LMFDB - Embedded newform 1425.2.c.i.799.4

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			799.4
Root			$-1.32288 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.i.799.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.64575i q^{2} +1.00000i q^{3} -5.00000 q^{4} -2.64575 q^{6} -3.64575i q^{7} -7.93725i q^{8} -1.00000 q^{9} +5.64575 q^{11} -5.00000i q^{12} +5.64575i q^{13} +9.64575 q^{14} +11.0000 q^{16} -4.00000i q^{17} -2.64575i q^{18} +1.00000 q^{19} +3.64575 q^{21} +14.9373i q^{22} +1.29150i q^{23} +7.93725 q^{24} -14.9373 q^{26} -1.00000i q^{27} +18.2288i q^{28} +6.93725 q^{29} +6.00000 q^{31} +13.2288i q^{32} +5.64575i q^{33} +10.5830 q^{34} +5.00000 q^{36} -1.64575i q^{37} +2.64575i q^{38} -5.64575 q^{39} -4.35425 q^{41} +9.64575i q^{42} -0.354249i q^{43} -28.2288 q^{44} -3.41699 q^{46} +9.29150i q^{47} +11.0000i q^{48} -6.29150 q^{49} +4.00000 q^{51} -28.2288i q^{52} -0.708497i q^{53} +2.64575 q^{54} -28.9373 q^{56} +1.00000i q^{57} +18.3542i q^{58} -0.708497 q^{59} -0.708497 q^{61} +15.8745i q^{62} +3.64575i q^{63} -13.0000 q^{64} -14.9373 q^{66} +14.5830i q^{67} +20.0000i q^{68} -1.29150 q^{69} -3.29150 q^{71} +7.93725i q^{72} -10.0000i q^{73} +4.35425 q^{74} -5.00000 q^{76} -20.5830i q^{77} -14.9373i q^{78} +14.5830 q^{79} +1.00000 q^{81} -11.5203i q^{82} -6.00000i q^{83} -18.2288 q^{84} +0.937254 q^{86} +6.93725i q^{87} -44.8118i q^{88} +1.06275 q^{89} +20.5830 q^{91} -6.45751i q^{92} +6.00000i q^{93} -24.5830 q^{94} -13.2288 q^{96} +12.9373i q^{97} -16.6458i q^{98} -5.64575 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	− 3.64575i	− 1.37796i	−0.724778	−	0.688982i	$-0.758058\pi$
$7$	− 3.64575i	− 1.37796i	0.724778	−	0.688982i	$-0.241942\pi$
$8$	− 7.93725i	− 2.80624i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.64575	1.70226	0.851129	−	0.524957i	$-0.175918\pi$
$11$	5.64575	1.70226	0.851129	+	0.524957i	$0.175918\pi$
$12$	− 5.00000i	− 1.44338i
$13$	5.64575i	1.56585i	0.622116	+	0.782925i	$0.286273\pi$
$13$	5.64575i	1.56585i	−0.622116	+	0.782925i	$0.713727\pi$
$14$	9.64575	2.57794
$15$	0	0
$16$	11.0000	2.75000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	− 2.64575i	− 0.623610i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.64575	0.795568
$22$	14.9373i	3.18463i
$23$	1.29150i	0.269297i	0.990893	+	0.134648i	$0.0429905\pi$
$23$	1.29150i	0.269297i	−0.990893	+	0.134648i	$0.957009\pi$
$24$	7.93725	1.62019
$25$	0	0
$26$	−14.9373	−2.92944
$27$	− 1.00000i	− 0.192450i
$28$	18.2288i	3.44491i
$29$	6.93725	1.28822	0.644108	−	0.764935i	$-0.277229\pi$
$29$	6.93725	1.28822	0.644108	+	0.764935i	$0.277229\pi$
$30$	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
$33$	5.64575i	0.982799i
$34$	10.5830	1.81497
$35$	0	0
$36$	5.00000	0.833333
$37$	− 1.64575i	− 0.270560i	−0.990807	−	0.135280i	$-0.956807\pi$
$37$	− 1.64575i	− 0.270560i	0.990807	−	0.135280i	$-0.0431934\pi$
$38$	2.64575i	0.429198i
$39$	−5.64575	−0.904044
$40$	0	0
$41$	−4.35425	−0.680019	−0.340010	−	0.940422i	$-0.610430\pi$
$41$	−4.35425	−0.680019	−0.340010	+	0.940422i	$0.610430\pi$
$42$	9.64575i	1.48837i
$43$	− 0.354249i	− 0.0540224i	−0.999635	−	0.0270112i	$-0.991401\pi$
$43$	− 0.354249i	− 0.0540224i	0.999635	−	0.0270112i	$-0.00859898\pi$
$44$	−28.2288	−4.25565
$45$	0	0
$46$	−3.41699	−0.503808
$47$	9.29150i	1.35530i	0.735382	+	0.677652i	$0.237003\pi$
$47$	9.29150i	1.35530i	−0.735382	+	0.677652i	$0.762997\pi$
$48$	11.0000i	1.58771i
$49$	−6.29150	−0.898786
$50$	0	0
$51$	4.00000	0.560112
$52$	− 28.2288i	− 3.91462i
$53$	− 0.708497i	− 0.0973196i	−0.998815	−	0.0486598i	$-0.984505\pi$
$53$	− 0.708497i	− 0.0973196i	0.998815	−	0.0486598i	$-0.0154950\pi$
$54$	2.64575	0.360041
$55$	0	0
$56$	−28.9373	−3.86690
$57$	1.00000i	0.132453i
$58$	18.3542i	2.41003i
$59$	−0.708497	−0.0922385	−0.0461193	−	0.998936i	$-0.514685\pi$
$59$	−0.708497	−0.0922385	−0.0461193	+	0.998936i	$0.514685\pi$
$60$	0	0
$61$	−0.708497	−0.0907138	−0.0453569	−	0.998971i	$-0.514443\pi$
$61$	−0.708497	−0.0907138	−0.0453569	+	0.998971i	$0.514443\pi$
$62$	15.8745i	2.01606i
$63$	3.64575i	0.459321i
$64$	−13.0000	−1.62500
$65$	0	0
$66$	−14.9373	−1.83865
$67$	14.5830i	1.78160i	0.454398	+	0.890799i	$0.349854\pi$
$67$	14.5830i	1.78160i	−0.454398	+	0.890799i	$0.650146\pi$
$68$	20.0000i	2.42536i
$69$	−1.29150	−0.155479
$70$	0	0
$71$	−3.29150	−0.390629	−0.195315	−	0.980741i	$-0.562573\pi$
$71$	−3.29150	−0.390629	−0.195315	+	0.980741i	$0.562573\pi$
$72$	7.93725i	0.935414i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	4.35425	0.506171
$75$	0	0
$76$	−5.00000	−0.573539
$77$	− 20.5830i	− 2.34565i
$78$	− 14.9373i	− 1.69131i
$79$	14.5830	1.64072	0.820358	−	0.571850i	$-0.193774\pi$
$79$	14.5830	1.64072	0.820358	+	0.571850i	$0.193774\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 11.5203i	− 1.27220i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	−18.2288	−1.98892
$85$	0	0
$86$	0.937254	0.101067
$87$	6.93725i	0.743752i
$88$	− 44.8118i	− 4.77695i
$89$	1.06275	0.112651	0.0563254	−	0.998412i	$-0.482062\pi$
$89$	1.06275	0.112651	0.0563254	+	0.998412i	$0.482062\pi$
$90$	0	0
$91$	20.5830	2.15769
$92$	− 6.45751i	− 0.673242i
$93$	6.00000i	0.622171i
$94$	−24.5830	−2.53554
$95$	0	0
$96$	−13.2288	−1.35015
$97$	12.9373i	1.31358i	0.754074	+	0.656790i	$0.228086\pi$
$97$	12.9373i	1.31358i	−0.754074	+	0.656790i	$0.771914\pi$
$98$	− 16.6458i	− 1.68147i
$99$	−5.64575	−0.567419
$100$	0	0
$101$	−1.29150	−0.128509	−0.0642547	−	0.997934i	$-0.520467\pi$
$101$	−1.29150	−0.128509	−0.0642547	+	0.997934i	$0.520467\pi$
$102$	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	$-0.174585\pi$
$104$	44.8118	4.39415
$105$	0	0
$106$	1.87451	0.182068
$107$	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	$-0.418446\pi$
$109$	5.29150	0.506834	0.253417	+	0.967357i	$0.418446\pi$
$110$	0	0
$111$	1.64575	0.156208
$112$	− 40.1033i	− 3.78940i
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	−2.64575	−0.247797
$115$	0	0
$116$	−34.6863	−3.22054
$117$	− 5.64575i	− 0.521950i
$118$	− 1.87451i	− 0.172562i
$119$	−14.5830	−1.33682
$120$	0	0
$121$	20.8745	1.89768
$122$	− 1.87451i	− 0.169710i
$123$	− 4.35425i	− 0.392609i
$124$	−30.0000	−2.69408
$125$	0	0
$126$	−9.64575	−0.859312
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 7.93725i	− 0.701561i
$129$	0.354249	0.0311899
$130$	0	0
$131$	12.2288	1.06843	0.534216	−	0.845348i	$-0.320607\pi$
$131$	12.2288	1.06843	0.534216	+	0.845348i	$0.320607\pi$
$132$	− 28.2288i	− 2.45700i
$133$	− 3.64575i	− 0.316127i
$134$	−38.5830	−3.33306
$135$	0	0
$136$	−31.7490	−2.72246
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	− 3.41699i	− 0.290874i
$139$	13.8745	1.17682	0.588410	−	0.808563i	$-0.299754\pi$
$139$	13.8745	1.17682	0.588410	+	0.808563i	$0.299754\pi$
$140$	0	0
$141$	−9.29150	−0.782486
$142$	− 8.70850i	− 0.730801i
$143$	31.8745i	2.66548i
$144$	−11.0000	−0.916667
$145$	0	0
$146$	26.4575	2.18964
$147$	− 6.29150i	− 0.518914i
$148$	8.22876i	0.676400i
$149$	−0.583005	−0.0477617	−0.0238808	−	0.999715i	$-0.507602\pi$
$149$	−0.583005	−0.0477617	−0.0238808	+	0.999715i	$0.507602\pi$
$150$	0	0
$151$	12.5830	1.02399	0.511995	−	0.858988i	$-0.328907\pi$
$151$	12.5830	1.02399	0.511995	+	0.858988i	$0.328907\pi$
$152$	− 7.93725i	− 0.643796i
$153$	4.00000i	0.323381i
$154$	54.4575	4.38831
$155$	0	0
$156$	28.2288	2.26011
$157$	− 2.70850i	− 0.216162i	−0.994142	−	0.108081i	$-0.965529\pi$
$157$	− 2.70850i	− 0.216162i	0.994142	−	0.108081i	$-0.0344705\pi$
$158$	38.5830i	3.06950i
$159$	0.708497	0.0561875
$160$	0	0
$161$	4.70850	0.371082
$162$	2.64575i	0.207870i
$163$	7.64575i	0.598861i	0.954118	+	0.299431i	$0.0967967\pi$
$163$	7.64575i	0.598861i	−0.954118	+	0.299431i	$0.903203\pi$
$164$	21.7712	1.70005
$165$	0	0
$166$	15.8745	1.23210
$167$	− 10.7085i	− 0.828648i	−0.910129	−	0.414324i	$-0.864018\pi$
$167$	− 10.7085i	− 0.828648i	0.910129	−	0.414324i	$-0.135982\pi$
$168$	− 28.9373i	− 2.23256i
$169$	−18.8745	−1.45189
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	1.77124i	0.135056i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−18.3542	−1.39143
$175$	0	0
$176$	62.1033	4.68121
$177$	− 0.708497i	− 0.0532539i
$178$	2.81176i	0.210750i
$179$	−3.29150	−0.246018	−0.123009	−	0.992406i	$-0.539254\pi$
$179$	−3.29150	−0.246018	−0.123009	+	0.992406i	$0.539254\pi$
$180$	0	0
$181$	6.70850	0.498639	0.249319	−	0.968421i	$-0.419793\pi$
$181$	6.70850	0.498639	0.249319	+	0.968421i	$0.419793\pi$
$182$	54.4575i	4.03666i
$183$	− 0.708497i	− 0.0523736i
$184$	10.2510	0.755713
$185$	0	0
$186$	−15.8745	−1.16398
$187$	− 22.5830i	− 1.65143i
$188$	− 46.4575i	− 3.38826i
$189$	−3.64575	−0.265189
$190$	0	0
$191$	−20.2288	−1.46370	−0.731851	−	0.681465i	$-0.761343\pi$
$191$	−20.2288	−1.46370	−0.731851	+	0.681465i	$0.761343\pi$
$192$	− 13.0000i	− 0.938194i
$193$	6.35425i	0.457389i	0.973498	+	0.228694i	$0.0734457\pi$
$193$	6.35425i	0.457389i	−0.973498	+	0.228694i	$0.926554\pi$
$194$	−34.2288	−2.45748
$195$	0	0
$196$	31.4575	2.24697
$197$	17.1660i	1.22303i	0.791234	+	0.611514i	$0.209439\pi$
$197$	17.1660i	1.22303i	−0.791234	+	0.611514i	$0.790561\pi$
$198$	− 14.9373i	− 1.06154i
$199$	3.29150	0.233328	0.116664	−	0.993171i	$-0.462780\pi$
$199$	3.29150	0.233328	0.116664	+	0.993171i	$0.462780\pi$
$200$	0	0
$201$	−14.5830	−1.02861
$202$	− 3.41699i	− 0.240419i
$203$	− 25.2915i	− 1.77512i
$204$	−20.0000	−1.40028
$205$	0	0
$206$	28.0000	1.95085
$207$	− 1.29150i	− 0.0897656i
$208$	62.1033i	4.30609i
$209$	5.64575	0.390525
$210$	0	0
$211$	−18.5830	−1.27931	−0.639653	−	0.768663i	$-0.720922\pi$
$211$	−18.5830	−1.27931	−0.639653	+	0.768663i	$0.720922\pi$
$212$	3.54249i	0.243299i
$213$	− 3.29150i	− 0.225530i
$214$	0	0
$215$	0	0
$216$	−7.93725	−0.540062
$217$	− 21.8745i	− 1.48494i
$218$	14.0000i	0.948200i
$219$	10.0000	0.675737
$220$	0	0
$221$	22.5830	1.51910
$222$	4.35425i	0.292238i
$223$	18.5830i	1.24441i	0.782854	+	0.622205i	$0.213763\pi$
$223$	18.5830i	1.24441i	−0.782854	+	0.622205i	$0.786237\pi$
$224$	48.2288	3.22242
$225$	0	0
$226$	−10.5830	−0.703971
$227$	− 21.2915i	− 1.41317i	−0.707630	−	0.706583i	$-0.750236\pi$
$227$	− 21.2915i	− 1.41317i	0.707630	−	0.706583i	$-0.249764\pi$
$228$	− 5.00000i	− 0.331133i
$229$	−19.2915	−1.27482	−0.637409	−	0.770525i	$-0.719994\pi$
$229$	−19.2915	−1.27482	−0.637409	+	0.770525i	$0.719994\pi$
$230$	0	0
$231$	20.5830	1.35426
$232$	− 55.0627i	− 3.61505i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	14.9373	0.976479
$235$	0	0
$236$	3.54249	0.230596
$237$	14.5830i	0.947268i
$238$	− 38.5830i	− 2.50096i
$239$	−10.3542	−0.669761	−0.334880	−	0.942261i	$-0.608696\pi$
$239$	−10.3542	−0.669761	−0.334880	+	0.942261i	$0.608696\pi$
$240$	0	0
$241$	−4.58301	−0.295217	−0.147609	−	0.989046i	$-0.547158\pi$
$241$	−4.58301	−0.295217	−0.147609	+	0.989046i	$0.547158\pi$
$242$	55.2288i	3.55024i
$243$	1.00000i	0.0641500i
$244$	3.54249	0.226784
$245$	0	0
$246$	11.5203	0.734505
$247$	5.64575i	0.359231i
$248$	− 47.6235i	− 3.02410i
$249$	6.00000	0.380235
$250$	0	0
$251$	−21.6458	−1.36627	−0.683134	−	0.730293i	$-0.739383\pi$
$251$	−21.6458	−1.36627	−0.683134	+	0.730293i	$0.739383\pi$
$252$	− 18.2288i	− 1.14830i
$253$	7.29150i	0.458413i
$254$	−10.5830	−0.664037
$255$	0	0
$256$	−5.00000	−0.312500
$257$	− 26.5830i	− 1.65820i	−0.559099	−	0.829101i	$-0.688853\pi$
$257$	− 26.5830i	− 1.65820i	0.559099	−	0.829101i	$-0.311147\pi$
$258$	0.937254i	0.0583509i
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−6.93725	−0.429405
$262$	32.3542i	1.99885i
$263$	16.5830i	1.02255i	0.859417	+	0.511276i	$0.170827\pi$
$263$	16.5830i	1.02255i	−0.859417	+	0.511276i	$0.829173\pi$
$264$	44.8118	2.75797
$265$	0	0
$266$	9.64575	0.591419
$267$	1.06275i	0.0650390i
$268$	− 72.9150i	− 4.45399i
$269$	22.2288	1.35531	0.677656	−	0.735379i	$-0.262996\pi$
$269$	22.2288	1.35531	0.677656	+	0.735379i	$0.262996\pi$
$270$	0	0
$271$	−15.2915	−0.928893	−0.464446	−	0.885601i	$-0.653747\pi$
$271$	−15.2915	−0.928893	−0.464446	+	0.885601i	$0.653747\pi$
$272$	− 44.0000i	− 2.66789i
$273$	20.5830i	1.24574i
$274$	−15.8745	−0.959014
$275$	0	0
$276$	6.45751	0.388697
$277$	− 20.5830i	− 1.23671i	−0.785898	−	0.618356i	$-0.787799\pi$
$277$	− 20.5830i	− 1.23671i	0.785898	−	0.618356i	$-0.212201\pi$
$278$	36.7085i	2.20163i
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−5.77124	−0.344284	−0.172142	−	0.985072i	$-0.555069\pi$
$281$	−5.77124	−0.344284	−0.172142	+	0.985072i	$0.555069\pi$
$282$	− 24.5830i	− 1.46390i
$283$	− 25.5203i	− 1.51702i	−0.651660	−	0.758511i	$-0.725927\pi$
$283$	− 25.5203i	− 1.51702i	0.651660	−	0.758511i	$-0.274073\pi$
$284$	16.4575	0.976574
$285$	0	0
$286$	−84.3320	−4.98666
$287$	15.8745i	0.937043i
$288$	− 13.2288i	− 0.779512i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.9373	−0.758395
$292$	50.0000i	2.92603i
$293$	− 26.5830i	− 1.55300i	−0.630120	−	0.776498i	$-0.716994\pi$
$293$	− 26.5830i	− 1.55300i	0.630120	−	0.776498i	$-0.283006\pi$
$294$	16.6458	0.970800
$295$	0	0
$296$	−13.0627	−0.759257
$297$	− 5.64575i	− 0.327600i
$298$	− 1.54249i	− 0.0893539i
$299$	−7.29150	−0.421678
$300$	0	0
$301$	−1.29150	−0.0744410
$302$	33.2915i	1.91571i
$303$	− 1.29150i	− 0.0741949i
$304$	11.0000	0.630893
$305$	0	0
$306$	−10.5830	−0.604990
$307$	28.4575i	1.62416i	0.583549	+	0.812078i	$0.301664\pi$
$307$	28.4575i	1.62416i	−0.583549	+	0.812078i	$0.698336\pi$
$308$	102.915i	5.86413i
$309$	10.5830	0.602046
$310$	0	0
$311$	7.06275	0.400492	0.200246	−	0.979746i	$-0.435826\pi$
$311$	7.06275	0.400492	0.200246	+	0.979746i	$0.435826\pi$
$312$	44.8118i	2.53697i
$313$	− 6.70850i	− 0.379187i	−0.981863	−	0.189593i	$-0.939283\pi$
$313$	− 6.70850i	− 0.379187i	0.981863	−	0.189593i	$-0.0607170\pi$
$314$	7.16601	0.404401
$315$	0	0
$316$	−72.9150	−4.10179
$317$	− 32.4575i	− 1.82300i	−0.411305	−	0.911498i	$-0.634927\pi$
$317$	− 32.4575i	− 1.82300i	0.411305	−	0.911498i	$-0.365073\pi$
$318$	1.87451i	0.105117i
$319$	39.1660	2.19288
$320$	0	0
$321$	0	0
$322$	12.4575i	0.694230i
$323$	− 4.00000i	− 0.222566i
$324$	−5.00000	−0.277778
$325$	0	0
$326$	−20.2288	−1.12037
$327$	5.29150i	0.292621i
$328$	34.5608i	1.90830i
$329$	33.8745	1.86756
$330$	0	0
$331$	32.5830	1.79092	0.895462	−	0.445138i	$-0.146845\pi$
$331$	32.5830	1.79092	0.895462	+	0.445138i	$0.146845\pi$
$332$	30.0000i	1.64646i
$333$	1.64575i	0.0901866i
$334$	28.3320	1.55026
$335$	0	0
$336$	40.1033	2.18781
$337$	− 0.937254i	− 0.0510555i	−0.999674	−	0.0255277i	$-0.991873\pi$
$337$	− 0.937254i	− 0.0510555i	0.999674	−	0.0255277i	$-0.00812661\pi$
$338$	− 49.9373i	− 2.71623i
$339$	−4.00000	−0.217250
$340$	0	0
$341$	33.8745	1.83441
$342$	− 2.64575i	− 0.143066i
$343$	− 2.58301i	− 0.139469i
$344$	−2.81176	−0.151600
$345$	0	0
$346$	0	0
$347$	26.0000i	1.39575i	0.716218	+	0.697877i	$0.245872\pi$
$347$	26.0000i	1.39575i	−0.716218	+	0.697877i	$0.754128\pi$
$348$	− 34.6863i	− 1.85938i
$349$	19.1660	1.02593	0.512967	−	0.858409i	$-0.328546\pi$
$349$	19.1660	1.02593	0.512967	+	0.858409i	$0.328546\pi$
$350$	0	0
$351$	5.64575	0.301348
$352$	74.6863i	3.98079i
$353$	20.5830i	1.09552i	0.836635	+	0.547761i	$0.184520\pi$
$353$	20.5830i	1.09552i	−0.836635	+	0.547761i	$0.815480\pi$
$354$	1.87451	0.0996290
$355$	0	0
$356$	−5.31373	−0.281627
$357$	− 14.5830i	− 0.771814i
$358$	− 8.70850i	− 0.460258i
$359$	−22.1033	−1.16657	−0.583283	−	0.812269i	$-0.698232\pi$
$359$	−22.1033	−1.16657	−0.583283	+	0.812269i	$0.698232\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	17.7490i	0.932868i
$363$	20.8745i	1.09563i
$364$	−102.915	−5.39421
$365$	0	0
$366$	1.87451	0.0979821
$367$	− 3.64575i	− 0.190307i	−0.995463	−	0.0951533i	$-0.969666\pi$
$367$	− 3.64575i	− 0.190307i	0.995463	−	0.0951533i	$-0.0303341\pi$
$368$	14.2065i	0.740567i
$369$	4.35425	0.226673
$370$	0	0
$371$	−2.58301	−0.134103
$372$	− 30.0000i	− 1.55543i
$373$	− 20.9373i	− 1.08409i	−0.840349	−	0.542045i	$-0.817650\pi$
$373$	− 20.9373i	− 1.08409i	0.840349	−	0.542045i	$-0.182350\pi$
$374$	59.7490	3.08955
$375$	0	0
$376$	73.7490	3.80332
$377$	39.1660i	2.01715i
$378$	− 9.64575i	− 0.496124i
$379$	−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
$382$	− 53.5203i	− 2.73833i
$383$	− 2.58301i	− 0.131985i	−0.997820	−	0.0659927i	$-0.978979\pi$
$383$	− 2.58301i	− 0.131985i	0.997820	−	0.0659927i	$-0.0210214\pi$
$384$	7.93725	0.405046
$385$	0	0
$386$	−16.8118	−0.855696
$387$	0.354249i	0.0180075i
$388$	− 64.6863i	− 3.28395i
$389$	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
$391$	5.16601	0.261256
$392$	49.9373i	2.52221i
$393$	12.2288i	0.616859i
$394$	−45.4170	−2.28808
$395$	0	0
$396$	28.2288	1.41855
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	8.70850i	0.436518i
$399$	3.64575	0.182516
$400$	0	0
$401$	10.9373	0.546180	0.273090	−	0.961988i	$-0.411954\pi$
$401$	10.9373	0.546180	0.273090	+	0.961988i	$0.411954\pi$
$402$	− 38.5830i	− 1.92435i
$403$	33.8745i	1.68741i
$404$	6.45751	0.321273
$405$	0	0
$406$	66.9150	3.32094
$407$	− 9.29150i	− 0.460563i
$408$	− 31.7490i	− 1.57181i
$409$	17.2915	0.855010	0.427505	−	0.904013i	$-0.359393\pi$
$409$	17.2915	0.855010	0.427505	+	0.904013i	$0.359393\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	52.9150i	2.60694i
$413$	2.58301i	0.127101i
$414$	3.41699	0.167936
$415$	0	0
$416$	−74.6863	−3.66180
$417$	13.8745i	0.679438i
$418$	14.9373i	0.730605i
$419$	−23.0627	−1.12669	−0.563344	−	0.826222i	$-0.690486\pi$
$419$	−23.0627	−1.12669	−0.563344	+	0.826222i	$0.690486\pi$
$420$	0	0
$421$	7.41699	0.361482	0.180741	−	0.983531i	$-0.442150\pi$
$421$	7.41699	0.361482	0.180741	+	0.983531i	$0.442150\pi$
$422$	− 49.1660i	− 2.39336i
$423$	− 9.29150i	− 0.451768i
$424$	−5.62352	−0.273102
$425$	0	0
$426$	8.70850	0.421928
$427$	2.58301i	0.125000i
$428$	0	0
$429$	−31.8745	−1.53892
$430$	0	0
$431$	−7.29150	−0.351219	−0.175610	−	0.984460i	$-0.556190\pi$
$431$	−7.29150	−0.351219	−0.175610	+	0.984460i	$0.556190\pi$
$432$	− 11.0000i	− 0.529238i
$433$	22.3542i	1.07428i	0.843494	+	0.537138i	$0.180495\pi$
$433$	22.3542i	1.07428i	−0.843494	+	0.537138i	$0.819505\pi$
$434$	57.8745	2.77807
$435$	0	0
$436$	−26.4575	−1.26709
$437$	1.29150i	0.0617809i
$438$	26.4575i	1.26419i
$439$	22.5830	1.07783	0.538914	−	0.842361i	$-0.318835\pi$
$439$	22.5830	1.07783	0.538914	+	0.842361i	$0.318835\pi$
$440$	0	0
$441$	6.29150	0.299595
$442$	59.7490i	2.84197i
$443$	− 37.2915i	− 1.77177i	−0.463902	−	0.885886i	$-0.653551\pi$
$443$	− 37.2915i	− 1.77177i	0.463902	−	0.885886i	$-0.346449\pi$
$444$	−8.22876	−0.390520
$445$	0	0
$446$	−49.1660	−2.32808
$447$	− 0.583005i	− 0.0275752i
$448$	47.3948i	2.23919i
$449$	9.77124	0.461133	0.230567	−	0.973057i	$-0.425942\pi$
$449$	9.77124	0.461133	0.230567	+	0.973057i	$0.425942\pi$
$450$	0	0
$451$	−24.5830	−1.15757
$452$	− 20.0000i	− 0.940721i
$453$	12.5830i	0.591201i
$454$	56.3320	2.64379
$455$	0	0
$456$	7.93725	0.371696
$457$	31.8745i	1.49103i	0.666491	+	0.745513i	$0.267796\pi$
$457$	31.8745i	1.49103i	−0.666491	+	0.745513i	$0.732204\pi$
$458$	− 51.0405i	− 2.38497i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	25.7490	1.19925	0.599626	−	0.800281i	$-0.295316\pi$
$461$	25.7490	1.19925	0.599626	+	0.800281i	$0.295316\pi$
$462$	54.4575i	2.53359i
$463$	− 1.52026i	− 0.0706524i	−0.999376	−	0.0353262i	$-0.988753\pi$
$463$	− 1.52026i	− 0.0706524i	0.999376	−	0.0353262i	$-0.0112470\pi$
$464$	76.3098	3.54259
$465$	0	0
$466$	−47.6235	−2.20612
$467$	− 11.8745i	− 0.549487i	−0.961518	−	0.274743i	$-0.911407\pi$
$467$	− 11.8745i	− 0.549487i	0.961518	−	0.274743i	$-0.0885929\pi$
$468$	28.2288i	1.30487i
$469$	53.1660	2.45498
$470$	0	0
$471$	2.70850	0.124801
$472$	5.62352i	0.258844i
$473$	− 2.00000i	− 0.0919601i
$474$	−38.5830	−1.77218
$475$	0	0
$476$	72.9150	3.34205
$477$	0.708497i	0.0324399i
$478$	− 27.3948i	− 1.25301i
$479$	−9.64575	−0.440726	−0.220363	−	0.975418i	$-0.570724\pi$
$479$	−9.64575	−0.440726	−0.220363	+	0.975418i	$0.570724\pi$
$480$	0	0
$481$	9.29150	0.423656
$482$	− 12.1255i	− 0.552301i
$483$	4.70850i	0.214244i
$484$	−104.373	−4.74421
$485$	0	0
$486$	−2.64575	−0.120014
$487$	− 13.8745i	− 0.628714i	−0.949305	−	0.314357i	$-0.898211\pi$
$487$	− 13.8745i	− 0.628714i	0.949305	−	0.314357i	$-0.101789\pi$
$488$	5.62352i	0.254565i
$489$	−7.64575	−0.345753
$490$	0	0
$491$	−32.2288	−1.45446	−0.727232	−	0.686392i	$-0.759193\pi$
$491$	−32.2288	−1.45446	−0.727232	+	0.686392i	$0.759193\pi$
$492$	21.7712i	0.981523i
$493$	− 27.7490i	− 1.24975i
$494$	−14.9373	−0.672059
$495$	0	0
$496$	66.0000	2.96349
$497$	12.0000i	0.538274i
$498$	15.8745i	0.711354i
$499$	−43.0405	−1.92676	−0.963379	−	0.268143i	$-0.913590\pi$
$499$	−43.0405	−1.92676	−0.963379	+	0.268143i	$0.913590\pi$
$500$	0	0
$501$	10.7085	0.478420
$502$	− 57.2693i	− 2.55605i
$503$	2.00000i	0.0891756i	0.999005	+	0.0445878i	$0.0141974\pi$
$503$	2.00000i	0.0891756i	−0.999005	+	0.0445878i	$0.985803\pi$
$504$	28.9373	1.28897
$505$	0	0
$506$	−19.2915	−0.857612
$507$	− 18.8745i	− 0.838246i
$508$	− 20.0000i	− 0.887357i
$509$	−24.1033	−1.06836	−0.534179	−	0.845371i	$-0.679379\pi$
$509$	−24.1033	−1.06836	−0.534179	+	0.845371i	$0.679379\pi$
$510$	0	0
$511$	−36.4575	−1.61279
$512$	− 29.1033i	− 1.28619i
$513$	− 1.00000i	− 0.0441511i
$514$	70.3320	3.10221
$515$	0	0
$516$	−1.77124	−0.0779746
$517$	52.4575i	2.30708i
$518$	− 15.8745i	− 0.697486i
$519$	0	0
$520$	0	0
$521$	−36.1033	−1.58171	−0.790856	−	0.612002i	$-0.790365\pi$
$521$	−36.1033	−1.58171	−0.790856	+	0.612002i	$0.790365\pi$
$522$	− 18.3542i	− 0.803344i
$523$	12.7085i	0.555704i	0.960624	+	0.277852i	$0.0896224\pi$
$523$	12.7085i	0.555704i	−0.960624	+	0.277852i	$0.910378\pi$
$524$	−61.1438	−2.67108
$525$	0	0
$526$	−43.8745	−1.91302
$527$	− 24.0000i	− 1.04546i
$528$	62.1033i	2.70270i
$529$	21.3320	0.927479
$530$	0	0
$531$	0.708497	0.0307462
$532$	18.2288i	0.790317i
$533$	− 24.5830i	− 1.06481i
$534$	−2.81176	−0.121677
$535$	0	0
$536$	115.749	4.99960
$537$	− 3.29150i	− 0.142039i
$538$	58.8118i	2.53556i
$539$	−35.5203	−1.52997
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	− 40.4575i	− 1.73780i
$543$	6.70850i	0.287889i
$544$	52.9150	2.26871
$545$	0	0
$546$	−54.4575	−2.33057
$547$	− 29.8745i	− 1.27734i	−0.769480	−	0.638671i	$-0.779485\pi$
$547$	− 29.8745i	− 1.27734i	0.769480	−	0.638671i	$-0.220515\pi$
$548$	− 30.0000i	− 1.28154i
$549$	0.708497	0.0302379
$550$	0	0
$551$	6.93725	0.295537
$552$	10.2510i	0.436311i
$553$	− 53.1660i	− 2.26085i
$554$	54.4575	2.31368
$555$	0	0
$556$	−69.3725	−2.94205
$557$	− 32.5830i	− 1.38059i	−0.723530	−	0.690293i	$-0.757482\pi$
$557$	− 32.5830i	− 1.38059i	0.723530	−	0.690293i	$-0.242518\pi$
$558$	− 15.8745i	− 0.672022i
$559$	2.00000	0.0845910
$560$	0	0
$561$	22.5830	0.953455
$562$	− 15.2693i	− 0.644095i
$563$	− 39.8745i	− 1.68051i	−0.542191	−	0.840255i	$-0.682405\pi$
$563$	− 39.8745i	− 1.68051i	0.542191	−	0.840255i	$-0.317595\pi$
$564$	46.4575	1.95621
$565$	0	0
$566$	67.5203	2.83809
$567$	− 3.64575i	− 0.153107i
$568$	26.1255i	1.09620i
$569$	22.9373	0.961580	0.480790	−	0.876836i	$-0.340350\pi$
$569$	22.9373	0.961580	0.480790	+	0.876836i	$0.340350\pi$
$570$	0	0
$571$	27.2915	1.14211	0.571057	−	0.820910i	$-0.306534\pi$
$571$	27.2915	1.14211	0.571057	+	0.820910i	$0.306534\pi$
$572$	− 159.373i	− 6.66370i
$573$	− 20.2288i	− 0.845068i
$574$	−42.0000	−1.75305
$575$	0	0
$576$	13.0000	0.541667
$577$	− 2.70850i	− 0.112756i	−0.998409	−	0.0563781i	$-0.982045\pi$
$577$	− 2.70850i	− 0.112756i	0.998409	−	0.0563781i	$-0.0179552\pi$
$578$	2.64575i	0.110049i
$579$	−6.35425	−0.264074
$580$	0	0
$581$	−21.8745	−0.907508
$582$	− 34.2288i	− 1.41883i
$583$	− 4.00000i	− 0.165663i
$584$	−79.3725	−3.28446
$585$	0	0
$586$	70.3320	2.90539
$587$	− 22.7085i	− 0.937280i	−0.883389	−	0.468640i	$-0.844744\pi$
$587$	− 22.7085i	− 0.937280i	0.883389	−	0.468640i	$-0.155256\pi$
$588$	31.4575i	1.29729i
$589$	6.00000	0.247226
$590$	0	0
$591$	−17.1660	−0.706115
$592$	− 18.1033i	− 0.744040i
$593$	− 24.5830i	− 1.00950i	−0.863265	−	0.504752i	$-0.831584\pi$
$593$	− 24.5830i	− 1.00950i	0.863265	−	0.504752i	$-0.168416\pi$
$594$	14.9373	0.612883
$595$	0	0
$596$	2.91503	0.119404
$597$	3.29150i	0.134712i
$598$	− 19.2915i	− 0.788888i
$599$	−30.5830	−1.24959	−0.624794	−	0.780790i	$-0.714817\pi$
$599$	−30.5830	−1.24959	−0.624794	+	0.780790i	$0.714817\pi$
$600$	0	0
$601$	33.2915	1.35799	0.678994	−	0.734143i	$-0.262416\pi$
$601$	33.2915	1.35799	0.678994	+	0.734143i	$0.262416\pi$
$602$	− 3.41699i	− 0.139266i
$603$	− 14.5830i	− 0.593866i
$604$	−62.9150	−2.55998
$605$	0	0
$606$	3.41699	0.138806
$607$	31.0405i	1.25990i	0.776637	+	0.629948i	$0.216924\pi$
$607$	31.0405i	1.25990i	−0.776637	+	0.629948i	$0.783076\pi$
$608$	13.2288i	0.536497i
$609$	25.2915	1.02486
$610$	0	0
$611$	−52.4575	−2.12220
$612$	− 20.0000i	− 0.808452i
$613$	− 22.4575i	− 0.907050i	−0.891243	−	0.453525i	$-0.850166\pi$
$613$	− 22.4575i	− 0.907050i	0.891243	−	0.453525i	$-0.149834\pi$
$614$	−75.2915	−3.03852
$615$	0	0
$616$	−163.373	−6.58247
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	28.0000i	1.12633i
$619$	−12.7085	−0.510798	−0.255399	−	0.966836i	$-0.582207\pi$
$619$	−12.7085	−0.510798	−0.255399	+	0.966836i	$0.582207\pi$
$620$	0	0
$621$	1.29150	0.0518262
$622$	18.6863i	0.749251i
$623$	− 3.87451i	− 0.155229i
$624$	−62.1033	−2.48612
$625$	0	0
$626$	17.7490	0.709393
$627$	5.64575i	0.225470i
$628$	13.5425i	0.540404i
$629$	−6.58301	−0.262482
$630$	0	0
$631$	6.58301	0.262065	0.131033	−	0.991378i	$-0.458171\pi$
$631$	6.58301	0.262065	0.131033	+	0.991378i	$0.458171\pi$
$632$	− 115.749i	− 4.60425i
$633$	− 18.5830i	− 0.738608i
$634$	85.8745	3.41051
$635$	0	0
$636$	−3.54249	−0.140469
$637$	− 35.5203i	− 1.40736i
$638$	103.624i	4.10249i
$639$	3.29150	0.130210
$640$	0	0
$641$	−1.06275	−0.0419759	−0.0209880	−	0.999780i	$-0.506681\pi$
$641$	−1.06275	−0.0419759	−0.0209880	+	0.999780i	$0.506681\pi$
$642$	0	0
$643$	8.35425i	0.329459i	0.986339	+	0.164730i	$0.0526752\pi$
$643$	8.35425i	0.329459i	−0.986339	+	0.164730i	$0.947325\pi$
$644$	−23.5425	−0.927704
$645$	0	0
$646$	10.5830	0.416383
$647$	− 24.5830i	− 0.966458i	−0.875494	−	0.483229i	$-0.839464\pi$
$647$	− 24.5830i	− 0.966458i	0.875494	−	0.483229i	$-0.160536\pi$
$648$	− 7.93725i	− 0.311805i
$649$	−4.00000	−0.157014
$650$	0	0
$651$	21.8745	0.857330
$652$	− 38.2288i	− 1.49715i
$653$	− 30.5830i	− 1.19681i	−0.801195	−	0.598403i	$-0.795802\pi$
$653$	− 30.5830i	− 1.19681i	0.801195	−	0.598403i	$-0.204198\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−47.8967	−1.87005
$657$	10.0000i	0.390137i
$658$	89.6235i	3.49389i
$659$	−46.5830	−1.81462	−0.907308	−	0.420466i	$-0.861866\pi$
$659$	−46.5830	−1.81462	−0.907308	+	0.420466i	$0.861866\pi$
$660$	0	0
$661$	−9.29150	−0.361398	−0.180699	−	0.983538i	$-0.557836\pi$
$661$	−9.29150	−0.361398	−0.180699	+	0.983538i	$0.557836\pi$
$662$	86.2065i	3.35051i
$663$	22.5830i	0.877051i
$664$	−47.6235	−1.84815
$665$	0	0
$666$	−4.35425	−0.168724
$667$	8.95948i	0.346913i
$668$	53.5425i	2.07162i
$669$	−18.5830	−0.718460
$670$	0	0
$671$	−4.00000	−0.154418
$672$	48.2288i	1.86046i
$673$	28.9373i	1.11545i	0.830026	+	0.557725i	$0.188325\pi$
$673$	28.9373i	1.11545i	−0.830026	+	0.557725i	$0.811675\pi$
$674$	2.47974	0.0955160
$675$	0	0
$676$	94.3725	3.62971
$677$	− 12.4575i	− 0.478781i	−0.970923	−	0.239391i	$-0.923052\pi$
$677$	− 12.4575i	− 0.478781i	0.970923	−	0.239391i	$-0.0769476\pi$
$678$	− 10.5830i	− 0.406438i
$679$	47.1660	1.81007
$680$	0	0
$681$	21.2915	0.815892
$682$	89.6235i	3.43186i
$683$	43.7490i	1.67401i	0.547196	+	0.837005i	$0.315695\pi$
$683$	43.7490i	1.67401i	−0.547196	+	0.837005i	$0.684305\pi$
$684$	5.00000	0.191180
$685$	0	0
$686$	6.83399	0.260923
$687$	− 19.2915i	− 0.736017i
$688$	− 3.89674i	− 0.148562i
$689$	4.00000	0.152388
$690$	0	0
$691$	31.0405	1.18084	0.590418	−	0.807097i	$-0.298963\pi$
$691$	31.0405	1.18084	0.590418	+	0.807097i	$0.298963\pi$
$692$	0	0
$693$	20.5830i	0.781884i
$694$	−68.7895	−2.61122
$695$	0	0
$696$	55.0627	2.08715
$697$	17.4170i	0.659716i
$698$	50.7085i	1.91934i
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−4.58301	−0.173098	−0.0865489	−	0.996248i	$-0.527584\pi$
$701$	−4.58301	−0.173098	−0.0865489	+	0.996248i	$0.527584\pi$
$702$	14.9373i	0.563770i
$703$	− 1.64575i	− 0.0620707i
$704$	−73.3948	−2.76617
$705$	0	0
$706$	−54.4575	−2.04954
$707$	4.70850i	0.177081i
$708$	3.54249i	0.133135i
$709$	15.2915	0.574284	0.287142	−	0.957888i	$-0.407295\pi$
$709$	15.2915	0.574284	0.287142	+	0.957888i	$0.407295\pi$
$710$	0	0
$711$	−14.5830	−0.546905
$712$	− 8.43529i	− 0.316126i
$713$	7.74902i	0.290203i
$714$	38.5830	1.44393
$715$	0	0
$716$	16.4575	0.615046
$717$	− 10.3542i	− 0.386687i
$718$	− 58.4797i	− 2.18244i
$719$	14.8118	0.552386	0.276193	−	0.961102i	$-0.410927\pi$
$719$	14.8118	0.552386	0.276193	+	0.961102i	$0.410927\pi$
$720$	0	0
$721$	−38.5830	−1.43691
$722$	2.64575i	0.0984647i
$723$	− 4.58301i	− 0.170444i
$724$	−33.5425	−1.24660
$725$	0	0
$726$	−55.2288	−2.04973
$727$	− 44.1033i	− 1.63570i	−0.575432	−	0.817850i	$-0.695166\pi$
$727$	− 44.1033i	− 1.63570i	0.575432	−	0.817850i	$-0.304834\pi$
$728$	− 163.373i	− 6.05499i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−1.41699	−0.0524094
$732$	3.54249i	0.130934i
$733$	− 28.5830i	− 1.05574i	−0.849326	−	0.527869i	$-0.822991\pi$
$733$	− 28.5830i	− 1.05574i	0.849326	−	0.527869i	$-0.177009\pi$
$734$	9.64575	0.356031
$735$	0	0
$736$	−17.0850	−0.629760
$737$	82.3320i	3.03274i
$738$	11.5203i	0.424067i
$739$	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
$741$	−5.64575	−0.207402
$742$	− 6.83399i	− 0.250884i
$743$	− 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
$746$	55.3948	2.02815
$747$	6.00000i	0.219529i
$748$	112.915i	4.12858i
$749$	0	0
$750$	0	0
$751$	−23.4170	−0.854498	−0.427249	−	0.904134i	$-0.640517\pi$
$751$	−23.4170	−0.854498	−0.427249	+	0.904134i	$0.640517\pi$
$752$	102.207i	3.72709i
$753$	− 21.6458i	− 0.788815i
$754$	−103.624	−3.77375
$755$	0	0
$756$	18.2288	0.662973
$757$	7.87451i	0.286204i	0.989708	+	0.143102i	$0.0457077\pi$
$757$	7.87451i	0.286204i	−0.989708	+	0.143102i	$0.954292\pi$
$758$	− 26.4575i	− 0.960980i
$759$	−7.29150	−0.264665
$760$	0	0
$761$	−37.7490	−1.36840	−0.684200	−	0.729294i	$-0.739849\pi$
$761$	−37.7490	−1.36840	−0.684200	+	0.729294i	$0.739849\pi$
$762$	− 10.5830i	− 0.383382i
$763$	− 19.2915i	− 0.698399i
$764$	101.144	3.65925
$765$	0	0
$766$	6.83399	0.246922
$767$	− 4.00000i	− 0.144432i
$768$	− 5.00000i	− 0.180422i
$769$	−45.7490	−1.64975	−0.824876	−	0.565314i	$-0.808755\pi$
$769$	−45.7490	−1.64975	−0.824876	+	0.565314i	$0.808755\pi$
$770$	0	0
$771$	26.5830	0.957364
$772$	− 31.7712i	− 1.14347i
$773$	19.2915i	0.693867i	0.937890	+	0.346934i	$0.112777\pi$
$773$	19.2915i	0.693867i	−0.937890	+	0.346934i	$0.887223\pi$
$774$	−0.937254	−0.0336889
$775$	0	0
$776$	102.686	3.68622
$777$	− 6.00000i	− 0.215249i
$778$	15.8745i	0.569129i
$779$	−4.35425	−0.156007
$780$	0	0
$781$	−18.5830	−0.664952
$782$	13.6680i	0.488766i
$783$	− 6.93725i	− 0.247917i
$784$	−69.2065	−2.47166
$785$	0	0
$786$	−32.3542	−1.15404
$787$	6.12549i	0.218350i	0.994023	+	0.109175i	$0.0348209\pi$
$787$	6.12549i	0.218350i	−0.994023	+	0.109175i	$0.965179\pi$
$788$	− 85.8301i	− 3.05757i
$789$	−16.5830	−0.590371
$790$	0	0
$791$	14.5830	0.518512
$792$	44.8118i	1.59232i
$793$	− 4.00000i	− 0.142044i
$794$	5.29150	0.187788
$795$	0	0
$796$	−16.4575	−0.583321
$797$	40.0000i	1.41687i	0.705775	+	0.708436i	$0.250599\pi$
$797$	40.0000i	1.41687i	−0.705775	+	0.708436i	$0.749401\pi$
$798$	9.64575i	0.341456i
$799$	37.1660	1.31484
$800$	0	0
$801$	−1.06275	−0.0375503
$802$	28.9373i	1.02181i
$803$	− 56.4575i	− 1.99234i
$804$	72.9150	2.57151
$805$	0	0
$806$	−89.6235	−3.15685
$807$	22.2288i	0.782489i
$808$	10.2510i	0.360628i
$809$	40.5830	1.42682	0.713411	−	0.700746i	$-0.247149\pi$
$809$	40.5830	1.42682	0.713411	+	0.700746i	$0.247149\pi$
$810$	0	0
$811$	−46.3320	−1.62694	−0.813469	−	0.581609i	$-0.802423\pi$
$811$	−46.3320	−1.62694	−0.813469	+	0.581609i	$0.802423\pi$
$812$	126.458i	4.43779i
$813$	− 15.2915i	− 0.536296i
$814$	24.5830	0.861634
$815$	0	0
$816$	44.0000	1.54031
$817$	− 0.354249i	− 0.0123936i
$818$	45.7490i	1.59958i
$819$	−20.5830	−0.719228
$820$	0	0
$821$	−47.6235	−1.66207	−0.831036	−	0.556218i	$-0.812252\pi$
$821$	−47.6235	−1.66207	−0.831036	+	0.556218i	$0.812252\pi$
$822$	− 15.8745i	− 0.553687i
$823$	22.2288i	0.774846i	0.921902	+	0.387423i	$0.126635\pi$
$823$	22.2288i	0.774846i	−0.921902	+	0.387423i	$0.873365\pi$
$824$	−84.0000	−2.92628
$825$	0	0
$826$	−6.83399	−0.237785
$827$	− 22.4575i	− 0.780924i	−0.920619	−	0.390462i	$-0.872315\pi$
$827$	− 22.4575i	− 0.780924i	0.920619	−	0.390462i	$-0.127685\pi$
$828$	6.45751i	0.224414i
$829$	−13.2915	−0.461633	−0.230816	−	0.972997i	$-0.574140\pi$
$829$	−13.2915	−0.461633	−0.230816	+	0.972997i	$0.574140\pi$
$830$	0	0
$831$	20.5830	0.714017
$832$	− 73.3948i	− 2.54451i
$833$	25.1660i	0.871951i
$834$	−36.7085	−1.27111
$835$	0	0
$836$	−28.2288	−0.976312
$837$	− 6.00000i	− 0.207390i
$838$	− 61.0183i	− 2.10784i
$839$	−40.4575	−1.39675	−0.698374	−	0.715733i	$-0.746093\pi$
$839$	−40.4575	−1.39675	−0.698374	+	0.715733i	$0.746093\pi$
$840$	0	0
$841$	19.1255	0.659500
$842$	19.6235i	0.676271i
$843$	− 5.77124i	− 0.198772i
$844$	92.9150	3.19827
$845$	0	0
$846$	24.5830	0.845181
$847$	− 76.1033i	− 2.61494i
$848$	− 7.79347i	− 0.267629i
$849$	25.5203	0.875853
$850$	0	0
$851$	2.12549	0.0728609
$852$	16.4575i	0.563825i
$853$	− 25.2915i	− 0.865965i	−0.901402	−	0.432982i	$-0.857461\pi$
$853$	− 25.2915i	− 0.865965i	0.901402	−	0.432982i	$-0.142539\pi$
$854$	−6.83399	−0.233854
$855$	0	0
$856$	0	0
$857$	− 43.0405i	− 1.47024i	−0.677939	−	0.735118i	$-0.737127\pi$
$857$	− 43.0405i	− 1.47024i	0.677939	−	0.735118i	$-0.262873\pi$
$858$	− 84.3320i	− 2.87905i
$859$	−50.3320	−1.71731	−0.858653	−	0.512557i	$-0.828698\pi$
$859$	−50.3320	−1.71731	−0.858653	+	0.512557i	$0.828698\pi$
$860$	0	0
$861$	−15.8745	−0.541002
$862$	− 19.2915i	− 0.657071i
$863$	20.0000i	0.680808i	0.940279	+	0.340404i	$0.110564\pi$
$863$	20.0000i	0.680808i	−0.940279	+	0.340404i	$0.889436\pi$
$864$	13.2288	0.450051
$865$	0	0
$866$	−59.1438	−2.00979
$867$	1.00000i	0.0339618i
$868$	109.373i	3.71235i
$869$	82.3320	2.79292
$870$	0	0
$871$	−82.3320	−2.78971
$872$	− 42.0000i	− 1.42230i
$873$	− 12.9373i	− 0.437860i
$874$	−3.41699	−0.115582
$875$	0	0
$876$	−50.0000	−1.68934
$877$	− 40.9373i	− 1.38235i	−0.722686	−	0.691176i	$-0.757093\pi$
$877$	− 40.9373i	− 1.38235i	0.722686	−	0.691176i	$-0.242907\pi$
$878$	59.7490i	2.01643i
$879$	26.5830	0.896623
$880$	0	0
$881$	−31.8745	−1.07388	−0.536940	−	0.843621i	$-0.680420\pi$
$881$	−31.8745	−1.07388	−0.536940	+	0.843621i	$0.680420\pi$
$882$	16.6458i	0.560492i
$883$	36.8118i	1.23881i	0.785070	+	0.619407i	$0.212627\pi$
$883$	36.8118i	1.23881i	−0.785070	+	0.619407i	$0.787373\pi$
$884$	−112.915	−3.79774
$885$	0	0
$886$	98.6640	3.31468
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	− 13.0627i	− 0.438357i
$889$	14.5830	0.489098
$890$	0	0
$891$	5.64575	0.189140
$892$	− 92.9150i	− 3.11103i
$893$	9.29150i	0.310928i
$894$	1.54249	0.0515885
$895$	0	0
$896$	−28.9373	−0.966726
$897$	− 7.29150i	− 0.243456i
$898$	25.8523i	0.862702i
$899$	41.6235	1.38822
$900$	0	0
$901$	−2.83399	−0.0944139
$902$	− 65.0405i	− 2.16561i
$903$	− 1.29150i	− 0.0429785i
$904$	31.7490	1.05596
$905$	0	0
$906$	−33.2915	−1.10604
$907$	27.0405i	0.897866i	0.893566	+	0.448933i	$0.148196\pi$
$907$	27.0405i	0.897866i	−0.893566	+	0.448933i	$0.851804\pi$
$908$	106.458i	3.53292i
$909$	1.29150	0.0428364
$910$	0	0
$911$	5.41699	0.179473	0.0897365	−	0.995966i	$-0.471398\pi$
$911$	5.41699	0.179473	0.0897365	+	0.995966i	$0.471398\pi$
$912$	11.0000i	0.364246i
$913$	− 33.8745i	− 1.12108i
$914$	−84.3320	−2.78946
$915$	0	0
$916$	96.4575	3.18705
$917$	− 44.5830i	− 1.47226i
$918$	− 10.5830i	− 0.349291i
$919$	−57.1660	−1.88573	−0.942866	−	0.333171i	$-0.891881\pi$
$919$	−57.1660	−1.88573	−0.942866	+	0.333171i	$0.891881\pi$
$920$	0	0
$921$	−28.4575	−0.937707
$922$	68.1255i	2.24359i
$923$	− 18.5830i	− 0.611667i
$924$	−102.915	−3.38566
$925$	0	0
$926$	4.02223	0.132179
$927$	10.5830i	0.347591i
$928$	91.7712i	3.01254i
$929$	−19.8745	−0.652061	−0.326031	−	0.945359i	$-0.605711\pi$
$929$	−19.8745	−0.652061	−0.326031	+	0.945359i	$0.605711\pi$
$930$	0	0
$931$	−6.29150	−0.206196
$932$	− 90.0000i	− 2.94805i
$933$	7.06275i	0.231224i
$934$	31.4170	1.02800
$935$	0	0
$936$	−44.8118	−1.46472
$937$	51.8745i	1.69467i	0.531062	+	0.847333i	$0.321793\pi$
$937$	51.8745i	1.69467i	−0.531062	+	0.847333i	$0.678207\pi$
$938$	140.664i	4.59284i
$939$	6.70850	0.218924
$940$	0	0
$941$	−38.2288	−1.24622	−0.623111	−	0.782133i	$-0.714131\pi$
$941$	−38.2288	−1.24622	−0.623111	+	0.782133i	$0.714131\pi$
$942$	7.16601i	0.233481i
$943$	− 5.62352i	− 0.183127i
$944$	−7.79347	−0.253656
$945$	0	0
$946$	5.29150	0.172042
$947$	32.5830i	1.05881i	0.848371	+	0.529403i	$0.177584\pi$
$947$	32.5830i	1.05881i	−0.848371	+	0.529403i	$0.822416\pi$
$948$	− 72.9150i	− 2.36817i
$949$	56.4575	1.83269
$950$	0	0
$951$	32.4575	1.05251
$952$	115.749i	3.75145i
$953$	− 10.5830i	− 0.342817i	−0.985200	−	0.171409i	$-0.945168\pi$
$953$	− 10.5830i	− 0.342817i	0.985200	−	0.171409i	$-0.0548318\pi$
$954$	−1.87451	−0.0606894
$955$	0	0
$956$	51.7712	1.67440
$957$	39.1660i	1.26606i
$958$	− 25.5203i	− 0.824522i
$959$	21.8745	0.706365
$960$	0	0
$961$	5.00000	0.161290
$962$	24.5830i	0.792588i
$963$	0	0
$964$	22.9150	0.738043
$965$	0	0
$966$	−12.4575	−0.400814
$967$	− 36.3542i	− 1.16907i	−0.811367	−	0.584537i	$-0.801276\pi$
$967$	− 36.3542i	− 1.16907i	0.811367	−	0.584537i	$-0.198724\pi$
$968$	− 165.686i	− 5.32536i
$969$	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
$973$	− 50.5830i	− 1.62162i
$974$	36.7085	1.17622
$975$	0	0
$976$	−7.79347	−0.249463
$977$	23.0405i	0.737131i	0.929602	+	0.368566i	$0.120151\pi$
$977$	23.0405i	0.737131i	−0.929602	+	0.368566i	$0.879849\pi$
$978$	− 20.2288i	− 0.646844i
$979$	6.00000	0.191761
$980$	0	0
$981$	−5.29150	−0.168945
$982$	− 85.2693i	− 2.72105i
$983$	18.4575i	0.588703i	0.955697	+	0.294352i	$0.0951037\pi$
$983$	18.4575i	0.588703i	−0.955697	+	0.294352i	$0.904896\pi$
$984$	−34.5608	−1.10176
$985$	0	0
$986$	73.4170	2.33807
$987$	33.8745i	1.07824i
$988$	− 28.2288i	− 0.898076i
$989$	0.457513	0.0145481
$990$	0	0
$991$	27.7490	0.881477	0.440738	−	0.897636i	$-0.354717\pi$
$991$	27.7490	0.881477	0.440738	+	0.897636i	$0.354717\pi$
$992$	79.3725i	2.52008i
$993$	32.5830i	1.03399i
$994$	−31.7490	−1.00702
$995$	0	0
$996$	−30.0000	−0.950586
$997$	37.2915i	1.18103i	0.807025	+	0.590517i	$0.201076\pi$
$997$	37.2915i	1.18103i	−0.807025	+	0.590517i	$0.798924\pi$
$998$	− 113.875i	− 3.60463i
$999$	−1.64575	−0.0520693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.i.799.4		4
5.2	odd	4		1425.2.a.p.1.1		2
5.3	odd	4		285.2.a.d.1.2	✓	2
5.4	even	2	inner	1425.2.c.i.799.1		4
15.2	even	4		4275.2.a.u.1.2		2
15.8	even	4		855.2.a.g.1.1		2
20.3	even	4		4560.2.a.bo.1.2		2
95.18	even	4		5415.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.d.1.2	✓	2	5.3	odd	4
855.2.a.g.1.1		2	15.8	even	4
1425.2.a.p.1.1		2	5.2	odd	4
1425.2.c.i.799.1		4	5.4	even	2	inner
1425.2.c.i.799.4		4	1.1	even	1	trivial
4275.2.a.u.1.2		2	15.2	even	4
4560.2.a.bo.1.2		2	20.3	even	4
5415.2.a.s.1.1		2	95.18	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)

\(f(q)\)	\(=\)	\(q+2.64575i q^{2} +1.00000i q^{3} -5.00000 q^{4} -2.64575 q^{6} -3.64575i q^{7} -7.93725i q^{8} -1.00000 q^{9} +5.64575 q^{11} -5.00000i q^{12} +5.64575i q^{13} +9.64575 q^{14} +11.0000 q^{16} -4.00000i q^{17} -2.64575i q^{18} +1.00000 q^{19} +3.64575 q^{21} +14.9373i q^{22} +1.29150i q^{23} +7.93725 q^{24} -14.9373 q^{26} -1.00000i q^{27} +18.2288i q^{28} +6.93725 q^{29} +6.00000 q^{31} +13.2288i q^{32} +5.64575i q^{33} +10.5830 q^{34} +5.00000 q^{36} -1.64575i q^{37} +2.64575i q^{38} -5.64575 q^{39} -4.35425 q^{41} +9.64575i q^{42} -0.354249i q^{43} -28.2288 q^{44} -3.41699 q^{46} +9.29150i q^{47} +11.0000i q^{48} -6.29150 q^{49} +4.00000 q^{51} -28.2288i q^{52} -0.708497i q^{53} +2.64575 q^{54} -28.9373 q^{56} +1.00000i q^{57} +18.3542i q^{58} -0.708497 q^{59} -0.708497 q^{61} +15.8745i q^{62} +3.64575i q^{63} -13.0000 q^{64} -14.9373 q^{66} +14.5830i q^{67} +20.0000i q^{68} -1.29150 q^{69} -3.29150 q^{71} +7.93725i q^{72} -10.0000i q^{73} +4.35425 q^{74} -5.00000 q^{76} -20.5830i q^{77} -14.9373i q^{78} +14.5830 q^{79} +1.00000 q^{81} -11.5203i q^{82} -6.00000i q^{83} -18.2288 q^{84} +0.937254 q^{86} +6.93725i q^{87} -44.8118i q^{88} +1.06275 q^{89} +20.5830 q^{91} -6.45751i q^{92} +6.00000i q^{93} -24.5830 q^{94} -13.2288 q^{96} +12.9373i q^{97} -16.6458i q^{98} -5.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{4} - 4 q^{9} + 12 q^{11} + 28 q^{14} + 44 q^{16} + 4 q^{19} + 4 q^{21} - 28 q^{26} - 4 q^{29} + 24 q^{31} + 20 q^{36} - 12 q^{39} - 28 q^{41} - 60 q^{44} - 56 q^{46} - 4 q^{49} + 16 q^{51} - 84 q^{56}+ \cdots - 12 q^{99}+O(q^{100}) \) 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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