LMFDB - Embedded newform 4016.2.a.k.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.33247$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.826533 q^{3} -1.37815 q^{5} -4.67534 q^{7} -2.31684 q^{9} +O(q^{10})$	$q+0.826533 q^{3} -1.37815 q^{5} -4.67534 q^{7} -2.31684 q^{9} -4.79545 q^{11} -1.48170 q^{13} -1.13908 q^{15} -3.39583 q^{17} +6.13909 q^{19} -3.86432 q^{21} +0.540532 q^{23} -3.10071 q^{25} -4.39454 q^{27} -2.60412 q^{29} +4.02908 q^{31} -3.96360 q^{33} +6.44330 q^{35} +8.19680 q^{37} -1.22467 q^{39} +2.41255 q^{41} -9.14924 q^{43} +3.19295 q^{45} +4.35755 q^{47} +14.8588 q^{49} -2.80676 q^{51} -11.0566 q^{53} +6.60884 q^{55} +5.07416 q^{57} -0.771780 q^{59} +14.2235 q^{61} +10.8320 q^{63} +2.04199 q^{65} -12.1606 q^{67} +0.446767 q^{69} +4.60854 q^{71} -8.84979 q^{73} -2.56284 q^{75} +22.4204 q^{77} +2.74719 q^{79} +3.31830 q^{81} -9.21185 q^{83} +4.67995 q^{85} -2.15239 q^{87} +1.72458 q^{89} +6.92743 q^{91} +3.33017 q^{93} -8.46057 q^{95} +10.8626 q^{97} +11.1103 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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 $

$2$	0	0
$2$	0	0
$3$	0.826533	0.477199	0.238599	−	0.971118i	$-0.423312\pi$
$3$	0.826533	0.477199	0.238599	+	0.971118i	$0.423312\pi$
$4$	0	0
$5$	−1.37815	−0.616326	−0.308163	−	0.951334i	$-0.599714\pi$
$5$	−1.37815	−0.616326	−0.308163	+	0.951334i	$0.599714\pi$
$6$	0	0
$7$	−4.67534	−1.76711	−0.883556	−	0.468326i	$-0.844857\pi$
$7$	−4.67534	−1.76711	−0.883556	+	0.468326i	$0.844857\pi$
$8$	0	0
$9$	−2.31684	−0.772281
$10$	0	0
$11$	−4.79545	−1.44588	−0.722942	−	0.690909i	$-0.757211\pi$
$11$	−4.79545	−1.44588	−0.722942	+	0.690909i	$0.757211\pi$
$12$	0	0
$13$	−1.48170	−0.410948	−0.205474	−	0.978663i	$-0.565874\pi$
$13$	−1.48170	−0.410948	−0.205474	+	0.978663i	$0.565874\pi$
$14$	0	0
$15$	−1.13908	−0.294110
$16$	0	0
$17$	−3.39583	−0.823610	−0.411805	−	0.911272i	$-0.635101\pi$
$17$	−3.39583	−0.823610	−0.411805	+	0.911272i	$0.635101\pi$
$18$	0	0
$19$	6.13909	1.40840	0.704202	−	0.710000i	$-0.251305\pi$
$19$	6.13909	1.40840	0.704202	+	0.710000i	$0.251305\pi$
$20$	0	0
$21$	−3.86432	−0.843264
$22$	0	0
$23$	0.540532	0.112709	0.0563543	−	0.998411i	$-0.482052\pi$
$23$	0.540532	0.112709	0.0563543	+	0.998411i	$0.482052\pi$
$24$	0	0
$25$	−3.10071	−0.620143
$26$	0	0
$27$	−4.39454	−0.845731
$28$	0	0
$29$	−2.60412	−0.483573	−0.241787	−	0.970329i	$-0.577733\pi$
$29$	−2.60412	−0.483573	−0.241787	+	0.970329i	$0.577733\pi$
$30$	0	0
$31$	4.02908	0.723645	0.361822	−	0.932247i	$-0.382155\pi$
$31$	4.02908	0.723645	0.361822	+	0.932247i	$0.382155\pi$
$32$	0	0
$33$	−3.96360	−0.689974
$34$	0	0
$35$	6.44330	1.08912
$36$	0	0
$37$	8.19680	1.34755	0.673773	−	0.738939i	$-0.264673\pi$
$37$	8.19680	1.34755	0.673773	+	0.738939i	$0.264673\pi$
$38$	0	0
$39$	−1.22467	−0.196104
$40$	0	0
$41$	2.41255	0.376777	0.188389	−	0.982095i	$-0.439674\pi$
$41$	2.41255	0.376777	0.188389	+	0.982095i	$0.439674\pi$
$42$	0	0
$43$	−9.14924	−1.39525	−0.697623	−	0.716465i	$-0.745759\pi$
$43$	−9.14924	−1.39525	−0.697623	+	0.716465i	$0.745759\pi$
$44$	0	0
$45$	3.19295	0.475977
$46$	0	0
$47$	4.35755	0.635614	0.317807	−	0.948155i	$-0.397054\pi$
$47$	4.35755	0.635614	0.317807	+	0.948155i	$0.397054\pi$
$48$	0	0
$49$	14.8588	2.12268
$50$	0	0
$51$	−2.80676	−0.393026
$52$	0	0
$53$	−11.0566	−1.51874	−0.759370	−	0.650659i	$-0.774493\pi$
$53$	−11.0566	−1.51874	−0.759370	+	0.650659i	$0.774493\pi$
$54$	0	0
$55$	6.60884	0.891135
$56$	0	0
$57$	5.07416	0.672089
$58$	0	0
$59$	−0.771780	−0.100477	−0.0502386	−	0.998737i	$-0.515998\pi$
$59$	−0.771780	−0.100477	−0.0502386	+	0.998737i	$0.515998\pi$
$60$	0	0
$61$	14.2235	1.82114	0.910568	−	0.413360i	$-0.135645\pi$
$61$	14.2235	1.82114	0.910568	+	0.413360i	$0.135645\pi$
$62$	0	0
$63$	10.8320	1.36471
$64$	0	0
$65$	2.04199	0.253278
$66$	0	0
$67$	−12.1606	−1.48566	−0.742829	−	0.669481i	$-0.766517\pi$
$67$	−12.1606	−1.48566	−0.742829	+	0.669481i	$0.766517\pi$
$68$	0	0
$69$	0.446767	0.0537844
$70$	0	0
$71$	4.60854	0.546934	0.273467	−	0.961881i	$-0.411830\pi$
$71$	4.60854	0.546934	0.273467	+	0.961881i	$0.411830\pi$
$72$	0	0
$73$	−8.84979	−1.03579	−0.517895	−	0.855444i	$-0.673284\pi$
$73$	−8.84979	−1.03579	−0.517895	+	0.855444i	$0.673284\pi$
$74$	0	0
$75$	−2.56284	−0.295931
$76$	0	0
$77$	22.4204	2.55504
$78$	0	0
$79$	2.74719	0.309083	0.154542	−	0.987986i	$-0.450610\pi$
$79$	2.74719	0.309083	0.154542	+	0.987986i	$0.450610\pi$
$80$	0	0
$81$	3.31830	0.368700
$82$	0	0
$83$	−9.21185	−1.01113	−0.505566	−	0.862788i	$-0.668716\pi$
$83$	−9.21185	−1.01113	−0.505566	+	0.862788i	$0.668716\pi$
$84$	0	0
$85$	4.67995	0.507612
$86$	0	0
$87$	−2.15239	−0.230761
$88$	0	0
$89$	1.72458	0.182805	0.0914026	−	0.995814i	$-0.470865\pi$
$89$	1.72458	0.182805	0.0914026	+	0.995814i	$0.470865\pi$
$90$	0	0
$91$	6.92743	0.726192
$92$	0	0
$93$	3.33017	0.345323
$94$	0	0
$95$	−8.46057	−0.868036
$96$	0	0
$97$	10.8626	1.10293	0.551463	−	0.834199i	$-0.314070\pi$
$97$	10.8626	1.10293	0.551463	+	0.834199i	$0.314070\pi$
$98$	0	0
$99$	11.1103	1.11663
$100$	0	0
$101$	−13.3192	−1.32531	−0.662653	−	0.748927i	$-0.730569\pi$
$101$	−13.3192	−1.32531	−0.662653	+	0.748927i	$0.730569\pi$
$102$	0	0
$103$	9.80900	0.966510	0.483255	−	0.875480i	$-0.339454\pi$
$103$	9.80900	0.966510	0.483255	+	0.875480i	$0.339454\pi$
$104$	0	0
$105$	5.32560	0.519725
$106$	0	0
$107$	11.6028	1.12169	0.560843	−	0.827922i	$-0.310477\pi$
$107$	11.6028	1.12169	0.560843	+	0.827922i	$0.310477\pi$
$108$	0	0
$109$	15.1738	1.45339	0.726695	−	0.686961i	$-0.241055\pi$
$109$	15.1738	1.45339	0.726695	+	0.686961i	$0.241055\pi$
$110$	0	0
$111$	6.77492	0.643047
$112$	0	0
$113$	1.62757	0.153109	0.0765544	−	0.997065i	$-0.475608\pi$
$113$	1.62757	0.153109	0.0765544	+	0.997065i	$0.475608\pi$
$114$	0	0
$115$	−0.744932	−0.0694652
$116$	0	0
$117$	3.43286	0.317368
$118$	0	0
$119$	15.8767	1.45541
$120$	0	0
$121$	11.9964	1.09058
$122$	0	0
$123$	1.99405	0.179798
$124$	0	0
$125$	11.1640	0.998536
$126$	0	0
$127$	−16.0786	−1.42674	−0.713372	−	0.700786i	$-0.752833\pi$
$127$	−16.0786	−1.42674	−0.713372	+	0.700786i	$0.752833\pi$
$128$	0	0
$129$	−7.56214	−0.665810
$130$	0	0
$131$	7.52003	0.657028	0.328514	−	0.944499i	$-0.393452\pi$
$131$	7.52003	0.657028	0.328514	+	0.944499i	$0.393452\pi$
$132$	0	0
$133$	−28.7023	−2.48881
$134$	0	0
$135$	6.05633	0.521246
$136$	0	0
$137$	−14.9089	−1.27375	−0.636875	−	0.770967i	$-0.719773\pi$
$137$	−14.9089	−1.27375	−0.636875	+	0.770967i	$0.719773\pi$
$138$	0	0
$139$	10.7358	0.910600	0.455300	−	0.890338i	$-0.349532\pi$
$139$	10.7358	0.910600	0.455300	+	0.890338i	$0.349532\pi$
$140$	0	0
$141$	3.60166	0.303314
$142$	0	0
$143$	7.10540	0.594184
$144$	0	0
$145$	3.58886	0.298039
$146$	0	0
$147$	12.2813	1.01294
$148$	0	0
$149$	18.1201	1.48446	0.742229	−	0.670146i	$-0.233768\pi$
$149$	18.1201	1.48446	0.742229	+	0.670146i	$0.233768\pi$
$150$	0	0
$151$	3.98269	0.324107	0.162053	−	0.986782i	$-0.448188\pi$
$151$	3.98269	0.324107	0.162053	+	0.986782i	$0.448188\pi$
$152$	0	0
$153$	7.86761	0.636058
$154$	0	0
$155$	−5.55267	−0.446001
$156$	0	0
$157$	3.79404	0.302797	0.151399	−	0.988473i	$-0.451622\pi$
$157$	3.79404	0.302797	0.151399	+	0.988473i	$0.451622\pi$
$158$	0	0
$159$	−9.13863	−0.724741
$160$	0	0
$161$	−2.52717	−0.199169
$162$	0	0
$163$	−5.96213	−0.466990	−0.233495	−	0.972358i	$-0.575016\pi$
$163$	−5.96213	−0.466990	−0.233495	+	0.972358i	$0.575016\pi$
$164$	0	0
$165$	5.46242	0.425249
$166$	0	0
$167$	−10.3015	−0.797154	−0.398577	−	0.917135i	$-0.630496\pi$
$167$	−10.3015	−0.797154	−0.398577	+	0.917135i	$0.630496\pi$
$168$	0	0
$169$	−10.8046	−0.831121
$170$	0	0
$171$	−14.2233	−1.08768
$172$	0	0
$173$	10.7472	0.817098	0.408549	−	0.912736i	$-0.366035\pi$
$173$	10.7472	0.817098	0.408549	+	0.912736i	$0.366035\pi$
$174$	0	0
$175$	14.4969	1.09586
$176$	0	0
$177$	−0.637901	−0.0479476
$178$	0	0
$179$	−4.25691	−0.318176	−0.159088	−	0.987264i	$-0.550855\pi$
$179$	−4.25691	−0.318176	−0.159088	+	0.987264i	$0.550855\pi$
$180$	0	0
$181$	7.91262	0.588141	0.294070	−	0.955784i	$-0.404990\pi$
$181$	7.91262	0.588141	0.294070	+	0.955784i	$0.404990\pi$
$182$	0	0
$183$	11.7562	0.869044
$184$	0	0
$185$	−11.2964	−0.830527
$186$	0	0
$187$	16.2845	1.19084
$188$	0	0
$189$	20.5460	1.49450
$190$	0	0
$191$	−2.65416	−0.192049	−0.0960243	−	0.995379i	$-0.530613\pi$
$191$	−2.65416	−0.192049	−0.0960243	+	0.995379i	$0.530613\pi$
$192$	0	0
$193$	10.9017	0.784719	0.392359	−	0.919812i	$-0.371659\pi$
$193$	10.9017	0.784719	0.392359	+	0.919812i	$0.371659\pi$
$194$	0	0
$195$	1.68777	0.120864
$196$	0	0
$197$	−9.45662	−0.673756	−0.336878	−	0.941548i	$-0.609371\pi$
$197$	−9.45662	−0.673756	−0.336878	+	0.941548i	$0.609371\pi$
$198$	0	0
$199$	−14.7022	−1.04221	−0.521105	−	0.853493i	$-0.674480\pi$
$199$	−14.7022	−1.04221	−0.521105	+	0.853493i	$0.674480\pi$
$200$	0	0
$201$	−10.0512	−0.708954
$202$	0	0
$203$	12.1751	0.854528
$204$	0	0
$205$	−3.32485	−0.232218
$206$	0	0
$207$	−1.25233	−0.0870428
$208$	0	0
$209$	−29.4397	−2.03639
$210$	0	0
$211$	−18.6771	−1.28578	−0.642891	−	0.765958i	$-0.722265\pi$
$211$	−18.6771	−1.28578	−0.642891	+	0.765958i	$0.722265\pi$
$212$	0	0
$213$	3.80911	0.260996
$214$	0	0
$215$	12.6090	0.859926
$216$	0	0
$217$	−18.8373	−1.27876
$218$	0	0
$219$	−7.31464	−0.494278
$220$	0	0
$221$	5.03159	0.338461
$222$	0	0
$223$	13.5689	0.908641	0.454321	−	0.890838i	$-0.349882\pi$
$223$	13.5689	0.908641	0.454321	+	0.890838i	$0.349882\pi$
$224$	0	0
$225$	7.18387	0.478924
$226$	0	0
$227$	−3.75762	−0.249402	−0.124701	−	0.992194i	$-0.539797\pi$
$227$	−3.75762	−0.249402	−0.124701	+	0.992194i	$0.539797\pi$
$228$	0	0
$229$	−19.1832	−1.26766	−0.633829	−	0.773473i	$-0.718518\pi$
$229$	−19.1832	−1.26766	−0.633829	+	0.773473i	$0.718518\pi$
$230$	0	0
$231$	18.5312	1.21926
$232$	0	0
$233$	−22.7427	−1.48992	−0.744962	−	0.667107i	$-0.767532\pi$
$233$	−22.7427	−1.48992	−0.744962	+	0.667107i	$0.767532\pi$
$234$	0	0
$235$	−6.00535	−0.391746
$236$	0	0
$237$	2.27064	0.147494
$238$	0	0
$239$	−2.77457	−0.179472	−0.0897360	−	0.995966i	$-0.528602\pi$
$239$	−2.77457	−0.179472	−0.0897360	+	0.995966i	$0.528602\pi$
$240$	0	0
$241$	6.73414	0.433784	0.216892	−	0.976196i	$-0.430408\pi$
$241$	6.73414	0.433784	0.216892	+	0.976196i	$0.430408\pi$
$242$	0	0
$243$	15.9263	1.02167
$244$	0	0
$245$	−20.4776	−1.30827
$246$	0	0
$247$	−9.09627	−0.578782
$248$	0	0
$249$	−7.61389	−0.482511
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−2.59209	−0.162964
$254$	0	0
$255$	3.86813	0.242232
$256$	0	0
$257$	3.78037	0.235813	0.117906	−	0.993025i	$-0.462382\pi$
$257$	3.78037	0.235813	0.117906	+	0.993025i	$0.462382\pi$
$258$	0	0
$259$	−38.3228	−2.38126
$260$	0	0
$261$	6.03334	0.373454
$262$	0	0
$263$	15.2806	0.942244	0.471122	−	0.882068i	$-0.343849\pi$
$263$	15.2806	0.942244	0.471122	+	0.882068i	$0.343849\pi$
$264$	0	0
$265$	15.2376	0.936038
$266$	0	0
$267$	1.42542	0.0872344
$268$	0	0
$269$	−12.7953	−0.780144	−0.390072	−	0.920784i	$-0.627550\pi$
$269$	−12.7953	−0.780144	−0.390072	+	0.920784i	$0.627550\pi$
$270$	0	0
$271$	12.6684	0.769553	0.384777	−	0.923010i	$-0.374279\pi$
$271$	12.6684	0.769553	0.384777	+	0.923010i	$0.374279\pi$
$272$	0	0
$273$	5.72575	0.346538
$274$	0	0
$275$	14.8693	0.896654
$276$	0	0
$277$	−23.3434	−1.40257	−0.701286	−	0.712880i	$-0.747390\pi$
$277$	−23.3434	−1.40257	−0.701286	+	0.712880i	$0.747390\pi$
$278$	0	0
$279$	−9.33476	−0.558857
$280$	0	0
$281$	4.24262	0.253093	0.126547	−	0.991961i	$-0.459611\pi$
$281$	4.24262	0.253093	0.126547	+	0.991961i	$0.459611\pi$
$282$	0	0
$283$	−5.57065	−0.331141	−0.165570	−	0.986198i	$-0.552946\pi$
$283$	−5.57065	−0.331141	−0.165570	+	0.986198i	$0.552946\pi$
$284$	0	0
$285$	−6.99293	−0.414226
$286$	0	0
$287$	−11.2795	−0.665808
$288$	0	0
$289$	−5.46834	−0.321667
$290$	0	0
$291$	8.97826	0.526315
$292$	0	0
$293$	21.4801	1.25488	0.627441	−	0.778664i	$-0.284102\pi$
$293$	21.4801	1.25488	0.627441	+	0.778664i	$0.284102\pi$
$294$	0	0
$295$	1.06363	0.0619267
$296$	0	0
$297$	21.0738	1.22283
$298$	0	0
$299$	−0.800903	−0.0463174
$300$	0	0
$301$	42.7758	2.46556
$302$	0	0
$303$	−11.0087	−0.632434
$304$	0	0
$305$	−19.6021	−1.12241
$306$	0	0
$307$	29.8544	1.70388	0.851940	−	0.523639i	$-0.175426\pi$
$307$	29.8544	1.70388	0.851940	+	0.523639i	$0.175426\pi$
$308$	0	0
$309$	8.10746	0.461217
$310$	0	0
$311$	6.31567	0.358129	0.179064	−	0.983837i	$-0.442693\pi$
$311$	6.31567	0.358129	0.179064	+	0.983837i	$0.442693\pi$
$312$	0	0
$313$	−8.82348	−0.498732	−0.249366	−	0.968409i	$-0.580222\pi$
$313$	−8.82348	−0.498732	−0.249366	+	0.968409i	$0.580222\pi$
$314$	0	0
$315$	−14.9281	−0.841104
$316$	0	0
$317$	−31.2074	−1.75278	−0.876392	−	0.481599i	$-0.840056\pi$
$317$	−31.2074	−1.75278	−0.876392	+	0.481599i	$0.840056\pi$
$318$	0	0
$319$	12.4879	0.699191
$320$	0	0
$321$	9.59011	0.535268
$322$	0	0
$323$	−20.8473	−1.15998
$324$	0	0
$325$	4.59431	0.254847
$326$	0	0
$327$	12.5417	0.693556
$328$	0	0
$329$	−20.3730	−1.12320
$330$	0	0
$331$	0.707563	0.0388912	0.0194456	−	0.999811i	$-0.493810\pi$
$331$	0.707563	0.0388912	0.0194456	+	0.999811i	$0.493810\pi$
$332$	0	0
$333$	−18.9907	−1.04068
$334$	0	0
$335$	16.7591	0.915650
$336$	0	0
$337$	−21.4847	−1.17035	−0.585174	−	0.810908i	$-0.698974\pi$
$337$	−21.4847	−1.17035	−0.585174	+	0.810908i	$0.698974\pi$
$338$	0	0
$339$	1.34524	0.0730634
$340$	0	0
$341$	−19.3213	−1.04631
$342$	0	0
$343$	−36.7425	−1.98391
$344$	0	0
$345$	−0.615710	−0.0331487
$346$	0	0
$347$	26.2472	1.40902	0.704512	−	0.709692i	$-0.251166\pi$
$347$	26.2472	1.40902	0.704512	+	0.709692i	$0.251166\pi$
$348$	0	0
$349$	8.26346	0.442333	0.221166	−	0.975236i	$-0.429014\pi$
$349$	8.26346	0.442333	0.221166	+	0.975236i	$0.429014\pi$
$350$	0	0
$351$	6.51138	0.347552
$352$	0	0
$353$	36.6488	1.95062	0.975308	−	0.220847i	$-0.0708822\pi$
$353$	36.6488	1.95062	0.975308	+	0.220847i	$0.0708822\pi$
$354$	0	0
$355$	−6.35125	−0.337089
$356$	0	0
$357$	13.1226	0.694520
$358$	0	0
$359$	0.216670	0.0114354	0.00571771	−	0.999984i	$-0.498180\pi$
$359$	0.216670	0.0114354	0.00571771	+	0.999984i	$0.498180\pi$
$360$	0	0
$361$	18.6884	0.983603
$362$	0	0
$363$	9.91540	0.520423
$364$	0	0
$365$	12.1963	0.638384
$366$	0	0
$367$	22.7098	1.18544	0.592722	−	0.805407i	$-0.298053\pi$
$367$	22.7098	1.18544	0.592722	+	0.805407i	$0.298053\pi$
$368$	0	0
$369$	−5.58951	−0.290978
$370$	0	0
$371$	51.6933	2.68378
$372$	0	0
$373$	12.9320	0.669591	0.334795	−	0.942291i	$-0.391333\pi$
$373$	12.9320	0.669591	0.334795	+	0.942291i	$0.391333\pi$
$374$	0	0
$375$	9.22738	0.476500
$376$	0	0
$377$	3.85851	0.198724
$378$	0	0
$379$	−34.1463	−1.75398	−0.876990	−	0.480508i	$-0.840452\pi$
$379$	−34.1463	−1.75398	−0.876990	+	0.480508i	$0.840452\pi$
$380$	0	0
$381$	−13.2895	−0.680840
$382$	0	0
$383$	−20.8466	−1.06521	−0.532605	−	0.846364i	$-0.678787\pi$
$383$	−20.8466	−1.06521	−0.532605	+	0.846364i	$0.678787\pi$
$384$	0	0
$385$	−30.8986	−1.57474
$386$	0	0
$387$	21.1974	1.07752
$388$	0	0
$389$	17.3699	0.880691	0.440345	−	0.897829i	$-0.354856\pi$
$389$	17.3699	0.880691	0.440345	+	0.897829i	$0.354856\pi$
$390$	0	0
$391$	−1.83555	−0.0928279
$392$	0	0
$393$	6.21555	0.313533
$394$	0	0
$395$	−3.78603	−0.190496
$396$	0	0
$397$	17.6032	0.883480	0.441740	−	0.897143i	$-0.354361\pi$
$397$	17.6032	0.883480	0.441740	+	0.897143i	$0.354361\pi$
$398$	0	0
$399$	−23.7234	−1.18766
$400$	0	0
$401$	−9.30165	−0.464502	−0.232251	−	0.972656i	$-0.574609\pi$
$401$	−9.30165	−0.464502	−0.232251	+	0.972656i	$0.574609\pi$
$402$	0	0
$403$	−5.96988	−0.297381
$404$	0	0
$405$	−4.57310	−0.227239
$406$	0	0
$407$	−39.3074	−1.94839
$408$	0	0
$409$	2.17955	0.107772	0.0538860	−	0.998547i	$-0.482839\pi$
$409$	2.17955	0.107772	0.0538860	+	0.998547i	$0.482839\pi$
$410$	0	0
$411$	−12.3227	−0.607832
$412$	0	0
$413$	3.60833	0.177554
$414$	0	0
$415$	12.6953	0.623186
$416$	0	0
$417$	8.87350	0.434537
$418$	0	0
$419$	−18.0142	−0.880052	−0.440026	−	0.897985i	$-0.645031\pi$
$419$	−18.0142	−0.880052	−0.440026	+	0.897985i	$0.645031\pi$
$420$	0	0
$421$	30.3245	1.47793	0.738964	−	0.673745i	$-0.235315\pi$
$421$	30.3245	1.47793	0.738964	+	0.673745i	$0.235315\pi$
$422$	0	0
$423$	−10.0958	−0.490873
$424$	0	0
$425$	10.5295	0.510755
$426$	0	0
$427$	−66.4998	−3.21815
$428$	0	0
$429$	5.87285	0.283544
$430$	0	0
$431$	−28.4048	−1.36821	−0.684106	−	0.729383i	$-0.739807\pi$
$431$	−28.4048	−1.36821	−0.684106	+	0.729383i	$0.739807\pi$
$432$	0	0
$433$	−23.4572	−1.12728	−0.563640	−	0.826020i	$-0.690600\pi$
$433$	−23.4572	−1.12728	−0.563640	+	0.826020i	$0.690600\pi$
$434$	0	0
$435$	2.96631	0.142224
$436$	0	0
$437$	3.31837	0.158739
$438$	0	0
$439$	18.8960	0.901857	0.450928	−	0.892560i	$-0.351093\pi$
$439$	18.8960	0.901857	0.450928	+	0.892560i	$0.351093\pi$
$440$	0	0
$441$	−34.4255	−1.63931
$442$	0	0
$443$	−25.8710	−1.22917	−0.614585	−	0.788851i	$-0.710676\pi$
$443$	−25.8710	−1.22917	−0.614585	+	0.788851i	$0.710676\pi$
$444$	0	0
$445$	−2.37672	−0.112668
$446$	0	0
$447$	14.9769	0.708382
$448$	0	0
$449$	−18.1943	−0.858640	−0.429320	−	0.903152i	$-0.641247\pi$
$449$	−18.1943	−0.858640	−0.429320	+	0.903152i	$0.641247\pi$
$450$	0	0
$451$	−11.5693	−0.544776
$452$	0	0
$453$	3.29182	0.154663
$454$	0	0
$455$	−9.54701	−0.447571
$456$	0	0
$457$	−25.4097	−1.18862	−0.594309	−	0.804237i	$-0.702574\pi$
$457$	−25.4097	−1.18862	−0.594309	+	0.804237i	$0.702574\pi$
$458$	0	0
$459$	14.9231	0.696552
$460$	0	0
$461$	31.0543	1.44635	0.723173	−	0.690667i	$-0.242683\pi$
$461$	31.0543	1.44635	0.723173	+	0.690667i	$0.242683\pi$
$462$	0	0
$463$	−26.3934	−1.22660	−0.613302	−	0.789848i	$-0.710159\pi$
$463$	−26.3934	−1.22660	−0.613302	+	0.789848i	$0.710159\pi$
$464$	0	0
$465$	−4.58946	−0.212831
$466$	0	0
$467$	29.3449	1.35792	0.678959	−	0.734176i	$-0.262431\pi$
$467$	29.3449	1.35792	0.678959	+	0.734176i	$0.262431\pi$
$468$	0	0
$469$	56.8551	2.62532
$470$	0	0
$471$	3.13589	0.144494
$472$	0	0
$473$	43.8748	2.01736
$474$	0	0
$475$	−19.0356	−0.873411
$476$	0	0
$477$	25.6164	1.17289
$478$	0	0
$479$	−38.2099	−1.74586	−0.872929	−	0.487848i	$-0.837782\pi$
$479$	−38.2099	−1.74586	−0.872929	+	0.487848i	$0.837782\pi$
$480$	0	0
$481$	−12.1452	−0.553772
$482$	0	0
$483$	−2.08879	−0.0950431
$484$	0	0
$485$	−14.9702	−0.679762
$486$	0	0
$487$	1.75387	0.0794753	0.0397377	−	0.999210i	$-0.487348\pi$
$487$	1.75387	0.0794753	0.0397377	+	0.999210i	$0.487348\pi$
$488$	0	0
$489$	−4.92790	−0.222847
$490$	0	0
$491$	27.9400	1.26091	0.630457	−	0.776224i	$-0.282867\pi$
$491$	27.9400	1.26091	0.630457	+	0.776224i	$0.282867\pi$
$492$	0	0
$493$	8.84315	0.398276
$494$	0	0
$495$	−15.3116	−0.688207
$496$	0	0
$497$	−21.5465	−0.966493
$498$	0	0
$499$	27.3791	1.22566	0.612828	−	0.790217i	$-0.290032\pi$
$499$	27.3791	1.22566	0.612828	+	0.790217i	$0.290032\pi$
$500$	0	0
$501$	−8.51452	−0.380401
$502$	0	0
$503$	36.2303	1.61543	0.807715	−	0.589573i	$-0.200704\pi$
$503$	36.2303	1.61543	0.807715	+	0.589573i	$0.200704\pi$
$504$	0	0
$505$	18.3557	0.816820
$506$	0	0
$507$	−8.93034	−0.396610
$508$	0	0
$509$	31.0374	1.37571	0.687855	−	0.725849i	$-0.258553\pi$
$509$	31.0374	1.37571	0.687855	+	0.725849i	$0.258553\pi$
$510$	0	0
$511$	41.3758	1.83036
$512$	0	0
$513$	−26.9785	−1.19113
$514$	0	0
$515$	−13.5182	−0.595685
$516$	0	0
$517$	−20.8964	−0.919025
$518$	0	0
$519$	8.88295	0.389918
$520$	0	0
$521$	29.5391	1.29413	0.647067	−	0.762433i	$-0.275995\pi$
$521$	29.5391	1.29413	0.647067	+	0.762433i	$0.275995\pi$
$522$	0	0
$523$	31.5481	1.37950	0.689751	−	0.724047i	$-0.257720\pi$
$523$	31.5481	1.37950	0.689751	+	0.724047i	$0.257720\pi$
$524$	0	0
$525$	11.9821	0.522944
$526$	0	0
$527$	−13.6821	−0.596001
$528$	0	0
$529$	−22.7078	−0.987297
$530$	0	0
$531$	1.78809	0.0775966
$532$	0	0
$533$	−3.57467	−0.154836
$534$	0	0
$535$	−15.9904	−0.691324
$536$	0	0
$537$	−3.51847	−0.151833
$538$	0	0
$539$	−71.2547	−3.06916
$540$	0	0
$541$	−21.0191	−0.903681	−0.451841	−	0.892099i	$-0.649232\pi$
$541$	−21.0191	−0.903681	−0.451841	+	0.892099i	$0.649232\pi$
$542$	0	0
$543$	6.54004	0.280660
$544$	0	0
$545$	−20.9118	−0.895761
$546$	0	0
$547$	12.4321	0.531559	0.265779	−	0.964034i	$-0.414371\pi$
$547$	12.4321	0.531559	0.265779	+	0.964034i	$0.414371\pi$
$548$	0	0
$549$	−32.9537	−1.40643
$550$	0	0
$551$	−15.9869	−0.681066
$552$	0	0
$553$	−12.8441	−0.546185
$554$	0	0
$555$	−9.33683	−0.396326
$556$	0	0
$557$	−20.0824	−0.850918	−0.425459	−	0.904978i	$-0.639887\pi$
$557$	−20.0824	−0.850918	−0.425459	+	0.904978i	$0.639887\pi$
$558$	0	0
$559$	13.5564	0.573374
$560$	0	0
$561$	13.4597	0.568269
$562$	0	0
$563$	33.4060	1.40789	0.703947	−	0.710253i	$-0.251419\pi$
$563$	33.4060	1.40789	0.703947	+	0.710253i	$0.251419\pi$
$564$	0	0
$565$	−2.24303	−0.0943650
$566$	0	0
$567$	−15.5142	−0.651534
$568$	0	0
$569$	4.41418	0.185052	0.0925260	−	0.995710i	$-0.470506\pi$
$569$	4.41418	0.185052	0.0925260	+	0.995710i	$0.470506\pi$
$570$	0	0
$571$	27.3257	1.14355	0.571773	−	0.820412i	$-0.306256\pi$
$571$	27.3257	1.14355	0.571773	+	0.820412i	$0.306256\pi$
$572$	0	0
$573$	−2.19375	−0.0916454
$574$	0	0
$575$	−1.67603	−0.0698954
$576$	0	0
$577$	34.6849	1.44395	0.721976	−	0.691918i	$-0.243234\pi$
$577$	34.6849	1.44395	0.721976	+	0.691918i	$0.243234\pi$
$578$	0	0
$579$	9.01058	0.374467
$580$	0	0
$581$	43.0685	1.78678
$582$	0	0
$583$	53.0214	2.19592
$584$	0	0
$585$	−4.73098	−0.195602
$586$	0	0
$587$	−3.80883	−0.157207	−0.0786036	−	0.996906i	$-0.525046\pi$
$587$	−3.80883	−0.157207	−0.0786036	+	0.996906i	$0.525046\pi$
$588$	0	0
$589$	24.7349	1.01918
$590$	0	0
$591$	−7.81620	−0.321516
$592$	0	0
$593$	−14.7444	−0.605479	−0.302740	−	0.953073i	$-0.597901\pi$
$593$	−14.7444	−0.605479	−0.302740	+	0.953073i	$0.597901\pi$
$594$	0	0
$595$	−21.8804	−0.897007
$596$	0	0
$597$	−12.1518	−0.497341
$598$	0	0
$599$	24.3311	0.994141	0.497070	−	0.867710i	$-0.334409\pi$
$599$	24.3311	0.994141	0.497070	+	0.867710i	$0.334409\pi$
$600$	0	0
$601$	42.4191	1.73031	0.865155	−	0.501504i	$-0.167220\pi$
$601$	42.4191	1.73031	0.865155	+	0.501504i	$0.167220\pi$
$602$	0	0
$603$	28.1743	1.14735
$604$	0	0
$605$	−16.5328	−0.672153
$606$	0	0
$607$	19.3021	0.783450	0.391725	−	0.920082i	$-0.371878\pi$
$607$	19.3021	0.783450	0.391725	+	0.920082i	$0.371878\pi$
$608$	0	0
$609$	10.0632	0.407780
$610$	0	0
$611$	−6.45657	−0.261205
$612$	0	0
$613$	−7.67537	−0.310005	−0.155003	−	0.987914i	$-0.549539\pi$
$613$	−7.67537	−0.310005	−0.155003	+	0.987914i	$0.549539\pi$
$614$	0	0
$615$	−2.74810	−0.110814
$616$	0	0
$617$	−6.82581	−0.274797	−0.137398	−	0.990516i	$-0.543874\pi$
$617$	−6.82581	−0.274797	−0.137398	+	0.990516i	$0.543874\pi$
$618$	0	0
$619$	−19.0207	−0.764507	−0.382253	−	0.924058i	$-0.624852\pi$
$619$	−19.0207	−0.764507	−0.382253	+	0.924058i	$0.624852\pi$
$620$	0	0
$621$	−2.37539	−0.0953211
$622$	0	0
$623$	−8.06300	−0.323037
$624$	0	0
$625$	0.117982	0.00471928
$626$	0	0
$627$	−24.3329	−0.971762
$628$	0	0
$629$	−27.8349	−1.10985
$630$	0	0
$631$	−29.6464	−1.18021	−0.590103	−	0.807328i	$-0.700913\pi$
$631$	−29.6464	−1.18021	−0.590103	+	0.807328i	$0.700913\pi$
$632$	0	0
$633$	−15.4372	−0.613574
$634$	0	0
$635$	22.1586	0.879339
$636$	0	0
$637$	−22.0162	−0.872314
$638$	0	0
$639$	−10.6773	−0.422387
$640$	0	0
$641$	7.26129	0.286804	0.143402	−	0.989665i	$-0.454196\pi$
$641$	7.26129	0.286804	0.143402	+	0.989665i	$0.454196\pi$
$642$	0	0
$643$	21.0418	0.829807	0.414904	−	0.909865i	$-0.363815\pi$
$643$	21.0418	0.829807	0.414904	+	0.909865i	$0.363815\pi$
$644$	0	0
$645$	10.4217	0.410356
$646$	0	0
$647$	−16.6025	−0.652714	−0.326357	−	0.945247i	$-0.605821\pi$
$647$	−16.6025	−0.652714	−0.326357	+	0.945247i	$0.605821\pi$
$648$	0	0
$649$	3.70103	0.145278
$650$	0	0
$651$	−15.5697	−0.610224
$652$	0	0
$653$	17.5973	0.688634	0.344317	−	0.938853i	$-0.388110\pi$
$653$	17.5973	0.688634	0.344317	+	0.938853i	$0.388110\pi$
$654$	0	0
$655$	−10.3637	−0.404944
$656$	0	0
$657$	20.5036	0.799921
$658$	0	0
$659$	9.37072	0.365032	0.182516	−	0.983203i	$-0.441576\pi$
$659$	9.37072	0.365032	0.182516	+	0.983203i	$0.441576\pi$
$660$	0	0
$661$	10.7131	0.416690	0.208345	−	0.978055i	$-0.433192\pi$
$661$	10.7131	0.416690	0.208345	+	0.978055i	$0.433192\pi$
$662$	0	0
$663$	4.15877	0.161513
$664$	0	0
$665$	39.5560	1.53392
$666$	0	0
$667$	−1.40761	−0.0545029
$668$	0	0
$669$	11.2151	0.433603
$670$	0	0
$671$	−68.2082	−2.63315
$672$	0	0
$673$	5.14194	0.198207	0.0991037	−	0.995077i	$-0.468402\pi$
$673$	5.14194	0.198207	0.0991037	+	0.995077i	$0.468402\pi$
$674$	0	0
$675$	13.6262	0.524473
$676$	0	0
$677$	−20.0704	−0.771367	−0.385684	−	0.922631i	$-0.626034\pi$
$677$	−20.0704	−0.771367	−0.385684	+	0.922631i	$0.626034\pi$
$678$	0	0
$679$	−50.7861	−1.94899
$680$	0	0
$681$	−3.10580	−0.119014
$682$	0	0
$683$	38.3195	1.46625	0.733127	−	0.680092i	$-0.238060\pi$
$683$	38.3195	1.46625	0.733127	+	0.680092i	$0.238060\pi$
$684$	0	0
$685$	20.5466	0.785045
$686$	0	0
$687$	−15.8555	−0.604925
$688$	0	0
$689$	16.3825	0.624124
$690$	0	0
$691$	−12.5976	−0.479236	−0.239618	−	0.970867i	$-0.577022\pi$
$691$	−12.5976	−0.479236	−0.239618	+	0.970867i	$0.577022\pi$
$692$	0	0
$693$	−51.9445	−1.97321
$694$	0	0
$695$	−14.7955	−0.561226
$696$	0	0
$697$	−8.19262	−0.310317
$698$	0	0
$699$	−18.7976	−0.710990
$700$	0	0
$701$	8.48130	0.320334	0.160167	−	0.987090i	$-0.448797\pi$
$701$	8.48130	0.320334	0.160167	+	0.987090i	$0.448797\pi$
$702$	0	0
$703$	50.3209	1.89789
$704$	0	0
$705$	−4.96361	−0.186941
$706$	0	0
$707$	62.2716	2.34196
$708$	0	0
$709$	−1.95421	−0.0733920	−0.0366960	−	0.999326i	$-0.511683\pi$
$709$	−1.95421	−0.0733920	−0.0366960	+	0.999326i	$0.511683\pi$
$710$	0	0
$711$	−6.36481	−0.238699
$712$	0	0
$713$	2.17785	0.0815610
$714$	0	0
$715$	−9.79229	−0.366211
$716$	0	0
$717$	−2.29327	−0.0856438
$718$	0	0
$719$	−43.9434	−1.63881	−0.819405	−	0.573214i	$-0.805696\pi$
$719$	−43.9434	−1.63881	−0.819405	+	0.573214i	$0.805696\pi$
$720$	0	0
$721$	−45.8604	−1.70793
$722$	0	0
$723$	5.56599	0.207001
$724$	0	0
$725$	8.07463	0.299884
$726$	0	0
$727$	9.24345	0.342821	0.171410	−	0.985200i	$-0.445168\pi$
$727$	9.24345	0.342821	0.171410	+	0.985200i	$0.445168\pi$
$728$	0	0
$729$	3.20873	0.118842
$730$	0	0
$731$	31.0693	1.14914
$732$	0	0
$733$	−23.9232	−0.883623	−0.441811	−	0.897108i	$-0.645664\pi$
$733$	−23.9232	−0.883623	−0.441811	+	0.897108i	$0.645664\pi$
$734$	0	0
$735$	−16.9254	−0.624303
$736$	0	0
$737$	58.3158	2.14809
$738$	0	0
$739$	9.97029	0.366763	0.183382	−	0.983042i	$-0.441296\pi$
$739$	9.97029	0.366763	0.183382	+	0.983042i	$0.441296\pi$
$740$	0	0
$741$	−7.51836	−0.276194
$742$	0	0
$743$	18.3014	0.671413	0.335706	−	0.941967i	$-0.391025\pi$
$743$	18.3014	0.671413	0.335706	+	0.941967i	$0.391025\pi$
$744$	0	0
$745$	−24.9722	−0.914910
$746$	0	0
$747$	21.3424	0.780878
$748$	0	0
$749$	−54.2471	−1.98215
$750$	0	0
$751$	44.3612	1.61876	0.809381	−	0.587284i	$-0.199803\pi$
$751$	44.3612	1.61876	0.809381	+	0.587284i	$0.199803\pi$
$752$	0	0
$753$	−0.826533	−0.0301205
$754$	0	0
$755$	−5.48873	−0.199755
$756$	0	0
$757$	−3.16492	−0.115031	−0.0575155	−	0.998345i	$-0.518318\pi$
$757$	−3.16492	−0.115031	−0.0575155	+	0.998345i	$0.518318\pi$
$758$	0	0
$759$	−2.14245	−0.0777660
$760$	0	0
$761$	−24.4739	−0.887178	−0.443589	−	0.896230i	$-0.646295\pi$
$761$	−24.4739	−0.887178	−0.443589	+	0.896230i	$0.646295\pi$
$762$	0	0
$763$	−70.9428	−2.56830
$764$	0	0
$765$	−10.8427	−0.392019
$766$	0	0
$767$	1.14354	0.0412909
$768$	0	0
$769$	21.2775	0.767287	0.383644	−	0.923481i	$-0.374669\pi$
$769$	21.2775	0.767287	0.383644	+	0.923481i	$0.374669\pi$
$770$	0	0
$771$	3.12460	0.112530
$772$	0	0
$773$	−15.5626	−0.559748	−0.279874	−	0.960037i	$-0.590293\pi$
$773$	−15.5626	−0.559748	−0.279874	+	0.960037i	$0.590293\pi$
$774$	0	0
$775$	−12.4930	−0.448763
$776$	0	0
$777$	−31.6750	−1.13634
$778$	0	0
$779$	14.8109	0.530655
$780$	0	0
$781$	−22.1001	−0.790802
$782$	0	0
$783$	11.4439	0.408973
$784$	0	0
$785$	−5.22874	−0.186622
$786$	0	0
$787$	−53.0601	−1.89139	−0.945694	−	0.325057i	$-0.894616\pi$
$787$	−53.0601	−1.89139	−0.945694	+	0.325057i	$0.894616\pi$
$788$	0	0
$789$	12.6299	0.449638
$790$	0	0
$791$	−7.60944	−0.270561
$792$	0	0
$793$	−21.0749	−0.748393
$794$	0	0
$795$	12.5944	0.446676
$796$	0	0
$797$	3.25592	0.115330	0.0576652	−	0.998336i	$-0.481634\pi$
$797$	3.25592	0.115330	0.0576652	+	0.998336i	$0.481634\pi$
$798$	0	0
$799$	−14.7975	−0.523498
$800$	0	0
$801$	−3.99558	−0.141177
$802$	0	0
$803$	42.4388	1.49763
$804$	0	0
$805$	3.48281	0.122753
$806$	0	0
$807$	−10.5757	−0.372284
$808$	0	0
$809$	−47.9691	−1.68650	−0.843251	−	0.537519i	$-0.819361\pi$
$809$	−47.9691	−1.68650	−0.843251	+	0.537519i	$0.819361\pi$
$810$	0	0
$811$	4.82090	0.169285	0.0846424	−	0.996411i	$-0.473025\pi$
$811$	4.82090	0.169285	0.0846424	+	0.996411i	$0.473025\pi$
$812$	0	0
$813$	10.4709	0.367230
$814$	0	0
$815$	8.21669	0.287818
$816$	0	0
$817$	−56.1680	−1.96507
$818$	0	0
$819$	−16.0498	−0.560824
$820$	0	0
$821$	13.7123	0.478563	0.239282	−	0.970950i	$-0.423088\pi$
$821$	13.7123	0.478563	0.239282	+	0.970950i	$0.423088\pi$
$822$	0	0
$823$	42.4079	1.47825	0.739123	−	0.673571i	$-0.235240\pi$
$823$	42.4079	1.47825	0.739123	+	0.673571i	$0.235240\pi$
$824$	0	0
$825$	12.2900	0.427882
$826$	0	0
$827$	18.9008	0.657245	0.328622	−	0.944461i	$-0.393416\pi$
$827$	18.9008	0.657245	0.328622	+	0.944461i	$0.393416\pi$
$828$	0	0
$829$	40.2622	1.39836	0.699181	−	0.714945i	$-0.253548\pi$
$829$	40.2622	1.39836	0.699181	+	0.714945i	$0.253548\pi$
$830$	0	0
$831$	−19.2941	−0.669305
$832$	0	0
$833$	−50.4579	−1.74826
$834$	0	0
$835$	14.1970	0.491306
$836$	0	0
$837$	−17.7060	−0.612009
$838$	0	0
$839$	−35.0091	−1.20865	−0.604323	−	0.796739i	$-0.706556\pi$
$839$	−35.0091	−1.20865	−0.604323	+	0.796739i	$0.706556\pi$
$840$	0	0
$841$	−22.2186	−0.766157
$842$	0	0
$843$	3.50666	0.120776
$844$	0	0
$845$	14.8903	0.512242
$846$	0	0
$847$	−56.0871	−1.92718
$848$	0	0
$849$	−4.60432	−0.158020
$850$	0	0
$851$	4.43063	0.151880
$852$	0	0
$853$	39.5842	1.35534	0.677668	−	0.735368i	$-0.262991\pi$
$853$	39.5842	1.35534	0.677668	+	0.735368i	$0.262991\pi$
$854$	0	0
$855$	19.6018	0.670368
$856$	0	0
$857$	15.7463	0.537884	0.268942	−	0.963156i	$-0.413326\pi$
$857$	15.7463	0.537884	0.268942	+	0.963156i	$0.413326\pi$
$858$	0	0
$859$	31.4857	1.07428	0.537140	−	0.843493i	$-0.319505\pi$
$859$	31.4857	1.07428	0.537140	+	0.843493i	$0.319505\pi$
$860$	0	0
$861$	−9.32287	−0.317723
$862$	0	0
$863$	−10.3171	−0.351198	−0.175599	−	0.984462i	$-0.556186\pi$
$863$	−10.3171	−0.351198	−0.175599	+	0.984462i	$0.556186\pi$
$864$	0	0
$865$	−14.8113	−0.503599
$866$	0	0
$867$	−4.51976	−0.153499
$868$	0	0
$869$	−13.1740	−0.446898
$870$	0	0
$871$	18.0184	0.610529
$872$	0	0
$873$	−25.1669	−0.851769
$874$	0	0
$875$	−52.1953	−1.76452
$876$	0	0
$877$	30.8259	1.04092	0.520459	−	0.853887i	$-0.325761\pi$
$877$	30.8259	1.04092	0.520459	+	0.853887i	$0.325761\pi$
$878$	0	0
$879$	17.7540	0.598828
$880$	0	0
$881$	−12.9657	−0.436826	−0.218413	−	0.975856i	$-0.570088\pi$
$881$	−12.9657	−0.436826	−0.218413	+	0.975856i	$0.570088\pi$
$882$	0	0
$883$	1.58580	0.0533663	0.0266831	−	0.999644i	$-0.491505\pi$
$883$	1.58580	0.0533663	0.0266831	+	0.999644i	$0.491505\pi$
$884$	0	0
$885$	0.879121	0.0295513
$886$	0	0
$887$	12.7748	0.428934	0.214467	−	0.976731i	$-0.431199\pi$
$887$	12.7748	0.428934	0.214467	+	0.976731i	$0.431199\pi$
$888$	0	0
$889$	75.1728	2.52122
$890$	0	0
$891$	−15.9127	−0.533097
$892$	0	0
$893$	26.7514	0.895202
$894$	0	0
$895$	5.86664	0.196100
$896$	0	0
$897$	−0.661973	−0.0221026
$898$	0	0
$899$	−10.4922	−0.349935
$900$	0	0
$901$	37.5463	1.25085
$902$	0	0
$903$	35.3556	1.17656
$904$	0	0
$905$	−10.9048	−0.362486
$906$	0	0
$907$	8.72375	0.289667	0.144834	−	0.989456i	$-0.453735\pi$
$907$	8.72375	0.289667	0.144834	+	0.989456i	$0.453735\pi$
$908$	0	0
$909$	30.8584	1.02351
$910$	0	0
$911$	−13.9591	−0.462487	−0.231244	−	0.972896i	$-0.574279\pi$
$911$	−13.9591	−0.462487	−0.231244	+	0.972896i	$0.574279\pi$
$912$	0	0
$913$	44.1750	1.46198
$914$	0	0
$915$	−16.2018	−0.535614
$916$	0	0
$917$	−35.1587	−1.16104
$918$	0	0
$919$	20.4640	0.675047	0.337523	−	0.941317i	$-0.390411\pi$
$919$	20.4640	0.675047	0.337523	+	0.941317i	$0.390411\pi$
$920$	0	0
$921$	24.6756	0.813090
$922$	0	0
$923$	−6.82846	−0.224761
$924$	0	0
$925$	−25.4159	−0.835670
$926$	0	0
$927$	−22.7259	−0.746417
$928$	0	0
$929$	−47.1277	−1.54621	−0.773105	−	0.634278i	$-0.781297\pi$
$929$	−47.1277	−1.54621	−0.773105	+	0.634278i	$0.781297\pi$
$930$	0	0
$931$	91.2195	2.98960
$932$	0	0
$933$	5.22011	0.170899
$934$	0	0
$935$	−22.4425	−0.733948
$936$	0	0
$937$	−11.2832	−0.368607	−0.184303	−	0.982869i	$-0.559003\pi$
$937$	−11.2832	−0.368607	−0.184303	+	0.982869i	$0.559003\pi$
$938$	0	0
$939$	−7.29289	−0.237994
$940$	0	0
$941$	−10.9960	−0.358458	−0.179229	−	0.983807i	$-0.557360\pi$
$941$	−10.9960	−0.358458	−0.179229	+	0.983807i	$0.557360\pi$
$942$	0	0
$943$	1.30406	0.0424661
$944$	0	0
$945$	−28.3154	−0.921099
$946$	0	0
$947$	−11.6560	−0.378770	−0.189385	−	0.981903i	$-0.560649\pi$
$947$	−11.6560	−0.378770	−0.189385	+	0.981903i	$0.560649\pi$
$948$	0	0
$949$	13.1127	0.425656
$950$	0	0
$951$	−25.7940	−0.836426
$952$	0	0
$953$	−49.3806	−1.59959	−0.799797	−	0.600270i	$-0.795060\pi$
$953$	−49.3806	−1.59959	−0.799797	+	0.600270i	$0.795060\pi$
$954$	0	0
$955$	3.65783	0.118365
$956$	0	0
$957$	10.3217	0.333653
$958$	0	0
$959$	69.7040	2.25086
$960$	0	0
$961$	−14.7665	−0.476338
$962$	0	0
$963$	−26.8819	−0.866258
$964$	0	0
$965$	−15.0241	−0.483642
$966$	0	0
$967$	−23.9305	−0.769553	−0.384777	−	0.923010i	$-0.625721\pi$
$967$	−23.9305	−0.769553	−0.384777	+	0.923010i	$0.625721\pi$
$968$	0	0
$969$	−17.2310	−0.553539
$970$	0	0
$971$	5.06740	0.162621	0.0813103	−	0.996689i	$-0.474090\pi$
$971$	5.06740	0.162621	0.0813103	+	0.996689i	$0.474090\pi$
$972$	0	0
$973$	−50.1936	−1.60913
$974$	0	0
$975$	3.79735	0.121613
$976$	0	0
$977$	−3.84517	−0.123018	−0.0615089	−	0.998107i	$-0.519591\pi$
$977$	−3.84517	−0.123018	−0.0615089	+	0.998107i	$0.519591\pi$
$978$	0	0
$979$	−8.27015	−0.264315
$980$	0	0
$981$	−35.1554	−1.12243
$982$	0	0
$983$	16.5742	0.528635	0.264317	−	0.964436i	$-0.414853\pi$
$983$	16.5742	0.528635	0.264317	+	0.964436i	$0.414853\pi$
$984$	0	0
$985$	13.0326	0.415253
$986$	0	0
$987$	−16.8390	−0.535991
$988$	0	0
$989$	−4.94545	−0.157256
$990$	0	0
$991$	−26.7043	−0.848291	−0.424145	−	0.905594i	$-0.639425\pi$
$991$	−26.7043	−0.848291	−0.424145	+	0.905594i	$0.639425\pi$
$992$	0	0
$993$	0.584824	0.0185588
$994$	0	0
$995$	20.2618	0.642341
$996$	0	0
$997$	23.8178	0.754316	0.377158	−	0.926149i	$-0.376901\pi$
$997$	23.8178	0.754316	0.377158	+	0.926149i	$0.376901\pi$
$998$	0	0
$999$	−36.0212	−1.13966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.11		17
4.3	odd	2		251.2.a.b.1.15	✓	17
12.11	even	2		2259.2.a.k.1.3		17
20.19	odd	2		6275.2.a.e.1.3		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.15	✓	17	4.3	odd	2
2259.2.a.k.1.3		17	12.11	even	2
4016.2.a.k.1.11		17	1.1	even	1	trivial
6275.2.a.e.1.3		17	20.19	odd	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+0.826533 q^{3} -1.37815 q^{5} -4.67534 q^{7} -2.31684 q^{9} +O(q^{10})\)	\(q+0.826533 q^{3} -1.37815 q^{5} -4.67534 q^{7} -2.31684 q^{9} -4.79545 q^{11} -1.48170 q^{13} -1.13908 q^{15} -3.39583 q^{17} +6.13909 q^{19} -3.86432 q^{21} +0.540532 q^{23} -3.10071 q^{25} -4.39454 q^{27} -2.60412 q^{29} +4.02908 q^{31} -3.96360 q^{33} +6.44330 q^{35} +8.19680 q^{37} -1.22467 q^{39} +2.41255 q^{41} -9.14924 q^{43} +3.19295 q^{45} +4.35755 q^{47} +14.8588 q^{49} -2.80676 q^{51} -11.0566 q^{53} +6.60884 q^{55} +5.07416 q^{57} -0.771780 q^{59} +14.2235 q^{61} +10.8320 q^{63} +2.04199 q^{65} -12.1606 q^{67} +0.446767 q^{69} +4.60854 q^{71} -8.84979 q^{73} -2.56284 q^{75} +22.4204 q^{77} +2.74719 q^{79} +3.31830 q^{81} -9.21185 q^{83} +4.67995 q^{85} -2.15239 q^{87} +1.72458 q^{89} +6.92743 q^{91} +3.33017 q^{93} -8.46057 q^{95} +10.8626 q^{97} +11.1103 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more