LMFDB - Embedded newform 2169.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	7.7.31056073.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 $ x^7 - 3x^6 - 3x^5 + 11x^4 + x^3 - 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.27758$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.277577 q^{2} -1.92295 q^{4} +1.23324 q^{5} +1.36627 q^{7} +1.08892 q^{8} +O(q^{10})$	\(q-0.277577 q^{2} -1.92295 q^{4} +1.23324 q^{5} +1.36627 q^{7} +1.08892 q^{8} -0.342320 q^{10} +4.69806 q^{11} -0.0431968 q^{13} -0.379244 q^{14} +3.54364 q^{16} +7.31430 q^{17} -0.697489 q^{19} -2.37146 q^{20} -1.30407 q^{22} -1.41195 q^{23} -3.47911 q^{25} +0.0119904 q^{26} -2.62726 q^{28} -8.30334 q^{29} +3.39655 q^{31} -3.16148 q^{32} -2.03028 q^{34} +1.68494 q^{35} +7.15948 q^{37} +0.193607 q^{38} +1.34290 q^{40} -5.45541 q^{41} -11.7568 q^{43} -9.03414 q^{44} +0.391925 q^{46} +5.24836 q^{47} -5.13332 q^{49} +0.965723 q^{50} +0.0830653 q^{52} +8.57769 q^{53} +5.79385 q^{55} +1.48776 q^{56} +2.30482 q^{58} +12.9925 q^{59} +10.1636 q^{61} -0.942804 q^{62} -6.20973 q^{64} -0.0532721 q^{65} +10.1259 q^{67} -14.0650 q^{68} -0.467700 q^{70} -1.86703 q^{71} +6.47826 q^{73} -1.98731 q^{74} +1.34124 q^{76} +6.41880 q^{77} -12.9436 q^{79} +4.37017 q^{80} +1.51430 q^{82} +2.32915 q^{83} +9.02030 q^{85} +3.26342 q^{86} +5.11582 q^{88} +14.5180 q^{89} -0.0590183 q^{91} +2.71511 q^{92} -1.45682 q^{94} -0.860173 q^{95} -2.23725 q^{97} +1.42489 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8	$ 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8} + 3 q^{10} + 18 q^{11} - q^{13} + 6 q^{14} + 4 q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 10 q^{22} + 22 q^{23} + 5 q^{25} - 8 q^{26} + 9 q^{28} + 16 q^{29} - 18 q^{31} + 6 q^{32} + 11 q^{34} - 7 q^{35} + 8 q^{37} - 16 q^{38} + 14 q^{40} + 15 q^{41} + 14 q^{43} + 4 q^{44} + 11 q^{46} + 10 q^{47} + 6 q^{49} + 4 q^{50} + 27 q^{52} - 15 q^{53} + 29 q^{55} - 13 q^{56} + 17 q^{58} + 18 q^{59} + 4 q^{61} - 13 q^{62} + 2 q^{64} + 7 q^{65} + 18 q^{67} + 15 q^{68} + 8 q^{70} + 50 q^{71} - 10 q^{74} - 20 q^{76} - 17 q^{77} - 15 q^{79} + 11 q^{80} + 45 q^{82} + 24 q^{83} - 2 q^{85} + 23 q^{86} + 8 q^{88} + 13 q^{89} - 12 q^{91} + 10 q^{92} - 32 q^{94} + 41 q^{95} + q^{97} - 9 q^{98}+O(q^{100}) $ 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8 + 3 * q^10 + 18 * q^11 - q^13 + 6 * q^14 + 4 * q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 10 * q^22 + 22 * q^23 + 5 * q^25 - 8 * q^26 + 9 * q^28 + 16 * q^29 - 18 * q^31 + 6 * q^32 + 11 * q^34 - 7 * q^35 + 8 * q^37 - 16 * q^38 + 14 * q^40 + 15 * q^41 + 14 * q^43 + 4 * q^44 + 11 * q^46 + 10 * q^47 + 6 * q^49 + 4 * q^50 + 27 * q^52 - 15 * q^53 + 29 * q^55 - 13 * q^56 + 17 * q^58 + 18 * q^59 + 4 * q^61 - 13 * q^62 + 2 * q^64 + 7 * q^65 + 18 * q^67 + 15 * q^68 + 8 * q^70 + 50 * q^71 - 10 * q^74 - 20 * q^76 - 17 * q^77 - 15 * q^79 + 11 * q^80 + 45 * q^82 + 24 * q^83 - 2 * q^85 + 23 * q^86 + 8 * q^88 + 13 * q^89 - 12 * q^91 + 10 * q^92 - 32 * q^94 + 41 * q^95 + q^97 - 9 * q^98

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 $

$2$	−0.277577	−0.196277	−0.0981383	−	0.995173i	$-0.531289\pi$
$2$	−0.277577	−0.196277	−0.0981383	+	0.995173i	$0.531289\pi$
$3$	0	0
$3$	0	0
$4$	−1.92295	−0.961475
$5$	1.23324	0.551523	0.275761	−	0.961226i	$-0.411070\pi$
$5$	1.23324	0.551523	0.275761	+	0.961226i	$0.411070\pi$
$6$	0	0
$7$	1.36627	0.516400	0.258200	−	0.966092i	$-0.416871\pi$
$7$	1.36627	0.516400	0.258200	+	0.966092i	$0.416871\pi$
$8$	1.08892	0.384992
$9$	0	0
$10$	−0.342320	−0.108251
$11$	4.69806	1.41652	0.708259	−	0.705952i	$-0.249481\pi$
$11$	4.69806	1.41652	0.708259	+	0.705952i	$0.249481\pi$
$12$	0	0
$13$	−0.0431968	−0.0119806	−0.00599032	−	0.999982i	$-0.501907\pi$
$13$	−0.0431968	−0.0119806	−0.00599032	+	0.999982i	$0.501907\pi$
$14$	−0.379244	−0.101357
$15$	0	0
$16$	3.54364	0.885911
$17$	7.31430	1.77398	0.886989	−	0.461790i	$-0.152793\pi$
$17$	7.31430	1.77398	0.886989	+	0.461790i	$0.152793\pi$
$18$	0	0
$19$	−0.697489	−0.160015	−0.0800075	−	0.996794i	$-0.525494\pi$
$19$	−0.697489	−0.160015	−0.0800075	+	0.996794i	$0.525494\pi$
$20$	−2.37146	−0.530275
$21$	0	0
$22$	−1.30407	−0.278030
$23$	−1.41195	−0.294412	−0.147206	−	0.989106i	$-0.547028\pi$
$23$	−1.41195	−0.294412	−0.147206	+	0.989106i	$0.547028\pi$
$24$	0	0
$25$	−3.47911	−0.695823
$26$	0.0119904	0.00235152
$27$	0	0
$28$	−2.62726	−0.496506
$29$	−8.30334	−1.54189	−0.770946	−	0.636901i	$-0.780216\pi$
$29$	−8.30334	−1.54189	−0.770946	+	0.636901i	$0.780216\pi$
$30$	0	0
$31$	3.39655	0.610038	0.305019	−	0.952346i	$-0.401337\pi$
$31$	3.39655	0.610038	0.305019	+	0.952346i	$0.401337\pi$
$32$	−3.16148	−0.558875
$33$	0	0
$34$	−2.03028	−0.348191
$35$	1.68494	0.284806
$36$	0	0
$37$	7.15948	1.17701	0.588506	−	0.808493i	$-0.299716\pi$
$37$	7.15948	1.17701	0.588506	+	0.808493i	$0.299716\pi$
$38$	0.193607	0.0314072
$39$	0	0
$40$	1.34290	0.212332
$41$	−5.45541	−0.851992	−0.425996	−	0.904725i	$-0.640076\pi$
$41$	−5.45541	−0.851992	−0.425996	+	0.904725i	$0.640076\pi$
$42$	0	0
$43$	−11.7568	−1.79290	−0.896448	−	0.443149i	$-0.853861\pi$
$43$	−11.7568	−1.79290	−0.896448	+	0.443149i	$0.853861\pi$
$44$	−9.03414	−1.36195
$45$	0	0
$46$	0.391925	0.0577862
$47$	5.24836	0.765551	0.382776	−	0.923841i	$-0.374968\pi$
$47$	5.24836	0.765551	0.382776	+	0.923841i	$0.374968\pi$
$48$	0	0
$49$	−5.13332	−0.733331
$50$	0.965723	0.136574
$51$	0	0
$52$	0.0830653	0.0115191
$53$	8.57769	1.17824	0.589118	−	0.808047i	$-0.299475\pi$
$53$	8.57769	1.17824	0.589118	+	0.808047i	$0.299475\pi$
$54$	0	0
$55$	5.79385	0.781242
$56$	1.48776	0.198810
$57$	0	0
$58$	2.30482	0.302637
$59$	12.9925	1.69148	0.845738	−	0.533598i	$-0.179160\pi$
$59$	12.9925	1.69148	0.845738	+	0.533598i	$0.179160\pi$
$60$	0	0
$61$	10.1636	1.30132	0.650658	−	0.759371i	$-0.274493\pi$
$61$	10.1636	1.30132	0.650658	+	0.759371i	$0.274493\pi$
$62$	−0.942804	−0.119736
$63$	0	0
$64$	−6.20973	−0.776216
$65$	−0.0532721	−0.00660759
$66$	0	0
$67$	10.1259	1.23707	0.618536	−	0.785757i	$-0.287726\pi$
$67$	10.1259	1.23707	0.618536	+	0.785757i	$0.287726\pi$
$68$	−14.0650	−1.70564
$69$	0	0
$70$	−0.467700	−0.0559008
$71$	−1.86703	−0.221576	−0.110788	−	0.993844i	$-0.535337\pi$
$71$	−1.86703	−0.221576	−0.110788	+	0.993844i	$0.535337\pi$
$72$	0	0
$73$	6.47826	0.758223	0.379111	−	0.925351i	$-0.376230\pi$
$73$	6.47826	0.758223	0.379111	+	0.925351i	$0.376230\pi$
$74$	−1.98731	−0.231020
$75$	0	0
$76$	1.34124	0.153850
$77$	6.41880	0.731490
$78$	0	0
$79$	−12.9436	−1.45627	−0.728136	−	0.685432i	$-0.759613\pi$
$79$	−12.9436	−1.45627	−0.728136	+	0.685432i	$0.759613\pi$
$80$	4.37017	0.488600
$81$	0	0
$82$	1.51430	0.167226
$83$	2.32915	0.255657	0.127829	−	0.991796i	$-0.459199\pi$
$83$	2.32915	0.255657	0.127829	+	0.991796i	$0.459199\pi$
$84$	0	0
$85$	9.02030	0.978389
$86$	3.26342	0.351904
$87$	0	0
$88$	5.11582	0.545348
$89$	14.5180	1.53890	0.769450	−	0.638707i	$-0.220530\pi$
$89$	14.5180	1.53890	0.769450	+	0.638707i	$0.220530\pi$
$90$	0	0
$91$	−0.0590183	−0.00618680
$92$	2.71511	0.283070
$93$	0	0
$94$	−1.45682	−0.150260
$95$	−0.860173	−0.0882519
$96$	0	0
$97$	−2.23725	−0.227158	−0.113579	−	0.993529i	$-0.536231\pi$
$97$	−2.23725	−0.227158	−0.113579	+	0.993529i	$0.536231\pi$
$98$	1.42489	0.143936
$99$	0	0
$100$	6.69017	0.669017
$101$	4.01607	0.399614	0.199807	−	0.979835i	$-0.435969\pi$
$101$	4.01607	0.399614	0.199807	+	0.979835i	$0.435969\pi$
$102$	0	0
$103$	−5.25187	−0.517482	−0.258741	−	0.965947i	$-0.583308\pi$
$103$	−5.25187	−0.517482	−0.258741	+	0.965947i	$0.583308\pi$
$104$	−0.0470379	−0.00461245
$105$	0	0
$106$	−2.38097	−0.231260
$107$	−5.60385	−0.541745	−0.270873	−	0.962615i	$-0.587312\pi$
$107$	−5.60385	−0.541745	−0.270873	+	0.962615i	$0.587312\pi$
$108$	0	0
$109$	−4.77557	−0.457416	−0.228708	−	0.973495i	$-0.573450\pi$
$109$	−4.77557	−0.457416	−0.228708	+	0.973495i	$0.573450\pi$
$110$	−1.60824	−0.153340
$111$	0	0
$112$	4.84155	0.457484
$113$	12.6186	1.18706	0.593531	−	0.804811i	$-0.297733\pi$
$113$	12.6186	1.18706	0.593531	+	0.804811i	$0.297733\pi$
$114$	0	0
$115$	−1.74127	−0.162375
$116$	15.9669	1.48249
$117$	0	0
$118$	−3.60642	−0.331997
$119$	9.99327	0.916082
$120$	0	0
$121$	11.0718	1.00653
$122$	−2.82118	−0.255418
$123$	0	0
$124$	−6.53139	−0.586536
$125$	−10.4568	−0.935285
$126$	0	0
$127$	0.427651	0.0379479	0.0189740	−	0.999820i	$-0.493960\pi$
$127$	0.427651	0.0379479	0.0189740	+	0.999820i	$0.493960\pi$
$128$	8.04663	0.711229
$129$	0	0
$130$	0.0147871	0.00129692
$131$	13.3633	1.16756	0.583780	−	0.811912i	$-0.301573\pi$
$131$	13.3633	1.16756	0.583780	+	0.811912i	$0.301573\pi$
$132$	0	0
$133$	−0.952955	−0.0826317
$134$	−2.81071	−0.242808
$135$	0	0
$136$	7.96470	0.682967
$137$	−11.2702	−0.962874	−0.481437	−	0.876481i	$-0.659885\pi$
$137$	−11.2702	−0.962874	−0.481437	+	0.876481i	$0.659885\pi$
$138$	0	0
$139$	−0.927184	−0.0786427	−0.0393214	−	0.999227i	$-0.512520\pi$
$139$	−0.927184	−0.0786427	−0.0393214	+	0.999227i	$0.512520\pi$
$140$	−3.24005	−0.273834
$141$	0	0
$142$	0.518245	0.0434902
$143$	−0.202941	−0.0169708
$144$	0	0
$145$	−10.2400	−0.850388
$146$	−1.79822	−0.148821
$147$	0	0
$148$	−13.7673	−1.13167
$149$	−12.8757	−1.05481	−0.527407	−	0.849612i	$-0.676836\pi$
$149$	−12.8757	−1.05481	−0.527407	+	0.849612i	$0.676836\pi$
$150$	0	0
$151$	−21.2436	−1.72878	−0.864390	−	0.502823i	$-0.832295\pi$
$151$	−21.2436	−1.72878	−0.864390	+	0.502823i	$0.832295\pi$
$152$	−0.759511	−0.0616045
$153$	0	0
$154$	−1.78171	−0.143574
$155$	4.18876	0.336450
$156$	0	0
$157$	17.1798	1.37110	0.685550	−	0.728026i	$-0.259562\pi$
$157$	17.1798	1.37110	0.685550	+	0.728026i	$0.259562\pi$
$158$	3.59286	0.285832
$159$	0	0
$160$	−3.89887	−0.308232
$161$	−1.92910	−0.152034
$162$	0	0
$163$	11.1081	0.870055	0.435028	−	0.900417i	$-0.356739\pi$
$163$	11.1081	0.870055	0.435028	+	0.900417i	$0.356739\pi$
$164$	10.4905	0.819170
$165$	0	0
$166$	−0.646519	−0.0501796
$167$	22.3791	1.73174	0.865872	−	0.500266i	$-0.166764\pi$
$167$	22.3791	1.73174	0.865872	+	0.500266i	$0.166764\pi$
$168$	0	0
$169$	−12.9981	−0.999856
$170$	−2.50383	−0.192035
$171$	0	0
$172$	22.6078	1.72383
$173$	−4.16100	−0.316355	−0.158178	−	0.987411i	$-0.550562\pi$
$173$	−4.16100	−0.316355	−0.158178	+	0.987411i	$0.550562\pi$
$174$	0	0
$175$	−4.75339	−0.359323
$176$	16.6482	1.25491
$177$	0	0
$178$	−4.02985	−0.302050
$179$	5.79009	0.432772	0.216386	−	0.976308i	$-0.430573\pi$
$179$	5.79009	0.432772	0.216386	+	0.976308i	$0.430573\pi$
$180$	0	0
$181$	17.4917	1.30015	0.650076	−	0.759869i	$-0.274737\pi$
$181$	17.4917	1.30015	0.650076	+	0.759869i	$0.274737\pi$
$182$	0.0163821	0.00121432
$183$	0	0
$184$	−1.53750	−0.113346
$185$	8.82937	0.649148
$186$	0	0
$187$	34.3630	2.51287
$188$	−10.0923	−0.736059
$189$	0	0
$190$	0.238764	0.0173218
$191$	21.6074	1.56346	0.781728	−	0.623620i	$-0.214339\pi$
$191$	21.6074	1.56346	0.781728	+	0.623620i	$0.214339\pi$
$192$	0	0
$193$	2.30886	0.166195	0.0830977	−	0.996541i	$-0.473519\pi$
$193$	2.30886	0.166195	0.0830977	+	0.996541i	$0.473519\pi$
$194$	0.621008	0.0445858
$195$	0	0
$196$	9.87112	0.705080
$197$	−1.91876	−0.136706	−0.0683531	−	0.997661i	$-0.521774\pi$
$197$	−1.91876	−0.136706	−0.0683531	+	0.997661i	$0.521774\pi$
$198$	0	0
$199$	21.2430	1.50588	0.752938	−	0.658091i	$-0.228636\pi$
$199$	21.2430	1.50588	0.752938	+	0.658091i	$0.228636\pi$
$200$	−3.78848	−0.267886
$201$	0	0
$202$	−1.11477	−0.0784349
$203$	−11.3446	−0.796232
$204$	0	0
$205$	−6.72784	−0.469893
$206$	1.45780	0.101570
$207$	0	0
$208$	−0.153074	−0.0106138
$209$	−3.27685	−0.226664
$210$	0	0
$211$	−19.0961	−1.31463	−0.657314	−	0.753616i	$-0.728308\pi$
$211$	−19.0961	−1.31463	−0.657314	+	0.753616i	$0.728308\pi$
$212$	−16.4945	−1.13285
$213$	0	0
$214$	1.55550	0.106332
$215$	−14.4990	−0.988823
$216$	0	0
$217$	4.64058	0.315023
$218$	1.32559	0.0897802
$219$	0	0
$220$	−11.1413	−0.751145
$221$	−0.315954	−0.0212534
$222$	0	0
$223$	8.41781	0.563698	0.281849	−	0.959459i	$-0.409052\pi$
$223$	8.41781	0.563698	0.281849	+	0.959459i	$0.409052\pi$
$224$	−4.31942	−0.288603
$225$	0	0
$226$	−3.50265	−0.232993
$227$	−1.82854	−0.121364	−0.0606821	−	0.998157i	$-0.519328\pi$
$227$	−1.82854	−0.121364	−0.0606821	+	0.998157i	$0.519328\pi$
$228$	0	0
$229$	2.52038	0.166551	0.0832756	−	0.996527i	$-0.473462\pi$
$229$	2.52038	0.166551	0.0832756	+	0.996527i	$0.473462\pi$
$230$	0.483338	0.0318704
$231$	0	0
$232$	−9.04169	−0.593616
$233$	−16.3812	−1.07317	−0.536585	−	0.843847i	$-0.680286\pi$
$233$	−16.3812	−1.07317	−0.536585	+	0.843847i	$0.680286\pi$
$234$	0	0
$235$	6.47249	0.422219
$236$	−24.9839	−1.62631
$237$	0	0
$238$	−2.77390	−0.179805
$239$	10.7430	0.694909	0.347455	−	0.937697i	$-0.387046\pi$
$239$	10.7430	0.694909	0.347455	+	0.937697i	$0.387046\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−3.07327	−0.197557
$243$	0	0
$244$	−19.5441	−1.25118
$245$	−6.33063	−0.404449
$246$	0	0
$247$	0.0301293	0.00191708
$248$	3.69857	0.234860
$249$	0	0
$250$	2.90257	0.183575
$251$	−23.8819	−1.50741	−0.753706	−	0.657212i	$-0.771736\pi$
$251$	−23.8819	−1.50741	−0.753706	+	0.657212i	$0.771736\pi$
$252$	0	0
$253$	−6.63342	−0.417040
$254$	−0.118706	−0.00744829
$255$	0	0
$256$	10.1859	0.636619
$257$	−5.25407	−0.327740	−0.163870	−	0.986482i	$-0.552398\pi$
$257$	−5.25407	−0.327740	−0.163870	+	0.986482i	$0.552398\pi$
$258$	0	0
$259$	9.78175	0.607808
$260$	0.102440	0.00635304
$261$	0	0
$262$	−3.70936	−0.229165
$263$	1.10964	0.0684235	0.0342118	−	0.999415i	$-0.489108\pi$
$263$	1.10964	0.0684235	0.0342118	+	0.999415i	$0.489108\pi$
$264$	0	0
$265$	10.5784	0.649824
$266$	0.264519	0.0162187
$267$	0	0
$268$	−19.4715	−1.18941
$269$	2.84634	0.173544	0.0867722	−	0.996228i	$-0.472345\pi$
$269$	2.84634	0.173544	0.0867722	+	0.996228i	$0.472345\pi$
$270$	0	0
$271$	13.1029	0.795946	0.397973	−	0.917397i	$-0.369714\pi$
$271$	13.1029	0.795946	0.397973	+	0.917397i	$0.369714\pi$
$272$	25.9193	1.57159
$273$	0	0
$274$	3.12834	0.188990
$275$	−16.3451	−0.985646
$276$	0	0
$277$	−23.4634	−1.40978	−0.704888	−	0.709318i	$-0.749003\pi$
$277$	−23.4634	−1.40978	−0.704888	+	0.709318i	$0.749003\pi$
$278$	0.257365	0.0154357
$279$	0	0
$280$	1.83476	0.109648
$281$	−8.91165	−0.531625	−0.265812	−	0.964025i	$-0.585640\pi$
$281$	−8.91165	−0.531625	−0.265812	+	0.964025i	$0.585640\pi$
$282$	0	0
$283$	−15.8040	−0.939453	−0.469726	−	0.882812i	$-0.655647\pi$
$283$	−15.8040	−0.939453	−0.469726	+	0.882812i	$0.655647\pi$
$284$	3.59021	0.213040
$285$	0	0
$286$	0.0563318	0.00333097
$287$	−7.45354	−0.439968
$288$	0	0
$289$	36.4990	2.14700
$290$	2.84240	0.166911
$291$	0	0
$292$	−12.4574	−0.729013
$293$	−32.9425	−1.92452	−0.962261	−	0.272127i	$-0.912273\pi$
$293$	−32.9425	−1.92452	−0.962261	+	0.272127i	$0.912273\pi$
$294$	0	0
$295$	16.0229	0.932888
$296$	7.79611	0.453140
$297$	0	0
$298$	3.57399	0.207036
$299$	0.0609917	0.00352724
$300$	0	0
$301$	−16.0629	−0.925851
$302$	5.89674	0.339319
$303$	0	0
$304$	−2.47165	−0.141759
$305$	12.5342	0.717705
$306$	0	0
$307$	20.2343	1.15483	0.577416	−	0.816450i	$-0.304061\pi$
$307$	20.2343	1.15483	0.577416	+	0.816450i	$0.304061\pi$
$308$	−12.3430	−0.703309
$309$	0	0
$310$	−1.16271	−0.0660372
$311$	−2.27289	−0.128884	−0.0644420	−	0.997921i	$-0.520527\pi$
$311$	−2.27289	−0.128884	−0.0644420	+	0.997921i	$0.520527\pi$
$312$	0	0
$313$	13.1142	0.741257	0.370629	−	0.928781i	$-0.379142\pi$
$313$	13.1142	0.741257	0.370629	+	0.928781i	$0.379142\pi$
$314$	−4.76873	−0.269115
$315$	0	0
$316$	24.8900	1.40017
$317$	7.86525	0.441757	0.220878	−	0.975301i	$-0.429108\pi$
$317$	7.86525	0.441757	0.220878	+	0.975301i	$0.429108\pi$
$318$	0	0
$319$	−39.0096	−2.18412
$320$	−7.65810	−0.428101
$321$	0	0
$322$	0.535473	0.0298407
$323$	−5.10164	−0.283863
$324$	0	0
$325$	0.150287	0.00833640
$326$	−3.08336	−0.170772
$327$	0	0
$328$	−5.94051	−0.328010
$329$	7.17065	0.395331
$330$	0	0
$331$	−19.3752	−1.06496	−0.532480	−	0.846443i	$-0.678740\pi$
$331$	−19.3752	−1.06496	−0.532480	+	0.846443i	$0.678740\pi$
$332$	−4.47884	−0.245808
$333$	0	0
$334$	−6.21192	−0.339901
$335$	12.4876	0.682273
$336$	0	0
$337$	−26.2459	−1.42970	−0.714852	−	0.699276i	$-0.753506\pi$
$337$	−26.2459	−1.42970	−0.714852	+	0.699276i	$0.753506\pi$
$338$	3.60798	0.196249
$339$	0	0
$340$	−17.3456	−0.940697
$341$	15.9572	0.864130
$342$	0	0
$343$	−16.5773	−0.895092
$344$	−12.8022	−0.690250
$345$	0	0
$346$	1.15500	0.0620931
$347$	3.65189	0.196044	0.0980219	−	0.995184i	$-0.468748\pi$
$347$	3.65189	0.196044	0.0980219	+	0.995184i	$0.468748\pi$
$348$	0	0
$349$	−0.818652	−0.0438214	−0.0219107	−	0.999760i	$-0.506975\pi$
$349$	−0.818652	−0.0438214	−0.0219107	+	0.999760i	$0.506975\pi$
$350$	1.31943	0.0705267
$351$	0	0
$352$	−14.8528	−0.791658
$353$	28.5367	1.51886	0.759428	−	0.650591i	$-0.225479\pi$
$353$	28.5367	1.51886	0.759428	+	0.650591i	$0.225479\pi$
$354$	0	0
$355$	−2.30250	−0.122204
$356$	−27.9173	−1.47961
$357$	0	0
$358$	−1.60720	−0.0849430
$359$	15.8623	0.837180	0.418590	−	0.908175i	$-0.362524\pi$
$359$	15.8623	0.837180	0.418590	+	0.908175i	$0.362524\pi$
$360$	0	0
$361$	−18.5135	−0.974395
$362$	−4.85531	−0.255189
$363$	0	0
$364$	0.113489	0.00594845
$365$	7.98926	0.418177
$366$	0	0
$367$	13.9937	0.730463	0.365232	−	0.930917i	$-0.380990\pi$
$367$	13.9937	0.730463	0.365232	+	0.930917i	$0.380990\pi$
$368$	−5.00344	−0.260822
$369$	0	0
$370$	−2.45083	−0.127413
$371$	11.7194	0.608441
$372$	0	0
$373$	12.2336	0.633434	0.316717	−	0.948520i	$-0.397419\pi$
$373$	12.2336	0.633434	0.316717	+	0.948520i	$0.397419\pi$
$374$	−9.53839	−0.493218
$375$	0	0
$376$	5.71505	0.294731
$377$	0.358678	0.0184728
$378$	0	0
$379$	0.284829	0.0146307	0.00731535	−	0.999973i	$-0.497671\pi$
$379$	0.284829	0.0146307	0.00731535	+	0.999973i	$0.497671\pi$
$380$	1.65407	0.0848520
$381$	0	0
$382$	−5.99772	−0.306870
$383$	−29.8268	−1.52408	−0.762038	−	0.647532i	$-0.775801\pi$
$383$	−29.8268	−1.52408	−0.762038	+	0.647532i	$0.775801\pi$
$384$	0	0
$385$	7.91593	0.403433
$386$	−0.640887	−0.0326203
$387$	0	0
$388$	4.30211	0.218407
$389$	8.88544	0.450510	0.225255	−	0.974300i	$-0.427679\pi$
$389$	8.88544	0.450510	0.225255	+	0.974300i	$0.427679\pi$
$390$	0	0
$391$	−10.3274	−0.522280
$392$	−5.58978	−0.282327
$393$	0	0
$394$	0.532605	0.0268322
$395$	−15.9626	−0.803167
$396$	0	0
$397$	−29.5831	−1.48473	−0.742367	−	0.669993i	$-0.766297\pi$
$397$	−29.5831	−1.48473	−0.742367	+	0.669993i	$0.766297\pi$
$398$	−5.89657	−0.295569
$399$	0	0
$400$	−12.3287	−0.616437
$401$	−6.30306	−0.314760	−0.157380	−	0.987538i	$-0.550305\pi$
$401$	−6.30306	−0.314760	−0.157380	+	0.987538i	$0.550305\pi$
$402$	0	0
$403$	−0.146720	−0.00730864
$404$	−7.72270	−0.384219
$405$	0	0
$406$	3.14899	0.156282
$407$	33.6357	1.66726
$408$	0	0
$409$	−7.63163	−0.377360	−0.188680	−	0.982039i	$-0.560421\pi$
$409$	−7.63163	−0.377360	−0.188680	+	0.982039i	$0.560421\pi$
$410$	1.86750	0.0922290
$411$	0	0
$412$	10.0991	0.497546
$413$	17.7512	0.873478
$414$	0	0
$415$	2.87240	0.141001
$416$	0.136566	0.00669568
$417$	0	0
$418$	0.909578	0.0444889
$419$	−2.34760	−0.114688	−0.0573439	−	0.998354i	$-0.518263\pi$
$419$	−2.34760	−0.114688	−0.0573439	+	0.998354i	$0.518263\pi$
$420$	0	0
$421$	8.61693	0.419963	0.209982	−	0.977705i	$-0.432660\pi$
$421$	8.61693	0.419963	0.209982	+	0.977705i	$0.432660\pi$
$422$	5.30064	0.258031
$423$	0	0
$424$	9.34043	0.453611
$425$	−25.4473	−1.23437
$426$	0	0
$427$	13.8862	0.671999
$428$	10.7759	0.520875
$429$	0	0
$430$	4.02459	0.194083
$431$	2.73295	0.131641	0.0658207	−	0.997831i	$-0.479033\pi$
$431$	2.73295	0.131641	0.0658207	+	0.997831i	$0.479033\pi$
$432$	0	0
$433$	14.8768	0.714934	0.357467	−	0.933926i	$-0.383640\pi$
$433$	14.8768	0.714934	0.357467	+	0.933926i	$0.383640\pi$
$434$	−1.28812	−0.0618317
$435$	0	0
$436$	9.18318	0.439795
$437$	0.984819	0.0471103
$438$	0	0
$439$	−19.8115	−0.945552	−0.472776	−	0.881183i	$-0.656748\pi$
$439$	−19.8115	−0.945552	−0.472776	+	0.881183i	$0.656748\pi$
$440$	6.30904	0.300772
$441$	0	0
$442$	0.0877017	0.00417154
$443$	39.5818	1.88059	0.940293	−	0.340366i	$-0.110551\pi$
$443$	39.5818	1.88059	0.940293	+	0.340366i	$0.110551\pi$
$444$	0	0
$445$	17.9041	0.848738
$446$	−2.33659	−0.110641
$447$	0	0
$448$	−8.48414	−0.400838
$449$	22.9205	1.08169	0.540843	−	0.841124i	$-0.318105\pi$
$449$	22.9205	1.08169	0.540843	+	0.841124i	$0.318105\pi$
$450$	0	0
$451$	−25.6299	−1.20686
$452$	−24.2650	−1.14133
$453$	0	0
$454$	0.507560	0.0238210
$455$	−0.0727838	−0.00341216
$456$	0	0
$457$	−9.66494	−0.452107	−0.226053	−	0.974115i	$-0.572582\pi$
$457$	−9.66494	−0.452107	−0.226053	+	0.974115i	$0.572582\pi$
$458$	−0.699599	−0.0326901
$459$	0	0
$460$	3.34839	0.156119
$461$	−6.28046	−0.292510	−0.146255	−	0.989247i	$-0.546722\pi$
$461$	−6.28046	−0.292510	−0.146255	+	0.989247i	$0.546722\pi$
$462$	0	0
$463$	−7.64337	−0.355218	−0.177609	−	0.984101i	$-0.556836\pi$
$463$	−7.64337	−0.355218	−0.177609	+	0.984101i	$0.556836\pi$
$464$	−29.4241	−1.36598
$465$	0	0
$466$	4.54705	0.210638
$467$	−25.8943	−1.19824	−0.599122	−	0.800658i	$-0.704484\pi$
$467$	−25.8943	−1.19824	−0.599122	+	0.800658i	$0.704484\pi$
$468$	0	0
$469$	13.8346	0.638823
$470$	−1.79662	−0.0828717
$471$	0	0
$472$	14.1478	0.651205
$473$	−55.2342	−2.53967
$474$	0	0
$475$	2.42664	0.111342
$476$	−19.2166	−0.880790
$477$	0	0
$478$	−2.98202	−0.136394
$479$	−12.6131	−0.576309	−0.288155	−	0.957584i	$-0.593042\pi$
$479$	−12.6131	−0.576309	−0.288155	+	0.957584i	$0.593042\pi$
$480$	0	0
$481$	−0.309267	−0.0141013
$482$	0.277577	0.0126433
$483$	0	0
$484$	−21.2905	−0.967749
$485$	−2.75907	−0.125283
$486$	0	0
$487$	31.7807	1.44012	0.720061	−	0.693910i	$-0.244114\pi$
$487$	31.7807	1.44012	0.720061	+	0.693910i	$0.244114\pi$
$488$	11.0674	0.500996
$489$	0	0
$490$	1.75724	0.0793839
$491$	10.2151	0.461001	0.230500	−	0.973072i	$-0.425964\pi$
$491$	10.2151	0.461001	0.230500	+	0.973072i	$0.425964\pi$
$492$	0	0
$493$	−60.7331	−2.73528
$494$	−0.00836321	−0.000376278 0
$495$	0	0
$496$	12.0361	0.540439
$497$	−2.55086	−0.114422
$498$	0	0
$499$	−9.35390	−0.418738	−0.209369	−	0.977837i	$-0.567141\pi$
$499$	−9.35390	−0.418738	−0.209369	+	0.977837i	$0.567141\pi$
$500$	20.1079	0.899253
$501$	0	0
$502$	6.62907	0.295870
$503$	13.7659	0.613792	0.306896	−	0.951743i	$-0.400710\pi$
$503$	13.7659	0.613792	0.306896	+	0.951743i	$0.400710\pi$
$504$	0	0
$505$	4.95278	0.220396
$506$	1.84129	0.0818552
$507$	0	0
$508$	−0.822353	−0.0364860
$509$	−22.6367	−1.00335	−0.501677	−	0.865055i	$-0.667283\pi$
$509$	−22.6367	−1.00335	−0.501677	+	0.865055i	$0.667283\pi$
$510$	0	0
$511$	8.85102	0.391546
$512$	−18.9206	−0.836182
$513$	0	0
$514$	1.45841	0.0643277
$515$	−6.47682	−0.285403
$516$	0	0
$517$	24.6571	1.08442
$518$	−2.71519	−0.119299
$519$	0	0
$520$	−0.0580091	−0.00254387
$521$	10.8738	0.476389	0.238194	−	0.971218i	$-0.423445\pi$
$521$	10.8738	0.476389	0.238194	+	0.971218i	$0.423445\pi$
$522$	0	0
$523$	−3.43232	−0.150085	−0.0750424	−	0.997180i	$-0.523909\pi$
$523$	−3.43232	−0.150085	−0.0750424	+	0.997180i	$0.523909\pi$
$524$	−25.6970	−1.12258
$525$	0	0
$526$	−0.308011	−0.0134299
$527$	24.8434	1.08219
$528$	0	0
$529$	−21.0064	−0.913322
$530$	−2.93631	−0.127545
$531$	0	0
$532$	1.83249	0.0794483
$533$	0.235656	0.0102074
$534$	0	0
$535$	−6.91091	−0.298785
$536$	11.0263	0.476263
$537$	0	0
$538$	−0.790079	−0.0340627
$539$	−24.1167	−1.03878
$540$	0	0
$541$	−1.00758	−0.0433193	−0.0216597	−	0.999765i	$-0.506895\pi$
$541$	−1.00758	−0.0433193	−0.0216597	+	0.999765i	$0.506895\pi$
$542$	−3.63707	−0.156226
$543$	0	0
$544$	−23.1240	−0.991433
$545$	−5.88943	−0.252275
$546$	0	0
$547$	10.6391	0.454894	0.227447	−	0.973791i	$-0.426962\pi$
$547$	10.6391	0.454894	0.227447	+	0.973791i	$0.426962\pi$
$548$	21.6720	0.925780
$549$	0	0
$550$	4.53702	0.193459
$551$	5.79149	0.246726
$552$	0	0
$553$	−17.6844	−0.752019
$554$	6.51289	0.276706
$555$	0	0
$556$	1.78293	0.0756130
$557$	−18.9092	−0.801207	−0.400603	−	0.916252i	$-0.631199\pi$
$557$	−18.9092	−0.801207	−0.400603	+	0.916252i	$0.631199\pi$
$558$	0	0
$559$	0.507856	0.0214800
$560$	5.97081	0.252313
$561$	0	0
$562$	2.47367	0.104346
$563$	−28.1768	−1.18751	−0.593756	−	0.804645i	$-0.702356\pi$
$563$	−28.1768	−1.18751	−0.593756	+	0.804645i	$0.702356\pi$
$564$	0	0
$565$	15.5618	0.654692
$566$	4.38684	0.184393
$567$	0	0
$568$	−2.03305	−0.0853049
$569$	0.499818	0.0209534	0.0104767	−	0.999945i	$-0.496665\pi$
$569$	0.499818	0.0209534	0.0104767	+	0.999945i	$0.496665\pi$
$570$	0	0
$571$	−28.2303	−1.18140	−0.590702	−	0.806890i	$-0.701149\pi$
$571$	−28.2303	−1.18140	−0.590702	+	0.806890i	$0.701149\pi$
$572$	0.390246	0.0163170
$573$	0	0
$574$	2.06893	0.0863555
$575$	4.91233	0.204858
$576$	0	0
$577$	−36.3728	−1.51422	−0.757111	−	0.653287i	$-0.773390\pi$
$577$	−36.3728	−1.51422	−0.757111	+	0.653287i	$0.773390\pi$
$578$	−10.1313	−0.421406
$579$	0	0
$580$	19.6911	0.817627
$581$	3.18224	0.132021
$582$	0	0
$583$	40.2985	1.66899
$584$	7.05432	0.291910
$585$	0	0
$586$	9.14409	0.377739
$587$	−16.1223	−0.665437	−0.332718	−	0.943026i	$-0.607966\pi$
$587$	−16.1223	−0.665437	−0.332718	+	0.943026i	$0.607966\pi$
$588$	0	0
$589$	−2.36905	−0.0976152
$590$	−4.44758	−0.183104
$591$	0	0
$592$	25.3706	1.04273
$593$	−24.8487	−1.02041	−0.510207	−	0.860052i	$-0.670431\pi$
$593$	−24.8487	−1.02041	−0.510207	+	0.860052i	$0.670431\pi$
$594$	0	0
$595$	12.3241	0.505240
$596$	24.7593	1.01418
$597$	0	0
$598$	−0.0169299	−0.000692315 0
$599$	1.50155	0.0613517	0.0306759	−	0.999529i	$-0.490234\pi$
$599$	1.50155	0.0613517	0.0306759	+	0.999529i	$0.490234\pi$
$600$	0	0
$601$	−24.0828	−0.982356	−0.491178	−	0.871059i	$-0.663434\pi$
$601$	−24.0828	−0.982356	−0.491178	+	0.871059i	$0.663434\pi$
$602$	4.45870	0.181723
$603$	0	0
$604$	40.8504	1.66218
$605$	13.6542	0.555121
$606$	0	0
$607$	16.0760	0.652506	0.326253	−	0.945283i	$-0.394214\pi$
$607$	16.0760	0.652506	0.326253	+	0.945283i	$0.394214\pi$
$608$	2.20510	0.0894284
$609$	0	0
$610$	−3.47920	−0.140869
$611$	−0.226712	−0.00917179
$612$	0	0
$613$	43.5233	1.75789	0.878945	−	0.476923i	$-0.158248\pi$
$613$	43.5233	1.75789	0.878945	+	0.476923i	$0.158248\pi$
$614$	−5.61658	−0.226667
$615$	0	0
$616$	6.98957	0.281618
$617$	3.82544	0.154006	0.0770032	−	0.997031i	$-0.475465\pi$
$617$	3.82544	0.154006	0.0770032	+	0.997031i	$0.475465\pi$
$618$	0	0
$619$	8.69881	0.349635	0.174817	−	0.984601i	$-0.444066\pi$
$619$	8.69881	0.349635	0.174817	+	0.984601i	$0.444066\pi$
$620$	−8.05479	−0.323488
$621$	0	0
$622$	0.630903	0.0252969
$623$	19.8354	0.794687
$624$	0	0
$625$	4.49981	0.179992
$626$	−3.64020	−0.145492
$627$	0	0
$628$	−33.0360	−1.31828
$629$	52.3666	2.08799
$630$	0	0
$631$	−28.0264	−1.11571	−0.557856	−	0.829938i	$-0.688376\pi$
$631$	−28.0264	−1.11571	−0.557856	+	0.829938i	$0.688376\pi$
$632$	−14.0946	−0.560653
$633$	0	0
$634$	−2.18321	−0.0867065
$635$	0.527398	0.0209291
$636$	0	0
$637$	0.221743	0.00878578
$638$	10.8282	0.428692
$639$	0	0
$640$	9.92345	0.392259
$641$	−22.7621	−0.899048	−0.449524	−	0.893268i	$-0.648406\pi$
$641$	−22.7621	−0.899048	−0.449524	+	0.893268i	$0.648406\pi$
$642$	0	0
$643$	35.9559	1.41796	0.708981	−	0.705228i	$-0.249155\pi$
$643$	35.9559	1.41796	0.708981	+	0.705228i	$0.249155\pi$
$644$	3.70956	0.146177
$645$	0	0
$646$	1.41610	0.0557157
$647$	1.94619	0.0765127	0.0382564	−	0.999268i	$-0.487820\pi$
$647$	1.94619	0.0765127	0.0382564	+	0.999268i	$0.487820\pi$
$648$	0	0
$649$	61.0395	2.39601
$650$	−0.0417161	−0.00163624
$651$	0	0
$652$	−21.3604	−0.836537
$653$	−24.3392	−0.952467	−0.476233	−	0.879319i	$-0.657998\pi$
$653$	−24.3392	−0.952467	−0.476233	+	0.879319i	$0.657998\pi$
$654$	0	0
$655$	16.4802	0.643936
$656$	−19.3320	−0.754789
$657$	0	0
$658$	−1.99041	−0.0775942
$659$	6.16907	0.240313	0.120156	−	0.992755i	$-0.461660\pi$
$659$	6.16907	0.240313	0.120156	+	0.992755i	$0.461660\pi$
$660$	0	0
$661$	−48.1704	−1.87361	−0.936807	−	0.349848i	$-0.886233\pi$
$661$	−48.1704	−1.87361	−0.936807	+	0.349848i	$0.886233\pi$
$662$	5.37812	0.209027
$663$	0	0
$664$	2.53626	0.0984260
$665$	−1.17522	−0.0455732
$666$	0	0
$667$	11.7239	0.453951
$668$	−43.0338	−1.66503
$669$	0	0
$670$	−3.46628	−0.133914
$671$	47.7492	1.84334
$672$	0	0
$673$	−19.2071	−0.740381	−0.370190	−	0.928956i	$-0.620708\pi$
$673$	−19.2071	−0.740381	−0.370190	+	0.928956i	$0.620708\pi$
$674$	7.28525	0.280617
$675$	0	0
$676$	24.9948	0.961337
$677$	14.9094	0.573014	0.286507	−	0.958078i	$-0.407506\pi$
$677$	14.9094	0.573014	0.286507	+	0.958078i	$0.407506\pi$
$678$	0	0
$679$	−3.05667	−0.117304
$680$	9.82240	0.376672
$681$	0	0
$682$	−4.42935	−0.169609
$683$	22.8682	0.875029	0.437515	−	0.899211i	$-0.355859\pi$
$683$	22.8682	0.875029	0.437515	+	0.899211i	$0.355859\pi$
$684$	0	0
$685$	−13.8988	−0.531047
$686$	4.60149	0.175686
$687$	0	0
$688$	−41.6619	−1.58835
$689$	−0.370529	−0.0141160
$690$	0	0
$691$	−35.5210	−1.35128	−0.675641	−	0.737231i	$-0.736133\pi$
$691$	−35.5210	−1.35128	−0.675641	+	0.737231i	$0.736133\pi$
$692$	8.00140	0.304168
$693$	0	0
$694$	−1.01368	−0.0384788
$695$	−1.14344	−0.0433732
$696$	0	0
$697$	−39.9025	−1.51142
$698$	0.227239	0.00860112
$699$	0	0
$700$	9.14054	0.345480
$701$	−9.22303	−0.348349	−0.174175	−	0.984715i	$-0.555726\pi$
$701$	−9.22303	−0.348349	−0.174175	+	0.984715i	$0.555726\pi$
$702$	0	0
$703$	−4.99366	−0.188339
$704$	−29.1737	−1.09952
$705$	0	0
$706$	−7.92114	−0.298116
$707$	5.48701	0.206360
$708$	0	0
$709$	28.7763	1.08072	0.540358	−	0.841435i	$-0.318289\pi$
$709$	28.7763	1.08072	0.540358	+	0.841435i	$0.318289\pi$
$710$	0.639122	0.0239858
$711$	0	0
$712$	15.8089	0.592464
$713$	−4.79575	−0.179602
$714$	0	0
$715$	−0.250276	−0.00935978
$716$	−11.1341	−0.416100
$717$	0	0
$718$	−4.40301	−0.164319
$719$	24.1878	0.902055	0.451027	−	0.892510i	$-0.351058\pi$
$719$	24.1878	0.902055	0.451027	+	0.892510i	$0.351058\pi$
$720$	0	0
$721$	−7.17544	−0.267227
$722$	5.13893	0.191251
$723$	0	0
$724$	−33.6358	−1.25006
$725$	28.8883	1.07288
$726$	0	0
$727$	9.73994	0.361235	0.180617	−	0.983553i	$-0.442190\pi$
$727$	9.73994	0.361235	0.180617	+	0.983553i	$0.442190\pi$
$728$	−0.0642663	−0.00238187
$729$	0	0
$730$	−2.21764	−0.0820784
$731$	−85.9928	−3.18056
$732$	0	0
$733$	−4.38060	−0.161801	−0.0809006	−	0.996722i	$-0.525780\pi$
$733$	−4.38060	−0.161801	−0.0809006	+	0.996722i	$0.525780\pi$
$734$	−3.88432	−0.143373
$735$	0	0
$736$	4.46384	0.164539
$737$	47.5719	1.75234
$738$	0	0
$739$	−6.21520	−0.228630	−0.114315	−	0.993445i	$-0.536467\pi$
$739$	−6.21520	−0.228630	−0.114315	+	0.993445i	$0.536467\pi$
$740$	−16.9784	−0.624140
$741$	0	0
$742$	−3.25304	−0.119423
$743$	−26.7097	−0.979885	−0.489942	−	0.871755i	$-0.662982\pi$
$743$	−26.7097	−0.979885	−0.489942	+	0.871755i	$0.662982\pi$
$744$	0	0
$745$	−15.8788	−0.581754
$746$	−3.39578	−0.124328
$747$	0	0
$748$	−66.0784	−2.41607
$749$	−7.65635	−0.279757
$750$	0	0
$751$	46.9924	1.71478	0.857388	−	0.514670i	$-0.172085\pi$
$751$	46.9924	1.71478	0.857388	+	0.514670i	$0.172085\pi$
$752$	18.5983	0.678210
$753$	0	0
$754$	−0.0995608	−0.00362579
$755$	−26.1985	−0.953461
$756$	0	0
$757$	−35.2425	−1.28091	−0.640456	−	0.767995i	$-0.721255\pi$
$757$	−35.2425	−1.28091	−0.640456	+	0.767995i	$0.721255\pi$
$758$	−0.0790621	−0.00287166
$759$	0	0
$760$	−0.936661	−0.0339763
$761$	47.1109	1.70777	0.853885	−	0.520461i	$-0.174240\pi$
$761$	47.1109	1.70777	0.853885	+	0.520461i	$0.174240\pi$
$762$	0	0
$763$	−6.52469	−0.236210
$764$	−41.5499	−1.50322
$765$	0	0
$766$	8.27923	0.299141
$767$	−0.561234	−0.0202650
$768$	0	0
$769$	−53.8827	−1.94306	−0.971530	−	0.236915i	$-0.923864\pi$
$769$	−53.8827	−1.94306	−0.971530	+	0.236915i	$0.923864\pi$
$770$	−2.19728	−0.0791845
$771$	0	0
$772$	−4.43983	−0.159793
$773$	7.33241	0.263728	0.131864	−	0.991268i	$-0.457904\pi$
$773$	7.33241	0.263728	0.131864	+	0.991268i	$0.457904\pi$
$774$	0	0
$775$	−11.8170	−0.424478
$776$	−2.43618	−0.0874539
$777$	0	0
$778$	−2.46639	−0.0884245
$779$	3.80509	0.136332
$780$	0	0
$781$	−8.77143	−0.313866
$782$	2.86666	0.102511
$783$	0	0
$784$	−18.1906	−0.649666
$785$	21.1869	0.756193
$786$	0	0
$787$	−0.274880	−0.00979841	−0.00489921	−	0.999988i	$-0.501559\pi$
$787$	−0.274880	−0.00979841	−0.00489921	+	0.999988i	$0.501559\pi$
$788$	3.68969	0.131440
$789$	0	0
$790$	4.43086	0.157643
$791$	17.2404	0.612998
$792$	0	0
$793$	−0.439035	−0.0155906
$794$	8.21160	0.291419
$795$	0	0
$796$	−40.8493	−1.44786
$797$	7.42858	0.263134	0.131567	−	0.991307i	$-0.457999\pi$
$797$	7.42858	0.263134	0.131567	+	0.991307i	$0.457999\pi$
$798$	0	0
$799$	38.3881	1.35807
$800$	10.9991	0.388878
$801$	0	0
$802$	1.74958	0.0617800
$803$	30.4353	1.07404
$804$	0	0
$805$	−2.37904	−0.0838502
$806$	0.0407261	0.00143452
$807$	0	0
$808$	4.37318	0.153848
$809$	−42.6491	−1.49946	−0.749730	−	0.661743i	$-0.769817\pi$
$809$	−42.6491	−1.49946	−0.749730	+	0.661743i	$0.769817\pi$
$810$	0	0
$811$	25.6139	0.899425	0.449713	−	0.893173i	$-0.351526\pi$
$811$	25.6139	0.899425	0.449713	+	0.893173i	$0.351526\pi$
$812$	21.8150	0.765558
$813$	0	0
$814$	−9.33649	−0.327244
$815$	13.6990	0.479855
$816$	0	0
$817$	8.20024	0.286890
$818$	2.11837	0.0740669
$819$	0	0
$820$	12.9373	0.451790
$821$	−21.8573	−0.762826	−0.381413	−	0.924405i	$-0.624562\pi$
$821$	−21.8573	−0.762826	−0.381413	+	0.924405i	$0.624562\pi$
$822$	0	0
$823$	1.58270	0.0551693	0.0275847	−	0.999619i	$-0.491218\pi$
$823$	1.58270	0.0551693	0.0275847	+	0.999619i	$0.491218\pi$
$824$	−5.71887	−0.199226
$825$	0	0
$826$	−4.92732	−0.171443
$827$	−36.2198	−1.25949	−0.629743	−	0.776803i	$-0.716840\pi$
$827$	−36.2198	−1.25949	−0.629743	+	0.776803i	$0.716840\pi$
$828$	0	0
$829$	30.5957	1.06263	0.531316	−	0.847174i	$-0.321698\pi$
$829$	30.5957	1.06263	0.531316	+	0.847174i	$0.321698\pi$
$830$	−0.797314	−0.0276752
$831$	0	0
$832$	0.268241	0.00929957
$833$	−37.5466	−1.30091
$834$	0	0
$835$	27.5988	0.955096
$836$	6.30122	0.217932
$837$	0	0
$838$	0.651640	0.0225105
$839$	9.85906	0.340373	0.170186	−	0.985412i	$-0.445563\pi$
$839$	9.85906	0.340373	0.170186	+	0.985412i	$0.445563\pi$
$840$	0	0
$841$	39.9455	1.37743
$842$	−2.39186	−0.0824290
$843$	0	0
$844$	36.7208	1.26398
$845$	−16.0298	−0.551443
$846$	0	0
$847$	15.1270	0.519769
$848$	30.3963	1.04381
$849$	0	0
$850$	7.06358	0.242279
$851$	−10.1088	−0.346526
$852$	0	0
$853$	20.6701	0.707730	0.353865	−	0.935296i	$-0.384867\pi$
$853$	20.6701	0.707730	0.353865	+	0.935296i	$0.384867\pi$
$854$	−3.85448	−0.131898
$855$	0	0
$856$	−6.10216	−0.208567
$857$	−36.0858	−1.23267	−0.616334	−	0.787485i	$-0.711383\pi$
$857$	−36.0858	−1.23267	−0.616334	+	0.787485i	$0.711383\pi$
$858$	0	0
$859$	40.7566	1.39060	0.695299	−	0.718720i	$-0.255272\pi$
$859$	40.7566	1.39060	0.695299	+	0.718720i	$0.255272\pi$
$860$	27.8808	0.950729
$861$	0	0
$862$	−0.758604	−0.0258381
$863$	−21.5125	−0.732295	−0.366148	−	0.930557i	$-0.619323\pi$
$863$	−21.5125	−0.732295	−0.366148	+	0.930557i	$0.619323\pi$
$864$	0	0
$865$	−5.13152	−0.174477
$866$	−4.12947	−0.140325
$867$	0	0
$868$	−8.92362	−0.302887
$869$	−60.8100	−2.06284
$870$	0	0
$871$	−0.437405	−0.0148209
$872$	−5.20022	−0.176102
$873$	0	0
$874$	−0.273363	−0.00924665
$875$	−14.2868	−0.482981
$876$	0	0
$877$	−47.8912	−1.61717	−0.808586	−	0.588378i	$-0.799767\pi$
$877$	−47.8912	−1.61717	−0.808586	+	0.588378i	$0.799767\pi$
$878$	5.49922	0.185590
$879$	0	0
$880$	20.5313	0.692111
$881$	18.2689	0.615496	0.307748	−	0.951468i	$-0.400425\pi$
$881$	18.2689	0.615496	0.307748	+	0.951468i	$0.400425\pi$
$882$	0	0
$883$	−37.0354	−1.24634	−0.623170	−	0.782087i	$-0.714155\pi$
$883$	−37.0354	−1.24634	−0.623170	+	0.782087i	$0.714155\pi$
$884$	0.607565	0.0204346
$885$	0	0
$886$	−10.9870	−0.369115
$887$	13.7619	0.462080	0.231040	−	0.972944i	$-0.425787\pi$
$887$	13.7619	0.462080	0.231040	+	0.972944i	$0.425787\pi$
$888$	0	0
$889$	0.584285	0.0195963
$890$	−4.96978	−0.166587
$891$	0	0
$892$	−16.1870	−0.541982
$893$	−3.66067	−0.122500
$894$	0	0
$895$	7.14059	0.238684
$896$	10.9938	0.367278
$897$	0	0
$898$	−6.36221	−0.212310
$899$	−28.2027	−0.940613
$900$	0	0
$901$	62.7398	2.09017
$902$	7.11426	0.236879
$903$	0	0
$904$	13.7407	0.457009
$905$	21.5716	0.717063
$906$	0	0
$907$	−27.1842	−0.902637	−0.451319	−	0.892363i	$-0.649046\pi$
$907$	−27.1842	−0.902637	−0.451319	+	0.892363i	$0.649046\pi$
$908$	3.51619	0.116689
$909$	0	0
$910$	0.0202031	0.000669727 0
$911$	−29.4894	−0.977027	−0.488513	−	0.872556i	$-0.662461\pi$
$911$	−29.4894	−0.977027	−0.488513	+	0.872556i	$0.662461\pi$
$912$	0	0
$913$	10.9425	0.362143
$914$	2.68277	0.0887380
$915$	0	0
$916$	−4.84656	−0.160135
$917$	18.2579	0.602928
$918$	0	0
$919$	−44.2354	−1.45919	−0.729596	−	0.683878i	$-0.760292\pi$
$919$	−44.2354	−1.45919	−0.729596	+	0.683878i	$0.760292\pi$
$920$	−1.89611	−0.0625129
$921$	0	0
$922$	1.74331	0.0574130
$923$	0.0806498	0.00265462
$924$	0	0
$925$	−24.9086	−0.818991
$926$	2.12163	0.0697210
$927$	0	0
$928$	26.2508	0.861726
$929$	−17.7425	−0.582112	−0.291056	−	0.956706i	$-0.594007\pi$
$929$	−17.7425	−0.582112	−0.291056	+	0.956706i	$0.594007\pi$
$930$	0	0
$931$	3.58043	0.117344
$932$	31.5003	1.03183
$933$	0	0
$934$	7.18766	0.235187
$935$	42.3779	1.38591
$936$	0	0
$937$	46.1680	1.50824	0.754121	−	0.656735i	$-0.228063\pi$
$937$	46.1680	1.50824	0.754121	+	0.656735i	$0.228063\pi$
$938$	−3.84017	−0.125386
$939$	0	0
$940$	−12.4463	−0.405953
$941$	22.0383	0.718428	0.359214	−	0.933255i	$-0.383045\pi$
$941$	22.0383	0.718428	0.359214	+	0.933255i	$0.383045\pi$
$942$	0	0
$943$	7.70276	0.250836
$944$	46.0407	1.49850
$945$	0	0
$946$	15.3317	0.498478
$947$	34.5940	1.12416	0.562078	−	0.827085i	$-0.310002\pi$
$947$	34.5940	1.12416	0.562078	+	0.827085i	$0.310002\pi$
$948$	0	0
$949$	−0.279840	−0.00908399
$950$	−0.673581	−0.0218539
$951$	0	0
$952$	10.8819	0.352684
$953$	20.9799	0.679607	0.339803	−	0.940496i	$-0.389639\pi$
$953$	20.9799	0.679607	0.339803	+	0.940496i	$0.389639\pi$
$954$	0	0
$955$	26.6471	0.862281
$956$	−20.6583	−0.668138
$957$	0	0
$958$	3.50112	0.113116
$959$	−15.3980	−0.497228
$960$	0	0
$961$	−19.4635	−0.627854
$962$	0.0858453	0.00276777
$963$	0	0
$964$	1.92295	0.0619341
$965$	2.84738	0.0916605
$966$	0	0
$967$	−0.0716187	−0.00230310	−0.00115155	−	0.999999i	$-0.500367\pi$
$967$	−0.0716187	−0.00230310	−0.00115155	+	0.999999i	$0.500367\pi$
$968$	12.0563	0.387504
$969$	0	0
$970$	0.765853	0.0245901
$971$	8.18205	0.262575	0.131287	−	0.991344i	$-0.458089\pi$
$971$	8.18205	0.262575	0.131287	+	0.991344i	$0.458089\pi$
$972$	0	0
$973$	−1.26678	−0.0406111
$974$	−8.82161	−0.282663
$975$	0	0
$976$	36.0162	1.15285
$977$	−22.6886	−0.725872	−0.362936	−	0.931814i	$-0.618226\pi$
$977$	−22.6886	−0.725872	−0.362936	+	0.931814i	$0.618226\pi$
$978$	0	0
$979$	68.2062	2.17988
$980$	12.1735	0.388868
$981$	0	0
$982$	−2.83548	−0.0904837
$983$	18.8463	0.601103	0.300551	−	0.953766i	$-0.402829\pi$
$983$	18.8463	0.601103	0.300551	+	0.953766i	$0.402829\pi$
$984$	0	0
$985$	−2.36630	−0.0753966
$986$	16.8581	0.536872
$987$	0	0
$988$	−0.0579372	−0.00184323
$989$	16.6000	0.527850
$990$	0	0
$991$	−16.8310	−0.534655	−0.267328	−	0.963606i	$-0.586141\pi$
$991$	−16.8310	−0.534655	−0.267328	+	0.963606i	$0.586141\pi$
$992$	−10.7381	−0.340935
$993$	0	0
$994$	0.708060	0.0224583
$995$	26.1978	0.830525
$996$	0	0
$997$	60.1343	1.90447	0.952236	−	0.305363i	$-0.0987779\pi$
$997$	60.1343	1.90447	0.952236	+	0.305363i	$0.0987779\pi$
$998$	2.59643	0.0821885
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.e.1.3		7
3.2	odd	2		241.2.a.a.1.5	✓	7
12.11	even	2		3856.2.a.j.1.4		7
15.14	odd	2		6025.2.a.f.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.5	✓	7	3.2	odd	2
2169.2.a.e.1.3		7	1.1	even	1	trivial
3856.2.a.j.1.4		7	12.11	even	2
6025.2.a.f.1.3		7	15.14	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-0.277577 q^{2} -1.92295 q^{4} +1.23324 q^{5} +1.36627 q^{7} +1.08892 q^{8} +O(q^{10})\)	\(q-0.277577 q^{2} -1.92295 q^{4} +1.23324 q^{5} +1.36627 q^{7} +1.08892 q^{8} -0.342320 q^{10} +4.69806 q^{11} -0.0431968 q^{13} -0.379244 q^{14} +3.54364 q^{16} +7.31430 q^{17} -0.697489 q^{19} -2.37146 q^{20} -1.30407 q^{22} -1.41195 q^{23} -3.47911 q^{25} +0.0119904 q^{26} -2.62726 q^{28} -8.30334 q^{29} +3.39655 q^{31} -3.16148 q^{32} -2.03028 q^{34} +1.68494 q^{35} +7.15948 q^{37} +0.193607 q^{38} +1.34290 q^{40} -5.45541 q^{41} -11.7568 q^{43} -9.03414 q^{44} +0.391925 q^{46} +5.24836 q^{47} -5.13332 q^{49} +0.965723 q^{50} +0.0830653 q^{52} +8.57769 q^{53} +5.79385 q^{55} +1.48776 q^{56} +2.30482 q^{58} +12.9925 q^{59} +10.1636 q^{61} -0.942804 q^{62} -6.20973 q^{64} -0.0532721 q^{65} +10.1259 q^{67} -14.0650 q^{68} -0.467700 q^{70} -1.86703 q^{71} +6.47826 q^{73} -1.98731 q^{74} +1.34124 q^{76} +6.41880 q^{77} -12.9436 q^{79} +4.37017 q^{80} +1.51430 q^{82} +2.32915 q^{83} +9.02030 q^{85} +3.26342 q^{86} +5.11582 q^{88} +14.5180 q^{89} -0.0590183 q^{91} +2.71511 q^{92} -1.45682 q^{94} -0.860173 q^{95} -2.23725 q^{97} +1.42489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8	\( 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8} + 3 q^{10} + 18 q^{11} - q^{13} + 6 q^{14} + 4 q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 10 q^{22} + 22 q^{23} + 5 q^{25} - 8 q^{26} + 9 q^{28} + 16 q^{29} - 18 q^{31} + 6 q^{32} + 11 q^{34} - 7 q^{35} + 8 q^{37} - 16 q^{38} + 14 q^{40} + 15 q^{41} + 14 q^{43} + 4 q^{44} + 11 q^{46} + 10 q^{47} + 6 q^{49} + 4 q^{50} + 27 q^{52} - 15 q^{53} + 29 q^{55} - 13 q^{56} + 17 q^{58} + 18 q^{59} + 4 q^{61} - 13 q^{62} + 2 q^{64} + 7 q^{65} + 18 q^{67} + 15 q^{68} + 8 q^{70} + 50 q^{71} - 10 q^{74} - 20 q^{76} - 17 q^{77} - 15 q^{79} + 11 q^{80} + 45 q^{82} + 24 q^{83} - 2 q^{85} + 23 q^{86} + 8 q^{88} + 13 q^{89} - 12 q^{91} + 10 q^{92} - 32 q^{94} + 41 q^{95} + q^{97} - 9 q^{98}+O(q^{100}) \) 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8 + 3 * q^10 + 18 * q^11 - q^13 + 6 * q^14 + 4 * q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 10 * q^22 + 22 * q^23 + 5 * q^25 - 8 * q^26 + 9 * q^28 + 16 * q^29 - 18 * q^31 + 6 * q^32 + 11 * q^34 - 7 * q^35 + 8 * q^37 - 16 * q^38 + 14 * q^40 + 15 * q^41 + 14 * q^43 + 4 * q^44 + 11 * q^46 + 10 * q^47 + 6 * q^49 + 4 * q^50 + 27 * q^52 - 15 * q^53 + 29 * q^55 - 13 * q^56 + 17 * q^58 + 18 * q^59 + 4 * q^61 - 13 * q^62 + 2 * q^64 + 7 * q^65 + 18 * q^67 + 15 * q^68 + 8 * q^70 + 50 * q^71 - 10 * q^74 - 20 * q^76 - 17 * q^77 - 15 * q^79 + 11 * q^80 + 45 * q^82 + 24 * q^83 - 2 * q^85 + 23 * q^86 + 8 * q^88 + 13 * q^89 - 12 * q^91 + 10 * q^92 - 32 * q^94 + 41 * q^95 + q^97 - 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more