LMFDB - Embedded newform 6864.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.481194 q^{5} +4.15633 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.481194 q^{5} +4.15633 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.481194 q^{15} -1.67513 q^{17} -4.15633 q^{19} -4.15633 q^{21} +5.50659 q^{23} -4.76845 q^{25} -1.00000 q^{27} +2.89938 q^{29} -6.48119 q^{31} -1.00000 q^{33} -2.00000 q^{35} -6.70052 q^{37} +1.00000 q^{39} +2.15633 q^{41} +7.86177 q^{43} -0.481194 q^{45} -8.57452 q^{47} +10.2750 q^{49} +1.67513 q^{51} -11.2750 q^{53} -0.481194 q^{55} +4.15633 q^{57} -12.9624 q^{59} +7.66291 q^{61} +4.15633 q^{63} +0.481194 q^{65} -10.0181 q^{67} -5.50659 q^{69} -9.47627 q^{71} -10.9321 q^{73} +4.76845 q^{75} +4.15633 q^{77} +1.75035 q^{79} +1.00000 q^{81} -8.96239 q^{83} +0.806063 q^{85} -2.89938 q^{87} +2.48119 q^{89} -4.15633 q^{91} +6.48119 q^{93} +2.00000 q^{95} +2.70052 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{15} - 2 q^{19} - 2 q^{21} - 4 q^{23} - 3 q^{25} - 3 q^{27} + 2 q^{29} - 14 q^{31} - 3 q^{33} - 6 q^{35} + 3 q^{39} - 4 q^{41} + 6 q^{43} + 4 q^{45} - 14 q^{47} - q^{49} - 2 q^{53} + 4 q^{55} + 2 q^{57} - 28 q^{59} - 8 q^{61} + 2 q^{63} - 4 q^{65} - 2 q^{67} + 4 q^{69} - 10 q^{71} - 24 q^{73} + 3 q^{75} + 2 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} + 2 q^{85} - 2 q^{87} + 2 q^{89} - 2 q^{91} + 14 q^{93} + 6 q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^15 - 2 * q^19 - 2 * q^21 - 4 * q^23 - 3 * q^25 - 3 * q^27 + 2 * q^29 - 14 * q^31 - 3 * q^33 - 6 * q^35 + 3 * q^39 - 4 * q^41 + 6 * q^43 + 4 * q^45 - 14 * q^47 - q^49 - 2 * q^53 + 4 * q^55 + 2 * q^57 - 28 * q^59 - 8 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 + 4 * q^69 - 10 * q^71 - 24 * q^73 + 3 * q^75 + 2 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 + 2 * q^85 - 2 * q^87 + 2 * q^89 - 2 * q^91 + 14 * q^93 + 6 * q^95 - 12 * q^97 + 3 * 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
$5$	−0.481194	−0.215197	−0.107598	−	0.994194i	$-0.534316\pi$
$5$	−0.481194	−0.215197	−0.107598	+	0.994194i	$0.534316\pi$
$6$	0	0
$7$	4.15633	1.57094	0.785472	−	0.618898i	$-0.212420\pi$
$7$	4.15633	1.57094	0.785472	+	0.618898i	$0.212420\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.481194	0.124244
$16$	0	0
$17$	−1.67513	−0.406279	−0.203139	−	0.979150i	$-0.565114\pi$
$17$	−1.67513	−0.406279	−0.203139	+	0.979150i	$0.565114\pi$
$18$	0	0
$19$	−4.15633	−0.953526	−0.476763	−	0.879032i	$-0.658190\pi$
$19$	−4.15633	−0.953526	−0.476763	+	0.879032i	$0.658190\pi$
$20$	0	0
$21$	−4.15633	−0.906985
$22$	0	0
$23$	5.50659	1.14820	0.574101	−	0.818784i	$-0.305352\pi$
$23$	5.50659	1.14820	0.574101	+	0.818784i	$0.305352\pi$
$24$	0	0
$25$	−4.76845	−0.953690
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.89938	0.538402	0.269201	−	0.963084i	$-0.413240\pi$
$29$	2.89938	0.538402	0.269201	+	0.963084i	$0.413240\pi$
$30$	0	0
$31$	−6.48119	−1.16406	−0.582028	−	0.813168i	$-0.697741\pi$
$31$	−6.48119	−1.16406	−0.582028	+	0.813168i	$0.697741\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−6.70052	−1.10156	−0.550780	−	0.834651i	$-0.685670\pi$
$37$	−6.70052	−1.10156	−0.550780	+	0.834651i	$0.685670\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.15633	0.336761	0.168381	−	0.985722i	$-0.446146\pi$
$41$	2.15633	0.336761	0.168381	+	0.985722i	$0.446146\pi$
$42$	0	0
$43$	7.86177	1.19891	0.599455	−	0.800409i	$-0.295384\pi$
$43$	7.86177	1.19891	0.599455	+	0.800409i	$0.295384\pi$
$44$	0	0
$45$	−0.481194	−0.0717322
$46$	0	0
$47$	−8.57452	−1.25072	−0.625361	−	0.780336i	$-0.715048\pi$
$47$	−8.57452	−1.25072	−0.625361	+	0.780336i	$0.715048\pi$
$48$	0	0
$49$	10.2750	1.46786
$50$	0	0
$51$	1.67513	0.234565
$52$	0	0
$53$	−11.2750	−1.54875	−0.774373	−	0.632730i	$-0.781934\pi$
$53$	−11.2750	−1.54875	−0.774373	+	0.632730i	$0.781934\pi$
$54$	0	0
$55$	−0.481194	−0.0648842
$56$	0	0
$57$	4.15633	0.550519
$58$	0	0
$59$	−12.9624	−1.68756	−0.843780	−	0.536690i	$-0.819675\pi$
$59$	−12.9624	−1.68756	−0.843780	+	0.536690i	$0.819675\pi$
$60$	0	0
$61$	7.66291	0.981135	0.490568	−	0.871403i	$-0.336789\pi$
$61$	7.66291	0.981135	0.490568	+	0.871403i	$0.336789\pi$
$62$	0	0
$63$	4.15633	0.523648
$64$	0	0
$65$	0.481194	0.0596848
$66$	0	0
$67$	−10.0181	−1.22391	−0.611953	−	0.790894i	$-0.709616\pi$
$67$	−10.0181	−1.22391	−0.611953	+	0.790894i	$0.709616\pi$
$68$	0	0
$69$	−5.50659	−0.662915
$70$	0	0
$71$	−9.47627	−1.12463	−0.562313	−	0.826924i	$-0.690088\pi$
$71$	−9.47627	−1.12463	−0.562313	+	0.826924i	$0.690088\pi$
$72$	0	0
$73$	−10.9321	−1.27950	−0.639751	−	0.768582i	$-0.720963\pi$
$73$	−10.9321	−1.27950	−0.639751	+	0.768582i	$0.720963\pi$
$74$	0	0
$75$	4.76845	0.550613
$76$	0	0
$77$	4.15633	0.473657
$78$	0	0
$79$	1.75035	0.196930	0.0984651	−	0.995141i	$-0.468607\pi$
$79$	1.75035	0.196930	0.0984651	+	0.995141i	$0.468607\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.96239	−0.983750	−0.491875	−	0.870666i	$-0.663688\pi$
$83$	−8.96239	−0.983750	−0.491875	+	0.870666i	$0.663688\pi$
$84$	0	0
$85$	0.806063	0.0874299
$86$	0	0
$87$	−2.89938	−0.310847
$88$	0	0
$89$	2.48119	0.263006	0.131503	−	0.991316i	$-0.458020\pi$
$89$	2.48119	0.263006	0.131503	+	0.991316i	$0.458020\pi$
$90$	0	0
$91$	−4.15633	−0.435701
$92$	0	0
$93$	6.48119	0.672069
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	2.70052	0.274197	0.137098	−	0.990557i	$-0.456222\pi$
$97$	2.70052	0.274197	0.137098	+	0.990557i	$0.456222\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−7.93700	−0.789761	−0.394880	−	0.918733i	$-0.629214\pi$
$101$	−7.93700	−0.789761	−0.394880	+	0.918733i	$0.629214\pi$
$102$	0	0
$103$	−0.775746	−0.0764366	−0.0382183	−	0.999269i	$-0.512168\pi$
$103$	−0.775746	−0.0764366	−0.0382183	+	0.999269i	$0.512168\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	8.18664	0.791433	0.395716	−	0.918373i	$-0.370496\pi$
$107$	8.18664	0.791433	0.395716	+	0.918373i	$0.370496\pi$
$108$	0	0
$109$	−20.3938	−1.95337	−0.976684	−	0.214684i	$-0.931128\pi$
$109$	−20.3938	−1.95337	−0.976684	+	0.214684i	$0.931128\pi$
$110$	0	0
$111$	6.70052	0.635986
$112$	0	0
$113$	16.3634	1.53934	0.769671	−	0.638440i	$-0.220420\pi$
$113$	16.3634	1.53934	0.769671	+	0.638440i	$0.220420\pi$
$114$	0	0
$115$	−2.64974	−0.247089
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−6.96239	−0.638241
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.15633	−0.194429
$124$	0	0
$125$	4.70052	0.420428
$126$	0	0
$127$	15.4739	1.37309	0.686543	−	0.727089i	$-0.259127\pi$
$127$	15.4739	1.37309	0.686543	+	0.727089i	$0.259127\pi$
$128$	0	0
$129$	−7.86177	−0.692191
$130$	0	0
$131$	3.53690	0.309021	0.154510	−	0.987991i	$-0.450620\pi$
$131$	3.53690	0.309021	0.154510	+	0.987991i	$0.450620\pi$
$132$	0	0
$133$	−17.2750	−1.49794
$134$	0	0
$135$	0.481194	0.0414146
$136$	0	0
$137$	15.2569	1.30349	0.651744	−	0.758439i	$-0.274038\pi$
$137$	15.2569	1.30349	0.651744	+	0.758439i	$0.274038\pi$
$138$	0	0
$139$	20.0386	1.69965	0.849825	−	0.527066i	$-0.176708\pi$
$139$	20.0386	1.69965	0.849825	+	0.527066i	$0.176708\pi$
$140$	0	0
$141$	8.57452	0.722104
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−1.39517	−0.115862
$146$	0	0
$147$	−10.2750	−0.847471
$148$	0	0
$149$	16.8568	1.38097	0.690483	−	0.723348i	$-0.257398\pi$
$149$	16.8568	1.38097	0.690483	+	0.723348i	$0.257398\pi$
$150$	0	0
$151$	−14.0811	−1.14590	−0.572952	−	0.819589i	$-0.694202\pi$
$151$	−14.0811	−1.14590	−0.572952	+	0.819589i	$0.694202\pi$
$152$	0	0
$153$	−1.67513	−0.135426
$154$	0	0
$155$	3.11871	0.250501
$156$	0	0
$157$	−20.7816	−1.65855	−0.829277	−	0.558838i	$-0.811248\pi$
$157$	−20.7816	−1.65855	−0.829277	+	0.558838i	$0.811248\pi$
$158$	0	0
$159$	11.2750	0.894169
$160$	0	0
$161$	22.8872	1.80376
$162$	0	0
$163$	−12.0181	−0.941330	−0.470665	−	0.882312i	$-0.655986\pi$
$163$	−12.0181	−0.941330	−0.470665	+	0.882312i	$0.655986\pi$
$164$	0	0
$165$	0.481194	0.0374609
$166$	0	0
$167$	18.2374	1.41125	0.705627	−	0.708583i	$-0.250665\pi$
$167$	18.2374	1.41125	0.705627	+	0.708583i	$0.250665\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.15633	−0.317842
$172$	0	0
$173$	16.2252	1.23358	0.616790	−	0.787128i	$-0.288433\pi$
$173$	16.2252	1.23358	0.616790	+	0.787128i	$0.288433\pi$
$174$	0	0
$175$	−19.8192	−1.49819
$176$	0	0
$177$	12.9624	0.974313
$178$	0	0
$179$	−9.38058	−0.701137	−0.350569	−	0.936537i	$-0.614012\pi$
$179$	−9.38058	−0.701137	−0.350569	+	0.936537i	$0.614012\pi$
$180$	0	0
$181$	6.23155	0.463187	0.231594	−	0.972813i	$-0.425606\pi$
$181$	6.23155	0.463187	0.231594	+	0.972813i	$0.425606\pi$
$182$	0	0
$183$	−7.66291	−0.566459
$184$	0	0
$185$	3.22425	0.237052
$186$	0	0
$187$	−1.67513	−0.122498
$188$	0	0
$189$	−4.15633	−0.302328
$190$	0	0
$191$	3.41090	0.246804	0.123402	−	0.992357i	$-0.460620\pi$
$191$	3.41090	0.246804	0.123402	+	0.992357i	$0.460620\pi$
$192$	0	0
$193$	−1.84367	−0.132711	−0.0663553	−	0.997796i	$-0.521137\pi$
$193$	−1.84367	−0.132711	−0.0663553	+	0.997796i	$0.521137\pi$
$194$	0	0
$195$	−0.481194	−0.0344590
$196$	0	0
$197$	−19.7440	−1.40670	−0.703351	−	0.710842i	$-0.748314\pi$
$197$	−19.7440	−1.40670	−0.703351	+	0.710842i	$0.748314\pi$
$198$	0	0
$199$	−3.22425	−0.228561	−0.114281	−	0.993449i	$-0.536456\pi$
$199$	−3.22425	−0.228561	−0.114281	+	0.993449i	$0.536456\pi$
$200$	0	0
$201$	10.0181	0.706622
$202$	0	0
$203$	12.0508	0.845799
$204$	0	0
$205$	−1.03761	−0.0724699
$206$	0	0
$207$	5.50659	0.382734
$208$	0	0
$209$	−4.15633	−0.287499
$210$	0	0
$211$	−12.6375	−0.870003	−0.435001	−	0.900430i	$-0.643252\pi$
$211$	−12.6375	−0.870003	−0.435001	+	0.900430i	$0.643252\pi$
$212$	0	0
$213$	9.47627	0.649303
$214$	0	0
$215$	−3.78304	−0.258001
$216$	0	0
$217$	−26.9380	−1.82867
$218$	0	0
$219$	10.9321	0.738721
$220$	0	0
$221$	1.67513	0.112681
$222$	0	0
$223$	1.70545	0.114205	0.0571026	−	0.998368i	$-0.481814\pi$
$223$	1.70545	0.114205	0.0571026	+	0.998368i	$0.481814\pi$
$224$	0	0
$225$	−4.76845	−0.317897
$226$	0	0
$227$	1.95509	0.129764	0.0648821	−	0.997893i	$-0.479333\pi$
$227$	1.95509	0.129764	0.0648821	+	0.997893i	$0.479333\pi$
$228$	0	0
$229$	−23.7235	−1.56770	−0.783848	−	0.620953i	$-0.786746\pi$
$229$	−23.7235	−1.56770	−0.783848	+	0.620953i	$0.786746\pi$
$230$	0	0
$231$	−4.15633	−0.273466
$232$	0	0
$233$	2.71274	0.177718	0.0888588	−	0.996044i	$-0.471678\pi$
$233$	2.71274	0.177718	0.0888588	+	0.996044i	$0.471678\pi$
$234$	0	0
$235$	4.12601	0.269151
$236$	0	0
$237$	−1.75035	−0.113698
$238$	0	0
$239$	22.5198	1.45668	0.728341	−	0.685215i	$-0.240292\pi$
$239$	22.5198	1.45668	0.728341	+	0.685215i	$0.240292\pi$
$240$	0	0
$241$	−10.4993	−0.676319	−0.338159	−	0.941089i	$-0.609804\pi$
$241$	−10.4993	−0.676319	−0.338159	+	0.941089i	$0.609804\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.94429	−0.315879
$246$	0	0
$247$	4.15633	0.264461
$248$	0	0
$249$	8.96239	0.567968
$250$	0	0
$251$	−17.3357	−1.09422	−0.547109	−	0.837061i	$-0.684272\pi$
$251$	−17.3357	−1.09422	−0.547109	+	0.837061i	$0.684272\pi$
$252$	0	0
$253$	5.50659	0.346196
$254$	0	0
$255$	−0.806063	−0.0504777
$256$	0	0
$257$	20.2374	1.26238	0.631188	−	0.775630i	$-0.282568\pi$
$257$	20.2374	1.26238	0.631188	+	0.775630i	$0.282568\pi$
$258$	0	0
$259$	−27.8496	−1.73049
$260$	0	0
$261$	2.89938	0.179467
$262$	0	0
$263$	−16.8119	−1.03667	−0.518334	−	0.855178i	$-0.673448\pi$
$263$	−16.8119	−1.03667	−0.518334	+	0.855178i	$0.673448\pi$
$264$	0	0
$265$	5.42548	0.333285
$266$	0	0
$267$	−2.48119	−0.151847
$268$	0	0
$269$	−0.201231	−0.0122693	−0.00613463	−	0.999981i	$-0.501953\pi$
$269$	−0.201231	−0.0122693	−0.00613463	+	0.999981i	$0.501953\pi$
$270$	0	0
$271$	−1.32979	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$271$	−1.32979	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$272$	0	0
$273$	4.15633	0.251552
$274$	0	0
$275$	−4.76845	−0.287548
$276$	0	0
$277$	1.01317	0.0608757	0.0304379	−	0.999537i	$-0.490310\pi$
$277$	1.01317	0.0608757	0.0304379	+	0.999537i	$0.490310\pi$
$278$	0	0
$279$	−6.48119	−0.388019
$280$	0	0
$281$	11.8945	0.709564	0.354782	−	0.934949i	$-0.384555\pi$
$281$	11.8945	0.709564	0.354782	+	0.934949i	$0.384555\pi$
$282$	0	0
$283$	−0.249646	−0.0148399	−0.00741997	−	0.999972i	$-0.502362\pi$
$283$	−0.249646	−0.0148399	−0.00741997	+	0.999972i	$0.502362\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	8.96239	0.529033
$288$	0	0
$289$	−14.1939	−0.834937
$290$	0	0
$291$	−2.70052	−0.158307
$292$	0	0
$293$	−1.56722	−0.0915580	−0.0457790	−	0.998952i	$-0.514577\pi$
$293$	−1.56722	−0.0915580	−0.0457790	+	0.998952i	$0.514577\pi$
$294$	0	0
$295$	6.23743	0.363157
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−5.50659	−0.318454
$300$	0	0
$301$	32.6761	1.88342
$302$	0	0
$303$	7.93700	0.455969
$304$	0	0
$305$	−3.68735	−0.211137
$306$	0	0
$307$	−27.1695	−1.55065	−0.775323	−	0.631565i	$-0.782413\pi$
$307$	−27.1695	−1.55065	−0.775323	+	0.631565i	$0.782413\pi$
$308$	0	0
$309$	0.775746	0.0441307
$310$	0	0
$311$	10.3430	0.586496	0.293248	−	0.956036i	$-0.405264\pi$
$311$	10.3430	0.586496	0.293248	+	0.956036i	$0.405264\pi$
$312$	0	0
$313$	−19.1041	−1.07983	−0.539915	−	0.841720i	$-0.681543\pi$
$313$	−19.1041	−1.07983	−0.539915	+	0.841720i	$0.681543\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−20.8446	−1.17075	−0.585376	−	0.810762i	$-0.699053\pi$
$317$	−20.8446	−1.17075	−0.585376	+	0.810762i	$0.699053\pi$
$318$	0	0
$319$	2.89938	0.162334
$320$	0	0
$321$	−8.18664	−0.456934
$322$	0	0
$323$	6.96239	0.387398
$324$	0	0
$325$	4.76845	0.264506
$326$	0	0
$327$	20.3938	1.12778
$328$	0	0
$329$	−35.6385	−1.96481
$330$	0	0
$331$	16.6678	0.916147	0.458074	−	0.888914i	$-0.348540\pi$
$331$	16.6678	0.916147	0.458074	+	0.888914i	$0.348540\pi$
$332$	0	0
$333$	−6.70052	−0.367186
$334$	0	0
$335$	4.82065	0.263380
$336$	0	0
$337$	6.18664	0.337008	0.168504	−	0.985701i	$-0.446106\pi$
$337$	6.18664	0.337008	0.168504	+	0.985701i	$0.446106\pi$
$338$	0	0
$339$	−16.3634	−0.888740
$340$	0	0
$341$	−6.48119	−0.350976
$342$	0	0
$343$	13.6121	0.734986
$344$	0	0
$345$	2.64974	0.142657
$346$	0	0
$347$	−27.4617	−1.47422	−0.737110	−	0.675773i	$-0.763810\pi$
$347$	−27.4617	−1.47422	−0.737110	+	0.675773i	$0.763810\pi$
$348$	0	0
$349$	−16.5745	−0.887213	−0.443607	−	0.896222i	$-0.646301\pi$
$349$	−16.5745	−0.887213	−0.443607	+	0.896222i	$0.646301\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−18.5926	−0.989585	−0.494792	−	0.869011i	$-0.664756\pi$
$353$	−18.5926	−0.989585	−0.494792	+	0.869011i	$0.664756\pi$
$354$	0	0
$355$	4.55993	0.242016
$356$	0	0
$357$	6.96239	0.368489
$358$	0	0
$359$	6.88129	0.363180	0.181590	−	0.983374i	$-0.441876\pi$
$359$	6.88129	0.363180	0.181590	+	0.983374i	$0.441876\pi$
$360$	0	0
$361$	−1.72496	−0.0907874
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	5.26045	0.275345
$366$	0	0
$367$	−17.7988	−0.929088	−0.464544	−	0.885550i	$-0.653782\pi$
$367$	−17.7988	−0.929088	−0.464544	+	0.885550i	$0.653782\pi$
$368$	0	0
$369$	2.15633	0.112254
$370$	0	0
$371$	−46.8627	−2.43299
$372$	0	0
$373$	7.33567	0.379827	0.189913	−	0.981801i	$-0.439179\pi$
$373$	7.33567	0.379827	0.189913	+	0.981801i	$0.439179\pi$
$374$	0	0
$375$	−4.70052	−0.242734
$376$	0	0
$377$	−2.89938	−0.149326
$378$	0	0
$379$	20.6434	1.06038	0.530190	−	0.847879i	$-0.322121\pi$
$379$	20.6434	1.06038	0.530190	+	0.847879i	$0.322121\pi$
$380$	0	0
$381$	−15.4739	−0.792752
$382$	0	0
$383$	−11.3503	−0.579971	−0.289986	−	0.957031i	$-0.593651\pi$
$383$	−11.3503	−0.579971	−0.289986	+	0.957031i	$0.593651\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	7.86177	0.399636
$388$	0	0
$389$	16.7367	0.848585	0.424293	−	0.905525i	$-0.360523\pi$
$389$	16.7367	0.848585	0.424293	+	0.905525i	$0.360523\pi$
$390$	0	0
$391$	−9.22425	−0.466491
$392$	0	0
$393$	−3.53690	−0.178413
$394$	0	0
$395$	−0.842260	−0.0423787
$396$	0	0
$397$	−32.8773	−1.65007	−0.825033	−	0.565085i	$-0.808843\pi$
$397$	−32.8773	−1.65007	−0.825033	+	0.565085i	$0.808843\pi$
$398$	0	0
$399$	17.2750	0.864834
$400$	0	0
$401$	−27.1573	−1.35617	−0.678085	−	0.734984i	$-0.737190\pi$
$401$	−27.1573	−1.35617	−0.678085	+	0.734984i	$0.737190\pi$
$402$	0	0
$403$	6.48119	0.322851
$404$	0	0
$405$	−0.481194	−0.0239107
$406$	0	0
$407$	−6.70052	−0.332133
$408$	0	0
$409$	−24.3331	−1.20320	−0.601598	−	0.798799i	$-0.705469\pi$
$409$	−24.3331	−1.20320	−0.601598	+	0.798799i	$0.705469\pi$
$410$	0	0
$411$	−15.2569	−0.752569
$412$	0	0
$413$	−53.8759	−2.65106
$414$	0	0
$415$	4.31265	0.211700
$416$	0	0
$417$	−20.0386	−0.981293
$418$	0	0
$419$	−15.5574	−0.760027	−0.380014	−	0.924981i	$-0.624081\pi$
$419$	−15.5574	−0.760027	−0.380014	+	0.924981i	$0.624081\pi$
$420$	0	0
$421$	30.9380	1.50782	0.753912	−	0.656976i	$-0.228165\pi$
$421$	30.9380	1.50782	0.753912	+	0.656976i	$0.228165\pi$
$422$	0	0
$423$	−8.57452	−0.416907
$424$	0	0
$425$	7.98778	0.387464
$426$	0	0
$427$	31.8496	1.54131
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	32.2638	1.55409	0.777046	−	0.629444i	$-0.216717\pi$
$431$	32.2638	1.55409	0.777046	+	0.629444i	$0.216717\pi$
$432$	0	0
$433$	−13.3747	−0.642747	−0.321374	−	0.946953i	$-0.604144\pi$
$433$	−13.3747	−0.642747	−0.321374	+	0.946953i	$0.604144\pi$
$434$	0	0
$435$	1.39517	0.0668932
$436$	0	0
$437$	−22.8872	−1.09484
$438$	0	0
$439$	−4.51151	−0.215323	−0.107661	−	0.994188i	$-0.534336\pi$
$439$	−4.51151	−0.215323	−0.107661	+	0.994188i	$0.534336\pi$
$440$	0	0
$441$	10.2750	0.489288
$442$	0	0
$443$	−4.37328	−0.207781	−0.103891	−	0.994589i	$-0.533129\pi$
$443$	−4.37328	−0.207781	−0.103891	+	0.994589i	$0.533129\pi$
$444$	0	0
$445$	−1.19394	−0.0565980
$446$	0	0
$447$	−16.8568	−0.797302
$448$	0	0
$449$	−32.1197	−1.51582	−0.757910	−	0.652359i	$-0.773780\pi$
$449$	−32.1197	−1.51582	−0.757910	+	0.652359i	$0.773780\pi$
$450$	0	0
$451$	2.15633	0.101537
$452$	0	0
$453$	14.0811	0.661588
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	4.86273	0.227469	0.113734	−	0.993511i	$-0.463719\pi$
$457$	4.86273	0.227469	0.113734	+	0.993511i	$0.463719\pi$
$458$	0	0
$459$	1.67513	0.0781884
$460$	0	0
$461$	−17.8437	−0.831063	−0.415531	−	0.909579i	$-0.636404\pi$
$461$	−17.8437	−0.831063	−0.415531	+	0.909579i	$0.636404\pi$
$462$	0	0
$463$	−36.0590	−1.67581	−0.837903	−	0.545820i	$-0.816218\pi$
$463$	−36.0590	−1.67581	−0.837903	+	0.545820i	$0.816218\pi$
$464$	0	0
$465$	−3.11871	−0.144627
$466$	0	0
$467$	22.0303	1.01944	0.509721	−	0.860340i	$-0.329749\pi$
$467$	22.0303	1.01944	0.509721	+	0.860340i	$0.329749\pi$
$468$	0	0
$469$	−41.6385	−1.92269
$470$	0	0
$471$	20.7816	0.957567
$472$	0	0
$473$	7.86177	0.361485
$474$	0	0
$475$	19.8192	0.909369
$476$	0	0
$477$	−11.2750	−0.516249
$478$	0	0
$479$	20.5343	0.938238	0.469119	−	0.883135i	$-0.344571\pi$
$479$	20.5343	0.938238	0.469119	+	0.883135i	$0.344571\pi$
$480$	0	0
$481$	6.70052	0.305518
$482$	0	0
$483$	−22.8872	−1.04140
$484$	0	0
$485$	−1.29948	−0.0590062
$486$	0	0
$487$	−1.50422	−0.0681626	−0.0340813	−	0.999419i	$-0.510851\pi$
$487$	−1.50422	−0.0681626	−0.0340813	+	0.999419i	$0.510851\pi$
$488$	0	0
$489$	12.0181	0.543477
$490$	0	0
$491$	15.2506	0.688250	0.344125	−	0.938924i	$-0.388176\pi$
$491$	15.2506	0.688250	0.344125	+	0.938924i	$0.388176\pi$
$492$	0	0
$493$	−4.85685	−0.218741
$494$	0	0
$495$	−0.481194	−0.0216281
$496$	0	0
$497$	−39.3865	−1.76672
$498$	0	0
$499$	12.6942	0.568270	0.284135	−	0.958784i	$-0.408294\pi$
$499$	12.6942	0.568270	0.284135	+	0.958784i	$0.408294\pi$
$500$	0	0
$501$	−18.2374	−0.814788
$502$	0	0
$503$	13.1128	0.584672	0.292336	−	0.956316i	$-0.405567\pi$
$503$	13.1128	0.584672	0.292336	+	0.956316i	$0.405567\pi$
$504$	0	0
$505$	3.81924	0.169954
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	25.3684	1.12443	0.562216	−	0.826990i	$-0.309949\pi$
$509$	25.3684	1.12443	0.562216	+	0.826990i	$0.309949\pi$
$510$	0	0
$511$	−45.4372	−2.01003
$512$	0	0
$513$	4.15633	0.183506
$514$	0	0
$515$	0.373285	0.0164489
$516$	0	0
$517$	−8.57452	−0.377107
$518$	0	0
$519$	−16.2252	−0.712208
$520$	0	0
$521$	4.85097	0.212525	0.106262	−	0.994338i	$-0.466112\pi$
$521$	4.85097	0.212525	0.106262	+	0.994338i	$0.466112\pi$
$522$	0	0
$523$	−44.7391	−1.95630	−0.978152	−	0.207891i	$-0.933340\pi$
$523$	−44.7391	−1.95630	−0.978152	+	0.207891i	$0.933340\pi$
$524$	0	0
$525$	19.8192	0.864982
$526$	0	0
$527$	10.8568	0.472932
$528$	0	0
$529$	7.32250	0.318370
$530$	0	0
$531$	−12.9624	−0.562520
$532$	0	0
$533$	−2.15633	−0.0934008
$534$	0	0
$535$	−3.93937	−0.170314
$536$	0	0
$537$	9.38058	0.404802
$538$	0	0
$539$	10.2750	0.442577
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−6.23155	−0.267421
$544$	0	0
$545$	9.81336	0.420358
$546$	0	0
$547$	37.4641	1.60185	0.800924	−	0.598767i	$-0.204342\pi$
$547$	37.4641	1.60185	0.800924	+	0.598767i	$0.204342\pi$
$548$	0	0
$549$	7.66291	0.327045
$550$	0	0
$551$	−12.0508	−0.513381
$552$	0	0
$553$	7.27504	0.309366
$554$	0	0
$555$	−3.22425	−0.136862
$556$	0	0
$557$	15.5574	0.659187	0.329594	−	0.944123i	$-0.393088\pi$
$557$	15.5574	0.659187	0.329594	+	0.944123i	$0.393088\pi$
$558$	0	0
$559$	−7.86177	−0.332518
$560$	0	0
$561$	1.67513	0.0707241
$562$	0	0
$563$	10.0870	0.425116	0.212558	−	0.977148i	$-0.431821\pi$
$563$	10.0870	0.425116	0.212558	+	0.977148i	$0.431821\pi$
$564$	0	0
$565$	−7.87399	−0.331261
$566$	0	0
$567$	4.15633	0.174549
$568$	0	0
$569$	−19.2022	−0.804998	−0.402499	−	0.915421i	$-0.631858\pi$
$569$	−19.2022	−0.804998	−0.402499	+	0.915421i	$0.631858\pi$
$570$	0	0
$571$	3.52469	0.147503	0.0737517	−	0.997277i	$-0.476503\pi$
$571$	3.52469	0.147503	0.0737517	+	0.997277i	$0.476503\pi$
$572$	0	0
$573$	−3.41090	−0.142492
$574$	0	0
$575$	−26.2579	−1.09503
$576$	0	0
$577$	30.3127	1.26193	0.630966	−	0.775810i	$-0.282659\pi$
$577$	30.3127	1.26193	0.630966	+	0.775810i	$0.282659\pi$
$578$	0	0
$579$	1.84367	0.0766205
$580$	0	0
$581$	−37.2506	−1.54542
$582$	0	0
$583$	−11.2750	−0.466964
$584$	0	0
$585$	0.481194	0.0198949
$586$	0	0
$587$	−8.06537	−0.332894	−0.166447	−	0.986050i	$-0.553229\pi$
$587$	−8.06537	−0.332894	−0.166447	+	0.986050i	$0.553229\pi$
$588$	0	0
$589$	26.9380	1.10996
$590$	0	0
$591$	19.7440	0.812160
$592$	0	0
$593$	35.2809	1.44881	0.724407	−	0.689373i	$-0.242114\pi$
$593$	35.2809	1.44881	0.724407	+	0.689373i	$0.242114\pi$
$594$	0	0
$595$	3.35026	0.137347
$596$	0	0
$597$	3.22425	0.131960
$598$	0	0
$599$	9.11283	0.372340	0.186170	−	0.982518i	$-0.440392\pi$
$599$	9.11283	0.372340	0.186170	+	0.982518i	$0.440392\pi$
$600$	0	0
$601$	−14.3733	−0.586299	−0.293149	−	0.956067i	$-0.594703\pi$
$601$	−14.3733	−0.586299	−0.293149	+	0.956067i	$0.594703\pi$
$602$	0	0
$603$	−10.0181	−0.407969
$604$	0	0
$605$	−0.481194	−0.0195633
$606$	0	0
$607$	−23.5999	−0.957891	−0.478945	−	0.877845i	$-0.658981\pi$
$607$	−23.5999	−0.957891	−0.478945	+	0.877845i	$0.658981\pi$
$608$	0	0
$609$	−12.0508	−0.488322
$610$	0	0
$611$	8.57452	0.346888
$612$	0	0
$613$	19.6180	0.792364	0.396182	−	0.918172i	$-0.370335\pi$
$613$	19.6180	0.792364	0.396182	+	0.918172i	$0.370335\pi$
$614$	0	0
$615$	1.03761	0.0418405
$616$	0	0
$617$	19.4582	0.783356	0.391678	−	0.920102i	$-0.371895\pi$
$617$	19.4582	0.783356	0.391678	+	0.920102i	$0.371895\pi$
$618$	0	0
$619$	−16.2946	−0.654933	−0.327467	−	0.944863i	$-0.606195\pi$
$619$	−16.2946	−0.654933	−0.327467	+	0.944863i	$0.606195\pi$
$620$	0	0
$621$	−5.50659	−0.220972
$622$	0	0
$623$	10.3127	0.413168
$624$	0	0
$625$	21.5804	0.863216
$626$	0	0
$627$	4.15633	0.165988
$628$	0	0
$629$	11.2243	0.447540
$630$	0	0
$631$	−12.4812	−0.496868	−0.248434	−	0.968649i	$-0.579916\pi$
$631$	−12.4812	−0.496868	−0.248434	+	0.968649i	$0.579916\pi$
$632$	0	0
$633$	12.6375	0.502296
$634$	0	0
$635$	−7.44595	−0.295484
$636$	0	0
$637$	−10.2750	−0.407112
$638$	0	0
$639$	−9.47627	−0.374875
$640$	0	0
$641$	10.0606	0.397371	0.198686	−	0.980063i	$-0.436333\pi$
$641$	10.0606	0.397371	0.198686	+	0.980063i	$0.436333\pi$
$642$	0	0
$643$	−23.3538	−0.920983	−0.460491	−	0.887664i	$-0.652327\pi$
$643$	−23.3538	−0.920983	−0.460491	+	0.887664i	$0.652327\pi$
$644$	0	0
$645$	3.78304	0.148957
$646$	0	0
$647$	10.3634	0.407429	0.203714	−	0.979030i	$-0.434699\pi$
$647$	10.3634	0.407429	0.203714	+	0.979030i	$0.434699\pi$
$648$	0	0
$649$	−12.9624	−0.508818
$650$	0	0
$651$	26.9380	1.05578
$652$	0	0
$653$	−3.98541	−0.155961	−0.0779806	−	0.996955i	$-0.524847\pi$
$653$	−3.98541	−0.155961	−0.0779806	+	0.996955i	$0.524847\pi$
$654$	0	0
$655$	−1.70194	−0.0665002
$656$	0	0
$657$	−10.9321	−0.426501
$658$	0	0
$659$	20.2130	0.787386	0.393693	−	0.919242i	$-0.371197\pi$
$659$	20.2130	0.787386	0.393693	+	0.919242i	$0.371197\pi$
$660$	0	0
$661$	−28.9478	−1.12594	−0.562969	−	0.826478i	$-0.690341\pi$
$661$	−28.9478	−1.12594	−0.562969	+	0.826478i	$0.690341\pi$
$662$	0	0
$663$	−1.67513	−0.0650567
$664$	0	0
$665$	8.31265	0.322351
$666$	0	0
$667$	15.9657	0.618195
$668$	0	0
$669$	−1.70545	−0.0659364
$670$	0	0
$671$	7.66291	0.295823
$672$	0	0
$673$	18.3390	0.706916	0.353458	−	0.935450i	$-0.385006\pi$
$673$	18.3390	0.706916	0.353458	+	0.935450i	$0.385006\pi$
$674$	0	0
$675$	4.76845	0.183538
$676$	0	0
$677$	26.2496	1.00886	0.504428	−	0.863454i	$-0.331703\pi$
$677$	26.2496	1.00886	0.504428	+	0.863454i	$0.331703\pi$
$678$	0	0
$679$	11.2243	0.430747
$680$	0	0
$681$	−1.95509	−0.0749194
$682$	0	0
$683$	0.225668	0.00863496	0.00431748	−	0.999991i	$-0.498626\pi$
$683$	0.225668	0.00863496	0.00431748	+	0.999991i	$0.498626\pi$
$684$	0	0
$685$	−7.34155	−0.280506
$686$	0	0
$687$	23.7235	0.905110
$688$	0	0
$689$	11.2750	0.429545
$690$	0	0
$691$	−42.4060	−1.61320	−0.806600	−	0.591098i	$-0.798695\pi$
$691$	−42.4060	−1.61320	−0.806600	+	0.591098i	$0.798695\pi$
$692$	0	0
$693$	4.15633	0.157886
$694$	0	0
$695$	−9.64244	−0.365759
$696$	0	0
$697$	−3.61213	−0.136819
$698$	0	0
$699$	−2.71274	−0.102605
$700$	0	0
$701$	−2.16266	−0.0816827	−0.0408414	−	0.999166i	$-0.513004\pi$
$701$	−2.16266	−0.0816827	−0.0408414	+	0.999166i	$0.513004\pi$
$702$	0	0
$703$	27.8496	1.05037
$704$	0	0
$705$	−4.12601	−0.155394
$706$	0	0
$707$	−32.9887	−1.24067
$708$	0	0
$709$	28.9624	1.08771	0.543853	−	0.839181i	$-0.316965\pi$
$709$	28.9624	1.08771	0.543853	+	0.839181i	$0.316965\pi$
$710$	0	0
$711$	1.75035	0.0656434
$712$	0	0
$713$	−35.6893	−1.33657
$714$	0	0
$715$	0.481194	0.0179956
$716$	0	0
$717$	−22.5198	−0.841016
$718$	0	0
$719$	3.22425	0.120244	0.0601222	−	0.998191i	$-0.480851\pi$
$719$	3.22425	0.120244	0.0601222	+	0.998191i	$0.480851\pi$
$720$	0	0
$721$	−3.22425	−0.120077
$722$	0	0
$723$	10.4993	0.390473
$724$	0	0
$725$	−13.8256	−0.513469
$726$	0	0
$727$	12.0244	0.445962	0.222981	−	0.974823i	$-0.428421\pi$
$727$	12.0244	0.445962	0.222981	+	0.974823i	$0.428421\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.1695	−0.487092
$732$	0	0
$733$	27.3952	1.01186	0.505932	−	0.862573i	$-0.331149\pi$
$733$	27.3952	1.01186	0.505932	+	0.862573i	$0.331149\pi$
$734$	0	0
$735$	4.94429	0.182373
$736$	0	0
$737$	−10.0181	−0.369021
$738$	0	0
$739$	27.7948	1.02245	0.511224	−	0.859447i	$-0.329192\pi$
$739$	27.7948	1.02245	0.511224	+	0.859447i	$0.329192\pi$
$740$	0	0
$741$	−4.15633	−0.152686
$742$	0	0
$743$	4.75131	0.174309	0.0871543	−	0.996195i	$-0.472223\pi$
$743$	4.75131	0.174309	0.0871543	+	0.996195i	$0.472223\pi$
$744$	0	0
$745$	−8.11142	−0.297179
$746$	0	0
$747$	−8.96239	−0.327917
$748$	0	0
$749$	34.0263	1.24330
$750$	0	0
$751$	−27.1998	−0.992535	−0.496268	−	0.868170i	$-0.665297\pi$
$751$	−27.1998	−0.992535	−0.496268	+	0.868170i	$0.665297\pi$
$752$	0	0
$753$	17.3357	0.631747
$754$	0	0
$755$	6.77575	0.246595
$756$	0	0
$757$	−27.0073	−0.981597	−0.490798	−	0.871273i	$-0.663295\pi$
$757$	−27.0073	−0.981597	−0.490798	+	0.871273i	$0.663295\pi$
$758$	0	0
$759$	−5.50659	−0.199876
$760$	0	0
$761$	41.2955	1.49696	0.748480	−	0.663157i	$-0.230784\pi$
$761$	41.2955	1.49696	0.748480	+	0.663157i	$0.230784\pi$
$762$	0	0
$763$	−84.7631	−3.06863
$764$	0	0
$765$	0.806063	0.0291433
$766$	0	0
$767$	12.9624	0.468045
$768$	0	0
$769$	−3.04746	−0.109894	−0.0549471	−	0.998489i	$-0.517499\pi$
$769$	−3.04746	−0.109894	−0.0549471	+	0.998489i	$0.517499\pi$
$770$	0	0
$771$	−20.2374	−0.728833
$772$	0	0
$773$	−21.0966	−0.758794	−0.379397	−	0.925234i	$-0.623868\pi$
$773$	−21.0966	−0.758794	−0.379397	+	0.925234i	$0.623868\pi$
$774$	0	0
$775$	30.9053	1.11015
$776$	0	0
$777$	27.8496	0.999097
$778$	0	0
$779$	−8.96239	−0.321111
$780$	0	0
$781$	−9.47627	−0.339088
$782$	0	0
$783$	−2.89938	−0.103616
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	−20.4083	−0.727479	−0.363739	−	0.931501i	$-0.618500\pi$
$787$	−20.4083	−0.727479	−0.363739	+	0.931501i	$0.618500\pi$
$788$	0	0
$789$	16.8119	0.598521
$790$	0	0
$791$	68.0118	2.41822
$792$	0	0
$793$	−7.66291	−0.272118
$794$	0	0
$795$	−5.42548	−0.192422
$796$	0	0
$797$	−32.1768	−1.13976	−0.569880	−	0.821728i	$-0.693010\pi$
$797$	−32.1768	−1.13976	−0.569880	+	0.821728i	$0.693010\pi$
$798$	0	0
$799$	14.3634	0.508142
$800$	0	0
$801$	2.48119	0.0876687
$802$	0	0
$803$	−10.9321	−0.385784
$804$	0	0
$805$	−11.0132	−0.388163
$806$	0	0
$807$	0.201231	0.00708366
$808$	0	0
$809$	37.5247	1.31930	0.659649	−	0.751574i	$-0.270705\pi$
$809$	37.5247	1.31930	0.659649	+	0.751574i	$0.270705\pi$
$810$	0	0
$811$	−3.21837	−0.113012	−0.0565062	−	0.998402i	$-0.517996\pi$
$811$	−3.21837	−0.113012	−0.0565062	+	0.998402i	$0.517996\pi$
$812$	0	0
$813$	1.32979	0.0466379
$814$	0	0
$815$	5.78304	0.202571
$816$	0	0
$817$	−32.6761	−1.14319
$818$	0	0
$819$	−4.15633	−0.145234
$820$	0	0
$821$	38.3430	1.33818	0.669089	−	0.743182i	$-0.266684\pi$
$821$	38.3430	1.33818	0.669089	+	0.743182i	$0.266684\pi$
$822$	0	0
$823$	2.82653	0.0985267	0.0492633	−	0.998786i	$-0.484313\pi$
$823$	2.82653	0.0985267	0.0492633	+	0.998786i	$0.484313\pi$
$824$	0	0
$825$	4.76845	0.166016
$826$	0	0
$827$	−42.1173	−1.46456	−0.732281	−	0.681003i	$-0.761544\pi$
$827$	−42.1173	−1.46456	−0.732281	+	0.681003i	$0.761544\pi$
$828$	0	0
$829$	49.1754	1.70793	0.853966	−	0.520329i	$-0.174191\pi$
$829$	49.1754	1.70793	0.853966	+	0.520329i	$0.174191\pi$
$830$	0	0
$831$	−1.01317	−0.0351466
$832$	0	0
$833$	−17.2120	−0.596362
$834$	0	0
$835$	−8.77575	−0.303697
$836$	0	0
$837$	6.48119	0.224023
$838$	0	0
$839$	14.5091	0.500911	0.250456	−	0.968128i	$-0.419420\pi$
$839$	14.5091	0.500911	0.250456	+	0.968128i	$0.419420\pi$
$840$	0	0
$841$	−20.5936	−0.710123
$842$	0	0
$843$	−11.8945	−0.409667
$844$	0	0
$845$	−0.481194	−0.0165536
$846$	0	0
$847$	4.15633	0.142813
$848$	0	0
$849$	0.249646	0.00856784
$850$	0	0
$851$	−36.8970	−1.26481
$852$	0	0
$853$	31.8350	1.09001	0.545004	−	0.838433i	$-0.316528\pi$
$853$	31.8350	1.09001	0.545004	+	0.838433i	$0.316528\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	1.49834	0.0511822	0.0255911	−	0.999672i	$-0.491853\pi$
$857$	1.49834	0.0511822	0.0255911	+	0.999672i	$0.491853\pi$
$858$	0	0
$859$	30.4241	1.03806	0.519028	−	0.854757i	$-0.326294\pi$
$859$	30.4241	1.03806	0.519028	+	0.854757i	$0.326294\pi$
$860$	0	0
$861$	−8.96239	−0.305437
$862$	0	0
$863$	−18.5256	−0.630620	−0.315310	−	0.948989i	$-0.602109\pi$
$863$	−18.5256	−0.630620	−0.315310	+	0.948989i	$0.602109\pi$
$864$	0	0
$865$	−7.80748	−0.265462
$866$	0	0
$867$	14.1939	0.482051
$868$	0	0
$869$	1.75035	0.0593767
$870$	0	0
$871$	10.0181	0.339450
$872$	0	0
$873$	2.70052	0.0913989
$874$	0	0
$875$	19.5369	0.660468
$876$	0	0
$877$	14.3488	0.484526	0.242263	−	0.970211i	$-0.422110\pi$
$877$	14.3488	0.484526	0.242263	+	0.970211i	$0.422110\pi$
$878$	0	0
$879$	1.56722	0.0528611
$880$	0	0
$881$	−16.8510	−0.567724	−0.283862	−	0.958865i	$-0.591616\pi$
$881$	−16.8510	−0.567724	−0.283862	+	0.958865i	$0.591616\pi$
$882$	0	0
$883$	−2.44851	−0.0823988	−0.0411994	−	0.999151i	$-0.513118\pi$
$883$	−2.44851	−0.0823988	−0.0411994	+	0.999151i	$0.513118\pi$
$884$	0	0
$885$	−6.23743	−0.209669
$886$	0	0
$887$	−26.8119	−0.900257	−0.450128	−	0.892964i	$-0.648622\pi$
$887$	−26.8119	−0.900257	−0.450128	+	0.892964i	$0.648622\pi$
$888$	0	0
$889$	64.3146	2.15704
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	35.6385	1.19260
$894$	0	0
$895$	4.51388	0.150882
$896$	0	0
$897$	5.50659	0.183860
$898$	0	0
$899$	−18.7915	−0.626731
$900$	0	0
$901$	18.8872	0.629223
$902$	0	0
$903$	−32.6761	−1.08739
$904$	0	0
$905$	−2.99859	−0.0996764
$906$	0	0
$907$	23.3503	0.775333	0.387666	−	0.921800i	$-0.373281\pi$
$907$	23.3503	0.775333	0.387666	+	0.921800i	$0.373281\pi$
$908$	0	0
$909$	−7.93700	−0.263254
$910$	0	0
$911$	−47.2408	−1.56516	−0.782578	−	0.622553i	$-0.786096\pi$
$911$	−47.2408	−1.56516	−0.782578	+	0.622553i	$0.786096\pi$
$912$	0	0
$913$	−8.96239	−0.296612
$914$	0	0
$915$	3.68735	0.121900
$916$	0	0
$917$	14.7005	0.485454
$918$	0	0
$919$	−0.152815	−0.00504091	−0.00252045	−	0.999997i	$-0.500802\pi$
$919$	−0.152815	−0.00504091	−0.00252045	+	0.999997i	$0.500802\pi$
$920$	0	0
$921$	27.1695	0.895265
$922$	0	0
$923$	9.47627	0.311915
$924$	0	0
$925$	31.9511	1.05055
$926$	0	0
$927$	−0.775746	−0.0254789
$928$	0	0
$929$	−0.0279475	−0.000916928 0	−0.000458464	−	1.00000i	$-0.500146\pi$
$929$	−0.0279475	−0.000916928 0	−0.000458464		1.00000i	$0.500146\pi$
$930$	0	0
$931$	−42.7064	−1.39965
$932$	0	0
$933$	−10.3430	−0.338614
$934$	0	0
$935$	0.806063	0.0263611
$936$	0	0
$937$	12.7466	0.416412	0.208206	−	0.978085i	$-0.433238\pi$
$937$	12.7466	0.416412	0.208206	+	0.978085i	$0.433238\pi$
$938$	0	0
$939$	19.1041	0.623440
$940$	0	0
$941$	24.7553	0.806999	0.403500	−	0.914980i	$-0.367794\pi$
$941$	24.7553	0.806999	0.403500	+	0.914980i	$0.367794\pi$
$942$	0	0
$943$	11.8740	0.386670
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	−15.9610	−0.518662	−0.259331	−	0.965789i	$-0.583502\pi$
$947$	−15.9610	−0.518662	−0.259331	+	0.965789i	$0.583502\pi$
$948$	0	0
$949$	10.9321	0.354870
$950$	0	0
$951$	20.8446	0.675933
$952$	0	0
$953$	−40.4382	−1.30992	−0.654961	−	0.755663i	$-0.727315\pi$
$953$	−40.4382	−1.30992	−0.654961	+	0.755663i	$0.727315\pi$
$954$	0	0
$955$	−1.64130	−0.0531113
$956$	0	0
$957$	−2.89938	−0.0937238
$958$	0	0
$959$	63.4128	2.04771
$960$	0	0
$961$	11.0059	0.355028
$962$	0	0
$963$	8.18664	0.263811
$964$	0	0
$965$	0.887166	0.0285589
$966$	0	0
$967$	−37.0698	−1.19209	−0.596043	−	0.802953i	$-0.703261\pi$
$967$	−37.0698	−1.19209	−0.596043	+	0.802953i	$0.703261\pi$
$968$	0	0
$969$	−6.96239	−0.223664
$970$	0	0
$971$	−35.5818	−1.14187	−0.570937	−	0.820994i	$-0.693420\pi$
$971$	−35.5818	−1.14187	−0.570937	+	0.820994i	$0.693420\pi$
$972$	0	0
$973$	83.2868	2.67005
$974$	0	0
$975$	−4.76845	−0.152713
$976$	0	0
$977$	10.5466	0.337415	0.168707	−	0.985666i	$-0.446041\pi$
$977$	10.5466	0.337415	0.168707	+	0.985666i	$0.446041\pi$
$978$	0	0
$979$	2.48119	0.0792993
$980$	0	0
$981$	−20.3938	−0.651122
$982$	0	0
$983$	−6.50914	−0.207609	−0.103805	−	0.994598i	$-0.533102\pi$
$983$	−6.50914	−0.207609	−0.103805	+	0.994598i	$0.533102\pi$
$984$	0	0
$985$	9.50071	0.302718
$986$	0	0
$987$	35.6385	1.13439
$988$	0	0
$989$	43.2915	1.37659
$990$	0	0
$991$	34.6155	1.09960	0.549798	−	0.835298i	$-0.314705\pi$
$991$	34.6155	1.09960	0.549798	+	0.835298i	$0.314705\pi$
$992$	0	0
$993$	−16.6678	−0.528938
$994$	0	0
$995$	1.55149	0.0491856
$996$	0	0
$997$	61.6629	1.95288	0.976442	−	0.215780i	$-0.0692294\pi$
$997$	61.6629	1.95288	0.976442	+	0.215780i	$0.0692294\pi$
$998$	0	0
$999$	6.70052	0.211995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bs.1.1		3
4.3	odd	2		429.2.a.g.1.3	✓	3
12.11	even	2		1287.2.a.h.1.1		3
44.43	even	2		4719.2.a.q.1.1		3
52.51	odd	2		5577.2.a.j.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.g.1.3	✓	3	4.3	odd	2
1287.2.a.h.1.1		3	12.11	even	2
4719.2.a.q.1.1		3	44.43	even	2
5577.2.a.j.1.1		3	52.51	odd	2
6864.2.a.bs.1.1		3	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.481194 q^{5} +4.15633 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.481194 q^{5} +4.15633 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.481194 q^{15} -1.67513 q^{17} -4.15633 q^{19} -4.15633 q^{21} +5.50659 q^{23} -4.76845 q^{25} -1.00000 q^{27} +2.89938 q^{29} -6.48119 q^{31} -1.00000 q^{33} -2.00000 q^{35} -6.70052 q^{37} +1.00000 q^{39} +2.15633 q^{41} +7.86177 q^{43} -0.481194 q^{45} -8.57452 q^{47} +10.2750 q^{49} +1.67513 q^{51} -11.2750 q^{53} -0.481194 q^{55} +4.15633 q^{57} -12.9624 q^{59} +7.66291 q^{61} +4.15633 q^{63} +0.481194 q^{65} -10.0181 q^{67} -5.50659 q^{69} -9.47627 q^{71} -10.9321 q^{73} +4.76845 q^{75} +4.15633 q^{77} +1.75035 q^{79} +1.00000 q^{81} -8.96239 q^{83} +0.806063 q^{85} -2.89938 q^{87} +2.48119 q^{89} -4.15633 q^{91} +6.48119 q^{93} +2.00000 q^{95} +2.70052 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{15} - 2 q^{19} - 2 q^{21} - 4 q^{23} - 3 q^{25} - 3 q^{27} + 2 q^{29} - 14 q^{31} - 3 q^{33} - 6 q^{35} + 3 q^{39} - 4 q^{41} + 6 q^{43} + 4 q^{45} - 14 q^{47} - q^{49} - 2 q^{53} + 4 q^{55} + 2 q^{57} - 28 q^{59} - 8 q^{61} + 2 q^{63} - 4 q^{65} - 2 q^{67} + 4 q^{69} - 10 q^{71} - 24 q^{73} + 3 q^{75} + 2 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} + 2 q^{85} - 2 q^{87} + 2 q^{89} - 2 q^{91} + 14 q^{93} + 6 q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^15 - 2 * q^19 - 2 * q^21 - 4 * q^23 - 3 * q^25 - 3 * q^27 + 2 * q^29 - 14 * q^31 - 3 * q^33 - 6 * q^35 + 3 * q^39 - 4 * q^41 + 6 * q^43 + 4 * q^45 - 14 * q^47 - q^49 - 2 * q^53 + 4 * q^55 + 2 * q^57 - 28 * q^59 - 8 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 + 4 * q^69 - 10 * q^71 - 24 * q^73 + 3 * q^75 + 2 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 + 2 * q^85 - 2 * q^87 + 2 * q^89 - 2 * q^91 + 14 * q^93 + 6 * q^95 - 12 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more