LMFDB - Embedded newform 431.2.a.f.1.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [431,2,Mod(1,431)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(431, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("431.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$431$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	431.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.44155232712$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	431.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.57370 q^{2} -2.57511 q^{3} +4.62395 q^{4} +2.28360 q^{5} -6.62758 q^{6} +0.357026 q^{7} +6.75328 q^{8} +3.63121 q^{9} +5.87731 q^{10} -3.46937 q^{11} -11.9072 q^{12} +3.92837 q^{13} +0.918878 q^{14} -5.88053 q^{15} +8.13303 q^{16} +6.75219 q^{17} +9.34567 q^{18} -7.25447 q^{19} +10.5593 q^{20} -0.919381 q^{21} -8.92912 q^{22} +2.04539 q^{23} -17.3905 q^{24} +0.214828 q^{25} +10.1105 q^{26} -1.62544 q^{27} +1.65087 q^{28} -5.43464 q^{29} -15.1347 q^{30} +5.78498 q^{31} +7.42547 q^{32} +8.93401 q^{33} +17.3781 q^{34} +0.815303 q^{35} +16.7906 q^{36} -4.94633 q^{37} -18.6709 q^{38} -10.1160 q^{39} +15.4218 q^{40} -1.78711 q^{41} -2.36622 q^{42} -3.93078 q^{43} -16.0422 q^{44} +8.29223 q^{45} +5.26423 q^{46} -5.17162 q^{47} -20.9435 q^{48} -6.87253 q^{49} +0.552903 q^{50} -17.3876 q^{51} +18.1646 q^{52} -14.1387 q^{53} -4.18341 q^{54} -7.92264 q^{55} +2.41109 q^{56} +18.6811 q^{57} -13.9872 q^{58} -8.33307 q^{59} -27.1913 q^{60} +9.86026 q^{61} +14.8888 q^{62} +1.29644 q^{63} +2.84489 q^{64} +8.97083 q^{65} +22.9935 q^{66} -4.19720 q^{67} +31.2218 q^{68} -5.26712 q^{69} +2.09835 q^{70} +11.5581 q^{71} +24.5226 q^{72} +4.14078 q^{73} -12.7304 q^{74} -0.553206 q^{75} -33.5443 q^{76} -1.23865 q^{77} -26.0356 q^{78} +4.16693 q^{79} +18.5726 q^{80} -6.70794 q^{81} -4.59949 q^{82} +8.38331 q^{83} -4.25118 q^{84} +15.4193 q^{85} -10.1167 q^{86} +13.9948 q^{87} -23.4296 q^{88} +0.930759 q^{89} +21.3418 q^{90} +1.40253 q^{91} +9.45779 q^{92} -14.8970 q^{93} -13.3102 q^{94} -16.5663 q^{95} -19.1214 q^{96} -13.8575 q^{97} -17.6879 q^{98} -12.5980 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q + q^{2} + q^{3} + 33 q^{4} + 13 q^{5} + 17 q^{6} + 8 q^{7} - 3 q^{8} + 31 q^{9} - 6 q^{10} + 15 q^{11} - 12 q^{12} + 11 q^{13} + 16 q^{14} - 5 q^{15} + 43 q^{16} + 6 q^{17} - 8 q^{18} + 18 q^{19}+ \cdots - 9 q^{99}+O(q^{100})$ 24 * q + q^2 + q^3 + 33 * q^4 + 13 * q^5 + 17 * q^6 + 8 * q^7 - 3 * q^8 + 31 * q^9 - 6 * q^10 + 15 * q^11 - 12 * q^12 + 11 * q^13 + 16 * q^14 - 5 * q^15 + 43 * q^16 + 6 * q^17 - 8 * q^18 + 18 * q^19 - 10 * q^20 - 7 * q^21 - 5 * q^22 - 8 * q^23 + 15 * q^24 + 51 * q^25 + 5 * q^26 + 13 * q^27 + 2 * q^28 + 17 * q^29 - 54 * q^30 + 37 * q^31 - 20 * q^32 + 16 * q^33 + 50 * q^34 + 3 * q^35 + 54 * q^36 + 7 * q^37 - 25 * q^38 - 38 * q^39 + 2 * q^40 + 24 * q^41 + 18 * q^42 + 10 * q^43 + 78 * q^45 + 9 * q^46 - 6 * q^47 - 25 * q^48 + 84 * q^49 - 65 * q^50 - 21 * q^51 - 31 * q^52 - 22 * q^53 + 23 * q^54 + 12 * q^55 - 45 * q^56 + 9 * q^57 - 12 * q^58 + 21 * q^59 - 82 * q^60 + 46 * q^61 - 68 * q^62 - 47 * q^63 + 53 * q^64 - 42 * q^65 - 56 * q^66 - 25 * q^67 + 3 * q^68 - 33 * q^69 - 18 * q^70 - 6 * q^71 - 115 * q^72 + 86 * q^73 - 60 * q^74 - 34 * q^75 + 48 * q^76 - 31 * q^77 - 84 * q^78 - 6 * q^79 - 24 * q^80 + 44 * q^81 - 3 * q^82 - 49 * q^83 - 126 * q^84 + 75 * q^85 - 56 * q^86 - 31 * q^87 - 91 * q^88 + 61 * q^89 - 161 * q^90 + 32 * q^91 - 130 * q^92 - 6 * q^93 + 8 * q^94 - 23 * q^95 - 5 * q^96 + 88 * q^97 - 120 * q^98 - 9 * 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

)

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

$2$	2.57370	1.81988	0.909942	−	0.414736i	$-0.136126\pi$
$2$	2.57370	1.81988	0.909942	+	0.414736i	$0.136126\pi$
$3$	−2.57511	−1.48674	−0.743371	−	0.668879i	$-0.766774\pi$
$3$	−2.57511	−1.48674	−0.743371	+	0.668879i	$0.766774\pi$
$4$	4.62395	2.31198
$5$	2.28360	1.02126	0.510628	−	0.859801i	$-0.329413\pi$
$5$	2.28360	1.02126	0.510628	+	0.859801i	$0.329413\pi$
$6$	−6.62758	−2.70570
$7$	0.357026	0.134943	0.0674715	−	0.997721i	$-0.478507\pi$
$7$	0.357026	0.134943	0.0674715	+	0.997721i	$0.478507\pi$
$8$	6.75328	2.38764
$9$	3.63121	1.21040
$10$	5.87731	1.85857
$11$	−3.46937	−1.04605	−0.523026	−	0.852316i	$-0.675197\pi$
$11$	−3.46937	−1.04605	−0.523026	+	0.852316i	$0.675197\pi$
$12$	−11.9072	−3.43731
$13$	3.92837	1.08953	0.544767	−	0.838587i	$-0.316618\pi$
$13$	3.92837	1.08953	0.544767	+	0.838587i	$0.316618\pi$
$14$	0.918878	0.245580
$15$	−5.88053	−1.51835
$16$	8.13303	2.03326
$17$	6.75219	1.63765	0.818823	−	0.574046i	$-0.194627\pi$
$17$	6.75219	1.63765	0.818823	+	0.574046i	$0.194627\pi$
$18$	9.34567	2.20279
$19$	−7.25447	−1.66429	−0.832144	−	0.554559i	$-0.812887\pi$
$19$	−7.25447	−1.66429	−0.832144	+	0.554559i	$0.812887\pi$
$20$	10.5593	2.36112
$21$	−0.919381	−0.200625
$22$	−8.92912	−1.90369
$23$	2.04539	0.426494	0.213247	−	0.976998i	$-0.431596\pi$
$23$	2.04539	0.426494	0.213247	+	0.976998i	$0.431596\pi$
$24$	−17.3905	−3.54981
$25$	0.214828	0.0429656
$26$	10.1105	1.98283
$27$	−1.62544	−0.312817
$28$	1.65087	0.311985
$29$	−5.43464	−1.00919	−0.504594	−	0.863357i	$-0.668358\pi$
$29$	−5.43464	−1.00919	−0.504594	+	0.863357i	$0.668358\pi$
$30$	−15.1347	−2.76321
$31$	5.78498	1.03901	0.519507	−	0.854466i	$-0.326116\pi$
$31$	5.78498	1.03901	0.519507	+	0.854466i	$0.326116\pi$
$32$	7.42547	1.31265
$33$	8.93401	1.55521
$34$	17.3781	2.98032
$35$	0.815303	0.137811
$36$	16.7906	2.79843
$37$	−4.94633	−0.813172	−0.406586	−	0.913613i	$-0.633281\pi$
$37$	−4.94633	−0.813172	−0.406586	+	0.913613i	$0.633281\pi$
$38$	−18.6709	−3.02881
$39$	−10.1160	−1.61986
$40$	15.4218	2.43840
$41$	−1.78711	−0.279099	−0.139550	−	0.990215i	$-0.544566\pi$
$41$	−1.78711	−0.279099	−0.139550	+	0.990215i	$0.544566\pi$
$42$	−2.36622	−0.365115
$43$	−3.93078	−0.599438	−0.299719	−	0.954028i	$-0.596893\pi$
$43$	−3.93078	−0.599438	−0.299719	+	0.954028i	$0.596893\pi$
$44$	−16.0422	−2.41845
$45$	8.29223	1.23613
$46$	5.26423	0.776169
$47$	−5.17162	−0.754358	−0.377179	−	0.926140i	$-0.623106\pi$
$47$	−5.17162	−0.754358	−0.377179	+	0.926140i	$0.623106\pi$
$48$	−20.9435	−3.02293
$49$	−6.87253	−0.981790
$50$	0.552903	0.0781923
$51$	−17.3876	−2.43476
$52$	18.1646	2.51898
$53$	−14.1387	−1.94210	−0.971051	−	0.238871i	$-0.923223\pi$
$53$	−14.1387	−1.94210	−0.971051	+	0.238871i	$0.923223\pi$
$54$	−4.18341	−0.569290
$55$	−7.92264	−1.06829
$56$	2.41109	0.322196
$57$	18.6811	2.47437
$58$	−13.9872	−1.83660
$59$	−8.33307	−1.08487	−0.542437	−	0.840097i	$-0.682498\pi$
$59$	−8.33307	−1.08487	−0.542437	+	0.840097i	$0.682498\pi$
$60$	−27.1913	−3.51038
$61$	9.86026	1.26248	0.631239	−	0.775589i	$-0.282547\pi$
$61$	9.86026	1.26248	0.631239	+	0.775589i	$0.282547\pi$
$62$	14.8888	1.89088
$63$	1.29644	0.163335
$64$	2.84489	0.355611
$65$	8.97083	1.11269
$66$	22.9935	2.83030
$67$	−4.19720	−0.512770	−0.256385	−	0.966575i	$-0.582532\pi$
$67$	−4.19720	−0.512770	−0.256385	+	0.966575i	$0.582532\pi$
$68$	31.2218	3.78620
$69$	−5.26712	−0.634086
$70$	2.09835	0.250801
$71$	11.5581	1.37169	0.685847	−	0.727746i	$-0.259432\pi$
$71$	11.5581	1.37169	0.685847	+	0.727746i	$0.259432\pi$
$72$	24.5226	2.89001
$73$	4.14078	0.484641	0.242321	−	0.970196i	$-0.422091\pi$
$73$	4.14078	0.484641	0.242321	+	0.970196i	$0.422091\pi$
$74$	−12.7304	−1.47988
$75$	−0.553206	−0.0638787
$76$	−33.5443	−3.84780
$77$	−1.23865	−0.141157
$78$	−26.0356	−2.94795
$79$	4.16693	0.468816	0.234408	−	0.972138i	$-0.424685\pi$
$79$	4.16693	0.468816	0.234408	+	0.972138i	$0.424685\pi$
$80$	18.5726	2.07648
$81$	−6.70794	−0.745326
$82$	−4.59949	−0.507928
$83$	8.38331	0.920187	0.460094	−	0.887870i	$-0.347816\pi$
$83$	8.38331	0.920187	0.460094	+	0.887870i	$0.347816\pi$
$84$	−4.25118	−0.463841
$85$	15.4193	1.67246
$86$	−10.1167	−1.09091
$87$	13.9948	1.50040
$88$	−23.4296	−2.49760
$89$	0.930759	0.0986602	0.0493301	−	0.998783i	$-0.484291\pi$
$89$	0.930759	0.0986602	0.0493301	+	0.998783i	$0.484291\pi$
$90$	21.3418	2.24962
$91$	1.40253	0.147025
$92$	9.45779	0.986043
$93$	−14.8970	−1.54475
$94$	−13.3102	−1.37284
$95$	−16.5663	−1.69967
$96$	−19.1214	−1.95157
$97$	−13.8575	−1.40701	−0.703507	−	0.710688i	$-0.748384\pi$
$97$	−13.8575	−1.40701	−0.703507	+	0.710688i	$0.748384\pi$
$98$	−17.6879	−1.78674
$99$	−12.5980	−1.26615
$100$	0.993353	0.0993353
$101$	6.91545	0.688113	0.344056	−	0.938949i	$-0.388199\pi$
$101$	6.91545	0.688113	0.344056	+	0.938949i	$0.388199\pi$
$102$	−44.7507	−4.43098
$103$	−2.27419	−0.224082	−0.112041	−	0.993704i	$-0.535739\pi$
$103$	−2.27419	−0.224082	−0.112041	+	0.993704i	$0.535739\pi$
$104$	26.5294	2.60142
$105$	−2.09950	−0.204890
$106$	−36.3889	−3.53440
$107$	17.3862	1.68079	0.840394	−	0.541976i	$-0.182323\pi$
$107$	17.3862	1.68079	0.840394	+	0.541976i	$0.182323\pi$
$108$	−7.51597	−0.723225
$109$	10.6882	1.02374	0.511871	−	0.859062i	$-0.328953\pi$
$109$	10.6882	1.02374	0.511871	+	0.859062i	$0.328953\pi$
$110$	−20.3905	−1.94416
$111$	12.7374	1.20898
$112$	2.90370	0.274374
$113$	−8.62652	−0.811515	−0.405757	−	0.913981i	$-0.632992\pi$
$113$	−8.62652	−0.811515	−0.405757	+	0.913981i	$0.632992\pi$
$114$	48.0796	4.50306
$115$	4.67086	0.435560
$116$	−25.1295	−2.33322
$117$	14.2648	1.31878
$118$	−21.4469	−1.97434
$119$	2.41070	0.220989
$120$	−39.7129	−3.62527
$121$	1.03649	0.0942268
$122$	25.3774	2.29756
$123$	4.60201	0.414949
$124$	26.7495	2.40217
$125$	−10.9274	−0.977378
$126$	3.33664	0.297252
$127$	14.9387	1.32559	0.662796	−	0.748800i	$-0.269370\pi$
$127$	14.9387	1.32559	0.662796	+	0.748800i	$0.269370\pi$
$128$	−7.52904	−0.665479
$129$	10.1222	0.891210
$130$	23.0883	2.02497
$131$	6.20221	0.541889	0.270945	−	0.962595i	$-0.412664\pi$
$131$	6.20221	0.541889	0.270945	+	0.962595i	$0.412664\pi$
$132$	41.3104	3.59561
$133$	−2.59003	−0.224584
$134$	−10.8024	−0.933182
$135$	−3.71186	−0.319466
$136$	45.5994	3.91012
$137$	5.11007	0.436583	0.218291	−	0.975884i	$-0.429952\pi$
$137$	5.11007	0.436583	0.218291	+	0.975884i	$0.429952\pi$
$138$	−13.5560	−1.15396
$139$	12.4468	1.05572	0.527860	−	0.849331i	$-0.322995\pi$
$139$	12.4468	1.05572	0.527860	+	0.849331i	$0.322995\pi$
$140$	3.76992	0.318617
$141$	13.3175	1.12154
$142$	29.7471	2.49632
$143$	−13.6290	−1.13971
$144$	29.5328	2.46106
$145$	−12.4105	−1.03064
$146$	10.6571	0.881990
$147$	17.6976	1.45967
$148$	−22.8716	−1.88003
$149$	−1.92002	−0.157294	−0.0786469	−	0.996903i	$-0.525060\pi$
$149$	−1.92002	−0.157294	−0.0786469	+	0.996903i	$0.525060\pi$
$150$	−1.42379	−0.116252
$151$	11.0283	0.897468	0.448734	−	0.893665i	$-0.351875\pi$
$151$	11.0283	0.897468	0.448734	+	0.893665i	$0.351875\pi$
$152$	−48.9914	−3.97373
$153$	24.5186	1.98221
$154$	−3.18792	−0.256890
$155$	13.2106	1.06110
$156$	−46.7759	−3.74507
$157$	−6.15407	−0.491148	−0.245574	−	0.969378i	$-0.578976\pi$
$157$	−6.15407	−0.491148	−0.245574	+	0.969378i	$0.578976\pi$
$158$	10.7244	0.853190
$159$	36.4088	2.88741
$160$	16.9568	1.34055
$161$	0.730257	0.0575523
$162$	−17.2642	−1.35641
$163$	6.77561	0.530706	0.265353	−	0.964151i	$-0.414511\pi$
$163$	6.77561	0.530706	0.265353	+	0.964151i	$0.414511\pi$
$164$	−8.26350	−0.645271
$165$	20.4017	1.58827
$166$	21.5761	1.67463
$167$	−18.1364	−1.40344	−0.701718	−	0.712455i	$-0.747583\pi$
$167$	−18.1364	−1.40344	−0.701718	+	0.712455i	$0.747583\pi$
$168$	−6.20884	−0.479022
$169$	2.43211	0.187086
$170$	39.6847	3.04368
$171$	−26.3425	−2.01446
$172$	−18.1757	−1.38589
$173$	22.1208	1.68181	0.840906	−	0.541182i	$-0.182023\pi$
$173$	22.1208	1.68181	0.840906	+	0.541182i	$0.182023\pi$
$174$	36.0185	2.73056
$175$	0.0766990	0.00579790
$176$	−28.2165	−2.12690
$177$	21.4586	1.61293
$178$	2.39550	0.179550
$179$	−23.5578	−1.76079	−0.880396	−	0.474239i	$-0.842723\pi$
$179$	−23.5578	−1.76079	−0.880396	+	0.474239i	$0.842723\pi$
$180$	38.3429	2.85791
$181$	18.3718	1.36556	0.682782	−	0.730623i	$-0.260770\pi$
$181$	18.3718	1.36556	0.682782	+	0.730623i	$0.260770\pi$
$182$	3.60970	0.267568
$183$	−25.3913	−1.87698
$184$	13.8131	1.01832
$185$	−11.2954	−0.830457
$186$	−38.3404	−2.81126
$187$	−23.4258	−1.71306
$188$	−23.9133	−1.74406
$189$	−0.580325	−0.0422124
$190$	−42.6368	−3.09320
$191$	22.4901	1.62732	0.813662	−	0.581338i	$-0.197470\pi$
$191$	22.4901	1.62732	0.813662	+	0.581338i	$0.197470\pi$
$192$	−7.32591	−0.528702
$193$	−20.9215	−1.50596	−0.752982	−	0.658041i	$-0.771385\pi$
$193$	−20.9215	−1.50596	−0.752982	+	0.658041i	$0.771385\pi$
$194$	−35.6651	−2.56060
$195$	−23.1009	−1.65429
$196$	−31.7783	−2.26988
$197$	14.8504	1.05805	0.529023	−	0.848608i	$-0.322559\pi$
$197$	14.8504	1.05805	0.529023	+	0.848608i	$0.322559\pi$
$198$	−32.4235	−2.30424
$199$	5.65028	0.400538	0.200269	−	0.979741i	$-0.435818\pi$
$199$	5.65028	0.400538	0.200269	+	0.979741i	$0.435818\pi$
$200$	1.45079	0.102586
$201$	10.8083	0.762357
$202$	17.7983	1.25228
$203$	−1.94030	−0.136183
$204$	−80.3997	−5.62910
$205$	−4.08104	−0.285032
$206$	−5.85308	−0.407804
$207$	7.42725	0.516230
$208$	31.9496	2.21531
$209$	25.1684	1.74093
$210$	−5.40349	−0.372876
$211$	−18.5688	−1.27833	−0.639163	−	0.769071i	$-0.720719\pi$
$211$	−18.5688	−1.27833	−0.639163	+	0.769071i	$0.720719\pi$
$212$	−65.3768	−4.49010
$213$	−29.7634	−2.03936
$214$	44.7470	3.05884
$215$	−8.97632	−0.612180
$216$	−10.9771	−0.746895
$217$	2.06539	0.140208
$218$	27.5082	1.86309
$219$	−10.6630	−0.720537
$220$	−36.6339	−2.46986
$221$	26.5251	1.78427
$222$	32.7822	2.20020
$223$	26.1851	1.75348	0.876741	−	0.480963i	$-0.159713\pi$
$223$	26.1851	1.75348	0.876741	+	0.480963i	$0.159713\pi$
$224$	2.65108	0.177133
$225$	0.780085	0.0520057
$226$	−22.2021	−1.47686
$227$	−10.2752	−0.681991	−0.340995	−	0.940065i	$-0.610764\pi$
$227$	−10.2752	−0.681991	−0.340995	+	0.940065i	$0.610764\pi$
$228$	86.3804	5.72068
$229$	−2.60508	−0.172149	−0.0860743	−	0.996289i	$-0.527432\pi$
$229$	−2.60508	−0.172149	−0.0860743	+	0.996289i	$0.527432\pi$
$230$	12.0214	0.792668
$231$	3.18967	0.209865
$232$	−36.7016	−2.40958
$233$	10.1213	0.663069	0.331535	−	0.943443i	$-0.392434\pi$
$233$	10.1213	0.663069	0.331535	+	0.943443i	$0.392434\pi$
$234$	36.7133	2.40002
$235$	−11.8099	−0.770393
$236$	−38.5317	−2.50820
$237$	−10.7303	−0.697009
$238$	6.20443	0.402174
$239$	0.429477	0.0277805	0.0138903	−	0.999904i	$-0.495578\pi$
$239$	0.429477	0.0277805	0.0138903	+	0.999904i	$0.495578\pi$
$240$	−47.8266	−3.08719
$241$	11.7589	0.757460	0.378730	−	0.925507i	$-0.376361\pi$
$241$	11.7589	0.757460	0.378730	+	0.925507i	$0.376361\pi$
$242$	2.66763	0.171482
$243$	22.1500	1.42092
$244$	45.5934	2.91882
$245$	−15.6941	−1.00266
$246$	11.8442	0.755159
$247$	−28.4983	−1.81330
$248$	39.0676	2.48079
$249$	−21.5880	−1.36808
$250$	−28.1239	−1.77871
$251$	−20.5272	−1.29566	−0.647831	−	0.761784i	$-0.724324\pi$
$251$	−20.5272	−1.29566	−0.647831	+	0.761784i	$0.724324\pi$
$252$	5.99466	0.377628
$253$	−7.09621	−0.446135
$254$	38.4477	2.41242
$255$	−39.7064	−2.48651
$256$	−25.0673	−1.56671
$257$	−8.29241	−0.517267	−0.258633	−	0.965976i	$-0.583272\pi$
$257$	−8.29241	−0.517267	−0.258633	+	0.965976i	$0.583272\pi$
$258$	26.0515	1.62190
$259$	−1.76597	−0.109732
$260$	41.4807	2.57252
$261$	−19.7343	−1.22152
$262$	15.9626	0.986175
$263$	−9.09964	−0.561108	−0.280554	−	0.959838i	$-0.590518\pi$
$263$	−9.09964	−0.561108	−0.280554	+	0.959838i	$0.590518\pi$
$264$	60.3339	3.71329
$265$	−32.2872	−1.98339
$266$	−6.66597	−0.408717
$267$	−2.39681	−0.146682
$268$	−19.4077	−1.18551
$269$	−16.7574	−1.02172	−0.510859	−	0.859665i	$-0.670672\pi$
$269$	−16.7574	−1.02172	−0.510859	+	0.859665i	$0.670672\pi$
$270$	−9.55323	−0.581391
$271$	3.42966	0.208337	0.104169	−	0.994560i	$-0.466782\pi$
$271$	3.42966	0.208337	0.104169	+	0.994560i	$0.466782\pi$
$272$	54.9158	3.32976
$273$	−3.61167	−0.218588
$274$	13.1518	0.794530
$275$	−0.745316	−0.0449442
$276$	−24.3549	−1.46599
$277$	27.9036	1.67656	0.838281	−	0.545238i	$-0.183561\pi$
$277$	27.9036	1.67656	0.838281	+	0.545238i	$0.183561\pi$
$278$	32.0343	1.92129
$279$	21.0065	1.25763
$280$	5.50597	0.329045
$281$	−0.416804	−0.0248644	−0.0124322	−	0.999923i	$-0.503957\pi$
$281$	−0.416804	−0.0248644	−0.0124322	+	0.999923i	$0.503957\pi$
$282$	34.2753	2.04107
$283$	−20.2605	−1.20436	−0.602180	−	0.798360i	$-0.705701\pi$
$283$	−20.2605	−1.20436	−0.602180	+	0.798360i	$0.705701\pi$
$284$	53.4441	3.17132
$285$	42.6601	2.52697
$286$	−35.0769	−2.07414
$287$	−0.638043	−0.0376625
$288$	26.9634	1.58884
$289$	28.5920	1.68188
$290$	−31.9411	−1.87564
$291$	35.6846	2.09187
$292$	19.1468	1.12048
$293$	−22.9900	−1.34309	−0.671544	−	0.740965i	$-0.734368\pi$
$293$	−22.9900	−1.34309	−0.671544	+	0.740965i	$0.734368\pi$
$294$	45.5483	2.65643
$295$	−19.0294	−1.10793
$296$	−33.4039	−1.94156
$297$	5.63925	0.327223
$298$	−4.94155	−0.286256
$299$	8.03506	0.464680
$300$	−2.55800	−0.147686
$301$	−1.40339	−0.0808899
$302$	28.3835	1.63329
$303$	−17.8081	−1.02305
$304$	−59.0008	−3.38393
$305$	22.5169	1.28931
$306$	63.1037	3.60740
$307$	0.973601	0.0555663	0.0277832	−	0.999614i	$-0.491155\pi$
$307$	0.973601	0.0555663	0.0277832	+	0.999614i	$0.491155\pi$
$308$	−5.72747	−0.326353
$309$	5.85629	0.333153
$310$	34.0001	1.93108
$311$	9.31814	0.528383	0.264192	−	0.964470i	$-0.414895\pi$
$311$	9.31814	0.528383	0.264192	+	0.964470i	$0.414895\pi$
$312$	−68.3162	−3.86764
$313$	7.32414	0.413985	0.206992	−	0.978343i	$-0.433632\pi$
$313$	7.32414	0.413985	0.206992	+	0.978343i	$0.433632\pi$
$314$	−15.8387	−0.893832
$315$	2.96054	0.166807
$316$	19.2677	1.08389
$317$	−24.9725	−1.40260	−0.701299	−	0.712867i	$-0.747396\pi$
$317$	−24.9725	−1.40260	−0.701299	+	0.712867i	$0.747396\pi$
$318$	93.7055	5.25475
$319$	18.8547	1.05566
$320$	6.49659	0.363170
$321$	−44.7715	−2.49890
$322$	1.87947	0.104739
$323$	−48.9835	−2.72552
$324$	−31.0172	−1.72318
$325$	0.843924	0.0468125
$326$	17.4384	0.965824
$327$	−27.5233	−1.52204
$328$	−12.0688	−0.666390
$329$	−1.84640	−0.101795
$330$	52.5080	2.89047
$331$	20.9861	1.15350	0.576750	−	0.816921i	$-0.304321\pi$
$331$	20.9861	1.15350	0.576750	+	0.816921i	$0.304321\pi$
$332$	38.7640	2.12745
$333$	−17.9612	−0.984266
$334$	−46.6777	−2.55409
$335$	−9.58474	−0.523670
$336$	−7.47736	−0.407923
$337$	−22.9966	−1.25270	−0.626352	−	0.779541i	$-0.715453\pi$
$337$	−22.9966	−1.25270	−0.626352	+	0.779541i	$0.715453\pi$
$338$	6.25954	0.340474
$339$	22.2143	1.20651
$340$	71.2981	3.86668
$341$	−20.0702	−1.08686
$342$	−67.7978	−3.66609
$343$	−4.95285	−0.267429
$344$	−26.5456	−1.43124
$345$	−12.0280	−0.647565
$346$	56.9323	3.06070
$347$	−28.4604	−1.52784	−0.763918	−	0.645314i	$-0.776727\pi$
$347$	−28.4604	−1.52784	−0.763918	+	0.645314i	$0.776727\pi$
$348$	64.7114	3.46889
$349$	15.7762	0.844483	0.422242	−	0.906483i	$-0.361243\pi$
$349$	15.7762	0.844483	0.422242	+	0.906483i	$0.361243\pi$
$350$	0.197401	0.0105515
$351$	−6.38535	−0.340825
$352$	−25.7617	−1.37310
$353$	14.0908	0.749976	0.374988	−	0.927030i	$-0.377647\pi$
$353$	14.0908	0.749976	0.374988	+	0.927030i	$0.377647\pi$
$354$	55.2281	2.93534
$355$	26.3941	1.40085
$356$	4.30379	0.228100
$357$	−6.20783	−0.328553
$358$	−60.6308	−3.20444
$359$	19.9187	1.05127	0.525635	−	0.850710i	$-0.323828\pi$
$359$	19.9187	1.05127	0.525635	+	0.850710i	$0.323828\pi$
$360$	55.9998	2.95145
$361$	33.6273	1.76986
$362$	47.2835	2.48517
$363$	−2.66909	−0.140091
$364$	6.48523	0.339918
$365$	9.45587	0.494943
$366$	−65.3497	−3.41588
$367$	−23.8968	−1.24740	−0.623701	−	0.781663i	$-0.714372\pi$
$367$	−23.8968	−1.24740	−0.623701	+	0.781663i	$0.714372\pi$
$368$	16.6352	0.867172
$369$	−6.48937	−0.337823
$370$	−29.0711	−1.51134
$371$	−5.04788	−0.262073
$372$	−68.8830	−3.57141
$373$	11.8632	0.614252	0.307126	−	0.951669i	$-0.400633\pi$
$373$	11.8632	0.614252	0.307126	+	0.951669i	$0.400633\pi$
$374$	−60.2911	−3.11758
$375$	28.1393	1.45311
$376$	−34.9254	−1.80114
$377$	−21.3493	−1.09954
$378$	−1.49358	−0.0768217
$379$	−0.367424	−0.0188733	−0.00943666	−	0.999955i	$-0.503004\pi$
$379$	−0.367424	−0.0188733	−0.00943666	+	0.999955i	$0.503004\pi$
$380$	−76.6018	−3.92959
$381$	−38.4687	−1.97081
$382$	57.8828	2.96154
$383$	8.56628	0.437717	0.218858	−	0.975757i	$-0.429767\pi$
$383$	8.56628	0.437717	0.218858	+	0.975757i	$0.429767\pi$
$384$	19.3881	0.989396
$385$	−2.82859	−0.144158
$386$	−53.8458	−2.74068
$387$	−14.2735	−0.725562
$388$	−64.0764	−3.25298
$389$	8.98279	0.455445	0.227723	−	0.973726i	$-0.426872\pi$
$389$	8.98279	0.455445	0.227723	+	0.973726i	$0.426872\pi$
$390$	−59.4549	−3.01062
$391$	13.8109	0.698445
$392$	−46.4121	−2.34417
$393$	−15.9714	−0.805650
$394$	38.2205	1.92552
$395$	9.51559	0.478781
$396$	−58.2526	−2.92730
$397$	17.3868	0.872619	0.436309	−	0.899797i	$-0.356285\pi$
$397$	17.3868	0.872619	0.436309	+	0.899797i	$0.356285\pi$
$398$	14.5422	0.728932
$399$	6.66962	0.333899
$400$	1.74720	0.0873601
$401$	0.584876	0.0292073	0.0146036	−	0.999893i	$-0.495351\pi$
$401$	0.584876	0.0292073	0.0146036	+	0.999893i	$0.495351\pi$
$402$	27.8173	1.38740
$403$	22.7256	1.13204
$404$	31.9767	1.59090
$405$	−15.3182	−0.761169
$406$	−4.99377	−0.247837
$407$	17.1606	0.850621
$408$	−117.424	−5.81334
$409$	39.3068	1.94360	0.971798	−	0.235814i	$-0.0757756\pi$
$409$	39.3068	1.94360	0.971798	+	0.235814i	$0.0757756\pi$
$410$	−10.5034	−0.518725
$411$	−13.1590	−0.649086
$412$	−10.5157	−0.518073
$413$	−2.97512	−0.146396
$414$	19.1155	0.939478
$415$	19.1441	0.939748
$416$	29.1700	1.43018
$417$	−32.0518	−1.56958
$418$	64.7760	3.16830
$419$	−32.5887	−1.59206	−0.796032	−	0.605254i	$-0.793071\pi$
$419$	−32.5887	−1.59206	−0.796032	+	0.605254i	$0.793071\pi$
$420$	−9.70799	−0.473701
$421$	21.1968	1.03307	0.516534	−	0.856267i	$-0.327222\pi$
$421$	21.1968	1.03307	0.516534	+	0.856267i	$0.327222\pi$
$422$	−47.7905	−2.32641
$423$	−18.7792	−0.913078
$424$	−95.4827	−4.63705
$425$	1.45056	0.0703623
$426$	−76.6022	−3.71139
$427$	3.52037	0.170362
$428$	80.3930	3.88594
$429$	35.0961	1.69446
$430$	−23.1024	−1.11410
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	−13.2198	−0.636037
$433$	20.5332	0.986762	0.493381	−	0.869813i	$-0.335761\pi$
$433$	20.5332	0.986762	0.493381	+	0.869813i	$0.335761\pi$
$434$	5.31569	0.255161
$435$	31.9586	1.53230
$436$	49.4217	2.36687
$437$	−14.8382	−0.709809
$438$	−27.4433	−1.31129
$439$	36.3028	1.73264	0.866319	−	0.499491i	$-0.166480\pi$
$439$	36.3028	1.73264	0.866319	+	0.499491i	$0.166480\pi$
$440$	−53.5038	−2.55069
$441$	−24.9556	−1.18836
$442$	68.2678	3.24717
$443$	−15.4351	−0.733344	−0.366672	−	0.930350i	$-0.619503\pi$
$443$	−15.4351	−0.733344	−0.366672	+	0.930350i	$0.619503\pi$
$444$	58.8970	2.79513
$445$	2.12548	0.100757
$446$	67.3926	3.19113
$447$	4.94426	0.233855
$448$	1.01570	0.0479872
$449$	−41.4986	−1.95844	−0.979221	−	0.202796i	$-0.934997\pi$
$449$	−41.4986	−1.95844	−0.979221	+	0.202796i	$0.934997\pi$
$450$	2.00771	0.0946443
$451$	6.20013	0.291953
$452$	−39.8886	−1.87620
$453$	−28.3990	−1.33430
$454$	−26.4454	−1.24114
$455$	3.20282	0.150150
$456$	126.159	5.90791
$457$	−7.90474	−0.369768	−0.184884	−	0.982760i	$-0.559191\pi$
$457$	−7.90474	−0.369768	−0.184884	+	0.982760i	$0.559191\pi$
$458$	−6.70471	−0.313290
$459$	−10.9753	−0.512283
$460$	21.5978	1.00700
$461$	−2.00122	−0.0932062	−0.0466031	−	0.998913i	$-0.514840\pi$
$461$	−2.00122	−0.0932062	−0.0466031	+	0.998913i	$0.514840\pi$
$462$	8.20927	0.381930
$463$	−18.9604	−0.881164	−0.440582	−	0.897712i	$-0.645228\pi$
$463$	−18.9604	−0.881164	−0.440582	+	0.897712i	$0.645228\pi$
$464$	−44.2001	−2.05194
$465$	−34.0188	−1.57758
$466$	26.0493	1.20671
$467$	6.42774	0.297440	0.148720	−	0.988879i	$-0.452485\pi$
$467$	6.42774	0.297440	0.148720	+	0.988879i	$0.452485\pi$
$468$	65.9596	3.04898
$469$	−1.49851	−0.0691947
$470$	−30.3952	−1.40203
$471$	15.8474	0.730211
$472$	−56.2755	−2.59029
$473$	13.6373	0.627044
$474$	−27.6166	−1.26847
$475$	−1.55846	−0.0715071
$476$	11.1470	0.510921
$477$	−51.3407	−2.35073
$478$	1.10535	0.0505573
$479$	−14.5170	−0.663300	−0.331650	−	0.943403i	$-0.607605\pi$
$479$	−14.5170	−0.663300	−0.331650	+	0.943403i	$0.607605\pi$
$480$	−43.6657	−1.99306
$481$	−19.4310	−0.885979
$482$	30.2640	1.37849
$483$	−1.88049	−0.0855655
$484$	4.79270	0.217850
$485$	−31.6450	−1.43692
$486$	57.0076	2.58592
$487$	29.9959	1.35925	0.679623	−	0.733561i	$-0.262143\pi$
$487$	29.9959	1.35925	0.679623	+	0.733561i	$0.262143\pi$
$488$	66.5891	3.01435
$489$	−17.4480	−0.789024
$490$	−40.3920	−1.82472
$491$	−31.7300	−1.43196	−0.715978	−	0.698123i	$-0.754019\pi$
$491$	−31.7300	−1.43196	−0.715978	+	0.698123i	$0.754019\pi$
$492$	21.2795	0.959352
$493$	−36.6957	−1.65269
$494$	−73.3461	−3.30000
$495$	−28.7688	−1.29306
$496$	47.0494	2.11258
$497$	4.12654	0.185100
$498$	−55.5610	−2.48975
$499$	20.3965	0.913072	0.456536	−	0.889705i	$-0.349090\pi$
$499$	20.3965	0.913072	0.456536	+	0.889705i	$0.349090\pi$
$500$	−50.5279	−2.25967
$501$	46.7032	2.08655
$502$	−52.8308	−2.35796
$503$	−13.7113	−0.611358	−0.305679	−	0.952135i	$-0.598883\pi$
$503$	−13.7113	−0.611358	−0.305679	+	0.952135i	$0.598883\pi$
$504$	8.75519	0.389987
$505$	15.7921	0.702740
$506$	−18.2635	−0.811914
$507$	−6.26297	−0.278148
$508$	69.0756	3.06474
$509$	19.8740	0.880901	0.440451	−	0.897777i	$-0.354819\pi$
$509$	19.8740	0.880901	0.440451	+	0.897777i	$0.354819\pi$
$510$	−102.193	−4.52516
$511$	1.47836	0.0653989
$512$	−49.4577	−2.18574
$513$	11.7917	0.520617
$514$	−21.3422	−0.941365
$515$	−5.19333	−0.228846
$516$	46.8046	2.06046
$517$	17.9422	0.789098
$518$	−4.54507	−0.199699
$519$	−56.9635	−2.50042
$520$	60.5825	2.65672
$521$	15.0333	0.658620	0.329310	−	0.944222i	$-0.393184\pi$
$521$	15.0333	0.658620	0.329310	+	0.944222i	$0.393184\pi$
$522$	−50.7903	−2.22303
$523$	−27.1598	−1.18761	−0.593807	−	0.804607i	$-0.702376\pi$
$523$	−27.1598	−1.18761	−0.593807	+	0.804607i	$0.702376\pi$
$524$	28.6787	1.25284
$525$	−0.197509	−0.00861998
$526$	−23.4198	−1.02115
$527$	39.0613	1.70154
$528$	72.6606	3.16215
$529$	−18.8164	−0.818103
$530$	−83.0977	−3.60953
$531$	−30.2591	−1.31313
$532$	−11.9762	−0.519233
$533$	−7.02043	−0.304088
$534$	−6.16868	−0.266945
$535$	39.7031	1.71652
$536$	−28.3449	−1.22431
$537$	60.6640	2.61785
$538$	−43.1286	−1.85941
$539$	23.8433	1.02700
$540$	−17.1635	−0.738598
$541$	4.52575	0.194577	0.0972886	−	0.995256i	$-0.468983\pi$
$541$	4.52575	0.194577	0.0972886	+	0.995256i	$0.468983\pi$
$542$	8.82694	0.379149
$543$	−47.3094	−2.03024
$544$	50.1381	2.14965
$545$	24.4075	1.04550
$546$	−9.29538	−0.397805
$547$	−24.6932	−1.05581	−0.527903	−	0.849304i	$-0.677022\pi$
$547$	−24.6932	−1.05581	−0.527903	+	0.849304i	$0.677022\pi$
$548$	23.6287	1.00937
$549$	35.8047	1.52811
$550$	−1.91822	−0.0817933
$551$	39.4254	1.67958
$552$	−35.5703	−1.51397
$553$	1.48770	0.0632634
$554$	71.8155	3.05115
$555$	29.0870	1.23468
$556$	57.5532	2.44080
$557$	28.0914	1.19027	0.595135	−	0.803626i	$-0.297099\pi$
$557$	28.0914	1.19027	0.595135	+	0.803626i	$0.297099\pi$
$558$	54.0645	2.28873
$559$	−15.4416	−0.653108
$560$	6.63089	0.280206
$561$	60.3241	2.54689
$562$	−1.07273	−0.0452504
$563$	21.6657	0.913100	0.456550	−	0.889698i	$-0.349085\pi$
$563$	21.6657	0.913100	0.456550	+	0.889698i	$0.349085\pi$
$564$	61.5795	2.59297
$565$	−19.6995	−0.828765
$566$	−52.1445	−2.19180
$567$	−2.39490	−0.100577
$568$	78.0550	3.27512
$569$	4.83832	0.202833	0.101416	−	0.994844i	$-0.467663\pi$
$569$	4.83832	0.202833	0.101416	+	0.994844i	$0.467663\pi$
$570$	109.795	4.59879
$571$	−38.9894	−1.63165	−0.815827	−	0.578296i	$-0.803718\pi$
$571$	−38.9894	−1.63165	−0.815827	+	0.578296i	$0.803718\pi$
$572$	−63.0197	−2.63498
$573$	−57.9145	−2.41941
$574$	−1.64213	−0.0685414
$575$	0.439407	0.0183245
$576$	10.3304	0.430433
$577$	19.7275	0.821269	0.410634	−	0.911800i	$-0.365307\pi$
$577$	19.7275	0.821269	0.410634	+	0.911800i	$0.365307\pi$
$578$	73.5874	3.06083
$579$	53.8753	2.23898
$580$	−57.3858	−2.38281
$581$	2.99305	0.124173
$582$	91.8416	3.80696
$583$	49.0524	2.03154
$584$	27.9638	1.15715
$585$	32.5750	1.34681
$586$	−59.1693	−2.44426
$587$	12.4387	0.513401	0.256701	−	0.966491i	$-0.417365\pi$
$587$	12.4387	0.513401	0.256701	+	0.966491i	$0.417365\pi$
$588$	81.8327	3.37472
$589$	−41.9670	−1.72922
$590$	−48.9760	−2.01631
$591$	−38.2414	−1.57304
$592$	−40.2287	−1.65339
$593$	−36.7614	−1.50961	−0.754806	−	0.655948i	$-0.772269\pi$
$593$	−36.7614	−1.50961	−0.754806	+	0.655948i	$0.772269\pi$
$594$	14.5138	0.595507
$595$	5.50508	0.225686
$596$	−8.87806	−0.363660
$597$	−14.5501	−0.595497
$598$	20.6799	0.845663
$599$	−11.1268	−0.454630	−0.227315	−	0.973821i	$-0.572995\pi$
$599$	−11.1268	−0.454630	−0.227315	+	0.973821i	$0.572995\pi$
$600$	−3.73595	−0.152520
$601$	3.94777	0.161033	0.0805165	−	0.996753i	$-0.474343\pi$
$601$	3.94777	0.161033	0.0805165	+	0.996753i	$0.474343\pi$
$602$	−3.61190	−0.147210
$603$	−15.2409	−0.620659
$604$	50.9942	2.07492
$605$	2.36694	0.0962297
$606$	−45.8327	−1.86183
$607$	36.7922	1.49335	0.746674	−	0.665190i	$-0.231649\pi$
$607$	36.7922	1.49335	0.746674	+	0.665190i	$0.231649\pi$
$608$	−53.8678	−2.18463
$609$	4.99651	0.202469
$610$	57.9518	2.34640
$611$	−20.3160	−0.821899
$612$	113.373	4.58283
$613$	15.3619	0.620462	0.310231	−	0.950661i	$-0.399594\pi$
$613$	15.3619	0.620462	0.310231	+	0.950661i	$0.399594\pi$
$614$	2.50576	0.101124
$615$	10.5091	0.423770
$616$	−8.36496	−0.337034
$617$	27.4366	1.10455	0.552277	−	0.833661i	$-0.313759\pi$
$617$	27.4366	1.10455	0.552277	+	0.833661i	$0.313759\pi$
$618$	15.0724	0.606299
$619$	−17.2553	−0.693549	−0.346775	−	0.937948i	$-0.612723\pi$
$619$	−17.2553	−0.693549	−0.346775	+	0.937948i	$0.612723\pi$
$620$	61.0851	2.45324
$621$	−3.32467	−0.133414
$622$	23.9821	0.961596
$623$	0.332305	0.0133135
$624$	−82.2738	−3.29359
$625$	−26.0280	−1.04112
$626$	18.8502	0.753404
$627$	−64.8115	−2.58832
$628$	−28.4561	−1.13552
$629$	−33.3985	−1.33169
$630$	7.61955	0.303570
$631$	17.3059	0.688935	0.344468	−	0.938798i	$-0.388059\pi$
$631$	17.3059	0.688935	0.344468	+	0.938798i	$0.388059\pi$
$632$	28.1404	1.11937
$633$	47.8167	1.90054
$634$	−64.2719	−2.55256
$635$	34.1139	1.35377
$636$	168.353	6.67562
$637$	−26.9979	−1.06969
$638$	48.5265	1.92118
$639$	41.9699	1.66030
$640$	−17.1933	−0.679625
$641$	−28.1003	−1.10989	−0.554947	−	0.831885i	$-0.687262\pi$
$641$	−28.1003	−1.10989	−0.554947	+	0.831885i	$0.687262\pi$
$642$	−115.229	−4.54771
$643$	30.7867	1.21411	0.607054	−	0.794661i	$-0.292351\pi$
$643$	30.7867	1.21411	0.607054	+	0.794661i	$0.292351\pi$
$644$	3.37667	0.133060
$645$	23.1150	0.910154
$646$	−126.069	−4.96012
$647$	18.4628	0.725847	0.362924	−	0.931819i	$-0.381779\pi$
$647$	18.4628	0.725847	0.362924	+	0.931819i	$0.381779\pi$
$648$	−45.3006	−1.77957
$649$	28.9105	1.13483
$650$	2.17201	0.0851932
$651$	−5.31860	−0.208453
$652$	31.3301	1.22698
$653$	20.0886	0.786129	0.393064	−	0.919511i	$-0.371415\pi$
$653$	20.0886	0.786129	0.393064	+	0.919511i	$0.371415\pi$
$654$	−70.8368	−2.76994
$655$	14.1634	0.553408
$656$	−14.5346	−0.567481
$657$	15.0360	0.586612
$658$	−4.75209	−0.185256
$659$	6.54611	0.255000	0.127500	−	0.991839i	$-0.459305\pi$
$659$	6.54611	0.255000	0.127500	+	0.991839i	$0.459305\pi$
$660$	94.3365	3.67204
$661$	−3.02612	−0.117702	−0.0588511	−	0.998267i	$-0.518744\pi$
$661$	−3.02612	−0.117702	−0.0588511	+	0.998267i	$0.518744\pi$
$662$	54.0120	2.09923
$663$	−68.3052	−2.65275
$664$	56.6148	2.19708
$665$	−5.91459	−0.229358
$666$	−46.2267	−1.79125
$667$	−11.1160	−0.430412
$668$	−83.8618	−3.24471
$669$	−67.4295	−2.60698
$670$	−24.6683	−0.953019
$671$	−34.2089	−1.32062
$672$	−6.82684	−0.263351
$673$	−15.0061	−0.578443	−0.289221	−	0.957262i	$-0.593396\pi$
$673$	−15.0061	−0.578443	−0.289221	+	0.957262i	$0.593396\pi$
$674$	−59.1864	−2.27977
$675$	−0.349190	−0.0134403
$676$	11.2460	0.432538
$677$	−24.4966	−0.941479	−0.470739	−	0.882272i	$-0.656013\pi$
$677$	−24.4966	−0.941479	−0.470739	+	0.882272i	$0.656013\pi$
$678$	57.1730	2.19571
$679$	−4.94748	−0.189867
$680$	104.131	3.99323
$681$	26.4599	1.01394
$682$	−51.6548	−1.97796
$683$	−34.9705	−1.33811	−0.669054	−	0.743214i	$-0.733301\pi$
$683$	−34.9705	−1.33811	−0.669054	+	0.743214i	$0.733301\pi$
$684$	−121.807	−4.65739
$685$	11.6694	0.445863
$686$	−12.7472	−0.486689
$687$	6.70838	0.255941
$688$	−31.9691	−1.21881
$689$	−55.5422	−2.11599
$690$	−30.9565	−1.17849
$691$	−17.5598	−0.668008	−0.334004	−	0.942572i	$-0.608400\pi$
$691$	−17.5598	−0.668008	−0.334004	+	0.942572i	$0.608400\pi$
$692$	102.285	3.88831
$693$	−4.49781	−0.170858
$694$	−73.2487	−2.78048
$695$	28.4234	1.07816
$696$	94.5109	3.58243
$697$	−12.0669	−0.457066
$698$	40.6034	1.53686
$699$	−26.0635	−0.985813
$700$	0.354653	0.0134046
$701$	4.20672	0.158886	0.0794428	−	0.996839i	$-0.474686\pi$
$701$	4.20672	0.158886	0.0794428	+	0.996839i	$0.474686\pi$
$702$	−16.4340	−0.620261
$703$	35.8830	1.35335
$704$	−9.86996	−0.371988
$705$	30.4118	1.14538
$706$	36.2655	1.36487
$707$	2.46899	0.0928559
$708$	99.2236	3.72905
$709$	13.4162	0.503856	0.251928	−	0.967746i	$-0.418935\pi$
$709$	13.4162	0.503856	0.251928	+	0.967746i	$0.418935\pi$
$710$	67.9305	2.54939
$711$	15.1310	0.567457
$712$	6.28567	0.235566
$713$	11.8326	0.443133
$714$	−15.9771	−0.597929
$715$	−31.1231	−1.16394
$716$	−108.930	−4.07091
$717$	−1.10595	−0.0413025
$718$	51.2649	1.91319
$719$	21.7997	0.812992	0.406496	−	0.913653i	$-0.366751\pi$
$719$	21.7997	0.812992	0.406496	+	0.913653i	$0.366751\pi$
$720$	67.4410	2.51338
$721$	−0.811942	−0.0302383
$722$	86.5467	3.22094
$723$	−30.2806	−1.12615
$724$	84.9502	3.15715
$725$	−1.16751	−0.0433603
$726$	−6.86945	−0.254949
$727$	0.197006	0.00730656	0.00365328	−	0.999993i	$-0.498837\pi$
$727$	0.197006	0.00730656	0.00365328	+	0.999993i	$0.498837\pi$
$728$	9.47167	0.351043
$729$	−36.9150	−1.36722
$730$	24.3366	0.900739
$731$	−26.5413	−0.981667
$732$	−117.408	−4.33953
$733$	17.8257	0.658406	0.329203	−	0.944259i	$-0.393220\pi$
$733$	17.8257	0.658406	0.329203	+	0.944259i	$0.393220\pi$
$734$	−61.5033	−2.27013
$735$	40.4141	1.49070
$736$	15.1880	0.559837
$737$	14.5616	0.536385
$738$	−16.7017	−0.614799
$739$	25.8894	0.952358	0.476179	−	0.879348i	$-0.342021\pi$
$739$	25.8894	0.952358	0.476179	+	0.879348i	$0.342021\pi$
$740$	−52.2296	−1.92000
$741$	73.3863	2.69591
$742$	−12.9918	−0.476943
$743$	−31.7529	−1.16490	−0.582451	−	0.812866i	$-0.697906\pi$
$743$	−31.7529	−1.16490	−0.582451	+	0.812866i	$0.697906\pi$
$744$	−100.603	−3.68830
$745$	−4.38455	−0.160637
$746$	30.5323	1.11787
$747$	30.4416	1.11380
$748$	−108.320	−3.96056
$749$	6.20732	0.226811
$750$	72.4224	2.64449
$751$	14.5689	0.531627	0.265813	−	0.964025i	$-0.414360\pi$
$751$	14.5689	0.531627	0.265813	+	0.964025i	$0.414360\pi$
$752$	−42.0609	−1.53380
$753$	52.8598	1.92632
$754$	−54.9468	−2.00104
$755$	25.1842	0.916545
$756$	−2.68339	−0.0975941
$757$	−20.7347	−0.753615	−0.376807	−	0.926292i	$-0.622978\pi$
$757$	−20.7347	−0.753615	−0.376807	+	0.926292i	$0.622978\pi$
$758$	−0.945641	−0.0343472
$759$	18.2735	0.663288
$760$	−111.877	−4.05820
$761$	−15.2610	−0.553211	−0.276605	−	0.960984i	$-0.589209\pi$
$761$	−15.2610	−0.553211	−0.276605	+	0.960984i	$0.589209\pi$
$762$	−99.0072	−3.58665
$763$	3.81595	0.138147
$764$	103.993	3.76234
$765$	55.9907	2.02435
$766$	22.0471	0.796593
$767$	−32.7354	−1.18201
$768$	64.5511	2.32929
$769$	−28.0157	−1.01027	−0.505136	−	0.863040i	$-0.668558\pi$
$769$	−28.0157	−1.01027	−0.505136	+	0.863040i	$0.668558\pi$
$770$	−7.27994	−0.262351
$771$	21.3539	0.769042
$772$	−96.7401	−3.48175
$773$	−48.4011	−1.74087	−0.870433	−	0.492286i	$-0.836161\pi$
$773$	−48.4011	−1.74087	−0.870433	+	0.492286i	$0.836161\pi$
$774$	−36.7357	−1.32044
$775$	1.24277	0.0446418
$776$	−93.5835	−3.35945
$777$	4.54756	0.163143
$778$	23.1190	0.828858
$779$	12.9645	0.464502
$780$	−106.818	−3.82468
$781$	−40.0993	−1.43486
$782$	35.5451	1.27109
$783$	8.83370	0.315691
$784$	−55.8945	−1.99623
$785$	−14.0534	−0.501588
$786$	−41.1056	−1.46619
$787$	−0.785954	−0.0280162	−0.0140081	−	0.999902i	$-0.504459\pi$
$787$	−0.785954	−0.0280162	−0.0140081	+	0.999902i	$0.504459\pi$
$788$	68.6674	2.44618
$789$	23.4326	0.834223
$790$	24.4903	0.871327
$791$	−3.07989	−0.109508
$792$	−85.0778	−3.02311
$793$	38.7348	1.37551
$794$	44.7485	1.58806
$795$	83.1432	2.94878
$796$	26.1266	0.926034
$797$	20.5907	0.729360	0.364680	−	0.931133i	$-0.381178\pi$
$797$	20.5907	0.729360	0.364680	+	0.931133i	$0.381178\pi$
$798$	17.1656	0.607657
$799$	−34.9197	−1.23537
$800$	1.59520	0.0563987
$801$	3.37978	0.119419
$802$	1.50530	0.0531539
$803$	−14.3659	−0.506960
$804$	49.9770	1.76255
$805$	1.66761	0.0587757
$806$	58.4889	2.06018
$807$	43.1522	1.51903
$808$	46.7019	1.64297
$809$	−51.1388	−1.79795	−0.898973	−	0.438005i	$-0.855685\pi$
$809$	−51.1388	−1.79795	−0.898973	+	0.438005i	$0.855685\pi$
$810$	−39.4246	−1.38524
$811$	24.9576	0.876380	0.438190	−	0.898882i	$-0.355620\pi$
$811$	24.9576	0.876380	0.438190	+	0.898882i	$0.355620\pi$
$812$	−8.97188	−0.314851
$813$	−8.83177	−0.309744
$814$	44.1664	1.54803
$815$	15.4728	0.541988
$816$	−141.414	−4.95049
$817$	28.5157	0.997638
$818$	101.164	3.53712
$819$	5.09288	0.177960
$820$	−18.8705	−0.658988
$821$	42.3879	1.47935	0.739673	−	0.672966i	$-0.234980\pi$
$821$	42.3879	1.47935	0.739673	+	0.672966i	$0.234980\pi$
$822$	−33.8674	−1.18126
$823$	28.6648	0.999192	0.499596	−	0.866259i	$-0.333482\pi$
$823$	28.6648	0.999192	0.499596	+	0.866259i	$0.333482\pi$
$824$	−15.3582	−0.535029
$825$	1.91927	0.0668205
$826$	−7.65707	−0.266424
$827$	28.1811	0.979952	0.489976	−	0.871736i	$-0.337006\pi$
$827$	28.1811	0.979952	0.489976	+	0.871736i	$0.337006\pi$
$828$	34.3433	1.19351
$829$	−0.632280	−0.0219600	−0.0109800	−	0.999940i	$-0.503495\pi$
$829$	−0.632280	−0.0219600	−0.0109800	+	0.999940i	$0.503495\pi$
$830$	49.2713	1.71023
$831$	−71.8548	−2.49262
$832$	11.1758	0.387450
$833$	−46.4046	−1.60782
$834$	−82.4919	−2.85646
$835$	−41.4162	−1.43327
$836$	116.377	4.02500
$837$	−9.40316	−0.325021
$838$	−83.8738	−2.89737
$839$	20.8027	0.718189	0.359095	−	0.933301i	$-0.383086\pi$
$839$	20.8027	0.718189	0.359095	+	0.933301i	$0.383086\pi$
$840$	−14.1785	−0.489205
$841$	0.535307	0.0184589
$842$	54.5542	1.88006
$843$	1.07332	0.0369670
$844$	−85.8611	−2.95546
$845$	5.55397	0.191063
$846$	−48.3322	−1.66170
$847$	0.370055	0.0127152
$848$	−114.991	−3.94880
$849$	52.1731	1.79057
$850$	3.73330	0.128051
$851$	−10.1172	−0.346813
$852$	−137.625	−4.71494
$853$	−28.6059	−0.979447	−0.489724	−	0.871878i	$-0.662902\pi$
$853$	−28.6059	−0.979447	−0.489724	+	0.871878i	$0.662902\pi$
$854$	9.06038	0.310040
$855$	−60.1557	−2.05728
$856$	117.414	4.01313
$857$	6.92375	0.236511	0.118255	−	0.992983i	$-0.462270\pi$
$857$	6.92375	0.236511	0.118255	+	0.992983i	$0.462270\pi$
$858$	90.3270	3.08371
$859$	27.6913	0.944816	0.472408	−	0.881380i	$-0.343385\pi$
$859$	27.6913	0.944816	0.472408	+	0.881380i	$0.343385\pi$
$860$	−41.5061	−1.41535
$861$	1.64303	0.0559945
$862$	2.57370	0.0876607
$863$	−28.3542	−0.965189	−0.482595	−	0.875844i	$-0.660306\pi$
$863$	−28.3542	−0.965189	−0.482595	+	0.875844i	$0.660306\pi$
$864$	−12.0697	−0.410619
$865$	50.5150	1.71756
$866$	52.8463	1.79579
$867$	−73.6277	−2.50053
$868$	9.55025	0.324156
$869$	−14.4566	−0.490406
$870$	82.2519	2.78860
$871$	−16.4882	−0.558681
$872$	72.1803	2.44433
$873$	−50.3195	−1.70306
$874$	−38.1892	−1.29177
$875$	−3.90137	−0.131890
$876$	−49.3051	−1.66586
$877$	31.3956	1.06016	0.530078	−	0.847949i	$-0.322163\pi$
$877$	31.3956	1.06016	0.530078	+	0.847949i	$0.322163\pi$
$878$	93.4327	3.15320
$879$	59.2017	1.99683
$880$	−64.4351	−2.17211
$881$	46.6917	1.57308	0.786542	−	0.617537i	$-0.211869\pi$
$881$	46.6917	1.57308	0.786542	+	0.617537i	$0.211869\pi$
$882$	−64.2284	−2.16268
$883$	45.3947	1.52765	0.763827	−	0.645421i	$-0.223318\pi$
$883$	45.3947	1.52765	0.763827	+	0.645421i	$0.223318\pi$
$884$	122.651	4.12519
$885$	49.0029	1.64721
$886$	−39.7254	−1.33460
$887$	21.0448	0.706614	0.353307	−	0.935507i	$-0.385057\pi$
$887$	21.0448	0.706614	0.353307	+	0.935507i	$0.385057\pi$
$888$	86.0190	2.88661
$889$	5.33348	0.178879
$890$	5.47036	0.183367
$891$	23.2723	0.779651
$892$	121.079	4.05401
$893$	37.5173	1.25547
$894$	12.7251	0.425590
$895$	−53.7966	−1.79822
$896$	−2.68806	−0.0898017
$897$	−20.6912	−0.690859
$898$	−106.805	−3.56414
$899$	−31.4393	−1.04856
$900$	3.60708	0.120236
$901$	−95.4673	−3.18048
$902$	15.9573	0.531320
$903$	3.61388	0.120262
$904$	−58.2573	−1.93761
$905$	41.9538	1.39459
$906$	−73.0908	−2.42828
$907$	−9.13393	−0.303287	−0.151644	−	0.988435i	$-0.548457\pi$
$907$	−9.13393	−0.303287	−0.151644	+	0.988435i	$0.548457\pi$
$908$	−47.5122	−1.57675
$909$	25.1114	0.832894
$910$	8.24310	0.273256
$911$	−56.0465	−1.85690	−0.928451	−	0.371454i	$-0.878859\pi$
$911$	−56.0465	−1.85690	−0.928451	+	0.371454i	$0.878859\pi$
$912$	151.934	5.03103
$913$	−29.0848	−0.962565
$914$	−20.3445	−0.672935
$915$	−57.9836	−1.91688
$916$	−12.0458	−0.398003
$917$	2.21435	0.0731241
$918$	−28.2472	−0.932295
$919$	8.36459	0.275923	0.137961	−	0.990438i	$-0.455945\pi$
$919$	8.36459	0.275923	0.137961	+	0.990438i	$0.455945\pi$
$920$	31.5436	1.03996
$921$	−2.50713	−0.0826128
$922$	−5.15055	−0.169624
$923$	45.4045	1.49451
$924$	14.7489	0.485203
$925$	−1.06261	−0.0349384
$926$	−48.7984	−1.60362
$927$	−8.25805	−0.271230
$928$	−40.3547	−1.32471
$929$	−44.1199	−1.44753	−0.723764	−	0.690048i	$-0.757589\pi$
$929$	−44.1199	−1.44753	−0.723764	+	0.690048i	$0.757589\pi$
$930$	−87.5542	−2.87102
$931$	49.8566	1.63398
$932$	46.8005	1.53300
$933$	−23.9953	−0.785570
$934$	16.5431	0.541307
$935$	−53.4951	−1.74948
$936$	96.3339	3.14877
$937$	−6.03479	−0.197148	−0.0985739	−	0.995130i	$-0.531428\pi$
$937$	−6.03479	−0.197148	−0.0985739	+	0.995130i	$0.531428\pi$
$938$	−3.85672	−0.125926
$939$	−18.8605	−0.615489
$940$	−54.6084	−1.78113
$941$	−16.1755	−0.527305	−0.263653	−	0.964618i	$-0.584927\pi$
$941$	−16.1755	−0.527305	−0.263653	+	0.964618i	$0.584927\pi$
$942$	40.7866	1.32890
$943$	−3.65534	−0.119034
$944$	−67.7731	−2.20583
$945$	−1.32523	−0.0431097
$946$	35.0984	1.14115
$947$	9.49983	0.308703	0.154351	−	0.988016i	$-0.450671\pi$
$947$	9.49983	0.308703	0.154351	+	0.988016i	$0.450671\pi$
$948$	−49.6165	−1.61147
$949$	16.2665	0.528033
$950$	−4.01102	−0.130135
$951$	64.3071	2.08530
$952$	16.2801	0.527643
$953$	7.46016	0.241658	0.120829	−	0.992673i	$-0.461445\pi$
$953$	7.46016	0.241658	0.120829	+	0.992673i	$0.461445\pi$
$954$	−132.136	−4.27805
$955$	51.3583	1.66192
$956$	1.98588	0.0642279
$957$	−48.5531	−1.56950
$958$	−37.3625	−1.20713
$959$	1.82443	0.0589138
$960$	−16.7294	−0.539941
$961$	2.46600	0.0795484
$962$	−50.0097	−1.61238
$963$	63.1330	2.03443
$964$	54.3728	1.75123
$965$	−47.7764	−1.53798
$966$	−4.83984	−0.155719
$967$	−21.3283	−0.685872	−0.342936	−	0.939359i	$-0.611421\pi$
$967$	−21.3283	−0.685872	−0.342936	+	0.939359i	$0.611421\pi$
$968$	6.99974	0.224980
$969$	126.138	4.05214
$970$	−81.4447	−2.61503
$971$	−24.7464	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$971$	−24.7464	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$972$	102.421	3.28514
$973$	4.44381	0.142462
$974$	77.2007	2.47367
$975$	−2.17320	−0.0695981
$976$	80.1939	2.56694
$977$	−7.20494	−0.230507	−0.115253	−	0.993336i	$-0.536768\pi$
$977$	−7.20494	−0.230507	−0.115253	+	0.993336i	$0.536768\pi$
$978$	−44.9059	−1.43593
$979$	−3.22914	−0.103204
$980$	−72.5688	−2.31813
$981$	38.8111	1.23914
$982$	−81.6637	−2.60599
$983$	−4.58079	−0.146104	−0.0730522	−	0.997328i	$-0.523274\pi$
$983$	−4.58079	−0.146104	−0.0730522	+	0.997328i	$0.523274\pi$
$984$	31.0786	0.990751
$985$	33.9123	1.08054
$986$	−94.4439	−3.00771
$987$	4.75469	0.151343
$988$	−131.775	−4.19231
$989$	−8.03998	−0.255656
$990$	−74.0424	−2.35322
$991$	11.3402	0.360233	0.180116	−	0.983645i	$-0.442353\pi$
$991$	11.3402	0.360233	0.180116	+	0.983645i	$0.442353\pi$
$992$	42.9562	1.36386
$993$	−54.0416	−1.71496
$994$	10.6205	0.336861
$995$	12.9030	0.409052
$996$	−99.8218	−3.16297
$997$	−8.45209	−0.267680	−0.133840	−	0.991003i	$-0.542731\pi$
$997$	−8.45209	−0.267680	−0.133840	+	0.991003i	$0.542731\pi$
$998$	52.4945	1.66168
$999$	8.03998	0.254374

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	431.2.a.f.1.23	✓	24
3.2	odd	2		3879.2.a.r.1.2		24
4.3	odd	2		6896.2.a.w.1.21		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.2.a.f.1.23	✓	24	1.1	even	1	trivial
3879.2.a.r.1.2		24	3.2	odd	2
6896.2.a.w.1.21		24	4.3	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 $\F_{q}$
Belyi maps

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Varieties

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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	431.2.a.f.1.23

Level	$431$
Weight	$2$
Character	431.1
Self dual	yes
Analytic conductor	$3.442$
Analytic rank	$0$
Dimension	$24$
CM	no
Inner twists	$1$