LMFDB - Embedded newform 8112.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.73205 q^{5} -2.73205 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.73205 q^{5} -2.73205 q^{7} +1.00000 q^{9} +1.26795 q^{11} +3.73205 q^{15} -5.73205 q^{17} +4.73205 q^{19} +2.73205 q^{21} -4.19615 q^{23} +8.92820 q^{25} -1.00000 q^{27} -4.46410 q^{29} +1.46410 q^{31} -1.26795 q^{33} +10.1962 q^{35} -3.53590 q^{37} +9.39230 q^{41} +9.66025 q^{43} -3.73205 q^{45} +2.19615 q^{47} +0.464102 q^{49} +5.73205 q^{51} -6.46410 q^{53} -4.73205 q^{55} -4.73205 q^{57} +8.00000 q^{59} -9.19615 q^{61} -2.73205 q^{63} +13.1244 q^{67} +4.19615 q^{69} +4.73205 q^{71} +6.26795 q^{73} -8.92820 q^{75} -3.46410 q^{77} +2.53590 q^{79} +1.00000 q^{81} -0.196152 q^{83} +21.3923 q^{85} +4.46410 q^{87} +9.46410 q^{89} -1.46410 q^{93} -17.6603 q^{95} +6.00000 q^{97} +1.26795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 4 q^{15} - 8 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} + 4 q^{25} - 2 q^{27} - 2 q^{29} - 4 q^{31} - 6 q^{33} + 10 q^{35} - 14 q^{37} - 2 q^{41} + 2 q^{43} - 4 q^{45} - 6 q^{47} - 6 q^{49} + 8 q^{51} - 6 q^{53} - 6 q^{55} - 6 q^{57} + 16 q^{59} - 8 q^{61} - 2 q^{63} + 2 q^{67} - 2 q^{69} + 6 q^{71} + 16 q^{73} - 4 q^{75} + 12 q^{79} + 2 q^{81} + 10 q^{83} + 22 q^{85} + 2 q^{87} + 12 q^{89} + 4 q^{93} - 18 q^{95} + 12 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 4 * q^15 - 8 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^23 + 4 * q^25 - 2 * q^27 - 2 * q^29 - 4 * q^31 - 6 * q^33 + 10 * q^35 - 14 * q^37 - 2 * q^41 + 2 * q^43 - 4 * q^45 - 6 * q^47 - 6 * q^49 + 8 * q^51 - 6 * q^53 - 6 * q^55 - 6 * q^57 + 16 * q^59 - 8 * q^61 - 2 * q^63 + 2 * q^67 - 2 * q^69 + 6 * q^71 + 16 * q^73 - 4 * q^75 + 12 * q^79 + 2 * q^81 + 10 * q^83 + 22 * q^85 + 2 * q^87 + 12 * q^89 + 4 * q^93 - 18 * q^95 + 12 * q^97 + 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.73205	−1.66902	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−3.73205	−1.66902	−0.834512	+	0.550990i	$0.814250\pi$
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.73205	0.963611
$16$	0	0
$17$	−5.73205	−1.39023	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	−5.73205	−1.39023	−0.695113	+	0.718900i	$0.744646\pi$
$18$	0	0
$19$	4.73205	1.08561	0.542803	−	0.839860i	$-0.317363\pi$
$19$	4.73205	1.08561	0.542803	+	0.839860i	$0.317363\pi$
$20$	0	0
$21$	2.73205	0.596182
$22$	0	0
$23$	−4.19615	−0.874958	−0.437479	−	0.899229i	$-0.644129\pi$
$23$	−4.19615	−0.874958	−0.437479	+	0.899229i	$0.644129\pi$
$24$	0	0
$25$	8.92820	1.78564
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.46410	−0.828963	−0.414481	−	0.910058i	$-0.636037\pi$
$29$	−4.46410	−0.828963	−0.414481	+	0.910058i	$0.636037\pi$
$30$	0	0
$31$	1.46410	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$31$	1.46410	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$32$	0	0
$33$	−1.26795	−0.220722
$34$	0	0
$35$	10.1962	1.72346
$36$	0	0
$37$	−3.53590	−0.581298	−0.290649	−	0.956830i	$-0.593871\pi$
$37$	−3.53590	−0.581298	−0.290649	+	0.956830i	$0.593871\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.39230	1.46683	0.733416	−	0.679780i	$-0.237925\pi$
$41$	9.39230	1.46683	0.733416	+	0.679780i	$0.237925\pi$
$42$	0	0
$43$	9.66025	1.47317	0.736587	−	0.676342i	$-0.236436\pi$
$43$	9.66025	1.47317	0.736587	+	0.676342i	$0.236436\pi$
$44$	0	0
$45$	−3.73205	−0.556341
$46$	0	0
$47$	2.19615	0.320342	0.160171	−	0.987089i	$-0.448795\pi$
$47$	2.19615	0.320342	0.160171	+	0.987089i	$0.448795\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	5.73205	0.802648
$52$	0	0
$53$	−6.46410	−0.887913	−0.443956	−	0.896048i	$-0.646425\pi$
$53$	−6.46410	−0.887913	−0.443956	+	0.896048i	$0.646425\pi$
$54$	0	0
$55$	−4.73205	−0.638070
$56$	0	0
$57$	−4.73205	−0.626775
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−9.19615	−1.17745	−0.588723	−	0.808335i	$-0.700369\pi$
$61$	−9.19615	−1.17745	−0.588723	+	0.808335i	$0.700369\pi$
$62$	0	0
$63$	−2.73205	−0.344206
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1244	1.60340	0.801698	−	0.597730i	$-0.203930\pi$
$67$	13.1244	1.60340	0.801698	+	0.597730i	$0.203930\pi$
$68$	0	0
$69$	4.19615	0.505157
$70$	0	0
$71$	4.73205	0.561591	0.280796	−	0.959768i	$-0.409402\pi$
$71$	4.73205	0.561591	0.280796	+	0.959768i	$0.409402\pi$
$72$	0	0
$73$	6.26795	0.733608	0.366804	−	0.930298i	$-0.380452\pi$
$73$	6.26795	0.733608	0.366804	+	0.930298i	$0.380452\pi$
$74$	0	0
$75$	−8.92820	−1.03094
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	2.53590	0.285311	0.142655	−	0.989772i	$-0.454436\pi$
$79$	2.53590	0.285311	0.142655	+	0.989772i	$0.454436\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.196152	−0.0215305	−0.0107653	−	0.999942i	$-0.503427\pi$
$83$	−0.196152	−0.0215305	−0.0107653	+	0.999942i	$0.503427\pi$
$84$	0	0
$85$	21.3923	2.32032
$86$	0	0
$87$	4.46410	0.478602
$88$	0	0
$89$	9.46410	1.00319	0.501596	−	0.865102i	$-0.332746\pi$
$89$	9.46410	1.00319	0.501596	+	0.865102i	$0.332746\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.46410	−0.151820
$94$	0	0
$95$	−17.6603	−1.81190
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	1.26795	0.127434
$100$	0	0
$101$	1.92820	0.191863	0.0959317	−	0.995388i	$-0.469417\pi$
$101$	1.92820	0.191863	0.0959317	+	0.995388i	$0.469417\pi$
$102$	0	0
$103$	15.2679	1.50440	0.752198	−	0.658937i	$-0.228994\pi$
$103$	15.2679	1.50440	0.752198	+	0.658937i	$0.228994\pi$
$104$	0	0
$105$	−10.1962	−0.995043
$106$	0	0
$107$	−10.1962	−0.985699	−0.492850	−	0.870114i	$-0.664045\pi$
$107$	−10.1962	−0.985699	−0.492850	+	0.870114i	$0.664045\pi$
$108$	0	0
$109$	−1.46410	−0.140236	−0.0701178	−	0.997539i	$-0.522338\pi$
$109$	−1.46410	−0.140236	−0.0701178	+	0.997539i	$0.522338\pi$
$110$	0	0
$111$	3.53590	0.335613
$112$	0	0
$113$	−1.33975	−0.126033	−0.0630163	−	0.998012i	$-0.520072\pi$
$113$	−1.33975	−0.126033	−0.0630163	+	0.998012i	$0.520072\pi$
$114$	0	0
$115$	15.6603	1.46033
$116$	0	0
$117$	0	0
$118$	0	0
$119$	15.6603	1.43557
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	−9.39230	−0.846876
$124$	0	0
$125$	−14.6603	−1.31125
$126$	0	0
$127$	9.85641	0.874615	0.437307	−	0.899312i	$-0.355932\pi$
$127$	9.85641	0.874615	0.437307	+	0.899312i	$0.355932\pi$
$128$	0	0
$129$	−9.66025	−0.850538
$130$	0	0
$131$	−6.53590	−0.571044	−0.285522	−	0.958372i	$-0.592167\pi$
$131$	−6.53590	−0.571044	−0.285522	+	0.958372i	$0.592167\pi$
$132$	0	0
$133$	−12.9282	−1.12102
$134$	0	0
$135$	3.73205	0.321204
$136$	0	0
$137$	−11.9282	−1.01910	−0.509548	−	0.860442i	$-0.670187\pi$
$137$	−11.9282	−1.01910	−0.509548	+	0.860442i	$0.670187\pi$
$138$	0	0
$139$	−17.8564	−1.51456	−0.757280	−	0.653090i	$-0.773472\pi$
$139$	−17.8564	−1.51456	−0.757280	+	0.653090i	$0.773472\pi$
$140$	0	0
$141$	−2.19615	−0.184949
$142$	0	0
$143$	0	0
$144$	0	0
$145$	16.6603	1.38356
$146$	0	0
$147$	−0.464102	−0.0382785
$148$	0	0
$149$	−13.1962	−1.08107	−0.540535	−	0.841321i	$-0.681778\pi$
$149$	−13.1962	−1.08107	−0.540535	+	0.841321i	$0.681778\pi$
$150$	0	0
$151$	6.73205	0.547847	0.273923	−	0.961752i	$-0.411679\pi$
$151$	6.73205	0.547847	0.273923	+	0.961752i	$0.411679\pi$
$152$	0	0
$153$	−5.73205	−0.463409
$154$	0	0
$155$	−5.46410	−0.438887
$156$	0	0
$157$	7.58846	0.605625	0.302812	−	0.953050i	$-0.402074\pi$
$157$	7.58846	0.605625	0.302812	+	0.953050i	$0.402074\pi$
$158$	0	0
$159$	6.46410	0.512637
$160$	0	0
$161$	11.4641	0.903498
$162$	0	0
$163$	−13.4641	−1.05459	−0.527295	−	0.849682i	$-0.676794\pi$
$163$	−13.4641	−1.05459	−0.527295	+	0.849682i	$0.676794\pi$
$164$	0	0
$165$	4.73205	0.368390
$166$	0	0
$167$	−9.46410	−0.732354	−0.366177	−	0.930545i	$-0.619334\pi$
$167$	−9.46410	−0.732354	−0.366177	+	0.930545i	$0.619334\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.73205	0.361869
$172$	0	0
$173$	4.39230	0.333941	0.166970	−	0.985962i	$-0.446602\pi$
$173$	4.39230	0.333941	0.166970	+	0.985962i	$0.446602\pi$
$174$	0	0
$175$	−24.3923	−1.84388
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	16.0526	1.19982	0.599912	−	0.800066i	$-0.295202\pi$
$179$	16.0526	1.19982	0.599912	+	0.800066i	$0.295202\pi$
$180$	0	0
$181$	19.1962	1.42684	0.713419	−	0.700737i	$-0.247145\pi$
$181$	19.1962	1.42684	0.713419	+	0.700737i	$0.247145\pi$
$182$	0	0
$183$	9.19615	0.679799
$184$	0	0
$185$	13.1962	0.970200
$186$	0	0
$187$	−7.26795	−0.531485
$188$	0	0
$189$	2.73205	0.198727
$190$	0	0
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	11.7321	0.844491	0.422246	−	0.906481i	$-0.361242\pi$
$193$	11.7321	0.844491	0.422246	+	0.906481i	$0.361242\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.8564	1.27222	0.636108	−	0.771600i	$-0.280543\pi$
$197$	17.8564	1.27222	0.636108	+	0.771600i	$0.280543\pi$
$198$	0	0
$199$	14.1962	1.00634	0.503169	−	0.864188i	$-0.332167\pi$
$199$	14.1962	1.00634	0.503169	+	0.864188i	$0.332167\pi$
$200$	0	0
$201$	−13.1244	−0.925721
$202$	0	0
$203$	12.1962	0.856002
$204$	0	0
$205$	−35.0526	−2.44818
$206$	0	0
$207$	−4.19615	−0.291653
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−16.3923	−1.12849	−0.564246	−	0.825606i	$-0.690833\pi$
$211$	−16.3923	−1.12849	−0.564246	+	0.825606i	$0.690833\pi$
$212$	0	0
$213$	−4.73205	−0.324235
$214$	0	0
$215$	−36.0526	−2.45876
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−6.26795	−0.423549
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−26.9282	−1.80325	−0.901623	−	0.432523i	$-0.857623\pi$
$223$	−26.9282	−1.80325	−0.901623	+	0.432523i	$0.857623\pi$
$224$	0	0
$225$	8.92820	0.595214
$226$	0	0
$227$	−12.1962	−0.809487	−0.404744	−	0.914430i	$-0.632639\pi$
$227$	−12.1962	−0.809487	−0.404744	+	0.914430i	$0.632639\pi$
$228$	0	0
$229$	11.8564	0.783493	0.391747	−	0.920073i	$-0.371871\pi$
$229$	11.8564	0.783493	0.391747	+	0.920073i	$0.371871\pi$
$230$	0	0
$231$	3.46410	0.227921
$232$	0	0
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	0	0
$235$	−8.19615	−0.534658
$236$	0	0
$237$	−2.53590	−0.164724
$238$	0	0
$239$	7.66025	0.495501	0.247750	−	0.968824i	$-0.420309\pi$
$239$	7.66025	0.495501	0.247750	+	0.968824i	$0.420309\pi$
$240$	0	0
$241$	−13.5885	−0.875309	−0.437655	−	0.899143i	$-0.644191\pi$
$241$	−13.5885	−0.875309	−0.437655	+	0.899143i	$0.644191\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.73205	−0.110657
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.196152	0.0124307
$250$	0	0
$251$	−13.4641	−0.849847	−0.424923	−	0.905229i	$-0.639699\pi$
$251$	−13.4641	−0.849847	−0.424923	+	0.905229i	$0.639699\pi$
$252$	0	0
$253$	−5.32051	−0.334497
$254$	0	0
$255$	−21.3923	−1.33964
$256$	0	0
$257$	9.33975	0.582597	0.291299	−	0.956632i	$-0.405913\pi$
$257$	9.33975	0.582597	0.291299	+	0.956632i	$0.405913\pi$
$258$	0	0
$259$	9.66025	0.600259
$260$	0	0
$261$	−4.46410	−0.276321
$262$	0	0
$263$	10.0526	0.619867	0.309934	−	0.950758i	$-0.399693\pi$
$263$	10.0526	0.619867	0.309934	+	0.950758i	$0.399693\pi$
$264$	0	0
$265$	24.1244	1.48195
$266$	0	0
$267$	−9.46410	−0.579194
$268$	0	0
$269$	5.46410	0.333152	0.166576	−	0.986029i	$-0.446729\pi$
$269$	5.46410	0.333152	0.166576	+	0.986029i	$0.446729\pi$
$270$	0	0
$271$	−21.8564	−1.32768	−0.663841	−	0.747874i	$-0.731075\pi$
$271$	−21.8564	−1.32768	−0.663841	+	0.747874i	$0.731075\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.3205	0.682652
$276$	0	0
$277$	−5.73205	−0.344406	−0.172203	−	0.985062i	$-0.555088\pi$
$277$	−5.73205	−0.344406	−0.172203	+	0.985062i	$0.555088\pi$
$278$	0	0
$279$	1.46410	0.0876535
$280$	0	0
$281$	−12.3205	−0.734980	−0.367490	−	0.930027i	$-0.619783\pi$
$281$	−12.3205	−0.734980	−0.367490	+	0.930027i	$0.619783\pi$
$282$	0	0
$283$	−25.6603	−1.52534	−0.762672	−	0.646786i	$-0.776113\pi$
$283$	−25.6603	−1.52534	−0.762672	+	0.646786i	$0.776113\pi$
$284$	0	0
$285$	17.6603	1.04610
$286$	0	0
$287$	−25.6603	−1.51468
$288$	0	0
$289$	15.8564	0.932730
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	−30.5167	−1.78280	−0.891401	−	0.453215i	$-0.850277\pi$
$293$	−30.5167	−1.78280	−0.891401	+	0.453215i	$0.850277\pi$
$294$	0	0
$295$	−29.8564	−1.73831
$296$	0	0
$297$	−1.26795	−0.0735739
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−26.3923	−1.52123
$302$	0	0
$303$	−1.92820	−0.110772
$304$	0	0
$305$	34.3205	1.96519
$306$	0	0
$307$	−22.5885	−1.28919	−0.644596	−	0.764524i	$-0.722974\pi$
$307$	−22.5885	−1.28919	−0.644596	+	0.764524i	$0.722974\pi$
$308$	0	0
$309$	−15.2679	−0.868563
$310$	0	0
$311$	1.66025	0.0941444	0.0470722	−	0.998891i	$-0.485011\pi$
$311$	1.66025	0.0941444	0.0470722	+	0.998891i	$0.485011\pi$
$312$	0	0
$313$	6.53590	0.369431	0.184715	−	0.982792i	$-0.440864\pi$
$313$	6.53590	0.369431	0.184715	+	0.982792i	$0.440864\pi$
$314$	0	0
$315$	10.1962	0.574488
$316$	0	0
$317$	−20.6603	−1.16040	−0.580198	−	0.814476i	$-0.697025\pi$
$317$	−20.6603	−1.16040	−0.580198	+	0.814476i	$0.697025\pi$
$318$	0	0
$319$	−5.66025	−0.316913
$320$	0	0
$321$	10.1962	0.569094
$322$	0	0
$323$	−27.1244	−1.50924
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.46410	0.0809650
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	−3.53590	−0.193766
$334$	0	0
$335$	−48.9808	−2.67610
$336$	0	0
$337$	−20.8564	−1.13612	−0.568060	−	0.822987i	$-0.692306\pi$
$337$	−20.8564	−1.13612	−0.568060	+	0.822987i	$0.692306\pi$
$338$	0	0
$339$	1.33975	0.0727650
$340$	0	0
$341$	1.85641	0.100530
$342$	0	0
$343$	17.8564	0.964155
$344$	0	0
$345$	−15.6603	−0.843120
$346$	0	0
$347$	−33.1244	−1.77821	−0.889104	−	0.457705i	$-0.848672\pi$
$347$	−33.1244	−1.77821	−0.889104	+	0.457705i	$0.848672\pi$
$348$	0	0
$349$	−15.3205	−0.820088	−0.410044	−	0.912066i	$-0.634487\pi$
$349$	−15.3205	−0.820088	−0.410044	+	0.912066i	$0.634487\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.7846	1.15948	0.579739	−	0.814802i	$-0.303155\pi$
$353$	21.7846	1.15948	0.579739	+	0.814802i	$0.303155\pi$
$354$	0	0
$355$	−17.6603	−0.937309
$356$	0	0
$357$	−15.6603	−0.828829
$358$	0	0
$359$	−1.12436	−0.0593412	−0.0296706	−	0.999560i	$-0.509446\pi$
$359$	−1.12436	−0.0593412	−0.0296706	+	0.999560i	$0.509446\pi$
$360$	0	0
$361$	3.39230	0.178542
$362$	0	0
$363$	9.39230	0.492968
$364$	0	0
$365$	−23.3923	−1.22441
$366$	0	0
$367$	11.2679	0.588182	0.294091	−	0.955777i	$-0.404983\pi$
$367$	11.2679	0.588182	0.294091	+	0.955777i	$0.404983\pi$
$368$	0	0
$369$	9.39230	0.488944
$370$	0	0
$371$	17.6603	0.916875
$372$	0	0
$373$	−13.7321	−0.711019	−0.355509	−	0.934673i	$-0.615693\pi$
$373$	−13.7321	−0.711019	−0.355509	+	0.934673i	$0.615693\pi$
$374$	0	0
$375$	14.6603	0.757052
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.46410	−0.280672	−0.140336	−	0.990104i	$-0.544818\pi$
$379$	−5.46410	−0.280672	−0.140336	+	0.990104i	$0.544818\pi$
$380$	0	0
$381$	−9.85641	−0.504959
$382$	0	0
$383$	−1.46410	−0.0748121	−0.0374060	−	0.999300i	$-0.511909\pi$
$383$	−1.46410	−0.0748121	−0.0374060	+	0.999300i	$0.511909\pi$
$384$	0	0
$385$	12.9282	0.658882
$386$	0	0
$387$	9.66025	0.491058
$388$	0	0
$389$	11.7846	0.597503	0.298752	−	0.954331i	$-0.403430\pi$
$389$	11.7846	0.597503	0.298752	+	0.954331i	$0.403430\pi$
$390$	0	0
$391$	24.0526	1.21639
$392$	0	0
$393$	6.53590	0.329692
$394$	0	0
$395$	−9.46410	−0.476191
$396$	0	0
$397$	−20.3923	−1.02346	−0.511730	−	0.859146i	$-0.670995\pi$
$397$	−20.3923	−1.02346	−0.511730	+	0.859146i	$0.670995\pi$
$398$	0	0
$399$	12.9282	0.647220
$400$	0	0
$401$	−8.07180	−0.403086	−0.201543	−	0.979480i	$-0.564596\pi$
$401$	−8.07180	−0.403086	−0.201543	+	0.979480i	$0.564596\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.73205	−0.185447
$406$	0	0
$407$	−4.48334	−0.222231
$408$	0	0
$409$	−17.7321	−0.876793	−0.438397	−	0.898782i	$-0.644454\pi$
$409$	−17.7321	−0.876793	−0.438397	+	0.898782i	$0.644454\pi$
$410$	0	0
$411$	11.9282	0.588375
$412$	0	0
$413$	−21.8564	−1.07548
$414$	0	0
$415$	0.732051	0.0359350
$416$	0	0
$417$	17.8564	0.874432
$418$	0	0
$419$	17.4641	0.853177	0.426589	−	0.904446i	$-0.359715\pi$
$419$	17.4641	0.853177	0.426589	+	0.904446i	$0.359715\pi$
$420$	0	0
$421$	22.7128	1.10695	0.553477	−	0.832864i	$-0.313301\pi$
$421$	22.7128	1.10695	0.553477	+	0.832864i	$0.313301\pi$
$422$	0	0
$423$	2.19615	0.106781
$424$	0	0
$425$	−51.1769	−2.48244
$426$	0	0
$427$	25.1244	1.21585
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.1244	0.632178	0.316089	−	0.948730i	$-0.397630\pi$
$431$	13.1244	0.632178	0.316089	+	0.948730i	$0.397630\pi$
$432$	0	0
$433$	12.8564	0.617839	0.308920	−	0.951088i	$-0.400033\pi$
$433$	12.8564	0.617839	0.308920	+	0.951088i	$0.400033\pi$
$434$	0	0
$435$	−16.6603	−0.798798
$436$	0	0
$437$	−19.8564	−0.949861
$438$	0	0
$439$	0.339746	0.0162152	0.00810760	−	0.999967i	$-0.497419\pi$
$439$	0.339746	0.0162152	0.00810760	+	0.999967i	$0.497419\pi$
$440$	0	0
$441$	0.464102	0.0221001
$442$	0	0
$443$	−15.6077	−0.741544	−0.370772	−	0.928724i	$-0.620907\pi$
$443$	−15.6077	−0.741544	−0.370772	+	0.928724i	$0.620907\pi$
$444$	0	0
$445$	−35.3205	−1.67435
$446$	0	0
$447$	13.1962	0.624157
$448$	0	0
$449$	−11.3205	−0.534248	−0.267124	−	0.963662i	$-0.586073\pi$
$449$	−11.3205	−0.534248	−0.267124	+	0.963662i	$0.586073\pi$
$450$	0	0
$451$	11.9090	0.560771
$452$	0	0
$453$	−6.73205	−0.316299
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.33975	−0.0626707	−0.0313353	−	0.999509i	$-0.509976\pi$
$457$	−1.33975	−0.0626707	−0.0313353	+	0.999509i	$0.509976\pi$
$458$	0	0
$459$	5.73205	0.267549
$460$	0	0
$461$	−22.2679	−1.03712	−0.518561	−	0.855041i	$-0.673532\pi$
$461$	−22.2679	−1.03712	−0.518561	+	0.855041i	$0.673532\pi$
$462$	0	0
$463$	10.0526	0.467182	0.233591	−	0.972335i	$-0.424952\pi$
$463$	10.0526	0.467182	0.233591	+	0.972335i	$0.424952\pi$
$464$	0	0
$465$	5.46410	0.253392
$466$	0	0
$467$	−18.5885	−0.860171	−0.430086	−	0.902788i	$-0.641517\pi$
$467$	−18.5885	−0.860171	−0.430086	+	0.902788i	$0.641517\pi$
$468$	0	0
$469$	−35.8564	−1.65570
$470$	0	0
$471$	−7.58846	−0.349658
$472$	0	0
$473$	12.2487	0.563196
$474$	0	0
$475$	42.2487	1.93850
$476$	0	0
$477$	−6.46410	−0.295971
$478$	0	0
$479$	33.4641	1.52901	0.764507	−	0.644616i	$-0.222983\pi$
$479$	33.4641	1.52901	0.764507	+	0.644616i	$0.222983\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−11.4641	−0.521635
$484$	0	0
$485$	−22.3923	−1.01678
$486$	0	0
$487$	−3.12436	−0.141578	−0.0707890	−	0.997491i	$-0.522552\pi$
$487$	−3.12436	−0.141578	−0.0707890	+	0.997491i	$0.522552\pi$
$488$	0	0
$489$	13.4641	0.608868
$490$	0	0
$491$	−8.73205	−0.394072	−0.197036	−	0.980396i	$-0.563132\pi$
$491$	−8.73205	−0.394072	−0.197036	+	0.980396i	$0.563132\pi$
$492$	0	0
$493$	25.5885	1.15245
$494$	0	0
$495$	−4.73205	−0.212690
$496$	0	0
$497$	−12.9282	−0.579909
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	9.46410	0.422825
$502$	0	0
$503$	−40.9808	−1.82724	−0.913621	−	0.406567i	$-0.866726\pi$
$503$	−40.9808	−1.82724	−0.913621	+	0.406567i	$0.866726\pi$
$504$	0	0
$505$	−7.19615	−0.320225
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.7321	−0.608662	−0.304331	−	0.952566i	$-0.598433\pi$
$509$	−13.7321	−0.608662	−0.304331	+	0.952566i	$0.598433\pi$
$510$	0	0
$511$	−17.1244	−0.757537
$512$	0	0
$513$	−4.73205	−0.208925
$514$	0	0
$515$	−56.9808	−2.51087
$516$	0	0
$517$	2.78461	0.122467
$518$	0	0
$519$	−4.39230	−0.192801
$520$	0	0
$521$	41.4449	1.81573	0.907866	−	0.419260i	$-0.137710\pi$
$521$	41.4449	1.81573	0.907866	+	0.419260i	$0.137710\pi$
$522$	0	0
$523$	22.4449	0.981445	0.490723	−	0.871316i	$-0.336733\pi$
$523$	22.4449	0.981445	0.490723	+	0.871316i	$0.336733\pi$
$524$	0	0
$525$	24.3923	1.06457
$526$	0	0
$527$	−8.39230	−0.365575
$528$	0	0
$529$	−5.39230	−0.234448
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	0	0
$534$	0	0
$535$	38.0526	1.64516
$536$	0	0
$537$	−16.0526	−0.692719
$538$	0	0
$539$	0.588457	0.0253466
$540$	0	0
$541$	−5.67949	−0.244180	−0.122090	−	0.992519i	$-0.538960\pi$
$541$	−5.67949	−0.244180	−0.122090	+	0.992519i	$0.538960\pi$
$542$	0	0
$543$	−19.1962	−0.823786
$544$	0	0
$545$	5.46410	0.234056
$546$	0	0
$547$	4.19615	0.179415	0.0897073	−	0.995968i	$-0.471407\pi$
$547$	4.19615	0.179415	0.0897073	+	0.995968i	$0.471407\pi$
$548$	0	0
$549$	−9.19615	−0.392482
$550$	0	0
$551$	−21.1244	−0.899928
$552$	0	0
$553$	−6.92820	−0.294617
$554$	0	0
$555$	−13.1962	−0.560145
$556$	0	0
$557$	−42.3731	−1.79540	−0.897702	−	0.440603i	$-0.854765\pi$
$557$	−42.3731	−1.79540	−0.897702	+	0.440603i	$0.854765\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	7.26795	0.306853
$562$	0	0
$563$	−34.9282	−1.47205	−0.736024	−	0.676955i	$-0.763299\pi$
$563$	−34.9282	−1.47205	−0.736024	+	0.676955i	$0.763299\pi$
$564$	0	0
$565$	5.00000	0.210352
$566$	0	0
$567$	−2.73205	−0.114735
$568$	0	0
$569$	30.6410	1.28454	0.642269	−	0.766479i	$-0.277993\pi$
$569$	30.6410	1.28454	0.642269	+	0.766479i	$0.277993\pi$
$570$	0	0
$571$	−14.0526	−0.588081	−0.294041	−	0.955793i	$-0.595000\pi$
$571$	−14.0526	−0.588081	−0.294041	+	0.955793i	$0.595000\pi$
$572$	0	0
$573$	−6.92820	−0.289430
$574$	0	0
$575$	−37.4641	−1.56236
$576$	0	0
$577$	−3.73205	−0.155367	−0.0776837	−	0.996978i	$-0.524752\pi$
$577$	−3.73205	−0.155367	−0.0776837	+	0.996978i	$0.524752\pi$
$578$	0	0
$579$	−11.7321	−0.487567
$580$	0	0
$581$	0.535898	0.0222328
$582$	0	0
$583$	−8.19615	−0.339450
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	6.92820	0.285472
$590$	0	0
$591$	−17.8564	−0.734514
$592$	0	0
$593$	9.14359	0.375482	0.187741	−	0.982219i	$-0.439883\pi$
$593$	9.14359	0.375482	0.187741	+	0.982219i	$0.439883\pi$
$594$	0	0
$595$	−58.4449	−2.39601
$596$	0	0
$597$	−14.1962	−0.581010
$598$	0	0
$599$	2.53590	0.103614	0.0518070	−	0.998657i	$-0.483502\pi$
$599$	2.53590	0.103614	0.0518070	+	0.998657i	$0.483502\pi$
$600$	0	0
$601$	7.92820	0.323398	0.161699	−	0.986840i	$-0.448303\pi$
$601$	7.92820	0.323398	0.161699	+	0.986840i	$0.448303\pi$
$602$	0	0
$603$	13.1244	0.534465
$604$	0	0
$605$	35.0526	1.42509
$606$	0	0
$607$	40.7846	1.65540	0.827698	−	0.561174i	$-0.189650\pi$
$607$	40.7846	1.65540	0.827698	+	0.561174i	$0.189650\pi$
$608$	0	0
$609$	−12.1962	−0.494213
$610$	0	0
$611$	0	0
$612$	0	0
$613$	9.39230	0.379352	0.189676	−	0.981847i	$-0.439256\pi$
$613$	9.39230	0.379352	0.189676	+	0.981847i	$0.439256\pi$
$614$	0	0
$615$	35.0526	1.41346
$616$	0	0
$617$	13.2487	0.533373	0.266687	−	0.963783i	$-0.414071\pi$
$617$	13.2487	0.533373	0.266687	+	0.963783i	$0.414071\pi$
$618$	0	0
$619$	17.4641	0.701942	0.350971	−	0.936386i	$-0.385852\pi$
$619$	17.4641	0.701942	0.350971	+	0.936386i	$0.385852\pi$
$620$	0	0
$621$	4.19615	0.168386
$622$	0	0
$623$	−25.8564	−1.03592
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	20.2679	0.808136
$630$	0	0
$631$	−7.71281	−0.307042	−0.153521	−	0.988145i	$-0.549061\pi$
$631$	−7.71281	−0.307042	−0.153521	+	0.988145i	$0.549061\pi$
$632$	0	0
$633$	16.3923	0.651536
$634$	0	0
$635$	−36.7846	−1.45975
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.73205	0.187197
$640$	0	0
$641$	−25.9808	−1.02618	−0.513089	−	0.858335i	$-0.671499\pi$
$641$	−25.9808	−1.02618	−0.513089	+	0.858335i	$0.671499\pi$
$642$	0	0
$643$	−13.8564	−0.546443	−0.273222	−	0.961951i	$-0.588089\pi$
$643$	−13.8564	−0.546443	−0.273222	+	0.961951i	$0.588089\pi$
$644$	0	0
$645$	36.0526	1.41957
$646$	0	0
$647$	22.2487	0.874687	0.437344	−	0.899295i	$-0.355919\pi$
$647$	22.2487	0.874687	0.437344	+	0.899295i	$0.355919\pi$
$648$	0	0
$649$	10.1436	0.398171
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	−17.4641	−0.683423	−0.341712	−	0.939805i	$-0.611007\pi$
$653$	−17.4641	−0.683423	−0.341712	+	0.939805i	$0.611007\pi$
$654$	0	0
$655$	24.3923	0.953086
$656$	0	0
$657$	6.26795	0.244536
$658$	0	0
$659$	10.2487	0.399233	0.199617	−	0.979874i	$-0.436030\pi$
$659$	10.2487	0.399233	0.199617	+	0.979874i	$0.436030\pi$
$660$	0	0
$661$	−11.3923	−0.443109	−0.221555	−	0.975148i	$-0.571113\pi$
$661$	−11.3923	−0.443109	−0.221555	+	0.975148i	$0.571113\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	48.2487	1.87100
$666$	0	0
$667$	18.7321	0.725308
$668$	0	0
$669$	26.9282	1.04110
$670$	0	0
$671$	−11.6603	−0.450139
$672$	0	0
$673$	27.9282	1.07655	0.538277	−	0.842768i	$-0.319076\pi$
$673$	27.9282	1.07655	0.538277	+	0.842768i	$0.319076\pi$
$674$	0	0
$675$	−8.92820	−0.343647
$676$	0	0
$677$	−45.4641	−1.74733	−0.873664	−	0.486530i	$-0.838262\pi$
$677$	−45.4641	−1.74733	−0.873664	+	0.486530i	$0.838262\pi$
$678$	0	0
$679$	−16.3923	−0.629079
$680$	0	0
$681$	12.1962	0.467358
$682$	0	0
$683$	−10.1436	−0.388134	−0.194067	−	0.980988i	$-0.562168\pi$
$683$	−10.1436	−0.388134	−0.194067	+	0.980988i	$0.562168\pi$
$684$	0	0
$685$	44.5167	1.70089
$686$	0	0
$687$	−11.8564	−0.452350
$688$	0	0
$689$	0	0
$690$	0	0
$691$	43.6603	1.66091	0.830457	−	0.557082i	$-0.188079\pi$
$691$	43.6603	1.66091	0.830457	+	0.557082i	$0.188079\pi$
$692$	0	0
$693$	−3.46410	−0.131590
$694$	0	0
$695$	66.6410	2.52784
$696$	0	0
$697$	−53.8372	−2.03923
$698$	0	0
$699$	7.85641	0.297157
$700$	0	0
$701$	−3.32051	−0.125414	−0.0627069	−	0.998032i	$-0.519973\pi$
$701$	−3.32051	−0.125414	−0.0627069	+	0.998032i	$0.519973\pi$
$702$	0	0
$703$	−16.7321	−0.631061
$704$	0	0
$705$	8.19615	0.308685
$706$	0	0
$707$	−5.26795	−0.198122
$708$	0	0
$709$	−13.1436	−0.493618	−0.246809	−	0.969064i	$-0.579382\pi$
$709$	−13.1436	−0.493618	−0.246809	+	0.969064i	$0.579382\pi$
$710$	0	0
$711$	2.53590	0.0951036
$712$	0	0
$713$	−6.14359	−0.230079
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.66025	−0.286077
$718$	0	0
$719$	29.4641	1.09883	0.549413	−	0.835551i	$-0.314851\pi$
$719$	29.4641	1.09883	0.549413	+	0.835551i	$0.314851\pi$
$720$	0	0
$721$	−41.7128	−1.55347
$722$	0	0
$723$	13.5885	0.505360
$724$	0	0
$725$	−39.8564	−1.48023
$726$	0	0
$727$	30.9808	1.14901	0.574506	−	0.818500i	$-0.305194\pi$
$727$	30.9808	1.14901	0.574506	+	0.818500i	$0.305194\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−55.3731	−2.04805
$732$	0	0
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	0	0
$735$	1.73205	0.0638877
$736$	0	0
$737$	16.6410	0.612980
$738$	0	0
$739$	2.92820	0.107716	0.0538578	−	0.998549i	$-0.482848\pi$
$739$	2.92820	0.107716	0.0538578	+	0.998549i	$0.482848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−48.3923	−1.77534	−0.887671	−	0.460479i	$-0.847678\pi$
$743$	−48.3923	−1.77534	−0.887671	+	0.460479i	$0.847678\pi$
$744$	0	0
$745$	49.2487	1.80433
$746$	0	0
$747$	−0.196152	−0.00717684
$748$	0	0
$749$	27.8564	1.01785
$750$	0	0
$751$	49.9090	1.82120	0.910602	−	0.413284i	$-0.135618\pi$
$751$	49.9090	1.82120	0.910602	+	0.413284i	$0.135618\pi$
$752$	0	0
$753$	13.4641	0.490659
$754$	0	0
$755$	−25.1244	−0.914369
$756$	0	0
$757$	20.9282	0.760648	0.380324	−	0.924853i	$-0.375812\pi$
$757$	20.9282	0.760648	0.380324	+	0.924853i	$0.375812\pi$
$758$	0	0
$759$	5.32051	0.193122
$760$	0	0
$761$	−11.3205	−0.410368	−0.205184	−	0.978723i	$-0.565779\pi$
$761$	−11.3205	−0.410368	−0.205184	+	0.978723i	$0.565779\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	21.3923	0.773440
$766$	0	0
$767$	0	0
$768$	0	0
$769$	43.8564	1.58150	0.790751	−	0.612138i	$-0.209690\pi$
$769$	43.8564	1.58150	0.790751	+	0.612138i	$0.209690\pi$
$770$	0	0
$771$	−9.33975	−0.336363
$772$	0	0
$773$	48.9282	1.75983	0.879913	−	0.475136i	$-0.157601\pi$
$773$	48.9282	1.75983	0.879913	+	0.475136i	$0.157601\pi$
$774$	0	0
$775$	13.0718	0.469553
$776$	0	0
$777$	−9.66025	−0.346560
$778$	0	0
$779$	44.4449	1.59240
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	4.46410	0.159534
$784$	0	0
$785$	−28.3205	−1.01080
$786$	0	0
$787$	4.67949	0.166806	0.0834029	−	0.996516i	$-0.473421\pi$
$787$	4.67949	0.166806	0.0834029	+	0.996516i	$0.473421\pi$
$788$	0	0
$789$	−10.0526	−0.357881
$790$	0	0
$791$	3.66025	0.130144
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−24.1244	−0.855603
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−12.5885	−0.445348
$800$	0	0
$801$	9.46410	0.334398
$802$	0	0
$803$	7.94744	0.280459
$804$	0	0
$805$	−42.7846	−1.50796
$806$	0	0
$807$	−5.46410	−0.192345
$808$	0	0
$809$	−53.5885	−1.88407	−0.942035	−	0.335515i	$-0.891090\pi$
$809$	−53.5885	−1.88407	−0.942035	+	0.335515i	$0.891090\pi$
$810$	0	0
$811$	17.1769	0.603163	0.301582	−	0.953440i	$-0.402485\pi$
$811$	17.1769	0.603163	0.301582	+	0.953440i	$0.402485\pi$
$812$	0	0
$813$	21.8564	0.766538
$814$	0	0
$815$	50.2487	1.76014
$816$	0	0
$817$	45.7128	1.59929
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.928203	0.0323945	0.0161973	−	0.999869i	$-0.494844\pi$
$821$	0.928203	0.0323945	0.0161973	+	0.999869i	$0.494844\pi$
$822$	0	0
$823$	−41.5692	−1.44901	−0.724506	−	0.689269i	$-0.757932\pi$
$823$	−41.5692	−1.44901	−0.724506	+	0.689269i	$0.757932\pi$
$824$	0	0
$825$	−11.3205	−0.394130
$826$	0	0
$827$	26.5359	0.922744	0.461372	−	0.887207i	$-0.347357\pi$
$827$	26.5359	0.922744	0.461372	+	0.887207i	$0.347357\pi$
$828$	0	0
$829$	−12.1244	−0.421096	−0.210548	−	0.977583i	$-0.567525\pi$
$829$	−12.1244	−0.421096	−0.210548	+	0.977583i	$0.567525\pi$
$830$	0	0
$831$	5.73205	0.198843
$832$	0	0
$833$	−2.66025	−0.0921723
$834$	0	0
$835$	35.3205	1.22232
$836$	0	0
$837$	−1.46410	−0.0506068
$838$	0	0
$839$	41.8564	1.44504	0.722522	−	0.691348i	$-0.242983\pi$
$839$	41.8564	1.44504	0.722522	+	0.691348i	$0.242983\pi$
$840$	0	0
$841$	−9.07180	−0.312821
$842$	0	0
$843$	12.3205	0.424341
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.6603	0.881697
$848$	0	0
$849$	25.6603	0.880658
$850$	0	0
$851$	14.8372	0.508612
$852$	0	0
$853$	54.1769	1.85498	0.927491	−	0.373845i	$-0.121961\pi$
$853$	54.1769	1.85498	0.927491	+	0.373845i	$0.121961\pi$
$854$	0	0
$855$	−17.6603	−0.603968
$856$	0	0
$857$	39.4449	1.34741	0.673705	−	0.739000i	$-0.264702\pi$
$857$	39.4449	1.34741	0.673705	+	0.739000i	$0.264702\pi$
$858$	0	0
$859$	47.1244	1.60786	0.803931	−	0.594722i	$-0.202738\pi$
$859$	47.1244	1.60786	0.803931	+	0.594722i	$0.202738\pi$
$860$	0	0
$861$	25.6603	0.874499
$862$	0	0
$863$	−17.1244	−0.582920	−0.291460	−	0.956583i	$-0.594141\pi$
$863$	−17.1244	−0.582920	−0.291460	+	0.956583i	$0.594141\pi$
$864$	0	0
$865$	−16.3923	−0.557355
$866$	0	0
$867$	−15.8564	−0.538512
$868$	0	0
$869$	3.21539	0.109075
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	40.0526	1.35402
$876$	0	0
$877$	−23.9282	−0.807998	−0.403999	−	0.914759i	$-0.632380\pi$
$877$	−23.9282	−0.807998	−0.403999	+	0.914759i	$0.632380\pi$
$878$	0	0
$879$	30.5167	1.02930
$880$	0	0
$881$	−27.8372	−0.937858	−0.468929	−	0.883236i	$-0.655360\pi$
$881$	−27.8372	−0.937858	−0.468929	+	0.883236i	$0.655360\pi$
$882$	0	0
$883$	−42.9282	−1.44465	−0.722325	−	0.691554i	$-0.756926\pi$
$883$	−42.9282	−1.44465	−0.722325	+	0.691554i	$0.756926\pi$
$884$	0	0
$885$	29.8564	1.00361
$886$	0	0
$887$	−37.8564	−1.27109	−0.635547	−	0.772062i	$-0.719225\pi$
$887$	−37.8564	−1.27109	−0.635547	+	0.772062i	$0.719225\pi$
$888$	0	0
$889$	−26.9282	−0.903143
$890$	0	0
$891$	1.26795	0.0424779
$892$	0	0
$893$	10.3923	0.347765
$894$	0	0
$895$	−59.9090	−2.00254
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.53590	−0.217984
$900$	0	0
$901$	37.0526	1.23440
$902$	0	0
$903$	26.3923	0.878281
$904$	0	0
$905$	−71.6410	−2.38143
$906$	0	0
$907$	−36.3923	−1.20839	−0.604193	−	0.796838i	$-0.706505\pi$
$907$	−36.3923	−1.20839	−0.604193	+	0.796838i	$0.706505\pi$
$908$	0	0
$909$	1.92820	0.0639545
$910$	0	0
$911$	2.53590	0.0840181	0.0420090	−	0.999117i	$-0.486624\pi$
$911$	2.53590	0.0840181	0.0420090	+	0.999117i	$0.486624\pi$
$912$	0	0
$913$	−0.248711	−0.00823114
$914$	0	0
$915$	−34.3205	−1.13460
$916$	0	0
$917$	17.8564	0.589670
$918$	0	0
$919$	−45.9615	−1.51613	−0.758065	−	0.652179i	$-0.773855\pi$
$919$	−45.9615	−1.51613	−0.758065	+	0.652179i	$0.773855\pi$
$920$	0	0
$921$	22.5885	0.744315
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−31.5692	−1.03799
$926$	0	0
$927$	15.2679	0.501465
$928$	0	0
$929$	39.2487	1.28771	0.643854	−	0.765148i	$-0.277334\pi$
$929$	39.2487	1.28771	0.643854	+	0.765148i	$0.277334\pi$
$930$	0	0
$931$	2.19615	0.0719760
$932$	0	0
$933$	−1.66025	−0.0543543
$934$	0	0
$935$	27.1244	0.887061
$936$	0	0
$937$	−5.24871	−0.171468	−0.0857340	−	0.996318i	$-0.527324\pi$
$937$	−5.24871	−0.171468	−0.0857340	+	0.996318i	$0.527324\pi$
$938$	0	0
$939$	−6.53590	−0.213291
$940$	0	0
$941$	−12.6410	−0.412085	−0.206043	−	0.978543i	$-0.566059\pi$
$941$	−12.6410	−0.412085	−0.206043	+	0.978543i	$0.566059\pi$
$942$	0	0
$943$	−39.4115	−1.28342
$944$	0	0
$945$	−10.1962	−0.331681
$946$	0	0
$947$	21.0718	0.684741	0.342371	−	0.939565i	$-0.388770\pi$
$947$	21.0718	0.684741	0.342371	+	0.939565i	$0.388770\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	20.6603	0.669955
$952$	0	0
$953$	41.5692	1.34656	0.673280	−	0.739388i	$-0.264885\pi$
$953$	41.5692	1.34656	0.673280	+	0.739388i	$0.264885\pi$
$954$	0	0
$955$	−25.8564	−0.836694
$956$	0	0
$957$	5.66025	0.182970
$958$	0	0
$959$	32.5885	1.05234
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	−10.1962	−0.328566
$964$	0	0
$965$	−43.7846	−1.40948
$966$	0	0
$967$	−43.1244	−1.38679	−0.693393	−	0.720560i	$-0.743885\pi$
$967$	−43.1244	−1.38679	−0.693393	+	0.720560i	$0.743885\pi$
$968$	0	0
$969$	27.1244	0.871360
$970$	0	0
$971$	−30.2487	−0.970727	−0.485364	−	0.874312i	$-0.661313\pi$
$971$	−30.2487	−0.970727	−0.485364	+	0.874312i	$0.661313\pi$
$972$	0	0
$973$	48.7846	1.56396
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45.9282	−1.46937	−0.734687	−	0.678407i	$-0.762671\pi$
$977$	−45.9282	−1.46937	−0.734687	+	0.678407i	$0.762671\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−1.46410	−0.0467452
$982$	0	0
$983$	20.7846	0.662926	0.331463	−	0.943468i	$-0.392458\pi$
$983$	20.7846	0.662926	0.331463	+	0.943468i	$0.392458\pi$
$984$	0	0
$985$	−66.6410	−2.12336
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	−40.5359	−1.28897
$990$	0	0
$991$	−22.5885	−0.717546	−0.358773	−	0.933425i	$-0.616805\pi$
$991$	−22.5885	−0.717546	−0.358773	+	0.933425i	$0.616805\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	−52.9808	−1.67960
$996$	0	0
$997$	21.3397	0.675837	0.337918	−	0.941175i	$-0.390277\pi$
$997$	21.3397	0.675837	0.337918	+	0.941175i	$0.390277\pi$
$998$	0	0
$999$	3.53590	0.111871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bj.1.1		2
4.3	odd	2		1014.2.a.i.1.1		2
12.11	even	2		3042.2.a.y.1.2		2
13.2	odd	12		624.2.bv.e.433.2		4
13.7	odd	12		624.2.bv.e.49.1		4
13.12	even	2		8112.2.a.bp.1.2		2
39.2	even	12		1872.2.by.h.433.1		4
39.20	even	12		1872.2.by.h.1297.2		4
52.3	odd	6		1014.2.e.i.529.1		4
52.7	even	12		78.2.i.a.49.1	yes	4
52.11	even	12		1014.2.i.a.823.2		4
52.15	even	12		78.2.i.a.43.1	✓	4
52.19	even	12		1014.2.i.a.361.2		4
52.23	odd	6		1014.2.e.g.529.2		4
52.31	even	4		1014.2.b.e.337.4		4
52.35	odd	6		1014.2.e.i.991.1		4
52.43	odd	6		1014.2.e.g.991.2		4
52.47	even	4		1014.2.b.e.337.1		4
52.51	odd	2		1014.2.a.k.1.2		2
156.47	odd	4		3042.2.b.i.1351.4		4
156.59	odd	12		234.2.l.c.127.2		4
156.83	odd	4		3042.2.b.i.1351.1		4
156.119	odd	12		234.2.l.c.199.2		4
156.155	even	2		3042.2.a.p.1.1		2
260.7	odd	12		1950.2.y.b.49.1		4
260.59	even	12		1950.2.bc.d.751.2		4
260.67	odd	12		1950.2.y.g.199.2		4
260.119	even	12		1950.2.bc.d.901.2		4
260.163	odd	12		1950.2.y.g.49.2		4
260.223	odd	12		1950.2.y.b.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.a.43.1	✓	4	52.15	even	12
78.2.i.a.49.1	yes	4	52.7	even	12
234.2.l.c.127.2		4	156.59	odd	12
234.2.l.c.199.2		4	156.119	odd	12
624.2.bv.e.49.1		4	13.7	odd	12
624.2.bv.e.433.2		4	13.2	odd	12
1014.2.a.i.1.1		2	4.3	odd	2
1014.2.a.k.1.2		2	52.51	odd	2
1014.2.b.e.337.1		4	52.47	even	4
1014.2.b.e.337.4		4	52.31	even	4
1014.2.e.g.529.2		4	52.23	odd	6
1014.2.e.g.991.2		4	52.43	odd	6
1014.2.e.i.529.1		4	52.3	odd	6
1014.2.e.i.991.1		4	52.35	odd	6
1014.2.i.a.361.2		4	52.19	even	12
1014.2.i.a.823.2		4	52.11	even	12
1872.2.by.h.433.1		4	39.2	even	12
1872.2.by.h.1297.2		4	39.20	even	12
1950.2.y.b.49.1		4	260.7	odd	12
1950.2.y.b.199.1		4	260.223	odd	12
1950.2.y.g.49.2		4	260.163	odd	12
1950.2.y.g.199.2		4	260.67	odd	12
1950.2.bc.d.751.2		4	260.59	even	12
1950.2.bc.d.901.2		4	260.119	even	12
3042.2.a.p.1.1		2	156.155	even	2
3042.2.a.y.1.2		2	12.11	even	2
3042.2.b.i.1351.1		4	156.83	odd	4
3042.2.b.i.1351.4		4	156.47	odd	4
8112.2.a.bj.1.1		2	1.1	even	1	trivial
8112.2.a.bp.1.2		2	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.73205 q^{5} -2.73205 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.73205 q^{5} -2.73205 q^{7} +1.00000 q^{9} +1.26795 q^{11} +3.73205 q^{15} -5.73205 q^{17} +4.73205 q^{19} +2.73205 q^{21} -4.19615 q^{23} +8.92820 q^{25} -1.00000 q^{27} -4.46410 q^{29} +1.46410 q^{31} -1.26795 q^{33} +10.1962 q^{35} -3.53590 q^{37} +9.39230 q^{41} +9.66025 q^{43} -3.73205 q^{45} +2.19615 q^{47} +0.464102 q^{49} +5.73205 q^{51} -6.46410 q^{53} -4.73205 q^{55} -4.73205 q^{57} +8.00000 q^{59} -9.19615 q^{61} -2.73205 q^{63} +13.1244 q^{67} +4.19615 q^{69} +4.73205 q^{71} +6.26795 q^{73} -8.92820 q^{75} -3.46410 q^{77} +2.53590 q^{79} +1.00000 q^{81} -0.196152 q^{83} +21.3923 q^{85} +4.46410 q^{87} +9.46410 q^{89} -1.46410 q^{93} -17.6603 q^{95} +6.00000 q^{97} +1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 4 q^{15} - 8 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} + 4 q^{25} - 2 q^{27} - 2 q^{29} - 4 q^{31} - 6 q^{33} + 10 q^{35} - 14 q^{37} - 2 q^{41} + 2 q^{43} - 4 q^{45} - 6 q^{47} - 6 q^{49} + 8 q^{51} - 6 q^{53} - 6 q^{55} - 6 q^{57} + 16 q^{59} - 8 q^{61} - 2 q^{63} + 2 q^{67} - 2 q^{69} + 6 q^{71} + 16 q^{73} - 4 q^{75} + 12 q^{79} + 2 q^{81} + 10 q^{83} + 22 q^{85} + 2 q^{87} + 12 q^{89} + 4 q^{93} - 18 q^{95} + 12 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 4 * q^15 - 8 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^23 + 4 * q^25 - 2 * q^27 - 2 * q^29 - 4 * q^31 - 6 * q^33 + 10 * q^35 - 14 * q^37 - 2 * q^41 + 2 * q^43 - 4 * q^45 - 6 * q^47 - 6 * q^49 + 8 * q^51 - 6 * q^53 - 6 * q^55 - 6 * q^57 + 16 * q^59 - 8 * q^61 - 2 * q^63 + 2 * q^67 - 2 * q^69 + 6 * q^71 + 16 * q^73 - 4 * q^75 + 12 * q^79 + 2 * q^81 + 10 * q^83 + 22 * q^85 + 2 * q^87 + 12 * q^89 + 4 * q^93 - 18 * q^95 + 12 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more