LMFDB - Embedded newform 4005.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
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:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.41421 q^{7} -4.41421 q^{8} +O(q^{10})$	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.41421 q^{7} -4.41421 q^{8} +2.41421 q^{10} -4.00000 q^{11} -0.828427 q^{13} -3.41421 q^{14} +3.00000 q^{16} -6.82843 q^{17} -2.24264 q^{19} -3.82843 q^{20} +9.65685 q^{22} +8.24264 q^{23} +1.00000 q^{25} +2.00000 q^{26} +5.41421 q^{28} +8.82843 q^{29} +3.41421 q^{31} +1.58579 q^{32} +16.4853 q^{34} -1.41421 q^{35} -3.65685 q^{37} +5.41421 q^{38} +4.41421 q^{40} +4.82843 q^{41} +7.07107 q^{43} -15.3137 q^{44} -19.8995 q^{46} +4.82843 q^{47} -5.00000 q^{49} -2.41421 q^{50} -3.17157 q^{52} +2.82843 q^{53} +4.00000 q^{55} -6.24264 q^{56} -21.3137 q^{58} -10.2426 q^{59} -10.0000 q^{61} -8.24264 q^{62} -9.82843 q^{64} +0.828427 q^{65} +2.00000 q^{67} -26.1421 q^{68} +3.41421 q^{70} +2.82843 q^{71} +11.6569 q^{73} +8.82843 q^{74} -8.58579 q^{76} -5.65685 q^{77} -10.8284 q^{79} -3.00000 q^{80} -11.6569 q^{82} -0.928932 q^{83} +6.82843 q^{85} -17.0711 q^{86} +17.6569 q^{88} +1.00000 q^{89} -1.17157 q^{91} +31.5563 q^{92} -11.6569 q^{94} +2.24264 q^{95} +2.82843 q^{97} +12.0711 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{8} + 2 q^{10} - 8 q^{11} + 4 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} + 4 q^{19} - 2 q^{20} + 8 q^{22} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 8 q^{28} + 12 q^{29} + 4 q^{31} + 6 q^{32} + 16 q^{34} + 4 q^{37} + 8 q^{38} + 6 q^{40} + 4 q^{41} - 8 q^{44} - 20 q^{46} + 4 q^{47} - 10 q^{49} - 2 q^{50} - 12 q^{52} + 8 q^{55} - 4 q^{56} - 20 q^{58} - 12 q^{59} - 20 q^{61} - 8 q^{62} - 14 q^{64} - 4 q^{65} + 4 q^{67} - 24 q^{68} + 4 q^{70} + 12 q^{73} + 12 q^{74} - 20 q^{76} - 16 q^{79} - 6 q^{80} - 12 q^{82} - 16 q^{83} + 8 q^{85} - 20 q^{86} + 24 q^{88} + 2 q^{89} - 8 q^{91} + 32 q^{92} - 12 q^{94} - 4 q^{95} + 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^8 + 2 * q^10 - 8 * q^11 + 4 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 + 4 * q^19 - 2 * q^20 + 8 * q^22 + 8 * q^23 + 2 * q^25 + 4 * q^26 + 8 * q^28 + 12 * q^29 + 4 * q^31 + 6 * q^32 + 16 * q^34 + 4 * q^37 + 8 * q^38 + 6 * q^40 + 4 * q^41 - 8 * q^44 - 20 * q^46 + 4 * q^47 - 10 * q^49 - 2 * q^50 - 12 * q^52 + 8 * q^55 - 4 * q^56 - 20 * q^58 - 12 * q^59 - 20 * q^61 - 8 * q^62 - 14 * q^64 - 4 * q^65 + 4 * q^67 - 24 * q^68 + 4 * q^70 + 12 * q^73 + 12 * q^74 - 20 * q^76 - 16 * q^79 - 6 * q^80 - 12 * q^82 - 16 * q^83 + 8 * q^85 - 20 * q^86 + 24 * q^88 + 2 * q^89 - 8 * q^91 + 32 * q^92 - 12 * q^94 - 4 * q^95 + 10 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	−4.41421	−1.56066
$9$	0	0
$10$	2.41421	0.763441
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	−3.41421	−0.912487
$15$	0	0
$16$	3.00000	0.750000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	−2.24264	−0.514497	−0.257249	−	0.966345i	$-0.582816\pi$
$19$	−2.24264	−0.514497	−0.257249	+	0.966345i	$0.582816\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	9.65685	2.05885
$23$	8.24264	1.71871	0.859355	−	0.511380i	$-0.170866\pi$
$23$	8.24264	1.71871	0.859355	+	0.511380i	$0.170866\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	5.41421	1.02319
$29$	8.82843	1.63940	0.819699	−	0.572795i	$-0.194141\pi$
$29$	8.82843	1.63940	0.819699	+	0.572795i	$0.194141\pi$
$30$	0	0
$31$	3.41421	0.613211	0.306605	−	0.951837i	$-0.400807\pi$
$31$	3.41421	0.613211	0.306605	+	0.951837i	$0.400807\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	16.4853	2.82720
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	5.41421	0.878301
$39$	0	0
$40$	4.41421	0.697948
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	0	0
$43$	7.07107	1.07833	0.539164	−	0.842201i	$-0.318740\pi$
$43$	7.07107	1.07833	0.539164	+	0.842201i	$0.318740\pi$
$44$	−15.3137	−2.30863
$45$	0	0
$46$	−19.8995	−2.93402
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	−2.41421	−0.341421
$51$	0	0
$52$	−3.17157	−0.439818
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	−6.24264	−0.834208
$57$	0	0
$58$	−21.3137	−2.79863
$59$	−10.2426	−1.33348	−0.666739	−	0.745291i	$-0.732310\pi$
$59$	−10.2426	−1.33348	−0.666739	+	0.745291i	$0.732310\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.24264	−1.04682
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0.828427	0.102754
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−26.1421	−3.17020
$69$	0	0
$70$	3.41421	0.408077
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	11.6569	1.36433	0.682166	−	0.731198i	$-0.261038\pi$
$73$	11.6569	1.36433	0.682166	+	0.731198i	$0.261038\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	−8.58579	−0.984857
$77$	−5.65685	−0.644658
$78$	0	0
$79$	−10.8284	−1.21829	−0.609147	−	0.793058i	$-0.708488\pi$
$79$	−10.8284	−1.21829	−0.609147	+	0.793058i	$0.708488\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−11.6569	−1.28728
$83$	−0.928932	−0.101964	−0.0509818	−	0.998700i	$-0.516235\pi$
$83$	−0.928932	−0.101964	−0.0509818	+	0.998700i	$0.516235\pi$
$84$	0	0
$85$	6.82843	0.740647
$86$	−17.0711	−1.84082
$87$	0	0
$88$	17.6569	1.88223
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−1.17157	−0.122814
$92$	31.5563	3.28998
$93$	0	0
$94$	−11.6569	−1.20231
$95$	2.24264	0.230090
$96$	0	0
$97$	2.82843	0.287183	0.143592	−	0.989637i	$-0.454135\pi$
$97$	2.82843	0.287183	0.143592	+	0.989637i	$0.454135\pi$
$98$	12.0711	1.21936
$99$	0	0
$100$	3.82843	0.382843
$101$	7.17157	0.713598	0.356799	−	0.934181i	$-0.383868\pi$
$101$	7.17157	0.713598	0.356799	+	0.934181i	$0.383868\pi$
$102$	0	0
$103$	0.242641	0.0239081	0.0119540	−	0.999929i	$-0.496195\pi$
$103$	0.242641	0.0239081	0.0119540	+	0.999929i	$0.496195\pi$
$104$	3.65685	0.358584
$105$	0	0
$106$	−6.82843	−0.663235
$107$	−0.828427	−0.0800871	−0.0400435	−	0.999198i	$-0.512750\pi$
$107$	−0.828427	−0.0800871	−0.0400435	+	0.999198i	$0.512750\pi$
$108$	0	0
$109$	−6.34315	−0.607563	−0.303782	−	0.952742i	$-0.598249\pi$
$109$	−6.34315	−0.607563	−0.303782	+	0.952742i	$0.598249\pi$
$110$	−9.65685	−0.920745
$111$	0	0
$112$	4.24264	0.400892
$113$	−8.82843	−0.830509	−0.415254	−	0.909705i	$-0.636307\pi$
$113$	−8.82843	−0.830509	−0.415254	+	0.909705i	$0.636307\pi$
$114$	0	0
$115$	−8.24264	−0.768630
$116$	33.7990	3.13816
$117$	0	0
$118$	24.7279	2.27639
$119$	−9.65685	−0.885242
$120$	0	0
$121$	5.00000	0.454545
$122$	24.1421	2.18573
$123$	0	0
$124$	13.0711	1.17382
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.0711	−0.982398	−0.491199	−	0.871047i	$-0.663441\pi$
$127$	−11.0711	−0.982398	−0.491199	+	0.871047i	$0.663441\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−19.7990	−1.72985	−0.864923	−	0.501905i	$-0.832633\pi$
$131$	−19.7990	−1.72985	−0.864923	+	0.501905i	$0.832633\pi$
$132$	0	0
$133$	−3.17157	−0.275010
$134$	−4.82843	−0.417113
$135$	0	0
$136$	30.1421	2.58467
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	21.6569	1.83691	0.918455	−	0.395525i	$-0.129437\pi$
$139$	21.6569	1.83691	0.918455	+	0.395525i	$0.129437\pi$
$140$	−5.41421	−0.457585
$141$	0	0
$142$	−6.82843	−0.573029
$143$	3.31371	0.277106
$144$	0	0
$145$	−8.82843	−0.733161
$146$	−28.1421	−2.32906
$147$	0	0
$148$	−14.0000	−1.15079
$149$	−17.3137	−1.41839	−0.709197	−	0.705010i	$-0.750942\pi$
$149$	−17.3137	−1.41839	−0.709197	+	0.705010i	$0.750942\pi$
$150$	0	0
$151$	20.3848	1.65889	0.829445	−	0.558589i	$-0.188657\pi$
$151$	20.3848	1.65889	0.829445	+	0.558589i	$0.188657\pi$
$152$	9.89949	0.802955
$153$	0	0
$154$	13.6569	1.10050
$155$	−3.41421	−0.274236
$156$	0	0
$157$	−2.82843	−0.225733	−0.112867	−	0.993610i	$-0.536003\pi$
$157$	−2.82843	−0.225733	−0.112867	+	0.993610i	$0.536003\pi$
$158$	26.1421	2.07976
$159$	0	0
$160$	−1.58579	−0.125367
$161$	11.6569	0.918689
$162$	0	0
$163$	19.5563	1.53177	0.765886	−	0.642977i	$-0.222301\pi$
$163$	19.5563	1.53177	0.765886	+	0.642977i	$0.222301\pi$
$164$	18.4853	1.44346
$165$	0	0
$166$	2.24264	0.174063
$167$	−12.3431	−0.955141	−0.477571	−	0.878593i	$-0.658483\pi$
$167$	−12.3431	−0.955141	−0.477571	+	0.878593i	$0.658483\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	−16.4853	−1.26436
$171$	0	0
$172$	27.0711	2.06415
$173$	−18.9706	−1.44231	−0.721153	−	0.692776i	$-0.756387\pi$
$173$	−18.9706	−1.44231	−0.721153	+	0.692776i	$0.756387\pi$
$174$	0	0
$175$	1.41421	0.106904
$176$	−12.0000	−0.904534
$177$	0	0
$178$	−2.41421	−0.180953
$179$	1.65685	0.123839	0.0619196	−	0.998081i	$-0.480278\pi$
$179$	1.65685	0.123839	0.0619196	+	0.998081i	$0.480278\pi$
$180$	0	0
$181$	−16.8284	−1.25085	−0.625424	−	0.780285i	$-0.715074\pi$
$181$	−16.8284	−1.25085	−0.625424	+	0.780285i	$0.715074\pi$
$182$	2.82843	0.209657
$183$	0	0
$184$	−36.3848	−2.68232
$185$	3.65685	0.268857
$186$	0	0
$187$	27.3137	1.99738
$188$	18.4853	1.34818
$189$	0	0
$190$	−5.41421	−0.392788
$191$	−3.41421	−0.247044	−0.123522	−	0.992342i	$-0.539419\pi$
$191$	−3.41421	−0.247044	−0.123522	+	0.992342i	$0.539419\pi$
$192$	0	0
$193$	4.34315	0.312626	0.156313	−	0.987708i	$-0.450039\pi$
$193$	4.34315	0.312626	0.156313	+	0.987708i	$0.450039\pi$
$194$	−6.82843	−0.490252
$195$	0	0
$196$	−19.1421	−1.36730
$197$	0.343146	0.0244481	0.0122241	−	0.999925i	$-0.496109\pi$
$197$	0.343146	0.0244481	0.0122241	+	0.999925i	$0.496109\pi$
$198$	0	0
$199$	−3.51472	−0.249152	−0.124576	−	0.992210i	$-0.539757\pi$
$199$	−3.51472	−0.249152	−0.124576	+	0.992210i	$0.539757\pi$
$200$	−4.41421	−0.312132
$201$	0	0
$202$	−17.3137	−1.21819
$203$	12.4853	0.876295
$204$	0	0
$205$	−4.82843	−0.337232
$206$	−0.585786	−0.0408137
$207$	0	0
$208$	−2.48528	−0.172323
$209$	8.97056	0.620507
$210$	0	0
$211$	−15.2132	−1.04732	−0.523660	−	0.851927i	$-0.675434\pi$
$211$	−15.2132	−1.04732	−0.523660	+	0.851927i	$0.675434\pi$
$212$	10.8284	0.743699
$213$	0	0
$214$	2.00000	0.136717
$215$	−7.07107	−0.482243
$216$	0	0
$217$	4.82843	0.327775
$218$	15.3137	1.03718
$219$	0	0
$220$	15.3137	1.03245
$221$	5.65685	0.380521
$222$	0	0
$223$	19.6569	1.31632	0.658160	−	0.752878i	$-0.271335\pi$
$223$	19.6569	1.31632	0.658160	+	0.752878i	$0.271335\pi$
$224$	2.24264	0.149843
$225$	0	0
$226$	21.3137	1.41777
$227$	0.828427	0.0549846	0.0274923	−	0.999622i	$-0.491248\pi$
$227$	0.828427	0.0549846	0.0274923	+	0.999622i	$0.491248\pi$
$228$	0	0
$229$	−20.8284	−1.37638	−0.688191	−	0.725530i	$-0.741595\pi$
$229$	−20.8284	−1.37638	−0.688191	+	0.725530i	$0.741595\pi$
$230$	19.8995	1.31213
$231$	0	0
$232$	−38.9706	−2.55854
$233$	18.9706	1.24280	0.621401	−	0.783492i	$-0.286564\pi$
$233$	18.9706	1.24280	0.621401	+	0.783492i	$0.286564\pi$
$234$	0	0
$235$	−4.82843	−0.314972
$236$	−39.2132	−2.55256
$237$	0	0
$238$	23.3137	1.51120
$239$	3.89949	0.252237	0.126119	−	0.992015i	$-0.459748\pi$
$239$	3.89949	0.252237	0.126119	+	0.992015i	$0.459748\pi$
$240$	0	0
$241$	−23.4558	−1.51092	−0.755462	−	0.655193i	$-0.772587\pi$
$241$	−23.4558	−1.51092	−0.755462	+	0.655193i	$0.772587\pi$
$242$	−12.0711	−0.775958
$243$	0	0
$244$	−38.2843	−2.45090
$245$	5.00000	0.319438
$246$	0	0
$247$	1.85786	0.118213
$248$	−15.0711	−0.957014
$249$	0	0
$250$	2.41421	0.152688
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	−32.9706	−2.07284
$254$	26.7279	1.67706
$255$	0	0
$256$	−29.9706	−1.87316
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	−5.17157	−0.321346
$260$	3.17157	0.196693
$261$	0	0
$262$	47.7990	2.95303
$263$	−22.0000	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$263$	−22.0000	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$264$	0	0
$265$	−2.82843	−0.173749
$266$	7.65685	0.469472
$267$	0	0
$268$	7.65685	0.467717
$269$	−2.34315	−0.142864	−0.0714321	−	0.997445i	$-0.522757\pi$
$269$	−2.34315	−0.142864	−0.0714321	+	0.997445i	$0.522757\pi$
$270$	0	0
$271$	−20.9706	−1.27387	−0.636935	−	0.770917i	$-0.719798\pi$
$271$	−20.9706	−1.27387	−0.636935	+	0.770917i	$0.719798\pi$
$272$	−20.4853	−1.24210
$273$	0	0
$274$	33.7990	2.04187
$275$	−4.00000	−0.241209
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−52.2843	−3.13580
$279$	0	0
$280$	6.24264	0.373069
$281$	9.51472	0.567601	0.283800	−	0.958883i	$-0.408405\pi$
$281$	9.51472	0.567601	0.283800	+	0.958883i	$0.408405\pi$
$282$	0	0
$283$	−6.68629	−0.397459	−0.198729	−	0.980054i	$-0.563681\pi$
$283$	−6.68629	−0.397459	−0.198729	+	0.980054i	$0.563681\pi$
$284$	10.8284	0.642549
$285$	0	0
$286$	−8.00000	−0.473050
$287$	6.82843	0.403069
$288$	0	0
$289$	29.6274	1.74279
$290$	21.3137	1.25158
$291$	0	0
$292$	44.6274	2.61162
$293$	−7.65685	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$293$	−7.65685	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$294$	0	0
$295$	10.2426	0.596350
$296$	16.1421	0.938243
$297$	0	0
$298$	41.7990	2.42135
$299$	−6.82843	−0.394898
$300$	0	0
$301$	10.0000	0.576390
$302$	−49.2132	−2.83190
$303$	0	0
$304$	−6.72792	−0.385873
$305$	10.0000	0.572598
$306$	0	0
$307$	8.82843	0.503865	0.251932	−	0.967745i	$-0.418934\pi$
$307$	8.82843	0.503865	0.251932	+	0.967745i	$0.418934\pi$
$308$	−21.6569	−1.23401
$309$	0	0
$310$	8.24264	0.468151
$311$	−28.9706	−1.64277	−0.821385	−	0.570374i	$-0.806798\pi$
$311$	−28.9706	−1.64277	−0.821385	+	0.570374i	$0.806798\pi$
$312$	0	0
$313$	23.4558	1.32580	0.662901	−	0.748707i	$-0.269325\pi$
$313$	23.4558	1.32580	0.662901	+	0.748707i	$0.269325\pi$
$314$	6.82843	0.385350
$315$	0	0
$316$	−41.4558	−2.33207
$317$	−23.6569	−1.32870	−0.664351	−	0.747421i	$-0.731292\pi$
$317$	−23.6569	−1.32870	−0.664351	+	0.747421i	$0.731292\pi$
$318$	0	0
$319$	−35.3137	−1.97719
$320$	9.82843	0.549426
$321$	0	0
$322$	−28.1421	−1.56830
$323$	15.3137	0.852078
$324$	0	0
$325$	−0.828427	−0.0459529
$326$	−47.2132	−2.61490
$327$	0	0
$328$	−21.3137	−1.17685
$329$	6.82843	0.376463
$330$	0	0
$331$	−21.6569	−1.19037	−0.595184	−	0.803589i	$-0.702921\pi$
$331$	−21.6569	−1.19037	−0.595184	+	0.803589i	$0.702921\pi$
$332$	−3.55635	−0.195180
$333$	0	0
$334$	29.7990	1.63053
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−2.68629	−0.146332	−0.0731658	−	0.997320i	$-0.523310\pi$
$337$	−2.68629	−0.146332	−0.0731658	+	0.997320i	$0.523310\pi$
$338$	29.7279	1.61699
$339$	0	0
$340$	26.1421	1.41776
$341$	−13.6569	−0.739560
$342$	0	0
$343$	−16.9706	−0.916324
$344$	−31.2132	−1.68290
$345$	0	0
$346$	45.7990	2.46217
$347$	6.48528	0.348148	0.174074	−	0.984733i	$-0.444307\pi$
$347$	6.48528	0.348148	0.174074	+	0.984733i	$0.444307\pi$
$348$	0	0
$349$	5.79899	0.310413	0.155206	−	0.987882i	$-0.450396\pi$
$349$	5.79899	0.310413	0.155206	+	0.987882i	$0.450396\pi$
$350$	−3.41421	−0.182497
$351$	0	0
$352$	−6.34315	−0.338091
$353$	16.3431	0.869858	0.434929	−	0.900465i	$-0.356773\pi$
$353$	16.3431	0.869858	0.434929	+	0.900465i	$0.356773\pi$
$354$	0	0
$355$	−2.82843	−0.150117
$356$	3.82843	0.202906
$357$	0	0
$358$	−4.00000	−0.211407
$359$	9.55635	0.504365	0.252182	−	0.967680i	$-0.418852\pi$
$359$	9.55635	0.504365	0.252182	+	0.967680i	$0.418852\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	40.6274	2.13533
$363$	0	0
$364$	−4.48528	−0.235093
$365$	−11.6569	−0.610148
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	24.7279	1.28903
$369$	0	0
$370$	−8.82843	−0.458968
$371$	4.00000	0.207670
$372$	0	0
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	−65.9411	−3.40973
$375$	0	0
$376$	−21.3137	−1.09917
$377$	−7.31371	−0.376675
$378$	0	0
$379$	3.41421	0.175376	0.0876882	−	0.996148i	$-0.472052\pi$
$379$	3.41421	0.175376	0.0876882	+	0.996148i	$0.472052\pi$
$380$	8.58579	0.440442
$381$	0	0
$382$	8.24264	0.421730
$383$	11.2721	0.575976	0.287988	−	0.957634i	$-0.407014\pi$
$383$	11.2721	0.575976	0.287988	+	0.957634i	$0.407014\pi$
$384$	0	0
$385$	5.65685	0.288300
$386$	−10.4853	−0.533687
$387$	0	0
$388$	10.8284	0.549730
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	−56.2843	−2.84642
$392$	22.0711	1.11476
$393$	0	0
$394$	−0.828427	−0.0417356
$395$	10.8284	0.544837
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	8.48528	0.425329
$399$	0	0
$400$	3.00000	0.150000
$401$	−3.02944	−0.151283	−0.0756414	−	0.997135i	$-0.524100\pi$
$401$	−3.02944	−0.151283	−0.0756414	+	0.997135i	$0.524100\pi$
$402$	0	0
$403$	−2.82843	−0.140894
$404$	27.4558	1.36598
$405$	0	0
$406$	−30.1421	−1.49593
$407$	14.6274	0.725054
$408$	0	0
$409$	−31.3137	−1.54836	−0.774182	−	0.632964i	$-0.781838\pi$
$409$	−31.3137	−1.54836	−0.774182	+	0.632964i	$0.781838\pi$
$410$	11.6569	0.575691
$411$	0	0
$412$	0.928932	0.0457652
$413$	−14.4853	−0.712774
$414$	0	0
$415$	0.928932	0.0455995
$416$	−1.31371	−0.0644099
$417$	0	0
$418$	−21.6569	−1.05927
$419$	−26.2426	−1.28204	−0.641018	−	0.767525i	$-0.721488\pi$
$419$	−26.2426	−1.28204	−0.641018	+	0.767525i	$0.721488\pi$
$420$	0	0
$421$	30.2843	1.47597	0.737983	−	0.674820i	$-0.235779\pi$
$421$	30.2843	1.47597	0.737983	+	0.674820i	$0.235779\pi$
$422$	36.7279	1.78789
$423$	0	0
$424$	−12.4853	−0.606339
$425$	−6.82843	−0.331227
$426$	0	0
$427$	−14.1421	−0.684386
$428$	−3.17157	−0.153304
$429$	0	0
$430$	17.0711	0.823240
$431$	−34.2426	−1.64941	−0.824705	−	0.565563i	$-0.808659\pi$
$431$	−34.2426	−1.64941	−0.824705	+	0.565563i	$0.808659\pi$
$432$	0	0
$433$	−25.7990	−1.23982	−0.619910	−	0.784673i	$-0.712831\pi$
$433$	−25.7990	−1.23982	−0.619910	+	0.784673i	$0.712831\pi$
$434$	−11.6569	−0.559547
$435$	0	0
$436$	−24.2843	−1.16301
$437$	−18.4853	−0.884271
$438$	0	0
$439$	8.38478	0.400184	0.200092	−	0.979777i	$-0.435876\pi$
$439$	8.38478	0.400184	0.200092	+	0.979777i	$0.435876\pi$
$440$	−17.6569	−0.841757
$441$	0	0
$442$	−13.6569	−0.649590
$443$	−33.3137	−1.58278	−0.791391	−	0.611310i	$-0.790643\pi$
$443$	−33.3137	−1.58278	−0.791391	+	0.611310i	$0.790643\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−47.4558	−2.24710
$447$	0	0
$448$	−13.8995	−0.656689
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−19.3137	−0.909447
$452$	−33.7990	−1.58977
$453$	0	0
$454$	−2.00000	−0.0938647
$455$	1.17157	0.0549242
$456$	0	0
$457$	−4.14214	−0.193761	−0.0968805	−	0.995296i	$-0.530886\pi$
$457$	−4.14214	−0.193761	−0.0968805	+	0.995296i	$0.530886\pi$
$458$	50.2843	2.34963
$459$	0	0
$460$	−31.5563	−1.47132
$461$	12.9706	0.604099	0.302050	−	0.953292i	$-0.402329\pi$
$461$	12.9706	0.604099	0.302050	+	0.953292i	$0.402329\pi$
$462$	0	0
$463$	4.14214	0.192501	0.0962507	−	0.995357i	$-0.469315\pi$
$463$	4.14214	0.192501	0.0962507	+	0.995357i	$0.469315\pi$
$464$	26.4853	1.22955
$465$	0	0
$466$	−45.7990	−2.12160
$467$	12.3431	0.571173	0.285586	−	0.958353i	$-0.407812\pi$
$467$	12.3431	0.571173	0.285586	+	0.958353i	$0.407812\pi$
$468$	0	0
$469$	2.82843	0.130605
$470$	11.6569	0.537691
$471$	0	0
$472$	45.2132	2.08111
$473$	−28.2843	−1.30051
$474$	0	0
$475$	−2.24264	−0.102899
$476$	−36.9706	−1.69454
$477$	0	0
$478$	−9.41421	−0.430596
$479$	2.82843	0.129234	0.0646171	−	0.997910i	$-0.479417\pi$
$479$	2.82843	0.129234	0.0646171	+	0.997910i	$0.479417\pi$
$480$	0	0
$481$	3.02944	0.138130
$482$	56.6274	2.57931
$483$	0	0
$484$	19.1421	0.870097
$485$	−2.82843	−0.128432
$486$	0	0
$487$	38.4853	1.74393	0.871967	−	0.489564i	$-0.162844\pi$
$487$	38.4853	1.74393	0.871967	+	0.489564i	$0.162844\pi$
$488$	44.1421	1.99822
$489$	0	0
$490$	−12.0711	−0.545315
$491$	−17.5563	−0.792307	−0.396153	−	0.918184i	$-0.629655\pi$
$491$	−17.5563	−0.792307	−0.396153	+	0.918184i	$0.629655\pi$
$492$	0	0
$493$	−60.2843	−2.71507
$494$	−4.48528	−0.201802
$495$	0	0
$496$	10.2426	0.459908
$497$	4.00000	0.179425
$498$	0	0
$499$	−19.2132	−0.860101	−0.430051	−	0.902805i	$-0.641504\pi$
$499$	−19.2132	−0.860101	−0.430051	+	0.902805i	$0.641504\pi$
$500$	−3.82843	−0.171212
$501$	0	0
$502$	9.65685	0.431006
$503$	13.2132	0.589148	0.294574	−	0.955629i	$-0.404822\pi$
$503$	13.2132	0.589148	0.294574	+	0.955629i	$0.404822\pi$
$504$	0	0
$505$	−7.17157	−0.319131
$506$	79.5980	3.53856
$507$	0	0
$508$	−42.3848	−1.88052
$509$	−29.9411	−1.32712	−0.663559	−	0.748124i	$-0.730955\pi$
$509$	−29.9411	−1.32712	−0.663559	+	0.748124i	$0.730955\pi$
$510$	0	0
$511$	16.4853	0.729266
$512$	31.2426	1.38074
$513$	0	0
$514$	−53.1127	−2.34270
$515$	−0.242641	−0.0106920
$516$	0	0
$517$	−19.3137	−0.849416
$518$	12.4853	0.548572
$519$	0	0
$520$	−3.65685	−0.160364
$521$	−25.3137	−1.10901	−0.554507	−	0.832179i	$-0.687093\pi$
$521$	−25.3137	−1.10901	−0.554507	+	0.832179i	$0.687093\pi$
$522$	0	0
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−75.7990	−3.31129
$525$	0	0
$526$	53.1127	2.31582
$527$	−23.3137	−1.01556
$528$	0	0
$529$	44.9411	1.95396
$530$	6.82843	0.296608
$531$	0	0
$532$	−12.1421	−0.526428
$533$	−4.00000	−0.173259
$534$	0	0
$535$	0.828427	0.0358160
$536$	−8.82843	−0.381330
$537$	0	0
$538$	5.65685	0.243884
$539$	20.0000	0.861461
$540$	0	0
$541$	−15.1716	−0.652277	−0.326138	−	0.945322i	$-0.605748\pi$
$541$	−15.1716	−0.652277	−0.326138	+	0.945322i	$0.605748\pi$
$542$	50.6274	2.17463
$543$	0	0
$544$	−10.8284	−0.464265
$545$	6.34315	0.271711
$546$	0	0
$547$	−29.2132	−1.24907	−0.624533	−	0.780998i	$-0.714711\pi$
$547$	−29.2132	−1.24907	−0.624533	+	0.780998i	$0.714711\pi$
$548$	−53.5980	−2.28959
$549$	0	0
$550$	9.65685	0.411770
$551$	−19.7990	−0.843465
$552$	0	0
$553$	−15.3137	−0.651205
$554$	−4.82843	−0.205140
$555$	0	0
$556$	82.9117	3.51624
$557$	−32.8284	−1.39099	−0.695493	−	0.718533i	$-0.744814\pi$
$557$	−32.8284	−1.39099	−0.695493	+	0.718533i	$0.744814\pi$
$558$	0	0
$559$	−5.85786	−0.247761
$560$	−4.24264	−0.179284
$561$	0	0
$562$	−22.9706	−0.968955
$563$	−37.8995	−1.59727	−0.798637	−	0.601814i	$-0.794445\pi$
$563$	−37.8995	−1.59727	−0.798637	+	0.601814i	$0.794445\pi$
$564$	0	0
$565$	8.82843	0.371415
$566$	16.1421	0.678505
$567$	0	0
$568$	−12.4853	−0.523871
$569$	−19.4558	−0.815631	−0.407816	−	0.913064i	$-0.633709\pi$
$569$	−19.4558	−0.815631	−0.407816	+	0.913064i	$0.633709\pi$
$570$	0	0
$571$	4.10051	0.171601	0.0858004	−	0.996312i	$-0.472655\pi$
$571$	4.10051	0.171601	0.0858004	+	0.996312i	$0.472655\pi$
$572$	12.6863	0.530440
$573$	0	0
$574$	−16.4853	−0.688082
$575$	8.24264	0.343742
$576$	0	0
$577$	3.17157	0.132034	0.0660172	−	0.997818i	$-0.478971\pi$
$577$	3.17157	0.132034	0.0660172	+	0.997818i	$0.478971\pi$
$578$	−71.5269	−2.97513
$579$	0	0
$580$	−33.7990	−1.40343
$581$	−1.31371	−0.0545018
$582$	0	0
$583$	−11.3137	−0.468566
$584$	−51.4558	−2.12926
$585$	0	0
$586$	18.4853	0.763620
$587$	−11.8579	−0.489426	−0.244713	−	0.969596i	$-0.578694\pi$
$587$	−11.8579	−0.489426	−0.244713	+	0.969596i	$0.578694\pi$
$588$	0	0
$589$	−7.65685	−0.315495
$590$	−24.7279	−1.01803
$591$	0	0
$592$	−10.9706	−0.450887
$593$	7.65685	0.314429	0.157215	−	0.987564i	$-0.449749\pi$
$593$	7.65685	0.314429	0.157215	+	0.987564i	$0.449749\pi$
$594$	0	0
$595$	9.65685	0.395892
$596$	−66.2843	−2.71511
$597$	0	0
$598$	16.4853	0.674133
$599$	35.8995	1.46681	0.733407	−	0.679790i	$-0.237929\pi$
$599$	35.8995	1.46681	0.733407	+	0.679790i	$0.237929\pi$
$600$	0	0
$601$	20.6274	0.841410	0.420705	−	0.907198i	$-0.361783\pi$
$601$	20.6274	0.841410	0.420705	+	0.907198i	$0.361783\pi$
$602$	−24.1421	−0.983960
$603$	0	0
$604$	78.0416	3.17547
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−43.2548	−1.75566	−0.877830	−	0.478973i	$-0.841009\pi$
$607$	−43.2548	−1.75566	−0.877830	+	0.478973i	$0.841009\pi$
$608$	−3.55635	−0.144229
$609$	0	0
$610$	−24.1421	−0.977486
$611$	−4.00000	−0.161823
$612$	0	0
$613$	−38.1421	−1.54055	−0.770273	−	0.637714i	$-0.779880\pi$
$613$	−38.1421	−1.54055	−0.770273	+	0.637714i	$0.779880\pi$
$614$	−21.3137	−0.860151
$615$	0	0
$616$	24.9706	1.00609
$617$	10.9706	0.441658	0.220829	−	0.975313i	$-0.429124\pi$
$617$	10.9706	0.441658	0.220829	+	0.975313i	$0.429124\pi$
$618$	0	0
$619$	32.2843	1.29761	0.648807	−	0.760953i	$-0.275268\pi$
$619$	32.2843	1.29761	0.648807	+	0.760953i	$0.275268\pi$
$620$	−13.0711	−0.524947
$621$	0	0
$622$	69.9411	2.80438
$623$	1.41421	0.0566593
$624$	0	0
$625$	1.00000	0.0400000
$626$	−56.6274	−2.26329
$627$	0	0
$628$	−10.8284	−0.432101
$629$	24.9706	0.995642
$630$	0	0
$631$	−29.6569	−1.18062	−0.590310	−	0.807176i	$-0.700995\pi$
$631$	−29.6569	−1.18062	−0.590310	+	0.807176i	$0.700995\pi$
$632$	47.7990	1.90134
$633$	0	0
$634$	57.1127	2.26824
$635$	11.0711	0.439342
$636$	0	0
$637$	4.14214	0.164117
$638$	85.2548	3.37527
$639$	0	0
$640$	−20.5563	−0.812561
$641$	38.9706	1.53924	0.769622	−	0.638499i	$-0.220445\pi$
$641$	38.9706	1.53924	0.769622	+	0.638499i	$0.220445\pi$
$642$	0	0
$643$	3.85786	0.152139	0.0760697	−	0.997103i	$-0.475763\pi$
$643$	3.85786	0.152139	0.0760697	+	0.997103i	$0.475763\pi$
$644$	44.6274	1.75857
$645$	0	0
$646$	−36.9706	−1.45459
$647$	39.0711	1.53604	0.768021	−	0.640425i	$-0.221242\pi$
$647$	39.0711	1.53604	0.768021	+	0.640425i	$0.221242\pi$
$648$	0	0
$649$	40.9706	1.60824
$650$	2.00000	0.0784465
$651$	0	0
$652$	74.8701	2.93214
$653$	−38.7696	−1.51717	−0.758585	−	0.651574i	$-0.774109\pi$
$653$	−38.7696	−1.51717	−0.758585	+	0.651574i	$0.774109\pi$
$654$	0	0
$655$	19.7990	0.773611
$656$	14.4853	0.565555
$657$	0	0
$658$	−16.4853	−0.642663
$659$	28.7696	1.12070	0.560351	−	0.828255i	$-0.310666\pi$
$659$	28.7696	1.12070	0.560351	+	0.828255i	$0.310666\pi$
$660$	0	0
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	52.2843	2.03209
$663$	0	0
$664$	4.10051	0.159130
$665$	3.17157	0.122988
$666$	0	0
$667$	72.7696	2.81765
$668$	−47.2548	−1.82834
$669$	0	0
$670$	4.82843	0.186538
$671$	40.0000	1.54418
$672$	0	0
$673$	−22.1421	−0.853517	−0.426758	−	0.904366i	$-0.640344\pi$
$673$	−22.1421	−0.853517	−0.426758	+	0.904366i	$0.640344\pi$
$674$	6.48528	0.249804
$675$	0	0
$676$	−47.1421	−1.81316
$677$	−3.45584	−0.132819	−0.0664094	−	0.997792i	$-0.521154\pi$
$677$	−3.45584	−0.132819	−0.0664094	+	0.997792i	$0.521154\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	−30.1421	−1.15590
$681$	0	0
$682$	32.9706	1.26251
$683$	−15.7574	−0.602939	−0.301469	−	0.953476i	$-0.597477\pi$
$683$	−15.7574	−0.602939	−0.301469	+	0.953476i	$0.597477\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	40.9706	1.56426
$687$	0	0
$688$	21.2132	0.808746
$689$	−2.34315	−0.0892667
$690$	0	0
$691$	27.1127	1.03142	0.515708	−	0.856765i	$-0.327529\pi$
$691$	27.1127	1.03142	0.515708	+	0.856765i	$0.327529\pi$
$692$	−72.6274	−2.76088
$693$	0	0
$694$	−15.6569	−0.594326
$695$	−21.6569	−0.821491
$696$	0	0
$697$	−32.9706	−1.24885
$698$	−14.0000	−0.529908
$699$	0	0
$700$	5.41421	0.204638
$701$	1.31371	0.0496181	0.0248090	−	0.999692i	$-0.492102\pi$
$701$	1.31371	0.0496181	0.0248090	+	0.999692i	$0.492102\pi$
$702$	0	0
$703$	8.20101	0.309307
$704$	39.3137	1.48169
$705$	0	0
$706$	−39.4558	−1.48494
$707$	10.1421	0.381434
$708$	0	0
$709$	40.4264	1.51825	0.759123	−	0.650947i	$-0.225628\pi$
$709$	40.4264	1.51825	0.759123	+	0.650947i	$0.225628\pi$
$710$	6.82843	0.256266
$711$	0	0
$712$	−4.41421	−0.165430
$713$	28.1421	1.05393
$714$	0	0
$715$	−3.31371	−0.123926
$716$	6.34315	0.237054
$717$	0	0
$718$	−23.0711	−0.861005
$719$	9.55635	0.356392	0.178196	−	0.983995i	$-0.442974\pi$
$719$	9.55635	0.356392	0.178196	+	0.983995i	$0.442974\pi$
$720$	0	0
$721$	0.343146	0.0127794
$722$	33.7279	1.25522
$723$	0	0
$724$	−64.4264	−2.39439
$725$	8.82843	0.327880
$726$	0	0
$727$	−40.7279	−1.51052	−0.755258	−	0.655428i	$-0.772488\pi$
$727$	−40.7279	−1.51052	−0.755258	+	0.655428i	$0.772488\pi$
$728$	5.17157	0.191671
$729$	0	0
$730$	28.1421	1.04159
$731$	−48.2843	−1.78586
$732$	0	0
$733$	−18.8284	−0.695444	−0.347722	−	0.937598i	$-0.613045\pi$
$733$	−18.8284	−0.695444	−0.347722	+	0.937598i	$0.613045\pi$
$734$	53.1127	1.96043
$735$	0	0
$736$	13.0711	0.481806
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−8.87006	−0.326290	−0.163145	−	0.986602i	$-0.552164\pi$
$739$	−8.87006	−0.326290	−0.163145	+	0.986602i	$0.552164\pi$
$740$	14.0000	0.514650
$741$	0	0
$742$	−9.65685	−0.354514
$743$	32.5269	1.19330	0.596648	−	0.802503i	$-0.296499\pi$
$743$	32.5269	1.19330	0.596648	+	0.802503i	$0.296499\pi$
$744$	0	0
$745$	17.3137	0.634325
$746$	43.4558	1.59103
$747$	0	0
$748$	104.569	3.82340
$749$	−1.17157	−0.0428083
$750$	0	0
$751$	−6.34315	−0.231465	−0.115732	−	0.993280i	$-0.536921\pi$
$751$	−6.34315	−0.231465	−0.115732	+	0.993280i	$0.536921\pi$
$752$	14.4853	0.528224
$753$	0	0
$754$	17.6569	0.643025
$755$	−20.3848	−0.741878
$756$	0	0
$757$	34.1421	1.24092	0.620459	−	0.784239i	$-0.286947\pi$
$757$	34.1421	1.24092	0.620459	+	0.784239i	$0.286947\pi$
$758$	−8.24264	−0.299386
$759$	0	0
$760$	−9.89949	−0.359092
$761$	−53.6569	−1.94506	−0.972530	−	0.232779i	$-0.925218\pi$
$761$	−53.6569	−1.94506	−0.972530	+	0.232779i	$0.925218\pi$
$762$	0	0
$763$	−8.97056	−0.324756
$764$	−13.0711	−0.472895
$765$	0	0
$766$	−27.2132	−0.983253
$767$	8.48528	0.306386
$768$	0	0
$769$	29.9411	1.07970	0.539852	−	0.841760i	$-0.318480\pi$
$769$	29.9411	1.07970	0.539852	+	0.841760i	$0.318480\pi$
$770$	−13.6569	−0.492159
$771$	0	0
$772$	16.6274	0.598434
$773$	−18.2843	−0.657640	−0.328820	−	0.944393i	$-0.606651\pi$
$773$	−18.2843	−0.657640	−0.328820	+	0.944393i	$0.606651\pi$
$774$	0	0
$775$	3.41421	0.122642
$776$	−12.4853	−0.448195
$777$	0	0
$778$	−53.1127	−1.90418
$779$	−10.8284	−0.387969
$780$	0	0
$781$	−11.3137	−0.404836
$782$	135.882	4.85914
$783$	0	0
$784$	−15.0000	−0.535714
$785$	2.82843	0.100951
$786$	0	0
$787$	−30.8701	−1.10040	−0.550199	−	0.835033i	$-0.685448\pi$
$787$	−30.8701	−1.10040	−0.550199	+	0.835033i	$0.685448\pi$
$788$	1.31371	0.0467989
$789$	0	0
$790$	−26.1421	−0.930095
$791$	−12.4853	−0.443925
$792$	0	0
$793$	8.28427	0.294183
$794$	−82.0833	−2.91303
$795$	0	0
$796$	−13.4558	−0.476930
$797$	−30.2843	−1.07272	−0.536362	−	0.843988i	$-0.680202\pi$
$797$	−30.2843	−1.07272	−0.536362	+	0.843988i	$0.680202\pi$
$798$	0	0
$799$	−32.9706	−1.16641
$800$	1.58579	0.0560660
$801$	0	0
$802$	7.31371	0.258256
$803$	−46.6274	−1.64545
$804$	0	0
$805$	−11.6569	−0.410850
$806$	6.82843	0.240521
$807$	0	0
$808$	−31.6569	−1.11368
$809$	48.6274	1.70965	0.854824	−	0.518917i	$-0.173665\pi$
$809$	48.6274	1.70965	0.854824	+	0.518917i	$0.173665\pi$
$810$	0	0
$811$	40.4853	1.42163	0.710815	−	0.703379i	$-0.248326\pi$
$811$	40.4853	1.42163	0.710815	+	0.703379i	$0.248326\pi$
$812$	47.7990	1.67742
$813$	0	0
$814$	−35.3137	−1.23774
$815$	−19.5563	−0.685029
$816$	0	0
$817$	−15.8579	−0.554796
$818$	75.5980	2.64322
$819$	0	0
$820$	−18.4853	−0.645534
$821$	−37.9411	−1.32415	−0.662077	−	0.749436i	$-0.730325\pi$
$821$	−37.9411	−1.32415	−0.662077	+	0.749436i	$0.730325\pi$
$822$	0	0
$823$	−26.6863	−0.930226	−0.465113	−	0.885251i	$-0.653986\pi$
$823$	−26.6863	−0.930226	−0.465113	+	0.885251i	$0.653986\pi$
$824$	−1.07107	−0.0373124
$825$	0	0
$826$	34.9706	1.21678
$827$	34.3848	1.19568	0.597838	−	0.801617i	$-0.296027\pi$
$827$	34.3848	1.19568	0.597838	+	0.801617i	$0.296027\pi$
$828$	0	0
$829$	−10.2843	−0.357188	−0.178594	−	0.983923i	$-0.557155\pi$
$829$	−10.2843	−0.357188	−0.178594	+	0.983923i	$0.557155\pi$
$830$	−2.24264	−0.0778432
$831$	0	0
$832$	8.14214	0.282278
$833$	34.1421	1.18295
$834$	0	0
$835$	12.3431	0.427152
$836$	34.3431	1.18778
$837$	0	0
$838$	63.3553	2.18857
$839$	−6.04163	−0.208580	−0.104290	−	0.994547i	$-0.533257\pi$
$839$	−6.04163	−0.208580	−0.104290	+	0.994547i	$0.533257\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	−73.1127	−2.51963
$843$	0	0
$844$	−58.2426	−2.00479
$845$	12.3137	0.423604
$846$	0	0
$847$	7.07107	0.242965
$848$	8.48528	0.291386
$849$	0	0
$850$	16.4853	0.565440
$851$	−30.1421	−1.03326
$852$	0	0
$853$	26.4853	0.906839	0.453419	−	0.891297i	$-0.350204\pi$
$853$	26.4853	0.906839	0.453419	+	0.891297i	$0.350204\pi$
$854$	34.1421	1.16832
$855$	0	0
$856$	3.65685	0.124989
$857$	−23.1716	−0.791526	−0.395763	−	0.918353i	$-0.629520\pi$
$857$	−23.1716	−0.791526	−0.395763	+	0.918353i	$0.629520\pi$
$858$	0	0
$859$	−3.21320	−0.109633	−0.0548165	−	0.998496i	$-0.517457\pi$
$859$	−3.21320	−0.109633	−0.0548165	+	0.998496i	$0.517457\pi$
$860$	−27.0711	−0.923116
$861$	0	0
$862$	82.6690	2.81572
$863$	49.8995	1.69860	0.849299	−	0.527912i	$-0.177025\pi$
$863$	49.8995	1.69860	0.849299	+	0.527912i	$0.177025\pi$
$864$	0	0
$865$	18.9706	0.645018
$866$	62.2843	2.11651
$867$	0	0
$868$	18.4853	0.627431
$869$	43.3137	1.46932
$870$	0	0
$871$	−1.65685	−0.0561404
$872$	28.0000	0.948200
$873$	0	0
$874$	44.6274	1.50954
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	50.2843	1.69798	0.848990	−	0.528410i	$-0.177211\pi$
$877$	50.2843	1.69798	0.848990	+	0.528410i	$0.177211\pi$
$878$	−20.2426	−0.683156
$879$	0	0
$880$	12.0000	0.404520
$881$	−52.9706	−1.78462	−0.892312	−	0.451420i	$-0.850918\pi$
$881$	−52.9706	−1.78462	−0.892312	+	0.451420i	$0.850918\pi$
$882$	0	0
$883$	−1.41421	−0.0475921	−0.0237960	−	0.999717i	$-0.507575\pi$
$883$	−1.41421	−0.0475921	−0.0237960	+	0.999717i	$0.507575\pi$
$884$	21.6569	0.728399
$885$	0	0
$886$	80.4264	2.70198
$887$	−15.5563	−0.522331	−0.261166	−	0.965294i	$-0.584107\pi$
$887$	−15.5563	−0.522331	−0.261166	+	0.965294i	$0.584107\pi$
$888$	0	0
$889$	−15.6569	−0.525114
$890$	2.41421	0.0809246
$891$	0	0
$892$	75.2548	2.51972
$893$	−10.8284	−0.362359
$894$	0	0
$895$	−1.65685	−0.0553825
$896$	29.0711	0.971196
$897$	0	0
$898$	−24.1421	−0.805634
$899$	30.1421	1.00530
$900$	0	0
$901$	−19.3137	−0.643433
$902$	46.6274	1.55252
$903$	0	0
$904$	38.9706	1.29614
$905$	16.8284	0.559396
$906$	0	0
$907$	15.4558	0.513203	0.256601	−	0.966517i	$-0.417397\pi$
$907$	15.4558	0.513203	0.256601	+	0.966517i	$0.417397\pi$
$908$	3.17157	0.105252
$909$	0	0
$910$	−2.82843	−0.0937614
$911$	−48.4853	−1.60639	−0.803195	−	0.595717i	$-0.796868\pi$
$911$	−48.4853	−1.60639	−0.803195	+	0.595717i	$0.796868\pi$
$912$	0	0
$913$	3.71573	0.122973
$914$	10.0000	0.330771
$915$	0	0
$916$	−79.7401	−2.63469
$917$	−28.0000	−0.924641
$918$	0	0
$919$	−37.0711	−1.22286	−0.611431	−	0.791298i	$-0.709406\pi$
$919$	−37.0711	−1.22286	−0.611431	+	0.791298i	$0.709406\pi$
$920$	36.3848	1.19957
$921$	0	0
$922$	−31.3137	−1.03126
$923$	−2.34315	−0.0771256
$924$	0	0
$925$	−3.65685	−0.120237
$926$	−10.0000	−0.328620
$927$	0	0
$928$	14.0000	0.459573
$929$	24.6274	0.807999	0.404000	−	0.914759i	$-0.367620\pi$
$929$	24.6274	0.807999	0.404000	+	0.914759i	$0.367620\pi$
$930$	0	0
$931$	11.2132	0.367498
$932$	72.6274	2.37899
$933$	0	0
$934$	−29.7990	−0.975053
$935$	−27.3137	−0.893254
$936$	0	0
$937$	−37.5980	−1.22827	−0.614136	−	0.789200i	$-0.710495\pi$
$937$	−37.5980	−1.22827	−0.614136	+	0.789200i	$0.710495\pi$
$938$	−6.82843	−0.222956
$939$	0	0
$940$	−18.4853	−0.602923
$941$	−25.1127	−0.818651	−0.409325	−	0.912389i	$-0.634236\pi$
$941$	−25.1127	−0.818651	−0.409325	+	0.912389i	$0.634236\pi$
$942$	0	0
$943$	39.7990	1.29603
$944$	−30.7279	−1.00011
$945$	0	0
$946$	68.2843	2.22011
$947$	46.9706	1.52634	0.763169	−	0.646199i	$-0.223642\pi$
$947$	46.9706	1.52634	0.763169	+	0.646199i	$0.223642\pi$
$948$	0	0
$949$	−9.65685	−0.313475
$950$	5.41421	0.175660
$951$	0	0
$952$	42.6274	1.38156
$953$	58.5685	1.89722	0.948611	−	0.316446i	$-0.102490\pi$
$953$	58.5685	1.89722	0.948611	+	0.316446i	$0.102490\pi$
$954$	0	0
$955$	3.41421	0.110481
$956$	14.9289	0.482836
$957$	0	0
$958$	−6.82843	−0.220616
$959$	−19.7990	−0.639343
$960$	0	0
$961$	−19.3431	−0.623972
$962$	−7.31371	−0.235803
$963$	0	0
$964$	−89.7990	−2.89223
$965$	−4.34315	−0.139811
$966$	0	0
$967$	6.58579	0.211785	0.105892	−	0.994378i	$-0.466230\pi$
$967$	6.58579	0.211785	0.105892	+	0.994378i	$0.466230\pi$
$968$	−22.0711	−0.709391
$969$	0	0
$970$	6.82843	0.219248
$971$	32.2843	1.03605	0.518026	−	0.855365i	$-0.326667\pi$
$971$	32.2843	1.03605	0.518026	+	0.855365i	$0.326667\pi$
$972$	0	0
$973$	30.6274	0.981870
$974$	−92.9117	−2.97708
$975$	0	0
$976$	−30.0000	−0.960277
$977$	7.65685	0.244964	0.122482	−	0.992471i	$-0.460915\pi$
$977$	7.65685	0.244964	0.122482	+	0.992471i	$0.460915\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	19.1421	0.611473
$981$	0	0
$982$	42.3848	1.35255
$983$	−48.3431	−1.54191	−0.770953	−	0.636892i	$-0.780220\pi$
$983$	−48.3431	−1.54191	−0.770953	+	0.636892i	$0.780220\pi$
$984$	0	0
$985$	−0.343146	−0.0109335
$986$	145.539	4.63491
$987$	0	0
$988$	7.11270	0.226285
$989$	58.2843	1.85333
$990$	0	0
$991$	−31.8995	−1.01332	−0.506660	−	0.862146i	$-0.669120\pi$
$991$	−31.8995	−1.01332	−0.506660	+	0.862146i	$0.669120\pi$
$992$	5.41421	0.171901
$993$	0	0
$994$	−9.65685	−0.306297
$995$	3.51472	0.111424
$996$	0	0
$997$	−16.6274	−0.526596	−0.263298	−	0.964715i	$-0.584810\pi$
$997$	−16.6274	−0.526596	−0.263298	+	0.964715i	$0.584810\pi$
$998$	46.3848	1.46828
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.e.1.1		2
3.2	odd	2		445.2.a.c.1.2	✓	2
12.11	even	2		7120.2.a.u.1.2		2
15.14	odd	2		2225.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.c.1.2	✓	2	3.2	odd	2
2225.2.a.c.1.1		2	15.14	odd	2
4005.2.a.e.1.1		2	1.1	even	1	trivial
7120.2.a.u.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.41421 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +1.41421 q^{7} -4.41421 q^{8} +2.41421 q^{10} -4.00000 q^{11} -0.828427 q^{13} -3.41421 q^{14} +3.00000 q^{16} -6.82843 q^{17} -2.24264 q^{19} -3.82843 q^{20} +9.65685 q^{22} +8.24264 q^{23} +1.00000 q^{25} +2.00000 q^{26} +5.41421 q^{28} +8.82843 q^{29} +3.41421 q^{31} +1.58579 q^{32} +16.4853 q^{34} -1.41421 q^{35} -3.65685 q^{37} +5.41421 q^{38} +4.41421 q^{40} +4.82843 q^{41} +7.07107 q^{43} -15.3137 q^{44} -19.8995 q^{46} +4.82843 q^{47} -5.00000 q^{49} -2.41421 q^{50} -3.17157 q^{52} +2.82843 q^{53} +4.00000 q^{55} -6.24264 q^{56} -21.3137 q^{58} -10.2426 q^{59} -10.0000 q^{61} -8.24264 q^{62} -9.82843 q^{64} +0.828427 q^{65} +2.00000 q^{67} -26.1421 q^{68} +3.41421 q^{70} +2.82843 q^{71} +11.6569 q^{73} +8.82843 q^{74} -8.58579 q^{76} -5.65685 q^{77} -10.8284 q^{79} -3.00000 q^{80} -11.6569 q^{82} -0.928932 q^{83} +6.82843 q^{85} -17.0711 q^{86} +17.6569 q^{88} +1.00000 q^{89} -1.17157 q^{91} +31.5563 q^{92} -11.6569 q^{94} +2.24264 q^{95} +2.82843 q^{97} +12.0711 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{8} + 2 q^{10} - 8 q^{11} + 4 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} + 4 q^{19} - 2 q^{20} + 8 q^{22} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 8 q^{28} + 12 q^{29} + 4 q^{31} + 6 q^{32} + 16 q^{34} + 4 q^{37} + 8 q^{38} + 6 q^{40} + 4 q^{41} - 8 q^{44} - 20 q^{46} + 4 q^{47} - 10 q^{49} - 2 q^{50} - 12 q^{52} + 8 q^{55} - 4 q^{56} - 20 q^{58} - 12 q^{59} - 20 q^{61} - 8 q^{62} - 14 q^{64} - 4 q^{65} + 4 q^{67} - 24 q^{68} + 4 q^{70} + 12 q^{73} + 12 q^{74} - 20 q^{76} - 16 q^{79} - 6 q^{80} - 12 q^{82} - 16 q^{83} + 8 q^{85} - 20 q^{86} + 24 q^{88} + 2 q^{89} - 8 q^{91} + 32 q^{92} - 12 q^{94} - 4 q^{95} + 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^8 + 2 * q^10 - 8 * q^11 + 4 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 + 4 * q^19 - 2 * q^20 + 8 * q^22 + 8 * q^23 + 2 * q^25 + 4 * q^26 + 8 * q^28 + 12 * q^29 + 4 * q^31 + 6 * q^32 + 16 * q^34 + 4 * q^37 + 8 * q^38 + 6 * q^40 + 4 * q^41 - 8 * q^44 - 20 * q^46 + 4 * q^47 - 10 * q^49 - 2 * q^50 - 12 * q^52 + 8 * q^55 - 4 * q^56 - 20 * q^58 - 12 * q^59 - 20 * q^61 - 8 * q^62 - 14 * q^64 - 4 * q^65 + 4 * q^67 - 24 * q^68 + 4 * q^70 + 12 * q^73 + 12 * q^74 - 20 * q^76 - 16 * q^79 - 6 * q^80 - 12 * q^82 - 16 * q^83 + 8 * q^85 - 20 * q^86 + 24 * q^88 + 2 * q^89 - 8 * q^91 + 32 * q^92 - 12 * q^94 - 4 * q^95 + 10 * q^98

Introduction

L-functions

Modular forms

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Database

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