$	−93.7542	−0.229228	−0.114614	−	0.993410i	$-0.536563\pi$
$409$	−93.7542	−0.229228	−0.114614	+	0.993410i	$0.536563\pi$
$410$	189.868i	0.463092i
$411$	0	0
$412$	−93.0642	−0.225884
$413$	− 151.100i	− 0.365859i
$414$	0	0
$415$	−144.447	−0.348066
$416$	132.538i	0.318600i
$417$	0	0
$418$	3.60250	0.00861843
$419$	480.847i	1.14761i	0.818993	+	0.573804i	$0.194533\pi$
$419$	480.847i	1.14761i	−0.818993	+	0.573804i	$0.805467\pi$
$420$	0	0
$421$	−209.796	−0.498328	−0.249164	−	0.968461i	$-0.580156\pi$
$421$	−209.796	−0.498328	−0.249164	+	0.968461i	$0.580156\pi$
$422$	288.512i	0.683678i
$423$	0	0
$424$	−225.998	−0.533014
$425$	− 52.1824i	− 0.122782i
$426$	0	0
$427$	−81.7244	−0.191392
$428$	215.874i	0.504379i
$429$	0	0
$430$	−247.770	−0.576209
$431$	323.120i	0.749698i	0.927086	+	0.374849i	$0.122305\pi$
$431$	323.120i	0.749698i	−0.927086	+	0.374849i	$0.877695\pi$
$432$	0	0
$433$	382.089	0.882422	0.441211	−	0.897403i	$-0.354549\pi$
$433$	382.089	0.882422	0.441211	+	0.897403i	$0.354549\pi$
$434$	148.426i	0.341995i
$435$	0	0
$436$	−122.788	−0.281623
$437$	− 185.090i	− 0.423547i
$438$	0	0
$439$	−280.589	−0.639154	−0.319577	−	0.947560i	$-0.603541\pi$
$439$	−280.589	−0.639154	−0.319577	+	0.947560i	$0.603541\pi$
$440$	− 5.10622i	− 0.0116050i
$441$	0	0
$442$	103.974	0.235236
$443$	− 90.2873i	− 0.203809i	−0.994794	−	0.101904i	$-0.967506\pi$
$443$	− 90.2873i	− 0.203809i	0.994794	−	0.101904i	$-0.0324936\pi$
$444$	0	0
$445$	371.270	0.834314
$446$	25.5005i	0.0571761i
$447$	0	0
$448$	18.5105	0.0413180
$449$	− 378.572i	− 0.843146i	−0.906795	−	0.421573i	$-0.861478\pi$
$449$	− 378.572i	− 0.843146i	0.906795	−	0.421573i	$-0.138522\pi$
$450$	0	0
$451$	−28.9558	−0.0642035
$452$	81.8393i	0.181060i
$453$	0	0
$454$	−152.131	−0.335090
$455$	− 156.844i	− 0.344713i
$456$	0	0
$457$	−397.770	−0.870395	−0.435197	−	0.900335i	$-0.643321\pi$
$457$	−397.770	−0.870395	−0.435197	+	0.900335i	$0.643321\pi$
$458$	197.079i	0.430302i
$459$	0	0
$460$	−262.348	−0.570321
$461$	401.965i	0.871941i	0.899961	+	0.435970i	$0.143595\pi$
$461$	401.965i	0.871941i	−0.899961	+	0.435970i	$0.856405\pi$
$462$	0	0
$463$	190.589	0.411638	0.205819	−	0.978590i	$-0.434014\pi$
$463$	190.589	0.411638	0.205819	+	0.978590i	$0.434014\pi$
$464$	− 79.8410i	− 0.172071i
$465$	0	0
$466$	−92.3438	−0.198163
$467$	861.682i	1.84514i	0.385825	+	0.922572i	$0.373917\pi$
$467$	861.682i	1.84514i	−0.385825	+	0.922572i	$0.626083\pi$
$468$	0	0
$469$	65.6769	0.140036
$470$	− 12.6906i	− 0.0270012i
$471$	0	0
$472$	−184.706	−0.391327
$473$	− 37.7862i	− 0.0798862i
$474$	0	0
$475$	−67.8875	−0.142921
$476$	− 14.5213i	− 0.0305068i
$477$	0	0
$478$	113.041	0.236487
$479$	− 474.105i	− 0.989782i	−0.868955	−	0.494891i	$-0.835208\pi$
$479$	− 474.105i	− 0.989782i	0.868955	−	0.494891i	$-0.164792\pi$
$480$	0	0
$481$	879.688	1.82887
$482$	− 343.343i	− 0.712330i
$483$	0	0
$484$	−241.221	−0.498391
$485$	− 407.781i	− 0.840786i
$486$	0	0
$487$	118.562	0.243454	0.121727	−	0.992564i	$-0.461157\pi$
$487$	118.562	0.243454	0.121727	+	0.992564i	$0.461157\pi$
$488$	99.9009i	0.204715i
$489$	0	0
$490$	178.583	0.364455
$491$	391.163i	0.796666i	0.917241	+	0.398333i	$0.130411\pi$
$491$	391.163i	0.796666i	−0.917241	+	0.398333i	$0.869589\pi$
$492$	0	0
$493$	−62.6344	−0.127047
$494$	− 135.267i	− 0.273820i
$495$	0	0
$496$	181.437	0.365801
$497$	− 188.963i	− 0.380207i
$498$	0	0
$499$	184.423	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$499$	184.423	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$500$	240.884i	0.481768i
$501$	0	0
$502$	−180.734	−0.360029
$503$	875.816i	1.74119i	0.492004	+	0.870593i	$0.336264\pi$
$503$	875.816i	1.74119i	−0.492004	+	0.870593i	$0.663736\pi$
$504$	0	0
$505$	−114.967	−0.227658
$506$	− 40.0094i	− 0.0790699i
$507$	0	0
$508$	427.366	0.841272
$509$	− 63.6850i	− 0.125118i	−0.998041	−	0.0625589i	$-0.980074\pi$
$509$	− 63.6850i	− 0.125118i	0.998041	−	0.0625589i	$-0.0199261\pi$
$510$	0	0
$511$	−58.3300	−0.114149
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	438.320	0.852762
$515$	− 134.626i	− 0.261410i
$516$	0	0
$517$	1.93538	0.00374348
$518$	− 122.859i	− 0.237180i
$519$	0	0
$520$	−191.728	−0.368708
$521$	483.372i	0.927778i	0.885893	+	0.463889i	$0.153546\pi$
$521$	483.372i	0.927778i	−0.885893	+	0.463889i	$0.846454\pi$
$522$	0	0
$523$	706.918	1.35166	0.675830	−	0.737058i	$-0.263785\pi$
$523$	706.918	1.35166	0.675830	+	0.737058i	$0.263785\pi$
$524$	− 29.3183i	− 0.0559509i
$525$	0	0
$526$	532.437	1.01224
$527$	− 142.336i	− 0.270086i
$528$	0	0
$529$	−1526.61	−2.88584
$530$	− 326.928i	− 0.616845i
$531$	0	0
$532$	−18.8917	−0.0355106
$533$	1087.23i	2.03983i
$534$	0	0
$535$	−312.283	−0.583706
$536$	− 80.2843i	− 0.149784i
$537$	0	0
$538$	374.789	0.696633
$539$	27.2348i	0.0505284i
$540$	0	0
$541$	−64.3040	−0.118861	−0.0594307	−	0.998232i	$-0.518929\pi$
$541$	−64.3040	−0.118861	−0.0594307	+	0.998232i	$0.518929\pi$
$542$	− 45.9533i	− 0.0847847i
$543$	0	0
$544$	−17.7510	−0.0326304
$545$	− 177.624i	− 0.325916i
$546$	0	0
$547$	172.930	0.316142	0.158071	−	0.987428i	$-0.449473\pi$
$547$	172.930	0.316142	0.158071	+	0.987428i	$0.449473\pi$
$548$	− 270.419i	− 0.493466i
$549$	0	0
$550$	−14.6747	−0.0266813
$551$	81.4852i	0.147886i
$552$	0	0
$553$	−153.974	−0.278434
$554$	660.988i	1.19312i
$555$	0	0
$556$	−149.296	−0.268518
$557$	664.213i	1.19248i	0.802805	+	0.596242i	$0.203340\pi$
$557$	664.213i	1.19248i	−0.802805	+	0.596242i	$0.796660\pi$
$558$	0	0
$559$	−1418.80	−2.53810
$560$	26.7772i	0.0478164i
$561$	0	0
$562$	−170.233	−0.302906
$563$	310.996i	0.552390i	0.961102	+	0.276195i	$0.0890736\pi$
$563$	310.996i	0.552390i	−0.961102	+	0.276195i	$0.910926\pi$
$564$	0	0
$565$	−118.388	−0.209537
$566$	− 452.707i	− 0.799837i
$567$	0	0
$568$	−230.991	−0.406673
$569$	327.687i	0.575899i	0.957646	+	0.287950i	$0.0929736\pi$
$569$	327.687i	0.575899i	−0.957646	+	0.287950i	$0.907026\pi$
$570$	0	0
$571$	−593.059	−1.03863	−0.519316	−	0.854582i	$-0.673813\pi$
$571$	−593.059	−1.03863	−0.519316	+	0.854582i	$0.673813\pi$
$572$	− 29.2395i	− 0.0511181i
$573$	0	0
$574$	151.845	0.264538
$575$	753.959i	1.31123i
$576$	0	0
$577$	174.098	0.301729	0.150864	−	0.988554i	$-0.451794\pi$
$577$	174.098	0.301729	0.150864	+	0.988554i	$0.451794\pi$
$578$	− 394.782i	− 0.683014i
$579$	0	0
$580$	115.498	0.199134
$581$	115.521i	0.198831i
$582$	0	0
$583$	49.8582	0.0855200
$584$	71.3033i	0.122095i
$585$	0	0
$586$	529.293	0.903230
$587$	− 627.083i	− 1.06828i	−0.845395	−	0.534142i	$-0.820635\pi$
$587$	− 627.083i	− 1.06828i	0.845395	−	0.534142i	$-0.179365\pi$
$588$	0	0
$589$	−185.174	−0.314387
$590$	− 267.195i	− 0.452874i
$591$	0	0
$592$	−150.184	−0.253690
$593$	− 495.118i	− 0.834938i	−0.908691	−	0.417469i	$-0.862917\pi$
$593$	− 495.118i	− 0.834938i	0.908691	−	0.417469i	$-0.137083\pi$
$594$	0	0
$595$	21.0064	0.0353049
$596$	− 439.387i	− 0.737226i
$597$	0	0
$598$	−1502.27	−2.51216
$599$	169.737i	0.283368i	0.989912	+	0.141684i	$0.0452516\pi$
$599$	169.737i	0.283368i	−0.989912	+	0.141684i	$0.954748\pi$
$600$	0	0
$601$	−338.065	−0.562504	−0.281252	−	0.959634i	$-0.590750\pi$
$601$	−338.065	−0.562504	−0.281252	+	0.959634i	$0.590750\pi$
$602$	198.152i	0.329156i
$603$	0	0
$604$	−242.351	−0.401244
$605$	− 348.950i	− 0.576777i
$606$	0	0
$607$	−924.685	−1.52337	−0.761685	−	0.647948i	$-0.775627\pi$
$607$	−924.685	−1.52337	−0.761685	+	0.647948i	$0.775627\pi$
$608$	23.0934i	0.0379825i
$609$	0	0
$610$	−144.516	−0.236912
$611$	− 72.6696i	− 0.118935i
$612$	0	0
$613$	−113.083	−0.184475	−0.0922375	−	0.995737i	$-0.529402\pi$
$613$	−113.083	−0.184475	−0.0922375	+	0.995737i	$0.529402\pi$
$614$	465.119i	0.757523i
$615$	0	0
$616$	−4.08366	−0.00662931
$617$	− 903.974i	− 1.46511i	−0.680707	−	0.732556i	$-0.738327\pi$
$617$	− 903.974i	− 1.46511i	0.680707	−	0.732556i	$-0.261673\pi$
$618$	0	0
$619$	−329.626	−0.532514	−0.266257	−	0.963902i	$-0.585787\pi$
$619$	−329.626	−0.532514	−0.266257	+	0.963902i	$0.585787\pi$
$620$	262.467i	0.423333i
$621$	0	0
$622$	−165.771	−0.266513
$623$	− 296.920i	− 0.476597i
$624$	0	0
$625$	67.2739	0.107638
$626$	528.824i	0.844766i
$627$	0	0
$628$	−139.218	−0.221685
$629$	117.818i	0.187310i
$630$	0	0
$631$	583.986	0.925492	0.462746	−	0.886491i	$-0.346864\pi$
$631$	583.986	0.925492	0.462746	+	0.886491i	$0.346864\pi$
$632$	188.220i	0.297816i
$633$	0	0
$634$	263.618	0.415801
$635$	618.226i	0.973585i
$636$	0	0
$637$	1022.61	1.60536
$638$	17.6140i	0.0276081i
$639$	0	0
$640$	32.7327	0.0511449
$641$	− 883.002i	− 1.37754i	−0.724981	−	0.688769i	$-0.758152\pi$
$641$	− 883.002i	− 1.37754i	0.724981	−	0.688769i	$-0.241848\pi$
$642$	0	0
$643$	−637.657	−0.991690	−0.495845	−	0.868411i	$-0.665142\pi$
$643$	−637.657	−0.991690	−0.495845	+	0.868411i	$0.665142\pi$
$644$	209.811i	0.325793i
$645$	0	0
$646$	18.1165	0.0280441
$647$	− 950.774i	− 1.46951i	−0.678332	−	0.734756i	$-0.737297\pi$
$647$	− 950.774i	− 1.46951i	0.678332	−	0.734756i	$-0.262703\pi$
$648$	0	0
$649$	40.7487	0.0627868
$650$	551.006i	0.847702i
$651$	0	0
$652$	−135.633	−0.208027
$653$	− 134.486i	− 0.205951i	−0.994684	−	0.102976i	$-0.967164\pi$
$653$	− 134.486i	− 0.205951i	0.994684	−	0.102976i	$-0.0328364\pi$
$654$	0	0
$655$	42.4117	0.0647507
$656$	− 185.617i	− 0.282953i
$657$	0	0
$658$	−10.1492	−0.0154243
$659$	− 72.1254i	− 0.109447i	−0.998502	−	0.0547234i	$-0.982572\pi$
$659$	− 72.1254i	− 0.109447i	0.998502	−	0.0547234i	$-0.0174277\pi$
$660$	0	0
$661$	682.102	1.03192	0.515962	−	0.856612i	$-0.327435\pi$
$661$	682.102	1.03192	0.515962	+	0.856612i	$0.327435\pi$
$662$	657.771i	0.993612i
$663$	0	0
$664$	141.214	0.212671
$665$	− 27.3286i	− 0.0410956i
$666$	0	0
$667$	904.974	1.35678
$668$	− 468.105i	− 0.700757i
$669$	0	0
$670$	116.139	0.173342
$671$	− 22.0395i	− 0.0328457i
$672$	0	0
$673$	−328.704	−0.488416	−0.244208	−	0.969723i	$-0.578528\pi$
$673$	−328.704	−0.488416	−0.244208	+	0.969723i	$0.578528\pi$
$674$	629.631i	0.934170i
$675$	0	0
$676$	−759.887	−1.12409
$677$	492.508i	0.727485i	0.931499	+	0.363743i	$0.118501\pi$
$677$	492.508i	0.727485i	−0.931499	+	0.363743i	$0.881499\pi$
$678$	0	0
$679$	−326.120	−0.480294
$680$	− 25.6785i	− 0.0377625i
$681$	0	0
$682$	−40.0275	−0.0586914
$683$	− 837.431i	− 1.22611i	−0.790042	−	0.613053i	$-0.789941\pi$
$683$	− 837.431i	− 1.22611i	0.790042	−	0.613053i	$-0.210059\pi$
$684$	0	0
$685$	391.188	0.571077
$686$	− 303.159i	− 0.441923i
$687$	0	0
$688$	242.223	0.352069
$689$	− 1872.08i	− 2.71709i
$690$	0	0
$691$	−163.612	−0.236776	−0.118388	−	0.992967i	$-0.537773\pi$
$691$	−163.612	−0.236776	−0.118388	+	0.992967i	$0.537773\pi$
$692$	107.168i	0.154867i
$693$	0	0
$694$	−686.153	−0.988692
$695$	− 215.971i	− 0.310749i
$696$	0	0
$697$	−145.615	−0.208916
$698$	− 336.493i	− 0.482082i
$699$	0	0
$700$	76.9546	0.109935
$701$	730.990i	1.04278i	0.853318	+	0.521391i	$0.174587\pi$
$701$	730.990i	1.04278i	−0.853318	+	0.521391i	$0.825413\pi$
$702$	0	0
$703$	153.277	0.218033
$704$	4.99191i	0.00709078i
$705$	0	0
$706$	718.984	1.01839
$707$	91.9443i	0.130048i
$708$	0	0
$709$	800.431	1.12896	0.564479	−	0.825447i	$-0.309077\pi$
$709$	800.431	1.12896	0.564479	+	0.825447i	$0.309077\pi$
$710$	− 334.150i	− 0.470634i
$711$	0	0
$712$	−362.959	−0.509773
$713$	2056.54i	2.88435i
$714$	0	0
$715$	42.2978	0.0591578
$716$	− 19.8384i	− 0.0277072i
$717$	0	0
$718$	−718.333	−1.00046
$719$	− 119.410i	− 0.166078i	−0.996546	−	0.0830392i	$-0.973537\pi$
$719$	− 119.410i	− 0.166078i	0.996546	−	0.0830392i	$-0.0264627\pi$
$720$	0	0
$721$	−107.666	−0.149329
$722$	486.962i	0.674463i
$723$	0	0
$724$	−216.623	−0.299203
$725$	− 331.928i	− 0.457831i
$726$	0	0
$727$	−135.695	−0.186651	−0.0933254	−	0.995636i	$-0.529750\pi$
$727$	−135.695	−0.186651	−0.0933254	+	0.995636i	$0.529750\pi$
$728$	153.333i	0.210623i
$729$	0	0
$730$	−103.147	−0.141297
$731$	− 190.022i	− 0.259947i
$732$	0	0
$733$	752.046	1.02598	0.512992	−	0.858393i	$-0.328537\pi$
$733$	752.046	1.02598	0.512992	+	0.858393i	$0.328537\pi$
$734$	− 129.504i	− 0.176436i
$735$	0	0
$736$	256.475	0.348471
$737$	17.7118i	0.0240323i
$738$	0	0
$739$	−1137.53	−1.53928	−0.769642	−	0.638476i	$-0.779565\pi$
$739$	−1137.53	−1.53928	−0.769642	+	0.638476i	$0.779565\pi$
$740$	− 217.256i	− 0.293589i
$741$	0	0
$742$	−261.458	−0.352369
$743$	− 718.359i	− 0.966835i	−0.875390	−	0.483418i	$-0.839395\pi$
$743$	− 718.359i	− 0.966835i	0.875390	−	0.483418i	$-0.160605\pi$
$744$	0	0
$745$	635.616	0.853175
$746$	323.875i	0.434149i
$747$	0	0
$748$	3.91610	0.00523542
$749$	249.746i	0.333439i
$750$	0	0
$751$	628.147	0.836414	0.418207	−	0.908352i	$-0.362659\pi$
$751$	628.147	0.836414	0.418207	+	0.908352i	$0.362659\pi$
$752$	12.4065i	0.0164980i
$753$	0	0
$754$	661.371	0.877149
$755$	− 350.584i	− 0.464350i
$756$	0	0
$757$	393.374	0.519648	0.259824	−	0.965656i	$-0.416335\pi$
$757$	393.374	0.519648	0.259824	+	0.965656i	$0.416335\pi$
$758$	202.892i	0.267667i
$759$	0	0
$760$	−33.4068	−0.0439563
$761$	− 808.968i	− 1.06303i	−0.847048	−	0.531517i	$-0.821622\pi$
$761$	− 808.968i	− 1.06303i	0.847048	−	0.531517i	$-0.178378\pi$
$762$	0	0
$763$	−142.054	−0.186178
$764$	− 312.090i	− 0.408495i
$765$	0	0
$766$	223.456	0.291717
$767$	− 1530.03i	− 1.99483i
$768$	0	0
$769$	89.5214	0.116413	0.0582064	−	0.998305i	$-0.481462\pi$
$769$	89.5214	0.116413	0.0582064	+	0.998305i	$0.481462\pi$
$770$	− 5.90740i	− 0.00767195i
$771$	0	0
$772$	526.727	0.682289
$773$	− 1216.61i	− 1.57388i	−0.617030	−	0.786939i	$-0.711664\pi$
$773$	− 1216.61i	− 1.57388i	0.617030	−	0.786939i	$-0.288336\pi$
$774$	0	0
$775$	754.300	0.973291
$776$	398.653i	0.513728i
$777$	0	0
$778$	310.217	0.398737
$779$	189.440i	0.243183i
$780$	0	0
$781$	50.9596	0.0652492
$782$	− 201.202i	− 0.257291i
$783$	0	0
$784$	−174.585	−0.222685
$785$	− 201.393i	− 0.256551i
$786$	0	0
$787$	820.609	1.04270	0.521352	−	0.853341i	$-0.325428\pi$
$787$	820.609	1.04270	0.521352	+	0.853341i	$0.325428\pi$
$788$	288.478i	0.366089i
$789$	0	0
$790$	−272.278	−0.344655
$791$	94.6802i	0.119697i
$792$	0	0
$793$	−827.539	−1.04355
$794$	− 113.093i	− 0.142435i
$795$	0	0
$796$	−532.548	−0.669030
$797$	957.478i	1.20135i	0.799492	+	0.600676i	$0.205102\pi$
$797$	957.478i	1.20135i	−0.799492	+	0.600676i	$0.794898\pi$
$798$	0	0
$799$	9.73275	0.0121812
$800$	− 94.0703i	− 0.117588i
$801$	0	0
$802$	847.017	1.05613
$803$	− 15.7305i	− 0.0195896i
$804$	0	0
$805$	−303.511	−0.377033
$806$	1502.95i	1.86471i
$807$	0	0
$808$	112.394	0.139101
$809$	− 180.576i	− 0.223209i	−0.993753	−	0.111605i	$-0.964401\pi$
$809$	− 180.576i	− 0.223209i	0.993753	−	0.111605i	$-0.0355990\pi$
$810$	0	0
$811$	518.424	0.639240	0.319620	−	0.947546i	$-0.396445\pi$
$811$	518.424	0.639240	0.319620	+	0.947546i	$0.396445\pi$
$812$	− 92.3684i	− 0.113754i
$813$	0	0
$814$	33.1327	0.0407035
$815$	− 196.207i	− 0.240745i
$816$	0	0
$817$	−247.212	−0.302585
$818$	132.588i	0.162089i
$819$	0	0
$820$	268.513	0.327455
$821$	274.490i	0.334336i	0.985928	+	0.167168i	$0.0534622\pi$
$821$	274.490i	0.334336i	−0.985928	+	0.167168i	$0.946538\pi$
$822$	0	0
$823$	−618.304	−0.751281	−0.375640	−	0.926766i	$-0.622577\pi$
$823$	−618.304	−0.751281	−0.375640	+	0.926766i	$0.622577\pi$
$824$	131.613i	0.159724i
$825$	0	0
$826$	−213.687	−0.258701
$827$	− 776.527i	− 0.938968i	−0.882941	−	0.469484i	$-0.844440\pi$
$827$	− 776.527i	− 0.938968i	0.882941	−	0.469484i	$-0.155560\pi$
$828$	0	0
$829$	−530.515	−0.639946	−0.319973	−	0.947427i	$-0.603674\pi$
$829$	−530.515	−0.639946	−0.319973	+	0.947427i	$0.603674\pi$
$830$	204.279i	0.246120i
$831$	0	0
$832$	187.436	0.225284
$833$	136.960i	0.164418i
$834$	0	0
$835$	677.160	0.810970
$836$	− 5.09471i	− 0.00609415i
$837$	0	0
$838$	680.021	0.811481
$839$	− 1007.88i	− 1.20129i	−0.799516	−	0.600645i	$-0.794911\pi$
$839$	− 1007.88i	− 1.20129i	0.799516	−	0.600645i	$-0.205089\pi$
$840$	0	0
$841$	442.588	0.526264
$842$	296.697i	0.352371i
$843$	0	0
$844$	408.018	0.483434
$845$	− 1099.25i	− 1.30089i
$846$	0	0
$847$	−279.070	−0.329480
$848$	319.609i	0.376898i
$849$	0	0
$850$	−73.7971	−0.0868201
$851$	− 1702.29i	− 2.00035i
$852$	0	0
$853$	872.111	1.02240	0.511202	−	0.859461i	$-0.329200\pi$
$853$	872.111	1.02240	0.511202	+	0.859461i	$0.329200\pi$
$854$	115.576i	0.135335i
$855$	0	0
$856$	305.292	0.356650
$857$	834.862i	0.974168i	0.873355	+	0.487084i	$0.161939\pi$
$857$	834.862i	0.974168i	−0.873355	+	0.487084i	$0.838061\pi$
$858$	0	0
$859$	−67.7280	−0.0788451	−0.0394226	−	0.999223i	$-0.512552\pi$
$859$	−67.7280	−0.0788451	−0.0394226	+	0.999223i	$0.512552\pi$
$860$	350.400i	0.407441i
$861$	0	0
$862$	456.961	0.530117
$863$	1191.58i	1.38074i	0.723456	+	0.690370i	$0.242552\pi$
$863$	1191.58i	1.38074i	−0.723456	+	0.690370i	$0.757448\pi$
$864$	0	0
$865$	−155.029	−0.179224
$866$	− 540.355i	− 0.623967i
$867$	0	0
$868$	209.906	0.241827
$869$	− 41.5238i	− 0.0477834i
$870$	0	0
$871$	665.042	0.763539
$872$	173.648i	0.199138i
$873$	0	0
$874$	−261.757	−0.299493
$875$	278.680i	0.318491i
$876$	0	0
$877$	−736.389	−0.839668	−0.419834	−	0.907601i	$-0.637912\pi$
$877$	−736.389	−0.839668	−0.419834	+	0.907601i	$0.637912\pi$
$878$	396.812i	0.451950i
$879$	0	0
$880$	−7.22128	−0.00820600
$881$	− 1188.05i	− 1.34853i	−0.738492	−	0.674263i	$-0.764462\pi$
$881$	− 1188.05i	− 1.34853i	0.738492	−	0.674263i	$-0.235538\pi$
$882$	0	0
$883$	−1018.97	−1.15399	−0.576993	−	0.816749i	$-0.695774\pi$
$883$	−1018.97	−1.15399	−0.576993	+	0.816749i	$0.695774\pi$
$884$	− 147.042i	− 0.166337i
$885$	0	0
$886$	−127.686	−0.144115
$887$	781.771i	0.881365i	0.897663	+	0.440683i	$0.145264\pi$
$887$	781.771i	0.881365i	−0.897663	+	0.440683i	$0.854736\pi$
$888$	0	0
$889$	494.422	0.556155
$890$	− 525.055i	− 0.589949i
$891$	0	0
$892$	36.0632	0.0404296
$893$	− 12.6620i	− 0.0141791i
$894$	0	0
$895$	28.6981	0.0320649
$896$	− 26.1778i	− 0.0292162i
$897$	0	0
$898$	−535.382	−0.596194
$899$	− 905.384i	− 1.00710i
$900$	0	0
$901$	250.730	0.278280
$902$	40.9496i	0.0453987i
$903$	0	0
$904$	115.738	0.128029
$905$	− 313.366i	− 0.346261i
$906$	0	0
$907$	479.654	0.528835	0.264418	−	0.964408i	$-0.414820\pi$
$907$	479.654	0.528835	0.264418	+	0.964408i	$0.414820\pi$
$908$	215.145i	0.236944i
$909$	0	0
$910$	−221.811	−0.243749
$911$	− 639.316i	− 0.701774i	−0.936418	−	0.350887i	$-0.885880\pi$
$911$	− 639.316i	− 0.701774i	0.936418	−	0.350887i	$-0.114120\pi$
$912$	0	0
$913$	−31.1537	−0.0341223
$914$	562.532i	0.615462i
$915$	0	0
$916$	278.711	0.304270
$917$	− 33.9184i	− 0.0369885i
$918$	0	0
$919$	−14.7768	−0.0160792	−0.00803959	−	0.999968i	$-0.502559\pi$
$919$	−14.7768	−0.0160792	−0.00803959	+	0.999968i	$0.502559\pi$
$920$	371.016i	0.403278i
$921$	0	0
$922$	568.464	0.616555
$923$	− 1913.43i	− 2.07306i
$924$	0	0
$925$	−624.370	−0.674995
$926$	− 269.533i	− 0.291072i
$927$	0	0
$928$	−112.912	−0.121673
$929$	1237.15i	1.33170i	0.746086	+	0.665849i	$0.231931\pi$
$929$	1237.15i	1.33170i	−0.746086	+	0.665849i	$0.768069\pi$
$930$	0	0
$931$	178.180	0.191386
$932$	130.594i	0.140122i
$933$	0	0
$934$	1218.60	1.30471
$935$	5.66501i	0.00605884i
$936$	0	0
$937$	1528.09	1.63084	0.815418	−	0.578873i	$-0.196507\pi$
$937$	1528.09	1.63084	0.815418	+	0.578873i	$0.196507\pi$
$938$	− 92.8812i	− 0.0990205i
$939$	0	0
$940$	−17.9472	−0.0190927
$941$	− 1029.72i	− 1.09428i	−0.837042	−	0.547139i	$-0.815717\pi$
$941$	− 1029.72i	− 1.09428i	0.837042	−	0.547139i	$-0.184283\pi$
$942$	0	0
$943$	2103.92	2.23109
$944$	261.214i	0.276710i
$945$	0	0
$946$	−53.4377	−0.0564881
$947$	− 131.201i	− 0.138544i	−0.997598	−	0.0692720i	$-0.977932\pi$
$947$	− 131.201i	− 0.138544i	0.997598	−	0.0692720i	$-0.0220676\pi$
$948$	0	0
$949$	−590.648	−0.622389
$950$	96.0075i	0.101061i
$951$	0	0
$952$	−20.5362	−0.0215716
$953$	1412.22i	1.48187i	0.671575	+	0.740936i	$0.265618\pi$
$953$	1412.22i	1.48187i	−0.671575	+	0.740936i	$0.734382\pi$
$954$	0	0
$955$	451.469	0.472742
$956$	− 159.864i	− 0.167222i
$957$	0	0
$958$	−670.486	−0.699881
$959$	− 312.849i	− 0.326224i
$960$	0	0
$961$	1096.47	1.14097
$962$	− 1244.07i	− 1.29321i
$963$	0	0
$964$	−485.560	−0.503693
$965$	761.962i	0.789597i
$966$	0	0
$967$	581.576	0.601423	0.300712	−	0.953715i	$-0.402776\pi$
$967$	581.576	0.601423	0.300712	+	0.953715i	$0.402776\pi$
$968$	341.138i	0.352416i
$969$	0	0
$970$	−576.690	−0.594525
$971$	− 208.668i	− 0.214900i	−0.994210	−	0.107450i	$-0.965731\pi$
$971$	− 208.668i	− 0.214900i	0.994210	−	0.107450i	$-0.0342686\pi$
$972$	0	0
$973$	−172.721	−0.177514
$974$	− 167.672i	− 0.172148i
$975$	0	0
$976$	141.281	0.144755
$977$	1758.66i	1.80006i	0.435826	+	0.900031i	$0.356456\pi$
$977$	1758.66i	1.80006i	−0.435826	+	0.900031i	$0.643544\pi$
$978$	0	0
$979$	80.0735	0.0817911
$980$	− 252.554i	− 0.257708i
$981$	0	0
$982$	553.188	0.563328
$983$	− 1237.15i	− 1.25854i	−0.777186	−	0.629270i	$-0.783354\pi$
$983$	− 1237.15i	− 1.25854i	0.777186	−	0.629270i	$-0.216646\pi$
$984$	0	0
$985$	−417.312	−0.423667
$986$	88.5784i	0.0898361i
$987$	0	0
$988$	−191.296	−0.193620
$989$	2745.53i	2.77607i
$990$	0	0
$991$	769.340	0.776327	0.388163	−	0.921591i	$-0.373110\pi$
$991$	769.340	0.776327	0.388163	+	0.921591i	$0.373110\pi$
$992$	− 256.591i	− 0.258660i
$993$	0	0
$994$	−267.234	−0.268847
$995$	− 770.382i	− 0.774253i
$996$	0	0
$997$	−88.9580	−0.0892257	−0.0446129	−	0.999004i	$-0.514205\pi$
$997$	−88.9580	−0.0892257	−0.0446129	+	0.999004i	$0.514205\pi$
$998$	− 260.814i	− 0.261337i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.6		36
3.2	odd	2	inner	1458.3.b.c.1457.31		36
27.5	odd	18		54.3.f.a.29.4	✓	36
27.11	odd	18		162.3.f.a.71.1		36
27.16	even	9		54.3.f.a.41.4	yes	36
27.22	even	9		162.3.f.a.89.1		36
108.43	odd	18		432.3.bc.c.257.4		36
108.59	even	18		432.3.bc.c.353.4		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.29.4	✓	36	27.5	odd	18
54.3.f.a.41.4	yes	36	27.16	even	9
162.3.f.a.71.1		36	27.11	odd	18
162.3.f.a.89.1		36	27.22	even	9
432.3.bc.c.257.4		36	108.43	odd	18
432.3.bc.c.353.4		36	108.59	even	18
1458.3.b.c.1457.6		36	1.1	even	1	trivial
1458.3.b.c.1457.31		36	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1458.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -2.89319i q^{5} -2.31381 q^{7} +2.82843i q^{8} -4.09159 q^{10} -0.623989i q^{11} -23.4295 q^{13} +3.27222i q^{14} +4.00000 q^{16} -3.13795i q^{17} -4.08237 q^{19} +5.78639i q^{20} -0.882453 q^{22} +45.3388i q^{23} +16.6294 q^{25} +33.1344i q^{26} +4.62762 q^{28} -19.9603i q^{29} +45.3593 q^{31} -5.65685i q^{32} -4.43774 q^{34} +6.69429i q^{35} -37.5461 q^{37} +5.77335i q^{38} +8.18319 q^{40} -46.4043i q^{41} +60.5559 q^{43} +1.24798i q^{44} +64.1187 q^{46} +3.10162i q^{47} -43.6463 q^{49} -23.5176i q^{50} +46.8591 q^{52} +79.9023i q^{53} -1.80532 q^{55} -6.54444i q^{56} -28.2281 q^{58} +65.3035i q^{59} +35.3203 q^{61} -64.1478i q^{62} -8.00000 q^{64} +67.7862i q^{65} -28.3848 q^{67} +6.27591i q^{68} +9.46716 q^{70} +81.6675i q^{71} +25.2095 q^{73} +53.0982i q^{74} +8.16474 q^{76} +1.44379i q^{77} +66.5457 q^{79} -11.5728i q^{80} -65.6256 q^{82} -49.9266i q^{83} -9.07871 q^{85} -85.6389i q^{86} +1.76491 q^{88} +128.325i q^{89} +54.2115 q^{91} -90.6776i q^{92} +4.38636 q^{94} +11.8111i q^{95} +140.945 q^{97} +61.7252i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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