LMFDB - Embedded newform 8040.2.a.t.1.2

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			1.2
Root			$1.51692$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.274078 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -0.274078 q^{7} +1.00000 q^{9} -4.75091 q^{11} -2.11982 q^{13} -1.00000 q^{15} +3.21588 q^{17} -1.31998 q^{19} -0.274078 q^{21} -4.76957 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.42263 q^{29} +3.56282 q^{31} -4.75091 q^{33} +0.274078 q^{35} +9.24113 q^{37} -2.11982 q^{39} -1.56282 q^{41} +2.64975 q^{43} -1.00000 q^{45} +7.55397 q^{47} -6.92488 q^{49} +3.21588 q^{51} -12.6509 q^{53} +4.75091 q^{55} -1.31998 q^{57} +11.9312 q^{59} +5.61957 q^{61} -0.274078 q^{63} +2.11982 q^{65} +1.00000 q^{67} -4.76957 q^{69} +5.03912 q^{71} +0.416872 q^{73} +1.00000 q^{75} +1.30212 q^{77} +15.6884 q^{79} +1.00000 q^{81} -8.57594 q^{83} -3.21588 q^{85} -6.42263 q^{87} +6.86302 q^{89} +0.580996 q^{91} +3.56282 q^{93} +1.31998 q^{95} -11.2626 q^{97} -4.75091 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.274078	−0.103592	−0.0517958	−	0.998658i	$-0.516494\pi$
$7$	−0.274078	−0.103592	−0.0517958	+	0.998658i	$0.516494\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.75091	−1.43245	−0.716227	−	0.697867i	$-0.754133\pi$
$11$	−4.75091	−1.43245	−0.716227	+	0.697867i	$0.754133\pi$
$12$	0	0
$13$	−2.11982	−0.587933	−0.293966	−	0.955816i	$-0.594975\pi$
$13$	−2.11982	−0.587933	−0.293966	+	0.955816i	$0.594975\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.21588	0.779964	0.389982	−	0.920822i	$-0.372481\pi$
$17$	3.21588	0.779964	0.389982	+	0.920822i	$0.372481\pi$
$18$	0	0
$19$	−1.31998	−0.302824	−0.151412	−	0.988471i	$-0.548382\pi$
$19$	−1.31998	−0.302824	−0.151412	+	0.988471i	$0.548382\pi$
$20$	0	0
$21$	−0.274078	−0.0598086
$22$	0	0
$23$	−4.76957	−0.994524	−0.497262	−	0.867600i	$-0.665661\pi$
$23$	−4.76957	−0.994524	−0.497262	+	0.867600i	$0.665661\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.42263	−1.19265	−0.596326	−	0.802742i	$-0.703373\pi$
$29$	−6.42263	−1.19265	−0.596326	+	0.802742i	$0.703373\pi$
$30$	0	0
$31$	3.56282	0.639901	0.319951	−	0.947434i	$-0.396334\pi$
$31$	3.56282	0.639901	0.319951	+	0.947434i	$0.396334\pi$
$32$	0	0
$33$	−4.75091	−0.827028
$34$	0	0
$35$	0.274078	0.0463276
$36$	0	0
$37$	9.24113	1.51923	0.759616	−	0.650372i	$-0.225387\pi$
$37$	9.24113	1.51923	0.759616	+	0.650372i	$0.225387\pi$
$38$	0	0
$39$	−2.11982	−0.339443
$40$	0	0
$41$	−1.56282	−0.244071	−0.122036	−	0.992526i	$-0.538942\pi$
$41$	−1.56282	−0.244071	−0.122036	+	0.992526i	$0.538942\pi$
$42$	0	0
$43$	2.64975	0.404083	0.202041	−	0.979377i	$-0.435242\pi$
$43$	2.64975	0.404083	0.202041	+	0.979377i	$0.435242\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.55397	1.10186	0.550930	−	0.834551i	$-0.314273\pi$
$47$	7.55397	1.10186	0.550930	+	0.834551i	$0.314273\pi$
$48$	0	0
$49$	−6.92488	−0.989269
$50$	0	0
$51$	3.21588	0.450313
$52$	0	0
$53$	−12.6509	−1.73774	−0.868868	−	0.495043i	$-0.835152\pi$
$53$	−12.6509	−1.73774	−0.868868	+	0.495043i	$0.835152\pi$
$54$	0	0
$55$	4.75091	0.640613
$56$	0	0
$57$	−1.31998	−0.174835
$58$	0	0
$59$	11.9312	1.55331	0.776657	−	0.629923i	$-0.216914\pi$
$59$	11.9312	1.55331	0.776657	+	0.629923i	$0.216914\pi$
$60$	0	0
$61$	5.61957	0.719512	0.359756	−	0.933046i	$-0.382860\pi$
$61$	5.61957	0.719512	0.359756	+	0.933046i	$0.382860\pi$
$62$	0	0
$63$	−0.274078	−0.0345305
$64$	0	0
$65$	2.11982	0.262932
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−4.76957	−0.574189
$70$	0	0
$71$	5.03912	0.598033	0.299017	−	0.954248i	$-0.403341\pi$
$71$	5.03912	0.598033	0.299017	+	0.954248i	$0.403341\pi$
$72$	0	0
$73$	0.416872	0.0487912	0.0243956	−	0.999702i	$-0.492234\pi$
$73$	0.416872	0.0487912	0.0243956	+	0.999702i	$0.492234\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	1.30212	0.148390
$78$	0	0
$79$	15.6884	1.76508	0.882541	−	0.470235i	$-0.155831\pi$
$79$	15.6884	1.76508	0.882541	+	0.470235i	$0.155831\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.57594	−0.941331	−0.470666	−	0.882312i	$-0.655986\pi$
$83$	−8.57594	−0.941331	−0.470666	+	0.882312i	$0.655986\pi$
$84$	0	0
$85$	−3.21588	−0.348811
$86$	0	0
$87$	−6.42263	−0.688578
$88$	0	0
$89$	6.86302	0.727479	0.363739	−	0.931501i	$-0.381500\pi$
$89$	6.86302	0.727479	0.363739	+	0.931501i	$0.381500\pi$
$90$	0	0
$91$	0.580996	0.0609049
$92$	0	0
$93$	3.56282	0.369447
$94$	0	0
$95$	1.31998	0.135427
$96$	0	0
$97$	−11.2626	−1.14354	−0.571772	−	0.820413i	$-0.693744\pi$
$97$	−11.2626	−1.14354	−0.571772	+	0.820413i	$0.693744\pi$
$98$	0	0
$99$	−4.75091	−0.477485
$100$	0	0
$101$	11.7316	1.16734	0.583670	−	0.811991i	$-0.301616\pi$
$101$	11.7316	1.16734	0.583670	+	0.811991i	$0.301616\pi$
$102$	0	0
$103$	5.29950	0.522175	0.261088	−	0.965315i	$-0.415919\pi$
$103$	5.29950	0.522175	0.261088	+	0.965315i	$0.415919\pi$
$104$	0	0
$105$	0.274078	0.0267472
$106$	0	0
$107$	7.02493	0.679126	0.339563	−	0.940583i	$-0.389721\pi$
$107$	7.02493	0.679126	0.339563	+	0.940583i	$0.389721\pi$
$108$	0	0
$109$	9.86380	0.944781	0.472390	−	0.881389i	$-0.343391\pi$
$109$	9.86380	0.944781	0.472390	+	0.881389i	$0.343391\pi$
$110$	0	0
$111$	9.24113	0.877129
$112$	0	0
$113$	−0.546353	−0.0513966	−0.0256983	−	0.999670i	$-0.508181\pi$
$113$	−0.546353	−0.0513966	−0.0256983	+	0.999670i	$0.508181\pi$
$114$	0	0
$115$	4.76957	0.444765
$116$	0	0
$117$	−2.11982	−0.195978
$118$	0	0
$119$	−0.881400	−0.0807978
$120$	0	0
$121$	11.5712	1.05192
$122$	0	0
$123$	−1.56282	−0.140915
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.53111	0.668278	0.334139	−	0.942524i	$-0.391554\pi$
$127$	7.53111	0.668278	0.334139	+	0.942524i	$0.391554\pi$
$128$	0	0
$129$	2.64975	0.233297
$130$	0	0
$131$	−12.9151	−1.12840	−0.564200	−	0.825638i	$-0.690815\pi$
$131$	−12.9151	−1.12840	−0.564200	+	0.825638i	$0.690815\pi$
$132$	0	0
$133$	0.361776	0.0313700
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	3.99615	0.341414	0.170707	−	0.985322i	$-0.445395\pi$
$137$	3.99615	0.341414	0.170707	+	0.985322i	$0.445395\pi$
$138$	0	0
$139$	−2.03032	−0.172210	−0.0861048	−	0.996286i	$-0.527442\pi$
$139$	−2.03032	−0.172210	−0.0861048	+	0.996286i	$0.527442\pi$
$140$	0	0
$141$	7.55397	0.636159
$142$	0	0
$143$	10.0711	0.842187
$144$	0	0
$145$	6.42263	0.533370
$146$	0	0
$147$	−6.92488	−0.571155
$148$	0	0
$149$	3.32105	0.272071	0.136036	−	0.990704i	$-0.456564\pi$
$149$	3.32105	0.272071	0.136036	+	0.990704i	$0.456564\pi$
$150$	0	0
$151$	−14.6664	−1.19354	−0.596769	−	0.802413i	$-0.703549\pi$
$151$	−14.6664	−1.19354	−0.596769	+	0.802413i	$0.703549\pi$
$152$	0	0
$153$	3.21588	0.259988
$154$	0	0
$155$	−3.56282	−0.286172
$156$	0	0
$157$	1.11832	0.0892518	0.0446259	−	0.999004i	$-0.485790\pi$
$157$	1.11832	0.0892518	0.0446259	+	0.999004i	$0.485790\pi$
$158$	0	0
$159$	−12.6509	−1.00328
$160$	0	0
$161$	1.30723	0.103024
$162$	0	0
$163$	1.93177	0.151308	0.0756538	−	0.997134i	$-0.475896\pi$
$163$	1.93177	0.151308	0.0756538	+	0.997134i	$0.475896\pi$
$164$	0	0
$165$	4.75091	0.369858
$166$	0	0
$167$	6.84403	0.529607	0.264803	−	0.964302i	$-0.414693\pi$
$167$	6.84403	0.529607	0.264803	+	0.964302i	$0.414693\pi$
$168$	0	0
$169$	−8.50636	−0.654335
$170$	0	0
$171$	−1.31998	−0.100941
$172$	0	0
$173$	4.23895	0.322282	0.161141	−	0.986931i	$-0.448483\pi$
$173$	4.23895	0.322282	0.161141	+	0.986931i	$0.448483\pi$
$174$	0	0
$175$	−0.274078	−0.0207183
$176$	0	0
$177$	11.9312	0.896807
$178$	0	0
$179$	17.5435	1.31126	0.655631	−	0.755081i	$-0.272403\pi$
$179$	17.5435	1.31126	0.655631	+	0.755081i	$0.272403\pi$
$180$	0	0
$181$	17.5358	1.30342	0.651712	−	0.758467i	$-0.274051\pi$
$181$	17.5358	1.30342	0.651712	+	0.758467i	$0.274051\pi$
$182$	0	0
$183$	5.61957	0.415410
$184$	0	0
$185$	−9.24113	−0.679421
$186$	0	0
$187$	−15.2783	−1.11726
$188$	0	0
$189$	−0.274078	−0.0199362
$190$	0	0
$191$	21.2082	1.53457	0.767285	−	0.641306i	$-0.221607\pi$
$191$	21.2082	1.53457	0.767285	+	0.641306i	$0.221607\pi$
$192$	0	0
$193$	15.2523	1.09789	0.548943	−	0.835860i	$-0.315030\pi$
$193$	15.2523	1.09789	0.548943	+	0.835860i	$0.315030\pi$
$194$	0	0
$195$	2.11982	0.151804
$196$	0	0
$197$	−14.0688	−1.00236	−0.501180	−	0.865343i	$-0.667101\pi$
$197$	−14.0688	−1.00236	−0.501180	+	0.865343i	$0.667101\pi$
$198$	0	0
$199$	13.2706	0.940728	0.470364	−	0.882472i	$-0.344123\pi$
$199$	13.2706	0.940728	0.470364	+	0.882472i	$0.344123\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	1.76030	0.123549
$204$	0	0
$205$	1.56282	0.109152
$206$	0	0
$207$	−4.76957	−0.331508
$208$	0	0
$209$	6.27110	0.433781
$210$	0	0
$211$	−3.60376	−0.248093	−0.124047	−	0.992276i	$-0.539587\pi$
$211$	−3.60376	−0.248093	−0.124047	+	0.992276i	$0.539587\pi$
$212$	0	0
$213$	5.03912	0.345275
$214$	0	0
$215$	−2.64975	−0.180711
$216$	0	0
$217$	−0.976489	−0.0662884
$218$	0	0
$219$	0.416872	0.0281696
$220$	0	0
$221$	−6.81708	−0.458567
$222$	0	0
$223$	10.1472	0.679507	0.339754	−	0.940514i	$-0.389656\pi$
$223$	10.1472	0.679507	0.339754	+	0.940514i	$0.389656\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−2.42806	−0.161156	−0.0805781	−	0.996748i	$-0.525677\pi$
$227$	−2.42806	−0.161156	−0.0805781	+	0.996748i	$0.525677\pi$
$228$	0	0
$229$	0.696447	0.0460225	0.0230113	−	0.999735i	$-0.492675\pi$
$229$	0.696447	0.0460225	0.0230113	+	0.999735i	$0.492675\pi$
$230$	0	0
$231$	1.30212	0.0856731
$232$	0	0
$233$	24.1181	1.58003	0.790016	−	0.613087i	$-0.210072\pi$
$233$	24.1181	1.58003	0.790016	+	0.613087i	$0.210072\pi$
$234$	0	0
$235$	−7.55397	−0.492767
$236$	0	0
$237$	15.6884	1.01907
$238$	0	0
$239$	−7.52635	−0.486839	−0.243420	−	0.969921i	$-0.578269\pi$
$239$	−7.52635	−0.486839	−0.243420	+	0.969921i	$0.578269\pi$
$240$	0	0
$241$	−8.96980	−0.577795	−0.288898	−	0.957360i	$-0.593289\pi$
$241$	−8.96980	−0.577795	−0.288898	+	0.957360i	$0.593289\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.92488	0.442414
$246$	0	0
$247$	2.79812	0.178040
$248$	0	0
$249$	−8.57594	−0.543478
$250$	0	0
$251$	−3.03755	−0.191729	−0.0958644	−	0.995394i	$-0.530562\pi$
$251$	−3.03755	−0.191729	−0.0958644	+	0.995394i	$0.530562\pi$
$252$	0	0
$253$	22.6598	1.42461
$254$	0	0
$255$	−3.21588	−0.201386
$256$	0	0
$257$	−3.68721	−0.230002	−0.115001	−	0.993365i	$-0.536687\pi$
$257$	−3.68721	−0.230002	−0.115001	+	0.993365i	$0.536687\pi$
$258$	0	0
$259$	−2.53279	−0.157380
$260$	0	0
$261$	−6.42263	−0.397551
$262$	0	0
$263$	0.242711	0.0149662	0.00748309	−	0.999972i	$-0.497618\pi$
$263$	0.242711	0.0149662	0.00748309	+	0.999972i	$0.497618\pi$
$264$	0	0
$265$	12.6509	0.777139
$266$	0	0
$267$	6.86302	0.420010
$268$	0	0
$269$	−28.3909	−1.73102	−0.865512	−	0.500888i	$-0.833007\pi$
$269$	−28.3909	−1.73102	−0.865512	+	0.500888i	$0.833007\pi$
$270$	0	0
$271$	2.93780	0.178459	0.0892294	−	0.996011i	$-0.471560\pi$
$271$	2.93780	0.178459	0.0892294	+	0.996011i	$0.471560\pi$
$272$	0	0
$273$	0.580996	0.0351635
$274$	0	0
$275$	−4.75091	−0.286491
$276$	0	0
$277$	3.37127	0.202560	0.101280	−	0.994858i	$-0.467706\pi$
$277$	3.37127	0.202560	0.101280	+	0.994858i	$0.467706\pi$
$278$	0	0
$279$	3.56282	0.213300
$280$	0	0
$281$	−5.98422	−0.356989	−0.178494	−	0.983941i	$-0.557123\pi$
$281$	−5.98422	−0.356989	−0.178494	+	0.983941i	$0.557123\pi$
$282$	0	0
$283$	17.8711	1.06233	0.531164	−	0.847269i	$-0.321755\pi$
$283$	17.8711	1.06233	0.531164	+	0.847269i	$0.321755\pi$
$284$	0	0
$285$	1.31998	0.0781887
$286$	0	0
$287$	0.428334	0.0252837
$288$	0	0
$289$	−6.65814	−0.391655
$290$	0	0
$291$	−11.2626	−0.660225
$292$	0	0
$293$	−21.8762	−1.27802	−0.639009	−	0.769199i	$-0.720655\pi$
$293$	−21.8762	−1.27802	−0.639009	+	0.769199i	$0.720655\pi$
$294$	0	0
$295$	−11.9312	−0.694663
$296$	0	0
$297$	−4.75091	−0.275676
$298$	0	0
$299$	10.1106	0.584713
$300$	0	0
$301$	−0.726237	−0.0418596
$302$	0	0
$303$	11.7316	0.673964
$304$	0	0
$305$	−5.61957	−0.321775
$306$	0	0
$307$	−19.4125	−1.10793	−0.553966	−	0.832539i	$-0.686886\pi$
$307$	−19.4125	−1.10793	−0.553966	+	0.832539i	$0.686886\pi$
$308$	0	0
$309$	5.29950	0.301478
$310$	0	0
$311$	32.7827	1.85894	0.929469	−	0.368899i	$-0.120265\pi$
$311$	32.7827	1.85894	0.929469	+	0.368899i	$0.120265\pi$
$312$	0	0
$313$	28.0757	1.58693	0.793467	−	0.608613i	$-0.208274\pi$
$313$	28.0757	1.58693	0.793467	+	0.608613i	$0.208274\pi$
$314$	0	0
$315$	0.274078	0.0154425
$316$	0	0
$317$	−0.862332	−0.0484334	−0.0242167	−	0.999707i	$-0.507709\pi$
$317$	−0.862332	−0.0484334	−0.0242167	+	0.999707i	$0.507709\pi$
$318$	0	0
$319$	30.5133	1.70842
$320$	0	0
$321$	7.02493	0.392093
$322$	0	0
$323$	−4.24488	−0.236192
$324$	0	0
$325$	−2.11982	−0.117587
$326$	0	0
$327$	9.86380	0.545469
$328$	0	0
$329$	−2.07037	−0.114143
$330$	0	0
$331$	12.9261	0.710484	0.355242	−	0.934774i	$-0.384398\pi$
$331$	12.9261	0.710484	0.355242	+	0.934774i	$0.384398\pi$
$332$	0	0
$333$	9.24113	0.506411
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	12.4358	0.677422	0.338711	−	0.940890i	$-0.390009\pi$
$337$	12.4358	0.677422	0.338711	+	0.940890i	$0.390009\pi$
$338$	0	0
$339$	−0.546353	−0.0296738
$340$	0	0
$341$	−16.9266	−0.916629
$342$	0	0
$343$	3.81650	0.206072
$344$	0	0
$345$	4.76957	0.256785
$346$	0	0
$347$	−11.5517	−0.620125	−0.310063	−	0.950716i	$-0.600350\pi$
$347$	−11.5517	−0.620125	−0.310063	+	0.950716i	$0.600350\pi$
$348$	0	0
$349$	−8.12813	−0.435089	−0.217544	−	0.976050i	$-0.569805\pi$
$349$	−8.12813	−0.435089	−0.217544	+	0.976050i	$0.569805\pi$
$350$	0	0
$351$	−2.11982	−0.113148
$352$	0	0
$353$	6.13851	0.326720	0.163360	−	0.986567i	$-0.447767\pi$
$353$	6.13851	0.326720	0.163360	+	0.986567i	$0.447767\pi$
$354$	0	0
$355$	−5.03912	−0.267449
$356$	0	0
$357$	−0.881400	−0.0466486
$358$	0	0
$359$	1.73600	0.0916223	0.0458112	−	0.998950i	$-0.485413\pi$
$359$	1.73600	0.0916223	0.0458112	+	0.998950i	$0.485413\pi$
$360$	0	0
$361$	−17.2577	−0.908298
$362$	0	0
$363$	11.5712	0.607329
$364$	0	0
$365$	−0.416872	−0.0218201
$366$	0	0
$367$	19.8185	1.03452	0.517260	−	0.855829i	$-0.326952\pi$
$367$	19.8185	1.03452	0.517260	+	0.855829i	$0.326952\pi$
$368$	0	0
$369$	−1.56282	−0.0813571
$370$	0	0
$371$	3.46733	0.180015
$372$	0	0
$373$	12.2367	0.633594	0.316797	−	0.948493i	$-0.397393\pi$
$373$	12.2367	0.633594	0.316797	+	0.948493i	$0.397393\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	13.6148	0.701199
$378$	0	0
$379$	10.5061	0.539662	0.269831	−	0.962908i	$-0.413032\pi$
$379$	10.5061	0.539662	0.269831	+	0.962908i	$0.413032\pi$
$380$	0	0
$381$	7.53111	0.385831
$382$	0	0
$383$	−16.0500	−0.820117	−0.410059	−	0.912059i	$-0.634492\pi$
$383$	−16.0500	−0.820117	−0.410059	+	0.912059i	$0.634492\pi$
$384$	0	0
$385$	−1.30212	−0.0663621
$386$	0	0
$387$	2.64975	0.134694
$388$	0	0
$389$	4.65662	0.236100	0.118050	−	0.993008i	$-0.462336\pi$
$389$	4.65662	0.236100	0.118050	+	0.993008i	$0.462336\pi$
$390$	0	0
$391$	−15.3383	−0.775693
$392$	0	0
$393$	−12.9151	−0.651483
$394$	0	0
$395$	−15.6884	−0.789369
$396$	0	0
$397$	8.51888	0.427551	0.213775	−	0.976883i	$-0.431424\pi$
$397$	8.51888	0.427551	0.213775	+	0.976883i	$0.431424\pi$
$398$	0	0
$399$	0.361776	0.0181115
$400$	0	0
$401$	5.21350	0.260350	0.130175	−	0.991491i	$-0.458446\pi$
$401$	5.21350	0.260350	0.130175	+	0.991491i	$0.458446\pi$
$402$	0	0
$403$	−7.55254	−0.376219
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−43.9038	−2.17623
$408$	0	0
$409$	−7.37518	−0.364679	−0.182340	−	0.983236i	$-0.558367\pi$
$409$	−7.37518	−0.364679	−0.182340	+	0.983236i	$0.558367\pi$
$410$	0	0
$411$	3.99615	0.197116
$412$	0	0
$413$	−3.27009	−0.160910
$414$	0	0
$415$	8.57594	0.420976
$416$	0	0
$417$	−2.03032	−0.0994252
$418$	0	0
$419$	−13.6802	−0.668320	−0.334160	−	0.942516i	$-0.608453\pi$
$419$	−13.6802	−0.668320	−0.334160	+	0.942516i	$0.608453\pi$
$420$	0	0
$421$	−20.9999	−1.02347	−0.511736	−	0.859143i	$-0.670997\pi$
$421$	−20.9999	−1.02347	−0.511736	+	0.859143i	$0.670997\pi$
$422$	0	0
$423$	7.55397	0.367287
$424$	0	0
$425$	3.21588	0.155993
$426$	0	0
$427$	−1.54020	−0.0745354
$428$	0	0
$429$	10.0711	0.486237
$430$	0	0
$431$	11.8306	0.569861	0.284930	−	0.958548i	$-0.408030\pi$
$431$	11.8306	0.569861	0.284930	+	0.958548i	$0.408030\pi$
$432$	0	0
$433$	−13.6077	−0.653944	−0.326972	−	0.945034i	$-0.606028\pi$
$433$	−13.6077	−0.653944	−0.326972	+	0.945034i	$0.606028\pi$
$434$	0	0
$435$	6.42263	0.307941
$436$	0	0
$437$	6.29572	0.301165
$438$	0	0
$439$	16.8746	0.805381	0.402691	−	0.915336i	$-0.368075\pi$
$439$	16.8746	0.805381	0.402691	+	0.915336i	$0.368075\pi$
$440$	0	0
$441$	−6.92488	−0.329756
$442$	0	0
$443$	−37.5322	−1.78321	−0.891604	−	0.452817i	$-0.850419\pi$
$443$	−37.5322	−1.78321	−0.891604	+	0.452817i	$0.850419\pi$
$444$	0	0
$445$	−6.86302	−0.325338
$446$	0	0
$447$	3.32105	0.157080
$448$	0	0
$449$	35.4315	1.67211	0.836057	−	0.548643i	$-0.184855\pi$
$449$	35.4315	1.67211	0.836057	+	0.548643i	$0.184855\pi$
$450$	0	0
$451$	7.42482	0.349621
$452$	0	0
$453$	−14.6664	−0.689089
$454$	0	0
$455$	−0.580996	−0.0272375
$456$	0	0
$457$	13.6692	0.639416	0.319708	−	0.947516i	$-0.396415\pi$
$457$	13.6692	0.639416	0.319708	+	0.947516i	$0.396415\pi$
$458$	0	0
$459$	3.21588	0.150104
$460$	0	0
$461$	20.9521	0.975837	0.487919	−	0.872889i	$-0.337756\pi$
$461$	20.9521	0.975837	0.487919	+	0.872889i	$0.337756\pi$
$462$	0	0
$463$	1.41264	0.0656508	0.0328254	−	0.999461i	$-0.489549\pi$
$463$	1.41264	0.0656508	0.0328254	+	0.999461i	$0.489549\pi$
$464$	0	0
$465$	−3.56282	−0.165222
$466$	0	0
$467$	−16.2423	−0.751603	−0.375801	−	0.926700i	$-0.622632\pi$
$467$	−16.2423	−0.751603	−0.375801	+	0.926700i	$0.622632\pi$
$468$	0	0
$469$	−0.274078	−0.0126557
$470$	0	0
$471$	1.11832	0.0515296
$472$	0	0
$473$	−12.5887	−0.578830
$474$	0	0
$475$	−1.31998	−0.0605647
$476$	0	0
$477$	−12.6509	−0.579245
$478$	0	0
$479$	−13.6332	−0.622917	−0.311459	−	0.950260i	$-0.600818\pi$
$479$	−13.6332	−0.622917	−0.311459	+	0.950260i	$0.600818\pi$
$480$	0	0
$481$	−19.5895	−0.893206
$482$	0	0
$483$	1.30723	0.0594811
$484$	0	0
$485$	11.2626	0.511408
$486$	0	0
$487$	34.3246	1.55540	0.777699	−	0.628637i	$-0.216387\pi$
$487$	34.3246	1.55540	0.777699	+	0.628637i	$0.216387\pi$
$488$	0	0
$489$	1.93177	0.0873575
$490$	0	0
$491$	−11.6081	−0.523865	−0.261933	−	0.965086i	$-0.584360\pi$
$491$	−11.6081	−0.523865	−0.261933	+	0.965086i	$0.584360\pi$
$492$	0	0
$493$	−20.6544	−0.930226
$494$	0	0
$495$	4.75091	0.213538
$496$	0	0
$497$	−1.38111	−0.0619512
$498$	0	0
$499$	43.5861	1.95118	0.975592	−	0.219591i	$-0.0704724\pi$
$499$	43.5861	1.95118	0.975592	+	0.219591i	$0.0704724\pi$
$500$	0	0
$501$	6.84403	0.305769
$502$	0	0
$503$	5.84233	0.260497	0.130248	−	0.991481i	$-0.458423\pi$
$503$	5.84233	0.260497	0.130248	+	0.991481i	$0.458423\pi$
$504$	0	0
$505$	−11.7316	−0.522050
$506$	0	0
$507$	−8.50636	−0.377781
$508$	0	0
$509$	−19.4123	−0.860435	−0.430217	−	0.902725i	$-0.641563\pi$
$509$	−19.4123	−0.860435	−0.430217	+	0.902725i	$0.641563\pi$
$510$	0	0
$511$	−0.114255	−0.00505436
$512$	0	0
$513$	−1.31998	−0.0582784
$514$	0	0
$515$	−5.29950	−0.233524
$516$	0	0
$517$	−35.8883	−1.57836
$518$	0	0
$519$	4.23895	0.186069
$520$	0	0
$521$	−43.9104	−1.92375	−0.961875	−	0.273491i	$-0.911822\pi$
$521$	−43.9104	−1.92375	−0.961875	+	0.273491i	$0.911822\pi$
$522$	0	0
$523$	15.9649	0.698096	0.349048	−	0.937105i	$-0.386505\pi$
$523$	15.9649	0.698096	0.349048	+	0.937105i	$0.386505\pi$
$524$	0	0
$525$	−0.274078	−0.0119617
$526$	0	0
$527$	11.4576	0.499100
$528$	0	0
$529$	−0.251210	−0.0109222
$530$	0	0
$531$	11.9312	0.517772
$532$	0	0
$533$	3.31290	0.143498
$534$	0	0
$535$	−7.02493	−0.303714
$536$	0	0
$537$	17.5435	0.757058
$538$	0	0
$539$	32.8995	1.41708
$540$	0	0
$541$	21.4941	0.924103	0.462051	−	0.886853i	$-0.347114\pi$
$541$	21.4941	0.924103	0.462051	+	0.886853i	$0.347114\pi$
$542$	0	0
$543$	17.5358	0.752532
$544$	0	0
$545$	−9.86380	−0.422519
$546$	0	0
$547$	−2.28962	−0.0978970	−0.0489485	−	0.998801i	$-0.515587\pi$
$547$	−2.28962	−0.0978970	−0.0489485	+	0.998801i	$0.515587\pi$
$548$	0	0
$549$	5.61957	0.239837
$550$	0	0
$551$	8.47772	0.361163
$552$	0	0
$553$	−4.29984	−0.182848
$554$	0	0
$555$	−9.24113	−0.392264
$556$	0	0
$557$	−4.50372	−0.190829	−0.0954144	−	0.995438i	$-0.530418\pi$
$557$	−4.50372	−0.190829	−0.0954144	+	0.995438i	$0.530418\pi$
$558$	0	0
$559$	−5.61699	−0.237573
$560$	0	0
$561$	−15.2783	−0.645052
$562$	0	0
$563$	−19.8701	−0.837425	−0.418712	−	0.908119i	$-0.637518\pi$
$563$	−19.8701	−0.837425	−0.418712	+	0.908119i	$0.637518\pi$
$564$	0	0
$565$	0.546353	0.0229853
$566$	0	0
$567$	−0.274078	−0.0115102
$568$	0	0
$569$	13.7633	0.576988	0.288494	−	0.957482i	$-0.406845\pi$
$569$	13.7633	0.576988	0.288494	+	0.957482i	$0.406845\pi$
$570$	0	0
$571$	−4.68946	−0.196248	−0.0981239	−	0.995174i	$-0.531284\pi$
$571$	−4.68946	−0.196248	−0.0981239	+	0.995174i	$0.531284\pi$
$572$	0	0
$573$	21.2082	0.885985
$574$	0	0
$575$	−4.76957	−0.198905
$576$	0	0
$577$	31.8293	1.32507	0.662536	−	0.749030i	$-0.269480\pi$
$577$	31.8293	1.32507	0.662536	+	0.749030i	$0.269480\pi$
$578$	0	0
$579$	15.2523	0.633865
$580$	0	0
$581$	2.35047	0.0975140
$582$	0	0
$583$	60.1034	2.48923
$584$	0	0
$585$	2.11982	0.0876438
$586$	0	0
$587$	19.6046	0.809167	0.404584	−	0.914501i	$-0.367416\pi$
$587$	19.6046	0.809167	0.404584	+	0.914501i	$0.367416\pi$
$588$	0	0
$589$	−4.70284	−0.193777
$590$	0	0
$591$	−14.0688	−0.578713
$592$	0	0
$593$	18.5848	0.763185	0.381593	−	0.924331i	$-0.375376\pi$
$593$	18.5848	0.763185	0.381593	+	0.924331i	$0.375376\pi$
$594$	0	0
$595$	0.881400	0.0361339
$596$	0	0
$597$	13.2706	0.543130
$598$	0	0
$599$	42.2678	1.72701	0.863507	−	0.504337i	$-0.168263\pi$
$599$	42.2678	1.72701	0.863507	+	0.504337i	$0.168263\pi$
$600$	0	0
$601$	−19.3968	−0.791211	−0.395606	−	0.918420i	$-0.629465\pi$
$601$	−19.3968	−0.791211	−0.395606	+	0.918420i	$0.629465\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−11.5712	−0.470435
$606$	0	0
$607$	−12.5784	−0.510540	−0.255270	−	0.966870i	$-0.582164\pi$
$607$	−12.5784	−0.510540	−0.255270	+	0.966870i	$0.582164\pi$
$608$	0	0
$609$	1.76030	0.0713309
$610$	0	0
$611$	−16.0131	−0.647820
$612$	0	0
$613$	−17.8994	−0.722950	−0.361475	−	0.932382i	$-0.617727\pi$
$613$	−17.8994	−0.722950	−0.361475	+	0.932382i	$0.617727\pi$
$614$	0	0
$615$	1.56282	0.0630189
$616$	0	0
$617$	−14.9521	−0.601949	−0.300974	−	0.953632i	$-0.597312\pi$
$617$	−14.9521	−0.601949	−0.300974	+	0.953632i	$0.597312\pi$
$618$	0	0
$619$	12.6389	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$619$	12.6389	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$620$	0	0
$621$	−4.76957	−0.191396
$622$	0	0
$623$	−1.88100	−0.0753607
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.27110	0.250444
$628$	0	0
$629$	29.7183	1.18495
$630$	0	0
$631$	1.95655	0.0778891	0.0389446	−	0.999241i	$-0.487600\pi$
$631$	1.95655	0.0778891	0.0389446	+	0.999241i	$0.487600\pi$
$632$	0	0
$633$	−3.60376	−0.143237
$634$	0	0
$635$	−7.53111	−0.298863
$636$	0	0
$637$	14.6795	0.581623
$638$	0	0
$639$	5.03912	0.199344
$640$	0	0
$641$	−15.6620	−0.618611	−0.309306	−	0.950963i	$-0.600097\pi$
$641$	−15.6620	−0.618611	−0.309306	+	0.950963i	$0.600097\pi$
$642$	0	0
$643$	31.8842	1.25739	0.628695	−	0.777652i	$-0.283589\pi$
$643$	31.8842	1.25739	0.628695	+	0.777652i	$0.283589\pi$
$644$	0	0
$645$	−2.64975	−0.104334
$646$	0	0
$647$	27.7142	1.08956	0.544779	−	0.838580i	$-0.316614\pi$
$647$	27.7142	1.08956	0.544779	+	0.838580i	$0.316614\pi$
$648$	0	0
$649$	−56.6843	−2.22505
$650$	0	0
$651$	−0.976489	−0.0382716
$652$	0	0
$653$	−29.2711	−1.14547	−0.572733	−	0.819742i	$-0.694117\pi$
$653$	−29.2711	−1.14547	−0.572733	+	0.819742i	$0.694117\pi$
$654$	0	0
$655$	12.9151	0.504636
$656$	0	0
$657$	0.416872	0.0162637
$658$	0	0
$659$	−28.8923	−1.12549	−0.562743	−	0.826632i	$-0.690254\pi$
$659$	−28.8923	−1.12549	−0.562743	+	0.826632i	$0.690254\pi$
$660$	0	0
$661$	−32.5189	−1.26484	−0.632419	−	0.774627i	$-0.717938\pi$
$661$	−32.5189	−1.26484	−0.632419	+	0.774627i	$0.717938\pi$
$662$	0	0
$663$	−6.81708	−0.264754
$664$	0	0
$665$	−0.361776	−0.0140291
$666$	0	0
$667$	30.6332	1.18612
$668$	0	0
$669$	10.1472	0.392314
$670$	0	0
$671$	−26.6981	−1.03067
$672$	0	0
$673$	42.9155	1.65427	0.827136	−	0.562002i	$-0.189969\pi$
$673$	42.9155	1.65427	0.827136	+	0.562002i	$0.189969\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	19.9407	0.766382	0.383191	−	0.923669i	$-0.374825\pi$
$677$	19.9407	0.766382	0.383191	+	0.923669i	$0.374825\pi$
$678$	0	0
$679$	3.08683	0.118462
$680$	0	0
$681$	−2.42806	−0.0930435
$682$	0	0
$683$	29.7954	1.14009	0.570045	−	0.821613i	$-0.306926\pi$
$683$	29.7954	1.14009	0.570045	+	0.821613i	$0.306926\pi$
$684$	0	0
$685$	−3.99615	−0.152685
$686$	0	0
$687$	0.696447	0.0265711
$688$	0	0
$689$	26.8177	1.02167
$690$	0	0
$691$	−48.5139	−1.84556	−0.922779	−	0.385330i	$-0.874088\pi$
$691$	−48.5139	−1.84556	−0.922779	+	0.385330i	$0.874088\pi$
$692$	0	0
$693$	1.30212	0.0494634
$694$	0	0
$695$	2.03032	0.0770145
$696$	0	0
$697$	−5.02583	−0.190367
$698$	0	0
$699$	24.1181	0.912232
$700$	0	0
$701$	51.7582	1.95488	0.977440	−	0.211214i	$-0.0677417\pi$
$701$	51.7582	1.95488	0.977440	+	0.211214i	$0.0677417\pi$
$702$	0	0
$703$	−12.1981	−0.460059
$704$	0	0
$705$	−7.55397	−0.284499
$706$	0	0
$707$	−3.21537	−0.120927
$708$	0	0
$709$	−26.0615	−0.978760	−0.489380	−	0.872071i	$-0.662777\pi$
$709$	−26.0615	−0.978760	−0.489380	+	0.872071i	$0.662777\pi$
$710$	0	0
$711$	15.6884	0.588361
$712$	0	0
$713$	−16.9931	−0.636397
$714$	0	0
$715$	−10.0711	−0.376637
$716$	0	0
$717$	−7.52635	−0.281077
$718$	0	0
$719$	13.0431	0.486425	0.243213	−	0.969973i	$-0.421799\pi$
$719$	13.0431	0.486425	0.243213	+	0.969973i	$0.421799\pi$
$720$	0	0
$721$	−1.45247	−0.0540930
$722$	0	0
$723$	−8.96980	−0.333590
$724$	0	0
$725$	−6.42263	−0.238530
$726$	0	0
$727$	−1.65637	−0.0614315	−0.0307157	−	0.999528i	$-0.509779\pi$
$727$	−1.65637	−0.0614315	−0.0307157	+	0.999528i	$0.509779\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.52126	0.315170
$732$	0	0
$733$	−35.7746	−1.32137	−0.660683	−	0.750665i	$-0.729733\pi$
$733$	−35.7746	−1.32137	−0.660683	+	0.750665i	$0.729733\pi$
$734$	0	0
$735$	6.92488	0.255428
$736$	0	0
$737$	−4.75091	−0.175002
$738$	0	0
$739$	43.9936	1.61833	0.809165	−	0.587581i	$-0.199920\pi$
$739$	43.9936	1.61833	0.809165	+	0.587581i	$0.199920\pi$
$740$	0	0
$741$	2.79812	0.102791
$742$	0	0
$743$	25.2341	0.925751	0.462875	−	0.886423i	$-0.346818\pi$
$743$	25.2341	0.925751	0.462875	+	0.886423i	$0.346818\pi$
$744$	0	0
$745$	−3.32105	−0.121674
$746$	0	0
$747$	−8.57594	−0.313777
$748$	0	0
$749$	−1.92538	−0.0703517
$750$	0	0
$751$	−29.4931	−1.07622	−0.538109	−	0.842875i	$-0.680861\pi$
$751$	−29.4931	−1.07622	−0.538109	+	0.842875i	$0.680861\pi$
$752$	0	0
$753$	−3.03755	−0.110695
$754$	0	0
$755$	14.6664	0.533766
$756$	0	0
$757$	16.9847	0.617318	0.308659	−	0.951173i	$-0.400120\pi$
$757$	16.9847	0.617318	0.308659	+	0.951173i	$0.400120\pi$
$758$	0	0
$759$	22.6598	0.822499
$760$	0	0
$761$	−45.0408	−1.63273	−0.816364	−	0.577538i	$-0.804014\pi$
$761$	−45.0408	−1.63273	−0.816364	+	0.577538i	$0.804014\pi$
$762$	0	0
$763$	−2.70345	−0.0978714
$764$	0	0
$765$	−3.21588	−0.116270
$766$	0	0
$767$	−25.2921	−0.913244
$768$	0	0
$769$	−24.3584	−0.878387	−0.439194	−	0.898392i	$-0.644736\pi$
$769$	−24.3584	−0.878387	−0.439194	+	0.898392i	$0.644736\pi$
$770$	0	0
$771$	−3.68721	−0.132792
$772$	0	0
$773$	24.1670	0.869227	0.434614	−	0.900617i	$-0.356885\pi$
$773$	24.1670	0.869227	0.434614	+	0.900617i	$0.356885\pi$
$774$	0	0
$775$	3.56282	0.127980
$776$	0	0
$777$	−2.53279	−0.0908632
$778$	0	0
$779$	2.06289	0.0739106
$780$	0	0
$781$	−23.9404	−0.856655
$782$	0	0
$783$	−6.42263	−0.229526
$784$	0	0
$785$	−1.11832	−0.0399146
$786$	0	0
$787$	−10.0855	−0.359508	−0.179754	−	0.983712i	$-0.557530\pi$
$787$	−10.0855	−0.359508	−0.179754	+	0.983712i	$0.557530\pi$
$788$	0	0
$789$	0.242711	0.00864073
$790$	0	0
$791$	0.149743	0.00532426
$792$	0	0
$793$	−11.9125	−0.423025
$794$	0	0
$795$	12.6509	0.448682
$796$	0	0
$797$	34.3548	1.21691	0.608454	−	0.793589i	$-0.291790\pi$
$797$	34.3548	1.21691	0.608454	+	0.793589i	$0.291790\pi$
$798$	0	0
$799$	24.2926	0.859412
$800$	0	0
$801$	6.86302	0.242493
$802$	0	0
$803$	−1.98052	−0.0698912
$804$	0	0
$805$	−1.30723	−0.0460739
$806$	0	0
$807$	−28.3909	−0.999407
$808$	0	0
$809$	10.1064	0.355322	0.177661	−	0.984092i	$-0.443147\pi$
$809$	10.1064	0.355322	0.177661	+	0.984092i	$0.443147\pi$
$810$	0	0
$811$	38.2166	1.34197	0.670983	−	0.741473i	$-0.265873\pi$
$811$	38.2166	1.34197	0.670983	+	0.741473i	$0.265873\pi$
$812$	0	0
$813$	2.93780	0.103033
$814$	0	0
$815$	−1.93177	−0.0676668
$816$	0	0
$817$	−3.49761	−0.122366
$818$	0	0
$819$	0.580996	0.0203016
$820$	0	0
$821$	48.9304	1.70768	0.853841	−	0.520534i	$-0.174267\pi$
$821$	48.9304	1.70768	0.853841	+	0.520534i	$0.174267\pi$
$822$	0	0
$823$	−44.0331	−1.53490	−0.767449	−	0.641110i	$-0.778474\pi$
$823$	−44.0331	−1.53490	−0.767449	+	0.641110i	$0.778474\pi$
$824$	0	0
$825$	−4.75091	−0.165406
$826$	0	0
$827$	−0.996143	−0.0346393	−0.0173196	−	0.999850i	$-0.505513\pi$
$827$	−0.996143	−0.0346393	−0.0173196	+	0.999850i	$0.505513\pi$
$828$	0	0
$829$	−18.1697	−0.631060	−0.315530	−	0.948916i	$-0.602182\pi$
$829$	−18.1697	−0.631060	−0.315530	+	0.948916i	$0.602182\pi$
$830$	0	0
$831$	3.37127	0.116948
$832$	0	0
$833$	−22.2696	−0.771594
$834$	0	0
$835$	−6.84403	−0.236847
$836$	0	0
$837$	3.56282	0.123149
$838$	0	0
$839$	22.2280	0.767395	0.383698	−	0.923459i	$-0.374650\pi$
$839$	22.2280	0.767395	0.383698	+	0.923459i	$0.374650\pi$
$840$	0	0
$841$	12.2501	0.422418
$842$	0	0
$843$	−5.98422	−0.206108
$844$	0	0
$845$	8.50636	0.292628
$846$	0	0
$847$	−3.17140	−0.108971
$848$	0	0
$849$	17.8711	0.613335
$850$	0	0
$851$	−44.0762	−1.51091
$852$	0	0
$853$	10.8813	0.372568	0.186284	−	0.982496i	$-0.440356\pi$
$853$	10.8813	0.372568	0.186284	+	0.982496i	$0.440356\pi$
$854$	0	0
$855$	1.31998	0.0451423
$856$	0	0
$857$	2.23751	0.0764320	0.0382160	−	0.999270i	$-0.487832\pi$
$857$	2.23751	0.0764320	0.0382160	+	0.999270i	$0.487832\pi$
$858$	0	0
$859$	21.2351	0.724532	0.362266	−	0.932075i	$-0.382003\pi$
$859$	21.2351	0.724532	0.362266	+	0.932075i	$0.382003\pi$
$860$	0	0
$861$	0.428334	0.0145976
$862$	0	0
$863$	−36.3874	−1.23864	−0.619321	−	0.785138i	$-0.712592\pi$
$863$	−36.3874	−1.23864	−0.619321	+	0.785138i	$0.712592\pi$
$864$	0	0
$865$	−4.23895	−0.144129
$866$	0	0
$867$	−6.65814	−0.226122
$868$	0	0
$869$	−74.5342	−2.52840
$870$	0	0
$871$	−2.11982	−0.0718274
$872$	0	0
$873$	−11.2626	−0.381181
$874$	0	0
$875$	0.274078	0.00926552
$876$	0	0
$877$	−12.3253	−0.416195	−0.208097	−	0.978108i	$-0.566727\pi$
$877$	−12.3253	−0.416195	−0.208097	+	0.978108i	$0.566727\pi$
$878$	0	0
$879$	−21.8762	−0.737864
$880$	0	0
$881$	−43.3712	−1.46121	−0.730606	−	0.682800i	$-0.760762\pi$
$881$	−43.3712	−1.46121	−0.730606	+	0.682800i	$0.760762\pi$
$882$	0	0
$883$	8.88396	0.298969	0.149485	−	0.988764i	$-0.452239\pi$
$883$	8.88396	0.298969	0.149485	+	0.988764i	$0.452239\pi$
$884$	0	0
$885$	−11.9312	−0.401064
$886$	0	0
$887$	−3.87622	−0.130151	−0.0650754	−	0.997880i	$-0.520729\pi$
$887$	−3.87622	−0.130151	−0.0650754	+	0.997880i	$0.520729\pi$
$888$	0	0
$889$	−2.06411	−0.0692280
$890$	0	0
$891$	−4.75091	−0.159162
$892$	0	0
$893$	−9.97107	−0.333669
$894$	0	0
$895$	−17.5435	−0.586415
$896$	0	0
$897$	10.1106	0.337584
$898$	0	0
$899$	−22.8827	−0.763179
$900$	0	0
$901$	−40.6838	−1.35537
$902$	0	0
$903$	−0.726237	−0.0241676
$904$	0	0
$905$	−17.5358	−0.582909
$906$	0	0
$907$	32.1024	1.06594	0.532972	−	0.846133i	$-0.321075\pi$
$907$	32.1024	1.06594	0.532972	+	0.846133i	$0.321075\pi$
$908$	0	0
$909$	11.7316	0.389113
$910$	0	0
$911$	−24.9479	−0.826562	−0.413281	−	0.910603i	$-0.635617\pi$
$911$	−24.9479	−0.826562	−0.413281	+	0.910603i	$0.635617\pi$
$912$	0	0
$913$	40.7435	1.34841
$914$	0	0
$915$	−5.61957	−0.185777
$916$	0	0
$917$	3.53975	0.116893
$918$	0	0
$919$	11.6271	0.383541	0.191771	−	0.981440i	$-0.438577\pi$
$919$	11.6271	0.383541	0.191771	+	0.981440i	$0.438577\pi$
$920$	0	0
$921$	−19.4125	−0.639665
$922$	0	0
$923$	−10.6820	−0.351603
$924$	0	0
$925$	9.24113	0.303846
$926$	0	0
$927$	5.29950	0.174058
$928$	0	0
$929$	−37.1907	−1.22019	−0.610094	−	0.792329i	$-0.708868\pi$
$929$	−37.1907	−1.22019	−0.610094	+	0.792329i	$0.708868\pi$
$930$	0	0
$931$	9.14069	0.299574
$932$	0	0
$933$	32.7827	1.07326
$934$	0	0
$935$	15.2783	0.499655
$936$	0	0
$937$	−35.7860	−1.16908	−0.584538	−	0.811366i	$-0.698724\pi$
$937$	−35.7860	−1.16908	−0.584538	+	0.811366i	$0.698724\pi$
$938$	0	0
$939$	28.0757	0.916217
$940$	0	0
$941$	−44.5118	−1.45104	−0.725521	−	0.688200i	$-0.758401\pi$
$941$	−44.5118	−1.45104	−0.725521	+	0.688200i	$0.758401\pi$
$942$	0	0
$943$	7.45397	0.242735
$944$	0	0
$945$	0.274078	0.00891575
$946$	0	0
$947$	−5.21198	−0.169366	−0.0846832	−	0.996408i	$-0.526988\pi$
$947$	−5.21198	−0.169366	−0.0846832	+	0.996408i	$0.526988\pi$
$948$	0	0
$949$	−0.883695	−0.0286859
$950$	0	0
$951$	−0.862332	−0.0279630
$952$	0	0
$953$	26.9135	0.871814	0.435907	−	0.899992i	$-0.356428\pi$
$953$	26.9135	0.871814	0.435907	+	0.899992i	$0.356428\pi$
$954$	0	0
$955$	−21.2082	−0.686281
$956$	0	0
$957$	30.5133	0.986356
$958$	0	0
$959$	−1.09526	−0.0353677
$960$	0	0
$961$	−18.3063	−0.590527
$962$	0	0
$963$	7.02493	0.226375
$964$	0	0
$965$	−15.2523	−0.490990
$966$	0	0
$967$	37.7612	1.21432	0.607159	−	0.794581i	$-0.292309\pi$
$967$	37.7612	1.21432	0.607159	+	0.794581i	$0.292309\pi$
$968$	0	0
$969$	−4.24488	−0.136365
$970$	0	0
$971$	11.1654	0.358316	0.179158	−	0.983820i	$-0.442663\pi$
$971$	11.1654	0.358316	0.179158	+	0.983820i	$0.442663\pi$
$972$	0	0
$973$	0.556466	0.0178395
$974$	0	0
$975$	−2.11982	−0.0678886
$976$	0	0
$977$	34.1644	1.09302	0.546508	−	0.837454i	$-0.315957\pi$
$977$	34.1644	1.09302	0.546508	+	0.837454i	$0.315957\pi$
$978$	0	0
$979$	−32.6056	−1.04208
$980$	0	0
$981$	9.86380	0.314927
$982$	0	0
$983$	−5.53628	−0.176580	−0.0882899	−	0.996095i	$-0.528140\pi$
$983$	−5.53628	−0.176580	−0.0882899	+	0.996095i	$0.528140\pi$
$984$	0	0
$985$	14.0688	0.448269
$986$	0	0
$987$	−2.07037	−0.0659008
$988$	0	0
$989$	−12.6382	−0.401870
$990$	0	0
$991$	0.895489	0.0284462	0.0142231	−	0.999899i	$-0.495473\pi$
$991$	0.895489	0.0284462	0.0142231	+	0.999899i	$0.495473\pi$
$992$	0	0
$993$	12.9261	0.410198
$994$	0	0
$995$	−13.2706	−0.420706
$996$	0	0
$997$	23.5066	0.744461	0.372230	−	0.928140i	$-0.378593\pi$
$997$	23.5066	0.744461	0.372230	+	0.928140i	$0.378593\pi$
$998$	0	0
$999$	9.24113	0.292376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.t.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.t.1.2	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.274078 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.274078 q^{7} +1.00000 q^{9} -4.75091 q^{11} -2.11982 q^{13} -1.00000 q^{15} +3.21588 q^{17} -1.31998 q^{19} -0.274078 q^{21} -4.76957 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.42263 q^{29} +3.56282 q^{31} -4.75091 q^{33} +0.274078 q^{35} +9.24113 q^{37} -2.11982 q^{39} -1.56282 q^{41} +2.64975 q^{43} -1.00000 q^{45} +7.55397 q^{47} -6.92488 q^{49} +3.21588 q^{51} -12.6509 q^{53} +4.75091 q^{55} -1.31998 q^{57} +11.9312 q^{59} +5.61957 q^{61} -0.274078 q^{63} +2.11982 q^{65} +1.00000 q^{67} -4.76957 q^{69} +5.03912 q^{71} +0.416872 q^{73} +1.00000 q^{75} +1.30212 q^{77} +15.6884 q^{79} +1.00000 q^{81} -8.57594 q^{83} -3.21588 q^{85} -6.42263 q^{87} +6.86302 q^{89} +0.580996 q^{91} +3.56282 q^{93} +1.31998 q^{95} -11.2626 q^{97} -4.75091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) \) 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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