LMFDB - Embedded newform 6160.2.a.bq.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.25488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 6x + 6 $ x^4 - 9x^2 - 6x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.20740$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} +0.347982 q^{13} +0.732051 q^{15} +2.60272 q^{17} -6.03074 q^{19} +0.732051 q^{21} +1.08003 q^{23} +1.00000 q^{25} -4.00000 q^{27} -9.49484 q^{29} +4.76279 q^{31} -0.732051 q^{33} +1.00000 q^{35} -4.28548 q^{37} +0.254741 q^{39} +9.84282 q^{41} -2.69596 q^{43} -2.46410 q^{45} -4.86138 q^{47} +1.00000 q^{49} +1.90533 q^{51} -3.77600 q^{53} -1.00000 q^{55} -4.41481 q^{57} -2.18792 q^{59} -0.883881 q^{61} -2.46410 q^{63} +0.347982 q^{65} -8.95071 q^{67} +0.790639 q^{69} -5.46410 q^{71} +3.29869 q^{73} +0.732051 q^{75} -1.00000 q^{77} -6.54413 q^{79} +4.46410 q^{81} -4.76814 q^{83} +2.60272 q^{85} -6.95071 q^{87} -3.46410 q^{89} +0.347982 q^{91} +3.48660 q^{93} -6.03074 q^{95} +2.12933 q^{97} +2.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} - 4 q^{21} - 6 q^{23} + 4 q^{25} - 16 q^{27} - 2 q^{29} - 10 q^{31} + 4 q^{33} + 4 q^{35} + 10 q^{37} - 4 q^{39} - 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{49} - 8 q^{51} + 2 q^{53} - 4 q^{55} + 8 q^{57} - 26 q^{59} - 14 q^{61} + 4 q^{63} - 2 q^{65} - 24 q^{67} + 12 q^{69} - 8 q^{71} - 2 q^{73} - 4 q^{75} - 4 q^{77} - 2 q^{79} + 4 q^{81} - 12 q^{83} + 2 q^{85} - 16 q^{87} - 2 q^{91} + 16 q^{93} - 2 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 - 4 * q^21 - 6 * q^23 + 4 * q^25 - 16 * q^27 - 2 * q^29 - 10 * q^31 + 4 * q^33 + 4 * q^35 + 10 * q^37 - 4 * q^39 - 4 * q^43 + 4 * q^45 - 14 * q^47 + 4 * q^49 - 8 * q^51 + 2 * q^53 - 4 * q^55 + 8 * q^57 - 26 * q^59 - 14 * q^61 + 4 * q^63 - 2 * q^65 - 24 * q^67 + 12 * q^69 - 8 * q^71 - 2 * q^73 - 4 * q^75 - 4 * q^77 - 2 * q^79 + 4 * q^81 - 12 * q^83 + 2 * q^85 - 16 * q^87 - 2 * q^91 + 16 * q^93 - 2 * q^95 + 10 * q^97 - 4 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.347982	0.0965129	0.0482565	−	0.998835i	$-0.484634\pi$
$13$	0.347982	0.0965129	0.0482565	+	0.998835i	$0.484634\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	2.60272	0.631253	0.315627	−	0.948884i	$-0.397785\pi$
$17$	2.60272	0.631253	0.315627	+	0.948884i	$0.397785\pi$
$18$	0	0
$19$	−6.03074	−1.38355	−0.691773	−	0.722115i	$-0.743170\pi$
$19$	−6.03074	−1.38355	−0.691773	+	0.722115i	$0.743170\pi$
$20$	0	0
$21$	0.732051	0.159747
$22$	0	0
$23$	1.08003	0.225202	0.112601	−	0.993640i	$-0.464082\pi$
$23$	1.08003	0.225202	0.112601	+	0.993640i	$0.464082\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−9.49484	−1.76315	−0.881574	−	0.472046i	$-0.843515\pi$
$29$	−9.49484	−1.76315	−0.881574	+	0.472046i	$0.843515\pi$
$30$	0	0
$31$	4.76279	0.855422	0.427711	−	0.903916i	$-0.359320\pi$
$31$	4.76279	0.855422	0.427711	+	0.903916i	$0.359320\pi$
$32$	0	0
$33$	−0.732051	−0.127434
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−4.28548	−0.704528	−0.352264	−	0.935901i	$-0.614588\pi$
$37$	−4.28548	−0.704528	−0.352264	+	0.935901i	$0.614588\pi$
$38$	0	0
$39$	0.254741	0.0407912
$40$	0	0
$41$	9.84282	1.53719	0.768595	−	0.639735i	$-0.220956\pi$
$41$	9.84282	1.53719	0.768595	+	0.639735i	$0.220956\pi$
$42$	0	0
$43$	−2.69596	−0.411131	−0.205565	−	0.978643i	$-0.565903\pi$
$43$	−2.69596	−0.411131	−0.205565	+	0.978643i	$0.565903\pi$
$44$	0	0
$45$	−2.46410	−0.367327
$46$	0	0
$47$	−4.86138	−0.709105	−0.354552	−	0.935036i	$-0.615367\pi$
$47$	−4.86138	−0.709105	−0.354552	+	0.935036i	$0.615367\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.90533	0.266799
$52$	0	0
$53$	−3.77600	−0.518673	−0.259337	−	0.965787i	$-0.583504\pi$
$53$	−3.77600	−0.518673	−0.259337	+	0.965787i	$0.583504\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−4.41481	−0.584755
$58$	0	0
$59$	−2.18792	−0.284842	−0.142421	−	0.989806i	$-0.545489\pi$
$59$	−2.18792	−0.284842	−0.142421	+	0.989806i	$0.545489\pi$
$60$	0	0
$61$	−0.883881	−0.113169	−0.0565847	−	0.998398i	$-0.518021\pi$
$61$	−0.883881	−0.113169	−0.0565847	+	0.998398i	$0.518021\pi$
$62$	0	0
$63$	−2.46410	−0.310448
$64$	0	0
$65$	0.347982	0.0431619
$66$	0	0
$67$	−8.95071	−1.09350	−0.546751	−	0.837295i	$-0.684136\pi$
$67$	−8.95071	−1.09350	−0.546751	+	0.837295i	$0.684136\pi$
$68$	0	0
$69$	0.790639	0.0951818
$70$	0	0
$71$	−5.46410	−0.648470	−0.324235	−	0.945977i	$-0.605107\pi$
$71$	−5.46410	−0.648470	−0.324235	+	0.945977i	$0.605107\pi$
$72$	0	0
$73$	3.29869	0.386082	0.193041	−	0.981191i	$-0.438165\pi$
$73$	3.29869	0.386082	0.193041	+	0.981191i	$0.438165\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−6.54413	−0.736273	−0.368136	−	0.929772i	$-0.620004\pi$
$79$	−6.54413	−0.736273	−0.368136	+	0.929772i	$0.620004\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−4.76814	−0.523371	−0.261686	−	0.965153i	$-0.584278\pi$
$83$	−4.76814	−0.523371	−0.261686	+	0.965153i	$0.584278\pi$
$84$	0	0
$85$	2.60272	0.282305
$86$	0	0
$87$	−6.95071	−0.745194
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	0.347982	0.0364784
$92$	0	0
$93$	3.48660	0.361544
$94$	0	0
$95$	−6.03074	−0.618741
$96$	0	0
$97$	2.12933	0.216200	0.108100	−	0.994140i	$-0.465523\pi$
$97$	2.12933	0.216200	0.108100	+	0.994140i	$0.465523\pi$
$98$	0	0
$99$	2.46410	0.247652
$100$	0	0
$101$	1.01753	0.101248	0.0506240	−	0.998718i	$-0.483879\pi$
$101$	1.01753	0.101248	0.0506240	+	0.998718i	$0.483879\pi$
$102$	0	0
$103$	−8.22689	−0.810620	−0.405310	−	0.914179i	$-0.632836\pi$
$103$	−8.22689	−0.810620	−0.405310	+	0.914179i	$0.632836\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	−17.1829	−1.66114	−0.830569	−	0.556916i	$-0.811985\pi$
$107$	−17.1829	−1.66114	−0.830569	+	0.556916i	$0.811985\pi$
$108$	0	0
$109$	4.72670	0.452736	0.226368	−	0.974042i	$-0.427315\pi$
$109$	4.72670	0.452736	0.226368	+	0.974042i	$0.427315\pi$
$110$	0	0
$111$	−3.13719	−0.297769
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	1.08003	0.100714
$116$	0	0
$117$	−0.857464	−0.0792725
$118$	0	0
$119$	2.60272	0.238591
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.20545	0.649693
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.92820	−0.614779	−0.307389	−	0.951584i	$-0.599455\pi$
$127$	−6.92820	−0.614779	−0.307389	+	0.951584i	$0.599455\pi$
$128$	0	0
$129$	−1.97358	−0.173764
$130$	0	0
$131$	10.7989	0.943502	0.471751	−	0.881732i	$-0.343622\pi$
$131$	10.7989	0.943502	0.471751	+	0.881732i	$0.343622\pi$
$132$	0	0
$133$	−6.03074	−0.522931
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	6.25474	0.534379	0.267189	−	0.963644i	$-0.413905\pi$
$137$	6.25474	0.534379	0.267189	+	0.963644i	$0.413905\pi$
$138$	0	0
$139$	0.566637	0.0480615	0.0240307	−	0.999711i	$-0.492350\pi$
$139$	0.566637	0.0480615	0.0240307	+	0.999711i	$0.492350\pi$
$140$	0	0
$141$	−3.55878	−0.299703
$142$	0	0
$143$	−0.347982	−0.0290997
$144$	0	0
$145$	−9.49484	−0.788504
$146$	0	0
$147$	0.732051	0.0603785
$148$	0	0
$149$	1.87067	0.153251	0.0766257	−	0.997060i	$-0.475585\pi$
$149$	1.87067	0.153251	0.0766257	+	0.997060i	$0.475585\pi$
$150$	0	0
$151$	1.00786	0.0820185	0.0410093	−	0.999159i	$-0.486943\pi$
$151$	1.00786	0.0820185	0.0410093	+	0.999159i	$0.486943\pi$
$152$	0	0
$153$	−6.41337	−0.518491
$154$	0	0
$155$	4.76279	0.382556
$156$	0	0
$157$	−4.49089	−0.358412	−0.179206	−	0.983812i	$-0.557353\pi$
$157$	−4.49089	−0.358412	−0.179206	+	0.983812i	$0.557353\pi$
$158$	0	0
$159$	−2.76422	−0.219217
$160$	0	0
$161$	1.08003	0.0851185
$162$	0	0
$163$	−11.3694	−0.890522	−0.445261	−	0.895401i	$-0.646889\pi$
$163$	−11.3694	−0.890522	−0.445261	+	0.895401i	$0.646889\pi$
$164$	0	0
$165$	−0.732051	−0.0569901
$166$	0	0
$167$	−10.2322	−0.791794	−0.395897	−	0.918295i	$-0.629566\pi$
$167$	−10.2322	−0.791794	−0.395897	+	0.918295i	$0.629566\pi$
$168$	0	0
$169$	−12.8789	−0.990685
$170$	0	0
$171$	14.8604	1.13640
$172$	0	0
$173$	−7.73600	−0.588157	−0.294078	−	0.955781i	$-0.595013\pi$
$173$	−7.73600	−0.588157	−0.294078	+	0.955781i	$0.595013\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−1.60167	−0.120389
$178$	0	0
$179$	16.4763	1.23150	0.615748	−	0.787943i	$-0.288854\pi$
$179$	16.4763	1.23150	0.615748	+	0.787943i	$0.288854\pi$
$180$	0	0
$181$	5.95500	0.442631	0.221316	−	0.975202i	$-0.428965\pi$
$181$	5.95500	0.442631	0.221316	+	0.975202i	$0.428965\pi$
$182$	0	0
$183$	−0.647046	−0.0478310
$184$	0	0
$185$	−4.28548	−0.315075
$186$	0	0
$187$	−2.60272	−0.190330
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−19.2483	−1.39276	−0.696380	−	0.717673i	$-0.745207\pi$
$191$	−19.2483	−1.39276	−0.696380	+	0.717673i	$0.745207\pi$
$192$	0	0
$193$	9.08827	0.654188	0.327094	−	0.944992i	$-0.393931\pi$
$193$	9.08827	0.654188	0.327094	+	0.944992i	$0.393931\pi$
$194$	0	0
$195$	0.254741	0.0182424
$196$	0	0
$197$	17.2444	1.22861	0.614307	−	0.789067i	$-0.289436\pi$
$197$	17.2444	1.22861	0.614307	+	0.789067i	$0.289436\pi$
$198$	0	0
$199$	−1.85709	−0.131645	−0.0658227	−	0.997831i	$-0.520967\pi$
$199$	−1.85709	−0.131645	−0.0658227	+	0.997831i	$0.520967\pi$
$200$	0	0
$201$	−6.55237	−0.462169
$202$	0	0
$203$	−9.49484	−0.666407
$204$	0	0
$205$	9.84282	0.687453
$206$	0	0
$207$	−2.66131	−0.184974
$208$	0	0
$209$	6.03074	0.417155
$210$	0	0
$211$	−27.5606	−1.89735	−0.948676	−	0.316249i	$-0.897577\pi$
$211$	−27.5606	−1.89735	−0.948676	+	0.316249i	$0.897577\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	−2.69596	−0.183863
$216$	0	0
$217$	4.76279	0.323319
$218$	0	0
$219$	2.41481	0.163178
$220$	0	0
$221$	0.905701	0.0609241
$222$	0	0
$223$	−9.25368	−0.619672	−0.309836	−	0.950790i	$-0.600274\pi$
$223$	−9.25368	−0.619672	−0.309836	+	0.950790i	$0.600274\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	−22.2215	−1.47490	−0.737448	−	0.675404i	$-0.763969\pi$
$227$	−22.2215	−1.47490	−0.737448	+	0.675404i	$0.763969\pi$
$228$	0	0
$229$	24.7700	1.63685	0.818424	−	0.574615i	$-0.194848\pi$
$229$	24.7700	1.63685	0.818424	+	0.574615i	$0.194848\pi$
$230$	0	0
$231$	−0.732051	−0.0481654
$232$	0	0
$233$	−18.3884	−1.20466	−0.602332	−	0.798246i	$-0.705762\pi$
$233$	−18.3884	−1.20466	−0.602332	+	0.798246i	$0.705762\pi$
$234$	0	0
$235$	−4.86138	−0.317121
$236$	0	0
$237$	−4.79064	−0.311185
$238$	0	0
$239$	1.77208	0.114626	0.0573132	−	0.998356i	$-0.481747\pi$
$239$	1.77208	0.114626	0.0573132	+	0.998356i	$0.481747\pi$
$240$	0	0
$241$	−24.9497	−1.60715	−0.803575	−	0.595203i	$-0.797071\pi$
$241$	−24.9497	−1.60715	−0.803575	+	0.595203i	$0.797071\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.09859	−0.133530
$248$	0	0
$249$	−3.49052	−0.221203
$250$	0	0
$251$	−20.7853	−1.31196	−0.655978	−	0.754780i	$-0.727744\pi$
$251$	−20.7853	−1.31196	−0.655978	+	0.754780i	$0.727744\pi$
$252$	0	0
$253$	−1.08003	−0.0679011
$254$	0	0
$255$	1.90533	0.119316
$256$	0	0
$257$	16.4230	1.02444	0.512221	−	0.858854i	$-0.328823\pi$
$257$	16.4230	1.02444	0.512221	+	0.858854i	$0.328823\pi$
$258$	0	0
$259$	−4.28548	−0.266287
$260$	0	0
$261$	23.3963	1.44819
$262$	0	0
$263$	6.09859	0.376055	0.188028	−	0.982164i	$-0.439791\pi$
$263$	6.09859	0.376055	0.188028	+	0.982164i	$0.439791\pi$
$264$	0	0
$265$	−3.77600	−0.231958
$266$	0	0
$267$	−2.53590	−0.155194
$268$	0	0
$269$	−21.7081	−1.32357	−0.661785	−	0.749694i	$-0.730201\pi$
$269$	−21.7081	−1.32357	−0.661785	+	0.749694i	$0.730201\pi$
$270$	0	0
$271$	25.2219	1.53212	0.766061	−	0.642768i	$-0.222214\pi$
$271$	25.2219	1.53212	0.766061	+	0.642768i	$0.222214\pi$
$272$	0	0
$273$	0.254741	0.0154176
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−2.61988	−0.157413	−0.0787066	−	0.996898i	$-0.525079\pi$
$277$	−2.61988	−0.157413	−0.0787066	+	0.996898i	$0.525079\pi$
$278$	0	0
$279$	−11.7360	−0.702616
$280$	0	0
$281$	15.2219	0.908064	0.454032	−	0.890986i	$-0.349985\pi$
$281$	15.2219	0.908064	0.454032	+	0.890986i	$0.349985\pi$
$282$	0	0
$283$	11.2933	0.671319	0.335660	−	0.941983i	$-0.391041\pi$
$283$	11.2933	0.671319	0.335660	+	0.941983i	$0.391041\pi$
$284$	0	0
$285$	−4.41481	−0.261511
$286$	0	0
$287$	9.84282	0.581003
$288$	0	0
$289$	−10.2258	−0.601520
$290$	0	0
$291$	1.55878	0.0913771
$292$	0	0
$293$	21.1326	1.23458	0.617290	−	0.786736i	$-0.288231\pi$
$293$	21.1326	1.23458	0.617290	+	0.786736i	$0.288231\pi$
$294$	0	0
$295$	−2.18792	−0.127385
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	0.375832	0.0217349
$300$	0	0
$301$	−2.69596	−0.155393
$302$	0	0
$303$	0.744883	0.0427924
$304$	0	0
$305$	−0.883881	−0.0506109
$306$	0	0
$307$	12.1951	0.696013	0.348006	−	0.937492i	$-0.386859\pi$
$307$	12.1951	0.696013	0.348006	+	0.937492i	$0.386859\pi$
$308$	0	0
$309$	−6.02250	−0.342608
$310$	0	0
$311$	0.397653	0.0225488	0.0112744	−	0.999936i	$-0.496411\pi$
$311$	0.397653	0.0225488	0.0112744	+	0.999936i	$0.496411\pi$
$312$	0	0
$313$	16.0922	0.909586	0.454793	−	0.890597i	$-0.349713\pi$
$313$	16.0922	0.909586	0.454793	+	0.890597i	$0.349713\pi$
$314$	0	0
$315$	−2.46410	−0.138836
$316$	0	0
$317$	1.87067	0.105067	0.0525337	−	0.998619i	$-0.483270\pi$
$317$	1.87067	0.105067	0.0525337	+	0.998619i	$0.483270\pi$
$318$	0	0
$319$	9.49484	0.531609
$320$	0	0
$321$	−12.5788	−0.702079
$322$	0	0
$323$	−15.6963	−0.873368
$324$	0	0
$325$	0.347982	0.0193026
$326$	0	0
$327$	3.46019	0.191349
$328$	0	0
$329$	−4.86138	−0.268016
$330$	0	0
$331$	−3.64667	−0.200439	−0.100220	−	0.994965i	$-0.531955\pi$
$331$	−3.64667	−0.200439	−0.100220	+	0.994965i	$0.531955\pi$
$332$	0	0
$333$	10.5599	0.578677
$334$	0	0
$335$	−8.95071	−0.489029
$336$	0	0
$337$	5.51377	0.300354	0.150177	−	0.988659i	$-0.452016\pi$
$337$	5.51377	0.300354	0.150177	+	0.988659i	$0.452016\pi$
$338$	0	0
$339$	−0.679492	−0.0369049
$340$	0	0
$341$	−4.76279	−0.257919
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.790639	0.0425666
$346$	0	0
$347$	−5.66917	−0.304337	−0.152169	−	0.988355i	$-0.548626\pi$
$347$	−5.66917	−0.304337	−0.152169	+	0.988355i	$0.548626\pi$
$348$	0	0
$349$	3.88426	0.207919	0.103960	−	0.994582i	$-0.466849\pi$
$349$	3.88426	0.207919	0.103960	+	0.994582i	$0.466849\pi$
$350$	0	0
$351$	−1.39193	−0.0742957
$352$	0	0
$353$	6.19080	0.329503	0.164752	−	0.986335i	$-0.447318\pi$
$353$	6.19080	0.329503	0.164752	+	0.986335i	$0.447318\pi$
$354$	0	0
$355$	−5.46410	−0.290004
$356$	0	0
$357$	1.90533	0.100841
$358$	0	0
$359$	15.8218	0.835040	0.417520	−	0.908668i	$-0.362899\pi$
$359$	15.8218	0.835040	0.417520	+	0.908668i	$0.362899\pi$
$360$	0	0
$361$	17.3698	0.914200
$362$	0	0
$363$	0.732051	0.0384227
$364$	0	0
$365$	3.29869	0.172661
$366$	0	0
$367$	18.1283	0.946290	0.473145	−	0.880985i	$-0.343119\pi$
$367$	18.1283	0.946290	0.473145	+	0.880985i	$0.343119\pi$
$368$	0	0
$369$	−24.2537	−1.26260
$370$	0	0
$371$	−3.77600	−0.196040
$372$	0	0
$373$	6.51340	0.337251	0.168625	−	0.985680i	$-0.446067\pi$
$373$	6.51340	0.337251	0.168625	+	0.985680i	$0.446067\pi$
$374$	0	0
$375$	0.732051	0.0378029
$376$	0	0
$377$	−3.30404	−0.170166
$378$	0	0
$379$	−29.4806	−1.51432	−0.757158	−	0.653232i	$-0.773413\pi$
$379$	−29.4806	−1.51432	−0.757158	+	0.653232i	$0.773413\pi$
$380$	0	0
$381$	−5.07180	−0.259836
$382$	0	0
$383$	−3.21079	−0.164064	−0.0820320	−	0.996630i	$-0.526141\pi$
$383$	−3.21079	−0.164064	−0.0820320	+	0.996630i	$0.526141\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	6.64313	0.337689
$388$	0	0
$389$	−15.8399	−0.803117	−0.401558	−	0.915833i	$-0.631531\pi$
$389$	−15.8399	−0.803117	−0.401558	+	0.915833i	$0.631531\pi$
$390$	0	0
$391$	2.81103	0.142160
$392$	0	0
$393$	7.90533	0.398771
$394$	0	0
$395$	−6.54413	−0.329271
$396$	0	0
$397$	24.8032	1.24484	0.622418	−	0.782685i	$-0.286150\pi$
$397$	24.8032	1.24484	0.622418	+	0.782685i	$0.286150\pi$
$398$	0	0
$399$	−4.41481	−0.221017
$400$	0	0
$401$	−8.31622	−0.415292	−0.207646	−	0.978204i	$-0.566580\pi$
$401$	−8.31622	−0.415292	−0.207646	+	0.978204i	$0.566580\pi$
$402$	0	0
$403$	1.65737	0.0825593
$404$	0	0
$405$	4.46410	0.221823
$406$	0	0
$407$	4.28548	0.212423
$408$	0	0
$409$	−10.9868	−0.543262	−0.271631	−	0.962401i	$-0.587563\pi$
$409$	−10.9868	−0.543262	−0.271631	+	0.962401i	$0.587563\pi$
$410$	0	0
$411$	4.57879	0.225855
$412$	0	0
$413$	−2.18792	−0.107660
$414$	0	0
$415$	−4.76814	−0.234059
$416$	0	0
$417$	0.414807	0.0203132
$418$	0	0
$419$	10.8282	0.528991	0.264496	−	0.964387i	$-0.414795\pi$
$419$	10.8282	0.528991	0.264496	+	0.964387i	$0.414795\pi$
$420$	0	0
$421$	−7.78032	−0.379190	−0.189595	−	0.981862i	$-0.560717\pi$
$421$	−7.78032	−0.379190	−0.189595	+	0.981862i	$0.560717\pi$
$422$	0	0
$423$	11.9789	0.582436
$424$	0	0
$425$	2.60272	0.126251
$426$	0	0
$427$	−0.883881	−0.0427740
$428$	0	0
$429$	−0.254741	−0.0122990
$430$	0	0
$431$	1.75952	0.0847533	0.0423767	−	0.999102i	$-0.486507\pi$
$431$	1.75952	0.0847533	0.0423767	+	0.999102i	$0.486507\pi$
$432$	0	0
$433$	−5.17471	−0.248681	−0.124340	−	0.992240i	$-0.539681\pi$
$433$	−5.17471	−0.248681	−0.124340	+	0.992240i	$0.539681\pi$
$434$	0	0
$435$	−6.95071	−0.333261
$436$	0	0
$437$	−6.51340	−0.311578
$438$	0	0
$439$	36.4703	1.74063	0.870315	−	0.492495i	$-0.163915\pi$
$439$	36.4703	1.74063	0.870315	+	0.492495i	$0.163915\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	−17.0844	−0.811702	−0.405851	−	0.913939i	$-0.633025\pi$
$443$	−17.0844	−0.811702	−0.405851	+	0.913939i	$0.633025\pi$
$444$	0	0
$445$	−3.46410	−0.164214
$446$	0	0
$447$	1.36943	0.0647717
$448$	0	0
$449$	−22.5359	−1.06353	−0.531767	−	0.846890i	$-0.678472\pi$
$449$	−22.5359	−1.06353	−0.531767	+	0.846890i	$0.678472\pi$
$450$	0	0
$451$	−9.84282	−0.463480
$452$	0	0
$453$	0.737805	0.0346651
$454$	0	0
$455$	0.347982	0.0163137
$456$	0	0
$457$	−10.0126	−0.468368	−0.234184	−	0.972192i	$-0.575242\pi$
$457$	−10.0126	−0.468368	−0.234184	+	0.972192i	$0.575242\pi$
$458$	0	0
$459$	−10.4109	−0.485939
$460$	0	0
$461$	−21.2762	−0.990931	−0.495465	−	0.868628i	$-0.665002\pi$
$461$	−21.2762	−0.990931	−0.495465	+	0.868628i	$0.665002\pi$
$462$	0	0
$463$	12.3734	0.575039	0.287520	−	0.957775i	$-0.407169\pi$
$463$	12.3734	0.575039	0.287520	+	0.957775i	$0.407169\pi$
$464$	0	0
$465$	3.48660	0.161687
$466$	0	0
$467$	7.32943	0.339165	0.169583	−	0.985516i	$-0.445758\pi$
$467$	7.32943	0.339165	0.169583	+	0.985516i	$0.445758\pi$
$468$	0	0
$469$	−8.95071	−0.413305
$470$	0	0
$471$	−3.28756	−0.151483
$472$	0	0
$473$	2.69596	0.123961
$474$	0	0
$475$	−6.03074	−0.276709
$476$	0	0
$477$	9.30444	0.426021
$478$	0	0
$479$	2.56307	0.117110	0.0585548	−	0.998284i	$-0.481351\pi$
$479$	2.56307	0.117110	0.0585548	+	0.998284i	$0.481351\pi$
$480$	0	0
$481$	−1.49127	−0.0679961
$482$	0	0
$483$	0.790639	0.0359753
$484$	0	0
$485$	2.12933	0.0966878
$486$	0	0
$487$	0.776373	0.0351808	0.0175904	−	0.999845i	$-0.494401\pi$
$487$	0.776373	0.0351808	0.0175904	+	0.999845i	$0.494401\pi$
$488$	0	0
$489$	−8.32300	−0.376379
$490$	0	0
$491$	38.2791	1.72751	0.863756	−	0.503910i	$-0.168106\pi$
$491$	38.2791	1.72751	0.863756	+	0.503910i	$0.168106\pi$
$492$	0	0
$493$	−24.7124	−1.11299
$494$	0	0
$495$	2.46410	0.110753
$496$	0	0
$497$	−5.46410	−0.245098
$498$	0	0
$499$	−21.5588	−0.965103	−0.482552	−	0.875867i	$-0.660290\pi$
$499$	−21.5588	−0.965103	−0.482552	+	0.875867i	$0.660290\pi$
$500$	0	0
$501$	−7.49052	−0.334652
$502$	0	0
$503$	−14.7310	−0.656824	−0.328412	−	0.944535i	$-0.606513\pi$
$503$	−14.7310	−0.656824	−0.328412	+	0.944535i	$0.606513\pi$
$504$	0	0
$505$	1.01753	0.0452795
$506$	0	0
$507$	−9.42802	−0.418713
$508$	0	0
$509$	−29.5713	−1.31073	−0.655363	−	0.755314i	$-0.727484\pi$
$509$	−29.5713	−1.31073	−0.655363	+	0.755314i	$0.727484\pi$
$510$	0	0
$511$	3.29869	0.145925
$512$	0	0
$513$	24.1230	1.06505
$514$	0	0
$515$	−8.22689	−0.362520
$516$	0	0
$517$	4.86138	0.213803
$518$	0	0
$519$	−5.66314	−0.248584
$520$	0	0
$521$	2.89109	0.126661	0.0633305	−	0.997993i	$-0.479828\pi$
$521$	2.89109	0.126661	0.0633305	+	0.997993i	$0.479828\pi$
$522$	0	0
$523$	8.94679	0.391216	0.195608	−	0.980682i	$-0.437332\pi$
$523$	8.94679	0.391216	0.195608	+	0.980682i	$0.437332\pi$
$524$	0	0
$525$	0.732051	0.0319493
$526$	0	0
$527$	12.3962	0.539988
$528$	0	0
$529$	−21.8335	−0.949284
$530$	0	0
$531$	5.39125	0.233960
$532$	0	0
$533$	3.42513	0.148359
$534$	0	0
$535$	−17.1829	−0.742883
$536$	0	0
$537$	12.0615	0.520491
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	7.98573	0.343334	0.171667	−	0.985155i	$-0.445085\pi$
$541$	7.98573	0.343334	0.171667	+	0.985155i	$0.445085\pi$
$542$	0	0
$543$	4.35936	0.187078
$544$	0	0
$545$	4.72670	0.202470
$546$	0	0
$547$	−34.7310	−1.48499	−0.742496	−	0.669851i	$-0.766358\pi$
$547$	−34.7310	−1.48499	−0.742496	+	0.669851i	$0.766358\pi$
$548$	0	0
$549$	2.17797	0.0929536
$550$	0	0
$551$	57.2609	2.43940
$552$	0	0
$553$	−6.54413	−0.278285
$554$	0	0
$555$	−3.13719	−0.133166
$556$	0	0
$557$	−31.1458	−1.31969	−0.659846	−	0.751401i	$-0.729378\pi$
$557$	−31.1458	−1.31969	−0.659846	+	0.751401i	$0.729378\pi$
$558$	0	0
$559$	−0.938148	−0.0396794
$560$	0	0
$561$	−1.90533	−0.0804429
$562$	0	0
$563$	35.7021	1.50466	0.752332	−	0.658784i	$-0.228929\pi$
$563$	35.7021	1.50466	0.752332	+	0.658784i	$0.228929\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	0	0
$567$	4.46410	0.187475
$568$	0	0
$569$	−36.1658	−1.51615	−0.758075	−	0.652167i	$-0.773860\pi$
$569$	−36.1658	−1.51615	−0.758075	+	0.652167i	$0.773860\pi$
$570$	0	0
$571$	5.65058	0.236470	0.118235	−	0.992986i	$-0.462276\pi$
$571$	5.65058	0.236470	0.118235	+	0.992986i	$0.462276\pi$
$572$	0	0
$573$	−14.0908	−0.588650
$574$	0	0
$575$	1.08003	0.0450405
$576$	0	0
$577$	1.87067	0.0778771	0.0389385	−	0.999242i	$-0.487602\pi$
$577$	1.87067	0.0778771	0.0389385	+	0.999242i	$0.487602\pi$
$578$	0	0
$579$	6.65307	0.276492
$580$	0	0
$581$	−4.76814	−0.197816
$582$	0	0
$583$	3.77600	0.156386
$584$	0	0
$585$	−0.857464	−0.0354518
$586$	0	0
$587$	−22.3826	−0.923830	−0.461915	−	0.886924i	$-0.652838\pi$
$587$	−22.3826	−0.923830	−0.461915	+	0.886924i	$0.652838\pi$
$588$	0	0
$589$	−28.7231	−1.18352
$590$	0	0
$591$	12.6238	0.519273
$592$	0	0
$593$	−41.6196	−1.70911	−0.854555	−	0.519360i	$-0.826170\pi$
$593$	−41.6196	−1.70911	−0.854555	+	0.519360i	$0.826170\pi$
$594$	0	0
$595$	2.60272	0.106701
$596$	0	0
$597$	−1.35948	−0.0556399
$598$	0	0
$599$	13.0990	0.535209	0.267605	−	0.963529i	$-0.413768\pi$
$599$	13.0990	0.535209	0.267605	+	0.963529i	$0.413768\pi$
$600$	0	0
$601$	21.0297	0.857819	0.428909	−	0.903348i	$-0.358898\pi$
$601$	21.0297	0.857819	0.428909	+	0.903348i	$0.358898\pi$
$602$	0	0
$603$	22.0554	0.898167
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−4.10045	−0.166432	−0.0832161	−	0.996532i	$-0.526519\pi$
$607$	−4.10045	−0.166432	−0.0832161	+	0.996532i	$0.526519\pi$
$608$	0	0
$609$	−6.95071	−0.281657
$610$	0	0
$611$	−1.69167	−0.0684378
$612$	0	0
$613$	29.2551	1.18160	0.590802	−	0.806817i	$-0.298812\pi$
$613$	29.2551	1.18160	0.590802	+	0.806817i	$0.298812\pi$
$614$	0	0
$615$	7.20545	0.290552
$616$	0	0
$617$	47.9833	1.93173	0.965867	−	0.259039i	$-0.0834057\pi$
$617$	47.9833	1.93173	0.965867	+	0.259039i	$0.0834057\pi$
$618$	0	0
$619$	−11.8471	−0.476177	−0.238088	−	0.971243i	$-0.576521\pi$
$619$	−11.8471	−0.476177	−0.238088	+	0.971243i	$0.576521\pi$
$620$	0	0
$621$	−4.32013	−0.173361
$622$	0	0
$623$	−3.46410	−0.138786
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.41481	0.176310
$628$	0	0
$629$	−11.1539	−0.444736
$630$	0	0
$631$	−10.8832	−0.433253	−0.216627	−	0.976255i	$-0.569505\pi$
$631$	−10.8832	−0.433253	−0.216627	+	0.976255i	$0.569505\pi$
$632$	0	0
$633$	−20.1758	−0.801916
$634$	0	0
$635$	−6.92820	−0.274937
$636$	0	0
$637$	0.347982	0.0137876
$638$	0	0
$639$	13.4641	0.532632
$640$	0	0
$641$	−8.85995	−0.349947	−0.174973	−	0.984573i	$-0.555984\pi$
$641$	−8.85995	−0.349947	−0.174973	+	0.984573i	$0.555984\pi$
$642$	0	0
$643$	4.48123	0.176722	0.0883611	−	0.996089i	$-0.471837\pi$
$643$	4.48123	0.176722	0.0883611	+	0.996089i	$0.471837\pi$
$644$	0	0
$645$	−1.97358	−0.0777098
$646$	0	0
$647$	5.94965	0.233905	0.116952	−	0.993138i	$-0.462688\pi$
$647$	5.94965	0.233905	0.116952	+	0.993138i	$0.462688\pi$
$648$	0	0
$649$	2.18792	0.0858832
$650$	0	0
$651$	3.48660	0.136651
$652$	0	0
$653$	−14.2523	−0.557735	−0.278867	−	0.960330i	$-0.589959\pi$
$653$	−14.2523	−0.557735	−0.278867	+	0.960330i	$0.589959\pi$
$654$	0	0
$655$	10.7989	0.421947
$656$	0	0
$657$	−8.12830	−0.317115
$658$	0	0
$659$	3.48660	0.135819	0.0679094	−	0.997691i	$-0.478367\pi$
$659$	3.48660	0.135819	0.0679094	+	0.997691i	$0.478367\pi$
$660$	0	0
$661$	20.6753	0.804177	0.402088	−	0.915601i	$-0.368284\pi$
$661$	20.6753	0.804177	0.402088	+	0.915601i	$0.368284\pi$
$662$	0	0
$663$	0.663019	0.0257495
$664$	0	0
$665$	−6.03074	−0.233862
$666$	0	0
$667$	−10.2547	−0.397065
$668$	0	0
$669$	−6.77417	−0.261904
$670$	0	0
$671$	0.883881	0.0341218
$672$	0	0
$673$	−44.8851	−1.73019	−0.865096	−	0.501606i	$-0.832743\pi$
$673$	−44.8851	−1.73019	−0.865096	+	0.501606i	$0.832743\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−39.6803	−1.52504	−0.762519	−	0.646966i	$-0.776038\pi$
$677$	−39.6803	−1.52504	−0.762519	+	0.646966i	$0.776038\pi$
$678$	0	0
$679$	2.12933	0.0817161
$680$	0	0
$681$	−16.2673	−0.623364
$682$	0	0
$683$	23.8067	0.910939	0.455470	−	0.890251i	$-0.349471\pi$
$683$	23.8067	0.910939	0.455470	+	0.890251i	$0.349471\pi$
$684$	0	0
$685$	6.25474	0.238981
$686$	0	0
$687$	18.1329	0.691813
$688$	0	0
$689$	−1.31398	−0.0500587
$690$	0	0
$691$	−21.3641	−0.812728	−0.406364	−	0.913711i	$-0.633203\pi$
$691$	−21.3641	−0.812728	−0.406364	+	0.913711i	$0.633203\pi$
$692$	0	0
$693$	2.46410	0.0936035
$694$	0	0
$695$	0.566637	0.0214937
$696$	0	0
$697$	25.6181	0.970356
$698$	0	0
$699$	−13.4612	−0.509151
$700$	0	0
$701$	−16.2259	−0.612842	−0.306421	−	0.951896i	$-0.599132\pi$
$701$	−16.2259	−0.612842	−0.306421	+	0.951896i	$0.599132\pi$
$702$	0	0
$703$	25.8446	0.974748
$704$	0	0
$705$	−3.55878	−0.134031
$706$	0	0
$707$	1.01753	0.0382681
$708$	0	0
$709$	50.1844	1.88472	0.942358	−	0.334607i	$-0.108603\pi$
$709$	50.1844	1.88472	0.942358	+	0.334607i	$0.108603\pi$
$710$	0	0
$711$	16.1254	0.604750
$712$	0	0
$713$	5.14397	0.192643
$714$	0	0
$715$	−0.347982	−0.0130138
$716$	0	0
$717$	1.29725	0.0484469
$718$	0	0
$719$	26.1322	0.974567	0.487284	−	0.873244i	$-0.337988\pi$
$719$	26.1322	0.974567	0.487284	+	0.873244i	$0.337988\pi$
$720$	0	0
$721$	−8.22689	−0.306385
$722$	0	0
$723$	−18.2644	−0.679262
$724$	0	0
$725$	−9.49484	−0.352629
$726$	0	0
$727$	−28.6535	−1.06270	−0.531350	−	0.847153i	$-0.678315\pi$
$727$	−28.6535	−1.06270	−0.531350	+	0.847153i	$0.678315\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−7.01685	−0.259528
$732$	0	0
$733$	−1.13862	−0.0420559	−0.0210280	−	0.999779i	$-0.506694\pi$
$733$	−1.13862	−0.0420559	−0.0210280	+	0.999779i	$0.506694\pi$
$734$	0	0
$735$	0.732051	0.0270021
$736$	0	0
$737$	8.95071	0.329703
$738$	0	0
$739$	32.4881	1.19509	0.597547	−	0.801834i	$-0.296142\pi$
$739$	32.4881	1.19509	0.597547	+	0.801834i	$0.296142\pi$
$740$	0	0
$741$	−1.53627	−0.0564364
$742$	0	0
$743$	−15.6135	−0.572803	−0.286401	−	0.958110i	$-0.592459\pi$
$743$	−15.6135	−0.572803	−0.286401	+	0.958110i	$0.592459\pi$
$744$	0	0
$745$	1.87067	0.0685361
$746$	0	0
$747$	11.7492	0.429880
$748$	0	0
$749$	−17.1829	−0.627851
$750$	0	0
$751$	−30.4695	−1.11185	−0.555924	−	0.831233i	$-0.687635\pi$
$751$	−30.4695	−1.11185	−0.555924	+	0.831233i	$0.687635\pi$
$752$	0	0
$753$	−15.2159	−0.554498
$754$	0	0
$755$	1.00786	0.0366798
$756$	0	0
$757$	19.1561	0.696241	0.348121	−	0.937450i	$-0.386820\pi$
$757$	19.1561	0.696241	0.348121	+	0.937450i	$0.386820\pi$
$758$	0	0
$759$	−0.790639	−0.0286984
$760$	0	0
$761$	10.9926	0.398480	0.199240	−	0.979951i	$-0.436153\pi$
$761$	10.9926	0.398480	0.199240	+	0.979951i	$0.436153\pi$
$762$	0	0
$763$	4.72670	0.171118
$764$	0	0
$765$	−6.41337	−0.231876
$766$	0	0
$767$	−0.761356	−0.0274910
$768$	0	0
$769$	2.02147	0.0728962	0.0364481	−	0.999336i	$-0.488396\pi$
$769$	2.02147	0.0728962	0.0364481	+	0.999336i	$0.488396\pi$
$770$	0	0
$771$	12.0225	0.432980
$772$	0	0
$773$	−49.0062	−1.76263	−0.881314	−	0.472530i	$-0.843341\pi$
$773$	−49.0062	−1.76263	−0.881314	+	0.472530i	$0.843341\pi$
$774$	0	0
$775$	4.76279	0.171084
$776$	0	0
$777$	−3.13719	−0.112546
$778$	0	0
$779$	−59.3595	−2.12677
$780$	0	0
$781$	5.46410	0.195521
$782$	0	0
$783$	37.9794	1.35727
$784$	0	0
$785$	−4.49089	−0.160287
$786$	0	0
$787$	11.4270	0.407328	0.203664	−	0.979041i	$-0.434715\pi$
$787$	11.4270	0.407328	0.203664	+	0.979041i	$0.434715\pi$
$788$	0	0
$789$	4.46448	0.158940
$790$	0	0
$791$	−0.928203	−0.0330031
$792$	0	0
$793$	−0.307575	−0.0109223
$794$	0	0
$795$	−2.76422	−0.0980369
$796$	0	0
$797$	39.5342	1.40037	0.700187	−	0.713960i	$-0.253100\pi$
$797$	39.5342	1.40037	0.700187	+	0.713960i	$0.253100\pi$
$798$	0	0
$799$	−12.6528	−0.447625
$800$	0	0
$801$	8.53590	0.301601
$802$	0	0
$803$	−3.29869	−0.116408
$804$	0	0
$805$	1.08003	0.0380662
$806$	0	0
$807$	−15.8915	−0.559406
$808$	0	0
$809$	2.62379	0.0922476	0.0461238	−	0.998936i	$-0.485313\pi$
$809$	2.62379	0.0922476	0.0461238	+	0.998936i	$0.485313\pi$
$810$	0	0
$811$	10.6124	0.372651	0.186326	−	0.982488i	$-0.440342\pi$
$811$	10.6124	0.372651	0.186326	+	0.982488i	$0.440342\pi$
$812$	0	0
$813$	18.4637	0.647551
$814$	0	0
$815$	−11.3694	−0.398254
$816$	0	0
$817$	16.2587	0.568818
$818$	0	0
$819$	−0.857464	−0.0299622
$820$	0	0
$821$	53.8343	1.87883	0.939415	−	0.342781i	$-0.111369\pi$
$821$	53.8343	1.87883	0.939415	+	0.342781i	$0.111369\pi$
$822$	0	0
$823$	13.7846	0.480502	0.240251	−	0.970711i	$-0.422770\pi$
$823$	13.7846	0.480502	0.240251	+	0.970711i	$0.422770\pi$
$824$	0	0
$825$	−0.732051	−0.0254867
$826$	0	0
$827$	−21.7803	−0.757376	−0.378688	−	0.925524i	$-0.623625\pi$
$827$	−21.7803	−0.757376	−0.378688	+	0.925524i	$0.623625\pi$
$828$	0	0
$829$	−26.9243	−0.935119	−0.467560	−	0.883962i	$-0.654867\pi$
$829$	−26.9243	−0.935119	−0.467560	+	0.883962i	$0.654867\pi$
$830$	0	0
$831$	−1.91788	−0.0665306
$832$	0	0
$833$	2.60272	0.0901790
$834$	0	0
$835$	−10.2322	−0.354101
$836$	0	0
$837$	−19.0512	−0.658504
$838$	0	0
$839$	15.9225	0.549705	0.274853	−	0.961486i	$-0.411371\pi$
$839$	15.9225	0.549705	0.274853	+	0.961486i	$0.411371\pi$
$840$	0	0
$841$	61.1520	2.10869
$842$	0	0
$843$	11.1432	0.383793
$844$	0	0
$845$	−12.8789	−0.443048
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	8.26730	0.283733
$850$	0	0
$851$	−4.62846	−0.158662
$852$	0	0
$853$	25.5474	0.874726	0.437363	−	0.899285i	$-0.355912\pi$
$853$	25.5474	0.874726	0.437363	+	0.899285i	$0.355912\pi$
$854$	0	0
$855$	14.8604	0.508213
$856$	0	0
$857$	−50.0190	−1.70862	−0.854308	−	0.519766i	$-0.826019\pi$
$857$	−50.0190	−1.70862	−0.854308	+	0.519766i	$0.826019\pi$
$858$	0	0
$859$	−28.8182	−0.983266	−0.491633	−	0.870803i	$-0.663600\pi$
$859$	−28.8182	−0.983266	−0.491633	+	0.870803i	$0.663600\pi$
$860$	0	0
$861$	7.20545	0.245561
$862$	0	0
$863$	32.7128	1.11356	0.556779	−	0.830661i	$-0.312037\pi$
$863$	32.7128	1.11356	0.556779	+	0.830661i	$0.312037\pi$
$864$	0	0
$865$	−7.73600	−0.263032
$866$	0	0
$867$	−7.48583	−0.254232
$868$	0	0
$869$	6.54413	0.221995
$870$	0	0
$871$	−3.11469	−0.105537
$872$	0	0
$873$	−5.24688	−0.177580
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	42.1826	1.42440	0.712202	−	0.701975i	$-0.247698\pi$
$877$	42.1826	1.42440	0.712202	+	0.701975i	$0.247698\pi$
$878$	0	0
$879$	15.4701	0.521795
$880$	0	0
$881$	−42.1230	−1.41916	−0.709579	−	0.704626i	$-0.751115\pi$
$881$	−42.1230	−1.41916	−0.709579	+	0.704626i	$0.751115\pi$
$882$	0	0
$883$	−1.12072	−0.0377151	−0.0188575	−	0.999822i	$-0.506003\pi$
$883$	−1.12072	−0.0377151	−0.0188575	+	0.999822i	$0.506003\pi$
$884$	0	0
$885$	−1.60167	−0.0538394
$886$	0	0
$887$	−8.17865	−0.274612	−0.137306	−	0.990529i	$-0.543844\pi$
$887$	−8.17865	−0.274612	−0.137306	+	0.990529i	$0.543844\pi$
$888$	0	0
$889$	−6.92820	−0.232364
$890$	0	0
$891$	−4.46410	−0.149553
$892$	0	0
$893$	29.3177	0.981079
$894$	0	0
$895$	16.4763	0.550741
$896$	0	0
$897$	0.275128	0.00918627
$898$	0	0
$899$	−45.2219	−1.50824
$900$	0	0
$901$	−9.82788	−0.327414
$902$	0	0
$903$	−1.97358	−0.0656767
$904$	0	0
$905$	5.95500	0.197951
$906$	0	0
$907$	4.84422	0.160850	0.0804249	−	0.996761i	$-0.474372\pi$
$907$	4.84422	0.160850	0.0804249	+	0.996761i	$0.474372\pi$
$908$	0	0
$909$	−2.50730	−0.0831618
$910$	0	0
$911$	−10.0186	−0.331931	−0.165965	−	0.986132i	$-0.553074\pi$
$911$	−10.0186	−0.331931	−0.165965	+	0.986132i	$0.553074\pi$
$912$	0	0
$913$	4.76814	0.157802
$914$	0	0
$915$	−0.647046	−0.0213907
$916$	0	0
$917$	10.7989	0.356610
$918$	0	0
$919$	33.0984	1.09181	0.545907	−	0.837846i	$-0.316185\pi$
$919$	33.0984	1.09181	0.545907	+	0.837846i	$0.316185\pi$
$920$	0	0
$921$	8.92745	0.294170
$922$	0	0
$923$	−1.90141	−0.0625857
$924$	0	0
$925$	−4.28548	−0.140906
$926$	0	0
$927$	20.2719	0.665816
$928$	0	0
$929$	−3.14110	−0.103056	−0.0515281	−	0.998672i	$-0.516409\pi$
$929$	−3.14110	−0.103056	−0.0515281	+	0.998672i	$0.516409\pi$
$930$	0	0
$931$	−6.03074	−0.197649
$932$	0	0
$933$	0.291102	0.00953026
$934$	0	0
$935$	−2.60272	−0.0851181
$936$	0	0
$937$	45.3502	1.48153	0.740764	−	0.671766i	$-0.234464\pi$
$937$	45.3502	1.48153	0.740764	+	0.671766i	$0.234464\pi$
$938$	0	0
$939$	11.7803	0.384436
$940$	0	0
$941$	−11.4920	−0.374627	−0.187313	−	0.982300i	$-0.559978\pi$
$941$	−11.4920	−0.374627	−0.187313	+	0.982300i	$0.559978\pi$
$942$	0	0
$943$	10.6306	0.346179
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−16.1483	−0.524748	−0.262374	−	0.964966i	$-0.584505\pi$
$947$	−16.1483	−0.524748	−0.262374	+	0.964966i	$0.584505\pi$
$948$	0	0
$949$	1.14788	0.0372619
$950$	0	0
$951$	1.36943	0.0444067
$952$	0	0
$953$	1.37546	0.0445554	0.0222777	−	0.999752i	$-0.492908\pi$
$953$	1.37546	0.0445554	0.0222777	+	0.999752i	$0.492908\pi$
$954$	0	0
$955$	−19.2483	−0.622862
$956$	0	0
$957$	6.95071	0.224684
$958$	0	0
$959$	6.25474	0.201976
$960$	0	0
$961$	−8.31584	−0.268253
$962$	0	0
$963$	42.3405	1.36440
$964$	0	0
$965$	9.08827	0.292562
$966$	0	0
$967$	34.5649	1.11153	0.555767	−	0.831338i	$-0.312425\pi$
$967$	34.5649	1.11153	0.555767	+	0.831338i	$0.312425\pi$
$968$	0	0
$969$	−11.4905	−0.369129
$970$	0	0
$971$	1.17182	0.0376055	0.0188027	−	0.999823i	$-0.494015\pi$
$971$	1.17182	0.0376055	0.0188027	+	0.999823i	$0.494015\pi$
$972$	0	0
$973$	0.566637	0.0181655
$974$	0	0
$975$	0.254741	0.00815823
$976$	0	0
$977$	42.0329	1.34475	0.672377	−	0.740209i	$-0.265274\pi$
$977$	42.0329	1.34475	0.672377	+	0.740209i	$0.265274\pi$
$978$	0	0
$979$	3.46410	0.110713
$980$	0	0
$981$	−11.6471	−0.371863
$982$	0	0
$983$	35.3973	1.12900	0.564499	−	0.825434i	$-0.309069\pi$
$983$	35.3973	1.12900	0.564499	+	0.825434i	$0.309069\pi$
$984$	0	0
$985$	17.2444	0.549453
$986$	0	0
$987$	−3.55878	−0.113277
$988$	0	0
$989$	−2.91173	−0.0925876
$990$	0	0
$991$	−13.9901	−0.444409	−0.222204	−	0.975000i	$-0.571325\pi$
$991$	−13.9901	−0.444409	−0.222204	+	0.975000i	$0.571325\pi$
$992$	0	0
$993$	−2.66955	−0.0847155
$994$	0	0
$995$	−1.85709	−0.0588736
$996$	0	0
$997$	15.7467	0.498703	0.249351	−	0.968413i	$-0.419783\pi$
$997$	15.7467	0.498703	0.249351	+	0.968413i	$0.419783\pi$
$998$	0	0
$999$	17.1419	0.542346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bq.1.4		4
4.3	odd	2		3080.2.a.q.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.q.1.2	✓	4	4.3	odd	2
6160.2.a.bq.1.4		4	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} +0.347982 q^{13} +0.732051 q^{15} +2.60272 q^{17} -6.03074 q^{19} +0.732051 q^{21} +1.08003 q^{23} +1.00000 q^{25} -4.00000 q^{27} -9.49484 q^{29} +4.76279 q^{31} -0.732051 q^{33} +1.00000 q^{35} -4.28548 q^{37} +0.254741 q^{39} +9.84282 q^{41} -2.69596 q^{43} -2.46410 q^{45} -4.86138 q^{47} +1.00000 q^{49} +1.90533 q^{51} -3.77600 q^{53} -1.00000 q^{55} -4.41481 q^{57} -2.18792 q^{59} -0.883881 q^{61} -2.46410 q^{63} +0.347982 q^{65} -8.95071 q^{67} +0.790639 q^{69} -5.46410 q^{71} +3.29869 q^{73} +0.732051 q^{75} -1.00000 q^{77} -6.54413 q^{79} +4.46410 q^{81} -4.76814 q^{83} +2.60272 q^{85} -6.95071 q^{87} -3.46410 q^{89} +0.347982 q^{91} +3.48660 q^{93} -6.03074 q^{95} +2.12933 q^{97} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} - 4 q^{21} - 6 q^{23} + 4 q^{25} - 16 q^{27} - 2 q^{29} - 10 q^{31} + 4 q^{33} + 4 q^{35} + 10 q^{37} - 4 q^{39} - 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{49} - 8 q^{51} + 2 q^{53} - 4 q^{55} + 8 q^{57} - 26 q^{59} - 14 q^{61} + 4 q^{63} - 2 q^{65} - 24 q^{67} + 12 q^{69} - 8 q^{71} - 2 q^{73} - 4 q^{75} - 4 q^{77} - 2 q^{79} + 4 q^{81} - 12 q^{83} + 2 q^{85} - 16 q^{87} - 2 q^{91} + 16 q^{93} - 2 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 - 4 * q^21 - 6 * q^23 + 4 * q^25 - 16 * q^27 - 2 * q^29 - 10 * q^31 + 4 * q^33 + 4 * q^35 + 10 * q^37 - 4 * q^39 - 4 * q^43 + 4 * q^45 - 14 * q^47 + 4 * q^49 - 8 * q^51 + 2 * q^53 - 4 * q^55 + 8 * q^57 - 26 * q^59 - 14 * q^61 + 4 * q^63 - 2 * q^65 - 24 * q^67 + 12 * q^69 - 8 * q^71 - 2 * q^73 - 4 * q^75 - 4 * q^77 - 2 * q^79 + 4 * q^81 - 12 * q^83 + 2 * q^85 - 16 * q^87 - 2 * q^91 + 16 * q^93 - 2 * q^95 + 10 * q^97 - 4 * q^99

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