LMFDB - Embedded newform 56.3.h.c.13.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [56,3,Mod(13,56)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(56, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("56.13");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 56 = 2^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	56.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$1.52588948042$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			13.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	56.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -2.82843 q^{3} +4.00000 q^{4} +8.48528 q^{5} -5.65685 q^{6} -7.00000 q^{7} +8.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -2.82843 q^{3} +4.00000 q^{4} +8.48528 q^{5} -5.65685 q^{6} -7.00000 q^{7} +8.00000 q^{8} -1.00000 q^{9} +16.9706 q^{10} -11.3137 q^{12} -25.4558 q^{13} -14.0000 q^{14} -24.0000 q^{15} +16.0000 q^{16} -2.00000 q^{18} +8.48528 q^{19} +33.9411 q^{20} +19.7990 q^{21} -10.0000 q^{23} -22.6274 q^{24} +47.0000 q^{25} -50.9117 q^{26} +28.2843 q^{27} -28.0000 q^{28} -48.0000 q^{30} +32.0000 q^{32} -59.3970 q^{35} -4.00000 q^{36} +16.9706 q^{38} +72.0000 q^{39} +67.8823 q^{40} +39.5980 q^{42} -8.48528 q^{45} -20.0000 q^{46} -45.2548 q^{48} +49.0000 q^{49} +94.0000 q^{50} -101.823 q^{52} +56.5685 q^{54} -56.0000 q^{56} -24.0000 q^{57} +76.3675 q^{59} -96.0000 q^{60} +8.48528 q^{61} +7.00000 q^{63} +64.0000 q^{64} -216.000 q^{65} +28.2843 q^{69} -118.794 q^{70} -110.000 q^{71} -8.00000 q^{72} -132.936 q^{75} +33.9411 q^{76} +144.000 q^{78} +130.000 q^{79} +135.765 q^{80} -71.0000 q^{81} -25.4558 q^{83} +79.1960 q^{84} -16.9706 q^{90} +178.191 q^{91} -40.0000 q^{92} +72.0000 q^{95} -90.5097 q^{96} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 - 14 * q^7 + 16 * q^8 - 2 * q^9	$ 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} - 2 q^{9} - 28 q^{14} - 48 q^{15} + 32 q^{16} - 4 q^{18} - 20 q^{23} + 94 q^{25} - 56 q^{28} - 96 q^{30} + 64 q^{32} - 8 q^{36} + 144 q^{39} - 40 q^{46} + 98 q^{49} + 188 q^{50} - 112 q^{56} - 48 q^{57} - 192 q^{60} + 14 q^{63} + 128 q^{64} - 432 q^{65} - 220 q^{71} - 16 q^{72} + 288 q^{78} + 260 q^{79} - 142 q^{81} - 80 q^{92} + 144 q^{95} + 196 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 - 14 * q^7 + 16 * q^8 - 2 * q^9 - 28 * q^14 - 48 * q^15 + 32 * q^16 - 4 * q^18 - 20 * q^23 + 94 * q^25 - 56 * q^28 - 96 * q^30 + 64 * q^32 - 8 * q^36 + 144 * q^39 - 40 * q^46 + 98 * q^49 + 188 * q^50 - 112 * q^56 - 48 * q^57 - 192 * q^60 + 14 * q^63 + 128 * q^64 - 432 * q^65 - 220 * q^71 - 16 * q^72 + 288 * q^78 + 260 * q^79 - 142 * q^81 - 80 * q^92 + 144 * q^95 + 196 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/56\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$29$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	2.00000	1.00000
$2$	2.00000	1.00000
$3$	−2.82843	−0.942809	−0.471405	−	0.881917i	$-0.656253\pi$
$3$	−2.82843	−0.942809	−0.471405	+	0.881917i	$0.656253\pi$
$4$	4.00000	1.00000
$5$	8.48528	1.69706	0.848528	−	0.529150i	$-0.177489\pi$
$5$	8.48528	1.69706	0.848528	+	0.529150i	$0.177489\pi$
$6$	−5.65685	−0.942809
$7$	−7.00000	−1.00000
$7$	−7.00000	−1.00000
$8$	8.00000	1.00000
$9$	−1.00000	−0.111111
$10$	16.9706	1.69706
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−11.3137	−0.942809
$13$	−25.4558	−1.95814	−0.979071	−	0.203519i	$-0.934762\pi$
$13$	−25.4558	−1.95814	−0.979071	+	0.203519i	$0.934762\pi$
$14$	−14.0000	−1.00000
$15$	−24.0000	−1.60000
$16$	16.0000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−2.00000	−0.111111
$19$	8.48528	0.446594	0.223297	−	0.974750i	$-0.428318\pi$
$19$	8.48528	0.446594	0.223297	+	0.974750i	$0.428318\pi$
$20$	33.9411	1.69706
$21$	19.7990	0.942809
$22$	0	0
$23$	−10.0000	−0.434783	−0.217391	−	0.976085i	$-0.569755\pi$
$23$	−10.0000	−0.434783	−0.217391	+	0.976085i	$0.569755\pi$
$24$	−22.6274	−0.942809
$25$	47.0000	1.88000
$26$	−50.9117	−1.95814
$27$	28.2843	1.04757
$28$	−28.0000	−1.00000
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−48.0000	−1.60000
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.0000	1.00000
$33$	0	0
$34$	0	0
$35$	−59.3970	−1.69706
$36$	−4.00000	−0.111111
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	16.9706	0.446594
$39$	72.0000	1.84615
$40$	67.8823	1.69706
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	39.5980	0.942809
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−8.48528	−0.188562
$46$	−20.0000	−0.434783
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−45.2548	−0.942809
$49$	49.0000	1.00000
$50$	94.0000	1.88000
$51$	0	0
$52$	−101.823	−1.95814
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	56.5685	1.04757
$55$	0	0
$56$	−56.0000	−1.00000
$57$	−24.0000	−0.421053
$58$	0	0
$59$	76.3675	1.29436	0.647182	−	0.762335i	$-0.275947\pi$
$59$	76.3675	1.29436	0.647182	+	0.762335i	$0.275947\pi$
$60$	−96.0000	−1.60000
$61$	8.48528	0.139103	0.0695515	−	0.997578i	$-0.477843\pi$
$61$	8.48528	0.139103	0.0695515	+	0.997578i	$0.477843\pi$
$62$	0	0
$63$	7.00000	0.111111
$64$	64.0000	1.00000
$65$	−216.000	−3.32308
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	28.2843	0.409917
$70$	−118.794	−1.69706
$71$	−110.000	−1.54930	−0.774648	−	0.632393i	$-0.782073\pi$
$71$	−110.000	−1.54930	−0.774648	+	0.632393i	$0.782073\pi$
$72$	−8.00000	−0.111111
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−132.936	−1.77248
$76$	33.9411	0.446594
$77$	0	0
$78$	144.000	1.84615
$79$	130.000	1.64557	0.822785	−	0.568353i	$-0.192419\pi$
$79$	130.000	1.64557	0.822785	+	0.568353i	$0.192419\pi$
$80$	135.765	1.69706
$81$	−71.0000	−0.876543
$82$	0	0
$83$	−25.4558	−0.306697	−0.153348	−	0.988172i	$-0.549006\pi$
$83$	−25.4558	−0.306697	−0.153348	+	0.988172i	$0.549006\pi$
$84$	79.1960	0.942809
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−16.9706	−0.188562
$91$	178.191	1.95814
$92$	−40.0000	−0.434783
$93$	0	0
$94$	0	0
$95$	72.0000	0.757895
$96$	−90.5097	−0.942809
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	98.0000	1.00000
$99$	0	0
$100$	188.000	1.88000
$101$	−161.220	−1.59624	−0.798121	−	0.602498i	$-0.794172\pi$
$101$	−161.220	−1.59624	−0.798121	+	0.602498i	$0.794172\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−203.647	−1.95814
$105$	168.000	1.60000
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	113.137	1.04757
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−112.000	−1.00000
$113$	−26.0000	−0.230088	−0.115044	−	0.993360i	$-0.536701\pi$
$113$	−26.0000	−0.230088	−0.115044	+	0.993360i	$0.536701\pi$
$114$	−48.0000	−0.421053
$115$	−84.8528	−0.737851
$116$	0	0
$117$	25.4558	0.217571
$118$	152.735	1.29436
$119$	0	0
$120$	−192.000	−1.60000
$121$	121.000	1.00000
$122$	16.9706	0.139103
$123$	0	0
$124$	0	0
$125$	186.676	1.49341
$126$	14.0000	0.111111
$127$	−250.000	−1.96850	−0.984252	−	0.176771i	$-0.943435\pi$
$127$	−250.000	−1.96850	−0.984252	+	0.176771i	$0.943435\pi$
$128$	128.000	1.00000
$129$	0	0
$130$	−432.000	−3.32308
$131$	246.073	1.87842	0.939211	−	0.343342i	$-0.111559\pi$
$131$	246.073	1.87842	0.939211	+	0.343342i	$0.111559\pi$
$132$	0	0
$133$	−59.3970	−0.446594
$134$	0	0
$135$	240.000	1.77778
$136$	0	0
$137$	50.0000	0.364964	0.182482	−	0.983209i	$-0.441587\pi$
$137$	50.0000	0.364964	0.182482	+	0.983209i	$0.441587\pi$
$138$	56.5685	0.409917
$139$	−263.044	−1.89240	−0.946200	−	0.323581i	$-0.895113\pi$
$139$	−263.044	−1.89240	−0.946200	+	0.323581i	$0.895113\pi$
$140$	−237.588	−1.69706
$141$	0	0
$142$	−220.000	−1.54930
$143$	0	0
$144$	−16.0000	−0.111111
$145$	0	0
$146$	0	0
$147$	−138.593	−0.942809
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−265.872	−1.77248
$151$	−202.000	−1.33775	−0.668874	−	0.743376i	$-0.733224\pi$
$151$	−202.000	−1.33775	−0.668874	+	0.743376i	$0.733224\pi$
$152$	67.8823	0.446594
$153$	0	0
$154$	0	0
$155$	0	0
$156$	288.000	1.84615
$157$	313.955	1.99972	0.999858	−	0.0168519i	$-0.00536439\pi$
$157$	313.955	1.99972	0.999858	+	0.0168519i	$0.00536439\pi$
$158$	260.000	1.64557
$159$	0	0
$160$	271.529	1.69706
$161$	70.0000	0.434783
$162$	−142.000	−0.876543
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−50.9117	−0.306697
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	158.392	0.942809
$169$	479.000	2.83432
$170$	0	0
$171$	−8.48528	−0.0496215
$172$	0	0
$173$	−229.103	−1.32429	−0.662146	−	0.749375i	$-0.730354\pi$
$173$	−229.103	−1.32429	−0.662146	+	0.749375i	$0.730354\pi$
$174$	0	0
$175$	−329.000	−1.88000
$176$	0	0
$177$	−216.000	−1.22034
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−33.9411	−0.188562
$181$	−263.044	−1.45328	−0.726640	−	0.687018i	$-0.758919\pi$
$181$	−263.044	−1.45328	−0.726640	+	0.687018i	$0.758919\pi$
$182$	356.382	1.95814
$183$	−24.0000	−0.131148
$184$	−80.0000	−0.434783
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−197.990	−1.04757
$190$	144.000	0.757895
$191$	130.000	0.680628	0.340314	−	0.940312i	$-0.389467\pi$
$191$	130.000	0.680628	0.340314	+	0.940312i	$0.389467\pi$
$192$	−181.019	−0.942809
$193$	−314.000	−1.62694	−0.813472	−	0.581605i	$-0.802425\pi$
$193$	−314.000	−1.62694	−0.813472	+	0.581605i	$0.802425\pi$
$194$	0	0
$195$	610.940	3.13303
$196$	196.000	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	376.000	1.88000
$201$	0	0
$202$	−322.441	−1.59624
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.0000	0.0483092
$208$	−407.294	−1.95814
$209$	0	0
$210$	336.000	1.60000
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	311.127	1.46069
$214$	0	0
$215$	0	0
$216$	226.274	1.04757
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−224.000	−1.00000
$225$	−47.0000	−0.208889
$226$	−52.0000	−0.230088
$227$	−398.808	−1.75686	−0.878432	−	0.477867i	$-0.841410\pi$
$227$	−398.808	−1.75686	−0.878432	+	0.477867i	$0.841410\pi$
$228$	−96.0000	−0.421053
$229$	246.073	1.07456	0.537278	−	0.843405i	$-0.319453\pi$
$229$	246.073	1.07456	0.537278	+	0.843405i	$0.319453\pi$
$230$	−169.706	−0.737851
$231$	0	0
$232$	0	0
$233$	−430.000	−1.84549	−0.922747	−	0.385407i	$-0.874061\pi$
$233$	−430.000	−1.84549	−0.922747	+	0.385407i	$0.874061\pi$
$234$	50.9117	0.217571
$235$	0	0
$236$	305.470	1.29436
$237$	−367.696	−1.55146
$238$	0	0
$239$	422.000	1.76569	0.882845	−	0.469664i	$-0.155625\pi$
$239$	422.000	1.76569	0.882845	+	0.469664i	$0.155625\pi$
$240$	−384.000	−1.60000
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	242.000	1.00000
$243$	−53.7401	−0.221153
$244$	33.9411	0.139103
$245$	415.779	1.69706
$246$	0	0
$247$	−216.000	−0.874494
$248$	0	0
$249$	72.0000	0.289157
$250$	373.352	1.49341
$251$	−500.632	−1.99455	−0.997274	−	0.0737859i	$-0.976492\pi$
$251$	−500.632	−1.99455	−0.997274	+	0.0737859i	$0.976492\pi$
$252$	28.0000	0.111111
$253$	0	0
$254$	−500.000	−1.96850
$255$	0	0
$256$	256.000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	−864.000	−3.32308
$261$	0	0
$262$	492.146	1.87842
$263$	274.000	1.04183	0.520913	−	0.853610i	$-0.325592\pi$
$263$	274.000	1.04183	0.520913	+	0.853610i	$0.325592\pi$
$264$	0	0
$265$	0	0
$266$	−118.794	−0.446594
$267$	0	0
$268$	0	0
$269$	−161.220	−0.599332	−0.299666	−	0.954044i	$-0.596875\pi$
$269$	−161.220	−0.599332	−0.299666	+	0.954044i	$0.596875\pi$
$270$	480.000	1.77778
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	−504.000	−1.84615
$274$	100.000	0.364964
$275$	0	0
$276$	113.137	0.409917
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−526.087	−1.89240
$279$	0	0
$280$	−475.176	−1.69706
$281$	338.000	1.20285	0.601423	−	0.798930i	$-0.294600\pi$
$281$	338.000	1.20285	0.601423	+	0.798930i	$0.294600\pi$
$282$	0	0
$283$	313.955	1.10938	0.554692	−	0.832056i	$-0.312836\pi$
$283$	313.955	1.10938	0.554692	+	0.832056i	$0.312836\pi$
$284$	−440.000	−1.54930
$285$	−203.647	−0.714550
$286$	0	0
$287$	0	0
$288$	−32.0000	−0.111111
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	449.720	1.53488	0.767440	−	0.641121i	$-0.221530\pi$
$293$	449.720	1.53488	0.767440	+	0.641121i	$0.221530\pi$
$294$	−277.186	−0.942809
$295$	648.000	2.19661
$296$	0	0
$297$	0	0
$298$	0	0
$299$	254.558	0.851366
$300$	−531.744	−1.77248
$301$	0	0
$302$	−404.000	−1.33775
$303$	456.000	1.50495
$304$	135.765	0.446594
$305$	72.0000	0.236066
$306$	0	0
$307$	449.720	1.46489	0.732443	−	0.680829i	$-0.238380\pi$
$307$	449.720	1.46489	0.732443	+	0.680829i	$0.238380\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	576.000	1.84615
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	627.911	1.99972
$315$	59.3970	0.188562
$316$	520.000	1.64557
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	543.058	1.69706
$321$	0	0
$322$	140.000	0.434783
$323$	0	0
$324$	−284.000	−0.876543
$325$	−1196.42	−3.68131
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−101.823	−0.306697
$333$	0	0
$334$	0	0
$335$	0	0
$336$	316.784	0.942809
$337$	−26.0000	−0.0771513	−0.0385757	−	0.999256i	$-0.512282\pi$
$337$	−26.0000	−0.0771513	−0.0385757	+	0.999256i	$0.512282\pi$
$338$	958.000	2.83432
$339$	73.5391	0.216930
$340$	0	0
$341$	0	0
$342$	−16.9706	−0.0496215
$343$	−343.000	−1.00000
$344$	0	0
$345$	240.000	0.695652
$346$	−458.205	−1.32429
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	687.308	1.96936	0.984682	−	0.174362i	$-0.0557862\pi$
$349$	687.308	1.96936	0.984682	+	0.174362i	$0.0557862\pi$
$350$	−658.000	−1.88000
$351$	−720.000	−2.05128
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−432.000	−1.22034
$355$	−933.381	−2.62924
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−682.000	−1.89972	−0.949861	−	0.312673i	$-0.898775\pi$
$359$	−682.000	−1.89972	−0.949861	+	0.312673i	$0.898775\pi$
$360$	−67.8823	−0.188562
$361$	−289.000	−0.800554
$362$	−526.087	−1.45328
$363$	−342.240	−0.942809
$364$	712.764	1.95814
$365$	0	0
$366$	−48.0000	−0.131148
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−160.000	−0.434783
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−528.000	−1.40800
$376$	0	0
$377$	0	0
$378$	−395.980	−1.04757
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	288.000	0.757895
$381$	707.107	1.85592
$382$	260.000	0.680628
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−362.039	−0.942809
$385$	0	0
$386$	−628.000	−1.62694
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	1221.88	3.13303
$391$	0	0
$392$	392.000	1.00000
$393$	−696.000	−1.77099
$394$	0	0
$395$	1103.09	2.79262
$396$	0	0
$397$	483.661	1.21829	0.609145	−	0.793059i	$-0.291513\pi$
$397$	483.661	1.21829	0.609145	+	0.793059i	$0.291513\pi$
$398$	0	0
$399$	168.000	0.421053
$400$	752.000	1.88000
$401$	550.000	1.37157	0.685786	−	0.727804i	$-0.259459\pi$
$401$	550.000	1.37157	0.685786	+	0.727804i	$0.259459\pi$
$402$	0	0
$403$	0	0
$404$	−644.881	−1.59624
$405$	−602.455	−1.48754
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−141.421	−0.344091
$412$	0	0
$413$	−534.573	−1.29436
$414$	20.0000	0.0483092
$415$	−216.000	−0.520482
$416$	−814.587	−1.95814
$417$	744.000	1.78417
$418$	0	0
$419$	−500.632	−1.19482	−0.597412	−	0.801934i	$-0.703804\pi$
$419$	−500.632	−1.19482	−0.597412	+	0.801934i	$0.703804\pi$
$420$	672.000	1.60000
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	622.254	1.46069
$427$	−59.3970	−0.139103
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−538.000	−1.24826	−0.624130	−	0.781321i	$-0.714546\pi$
$431$	−538.000	−1.24826	−0.624130	+	0.781321i	$0.714546\pi$
$432$	452.548	1.04757
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−84.8528	−0.194171
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−49.0000	−0.111111
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−448.000	−1.00000
$449$	2.00000	0.00445434	0.00222717	−	0.999998i	$-0.499291\pi$
$449$	2.00000	0.00445434	0.00222717	+	0.999998i	$0.499291\pi$
$450$	−94.0000	−0.208889
$451$	0	0
$452$	−104.000	−0.230088
$453$	571.342	1.26124
$454$	−797.616	−1.75686
$455$	1512.00	3.32308
$456$	−192.000	−0.421053
$457$	886.000	1.93873	0.969365	−	0.245623i	$-0.0789925\pi$
$457$	886.000	1.93873	0.969365	+	0.245623i	$0.0789925\pi$
$458$	492.146	1.07456
$459$	0	0
$460$	−339.411	−0.737851
$461$	−263.044	−0.570594	−0.285297	−	0.958439i	$-0.592092\pi$
$461$	−263.044	−0.570594	−0.285297	+	0.958439i	$0.592092\pi$
$462$	0	0
$463$	226.000	0.488121	0.244060	−	0.969760i	$-0.421520\pi$
$463$	226.000	0.488121	0.244060	+	0.969760i	$0.421520\pi$
$464$	0	0
$465$	0	0
$466$	−860.000	−1.84549
$467$	483.661	1.03568	0.517838	−	0.855478i	$-0.326737\pi$
$467$	483.661	1.03568	0.517838	+	0.855478i	$0.326737\pi$
$468$	101.823	0.217571
$469$	0	0
$470$	0	0
$471$	−888.000	−1.88535
$472$	610.940	1.29436
$473$	0	0
$474$	−735.391	−1.55146
$475$	398.808	0.839596
$476$	0	0
$477$	0	0
$478$	844.000	1.76569
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−768.000	−1.60000
$481$	0	0
$482$	0	0
$483$	−197.990	−0.409917
$484$	484.000	1.00000
$485$	0	0
$486$	−107.480	−0.221153
$487$	470.000	0.965092	0.482546	−	0.875871i	$-0.339712\pi$
$487$	470.000	0.965092	0.482546	+	0.875871i	$0.339712\pi$
$488$	67.8823	0.139103
$489$	0	0
$490$	831.558	1.69706
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	−432.000	−0.874494
$495$	0	0
$496$	0	0
$497$	770.000	1.54930
$498$	144.000	0.289157
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	746.705	1.49341
$501$	0	0
$502$	−1001.26	−1.99455
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	56.0000	0.111111
$505$	−1368.00	−2.70891
$506$	0	0
$507$	−1354.82	−2.67222
$508$	−1000.00	−1.96850
$509$	−941.866	−1.85042	−0.925212	−	0.379450i	$-0.876113\pi$
$509$	−941.866	−1.85042	−0.925212	+	0.379450i	$0.876113\pi$
$510$	0	0
$511$	0	0
$512$	512.000	1.00000
$513$	240.000	0.467836
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	648.000	1.24855
$520$	−1728.00	−3.32308
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	958.837	1.83334	0.916670	−	0.399645i	$-0.130867\pi$
$523$	958.837	1.83334	0.916670	+	0.399645i	$0.130867\pi$
$524$	984.293	1.87842
$525$	930.553	1.77248
$526$	548.000	1.04183
$527$	0	0
$528$	0	0
$529$	−429.000	−0.810964
$530$	0	0
$531$	−76.3675	−0.143818
$532$	−237.588	−0.446594
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−322.441	−0.599332
$539$	0	0
$540$	960.000	1.77778
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	744.000	1.37017
$544$	0	0
$545$	0	0
$546$	−1008.00	−1.84615
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	200.000	0.364964
$549$	−8.48528	−0.0154559
$550$	0	0
$551$	0	0
$552$	226.274	0.409917
$553$	−910.000	−1.64557
$554$	0	0
$555$	0	0
$556$	−1052.17	−1.89240
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−950.352	−1.69706
$561$	0	0
$562$	676.000	1.20285
$563$	−398.808	−0.708363	−0.354181	−	0.935177i	$-0.615240\pi$
$563$	−398.808	−0.708363	−0.354181	+	0.935177i	$0.615240\pi$
$564$	0	0
$565$	−220.617	−0.390473
$566$	627.911	1.10938
$567$	497.000	0.876543
$568$	−880.000	−1.54930
$569$	−1130.00	−1.98594	−0.992970	−	0.118365i	$-0.962235\pi$
$569$	−1130.00	−1.98594	−0.992970	+	0.118365i	$0.962235\pi$
$570$	−407.294	−0.714550
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	−367.696	−0.641702
$574$	0	0
$575$	−470.000	−0.817391
$576$	−64.0000	−0.111111
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	578.000	1.00000
$579$	888.126	1.53390
$580$	0	0
$581$	178.191	0.306697
$582$	0	0
$583$	0	0
$584$	0	0
$585$	216.000	0.369231
$586$	899.440	1.53488
$587$	1162.48	1.98038	0.990190	−	0.139725i	$-0.0446217\pi$
$587$	1162.48	1.98038	0.990190	+	0.139725i	$0.0446217\pi$
$588$	−554.372	−0.942809
$589$	0	0
$590$	1296.00	2.19661
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	509.117	0.851366
$599$	−1070.00	−1.78631	−0.893155	−	0.449748i	$-0.851514\pi$
$599$	−1070.00	−1.78631	−0.893155	+	0.449748i	$0.851514\pi$
$600$	−1063.49	−1.77248
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	−808.000	−1.33775
$605$	1026.72	1.69706
$606$	912.000	1.50495
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	271.529	0.446594
$609$	0	0
$610$	144.000	0.236066
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	899.440	1.46489
$615$	0	0
$616$	0	0
$617$	−1034.00	−1.67585	−0.837925	−	0.545785i	$-0.816232\pi$
$617$	−1034.00	−1.67585	−0.837925	+	0.545785i	$0.816232\pi$
$618$	0	0
$619$	−1111.57	−1.79575	−0.897877	−	0.440246i	$-0.854891\pi$
$619$	−1111.57	−1.79575	−0.897877	+	0.440246i	$0.854891\pi$
$620$	0	0
$621$	−282.843	−0.455463
$622$	0	0
$623$	0	0
$624$	1152.00	1.84615
$625$	409.000	0.654400
$626$	0	0
$627$	0	0
$628$	1255.82	1.99972
$629$	0	0
$630$	118.794	0.188562
$631$	−110.000	−0.174326	−0.0871632	−	0.996194i	$-0.527780\pi$
$631$	−110.000	−0.174326	−0.0871632	+	0.996194i	$0.527780\pi$
$632$	1040.00	1.64557
$633$	0	0
$634$	0	0
$635$	−2121.32	−3.34066
$636$	0	0
$637$	−1247.34	−1.95814
$638$	0	0
$639$	110.000	0.172144
$640$	1086.12	1.69706
$641$	1030.00	1.60686	0.803432	−	0.595396i	$-0.203005\pi$
$641$	1030.00	1.60686	0.803432	+	0.595396i	$0.203005\pi$
$642$	0	0
$643$	−738.219	−1.14809	−0.574043	−	0.818825i	$-0.694626\pi$
$643$	−738.219	−1.14809	−0.574043	+	0.818825i	$0.694626\pi$
$644$	280.000	0.434783
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−568.000	−0.876543
$649$	0	0
$650$	−2392.85	−3.68131
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	2088.00	3.18779
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1264.31	1.91272	0.956359	−	0.292193i	$-0.0943851\pi$
$661$	1264.31	1.91272	0.956359	+	0.292193i	$0.0943851\pi$
$662$	0	0
$663$	0	0
$664$	−203.647	−0.306697
$665$	−504.000	−0.757895
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	633.568	0.942809
$673$	−670.000	−0.995542	−0.497771	−	0.867308i	$-0.665848\pi$
$673$	−670.000	−0.995542	−0.497771	+	0.867308i	$0.665848\pi$
$674$	−52.0000	−0.0771513
$675$	1329.36	1.96942
$676$	1916.00	2.83432
$677$	483.661	0.714418	0.357209	−	0.934024i	$-0.383728\pi$
$677$	483.661	0.714418	0.357209	+	0.934024i	$0.383728\pi$
$678$	147.078	0.216930
$679$	0	0
$680$	0	0
$681$	1128.00	1.65639
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−33.9411	−0.0496215
$685$	424.264	0.619364
$686$	−686.000	−1.00000
$687$	−696.000	−1.01310
$688$	0	0
$689$	0	0
$690$	480.000	0.695652
$691$	246.073	0.356112	0.178056	−	0.984020i	$-0.443019\pi$
$691$	246.073	0.356112	0.178056	+	0.984020i	$0.443019\pi$
$692$	−916.410	−1.32429
$693$	0	0
$694$	0	0
$695$	−2232.00	−3.21151
$696$	0	0
$697$	0	0
$698$	1374.62	1.96936
$699$	1216.22	1.73995
$700$	−1316.00	−1.88000
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−1440.00	−2.05128
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1128.54	1.59624
$708$	−864.000	−1.22034
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−1866.76	−2.62924
$711$	−130.000	−0.182841
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1193.60	−1.66471
$718$	−1364.00	−1.89972
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−135.765	−0.188562
$721$	0	0
$722$	−578.000	−0.800554
$723$	0	0
$724$	−1052.17	−1.45328
$725$	0	0
$726$	−684.479	−0.942809
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	1425.53	1.95814
$729$	791.000	1.08505
$730$	0	0
$731$	0	0
$732$	−96.0000	−0.131148
$733$	−1417.04	−1.93321	−0.966604	−	0.256273i	$-0.917505\pi$
$733$	−1417.04	−1.93321	−0.966604	+	0.256273i	$0.917505\pi$
$734$	0	0
$735$	−1176.00	−1.60000
$736$	−320.000	−0.434783
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	610.940	0.824481
$742$	0	0
$743$	1430.00	1.92463	0.962315	−	0.271937i	$-0.0876644\pi$
$743$	1430.00	1.92463	0.962315	+	0.271937i	$0.0876644\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	25.4558	0.0340774
$748$	0	0
$749$	0	0
$750$	−1056.00	−1.40800
$751$	998.000	1.32889	0.664447	−	0.747335i	$-0.268667\pi$
$751$	998.000	1.32889	0.664447	+	0.747335i	$0.268667\pi$
$752$	0	0
$753$	1416.00	1.88048
$754$	0	0
$755$	−1714.03	−2.27023
$756$	−791.960	−1.04757
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	576.000	0.757895
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	1414.21	1.85592
$763$	0	0
$764$	520.000	0.680628
$765$	0	0
$766$	0	0
$767$	−1944.00	−2.53455
$768$	−724.077	−0.942809
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−1256.00	−1.62694
$773$	−873.984	−1.13064	−0.565320	−	0.824872i	$-0.691247\pi$
$773$	−873.984	−1.13064	−0.565320	+	0.824872i	$0.691247\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	2443.76	3.13303
$781$	0	0
$782$	0	0
$783$	0	0
$784$	784.000	1.00000
$785$	2664.00	3.39363
$786$	−1392.00	−1.77099
$787$	1501.89	1.90838	0.954190	−	0.299202i	$-0.0967204\pi$
$787$	1501.89	1.90838	0.954190	+	0.299202i	$0.0967204\pi$
$788$	0	0
$789$	−774.989	−0.982242
$790$	2206.17	2.79262
$791$	182.000	0.230088
$792$	0	0
$793$	−216.000	−0.272383
$794$	967.322	1.21829
$795$	0	0
$796$	0	0
$797$	449.720	0.564266	0.282133	−	0.959375i	$-0.408958\pi$
$797$	449.720	0.564266	0.282133	+	0.959375i	$0.408958\pi$
$798$	336.000	0.421053
$799$	0	0
$800$	1504.00	1.88000
$801$	0	0
$802$	1100.00	1.37157
$803$	0	0
$804$	0	0
$805$	593.970	0.737851
$806$	0	0
$807$	456.000	0.565056
$808$	−1289.76	−1.59624
$809$	−650.000	−0.803461	−0.401731	−	0.915758i	$-0.631591\pi$
$809$	−650.000	−0.803461	−0.401731	+	0.915758i	$0.631591\pi$
$810$	−1204.91	−1.48754
$811$	−1450.98	−1.78913	−0.894564	−	0.446939i	$-0.852514\pi$
$811$	−1450.98	−1.78913	−0.894564	+	0.446939i	$0.852514\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−178.191	−0.217571
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−282.843	−0.344091
$823$	946.000	1.14945	0.574727	−	0.818345i	$-0.305108\pi$
$823$	946.000	1.14945	0.574727	+	0.818345i	$0.305108\pi$
$824$	0	0
$825$	0	0
$826$	−1069.15	−1.29436
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	40.0000	0.0483092
$829$	76.3675	0.0921201	0.0460600	−	0.998939i	$-0.485333\pi$
$829$	76.3675	0.0921201	0.0460600	+	0.998939i	$0.485333\pi$
$830$	−432.000	−0.520482
$831$	0	0
$832$	−1629.17	−1.95814
$833$	0	0
$834$	1488.00	1.78417
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−1001.26	−1.19482
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	1344.00	1.60000
$841$	841.000	1.00000
$842$	0	0
$843$	−956.008	−1.13406
$844$	0	0
$845$	4064.45	4.81000
$846$	0	0
$847$	−847.000	−1.00000
$848$	0	0
$849$	−888.000	−1.04594
$850$	0	0
$851$	0	0
$852$	1244.51	1.46069
$853$	449.720	0.527221	0.263611	−	0.964629i	$-0.415087\pi$
$853$	449.720	0.527221	0.263611	+	0.964629i	$0.415087\pi$
$854$	−118.794	−0.139103
$855$	−72.0000	−0.0842105
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1196.42	1.39281	0.696406	−	0.717648i	$-0.254782\pi$
$859$	1196.42	1.39281	0.696406	+	0.717648i	$0.254782\pi$
$860$	0	0
$861$	0	0
$862$	−1076.00	−1.24826
$863$	1474.00	1.70800	0.853998	−	0.520277i	$-0.174171\pi$
$863$	1474.00	1.70800	0.853998	+	0.520277i	$0.174171\pi$
$864$	905.097	1.04757
$865$	−1944.00	−2.24740
$866$	0	0
$867$	−817.415	−0.942809
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	−169.706	−0.194171
$875$	−1306.73	−1.49341
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−1272.00	−1.44710
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−98.0000	−0.111111
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−1832.82	−2.07098
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	1750.00	1.96850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−896.000	−1.00000
$897$	−720.000	−0.802676
$898$	4.00000	0.00445434
$899$	0	0
$900$	−188.000	−0.208889
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−208.000	−0.230088
$905$	−2232.00	−2.46630
$906$	1142.68	1.26124
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−1595.23	−1.75686
$909$	161.220	0.177360
$910$	3024.00	3.32308
$911$	−922.000	−1.01207	−0.506037	−	0.862512i	$-0.668890\pi$
$911$	−922.000	−1.01207	−0.506037	+	0.862512i	$0.668890\pi$
$912$	−384.000	−0.421053
$913$	0	0
$914$	1772.00	1.93873
$915$	−203.647	−0.222565
$916$	984.293	1.07456
$917$	−1722.51	−1.87842
$918$	0	0
$919$	−1550.00	−1.68662	−0.843308	−	0.537431i	$-0.819395\pi$
$919$	−1550.00	−1.68662	−0.843308	+	0.537431i	$0.819395\pi$
$920$	−678.823	−0.737851
$921$	−1272.00	−1.38111
$922$	−526.087	−0.570594
$923$	2800.14	3.03374
$924$	0	0
$925$	0	0
$926$	452.000	0.488121
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	415.779	0.446594
$932$	−1720.00	−1.84549
$933$	0	0
$934$	967.322	1.03568
$935$	0	0
$936$	203.647	0.217571
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1111.57	−1.18127	−0.590633	−	0.806940i	$-0.701122\pi$
$941$	−1111.57	−1.18127	−0.590633	+	0.806940i	$0.701122\pi$
$942$	−1776.00	−1.88535
$943$	0	0
$944$	1221.88	1.29436
$945$	−1680.00	−1.77778
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−1470.78	−1.55146
$949$	0	0
$950$	797.616	0.839596
$951$	0	0
$952$	0	0
$953$	1010.00	1.05981	0.529906	−	0.848057i	$-0.322227\pi$
$953$	1010.00	1.05981	0.529906	+	0.848057i	$0.322227\pi$
$954$	0	0
$955$	1103.09	1.15506
$956$	1688.00	1.76569
$957$	0	0
$958$	0	0
$959$	−350.000	−0.364964
$960$	−1536.00	−1.60000
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2664.38	−2.76101
$966$	−395.980	−0.409917
$967$	1430.00	1.47880	0.739400	−	0.673266i	$-0.235109\pi$
$967$	1430.00	1.47880	0.739400	+	0.673266i	$0.235109\pi$
$968$	968.000	1.00000
$969$	0	0
$970$	0	0
$971$	8.48528	0.00873870	0.00436935	−	0.999990i	$-0.498609\pi$
$971$	8.48528	0.00873870	0.00436935	+	0.999990i	$0.498609\pi$
$972$	−214.960	−0.221153
$973$	1841.31	1.89240
$974$	940.000	0.965092
$975$	3384.00	3.47077
$976$	135.765	0.139103
$977$	−1630.00	−1.66837	−0.834186	−	0.551483i	$-0.814062\pi$
$977$	−1630.00	−1.66837	−0.834186	+	0.551483i	$0.814062\pi$
$978$	0	0
$979$	0	0
$980$	1663.12	1.69706
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−864.000	−0.874494
$989$	0	0
$990$	0	0
$991$	610.000	0.615540	0.307770	−	0.951461i	$-0.400417\pi$
$991$	610.000	0.615540	0.307770	+	0.951461i	$0.400417\pi$
$992$	0	0
$993$	0	0
$994$	1540.00	1.54930
$995$	0	0
$996$	288.000	0.289157
$997$	1977.07	1.98302	0.991510	−	0.130032i	$-0.0415080\pi$
$997$	1977.07	1.98302	0.991510	+	0.130032i	$0.0415080\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.3.h.c.13.1	✓	2
3.2	odd	2		504.3.l.a.181.1		2
4.3	odd	2		224.3.h.c.209.2		2
7.2	even	3		392.3.j.a.325.2		4
7.3	odd	6		392.3.j.a.117.1		4
7.4	even	3		392.3.j.a.117.2		4
7.5	odd	6		392.3.j.a.325.1		4
7.6	odd	2	inner	56.3.h.c.13.2	yes	2
8.3	odd	2		224.3.h.c.209.1		2
8.5	even	2	inner	56.3.h.c.13.2	yes	2
12.11	even	2		2016.3.l.c.433.1		2
21.20	even	2		504.3.l.a.181.2		2
24.5	odd	2		504.3.l.a.181.2		2
24.11	even	2		2016.3.l.c.433.2		2
28.27	even	2		224.3.h.c.209.1		2
56.5	odd	6		392.3.j.a.325.2		4
56.13	odd	2	CM	56.3.h.c.13.1	✓	2
56.27	even	2		224.3.h.c.209.2		2
56.37	even	6		392.3.j.a.325.1		4
56.45	odd	6		392.3.j.a.117.2		4
56.53	even	6		392.3.j.a.117.1		4
84.83	odd	2		2016.3.l.c.433.2		2
168.83	odd	2		2016.3.l.c.433.1		2
168.125	even	2		504.3.l.a.181.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.h.c.13.1	✓	2	1.1	even	1	trivial
56.3.h.c.13.1	✓	2	56.13	odd	2	CM
56.3.h.c.13.2	yes	2	7.6	odd	2	inner
56.3.h.c.13.2	yes	2	8.5	even	2	inner
224.3.h.c.209.1		2	8.3	odd	2
224.3.h.c.209.1		2	28.27	even	2
224.3.h.c.209.2		2	4.3	odd	2
224.3.h.c.209.2		2	56.27	even	2
392.3.j.a.117.1		4	7.3	odd	6
392.3.j.a.117.1		4	56.53	even	6
392.3.j.a.117.2		4	7.4	even	3
392.3.j.a.117.2		4	56.45	odd	6
392.3.j.a.325.1		4	7.5	odd	6
392.3.j.a.325.1		4	56.37	even	6
392.3.j.a.325.2		4	7.2	even	3
392.3.j.a.325.2		4	56.5	odd	6
504.3.l.a.181.1		2	3.2	odd	2
504.3.l.a.181.1		2	168.125	even	2
504.3.l.a.181.2		2	21.20	even	2
504.3.l.a.181.2		2	24.5	odd	2
2016.3.l.c.433.1		2	12.11	even	2
2016.3.l.c.433.1		2	168.83	odd	2
2016.3.l.c.433.2		2	24.11	even	2
2016.3.l.c.433.2		2	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	56.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -2.82843 q^{3} +4.00000 q^{4} +8.48528 q^{5} -5.65685 q^{6} -7.00000 q^{7} +8.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -2.82843 q^{3} +4.00000 q^{4} +8.48528 q^{5} -5.65685 q^{6} -7.00000 q^{7} +8.00000 q^{8} -1.00000 q^{9} +16.9706 q^{10} -11.3137 q^{12} -25.4558 q^{13} -14.0000 q^{14} -24.0000 q^{15} +16.0000 q^{16} -2.00000 q^{18} +8.48528 q^{19} +33.9411 q^{20} +19.7990 q^{21} -10.0000 q^{23} -22.6274 q^{24} +47.0000 q^{25} -50.9117 q^{26} +28.2843 q^{27} -28.0000 q^{28} -48.0000 q^{30} +32.0000 q^{32} -59.3970 q^{35} -4.00000 q^{36} +16.9706 q^{38} +72.0000 q^{39} +67.8823 q^{40} +39.5980 q^{42} -8.48528 q^{45} -20.0000 q^{46} -45.2548 q^{48} +49.0000 q^{49} +94.0000 q^{50} -101.823 q^{52} +56.5685 q^{54} -56.0000 q^{56} -24.0000 q^{57} +76.3675 q^{59} -96.0000 q^{60} +8.48528 q^{61} +7.00000 q^{63} +64.0000 q^{64} -216.000 q^{65} +28.2843 q^{69} -118.794 q^{70} -110.000 q^{71} -8.00000 q^{72} -132.936 q^{75} +33.9411 q^{76} +144.000 q^{78} +130.000 q^{79} +135.765 q^{80} -71.0000 q^{81} -25.4558 q^{83} +79.1960 q^{84} -16.9706 q^{90} +178.191 q^{91} -40.0000 q^{92} +72.0000 q^{95} -90.5097 q^{96} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 - 14 * q^7 + 16 * q^8 - 2 * q^9	\( 2 q + 4 q^{2} + 8 q^{4} - 14 q^{7} + 16 q^{8} - 2 q^{9} - 28 q^{14} - 48 q^{15} + 32 q^{16} - 4 q^{18} - 20 q^{23} + 94 q^{25} - 56 q^{28} - 96 q^{30} + 64 q^{32} - 8 q^{36} + 144 q^{39} - 40 q^{46} + 98 q^{49} + 188 q^{50} - 112 q^{56} - 48 q^{57} - 192 q^{60} + 14 q^{63} + 128 q^{64} - 432 q^{65} - 220 q^{71} - 16 q^{72} + 288 q^{78} + 260 q^{79} - 142 q^{81} - 80 q^{92} + 144 q^{95} + 196 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 - 14 * q^7 + 16 * q^8 - 2 * q^9 - 28 * q^14 - 48 * q^15 + 32 * q^16 - 4 * q^18 - 20 * q^23 + 94 * q^25 - 56 * q^28 - 96 * q^30 + 64 * q^32 - 8 * q^36 + 144 * q^39 - 40 * q^46 + 98 * q^49 + 188 * q^50 - 112 * q^56 - 48 * q^57 - 192 * q^60 + 14 * q^63 + 128 * q^64 - 432 * q^65 - 220 * q^71 - 16 * q^72 + 288 * q^78 + 260 * q^79 - 142 * q^81 - 80 * q^92 + 144 * q^95 + 196 * q^98

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