LMFDB - Embedded newform 414.3.c.b.323.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,3,Mod(323,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("414.323");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	414.c (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:	$11.2806829445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 $ x^8 - 2x^7 + 19x^6 - 88x^5 + 301x^4 - 1010x^3 + 2713x^2 - 7044*x + 9558
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.4
Root			$-1.93130 + 3.33657i$ of defining polynomial
Character	$\chi$	$=$	414.323
Dual form			414.3.c.b.323.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +5.46255i q^{5} +2.65490 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +5.46255i q^{5} +2.65490 q^{7} +2.82843i q^{8} +7.72521 q^{10} +3.75460i q^{11} -23.2899 q^{13} -3.75460i q^{14} +4.00000 q^{16} -19.2161i q^{17} -35.0940 q^{19} -10.9251i q^{20} +5.30981 q^{22} +4.79583i q^{23} -4.83945 q^{25} +32.9369i q^{26} -5.30981 q^{28} +45.2762i q^{29} -32.4391 q^{31} -5.65685i q^{32} -27.1756 q^{34} +14.5025i q^{35} -40.5797 q^{37} +49.6305i q^{38} -15.4504 q^{40} -41.1112i q^{41} +57.0062 q^{43} -7.50920i q^{44} +6.78233 q^{46} +58.8804i q^{47} -41.9515 q^{49} +6.84401i q^{50} +46.5797 q^{52} +32.3166i q^{53} -20.5097 q^{55} +7.50920i q^{56} +64.0302 q^{58} +41.6865i q^{59} +97.7791 q^{61} +45.8759i q^{62} -8.00000 q^{64} -127.222i q^{65} -35.0235 q^{67} +38.4322i q^{68} +20.5097 q^{70} -60.7708i q^{71} +33.2521 q^{73} +57.3884i q^{74} +70.1881 q^{76} +9.96811i q^{77} -139.994 q^{79} +21.8502i q^{80} -58.1401 q^{82} +67.4528i q^{83} +104.969 q^{85} -80.6189i q^{86} -10.6196 q^{88} +45.1603i q^{89} -61.8324 q^{91} -9.59166i q^{92} +83.2694 q^{94} -191.703i q^{95} +12.2522 q^{97} +59.3284i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 16 * q^7	$ 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$-1$	$1$

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
$5$	5.46255i	1.09251i	0.837619	+	0.546255i	$0.183947\pi$
$5$	5.46255i	1.09251i	−0.837619	+	0.546255i	$0.816053\pi$
$6$	0	0
$7$	2.65490	0.379272	0.189636	−	0.981854i	$-0.439269\pi$
$7$	2.65490	0.379272	0.189636	+	0.981854i	$0.439269\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	7.72521	0.772521
$11$	3.75460i	0.341327i	0.985329	+	0.170664i	$0.0545912\pi$
$11$	3.75460i	0.341327i	−0.985329	+	0.170664i	$0.945409\pi$
$12$	0	0
$13$	−23.2899	−1.79153	−0.895764	−	0.444529i	$-0.853371\pi$
$13$	−23.2899	−1.79153	−0.895764	+	0.444529i	$0.853371\pi$
$14$	− 3.75460i	− 0.268186i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 19.2161i	− 1.13036i	−0.824969	−	0.565179i	$-0.808807\pi$
$17$	− 19.2161i	− 1.13036i	0.824969	−	0.565179i	$-0.191193\pi$
$18$	0	0
$19$	−35.0940	−1.84705	−0.923527	−	0.383533i	$-0.874707\pi$
$19$	−35.0940	−1.84705	−0.923527	+	0.383533i	$0.874707\pi$
$20$	− 10.9251i	− 0.546255i
$21$	0	0
$22$	5.30981	0.241355
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	−4.83945	−0.193578
$26$	32.9369i	1.26680i
$27$	0	0
$28$	−5.30981	−0.189636
$29$	45.2762i	1.56125i	0.625002	+	0.780624i	$0.285098\pi$
$29$	45.2762i	1.56125i	−0.625002	+	0.780624i	$0.714902\pi$
$30$	0	0
$31$	−32.4391	−1.04642	−0.523212	−	0.852203i	$-0.675266\pi$
$31$	−32.4391	−1.04642	−0.523212	+	0.852203i	$0.675266\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−27.1756	−0.799283
$35$	14.5025i	0.414359i
$36$	0	0
$37$	−40.5797	−1.09675	−0.548375	−	0.836233i	$-0.684753\pi$
$37$	−40.5797	−1.09675	−0.548375	+	0.836233i	$0.684753\pi$
$38$	49.6305i	1.30606i
$39$	0	0
$40$	−15.4504	−0.386261
$41$	− 41.1112i	− 1.00271i	−0.865241	−	0.501357i	$-0.832834\pi$
$41$	− 41.1112i	− 1.00271i	0.865241	−	0.501357i	$-0.167166\pi$
$42$	0	0
$43$	57.0062	1.32573	0.662863	−	0.748741i	$-0.269341\pi$
$43$	57.0062	1.32573	0.662863	+	0.748741i	$0.269341\pi$
$44$	− 7.50920i	− 0.170664i
$45$	0	0
$46$	6.78233	0.147442
$47$	58.8804i	1.25277i	0.779512	+	0.626387i	$0.215467\pi$
$47$	58.8804i	1.25277i	−0.779512	+	0.626387i	$0.784533\pi$
$48$	0	0
$49$	−41.9515	−0.856153
$50$	6.84401i	0.136880i
$51$	0	0
$52$	46.5797	0.895764
$53$	32.3166i	0.609748i	0.952393	+	0.304874i	$0.0986143\pi$
$53$	32.3166i	0.609748i	−0.952393	+	0.304874i	$0.901386\pi$
$54$	0	0
$55$	−20.5097	−0.372904
$56$	7.50920i	0.134093i
$57$	0	0
$58$	64.0302	1.10397
$59$	41.6865i	0.706551i	0.935519	+	0.353275i	$0.114932\pi$
$59$	41.6865i	0.706551i	−0.935519	+	0.353275i	$0.885068\pi$
$60$	0	0
$61$	97.7791	1.60294	0.801468	−	0.598038i	$-0.204053\pi$
$61$	97.7791	1.60294	0.801468	+	0.598038i	$0.204053\pi$
$62$	45.8759i	0.739933i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 127.222i	− 1.95726i
$66$	0	0
$67$	−35.0235	−0.522738	−0.261369	−	0.965239i	$-0.584174\pi$
$67$	−35.0235	−0.522738	−0.261369	+	0.965239i	$0.584174\pi$
$68$	38.4322i	0.565179i
$69$	0	0
$70$	20.5097	0.292996
$71$	− 60.7708i	− 0.855926i	−0.903796	−	0.427963i	$-0.859231\pi$
$71$	− 60.7708i	− 0.855926i	0.903796	−	0.427963i	$-0.140769\pi$
$72$	0	0
$73$	33.2521	0.455508	0.227754	−	0.973719i	$-0.426862\pi$
$73$	33.2521	0.455508	0.227754	+	0.973719i	$0.426862\pi$
$74$	57.3884i	0.775519i
$75$	0	0
$76$	70.1881	0.923527
$77$	9.96811i	0.129456i
$78$	0	0
$79$	−139.994	−1.77208	−0.886040	−	0.463608i	$-0.846554\pi$
$79$	−139.994	−1.77208	−0.886040	+	0.463608i	$0.846554\pi$
$80$	21.8502i	0.273127i
$81$	0	0
$82$	−58.1401	−0.709025
$83$	67.4528i	0.812685i	0.913721	+	0.406342i	$0.133196\pi$
$83$	67.4528i	0.812685i	−0.913721	+	0.406342i	$0.866804\pi$
$84$	0	0
$85$	104.969	1.23493
$86$	− 80.6189i	− 0.937429i
$87$	0	0
$88$	−10.6196	−0.120677
$89$	45.1603i	0.507419i	0.967280	+	0.253710i	$0.0816508\pi$
$89$	45.1603i	0.507419i	−0.967280	+	0.253710i	$0.918349\pi$
$90$	0	0
$91$	−61.8324	−0.679477
$92$	− 9.59166i	− 0.104257i
$93$	0	0
$94$	83.2694	0.885845
$95$	− 191.703i	− 2.01793i
$96$	0	0
$97$	12.2522	0.126311	0.0631557	−	0.998004i	$-0.479884\pi$
$97$	12.2522	0.126311	0.0631557	+	0.998004i	$0.479884\pi$
$98$	59.3284i	0.605391i
$99$	0	0
$100$	9.67890	0.0967890
$101$	25.4147i	0.251630i	0.992054	+	0.125815i	$0.0401546\pi$
$101$	25.4147i	0.251630i	−0.992054	+	0.125815i	$0.959845\pi$
$102$	0	0
$103$	−149.727	−1.45366	−0.726828	−	0.686820i	$-0.759006\pi$
$103$	−149.727	−1.45366	−0.726828	+	0.686820i	$0.759006\pi$
$104$	− 65.8737i	− 0.633401i
$105$	0	0
$106$	45.7026	0.431157
$107$	− 103.893i	− 0.970958i	−0.874248	−	0.485479i	$-0.838645\pi$
$107$	− 103.893i	− 0.970958i	0.874248	−	0.485479i	$-0.161355\pi$
$108$	0	0
$109$	24.2974	0.222912	0.111456	−	0.993769i	$-0.464449\pi$
$109$	24.2974	0.222912	0.111456	+	0.993769i	$0.464449\pi$
$110$	29.0051i	0.263683i
$111$	0	0
$112$	10.6196	0.0948180
$113$	− 136.200i	− 1.20531i	−0.798001	−	0.602656i	$-0.794109\pi$
$113$	− 136.200i	− 1.20531i	0.798001	−	0.602656i	$-0.205891\pi$
$114$	0	0
$115$	−26.1975	−0.227804
$116$	− 90.5523i	− 0.780624i
$117$	0	0
$118$	58.9536	0.499607
$119$	− 51.0169i	− 0.428713i
$120$	0	0
$121$	106.903	0.883496
$122$	− 138.281i	− 1.13345i
$123$	0	0
$124$	64.8783	0.523212
$125$	110.128i	0.881024i
$126$	0	0
$127$	137.362	1.08159	0.540793	−	0.841155i	$-0.318124\pi$
$127$	137.362	1.08159	0.540793	+	0.841155i	$0.318124\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−179.919	−1.38399
$131$	43.8314i	0.334591i	0.985907	+	0.167295i	$0.0535034\pi$
$131$	43.8314i	0.334591i	−0.985907	+	0.167295i	$0.946497\pi$
$132$	0	0
$133$	−93.1713	−0.700536
$134$	49.5307i	0.369632i
$135$	0	0
$136$	54.3513	0.399642
$137$	129.400i	0.944524i	0.881458	+	0.472262i	$0.156562\pi$
$137$	129.400i	0.944524i	−0.881458	+	0.472262i	$0.843438\pi$
$138$	0	0
$139$	−19.3050	−0.138885	−0.0694426	−	0.997586i	$-0.522122\pi$
$139$	−19.3050	−0.138885	−0.0694426	+	0.997586i	$0.522122\pi$
$140$	− 29.0051i	− 0.207179i
$141$	0	0
$142$	−85.9428	−0.605231
$143$	− 87.4442i	− 0.611498i
$144$	0	0
$145$	−247.323	−1.70568
$146$	− 47.0256i	− 0.322093i
$147$	0	0
$148$	81.1595	0.548375
$149$	49.5264i	0.332392i	0.986093	+	0.166196i	$0.0531484\pi$
$149$	49.5264i	0.332392i	−0.986093	+	0.166196i	$0.946852\pi$
$150$	0	0
$151$	52.3158	0.346462	0.173231	−	0.984881i	$-0.444579\pi$
$151$	52.3158	0.346462	0.173231	+	0.984881i	$0.444579\pi$
$152$	− 99.2609i	− 0.653032i
$153$	0	0
$154$	14.0970	0.0915392
$155$	− 177.200i	− 1.14323i
$156$	0	0
$157$	109.378	0.696677	0.348338	−	0.937369i	$-0.386746\pi$
$157$	109.378	0.696677	0.348338	+	0.937369i	$0.386746\pi$
$158$	197.982i	1.25305i
$159$	0	0
$160$	30.9008	0.193130
$161$	12.7325i	0.0790837i
$162$	0	0
$163$	175.252	1.07516	0.537582	−	0.843212i	$-0.319338\pi$
$163$	175.252	1.07516	0.537582	+	0.843212i	$0.319338\pi$
$164$	82.2225i	0.501357i
$165$	0	0
$166$	95.3927	0.574655
$167$	− 188.781i	− 1.13042i	−0.824946	−	0.565212i	$-0.808794\pi$
$167$	− 188.781i	− 1.13042i	0.824946	−	0.565212i	$-0.191206\pi$
$168$	0	0
$169$	373.418	2.20957
$170$	− 148.448i	− 0.873225i
$171$	0	0
$172$	−114.012	−0.662863
$173$	− 239.909i	− 1.38676i	−0.720574	−	0.693378i	$-0.756122\pi$
$173$	− 239.909i	− 1.38676i	0.720574	−	0.693378i	$-0.243878\pi$
$174$	0	0
$175$	−12.8483	−0.0734187
$176$	15.0184i	0.0853319i
$177$	0	0
$178$	63.8664	0.358800
$179$	− 233.807i	− 1.30618i	−0.757279	−	0.653092i	$-0.773471\pi$
$179$	− 233.807i	− 1.30618i	0.757279	−	0.653092i	$-0.226529\pi$
$180$	0	0
$181$	−73.0872	−0.403797	−0.201898	−	0.979406i	$-0.564711\pi$
$181$	−73.0872	−0.403797	−0.201898	+	0.979406i	$0.564711\pi$
$182$	87.4442i	0.480463i
$183$	0	0
$184$	−13.5647	−0.0737210
$185$	− 221.669i	− 1.19821i
$186$	0	0
$187$	72.1487	0.385822
$188$	− 117.761i	− 0.626387i
$189$	0	0
$190$	−271.109	−1.42689
$191$	104.577i	0.547526i	0.961797	+	0.273763i	$0.0882684\pi$
$191$	104.577i	0.547526i	−0.961797	+	0.273763i	$0.911732\pi$
$192$	0	0
$193$	−90.8156	−0.470547	−0.235274	−	0.971929i	$-0.575599\pi$
$193$	−90.8156	−0.470547	−0.235274	+	0.971929i	$0.575599\pi$
$194$	− 17.3272i	− 0.0893156i
$195$	0	0
$196$	83.9030	0.428076
$197$	285.721i	1.45036i	0.688558	+	0.725181i	$0.258244\pi$
$197$	285.721i	1.45036i	−0.688558	+	0.725181i	$0.741756\pi$
$198$	0	0
$199$	−158.601	−0.796988	−0.398494	−	0.917171i	$-0.630467\pi$
$199$	−158.601	−0.796988	−0.398494	+	0.917171i	$0.630467\pi$
$200$	− 13.6880i	− 0.0684401i
$201$	0	0
$202$	35.9418	0.177930
$203$	120.204i	0.592137i
$204$	0	0
$205$	224.572	1.09547
$206$	211.745i	1.02789i
$207$	0	0
$208$	−93.1595	−0.447882
$209$	− 131.764i	− 0.630450i
$210$	0	0
$211$	−140.519	−0.665966	−0.332983	−	0.942933i	$-0.608055\pi$
$211$	−140.519	−0.665966	−0.332983	+	0.942933i	$0.608055\pi$
$212$	− 64.6333i	− 0.304874i
$213$	0	0
$214$	−146.926	−0.686571
$215$	311.399i	1.44837i
$216$	0	0
$217$	−86.1228	−0.396879
$218$	− 34.3617i	− 0.157622i
$219$	0	0
$220$	41.0194	0.186452
$221$	447.540i	2.02507i
$222$	0	0
$223$	45.7694	0.205244	0.102622	−	0.994720i	$-0.467277\pi$
$223$	45.7694	0.205244	0.102622	+	0.994720i	$0.467277\pi$
$224$	− 15.0184i	− 0.0670465i
$225$	0	0
$226$	−192.616	−0.852285
$227$	13.9578i	0.0614879i	0.999527	+	0.0307440i	$0.00978765\pi$
$227$	13.9578i	0.0614879i	−0.999527	+	0.0307440i	$0.990212\pi$
$228$	0	0
$229$	−29.5010	−0.128825	−0.0644126	−	0.997923i	$-0.520517\pi$
$229$	−29.5010	−0.128825	−0.0644126	+	0.997923i	$0.520517\pi$
$230$	37.0488i	0.161082i
$231$	0	0
$232$	−128.060	−0.551984
$233$	350.404i	1.50388i	0.659233	+	0.751939i	$0.270881\pi$
$233$	350.404i	1.50388i	−0.659233	+	0.751939i	$0.729119\pi$
$234$	0	0
$235$	−321.637	−1.36867
$236$	− 83.3730i	− 0.353275i
$237$	0	0
$238$	−72.1487	−0.303146
$239$	− 27.0749i	− 0.113284i	−0.998395	−	0.0566421i	$-0.981961\pi$
$239$	− 27.0749i	− 0.113284i	0.998395	−	0.0566421i	$-0.0180394\pi$
$240$	0	0
$241$	370.823	1.53869	0.769343	−	0.638836i	$-0.220584\pi$
$241$	370.823	1.53869	0.769343	+	0.638836i	$0.220584\pi$
$242$	− 151.184i	− 0.624726i
$243$	0	0
$244$	−195.558	−0.801468
$245$	− 229.162i	− 0.935355i
$246$	0	0
$247$	817.336	3.30905
$248$	− 91.7517i	− 0.369967i
$249$	0	0
$250$	155.745	0.622978
$251$	− 183.177i	− 0.729789i	−0.931049	−	0.364895i	$-0.881105\pi$
$251$	− 183.177i	− 0.729789i	0.931049	−	0.364895i	$-0.118895\pi$
$252$	0	0
$253$	−18.0064	−0.0711717
$254$	− 194.259i	− 0.764797i
$255$	0	0
$256$	16.0000	0.0625000
$257$	179.627i	0.698936i	0.936948	+	0.349468i	$0.113638\pi$
$257$	179.627i	0.698936i	−0.936948	+	0.349468i	$0.886362\pi$
$258$	0	0
$259$	−107.735	−0.415967
$260$	254.444i	0.978631i
$261$	0	0
$262$	61.9870	0.236591
$263$	372.423i	1.41606i	0.706183	+	0.708029i	$0.250415\pi$
$263$	372.423i	1.41606i	−0.706183	+	0.708029i	$0.749585\pi$
$264$	0	0
$265$	−176.531	−0.666156
$266$	131.764i	0.495354i
$267$	0	0
$268$	70.0469	0.261369
$269$	436.192i	1.62153i	0.585371	+	0.810766i	$0.300949\pi$
$269$	436.192i	1.62153i	−0.585371	+	0.810766i	$0.699051\pi$
$270$	0	0
$271$	−361.492	−1.33392	−0.666959	−	0.745094i	$-0.732405\pi$
$271$	−361.492	−1.33392	−0.666959	+	0.745094i	$0.732405\pi$
$272$	− 76.8643i	− 0.282589i
$273$	0	0
$274$	182.999	0.667879
$275$	− 18.1702i	− 0.0660735i
$276$	0	0
$277$	218.004	0.787019	0.393510	−	0.919320i	$-0.371261\pi$
$277$	218.004	0.787019	0.393510	+	0.919320i	$0.371261\pi$
$278$	27.3014i	0.0982066i
$279$	0	0
$280$	−41.0194	−0.146498
$281$	195.789i	0.696756i	0.937354	+	0.348378i	$0.113268\pi$
$281$	195.789i	0.696756i	−0.937354	+	0.348378i	$0.886732\pi$
$282$	0	0
$283$	−36.7055	−0.129702	−0.0648508	−	0.997895i	$-0.520657\pi$
$283$	−36.7055	−0.129702	−0.0648508	+	0.997895i	$0.520657\pi$
$284$	121.542i	0.427963i
$285$	0	0
$286$	−123.665	−0.432394
$287$	− 109.146i	− 0.380301i
$288$	0	0
$289$	−80.2576	−0.277708
$290$	349.768i	1.20610i
$291$	0	0
$292$	−66.5042	−0.227754
$293$	474.261i	1.61864i	0.587370	+	0.809319i	$0.300164\pi$
$293$	474.261i	1.61864i	−0.587370	+	0.809319i	$0.699836\pi$
$294$	0	0
$295$	−227.715	−0.771914
$296$	− 114.777i	− 0.387760i
$297$	0	0
$298$	70.0408	0.235036
$299$	− 111.694i	− 0.373560i
$300$	0	0
$301$	151.346	0.502811
$302$	− 73.9857i	− 0.244986i
$303$	0	0
$304$	−140.376	−0.461764
$305$	534.123i	1.75122i
$306$	0	0
$307$	−70.4059	−0.229335	−0.114668	−	0.993404i	$-0.536580\pi$
$307$	−70.4059	−0.229335	−0.114668	+	0.993404i	$0.536580\pi$
$308$	− 19.9362i	− 0.0647280i
$309$	0	0
$310$	−250.599	−0.808384
$311$	− 375.724i	− 1.20812i	−0.796941	−	0.604058i	$-0.793550\pi$
$311$	− 375.724i	− 1.20812i	0.796941	−	0.604058i	$-0.206450\pi$
$312$	0	0
$313$	317.485	1.01433	0.507164	−	0.861849i	$-0.330694\pi$
$313$	317.485	1.01433	0.507164	+	0.861849i	$0.330694\pi$
$314$	− 154.684i	− 0.492625i
$315$	0	0
$316$	279.989	0.886040
$317$	228.905i	0.722097i	0.932547	+	0.361048i	$0.117581\pi$
$317$	228.905i	0.722097i	−0.932547	+	0.361048i	$0.882419\pi$
$318$	0	0
$319$	−169.994	−0.532897
$320$	− 43.7004i	− 0.136564i
$321$	0	0
$322$	18.0064	0.0559206
$323$	674.370i	2.08783i
$324$	0	0
$325$	112.710	0.346800
$326$	− 247.843i	− 0.760255i
$327$	0	0
$328$	116.280	0.354513
$329$	156.322i	0.475142i
$330$	0	0
$331$	−643.371	−1.94372	−0.971860	−	0.235560i	$-0.924308\pi$
$331$	−643.371	−1.94372	−0.971860	+	0.235560i	$0.924308\pi$
$332$	− 134.906i	− 0.406342i
$333$	0	0
$334$	−266.976	−0.799330
$335$	− 191.317i	− 0.571097i
$336$	0	0
$337$	217.354	0.644966	0.322483	−	0.946575i	$-0.395482\pi$
$337$	217.354	0.644966	0.322483	+	0.946575i	$0.395482\pi$
$338$	− 528.093i	− 1.56241i
$339$	0	0
$340$	−209.938	−0.617463
$341$	− 121.796i	− 0.357173i
$342$	0	0
$343$	−241.468	−0.703987
$344$	161.238i	0.468715i
$345$	0	0
$346$	−339.282	−0.980584
$347$	− 376.022i	− 1.08364i	−0.840496	−	0.541818i	$-0.817736\pi$
$347$	− 376.022i	− 1.08364i	0.840496	−	0.541818i	$-0.182264\pi$
$348$	0	0
$349$	−431.406	−1.23612	−0.618061	−	0.786130i	$-0.712081\pi$
$349$	−431.406	−1.23612	−0.618061	+	0.786130i	$0.712081\pi$
$350$	18.1702i	0.0519149i
$351$	0	0
$352$	21.2392	0.0603387
$353$	− 335.644i	− 0.950834i	−0.879761	−	0.475417i	$-0.842297\pi$
$353$	− 335.644i	− 0.950834i	0.879761	−	0.475417i	$-0.157703\pi$
$354$	0	0
$355$	331.963	0.935108
$356$	− 90.3207i	− 0.253710i
$357$	0	0
$358$	−330.653	−0.923612
$359$	− 250.032i	− 0.696468i	−0.937408	−	0.348234i	$-0.886781\pi$
$359$	− 250.032i	− 0.696468i	0.937408	−	0.348234i	$-0.113219\pi$
$360$	0	0
$361$	870.591	2.41161
$362$	103.361i	0.285528i
$363$	0	0
$364$	123.665	0.339738
$365$	181.641i	0.497648i
$366$	0	0
$367$	−460.168	−1.25386	−0.626932	−	0.779074i	$-0.715690\pi$
$367$	−460.168	−1.25386	−0.626932	+	0.779074i	$0.715690\pi$
$368$	19.1833i	0.0521286i
$369$	0	0
$370$	−313.487	−0.847262
$371$	85.7976i	0.231260i
$372$	0	0
$373$	−162.929	−0.436806	−0.218403	−	0.975859i	$-0.570085\pi$
$373$	−162.929	−0.436806	−0.218403	+	0.975859i	$0.570085\pi$
$374$	− 102.034i	− 0.272817i
$375$	0	0
$376$	−166.539	−0.442922
$377$	− 1054.48i	− 2.79702i
$378$	0	0
$379$	−369.874	−0.975920	−0.487960	−	0.872866i	$-0.662259\pi$
$379$	−369.874	−0.975920	−0.487960	+	0.872866i	$0.662259\pi$
$380$	383.406i	1.00896i
$381$	0	0
$382$	147.895	0.387159
$383$	579.820i	1.51389i	0.653479	+	0.756945i	$0.273309\pi$
$383$	579.820i	1.51389i	−0.653479	+	0.756945i	$0.726691\pi$
$384$	0	0
$385$	−54.4513	−0.141432
$386$	128.433i	0.332727i
$387$	0	0
$388$	−24.5044	−0.0631557
$389$	− 415.995i	− 1.06939i	−0.845044	−	0.534697i	$-0.820426\pi$
$389$	− 415.995i	− 1.06939i	0.845044	−	0.534697i	$-0.179574\pi$
$390$	0	0
$391$	92.1571	0.235696
$392$	− 118.657i	− 0.302696i
$393$	0	0
$394$	404.071	1.02556
$395$	− 764.726i	− 1.93602i
$396$	0	0
$397$	−144.905	−0.365000	−0.182500	−	0.983206i	$-0.558419\pi$
$397$	−144.905	−0.365000	−0.182500	+	0.983206i	$0.558419\pi$
$398$	224.295i	0.563555i
$399$	0	0
$400$	−19.3578	−0.0483945
$401$	− 181.675i	− 0.453056i	−0.974005	−	0.226528i	$-0.927263\pi$
$401$	− 181.675i	− 0.453056i	0.974005	−	0.226528i	$-0.0727375\pi$
$402$	0	0
$403$	755.503	1.87470
$404$	− 50.8294i	− 0.125815i
$405$	0	0
$406$	169.994	0.418704
$407$	− 152.361i	− 0.374351i
$408$	0	0
$409$	−30.3610	−0.0742322	−0.0371161	−	0.999311i	$-0.511817\pi$
$409$	−30.3610	−0.0742322	−0.0371161	+	0.999311i	$0.511817\pi$
$410$	− 317.593i	− 0.774617i
$411$	0	0
$412$	299.453	0.726828
$413$	110.674i	0.267975i
$414$	0	0
$415$	−368.465	−0.887866
$416$	131.747i	0.316701i
$417$	0	0
$418$	−186.343	−0.445796
$419$	653.899i	1.56062i	0.625394	+	0.780309i	$0.284938\pi$
$419$	653.899i	1.56062i	−0.625394	+	0.780309i	$0.715062\pi$
$420$	0	0
$421$	516.420	1.22665	0.613326	−	0.789830i	$-0.289831\pi$
$421$	516.420	1.22665	0.613326	+	0.789830i	$0.289831\pi$
$422$	198.724i	0.470909i
$423$	0	0
$424$	−91.4052	−0.215578
$425$	92.9952i	0.218812i
$426$	0	0
$427$	259.594	0.607949
$428$	207.785i	0.485479i
$429$	0	0
$430$	440.385	1.02415
$431$	− 59.2387i	− 0.137445i	−0.997636	−	0.0687224i	$-0.978108\pi$
$431$	− 59.2387i	− 0.137445i	0.997636	−	0.0687224i	$-0.0218923\pi$
$432$	0	0
$433$	−785.285	−1.81359	−0.906796	−	0.421569i	$-0.861479\pi$
$433$	−785.285	−1.81359	−0.906796	+	0.421569i	$0.861479\pi$
$434$	121.796i	0.280636i
$435$	0	0
$436$	−48.5948	−0.111456
$437$	− 168.305i	− 0.385137i
$438$	0	0
$439$	322.100	0.733713	0.366857	−	0.930278i	$-0.380434\pi$
$439$	322.100	0.733713	0.366857	+	0.930278i	$0.380434\pi$
$440$	− 58.0102i	− 0.131841i
$441$	0	0
$442$	632.917	1.43194
$443$	− 112.614i	− 0.254207i	−0.991889	−	0.127103i	$-0.959432\pi$
$443$	− 112.614i	− 0.254207i	0.991889	−	0.127103i	$-0.0405680\pi$
$444$	0	0
$445$	−246.691	−0.554361
$446$	− 64.7277i	− 0.145129i
$447$	0	0
$448$	−21.2392	−0.0474090
$449$	32.4842i	0.0723480i	0.999346	+	0.0361740i	$0.0115170\pi$
$449$	32.4842i	0.0723480i	−0.999346	+	0.0361740i	$0.988483\pi$
$450$	0	0
$451$	154.356	0.342254
$452$	272.401i	0.602656i
$453$	0	0
$454$	19.7392	0.0434785
$455$	− 337.762i	− 0.742335i
$456$	0	0
$457$	−63.4223	−0.138780	−0.0693898	−	0.997590i	$-0.522105\pi$
$457$	−63.4223	−0.138780	−0.0693898	+	0.997590i	$0.522105\pi$
$458$	41.7207i	0.0910932i
$459$	0	0
$460$	52.3949	0.113902
$461$	− 768.616i	− 1.66728i	−0.552308	−	0.833640i	$-0.686253\pi$
$461$	− 768.616i	− 1.66728i	0.552308	−	0.833640i	$-0.313747\pi$
$462$	0	0
$463$	−453.446	−0.979365	−0.489683	−	0.871901i	$-0.662887\pi$
$463$	−453.446	−0.979365	−0.489683	+	0.871901i	$0.662887\pi$
$464$	181.105i	0.390312i
$465$	0	0
$466$	495.545	1.06340
$467$	− 65.6087i	− 0.140490i	−0.997530	−	0.0702449i	$-0.977622\pi$
$467$	− 65.6087i	− 0.140490i	0.997530	−	0.0702449i	$-0.0223781\pi$
$468$	0	0
$469$	−92.9840	−0.198260
$470$	454.863i	0.967794i
$471$	0	0
$472$	−117.907	−0.249803
$473$	214.036i	0.452506i
$474$	0	0
$475$	169.836	0.357549
$476$	102.034i	0.214357i
$477$	0	0
$478$	−38.2897	−0.0801040
$479$	− 157.516i	− 0.328843i	−0.986390	−	0.164421i	$-0.947424\pi$
$479$	− 157.516i	− 0.328843i	0.986390	−	0.164421i	$-0.0525757\pi$
$480$	0	0
$481$	945.097	1.96486
$482$	− 524.423i	− 1.08802i
$483$	0	0
$484$	−213.806	−0.441748
$485$	66.9283i	0.137996i
$486$	0	0
$487$	196.013	0.402491	0.201245	−	0.979541i	$-0.435501\pi$
$487$	196.013	0.402491	0.201245	+	0.979541i	$0.435501\pi$
$488$	276.561i	0.566724i
$489$	0	0
$490$	−324.084	−0.661396
$491$	897.356i	1.82761i	0.406155	+	0.913804i	$0.366869\pi$
$491$	897.356i	1.82761i	−0.406155	+	0.913804i	$0.633131\pi$
$492$	0	0
$493$	870.030	1.76477
$494$	− 1155.89i	− 2.33985i
$495$	0	0
$496$	−129.757	−0.261606
$497$	− 161.341i	− 0.324629i
$498$	0	0
$499$	521.387	1.04486	0.522432	−	0.852681i	$-0.325025\pi$
$499$	521.387	1.04486	0.522432	+	0.852681i	$0.325025\pi$
$500$	− 220.256i	− 0.440512i
$501$	0	0
$502$	−259.052	−0.516039
$503$	830.919i	1.65193i	0.563724	+	0.825963i	$0.309368\pi$
$503$	830.919i	1.65193i	−0.563724	+	0.825963i	$0.690632\pi$
$504$	0	0
$505$	−138.829	−0.274909
$506$	25.4650i	0.0503260i
$507$	0	0
$508$	−274.723	−0.540793
$509$	− 1001.93i	− 1.96843i	−0.176986	−	0.984213i	$-0.556635\pi$
$509$	− 1001.93i	− 1.96843i	0.176986	−	0.984213i	$-0.443365\pi$
$510$	0	0
$511$	88.2812	0.172762
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	254.030	0.494222
$515$	− 817.889i	− 1.58813i
$516$	0	0
$517$	−221.072	−0.427606
$518$	152.361i	0.294133i
$519$	0	0
$520$	359.838	0.691997
$521$	− 726.690i	− 1.39480i	−0.716683	−	0.697399i	$-0.754341\pi$
$521$	− 726.690i	− 1.39480i	0.716683	−	0.697399i	$-0.245659\pi$
$522$	0	0
$523$	−394.541	−0.754380	−0.377190	−	0.926136i	$-0.623110\pi$
$523$	−394.541	−0.754380	−0.377190	+	0.926136i	$0.623110\pi$
$524$	− 87.6628i	− 0.167295i
$525$	0	0
$526$	526.686	1.00130
$527$	623.353i	1.18283i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	249.653i	0.471043i
$531$	0	0
$532$	186.343	0.350268
$533$	957.476i	1.79639i
$534$	0	0
$535$	567.518	1.06078
$536$	− 99.0613i	− 0.184816i
$537$	0	0
$538$	616.869	1.14660
$539$	− 157.511i	− 0.292228i
$540$	0	0
$541$	116.598	0.215524	0.107762	−	0.994177i	$-0.465632\pi$
$541$	116.598	0.215524	0.107762	+	0.994177i	$0.465632\pi$
$542$	511.227i	0.943223i
$543$	0	0
$544$	−108.703	−0.199821
$545$	132.726i	0.243533i
$546$	0	0
$547$	−344.108	−0.629082	−0.314541	−	0.949244i	$-0.601851\pi$
$547$	−344.108	−0.629082	−0.314541	+	0.949244i	$0.601851\pi$
$548$	− 258.799i	− 0.472262i
$549$	0	0
$550$	−25.6965	−0.0467210
$551$	− 1588.92i	− 2.88371i
$552$	0	0
$553$	−371.672	−0.672101
$554$	− 308.305i	− 0.556507i
$555$	0	0
$556$	38.6101	0.0694426
$557$	401.523i	0.720867i	0.932785	+	0.360434i	$0.117371\pi$
$557$	401.523i	0.720867i	−0.932785	+	0.360434i	$0.882629\pi$
$558$	0	0
$559$	−1327.67	−2.37507
$560$	58.0102i	0.103590i
$561$	0	0
$562$	276.887	0.492681
$563$	− 605.143i	− 1.07485i	−0.843310	−	0.537427i	$-0.819396\pi$
$563$	− 605.143i	− 1.07485i	0.843310	−	0.537427i	$-0.180604\pi$
$564$	0	0
$565$	744.001	1.31682
$566$	51.9095i	0.0917129i
$567$	0	0
$568$	171.886	0.302616
$569$	129.848i	0.228204i	0.993469	+	0.114102i	$0.0363991\pi$
$569$	129.848i	0.228204i	−0.993469	+	0.114102i	$0.963601\pi$
$570$	0	0
$571$	−540.155	−0.945981	−0.472991	−	0.881067i	$-0.656826\pi$
$571$	−540.155	−0.945981	−0.472991	+	0.881067i	$0.656826\pi$
$572$	174.888i	0.305749i
$573$	0	0
$574$	−154.356	−0.268914
$575$	− 23.2092i	− 0.0403638i
$576$	0	0
$577$	−767.951	−1.33094	−0.665469	−	0.746425i	$-0.731769\pi$
$577$	−767.951	−1.33094	−0.665469	+	0.746425i	$0.731769\pi$
$578$	113.501i	0.196369i
$579$	0	0
$580$	494.647	0.852839
$581$	179.081i	0.308229i
$582$	0	0
$583$	−121.336	−0.208124
$584$	94.0512i	0.161047i
$585$	0	0
$586$	670.706	1.14455
$587$	361.967i	0.616639i	0.951283	+	0.308319i	$0.0997665\pi$
$587$	361.967i	0.616639i	−0.951283	+	0.308319i	$0.900233\pi$
$588$	0	0
$589$	1138.42	1.93280
$590$	322.037i	0.545825i
$591$	0	0
$592$	−162.319	−0.274187
$593$	− 685.417i	− 1.15585i	−0.816091	−	0.577923i	$-0.803863\pi$
$593$	− 685.417i	− 1.15585i	0.816091	−	0.577923i	$-0.196137\pi$
$594$	0	0
$595$	278.682	0.468373
$596$	− 99.0527i	− 0.166196i
$597$	0	0
$598$	−157.960	−0.264146
$599$	− 888.738i	− 1.48370i	−0.670564	−	0.741851i	$-0.733948\pi$
$599$	− 888.738i	− 1.48370i	0.670564	−	0.741851i	$-0.266052\pi$
$600$	0	0
$601$	−911.130	−1.51602	−0.758012	−	0.652241i	$-0.773829\pi$
$601$	−911.130	−1.51602	−0.758012	+	0.652241i	$0.773829\pi$
$602$	− 214.036i	− 0.355541i
$603$	0	0
$604$	−104.632	−0.173231
$605$	583.963i	0.965228i
$606$	0	0
$607$	−331.088	−0.545450	−0.272725	−	0.962092i	$-0.587925\pi$
$607$	−331.088	−0.545450	−0.272725	+	0.962092i	$0.587925\pi$
$608$	198.522i	0.326516i
$609$	0	0
$610$	755.364	1.23830
$611$	− 1371.32i	− 2.24438i
$612$	0	0
$613$	516.088	0.841906	0.420953	−	0.907083i	$-0.361696\pi$
$613$	516.088	0.841906	0.420953	+	0.907083i	$0.361696\pi$
$614$	99.5690i	0.162165i
$615$	0	0
$616$	−28.1941	−0.0457696
$617$	53.3362i	0.0864444i	0.999065	+	0.0432222i	$0.0137623\pi$
$617$	53.3362i	0.0864444i	−0.999065	+	0.0432222i	$0.986238\pi$
$618$	0	0
$619$	−568.724	−0.918778	−0.459389	−	0.888235i	$-0.651932\pi$
$619$	−568.724	−0.918778	−0.459389	+	0.888235i	$0.651932\pi$
$620$	354.401i	0.571614i
$621$	0	0
$622$	−531.354	−0.854266
$623$	119.896i	0.192450i
$624$	0	0
$625$	−722.566	−1.15611
$626$	− 448.991i	− 0.717239i
$627$	0	0
$628$	−218.757	−0.348338
$629$	779.783i	1.23972i
$630$	0	0
$631$	65.0487	0.103088	0.0515441	−	0.998671i	$-0.483586\pi$
$631$	65.0487	0.103088	0.0515441	+	0.998671i	$0.483586\pi$
$632$	− 395.964i	− 0.626525i
$633$	0	0
$634$	323.720	0.510599
$635$	750.344i	1.18164i
$636$	0	0
$637$	977.045	1.53382
$638$	240.408i	0.376815i
$639$	0	0
$640$	−61.8017	−0.0965651
$641$	149.022i	0.232483i	0.993221	+	0.116242i	$0.0370847\pi$
$641$	149.022i	0.232483i	−0.993221	+	0.116242i	$0.962915\pi$
$642$	0	0
$643$	−500.221	−0.777948	−0.388974	−	0.921249i	$-0.627170\pi$
$643$	−500.221	−0.777948	−0.388974	+	0.921249i	$0.627170\pi$
$644$	− 25.4650i	− 0.0395419i
$645$	0	0
$646$	953.703	1.47632
$647$	805.794i	1.24543i	0.782448	+	0.622716i	$0.213971\pi$
$647$	805.794i	1.24543i	−0.782448	+	0.622716i	$0.786029\pi$
$648$	0	0
$649$	−156.516	−0.241165
$650$	− 159.396i	− 0.245225i
$651$	0	0
$652$	−350.503	−0.537582
$653$	601.044i	0.920435i	0.887806	+	0.460217i	$0.152229\pi$
$653$	601.044i	0.920435i	−0.887806	+	0.460217i	$0.847771\pi$
$654$	0	0
$655$	−239.431	−0.365544
$656$	− 164.445i	− 0.250678i
$657$	0	0
$658$	221.072	0.335976
$659$	607.968i	0.922562i	0.887254	+	0.461281i	$0.152610\pi$
$659$	607.968i	0.922562i	−0.887254	+	0.461281i	$0.847390\pi$
$660$	0	0
$661$	290.913	0.440110	0.220055	−	0.975487i	$-0.429376\pi$
$661$	290.913	0.440110	0.220055	+	0.975487i	$0.429376\pi$
$662$	909.864i	1.37442i
$663$	0	0
$664$	−190.785	−0.287328
$665$	− 508.953i	− 0.765343i
$666$	0	0
$667$	−217.137	−0.325543
$668$	377.562i	0.565212i
$669$	0	0
$670$	−270.564	−0.403826
$671$	367.122i	0.547126i
$672$	0	0
$673$	−481.374	−0.715267	−0.357633	−	0.933862i	$-0.616416\pi$
$673$	−481.374	−0.715267	−0.357633	+	0.933862i	$0.616416\pi$
$674$	− 307.384i	− 0.456060i
$675$	0	0
$676$	−746.836	−1.10479
$677$	870.332i	1.28557i	0.766046	+	0.642786i	$0.222222\pi$
$677$	870.332i	1.28557i	−0.766046	+	0.642786i	$0.777778\pi$
$678$	0	0
$679$	32.5284	0.0479064
$680$	296.897i	0.436613i
$681$	0	0
$682$	−172.246	−0.252560
$683$	102.178i	0.149602i	0.997198	+	0.0748008i	$0.0238321\pi$
$683$	102.178i	0.149602i	−0.997198	+	0.0748008i	$0.976168\pi$
$684$	0	0
$685$	−706.852	−1.03190
$686$	341.487i	0.497794i
$687$	0	0
$688$	228.025	0.331431
$689$	− 752.650i	− 1.09238i
$690$	0	0
$691$	−1291.14	−1.86851	−0.934256	−	0.356603i	$-0.883935\pi$
$691$	−1291.14	−1.86851	−0.934256	+	0.356603i	$0.883935\pi$
$692$	479.817i	0.693378i
$693$	0	0
$694$	−531.775	−0.766246
$695$	− 105.455i	− 0.151733i
$696$	0	0
$697$	−789.997	−1.13342
$698$	610.101i	0.874070i
$699$	0	0
$700$	25.6965	0.0367094
$701$	− 412.010i	− 0.587747i	−0.955844	−	0.293873i	$-0.905056\pi$
$701$	− 412.010i	− 0.587747i	0.955844	−	0.293873i	$-0.0949444\pi$
$702$	0	0
$703$	1424.11	2.02576
$704$	− 30.0368i	− 0.0426659i
$705$	0	0
$706$	−474.673	−0.672341
$707$	67.4735i	0.0954364i
$708$	0	0
$709$	−937.381	−1.32212	−0.661059	−	0.750334i	$-0.729893\pi$
$709$	−937.381	−1.32212	−0.661059	+	0.750334i	$0.729893\pi$
$710$	− 469.467i	− 0.661221i
$711$	0	0
$712$	−127.733	−0.179400
$713$	− 155.573i	− 0.218194i
$714$	0	0
$715$	477.668	0.668068
$716$	467.614i	0.653092i
$717$	0	0
$718$	−353.599	−0.492477
$719$	352.526i	0.490300i	0.969485	+	0.245150i	$0.0788372\pi$
$719$	352.526i	0.490300i	−0.969485	+	0.245150i	$0.921163\pi$
$720$	0	0
$721$	−397.510	−0.551331
$722$	− 1231.20i	− 1.70527i
$723$	0	0
$724$	146.174	0.201898
$725$	− 219.112i	− 0.302223i
$726$	0	0
$727$	665.087	0.914838	0.457419	−	0.889251i	$-0.348774\pi$
$727$	665.087	0.914838	0.457419	+	0.889251i	$0.348774\pi$
$728$	− 174.888i	− 0.240231i
$729$	0	0
$730$	256.880	0.351890
$731$	− 1095.43i	− 1.49854i
$732$	0	0
$733$	−821.098	−1.12019	−0.560094	−	0.828429i	$-0.689235\pi$
$733$	−821.098	−1.12019	−0.560094	+	0.828429i	$0.689235\pi$
$734$	650.776i	0.886616i
$735$	0	0
$736$	27.1293	0.0368605
$737$	− 131.499i	− 0.178425i
$738$	0	0
$739$	1282.72	1.73574	0.867872	−	0.496787i	$-0.165487\pi$
$739$	1282.72	1.73574	0.867872	+	0.496787i	$0.165487\pi$
$740$	443.338i	0.599105i
$741$	0	0
$742$	121.336	0.163526
$743$	− 45.8919i	− 0.0617657i	−0.999523	−	0.0308828i	$-0.990168\pi$
$743$	− 45.8919i	− 0.0617657i	0.999523	−	0.0308828i	$-0.00983187\pi$
$744$	0	0
$745$	−270.540	−0.363141
$746$	230.416i	0.308869i
$747$	0	0
$748$	−144.297	−0.192911
$749$	− 275.825i	− 0.368257i
$750$	0	0
$751$	434.621	0.578724	0.289362	−	0.957220i	$-0.406557\pi$
$751$	434.621	0.578724	0.289362	+	0.957220i	$0.406557\pi$
$752$	235.521i	0.313193i
$753$	0	0
$754$	−1491.25	−1.97779
$755$	285.778i	0.378513i
$756$	0	0
$757$	517.650	0.683817	0.341909	−	0.939733i	$-0.388927\pi$
$757$	517.650	0.683817	0.341909	+	0.939733i	$0.388927\pi$
$758$	523.081i	0.690080i
$759$	0	0
$760$	542.218	0.713444
$761$	1191.22i	1.56534i	0.622437	+	0.782670i	$0.286143\pi$
$761$	1191.22i	1.56534i	−0.622437	+	0.782670i	$0.713857\pi$
$762$	0	0
$763$	64.5072	0.0845442
$764$	− 209.155i	− 0.273763i
$765$	0	0
$766$	819.989	1.07048
$767$	− 970.873i	− 1.26581i
$768$	0	0
$769$	295.214	0.383894	0.191947	−	0.981405i	$-0.438520\pi$
$769$	295.214	0.383894	0.191947	+	0.981405i	$0.438520\pi$
$770$	77.0058i	0.100007i
$771$	0	0
$772$	181.631	0.235274
$773$	582.034i	0.752955i	0.926426	+	0.376478i	$0.122865\pi$
$773$	582.034i	0.752955i	−0.926426	+	0.376478i	$0.877135\pi$
$774$	0	0
$775$	156.987	0.202564
$776$	34.6545i	0.0446578i
$777$	0	0
$778$	−588.305	−0.756176
$779$	1442.76i	1.85207i
$780$	0	0
$781$	228.170	0.292151
$782$	− 130.330i	− 0.166662i
$783$	0	0
$784$	−167.806	−0.214038
$785$	597.484i	0.761126i
$786$	0	0
$787$	−466.725	−0.593043	−0.296522	−	0.955026i	$-0.595827\pi$
$787$	−466.725	−0.593043	−0.296522	+	0.955026i	$0.595827\pi$
$788$	− 571.443i	− 0.725181i
$789$	0	0
$790$	−1081.49	−1.36897
$791$	− 361.599i	− 0.457141i
$792$	0	0
$793$	−2277.26	−2.87171
$794$	204.927i	0.258094i
$795$	0	0
$796$	317.201	0.398494
$797$	− 631.543i	− 0.792401i	−0.918164	−	0.396200i	$-0.870329\pi$
$797$	− 631.543i	− 0.792401i	0.918164	−	0.396200i	$-0.129671\pi$
$798$	0	0
$799$	1131.45	1.41608
$800$	27.3761i	0.0342201i
$801$	0	0
$802$	−256.928	−0.320359
$803$	124.848i	0.155478i
$804$	0	0
$805$	−69.5518	−0.0863997
$806$	− 1068.44i	− 1.32561i
$807$	0	0
$808$	−71.8836	−0.0889648
$809$	− 617.910i	− 0.763794i	−0.924205	−	0.381897i	$-0.875271\pi$
$809$	− 617.910i	− 0.763794i	0.924205	−	0.381897i	$-0.124729\pi$
$810$	0	0
$811$	−674.261	−0.831394	−0.415697	−	0.909503i	$-0.636462\pi$
$811$	−674.261	−0.831394	−0.415697	+	0.909503i	$0.636462\pi$
$812$	− 240.408i	− 0.296069i
$813$	0	0
$814$	−215.471	−0.264706
$815$	957.320i	1.17463i
$816$	0	0
$817$	−2000.58	−2.44869
$818$	42.9369i	0.0524901i
$819$	0	0
$820$	−449.144	−0.547737
$821$	− 1164.86i	− 1.41883i	−0.704790	−	0.709416i	$-0.748959\pi$
$821$	− 1164.86i	− 1.41883i	0.704790	−	0.709416i	$-0.251041\pi$
$822$	0	0
$823$	−1028.04	−1.24913	−0.624566	−	0.780972i	$-0.714724\pi$
$823$	−1028.04	−1.24913	−0.624566	+	0.780972i	$0.714724\pi$
$824$	− 423.491i	− 0.513945i
$825$	0	0
$826$	156.516	0.189487
$827$	789.195i	0.954287i	0.878825	+	0.477143i	$0.158328\pi$
$827$	789.195i	0.954287i	−0.878825	+	0.477143i	$0.841672\pi$
$828$	0	0
$829$	475.254	0.573285	0.286643	−	0.958038i	$-0.407461\pi$
$829$	475.254	0.573285	0.286643	+	0.958038i	$0.407461\pi$
$830$	521.088i	0.627816i
$831$	0	0
$832$	186.319	0.223941
$833$	806.143i	0.967759i
$834$	0	0
$835$	1031.22	1.23500
$836$	263.528i	0.315225i
$837$	0	0
$838$	924.753	1.10352
$839$	217.224i	0.258908i	0.991585	+	0.129454i	$0.0413224\pi$
$839$	217.224i	0.258908i	−0.991585	+	0.129454i	$0.958678\pi$
$840$	0	0
$841$	−1208.93	−1.43749
$842$	− 730.329i	− 0.867374i
$843$	0	0
$844$	281.038	0.332983
$845$	2039.82i	2.41398i
$846$	0	0
$847$	283.817	0.335085
$848$	129.267i	0.152437i
$849$	0	0
$850$	131.515	0.154724
$851$	− 194.614i	− 0.228688i
$852$	0	0
$853$	341.028	0.399798	0.199899	−	0.979816i	$-0.435939\pi$
$853$	341.028	0.399798	0.199899	+	0.979816i	$0.435939\pi$
$854$	− 367.122i	− 0.429885i
$855$	0	0
$856$	293.852	0.343286
$857$	− 897.544i	− 1.04731i	−0.851931	−	0.523655i	$-0.824568\pi$
$857$	− 897.544i	− 1.04731i	0.851931	−	0.523655i	$-0.175432\pi$
$858$	0	0
$859$	180.587	0.210230	0.105115	−	0.994460i	$-0.466479\pi$
$859$	180.587	0.210230	0.105115	+	0.994460i	$0.466479\pi$
$860$	− 622.798i	− 0.724184i
$861$	0	0
$862$	−83.7762	−0.0971882
$863$	1121.05i	1.29902i	0.760353	+	0.649510i	$0.225026\pi$
$863$	1121.05i	1.29902i	−0.760353	+	0.649510i	$0.774974\pi$
$864$	0	0
$865$	1310.51	1.51504
$866$	1110.56i	1.28240i
$867$	0	0
$868$	172.246	0.198440
$869$	− 525.623i	− 0.604860i
$870$	0	0
$871$	815.692	0.936501
$872$	68.7234i	0.0788112i
$873$	0	0
$874$	−238.019	−0.272333
$875$	292.379i	0.334148i
$876$	0	0
$877$	1613.95	1.84031	0.920156	−	0.391552i	$-0.128062\pi$
$877$	1613.95	1.84031	0.920156	+	0.391552i	$0.128062\pi$
$878$	− 455.518i	− 0.518813i
$879$	0	0
$880$	−82.0388	−0.0932259
$881$	− 26.7686i	− 0.0303843i	−0.999885	−	0.0151921i	$-0.995164\pi$
$881$	− 26.7686i	− 0.0303843i	0.999885	−	0.0151921i	$-0.00483600\pi$
$882$	0	0
$883$	−104.854	−0.118747	−0.0593737	−	0.998236i	$-0.518910\pi$
$883$	−104.854	−0.118747	−0.0593737	+	0.998236i	$0.518910\pi$
$884$	− 895.080i	− 1.01253i
$885$	0	0
$886$	−159.260	−0.179751
$887$	463.704i	0.522778i	0.965234	+	0.261389i	$0.0841806\pi$
$887$	463.704i	0.522778i	−0.965234	+	0.261389i	$0.915819\pi$
$888$	0	0
$889$	364.682	0.410216
$890$	348.873i	0.391992i
$891$	0	0
$892$	−91.5388	−0.102622
$893$	− 2066.35i	− 2.31394i
$894$	0	0
$895$	1277.18	1.42702
$896$	30.0368i	0.0335232i
$897$	0	0
$898$	45.9396	0.0511577
$899$	− 1468.72i	− 1.63373i
$900$	0	0
$901$	620.999	0.689233
$902$	− 218.293i	− 0.242010i
$903$	0	0
$904$	385.233	0.426142
$905$	− 399.243i	− 0.441152i
$906$	0	0
$907$	−93.5538	−0.103146	−0.0515732	−	0.998669i	$-0.516424\pi$
$907$	−93.5538	−0.103146	−0.0515732	+	0.998669i	$0.516424\pi$
$908$	− 27.9155i	− 0.0307440i
$909$	0	0
$910$	−477.668	−0.524910
$911$	1308.79i	1.43665i	0.695705	+	0.718327i	$0.255092\pi$
$911$	1308.79i	1.43665i	−0.695705	+	0.718327i	$0.744908\pi$
$912$	0	0
$913$	−253.259	−0.277392
$914$	89.6927i	0.0981320i
$915$	0	0
$916$	59.0020	0.0644126
$917$	116.368i	0.126901i
$918$	0	0
$919$	1146.21	1.24723	0.623616	−	0.781731i	$-0.285663\pi$
$919$	1146.21	1.24723	0.623616	+	0.781731i	$0.285663\pi$
$920$	− 74.0976i	− 0.0805409i
$921$	0	0
$922$	−1086.99	−1.17895
$923$	1415.34i	1.53342i
$924$	0	0
$925$	196.384	0.212307
$926$	641.269i	0.692516i
$927$	0	0
$928$	256.121	0.275992
$929$	402.392i	0.433145i	0.976267	+	0.216573i	$0.0694878\pi$
$929$	402.392i	0.433145i	−0.976267	+	0.216573i	$0.930512\pi$
$930$	0	0
$931$	1472.25	1.58136
$932$	− 700.807i	− 0.751939i
$933$	0	0
$934$	−92.7847	−0.0993413
$935$	394.116i	0.421514i
$936$	0	0
$937$	1315.29	1.40373	0.701863	−	0.712312i	$-0.252352\pi$
$937$	1315.29	1.40373	0.701863	+	0.712312i	$0.252352\pi$
$938$	131.499i	0.140191i
$939$	0	0
$940$	643.274	0.684334
$941$	1057.80i	1.12412i	0.827095	+	0.562062i	$0.189992\pi$
$941$	1057.80i	1.12412i	−0.827095	+	0.562062i	$0.810008\pi$
$942$	0	0
$943$	197.163	0.209080
$944$	166.746i	0.176638i
$945$	0	0
$946$	302.692	0.319970
$947$	− 942.801i	− 0.995566i	−0.867302	−	0.497783i	$-0.834148\pi$
$947$	− 942.801i	− 0.995566i	0.867302	−	0.497783i	$-0.165852\pi$
$948$	0	0
$949$	−774.438	−0.816056
$950$	− 240.184i	− 0.252825i
$951$	0	0
$952$	144.297	0.151573
$953$	1019.03i	1.06928i	0.845079	+	0.534642i	$0.179554\pi$
$953$	1019.03i	1.06928i	−0.845079	+	0.534642i	$0.820446\pi$
$954$	0	0
$955$	−571.260	−0.598178
$956$	54.1498i	0.0566421i
$957$	0	0
$958$	−222.761	−0.232527
$959$	343.544i	0.358231i
$960$	0	0
$961$	91.2971	0.0950022
$962$	− 1336.57i	− 1.38936i
$963$	0	0
$964$	−741.647	−0.769343
$965$	− 496.085i	− 0.514078i
$966$	0	0
$967$	−1197.09	−1.23795	−0.618974	−	0.785412i	$-0.712451\pi$
$967$	−1197.09	−1.23795	−0.618974	+	0.785412i	$0.712451\pi$
$968$	302.367i	0.312363i
$969$	0	0
$970$	94.6508	0.0975782
$971$	− 282.153i	− 0.290580i	−0.989389	−	0.145290i	$-0.953589\pi$
$971$	− 282.153i	− 0.290580i	0.989389	−	0.145290i	$-0.0464114\pi$
$972$	0	0
$973$	−51.2530	−0.0526753
$974$	− 277.204i	− 0.284604i
$975$	0	0
$976$	391.116	0.400734
$977$	72.9646i	0.0746823i	0.999303	+	0.0373411i	$0.0118888\pi$
$977$	72.9646i	0.0746823i	−0.999303	+	0.0373411i	$0.988111\pi$
$978$	0	0
$979$	−169.559	−0.173196
$980$	458.324i	0.467678i
$981$	0	0
$982$	1269.05	1.29231
$983$	1345.86i	1.36913i	0.728951	+	0.684566i	$0.240008\pi$
$983$	1345.86i	1.36913i	−0.728951	+	0.684566i	$0.759992\pi$
$984$	0	0
$985$	−1560.77	−1.58454
$986$	− 1230.41i	− 1.24788i
$987$	0	0
$988$	−1634.67	−1.65453
$989$	273.392i	0.276433i
$990$	0	0
$991$	1316.06	1.32802	0.664008	−	0.747726i	$-0.268854\pi$
$991$	1316.06	1.32802	0.664008	+	0.747726i	$0.268854\pi$
$992$	183.503i	0.184983i
$993$	0	0
$994$	−228.170	−0.229547
$995$	− 866.363i	− 0.870717i
$996$	0	0
$997$	1880.80	1.88646	0.943229	−	0.332144i	$-0.107772\pi$
$997$	1880.80	1.88646	0.943229	+	0.332144i	$0.107772\pi$
$998$	− 737.353i	− 0.738831i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.c.b.323.4	✓	8
3.2	odd	2	inner	414.3.c.b.323.5	yes	8
4.3	odd	2		3312.3.g.b.737.7		8
12.11	even	2		3312.3.g.b.737.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.c.b.323.4	✓	8	1.1	even	1	trivial
414.3.c.b.323.5	yes	8	3.2	odd	2	inner
3312.3.g.b.737.2		8	12.11	even	2
3312.3.g.b.737.7		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +5.46255i q^{5} +2.65490 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +5.46255i q^{5} +2.65490 q^{7} +2.82843i q^{8} +7.72521 q^{10} +3.75460i q^{11} -23.2899 q^{13} -3.75460i q^{14} +4.00000 q^{16} -19.2161i q^{17} -35.0940 q^{19} -10.9251i q^{20} +5.30981 q^{22} +4.79583i q^{23} -4.83945 q^{25} +32.9369i q^{26} -5.30981 q^{28} +45.2762i q^{29} -32.4391 q^{31} -5.65685i q^{32} -27.1756 q^{34} +14.5025i q^{35} -40.5797 q^{37} +49.6305i q^{38} -15.4504 q^{40} -41.1112i q^{41} +57.0062 q^{43} -7.50920i q^{44} +6.78233 q^{46} +58.8804i q^{47} -41.9515 q^{49} +6.84401i q^{50} +46.5797 q^{52} +32.3166i q^{53} -20.5097 q^{55} +7.50920i q^{56} +64.0302 q^{58} +41.6865i q^{59} +97.7791 q^{61} +45.8759i q^{62} -8.00000 q^{64} -127.222i q^{65} -35.0235 q^{67} +38.4322i q^{68} +20.5097 q^{70} -60.7708i q^{71} +33.2521 q^{73} +57.3884i q^{74} +70.1881 q^{76} +9.96811i q^{77} -139.994 q^{79} +21.8502i q^{80} -58.1401 q^{82} +67.4528i q^{83} +104.969 q^{85} -80.6189i q^{86} -10.6196 q^{88} +45.1603i q^{89} -61.8324 q^{91} -9.59166i q^{92} +83.2694 q^{94} -191.703i q^{95} +12.2522 q^{97} +59.3284i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 16 * q^7	\( 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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