LMFDB - Embedded newform 6080.2.a.ck.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2363492.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10 $ x^5 - x^4 - 11x^3 - 6x^2 + 14*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.19018$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78093 q^{3} -1.00000 q^{5} -0.646760 q^{7} +4.73359 q^{9} +O(q^{10})$	$q-2.78093 q^{3} -1.00000 q^{5} -0.646760 q^{7} +4.73359 q^{9} +2.99272 q^{11} -6.15401 q^{13} +2.78093 q^{15} +2.72632 q^{17} +1.00000 q^{19} +1.79860 q^{21} -0.646760 q^{23} +1.00000 q^{25} -4.82100 q^{27} -3.81315 q^{29} +0.0145507 q^{31} -8.32257 q^{33} +0.646760 q^{35} -6.49997 q^{37} +17.1139 q^{39} +0.532818 q^{41} +2.45991 q^{43} -4.73359 q^{45} +11.8627 q^{47} -6.58170 q^{49} -7.58170 q^{51} +1.29862 q^{53} -2.99272 q^{55} -2.78093 q^{57} +13.0489 q^{59} +3.75343 q^{61} -3.06150 q^{63} +6.15401 q^{65} -13.2481 q^{67} +1.79860 q^{69} -1.46718 q^{71} -8.82100 q^{73} -2.78093 q^{75} -1.93557 q^{77} -9.17361 q^{79} -0.793890 q^{81} -11.3797 q^{83} -2.72632 q^{85} +10.6041 q^{87} +14.8408 q^{89} +3.98017 q^{91} -0.0404646 q^{93} -1.00000 q^{95} -6.34086 q^{97} +14.1663 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9	$ 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9} + 10 q^{11} - 2 q^{15} + 4 q^{17} + 5 q^{19} - 10 q^{21} + 2 q^{23} + 5 q^{25} + 8 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{33} - 2 q^{35} - 2 q^{37} + 10 q^{39} + 12 q^{41} - 2 q^{43} - 19 q^{45} + 22 q^{47} + 19 q^{49} + 14 q^{51} + 18 q^{53} - 10 q^{55} + 2 q^{57} + 4 q^{59} - 6 q^{61} + 10 q^{63} + 20 q^{67} - 10 q^{69} + 2 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 14 q^{79} + 13 q^{81} - 14 q^{83} - 4 q^{85} - 12 q^{87} + 22 q^{89} + 40 q^{91} - 8 q^{93} - 5 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9 + 10 * q^11 - 2 * q^15 + 4 * q^17 + 5 * q^19 - 10 * q^21 + 2 * q^23 + 5 * q^25 + 8 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^33 - 2 * q^35 - 2 * q^37 + 10 * q^39 + 12 * q^41 - 2 * q^43 - 19 * q^45 + 22 * q^47 + 19 * q^49 + 14 * q^51 + 18 * q^53 - 10 * q^55 + 2 * q^57 + 4 * q^59 - 6 * q^61 + 10 * q^63 + 20 * q^67 - 10 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 + 13 * q^81 - 14 * q^83 - 4 * q^85 - 12 * q^87 + 22 * q^89 + 40 * q^91 - 8 * q^93 - 5 * q^95 - 10 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.78093	−1.60557	−0.802786	−	0.596267i	$-0.796650\pi$
$3$	−2.78093	−1.60557	−0.802786	+	0.596267i	$0.796650\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.646760	−0.244452	−0.122226	−	0.992502i	$-0.539003\pi$
$7$	−0.646760	−0.244452	−0.122226	+	0.992502i	$0.539003\pi$
$8$	0	0
$9$	4.73359	1.57786
$10$	0	0
$11$	2.99272	0.902340	0.451170	−	0.892438i	$-0.351007\pi$
$11$	2.99272	0.902340	0.451170	+	0.892438i	$0.351007\pi$
$12$	0	0
$13$	−6.15401	−1.70681	−0.853407	−	0.521244i	$-0.825468\pi$
$13$	−6.15401	−1.70681	−0.853407	+	0.521244i	$0.825468\pi$
$14$	0	0
$15$	2.78093	0.718034
$16$	0	0
$17$	2.72632	0.661229	0.330614	−	0.943766i	$-0.392744\pi$
$17$	2.72632	0.661229	0.330614	+	0.943766i	$0.392744\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.79860	0.392486
$22$	0	0
$23$	−0.646760	−0.134859	−0.0674294	−	0.997724i	$-0.521480\pi$
$23$	−0.646760	−0.134859	−0.0674294	+	0.997724i	$0.521480\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.82100	−0.927802
$28$	0	0
$29$	−3.81315	−0.708084	−0.354042	−	0.935230i	$-0.615193\pi$
$29$	−3.81315	−0.708084	−0.354042	+	0.935230i	$0.615193\pi$
$30$	0	0
$31$	0.0145507	0.00261339	0.00130669	−	0.999999i	$-0.499584\pi$
$31$	0.0145507	0.00261339	0.00130669	+	0.999999i	$0.499584\pi$
$32$	0	0
$33$	−8.32257	−1.44877
$34$	0	0
$35$	0.646760	0.109322
$36$	0	0
$37$	−6.49997	−1.06859	−0.534295	−	0.845298i	$-0.679423\pi$
$37$	−6.49997	−1.06859	−0.534295	+	0.845298i	$0.679423\pi$
$38$	0	0
$39$	17.1139	2.74042
$40$	0	0
$41$	0.532818	0.0832122	0.0416061	−	0.999134i	$-0.486753\pi$
$41$	0.532818	0.0832122	0.0416061	+	0.999134i	$0.486753\pi$
$42$	0	0
$43$	2.45991	0.375132	0.187566	−	0.982252i	$-0.439940\pi$
$43$	2.45991	0.375132	0.187566	+	0.982252i	$0.439940\pi$
$44$	0	0
$45$	−4.73359	−0.705642
$46$	0	0
$47$	11.8627	1.73035	0.865174	−	0.501473i	$-0.167208\pi$
$47$	11.8627	1.73035	0.865174	+	0.501473i	$0.167208\pi$
$48$	0	0
$49$	−6.58170	−0.940243
$50$	0	0
$51$	−7.58170	−1.06165
$52$	0	0
$53$	1.29862	0.178379	0.0891897	−	0.996015i	$-0.471572\pi$
$53$	1.29862	0.178379	0.0891897	+	0.996015i	$0.471572\pi$
$54$	0	0
$55$	−2.99272	−0.403539
$56$	0	0
$57$	−2.78093	−0.368344
$58$	0	0
$59$	13.0489	1.69882	0.849410	−	0.527734i	$-0.176958\pi$
$59$	13.0489	1.69882	0.849410	+	0.527734i	$0.176958\pi$
$60$	0	0
$61$	3.75343	0.480577	0.240288	−	0.970702i	$-0.422758\pi$
$61$	3.75343	0.480577	0.240288	+	0.970702i	$0.422758\pi$
$62$	0	0
$63$	−3.06150	−0.385712
$64$	0	0
$65$	6.15401	0.763311
$66$	0	0
$67$	−13.2481	−1.61851	−0.809254	−	0.587459i	$-0.800128\pi$
$67$	−13.2481	−1.61851	−0.809254	+	0.587459i	$0.800128\pi$
$68$	0	0
$69$	1.79860	0.216525
$70$	0	0
$71$	−1.46718	−0.174122	−0.0870612	−	0.996203i	$-0.527748\pi$
$71$	−1.46718	−0.174122	−0.0870612	+	0.996203i	$0.527748\pi$
$72$	0	0
$73$	−8.82100	−1.03242	−0.516210	−	0.856462i	$-0.672658\pi$
$73$	−8.82100	−1.03242	−0.516210	+	0.856462i	$0.672658\pi$
$74$	0	0
$75$	−2.78093	−0.321115
$76$	0	0
$77$	−1.93557	−0.220579
$78$	0	0
$79$	−9.17361	−1.03211	−0.516056	−	0.856555i	$-0.672600\pi$
$79$	−9.17361	−1.03211	−0.516056	+	0.856555i	$0.672600\pi$
$80$	0	0
$81$	−0.793890	−0.0882100
$82$	0	0
$83$	−11.3797	−1.24909	−0.624543	−	0.780990i	$-0.714715\pi$
$83$	−11.3797	−1.24909	−0.624543	+	0.780990i	$0.714715\pi$
$84$	0	0
$85$	−2.72632	−0.295710
$86$	0	0
$87$	10.6041	1.13688
$88$	0	0
$89$	14.8408	1.57313	0.786563	−	0.617510i	$-0.211859\pi$
$89$	14.8408	1.57313	0.786563	+	0.617510i	$0.211859\pi$
$90$	0	0
$91$	3.98017	0.417235
$92$	0	0
$93$	−0.0404646	−0.00419599
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−6.34086	−0.643817	−0.321909	−	0.946771i	$-0.604324\pi$
$97$	−6.34086	−0.643817	−0.321909	+	0.946771i	$0.604324\pi$
$98$	0	0
$99$	14.1663	1.42377
$100$	0	0
$101$	−18.9270	−1.88331	−0.941655	−	0.336579i	$-0.890730\pi$
$101$	−18.9270	−1.88331	−0.941655	+	0.336579i	$0.890730\pi$
$102$	0	0
$103$	−2.76110	−0.272059	−0.136030	−	0.990705i	$-0.543434\pi$
$103$	−2.76110	−0.272059	−0.136030	+	0.990705i	$0.543434\pi$
$104$	0	0
$105$	−1.79860	−0.175525
$106$	0	0
$107$	−8.78093	−0.848885	−0.424442	−	0.905455i	$-0.639530\pi$
$107$	−8.78093	−0.848885	−0.424442	+	0.905455i	$0.639530\pi$
$108$	0	0
$109$	−7.63948	−0.731730	−0.365865	−	0.930668i	$-0.619227\pi$
$109$	−7.63948	−0.731730	−0.365865	+	0.930668i	$0.619227\pi$
$110$	0	0
$111$	18.0760	1.71570
$112$	0	0
$113$	−6.78287	−0.638079	−0.319039	−	0.947741i	$-0.603360\pi$
$113$	−6.78287	−0.638079	−0.319039	+	0.947741i	$0.603360\pi$
$114$	0	0
$115$	0.646760	0.0603107
$116$	0	0
$117$	−29.1306	−2.69312
$118$	0	0
$119$	−1.76327	−0.161639
$120$	0	0
$121$	−2.04360	−0.185782
$122$	0	0
$123$	−1.48173	−0.133603
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.68091	0.859042	0.429521	−	0.903057i	$-0.358682\pi$
$127$	9.68091	0.859042	0.429521	+	0.903057i	$0.358682\pi$
$128$	0	0
$129$	−6.84084	−0.602302
$130$	0	0
$131$	13.7897	1.20482	0.602408	−	0.798188i	$-0.294208\pi$
$131$	13.7897	1.20482	0.602408	+	0.798188i	$0.294208\pi$
$132$	0	0
$133$	−0.646760	−0.0560812
$134$	0	0
$135$	4.82100	0.414926
$136$	0	0
$137$	−3.25913	−0.278447	−0.139223	−	0.990261i	$-0.544461\pi$
$137$	−3.25913	−0.278447	−0.139223	+	0.990261i	$0.544461\pi$
$138$	0	0
$139$	9.19665	0.780049	0.390025	−	0.920804i	$-0.372466\pi$
$139$	9.19665	0.780049	0.390025	+	0.920804i	$0.372466\pi$
$140$	0	0
$141$	−32.9893	−2.77820
$142$	0	0
$143$	−18.4173	−1.54013
$144$	0	0
$145$	3.81315	0.316665
$146$	0	0
$147$	18.3033	1.50963
$148$	0	0
$149$	14.8772	1.21878	0.609392	−	0.792869i	$-0.291414\pi$
$149$	14.8772	1.21878	0.609392	+	0.792869i	$0.291414\pi$
$150$	0	0
$151$	−14.4817	−1.17850	−0.589251	−	0.807950i	$-0.700577\pi$
$151$	−14.4817	−1.17850	−0.589251	+	0.807950i	$0.700577\pi$
$152$	0	0
$153$	12.9053	1.04333
$154$	0	0
$155$	−0.0145507	−0.00116874
$156$	0	0
$157$	−6.76070	−0.539563	−0.269781	−	0.962922i	$-0.586951\pi$
$157$	−6.76070	−0.539563	−0.269781	+	0.962922i	$0.586951\pi$
$158$	0	0
$159$	−3.61138	−0.286401
$160$	0	0
$161$	0.418298	0.0329665
$162$	0	0
$163$	15.5691	1.21947	0.609733	−	0.792607i	$-0.291277\pi$
$163$	15.5691	1.21947	0.609733	+	0.792607i	$0.291277\pi$
$164$	0	0
$165$	8.32257	0.647911
$166$	0	0
$167$	8.34808	0.645994	0.322997	−	0.946400i	$-0.395310\pi$
$167$	8.34808	0.645994	0.322997	+	0.946400i	$0.395310\pi$
$168$	0	0
$169$	24.8718	1.91322
$170$	0	0
$171$	4.73359	0.361987
$172$	0	0
$173$	−19.1200	−1.45367	−0.726833	−	0.686814i	$-0.759009\pi$
$173$	−19.1200	−1.45367	−0.726833	+	0.686814i	$0.759009\pi$
$174$	0	0
$175$	−0.646760	−0.0488904
$176$	0	0
$177$	−36.2881	−2.72758
$178$	0	0
$179$	6.64205	0.496450	0.248225	−	0.968702i	$-0.420153\pi$
$179$	6.64205	0.496450	0.248225	+	0.968702i	$0.420153\pi$
$180$	0	0
$181$	16.7897	1.24797	0.623984	−	0.781437i	$-0.285513\pi$
$181$	16.7897	1.24797	0.623984	+	0.781437i	$0.285513\pi$
$182$	0	0
$183$	−10.4380	−0.771601
$184$	0	0
$185$	6.49997	0.477888
$186$	0	0
$187$	8.15911	0.596653
$188$	0	0
$189$	3.11803	0.226803
$190$	0	0
$191$	5.54480	0.401208	0.200604	−	0.979672i	$-0.435710\pi$
$191$	5.54480	0.401208	0.200604	+	0.979672i	$0.435710\pi$
$192$	0	0
$193$	17.1962	1.23781	0.618907	−	0.785465i	$-0.287576\pi$
$193$	17.1962	1.23781	0.618907	+	0.785465i	$0.287576\pi$
$194$	0	0
$195$	−17.1139	−1.22555
$196$	0	0
$197$	18.8844	1.34546	0.672728	−	0.739890i	$-0.265122\pi$
$197$	18.8844	1.34546	0.672728	+	0.739890i	$0.265122\pi$
$198$	0	0
$199$	15.5302	1.10091	0.550455	−	0.834865i	$-0.314454\pi$
$199$	15.5302	1.10091	0.550455	+	0.834865i	$0.314454\pi$
$200$	0	0
$201$	36.8420	2.59863
$202$	0	0
$203$	2.46619	0.173093
$204$	0	0
$205$	−0.532818	−0.0372136
$206$	0	0
$207$	−3.06150	−0.212789
$208$	0	0
$209$	2.99272	0.207011
$210$	0	0
$211$	8.07485	0.555896	0.277948	−	0.960596i	$-0.410346\pi$
$211$	8.07485	0.555896	0.277948	+	0.960596i	$0.410346\pi$
$212$	0	0
$213$	4.08013	0.279566
$214$	0	0
$215$	−2.45991	−0.167764
$216$	0	0
$217$	−0.00941083	−0.000638849 0
$218$	0	0
$219$	24.5306	1.65763
$220$	0	0
$221$	−16.7778	−1.12859
$222$	0	0
$223$	−16.6455	−1.11467	−0.557334	−	0.830289i	$-0.688176\pi$
$223$	−16.6455	−1.11467	−0.557334	+	0.830289i	$0.688176\pi$
$224$	0	0
$225$	4.73359	0.315573
$226$	0	0
$227$	−4.24812	−0.281957	−0.140979	−	0.990013i	$-0.545025\pi$
$227$	−4.24812	−0.281957	−0.140979	+	0.990013i	$0.545025\pi$
$228$	0	0
$229$	7.63478	0.504520	0.252260	−	0.967659i	$-0.418826\pi$
$229$	7.63478	0.504520	0.252260	+	0.967659i	$0.418826\pi$
$230$	0	0
$231$	5.38270	0.354156
$232$	0	0
$233$	−16.5213	−1.08235	−0.541175	−	0.840910i	$-0.682020\pi$
$233$	−16.5213	−1.08235	−0.541175	+	0.840910i	$0.682020\pi$
$234$	0	0
$235$	−11.8627	−0.773835
$236$	0	0
$237$	25.5112	1.65713
$238$	0	0
$239$	−14.0713	−0.910196	−0.455098	−	0.890441i	$-0.650396\pi$
$239$	−14.0713	−0.910196	−0.455098	+	0.890441i	$0.650396\pi$
$240$	0	0
$241$	−18.0290	−1.16135	−0.580675	−	0.814136i	$-0.697211\pi$
$241$	−18.0290	−1.16135	−0.580675	+	0.814136i	$0.697211\pi$
$242$	0	0
$243$	16.6708	1.06943
$244$	0	0
$245$	6.58170	0.420490
$246$	0	0
$247$	−6.15401	−0.391570
$248$	0	0
$249$	31.6462	2.00550
$250$	0	0
$251$	15.7253	0.992574	0.496287	−	0.868159i	$-0.334696\pi$
$251$	15.7253	0.992574	0.496287	+	0.868159i	$0.334696\pi$
$252$	0	0
$253$	−1.93557	−0.121688
$254$	0	0
$255$	7.58170	0.474785
$256$	0	0
$257$	21.4052	1.33522	0.667611	−	0.744510i	$-0.267317\pi$
$257$	21.4052	1.33522	0.667611	+	0.744510i	$0.267317\pi$
$258$	0	0
$259$	4.20392	0.261219
$260$	0	0
$261$	−18.0499	−1.11726
$262$	0	0
$263$	−13.7023	−0.844919	−0.422460	−	0.906382i	$-0.638833\pi$
$263$	−13.7023	−0.844919	−0.422460	+	0.906382i	$0.638833\pi$
$264$	0	0
$265$	−1.29862	−0.0797737
$266$	0	0
$267$	−41.2714	−2.52577
$268$	0	0
$269$	−1.37365	−0.0837532	−0.0418766	−	0.999123i	$-0.513334\pi$
$269$	−1.37365	−0.0837532	−0.0418766	+	0.999123i	$0.513334\pi$
$270$	0	0
$271$	0.871561	0.0529435	0.0264718	−	0.999650i	$-0.491573\pi$
$271$	0.871561	0.0529435	0.0264718	+	0.999650i	$0.491573\pi$
$272$	0	0
$273$	−11.0686	−0.669901
$274$	0	0
$275$	2.99272	0.180468
$276$	0	0
$277$	28.7500	1.72742	0.863711	−	0.503988i	$-0.168134\pi$
$277$	28.7500	1.72742	0.863711	+	0.503988i	$0.168134\pi$
$278$	0	0
$279$	0.0688772	0.00412357
$280$	0	0
$281$	0.343448	0.0204884	0.0102442	−	0.999948i	$-0.496739\pi$
$281$	0.343448	0.0204884	0.0102442	+	0.999948i	$0.496739\pi$
$282$	0	0
$283$	6.95305	0.413316	0.206658	−	0.978413i	$-0.433741\pi$
$283$	6.95305	0.413316	0.206658	+	0.978413i	$0.433741\pi$
$284$	0	0
$285$	2.78093	0.164728
$286$	0	0
$287$	−0.344605	−0.0203414
$288$	0	0
$289$	−9.56720	−0.562777
$290$	0	0
$291$	17.6335	1.03369
$292$	0	0
$293$	−4.02400	−0.235085	−0.117542	−	0.993068i	$-0.537502\pi$
$293$	−4.02400	−0.235085	−0.117542	+	0.993068i	$0.537502\pi$
$294$	0	0
$295$	−13.0489	−0.759735
$296$	0	0
$297$	−14.4279	−0.837193
$298$	0	0
$299$	3.98017	0.230179
$300$	0	0
$301$	−1.59097	−0.0917019
$302$	0	0
$303$	52.6348	3.02379
$304$	0	0
$305$	−3.75343	−0.214921
$306$	0	0
$307$	18.3178	1.04545	0.522727	−	0.852500i	$-0.324915\pi$
$307$	18.3178	1.04545	0.522727	+	0.852500i	$0.324915\pi$
$308$	0	0
$309$	7.67843	0.436811
$310$	0	0
$311$	7.80592	0.442633	0.221317	−	0.975202i	$-0.428965\pi$
$311$	7.80592	0.442633	0.221317	+	0.975202i	$0.428965\pi$
$312$	0	0
$313$	12.3173	0.696214	0.348107	−	0.937455i	$-0.386825\pi$
$313$	12.3173	0.696214	0.348107	+	0.937455i	$0.386825\pi$
$314$	0	0
$315$	3.06150	0.172496
$316$	0	0
$317$	7.35798	0.413265	0.206633	−	0.978419i	$-0.433749\pi$
$317$	7.35798	0.413265	0.206633	+	0.978419i	$0.433749\pi$
$318$	0	0
$319$	−11.4117	−0.638932
$320$	0	0
$321$	24.4192	1.36295
$322$	0	0
$323$	2.72632	0.151696
$324$	0	0
$325$	−6.15401	−0.341363
$326$	0	0
$327$	21.2449	1.17485
$328$	0	0
$329$	−7.67229	−0.422987
$330$	0	0
$331$	−12.9656	−0.712652	−0.356326	−	0.934362i	$-0.615971\pi$
$331$	−12.9656	−0.712652	−0.356326	+	0.934362i	$0.615971\pi$
$332$	0	0
$333$	−30.7682	−1.68609
$334$	0	0
$335$	13.2481	0.723819
$336$	0	0
$337$	−5.17113	−0.281689	−0.140845	−	0.990032i	$-0.544982\pi$
$337$	−5.17113	−0.281689	−0.140845	+	0.990032i	$0.544982\pi$
$338$	0	0
$339$	18.8627	1.02448
$340$	0	0
$341$	0.0435463	0.00235817
$342$	0	0
$343$	8.78410	0.474297
$344$	0	0
$345$	−1.79860	−0.0968331
$346$	0	0
$347$	−7.79389	−0.418398	−0.209199	−	0.977873i	$-0.567086\pi$
$347$	−7.79389	−0.418398	−0.209199	+	0.977873i	$0.567086\pi$
$348$	0	0
$349$	15.4066	0.824698	0.412349	−	0.911026i	$-0.364708\pi$
$349$	15.4066	0.824698	0.412349	+	0.911026i	$0.364708\pi$
$350$	0	0
$351$	29.6685	1.58359
$352$	0	0
$353$	13.2446	0.704938	0.352469	−	0.935823i	$-0.385342\pi$
$353$	13.2446	0.704938	0.352469	+	0.935823i	$0.385342\pi$
$354$	0	0
$355$	1.46718	0.0778699
$356$	0	0
$357$	4.90354	0.259523
$358$	0	0
$359$	−23.5166	−1.24116	−0.620579	−	0.784144i	$-0.713102\pi$
$359$	−23.5166	−1.24116	−0.620579	+	0.784144i	$0.713102\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.68311	0.298286
$364$	0	0
$365$	8.82100	0.461712
$366$	0	0
$367$	18.3047	0.955496	0.477748	−	0.878497i	$-0.341453\pi$
$367$	18.3047	0.955496	0.477748	+	0.878497i	$0.341453\pi$
$368$	0	0
$369$	2.52214	0.131298
$370$	0	0
$371$	−0.839897	−0.0436053
$372$	0	0
$373$	−21.9948	−1.13885	−0.569424	−	0.822044i	$-0.692834\pi$
$373$	−21.9948	−1.13885	−0.569424	+	0.822044i	$0.692834\pi$
$374$	0	0
$375$	2.78093	0.143607
$376$	0	0
$377$	23.4661	1.20857
$378$	0	0
$379$	1.98017	0.101714	0.0508571	−	0.998706i	$-0.483805\pi$
$379$	1.98017	0.101714	0.0508571	+	0.998706i	$0.483805\pi$
$380$	0	0
$381$	−26.9220	−1.37925
$382$	0	0
$383$	−1.48647	−0.0759553	−0.0379777	−	0.999279i	$-0.512092\pi$
$383$	−1.48647	−0.0759553	−0.0379777	+	0.999279i	$0.512092\pi$
$384$	0	0
$385$	1.93557	0.0986460
$386$	0	0
$387$	11.6442	0.591907
$388$	0	0
$389$	34.3807	1.74317	0.871584	−	0.490245i	$-0.163093\pi$
$389$	34.3807	1.74317	0.871584	+	0.490245i	$0.163093\pi$
$390$	0	0
$391$	−1.76327	−0.0891725
$392$	0	0
$393$	−38.3484	−1.93442
$394$	0	0
$395$	9.17361	0.461574
$396$	0	0
$397$	1.63378	0.0819971	0.0409985	−	0.999159i	$-0.486946\pi$
$397$	1.63378	0.0819971	0.0409985	+	0.999159i	$0.486946\pi$
$398$	0	0
$399$	1.79860	0.0900424
$400$	0	0
$401$	25.6962	1.28321	0.641604	−	0.767036i	$-0.278269\pi$
$401$	25.6962	1.28321	0.641604	+	0.767036i	$0.278269\pi$
$402$	0	0
$403$	−0.0895453	−0.00446057
$404$	0	0
$405$	0.793890	0.0394487
$406$	0	0
$407$	−19.4526	−0.964231
$408$	0	0
$409$	−11.7713	−0.582054	−0.291027	−	0.956715i	$-0.593997\pi$
$409$	−11.7713	−0.582054	−0.291027	+	0.956715i	$0.593997\pi$
$410$	0	0
$411$	9.06343	0.447066
$412$	0	0
$413$	−8.43949	−0.415280
$414$	0	0
$415$	11.3797	0.558608
$416$	0	0
$417$	−25.5753	−1.25243
$418$	0	0
$419$	33.3881	1.63112	0.815558	−	0.578676i	$-0.196430\pi$
$419$	33.3881	1.63112	0.815558	+	0.578676i	$0.196430\pi$
$420$	0	0
$421$	17.5592	0.855785	0.427893	−	0.903830i	$-0.359256\pi$
$421$	17.5592	0.855785	0.427893	+	0.903830i	$0.359256\pi$
$422$	0	0
$423$	56.1530	2.73025
$424$	0	0
$425$	2.72632	0.132246
$426$	0	0
$427$	−2.42756	−0.117478
$428$	0	0
$429$	51.2172	2.47279
$430$	0	0
$431$	16.7253	0.805628	0.402814	−	0.915282i	$-0.368032\pi$
$431$	16.7253	0.805628	0.402814	+	0.915282i	$0.368032\pi$
$432$	0	0
$433$	−9.67485	−0.464943	−0.232472	−	0.972603i	$-0.574681\pi$
$433$	−9.67485	−0.464943	−0.232472	+	0.972603i	$0.574681\pi$
$434$	0	0
$435$	−10.6041	−0.508428
$436$	0	0
$437$	−0.646760	−0.0309387
$438$	0	0
$439$	15.1882	0.724891	0.362446	−	0.932005i	$-0.381942\pi$
$439$	15.1882	0.724891	0.362446	+	0.932005i	$0.381942\pi$
$440$	0	0
$441$	−31.1551	−1.48358
$442$	0	0
$443$	30.3690	1.44287	0.721437	−	0.692480i	$-0.243482\pi$
$443$	30.3690	1.44287	0.721437	+	0.692480i	$0.243482\pi$
$444$	0	0
$445$	−14.8408	−0.703523
$446$	0	0
$447$	−41.3724	−1.95685
$448$	0	0
$449$	25.4463	1.20089	0.600443	−	0.799668i	$-0.294991\pi$
$449$	25.4463	1.20089	0.600443	+	0.799668i	$0.294991\pi$
$450$	0	0
$451$	1.59458	0.0750858
$452$	0	0
$453$	40.2726	1.89217
$454$	0	0
$455$	−3.98017	−0.186593
$456$	0	0
$457$	32.7592	1.53241	0.766205	−	0.642596i	$-0.222143\pi$
$457$	32.7592	1.53241	0.766205	+	0.642596i	$0.222143\pi$
$458$	0	0
$459$	−13.1436	−0.613489
$460$	0	0
$461$	20.7606	0.966917	0.483459	−	0.875367i	$-0.339380\pi$
$461$	20.7606	0.966917	0.483459	+	0.875367i	$0.339380\pi$
$462$	0	0
$463$	17.3550	0.806554	0.403277	−	0.915078i	$-0.367871\pi$
$463$	17.3550	0.806554	0.403277	+	0.915078i	$0.367871\pi$
$464$	0	0
$465$	0.0404646	0.00187650
$466$	0	0
$467$	−7.83356	−0.362494	−0.181247	−	0.983438i	$-0.558013\pi$
$467$	−7.83356	−0.362494	−0.181247	+	0.983438i	$0.558013\pi$
$468$	0	0
$469$	8.56831	0.395648
$470$	0	0
$471$	18.8011	0.866307
$472$	0	0
$473$	7.36182	0.338497
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	6.14715	0.281459
$478$	0	0
$479$	−2.58604	−0.118159	−0.0590796	−	0.998253i	$-0.518817\pi$
$479$	−2.58604	−0.118159	−0.0590796	+	0.998253i	$0.518817\pi$
$480$	0	0
$481$	40.0009	1.82388
$482$	0	0
$483$	−1.16326	−0.0529301
$484$	0	0
$485$	6.34086	0.287924
$486$	0	0
$487$	21.9744	0.995754	0.497877	−	0.867248i	$-0.334113\pi$
$487$	21.9744	0.995754	0.497877	+	0.867248i	$0.334113\pi$
$488$	0	0
$489$	−43.2966	−1.95794
$490$	0	0
$491$	−19.8145	−0.894216	−0.447108	−	0.894480i	$-0.647546\pi$
$491$	−19.8145	−0.894216	−0.447108	+	0.894480i	$0.647546\pi$
$492$	0	0
$493$	−10.3958	−0.468205
$494$	0	0
$495$	−14.1663	−0.636729
$496$	0	0
$497$	0.948914	0.0425646
$498$	0	0
$499$	4.50356	0.201607	0.100803	−	0.994906i	$-0.467859\pi$
$499$	4.50356	0.201607	0.100803	+	0.994906i	$0.467859\pi$
$500$	0	0
$501$	−23.2155	−1.03719
$502$	0	0
$503$	8.22951	0.366936	0.183468	−	0.983026i	$-0.441268\pi$
$503$	8.22951	0.366936	0.183468	+	0.983026i	$0.441268\pi$
$504$	0	0
$505$	18.9270	0.842242
$506$	0	0
$507$	−69.1669	−3.07181
$508$	0	0
$509$	32.1685	1.42584	0.712922	−	0.701244i	$-0.247372\pi$
$509$	32.1685	1.42584	0.712922	+	0.701244i	$0.247372\pi$
$510$	0	0
$511$	5.70507	0.252377
$512$	0	0
$513$	−4.82100	−0.212852
$514$	0	0
$515$	2.76110	0.121669
$516$	0	0
$517$	35.5017	1.56136
$518$	0	0
$519$	53.1714	2.33397
$520$	0	0
$521$	30.3308	1.32882	0.664408	−	0.747370i	$-0.268684\pi$
$521$	30.3308	1.32882	0.664408	+	0.747370i	$0.268684\pi$
$522$	0	0
$523$	20.5915	0.900402	0.450201	−	0.892927i	$-0.351352\pi$
$523$	20.5915	0.900402	0.450201	+	0.892927i	$0.351352\pi$
$524$	0	0
$525$	1.79860	0.0784972
$526$	0	0
$527$	0.0396699	0.00172805
$528$	0	0
$529$	−22.5817	−0.981813
$530$	0	0
$531$	61.7681	2.68051
$532$	0	0
$533$	−3.27897	−0.142028
$534$	0	0
$535$	8.78093	0.379633
$536$	0	0
$537$	−18.4711	−0.797087
$538$	0	0
$539$	−19.6972	−0.848419
$540$	0	0
$541$	−33.5576	−1.44275	−0.721377	−	0.692542i	$-0.756491\pi$
$541$	−33.5576	−1.44275	−0.721377	+	0.692542i	$0.756491\pi$
$542$	0	0
$543$	−46.6910	−2.00370
$544$	0	0
$545$	7.63948	0.327240
$546$	0	0
$547$	20.2577	0.866159	0.433079	−	0.901356i	$-0.357427\pi$
$547$	20.2577	0.866159	0.433079	+	0.901356i	$0.357427\pi$
$548$	0	0
$549$	17.7672	0.758285
$550$	0	0
$551$	−3.81315	−0.162446
$552$	0	0
$553$	5.93312	0.252302
$554$	0	0
$555$	−18.0760	−0.767283
$556$	0	0
$557$	−0.531555	−0.0225227	−0.0112614	−	0.999937i	$-0.503585\pi$
$557$	−0.531555	−0.0225227	−0.0112614	+	0.999937i	$0.503585\pi$
$558$	0	0
$559$	−15.1383	−0.640281
$560$	0	0
$561$	−22.6899	−0.957970
$562$	0	0
$563$	−9.64160	−0.406345	−0.203173	−	0.979143i	$-0.565125\pi$
$563$	−9.64160	−0.406345	−0.203173	+	0.979143i	$0.565125\pi$
$564$	0	0
$565$	6.78287	0.285358
$566$	0	0
$567$	0.513456	0.0215631
$568$	0	0
$569$	2.57955	0.108140	0.0540702	−	0.998537i	$-0.482781\pi$
$569$	2.57955	0.108140	0.0540702	+	0.998537i	$0.482781\pi$
$570$	0	0
$571$	−27.6284	−1.15621	−0.578107	−	0.815961i	$-0.696209\pi$
$571$	−27.6284	−1.15621	−0.578107	+	0.815961i	$0.696209\pi$
$572$	0	0
$573$	−15.4197	−0.644168
$574$	0	0
$575$	−0.646760	−0.0269717
$576$	0	0
$577$	−47.3828	−1.97257	−0.986286	−	0.165044i	$-0.947223\pi$
$577$	−47.3828	−1.97257	−0.986286	+	0.165044i	$0.947223\pi$
$578$	0	0
$579$	−47.8216	−1.98740
$580$	0	0
$581$	7.35994	0.305342
$582$	0	0
$583$	3.88642	0.160959
$584$	0	0
$585$	29.1306	1.20440
$586$	0	0
$587$	−27.2206	−1.12351	−0.561757	−	0.827302i	$-0.689875\pi$
$587$	−27.2206	−1.12351	−0.561757	+	0.827302i	$0.689875\pi$
$588$	0	0
$589$	0.0145507	0.000599553 0
$590$	0	0
$591$	−52.5162	−2.16023
$592$	0	0
$593$	−11.1818	−0.459183	−0.229591	−	0.973287i	$-0.573739\pi$
$593$	−11.1818	−0.459183	−0.229591	+	0.973287i	$0.573739\pi$
$594$	0	0
$595$	1.76327	0.0722871
$596$	0	0
$597$	−43.1886	−1.76759
$598$	0	0
$599$	45.2965	1.85076	0.925382	−	0.379035i	$-0.123744\pi$
$599$	45.2965	1.85076	0.925382	+	0.379035i	$0.123744\pi$
$600$	0	0
$601$	−30.8585	−1.25874	−0.629372	−	0.777105i	$-0.716688\pi$
$601$	−30.8585	−1.25874	−0.629372	+	0.777105i	$0.716688\pi$
$602$	0	0
$603$	−62.7109	−2.55379
$604$	0	0
$605$	2.04360	0.0830841
$606$	0	0
$607$	26.9100	1.09224	0.546122	−	0.837706i	$-0.316104\pi$
$607$	26.9100	1.09224	0.546122	+	0.837706i	$0.316104\pi$
$608$	0	0
$609$	−6.85831	−0.277913
$610$	0	0
$611$	−73.0029	−2.95338
$612$	0	0
$613$	−7.22783	−0.291929	−0.145965	−	0.989290i	$-0.546629\pi$
$613$	−7.22783	−0.291929	−0.145965	+	0.989290i	$0.546629\pi$
$614$	0	0
$615$	1.48173	0.0597492
$616$	0	0
$617$	24.1382	0.971769	0.485884	−	0.874023i	$-0.338498\pi$
$617$	24.1382	0.971769	0.485884	+	0.874023i	$0.338498\pi$
$618$	0	0
$619$	−44.8480	−1.80259	−0.901296	−	0.433204i	$-0.857383\pi$
$619$	−44.8480	−1.80259	−0.901296	+	0.433204i	$0.857383\pi$
$620$	0	0
$621$	3.11803	0.125122
$622$	0	0
$623$	−9.59846	−0.384554
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.32257	−0.332371
$628$	0	0
$629$	−17.7210	−0.706582
$630$	0	0
$631$	−21.2457	−0.845779	−0.422890	−	0.906181i	$-0.638984\pi$
$631$	−21.2457	−0.845779	−0.422890	+	0.906181i	$0.638984\pi$
$632$	0	0
$633$	−22.4556	−0.892531
$634$	0	0
$635$	−9.68091	−0.384175
$636$	0	0
$637$	40.5038	1.60482
$638$	0	0
$639$	−6.94504	−0.274741
$640$	0	0
$641$	−26.5712	−1.04950	−0.524750	−	0.851256i	$-0.675841\pi$
$641$	−26.5712	−1.04950	−0.524750	+	0.851256i	$0.675841\pi$
$642$	0	0
$643$	47.8519	1.88709	0.943547	−	0.331239i	$-0.107467\pi$
$643$	47.8519	1.88709	0.943547	+	0.331239i	$0.107467\pi$
$644$	0	0
$645$	6.84084	0.269358
$646$	0	0
$647$	−37.4718	−1.47317	−0.736584	−	0.676346i	$-0.763562\pi$
$647$	−37.4718	−1.47317	−0.736584	+	0.676346i	$0.763562\pi$
$648$	0	0
$649$	39.0517	1.53291
$650$	0	0
$651$	0.0261709	0.00102572
$652$	0	0
$653$	37.5750	1.47043	0.735213	−	0.677837i	$-0.237082\pi$
$653$	37.5750	1.47043	0.735213	+	0.677837i	$0.237082\pi$
$654$	0	0
$655$	−13.7897	−0.538810
$656$	0	0
$657$	−41.7550	−1.62902
$658$	0	0
$659$	3.03046	0.118050	0.0590249	−	0.998257i	$-0.481201\pi$
$659$	3.03046	0.118050	0.0590249	+	0.998257i	$0.481201\pi$
$660$	0	0
$661$	40.2386	1.56510	0.782550	−	0.622587i	$-0.213918\pi$
$661$	40.2386	1.56510	0.782550	+	0.622587i	$0.213918\pi$
$662$	0	0
$663$	46.6579	1.81204
$664$	0	0
$665$	0.646760	0.0250803
$666$	0	0
$667$	2.46619	0.0954912
$668$	0	0
$669$	46.2901	1.78968
$670$	0	0
$671$	11.2330	0.433644
$672$	0	0
$673$	−18.0372	−0.695283	−0.347641	−	0.937628i	$-0.613017\pi$
$673$	−18.0372	−0.695283	−0.347641	+	0.937628i	$0.613017\pi$
$674$	0	0
$675$	−4.82100	−0.185560
$676$	0	0
$677$	1.17918	0.0453196	0.0226598	−	0.999743i	$-0.492787\pi$
$677$	1.17918	0.0453196	0.0226598	+	0.999743i	$0.492787\pi$
$678$	0	0
$679$	4.10101	0.157382
$680$	0	0
$681$	11.8137	0.452703
$682$	0	0
$683$	10.8165	0.413881	0.206940	−	0.978354i	$-0.433649\pi$
$683$	10.8165	0.413881	0.206940	+	0.978354i	$0.433649\pi$
$684$	0	0
$685$	3.25913	0.124525
$686$	0	0
$687$	−21.2318	−0.810044
$688$	0	0
$689$	−7.99173	−0.304461
$690$	0	0
$691$	21.3663	0.812813	0.406407	−	0.913692i	$-0.366782\pi$
$691$	21.3663	0.812813	0.406407	+	0.913692i	$0.366782\pi$
$692$	0	0
$693$	−9.16221	−0.348044
$694$	0	0
$695$	−9.19665	−0.348849
$696$	0	0
$697$	1.45263	0.0550223
$698$	0	0
$699$	45.9448	1.73779
$700$	0	0
$701$	−39.6032	−1.49579	−0.747897	−	0.663815i	$-0.768936\pi$
$701$	−39.6032	−1.49579	−0.747897	+	0.663815i	$0.768936\pi$
$702$	0	0
$703$	−6.49997	−0.245151
$704$	0	0
$705$	32.9893	1.24245
$706$	0	0
$707$	12.2412	0.460379
$708$	0	0
$709$	−36.8989	−1.38577	−0.692884	−	0.721049i	$-0.743660\pi$
$709$	−36.8989	−1.38577	−0.692884	+	0.721049i	$0.743660\pi$
$710$	0	0
$711$	−43.4241	−1.62853
$712$	0	0
$713$	−0.00941083	−0.000352438 0
$714$	0	0
$715$	18.4173	0.688766
$716$	0	0
$717$	39.1313	1.46139
$718$	0	0
$719$	−27.1284	−1.01172	−0.505860	−	0.862616i	$-0.668825\pi$
$719$	−27.1284	−1.01172	−0.505860	+	0.862616i	$0.668825\pi$
$720$	0	0
$721$	1.78577	0.0665055
$722$	0	0
$723$	50.1374	1.86463
$724$	0	0
$725$	−3.81315	−0.141617
$726$	0	0
$727$	−45.2262	−1.67735	−0.838674	−	0.544634i	$-0.816668\pi$
$727$	−45.2262	−1.67735	−0.838674	+	0.544634i	$0.816668\pi$
$728$	0	0
$729$	−43.9786	−1.62884
$730$	0	0
$731$	6.70648	0.248048
$732$	0	0
$733$	25.7064	0.949489	0.474744	−	0.880124i	$-0.342541\pi$
$733$	25.7064	0.949489	0.474744	+	0.880124i	$0.342541\pi$
$734$	0	0
$735$	−18.3033	−0.675126
$736$	0	0
$737$	−39.6478	−1.46045
$738$	0	0
$739$	−24.4766	−0.900387	−0.450193	−	0.892931i	$-0.648645\pi$
$739$	−24.4766	−0.900387	−0.450193	+	0.892931i	$0.648645\pi$
$740$	0	0
$741$	17.1139	0.628694
$742$	0	0
$743$	42.8706	1.57277	0.786385	−	0.617737i	$-0.211950\pi$
$743$	42.8706	1.57277	0.786385	+	0.617737i	$0.211950\pi$
$744$	0	0
$745$	−14.8772	−0.545057
$746$	0	0
$747$	−53.8669	−1.97089
$748$	0	0
$749$	5.67915	0.207512
$750$	0	0
$751$	2.13320	0.0778415	0.0389207	−	0.999242i	$-0.487608\pi$
$751$	2.13320	0.0778415	0.0389207	+	0.999242i	$0.487608\pi$
$752$	0	0
$753$	−43.7311	−1.59365
$754$	0	0
$755$	14.4817	0.527042
$756$	0	0
$757$	27.5713	1.00210	0.501048	−	0.865420i	$-0.332948\pi$
$757$	27.5713	1.00210	0.501048	+	0.865420i	$0.332948\pi$
$758$	0	0
$759$	5.38270	0.195380
$760$	0	0
$761$	28.1465	1.02031	0.510155	−	0.860083i	$-0.329588\pi$
$761$	28.1465	1.02031	0.510155	+	0.860083i	$0.329588\pi$
$762$	0	0
$763$	4.94091	0.178873
$764$	0	0
$765$	−12.9053	−0.466591
$766$	0	0
$767$	−80.3029	−2.89957
$768$	0	0
$769$	44.8313	1.61666	0.808329	−	0.588731i	$-0.200372\pi$
$769$	44.8313	1.61666	0.808329	+	0.588731i	$0.200372\pi$
$770$	0	0
$771$	−59.5265	−2.14380
$772$	0	0
$773$	42.0321	1.51179	0.755895	−	0.654693i	$-0.227202\pi$
$773$	42.0321	1.51179	0.755895	+	0.654693i	$0.227202\pi$
$774$	0	0
$775$	0.0145507	0.000522678 0
$776$	0	0
$777$	−11.6908	−0.419406
$778$	0	0
$779$	0.532818	0.0190902
$780$	0	0
$781$	−4.39087	−0.157118
$782$	0	0
$783$	18.3832	0.656961
$784$	0	0
$785$	6.76070	0.241300
$786$	0	0
$787$	2.76957	0.0987245	0.0493623	−	0.998781i	$-0.484281\pi$
$787$	2.76957	0.0987245	0.0493623	+	0.998781i	$0.484281\pi$
$788$	0	0
$789$	38.1051	1.35658
$790$	0	0
$791$	4.38689	0.155980
$792$	0	0
$793$	−23.0986	−0.820256
$794$	0	0
$795$	3.61138	0.128083
$796$	0	0
$797$	−39.1280	−1.38599	−0.692993	−	0.720944i	$-0.743708\pi$
$797$	−39.1280	−1.38599	−0.692993	+	0.720944i	$0.743708\pi$
$798$	0	0
$799$	32.3414	1.14416
$800$	0	0
$801$	70.2504	2.48218
$802$	0	0
$803$	−26.3988	−0.931594
$804$	0	0
$805$	−0.418298	−0.0147431
$806$	0	0
$807$	3.82004	0.134472
$808$	0	0
$809$	14.0375	0.493531	0.246766	−	0.969075i	$-0.420632\pi$
$809$	14.0375	0.493531	0.246766	+	0.969075i	$0.420632\pi$
$810$	0	0
$811$	23.1474	0.812817	0.406408	−	0.913692i	$-0.366781\pi$
$811$	23.1474	0.812817	0.406408	+	0.913692i	$0.366781\pi$
$812$	0	0
$813$	−2.42375	−0.0850047
$814$	0	0
$815$	−15.5691	−0.545361
$816$	0	0
$817$	2.45991	0.0860612
$818$	0	0
$819$	18.8405	0.658339
$820$	0	0
$821$	−54.1127	−1.88855	−0.944273	−	0.329164i	$-0.893233\pi$
$821$	−54.1127	−1.88855	−0.944273	+	0.329164i	$0.893233\pi$
$822$	0	0
$823$	46.1025	1.60703	0.803516	−	0.595284i	$-0.202960\pi$
$823$	46.1025	1.60703	0.803516	+	0.595284i	$0.202960\pi$
$824$	0	0
$825$	−8.32257	−0.289755
$826$	0	0
$827$	−28.3825	−0.986955	−0.493478	−	0.869759i	$-0.664275\pi$
$827$	−28.3825	−0.986955	−0.493478	+	0.869759i	$0.664275\pi$
$828$	0	0
$829$	−11.6104	−0.403245	−0.201623	−	0.979463i	$-0.564621\pi$
$829$	−11.6104	−0.403245	−0.201623	+	0.979463i	$0.564621\pi$
$830$	0	0
$831$	−79.9519	−2.77350
$832$	0	0
$833$	−17.9438	−0.621716
$834$	0	0
$835$	−8.34808	−0.288897
$836$	0	0
$837$	−0.0701491	−0.00242471
$838$	0	0
$839$	−26.1331	−0.902215	−0.451107	−	0.892470i	$-0.648971\pi$
$839$	−26.1331	−0.902215	−0.451107	+	0.892470i	$0.648971\pi$
$840$	0	0
$841$	−14.4599	−0.498618
$842$	0	0
$843$	−0.955106	−0.0328956
$844$	0	0
$845$	−24.8718	−0.855617
$846$	0	0
$847$	1.32172	0.0454148
$848$	0	0
$849$	−19.3360	−0.663609
$850$	0	0
$851$	4.20392	0.144109
$852$	0	0
$853$	17.0511	0.583818	0.291909	−	0.956446i	$-0.405710\pi$
$853$	17.0511	0.583818	0.291909	+	0.956446i	$0.405710\pi$
$854$	0	0
$855$	−4.73359	−0.161885
$856$	0	0
$857$	14.0556	0.480131	0.240065	−	0.970757i	$-0.422831\pi$
$857$	14.0556	0.480131	0.240065	+	0.970757i	$0.422831\pi$
$858$	0	0
$859$	40.8195	1.39274	0.696371	−	0.717682i	$-0.254797\pi$
$859$	40.8195	1.39274	0.696371	+	0.717682i	$0.254797\pi$
$860$	0	0
$861$	0.958325	0.0326596
$862$	0	0
$863$	−32.2655	−1.09833	−0.549165	−	0.835714i	$-0.685054\pi$
$863$	−32.2655	−1.09833	−0.549165	+	0.835714i	$0.685054\pi$
$864$	0	0
$865$	19.1200	0.650099
$866$	0	0
$867$	26.6058	0.903579
$868$	0	0
$869$	−27.4541	−0.931316
$870$	0	0
$871$	81.5287	2.76249
$872$	0	0
$873$	−30.0150	−1.01586
$874$	0	0
$875$	0.646760	0.0218645
$876$	0	0
$877$	21.4974	0.725916	0.362958	−	0.931806i	$-0.381767\pi$
$877$	21.4974	0.725916	0.362958	+	0.931806i	$0.381767\pi$
$878$	0	0
$879$	11.1905	0.377445
$880$	0	0
$881$	−40.5627	−1.36659	−0.683296	−	0.730141i	$-0.739454\pi$
$881$	−40.5627	−1.36659	−0.683296	+	0.730141i	$0.739454\pi$
$882$	0	0
$883$	9.41504	0.316841	0.158421	−	0.987372i	$-0.449360\pi$
$883$	9.41504	0.316841	0.158421	+	0.987372i	$0.449360\pi$
$884$	0	0
$885$	36.2881	1.21981
$886$	0	0
$887$	−24.2113	−0.812935	−0.406468	−	0.913665i	$-0.633240\pi$
$887$	−24.2113	−0.812935	−0.406468	+	0.913665i	$0.633240\pi$
$888$	0	0
$889$	−6.26122	−0.209995
$890$	0	0
$891$	−2.37590	−0.0795955
$892$	0	0
$893$	11.8627	0.396969
$894$	0	0
$895$	−6.64205	−0.222019
$896$	0	0
$897$	−11.0686	−0.369569
$898$	0	0
$899$	−0.0554841	−0.00185050
$900$	0	0
$901$	3.54045	0.117950
$902$	0	0
$903$	4.42438	0.147234
$904$	0	0
$905$	−16.7897	−0.558108
$906$	0	0
$907$	15.8060	0.524830	0.262415	−	0.964955i	$-0.415481\pi$
$907$	15.8060	0.524830	0.262415	+	0.964955i	$0.415481\pi$
$908$	0	0
$909$	−89.5928	−2.97161
$910$	0	0
$911$	−49.8333	−1.65105	−0.825525	−	0.564365i	$-0.809121\pi$
$911$	−49.8333	−1.65105	−0.825525	+	0.564365i	$0.809121\pi$
$912$	0	0
$913$	−34.0564	−1.12710
$914$	0	0
$915$	10.4380	0.345071
$916$	0	0
$917$	−8.91865	−0.294520
$918$	0	0
$919$	−21.6976	−0.715737	−0.357869	−	0.933772i	$-0.616496\pi$
$919$	−21.6976	−0.715737	−0.357869	+	0.933772i	$0.616496\pi$
$920$	0	0
$921$	−50.9407	−1.67855
$922$	0	0
$923$	9.02905	0.297195
$924$	0	0
$925$	−6.49997	−0.213718
$926$	0	0
$927$	−13.0699	−0.429272
$928$	0	0
$929$	35.6115	1.16838	0.584188	−	0.811619i	$-0.301413\pi$
$929$	35.6115	1.16838	0.584188	+	0.811619i	$0.301413\pi$
$930$	0	0
$931$	−6.58170	−0.215707
$932$	0	0
$933$	−21.7078	−0.710680
$934$	0	0
$935$	−8.15911	−0.266831
$936$	0	0
$937$	38.0134	1.24184	0.620922	−	0.783872i	$-0.286758\pi$
$937$	38.0134	1.24184	0.620922	+	0.783872i	$0.286758\pi$
$938$	0	0
$939$	−34.2535	−1.11782
$940$	0	0
$941$	−15.9582	−0.520222	−0.260111	−	0.965579i	$-0.583759\pi$
$941$	−15.9582	−0.520222	−0.260111	+	0.965579i	$0.583759\pi$
$942$	0	0
$943$	−0.344605	−0.0112219
$944$	0	0
$945$	−3.11803	−0.101429
$946$	0	0
$947$	60.5729	1.96836	0.984178	−	0.177185i	$-0.0566990\pi$
$947$	60.5729	1.96836	0.984178	+	0.177185i	$0.0566990\pi$
$948$	0	0
$949$	54.2845	1.76215
$950$	0	0
$951$	−20.4621	−0.663528
$952$	0	0
$953$	29.3094	0.949423	0.474712	−	0.880141i	$-0.342552\pi$
$953$	29.3094	0.949423	0.474712	+	0.880141i	$0.342552\pi$
$954$	0	0
$955$	−5.54480	−0.179425
$956$	0	0
$957$	31.7352	1.02585
$958$	0	0
$959$	2.10788	0.0680669
$960$	0	0
$961$	−30.9998	−0.999993
$962$	0	0
$963$	−41.5653	−1.33942
$964$	0	0
$965$	−17.1962	−0.553567
$966$	0	0
$967$	−14.7396	−0.473994	−0.236997	−	0.971510i	$-0.576163\pi$
$967$	−14.7396	−0.473994	−0.236997	+	0.971510i	$0.576163\pi$
$968$	0	0
$969$	−7.58170	−0.243559
$970$	0	0
$971$	58.1559	1.86631	0.933155	−	0.359474i	$-0.117044\pi$
$971$	58.1559	1.86631	0.933155	+	0.359474i	$0.117044\pi$
$972$	0	0
$973$	−5.94802	−0.190685
$974$	0	0
$975$	17.1139	0.548083
$976$	0	0
$977$	1.56869	0.0501869	0.0250935	−	0.999685i	$-0.492012\pi$
$977$	1.56869	0.0501869	0.0250935	+	0.999685i	$0.492012\pi$
$978$	0	0
$979$	44.4145	1.41949
$980$	0	0
$981$	−36.1622	−1.15457
$982$	0	0
$983$	11.5312	0.367788	0.183894	−	0.982946i	$-0.441130\pi$
$983$	11.5312	0.367788	0.183894	+	0.982946i	$0.441130\pi$
$984$	0	0
$985$	−18.8844	−0.601706
$986$	0	0
$987$	21.3361	0.679137
$988$	0	0
$989$	−1.59097	−0.0505899
$990$	0	0
$991$	19.3870	0.615850	0.307925	−	0.951411i	$-0.400365\pi$
$991$	19.3870	0.615850	0.307925	+	0.951411i	$0.400365\pi$
$992$	0	0
$993$	36.0564	1.14421
$994$	0	0
$995$	−15.5302	−0.492342
$996$	0	0
$997$	36.5724	1.15826	0.579129	−	0.815236i	$-0.303393\pi$
$997$	36.5724	1.15826	0.579129	+	0.815236i	$0.303393\pi$
$998$	0	0
$999$	31.3364	0.991439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.ck.1.1		5
4.3	odd	2		6080.2.a.cj.1.5		5
8.3	odd	2		3040.2.a.w.1.1	yes	5
8.5	even	2		3040.2.a.v.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.v.1.5	✓	5	8.5	even	2
3040.2.a.w.1.1	yes	5	8.3	odd	2
6080.2.a.cj.1.5		5	4.3	odd	2
6080.2.a.ck.1.1		5	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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78093 q^{3} -1.00000 q^{5} -0.646760 q^{7} +4.73359 q^{9} +O(q^{10})\)	\(q-2.78093 q^{3} -1.00000 q^{5} -0.646760 q^{7} +4.73359 q^{9} +2.99272 q^{11} -6.15401 q^{13} +2.78093 q^{15} +2.72632 q^{17} +1.00000 q^{19} +1.79860 q^{21} -0.646760 q^{23} +1.00000 q^{25} -4.82100 q^{27} -3.81315 q^{29} +0.0145507 q^{31} -8.32257 q^{33} +0.646760 q^{35} -6.49997 q^{37} +17.1139 q^{39} +0.532818 q^{41} +2.45991 q^{43} -4.73359 q^{45} +11.8627 q^{47} -6.58170 q^{49} -7.58170 q^{51} +1.29862 q^{53} -2.99272 q^{55} -2.78093 q^{57} +13.0489 q^{59} +3.75343 q^{61} -3.06150 q^{63} +6.15401 q^{65} -13.2481 q^{67} +1.79860 q^{69} -1.46718 q^{71} -8.82100 q^{73} -2.78093 q^{75} -1.93557 q^{77} -9.17361 q^{79} -0.793890 q^{81} -11.3797 q^{83} -2.72632 q^{85} +10.6041 q^{87} +14.8408 q^{89} +3.98017 q^{91} -0.0404646 q^{93} -1.00000 q^{95} -6.34086 q^{97} +14.1663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9	\( 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9} + 10 q^{11} - 2 q^{15} + 4 q^{17} + 5 q^{19} - 10 q^{21} + 2 q^{23} + 5 q^{25} + 8 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{33} - 2 q^{35} - 2 q^{37} + 10 q^{39} + 12 q^{41} - 2 q^{43} - 19 q^{45} + 22 q^{47} + 19 q^{49} + 14 q^{51} + 18 q^{53} - 10 q^{55} + 2 q^{57} + 4 q^{59} - 6 q^{61} + 10 q^{63} + 20 q^{67} - 10 q^{69} + 2 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 14 q^{79} + 13 q^{81} - 14 q^{83} - 4 q^{85} - 12 q^{87} + 22 q^{89} + 40 q^{91} - 8 q^{93} - 5 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9 + 10 * q^11 - 2 * q^15 + 4 * q^17 + 5 * q^19 - 10 * q^21 + 2 * q^23 + 5 * q^25 + 8 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^33 - 2 * q^35 - 2 * q^37 + 10 * q^39 + 12 * q^41 - 2 * q^43 - 19 * q^45 + 22 * q^47 + 19 * q^49 + 14 * q^51 + 18 * q^53 - 10 * q^55 + 2 * q^57 + 4 * q^59 - 6 * q^61 + 10 * q^63 + 20 * q^67 - 10 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 + 13 * q^81 - 14 * q^83 - 4 * q^85 - 12 * q^87 + 22 * q^89 + 40 * q^91 - 8 * q^93 - 5 * q^95 - 10 * q^97 + 6 * q^99

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