LMFDB - Embedded newform 133.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [133,2,Mod(1,133)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(133, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("133.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$133 = 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	133.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:	$1.06201034688$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 4x - 1$ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	133.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46260 q^{2} +2.86081 q^{3} +0.139194 q^{4} -3.32340 q^{5} +4.18421 q^{6} -1.00000 q^{7} -2.72161 q^{8} +5.18421 q^{9} -4.86081 q^{10} +1.53740 q^{11} +0.398207 q^{12} -3.32340 q^{13} -1.46260 q^{14} -9.50761 q^{15} -4.25901 q^{16} +5.25901 q^{17} +7.58242 q^{18} +1.00000 q^{19} -0.462598 q^{20} -2.86081 q^{21} +2.24860 q^{22} +3.60179 q^{23} -7.78600 q^{24} +6.04502 q^{25} -4.86081 q^{26} +6.24860 q^{27} -0.139194 q^{28} +8.83102 q^{29} -13.9058 q^{30} -8.18421 q^{31} -0.786003 q^{32} +4.39821 q^{33} +7.69182 q^{34} +3.32340 q^{35} +0.721612 q^{36} +4.24860 q^{37} +1.46260 q^{38} -9.50761 q^{39} +9.04502 q^{40} -12.7022 q^{41} -4.18421 q^{42} -6.64681 q^{43} +0.213997 q^{44} -17.2292 q^{45} +5.26798 q^{46} -2.11982 q^{47} -12.1842 q^{48} +1.00000 q^{49} +8.84143 q^{50} +15.0450 q^{51} -0.462598 q^{52} -0.0643910 q^{53} +9.13919 q^{54} -5.10941 q^{55} +2.72161 q^{56} +2.86081 q^{57} +12.9162 q^{58} +3.04502 q^{59} -1.32340 q^{60} -1.47301 q^{61} -11.9702 q^{62} -5.18421 q^{63} +7.36842 q^{64} +11.0450 q^{65} +6.43281 q^{66} -2.33382 q^{67} +0.732024 q^{68} +10.3040 q^{69} +4.86081 q^{70} -0.526989 q^{71} -14.1094 q^{72} -0.989588 q^{73} +6.21400 q^{74} +17.2936 q^{75} +0.139194 q^{76} -1.53740 q^{77} -13.9058 q^{78} +0.796415 q^{79} +14.1544 q^{80} +2.32340 q^{81} -18.5783 q^{82} +10.0644 q^{83} -0.398207 q^{84} -17.4778 q^{85} -9.72161 q^{86} +25.2638 q^{87} -4.18421 q^{88} +5.01523 q^{89} -25.1994 q^{90} +3.32340 q^{91} +0.501348 q^{92} -23.4134 q^{93} -3.10044 q^{94} -3.32340 q^{95} -2.24860 q^{96} -17.9702 q^{97} +1.46260 q^{98} +7.97021 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 2 q^{9} - 9 q^{10} + 7 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{14} - 7 q^{15} - 4 q^{16} + 7 q^{17} + 6 q^{18} + 3 q^{19} + q^{20}+ \cdots + 2 q^{98}+O(q^{100})$ 3 * q + 2 * q^2 + 3 * q^3 + 6 * q^4 - 2 * q^5 - q^6 - 3 * q^7 + 3 * q^8 + 2 * q^9 - 9 * q^10 + 7 * q^11 - 2 * q^12 - 2 * q^13 - 2 * q^14 - 7 * q^15 - 4 * q^16 + 7 * q^17 + 6 * q^18 + 3 * q^19 + q^20 - 3 * q^21 - 6 * q^22 + 14 * q^23 - 13 * q^24 - q^25 - 9 * q^26 + 6 * q^27 - 6 * q^28 - 3 * q^29 - 17 * q^30 - 11 * q^31 + 8 * q^32 + 10 * q^33 - 12 * q^34 + 2 * q^35 - 9 * q^36 + 2 * q^38 - 7 * q^39 + 8 * q^40 - 7 * q^41 + q^42 - 4 * q^43 + 11 * q^44 - 19 * q^45 + 23 * q^46 + 8 * q^47 - 23 * q^48 + 3 * q^49 + q^50 + 26 * q^51 + q^52 - q^53 + 33 * q^54 + 3 * q^55 - 3 * q^56 + 3 * q^57 + 18 * q^58 - 10 * q^59 + 4 * q^60 - 6 * q^61 - 12 * q^62 - 2 * q^63 - 5 * q^64 + 14 * q^65 - 7 * q^66 - 3 * q^67 - 5 * q^68 + 3 * q^69 + 9 * q^70 - 24 * q^72 + q^73 + 29 * q^74 + 20 * q^75 + 6 * q^76 - 7 * q^77 - 17 * q^78 - 4 * q^79 + 5 * q^80 - q^81 + 24 * q^82 + 31 * q^83 + 2 * q^84 - 7 * q^85 - 18 * q^86 + 20 * q^87 + q^88 - 28 * q^89 - 19 * q^90 + 2 * q^91 + 39 * q^92 - 24 * q^93 + 25 * q^94 - 2 * q^95 + 6 * q^96 - 30 * q^97 + 2 * q^98

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

)

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

$2$	1.46260	1.03421	0.517107	−	0.855921i	$-0.327009\pi$
$2$	1.46260	1.03421	0.517107	+	0.855921i	$0.327009\pi$
$3$	2.86081	1.65169	0.825844	−	0.563899i	$-0.190699\pi$
$3$	2.86081	1.65169	0.825844	+	0.563899i	$0.190699\pi$
$4$	0.139194	0.0695971
$5$	−3.32340	−1.48627	−0.743136	−	0.669141i	$-0.766662\pi$
$5$	−3.32340	−1.48627	−0.743136	+	0.669141i	$0.766662\pi$
$6$	4.18421	1.70820
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.72161	−0.962235
$9$	5.18421	1.72807
$10$	−4.86081	−1.53712
$11$	1.53740	0.463544	0.231772	−	0.972770i	$-0.425548\pi$
$11$	1.53740	0.463544	0.231772	+	0.972770i	$0.425548\pi$
$12$	0.398207	0.114953
$13$	−3.32340	−0.921747	−0.460873	−	0.887466i	$-0.652464\pi$
$13$	−3.32340	−0.921747	−0.460873	+	0.887466i	$0.652464\pi$
$14$	−1.46260	−0.390896
$15$	−9.50761	−2.45486
$16$	−4.25901	−1.06475
$17$	5.25901	1.27550	0.637749	−	0.770244i	$-0.279866\pi$
$17$	5.25901	1.27550	0.637749	+	0.770244i	$0.279866\pi$
$18$	7.58242	1.78719
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.462598	−0.103440
$21$	−2.86081	−0.624279
$22$	2.24860	0.479403
$23$	3.60179	0.751026	0.375513	−	0.926817i	$-0.377467\pi$
$23$	3.60179	0.751026	0.375513	+	0.926817i	$0.377467\pi$
$24$	−7.78600	−1.58931
$25$	6.04502	1.20900
$26$	−4.86081	−0.953282
$27$	6.24860	1.20254
$28$	−0.139194	−0.0263052
$29$	8.83102	1.63988	0.819940	−	0.572450i	$-0.194007\pi$
$29$	8.83102	1.63988	0.819940	+	0.572450i	$0.194007\pi$
$30$	−13.9058	−2.53884
$31$	−8.18421	−1.46993	−0.734964	−	0.678106i	$-0.762801\pi$
$31$	−8.18421	−1.46993	−0.734964	+	0.678106i	$0.762801\pi$
$32$	−0.786003	−0.138947
$33$	4.39821	0.765630
$34$	7.69182	1.31914
$35$	3.32340	0.561758
$36$	0.721612	0.120269
$37$	4.24860	0.698466	0.349233	−	0.937036i	$-0.386442\pi$
$37$	4.24860	0.698466	0.349233	+	0.937036i	$0.386442\pi$
$38$	1.46260	0.237265
$39$	−9.50761	−1.52244
$40$	9.04502	1.43014
$41$	−12.7022	−1.98376	−0.991878	−	0.127193i	$-0.959403\pi$
$41$	−12.7022	−1.98376	−0.991878	+	0.127193i	$0.959403\pi$
$42$	−4.18421	−0.645638
$43$	−6.64681	−1.01363	−0.506814	−	0.862055i	$-0.669177\pi$
$43$	−6.64681	−1.01363	−0.506814	+	0.862055i	$0.669177\pi$
$44$	0.213997	0.0322613
$45$	−17.2292	−2.56838
$46$	5.26798	0.776721
$47$	−2.11982	−0.309207	−0.154604	−	0.987977i	$-0.549410\pi$
$47$	−2.11982	−0.309207	−0.154604	+	0.987977i	$0.549410\pi$
$48$	−12.1842	−1.75864
$49$	1.00000	0.142857
$50$	8.84143	1.25037
$51$	15.0450	2.10672
$52$	−0.462598	−0.0641509
$53$	−0.0643910	−0.00884478	−0.00442239	−	0.999990i	$-0.501408\pi$
$53$	−0.0643910	−0.00884478	−0.00442239	+	0.999990i	$0.501408\pi$
$54$	9.13919	1.24369
$55$	−5.10941	−0.688952
$56$	2.72161	0.363691
$57$	2.86081	0.378923
$58$	12.9162	1.69598
$59$	3.04502	0.396427	0.198214	−	0.980159i	$-0.436486\pi$
$59$	3.04502	0.396427	0.198214	+	0.980159i	$0.436486\pi$
$60$	−1.32340	−0.170851
$61$	−1.47301	−0.188600	−0.0942998	−	0.995544i	$-0.530061\pi$
$61$	−1.47301	−0.188600	−0.0942998	+	0.995544i	$0.530061\pi$
$62$	−11.9702	−1.52022
$63$	−5.18421	−0.653149
$64$	7.36842	0.921053
$65$	11.0450	1.36997
$66$	6.43281	0.791824
$67$	−2.33382	−0.285121	−0.142561	−	0.989786i	$-0.545534\pi$
$67$	−2.33382	−0.285121	−0.142561	+	0.989786i	$0.545534\pi$
$68$	0.732024	0.0887709
$69$	10.3040	1.24046
$70$	4.86081	0.580977
$71$	−0.526989	−0.0625421	−0.0312711	−	0.999511i	$-0.509956\pi$
$71$	−0.526989	−0.0625421	−0.0312711	+	0.999511i	$0.509956\pi$
$72$	−14.1094	−1.66281
$73$	−0.989588	−0.115822	−0.0579112	−	0.998322i	$-0.518444\pi$
$73$	−0.989588	−0.115822	−0.0579112	+	0.998322i	$0.518444\pi$
$74$	6.21400	0.722363
$75$	17.2936	1.99689
$76$	0.139194	0.0159667
$77$	−1.53740	−0.175203
$78$	−13.9058	−1.57452
$79$	0.796415	0.0896037	0.0448018	−	0.998996i	$-0.485734\pi$
$79$	0.796415	0.0896037	0.0448018	+	0.998996i	$0.485734\pi$
$80$	14.1544	1.58251
$81$	2.32340	0.258156
$82$	−18.5783	−2.05163
$83$	10.0644	1.10471	0.552355	−	0.833609i	$-0.313729\pi$
$83$	10.0644	1.10471	0.552355	+	0.833609i	$0.313729\pi$
$84$	−0.398207	−0.0434480
$85$	−17.4778	−1.89574
$86$	−9.72161	−1.04831
$87$	25.2638	2.70857
$88$	−4.18421	−0.446038
$89$	5.01523	0.531613	0.265807	−	0.964026i	$-0.414362\pi$
$89$	5.01523	0.531613	0.265807	+	0.964026i	$0.414362\pi$
$90$	−25.1994	−2.65625
$91$	3.32340	0.348387
$92$	0.501348	0.0522692
$93$	−23.4134	−2.42786
$94$	−3.10044	−0.319786
$95$	−3.32340	−0.340974
$96$	−2.24860	−0.229497
$97$	−17.9702	−1.82460	−0.912299	−	0.409524i	$-0.865695\pi$
$97$	−17.9702	−1.82460	−0.912299	+	0.409524i	$0.865695\pi$
$98$	1.46260	0.147745
$99$	7.97021	0.801037
$100$	0.841431	0.0841431
$101$	6.89541	0.686119	0.343059	−	0.939314i	$-0.388537\pi$
$101$	6.89541	0.686119	0.343059	+	0.939314i	$0.388537\pi$
$102$	22.0048	2.17880
$103$	13.1350	1.29423	0.647117	−	0.762390i	$-0.275974\pi$
$103$	13.1350	1.29423	0.647117	+	0.762390i	$0.275974\pi$
$104$	9.04502	0.886937
$105$	9.50761	0.927848
$106$	−0.0941782	−0.00914739
$107$	10.6766	1.03215	0.516073	−	0.856545i	$-0.327393\pi$
$107$	10.6766	1.03215	0.516073	+	0.856545i	$0.327393\pi$
$108$	0.869769	0.0836935
$109$	9.97021	0.954973	0.477487	−	0.878639i	$-0.341548\pi$
$109$	9.97021	0.954973	0.477487	+	0.878639i	$0.341548\pi$
$110$	−7.47301	−0.712524
$111$	12.1544	1.15365
$112$	4.25901	0.402439
$113$	−18.5228	−1.74248	−0.871241	−	0.490855i	$-0.836684\pi$
$113$	−18.5228	−1.74248	−0.871241	+	0.490855i	$0.836684\pi$
$114$	4.18421	0.391887
$115$	−11.9702	−1.11623
$116$	1.22923	0.114131
$117$	−17.2292	−1.59284
$118$	4.45364	0.409990
$119$	−5.25901	−0.482093
$120$	25.8760	2.36215
$121$	−8.63640	−0.785127
$122$	−2.15442	−0.195052
$123$	−36.3386	−3.27654
$124$	−1.13919	−0.102303
$125$	−3.47301	−0.310636
$126$	−7.58242	−0.675495
$127$	−6.77559	−0.601236	−0.300618	−	0.953745i	$-0.597193\pi$
$127$	−6.77559	−0.601236	−0.300618	+	0.953745i	$0.597193\pi$
$128$	12.3490	1.09151
$129$	−19.0152	−1.67420
$130$	16.1544	1.41684
$131$	7.25901	0.634223	0.317111	−	0.948388i	$-0.397287\pi$
$131$	7.25901	0.634223	0.317111	+	0.948388i	$0.397287\pi$
$132$	0.612205	0.0532856
$133$	−1.00000	−0.0867110
$134$	−3.41344	−0.294876
$135$	−20.7666	−1.78731
$136$	−14.3130	−1.22733
$137$	15.2936	1.30662	0.653311	−	0.757090i	$-0.273379\pi$
$137$	15.2936	1.30662	0.653311	+	0.757090i	$0.273379\pi$
$138$	15.0707	1.28290
$139$	−9.82061	−0.832973	−0.416486	−	0.909142i	$-0.636739\pi$
$139$	−9.82061	−0.832973	−0.416486	+	0.909142i	$0.636739\pi$
$140$	0.462598	0.0390967
$141$	−6.06439	−0.510714
$142$	−0.770774	−0.0646819
$143$	−5.10941	−0.427270
$144$	−22.0796	−1.83997
$145$	−29.3490	−2.43731
$146$	−1.44737	−0.119785
$147$	2.86081	0.235955
$148$	0.591380	0.0486112
$149$	−13.5422	−1.10942	−0.554711	−	0.832043i	$-0.687171\pi$
$149$	−13.5422	−1.10942	−0.554711	+	0.832043i	$0.687171\pi$
$150$	25.2936	2.06522
$151$	8.18421	0.666022	0.333011	−	0.942923i	$-0.391935\pi$
$151$	8.18421	0.666022	0.333011	+	0.942923i	$0.391935\pi$
$152$	−2.72161	−0.220752
$153$	27.2638	2.20415
$154$	−2.24860	−0.181197
$155$	27.1994	2.18471
$156$	−1.32340	−0.105957
$157$	5.13023	0.409437	0.204719	−	0.978821i	$-0.434372\pi$
$157$	5.13023	0.409437	0.204719	+	0.978821i	$0.434372\pi$
$158$	1.16484	0.0926693
$159$	−0.184210	−0.0146088
$160$	2.61220	0.206513
$161$	−3.60179	−0.283861
$162$	3.39821	0.266988
$163$	−1.37883	−0.107998	−0.0539992	−	0.998541i	$-0.517197\pi$
$163$	−1.37883	−0.107998	−0.0539992	+	0.998541i	$0.517197\pi$
$164$	−1.76808	−0.138064
$165$	−14.6170	−1.13793
$166$	14.7202	1.14251
$167$	21.5630	1.66860	0.834299	−	0.551312i	$-0.185873\pi$
$167$	21.5630	1.66860	0.834299	+	0.551312i	$0.185873\pi$
$168$	7.78600	0.600703
$169$	−1.95498	−0.150383
$170$	−25.5630	−1.96060
$171$	5.18421	0.396446
$172$	−0.925197	−0.0705456
$173$	−19.1648	−1.45708	−0.728538	−	0.685006i	$-0.759800\pi$
$173$	−19.1648	−1.45708	−0.728538	+	0.685006i	$0.759800\pi$
$174$	36.9508	2.80124
$175$	−6.04502	−0.456960
$176$	−6.54781	−0.493560
$177$	8.71120	0.654774
$178$	7.33527	0.549801
$179$	−15.1994	−1.13606	−0.568030	−	0.823008i	$-0.692294\pi$
$179$	−15.1994	−1.13606	−0.568030	+	0.823008i	$0.692294\pi$
$180$	−2.39821	−0.178752
$181$	2.31299	0.171923	0.0859617	−	0.996298i	$-0.472604\pi$
$181$	2.31299	0.171923	0.0859617	+	0.996298i	$0.472604\pi$
$182$	4.86081	0.360307
$183$	−4.21400	−0.311508
$184$	−9.80268	−0.722663
$185$	−14.1198	−1.03811
$186$	−34.2445	−2.51093
$187$	8.08522	0.591250
$188$	−0.295066	−0.0215199
$189$	−6.24860	−0.454519
$190$	−4.86081	−0.352640
$191$	−0.0852153	−0.00616596	−0.00308298	−	0.999995i	$-0.500981\pi$
$191$	−0.0852153	−0.00616596	−0.00308298	+	0.999995i	$0.500981\pi$
$192$	21.0796	1.52129
$193$	1.75622	0.126415	0.0632076	−	0.998000i	$-0.479867\pi$
$193$	1.75622	0.126415	0.0632076	+	0.998000i	$0.479867\pi$
$194$	−26.2832	−1.88702
$195$	31.5976	2.26275
$196$	0.139194	0.00994244
$197$	10.8400	0.772317	0.386158	−	0.922433i	$-0.373802\pi$
$197$	10.8400	0.772317	0.386158	+	0.922433i	$0.373802\pi$
$198$	11.6572	0.828443
$199$	−4.82620	−0.342120	−0.171060	−	0.985261i	$-0.554719\pi$
$199$	−4.82620	−0.342120	−0.171060	+	0.985261i	$0.554719\pi$
$200$	−16.4522	−1.16335
$201$	−6.67660	−0.470931
$202$	10.0852	0.709593
$203$	−8.83102	−0.619816
$204$	2.09418	0.146622
$205$	42.2147	2.94840
$206$	19.2113	1.33851
$207$	18.6724	1.29782
$208$	14.1544	0.981433
$209$	1.53740	0.106344
$210$	13.9058	0.959593
$211$	−3.91478	−0.269505	−0.134752	−	0.990879i	$-0.543024\pi$
$211$	−3.91478	−0.269505	−0.134752	+	0.990879i	$0.543024\pi$
$212$	−0.00896285	−0.000615571 0
$213$	−1.50761	−0.103300
$214$	15.6156	1.06746
$215$	22.0900	1.50653
$216$	−17.0063	−1.15713
$217$	8.18421	0.555580
$218$	14.5824	0.987646
$219$	−2.83102	−0.191303
$220$	−0.711200	−0.0479491
$221$	−17.4778	−1.17569
$222$	17.7770	1.19312
$223$	−2.11982	−0.141954	−0.0709768	−	0.997478i	$-0.522612\pi$
$223$	−2.11982	−0.141954	−0.0709768	+	0.997478i	$0.522612\pi$
$224$	0.786003	0.0525170
$225$	31.3386	2.08924
$226$	−27.0915	−1.80210
$227$	10.9598	0.727428	0.363714	−	0.931511i	$-0.381509\pi$
$227$	10.9598	0.727428	0.363714	+	0.931511i	$0.381509\pi$
$228$	0.398207	0.0263719
$229$	9.70079	0.641046	0.320523	−	0.947241i	$-0.396141\pi$
$229$	9.70079	0.641046	0.320523	+	0.947241i	$0.396141\pi$
$230$	−17.5076	−1.15442
$231$	−4.39821	−0.289381
$232$	−24.0346	−1.57795
$233$	−9.62743	−0.630714	−0.315357	−	0.948973i	$-0.602124\pi$
$233$	−9.62743	−0.630714	−0.315357	+	0.948973i	$0.602124\pi$
$234$	−25.1994	−1.64734
$235$	7.04502	0.459566
$236$	0.423848	0.0275902
$237$	2.27839	0.147997
$238$	−7.69182	−0.498587
$239$	4.39821	0.284496	0.142248	−	0.989831i	$-0.454567\pi$
$239$	4.39821	0.284496	0.142248	+	0.989831i	$0.454567\pi$
$240$	40.4931	2.61382
$241$	26.3684	1.69854	0.849270	−	0.527959i	$-0.177043\pi$
$241$	26.3684	1.69854	0.849270	+	0.527959i	$0.177043\pi$
$242$	−12.6316	−0.811989
$243$	−12.0990	−0.776151
$244$	−0.205034	−0.0131260
$245$	−3.32340	−0.212325
$246$	−53.1488	−3.38865
$247$	−3.32340	−0.211463
$248$	22.2742	1.41442
$249$	28.7923	1.82464
$250$	−5.07962	−0.321263
$251$	−5.34905	−0.337629	−0.168814	−	0.985648i	$-0.553994\pi$
$251$	−5.34905	−0.337629	−0.168814	+	0.985648i	$0.553994\pi$
$252$	−0.721612	−0.0454573
$253$	5.53740	0.348133
$254$	−9.90997	−0.621807
$255$	−50.0007	−3.13116
$256$	3.32485	0.207803
$257$	−24.9300	−1.55509	−0.777546	−	0.628826i	$-0.783536\pi$
$257$	−24.9300	−1.55509	−0.777546	+	0.628826i	$0.783536\pi$
$258$	−27.8116	−1.73148
$259$	−4.24860	−0.263995
$260$	1.53740	0.0953456
$261$	45.7819	2.83383
$262$	10.6170	0.655922
$263$	−4.21400	−0.259846	−0.129923	−	0.991524i	$-0.541473\pi$
$263$	−4.21400	−0.259846	−0.129923	+	0.991524i	$0.541473\pi$
$264$	−11.9702	−0.736716
$265$	0.213997	0.0131457
$266$	−1.46260	−0.0896777
$267$	14.3476	0.878059
$268$	−0.324854	−0.0198436
$269$	−4.98959	−0.304221	−0.152110	−	0.988364i	$-0.548607\pi$
$269$	−4.98959	−0.304221	−0.152110	+	0.988364i	$0.548607\pi$
$270$	−30.3732	−1.84846
$271$	−31.0707	−1.88741	−0.943704	−	0.330791i	$-0.892684\pi$
$271$	−31.0707	−1.88741	−0.943704	+	0.330791i	$0.892684\pi$
$272$	−22.3982	−1.35809
$273$	9.50761	0.575427
$274$	22.3684	1.35133
$275$	9.29362	0.560426
$276$	1.43426	0.0863323
$277$	−17.2549	−1.03674	−0.518372	−	0.855155i	$-0.673462\pi$
$277$	−17.2549	−1.03674	−0.518372	+	0.855155i	$0.673462\pi$
$278$	−14.3636	−0.861472
$279$	−42.4287	−2.54014
$280$	−9.04502	−0.540543
$281$	−4.61702	−0.275428	−0.137714	−	0.990472i	$-0.543976\pi$
$281$	−4.61702	−0.275428	−0.137714	+	0.990472i	$0.543976\pi$
$282$	−8.86977	−0.528187
$283$	24.7804	1.47304	0.736521	−	0.676415i	$-0.236467\pi$
$283$	24.7804	1.47304	0.736521	+	0.676415i	$0.236467\pi$
$284$	−0.0733538	−0.00435275
$285$	−9.50761	−0.563182
$286$	−7.47301	−0.441888
$287$	12.7022	0.749789
$288$	−4.07480	−0.240110
$289$	10.6572	0.626895
$290$	−42.9259	−2.52069
$291$	−51.4093	−3.01367
$292$	−0.137745	−0.00806091
$293$	9.28465	0.542415	0.271208	−	0.962521i	$-0.412577\pi$
$293$	9.28465	0.542415	0.271208	+	0.962521i	$0.412577\pi$
$294$	4.18421	0.244028
$295$	−10.1198	−0.589199
$296$	−11.5630	−0.672088
$297$	9.60661	0.557432
$298$	−19.8068	−1.14738
$299$	−11.9702	−0.692255
$300$	2.40717	0.138978
$301$	6.64681	0.383116
$302$	11.9702	0.688808
$303$	19.7264	1.13325
$304$	−4.25901	−0.244271
$305$	4.89541	0.280310
$306$	39.8760	2.27956
$307$	−3.81579	−0.217779	−0.108889	−	0.994054i	$-0.534729\pi$
$307$	−3.81579	−0.217779	−0.108889	+	0.994054i	$0.534729\pi$
$308$	−0.213997	−0.0121936
$309$	37.5768	2.13767
$310$	39.7819	2.25946
$311$	0.354641	0.0201098	0.0100549	−	0.999949i	$-0.496799\pi$
$311$	0.354641	0.0201098	0.0100549	+	0.999949i	$0.496799\pi$
$312$	25.8760	1.46494
$313$	9.63158	0.544409	0.272205	−	0.962239i	$-0.412247\pi$
$313$	9.63158	0.544409	0.272205	+	0.962239i	$0.412247\pi$
$314$	7.50347	0.423445
$315$	17.2292	0.970757
$316$	0.110856	0.00623615
$317$	−13.4432	−0.755047	−0.377523	−	0.926000i	$-0.623224\pi$
$317$	−13.4432	−0.755047	−0.377523	+	0.926000i	$0.623224\pi$
$318$	−0.269425	−0.0151086
$319$	13.5768	0.760156
$320$	−24.4882	−1.36893
$321$	30.5437	1.70478
$322$	−5.26798	−0.293573
$323$	5.25901	0.292619
$324$	0.323404	0.0179669
$325$	−20.0900	−1.11439
$326$	−2.01668	−0.111693
$327$	28.5228	1.57732
$328$	34.5706	1.90884
$329$	2.11982	0.116869
$330$	−21.3788	−1.17687
$331$	11.6572	0.640739	0.320369	−	0.947293i	$-0.396193\pi$
$331$	11.6572	0.640739	0.320369	+	0.947293i	$0.396193\pi$
$332$	1.40090	0.0768846
$333$	22.0256	1.20700
$334$	31.5381	1.72569
$335$	7.75622	0.423767
$336$	12.1842	0.664703
$337$	19.6662	1.07129	0.535643	−	0.844445i	$-0.320069\pi$
$337$	19.6662	1.07129	0.535643	+	0.844445i	$0.320069\pi$
$338$	−2.85936	−0.155528
$339$	−52.9903	−2.87804
$340$	−2.43281	−0.131938
$341$	−12.5824	−0.681376
$342$	7.58242	0.410010
$343$	−1.00000	−0.0539949
$344$	18.0900	0.975349
$345$	−34.2445	−1.84366
$346$	−28.0305	−1.50693
$347$	34.4328	1.84845	0.924225	−	0.381848i	$-0.124712\pi$
$347$	34.4328	1.84845	0.924225	+	0.381848i	$0.124712\pi$
$348$	3.51658	0.188508
$349$	−9.48679	−0.507816	−0.253908	−	0.967228i	$-0.581716\pi$
$349$	−9.48679	−0.507816	−0.253908	+	0.967228i	$0.581716\pi$
$350$	−8.84143	−0.472594
$351$	−20.7666	−1.10844
$352$	−1.20840	−0.0644080
$353$	22.3130	1.18760	0.593800	−	0.804612i	$-0.297627\pi$
$353$	22.3130	1.18760	0.593800	+	0.804612i	$0.297627\pi$
$354$	12.7410	0.677176
$355$	1.75140	0.0929546
$356$	0.698090	0.0369987
$357$	−15.0450	−0.796267
$358$	−22.2307	−1.17493
$359$	−8.61220	−0.454535	−0.227267	−	0.973832i	$-0.572979\pi$
$359$	−8.61220	−0.454535	−0.227267	+	0.973832i	$0.572979\pi$
$360$	46.8913	2.47139
$361$	1.00000	0.0526316
$362$	3.38298	0.177805
$363$	−24.7071	−1.29678
$364$	0.462598	0.0242467
$365$	3.28880	0.172144
$366$	−6.16339	−0.322165
$367$	13.3926	0.699089	0.349544	−	0.936920i	$-0.386336\pi$
$367$	13.3926	0.699089	0.349544	+	0.936920i	$0.386336\pi$
$368$	−15.3401	−0.799657
$369$	−65.8511	−3.42807
$370$	−20.6516	−1.07363
$371$	0.0643910	0.00334301
$372$	−3.25901	−0.168972
$373$	−12.1932	−0.631339	−0.315669	−	0.948869i	$-0.602229\pi$
$373$	−12.1932	−0.631339	−0.315669	+	0.948869i	$0.602229\pi$
$374$	11.8254	0.611478
$375$	−9.93561	−0.513073
$376$	5.76932	0.297530
$377$	−29.3490	−1.51155
$378$	−9.13919	−0.470069
$379$	15.2847	0.785120	0.392560	−	0.919726i	$-0.371590\pi$
$379$	15.2847	0.785120	0.392560	+	0.919726i	$0.371590\pi$
$380$	−0.462598	−0.0237308
$381$	−19.3836	−0.993054
$382$	−0.124636	−0.00637692
$383$	7.04502	0.359984	0.179992	−	0.983668i	$-0.442393\pi$
$383$	7.04502	0.359984	0.179992	+	0.983668i	$0.442393\pi$
$384$	35.3282	1.80284
$385$	5.10941	0.260399
$386$	2.56864	0.130740
$387$	−34.4585	−1.75162
$388$	−2.50135	−0.126987
$389$	17.0014	0.862008	0.431004	−	0.902350i	$-0.358160\pi$
$389$	17.0014	0.862008	0.431004	+	0.902350i	$0.358160\pi$
$390$	46.2147	2.34017
$391$	18.9419	0.957932
$392$	−2.72161	−0.137462
$393$	20.7666	1.04754
$394$	15.8545	0.798740
$395$	−2.64681	−0.133175
$396$	1.10941	0.0557498
$397$	29.1953	1.46527	0.732635	−	0.680622i	$-0.238290\pi$
$397$	29.1953	1.46527	0.732635	+	0.680622i	$0.238290\pi$
$398$	−7.05880	−0.353825
$399$	−2.86081	−0.143219
$400$	−25.7458	−1.28729
$401$	−13.3878	−0.668555	−0.334277	−	0.942475i	$-0.608492\pi$
$401$	−13.3878	−0.668555	−0.334277	+	0.942475i	$0.608492\pi$
$402$	−9.76518	−0.487043
$403$	27.1994	1.35490
$404$	0.959801	0.0477519
$405$	−7.72161	−0.383690
$406$	−12.9162	−0.641022
$407$	6.53181	0.323770
$408$	−40.9467	−2.02716
$409$	20.9300	1.03492	0.517461	−	0.855707i	$-0.326877\pi$
$409$	20.9300	1.03492	0.517461	+	0.855707i	$0.326877\pi$
$410$	61.7431	3.04927
$411$	43.7521	2.15813
$412$	1.82832	0.0900749
$413$	−3.04502	−0.149835
$414$	27.3103	1.34223
$415$	−33.4480	−1.64190
$416$	2.61220	0.128074
$417$	−28.0948	−1.37581
$418$	2.24860	0.109983
$419$	−16.7666	−0.819103	−0.409552	−	0.912287i	$-0.634315\pi$
$419$	−16.7666	−0.819103	−0.409552	+	0.912287i	$0.634315\pi$
$420$	1.32340	0.0645755
$421$	−17.7819	−0.866635	−0.433317	−	0.901241i	$-0.642657\pi$
$421$	−17.7819	−0.866635	−0.433317	+	0.901241i	$0.642657\pi$
$422$	−5.72576	−0.278726
$423$	−10.9896	−0.534332
$424$	0.175247	0.00851076
$425$	31.7908	1.54208
$426$	−2.20503	−0.106834
$427$	1.47301	0.0712840
$428$	1.48612	0.0718343
$429$	−14.6170	−0.705716
$430$	32.3088	1.55807
$431$	29.8325	1.43698	0.718490	−	0.695538i	$-0.244834\pi$
$431$	29.8325	1.43698	0.718490	+	0.695538i	$0.244834\pi$
$432$	−26.6129	−1.28041
$433$	−11.1350	−0.535116	−0.267558	−	0.963542i	$-0.586217\pi$
$433$	−11.1350	−0.535116	−0.267558	+	0.963542i	$0.586217\pi$
$434$	11.9702	0.574589
$435$	−83.9619	−4.02567
$436$	1.38780	0.0664633
$437$	3.60179	0.172297
$438$	−4.14064	−0.197848
$439$	14.8864	0.710491	0.355246	−	0.934773i	$-0.384397\pi$
$439$	14.8864	0.710491	0.355246	+	0.934773i	$0.384397\pi$
$440$	13.9058	0.662934
$441$	5.18421	0.246867
$442$	−25.5630	−1.21591
$443$	30.4149	1.44505	0.722527	−	0.691342i	$-0.242980\pi$
$443$	30.4149	1.44505	0.722527	+	0.691342i	$0.242980\pi$
$444$	1.69182	0.0802904
$445$	−16.6676	−0.790122
$446$	−3.10044	−0.146810
$447$	−38.7417	−1.83242
$448$	−7.36842	−0.348125
$449$	−9.35801	−0.441632	−0.220816	−	0.975316i	$-0.570872\pi$
$449$	−9.35801	−0.441632	−0.220816	+	0.975316i	$0.570872\pi$
$450$	45.8358	2.16072
$451$	−19.5284	−0.919558
$452$	−2.57827	−0.121272
$453$	23.4134	1.10006
$454$	16.0298	0.752315
$455$	−11.0450	−0.517798
$456$	−7.78600	−0.364613
$457$	−29.8552	−1.39657	−0.698284	−	0.715821i	$-0.746053\pi$
$457$	−29.8552	−1.39657	−0.698284	+	0.715821i	$0.746053\pi$
$458$	14.1884	0.662978
$459$	32.8615	1.53384
$460$	−1.66618	−0.0776862
$461$	−16.8608	−0.785286	−0.392643	−	0.919691i	$-0.628439\pi$
$461$	−16.8608	−0.785286	−0.392643	+	0.919691i	$0.628439\pi$
$462$	−6.43281	−0.299281
$463$	−9.86226	−0.458338	−0.229169	−	0.973387i	$-0.573601\pi$
$463$	−9.86226	−0.458338	−0.229169	+	0.973387i	$0.573601\pi$
$464$	−37.6114	−1.74607
$465$	77.8123	3.60846
$466$	−14.0811	−0.652293
$467$	18.4924	0.855726	0.427863	−	0.903844i	$-0.359267\pi$
$467$	18.4924	0.855726	0.427863	+	0.903844i	$0.359267\pi$
$468$	−2.39821	−0.110857
$469$	2.33382	0.107766
$470$	10.3040	0.475289
$471$	14.6766	0.676262
$472$	−8.28735	−0.381456
$473$	−10.2188	−0.469862
$474$	3.33237	0.153061
$475$	6.04502	0.277364
$476$	−0.732024	−0.0335523
$477$	−0.333816	−0.0152844
$478$	6.43281	0.294230
$479$	−32.2951	−1.47560	−0.737800	−	0.675020i	$-0.764135\pi$
$479$	−32.2951	−1.47560	−0.737800	+	0.675020i	$0.764135\pi$
$480$	7.47301	0.341095
$481$	−14.1198	−0.643808
$482$	38.5664	1.75665
$483$	−10.3040	−0.468850
$484$	−1.20214	−0.0546425
$485$	59.7223	2.71185
$486$	−17.6960	−0.802706
$487$	−27.1142	−1.22866	−0.614331	−	0.789048i	$-0.710574\pi$
$487$	−27.1142	−1.22866	−0.614331	+	0.789048i	$0.710574\pi$
$488$	4.00896	0.181477
$489$	−3.94457	−0.178380
$490$	−4.86081	−0.219589
$491$	6.58723	0.297278	0.148639	−	0.988892i	$-0.452511\pi$
$491$	6.58723	0.297278	0.148639	+	0.988892i	$0.452511\pi$
$492$	−5.05813	−0.228038
$493$	46.4424	2.09166
$494$	−4.86081	−0.218698
$495$	−26.4882	−1.19056
$496$	34.8567	1.56511
$497$	0.526989	0.0236387
$498$	42.1115	1.88706
$499$	23.4689	1.05061	0.525305	−	0.850914i	$-0.323951\pi$
$499$	23.4689	1.05061	0.525305	+	0.850914i	$0.323951\pi$
$500$	−0.483423	−0.0216193
$501$	61.6877	2.75600
$502$	−7.82351	−0.349180
$503$	20.1801	0.899785	0.449892	−	0.893083i	$-0.351462\pi$
$503$	20.1801	0.899785	0.449892	+	0.893083i	$0.351462\pi$
$504$	14.1094	0.628483
$505$	−22.9162	−1.01976
$506$	8.09899	0.360044
$507$	−5.59283	−0.248386
$508$	−0.943123	−0.0418443
$509$	−28.8448	−1.27852	−0.639262	−	0.768989i	$-0.720760\pi$
$509$	−28.8448	−1.27852	−0.639262	+	0.768989i	$0.720760\pi$
$510$	−73.1309	−3.23829
$511$	0.989588	0.0437768
$512$	−19.8352	−0.876599
$513$	6.24860	0.275882
$514$	−36.4626	−1.60830
$515$	−43.6531	−1.92358
$516$	−2.64681	−0.116519
$517$	−3.25901	−0.143331
$518$	−6.21400	−0.273027
$519$	−54.8269	−2.40663
$520$	−30.0602	−1.31823
$521$	−5.36215	−0.234920	−0.117460	−	0.993078i	$-0.537475\pi$
$521$	−5.36215	−0.234920	−0.117460	+	0.993078i	$0.537475\pi$
$522$	66.9605	2.93078
$523$	−1.56304	−0.0683471	−0.0341735	−	0.999416i	$-0.510880\pi$
$523$	−1.56304	−0.0683471	−0.0341735	+	0.999416i	$0.510880\pi$
$524$	1.01041	0.0441401
$525$	−17.2936	−0.754755
$526$	−6.16339	−0.268736
$527$	−43.0409	−1.87489
$528$	−18.7320	−0.815207
$529$	−10.0271	−0.435960
$530$	0.312992	0.0135955
$531$	15.7860	0.685054
$532$	−0.139194	−0.00603483
$533$	42.2147	1.82852
$534$	20.9848	0.908100
$535$	−35.4826	−1.53405
$536$	6.35174	0.274353
$537$	−43.4826	−1.87641
$538$	−7.29776	−0.314629
$539$	1.53740	0.0662206
$540$	−2.89059	−0.124391
$541$	20.8864	0.897978	0.448989	−	0.893537i	$-0.351784\pi$
$541$	20.8864	0.897978	0.448989	+	0.893537i	$0.351784\pi$
$542$	−45.4439	−1.95198
$543$	6.61702	0.283964
$544$	−4.13360	−0.177227
$545$	−33.1350	−1.41935
$546$	13.9058	0.595114
$547$	21.8165	0.932804	0.466402	−	0.884573i	$-0.345550\pi$
$547$	21.8165	0.932804	0.466402	+	0.884573i	$0.345550\pi$
$548$	2.12878	0.0909371
$549$	−7.63640	−0.325913
$550$	13.5928	0.579600
$551$	8.83102	0.376214
$552$	−28.0436	−1.19361
$553$	−0.796415	−0.0338670
$554$	−25.2369	−1.07221
$555$	−40.3941	−1.71463
$556$	−1.36697	−0.0579725
$557$	11.7950	0.499769	0.249884	−	0.968276i	$-0.419607\pi$
$557$	11.7950	0.499769	0.249884	+	0.968276i	$0.419607\pi$
$558$	−62.0561	−2.62704
$559$	22.0900	0.934309
$560$	−14.1544	−0.598134
$561$	23.1302	0.976559
$562$	−6.75285	−0.284852
$563$	35.8414	1.51054	0.755268	−	0.655416i	$-0.227507\pi$
$563$	35.8414	1.51054	0.755268	+	0.655416i	$0.227507\pi$
$564$	−0.844128	−0.0355442
$565$	61.5589	2.58980
$566$	36.2438	1.52344
$567$	−2.32340	−0.0975738
$568$	1.43426	0.0601802
$569$	−8.29025	−0.347545	−0.173773	−	0.984786i	$-0.555596\pi$
$569$	−8.29025	−0.347545	−0.173773	+	0.984786i	$0.555596\pi$
$570$	−13.9058	−0.582451
$571$	8.50906	0.356093	0.178047	−	0.984022i	$-0.443022\pi$
$571$	8.50906	0.356093	0.178047	+	0.984022i	$0.443022\pi$
$572$	−0.711200	−0.0297367
$573$	−0.243784	−0.0101842
$574$	18.5783	0.775442
$575$	21.7729	0.907992
$576$	38.1994	1.59164
$577$	−12.3732	−0.515105	−0.257552	−	0.966264i	$-0.582916\pi$
$577$	−12.3732	−0.515105	−0.257552	+	0.966264i	$0.582916\pi$
$578$	15.5872	0.648343
$579$	5.02419	0.208798
$580$	−4.08522	−0.169629
$581$	−10.0644	−0.417541
$582$	−75.1911	−3.11677
$583$	−0.0989948	−0.00409995
$584$	2.69327	0.111448
$585$	57.2597	2.36740
$586$	13.5797	0.560973
$587$	−44.6773	−1.84403	−0.922014	−	0.387156i	$-0.873457\pi$
$587$	−44.6773	−1.84403	−0.922014	+	0.387156i	$0.873457\pi$
$588$	0.398207	0.0164218
$589$	−8.18421	−0.337225
$590$	−14.8012	−0.609357
$591$	31.0111	1.27563
$592$	−18.0948	−0.743694
$593$	1.84143	0.0756185	0.0378093	−	0.999285i	$-0.487962\pi$
$593$	1.84143	0.0756185	0.0378093	+	0.999285i	$0.487962\pi$
$594$	14.0506	0.576504
$595$	17.4778	0.716521
$596$	−1.88500	−0.0772125
$597$	−13.8068	−0.565076
$598$	−17.5076	−0.715940
$599$	−30.5437	−1.24798	−0.623990	−	0.781432i	$-0.714489\pi$
$599$	−30.5437	−1.24798	−0.623990	+	0.781432i	$0.714489\pi$
$600$	−47.0665	−1.92148
$601$	−34.8913	−1.42324	−0.711622	−	0.702562i	$-0.752039\pi$
$601$	−34.8913	−1.42324	−0.711622	+	0.702562i	$0.752039\pi$
$602$	9.72161	0.396223
$603$	−12.0990	−0.492709
$604$	1.13919	0.0463532
$605$	28.7022	1.16691
$606$	28.8518	1.17203
$607$	−29.3926	−1.19301	−0.596505	−	0.802610i	$-0.703444\pi$
$607$	−29.3926	−1.19301	−0.596505	+	0.802610i	$0.703444\pi$
$608$	−0.786003	−0.0318766
$609$	−25.2638	−1.02374
$610$	7.16002	0.289901
$611$	7.04502	0.285011
$612$	3.79497	0.153402
$613$	−40.0561	−1.61785	−0.808925	−	0.587911i	$-0.799950\pi$
$613$	−40.0561	−1.61785	−0.808925	+	0.587911i	$0.799950\pi$
$614$	−5.58097	−0.225230
$615$	120.768	4.86983
$616$	4.18421	0.168587
$617$	24.9002	1.00245	0.501223	−	0.865318i	$-0.332884\pi$
$617$	24.9002	1.00245	0.501223	+	0.865318i	$0.332884\pi$
$618$	54.9598	2.21081
$619$	−15.7264	−0.632099	−0.316049	−	0.948743i	$-0.602356\pi$
$619$	−15.7264	−0.632099	−0.316049	+	0.948743i	$0.602356\pi$
$620$	3.78600	0.152050
$621$	22.5062	0.903141
$622$	0.518697	0.0207979
$623$	−5.01523	−0.200931
$624$	40.4931	1.62102
$625$	−18.6829	−0.747314
$626$	14.0871	0.563035
$627$	4.39821	0.175647
$628$	0.714098	0.0284956
$629$	22.3434	0.890892
$630$	25.1994	1.00397
$631$	−38.2999	−1.52469	−0.762347	−	0.647168i	$-0.775953\pi$
$631$	−38.2999	−1.52469	−0.762347	+	0.647168i	$0.775953\pi$
$632$	−2.16753	−0.0862198
$633$	−11.1994	−0.445138
$634$	−19.6620	−0.780879
$635$	22.5180	0.893601
$636$	−0.0256410	−0.00101673
$637$	−3.32340	−0.131678
$638$	19.8574	0.786164
$639$	−2.73202	−0.108077
$640$	−41.0409	−1.62228
$641$	−41.9363	−1.65638	−0.828192	−	0.560445i	$-0.810630\pi$
$641$	−41.9363	−1.65638	−0.828192	+	0.560445i	$0.810630\pi$
$642$	44.6731	1.76311
$643$	−16.9371	−0.667932	−0.333966	−	0.942585i	$-0.608387\pi$
$643$	−16.9371	−0.667932	−0.333966	+	0.942585i	$0.608387\pi$
$644$	−0.501348	−0.0197559
$645$	63.1953	2.48831
$646$	7.69182	0.302631
$647$	4.50906	0.177270	0.0886348	−	0.996064i	$-0.471750\pi$
$647$	4.50906	0.177270	0.0886348	+	0.996064i	$0.471750\pi$
$648$	−6.32340	−0.248407
$649$	4.68141	0.183762
$650$	−29.3836	−1.15252
$651$	23.4134	0.917645
$652$	−0.191925	−0.00751638
$653$	−15.0242	−0.587942	−0.293971	−	0.955814i	$-0.594977\pi$
$653$	−15.0242	−0.587942	−0.293971	+	0.955814i	$0.594977\pi$
$654$	41.7175	1.63128
$655$	−24.1246	−0.942628
$656$	54.0990	2.11221
$657$	−5.13023	−0.200149
$658$	3.10044	0.120868
$659$	24.7410	0.963772	0.481886	−	0.876234i	$-0.339952\pi$
$659$	24.7410	0.963772	0.481886	+	0.876234i	$0.339952\pi$
$660$	−2.03460	−0.0791968
$661$	−9.62329	−0.374302	−0.187151	−	0.982331i	$-0.559925\pi$
$661$	−9.62329	−0.374302	−0.187151	+	0.982331i	$0.559925\pi$
$662$	17.0498	0.662661
$663$	−50.0007	−1.94187
$664$	−27.3914	−1.06299
$665$	3.32340	0.128876
$666$	32.2147	1.24829
$667$	31.8075	1.23159
$668$	3.00145	0.116130
$669$	−6.06439	−0.234463
$670$	11.3442	0.438266
$671$	−2.26461	−0.0874243
$672$	2.24860	0.0867417
$673$	−11.1213	−0.428693	−0.214347	−	0.976758i	$-0.568762\pi$
$673$	−11.1213	−0.428693	−0.214347	+	0.976758i	$0.568762\pi$
$674$	28.7637	1.10794
$675$	37.7729	1.45388
$676$	−0.272122	−0.0104662
$677$	16.5616	0.636514	0.318257	−	0.948005i	$-0.396903\pi$
$677$	16.5616	0.636514	0.318257	+	0.948005i	$0.396903\pi$
$678$	−77.5035	−2.97650
$679$	17.9702	0.689633
$680$	47.5679	1.82414
$681$	31.3539	1.20148
$682$	−18.4030	−0.704688
$683$	9.62262	0.368199	0.184100	−	0.982908i	$-0.441063\pi$
$683$	9.62262	0.368199	0.184100	+	0.982908i	$0.441063\pi$
$684$	0.721612	0.0275915
$685$	−50.8269	−1.94199
$686$	−1.46260	−0.0558423
$687$	27.7521	1.05881
$688$	28.3088	1.07926
$689$	0.213997	0.00815265
$690$	−50.0859	−1.90674
$691$	11.5720	0.440220	0.220110	−	0.975475i	$-0.429358\pi$
$691$	11.5720	0.440220	0.220110	+	0.975475i	$0.429358\pi$
$692$	−2.66763	−0.101408
$693$	−7.97021	−0.302763
$694$	50.3614	1.91169
$695$	32.6378	1.23802
$696$	−68.7583	−2.60628
$697$	−66.8012	−2.53028
$698$	−13.8754	−0.525190
$699$	−27.5422	−1.04174
$700$	−0.841431	−0.0318031
$701$	−46.3781	−1.75167	−0.875837	−	0.482606i	$-0.839690\pi$
$701$	−46.3781	−1.75167	−0.875837	+	0.482606i	$0.839690\pi$
$702$	−30.3732	−1.14636
$703$	4.24860	0.160239
$704$	11.3282	0.426948
$705$	20.1544	0.759059
$706$	32.6349	1.22823
$707$	−6.89541	−0.259329
$708$	1.21255	0.0455703
$709$	−26.2099	−0.984332	−0.492166	−	0.870501i	$-0.663795\pi$
$709$	−26.2099	−0.984332	−0.492166	+	0.870501i	$0.663795\pi$
$710$	2.56159	0.0961349
$711$	4.12878	0.154841
$712$	−13.6495	−0.511537
$713$	−29.4778	−1.10395
$714$	−22.0048	−0.823510
$715$	16.9806	0.635039
$716$	−2.11567	−0.0790664
$717$	12.5824	0.469899
$718$	−12.5962	−0.470086
$719$	−11.4432	−0.426760	−0.213380	−	0.976969i	$-0.568447\pi$
$719$	−11.4432	−0.426760	−0.213380	+	0.976969i	$0.568447\pi$
$720$	73.3795	2.73469
$721$	−13.1350	−0.489175
$722$	1.46260	0.0544323
$723$	75.4349	2.80546
$724$	0.321955	0.0119654
$725$	53.3836	1.98262
$726$	−36.1365	−1.34115
$727$	26.8054	0.994156	0.497078	−	0.867706i	$-0.334406\pi$
$727$	26.8054	0.994156	0.497078	+	0.867706i	$0.334406\pi$
$728$	−9.04502	−0.335231
$729$	−41.5831	−1.54011
$730$	4.81019	0.178033
$731$	−34.9557	−1.29288
$732$	−0.586564	−0.0216800
$733$	42.7189	1.57786	0.788930	−	0.614484i	$-0.210636\pi$
$733$	42.7189	1.57786	0.788930	+	0.614484i	$0.210636\pi$
$734$	19.5880	0.723007
$735$	−9.50761	−0.350694
$736$	−2.83102	−0.104353
$737$	−3.58801	−0.132166
$738$	−96.3137	−3.54535
$739$	4.35656	0.160259	0.0801293	−	0.996784i	$-0.474467\pi$
$739$	4.35656	0.160259	0.0801293	+	0.996784i	$0.474467\pi$
$740$	−1.96540	−0.0722494
$741$	−9.50761	−0.349271
$742$	0.0941782	0.00345739
$743$	−2.11982	−0.0777686	−0.0388843	−	0.999244i	$-0.512380\pi$
$743$	−2.11982	−0.0777686	−0.0388843	+	0.999244i	$0.512380\pi$
$744$	63.7223	2.33617
$745$	45.0063	1.64890
$746$	−17.8337	−0.652939
$747$	52.1759	1.90902
$748$	1.12541	0.0411492
$749$	−10.6766	−0.390114
$750$	−14.5318	−0.530627
$751$	−27.1004	−0.988909	−0.494455	−	0.869203i	$-0.664632\pi$
$751$	−27.1004	−0.988909	−0.494455	+	0.869203i	$0.664632\pi$
$752$	9.02834	0.329230
$753$	−15.3026	−0.557657
$754$	−42.9259	−1.56327
$755$	−27.1994	−0.989889
$756$	−0.869769	−0.0316332
$757$	40.3297	1.46581	0.732903	−	0.680333i	$-0.238165\pi$
$757$	40.3297	1.46581	0.732903	+	0.680333i	$0.238165\pi$
$758$	22.3553	0.811981
$759$	15.8414	0.575008
$760$	9.04502	0.328097
$761$	15.8206	0.573497	0.286748	−	0.958006i	$-0.407426\pi$
$761$	15.8206	0.573497	0.286748	+	0.958006i	$0.407426\pi$
$762$	−28.3505	−1.02703
$763$	−9.97021	−0.360946
$764$	−0.0118615	−0.000429133 0
$765$	−90.6087	−3.27597
$766$	10.3040	0.372300
$767$	−10.1198	−0.365405
$768$	9.51176	0.343226
$769$	33.6829	1.21464	0.607318	−	0.794459i	$-0.292246\pi$
$769$	33.6829	1.21464	0.607318	+	0.794459i	$0.292246\pi$
$770$	7.47301	0.269309
$771$	−71.3199	−2.56852
$772$	0.244455	0.00879813
$773$	−22.9252	−0.824562	−0.412281	−	0.911057i	$-0.635268\pi$
$773$	−22.9252	−0.824562	−0.412281	+	0.911057i	$0.635268\pi$
$774$	−50.3989	−1.81155
$775$	−49.4737	−1.77715
$776$	48.9079	1.75569
$777$	−12.1544	−0.436037
$778$	24.8663	0.891500
$779$	−12.7022	−0.455105
$780$	4.39821	0.157481
$781$	−0.810194	−0.0289910
$782$	27.7044	0.990706
$783$	55.1815	1.97203
$784$	−4.25901	−0.152108
$785$	−17.0498	−0.608535
$786$	30.3732	1.08338
$787$	7.70079	0.274503	0.137252	−	0.990536i	$-0.456173\pi$
$787$	7.70079	0.274503	0.137252	+	0.990536i	$0.456173\pi$
$788$	1.50886	0.0537510
$789$	−12.0554	−0.429185
$790$	−3.87122	−0.137732
$791$	18.5228	0.658596
$792$	−21.6918	−0.770785
$793$	4.89541	0.173841
$794$	42.7010	1.51540
$795$	0.612205	0.0217127
$796$	−0.671779	−0.0238106
$797$	−16.5914	−0.587697	−0.293848	−	0.955852i	$-0.594936\pi$
$797$	−16.5914	−0.587697	−0.293848	+	0.955852i	$0.594936\pi$
$798$	−4.18421	−0.148119
$799$	−11.1482	−0.394393
$800$	−4.75140	−0.167987
$801$	26.0000	0.918665
$802$	−19.5810	−0.691428
$803$	−1.52139	−0.0536888
$804$	−0.929343	−0.0327754
$805$	11.9702	0.421895
$806$	39.7819	1.40126
$807$	−14.2742	−0.502477
$808$	−18.7666	−0.660208
$809$	1.41344	0.0496938	0.0248469	−	0.999691i	$-0.492090\pi$
$809$	1.41344	0.0496938	0.0248469	+	0.999691i	$0.492090\pi$
$810$	−11.2936	−0.396817
$811$	28.1980	0.990165	0.495083	−	0.868846i	$-0.335138\pi$
$811$	28.1980	0.990165	0.495083	+	0.868846i	$0.335138\pi$
$812$	−1.22923	−0.0431374
$813$	−88.8871	−3.11741
$814$	9.55341	0.334847
$815$	4.58242	0.160515
$816$	−64.0769	−2.24314
$817$	−6.64681	−0.232542
$818$	30.6122	1.07033
$819$	17.2292	0.602038
$820$	5.87603	0.205200
$821$	26.9765	0.941486	0.470743	−	0.882270i	$-0.343986\pi$
$821$	26.9765	0.941486	0.470743	+	0.882270i	$0.343986\pi$
$822$	63.9917	2.23197
$823$	25.6441	0.893898	0.446949	−	0.894560i	$-0.352511\pi$
$823$	25.6441	0.893898	0.446949	+	0.894560i	$0.352511\pi$
$824$	−35.7485	−1.24536
$825$	26.5872	0.925649
$826$	−4.45364	−0.154962
$827$	29.5451	1.02738	0.513692	−	0.857975i	$-0.328277\pi$
$827$	29.5451	1.02738	0.513692	+	0.857975i	$0.328277\pi$
$828$	2.59910	0.0903248
$829$	4.82687	0.167644	0.0838221	−	0.996481i	$-0.473287\pi$
$829$	4.82687	0.167644	0.0838221	+	0.996481i	$0.473287\pi$
$830$	−48.9211	−1.69807
$831$	−49.3628	−1.71238
$832$	−24.4882	−0.848977
$833$	5.25901	0.182214
$834$	−41.0915	−1.42288
$835$	−71.6627	−2.47999
$836$	0.213997	0.00740125
$837$	−51.1399	−1.76765
$838$	−24.5228	−0.847128
$839$	−35.3241	−1.21952	−0.609761	−	0.792585i	$-0.708735\pi$
$839$	−35.3241	−1.21952	−0.609761	+	0.792585i	$0.708735\pi$
$840$	−25.8760	−0.892808
$841$	48.9869	1.68920
$842$	−26.0077	−0.896285
$843$	−13.2084	−0.454922
$844$	−0.544915	−0.0187568
$845$	6.49720	0.223511
$846$	−16.0734	−0.552613
$847$	8.63640	0.296750
$848$	0.274242	0.00941751
$849$	70.8919	2.43300
$850$	46.4972	1.59484
$851$	15.3026	0.524566
$852$	−0.209851	−0.00718938
$853$	40.6129	1.39056	0.695279	−	0.718740i	$-0.255281\pi$
$853$	40.6129	1.39056	0.695279	+	0.718740i	$0.255281\pi$
$854$	2.15442	0.0737228
$855$	−17.2292	−0.589227
$856$	−29.0575	−0.993167
$857$	21.4570	0.732957	0.366479	−	0.930426i	$-0.380563\pi$
$857$	21.4570	0.732957	0.366479	+	0.930426i	$0.380563\pi$
$858$	−21.3788	−0.729861
$859$	21.9237	0.748029	0.374014	−	0.927423i	$-0.377981\pi$
$859$	21.9237	0.748029	0.374014	+	0.927423i	$0.377981\pi$
$860$	3.07480	0.104850
$861$	36.3386	1.23842
$862$	43.6329	1.48614
$863$	−27.1004	−0.922510	−0.461255	−	0.887268i	$-0.652601\pi$
$863$	−27.1004	−0.922510	−0.461255	+	0.887268i	$0.652601\pi$
$864$	−4.91142	−0.167090
$865$	63.6925	2.16561
$866$	−16.2861	−0.553424
$867$	30.4882	1.03543
$868$	1.13919	0.0386668
$869$	1.22441	0.0415352
$870$	−122.803	−4.16340
$871$	7.75622	0.262809
$872$	−27.1350	−0.918909
$873$	−93.1614	−3.15303
$874$	5.26798	0.178192
$875$	3.47301	0.117409
$876$	−0.394061	−0.0133141
$877$	−40.7160	−1.37488	−0.687441	−	0.726240i	$-0.741266\pi$
$877$	−40.7160	−1.37488	−0.687441	+	0.726240i	$0.741266\pi$
$878$	21.7729	0.734800
$879$	26.5616	0.895900
$880$	21.7610	0.733564
$881$	−10.2653	−0.345846	−0.172923	−	0.984935i	$-0.555321\pi$
$881$	−10.2653	−0.345846	−0.172923	+	0.984935i	$0.555321\pi$
$882$	7.58242	0.255313
$883$	−13.7514	−0.462771	−0.231386	−	0.972862i	$-0.574326\pi$
$883$	−13.7514	−0.462771	−0.231386	+	0.972862i	$0.574326\pi$
$884$	−2.43281	−0.0818243
$885$	−28.9508	−0.973172
$886$	44.4848	1.49449
$887$	55.6710	1.86925	0.934625	−	0.355636i	$-0.115736\pi$
$887$	55.6710	1.86925	0.934625	+	0.355636i	$0.115736\pi$
$888$	−33.0796	−1.11008
$889$	6.77559	0.227246
$890$	−24.3781	−0.817154
$891$	3.57201	0.119667
$892$	−0.295066	−0.00987955
$893$	−2.11982	−0.0709370
$894$	−56.6635	−1.89511
$895$	50.5139	1.68849
$896$	−12.3490	−0.412553
$897$	−34.2445	−1.14339
$898$	−13.6870	−0.456741
$899$	−72.2749	−2.41050
$900$	4.36215	0.145405
$901$	−0.338633	−0.0112815
$902$	−28.5623	−0.951019
$903$	19.0152	0.632787
$904$	50.4120	1.67668
$905$	−7.68701	−0.255525
$906$	34.2445	1.13770
$907$	5.21255	0.173080	0.0865399	−	0.996248i	$-0.472419\pi$
$907$	5.21255	0.173080	0.0865399	+	0.996248i	$0.472419\pi$
$908$	1.52554	0.0506268
$909$	35.7473	1.18566
$910$	−16.1544	−0.535514
$911$	−38.0900	−1.26198	−0.630990	−	0.775791i	$-0.717351\pi$
$911$	−38.0900	−1.26198	−0.630990	+	0.775791i	$0.717351\pi$
$912$	−12.1842	−0.403460
$913$	15.4730	0.512082
$914$	−43.6662	−1.44435
$915$	14.0048	0.462985
$916$	1.35029	0.0446149
$917$	−7.25901	−0.239714
$918$	48.0631	1.58632
$919$	43.3955	1.43149	0.715743	−	0.698364i	$-0.246088\pi$
$919$	43.3955	1.43149	0.715743	+	0.698364i	$0.246088\pi$
$920$	32.5783	1.07407
$921$	−10.9162	−0.359702
$922$	−24.6606	−0.812153
$923$	1.75140	0.0576480
$924$	−0.612205	−0.0201401
$925$	25.6829	0.844447
$926$	−14.4245	−0.474019
$927$	68.0948	2.23653
$928$	−6.94120	−0.227856
$929$	−6.34278	−0.208100	−0.104050	−	0.994572i	$-0.533180\pi$
$929$	−6.34278	−0.208100	−0.104050	+	0.994572i	$0.533180\pi$
$930$	113.808	3.73192
$931$	1.00000	0.0327737
$932$	−1.34008	−0.0438959
$933$	1.01456	0.0332151
$934$	27.0469	0.885003
$935$	−26.8704	−0.878757
$936$	46.8913	1.53269
$937$	0.444673	0.0145268	0.00726341	−	0.999974i	$-0.497688\pi$
$937$	0.444673	0.0145268	0.00726341	+	0.999974i	$0.497688\pi$
$938$	3.41344	0.111453
$939$	27.5541	0.899193
$940$	0.980625	0.0319845
$941$	−15.2936	−0.498558	−0.249279	−	0.968432i	$-0.580194\pi$
$941$	−15.2936	−0.498558	−0.249279	+	0.968432i	$0.580194\pi$
$942$	21.4660	0.699399
$943$	−45.7508	−1.48985
$944$	−12.9688	−0.422097
$945$	20.7666	0.675538
$946$	−14.9460	−0.485937
$947$	44.7112	1.45292	0.726459	−	0.687209i	$-0.241165\pi$
$947$	44.7112	1.45292	0.726459	+	0.687209i	$0.241165\pi$
$948$	0.317138	0.0103002
$949$	3.28880	0.106759
$950$	8.84143	0.286854
$951$	−38.4585	−1.24710
$952$	14.3130	0.463887
$953$	−48.3434	−1.56600	−0.782999	−	0.622023i	$-0.786311\pi$
$953$	−48.3434	−1.56600	−0.782999	+	0.622023i	$0.786311\pi$
$954$	−0.488239	−0.0158073
$955$	0.283205	0.00916430
$956$	0.612205	0.0198001
$957$	38.8407	1.25554
$958$	−47.2347	−1.52608
$959$	−15.2936	−0.493857
$960$	−70.0561	−2.26105
$961$	35.9813	1.16069
$962$	−20.6516	−0.665835
$963$	55.3497	1.78362
$964$	3.67033	0.118213
$965$	−5.83661	−0.187887
$966$	−15.0707	−0.484890
$967$	56.1759	1.80650	0.903248	−	0.429119i	$-0.141176\pi$
$967$	56.1759	1.80650	0.903248	+	0.429119i	$0.141176\pi$
$968$	23.5049	0.755477
$969$	15.0450	0.483316
$970$	87.3497	2.80463
$971$	−48.5603	−1.55838	−0.779188	−	0.626791i	$-0.784368\pi$
$971$	−48.5603	−1.55838	−0.779188	+	0.626791i	$0.784368\pi$
$972$	−1.68411	−0.0540178
$973$	9.82061	0.314834
$974$	−39.6572	−1.27070
$975$	−57.4737	−1.84063
$976$	6.27357	0.200812
$977$	29.6925	0.949947	0.474974	−	0.880000i	$-0.342458\pi$
$977$	29.6925	0.949947	0.474974	+	0.880000i	$0.342458\pi$
$978$	−5.76932	−0.184483
$979$	7.71042	0.246426
$980$	−0.462598	−0.0147772
$981$	51.6877	1.65026
$982$	9.63448	0.307449
$983$	16.8358	0.536980	0.268490	−	0.963283i	$-0.413475\pi$
$983$	16.8358	0.536980	0.268490	+	0.963283i	$0.413475\pi$
$984$	98.8996	3.15281
$985$	−36.0256	−1.14787
$986$	67.9266	2.16323
$987$	6.06439	0.193032
$988$	−0.462598	−0.0147172
$989$	−23.9404	−0.761261
$990$	−38.7417	−1.23129
$991$	−2.53595	−0.0805572	−0.0402786	−	0.999188i	$-0.512825\pi$
$991$	−2.53595	−0.0805572	−0.0402786	+	0.999188i	$0.512825\pi$
$992$	6.43281	0.204242
$993$	33.3490	1.05830
$994$	0.770774	0.0244475
$995$	16.0394	0.508484
$996$	4.00772	0.126989
$997$	−36.9217	−1.16932	−0.584661	−	0.811277i	$-0.698773\pi$
$997$	−36.9217	−1.16932	−0.584661	+	0.811277i	$0.698773\pi$
$998$	34.3255	1.08656
$999$	26.5478	0.839936

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	133.2.a.d.1.2	✓	3
3.2	odd	2		1197.2.a.k.1.2		3
4.3	odd	2		2128.2.a.p.1.1		3
5.4	even	2		3325.2.a.r.1.2		3
7.2	even	3		931.2.f.l.704.2		6
7.3	odd	6		931.2.f.m.324.2		6
7.4	even	3		931.2.f.l.324.2		6
7.5	odd	6		931.2.f.m.704.2		6
7.6	odd	2		931.2.a.k.1.2		3
8.3	odd	2		8512.2.a.bp.1.3		3
8.5	even	2		8512.2.a.bi.1.1		3
19.18	odd	2		2527.2.a.f.1.2		3
21.20	even	2		8379.2.a.bo.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.a.d.1.2	✓	3	1.1	even	1	trivial
931.2.a.k.1.2		3	7.6	odd	2
931.2.f.l.324.2		6	7.4	even	3
931.2.f.l.704.2		6	7.2	even	3
931.2.f.m.324.2		6	7.3	odd	6
931.2.f.m.704.2		6	7.5	odd	6
1197.2.a.k.1.2		3	3.2	odd	2
2128.2.a.p.1.1		3	4.3	odd	2
2527.2.a.f.1.2		3	19.18	odd	2
3325.2.a.r.1.2		3	5.4	even	2
8379.2.a.bo.1.2		3	21.20	even	2
8512.2.a.bi.1.1		3	8.5	even	2
8512.2.a.bp.1.3		3	8.3	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

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	133.2.a.d.1.2

Level	$133$
Weight	$2$
Character	133.1
Self dual	yes
Analytic conductor	$1.062$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$