LMFDB - Embedded newform 2224.2.a.o.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2224,2,Mod(1,2224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2224, 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("2224.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2224 = 2^{4} \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2224.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:	$17.7587294095$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 $ x^7 - x^6 - 11x^5 + 8x^4 + 35x^3 - 10x^2 - 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 139)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.308806$ of defining polynomial
Character	$\chi$	$=$	2224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.39811 q^{3} +2.83261 q^{5} -4.16776 q^{7} -1.04529 q^{9} +O(q^{10})$	$q+1.39811 q^{3} +2.83261 q^{5} -4.16776 q^{7} -1.04529 q^{9} -1.93164 q^{11} +4.12937 q^{13} +3.96031 q^{15} +1.03363 q^{17} +7.34157 q^{19} -5.82699 q^{21} +4.09697 q^{23} +3.02371 q^{25} -5.65576 q^{27} +4.49067 q^{29} +1.28386 q^{31} -2.70065 q^{33} -11.8056 q^{35} -7.77849 q^{37} +5.77331 q^{39} +11.5603 q^{41} +7.87867 q^{43} -2.96089 q^{45} +6.84669 q^{47} +10.3702 q^{49} +1.44513 q^{51} +3.68473 q^{53} -5.47160 q^{55} +10.2643 q^{57} -1.24534 q^{59} +0.281096 q^{61} +4.35649 q^{63} +11.6969 q^{65} +8.51902 q^{67} +5.72803 q^{69} +6.40133 q^{71} -5.92608 q^{73} +4.22748 q^{75} +8.05062 q^{77} +8.65375 q^{79} -4.77152 q^{81} -15.8366 q^{83} +2.92789 q^{85} +6.27846 q^{87} -3.23844 q^{89} -17.2102 q^{91} +1.79497 q^{93} +20.7958 q^{95} -7.27325 q^{97} +2.01912 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) $ 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9	$ 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9} - 2 q^{11} + 6 q^{13} + 3 q^{15} + 5 q^{17} + 10 q^{19} - 5 q^{21} + q^{23} + 14 q^{25} + 11 q^{27} + 30 q^{29} + 20 q^{31} - 20 q^{33} + 7 q^{35} + 6 q^{37} - 11 q^{39} + 19 q^{41} + 12 q^{43} + 27 q^{45} + 3 q^{47} - 8 q^{49} - 23 q^{51} + 38 q^{53} - 7 q^{55} - 19 q^{57} + 14 q^{59} + 4 q^{61} + 18 q^{63} + 10 q^{65} - 9 q^{67} + 9 q^{69} - 24 q^{71} - 5 q^{73} + 21 q^{75} - 13 q^{77} - 8 q^{79} + 39 q^{81} + 9 q^{83} - 22 q^{85} + 25 q^{87} + 10 q^{89} - 7 q^{91} - 15 q^{93} + 21 q^{95} - 5 q^{97} + 31 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9 - 2 * q^11 + 6 * q^13 + 3 * q^15 + 5 * q^17 + 10 * q^19 - 5 * q^21 + q^23 + 14 * q^25 + 11 * q^27 + 30 * q^29 + 20 * q^31 - 20 * q^33 + 7 * q^35 + 6 * q^37 - 11 * q^39 + 19 * q^41 + 12 * q^43 + 27 * q^45 + 3 * q^47 - 8 * q^49 - 23 * q^51 + 38 * q^53 - 7 * q^55 - 19 * q^57 + 14 * q^59 + 4 * q^61 + 18 * q^63 + 10 * q^65 - 9 * q^67 + 9 * q^69 - 24 * q^71 - 5 * q^73 + 21 * q^75 - 13 * q^77 - 8 * q^79 + 39 * q^81 + 9 * q^83 - 22 * q^85 + 25 * q^87 + 10 * q^89 - 7 * q^91 - 15 * q^93 + 21 * q^95 - 5 * q^97 + 31 * 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
$3$	1.39811	0.807200	0.403600	−	0.914936i	$-0.367759\pi$
$3$	1.39811	0.807200	0.403600	+	0.914936i	$0.367759\pi$
$4$	0	0
$5$	2.83261	1.26678	0.633392	−	0.773831i	$-0.281662\pi$
$5$	2.83261	1.26678	0.633392	+	0.773831i	$0.281662\pi$
$6$	0	0
$7$	−4.16776	−1.57526	−0.787632	−	0.616146i	$-0.788693\pi$
$7$	−4.16776	−1.57526	−0.787632	+	0.616146i	$0.788693\pi$
$8$	0	0
$9$	−1.04529	−0.348428
$10$	0	0
$11$	−1.93164	−0.582412	−0.291206	−	0.956660i	$-0.594057\pi$
$11$	−1.93164	−0.582412	−0.291206	+	0.956660i	$0.594057\pi$
$12$	0	0
$13$	4.12937	1.14528	0.572640	−	0.819807i	$-0.305919\pi$
$13$	4.12937	1.14528	0.572640	+	0.819807i	$0.305919\pi$
$14$	0	0
$15$	3.96031	1.02255
$16$	0	0
$17$	1.03363	0.250693	0.125347	−	0.992113i	$-0.459996\pi$
$17$	1.03363	0.250693	0.125347	+	0.992113i	$0.459996\pi$
$18$	0	0
$19$	7.34157	1.68427	0.842135	−	0.539266i	$-0.181298\pi$
$19$	7.34157	1.68427	0.842135	+	0.539266i	$0.181298\pi$
$20$	0	0
$21$	−5.82699	−1.27155
$22$	0	0
$23$	4.09697	0.854278	0.427139	−	0.904186i	$-0.359521\pi$
$23$	4.09697	0.854278	0.427139	+	0.904186i	$0.359521\pi$
$24$	0	0
$25$	3.02371	0.604741
$26$	0	0
$27$	−5.65576	−1.08845
$28$	0	0
$29$	4.49067	0.833897	0.416948	−	0.908930i	$-0.363100\pi$
$29$	4.49067	0.833897	0.416948	+	0.908930i	$0.363100\pi$
$30$	0	0
$31$	1.28386	0.230587	0.115294	−	0.993331i	$-0.463219\pi$
$31$	1.28386	0.230587	0.115294	+	0.993331i	$0.463219\pi$
$32$	0	0
$33$	−2.70065	−0.470123
$34$	0	0
$35$	−11.8056	−1.99552
$36$	0	0
$37$	−7.77849	−1.27878	−0.639388	−	0.768884i	$-0.720812\pi$
$37$	−7.77849	−1.27878	−0.639388	+	0.768884i	$0.720812\pi$
$38$	0	0
$39$	5.77331	0.924470
$40$	0	0
$41$	11.5603	1.80541	0.902706	−	0.430258i	$-0.141577\pi$
$41$	11.5603	1.80541	0.902706	+	0.430258i	$0.141577\pi$
$42$	0	0
$43$	7.87867	1.20149	0.600743	−	0.799442i	$-0.294871\pi$
$43$	7.87867	1.20149	0.600743	+	0.799442i	$0.294871\pi$
$44$	0	0
$45$	−2.96089	−0.441383
$46$	0	0
$47$	6.84669	0.998692	0.499346	−	0.866403i	$-0.333574\pi$
$47$	6.84669	0.998692	0.499346	+	0.866403i	$0.333574\pi$
$48$	0	0
$49$	10.3702	1.48145
$50$	0	0
$51$	1.44513	0.202359
$52$	0	0
$53$	3.68473	0.506136	0.253068	−	0.967449i	$-0.418560\pi$
$53$	3.68473	0.506136	0.253068	+	0.967449i	$0.418560\pi$
$54$	0	0
$55$	−5.47160	−0.737791
$56$	0	0
$57$	10.2643	1.35954
$58$	0	0
$59$	−1.24534	−0.162129	−0.0810647	−	0.996709i	$-0.525832\pi$
$59$	−1.24534	−0.162129	−0.0810647	+	0.996709i	$0.525832\pi$
$60$	0	0
$61$	0.281096	0.0359907	0.0179953	−	0.999838i	$-0.494272\pi$
$61$	0.281096	0.0359907	0.0179953	+	0.999838i	$0.494272\pi$
$62$	0	0
$63$	4.35649	0.548867
$64$	0	0
$65$	11.6969	1.45082
$66$	0	0
$67$	8.51902	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$67$	8.51902	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$68$	0	0
$69$	5.72803	0.689573
$70$	0	0
$71$	6.40133	0.759698	0.379849	−	0.925048i	$-0.375976\pi$
$71$	6.40133	0.759698	0.379849	+	0.925048i	$0.375976\pi$
$72$	0	0
$73$	−5.92608	−0.693595	−0.346798	−	0.937940i	$-0.612731\pi$
$73$	−5.92608	−0.693595	−0.346798	+	0.937940i	$0.612731\pi$
$74$	0	0
$75$	4.22748	0.488147
$76$	0	0
$77$	8.05062	0.917453
$78$	0	0
$79$	8.65375	0.973623	0.486811	−	0.873507i	$-0.338160\pi$
$79$	8.65375	0.973623	0.486811	+	0.873507i	$0.338160\pi$
$80$	0	0
$81$	−4.77152	−0.530169
$82$	0	0
$83$	−15.8366	−1.73829	−0.869146	−	0.494555i	$-0.835331\pi$
$83$	−15.8366	−1.73829	−0.869146	+	0.494555i	$0.835331\pi$
$84$	0	0
$85$	2.92789	0.317574
$86$	0	0
$87$	6.27846	0.673121
$88$	0	0
$89$	−3.23844	−0.343274	−0.171637	−	0.985160i	$-0.554906\pi$
$89$	−3.23844	−0.343274	−0.171637	+	0.985160i	$0.554906\pi$
$90$	0	0
$91$	−17.2102	−1.80412
$92$	0	0
$93$	1.79497	0.186130
$94$	0	0
$95$	20.7958	2.13361
$96$	0	0
$97$	−7.27325	−0.738487	−0.369243	−	0.929333i	$-0.620383\pi$
$97$	−7.27325	−0.738487	−0.369243	+	0.929333i	$0.620383\pi$
$98$	0	0
$99$	2.01912	0.202929
$100$	0	0
$101$	−13.0456	−1.29809	−0.649044	−	0.760751i	$-0.724831\pi$
$101$	−13.0456	−1.29809	−0.649044	+	0.760751i	$0.724831\pi$
$102$	0	0
$103$	11.9048	1.17302	0.586510	−	0.809942i	$-0.300502\pi$
$103$	11.9048	1.17302	0.586510	+	0.809942i	$0.300502\pi$
$104$	0	0
$105$	−16.5056	−1.61078
$106$	0	0
$107$	−16.1058	−1.55700	−0.778502	−	0.627642i	$-0.784020\pi$
$107$	−16.1058	−1.55700	−0.778502	+	0.627642i	$0.784020\pi$
$108$	0	0
$109$	13.1464	1.25919	0.629597	−	0.776922i	$-0.283220\pi$
$109$	13.1464	1.25919	0.629597	+	0.776922i	$0.283220\pi$
$110$	0	0
$111$	−10.8752	−1.03223
$112$	0	0
$113$	−14.7595	−1.38846	−0.694228	−	0.719755i	$-0.744254\pi$
$113$	−14.7595	−1.38846	−0.694228	+	0.719755i	$0.744254\pi$
$114$	0	0
$115$	11.6052	1.08219
$116$	0	0
$117$	−4.31636	−0.399048
$118$	0	0
$119$	−4.30793	−0.394908
$120$	0	0
$121$	−7.26876	−0.660796
$122$	0	0
$123$	16.1626	1.45733
$124$	0	0
$125$	−5.59808	−0.500707
$126$	0	0
$127$	−11.0142	−0.977350	−0.488675	−	0.872466i	$-0.662520\pi$
$127$	−11.0142	−0.977350	−0.488675	+	0.872466i	$0.662520\pi$
$128$	0	0
$129$	11.0153	0.969840
$130$	0	0
$131$	5.25402	0.459046	0.229523	−	0.973303i	$-0.426283\pi$
$131$	5.25402	0.459046	0.229523	+	0.973303i	$0.426283\pi$
$132$	0	0
$133$	−30.5979	−2.65317
$134$	0	0
$135$	−16.0206	−1.37883
$136$	0	0
$137$	−1.68115	−0.143630	−0.0718151	−	0.997418i	$-0.522879\pi$
$137$	−1.68115	−0.143630	−0.0718151	+	0.997418i	$0.522879\pi$
$138$	0	0
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	0	0
$141$	9.57243	0.806144
$142$	0	0
$143$	−7.97646	−0.667025
$144$	0	0
$145$	12.7203	1.05637
$146$	0	0
$147$	14.4987	1.19583
$148$	0	0
$149$	13.9257	1.14084	0.570421	−	0.821353i	$-0.306780\pi$
$149$	13.9257	1.14084	0.570421	+	0.821353i	$0.306780\pi$
$150$	0	0
$151$	−8.63888	−0.703022	−0.351511	−	0.936184i	$-0.614332\pi$
$151$	−8.63888	−0.703022	−0.351511	+	0.936184i	$0.614332\pi$
$152$	0	0
$153$	−1.08044	−0.0873486
$154$	0	0
$155$	3.63667	0.292104
$156$	0	0
$157$	10.4362	0.832900	0.416450	−	0.909159i	$-0.363274\pi$
$157$	10.4362	0.832900	0.416450	+	0.909159i	$0.363274\pi$
$158$	0	0
$159$	5.15166	0.408553
$160$	0	0
$161$	−17.0752	−1.34571
$162$	0	0
$163$	3.73406	0.292474	0.146237	−	0.989250i	$-0.453284\pi$
$163$	3.73406	0.292474	0.146237	+	0.989250i	$0.453284\pi$
$164$	0	0
$165$	−7.64991	−0.595544
$166$	0	0
$167$	19.7503	1.52832	0.764161	−	0.645025i	$-0.223153\pi$
$167$	19.7503	1.52832	0.764161	+	0.645025i	$0.223153\pi$
$168$	0	0
$169$	4.05166	0.311666
$170$	0	0
$171$	−7.67403	−0.586848
$172$	0	0
$173$	−7.51428	−0.571300	−0.285650	−	0.958334i	$-0.592210\pi$
$173$	−7.51428	−0.571300	−0.285650	+	0.958334i	$0.592210\pi$
$174$	0	0
$175$	−12.6021	−0.952627
$176$	0	0
$177$	−1.74112	−0.130871
$178$	0	0
$179$	−24.2625	−1.81346	−0.906732	−	0.421708i	$-0.861431\pi$
$179$	−24.2625	−1.81346	−0.906732	+	0.421708i	$0.861431\pi$
$180$	0	0
$181$	−3.06020	−0.227463	−0.113732	−	0.993512i	$-0.536280\pi$
$181$	−3.06020	−0.227463	−0.113732	+	0.993512i	$0.536280\pi$
$182$	0	0
$183$	0.393004	0.0290517
$184$	0	0
$185$	−22.0335	−1.61993
$186$	0	0
$187$	−1.99661	−0.146007
$188$	0	0
$189$	23.5718	1.71460
$190$	0	0
$191$	−8.83544	−0.639310	−0.319655	−	0.947534i	$-0.603567\pi$
$191$	−8.83544	−0.639310	−0.319655	+	0.947534i	$0.603567\pi$
$192$	0	0
$193$	14.9309	1.07475	0.537374	−	0.843344i	$-0.319416\pi$
$193$	14.9309	1.07475	0.537374	+	0.843344i	$0.319416\pi$
$194$	0	0
$195$	16.3536	1.17110
$196$	0	0
$197$	−11.8981	−0.847702	−0.423851	−	0.905732i	$-0.639322\pi$
$197$	−11.8981	−0.847702	−0.423851	+	0.905732i	$0.639322\pi$
$198$	0	0
$199$	4.40723	0.312420	0.156210	−	0.987724i	$-0.450072\pi$
$199$	4.40723	0.312420	0.156210	+	0.987724i	$0.450072\pi$
$200$	0	0
$201$	11.9105	0.840104
$202$	0	0
$203$	−18.7160	−1.31361
$204$	0	0
$205$	32.7458	2.28707
$206$	0	0
$207$	−4.28251	−0.297655
$208$	0	0
$209$	−14.1813	−0.980940
$210$	0	0
$211$	−21.4218	−1.47474	−0.737368	−	0.675492i	$-0.763931\pi$
$211$	−21.4218	−1.47474	−0.737368	+	0.675492i	$0.763931\pi$
$212$	0	0
$213$	8.94977	0.613228
$214$	0	0
$215$	22.3173	1.52202
$216$	0	0
$217$	−5.35079	−0.363236
$218$	0	0
$219$	−8.28532	−0.559870
$220$	0	0
$221$	4.26825	0.287114
$222$	0	0
$223$	7.93581	0.531421	0.265711	−	0.964053i	$-0.414393\pi$
$223$	7.93581	0.531421	0.265711	+	0.964053i	$0.414393\pi$
$224$	0	0
$225$	−3.16064	−0.210709
$226$	0	0
$227$	−15.9354	−1.05767	−0.528836	−	0.848724i	$-0.677371\pi$
$227$	−15.9354	−1.05767	−0.528836	+	0.848724i	$0.677371\pi$
$228$	0	0
$229$	13.4712	0.890203	0.445102	−	0.895480i	$-0.353168\pi$
$229$	13.4712	0.890203	0.445102	+	0.895480i	$0.353168\pi$
$230$	0	0
$231$	11.2557	0.740568
$232$	0	0
$233$	18.4315	1.20749	0.603745	−	0.797177i	$-0.293674\pi$
$233$	18.4315	1.20749	0.603745	+	0.797177i	$0.293674\pi$
$234$	0	0
$235$	19.3940	1.26513
$236$	0	0
$237$	12.0989	0.785908
$238$	0	0
$239$	−0.889441	−0.0575331	−0.0287666	−	0.999586i	$-0.509158\pi$
$239$	−0.889441	−0.0575331	−0.0287666	+	0.999586i	$0.509158\pi$
$240$	0	0
$241$	−11.7554	−0.757233	−0.378616	−	0.925554i	$-0.623600\pi$
$241$	−11.7554	−0.757233	−0.378616	+	0.925554i	$0.623600\pi$
$242$	0	0
$243$	10.2962	0.660499
$244$	0	0
$245$	29.3747	1.87668
$246$	0	0
$247$	30.3160	1.92896
$248$	0	0
$249$	−22.1413	−1.40315
$250$	0	0
$251$	14.4551	0.912397	0.456199	−	0.889878i	$-0.349211\pi$
$251$	14.4551	0.912397	0.456199	+	0.889878i	$0.349211\pi$
$252$	0	0
$253$	−7.91389	−0.497542
$254$	0	0
$255$	4.09351	0.256346
$256$	0	0
$257$	17.2989	1.07907	0.539537	−	0.841962i	$-0.318599\pi$
$257$	17.2989	1.07907	0.539537	+	0.841962i	$0.318599\pi$
$258$	0	0
$259$	32.4189	2.01441
$260$	0	0
$261$	−4.69403	−0.290553
$262$	0	0
$263$	−4.00876	−0.247191	−0.123595	−	0.992333i	$-0.539442\pi$
$263$	−4.00876	−0.247191	−0.123595	+	0.992333i	$0.539442\pi$
$264$	0	0
$265$	10.4374	0.641165
$266$	0	0
$267$	−4.52770	−0.277091
$268$	0	0
$269$	28.0301	1.70902	0.854511	−	0.519433i	$-0.173857\pi$
$269$	28.0301	1.70902	0.854511	+	0.519433i	$0.173857\pi$
$270$	0	0
$271$	−16.3058	−0.990507	−0.495253	−	0.868749i	$-0.664925\pi$
$271$	−16.3058	−0.990507	−0.495253	+	0.868749i	$0.664925\pi$
$272$	0	0
$273$	−24.0617	−1.45628
$274$	0	0
$275$	−5.84072	−0.352209
$276$	0	0
$277$	−26.3241	−1.58166	−0.790831	−	0.612035i	$-0.790351\pi$
$277$	−26.3241	−1.58166	−0.790831	+	0.612035i	$0.790351\pi$
$278$	0	0
$279$	−1.34199	−0.0803431
$280$	0	0
$281$	−7.58167	−0.452285	−0.226142	−	0.974094i	$-0.572611\pi$
$281$	−7.58167	−0.452285	−0.226142	+	0.974094i	$0.572611\pi$
$282$	0	0
$283$	14.2221	0.845417	0.422709	−	0.906266i	$-0.361079\pi$
$283$	14.2221	0.845417	0.422709	+	0.906266i	$0.361079\pi$
$284$	0	0
$285$	29.0749	1.72225
$286$	0	0
$287$	−48.1804	−2.84400
$288$	0	0
$289$	−15.9316	−0.937153
$290$	0	0
$291$	−10.1688	−0.596107
$292$	0	0
$293$	−4.68983	−0.273983	−0.136991	−	0.990572i	$-0.543743\pi$
$293$	−4.68983	−0.273983	−0.136991	+	0.990572i	$0.543743\pi$
$294$	0	0
$295$	−3.52757	−0.205383
$296$	0	0
$297$	10.9249	0.633927
$298$	0	0
$299$	16.9179	0.978388
$300$	0	0
$301$	−32.8364	−1.89266
$302$	0	0
$303$	−18.2392	−1.04782
$304$	0	0
$305$	0.796238	0.0455924
$306$	0	0
$307$	−10.6916	−0.610199	−0.305100	−	0.952320i	$-0.598690\pi$
$307$	−10.6916	−0.610199	−0.305100	+	0.952320i	$0.598690\pi$
$308$	0	0
$309$	16.6443	0.946861
$310$	0	0
$311$	21.0455	1.19338	0.596690	−	0.802472i	$-0.296482\pi$
$311$	21.0455	1.19338	0.596690	+	0.802472i	$0.296482\pi$
$312$	0	0
$313$	−14.2033	−0.802819	−0.401409	−	0.915899i	$-0.631480\pi$
$313$	−14.2033	−0.802819	−0.401409	+	0.915899i	$0.631480\pi$
$314$	0	0
$315$	12.3403	0.695295
$316$	0	0
$317$	2.85738	0.160487	0.0802433	−	0.996775i	$-0.474430\pi$
$317$	2.85738	0.160487	0.0802433	+	0.996775i	$0.474430\pi$
$318$	0	0
$319$	−8.67437	−0.485672
$320$	0	0
$321$	−22.5177	−1.25681
$322$	0	0
$323$	7.58849	0.422235
$324$	0	0
$325$	12.4860	0.692598
$326$	0	0
$327$	18.3801	1.01642
$328$	0	0
$329$	−28.5353	−1.57320
$330$	0	0
$331$	−12.1172	−0.666019	−0.333010	−	0.942923i	$-0.608064\pi$
$331$	−12.1172	−0.666019	−0.333010	+	0.942923i	$0.608064\pi$
$332$	0	0
$333$	8.13074	0.445562
$334$	0	0
$335$	24.1311	1.31842
$336$	0	0
$337$	−12.4929	−0.680532	−0.340266	−	0.940329i	$-0.610517\pi$
$337$	−12.4929	−0.680532	−0.340266	+	0.940329i	$0.610517\pi$
$338$	0	0
$339$	−20.6354	−1.12076
$340$	0	0
$341$	−2.47995	−0.134297
$342$	0	0
$343$	−14.0461	−0.758418
$344$	0	0
$345$	16.2253	0.873540
$346$	0	0
$347$	11.1030	0.596039	0.298020	−	0.954560i	$-0.403674\pi$
$347$	11.1030	0.596039	0.298020	+	0.954560i	$0.403674\pi$
$348$	0	0
$349$	16.3372	0.874509	0.437254	−	0.899338i	$-0.355951\pi$
$349$	16.3372	0.874509	0.437254	+	0.899338i	$0.355951\pi$
$350$	0	0
$351$	−23.3547	−1.24658
$352$	0	0
$353$	−26.2175	−1.39542	−0.697709	−	0.716382i	$-0.745797\pi$
$353$	−26.2175	−1.39542	−0.697709	+	0.716382i	$0.745797\pi$
$354$	0	0
$355$	18.1325	0.962373
$356$	0	0
$357$	−6.02297	−0.318769
$358$	0	0
$359$	−18.6480	−0.984204	−0.492102	−	0.870537i	$-0.663771\pi$
$359$	−18.6480	−0.984204	−0.492102	+	0.870537i	$0.663771\pi$
$360$	0	0
$361$	34.8986	1.83677
$362$	0	0
$363$	−10.1625	−0.533394
$364$	0	0
$365$	−16.7863	−0.878636
$366$	0	0
$367$	3.92562	0.204916	0.102458	−	0.994737i	$-0.467329\pi$
$367$	3.92562	0.204916	0.102458	+	0.994737i	$0.467329\pi$
$368$	0	0
$369$	−12.0838	−0.629057
$370$	0	0
$371$	−15.3570	−0.797297
$372$	0	0
$373$	14.3279	0.741870	0.370935	−	0.928659i	$-0.379037\pi$
$373$	14.3279	0.741870	0.370935	+	0.928659i	$0.379037\pi$
$374$	0	0
$375$	−7.82673	−0.404171
$376$	0	0
$377$	18.5436	0.955045
$378$	0	0
$379$	−5.65835	−0.290650	−0.145325	−	0.989384i	$-0.546423\pi$
$379$	−5.65835	−0.290650	−0.145325	+	0.989384i	$0.546423\pi$
$380$	0	0
$381$	−15.3990	−0.788917
$382$	0	0
$383$	−4.23323	−0.216308	−0.108154	−	0.994134i	$-0.534494\pi$
$383$	−4.23323	−0.216308	−0.108154	+	0.994134i	$0.534494\pi$
$384$	0	0
$385$	22.8043	1.16221
$386$	0	0
$387$	−8.23546	−0.418632
$388$	0	0
$389$	−5.21409	−0.264365	−0.132182	−	0.991225i	$-0.542198\pi$
$389$	−5.21409	−0.264365	−0.132182	+	0.991225i	$0.542198\pi$
$390$	0	0
$391$	4.23477	0.214162
$392$	0	0
$393$	7.34570	0.370542
$394$	0	0
$395$	24.5127	1.23337
$396$	0	0
$397$	9.63263	0.483448	0.241724	−	0.970345i	$-0.422287\pi$
$397$	9.63263	0.483448	0.241724	+	0.970345i	$0.422287\pi$
$398$	0	0
$399$	−42.7792	−2.14164
$400$	0	0
$401$	24.8722	1.24206	0.621030	−	0.783787i	$-0.286714\pi$
$401$	24.8722	1.24206	0.621030	+	0.783787i	$0.286714\pi$
$402$	0	0
$403$	5.30151	0.264087
$404$	0	0
$405$	−13.5159	−0.671610
$406$	0	0
$407$	15.0253	0.744775
$408$	0	0
$409$	−28.4449	−1.40651	−0.703255	−	0.710938i	$-0.748271\pi$
$409$	−28.4449	−1.40651	−0.703255	+	0.710938i	$0.748271\pi$
$410$	0	0
$411$	−2.35043	−0.115938
$412$	0	0
$413$	5.19027	0.255397
$414$	0	0
$415$	−44.8590	−2.20204
$416$	0	0
$417$	−1.39811	−0.0684658
$418$	0	0
$419$	4.59075	0.224273	0.112136	−	0.993693i	$-0.464231\pi$
$419$	4.59075	0.224273	0.112136	+	0.993693i	$0.464231\pi$
$420$	0	0
$421$	20.9586	1.02146	0.510729	−	0.859742i	$-0.329375\pi$
$421$	20.9586	1.02146	0.510729	+	0.859742i	$0.329375\pi$
$422$	0	0
$423$	−7.15674	−0.347973
$424$	0	0
$425$	3.12541	0.151604
$426$	0	0
$427$	−1.17154	−0.0566948
$428$	0	0
$429$	−11.1520	−0.538423
$430$	0	0
$431$	−14.1727	−0.682676	−0.341338	−	0.939941i	$-0.610880\pi$
$431$	−14.1727	−0.682676	−0.341338	+	0.939941i	$0.610880\pi$
$432$	0	0
$433$	7.32714	0.352120	0.176060	−	0.984379i	$-0.443665\pi$
$433$	7.32714	0.352120	0.176060	+	0.984379i	$0.443665\pi$
$434$	0	0
$435$	17.7844	0.852699
$436$	0	0
$437$	30.0782	1.43884
$438$	0	0
$439$	−3.07088	−0.146565	−0.0732825	−	0.997311i	$-0.523347\pi$
$439$	−3.07088	−0.146565	−0.0732825	+	0.997311i	$0.523347\pi$
$440$	0	0
$441$	−10.8398	−0.516181
$442$	0	0
$443$	−14.9982	−0.712586	−0.356293	−	0.934374i	$-0.615960\pi$
$443$	−14.9982	−0.712586	−0.356293	+	0.934374i	$0.615960\pi$
$444$	0	0
$445$	−9.17325	−0.434854
$446$	0	0
$447$	19.4697	0.920887
$448$	0	0
$449$	−22.4064	−1.05742	−0.528711	−	0.848802i	$-0.677324\pi$
$449$	−22.4064	−1.05742	−0.528711	+	0.848802i	$0.677324\pi$
$450$	0	0
$451$	−22.3303	−1.05149
$452$	0	0
$453$	−12.0781	−0.567479
$454$	0	0
$455$	−48.7498	−2.28543
$456$	0	0
$457$	−12.9317	−0.604919	−0.302460	−	0.953162i	$-0.597808\pi$
$457$	−12.9317	−0.604919	−0.302460	+	0.953162i	$0.597808\pi$
$458$	0	0
$459$	−5.84598	−0.272867
$460$	0	0
$461$	−39.7115	−1.84955	−0.924774	−	0.380517i	$-0.875746\pi$
$461$	−39.7115	−1.84955	−0.924774	+	0.380517i	$0.875746\pi$
$462$	0	0
$463$	−21.4987	−0.999130	−0.499565	−	0.866276i	$-0.666507\pi$
$463$	−21.4987	−0.999130	−0.499565	+	0.866276i	$0.666507\pi$
$464$	0	0
$465$	5.08447	0.235786
$466$	0	0
$467$	19.2688	0.891655	0.445827	−	0.895119i	$-0.352910\pi$
$467$	19.2688	0.891655	0.445827	+	0.895119i	$0.352910\pi$
$468$	0	0
$469$	−35.5052	−1.63948
$470$	0	0
$471$	14.5910	0.672317
$472$	0	0
$473$	−15.2188	−0.699761
$474$	0	0
$475$	22.1987	1.01855
$476$	0	0
$477$	−3.85159	−0.176352
$478$	0	0
$479$	23.7465	1.08500	0.542502	−	0.840055i	$-0.317477\pi$
$479$	23.7465	1.08500	0.542502	+	0.840055i	$0.317477\pi$
$480$	0	0
$481$	−32.1202	−1.46456
$482$	0	0
$483$	−23.8730	−1.08626
$484$	0	0
$485$	−20.6023	−0.935503
$486$	0	0
$487$	27.3504	1.23937	0.619683	−	0.784852i	$-0.287261\pi$
$487$	27.3504	1.23937	0.619683	+	0.784852i	$0.287261\pi$
$488$	0	0
$489$	5.22062	0.236085
$490$	0	0
$491$	29.0333	1.31025	0.655126	−	0.755519i	$-0.272615\pi$
$491$	29.0333	1.31025	0.655126	+	0.755519i	$0.272615\pi$
$492$	0	0
$493$	4.64171	0.209052
$494$	0	0
$495$	5.71938	0.257067
$496$	0	0
$497$	−26.6792	−1.19672
$498$	0	0
$499$	−26.6880	−1.19472	−0.597359	−	0.801974i	$-0.703783\pi$
$499$	−26.6880	−1.19472	−0.597359	+	0.801974i	$0.703783\pi$
$500$	0	0
$501$	27.6131	1.23366
$502$	0	0
$503$	−7.41588	−0.330658	−0.165329	−	0.986238i	$-0.552869\pi$
$503$	−7.41588	−0.330658	−0.165329	+	0.986238i	$0.552869\pi$
$504$	0	0
$505$	−36.9532	−1.64440
$506$	0	0
$507$	5.66466	0.251577
$508$	0	0
$509$	36.1455	1.60212	0.801060	−	0.598583i	$-0.204270\pi$
$509$	36.1455	1.60212	0.801060	+	0.598583i	$0.204270\pi$
$510$	0	0
$511$	24.6985	1.09260
$512$	0	0
$513$	−41.5221	−1.83325
$514$	0	0
$515$	33.7219	1.48596
$516$	0	0
$517$	−13.2254	−0.581651
$518$	0	0
$519$	−10.5058	−0.461153
$520$	0	0
$521$	33.9636	1.48797	0.743986	−	0.668196i	$-0.232933\pi$
$521$	33.9636	1.48797	0.743986	+	0.668196i	$0.232933\pi$
$522$	0	0
$523$	11.7578	0.514134	0.257067	−	0.966394i	$-0.417244\pi$
$523$	11.7578	0.514134	0.257067	+	0.966394i	$0.417244\pi$
$524$	0	0
$525$	−17.6191	−0.768960
$526$	0	0
$527$	1.32704	0.0578066
$528$	0	0
$529$	−6.21480	−0.270209
$530$	0	0
$531$	1.30174	0.0564905
$532$	0	0
$533$	47.7366	2.06770
$534$	0	0
$535$	−45.6214	−1.97239
$536$	0	0
$537$	−33.9217	−1.46383
$538$	0	0
$539$	−20.0315	−0.862818
$540$	0	0
$541$	0.804094	0.0345707	0.0172854	−	0.999851i	$-0.494498\pi$
$541$	0.804094	0.0345707	0.0172854	+	0.999851i	$0.494498\pi$
$542$	0	0
$543$	−4.27851	−0.183608
$544$	0	0
$545$	37.2386	1.59513
$546$	0	0
$547$	4.34094	0.185605	0.0928027	−	0.995685i	$-0.470417\pi$
$547$	4.34094	0.185605	0.0928027	+	0.995685i	$0.470417\pi$
$548$	0	0
$549$	−0.293826	−0.0125402
$550$	0	0
$551$	32.9686	1.40451
$552$	0	0
$553$	−36.0667	−1.53371
$554$	0	0
$555$	−30.8053	−1.30761
$556$	0	0
$557$	−29.2321	−1.23861	−0.619303	−	0.785152i	$-0.712585\pi$
$557$	−29.2321	−1.23861	−0.619303	+	0.785152i	$0.712585\pi$
$558$	0	0
$559$	32.5339	1.37604
$560$	0	0
$561$	−2.79148	−0.117857
$562$	0	0
$563$	−27.9325	−1.17721	−0.588606	−	0.808420i	$-0.700323\pi$
$563$	−27.9325	−1.17721	−0.588606	+	0.808420i	$0.700323\pi$
$564$	0	0
$565$	−41.8079	−1.75887
$566$	0	0
$567$	19.8865	0.835156
$568$	0	0
$569$	21.5023	0.901424	0.450712	−	0.892669i	$-0.351170\pi$
$569$	21.5023	0.901424	0.450712	+	0.892669i	$0.351170\pi$
$570$	0	0
$571$	−19.6384	−0.821843	−0.410922	−	0.911671i	$-0.634793\pi$
$571$	−19.6384	−0.821843	−0.410922	+	0.911671i	$0.634793\pi$
$572$	0	0
$573$	−12.3529	−0.516051
$574$	0	0
$575$	12.3881	0.516617
$576$	0	0
$577$	14.7632	0.614599	0.307299	−	0.951613i	$-0.400575\pi$
$577$	14.7632	0.614599	0.307299	+	0.951613i	$0.400575\pi$
$578$	0	0
$579$	20.8750	0.867536
$580$	0	0
$581$	66.0031	2.73827
$582$	0	0
$583$	−7.11757	−0.294780
$584$	0	0
$585$	−12.2266	−0.505508
$586$	0	0
$587$	6.27328	0.258926	0.129463	−	0.991584i	$-0.458675\pi$
$587$	6.27328	0.258926	0.129463	+	0.991584i	$0.458675\pi$
$588$	0	0
$589$	9.42551	0.388371
$590$	0	0
$591$	−16.6348	−0.684265
$592$	0	0
$593$	40.8082	1.67579	0.837895	−	0.545831i	$-0.183786\pi$
$593$	40.8082	1.67579	0.837895	+	0.545831i	$0.183786\pi$
$594$	0	0
$595$	−12.2027	−0.500263
$596$	0	0
$597$	6.16179	0.252185
$598$	0	0
$599$	10.3330	0.422193	0.211096	−	0.977465i	$-0.432297\pi$
$599$	10.3330	0.422193	0.211096	+	0.977465i	$0.432297\pi$
$600$	0	0
$601$	15.7883	0.644019	0.322009	−	0.946737i	$-0.395642\pi$
$601$	15.7883	0.644019	0.322009	+	0.946737i	$0.395642\pi$
$602$	0	0
$603$	−8.90480	−0.362632
$604$	0	0
$605$	−20.5896	−0.837086
$606$	0	0
$607$	−12.8843	−0.522958	−0.261479	−	0.965209i	$-0.584210\pi$
$607$	−12.8843	−0.522958	−0.261479	+	0.965209i	$0.584210\pi$
$608$	0	0
$609$	−26.1671	−1.06034
$610$	0	0
$611$	28.2725	1.14378
$612$	0	0
$613$	6.99917	0.282694	0.141347	−	0.989960i	$-0.454857\pi$
$613$	6.99917	0.282694	0.141347	+	0.989960i	$0.454857\pi$
$614$	0	0
$615$	45.7823	1.84612
$616$	0	0
$617$	−8.16208	−0.328593	−0.164297	−	0.986411i	$-0.552535\pi$
$617$	−8.16208	−0.328593	−0.164297	+	0.986411i	$0.552535\pi$
$618$	0	0
$619$	12.6031	0.506559	0.253280	−	0.967393i	$-0.418491\pi$
$619$	12.6031	0.506559	0.253280	+	0.967393i	$0.418491\pi$
$620$	0	0
$621$	−23.1715	−0.929840
$622$	0	0
$623$	13.4970	0.540747
$624$	0	0
$625$	−30.9757	−1.23903
$626$	0	0
$627$	−19.8270	−0.791815
$628$	0	0
$629$	−8.04011	−0.320580
$630$	0	0
$631$	−45.4769	−1.81041	−0.905203	−	0.424980i	$-0.860281\pi$
$631$	−45.4769	−1.81041	−0.905203	+	0.424980i	$0.860281\pi$
$632$	0	0
$633$	−29.9500	−1.19041
$634$	0	0
$635$	−31.1989	−1.23809
$636$	0	0
$637$	42.8223	1.69668
$638$	0	0
$639$	−6.69122	−0.264700
$640$	0	0
$641$	12.1865	0.481339	0.240669	−	0.970607i	$-0.422633\pi$
$641$	12.1865	0.481339	0.240669	+	0.970607i	$0.422633\pi$
$642$	0	0
$643$	−29.7340	−1.17259	−0.586296	−	0.810097i	$-0.699415\pi$
$643$	−29.7340	−1.17259	−0.586296	+	0.810097i	$0.699415\pi$
$644$	0	0
$645$	31.2020	1.22858
$646$	0	0
$647$	32.8228	1.29040	0.645199	−	0.764015i	$-0.276774\pi$
$647$	32.8228	1.29040	0.645199	+	0.764015i	$0.276774\pi$
$648$	0	0
$649$	2.40555	0.0944262
$650$	0	0
$651$	−7.48100	−0.293204
$652$	0	0
$653$	10.3258	0.404078	0.202039	−	0.979377i	$-0.435243\pi$
$653$	10.3258	0.404078	0.202039	+	0.979377i	$0.435243\pi$
$654$	0	0
$655$	14.8826	0.581512
$656$	0	0
$657$	6.19445	0.241668
$658$	0	0
$659$	−25.8076	−1.00532	−0.502661	−	0.864484i	$-0.667646\pi$
$659$	−25.8076	−1.00532	−0.502661	+	0.864484i	$0.667646\pi$
$660$	0	0
$661$	−13.3955	−0.521025	−0.260513	−	0.965470i	$-0.583892\pi$
$661$	−13.3955	−0.521025	−0.260513	+	0.965470i	$0.583892\pi$
$662$	0	0
$663$	5.96749	0.231758
$664$	0	0
$665$	−86.6719	−3.36099
$666$	0	0
$667$	18.3982	0.712380
$668$	0	0
$669$	11.0952	0.428963
$670$	0	0
$671$	−0.542978	−0.0209614
$672$	0	0
$673$	−6.82079	−0.262922	−0.131461	−	0.991321i	$-0.541967\pi$
$673$	−6.82079	−0.262922	−0.131461	+	0.991321i	$0.541967\pi$
$674$	0	0
$675$	−17.1014	−0.658232
$676$	0	0
$677$	10.5869	0.406888	0.203444	−	0.979087i	$-0.434787\pi$
$677$	10.5869	0.406888	0.203444	+	0.979087i	$0.434787\pi$
$678$	0	0
$679$	30.3131	1.16331
$680$	0	0
$681$	−22.2795	−0.853753
$682$	0	0
$683$	−43.3107	−1.65724	−0.828620	−	0.559812i	$-0.810873\pi$
$683$	−43.3107	−1.65724	−0.828620	+	0.559812i	$0.810873\pi$
$684$	0	0
$685$	−4.76205	−0.181948
$686$	0	0
$687$	18.8343	0.718572
$688$	0	0
$689$	15.2156	0.579667
$690$	0	0
$691$	27.5238	1.04706	0.523528	−	0.852009i	$-0.324616\pi$
$691$	27.5238	1.04706	0.523528	+	0.852009i	$0.324616\pi$
$692$	0	0
$693$	−8.41519	−0.319667
$694$	0	0
$695$	−2.83261	−0.107447
$696$	0	0
$697$	11.9491	0.452604
$698$	0	0
$699$	25.7694	0.974686
$700$	0	0
$701$	51.4490	1.94320	0.971601	−	0.236625i	$-0.0760413\pi$
$701$	51.4490	1.94320	0.971601	+	0.236625i	$0.0760413\pi$
$702$	0	0
$703$	−57.1063	−2.15381
$704$	0	0
$705$	27.1150	1.02121
$706$	0	0
$707$	54.3710	2.04483
$708$	0	0
$709$	16.8255	0.631894	0.315947	−	0.948777i	$-0.397678\pi$
$709$	16.8255	0.631894	0.315947	+	0.948777i	$0.397678\pi$
$710$	0	0
$711$	−9.04563	−0.339238
$712$	0	0
$713$	5.25992	0.196986
$714$	0	0
$715$	−22.5942	−0.844977
$716$	0	0
$717$	−1.24354	−0.0464407
$718$	0	0
$719$	−26.2933	−0.980574	−0.490287	−	0.871561i	$-0.663108\pi$
$719$	−26.2933	−0.980574	−0.490287	+	0.871561i	$0.663108\pi$
$720$	0	0
$721$	−49.6165	−1.84781
$722$	0	0
$723$	−16.4354	−0.611238
$724$	0	0
$725$	13.5785	0.504292
$726$	0	0
$727$	−23.5378	−0.872967	−0.436484	−	0.899712i	$-0.643776\pi$
$727$	−23.5378	−0.872967	−0.436484	+	0.899712i	$0.643776\pi$
$728$	0	0
$729$	28.7097	1.06332
$730$	0	0
$731$	8.14366	0.301204
$732$	0	0
$733$	−10.0422	−0.370919	−0.185459	−	0.982652i	$-0.559377\pi$
$733$	−10.0422	−0.370919	−0.185459	+	0.982652i	$0.559377\pi$
$734$	0	0
$735$	41.0691	1.51486
$736$	0	0
$737$	−16.4557	−0.606154
$738$	0	0
$739$	−8.15975	−0.300161	−0.150081	−	0.988674i	$-0.547953\pi$
$739$	−8.15975	−0.300161	−0.150081	+	0.988674i	$0.547953\pi$
$740$	0	0
$741$	42.3851	1.55706
$742$	0	0
$743$	−7.78206	−0.285496	−0.142748	−	0.989759i	$-0.545594\pi$
$743$	−7.78206	−0.285496	−0.142748	+	0.989759i	$0.545594\pi$
$744$	0	0
$745$	39.4463	1.44520
$746$	0	0
$747$	16.5538	0.605670
$748$	0	0
$749$	67.1249	2.45269
$750$	0	0
$751$	8.30887	0.303195	0.151598	−	0.988442i	$-0.451558\pi$
$751$	8.30887	0.303195	0.151598	+	0.988442i	$0.451558\pi$
$752$	0	0
$753$	20.2098	0.736487
$754$	0	0
$755$	−24.4706	−0.890577
$756$	0	0
$757$	−38.8376	−1.41158	−0.705788	−	0.708424i	$-0.749407\pi$
$757$	−38.8376	−1.41158	−0.705788	+	0.708424i	$0.749407\pi$
$758$	0	0
$759$	−11.0645	−0.401616
$760$	0	0
$761$	−25.3906	−0.920408	−0.460204	−	0.887813i	$-0.652224\pi$
$761$	−25.3906	−0.920408	−0.460204	+	0.887813i	$0.652224\pi$
$762$	0	0
$763$	−54.7909	−1.98356
$764$	0	0
$765$	−3.06048	−0.110652
$766$	0	0
$767$	−5.14246	−0.185684
$768$	0	0
$769$	39.8098	1.43558	0.717788	−	0.696261i	$-0.245155\pi$
$769$	39.8098	1.43558	0.717788	+	0.696261i	$0.245155\pi$
$770$	0	0
$771$	24.1857	0.871028
$772$	0	0
$773$	9.95735	0.358141	0.179070	−	0.983836i	$-0.442691\pi$
$773$	9.95735	0.358141	0.179070	+	0.983836i	$0.442691\pi$
$774$	0	0
$775$	3.88200	0.139446
$776$	0	0
$777$	45.3252	1.62603
$778$	0	0
$779$	84.8706	3.04080
$780$	0	0
$781$	−12.3651	−0.442458
$782$	0	0
$783$	−25.3981	−0.907656
$784$	0	0
$785$	29.5618	1.05510
$786$	0	0
$787$	−30.2211	−1.07727	−0.538633	−	0.842541i	$-0.681059\pi$
$787$	−30.2211	−1.07727	−0.538633	+	0.842541i	$0.681059\pi$
$788$	0	0
$789$	−5.60469	−0.199532
$790$	0	0
$791$	61.5139	2.18718
$792$	0	0
$793$	1.16075	0.0412194
$794$	0	0
$795$	14.5927	0.517548
$796$	0	0
$797$	−10.6318	−0.376597	−0.188298	−	0.982112i	$-0.560297\pi$
$797$	−10.6318	−0.376597	−0.188298	+	0.982112i	$0.560297\pi$
$798$	0	0
$799$	7.07697	0.250365
$800$	0	0
$801$	3.38509	0.119606
$802$	0	0
$803$	11.4471	0.403959
$804$	0	0
$805$	−48.3674	−1.70473
$806$	0	0
$807$	39.1891	1.37952
$808$	0	0
$809$	−15.9047	−0.559178	−0.279589	−	0.960120i	$-0.590198\pi$
$809$	−15.9047	−0.559178	−0.279589	+	0.960120i	$0.590198\pi$
$810$	0	0
$811$	−53.3099	−1.87196	−0.935981	−	0.352049i	$-0.885485\pi$
$811$	−53.3099	−1.87196	−0.935981	+	0.352049i	$0.885485\pi$
$812$	0	0
$813$	−22.7973	−0.799537
$814$	0	0
$815$	10.5771	0.370501
$816$	0	0
$817$	57.8418	2.02363
$818$	0	0
$819$	17.9895	0.628606
$820$	0	0
$821$	−16.7614	−0.584975	−0.292488	−	0.956269i	$-0.594483\pi$
$821$	−16.7614	−0.584975	−0.292488	+	0.956269i	$0.594483\pi$
$822$	0	0
$823$	−38.0271	−1.32554	−0.662770	−	0.748823i	$-0.730619\pi$
$823$	−38.0271	−1.32554	−0.662770	+	0.748823i	$0.730619\pi$
$824$	0	0
$825$	−8.16598	−0.284303
$826$	0	0
$827$	−40.6301	−1.41285	−0.706424	−	0.707789i	$-0.749693\pi$
$827$	−40.6301	−1.41285	−0.706424	+	0.707789i	$0.749693\pi$
$828$	0	0
$829$	−48.8098	−1.69523	−0.847617	−	0.530608i	$-0.821964\pi$
$829$	−48.8098	−1.69523	−0.847617	+	0.530608i	$0.821964\pi$
$830$	0	0
$831$	−36.8040	−1.27672
$832$	0	0
$833$	10.7190	0.371390
$834$	0	0
$835$	55.9449	1.93605
$836$	0	0
$837$	−7.26117	−0.250983
$838$	0	0
$839$	13.9510	0.481641	0.240821	−	0.970570i	$-0.422583\pi$
$839$	13.9510	0.481641	0.240821	+	0.970570i	$0.422583\pi$
$840$	0	0
$841$	−8.83388	−0.304617
$842$	0	0
$843$	−10.6000	−0.365084
$844$	0	0
$845$	11.4768	0.394813
$846$	0	0
$847$	30.2944	1.04093
$848$	0	0
$849$	19.8841	0.682421
$850$	0	0
$851$	−31.8683	−1.09243
$852$	0	0
$853$	−48.3304	−1.65480	−0.827401	−	0.561612i	$-0.810181\pi$
$853$	−48.3304	−1.65480	−0.827401	+	0.561612i	$0.810181\pi$
$854$	0	0
$855$	−21.7376	−0.743409
$856$	0	0
$857$	−47.8068	−1.63305	−0.816525	−	0.577310i	$-0.804103\pi$
$857$	−47.8068	−1.63305	−0.816525	+	0.577310i	$0.804103\pi$
$858$	0	0
$859$	−57.7110	−1.96907	−0.984537	−	0.175179i	$-0.943950\pi$
$859$	−57.7110	−1.96907	−0.984537	+	0.175179i	$0.943950\pi$
$860$	0	0
$861$	−67.3616	−2.29568
$862$	0	0
$863$	−22.5913	−0.769018	−0.384509	−	0.923121i	$-0.625629\pi$
$863$	−22.5913	−0.769018	−0.384509	+	0.923121i	$0.625629\pi$
$864$	0	0
$865$	−21.2851	−0.723714
$866$	0	0
$867$	−22.2741	−0.756470
$868$	0	0
$869$	−16.7160	−0.567050
$870$	0	0
$871$	35.1781	1.19197
$872$	0	0
$873$	7.60262	0.257310
$874$	0	0
$875$	23.3314	0.788746
$876$	0	0
$877$	8.20763	0.277152	0.138576	−	0.990352i	$-0.455748\pi$
$877$	8.20763	0.277152	0.138576	+	0.990352i	$0.455748\pi$
$878$	0	0
$879$	−6.55690	−0.221159
$880$	0	0
$881$	54.9341	1.85078	0.925388	−	0.379020i	$-0.123739\pi$
$881$	54.9341	1.85078	0.925388	+	0.379020i	$0.123739\pi$
$882$	0	0
$883$	42.9747	1.44621	0.723107	−	0.690736i	$-0.242714\pi$
$883$	42.9747	1.44621	0.723107	+	0.690736i	$0.242714\pi$
$884$	0	0
$885$	−4.93193	−0.165785
$886$	0	0
$887$	16.0125	0.537647	0.268824	−	0.963189i	$-0.413365\pi$
$887$	16.0125	0.537647	0.268824	+	0.963189i	$0.413365\pi$
$888$	0	0
$889$	45.9044	1.53958
$890$	0	0
$891$	9.21688	0.308777
$892$	0	0
$893$	50.2654	1.68207
$894$	0	0
$895$	−68.7263	−2.29727
$896$	0	0
$897$	23.6531	0.789754
$898$	0	0
$899$	5.76537	0.192286
$900$	0	0
$901$	3.80866	0.126885
$902$	0	0
$903$	−45.9089	−1.52775
$904$	0	0
$905$	−8.66838	−0.288147
$906$	0	0
$907$	−17.4683	−0.580026	−0.290013	−	0.957023i	$-0.593660\pi$
$907$	−17.4683	−0.580026	−0.290013	+	0.957023i	$0.593660\pi$
$908$	0	0
$909$	13.6364	0.452291
$910$	0	0
$911$	−16.6656	−0.552157	−0.276078	−	0.961135i	$-0.589035\pi$
$911$	−16.6656	−0.552157	−0.276078	+	0.961135i	$0.589035\pi$
$912$	0	0
$913$	30.5907	1.01240
$914$	0	0
$915$	1.11323	0.0368022
$916$	0	0
$917$	−21.8975	−0.723118
$918$	0	0
$919$	−18.5901	−0.613230	−0.306615	−	0.951834i	$-0.599196\pi$
$919$	−18.5901	−0.613230	−0.306615	+	0.951834i	$0.599196\pi$
$920$	0	0
$921$	−14.9480	−0.492553
$922$	0	0
$923$	26.4334	0.870067
$924$	0	0
$925$	−23.5199	−0.773329
$926$	0	0
$927$	−12.4440	−0.408713
$928$	0	0
$929$	6.08296	0.199576	0.0997878	−	0.995009i	$-0.468184\pi$
$929$	6.08296	0.199576	0.0997878	+	0.995009i	$0.468184\pi$
$930$	0	0
$931$	76.1334	2.49517
$932$	0	0
$933$	29.4239	0.963296
$934$	0	0
$935$	−5.65563	−0.184959
$936$	0	0
$937$	−50.4851	−1.64928	−0.824639	−	0.565660i	$-0.808622\pi$
$937$	−50.4851	−1.64928	−0.824639	+	0.565660i	$0.808622\pi$
$938$	0	0
$939$	−19.8578	−0.648035
$940$	0	0
$941$	−26.6303	−0.868123	−0.434061	−	0.900883i	$-0.642920\pi$
$941$	−26.6303	−0.868123	−0.434061	+	0.900883i	$0.642920\pi$
$942$	0	0
$943$	47.3622	1.54232
$944$	0	0
$945$	66.7699	2.17202
$946$	0	0
$947$	41.7464	1.35658	0.678289	−	0.734796i	$-0.262722\pi$
$947$	41.7464	1.35658	0.678289	+	0.734796i	$0.262722\pi$
$948$	0	0
$949$	−24.4710	−0.794361
$950$	0	0
$951$	3.99494	0.129545
$952$	0	0
$953$	−45.7416	−1.48172	−0.740858	−	0.671662i	$-0.765581\pi$
$953$	−45.7416	−1.48172	−0.740858	+	0.671662i	$0.765581\pi$
$954$	0	0
$955$	−25.0274	−0.809867
$956$	0	0
$957$	−12.1277	−0.392034
$958$	0	0
$959$	7.00662	0.226255
$960$	0	0
$961$	−29.3517	−0.946830
$962$	0	0
$963$	16.8351	0.542504
$964$	0	0
$965$	42.2934	1.36147
$966$	0	0
$967$	−29.9836	−0.964207	−0.482104	−	0.876114i	$-0.660127\pi$
$967$	−29.9836	−0.964207	−0.482104	+	0.876114i	$0.660127\pi$
$968$	0	0
$969$	10.6096	0.340828
$970$	0	0
$971$	32.2248	1.03414	0.517071	−	0.855942i	$-0.327022\pi$
$971$	32.2248	1.03414	0.517071	+	0.855942i	$0.327022\pi$
$972$	0	0
$973$	4.16776	0.133612
$974$	0	0
$975$	17.4568	0.559065
$976$	0	0
$977$	−1.45628	−0.0465907	−0.0232953	−	0.999729i	$-0.507416\pi$
$977$	−1.45628	−0.0465907	−0.0232953	+	0.999729i	$0.507416\pi$
$978$	0	0
$979$	6.25551	0.199927
$980$	0	0
$981$	−13.7417	−0.438739
$982$	0	0
$983$	35.1255	1.12033	0.560164	−	0.828382i	$-0.310738\pi$
$983$	35.1255	1.12033	0.560164	+	0.828382i	$0.310738\pi$
$984$	0	0
$985$	−33.7026	−1.07385
$986$	0	0
$987$	−39.8956	−1.26989
$988$	0	0
$989$	32.2787	1.02640
$990$	0	0
$991$	9.58149	0.304366	0.152183	−	0.988352i	$-0.451370\pi$
$991$	9.58149	0.304366	0.152183	+	0.988352i	$0.451370\pi$
$992$	0	0
$993$	−16.9411	−0.537611
$994$	0	0
$995$	12.4840	0.395769
$996$	0	0
$997$	−16.5332	−0.523613	−0.261807	−	0.965120i	$-0.584318\pi$
$997$	−16.5332	−0.523613	−0.261807	+	0.965120i	$0.584318\pi$
$998$	0	0
$999$	43.9933	1.39189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2224.2.a.o.1.5		7
4.3	odd	2		139.2.a.c.1.4	✓	7
8.3	odd	2		8896.2.a.be.1.5		7
8.5	even	2		8896.2.a.bd.1.3		7
12.11	even	2		1251.2.a.k.1.4		7
20.19	odd	2		3475.2.a.e.1.4		7
28.27	even	2		6811.2.a.p.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
139.2.a.c.1.4	✓	7	4.3	odd	2
1251.2.a.k.1.4		7	12.11	even	2
2224.2.a.o.1.5		7	1.1	even	1	trivial
3475.2.a.e.1.4		7	20.19	odd	2
6811.2.a.p.1.4		7	28.27	even	2
8896.2.a.bd.1.3		7	8.5	even	2
8896.2.a.be.1.5		7	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2224 = 2^{4} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2224.a (trivial)

\(f(q)\)	\(=\)	\(q+1.39811 q^{3} +2.83261 q^{5} -4.16776 q^{7} -1.04529 q^{9} +O(q^{10})\)	\(q+1.39811 q^{3} +2.83261 q^{5} -4.16776 q^{7} -1.04529 q^{9} -1.93164 q^{11} +4.12937 q^{13} +3.96031 q^{15} +1.03363 q^{17} +7.34157 q^{19} -5.82699 q^{21} +4.09697 q^{23} +3.02371 q^{25} -5.65576 q^{27} +4.49067 q^{29} +1.28386 q^{31} -2.70065 q^{33} -11.8056 q^{35} -7.77849 q^{37} +5.77331 q^{39} +11.5603 q^{41} +7.87867 q^{43} -2.96089 q^{45} +6.84669 q^{47} +10.3702 q^{49} +1.44513 q^{51} +3.68473 q^{53} -5.47160 q^{55} +10.2643 q^{57} -1.24534 q^{59} +0.281096 q^{61} +4.35649 q^{63} +11.6969 q^{65} +8.51902 q^{67} +5.72803 q^{69} +6.40133 q^{71} -5.92608 q^{73} +4.22748 q^{75} +8.05062 q^{77} +8.65375 q^{79} -4.77152 q^{81} -15.8366 q^{83} +2.92789 q^{85} +6.27846 q^{87} -3.23844 q^{89} -17.2102 q^{91} +1.79497 q^{93} +20.7958 q^{95} -7.27325 q^{97} +2.01912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) \) 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9	\( 7 q + 2 q^{3} + 11 q^{5} + 5 q^{7} + 13 q^{9} - 2 q^{11} + 6 q^{13} + 3 q^{15} + 5 q^{17} + 10 q^{19} - 5 q^{21} + q^{23} + 14 q^{25} + 11 q^{27} + 30 q^{29} + 20 q^{31} - 20 q^{33} + 7 q^{35} + 6 q^{37} - 11 q^{39} + 19 q^{41} + 12 q^{43} + 27 q^{45} + 3 q^{47} - 8 q^{49} - 23 q^{51} + 38 q^{53} - 7 q^{55} - 19 q^{57} + 14 q^{59} + 4 q^{61} + 18 q^{63} + 10 q^{65} - 9 q^{67} + 9 q^{69} - 24 q^{71} - 5 q^{73} + 21 q^{75} - 13 q^{77} - 8 q^{79} + 39 q^{81} + 9 q^{83} - 22 q^{85} + 25 q^{87} + 10 q^{89} - 7 q^{91} - 15 q^{93} + 21 q^{95} - 5 q^{97} + 31 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 11 * q^5 + 5 * q^7 + 13 * q^9 - 2 * q^11 + 6 * q^13 + 3 * q^15 + 5 * q^17 + 10 * q^19 - 5 * q^21 + q^23 + 14 * q^25 + 11 * q^27 + 30 * q^29 + 20 * q^31 - 20 * q^33 + 7 * q^35 + 6 * q^37 - 11 * q^39 + 19 * q^41 + 12 * q^43 + 27 * q^45 + 3 * q^47 - 8 * q^49 - 23 * q^51 + 38 * q^53 - 7 * q^55 - 19 * q^57 + 14 * q^59 + 4 * q^61 + 18 * q^63 + 10 * q^65 - 9 * q^67 + 9 * q^69 - 24 * q^71 - 5 * q^73 + 21 * q^75 - 13 * q^77 - 8 * q^79 + 39 * q^81 + 9 * q^83 - 22 * q^85 + 25 * q^87 + 10 * q^89 - 7 * q^91 - 15 * q^93 + 21 * q^95 - 5 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more