LMFDB - Embedded newform 2576.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.a (trivial)

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:	$20.5694635607$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2147108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 $ x^5 - 2x^4 - 9x^3 + 17x^2 + 16x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.54577$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46268 q^{3} +2.78847 q^{5} -1.00000 q^{7} +3.06481 q^{9} +O(q^{10})$	$q-2.46268 q^{3} +2.78847 q^{5} -1.00000 q^{7} +3.06481 q^{9} +4.70095 q^{11} -2.32579 q^{13} -6.86713 q^{15} -1.82655 q^{17} -7.09155 q^{19} +2.46268 q^{21} +1.00000 q^{23} +2.77558 q^{25} -0.159610 q^{27} -9.98866 q^{29} -3.53732 q^{31} -11.5769 q^{33} -2.78847 q^{35} -0.166179 q^{37} +5.72768 q^{39} +7.25116 q^{41} +9.57695 q^{43} +8.54614 q^{45} -4.66542 q^{47} +1.00000 q^{49} +4.49821 q^{51} -0.961924 q^{53} +13.1085 q^{55} +17.4642 q^{57} -13.8800 q^{59} +0.954652 q^{61} -3.06481 q^{63} -6.48540 q^{65} +11.9221 q^{67} -2.46268 q^{69} -4.59958 q^{71} -7.59806 q^{73} -6.83537 q^{75} -4.70095 q^{77} -5.73902 q^{79} -8.80137 q^{81} +5.57695 q^{83} -5.09328 q^{85} +24.5989 q^{87} -11.2284 q^{89} +2.32579 q^{91} +8.71129 q^{93} -19.7746 q^{95} -1.68805 q^{97} +14.4075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9}+O(q^{10}) $ 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9	$ 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9} + 4 q^{11} - 6 q^{13} - 10 q^{15} - 12 q^{17} - 6 q^{19} + 5 q^{23} + 19 q^{25} - 4 q^{29} - 30 q^{31} - 22 q^{33} + 4 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 12 q^{43} - 12 q^{45} - 10 q^{47} + 5 q^{49} + 4 q^{51} + 16 q^{53} - 18 q^{55} + 6 q^{57} - 22 q^{59} - 18 q^{61} - 11 q^{63} - 26 q^{65} + 2 q^{67} - 4 q^{71} - 2 q^{73} + 30 q^{75} - 4 q^{77} - 30 q^{79} - 3 q^{81} - 8 q^{83} - 12 q^{85} + 12 q^{87} - 20 q^{89} + 6 q^{91} - 26 q^{93} - 8 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9 + 4 * q^11 - 6 * q^13 - 10 * q^15 - 12 * q^17 - 6 * q^19 + 5 * q^23 + 19 * q^25 - 4 * q^29 - 30 * q^31 - 22 * q^33 + 4 * q^35 + 4 * q^37 - 16 * q^39 + 6 * q^41 + 12 * q^43 - 12 * q^45 - 10 * q^47 + 5 * q^49 + 4 * q^51 + 16 * q^53 - 18 * q^55 + 6 * q^57 - 22 * q^59 - 18 * q^61 - 11 * q^63 - 26 * q^65 + 2 * q^67 - 4 * q^71 - 2 * q^73 + 30 * q^75 - 4 * q^77 - 30 * q^79 - 3 * q^81 - 8 * q^83 - 12 * q^85 + 12 * q^87 - 20 * q^89 + 6 * q^91 - 26 * q^93 - 8 * q^95 - 12 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−2.46268	−1.42183	−0.710916	−	0.703277i	$-0.751719\pi$
$3$	−2.46268	−1.42183	−0.710916	+	0.703277i	$0.751719\pi$
$4$	0	0
$5$	2.78847	1.24704	0.623521	−	0.781806i	$-0.285701\pi$
$5$	2.78847	1.24704	0.623521	+	0.781806i	$0.285701\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.06481	1.02160
$10$	0	0
$11$	4.70095	1.41739	0.708694	−	0.705516i	$-0.249285\pi$
$11$	4.70095	1.41739	0.708694	+	0.705516i	$0.249285\pi$
$12$	0	0
$13$	−2.32579	−0.645058	−0.322529	−	0.946560i	$-0.604533\pi$
$13$	−2.32579	−0.645058	−0.322529	+	0.946560i	$0.604533\pi$
$14$	0	0
$15$	−6.86713	−1.77308
$16$	0	0
$17$	−1.82655	−0.443003	−0.221502	−	0.975160i	$-0.571096\pi$
$17$	−1.82655	−0.443003	−0.221502	+	0.975160i	$0.571096\pi$
$18$	0	0
$19$	−7.09155	−1.62691	−0.813456	−	0.581626i	$-0.802417\pi$
$19$	−7.09155	−1.62691	−0.813456	+	0.581626i	$0.802417\pi$
$20$	0	0
$21$	2.46268	0.537402
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.77558	0.555116
$26$	0	0
$27$	−0.159610	−0.0307169
$28$	0	0
$29$	−9.98866	−1.85485	−0.927424	−	0.374012i	$-0.877982\pi$
$29$	−9.98866	−1.85485	−0.927424	+	0.374012i	$0.877982\pi$
$30$	0	0
$31$	−3.53732	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$31$	−3.53732	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$32$	0	0
$33$	−11.5769	−2.01529
$34$	0	0
$35$	−2.78847	−0.471338
$36$	0	0
$37$	−0.166179	−0.0273197	−0.0136599	−	0.999907i	$-0.504348\pi$
$37$	−0.166179	−0.0273197	−0.0136599	+	0.999907i	$0.504348\pi$
$38$	0	0
$39$	5.72768	0.917163
$40$	0	0
$41$	7.25116	1.13244	0.566220	−	0.824254i	$-0.308405\pi$
$41$	7.25116	1.13244	0.566220	+	0.824254i	$0.308405\pi$
$42$	0	0
$43$	9.57695	1.46047	0.730235	−	0.683196i	$-0.239410\pi$
$43$	9.57695	1.46047	0.730235	+	0.683196i	$0.239410\pi$
$44$	0	0
$45$	8.54614	1.27398
$46$	0	0
$47$	−4.66542	−0.680521	−0.340261	−	0.940331i	$-0.610515\pi$
$47$	−4.66542	−0.680521	−0.340261	+	0.940331i	$0.610515\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.49821	0.629875
$52$	0	0
$53$	−0.961924	−0.132130	−0.0660652	−	0.997815i	$-0.521045\pi$
$53$	−0.961924	−0.132130	−0.0660652	+	0.997815i	$0.521045\pi$
$54$	0	0
$55$	13.1085	1.76754
$56$	0	0
$57$	17.4642	2.31319
$58$	0	0
$59$	−13.8800	−1.80702	−0.903512	−	0.428562i	$-0.859020\pi$
$59$	−13.8800	−1.80702	−0.903512	+	0.428562i	$0.859020\pi$
$60$	0	0
$61$	0.954652	0.122231	0.0611153	−	0.998131i	$-0.480534\pi$
$61$	0.954652	0.122231	0.0611153	+	0.998131i	$0.480534\pi$
$62$	0	0
$63$	−3.06481	−0.386130
$64$	0	0
$65$	−6.48540	−0.804415
$66$	0	0
$67$	11.9221	1.45652	0.728259	−	0.685302i	$-0.240330\pi$
$67$	11.9221	1.45652	0.728259	+	0.685302i	$0.240330\pi$
$68$	0	0
$69$	−2.46268	−0.296472
$70$	0	0
$71$	−4.59958	−0.545870	−0.272935	−	0.962033i	$-0.587994\pi$
$71$	−4.59958	−0.545870	−0.272935	+	0.962033i	$0.587994\pi$
$72$	0	0
$73$	−7.59806	−0.889286	−0.444643	−	0.895708i	$-0.646669\pi$
$73$	−7.59806	−0.889286	−0.444643	+	0.895708i	$0.646669\pi$
$74$	0	0
$75$	−6.83537	−0.789281
$76$	0	0
$77$	−4.70095	−0.535723
$78$	0	0
$79$	−5.73902	−0.645690	−0.322845	−	0.946452i	$-0.604639\pi$
$79$	−5.73902	−0.645690	−0.322845	+	0.946452i	$0.604639\pi$
$80$	0	0
$81$	−8.80137	−0.977930
$82$	0	0
$83$	5.57695	0.612149	0.306075	−	0.952008i	$-0.400984\pi$
$83$	5.57695	0.612149	0.306075	+	0.952008i	$0.400984\pi$
$84$	0	0
$85$	−5.09328	−0.552444
$86$	0	0
$87$	24.5989	2.63728
$88$	0	0
$89$	−11.2284	−1.19021	−0.595106	−	0.803647i	$-0.702890\pi$
$89$	−11.2284	−1.19021	−0.595106	+	0.803647i	$0.702890\pi$
$90$	0	0
$91$	2.32579	0.243809
$92$	0	0
$93$	8.71129	0.903319
$94$	0	0
$95$	−19.7746	−2.02883
$96$	0	0
$97$	−1.68805	−0.171396	−0.0856979	−	0.996321i	$-0.527312\pi$
$97$	−1.68805	−0.171396	−0.0856979	+	0.996321i	$0.527312\pi$
$98$	0	0
$99$	14.4075	1.44801
$100$	0	0
$101$	3.12810	0.311258	0.155629	−	0.987816i	$-0.450260\pi$
$101$	3.12810	0.311258	0.155629	+	0.987816i	$0.450260\pi$
$102$	0	0
$103$	−1.03808	−0.102285	−0.0511423	−	0.998691i	$-0.516286\pi$
$103$	−1.03808	−0.102285	−0.0511423	+	0.998691i	$0.516286\pi$
$104$	0	0
$105$	6.86713	0.670163
$106$	0	0
$107$	−8.21965	−0.794624	−0.397312	−	0.917684i	$-0.630057\pi$
$107$	−8.21965	−0.794624	−0.397312	+	0.917684i	$0.630057\pi$
$108$	0	0
$109$	6.99848	0.670333	0.335166	−	0.942159i	$-0.391207\pi$
$109$	6.99848	0.670333	0.335166	+	0.942159i	$0.391207\pi$
$110$	0	0
$111$	0.409247	0.0388440
$112$	0	0
$113$	2.65158	0.249439	0.124720	−	0.992192i	$-0.460197\pi$
$113$	2.65158	0.249439	0.124720	+	0.992192i	$0.460197\pi$
$114$	0	0
$115$	2.78847	0.260026
$116$	0	0
$117$	−7.12810	−0.658993
$118$	0	0
$119$	1.82655	0.167439
$120$	0	0
$121$	11.0989	1.00899
$122$	0	0
$123$	−17.8573	−1.61014
$124$	0	0
$125$	−6.20274	−0.554790
$126$	0	0
$127$	0.847744	0.0752251	0.0376126	−	0.999292i	$-0.488025\pi$
$127$	0.847744	0.0752251	0.0376126	+	0.999292i	$0.488025\pi$
$128$	0	0
$129$	−23.5850	−2.07654
$130$	0	0
$131$	3.40769	0.297732	0.148866	−	0.988857i	$-0.452438\pi$
$131$	3.40769	0.297732	0.148866	+	0.988857i	$0.452438\pi$
$132$	0	0
$133$	7.09155	0.614915
$134$	0	0
$135$	−0.445067	−0.0383052
$136$	0	0
$137$	−14.4092	−1.23107	−0.615533	−	0.788111i	$-0.711059\pi$
$137$	−14.4092	−1.23107	−0.615533	+	0.788111i	$0.711059\pi$
$138$	0	0
$139$	−2.60685	−0.221110	−0.110555	−	0.993870i	$-0.535263\pi$
$139$	−2.60685	−0.221110	−0.110555	+	0.993870i	$0.535263\pi$
$140$	0	0
$141$	11.4895	0.967587
$142$	0	0
$143$	−10.9334	−0.914298
$144$	0	0
$145$	−27.8531	−2.31307
$146$	0	0
$147$	−2.46268	−0.203119
$148$	0	0
$149$	−0.00887233	−0.000726849 0	−0.000363425	−	1.00000i	$-0.500116\pi$
$149$	−0.00887233	−0.000726849 0	−0.000363425		1.00000i	$0.500116\pi$
$150$	0	0
$151$	−13.1070	−1.06663	−0.533316	−	0.845916i	$-0.679054\pi$
$151$	−13.1070	−1.06663	−0.533316	+	0.845916i	$0.679054\pi$
$152$	0	0
$153$	−5.59803	−0.452574
$154$	0	0
$155$	−9.86371	−0.792272
$156$	0	0
$157$	−0.249603	−0.0199205	−0.00996025	−	0.999950i	$-0.503170\pi$
$157$	−0.249603	−0.0199205	−0.00996025	+	0.999950i	$0.503170\pi$
$158$	0	0
$159$	2.36892	0.187867
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	0.150737	0.0118066	0.00590332	−	0.999983i	$-0.498121\pi$
$163$	0.150737	0.0118066	0.00590332	+	0.999983i	$0.498121\pi$
$164$	0	0
$165$	−32.2820	−2.51315
$166$	0	0
$167$	−11.8235	−0.914931	−0.457465	−	0.889227i	$-0.651243\pi$
$167$	−11.8235	−0.914931	−0.457465	+	0.889227i	$0.651243\pi$
$168$	0	0
$169$	−7.59071	−0.583900
$170$	0	0
$171$	−21.7343	−1.66206
$172$	0	0
$173$	6.78120	0.515565	0.257783	−	0.966203i	$-0.417008\pi$
$173$	6.78120	0.515565	0.257783	+	0.966203i	$0.417008\pi$
$174$	0	0
$175$	−2.77558	−0.209814
$176$	0	0
$177$	34.1821	2.56928
$178$	0	0
$179$	−20.5624	−1.53690	−0.768451	−	0.639908i	$-0.778972\pi$
$179$	−20.5624	−1.53690	−0.768451	+	0.639908i	$0.778972\pi$
$180$	0	0
$181$	−1.69693	−0.126131	−0.0630657	−	0.998009i	$-0.520088\pi$
$181$	−1.69693	−0.126131	−0.0630657	+	0.998009i	$0.520088\pi$
$182$	0	0
$183$	−2.35101	−0.173791
$184$	0	0
$185$	−0.463386	−0.0340688
$186$	0	0
$187$	−8.58651	−0.627907
$188$	0	0
$189$	0.159610	0.0116099
$190$	0	0
$191$	−15.8873	−1.14956	−0.574782	−	0.818307i	$-0.694913\pi$
$191$	−15.8873	−1.14956	−0.574782	+	0.818307i	$0.694913\pi$
$192$	0	0
$193$	−21.1919	−1.52543	−0.762714	−	0.646736i	$-0.776134\pi$
$193$	−21.1919	−1.52543	−0.762714	+	0.646736i	$0.776134\pi$
$194$	0	0
$195$	15.9715	1.14374
$196$	0	0
$197$	18.5146	1.31911	0.659554	−	0.751657i	$-0.270745\pi$
$197$	18.5146	1.31911	0.659554	+	0.751657i	$0.270745\pi$
$198$	0	0
$199$	−12.3192	−0.873286	−0.436643	−	0.899635i	$-0.643833\pi$
$199$	−12.3192	−0.873286	−0.436643	+	0.899635i	$0.643833\pi$
$200$	0	0
$201$	−29.3604	−2.07092
$202$	0	0
$203$	9.98866	0.701066
$204$	0	0
$205$	20.2197	1.41220
$206$	0	0
$207$	3.06481	0.213019
$208$	0	0
$209$	−33.3370	−2.30597
$210$	0	0
$211$	3.49259	0.240440	0.120220	−	0.992747i	$-0.461640\pi$
$211$	3.49259	0.240440	0.120220	+	0.992747i	$0.461640\pi$
$212$	0	0
$213$	11.3273	0.776134
$214$	0	0
$215$	26.7050	1.82127
$216$	0	0
$217$	3.53732	0.240129
$218$	0	0
$219$	18.7116	1.26441
$220$	0	0
$221$	4.24817	0.285763
$222$	0	0
$223$	−9.77808	−0.654789	−0.327394	−	0.944888i	$-0.606171\pi$
$223$	−9.77808	−0.654789	−0.327394	+	0.944888i	$0.606171\pi$
$224$	0	0
$225$	8.50663	0.567108
$226$	0	0
$227$	−8.25924	−0.548185	−0.274093	−	0.961703i	$-0.588378\pi$
$227$	−8.25924	−0.548185	−0.274093	+	0.961703i	$0.588378\pi$
$228$	0	0
$229$	3.13841	0.207392	0.103696	−	0.994609i	$-0.466933\pi$
$229$	3.13841	0.207392	0.103696	+	0.994609i	$0.466933\pi$
$230$	0	0
$231$	11.5769	0.761707
$232$	0	0
$233$	8.84601	0.579521	0.289761	−	0.957099i	$-0.406424\pi$
$233$	8.84601	0.579521	0.289761	+	0.957099i	$0.406424\pi$
$234$	0	0
$235$	−13.0094	−0.848639
$236$	0	0
$237$	14.1334	0.918063
$238$	0	0
$239$	10.3793	0.671379	0.335689	−	0.941973i	$-0.391031\pi$
$239$	10.3793	0.671379	0.335689	+	0.941973i	$0.391031\pi$
$240$	0	0
$241$	−2.47813	−0.159630	−0.0798151	−	0.996810i	$-0.525433\pi$
$241$	−2.47813	−0.159630	−0.0798151	+	0.996810i	$0.525433\pi$
$242$	0	0
$243$	22.1538	1.42117
$244$	0	0
$245$	2.78847	0.178149
$246$	0	0
$247$	16.4934	1.04945
$248$	0	0
$249$	−13.7343	−0.870373
$250$	0	0
$251$	9.66697	0.610174	0.305087	−	0.952324i	$-0.401314\pi$
$251$	9.66697	0.610174	0.305087	+	0.952324i	$0.401314\pi$
$252$	0	0
$253$	4.70095	0.295546
$254$	0	0
$255$	12.5431	0.785482
$256$	0	0
$257$	16.2773	1.01535	0.507676	−	0.861548i	$-0.330505\pi$
$257$	16.2773	1.01535	0.507676	+	0.861548i	$0.330505\pi$
$258$	0	0
$259$	0.166179	0.0103259
$260$	0	0
$261$	−30.6134	−1.89492
$262$	0	0
$263$	9.03331	0.557017	0.278509	−	0.960434i	$-0.410160\pi$
$263$	9.03331	0.557017	0.278509	+	0.960434i	$0.410160\pi$
$264$	0	0
$265$	−2.68230	−0.164772
$266$	0	0
$267$	27.6521	1.69228
$268$	0	0
$269$	−17.6238	−1.07454	−0.537272	−	0.843409i	$-0.680545\pi$
$269$	−17.6238	−1.07454	−0.537272	+	0.843409i	$0.680545\pi$
$270$	0	0
$271$	−23.9935	−1.45750	−0.728750	−	0.684780i	$-0.759898\pi$
$271$	−23.9935	−1.45750	−0.728750	+	0.684780i	$0.759898\pi$
$272$	0	0
$273$	−5.72768	−0.346655
$274$	0	0
$275$	13.0479	0.786815
$276$	0	0
$277$	16.5641	0.995239	0.497620	−	0.867395i	$-0.334208\pi$
$277$	16.5641	0.995239	0.497620	+	0.867395i	$0.334208\pi$
$278$	0	0
$279$	−10.8412	−0.649046
$280$	0	0
$281$	−19.5228	−1.16463	−0.582316	−	0.812962i	$-0.697854\pi$
$281$	−19.5228	−1.16463	−0.582316	+	0.812962i	$0.697854\pi$
$282$	0	0
$283$	15.7324	0.935191	0.467596	−	0.883942i	$-0.345120\pi$
$283$	15.7324	0.935191	0.467596	+	0.883942i	$0.345120\pi$
$284$	0	0
$285$	48.6985	2.88465
$286$	0	0
$287$	−7.25116	−0.428022
$288$	0	0
$289$	−13.6637	−0.803748
$290$	0	0
$291$	4.15714	0.243696
$292$	0	0
$293$	−8.16268	−0.476869	−0.238434	−	0.971159i	$-0.576634\pi$
$293$	−8.16268	−0.476869	−0.238434	+	0.971159i	$0.576634\pi$
$294$	0	0
$295$	−38.7041	−2.25344
$296$	0	0
$297$	−0.750316	−0.0435377
$298$	0	0
$299$	−2.32579	−0.134504
$300$	0	0
$301$	−9.57695	−0.552006
$302$	0	0
$303$	−7.70353	−0.442556
$304$	0	0
$305$	2.66202	0.152427
$306$	0	0
$307$	−30.5542	−1.74382	−0.871910	−	0.489667i	$-0.837118\pi$
$307$	−30.5542	−1.74382	−0.871910	+	0.489667i	$0.837118\pi$
$308$	0	0
$309$	2.55645	0.145431
$310$	0	0
$311$	−25.3573	−1.43788	−0.718941	−	0.695071i	$-0.755373\pi$
$311$	−25.3573	−1.43788	−0.718941	+	0.695071i	$0.755373\pi$
$312$	0	0
$313$	−6.15654	−0.347988	−0.173994	−	0.984747i	$-0.555667\pi$
$313$	−6.15654	−0.347988	−0.173994	+	0.984747i	$0.555667\pi$
$314$	0	0
$315$	−8.54614	−0.481521
$316$	0	0
$317$	−4.48063	−0.251657	−0.125829	−	0.992052i	$-0.540159\pi$
$317$	−4.48063	−0.251657	−0.125829	+	0.992052i	$0.540159\pi$
$318$	0	0
$319$	−46.9562	−2.62904
$320$	0	0
$321$	20.2424	1.12982
$322$	0	0
$323$	12.9531	0.720727
$324$	0	0
$325$	−6.45541	−0.358082
$326$	0	0
$327$	−17.2350	−0.953100
$328$	0	0
$329$	4.66542	0.257213
$330$	0	0
$331$	1.62894	0.0895349	0.0447674	−	0.998997i	$-0.485745\pi$
$331$	1.62894	0.0895349	0.0447674	+	0.998997i	$0.485745\pi$
$332$	0	0
$333$	−0.509308	−0.0279099
$334$	0	0
$335$	33.2445	1.81634
$336$	0	0
$337$	4.91650	0.267819	0.133909	−	0.990994i	$-0.457247\pi$
$337$	4.91650	0.267819	0.133909	+	0.990994i	$0.457247\pi$
$338$	0	0
$339$	−6.53000	−0.354661
$340$	0	0
$341$	−16.6287	−0.900497
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.86713	−0.369714
$346$	0	0
$347$	−33.8004	−1.81450	−0.907249	−	0.420593i	$-0.861822\pi$
$347$	−33.8004	−1.81450	−0.907249	+	0.420593i	$0.861822\pi$
$348$	0	0
$349$	3.26430	0.174734	0.0873669	−	0.996176i	$-0.472155\pi$
$349$	3.26430	0.174734	0.0873669	+	0.996176i	$0.472155\pi$
$350$	0	0
$351$	0.371218	0.0198142
$352$	0	0
$353$	11.1070	0.591165	0.295583	−	0.955317i	$-0.404486\pi$
$353$	11.1070	0.591165	0.295583	+	0.955317i	$0.404486\pi$
$354$	0	0
$355$	−12.8258	−0.680723
$356$	0	0
$357$	−4.49821	−0.238071
$358$	0	0
$359$	19.0760	1.00679	0.503397	−	0.864056i	$-0.332084\pi$
$359$	19.0760	1.00679	0.503397	+	0.864056i	$0.332084\pi$
$360$	0	0
$361$	31.2900	1.64684
$362$	0	0
$363$	−27.3331	−1.43461
$364$	0	0
$365$	−21.1870	−1.10898
$366$	0	0
$367$	31.9319	1.66683	0.833416	−	0.552647i	$-0.186382\pi$
$367$	31.9319	1.66683	0.833416	+	0.552647i	$0.186382\pi$
$368$	0	0
$369$	22.2234	1.15691
$370$	0	0
$371$	0.961924	0.0499406
$372$	0	0
$373$	20.8038	1.07718	0.538590	−	0.842568i	$-0.318957\pi$
$373$	20.8038	1.07718	0.538590	+	0.842568i	$0.318957\pi$
$374$	0	0
$375$	15.2754	0.788817
$376$	0	0
$377$	23.2315	1.19648
$378$	0	0
$379$	26.5314	1.36282	0.681412	−	0.731900i	$-0.261366\pi$
$379$	26.5314	1.36282	0.681412	+	0.731900i	$0.261366\pi$
$380$	0	0
$381$	−2.08773	−0.106957
$382$	0	0
$383$	20.4224	1.04354	0.521768	−	0.853088i	$-0.325273\pi$
$383$	20.4224	1.04354	0.521768	+	0.853088i	$0.325273\pi$
$384$	0	0
$385$	−13.1085	−0.668069
$386$	0	0
$387$	29.3515	1.49202
$388$	0	0
$389$	1.10846	0.0562012	0.0281006	−	0.999605i	$-0.491054\pi$
$389$	1.10846	0.0562012	0.0281006	+	0.999605i	$0.491054\pi$
$390$	0	0
$391$	−1.82655	−0.0923725
$392$	0	0
$393$	−8.39207	−0.423324
$394$	0	0
$395$	−16.0031	−0.805204
$396$	0	0
$397$	−19.3304	−0.970166	−0.485083	−	0.874468i	$-0.661211\pi$
$397$	−19.3304	−0.970166	−0.485083	+	0.874468i	$0.661211\pi$
$398$	0	0
$399$	−17.4642	−0.874305
$400$	0	0
$401$	34.3076	1.71324	0.856620	−	0.515947i	$-0.172560\pi$
$401$	34.3076	1.71324	0.856620	+	0.515947i	$0.172560\pi$
$402$	0	0
$403$	8.22705	0.409819
$404$	0	0
$405$	−24.5424	−1.21952
$406$	0	0
$407$	−0.781200	−0.0387227
$408$	0	0
$409$	−11.1492	−0.551293	−0.275647	−	0.961259i	$-0.588892\pi$
$409$	−11.1492	−0.551293	−0.275647	+	0.961259i	$0.588892\pi$
$410$	0	0
$411$	35.4854	1.75037
$412$	0	0
$413$	13.8800	0.682991
$414$	0	0
$415$	15.5512	0.763376
$416$	0	0
$417$	6.41985	0.314381
$418$	0	0
$419$	36.8881	1.80210	0.901052	−	0.433711i	$-0.142796\pi$
$419$	36.8881	1.80210	0.901052	+	0.433711i	$0.142796\pi$
$420$	0	0
$421$	21.8055	1.06273	0.531367	−	0.847142i	$-0.321679\pi$
$421$	21.8055	1.06273	0.531367	+	0.847142i	$0.321679\pi$
$422$	0	0
$423$	−14.2986	−0.695223
$424$	0	0
$425$	−5.06973	−0.245918
$426$	0	0
$427$	−0.954652	−0.0461988
$428$	0	0
$429$	26.9255	1.29998
$430$	0	0
$431$	−17.1450	−0.825846	−0.412923	−	0.910766i	$-0.635492\pi$
$431$	−17.1450	−0.825846	−0.412923	+	0.910766i	$0.635492\pi$
$432$	0	0
$433$	2.68890	0.129220	0.0646102	−	0.997911i	$-0.479420\pi$
$433$	2.68890	0.129220	0.0646102	+	0.997911i	$0.479420\pi$
$434$	0	0
$435$	68.5934	3.28880
$436$	0	0
$437$	−7.09155	−0.339235
$438$	0	0
$439$	−23.9686	−1.14396	−0.571979	−	0.820268i	$-0.693824\pi$
$439$	−23.9686	−1.14396	−0.571979	+	0.820268i	$0.693824\pi$
$440$	0	0
$441$	3.06481	0.145943
$442$	0	0
$443$	28.5038	1.35426	0.677128	−	0.735865i	$-0.263224\pi$
$443$	28.5038	1.35426	0.677128	+	0.735865i	$0.263224\pi$
$444$	0	0
$445$	−31.3102	−1.48425
$446$	0	0
$447$	0.0218497	0.00103346
$448$	0	0
$449$	1.18398	0.0558753	0.0279376	−	0.999610i	$-0.491106\pi$
$449$	1.18398	0.0558753	0.0279376	+	0.999610i	$0.491106\pi$
$450$	0	0
$451$	34.0873	1.60511
$452$	0	0
$453$	32.2784	1.51657
$454$	0	0
$455$	6.48540	0.304040
$456$	0	0
$457$	33.3173	1.55852	0.779260	−	0.626701i	$-0.215595\pi$
$457$	33.3173	1.55852	0.779260	+	0.626701i	$0.215595\pi$
$458$	0	0
$459$	0.291534	0.0136077
$460$	0	0
$461$	−1.59002	−0.0740545	−0.0370273	−	0.999314i	$-0.511789\pi$
$461$	−1.59002	−0.0740545	−0.0370273	+	0.999314i	$0.511789\pi$
$462$	0	0
$463$	−30.4569	−1.41545	−0.707726	−	0.706487i	$-0.750279\pi$
$463$	−30.4569	−1.41545	−0.707726	+	0.706487i	$0.750279\pi$
$464$	0	0
$465$	24.2912	1.12648
$466$	0	0
$467$	−27.6450	−1.27926	−0.639628	−	0.768685i	$-0.720912\pi$
$467$	−27.6450	−1.27926	−0.639628	+	0.768685i	$0.720912\pi$
$468$	0	0
$469$	−11.9221	−0.550512
$470$	0	0
$471$	0.614693	0.0283236
$472$	0	0
$473$	45.0207	2.07005
$474$	0	0
$475$	−19.6832	−0.903125
$476$	0	0
$477$	−2.94812	−0.134985
$478$	0	0
$479$	−36.3245	−1.65971	−0.829855	−	0.557979i	$-0.811577\pi$
$479$	−36.3245	−1.65971	−0.829855	+	0.557979i	$0.811577\pi$
$480$	0	0
$481$	0.386498	0.0176228
$482$	0	0
$483$	2.46268	0.112056
$484$	0	0
$485$	−4.70709	−0.213738
$486$	0	0
$487$	−31.6204	−1.43286	−0.716428	−	0.697661i	$-0.754224\pi$
$487$	−31.6204	−1.43286	−0.716428	+	0.697661i	$0.754224\pi$
$488$	0	0
$489$	−0.371218	−0.0167871
$490$	0	0
$491$	41.6773	1.88087	0.940436	−	0.339972i	$-0.110418\pi$
$491$	41.6773	1.88087	0.940436	+	0.339972i	$0.110418\pi$
$492$	0	0
$493$	18.2448	0.821703
$494$	0	0
$495$	40.1750	1.80573
$496$	0	0
$497$	4.59958	0.206319
$498$	0	0
$499$	27.7258	1.24118	0.620588	−	0.784137i	$-0.286894\pi$
$499$	27.7258	1.24118	0.620588	+	0.784137i	$0.286894\pi$
$500$	0	0
$501$	29.1176	1.30088
$502$	0	0
$503$	29.0492	1.29524	0.647620	−	0.761963i	$-0.275764\pi$
$503$	29.0492	1.29524	0.647620	+	0.761963i	$0.275764\pi$
$504$	0	0
$505$	8.72263	0.388152
$506$	0	0
$507$	18.6935	0.830208
$508$	0	0
$509$	−8.20421	−0.363645	−0.181823	−	0.983331i	$-0.558200\pi$
$509$	−8.20421	−0.363645	−0.181823	+	0.983331i	$0.558200\pi$
$510$	0	0
$511$	7.59806	0.336118
$512$	0	0
$513$	1.13188	0.0499737
$514$	0	0
$515$	−2.89465	−0.127553
$516$	0	0
$517$	−21.9319	−0.964563
$518$	0	0
$519$	−16.7000	−0.733046
$520$	0	0
$521$	−9.59168	−0.420219	−0.210110	−	0.977678i	$-0.567382\pi$
$521$	−9.59168	−0.420219	−0.210110	+	0.977678i	$0.567382\pi$
$522$	0	0
$523$	6.06424	0.265171	0.132585	−	0.991172i	$-0.457672\pi$
$523$	6.06424	0.265171	0.132585	+	0.991172i	$0.457672\pi$
$524$	0	0
$525$	6.83537	0.298320
$526$	0	0
$527$	6.46108	0.281449
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−42.5396	−1.84606
$532$	0	0
$533$	−16.8647	−0.730489
$534$	0	0
$535$	−22.9203	−0.990930
$536$	0	0
$537$	50.6386	2.18522
$538$	0	0
$539$	4.70095	0.202484
$540$	0	0
$541$	−28.6760	−1.23288	−0.616438	−	0.787403i	$-0.711425\pi$
$541$	−28.6760	−1.23288	−0.616438	+	0.787403i	$0.711425\pi$
$542$	0	0
$543$	4.17899	0.179338
$544$	0	0
$545$	19.5151	0.835934
$546$	0	0
$547$	24.5492	1.04965	0.524824	−	0.851211i	$-0.324131\pi$
$547$	24.5492	1.04965	0.524824	+	0.851211i	$0.324131\pi$
$548$	0	0
$549$	2.92583	0.124871
$550$	0	0
$551$	70.8350	3.01767
$552$	0	0
$553$	5.73902	0.244048
$554$	0	0
$555$	1.14117	0.0484401
$556$	0	0
$557$	−5.82872	−0.246971	−0.123485	−	0.992346i	$-0.539407\pi$
$557$	−5.82872	−0.246971	−0.123485	+	0.992346i	$0.539407\pi$
$558$	0	0
$559$	−22.2740	−0.942088
$560$	0	0
$561$	21.1458	0.892778
$562$	0	0
$563$	30.9619	1.30489	0.652445	−	0.757836i	$-0.273743\pi$
$563$	30.9619	1.30489	0.652445	+	0.757836i	$0.273743\pi$
$564$	0	0
$565$	7.39385	0.311062
$566$	0	0
$567$	8.80137	0.369623
$568$	0	0
$569$	6.77249	0.283918	0.141959	−	0.989873i	$-0.454660\pi$
$569$	6.77249	0.283918	0.141959	+	0.989873i	$0.454660\pi$
$570$	0	0
$571$	26.7220	1.11828	0.559140	−	0.829073i	$-0.311132\pi$
$571$	26.7220	1.11828	0.559140	+	0.829073i	$0.311132\pi$
$572$	0	0
$573$	39.1254	1.63449
$574$	0	0
$575$	2.77558	0.115750
$576$	0	0
$577$	−3.13787	−0.130631	−0.0653157	−	0.997865i	$-0.520805\pi$
$577$	−3.13787	−0.130631	−0.0653157	+	0.997865i	$0.520805\pi$
$578$	0	0
$579$	52.1890	2.16890
$580$	0	0
$581$	−5.57695	−0.231371
$582$	0	0
$583$	−4.52196	−0.187280
$584$	0	0
$585$	−19.8765	−0.821793
$586$	0	0
$587$	−31.4784	−1.29925	−0.649626	−	0.760254i	$-0.725075\pi$
$587$	−31.4784	−1.29925	−0.649626	+	0.760254i	$0.725075\pi$
$588$	0	0
$589$	25.0850	1.03361
$590$	0	0
$591$	−45.5955	−1.87555
$592$	0	0
$593$	15.4196	0.633209	0.316604	−	0.948558i	$-0.397457\pi$
$593$	15.4196	0.633209	0.316604	+	0.948558i	$0.397457\pi$
$594$	0	0
$595$	5.09328	0.208804
$596$	0	0
$597$	30.3383	1.24167
$598$	0	0
$599$	18.7196	0.764862	0.382431	−	0.923984i	$-0.375087\pi$
$599$	18.7196	0.764862	0.382431	+	0.923984i	$0.375087\pi$
$600$	0	0
$601$	12.5523	0.512017	0.256009	−	0.966675i	$-0.417592\pi$
$601$	12.5523	0.512017	0.256009	+	0.966675i	$0.417592\pi$
$602$	0	0
$603$	36.5390	1.48798
$604$	0	0
$605$	30.9490	1.25825
$606$	0	0
$607$	−20.5171	−0.832761	−0.416381	−	0.909190i	$-0.636702\pi$
$607$	−20.5171	−0.832761	−0.416381	+	0.909190i	$0.636702\pi$
$608$	0	0
$609$	−24.5989	−0.996798
$610$	0	0
$611$	10.8508	0.438976
$612$	0	0
$613$	−24.9165	−1.00637	−0.503184	−	0.864179i	$-0.667838\pi$
$613$	−24.9165	−1.00637	−0.503184	+	0.864179i	$0.667838\pi$
$614$	0	0
$615$	−49.7946	−2.00791
$616$	0	0
$617$	−8.77553	−0.353289	−0.176645	−	0.984275i	$-0.556524\pi$
$617$	−8.77553	−0.353289	−0.176645	+	0.984275i	$0.556524\pi$
$618$	0	0
$619$	14.6150	0.587427	0.293714	−	0.955893i	$-0.405109\pi$
$619$	14.6150	0.587427	0.293714	+	0.955893i	$0.405109\pi$
$620$	0	0
$621$	−0.159610	−0.00640491
$622$	0	0
$623$	11.2284	0.449858
$624$	0	0
$625$	−31.1741	−1.24696
$626$	0	0
$627$	82.0984	3.27870
$628$	0	0
$629$	0.303535	0.0121027
$630$	0	0
$631$	−18.5826	−0.739760	−0.369880	−	0.929079i	$-0.620601\pi$
$631$	−18.5826	−0.739760	−0.369880	+	0.929079i	$0.620601\pi$
$632$	0	0
$633$	−8.60114	−0.341865
$634$	0	0
$635$	2.36391	0.0938089
$636$	0	0
$637$	−2.32579	−0.0921511
$638$	0	0
$639$	−14.0968	−0.557662
$640$	0	0
$641$	9.57505	0.378192	0.189096	−	0.981959i	$-0.439444\pi$
$641$	9.57505	0.378192	0.189096	+	0.981959i	$0.439444\pi$
$642$	0	0
$643$	−23.7204	−0.935443	−0.467722	−	0.883876i	$-0.654925\pi$
$643$	−23.7204	−0.935443	−0.467722	+	0.883876i	$0.654925\pi$
$644$	0	0
$645$	−65.7661	−2.58954
$646$	0	0
$647$	20.1385	0.791727	0.395864	−	0.918309i	$-0.370445\pi$
$647$	20.1385	0.791727	0.395864	+	0.918309i	$0.370445\pi$
$648$	0	0
$649$	−65.2492	−2.56126
$650$	0	0
$651$	−8.71129	−0.341422
$652$	0	0
$653$	−32.6863	−1.27911	−0.639557	−	0.768744i	$-0.720882\pi$
$653$	−32.6863	−1.27911	−0.639557	+	0.768744i	$0.720882\pi$
$654$	0	0
$655$	9.50226	0.371284
$656$	0	0
$657$	−23.2866	−0.908498
$658$	0	0
$659$	−37.3251	−1.45398	−0.726989	−	0.686649i	$-0.759081\pi$
$659$	−37.3251	−1.45398	−0.726989	+	0.686649i	$0.759081\pi$
$660$	0	0
$661$	−9.68312	−0.376630	−0.188315	−	0.982109i	$-0.560303\pi$
$661$	−9.68312	−0.376630	−0.188315	+	0.982109i	$0.560303\pi$
$662$	0	0
$663$	−10.4619	−0.406306
$664$	0	0
$665$	19.7746	0.766825
$666$	0	0
$667$	−9.98866	−0.386762
$668$	0	0
$669$	24.0803	0.930999
$670$	0	0
$671$	4.48777	0.173248
$672$	0	0
$673$	38.5622	1.48646	0.743232	−	0.669034i	$-0.233292\pi$
$673$	38.5622	1.48646	0.743232	+	0.669034i	$0.233292\pi$
$674$	0	0
$675$	−0.443009	−0.0170514
$676$	0	0
$677$	2.70428	0.103934	0.0519669	−	0.998649i	$-0.483451\pi$
$677$	2.70428	0.103934	0.0519669	+	0.998649i	$0.483451\pi$
$678$	0	0
$679$	1.68805	0.0647815
$680$	0	0
$681$	20.3399	0.779427
$682$	0	0
$683$	10.6222	0.406446	0.203223	−	0.979132i	$-0.434858\pi$
$683$	10.6222	0.406446	0.203223	+	0.979132i	$0.434858\pi$
$684$	0	0
$685$	−40.1798	−1.53519
$686$	0	0
$687$	−7.72892	−0.294877
$688$	0	0
$689$	2.23723	0.0852318
$690$	0	0
$691$	−23.7327	−0.902833	−0.451416	−	0.892313i	$-0.649081\pi$
$691$	−23.7327	−0.902833	−0.451416	+	0.892313i	$0.649081\pi$
$692$	0	0
$693$	−14.4075	−0.547296
$694$	0	0
$695$	−7.26913	−0.275734
$696$	0	0
$697$	−13.2446	−0.501674
$698$	0	0
$699$	−21.7849	−0.823982
$700$	0	0
$701$	22.5615	0.852138	0.426069	−	0.904691i	$-0.359898\pi$
$701$	22.5615	0.852138	0.426069	+	0.904691i	$0.359898\pi$
$702$	0	0
$703$	1.17847	0.0444468
$704$	0	0
$705$	32.0380	1.20662
$706$	0	0
$707$	−3.12810	−0.117644
$708$	0	0
$709$	−16.7030	−0.627294	−0.313647	−	0.949540i	$-0.601551\pi$
$709$	−16.7030	−0.627294	−0.313647	+	0.949540i	$0.601551\pi$
$710$	0	0
$711$	−17.5890	−0.659640
$712$	0	0
$713$	−3.53732	−0.132474
$714$	0	0
$715$	−30.4875	−1.14017
$716$	0	0
$717$	−25.5608	−0.954587
$718$	0	0
$719$	−27.7469	−1.03479	−0.517393	−	0.855748i	$-0.673097\pi$
$719$	−27.7469	−1.03479	−0.517393	+	0.855748i	$0.673097\pi$
$720$	0	0
$721$	1.03808	0.0386600
$722$	0	0
$723$	6.10284	0.226967
$724$	0	0
$725$	−27.7243	−1.02966
$726$	0	0
$727$	41.4981	1.53908	0.769539	−	0.638600i	$-0.220486\pi$
$727$	41.4981	1.53908	0.769539	+	0.638600i	$0.220486\pi$
$728$	0	0
$729$	−28.1537	−1.04273
$730$	0	0
$731$	−17.4928	−0.646993
$732$	0	0
$733$	35.7212	1.31939	0.659697	−	0.751532i	$-0.270685\pi$
$733$	35.7212	1.31939	0.659697	+	0.751532i	$0.270685\pi$
$734$	0	0
$735$	−6.86713	−0.253298
$736$	0	0
$737$	56.0452	2.06445
$738$	0	0
$739$	44.4520	1.63519	0.817597	−	0.575791i	$-0.195306\pi$
$739$	44.4520	1.63519	0.817597	+	0.575791i	$0.195306\pi$
$740$	0	0
$741$	−40.6181	−1.49214
$742$	0	0
$743$	17.0898	0.626964	0.313482	−	0.949594i	$-0.398504\pi$
$743$	17.0898	0.626964	0.313482	+	0.949594i	$0.398504\pi$
$744$	0	0
$745$	−0.0247402	−0.000906412 0
$746$	0	0
$747$	17.0923	0.625374
$748$	0	0
$749$	8.21965	0.300339
$750$	0	0
$751$	40.9748	1.49519	0.747596	−	0.664154i	$-0.231208\pi$
$751$	40.9748	1.49519	0.747596	+	0.664154i	$0.231208\pi$
$752$	0	0
$753$	−23.8067	−0.867564
$754$	0	0
$755$	−36.5485	−1.33014
$756$	0	0
$757$	−23.7095	−0.861737	−0.430868	−	0.902415i	$-0.641793\pi$
$757$	−23.7095	−0.861737	−0.430868	+	0.902415i	$0.641793\pi$
$758$	0	0
$759$	−11.5769	−0.420216
$760$	0	0
$761$	−42.7680	−1.55034	−0.775170	−	0.631753i	$-0.782336\pi$
$761$	−42.7680	−1.55034	−0.775170	+	0.631753i	$0.782336\pi$
$762$	0	0
$763$	−6.99848	−0.253362
$764$	0	0
$765$	−15.6099	−0.564379
$766$	0	0
$767$	32.2820	1.16564
$768$	0	0
$769$	35.6281	1.28478	0.642392	−	0.766376i	$-0.277942\pi$
$769$	35.6281	1.28478	0.642392	+	0.766376i	$0.277942\pi$
$770$	0	0
$771$	−40.0859	−1.44366
$772$	0	0
$773$	16.9388	0.609246	0.304623	−	0.952473i	$-0.401470\pi$
$773$	16.9388	0.609246	0.304623	+	0.952473i	$0.401470\pi$
$774$	0	0
$775$	−9.81810	−0.352677
$776$	0	0
$777$	−0.409247	−0.0146817
$778$	0	0
$779$	−51.4219	−1.84238
$780$	0	0
$781$	−21.6224	−0.773709
$782$	0	0
$783$	1.59429	0.0569751
$784$	0	0
$785$	−0.696011	−0.0248417
$786$	0	0
$787$	15.4748	0.551619	0.275809	−	0.961212i	$-0.411054\pi$
$787$	15.4748	0.551619	0.275809	+	0.961212i	$0.411054\pi$
$788$	0	0
$789$	−22.2462	−0.791985
$790$	0	0
$791$	−2.65158	−0.0942792
$792$	0	0
$793$	−2.22032	−0.0788458
$794$	0	0
$795$	6.60566	0.234278
$796$	0	0
$797$	28.1935	0.998664	0.499332	−	0.866411i	$-0.333579\pi$
$797$	28.1935	0.998664	0.499332	+	0.866411i	$0.333579\pi$
$798$	0	0
$799$	8.52161	0.301473
$800$	0	0
$801$	−34.4131	−1.21593
$802$	0	0
$803$	−35.7181	−1.26046
$804$	0	0
$805$	−2.78847	−0.0982807
$806$	0	0
$807$	43.4020	1.52782
$808$	0	0
$809$	−30.7565	−1.08134	−0.540670	−	0.841235i	$-0.681829\pi$
$809$	−30.7565	−1.08134	−0.540670	+	0.841235i	$0.681829\pi$
$810$	0	0
$811$	−1.27477	−0.0447632	−0.0223816	−	0.999750i	$-0.507125\pi$
$811$	−1.27477	−0.0447632	−0.0223816	+	0.999750i	$0.507125\pi$
$812$	0	0
$813$	59.0883	2.07232
$814$	0	0
$815$	0.420327	0.0147234
$816$	0	0
$817$	−67.9154	−2.37606
$818$	0	0
$819$	7.12810	0.249076
$820$	0	0
$821$	−5.88860	−0.205513	−0.102757	−	0.994707i	$-0.532766\pi$
$821$	−5.88860	−0.205513	−0.102757	+	0.994707i	$0.532766\pi$
$822$	0	0
$823$	−13.3985	−0.467043	−0.233522	−	0.972352i	$-0.575025\pi$
$823$	−13.3985	−0.467043	−0.233522	+	0.972352i	$0.575025\pi$
$824$	0	0
$825$	−32.1327	−1.11872
$826$	0	0
$827$	−30.5493	−1.06230	−0.531151	−	0.847277i	$-0.678240\pi$
$827$	−30.5493	−1.06230	−0.531151	+	0.847277i	$0.678240\pi$
$828$	0	0
$829$	−33.2608	−1.15520	−0.577598	−	0.816321i	$-0.696010\pi$
$829$	−33.2608	−1.15520	−0.577598	+	0.816321i	$0.696010\pi$
$830$	0	0
$831$	−40.7921	−1.41506
$832$	0	0
$833$	−1.82655	−0.0632861
$834$	0	0
$835$	−32.9695	−1.14096
$836$	0	0
$837$	0.564589	0.0195151
$838$	0	0
$839$	−19.4516	−0.671543	−0.335772	−	0.941943i	$-0.608997\pi$
$839$	−19.4516	−0.671543	−0.335772	+	0.941943i	$0.608997\pi$
$840$	0	0
$841$	70.7733	2.44046
$842$	0	0
$843$	48.0785	1.65591
$844$	0	0
$845$	−21.1665	−0.728149
$846$	0	0
$847$	−11.0989	−0.381363
$848$	0	0
$849$	−38.7438	−1.32968
$850$	0	0
$851$	−0.166179	−0.00569655
$852$	0	0
$853$	−43.9900	−1.50619	−0.753094	−	0.657913i	$-0.771439\pi$
$853$	−43.9900	−1.50619	−0.753094	+	0.657913i	$0.771439\pi$
$854$	0	0
$855$	−60.6054	−2.07266
$856$	0	0
$857$	−50.9265	−1.73962	−0.869808	−	0.493391i	$-0.835757\pi$
$857$	−50.9265	−1.73962	−0.869808	+	0.493391i	$0.835757\pi$
$858$	0	0
$859$	−19.0473	−0.649885	−0.324943	−	0.945734i	$-0.605345\pi$
$859$	−19.0473	−0.649885	−0.324943	+	0.945734i	$0.605345\pi$
$860$	0	0
$861$	17.8573	0.608575
$862$	0	0
$863$	30.6481	1.04327	0.521637	−	0.853168i	$-0.325322\pi$
$863$	30.6481	1.04327	0.521637	+	0.853168i	$0.325322\pi$
$864$	0	0
$865$	18.9092	0.642932
$866$	0	0
$867$	33.6494	1.14279
$868$	0	0
$869$	−26.9788	−0.915194
$870$	0	0
$871$	−27.7283	−0.939538
$872$	0	0
$873$	−5.17357	−0.175099
$874$	0	0
$875$	6.20274	0.209691
$876$	0	0
$877$	15.6561	0.528668	0.264334	−	0.964431i	$-0.414848\pi$
$877$	15.6561	0.528668	0.264334	+	0.964431i	$0.414848\pi$
$878$	0	0
$879$	20.1021	0.678027
$880$	0	0
$881$	−51.9258	−1.74943	−0.874713	−	0.484642i	$-0.838950\pi$
$881$	−51.9258	−1.74943	−0.874713	+	0.484642i	$0.838950\pi$
$882$	0	0
$883$	−18.1603	−0.611144	−0.305572	−	0.952169i	$-0.598848\pi$
$883$	−18.1603	−0.611144	−0.305572	+	0.952169i	$0.598848\pi$
$884$	0	0
$885$	95.3158	3.20401
$886$	0	0
$887$	29.5677	0.992786	0.496393	−	0.868098i	$-0.334658\pi$
$887$	29.5677	0.992786	0.496393	+	0.868098i	$0.334658\pi$
$888$	0	0
$889$	−0.847744	−0.0284324
$890$	0	0
$891$	−41.3748	−1.38611
$892$	0	0
$893$	33.0850	1.10715
$894$	0	0
$895$	−57.3376	−1.91658
$896$	0	0
$897$	5.72768	0.191242
$898$	0	0
$899$	35.3330	1.17842
$900$	0	0
$901$	1.75700	0.0585342
$902$	0	0
$903$	23.5850	0.784859
$904$	0	0
$905$	−4.73183	−0.157291
$906$	0	0
$907$	−30.4350	−1.01058	−0.505289	−	0.862950i	$-0.668614\pi$
$907$	−30.4350	−1.01058	−0.505289	+	0.862950i	$0.668614\pi$
$908$	0	0
$909$	9.58705	0.317982
$910$	0	0
$911$	48.2859	1.59978	0.799892	−	0.600144i	$-0.204890\pi$
$911$	48.2859	1.59978	0.799892	+	0.600144i	$0.204890\pi$
$912$	0	0
$913$	26.2169	0.867653
$914$	0	0
$915$	−6.55571	−0.216725
$916$	0	0
$917$	−3.40769	−0.112532
$918$	0	0
$919$	28.4097	0.937150	0.468575	−	0.883424i	$-0.344768\pi$
$919$	28.4097	0.937150	0.468575	+	0.883424i	$0.344768\pi$
$920$	0	0
$921$	75.2453	2.47942
$922$	0	0
$923$	10.6976	0.352117
$924$	0	0
$925$	−0.461244	−0.0151656
$926$	0	0
$927$	−3.18151	−0.104494
$928$	0	0
$929$	27.0080	0.886104	0.443052	−	0.896496i	$-0.353896\pi$
$929$	27.0080	0.886104	0.443052	+	0.896496i	$0.353896\pi$
$930$	0	0
$931$	−7.09155	−0.232416
$932$	0	0
$933$	62.4471	2.04443
$934$	0	0
$935$	−23.9432	−0.783028
$936$	0	0
$937$	15.3639	0.501917	0.250958	−	0.967998i	$-0.419254\pi$
$937$	15.3639	0.501917	0.250958	+	0.967998i	$0.419254\pi$
$938$	0	0
$939$	15.1616	0.494780
$940$	0	0
$941$	−46.8015	−1.52569	−0.762843	−	0.646584i	$-0.776197\pi$
$941$	−46.8015	−1.52569	−0.762843	+	0.646584i	$0.776197\pi$
$942$	0	0
$943$	7.25116	0.236130
$944$	0	0
$945$	0.445067	0.0144780
$946$	0	0
$947$	−13.1326	−0.426753	−0.213376	−	0.976970i	$-0.568446\pi$
$947$	−13.1326	−0.426753	−0.213376	+	0.976970i	$0.568446\pi$
$948$	0	0
$949$	17.6715	0.573641
$950$	0	0
$951$	11.0344	0.357814
$952$	0	0
$953$	3.29495	0.106734	0.0533670	−	0.998575i	$-0.483005\pi$
$953$	3.29495	0.106734	0.0533670	+	0.998575i	$0.483005\pi$
$954$	0	0
$955$	−44.3013	−1.43356
$956$	0	0
$957$	115.638	3.73805
$958$	0	0
$959$	14.4092	0.465299
$960$	0	0
$961$	−18.4874	−0.596368
$962$	0	0
$963$	−25.1917	−0.811790
$964$	0	0
$965$	−59.0931	−1.90227
$966$	0	0
$967$	−1.39392	−0.0448254	−0.0224127	−	0.999749i	$-0.507135\pi$
$967$	−1.39392	−0.0448254	−0.0224127	+	0.999749i	$0.507135\pi$
$968$	0	0
$969$	−31.8993	−1.02475
$970$	0	0
$971$	10.8178	0.347158	0.173579	−	0.984820i	$-0.444467\pi$
$971$	10.8178	0.347158	0.173579	+	0.984820i	$0.444467\pi$
$972$	0	0
$973$	2.60685	0.0835718
$974$	0	0
$975$	15.8976	0.509132
$976$	0	0
$977$	23.4628	0.750642	0.375321	−	0.926895i	$-0.377532\pi$
$977$	23.4628	0.750642	0.375321	+	0.926895i	$0.377532\pi$
$978$	0	0
$979$	−52.7843	−1.68699
$980$	0	0
$981$	21.4490	0.684815
$982$	0	0
$983$	−57.3568	−1.82940	−0.914700	−	0.404134i	$-0.867573\pi$
$983$	−57.3568	−1.82940	−0.914700	+	0.404134i	$0.867573\pi$
$984$	0	0
$985$	51.6273	1.64498
$986$	0	0
$987$	−11.4895	−0.365713
$988$	0	0
$989$	9.57695	0.304529
$990$	0	0
$991$	−20.1069	−0.638718	−0.319359	−	0.947634i	$-0.603468\pi$
$991$	−20.1069	−0.638718	−0.319359	+	0.947634i	$0.603468\pi$
$992$	0	0
$993$	−4.01157	−0.127303
$994$	0	0
$995$	−34.3518	−1.08903
$996$	0	0
$997$	−12.2818	−0.388970	−0.194485	−	0.980906i	$-0.562303\pi$
$997$	−12.2818	−0.388970	−0.194485	+	0.980906i	$0.562303\pi$
$998$	0	0
$999$	0.0265238	0.000839176 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.bd.1.2		5
4.3	odd	2		161.2.a.d.1.1	✓	5
12.11	even	2		1449.2.a.r.1.5		5
20.19	odd	2		4025.2.a.p.1.5		5
28.27	even	2		1127.2.a.h.1.1		5
92.91	even	2		3703.2.a.j.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.1	✓	5	4.3	odd	2
1127.2.a.h.1.1		5	28.27	even	2
1449.2.a.r.1.5		5	12.11	even	2
2576.2.a.bd.1.2		5	1.1	even	1	trivial
3703.2.a.j.1.1		5	92.91	even	2
4025.2.a.p.1.5		5	20.19	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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46268 q^{3} +2.78847 q^{5} -1.00000 q^{7} +3.06481 q^{9} +O(q^{10})\)	\(q-2.46268 q^{3} +2.78847 q^{5} -1.00000 q^{7} +3.06481 q^{9} +4.70095 q^{11} -2.32579 q^{13} -6.86713 q^{15} -1.82655 q^{17} -7.09155 q^{19} +2.46268 q^{21} +1.00000 q^{23} +2.77558 q^{25} -0.159610 q^{27} -9.98866 q^{29} -3.53732 q^{31} -11.5769 q^{33} -2.78847 q^{35} -0.166179 q^{37} +5.72768 q^{39} +7.25116 q^{41} +9.57695 q^{43} +8.54614 q^{45} -4.66542 q^{47} +1.00000 q^{49} +4.49821 q^{51} -0.961924 q^{53} +13.1085 q^{55} +17.4642 q^{57} -13.8800 q^{59} +0.954652 q^{61} -3.06481 q^{63} -6.48540 q^{65} +11.9221 q^{67} -2.46268 q^{69} -4.59958 q^{71} -7.59806 q^{73} -6.83537 q^{75} -4.70095 q^{77} -5.73902 q^{79} -8.80137 q^{81} +5.57695 q^{83} -5.09328 q^{85} +24.5989 q^{87} -11.2284 q^{89} +2.32579 q^{91} +8.71129 q^{93} -19.7746 q^{95} -1.68805 q^{97} +14.4075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9}+O(q^{10}) \) 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9	\( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9} + 4 q^{11} - 6 q^{13} - 10 q^{15} - 12 q^{17} - 6 q^{19} + 5 q^{23} + 19 q^{25} - 4 q^{29} - 30 q^{31} - 22 q^{33} + 4 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 12 q^{43} - 12 q^{45} - 10 q^{47} + 5 q^{49} + 4 q^{51} + 16 q^{53} - 18 q^{55} + 6 q^{57} - 22 q^{59} - 18 q^{61} - 11 q^{63} - 26 q^{65} + 2 q^{67} - 4 q^{71} - 2 q^{73} + 30 q^{75} - 4 q^{77} - 30 q^{79} - 3 q^{81} - 8 q^{83} - 12 q^{85} + 12 q^{87} - 20 q^{89} + 6 q^{91} - 26 q^{93} - 8 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9 + 4 * q^11 - 6 * q^13 - 10 * q^15 - 12 * q^17 - 6 * q^19 + 5 * q^23 + 19 * q^25 - 4 * q^29 - 30 * q^31 - 22 * q^33 + 4 * q^35 + 4 * q^37 - 16 * q^39 + 6 * q^41 + 12 * q^43 - 12 * q^45 - 10 * q^47 + 5 * q^49 + 4 * q^51 + 16 * q^53 - 18 * q^55 + 6 * q^57 - 22 * q^59 - 18 * q^61 - 11 * q^63 - 26 * q^65 + 2 * q^67 - 4 * q^71 - 2 * q^73 + 30 * q^75 - 4 * q^77 - 30 * q^79 - 3 * q^81 - 8 * q^83 - 12 * q^85 + 12 * q^87 - 20 * q^89 + 6 * q^91 - 26 * q^93 - 8 * q^95 - 12 * q^97 + 8 * q^99

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