LMFDB - Embedded newform 1458.3.b.c.1457.18

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			1457.18
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} +8.63280i q^{5} -8.33391 q^{7} +2.82843i q^{8} +12.2086 q^{10} +16.8315i q^{11} +3.02312 q^{13} +11.7859i q^{14} +4.00000 q^{16} +9.74101i q^{17} -7.69355 q^{19} -17.2656i q^{20} +23.8033 q^{22} +29.6574i q^{23} -49.5253 q^{25} -4.27534i q^{26} +16.6678 q^{28} -10.7441i q^{29} +11.4949 q^{31} -5.65685i q^{32} +13.7759 q^{34} -71.9450i q^{35} +23.1849 q^{37} +10.8803i q^{38} -24.4173 q^{40} +16.9844i q^{41} +23.6124 q^{43} -33.6629i q^{44} +41.9419 q^{46} +16.6043i q^{47} +20.4540 q^{49} +70.0393i q^{50} -6.04625 q^{52} -75.3383i q^{53} -145.303 q^{55} -23.5718i q^{56} -15.1944 q^{58} -60.2015i q^{59} -34.1626 q^{61} -16.2563i q^{62} -8.00000 q^{64} +26.0980i q^{65} +37.6326 q^{67} -19.4820i q^{68} -101.746 q^{70} +50.8443i q^{71} -98.4906 q^{73} -32.7884i q^{74} +15.3871 q^{76} -140.272i q^{77} -81.1487 q^{79} +34.5312i q^{80} +24.0196 q^{82} -16.1319i q^{83} -84.0922 q^{85} -33.3930i q^{86} -47.6066 q^{88} -16.1844i q^{89} -25.1944 q^{91} -59.3149i q^{92} +23.4820 q^{94} -66.4169i q^{95} -132.692 q^{97} -28.9263i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	8.63280i	1.72656i	0.504725	+	0.863280i	$0.331594\pi$
$5$	8.63280i	1.72656i	−0.504725	+	0.863280i	$0.668406\pi$
$6$	0	0
$7$	−8.33391	−1.19056	−0.595279	−	0.803519i	$-0.702958\pi$
$7$	−8.33391	−1.19056	−0.595279	+	0.803519i	$0.702958\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	12.2086	1.22086
$11$	16.8315i	1.53013i	0.643951	+	0.765067i	$0.277294\pi$
$11$	16.8315i	1.53013i	−0.643951	+	0.765067i	$0.722706\pi$
$12$	0	0
$13$	3.02312	0.232548	0.116274	−	0.993217i	$-0.462905\pi$
$13$	3.02312	0.232548	0.116274	+	0.993217i	$0.462905\pi$
$14$	11.7859i	0.841852i
$15$	0	0
$16$	4.00000	0.250000
$17$	9.74101i	0.573001i	0.958080	+	0.286500i	$0.0924920\pi$
$17$	9.74101i	0.573001i	−0.958080	+	0.286500i	$0.907508\pi$
$18$	0	0
$19$	−7.69355	−0.404923	−0.202462	−	0.979290i	$-0.564894\pi$
$19$	−7.69355	−0.404923	−0.202462	+	0.979290i	$0.564894\pi$
$20$	− 17.2656i	− 0.863280i
$21$	0	0
$22$	23.8033	1.08197
$23$	29.6574i	1.28945i	0.764413	+	0.644727i	$0.223029\pi$
$23$	29.6574i	1.28945i	−0.764413	+	0.644727i	$0.776971\pi$
$24$	0	0
$25$	−49.5253	−1.98101
$26$	− 4.27534i	− 0.164436i
$27$	0	0
$28$	16.6678	0.595279
$29$	− 10.7441i	− 0.370486i	−0.982693	−	0.185243i	$-0.940693\pi$
$29$	− 10.7441i	− 0.370486i	0.982693	−	0.185243i	$-0.0593072\pi$
$30$	0	0
$31$	11.4949	0.370803	0.185402	−	0.982663i	$-0.440641\pi$
$31$	11.4949	0.370803	0.185402	+	0.982663i	$0.440641\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	13.7759	0.405173
$35$	− 71.9450i	− 2.05557i
$36$	0	0
$37$	23.1849	0.626619	0.313309	−	0.949651i	$-0.398562\pi$
$37$	23.1849	0.626619	0.313309	+	0.949651i	$0.398562\pi$
$38$	10.8803i	0.286324i
$39$	0	0
$40$	−24.4173	−0.610431
$41$	16.9844i	0.414254i	0.978314	+	0.207127i	$0.0664113\pi$
$41$	16.9844i	0.414254i	−0.978314	+	0.207127i	$0.933589\pi$
$42$	0	0
$43$	23.6124	0.549125	0.274563	−	0.961569i	$-0.411467\pi$
$43$	23.6124	0.549125	0.274563	+	0.961569i	$0.411467\pi$
$44$	− 33.6629i	− 0.765067i
$45$	0	0
$46$	41.9419	0.911781
$47$	16.6043i	0.353283i	0.984275	+	0.176642i	$0.0565233\pi$
$47$	16.6043i	0.353283i	−0.984275	+	0.176642i	$0.943477\pi$
$48$	0	0
$49$	20.4540	0.417429
$50$	70.0393i	1.40079i
$51$	0	0
$52$	−6.04625	−0.116274
$53$	− 75.3383i	− 1.42148i	−0.703456	−	0.710739i	$-0.748361\pi$
$53$	− 75.3383i	− 1.42148i	0.703456	−	0.710739i	$-0.251639\pi$
$54$	0	0
$55$	−145.303	−2.64187
$56$	− 23.5718i	− 0.420926i
$57$	0	0
$58$	−15.1944	−0.261973
$59$	− 60.2015i	− 1.02036i	−0.860066	−	0.510182i	$-0.829578\pi$
$59$	− 60.2015i	− 1.02036i	0.860066	−	0.510182i	$-0.170422\pi$
$60$	0	0
$61$	−34.1626	−0.560042	−0.280021	−	0.959994i	$-0.590341\pi$
$61$	−34.1626	−0.560042	−0.280021	+	0.959994i	$0.590341\pi$
$62$	− 16.2563i	− 0.262198i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	26.0980i	0.401508i
$66$	0	0
$67$	37.6326	0.561680	0.280840	−	0.959755i	$-0.409387\pi$
$67$	37.6326	0.561680	0.280840	+	0.959755i	$0.409387\pi$
$68$	− 19.4820i	− 0.286500i
$69$	0	0
$70$	−101.746	−1.45351
$71$	50.8443i	0.716117i	0.933699	+	0.358058i	$0.116561\pi$
$71$	50.8443i	0.716117i	−0.933699	+	0.358058i	$0.883439\pi$
$72$	0	0
$73$	−98.4906	−1.34919	−0.674593	−	0.738189i	$-0.735681\pi$
$73$	−98.4906	−1.34919	−0.674593	+	0.738189i	$0.735681\pi$
$74$	− 32.7884i	− 0.443087i
$75$	0	0
$76$	15.3871	0.202462
$77$	− 140.272i	− 1.82171i
$78$	0	0
$79$	−81.1487	−1.02720	−0.513599	−	0.858030i	$-0.671688\pi$
$79$	−81.1487	−1.02720	−0.513599	+	0.858030i	$0.671688\pi$
$80$	34.5312i	0.431640i
$81$	0	0
$82$	24.0196	0.292921
$83$	− 16.1319i	− 0.194360i	−0.995267	−	0.0971800i	$-0.969018\pi$
$83$	− 16.1319i	− 0.194360i	0.995267	−	0.0971800i	$-0.0309823\pi$
$84$	0	0
$85$	−84.0922	−0.989320
$86$	− 33.3930i	− 0.388290i
$87$	0	0
$88$	−47.6066	−0.540984
$89$	− 16.1844i	− 0.181848i	−0.995858	−	0.0909238i	$-0.971018\pi$
$89$	− 16.1844i	− 0.181848i	0.995858	−	0.0909238i	$-0.0289820\pi$
$90$	0	0
$91$	−25.1944	−0.276862
$92$	− 59.3149i	− 0.644727i
$93$	0	0
$94$	23.4820	0.249809
$95$	− 66.4169i	− 0.699125i
$96$	0	0
$97$	−132.692	−1.36796	−0.683980	−	0.729501i	$-0.739753\pi$
$97$	−132.692	−1.36796	−0.683980	+	0.729501i	$0.739753\pi$
$98$	− 28.9263i	− 0.295167i
$99$	0	0
$100$	99.0506	0.990506
$101$	15.2221i	0.150714i	0.997157	+	0.0753569i	$0.0240096\pi$
$101$	15.2221i	0.150714i	−0.997157	+	0.0753569i	$0.975990\pi$
$102$	0	0
$103$	−37.8526	−0.367501	−0.183750	−	0.982973i	$-0.558824\pi$
$103$	−37.8526	−0.367501	−0.183750	+	0.982973i	$0.558824\pi$
$104$	8.55069i	0.0822181i
$105$	0	0
$106$	−106.544	−1.00514
$107$	118.659i	1.10897i	0.832195	+	0.554483i	$0.187084\pi$
$107$	118.659i	1.10897i	−0.832195	+	0.554483i	$0.812916\pi$
$108$	0	0
$109$	206.055	1.89042	0.945208	−	0.326468i	$-0.105859\pi$
$109$	206.055	1.89042	0.945208	+	0.326468i	$0.105859\pi$
$110$	205.489i	1.86808i
$111$	0	0
$112$	−33.3356	−0.297640
$113$	− 71.0293i	− 0.628578i	−0.949327	−	0.314289i	$-0.898234\pi$
$113$	− 71.0293i	− 0.628578i	0.949327	−	0.314289i	$-0.101766\pi$
$114$	0	0
$115$	−256.027	−2.22632
$116$	21.4882i	0.185243i
$117$	0	0
$118$	−85.1378	−0.721507
$119$	− 81.1807i	− 0.682191i
$120$	0	0
$121$	−162.298	−1.34131
$122$	48.3131i	0.396009i
$123$	0	0
$124$	−22.9898	−0.185402
$125$	− 211.722i	− 1.69378i
$126$	0	0
$127$	38.2694	0.301334	0.150667	−	0.988585i	$-0.451858\pi$
$127$	38.2694	0.301334	0.150667	+	0.988585i	$0.451858\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	36.9082	0.283909
$131$	− 217.927i	− 1.66357i	−0.555100	−	0.831783i	$-0.687320\pi$
$131$	− 217.927i	− 1.66357i	0.555100	−	0.831783i	$-0.312680\pi$
$132$	0	0
$133$	64.1173	0.482085
$134$	− 53.2205i	− 0.397168i
$135$	0	0
$136$	−27.5517	−0.202586
$137$	− 184.092i	− 1.34374i	−0.740670	−	0.671868i	$-0.765492\pi$
$137$	− 184.092i	− 1.34374i	0.740670	−	0.671868i	$-0.234508\pi$
$138$	0	0
$139$	243.150	1.74928	0.874641	−	0.484771i	$-0.161097\pi$
$139$	243.150	1.74928	0.874641	+	0.484771i	$0.161097\pi$
$140$	143.890i	1.02779i
$141$	0	0
$142$	71.9047	0.506371
$143$	50.8836i	0.355830i
$144$	0	0
$145$	92.7517	0.639667
$146$	139.287i	0.954019i
$147$	0	0
$148$	−46.3698	−0.313309
$149$	− 144.075i	− 0.966945i	−0.875360	−	0.483472i	$-0.839375\pi$
$149$	− 144.075i	− 0.966945i	0.875360	−	0.483472i	$-0.160625\pi$
$150$	0	0
$151$	−101.609	−0.672906	−0.336453	−	0.941700i	$-0.609227\pi$
$151$	−101.609	−0.672906	−0.336453	+	0.941700i	$0.609227\pi$
$152$	− 21.7606i	− 0.143162i
$153$	0	0
$154$	−198.374	−1.28815
$155$	99.2333i	0.640215i
$156$	0	0
$157$	−309.535	−1.97156	−0.985781	−	0.168033i	$-0.946258\pi$
$157$	−309.535	−1.97156	−0.985781	+	0.168033i	$0.946258\pi$
$158$	114.762i	0.726339i
$159$	0	0
$160$	48.8345	0.305216
$161$	− 247.162i	− 1.53517i
$162$	0	0
$163$	160.056	0.981937	0.490969	−	0.871177i	$-0.336643\pi$
$163$	160.056	0.981937	0.490969	+	0.871177i	$0.336643\pi$
$164$	− 33.9688i	− 0.207127i
$165$	0	0
$166$	−22.8139	−0.137433
$167$	248.945i	1.49069i	0.666679	+	0.745345i	$0.267715\pi$
$167$	248.945i	1.49069i	−0.666679	+	0.745345i	$0.732285\pi$
$168$	0	0
$169$	−159.861	−0.945921
$170$	118.924i	0.699555i
$171$	0	0
$172$	−47.2248	−0.274563
$173$	− 67.5780i	− 0.390624i	−0.980741	−	0.195312i	$-0.937428\pi$
$173$	− 67.5780i	− 0.390624i	0.980741	−	0.195312i	$-0.0625720\pi$
$174$	0	0
$175$	412.739	2.35851
$176$	67.3259i	0.382533i
$177$	0	0
$178$	−22.8883	−0.128586
$179$	235.232i	1.31415i	0.753827	+	0.657073i	$0.228206\pi$
$179$	235.232i	1.31415i	−0.753827	+	0.657073i	$0.771794\pi$
$180$	0	0
$181$	217.831	1.20349	0.601744	−	0.798689i	$-0.294473\pi$
$181$	217.831	1.20349	0.601744	+	0.798689i	$0.294473\pi$
$182$	35.6303i	0.195771i
$183$	0	0
$184$	−83.8839	−0.455891
$185$	200.151i	1.08190i
$186$	0	0
$187$	−163.956	−0.876768
$188$	− 33.2086i	− 0.176642i
$189$	0	0
$190$	−93.9276	−0.494356
$191$	257.327i	1.34726i	0.739067	+	0.673631i	$0.235266\pi$
$191$	257.327i	1.34726i	−0.739067	+	0.673631i	$0.764734\pi$
$192$	0	0
$193$	−76.6710	−0.397259	−0.198630	−	0.980075i	$-0.563649\pi$
$193$	−76.6710	−0.397259	−0.198630	+	0.980075i	$0.563649\pi$
$194$	187.655i	0.967294i
$195$	0	0
$196$	−40.9080	−0.208714
$197$	243.988i	1.23852i	0.785186	+	0.619260i	$0.212567\pi$
$197$	243.988i	1.23852i	−0.785186	+	0.619260i	$0.787433\pi$
$198$	0	0
$199$	82.1079	0.412602	0.206301	−	0.978489i	$-0.433857\pi$
$199$	82.1079	0.412602	0.206301	+	0.978489i	$0.433857\pi$
$200$	− 140.079i	− 0.700393i
$201$	0	0
$202$	21.5273	0.106571
$203$	89.5403i	0.441085i
$204$	0	0
$205$	−146.623	−0.715234
$206$	53.5316i	0.259862i
$207$	0	0
$208$	12.0925	0.0581370
$209$	− 129.494i	− 0.619587i
$210$	0	0
$211$	57.4937	0.272482	0.136241	−	0.990676i	$-0.456498\pi$
$211$	57.4937	0.272482	0.136241	+	0.990676i	$0.456498\pi$
$212$	150.677i	0.710739i
$213$	0	0
$214$	167.810	0.784158
$215$	203.841i	0.948098i
$216$	0	0
$217$	−95.7975	−0.441463
$218$	− 291.406i	− 1.33673i
$219$	0	0
$220$	290.606	1.32093
$221$	29.4483i	0.133250i
$222$	0	0
$223$	231.988	1.04030	0.520152	−	0.854074i	$-0.325875\pi$
$223$	231.988	1.04030	0.520152	+	0.854074i	$0.325875\pi$
$224$	47.1437i	0.210463i
$225$	0	0
$226$	−100.451	−0.444472
$227$	− 214.533i	− 0.945080i	−0.881309	−	0.472540i	$-0.843337\pi$
$227$	− 214.533i	− 0.945080i	0.881309	−	0.472540i	$-0.156663\pi$
$228$	0	0
$229$	−395.410	−1.72668	−0.863341	−	0.504620i	$-0.831633\pi$
$229$	−395.410	−1.72668	−0.863341	+	0.504620i	$0.831633\pi$
$230$	362.077i	1.57425i
$231$	0	0
$232$	30.3889	0.130987
$233$	95.7591i	0.410983i	0.978659	+	0.205492i	$0.0658794\pi$
$233$	95.7591i	0.410983i	−0.978659	+	0.205492i	$0.934121\pi$
$234$	0	0
$235$	−143.342	−0.609965
$236$	120.403i	0.510182i
$237$	0	0
$238$	−114.807	−0.482382
$239$	− 88.1870i	− 0.368983i	−0.982834	−	0.184492i	$-0.940936\pi$
$239$	− 88.1870i	− 0.368983i	0.982834	−	0.184492i	$-0.0590638\pi$
$240$	0	0
$241$	−385.135	−1.59807	−0.799035	−	0.601284i	$-0.794656\pi$
$241$	−385.135	−1.59807	−0.799035	+	0.601284i	$0.794656\pi$
$242$	229.525i	0.948449i
$243$	0	0
$244$	68.3251	0.280021
$245$	176.575i	0.720716i
$246$	0	0
$247$	−23.2585	−0.0941641
$248$	32.5125i	0.131099i
$249$	0	0
$250$	−299.420	−1.19768
$251$	146.474i	0.583561i	0.956485	+	0.291780i	$0.0942476\pi$
$251$	146.474i	0.583561i	−0.956485	+	0.291780i	$0.905752\pi$
$252$	0	0
$253$	−499.178	−1.97304
$254$	− 54.1211i	− 0.213075i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 207.610i	− 0.807821i	−0.914799	−	0.403910i	$-0.867651\pi$
$257$	− 207.610i	− 0.807821i	0.914799	−	0.403910i	$-0.132349\pi$
$258$	0	0
$259$	−193.221	−0.746026
$260$	− 52.1961i	− 0.200754i
$261$	0	0
$262$	−308.196	−1.17632
$263$	33.2825i	0.126550i	0.997996	+	0.0632748i	$0.0201544\pi$
$263$	33.2825i	0.126550i	−0.997996	+	0.0632748i	$0.979846\pi$
$264$	0	0
$265$	650.381	2.45427
$266$	− 90.6756i	− 0.340886i
$267$	0	0
$268$	−75.2652	−0.280840
$269$	399.099i	1.48364i	0.670599	+	0.741820i	$0.266037\pi$
$269$	399.099i	1.48364i	−0.670599	+	0.741820i	$0.733963\pi$
$270$	0	0
$271$	−83.9974	−0.309954	−0.154977	−	0.987918i	$-0.549530\pi$
$271$	−83.9974	−0.309954	−0.154977	+	0.987918i	$0.549530\pi$
$272$	38.9640i	0.143250i
$273$	0	0
$274$	−260.345	−0.950166
$275$	− 833.584i	− 3.03121i
$276$	0	0
$277$	179.211	0.646970	0.323485	−	0.946233i	$-0.395146\pi$
$277$	179.211	0.646970	0.323485	+	0.946233i	$0.395146\pi$
$278$	− 343.866i	− 1.23693i
$279$	0	0
$280$	203.491	0.726754
$281$	464.964i	1.65468i	0.561704	+	0.827338i	$0.310146\pi$
$281$	464.964i	1.65468i	−0.561704	+	0.827338i	$0.689854\pi$
$282$	0	0
$283$	261.936	0.925568	0.462784	−	0.886471i	$-0.346851\pi$
$283$	261.936	0.925568	0.462784	+	0.886471i	$0.346851\pi$
$284$	− 101.689i	− 0.358058i
$285$	0	0
$286$	71.9603	0.251610
$287$	− 141.546i	− 0.493193i
$288$	0	0
$289$	194.113	0.671670
$290$	− 131.171i	− 0.452313i
$291$	0	0
$292$	196.981	0.674593
$293$	− 98.2311i	− 0.335260i	−0.985850	−	0.167630i	$-0.946389\pi$
$293$	− 98.2311i	− 0.335260i	0.985850	−	0.167630i	$-0.0536114\pi$
$294$	0	0
$295$	519.708	1.76172
$296$	65.5768i	0.221543i
$297$	0	0
$298$	−203.753	−0.683733
$299$	89.6581i	0.299860i
$300$	0	0
$301$	−196.784	−0.653766
$302$	143.696i	0.475816i
$303$	0	0
$304$	−30.7742	−0.101231
$305$	− 294.919i	− 0.966946i
$306$	0	0
$307$	492.852	1.60538	0.802690	−	0.596397i	$-0.203401\pi$
$307$	492.852	1.60538	0.802690	+	0.596397i	$0.203401\pi$
$308$	280.544i	0.910857i
$309$	0	0
$310$	140.337	0.452700
$311$	16.9054i	0.0543582i	0.999631	+	0.0271791i	$0.00865245\pi$
$311$	16.9054i	0.0543582i	−0.999631	+	0.0271791i	$0.991348\pi$
$312$	0	0
$313$	417.110	1.33262	0.666310	−	0.745675i	$-0.267873\pi$
$313$	417.110	1.33262	0.666310	+	0.745675i	$0.267873\pi$
$314$	437.749i	1.39411i
$315$	0	0
$316$	162.297	0.513599
$317$	− 453.753i	− 1.43140i	−0.698410	−	0.715698i	$-0.746109\pi$
$317$	− 453.753i	− 1.43140i	0.698410	−	0.715698i	$-0.253891\pi$
$318$	0	0
$319$	180.839	0.566893
$320$	− 69.0624i	− 0.215820i
$321$	0	0
$322$	−349.540	−1.08553
$323$	− 74.9429i	− 0.232021i
$324$	0	0
$325$	−149.721	−0.460680
$326$	− 226.353i	− 0.694334i
$327$	0	0
$328$	−48.0391	−0.146461
$329$	− 138.379i	− 0.420604i
$330$	0	0
$331$	−234.775	−0.709290	−0.354645	−	0.935001i	$-0.615398\pi$
$331$	−234.775	−0.709290	−0.354645	+	0.935001i	$0.615398\pi$
$332$	32.2638i	0.0971800i
$333$	0	0
$334$	352.062	1.05408
$335$	324.875i	0.969775i
$336$	0	0
$337$	−317.247	−0.941387	−0.470693	−	0.882297i	$-0.655996\pi$
$337$	−317.247	−0.941387	−0.470693	+	0.882297i	$0.655996\pi$
$338$	226.077i	0.668867i
$339$	0	0
$340$	168.184	0.494660
$341$	193.476i	0.567379i
$342$	0	0
$343$	237.900	0.693585
$344$	66.7859i	0.194145i
$345$	0	0
$346$	−95.5697	−0.276213
$347$	− 44.1393i	− 0.127203i	−0.997975	−	0.0636013i	$-0.979741\pi$
$347$	− 44.1393i	− 0.127203i	0.997975	−	0.0636013i	$-0.0202586\pi$
$348$	0	0
$349$	330.545	0.947121	0.473560	−	0.880761i	$-0.342969\pi$
$349$	330.545	0.947121	0.473560	+	0.880761i	$0.342969\pi$
$350$	− 583.701i	− 1.66772i
$351$	0	0
$352$	95.2132	0.270492
$353$	− 473.491i	− 1.34133i	−0.741758	−	0.670667i	$-0.766008\pi$
$353$	− 473.491i	− 1.34133i	0.741758	−	0.670667i	$-0.233992\pi$
$354$	0	0
$355$	−438.929	−1.23642
$356$	32.3689i	0.0909238i
$357$	0	0
$358$	332.668	0.929241
$359$	− 436.540i	− 1.21599i	−0.793941	−	0.607994i	$-0.791974\pi$
$359$	− 436.540i	− 1.21599i	0.793941	−	0.607994i	$-0.208026\pi$
$360$	0	0
$361$	−301.809	−0.836037
$362$	− 308.060i	− 0.850994i
$363$	0	0
$364$	50.3889	0.138431
$365$	− 850.250i	− 2.32945i
$366$	0	0
$367$	476.675	1.29884	0.649421	−	0.760429i	$-0.275011\pi$
$367$	476.675	1.29884	0.649421	+	0.760429i	$0.275011\pi$
$368$	118.630i	0.322363i
$369$	0	0
$370$	283.056	0.765016
$371$	627.863i	1.69235i
$372$	0	0
$373$	−369.120	−0.989599	−0.494799	−	0.869007i	$-0.664758\pi$
$373$	−369.120	−0.989599	−0.494799	+	0.869007i	$0.664758\pi$
$374$	231.868i	0.619968i
$375$	0	0
$376$	−46.9641	−0.124904
$377$	− 32.4807i	− 0.0861558i
$378$	0	0
$379$	−216.903	−0.572304	−0.286152	−	0.958184i	$-0.592376\pi$
$379$	−216.903	−0.572304	−0.286152	+	0.958184i	$0.592376\pi$
$380$	132.834i	0.349562i
$381$	0	0
$382$	363.916	0.952659
$383$	− 539.347i	− 1.40822i	−0.710092	−	0.704109i	$-0.751347\pi$
$383$	− 539.347i	− 1.40822i	0.710092	−	0.704109i	$-0.248653\pi$
$384$	0	0
$385$	1210.94	3.14530
$386$	108.429i	0.280905i
$387$	0	0
$388$	265.384	0.683980
$389$	− 147.663i	− 0.379597i	−0.981823	−	0.189799i	$-0.939217\pi$
$389$	− 147.663i	− 0.379597i	0.981823	−	0.189799i	$-0.0607835\pi$
$390$	0	0
$391$	−288.893	−0.738858
$392$	57.8527i	0.147583i
$393$	0	0
$394$	345.052	0.875766
$395$	− 700.541i	− 1.77352i
$396$	0	0
$397$	−502.615	−1.26603	−0.633017	−	0.774138i	$-0.718184\pi$
$397$	−502.615	−1.26603	−0.633017	+	0.774138i	$0.718184\pi$
$398$	− 116.118i	− 0.291754i
$399$	0	0
$400$	−198.101	−0.495253
$401$	231.445i	0.577170i	0.957454	+	0.288585i	$0.0931847\pi$
$401$	231.445i	0.577170i	−0.957454	+	0.288585i	$0.906815\pi$
$402$	0	0
$403$	34.7505	0.0862296
$404$	− 30.4442i	− 0.0753569i
$405$	0	0
$406$	126.629	0.311894
$407$	390.236i	0.958811i
$408$	0	0
$409$	450.444	1.10133	0.550665	−	0.834726i	$-0.314374\pi$
$409$	450.444	1.10133	0.550665	+	0.834726i	$0.314374\pi$
$410$	207.356i	0.505747i
$411$	0	0
$412$	75.7051	0.183750
$413$	501.714i	1.21480i
$414$	0	0
$415$	139.263	0.335574
$416$	− 17.1014i	− 0.0411091i
$417$	0	0
$418$	−183.132	−0.438114
$419$	329.142i	0.785542i	0.919636	+	0.392771i	$0.128484\pi$
$419$	329.142i	0.785542i	−0.919636	+	0.392771i	$0.871516\pi$
$420$	0	0
$421$	−204.609	−0.486007	−0.243003	−	0.970025i	$-0.578133\pi$
$421$	−204.609	−0.486007	−0.243003	+	0.970025i	$0.578133\pi$
$422$	− 81.3083i	− 0.192674i
$423$	0	0
$424$	213.089	0.502568
$425$	− 482.426i	− 1.13512i
$426$	0	0
$427$	284.708	0.666762
$428$	− 237.319i	− 0.554483i
$429$	0	0
$430$	288.275	0.670407
$431$	− 5.85402i	− 0.0135824i	−0.999977	−	0.00679120i	$-0.997838\pi$
$431$	− 5.85402i	− 0.0135824i	0.999977	−	0.00679120i	$-0.00216172\pi$
$432$	0	0
$433$	233.310	0.538821	0.269411	−	0.963025i	$-0.413171\pi$
$433$	233.310	0.538821	0.269411	+	0.963025i	$0.413171\pi$
$434$	135.478i	0.312162i
$435$	0	0
$436$	−412.111	−0.945208
$437$	− 228.171i	− 0.522130i
$438$	0	0
$439$	−134.791	−0.307040	−0.153520	−	0.988146i	$-0.549061\pi$
$439$	−134.791	−0.307040	−0.153520	+	0.988146i	$0.549061\pi$
$440$	− 410.978i	− 0.934042i
$441$	0	0
$442$	41.6462	0.0942221
$443$	− 28.6441i	− 0.0646593i	−0.999477	−	0.0323297i	$-0.989707\pi$
$443$	− 28.6441i	− 0.0646593i	0.999477	−	0.0323297i	$-0.0102926\pi$
$444$	0	0
$445$	139.717	0.313971
$446$	− 328.080i	− 0.735606i
$447$	0	0
$448$	66.6713	0.148820
$449$	496.782i	1.10642i	0.833042	+	0.553210i	$0.186597\pi$
$449$	496.782i	1.10642i	−0.833042	+	0.553210i	$0.813403\pi$
$450$	0	0
$451$	−285.872	−0.633863
$452$	142.059i	0.314289i
$453$	0	0
$454$	−303.396	−0.668272
$455$	− 217.499i	− 0.478019i
$456$	0	0
$457$	−463.804	−1.01489	−0.507444	−	0.861685i	$-0.669410\pi$
$457$	−463.804	−1.01489	−0.507444	+	0.861685i	$0.669410\pi$
$458$	559.195i	1.22095i
$459$	0	0
$460$	512.054	1.11316
$461$	− 266.382i	− 0.577836i	−0.957354	−	0.288918i	$-0.906705\pi$
$461$	− 266.382i	− 0.577836i	0.957354	−	0.288918i	$-0.0932954\pi$
$462$	0	0
$463$	239.371	0.517000	0.258500	−	0.966011i	$-0.416772\pi$
$463$	239.371	0.517000	0.258500	+	0.966011i	$0.416772\pi$
$464$	− 42.9764i	− 0.0926215i
$465$	0	0
$466$	135.424	0.290609
$467$	213.118i	0.456355i	0.973620	+	0.228177i	$0.0732766\pi$
$467$	213.118i	0.456355i	−0.973620	+	0.228177i	$0.926723\pi$
$468$	0	0
$469$	−313.626	−0.668713
$470$	202.716i	0.431310i
$471$	0	0
$472$	170.276	0.360753
$473$	397.431i	0.840236i
$474$	0	0
$475$	381.025	0.802158
$476$	162.361i	0.341095i
$477$	0	0
$478$	−124.715	−0.260911
$479$	− 400.215i	− 0.835522i	−0.908557	−	0.417761i	$-0.862815\pi$
$479$	− 400.215i	− 0.835522i	0.908557	−	0.417761i	$-0.137185\pi$
$480$	0	0
$481$	70.0908	0.145719
$482$	544.663i	1.13001i
$483$	0	0
$484$	324.597	0.670655
$485$	− 1145.51i	− 2.36187i
$486$	0	0
$487$	−323.676	−0.664632	−0.332316	−	0.943168i	$-0.607830\pi$
$487$	−323.676	−0.664632	−0.332316	+	0.943168i	$0.607830\pi$
$488$	− 96.6263i	− 0.198005i
$489$	0	0
$490$	249.715	0.509623
$491$	917.662i	1.86897i	0.356008	+	0.934483i	$0.384138\pi$
$491$	917.662i	1.86897i	−0.356008	+	0.934483i	$0.615862\pi$
$492$	0	0
$493$	104.658	0.212289
$494$	32.8925i	0.0665841i
$495$	0	0
$496$	45.9796	0.0927009
$497$	− 423.732i	− 0.852579i
$498$	0	0
$499$	−216.070	−0.433005	−0.216503	−	0.976282i	$-0.569465\pi$
$499$	−216.070	−0.433005	−0.216503	+	0.976282i	$0.569465\pi$
$500$	423.444i	0.846888i
$501$	0	0
$502$	207.145	0.412640
$503$	36.2064i	0.0719810i	0.999352	+	0.0359905i	$0.0114586\pi$
$503$	36.2064i	0.0719810i	−0.999352	+	0.0359905i	$0.988541\pi$
$504$	0	0
$505$	−131.409	−0.260216
$506$	705.945i	1.39515i
$507$	0	0
$508$	−76.5387	−0.150667
$509$	− 233.360i	− 0.458467i	−0.973371	−	0.229234i	$-0.926378\pi$
$509$	− 233.360i	− 0.458467i	0.973371	−	0.229234i	$-0.0736220\pi$
$510$	0	0
$511$	820.812	1.60629
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−293.605	−0.571216
$515$	− 326.774i	− 0.634512i
$516$	0	0
$517$	−279.475	−0.540570
$518$	273.255i	0.527520i
$519$	0	0
$520$	−73.8164	−0.141955
$521$	138.340i	0.265528i	0.991148	+	0.132764i	$0.0423852\pi$
$521$	138.340i	0.265528i	−0.991148	+	0.132764i	$0.957615\pi$
$522$	0	0
$523$	−196.869	−0.376423	−0.188212	−	0.982129i	$-0.560269\pi$
$523$	−196.869	−0.376423	−0.188212	+	0.982129i	$0.560269\pi$
$524$	435.854i	0.831783i
$525$	0	0
$526$	47.0686	0.0894840
$527$	111.972i	0.212471i
$528$	0	0
$529$	−350.563	−0.662691
$530$	− 919.778i	− 1.73543i
$531$	0	0
$532$	−128.235	−0.241042
$533$	51.3459i	0.0963338i
$534$	0	0
$535$	−1024.36	−1.91470
$536$	106.441i	0.198584i
$537$	0	0
$538$	564.412	1.04909
$539$	344.271i	0.638722i
$540$	0	0
$541$	−136.735	−0.252745	−0.126373	−	0.991983i	$-0.540334\pi$
$541$	−136.735	−0.252745	−0.126373	+	0.991983i	$0.540334\pi$
$542$	118.790i	0.219170i
$543$	0	0
$544$	55.1035	0.101293
$545$	1778.84i	3.26392i
$546$	0	0
$547$	−665.853	−1.21728	−0.608641	−	0.793446i	$-0.708285\pi$
$547$	−665.853	−1.21728	−0.608641	+	0.793446i	$0.708285\pi$
$548$	368.184i	0.671868i
$549$	0	0
$550$	−1178.87	−2.14339
$551$	82.6602i	0.150019i
$552$	0	0
$553$	676.285	1.22294
$554$	− 253.442i	− 0.457477i
$555$	0	0
$556$	−486.300	−0.874641
$557$	− 1089.88i	− 1.95669i	−0.206977	−	0.978346i	$-0.566362\pi$
$557$	− 1089.88i	− 1.95669i	0.206977	−	0.978346i	$-0.433638\pi$
$558$	0	0
$559$	71.3832	0.127698
$560$	− 287.780i	− 0.513893i
$561$	0	0
$562$	657.559	1.17003
$563$	612.246i	1.08747i	0.839257	+	0.543735i	$0.182990\pi$
$563$	612.246i	1.08747i	−0.839257	+	0.543735i	$0.817010\pi$
$564$	0	0
$565$	613.182	1.08528
$566$	− 370.433i	− 0.654475i
$567$	0	0
$568$	−143.809	−0.253186
$569$	499.370i	0.877628i	0.898578	+	0.438814i	$0.144601\pi$
$569$	499.370i	0.877628i	−0.898578	+	0.438814i	$0.855399\pi$
$570$	0	0
$571$	−302.814	−0.530323	−0.265161	−	0.964204i	$-0.585425\pi$
$571$	−302.814	−0.530323	−0.265161	+	0.964204i	$0.585425\pi$
$572$	− 101.767i	− 0.177915i
$573$	0	0
$574$	−200.177	−0.348740
$575$	− 1468.79i	− 2.55442i
$576$	0	0
$577$	37.1228	0.0643377	0.0321688	−	0.999482i	$-0.489759\pi$
$577$	37.1228	0.0643377	0.0321688	+	0.999482i	$0.489759\pi$
$578$	− 274.517i	− 0.474943i
$579$	0	0
$580$	−185.503	−0.319833
$581$	134.442i	0.231397i
$582$	0	0
$583$	1268.05	2.17505
$584$	− 278.574i	− 0.477010i
$585$	0	0
$586$	−138.920	−0.237065
$587$	877.106i	1.49422i	0.664701	+	0.747109i	$0.268559\pi$
$587$	877.106i	1.49422i	−0.664701	+	0.747109i	$0.731441\pi$
$588$	0	0
$589$	−88.4366	−0.150147
$590$	− 734.978i	− 1.24573i
$591$	0	0
$592$	92.7396	0.156655
$593$	284.591i	0.479917i	0.970783	+	0.239958i	$0.0771338\pi$
$593$	284.591i	0.479917i	−0.970783	+	0.239958i	$0.922866\pi$
$594$	0	0
$595$	700.817	1.17784
$596$	288.150i	0.483472i
$597$	0	0
$598$	126.796	0.212033
$599$	− 329.061i	− 0.549350i	−0.961537	−	0.274675i	$-0.911430\pi$
$599$	− 329.061i	− 0.549350i	0.961537	−	0.274675i	$-0.0885703\pi$
$600$	0	0
$601$	7.07343	0.0117694	0.00588472	−	0.999983i	$-0.498127\pi$
$601$	7.07343	0.0117694	0.00588472	+	0.999983i	$0.498127\pi$
$602$	278.294i	0.462282i
$603$	0	0
$604$	203.218	0.336453
$605$	− 1401.09i	− 2.31585i
$606$	0	0
$607$	483.117	0.795910	0.397955	−	0.917405i	$-0.369720\pi$
$607$	483.117	0.795910	0.397955	+	0.917405i	$0.369720\pi$
$608$	43.5213i	0.0715810i
$609$	0	0
$610$	−417.078	−0.683734
$611$	50.1969i	0.0821553i
$612$	0	0
$613$	575.865	0.939420	0.469710	−	0.882821i	$-0.344358\pi$
$613$	575.865	0.939420	0.469710	+	0.882821i	$0.344358\pi$
$614$	− 696.997i	− 1.13517i
$615$	0	0
$616$	396.749	0.644073
$617$	− 701.686i	− 1.13725i	−0.822595	−	0.568627i	$-0.807475\pi$
$617$	− 701.686i	− 1.13725i	0.822595	−	0.568627i	$-0.192525\pi$
$618$	0	0
$619$	451.135	0.728813	0.364407	−	0.931240i	$-0.381272\pi$
$619$	451.135	0.728813	0.364407	+	0.931240i	$0.381272\pi$
$620$	− 198.467i	− 0.320107i
$621$	0	0
$622$	23.9079	0.0384371
$623$	134.880i	0.216500i
$624$	0	0
$625$	589.622	0.943396
$626$	− 589.882i	− 0.942304i
$627$	0	0
$628$	619.071	0.985781
$629$	225.844i	0.359053i
$630$	0	0
$631$	−567.433	−0.899260	−0.449630	−	0.893215i	$-0.648444\pi$
$631$	−567.433	−0.899260	−0.449630	+	0.893215i	$0.648444\pi$
$632$	− 229.523i	− 0.363169i
$633$	0	0
$634$	−641.703	−1.01215
$635$	330.372i	0.520271i
$636$	0	0
$637$	61.8350	0.0970722
$638$	− 255.745i	− 0.400854i
$639$	0	0
$640$	−97.6690	−0.152608
$641$	− 149.037i	− 0.232506i	−0.993220	−	0.116253i	$-0.962912\pi$
$641$	− 149.037i	− 0.232506i	0.993220	−	0.116253i	$-0.0370884\pi$
$642$	0	0
$643$	−27.2824	−0.0424299	−0.0212150	−	0.999775i	$-0.506753\pi$
$643$	−27.2824	−0.0424299	−0.0212150	+	0.999775i	$0.506753\pi$
$644$	494.325i	0.767585i
$645$	0	0
$646$	−105.985	−0.164064
$647$	− 263.970i	− 0.407991i	−0.978972	−	0.203996i	$-0.934607\pi$
$647$	− 263.970i	− 0.407991i	0.978972	−	0.203996i	$-0.0653929\pi$
$648$	0	0
$649$	1013.28	1.56129
$650$	211.738i	0.325750i
$651$	0	0
$652$	−320.111	−0.490969
$653$	− 837.516i	− 1.28257i	−0.767304	−	0.641283i	$-0.778402\pi$
$653$	− 837.516i	− 1.28257i	0.767304	−	0.641283i	$-0.221598\pi$
$654$	0	0
$655$	1881.32	2.87225
$656$	67.9376i	0.103563i
$657$	0	0
$658$	−195.697	−0.297412
$659$	743.210i	1.12778i	0.825848	+	0.563892i	$0.190697\pi$
$659$	743.210i	1.12778i	−0.825848	+	0.563892i	$0.809303\pi$
$660$	0	0
$661$	268.581	0.406325	0.203163	−	0.979145i	$-0.434878\pi$
$661$	268.581	0.406325	0.203163	+	0.979145i	$0.434878\pi$
$662$	332.022i	0.501544i
$663$	0	0
$664$	45.6279	0.0687167
$665$	553.512i	0.832349i
$666$	0	0
$667$	318.642	0.477725
$668$	− 497.890i	− 0.745345i
$669$	0	0
$670$	459.442	0.685735
$671$	− 575.006i	− 0.856939i
$672$	0	0
$673$	659.800	0.980386	0.490193	−	0.871614i	$-0.336926\pi$
$673$	659.800	0.980386	0.490193	+	0.871614i	$0.336926\pi$
$674$	448.656i	0.665661i
$675$	0	0
$676$	319.721	0.472961
$677$	254.990i	0.376647i	0.982107	+	0.188324i	$0.0603054\pi$
$677$	254.990i	0.376647i	−0.982107	+	0.188324i	$0.939695\pi$
$678$	0	0
$679$	1105.84	1.62864
$680$	− 237.849i	− 0.349778i
$681$	0	0
$682$	273.617	0.401197
$683$	434.644i	0.636375i	0.948028	+	0.318187i	$0.103074\pi$
$683$	434.644i	0.636375i	−0.948028	+	0.318187i	$0.896926\pi$
$684$	0	0
$685$	1589.23	2.32004
$686$	− 336.441i	− 0.490439i
$687$	0	0
$688$	94.4496	0.137281
$689$	− 227.757i	− 0.330562i
$690$	0	0
$691$	−1334.94	−1.93190	−0.965951	−	0.258725i	$-0.916698\pi$
$691$	−1334.94	−1.93190	−0.965951	+	0.258725i	$0.916698\pi$
$692$	135.156i	0.195312i
$693$	0	0
$694$	−62.4224	−0.0899459
$695$	2099.07i	3.02024i
$696$	0	0
$697$	−165.445	−0.237368
$698$	− 467.461i	− 0.669716i
$699$	0	0
$700$	−825.478	−1.17925
$701$	1001.61i	1.42883i	0.699722	+	0.714415i	$0.253307\pi$
$701$	1001.61i	1.42883i	−0.699722	+	0.714415i	$0.746693\pi$
$702$	0	0
$703$	−178.374	−0.253733
$704$	− 134.652i	− 0.191267i
$705$	0	0
$706$	−669.618	−0.948467
$707$	− 126.859i	− 0.179433i
$708$	0	0
$709$	−766.266	−1.08077	−0.540385	−	0.841418i	$-0.681721\pi$
$709$	−766.266	−1.08077	−0.540385	+	0.841418i	$0.681721\pi$
$710$	620.739i	0.874280i
$711$	0	0
$712$	45.7765	0.0642928
$713$	340.909i	0.478134i
$714$	0	0
$715$	−439.268	−0.614361
$716$	− 470.464i	− 0.657073i
$717$	0	0
$718$	−617.361	−0.859834
$719$	628.844i	0.874609i	0.899313	+	0.437305i	$0.144067\pi$
$719$	628.844i	0.874609i	−0.899313	+	0.437305i	$0.855933\pi$
$720$	0	0
$721$	315.460	0.437531
$722$	426.823i	0.591167i
$723$	0	0
$724$	−435.662	−0.601744
$725$	532.105i	0.733937i
$726$	0	0
$727$	−668.852	−0.920016	−0.460008	−	0.887915i	$-0.652153\pi$
$727$	−668.852	−0.920016	−0.460008	+	0.887915i	$0.652153\pi$
$728$	− 71.2606i	− 0.0978855i
$729$	0	0
$730$	−1202.44	−1.64717
$731$	230.009i	0.314649i
$732$	0	0
$733$	110.981	0.151407	0.0757034	−	0.997130i	$-0.475880\pi$
$733$	110.981	0.151407	0.0757034	+	0.997130i	$0.475880\pi$
$734$	− 674.120i	− 0.918419i
$735$	0	0
$736$	167.768	0.227945
$737$	633.412i	0.859446i
$738$	0	0
$739$	462.595	0.625975	0.312987	−	0.949757i	$-0.398670\pi$
$739$	462.595	0.625975	0.312987	+	0.949757i	$0.398670\pi$
$740$	− 400.301i	− 0.540948i
$741$	0	0
$742$	887.932	1.19667
$743$	− 966.907i	− 1.30136i	−0.759354	−	0.650678i	$-0.774485\pi$
$743$	− 966.907i	− 1.30136i	0.759354	−	0.650678i	$-0.225515\pi$
$744$	0	0
$745$	1243.77	1.66949
$746$	522.015i	0.699752i
$747$	0	0
$748$	327.911	0.438384
$749$	− 988.897i	− 1.32029i
$750$	0	0
$751$	−1051.00	−1.39947	−0.699733	−	0.714404i	$-0.746698\pi$
$751$	−1051.00	−1.39947	−0.699733	+	0.714404i	$0.746698\pi$
$752$	66.4172i	0.0883208i
$753$	0	0
$754$	−45.9347	−0.0609214
$755$	− 877.168i	− 1.16181i
$756$	0	0
$757$	−266.724	−0.352343	−0.176172	−	0.984359i	$-0.556371\pi$
$757$	−266.724	−0.352343	−0.176172	+	0.984359i	$0.556371\pi$
$758$	306.748i	0.404680i
$759$	0	0
$760$	187.855	0.247178
$761$	− 0.356877i	0 0.000468958i	−1.00000		0.000234479i	$-0.999925\pi$
$761$	− 0.356877i	0 0.000468958i	1.00000		0.000234479i	$-7.46369e-5\pi$
$762$	0	0
$763$	−1717.25	−2.25065
$764$	− 514.654i	− 0.673631i
$765$	0	0
$766$	−762.752	−0.995760
$767$	− 181.997i	− 0.237284i
$768$	0	0
$769$	448.395	0.583089	0.291544	−	0.956557i	$-0.405831\pi$
$769$	448.395	0.583089	0.291544	+	0.956557i	$0.405831\pi$
$770$	− 1712.53i	− 2.22406i
$771$	0	0
$772$	153.342	0.198630
$773$	670.666i	0.867614i	0.901006	+	0.433807i	$0.142830\pi$
$773$	670.666i	0.867614i	−0.901006	+	0.433807i	$0.857170\pi$
$774$	0	0
$775$	−569.289	−0.734566
$776$	− 375.310i	− 0.483647i
$777$	0	0
$778$	−208.827	−0.268416
$779$	− 130.670i	− 0.167741i
$780$	0	0
$781$	−855.785	−1.09575
$782$	408.557i	0.522451i
$783$	0	0
$784$	81.8160	0.104357
$785$	− 2672.16i	− 3.40402i
$786$	0	0
$787$	−1226.18	−1.55804	−0.779019	−	0.627000i	$-0.784283\pi$
$787$	−1226.18	−1.55804	−0.779019	+	0.627000i	$0.784283\pi$
$788$	− 487.977i	− 0.619260i
$789$	0	0
$790$	−990.714	−1.25407
$791$	591.952i	0.748358i
$792$	0	0
$793$	−103.278	−0.130237
$794$	710.805i	0.895221i
$795$	0	0
$796$	−164.216	−0.206301
$797$	397.802i	0.499124i	0.968359	+	0.249562i	$0.0802866\pi$
$797$	397.802i	0.499124i	−0.968359	+	0.249562i	$0.919713\pi$
$798$	0	0
$799$	−161.743	−0.202431
$800$	280.157i	0.350197i
$801$	0	0
$802$	327.313	0.408121
$803$	− 1657.74i	− 2.06444i
$804$	0	0
$805$	2133.70	2.65056
$806$	− 49.1447i	− 0.0609735i
$807$	0	0
$808$	−43.0546	−0.0532853
$809$	− 438.654i	− 0.542218i	−0.962549	−	0.271109i	$-0.912610\pi$
$809$	− 438.654i	− 0.542218i	0.962549	−	0.271109i	$-0.0873903\pi$
$810$	0	0
$811$	−1037.62	−1.27943	−0.639716	−	0.768611i	$-0.720948\pi$
$811$	−1037.62	−1.27943	−0.639716	+	0.768611i	$0.720948\pi$
$812$	− 179.081i	− 0.220543i
$813$	0	0
$814$	551.877	0.677982
$815$	1381.73i	1.69537i
$816$	0	0
$817$	−181.663	−0.222354
$818$	− 637.024i	− 0.778758i
$819$	0	0
$820$	293.246	0.357617
$821$	1429.95i	1.74172i	0.491530	+	0.870861i	$0.336438\pi$
$821$	1429.95i	1.74172i	−0.491530	+	0.870861i	$0.663562\pi$
$822$	0	0
$823$	−1415.06	−1.71939	−0.859694	−	0.510810i	$-0.829346\pi$
$823$	−1415.06	−1.71939	−0.859694	+	0.510810i	$0.829346\pi$
$824$	− 107.063i	− 0.129931i
$825$	0	0
$826$	709.531	0.858996
$827$	704.209i	0.851523i	0.904836	+	0.425761i	$0.139994\pi$
$827$	704.209i	0.851523i	−0.904836	+	0.425761i	$0.860006\pi$
$828$	0	0
$829$	291.158	0.351216	0.175608	−	0.984460i	$-0.443811\pi$
$829$	291.158	0.351216	0.175608	+	0.984460i	$0.443811\pi$
$830$	− 196.948i	− 0.237287i
$831$	0	0
$832$	−24.1850	−0.0290685
$833$	199.243i	0.239187i
$834$	0	0
$835$	−2149.09	−2.57377
$836$	258.987i	0.309794i
$837$	0	0
$838$	465.477	0.555462
$839$	865.355i	1.03141i	0.856766	+	0.515706i	$0.172470\pi$
$839$	865.355i	1.03141i	−0.856766	+	0.515706i	$0.827530\pi$
$840$	0	0
$841$	725.564	0.862740
$842$	289.361i	0.343659i
$843$	0	0
$844$	−114.987	−0.136241
$845$	− 1380.05i	− 1.63319i
$846$	0	0
$847$	1352.58	1.59691
$848$	− 301.353i	− 0.355369i
$849$	0	0
$850$	−682.254	−0.802652
$851$	687.605i	0.807996i
$852$	0	0
$853$	−108.638	−0.127360	−0.0636799	−	0.997970i	$-0.520284\pi$
$853$	−108.638	−0.127360	−0.0636799	+	0.997970i	$0.520284\pi$
$854$	− 402.637i	− 0.471472i
$855$	0	0
$856$	−335.620	−0.392079
$857$	1066.28i	1.24420i	0.782937	+	0.622102i	$0.213721\pi$
$857$	1066.28i	1.24420i	−0.782937	+	0.622102i	$0.786279\pi$
$858$	0	0
$859$	−55.4656	−0.0645700	−0.0322850	−	0.999479i	$-0.510278\pi$
$859$	−55.4656	−0.0645700	−0.0322850	+	0.999479i	$0.510278\pi$
$860$	− 407.682i	− 0.474049i
$861$	0	0
$862$	−8.27883	−0.00960421
$863$	75.8244i	0.0878614i	0.999035	+	0.0439307i	$0.0139881\pi$
$863$	75.8244i	0.0878614i	−0.999035	+	0.0439307i	$0.986012\pi$
$864$	0	0
$865$	583.388	0.674437
$866$	− 329.950i	− 0.381004i
$867$	0	0
$868$	191.595	0.220732
$869$	− 1365.85i	− 1.57175i
$870$	0	0
$871$	113.768	0.130618
$872$	582.813i	0.668363i
$873$	0	0
$874$	−322.682	−0.369202
$875$	1764.47i	2.01654i
$876$	0	0
$877$	577.772	0.658805	0.329403	−	0.944190i	$-0.393153\pi$
$877$	577.772	0.658805	0.329403	+	0.944190i	$0.393153\pi$
$878$	190.623i	0.217110i
$879$	0	0
$880$	−581.211	−0.660467
$881$	136.922i	0.155416i	0.996976	+	0.0777082i	$0.0247602\pi$
$881$	136.922i	0.155416i	−0.996976	+	0.0777082i	$0.975240\pi$
$882$	0	0
$883$	−1269.49	−1.43770	−0.718851	−	0.695165i	$-0.755332\pi$
$883$	−1269.49	−1.43770	−0.718851	+	0.695165i	$0.755332\pi$
$884$	− 58.8966i	− 0.0666251i
$885$	0	0
$886$	−40.5089	−0.0457211
$887$	644.315i	0.726398i	0.931712	+	0.363199i	$0.118316\pi$
$887$	644.315i	0.726398i	−0.931712	+	0.363199i	$0.881684\pi$
$888$	0	0
$889$	−318.933	−0.358755
$890$	− 197.590i	− 0.222011i
$891$	0	0
$892$	−463.975	−0.520152
$893$	− 127.746i	− 0.143053i
$894$	0	0
$895$	−2030.71	−2.26895
$896$	− 94.2874i	− 0.105231i
$897$	0	0
$898$	702.556	0.782357
$899$	− 123.502i	− 0.137378i
$900$	0	0
$901$	733.871	0.814508
$902$	404.285i	0.448209i
$903$	0	0
$904$	200.901	0.222236
$905$	1880.49i	2.07789i
$906$	0	0
$907$	−611.492	−0.674192	−0.337096	−	0.941470i	$-0.609445\pi$
$907$	−611.492	−0.674192	−0.337096	+	0.941470i	$0.609445\pi$
$908$	429.066i	0.472540i
$909$	0	0
$910$	−307.589	−0.338010
$911$	− 1050.27i	− 1.15287i	−0.817143	−	0.576435i	$-0.804443\pi$
$911$	− 1050.27i	− 1.15287i	0.817143	−	0.576435i	$-0.195557\pi$
$912$	0	0
$913$	271.523	0.297397
$914$	655.918i	0.717634i
$915$	0	0
$916$	790.821	0.863341
$917$	1816.19i	1.98057i
$918$	0	0
$919$	133.483	0.145248	0.0726238	−	0.997359i	$-0.476863\pi$
$919$	133.483	0.145248	0.0726238	+	0.997359i	$0.476863\pi$
$920$	− 724.153i	− 0.787123i
$921$	0	0
$922$	−376.721	−0.408591
$923$	153.709i	0.166532i
$924$	0	0
$925$	−1148.24	−1.24134
$926$	− 338.522i	− 0.365574i
$927$	0	0
$928$	−60.7778	−0.0654933
$929$	− 1111.09i	− 1.19600i	−0.801496	−	0.598001i	$-0.795962\pi$
$929$	− 1111.09i	− 1.19600i	0.801496	−	0.598001i	$-0.204038\pi$
$930$	0	0
$931$	−157.364	−0.169027
$932$	− 191.518i	− 0.205492i
$933$	0	0
$934$	301.394	0.322691
$935$	− 1415.40i	− 1.51379i
$936$	0	0
$937$	−1725.54	−1.84156	−0.920781	−	0.390079i	$-0.872448\pi$
$937$	−1725.54	−1.84156	−0.920781	+	0.390079i	$0.872448\pi$
$938$	443.535i	0.472852i
$939$	0	0
$940$	286.683	0.304982
$941$	1360.15i	1.44544i	0.691143	+	0.722718i	$0.257107\pi$
$941$	1360.15i	1.44544i	−0.691143	+	0.722718i	$0.742893\pi$
$942$	0	0
$943$	−503.714	−0.534161
$944$	− 240.806i	− 0.255091i
$945$	0	0
$946$	562.053	0.594136
$947$	1614.67i	1.70504i	0.522694	+	0.852521i	$0.324927\pi$
$947$	1614.67i	1.70504i	−0.522694	+	0.852521i	$0.675073\pi$
$948$	0	0
$949$	−297.749	−0.313751
$950$	− 538.851i	− 0.567211i
$951$	0	0
$952$	229.614	0.241191
$953$	209.170i	0.219485i	0.993960	+	0.109743i	$0.0350027\pi$
$953$	209.170i	0.219485i	−0.993960	+	0.109743i	$0.964997\pi$
$954$	0	0
$955$	−2221.46	−2.32613
$956$	176.374i	0.184492i
$957$	0	0
$958$	−565.989	−0.590803
$959$	1534.21i	1.59980i
$960$	0	0
$961$	−828.867	−0.862505
$962$	− 99.1234i	− 0.103039i
$963$	0	0
$964$	770.270	0.799035
$965$	− 661.886i	− 0.685892i
$966$	0	0
$967$	877.370	0.907311	0.453655	−	0.891177i	$-0.350120\pi$
$967$	877.370	0.907311	0.453655	+	0.891177i	$0.350120\pi$
$968$	− 459.049i	− 0.474225i
$969$	0	0
$970$	−1619.99	−1.67009
$971$	686.372i	0.706871i	0.935459	+	0.353435i	$0.114987\pi$
$971$	686.372i	0.706871i	−0.935459	+	0.353435i	$0.885013\pi$
$972$	0	0
$973$	−2026.39	−2.08262
$974$	457.747i	0.469966i
$975$	0	0
$976$	−136.650	−0.140010
$977$	1408.93i	1.44209i	0.692886	+	0.721047i	$0.256339\pi$
$977$	1408.93i	1.44209i	−0.692886	+	0.721047i	$0.743661\pi$
$978$	0	0
$979$	272.408	0.278251
$980$	− 353.151i	− 0.360358i
$981$	0	0
$982$	1297.77	1.32156
$983$	− 943.816i	− 0.960138i	−0.877231	−	0.480069i	$-0.840612\pi$
$983$	− 943.816i	− 0.960138i	0.877231	−	0.480069i	$-0.159388\pi$
$984$	0	0
$985$	−2106.30	−2.13838
$986$	− 148.009i	− 0.150111i
$987$	0	0
$988$	46.5171	0.0470821
$989$	700.283i	0.708072i
$990$	0	0
$991$	848.305	0.856009	0.428005	−	0.903777i	$-0.359217\pi$
$991$	848.305	0.856009	0.428005	+	0.903777i	$0.359217\pi$
$992$	− 65.0250i	− 0.0655494i
$993$	0	0
$994$	−599.247	−0.602864
$995$	708.821i	0.712383i
$996$	0	0
$997$	−168.711	−0.169218	−0.0846092	−	0.996414i	$-0.526964\pi$
$997$	−168.711	−0.169218	−0.0846092	+	0.996414i	$0.526964\pi$
$998$	305.569i	0.306181i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.18		36
3.2	odd	2	inner	1458.3.b.c.1457.19		36
27.5	odd	18		162.3.f.a.89.4		36
27.11	odd	18		54.3.f.a.41.1	yes	36
27.16	even	9		162.3.f.a.71.4		36
27.22	even	9		54.3.f.a.29.1	✓	36
108.11	even	18		432.3.bc.c.257.6		36
108.103	odd	18		432.3.bc.c.353.6		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.29.1	✓	36	27.22	even	9
54.3.f.a.41.1	yes	36	27.11	odd	18
162.3.f.a.71.4		36	27.16	even	9
162.3.f.a.89.4		36	27.5	odd	18
432.3.bc.c.257.6		36	108.11	even	18
432.3.bc.c.353.6		36	108.103	odd	18
1458.3.b.c.1457.18		36	1.1	even	1	trivial
1458.3.b.c.1457.19		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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +8.63280i q^{5} -8.33391 q^{7} +2.82843i q^{8} +12.2086 q^{10} +16.8315i q^{11} +3.02312 q^{13} +11.7859i q^{14} +4.00000 q^{16} +9.74101i q^{17} -7.69355 q^{19} -17.2656i q^{20} +23.8033 q^{22} +29.6574i q^{23} -49.5253 q^{25} -4.27534i q^{26} +16.6678 q^{28} -10.7441i q^{29} +11.4949 q^{31} -5.65685i q^{32} +13.7759 q^{34} -71.9450i q^{35} +23.1849 q^{37} +10.8803i q^{38} -24.4173 q^{40} +16.9844i q^{41} +23.6124 q^{43} -33.6629i q^{44} +41.9419 q^{46} +16.6043i q^{47} +20.4540 q^{49} +70.0393i q^{50} -6.04625 q^{52} -75.3383i q^{53} -145.303 q^{55} -23.5718i q^{56} -15.1944 q^{58} -60.2015i q^{59} -34.1626 q^{61} -16.2563i q^{62} -8.00000 q^{64} +26.0980i q^{65} +37.6326 q^{67} -19.4820i q^{68} -101.746 q^{70} +50.8443i q^{71} -98.4906 q^{73} -32.7884i q^{74} +15.3871 q^{76} -140.272i q^{77} -81.1487 q^{79} +34.5312i q^{80} +24.0196 q^{82} -16.1319i q^{83} -84.0922 q^{85} -33.3930i q^{86} -47.6066 q^{88} -16.1844i q^{89} -25.1944 q^{91} -59.3149i q^{92} +23.4820 q^{94} -66.4169i q^{95} -132.692 q^{97} -28.9263i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 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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