LMFDB - Embedded newform 1425.2.c.i.799.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1425,2,Mod(799,1425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-20,0,0,0,0,-4,0,12,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.3
Root			$1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.i.799.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.64575i q^{2} -1.00000i q^{3} -5.00000 q^{4} +2.64575 q^{6} -1.64575i q^{7} -7.93725i q^{8} -1.00000 q^{9} +0.354249 q^{11} +5.00000i q^{12} -0.354249i q^{13} +4.35425 q^{14} +11.0000 q^{16} +4.00000i q^{17} -2.64575i q^{18} +1.00000 q^{19} -1.64575 q^{21} +0.937254i q^{22} +9.29150i q^{23} -7.93725 q^{24} +0.937254 q^{26} +1.00000i q^{27} +8.22876i q^{28} -8.93725 q^{29} +6.00000 q^{31} +13.2288i q^{32} -0.354249i q^{33} -10.5830 q^{34} +5.00000 q^{36} -3.64575i q^{37} +2.64575i q^{38} -0.354249 q^{39} -9.64575 q^{41} -4.35425i q^{42} +5.64575i q^{43} -1.77124 q^{44} -24.5830 q^{46} +1.29150i q^{47} -11.0000i q^{48} +4.29150 q^{49} +4.00000 q^{51} +1.77124i q^{52} +11.2915i q^{53} -2.64575 q^{54} -13.0627 q^{56} -1.00000i q^{57} -23.6458i q^{58} -11.2915 q^{59} -11.2915 q^{61} +15.8745i q^{62} +1.64575i q^{63} -13.0000 q^{64} +0.937254 q^{66} +6.58301i q^{67} -20.0000i q^{68} +9.29150 q^{69} +7.29150 q^{71} +7.93725i q^{72} +10.0000i q^{73} +9.64575 q^{74} -5.00000 q^{76} -0.583005i q^{77} -0.937254i q^{78} -6.58301 q^{79} +1.00000 q^{81} -25.5203i q^{82} +6.00000i q^{83} +8.22876 q^{84} -14.9373 q^{86} +8.93725i q^{87} -2.81176i q^{88} +16.9373 q^{89} -0.583005 q^{91} -46.4575i q^{92} -6.00000i q^{93} -3.41699 q^{94} +13.2288 q^{96} +2.93725i q^{97} +11.3542i q^{98} -0.354249 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{4} - 4 q^{9} + 12 q^{11} + 28 q^{14} + 44 q^{16} + 4 q^{19} + 4 q^{21} - 28 q^{26} - 4 q^{29} + 24 q^{31} + 20 q^{36} - 12 q^{39} - 28 q^{41} - 60 q^{44} - 56 q^{46} - 4 q^{49} + 16 q^{51} - 84 q^{56}+ \cdots - 12 q^{99}+O(q^{100}) $ 4 * q - 20 * q^4 - 4 * q^9 + 12 * q^11 + 28 * q^14 + 44 * q^16 + 4 * q^19 + 4 * q^21 - 28 * q^26 - 4 * q^29 + 24 * q^31 + 20 * q^36 - 12 * q^39 - 28 * q^41 - 60 * q^44 - 56 * q^46 - 4 * q^49 + 16 * q^51 - 84 * q^56 - 24 * q^59 - 24 * q^61 - 52 * q^64 - 28 * q^66 + 16 * q^69 + 8 * q^71 + 28 * q^74 - 20 * q^76 + 16 * q^79 + 4 * q^81 - 20 * q^84 - 28 * q^86 + 36 * q^89 + 40 * q^91 - 56 * q^94 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	2.64575i	1.87083i	0.353553	+	0.935414i	$0.384973\pi$
$2$	2.64575i	1.87083i	−0.353553	+	0.935414i	$0.615027\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.00000	−2.50000
$5$	0	0
$5$	0	0
$6$	2.64575	1.08012
$7$	− 1.64575i	− 0.622036i	−0.950404	−	0.311018i	$-0.899330\pi$
$7$	− 1.64575i	− 0.622036i	0.950404	−	0.311018i	$-0.100670\pi$
$8$	− 7.93725i	− 2.80624i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.354249	0.106810	0.0534050	−	0.998573i	$-0.482993\pi$
$11$	0.354249	0.106810	0.0534050	+	0.998573i	$0.482993\pi$
$12$	5.00000i	1.44338i
$13$	− 0.354249i	− 0.0982509i	−0.998793	−	0.0491255i	$-0.984357\pi$
$13$	− 0.354249i	− 0.0982509i	0.998793	−	0.0491255i	$-0.0156434\pi$
$14$	4.35425	1.16372
$15$	0	0
$16$	11.0000	2.75000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	− 2.64575i	− 0.623610i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.64575	−0.359132
$22$	0.937254i	0.199823i
$23$	9.29150i	1.93741i	0.248211	+	0.968706i	$0.420158\pi$
$23$	9.29150i	1.93741i	−0.248211	+	0.968706i	$0.579842\pi$
$24$	−7.93725	−1.62019
$25$	0	0
$26$	0.937254	0.183811
$27$	1.00000i	0.192450i
$28$	8.22876i	1.55509i
$29$	−8.93725	−1.65961	−0.829803	−	0.558056i	$-0.811547\pi$
$29$	−8.93725	−1.65961	−0.829803	+	0.558056i	$0.811547\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	13.2288i	2.33854i
$33$	− 0.354249i	− 0.0616668i
$34$	−10.5830	−1.81497
$35$	0	0
$36$	5.00000	0.833333
$37$	− 3.64575i	− 0.599358i	−0.954040	−	0.299679i	$-0.903120\pi$
$37$	− 3.64575i	− 0.599358i	0.954040	−	0.299679i	$-0.0968795\pi$
$38$	2.64575i	0.429198i
$39$	−0.354249	−0.0567252
$40$	0	0
$41$	−9.64575	−1.50641	−0.753207	−	0.657784i	$-0.771494\pi$
$41$	−9.64575	−1.50641	−0.753207	+	0.657784i	$0.771494\pi$
$42$	− 4.35425i	− 0.671875i
$43$	5.64575i	0.860969i	0.902598	+	0.430485i	$0.141657\pi$
$43$	5.64575i	0.860969i	−0.902598	+	0.430485i	$0.858343\pi$
$44$	−1.77124	−0.267025
$45$	0	0
$46$	−24.5830	−3.62457
$47$	1.29150i	0.188385i	0.995554	+	0.0941925i	$0.0300269\pi$
$47$	1.29150i	0.188385i	−0.995554	+	0.0941925i	$0.969973\pi$
$48$	− 11.0000i	− 1.58771i
$49$	4.29150	0.613072
$50$	0	0
$51$	4.00000	0.560112
$52$	1.77124i	0.245627i
$53$	11.2915i	1.55101i	0.631343	+	0.775504i	$0.282504\pi$
$53$	11.2915i	1.55101i	−0.631343	+	0.775504i	$0.717496\pi$
$54$	−2.64575	−0.360041
$55$	0	0
$56$	−13.0627	−1.74558
$57$	− 1.00000i	− 0.132453i
$58$	− 23.6458i	− 3.10484i
$59$	−11.2915	−1.47003	−0.735014	−	0.678052i	$-0.762825\pi$
$59$	−11.2915	−1.47003	−0.735014	+	0.678052i	$0.762825\pi$
$60$	0	0
$61$	−11.2915	−1.44573	−0.722864	−	0.690990i	$-0.757175\pi$
$61$	−11.2915	−1.44573	−0.722864	+	0.690990i	$0.757175\pi$
$62$	15.8745i	2.01606i
$63$	1.64575i	0.207345i
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0.937254	0.115368
$67$	6.58301i	0.804242i	0.915587	+	0.402121i	$0.131727\pi$
$67$	6.58301i	0.804242i	−0.915587	+	0.402121i	$0.868273\pi$
$68$	− 20.0000i	− 2.42536i
$69$	9.29150	1.11857
$70$	0	0
$71$	7.29150	0.865342	0.432671	−	0.901552i	$-0.357571\pi$
$71$	7.29150	0.865342	0.432671	+	0.901552i	$0.357571\pi$
$72$	7.93725i	0.935414i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	9.64575	1.12130
$75$	0	0
$76$	−5.00000	−0.573539
$77$	− 0.583005i	− 0.0664396i
$78$	− 0.937254i	− 0.106123i
$79$	−6.58301	−0.740646	−0.370323	−	0.928903i	$-0.620753\pi$
$79$	−6.58301	−0.740646	−0.370323	+	0.928903i	$0.620753\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 25.5203i	− 2.81824i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	8.22876	0.897831
$85$	0	0
$86$	−14.9373	−1.61073
$87$	8.93725i	0.958174i
$88$	− 2.81176i	− 0.299735i
$89$	16.9373	1.79535	0.897673	−	0.440663i	$-0.145257\pi$
$89$	16.9373	1.79535	0.897673	+	0.440663i	$0.145257\pi$
$90$	0	0
$91$	−0.583005	−0.0611156
$92$	− 46.4575i	− 4.84353i
$93$	− 6.00000i	− 0.622171i
$94$	−3.41699	−0.352436
$95$	0	0
$96$	13.2288	1.35015
$97$	2.93725i	0.298233i	0.988820	+	0.149116i	$0.0476429\pi$
$97$	2.93725i	0.298233i	−0.988820	+	0.149116i	$0.952357\pi$
$98$	11.3542i	1.14695i
$99$	−0.354249	−0.0356033
$100$	0	0
$101$	9.29150	0.924539	0.462270	−	0.886739i	$-0.347035\pi$
$101$	9.29150	0.924539	0.462270	+	0.886739i	$0.347035\pi$
$102$	10.5830i	1.04787i
$103$	− 10.5830i	− 1.04277i	−0.853320	−	0.521387i	$-0.825415\pi$
$103$	− 10.5830i	− 1.04277i	0.853320	−	0.521387i	$-0.174585\pi$
$104$	−2.81176	−0.275716
$105$	0	0
$106$	−29.8745	−2.90167
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 5.00000i	− 0.481125i
$109$	−5.29150	−0.506834	−0.253417	−	0.967357i	$-0.581554\pi$
$109$	−5.29150	−0.506834	−0.253417	+	0.967357i	$0.581554\pi$
$110$	0	0
$111$	−3.64575	−0.346039
$112$	− 18.1033i	− 1.71060i
$113$	− 4.00000i	− 0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$	− 4.00000i	− 0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$	2.64575	0.247797
$115$	0	0
$116$	44.6863	4.14902
$117$	0.354249i	0.0327503i
$118$	− 29.8745i	− 2.75017i
$119$	6.58301	0.603463
$120$	0	0
$121$	−10.8745	−0.988592
$122$	− 29.8745i	− 2.70471i
$123$	9.64575i	0.869728i
$124$	−30.0000	−2.69408
$125$	0	0
$126$	−4.35425	−0.387907
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	− 7.93725i	− 0.701561i
$129$	5.64575	0.497081
$130$	0	0
$131$	−14.2288	−1.24317	−0.621586	−	0.783346i	$-0.713511\pi$
$131$	−14.2288	−1.24317	−0.621586	+	0.783346i	$0.713511\pi$
$132$	1.77124i	0.154167i
$133$	− 1.64575i	− 0.142705i
$134$	−17.4170	−1.50460
$135$	0	0
$136$	31.7490	2.72246
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	24.5830i	2.09264i
$139$	−17.8745	−1.51610	−0.758048	−	0.652199i	$-0.773847\pi$
$139$	−17.8745	−1.51610	−0.758048	+	0.652199i	$0.773847\pi$
$140$	0	0
$141$	1.29150	0.108764
$142$	19.2915i	1.61891i
$143$	− 0.125492i	− 0.0104942i
$144$	−11.0000	−0.916667
$145$	0	0
$146$	−26.4575	−2.18964
$147$	− 4.29150i	− 0.353957i
$148$	18.2288i	1.49839i
$149$	20.5830	1.68623	0.843113	−	0.537737i	$-0.180721\pi$
$149$	20.5830	1.68623	0.843113	+	0.537737i	$0.180721\pi$
$150$	0	0
$151$	−8.58301	−0.698475	−0.349238	−	0.937034i	$-0.613559\pi$
$151$	−8.58301	−0.698475	−0.349238	+	0.937034i	$0.613559\pi$
$152$	− 7.93725i	− 0.643796i
$153$	− 4.00000i	− 0.323381i
$154$	1.54249	0.124297
$155$	0	0
$156$	1.77124	0.141813
$157$	13.2915i	1.06078i	0.847755	+	0.530389i	$0.177954\pi$
$157$	13.2915i	1.06078i	−0.847755	+	0.530389i	$0.822046\pi$
$158$	− 17.4170i	− 1.38562i
$159$	11.2915	0.895474
$160$	0	0
$161$	15.2915	1.20514
$162$	2.64575i	0.207870i
$163$	− 2.35425i	− 0.184399i	−0.995741	−	0.0921995i	$-0.970610\pi$
$163$	− 2.35425i	− 0.184399i	0.995741	−	0.0921995i	$-0.0293898\pi$
$164$	48.2288	3.76603
$165$	0	0
$166$	−15.8745	−1.23210
$167$	21.2915i	1.64759i	0.566891	+	0.823793i	$0.308146\pi$
$167$	21.2915i	1.64759i	−0.566891	+	0.823793i	$0.691854\pi$
$168$	13.0627i	1.00781i
$169$	12.8745	0.990347
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 28.2288i	− 2.15242i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−23.6458	−1.79258
$175$	0	0
$176$	3.89674	0.293727
$177$	11.2915i	0.848721i
$178$	44.8118i	3.35878i
$179$	7.29150	0.544992	0.272496	−	0.962157i	$-0.412151\pi$
$179$	7.29150	0.544992	0.272496	+	0.962157i	$0.412151\pi$
$180$	0	0
$181$	17.2915	1.28527	0.642634	−	0.766174i	$-0.277842\pi$
$181$	17.2915	1.28527	0.642634	+	0.766174i	$0.277842\pi$
$182$	− 1.54249i	− 0.114337i
$183$	11.2915i	0.834692i
$184$	73.7490	5.43685
$185$	0	0
$186$	15.8745	1.16398
$187$	1.41699i	0.103621i
$188$	− 6.45751i	− 0.470963i
$189$	1.64575	0.119711
$190$	0	0
$191$	6.22876	0.450697	0.225349	−	0.974278i	$-0.427648\pi$
$191$	6.22876	0.450697	0.225349	+	0.974278i	$0.427648\pi$
$192$	13.0000i	0.938194i
$193$	− 11.6458i	− 0.838280i	−0.907922	−	0.419140i	$-0.862332\pi$
$193$	− 11.6458i	− 0.838280i	0.907922	−	0.419140i	$-0.137668\pi$
$194$	−7.77124	−0.557943
$195$	0	0
$196$	−21.4575	−1.53268
$197$	25.1660i	1.79300i	0.443040	+	0.896502i	$0.353900\pi$
$197$	25.1660i	1.79300i	−0.443040	+	0.896502i	$0.646100\pi$
$198$	− 0.937254i	− 0.0666077i
$199$	−7.29150	−0.516881	−0.258440	−	0.966027i	$-0.583209\pi$
$199$	−7.29150	−0.516881	−0.258440	+	0.966027i	$0.583209\pi$
$200$	0	0
$201$	6.58301	0.464329
$202$	24.5830i	1.72965i
$203$	14.7085i	1.03233i
$204$	−20.0000	−1.40028
$205$	0	0
$206$	28.0000	1.95085
$207$	− 9.29150i	− 0.645804i
$208$	− 3.89674i	− 0.270190i
$209$	0.354249	0.0245039
$210$	0	0
$211$	2.58301	0.177821	0.0889107	−	0.996040i	$-0.471661\pi$
$211$	2.58301	0.177821	0.0889107	+	0.996040i	$0.471661\pi$
$212$	− 56.4575i	− 3.87752i
$213$	− 7.29150i	− 0.499606i
$214$	0	0
$215$	0	0
$216$	7.93725	0.540062
$217$	− 9.87451i	− 0.670325i
$218$	− 14.0000i	− 0.948200i
$219$	10.0000	0.675737
$220$	0	0
$221$	1.41699	0.0953174
$222$	− 9.64575i	− 0.647380i
$223$	2.58301i	0.172971i	0.996253	+	0.0864854i	$0.0275636\pi$
$223$	2.58301i	0.172971i	−0.996253	+	0.0864854i	$0.972436\pi$
$224$	21.7712	1.45465
$225$	0	0
$226$	10.5830	0.703971
$227$	10.7085i	0.710748i	0.934724	+	0.355374i	$0.115646\pi$
$227$	10.7085i	0.710748i	−0.934724	+	0.355374i	$0.884354\pi$
$228$	5.00000i	0.331133i
$229$	−8.70850	−0.575474	−0.287737	−	0.957710i	$-0.592903\pi$
$229$	−8.70850	−0.575474	−0.287737	+	0.957710i	$0.592903\pi$
$230$	0	0
$231$	−0.583005	−0.0383589
$232$	70.9373i	4.65726i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−0.937254	−0.0612702
$235$	0	0
$236$	56.4575	3.67507
$237$	6.58301i	0.427612i
$238$	17.4170i	1.12898i
$239$	−15.6458	−1.01204	−0.506020	−	0.862522i	$-0.668884\pi$
$239$	−15.6458	−1.01204	−0.506020	+	0.862522i	$0.668884\pi$
$240$	0	0
$241$	16.5830	1.06821	0.534103	−	0.845420i	$-0.320650\pi$
$241$	16.5830	1.06821	0.534103	+	0.845420i	$0.320650\pi$
$242$	− 28.7712i	− 1.84949i
$243$	− 1.00000i	− 0.0641500i
$244$	56.4575	3.61432
$245$	0	0
$246$	−25.5203	−1.62711
$247$	− 0.354249i	− 0.0225403i
$248$	− 47.6235i	− 3.02410i
$249$	6.00000	0.380235
$250$	0	0
$251$	−16.3542	−1.03227	−0.516136	−	0.856507i	$-0.672630\pi$
$251$	−16.3542	−1.03227	−0.516136	+	0.856507i	$0.672630\pi$
$252$	− 8.22876i	− 0.518363i
$253$	3.29150i	0.206935i
$254$	10.5830	0.664037
$255$	0	0
$256$	−5.00000	−0.312500
$257$	5.41699i	0.337903i	0.985624	+	0.168951i	$0.0540381\pi$
$257$	5.41699i	0.337903i	−0.985624	+	0.168951i	$0.945962\pi$
$258$	14.9373i	0.929953i
$259$	−6.00000	−0.372822
$260$	0	0
$261$	8.93725	0.553202
$262$	− 37.6458i	− 2.32576i
$263$	4.58301i	0.282600i	0.989967	+	0.141300i	$0.0451282\pi$
$263$	4.58301i	0.282600i	−0.989967	+	0.141300i	$0.954872\pi$
$264$	−2.81176	−0.173052
$265$	0	0
$266$	4.35425	0.266976
$267$	− 16.9373i	− 1.03654i
$268$	− 32.9150i	− 2.01061i
$269$	−4.22876	−0.257832	−0.128916	−	0.991656i	$-0.541150\pi$
$269$	−4.22876	−0.257832	−0.128916	+	0.991656i	$0.541150\pi$
$270$	0	0
$271$	−4.70850	−0.286021	−0.143010	−	0.989721i	$-0.545678\pi$
$271$	−4.70850	−0.286021	−0.143010	+	0.989721i	$0.545678\pi$
$272$	44.0000i	2.66789i
$273$	0.583005i	0.0352851i
$274$	15.8745	0.959014
$275$	0	0
$276$	−46.4575	−2.79641
$277$	− 0.583005i	− 0.0350294i	−0.999847	−	0.0175147i	$-0.994425\pi$
$277$	− 0.583005i	− 0.0350294i	0.999847	−	0.0175147i	$-0.00557539\pi$
$278$	− 47.2915i	− 2.83636i
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−32.2288	−1.92261	−0.961303	−	0.275493i	$-0.911159\pi$
$281$	−32.2288	−1.92261	−0.961303	+	0.275493i	$0.911159\pi$
$282$	3.41699i	0.203479i
$283$	− 11.5203i	− 0.684808i	−0.939553	−	0.342404i	$-0.888759\pi$
$283$	− 11.5203i	− 0.684808i	0.939553	−	0.342404i	$-0.111241\pi$
$284$	−36.4575	−2.16336
$285$	0	0
$286$	0.332021	0.0196328
$287$	15.8745i	0.937043i
$288$	− 13.2288i	− 0.779512i
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.93725	0.172185
$292$	− 50.0000i	− 2.92603i
$293$	5.41699i	0.316464i	0.987402	+	0.158232i	$0.0505794\pi$
$293$	5.41699i	0.316464i	−0.987402	+	0.158232i	$0.949421\pi$
$294$	11.3542	0.662193
$295$	0	0
$296$	−28.9373	−1.68194
$297$	0.354249i	0.0205556i
$298$	54.4575i	3.15464i
$299$	3.29150	0.190353
$300$	0	0
$301$	9.29150	0.535553
$302$	− 22.7085i	− 1.30673i
$303$	− 9.29150i	− 0.533783i
$304$	11.0000	0.630893
$305$	0	0
$306$	10.5830	0.604990
$307$	24.4575i	1.39586i	0.716164	+	0.697932i	$0.245896\pi$
$307$	24.4575i	1.39586i	−0.716164	+	0.697932i	$0.754104\pi$
$308$	2.91503i	0.166099i
$309$	−10.5830	−0.602046
$310$	0	0
$311$	22.9373	1.30065	0.650326	−	0.759655i	$-0.274632\pi$
$311$	22.9373	1.30065	0.650326	+	0.759655i	$0.274632\pi$
$312$	2.81176i	0.159185i
$313$	17.2915i	0.977374i	0.872459	+	0.488687i	$0.162524\pi$
$313$	17.2915i	0.977374i	−0.872459	+	0.488687i	$0.837476\pi$
$314$	−35.1660	−1.98453
$315$	0	0
$316$	32.9150	1.85161
$317$	− 20.4575i	− 1.14901i	−0.818502	−	0.574504i	$-0.805195\pi$
$317$	− 20.4575i	− 1.14901i	0.818502	−	0.574504i	$-0.194805\pi$
$318$	29.8745i	1.67528i
$319$	−3.16601	−0.177263
$320$	0	0
$321$	0	0
$322$	40.4575i	2.25461i
$323$	4.00000i	0.222566i
$324$	−5.00000	−0.277778
$325$	0	0
$326$	6.22876	0.344979
$327$	5.29150i	0.292621i
$328$	76.5608i	4.22736i
$329$	2.12549	0.117182
$330$	0	0
$331$	11.4170	0.627535	0.313767	−	0.949500i	$-0.398409\pi$
$331$	11.4170	0.627535	0.313767	+	0.949500i	$0.398409\pi$
$332$	− 30.0000i	− 1.64646i
$333$	3.64575i	0.199786i
$334$	−56.3320	−3.08235
$335$	0	0
$336$	−18.1033	−0.987614
$337$	− 14.9373i	− 0.813684i	−0.913499	−	0.406842i	$-0.866630\pi$
$337$	− 14.9373i	− 0.813684i	0.913499	−	0.406842i	$-0.133370\pi$
$338$	34.0627i	1.85277i
$339$	−4.00000	−0.217250
$340$	0	0
$341$	2.12549	0.115102
$342$	− 2.64575i	− 0.143066i
$343$	− 18.5830i	− 1.00339i
$344$	44.8118	2.41609
$345$	0	0
$346$	0	0
$347$	− 26.0000i	− 1.39575i	−0.716218	−	0.697877i	$-0.754128\pi$
$347$	− 26.0000i	− 1.39575i	0.716218	−	0.697877i	$-0.245872\pi$
$348$	− 44.6863i	− 2.39544i
$349$	−23.1660	−1.24005	−0.620024	−	0.784583i	$-0.712877\pi$
$349$	−23.1660	−1.24005	−0.620024	+	0.784583i	$0.712877\pi$
$350$	0	0
$351$	0.354249	0.0189084
$352$	4.68627i	0.249779i
$353$	0.583005i	0.0310302i	0.999880	+	0.0155151i	$0.00493881\pi$
$353$	0.583005i	0.0310302i	−0.999880	+	0.0155151i	$0.995061\pi$
$354$	−29.8745	−1.58781
$355$	0	0
$356$	−84.6863	−4.48836
$357$	− 6.58301i	− 0.348410i
$358$	19.2915i	1.01959i
$359$	36.1033	1.90546	0.952729	−	0.303822i	$-0.0982629\pi$
$359$	36.1033	1.90546	0.952729	+	0.303822i	$0.0982629\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	45.7490i	2.40451i
$363$	10.8745i	0.570764i
$364$	2.91503	0.152789
$365$	0	0
$366$	−29.8745	−1.56157
$367$	− 1.64575i	− 0.0859075i	−0.999077	−	0.0429538i	$-0.986323\pi$
$367$	− 1.64575i	− 0.0859075i	0.999077	−	0.0429538i	$-0.0136768\pi$
$368$	102.207i	5.32788i
$369$	9.64575	0.502138
$370$	0	0
$371$	18.5830	0.964782
$372$	30.0000i	1.55543i
$373$	5.06275i	0.262139i	0.991373	+	0.131070i	$0.0418411\pi$
$373$	5.06275i	0.262139i	−0.991373	+	0.131070i	$0.958159\pi$
$374$	−3.74902	−0.193857
$375$	0	0
$376$	10.2510	0.528654
$377$	3.16601i	0.163058i
$378$	4.35425i	0.223958i
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	16.4797i	0.843177i
$383$	− 18.5830i	− 0.949547i	−0.880108	−	0.474774i	$-0.842530\pi$
$383$	− 18.5830i	− 0.949547i	0.880108	−	0.474774i	$-0.157470\pi$
$384$	−7.93725	−0.405046
$385$	0	0
$386$	30.8118	1.56828
$387$	− 5.64575i	− 0.286990i
$388$	− 14.6863i	− 0.745582i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−37.1660	−1.87957
$392$	− 34.0627i	− 1.72043i
$393$	14.2288i	0.717746i
$394$	−66.5830	−3.35440
$395$	0	0
$396$	1.77124	0.0890083
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 19.2915i	− 0.966996i
$399$	−1.64575	−0.0823906
$400$	0	0
$401$	−4.93725	−0.246555	−0.123277	−	0.992372i	$-0.539340\pi$
$401$	−4.93725	−0.246555	−0.123277	+	0.992372i	$0.539340\pi$
$402$	17.4170i	0.868681i
$403$	− 2.12549i	− 0.105878i
$404$	−46.4575	−2.31135
$405$	0	0
$406$	−38.9150	−1.93132
$407$	− 1.29150i	− 0.0640174i
$408$	− 31.7490i	− 1.57181i
$409$	6.70850	0.331714	0.165857	−	0.986150i	$-0.446961\pi$
$409$	6.70850	0.331714	0.165857	+	0.986150i	$0.446961\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	52.9150i	2.60694i
$413$	18.5830i	0.914410i
$414$	24.5830	1.20819
$415$	0	0
$416$	4.68627	0.229763
$417$	17.8745i	0.875318i
$418$	0.937254i	0.0458426i
$419$	−38.9373	−1.90221	−0.951105	−	0.308869i	$-0.900050\pi$
$419$	−38.9373	−1.90221	−0.951105	+	0.308869i	$0.900050\pi$
$420$	0	0
$421$	28.5830	1.39305	0.696525	−	0.717532i	$-0.254728\pi$
$421$	28.5830	1.39305	0.696525	+	0.717532i	$0.254728\pi$
$422$	6.83399i	0.332673i
$423$	− 1.29150i	− 0.0627950i
$424$	89.6235	4.35250
$425$	0	0
$426$	19.2915	0.934676
$427$	18.5830i	0.899295i
$428$	0	0
$429$	−0.125492	−0.00605882
$430$	0	0
$431$	3.29150	0.158546	0.0792731	−	0.996853i	$-0.474740\pi$
$431$	3.29150	0.158546	0.0792731	+	0.996853i	$0.474740\pi$
$432$	11.0000i	0.529238i
$433$	− 27.6458i	− 1.32857i	−0.747479	−	0.664285i	$-0.768736\pi$
$433$	− 27.6458i	− 1.32857i	0.747479	−	0.664285i	$-0.231264\pi$
$434$	26.1255	1.25406
$435$	0	0
$436$	26.4575	1.26709
$437$	9.29150i	0.444473i
$438$	26.4575i	1.26419i
$439$	1.41699	0.0676295	0.0338147	−	0.999428i	$-0.489234\pi$
$439$	1.41699	0.0676295	0.0338147	+	0.999428i	$0.489234\pi$
$440$	0	0
$441$	−4.29150	−0.204357
$442$	3.74902i	0.178322i
$443$	26.7085i	1.26896i	0.772940	+	0.634480i	$0.218786\pi$
$443$	26.7085i	1.26896i	−0.772940	+	0.634480i	$0.781214\pi$
$444$	18.2288	0.865099
$445$	0	0
$446$	−6.83399	−0.323599
$447$	− 20.5830i	− 0.973543i
$448$	21.3948i	1.01081i
$449$	36.2288	1.70974	0.854870	−	0.518842i	$-0.173637\pi$
$449$	36.2288	1.70974	0.854870	+	0.518842i	$0.173637\pi$
$450$	0	0
$451$	−3.41699	−0.160900
$452$	20.0000i	0.940721i
$453$	8.58301i	0.403265i
$454$	−28.3320	−1.32969
$455$	0	0
$456$	−7.93725	−0.371696
$457$	− 0.125492i	− 0.00587027i	−0.999996	−	0.00293514i	$-0.999066\pi$
$457$	− 0.125492i	− 0.00587027i	0.999996	−	0.00293514i	$-0.000934285\pi$
$458$	− 23.0405i	− 1.07661i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−37.7490	−1.75815	−0.879073	−	0.476686i	$-0.841838\pi$
$461$	−37.7490	−1.75815	−0.879073	+	0.476686i	$0.841838\pi$
$462$	− 1.54249i	− 0.0717630i
$463$	− 35.5203i	− 1.65077i	−0.564573	−	0.825383i	$-0.690959\pi$
$463$	− 35.5203i	− 1.65077i	0.564573	−	0.825383i	$-0.309041\pi$
$464$	−98.3098	−4.56392
$465$	0	0
$466$	47.6235	2.20612
$467$	− 19.8745i	− 0.919683i	−0.888001	−	0.459841i	$-0.847906\pi$
$467$	− 19.8745i	− 0.919683i	0.888001	−	0.459841i	$-0.152094\pi$
$468$	− 1.77124i	− 0.0818758i
$469$	10.8340	0.500267
$470$	0	0
$471$	13.2915	0.612440
$472$	89.6235i	4.12526i
$473$	2.00000i	0.0919601i
$474$	−17.4170	−0.799989
$475$	0	0
$476$	−32.9150	−1.50866
$477$	− 11.2915i	− 0.517002i
$478$	− 41.3948i	− 1.89335i
$479$	−4.35425	−0.198951	−0.0994754	−	0.995040i	$-0.531716\pi$
$479$	−4.35425	−0.198951	−0.0994754	+	0.995040i	$0.531716\pi$
$480$	0	0
$481$	−1.29150	−0.0588875
$482$	43.8745i	1.99843i
$483$	− 15.2915i	− 0.695787i
$484$	54.3725	2.47148
$485$	0	0
$486$	2.64575	0.120014
$487$	− 17.8745i	− 0.809971i	−0.914323	−	0.404986i	$-0.867277\pi$
$487$	− 17.8745i	− 0.809971i	0.914323	−	0.404986i	$-0.132723\pi$
$488$	89.6235i	4.05707i
$489$	−2.35425	−0.106463
$490$	0	0
$491$	−5.77124	−0.260453	−0.130226	−	0.991484i	$-0.541570\pi$
$491$	−5.77124	−0.260453	−0.130226	+	0.991484i	$0.541570\pi$
$492$	− 48.2288i	− 2.17432i
$493$	− 35.7490i	− 1.61005i
$494$	0.937254	0.0421690
$495$	0	0
$496$	66.0000	2.96349
$497$	− 12.0000i	− 0.538274i
$498$	15.8745i	0.711354i
$499$	31.0405	1.38956	0.694782	−	0.719220i	$-0.255501\pi$
$499$	31.0405	1.38956	0.694782	+	0.719220i	$0.255501\pi$
$500$	0	0
$501$	21.2915	0.951234
$502$	− 43.2693i	− 1.93120i
$503$	− 2.00000i	− 0.0891756i	−0.999005	−	0.0445878i	$-0.985803\pi$
$503$	− 2.00000i	− 0.0891756i	0.999005	−	0.0445878i	$-0.0141974\pi$
$504$	13.0627	0.581861
$505$	0	0
$506$	−8.70850	−0.387140
$507$	− 12.8745i	− 0.571777i
$508$	20.0000i	0.887357i
$509$	34.1033	1.51160	0.755800	−	0.654802i	$-0.227248\pi$
$509$	34.1033	1.51160	0.755800	+	0.654802i	$0.227248\pi$
$510$	0	0
$511$	16.4575	0.728038
$512$	− 29.1033i	− 1.28619i
$513$	1.00000i	0.0441511i
$514$	−14.3320	−0.632158
$515$	0	0
$516$	−28.2288	−1.24270
$517$	0.457513i	0.0201214i
$518$	− 15.8745i	− 0.697486i
$519$	0	0
$520$	0	0
$521$	22.1033	0.968362	0.484181	−	0.874968i	$-0.339118\pi$
$521$	22.1033	0.968362	0.484181	+	0.874968i	$0.339118\pi$
$522$	23.6458i	1.03495i
$523$	− 23.2915i	− 1.01847i	−0.860629	−	0.509233i	$-0.829929\pi$
$523$	− 23.2915i	− 1.01847i	0.860629	−	0.509233i	$-0.170071\pi$
$524$	71.1438	3.10793
$525$	0	0
$526$	−12.1255	−0.528697
$527$	24.0000i	1.04546i
$528$	− 3.89674i	− 0.169584i
$529$	−63.3320	−2.75357
$530$	0	0
$531$	11.2915	0.490009
$532$	8.22876i	0.356762i
$533$	3.41699i	0.148006i
$534$	44.8118	1.93919
$535$	0	0
$536$	52.2510	2.25690
$537$	− 7.29150i	− 0.314652i
$538$	− 11.1882i	− 0.482359i
$539$	1.52026	0.0654822
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	− 12.4575i	− 0.535096i
$543$	− 17.2915i	− 0.742049i
$544$	−52.9150	−2.26871
$545$	0	0
$546$	−1.54249	−0.0660123
$547$	− 1.87451i	− 0.0801482i	−0.999197	−	0.0400741i	$-0.987241\pi$
$547$	− 1.87451i	− 0.0801482i	0.999197	−	0.0400741i	$-0.0127594\pi$
$548$	30.0000i	1.28154i
$549$	11.2915	0.481910
$550$	0	0
$551$	−8.93725	−0.380740
$552$	− 73.7490i	− 3.13897i
$553$	10.8340i	0.460708i
$554$	1.54249	0.0655340
$555$	0	0
$556$	89.3725	3.79024
$557$	11.4170i	0.483754i	0.970307	+	0.241877i	$0.0777630\pi$
$557$	11.4170i	0.483754i	−0.970307	+	0.241877i	$0.922237\pi$
$558$	− 15.8745i	− 0.672022i
$559$	2.00000	0.0845910
$560$	0	0
$561$	1.41699	0.0598256
$562$	− 85.2693i	− 3.59687i
$563$	8.12549i	0.342449i	0.985232	+	0.171224i	$0.0547723\pi$
$563$	8.12549i	0.342449i	−0.985232	+	0.171224i	$0.945228\pi$
$564$	−6.45751	−0.271910
$565$	0	0
$566$	30.4797	1.28116
$567$	− 1.64575i	− 0.0691151i
$568$	− 57.8745i	− 2.42836i
$569$	7.06275	0.296086	0.148043	−	0.988981i	$-0.452703\pi$
$569$	7.06275	0.296086	0.148043	+	0.988981i	$0.452703\pi$
$570$	0	0
$571$	16.7085	0.699229	0.349614	−	0.936894i	$-0.386313\pi$
$571$	16.7085	0.699229	0.349614	+	0.936894i	$0.386313\pi$
$572$	0.627461i	0.0262354i
$573$	− 6.22876i	− 0.260210i
$574$	−42.0000	−1.75305
$575$	0	0
$576$	13.0000	0.541667
$577$	13.2915i	0.553332i	0.960966	+	0.276666i	$0.0892296\pi$
$577$	13.2915i	0.553332i	−0.960966	+	0.276666i	$0.910770\pi$
$578$	2.64575i	0.110049i
$579$	−11.6458	−0.483981
$580$	0	0
$581$	9.87451	0.409664
$582$	7.77124i	0.322128i
$583$	4.00000i	0.165663i
$584$	79.3725	3.28446
$585$	0	0
$586$	−14.3320	−0.592050
$587$	33.2915i	1.37409i	0.726616	+	0.687044i	$0.241092\pi$
$587$	33.2915i	1.37409i	−0.726616	+	0.687044i	$0.758908\pi$
$588$	21.4575i	0.884893i
$589$	6.00000	0.247226
$590$	0	0
$591$	25.1660	1.03519
$592$	− 40.1033i	− 1.64823i
$593$	3.41699i	0.140319i	0.997536	+	0.0701596i	$0.0223508\pi$
$593$	3.41699i	0.140319i	−0.997536	+	0.0701596i	$0.977649\pi$
$594$	−0.937254	−0.0384560
$595$	0	0
$596$	−102.915	−4.21556
$597$	7.29150i	0.298421i
$598$	8.70850i	0.356117i
$599$	−9.41699	−0.384768	−0.192384	−	0.981320i	$-0.561622\pi$
$599$	−9.41699	−0.384768	−0.192384	+	0.981320i	$0.561622\pi$
$600$	0	0
$601$	22.7085	0.926299	0.463149	−	0.886280i	$-0.346719\pi$
$601$	22.7085	0.926299	0.463149	+	0.886280i	$0.346719\pi$
$602$	24.5830i	1.00193i
$603$	− 6.58301i	− 0.268081i
$604$	42.9150	1.74619
$605$	0	0
$606$	24.5830	0.998616
$607$	43.0405i	1.74696i	0.486859	+	0.873480i	$0.338142\pi$
$607$	43.0405i	1.74696i	−0.486859	+	0.873480i	$0.661858\pi$
$608$	13.2288i	0.536497i
$609$	14.7085	0.596018
$610$	0	0
$611$	0.457513	0.0185090
$612$	20.0000i	0.808452i
$613$	− 30.4575i	− 1.23017i	−0.788462	−	0.615084i	$-0.789122\pi$
$613$	− 30.4575i	− 1.23017i	0.788462	−	0.615084i	$-0.210878\pi$
$614$	−64.7085	−2.61142
$615$	0	0
$616$	−4.62746	−0.186446
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	− 28.0000i	− 1.12633i
$619$	−23.2915	−0.936165	−0.468082	−	0.883685i	$-0.655055\pi$
$619$	−23.2915	−0.936165	−0.468082	+	0.883685i	$0.655055\pi$
$620$	0	0
$621$	−9.29150	−0.372855
$622$	60.6863i	2.43330i
$623$	− 27.8745i	− 1.11677i
$624$	−3.89674	−0.155994
$625$	0	0
$626$	−45.7490	−1.82850
$627$	− 0.354249i	− 0.0141473i
$628$	− 66.4575i	− 2.65194i
$629$	14.5830	0.581462
$630$	0	0
$631$	−14.5830	−0.580540	−0.290270	−	0.956945i	$-0.593745\pi$
$631$	−14.5830	−0.580540	−0.290270	+	0.956945i	$0.593745\pi$
$632$	52.2510i	2.07843i
$633$	− 2.58301i	− 0.102665i
$634$	54.1255	2.14960
$635$	0	0
$636$	−56.4575	−2.23869
$637$	− 1.52026i	− 0.0602349i
$638$	− 8.37648i	− 0.331628i
$639$	−7.29150	−0.288447
$640$	0	0
$641$	−16.9373	−0.668981	−0.334491	−	0.942399i	$-0.608564\pi$
$641$	−16.9373	−0.668981	−0.334491	+	0.942399i	$0.608564\pi$
$642$	0	0
$643$	− 13.6458i	− 0.538136i	−0.963121	−	0.269068i	$-0.913284\pi$
$643$	− 13.6458i	− 0.538136i	0.963121	−	0.269068i	$-0.0867156\pi$
$644$	−76.4575	−3.01285
$645$	0	0
$646$	−10.5830	−0.416383
$647$	3.41699i	0.134336i	0.997742	+	0.0671680i	$0.0213963\pi$
$647$	3.41699i	0.134336i	−0.997742	+	0.0671680i	$0.978604\pi$
$648$	− 7.93725i	− 0.311805i
$649$	−4.00000	−0.157014
$650$	0	0
$651$	−9.87451	−0.387012
$652$	11.7712i	0.460997i
$653$	9.41699i	0.368515i	0.982878	+	0.184258i	$0.0589881\pi$
$653$	9.41699i	0.368515i	−0.982878	+	0.184258i	$0.941012\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−106.103	−4.14264
$657$	− 10.0000i	− 0.390137i
$658$	5.62352i	0.219228i
$659$	−25.4170	−0.990106	−0.495053	−	0.868863i	$-0.664851\pi$
$659$	−25.4170	−0.990106	−0.495053	+	0.868863i	$0.664851\pi$
$660$	0	0
$661$	1.29150	0.0502336	0.0251168	−	0.999685i	$-0.492004\pi$
$661$	1.29150	0.0502336	0.0251168	+	0.999685i	$0.492004\pi$
$662$	30.2065i	1.17401i
$663$	− 1.41699i	− 0.0550315i
$664$	47.6235	1.84815
$665$	0	0
$666$	−9.64575	−0.373765
$667$	− 83.0405i	− 3.21534i
$668$	− 106.458i	− 4.11896i
$669$	2.58301	0.0998648
$670$	0	0
$671$	−4.00000	−0.154418
$672$	− 21.7712i	− 0.839844i
$673$	− 13.0627i	− 0.503532i	−0.967788	−	0.251766i	$-0.918989\pi$
$673$	− 13.0627i	− 0.503532i	0.967788	−	0.251766i	$-0.0810113\pi$
$674$	39.5203	1.52226
$675$	0	0
$676$	−64.3725	−2.47587
$677$	− 40.4575i	− 1.55491i	−0.628939	−	0.777454i	$-0.716511\pi$
$677$	− 40.4575i	− 1.55491i	0.628939	−	0.777454i	$-0.283489\pi$
$678$	− 10.5830i	− 0.406438i
$679$	4.83399	0.185511
$680$	0	0
$681$	10.7085	0.410351
$682$	5.62352i	0.215336i
$683$	19.7490i	0.755675i	0.925872	+	0.377838i	$0.123332\pi$
$683$	19.7490i	0.755675i	−0.925872	+	0.377838i	$0.876668\pi$
$684$	5.00000	0.191180
$685$	0	0
$686$	49.1660	1.87717
$687$	8.70850i	0.332250i
$688$	62.1033i	2.36766i
$689$	4.00000	0.152388
$690$	0	0
$691$	−43.0405	−1.63734	−0.818669	−	0.574265i	$-0.805288\pi$
$691$	−43.0405	−1.63734	−0.818669	+	0.574265i	$0.805288\pi$
$692$	0	0
$693$	0.583005i	0.0221465i
$694$	68.7895	2.61122
$695$	0	0
$696$	70.9373	2.68887
$697$	− 38.5830i	− 1.46144i
$698$	− 61.2915i	− 2.31992i
$699$	−18.0000	−0.680823
$700$	0	0
$701$	16.5830	0.626331	0.313166	−	0.949698i	$-0.398610\pi$
$701$	16.5830	0.626331	0.313166	+	0.949698i	$0.398610\pi$
$702$	0.937254i	0.0353744i
$703$	− 3.64575i	− 0.137502i
$704$	−4.60523	−0.173566
$705$	0	0
$706$	−1.54249	−0.0580523
$707$	− 15.2915i	− 0.575096i
$708$	− 56.4575i	− 2.12180i
$709$	4.70850	0.176831	0.0884157	−	0.996084i	$-0.471820\pi$
$709$	4.70850	0.176831	0.0884157	+	0.996084i	$0.471820\pi$
$710$	0	0
$711$	6.58301	0.246882
$712$	− 134.435i	− 5.03818i
$713$	55.7490i	2.08782i
$714$	17.4170	0.651815
$715$	0	0
$716$	−36.4575	−1.36248
$717$	15.6458i	0.584301i
$718$	95.5203i	3.56478i
$719$	−32.8118	−1.22367	−0.611836	−	0.790985i	$-0.709569\pi$
$719$	−32.8118	−1.22367	−0.611836	+	0.790985i	$0.709569\pi$
$720$	0	0
$721$	−17.4170	−0.648643
$722$	2.64575i	0.0984647i
$723$	− 16.5830i	− 0.616729i
$724$	−86.4575	−3.21317
$725$	0	0
$726$	−28.7712	−1.06780
$727$	− 14.1033i	− 0.523061i	−0.965195	−	0.261531i	$-0.915773\pi$
$727$	− 14.1033i	− 0.523061i	0.965195	−	0.261531i	$-0.0842272\pi$
$728$	4.62746i	0.171505i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−22.5830	−0.835263
$732$	− 56.4575i	− 2.08673i
$733$	7.41699i	0.273953i	0.990574	+	0.136976i	$0.0437385\pi$
$733$	7.41699i	0.273953i	−0.990574	+	0.136976i	$0.956262\pi$
$734$	4.35425	0.160718
$735$	0	0
$736$	−122.915	−4.53071
$737$	2.33202i	0.0859011i
$738$	25.5203i	0.939414i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−0.354249	−0.0130137
$742$	49.1660i	1.80494i
$743$	− 10.5830i	− 0.388253i	−0.980977	−	0.194126i	$-0.937813\pi$
$743$	− 10.5830i	− 0.388253i	0.980977	−	0.194126i	$-0.0621872\pi$
$744$	−47.6235	−1.74596
$745$	0	0
$746$	−13.3948	−0.490417
$747$	− 6.00000i	− 0.219529i
$748$	− 7.08497i	− 0.259052i
$749$	0	0
$750$	0	0
$751$	−44.5830	−1.62686	−0.813428	−	0.581665i	$-0.802401\pi$
$751$	−44.5830	−1.62686	−0.813428	+	0.581665i	$0.802401\pi$
$752$	14.2065i	0.518059i
$753$	16.3542i	0.595982i
$754$	−8.37648	−0.305053
$755$	0	0
$756$	−8.22876	−0.299277
$757$	23.8745i	0.867734i	0.900977	+	0.433867i	$0.142851\pi$
$757$	23.8745i	0.867734i	−0.900977	+	0.433867i	$0.857149\pi$
$758$	− 26.4575i	− 0.960980i
$759$	3.29150	0.119474
$760$	0	0
$761$	25.7490	0.933401	0.466701	−	0.884415i	$-0.345443\pi$
$761$	25.7490	0.933401	0.466701	+	0.884415i	$0.345443\pi$
$762$	− 10.5830i	− 0.383382i
$763$	8.70850i	0.315269i
$764$	−31.1438	−1.12674
$765$	0	0
$766$	49.1660	1.77644
$767$	4.00000i	0.144432i
$768$	5.00000i	0.180422i
$769$	17.7490	0.640046	0.320023	−	0.947410i	$-0.396309\pi$
$769$	17.7490	0.640046	0.320023	+	0.947410i	$0.396309\pi$
$770$	0	0
$771$	5.41699	0.195088
$772$	58.2288i	2.09570i
$773$	− 8.70850i	− 0.313223i	−0.987660	−	0.156611i	$-0.949943\pi$
$773$	− 8.70850i	− 0.313223i	0.987660	−	0.156611i	$-0.0500570\pi$
$774$	14.9373	0.536909
$775$	0	0
$776$	23.3137	0.836914
$777$	6.00000i	0.215249i
$778$	15.8745i	0.569129i
$779$	−9.64575	−0.345595
$780$	0	0
$781$	2.58301	0.0924272
$782$	− 98.3320i	− 3.51635i
$783$	− 8.93725i	− 0.319391i
$784$	47.2065	1.68595
$785$	0	0
$786$	−37.6458	−1.34278
$787$	− 37.8745i	− 1.35008i	−0.737781	−	0.675040i	$-0.764126\pi$
$787$	− 37.8745i	− 1.35008i	0.737781	−	0.675040i	$-0.235874\pi$
$788$	− 125.830i	− 4.48251i
$789$	4.58301	0.163159
$790$	0	0
$791$	−6.58301	−0.234065
$792$	2.81176i	0.0999116i
$793$	4.00000i	0.142044i
$794$	−5.29150	−0.187788
$795$	0	0
$796$	36.4575	1.29220
$797$	− 40.0000i	− 1.41687i	−0.705775	−	0.708436i	$-0.749401\pi$
$797$	− 40.0000i	− 1.41687i	0.705775	−	0.708436i	$-0.250599\pi$
$798$	− 4.35425i	− 0.154139i
$799$	−5.16601	−0.182760
$800$	0	0
$801$	−16.9373	−0.598448
$802$	− 13.0627i	− 0.461262i
$803$	3.54249i	0.125012i
$804$	−32.9150	−1.16082
$805$	0	0
$806$	5.62352	0.198080
$807$	4.22876i	0.148859i
$808$	− 73.7490i	− 2.59448i
$809$	19.4170	0.682665	0.341333	−	0.939943i	$-0.389122\pi$
$809$	19.4170	0.682665	0.341333	+	0.939943i	$0.389122\pi$
$810$	0	0
$811$	38.3320	1.34602	0.673010	−	0.739634i	$-0.265001\pi$
$811$	38.3320	1.34602	0.673010	+	0.739634i	$0.265001\pi$
$812$	− 73.5425i	− 2.58084i
$813$	4.70850i	0.165134i
$814$	3.41699	0.119766
$815$	0	0
$816$	44.0000	1.54031
$817$	5.64575i	0.197520i
$818$	17.7490i	0.620580i
$819$	0.583005	0.0203719
$820$	0	0
$821$	47.6235	1.66207	0.831036	−	0.556218i	$-0.187748\pi$
$821$	47.6235	1.66207	0.831036	+	0.556218i	$0.187748\pi$
$822$	− 15.8745i	− 0.553687i
$823$	4.22876i	0.147405i	0.997280	+	0.0737026i	$0.0234816\pi$
$823$	4.22876i	0.147405i	−0.997280	+	0.0737026i	$0.976518\pi$
$824$	−84.0000	−2.92628
$825$	0	0
$826$	−49.1660	−1.71070
$827$	− 30.4575i	− 1.05911i	−0.848275	−	0.529556i	$-0.822359\pi$
$827$	− 30.4575i	− 1.05911i	0.848275	−	0.529556i	$-0.177641\pi$
$828$	46.4575i	1.61451i
$829$	−2.70850	−0.0940700	−0.0470350	−	0.998893i	$-0.514977\pi$
$829$	−2.70850	−0.0940700	−0.0470350	+	0.998893i	$0.514977\pi$
$830$	0	0
$831$	−0.583005	−0.0202242
$832$	4.60523i	0.159658i
$833$	17.1660i	0.594767i
$834$	−47.2915	−1.63757
$835$	0	0
$836$	−1.77124	−0.0612597
$837$	6.00000i	0.207390i
$838$	− 103.018i	− 3.55871i
$839$	12.4575	0.430081	0.215041	−	0.976605i	$-0.431012\pi$
$839$	12.4575	0.430081	0.215041	+	0.976605i	$0.431012\pi$
$840$	0	0
$841$	50.8745	1.75429
$842$	75.6235i	2.60616i
$843$	32.2288i	1.11002i
$844$	−12.9150	−0.444554
$845$	0	0
$846$	3.41699	0.117479
$847$	17.8967i	0.614939i
$848$	124.207i	4.26527i
$849$	−11.5203	−0.395374
$850$	0	0
$851$	33.8745	1.16120
$852$	36.4575i	1.24901i
$853$	14.7085i	0.503609i	0.967778	+	0.251805i	$0.0810241\pi$
$853$	14.7085i	0.503609i	−0.967778	+	0.251805i	$0.918976\pi$
$854$	−49.1660	−1.68243
$855$	0	0
$856$	0	0
$857$	− 31.0405i	− 1.06032i	−0.847896	−	0.530162i	$-0.822131\pi$
$857$	− 31.0405i	− 1.06032i	0.847896	−	0.530162i	$-0.177869\pi$
$858$	− 0.332021i	− 0.0113350i
$859$	34.3320	1.17139	0.585697	−	0.810530i	$-0.300821\pi$
$859$	34.3320	1.17139	0.585697	+	0.810530i	$0.300821\pi$
$860$	0	0
$861$	15.8745	0.541002
$862$	8.70850i	0.296613i
$863$	− 20.0000i	− 0.680808i	−0.940279	−	0.340404i	$-0.889436\pi$
$863$	− 20.0000i	− 0.680808i	0.940279	−	0.340404i	$-0.110564\pi$
$864$	−13.2288	−0.450051
$865$	0	0
$866$	73.1438	2.48553
$867$	− 1.00000i	− 0.0339618i
$868$	49.3725i	1.67581i
$869$	−2.33202	−0.0791084
$870$	0	0
$871$	2.33202	0.0790175
$872$	42.0000i	1.42230i
$873$	− 2.93725i	− 0.0994110i
$874$	−24.5830	−0.831533
$875$	0	0
$876$	−50.0000	−1.68934
$877$	25.0627i	0.846309i	0.906058	+	0.423154i	$0.139077\pi$
$877$	25.0627i	0.846309i	−0.906058	+	0.423154i	$0.860923\pi$
$878$	3.74902i	0.126523i
$879$	5.41699	0.182711
$880$	0	0
$881$	−0.125492	−0.00422794	−0.00211397	−	0.999998i	$-0.500673\pi$
$881$	−0.125492	−0.00422794	−0.00211397	+	0.999998i	$0.500673\pi$
$882$	− 11.3542i	− 0.382317i
$883$	10.8118i	0.363845i	0.983313	+	0.181922i	$0.0582320\pi$
$883$	10.8118i	0.363845i	−0.983313	+	0.181922i	$0.941768\pi$
$884$	−7.08497	−0.238293
$885$	0	0
$886$	−70.6640	−2.37400
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	28.9373i	0.971071i
$889$	−6.58301	−0.220787
$890$	0	0
$891$	0.354249	0.0118678
$892$	− 12.9150i	− 0.432427i
$893$	1.29150i	0.0432185i
$894$	54.4575	1.82133
$895$	0	0
$896$	−13.0627	−0.436396
$897$	− 3.29150i	− 0.109900i
$898$	95.8523i	3.19863i
$899$	−53.6235	−1.78844
$900$	0	0
$901$	−45.1660	−1.50470
$902$	− 9.04052i	− 0.301016i
$903$	− 9.29150i	− 0.309202i
$904$	−31.7490	−1.05596
$905$	0	0
$906$	−22.7085	−0.754439
$907$	47.0405i	1.56195i	0.624559	+	0.780977i	$0.285279\pi$
$907$	47.0405i	1.56195i	−0.624559	+	0.780977i	$0.714721\pi$
$908$	− 53.5425i	− 1.77687i
$909$	−9.29150	−0.308180
$910$	0	0
$911$	26.5830	0.880734	0.440367	−	0.897818i	$-0.354848\pi$
$911$	26.5830	0.880734	0.440367	+	0.897818i	$0.354848\pi$
$912$	− 11.0000i	− 0.364246i
$913$	2.12549i	0.0703435i
$914$	0.332021	0.0109823
$915$	0	0
$916$	43.5425	1.43868
$917$	23.4170i	0.773297i
$918$	− 10.5830i	− 0.349291i
$919$	−14.8340	−0.489328	−0.244664	−	0.969608i	$-0.578678\pi$
$919$	−14.8340	−0.489328	−0.244664	+	0.969608i	$0.578678\pi$
$920$	0	0
$921$	24.4575	0.805902
$922$	− 99.8745i	− 3.28919i
$923$	− 2.58301i	− 0.0850207i
$924$	2.91503	0.0958973
$925$	0	0
$926$	93.9778	3.08830
$927$	10.5830i	0.347591i
$928$	− 118.229i	− 3.88105i
$929$	11.8745	0.389590	0.194795	−	0.980844i	$-0.437596\pi$
$929$	11.8745	0.389590	0.194795	+	0.980844i	$0.437596\pi$
$930$	0	0
$931$	4.29150	0.140648
$932$	90.0000i	2.94805i
$933$	− 22.9373i	− 0.750932i
$934$	52.5830	1.72057
$935$	0	0
$936$	2.81176	0.0919053
$937$	− 20.1255i	− 0.657471i	−0.944422	−	0.328736i	$-0.893378\pi$
$937$	− 20.1255i	− 0.657471i	0.944422	−	0.328736i	$-0.106622\pi$
$938$	28.6640i	0.935914i
$939$	17.2915	0.564287
$940$	0	0
$941$	−11.7712	−0.383732	−0.191866	−	0.981421i	$-0.561454\pi$
$941$	−11.7712	−0.383732	−0.191866	+	0.981421i	$0.561454\pi$
$942$	35.1660i	1.14577i
$943$	− 89.6235i	− 2.91854i
$944$	−124.207	−4.04258
$945$	0	0
$946$	−5.29150	−0.172042
$947$	− 11.4170i	− 0.371002i	−0.982644	−	0.185501i	$-0.940609\pi$
$947$	− 11.4170i	− 0.371002i	0.982644	−	0.185501i	$-0.0593909\pi$
$948$	− 32.9150i	− 1.06903i
$949$	3.54249	0.114994
$950$	0	0
$951$	−20.4575	−0.663380
$952$	− 52.2510i	− 1.69346i
$953$	− 10.5830i	− 0.342817i	−0.985200	−	0.171409i	$-0.945168\pi$
$953$	− 10.5830i	− 0.342817i	0.985200	−	0.171409i	$-0.0548318\pi$
$954$	29.8745	0.967223
$955$	0	0
$956$	78.2288	2.53010
$957$	3.16601i	0.102343i
$958$	− 11.5203i	− 0.372203i
$959$	−9.87451	−0.318864
$960$	0	0
$961$	5.00000	0.161290
$962$	− 3.41699i	− 0.110168i
$963$	0	0
$964$	−82.9150	−2.67051
$965$	0	0
$966$	40.4575	1.30170
$967$	41.6458i	1.33924i	0.742705	+	0.669619i	$0.233542\pi$
$967$	41.6458i	1.33924i	−0.742705	+	0.669619i	$0.766458\pi$
$968$	86.3137i	2.77423i
$969$	4.00000	0.128499
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	5.00000i	0.160375i
$973$	29.4170i	0.943066i
$974$	47.2915	1.51532
$975$	0	0
$976$	−124.207	−3.97575
$977$	51.0405i	1.63293i	0.577394	+	0.816465i	$0.304070\pi$
$977$	51.0405i	1.63293i	−0.577394	+	0.816465i	$0.695930\pi$
$978$	− 6.22876i	− 0.199174i
$979$	6.00000	0.191761
$980$	0	0
$981$	5.29150	0.168945
$982$	− 15.2693i	− 0.487262i
$983$	34.4575i	1.09902i	0.835486	+	0.549512i	$0.185186\pi$
$983$	34.4575i	1.09902i	−0.835486	+	0.549512i	$0.814814\pi$
$984$	76.5608	2.44067
$985$	0	0
$986$	94.5830	3.01214
$987$	− 2.12549i	− 0.0676552i
$988$	1.77124i	0.0563508i
$989$	−52.4575	−1.66805
$990$	0	0
$991$	−35.7490	−1.13560	−0.567802	−	0.823165i	$-0.692206\pi$
$991$	−35.7490	−1.13560	−0.567802	+	0.823165i	$0.692206\pi$
$992$	79.3725i	2.52008i
$993$	− 11.4170i	− 0.362307i
$994$	31.7490	1.00702
$995$	0	0
$996$	−30.0000	−0.950586
$997$	− 26.7085i	− 0.845867i	−0.906161	−	0.422933i	$-0.861000\pi$
$997$	− 26.7085i	− 0.845867i	0.906161	−	0.422933i	$-0.139000\pi$
$998$	82.1255i	2.59964i
$999$	3.64575	0.115346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.i.799.3		4
5.2	odd	4		285.2.a.d.1.1	✓	2
5.3	odd	4		1425.2.a.p.1.2		2
5.4	even	2	inner	1425.2.c.i.799.2		4
15.2	even	4		855.2.a.g.1.2		2
15.8	even	4		4275.2.a.u.1.1		2
20.7	even	4		4560.2.a.bo.1.1		2
95.37	even	4		5415.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.d.1.1	✓	2	5.2	odd	4
855.2.a.g.1.2		2	15.2	even	4
1425.2.a.p.1.2		2	5.3	odd	4
1425.2.c.i.799.2		4	5.4	even	2	inner
1425.2.c.i.799.3		4	1.1	even	1	trivial
4275.2.a.u.1.1		2	15.8	even	4
4560.2.a.bo.1.1		2	20.7	even	4
5415.2.a.s.1.2		2	95.37	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.64575i q^{2} -1.00000i q^{3} -5.00000 q^{4} +2.64575 q^{6} -1.64575i q^{7} -7.93725i q^{8} -1.00000 q^{9} +0.354249 q^{11} +5.00000i q^{12} -0.354249i q^{13} +4.35425 q^{14} +11.0000 q^{16} +4.00000i q^{17} -2.64575i q^{18} +1.00000 q^{19} -1.64575 q^{21} +0.937254i q^{22} +9.29150i q^{23} -7.93725 q^{24} +0.937254 q^{26} +1.00000i q^{27} +8.22876i q^{28} -8.93725 q^{29} +6.00000 q^{31} +13.2288i q^{32} -0.354249i q^{33} -10.5830 q^{34} +5.00000 q^{36} -3.64575i q^{37} +2.64575i q^{38} -0.354249 q^{39} -9.64575 q^{41} -4.35425i q^{42} +5.64575i q^{43} -1.77124 q^{44} -24.5830 q^{46} +1.29150i q^{47} -11.0000i q^{48} +4.29150 q^{49} +4.00000 q^{51} +1.77124i q^{52} +11.2915i q^{53} -2.64575 q^{54} -13.0627 q^{56} -1.00000i q^{57} -23.6458i q^{58} -11.2915 q^{59} -11.2915 q^{61} +15.8745i q^{62} +1.64575i q^{63} -13.0000 q^{64} +0.937254 q^{66} +6.58301i q^{67} -20.0000i q^{68} +9.29150 q^{69} +7.29150 q^{71} +7.93725i q^{72} +10.0000i q^{73} +9.64575 q^{74} -5.00000 q^{76} -0.583005i q^{77} -0.937254i q^{78} -6.58301 q^{79} +1.00000 q^{81} -25.5203i q^{82} +6.00000i q^{83} +8.22876 q^{84} -14.9373 q^{86} +8.93725i q^{87} -2.81176i q^{88} +16.9373 q^{89} -0.583005 q^{91} -46.4575i q^{92} -6.00000i q^{93} -3.41699 q^{94} +13.2288 q^{96} +2.93725i q^{97} +11.3542i q^{98} -0.354249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{4} - 4 q^{9} + 12 q^{11} + 28 q^{14} + 44 q^{16} + 4 q^{19} + 4 q^{21} - 28 q^{26} - 4 q^{29} + 24 q^{31} + 20 q^{36} - 12 q^{39} - 28 q^{41} - 60 q^{44} - 56 q^{46} - 4 q^{49} + 16 q^{51} - 84 q^{56}+ \cdots - 12 q^{99}+O(q^{100}) \) 4 * q - 20 * q^4 - 4 * q^9 + 12 * q^11 + 28 * q^14 + 44 * q^16 + 4 * q^19 + 4 * q^21 - 28 * q^26 - 4 * q^29 + 24 * q^31 + 20 * q^36 - 12 * q^39 - 28 * q^41 - 60 * q^44 - 56 * q^46 - 4 * q^49 + 16 * q^51 - 84 * q^56 - 24 * q^59 - 24 * q^61 - 52 * q^64 - 28 * q^66 + 16 * q^69 + 8 * q^71 + 28 * q^74 - 20 * q^76 + 16 * q^79 + 4 * q^81 - 20 * q^84 - 28 * q^86 + 36 * q^89 + 40 * q^91 - 56 * q^94 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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