LMFDB - Embedded newform 8752.2.a.t.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.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:	$69.8850718490$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 8 x^{18} - 6 x^{17} + 187 x^{16} - 201 x^{15} - 1757 x^{14} + 3000 x^{13} + 8703 x^{12} + \cdots - 26 $ x^19 - 8x^18 - 6x^17 + 187x^16 - 201x^15 - 1757x^14 + 3000x^13 + 8703x^12 - 17349x^11 - 25437x^10 + 50901x^9 + 47173x^8 - 77021x^7 - 55807x^6 + 51357x^5 + 33800x^4 - 6455x^3 - 2126x^2 + 502x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 2188)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$2.70555$ of defining polynomial
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70555 q^{3} +2.14706 q^{5} -4.69407 q^{7} +4.32000 q^{9} +O(q^{10})$	$q+2.70555 q^{3} +2.14706 q^{5} -4.69407 q^{7} +4.32000 q^{9} -0.590837 q^{11} -5.74905 q^{13} +5.80897 q^{15} -4.15474 q^{17} +1.46951 q^{19} -12.7000 q^{21} +9.20632 q^{23} -0.390142 q^{25} +3.57132 q^{27} +7.19187 q^{29} +9.61556 q^{31} -1.59854 q^{33} -10.0784 q^{35} +1.37912 q^{37} -15.5544 q^{39} -1.27885 q^{41} +9.14522 q^{43} +9.27529 q^{45} +8.01716 q^{47} +15.0343 q^{49} -11.2409 q^{51} +8.13076 q^{53} -1.26856 q^{55} +3.97582 q^{57} +4.91482 q^{59} -5.28465 q^{61} -20.2784 q^{63} -12.3436 q^{65} -4.67199 q^{67} +24.9082 q^{69} +1.33994 q^{71} -3.84901 q^{73} -1.05555 q^{75} +2.77343 q^{77} -15.7596 q^{79} -3.29761 q^{81} +14.8295 q^{83} -8.92047 q^{85} +19.4580 q^{87} -12.7019 q^{89} +26.9864 q^{91} +26.0154 q^{93} +3.15511 q^{95} +7.39929 q^{97} -2.55242 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9	$ 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9} + 10 q^{11} - 4 q^{13} + 11 q^{15} - 32 q^{17} + 14 q^{19} - 8 q^{21} + 48 q^{23} + 14 q^{25} + 47 q^{27} - 10 q^{29} - 7 q^{31} - 16 q^{33} + 25 q^{35} + 8 q^{37} + 11 q^{39} - 9 q^{41} + 4 q^{43} + 8 q^{45} + 53 q^{47} + 6 q^{49} + q^{51} + 4 q^{53} + 31 q^{55} - 22 q^{57} + 14 q^{59} + 14 q^{61} + 45 q^{63} - 36 q^{65} + 29 q^{67} + 18 q^{69} + 41 q^{71} - 34 q^{73} + 61 q^{75} + 13 q^{77} + 12 q^{79} + 11 q^{81} + 87 q^{83} - 25 q^{85} + 39 q^{87} - 46 q^{89} + 35 q^{91} - 24 q^{93} + 44 q^{95} - 46 q^{97} + 51 q^{99}+O(q^{100}) $ 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9 + 10 * q^11 - 4 * q^13 + 11 * q^15 - 32 * q^17 + 14 * q^19 - 8 * q^21 + 48 * q^23 + 14 * q^25 + 47 * q^27 - 10 * q^29 - 7 * q^31 - 16 * q^33 + 25 * q^35 + 8 * q^37 + 11 * q^39 - 9 * q^41 + 4 * q^43 + 8 * q^45 + 53 * q^47 + 6 * q^49 + q^51 + 4 * q^53 + 31 * q^55 - 22 * q^57 + 14 * q^59 + 14 * q^61 + 45 * q^63 - 36 * q^65 + 29 * q^67 + 18 * q^69 + 41 * q^71 - 34 * q^73 + 61 * q^75 + 13 * q^77 + 12 * q^79 + 11 * q^81 + 87 * q^83 - 25 * q^85 + 39 * q^87 - 46 * q^89 + 35 * q^91 - 24 * q^93 + 44 * q^95 - 46 * q^97 + 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	2.70555	1.56205	0.781025	−	0.624500i	$-0.214697\pi$
$3$	2.70555	1.56205	0.781025	+	0.624500i	$0.214697\pi$
$4$	0	0
$5$	2.14706	0.960194	0.480097	−	0.877216i	$-0.340602\pi$
$5$	2.14706	0.960194	0.480097	+	0.877216i	$0.340602\pi$
$6$	0	0
$7$	−4.69407	−1.77419	−0.887095	−	0.461586i	$-0.847280\pi$
$7$	−4.69407	−1.77419	−0.887095	+	0.461586i	$0.847280\pi$
$8$	0	0
$9$	4.32000	1.44000
$10$	0	0
$11$	−0.590837	−0.178144	−0.0890721	−	0.996025i	$-0.528390\pi$
$11$	−0.590837	−0.178144	−0.0890721	+	0.996025i	$0.528390\pi$
$12$	0	0
$13$	−5.74905	−1.59450	−0.797250	−	0.603649i	$-0.793713\pi$
$13$	−5.74905	−1.59450	−0.797250	+	0.603649i	$0.793713\pi$
$14$	0	0
$15$	5.80897	1.49987
$16$	0	0
$17$	−4.15474	−1.00767	−0.503836	−	0.863799i	$-0.668078\pi$
$17$	−4.15474	−1.00767	−0.503836	+	0.863799i	$0.668078\pi$
$18$	0	0
$19$	1.46951	0.337128	0.168564	−	0.985691i	$-0.446087\pi$
$19$	1.46951	0.337128	0.168564	+	0.985691i	$0.446087\pi$
$20$	0	0
$21$	−12.7000	−2.77137
$22$	0	0
$23$	9.20632	1.91965	0.959825	−	0.280598i	$-0.0905328\pi$
$23$	9.20632	1.91965	0.959825	+	0.280598i	$0.0905328\pi$
$24$	0	0
$25$	−0.390142	−0.0780284
$26$	0	0
$27$	3.57132	0.687301
$28$	0	0
$29$	7.19187	1.33550	0.667749	−	0.744387i	$-0.267258\pi$
$29$	7.19187	1.33550	0.667749	+	0.744387i	$0.267258\pi$
$30$	0	0
$31$	9.61556	1.72701	0.863503	−	0.504344i	$-0.168266\pi$
$31$	9.61556	1.72701	0.863503	+	0.504344i	$0.168266\pi$
$32$	0	0
$33$	−1.59854	−0.278270
$34$	0	0
$35$	−10.0784	−1.70357
$36$	0	0
$37$	1.37912	0.226727	0.113363	−	0.993554i	$-0.463838\pi$
$37$	1.37912	0.226727	0.113363	+	0.993554i	$0.463838\pi$
$38$	0	0
$39$	−15.5544	−2.49069
$40$	0	0
$41$	−1.27885	−0.199722	−0.0998611	−	0.995001i	$-0.531840\pi$
$41$	−1.27885	−0.199722	−0.0998611	+	0.995001i	$0.531840\pi$
$42$	0	0
$43$	9.14522	1.39463	0.697317	−	0.716763i	$-0.254377\pi$
$43$	9.14522	1.39463	0.697317	+	0.716763i	$0.254377\pi$
$44$	0	0
$45$	9.27529	1.38268
$46$	0	0
$47$	8.01716	1.16942	0.584711	−	0.811241i	$-0.301208\pi$
$47$	8.01716	1.16942	0.584711	+	0.811241i	$0.301208\pi$
$48$	0	0
$49$	15.0343	2.14775
$50$	0	0
$51$	−11.2409	−1.57403
$52$	0	0
$53$	8.13076	1.11685	0.558423	−	0.829557i	$-0.311407\pi$
$53$	8.13076	1.11685	0.558423	+	0.829557i	$0.311407\pi$
$54$	0	0
$55$	−1.26856	−0.171053
$56$	0	0
$57$	3.97582	0.526610
$58$	0	0
$59$	4.91482	0.639855	0.319928	−	0.947442i	$-0.396341\pi$
$59$	4.91482	0.639855	0.319928	+	0.947442i	$0.396341\pi$
$60$	0	0
$61$	−5.28465	−0.676630	−0.338315	−	0.941033i	$-0.609857\pi$
$61$	−5.28465	−0.676630	−0.338315	+	0.941033i	$0.609857\pi$
$62$	0	0
$63$	−20.2784	−2.55483
$64$	0	0
$65$	−12.3436	−1.53103
$66$	0	0
$67$	−4.67199	−0.570774	−0.285387	−	0.958412i	$-0.592122\pi$
$67$	−4.67199	−0.570774	−0.285387	+	0.958412i	$0.592122\pi$
$68$	0	0
$69$	24.9082	2.99859
$70$	0	0
$71$	1.33994	0.159021	0.0795107	−	0.996834i	$-0.474664\pi$
$71$	1.33994	0.159021	0.0795107	+	0.996834i	$0.474664\pi$
$72$	0	0
$73$	−3.84901	−0.450493	−0.225246	−	0.974302i	$-0.572319\pi$
$73$	−3.84901	−0.450493	−0.225246	+	0.974302i	$0.572319\pi$
$74$	0	0
$75$	−1.05555	−0.121884
$76$	0	0
$77$	2.77343	0.316062
$78$	0	0
$79$	−15.7596	−1.77310	−0.886549	−	0.462634i	$-0.846904\pi$
$79$	−15.7596	−1.77310	−0.886549	+	0.462634i	$0.846904\pi$
$80$	0	0
$81$	−3.29761	−0.366401
$82$	0	0
$83$	14.8295	1.62775	0.813875	−	0.581040i	$-0.197354\pi$
$83$	14.8295	1.62775	0.813875	+	0.581040i	$0.197354\pi$
$84$	0	0
$85$	−8.92047	−0.967560
$86$	0	0
$87$	19.4580	2.08611
$88$	0	0
$89$	−12.7019	−1.34639	−0.673197	−	0.739463i	$-0.735080\pi$
$89$	−12.7019	−1.34639	−0.673197	+	0.739463i	$0.735080\pi$
$90$	0	0
$91$	26.9864	2.82895
$92$	0	0
$93$	26.0154	2.69767
$94$	0	0
$95$	3.15511	0.323708
$96$	0	0
$97$	7.39929	0.751284	0.375642	−	0.926765i	$-0.377422\pi$
$97$	7.39929	0.751284	0.375642	+	0.926765i	$0.377422\pi$
$98$	0	0
$99$	−2.55242	−0.256528
$100$	0	0
$101$	15.9619	1.58827	0.794135	−	0.607741i	$-0.207924\pi$
$101$	15.9619	1.58827	0.794135	+	0.607741i	$0.207924\pi$
$102$	0	0
$103$	13.8375	1.36345	0.681725	−	0.731609i	$-0.261230\pi$
$103$	13.8375	1.36345	0.681725	+	0.731609i	$0.261230\pi$
$104$	0	0
$105$	−27.2677	−2.66106
$106$	0	0
$107$	−5.31245	−0.513574	−0.256787	−	0.966468i	$-0.582664\pi$
$107$	−5.31245	−0.513574	−0.256787	+	0.966468i	$0.582664\pi$
$108$	0	0
$109$	9.15861	0.877236	0.438618	−	0.898674i	$-0.355468\pi$
$109$	9.15861	0.877236	0.438618	+	0.898674i	$0.355468\pi$
$110$	0	0
$111$	3.73129	0.354158
$112$	0	0
$113$	−3.84691	−0.361886	−0.180943	−	0.983494i	$-0.557915\pi$
$113$	−3.84691	−0.361886	−0.180943	+	0.983494i	$0.557915\pi$
$114$	0	0
$115$	19.7665	1.84324
$116$	0	0
$117$	−24.8359	−2.29608
$118$	0	0
$119$	19.5026	1.78780
$120$	0	0
$121$	−10.6509	−0.968265
$122$	0	0
$123$	−3.45998	−0.311976
$124$	0	0
$125$	−11.5729	−1.03512
$126$	0	0
$127$	8.37675	0.743317	0.371658	−	0.928370i	$-0.378789\pi$
$127$	8.37675	0.743317	0.371658	+	0.928370i	$0.378789\pi$
$128$	0	0
$129$	24.7428	2.17849
$130$	0	0
$131$	−9.70641	−0.848053	−0.424027	−	0.905650i	$-0.639384\pi$
$131$	−9.70641	−0.848053	−0.424027	+	0.905650i	$0.639384\pi$
$132$	0	0
$133$	−6.89796	−0.598129
$134$	0	0
$135$	7.66783	0.659942
$136$	0	0
$137$	1.99461	0.170411	0.0852055	−	0.996363i	$-0.472845\pi$
$137$	1.99461	0.170411	0.0852055	+	0.996363i	$0.472845\pi$
$138$	0	0
$139$	−10.6719	−0.905181	−0.452590	−	0.891718i	$-0.649500\pi$
$139$	−10.6719	−0.905181	−0.452590	+	0.891718i	$0.649500\pi$
$140$	0	0
$141$	21.6908	1.82670
$142$	0	0
$143$	3.39676	0.284051
$144$	0	0
$145$	15.4414	1.28234
$146$	0	0
$147$	40.6759	3.35489
$148$	0	0
$149$	−8.88883	−0.728201	−0.364101	−	0.931360i	$-0.618624\pi$
$149$	−8.88883	−0.728201	−0.364101	+	0.931360i	$0.618624\pi$
$150$	0	0
$151$	13.8575	1.12771	0.563855	−	0.825874i	$-0.309318\pi$
$151$	13.8575	1.12771	0.563855	+	0.825874i	$0.309318\pi$
$152$	0	0
$153$	−17.9485	−1.45105
$154$	0	0
$155$	20.6452	1.65826
$156$	0	0
$157$	19.6167	1.56558	0.782791	−	0.622285i	$-0.213796\pi$
$157$	19.6167	1.56558	0.782791	+	0.622285i	$0.213796\pi$
$158$	0	0
$159$	21.9982	1.74457
$160$	0	0
$161$	−43.2151	−3.40583
$162$	0	0
$163$	−6.86906	−0.538027	−0.269013	−	0.963136i	$-0.586698\pi$
$163$	−6.86906	−0.538027	−0.269013	+	0.963136i	$0.586698\pi$
$164$	0	0
$165$	−3.43216	−0.267193
$166$	0	0
$167$	4.02765	0.311669	0.155834	−	0.987783i	$-0.450193\pi$
$167$	4.02765	0.311669	0.155834	+	0.987783i	$0.450193\pi$
$168$	0	0
$169$	20.0516	1.54243
$170$	0	0
$171$	6.34826	0.485464
$172$	0	0
$173$	−16.9875	−1.29154	−0.645769	−	0.763533i	$-0.723463\pi$
$173$	−16.9875	−1.29154	−0.645769	+	0.763533i	$0.723463\pi$
$174$	0	0
$175$	1.83135	0.138437
$176$	0	0
$177$	13.2973	0.999485
$178$	0	0
$179$	13.1553	0.983276	0.491638	−	0.870800i	$-0.336398\pi$
$179$	13.1553	0.983276	0.491638	+	0.870800i	$0.336398\pi$
$180$	0	0
$181$	8.47748	0.630126	0.315063	−	0.949071i	$-0.397974\pi$
$181$	8.47748	0.630126	0.315063	+	0.949071i	$0.397974\pi$
$182$	0	0
$183$	−14.2979	−1.05693
$184$	0	0
$185$	2.96106	0.217701
$186$	0	0
$187$	2.45478	0.179511
$188$	0	0
$189$	−16.7640	−1.21940
$190$	0	0
$191$	4.26070	0.308293	0.154147	−	0.988048i	$-0.450737\pi$
$191$	4.26070	0.308293	0.154147	+	0.988048i	$0.450737\pi$
$192$	0	0
$193$	10.0626	0.724319	0.362160	−	0.932116i	$-0.382040\pi$
$193$	10.0626	0.724319	0.362160	+	0.932116i	$0.382040\pi$
$194$	0	0
$195$	−33.3961	−2.39154
$196$	0	0
$197$	6.79741	0.484296	0.242148	−	0.970239i	$-0.422148\pi$
$197$	6.79741	0.484296	0.242148	+	0.970239i	$0.422148\pi$
$198$	0	0
$199$	2.42106	0.171624	0.0858122	−	0.996311i	$-0.472652\pi$
$199$	2.42106	0.171624	0.0858122	+	0.996311i	$0.472652\pi$
$200$	0	0
$201$	−12.6403	−0.891578
$202$	0	0
$203$	−33.7591	−2.36943
$204$	0	0
$205$	−2.74576	−0.191772
$206$	0	0
$207$	39.7713	2.76430
$208$	0	0
$209$	−0.868239	−0.0600573
$210$	0	0
$211$	3.79643	0.261357	0.130679	−	0.991425i	$-0.458284\pi$
$211$	3.79643	0.261357	0.130679	+	0.991425i	$0.458284\pi$
$212$	0	0
$213$	3.62527	0.248399
$214$	0	0
$215$	19.6353	1.33912
$216$	0	0
$217$	−45.1361	−3.06404
$218$	0	0
$219$	−10.4137	−0.703692
$220$	0	0
$221$	23.8858	1.60673
$222$	0	0
$223$	−12.2596	−0.820966	−0.410483	−	0.911868i	$-0.634640\pi$
$223$	−12.2596	−0.820966	−0.410483	+	0.911868i	$0.634640\pi$
$224$	0	0
$225$	−1.68541	−0.112361
$226$	0	0
$227$	18.5387	1.23045	0.615227	−	0.788350i	$-0.289064\pi$
$227$	18.5387	1.23045	0.615227	+	0.788350i	$0.289064\pi$
$228$	0	0
$229$	2.50004	0.165207	0.0826036	−	0.996582i	$-0.473676\pi$
$229$	2.50004	0.165207	0.0826036	+	0.996582i	$0.473676\pi$
$230$	0	0
$231$	7.50365	0.493704
$232$	0	0
$233$	−3.32097	−0.217564	−0.108782	−	0.994066i	$-0.534695\pi$
$233$	−3.32097	−0.217564	−0.108782	+	0.994066i	$0.534695\pi$
$234$	0	0
$235$	17.2133	1.12287
$236$	0	0
$237$	−42.6385	−2.76967
$238$	0	0
$239$	−1.26627	−0.0819080	−0.0409540	−	0.999161i	$-0.513040\pi$
$239$	−1.26627	−0.0819080	−0.0409540	+	0.999161i	$0.513040\pi$
$240$	0	0
$241$	17.3866	1.11997	0.559986	−	0.828502i	$-0.310806\pi$
$241$	17.3866	1.11997	0.559986	+	0.828502i	$0.310806\pi$
$242$	0	0
$243$	−19.6358	−1.25964
$244$	0	0
$245$	32.2794	2.06226
$246$	0	0
$247$	−8.44827	−0.537550
$248$	0	0
$249$	40.1220	2.54263
$250$	0	0
$251$	−24.6180	−1.55388	−0.776938	−	0.629577i	$-0.783228\pi$
$251$	−24.6180	−1.55388	−0.776938	+	0.629577i	$0.783228\pi$
$252$	0	0
$253$	−5.43944	−0.341975
$254$	0	0
$255$	−24.1348	−1.51138
$256$	0	0
$257$	15.1970	0.947961	0.473981	−	0.880535i	$-0.342817\pi$
$257$	15.1970	0.947961	0.473981	+	0.880535i	$0.342817\pi$
$258$	0	0
$259$	−6.47370	−0.402256
$260$	0	0
$261$	31.0689	1.92312
$262$	0	0
$263$	11.4014	0.703038	0.351519	−	0.936181i	$-0.385665\pi$
$263$	11.4014	0.703038	0.351519	+	0.936181i	$0.385665\pi$
$264$	0	0
$265$	17.4572	1.07239
$266$	0	0
$267$	−34.3655	−2.10314
$268$	0	0
$269$	−18.2805	−1.11458	−0.557291	−	0.830317i	$-0.688159\pi$
$269$	−18.2805	−1.11458	−0.557291	+	0.830317i	$0.688159\pi$
$270$	0	0
$271$	24.6033	1.49454	0.747272	−	0.664518i	$-0.231363\pi$
$271$	24.6033	1.49454	0.747272	+	0.664518i	$0.231363\pi$
$272$	0	0
$273$	73.0132	4.41896
$274$	0	0
$275$	0.230511	0.0139003
$276$	0	0
$277$	−13.8532	−0.832360	−0.416180	−	0.909282i	$-0.636631\pi$
$277$	−13.8532	−0.832360	−0.416180	+	0.909282i	$0.636631\pi$
$278$	0	0
$279$	41.5392	2.48689
$280$	0	0
$281$	1.38849	0.0828304	0.0414152	−	0.999142i	$-0.486813\pi$
$281$	1.38849	0.0828304	0.0414152	+	0.999142i	$0.486813\pi$
$282$	0	0
$283$	5.99481	0.356355	0.178177	−	0.983998i	$-0.442980\pi$
$283$	5.99481	0.356355	0.178177	+	0.983998i	$0.442980\pi$
$284$	0	0
$285$	8.53632	0.505648
$286$	0	0
$287$	6.00299	0.354345
$288$	0	0
$289$	0.261861	0.0154036
$290$	0	0
$291$	20.0192	1.17354
$292$	0	0
$293$	10.9440	0.639356	0.319678	−	0.947526i	$-0.396425\pi$
$293$	10.9440	0.639356	0.319678	+	0.947526i	$0.396425\pi$
$294$	0	0
$295$	10.5524	0.614385
$296$	0	0
$297$	−2.11007	−0.122439
$298$	0	0
$299$	−52.9276	−3.06088
$300$	0	0
$301$	−42.9283	−2.47434
$302$	0	0
$303$	43.1858	2.48096
$304$	0	0
$305$	−11.3465	−0.649696
$306$	0	0
$307$	−10.3830	−0.592588	−0.296294	−	0.955097i	$-0.595751\pi$
$307$	−10.3830	−0.592588	−0.296294	+	0.955097i	$0.595751\pi$
$308$	0	0
$309$	37.4381	2.12978
$310$	0	0
$311$	−10.4132	−0.590477	−0.295239	−	0.955424i	$-0.595399\pi$
$311$	−10.4132	−0.590477	−0.295239	+	0.955424i	$0.595399\pi$
$312$	0	0
$313$	−23.1140	−1.30648	−0.653240	−	0.757151i	$-0.726591\pi$
$313$	−23.1140	−1.30648	−0.653240	+	0.757151i	$0.726591\pi$
$314$	0	0
$315$	−43.5388	−2.45313
$316$	0	0
$317$	17.9414	1.00769	0.503846	−	0.863794i	$-0.331918\pi$
$317$	17.9414	1.00769	0.503846	+	0.863794i	$0.331918\pi$
$318$	0	0
$319$	−4.24923	−0.237911
$320$	0	0
$321$	−14.3731	−0.802228
$322$	0	0
$323$	−6.10541	−0.339714
$324$	0	0
$325$	2.24295	0.124416
$326$	0	0
$327$	24.7791	1.37029
$328$	0	0
$329$	−37.6331	−2.07478
$330$	0	0
$331$	−12.6507	−0.695347	−0.347674	−	0.937616i	$-0.613028\pi$
$331$	−12.6507	−0.695347	−0.347674	+	0.937616i	$0.613028\pi$
$332$	0	0
$333$	5.95781	0.326486
$334$	0	0
$335$	−10.0310	−0.548054
$336$	0	0
$337$	34.9183	1.90212	0.951061	−	0.309002i	$-0.0999951\pi$
$337$	34.9183	1.90212	0.951061	+	0.309002i	$0.0999951\pi$
$338$	0	0
$339$	−10.4080	−0.565285
$340$	0	0
$341$	−5.68123	−0.307656
$342$	0	0
$343$	−37.7134	−2.03633
$344$	0	0
$345$	53.4793	2.87923
$346$	0	0
$347$	30.8185	1.65442	0.827212	−	0.561890i	$-0.189926\pi$
$347$	30.8185	1.65442	0.827212	+	0.561890i	$0.189926\pi$
$348$	0	0
$349$	−8.68565	−0.464932	−0.232466	−	0.972605i	$-0.574679\pi$
$349$	−8.68565	−0.464932	−0.232466	+	0.972605i	$0.574679\pi$
$350$	0	0
$351$	−20.5317	−1.09590
$352$	0	0
$353$	−29.0976	−1.54871	−0.774356	−	0.632751i	$-0.781926\pi$
$353$	−29.0976	−1.54871	−0.774356	+	0.632751i	$0.781926\pi$
$354$	0	0
$355$	2.87692	0.152691
$356$	0	0
$357$	52.7653	2.79264
$358$	0	0
$359$	22.5265	1.18890	0.594452	−	0.804131i	$-0.297369\pi$
$359$	22.5265	1.18890	0.594452	+	0.804131i	$0.297369\pi$
$360$	0	0
$361$	−16.8406	−0.886345
$362$	0	0
$363$	−28.8166	−1.51248
$364$	0	0
$365$	−8.26405	−0.432560
$366$	0	0
$367$	−13.4363	−0.701370	−0.350685	−	0.936493i	$-0.614051\pi$
$367$	−13.4363	−0.701370	−0.350685	+	0.936493i	$0.614051\pi$
$368$	0	0
$369$	−5.52461	−0.287600
$370$	0	0
$371$	−38.1663	−1.98150
$372$	0	0
$373$	−12.0426	−0.623541	−0.311771	−	0.950157i	$-0.600922\pi$
$373$	−12.0426	−0.623541	−0.311771	+	0.950157i	$0.600922\pi$
$374$	0	0
$375$	−31.3112	−1.61690
$376$	0	0
$377$	−41.3465	−2.12945
$378$	0	0
$379$	28.4063	1.45913	0.729567	−	0.683909i	$-0.239721\pi$
$379$	28.4063	1.45913	0.729567	+	0.683909i	$0.239721\pi$
$380$	0	0
$381$	22.6637	1.16110
$382$	0	0
$383$	−0.299679	−0.0153129	−0.00765645	−	0.999971i	$-0.502437\pi$
$383$	−0.299679	−0.0153129	−0.00765645	+	0.999971i	$0.502437\pi$
$384$	0	0
$385$	5.95472	0.303480
$386$	0	0
$387$	39.5073	2.00827
$388$	0	0
$389$	−21.8178	−1.10621	−0.553103	−	0.833113i	$-0.686556\pi$
$389$	−21.8178	−1.10621	−0.553103	+	0.833113i	$0.686556\pi$
$390$	0	0
$391$	−38.2499	−1.93438
$392$	0	0
$393$	−26.2612	−1.32470
$394$	0	0
$395$	−33.8369	−1.70252
$396$	0	0
$397$	7.00401	0.351521	0.175760	−	0.984433i	$-0.443762\pi$
$397$	7.00401	0.351521	0.175760	+	0.984433i	$0.443762\pi$
$398$	0	0
$399$	−18.6628	−0.934307
$400$	0	0
$401$	−8.06453	−0.402723	−0.201362	−	0.979517i	$-0.564537\pi$
$401$	−8.06453	−0.402723	−0.201362	+	0.979517i	$0.564537\pi$
$402$	0	0
$403$	−55.2804	−2.75371
$404$	0	0
$405$	−7.08016	−0.351816
$406$	0	0
$407$	−0.814838	−0.0403900
$408$	0	0
$409$	19.3853	0.958544	0.479272	−	0.877666i	$-0.340901\pi$
$409$	19.3853	0.958544	0.479272	+	0.877666i	$0.340901\pi$
$410$	0	0
$411$	5.39651	0.266190
$412$	0	0
$413$	−23.0705	−1.13522
$414$	0	0
$415$	31.8398	1.56295
$416$	0	0
$417$	−28.8734	−1.41394
$418$	0	0
$419$	21.0266	1.02722	0.513609	−	0.858024i	$-0.328308\pi$
$419$	21.0266	1.02722	0.513609	+	0.858024i	$0.328308\pi$
$420$	0	0
$421$	36.2193	1.76522	0.882611	−	0.470105i	$-0.155784\pi$
$421$	36.2193	1.76522	0.882611	+	0.470105i	$0.155784\pi$
$422$	0	0
$423$	34.6341	1.68397
$424$	0	0
$425$	1.62094	0.0786271
$426$	0	0
$427$	24.8065	1.20047
$428$	0	0
$429$	9.19009	0.443702
$430$	0	0
$431$	−23.9972	−1.15591	−0.577953	−	0.816070i	$-0.696148\pi$
$431$	−23.9972	−1.15591	−0.577953	+	0.816070i	$0.696148\pi$
$432$	0	0
$433$	22.2587	1.06969	0.534843	−	0.844952i	$-0.320371\pi$
$433$	22.2587	1.06969	0.534843	+	0.844952i	$0.320371\pi$
$434$	0	0
$435$	41.7774	2.00307
$436$	0	0
$437$	13.5287	0.647167
$438$	0	0
$439$	35.9300	1.71485	0.857423	−	0.514612i	$-0.172064\pi$
$439$	35.9300	1.71485	0.857423	+	0.514612i	$0.172064\pi$
$440$	0	0
$441$	64.9480	3.09276
$442$	0	0
$443$	−18.0219	−0.856245	−0.428122	−	0.903721i	$-0.640825\pi$
$443$	−18.0219	−0.856245	−0.428122	+	0.903721i	$0.640825\pi$
$444$	0	0
$445$	−27.2716	−1.29280
$446$	0	0
$447$	−24.0492	−1.13749
$448$	0	0
$449$	15.4794	0.730517	0.365258	−	0.930906i	$-0.380981\pi$
$449$	15.4794	0.730517	0.365258	+	0.930906i	$0.380981\pi$
$450$	0	0
$451$	0.755590	0.0355794
$452$	0	0
$453$	37.4922	1.76154
$454$	0	0
$455$	57.9415	2.71634
$456$	0	0
$457$	8.50970	0.398067	0.199033	−	0.979993i	$-0.436220\pi$
$457$	8.50970	0.398067	0.199033	+	0.979993i	$0.436220\pi$
$458$	0	0
$459$	−14.8379	−0.692574
$460$	0	0
$461$	−28.9586	−1.34873	−0.674367	−	0.738396i	$-0.735584\pi$
$461$	−28.9586	−1.34873	−0.674367	+	0.738396i	$0.735584\pi$
$462$	0	0
$463$	−1.09061	−0.0506850	−0.0253425	−	0.999679i	$-0.508068\pi$
$463$	−1.09061	−0.0506850	−0.0253425	+	0.999679i	$0.508068\pi$
$464$	0	0
$465$	55.8565	2.59028
$466$	0	0
$467$	−13.3623	−0.618333	−0.309167	−	0.951008i	$-0.600050\pi$
$467$	−13.3623	−0.618333	−0.309167	+	0.951008i	$0.600050\pi$
$468$	0	0
$469$	21.9306	1.01266
$470$	0	0
$471$	53.0739	2.44552
$472$	0	0
$473$	−5.40334	−0.248446
$474$	0	0
$475$	−0.573316	−0.0263055
$476$	0	0
$477$	35.1249	1.60826
$478$	0	0
$479$	−5.84759	−0.267183	−0.133592	−	0.991036i	$-0.542651\pi$
$479$	−5.84759	−0.267183	−0.133592	+	0.991036i	$0.542651\pi$
$480$	0	0
$481$	−7.92866	−0.361516
$482$	0	0
$483$	−116.921	−5.32007
$484$	0	0
$485$	15.8867	0.721378
$486$	0	0
$487$	9.40839	0.426335	0.213168	−	0.977016i	$-0.431622\pi$
$487$	9.40839	0.426335	0.213168	+	0.977016i	$0.431622\pi$
$488$	0	0
$489$	−18.5846	−0.840424
$490$	0	0
$491$	−23.8787	−1.07763	−0.538814	−	0.842425i	$-0.681128\pi$
$491$	−23.8787	−1.07763	−0.538814	+	0.842425i	$0.681128\pi$
$492$	0	0
$493$	−29.8804	−1.34574
$494$	0	0
$495$	−5.48019	−0.246316
$496$	0	0
$497$	−6.28976	−0.282134
$498$	0	0
$499$	−29.2196	−1.30805	−0.654025	−	0.756473i	$-0.726921\pi$
$499$	−29.2196	−1.30805	−0.654025	+	0.756473i	$0.726921\pi$
$500$	0	0
$501$	10.8970	0.486842
$502$	0	0
$503$	−38.9542	−1.73688	−0.868440	−	0.495794i	$-0.834877\pi$
$503$	−38.9542	−1.73688	−0.868440	+	0.495794i	$0.834877\pi$
$504$	0	0
$505$	34.2712	1.52505
$506$	0	0
$507$	54.2507	2.40936
$508$	0	0
$509$	−0.811469	−0.0359677	−0.0179839	−	0.999838i	$-0.505725\pi$
$509$	−0.811469	−0.0359677	−0.0179839	+	0.999838i	$0.505725\pi$
$510$	0	0
$511$	18.0675	0.799260
$512$	0	0
$513$	5.24807	0.231708
$514$	0	0
$515$	29.7099	1.30918
$516$	0	0
$517$	−4.73684	−0.208326
$518$	0	0
$519$	−45.9606	−2.01745
$520$	0	0
$521$	−17.3710	−0.761038	−0.380519	−	0.924773i	$-0.624255\pi$
$521$	−17.3710	−0.761038	−0.380519	+	0.924773i	$0.624255\pi$
$522$	0	0
$523$	37.4711	1.63850	0.819249	−	0.573438i	$-0.194391\pi$
$523$	37.4711	1.63850	0.819249	+	0.573438i	$0.194391\pi$
$524$	0	0
$525$	4.95482	0.216246
$526$	0	0
$527$	−39.9501	−1.74026
$528$	0	0
$529$	61.7563	2.68506
$530$	0	0
$531$	21.2320	0.921391
$532$	0	0
$533$	7.35216	0.318457
$534$	0	0
$535$	−11.4061	−0.493130
$536$	0	0
$537$	35.5924	1.53593
$538$	0	0
$539$	−8.88280	−0.382609
$540$	0	0
$541$	−15.9974	−0.687780	−0.343890	−	0.939010i	$-0.611745\pi$
$541$	−15.9974	−0.687780	−0.343890	+	0.939010i	$0.611745\pi$
$542$	0	0
$543$	22.9362	0.984289
$544$	0	0
$545$	19.6641	0.842316
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	0	0
$549$	−22.8297	−0.974347
$550$	0	0
$551$	10.5685	0.450233
$552$	0	0
$553$	73.9768	3.14581
$554$	0	0
$555$	8.01129	0.340060
$556$	0	0
$557$	−39.9182	−1.69139	−0.845695	−	0.533667i	$-0.820814\pi$
$557$	−39.9182	−1.69139	−0.845695	+	0.533667i	$0.820814\pi$
$558$	0	0
$559$	−52.5764	−2.22374
$560$	0	0
$561$	6.64152	0.280405
$562$	0	0
$563$	31.8813	1.34363	0.671817	−	0.740717i	$-0.265514\pi$
$563$	31.8813	1.34363	0.671817	+	0.740717i	$0.265514\pi$
$564$	0	0
$565$	−8.25953	−0.347481
$566$	0	0
$567$	15.4792	0.650065
$568$	0	0
$569$	−4.19999	−0.176073	−0.0880363	−	0.996117i	$-0.528059\pi$
$569$	−4.19999	−0.176073	−0.0880363	+	0.996117i	$0.528059\pi$
$570$	0	0
$571$	−15.3317	−0.641611	−0.320805	−	0.947145i	$-0.603954\pi$
$571$	−15.3317	−0.641611	−0.320805	+	0.947145i	$0.603954\pi$
$572$	0	0
$573$	11.5275	0.481570
$574$	0	0
$575$	−3.59177	−0.149787
$576$	0	0
$577$	−44.1755	−1.83905	−0.919526	−	0.393030i	$-0.871427\pi$
$577$	−44.1755	−1.83905	−0.919526	+	0.393030i	$0.871427\pi$
$578$	0	0
$579$	27.2248	1.13142
$580$	0	0
$581$	−69.6107	−2.88794
$582$	0	0
$583$	−4.80396	−0.198960
$584$	0	0
$585$	−53.3241	−2.20468
$586$	0	0
$587$	7.59967	0.313672	0.156836	−	0.987625i	$-0.449871\pi$
$587$	7.59967	0.313672	0.156836	+	0.987625i	$0.449871\pi$
$588$	0	0
$589$	14.1301	0.582221
$590$	0	0
$591$	18.3907	0.756494
$592$	0	0
$593$	−37.1183	−1.52426	−0.762132	−	0.647422i	$-0.775847\pi$
$593$	−37.1183	−1.52426	−0.762132	+	0.647422i	$0.775847\pi$
$594$	0	0
$595$	41.8733	1.71664
$596$	0	0
$597$	6.55029	0.268086
$598$	0	0
$599$	5.78766	0.236477	0.118239	−	0.992985i	$-0.462275\pi$
$599$	5.78766	0.236477	0.118239	+	0.992985i	$0.462275\pi$
$600$	0	0
$601$	40.0019	1.63171	0.815856	−	0.578255i	$-0.196266\pi$
$601$	40.0019	1.63171	0.815856	+	0.578255i	$0.196266\pi$
$602$	0	0
$603$	−20.1830	−0.821915
$604$	0	0
$605$	−22.8681	−0.929721
$606$	0	0
$607$	9.15068	0.371415	0.185707	−	0.982605i	$-0.440542\pi$
$607$	9.15068	0.371415	0.185707	+	0.982605i	$0.440542\pi$
$608$	0	0
$609$	−91.3370	−3.70116
$610$	0	0
$611$	−46.0911	−1.86465
$612$	0	0
$613$	−38.0698	−1.53763	−0.768813	−	0.639474i	$-0.779152\pi$
$613$	−38.0698	−1.53763	−0.768813	+	0.639474i	$0.779152\pi$
$614$	0	0
$615$	−7.42878	−0.299557
$616$	0	0
$617$	−24.2439	−0.976023	−0.488011	−	0.872837i	$-0.662278\pi$
$617$	−24.2439	−0.976023	−0.488011	+	0.872837i	$0.662278\pi$
$618$	0	0
$619$	−0.304762	−0.0122494	−0.00612471	−	0.999981i	$-0.501950\pi$
$619$	−0.304762	−0.0122494	−0.00612471	+	0.999981i	$0.501950\pi$
$620$	0	0
$621$	32.8787	1.31938
$622$	0	0
$623$	59.6234	2.38876
$624$	0	0
$625$	−22.8971	−0.915883
$626$	0	0
$627$	−2.34906	−0.0938125
$628$	0	0
$629$	−5.72990	−0.228466
$630$	0	0
$631$	−11.9553	−0.475935	−0.237967	−	0.971273i	$-0.576481\pi$
$631$	−11.9553	−0.475935	−0.237967	+	0.971273i	$0.576481\pi$
$632$	0	0
$633$	10.2714	0.408253
$634$	0	0
$635$	17.9854	0.713728
$636$	0	0
$637$	−86.4328	−3.42459
$638$	0	0
$639$	5.78853	0.228991
$640$	0	0
$641$	12.7547	0.503782	0.251891	−	0.967756i	$-0.418948\pi$
$641$	12.7547	0.503782	0.251891	+	0.967756i	$0.418948\pi$
$642$	0	0
$643$	−22.1093	−0.871907	−0.435954	−	0.899969i	$-0.643589\pi$
$643$	−22.1093	−0.871907	−0.435954	+	0.899969i	$0.643589\pi$
$644$	0	0
$645$	53.1243	2.09177
$646$	0	0
$647$	42.8684	1.68533	0.842666	−	0.538436i	$-0.180985\pi$
$647$	42.8684	1.68533	0.842666	+	0.538436i	$0.180985\pi$
$648$	0	0
$649$	−2.90386	−0.113986
$650$	0	0
$651$	−122.118	−4.78618
$652$	0	0
$653$	−23.0620	−0.902484	−0.451242	−	0.892402i	$-0.649019\pi$
$653$	−23.0620	−0.902484	−0.451242	+	0.892402i	$0.649019\pi$
$654$	0	0
$655$	−20.8402	−0.814295
$656$	0	0
$657$	−16.6277	−0.648709
$658$	0	0
$659$	−24.7761	−0.965138	−0.482569	−	0.875858i	$-0.660296\pi$
$659$	−24.7761	−0.965138	−0.482569	+	0.875858i	$0.660296\pi$
$660$	0	0
$661$	29.6199	1.15208	0.576039	−	0.817422i	$-0.304597\pi$
$661$	29.6199	1.15208	0.576039	+	0.817422i	$0.304597\pi$
$662$	0	0
$663$	64.6243	2.50980
$664$	0	0
$665$	−14.8103	−0.574319
$666$	0	0
$667$	66.2107	2.56369
$668$	0	0
$669$	−33.1690	−1.28239
$670$	0	0
$671$	3.12237	0.120538
$672$	0	0
$673$	−27.5103	−1.06045	−0.530223	−	0.847858i	$-0.677892\pi$
$673$	−27.5103	−1.06045	−0.530223	+	0.847858i	$0.677892\pi$
$674$	0	0
$675$	−1.39332	−0.0536290
$676$	0	0
$677$	43.7448	1.68125	0.840624	−	0.541619i	$-0.182188\pi$
$677$	43.7448	1.68125	0.840624	+	0.541619i	$0.182188\pi$
$678$	0	0
$679$	−34.7328	−1.33292
$680$	0	0
$681$	50.1573	1.92203
$682$	0	0
$683$	−34.9503	−1.33734	−0.668668	−	0.743561i	$-0.733135\pi$
$683$	−34.9503	−1.33734	−0.668668	+	0.743561i	$0.733135\pi$
$684$	0	0
$685$	4.28254	0.163627
$686$	0	0
$687$	6.76398	0.258062
$688$	0	0
$689$	−46.7442	−1.78081
$690$	0	0
$691$	−35.4820	−1.34980	−0.674899	−	0.737910i	$-0.735813\pi$
$691$	−35.4820	−1.34980	−0.674899	+	0.737910i	$0.735813\pi$
$692$	0	0
$693$	11.9812	0.455129
$694$	0	0
$695$	−22.9132	−0.869149
$696$	0	0
$697$	5.31327	0.201255
$698$	0	0
$699$	−8.98505	−0.339846
$700$	0	0
$701$	−45.8462	−1.73159	−0.865794	−	0.500400i	$-0.833186\pi$
$701$	−45.8462	−1.73159	−0.865794	+	0.500400i	$0.833186\pi$
$702$	0	0
$703$	2.02663	0.0764358
$704$	0	0
$705$	46.5714	1.75398
$706$	0	0
$707$	−74.9263	−2.81789
$708$	0	0
$709$	19.1166	0.717939	0.358969	−	0.933349i	$-0.383128\pi$
$709$	19.1166	0.717939	0.358969	+	0.933349i	$0.383128\pi$
$710$	0	0
$711$	−68.0816	−2.55326
$712$	0	0
$713$	88.5239	3.31525
$714$	0	0
$715$	7.29303	0.272744
$716$	0	0
$717$	−3.42595	−0.127944
$718$	0	0
$719$	−10.4573	−0.389993	−0.194997	−	0.980804i	$-0.562470\pi$
$719$	−10.4573	−0.389993	−0.194997	+	0.980804i	$0.562470\pi$
$720$	0	0
$721$	−64.9542	−2.41902
$722$	0	0
$723$	47.0404	1.74945
$724$	0	0
$725$	−2.80585	−0.104207
$726$	0	0
$727$	13.4503	0.498844	0.249422	−	0.968395i	$-0.419759\pi$
$727$	13.4503	0.498844	0.249422	+	0.968395i	$0.419759\pi$
$728$	0	0
$729$	−43.2328	−1.60122
$730$	0	0
$731$	−37.9960	−1.40533
$732$	0	0
$733$	−40.8592	−1.50917	−0.754584	−	0.656203i	$-0.772161\pi$
$733$	−40.8592	−1.50917	−0.754584	+	0.656203i	$0.772161\pi$
$734$	0	0
$735$	87.3336	3.22135
$736$	0	0
$737$	2.76039	0.101680
$738$	0	0
$739$	−32.2376	−1.18588	−0.592939	−	0.805247i	$-0.702033\pi$
$739$	−32.2376	−1.18588	−0.592939	+	0.805247i	$0.702033\pi$
$740$	0	0
$741$	−22.8572	−0.839680
$742$	0	0
$743$	−15.5927	−0.572040	−0.286020	−	0.958224i	$-0.592332\pi$
$743$	−15.5927	−0.572040	−0.286020	+	0.958224i	$0.592332\pi$
$744$	0	0
$745$	−19.0848	−0.699214
$746$	0	0
$747$	64.0634	2.34396
$748$	0	0
$749$	24.9370	0.911178
$750$	0	0
$751$	−43.8336	−1.59951	−0.799755	−	0.600327i	$-0.795037\pi$
$751$	−43.8336	−1.59951	−0.799755	+	0.600327i	$0.795037\pi$
$752$	0	0
$753$	−66.6053	−2.42723
$754$	0	0
$755$	29.7529	1.08282
$756$	0	0
$757$	−9.28134	−0.337336	−0.168668	−	0.985673i	$-0.553947\pi$
$757$	−9.28134	−0.337336	−0.168668	+	0.985673i	$0.553947\pi$
$758$	0	0
$759$	−14.7167	−0.534181
$760$	0	0
$761$	−42.5353	−1.54190	−0.770952	−	0.636893i	$-0.780219\pi$
$761$	−42.5353	−1.54190	−0.770952	+	0.636893i	$0.780219\pi$
$762$	0	0
$763$	−42.9911	−1.55638
$764$	0	0
$765$	−38.5364	−1.39329
$766$	0	0
$767$	−28.2556	−1.02025
$768$	0	0
$769$	40.0893	1.44566	0.722829	−	0.691027i	$-0.242842\pi$
$769$	40.0893	1.44566	0.722829	+	0.691027i	$0.242842\pi$
$770$	0	0
$771$	41.1162	1.48076
$772$	0	0
$773$	−19.6626	−0.707216	−0.353608	−	0.935394i	$-0.615045\pi$
$773$	−19.6626	−0.707216	−0.353608	+	0.935394i	$0.615045\pi$
$774$	0	0
$775$	−3.75143	−0.134756
$776$	0	0
$777$	−17.5149	−0.628344
$778$	0	0
$779$	−1.87927	−0.0673319
$780$	0	0
$781$	−0.791686	−0.0283287
$782$	0	0
$783$	25.6845	0.917889
$784$	0	0
$785$	42.1181	1.50326
$786$	0	0
$787$	−23.6239	−0.842103	−0.421051	−	0.907037i	$-0.638339\pi$
$787$	−23.6239	−0.842103	−0.421051	+	0.907037i	$0.638339\pi$
$788$	0	0
$789$	30.8470	1.09818
$790$	0	0
$791$	18.0576	0.642055
$792$	0	0
$793$	30.3818	1.07889
$794$	0	0
$795$	47.2313	1.67512
$796$	0	0
$797$	9.45081	0.334765	0.167382	−	0.985892i	$-0.446469\pi$
$797$	9.45081	0.334765	0.167382	+	0.985892i	$0.446469\pi$
$798$	0	0
$799$	−33.3092	−1.17839
$800$	0	0
$801$	−54.8720	−1.93881
$802$	0	0
$803$	2.27414	0.0802527
$804$	0	0
$805$	−92.7853	−3.27025
$806$	0	0
$807$	−49.4588	−1.74103
$808$	0	0
$809$	31.8431	1.11955	0.559773	−	0.828646i	$-0.310888\pi$
$809$	31.8431	1.11955	0.559773	+	0.828646i	$0.310888\pi$
$810$	0	0
$811$	−28.3073	−0.994003	−0.497001	−	0.867750i	$-0.665566\pi$
$811$	−28.3073	−0.994003	−0.497001	+	0.867750i	$0.665566\pi$
$812$	0	0
$813$	66.5655	2.33455
$814$	0	0
$815$	−14.7483	−0.516610
$816$	0	0
$817$	13.4390	0.470169
$818$	0	0
$819$	116.581	4.07368
$820$	0	0
$821$	−42.9066	−1.49745	−0.748726	−	0.662880i	$-0.769334\pi$
$821$	−42.9066	−1.49745	−0.748726	+	0.662880i	$0.769334\pi$
$822$	0	0
$823$	−18.5656	−0.647157	−0.323579	−	0.946201i	$-0.604886\pi$
$823$	−18.5656	−0.647157	−0.323579	+	0.946201i	$0.604886\pi$
$824$	0	0
$825$	0.623658	0.0217130
$826$	0	0
$827$	−6.00267	−0.208733	−0.104367	−	0.994539i	$-0.533282\pi$
$827$	−6.00267	−0.208733	−0.104367	+	0.994539i	$0.533282\pi$
$828$	0	0
$829$	1.15215	0.0400159	0.0200080	−	0.999800i	$-0.493631\pi$
$829$	1.15215	0.0400159	0.0200080	+	0.999800i	$0.493631\pi$
$830$	0	0
$831$	−37.4806	−1.30019
$832$	0	0
$833$	−62.4634	−2.16423
$834$	0	0
$835$	8.64759	0.299262
$836$	0	0
$837$	34.3402	1.18697
$838$	0	0
$839$	−13.7860	−0.475947	−0.237973	−	0.971272i	$-0.576483\pi$
$839$	−13.7860	−0.475947	−0.237973	+	0.971272i	$0.576483\pi$
$840$	0	0
$841$	22.7231	0.783554
$842$	0	0
$843$	3.75663	0.129385
$844$	0	0
$845$	43.0520	1.48103
$846$	0	0
$847$	49.9961	1.71789
$848$	0	0
$849$	16.2193	0.556644
$850$	0	0
$851$	12.6967	0.435236
$852$	0	0
$853$	42.9006	1.46889	0.734445	−	0.678668i	$-0.237443\pi$
$853$	42.9006	1.46889	0.734445	+	0.678668i	$0.237443\pi$
$854$	0	0
$855$	13.6301	0.466139
$856$	0	0
$857$	14.4266	0.492802	0.246401	−	0.969168i	$-0.420752\pi$
$857$	14.4266	0.492802	0.246401	+	0.969168i	$0.420752\pi$
$858$	0	0
$859$	29.6901	1.01301	0.506506	−	0.862236i	$-0.330937\pi$
$859$	29.6901	1.01301	0.506506	+	0.862236i	$0.330937\pi$
$860$	0	0
$861$	16.2414	0.553505
$862$	0	0
$863$	17.9177	0.609925	0.304963	−	0.952364i	$-0.401356\pi$
$863$	17.9177	0.609925	0.304963	+	0.952364i	$0.401356\pi$
$864$	0	0
$865$	−36.4732	−1.24013
$866$	0	0
$867$	0.708479	0.0240612
$868$	0	0
$869$	9.31139	0.315867
$870$	0	0
$871$	26.8595	0.910100
$872$	0	0
$873$	31.9649	1.08185
$874$	0	0
$875$	54.3242	1.83649
$876$	0	0
$877$	40.1479	1.35570	0.677849	−	0.735201i	$-0.262912\pi$
$877$	40.1479	1.35570	0.677849	+	0.735201i	$0.262912\pi$
$878$	0	0
$879$	29.6096	0.998706
$880$	0	0
$881$	32.1429	1.08292	0.541461	−	0.840726i	$-0.317871\pi$
$881$	32.1429	1.08292	0.541461	+	0.840726i	$0.317871\pi$
$882$	0	0
$883$	43.7505	1.47232	0.736161	−	0.676807i	$-0.236637\pi$
$883$	43.7505	1.47232	0.736161	+	0.676807i	$0.236637\pi$
$884$	0	0
$885$	28.5500	0.959699
$886$	0	0
$887$	−26.7741	−0.898986	−0.449493	−	0.893284i	$-0.648395\pi$
$887$	−26.7741	−0.898986	−0.449493	+	0.893284i	$0.648395\pi$
$888$	0	0
$889$	−39.3210	−1.31879
$890$	0	0
$891$	1.94835	0.0652722
$892$	0	0
$893$	11.7813	0.394245
$894$	0	0
$895$	28.2453	0.944136
$896$	0	0
$897$	−143.198	−4.78125
$898$	0	0
$899$	69.1539	2.30641
$900$	0	0
$901$	−33.7812	−1.12541
$902$	0	0
$903$	−116.145	−3.86505
$904$	0	0
$905$	18.2016	0.605043
$906$	0	0
$907$	17.6942	0.587525	0.293762	−	0.955878i	$-0.405093\pi$
$907$	17.6942	0.587525	0.293762	+	0.955878i	$0.405093\pi$
$908$	0	0
$909$	68.9555	2.28711
$910$	0	0
$911$	41.8408	1.38625	0.693124	−	0.720818i	$-0.256234\pi$
$911$	41.8408	1.38625	0.693124	+	0.720818i	$0.256234\pi$
$912$	0	0
$913$	−8.76183	−0.289974
$914$	0	0
$915$	−30.6984	−1.01486
$916$	0	0
$917$	45.5626	1.50461
$918$	0	0
$919$	−1.91865	−0.0632905	−0.0316452	−	0.999499i	$-0.510075\pi$
$919$	−1.91865	−0.0632905	−0.0316452	+	0.999499i	$0.510075\pi$
$920$	0	0
$921$	−28.0917	−0.925652
$922$	0	0
$923$	−7.70338	−0.253560
$924$	0	0
$925$	−0.538055	−0.0176911
$926$	0	0
$927$	59.7780	1.96337
$928$	0	0
$929$	52.9575	1.73748	0.868740	−	0.495268i	$-0.164930\pi$
$929$	52.9575	1.73748	0.868740	+	0.495268i	$0.164930\pi$
$930$	0	0
$931$	22.0929	0.724067
$932$	0	0
$933$	−28.1734	−0.922355
$934$	0	0
$935$	5.27055	0.172365
$936$	0	0
$937$	17.6093	0.575270	0.287635	−	0.957740i	$-0.407131\pi$
$937$	17.6093	0.575270	0.287635	+	0.957740i	$0.407131\pi$
$938$	0	0
$939$	−62.5361	−2.04079
$940$	0	0
$941$	−21.4295	−0.698583	−0.349292	−	0.937014i	$-0.613578\pi$
$941$	−21.4295	−0.698583	−0.349292	+	0.937014i	$0.613578\pi$
$942$	0	0
$943$	−11.7735	−0.383397
$944$	0	0
$945$	−35.9933	−1.17086
$946$	0	0
$947$	35.2679	1.14605	0.573027	−	0.819537i	$-0.305769\pi$
$947$	35.2679	1.14605	0.573027	+	0.819537i	$0.305769\pi$
$948$	0	0
$949$	22.1282	0.718311
$950$	0	0
$951$	48.5414	1.57406
$952$	0	0
$953$	36.9565	1.19714	0.598570	−	0.801071i	$-0.295736\pi$
$953$	36.9565	1.19714	0.598570	+	0.801071i	$0.295736\pi$
$954$	0	0
$955$	9.14797	0.296021
$956$	0	0
$957$	−11.4965	−0.371629
$958$	0	0
$959$	−9.36283	−0.302341
$960$	0	0
$961$	61.4590	1.98255
$962$	0	0
$963$	−22.9498	−0.739546
$964$	0	0
$965$	21.6049	0.695486
$966$	0	0
$967$	−18.3859	−0.591250	−0.295625	−	0.955304i	$-0.595528\pi$
$967$	−18.3859	−0.591250	−0.295625	+	0.955304i	$0.595528\pi$
$968$	0	0
$969$	−16.5185	−0.530651
$970$	0	0
$971$	0.173527	0.00556874	0.00278437	−	0.999996i	$-0.499114\pi$
$971$	0.173527	0.00556874	0.00278437	+	0.999996i	$0.499114\pi$
$972$	0	0
$973$	50.0947	1.60596
$974$	0	0
$975$	6.06841	0.194345
$976$	0	0
$977$	9.78931	0.313188	0.156594	−	0.987663i	$-0.449949\pi$
$977$	9.78931	0.313188	0.156594	+	0.987663i	$0.449949\pi$
$978$	0	0
$979$	7.50474	0.239852
$980$	0	0
$981$	39.5652	1.26322
$982$	0	0
$983$	−47.3555	−1.51040	−0.755202	−	0.655492i	$-0.772461\pi$
$983$	−47.3555	−1.51040	−0.755202	+	0.655492i	$0.772461\pi$
$984$	0	0
$985$	14.5944	0.465018
$986$	0	0
$987$	−101.818	−3.24091
$988$	0	0
$989$	84.1938	2.67721
$990$	0	0
$991$	34.1753	1.08562	0.542808	−	0.839857i	$-0.317361\pi$
$991$	34.1753	1.08562	0.542808	+	0.839857i	$0.317361\pi$
$992$	0	0
$993$	−34.2272	−1.08617
$994$	0	0
$995$	5.19815	0.164793
$996$	0	0
$997$	53.5988	1.69749	0.848746	−	0.528800i	$-0.177358\pi$
$997$	53.5988	1.69749	0.848746	+	0.528800i	$0.177358\pi$
$998$	0	0
$999$	4.92529	0.155829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.t.1.17		19
4.3	odd	2		2188.2.a.d.1.3	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2188.2.a.d.1.3	✓	19	4.3	odd	2
8752.2.a.t.1.17		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70555 q^{3} +2.14706 q^{5} -4.69407 q^{7} +4.32000 q^{9} +O(q^{10})\)	\(q+2.70555 q^{3} +2.14706 q^{5} -4.69407 q^{7} +4.32000 q^{9} -0.590837 q^{11} -5.74905 q^{13} +5.80897 q^{15} -4.15474 q^{17} +1.46951 q^{19} -12.7000 q^{21} +9.20632 q^{23} -0.390142 q^{25} +3.57132 q^{27} +7.19187 q^{29} +9.61556 q^{31} -1.59854 q^{33} -10.0784 q^{35} +1.37912 q^{37} -15.5544 q^{39} -1.27885 q^{41} +9.14522 q^{43} +9.27529 q^{45} +8.01716 q^{47} +15.0343 q^{49} -11.2409 q^{51} +8.13076 q^{53} -1.26856 q^{55} +3.97582 q^{57} +4.91482 q^{59} -5.28465 q^{61} -20.2784 q^{63} -12.3436 q^{65} -4.67199 q^{67} +24.9082 q^{69} +1.33994 q^{71} -3.84901 q^{73} -1.05555 q^{75} +2.77343 q^{77} -15.7596 q^{79} -3.29761 q^{81} +14.8295 q^{83} -8.92047 q^{85} +19.4580 q^{87} -12.7019 q^{89} +26.9864 q^{91} +26.0154 q^{93} +3.15511 q^{95} +7.39929 q^{97} -2.55242 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9	\( 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9} + 10 q^{11} - 4 q^{13} + 11 q^{15} - 32 q^{17} + 14 q^{19} - 8 q^{21} + 48 q^{23} + 14 q^{25} + 47 q^{27} - 10 q^{29} - 7 q^{31} - 16 q^{33} + 25 q^{35} + 8 q^{37} + 11 q^{39} - 9 q^{41} + 4 q^{43} + 8 q^{45} + 53 q^{47} + 6 q^{49} + q^{51} + 4 q^{53} + 31 q^{55} - 22 q^{57} + 14 q^{59} + 14 q^{61} + 45 q^{63} - 36 q^{65} + 29 q^{67} + 18 q^{69} + 41 q^{71} - 34 q^{73} + 61 q^{75} + 13 q^{77} + 12 q^{79} + 11 q^{81} + 87 q^{83} - 25 q^{85} + 39 q^{87} - 46 q^{89} + 35 q^{91} - 24 q^{93} + 44 q^{95} - 46 q^{97} + 51 q^{99}+O(q^{100}) \) 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9 + 10 * q^11 - 4 * q^13 + 11 * q^15 - 32 * q^17 + 14 * q^19 - 8 * q^21 + 48 * q^23 + 14 * q^25 + 47 * q^27 - 10 * q^29 - 7 * q^31 - 16 * q^33 + 25 * q^35 + 8 * q^37 + 11 * q^39 - 9 * q^41 + 4 * q^43 + 8 * q^45 + 53 * q^47 + 6 * q^49 + q^51 + 4 * q^53 + 31 * q^55 - 22 * q^57 + 14 * q^59 + 14 * q^61 + 45 * q^63 - 36 * q^65 + 29 * q^67 + 18 * q^69 + 41 * q^71 - 34 * q^73 + 61 * q^75 + 13 * q^77 + 12 * q^79 + 11 * q^81 + 87 * q^83 - 25 * q^85 + 39 * q^87 - 46 * q^89 + 35 * q^91 - 24 * q^93 + 44 * q^95 - 46 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more