LMFDB - Embedded newform 8040.2.a.t.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 3x^{4} + 43x^{3} - 6x^{2} - 29x + 6 $ x^7 - x^6 - 15x^5 + 3x^4 + 43x^3 - 6x^2 - 29*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.84959$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +4.47701 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +4.47701 q^{7} +1.00000 q^{9} +3.66054 q^{11} +3.81438 q^{13} -1.00000 q^{15} -6.96978 q^{17} -0.0738657 q^{19} +4.47701 q^{21} -7.13234 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.538723 q^{29} +2.70128 q^{31} +3.66054 q^{33} -4.47701 q^{35} +8.36592 q^{37} +3.81438 q^{39} -0.701282 q^{41} +10.9467 q^{43} -1.00000 q^{45} +2.26292 q^{47} +13.0436 q^{49} -6.96978 q^{51} +1.74875 q^{53} -3.66054 q^{55} -0.0738657 q^{57} +0.807367 q^{59} -4.46218 q^{61} +4.47701 q^{63} -3.81438 q^{65} +1.00000 q^{67} -7.13234 q^{69} -4.11630 q^{71} -1.83177 q^{73} +1.00000 q^{75} +16.3883 q^{77} +4.17995 q^{79} +1.00000 q^{81} +3.56198 q^{83} +6.96978 q^{85} +0.538723 q^{87} +8.57170 q^{89} +17.0770 q^{91} +2.70128 q^{93} +0.0738657 q^{95} +9.19906 q^{97} +3.66054 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.47701	1.69215	0.846075	−	0.533063i	$-0.178959\pi$
$7$	4.47701	1.69215	0.846075	+	0.533063i	$0.178959\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.66054	1.10369	0.551847	−	0.833945i	$-0.313923\pi$
$11$	3.66054	1.10369	0.551847	+	0.833945i	$0.313923\pi$
$12$	0	0
$13$	3.81438	1.05792	0.528959	−	0.848648i	$-0.322583\pi$
$13$	3.81438	1.05792	0.528959	+	0.848648i	$0.322583\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.96978	−1.69042	−0.845210	−	0.534434i	$-0.820525\pi$
$17$	−6.96978	−1.69042	−0.845210	+	0.534434i	$0.820525\pi$
$18$	0	0
$19$	−0.0738657	−0.0169459	−0.00847297	−	0.999964i	$-0.502697\pi$
$19$	−0.0738657	−0.0169459	−0.00847297	+	0.999964i	$0.502697\pi$
$20$	0	0
$21$	4.47701	0.976964
$22$	0	0
$23$	−7.13234	−1.48720	−0.743598	−	0.668627i	$-0.766882\pi$
$23$	−7.13234	−1.48720	−0.743598	+	0.668627i	$0.766882\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.538723	0.100038	0.0500192	−	0.998748i	$-0.484072\pi$
$29$	0.538723	0.100038	0.0500192	+	0.998748i	$0.484072\pi$
$30$	0	0
$31$	2.70128	0.485165	0.242582	−	0.970131i	$-0.422006\pi$
$31$	2.70128	0.485165	0.242582	+	0.970131i	$0.422006\pi$
$32$	0	0
$33$	3.66054	0.637218
$34$	0	0
$35$	−4.47701	−0.756753
$36$	0	0
$37$	8.36592	1.37535	0.687674	−	0.726020i	$-0.258632\pi$
$37$	8.36592	1.37535	0.687674	+	0.726020i	$0.258632\pi$
$38$	0	0
$39$	3.81438	0.610789
$40$	0	0
$41$	−0.701282	−0.109522	−0.0547609	−	0.998499i	$-0.517440\pi$
$41$	−0.701282	−0.109522	−0.0547609	+	0.998499i	$0.517440\pi$
$42$	0	0
$43$	10.9467	1.66936	0.834679	−	0.550737i	$-0.185653\pi$
$43$	10.9467	1.66936	0.834679	+	0.550737i	$0.185653\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	2.26292	0.330081	0.165040	−	0.986287i	$-0.447225\pi$
$47$	2.26292	0.330081	0.165040	+	0.986287i	$0.447225\pi$
$48$	0	0
$49$	13.0436	1.86337
$50$	0	0
$51$	−6.96978	−0.975964
$52$	0	0
$53$	1.74875	0.240209	0.120105	−	0.992761i	$-0.461677\pi$
$53$	1.74875	0.240209	0.120105	+	0.992761i	$0.461677\pi$
$54$	0	0
$55$	−3.66054	−0.493587
$56$	0	0
$57$	−0.0738657	−0.00978375
$58$	0	0
$59$	0.807367	0.105110	0.0525551	−	0.998618i	$-0.483263\pi$
$59$	0.807367	0.105110	0.0525551	+	0.998618i	$0.483263\pi$
$60$	0	0
$61$	−4.46218	−0.571324	−0.285662	−	0.958330i	$-0.592213\pi$
$61$	−4.46218	−0.571324	−0.285662	+	0.958330i	$0.592213\pi$
$62$	0	0
$63$	4.47701	0.564050
$64$	0	0
$65$	−3.81438	−0.473115
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−7.13234	−0.858633
$70$	0	0
$71$	−4.11630	−0.488515	−0.244257	−	0.969710i	$-0.578544\pi$
$71$	−4.11630	−0.488515	−0.244257	+	0.969710i	$0.578544\pi$
$72$	0	0
$73$	−1.83177	−0.214392	−0.107196	−	0.994238i	$-0.534187\pi$
$73$	−1.83177	−0.214392	−0.107196	+	0.994238i	$0.534187\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	16.3883	1.86762
$78$	0	0
$79$	4.17995	0.470281	0.235141	−	0.971961i	$-0.424445\pi$
$79$	4.17995	0.470281	0.235141	+	0.971961i	$0.424445\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.56198	0.390978	0.195489	−	0.980706i	$-0.437371\pi$
$83$	3.56198	0.390978	0.195489	+	0.980706i	$0.437371\pi$
$84$	0	0
$85$	6.96978	0.755979
$86$	0	0
$87$	0.538723	0.0577572
$88$	0	0
$89$	8.57170	0.908598	0.454299	−	0.890849i	$-0.349890\pi$
$89$	8.57170	0.908598	0.454299	+	0.890849i	$0.349890\pi$
$90$	0	0
$91$	17.0770	1.79016
$92$	0	0
$93$	2.70128	0.280110
$94$	0	0
$95$	0.0738657	0.00757846
$96$	0	0
$97$	9.19906	0.934023	0.467012	−	0.884251i	$-0.345331\pi$
$97$	9.19906	0.934023	0.467012	+	0.884251i	$0.345331\pi$
$98$	0	0
$99$	3.66054	0.367898
$100$	0	0
$101$	0.553173	0.0550428	0.0275214	−	0.999621i	$-0.491239\pi$
$101$	0.553173	0.0550428	0.0275214	+	0.999621i	$0.491239\pi$
$102$	0	0
$103$	−8.79837	−0.866930	−0.433465	−	0.901170i	$-0.642709\pi$
$103$	−8.79837	−0.866930	−0.433465	+	0.901170i	$0.642709\pi$
$104$	0	0
$105$	−4.47701	−0.436911
$106$	0	0
$107$	−17.3544	−1.67772	−0.838858	−	0.544350i	$-0.816776\pi$
$107$	−17.3544	−1.67772	−0.838858	+	0.544350i	$0.816776\pi$
$108$	0	0
$109$	5.89757	0.564885	0.282443	−	0.959284i	$-0.408855\pi$
$109$	5.89757	0.564885	0.282443	+	0.959284i	$0.408855\pi$
$110$	0	0
$111$	8.36592	0.794058
$112$	0	0
$113$	14.1199	1.32829	0.664146	−	0.747603i	$-0.268795\pi$
$113$	14.1199	1.32829	0.664146	+	0.747603i	$0.268795\pi$
$114$	0	0
$115$	7.13234	0.665094
$116$	0	0
$117$	3.81438	0.352639
$118$	0	0
$119$	−31.2038	−2.86045
$120$	0	0
$121$	2.39956	0.218142
$122$	0	0
$123$	−0.701282	−0.0632324
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.15524	0.102511	0.0512556	−	0.998686i	$-0.483678\pi$
$127$	1.15524	0.102511	0.0512556	+	0.998686i	$0.483678\pi$
$128$	0	0
$129$	10.9467	0.963804
$130$	0	0
$131$	−11.2772	−0.985298	−0.492649	−	0.870228i	$-0.663971\pi$
$131$	−11.2772	−0.985298	−0.492649	+	0.870228i	$0.663971\pi$
$132$	0	0
$133$	−0.330697	−0.0286751
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	16.9837	1.45102	0.725509	−	0.688213i	$-0.241604\pi$
$137$	16.9837	1.45102	0.725509	+	0.688213i	$0.241604\pi$
$138$	0	0
$139$	13.1266	1.11338	0.556692	−	0.830719i	$-0.312070\pi$
$139$	13.1266	1.11338	0.556692	+	0.830719i	$0.312070\pi$
$140$	0	0
$141$	2.26292	0.190572
$142$	0	0
$143$	13.9627	1.16762
$144$	0	0
$145$	−0.538723	−0.0447385
$146$	0	0
$147$	13.0436	1.07582
$148$	0	0
$149$	11.3420	0.929170	0.464585	−	0.885529i	$-0.346204\pi$
$149$	11.3420	0.929170	0.464585	+	0.885529i	$0.346204\pi$
$150$	0	0
$151$	−10.0048	−0.814183	−0.407091	−	0.913387i	$-0.633457\pi$
$151$	−10.0048	−0.814183	−0.407091	+	0.913387i	$0.633457\pi$
$152$	0	0
$153$	−6.96978	−0.563473
$154$	0	0
$155$	−2.70128	−0.216972
$156$	0	0
$157$	−11.3646	−0.906992	−0.453496	−	0.891258i	$-0.649823\pi$
$157$	−11.3646	−0.906992	−0.453496	+	0.891258i	$0.649823\pi$
$158$	0	0
$159$	1.74875	0.138685
$160$	0	0
$161$	−31.9316	−2.51656
$162$	0	0
$163$	−18.0171	−1.41121	−0.705604	−	0.708607i	$-0.749324\pi$
$163$	−18.0171	−1.41121	−0.705604	+	0.708607i	$0.749324\pi$
$164$	0	0
$165$	−3.66054	−0.284973
$166$	0	0
$167$	4.40114	0.340570	0.170285	−	0.985395i	$-0.445531\pi$
$167$	4.40114	0.340570	0.170285	+	0.985395i	$0.445531\pi$
$168$	0	0
$169$	1.54946	0.119189
$170$	0	0
$171$	−0.0738657	−0.00564865
$172$	0	0
$173$	10.7769	0.819350	0.409675	−	0.912232i	$-0.365642\pi$
$173$	10.7769	0.819350	0.409675	+	0.912232i	$0.365642\pi$
$174$	0	0
$175$	4.47701	0.338430
$176$	0	0
$177$	0.807367	0.0606854
$178$	0	0
$179$	11.8254	0.883870	0.441935	−	0.897047i	$-0.354292\pi$
$179$	11.8254	0.883870	0.441935	+	0.897047i	$0.354292\pi$
$180$	0	0
$181$	−7.20241	−0.535351	−0.267675	−	0.963509i	$-0.586256\pi$
$181$	−7.20241	−0.535351	−0.267675	+	0.963509i	$0.586256\pi$
$182$	0	0
$183$	−4.46218	−0.329854
$184$	0	0
$185$	−8.36592	−0.615074
$186$	0	0
$187$	−25.5132	−1.86571
$188$	0	0
$189$	4.47701	0.325655
$190$	0	0
$191$	−16.1036	−1.16522	−0.582608	−	0.812753i	$-0.697968\pi$
$191$	−16.1036	−1.16522	−0.582608	+	0.812753i	$0.697968\pi$
$192$	0	0
$193$	−11.7082	−0.842774	−0.421387	−	0.906881i	$-0.638457\pi$
$193$	−11.7082	−0.842774	−0.421387	+	0.906881i	$0.638457\pi$
$194$	0	0
$195$	−3.81438	−0.273153
$196$	0	0
$197$	2.23632	0.159331	0.0796655	−	0.996822i	$-0.474615\pi$
$197$	2.23632	0.159331	0.0796655	+	0.996822i	$0.474615\pi$
$198$	0	0
$199$	−21.7620	−1.54266	−0.771332	−	0.636433i	$-0.780409\pi$
$199$	−21.7620	−1.54266	−0.771332	+	0.636433i	$0.780409\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	2.41187	0.169280
$204$	0	0
$205$	0.701282	0.0489796
$206$	0	0
$207$	−7.13234	−0.495732
$208$	0	0
$209$	−0.270388	−0.0187032
$210$	0	0
$211$	−0.840526	−0.0578642	−0.0289321	−	0.999581i	$-0.509211\pi$
$211$	−0.840526	−0.0578642	−0.0289321	+	0.999581i	$0.509211\pi$
$212$	0	0
$213$	−4.11630	−0.282044
$214$	0	0
$215$	−10.9467	−0.746560
$216$	0	0
$217$	12.0937	0.820971
$218$	0	0
$219$	−1.83177	−0.123779
$220$	0	0
$221$	−26.5854	−1.78832
$222$	0	0
$223$	12.0649	0.807927	0.403964	−	0.914775i	$-0.367632\pi$
$223$	12.0649	0.807927	0.403964	+	0.914775i	$0.367632\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	21.4614	1.42444	0.712222	−	0.701954i	$-0.247689\pi$
$227$	21.4614	1.42444	0.712222	+	0.701954i	$0.247689\pi$
$228$	0	0
$229$	3.73690	0.246941	0.123471	−	0.992348i	$-0.460598\pi$
$229$	3.73690	0.246941	0.123471	+	0.992348i	$0.460598\pi$
$230$	0	0
$231$	16.3883	1.07827
$232$	0	0
$233$	−23.9633	−1.56989	−0.784945	−	0.619566i	$-0.787309\pi$
$233$	−23.9633	−1.56989	−0.784945	+	0.619566i	$0.787309\pi$
$234$	0	0
$235$	−2.26292	−0.147617
$236$	0	0
$237$	4.17995	0.271517
$238$	0	0
$239$	−4.11678	−0.266293	−0.133146	−	0.991096i	$-0.542508\pi$
$239$	−4.11678	−0.266293	−0.133146	+	0.991096i	$0.542508\pi$
$240$	0	0
$241$	−24.3695	−1.56978	−0.784888	−	0.619638i	$-0.787279\pi$
$241$	−24.3695	−1.56978	−0.784888	+	0.619638i	$0.787279\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−13.0436	−0.833326
$246$	0	0
$247$	−0.281751	−0.0179274
$248$	0	0
$249$	3.56198	0.225731
$250$	0	0
$251$	21.5904	1.36277	0.681387	−	0.731924i	$-0.261377\pi$
$251$	21.5904	1.36277	0.681387	+	0.731924i	$0.261377\pi$
$252$	0	0
$253$	−26.1082	−1.64141
$254$	0	0
$255$	6.96978	0.436465
$256$	0	0
$257$	20.1990	1.25998	0.629989	−	0.776604i	$-0.283059\pi$
$257$	20.1990	1.25998	0.629989	+	0.776604i	$0.283059\pi$
$258$	0	0
$259$	37.4543	2.32730
$260$	0	0
$261$	0.538723	0.0333461
$262$	0	0
$263$	−2.28006	−0.140594	−0.0702972	−	0.997526i	$-0.522395\pi$
$263$	−2.28006	−0.140594	−0.0702972	+	0.997526i	$0.522395\pi$
$264$	0	0
$265$	−1.74875	−0.107425
$266$	0	0
$267$	8.57170	0.524579
$268$	0	0
$269$	3.40768	0.207770	0.103885	−	0.994589i	$-0.466873\pi$
$269$	3.40768	0.207770	0.103885	+	0.994589i	$0.466873\pi$
$270$	0	0
$271$	31.2376	1.89755	0.948776	−	0.315951i	$-0.102323\pi$
$271$	31.2376	1.89755	0.948776	+	0.315951i	$0.102323\pi$
$272$	0	0
$273$	17.0770	1.03355
$274$	0	0
$275$	3.66054	0.220739
$276$	0	0
$277$	14.8526	0.892404	0.446202	−	0.894932i	$-0.352776\pi$
$277$	14.8526	0.892404	0.446202	+	0.894932i	$0.352776\pi$
$278$	0	0
$279$	2.70128	0.161722
$280$	0	0
$281$	−9.64114	−0.575142	−0.287571	−	0.957759i	$-0.592848\pi$
$281$	−9.64114	−0.575142	−0.287571	+	0.957759i	$0.592848\pi$
$282$	0	0
$283$	−13.4963	−0.802269	−0.401135	−	0.916019i	$-0.631384\pi$
$283$	−13.4963	−0.802269	−0.401135	+	0.916019i	$0.631384\pi$
$284$	0	0
$285$	0.0738657	0.00437542
$286$	0	0
$287$	−3.13965	−0.185327
$288$	0	0
$289$	31.5778	1.85752
$290$	0	0
$291$	9.19906	0.539258
$292$	0	0
$293$	26.4275	1.54391	0.771957	−	0.635675i	$-0.219278\pi$
$293$	26.4275	1.54391	0.771957	+	0.635675i	$0.219278\pi$
$294$	0	0
$295$	−0.807367	−0.0470067
$296$	0	0
$297$	3.66054	0.212406
$298$	0	0
$299$	−27.2054	−1.57333
$300$	0	0
$301$	49.0085	2.82481
$302$	0	0
$303$	0.553173	0.0317790
$304$	0	0
$305$	4.46218	0.255504
$306$	0	0
$307$	−12.9674	−0.740089	−0.370045	−	0.929014i	$-0.620658\pi$
$307$	−12.9674	−0.740089	−0.370045	+	0.929014i	$0.620658\pi$
$308$	0	0
$309$	−8.79837	−0.500522
$310$	0	0
$311$	12.8951	0.731212	0.365606	−	0.930770i	$-0.380862\pi$
$311$	12.8951	0.731212	0.365606	+	0.930770i	$0.380862\pi$
$312$	0	0
$313$	−0.760285	−0.0429739	−0.0214869	−	0.999769i	$-0.506840\pi$
$313$	−0.760285	−0.0429739	−0.0214869	+	0.999769i	$0.506840\pi$
$314$	0	0
$315$	−4.47701	−0.252251
$316$	0	0
$317$	−20.9773	−1.17820	−0.589102	−	0.808059i	$-0.700518\pi$
$317$	−20.9773	−1.17820	−0.589102	+	0.808059i	$0.700518\pi$
$318$	0	0
$319$	1.97202	0.110412
$320$	0	0
$321$	−17.3544	−0.968630
$322$	0	0
$323$	0.514827	0.0286458
$324$	0	0
$325$	3.81438	0.211583
$326$	0	0
$327$	5.89757	0.326137
$328$	0	0
$329$	10.1311	0.558546
$330$	0	0
$331$	14.2592	0.783757	0.391879	−	0.920017i	$-0.371825\pi$
$331$	14.2592	0.783757	0.391879	+	0.920017i	$0.371825\pi$
$332$	0	0
$333$	8.36592	0.458449
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	2.54691	0.138739	0.0693695	−	0.997591i	$-0.477901\pi$
$337$	2.54691	0.138739	0.0693695	+	0.997591i	$0.477901\pi$
$338$	0	0
$339$	14.1199	0.766890
$340$	0	0
$341$	9.88815	0.535474
$342$	0	0
$343$	27.0573	1.46096
$344$	0	0
$345$	7.13234	0.383992
$346$	0	0
$347$	−29.2444	−1.56992	−0.784960	−	0.619547i	$-0.787316\pi$
$347$	−29.2444	−1.56992	−0.784960	+	0.619547i	$0.787316\pi$
$348$	0	0
$349$	4.54414	0.243242	0.121621	−	0.992577i	$-0.461191\pi$
$349$	4.54414	0.243242	0.121621	+	0.992577i	$0.461191\pi$
$350$	0	0
$351$	3.81438	0.203596
$352$	0	0
$353$	−23.3749	−1.24412	−0.622059	−	0.782970i	$-0.713704\pi$
$353$	−23.3749	−1.24412	−0.622059	+	0.782970i	$0.713704\pi$
$354$	0	0
$355$	4.11630	0.218470
$356$	0	0
$357$	−31.2038	−1.65148
$358$	0	0
$359$	−31.2550	−1.64958	−0.824788	−	0.565442i	$-0.808706\pi$
$359$	−31.2550	−1.64958	−0.824788	+	0.565442i	$0.808706\pi$
$360$	0	0
$361$	−18.9945	−0.999713
$362$	0	0
$363$	2.39956	0.125944
$364$	0	0
$365$	1.83177	0.0958791
$366$	0	0
$367$	−4.48696	−0.234218	−0.117109	−	0.993119i	$-0.537363\pi$
$367$	−4.48696	−0.234218	−0.117109	+	0.993119i	$0.537363\pi$
$368$	0	0
$369$	−0.701282	−0.0365073
$370$	0	0
$371$	7.82917	0.406470
$372$	0	0
$373$	5.35029	0.277027	0.138514	−	0.990361i	$-0.455768\pi$
$373$	5.35029	0.277027	0.138514	+	0.990361i	$0.455768\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	2.05489	0.105832
$378$	0	0
$379$	20.3794	1.04682	0.523409	−	0.852082i	$-0.324660\pi$
$379$	20.3794	1.04682	0.523409	+	0.852082i	$0.324660\pi$
$380$	0	0
$381$	1.15524	0.0591848
$382$	0	0
$383$	13.9367	0.712131	0.356065	−	0.934461i	$-0.384118\pi$
$383$	13.9367	0.712131	0.356065	+	0.934461i	$0.384118\pi$
$384$	0	0
$385$	−16.3883	−0.835224
$386$	0	0
$387$	10.9467	0.556453
$388$	0	0
$389$	−6.34967	−0.321941	−0.160971	−	0.986959i	$-0.551462\pi$
$389$	−6.34967	−0.321941	−0.160971	+	0.986959i	$0.551462\pi$
$390$	0	0
$391$	49.7108	2.51398
$392$	0	0
$393$	−11.2772	−0.568862
$394$	0	0
$395$	−4.17995	−0.210316
$396$	0	0
$397$	−22.1296	−1.11065	−0.555325	−	0.831633i	$-0.687406\pi$
$397$	−22.1296	−1.11065	−0.555325	+	0.831633i	$0.687406\pi$
$398$	0	0
$399$	−0.330697	−0.0165556
$400$	0	0
$401$	6.79478	0.339315	0.169658	−	0.985503i	$-0.445734\pi$
$401$	6.79478	0.339315	0.169658	+	0.985503i	$0.445734\pi$
$402$	0	0
$403$	10.3037	0.513264
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	30.6238	1.51796
$408$	0	0
$409$	−7.68364	−0.379932	−0.189966	−	0.981791i	$-0.560838\pi$
$409$	−7.68364	−0.379932	−0.189966	+	0.981791i	$0.560838\pi$
$410$	0	0
$411$	16.9837	0.837746
$412$	0	0
$413$	3.61459	0.177862
$414$	0	0
$415$	−3.56198	−0.174851
$416$	0	0
$417$	13.1266	0.642813
$418$	0	0
$419$	27.4364	1.34036	0.670179	−	0.742200i	$-0.266217\pi$
$419$	27.4364	1.34036	0.670179	+	0.742200i	$0.266217\pi$
$420$	0	0
$421$	−0.859225	−0.0418761	−0.0209380	−	0.999781i	$-0.506665\pi$
$421$	−0.859225	−0.0418761	−0.0209380	+	0.999781i	$0.506665\pi$
$422$	0	0
$423$	2.26292	0.110027
$424$	0	0
$425$	−6.96978	−0.338084
$426$	0	0
$427$	−19.9772	−0.966766
$428$	0	0
$429$	13.9627	0.674124
$430$	0	0
$431$	34.8375	1.67806	0.839032	−	0.544082i	$-0.183122\pi$
$431$	34.8375	1.67806	0.839032	+	0.544082i	$0.183122\pi$
$432$	0	0
$433$	−21.1636	−1.01706	−0.508529	−	0.861045i	$-0.669810\pi$
$433$	−21.1636	−1.01706	−0.508529	+	0.861045i	$0.669810\pi$
$434$	0	0
$435$	−0.538723	−0.0258298
$436$	0	0
$437$	0.526835	0.0252019
$438$	0	0
$439$	35.1116	1.67579	0.837893	−	0.545835i	$-0.183787\pi$
$439$	35.1116	1.67579	0.837893	+	0.545835i	$0.183787\pi$
$440$	0	0
$441$	13.0436	0.621125
$442$	0	0
$443$	−0.532835	−0.0253158	−0.0126579	−	0.999920i	$-0.504029\pi$
$443$	−0.532835	−0.0253158	−0.0126579	+	0.999920i	$0.504029\pi$
$444$	0	0
$445$	−8.57170	−0.406337
$446$	0	0
$447$	11.3420	0.536456
$448$	0	0
$449$	7.04264	0.332363	0.166181	−	0.986095i	$-0.446856\pi$
$449$	7.04264	0.332363	0.166181	+	0.986095i	$0.446856\pi$
$450$	0	0
$451$	−2.56707	−0.120879
$452$	0	0
$453$	−10.0048	−0.470069
$454$	0	0
$455$	−17.0770	−0.800582
$456$	0	0
$457$	8.31756	0.389079	0.194540	−	0.980895i	$-0.437679\pi$
$457$	8.31756	0.389079	0.194540	+	0.980895i	$0.437679\pi$
$458$	0	0
$459$	−6.96978	−0.325321
$460$	0	0
$461$	23.0483	1.07347	0.536733	−	0.843752i	$-0.319658\pi$
$461$	23.0483	1.07347	0.536733	+	0.843752i	$0.319658\pi$
$462$	0	0
$463$	14.8080	0.688186	0.344093	−	0.938936i	$-0.388186\pi$
$463$	14.8080	0.688186	0.344093	+	0.938936i	$0.388186\pi$
$464$	0	0
$465$	−2.70128	−0.125269
$466$	0	0
$467$	40.1110	1.85611	0.928057	−	0.372438i	$-0.121478\pi$
$467$	40.1110	1.85611	0.928057	+	0.372438i	$0.121478\pi$
$468$	0	0
$469$	4.47701	0.206729
$470$	0	0
$471$	−11.3646	−0.523652
$472$	0	0
$473$	40.0709	1.84246
$474$	0	0
$475$	−0.0738657	−0.00338919
$476$	0	0
$477$	1.74875	0.0800697
$478$	0	0
$479$	−26.8175	−1.22532	−0.612662	−	0.790345i	$-0.709901\pi$
$479$	−26.8175	−1.22532	−0.612662	+	0.790345i	$0.709901\pi$
$480$	0	0
$481$	31.9107	1.45500
$482$	0	0
$483$	−31.9316	−1.45294
$484$	0	0
$485$	−9.19906	−0.417708
$486$	0	0
$487$	37.0018	1.67671	0.838355	−	0.545124i	$-0.183517\pi$
$487$	37.0018	1.67671	0.838355	+	0.545124i	$0.183517\pi$
$488$	0	0
$489$	−18.0171	−0.814761
$490$	0	0
$491$	8.43305	0.380578	0.190289	−	0.981728i	$-0.439057\pi$
$491$	8.43305	0.380578	0.190289	+	0.981728i	$0.439057\pi$
$492$	0	0
$493$	−3.75478	−0.169107
$494$	0	0
$495$	−3.66054	−0.164529
$496$	0	0
$497$	−18.4287	−0.826640
$498$	0	0
$499$	−17.9648	−0.804214	−0.402107	−	0.915593i	$-0.631722\pi$
$499$	−17.9648	−0.804214	−0.402107	+	0.915593i	$0.631722\pi$
$500$	0	0
$501$	4.40114	0.196628
$502$	0	0
$503$	−16.6984	−0.744546	−0.372273	−	0.928123i	$-0.621421\pi$
$503$	−16.6984	−0.744546	−0.372273	+	0.928123i	$0.621421\pi$
$504$	0	0
$505$	−0.553173	−0.0246159
$506$	0	0
$507$	1.54946	0.0688138
$508$	0	0
$509$	20.9368	0.928006	0.464003	−	0.885834i	$-0.346413\pi$
$509$	20.9368	0.928006	0.464003	+	0.885834i	$0.346413\pi$
$510$	0	0
$511$	−8.20084	−0.362784
$512$	0	0
$513$	−0.0738657	−0.00326125
$514$	0	0
$515$	8.79837	0.387703
$516$	0	0
$517$	8.28351	0.364308
$518$	0	0
$519$	10.7769	0.473052
$520$	0	0
$521$	−3.56844	−0.156336	−0.0781681	−	0.996940i	$-0.524907\pi$
$521$	−3.56844	−0.156336	−0.0781681	+	0.996940i	$0.524907\pi$
$522$	0	0
$523$	−11.4821	−0.502077	−0.251039	−	0.967977i	$-0.580772\pi$
$523$	−11.4821	−0.502077	−0.251039	+	0.967977i	$0.580772\pi$
$524$	0	0
$525$	4.47701	0.195393
$526$	0	0
$527$	−18.8273	−0.820132
$528$	0	0
$529$	27.8703	1.21175
$530$	0	0
$531$	0.807367	0.0350367
$532$	0	0
$533$	−2.67495	−0.115865
$534$	0	0
$535$	17.3544	0.750298
$536$	0	0
$537$	11.8254	0.510303
$538$	0	0
$539$	47.7467	2.05660
$540$	0	0
$541$	−33.6718	−1.44766	−0.723832	−	0.689977i	$-0.757621\pi$
$541$	−33.6718	−1.44766	−0.723832	+	0.689977i	$0.757621\pi$
$542$	0	0
$543$	−7.20241	−0.309085
$544$	0	0
$545$	−5.89757	−0.252624
$546$	0	0
$547$	−37.7650	−1.61472	−0.807358	−	0.590062i	$-0.799103\pi$
$547$	−37.7650	−1.61472	−0.807358	+	0.590062i	$0.799103\pi$
$548$	0	0
$549$	−4.46218	−0.190441
$550$	0	0
$551$	−0.0397931	−0.00169525
$552$	0	0
$553$	18.7137	0.795787
$554$	0	0
$555$	−8.36592	−0.355113
$556$	0	0
$557$	−32.3987	−1.37278	−0.686388	−	0.727236i	$-0.740805\pi$
$557$	−32.3987	−1.37278	−0.686388	+	0.727236i	$0.740805\pi$
$558$	0	0
$559$	41.7549	1.76604
$560$	0	0
$561$	−25.5132	−1.07717
$562$	0	0
$563$	45.1969	1.90482	0.952411	−	0.304818i	$-0.0985956\pi$
$563$	45.1969	1.90482	0.952411	+	0.304818i	$0.0985956\pi$
$564$	0	0
$565$	−14.1199	−0.594031
$566$	0	0
$567$	4.47701	0.188017
$568$	0	0
$569$	−26.9752	−1.13086	−0.565429	−	0.824797i	$-0.691290\pi$
$569$	−26.9752	−1.13086	−0.565429	+	0.824797i	$0.691290\pi$
$570$	0	0
$571$	−19.6813	−0.823639	−0.411819	−	0.911266i	$-0.635107\pi$
$571$	−19.6813	−0.823639	−0.411819	+	0.911266i	$0.635107\pi$
$572$	0	0
$573$	−16.1036	−0.672738
$574$	0	0
$575$	−7.13234	−0.297439
$576$	0	0
$577$	35.2013	1.46545	0.732724	−	0.680526i	$-0.238248\pi$
$577$	35.2013	1.46545	0.732724	+	0.680526i	$0.238248\pi$
$578$	0	0
$579$	−11.7082	−0.486576
$580$	0	0
$581$	15.9470	0.661593
$582$	0	0
$583$	6.40137	0.265118
$584$	0	0
$585$	−3.81438	−0.157705
$586$	0	0
$587$	−36.4737	−1.50543	−0.752715	−	0.658346i	$-0.771256\pi$
$587$	−36.4737	−1.50543	−0.752715	+	0.658346i	$0.771256\pi$
$588$	0	0
$589$	−0.199532	−0.00822157
$590$	0	0
$591$	2.23632	0.0919897
$592$	0	0
$593$	−8.44653	−0.346857	−0.173429	−	0.984846i	$-0.555485\pi$
$593$	−8.44653	−0.346857	−0.173429	+	0.984846i	$0.555485\pi$
$594$	0	0
$595$	31.2038	1.27923
$596$	0	0
$597$	−21.7620	−0.890658
$598$	0	0
$599$	1.24306	0.0507899	0.0253949	−	0.999677i	$-0.491916\pi$
$599$	1.24306	0.0507899	0.0253949	+	0.999677i	$0.491916\pi$
$600$	0	0
$601$	−22.9241	−0.935092	−0.467546	−	0.883969i	$-0.654862\pi$
$601$	−22.9241	−0.935092	−0.467546	+	0.883969i	$0.654862\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−2.39956	−0.0975561
$606$	0	0
$607$	18.8576	0.765405	0.382703	−	0.923872i	$-0.374993\pi$
$607$	18.8576	0.765405	0.382703	+	0.923872i	$0.374993\pi$
$608$	0	0
$609$	2.41187	0.0977339
$610$	0	0
$611$	8.63162	0.349198
$612$	0	0
$613$	−31.9288	−1.28959	−0.644797	−	0.764354i	$-0.723058\pi$
$613$	−31.9288	−1.28959	−0.644797	+	0.764354i	$0.723058\pi$
$614$	0	0
$615$	0.701282	0.0282784
$616$	0	0
$617$	27.1641	1.09358	0.546792	−	0.837268i	$-0.315849\pi$
$617$	27.1641	1.09358	0.546792	+	0.837268i	$0.315849\pi$
$618$	0	0
$619$	15.1831	0.610260	0.305130	−	0.952311i	$-0.401300\pi$
$619$	15.1831	0.610260	0.305130	+	0.952311i	$0.401300\pi$
$620$	0	0
$621$	−7.13234	−0.286211
$622$	0	0
$623$	38.3756	1.53748
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.270388	−0.0107983
$628$	0	0
$629$	−58.3086	−2.32492
$630$	0	0
$631$	−17.9740	−0.715532	−0.357766	−	0.933811i	$-0.616461\pi$
$631$	−17.9740	−0.715532	−0.357766	+	0.933811i	$0.616461\pi$
$632$	0	0
$633$	−0.840526	−0.0334079
$634$	0	0
$635$	−1.15524	−0.0458444
$636$	0	0
$637$	49.7533	1.97130
$638$	0	0
$639$	−4.11630	−0.162838
$640$	0	0
$641$	−16.9647	−0.670066	−0.335033	−	0.942206i	$-0.608747\pi$
$641$	−16.9647	−0.670066	−0.335033	+	0.942206i	$0.608747\pi$
$642$	0	0
$643$	−40.4544	−1.59536	−0.797682	−	0.603078i	$-0.793941\pi$
$643$	−40.4544	−1.59536	−0.797682	+	0.603078i	$0.793941\pi$
$644$	0	0
$645$	−10.9467	−0.431026
$646$	0	0
$647$	43.7214	1.71887	0.859433	−	0.511248i	$-0.170817\pi$
$647$	43.7214	1.71887	0.859433	+	0.511248i	$0.170817\pi$
$648$	0	0
$649$	2.95540	0.116010
$650$	0	0
$651$	12.0937	0.473988
$652$	0	0
$653$	15.0713	0.589785	0.294892	−	0.955531i	$-0.404716\pi$
$653$	15.0713	0.589785	0.294892	+	0.955531i	$0.404716\pi$
$654$	0	0
$655$	11.2772	0.440639
$656$	0	0
$657$	−1.83177	−0.0714640
$658$	0	0
$659$	6.37351	0.248277	0.124138	−	0.992265i	$-0.460383\pi$
$659$	6.37351	0.248277	0.124138	+	0.992265i	$0.460383\pi$
$660$	0	0
$661$	9.83983	0.382725	0.191362	−	0.981519i	$-0.438709\pi$
$661$	9.83983	0.382725	0.191362	+	0.981519i	$0.438709\pi$
$662$	0	0
$663$	−26.5854	−1.03249
$664$	0	0
$665$	0.330697	0.0128239
$666$	0	0
$667$	−3.84236	−0.148777
$668$	0	0
$669$	12.0649	0.466457
$670$	0	0
$671$	−16.3340	−0.630567
$672$	0	0
$673$	17.1468	0.660961	0.330481	−	0.943813i	$-0.392789\pi$
$673$	17.1468	0.660961	0.330481	+	0.943813i	$0.392789\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−12.5118	−0.480869	−0.240434	−	0.970665i	$-0.577290\pi$
$677$	−12.5118	−0.480869	−0.240434	+	0.970665i	$0.577290\pi$
$678$	0	0
$679$	41.1843	1.58051
$680$	0	0
$681$	21.4614	0.822403
$682$	0	0
$683$	−38.7594	−1.48309	−0.741543	−	0.670905i	$-0.765906\pi$
$683$	−38.7594	−1.48309	−0.741543	+	0.670905i	$0.765906\pi$
$684$	0	0
$685$	−16.9837	−0.648915
$686$	0	0
$687$	3.73690	0.142572
$688$	0	0
$689$	6.67038	0.254121
$690$	0	0
$691$	−39.2097	−1.49161	−0.745804	−	0.666166i	$-0.767934\pi$
$691$	−39.2097	−1.49161	−0.745804	+	0.666166i	$0.767934\pi$
$692$	0	0
$693$	16.3883	0.622539
$694$	0	0
$695$	−13.1266	−0.497921
$696$	0	0
$697$	4.88778	0.185138
$698$	0	0
$699$	−23.9633	−0.906376
$700$	0	0
$701$	−17.3206	−0.654192	−0.327096	−	0.944991i	$-0.606070\pi$
$701$	−17.3206	−0.654192	−0.327096	+	0.944991i	$0.606070\pi$
$702$	0	0
$703$	−0.617954	−0.0233066
$704$	0	0
$705$	−2.26292	−0.0852264
$706$	0	0
$707$	2.47656	0.0931407
$708$	0	0
$709$	−24.6689	−0.926459	−0.463230	−	0.886238i	$-0.653309\pi$
$709$	−24.6689	−0.926459	−0.463230	+	0.886238i	$0.653309\pi$
$710$	0	0
$711$	4.17995	0.156760
$712$	0	0
$713$	−19.2665	−0.721534
$714$	0	0
$715$	−13.9627	−0.522175
$716$	0	0
$717$	−4.11678	−0.153744
$718$	0	0
$719$	24.4993	0.913668	0.456834	−	0.889552i	$-0.348983\pi$
$719$	24.4993	0.913668	0.456834	+	0.889552i	$0.348983\pi$
$720$	0	0
$721$	−39.3904	−1.46698
$722$	0	0
$723$	−24.3695	−0.906310
$724$	0	0
$725$	0.538723	0.0200077
$726$	0	0
$727$	11.8303	0.438763	0.219381	−	0.975639i	$-0.429596\pi$
$727$	11.8303	0.438763	0.219381	+	0.975639i	$0.429596\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−76.2962	−2.82192
$732$	0	0
$733$	7.80437	0.288261	0.144130	−	0.989559i	$-0.453962\pi$
$733$	7.80437	0.288261	0.144130	+	0.989559i	$0.453962\pi$
$734$	0	0
$735$	−13.0436	−0.481121
$736$	0	0
$737$	3.66054	0.134838
$738$	0	0
$739$	7.20240	0.264945	0.132472	−	0.991187i	$-0.457708\pi$
$739$	7.20240	0.264945	0.132472	+	0.991187i	$0.457708\pi$
$740$	0	0
$741$	−0.281751	−0.0103504
$742$	0	0
$743$	13.1180	0.481252	0.240626	−	0.970618i	$-0.422647\pi$
$743$	13.1180	0.481252	0.240626	+	0.970618i	$0.422647\pi$
$744$	0	0
$745$	−11.3420	−0.415537
$746$	0	0
$747$	3.56198	0.130326
$748$	0	0
$749$	−77.6960	−2.83895
$750$	0	0
$751$	41.7499	1.52347	0.761737	−	0.647886i	$-0.224347\pi$
$751$	41.7499	1.52347	0.761737	+	0.647886i	$0.224347\pi$
$752$	0	0
$753$	21.5904	0.786798
$754$	0	0
$755$	10.0048	0.364114
$756$	0	0
$757$	−4.48623	−0.163055	−0.0815275	−	0.996671i	$-0.525980\pi$
$757$	−4.48623	−0.163055	−0.0815275	+	0.996671i	$0.525980\pi$
$758$	0	0
$759$	−26.1082	−0.947668
$760$	0	0
$761$	19.4036	0.703380	0.351690	−	0.936117i	$-0.385607\pi$
$761$	19.4036	0.703380	0.351690	+	0.936117i	$0.385607\pi$
$762$	0	0
$763$	26.4035	0.955871
$764$	0	0
$765$	6.96978	0.251993
$766$	0	0
$767$	3.07960	0.111198
$768$	0	0
$769$	−27.3437	−0.986040	−0.493020	−	0.870018i	$-0.664107\pi$
$769$	−27.3437	−0.986040	−0.493020	+	0.870018i	$0.664107\pi$
$770$	0	0
$771$	20.1990	0.727449
$772$	0	0
$773$	−5.16057	−0.185613	−0.0928064	−	0.995684i	$-0.529584\pi$
$773$	−5.16057	−0.185613	−0.0928064	+	0.995684i	$0.529584\pi$
$774$	0	0
$775$	2.70128	0.0970329
$776$	0	0
$777$	37.4543	1.34367
$778$	0	0
$779$	0.0518006	0.00185595
$780$	0	0
$781$	−15.0679	−0.539171
$782$	0	0
$783$	0.538723	0.0192524
$784$	0	0
$785$	11.3646	0.405619
$786$	0	0
$787$	−42.7502	−1.52388	−0.761939	−	0.647648i	$-0.775753\pi$
$787$	−42.7502	−1.52388	−0.761939	+	0.647648i	$0.775753\pi$
$788$	0	0
$789$	−2.28006	−0.0811723
$790$	0	0
$791$	63.2151	2.24767
$792$	0	0
$793$	−17.0204	−0.604414
$794$	0	0
$795$	−1.74875	−0.0620217
$796$	0	0
$797$	37.6597	1.33398	0.666988	−	0.745069i	$-0.267583\pi$
$797$	37.6597	1.33398	0.666988	+	0.745069i	$0.267583\pi$
$798$	0	0
$799$	−15.7720	−0.557975
$800$	0	0
$801$	8.57170	0.302866
$802$	0	0
$803$	−6.70526	−0.236623
$804$	0	0
$805$	31.9316	1.12544
$806$	0	0
$807$	3.40768	0.119956
$808$	0	0
$809$	−18.2412	−0.641328	−0.320664	−	0.947193i	$-0.603906\pi$
$809$	−18.2412	−0.641328	−0.320664	+	0.947193i	$0.603906\pi$
$810$	0	0
$811$	36.8070	1.29247	0.646235	−	0.763139i	$-0.276343\pi$
$811$	36.8070	1.29247	0.646235	+	0.763139i	$0.276343\pi$
$812$	0	0
$813$	31.2376	1.09555
$814$	0	0
$815$	18.0171	0.631111
$816$	0	0
$817$	−0.808586	−0.0282889
$818$	0	0
$819$	17.0770	0.596719
$820$	0	0
$821$	−26.9488	−0.940520	−0.470260	−	0.882528i	$-0.655840\pi$
$821$	−26.9488	−0.940520	−0.470260	+	0.882528i	$0.655840\pi$
$822$	0	0
$823$	25.8314	0.900425	0.450213	−	0.892921i	$-0.351348\pi$
$823$	25.8314	0.900425	0.450213	+	0.892921i	$0.351348\pi$
$824$	0	0
$825$	3.66054	0.127444
$826$	0	0
$827$	−24.9623	−0.868023	−0.434012	−	0.900907i	$-0.642902\pi$
$827$	−24.9623	−0.868023	−0.434012	+	0.900907i	$0.642902\pi$
$828$	0	0
$829$	16.4174	0.570201	0.285100	−	0.958498i	$-0.407973\pi$
$829$	16.4174	0.570201	0.285100	+	0.958498i	$0.407973\pi$
$830$	0	0
$831$	14.8526	0.515230
$832$	0	0
$833$	−90.9112	−3.14988
$834$	0	0
$835$	−4.40114	−0.152308
$836$	0	0
$837$	2.70128	0.0933700
$838$	0	0
$839$	28.9631	0.999917	0.499958	−	0.866049i	$-0.333349\pi$
$839$	28.9631	0.999917	0.499958	+	0.866049i	$0.333349\pi$
$840$	0	0
$841$	−28.7098	−0.989992
$842$	0	0
$843$	−9.64114	−0.332059
$844$	0	0
$845$	−1.54946	−0.0533029
$846$	0	0
$847$	10.7429	0.369129
$848$	0	0
$849$	−13.4963	−0.463190
$850$	0	0
$851$	−59.6685	−2.04541
$852$	0	0
$853$	−45.4968	−1.55778	−0.778890	−	0.627160i	$-0.784217\pi$
$853$	−45.4968	−1.55778	−0.778890	+	0.627160i	$0.784217\pi$
$854$	0	0
$855$	0.0738657	0.00252615
$856$	0	0
$857$	42.5589	1.45378	0.726892	−	0.686752i	$-0.240964\pi$
$857$	42.5589	1.45378	0.726892	+	0.686752i	$0.240964\pi$
$858$	0	0
$859$	−30.1818	−1.02979	−0.514895	−	0.857253i	$-0.672169\pi$
$859$	−30.1818	−1.02979	−0.514895	+	0.857253i	$0.672169\pi$
$860$	0	0
$861$	−3.13965	−0.106999
$862$	0	0
$863$	−36.4846	−1.24195	−0.620974	−	0.783831i	$-0.713263\pi$
$863$	−36.4846	−1.24195	−0.620974	+	0.783831i	$0.713263\pi$
$864$	0	0
$865$	−10.7769	−0.366424
$866$	0	0
$867$	31.5778	1.07244
$868$	0	0
$869$	15.3009	0.519047
$870$	0	0
$871$	3.81438	0.129245
$872$	0	0
$873$	9.19906	0.311341
$874$	0	0
$875$	−4.47701	−0.151351
$876$	0	0
$877$	22.9030	0.773381	0.386690	−	0.922210i	$-0.373618\pi$
$877$	22.9030	0.773381	0.386690	+	0.922210i	$0.373618\pi$
$878$	0	0
$879$	26.4275	0.891379
$880$	0	0
$881$	−23.0167	−0.775453	−0.387727	−	0.921774i	$-0.626740\pi$
$881$	−23.0167	−0.775453	−0.387727	+	0.921774i	$0.626740\pi$
$882$	0	0
$883$	−46.6121	−1.56862	−0.784311	−	0.620367i	$-0.786983\pi$
$883$	−46.6121	−1.56862	−0.784311	+	0.620367i	$0.786983\pi$
$884$	0	0
$885$	−0.807367	−0.0271393
$886$	0	0
$887$	−36.0137	−1.20922	−0.604612	−	0.796520i	$-0.706672\pi$
$887$	−36.0137	−1.20922	−0.604612	+	0.796520i	$0.706672\pi$
$888$	0	0
$889$	5.17203	0.173464
$890$	0	0
$891$	3.66054	0.122633
$892$	0	0
$893$	−0.167152	−0.00559353
$894$	0	0
$895$	−11.8254	−0.395279
$896$	0	0
$897$	−27.2054	−0.908362
$898$	0	0
$899$	1.45524	0.0485351
$900$	0	0
$901$	−12.1884	−0.406054
$902$	0	0
$903$	49.0085	1.63090
$904$	0	0
$905$	7.20241	0.239416
$906$	0	0
$907$	−10.5074	−0.348892	−0.174446	−	0.984667i	$-0.555813\pi$
$907$	−10.5074	−0.348892	−0.174446	+	0.984667i	$0.555813\pi$
$908$	0	0
$909$	0.553173	0.0183476
$910$	0	0
$911$	−18.6783	−0.618841	−0.309420	−	0.950925i	$-0.600135\pi$
$911$	−18.6783	−0.618841	−0.309420	+	0.950925i	$0.600135\pi$
$912$	0	0
$913$	13.0388	0.431520
$914$	0	0
$915$	4.46218	0.147515
$916$	0	0
$917$	−50.4883	−1.66727
$918$	0	0
$919$	30.8738	1.01843	0.509216	−	0.860639i	$-0.329935\pi$
$919$	30.8738	1.01843	0.509216	+	0.860639i	$0.329935\pi$
$920$	0	0
$921$	−12.9674	−0.427291
$922$	0	0
$923$	−15.7011	−0.516808
$924$	0	0
$925$	8.36592	0.275070
$926$	0	0
$927$	−8.79837	−0.288977
$928$	0	0
$929$	−0.907452	−0.0297725	−0.0148863	−	0.999889i	$-0.504739\pi$
$929$	−0.907452	−0.0297725	−0.0148863	+	0.999889i	$0.504739\pi$
$930$	0	0
$931$	−0.963476	−0.0315766
$932$	0	0
$933$	12.8951	0.422165
$934$	0	0
$935$	25.5132	0.834370
$936$	0	0
$937$	−18.5306	−0.605368	−0.302684	−	0.953091i	$-0.597883\pi$
$937$	−18.5306	−0.605368	−0.302684	+	0.953091i	$0.597883\pi$
$938$	0	0
$939$	−0.760285	−0.0248110
$940$	0	0
$941$	−46.1583	−1.50472	−0.752359	−	0.658753i	$-0.771084\pi$
$941$	−46.1583	−1.50472	−0.752359	+	0.658753i	$0.771084\pi$
$942$	0	0
$943$	5.00178	0.162880
$944$	0	0
$945$	−4.47701	−0.145637
$946$	0	0
$947$	46.9021	1.52411	0.762057	−	0.647509i	$-0.224189\pi$
$947$	46.9021	1.52411	0.762057	+	0.647509i	$0.224189\pi$
$948$	0	0
$949$	−6.98705	−0.226809
$950$	0	0
$951$	−20.9773	−0.680236
$952$	0	0
$953$	4.84102	0.156816	0.0784081	−	0.996921i	$-0.475016\pi$
$953$	4.84102	0.156816	0.0784081	+	0.996921i	$0.475016\pi$
$954$	0	0
$955$	16.1036	0.521100
$956$	0	0
$957$	1.97202	0.0637463
$958$	0	0
$959$	76.0363	2.45534
$960$	0	0
$961$	−23.7031	−0.764615
$962$	0	0
$963$	−17.3544	−0.559239
$964$	0	0
$965$	11.7082	0.376900
$966$	0	0
$967$	14.9492	0.480734	0.240367	−	0.970682i	$-0.422732\pi$
$967$	14.9492	0.480734	0.240367	+	0.970682i	$0.422732\pi$
$968$	0	0
$969$	0.514827	0.0165386
$970$	0	0
$971$	−6.32963	−0.203127	−0.101564	−	0.994829i	$-0.532385\pi$
$971$	−6.32963	−0.203127	−0.101564	+	0.994829i	$0.532385\pi$
$972$	0	0
$973$	58.7680	1.88401
$974$	0	0
$975$	3.81438	0.122158
$976$	0	0
$977$	−23.5643	−0.753890	−0.376945	−	0.926236i	$-0.623025\pi$
$977$	−23.5643	−0.753890	−0.376945	+	0.926236i	$0.623025\pi$
$978$	0	0
$979$	31.3770	1.00281
$980$	0	0
$981$	5.89757	0.188295
$982$	0	0
$983$	8.61381	0.274738	0.137369	−	0.990520i	$-0.456135\pi$
$983$	8.61381	0.274738	0.137369	+	0.990520i	$0.456135\pi$
$984$	0	0
$985$	−2.23632	−0.0712549
$986$	0	0
$987$	10.1311	0.322477
$988$	0	0
$989$	−78.0757	−2.48266
$990$	0	0
$991$	−59.6434	−1.89464	−0.947318	−	0.320295i	$-0.896218\pi$
$991$	−59.6434	−1.89464	−0.947318	+	0.320295i	$0.896218\pi$
$992$	0	0
$993$	14.2592	0.452503
$994$	0	0
$995$	21.7620	0.689900
$996$	0	0
$997$	−49.7555	−1.57577	−0.787887	−	0.615820i	$-0.788825\pi$
$997$	−49.7555	−1.57577	−0.787887	+	0.615820i	$0.788825\pi$
$998$	0	0
$999$	8.36592	0.264686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.t.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.t.1.6	✓	7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.47701 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.47701 q^{7} +1.00000 q^{9} +3.66054 q^{11} +3.81438 q^{13} -1.00000 q^{15} -6.96978 q^{17} -0.0738657 q^{19} +4.47701 q^{21} -7.13234 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.538723 q^{29} +2.70128 q^{31} +3.66054 q^{33} -4.47701 q^{35} +8.36592 q^{37} +3.81438 q^{39} -0.701282 q^{41} +10.9467 q^{43} -1.00000 q^{45} +2.26292 q^{47} +13.0436 q^{49} -6.96978 q^{51} +1.74875 q^{53} -3.66054 q^{55} -0.0738657 q^{57} +0.807367 q^{59} -4.46218 q^{61} +4.47701 q^{63} -3.81438 q^{65} +1.00000 q^{67} -7.13234 q^{69} -4.11630 q^{71} -1.83177 q^{73} +1.00000 q^{75} +16.3883 q^{77} +4.17995 q^{79} +1.00000 q^{81} +3.56198 q^{83} +6.96978 q^{85} +0.538723 q^{87} +8.57170 q^{89} +17.0770 q^{91} +2.70128 q^{93} +0.0738657 q^{95} +9.19906 q^{97} +3.66054 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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