LMFDB - Embedded newform 1425.2.c.i.799.1

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

$q$-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	$-0.615027\pi$
$2$	− 2.64575i	− 1.87083i	0.353553	−	0.935414i	$-0.384973\pi$
$3$	− 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	$0.241942\pi$
$7$	3.64575i	1.37796i	−0.724778	+	0.688982i	$0.758058\pi$
$8$	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	$-0.713727\pi$
$13$	− 5.64575i	− 1.56585i	0.622116	−	0.782925i	$-0.286273\pi$
$14$	9.64575	2.57794
$15$	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
$21$	3.64575	0.795568
$22$	− 14.9373i	− 3.18463i
$23$	− 1.29150i	− 0.269297i	−0.990893	−	0.134648i	$-0.957009\pi$
$23$	− 1.29150i	− 0.269297i	0.990893	−	0.134648i	$-0.0429905\pi$
$24$	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	$0.0431934\pi$
$37$	1.64575i	0.270560i	−0.990807	+	0.135280i	$0.956807\pi$
$38$	− 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	$0.00859898\pi$
$43$	0.354249i	0.0540224i	−0.999635	+	0.0270112i	$0.991401\pi$
$44$	−28.2288	−4.25565
$45$	0	0
$46$	−3.41699	−0.503808
$47$	− 9.29150i	− 1.35530i	−0.735382	−	0.677652i	$-0.762997\pi$
$47$	− 9.29150i	− 1.35530i	0.735382	−	0.677652i	$-0.237003\pi$
$48$	− 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	$0.0154950\pi$
$53$	0.708497i	0.0973196i	−0.998815	+	0.0486598i	$0.984505\pi$
$54$	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	$-0.650146\pi$
$67$	− 14.5830i	− 1.78160i	0.454398	−	0.890799i	$-0.349854\pi$
$68$	− 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	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	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	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	−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	$-0.771914\pi$
$97$	− 12.9373i	− 1.31358i	0.754074	−	0.656790i	$-0.228086\pi$
$98$	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	$0.174585\pi$
$103$	10.5830i	1.04277i	−0.853320	+	0.521387i	$0.825415\pi$
$104$	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	$-0.939753\pi$
$113$	− 4.00000i	− 0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$	−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	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	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	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	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	$0.0344705\pi$
$157$	2.70850i	0.216162i	−0.994142	+	0.108081i	$0.965529\pi$
$158$	− 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	$-0.903203\pi$
$163$	− 7.64575i	− 0.598861i	0.954118	−	0.299431i	$-0.0967967\pi$
$164$	21.7712	1.70005
$165$	0	0
$166$	15.8745	1.23210
$167$	10.7085i	0.828648i	0.910129	+	0.414324i	$0.135982\pi$
$167$	10.7085i	0.828648i	−0.910129	+	0.414324i	$0.864018\pi$
$168$	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	$-0.926554\pi$
$193$	− 6.35425i	− 0.457389i	0.973498	−	0.228694i	$-0.0734457\pi$
$194$	−34.2288	−2.45748
$195$	0	0
$196$	31.4575	2.24697
$197$	− 17.1660i	− 1.22303i	−0.791234	−	0.611514i	$-0.790561\pi$
$197$	− 17.1660i	− 1.22303i	0.791234	−	0.611514i	$-0.209439\pi$
$198$	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	$-0.786237\pi$
$223$	− 18.5830i	− 1.24441i	0.782854	−	0.622205i	$-0.213763\pi$
$224$	48.2288	3.22242
$225$	0	0
$226$	−10.5830	−0.703971
$227$	21.2915i	1.41317i	0.707630	+	0.706583i	$0.249764\pi$
$227$	21.2915i	1.41317i	−0.707630	+	0.706583i	$0.750236\pi$
$228$	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	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	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	$0.311147\pi$
$257$	26.5830i	1.65820i	−0.559099	+	0.829101i	$0.688853\pi$
$258$	− 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	$-0.829173\pi$
$263$	− 16.5830i	− 1.02255i	0.859417	−	0.511276i	$-0.170827\pi$
$264$	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	$0.212201\pi$
$277$	20.5830i	1.23671i	−0.785898	+	0.618356i	$0.787799\pi$
$278$	− 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	$0.274073\pi$
$283$	25.5203i	1.51702i	−0.651660	+	0.758511i	$0.725927\pi$
$284$	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	$0.283006\pi$
$293$	26.5830i	1.55300i	−0.630120	+	0.776498i	$0.716994\pi$
$294$	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	$-0.698336\pi$
$307$	− 28.4575i	− 1.62416i	0.583549	−	0.812078i	$-0.301664\pi$
$308$	− 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	$0.0607170\pi$
$313$	6.70850i	0.379187i	−0.981863	+	0.189593i	$0.939283\pi$
$314$	7.16601	0.404401
$315$	0	0
$316$	−72.9150	−4.10179
$317$	32.4575i	1.82300i	0.411305	+	0.911498i	$0.365073\pi$
$317$	32.4575i	1.82300i	−0.411305	+	0.911498i	$0.634927\pi$
$318$	− 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	$0.00812661\pi$
$337$	0.937254i	0.0510555i	−0.999674	+	0.0255277i	$0.991873\pi$
$338$	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	$-0.754128\pi$
$347$	− 26.0000i	− 1.39575i	0.716218	−	0.697877i	$-0.245872\pi$
$348$	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	$-0.815480\pi$
$353$	− 20.5830i	− 1.09552i	0.836635	−	0.547761i	$-0.184520\pi$
$354$	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	$0.0303341\pi$
$367$	3.64575i	0.190307i	−0.995463	+	0.0951533i	$0.969666\pi$
$368$	− 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	$0.182350\pi$
$373$	20.9373i	1.08409i	−0.840349	+	0.542045i	$0.817650\pi$
$374$	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	$0.0210214\pi$
$383$	2.58301i	0.131985i	−0.997820	+	0.0659927i	$0.978979\pi$
$384$	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	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 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	$-0.819505\pi$
$433$	− 22.3542i	− 1.07428i	0.843494	−	0.537138i	$-0.180495\pi$
$434$	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	$0.346449\pi$
$443$	37.2915i	1.77177i	−0.463902	+	0.885886i	$0.653551\pi$
$444$	−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	$-0.732204\pi$
$457$	− 31.8745i	− 1.49103i	0.666491	−	0.745513i	$-0.267796\pi$
$458$	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	$0.0112470\pi$
$463$	1.52026i	0.0706524i	−0.999376	+	0.0353262i	$0.988753\pi$
$464$	76.3098	3.54259
$465$	0	0
$466$	−47.6235	−2.20612
$467$	11.8745i	0.549487i	0.961518	+	0.274743i	$0.0885929\pi$
$467$	11.8745i	0.549487i	−0.961518	+	0.274743i	$0.911407\pi$
$468$	− 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	$0.101789\pi$
$487$	13.8745i	0.628714i	−0.949305	+	0.314357i	$0.898211\pi$
$488$	− 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	$-0.985803\pi$
$503$	− 2.00000i	− 0.0891756i	0.999005	−	0.0445878i	$-0.0141974\pi$
$504$	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	$-0.910378\pi$
$523$	− 12.7085i	− 0.555704i	0.960624	−	0.277852i	$-0.0896224\pi$
$524$	−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	$0.220515\pi$
$547$	29.8745i	1.27734i	−0.769480	+	0.638671i	$0.779485\pi$
$548$	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	$0.242518\pi$
$557$	32.5830i	1.38059i	−0.723530	+	0.690293i	$0.757482\pi$
$558$	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	$0.317595\pi$
$563$	39.8745i	1.68051i	−0.542191	+	0.840255i	$0.682405\pi$
$564$	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	$0.0179552\pi$
$577$	2.70850i	0.112756i	−0.998409	+	0.0563781i	$0.982045\pi$
$578$	− 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	$0.155256\pi$
$587$	22.7085i	0.937280i	−0.883389	+	0.468640i	$0.844744\pi$
$588$	− 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	$0.168416\pi$
$593$	24.5830i	1.00950i	−0.863265	+	0.504752i	$0.831584\pi$
$594$	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	$-0.783076\pi$
$607$	− 31.0405i	− 1.25990i	0.776637	−	0.629948i	$-0.216924\pi$
$608$	− 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	$0.149834\pi$
$613$	22.4575i	0.907050i	−0.891243	+	0.453525i	$0.850166\pi$
$614$	−75.2915	−3.03852
$615$	0	0
$616$	−163.373	−6.58247
$617$	− 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
$619$	−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	$-0.947325\pi$
$643$	− 8.35425i	− 0.329459i	0.986339	−	0.164730i	$-0.0526752\pi$
$644$	−23.5425	−0.927704
$645$	0	0
$646$	10.5830	0.416383
$647$	24.5830i	0.966458i	0.875494	+	0.483229i	$0.160536\pi$
$647$	24.5830i	0.966458i	−0.875494	+	0.483229i	$0.839464\pi$
$648$	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	$0.204198\pi$
$653$	30.5830i	1.19681i	−0.801195	+	0.598403i	$0.795802\pi$
$654$	−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	$-0.811675\pi$
$673$	− 28.9373i	− 1.11545i	0.830026	−	0.557725i	$-0.188325\pi$
$674$	2.47974	0.0955160
$675$	0	0
$676$	94.3725	3.62971
$677$	12.4575i	0.478781i	0.970923	+	0.239391i	$0.0769476\pi$
$677$	12.4575i	0.478781i	−0.970923	+	0.239391i	$0.923052\pi$
$678$	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	$-0.684305\pi$
$683$	− 43.7490i	− 1.67401i	0.547196	−	0.837005i	$-0.315695\pi$
$684$	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	$0.304834\pi$
$727$	44.1033i	1.63570i	−0.575432	+	0.817850i	$0.695166\pi$
$728$	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	$0.177009\pi$
$733$	28.5830i	1.05574i	−0.849326	+	0.527869i	$0.822991\pi$
$734$	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	$0.0621872\pi$
$743$	10.5830i	0.388253i	−0.980977	+	0.194126i	$0.937813\pi$
$744$	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	$-0.954292\pi$
$757$	− 7.87451i	− 0.286204i	0.989708	−	0.143102i	$-0.0457077\pi$
$758$	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	$-0.887223\pi$
$773$	− 19.2915i	− 0.693867i	0.937890	−	0.346934i	$-0.112777\pi$
$774$	−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	$-0.965179\pi$
$787$	− 6.12549i	− 0.218350i	0.994023	−	0.109175i	$-0.0348209\pi$
$788$	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	$-0.749401\pi$
$797$	− 40.0000i	− 1.41687i	0.705775	−	0.708436i	$-0.250599\pi$
$798$	− 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	$-0.873365\pi$
$823$	− 22.2288i	− 0.774846i	0.921902	−	0.387423i	$-0.126635\pi$
$824$	−84.0000	−2.92628
$825$	0	0
$826$	−6.83399	−0.237785
$827$	22.4575i	0.780924i	0.920619	+	0.390462i	$0.127685\pi$
$827$	22.4575i	0.780924i	−0.920619	+	0.390462i	$0.872315\pi$
$828$	− 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	$0.142539\pi$
$853$	25.2915i	0.865965i	−0.901402	+	0.432982i	$0.857461\pi$
$854$	−6.83399	−0.233854
$855$	0	0
$856$	0	0
$857$	43.0405i	1.47024i	0.677939	+	0.735118i	$0.262873\pi$
$857$	43.0405i	1.47024i	−0.677939	+	0.735118i	$0.737127\pi$
$858$	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	$-0.889436\pi$
$863$	− 20.0000i	− 0.680808i	0.940279	−	0.340404i	$-0.110564\pi$
$864$	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	$0.242907\pi$
$877$	40.9373i	1.38235i	−0.722686	+	0.691176i	$0.757093\pi$
$878$	− 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	$-0.787373\pi$
$883$	− 36.8118i	− 1.23881i	0.785070	−	0.619407i	$-0.212627\pi$
$884$	−112.915	−3.79774
$885$	0	0
$886$	98.6640	3.31468
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	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	$-0.851804\pi$
$907$	− 27.0405i	− 0.897866i	0.893566	−	0.448933i	$-0.148196\pi$
$908$	− 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	$-0.678207\pi$
$937$	− 51.8745i	− 1.69467i	0.531062	−	0.847333i	$-0.321793\pi$
$938$	− 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	$-0.822416\pi$
$947$	− 32.5830i	− 1.05881i	0.848371	−	0.529403i	$-0.177584\pi$
$948$	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	$0.0548318\pi$
$953$	10.5830i	0.342817i	−0.985200	+	0.171409i	$0.945168\pi$
$954$	−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	$0.198724\pi$
$967$	36.3542i	1.16907i	−0.811367	+	0.584537i	$0.801276\pi$
$968$	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	$-0.879849\pi$
$977$	− 23.0405i	− 0.737131i	0.929602	−	0.368566i	$-0.120151\pi$
$978$	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	$-0.904896\pi$
$983$	− 18.4575i	− 0.588703i	0.955697	−	0.294352i	$-0.0951037\pi$
$984$	−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	$-0.798924\pi$
$997$	− 37.2915i	− 1.18103i	0.807025	−	0.590517i	$-0.201076\pi$
$998$	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.799.1		4
5.2	odd	4		285.2.a.d.1.2	✓	2
5.3	odd	4		1425.2.a.p.1.1		2
5.4	even	2	inner	1425.2.c.i.799.4		4
15.2	even	4		855.2.a.g.1.1		2
15.8	even	4		4275.2.a.u.1.2		2
20.7	even	4		4560.2.a.bo.1.2		2
95.37	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	5.2	odd	4
855.2.a.g.1.1		2	15.2	even	4
1425.2.a.p.1.1		2	5.3	odd	4
1425.2.c.i.799.1		4	1.1	even	1	trivial
1425.2.c.i.799.4		4	5.4	even	2	inner
4275.2.a.u.1.2		2	15.8	even	4
4560.2.a.bo.1.2		2	20.7	even	4
5415.2.a.s.1.1		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)

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

Related objects

Downloads

Learn more