LMFDB - Embedded newform 4029.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.00000 q^{3} -2.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.00000 q^{3} -2.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} +2.82843 q^{8} +1.00000 q^{9} +3.41421 q^{10} +5.65685 q^{11} -1.41421 q^{14} -2.41421 q^{15} -4.00000 q^{16} +1.00000 q^{17} -1.41421 q^{18} -7.82843 q^{19} +1.00000 q^{21} -8.00000 q^{22} -7.24264 q^{23} +2.82843 q^{24} +0.828427 q^{25} +1.00000 q^{27} -0.828427 q^{29} +3.41421 q^{30} +9.65685 q^{31} +5.65685 q^{33} -1.41421 q^{34} -2.41421 q^{35} -10.6569 q^{37} +11.0711 q^{38} -6.82843 q^{40} -3.65685 q^{41} -1.41421 q^{42} +10.4853 q^{43} -2.41421 q^{45} +10.2426 q^{46} +4.07107 q^{47} -4.00000 q^{48} -6.00000 q^{49} -1.17157 q^{50} +1.00000 q^{51} -9.58579 q^{53} -1.41421 q^{54} -13.6569 q^{55} +2.82843 q^{56} -7.82843 q^{57} +1.17157 q^{58} -6.41421 q^{59} +10.6569 q^{61} -13.6569 q^{62} +1.00000 q^{63} +8.00000 q^{64} -8.00000 q^{66} -10.0000 q^{67} -7.24264 q^{69} +3.41421 q^{70} +1.07107 q^{71} +2.82843 q^{72} +14.7279 q^{73} +15.0711 q^{74} +0.828427 q^{75} +5.65685 q^{77} -1.00000 q^{79} +9.65685 q^{80} +1.00000 q^{81} +5.17157 q^{82} +2.58579 q^{83} -2.41421 q^{85} -14.8284 q^{86} -0.828427 q^{87} +16.0000 q^{88} -17.6569 q^{89} +3.41421 q^{90} +9.65685 q^{93} -5.75736 q^{94} +18.8995 q^{95} +14.5858 q^{97} +8.48528 q^{98} +5.65685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{15} - 8 q^{16} + 2 q^{17} - 10 q^{19} + 2 q^{21} - 16 q^{22} - 6 q^{23} - 4 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{31} - 2 q^{35} - 10 q^{37} + 8 q^{38} - 8 q^{40} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 12 q^{46} - 6 q^{47} - 8 q^{48} - 12 q^{49} - 8 q^{50} + 2 q^{51} - 22 q^{53} - 16 q^{55} - 10 q^{57} + 8 q^{58} - 10 q^{59} + 10 q^{61} - 16 q^{62} + 2 q^{63} + 16 q^{64} - 16 q^{66} - 20 q^{67} - 6 q^{69} + 4 q^{70} - 12 q^{71} + 4 q^{73} + 16 q^{74} - 4 q^{75} - 2 q^{79} + 8 q^{80} + 2 q^{81} + 16 q^{82} + 8 q^{83} - 2 q^{85} - 24 q^{86} + 4 q^{87} + 32 q^{88} - 24 q^{89} + 4 q^{90} + 8 q^{93} - 20 q^{94} + 18 q^{95} + 32 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^10 - 2 * q^15 - 8 * q^16 + 2 * q^17 - 10 * q^19 + 2 * q^21 - 16 * q^22 - 6 * q^23 - 4 * q^25 + 2 * q^27 + 4 * q^29 + 4 * q^30 + 8 * q^31 - 2 * q^35 - 10 * q^37 + 8 * q^38 - 8 * q^40 + 4 * q^41 + 4 * q^43 - 2 * q^45 + 12 * q^46 - 6 * q^47 - 8 * q^48 - 12 * q^49 - 8 * q^50 + 2 * q^51 - 22 * q^53 - 16 * q^55 - 10 * q^57 + 8 * q^58 - 10 * q^59 + 10 * q^61 - 16 * q^62 + 2 * q^63 + 16 * q^64 - 16 * q^66 - 20 * q^67 - 6 * q^69 + 4 * q^70 - 12 * q^71 + 4 * q^73 + 16 * q^74 - 4 * q^75 - 2 * q^79 + 8 * q^80 + 2 * q^81 + 16 * q^82 + 8 * q^83 - 2 * q^85 - 24 * q^86 + 4 * q^87 + 32 * q^88 - 24 * q^89 + 4 * q^90 + 8 * q^93 - 20 * q^94 + 18 * q^95 + 32 * q^97

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.41421	−1.07967	−0.539835	−	0.841771i	$-0.681513\pi$
$5$	−2.41421	−1.07967	−0.539835	+	0.841771i	$0.681513\pi$
$6$	−1.41421	−0.577350
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	2.82843	1.00000
$9$	1.00000	0.333333
$10$	3.41421	1.07967
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−1.41421	−0.377964
$15$	−2.41421	−0.623347
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.41421	−0.333333
$19$	−7.82843	−1.79596	−0.897982	−	0.440032i	$-0.854967\pi$
$19$	−7.82843	−1.79596	−0.897982	+	0.440032i	$0.854967\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−8.00000	−1.70561
$23$	−7.24264	−1.51019	−0.755097	−	0.655613i	$-0.772410\pi$
$23$	−7.24264	−1.51019	−0.755097	+	0.655613i	$0.772410\pi$
$24$	2.82843	0.577350
$25$	0.828427	0.165685
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	3.41421	0.623347
$31$	9.65685	1.73442	0.867211	−	0.497941i	$-0.165910\pi$
$31$	9.65685	1.73442	0.867211	+	0.497941i	$0.165910\pi$
$32$	0	0
$33$	5.65685	0.984732
$34$	−1.41421	−0.242536
$35$	−2.41421	−0.408077
$36$	0	0
$37$	−10.6569	−1.75198	−0.875988	−	0.482333i	$-0.839790\pi$
$37$	−10.6569	−1.75198	−0.875988	+	0.482333i	$0.839790\pi$
$38$	11.0711	1.79596
$39$	0	0
$40$	−6.82843	−1.07967
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	−1.41421	−0.218218
$43$	10.4853	1.59899	0.799495	−	0.600672i	$-0.205100\pi$
$43$	10.4853	1.59899	0.799495	+	0.600672i	$0.205100\pi$
$44$	0	0
$45$	−2.41421	−0.359890
$46$	10.2426	1.51019
$47$	4.07107	0.593826	0.296913	−	0.954904i	$-0.404043\pi$
$47$	4.07107	0.593826	0.296913	+	0.954904i	$0.404043\pi$
$48$	−4.00000	−0.577350
$49$	−6.00000	−0.857143
$50$	−1.17157	−0.165685
$51$	1.00000	0.140028
$52$	0	0
$53$	−9.58579	−1.31671	−0.658355	−	0.752708i	$-0.728747\pi$
$53$	−9.58579	−1.31671	−0.658355	+	0.752708i	$0.728747\pi$
$54$	−1.41421	−0.192450
$55$	−13.6569	−1.84149
$56$	2.82843	0.377964
$57$	−7.82843	−1.03690
$58$	1.17157	0.153835
$59$	−6.41421	−0.835059	−0.417530	−	0.908663i	$-0.637104\pi$
$59$	−6.41421	−0.835059	−0.417530	+	0.908663i	$0.637104\pi$
$60$	0	0
$61$	10.6569	1.36447	0.682235	−	0.731133i	$-0.261008\pi$
$61$	10.6569	1.36447	0.682235	+	0.731133i	$0.261008\pi$
$62$	−13.6569	−1.73442
$63$	1.00000	0.125988
$64$	8.00000	1.00000
$65$	0	0
$66$	−8.00000	−0.984732
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	−7.24264	−0.871911
$70$	3.41421	0.408077
$71$	1.07107	0.127112	0.0635562	−	0.997978i	$-0.479756\pi$
$71$	1.07107	0.127112	0.0635562	+	0.997978i	$0.479756\pi$
$72$	2.82843	0.333333
$73$	14.7279	1.72377	0.861886	−	0.507101i	$-0.169283\pi$
$73$	14.7279	1.72377	0.861886	+	0.507101i	$0.169283\pi$
$74$	15.0711	1.75198
$75$	0.828427	0.0956585
$76$	0	0
$77$	5.65685	0.644658
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	9.65685	1.07967
$81$	1.00000	0.111111
$82$	5.17157	0.571105
$83$	2.58579	0.283827	0.141913	−	0.989879i	$-0.454675\pi$
$83$	2.58579	0.283827	0.141913	+	0.989879i	$0.454675\pi$
$84$	0	0
$85$	−2.41421	−0.261858
$86$	−14.8284	−1.59899
$87$	−0.828427	−0.0888167
$88$	16.0000	1.70561
$89$	−17.6569	−1.87162	−0.935811	−	0.352501i	$-0.885331\pi$
$89$	−17.6569	−1.87162	−0.935811	+	0.352501i	$0.885331\pi$
$90$	3.41421	0.359890
$91$	0	0
$92$	0	0
$93$	9.65685	1.00137
$94$	−5.75736	−0.593826
$95$	18.8995	1.93905
$96$	0	0
$97$	14.5858	1.48096	0.740481	−	0.672077i	$-0.234598\pi$
$97$	14.5858	1.48096	0.740481	+	0.672077i	$0.234598\pi$
$98$	8.48528	0.857143
$99$	5.65685	0.568535
$100$	0	0
$101$	−4.24264	−0.422159	−0.211079	−	0.977469i	$-0.567698\pi$
$101$	−4.24264	−0.422159	−0.211079	+	0.977469i	$0.567698\pi$
$102$	−1.41421	−0.140028
$103$	6.24264	0.615106	0.307553	−	0.951531i	$-0.400490\pi$
$103$	6.24264	0.615106	0.307553	+	0.951531i	$0.400490\pi$
$104$	0	0
$105$	−2.41421	−0.235603
$106$	13.5563	1.31671
$107$	−0.585786	−0.0566301	−0.0283151	−	0.999599i	$-0.509014\pi$
$107$	−0.585786	−0.0566301	−0.0283151	+	0.999599i	$0.509014\pi$
$108$	0	0
$109$	−8.48528	−0.812743	−0.406371	−	0.913708i	$-0.633206\pi$
$109$	−8.48528	−0.812743	−0.406371	+	0.913708i	$0.633206\pi$
$110$	19.3137	1.84149
$111$	−10.6569	−1.01150
$112$	−4.00000	−0.377964
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	11.0711	1.03690
$115$	17.4853	1.63051
$116$	0	0
$117$	0	0
$118$	9.07107	0.835059
$119$	1.00000	0.0916698
$120$	−6.82843	−0.623347
$121$	21.0000	1.90909
$122$	−15.0711	−1.36447
$123$	−3.65685	−0.329727
$124$	0	0
$125$	10.0711	0.900784
$126$	−1.41421	−0.125988
$127$	11.7574	1.04330	0.521648	−	0.853161i	$-0.325317\pi$
$127$	11.7574	1.04330	0.521648	+	0.853161i	$0.325317\pi$
$128$	−11.3137	−1.00000
$129$	10.4853	0.923178
$130$	0	0
$131$	−2.41421	−0.210931	−0.105465	−	0.994423i	$-0.533633\pi$
$131$	−2.41421	−0.210931	−0.105465	+	0.994423i	$0.533633\pi$
$132$	0	0
$133$	−7.82843	−0.678811
$134$	14.1421	1.22169
$135$	−2.41421	−0.207782
$136$	2.82843	0.242536
$137$	−12.5563	−1.07276	−0.536381	−	0.843976i	$-0.680209\pi$
$137$	−12.5563	−1.07276	−0.536381	+	0.843976i	$0.680209\pi$
$138$	10.2426	0.871911
$139$	−14.4853	−1.22863	−0.614313	−	0.789063i	$-0.710567\pi$
$139$	−14.4853	−1.22863	−0.614313	+	0.789063i	$0.710567\pi$
$140$	0	0
$141$	4.07107	0.342846
$142$	−1.51472	−0.127112
$143$	0	0
$144$	−4.00000	−0.333333
$145$	2.00000	0.166091
$146$	−20.8284	−1.72377
$147$	−6.00000	−0.494872
$148$	0	0
$149$	−1.92893	−0.158024	−0.0790121	−	0.996874i	$-0.525177\pi$
$149$	−1.92893	−0.158024	−0.0790121	+	0.996874i	$0.525177\pi$
$150$	−1.17157	−0.0956585
$151$	17.4853	1.42293	0.711466	−	0.702721i	$-0.248032\pi$
$151$	17.4853	1.42293	0.711466	+	0.702721i	$0.248032\pi$
$152$	−22.1421	−1.79596
$153$	1.00000	0.0808452
$154$	−8.00000	−0.644658
$155$	−23.3137	−1.87260
$156$	0	0
$157$	−14.7279	−1.17542	−0.587708	−	0.809073i	$-0.699970\pi$
$157$	−14.7279	−1.17542	−0.587708	+	0.809073i	$0.699970\pi$
$158$	1.41421	0.112509
$159$	−9.58579	−0.760202
$160$	0	0
$161$	−7.24264	−0.570800
$162$	−1.41421	−0.111111
$163$	−12.2426	−0.958918	−0.479459	−	0.877564i	$-0.659167\pi$
$163$	−12.2426	−0.958918	−0.479459	+	0.877564i	$0.659167\pi$
$164$	0	0
$165$	−13.6569	−1.06318
$166$	−3.65685	−0.283827
$167$	−4.07107	−0.315029	−0.157514	−	0.987517i	$-0.550348\pi$
$167$	−4.07107	−0.315029	−0.157514	+	0.987517i	$0.550348\pi$
$168$	2.82843	0.218218
$169$	−13.0000	−1.00000
$170$	3.41421	0.261858
$171$	−7.82843	−0.598655
$172$	0	0
$173$	−9.07107	−0.689661	−0.344830	−	0.938665i	$-0.612064\pi$
$173$	−9.07107	−0.689661	−0.344830	+	0.938665i	$0.612064\pi$
$174$	1.17157	0.0888167
$175$	0.828427	0.0626232
$176$	−22.6274	−1.70561
$177$	−6.41421	−0.482122
$178$	24.9706	1.87162
$179$	−1.75736	−0.131351	−0.0656756	−	0.997841i	$-0.520920\pi$
$179$	−1.75736	−0.131351	−0.0656756	+	0.997841i	$0.520920\pi$
$180$	0	0
$181$	3.41421	0.253776	0.126888	−	0.991917i	$-0.459501\pi$
$181$	3.41421	0.253776	0.126888	+	0.991917i	$0.459501\pi$
$182$	0	0
$183$	10.6569	0.787777
$184$	−20.4853	−1.51019
$185$	25.7279	1.89155
$186$	−13.6569	−1.00137
$187$	5.65685	0.413670
$188$	0	0
$189$	1.00000	0.0727393
$190$	−26.7279	−1.93905
$191$	−22.4142	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$191$	−22.4142	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$192$	8.00000	0.577350
$193$	−1.48528	−0.106913	−0.0534564	−	0.998570i	$-0.517024\pi$
$193$	−1.48528	−0.106913	−0.0534564	+	0.998570i	$0.517024\pi$
$194$	−20.6274	−1.48096
$195$	0	0
$196$	0	0
$197$	20.4853	1.45952	0.729758	−	0.683706i	$-0.239633\pi$
$197$	20.4853	1.45952	0.729758	+	0.683706i	$0.239633\pi$
$198$	−8.00000	−0.568535
$199$	3.82843	0.271390	0.135695	−	0.990751i	$-0.456673\pi$
$199$	3.82843	0.271390	0.135695	+	0.990751i	$0.456673\pi$
$200$	2.34315	0.165685
$201$	−10.0000	−0.705346
$202$	6.00000	0.422159
$203$	−0.828427	−0.0581442
$204$	0	0
$205$	8.82843	0.616604
$206$	−8.82843	−0.615106
$207$	−7.24264	−0.503398
$208$	0	0
$209$	−44.2843	−3.06321
$210$	3.41421	0.235603
$211$	−19.9706	−1.37483	−0.687415	−	0.726265i	$-0.741255\pi$
$211$	−19.9706	−1.37483	−0.687415	+	0.726265i	$0.741255\pi$
$212$	0	0
$213$	1.07107	0.0733884
$214$	0.828427	0.0566301
$215$	−25.3137	−1.72638
$216$	2.82843	0.192450
$217$	9.65685	0.655550
$218$	12.0000	0.812743
$219$	14.7279	0.995221
$220$	0	0
$221$	0	0
$222$	15.0711	1.01150
$223$	−9.82843	−0.658160	−0.329080	−	0.944302i	$-0.606739\pi$
$223$	−9.82843	−0.658160	−0.329080	+	0.944302i	$0.606739\pi$
$224$	0	0
$225$	0.828427	0.0552285
$226$	16.9706	1.12887
$227$	−13.1716	−0.874228	−0.437114	−	0.899406i	$-0.643999\pi$
$227$	−13.1716	−0.874228	−0.437114	+	0.899406i	$0.643999\pi$
$228$	0	0
$229$	−5.41421	−0.357781	−0.178891	−	0.983869i	$-0.557251\pi$
$229$	−5.41421	−0.357781	−0.178891	+	0.983869i	$0.557251\pi$
$230$	−24.7279	−1.63051
$231$	5.65685	0.372194
$232$	−2.34315	−0.153835
$233$	−12.3848	−0.811354	−0.405677	−	0.914017i	$-0.632964\pi$
$233$	−12.3848	−0.811354	−0.405677	+	0.914017i	$0.632964\pi$
$234$	0	0
$235$	−9.82843	−0.641136
$236$	0	0
$237$	−1.00000	−0.0649570
$238$	−1.41421	−0.0916698
$239$	18.7279	1.21141	0.605704	−	0.795690i	$-0.292891\pi$
$239$	18.7279	1.21141	0.605704	+	0.795690i	$0.292891\pi$
$240$	9.65685	0.623347
$241$	−5.41421	−0.348760	−0.174380	−	0.984678i	$-0.555792\pi$
$241$	−5.41421	−0.348760	−0.174380	+	0.984678i	$0.555792\pi$
$242$	−29.6985	−1.90909
$243$	1.00000	0.0641500
$244$	0	0
$245$	14.4853	0.925431
$246$	5.17157	0.329727
$247$	0	0
$248$	27.3137	1.73442
$249$	2.58579	0.163868
$250$	−14.2426	−0.900784
$251$	−18.5563	−1.17127	−0.585633	−	0.810576i	$-0.699154\pi$
$251$	−18.5563	−1.17127	−0.585633	+	0.810576i	$0.699154\pi$
$252$	0	0
$253$	−40.9706	−2.57580
$254$	−16.6274	−1.04330
$255$	−2.41421	−0.151184
$256$	0	0
$257$	−24.6274	−1.53622	−0.768108	−	0.640320i	$-0.778802\pi$
$257$	−24.6274	−1.53622	−0.768108	+	0.640320i	$0.778802\pi$
$258$	−14.8284	−0.923178
$259$	−10.6569	−0.662185
$260$	0	0
$261$	−0.828427	−0.0512784
$262$	3.41421	0.210931
$263$	12.3848	0.763678	0.381839	−	0.924229i	$-0.375291\pi$
$263$	12.3848	0.763678	0.381839	+	0.924229i	$0.375291\pi$
$264$	16.0000	0.984732
$265$	23.1421	1.42161
$266$	11.0711	0.678811
$267$	−17.6569	−1.08058
$268$	0	0
$269$	2.48528	0.151530	0.0757651	−	0.997126i	$-0.475860\pi$
$269$	2.48528	0.151530	0.0757651	+	0.997126i	$0.475860\pi$
$270$	3.41421	0.207782
$271$	−2.72792	−0.165709	−0.0828547	−	0.996562i	$-0.526404\pi$
$271$	−2.72792	−0.165709	−0.0828547	+	0.996562i	$0.526404\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	17.7574	1.07276
$275$	4.68629	0.282594
$276$	0	0
$277$	16.9706	1.01966	0.509831	−	0.860274i	$-0.329708\pi$
$277$	16.9706	1.01966	0.509831	+	0.860274i	$0.329708\pi$
$278$	20.4853	1.22863
$279$	9.65685	0.578141
$280$	−6.82843	−0.408077
$281$	−9.41421	−0.561605	−0.280802	−	0.959766i	$-0.590601\pi$
$281$	−9.41421	−0.561605	−0.280802	+	0.959766i	$0.590601\pi$
$282$	−5.75736	−0.342846
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	18.8995	1.11951
$286$	0	0
$287$	−3.65685	−0.215857
$288$	0	0
$289$	1.00000	0.0588235
$290$	−2.82843	−0.166091
$291$	14.5858	0.855034
$292$	0	0
$293$	−29.8701	−1.74503	−0.872514	−	0.488590i	$-0.837512\pi$
$293$	−29.8701	−1.74503	−0.872514	+	0.488590i	$0.837512\pi$
$294$	8.48528	0.494872
$295$	15.4853	0.901588
$296$	−30.1421	−1.75198
$297$	5.65685	0.328244
$298$	2.72792	0.158024
$299$	0	0
$300$	0	0
$301$	10.4853	0.604362
$302$	−24.7279	−1.42293
$303$	−4.24264	−0.243733
$304$	31.3137	1.79596
$305$	−25.7279	−1.47318
$306$	−1.41421	−0.0808452
$307$	−1.55635	−0.0888255	−0.0444128	−	0.999013i	$-0.514142\pi$
$307$	−1.55635	−0.0888255	−0.0444128	+	0.999013i	$0.514142\pi$
$308$	0	0
$309$	6.24264	0.355131
$310$	32.9706	1.87260
$311$	25.7990	1.46293	0.731463	−	0.681881i	$-0.238838\pi$
$311$	25.7990	1.46293	0.731463	+	0.681881i	$0.238838\pi$
$312$	0	0
$313$	−27.3137	−1.54386	−0.771931	−	0.635706i	$-0.780709\pi$
$313$	−27.3137	−1.54386	−0.771931	+	0.635706i	$0.780709\pi$
$314$	20.8284	1.17542
$315$	−2.41421	−0.136026
$316$	0	0
$317$	−2.27208	−0.127613	−0.0638063	−	0.997962i	$-0.520324\pi$
$317$	−2.27208	−0.127613	−0.0638063	+	0.997962i	$0.520324\pi$
$318$	13.5563	0.760202
$319$	−4.68629	−0.262382
$320$	−19.3137	−1.07967
$321$	−0.585786	−0.0326954
$322$	10.2426	0.570800
$323$	−7.82843	−0.435585
$324$	0	0
$325$	0	0
$326$	17.3137	0.958918
$327$	−8.48528	−0.469237
$328$	−10.3431	−0.571105
$329$	4.07107	0.224445
$330$	19.3137	1.06318
$331$	−22.2426	−1.22257	−0.611283	−	0.791412i	$-0.709346\pi$
$331$	−22.2426	−1.22257	−0.611283	+	0.791412i	$0.709346\pi$
$332$	0	0
$333$	−10.6569	−0.583992
$334$	5.75736	0.315029
$335$	24.1421	1.31903
$336$	−4.00000	−0.218218
$337$	12.7279	0.693334	0.346667	−	0.937988i	$-0.387313\pi$
$337$	12.7279	0.693334	0.346667	+	0.937988i	$0.387313\pi$
$338$	18.3848	1.00000
$339$	−12.0000	−0.651751
$340$	0	0
$341$	54.6274	2.95824
$342$	11.0711	0.598655
$343$	−13.0000	−0.701934
$344$	29.6569	1.59899
$345$	17.4853	0.941376
$346$	12.8284	0.689661
$347$	31.3848	1.68482	0.842412	−	0.538835i	$-0.181135\pi$
$347$	31.3848	1.68482	0.842412	+	0.538835i	$0.181135\pi$
$348$	0	0
$349$	2.24264	0.120046	0.0600229	−	0.998197i	$-0.480883\pi$
$349$	2.24264	0.120046	0.0600229	+	0.998197i	$0.480883\pi$
$350$	−1.17157	−0.0626232
$351$	0	0
$352$	0	0
$353$	−15.7279	−0.837113	−0.418556	−	0.908191i	$-0.637464\pi$
$353$	−15.7279	−0.837113	−0.418556	+	0.908191i	$0.637464\pi$
$354$	9.07107	0.482122
$355$	−2.58579	−0.137239
$356$	0	0
$357$	1.00000	0.0529256
$358$	2.48528	0.131351
$359$	−4.34315	−0.229222	−0.114611	−	0.993410i	$-0.536562\pi$
$359$	−4.34315	−0.229222	−0.114611	+	0.993410i	$0.536562\pi$
$360$	−6.82843	−0.359890
$361$	42.2843	2.22549
$362$	−4.82843	−0.253776
$363$	21.0000	1.10221
$364$	0	0
$365$	−35.5563	−1.86110
$366$	−15.0711	−0.787777
$367$	7.55635	0.394438	0.197219	−	0.980359i	$-0.436809\pi$
$367$	7.55635	0.394438	0.197219	+	0.980359i	$0.436809\pi$
$368$	28.9706	1.51019
$369$	−3.65685	−0.190368
$370$	−36.3848	−1.89155
$371$	−9.58579	−0.497669
$372$	0	0
$373$	5.75736	0.298105	0.149052	−	0.988829i	$-0.452378\pi$
$373$	5.75736	0.298105	0.149052	+	0.988829i	$0.452378\pi$
$374$	−8.00000	−0.413670
$375$	10.0711	0.520068
$376$	11.5147	0.593826
$377$	0	0
$378$	−1.41421	−0.0727393
$379$	−12.4558	−0.639814	−0.319907	−	0.947449i	$-0.603652\pi$
$379$	−12.4558	−0.639814	−0.319907	+	0.947449i	$0.603652\pi$
$380$	0	0
$381$	11.7574	0.602348
$382$	31.6985	1.62184
$383$	22.2426	1.13655	0.568273	−	0.822840i	$-0.307612\pi$
$383$	22.2426	1.13655	0.568273	+	0.822840i	$0.307612\pi$
$384$	−11.3137	−0.577350
$385$	−13.6569	−0.696018
$386$	2.10051	0.106913
$387$	10.4853	0.532997
$388$	0	0
$389$	14.5858	0.739529	0.369764	−	0.929126i	$-0.379438\pi$
$389$	14.5858	0.739529	0.369764	+	0.929126i	$0.379438\pi$
$390$	0	0
$391$	−7.24264	−0.366276
$392$	−16.9706	−0.857143
$393$	−2.41421	−0.121781
$394$	−28.9706	−1.45952
$395$	2.41421	0.121472
$396$	0	0
$397$	−36.1421	−1.81392	−0.906961	−	0.421215i	$-0.861604\pi$
$397$	−36.1421	−1.81392	−0.906961	+	0.421215i	$0.861604\pi$
$398$	−5.41421	−0.271390
$399$	−7.82843	−0.391912
$400$	−3.31371	−0.165685
$401$	−4.24264	−0.211867	−0.105934	−	0.994373i	$-0.533783\pi$
$401$	−4.24264	−0.211867	−0.105934	+	0.994373i	$0.533783\pi$
$402$	14.1421	0.705346
$403$	0	0
$404$	0	0
$405$	−2.41421	−0.119963
$406$	1.17157	0.0581442
$407$	−60.2843	−2.98818
$408$	2.82843	0.140028
$409$	−38.2843	−1.89304	−0.946518	−	0.322652i	$-0.895426\pi$
$409$	−38.2843	−1.89304	−0.946518	+	0.322652i	$0.895426\pi$
$410$	−12.4853	−0.616604
$411$	−12.5563	−0.619359
$412$	0	0
$413$	−6.41421	−0.315623
$414$	10.2426	0.503398
$415$	−6.24264	−0.306439
$416$	0	0
$417$	−14.4853	−0.709347
$418$	62.6274	3.06321
$419$	0.928932	0.0453813	0.0226907	−	0.999743i	$-0.492777\pi$
$419$	0.928932	0.0453813	0.0226907	+	0.999743i	$0.492777\pi$
$420$	0	0
$421$	−21.9706	−1.07078	−0.535390	−	0.844605i	$-0.679835\pi$
$421$	−21.9706	−1.07078	−0.535390	+	0.844605i	$0.679835\pi$
$422$	28.2426	1.37483
$423$	4.07107	0.197942
$424$	−27.1127	−1.31671
$425$	0.828427	0.0401846
$426$	−1.51472	−0.0733884
$427$	10.6569	0.515721
$428$	0	0
$429$	0	0
$430$	35.7990	1.72638
$431$	−14.2721	−0.687462	−0.343731	−	0.939068i	$-0.611691\pi$
$431$	−14.2721	−0.687462	−0.343731	+	0.939068i	$0.611691\pi$
$432$	−4.00000	−0.192450
$433$	−14.3431	−0.689288	−0.344644	−	0.938734i	$-0.612000\pi$
$433$	−14.3431	−0.689288	−0.344644	+	0.938734i	$0.612000\pi$
$434$	−13.6569	−0.655550
$435$	2.00000	0.0958927
$436$	0	0
$437$	56.6985	2.71226
$438$	−20.8284	−0.995221
$439$	35.0711	1.67385	0.836925	−	0.547317i	$-0.184351\pi$
$439$	35.0711	1.67385	0.836925	+	0.547317i	$0.184351\pi$
$440$	−38.6274	−1.84149
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−29.5858	−1.40566	−0.702832	−	0.711356i	$-0.748081\pi$
$443$	−29.5858	−1.40566	−0.702832	+	0.711356i	$0.748081\pi$
$444$	0	0
$445$	42.6274	2.02073
$446$	13.8995	0.658160
$447$	−1.92893	−0.0912354
$448$	8.00000	0.377964
$449$	2.48528	0.117288	0.0586438	−	0.998279i	$-0.481322\pi$
$449$	2.48528	0.117288	0.0586438	+	0.998279i	$0.481322\pi$
$450$	−1.17157	−0.0552285
$451$	−20.6863	−0.974079
$452$	0	0
$453$	17.4853	0.821530
$454$	18.6274	0.874228
$455$	0	0
$456$	−22.1421	−1.03690
$457$	−34.4558	−1.61178	−0.805888	−	0.592068i	$-0.798312\pi$
$457$	−34.4558	−1.61178	−0.805888	+	0.592068i	$0.798312\pi$
$458$	7.65685	0.357781
$459$	1.00000	0.0466760
$460$	0	0
$461$	−7.79899	−0.363235	−0.181618	−	0.983369i	$-0.558133\pi$
$461$	−7.79899	−0.363235	−0.181618	+	0.983369i	$0.558133\pi$
$462$	−8.00000	−0.372194
$463$	1.75736	0.0816714	0.0408357	−	0.999166i	$-0.486998\pi$
$463$	1.75736	0.0816714	0.0408357	+	0.999166i	$0.486998\pi$
$464$	3.31371	0.153835
$465$	−23.3137	−1.08115
$466$	17.5147	0.811354
$467$	37.4558	1.73325	0.866625	−	0.498960i	$-0.166285\pi$
$467$	37.4558	1.73325	0.866625	+	0.498960i	$0.166285\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	13.8995	0.641136
$471$	−14.7279	−0.678627
$472$	−18.1421	−0.835059
$473$	59.3137	2.72725
$474$	1.41421	0.0649570
$475$	−6.48528	−0.297565
$476$	0	0
$477$	−9.58579	−0.438903
$478$	−26.4853	−1.21141
$479$	15.1005	0.689960	0.344980	−	0.938610i	$-0.387886\pi$
$479$	15.1005	0.689960	0.344980	+	0.938610i	$0.387886\pi$
$480$	0	0
$481$	0	0
$482$	7.65685	0.348760
$483$	−7.24264	−0.329552
$484$	0	0
$485$	−35.2132	−1.59895
$486$	−1.41421	−0.0641500
$487$	−1.21320	−0.0549755	−0.0274877	−	0.999622i	$-0.508751\pi$
$487$	−1.21320	−0.0549755	−0.0274877	+	0.999622i	$0.508751\pi$
$488$	30.1421	1.36447
$489$	−12.2426	−0.553631
$490$	−20.4853	−0.925431
$491$	27.7990	1.25455	0.627275	−	0.778797i	$-0.284170\pi$
$491$	27.7990	1.25455	0.627275	+	0.778797i	$0.284170\pi$
$492$	0	0
$493$	−0.828427	−0.0373105
$494$	0	0
$495$	−13.6569	−0.613830
$496$	−38.6274	−1.73442
$497$	1.07107	0.0480440
$498$	−3.65685	−0.163868
$499$	39.9411	1.78801	0.894005	−	0.448057i	$-0.147884\pi$
$499$	39.9411	1.78801	0.894005	+	0.448057i	$0.147884\pi$
$500$	0	0
$501$	−4.07107	−0.181882
$502$	26.2426	1.17127
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	2.82843	0.125988
$505$	10.2426	0.455792
$506$	57.9411	2.57580
$507$	−13.0000	−0.577350
$508$	0	0
$509$	−11.3848	−0.504621	−0.252311	−	0.967646i	$-0.581190\pi$
$509$	−11.3848	−0.504621	−0.252311	+	0.967646i	$0.581190\pi$
$510$	3.41421	0.151184
$511$	14.7279	0.651525
$512$	22.6274	1.00000
$513$	−7.82843	−0.345634
$514$	34.8284	1.53622
$515$	−15.0711	−0.664111
$516$	0	0
$517$	23.0294	1.01283
$518$	15.0711	0.662185
$519$	−9.07107	−0.398176
$520$	0	0
$521$	−11.3137	−0.495663	−0.247831	−	0.968803i	$-0.579718\pi$
$521$	−11.3137	−0.495663	−0.247831	+	0.968803i	$0.579718\pi$
$522$	1.17157	0.0512784
$523$	23.4558	1.02565	0.512826	−	0.858492i	$-0.328599\pi$
$523$	23.4558	1.02565	0.512826	+	0.858492i	$0.328599\pi$
$524$	0	0
$525$	0.828427	0.0361555
$526$	−17.5147	−0.763678
$527$	9.65685	0.420659
$528$	−22.6274	−0.984732
$529$	29.4558	1.28069
$530$	−32.7279	−1.42161
$531$	−6.41421	−0.278353
$532$	0	0
$533$	0	0
$534$	24.9706	1.08058
$535$	1.41421	0.0611418
$536$	−28.2843	−1.22169
$537$	−1.75736	−0.0758357
$538$	−3.51472	−0.151530
$539$	−33.9411	−1.46195
$540$	0	0
$541$	−28.7279	−1.23511	−0.617555	−	0.786528i	$-0.711877\pi$
$541$	−28.7279	−1.23511	−0.617555	+	0.786528i	$0.711877\pi$
$542$	3.85786	0.165709
$543$	3.41421	0.146518
$544$	0	0
$545$	20.4853	0.877493
$546$	0	0
$547$	12.4437	0.532052	0.266026	−	0.963966i	$-0.414289\pi$
$547$	12.4437	0.532052	0.266026	+	0.963966i	$0.414289\pi$
$548$	0	0
$549$	10.6569	0.454823
$550$	−6.62742	−0.282594
$551$	6.48528	0.276282
$552$	−20.4853	−0.871911
$553$	−1.00000	−0.0425243
$554$	−24.0000	−1.01966
$555$	25.7279	1.09209
$556$	0	0
$557$	0.142136	0.00602248	0.00301124	−	0.999995i	$-0.499041\pi$
$557$	0.142136	0.00602248	0.00301124	+	0.999995i	$0.499041\pi$
$558$	−13.6569	−0.578141
$559$	0	0
$560$	9.65685	0.408077
$561$	5.65685	0.238833
$562$	13.3137	0.561605
$563$	−37.4142	−1.57682	−0.788411	−	0.615149i	$-0.789096\pi$
$563$	−37.4142	−1.57682	−0.788411	+	0.615149i	$0.789096\pi$
$564$	0	0
$565$	28.9706	1.21880
$566$	−16.9706	−0.713326
$567$	1.00000	0.0419961
$568$	3.02944	0.127112
$569$	1.61522	0.0677137	0.0338568	−	0.999427i	$-0.489221\pi$
$569$	1.61522	0.0677137	0.0338568	+	0.999427i	$0.489221\pi$
$570$	−26.7279	−1.11951
$571$	−24.6274	−1.03063	−0.515313	−	0.857002i	$-0.672324\pi$
$571$	−24.6274	−1.03063	−0.515313	+	0.857002i	$0.672324\pi$
$572$	0	0
$573$	−22.4142	−0.936367
$574$	5.17157	0.215857
$575$	−6.00000	−0.250217
$576$	8.00000	0.333333
$577$	18.7279	0.779654	0.389827	−	0.920888i	$-0.372535\pi$
$577$	18.7279	0.779654	0.389827	+	0.920888i	$0.372535\pi$
$578$	−1.41421	−0.0588235
$579$	−1.48528	−0.0617262
$580$	0	0
$581$	2.58579	0.107276
$582$	−20.6274	−0.855034
$583$	−54.2254	−2.24579
$584$	41.6569	1.72377
$585$	0	0
$586$	42.2426	1.74503
$587$	12.8995	0.532419	0.266210	−	0.963915i	$-0.414229\pi$
$587$	12.8995	0.532419	0.266210	+	0.963915i	$0.414229\pi$
$588$	0	0
$589$	−75.5980	−3.11496
$590$	−21.8995	−0.901588
$591$	20.4853	0.842652
$592$	42.6274	1.75198
$593$	−16.1005	−0.661168	−0.330584	−	0.943776i	$-0.607246\pi$
$593$	−16.1005	−0.661168	−0.330584	+	0.943776i	$0.607246\pi$
$594$	−8.00000	−0.328244
$595$	−2.41421	−0.0989731
$596$	0	0
$597$	3.82843	0.156687
$598$	0	0
$599$	−47.5563	−1.94310	−0.971550	−	0.236835i	$-0.923890\pi$
$599$	−47.5563	−1.94310	−0.971550	+	0.236835i	$0.923890\pi$
$600$	2.34315	0.0956585
$601$	−1.97056	−0.0803809	−0.0401905	−	0.999192i	$-0.512796\pi$
$601$	−1.97056	−0.0803809	−0.0401905	+	0.999192i	$0.512796\pi$
$602$	−14.8284	−0.604362
$603$	−10.0000	−0.407231
$604$	0	0
$605$	−50.6985	−2.06119
$606$	6.00000	0.243733
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	−0.828427	−0.0335696
$610$	36.3848	1.47318
$611$	0	0
$612$	0	0
$613$	18.4853	0.746613	0.373307	−	0.927708i	$-0.378224\pi$
$613$	18.4853	0.746613	0.373307	+	0.927708i	$0.378224\pi$
$614$	2.20101	0.0888255
$615$	8.82843	0.355997
$616$	16.0000	0.644658
$617$	13.3137	0.535990	0.267995	−	0.963420i	$-0.413639\pi$
$617$	13.3137	0.535990	0.267995	+	0.963420i	$0.413639\pi$
$618$	−8.82843	−0.355131
$619$	21.0000	0.844061	0.422031	−	0.906582i	$-0.361317\pi$
$619$	21.0000	0.844061	0.422031	+	0.906582i	$0.361317\pi$
$620$	0	0
$621$	−7.24264	−0.290637
$622$	−36.4853	−1.46293
$623$	−17.6569	−0.707407
$624$	0	0
$625$	−28.4558	−1.13823
$626$	38.6274	1.54386
$627$	−44.2843	−1.76854
$628$	0	0
$629$	−10.6569	−0.424917
$630$	3.41421	0.136026
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	−2.82843	−0.112509
$633$	−19.9706	−0.793759
$634$	3.21320	0.127613
$635$	−28.3848	−1.12642
$636$	0	0
$637$	0	0
$638$	6.62742	0.262382
$639$	1.07107	0.0423708
$640$	27.3137	1.07967
$641$	−24.5563	−0.969917	−0.484959	−	0.874537i	$-0.661165\pi$
$641$	−24.5563	−0.969917	−0.484959	+	0.874537i	$0.661165\pi$
$642$	0.828427	0.0326954
$643$	9.21320	0.363333	0.181667	−	0.983360i	$-0.441851\pi$
$643$	9.21320	0.363333	0.181667	+	0.983360i	$0.441851\pi$
$644$	0	0
$645$	−25.3137	−0.996726
$646$	11.0711	0.435585
$647$	−4.41421	−0.173541	−0.0867703	−	0.996228i	$-0.527655\pi$
$647$	−4.41421	−0.173541	−0.0867703	+	0.996228i	$0.527655\pi$
$648$	2.82843	0.111111
$649$	−36.2843	−1.42428
$650$	0	0
$651$	9.65685	0.378482
$652$	0	0
$653$	−17.9289	−0.701613	−0.350807	−	0.936448i	$-0.614093\pi$
$653$	−17.9289	−0.701613	−0.350807	+	0.936448i	$0.614093\pi$
$654$	12.0000	0.469237
$655$	5.82843	0.227735
$656$	14.6274	0.571105
$657$	14.7279	0.574591
$658$	−5.75736	−0.224445
$659$	−21.7279	−0.846400	−0.423200	−	0.906036i	$-0.639093\pi$
$659$	−21.7279	−0.846400	−0.423200	+	0.906036i	$0.639093\pi$
$660$	0	0
$661$	−13.6985	−0.532809	−0.266405	−	0.963861i	$-0.585836\pi$
$661$	−13.6985	−0.532809	−0.266405	+	0.963861i	$0.585836\pi$
$662$	31.4558	1.22257
$663$	0	0
$664$	7.31371	0.283827
$665$	18.8995	0.732891
$666$	15.0711	0.583992
$667$	6.00000	0.232321
$668$	0	0
$669$	−9.82843	−0.379989
$670$	−34.1421	−1.31903
$671$	60.2843	2.32725
$672$	0	0
$673$	−16.9706	−0.654167	−0.327084	−	0.944995i	$-0.606066\pi$
$673$	−16.9706	−0.654167	−0.327084	+	0.944995i	$0.606066\pi$
$674$	−18.0000	−0.693334
$675$	0.828427	0.0318862
$676$	0	0
$677$	−24.4142	−0.938314	−0.469157	−	0.883115i	$-0.655442\pi$
$677$	−24.4142	−0.938314	−0.469157	+	0.883115i	$0.655442\pi$
$678$	16.9706	0.651751
$679$	14.5858	0.559751
$680$	−6.82843	−0.261858
$681$	−13.1716	−0.504736
$682$	−77.2548	−2.95824
$683$	−27.8701	−1.06642	−0.533209	−	0.845984i	$-0.679014\pi$
$683$	−27.8701	−1.06642	−0.533209	+	0.845984i	$0.679014\pi$
$684$	0	0
$685$	30.3137	1.15823
$686$	18.3848	0.701934
$687$	−5.41421	−0.206565
$688$	−41.9411	−1.59899
$689$	0	0
$690$	−24.7279	−0.941376
$691$	−8.51472	−0.323915	−0.161958	−	0.986798i	$-0.551781\pi$
$691$	−8.51472	−0.323915	−0.161958	+	0.986798i	$0.551781\pi$
$692$	0	0
$693$	5.65685	0.214886
$694$	−44.3848	−1.68482
$695$	34.9706	1.32651
$696$	−2.34315	−0.0888167
$697$	−3.65685	−0.138513
$698$	−3.17157	−0.120046
$699$	−12.3848	−0.468435
$700$	0	0
$701$	−7.58579	−0.286511	−0.143256	−	0.989686i	$-0.545757\pi$
$701$	−7.58579	−0.286511	−0.143256	+	0.989686i	$0.545757\pi$
$702$	0	0
$703$	83.4264	3.14649
$704$	45.2548	1.70561
$705$	−9.82843	−0.370160
$706$	22.2426	0.837113
$707$	−4.24264	−0.159561
$708$	0	0
$709$	−5.20101	−0.195328	−0.0976640	−	0.995219i	$-0.531137\pi$
$709$	−5.20101	−0.195328	−0.0976640	+	0.995219i	$0.531137\pi$
$710$	3.65685	0.137239
$711$	−1.00000	−0.0375029
$712$	−49.9411	−1.87162
$713$	−69.9411	−2.61932
$714$	−1.41421	−0.0529256
$715$	0	0
$716$	0	0
$717$	18.7279	0.699407
$718$	6.14214	0.229222
$719$	−7.31371	−0.272755	−0.136378	−	0.990657i	$-0.543546\pi$
$719$	−7.31371	−0.272755	−0.136378	+	0.990657i	$0.543546\pi$
$720$	9.65685	0.359890
$721$	6.24264	0.232488
$722$	−59.7990	−2.22549
$723$	−5.41421	−0.201357
$724$	0	0
$725$	−0.686292	−0.0254882
$726$	−29.6985	−1.10221
$727$	5.00000	0.185440	0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000	0.185440	0.0927199	+	0.995692i	$0.470444\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	50.2843	1.86110
$731$	10.4853	0.387812
$732$	0	0
$733$	15.2843	0.564537	0.282269	−	0.959335i	$-0.408913\pi$
$733$	15.2843	0.564537	0.282269	+	0.959335i	$0.408913\pi$
$734$	−10.6863	−0.394438
$735$	14.4853	0.534298
$736$	0	0
$737$	−56.5685	−2.08373
$738$	5.17157	0.190368
$739$	48.6274	1.78879	0.894394	−	0.447280i	$-0.147607\pi$
$739$	48.6274	1.78879	0.894394	+	0.447280i	$0.147607\pi$
$740$	0	0
$741$	0	0
$742$	13.5563	0.497669
$743$	0.213203	0.00782168	0.00391084	−	0.999992i	$-0.498755\pi$
$743$	0.213203	0.00782168	0.00391084	+	0.999992i	$0.498755\pi$
$744$	27.3137	1.00137
$745$	4.65685	0.170614
$746$	−8.14214	−0.298105
$747$	2.58579	0.0946090
$748$	0	0
$749$	−0.585786	−0.0214042
$750$	−14.2426	−0.520068
$751$	24.2843	0.886146	0.443073	−	0.896486i	$-0.353888\pi$
$751$	24.2843	0.886146	0.443073	+	0.896486i	$0.353888\pi$
$752$	−16.2843	−0.593826
$753$	−18.5563	−0.676231
$754$	0	0
$755$	−42.2132	−1.53630
$756$	0	0
$757$	21.9706	0.798534	0.399267	−	0.916835i	$-0.369265\pi$
$757$	21.9706	0.798534	0.399267	+	0.916835i	$0.369265\pi$
$758$	17.6152	0.639814
$759$	−40.9706	−1.48714
$760$	53.4558	1.93905
$761$	27.5563	0.998917	0.499458	−	0.866338i	$-0.333532\pi$
$761$	27.5563	0.998917	0.499458	+	0.866338i	$0.333532\pi$
$762$	−16.6274	−0.602348
$763$	−8.48528	−0.307188
$764$	0	0
$765$	−2.41421	−0.0872861
$766$	−31.4558	−1.13655
$767$	0	0
$768$	0	0
$769$	23.5147	0.847962	0.423981	−	0.905671i	$-0.360632\pi$
$769$	23.5147	0.847962	0.423981	+	0.905671i	$0.360632\pi$
$770$	19.3137	0.696018
$771$	−24.6274	−0.886935
$772$	0	0
$773$	−6.34315	−0.228147	−0.114074	−	0.993472i	$-0.536390\pi$
$773$	−6.34315	−0.228147	−0.114074	+	0.993472i	$0.536390\pi$
$774$	−14.8284	−0.532997
$775$	8.00000	0.287368
$776$	41.2548	1.48096
$777$	−10.6569	−0.382313
$778$	−20.6274	−0.739529
$779$	28.6274	1.02568
$780$	0	0
$781$	6.05887	0.216804
$782$	10.2426	0.366276
$783$	−0.828427	−0.0296056
$784$	24.0000	0.857143
$785$	35.5563	1.26906
$786$	3.41421	0.121781
$787$	−28.7279	−1.02404	−0.512020	−	0.858974i	$-0.671103\pi$
$787$	−28.7279	−1.02404	−0.512020	+	0.858974i	$0.671103\pi$
$788$	0	0
$789$	12.3848	0.440910
$790$	−3.41421	−0.121472
$791$	−12.0000	−0.426671
$792$	16.0000	0.568535
$793$	0	0
$794$	51.1127	1.81392
$795$	23.1421	0.820767
$796$	0	0
$797$	−36.4142	−1.28986	−0.644929	−	0.764243i	$-0.723113\pi$
$797$	−36.4142	−1.28986	−0.644929	+	0.764243i	$0.723113\pi$
$798$	11.0711	0.391912
$799$	4.07107	0.144024
$800$	0	0
$801$	−17.6569	−0.623874
$802$	6.00000	0.211867
$803$	83.3137	2.94008
$804$	0	0
$805$	17.4853	0.616275
$806$	0	0
$807$	2.48528	0.0874860
$808$	−12.0000	−0.422159
$809$	7.65685	0.269201	0.134600	−	0.990900i	$-0.457025\pi$
$809$	7.65685	0.269201	0.134600	+	0.990900i	$0.457025\pi$
$810$	3.41421	0.119963
$811$	−22.7279	−0.798085	−0.399043	−	0.916932i	$-0.630657\pi$
$811$	−22.7279	−0.798085	−0.399043	+	0.916932i	$0.630657\pi$
$812$	0	0
$813$	−2.72792	−0.0956724
$814$	85.2548	2.98818
$815$	29.5563	1.03531
$816$	−4.00000	−0.140028
$817$	−82.0833	−2.87173
$818$	54.1421	1.89304
$819$	0	0
$820$	0	0
$821$	19.1127	0.667038	0.333519	−	0.942743i	$-0.391764\pi$
$821$	19.1127	0.667038	0.333519	+	0.942743i	$0.391764\pi$
$822$	17.7574	0.619359
$823$	−14.5147	−0.505951	−0.252976	−	0.967473i	$-0.581409\pi$
$823$	−14.5147	−0.505951	−0.252976	+	0.967473i	$0.581409\pi$
$824$	17.6569	0.615106
$825$	4.68629	0.163156
$826$	9.07107	0.315623
$827$	28.2426	0.982093	0.491046	−	0.871133i	$-0.336615\pi$
$827$	28.2426	0.982093	0.491046	+	0.871133i	$0.336615\pi$
$828$	0	0
$829$	1.31371	0.0456270	0.0228135	−	0.999740i	$-0.492738\pi$
$829$	1.31371	0.0456270	0.0228135	+	0.999740i	$0.492738\pi$
$830$	8.82843	0.306439
$831$	16.9706	0.588702
$832$	0	0
$833$	−6.00000	−0.207888
$834$	20.4853	0.709347
$835$	9.82843	0.340127
$836$	0	0
$837$	9.65685	0.333790
$838$	−1.31371	−0.0453813
$839$	−17.5269	−0.605096	−0.302548	−	0.953134i	$-0.597837\pi$
$839$	−17.5269	−0.605096	−0.302548	+	0.953134i	$0.597837\pi$
$840$	−6.82843	−0.235603
$841$	−28.3137	−0.976335
$842$	31.0711	1.07078
$843$	−9.41421	−0.324243
$844$	0	0
$845$	31.3848	1.07967
$846$	−5.75736	−0.197942
$847$	21.0000	0.721569
$848$	38.3431	1.31671
$849$	12.0000	0.411839
$850$	−1.17157	−0.0401846
$851$	77.1838	2.64583
$852$	0	0
$853$	43.9706	1.50552	0.752762	−	0.658293i	$-0.228721\pi$
$853$	43.9706	1.50552	0.752762	+	0.658293i	$0.228721\pi$
$854$	−15.0711	−0.515721
$855$	18.8995	0.646349
$856$	−1.65685	−0.0566301
$857$	−13.3137	−0.454788	−0.227394	−	0.973803i	$-0.573020\pi$
$857$	−13.3137	−0.454788	−0.227394	+	0.973803i	$0.573020\pi$
$858$	0	0
$859$	29.4558	1.00502	0.502510	−	0.864571i	$-0.332410\pi$
$859$	29.4558	1.00502	0.502510	+	0.864571i	$0.332410\pi$
$860$	0	0
$861$	−3.65685	−0.124625
$862$	20.1838	0.687462
$863$	14.6274	0.497923	0.248962	−	0.968513i	$-0.419911\pi$
$863$	14.6274	0.497923	0.248962	+	0.968513i	$0.419911\pi$
$864$	0	0
$865$	21.8995	0.744605
$866$	20.2843	0.689288
$867$	1.00000	0.0339618
$868$	0	0
$869$	−5.65685	−0.191896
$870$	−2.82843	−0.0958927
$871$	0	0
$872$	−24.0000	−0.812743
$873$	14.5858	0.493654
$874$	−80.1838	−2.71226
$875$	10.0711	0.340464
$876$	0	0
$877$	−17.2721	−0.583237	−0.291618	−	0.956535i	$-0.594194\pi$
$877$	−17.2721	−0.583237	−0.291618	+	0.956535i	$0.594194\pi$
$878$	−49.5980	−1.67385
$879$	−29.8701	−1.00749
$880$	54.6274	1.84149
$881$	−33.9411	−1.14351	−0.571753	−	0.820426i	$-0.693736\pi$
$881$	−33.9411	−1.14351	−0.571753	+	0.820426i	$0.693736\pi$
$882$	8.48528	0.285714
$883$	−58.9706	−1.98452	−0.992259	−	0.124188i	$-0.960367\pi$
$883$	−58.9706	−1.98452	−0.992259	+	0.124188i	$0.960367\pi$
$884$	0	0
$885$	15.4853	0.520532
$886$	41.8406	1.40566
$887$	21.5858	0.724780	0.362390	−	0.932027i	$-0.381961\pi$
$887$	21.5858	0.724780	0.362390	+	0.932027i	$0.381961\pi$
$888$	−30.1421	−1.01150
$889$	11.7574	0.394329
$890$	−60.2843	−2.02073
$891$	5.65685	0.189512
$892$	0	0
$893$	−31.8701	−1.06649
$894$	2.72792	0.0912354
$895$	4.24264	0.141816
$896$	−11.3137	−0.377964
$897$	0	0
$898$	−3.51472	−0.117288
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−9.58579	−0.319349
$902$	29.2548	0.974079
$903$	10.4853	0.348928
$904$	−33.9411	−1.12887
$905$	−8.24264	−0.273995
$906$	−24.7279	−0.821530
$907$	−41.8995	−1.39125	−0.695625	−	0.718405i	$-0.744872\pi$
$907$	−41.8995	−1.39125	−0.695625	+	0.718405i	$0.744872\pi$
$908$	0	0
$909$	−4.24264	−0.140720
$910$	0	0
$911$	−10.6863	−0.354053	−0.177026	−	0.984206i	$-0.556648\pi$
$911$	−10.6863	−0.354053	−0.177026	+	0.984206i	$0.556648\pi$
$912$	31.3137	1.03690
$913$	14.6274	0.484097
$914$	48.7279	1.61178
$915$	−25.7279	−0.850539
$916$	0	0
$917$	−2.41421	−0.0797244
$918$	−1.41421	−0.0466760
$919$	−26.9706	−0.889677	−0.444838	−	0.895611i	$-0.646739\pi$
$919$	−26.9706	−0.889677	−0.444838	+	0.895611i	$0.646739\pi$
$920$	49.4558	1.63051
$921$	−1.55635	−0.0512834
$922$	11.0294	0.363235
$923$	0	0
$924$	0	0
$925$	−8.82843	−0.290277
$926$	−2.48528	−0.0816714
$927$	6.24264	0.205035
$928$	0	0
$929$	−43.0711	−1.41312	−0.706558	−	0.707655i	$-0.749753\pi$
$929$	−43.0711	−1.41312	−0.706558	+	0.707655i	$0.749753\pi$
$930$	32.9706	1.08115
$931$	46.9706	1.53940
$932$	0	0
$933$	25.7990	0.844621
$934$	−52.9706	−1.73325
$935$	−13.6569	−0.446627
$936$	0	0
$937$	48.7279	1.59187	0.795936	−	0.605381i	$-0.206979\pi$
$937$	48.7279	1.59187	0.795936	+	0.605381i	$0.206979\pi$
$938$	14.1421	0.461757
$939$	−27.3137	−0.891349
$940$	0	0
$941$	−48.4975	−1.58097	−0.790486	−	0.612480i	$-0.790172\pi$
$941$	−48.4975	−1.58097	−0.790486	+	0.612480i	$0.790172\pi$
$942$	20.8284	0.678627
$943$	26.4853	0.862479
$944$	25.6569	0.835059
$945$	−2.41421	−0.0785344
$946$	−83.8823	−2.72725
$947$	29.3137	0.952568	0.476284	−	0.879292i	$-0.341984\pi$
$947$	29.3137	0.952568	0.476284	+	0.879292i	$0.341984\pi$
$948$	0	0
$949$	0	0
$950$	9.17157	0.297565
$951$	−2.27208	−0.0736772
$952$	2.82843	0.0916698
$953$	23.6569	0.766321	0.383160	−	0.923682i	$-0.374836\pi$
$953$	23.6569	0.766321	0.383160	+	0.923682i	$0.374836\pi$
$954$	13.5563	0.438903
$955$	54.1127	1.75105
$956$	0	0
$957$	−4.68629	−0.151486
$958$	−21.3553	−0.689960
$959$	−12.5563	−0.405466
$960$	−19.3137	−0.623347
$961$	62.2548	2.00822
$962$	0	0
$963$	−0.585786	−0.0188767
$964$	0	0
$965$	3.58579	0.115431
$966$	10.2426	0.329552
$967$	−11.6863	−0.375806	−0.187903	−	0.982188i	$-0.560169\pi$
$967$	−11.6863	−0.375806	−0.187903	+	0.982188i	$0.560169\pi$
$968$	59.3970	1.90909
$969$	−7.82843	−0.251485
$970$	49.7990	1.59895
$971$	8.28427	0.265855	0.132927	−	0.991126i	$-0.457562\pi$
$971$	8.28427	0.265855	0.132927	+	0.991126i	$0.457562\pi$
$972$	0	0
$973$	−14.4853	−0.464377
$974$	1.71573	0.0549755
$975$	0	0
$976$	−42.6274	−1.36447
$977$	−25.5858	−0.818562	−0.409281	−	0.912408i	$-0.634220\pi$
$977$	−25.5858	−0.818562	−0.409281	+	0.912408i	$0.634220\pi$
$978$	17.3137	0.553631
$979$	−99.8823	−3.19225
$980$	0	0
$981$	−8.48528	−0.270914
$982$	−39.3137	−1.25455
$983$	36.7279	1.17144	0.585719	−	0.810514i	$-0.300812\pi$
$983$	36.7279	1.17144	0.585719	+	0.810514i	$0.300812\pi$
$984$	−10.3431	−0.329727
$985$	−49.4558	−1.57579
$986$	1.17157	0.0373105
$987$	4.07107	0.129584
$988$	0	0
$989$	−75.9411	−2.41479
$990$	19.3137	0.613830
$991$	−8.51472	−0.270479	−0.135239	−	0.990813i	$-0.543180\pi$
$991$	−8.51472	−0.270479	−0.135239	+	0.990813i	$0.543180\pi$
$992$	0	0
$993$	−22.2426	−0.705849
$994$	−1.51472	−0.0480440
$995$	−9.24264	−0.293011
$996$	0	0
$997$	25.5147	0.808059	0.404030	−	0.914746i	$-0.367609\pi$
$997$	25.5147	0.808059	0.404030	+	0.914746i	$0.367609\pi$
$998$	−56.4853	−1.78801
$999$	−10.6569	−0.337168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.c.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.c.1.1	✓	2	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.00000 q^{3} -2.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.00000 q^{3} -2.41421 q^{5} -1.41421 q^{6} +1.00000 q^{7} +2.82843 q^{8} +1.00000 q^{9} +3.41421 q^{10} +5.65685 q^{11} -1.41421 q^{14} -2.41421 q^{15} -4.00000 q^{16} +1.00000 q^{17} -1.41421 q^{18} -7.82843 q^{19} +1.00000 q^{21} -8.00000 q^{22} -7.24264 q^{23} +2.82843 q^{24} +0.828427 q^{25} +1.00000 q^{27} -0.828427 q^{29} +3.41421 q^{30} +9.65685 q^{31} +5.65685 q^{33} -1.41421 q^{34} -2.41421 q^{35} -10.6569 q^{37} +11.0711 q^{38} -6.82843 q^{40} -3.65685 q^{41} -1.41421 q^{42} +10.4853 q^{43} -2.41421 q^{45} +10.2426 q^{46} +4.07107 q^{47} -4.00000 q^{48} -6.00000 q^{49} -1.17157 q^{50} +1.00000 q^{51} -9.58579 q^{53} -1.41421 q^{54} -13.6569 q^{55} +2.82843 q^{56} -7.82843 q^{57} +1.17157 q^{58} -6.41421 q^{59} +10.6569 q^{61} -13.6569 q^{62} +1.00000 q^{63} +8.00000 q^{64} -8.00000 q^{66} -10.0000 q^{67} -7.24264 q^{69} +3.41421 q^{70} +1.07107 q^{71} +2.82843 q^{72} +14.7279 q^{73} +15.0711 q^{74} +0.828427 q^{75} +5.65685 q^{77} -1.00000 q^{79} +9.65685 q^{80} +1.00000 q^{81} +5.17157 q^{82} +2.58579 q^{83} -2.41421 q^{85} -14.8284 q^{86} -0.828427 q^{87} +16.0000 q^{88} -17.6569 q^{89} +3.41421 q^{90} +9.65685 q^{93} -5.75736 q^{94} +18.8995 q^{95} +14.5858 q^{97} +8.48528 q^{98} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{15} - 8 q^{16} + 2 q^{17} - 10 q^{19} + 2 q^{21} - 16 q^{22} - 6 q^{23} - 4 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{31} - 2 q^{35} - 10 q^{37} + 8 q^{38} - 8 q^{40} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 12 q^{46} - 6 q^{47} - 8 q^{48} - 12 q^{49} - 8 q^{50} + 2 q^{51} - 22 q^{53} - 16 q^{55} - 10 q^{57} + 8 q^{58} - 10 q^{59} + 10 q^{61} - 16 q^{62} + 2 q^{63} + 16 q^{64} - 16 q^{66} - 20 q^{67} - 6 q^{69} + 4 q^{70} - 12 q^{71} + 4 q^{73} + 16 q^{74} - 4 q^{75} - 2 q^{79} + 8 q^{80} + 2 q^{81} + 16 q^{82} + 8 q^{83} - 2 q^{85} - 24 q^{86} + 4 q^{87} + 32 q^{88} - 24 q^{89} + 4 q^{90} + 8 q^{93} - 20 q^{94} + 18 q^{95} + 32 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^10 - 2 * q^15 - 8 * q^16 + 2 * q^17 - 10 * q^19 + 2 * q^21 - 16 * q^22 - 6 * q^23 - 4 * q^25 + 2 * q^27 + 4 * q^29 + 4 * q^30 + 8 * q^31 - 2 * q^35 - 10 * q^37 + 8 * q^38 - 8 * q^40 + 4 * q^41 + 4 * q^43 - 2 * q^45 + 12 * q^46 - 6 * q^47 - 8 * q^48 - 12 * q^49 - 8 * q^50 + 2 * q^51 - 22 * q^53 - 16 * q^55 - 10 * q^57 + 8 * q^58 - 10 * q^59 + 10 * q^61 - 16 * q^62 + 2 * q^63 + 16 * q^64 - 16 * q^66 - 20 * q^67 - 6 * q^69 + 4 * q^70 - 12 * q^71 + 4 * q^73 + 16 * q^74 - 4 * q^75 - 2 * q^79 + 8 * q^80 + 2 * q^81 + 16 * q^82 + 8 * q^83 - 2 * q^85 - 24 * q^86 + 4 * q^87 + 32 * q^88 - 24 * q^89 + 4 * q^90 + 8 * q^93 - 20 * q^94 + 18 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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