LMFDB - Embedded newform 2750.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2750,2,Mod(1,2750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2750, 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("2750.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2750 = 2 \cdot 5^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2750.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:	$21.9588605559$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2750.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} +0.381966 q^{6} -0.618034 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} +0.381966 q^{6} -0.618034 q^{7} +1.00000 q^{8} -2.85410 q^{9} +1.00000 q^{11} +0.381966 q^{12} -3.23607 q^{13} -0.618034 q^{14} +1.00000 q^{16} +5.23607 q^{17} -2.85410 q^{18} -0.236068 q^{21} +1.00000 q^{22} +2.61803 q^{23} +0.381966 q^{24} -3.23607 q^{26} -2.23607 q^{27} -0.618034 q^{28} +8.09017 q^{29} +2.00000 q^{31} +1.00000 q^{32} +0.381966 q^{33} +5.23607 q^{34} -2.85410 q^{36} +9.70820 q^{37} -1.23607 q^{39} +10.0902 q^{41} -0.236068 q^{42} +12.0902 q^{43} +1.00000 q^{44} +2.61803 q^{46} +3.32624 q^{47} +0.381966 q^{48} -6.61803 q^{49} +2.00000 q^{51} -3.23607 q^{52} -0.472136 q^{53} -2.23607 q^{54} -0.618034 q^{56} +8.09017 q^{58} +2.76393 q^{59} -11.0902 q^{61} +2.00000 q^{62} +1.76393 q^{63} +1.00000 q^{64} +0.381966 q^{66} +2.47214 q^{67} +5.23607 q^{68} +1.00000 q^{69} +12.0000 q^{71} -2.85410 q^{72} -7.70820 q^{73} +9.70820 q^{74} -0.618034 q^{77} -1.23607 q^{78} -11.7082 q^{79} +7.70820 q^{81} +10.0902 q^{82} -14.6180 q^{83} -0.236068 q^{84} +12.0902 q^{86} +3.09017 q^{87} +1.00000 q^{88} -8.61803 q^{89} +2.00000 q^{91} +2.61803 q^{92} +0.763932 q^{93} +3.32624 q^{94} +0.381966 q^{96} +16.9443 q^{97} -6.61803 q^{98} -2.85410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 3 * q^6 + q^7 + 2 * q^8 + q^9	$ 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{6} + q^{7} + 2 q^{8} + q^{9} + 2 q^{11} + 3 q^{12} - 2 q^{13} + q^{14} + 2 q^{16} + 6 q^{17} + q^{18} + 4 q^{21} + 2 q^{22} + 3 q^{23} + 3 q^{24} - 2 q^{26} + q^{28} + 5 q^{29} + 4 q^{31} + 2 q^{32} + 3 q^{33} + 6 q^{34} + q^{36} + 6 q^{37} + 2 q^{39} + 9 q^{41} + 4 q^{42} + 13 q^{43} + 2 q^{44} + 3 q^{46} - 9 q^{47} + 3 q^{48} - 11 q^{49} + 4 q^{51} - 2 q^{52} + 8 q^{53} + q^{56} + 5 q^{58} + 10 q^{59} - 11 q^{61} + 4 q^{62} + 8 q^{63} + 2 q^{64} + 3 q^{66} - 4 q^{67} + 6 q^{68} + 2 q^{69} + 24 q^{71} + q^{72} - 2 q^{73} + 6 q^{74} + q^{77} + 2 q^{78} - 10 q^{79} + 2 q^{81} + 9 q^{82} - 27 q^{83} + 4 q^{84} + 13 q^{86} - 5 q^{87} + 2 q^{88} - 15 q^{89} + 4 q^{91} + 3 q^{92} + 6 q^{93} - 9 q^{94} + 3 q^{96} + 16 q^{97} - 11 q^{98} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 3 * q^6 + q^7 + 2 * q^8 + q^9 + 2 * q^11 + 3 * q^12 - 2 * q^13 + q^14 + 2 * q^16 + 6 * q^17 + q^18 + 4 * q^21 + 2 * q^22 + 3 * q^23 + 3 * q^24 - 2 * q^26 + q^28 + 5 * q^29 + 4 * q^31 + 2 * q^32 + 3 * q^33 + 6 * q^34 + q^36 + 6 * q^37 + 2 * q^39 + 9 * q^41 + 4 * q^42 + 13 * q^43 + 2 * q^44 + 3 * q^46 - 9 * q^47 + 3 * q^48 - 11 * q^49 + 4 * q^51 - 2 * q^52 + 8 * q^53 + q^56 + 5 * q^58 + 10 * q^59 - 11 * q^61 + 4 * q^62 + 8 * q^63 + 2 * q^64 + 3 * q^66 - 4 * q^67 + 6 * q^68 + 2 * q^69 + 24 * q^71 + q^72 - 2 * q^73 + 6 * q^74 + q^77 + 2 * q^78 - 10 * q^79 + 2 * q^81 + 9 * q^82 - 27 * q^83 + 4 * q^84 + 13 * q^86 - 5 * q^87 + 2 * q^88 - 15 * q^89 + 4 * q^91 + 3 * q^92 + 6 * q^93 - 9 * q^94 + 3 * q^96 + 16 * q^97 - 11 * q^98 + 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.381966	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$3$	0.381966	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.381966	0.155937
$7$	−0.618034	−0.233595	−0.116797	−	0.993156i	$-0.537263\pi$
$7$	−0.618034	−0.233595	−0.116797	+	0.993156i	$0.537263\pi$
$8$	1.00000	0.353553
$9$	−2.85410	−0.951367
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.381966	0.110264
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.00000	0.250000
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	−2.85410	−0.672718
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−0.236068	−0.0515143
$22$	1.00000	0.213201
$23$	2.61803	0.545898	0.272949	−	0.962029i	$-0.412001\pi$
$23$	2.61803	0.545898	0.272949	+	0.962029i	$0.412001\pi$
$24$	0.381966	0.0779685
$25$	0	0
$26$	−3.23607	−0.634645
$27$	−2.23607	−0.430331
$28$	−0.618034	−0.116797
$29$	8.09017	1.50231	0.751153	−	0.660128i	$-0.229498\pi$
$29$	8.09017	1.50231	0.751153	+	0.660128i	$0.229498\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0.381966	0.0664917
$34$	5.23607	0.897978
$35$	0	0
$36$	−2.85410	−0.475684
$37$	9.70820	1.59602	0.798009	−	0.602645i	$-0.205886\pi$
$37$	9.70820	1.59602	0.798009	+	0.602645i	$0.205886\pi$
$38$	0	0
$39$	−1.23607	−0.197929
$40$	0	0
$41$	10.0902	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$41$	10.0902	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$42$	−0.236068	−0.0364261
$43$	12.0902	1.84373	0.921867	−	0.387507i	$-0.126664\pi$
$43$	12.0902	1.84373	0.921867	+	0.387507i	$0.126664\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	2.61803	0.386008
$47$	3.32624	0.485182	0.242591	−	0.970129i	$-0.422003\pi$
$47$	3.32624	0.485182	0.242591	+	0.970129i	$0.422003\pi$
$48$	0.381966	0.0551320
$49$	−6.61803	−0.945433
$50$	0	0
$51$	2.00000	0.280056
$52$	−3.23607	−0.448762
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	−0.618034	−0.0825883
$57$	0	0
$58$	8.09017	1.06229
$59$	2.76393	0.359833	0.179917	−	0.983682i	$-0.442417\pi$
$59$	2.76393	0.359833	0.179917	+	0.983682i	$0.442417\pi$
$60$	0	0
$61$	−11.0902	−1.41995	−0.709975	−	0.704226i	$-0.751294\pi$
$61$	−11.0902	−1.41995	−0.709975	+	0.704226i	$0.751294\pi$
$62$	2.00000	0.254000
$63$	1.76393	0.222235
$64$	1.00000	0.125000
$65$	0	0
$66$	0.381966	0.0470168
$67$	2.47214	0.302019	0.151010	−	0.988532i	$-0.451748\pi$
$67$	2.47214	0.302019	0.151010	+	0.988532i	$0.451748\pi$
$68$	5.23607	0.634967
$69$	1.00000	0.120386
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−2.85410	−0.336359
$73$	−7.70820	−0.902177	−0.451089	−	0.892479i	$-0.648964\pi$
$73$	−7.70820	−0.902177	−0.451089	+	0.892479i	$0.648964\pi$
$74$	9.70820	1.12856
$75$	0	0
$76$	0	0
$77$	−0.618034	−0.0704315
$78$	−1.23607	−0.139957
$79$	−11.7082	−1.31728	−0.658638	−	0.752460i	$-0.728867\pi$
$79$	−11.7082	−1.31728	−0.658638	+	0.752460i	$0.728867\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	10.0902	1.11427
$83$	−14.6180	−1.60454	−0.802269	−	0.596963i	$-0.796374\pi$
$83$	−14.6180	−1.60454	−0.802269	+	0.596963i	$0.796374\pi$
$84$	−0.236068	−0.0257571
$85$	0	0
$86$	12.0902	1.30372
$87$	3.09017	0.331301
$88$	1.00000	0.106600
$89$	−8.61803	−0.913510	−0.456755	−	0.889593i	$-0.650988\pi$
$89$	−8.61803	−0.913510	−0.456755	+	0.889593i	$0.650988\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	2.61803	0.272949
$93$	0.763932	0.0792161
$94$	3.32624	0.343075
$95$	0	0
$96$	0.381966	0.0389842
$97$	16.9443	1.72043	0.860215	−	0.509931i	$-0.170329\pi$
$97$	16.9443	1.72043	0.860215	+	0.509931i	$0.170329\pi$
$98$	−6.61803	−0.668522
$99$	−2.85410	−0.286848
$100$	0	0
$101$	−4.90983	−0.488546	−0.244273	−	0.969706i	$-0.578549\pi$
$101$	−4.90983	−0.488546	−0.244273	+	0.969706i	$0.578549\pi$
$102$	2.00000	0.198030
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−3.23607	−0.317323
$105$	0	0
$106$	−0.472136	−0.0458579
$107$	17.7984	1.72063	0.860317	−	0.509760i	$-0.170266\pi$
$107$	17.7984	1.72063	0.860317	+	0.509760i	$0.170266\pi$
$108$	−2.23607	−0.215166
$109$	−2.56231	−0.245424	−0.122712	−	0.992442i	$-0.539159\pi$
$109$	−2.56231	−0.245424	−0.122712	+	0.992442i	$0.539159\pi$
$110$	0	0
$111$	3.70820	0.351967
$112$	−0.618034	−0.0583987
$113$	−0.472136	−0.0444148	−0.0222074	−	0.999753i	$-0.507069\pi$
$113$	−0.472136	−0.0444148	−0.0222074	+	0.999753i	$0.507069\pi$
$114$	0	0
$115$	0	0
$116$	8.09017	0.751153
$117$	9.23607	0.853875
$118$	2.76393	0.254441
$119$	−3.23607	−0.296650
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.0902	−1.00406
$123$	3.85410	0.347513
$124$	2.00000	0.179605
$125$	0	0
$126$	1.76393	0.157144
$127$	−4.56231	−0.404839	−0.202420	−	0.979299i	$-0.564880\pi$
$127$	−4.56231	−0.404839	−0.202420	+	0.979299i	$0.564880\pi$
$128$	1.00000	0.0883883
$129$	4.61803	0.406595
$130$	0	0
$131$	−14.1803	−1.23894	−0.619471	−	0.785020i	$-0.712653\pi$
$131$	−14.1803	−1.23894	−0.619471	+	0.785020i	$0.712653\pi$
$132$	0.381966	0.0332459
$133$	0	0
$134$	2.47214	0.213560
$135$	0	0
$136$	5.23607	0.448989
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	1.00000	0.0851257
$139$	−14.4721	−1.22751	−0.613755	−	0.789496i	$-0.710342\pi$
$139$	−14.4721	−1.22751	−0.613755	+	0.789496i	$0.710342\pi$
$140$	0	0
$141$	1.27051	0.106996
$142$	12.0000	1.00702
$143$	−3.23607	−0.270614
$144$	−2.85410	−0.237842
$145$	0	0
$146$	−7.70820	−0.637935
$147$	−2.52786	−0.208495
$148$	9.70820	0.798009
$149$	17.5623	1.43876	0.719380	−	0.694617i	$-0.244426\pi$
$149$	17.5623	1.43876	0.719380	+	0.694617i	$0.244426\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	−14.9443	−1.20817
$154$	−0.618034	−0.0498026
$155$	0	0
$156$	−1.23607	−0.0989646
$157$	−0.291796	−0.0232879	−0.0116439	−	0.999932i	$-0.503706\pi$
$157$	−0.291796	−0.0232879	−0.0116439	+	0.999932i	$0.503706\pi$
$158$	−11.7082	−0.931455
$159$	−0.180340	−0.0143019
$160$	0	0
$161$	−1.61803	−0.127519
$162$	7.70820	0.605614
$163$	−22.5066	−1.76285	−0.881426	−	0.472323i	$-0.843416\pi$
$163$	−22.5066	−1.76285	−0.881426	+	0.472323i	$0.843416\pi$
$164$	10.0902	0.787910
$165$	0	0
$166$	−14.6180	−1.13458
$167$	11.6180	0.899030	0.449515	−	0.893273i	$-0.351597\pi$
$167$	11.6180	0.899030	0.449515	+	0.893273i	$0.351597\pi$
$168$	−0.236068	−0.0182130
$169$	−2.52786	−0.194451
$170$	0	0
$171$	0	0
$172$	12.0902	0.921867
$173$	16.7639	1.27454	0.637269	−	0.770641i	$-0.280064\pi$
$173$	16.7639	1.27454	0.637269	+	0.770641i	$0.280064\pi$
$174$	3.09017	0.234265
$175$	0	0
$176$	1.00000	0.0753778
$177$	1.05573	0.0793534
$178$	−8.61803	−0.645949
$179$	−14.4721	−1.08170	−0.540849	−	0.841120i	$-0.681897\pi$
$179$	−14.4721	−1.08170	−0.540849	+	0.841120i	$0.681897\pi$
$180$	0	0
$181$	−6.09017	−0.452679	−0.226339	−	0.974049i	$-0.572676\pi$
$181$	−6.09017	−0.452679	−0.226339	+	0.974049i	$0.572676\pi$
$182$	2.00000	0.148250
$183$	−4.23607	−0.313139
$184$	2.61803	0.193004
$185$	0	0
$186$	0.763932	0.0560142
$187$	5.23607	0.382899
$188$	3.32624	0.242591
$189$	1.38197	0.100523
$190$	0	0
$191$	8.18034	0.591909	0.295954	−	0.955202i	$-0.404362\pi$
$191$	8.18034	0.591909	0.295954	+	0.955202i	$0.404362\pi$
$192$	0.381966	0.0275660
$193$	−0.472136	−0.0339851	−0.0169925	−	0.999856i	$-0.505409\pi$
$193$	−0.472136	−0.0339851	−0.0169925	+	0.999856i	$0.505409\pi$
$194$	16.9443	1.21653
$195$	0	0
$196$	−6.61803	−0.472717
$197$	9.70820	0.691681	0.345840	−	0.938293i	$-0.387594\pi$
$197$	9.70820	0.691681	0.345840	+	0.938293i	$0.387594\pi$
$198$	−2.85410	−0.202832
$199$	6.18034	0.438113	0.219056	−	0.975712i	$-0.429702\pi$
$199$	6.18034	0.438113	0.219056	+	0.975712i	$0.429702\pi$
$200$	0	0
$201$	0.944272	0.0666038
$202$	−4.90983	−0.345454
$203$	−5.00000	−0.350931
$204$	2.00000	0.140028
$205$	0	0
$206$	4.00000	0.278693
$207$	−7.47214	−0.519349
$208$	−3.23607	−0.224381
$209$	0	0
$210$	0	0
$211$	24.3607	1.67706	0.838529	−	0.544857i	$-0.183416\pi$
$211$	24.3607	1.67706	0.838529	+	0.544857i	$0.183416\pi$
$212$	−0.472136	−0.0324264
$213$	4.58359	0.314063
$214$	17.7984	1.21667
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−1.23607	−0.0839098
$218$	−2.56231	−0.173541
$219$	−2.94427	−0.198955
$220$	0	0
$221$	−16.9443	−1.13980
$222$	3.70820	0.248878
$223$	4.85410	0.325055	0.162527	−	0.986704i	$-0.448035\pi$
$223$	4.85410	0.325055	0.162527	+	0.986704i	$0.448035\pi$
$224$	−0.618034	−0.0412941
$225$	0	0
$226$	−0.472136	−0.0314060
$227$	−7.85410	−0.521295	−0.260648	−	0.965434i	$-0.583936\pi$
$227$	−7.85410	−0.521295	−0.260648	+	0.965434i	$0.583936\pi$
$228$	0	0
$229$	−24.7984	−1.63872	−0.819361	−	0.573277i	$-0.805672\pi$
$229$	−24.7984	−1.63872	−0.819361	+	0.573277i	$0.805672\pi$
$230$	0	0
$231$	−0.236068	−0.0155321
$232$	8.09017	0.531146
$233$	11.2361	0.736099	0.368050	−	0.929806i	$-0.380026\pi$
$233$	11.2361	0.736099	0.368050	+	0.929806i	$0.380026\pi$
$234$	9.23607	0.603781
$235$	0	0
$236$	2.76393	0.179917
$237$	−4.47214	−0.290496
$238$	−3.23607	−0.209763
$239$	−12.7639	−0.825630	−0.412815	−	0.910815i	$-0.635454\pi$
$239$	−12.7639	−0.825630	−0.412815	+	0.910815i	$0.635454\pi$
$240$	0	0
$241$	5.09017	0.327887	0.163943	−	0.986470i	$-0.447579\pi$
$241$	5.09017	0.327887	0.163943	+	0.986470i	$0.447579\pi$
$242$	1.00000	0.0642824
$243$	9.65248	0.619207
$244$	−11.0902	−0.709975
$245$	0	0
$246$	3.85410	0.245729
$247$	0	0
$248$	2.00000	0.127000
$249$	−5.58359	−0.353846
$250$	0	0
$251$	−1.81966	−0.114856	−0.0574280	−	0.998350i	$-0.518290\pi$
$251$	−1.81966	−0.114856	−0.0574280	+	0.998350i	$0.518290\pi$
$252$	1.76393	0.111117
$253$	2.61803	0.164594
$254$	−4.56231	−0.286265
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.6525	1.16351	0.581755	−	0.813364i	$-0.302366\pi$
$257$	18.6525	1.16351	0.581755	+	0.813364i	$0.302366\pi$
$258$	4.61803	0.287506
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−23.0902	−1.42925
$262$	−14.1803	−0.876064
$263$	0.381966	0.0235530	0.0117765	−	0.999931i	$-0.496251\pi$
$263$	0.381966	0.0235530	0.0117765	+	0.999931i	$0.496251\pi$
$264$	0.381966	0.0235084
$265$	0	0
$266$	0	0
$267$	−3.29180	−0.201455
$268$	2.47214	0.151010
$269$	−4.47214	−0.272671	−0.136335	−	0.990663i	$-0.543533\pi$
$269$	−4.47214	−0.272671	−0.136335	+	0.990663i	$0.543533\pi$
$270$	0	0
$271$	5.81966	0.353519	0.176760	−	0.984254i	$-0.443438\pi$
$271$	5.81966	0.353519	0.176760	+	0.984254i	$0.443438\pi$
$272$	5.23607	0.317483
$273$	0.763932	0.0462353
$274$	−12.0000	−0.724947
$275$	0	0
$276$	1.00000	0.0601929
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−14.4721	−0.867981
$279$	−5.70820	−0.341741
$280$	0	0
$281$	−27.2705	−1.62682	−0.813411	−	0.581689i	$-0.802392\pi$
$281$	−27.2705	−1.62682	−0.813411	+	0.581689i	$0.802392\pi$
$282$	1.27051	0.0756578
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−3.23607	−0.191353
$287$	−6.23607	−0.368103
$288$	−2.85410	−0.168180
$289$	10.4164	0.612730
$290$	0	0
$291$	6.47214	0.379403
$292$	−7.70820	−0.451089
$293$	12.9443	0.756212	0.378106	−	0.925762i	$-0.376575\pi$
$293$	12.9443	0.756212	0.378106	+	0.925762i	$0.376575\pi$
$294$	−2.52786	−0.147428
$295$	0	0
$296$	9.70820	0.564278
$297$	−2.23607	−0.129750
$298$	17.5623	1.01736
$299$	−8.47214	−0.489956
$300$	0	0
$301$	−7.47214	−0.430687
$302$	−18.0000	−1.03578
$303$	−1.87539	−0.107738
$304$	0	0
$305$	0	0
$306$	−14.9443	−0.854307
$307$	27.2705	1.55641	0.778205	−	0.628010i	$-0.216130\pi$
$307$	27.2705	1.55641	0.778205	+	0.628010i	$0.216130\pi$
$308$	−0.618034	−0.0352158
$309$	1.52786	0.0869171
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−1.23607	−0.0699786
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−0.291796	−0.0164670
$315$	0	0
$316$	−11.7082	−0.658638
$317$	9.70820	0.545267	0.272634	−	0.962118i	$-0.412105\pi$
$317$	9.70820	0.545267	0.272634	+	0.962118i	$0.412105\pi$
$318$	−0.180340	−0.0101130
$319$	8.09017	0.452963
$320$	0	0
$321$	6.79837	0.379448
$322$	−1.61803	−0.0901695
$323$	0	0
$324$	7.70820	0.428234
$325$	0	0
$326$	−22.5066	−1.24652
$327$	−0.978714	−0.0541230
$328$	10.0902	0.557136
$329$	−2.05573	−0.113336
$330$	0	0
$331$	24.3607	1.33898	0.669492	−	0.742819i	$-0.266512\pi$
$331$	24.3607	1.33898	0.669492	+	0.742819i	$0.266512\pi$
$332$	−14.6180	−0.802269
$333$	−27.7082	−1.51840
$334$	11.6180	0.635711
$335$	0	0
$336$	−0.236068	−0.0128786
$337$	−18.8328	−1.02589	−0.512944	−	0.858422i	$-0.671445\pi$
$337$	−18.8328	−1.02589	−0.512944	+	0.858422i	$0.671445\pi$
$338$	−2.52786	−0.137498
$339$	−0.180340	−0.00979472
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	8.41641	0.454443
$344$	12.0902	0.651858
$345$	0	0
$346$	16.7639	0.901235
$347$	−10.6180	−0.570006	−0.285003	−	0.958527i	$-0.591995\pi$
$347$	−10.6180	−0.570006	−0.285003	+	0.958527i	$0.591995\pi$
$348$	3.09017	0.165650
$349$	−14.1459	−0.757213	−0.378606	−	0.925558i	$-0.623597\pi$
$349$	−14.1459	−0.757213	−0.378606	+	0.925558i	$0.623597\pi$
$350$	0	0
$351$	7.23607	0.386233
$352$	1.00000	0.0533002
$353$	−18.7639	−0.998703	−0.499352	−	0.866399i	$-0.666428\pi$
$353$	−18.7639	−0.998703	−0.499352	+	0.866399i	$0.666428\pi$
$354$	1.05573	0.0561113
$355$	0	0
$356$	−8.61803	−0.456755
$357$	−1.23607	−0.0654197
$358$	−14.4721	−0.764876
$359$	8.94427	0.472061	0.236030	−	0.971746i	$-0.424154\pi$
$359$	8.94427	0.472061	0.236030	+	0.971746i	$0.424154\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−6.09017	−0.320092
$363$	0.381966	0.0200480
$364$	2.00000	0.104828
$365$	0	0
$366$	−4.23607	−0.221423
$367$	23.8541	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$367$	23.8541	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$368$	2.61803	0.136474
$369$	−28.7984	−1.49918
$370$	0	0
$371$	0.291796	0.0151493
$372$	0.763932	0.0396080
$373$	25.7082	1.33112	0.665560	−	0.746344i	$-0.268193\pi$
$373$	25.7082	1.33112	0.665560	+	0.746344i	$0.268193\pi$
$374$	5.23607	0.270751
$375$	0	0
$376$	3.32624	0.171538
$377$	−26.1803	−1.34836
$378$	1.38197	0.0710807
$379$	7.23607	0.371692	0.185846	−	0.982579i	$-0.440497\pi$
$379$	7.23607	0.371692	0.185846	+	0.982579i	$0.440497\pi$
$380$	0	0
$381$	−1.74265	−0.0892784
$382$	8.18034	0.418543
$383$	−16.8541	−0.861204	−0.430602	−	0.902542i	$-0.641699\pi$
$383$	−16.8541	−0.861204	−0.430602	+	0.902542i	$0.641699\pi$
$384$	0.381966	0.0194921
$385$	0	0
$386$	−0.472136	−0.0240311
$387$	−34.5066	−1.75407
$388$	16.9443	0.860215
$389$	−32.5623	−1.65097	−0.825487	−	0.564421i	$-0.809100\pi$
$389$	−32.5623	−1.65097	−0.825487	+	0.564421i	$0.809100\pi$
$390$	0	0
$391$	13.7082	0.693254
$392$	−6.61803	−0.334261
$393$	−5.41641	−0.273222
$394$	9.70820	0.489092
$395$	0	0
$396$	−2.85410	−0.143424
$397$	−18.8328	−0.945192	−0.472596	−	0.881279i	$-0.656683\pi$
$397$	−18.8328	−0.945192	−0.472596	+	0.881279i	$0.656683\pi$
$398$	6.18034	0.309792
$399$	0	0
$400$	0	0
$401$	2.72949	0.136304	0.0681521	−	0.997675i	$-0.478290\pi$
$401$	2.72949	0.136304	0.0681521	+	0.997675i	$0.478290\pi$
$402$	0.944272	0.0470960
$403$	−6.47214	−0.322400
$404$	−4.90983	−0.244273
$405$	0	0
$406$	−5.00000	−0.248146
$407$	9.70820	0.481218
$408$	2.00000	0.0990148
$409$	10.3262	0.510600	0.255300	−	0.966862i	$-0.417826\pi$
$409$	10.3262	0.510600	0.255300	+	0.966862i	$0.417826\pi$
$410$	0	0
$411$	−4.58359	−0.226092
$412$	4.00000	0.197066
$413$	−1.70820	−0.0840552
$414$	−7.47214	−0.367235
$415$	0	0
$416$	−3.23607	−0.158661
$417$	−5.52786	−0.270701
$418$	0	0
$419$	29.5967	1.44590	0.722948	−	0.690903i	$-0.242787\pi$
$419$	29.5967	1.44590	0.722948	+	0.690903i	$0.242787\pi$
$420$	0	0
$421$	−22.2705	−1.08540	−0.542699	−	0.839927i	$-0.682598\pi$
$421$	−22.2705	−1.08540	−0.542699	+	0.839927i	$0.682598\pi$
$422$	24.3607	1.18586
$423$	−9.49342	−0.461586
$424$	−0.472136	−0.0229289
$425$	0	0
$426$	4.58359	0.222076
$427$	6.85410	0.331693
$428$	17.7984	0.860317
$429$	−1.23607	−0.0596779
$430$	0	0
$431$	15.8197	0.762006	0.381003	−	0.924574i	$-0.375579\pi$
$431$	15.8197	0.762006	0.381003	+	0.924574i	$0.375579\pi$
$432$	−2.23607	−0.107583
$433$	−20.4721	−0.983828	−0.491914	−	0.870644i	$-0.663703\pi$
$433$	−20.4721	−0.983828	−0.491914	+	0.870644i	$0.663703\pi$
$434$	−1.23607	−0.0593332
$435$	0	0
$436$	−2.56231	−0.122712
$437$	0	0
$438$	−2.94427	−0.140683
$439$	14.4721	0.690717	0.345359	−	0.938471i	$-0.387757\pi$
$439$	14.4721	0.690717	0.345359	+	0.938471i	$0.387757\pi$
$440$	0	0
$441$	18.8885	0.899454
$442$	−16.9443	−0.805957
$443$	10.5066	0.499183	0.249591	−	0.968351i	$-0.419704\pi$
$443$	10.5066	0.499183	0.249591	+	0.968351i	$0.419704\pi$
$444$	3.70820	0.175984
$445$	0	0
$446$	4.85410	0.229848
$447$	6.70820	0.317287
$448$	−0.618034	−0.0291994
$449$	7.88854	0.372283	0.186142	−	0.982523i	$-0.440402\pi$
$449$	7.88854	0.372283	0.186142	+	0.982523i	$0.440402\pi$
$450$	0	0
$451$	10.0902	0.475128
$452$	−0.472136	−0.0222074
$453$	−6.87539	−0.323034
$454$	−7.85410	−0.368611
$455$	0	0
$456$	0	0
$457$	−25.4164	−1.18893	−0.594465	−	0.804122i	$-0.702636\pi$
$457$	−25.4164	−1.18893	−0.594465	+	0.804122i	$0.702636\pi$
$458$	−24.7984	−1.15875
$459$	−11.7082	−0.546492
$460$	0	0
$461$	2.72949	0.127125	0.0635625	−	0.997978i	$-0.479754\pi$
$461$	2.72949	0.127125	0.0635625	+	0.997978i	$0.479754\pi$
$462$	−0.236068	−0.0109829
$463$	11.5623	0.537346	0.268673	−	0.963231i	$-0.413415\pi$
$463$	11.5623	0.537346	0.268673	+	0.963231i	$0.413415\pi$
$464$	8.09017	0.375577
$465$	0	0
$466$	11.2361	0.520501
$467$	−30.2148	−1.39817	−0.699087	−	0.715037i	$-0.746410\pi$
$467$	−30.2148	−1.39817	−0.699087	+	0.715037i	$0.746410\pi$
$468$	9.23607	0.426937
$469$	−1.52786	−0.0705502
$470$	0	0
$471$	−0.111456	−0.00513563
$472$	2.76393	0.127220
$473$	12.0902	0.555907
$474$	−4.47214	−0.205412
$475$	0	0
$476$	−3.23607	−0.148325
$477$	1.34752	0.0616989
$478$	−12.7639	−0.583809
$479$	17.2361	0.787536	0.393768	−	0.919210i	$-0.371171\pi$
$479$	17.2361	0.787536	0.393768	+	0.919210i	$0.371171\pi$
$480$	0	0
$481$	−31.4164	−1.43246
$482$	5.09017	0.231851
$483$	−0.618034	−0.0281215
$484$	1.00000	0.0454545
$485$	0	0
$486$	9.65248	0.437845
$487$	−39.6869	−1.79839	−0.899193	−	0.437552i	$-0.855845\pi$
$487$	−39.6869	−1.79839	−0.899193	+	0.437552i	$0.855845\pi$
$488$	−11.0902	−0.502028
$489$	−8.59675	−0.388758
$490$	0	0
$491$	−10.3607	−0.467571	−0.233785	−	0.972288i	$-0.575111\pi$
$491$	−10.3607	−0.467571	−0.233785	+	0.972288i	$0.575111\pi$
$492$	3.85410	0.173756
$493$	42.3607	1.90783
$494$	0	0
$495$	0	0
$496$	2.00000	0.0898027
$497$	−7.41641	−0.332671
$498$	−5.58359	−0.250207
$499$	27.2361	1.21925	0.609627	−	0.792688i	$-0.291319\pi$
$499$	27.2361	1.21925	0.609627	+	0.792688i	$0.291319\pi$
$500$	0	0
$501$	4.43769	0.198262
$502$	−1.81966	−0.0812154
$503$	−18.4377	−0.822096	−0.411048	−	0.911614i	$-0.634837\pi$
$503$	−18.4377	−0.822096	−0.411048	+	0.911614i	$0.634837\pi$
$504$	1.76393	0.0785718
$505$	0	0
$506$	2.61803	0.116386
$507$	−0.965558	−0.0428819
$508$	−4.56231	−0.202420
$509$	4.47214	0.198224	0.0991120	−	0.995076i	$-0.468400\pi$
$509$	4.47214	0.198224	0.0991120	+	0.995076i	$0.468400\pi$
$510$	0	0
$511$	4.76393	0.210744
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.6525	0.822725
$515$	0	0
$516$	4.61803	0.203298
$517$	3.32624	0.146288
$518$	−6.00000	−0.263625
$519$	6.40325	0.281072
$520$	0	0
$521$	3.90983	0.171293	0.0856464	−	0.996326i	$-0.472704\pi$
$521$	3.90983	0.171293	0.0856464	+	0.996326i	$0.472704\pi$
$522$	−23.0902	−1.01063
$523$	15.5066	0.678055	0.339028	−	0.940776i	$-0.389902\pi$
$523$	15.5066	0.678055	0.339028	+	0.940776i	$0.389902\pi$
$524$	−14.1803	−0.619471
$525$	0	0
$526$	0.381966	0.0166545
$527$	10.4721	0.456173
$528$	0.381966	0.0166229
$529$	−16.1459	−0.701996
$530$	0	0
$531$	−7.88854	−0.342334
$532$	0	0
$533$	−32.6525	−1.41434
$534$	−3.29180	−0.142450
$535$	0	0
$536$	2.47214	0.106780
$537$	−5.52786	−0.238545
$538$	−4.47214	−0.192807
$539$	−6.61803	−0.285059
$540$	0	0
$541$	16.2705	0.699524	0.349762	−	0.936839i	$-0.386262\pi$
$541$	16.2705	0.699524	0.349762	+	0.936839i	$0.386262\pi$
$542$	5.81966	0.249976
$543$	−2.32624	−0.0998284
$544$	5.23607	0.224495
$545$	0	0
$546$	0.763932	0.0326933
$547$	−14.0344	−0.600069	−0.300035	−	0.953928i	$-0.596998\pi$
$547$	−14.0344	−0.600069	−0.300035	+	0.953928i	$0.596998\pi$
$548$	−12.0000	−0.512615
$549$	31.6525	1.35089
$550$	0	0
$551$	0	0
$552$	1.00000	0.0425628
$553$	7.23607	0.307709
$554$	8.00000	0.339887
$555$	0	0
$556$	−14.4721	−0.613755
$557$	18.6525	0.790331	0.395165	−	0.918610i	$-0.370687\pi$
$557$	18.6525	0.790331	0.395165	+	0.918610i	$0.370687\pi$
$558$	−5.70820	−0.241648
$559$	−39.1246	−1.65479
$560$	0	0
$561$	2.00000	0.0844401
$562$	−27.2705	−1.15034
$563$	−19.4164	−0.818304	−0.409152	−	0.912466i	$-0.634175\pi$
$563$	−19.4164	−0.818304	−0.409152	+	0.912466i	$0.634175\pi$
$564$	1.27051	0.0534981
$565$	0	0
$566$	24.0000	1.00880
$567$	−4.76393	−0.200066
$568$	12.0000	0.503509
$569$	−18.0902	−0.758379	−0.379190	−	0.925319i	$-0.623797\pi$
$569$	−18.0902	−0.758379	−0.379190	+	0.925319i	$0.623797\pi$
$570$	0	0
$571$	−4.18034	−0.174942	−0.0874709	−	0.996167i	$-0.527878\pi$
$571$	−4.18034	−0.174942	−0.0874709	+	0.996167i	$0.527878\pi$
$572$	−3.23607	−0.135307
$573$	3.12461	0.130533
$574$	−6.23607	−0.260288
$575$	0	0
$576$	−2.85410	−0.118921
$577$	−38.8328	−1.61663	−0.808316	−	0.588749i	$-0.799620\pi$
$577$	−38.8328	−1.61663	−0.808316	+	0.588749i	$0.799620\pi$
$578$	10.4164	0.433265
$579$	−0.180340	−0.00749467
$580$	0	0
$581$	9.03444	0.374812
$582$	6.47214	0.268279
$583$	−0.472136	−0.0195539
$584$	−7.70820	−0.318968
$585$	0	0
$586$	12.9443	0.534723
$587$	−38.8328	−1.60280	−0.801401	−	0.598128i	$-0.795912\pi$
$587$	−38.8328	−1.60280	−0.801401	+	0.598128i	$0.795912\pi$
$588$	−2.52786	−0.104247
$589$	0	0
$590$	0	0
$591$	3.70820	0.152535
$592$	9.70820	0.399005
$593$	−24.5410	−1.00778	−0.503889	−	0.863768i	$-0.668098\pi$
$593$	−24.5410	−1.00778	−0.503889	+	0.863768i	$0.668098\pi$
$594$	−2.23607	−0.0917470
$595$	0	0
$596$	17.5623	0.719380
$597$	2.36068	0.0966162
$598$	−8.47214	−0.346451
$599$	34.4721	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$599$	34.4721	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$600$	0	0
$601$	−22.2705	−0.908433	−0.454217	−	0.890891i	$-0.650081\pi$
$601$	−22.2705	−0.908433	−0.454217	+	0.890891i	$0.650081\pi$
$602$	−7.47214	−0.304542
$603$	−7.05573	−0.287331
$604$	−18.0000	−0.732410
$605$	0	0
$606$	−1.87539	−0.0761824
$607$	19.0557	0.773448	0.386724	−	0.922195i	$-0.373607\pi$
$607$	19.0557	0.773448	0.386724	+	0.922195i	$0.373607\pi$
$608$	0	0
$609$	−1.90983	−0.0773902
$610$	0	0
$611$	−10.7639	−0.435462
$612$	−14.9443	−0.604086
$613$	9.52786	0.384827	0.192413	−	0.981314i	$-0.438369\pi$
$613$	9.52786	0.384827	0.192413	+	0.981314i	$0.438369\pi$
$614$	27.2705	1.10055
$615$	0	0
$616$	−0.618034	−0.0249013
$617$	−37.5279	−1.51081	−0.755407	−	0.655255i	$-0.772561\pi$
$617$	−37.5279	−1.51081	−0.755407	+	0.655255i	$0.772561\pi$
$618$	1.52786	0.0614597
$619$	−28.5410	−1.14716	−0.573580	−	0.819149i	$-0.694446\pi$
$619$	−28.5410	−1.14716	−0.573580	+	0.819149i	$0.694446\pi$
$620$	0	0
$621$	−5.85410	−0.234917
$622$	−18.0000	−0.721734
$623$	5.32624	0.213391
$624$	−1.23607	−0.0494823
$625$	0	0
$626$	14.0000	0.559553
$627$	0	0
$628$	−0.291796	−0.0116439
$629$	50.8328	2.02684
$630$	0	0
$631$	30.5410	1.21582	0.607909	−	0.794006i	$-0.292008\pi$
$631$	30.5410	1.21582	0.607909	+	0.794006i	$0.292008\pi$
$632$	−11.7082	−0.465727
$633$	9.30495	0.369839
$634$	9.70820	0.385562
$635$	0	0
$636$	−0.180340	−0.00715094
$637$	21.4164	0.848549
$638$	8.09017	0.320293
$639$	−34.2492	−1.35488
$640$	0	0
$641$	−12.2705	−0.484656	−0.242328	−	0.970194i	$-0.577911\pi$
$641$	−12.2705	−0.484656	−0.242328	+	0.970194i	$0.577911\pi$
$642$	6.79837	0.268310
$643$	24.9787	0.985064	0.492532	−	0.870294i	$-0.336071\pi$
$643$	24.9787	0.985064	0.492532	+	0.870294i	$0.336071\pi$
$644$	−1.61803	−0.0637595
$645$	0	0
$646$	0	0
$647$	−23.0557	−0.906414	−0.453207	−	0.891405i	$-0.649720\pi$
$647$	−23.0557	−0.906414	−0.453207	+	0.891405i	$0.649720\pi$
$648$	7.70820	0.302807
$649$	2.76393	0.108494
$650$	0	0
$651$	−0.472136	−0.0185045
$652$	−22.5066	−0.881426
$653$	−8.36068	−0.327179	−0.163589	−	0.986529i	$-0.552307\pi$
$653$	−8.36068	−0.327179	−0.163589	+	0.986529i	$0.552307\pi$
$654$	−0.978714	−0.0382707
$655$	0	0
$656$	10.0902	0.393955
$657$	22.0000	0.858302
$658$	−2.05573	−0.0801406
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	−2.27051	−0.0883126	−0.0441563	−	0.999025i	$-0.514060\pi$
$661$	−2.27051	−0.0883126	−0.0441563	+	0.999025i	$0.514060\pi$
$662$	24.3607	0.946805
$663$	−6.47214	−0.251357
$664$	−14.6180	−0.567290
$665$	0	0
$666$	−27.7082	−1.07367
$667$	21.1803	0.820106
$668$	11.6180	0.449515
$669$	1.85410	0.0716837
$670$	0	0
$671$	−11.0902	−0.428131
$672$	−0.236068	−0.00910652
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	−18.8328	−0.725413
$675$	0	0
$676$	−2.52786	−0.0972255
$677$	−20.2918	−0.779877	−0.389939	−	0.920841i	$-0.627504\pi$
$677$	−20.2918	−0.779877	−0.389939	+	0.920841i	$0.627504\pi$
$678$	−0.180340	−0.00692591
$679$	−10.4721	−0.401884
$680$	0	0
$681$	−3.00000	−0.114960
$682$	2.00000	0.0765840
$683$	2.49342	0.0954081	0.0477041	−	0.998862i	$-0.484810\pi$
$683$	2.49342	0.0954081	0.0477041	+	0.998862i	$0.484810\pi$
$684$	0	0
$685$	0	0
$686$	8.41641	0.321340
$687$	−9.47214	−0.361385
$688$	12.0902	0.460933
$689$	1.52786	0.0582070
$690$	0	0
$691$	24.3607	0.926724	0.463362	−	0.886169i	$-0.346643\pi$
$691$	24.3607	0.926724	0.463362	+	0.886169i	$0.346643\pi$
$692$	16.7639	0.637269
$693$	1.76393	0.0670062
$694$	−10.6180	−0.403055
$695$	0	0
$696$	3.09017	0.117133
$697$	52.8328	2.00119
$698$	−14.1459	−0.535430
$699$	4.29180	0.162331
$700$	0	0
$701$	−50.3607	−1.90210	−0.951048	−	0.309042i	$-0.899992\pi$
$701$	−50.3607	−1.90210	−0.951048	+	0.309042i	$0.899992\pi$
$702$	7.23607	0.273108
$703$	0	0
$704$	1.00000	0.0376889
$705$	0	0
$706$	−18.7639	−0.706190
$707$	3.03444	0.114122
$708$	1.05573	0.0396767
$709$	0.729490	0.0273966	0.0136983	−	0.999906i	$-0.495640\pi$
$709$	0.729490	0.0273966	0.0136983	+	0.999906i	$0.495640\pi$
$710$	0	0
$711$	33.4164	1.25321
$712$	−8.61803	−0.322974
$713$	5.23607	0.196092
$714$	−1.23607	−0.0462587
$715$	0	0
$716$	−14.4721	−0.540849
$717$	−4.87539	−0.182075
$718$	8.94427	0.333797
$719$	8.29180	0.309232	0.154616	−	0.987975i	$-0.450586\pi$
$719$	8.29180	0.309232	0.154616	+	0.987975i	$0.450586\pi$
$720$	0	0
$721$	−2.47214	−0.0920672
$722$	−19.0000	−0.707107
$723$	1.94427	0.0723083
$724$	−6.09017	−0.226339
$725$	0	0
$726$	0.381966	0.0141761
$727$	−52.0476	−1.93034	−0.965169	−	0.261626i	$-0.915741\pi$
$727$	−52.0476	−1.93034	−0.965169	+	0.261626i	$0.915741\pi$
$728$	2.00000	0.0741249
$729$	−19.4377	−0.719915
$730$	0	0
$731$	63.3050	2.34142
$732$	−4.23607	−0.156570
$733$	−14.2918	−0.527880	−0.263940	−	0.964539i	$-0.585022\pi$
$733$	−14.2918	−0.527880	−0.263940	+	0.964539i	$0.585022\pi$
$734$	23.8541	0.880471
$735$	0	0
$736$	2.61803	0.0965020
$737$	2.47214	0.0910623
$738$	−28.7984	−1.06008
$739$	−36.1803	−1.33092	−0.665458	−	0.746436i	$-0.731764\pi$
$739$	−36.1803	−1.33092	−0.665458	+	0.746436i	$0.731764\pi$
$740$	0	0
$741$	0	0
$742$	0.291796	0.0107122
$743$	9.52786	0.349543	0.174772	−	0.984609i	$-0.444081\pi$
$743$	9.52786	0.349543	0.174772	+	0.984609i	$0.444081\pi$
$744$	0.763932	0.0280071
$745$	0	0
$746$	25.7082	0.941244
$747$	41.7214	1.52650
$748$	5.23607	0.191450
$749$	−11.0000	−0.401931
$750$	0	0
$751$	50.5410	1.84427	0.922134	−	0.386871i	$-0.126444\pi$
$751$	50.5410	1.84427	0.922134	+	0.386871i	$0.126444\pi$
$752$	3.32624	0.121295
$753$	−0.695048	−0.0253290
$754$	−26.1803	−0.953432
$755$	0	0
$756$	1.38197	0.0502616
$757$	−37.1246	−1.34932	−0.674658	−	0.738130i	$-0.735709\pi$
$757$	−37.1246	−1.34932	−0.674658	+	0.738130i	$0.735709\pi$
$758$	7.23607	0.262826
$759$	1.00000	0.0362977
$760$	0	0
$761$	−8.72949	−0.316444	−0.158222	−	0.987404i	$-0.550576\pi$
$761$	−8.72949	−0.316444	−0.158222	+	0.987404i	$0.550576\pi$
$762$	−1.74265	−0.0631294
$763$	1.58359	0.0573299
$764$	8.18034	0.295954
$765$	0	0
$766$	−16.8541	−0.608963
$767$	−8.94427	−0.322959
$768$	0.381966	0.0137830
$769$	−10.8541	−0.391409	−0.195704	−	0.980663i	$-0.562699\pi$
$769$	−10.8541	−0.391409	−0.195704	+	0.980663i	$0.562699\pi$
$770$	0	0
$771$	7.12461	0.256587
$772$	−0.472136	−0.0169925
$773$	−49.4164	−1.77738	−0.888692	−	0.458504i	$-0.848385\pi$
$773$	−49.4164	−1.77738	−0.888692	+	0.458504i	$0.848385\pi$
$774$	−34.5066	−1.24031
$775$	0	0
$776$	16.9443	0.608264
$777$	−2.29180	−0.0822177
$778$	−32.5623	−1.16742
$779$	0	0
$780$	0	0
$781$	12.0000	0.429394
$782$	13.7082	0.490204
$783$	−18.0902	−0.646490
$784$	−6.61803	−0.236358
$785$	0	0
$786$	−5.41641	−0.193197
$787$	−23.9098	−0.852293	−0.426147	−	0.904654i	$-0.640129\pi$
$787$	−23.9098	−0.852293	−0.426147	+	0.904654i	$0.640129\pi$
$788$	9.70820	0.345840
$789$	0.145898	0.00519411
$790$	0	0
$791$	0.291796	0.0103751
$792$	−2.85410	−0.101416
$793$	35.8885	1.27444
$794$	−18.8328	−0.668352
$795$	0	0
$796$	6.18034	0.219056
$797$	−39.8885	−1.41292	−0.706462	−	0.707751i	$-0.749710\pi$
$797$	−39.8885	−1.41292	−0.706462	+	0.707751i	$0.749710\pi$
$798$	0	0
$799$	17.4164	0.616148
$800$	0	0
$801$	24.5967	0.869083
$802$	2.72949	0.0963816
$803$	−7.70820	−0.272017
$804$	0.944272	0.0333019
$805$	0	0
$806$	−6.47214	−0.227971
$807$	−1.70820	−0.0601316
$808$	−4.90983	−0.172727
$809$	5.45085	0.191642	0.0958208	−	0.995399i	$-0.469452\pi$
$809$	5.45085	0.191642	0.0958208	+	0.995399i	$0.469452\pi$
$810$	0	0
$811$	24.3607	0.855419	0.427710	−	0.903916i	$-0.359321\pi$
$811$	24.3607	0.855419	0.427710	+	0.903916i	$0.359321\pi$
$812$	−5.00000	−0.175466
$813$	2.22291	0.0779609
$814$	9.70820	0.340272
$815$	0	0
$816$	2.00000	0.0700140
$817$	0	0
$818$	10.3262	0.361048
$819$	−5.70820	−0.199461
$820$	0	0
$821$	−31.0902	−1.08505	−0.542527	−	0.840038i	$-0.682532\pi$
$821$	−31.0902	−1.08505	−0.542527	+	0.840038i	$0.682532\pi$
$822$	−4.58359	−0.159871
$823$	36.3607	1.26745	0.633727	−	0.773557i	$-0.281524\pi$
$823$	36.3607	1.26745	0.633727	+	0.773557i	$0.281524\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−1.70820	−0.0594360
$827$	−56.7214	−1.97239	−0.986197	−	0.165573i	$-0.947053\pi$
$827$	−56.7214	−1.97239	−0.986197	+	0.165573i	$0.947053\pi$
$828$	−7.47214	−0.259675
$829$	34.2705	1.19026	0.595132	−	0.803628i	$-0.297100\pi$
$829$	34.2705	1.19026	0.595132	+	0.803628i	$0.297100\pi$
$830$	0	0
$831$	3.05573	0.106002
$832$	−3.23607	−0.112190
$833$	−34.6525	−1.20064
$834$	−5.52786	−0.191414
$835$	0	0
$836$	0	0
$837$	−4.47214	−0.154580
$838$	29.5967	1.02240
$839$	25.7771	0.889924	0.444962	−	0.895549i	$-0.353217\pi$
$839$	25.7771	0.889924	0.444962	+	0.895549i	$0.353217\pi$
$840$	0	0
$841$	36.4508	1.25693
$842$	−22.2705	−0.767492
$843$	−10.4164	−0.358760
$844$	24.3607	0.838529
$845$	0	0
$846$	−9.49342	−0.326391
$847$	−0.618034	−0.0212359
$848$	−0.472136	−0.0162132
$849$	9.16718	0.314617
$850$	0	0
$851$	25.4164	0.871263
$852$	4.58359	0.157031
$853$	19.1246	0.654814	0.327407	−	0.944883i	$-0.393825\pi$
$853$	19.1246	0.654814	0.327407	+	0.944883i	$0.393825\pi$
$854$	6.85410	0.234543
$855$	0	0
$856$	17.7984	0.608336
$857$	50.3607	1.72029	0.860144	−	0.510051i	$-0.170374\pi$
$857$	50.3607	1.72029	0.860144	+	0.510051i	$0.170374\pi$
$858$	−1.23607	−0.0421987
$859$	31.7082	1.08187	0.540935	−	0.841064i	$-0.318071\pi$
$859$	31.7082	1.08187	0.540935	+	0.841064i	$0.318071\pi$
$860$	0	0
$861$	−2.38197	−0.0811772
$862$	15.8197	0.538820
$863$	29.4508	1.00252	0.501259	−	0.865297i	$-0.332870\pi$
$863$	29.4508	1.00252	0.501259	+	0.865297i	$0.332870\pi$
$864$	−2.23607	−0.0760726
$865$	0	0
$866$	−20.4721	−0.695671
$867$	3.97871	0.135124
$868$	−1.23607	−0.0419549
$869$	−11.7082	−0.397174
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−2.56231	−0.0867706
$873$	−48.3607	−1.63676
$874$	0	0
$875$	0	0
$876$	−2.94427	−0.0994777
$877$	26.2918	0.887811	0.443905	−	0.896074i	$-0.353593\pi$
$877$	26.2918	0.887811	0.443905	+	0.896074i	$0.353593\pi$
$878$	14.4721	0.488411
$879$	4.94427	0.166766
$880$	0	0
$881$	−51.0902	−1.72127	−0.860636	−	0.509221i	$-0.829934\pi$
$881$	−51.0902	−1.72127	−0.860636	+	0.509221i	$0.829934\pi$
$882$	18.8885	0.636010
$883$	−25.2705	−0.850420	−0.425210	−	0.905095i	$-0.639800\pi$
$883$	−25.2705	−0.850420	−0.425210	+	0.905095i	$0.639800\pi$
$884$	−16.9443	−0.569898
$885$	0	0
$886$	10.5066	0.352975
$887$	56.6180	1.90105	0.950524	−	0.310652i	$-0.100547\pi$
$887$	56.6180	1.90105	0.950524	+	0.310652i	$0.100547\pi$
$888$	3.70820	0.124439
$889$	2.81966	0.0945684
$890$	0	0
$891$	7.70820	0.258235
$892$	4.85410	0.162527
$893$	0	0
$894$	6.70820	0.224356
$895$	0	0
$896$	−0.618034	−0.0206471
$897$	−3.23607	−0.108049
$898$	7.88854	0.263244
$899$	16.1803	0.539645
$900$	0	0
$901$	−2.47214	−0.0823588
$902$	10.0902	0.335966
$903$	−2.85410	−0.0949786
$904$	−0.472136	−0.0157030
$905$	0	0
$906$	−6.87539	−0.228419
$907$	−6.14590	−0.204071	−0.102036	−	0.994781i	$-0.532536\pi$
$907$	−6.14590	−0.204071	−0.102036	+	0.994781i	$0.532536\pi$
$908$	−7.85410	−0.260648
$909$	14.0132	0.464787
$910$	0	0
$911$	5.81966	0.192814	0.0964070	−	0.995342i	$-0.469265\pi$
$911$	5.81966	0.192814	0.0964070	+	0.995342i	$0.469265\pi$
$912$	0	0
$913$	−14.6180	−0.483786
$914$	−25.4164	−0.840700
$915$	0	0
$916$	−24.7984	−0.819361
$917$	8.76393	0.289411
$918$	−11.7082	−0.386428
$919$	21.7082	0.716088	0.358044	−	0.933705i	$-0.383444\pi$
$919$	21.7082	0.716088	0.358044	+	0.933705i	$0.383444\pi$
$920$	0	0
$921$	10.4164	0.343232
$922$	2.72949	0.0898910
$923$	−38.8328	−1.27820
$924$	−0.236068	−0.00776607
$925$	0	0
$926$	11.5623	0.379961
$927$	−11.4164	−0.374964
$928$	8.09017	0.265573
$929$	−13.4934	−0.442705	−0.221352	−	0.975194i	$-0.571047\pi$
$929$	−13.4934	−0.442705	−0.221352	+	0.975194i	$0.571047\pi$
$930$	0	0
$931$	0	0
$932$	11.2361	0.368050
$933$	−6.87539	−0.225090
$934$	−30.2148	−0.988658
$935$	0	0
$936$	9.23607	0.301890
$937$	−3.70820	−0.121142	−0.0605709	−	0.998164i	$-0.519292\pi$
$937$	−3.70820	−0.121142	−0.0605709	+	0.998164i	$0.519292\pi$
$938$	−1.52786	−0.0498865
$939$	5.34752	0.174510
$940$	0	0
$941$	−10.3607	−0.337749	−0.168874	−	0.985638i	$-0.554013\pi$
$941$	−10.3607	−0.337749	−0.168874	+	0.985638i	$0.554013\pi$
$942$	−0.111456	−0.00363144
$943$	26.4164	0.860237
$944$	2.76393	0.0899583
$945$	0	0
$946$	12.0902	0.393085
$947$	−8.38197	−0.272377	−0.136189	−	0.990683i	$-0.543485\pi$
$947$	−8.38197	−0.272377	−0.136189	+	0.990683i	$0.543485\pi$
$948$	−4.47214	−0.145248
$949$	24.9443	0.809725
$950$	0	0
$951$	3.70820	0.120247
$952$	−3.23607	−0.104882
$953$	50.1803	1.62550	0.812750	−	0.582612i	$-0.197969\pi$
$953$	50.1803	1.62550	0.812750	+	0.582612i	$0.197969\pi$
$954$	1.34752	0.0436277
$955$	0	0
$956$	−12.7639	−0.412815
$957$	3.09017	0.0998910
$958$	17.2361	0.556872
$959$	7.41641	0.239488
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−31.4164	−1.01291
$963$	−50.7984	−1.63695
$964$	5.09017	0.163943
$965$	0	0
$966$	−0.618034	−0.0198849
$967$	8.20163	0.263747	0.131873	−	0.991267i	$-0.457901\pi$
$967$	8.20163	0.263747	0.131873	+	0.991267i	$0.457901\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	9.65248	0.309603
$973$	8.94427	0.286740
$974$	−39.6869	−1.27165
$975$	0	0
$976$	−11.0902	−0.354988
$977$	20.3607	0.651396	0.325698	−	0.945474i	$-0.394401\pi$
$977$	20.3607	0.651396	0.325698	+	0.945474i	$0.394401\pi$
$978$	−8.59675	−0.274894
$979$	−8.61803	−0.275434
$980$	0	0
$981$	7.31308	0.233489
$982$	−10.3607	−0.330623
$983$	9.52786	0.303892	0.151946	−	0.988389i	$-0.451446\pi$
$983$	9.52786	0.303892	0.151946	+	0.988389i	$0.451446\pi$
$984$	3.85410	0.122864
$985$	0	0
$986$	42.3607	1.34904
$987$	−0.785218	−0.0249938
$988$	0	0
$989$	31.6525	1.00649
$990$	0	0
$991$	−36.5410	−1.16076	−0.580382	−	0.814344i	$-0.697097\pi$
$991$	−36.5410	−1.16076	−0.580382	+	0.814344i	$0.697097\pi$
$992$	2.00000	0.0635001
$993$	9.30495	0.295284
$994$	−7.41641	−0.235234
$995$	0	0
$996$	−5.58359	−0.176923
$997$	−50.5410	−1.60065	−0.800325	−	0.599566i	$-0.795340\pi$
$997$	−50.5410	−1.60065	−0.800325	+	0.599566i	$0.795340\pi$
$998$	27.2361	0.862143
$999$	−21.7082	−0.686817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2750.2.a.f.1.1	yes	2
5.2	odd	4		2750.2.b.c.749.4		4
5.3	odd	4		2750.2.b.c.749.1		4
5.4	even	2		2750.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2750.2.a.a.1.2	✓	2	5.4	even	2
2750.2.a.f.1.1	yes	2	1.1	even	1	trivial
2750.2.b.c.749.1		4	5.3	odd	4
2750.2.b.c.749.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 2750 = 2 \cdot 5^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2750.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} +0.381966 q^{6} -0.618034 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} +0.381966 q^{6} -0.618034 q^{7} +1.00000 q^{8} -2.85410 q^{9} +1.00000 q^{11} +0.381966 q^{12} -3.23607 q^{13} -0.618034 q^{14} +1.00000 q^{16} +5.23607 q^{17} -2.85410 q^{18} -0.236068 q^{21} +1.00000 q^{22} +2.61803 q^{23} +0.381966 q^{24} -3.23607 q^{26} -2.23607 q^{27} -0.618034 q^{28} +8.09017 q^{29} +2.00000 q^{31} +1.00000 q^{32} +0.381966 q^{33} +5.23607 q^{34} -2.85410 q^{36} +9.70820 q^{37} -1.23607 q^{39} +10.0902 q^{41} -0.236068 q^{42} +12.0902 q^{43} +1.00000 q^{44} +2.61803 q^{46} +3.32624 q^{47} +0.381966 q^{48} -6.61803 q^{49} +2.00000 q^{51} -3.23607 q^{52} -0.472136 q^{53} -2.23607 q^{54} -0.618034 q^{56} +8.09017 q^{58} +2.76393 q^{59} -11.0902 q^{61} +2.00000 q^{62} +1.76393 q^{63} +1.00000 q^{64} +0.381966 q^{66} +2.47214 q^{67} +5.23607 q^{68} +1.00000 q^{69} +12.0000 q^{71} -2.85410 q^{72} -7.70820 q^{73} +9.70820 q^{74} -0.618034 q^{77} -1.23607 q^{78} -11.7082 q^{79} +7.70820 q^{81} +10.0902 q^{82} -14.6180 q^{83} -0.236068 q^{84} +12.0902 q^{86} +3.09017 q^{87} +1.00000 q^{88} -8.61803 q^{89} +2.00000 q^{91} +2.61803 q^{92} +0.763932 q^{93} +3.32624 q^{94} +0.381966 q^{96} +16.9443 q^{97} -6.61803 q^{98} -2.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 3 * q^6 + q^7 + 2 * q^8 + q^9	\( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{6} + q^{7} + 2 q^{8} + q^{9} + 2 q^{11} + 3 q^{12} - 2 q^{13} + q^{14} + 2 q^{16} + 6 q^{17} + q^{18} + 4 q^{21} + 2 q^{22} + 3 q^{23} + 3 q^{24} - 2 q^{26} + q^{28} + 5 q^{29} + 4 q^{31} + 2 q^{32} + 3 q^{33} + 6 q^{34} + q^{36} + 6 q^{37} + 2 q^{39} + 9 q^{41} + 4 q^{42} + 13 q^{43} + 2 q^{44} + 3 q^{46} - 9 q^{47} + 3 q^{48} - 11 q^{49} + 4 q^{51} - 2 q^{52} + 8 q^{53} + q^{56} + 5 q^{58} + 10 q^{59} - 11 q^{61} + 4 q^{62} + 8 q^{63} + 2 q^{64} + 3 q^{66} - 4 q^{67} + 6 q^{68} + 2 q^{69} + 24 q^{71} + q^{72} - 2 q^{73} + 6 q^{74} + q^{77} + 2 q^{78} - 10 q^{79} + 2 q^{81} + 9 q^{82} - 27 q^{83} + 4 q^{84} + 13 q^{86} - 5 q^{87} + 2 q^{88} - 15 q^{89} + 4 q^{91} + 3 q^{92} + 6 q^{93} - 9 q^{94} + 3 q^{96} + 16 q^{97} - 11 q^{98} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 3 * q^6 + q^7 + 2 * q^8 + q^9 + 2 * q^11 + 3 * q^12 - 2 * q^13 + q^14 + 2 * q^16 + 6 * q^17 + q^18 + 4 * q^21 + 2 * q^22 + 3 * q^23 + 3 * q^24 - 2 * q^26 + q^28 + 5 * q^29 + 4 * q^31 + 2 * q^32 + 3 * q^33 + 6 * q^34 + q^36 + 6 * q^37 + 2 * q^39 + 9 * q^41 + 4 * q^42 + 13 * q^43 + 2 * q^44 + 3 * q^46 - 9 * q^47 + 3 * q^48 - 11 * q^49 + 4 * q^51 - 2 * q^52 + 8 * q^53 + q^56 + 5 * q^58 + 10 * q^59 - 11 * q^61 + 4 * q^62 + 8 * q^63 + 2 * q^64 + 3 * q^66 - 4 * q^67 + 6 * q^68 + 2 * q^69 + 24 * q^71 + q^72 - 2 * q^73 + 6 * q^74 + q^77 + 2 * q^78 - 10 * q^79 + 2 * q^81 + 9 * q^82 - 27 * q^83 + 4 * q^84 + 13 * q^86 - 5 * q^87 + 2 * q^88 - 15 * q^89 + 4 * q^91 + 3 * q^92 + 6 * q^93 - 9 * q^94 + 3 * q^96 + 16 * q^97 - 11 * q^98 + q^99

Introduction

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