LMFDB - Embedded newform 862.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [862,2,Mod(1,862)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(862, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("862.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$862 = 2 \cdot 431$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	862.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:	$6.88310465423$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.9783113.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 8x^{4} + 7x^{3} + 11x^{2} - 9x + 1$ x^6 - x^5 - 8x^4 + 7x^3 + 11x^2 - 9x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.34828$ of defining polynomial
Character	$\chi$	$=$	862.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.63132 q^{3} +1.00000 q^{4} -3.84124 q^{5} +1.63132 q^{6} -4.15570 q^{7} +1.00000 q^{8} -0.338810 q^{9} -3.84124 q^{10} -0.276147 q^{11} +1.63132 q^{12} -1.67873 q^{13} -4.15570 q^{14} -6.26627 q^{15} +1.00000 q^{16} -5.94165 q^{17} -0.338810 q^{18} +4.04484 q^{19} -3.84124 q^{20} -6.77925 q^{21} -0.276147 q^{22} +4.98236 q^{23} +1.63132 q^{24} +9.75511 q^{25} -1.67873 q^{26} -5.44665 q^{27} -4.15570 q^{28} -8.14390 q^{29} -6.26627 q^{30} -4.97939 q^{31} +1.00000 q^{32} -0.450483 q^{33} -5.94165 q^{34} +15.9630 q^{35} -0.338810 q^{36} +6.04809 q^{37} +4.04484 q^{38} -2.73854 q^{39} -3.84124 q^{40} -5.16989 q^{41} -6.77925 q^{42} +5.45886 q^{43} -0.276147 q^{44} +1.30145 q^{45} +4.98236 q^{46} -9.49305 q^{47} +1.63132 q^{48} +10.2698 q^{49} +9.75511 q^{50} -9.69271 q^{51} -1.67873 q^{52} +5.04713 q^{53} -5.44665 q^{54} +1.06075 q^{55} -4.15570 q^{56} +6.59842 q^{57} -8.14390 q^{58} +0.0328732 q^{59} -6.26627 q^{60} -15.2993 q^{61} -4.97939 q^{62} +1.40799 q^{63} +1.00000 q^{64} +6.44841 q^{65} -0.450483 q^{66} +8.68500 q^{67} -5.94165 q^{68} +8.12781 q^{69} +15.9630 q^{70} +12.8570 q^{71} -0.338810 q^{72} -8.66551 q^{73} +6.04809 q^{74} +15.9137 q^{75} +4.04484 q^{76} +1.14758 q^{77} -2.73854 q^{78} -0.311274 q^{79} -3.84124 q^{80} -7.86878 q^{81} -5.16989 q^{82} +12.4146 q^{83} -6.77925 q^{84} +22.8233 q^{85} +5.45886 q^{86} -13.2853 q^{87} -0.276147 q^{88} +17.1312 q^{89} +1.30145 q^{90} +6.97630 q^{91} +4.98236 q^{92} -8.12296 q^{93} -9.49305 q^{94} -15.5372 q^{95} +1.63132 q^{96} +8.80436 q^{97} +10.2698 q^{98} +0.0935612 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 6 q^{2} - 8 q^{3} + 6 q^{4} - 8 q^{5} - 8 q^{6} - 9 q^{7} + 6 q^{8} + 14 q^{9} - 8 q^{10} - 10 q^{11} - 8 q^{12} + 3 q^{13} - 9 q^{14} - q^{15} + 6 q^{16} - 11 q^{17} + 14 q^{18} - 3 q^{19} - 8 q^{20}+ \cdots + 18 q^{99}+O(q^{100})$ 6 * q + 6 * q^2 - 8 * q^3 + 6 * q^4 - 8 * q^5 - 8 * q^6 - 9 * q^7 + 6 * q^8 + 14 * q^9 - 8 * q^10 - 10 * q^11 - 8 * q^12 + 3 * q^13 - 9 * q^14 - q^15 + 6 * q^16 - 11 * q^17 + 14 * q^18 - 3 * q^19 - 8 * q^20 - 2 * q^21 - 10 * q^22 - 20 * q^23 - 8 * q^24 + 10 * q^25 + 3 * q^26 - 41 * q^27 - 9 * q^28 - 21 * q^29 - q^30 - 28 * q^31 + 6 * q^32 + 7 * q^33 - 11 * q^34 - 6 * q^35 + 14 * q^36 - 7 * q^37 - 3 * q^38 - 3 * q^39 - 8 * q^40 - 11 * q^41 - 2 * q^42 + 3 * q^43 - 10 * q^44 - 8 * q^45 - 20 * q^46 - 12 * q^47 - 8 * q^48 + 33 * q^49 + 10 * q^50 + 14 * q^51 + 3 * q^52 + 11 * q^53 - 41 * q^54 + 15 * q^55 - 9 * q^56 + 9 * q^57 - 21 * q^58 - 9 * q^59 - q^60 - 16 * q^61 - 28 * q^62 + 8 * q^63 + 6 * q^64 - 4 * q^65 + 7 * q^66 + 2 * q^67 - 11 * q^68 + 40 * q^69 - 6 * q^70 - 4 * q^71 + 14 * q^72 - 19 * q^73 - 7 * q^74 + 7 * q^75 - 3 * q^76 - 25 * q^77 - 3 * q^78 - 4 * q^79 - 8 * q^80 + 74 * q^81 - 11 * q^82 + 10 * q^83 - 2 * q^84 + 18 * q^85 + 3 * q^86 + 11 * q^87 - 10 * q^88 + 23 * q^89 - 8 * q^90 + 4 * q^91 - 20 * q^92 + 57 * q^93 - 12 * q^94 + 18 * q^95 - 8 * q^96 + 7 * q^97 + 33 * q^98 + 18 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.63132	0.941840	0.470920	−	0.882176i	$-0.343922\pi$
$3$	1.63132	0.941840	0.470920	+	0.882176i	$0.343922\pi$
$4$	1.00000	0.500000
$5$	−3.84124	−1.71785	−0.858927	−	0.512098i	$-0.828868\pi$
$5$	−3.84124	−1.71785	−0.858927	+	0.512098i	$0.828868\pi$
$6$	1.63132	0.665982
$7$	−4.15570	−1.57071	−0.785353	−	0.619049i	$-0.787518\pi$
$7$	−4.15570	−1.57071	−0.785353	+	0.619049i	$0.787518\pi$
$8$	1.00000	0.353553
$9$	−0.338810	−0.112937
$10$	−3.84124	−1.21471
$11$	−0.276147	−0.0832614	−0.0416307	−	0.999133i	$-0.513255\pi$
$11$	−0.276147	−0.0832614	−0.0416307	+	0.999133i	$0.513255\pi$
$12$	1.63132	0.470920
$13$	−1.67873	−0.465596	−0.232798	−	0.972525i	$-0.574788\pi$
$13$	−1.67873	−0.465596	−0.232798	+	0.972525i	$0.574788\pi$
$14$	−4.15570	−1.11066
$15$	−6.26627	−1.61794
$16$	1.00000	0.250000
$17$	−5.94165	−1.44106	−0.720531	−	0.693422i	$-0.756102\pi$
$17$	−5.94165	−1.44106	−0.720531	+	0.693422i	$0.756102\pi$
$18$	−0.338810	−0.0798582
$19$	4.04484	0.927951	0.463976	−	0.885848i	$-0.346423\pi$
$19$	4.04484	0.927951	0.463976	+	0.885848i	$0.346423\pi$
$20$	−3.84124	−0.858927
$21$	−6.77925	−1.47935
$22$	−0.276147	−0.0588747
$23$	4.98236	1.03889	0.519447	−	0.854503i	$-0.326138\pi$
$23$	4.98236	1.03889	0.519447	+	0.854503i	$0.326138\pi$
$24$	1.63132	0.332991
$25$	9.75511	1.95102
$26$	−1.67873	−0.329226
$27$	−5.44665	−1.04821
$28$	−4.15570	−0.785353
$29$	−8.14390	−1.51228	−0.756142	−	0.654407i	$-0.772918\pi$
$29$	−8.14390	−1.51228	−0.756142	+	0.654407i	$0.772918\pi$
$30$	−6.26627	−1.14406
$31$	−4.97939	−0.894325	−0.447163	−	0.894453i	$-0.647565\pi$
$31$	−4.97939	−0.894325	−0.447163	+	0.894453i	$0.647565\pi$
$32$	1.00000	0.176777
$33$	−0.450483	−0.0784190
$34$	−5.94165	−1.01899
$35$	15.9630	2.69824
$36$	−0.338810	−0.0564683
$37$	6.04809	0.994300	0.497150	−	0.867665i	$-0.334380\pi$
$37$	6.04809	0.994300	0.497150	+	0.867665i	$0.334380\pi$
$38$	4.04484	0.656160
$39$	−2.73854	−0.438518
$40$	−3.84124	−0.607353
$41$	−5.16989	−0.807401	−0.403700	−	0.914891i	$-0.632276\pi$
$41$	−5.16989	−0.807401	−0.403700	+	0.914891i	$0.632276\pi$
$42$	−6.77925	−1.04606
$43$	5.45886	0.832468	0.416234	−	0.909258i	$-0.363350\pi$
$43$	5.45886	0.832468	0.416234	+	0.909258i	$0.363350\pi$
$44$	−0.276147	−0.0416307
$45$	1.30145	0.194008
$46$	4.98236	0.734609
$47$	−9.49305	−1.38470	−0.692352	−	0.721560i	$-0.743425\pi$
$47$	−9.49305	−1.38470	−0.692352	+	0.721560i	$0.743425\pi$
$48$	1.63132	0.235460
$49$	10.2698	1.46712
$50$	9.75511	1.37958
$51$	−9.69271	−1.35725
$52$	−1.67873	−0.232798
$53$	5.04713	0.693276	0.346638	−	0.937999i	$-0.387323\pi$
$53$	5.04713	0.693276	0.346638	+	0.937999i	$0.387323\pi$
$54$	−5.44665	−0.741195
$55$	1.06075	0.143031
$56$	−4.15570	−0.555328
$57$	6.59842	0.873982
$58$	−8.14390	−1.06935
$59$	0.0328732	0.00427973	0.00213986	−	0.999998i	$-0.499319\pi$
$59$	0.0328732	0.00427973	0.00213986	+	0.999998i	$0.499319\pi$
$60$	−6.26627	−0.808972
$61$	−15.2993	−1.95888	−0.979439	−	0.201742i	$-0.935340\pi$
$61$	−15.2993	−1.95888	−0.979439	+	0.201742i	$0.935340\pi$
$62$	−4.97939	−0.632383
$63$	1.40799	0.177390
$64$	1.00000	0.125000
$65$	6.44841	0.799826
$66$	−0.450483	−0.0554506
$67$	8.68500	1.06104	0.530521	−	0.847672i	$-0.321996\pi$
$67$	8.68500	1.06104	0.530521	+	0.847672i	$0.321996\pi$
$68$	−5.94165	−0.720531
$69$	8.12781	0.978473
$70$	15.9630	1.90795
$71$	12.8570	1.52585	0.762924	−	0.646488i	$-0.223763\pi$
$71$	12.8570	1.52585	0.762924	+	0.646488i	$0.223763\pi$
$72$	−0.338810	−0.0399291
$73$	−8.66551	−1.01422	−0.507110	−	0.861881i	$-0.669286\pi$
$73$	−8.66551	−1.01422	−0.507110	+	0.861881i	$0.669286\pi$
$74$	6.04809	0.703076
$75$	15.9137	1.83755
$76$	4.04484	0.463976
$77$	1.14758	0.130779
$78$	−2.73854	−0.310079
$79$	−0.311274	−0.0350210	−0.0175105	−	0.999847i	$-0.505574\pi$
$79$	−0.311274	−0.0350210	−0.0175105	+	0.999847i	$0.505574\pi$
$80$	−3.84124	−0.429463
$81$	−7.86878	−0.874309
$82$	−5.16989	−0.570919
$83$	12.4146	1.36268	0.681338	−	0.731969i	$-0.261398\pi$
$83$	12.4146	1.36268	0.681338	+	0.731969i	$0.261398\pi$
$84$	−6.77925	−0.739677
$85$	22.8233	2.47553
$86$	5.45886	0.588644
$87$	−13.2853	−1.42433
$88$	−0.276147	−0.0294374
$89$	17.1312	1.81591	0.907954	−	0.419069i	$-0.137643\pi$
$89$	17.1312	1.81591	0.907954	+	0.419069i	$0.137643\pi$
$90$	1.30145	0.137185
$91$	6.97630	0.731315
$92$	4.98236	0.519447
$93$	−8.12296	−0.842312
$94$	−9.49305	−0.979133
$95$	−15.5372	−1.59408
$96$	1.63132	0.166495
$97$	8.80436	0.893947	0.446974	−	0.894547i	$-0.352502\pi$
$97$	8.80436	0.893947	0.446974	+	0.894547i	$0.352502\pi$
$98$	10.2698	1.03741
$99$	0.0935612	0.00940326
$100$	9.75511	0.975511
$101$	−3.45921	−0.344204	−0.172102	−	0.985079i	$-0.555056\pi$
$101$	−3.45921	−0.344204	−0.172102	+	0.985079i	$0.555056\pi$
$102$	−9.69271	−0.959721
$103$	−7.23292	−0.712681	−0.356340	−	0.934356i	$-0.615976\pi$
$103$	−7.23292	−0.712681	−0.356340	+	0.934356i	$0.615976\pi$
$104$	−1.67873	−0.164613
$105$	26.0407	2.54131
$106$	5.04713	0.490220
$107$	−12.7985	−1.23728	−0.618638	−	0.785676i	$-0.712315\pi$
$107$	−12.7985	−1.23728	−0.618638	+	0.785676i	$0.712315\pi$
$108$	−5.44665	−0.524104
$109$	−7.07598	−0.677756	−0.338878	−	0.940830i	$-0.610047\pi$
$109$	−7.07598	−0.677756	−0.338878	+	0.940830i	$0.610047\pi$
$110$	1.06075	0.101138
$111$	9.86635	0.936472
$112$	−4.15570	−0.392676
$113$	−10.1828	−0.957918	−0.478959	−	0.877837i	$-0.658986\pi$
$113$	−10.1828	−0.957918	−0.478959	+	0.877837i	$0.658986\pi$
$114$	6.59842	0.617999
$115$	−19.1384	−1.78467
$116$	−8.14390	−0.756142
$117$	0.568770	0.0525829
$118$	0.0328732	0.00302622
$119$	24.6917	2.26348
$120$	−6.26627	−0.572030
$121$	−10.9237	−0.993068
$122$	−15.2993	−1.38514
$123$	−8.43372	−0.760443
$124$	−4.97939	−0.447163
$125$	−18.2655	−1.63371
$126$	1.40799	0.125434
$127$	−4.34187	−0.385278	−0.192639	−	0.981270i	$-0.561705\pi$
$127$	−4.34187	−0.385278	−0.192639	+	0.981270i	$0.561705\pi$
$128$	1.00000	0.0883883
$129$	8.90512	0.784052
$130$	6.44841	0.565563
$131$	4.05026	0.353873	0.176936	−	0.984222i	$-0.443381\pi$
$131$	4.05026	0.353873	0.176936	+	0.984222i	$0.443381\pi$
$132$	−0.450483	−0.0392095
$133$	−16.8091	−1.45754
$134$	8.68500	0.750270
$135$	20.9219	1.80067
$136$	−5.94165	−0.509493
$137$	−18.7826	−1.60470	−0.802352	−	0.596850i	$-0.796419\pi$
$137$	−18.7826	−1.60470	−0.802352	+	0.596850i	$0.796419\pi$
$138$	8.12781	0.691885
$139$	−0.403657	−0.0342378	−0.0171189	−	0.999853i	$-0.505449\pi$
$139$	−0.403657	−0.0342378	−0.0171189	+	0.999853i	$0.505449\pi$
$140$	15.9630	1.34912
$141$	−15.4862	−1.30417
$142$	12.8570	1.07894
$143$	0.463576	0.0387662
$144$	−0.338810	−0.0282341
$145$	31.2827	2.59788
$146$	−8.66551	−0.717162
$147$	16.7533	1.38179
$148$	6.04809	0.497150
$149$	−15.6313	−1.28057	−0.640283	−	0.768139i	$-0.721183\pi$
$149$	−15.6313	−1.28057	−0.640283	+	0.768139i	$0.721183\pi$
$150$	15.9137	1.29934
$151$	10.7772	0.877038	0.438519	−	0.898722i	$-0.355503\pi$
$151$	10.7772	0.877038	0.438519	+	0.898722i	$0.355503\pi$
$152$	4.04484	0.328080
$153$	2.01309	0.162749
$154$	1.14758	0.0924748
$155$	19.1270	1.53632
$156$	−2.73854	−0.219259
$157$	7.11275	0.567660	0.283830	−	0.958875i	$-0.408395\pi$
$157$	7.11275	0.567660	0.283830	+	0.958875i	$0.408395\pi$
$158$	−0.311274	−0.0247636
$159$	8.23345	0.652955
$160$	−3.84124	−0.303676
$161$	−20.7052	−1.63180
$162$	−7.86878	−0.618230
$163$	−11.6073	−0.909151	−0.454576	−	0.890708i	$-0.650209\pi$
$163$	−11.6073	−0.909151	−0.454576	+	0.890708i	$0.650209\pi$
$164$	−5.16989	−0.403700
$165$	1.73041	0.134712
$166$	12.4146	0.963558
$167$	−22.1423	−1.71342	−0.856711	−	0.515797i	$-0.827496\pi$
$167$	−22.1423	−1.71342	−0.856711	+	0.515797i	$0.827496\pi$
$168$	−6.77925	−0.523031
$169$	−10.1819	−0.783220
$170$	22.8233	1.75047
$171$	−1.37043	−0.104800
$172$	5.45886	0.416234
$173$	−21.5519	−1.63856	−0.819279	−	0.573395i	$-0.805626\pi$
$173$	−21.5519	−1.63856	−0.819279	+	0.573395i	$0.805626\pi$
$174$	−13.2853	−1.00715
$175$	−40.5393	−3.06448
$176$	−0.276147	−0.0208154
$177$	0.0536266	0.00403082
$178$	17.1312	1.28404
$179$	−4.84364	−0.362030	−0.181015	−	0.983480i	$-0.557938\pi$
$179$	−4.84364	−0.362030	−0.181015	+	0.983480i	$0.557938\pi$
$180$	1.30145	0.0970042
$181$	17.5445	1.30407	0.652036	−	0.758188i	$-0.273915\pi$
$181$	17.5445	1.30407	0.652036	+	0.758188i	$0.273915\pi$
$182$	6.97630	0.517118
$183$	−24.9580	−1.84495
$184$	4.98236	0.367305
$185$	−23.2322	−1.70806
$186$	−8.12296	−0.595604
$187$	1.64077	0.119985
$188$	−9.49305	−0.692352
$189$	22.6346	1.64643
$190$	−15.5372	−1.12719
$191$	−9.30170	−0.673048	−0.336524	−	0.941675i	$-0.609251\pi$
$191$	−9.30170	−0.673048	−0.336524	+	0.941675i	$0.609251\pi$
$192$	1.63132	0.117730
$193$	11.0308	0.794012	0.397006	−	0.917816i	$-0.370049\pi$
$193$	11.0308	0.794012	0.397006	+	0.917816i	$0.370049\pi$
$194$	8.80436	0.632116
$195$	10.5194	0.753309
$196$	10.2698	0.733558
$197$	7.28400	0.518964	0.259482	−	0.965748i	$-0.416448\pi$
$197$	7.28400	0.518964	0.259482	+	0.965748i	$0.416448\pi$
$198$	0.0935612	0.00664911
$199$	−0.786893	−0.0557814	−0.0278907	−	0.999611i	$-0.508879\pi$
$199$	−0.786893	−0.0557814	−0.0278907	+	0.999611i	$0.508879\pi$
$200$	9.75511	0.689790
$201$	14.1680	0.999332
$202$	−3.45921	−0.243389
$203$	33.8436	2.37535
$204$	−9.69271	−0.678625
$205$	19.8588	1.38700
$206$	−7.23292	−0.503941
$207$	−1.68807	−0.117329
$208$	−1.67873	−0.116399
$209$	−1.11697	−0.0772625
$210$	26.0407	1.79698
$211$	3.83957	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$211$	3.83957	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$212$	5.04713	0.346638
$213$	20.9739	1.43711
$214$	−12.7985	−0.874886
$215$	−20.9688	−1.43006
$216$	−5.44665	−0.370598
$217$	20.6928	1.40472
$218$	−7.07598	−0.479246
$219$	−14.1362	−0.955234
$220$	1.06075	0.0715155
$221$	9.97444	0.670954
$222$	9.86635	0.662186
$223$	0.306499	0.0205247	0.0102623	−	0.999947i	$-0.496733\pi$
$223$	0.306499	0.0205247	0.0102623	+	0.999947i	$0.496733\pi$
$224$	−4.15570	−0.277664
$225$	−3.30512	−0.220342
$226$	−10.1828	−0.677351
$227$	−14.8578	−0.986147	−0.493073	−	0.869988i	$-0.664127\pi$
$227$	−14.8578	−0.986147	−0.493073	+	0.869988i	$0.664127\pi$
$228$	6.59842	0.436991
$229$	25.3876	1.67766	0.838831	−	0.544391i	$-0.183239\pi$
$229$	25.3876	1.67766	0.838831	+	0.544391i	$0.183239\pi$
$230$	−19.1384	−1.26195
$231$	1.87207	0.123173
$232$	−8.14390	−0.534673
$233$	−17.0575	−1.11747	−0.558735	−	0.829346i	$-0.688713\pi$
$233$	−17.0575	−1.11747	−0.558735	+	0.829346i	$0.688713\pi$
$234$	0.568770	0.0371817
$235$	36.4650	2.37872
$236$	0.0328732	0.00213986
$237$	−0.507786	−0.0329842
$238$	24.6917	1.60053
$239$	24.8233	1.60568	0.802842	−	0.596191i	$-0.203320\pi$
$239$	24.8233	1.60568	0.802842	+	0.596191i	$0.203320\pi$
$240$	−6.26627	−0.404486
$241$	22.7265	1.46394	0.731972	−	0.681334i	$-0.238600\pi$
$241$	22.7265	1.46394	0.731972	+	0.681334i	$0.238600\pi$
$242$	−10.9237	−0.702205
$243$	3.50349	0.224749
$244$	−15.2993	−0.979439
$245$	−39.4488	−2.52029
$246$	−8.43372	−0.537714
$247$	−6.79021	−0.432051
$248$	−4.97939	−0.316192
$249$	20.2521	1.28342
$250$	−18.2655	−1.15521
$251$	−18.5322	−1.16974	−0.584870	−	0.811127i	$-0.698854\pi$
$251$	−18.5322	−1.16974	−0.584870	+	0.811127i	$0.698854\pi$
$252$	1.40799	0.0886950
$253$	−1.37586	−0.0864998
$254$	−4.34187	−0.272433
$255$	37.2320	2.33156
$256$	1.00000	0.0625000
$257$	−9.16281	−0.571560	−0.285780	−	0.958295i	$-0.592253\pi$
$257$	−9.16281	−0.571560	−0.285780	+	0.958295i	$0.592253\pi$
$258$	8.90512	0.554408
$259$	−25.1340	−1.56175
$260$	6.44841	0.399913
$261$	2.75923	0.170792
$262$	4.05026	0.250226
$263$	−24.4845	−1.50978	−0.754888	−	0.655853i	$-0.772309\pi$
$263$	−24.4845	−1.50978	−0.754888	+	0.655853i	$0.772309\pi$
$264$	−0.450483	−0.0277253
$265$	−19.3872	−1.19095
$266$	−16.8091	−1.03063
$267$	27.9465	1.71030
$268$	8.68500	0.530521
$269$	9.68210	0.590328	0.295164	−	0.955447i	$-0.404626\pi$
$269$	9.68210	0.590328	0.295164	+	0.955447i	$0.404626\pi$
$270$	20.9219	1.27327
$271$	−13.0895	−0.795132	−0.397566	−	0.917574i	$-0.630145\pi$
$271$	−13.0895	−0.795132	−0.397566	+	0.917574i	$0.630145\pi$
$272$	−5.94165	−0.360266
$273$	11.3805	0.688782
$274$	−18.7826	−1.13470
$275$	−2.69384	−0.162445
$276$	8.12781	0.489236
$277$	−7.89689	−0.474478	−0.237239	−	0.971451i	$-0.576242\pi$
$277$	−7.89689	−0.474478	−0.237239	+	0.971451i	$0.576242\pi$
$278$	−0.403657	−0.0242098
$279$	1.68707	0.101002
$280$	15.9630	0.953973
$281$	1.49322	0.0890782	0.0445391	−	0.999008i	$-0.485818\pi$
$281$	1.49322	0.0890782	0.0445391	+	0.999008i	$0.485818\pi$
$282$	−15.4862	−0.922187
$283$	16.5837	0.985797	0.492899	−	0.870087i	$-0.335937\pi$
$283$	16.5837	0.985797	0.492899	+	0.870087i	$0.335937\pi$
$284$	12.8570	0.762924
$285$	−25.3461	−1.50137
$286$	0.463576	0.0274118
$287$	21.4845	1.26819
$288$	−0.338810	−0.0199646
$289$	18.3032	1.07666
$290$	31.2827	1.83698
$291$	14.3627	0.841956
$292$	−8.66551	−0.507110
$293$	12.5499	0.733176	0.366588	−	0.930383i	$-0.380526\pi$
$293$	12.5499	0.733176	0.366588	+	0.930383i	$0.380526\pi$
$294$	16.7533	0.977072
$295$	−0.126274	−0.00735194
$296$	6.04809	0.351538
$297$	1.50408	0.0872753
$298$	−15.6313	−0.905497
$299$	−8.36405	−0.483706
$300$	15.9137	0.918775
$301$	−22.6853	−1.30756
$302$	10.7772	0.620159
$303$	−5.64306	−0.324185
$304$	4.04484	0.231988
$305$	58.7683	3.36506
$306$	2.01309	0.115081
$307$	−1.67505	−0.0956002	−0.0478001	−	0.998857i	$-0.515221\pi$
$307$	−1.67505	−0.0956002	−0.0478001	+	0.998857i	$0.515221\pi$
$308$	1.14758	0.0653896
$309$	−11.7992	−0.671231
$310$	19.1270	1.08634
$311$	1.96842	0.111619	0.0558094	−	0.998441i	$-0.482226\pi$
$311$	1.96842	0.111619	0.0558094	+	0.998441i	$0.482226\pi$
$312$	−2.73854	−0.155039
$313$	26.8085	1.51531	0.757654	−	0.652656i	$-0.226345\pi$
$313$	26.8085	1.51531	0.757654	+	0.652656i	$0.226345\pi$
$314$	7.11275	0.401396
$315$	−5.40842	−0.304730
$316$	−0.311274	−0.0175105
$317$	−9.28552	−0.521527	−0.260763	−	0.965403i	$-0.583974\pi$
$317$	−9.28552	−0.521527	−0.260763	+	0.965403i	$0.583974\pi$
$318$	8.23345	0.461709
$319$	2.24891	0.125915
$320$	−3.84124	−0.214732
$321$	−20.8784	−1.16532
$322$	−20.7052	−1.15385
$323$	−24.0331	−1.33724
$324$	−7.86878	−0.437154
$325$	−16.3762	−0.908388
$326$	−11.6073	−0.642867
$327$	−11.5432	−0.638338
$328$	−5.16989	−0.285459
$329$	39.4502	2.17496
$330$	1.73041	0.0952560
$331$	−10.9258	−0.600538	−0.300269	−	0.953854i	$-0.597077\pi$
$331$	−10.9258	−0.600538	−0.300269	+	0.953854i	$0.597077\pi$
$332$	12.4146	0.681338
$333$	−2.04915	−0.112293
$334$	−22.1423	−1.21157
$335$	−33.3612	−1.82271
$336$	−6.77925	−0.369838
$337$	3.84795	0.209611	0.104806	−	0.994493i	$-0.466578\pi$
$337$	3.84795	0.209611	0.104806	+	0.994493i	$0.466578\pi$
$338$	−10.1819	−0.553820
$339$	−16.6114	−0.902206
$340$	22.8233	1.23777
$341$	1.37504	0.0744628
$342$	−1.37043	−0.0741045
$343$	−13.5883	−0.733701
$344$	5.45886	0.294322
$345$	−31.2208	−1.68087
$346$	−21.5519	−1.15864
$347$	−14.2307	−0.763942	−0.381971	−	0.924174i	$-0.624755\pi$
$347$	−14.2307	−0.763942	−0.381971	+	0.924174i	$0.624755\pi$
$348$	−13.2853	−0.712165
$349$	10.1198	0.541698	0.270849	−	0.962622i	$-0.412696\pi$
$349$	10.1198	0.541698	0.270849	+	0.962622i	$0.412696\pi$
$350$	−40.5393	−2.16691
$351$	9.14347	0.488042
$352$	−0.276147	−0.0147187
$353$	−8.64445	−0.460098	−0.230049	−	0.973179i	$-0.573889\pi$
$353$	−8.64445	−0.460098	−0.230049	+	0.973179i	$0.573889\pi$
$354$	0.0536266	0.00285022
$355$	−49.3869	−2.62118
$356$	17.1312	0.907954
$357$	40.2800	2.13184
$358$	−4.84364	−0.255994
$359$	16.3812	0.864568	0.432284	−	0.901738i	$-0.357708\pi$
$359$	16.3812	0.864568	0.432284	+	0.901738i	$0.357708\pi$
$360$	1.30145	0.0685923
$361$	−2.63923	−0.138907
$362$	17.5445	0.922119
$363$	−17.8201	−0.935311
$364$	6.97630	0.365657
$365$	33.2863	1.74228
$366$	−24.9580	−1.30458
$367$	10.0760	0.525965	0.262982	−	0.964801i	$-0.415294\pi$
$367$	10.0760	0.525965	0.262982	+	0.964801i	$0.415294\pi$
$368$	4.98236	0.259724
$369$	1.75161	0.0911851
$370$	−23.2322	−1.20778
$371$	−20.9743	−1.08893
$372$	−8.12296	−0.421156
$373$	−10.7310	−0.555628	−0.277814	−	0.960635i	$-0.589610\pi$
$373$	−10.7310	−0.555628	−0.277814	+	0.960635i	$0.589610\pi$
$374$	1.64077	0.0848421
$375$	−29.7968	−1.53870
$376$	−9.49305	−0.489567
$377$	13.6714	0.704114
$378$	22.6346	1.16420
$379$	−15.8594	−0.814644	−0.407322	−	0.913285i	$-0.633537\pi$
$379$	−15.8594	−0.814644	−0.407322	+	0.913285i	$0.633537\pi$
$380$	−15.5372	−0.797042
$381$	−7.08295	−0.362871
$382$	−9.30170	−0.475917
$383$	5.47574	0.279797	0.139899	−	0.990166i	$-0.455322\pi$
$383$	5.47574	0.279797	0.139899	+	0.990166i	$0.455322\pi$
$384$	1.63132	0.0832477
$385$	−4.40814	−0.224659
$386$	11.0308	0.561451
$387$	−1.84951	−0.0940160
$388$	8.80436	0.446974
$389$	−5.64583	−0.286255	−0.143127	−	0.989704i	$-0.545716\pi$
$389$	−5.64583	−0.286255	−0.143127	+	0.989704i	$0.545716\pi$
$390$	10.5194	0.532670
$391$	−29.6035	−1.49711
$392$	10.2698	0.518704
$393$	6.60725	0.333292
$394$	7.28400	0.366963
$395$	1.19568	0.0601610
$396$	0.0935612	0.00470163
$397$	−31.3006	−1.57093	−0.785465	−	0.618906i	$-0.787576\pi$
$397$	−31.3006	−1.57093	−0.785465	+	0.618906i	$0.787576\pi$
$398$	−0.786893	−0.0394434
$399$	−27.4210	−1.37277
$400$	9.75511	0.487755
$401$	39.7974	1.98739	0.993694	−	0.112126i	$-0.0357659\pi$
$401$	39.7974	1.98739	0.993694	+	0.112126i	$0.0357659\pi$
$402$	14.1680	0.706635
$403$	8.35906	0.416395
$404$	−3.45921	−0.172102
$405$	30.2258	1.50193
$406$	33.8436	1.67963
$407$	−1.67016	−0.0827868
$408$	−9.69271	−0.479861
$409$	25.8648	1.27893	0.639467	−	0.768818i	$-0.279155\pi$
$409$	25.8648	1.27893	0.639467	+	0.768818i	$0.279155\pi$
$410$	19.8588	0.980755
$411$	−30.6403	−1.51138
$412$	−7.23292	−0.356340
$413$	−0.136611	−0.00672219
$414$	−1.68807	−0.0829643
$415$	−47.6873	−2.34088
$416$	−1.67873	−0.0823066
$417$	−0.658493	−0.0322465
$418$	−1.11697	−0.0546328
$419$	−21.1908	−1.03524	−0.517618	−	0.855612i	$-0.673181\pi$
$419$	−21.1908	−1.03524	−0.517618	+	0.855612i	$0.673181\pi$
$420$	26.0407	1.27066
$421$	18.2822	0.891018	0.445509	−	0.895278i	$-0.353023\pi$
$421$	18.2822	0.891018	0.445509	+	0.895278i	$0.353023\pi$
$422$	3.83957	0.186908
$423$	3.21634	0.156384
$424$	5.04713	0.245110
$425$	−57.9615	−2.81154
$426$	20.9739	1.01619
$427$	63.5793	3.07682
$428$	−12.7985	−0.618638
$429$	0.756239	0.0365116
$430$	−20.9688	−1.01120
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	−5.44665	−0.262052
$433$	6.14415	0.295269	0.147635	−	0.989042i	$-0.452834\pi$
$433$	6.14415	0.295269	0.147635	+	0.989042i	$0.452834\pi$
$434$	20.6928	0.993288
$435$	51.0319	2.44679
$436$	−7.07598	−0.338878
$437$	20.1529	0.964043
$438$	−14.1362	−0.675453
$439$	−36.1274	−1.72427	−0.862134	−	0.506680i	$-0.830873\pi$
$439$	−36.1274	−1.72427	−0.862134	+	0.506680i	$0.830873\pi$
$440$	1.06075	0.0505691
$441$	−3.47951	−0.165691
$442$	9.97444	0.474436
$443$	−8.89045	−0.422398	−0.211199	−	0.977443i	$-0.567737\pi$
$443$	−8.89045	−0.422398	−0.211199	+	0.977443i	$0.567737\pi$
$444$	9.86635	0.468236
$445$	−65.8052	−3.11947
$446$	0.306499	0.0145131
$447$	−25.4996	−1.20609
$448$	−4.15570	−0.196338
$449$	6.53770	0.308533	0.154266	−	0.988029i	$-0.450699\pi$
$449$	6.53770	0.308533	0.154266	+	0.988029i	$0.450699\pi$
$450$	−3.30512	−0.155805
$451$	1.42765	0.0672253
$452$	−10.1828	−0.478959
$453$	17.5810	0.826030
$454$	−14.8578	−0.697311
$455$	−26.7976	−1.25629
$456$	6.59842	0.308999
$457$	33.0933	1.54804	0.774019	−	0.633163i	$-0.218244\pi$
$457$	33.0933	1.54804	0.774019	+	0.633163i	$0.218244\pi$
$458$	25.3876	1.18629
$459$	32.3621	1.51053
$460$	−19.1384	−0.892334
$461$	−3.64501	−0.169765	−0.0848825	−	0.996391i	$-0.527051\pi$
$461$	−3.64501	−0.169765	−0.0848825	+	0.996391i	$0.527051\pi$
$462$	1.87207	0.0870965
$463$	−20.6250	−0.958526	−0.479263	−	0.877671i	$-0.659096\pi$
$463$	−20.6250	−0.958526	−0.479263	+	0.877671i	$0.659096\pi$
$464$	−8.14390	−0.378071
$465$	31.2022	1.44697
$466$	−17.0575	−0.790171
$467$	−28.0985	−1.30025	−0.650123	−	0.759829i	$-0.725282\pi$
$467$	−28.0985	−1.30025	−0.650123	+	0.759829i	$0.725282\pi$
$468$	0.568770	0.0262914
$469$	−36.0922	−1.66658
$470$	36.4650	1.68201
$471$	11.6031	0.534645
$472$	0.0328732	0.00151311
$473$	−1.50745	−0.0693124
$474$	−0.507786	−0.0233234
$475$	39.4579	1.81045
$476$	24.6917	1.13174
$477$	−1.71001	−0.0782962
$478$	24.8233	1.13539
$479$	−12.3551	−0.564518	−0.282259	−	0.959338i	$-0.591084\pi$
$479$	−12.3551	−0.564518	−0.282259	+	0.959338i	$0.591084\pi$
$480$	−6.26627	−0.286015
$481$	−10.1531	−0.462943
$482$	22.7265	1.03517
$483$	−33.7767	−1.53689
$484$	−10.9237	−0.496534
$485$	−33.8196	−1.53567
$486$	3.50349	0.158922
$487$	4.08488	0.185104	0.0925518	−	0.995708i	$-0.470498\pi$
$487$	4.08488	0.185104	0.0925518	+	0.995708i	$0.470498\pi$
$488$	−15.2993	−0.692568
$489$	−18.9351	−0.856275
$490$	−39.4488	−1.78211
$491$	22.2836	1.00565	0.502823	−	0.864389i	$-0.332295\pi$
$491$	22.2836	1.00565	0.502823	+	0.864389i	$0.332295\pi$
$492$	−8.43372	−0.380221
$493$	48.3882	2.17930
$494$	−6.79021	−0.305506
$495$	−0.359391	−0.0161534
$496$	−4.97939	−0.223581
$497$	−53.4299	−2.39666
$498$	20.2521	0.907518
$499$	9.36238	0.419118	0.209559	−	0.977796i	$-0.432797\pi$
$499$	9.36238	0.419118	0.209559	+	0.977796i	$0.432797\pi$
$500$	−18.2655	−0.816857
$501$	−36.1211	−1.61377
$502$	−18.5322	−0.827131
$503$	−40.7628	−1.81753	−0.908763	−	0.417313i	$-0.862972\pi$
$503$	−40.7628	−1.81753	−0.908763	+	0.417313i	$0.862972\pi$
$504$	1.40799	0.0627169
$505$	13.2876	0.591292
$506$	−1.37586	−0.0611646
$507$	−16.6098	−0.737668
$508$	−4.34187	−0.192639
$509$	−9.44020	−0.418430	−0.209215	−	0.977870i	$-0.567091\pi$
$509$	−9.44020	−0.418430	−0.209215	+	0.977870i	$0.567091\pi$
$510$	37.2320	1.64866
$511$	36.0112	1.59304
$512$	1.00000	0.0441942
$513$	−22.0309	−0.972686
$514$	−9.16281	−0.404154
$515$	27.7834	1.22428
$516$	8.90512	0.392026
$517$	2.62147	0.115292
$518$	−25.1340	−1.10433
$519$	−35.1579	−1.54326
$520$	6.44841	0.282781
$521$	29.0149	1.27116	0.635582	−	0.772033i	$-0.280760\pi$
$521$	29.0149	1.27116	0.635582	+	0.772033i	$0.280760\pi$
$522$	2.75923	0.120768
$523$	−8.17788	−0.357594	−0.178797	−	0.983886i	$-0.557221\pi$
$523$	−8.17788	−0.357594	−0.178797	+	0.983886i	$0.557221\pi$
$524$	4.05026	0.176936
$525$	−66.1323	−2.88625
$526$	−24.4845	−1.06757
$527$	29.5858	1.28878
$528$	−0.450483	−0.0196047
$529$	1.82394	0.0793019
$530$	−19.3872	−0.842127
$531$	−0.0111378	−0.000483337 0
$532$	−16.8091	−0.728769
$533$	8.67886	0.375923
$534$	27.9465	1.20936
$535$	49.1620	2.12546
$536$	8.68500	0.375135
$537$	−7.90150	−0.340975
$538$	9.68210	0.417425
$539$	−2.83598	−0.122154
$540$	20.9219	0.900335
$541$	38.2930	1.64634	0.823172	−	0.567791i	$-0.192202\pi$
$541$	38.2930	1.64634	0.823172	+	0.567791i	$0.192202\pi$
$542$	−13.0895	−0.562243
$543$	28.6206	1.22823
$544$	−5.94165	−0.254746
$545$	27.1805	1.16429
$546$	11.3805	0.487042
$547$	−0.158405	−0.00677293	−0.00338646	−	0.999994i	$-0.501078\pi$
$547$	−0.158405	−0.00677293	−0.00338646	+	0.999994i	$0.501078\pi$
$548$	−18.7826	−0.802352
$549$	5.18356	0.221229
$550$	−2.69384	−0.114866
$551$	−32.9408	−1.40333
$552$	8.12781	0.345942
$553$	1.29356	0.0550077
$554$	−7.89689	−0.335507
$555$	−37.8990	−1.60872
$556$	−0.403657	−0.0171189
$557$	6.35752	0.269377	0.134688	−	0.990888i	$-0.456997\pi$
$557$	6.35752	0.269377	0.134688	+	0.990888i	$0.456997\pi$
$558$	1.68707	0.0714192
$559$	−9.16395	−0.387594
$560$	15.9630	0.674560
$561$	2.67661	0.113007
$562$	1.49322	0.0629878
$563$	−11.3821	−0.479697	−0.239848	−	0.970810i	$-0.577098\pi$
$563$	−11.3821	−0.479697	−0.239848	+	0.970810i	$0.577098\pi$
$564$	−15.4862	−0.652085
$565$	39.1146	1.64556
$566$	16.5837	0.697064
$567$	32.7003	1.37328
$568$	12.8570	0.539469
$569$	4.04360	0.169516	0.0847582	−	0.996402i	$-0.472988\pi$
$569$	4.04360	0.169516	0.0847582	+	0.996402i	$0.472988\pi$
$570$	−25.3461	−1.06163
$571$	27.7431	1.16101	0.580506	−	0.814256i	$-0.302855\pi$
$571$	27.7431	1.16101	0.580506	+	0.814256i	$0.302855\pi$
$572$	0.463576	0.0193831
$573$	−15.1740	−0.633904
$574$	21.4845	0.896745
$575$	48.6035	2.02691
$576$	−0.338810	−0.0141171
$577$	10.7599	0.447939	0.223969	−	0.974596i	$-0.428098\pi$
$577$	10.7599	0.447939	0.223969	+	0.974596i	$0.428098\pi$
$578$	18.3032	0.761314
$579$	17.9947	0.747833
$580$	31.2827	1.29894
$581$	−51.5912	−2.14036
$582$	14.3627	0.595353
$583$	−1.39375	−0.0577231
$584$	−8.66551	−0.358581
$585$	−2.18478	−0.0903296
$586$	12.5499	0.518433
$587$	−42.8169	−1.76724	−0.883622	−	0.468202i	$-0.844902\pi$
$587$	−42.8169	−1.76724	−0.883622	+	0.468202i	$0.844902\pi$
$588$	16.7533	0.690894
$589$	−20.1409	−0.829890
$590$	−0.126274	−0.00519861
$591$	11.8825	0.488781
$592$	6.04809	0.248575
$593$	19.4319	0.797974	0.398987	−	0.916957i	$-0.369362\pi$
$593$	19.4319	0.797974	0.398987	+	0.916957i	$0.369362\pi$
$594$	1.50408	0.0617130
$595$	−94.8467	−3.88834
$596$	−15.6313	−0.640283
$597$	−1.28367	−0.0525372
$598$	−8.36405	−0.342031
$599$	10.9336	0.446736	0.223368	−	0.974734i	$-0.428295\pi$
$599$	10.9336	0.446736	0.223368	+	0.974734i	$0.428295\pi$
$600$	15.9137	0.649672
$601$	−45.3393	−1.84943	−0.924715	−	0.380661i	$-0.875696\pi$
$601$	−45.3393	−1.84943	−0.924715	+	0.380661i	$0.875696\pi$
$602$	−22.6853	−0.924586
$603$	−2.94256	−0.119830
$604$	10.7772	0.438519
$605$	41.9607	1.70594
$606$	−5.64306	−0.229234
$607$	30.2535	1.22795	0.613976	−	0.789325i	$-0.289569\pi$
$607$	30.2535	1.22795	0.613976	+	0.789325i	$0.289569\pi$
$608$	4.04484	0.164040
$609$	55.2096	2.23720
$610$	58.7683	2.37946
$611$	15.9363	0.644713
$612$	2.01309	0.0813743
$613$	−35.3247	−1.42675	−0.713376	−	0.700782i	$-0.752835\pi$
$613$	−35.3247	−1.42675	−0.713376	+	0.700782i	$0.752835\pi$
$614$	−1.67505	−0.0675996
$615$	32.3959	1.30633
$616$	1.14758	0.0462374
$617$	−39.8514	−1.60436	−0.802179	−	0.597084i	$-0.796326\pi$
$617$	−39.8514	−1.60436	−0.802179	+	0.597084i	$0.796326\pi$
$618$	−11.7992	−0.474632
$619$	8.23400	0.330952	0.165476	−	0.986214i	$-0.447084\pi$
$619$	8.23400	0.330952	0.165476	+	0.986214i	$0.447084\pi$
$620$	19.1270	0.768160
$621$	−27.1372	−1.08898
$622$	1.96842	0.0789265
$623$	−71.1923	−2.85226
$624$	−2.73854	−0.109629
$625$	21.3866	0.855462
$626$	26.8085	1.07149
$627$	−1.82213	−0.0727690
$628$	7.11275	0.283830
$629$	−35.9357	−1.43285
$630$	−5.40842	−0.215477
$631$	22.0301	0.877003	0.438501	−	0.898731i	$-0.355509\pi$
$631$	22.0301	0.877003	0.438501	+	0.898731i	$0.355509\pi$
$632$	−0.311274	−0.0123818
$633$	6.26356	0.248954
$634$	−9.28552	−0.368775
$635$	16.6781	0.661852
$636$	8.23345	0.326478
$637$	−17.2403	−0.683084
$638$	2.24891	0.0890353
$639$	−4.35609	−0.172324
$640$	−3.84124	−0.151838
$641$	27.4655	1.08482	0.542412	−	0.840113i	$-0.317511\pi$
$641$	27.4655	1.08482	0.542412	+	0.840113i	$0.317511\pi$
$642$	−20.8784	−0.824003
$643$	−26.5753	−1.04803	−0.524014	−	0.851710i	$-0.675566\pi$
$643$	−26.5753	−1.04803	−0.524014	+	0.851710i	$0.675566\pi$
$644$	−20.7052	−0.815899
$645$	−34.2067	−1.34689
$646$	−24.0331	−0.945568
$647$	50.6370	1.99075	0.995373	−	0.0960843i	$-0.0306318\pi$
$647$	50.6370	1.99075	0.995373	+	0.0960843i	$0.0306318\pi$
$648$	−7.86878	−0.309115
$649$	−0.00907783	−0.000356336 0
$650$	−16.3762	−0.642328
$651$	33.7565	1.32302
$652$	−11.6073	−0.454576
$653$	32.4939	1.27158	0.635792	−	0.771861i	$-0.280674\pi$
$653$	32.4939	1.27158	0.635792	+	0.771861i	$0.280674\pi$
$654$	−11.5432	−0.451373
$655$	−15.5580	−0.607902
$656$	−5.16989	−0.201850
$657$	2.93596	0.114543
$658$	39.4502	1.53793
$659$	6.50088	0.253238	0.126619	−	0.991951i	$-0.459587\pi$
$659$	6.50088	0.253238	0.126619	+	0.991951i	$0.459587\pi$
$660$	1.73041	0.0673561
$661$	−15.5197	−0.603648	−0.301824	−	0.953364i	$-0.597595\pi$
$661$	−15.5197	−0.603648	−0.301824	+	0.953364i	$0.597595\pi$
$662$	−10.9258	−0.424645
$663$	16.2715	0.631931
$664$	12.4146	0.481779
$665$	64.5679	2.50384
$666$	−2.04915	−0.0794030
$667$	−40.5759	−1.57110
$668$	−22.1423	−0.856711
$669$	0.499996	0.0193310
$670$	−33.3612	−1.28885
$671$	4.22486	0.163099
$672$	−6.77925	−0.261515
$673$	−6.03387	−0.232589	−0.116294	−	0.993215i	$-0.537102\pi$
$673$	−6.03387	−0.232589	−0.116294	+	0.993215i	$0.537102\pi$
$674$	3.84795	0.148218
$675$	−53.1327	−2.04508
$676$	−10.1819	−0.391610
$677$	7.31202	0.281024	0.140512	−	0.990079i	$-0.455125\pi$
$677$	7.31202	0.281024	0.140512	+	0.990079i	$0.455125\pi$
$678$	−16.6114	−0.637956
$679$	−36.5882	−1.40413
$680$	22.8233	0.875234
$681$	−24.2378	−0.928793
$682$	1.37504	0.0526531
$683$	15.9625	0.610787	0.305393	−	0.952226i	$-0.401212\pi$
$683$	15.9625	0.610787	0.305393	+	0.952226i	$0.401212\pi$
$684$	−1.37043	−0.0523998
$685$	72.1484	2.75665
$686$	−13.5883	−0.518805
$687$	41.4153	1.58009
$688$	5.45886	0.208117
$689$	−8.47277	−0.322787
$690$	−31.2208	−1.18856
$691$	−25.4080	−0.966564	−0.483282	−	0.875465i	$-0.660556\pi$
$691$	−25.4080	−0.966564	−0.483282	+	0.875465i	$0.660556\pi$
$692$	−21.5519	−0.819279
$693$	−0.388812	−0.0147697
$694$	−14.2307	−0.540189
$695$	1.55054	0.0588155
$696$	−13.2853	−0.503577
$697$	30.7177	1.16352
$698$	10.1198	0.383038
$699$	−27.8261	−1.05248
$700$	−40.5393	−1.53224
$701$	−4.74023	−0.179036	−0.0895180	−	0.995985i	$-0.528533\pi$
$701$	−4.74023	−0.179036	−0.0895180	+	0.995985i	$0.528533\pi$
$702$	9.14347	0.345098
$703$	24.4636	0.922662
$704$	−0.276147	−0.0104077
$705$	59.4860	2.24037
$706$	−8.64445	−0.325338
$707$	14.3754	0.540643
$708$	0.0536266	0.00201541
$709$	−0.120576	−0.00452833	−0.00226417	−	0.999997i	$-0.500721\pi$
$709$	−0.120576	−0.00452833	−0.00226417	+	0.999997i	$0.500721\pi$
$710$	−49.3869	−1.85346
$711$	0.105463	0.00395515
$712$	17.1312	0.642021
$713$	−24.8091	−0.929110
$714$	40.2800	1.50744
$715$	−1.78071	−0.0665947
$716$	−4.84364	−0.181015
$717$	40.4946	1.51230
$718$	16.3812	0.611342
$719$	42.5762	1.58783	0.793913	−	0.608032i	$-0.208041\pi$
$719$	42.5762	1.58783	0.793913	+	0.608032i	$0.208041\pi$
$720$	1.30145	0.0485021
$721$	30.0578	1.11941
$722$	−2.63923	−0.0982219
$723$	37.0741	1.37880
$724$	17.5445	0.652036
$725$	−79.4446	−2.95050
$726$	−17.8201	−0.661365
$727$	26.2967	0.975292	0.487646	−	0.873041i	$-0.337856\pi$
$727$	26.2967	0.975292	0.487646	+	0.873041i	$0.337856\pi$
$728$	6.97630	0.258559
$729$	29.3216	1.08599
$730$	33.2863	1.23198
$731$	−32.4346	−1.19964
$732$	−24.9580	−0.922475
$733$	−46.8153	−1.72916	−0.864582	−	0.502492i	$-0.832417\pi$
$733$	−46.8153	−1.72916	−0.864582	+	0.502492i	$0.832417\pi$
$734$	10.0760	0.371913
$735$	−64.3534	−2.37371
$736$	4.98236	0.183652
$737$	−2.39834	−0.0883438
$738$	1.75161	0.0644776
$739$	42.9520	1.58002	0.790008	−	0.613096i	$-0.210076\pi$
$739$	42.9520	1.58002	0.790008	+	0.613096i	$0.210076\pi$
$740$	−23.2322	−0.854031
$741$	−11.0770	−0.406923
$742$	−20.9743	−0.769992
$743$	8.12998	0.298260	0.149130	−	0.988818i	$-0.452353\pi$
$743$	8.12998	0.298260	0.149130	+	0.988818i	$0.452353\pi$
$744$	−8.12296	−0.297802
$745$	60.0435	2.19982
$746$	−10.7310	−0.392888
$747$	−4.20618	−0.153896
$748$	1.64077	0.0599924
$749$	53.1866	1.94340
$750$	−29.7968	−1.08802
$751$	−29.3449	−1.07081	−0.535406	−	0.844595i	$-0.679841\pi$
$751$	−29.3449	−1.07081	−0.535406	+	0.844595i	$0.679841\pi$
$752$	−9.49305	−0.346176
$753$	−30.2318	−1.10171
$754$	13.6714	0.497884
$755$	−41.3979	−1.50662
$756$	22.6346	0.823214
$757$	23.4338	0.851717	0.425859	−	0.904790i	$-0.359972\pi$
$757$	23.4338	0.851717	0.425859	+	0.904790i	$0.359972\pi$
$758$	−15.8594	−0.576040
$759$	−2.24447	−0.0814690
$760$	−15.5372	−0.563594
$761$	−41.7728	−1.51426	−0.757132	−	0.653262i	$-0.773400\pi$
$761$	−41.7728	−1.51426	−0.757132	+	0.653262i	$0.773400\pi$
$762$	−7.08295	−0.256588
$763$	29.4056	1.06455
$764$	−9.30170	−0.336524
$765$	−7.73275	−0.279578
$766$	5.47574	0.197847
$767$	−0.0551853	−0.00199262
$768$	1.63132	0.0588650
$769$	−28.3426	−1.02206	−0.511029	−	0.859563i	$-0.670736\pi$
$769$	−28.3426	−1.02206	−0.511029	+	0.859563i	$0.670736\pi$
$770$	−4.40814	−0.158858
$771$	−14.9474	−0.538318
$772$	11.0308	0.397006
$773$	−1.24443	−0.0447590	−0.0223795	−	0.999750i	$-0.507124\pi$
$773$	−1.24443	−0.0447590	−0.0223795	+	0.999750i	$0.507124\pi$
$774$	−1.84951	−0.0664794
$775$	−48.5745	−1.74485
$776$	8.80436	0.316058
$777$	−41.0015	−1.47092
$778$	−5.64583	−0.202413
$779$	−20.9114	−0.749229
$780$	10.5194	0.376654
$781$	−3.55043	−0.127044
$782$	−29.6035	−1.05862
$783$	44.3570	1.58519
$784$	10.2698	0.366779
$785$	−27.3218	−0.975156
$786$	6.60725	0.235673
$787$	−17.9016	−0.638124	−0.319062	−	0.947734i	$-0.603368\pi$
$787$	−17.9016	−0.638124	−0.319062	+	0.947734i	$0.603368\pi$
$788$	7.28400	0.259482
$789$	−39.9419	−1.42197
$790$	1.19568	0.0425403
$791$	42.3167	1.50461
$792$	0.0935612	0.00332455
$793$	25.6835	0.912046
$794$	−31.3006	−1.11082
$795$	−31.6267	−1.12168
$796$	−0.786893	−0.0278907
$797$	9.40528	0.333152	0.166576	−	0.986029i	$-0.446729\pi$
$797$	9.40528	0.333152	0.166576	+	0.986029i	$0.446729\pi$
$798$	−27.4210	−0.970694
$799$	56.4044	1.99544
$800$	9.75511	0.344895
$801$	−5.80423	−0.205082
$802$	39.7974	1.40530
$803$	2.39295	0.0844454
$804$	14.1680	0.499666
$805$	79.5335	2.80319
$806$	8.35906	0.294435
$807$	15.7946	0.555995
$808$	−3.45921	−0.121695
$809$	−17.4796	−0.614549	−0.307274	−	0.951621i	$-0.599417\pi$
$809$	−17.4796	−0.614549	−0.307274	+	0.951621i	$0.599417\pi$
$810$	30.2258	1.06203
$811$	31.4888	1.10572	0.552860	−	0.833274i	$-0.313536\pi$
$811$	31.4888	1.10572	0.552860	+	0.833274i	$0.313536\pi$
$812$	33.8436	1.18768
$813$	−21.3531	−0.748888
$814$	−1.67016	−0.0585391
$815$	44.5863	1.56179
$816$	−9.69271	−0.339313
$817$	22.0802	0.772489
$818$	25.8648	0.904343
$819$	−2.36364	−0.0825922
$820$	19.8588	0.693498
$821$	−38.3261	−1.33759	−0.668795	−	0.743447i	$-0.733190\pi$
$821$	−38.3261	−1.33759	−0.668795	+	0.743447i	$0.733190\pi$
$822$	−30.6403	−1.06870
$823$	−19.1836	−0.668699	−0.334350	−	0.942449i	$-0.608517\pi$
$823$	−19.1836	−0.668699	−0.334350	+	0.942449i	$0.608517\pi$
$824$	−7.23292	−0.251971
$825$	−4.39451	−0.152997
$826$	−0.136611	−0.00475331
$827$	−12.6443	−0.439686	−0.219843	−	0.975535i	$-0.570555\pi$
$827$	−12.6443	−0.439686	−0.219843	+	0.975535i	$0.570555\pi$
$828$	−1.68807	−0.0586646
$829$	−32.1585	−1.11691	−0.558455	−	0.829535i	$-0.688606\pi$
$829$	−32.1585	−1.11691	−0.558455	+	0.829535i	$0.688606\pi$
$830$	−47.6873	−1.65525
$831$	−12.8823	−0.446882
$832$	−1.67873	−0.0581995
$833$	−61.0196	−2.11421
$834$	−0.658493	−0.0228017
$835$	85.0538	2.94341
$836$	−1.11697	−0.0386313
$837$	27.1210	0.937439
$838$	−21.1908	−0.732022
$839$	−32.6744	−1.12804	−0.564022	−	0.825760i	$-0.690747\pi$
$839$	−32.6744	−1.12804	−0.564022	+	0.825760i	$0.690747\pi$
$840$	26.0407	0.898490
$841$	37.3231	1.28700
$842$	18.2822	0.630045
$843$	2.43592	0.0838974
$844$	3.83957	0.132164
$845$	39.1109	1.34546
$846$	3.21634	0.110580
$847$	45.3958	1.55982
$848$	5.04713	0.173319
$849$	27.0532	0.928464
$850$	−57.9615	−1.98806
$851$	30.1338	1.03297
$852$	20.9739	0.718553
$853$	31.9590	1.09426	0.547128	−	0.837049i	$-0.315721\pi$
$853$	31.9590	1.09426	0.547128	+	0.837049i	$0.315721\pi$
$854$	63.5793	2.17564
$855$	5.26416	0.180030
$856$	−12.7985	−0.437443
$857$	45.1022	1.54066	0.770332	−	0.637643i	$-0.220091\pi$
$857$	45.1022	1.54066	0.770332	+	0.637643i	$0.220091\pi$
$858$	0.756239	0.0258176
$859$	10.6491	0.363343	0.181672	−	0.983359i	$-0.441849\pi$
$859$	10.6491	0.363343	0.181672	+	0.983359i	$0.441849\pi$
$860$	−20.9688	−0.715029
$861$	35.0480	1.19443
$862$	1.00000	0.0340601
$863$	−34.5080	−1.17467	−0.587333	−	0.809346i	$-0.699822\pi$
$863$	−34.5080	−1.17467	−0.587333	+	0.809346i	$0.699822\pi$
$864$	−5.44665	−0.185299
$865$	82.7858	2.81480
$866$	6.14415	0.208787
$867$	29.8584	1.01404
$868$	20.6928	0.702361
$869$	0.0859573	0.00291590
$870$	51.0319	1.73014
$871$	−14.5798	−0.494017
$872$	−7.07598	−0.239623
$873$	−2.98300	−0.100959
$874$	20.1529	0.681682
$875$	75.9058	2.56608
$876$	−14.1362	−0.477617
$877$	24.1703	0.816173	0.408087	−	0.912943i	$-0.366196\pi$
$877$	24.1703	0.816173	0.408087	+	0.912943i	$0.366196\pi$
$878$	−36.1274	−1.21924
$879$	20.4729	0.690534
$880$	1.06075	0.0357577
$881$	35.8442	1.20762	0.603810	−	0.797128i	$-0.293648\pi$
$881$	35.8442	1.20762	0.603810	+	0.797128i	$0.293648\pi$
$882$	−3.47951	−0.117161
$883$	−14.5189	−0.488600	−0.244300	−	0.969700i	$-0.578558\pi$
$883$	−14.5189	−0.488600	−0.244300	+	0.969700i	$0.578558\pi$
$884$	9.97444	0.335477
$885$	−0.205992	−0.00692436
$886$	−8.89045	−0.298680
$887$	20.5709	0.690703	0.345351	−	0.938473i	$-0.387760\pi$
$887$	20.5709	0.690703	0.345351	+	0.938473i	$0.387760\pi$
$888$	9.86635	0.331093
$889$	18.0435	0.605159
$890$	−65.8052	−2.20580
$891$	2.17294	0.0727962
$892$	0.306499	0.0102623
$893$	−38.3979	−1.28494
$894$	−25.4996	−0.852833
$895$	18.6056	0.621915
$896$	−4.15570	−0.138832
$897$	−13.6444	−0.455573
$898$	6.53770	0.218166
$899$	40.5517	1.35247
$900$	−3.30512	−0.110171
$901$	−29.9883	−0.999054
$902$	1.42765	0.0475355
$903$	−37.0070	−1.23151
$904$	−10.1828	−0.338675
$905$	−67.3926	−2.24021
$906$	17.5810	0.584091
$907$	−4.98294	−0.165456	−0.0827280	−	0.996572i	$-0.526363\pi$
$907$	−4.98294	−0.165456	−0.0827280	+	0.996572i	$0.526363\pi$
$908$	−14.8578	−0.493073
$909$	1.17201	0.0388732
$910$	−26.7976	−0.888332
$911$	34.1001	1.12979	0.564893	−	0.825164i	$-0.308918\pi$
$911$	34.1001	1.12979	0.564893	+	0.825164i	$0.308918\pi$
$912$	6.59842	0.218495
$913$	−3.42825	−0.113458
$914$	33.0933	1.09463
$915$	95.8697	3.16935
$916$	25.3876	0.838831
$917$	−16.8316	−0.555830
$918$	32.3621	1.06811
$919$	42.9521	1.41686	0.708430	−	0.705781i	$-0.249404\pi$
$919$	42.9521	1.41686	0.708430	+	0.705781i	$0.249404\pi$
$920$	−19.1384	−0.630976
$921$	−2.73254	−0.0900402
$922$	−3.64501	−0.120042
$923$	−21.5835	−0.710430
$924$	1.87207	0.0615865
$925$	58.9998	1.93990
$926$	−20.6250	−0.677780
$927$	2.45058	0.0804877
$928$	−8.14390	−0.267337
$929$	−19.9917	−0.655906	−0.327953	−	0.944694i	$-0.606359\pi$
$929$	−19.9917	−0.655906	−0.327953	+	0.944694i	$0.606359\pi$
$930$	31.2022	1.02316
$931$	41.5398	1.36141
$932$	−17.0575	−0.558735
$933$	3.21111	0.105127
$934$	−28.0985	−0.919412
$935$	−6.30258	−0.206116
$936$	0.568770	0.0185908
$937$	5.33935	0.174429	0.0872144	−	0.996190i	$-0.472203\pi$
$937$	5.33935	0.174429	0.0872144	+	0.996190i	$0.472203\pi$
$938$	−36.0922	−1.17845
$939$	43.7332	1.42718
$940$	36.4650	1.18936
$941$	4.96644	0.161901	0.0809506	−	0.996718i	$-0.474204\pi$
$941$	4.96644	0.161901	0.0809506	+	0.996718i	$0.474204\pi$
$942$	11.6031	0.378051
$943$	−25.7583	−0.838804
$944$	0.0328732	0.00106993
$945$	−86.9450	−2.82832
$946$	−1.50745	−0.0490113
$947$	32.5717	1.05844	0.529220	−	0.848485i	$-0.322485\pi$
$947$	32.5717	1.05844	0.529220	+	0.848485i	$0.322485\pi$
$948$	−0.507786	−0.0164921
$949$	14.5471	0.472218
$950$	39.4579	1.28018
$951$	−15.1476	−0.491195
$952$	24.6917	0.800263
$953$	−13.3712	−0.433137	−0.216568	−	0.976267i	$-0.569486\pi$
$953$	−13.3712	−0.433137	−0.216568	+	0.976267i	$0.569486\pi$
$954$	−1.71001	−0.0553638
$955$	35.7301	1.15620
$956$	24.8233	0.802842
$957$	3.66869	0.118592
$958$	−12.3551	−0.399175
$959$	78.0547	2.52052
$960$	−6.26627	−0.202243
$961$	−6.20565	−0.200182
$962$	−10.1531	−0.327350
$963$	4.33625	0.139734
$964$	22.7265	0.731972
$965$	−42.3718	−1.36400
$966$	−33.7767	−1.08675
$967$	5.70566	0.183482	0.0917408	−	0.995783i	$-0.470757\pi$
$967$	5.70566	0.183482	0.0917408	+	0.995783i	$0.470757\pi$
$968$	−10.9237	−0.351102
$969$	−39.2055	−1.25946
$970$	−33.8196	−1.08588
$971$	−14.7811	−0.474348	−0.237174	−	0.971467i	$-0.576221\pi$
$971$	−14.7811	−0.474348	−0.237174	+	0.971467i	$0.576221\pi$
$972$	3.50349	0.112375
$973$	1.67748	0.0537775
$974$	4.08488	0.130888
$975$	−26.7148	−0.855557
$976$	−15.2993	−0.489719
$977$	17.1592	0.548971	0.274486	−	0.961591i	$-0.411492\pi$
$977$	17.1592	0.548971	0.274486	+	0.961591i	$0.411492\pi$
$978$	−18.9351	−0.605478
$979$	−4.73074	−0.151195
$980$	−39.4488	−1.26014
$981$	2.39741	0.0765434
$982$	22.2836	0.711099
$983$	39.3412	1.25479	0.627395	−	0.778701i	$-0.284121\pi$
$983$	39.3412	1.25479	0.627395	+	0.778701i	$0.284121\pi$
$984$	−8.43372	−0.268857
$985$	−27.9796	−0.891503
$986$	48.3882	1.54100
$987$	64.3557	2.04847
$988$	−6.79021	−0.216025
$989$	27.1980	0.864846
$990$	−0.359391	−0.0114222
$991$	−3.45230	−0.109666	−0.0548330	−	0.998496i	$-0.517463\pi$
$991$	−3.45230	−0.109666	−0.0548330	+	0.998496i	$0.517463\pi$
$992$	−4.97939	−0.158096
$993$	−17.8235	−0.565611
$994$	−53.4299	−1.69469
$995$	3.02264	0.0958243
$996$	20.2521	0.641712
$997$	5.40355	0.171132	0.0855661	−	0.996332i	$-0.472730\pi$
$997$	5.40355	0.171132	0.0855661	+	0.996332i	$0.472730\pi$
$998$	9.36238	0.296361
$999$	−32.9419	−1.04223

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	862.2.a.j.1.6	✓	6
3.2	odd	2		7758.2.a.r.1.6		6
4.3	odd	2		6896.2.a.r.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
862.2.a.j.1.6	✓	6	1.1	even	1	trivial
6896.2.a.r.1.1		6	4.3	odd	2
7758.2.a.r.1.6		6	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	862.2.a.j.1.6

Level	$862$
Weight	$2$
Character	862.1
Self dual	yes
Analytic conductor	$6.883$
Analytic rank	$1$
Dimension	$6$
CM	no
Inner twists	$1$