LMFDB - Embedded newform 5225.2.a.ba.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 26 x^{18} + 281 x^{16} - 1640 x^{14} + 5623 x^{12} - 11551 x^{10} + 13894 x^{8} - 9095 x^{6} + \cdots + 4 $ x^20 - 26x^18 + 281x^16 - 1640x^14 + 5623x^12 - 11551x^10 + 13894x^8 - 9095x^6 + 2753x^4 - 276*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$1.51095$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51095 q^{2} +0.900785 q^{3} +0.282982 q^{4} +1.36104 q^{6} -0.603482 q^{7} -2.59434 q^{8} -2.18859 q^{9} +O(q^{10})$	\(q+1.51095 q^{2} +0.900785 q^{3} +0.282982 q^{4} +1.36104 q^{6} -0.603482 q^{7} -2.59434 q^{8} -2.18859 q^{9} +1.00000 q^{11} +0.254906 q^{12} +3.06772 q^{13} -0.911833 q^{14} -4.48589 q^{16} +4.08319 q^{17} -3.30685 q^{18} -1.00000 q^{19} -0.543607 q^{21} +1.51095 q^{22} -6.19344 q^{23} -2.33694 q^{24} +4.63519 q^{26} -4.67380 q^{27} -0.170774 q^{28} -2.92797 q^{29} +7.98044 q^{31} -1.58930 q^{32} +0.900785 q^{33} +6.16951 q^{34} -0.619330 q^{36} -10.2773 q^{37} -1.51095 q^{38} +2.76336 q^{39} -7.31303 q^{41} -0.821366 q^{42} -0.0821817 q^{43} +0.282982 q^{44} -9.35800 q^{46} +2.77574 q^{47} -4.04082 q^{48} -6.63581 q^{49} +3.67807 q^{51} +0.868110 q^{52} -8.01831 q^{53} -7.06190 q^{54} +1.56563 q^{56} -0.900785 q^{57} -4.42403 q^{58} +3.21020 q^{59} -11.1879 q^{61} +12.0581 q^{62} +1.32077 q^{63} +6.57042 q^{64} +1.36104 q^{66} +3.52951 q^{67} +1.15547 q^{68} -5.57896 q^{69} +1.51928 q^{71} +5.67793 q^{72} +3.04579 q^{73} -15.5285 q^{74} -0.282982 q^{76} -0.603482 q^{77} +4.17531 q^{78} -3.57318 q^{79} +2.35567 q^{81} -11.0497 q^{82} -9.37109 q^{83} -0.153831 q^{84} -0.124173 q^{86} -2.63747 q^{87} -2.59434 q^{88} -0.629618 q^{89} -1.85132 q^{91} -1.75263 q^{92} +7.18867 q^{93} +4.19402 q^{94} -1.43161 q^{96} +5.71214 q^{97} -10.0264 q^{98} -2.18859 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9}+O(q^{10}) $ 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9	$ 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9} + 20 q^{11} - 24 q^{14} - 4 q^{16} - 20 q^{19} - 30 q^{21} - 38 q^{24} + 8 q^{26} - 50 q^{29} - 50 q^{31} - 28 q^{34} - 12 q^{36} - 48 q^{39} - 34 q^{41} + 12 q^{44} - 36 q^{46} + 6 q^{49} - 40 q^{51} + 6 q^{54} - 40 q^{56} - 30 q^{59} - 14 q^{61} - 36 q^{64} - 8 q^{66} + 12 q^{69} - 40 q^{71} - 50 q^{74} - 12 q^{76} - 106 q^{79} + 30 q^{84} + 56 q^{86} - 36 q^{89} - 56 q^{91} - 28 q^{94} + 66 q^{96} + 10 q^{99}+O(q^{100}) $ 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9 + 20 * q^11 - 24 * q^14 - 4 * q^16 - 20 * q^19 - 30 * q^21 - 38 * q^24 + 8 * q^26 - 50 * q^29 - 50 * q^31 - 28 * q^34 - 12 * q^36 - 48 * q^39 - 34 * q^41 + 12 * q^44 - 36 * q^46 + 6 * q^49 - 40 * q^51 + 6 * q^54 - 40 * q^56 - 30 * q^59 - 14 * q^61 - 36 * q^64 - 8 * q^66 + 12 * q^69 - 40 * q^71 - 50 * q^74 - 12 * q^76 - 106 * q^79 + 30 * q^84 + 56 * q^86 - 36 * q^89 - 56 * q^91 - 28 * q^94 + 66 * q^96 + 10 * 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 $

$2$	1.51095	1.06841	0.534203	−	0.845356i	$-0.320612\pi$
$2$	1.51095	1.06841	0.534203	+	0.845356i	$0.320612\pi$
$3$	0.900785	0.520069	0.260034	−	0.965599i	$-0.416266\pi$
$3$	0.900785	0.520069	0.260034	+	0.965599i	$0.416266\pi$
$4$	0.282982	0.141491
$5$	0	0
$5$	0	0
$6$	1.36104	0.555644
$7$	−0.603482	−0.228095	−0.114047	−	0.993475i	$-0.536382\pi$
$7$	−0.603482	−0.228095	−0.114047	+	0.993475i	$0.536382\pi$
$8$	−2.59434	−0.917236
$9$	−2.18859	−0.729529
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.254906	0.0735850
$13$	3.06772	0.850833	0.425417	−	0.904998i	$-0.360128\pi$
$13$	3.06772	0.850833	0.425417	+	0.904998i	$0.360128\pi$
$14$	−0.911833	−0.243698
$15$	0	0
$16$	−4.48589	−1.12147
$17$	4.08319	0.990318	0.495159	−	0.868802i	$-0.335110\pi$
$17$	4.08319	0.990318	0.495159	+	0.868802i	$0.335110\pi$
$18$	−3.30685	−0.779433
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.543607	−0.118625
$22$	1.51095	0.322136
$23$	−6.19344	−1.29142	−0.645710	−	0.763582i	$-0.723439\pi$
$23$	−6.19344	−1.29142	−0.645710	+	0.763582i	$0.723439\pi$
$24$	−2.33694	−0.477026
$25$	0	0
$26$	4.63519	0.909035
$27$	−4.67380	−0.899473
$28$	−0.170774	−0.0322733
$29$	−2.92797	−0.543711	−0.271855	−	0.962338i	$-0.587637\pi$
$29$	−2.92797	−0.543711	−0.271855	+	0.962338i	$0.587637\pi$
$30$	0	0
$31$	7.98044	1.43333	0.716665	−	0.697418i	$-0.245668\pi$
$31$	7.98044	1.43333	0.716665	+	0.697418i	$0.245668\pi$
$32$	−1.58930	−0.280950
$33$	0.900785	0.156807
$34$	6.16951	1.05806
$35$	0	0
$36$	−0.619330	−0.103222
$37$	−10.2773	−1.68957	−0.844785	−	0.535106i	$-0.820272\pi$
$37$	−10.2773	−1.68957	−0.844785	+	0.535106i	$0.820272\pi$
$38$	−1.51095	−0.245109
$39$	2.76336	0.442492
$40$	0	0
$41$	−7.31303	−1.14210	−0.571052	−	0.820914i	$-0.693465\pi$
$41$	−7.31303	−1.14210	−0.571052	+	0.820914i	$0.693465\pi$
$42$	−0.821366	−0.126740
$43$	−0.0821817	−0.0125326	−0.00626629	−	0.999980i	$-0.501995\pi$
$43$	−0.0821817	−0.0125326	−0.00626629	+	0.999980i	$0.501995\pi$
$44$	0.282982	0.0426611
$45$	0	0
$46$	−9.35800	−1.37976
$47$	2.77574	0.404883	0.202442	−	0.979294i	$-0.435112\pi$
$47$	2.77574	0.404883	0.202442	+	0.979294i	$0.435112\pi$
$48$	−4.04082	−0.583242
$49$	−6.63581	−0.947973
$50$	0	0
$51$	3.67807	0.515033
$52$	0.868110	0.120385
$53$	−8.01831	−1.10140	−0.550700	−	0.834703i	$-0.685639\pi$
$53$	−8.01831	−1.10140	−0.550700	+	0.834703i	$0.685639\pi$
$54$	−7.06190	−0.961003
$55$	0	0
$56$	1.56563	0.209217
$57$	−0.900785	−0.119312
$58$	−4.42403	−0.580904
$59$	3.21020	0.417933	0.208966	−	0.977923i	$-0.432990\pi$
$59$	3.21020	0.417933	0.208966	+	0.977923i	$0.432990\pi$
$60$	0	0
$61$	−11.1879	−1.43247	−0.716233	−	0.697862i	$-0.754135\pi$
$61$	−11.1879	−1.43247	−0.716233	+	0.697862i	$0.754135\pi$
$62$	12.0581	1.53138
$63$	1.32077	0.166402
$64$	6.57042	0.821302
$65$	0	0
$66$	1.36104	0.167533
$67$	3.52951	0.431198	0.215599	−	0.976482i	$-0.430830\pi$
$67$	3.52951	0.431198	0.215599	+	0.976482i	$0.430830\pi$
$68$	1.15547	0.140121
$69$	−5.57896	−0.671627
$70$	0	0
$71$	1.51928	0.180305	0.0901525	−	0.995928i	$-0.471265\pi$
$71$	1.51928	0.180305	0.0901525	+	0.995928i	$0.471265\pi$
$72$	5.67793	0.669150
$73$	3.04579	0.356483	0.178242	−	0.983987i	$-0.442959\pi$
$73$	3.04579	0.356483	0.178242	+	0.983987i	$0.442959\pi$
$74$	−15.5285	−1.80515
$75$	0	0
$76$	−0.282982	−0.0324603
$77$	−0.603482	−0.0687731
$78$	4.17531	0.472761
$79$	−3.57318	−0.402014	−0.201007	−	0.979590i	$-0.564421\pi$
$79$	−3.57318	−0.402014	−0.201007	+	0.979590i	$0.564421\pi$
$80$	0	0
$81$	2.35567	0.261741
$82$	−11.0497	−1.22023
$83$	−9.37109	−1.02861	−0.514305	−	0.857607i	$-0.671950\pi$
$83$	−9.37109	−1.02861	−0.514305	+	0.857607i	$0.671950\pi$
$84$	−0.153831	−0.0167843
$85$	0	0
$86$	−0.124173	−0.0133899
$87$	−2.63747	−0.282767
$88$	−2.59434	−0.276557
$89$	−0.629618	−0.0667394	−0.0333697	−	0.999443i	$-0.510624\pi$
$89$	−0.629618	−0.0667394	−0.0333697	+	0.999443i	$0.510624\pi$
$90$	0	0
$91$	−1.85132	−0.194071
$92$	−1.75263	−0.182724
$93$	7.18867	0.745430
$94$	4.19402	0.432580
$95$	0	0
$96$	−1.43161	−0.146113
$97$	5.71214	0.579980	0.289990	−	0.957030i	$-0.406348\pi$
$97$	5.71214	0.579980	0.289990	+	0.957030i	$0.406348\pi$
$98$	−10.0264	−1.01282
$99$	−2.18859	−0.219961
$100$	0	0
$101$	−15.0319	−1.49573	−0.747865	−	0.663850i	$-0.768921\pi$
$101$	−15.0319	−1.49573	−0.747865	+	0.663850i	$0.768921\pi$
$102$	5.55740	0.550264
$103$	−15.9710	−1.57367	−0.786836	−	0.617162i	$-0.788282\pi$
$103$	−15.9710	−1.57367	−0.786836	+	0.617162i	$0.788282\pi$
$104$	−7.95870	−0.780415
$105$	0	0
$106$	−12.1153	−1.17674
$107$	−0.124523	−0.0120381	−0.00601903	−	0.999982i	$-0.501916\pi$
$107$	−0.124523	−0.0120381	−0.00601903	+	0.999982i	$0.501916\pi$
$108$	−1.32260	−0.127267
$109$	−14.3549	−1.37495	−0.687477	−	0.726206i	$-0.741282\pi$
$109$	−14.3549	−1.37495	−0.687477	+	0.726206i	$0.741282\pi$
$110$	0	0
$111$	−9.25760	−0.878692
$112$	2.70715	0.255802
$113$	11.5904	1.09033	0.545166	−	0.838328i	$-0.316467\pi$
$113$	11.5904	1.09033	0.545166	+	0.838328i	$0.316467\pi$
$114$	−1.36104	−0.127474
$115$	0	0
$116$	−0.828563	−0.0769302
$117$	−6.71398	−0.620707
$118$	4.85047	0.446522
$119$	−2.46413	−0.225886
$120$	0	0
$121$	1.00000	0.0909091
$122$	−16.9044	−1.53045
$123$	−6.58747	−0.593972
$124$	2.25832	0.202803
$125$	0	0
$126$	1.99563	0.177784
$127$	−18.8000	−1.66823	−0.834115	−	0.551591i	$-0.814021\pi$
$127$	−18.8000	−1.66823	−0.834115	+	0.551591i	$0.814021\pi$
$128$	13.1062	1.15843
$129$	−0.0740280	−0.00651780
$130$	0	0
$131$	16.2223	1.41735	0.708674	−	0.705537i	$-0.249294\pi$
$131$	16.2223	1.41735	0.708674	+	0.705537i	$0.249294\pi$
$132$	0.254906	0.0221867
$133$	0.603482	0.0523285
$134$	5.33292	0.460694
$135$	0	0
$136$	−10.5932	−0.908355
$137$	−21.2561	−1.81603	−0.908017	−	0.418932i	$-0.862404\pi$
$137$	−21.2561	−1.81603	−0.908017	+	0.418932i	$0.862404\pi$
$138$	−8.42955	−0.717571
$139$	21.9041	1.85788	0.928942	−	0.370226i	$-0.120720\pi$
$139$	21.9041	1.85788	0.928942	+	0.370226i	$0.120720\pi$
$140$	0	0
$141$	2.50035	0.210567
$142$	2.29556	0.192639
$143$	3.06772	0.256536
$144$	9.81775	0.818145
$145$	0	0
$146$	4.60206	0.380869
$147$	−5.97744	−0.493011
$148$	−2.90828	−0.239059
$149$	11.0150	0.902388	0.451194	−	0.892426i	$-0.350998\pi$
$149$	11.0150	0.902388	0.451194	+	0.892426i	$0.350998\pi$
$150$	0	0
$151$	15.7653	1.28296	0.641481	−	0.767139i	$-0.278320\pi$
$151$	15.7653	1.28296	0.641481	+	0.767139i	$0.278320\pi$
$152$	2.59434	0.210428
$153$	−8.93640	−0.722465
$154$	−0.911833	−0.0734776
$155$	0	0
$156$	0.781981	0.0626086
$157$	−23.8820	−1.90599	−0.952997	−	0.302981i	$-0.902018\pi$
$157$	−23.8820	−1.90599	−0.952997	+	0.302981i	$0.902018\pi$
$158$	−5.39890	−0.429514
$159$	−7.22277	−0.572803
$160$	0	0
$161$	3.73763	0.294566
$162$	3.55930	0.279645
$163$	18.1018	1.41784	0.708919	−	0.705290i	$-0.249183\pi$
$163$	18.1018	1.41784	0.708919	+	0.705290i	$0.249183\pi$
$164$	−2.06946	−0.161597
$165$	0	0
$166$	−14.1593	−1.09897
$167$	−8.11307	−0.627808	−0.313904	−	0.949455i	$-0.601637\pi$
$167$	−8.11307	−0.627808	−0.313904	+	0.949455i	$0.601637\pi$
$168$	1.41030	0.108807
$169$	−3.58907	−0.276082
$170$	0	0
$171$	2.18859	0.167365
$172$	−0.0232559	−0.00177325
$173$	17.9368	1.36371	0.681853	−	0.731489i	$-0.261174\pi$
$173$	17.9368	1.36371	0.681853	+	0.731489i	$0.261174\pi$
$174$	−3.98510	−0.302110
$175$	0	0
$176$	−4.48589	−0.338136
$177$	2.89170	0.217354
$178$	−0.951324	−0.0713048
$179$	−22.6454	−1.69260	−0.846300	−	0.532707i	$-0.821175\pi$
$179$	−22.6454	−1.69260	−0.846300	+	0.532707i	$0.821175\pi$
$180$	0	0
$181$	−2.12283	−0.157789	−0.0788943	−	0.996883i	$-0.525139\pi$
$181$	−2.12283	−0.157789	−0.0788943	+	0.996883i	$0.525139\pi$
$182$	−2.79725	−0.207346
$183$	−10.0779	−0.744980
$184$	16.0679	1.18454
$185$	0	0
$186$	10.8617	0.796422
$187$	4.08319	0.298592
$188$	0.785485	0.0572873
$189$	2.82055	0.205165
$190$	0	0
$191$	−5.41877	−0.392088	−0.196044	−	0.980595i	$-0.562810\pi$
$191$	−5.41877	−0.392088	−0.196044	+	0.980595i	$0.562810\pi$
$192$	5.91854	0.427133
$193$	2.89582	0.208446	0.104223	−	0.994554i	$-0.466764\pi$
$193$	2.89582	0.208446	0.104223	+	0.994554i	$0.466764\pi$
$194$	8.63078	0.619654
$195$	0	0
$196$	−1.87781	−0.134130
$197$	−3.06581	−0.218430	−0.109215	−	0.994018i	$-0.534834\pi$
$197$	−3.06581	−0.218430	−0.109215	+	0.994018i	$0.534834\pi$
$198$	−3.30685	−0.235008
$199$	−23.6145	−1.67399	−0.836993	−	0.547214i	$-0.815688\pi$
$199$	−23.6145	−1.67399	−0.836993	+	0.547214i	$0.815688\pi$
$200$	0	0
$201$	3.17933	0.224252
$202$	−22.7125	−1.59805
$203$	1.76698	0.124018
$204$	1.04083	0.0728725
$205$	0	0
$206$	−24.1315	−1.68132
$207$	13.5549	0.942129
$208$	−13.7615	−0.954185
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−11.5535	−0.795377	−0.397688	−	0.917521i	$-0.630187\pi$
$211$	−11.5535	−0.795377	−0.397688	+	0.917521i	$0.630187\pi$
$212$	−2.26904	−0.155838
$213$	1.36854	0.0937709
$214$	−0.188148	−0.0128615
$215$	0	0
$216$	12.1254	0.825030
$217$	−4.81605	−0.326935
$218$	−21.6897	−1.46901
$219$	2.74361	0.185396
$220$	0	0
$221$	12.5261	0.842596
$222$	−13.9878	−0.938800
$223$	−9.59966	−0.642841	−0.321420	−	0.946937i	$-0.604160\pi$
$223$	−9.59966	−0.642841	−0.321420	+	0.946937i	$0.604160\pi$
$224$	0.959111	0.0640833
$225$	0	0
$226$	17.5125	1.16492
$227$	−5.29283	−0.351298	−0.175649	−	0.984453i	$-0.556202\pi$
$227$	−5.29283	−0.351298	−0.175649	+	0.984453i	$0.556202\pi$
$228$	−0.254906	−0.0168816
$229$	−1.45160	−0.0959243	−0.0479622	−	0.998849i	$-0.515273\pi$
$229$	−1.45160	−0.0959243	−0.0479622	+	0.998849i	$0.515273\pi$
$230$	0	0
$231$	−0.543607	−0.0357667
$232$	7.59614	0.498711
$233$	25.0472	1.64090	0.820448	−	0.571721i	$-0.193724\pi$
$233$	25.0472	1.64090	0.820448	+	0.571721i	$0.193724\pi$
$234$	−10.1445	−0.663167
$235$	0	0
$236$	0.908429	0.0591337
$237$	−3.21866	−0.209075
$238$	−3.72318	−0.241338
$239$	1.53900	0.0995499	0.0497749	−	0.998760i	$-0.484150\pi$
$239$	1.53900	0.0995499	0.0497749	+	0.998760i	$0.484150\pi$
$240$	0	0
$241$	−4.82341	−0.310703	−0.155352	−	0.987859i	$-0.549651\pi$
$241$	−4.82341	−0.310703	−0.155352	+	0.987859i	$0.549651\pi$
$242$	1.51095	0.0971278
$243$	16.1434	1.03560
$244$	−3.16598	−0.202681
$245$	0	0
$246$	−9.95337	−0.634603
$247$	−3.06772	−0.195195
$248$	−20.7039	−1.31470
$249$	−8.44134	−0.534948
$250$	0	0
$251$	−11.4453	−0.722423	−0.361211	−	0.932484i	$-0.617637\pi$
$251$	−11.4453	−0.722423	−0.361211	+	0.932484i	$0.617637\pi$
$252$	0.373755	0.0235443
$253$	−6.19344	−0.389378
$254$	−28.4059	−1.78235
$255$	0	0
$256$	6.66201	0.416376
$257$	−4.14766	−0.258724	−0.129362	−	0.991597i	$-0.541293\pi$
$257$	−4.14766	−0.258724	−0.129362	+	0.991597i	$0.541293\pi$
$258$	−0.111853	−0.00696366
$259$	6.20214	0.385382
$260$	0	0
$261$	6.40812	0.396653
$262$	24.5111	1.51430
$263$	4.43431	0.273431	0.136716	−	0.990610i	$-0.456345\pi$
$263$	4.43431	0.273431	0.136716	+	0.990610i	$0.456345\pi$
$264$	−2.33694	−0.143829
$265$	0	0
$266$	0.911833	0.0559081
$267$	−0.567151	−0.0347091
$268$	0.998787	0.0610106
$269$	4.65270	0.283680	0.141840	−	0.989890i	$-0.454698\pi$
$269$	4.65270	0.283680	0.141840	+	0.989890i	$0.454698\pi$
$270$	0	0
$271$	−19.7128	−1.19746	−0.598732	−	0.800949i	$-0.704329\pi$
$271$	−19.7128	−1.19746	−0.598732	+	0.800949i	$0.704329\pi$
$272$	−18.3167	−1.11061
$273$	−1.66764	−0.100930
$274$	−32.1171	−1.94026
$275$	0	0
$276$	−1.57874	−0.0950292
$277$	21.9425	1.31840	0.659198	−	0.751970i	$-0.270896\pi$
$277$	21.9425	1.31840	0.659198	+	0.751970i	$0.270896\pi$
$278$	33.0961	1.98497
$279$	−17.4659	−1.04566
$280$	0	0
$281$	−4.67060	−0.278625	−0.139312	−	0.990248i	$-0.544489\pi$
$281$	−4.67060	−0.278625	−0.139312	+	0.990248i	$0.544489\pi$
$282$	3.77791	0.224971
$283$	10.2046	0.606598	0.303299	−	0.952895i	$-0.401912\pi$
$283$	10.2046	0.606598	0.303299	+	0.952895i	$0.401912\pi$
$284$	0.429928	0.0255115
$285$	0	0
$286$	4.63519	0.274084
$287$	4.41328	0.260508
$288$	3.47831	0.204961
$289$	−0.327596	−0.0192703
$290$	0	0
$291$	5.14541	0.301629
$292$	0.861905	0.0504392
$293$	19.0508	1.11296	0.556478	−	0.830862i	$-0.312152\pi$
$293$	19.0508	1.11296	0.556478	+	0.830862i	$0.312152\pi$
$294$	−9.03164	−0.526736
$295$	0	0
$296$	26.6626	1.54973
$297$	−4.67380	−0.271201
$298$	16.6432	0.964117
$299$	−18.9998	−1.09878
$300$	0	0
$301$	0.0495951	0.00285862
$302$	23.8206	1.37072
$303$	−13.5405	−0.777883
$304$	4.48589	0.257283
$305$	0	0
$306$	−13.5025	−0.771886
$307$	−1.26498	−0.0721961	−0.0360981	−	0.999348i	$-0.511493\pi$
$307$	−1.26498	−0.0721961	−0.0360981	+	0.999348i	$0.511493\pi$
$308$	−0.170774	−0.00973078
$309$	−14.3865	−0.818417
$310$	0	0
$311$	34.2924	1.94455	0.972273	−	0.233849i	$-0.0751322\pi$
$311$	34.2924	1.94455	0.972273	+	0.233849i	$0.0751322\pi$
$312$	−7.16908	−0.405869
$313$	−0.755401	−0.0426978	−0.0213489	−	0.999772i	$-0.506796\pi$
$313$	−0.755401	−0.0426978	−0.0213489	+	0.999772i	$0.506796\pi$
$314$	−36.0846	−2.03637
$315$	0	0
$316$	−1.01114	−0.0568813
$317$	19.4248	1.09101	0.545504	−	0.838108i	$-0.316338\pi$
$317$	19.4248	1.09101	0.545504	+	0.838108i	$0.316338\pi$
$318$	−10.9133	−0.611986
$319$	−2.92797	−0.163935
$320$	0	0
$321$	−0.112168	−0.00626062
$322$	5.64738	0.314716
$323$	−4.08319	−0.227195
$324$	0.666611	0.0370340
$325$	0	0
$326$	27.3509	1.51483
$327$	−12.9307	−0.715071
$328$	18.9725	1.04758
$329$	−1.67511	−0.0923518
$330$	0	0
$331$	26.3233	1.44686	0.723429	−	0.690399i	$-0.242565\pi$
$331$	26.3233	1.44686	0.723429	+	0.690399i	$0.242565\pi$
$332$	−2.65185	−0.145539
$333$	22.4927	1.23259
$334$	−12.2585	−0.670754
$335$	0	0
$336$	2.43856	0.133034
$337$	28.4082	1.54749	0.773746	−	0.633495i	$-0.218380\pi$
$337$	28.4082	1.54749	0.773746	+	0.633495i	$0.218380\pi$
$338$	−5.42292	−0.294968
$339$	10.4404	0.567047
$340$	0	0
$341$	7.98044	0.432165
$342$	3.30685	0.178814
$343$	8.22896	0.444322
$344$	0.213207	0.0114953
$345$	0	0
$346$	27.1016	1.45699
$347$	15.4328	0.828475	0.414237	−	0.910169i	$-0.364048\pi$
$347$	15.4328	0.828475	0.414237	+	0.910169i	$0.364048\pi$
$348$	−0.746358	−0.0400090
$349$	9.55388	0.511408	0.255704	−	0.966755i	$-0.417693\pi$
$349$	9.55388	0.511408	0.255704	+	0.966755i	$0.417693\pi$
$350$	0	0
$351$	−14.3379	−0.765302
$352$	−1.58930	−0.0847097
$353$	6.07556	0.323370	0.161685	−	0.986842i	$-0.448307\pi$
$353$	6.07556	0.323370	0.161685	+	0.986842i	$0.448307\pi$
$354$	4.36923	0.232222
$355$	0	0
$356$	−0.178171	−0.00944302
$357$	−2.21965	−0.117476
$358$	−34.2162	−1.80838
$359$	19.9636	1.05364	0.526820	−	0.849977i	$-0.323384\pi$
$359$	19.9636	1.05364	0.526820	+	0.849977i	$0.323384\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−3.20750	−0.168582
$363$	0.900785	0.0472790
$364$	−0.523889	−0.0274592
$365$	0	0
$366$	−15.2273	−0.795941
$367$	0.193444	0.0100977	0.00504886	−	0.999987i	$-0.498393\pi$
$367$	0.193444	0.0100977	0.00504886	+	0.999987i	$0.498393\pi$
$368$	27.7830	1.44829
$369$	16.0052	0.833198
$370$	0	0
$371$	4.83890	0.251223
$372$	2.03426	0.105472
$373$	33.8918	1.75485	0.877426	−	0.479712i	$-0.159259\pi$
$373$	33.8918	1.75485	0.877426	+	0.479712i	$0.159259\pi$
$374$	6.16951	0.319018
$375$	0	0
$376$	−7.20120	−0.371374
$377$	−8.98221	−0.462607
$378$	4.26173	0.219200
$379$	−9.70742	−0.498637	−0.249318	−	0.968422i	$-0.580207\pi$
$379$	−9.70742	−0.498637	−0.249318	+	0.968422i	$0.580207\pi$
$380$	0	0
$381$	−16.9348	−0.867594
$382$	−8.18751	−0.418909
$383$	−12.2664	−0.626783	−0.313391	−	0.949624i	$-0.601465\pi$
$383$	−12.2664	−0.626783	−0.313391	+	0.949624i	$0.601465\pi$
$384$	11.8059	0.602465
$385$	0	0
$386$	4.37546	0.222705
$387$	0.179862	0.00914288
$388$	1.61643	0.0820619
$389$	22.1011	1.12057	0.560285	−	0.828300i	$-0.310692\pi$
$389$	22.1011	1.12057	0.560285	+	0.828300i	$0.310692\pi$
$390$	0	0
$391$	−25.2890	−1.27892
$392$	17.2155	0.869515
$393$	14.6128	0.737118
$394$	−4.63229	−0.233372
$395$	0	0
$396$	−0.619330	−0.0311225
$397$	−1.76411	−0.0885380	−0.0442690	−	0.999020i	$-0.514096\pi$
$397$	−1.76411	−0.0885380	−0.0442690	+	0.999020i	$0.514096\pi$
$398$	−35.6804	−1.78850
$399$	0.543607	0.0272144
$400$	0	0
$401$	−37.7407	−1.88468	−0.942339	−	0.334659i	$-0.891379\pi$
$401$	−37.7407	−1.88468	−0.942339	+	0.334659i	$0.891379\pi$
$402$	4.80382	0.239593
$403$	24.4818	1.21953
$404$	−4.25376	−0.211632
$405$	0	0
$406$	2.66982	0.132501
$407$	−10.2773	−0.509425
$408$	−9.54215	−0.472407
$409$	−15.2172	−0.752440	−0.376220	−	0.926530i	$-0.622776\pi$
$409$	−15.2172	−0.752440	−0.376220	+	0.926530i	$0.622776\pi$
$410$	0	0
$411$	−19.1472	−0.944463
$412$	−4.51951	−0.222660
$413$	−1.93730	−0.0953282
$414$	20.4808	1.00658
$415$	0	0
$416$	−4.87552	−0.239042
$417$	19.7309	0.966227
$418$	−1.51095	−0.0739032
$419$	−23.6883	−1.15725	−0.578624	−	0.815595i	$-0.696410\pi$
$419$	−23.6883	−1.15725	−0.578624	+	0.815595i	$0.696410\pi$
$420$	0	0
$421$	−14.2242	−0.693244	−0.346622	−	0.938005i	$-0.612671\pi$
$421$	−14.2242	−0.693244	−0.346622	+	0.938005i	$0.612671\pi$
$422$	−17.4568	−0.849785
$423$	−6.07495	−0.295374
$424$	20.8022	1.01024
$425$	0	0
$426$	2.06780	0.100185
$427$	6.75170	0.326738
$428$	−0.0352377	−0.00170328
$429$	2.76336	0.133416
$430$	0	0
$431$	−23.9472	−1.15350	−0.576749	−	0.816922i	$-0.695679\pi$
$431$	−23.9472	−1.15350	−0.576749	+	0.816922i	$0.695679\pi$
$432$	20.9661	1.00873
$433$	26.4436	1.27080	0.635398	−	0.772185i	$-0.280836\pi$
$433$	26.4436	1.27080	0.635398	+	0.772185i	$0.280836\pi$
$434$	−7.27683	−0.349299
$435$	0	0
$436$	−4.06219	−0.194544
$437$	6.19344	0.296272
$438$	4.14546	0.198078
$439$	22.3823	1.06825	0.534124	−	0.845406i	$-0.320641\pi$
$439$	22.3823	1.06825	0.534124	+	0.845406i	$0.320641\pi$
$440$	0	0
$441$	14.5230	0.691573
$442$	18.9263	0.900234
$443$	10.3295	0.490771	0.245386	−	0.969426i	$-0.421085\pi$
$443$	10.3295	0.490771	0.245386	+	0.969426i	$0.421085\pi$
$444$	−2.61973	−0.124327
$445$	0	0
$446$	−14.5046	−0.686815
$447$	9.92219	0.469304
$448$	−3.96513	−0.187335
$449$	−14.4952	−0.684068	−0.342034	−	0.939687i	$-0.611116\pi$
$449$	−14.4952	−0.684068	−0.342034	+	0.939687i	$0.611116\pi$
$450$	0	0
$451$	−7.31303	−0.344357
$452$	3.27987	0.154272
$453$	14.2012	0.667228
$454$	−7.99723	−0.375328
$455$	0	0
$456$	2.33694	0.109437
$457$	16.8399	0.787737	0.393869	−	0.919167i	$-0.371136\pi$
$457$	16.8399	0.787737	0.393869	+	0.919167i	$0.371136\pi$
$458$	−2.19330	−0.102486
$459$	−19.0840	−0.890765
$460$	0	0
$461$	10.7768	0.501928	0.250964	−	0.967996i	$-0.419252\pi$
$461$	10.7768	0.501928	0.250964	+	0.967996i	$0.419252\pi$
$462$	−0.821366	−0.0382134
$463$	−13.0560	−0.606764	−0.303382	−	0.952869i	$-0.598116\pi$
$463$	−13.0560	−0.606764	−0.303382	+	0.952869i	$0.598116\pi$
$464$	13.1345	0.609756
$465$	0	0
$466$	37.8451	1.75314
$467$	−13.1491	−0.608470	−0.304235	−	0.952597i	$-0.598401\pi$
$467$	−13.1491	−0.608470	−0.304235	+	0.952597i	$0.598401\pi$
$468$	−1.89993	−0.0878245
$469$	−2.12999	−0.0983540
$470$	0	0
$471$	−21.5126	−0.991247
$472$	−8.32834	−0.383343
$473$	−0.0821817	−0.00377872
$474$	−4.86325	−0.223377
$475$	0	0
$476$	−0.697304	−0.0319609
$477$	17.5488	0.803502
$478$	2.32536	0.106360
$479$	6.76244	0.308984	0.154492	−	0.987994i	$-0.450626\pi$
$479$	6.76244	0.308984	0.154492	+	0.987994i	$0.450626\pi$
$480$	0	0
$481$	−31.5278	−1.43754
$482$	−7.28795	−0.331957
$483$	3.36680	0.153195
$484$	0.282982	0.0128628
$485$	0	0
$486$	24.3919	1.10644
$487$	28.4833	1.29070	0.645351	−	0.763886i	$-0.276711\pi$
$487$	28.4833	1.29070	0.645351	+	0.763886i	$0.276711\pi$
$488$	29.0252	1.31391
$489$	16.3058	0.737373
$490$	0	0
$491$	−0.746198	−0.0336754	−0.0168377	−	0.999858i	$-0.505360\pi$
$491$	−0.746198	−0.0336754	−0.0168377	+	0.999858i	$0.505360\pi$
$492$	−1.86414	−0.0840417
$493$	−11.9555	−0.538447
$494$	−4.63519	−0.208547
$495$	0	0
$496$	−35.7994	−1.60744
$497$	−0.916856	−0.0411266
$498$	−12.7545	−0.571541
$499$	−10.7995	−0.483452	−0.241726	−	0.970345i	$-0.577713\pi$
$499$	−10.7995	−0.483452	−0.241726	+	0.970345i	$0.577713\pi$
$500$	0	0
$501$	−7.30813	−0.326503
$502$	−17.2934	−0.771841
$503$	−23.2195	−1.03530	−0.517652	−	0.855591i	$-0.673194\pi$
$503$	−23.2195	−1.03530	−0.517652	+	0.855591i	$0.673194\pi$
$504$	−3.42653	−0.152630
$505$	0	0
$506$	−9.35800	−0.416014
$507$	−3.23298	−0.143582
$508$	−5.32006	−0.236039
$509$	35.1577	1.55834	0.779169	−	0.626814i	$-0.215641\pi$
$509$	35.1577	1.55834	0.779169	+	0.626814i	$0.215641\pi$
$510$	0	0
$511$	−1.83808	−0.0813119
$512$	−16.1464	−0.713576
$513$	4.67380	0.206353
$514$	−6.26692	−0.276422
$515$	0	0
$516$	−0.0209486	−0.000922211 0
$517$	2.77574	0.122077
$518$	9.37114	0.411744
$519$	16.1572	0.709221
$520$	0	0
$521$	−6.74136	−0.295344	−0.147672	−	0.989036i	$-0.547178\pi$
$521$	−6.74136	−0.295344	−0.147672	+	0.989036i	$0.547178\pi$
$522$	9.68237	0.423786
$523$	−17.7769	−0.777329	−0.388664	−	0.921379i	$-0.627063\pi$
$523$	−17.7769	−0.777329	−0.388664	+	0.921379i	$0.627063\pi$
$524$	4.59061	0.200542
$525$	0	0
$526$	6.70004	0.292135
$527$	32.5856	1.41945
$528$	−4.04082	−0.175854
$529$	15.3587	0.667768
$530$	0	0
$531$	−7.02580	−0.304894
$532$	0.170774	0.00740401
$533$	−22.4344	−0.971740
$534$	−0.856939	−0.0370834
$535$	0	0
$536$	−9.15673	−0.395510
$537$	−20.3987	−0.880268
$538$	7.03001	0.303085
$539$	−6.63581	−0.285825
$540$	0	0
$541$	8.59351	0.369464	0.184732	−	0.982789i	$-0.440858\pi$
$541$	8.59351	0.369464	0.184732	+	0.982789i	$0.440858\pi$
$542$	−29.7851	−1.27938
$543$	−1.91221	−0.0820609
$544$	−6.48939	−0.278230
$545$	0	0
$546$	−2.51972	−0.107834
$547$	8.81086	0.376725	0.188363	−	0.982100i	$-0.439682\pi$
$547$	8.81086	0.376725	0.188363	+	0.982100i	$0.439682\pi$
$548$	−6.01510	−0.256953
$549$	24.4857	1.04502
$550$	0	0
$551$	2.92797	0.124736
$552$	14.4737	0.616041
$553$	2.15635	0.0916972
$554$	33.1541	1.40858
$555$	0	0
$556$	6.19847	0.262874
$557$	27.6651	1.17221	0.586105	−	0.810235i	$-0.300661\pi$
$557$	27.6651	1.17221	0.586105	+	0.810235i	$0.300661\pi$
$558$	−26.3902	−1.11718
$559$	−0.252111	−0.0106631
$560$	0	0
$561$	3.67807	0.155288
$562$	−7.05707	−0.297684
$563$	9.49474	0.400156	0.200078	−	0.979780i	$-0.435880\pi$
$563$	9.49474	0.400156	0.200078	+	0.979780i	$0.435880\pi$
$564$	0.707553	0.0297933
$565$	0	0
$566$	15.4186	0.648093
$567$	−1.42160	−0.0597017
$568$	−3.94151	−0.165382
$569$	0.862737	0.0361678	0.0180839	−	0.999836i	$-0.494243\pi$
$569$	0.862737	0.0361678	0.0180839	+	0.999836i	$0.494243\pi$
$570$	0	0
$571$	−31.5625	−1.32085	−0.660424	−	0.750892i	$-0.729624\pi$
$571$	−31.5625	−1.32085	−0.660424	+	0.750892i	$0.729624\pi$
$572$	0.868110	0.0362975
$573$	−4.88114	−0.203913
$574$	6.66827	0.278328
$575$	0	0
$576$	−14.3799	−0.599164
$577$	19.9673	0.831250	0.415625	−	0.909536i	$-0.363563\pi$
$577$	19.9673	0.831250	0.415625	+	0.909536i	$0.363563\pi$
$578$	−0.494982	−0.0205886
$579$	2.60851	0.108406
$580$	0	0
$581$	5.65528	0.234621
$582$	7.77448	0.322262
$583$	−8.01831	−0.332084
$584$	−7.90181	−0.326979
$585$	0	0
$586$	28.7848	1.18909
$587$	−33.3785	−1.37768	−0.688840	−	0.724913i	$-0.741880\pi$
$587$	−33.3785	−1.37768	−0.688840	+	0.724913i	$0.741880\pi$
$588$	−1.69151	−0.0697566
$589$	−7.98044	−0.328828
$590$	0	0
$591$	−2.76163	−0.113598
$592$	46.1026	1.89480
$593$	−30.5439	−1.25429	−0.627143	−	0.778904i	$-0.715776\pi$
$593$	−30.5439	−1.25429	−0.627143	+	0.778904i	$0.715776\pi$
$594$	−7.06190	−0.289753
$595$	0	0
$596$	3.11706	0.127680
$597$	−21.2716	−0.870587
$598$	−28.7078	−1.17395
$599$	−0.919784	−0.0375814	−0.0187907	−	0.999823i	$-0.505982\pi$
$599$	−0.919784	−0.0375814	−0.0187907	+	0.999823i	$0.505982\pi$
$600$	0	0
$601$	−21.5390	−0.878592	−0.439296	−	0.898342i	$-0.644772\pi$
$601$	−21.5390	−0.878592	−0.439296	+	0.898342i	$0.644772\pi$
$602$	0.0749360	0.00305416
$603$	−7.72463	−0.314571
$604$	4.46130	0.181528
$605$	0	0
$606$	−20.4591	−0.831094
$607$	21.9638	0.891485	0.445742	−	0.895161i	$-0.352940\pi$
$607$	21.9638	0.891485	0.445742	+	0.895161i	$0.352940\pi$
$608$	1.58930	0.0644544
$609$	1.59167	0.0644976
$610$	0	0
$611$	8.51521	0.344488
$612$	−2.52884	−0.102222
$613$	−34.6490	−1.39946	−0.699730	−	0.714408i	$-0.746696\pi$
$613$	−34.6490	−1.39946	−0.699730	+	0.714408i	$0.746696\pi$
$614$	−1.91132	−0.0771347
$615$	0	0
$616$	1.56563	0.0630812
$617$	10.0214	0.403448	0.201724	−	0.979442i	$-0.435346\pi$
$617$	10.0214	0.403448	0.201724	+	0.979442i	$0.435346\pi$
$618$	−21.7373	−0.874402
$619$	−41.5032	−1.66815	−0.834077	−	0.551648i	$-0.813999\pi$
$619$	−41.5032	−1.66815	−0.834077	+	0.551648i	$0.813999\pi$
$620$	0	0
$621$	28.9469	1.16160
$622$	51.8143	2.07756
$623$	0.379963	0.0152229
$624$	−12.3961	−0.496242
$625$	0	0
$626$	−1.14138	−0.0456186
$627$	−0.900785	−0.0359739
$628$	−6.75818	−0.269681
$629$	−41.9639	−1.67321
$630$	0	0
$631$	−38.1373	−1.51822	−0.759112	−	0.650960i	$-0.774367\pi$
$631$	−38.1373	−1.51822	−0.759112	+	0.650960i	$0.774367\pi$
$632$	9.27002	0.368741
$633$	−10.4072	−0.413650
$634$	29.3500	1.16564
$635$	0	0
$636$	−2.04391	−0.0810465
$637$	−20.3568	−0.806567
$638$	−4.42403	−0.175149
$639$	−3.32507	−0.131538
$640$	0	0
$641$	35.1370	1.38783	0.693914	−	0.720058i	$-0.255885\pi$
$641$	35.1370	1.38783	0.693914	+	0.720058i	$0.255885\pi$
$642$	−0.169481	−0.00668888
$643$	32.6958	1.28940	0.644698	−	0.764437i	$-0.276983\pi$
$643$	32.6958	1.28940	0.644698	+	0.764437i	$0.276983\pi$
$644$	1.05768	0.0416785
$645$	0	0
$646$	−6.16951	−0.242736
$647$	6.81340	0.267863	0.133931	−	0.990991i	$-0.457240\pi$
$647$	6.81340	0.267863	0.133931	+	0.990991i	$0.457240\pi$
$648$	−6.11139	−0.240078
$649$	3.21020	0.126011
$650$	0	0
$651$	−4.33823	−0.170029
$652$	5.12247	0.200611
$653$	19.1447	0.749190	0.374595	−	0.927188i	$-0.377782\pi$
$653$	19.1447	0.749190	0.374595	+	0.927188i	$0.377782\pi$
$654$	−19.5377	−0.763986
$655$	0	0
$656$	32.8054	1.28084
$657$	−6.66598	−0.260065
$658$	−2.53101	−0.0986692
$659$	−10.4111	−0.405559	−0.202779	−	0.979224i	$-0.564997\pi$
$659$	−10.4111	−0.405559	−0.202779	+	0.979224i	$0.564997\pi$
$660$	0	0
$661$	−18.8172	−0.731903	−0.365951	−	0.930634i	$-0.619256\pi$
$661$	−18.8172	−0.731903	−0.365951	+	0.930634i	$0.619256\pi$
$662$	39.7733	1.54583
$663$	11.2833	0.438208
$664$	24.3117	0.943479
$665$	0	0
$666$	33.9854	1.31691
$667$	18.1342	0.702160
$668$	−2.29585	−0.0888292
$669$	−8.64723	−0.334321
$670$	0	0
$671$	−11.1879	−0.431905
$672$	0.863953	0.0333277
$673$	48.1251	1.85508	0.927542	−	0.373718i	$-0.121917\pi$
$673$	48.1251	1.85508	0.927542	+	0.373718i	$0.121917\pi$
$674$	42.9235	1.65335
$675$	0	0
$676$	−1.01564	−0.0390632
$677$	−35.9017	−1.37981	−0.689907	−	0.723898i	$-0.742349\pi$
$677$	−35.9017	−1.37981	−0.689907	+	0.723898i	$0.742349\pi$
$678$	15.7750	0.605837
$679$	−3.44717	−0.132290
$680$	0	0
$681$	−4.76771	−0.182699
$682$	12.0581	0.461728
$683$	−10.6585	−0.407837	−0.203918	−	0.978988i	$-0.565368\pi$
$683$	−10.6585	−0.407837	−0.203918	+	0.978988i	$0.565368\pi$
$684$	0.619330	0.0236807
$685$	0	0
$686$	12.4336	0.474716
$687$	−1.30758	−0.0498872
$688$	0.368658	0.0140549
$689$	−24.5980	−0.937108
$690$	0	0
$691$	−4.88607	−0.185875	−0.0929375	−	0.995672i	$-0.529626\pi$
$691$	−4.88607	−0.185875	−0.0929375	+	0.995672i	$0.529626\pi$
$692$	5.07578	0.192952
$693$	1.32077	0.0501720
$694$	23.3182	0.885147
$695$	0	0
$696$	6.84249	0.259364
$697$	−29.8605	−1.13105
$698$	14.4355	0.546391
$699$	22.5621	0.853378
$700$	0	0
$701$	22.9415	0.866490	0.433245	−	0.901276i	$-0.357368\pi$
$701$	22.9415	0.866490	0.433245	+	0.901276i	$0.357368\pi$
$702$	−21.6640	−0.817653
$703$	10.2773	0.387614
$704$	6.57042	0.247632
$705$	0	0
$706$	9.17990	0.345490
$707$	9.07148	0.341168
$708$	0.818299	0.0307536
$709$	−35.0072	−1.31472	−0.657361	−	0.753576i	$-0.728327\pi$
$709$	−35.0072	−1.31472	−0.657361	+	0.753576i	$0.728327\pi$
$710$	0	0
$711$	7.82020	0.293281
$712$	1.63344	0.0612158
$713$	−49.4264	−1.85103
$714$	−3.35379	−0.125512
$715$	0	0
$716$	−6.40825	−0.239487
$717$	1.38631	0.0517728
$718$	30.1641	1.12571
$719$	48.2732	1.80029	0.900144	−	0.435592i	$-0.143461\pi$
$719$	48.2732	1.80029	0.900144	+	0.435592i	$0.143461\pi$
$720$	0	0
$721$	9.63822	0.358946
$722$	1.51095	0.0562319
$723$	−4.34486	−0.161587
$724$	−0.600722	−0.0223257
$725$	0	0
$726$	1.36104	0.0505131
$727$	14.0424	0.520803	0.260402	−	0.965500i	$-0.416145\pi$
$727$	14.0424	0.520803	0.260402	+	0.965500i	$0.416145\pi$
$728$	4.80293	0.178009
$729$	7.47469	0.276840
$730$	0	0
$731$	−0.335563	−0.0124112
$732$	−2.85186	−0.105408
$733$	−41.9169	−1.54824	−0.774119	−	0.633040i	$-0.781807\pi$
$733$	−41.9169	−1.54824	−0.774119	+	0.633040i	$0.781807\pi$
$734$	0.292286	0.0107885
$735$	0	0
$736$	9.84320	0.362825
$737$	3.52951	0.130011
$738$	24.1831	0.890193
$739$	−43.9013	−1.61493	−0.807467	−	0.589912i	$-0.799162\pi$
$739$	−43.9013	−1.61493	−0.807467	+	0.589912i	$0.799162\pi$
$740$	0	0
$741$	−2.76336	−0.101515
$742$	7.31136	0.268409
$743$	16.7953	0.616161	0.308080	−	0.951360i	$-0.400313\pi$
$743$	16.7953	0.616161	0.308080	+	0.951360i	$0.400313\pi$
$744$	−18.6498	−0.683735
$745$	0	0
$746$	51.2090	1.87489
$747$	20.5094	0.750401
$748$	1.15547	0.0422481
$749$	0.0751472	0.00274582
$750$	0	0
$751$	−31.0266	−1.13218	−0.566089	−	0.824344i	$-0.691544\pi$
$751$	−31.0266	−1.13218	−0.566089	+	0.824344i	$0.691544\pi$
$752$	−12.4517	−0.454065
$753$	−10.3098	−0.375709
$754$	−13.5717	−0.494252
$755$	0	0
$756$	0.798166	0.0290290
$757$	37.0243	1.34567	0.672835	−	0.739793i	$-0.265076\pi$
$757$	37.0243	1.34567	0.672835	+	0.739793i	$0.265076\pi$
$758$	−14.6675	−0.532746
$759$	−5.57896	−0.202503
$760$	0	0
$761$	−5.37641	−0.194895	−0.0974473	−	0.995241i	$-0.531068\pi$
$761$	−5.37641	−0.194895	−0.0974473	+	0.995241i	$0.531068\pi$
$762$	−25.5876	−0.926942
$763$	8.66295	0.313620
$764$	−1.53341	−0.0554769
$765$	0	0
$766$	−18.5339	−0.669658
$767$	9.84801	0.355591
$768$	6.00104	0.216544
$769$	−45.0438	−1.62432	−0.812161	−	0.583433i	$-0.801709\pi$
$769$	−45.0438	−1.62432	−0.812161	+	0.583433i	$0.801709\pi$
$770$	0	0
$771$	−3.73615	−0.134554
$772$	0.819466	0.0294932
$773$	8.86055	0.318692	0.159346	−	0.987223i	$-0.449061\pi$
$773$	8.86055	0.318692	0.159346	+	0.987223i	$0.449061\pi$
$774$	0.271763	0.00976831
$775$	0	0
$776$	−14.8192	−0.531978
$777$	5.58679	0.200425
$778$	33.3937	1.19722
$779$	7.31303	0.262017
$780$	0	0
$781$	1.51928	0.0543640
$782$	−38.2104	−1.36640
$783$	13.6848	0.489054
$784$	29.7675	1.06312
$785$	0	0
$786$	22.0793	0.787541
$787$	7.51142	0.267753	0.133877	−	0.990998i	$-0.457257\pi$
$787$	7.51142	0.267753	0.133877	+	0.990998i	$0.457257\pi$
$788$	−0.867568	−0.0309058
$789$	3.99436	0.142203
$790$	0	0
$791$	−6.99459	−0.248699
$792$	5.67793	0.201756
$793$	−34.3214	−1.21879
$794$	−2.66548	−0.0945945
$795$	0	0
$796$	−6.68247	−0.236854
$797$	−20.2219	−0.716295	−0.358148	−	0.933665i	$-0.616592\pi$
$797$	−20.2219	−0.716295	−0.358148	+	0.933665i	$0.616592\pi$
$798$	0.821366	0.0290760
$799$	11.3339	0.400963
$800$	0	0
$801$	1.37797	0.0486883
$802$	−57.0244	−2.01360
$803$	3.04579	0.107484
$804$	0.899692	0.0317297
$805$	0	0
$806$	36.9909	1.30295
$807$	4.19108	0.147533
$808$	38.9978	1.37194
$809$	−27.4034	−0.963453	−0.481727	−	0.876322i	$-0.659990\pi$
$809$	−27.4034	−0.963453	−0.481727	+	0.876322i	$0.659990\pi$
$810$	0	0
$811$	−27.4966	−0.965535	−0.482768	−	0.875748i	$-0.660368\pi$
$811$	−27.4966	−0.965535	−0.482768	+	0.875748i	$0.660368\pi$
$812$	0.500023	0.0175474
$813$	−17.7570	−0.622764
$814$	−15.5285	−0.544272
$815$	0	0
$816$	−16.4994	−0.577595
$817$	0.0821817	0.00287517
$818$	−22.9924	−0.803911
$819$	4.05176	0.141580
$820$	0	0
$821$	24.3693	0.850495	0.425247	−	0.905077i	$-0.360187\pi$
$821$	24.3693	0.850495	0.425247	+	0.905077i	$0.360187\pi$
$822$	−28.9306	−1.00907
$823$	−38.8757	−1.35512	−0.677561	−	0.735467i	$-0.736963\pi$
$823$	−38.8757	−1.35512	−0.677561	+	0.735467i	$0.736963\pi$
$824$	41.4342	1.44343
$825$	0	0
$826$	−2.92717	−0.101849
$827$	33.7426	1.17335	0.586673	−	0.809824i	$-0.300437\pi$
$827$	33.7426	1.17335	0.586673	+	0.809824i	$0.300437\pi$
$828$	3.83578	0.133303
$829$	0.348062	0.0120887	0.00604435	−	0.999982i	$-0.498076\pi$
$829$	0.348062	0.0120887	0.00604435	+	0.999982i	$0.498076\pi$
$830$	0	0
$831$	19.7655	0.685656
$832$	20.1562	0.698791
$833$	−27.0952	−0.938794
$834$	29.8125	1.03232
$835$	0	0
$836$	−0.282982	−0.00978713
$837$	−37.2990	−1.28924
$838$	−35.7919	−1.23641
$839$	16.5097	0.569979	0.284989	−	0.958531i	$-0.408010\pi$
$839$	16.5097	0.569979	0.284989	+	0.958531i	$0.408010\pi$
$840$	0	0
$841$	−20.4270	−0.704378
$842$	−21.4921	−0.740666
$843$	−4.20721	−0.144904
$844$	−3.26944	−0.112539
$845$	0	0
$846$	−9.17897	−0.315579
$847$	−0.603482	−0.0207359
$848$	35.9692	1.23519
$849$	9.19212	0.315473
$850$	0	0
$851$	63.6515	2.18195
$852$	0.387273	0.0132677
$853$	7.46786	0.255695	0.127847	−	0.991794i	$-0.459193\pi$
$853$	7.46786	0.255695	0.127847	+	0.991794i	$0.459193\pi$
$854$	10.2015	0.349088
$855$	0	0
$856$	0.323054	0.0110418
$857$	−11.9526	−0.408294	−0.204147	−	0.978940i	$-0.565442\pi$
$857$	−11.9526	−0.408294	−0.204147	+	0.978940i	$0.565442\pi$
$858$	4.17531	0.142543
$859$	33.9395	1.15800	0.579001	−	0.815327i	$-0.303443\pi$
$859$	33.9395	1.15800	0.579001	+	0.815327i	$0.303443\pi$
$860$	0	0
$861$	3.97542	0.135482
$862$	−36.1832	−1.23240
$863$	42.3256	1.44078	0.720391	−	0.693569i	$-0.243963\pi$
$863$	42.3256	1.44078	0.720391	+	0.693569i	$0.243963\pi$
$864$	7.42805	0.252707
$865$	0	0
$866$	39.9550	1.35773
$867$	−0.295094	−0.0100219
$868$	−1.36286	−0.0462583
$869$	−3.57318	−0.121212
$870$	0	0
$871$	10.8276	0.366878
$872$	37.2416	1.26116
$873$	−12.5015	−0.423112
$874$	9.35800	0.316539
$875$	0	0
$876$	0.776391	0.0262318
$877$	−24.5946	−0.830501	−0.415251	−	0.909707i	$-0.636306\pi$
$877$	−24.5946	−0.830501	−0.415251	+	0.909707i	$0.636306\pi$
$878$	33.8186	1.14132
$879$	17.1606	0.578814
$880$	0	0
$881$	−11.3838	−0.383529	−0.191765	−	0.981441i	$-0.561421\pi$
$881$	−11.3838	−0.383529	−0.191765	+	0.981441i	$0.561421\pi$
$882$	21.9436	0.738881
$883$	32.2722	1.08605	0.543024	−	0.839717i	$-0.317279\pi$
$883$	32.2722	1.08605	0.543024	+	0.839717i	$0.317279\pi$
$884$	3.54466	0.119220
$885$	0	0
$886$	15.6075	0.524343
$887$	−29.2459	−0.981980	−0.490990	−	0.871165i	$-0.663365\pi$
$887$	−29.2459	−0.981980	−0.490990	+	0.871165i	$0.663365\pi$
$888$	24.0173	0.805968
$889$	11.3455	0.380514
$890$	0	0
$891$	2.35567	0.0789178
$892$	−2.71653	−0.0909562
$893$	−2.77574	−0.0928866
$894$	14.9920	0.501407
$895$	0	0
$896$	−7.90935	−0.264233
$897$	−17.1147	−0.571443
$898$	−21.9015	−0.730863
$899$	−23.3665	−0.779317
$900$	0	0
$901$	−32.7402	−1.09074
$902$	−11.0497	−0.367913
$903$	0.0446746	0.00148668
$904$	−30.0693	−1.00009
$905$	0	0
$906$	21.4573	0.712871
$907$	24.0627	0.798990	0.399495	−	0.916735i	$-0.369185\pi$
$907$	24.0627	0.798990	0.399495	+	0.916735i	$0.369185\pi$
$908$	−1.49778	−0.0497054
$909$	32.8986	1.09118
$910$	0	0
$911$	−8.77047	−0.290579	−0.145289	−	0.989389i	$-0.546411\pi$
$911$	−8.77047	−0.290579	−0.145289	+	0.989389i	$0.546411\pi$
$912$	4.04082	0.133805
$913$	−9.37109	−0.310138
$914$	25.4443	0.841623
$915$	0	0
$916$	−0.410776	−0.0135724
$917$	−9.78985	−0.323289
$918$	−28.8350	−0.951698
$919$	−9.19702	−0.303382	−0.151691	−	0.988428i	$-0.548472\pi$
$919$	−9.19702	−0.303382	−0.151691	+	0.988428i	$0.548472\pi$
$920$	0	0
$921$	−1.13947	−0.0375469
$922$	16.2833	0.536263
$923$	4.66072	0.153410
$924$	−0.153831	−0.00506067
$925$	0	0
$926$	−19.7270	−0.648270
$927$	34.9540	1.14804
$928$	4.65341	0.152756
$929$	40.7749	1.33778	0.668891	−	0.743360i	$-0.266769\pi$
$929$	40.7749	1.33778	0.668891	+	0.743360i	$0.266769\pi$
$930$	0	0
$931$	6.63581	0.217480
$932$	7.08790	0.232172
$933$	30.8901	1.01130
$934$	−19.8678	−0.650093
$935$	0	0
$936$	17.4183	0.569335
$937$	18.4291	0.602052	0.301026	−	0.953616i	$-0.402671\pi$
$937$	18.4291	0.602052	0.301026	+	0.953616i	$0.402671\pi$
$938$	−3.21832	−0.105082
$939$	−0.680454	−0.0222058
$940$	0	0
$941$	−36.4393	−1.18789	−0.593943	−	0.804507i	$-0.702430\pi$
$941$	−36.4393	−1.18789	−0.593943	+	0.804507i	$0.702430\pi$
$942$	−32.5045	−1.05905
$943$	45.2928	1.47494
$944$	−14.4006	−0.468699
$945$	0	0
$946$	−0.124173	−0.00403720
$947$	35.2476	1.14539	0.572697	−	0.819767i	$-0.305897\pi$
$947$	35.2476	1.14539	0.572697	+	0.819767i	$0.305897\pi$
$948$	−0.910824	−0.0295822
$949$	9.34366	0.303308
$950$	0	0
$951$	17.4976	0.567399
$952$	6.39278	0.207191
$953$	−21.0701	−0.682527	−0.341263	−	0.939968i	$-0.610855\pi$
$953$	−21.0701	−0.682527	−0.341263	+	0.939968i	$0.610855\pi$
$954$	26.5154	0.858467
$955$	0	0
$956$	0.435510	0.0140854
$957$	−2.63747	−0.0852574
$958$	10.2177	0.330120
$959$	12.8277	0.414228
$960$	0	0
$961$	32.6875	1.05443
$962$	−47.6370	−1.53588
$963$	0.272529	0.00878212
$964$	−1.36494	−0.0439617
$965$	0	0
$966$	5.08708	0.163674
$967$	−3.18321	−0.102365	−0.0511826	−	0.998689i	$-0.516299\pi$
$967$	−3.18321	−0.102365	−0.0511826	+	0.998689i	$0.516299\pi$
$968$	−2.59434	−0.0833851
$969$	−3.67807	−0.118157
$970$	0	0
$971$	−3.50130	−0.112362	−0.0561810	−	0.998421i	$-0.517892\pi$
$971$	−3.50130	−0.112362	−0.0561810	+	0.998421i	$0.517892\pi$
$972$	4.56828	0.146528
$973$	−13.2187	−0.423773
$974$	43.0370	1.37899
$975$	0	0
$976$	50.1877	1.60647
$977$	19.0989	0.611027	0.305514	−	0.952188i	$-0.401172\pi$
$977$	19.0989	0.611027	0.305514	+	0.952188i	$0.401172\pi$
$978$	24.6373	0.787814
$979$	−0.629618	−0.0201227
$980$	0	0
$981$	31.4170	1.00307
$982$	−1.12747	−0.0359790
$983$	−48.2760	−1.53976	−0.769882	−	0.638186i	$-0.779685\pi$
$983$	−48.2760	−1.53976	−0.769882	+	0.638186i	$0.779685\pi$
$984$	17.0901	0.544813
$985$	0	0
$986$	−18.0641	−0.575280
$987$	−1.50891	−0.0480293
$988$	−0.868110	−0.0276183
$989$	0.508987	0.0161848
$990$	0	0
$991$	23.8140	0.756477	0.378238	−	0.925708i	$-0.376530\pi$
$991$	23.8140	0.756477	0.378238	+	0.925708i	$0.376530\pi$
$992$	−12.6833	−0.402695
$993$	23.7116	0.752465
$994$	−1.38533	−0.0439399
$995$	0	0
$996$	−2.38875	−0.0756903
$997$	1.80923	0.0572988	0.0286494	−	0.999590i	$-0.490879\pi$
$997$	1.80923	0.0572988	0.0286494	+	0.999590i	$0.490879\pi$
$998$	−16.3175	−0.516522
$999$	48.0338	1.51972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.ba.1.16		20
5.2	odd	4		1045.2.b.c.419.16	yes	20
5.3	odd	4		1045.2.b.c.419.5	✓	20
5.4	even	2	inner	5225.2.a.ba.1.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.c.419.5	✓	20	5.3	odd	4
1045.2.b.c.419.16	yes	20	5.2	odd	4
5225.2.a.ba.1.5		20	5.4	even	2	inner
5225.2.a.ba.1.16		20	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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.51095 q^{2} +0.900785 q^{3} +0.282982 q^{4} +1.36104 q^{6} -0.603482 q^{7} -2.59434 q^{8} -2.18859 q^{9} +O(q^{10})\)	\(q+1.51095 q^{2} +0.900785 q^{3} +0.282982 q^{4} +1.36104 q^{6} -0.603482 q^{7} -2.59434 q^{8} -2.18859 q^{9} +1.00000 q^{11} +0.254906 q^{12} +3.06772 q^{13} -0.911833 q^{14} -4.48589 q^{16} +4.08319 q^{17} -3.30685 q^{18} -1.00000 q^{19} -0.543607 q^{21} +1.51095 q^{22} -6.19344 q^{23} -2.33694 q^{24} +4.63519 q^{26} -4.67380 q^{27} -0.170774 q^{28} -2.92797 q^{29} +7.98044 q^{31} -1.58930 q^{32} +0.900785 q^{33} +6.16951 q^{34} -0.619330 q^{36} -10.2773 q^{37} -1.51095 q^{38} +2.76336 q^{39} -7.31303 q^{41} -0.821366 q^{42} -0.0821817 q^{43} +0.282982 q^{44} -9.35800 q^{46} +2.77574 q^{47} -4.04082 q^{48} -6.63581 q^{49} +3.67807 q^{51} +0.868110 q^{52} -8.01831 q^{53} -7.06190 q^{54} +1.56563 q^{56} -0.900785 q^{57} -4.42403 q^{58} +3.21020 q^{59} -11.1879 q^{61} +12.0581 q^{62} +1.32077 q^{63} +6.57042 q^{64} +1.36104 q^{66} +3.52951 q^{67} +1.15547 q^{68} -5.57896 q^{69} +1.51928 q^{71} +5.67793 q^{72} +3.04579 q^{73} -15.5285 q^{74} -0.282982 q^{76} -0.603482 q^{77} +4.17531 q^{78} -3.57318 q^{79} +2.35567 q^{81} -11.0497 q^{82} -9.37109 q^{83} -0.153831 q^{84} -0.124173 q^{86} -2.63747 q^{87} -2.59434 q^{88} -0.629618 q^{89} -1.85132 q^{91} -1.75263 q^{92} +7.18867 q^{93} +4.19402 q^{94} -1.43161 q^{96} +5.71214 q^{97} -10.0264 q^{98} -2.18859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9}+O(q^{10}) \) 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9	\( 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9} + 20 q^{11} - 24 q^{14} - 4 q^{16} - 20 q^{19} - 30 q^{21} - 38 q^{24} + 8 q^{26} - 50 q^{29} - 50 q^{31} - 28 q^{34} - 12 q^{36} - 48 q^{39} - 34 q^{41} + 12 q^{44} - 36 q^{46} + 6 q^{49} - 40 q^{51} + 6 q^{54} - 40 q^{56} - 30 q^{59} - 14 q^{61} - 36 q^{64} - 8 q^{66} + 12 q^{69} - 40 q^{71} - 50 q^{74} - 12 q^{76} - 106 q^{79} + 30 q^{84} + 56 q^{86} - 36 q^{89} - 56 q^{91} - 28 q^{94} + 66 q^{96} + 10 q^{99}+O(q^{100}) \) 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9 + 20 * q^11 - 24 * q^14 - 4 * q^16 - 20 * q^19 - 30 * q^21 - 38 * q^24 + 8 * q^26 - 50 * q^29 - 50 * q^31 - 28 * q^34 - 12 * q^36 - 48 * q^39 - 34 * q^41 + 12 * q^44 - 36 * q^46 + 6 * q^49 - 40 * q^51 + 6 * q^54 - 40 * q^56 - 30 * q^59 - 14 * q^61 - 36 * q^64 - 8 * q^66 + 12 * q^69 - 40 * q^71 - 50 * q^74 - 12 * q^76 - 106 * q^79 + 30 * q^84 + 56 * q^86 - 36 * q^89 - 56 * q^91 - 28 * q^94 + 66 * q^96 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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