LMFDB - Embedded newform 6240.2.a.bz.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 8 $ x^3 - x^2 - 14*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.88824$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.88824 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +4.88824 q^{7} +1.00000 q^{9} +4.88824 q^{11} -1.00000 q^{13} -1.00000 q^{15} -8.11838 q^{17} -7.00662 q^{19} -4.88824 q^{21} +6.11838 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.77647 q^{29} -5.23014 q^{31} -4.88824 q^{33} +4.88824 q^{35} +8.11838 q^{37} +1.00000 q^{39} -5.89485 q^{41} +5.77647 q^{43} +1.00000 q^{45} +1.23014 q^{47} +16.8949 q^{49} +8.11838 q^{51} -0.118379 q^{53} +4.88824 q^{55} +7.00662 q^{57} -1.23014 q^{59} +12.1184 q^{61} +4.88824 q^{63} -1.00000 q^{65} -2.76986 q^{67} -6.11838 q^{69} -3.11176 q^{71} +2.00000 q^{73} -1.00000 q^{75} +23.8949 q^{77} +2.11838 q^{79} +1.00000 q^{81} +2.76986 q^{83} -8.11838 q^{85} -7.77647 q^{87} +6.34191 q^{89} -4.88824 q^{91} +5.23014 q^{93} -7.00662 q^{95} -15.6713 q^{97} +4.88824 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9} + 5 q^{11} - 3 q^{13} - 3 q^{15} - 9 q^{17} + 4 q^{19} - 5 q^{21} + 3 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} - 10 q^{31} - 5 q^{33} + 5 q^{35} + 9 q^{37} + 3 q^{39} + 17 q^{41} - 2 q^{43} + 3 q^{45} - 2 q^{47} + 16 q^{49} + 9 q^{51} + 15 q^{53} + 5 q^{55} - 4 q^{57} + 2 q^{59} + 21 q^{61} + 5 q^{63} - 3 q^{65} - 14 q^{67} - 3 q^{69} - 19 q^{71} + 6 q^{73} - 3 q^{75} + 37 q^{77} - 9 q^{79} + 3 q^{81} + 14 q^{83} - 9 q^{85} - 4 q^{87} + 23 q^{89} - 5 q^{91} + 10 q^{93} + 4 q^{95} + 7 q^{97} + 5 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9 + 5 * q^11 - 3 * q^13 - 3 * q^15 - 9 * q^17 + 4 * q^19 - 5 * q^21 + 3 * q^23 + 3 * q^25 - 3 * q^27 + 4 * q^29 - 10 * q^31 - 5 * q^33 + 5 * q^35 + 9 * q^37 + 3 * q^39 + 17 * q^41 - 2 * q^43 + 3 * q^45 - 2 * q^47 + 16 * q^49 + 9 * q^51 + 15 * q^53 + 5 * q^55 - 4 * q^57 + 2 * q^59 + 21 * q^61 + 5 * q^63 - 3 * q^65 - 14 * q^67 - 3 * q^69 - 19 * q^71 + 6 * q^73 - 3 * q^75 + 37 * q^77 - 9 * q^79 + 3 * q^81 + 14 * q^83 - 9 * q^85 - 4 * q^87 + 23 * q^89 - 5 * q^91 + 10 * q^93 + 4 * q^95 + 7 * q^97 + 5 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.88824	1.84758	0.923790	−	0.382900i	$-0.125075\pi$
$7$	4.88824	1.84758	0.923790	+	0.382900i	$0.125075\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.88824	1.47386	0.736929	−	0.675970i	$-0.236275\pi$
$11$	4.88824	1.47386	0.736929	+	0.675970i	$0.236275\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−8.11838	−1.96900	−0.984498	−	0.175395i	$-0.943880\pi$
$17$	−8.11838	−1.96900	−0.984498	+	0.175395i	$0.943880\pi$
$18$	0	0
$19$	−7.00662	−1.60743	−0.803714	−	0.595016i	$-0.797146\pi$
$19$	−7.00662	−1.60743	−0.803714	+	0.595016i	$0.797146\pi$
$20$	0	0
$21$	−4.88824	−1.06670
$22$	0	0
$23$	6.11838	1.27577	0.637885	−	0.770132i	$-0.279809\pi$
$23$	6.11838	1.27577	0.637885	+	0.770132i	$0.279809\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.77647	1.44405	0.722027	−	0.691865i	$-0.243210\pi$
$29$	7.77647	1.44405	0.722027	+	0.691865i	$0.243210\pi$
$30$	0	0
$31$	−5.23014	−0.939361	−0.469681	−	0.882836i	$-0.655631\pi$
$31$	−5.23014	−0.939361	−0.469681	+	0.882836i	$0.655631\pi$
$32$	0	0
$33$	−4.88824	−0.850933
$34$	0	0
$35$	4.88824	0.826263
$36$	0	0
$37$	8.11838	1.33465	0.667327	−	0.744765i	$-0.267439\pi$
$37$	8.11838	1.33465	0.667327	+	0.744765i	$0.267439\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−5.89485	−0.920621	−0.460311	−	0.887758i	$-0.652262\pi$
$41$	−5.89485	−0.920621	−0.460311	+	0.887758i	$0.652262\pi$
$42$	0	0
$43$	5.77647	0.880904	0.440452	−	0.897776i	$-0.354818\pi$
$43$	5.77647	0.880904	0.440452	+	0.897776i	$0.354818\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	1.23014	0.179435	0.0897174	−	0.995967i	$-0.471404\pi$
$47$	1.23014	0.179435	0.0897174	+	0.995967i	$0.471404\pi$
$48$	0	0
$49$	16.8949	2.41355
$50$	0	0
$51$	8.11838	1.13680
$52$	0	0
$53$	−0.118379	−0.0162606	−0.00813032	−	0.999967i	$-0.502588\pi$
$53$	−0.118379	−0.0162606	−0.00813032	+	0.999967i	$0.502588\pi$
$54$	0	0
$55$	4.88824	0.659130
$56$	0	0
$57$	7.00662	0.928049
$58$	0	0
$59$	−1.23014	−0.160151	−0.0800755	−	0.996789i	$-0.525516\pi$
$59$	−1.23014	−0.160151	−0.0800755	+	0.996789i	$0.525516\pi$
$60$	0	0
$61$	12.1184	1.55160	0.775800	−	0.630979i	$-0.217347\pi$
$61$	12.1184	1.55160	0.775800	+	0.630979i	$0.217347\pi$
$62$	0	0
$63$	4.88824	0.615860
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−2.76986	−0.338392	−0.169196	−	0.985582i	$-0.554117\pi$
$67$	−2.76986	−0.338392	−0.169196	+	0.985582i	$0.554117\pi$
$68$	0	0
$69$	−6.11838	−0.736566
$70$	0	0
$71$	−3.11176	−0.369298	−0.184649	−	0.982804i	$-0.559115\pi$
$71$	−3.11176	−0.369298	−0.184649	+	0.982804i	$0.559115\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	23.8949	2.72307
$78$	0	0
$79$	2.11838	0.238336	0.119168	−	0.992874i	$-0.461977\pi$
$79$	2.11838	0.238336	0.119168	+	0.992874i	$0.461977\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.76986	0.304031	0.152016	−	0.988378i	$-0.451424\pi$
$83$	2.76986	0.304031	0.152016	+	0.988378i	$0.451424\pi$
$84$	0	0
$85$	−8.11838	−0.880562
$86$	0	0
$87$	−7.77647	−0.833725
$88$	0	0
$89$	6.34191	0.672241	0.336120	−	0.941819i	$-0.390885\pi$
$89$	6.34191	0.672241	0.336120	+	0.941819i	$0.390885\pi$
$90$	0	0
$91$	−4.88824	−0.512426
$92$	0	0
$93$	5.23014	0.542341
$94$	0	0
$95$	−7.00662	−0.718864
$96$	0	0
$97$	−15.6713	−1.59118	−0.795591	−	0.605834i	$-0.792839\pi$
$97$	−15.6713	−1.59118	−0.795591	+	0.605834i	$0.792839\pi$
$98$	0	0
$99$	4.88824	0.491286
$100$	0	0
$101$	18.2368	1.81463	0.907313	−	0.420457i	$-0.138130\pi$
$101$	18.2368	1.81463	0.907313	+	0.420457i	$0.138130\pi$
$102$	0	0
$103$	11.5529	1.13835	0.569173	−	0.822218i	$-0.307264\pi$
$103$	11.5529	1.13835	0.569173	+	0.822218i	$0.307264\pi$
$104$	0	0
$105$	−4.88824	−0.477043
$106$	0	0
$107$	−5.88162	−0.568598	−0.284299	−	0.958736i	$-0.591761\pi$
$107$	−5.88162	−0.568598	−0.284299	+	0.958736i	$0.591761\pi$
$108$	0	0
$109$	15.7765	1.51111	0.755556	−	0.655084i	$-0.227367\pi$
$109$	15.7765	1.51111	0.755556	+	0.655084i	$0.227367\pi$
$110$	0	0
$111$	−8.11838	−0.770562
$112$	0	0
$113$	9.55294	0.898665	0.449333	−	0.893365i	$-0.351662\pi$
$113$	9.55294	0.898665	0.449333	+	0.893365i	$0.351662\pi$
$114$	0	0
$115$	6.11838	0.570542
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−39.6846	−3.63788
$120$	0	0
$121$	12.8949	1.17226
$122$	0	0
$123$	5.89485	0.531521
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.5529	−1.73504	−0.867522	−	0.497399i	$-0.834288\pi$
$127$	−19.5529	−1.73504	−0.867522	+	0.497399i	$0.834288\pi$
$128$	0	0
$129$	−5.77647	−0.508590
$130$	0	0
$131$	10.4603	0.913919	0.456960	−	0.889487i	$-0.348938\pi$
$131$	10.4603	0.913919	0.456960	+	0.889487i	$0.348938\pi$
$132$	0	0
$133$	−34.2500	−2.96985
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−13.6713	−1.15959	−0.579793	−	0.814764i	$-0.696867\pi$
$139$	−13.6713	−1.15959	−0.579793	+	0.814764i	$0.696867\pi$
$140$	0	0
$141$	−1.23014	−0.103597
$142$	0	0
$143$	−4.88824	−0.408775
$144$	0	0
$145$	7.77647	0.645801
$146$	0	0
$147$	−16.8949	−1.39346
$148$	0	0
$149$	−11.6713	−0.956152	−0.478076	−	0.878318i	$-0.658666\pi$
$149$	−11.6713	−0.956152	−0.478076	+	0.878318i	$0.658666\pi$
$150$	0	0
$151$	−2.76986	−0.225408	−0.112704	−	0.993629i	$-0.535951\pi$
$151$	−2.76986	−0.225408	−0.112704	+	0.993629i	$0.535951\pi$
$152$	0	0
$153$	−8.11838	−0.656332
$154$	0	0
$155$	−5.23014	−0.420095
$156$	0	0
$157$	13.5529	1.08164	0.540821	−	0.841137i	$-0.318114\pi$
$157$	13.5529	1.08164	0.540821	+	0.841137i	$0.318114\pi$
$158$	0	0
$159$	0.118379	0.00938809
$160$	0	0
$161$	29.9081	2.35709
$162$	0	0
$163$	−2.66471	−0.208716	−0.104358	−	0.994540i	$-0.533279\pi$
$163$	−2.66471	−0.208716	−0.104358	+	0.994540i	$0.533279\pi$
$164$	0	0
$165$	−4.88824	−0.380549
$166$	0	0
$167$	12.7831	0.989185	0.494592	−	0.869125i	$-0.335317\pi$
$167$	12.7831	0.989185	0.494592	+	0.869125i	$0.335317\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.00662	−0.535809
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	4.88824	0.369516
$176$	0	0
$177$	1.23014	0.0924632
$178$	0	0
$179$	12.2368	0.914618	0.457309	−	0.889308i	$-0.348813\pi$
$179$	12.2368	0.914618	0.457309	+	0.889308i	$0.348813\pi$
$180$	0	0
$181$	−9.89485	−0.735479	−0.367739	−	0.929929i	$-0.619868\pi$
$181$	−9.89485	−0.735479	−0.367739	+	0.929929i	$0.619868\pi$
$182$	0	0
$183$	−12.1184	−0.895816
$184$	0	0
$185$	8.11838	0.596875
$186$	0	0
$187$	−39.6846	−2.90202
$188$	0	0
$189$	−4.88824	−0.355567
$190$	0	0
$191$	−1.77647	−0.128541	−0.0642705	−	0.997933i	$-0.520472\pi$
$191$	−1.77647	−0.128541	−0.0642705	+	0.997933i	$0.520472\pi$
$192$	0	0
$193$	−11.4346	−0.823078	−0.411539	−	0.911392i	$-0.635009\pi$
$193$	−11.4346	−0.823078	−0.411539	+	0.911392i	$0.635009\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	17.5529	1.25060	0.625298	−	0.780386i	$-0.284978\pi$
$197$	17.5529	1.25060	0.625298	+	0.780386i	$0.284978\pi$
$198$	0	0
$199$	18.0132	1.27692	0.638462	−	0.769653i	$-0.279571\pi$
$199$	18.0132	1.27692	0.638462	+	0.769653i	$0.279571\pi$
$200$	0	0
$201$	2.76986	0.195371
$202$	0	0
$203$	38.0132	2.66801
$204$	0	0
$205$	−5.89485	−0.411714
$206$	0	0
$207$	6.11838	0.425257
$208$	0	0
$209$	−34.2500	−2.36912
$210$	0	0
$211$	−23.5529	−1.62145	−0.810726	−	0.585426i	$-0.800927\pi$
$211$	−23.5529	−1.62145	−0.810726	+	0.585426i	$0.800927\pi$
$212$	0	0
$213$	3.11176	0.213215
$214$	0	0
$215$	5.77647	0.393952
$216$	0	0
$217$	−25.5662	−1.73555
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	8.11838	0.546101
$222$	0	0
$223$	−15.2434	−1.02077	−0.510386	−	0.859945i	$-0.670497\pi$
$223$	−15.2434	−1.02077	−0.510386	+	0.859945i	$0.670497\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−21.0198	−1.39514	−0.697568	−	0.716519i	$-0.745735\pi$
$227$	−21.0198	−1.39514	−0.697568	+	0.716519i	$0.745735\pi$
$228$	0	0
$229$	−19.7765	−1.30687	−0.653433	−	0.756984i	$-0.726672\pi$
$229$	−19.7765	−1.30687	−0.653433	+	0.756984i	$0.726672\pi$
$230$	0	0
$231$	−23.8949	−1.57217
$232$	0	0
$233$	21.2110	1.38958	0.694791	−	0.719212i	$-0.255497\pi$
$233$	21.2110	1.38958	0.694791	+	0.719212i	$0.255497\pi$
$234$	0	0
$235$	1.23014	0.0802457
$236$	0	0
$237$	−2.11838	−0.137604
$238$	0	0
$239$	−3.11176	−0.201283	−0.100642	−	0.994923i	$-0.532090\pi$
$239$	−3.11176	−0.201283	−0.100642	+	0.994923i	$0.532090\pi$
$240$	0	0
$241$	−13.7897	−0.888273	−0.444136	−	0.895959i	$-0.646489\pi$
$241$	−13.7897	−0.888273	−0.444136	+	0.895959i	$0.646489\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	16.8949	1.07937
$246$	0	0
$247$	7.00662	0.445820
$248$	0	0
$249$	−2.76986	−0.175533
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	29.9081	1.88030
$254$	0	0
$255$	8.11838	0.508393
$256$	0	0
$257$	−7.53971	−0.470314	−0.235157	−	0.971957i	$-0.575560\pi$
$257$	−7.53971	−0.470314	−0.235157	+	0.971957i	$0.575560\pi$
$258$	0	0
$259$	39.6846	2.46588
$260$	0	0
$261$	7.77647	0.481352
$262$	0	0
$263$	2.46029	0.151708	0.0758539	−	0.997119i	$-0.475832\pi$
$263$	2.46029	0.151708	0.0758539	+	0.997119i	$0.475832\pi$
$264$	0	0
$265$	−0.118379	−0.00727198
$266$	0	0
$267$	−6.34191	−0.388118
$268$	0	0
$269$	7.77647	0.474140	0.237070	−	0.971493i	$-0.423813\pi$
$269$	7.77647	0.474140	0.237070	+	0.971493i	$0.423813\pi$
$270$	0	0
$271$	4.54633	0.276170	0.138085	−	0.990420i	$-0.455905\pi$
$271$	4.54633	0.276170	0.138085	+	0.990420i	$0.455905\pi$
$272$	0	0
$273$	4.88824	0.295850
$274$	0	0
$275$	4.88824	0.294772
$276$	0	0
$277$	−7.77647	−0.467243	−0.233621	−	0.972328i	$-0.575058\pi$
$277$	−7.77647	−0.467243	−0.233621	+	0.972328i	$0.575058\pi$
$278$	0	0
$279$	−5.23014	−0.313120
$280$	0	0
$281$	−0.460287	−0.0274584	−0.0137292	−	0.999906i	$-0.504370\pi$
$281$	−0.460287	−0.0274584	−0.0137292	+	0.999906i	$0.504370\pi$
$282$	0	0
$283$	−6.46029	−0.384024	−0.192012	−	0.981393i	$-0.561501\pi$
$283$	−6.46029	−0.384024	−0.192012	+	0.981393i	$0.561501\pi$
$284$	0	0
$285$	7.00662	0.415036
$286$	0	0
$287$	−28.8154	−1.70092
$288$	0	0
$289$	48.9081	2.87695
$290$	0	0
$291$	15.6713	0.918669
$292$	0	0
$293$	29.7897	1.74033	0.870167	−	0.492758i	$-0.164011\pi$
$293$	29.7897	1.74033	0.870167	+	0.492758i	$0.164011\pi$
$294$	0	0
$295$	−1.23014	−0.0716217
$296$	0	0
$297$	−4.88824	−0.283644
$298$	0	0
$299$	−6.11838	−0.353835
$300$	0	0
$301$	28.2368	1.62754
$302$	0	0
$303$	−18.2368	−1.04767
$304$	0	0
$305$	12.1184	0.693896
$306$	0	0
$307$	4.44118	0.253472	0.126736	−	0.991937i	$-0.459550\pi$
$307$	4.44118	0.253472	0.126736	+	0.991937i	$0.459550\pi$
$308$	0	0
$309$	−11.5529	−0.657224
$310$	0	0
$311$	8.68381	0.492414	0.246207	−	0.969217i	$-0.420816\pi$
$311$	8.68381	0.492414	0.246207	+	0.969217i	$0.420816\pi$
$312$	0	0
$313$	−11.5397	−0.652263	−0.326132	−	0.945324i	$-0.605745\pi$
$313$	−11.5397	−0.652263	−0.326132	+	0.945324i	$0.605745\pi$
$314$	0	0
$315$	4.88824	0.275421
$316$	0	0
$317$	−18.4735	−1.03758	−0.518788	−	0.854903i	$-0.673617\pi$
$317$	−18.4735	−1.03758	−0.518788	+	0.854903i	$0.673617\pi$
$318$	0	0
$319$	38.0132	2.12833
$320$	0	0
$321$	5.88162	0.328280
$322$	0	0
$323$	56.8824	3.16502
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−15.7765	−0.872441
$328$	0	0
$329$	6.01323	0.331520
$330$	0	0
$331$	33.4669	1.83951	0.919754	−	0.392496i	$-0.128388\pi$
$331$	33.4669	1.83951	0.919754	+	0.392496i	$0.128388\pi$
$332$	0	0
$333$	8.11838	0.444884
$334$	0	0
$335$	−2.76986	−0.151333
$336$	0	0
$337$	−8.46029	−0.460861	−0.230431	−	0.973089i	$-0.574013\pi$
$337$	−8.46029	−0.460861	−0.230431	+	0.973089i	$0.574013\pi$
$338$	0	0
$339$	−9.55294	−0.518845
$340$	0	0
$341$	−25.5662	−1.38449
$342$	0	0
$343$	48.3684	2.61165
$344$	0	0
$345$	−6.11838	−0.329402
$346$	0	0
$347$	6.56544	0.352451	0.176226	−	0.984350i	$-0.443611\pi$
$347$	6.56544	0.352451	0.176226	+	0.984350i	$0.443611\pi$
$348$	0	0
$349$	20.0132	1.07128	0.535642	−	0.844445i	$-0.320070\pi$
$349$	20.0132	1.07128	0.535642	+	0.844445i	$0.320070\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	6.68381	0.355744	0.177872	−	0.984054i	$-0.443079\pi$
$353$	6.68381	0.355744	0.177872	+	0.984054i	$0.443079\pi$
$354$	0	0
$355$	−3.11176	−0.165155
$356$	0	0
$357$	39.6846	2.10033
$358$	0	0
$359$	−15.2434	−0.804514	−0.402257	−	0.915527i	$-0.631774\pi$
$359$	−15.2434	−0.804514	−0.402257	+	0.915527i	$0.631774\pi$
$360$	0	0
$361$	30.0927	1.58382
$362$	0	0
$363$	−12.8949	−0.676804
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	−17.7765	−0.927924	−0.463962	−	0.885855i	$-0.653573\pi$
$367$	−17.7765	−0.927924	−0.463962	+	0.885855i	$0.653573\pi$
$368$	0	0
$369$	−5.89485	−0.306874
$370$	0	0
$371$	−0.578666	−0.0300428
$372$	0	0
$373$	6.92057	0.358334	0.179167	−	0.983819i	$-0.442660\pi$
$373$	6.92057	0.358334	0.179167	+	0.983819i	$0.442660\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−7.77647	−0.400509
$378$	0	0
$379$	24.7831	1.27302	0.636511	−	0.771268i	$-0.280377\pi$
$379$	24.7831	1.27302	0.636511	+	0.771268i	$0.280377\pi$
$380$	0	0
$381$	19.5529	1.00173
$382$	0	0
$383$	−0.546329	−0.0279161	−0.0139580	−	0.999903i	$-0.504443\pi$
$383$	−0.546329	−0.0279161	−0.0139580	+	0.999903i	$0.504443\pi$
$384$	0	0
$385$	23.8949	1.21779
$386$	0	0
$387$	5.77647	0.293635
$388$	0	0
$389$	20.6970	1.04938	0.524691	−	0.851293i	$-0.324181\pi$
$389$	20.6970	1.04938	0.524691	+	0.851293i	$0.324181\pi$
$390$	0	0
$391$	−49.6713	−2.51199
$392$	0	0
$393$	−10.4603	−0.527652
$394$	0	0
$395$	2.11838	0.106587
$396$	0	0
$397$	−4.80219	−0.241015	−0.120508	−	0.992712i	$-0.538452\pi$
$397$	−4.80219	−0.241015	−0.120508	+	0.992712i	$0.538452\pi$
$398$	0	0
$399$	34.2500	1.71464
$400$	0	0
$401$	15.5397	0.776016	0.388008	−	0.921656i	$-0.373163\pi$
$401$	15.5397	0.776016	0.388008	+	0.921656i	$0.373163\pi$
$402$	0	0
$403$	5.23014	0.260532
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	39.6846	1.96709
$408$	0	0
$409$	20.8691	1.03191	0.515956	−	0.856615i	$-0.327437\pi$
$409$	20.8691	1.03191	0.515956	+	0.856615i	$0.327437\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	−6.01323	−0.295892
$414$	0	0
$415$	2.76986	0.135967
$416$	0	0
$417$	13.6713	0.669488
$418$	0	0
$419$	22.2235	1.08569	0.542845	−	0.839833i	$-0.317347\pi$
$419$	22.2235	1.08569	0.542845	+	0.839833i	$0.317347\pi$
$420$	0	0
$421$	−23.3294	−1.13701	−0.568503	−	0.822681i	$-0.692477\pi$
$421$	−23.3294	−1.13701	−0.568503	+	0.822681i	$0.692477\pi$
$422$	0	0
$423$	1.23014	0.0598116
$424$	0	0
$425$	−8.11838	−0.393799
$426$	0	0
$427$	59.2375	2.86670
$428$	0	0
$429$	4.88824	0.236006
$430$	0	0
$431$	23.2434	1.11959	0.559797	−	0.828630i	$-0.310879\pi$
$431$	23.2434	1.11959	0.559797	+	0.828630i	$0.310879\pi$
$432$	0	0
$433$	6.44706	0.309826	0.154913	−	0.987928i	$-0.450490\pi$
$433$	6.44706	0.309826	0.154913	+	0.987928i	$0.450490\pi$
$434$	0	0
$435$	−7.77647	−0.372853
$436$	0	0
$437$	−42.8691	−2.05071
$438$	0	0
$439$	−31.6581	−1.51096	−0.755479	−	0.655173i	$-0.772596\pi$
$439$	−31.6581	−1.51096	−0.755479	+	0.655173i	$0.772596\pi$
$440$	0	0
$441$	16.8949	0.804517
$442$	0	0
$443$	6.56544	0.311933	0.155967	−	0.987762i	$-0.450151\pi$
$443$	6.56544	0.311933	0.155967	+	0.987762i	$0.450151\pi$
$444$	0	0
$445$	6.34191	0.300635
$446$	0	0
$447$	11.6713	0.552035
$448$	0	0
$449$	−30.3684	−1.43317	−0.716586	−	0.697499i	$-0.754296\pi$
$449$	−30.3684	−1.43317	−0.716586	+	0.697499i	$0.754296\pi$
$450$	0	0
$451$	−28.8154	−1.35687
$452$	0	0
$453$	2.76986	0.130139
$454$	0	0
$455$	−4.88824	−0.229164
$456$	0	0
$457$	0.802194	0.0375250	0.0187625	−	0.999824i	$-0.494027\pi$
$457$	0.802194	0.0375250	0.0187625	+	0.999824i	$0.494027\pi$
$458$	0	0
$459$	8.11838	0.378933
$460$	0	0
$461$	−35.4610	−1.65158	−0.825792	−	0.563974i	$-0.809272\pi$
$461$	−35.4610	−1.65158	−0.825792	+	0.563974i	$0.809272\pi$
$462$	0	0
$463$	−8.03234	−0.373294	−0.186647	−	0.982427i	$-0.559762\pi$
$463$	−8.03234	−0.373294	−0.186647	+	0.982427i	$0.559762\pi$
$464$	0	0
$465$	5.23014	0.242542
$466$	0	0
$467$	4.10515	0.189964	0.0949818	−	0.995479i	$-0.469721\pi$
$467$	4.10515	0.189964	0.0949818	+	0.995479i	$0.469721\pi$
$468$	0	0
$469$	−13.5397	−0.625206
$470$	0	0
$471$	−13.5529	−0.624487
$472$	0	0
$473$	28.2368	1.29833
$474$	0	0
$475$	−7.00662	−0.321486
$476$	0	0
$477$	−0.118379	−0.00542021
$478$	0	0
$479$	30.6647	1.40111	0.700553	−	0.713600i	$-0.252937\pi$
$479$	30.6647	1.40111	0.700553	+	0.713600i	$0.252937\pi$
$480$	0	0
$481$	−8.11838	−0.370166
$482$	0	0
$483$	−29.9081	−1.36086
$484$	0	0
$485$	−15.6713	−0.711598
$486$	0	0
$487$	−37.3618	−1.69302	−0.846511	−	0.532371i	$-0.821301\pi$
$487$	−37.3618	−1.69302	−0.846511	+	0.532371i	$0.821301\pi$
$488$	0	0
$489$	2.66471	0.120502
$490$	0	0
$491$	0.210297	0.00949059	0.00474530	−	0.999989i	$-0.498490\pi$
$491$	0.210297	0.00949059	0.00474530	+	0.999989i	$0.498490\pi$
$492$	0	0
$493$	−63.1323	−2.84334
$494$	0	0
$495$	4.88824	0.219710
$496$	0	0
$497$	−15.2110	−0.682308
$498$	0	0
$499$	−18.5596	−0.830840	−0.415420	−	0.909630i	$-0.636365\pi$
$499$	−18.5596	−0.830840	−0.415420	+	0.909630i	$0.636365\pi$
$500$	0	0
$501$	−12.7831	−0.571106
$502$	0	0
$503$	−2.46029	−0.109699	−0.0548494	−	0.998495i	$-0.517468\pi$
$503$	−2.46029	−0.109699	−0.0548494	+	0.998495i	$0.517468\pi$
$504$	0	0
$505$	18.2368	0.811525
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−40.5919	−1.79920	−0.899602	−	0.436711i	$-0.856143\pi$
$509$	−40.5919	−1.79920	−0.899602	+	0.436711i	$0.856143\pi$
$510$	0	0
$511$	9.77647	0.432486
$512$	0	0
$513$	7.00662	0.309350
$514$	0	0
$515$	11.5529	0.509084
$516$	0	0
$517$	6.01323	0.264462
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	16.0132	0.701552	0.350776	−	0.936459i	$-0.385918\pi$
$521$	16.0132	0.701552	0.350776	+	0.936459i	$0.385918\pi$
$522$	0	0
$523$	−11.7897	−0.515528	−0.257764	−	0.966208i	$-0.582986\pi$
$523$	−11.7897	−0.515528	−0.257764	+	0.966208i	$0.582986\pi$
$524$	0	0
$525$	−4.88824	−0.213340
$526$	0	0
$527$	42.4603	1.84960
$528$	0	0
$529$	14.4346	0.627590
$530$	0	0
$531$	−1.23014	−0.0533837
$532$	0	0
$533$	5.89485	0.255334
$534$	0	0
$535$	−5.88162	−0.254285
$536$	0	0
$537$	−12.2368	−0.528055
$538$	0	0
$539$	82.5860	3.55723
$540$	0	0
$541$	−32.0132	−1.37636	−0.688178	−	0.725542i	$-0.741589\pi$
$541$	−32.0132	−1.37636	−0.688178	+	0.725542i	$0.741589\pi$
$542$	0	0
$543$	9.89485	0.424629
$544$	0	0
$545$	15.7765	0.675790
$546$	0	0
$547$	−15.5529	−0.664996	−0.332498	−	0.943104i	$-0.607891\pi$
$547$	−15.5529	−0.664996	−0.332498	+	0.943104i	$0.607891\pi$
$548$	0	0
$549$	12.1184	0.517200
$550$	0	0
$551$	−54.4867	−2.32121
$552$	0	0
$553$	10.3551	0.440345
$554$	0	0
$555$	−8.11838	−0.344606
$556$	0	0
$557$	24.8691	1.05374	0.526869	−	0.849946i	$-0.323366\pi$
$557$	24.8691	1.05374	0.526869	+	0.849946i	$0.323366\pi$
$558$	0	0
$559$	−5.77647	−0.244319
$560$	0	0
$561$	39.6846	1.67548
$562$	0	0
$563$	−44.3684	−1.86990	−0.934952	−	0.354775i	$-0.884558\pi$
$563$	−44.3684	−1.86990	−0.934952	+	0.354775i	$0.884558\pi$
$564$	0	0
$565$	9.55294	0.401895
$566$	0	0
$567$	4.88824	0.205287
$568$	0	0
$569$	14.9206	0.625503	0.312751	−	0.949835i	$-0.398749\pi$
$569$	14.9206	0.625503	0.312751	+	0.949835i	$0.398749\pi$
$570$	0	0
$571$	−27.8949	−1.16736	−0.583682	−	0.811983i	$-0.698388\pi$
$571$	−27.8949	−1.16736	−0.583682	+	0.811983i	$0.698388\pi$
$572$	0	0
$573$	1.77647	0.0742132
$574$	0	0
$575$	6.11838	0.255154
$576$	0	0
$577$	−3.64486	−0.151738	−0.0758688	−	0.997118i	$-0.524173\pi$
$577$	−3.64486	−0.151738	−0.0758688	+	0.997118i	$0.524173\pi$
$578$	0	0
$579$	11.4346	0.475204
$580$	0	0
$581$	13.5397	0.561722
$582$	0	0
$583$	−0.578666	−0.0239659
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−4.54633	−0.187647	−0.0938235	−	0.995589i	$-0.529909\pi$
$587$	−4.54633	−0.187647	−0.0938235	+	0.995589i	$0.529909\pi$
$588$	0	0
$589$	36.6456	1.50996
$590$	0	0
$591$	−17.5529	−0.722032
$592$	0	0
$593$	16.8691	0.692732	0.346366	−	0.938099i	$-0.387416\pi$
$593$	16.8691	0.692732	0.346366	+	0.938099i	$0.387416\pi$
$594$	0	0
$595$	−39.6846	−1.62691
$596$	0	0
$597$	−18.0132	−0.737232
$598$	0	0
$599$	25.5662	1.04461	0.522303	−	0.852760i	$-0.325073\pi$
$599$	25.5662	1.04461	0.522303	+	0.852760i	$0.325073\pi$
$600$	0	0
$601$	12.9743	0.529232	0.264616	−	0.964354i	$-0.414755\pi$
$601$	12.9743	0.529232	0.264616	+	0.964354i	$0.414755\pi$
$602$	0	0
$603$	−2.76986	−0.112797
$604$	0	0
$605$	12.8949	0.524250
$606$	0	0
$607$	−37.3294	−1.51515	−0.757577	−	0.652746i	$-0.773617\pi$
$607$	−37.3294	−1.51515	−0.757577	+	0.652746i	$0.773617\pi$
$608$	0	0
$609$	−38.0132	−1.54037
$610$	0	0
$611$	−1.23014	−0.0497663
$612$	0	0
$613$	34.9875	1.41313	0.706566	−	0.707647i	$-0.250243\pi$
$613$	34.9875	1.41313	0.706566	+	0.707647i	$0.250243\pi$
$614$	0	0
$615$	5.89485	0.237703
$616$	0	0
$617$	−13.5529	−0.545621	−0.272810	−	0.962068i	$-0.587953\pi$
$617$	−13.5529	−0.545621	−0.272810	+	0.962068i	$0.587953\pi$
$618$	0	0
$619$	−13.7037	−0.550797	−0.275398	−	0.961330i	$-0.588810\pi$
$619$	−13.7037	−0.550797	−0.275398	+	0.961330i	$0.588810\pi$
$620$	0	0
$621$	−6.11838	−0.245522
$622$	0	0
$623$	31.0007	1.24202
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	34.2500	1.36781
$628$	0	0
$629$	−65.9081	−2.62793
$630$	0	0
$631$	−7.21691	−0.287301	−0.143650	−	0.989629i	$-0.545884\pi$
$631$	−7.21691	−0.287301	−0.143650	+	0.989629i	$0.545884\pi$
$632$	0	0
$633$	23.5529	0.936145
$634$	0	0
$635$	−19.5529	−0.775935
$636$	0	0
$637$	−16.8949	−0.669398
$638$	0	0
$639$	−3.11176	−0.123099
$640$	0	0
$641$	−26.6456	−1.05244	−0.526219	−	0.850349i	$-0.676391\pi$
$641$	−26.6456	−1.05244	−0.526219	+	0.850349i	$0.676391\pi$
$642$	0	0
$643$	24.2691	0.957080	0.478540	−	0.878066i	$-0.341166\pi$
$643$	24.2691	0.957080	0.478540	+	0.878066i	$0.341166\pi$
$644$	0	0
$645$	−5.77647	−0.227448
$646$	0	0
$647$	−37.2243	−1.46344	−0.731718	−	0.681607i	$-0.761281\pi$
$647$	−37.2243	−1.46344	−0.731718	+	0.681607i	$0.761281\pi$
$648$	0	0
$649$	−6.01323	−0.236040
$650$	0	0
$651$	25.5662	1.00202
$652$	0	0
$653$	20.0132	0.783178	0.391589	−	0.920140i	$-0.371925\pi$
$653$	20.0132	0.783178	0.391589	+	0.920140i	$0.371925\pi$
$654$	0	0
$655$	10.4603	0.408717
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	10.4603	0.407475	0.203737	−	0.979026i	$-0.434691\pi$
$659$	10.4603	0.407475	0.203737	+	0.979026i	$0.434691\pi$
$660$	0	0
$661$	28.0132	1.08959	0.544794	−	0.838570i	$-0.316608\pi$
$661$	28.0132	1.08959	0.544794	+	0.838570i	$0.316608\pi$
$662$	0	0
$663$	−8.11838	−0.315292
$664$	0	0
$665$	−34.2500	−1.32816
$666$	0	0
$667$	47.5794	1.84228
$668$	0	0
$669$	15.2434	0.589343
$670$	0	0
$671$	59.2375	2.28684
$672$	0	0
$673$	−3.32942	−0.128340	−0.0641698	−	0.997939i	$-0.520440\pi$
$673$	−3.32942	−0.128340	−0.0641698	+	0.997939i	$0.520440\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−27.6713	−1.06350	−0.531748	−	0.846903i	$-0.678465\pi$
$677$	−27.6713	−1.06350	−0.531748	+	0.846903i	$0.678465\pi$
$678$	0	0
$679$	−76.6051	−2.93983
$680$	0	0
$681$	21.0198	0.805482
$682$	0	0
$683$	−13.6390	−0.521881	−0.260941	−	0.965355i	$-0.584033\pi$
$683$	−13.6390	−0.521881	−0.260941	+	0.965355i	$0.584033\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	19.7765	0.754519
$688$	0	0
$689$	0.118379	0.00450989
$690$	0	0
$691$	41.9404	1.59549	0.797744	−	0.602996i	$-0.206026\pi$
$691$	41.9404	1.59549	0.797744	+	0.602996i	$0.206026\pi$
$692$	0	0
$693$	23.8949	0.907690
$694$	0	0
$695$	−13.6713	−0.518583
$696$	0	0
$697$	47.8566	1.81270
$698$	0	0
$699$	−21.2110	−0.802275
$700$	0	0
$701$	20.2235	0.763832	0.381916	−	0.924197i	$-0.375264\pi$
$701$	20.2235	0.763832	0.381916	+	0.924197i	$0.375264\pi$
$702$	0	0
$703$	−56.8824	−2.14536
$704$	0	0
$705$	−1.23014	−0.0463299
$706$	0	0
$707$	89.1456	3.35266
$708$	0	0
$709$	−16.2235	−0.609287	−0.304644	−	0.952466i	$-0.598537\pi$
$709$	−16.2235	−0.609287	−0.304644	+	0.952466i	$0.598537\pi$
$710$	0	0
$711$	2.11838	0.0794454
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−4.88824	−0.182810
$716$	0	0
$717$	3.11176	0.116211
$718$	0	0
$719$	−16.4735	−0.614359	−0.307179	−	0.951652i	$-0.599385\pi$
$719$	−16.4735	−0.614359	−0.307179	+	0.951652i	$0.599385\pi$
$720$	0	0
$721$	56.4735	2.10318
$722$	0	0
$723$	13.7897	0.512845
$724$	0	0
$725$	7.77647	0.288811
$726$	0	0
$727$	−44.7103	−1.65821	−0.829106	−	0.559091i	$-0.811150\pi$
$727$	−44.7103	−1.65821	−0.829106	+	0.559091i	$0.811150\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−46.8956	−1.73450
$732$	0	0
$733$	−30.9875	−1.14455	−0.572275	−	0.820062i	$-0.693939\pi$
$733$	−30.9875	−1.14455	−0.572275	+	0.820062i	$0.693939\pi$
$734$	0	0
$735$	−16.8949	−0.623176
$736$	0	0
$737$	−13.5397	−0.498742
$738$	0	0
$739$	−24.5728	−0.903925	−0.451962	−	0.892037i	$-0.649276\pi$
$739$	−24.5728	−0.903925	−0.451962	+	0.892037i	$0.649276\pi$
$740$	0	0
$741$	−7.00662	−0.257394
$742$	0	0
$743$	−36.9934	−1.35716	−0.678578	−	0.734529i	$-0.737403\pi$
$743$	−36.9934	−1.35716	−0.678578	+	0.734529i	$0.737403\pi$
$744$	0	0
$745$	−11.6713	−0.427604
$746$	0	0
$747$	2.76986	0.101344
$748$	0	0
$749$	−28.7508	−1.05053
$750$	0	0
$751$	−23.6581	−0.863296	−0.431648	−	0.902042i	$-0.642068\pi$
$751$	−23.6581	−0.863296	−0.431648	+	0.902042i	$0.642068\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	−2.76986	−0.100805
$756$	0	0
$757$	−4.22353	−0.153507	−0.0767534	−	0.997050i	$-0.524455\pi$
$757$	−4.22353	−0.153507	−0.0767534	+	0.997050i	$0.524455\pi$
$758$	0	0
$759$	−29.9081	−1.08559
$760$	0	0
$761$	40.4867	1.46764	0.733822	−	0.679342i	$-0.237735\pi$
$761$	40.4867	1.46764	0.733822	+	0.679342i	$0.237735\pi$
$762$	0	0
$763$	77.1191	2.79190
$764$	0	0
$765$	−8.11838	−0.293521
$766$	0	0
$767$	1.23014	0.0444179
$768$	0	0
$769$	−6.47352	−0.233441	−0.116720	−	0.993165i	$-0.537238\pi$
$769$	−6.47352	−0.233441	−0.116720	+	0.993165i	$0.537238\pi$
$770$	0	0
$771$	7.53971	0.271536
$772$	0	0
$773$	24.8691	0.894480	0.447240	−	0.894414i	$-0.352407\pi$
$773$	24.8691	0.894480	0.447240	+	0.894414i	$0.352407\pi$
$774$	0	0
$775$	−5.23014	−0.187872
$776$	0	0
$777$	−39.6846	−1.42368
$778$	0	0
$779$	41.3030	1.47983
$780$	0	0
$781$	−15.2110	−0.544294
$782$	0	0
$783$	−7.77647	−0.277908
$784$	0	0
$785$	13.5529	0.483725
$786$	0	0
$787$	14.7963	0.527432	0.263716	−	0.964600i	$-0.415052\pi$
$787$	14.7963	0.527432	0.263716	+	0.964600i	$0.415052\pi$
$788$	0	0
$789$	−2.46029	−0.0875885
$790$	0	0
$791$	46.6970	1.66036
$792$	0	0
$793$	−12.1184	−0.430336
$794$	0	0
$795$	0.118379	0.00419848
$796$	0	0
$797$	19.4992	0.690698	0.345349	−	0.938474i	$-0.387760\pi$
$797$	19.4992	0.690698	0.345349	+	0.938474i	$0.387760\pi$
$798$	0	0
$799$	−9.98677	−0.353307
$800$	0	0
$801$	6.34191	0.224080
$802$	0	0
$803$	9.77647	0.345004
$804$	0	0
$805$	29.9081	1.05412
$806$	0	0
$807$	−7.77647	−0.273745
$808$	0	0
$809$	44.9338	1.57979	0.789894	−	0.613243i	$-0.210135\pi$
$809$	44.9338	1.57979	0.789894	+	0.613243i	$0.210135\pi$
$810$	0	0
$811$	44.1257	1.54946	0.774732	−	0.632290i	$-0.217885\pi$
$811$	44.1257	1.54946	0.774732	+	0.632290i	$0.217885\pi$
$812$	0	0
$813$	−4.54633	−0.159447
$814$	0	0
$815$	−2.66471	−0.0933407
$816$	0	0
$817$	−40.4735	−1.41599
$818$	0	0
$819$	−4.88824	−0.170809
$820$	0	0
$821$	36.1184	1.26054	0.630270	−	0.776376i	$-0.282944\pi$
$821$	36.1184	1.26054	0.630270	+	0.776376i	$0.282944\pi$
$822$	0	0
$823$	12.9206	0.450383	0.225191	−	0.974315i	$-0.427699\pi$
$823$	12.9206	0.450383	0.225191	+	0.974315i	$0.427699\pi$
$824$	0	0
$825$	−4.88824	−0.170187
$826$	0	0
$827$	−22.3228	−0.776240	−0.388120	−	0.921609i	$-0.626875\pi$
$827$	−22.3228	−0.776240	−0.388120	+	0.921609i	$0.626875\pi$
$828$	0	0
$829$	32.6588	1.13429	0.567144	−	0.823619i	$-0.308048\pi$
$829$	32.6588	1.13429	0.567144	+	0.823619i	$0.308048\pi$
$830$	0	0
$831$	7.77647	0.269763
$832$	0	0
$833$	−137.159	−4.75227
$834$	0	0
$835$	12.7831	0.442377
$836$	0	0
$837$	5.23014	0.180780
$838$	0	0
$839$	50.0074	1.72645	0.863223	−	0.504823i	$-0.168442\pi$
$839$	50.0074	1.72645	0.863223	+	0.504823i	$0.168442\pi$
$840$	0	0
$841$	31.4735	1.08529
$842$	0	0
$843$	0.460287	0.0158531
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	63.0331	2.16584
$848$	0	0
$849$	6.46029	0.221716
$850$	0	0
$851$	49.6713	1.70271
$852$	0	0
$853$	−16.5654	−0.567190	−0.283595	−	0.958944i	$-0.591527\pi$
$853$	−16.5654	−0.567190	−0.283595	+	0.958944i	$0.591527\pi$
$854$	0	0
$855$	−7.00662	−0.239621
$856$	0	0
$857$	32.7640	1.11920	0.559598	−	0.828764i	$-0.310955\pi$
$857$	32.7640	1.11920	0.559598	+	0.828764i	$0.310955\pi$
$858$	0	0
$859$	−7.03895	−0.240166	−0.120083	−	0.992764i	$-0.538316\pi$
$859$	−7.03895	−0.240166	−0.120083	+	0.992764i	$0.538316\pi$
$860$	0	0
$861$	28.8154	0.982027
$862$	0	0
$863$	−28.0993	−0.956510	−0.478255	−	0.878221i	$-0.658731\pi$
$863$	−28.0993	−0.956510	−0.478255	+	0.878221i	$0.658731\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	−48.9081	−1.66101
$868$	0	0
$869$	10.3551	0.351274
$870$	0	0
$871$	2.76986	0.0938530
$872$	0	0
$873$	−15.6713	−0.530394
$874$	0	0
$875$	4.88824	0.165253
$876$	0	0
$877$	−9.55294	−0.322580	−0.161290	−	0.986907i	$-0.551565\pi$
$877$	−9.55294	−0.322580	−0.161290	+	0.986907i	$0.551565\pi$
$878$	0	0
$879$	−29.7897	−1.00478
$880$	0	0
$881$	−34.2368	−1.15347	−0.576733	−	0.816933i	$-0.695673\pi$
$881$	−34.2368	−1.15347	−0.576733	+	0.816933i	$0.695673\pi$
$882$	0	0
$883$	−45.5662	−1.53342	−0.766712	−	0.641991i	$-0.778109\pi$
$883$	−45.5662	−1.53342	−0.766712	+	0.641991i	$0.778109\pi$
$884$	0	0
$885$	1.23014	0.0413508
$886$	0	0
$887$	7.89485	0.265083	0.132542	−	0.991177i	$-0.457686\pi$
$887$	7.89485	0.265083	0.132542	+	0.991177i	$0.457686\pi$
$888$	0	0
$889$	−95.5794	−3.20563
$890$	0	0
$891$	4.88824	0.163762
$892$	0	0
$893$	−8.61914	−0.288429
$894$	0	0
$895$	12.2368	0.409030
$896$	0	0
$897$	6.11838	0.204287
$898$	0	0
$899$	−40.6721	−1.35649
$900$	0	0
$901$	0.961048	0.0320171
$902$	0	0
$903$	−28.2368	−0.939660
$904$	0	0
$905$	−9.89485	−0.328916
$906$	0	0
$907$	5.77647	0.191805	0.0959023	−	0.995391i	$-0.469426\pi$
$907$	5.77647	0.191805	0.0959023	+	0.995391i	$0.469426\pi$
$908$	0	0
$909$	18.2368	0.604875
$910$	0	0
$911$	22.2235	0.736298	0.368149	−	0.929767i	$-0.379992\pi$
$911$	22.2235	0.736298	0.368149	+	0.929767i	$0.379992\pi$
$912$	0	0
$913$	13.5397	0.448099
$914$	0	0
$915$	−12.1184	−0.400621
$916$	0	0
$917$	51.1323	1.68854
$918$	0	0
$919$	−2.32868	−0.0768160	−0.0384080	−	0.999262i	$-0.512229\pi$
$919$	−2.32868	−0.0768160	−0.0384080	+	0.999262i	$0.512229\pi$
$920$	0	0
$921$	−4.44118	−0.146342
$922$	0	0
$923$	3.11176	0.102425
$924$	0	0
$925$	8.11838	0.266931
$926$	0	0
$927$	11.5529	0.379448
$928$	0	0
$929$	−45.2110	−1.48333	−0.741663	−	0.670773i	$-0.765963\pi$
$929$	−45.2110	−1.48333	−0.741663	+	0.670773i	$0.765963\pi$
$930$	0	0
$931$	−118.376	−3.87961
$932$	0	0
$933$	−8.68381	−0.284295
$934$	0	0
$935$	−39.6846	−1.29782
$936$	0	0
$937$	14.7103	0.480564	0.240282	−	0.970703i	$-0.422760\pi$
$937$	14.7103	0.480564	0.240282	+	0.970703i	$0.422760\pi$
$938$	0	0
$939$	11.5397	0.376584
$940$	0	0
$941$	38.9875	1.27096	0.635478	−	0.772119i	$-0.280803\pi$
$941$	38.9875	1.27096	0.635478	+	0.772119i	$0.280803\pi$
$942$	0	0
$943$	−36.0669	−1.17450
$944$	0	0
$945$	−4.88824	−0.159014
$946$	0	0
$947$	−59.7169	−1.94054	−0.970269	−	0.242029i	$-0.922187\pi$
$947$	−59.7169	−1.94054	−0.970269	+	0.242029i	$0.922187\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	18.4735	0.599045
$952$	0	0
$953$	−46.1316	−1.49435	−0.747175	−	0.664628i	$-0.768590\pi$
$953$	−46.1316	−1.49435	−0.747175	+	0.664628i	$0.768590\pi$
$954$	0	0
$955$	−1.77647	−0.0574853
$956$	0	0
$957$	−38.0132	−1.22879
$958$	0	0
$959$	−48.8824	−1.57849
$960$	0	0
$961$	−3.64560	−0.117600
$962$	0	0
$963$	−5.88162	−0.189533
$964$	0	0
$965$	−11.4346	−0.368092
$966$	0	0
$967$	−27.2169	−0.875237	−0.437618	−	0.899161i	$-0.644178\pi$
$967$	−27.2169	−0.875237	−0.437618	+	0.899161i	$0.644178\pi$
$968$	0	0
$969$	−56.8824	−1.82732
$970$	0	0
$971$	17.7765	0.570474	0.285237	−	0.958457i	$-0.407928\pi$
$971$	17.7765	0.570474	0.285237	+	0.958457i	$0.407928\pi$
$972$	0	0
$973$	−66.8287	−2.14243
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−18.4735	−0.591020	−0.295510	−	0.955340i	$-0.595490\pi$
$977$	−18.4735	−0.591020	−0.295510	+	0.955340i	$0.595490\pi$
$978$	0	0
$979$	31.0007	0.990788
$980$	0	0
$981$	15.7765	0.503704
$982$	0	0
$983$	8.75663	0.279293	0.139647	−	0.990201i	$-0.455403\pi$
$983$	8.75663	0.279293	0.139647	+	0.990201i	$0.455403\pi$
$984$	0	0
$985$	17.5529	0.559283
$986$	0	0
$987$	−6.01323	−0.191403
$988$	0	0
$989$	35.3426	1.12383
$990$	0	0
$991$	−21.6713	−0.688412	−0.344206	−	0.938894i	$-0.611852\pi$
$991$	−21.6713	−0.688412	−0.344206	+	0.938894i	$0.611852\pi$
$992$	0	0
$993$	−33.4669	−1.06204
$994$	0	0
$995$	18.0132	0.571058
$996$	0	0
$997$	18.8824	0.598010	0.299005	−	0.954251i	$-0.403345\pi$
$997$	18.8824	0.598010	0.299005	+	0.954251i	$0.403345\pi$
$998$	0	0
$999$	−8.11838	−0.256854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bz.1.3	✓	3
4.3	odd	2		6240.2.a.cc.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bz.1.3	✓	3	1.1	even	1	trivial
6240.2.a.cc.1.1	yes	3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.88824 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.88824 q^{7} +1.00000 q^{9} +4.88824 q^{11} -1.00000 q^{13} -1.00000 q^{15} -8.11838 q^{17} -7.00662 q^{19} -4.88824 q^{21} +6.11838 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.77647 q^{29} -5.23014 q^{31} -4.88824 q^{33} +4.88824 q^{35} +8.11838 q^{37} +1.00000 q^{39} -5.89485 q^{41} +5.77647 q^{43} +1.00000 q^{45} +1.23014 q^{47} +16.8949 q^{49} +8.11838 q^{51} -0.118379 q^{53} +4.88824 q^{55} +7.00662 q^{57} -1.23014 q^{59} +12.1184 q^{61} +4.88824 q^{63} -1.00000 q^{65} -2.76986 q^{67} -6.11838 q^{69} -3.11176 q^{71} +2.00000 q^{73} -1.00000 q^{75} +23.8949 q^{77} +2.11838 q^{79} +1.00000 q^{81} +2.76986 q^{83} -8.11838 q^{85} -7.77647 q^{87} +6.34191 q^{89} -4.88824 q^{91} +5.23014 q^{93} -7.00662 q^{95} -15.6713 q^{97} +4.88824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} + 5 q^{7} + 3 q^{9} + 5 q^{11} - 3 q^{13} - 3 q^{15} - 9 q^{17} + 4 q^{19} - 5 q^{21} + 3 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} - 10 q^{31} - 5 q^{33} + 5 q^{35} + 9 q^{37} + 3 q^{39} + 17 q^{41} - 2 q^{43} + 3 q^{45} - 2 q^{47} + 16 q^{49} + 9 q^{51} + 15 q^{53} + 5 q^{55} - 4 q^{57} + 2 q^{59} + 21 q^{61} + 5 q^{63} - 3 q^{65} - 14 q^{67} - 3 q^{69} - 19 q^{71} + 6 q^{73} - 3 q^{75} + 37 q^{77} - 9 q^{79} + 3 q^{81} + 14 q^{83} - 9 q^{85} - 4 q^{87} + 23 q^{89} - 5 q^{91} + 10 q^{93} + 4 q^{95} + 7 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 5 * q^7 + 3 * q^9 + 5 * q^11 - 3 * q^13 - 3 * q^15 - 9 * q^17 + 4 * q^19 - 5 * q^21 + 3 * q^23 + 3 * q^25 - 3 * q^27 + 4 * q^29 - 10 * q^31 - 5 * q^33 + 5 * q^35 + 9 * q^37 + 3 * q^39 + 17 * q^41 - 2 * q^43 + 3 * q^45 - 2 * q^47 + 16 * q^49 + 9 * q^51 + 15 * q^53 + 5 * q^55 - 4 * q^57 + 2 * q^59 + 21 * q^61 + 5 * q^63 - 3 * q^65 - 14 * q^67 - 3 * q^69 - 19 * q^71 + 6 * q^73 - 3 * q^75 + 37 * q^77 - 9 * q^79 + 3 * q^81 + 14 * q^83 - 9 * q^85 - 4 * q^87 + 23 * q^89 - 5 * q^91 + 10 * q^93 + 4 * q^95 + 7 * q^97 + 5 * q^99

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