LMFDB - Embedded newform 4425.2.a.bg.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4425,2,Mod(1,4425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4425, 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("4425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4425 = 3 \cdot 5^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4425.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:	$35.3338028944$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.22298624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 8x^{3} + 14x^{2} - 4x - 2 $ x^6 - 2x^5 - 7x^4 + 8x^3 + 14x^2 - 4*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 885)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.69804$ of defining polynomial
Character	$\chi$	$=$	4425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69804 q^{2} +1.00000 q^{3} +5.27940 q^{4} +2.69804 q^{6} +2.82843 q^{7} +8.84793 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.69804 q^{2} +1.00000 q^{3} +5.27940 q^{4} +2.69804 q^{6} +2.82843 q^{7} +8.84793 q^{8} +1.00000 q^{9} +1.04833 q^{11} +5.27940 q^{12} -5.59570 q^{13} +7.63120 q^{14} +13.3132 q^{16} +3.04833 q^{17} +2.69804 q^{18} -3.52109 q^{19} +2.82843 q^{21} +2.82843 q^{22} -1.81560 q^{23} +8.84793 q^{24} -15.0974 q^{26} +1.00000 q^{27} +14.9324 q^{28} -6.56942 q^{29} +7.13706 q^{31} +18.2237 q^{32} +1.04833 q^{33} +8.22450 q^{34} +5.27940 q^{36} -5.59570 q^{37} -9.50003 q^{38} -5.59570 q^{39} +11.8600 q^{41} +7.63120 q^{42} +6.17397 q^{43} +5.53454 q^{44} -4.89855 q^{46} +3.71894 q^{47} +13.3132 q^{48} +1.00000 q^{49} +3.04833 q^{51} -29.5419 q^{52} -1.51046 q^{53} +2.69804 q^{54} +25.0257 q^{56} -3.52109 q^{57} -17.7245 q^{58} -1.00000 q^{59} -3.60805 q^{61} +19.2561 q^{62} +2.82843 q^{63} +22.5418 q^{64} +2.82843 q^{66} +3.80037 q^{67} +16.0933 q^{68} -1.81560 q^{69} -11.8645 q^{71} +8.84793 q^{72} +3.49904 q^{73} -15.0974 q^{74} -18.5892 q^{76} +2.96512 q^{77} -15.0974 q^{78} -14.4099 q^{79} +1.00000 q^{81} +31.9987 q^{82} -12.0897 q^{83} +14.9324 q^{84} +16.6576 q^{86} -6.56942 q^{87} +9.27553 q^{88} -11.0032 q^{89} -15.8270 q^{91} -9.58527 q^{92} +7.13706 q^{93} +10.0338 q^{94} +18.2237 q^{96} +11.5353 q^{97} +2.69804 q^{98} +1.04833 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9	$ 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9} - 2 q^{11} + 8 q^{12} - 2 q^{13} + 12 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 12 q^{23} + 12 q^{24} - 8 q^{26} + 6 q^{27} + 16 q^{28} - 12 q^{29} + 8 q^{31} + 28 q^{32} - 2 q^{33} + 8 q^{34} + 8 q^{36} - 2 q^{37} + 24 q^{38} - 2 q^{39} - 4 q^{41} + 18 q^{43} + 4 q^{44} + 16 q^{47} + 12 q^{48} + 6 q^{49} + 10 q^{51} - 12 q^{52} + 30 q^{53} + 4 q^{54} - 2 q^{57} + 16 q^{58} - 6 q^{59} + 4 q^{61} + 16 q^{62} + 20 q^{64} + 30 q^{67} + 20 q^{68} + 12 q^{69} - 24 q^{71} + 12 q^{72} + 6 q^{73} - 8 q^{74} - 4 q^{76} + 16 q^{77} - 8 q^{78} - 2 q^{79} + 6 q^{81} + 20 q^{83} + 16 q^{84} - 12 q^{87} + 16 q^{88} + 14 q^{89} - 48 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 28 q^{96} - 4 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9 - 2 * q^11 + 8 * q^12 - 2 * q^13 + 12 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 12 * q^23 + 12 * q^24 - 8 * q^26 + 6 * q^27 + 16 * q^28 - 12 * q^29 + 8 * q^31 + 28 * q^32 - 2 * q^33 + 8 * q^34 + 8 * q^36 - 2 * q^37 + 24 * q^38 - 2 * q^39 - 4 * q^41 + 18 * q^43 + 4 * q^44 + 16 * q^47 + 12 * q^48 + 6 * q^49 + 10 * q^51 - 12 * q^52 + 30 * q^53 + 4 * q^54 - 2 * q^57 + 16 * q^58 - 6 * q^59 + 4 * q^61 + 16 * q^62 + 20 * q^64 + 30 * q^67 + 20 * q^68 + 12 * q^69 - 24 * q^71 + 12 * q^72 + 6 * q^73 - 8 * q^74 - 4 * q^76 + 16 * q^77 - 8 * q^78 - 2 * q^79 + 6 * q^81 + 20 * q^83 + 16 * q^84 - 12 * q^87 + 16 * q^88 + 14 * q^89 - 48 * q^91 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 28 * q^96 - 4 * q^97 + 4 * q^98 - 2 * 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 $

$2$	2.69804	1.90780	0.953900	−	0.300126i	$-0.0970286\pi$
$2$	2.69804	1.90780	0.953900	+	0.300126i	$0.0970286\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.27940	2.63970
$5$	0	0
$5$	0	0
$6$	2.69804	1.10147
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	8.84793	3.12822
$9$	1.00000	0.333333
$10$	0	0
$11$	1.04833	0.316083	0.158041	−	0.987432i	$-0.449482\pi$
$11$	1.04833	0.316083	0.158041	+	0.987432i	$0.449482\pi$
$12$	5.27940	1.52403
$13$	−5.59570	−1.55197	−0.775984	−	0.630753i	$-0.782746\pi$
$13$	−5.59570	−1.55197	−0.775984	+	0.630753i	$0.782746\pi$
$14$	7.63120	2.03952
$15$	0	0
$16$	13.3132	3.32831
$17$	3.04833	0.739328	0.369664	−	0.929165i	$-0.379473\pi$
$17$	3.04833	0.739328	0.369664	+	0.929165i	$0.379473\pi$
$18$	2.69804	0.635933
$19$	−3.52109	−0.807794	−0.403897	−	0.914804i	$-0.632345\pi$
$19$	−3.52109	−0.807794	−0.403897	+	0.914804i	$0.632345\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	2.82843	0.603023
$23$	−1.81560	−0.378578	−0.189289	−	0.981921i	$-0.560618\pi$
$23$	−1.81560	−0.378578	−0.189289	+	0.981921i	$0.560618\pi$
$24$	8.84793	1.80608
$25$	0	0
$26$	−15.0974	−2.96084
$27$	1.00000	0.192450
$28$	14.9324	2.82196
$29$	−6.56942	−1.21991	−0.609955	−	0.792436i	$-0.708813\pi$
$29$	−6.56942	−1.21991	−0.609955	+	0.792436i	$0.708813\pi$
$30$	0	0
$31$	7.13706	1.28185	0.640927	−	0.767602i	$-0.278550\pi$
$31$	7.13706	1.28185	0.640927	+	0.767602i	$0.278550\pi$
$32$	18.2237	3.22153
$33$	1.04833	0.182491
$34$	8.22450	1.41049
$35$	0	0
$36$	5.27940	0.879899
$37$	−5.59570	−0.919927	−0.459964	−	0.887938i	$-0.652138\pi$
$37$	−5.59570	−0.919927	−0.459964	+	0.887938i	$0.652138\pi$
$38$	−9.50003	−1.54111
$39$	−5.59570	−0.896029
$40$	0	0
$41$	11.8600	1.85222	0.926109	−	0.377256i	$-0.123132\pi$
$41$	11.8600	1.85222	0.926109	+	0.377256i	$0.123132\pi$
$42$	7.63120	1.17752
$43$	6.17397	0.941521	0.470761	−	0.882261i	$-0.343980\pi$
$43$	6.17397	0.941521	0.470761	+	0.882261i	$0.343980\pi$
$44$	5.53454	0.834363
$45$	0	0
$46$	−4.89855	−0.722252
$47$	3.71894	0.542463	0.271232	−	0.962514i	$-0.412569\pi$
$47$	3.71894	0.542463	0.271232	+	0.962514i	$0.412569\pi$
$48$	13.3132	1.92160
$49$	1.00000	0.142857
$50$	0	0
$51$	3.04833	0.426851
$52$	−29.5419	−4.09673
$53$	−1.51046	−0.207478	−0.103739	−	0.994605i	$-0.533081\pi$
$53$	−1.51046	−0.207478	−0.103739	+	0.994605i	$0.533081\pi$
$54$	2.69804	0.367156
$55$	0	0
$56$	25.0257	3.34420
$57$	−3.52109	−0.466380
$58$	−17.7245	−2.32734
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−3.60805	−0.461964	−0.230982	−	0.972958i	$-0.574194\pi$
$61$	−3.60805	−0.461964	−0.230982	+	0.972958i	$0.574194\pi$
$62$	19.2561	2.44552
$63$	2.82843	0.356348
$64$	22.5418	2.81772
$65$	0	0
$66$	2.82843	0.348155
$67$	3.80037	0.464290	0.232145	−	0.972681i	$-0.425426\pi$
$67$	3.80037	0.464290	0.232145	+	0.972681i	$0.425426\pi$
$68$	16.0933	1.95160
$69$	−1.81560	−0.218572
$70$	0	0
$71$	−11.8645	−1.40806	−0.704031	−	0.710169i	$-0.748619\pi$
$71$	−11.8645	−1.40806	−0.704031	+	0.710169i	$0.748619\pi$
$72$	8.84793	1.04274
$73$	3.49904	0.409532	0.204766	−	0.978811i	$-0.434357\pi$
$73$	3.49904	0.409532	0.204766	+	0.978811i	$0.434357\pi$
$74$	−15.0974	−1.75504
$75$	0	0
$76$	−18.5892	−2.13233
$77$	2.96512	0.337907
$78$	−15.0974	−1.70944
$79$	−14.4099	−1.62124	−0.810620	−	0.585573i	$-0.800870\pi$
$79$	−14.4099	−1.62124	−0.810620	+	0.585573i	$0.800870\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	31.9987	3.53366
$83$	−12.0897	−1.32702	−0.663510	−	0.748167i	$-0.730934\pi$
$83$	−12.0897	−1.32702	−0.663510	+	0.748167i	$0.730934\pi$
$84$	14.9324	1.62926
$85$	0	0
$86$	16.6576	1.79623
$87$	−6.56942	−0.704316
$88$	9.27553	0.988775
$89$	−11.0032	−1.16634	−0.583168	−	0.812352i	$-0.698187\pi$
$89$	−11.0032	−1.16634	−0.583168	+	0.812352i	$0.698187\pi$
$90$	0	0
$91$	−15.8270	−1.65912
$92$	−9.58527	−0.999333
$93$	7.13706	0.740079
$94$	10.0338	1.03491
$95$	0	0
$96$	18.2237	1.85995
$97$	11.5353	1.17124	0.585618	−	0.810587i	$-0.300852\pi$
$97$	11.5353	1.17124	0.585618	+	0.810587i	$0.300852\pi$
$98$	2.69804	0.272543
$99$	1.04833	0.105361
$100$	0	0
$101$	−14.2318	−1.41612	−0.708059	−	0.706154i	$-0.750429\pi$
$101$	−14.2318	−1.41612	−0.708059	+	0.706154i	$0.750429\pi$
$102$	8.22450	0.814347
$103$	6.43204	0.633768	0.316884	−	0.948464i	$-0.397363\pi$
$103$	6.43204	0.633768	0.316884	+	0.948464i	$0.397363\pi$
$104$	−49.5103	−4.85489
$105$	0	0
$106$	−4.07529	−0.395827
$107$	0.212596	0.0205525	0.0102762	−	0.999947i	$-0.496729\pi$
$107$	0.212596	0.0205525	0.0102762	+	0.999947i	$0.496729\pi$
$108$	5.27940	0.508010
$109$	8.37360	0.802045	0.401022	−	0.916068i	$-0.368655\pi$
$109$	8.37360	0.802045	0.401022	+	0.916068i	$0.368655\pi$
$110$	0	0
$111$	−5.59570	−0.531120
$112$	37.6555	3.55811
$113$	−6.22089	−0.585212	−0.292606	−	0.956233i	$-0.594522\pi$
$113$	−6.22089	−0.585212	−0.292606	+	0.956233i	$0.594522\pi$
$114$	−9.50003	−0.889760
$115$	0	0
$116$	−34.6826	−3.22020
$117$	−5.59570	−0.517322
$118$	−2.69804	−0.248374
$119$	8.62197	0.790375
$120$	0	0
$121$	−9.90101	−0.900092
$122$	−9.73465	−0.881334
$123$	11.8600	1.06938
$124$	37.6794	3.38371
$125$	0	0
$126$	7.63120	0.679841
$127$	8.81780	0.782453	0.391227	−	0.920294i	$-0.372051\pi$
$127$	8.81780	0.782453	0.391227	+	0.920294i	$0.372051\pi$
$128$	24.3711	2.15412
$129$	6.17397	0.543588
$130$	0	0
$131$	−15.9469	−1.39329	−0.696646	−	0.717416i	$-0.745325\pi$
$131$	−15.9469	−1.39329	−0.696646	+	0.717416i	$0.745325\pi$
$132$	5.53454	0.481720
$133$	−9.95915	−0.863568
$134$	10.2535	0.885771
$135$	0	0
$136$	26.9714	2.31278
$137$	−18.8855	−1.61350	−0.806748	−	0.590896i	$-0.798774\pi$
$137$	−18.8855	−1.61350	−0.806748	+	0.590896i	$0.798774\pi$
$138$	−4.89855	−0.416992
$139$	−22.1059	−1.87499	−0.937497	−	0.347994i	$-0.886863\pi$
$139$	−22.1059	−1.87499	−0.937497	+	0.347994i	$0.886863\pi$
$140$	0	0
$141$	3.71894	0.313191
$142$	−32.0110	−2.68630
$143$	−5.86613	−0.490550
$144$	13.3132	1.10944
$145$	0	0
$146$	9.44054	0.781304
$147$	1.00000	0.0824786
$148$	−29.5419	−2.42833
$149$	16.9775	1.39085	0.695427	−	0.718597i	$-0.255215\pi$
$149$	16.9775	1.39085	0.695427	+	0.718597i	$0.255215\pi$
$150$	0	0
$151$	−12.7964	−1.04136	−0.520679	−	0.853752i	$-0.674321\pi$
$151$	−12.7964	−1.04136	−0.520679	+	0.853752i	$0.674321\pi$
$152$	−31.1544	−2.52695
$153$	3.04833	0.246443
$154$	8.00000	0.644658
$155$	0	0
$156$	−29.5419	−2.36525
$157$	23.4435	1.87099	0.935496	−	0.353336i	$-0.114953\pi$
$157$	23.4435	1.87099	0.935496	+	0.353336i	$0.114953\pi$
$158$	−38.8784	−3.09300
$159$	−1.51046	−0.119788
$160$	0	0
$161$	−5.13529	−0.404717
$162$	2.69804	0.211978
$163$	−3.40241	−0.266497	−0.133249	−	0.991083i	$-0.542541\pi$
$163$	−3.40241	−0.266497	−0.133249	+	0.991083i	$0.542541\pi$
$164$	62.6136	4.88930
$165$	0	0
$166$	−32.6185	−2.53169
$167$	21.8327	1.68947	0.844734	−	0.535186i	$-0.179758\pi$
$167$	21.8327	1.68947	0.844734	+	0.535186i	$0.179758\pi$
$168$	25.0257	1.93078
$169$	18.3118	1.40860
$170$	0	0
$171$	−3.52109	−0.269265
$172$	32.5948	2.48533
$173$	18.9224	1.43865	0.719323	−	0.694675i	$-0.244452\pi$
$173$	18.9224	1.43865	0.719323	+	0.694675i	$0.244452\pi$
$174$	−17.7245	−1.34369
$175$	0	0
$176$	13.9566	1.05202
$177$	−1.00000	−0.0751646
$178$	−29.6870	−2.22514
$179$	14.2141	1.06241	0.531205	−	0.847244i	$-0.321740\pi$
$179$	14.2141	1.06241	0.531205	+	0.847244i	$0.321740\pi$
$180$	0	0
$181$	17.8240	1.32485	0.662424	−	0.749129i	$-0.269528\pi$
$181$	17.8240	1.32485	0.662424	+	0.749129i	$0.269528\pi$
$182$	−42.7019	−3.16527
$183$	−3.60805	−0.266715
$184$	−16.0643	−1.18428
$185$	0	0
$186$	19.2561	1.41192
$187$	3.19565	0.233689
$188$	19.6338	1.43194
$189$	2.82843	0.205738
$190$	0	0
$191$	0.438061	0.0316970	0.0158485	−	0.999874i	$-0.494955\pi$
$191$	0.438061	0.0316970	0.0158485	+	0.999874i	$0.494955\pi$
$192$	22.5418	1.62681
$193$	−13.2799	−0.955911	−0.477955	−	0.878384i	$-0.658622\pi$
$193$	−13.2799	−0.955911	−0.477955	+	0.878384i	$0.658622\pi$
$194$	31.1228	2.23448
$195$	0	0
$196$	5.27940	0.377100
$197$	22.0517	1.57112	0.785560	−	0.618786i	$-0.212375\pi$
$197$	22.0517	1.57112	0.785560	+	0.618786i	$0.212375\pi$
$198$	2.82843	0.201008
$199$	1.12060	0.0794372	0.0397186	−	0.999211i	$-0.487354\pi$
$199$	1.12060	0.0794372	0.0397186	+	0.999211i	$0.487354\pi$
$200$	0	0
$201$	3.80037	0.268058
$202$	−38.3979	−2.70167
$203$	−18.5811	−1.30414
$204$	16.0933	1.12676
$205$	0	0
$206$	17.3539	1.20910
$207$	−1.81560	−0.126193
$208$	−74.4968	−5.16543
$209$	−3.69126	−0.255330
$210$	0	0
$211$	−11.5303	−0.793775	−0.396888	−	0.917867i	$-0.629910\pi$
$211$	−11.5303	−0.793775	−0.396888	+	0.917867i	$0.629910\pi$
$212$	−7.97434	−0.547680
$213$	−11.8645	−0.812945
$214$	0.573593	0.0392100
$215$	0	0
$216$	8.84793	0.602025
$217$	20.1867	1.37036
$218$	22.5923	1.53014
$219$	3.49904	0.236443
$220$	0	0
$221$	−17.0575	−1.14741
$222$	−15.0974	−1.01327
$223$	5.38282	0.360460	0.180230	−	0.983624i	$-0.442316\pi$
$223$	5.38282	0.360460	0.180230	+	0.983624i	$0.442316\pi$
$224$	51.5445	3.44396
$225$	0	0
$226$	−16.7842	−1.11647
$227$	−10.2671	−0.681452	−0.340726	−	0.940163i	$-0.610673\pi$
$227$	−10.2671	−0.681452	−0.340726	+	0.940163i	$0.610673\pi$
$228$	−18.5892	−1.23110
$229$	−13.9604	−0.922529	−0.461264	−	0.887263i	$-0.652604\pi$
$229$	−13.9604	−0.922529	−0.461264	+	0.887263i	$0.652604\pi$
$230$	0	0
$231$	2.96512	0.195091
$232$	−58.1258	−3.81614
$233$	15.8838	1.04058	0.520292	−	0.853989i	$-0.325823\pi$
$233$	15.8838	1.04058	0.520292	+	0.853989i	$0.325823\pi$
$234$	−15.0974	−0.986947
$235$	0	0
$236$	−5.27940	−0.343659
$237$	−14.4099	−0.936023
$238$	23.2624	1.50788
$239$	10.9443	0.707925	0.353963	−	0.935260i	$-0.384834\pi$
$239$	10.9443	0.707925	0.353963	+	0.935260i	$0.384834\pi$
$240$	0	0
$241$	8.70769	0.560912	0.280456	−	0.959867i	$-0.409514\pi$
$241$	8.70769	0.560912	0.280456	+	0.959867i	$0.409514\pi$
$242$	−26.7133	−1.71719
$243$	1.00000	0.0641500
$244$	−19.0483	−1.21945
$245$	0	0
$246$	31.9987	2.04016
$247$	19.7030	1.25367
$248$	63.1482	4.00992
$249$	−12.0897	−0.766155
$250$	0	0
$251$	10.3887	0.655730	0.327865	−	0.944725i	$-0.393671\pi$
$251$	10.3887	0.655730	0.327865	+	0.944725i	$0.393671\pi$
$252$	14.9324	0.940652
$253$	−1.90334	−0.119662
$254$	23.7907	1.49276
$255$	0	0
$256$	20.6705	1.29191
$257$	−28.8793	−1.80144	−0.900722	−	0.434397i	$-0.856962\pi$
$257$	−28.8793	−1.80144	−0.900722	+	0.434397i	$0.856962\pi$
$258$	16.6576	1.03706
$259$	−15.8270	−0.983443
$260$	0	0
$261$	−6.56942	−0.406637
$262$	−43.0254	−2.65812
$263$	5.52360	0.340600	0.170300	−	0.985392i	$-0.445526\pi$
$263$	5.52360	0.340600	0.170300	+	0.985392i	$0.445526\pi$
$264$	9.27553	0.570870
$265$	0	0
$266$	−26.8702	−1.64751
$267$	−11.0032	−0.673384
$268$	20.0637	1.22558
$269$	−2.24212	−0.136705	−0.0683523	−	0.997661i	$-0.521774\pi$
$269$	−2.24212	−0.136705	−0.0683523	+	0.997661i	$0.521774\pi$
$270$	0	0
$271$	14.4544	0.878043	0.439021	−	0.898477i	$-0.355325\pi$
$271$	14.4544	0.878043	0.439021	+	0.898477i	$0.355325\pi$
$272$	40.5831	2.46071
$273$	−15.8270	−0.957895
$274$	−50.9537	−3.07822
$275$	0	0
$276$	−9.58527	−0.576965
$277$	−16.3333	−0.981371	−0.490685	−	0.871337i	$-0.663254\pi$
$277$	−16.3333	−0.981371	−0.490685	+	0.871337i	$0.663254\pi$
$278$	−59.6424	−3.57711
$279$	7.13706	0.427285
$280$	0	0
$281$	−3.74621	−0.223480	−0.111740	−	0.993737i	$-0.535642\pi$
$281$	−3.74621	−0.223480	−0.111740	+	0.993737i	$0.535642\pi$
$282$	10.0338	0.597506
$283$	−15.7201	−0.934463	−0.467231	−	0.884135i	$-0.654748\pi$
$283$	−15.7201	−0.934463	−0.467231	+	0.884135i	$0.654748\pi$
$284$	−62.6376	−3.71686
$285$	0	0
$286$	−15.8270	−0.935871
$287$	33.5451	1.98010
$288$	18.2237	1.07384
$289$	−7.70769	−0.453394
$290$	0	0
$291$	11.5353	0.676213
$292$	18.4728	1.08104
$293$	3.32330	0.194149	0.0970745	−	0.995277i	$-0.469051\pi$
$293$	3.32330	0.194149	0.0970745	+	0.995277i	$0.469051\pi$
$294$	2.69804	0.157353
$295$	0	0
$296$	−49.5103	−2.87773
$297$	1.04833	0.0608302
$298$	45.8060	2.65347
$299$	10.1595	0.587541
$300$	0	0
$301$	17.4626	1.00653
$302$	−34.5252	−1.98670
$303$	−14.2318	−0.817596
$304$	−46.8771	−2.68859
$305$	0	0
$306$	8.22450	0.470163
$307$	3.30878	0.188842	0.0944210	−	0.995532i	$-0.469900\pi$
$307$	3.30878	0.188842	0.0944210	+	0.995532i	$0.469900\pi$
$308$	15.6540	0.891972
$309$	6.43204	0.365906
$310$	0	0
$311$	31.0001	1.75785	0.878926	−	0.476958i	$-0.158261\pi$
$311$	31.0001	1.75785	0.878926	+	0.476958i	$0.158261\pi$
$312$	−49.5103	−2.80297
$313$	−25.0430	−1.41551	−0.707756	−	0.706456i	$-0.750293\pi$
$313$	−25.0430	−1.41551	−0.707756	+	0.706456i	$0.750293\pi$
$314$	63.2513	3.56948
$315$	0	0
$316$	−76.0755	−4.27958
$317$	−6.70424	−0.376547	−0.188274	−	0.982117i	$-0.560289\pi$
$317$	−6.70424	−0.376547	−0.188274	+	0.982117i	$0.560289\pi$
$318$	−4.07529	−0.228531
$319$	−6.88691	−0.385593
$320$	0	0
$321$	0.212596	0.0118660
$322$	−13.8552	−0.772120
$323$	−10.7334	−0.597225
$324$	5.27940	0.293300
$325$	0	0
$326$	−9.17982	−0.508423
$327$	8.37360	0.463061
$328$	104.936	5.79414
$329$	10.5188	0.579918
$330$	0	0
$331$	−31.8522	−1.75075	−0.875377	−	0.483441i	$-0.839387\pi$
$331$	−31.8522	−1.75075	−0.875377	+	0.483441i	$0.839387\pi$
$332$	−63.8265	−3.50293
$333$	−5.59570	−0.306642
$334$	58.9055	3.22317
$335$	0	0
$336$	37.6555	2.05428
$337$	10.5641	0.575462	0.287731	−	0.957711i	$-0.407099\pi$
$337$	10.5641	0.575462	0.287731	+	0.957711i	$0.407099\pi$
$338$	49.4060	2.68733
$339$	−6.22089	−0.337873
$340$	0	0
$341$	7.48199	0.405172
$342$	−9.50003	−0.513703
$343$	−16.9706	−0.916324
$344$	54.6268	2.94528
$345$	0	0
$346$	51.0534	2.74465
$347$	−3.61696	−0.194169	−0.0970844	−	0.995276i	$-0.530952\pi$
$347$	−3.61696	−0.194169	−0.0970844	+	0.995276i	$0.530952\pi$
$348$	−34.6826	−1.85918
$349$	22.6807	1.21407	0.607035	−	0.794675i	$-0.292359\pi$
$349$	22.6807	1.21407	0.607035	+	0.794675i	$0.292359\pi$
$350$	0	0
$351$	−5.59570	−0.298676
$352$	19.1044	1.01827
$353$	17.7405	0.944234	0.472117	−	0.881536i	$-0.343490\pi$
$353$	17.7405	0.944234	0.472117	+	0.881536i	$0.343490\pi$
$354$	−2.69804	−0.143399
$355$	0	0
$356$	−58.0902	−3.07878
$357$	8.62197	0.456323
$358$	38.3501	2.02686
$359$	−2.20062	−0.116144	−0.0580721	−	0.998312i	$-0.518495\pi$
$359$	−2.20062	−0.116144	−0.0580721	+	0.998312i	$0.518495\pi$
$360$	0	0
$361$	−6.60191	−0.347469
$362$	48.0898	2.52754
$363$	−9.90101	−0.519668
$364$	−83.5571	−4.37958
$365$	0	0
$366$	−9.73465	−0.508839
$367$	−29.0954	−1.51877	−0.759385	−	0.650642i	$-0.774500\pi$
$367$	−29.0954	−1.51877	−0.759385	+	0.650642i	$0.774500\pi$
$368$	−24.1715	−1.26003
$369$	11.8600	0.617406
$370$	0	0
$371$	−4.27224	−0.221804
$372$	37.6794	1.95359
$373$	0.162350	0.00840616	0.00420308	−	0.999991i	$-0.498662\pi$
$373$	0.162350	0.00840616	0.00420308	+	0.999991i	$0.498662\pi$
$374$	8.62197	0.445832
$375$	0	0
$376$	32.9049	1.69694
$377$	36.7605	1.89326
$378$	7.63120	0.392506
$379$	−10.0122	−0.514293	−0.257146	−	0.966372i	$-0.582782\pi$
$379$	−10.0122	−0.514293	−0.257146	+	0.966372i	$0.582782\pi$
$380$	0	0
$381$	8.81780	0.451750
$382$	1.18190	0.0604714
$383$	33.8276	1.72851	0.864254	−	0.503056i	$-0.167791\pi$
$383$	33.8276	1.72851	0.864254	+	0.503056i	$0.167791\pi$
$384$	24.3711	1.24368
$385$	0	0
$386$	−35.8297	−1.82369
$387$	6.17397	0.313840
$388$	60.8996	3.09171
$389$	15.4582	0.783763	0.391881	−	0.920016i	$-0.371824\pi$
$389$	15.4582	0.783763	0.391881	+	0.920016i	$0.371824\pi$
$390$	0	0
$391$	−5.53454	−0.279894
$392$	8.84793	0.446888
$393$	−15.9469	−0.804417
$394$	59.4963	2.99738
$395$	0	0
$396$	5.53454	0.278121
$397$	−11.7547	−0.589953	−0.294976	−	0.955505i	$-0.595312\pi$
$397$	−11.7547	−0.589953	−0.294976	+	0.955505i	$0.595312\pi$
$398$	3.02342	0.151550
$399$	−9.95915	−0.498581
$400$	0	0
$401$	−3.54580	−0.177069	−0.0885343	−	0.996073i	$-0.528218\pi$
$401$	−3.54580	−0.177069	−0.0885343	+	0.996073i	$0.528218\pi$
$402$	10.2535	0.511400
$403$	−39.9369	−1.98940
$404$	−75.1353	−3.73812
$405$	0	0
$406$	−50.1325	−2.48804
$407$	−5.86613	−0.290773
$408$	26.9714	1.33528
$409$	−22.5175	−1.11342	−0.556709	−	0.830708i	$-0.687936\pi$
$409$	−22.5175	−1.11342	−0.556709	+	0.830708i	$0.687936\pi$
$410$	0	0
$411$	−18.8855	−0.931552
$412$	33.9573	1.67296
$413$	−2.82843	−0.139178
$414$	−4.89855	−0.240751
$415$	0	0
$416$	−101.974	−4.99971
$417$	−22.1059	−1.08253
$418$	−9.95915	−0.487118
$419$	−26.3187	−1.28575	−0.642876	−	0.765970i	$-0.722259\pi$
$419$	−26.3187	−1.28575	−0.642876	+	0.765970i	$0.722259\pi$
$420$	0	0
$421$	6.75543	0.329239	0.164620	−	0.986357i	$-0.447360\pi$
$421$	6.75543	0.329239	0.164620	+	0.986357i	$0.447360\pi$
$422$	−31.1090	−1.51436
$423$	3.71894	0.180821
$424$	−13.3645	−0.649037
$425$	0	0
$426$	−32.0110	−1.55094
$427$	−10.2051	−0.493860
$428$	1.12238	0.0542523
$429$	−5.86613	−0.283219
$430$	0	0
$431$	−29.5829	−1.42496	−0.712480	−	0.701693i	$-0.752428\pi$
$431$	−29.5829	−1.42496	−0.712480	+	0.701693i	$0.752428\pi$
$432$	13.3132	0.640533
$433$	31.1203	1.49555	0.747773	−	0.663954i	$-0.231123\pi$
$433$	31.1203	1.49555	0.747773	+	0.663954i	$0.231123\pi$
$434$	54.4644	2.61437
$435$	0	0
$436$	44.2075	2.11716
$437$	6.39289	0.305813
$438$	9.44054	0.451086
$439$	28.6840	1.36901	0.684507	−	0.729006i	$-0.260018\pi$
$439$	28.6840	1.36901	0.684507	+	0.729006i	$0.260018\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−46.0218	−2.18903
$443$	41.3052	1.96247	0.981235	−	0.192816i	$-0.0617621\pi$
$443$	41.3052	1.96247	0.981235	+	0.192816i	$0.0617621\pi$
$444$	−29.5419	−1.40200
$445$	0	0
$446$	14.5230	0.687686
$447$	16.9775	0.803010
$448$	63.7578	3.01227
$449$	−16.4734	−0.777427	−0.388713	−	0.921359i	$-0.627080\pi$
$449$	−16.4734	−0.777427	−0.388713	+	0.921359i	$0.627080\pi$
$450$	0	0
$451$	12.4332	0.585454
$452$	−32.8426	−1.54478
$453$	−12.7964	−0.601229
$454$	−27.7010	−1.30007
$455$	0	0
$456$	−31.1544	−1.45894
$457$	22.0200	1.03005	0.515026	−	0.857175i	$-0.327782\pi$
$457$	22.0200	1.03005	0.515026	+	0.857175i	$0.327782\pi$
$458$	−37.6656	−1.76000
$459$	3.04833	0.142284
$460$	0	0
$461$	23.3176	1.08601	0.543003	−	0.839731i	$-0.317287\pi$
$461$	23.3176	1.08601	0.543003	+	0.839731i	$0.317287\pi$
$462$	8.00000	0.372194
$463$	0.155728	0.00723730	0.00361865	−	0.999993i	$-0.498848\pi$
$463$	0.155728	0.00723730	0.00361865	+	0.999993i	$0.498848\pi$
$464$	−87.4602	−4.06024
$465$	0	0
$466$	42.8551	1.98522
$467$	35.4059	1.63839	0.819196	−	0.573514i	$-0.194420\pi$
$467$	35.4059	1.63839	0.819196	+	0.573514i	$0.194420\pi$
$468$	−29.5419	−1.36558
$469$	10.7491	0.496346
$470$	0	0
$471$	23.4435	1.08022
$472$	−8.84793	−0.407259
$473$	6.47235	0.297599
$474$	−38.8784	−1.78574
$475$	0	0
$476$	45.5188	2.08635
$477$	−1.51046	−0.0691594
$478$	29.5280	1.35058
$479$	33.6776	1.53877	0.769385	−	0.638786i	$-0.220563\pi$
$479$	33.6776	1.53877	0.769385	+	0.638786i	$0.220563\pi$
$480$	0	0
$481$	31.3118	1.42770
$482$	23.4937	1.07011
$483$	−5.13529	−0.233664
$484$	−52.2713	−2.37597
$485$	0	0
$486$	2.69804	0.122385
$487$	−15.2727	−0.692073	−0.346037	−	0.938221i	$-0.612473\pi$
$487$	−15.2727	−0.692073	−0.346037	+	0.938221i	$0.612473\pi$
$488$	−31.9238	−1.44512
$489$	−3.40241	−0.153862
$490$	0	0
$491$	−28.0564	−1.26617	−0.633084	−	0.774083i	$-0.718211\pi$
$491$	−28.0564	−1.26617	−0.633084	+	0.774083i	$0.718211\pi$
$492$	62.6136	2.82284
$493$	−20.0258	−0.901914
$494$	53.1593	2.39175
$495$	0	0
$496$	95.0174	4.26641
$497$	−33.5580	−1.50528
$498$	−32.6185	−1.46167
$499$	−20.8976	−0.935507	−0.467753	−	0.883859i	$-0.654936\pi$
$499$	−20.8976	−0.935507	−0.467753	+	0.883859i	$0.654936\pi$
$500$	0	0
$501$	21.8327	0.975415
$502$	28.0291	1.25100
$503$	−6.77406	−0.302041	−0.151020	−	0.988531i	$-0.548256\pi$
$503$	−6.77406	−0.302041	−0.151020	+	0.988531i	$0.548256\pi$
$504$	25.0257	1.11473
$505$	0	0
$506$	−5.13529	−0.228291
$507$	18.3118	0.813257
$508$	46.5527	2.06544
$509$	1.67104	0.0740675	0.0370337	−	0.999314i	$-0.488209\pi$
$509$	1.67104	0.0740675	0.0370337	+	0.999314i	$0.488209\pi$
$510$	0	0
$511$	9.89678	0.437808
$512$	7.02762	0.310580
$513$	−3.52109	−0.155460
$514$	−77.9175	−3.43679
$515$	0	0
$516$	32.5948	1.43491
$517$	3.89867	0.171463
$518$	−42.7019	−1.87621
$519$	18.9224	0.830603
$520$	0	0
$521$	11.3162	0.495773	0.247886	−	0.968789i	$-0.420264\pi$
$521$	11.3162	0.495773	0.247886	+	0.968789i	$0.420264\pi$
$522$	−17.7245	−0.775782
$523$	−23.5848	−1.03129	−0.515646	−	0.856802i	$-0.672448\pi$
$523$	−23.5848	−1.03129	−0.515646	+	0.856802i	$0.672448\pi$
$524$	−84.1902	−3.67787
$525$	0	0
$526$	14.9029	0.649797
$527$	21.7561	0.947711
$528$	13.9566	0.607385
$529$	−19.7036	−0.856678
$530$	0	0
$531$	−1.00000	−0.0433963
$532$	−52.5783	−2.27956
$533$	−66.3649	−2.87458
$534$	−29.6870	−1.28468
$535$	0	0
$536$	33.6254	1.45240
$537$	14.2141	0.613382
$538$	−6.04932	−0.260805
$539$	1.04833	0.0451547
$540$	0	0
$541$	−19.2951	−0.829559	−0.414780	−	0.909922i	$-0.636141\pi$
$541$	−19.2951	−0.829559	−0.414780	+	0.909922i	$0.636141\pi$
$542$	38.9985	1.67513
$543$	17.8240	0.764901
$544$	55.5519	2.38177
$545$	0	0
$546$	−42.7019	−1.82747
$547$	24.5303	1.04884	0.524421	−	0.851459i	$-0.324282\pi$
$547$	24.5303	1.04884	0.524421	+	0.851459i	$0.324282\pi$
$548$	−99.7039	−4.25914
$549$	−3.60805	−0.153988
$550$	0	0
$551$	23.1315	0.985437
$552$	−16.0643	−0.683742
$553$	−40.7573	−1.73318
$554$	−44.0677	−1.87226
$555$	0	0
$556$	−116.706	−4.94942
$557$	−8.01315	−0.339528	−0.169764	−	0.985485i	$-0.554301\pi$
$557$	−8.01315	−0.339528	−0.169764	+	0.985485i	$0.554301\pi$
$558$	19.2561	0.815174
$559$	−34.5477	−1.46121
$560$	0	0
$561$	3.19565	0.134920
$562$	−10.1074	−0.426355
$563$	20.2329	0.852715	0.426358	−	0.904555i	$-0.359797\pi$
$563$	20.2329	0.852715	0.426358	+	0.904555i	$0.359797\pi$
$564$	19.6338	0.826731
$565$	0	0
$566$	−42.4134	−1.78277
$567$	2.82843	0.118783
$568$	−104.977	−4.40472
$569$	20.4863	0.858830	0.429415	−	0.903107i	$-0.358720\pi$
$569$	20.4863	0.858830	0.429415	+	0.903107i	$0.358720\pi$
$570$	0	0
$571$	−2.38326	−0.0997362	−0.0498681	−	0.998756i	$-0.515880\pi$
$571$	−2.38326	−0.0997362	−0.0498681	+	0.998756i	$0.515880\pi$
$572$	−30.9696	−1.29490
$573$	0.438061	0.0183002
$574$	90.5059	3.77764
$575$	0	0
$576$	22.5418	0.939241
$577$	−31.5226	−1.31230	−0.656152	−	0.754628i	$-0.727817\pi$
$577$	−31.5226	−1.31230	−0.656152	+	0.754628i	$0.727817\pi$
$578$	−20.7956	−0.864984
$579$	−13.2799	−0.551895
$580$	0	0
$581$	−34.1949	−1.41864
$582$	31.1228	1.29008
$583$	−1.58346	−0.0655803
$584$	30.9593	1.28110
$585$	0	0
$586$	8.96637	0.370397
$587$	−9.80702	−0.404779	−0.202390	−	0.979305i	$-0.564871\pi$
$587$	−9.80702	−0.404779	−0.202390	+	0.979305i	$0.564871\pi$
$588$	5.27940	0.217719
$589$	−25.1303	−1.03547
$590$	0	0
$591$	22.0517	0.907086
$592$	−74.4968	−3.06180
$593$	15.9282	0.654091	0.327046	−	0.945009i	$-0.393947\pi$
$593$	15.9282	0.654091	0.327046	+	0.945009i	$0.393947\pi$
$594$	2.82843	0.116052
$595$	0	0
$596$	89.6311	3.67143
$597$	1.12060	0.0458631
$598$	27.4108	1.12091
$599$	14.9840	0.612230	0.306115	−	0.951995i	$-0.400971\pi$
$599$	14.9840	0.612230	0.306115	+	0.951995i	$0.400971\pi$
$600$	0	0
$601$	22.8067	0.930303	0.465152	−	0.885231i	$-0.346000\pi$
$601$	22.8067	0.930303	0.465152	+	0.885231i	$0.346000\pi$
$602$	47.1148	1.92025
$603$	3.80037	0.154763
$604$	−67.5574	−2.74887
$605$	0	0
$606$	−38.3979	−1.55981
$607$	27.7092	1.12468	0.562341	−	0.826905i	$-0.309901\pi$
$607$	27.7092	1.12468	0.562341	+	0.826905i	$0.309901\pi$
$608$	−64.1674	−2.60233
$609$	−18.5811	−0.752945
$610$	0	0
$611$	−20.8101	−0.841886
$612$	16.0933	0.650534
$613$	−43.6992	−1.76499	−0.882496	−	0.470320i	$-0.844139\pi$
$613$	−43.6992	−1.76499	−0.882496	+	0.470320i	$0.844139\pi$
$614$	8.92721	0.360273
$615$	0	0
$616$	26.2352	1.05705
$617$	−16.9500	−0.682383	−0.341191	−	0.939994i	$-0.610830\pi$
$617$	−16.9500	−0.682383	−0.341191	+	0.939994i	$0.610830\pi$
$618$	17.3539	0.698076
$619$	−39.1125	−1.57206	−0.786032	−	0.618186i	$-0.787868\pi$
$619$	−39.1125	−1.57206	−0.786032	+	0.618186i	$0.787868\pi$
$620$	0	0
$621$	−1.81560	−0.0728575
$622$	83.6393	3.35363
$623$	−31.1217	−1.24687
$624$	−74.4968	−2.98226
$625$	0	0
$626$	−67.5669	−2.70051
$627$	−3.69126	−0.147415
$628$	123.767	4.93886
$629$	−17.0575	−0.680128
$630$	0	0
$631$	−14.7534	−0.587325	−0.293662	−	0.955909i	$-0.594874\pi$
$631$	−14.7534	−0.587325	−0.293662	+	0.955909i	$0.594874\pi$
$632$	−127.498	−5.07159
$633$	−11.5303	−0.458286
$634$	−18.0883	−0.718377
$635$	0	0
$636$	−7.97434	−0.316203
$637$	−5.59570	−0.221710
$638$	−18.5811	−0.735634
$639$	−11.8645	−0.469354
$640$	0	0
$641$	−0.591325	−0.0233560	−0.0116780	−	0.999932i	$-0.503717\pi$
$641$	−0.591325	−0.0233560	−0.0116780	+	0.999932i	$0.503717\pi$
$642$	0.573593	0.0226379
$643$	−27.3003	−1.07662	−0.538309	−	0.842748i	$-0.680937\pi$
$643$	−27.3003	−1.07662	−0.538309	+	0.842748i	$0.680937\pi$
$644$	−27.1112	−1.06833
$645$	0	0
$646$	−28.9592	−1.13939
$647$	−33.4517	−1.31512	−0.657560	−	0.753402i	$-0.728411\pi$
$647$	−33.4517	−1.31512	−0.657560	+	0.753402i	$0.728411\pi$
$648$	8.84793	0.347579
$649$	−1.04833	−0.0411505
$650$	0	0
$651$	20.1867	0.791178
$652$	−17.9627	−0.703473
$653$	8.77472	0.343381	0.171691	−	0.985151i	$-0.445077\pi$
$653$	8.77472	0.343381	0.171691	+	0.985151i	$0.445077\pi$
$654$	22.5923	0.883427
$655$	0	0
$656$	157.895	6.16475
$657$	3.49904	0.136511
$658$	28.3800	1.10637
$659$	−32.3782	−1.26127	−0.630637	−	0.776078i	$-0.717206\pi$
$659$	−32.3782	−1.26127	−0.630637	+	0.776078i	$0.717206\pi$
$660$	0	0
$661$	−8.12227	−0.315920	−0.157960	−	0.987446i	$-0.550492\pi$
$661$	−8.12227	−0.315920	−0.157960	+	0.987446i	$0.550492\pi$
$662$	−85.9383	−3.34009
$663$	−17.0575	−0.662459
$664$	−106.969	−4.15120
$665$	0	0
$666$	−15.0974	−0.585012
$667$	11.9274	0.461832
$668$	115.264	4.45969
$669$	5.38282	0.208112
$670$	0	0
$671$	−3.78242	−0.146019
$672$	51.5445	1.98837
$673$	35.9752	1.38674	0.693371	−	0.720581i	$-0.256125\pi$
$673$	35.9752	1.38674	0.693371	+	0.720581i	$0.256125\pi$
$674$	28.5022	1.09787
$675$	0	0
$676$	96.6754	3.71829
$677$	44.5437	1.71195	0.855977	−	0.517014i	$-0.172957\pi$
$677$	44.5437	1.71195	0.855977	+	0.517014i	$0.172957\pi$
$678$	−16.7842	−0.644593
$679$	32.6269	1.25210
$680$	0	0
$681$	−10.2671	−0.393436
$682$	20.1867	0.772987
$683$	20.9335	0.800997	0.400499	−	0.916297i	$-0.368837\pi$
$683$	20.9335	0.800997	0.400499	+	0.916297i	$0.368837\pi$
$684$	−18.5892	−0.710777
$685$	0	0
$686$	−45.7872	−1.74816
$687$	−13.9604	−0.532622
$688$	82.1955	3.13367
$689$	8.45210	0.322000
$690$	0	0
$691$	1.75327	0.0666976	0.0333488	−	0.999444i	$-0.489383\pi$
$691$	1.75327	0.0666976	0.0333488	+	0.999444i	$0.489383\pi$
$692$	99.8991	3.79759
$693$	2.96512	0.112636
$694$	−9.75870	−0.370435
$695$	0	0
$696$	−58.1258	−2.20325
$697$	36.1531	1.36940
$698$	61.1933	2.31620
$699$	15.8838	0.600781
$700$	0	0
$701$	−5.28961	−0.199786	−0.0998929	−	0.994998i	$-0.531850\pi$
$701$	−5.28961	−0.199786	−0.0998929	+	0.994998i	$0.531850\pi$
$702$	−15.0974	−0.569814
$703$	19.7030	0.743111
$704$	23.6312	0.890634
$705$	0	0
$706$	47.8646	1.80141
$707$	−40.2536	−1.51389
$708$	−5.27940	−0.198412
$709$	15.4477	0.580149	0.290075	−	0.957004i	$-0.406320\pi$
$709$	15.4477	0.580149	0.290075	+	0.957004i	$0.406320\pi$
$710$	0	0
$711$	−14.4099	−0.540413
$712$	−97.3555	−3.64855
$713$	−12.9580	−0.485283
$714$	23.2624	0.870573
$715$	0	0
$716$	75.0417	2.80444
$717$	10.9443	0.408721
$718$	−5.93735	−0.221580
$719$	15.0810	0.562426	0.281213	−	0.959645i	$-0.409263\pi$
$719$	15.0810	0.562426	0.281213	+	0.959645i	$0.409263\pi$
$720$	0	0
$721$	18.1926	0.677527
$722$	−17.8122	−0.662901
$723$	8.70769	0.323843
$724$	94.0999	3.49720
$725$	0	0
$726$	−26.7133	−0.991423
$727$	26.6948	0.990054	0.495027	−	0.868878i	$-0.335158\pi$
$727$	26.6948	0.990054	0.495027	+	0.868878i	$0.335158\pi$
$728$	−140.036	−5.19009
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.8203	0.696093
$732$	−19.0483	−0.704047
$733$	−32.8282	−1.21254	−0.606269	−	0.795259i	$-0.707335\pi$
$733$	−32.8282	−1.21254	−0.606269	+	0.795259i	$0.707335\pi$
$734$	−78.5005	−2.89751
$735$	0	0
$736$	−33.0870	−1.21960
$737$	3.98404	0.146754
$738$	31.9987	1.17789
$739$	−39.6606	−1.45894	−0.729470	−	0.684013i	$-0.760233\pi$
$739$	−39.6606	−1.45894	−0.729470	+	0.684013i	$0.760233\pi$
$740$	0	0
$741$	19.7030	0.723807
$742$	−11.5267	−0.423157
$743$	−16.6562	−0.611056	−0.305528	−	0.952183i	$-0.598833\pi$
$743$	−16.6562	−0.611056	−0.305528	+	0.952183i	$0.598833\pi$
$744$	63.1482	2.31513
$745$	0	0
$746$	0.438026	0.0160373
$747$	−12.0897	−0.442340
$748$	16.8711	0.616868
$749$	0.601313	0.0219715
$750$	0	0
$751$	40.4394	1.47565	0.737827	−	0.674990i	$-0.235852\pi$
$751$	40.4394	1.47565	0.737827	+	0.674990i	$0.235852\pi$
$752$	49.5112	1.80549
$753$	10.3887	0.378586
$754$	99.1811	3.61196
$755$	0	0
$756$	14.9324	0.543086
$757$	30.6063	1.11240	0.556202	−	0.831047i	$-0.312258\pi$
$757$	30.6063	1.11240	0.556202	+	0.831047i	$0.312258\pi$
$758$	−27.0133	−0.981167
$759$	−1.90334	−0.0690870
$760$	0	0
$761$	21.4975	0.779285	0.389642	−	0.920966i	$-0.372599\pi$
$761$	21.4975	0.779285	0.389642	+	0.920966i	$0.372599\pi$
$762$	23.7907	0.861848
$763$	23.6841	0.857422
$764$	2.31270	0.0836704
$765$	0	0
$766$	91.2680	3.29765
$767$	5.59570	0.202049
$768$	20.6705	0.745884
$769$	34.6059	1.24792	0.623960	−	0.781457i	$-0.285523\pi$
$769$	34.6059	1.24792	0.623960	+	0.781457i	$0.285523\pi$
$770$	0	0
$771$	−28.8793	−1.04006
$772$	−70.1101	−2.52332
$773$	2.50793	0.0902041	0.0451020	−	0.998982i	$-0.485639\pi$
$773$	2.50793	0.0902041	0.0451020	+	0.998982i	$0.485639\pi$
$774$	16.6576	0.598745
$775$	0	0
$776$	102.064	3.66388
$777$	−15.8270	−0.567791
$778$	41.7068	1.49526
$779$	−41.7601	−1.49621
$780$	0	0
$781$	−12.4379	−0.445064
$782$	−14.9324	−0.533981
$783$	−6.56942	−0.234772
$784$	13.3132	0.475473
$785$	0	0
$786$	−43.0254	−1.53467
$787$	−40.1842	−1.43241	−0.716205	−	0.697890i	$-0.754123\pi$
$787$	−40.1842	−1.43241	−0.716205	+	0.697890i	$0.754123\pi$
$788$	116.420	4.14728
$789$	5.52360	0.196646
$790$	0	0
$791$	−17.5953	−0.625618
$792$	9.27553	0.329592
$793$	20.1896	0.716953
$794$	−31.7147	−1.12551
$795$	0	0
$796$	5.91609	0.209690
$797$	30.4544	1.07875	0.539375	−	0.842066i	$-0.318660\pi$
$797$	30.4544	1.07875	0.539375	+	0.842066i	$0.318660\pi$
$798$	−26.8702	−0.951193
$799$	11.3366	0.401059
$800$	0	0
$801$	−11.0032	−0.388779
$802$	−9.56668	−0.337811
$803$	3.66814	0.129446
$804$	20.0637	0.707591
$805$	0	0
$806$	−107.751	−3.79537
$807$	−2.24212	−0.0789264
$808$	−125.922	−4.42992
$809$	20.2061	0.710410	0.355205	−	0.934788i	$-0.384411\pi$
$809$	20.2061	0.710410	0.355205	+	0.934788i	$0.384411\pi$
$810$	0	0
$811$	−19.2204	−0.674920	−0.337460	−	0.941340i	$-0.609568\pi$
$811$	−19.2204	−0.674920	−0.337460	+	0.941340i	$0.609568\pi$
$812$	−98.0971	−3.44253
$813$	14.4544	0.506938
$814$	−15.8270	−0.554737
$815$	0	0
$816$	40.5831	1.42069
$817$	−21.7391	−0.760555
$818$	−60.7530	−2.12418
$819$	−15.8270	−0.553041
$820$	0	0
$821$	25.0858	0.875499	0.437749	−	0.899097i	$-0.355776\pi$
$821$	25.0858	0.875499	0.437749	+	0.899097i	$0.355776\pi$
$822$	−50.9537	−1.77721
$823$	8.82767	0.307713	0.153857	−	0.988093i	$-0.450831\pi$
$823$	8.82767	0.307713	0.153857	+	0.988093i	$0.450831\pi$
$824$	56.9103	1.98256
$825$	0	0
$826$	−7.63120	−0.265523
$827$	43.4042	1.50931	0.754657	−	0.656120i	$-0.227804\pi$
$827$	43.4042	1.50931	0.754657	+	0.656120i	$0.227804\pi$
$828$	−9.58527	−0.333111
$829$	27.4263	0.952553	0.476277	−	0.879295i	$-0.341986\pi$
$829$	27.4263	0.952553	0.476277	+	0.879295i	$0.341986\pi$
$830$	0	0
$831$	−16.3333	−0.566595
$832$	−126.137	−4.37302
$833$	3.04833	0.105618
$834$	−59.6424	−2.06525
$835$	0	0
$836$	−19.4876	−0.673994
$837$	7.13706	0.246693
$838$	−71.0087	−2.45296
$839$	10.9694	0.378705	0.189352	−	0.981909i	$-0.439361\pi$
$839$	10.9694	0.378705	0.189352	+	0.981909i	$0.439361\pi$
$840$	0	0
$841$	14.1573	0.488182
$842$	18.2264	0.628123
$843$	−3.74621	−0.129026
$844$	−60.8728	−2.09533
$845$	0	0
$846$	10.0338	0.344970
$847$	−28.0043	−0.962238
$848$	−20.1092	−0.690552
$849$	−15.7201	−0.539512
$850$	0	0
$851$	10.1595	0.348265
$852$	−62.6376	−2.14593
$853$	35.0392	1.19972	0.599859	−	0.800105i	$-0.295223\pi$
$853$	35.0392	1.19972	0.599859	+	0.800105i	$0.295223\pi$
$854$	−27.5338	−0.942186
$855$	0	0
$856$	1.88104	0.0642925
$857$	15.3315	0.523713	0.261857	−	0.965107i	$-0.415665\pi$
$857$	15.3315	0.523713	0.261857	+	0.965107i	$0.415665\pi$
$858$	−15.8270	−0.540326
$859$	40.8101	1.39242	0.696212	−	0.717837i	$-0.254867\pi$
$859$	40.8101	1.39242	0.696212	+	0.717837i	$0.254867\pi$
$860$	0	0
$861$	33.5451	1.14321
$862$	−79.8158	−2.71854
$863$	−14.3565	−0.488700	−0.244350	−	0.969687i	$-0.578575\pi$
$863$	−14.3565	−0.488700	−0.244350	+	0.969687i	$0.578575\pi$
$864$	18.2237	0.619984
$865$	0	0
$866$	83.9637	2.85320
$867$	−7.70769	−0.261767
$868$	106.573	3.61734
$869$	−15.1063	−0.512446
$870$	0	0
$871$	−21.2657	−0.720562
$872$	74.0890	2.50897
$873$	11.5353	0.390412
$874$	17.2482	0.583431
$875$	0	0
$876$	18.4728	0.624139
$877$	−3.54782	−0.119801	−0.0599006	−	0.998204i	$-0.519078\pi$
$877$	−3.54782	−0.119801	−0.0599006	+	0.998204i	$0.519078\pi$
$878$	77.3905	2.61180
$879$	3.32330	0.112092
$880$	0	0
$881$	28.7404	0.968288	0.484144	−	0.874988i	$-0.339131\pi$
$881$	28.7404	0.968288	0.484144	+	0.874988i	$0.339131\pi$
$882$	2.69804	0.0908476
$883$	6.33868	0.213313	0.106657	−	0.994296i	$-0.465985\pi$
$883$	6.33868	0.213313	0.106657	+	0.994296i	$0.465985\pi$
$884$	−90.0534	−3.02882
$885$	0	0
$886$	111.443	3.74400
$887$	−36.2827	−1.21825	−0.609126	−	0.793073i	$-0.708480\pi$
$887$	−36.2827	−1.21825	−0.609126	+	0.793073i	$0.708480\pi$
$888$	−49.5103	−1.66146
$889$	24.9405	0.836478
$890$	0	0
$891$	1.04833	0.0351203
$892$	28.4180	0.951506
$893$	−13.0947	−0.438199
$894$	45.8060	1.53198
$895$	0	0
$896$	68.9319	2.30285
$897$	10.1595	0.339217
$898$	−44.4458	−1.48317
$899$	−46.8864	−1.56375
$900$	0	0
$901$	−4.60439	−0.153395
$902$	33.5451	1.11693
$903$	17.4626	0.581120
$904$	−55.0420	−1.83067
$905$	0	0
$906$	−34.5252	−1.14702
$907$	18.7566	0.622802	0.311401	−	0.950279i	$-0.399202\pi$
$907$	18.7566	0.622802	0.311401	+	0.950279i	$0.399202\pi$
$908$	−54.2041	−1.79883
$909$	−14.2318	−0.472039
$910$	0	0
$911$	7.04329	0.233355	0.116677	−	0.993170i	$-0.462776\pi$
$911$	7.04329	0.233355	0.116677	+	0.993170i	$0.462776\pi$
$912$	−46.8771	−1.55226
$913$	−12.6740	−0.419448
$914$	59.4107	1.96513
$915$	0	0
$916$	−73.7025	−2.43520
$917$	−45.1048	−1.48949
$918$	8.22450	0.271449
$919$	23.5646	0.777324	0.388662	−	0.921380i	$-0.372937\pi$
$919$	23.5646	0.777324	0.388662	+	0.921380i	$0.372937\pi$
$920$	0	0
$921$	3.30878	0.109028
$922$	62.9116	2.07188
$923$	66.3904	2.18527
$924$	15.6540	0.514980
$925$	0	0
$926$	0.420160	0.0138073
$927$	6.43204	0.211256
$928$	−119.719	−3.92998
$929$	−1.08356	−0.0355506	−0.0177753	−	0.999842i	$-0.505658\pi$
$929$	−1.08356	−0.0355506	−0.0177753	+	0.999842i	$0.505658\pi$
$930$	0	0
$931$	−3.52109	−0.115399
$932$	83.8569	2.74683
$933$	31.0001	1.01490
$934$	95.5265	3.12572
$935$	0	0
$936$	−49.5103	−1.61830
$937$	−37.0120	−1.20913	−0.604565	−	0.796556i	$-0.706653\pi$
$937$	−37.0120	−1.20913	−0.604565	+	0.796556i	$0.706653\pi$
$938$	29.0014	0.946929
$939$	−25.0430	−0.817247
$940$	0	0
$941$	−21.9744	−0.716344	−0.358172	−	0.933656i	$-0.616600\pi$
$941$	−21.9744	−0.716344	−0.358172	+	0.933656i	$0.616600\pi$
$942$	63.2513	2.06084
$943$	−21.5330	−0.701210
$944$	−13.3132	−0.433309
$945$	0	0
$946$	17.4626	0.567759
$947$	5.99249	0.194730	0.0973649	−	0.995249i	$-0.468959\pi$
$947$	5.99249	0.194730	0.0973649	+	0.995249i	$0.468959\pi$
$948$	−76.0755	−2.47082
$949$	−19.5796	−0.635580
$950$	0	0
$951$	−6.70424	−0.217400
$952$	76.2866	2.47246
$953$	25.9967	0.842115	0.421058	−	0.907034i	$-0.361659\pi$
$953$	25.9967	0.842115	0.421058	+	0.907034i	$0.361659\pi$
$954$	−4.07529	−0.131942
$955$	0	0
$956$	57.7791	1.86871
$957$	−6.88691	−0.222622
$958$	90.8634	2.93566
$959$	−53.4162	−1.72490
$960$	0	0
$961$	19.9377	0.643151
$962$	84.4804	2.72376
$963$	0.212596	0.00685082
$964$	45.9714	1.48064
$965$	0	0
$966$	−13.8552	−0.445783
$967$	28.6419	0.921062	0.460531	−	0.887644i	$-0.347659\pi$
$967$	28.6419	0.921062	0.460531	+	0.887644i	$0.347659\pi$
$968$	−87.6034	−2.81568
$969$	−10.7334	−0.344808
$970$	0	0
$971$	−24.3952	−0.782879	−0.391440	−	0.920204i	$-0.628023\pi$
$971$	−24.3952	−0.782879	−0.391440	+	0.920204i	$0.628023\pi$
$972$	5.27940	0.169337
$973$	−62.5248	−2.00445
$974$	−41.2064	−1.32034
$975$	0	0
$976$	−48.0349	−1.53756
$977$	22.3923	0.716393	0.358197	−	0.933646i	$-0.383392\pi$
$977$	22.3923	0.716393	0.358197	+	0.933646i	$0.383392\pi$
$978$	−9.17982	−0.293538
$979$	−11.5350	−0.368659
$980$	0	0
$981$	8.37360	0.267348
$982$	−75.6972	−2.41560
$983$	−0.352396	−0.0112397	−0.00561985	−	0.999984i	$-0.501789\pi$
$983$	−0.352396	−0.0112397	−0.00561985	+	0.999984i	$0.501789\pi$
$984$	104.936	3.34525
$985$	0	0
$986$	−54.0302	−1.72067
$987$	10.5188	0.334816
$988$	104.020	3.30931
$989$	−11.2095	−0.356440
$990$	0	0
$991$	−19.5983	−0.622562	−0.311281	−	0.950318i	$-0.600758\pi$
$991$	−19.5983	−0.622562	−0.311281	+	0.950318i	$0.600758\pi$
$992$	130.064	4.12953
$993$	−31.8522	−1.01080
$994$	−90.5407	−2.87178
$995$	0	0
$996$	−63.8265	−2.02242
$997$	33.8104	1.07078	0.535392	−	0.844603i	$-0.320164\pi$
$997$	33.8104	1.07078	0.535392	+	0.844603i	$0.320164\pi$
$998$	−56.3826	−1.78476
$999$	−5.59570	−0.177040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4425.2.a.bg.1.6		6
5.4	even	2		885.2.a.j.1.1	✓	6
15.14	odd	2		2655.2.a.v.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
885.2.a.j.1.1	✓	6	5.4	even	2
2655.2.a.v.1.6		6	15.14	odd	2
4425.2.a.bg.1.6		6	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:	\( N \)	\(=\)	\( 4425 = 3 \cdot 5^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4425.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69804 q^{2} +1.00000 q^{3} +5.27940 q^{4} +2.69804 q^{6} +2.82843 q^{7} +8.84793 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.69804 q^{2} +1.00000 q^{3} +5.27940 q^{4} +2.69804 q^{6} +2.82843 q^{7} +8.84793 q^{8} +1.00000 q^{9} +1.04833 q^{11} +5.27940 q^{12} -5.59570 q^{13} +7.63120 q^{14} +13.3132 q^{16} +3.04833 q^{17} +2.69804 q^{18} -3.52109 q^{19} +2.82843 q^{21} +2.82843 q^{22} -1.81560 q^{23} +8.84793 q^{24} -15.0974 q^{26} +1.00000 q^{27} +14.9324 q^{28} -6.56942 q^{29} +7.13706 q^{31} +18.2237 q^{32} +1.04833 q^{33} +8.22450 q^{34} +5.27940 q^{36} -5.59570 q^{37} -9.50003 q^{38} -5.59570 q^{39} +11.8600 q^{41} +7.63120 q^{42} +6.17397 q^{43} +5.53454 q^{44} -4.89855 q^{46} +3.71894 q^{47} +13.3132 q^{48} +1.00000 q^{49} +3.04833 q^{51} -29.5419 q^{52} -1.51046 q^{53} +2.69804 q^{54} +25.0257 q^{56} -3.52109 q^{57} -17.7245 q^{58} -1.00000 q^{59} -3.60805 q^{61} +19.2561 q^{62} +2.82843 q^{63} +22.5418 q^{64} +2.82843 q^{66} +3.80037 q^{67} +16.0933 q^{68} -1.81560 q^{69} -11.8645 q^{71} +8.84793 q^{72} +3.49904 q^{73} -15.0974 q^{74} -18.5892 q^{76} +2.96512 q^{77} -15.0974 q^{78} -14.4099 q^{79} +1.00000 q^{81} +31.9987 q^{82} -12.0897 q^{83} +14.9324 q^{84} +16.6576 q^{86} -6.56942 q^{87} +9.27553 q^{88} -11.0032 q^{89} -15.8270 q^{91} -9.58527 q^{92} +7.13706 q^{93} +10.0338 q^{94} +18.2237 q^{96} +11.5353 q^{97} +2.69804 q^{98} +1.04833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9	\( 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9} - 2 q^{11} + 8 q^{12} - 2 q^{13} + 12 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 12 q^{23} + 12 q^{24} - 8 q^{26} + 6 q^{27} + 16 q^{28} - 12 q^{29} + 8 q^{31} + 28 q^{32} - 2 q^{33} + 8 q^{34} + 8 q^{36} - 2 q^{37} + 24 q^{38} - 2 q^{39} - 4 q^{41} + 18 q^{43} + 4 q^{44} + 16 q^{47} + 12 q^{48} + 6 q^{49} + 10 q^{51} - 12 q^{52} + 30 q^{53} + 4 q^{54} - 2 q^{57} + 16 q^{58} - 6 q^{59} + 4 q^{61} + 16 q^{62} + 20 q^{64} + 30 q^{67} + 20 q^{68} + 12 q^{69} - 24 q^{71} + 12 q^{72} + 6 q^{73} - 8 q^{74} - 4 q^{76} + 16 q^{77} - 8 q^{78} - 2 q^{79} + 6 q^{81} + 20 q^{83} + 16 q^{84} - 12 q^{87} + 16 q^{88} + 14 q^{89} - 48 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 28 q^{96} - 4 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9 - 2 * q^11 + 8 * q^12 - 2 * q^13 + 12 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 12 * q^23 + 12 * q^24 - 8 * q^26 + 6 * q^27 + 16 * q^28 - 12 * q^29 + 8 * q^31 + 28 * q^32 - 2 * q^33 + 8 * q^34 + 8 * q^36 - 2 * q^37 + 24 * q^38 - 2 * q^39 - 4 * q^41 + 18 * q^43 + 4 * q^44 + 16 * q^47 + 12 * q^48 + 6 * q^49 + 10 * q^51 - 12 * q^52 + 30 * q^53 + 4 * q^54 - 2 * q^57 + 16 * q^58 - 6 * q^59 + 4 * q^61 + 16 * q^62 + 20 * q^64 + 30 * q^67 + 20 * q^68 + 12 * q^69 - 24 * q^71 + 12 * q^72 + 6 * q^73 - 8 * q^74 - 4 * q^76 + 16 * q^77 - 8 * q^78 - 2 * q^79 + 6 * q^81 + 20 * q^83 + 16 * q^84 - 12 * q^87 + 16 * q^88 + 14 * q^89 - 48 * q^91 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 28 * q^96 - 4 * q^97 + 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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