LMFDB - Embedded newform 845.2.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$845 = 5 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1$ x^9 - 17x^7 - 9x^6 + 59x^5 + 32x^4 - 44x^3 - 23x^2 + x + 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			1.5
Root			$-0.421015$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0240266 q^{2} +2.93017 q^{3} -1.99942 q^{4} -1.00000 q^{5} -0.0704021 q^{6} +1.66541 q^{7} +0.0960927 q^{8} +5.58589 q^{9} +0.0240266 q^{10} -3.33397 q^{11} -5.85865 q^{12} -0.0400143 q^{14} -2.93017 q^{15} +3.99654 q^{16} +7.07621 q^{17} -0.134210 q^{18} +6.68286 q^{19} +1.99942 q^{20} +4.87994 q^{21} +0.0801041 q^{22} +4.02860 q^{23} +0.281568 q^{24} +1.00000 q^{25} +7.57710 q^{27} -3.32987 q^{28} -0.000401110 q^{29} +0.0704021 q^{30} -4.14759 q^{31} -0.288209 q^{32} -9.76910 q^{33} -0.170018 q^{34} -1.66541 q^{35} -11.1686 q^{36} +8.96459 q^{37} -0.160567 q^{38} -0.0960927 q^{40} -9.60848 q^{41} -0.117249 q^{42} +2.78606 q^{43} +6.66601 q^{44} -5.58589 q^{45} -0.0967937 q^{46} -0.958374 q^{47} +11.7105 q^{48} -4.22640 q^{49} -0.0240266 q^{50} +20.7345 q^{51} -3.68268 q^{53} -0.182052 q^{54} +3.33397 q^{55} +0.160034 q^{56} +19.5819 q^{57} +9.63732e-6 q^{58} -7.99148 q^{59} +5.85865 q^{60} -9.14117 q^{61} +0.0996526 q^{62} +9.30282 q^{63} -7.98615 q^{64} +0.234719 q^{66} +6.72871 q^{67} -14.1483 q^{68} +11.8045 q^{69} +0.0400143 q^{70} +4.38321 q^{71} +0.536763 q^{72} +1.97245 q^{73} -0.215389 q^{74} +2.93017 q^{75} -13.3619 q^{76} -5.55244 q^{77} -3.77988 q^{79} -3.99654 q^{80} +5.44451 q^{81} +0.230860 q^{82} -3.64818 q^{83} -9.75707 q^{84} -7.07621 q^{85} -0.0669397 q^{86} -0.00117532 q^{87} -0.320370 q^{88} +0.989573 q^{89} +0.134210 q^{90} -8.05487 q^{92} -12.1531 q^{93} +0.0230265 q^{94} -6.68286 q^{95} -0.844500 q^{96} +18.4460 q^{97} +0.101546 q^{98} -18.6232 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9} + 3 q^{10} + 9 q^{11} + 12 q^{12} - 2 q^{14} - 7 q^{15} + 37 q^{16} - q^{17} + 10 q^{18} + 4 q^{19} - 17 q^{20} + q^{21}+ \cdots + q^{99}+O(q^{100})$ 9 * q - 3 * q^2 + 7 * q^3 + 17 * q^4 - 9 * q^5 + 2 * q^6 - 7 * q^7 - 12 * q^8 + 16 * q^9 + 3 * q^10 + 9 * q^11 + 12 * q^12 - 2 * q^14 - 7 * q^15 + 37 * q^16 - q^17 + 10 * q^18 + 4 * q^19 - 17 * q^20 + q^21 + 12 * q^22 + 14 * q^23 + 35 * q^24 + 9 * q^25 + 22 * q^27 - 18 * q^28 + 12 * q^29 - 2 * q^30 + 7 * q^31 - 22 * q^32 + 8 * q^33 + 30 * q^34 + 7 * q^35 + 3 * q^36 + 5 * q^37 - 47 * q^38 + 12 * q^40 + 10 * q^41 - 11 * q^42 + 39 * q^43 + 25 * q^44 - 16 * q^45 - 6 * q^46 - 36 * q^47 - 3 * q^48 + 16 * q^49 - 3 * q^50 + 43 * q^51 - 8 * q^53 + 2 * q^54 - 9 * q^55 - 29 * q^56 + 32 * q^57 - 21 * q^58 + 21 * q^59 - 12 * q^60 - 3 * q^61 - 10 * q^62 - 35 * q^63 + 34 * q^64 - 49 * q^66 - q^67 - 20 * q^68 - 13 * q^69 + 2 * q^70 + q^71 + 3 * q^72 - 15 * q^74 + 7 * q^75 + 5 * q^76 - 4 * q^77 + 39 * q^79 - 37 * q^80 + 29 * q^81 - 4 * q^82 - 7 * q^83 - 12 * q^84 + q^85 + 24 * q^86 + 16 * q^87 + 42 * q^88 + 19 * q^89 - 10 * q^90 - 27 * q^92 - 31 * q^93 + 16 * q^94 - 4 * q^95 + 7 * q^96 + 34 * q^97 - 48 * q^98 + 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

)

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

$2$	−0.0240266	−0.0169894	−0.00849470	−	0.999964i	$-0.502704\pi$
$2$	−0.0240266	−0.0169894	−0.00849470	+	0.999964i	$0.502704\pi$
$3$	2.93017	1.69173	0.845867	−	0.533394i	$-0.179084\pi$
$3$	2.93017	1.69173	0.845867	+	0.533394i	$0.179084\pi$
$4$	−1.99942	−0.999711
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.0704021	−0.0287415
$7$	1.66541	0.629467	0.314734	−	0.949180i	$-0.398085\pi$
$7$	1.66541	0.629467	0.314734	+	0.949180i	$0.398085\pi$
$8$	0.0960927	0.0339739
$9$	5.58589	1.86196
$10$	0.0240266	0.00759789
$11$	−3.33397	−1.00523	−0.502615	−	0.864510i	$-0.667629\pi$
$11$	−3.33397	−1.00523	−0.502615	+	0.864510i	$0.667629\pi$
$12$	−5.85865	−1.69125
$13$	0	0
$13$	0	0
$14$	−0.0400143	−0.0106943
$15$	−2.93017	−0.756566
$16$	3.99654	0.999134
$17$	7.07621	1.71623	0.858116	−	0.513455i	$-0.171635\pi$
$17$	7.07621	1.71623	0.858116	+	0.513455i	$0.171635\pi$
$18$	−0.134210	−0.0316337
$19$	6.68286	1.53315	0.766577	−	0.642153i	$-0.221958\pi$
$19$	6.68286	1.53315	0.766577	+	0.642153i	$0.221958\pi$
$20$	1.99942	0.447085
$21$	4.87994	1.06489
$22$	0.0801041	0.0170783
$23$	4.02860	0.840021	0.420011	−	0.907519i	$-0.362026\pi$
$23$	4.02860	0.840021	0.420011	+	0.907519i	$0.362026\pi$
$24$	0.281568	0.0574748
$25$	1.00000	0.200000
$26$	0	0
$27$	7.57710	1.45821
$28$	−3.32987	−0.629286
$29$	−0.000401110 0	−7.44842e−5 0	−3.72421e−5	−	1.00000i	$-0.500012\pi$
$29$	−0.000401110 0	−7.44842e−5 0	−3.72421e−5		1.00000i	$0.500012\pi$
$30$	0.0704021	0.0128536
$31$	−4.14759	−0.744929	−0.372464	−	0.928046i	$-0.621487\pi$
$31$	−4.14759	−0.744929	−0.372464	+	0.928046i	$0.621487\pi$
$32$	−0.288209	−0.0509486
$33$	−9.76910	−1.70058
$34$	−0.170018	−0.0291578
$35$	−1.66541	−0.281506
$36$	−11.1686	−1.86143
$37$	8.96459	1.47377	0.736885	−	0.676018i	$-0.236296\pi$
$37$	8.96459	1.47377	0.736885	+	0.676018i	$0.236296\pi$
$38$	−0.160567	−0.0260474
$39$	0	0
$40$	−0.0960927	−0.0151936
$41$	−9.60848	−1.50059	−0.750296	−	0.661102i	$-0.770089\pi$
$41$	−9.60848	−1.50059	−0.750296	+	0.661102i	$0.770089\pi$
$42$	−0.117249	−0.0180919
$43$	2.78606	0.424870	0.212435	−	0.977175i	$-0.431861\pi$
$43$	2.78606	0.424870	0.212435	+	0.977175i	$0.431861\pi$
$44$	6.66601	1.00494
$45$	−5.58589	−0.832696
$46$	−0.0967937	−0.0142715
$47$	−0.958374	−0.139793	−0.0698966	−	0.997554i	$-0.522267\pi$
$47$	−0.958374	−0.139793	−0.0698966	+	0.997554i	$0.522267\pi$
$48$	11.7105	1.69027
$49$	−4.22640	−0.603771
$50$	−0.0240266	−0.00339788
$51$	20.7345	2.90341
$52$	0	0
$53$	−3.68268	−0.505855	−0.252928	−	0.967485i	$-0.581393\pi$
$53$	−3.68268	−0.505855	−0.252928	+	0.967485i	$0.581393\pi$
$54$	−0.182052	−0.0247742
$55$	3.33397	0.449552
$56$	0.160034	0.0213855
$57$	19.5819	2.59369
$58$	9.63732e−6 0	1.26544e−6 0
$59$	−7.99148	−1.04040	−0.520201	−	0.854044i	$-0.674143\pi$
$59$	−7.99148	−1.04040	−0.520201	+	0.854044i	$0.674143\pi$
$60$	5.85865	0.756348
$61$	−9.14117	−1.17041	−0.585203	−	0.810886i	$-0.698985\pi$
$61$	−9.14117	−1.17041	−0.585203	+	0.810886i	$0.698985\pi$
$62$	0.0996526	0.0126559
$63$	9.30282	1.17205
$64$	−7.98615	−0.998269
$65$	0	0
$66$	0.234719	0.0288919
$67$	6.72871	0.822042	0.411021	−	0.911626i	$-0.365172\pi$
$67$	6.72871	0.822042	0.411021	+	0.911626i	$0.365172\pi$
$68$	−14.1483	−1.71574
$69$	11.8045	1.42109
$70$	0.0400143	0.00478262
$71$	4.38321	0.520191	0.260096	−	0.965583i	$-0.416246\pi$
$71$	4.38321	0.520191	0.260096	+	0.965583i	$0.416246\pi$
$72$	0.536763	0.0632582
$73$	1.97245	0.230858	0.115429	−	0.993316i	$-0.463176\pi$
$73$	1.97245	0.230858	0.115429	+	0.993316i	$0.463176\pi$
$74$	−0.215389	−0.0250385
$75$	2.93017	0.338347
$76$	−13.3619	−1.53271
$77$	−5.55244	−0.632759
$78$	0	0
$79$	−3.77988	−0.425269	−0.212635	−	0.977132i	$-0.568204\pi$
$79$	−3.77988	−0.425269	−0.212635	+	0.977132i	$0.568204\pi$
$80$	−3.99654	−0.446826
$81$	5.44451	0.604945
$82$	0.230860	0.0254942
$83$	−3.64818	−0.400440	−0.200220	−	0.979751i	$-0.564166\pi$
$83$	−3.64818	−0.400440	−0.200220	+	0.979751i	$0.564166\pi$
$84$	−9.75707	−1.06458
$85$	−7.07621	−0.767523
$86$	−0.0669397	−0.00721830
$87$	−0.00117532	−0.000126007 0
$88$	−0.320370	−0.0341516
$89$	0.989573	0.104895	0.0524473	−	0.998624i	$-0.483298\pi$
$89$	0.989573	0.104895	0.0524473	+	0.998624i	$0.483298\pi$
$90$	0.134210	0.0141470
$91$	0	0
$92$	−8.05487	−0.839779
$93$	−12.1531	−1.26022
$94$	0.0230265	0.00237500
$95$	−6.68286	−0.685647
$96$	−0.844500	−0.0861915
$97$	18.4460	1.87291	0.936454	−	0.350790i	$-0.114087\pi$
$97$	18.4460	1.87291	0.936454	+	0.350790i	$0.114087\pi$
$98$	0.101546	0.0102577
$99$	−18.6232	−1.87170
$100$	−1.99942	−0.199942
$101$	−9.86892	−0.981994	−0.490997	−	0.871161i	$-0.663367\pi$
$101$	−9.86892	−0.981994	−0.490997	+	0.871161i	$0.663367\pi$
$102$	−0.498180	−0.0493272
$103$	−15.1288	−1.49069	−0.745344	−	0.666680i	$-0.767715\pi$
$103$	−15.1288	−1.49069	−0.745344	+	0.666680i	$0.767715\pi$
$104$	0	0
$105$	−4.87994	−0.476234
$106$	0.0884825	0.00859418
$107$	2.99312	0.289356	0.144678	−	0.989479i	$-0.453785\pi$
$107$	2.99312	0.289356	0.144678	+	0.989479i	$0.453785\pi$
$108$	−15.1498	−1.45779
$109$	−13.4179	−1.28520	−0.642602	−	0.766200i	$-0.722145\pi$
$109$	−13.4179	−1.28520	−0.642602	+	0.766200i	$0.722145\pi$
$110$	−0.0801041	−0.00763763
$111$	26.2678	2.49323
$112$	6.65589	0.628922
$113$	−3.38444	−0.318382	−0.159191	−	0.987248i	$-0.550889\pi$
$113$	−3.38444	−0.318382	−0.159191	+	0.987248i	$0.550889\pi$
$114$	−0.470488	−0.0440652
$115$	−4.02860	−0.375669
$116$	0.000801988 0	7.44627e−5 0
$117$	0	0
$118$	0.192009	0.0176758
$119$	11.7848	1.08031
$120$	−0.281568	−0.0257035
$121$	0.115354	0.0104868
$122$	0.219632	0.0198845
$123$	−28.1545	−2.53860
$124$	8.29278	0.744714
$125$	−1.00000	−0.0894427
$126$	−0.223516	−0.0199124
$127$	8.29226	0.735819	0.367910	−	0.929862i	$-0.380074\pi$
$127$	8.29226	0.735819	0.367910	+	0.929862i	$0.380074\pi$
$128$	0.768298	0.0679086
$129$	8.16363	0.718768
$130$	0	0
$131$	4.86438	0.425003	0.212501	−	0.977161i	$-0.431839\pi$
$131$	4.86438	0.425003	0.212501	+	0.977161i	$0.431839\pi$
$132$	19.5326	1.70009
$133$	11.1297	0.965070
$134$	−0.161668	−0.0139660
$135$	−7.57710	−0.652133
$136$	0.679972	0.0583071
$137$	4.43432	0.378850	0.189425	−	0.981895i	$-0.439338\pi$
$137$	4.43432	0.378850	0.189425	+	0.981895i	$0.439338\pi$
$138$	−0.283622	−0.0241435
$139$	6.07076	0.514915	0.257457	−	0.966290i	$-0.417115\pi$
$139$	6.07076	0.514915	0.257457	+	0.966290i	$0.417115\pi$
$140$	3.32987	0.281425
$141$	−2.80820	−0.236493
$142$	−0.105314	−0.00883773
$143$	0	0
$144$	22.3242	1.86035
$145$	0.000401110 0	3.33104e−5 0
$146$	−0.0473914	−0.00392214
$147$	−12.3841	−1.02142
$148$	−17.9240	−1.47334
$149$	12.0203	0.984741	0.492370	−	0.870386i	$-0.336131\pi$
$149$	12.0203	0.984741	0.492370	+	0.870386i	$0.336131\pi$
$150$	−0.0704021	−0.00574831
$151$	−6.64251	−0.540560	−0.270280	−	0.962782i	$-0.587116\pi$
$151$	−6.64251	−0.540560	−0.270280	+	0.962782i	$0.587116\pi$
$152$	0.642174	0.0520872
$153$	39.5269	3.19556
$154$	0.133406	0.0107502
$155$	4.14759	0.333142
$156$	0	0
$157$	8.94035	0.713517	0.356759	−	0.934197i	$-0.383882\pi$
$157$	8.94035	0.713517	0.356759	+	0.934197i	$0.383882\pi$
$158$	0.0908177	0.00722507
$159$	−10.7909	−0.855773
$160$	0.288209	0.0227849
$161$	6.70929	0.528766
$162$	−0.130813	−0.0102777
$163$	4.49497	0.352073	0.176037	−	0.984384i	$-0.443672\pi$
$163$	4.49497	0.352073	0.176037	+	0.984384i	$0.443672\pi$
$164$	19.2114	1.50016
$165$	9.76910	0.760523
$166$	0.0876536	0.00680324
$167$	−18.1721	−1.40620	−0.703098	−	0.711093i	$-0.748201\pi$
$167$	−18.1721	−1.40620	−0.703098	+	0.711093i	$0.748201\pi$
$168$	0.468927	0.0361785
$169$	0	0
$170$	0.170018	0.0130398
$171$	37.3297	2.85468
$172$	−5.57052	−0.424748
$173$	0.0851909	0.00647694	0.00323847	−	0.999995i	$-0.498969\pi$
$173$	0.0851909	0.00647694	0.00323847	+	0.999995i	$0.498969\pi$
$174$	2.82390e−5 0	2.14079e−6 0
$175$	1.66541	0.125893
$176$	−13.3243	−1.00436
$177$	−23.4164	−1.76008
$178$	−0.0237761	−0.00178210
$179$	9.57903	0.715970	0.357985	−	0.933727i	$-0.383464\pi$
$179$	9.57903	0.715970	0.357985	+	0.933727i	$0.383464\pi$
$180$	11.1686	0.832455
$181$	−11.0716	−0.822946	−0.411473	−	0.911422i	$-0.634986\pi$
$181$	−11.0716	−0.822946	−0.411473	+	0.911422i	$0.634986\pi$
$182$	0	0
$183$	−26.7852	−1.98002
$184$	0.387119	0.0285388
$185$	−8.96459	−0.659090
$186$	0.291999	0.0214104
$187$	−23.5919	−1.72521
$188$	1.91619	0.139753
$189$	12.6190	0.917898
$190$	0.160567	0.0116487
$191$	−19.6800	−1.42399	−0.711996	−	0.702183i	$-0.752209\pi$
$191$	−19.6800	−1.42399	−0.711996	+	0.702183i	$0.752209\pi$
$192$	−23.4008	−1.68880
$193$	2.22851	0.160412	0.0802060	−	0.996778i	$-0.474442\pi$
$193$	2.22851	0.160412	0.0802060	+	0.996778i	$0.474442\pi$
$194$	−0.443196	−0.0318196
$195$	0	0
$196$	8.45035	0.603597
$197$	−9.53209	−0.679134	−0.339567	−	0.940582i	$-0.610280\pi$
$197$	−9.53209	−0.679134	−0.339567	+	0.940582i	$0.610280\pi$
$198$	0.447453	0.0317991
$199$	−20.4192	−1.44748	−0.723741	−	0.690072i	$-0.757579\pi$
$199$	−20.4192	−1.44748	−0.723741	+	0.690072i	$0.757579\pi$
$200$	0.0960927	0.00679478
$201$	19.7162	1.39068
$202$	0.237117	0.0166835
$203$	−0.000668014 0	−4.68854e−5 0
$204$	−41.4570	−2.90257
$205$	9.60848	0.671085
$206$	0.363495	0.0253259
$207$	22.5033	1.56409
$208$	0	0
$209$	−22.2805	−1.54117
$210$	0.117249	0.00809093
$211$	−6.41514	−0.441637	−0.220818	−	0.975315i	$-0.570873\pi$
$211$	−6.41514	−0.441637	−0.220818	+	0.975315i	$0.570873\pi$
$212$	7.36324	0.505709
$213$	12.8435	0.880025
$214$	−0.0719146	−0.00491598
$215$	−2.78606	−0.190008
$216$	0.728104	0.0495412
$217$	−6.90745	−0.468908
$218$	0.322388	0.0218349
$219$	5.77962	0.390550
$220$	−6.66601	−0.449423
$221$	0	0
$222$	−0.631127	−0.0423584
$223$	−2.85750	−0.191353	−0.0956763	−	0.995412i	$-0.530501\pi$
$223$	−2.85750	−0.191353	−0.0956763	+	0.995412i	$0.530501\pi$
$224$	−0.479987	−0.0320705
$225$	5.58589	0.372393
$226$	0.0813168	0.00540911
$227$	−22.2069	−1.47393	−0.736963	−	0.675933i	$-0.763741\pi$
$227$	−22.2069	−1.47393	−0.736963	+	0.675933i	$0.763741\pi$
$228$	−39.1525	−2.59294
$229$	20.0068	1.32209	0.661045	−	0.750347i	$-0.270113\pi$
$229$	20.0068	1.32209	0.661045	+	0.750347i	$0.270113\pi$
$230$	0.0967937	0.00638239
$231$	−16.2696	−1.07046
$232$	−3.85437e−5 0	−2.53052e−6 0
$233$	2.11070	0.138276	0.0691381	−	0.997607i	$-0.477975\pi$
$233$	2.11070	0.138276	0.0691381	+	0.997607i	$0.477975\pi$
$234$	0	0
$235$	0.958374	0.0625174
$236$	15.9784	1.04010
$237$	−11.0757	−0.719442
$238$	−0.283150	−0.0183539
$239$	18.8655	1.22031	0.610154	−	0.792283i	$-0.291108\pi$
$239$	18.8655	1.22031	0.610154	+	0.792283i	$0.291108\pi$
$240$	−11.7105	−0.755911
$241$	13.0649	0.841584	0.420792	−	0.907157i	$-0.361752\pi$
$241$	13.0649	0.841584	0.420792	+	0.907157i	$0.361752\pi$
$242$	−0.00277158	−0.000178164 0
$243$	−6.77797	−0.434807
$244$	18.2771	1.17007
$245$	4.22640	0.270015
$246$	0.676458	0.0431294
$247$	0	0
$248$	−0.398553	−0.0253081
$249$	−10.6898	−0.677438
$250$	0.0240266	0.00151958
$251$	11.9931	0.757001	0.378500	−	0.925601i	$-0.376440\pi$
$251$	11.9931	0.757001	0.378500	+	0.925601i	$0.376440\pi$
$252$	−18.6003	−1.17171
$253$	−13.4312	−0.844414
$254$	−0.199235	−0.0125011
$255$	−20.7345	−1.29844
$256$	15.9538	0.997115
$257$	−26.6130	−1.66007	−0.830036	−	0.557709i	$-0.811680\pi$
$257$	−26.6130	−1.66007	−0.830036	+	0.557709i	$0.811680\pi$
$258$	−0.196145	−0.0122114
$259$	14.9298	0.927690
$260$	0	0
$261$	−0.00224056	−0.000138687 0
$262$	−0.116875	−0.00722055
$263$	−17.0235	−1.04971	−0.524856	−	0.851191i	$-0.675881\pi$
$263$	−17.0235	−1.04971	−0.524856	+	0.851191i	$0.675881\pi$
$264$	−0.938739	−0.0577754
$265$	3.68268	0.226225
$266$	−0.267410	−0.0163960
$267$	2.89962	0.177454
$268$	−13.4535	−0.821805
$269$	5.98522	0.364925	0.182463	−	0.983213i	$-0.441593\pi$
$269$	5.98522	0.364925	0.182463	+	0.983213i	$0.441593\pi$
$270$	0.182052	0.0110793
$271$	7.11684	0.432317	0.216159	−	0.976358i	$-0.430647\pi$
$271$	7.11684	0.432317	0.216159	+	0.976358i	$0.430647\pi$
$272$	28.2803	1.71475
$273$	0	0
$274$	−0.106542	−0.00643643
$275$	−3.33397	−0.201046
$276$	−23.6021	−1.42068
$277$	7.48738	0.449873	0.224936	−	0.974373i	$-0.427783\pi$
$277$	7.48738	0.449873	0.224936	+	0.974373i	$0.427783\pi$
$278$	−0.145860	−0.00874810
$279$	−23.1680	−1.38703
$280$	−0.160034	−0.00956387
$281$	−24.8752	−1.48393	−0.741964	−	0.670440i	$-0.766105\pi$
$281$	−24.8752	−1.48393	−0.741964	+	0.670440i	$0.766105\pi$
$282$	0.0674716	0.00401787
$283$	12.7413	0.757390	0.378695	−	0.925522i	$-0.376373\pi$
$283$	12.7413	0.757390	0.378695	+	0.925522i	$0.376373\pi$
$284$	−8.76388	−0.520041
$285$	−19.5819	−1.15993
$286$	0	0
$287$	−16.0021	−0.944574
$288$	−1.60990	−0.0948644
$289$	33.0727	1.94545
$290$	−9.63732e−6 0	−5.65923e−7 0
$291$	54.0499	3.16846
$292$	−3.94377	−0.230791
$293$	17.5748	1.02673	0.513365	−	0.858171i	$-0.328399\pi$
$293$	17.5748	1.02673	0.513365	+	0.858171i	$0.328399\pi$
$294$	0.297547	0.0173533
$295$	7.99148	0.465282
$296$	0.861432	0.0500697
$297$	−25.2618	−1.46584
$298$	−0.288807	−0.0167302
$299$	0	0
$300$	−5.85865	−0.338249
$301$	4.63995	0.267442
$302$	0.159597	0.00918379
$303$	−28.9176	−1.66127
$304$	26.7083	1.53183
$305$	9.14117	0.523422
$306$	−0.949700	−0.0542907
$307$	−0.592094	−0.0337926	−0.0168963	−	0.999857i	$-0.505379\pi$
$307$	−0.592094	−0.0337926	−0.0168963	+	0.999857i	$0.505379\pi$
$308$	11.1017	0.632577
$309$	−44.3300	−2.52185
$310$	−0.0996526	−0.00565989
$311$	−23.6771	−1.34260	−0.671301	−	0.741185i	$-0.734264\pi$
$311$	−23.6771	−1.34260	−0.671301	+	0.741185i	$0.734264\pi$
$312$	0	0
$313$	9.09689	0.514187	0.257093	−	0.966387i	$-0.417235\pi$
$313$	9.09689	0.514187	0.257093	+	0.966387i	$0.417235\pi$
$314$	−0.214807	−0.0121222
$315$	−9.30282	−0.524155
$316$	7.55757	0.425146
$317$	−4.34397	−0.243982	−0.121991	−	0.992531i	$-0.538928\pi$
$317$	−4.34397	−0.243982	−0.121991	+	0.992531i	$0.538928\pi$
$318$	0.259269	0.0145391
$319$	0.00133729	7.48737e−5 0
$320$	7.98615	0.446439
$321$	8.77034	0.489513
$322$	−0.161202	−0.00898342
$323$	47.2893	2.63125
$324$	−10.8859	−0.604771
$325$	0	0
$326$	−0.107999	−0.00598152
$327$	−39.3168	−2.17422
$328$	−0.923305	−0.0509810
$329$	−1.59609	−0.0879953
$330$	−0.234719	−0.0129208
$331$	19.3221	1.06204	0.531019	−	0.847360i	$-0.321809\pi$
$331$	19.3221	1.06204	0.531019	+	0.847360i	$0.321809\pi$
$332$	7.29426	0.400324
$333$	50.0752	2.74411
$334$	0.436614	0.0238904
$335$	−6.72871	−0.367628
$336$	19.5029	1.06397
$337$	−5.22779	−0.284776	−0.142388	−	0.989811i	$-0.545478\pi$
$337$	−5.22779	−0.284776	−0.142388	+	0.989811i	$0.545478\pi$
$338$	0	0
$339$	−9.91699	−0.538617
$340$	14.1483	0.767301
$341$	13.8279	0.748825
$342$	−0.896908	−0.0484992
$343$	−18.6966	−1.00952
$344$	0.267720	0.0144345
$345$	−11.8045	−0.635532
$346$	−0.00204685	−0.000110039 0
$347$	−18.8624	−1.01259	−0.506293	−	0.862361i	$-0.668985\pi$
$347$	−18.8624	−1.01259	−0.506293	+	0.862361i	$0.668985\pi$
$348$	0.00234996	0.000125971 0
$349$	−27.8225	−1.48930	−0.744652	−	0.667453i	$-0.767384\pi$
$349$	−27.8225	−1.48930	−0.744652	+	0.667453i	$0.767384\pi$
$350$	−0.0400143	−0.00213885
$351$	0	0
$352$	0.960879	0.0512150
$353$	−34.6884	−1.84627	−0.923137	−	0.384470i	$-0.874384\pi$
$353$	−34.6884	−1.84627	−0.923137	+	0.384470i	$0.874384\pi$
$354$	0.562617	0.0299028
$355$	−4.38321	−0.232636
$356$	−1.97858	−0.104864
$357$	34.5315	1.82760
$358$	−0.230152	−0.0121639
$359$	−27.1511	−1.43298	−0.716490	−	0.697597i	$-0.754253\pi$
$359$	−27.1511	−1.43298	−0.716490	+	0.697597i	$0.754253\pi$
$360$	−0.536763	−0.0282899
$361$	25.6606	1.35056
$362$	0.266014	0.0139814
$363$	0.338008	0.0177408
$364$	0	0
$365$	−1.97245	−0.103243
$366$	0.643558	0.0336393
$367$	25.2613	1.31863	0.659315	−	0.751867i	$-0.270847\pi$
$367$	25.2613	1.31863	0.659315	+	0.751867i	$0.270847\pi$
$368$	16.1004	0.839294
$369$	−53.6719	−2.79405
$370$	0.215389	0.0111975
$371$	−6.13319	−0.318419
$372$	24.2993	1.25986
$373$	4.53353	0.234737	0.117369	−	0.993088i	$-0.462554\pi$
$373$	4.53353	0.234737	0.117369	+	0.993088i	$0.462554\pi$
$374$	0.566833	0.0293103
$375$	−2.93017	−0.151313
$376$	−0.0920928	−0.00474932
$377$	0	0
$378$	−0.303192	−0.0155945
$379$	5.35880	0.275263	0.137632	−	0.990483i	$-0.456051\pi$
$379$	5.35880	0.275263	0.137632	+	0.990483i	$0.456051\pi$
$380$	13.3619	0.685449
$381$	24.2977	1.24481
$382$	0.472843	0.0241928
$383$	−5.61342	−0.286832	−0.143416	−	0.989662i	$-0.545809\pi$
$383$	−5.61342	−0.286832	−0.143416	+	0.989662i	$0.545809\pi$
$384$	2.25124	0.114883
$385$	5.55244	0.282979
$386$	−0.0535437	−0.00272530
$387$	15.5626	0.791093
$388$	−36.8814	−1.87237
$389$	2.71877	0.137847	0.0689236	−	0.997622i	$-0.478044\pi$
$389$	2.71877	0.137847	0.0689236	+	0.997622i	$0.478044\pi$
$390$	0	0
$391$	28.5072	1.44167
$392$	−0.406126	−0.0205125
$393$	14.2535	0.718992
$394$	0.229024	0.0115381
$395$	3.77988	0.190186
$396$	37.2356	1.87116
$397$	−5.01013	−0.251451	−0.125726	−	0.992065i	$-0.540126\pi$
$397$	−5.01013	−0.251451	−0.125726	+	0.992065i	$0.540126\pi$
$398$	0.490606	0.0245919
$399$	32.6120	1.63264
$400$	3.99654	0.199827
$401$	19.0325	0.950435	0.475218	−	0.879868i	$-0.342369\pi$
$401$	19.0325	0.950435	0.475218	+	0.879868i	$0.342369\pi$
$402$	−0.473715	−0.0236268
$403$	0	0
$404$	19.7321	0.981711
$405$	−5.44451	−0.270540
$406$	1.60501e−5 0	7.96555e−7 0
$407$	−29.8877	−1.48148
$408$	1.99243	0.0986401
$409$	11.9424	0.590513	0.295256	−	0.955418i	$-0.404595\pi$
$409$	11.9424	0.590513	0.295256	+	0.955418i	$0.404595\pi$
$410$	−0.230860	−0.0114013
$411$	12.9933	0.640913
$412$	30.2489	1.49026
$413$	−13.3091	−0.654899
$414$	−0.540679	−0.0265729
$415$	3.64818	0.179082
$416$	0	0
$417$	17.7883	0.871099
$418$	0.535325	0.0261836
$419$	−2.49593	−0.121934	−0.0609671	−	0.998140i	$-0.519418\pi$
$419$	−2.49593	−0.121934	−0.0609671	+	0.998140i	$0.519418\pi$
$420$	9.75707	0.476096
$421$	−23.2220	−1.13177	−0.565885	−	0.824484i	$-0.691465\pi$
$421$	−23.2220	−1.13177	−0.565885	+	0.824484i	$0.691465\pi$
$422$	0.154134	0.00750314
$423$	−5.35337	−0.260290
$424$	−0.353879	−0.0171859
$425$	7.07621	0.343247
$426$	−0.308587	−0.0149511
$427$	−15.2238	−0.736733
$428$	−5.98451	−0.289272
$429$	0	0
$430$	0.0669397	0.00322812
$431$	18.6779	0.899682	0.449841	−	0.893109i	$-0.351481\pi$
$431$	18.6779	0.899682	0.449841	+	0.893109i	$0.351481\pi$
$432$	30.2822	1.45695
$433$	26.2912	1.26348	0.631738	−	0.775182i	$-0.282342\pi$
$433$	26.2912	1.26348	0.631738	+	0.775182i	$0.282342\pi$
$434$	0.165963	0.00796647
$435$	0.00117532	5.63523e−5 0
$436$	26.8281	1.28483
$437$	26.9226	1.28788
$438$	−0.138865	−0.00663522
$439$	7.98298	0.381007	0.190503	−	0.981687i	$-0.438988\pi$
$439$	7.98298	0.381007	0.190503	+	0.981687i	$0.438988\pi$
$440$	0.320370	0.0152730
$441$	−23.6082	−1.12420
$442$	0	0
$443$	40.4743	1.92299	0.961495	−	0.274823i	$-0.0886192\pi$
$443$	40.4743	1.92299	0.961495	+	0.274823i	$0.0886192\pi$
$444$	−52.5204	−2.49251
$445$	−0.989573	−0.0469103
$446$	0.0686562	0.00325097
$447$	35.2215	1.66592
$448$	−13.3002	−0.628377
$449$	−15.7278	−0.742241	−0.371121	−	0.928585i	$-0.621026\pi$
$449$	−15.7278	−0.742241	−0.371121	+	0.928585i	$0.621026\pi$
$450$	−0.134210	−0.00632673
$451$	32.0344	1.50844
$452$	6.76693	0.318290
$453$	−19.4637	−0.914483
$454$	0.533558	0.0250411
$455$	0	0
$456$	1.88168	0.0881177
$457$	21.2829	0.995573	0.497786	−	0.867300i	$-0.334146\pi$
$457$	21.2829	0.995573	0.497786	+	0.867300i	$0.334146\pi$
$458$	−0.480697	−0.0224615
$459$	53.6171	2.50263
$460$	8.05487	0.375560
$461$	−38.2892	−1.78331	−0.891653	−	0.452720i	$-0.850454\pi$
$461$	−38.2892	−1.78331	−0.891653	+	0.452720i	$0.850454\pi$
$462$	0.390904	0.0181865
$463$	−12.1131	−0.562944	−0.281472	−	0.959569i	$-0.590823\pi$
$463$	−12.1131	−0.562944	−0.281472	+	0.959569i	$0.590823\pi$
$464$	−0.00160305	−7.44197e−5 0
$465$	12.1531	0.563588
$466$	−0.0507129	−0.00234923
$467$	23.2313	1.07502	0.537509	−	0.843258i	$-0.319365\pi$
$467$	23.2313	1.07502	0.537509	+	0.843258i	$0.319365\pi$
$468$	0	0
$469$	11.2061	0.517449
$470$	−0.0230265	−0.00106213
$471$	26.1967	1.20708
$472$	−0.767923	−0.0353465
$473$	−9.28865	−0.427092
$474$	0.266111	0.0122229
$475$	6.68286	0.306631
$476$	−23.5628	−1.08000
$477$	−20.5711	−0.941884
$478$	−0.453275	−0.0207323
$479$	37.9716	1.73497	0.867483	−	0.497466i	$-0.165736\pi$
$479$	37.9716	1.73497	0.867483	+	0.497466i	$0.165736\pi$
$480$	0.844500	0.0385460
$481$	0	0
$482$	−0.313906	−0.0142980
$483$	19.6593	0.894531
$484$	−0.230642	−0.0104837
$485$	−18.4460	−0.837590
$486$	0.162852	0.00738711
$487$	−22.1216	−1.00242	−0.501212	−	0.865324i	$-0.667112\pi$
$487$	−22.1216	−1.00242	−0.501212	+	0.865324i	$0.667112\pi$
$488$	−0.878400	−0.0397633
$489$	13.1710	0.595615
$490$	−0.101546	−0.00458739
$491$	−26.1440	−1.17986	−0.589931	−	0.807453i	$-0.700845\pi$
$491$	−26.1440	−1.17986	−0.589931	+	0.807453i	$0.700845\pi$
$492$	56.2927	2.53787
$493$	−0.00283834	−0.000127832 0
$494$	0	0
$495$	18.6232	0.837050
$496$	−16.5760	−0.744284
$497$	7.29986	0.327443
$498$	0.256840	0.0115093
$499$	−22.5180	−1.00805	−0.504023	−	0.863690i	$-0.668147\pi$
$499$	−22.5180	−1.00805	−0.504023	+	0.863690i	$0.668147\pi$
$500$	1.99942	0.0894169
$501$	−53.2472	−2.37891
$502$	−0.288155	−0.0128610
$503$	20.4701	0.912716	0.456358	−	0.889796i	$-0.349154\pi$
$503$	20.4701	0.912716	0.456358	+	0.889796i	$0.349154\pi$
$504$	0.893933	0.0398190
$505$	9.86892	0.439161
$506$	0.322707	0.0143461
$507$	0	0
$508$	−16.5797	−0.735607
$509$	20.2724	0.898557	0.449278	−	0.893392i	$-0.351681\pi$
$509$	20.2724	0.898557	0.449278	+	0.893392i	$0.351681\pi$
$510$	0.498180	0.0220598
$511$	3.28495	0.145318
$512$	−1.91991	−0.0848490
$513$	50.6367	2.23567
$514$	0.639421	0.0282036
$515$	15.1288	0.666656
$516$	−16.3226	−0.718560
$517$	3.19519	0.140524
$518$	−0.358712	−0.0157609
$519$	0.249624	0.0109573
$520$	0	0
$521$	−30.1602	−1.32134	−0.660670	−	0.750676i	$-0.729728\pi$
$521$	−30.1602	−1.32134	−0.660670	+	0.750676i	$0.729728\pi$
$522$	5.38330e−5 0	2.35621e−6 0
$523$	14.1082	0.616910	0.308455	−	0.951239i	$-0.400188\pi$
$523$	14.1082	0.616910	0.308455	+	0.951239i	$0.400188\pi$
$524$	−9.72595	−0.424880
$525$	4.87994	0.212978
$526$	0.409017	0.0178340
$527$	−29.3492	−1.27847
$528$	−39.0425	−1.69911
$529$	−6.77038	−0.294364
$530$	−0.0884825	−0.00384343
$531$	−44.6396	−1.93719
$532$	−22.2530	−0.964792
$533$	0	0
$534$	−0.0696681	−0.00301483
$535$	−2.99312	−0.129404
$536$	0.646580	0.0279280
$537$	28.0682	1.21123
$538$	−0.143805	−0.00619986
$539$	14.0907	0.606928
$540$	15.1498	0.651945
$541$	33.5648	1.44306	0.721532	−	0.692381i	$-0.243438\pi$
$541$	33.5648	1.44306	0.721532	+	0.692381i	$0.243438\pi$
$542$	−0.170994	−0.00734481
$543$	−32.4417	−1.39221
$544$	−2.03943	−0.0874396
$545$	13.4179	0.574761
$546$	0	0
$547$	34.8045	1.48813	0.744067	−	0.668106i	$-0.232895\pi$
$547$	34.8045	1.48813	0.744067	+	0.668106i	$0.232895\pi$
$548$	−8.86609	−0.378740
$549$	−51.0616	−2.17926
$550$	0.0801041	0.00341565
$551$	−0.00268056	−0.000114196 0
$552$	1.13432	0.0482801
$553$	−6.29506	−0.267693
$554$	−0.179897	−0.00764307
$555$	−26.2678	−1.11501
$556$	−12.1380	−0.514766
$557$	−18.3989	−0.779587	−0.389794	−	0.920902i	$-0.627454\pi$
$557$	−18.3989	−0.779587	−0.389794	+	0.920902i	$0.627454\pi$
$558$	0.556649	0.0235648
$559$	0	0
$560$	−6.65589	−0.281263
$561$	−69.1282	−2.91859
$562$	0.597667	0.0252110
$563$	18.6109	0.784357	0.392178	−	0.919889i	$-0.371722\pi$
$563$	18.6109	0.784357	0.392178	+	0.919889i	$0.371722\pi$
$564$	5.61477	0.236425
$565$	3.38444	0.142385
$566$	−0.306130	−0.0128676
$567$	9.06736	0.380793
$568$	0.421194	0.0176729
$569$	−10.8666	−0.455550	−0.227775	−	0.973714i	$-0.573145\pi$
$569$	−10.8666	−0.455550	−0.227775	+	0.973714i	$0.573145\pi$
$570$	0.470488	0.0197066
$571$	38.9376	1.62949	0.814743	−	0.579822i	$-0.196878\pi$
$571$	38.9376	1.62949	0.814743	+	0.579822i	$0.196878\pi$
$572$	0	0
$573$	−57.6656	−2.40902
$574$	0.384477	0.0160477
$575$	4.02860	0.168004
$576$	−44.6098	−1.85874
$577$	33.9986	1.41538	0.707691	−	0.706522i	$-0.249737\pi$
$577$	33.9986	1.41538	0.707691	+	0.706522i	$0.249737\pi$
$578$	−0.794627	−0.0330521
$579$	6.52992	0.271374
$580$	−0.000801988 0	−3.33007e−5 0
$581$	−6.07573	−0.252064
$582$	−1.29864	−0.0538303
$583$	12.2780	0.508501
$584$	0.189538	0.00784315
$585$	0	0
$586$	−0.422263	−0.0174435
$587$	−17.9777	−0.742018	−0.371009	−	0.928629i	$-0.620988\pi$
$587$	−17.9777	−0.742018	−0.371009	+	0.928629i	$0.620988\pi$
$588$	24.7610	1.02112
$589$	−27.7178	−1.14209
$590$	−0.192009	−0.00790487
$591$	−27.9306	−1.14891
$592$	35.8273	1.47249
$593$	−26.9189	−1.10542	−0.552712	−	0.833372i	$-0.686407\pi$
$593$	−26.9189	−1.10542	−0.552712	+	0.833372i	$0.686407\pi$
$594$	0.606957	0.0249037
$595$	−11.7848	−0.483130
$596$	−24.0336	−0.984457
$597$	−59.8318	−2.44875
$598$	0	0
$599$	−8.68542	−0.354877	−0.177438	−	0.984132i	$-0.556781\pi$
$599$	−8.68542	−0.354877	−0.177438	+	0.984132i	$0.556781\pi$
$600$	0.281568	0.0114950
$601$	−31.3757	−1.27984	−0.639921	−	0.768440i	$-0.721033\pi$
$601$	−31.3757	−1.27984	−0.639921	+	0.768440i	$0.721033\pi$
$602$	−0.111482	−0.00454368
$603$	37.5858	1.53061
$604$	13.2812	0.540404
$605$	−0.115354	−0.00468982
$606$	0.694793	0.0282240
$607$	3.77489	0.153218	0.0766091	−	0.997061i	$-0.475591\pi$
$607$	3.77489	0.153218	0.0766091	+	0.997061i	$0.475591\pi$
$608$	−1.92606	−0.0781120
$609$	−0.00195739	−7.93176e−5 0
$610$	−0.219632	−0.00889263
$611$	0	0
$612$	−79.0310	−3.19464
$613$	28.6074	1.15544	0.577722	−	0.816234i	$-0.303942\pi$
$613$	28.6074	1.15544	0.577722	+	0.816234i	$0.303942\pi$
$614$	0.0142260	0.000574116 0
$615$	28.1545	1.13530
$616$	−0.533549	−0.0214973
$617$	−12.6947	−0.511071	−0.255535	−	0.966800i	$-0.582252\pi$
$617$	−12.6947	−0.511071	−0.255535	+	0.966800i	$0.582252\pi$
$618$	1.06510	0.0428447
$619$	24.5421	0.986430	0.493215	−	0.869907i	$-0.335822\pi$
$619$	24.5421	0.986430	0.493215	+	0.869907i	$0.335822\pi$
$620$	−8.29278	−0.333046
$621$	30.5251	1.22493
$622$	0.568880	0.0228100
$623$	1.64805	0.0660277
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.218568	−0.00873573
$627$	−65.2855	−2.60725
$628$	−17.8755	−0.713311
$629$	63.4353	2.52933
$630$	0.223516	0.00890507
$631$	7.66905	0.305300	0.152650	−	0.988280i	$-0.451219\pi$
$631$	7.66905	0.305300	0.152650	+	0.988280i	$0.451219\pi$
$632$	−0.363218	−0.0144481
$633$	−18.7974	−0.747132
$634$	0.104371	0.00414510
$635$	−8.29226	−0.329068
$636$	21.5755	0.855526
$637$	0	0
$638$	−3.21305e−5 0	−1.27206e−6 0
$639$	24.4841	0.968577
$640$	−0.768298	−0.0303696
$641$	−8.98009	−0.354692	−0.177346	−	0.984149i	$-0.556751\pi$
$641$	−8.98009	−0.354692	−0.177346	+	0.984149i	$0.556751\pi$
$642$	−0.210722	−0.00831653
$643$	−14.1503	−0.558032	−0.279016	−	0.960287i	$-0.590008\pi$
$643$	−14.1503	−0.558032	−0.279016	+	0.960287i	$0.590008\pi$
$644$	−13.4147	−0.528613
$645$	−8.16363	−0.321443
$646$	−1.13620	−0.0447033
$647$	21.8757	0.860024	0.430012	−	0.902823i	$-0.358509\pi$
$647$	21.8757	0.860024	0.430012	+	0.902823i	$0.358509\pi$
$648$	0.523177	0.0205524
$649$	26.6434	1.04584
$650$	0	0
$651$	−20.2400	−0.793268
$652$	−8.98735	−0.351972
$653$	−45.1455	−1.76668	−0.883340	−	0.468733i	$-0.844711\pi$
$653$	−45.1455	−1.76668	−0.883340	+	0.468733i	$0.844711\pi$
$654$	0.944651	0.0369388
$655$	−4.86438	−0.190067
$656$	−38.4006	−1.49929
$657$	11.0179	0.429849
$658$	0.0383487	0.00149499
$659$	32.9439	1.28331	0.641656	−	0.766993i	$-0.278248\pi$
$659$	32.9439	1.28331	0.641656	+	0.766993i	$0.278248\pi$
$660$	−19.5326	−0.760304
$661$	29.7714	1.15798	0.578988	−	0.815336i	$-0.303448\pi$
$661$	29.7714	1.15798	0.578988	+	0.815336i	$0.303448\pi$
$662$	−0.464245	−0.0180434
$663$	0	0
$664$	−0.350564	−0.0136045
$665$	−11.1297	−0.431592
$666$	−1.20314	−0.0466207
$667$	−0.00161591	−6.25683e−5 0
$668$	36.3336	1.40579
$669$	−8.37297	−0.323718
$670$	0.161668	0.00624579
$671$	30.4764	1.17653
$672$	−1.40644	−0.0542547
$673$	−24.7144	−0.952671	−0.476336	−	0.879264i	$-0.658035\pi$
$673$	−24.7144	−0.952671	−0.476336	+	0.879264i	$0.658035\pi$
$674$	0.125606	0.00483817
$675$	7.57710	0.291643
$676$	0	0
$677$	47.2303	1.81521	0.907605	−	0.419826i	$-0.137909\pi$
$677$	47.2303	1.81521	0.907605	+	0.419826i	$0.137909\pi$
$678$	0.238272	0.00915078
$679$	30.7202	1.17893
$680$	−0.679972	−0.0260757
$681$	−65.0700	−2.49349
$682$	−0.332239	−0.0127221
$683$	32.6572	1.24959	0.624797	−	0.780787i	$-0.285182\pi$
$683$	32.6572	1.24959	0.624797	+	0.780787i	$0.285182\pi$
$684$	−74.6379	−2.85385
$685$	−4.43432	−0.169427
$686$	0.449216	0.0171512
$687$	58.6234	2.23662
$688$	11.1346	0.424503
$689$	0	0
$690$	0.283622	0.0107973
$691$	2.93042	0.111479	0.0557393	−	0.998445i	$-0.482248\pi$
$691$	2.93042	0.111479	0.0557393	+	0.998445i	$0.482248\pi$
$692$	−0.170333	−0.00647507
$693$	−31.0153	−1.17817
$694$	0.453200	0.0172032
$695$	−6.07076	−0.230277
$696$	−0.000112940 0	−4.28097e−6 0
$697$	−67.9916	−2.57537
$698$	0.668481	0.0253024
$699$	6.18469	0.233927
$700$	−3.32987	−0.125857
$701$	14.4881	0.547210	0.273605	−	0.961842i	$-0.411784\pi$
$701$	14.4881	0.547210	0.273605	+	0.961842i	$0.411784\pi$
$702$	0	0
$703$	59.9091	2.25952
$704$	26.6256	1.00349
$705$	2.80820	0.105763
$706$	0.833445	0.0313671
$707$	−16.4358	−0.618133
$708$	46.8193	1.75958
$709$	−49.2943	−1.85129	−0.925643	−	0.378397i	$-0.876475\pi$
$709$	−49.2943	−1.85129	−0.925643	+	0.378397i	$0.876475\pi$
$710$	0.105314	0.00395236
$711$	−21.1140	−0.791836
$712$	0.0950908	0.00356368
$713$	−16.7090	−0.625756
$714$	−0.829676	−0.0310499
$715$	0	0
$716$	−19.1525	−0.715764
$717$	55.2791	2.06444
$718$	0.652350	0.0243455
$719$	−26.0709	−0.972282	−0.486141	−	0.873880i	$-0.661596\pi$
$719$	−26.0709	−0.972282	−0.486141	+	0.873880i	$0.661596\pi$
$720$	−22.3242	−0.831975
$721$	−25.1958	−0.938339
$722$	−0.616539	−0.0229452
$723$	38.2823	1.42374
$724$	22.1368	0.822709
$725$	−0.000401110 0	−1.48968e−5 0
$726$	−0.00812119	−0.000301405 0
$727$	−53.8109	−1.99574	−0.997868	−	0.0652593i	$-0.979213\pi$
$727$	−53.8109	−1.99574	−0.997868	+	0.0652593i	$0.979213\pi$
$728$	0	0
$729$	−36.1941	−1.34052
$730$	0.0473914	0.00175403
$731$	19.7148	0.729177
$732$	53.5549	1.97945
$733$	−16.4544	−0.607758	−0.303879	−	0.952711i	$-0.598282\pi$
$733$	−16.4544	−0.607758	−0.303879	+	0.952711i	$0.598282\pi$
$734$	−0.606944	−0.0224027
$735$	12.3841	0.456793
$736$	−1.16108	−0.0427979
$737$	−22.4333	−0.826341
$738$	1.28956	0.0474692
$739$	8.56267	0.314983	0.157491	−	0.987520i	$-0.449659\pi$
$739$	8.56267	0.314983	0.157491	+	0.987520i	$0.449659\pi$
$740$	17.9240	0.658900
$741$	0	0
$742$	0.147360	0.00540976
$743$	36.8946	1.35353	0.676766	−	0.736199i	$-0.263381\pi$
$743$	36.8946	1.35353	0.676766	+	0.736199i	$0.263381\pi$
$744$	−1.16783	−0.0428146
$745$	−12.0203	−0.440389
$746$	−0.108926	−0.00398805
$747$	−20.3783	−0.745605
$748$	47.1701	1.72471
$749$	4.98478	0.182140
$750$	0.0704021	0.00257072
$751$	−19.2206	−0.701369	−0.350685	−	0.936494i	$-0.614051\pi$
$751$	−19.2206	−0.701369	−0.350685	+	0.936494i	$0.614051\pi$
$752$	−3.83018	−0.139672
$753$	35.1420	1.28064
$754$	0	0
$755$	6.64251	0.241746
$756$	−25.2307	−0.917633
$757$	−36.7110	−1.33428	−0.667142	−	0.744931i	$-0.732482\pi$
$757$	−36.7110	−1.33428	−0.667142	+	0.744931i	$0.732482\pi$
$758$	−0.128754	−0.00467656
$759$	−39.3558	−1.42852
$760$	−0.642174	−0.0232941
$761$	12.8270	0.464979	0.232489	−	0.972599i	$-0.425313\pi$
$761$	12.8270	0.464979	0.232489	+	0.972599i	$0.425313\pi$
$762$	−0.583793	−0.0211486
$763$	−22.3464	−0.808994
$764$	39.3486	1.42358
$765$	−39.5269	−1.42910
$766$	0.134872	0.00487311
$767$	0	0
$768$	46.7474	1.68685
$769$	34.6300	1.24879	0.624396	−	0.781108i	$-0.285345\pi$
$769$	34.6300	1.24879	0.624396	+	0.781108i	$0.285345\pi$
$770$	−0.133406	−0.00480764
$771$	−77.9806	−2.80840
$772$	−4.45574	−0.160366
$773$	21.1483	0.760653	0.380326	−	0.924852i	$-0.375812\pi$
$773$	21.1483	0.760653	0.380326	+	0.924852i	$0.375812\pi$
$774$	−0.373918	−0.0134402
$775$	−4.14759	−0.148986
$776$	1.77253	0.0636300
$777$	43.7467	1.56941
$778$	−0.0653229	−0.00234194
$779$	−64.2122	−2.30064
$780$	0	0
$781$	−14.6135	−0.522911
$782$	−0.684933	−0.0244931
$783$	−0.00303925	−0.000108614 0
$784$	−16.8909	−0.603248
$785$	−8.94035	−0.319095
$786$	−0.342463	−0.0122152
$787$	−20.7662	−0.740235	−0.370117	−	0.928985i	$-0.620683\pi$
$787$	−20.7662	−0.740235	−0.370117	+	0.928985i	$0.620683\pi$
$788$	19.0587	0.678938
$789$	−49.8816	−1.77583
$790$	−0.0908177	−0.00323115
$791$	−5.63650	−0.200411
$792$	−1.78955	−0.0635890
$793$	0	0
$794$	0.120377	0.00427200
$795$	10.7909	0.382713
$796$	40.8267	1.44706
$797$	−15.7183	−0.556770	−0.278385	−	0.960470i	$-0.589799\pi$
$797$	−15.7183	−0.556770	−0.278385	+	0.960470i	$0.589799\pi$
$798$	−0.783557	−0.0277376
$799$	−6.78165	−0.239918
$800$	−0.288209	−0.0101897
$801$	5.52765	0.195310
$802$	−0.457286	−0.0161473
$803$	−6.57610	−0.232065
$804$	−39.4211	−1.39028
$805$	−6.70929	−0.236471
$806$	0	0
$807$	17.5377	0.617356
$808$	−0.948331	−0.0333622
$809$	−5.08807	−0.178887	−0.0894435	−	0.995992i	$-0.528509\pi$
$809$	−5.08807	−0.178887	−0.0894435	+	0.995992i	$0.528509\pi$
$810$	0.130813	0.00459631
$811$	−35.0950	−1.23235	−0.616176	−	0.787608i	$-0.711319\pi$
$811$	−35.0950	−1.23235	−0.616176	+	0.787608i	$0.711319\pi$
$812$	0.00133564	4.68718e−5 0
$813$	20.8536	0.731366
$814$	0.718101	0.0251694
$815$	−4.49497	−0.157452
$816$	82.8661	2.90090
$817$	18.6189	0.651392
$818$	−0.286935	−0.0100325
$819$	0	0
$820$	−19.2114	−0.670892
$821$	−26.4352	−0.922596	−0.461298	−	0.887245i	$-0.652616\pi$
$821$	−26.4352	−0.922596	−0.461298	+	0.887245i	$0.652616\pi$
$822$	−0.312186	−0.0108887
$823$	45.9498	1.60171	0.800854	−	0.598859i	$-0.204379\pi$
$823$	45.9498	1.60171	0.800854	+	0.598859i	$0.204379\pi$
$824$	−1.45377	−0.0506445
$825$	−9.76910	−0.340116
$826$	0.319774	0.0111263
$827$	−25.1837	−0.875723	−0.437861	−	0.899042i	$-0.644264\pi$
$827$	−25.1837	−0.875723	−0.437861	+	0.899042i	$0.644264\pi$
$828$	−44.9937	−1.56364
$829$	19.5773	0.679947	0.339973	−	0.940435i	$-0.389582\pi$
$829$	19.5773	0.679947	0.339973	+	0.940435i	$0.389582\pi$
$830$	−0.0876536	−0.00304250
$831$	21.9393	0.761065
$832$	0	0
$833$	−29.9069	−1.03621
$834$	−0.427394	−0.0147995
$835$	18.1721	0.628870
$836$	44.5481	1.54073
$837$	−31.4267	−1.08627
$838$	0.0599689	0.00207159
$839$	14.1467	0.488398	0.244199	−	0.969725i	$-0.421475\pi$
$839$	14.1467	0.488398	0.244199	+	0.969725i	$0.421475\pi$
$840$	−0.468927	−0.0161795
$841$	−29.0000	−1.00000
$842$	0.557946	0.0192281
$843$	−72.8884	−2.51041
$844$	12.8266	0.441509
$845$	0	0
$846$	0.128624	0.00442217
$847$	0.192113	0.00660107
$848$	−14.7180	−0.505417
$849$	37.3341	1.28130
$850$	−0.170018	−0.00583155
$851$	36.1148	1.23800
$852$	−25.6797	−0.879771
$853$	2.86179	0.0979860	0.0489930	−	0.998799i	$-0.484399\pi$
$853$	2.86179	0.0979860	0.0489930	+	0.998799i	$0.484399\pi$
$854$	0.365778	0.0125167
$855$	−37.3297	−1.27665
$856$	0.287617	0.00983054
$857$	28.2816	0.966082	0.483041	−	0.875598i	$-0.339532\pi$
$857$	28.2816	0.966082	0.483041	+	0.875598i	$0.339532\pi$
$858$	0	0
$859$	−8.52025	−0.290707	−0.145354	−	0.989380i	$-0.546432\pi$
$859$	−8.52025	−0.290707	−0.145354	+	0.989380i	$0.546432\pi$
$860$	5.57052	0.189953
$861$	−46.8889	−1.59797
$862$	−0.448767	−0.0152851
$863$	−36.8913	−1.25579	−0.627897	−	0.778297i	$-0.716084\pi$
$863$	−36.8913	−1.25579	−0.627897	+	0.778297i	$0.716084\pi$
$864$	−2.18379	−0.0742939
$865$	−0.0851909	−0.00289658
$866$	−0.631690	−0.0214657
$867$	96.9087	3.29119
$868$	13.8109	0.468773
$869$	12.6020	0.427493
$870$	−2.82390e−5 0	−9.57391e−7 0
$871$	0	0
$872$	−1.28936	−0.0436634
$873$	103.037	3.48729
$874$	−0.646859	−0.0218803
$875$	−1.66541	−0.0563013
$876$	−11.5559	−0.390438
$877$	37.6091	1.26997	0.634984	−	0.772525i	$-0.281006\pi$
$877$	37.6091	1.26997	0.634984	+	0.772525i	$0.281006\pi$
$878$	−0.191804	−0.00647308
$879$	51.4971	1.73695
$880$	13.3243	0.449163
$881$	39.3542	1.32588	0.662938	−	0.748675i	$-0.269309\pi$
$881$	39.3542	1.32588	0.662938	+	0.748675i	$0.269309\pi$
$882$	0.567226	0.0190995
$883$	−9.17085	−0.308624	−0.154312	−	0.988022i	$-0.549316\pi$
$883$	−9.17085	−0.308624	−0.154312	+	0.988022i	$0.549316\pi$
$884$	0	0
$885$	23.4164	0.787134
$886$	−0.972460	−0.0326704
$887$	35.6093	1.19564	0.597822	−	0.801629i	$-0.296033\pi$
$887$	35.6093	1.19564	0.597822	+	0.801629i	$0.296033\pi$
$888$	2.52414	0.0847047
$889$	13.8100	0.463174
$890$	0.0237761	0.000796977 0
$891$	−18.1518	−0.608109
$892$	5.71336	0.191297
$893$	−6.40468	−0.214324
$894$	−0.846254	−0.0283030
$895$	−9.57903	−0.320192
$896$	1.27953	0.0427462
$897$	0	0
$898$	0.377886	0.0126102
$899$	0.00166364	5.54855e−5 0
$900$	−11.1686	−0.372285
$901$	−26.0594	−0.868165
$902$	−0.769679	−0.0256275
$903$	13.5958	0.452441
$904$	−0.325220	−0.0108167
$905$	11.0716	0.368033
$906$	0.467647	0.0155365
$907$	32.1739	1.06832	0.534158	−	0.845385i	$-0.320629\pi$
$907$	32.1739	1.06832	0.534158	+	0.845385i	$0.320629\pi$
$908$	44.4010	1.47350
$909$	−55.1267	−1.82844
$910$	0	0
$911$	41.5764	1.37749	0.688743	−	0.725005i	$-0.258163\pi$
$911$	41.5764	1.37749	0.688743	+	0.725005i	$0.258163\pi$
$912$	78.2598	2.59144
$913$	12.1629	0.402534
$914$	−0.511357	−0.0169142
$915$	26.7852	0.885491
$916$	−40.0021	−1.32171
$917$	8.10121	0.267525
$918$	−1.28824	−0.0425182
$919$	−29.1810	−0.962592	−0.481296	−	0.876558i	$-0.659834\pi$
$919$	−29.1810	−0.962592	−0.481296	+	0.876558i	$0.659834\pi$
$920$	−0.387119	−0.0127629
$921$	−1.73493	−0.0571680
$922$	0.919961	0.0302973
$923$	0	0
$924$	32.5298	1.07015
$925$	8.96459	0.294754
$926$	0.291037	0.00956408
$927$	−84.5080	−2.77561
$928$	0.000115603 0	3.79487e−6 0
$929$	−0.0200966	−0.000659349 0	−0.000329675	−	1.00000i	$-0.500105\pi$
$929$	−0.0200966	−0.000659349 0	−0.000329675		1.00000i	$0.500105\pi$
$930$	−0.291999	−0.00957503
$931$	−28.2444	−0.925674
$932$	−4.22017	−0.138236
$933$	−69.3778	−2.27133
$934$	−0.558171	−0.0182639
$935$	23.5919	0.771537
$936$	0	0
$937$	−51.2388	−1.67390	−0.836949	−	0.547281i	$-0.815663\pi$
$937$	−51.2388	−1.67390	−0.836949	+	0.547281i	$0.815663\pi$
$938$	−0.269245	−0.00879114
$939$	26.6554	0.869867
$940$	−1.91619	−0.0624994
$941$	−32.4122	−1.05661	−0.528304	−	0.849055i	$-0.677172\pi$
$941$	−32.4122	−1.05661	−0.528304	+	0.849055i	$0.677172\pi$
$942$	−0.629419	−0.0205076
$943$	−38.7087	−1.26053
$944$	−31.9383	−1.03950
$945$	−12.6190	−0.410496
$946$	0.223175	0.00725605
$947$	−29.2152	−0.949365	−0.474683	−	0.880157i	$-0.657437\pi$
$947$	−29.2152	−0.949365	−0.474683	+	0.880157i	$0.657437\pi$
$948$	22.1450	0.719235
$949$	0	0
$950$	−0.160567	−0.00520947
$951$	−12.7286	−0.412752
$952$	1.13243	0.0367024
$953$	−13.0955	−0.424204	−0.212102	−	0.977248i	$-0.568031\pi$
$953$	−13.0955	−0.424204	−0.212102	+	0.977248i	$0.568031\pi$
$954$	0.494254	0.0160021
$955$	19.6800	0.636829
$956$	−37.7201	−1.21996
$957$	0.00391848	0.000126666 0
$958$	−0.912330	−0.0294761
$959$	7.38498	0.238474
$960$	23.4008	0.755256
$961$	−13.7975	−0.445081
$962$	0	0
$963$	16.7192	0.538770
$964$	−26.1222	−0.841341
$965$	−2.22851	−0.0717384
$966$	−0.472348	−0.0151976
$967$	28.4227	0.914012	0.457006	−	0.889464i	$-0.348922\pi$
$967$	28.4227	0.914012	0.457006	+	0.889464i	$0.348922\pi$
$968$	0.0110847	0.000356276 0
$969$	138.566	4.45137
$970$	0.443196	0.0142302
$971$	39.1760	1.25722	0.628608	−	0.777722i	$-0.283625\pi$
$971$	39.1760	1.25722	0.628608	+	0.777722i	$0.283625\pi$
$972$	13.5520	0.434681
$973$	10.1103	0.324122
$974$	0.531507	0.0170306
$975$	0	0
$976$	−36.5330	−1.16939
$977$	−51.4703	−1.64668	−0.823340	−	0.567549i	$-0.807892\pi$
$977$	−51.4703	−1.64668	−0.823340	+	0.567549i	$0.807892\pi$
$978$	−0.316456	−0.0101191
$979$	−3.29921	−0.105443
$980$	−8.45035	−0.269937
$981$	−74.9511	−2.39300
$982$	0.628153	0.0200452
$983$	−62.1330	−1.98173	−0.990867	−	0.134841i	$-0.956948\pi$
$983$	−62.1330	−1.98173	−0.990867	+	0.134841i	$0.956948\pi$
$984$	−2.70544	−0.0862463
$985$	9.53209	0.303718
$986$	6.81957e−5 0	2.17179e−6 0
$987$	−4.67681	−0.148865
$988$	0	0
$989$	11.2239	0.356900
$990$	−0.447453	−0.0142210
$991$	8.63773	0.274387	0.137193	−	0.990544i	$-0.456192\pi$
$991$	8.63773	0.274387	0.137193	+	0.990544i	$0.456192\pi$
$992$	1.19537	0.0379531
$993$	56.6170	1.79669
$994$	−0.175391	−0.00556307
$995$	20.4192	0.647334
$996$	21.3734	0.677242
$997$	5.36590	0.169940	0.0849698	−	0.996384i	$-0.472921\pi$
$997$	5.36590	0.169940	0.0849698	+	0.996384i	$0.472921\pi$
$998$	0.541033	0.0171261
$999$	67.9256	2.14907

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	845.2.a.n.1.5	✓	9
3.2	odd	2		7605.2.a.cs.1.5		9
5.4	even	2		4225.2.a.bt.1.5		9
13.2	odd	12		845.2.m.j.316.9		36
13.3	even	3		845.2.e.p.191.5		18
13.4	even	6		845.2.e.o.146.5		18
13.5	odd	4		845.2.c.h.506.10		18
13.6	odd	12		845.2.m.j.361.10		36
13.7	odd	12		845.2.m.j.361.9		36
13.8	odd	4		845.2.c.h.506.9		18
13.9	even	3		845.2.e.p.146.5		18
13.10	even	6		845.2.e.o.191.5		18
13.11	odd	12		845.2.m.j.316.10		36
13.12	even	2		845.2.a.o.1.5	yes	9
39.38	odd	2		7605.2.a.cp.1.5		9
65.64	even	2		4225.2.a.bs.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.n.1.5	✓	9	1.1	even	1	trivial
845.2.a.o.1.5	yes	9	13.12	even	2
845.2.c.h.506.9		18	13.8	odd	4
845.2.c.h.506.10		18	13.5	odd	4
845.2.e.o.146.5		18	13.4	even	6
845.2.e.o.191.5		18	13.10	even	6
845.2.e.p.146.5		18	13.9	even	3
845.2.e.p.191.5		18	13.3	even	3
845.2.m.j.316.9		36	13.2	odd	12
845.2.m.j.316.10		36	13.11	odd	12
845.2.m.j.361.9		36	13.7	odd	12
845.2.m.j.361.10		36	13.6	odd	12
4225.2.a.bs.1.5		9	65.64	even	2
4225.2.a.bt.1.5		9	5.4	even	2
7605.2.a.cp.1.5		9	39.38	odd	2
7605.2.a.cs.1.5		9	3.2	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	845.2.a.n.1.5

Level	$845$
Weight	$2$
Character	845.1
Self dual	yes
Analytic conductor	$6.747$
Analytic rank	$0$
Dimension	$9$
CM	no
Inner twists	$1$