LMFDB - Embedded newform 6864.2.a.bn.1.3

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.210756$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.74483 q^{5} +4.53407 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.74483 q^{5} +4.53407 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.74483 q^{15} -1.63227 q^{17} -0.534070 q^{19} -4.53407 q^{21} +7.60221 q^{23} -1.95558 q^{25} -1.00000 q^{27} -1.21076 q^{29} +2.67669 q^{31} +1.00000 q^{33} +7.91116 q^{35} -9.91116 q^{37} -1.00000 q^{39} +3.37709 q^{41} -3.63227 q^{43} +1.74483 q^{45} +9.48965 q^{47} +13.5578 q^{49} +1.63227 q^{51} -3.57849 q^{53} -1.74483 q^{55} +0.534070 q^{57} +2.64663 q^{59} +12.5578 q^{61} +4.53407 q^{63} +1.74483 q^{65} +12.5878 q^{67} -7.60221 q^{69} -3.06814 q^{71} +4.95558 q^{73} +1.95558 q^{75} -4.53407 q^{77} +1.72110 q^{79} +1.00000 q^{81} -1.35337 q^{83} -2.84802 q^{85} +1.21076 q^{87} +6.39145 q^{89} +4.53407 q^{91} -2.67669 q^{93} -0.931860 q^{95} -2.93186 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 4 q^{15} + 6 q^{19} - 6 q^{21} + 5 q^{25} - 3 q^{27} - 2 q^{29} + 14 q^{31} + 3 q^{33} + 2 q^{35} - 8 q^{37} - 3 q^{39} - 4 q^{41} - 6 q^{43} - 4 q^{45} + 10 q^{47} + 7 q^{49} - 14 q^{53} + 4 q^{55} - 6 q^{57} - 4 q^{59} + 4 q^{61} + 6 q^{63} - 4 q^{65} + 22 q^{67} + 6 q^{71} + 4 q^{73} - 5 q^{75} - 6 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} - 14 q^{85} + 2 q^{87} - 2 q^{89} + 6 q^{91} - 14 q^{93} - 18 q^{95} - 24 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 4 * q^15 + 6 * q^19 - 6 * q^21 + 5 * q^25 - 3 * q^27 - 2 * q^29 + 14 * q^31 + 3 * q^33 + 2 * q^35 - 8 * q^37 - 3 * q^39 - 4 * q^41 - 6 * q^43 - 4 * q^45 + 10 * q^47 + 7 * q^49 - 14 * q^53 + 4 * q^55 - 6 * q^57 - 4 * q^59 + 4 * q^61 + 6 * q^63 - 4 * q^65 + 22 * q^67 + 6 * q^71 + 4 * q^73 - 5 * q^75 - 6 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 - 14 * q^85 + 2 * q^87 - 2 * q^89 + 6 * q^91 - 14 * q^93 - 18 * q^95 - 24 * 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
$5$	1.74483	0.780310	0.390155	−	0.920749i	$-0.372421\pi$
$5$	1.74483	0.780310	0.390155	+	0.920749i	$0.372421\pi$
$6$	0	0
$7$	4.53407	1.71372	0.856859	−	0.515551i	$-0.172413\pi$
$7$	4.53407	1.71372	0.856859	+	0.515551i	$0.172413\pi$
$8$	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
$15$	−1.74483	−0.450512
$16$	0	0
$17$	−1.63227	−0.395883	−0.197942	−	0.980214i	$-0.563426\pi$
$17$	−1.63227	−0.395883	−0.197942	+	0.980214i	$0.563426\pi$
$18$	0	0
$19$	−0.534070	−0.122524	−0.0612621	−	0.998122i	$-0.519513\pi$
$19$	−0.534070	−0.122524	−0.0612621	+	0.998122i	$0.519513\pi$
$20$	0	0
$21$	−4.53407	−0.989415
$22$	0	0
$23$	7.60221	1.58517	0.792585	−	0.609761i	$-0.208735\pi$
$23$	7.60221	1.58517	0.792585	+	0.609761i	$0.208735\pi$
$24$	0	0
$25$	−1.95558	−0.391116
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.21076	−0.224832	−0.112416	−	0.993661i	$-0.535859\pi$
$29$	−1.21076	−0.224832	−0.112416	+	0.993661i	$0.535859\pi$
$30$	0	0
$31$	2.67669	0.480747	0.240373	−	0.970680i	$-0.422730\pi$
$31$	2.67669	0.480747	0.240373	+	0.970680i	$0.422730\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	7.91116	1.33723
$36$	0	0
$37$	−9.91116	−1.62939	−0.814693	−	0.579893i	$-0.803094\pi$
$37$	−9.91116	−1.62939	−0.814693	+	0.579893i	$0.803094\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.37709	0.527413	0.263707	−	0.964603i	$-0.415055\pi$
$41$	3.37709	0.527413	0.263707	+	0.964603i	$0.415055\pi$
$42$	0	0
$43$	−3.63227	−0.553916	−0.276958	−	0.960882i	$-0.589326\pi$
$43$	−3.63227	−0.553916	−0.276958	+	0.960882i	$0.589326\pi$
$44$	0	0
$45$	1.74483	0.260103
$46$	0	0
$47$	9.48965	1.38421	0.692104	−	0.721798i	$-0.256684\pi$
$47$	9.48965	1.38421	0.692104	+	0.721798i	$0.256684\pi$
$48$	0	0
$49$	13.5578	1.93683
$50$	0	0
$51$	1.63227	0.228563
$52$	0	0
$53$	−3.57849	−0.491543	−0.245772	−	0.969328i	$-0.579041\pi$
$53$	−3.57849	−0.491543	−0.245772	+	0.969328i	$0.579041\pi$
$54$	0	0
$55$	−1.74483	−0.235272
$56$	0	0
$57$	0.534070	0.0707393
$58$	0	0
$59$	2.64663	0.344562	0.172281	−	0.985048i	$-0.444886\pi$
$59$	2.64663	0.344562	0.172281	+	0.985048i	$0.444886\pi$
$60$	0	0
$61$	12.5578	1.60786	0.803930	−	0.594724i	$-0.202738\pi$
$61$	12.5578	1.60786	0.803930	+	0.594724i	$0.202738\pi$
$62$	0	0
$63$	4.53407	0.571239
$64$	0	0
$65$	1.74483	0.216419
$66$	0	0
$67$	12.5878	1.53785	0.768925	−	0.639339i	$-0.220792\pi$
$67$	12.5878	1.53785	0.768925	+	0.639339i	$0.220792\pi$
$68$	0	0
$69$	−7.60221	−0.915199
$70$	0	0
$71$	−3.06814	−0.364121	−0.182061	−	0.983287i	$-0.558277\pi$
$71$	−3.06814	−0.364121	−0.182061	+	0.983287i	$0.558277\pi$
$72$	0	0
$73$	4.95558	0.580007	0.290003	−	0.957026i	$-0.406344\pi$
$73$	4.95558	0.580007	0.290003	+	0.957026i	$0.406344\pi$
$74$	0	0
$75$	1.95558	0.225811
$76$	0	0
$77$	−4.53407	−0.516705
$78$	0	0
$79$	1.72110	0.193639	0.0968196	−	0.995302i	$-0.469133\pi$
$79$	1.72110	0.193639	0.0968196	+	0.995302i	$0.469133\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.35337	−0.148552	−0.0742759	−	0.997238i	$-0.523665\pi$
$83$	−1.35337	−0.148552	−0.0742759	+	0.997238i	$0.523665\pi$
$84$	0	0
$85$	−2.84802	−0.308911
$86$	0	0
$87$	1.21076	0.129807
$88$	0	0
$89$	6.39145	0.677493	0.338746	−	0.940878i	$-0.389997\pi$
$89$	6.39145	0.677493	0.338746	+	0.940878i	$0.389997\pi$
$90$	0	0
$91$	4.53407	0.475300
$92$	0	0
$93$	−2.67669	−0.277559
$94$	0	0
$95$	−0.931860	−0.0956068
$96$	0	0
$97$	−2.93186	−0.297685	−0.148843	−	0.988861i	$-0.547555\pi$
$97$	−2.93186	−0.297685	−0.148843	+	0.988861i	$0.547555\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−13.1219	−1.30568	−0.652840	−	0.757496i	$-0.726423\pi$
$101$	−13.1219	−1.30568	−0.652840	+	0.757496i	$0.726423\pi$
$102$	0	0
$103$	3.15698	0.311066	0.155533	−	0.987831i	$-0.450290\pi$
$103$	3.15698	0.311066	0.155533	+	0.987831i	$0.450290\pi$
$104$	0	0
$105$	−7.91116	−0.772051
$106$	0	0
$107$	18.4690	1.78546	0.892731	−	0.450591i	$-0.148787\pi$
$107$	18.4690	1.78546	0.892731	+	0.450591i	$0.148787\pi$
$108$	0	0
$109$	6.86675	0.657715	0.328857	−	0.944380i	$-0.393336\pi$
$109$	6.86675	0.657715	0.328857	+	0.944380i	$0.393336\pi$
$110$	0	0
$111$	9.91116	0.940726
$112$	0	0
$113$	−0.646629	−0.0608297	−0.0304149	−	0.999537i	$-0.509683\pi$
$113$	−0.646629	−0.0608297	−0.0304149	+	0.999537i	$0.509683\pi$
$114$	0	0
$115$	13.2645	1.23692
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−7.40082	−0.678432
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.37709	−0.304502
$124$	0	0
$125$	−12.1363	−1.08550
$126$	0	0
$127$	7.43587	0.659827	0.329914	−	0.944011i	$-0.392980\pi$
$127$	7.43587	0.659827	0.329914	+	0.944011i	$0.392980\pi$
$128$	0	0
$129$	3.63227	0.319803
$130$	0	0
$131$	−15.2044	−1.32842	−0.664208	−	0.747548i	$-0.731231\pi$
$131$	−15.2044	−1.32842	−0.664208	+	0.747548i	$0.731231\pi$
$132$	0	0
$133$	−2.42151	−0.209972
$134$	0	0
$135$	−1.74483	−0.150171
$136$	0	0
$137$	−16.5878	−1.41720	−0.708598	−	0.705613i	$-0.750672\pi$
$137$	−16.5878	−1.41720	−0.708598	+	0.705613i	$0.750672\pi$
$138$	0	0
$139$	3.21076	0.272333	0.136166	−	0.990686i	$-0.456522\pi$
$139$	3.21076	0.272333	0.136166	+	0.990686i	$0.456522\pi$
$140$	0	0
$141$	−9.48965	−0.799173
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−2.11256	−0.175438
$146$	0	0
$147$	−13.5578	−1.11823
$148$	0	0
$149$	−10.7592	−0.881427	−0.440713	−	0.897648i	$-0.645275\pi$
$149$	−10.7592	−0.881427	−0.440713	+	0.897648i	$0.645275\pi$
$150$	0	0
$151$	0.486625	0.0396010	0.0198005	−	0.999804i	$-0.493697\pi$
$151$	0.486625	0.0396010	0.0198005	+	0.999804i	$0.493697\pi$
$152$	0	0
$153$	−1.63227	−0.131961
$154$	0	0
$155$	4.67035	0.375132
$156$	0	0
$157$	2.95558	0.235881	0.117941	−	0.993021i	$-0.462371\pi$
$157$	2.95558	0.235881	0.117941	+	0.993021i	$0.462371\pi$
$158$	0	0
$159$	3.57849	0.283793
$160$	0	0
$161$	34.4690	2.71653
$162$	0	0
$163$	23.8811	1.87051	0.935256	−	0.353971i	$-0.115169\pi$
$163$	23.8811	1.87051	0.935256	+	0.353971i	$0.115169\pi$
$164$	0	0
$165$	1.74483	0.135835
$166$	0	0
$167$	−12.8905	−0.997494	−0.498747	−	0.866748i	$-0.666206\pi$
$167$	−12.8905	−0.997494	−0.498747	+	0.866748i	$0.666206\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.534070	−0.0408414
$172$	0	0
$173$	−4.89680	−0.372297	−0.186149	−	0.982522i	$-0.559601\pi$
$173$	−4.89680	−0.372297	−0.186149	+	0.982522i	$0.559601\pi$
$174$	0	0
$175$	−8.86675	−0.670263
$176$	0	0
$177$	−2.64663	−0.198933
$178$	0	0
$179$	9.79861	0.732382	0.366191	−	0.930540i	$-0.380662\pi$
$179$	9.79861	0.732382	0.366191	+	0.930540i	$0.380662\pi$
$180$	0	0
$181$	8.02372	0.596399	0.298199	−	0.954504i	$-0.403614\pi$
$181$	8.02372	0.596399	0.298199	+	0.954504i	$0.403614\pi$
$182$	0	0
$183$	−12.5578	−0.928299
$184$	0	0
$185$	−17.2933	−1.27143
$186$	0	0
$187$	1.63227	0.119363
$188$	0	0
$189$	−4.53407	−0.329805
$190$	0	0
$191$	12.3327	0.892361	0.446181	−	0.894943i	$-0.352784\pi$
$191$	12.3327	0.892361	0.446181	+	0.894943i	$0.352784\pi$
$192$	0	0
$193$	8.33768	0.600159	0.300080	−	0.953914i	$-0.402987\pi$
$193$	8.33768	0.600159	0.300080	+	0.953914i	$0.402987\pi$
$194$	0	0
$195$	−1.74483	−0.124950
$196$	0	0
$197$	−3.04442	−0.216906	−0.108453	−	0.994102i	$-0.534590\pi$
$197$	−3.04442	−0.216906	−0.108453	+	0.994102i	$0.534590\pi$
$198$	0	0
$199$	6.13628	0.434989	0.217495	−	0.976062i	$-0.430212\pi$
$199$	6.13628	0.434989	0.217495	+	0.976062i	$0.430212\pi$
$200$	0	0
$201$	−12.5878	−0.887878
$202$	0	0
$203$	−5.48965	−0.385298
$204$	0	0
$205$	5.89244	0.411546
$206$	0	0
$207$	7.60221	0.528390
$208$	0	0
$209$	0.534070	0.0369424
$210$	0	0
$211$	1.94622	0.133983	0.0669917	−	0.997754i	$-0.478660\pi$
$211$	1.94622	0.133983	0.0669917	+	0.997754i	$0.478660\pi$
$212$	0	0
$213$	3.06814	0.210226
$214$	0	0
$215$	−6.33768	−0.432226
$216$	0	0
$217$	12.1363	0.823864
$218$	0	0
$219$	−4.95558	−0.334867
$220$	0	0
$221$	−1.63227	−0.109798
$222$	0	0
$223$	−11.7923	−0.789669	−0.394834	−	0.918752i	$-0.629198\pi$
$223$	−11.7923	−0.789669	−0.394834	+	0.918752i	$0.629198\pi$
$224$	0	0
$225$	−1.95558	−0.130372
$226$	0	0
$227$	−22.6704	−1.50468	−0.752342	−	0.658773i	$-0.771076\pi$
$227$	−22.6704	−1.50468	−0.752342	+	0.658773i	$0.771076\pi$
$228$	0	0
$229$	−8.51035	−0.562380	−0.281190	−	0.959652i	$-0.590729\pi$
$229$	−8.51035	−0.562380	−0.281190	+	0.959652i	$0.590729\pi$
$230$	0	0
$231$	4.53407	0.298320
$232$	0	0
$233$	21.4359	1.40431	0.702155	−	0.712024i	$-0.252221\pi$
$233$	21.4359	1.40431	0.702155	+	0.712024i	$0.252221\pi$
$234$	0	0
$235$	16.5578	1.08011
$236$	0	0
$237$	−1.72110	−0.111798
$238$	0	0
$239$	−7.91616	−0.512054	−0.256027	−	0.966670i	$-0.582414\pi$
$239$	−7.91616	−0.512054	−0.256027	+	0.966670i	$0.582414\pi$
$240$	0	0
$241$	−24.6941	−1.59069	−0.795343	−	0.606160i	$-0.792709\pi$
$241$	−24.6941	−1.59069	−0.795343	+	0.606160i	$0.792709\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	23.6560	1.51133
$246$	0	0
$247$	−0.534070	−0.0339821
$248$	0	0
$249$	1.35337	0.0857664
$250$	0	0
$251$	−29.6259	−1.86997	−0.934986	−	0.354684i	$-0.884588\pi$
$251$	−29.6259	−1.86997	−0.934986	+	0.354684i	$0.884588\pi$
$252$	0	0
$253$	−7.60221	−0.477947
$254$	0	0
$255$	2.84802	0.178350
$256$	0	0
$257$	6.61791	0.412814	0.206407	−	0.978466i	$-0.433823\pi$
$257$	6.61791	0.412814	0.206407	+	0.978466i	$0.433823\pi$
$258$	0	0
$259$	−44.9379	−2.79231
$260$	0	0
$261$	−1.21076	−0.0749439
$262$	0	0
$263$	−27.7622	−1.71189	−0.855946	−	0.517066i	$-0.827024\pi$
$263$	−27.7622	−1.71189	−0.855946	+	0.517066i	$0.827024\pi$
$264$	0	0
$265$	−6.24384	−0.383556
$266$	0	0
$267$	−6.39145	−0.391151
$268$	0	0
$269$	7.57849	0.462069	0.231034	−	0.972946i	$-0.425789\pi$
$269$	7.57849	0.462069	0.231034	+	0.972946i	$0.425789\pi$
$270$	0	0
$271$	19.1807	1.16514	0.582572	−	0.812779i	$-0.302046\pi$
$271$	19.1807	1.16514	0.582572	+	0.812779i	$0.302046\pi$
$272$	0	0
$273$	−4.53407	−0.274414
$274$	0	0
$275$	1.95558	0.117926
$276$	0	0
$277$	−13.2044	−0.793377	−0.396688	−	0.917953i	$-0.629841\pi$
$277$	−13.2044	−0.793377	−0.396688	+	0.917953i	$0.629841\pi$
$278$	0	0
$279$	2.67669	0.160249
$280$	0	0
$281$	19.7098	1.17579	0.587893	−	0.808939i	$-0.299958\pi$
$281$	19.7098	1.17579	0.587893	+	0.808939i	$0.299958\pi$
$282$	0	0
$283$	26.1299	1.55326	0.776632	−	0.629955i	$-0.216926\pi$
$283$	26.1299	1.55326	0.776632	+	0.629955i	$0.216926\pi$
$284$	0	0
$285$	0.931860	0.0551986
$286$	0	0
$287$	15.3120	0.903838
$288$	0	0
$289$	−14.3357	−0.843277
$290$	0	0
$291$	2.93186	0.171869
$292$	0	0
$293$	5.97628	0.349138	0.174569	−	0.984645i	$-0.444147\pi$
$293$	5.97628	0.349138	0.174569	+	0.984645i	$0.444147\pi$
$294$	0	0
$295$	4.61791	0.268865
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	7.60221	0.439647
$300$	0	0
$301$	−16.4690	−0.949255
$302$	0	0
$303$	13.1219	0.753835
$304$	0	0
$305$	21.9112	1.25463
$306$	0	0
$307$	13.3771	0.763471	0.381736	−	0.924272i	$-0.375327\pi$
$307$	13.3771	0.763471	0.381736	+	0.924272i	$0.375327\pi$
$308$	0	0
$309$	−3.15698	−0.179594
$310$	0	0
$311$	−25.5608	−1.44942	−0.724711	−	0.689053i	$-0.758027\pi$
$311$	−25.5608	−1.44942	−0.724711	+	0.689053i	$0.758027\pi$
$312$	0	0
$313$	29.1994	1.65045	0.825224	−	0.564805i	$-0.191049\pi$
$313$	29.1994	1.65045	0.825224	+	0.564805i	$0.191049\pi$
$314$	0	0
$315$	7.91116	0.445744
$316$	0	0
$317$	9.94122	0.558355	0.279177	−	0.960240i	$-0.409938\pi$
$317$	9.94122	0.558355	0.279177	+	0.960240i	$0.409938\pi$
$318$	0	0
$319$	1.21076	0.0677893
$320$	0	0
$321$	−18.4690	−1.03084
$322$	0	0
$323$	0.871746	0.0485052
$324$	0	0
$325$	−1.95558	−0.108476
$326$	0	0
$327$	−6.86675	−0.379732
$328$	0	0
$329$	43.0267	2.37214
$330$	0	0
$331$	−6.87308	−0.377779	−0.188889	−	0.981998i	$-0.560489\pi$
$331$	−6.87308	−0.377779	−0.188889	+	0.981998i	$0.560489\pi$
$332$	0	0
$333$	−9.91116	−0.543128
$334$	0	0
$335$	21.9636	1.20000
$336$	0	0
$337$	9.66732	0.526613	0.263306	−	0.964712i	$-0.415187\pi$
$337$	9.66732	0.526613	0.263306	+	0.964712i	$0.415187\pi$
$338$	0	0
$339$	0.646629	0.0351200
$340$	0	0
$341$	−2.67669	−0.144951
$342$	0	0
$343$	29.7335	1.60546
$344$	0	0
$345$	−13.2645	−0.714139
$346$	0	0
$347$	10.4502	0.560998	0.280499	−	0.959854i	$-0.409500\pi$
$347$	10.4502	0.560998	0.280499	+	0.959854i	$0.409500\pi$
$348$	0	0
$349$	1.44221	0.0771996	0.0385998	−	0.999255i	$-0.487710\pi$
$349$	1.44221	0.0771996	0.0385998	+	0.999255i	$0.487710\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−16.7241	−0.890136	−0.445068	−	0.895497i	$-0.646820\pi$
$353$	−16.7241	−0.890136	−0.445068	+	0.895497i	$0.646820\pi$
$354$	0	0
$355$	−5.35337	−0.284127
$356$	0	0
$357$	7.40082	0.391693
$358$	0	0
$359$	18.7178	0.987887	0.493944	−	0.869494i	$-0.335555\pi$
$359$	18.7178	0.987887	0.493944	+	0.869494i	$0.335555\pi$
$360$	0	0
$361$	−18.7148	−0.984988
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	8.64663	0.452585
$366$	0	0
$367$	−4.55779	−0.237915	−0.118957	−	0.992899i	$-0.537955\pi$
$367$	−4.55779	−0.237915	−0.118957	+	0.992899i	$0.537955\pi$
$368$	0	0
$369$	3.37709	0.175804
$370$	0	0
$371$	−16.2251	−0.842366
$372$	0	0
$373$	−29.3120	−1.51772	−0.758858	−	0.651256i	$-0.774243\pi$
$373$	−29.3120	−1.51772	−0.758858	+	0.651256i	$0.774243\pi$
$374$	0	0
$375$	12.1363	0.626715
$376$	0	0
$377$	−1.21076	−0.0623571
$378$	0	0
$379$	19.9412	1.02431	0.512156	−	0.858893i	$-0.328847\pi$
$379$	19.9412	1.02431	0.512156	+	0.858893i	$0.328847\pi$
$380$	0	0
$381$	−7.43587	−0.380951
$382$	0	0
$383$	15.2645	0.779981	0.389991	−	0.920819i	$-0.372478\pi$
$383$	15.2645	0.779981	0.389991	+	0.920819i	$0.372478\pi$
$384$	0	0
$385$	−7.91116	−0.403190
$386$	0	0
$387$	−3.63227	−0.184639
$388$	0	0
$389$	−3.17570	−0.161014	−0.0805072	−	0.996754i	$-0.525654\pi$
$389$	−3.17570	−0.161014	−0.0805072	+	0.996754i	$0.525654\pi$
$390$	0	0
$391$	−12.4088	−0.627542
$392$	0	0
$393$	15.2044	0.766962
$394$	0	0
$395$	3.00303	0.151099
$396$	0	0
$397$	−9.91116	−0.497427	−0.248714	−	0.968577i	$-0.580008\pi$
$397$	−9.91116	−0.497427	−0.248714	+	0.968577i	$0.580008\pi$
$398$	0	0
$399$	2.42151	0.121227
$400$	0	0
$401$	19.1456	0.956088	0.478044	−	0.878336i	$-0.341346\pi$
$401$	19.1456	0.956088	0.478044	+	0.878336i	$0.341346\pi$
$402$	0	0
$403$	2.67669	0.133335
$404$	0	0
$405$	1.74483	0.0867011
$406$	0	0
$407$	9.91116	0.491278
$408$	0	0
$409$	−5.18070	−0.256169	−0.128085	−	0.991763i	$-0.540883\pi$
$409$	−5.18070	−0.256169	−0.128085	+	0.991763i	$0.540883\pi$
$410$	0	0
$411$	16.5878	0.818218
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	−2.36140	−0.115916
$416$	0	0
$417$	−3.21076	−0.157231
$418$	0	0
$419$	−14.0237	−0.685104	−0.342552	−	0.939499i	$-0.611291\pi$
$419$	−14.0237	−0.685104	−0.342552	+	0.939499i	$0.611291\pi$
$420$	0	0
$421$	27.0681	1.31922	0.659610	−	0.751608i	$-0.270721\pi$
$421$	27.0681	1.31922	0.659610	+	0.751608i	$0.270721\pi$
$422$	0	0
$423$	9.48965	0.461403
$424$	0	0
$425$	3.19203	0.154836
$426$	0	0
$427$	56.9379	2.75542
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	35.2044	1.69574	0.847869	−	0.530206i	$-0.177886\pi$
$431$	35.2044	1.69574	0.847869	+	0.530206i	$0.177886\pi$
$432$	0	0
$433$	23.9586	1.15138	0.575689	−	0.817669i	$-0.304734\pi$
$433$	23.9586	1.15138	0.575689	+	0.817669i	$0.304734\pi$
$434$	0	0
$435$	2.11256	0.101289
$436$	0	0
$437$	−4.06011	−0.194222
$438$	0	0
$439$	−0.260174	−0.0124174	−0.00620870	−	0.999981i	$-0.501976\pi$
$439$	−0.260174	−0.0124174	−0.00620870	+	0.999981i	$0.501976\pi$
$440$	0	0
$441$	13.5578	0.645609
$442$	0	0
$443$	−15.8223	−0.751741	−0.375871	−	0.926672i	$-0.622656\pi$
$443$	−15.8223	−0.751741	−0.375871	+	0.926672i	$0.622656\pi$
$444$	0	0
$445$	11.1520	0.528654
$446$	0	0
$447$	10.7592	0.508892
$448$	0	0
$449$	−26.3026	−1.24130	−0.620649	−	0.784089i	$-0.713131\pi$
$449$	−26.3026	−1.24130	−0.620649	+	0.784089i	$0.713131\pi$
$450$	0	0
$451$	−3.37709	−0.159021
$452$	0	0
$453$	−0.486625	−0.0228637
$454$	0	0
$455$	7.91116	0.370881
$456$	0	0
$457$	31.0681	1.45331	0.726653	−	0.687005i	$-0.241075\pi$
$457$	31.0681	1.45331	0.726653	+	0.687005i	$0.241075\pi$
$458$	0	0
$459$	1.63227	0.0761877
$460$	0	0
$461$	22.8066	1.06221	0.531105	−	0.847306i	$-0.321777\pi$
$461$	22.8066	1.06221	0.531105	+	0.847306i	$0.321777\pi$
$462$	0	0
$463$	−11.9887	−0.557161	−0.278580	−	0.960413i	$-0.589864\pi$
$463$	−11.9887	−0.557161	−0.278580	+	0.960413i	$0.589864\pi$
$464$	0	0
$465$	−4.67035	−0.216582
$466$	0	0
$467$	−12.4927	−0.578092	−0.289046	−	0.957315i	$-0.593338\pi$
$467$	−12.4927	−0.578092	−0.289046	+	0.957315i	$0.593338\pi$
$468$	0	0
$469$	57.0742	2.63544
$470$	0	0
$471$	−2.95558	−0.136186
$472$	0	0
$473$	3.63227	0.167012
$474$	0	0
$475$	1.04442	0.0479212
$476$	0	0
$477$	−3.57849	−0.163848
$478$	0	0
$479$	8.48663	0.387764	0.193882	−	0.981025i	$-0.437892\pi$
$479$	8.48663	0.387764	0.193882	+	0.981025i	$0.437892\pi$
$480$	0	0
$481$	−9.91116	−0.451910
$482$	0	0
$483$	−34.4690	−1.56839
$484$	0	0
$485$	−5.11559	−0.232287
$486$	0	0
$487$	35.1931	1.59475	0.797375	−	0.603483i	$-0.206221\pi$
$487$	35.1931	1.59475	0.797375	+	0.603483i	$0.206221\pi$
$488$	0	0
$489$	−23.8811	−1.07994
$490$	0	0
$491$	−4.08884	−0.184527	−0.0922633	−	0.995735i	$-0.529410\pi$
$491$	−4.08884	−0.184527	−0.0922633	+	0.995735i	$0.529410\pi$
$492$	0	0
$493$	1.97628	0.0890071
$494$	0	0
$495$	−1.74483	−0.0784241
$496$	0	0
$497$	−13.9112	−0.624001
$498$	0	0
$499$	9.92855	0.444463	0.222232	−	0.974994i	$-0.428666\pi$
$499$	9.92855	0.444463	0.222232	+	0.974994i	$0.428666\pi$
$500$	0	0
$501$	12.8905	0.575904
$502$	0	0
$503$	−37.8411	−1.68725	−0.843625	−	0.536934i	$-0.819583\pi$
$503$	−37.8411	−1.68725	−0.843625	+	0.536934i	$0.819583\pi$
$504$	0	0
$505$	−22.8955	−1.01883
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−10.3915	−0.460593	−0.230297	−	0.973120i	$-0.573970\pi$
$509$	−10.3915	−0.460593	−0.230297	+	0.973120i	$0.573970\pi$
$510$	0	0
$511$	22.4690	0.993968
$512$	0	0
$513$	0.534070	0.0235798
$514$	0	0
$515$	5.50837	0.242728
$516$	0	0
$517$	−9.48965	−0.417354
$518$	0	0
$519$	4.89680	0.214946
$520$	0	0
$521$	30.9379	1.35541	0.677707	−	0.735332i	$-0.262974\pi$
$521$	30.9379	1.35541	0.677707	+	0.735332i	$0.262974\pi$
$522$	0	0
$523$	−12.1901	−0.533034	−0.266517	−	0.963830i	$-0.585873\pi$
$523$	−12.1901	−0.533034	−0.266517	+	0.963830i	$0.585873\pi$
$524$	0	0
$525$	8.86675	0.386977
$526$	0	0
$527$	−4.36907	−0.190320
$528$	0	0
$529$	34.7936	1.51277
$530$	0	0
$531$	2.64663	0.114854
$532$	0	0
$533$	3.37709	0.146278
$534$	0	0
$535$	32.2251	1.39321
$536$	0	0
$537$	−9.79861	−0.422841
$538$	0	0
$539$	−13.5578	−0.583975
$540$	0	0
$541$	−37.8223	−1.62611	−0.813054	−	0.582188i	$-0.802197\pi$
$541$	−37.8223	−1.62611	−0.813054	+	0.582188i	$0.802197\pi$
$542$	0	0
$543$	−8.02372	−0.344331
$544$	0	0
$545$	11.9813	0.513222
$546$	0	0
$547$	−29.5622	−1.26399	−0.631993	−	0.774974i	$-0.717763\pi$
$547$	−29.5622	−1.26399	−0.631993	+	0.774974i	$0.717763\pi$
$548$	0	0
$549$	12.5578	0.535954
$550$	0	0
$551$	0.646629	0.0275473
$552$	0	0
$553$	7.80361	0.331843
$554$	0	0
$555$	17.2933	0.734058
$556$	0	0
$557$	−11.4245	−0.484073	−0.242037	−	0.970267i	$-0.577815\pi$
$557$	−11.4245	−0.484073	−0.242037	+	0.970267i	$0.577815\pi$
$558$	0	0
$559$	−3.63227	−0.153629
$560$	0	0
$561$	−1.63227	−0.0689144
$562$	0	0
$563$	−9.06814	−0.382177	−0.191088	−	0.981573i	$-0.561202\pi$
$563$	−9.06814	−0.382177	−0.191088	+	0.981573i	$0.561202\pi$
$564$	0	0
$565$	−1.12825	−0.0474660
$566$	0	0
$567$	4.53407	0.190413
$568$	0	0
$569$	21.6323	0.906872	0.453436	−	0.891289i	$-0.350198\pi$
$569$	21.6323	0.906872	0.453436	+	0.891289i	$0.350198\pi$
$570$	0	0
$571$	44.9730	1.88206	0.941030	−	0.338323i	$-0.109860\pi$
$571$	44.9730	1.88206	0.941030	+	0.338323i	$0.109860\pi$
$572$	0	0
$573$	−12.3327	−0.515205
$574$	0	0
$575$	−14.8667	−0.619986
$576$	0	0
$577$	9.20442	0.383185	0.191593	−	0.981475i	$-0.438635\pi$
$577$	9.20442	0.383185	0.191593	+	0.981475i	$0.438635\pi$
$578$	0	0
$579$	−8.33768	−0.346502
$580$	0	0
$581$	−6.13628	−0.254576
$582$	0	0
$583$	3.57849	0.148206
$584$	0	0
$585$	1.74483	0.0721397
$586$	0	0
$587$	33.6259	1.38789	0.693945	−	0.720028i	$-0.255871\pi$
$587$	33.6259	1.38789	0.693945	+	0.720028i	$0.255871\pi$
$588$	0	0
$589$	−1.42954	−0.0589031
$590$	0	0
$591$	3.04442	0.125231
$592$	0	0
$593$	20.3851	0.837117	0.418558	−	0.908190i	$-0.362536\pi$
$593$	20.3851	0.837117	0.418558	+	0.908190i	$0.362536\pi$
$594$	0	0
$595$	−12.9131	−0.529387
$596$	0	0
$597$	−6.13628	−0.251141
$598$	0	0
$599$	5.80361	0.237129	0.118564	−	0.992946i	$-0.462171\pi$
$599$	5.80361	0.237129	0.118564	+	0.992946i	$0.462171\pi$
$600$	0	0
$601$	−30.0949	−1.22760	−0.613798	−	0.789463i	$-0.710359\pi$
$601$	−30.0949	−1.22760	−0.613798	+	0.789463i	$0.710359\pi$
$602$	0	0
$603$	12.5878	0.512617
$604$	0	0
$605$	1.74483	0.0709373
$606$	0	0
$607$	4.84936	0.196829	0.0984147	−	0.995145i	$-0.468623\pi$
$607$	4.84936	0.196829	0.0984147	+	0.995145i	$0.468623\pi$
$608$	0	0
$609$	5.48965	0.222452
$610$	0	0
$611$	9.48965	0.383910
$612$	0	0
$613$	35.7572	1.44422	0.722110	−	0.691778i	$-0.243172\pi$
$613$	35.7572	1.44422	0.722110	+	0.691778i	$0.243172\pi$
$614$	0	0
$615$	−5.89244	−0.237606
$616$	0	0
$617$	6.63529	0.267127	0.133563	−	0.991040i	$-0.457358\pi$
$617$	6.63529	0.267127	0.133563	+	0.991040i	$0.457358\pi$
$618$	0	0
$619$	−32.1062	−1.29046	−0.645229	−	0.763989i	$-0.723238\pi$
$619$	−32.1062	−1.29046	−0.645229	+	0.763989i	$0.723238\pi$
$620$	0	0
$621$	−7.60221	−0.305066
$622$	0	0
$623$	28.9793	1.16103
$624$	0	0
$625$	−11.3978	−0.455912
$626$	0	0
$627$	−0.534070	−0.0213287
$628$	0	0
$629$	16.1777	0.645046
$630$	0	0
$631$	4.67669	0.186176	0.0930880	−	0.995658i	$-0.470326\pi$
$631$	4.67669	0.186176	0.0930880	+	0.995658i	$0.470326\pi$
$632$	0	0
$633$	−1.94622	−0.0773553
$634$	0	0
$635$	12.9743	0.514870
$636$	0	0
$637$	13.5578	0.537179
$638$	0	0
$639$	−3.06814	−0.121374
$640$	0	0
$641$	5.77488	0.228094	0.114047	−	0.993475i	$-0.463619\pi$
$641$	5.77488	0.228094	0.114047	+	0.993475i	$0.463619\pi$
$642$	0	0
$643$	36.0775	1.42276	0.711379	−	0.702809i	$-0.248071\pi$
$643$	36.0775	1.42276	0.711379	+	0.702809i	$0.248071\pi$
$644$	0	0
$645$	6.33768	0.249546
$646$	0	0
$647$	−23.7048	−0.931931	−0.465965	−	0.884803i	$-0.654293\pi$
$647$	−23.7048	−0.931931	−0.465965	+	0.884803i	$0.654293\pi$
$648$	0	0
$649$	−2.64663	−0.103889
$650$	0	0
$651$	−12.1363	−0.475658
$652$	0	0
$653$	−1.77488	−0.0694565	−0.0347283	−	0.999397i	$-0.511057\pi$
$653$	−1.77488	−0.0694565	−0.0347283	+	0.999397i	$0.511057\pi$
$654$	0	0
$655$	−26.5291	−1.03658
$656$	0	0
$657$	4.95558	0.193336
$658$	0	0
$659$	−23.4008	−0.911566	−0.455783	−	0.890091i	$-0.650641\pi$
$659$	−23.4008	−0.911566	−0.455783	+	0.890091i	$0.650641\pi$
$660$	0	0
$661$	23.7936	0.925464	0.462732	−	0.886498i	$-0.346869\pi$
$661$	23.7936	0.925464	0.462732	+	0.886498i	$0.346869\pi$
$662$	0	0
$663$	1.63227	0.0633920
$664$	0	0
$665$	−4.22512	−0.163843
$666$	0	0
$667$	−9.20442	−0.356397
$668$	0	0
$669$	11.7923	0.455916
$670$	0	0
$671$	−12.5578	−0.484788
$672$	0	0
$673$	22.9793	0.885787	0.442894	−	0.896574i	$-0.353952\pi$
$673$	22.9793	0.885787	0.442894	+	0.896574i	$0.353952\pi$
$674$	0	0
$675$	1.95558	0.0752704
$676$	0	0
$677$	40.8554	1.57020	0.785101	−	0.619368i	$-0.212611\pi$
$677$	40.8554	1.57020	0.785101	+	0.619368i	$0.212611\pi$
$678$	0	0
$679$	−13.2933	−0.510148
$680$	0	0
$681$	22.6704	0.868730
$682$	0	0
$683$	−7.24581	−0.277253	−0.138627	−	0.990345i	$-0.544269\pi$
$683$	−7.24581	−0.277253	−0.138627	+	0.990345i	$0.544269\pi$
$684$	0	0
$685$	−28.9429	−1.10585
$686$	0	0
$687$	8.51035	0.324690
$688$	0	0
$689$	−3.57849	−0.136330
$690$	0	0
$691$	−34.5177	−1.31312	−0.656558	−	0.754275i	$-0.727988\pi$
$691$	−34.5177	−1.31312	−0.656558	+	0.754275i	$0.727988\pi$
$692$	0	0
$693$	−4.53407	−0.172235
$694$	0	0
$695$	5.60221	0.212504
$696$	0	0
$697$	−5.51232	−0.208794
$698$	0	0
$699$	−21.4359	−0.810779
$700$	0	0
$701$	−29.2168	−1.10350	−0.551752	−	0.834008i	$-0.686040\pi$
$701$	−29.2168	−1.10350	−0.551752	+	0.834008i	$0.686040\pi$
$702$	0	0
$703$	5.29326	0.199639
$704$	0	0
$705$	−16.5578	−0.623603
$706$	0	0
$707$	−59.4957	−2.23757
$708$	0	0
$709$	17.8510	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$709$	17.8510	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$710$	0	0
$711$	1.72110	0.0645464
$712$	0	0
$713$	20.3487	0.762066
$714$	0	0
$715$	−1.74483	−0.0652528
$716$	0	0
$717$	7.91616	0.295635
$718$	0	0
$719$	−24.4502	−0.911840	−0.455920	−	0.890021i	$-0.650690\pi$
$719$	−24.4502	−0.911840	−0.455920	+	0.890021i	$0.650690\pi$
$720$	0	0
$721$	14.3140	0.533079
$722$	0	0
$723$	24.6941	0.918382
$724$	0	0
$725$	2.36773	0.0879354
$726$	0	0
$727$	−9.00803	−0.334089	−0.167045	−	0.985949i	$-0.553422\pi$
$727$	−9.00803	−0.334089	−0.167045	+	0.985949i	$0.553422\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.92883	0.219286
$732$	0	0
$733$	44.3851	1.63940	0.819701	−	0.572792i	$-0.194140\pi$
$733$	44.3851	1.63940	0.819701	+	0.572792i	$0.194140\pi$
$734$	0	0
$735$	−23.6560	−0.872564
$736$	0	0
$737$	−12.5878	−0.463679
$738$	0	0
$739$	−40.4513	−1.48802	−0.744012	−	0.668166i	$-0.767080\pi$
$739$	−40.4513	−1.48802	−0.744012	+	0.668166i	$0.767080\pi$
$740$	0	0
$741$	0.534070	0.0196196
$742$	0	0
$743$	−7.71477	−0.283027	−0.141514	−	0.989936i	$-0.545197\pi$
$743$	−7.71477	−0.283027	−0.141514	+	0.989936i	$0.545197\pi$
$744$	0	0
$745$	−18.7729	−0.687786
$746$	0	0
$747$	−1.35337	−0.0495173
$748$	0	0
$749$	83.7395	3.05978
$750$	0	0
$751$	−17.1283	−0.625019	−0.312509	−	0.949915i	$-0.601170\pi$
$751$	−17.1283	−0.625019	−0.312509	+	0.949915i	$0.601170\pi$
$752$	0	0
$753$	29.6259	1.07963
$754$	0	0
$755$	0.849077	0.0309011
$756$	0	0
$757$	−6.33768	−0.230347	−0.115173	−	0.993345i	$-0.536742\pi$
$757$	−6.33768	−0.230347	−0.115173	+	0.993345i	$0.536742\pi$
$758$	0	0
$759$	7.60221	0.275943
$760$	0	0
$761$	−1.40582	−0.0509608	−0.0254804	−	0.999675i	$-0.508112\pi$
$761$	−1.40582	−0.0509608	−0.0254804	+	0.999675i	$0.508112\pi$
$762$	0	0
$763$	31.1343	1.12714
$764$	0	0
$765$	−2.84802	−0.102970
$766$	0	0
$767$	2.64663	0.0955642
$768$	0	0
$769$	−14.2151	−0.512610	−0.256305	−	0.966596i	$-0.582505\pi$
$769$	−14.2151	−0.512610	−0.256305	+	0.966596i	$0.582505\pi$
$770$	0	0
$771$	−6.61791	−0.238338
$772$	0	0
$773$	−36.0174	−1.29546	−0.647728	−	0.761872i	$-0.724281\pi$
$773$	−36.0174	−1.29546	−0.647728	+	0.761872i	$0.724281\pi$
$774$	0	0
$775$	−5.23448	−0.188028
$776$	0	0
$777$	44.9379	1.61214
$778$	0	0
$779$	−1.80361	−0.0646209
$780$	0	0
$781$	3.06814	0.109787
$782$	0	0
$783$	1.21076	0.0432689
$784$	0	0
$785$	5.15698	0.184060
$786$	0	0
$787$	30.2676	1.07892	0.539461	−	0.842011i	$-0.318628\pi$
$787$	30.2676	1.07892	0.539461	+	0.842011i	$0.318628\pi$
$788$	0	0
$789$	27.7622	0.988361
$790$	0	0
$791$	−2.93186	−0.104245
$792$	0	0
$793$	12.5578	0.445940
$794$	0	0
$795$	6.24384	0.221446
$796$	0	0
$797$	−43.7809	−1.55080	−0.775400	−	0.631470i	$-0.782452\pi$
$797$	−43.7809	−1.55080	−0.775400	+	0.631470i	$0.782452\pi$
$798$	0	0
$799$	−15.4897	−0.547985
$800$	0	0
$801$	6.39145	0.225831
$802$	0	0
$803$	−4.95558	−0.174879
$804$	0	0
$805$	60.1423	2.11974
$806$	0	0
$807$	−7.57849	−0.266775
$808$	0	0
$809$	−33.6196	−1.18200	−0.591001	−	0.806671i	$-0.701267\pi$
$809$	−33.6196	−1.18200	−0.591001	+	0.806671i	$0.701267\pi$
$810$	0	0
$811$	24.6290	0.864840	0.432420	−	0.901672i	$-0.357660\pi$
$811$	24.6290	0.864840	0.432420	+	0.901672i	$0.357660\pi$
$812$	0	0
$813$	−19.1807	−0.672696
$814$	0	0
$815$	41.6684	1.45958
$816$	0	0
$817$	1.93989	0.0678680
$818$	0	0
$819$	4.53407	0.158433
$820$	0	0
$821$	−38.9142	−1.35811	−0.679057	−	0.734085i	$-0.737611\pi$
$821$	−38.9142	−1.35811	−0.679057	+	0.734085i	$0.737611\pi$
$822$	0	0
$823$	−30.6941	−1.06993	−0.534964	−	0.844875i	$-0.679675\pi$
$823$	−30.6941	−1.06993	−0.534964	+	0.844875i	$0.679675\pi$
$824$	0	0
$825$	−1.95558	−0.0680846
$826$	0	0
$827$	12.4038	0.431324	0.215662	−	0.976468i	$-0.430809\pi$
$827$	12.4038	0.431324	0.215662	+	0.976468i	$0.430809\pi$
$828$	0	0
$829$	37.4295	1.29998	0.649991	−	0.759942i	$-0.274773\pi$
$829$	37.4295	1.29998	0.649991	+	0.759942i	$0.274773\pi$
$830$	0	0
$831$	13.2044	0.458056
$832$	0	0
$833$	−22.1299	−0.766757
$834$	0	0
$835$	−22.4916	−0.778355
$836$	0	0
$837$	−2.67669	−0.0925198
$838$	0	0
$839$	−24.8905	−0.859314	−0.429657	−	0.902992i	$-0.641366\pi$
$839$	−24.8905	−0.859314	−0.429657	+	0.902992i	$0.641366\pi$
$840$	0	0
$841$	−27.5341	−0.949451
$842$	0	0
$843$	−19.7098	−0.678841
$844$	0	0
$845$	1.74483	0.0600238
$846$	0	0
$847$	4.53407	0.155792
$848$	0	0
$849$	−26.1299	−0.896777
$850$	0	0
$851$	−75.3468	−2.58285
$852$	0	0
$853$	−27.1630	−0.930044	−0.465022	−	0.885299i	$-0.653954\pi$
$853$	−27.1630	−0.930044	−0.465022	+	0.885299i	$0.653954\pi$
$854$	0	0
$855$	−0.931860	−0.0318689
$856$	0	0
$857$	15.0044	0.512539	0.256270	−	0.966605i	$-0.417507\pi$
$857$	15.0044	0.512539	0.256270	+	0.966605i	$0.417507\pi$
$858$	0	0
$859$	−54.1236	−1.84667	−0.923337	−	0.383991i	$-0.874549\pi$
$859$	−54.1236	−1.84667	−0.923337	+	0.383991i	$0.874549\pi$
$860$	0	0
$861$	−15.3120	−0.521831
$862$	0	0
$863$	26.2438	0.893351	0.446675	−	0.894696i	$-0.352608\pi$
$863$	26.2438	0.893351	0.446675	+	0.894696i	$0.352608\pi$
$864$	0	0
$865$	−8.54407	−0.290507
$866$	0	0
$867$	14.3357	0.486866
$868$	0	0
$869$	−1.72110	−0.0583844
$870$	0	0
$871$	12.5878	0.426523
$872$	0	0
$873$	−2.93186	−0.0992284
$874$	0	0
$875$	−55.0267	−1.86024
$876$	0	0
$877$	8.02872	0.271111	0.135555	−	0.990770i	$-0.456718\pi$
$877$	8.02872	0.271111	0.135555	+	0.990770i	$0.456718\pi$
$878$	0	0
$879$	−5.97628	−0.201575
$880$	0	0
$881$	−29.1944	−0.983585	−0.491793	−	0.870712i	$-0.663658\pi$
$881$	−29.1944	−0.983585	−0.491793	+	0.870712i	$0.663658\pi$
$882$	0	0
$883$	11.8223	0.397853	0.198927	−	0.980014i	$-0.436254\pi$
$883$	11.8223	0.397853	0.198927	+	0.980014i	$0.436254\pi$
$884$	0	0
$885$	−4.61791	−0.155229
$886$	0	0
$887$	52.1236	1.75014	0.875070	−	0.483997i	$-0.160815\pi$
$887$	52.1236	1.75014	0.875070	+	0.483997i	$0.160815\pi$
$888$	0	0
$889$	33.7148	1.13076
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−5.06814	−0.169599
$894$	0	0
$895$	17.0969	0.571485
$896$	0	0
$897$	−7.60221	−0.253830
$898$	0	0
$899$	−3.24081	−0.108087
$900$	0	0
$901$	5.84105	0.194594
$902$	0	0
$903$	16.4690	0.548053
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	−56.6527	−1.88112	−0.940561	−	0.339626i	$-0.889700\pi$
$907$	−56.6527	−1.88112	−0.940561	+	0.339626i	$0.889700\pi$
$908$	0	0
$909$	−13.1219	−0.435227
$910$	0	0
$911$	24.4276	0.809321	0.404661	−	0.914467i	$-0.367390\pi$
$911$	24.4276	0.809321	0.404661	+	0.914467i	$0.367390\pi$
$912$	0	0
$913$	1.35337	0.0447901
$914$	0	0
$915$	−21.9112	−0.724361
$916$	0	0
$917$	−68.9379	−2.27653
$918$	0	0
$919$	−33.0144	−1.08904	−0.544522	−	0.838747i	$-0.683289\pi$
$919$	−33.0144	−1.08904	−0.544522	+	0.838747i	$0.683289\pi$
$920$	0	0
$921$	−13.3771	−0.440790
$922$	0	0
$923$	−3.06814	−0.100989
$924$	0	0
$925$	19.3821	0.637279
$926$	0	0
$927$	3.15698	0.103689
$928$	0	0
$929$	−41.9286	−1.37563	−0.687816	−	0.725885i	$-0.741430\pi$
$929$	−41.9286	−1.37563	−0.687816	+	0.725885i	$0.741430\pi$
$930$	0	0
$931$	−7.24081	−0.237308
$932$	0	0
$933$	25.5608	0.836824
$934$	0	0
$935$	2.84802	0.0931403
$936$	0	0
$937$	−38.3614	−1.25321	−0.626606	−	0.779336i	$-0.715556\pi$
$937$	−38.3614	−1.25321	−0.626606	+	0.779336i	$0.715556\pi$
$938$	0	0
$939$	−29.1994	−0.952887
$940$	0	0
$941$	32.2776	1.05222	0.526109	−	0.850417i	$-0.323650\pi$
$941$	32.2776	1.05222	0.526109	+	0.850417i	$0.323650\pi$
$942$	0	0
$943$	25.6734	0.836040
$944$	0	0
$945$	−7.91116	−0.257350
$946$	0	0
$947$	0.243840	0.00792372	0.00396186	−	0.999992i	$-0.498739\pi$
$947$	0.243840	0.00792372	0.00396186	+	0.999992i	$0.498739\pi$
$948$	0	0
$949$	4.95558	0.160865
$950$	0	0
$951$	−9.94122	−0.322366
$952$	0	0
$953$	40.0725	1.29808	0.649038	−	0.760756i	$-0.275172\pi$
$953$	40.0725	1.29808	0.649038	+	0.760756i	$0.275172\pi$
$954$	0	0
$955$	21.5184	0.696318
$956$	0	0
$957$	−1.21076	−0.0391382
$958$	0	0
$959$	−75.2105	−2.42867
$960$	0	0
$961$	−23.8354	−0.768882
$962$	0	0
$963$	18.4690	0.595154
$964$	0	0
$965$	14.5478	0.468310
$966$	0	0
$967$	51.5007	1.65615	0.828076	−	0.560617i	$-0.189436\pi$
$967$	51.5007	1.65615	0.828076	+	0.560617i	$0.189436\pi$
$968$	0	0
$969$	−0.871746	−0.0280045
$970$	0	0
$971$	−2.97430	−0.0954500	−0.0477250	−	0.998861i	$-0.515197\pi$
$971$	−2.97430	−0.0954500	−0.0477250	+	0.998861i	$0.515197\pi$
$972$	0	0
$973$	14.5578	0.466701
$974$	0	0
$975$	1.95558	0.0626287
$976$	0	0
$977$	−14.2966	−0.457388	−0.228694	−	0.973498i	$-0.573445\pi$
$977$	−14.2966	−0.457388	−0.228694	+	0.973498i	$0.573445\pi$
$978$	0	0
$979$	−6.39145	−0.204272
$980$	0	0
$981$	6.86675	0.219238
$982$	0	0
$983$	−4.61791	−0.147288	−0.0736442	−	0.997285i	$-0.523463\pi$
$983$	−4.61791	−0.147288	−0.0736442	+	0.997285i	$0.523463\pi$
$984$	0	0
$985$	−5.31198	−0.169254
$986$	0	0
$987$	−43.0267	−1.36956
$988$	0	0
$989$	−27.6133	−0.878051
$990$	0	0
$991$	32.1550	1.02144	0.510719	−	0.859748i	$-0.329379\pi$
$991$	32.1550	1.02144	0.510719	+	0.859748i	$0.329379\pi$
$992$	0	0
$993$	6.87308	0.218111
$994$	0	0
$995$	10.7067	0.339427
$996$	0	0
$997$	−50.6052	−1.60268	−0.801342	−	0.598207i	$-0.795880\pi$
$997$	−50.6052	−1.60268	−0.801342	+	0.598207i	$0.795880\pi$
$998$	0	0
$999$	9.91116	0.313575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bn.1.3		3
4.3	odd	2		3432.2.a.n.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.n.1.3	✓	3	4.3	odd	2
6864.2.a.bn.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.74483 q^{5} +4.53407 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.74483 q^{5} +4.53407 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.74483 q^{15} -1.63227 q^{17} -0.534070 q^{19} -4.53407 q^{21} +7.60221 q^{23} -1.95558 q^{25} -1.00000 q^{27} -1.21076 q^{29} +2.67669 q^{31} +1.00000 q^{33} +7.91116 q^{35} -9.91116 q^{37} -1.00000 q^{39} +3.37709 q^{41} -3.63227 q^{43} +1.74483 q^{45} +9.48965 q^{47} +13.5578 q^{49} +1.63227 q^{51} -3.57849 q^{53} -1.74483 q^{55} +0.534070 q^{57} +2.64663 q^{59} +12.5578 q^{61} +4.53407 q^{63} +1.74483 q^{65} +12.5878 q^{67} -7.60221 q^{69} -3.06814 q^{71} +4.95558 q^{73} +1.95558 q^{75} -4.53407 q^{77} +1.72110 q^{79} +1.00000 q^{81} -1.35337 q^{83} -2.84802 q^{85} +1.21076 q^{87} +6.39145 q^{89} +4.53407 q^{91} -2.67669 q^{93} -0.931860 q^{95} -2.93186 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 4 q^{15} + 6 q^{19} - 6 q^{21} + 5 q^{25} - 3 q^{27} - 2 q^{29} + 14 q^{31} + 3 q^{33} + 2 q^{35} - 8 q^{37} - 3 q^{39} - 4 q^{41} - 6 q^{43} - 4 q^{45} + 10 q^{47} + 7 q^{49} - 14 q^{53} + 4 q^{55} - 6 q^{57} - 4 q^{59} + 4 q^{61} + 6 q^{63} - 4 q^{65} + 22 q^{67} + 6 q^{71} + 4 q^{73} - 5 q^{75} - 6 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} - 14 q^{85} + 2 q^{87} - 2 q^{89} + 6 q^{91} - 14 q^{93} - 18 q^{95} - 24 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 4 * q^15 + 6 * q^19 - 6 * q^21 + 5 * q^25 - 3 * q^27 - 2 * q^29 + 14 * q^31 + 3 * q^33 + 2 * q^35 - 8 * q^37 - 3 * q^39 - 4 * q^41 - 6 * q^43 - 4 * q^45 + 10 * q^47 + 7 * q^49 - 14 * q^53 + 4 * q^55 - 6 * q^57 - 4 * q^59 + 4 * q^61 + 6 * q^63 - 4 * q^65 + 22 * q^67 + 6 * q^71 + 4 * q^73 - 5 * q^75 - 6 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 - 14 * q^85 + 2 * q^87 - 2 * q^89 + 6 * q^91 - 14 * q^93 - 18 * q^95 - 24 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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