LMFDB - Embedded newform 1300.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,4,Mod(1,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1300.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1300.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:	$76.7024830075$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2785296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 30x^{2} + 2x + 65 $ x^4 - 2x^3 - 30x^2 + 2*x + 65
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.28450$ of defining polynomial
Character	$\chi$	$=$	1300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14910 q^{3} -26.4482 q^{7} -22.3814 q^{9} +O(q^{10})$	$q-2.14910 q^{3} -26.4482 q^{7} -22.3814 q^{9} +34.8167 q^{11} -13.0000 q^{13} +71.4398 q^{17} +104.952 q^{19} +56.8398 q^{21} -132.324 q^{23} +106.125 q^{27} +71.4534 q^{29} +289.609 q^{31} -74.8245 q^{33} +51.1500 q^{37} +27.9383 q^{39} +79.7173 q^{41} -119.450 q^{43} -266.396 q^{47} +356.507 q^{49} -153.531 q^{51} +204.626 q^{53} -225.552 q^{57} -464.253 q^{59} -284.914 q^{61} +591.947 q^{63} -591.976 q^{67} +284.377 q^{69} +260.648 q^{71} -1083.91 q^{73} -920.838 q^{77} -676.063 q^{79} +376.223 q^{81} +266.762 q^{83} -153.561 q^{87} -1532.75 q^{89} +343.826 q^{91} -622.398 q^{93} +871.393 q^{97} -779.245 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9	$ 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9} + 28 q^{11} - 52 q^{13} - 200 q^{17} + 132 q^{19} + 160 q^{21} - 244 q^{23} - 496 q^{27} - 8 q^{29} + 292 q^{31} - 128 q^{33} - 168 q^{37} + 52 q^{39} + 280 q^{41} - 452 q^{43} + 280 q^{47} + 340 q^{49} + 792 q^{51} - 584 q^{53} - 1200 q^{57} - 708 q^{59} + 1128 q^{61} - 232 q^{63} - 176 q^{67} + 160 q^{69} - 1028 q^{71} - 664 q^{73} + 864 q^{77} - 728 q^{79} + 2244 q^{81} - 552 q^{83} + 2064 q^{87} - 2824 q^{89} - 104 q^{91} - 1392 q^{93} - 1160 q^{97} - 4220 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9 + 28 * q^11 - 52 * q^13 - 200 * q^17 + 132 * q^19 + 160 * q^21 - 244 * q^23 - 496 * q^27 - 8 * q^29 + 292 * q^31 - 128 * q^33 - 168 * q^37 + 52 * q^39 + 280 * q^41 - 452 * q^43 + 280 * q^47 + 340 * q^49 + 792 * q^51 - 584 * q^53 - 1200 * q^57 - 708 * q^59 + 1128 * q^61 - 232 * q^63 - 176 * q^67 + 160 * q^69 - 1028 * q^71 - 664 * q^73 + 864 * q^77 - 728 * q^79 + 2244 * q^81 - 552 * q^83 + 2064 * q^87 - 2824 * q^89 - 104 * q^91 - 1392 * q^93 - 1160 * q^97 - 4220 * 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
$3$	−2.14910	−0.413594	−0.206797	−	0.978384i	$-0.566304\pi$
$3$	−2.14910	−0.413594	−0.206797	+	0.978384i	$0.566304\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−26.4482	−1.42807	−0.714034	−	0.700111i	$-0.753134\pi$
$7$	−26.4482	−1.42807	−0.714034	+	0.700111i	$0.753134\pi$
$8$	0	0
$9$	−22.3814	−0.828940
$10$	0	0
$11$	34.8167	0.954329	0.477164	−	0.878814i	$-0.341665\pi$
$11$	34.8167	0.954329	0.477164	+	0.878814i	$0.341665\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	71.4398	1.01922	0.509608	−	0.860406i	$-0.329790\pi$
$17$	71.4398	1.01922	0.509608	+	0.860406i	$0.329790\pi$
$18$	0	0
$19$	104.952	1.26725	0.633623	−	0.773642i	$-0.281567\pi$
$19$	104.952	1.26725	0.633623	+	0.773642i	$0.281567\pi$
$20$	0	0
$21$	56.8398	0.590641
$22$	0	0
$23$	−132.324	−1.19963	−0.599814	−	0.800139i	$-0.704759\pi$
$23$	−132.324	−1.19963	−0.599814	+	0.800139i	$0.704759\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	106.125	0.756439
$28$	0	0
$29$	71.4534	0.457537	0.228768	−	0.973481i	$-0.426530\pi$
$29$	71.4534	0.457537	0.228768	+	0.973481i	$0.426530\pi$
$30$	0	0
$31$	289.609	1.67791	0.838956	−	0.544200i	$-0.183167\pi$
$31$	289.609	1.67791	0.838956	+	0.544200i	$0.183167\pi$
$32$	0	0
$33$	−74.8245	−0.394705
$34$	0	0
$35$	0	0
$36$	0	0
$37$	51.1500	0.227270	0.113635	−	0.993523i	$-0.463750\pi$
$37$	51.1500	0.227270	0.113635	+	0.993523i	$0.463750\pi$
$38$	0	0
$39$	27.9383	0.114710
$40$	0	0
$41$	79.7173	0.303652	0.151826	−	0.988407i	$-0.451485\pi$
$41$	79.7173	0.303652	0.151826	+	0.988407i	$0.451485\pi$
$42$	0	0
$43$	−119.450	−0.423627	−0.211813	−	0.977310i	$-0.567937\pi$
$43$	−119.450	−0.423627	−0.211813	+	0.977310i	$0.567937\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−266.396	−0.826761	−0.413381	−	0.910558i	$-0.635652\pi$
$47$	−266.396	−0.826761	−0.413381	+	0.910558i	$0.635652\pi$
$48$	0	0
$49$	356.507	1.03938
$50$	0	0
$51$	−153.531	−0.421542
$52$	0	0
$53$	204.626	0.530331	0.265165	−	0.964203i	$-0.414573\pi$
$53$	204.626	0.530331	0.265165	+	0.964203i	$0.414573\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−225.552	−0.524125
$58$	0	0
$59$	−464.253	−1.02442	−0.512209	−	0.858861i	$-0.671173\pi$
$59$	−464.253	−1.02442	−0.512209	+	0.858861i	$0.671173\pi$
$60$	0	0
$61$	−284.914	−0.598025	−0.299013	−	0.954249i	$-0.596657\pi$
$61$	−284.914	−0.598025	−0.299013	+	0.954249i	$0.596657\pi$
$62$	0	0
$63$	591.947	1.18378
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−591.976	−1.07942	−0.539712	−	0.841850i	$-0.681467\pi$
$67$	−591.976	−1.07942	−0.539712	+	0.841850i	$0.681467\pi$
$68$	0	0
$69$	284.377	0.496160
$70$	0	0
$71$	260.648	0.435679	0.217839	−	0.975985i	$-0.430099\pi$
$71$	260.648	0.435679	0.217839	+	0.975985i	$0.430099\pi$
$72$	0	0
$73$	−1083.91	−1.73784	−0.868922	−	0.494950i	$-0.835186\pi$
$73$	−1083.91	−1.73784	−0.868922	+	0.494950i	$0.835186\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−920.838	−1.36285
$78$	0	0
$79$	−676.063	−0.962823	−0.481411	−	0.876495i	$-0.659876\pi$
$79$	−676.063	−0.962823	−0.481411	+	0.876495i	$0.659876\pi$
$80$	0	0
$81$	376.223	0.516081
$82$	0	0
$83$	266.762	0.352783	0.176391	−	0.984320i	$-0.443558\pi$
$83$	266.762	0.352783	0.176391	+	0.984320i	$0.443558\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−153.561	−0.189235
$88$	0	0
$89$	−1532.75	−1.82552	−0.912759	−	0.408498i	$-0.866053\pi$
$89$	−1532.75	−1.82552	−0.912759	+	0.408498i	$0.866053\pi$
$90$	0	0
$91$	343.826	0.396075
$92$	0	0
$93$	−622.398	−0.693975
$94$	0	0
$95$	0	0
$96$	0	0
$97$	871.393	0.912130	0.456065	−	0.889947i	$-0.349259\pi$
$97$	871.393	0.912130	0.456065	+	0.889947i	$0.349259\pi$
$98$	0	0
$99$	−779.245	−0.791081
$100$	0	0
$101$	−190.494	−0.187672	−0.0938359	−	0.995588i	$-0.529913\pi$
$101$	−190.494	−0.187672	−0.0938359	+	0.995588i	$0.529913\pi$
$102$	0	0
$103$	551.837	0.527904	0.263952	−	0.964536i	$-0.414974\pi$
$103$	551.837	0.527904	0.263952	+	0.964536i	$0.414974\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	803.798	0.726225	0.363113	−	0.931745i	$-0.381714\pi$
$107$	803.798	0.726225	0.363113	+	0.931745i	$0.381714\pi$
$108$	0	0
$109$	−1178.03	−1.03518	−0.517592	−	0.855628i	$-0.673171\pi$
$109$	−1178.03	−1.03518	−0.517592	+	0.855628i	$0.673171\pi$
$110$	0	0
$111$	−109.926	−0.0939978
$112$	0	0
$113$	1312.70	1.09282	0.546409	−	0.837519i	$-0.315995\pi$
$113$	1312.70	1.09282	0.546409	+	0.837519i	$0.315995\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	290.958	0.229907
$118$	0	0
$119$	−1889.45	−1.45551
$120$	0	0
$121$	−118.800	−0.0892563
$122$	0	0
$123$	−171.320	−0.125589
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1063.52	0.743087	0.371544	−	0.928415i	$-0.378829\pi$
$127$	1063.52	0.743087	0.371544	+	0.928415i	$0.378829\pi$
$128$	0	0
$129$	256.710	0.175210
$130$	0	0
$131$	1501.12	1.00117	0.500587	−	0.865686i	$-0.333117\pi$
$131$	1501.12	1.00117	0.500587	+	0.865686i	$0.333117\pi$
$132$	0	0
$133$	−2775.79	−1.80971
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1638.53	1.02182	0.510910	−	0.859634i	$-0.329308\pi$
$137$	1638.53	1.02182	0.510910	+	0.859634i	$0.329308\pi$
$138$	0	0
$139$	−89.1606	−0.0544065	−0.0272033	−	0.999630i	$-0.508660\pi$
$139$	−89.1606	−0.0544065	−0.0272033	+	0.999630i	$0.508660\pi$
$140$	0	0
$141$	572.510	0.341944
$142$	0	0
$143$	−452.617	−0.264683
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−766.169	−0.429881
$148$	0	0
$149$	−251.538	−0.138301	−0.0691504	−	0.997606i	$-0.522029\pi$
$149$	−251.538	−0.138301	−0.0691504	+	0.997606i	$0.522029\pi$
$150$	0	0
$151$	−3308.90	−1.78327	−0.891637	−	0.452751i	$-0.850443\pi$
$151$	−3308.90	−1.78327	−0.891637	+	0.452751i	$0.850443\pi$
$152$	0	0
$153$	−1598.92	−0.844869
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2810.03	−1.42844	−0.714220	−	0.699921i	$-0.753218\pi$
$157$	−2810.03	−1.42844	−0.714220	+	0.699921i	$0.753218\pi$
$158$	0	0
$159$	−439.761	−0.219342
$160$	0	0
$161$	3499.73	1.71315
$162$	0	0
$163$	−1474.36	−0.708473	−0.354236	−	0.935156i	$-0.615259\pi$
$163$	−1474.36	−0.708473	−0.354236	+	0.935156i	$0.615259\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2459.35	1.13958	0.569792	−	0.821789i	$-0.307024\pi$
$167$	2459.35	1.13958	0.569792	+	0.821789i	$0.307024\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−2348.97	−1.05047
$172$	0	0
$173$	−1791.09	−0.787133	−0.393567	−	0.919296i	$-0.628759\pi$
$173$	−1791.09	−0.787133	−0.393567	+	0.919296i	$0.628759\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	997.726	0.423693
$178$	0	0
$179$	−124.760	−0.0520951	−0.0260475	−	0.999661i	$-0.508292\pi$
$179$	−124.760	−0.0520951	−0.0260475	+	0.999661i	$0.508292\pi$
$180$	0	0
$181$	−3516.32	−1.44401	−0.722005	−	0.691888i	$-0.756779\pi$
$181$	−3516.32	−1.44401	−0.722005	+	0.691888i	$0.756779\pi$
$182$	0	0
$183$	612.309	0.247340
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2487.29	0.972668
$188$	0	0
$189$	−2806.83	−1.08025
$190$	0	0
$191$	4120.68	1.56106	0.780529	−	0.625119i	$-0.214950\pi$
$191$	4120.68	1.56106	0.780529	+	0.625119i	$0.214950\pi$
$192$	0	0
$193$	−1283.12	−0.478556	−0.239278	−	0.970951i	$-0.576911\pi$
$193$	−1283.12	−0.478556	−0.239278	+	0.970951i	$0.576911\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3952.45	−1.42944	−0.714722	−	0.699408i	$-0.753447\pi$
$197$	−3952.45	−1.42944	−0.714722	+	0.699408i	$0.753447\pi$
$198$	0	0
$199$	−599.586	−0.213586	−0.106793	−	0.994281i	$-0.534058\pi$
$199$	−599.586	−0.213586	−0.106793	+	0.994281i	$0.534058\pi$
$200$	0	0
$201$	1272.21	0.446443
$202$	0	0
$203$	−1889.81	−0.653394
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2961.59	0.994420
$208$	0	0
$209$	3654.08	1.20937
$210$	0	0
$211$	5140.93	1.67733	0.838664	−	0.544649i	$-0.183337\pi$
$211$	5140.93	1.67733	0.838664	+	0.544649i	$0.183337\pi$
$212$	0	0
$213$	−560.158	−0.180194
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7659.63	−2.39617
$218$	0	0
$219$	2329.44	0.718762
$220$	0	0
$221$	−928.717	−0.282680
$222$	0	0
$223$	−3266.57	−0.980923	−0.490462	−	0.871463i	$-0.663172\pi$
$223$	−3266.57	−0.980923	−0.490462	+	0.871463i	$0.663172\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1041.44	−0.304505	−0.152252	−	0.988342i	$-0.548653\pi$
$227$	−1041.44	−0.304505	−0.152252	+	0.988342i	$0.548653\pi$
$228$	0	0
$229$	−3273.34	−0.944577	−0.472288	−	0.881444i	$-0.656572\pi$
$229$	−3273.34	−0.944577	−0.472288	+	0.881444i	$0.656572\pi$
$230$	0	0
$231$	1978.97	0.563666
$232$	0	0
$233$	−3061.68	−0.860846	−0.430423	−	0.902627i	$-0.641636\pi$
$233$	−3061.68	−0.860846	−0.430423	+	0.902627i	$0.641636\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1452.93	0.398218
$238$	0	0
$239$	−7212.12	−1.95194	−0.975969	−	0.217908i	$-0.930077\pi$
$239$	−7212.12	−1.95194	−0.975969	+	0.217908i	$0.930077\pi$
$240$	0	0
$241$	−1216.92	−0.325265	−0.162633	−	0.986687i	$-0.551999\pi$
$241$	−1216.92	−0.325265	−0.162633	+	0.986687i	$0.551999\pi$
$242$	0	0
$243$	−3673.93	−0.969887
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1364.38	−0.351471
$248$	0	0
$249$	−573.298	−0.145909
$250$	0	0
$251$	−2538.22	−0.638291	−0.319145	−	0.947706i	$-0.603396\pi$
$251$	−2538.22	−0.638291	−0.319145	+	0.947706i	$0.603396\pi$
$252$	0	0
$253$	−4607.08	−1.14484
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3216.79	−0.780771	−0.390385	−	0.920652i	$-0.627658\pi$
$257$	−3216.79	−0.780771	−0.390385	+	0.920652i	$0.627658\pi$
$258$	0	0
$259$	−1352.82	−0.324558
$260$	0	0
$261$	−1599.23	−0.379270
$262$	0	0
$263$	3753.22	0.879976	0.439988	−	0.898004i	$-0.354983\pi$
$263$	3753.22	0.879976	0.439988	+	0.898004i	$0.354983\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3294.03	0.755024
$268$	0	0
$269$	−5989.67	−1.35761	−0.678804	−	0.734320i	$-0.737501\pi$
$269$	−5989.67	−1.35761	−0.678804	+	0.734320i	$0.737501\pi$
$270$	0	0
$271$	7639.65	1.71246	0.856228	−	0.516598i	$-0.172802\pi$
$271$	7639.65	1.71246	0.856228	+	0.516598i	$0.172802\pi$
$272$	0	0
$273$	−738.917	−0.163814
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8802.50	−1.90935	−0.954676	−	0.297646i	$-0.903798\pi$
$277$	−8802.50	−1.90935	−0.954676	+	0.297646i	$0.903798\pi$
$278$	0	0
$279$	−6481.84	−1.39089
$280$	0	0
$281$	−2402.18	−0.509971	−0.254985	−	0.966945i	$-0.582071\pi$
$281$	−2402.18	−0.509971	−0.254985	+	0.966945i	$0.582071\pi$
$282$	0	0
$283$	−4606.51	−0.967592	−0.483796	−	0.875181i	$-0.660742\pi$
$283$	−4606.51	−0.967592	−0.483796	+	0.875181i	$0.660742\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2108.38	−0.433636
$288$	0	0
$289$	190.639	0.0388030
$290$	0	0
$291$	−1872.71	−0.377252
$292$	0	0
$293$	6989.55	1.39363	0.696815	−	0.717251i	$-0.254600\pi$
$293$	6989.55	1.39363	0.696815	+	0.717251i	$0.254600\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3694.93	0.721892
$298$	0	0
$299$	1720.21	0.332717
$300$	0	0
$301$	3159.24	0.604968
$302$	0	0
$303$	409.390	0.0776200
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5853.68	−1.08823	−0.544116	−	0.839010i	$-0.683135\pi$
$307$	−5853.68	−1.08823	−0.544116	+	0.839010i	$0.683135\pi$
$308$	0	0
$309$	−1185.95	−0.218338
$310$	0	0
$311$	−8331.55	−1.51910	−0.759548	−	0.650451i	$-0.774580\pi$
$311$	−8331.55	−1.51910	−0.759548	+	0.650451i	$0.774580\pi$
$312$	0	0
$313$	−1104.27	−0.199415	−0.0997075	−	0.995017i	$-0.531791\pi$
$313$	−1104.27	−0.199415	−0.0997075	+	0.995017i	$0.531791\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6856.94	1.21490	0.607452	−	0.794357i	$-0.292192\pi$
$317$	6856.94	1.21490	0.607452	+	0.794357i	$0.292192\pi$
$318$	0	0
$319$	2487.77	0.436641
$320$	0	0
$321$	−1727.44	−0.300363
$322$	0	0
$323$	7497.75	1.29160
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2531.71	0.428146
$328$	0	0
$329$	7045.68	1.18067
$330$	0	0
$331$	10914.5	1.81243	0.906215	−	0.422818i	$-0.138959\pi$
$331$	10914.5	1.81243	0.906215	+	0.422818i	$0.138959\pi$
$332$	0	0
$333$	−1144.81	−0.188393
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9178.45	1.48363	0.741813	−	0.670607i	$-0.233966\pi$
$337$	9178.45	1.48363	0.741813	+	0.670607i	$0.233966\pi$
$338$	0	0
$339$	−2821.12	−0.451983
$340$	0	0
$341$	10083.2	1.60128
$342$	0	0
$343$	−357.232	−0.0562353
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2346.09	−0.362952	−0.181476	−	0.983395i	$-0.558088\pi$
$347$	−2346.09	−0.362952	−0.181476	+	0.983395i	$0.558088\pi$
$348$	0	0
$349$	−5646.28	−0.866012	−0.433006	−	0.901391i	$-0.642547\pi$
$349$	−5646.28	−0.866012	−0.433006	+	0.901391i	$0.642547\pi$
$350$	0	0
$351$	−1379.63	−0.209798
$352$	0	0
$353$	11529.9	1.73846	0.869228	−	0.494412i	$-0.164616\pi$
$353$	11529.9	1.73846	0.869228	+	0.494412i	$0.164616\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4060.62	0.601991
$358$	0	0
$359$	9944.14	1.46193	0.730963	−	0.682417i	$-0.239071\pi$
$359$	9944.14	1.46193	0.730963	+	0.682417i	$0.239071\pi$
$360$	0	0
$361$	4155.94	0.605911
$362$	0	0
$363$	255.313	0.0369159
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11042.4	1.57060	0.785301	−	0.619114i	$-0.212508\pi$
$367$	11042.4	1.57060	0.785301	+	0.619114i	$0.212508\pi$
$368$	0	0
$369$	−1784.18	−0.251709
$370$	0	0
$371$	−5411.99	−0.757348
$372$	0	0
$373$	7282.92	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$373$	7282.92	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−928.895	−0.126898
$378$	0	0
$379$	3200.93	0.433828	0.216914	−	0.976191i	$-0.430401\pi$
$379$	3200.93	0.433828	0.216914	+	0.976191i	$0.430401\pi$
$380$	0	0
$381$	−2285.61	−0.307337
$382$	0	0
$383$	−13706.1	−1.82859	−0.914295	−	0.405049i	$-0.867254\pi$
$383$	−13706.1	−1.82859	−0.914295	+	0.405049i	$0.867254\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2673.45	0.351161
$388$	0	0
$389$	−12073.1	−1.57360	−0.786799	−	0.617209i	$-0.788263\pi$
$389$	−12073.1	−1.57360	−0.786799	+	0.617209i	$0.788263\pi$
$390$	0	0
$391$	−9453.20	−1.22268
$392$	0	0
$393$	−3226.07	−0.414080
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6864.21	−0.867770	−0.433885	−	0.900968i	$-0.642858\pi$
$397$	−6864.21	−0.867770	−0.433885	+	0.900968i	$0.642858\pi$
$398$	0	0
$399$	5965.45	0.748487
$400$	0	0
$401$	−12392.7	−1.54330	−0.771649	−	0.636048i	$-0.780568\pi$
$401$	−12392.7	−1.54330	−0.771649	+	0.636048i	$0.780568\pi$
$402$	0	0
$403$	−3764.91	−0.465369
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1780.87	0.216891
$408$	0	0
$409$	16059.3	1.94152	0.970760	−	0.240051i	$-0.0771643\pi$
$409$	16059.3	1.94152	0.970760	+	0.240051i	$0.0771643\pi$
$410$	0	0
$411$	−3521.37	−0.422619
$412$	0	0
$413$	12278.7	1.46294
$414$	0	0
$415$	0	0
$416$	0	0
$417$	191.615	0.0225022
$418$	0	0
$419$	7177.03	0.836804	0.418402	−	0.908262i	$-0.362590\pi$
$419$	7177.03	0.836804	0.418402	+	0.908262i	$0.362590\pi$
$420$	0	0
$421$	−2187.72	−0.253261	−0.126631	−	0.991950i	$-0.540416\pi$
$421$	−2187.72	−0.253261	−0.126631	+	0.991950i	$0.540416\pi$
$422$	0	0
$423$	5962.30	0.685335
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7535.47	0.854021
$428$	0	0
$429$	972.718	0.109471
$430$	0	0
$431$	1709.68	0.191073	0.0955364	−	0.995426i	$-0.469543\pi$
$431$	1709.68	0.191073	0.0955364	+	0.995426i	$0.469543\pi$
$432$	0	0
$433$	−6064.85	−0.673113	−0.336557	−	0.941663i	$-0.609262\pi$
$433$	−6064.85	−0.673113	−0.336557	+	0.941663i	$0.609262\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13887.7	−1.52022
$438$	0	0
$439$	−8690.16	−0.944780	−0.472390	−	0.881390i	$-0.656609\pi$
$439$	−8690.16	−0.944780	−0.472390	+	0.881390i	$0.656609\pi$
$440$	0	0
$441$	−7979.11	−0.861582
$442$	0	0
$443$	9289.01	0.996240	0.498120	−	0.867108i	$-0.334024\pi$
$443$	9289.01	0.996240	0.498120	+	0.867108i	$0.334024\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	540.581	0.0572004
$448$	0	0
$449$	−6189.24	−0.650530	−0.325265	−	0.945623i	$-0.605454\pi$
$449$	−6189.24	−0.650530	−0.325265	+	0.945623i	$0.605454\pi$
$450$	0	0
$451$	2775.49	0.289784
$452$	0	0
$453$	7111.15	0.737552
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10805.0	1.10598	0.552992	−	0.833187i	$-0.313486\pi$
$457$	10805.0	1.10598	0.552992	+	0.833187i	$0.313486\pi$
$458$	0	0
$459$	7581.58	0.770975
$460$	0	0
$461$	7220.93	0.729528	0.364764	−	0.931100i	$-0.381150\pi$
$461$	7220.93	0.729528	0.364764	+	0.931100i	$0.381150\pi$
$462$	0	0
$463$	5721.35	0.574284	0.287142	−	0.957888i	$-0.407295\pi$
$463$	5721.35	0.574284	0.287142	+	0.957888i	$0.407295\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10953.8	−1.08540	−0.542701	−	0.839926i	$-0.682598\pi$
$467$	−10953.8	−1.08540	−0.542701	+	0.839926i	$0.682598\pi$
$468$	0	0
$469$	15656.7	1.54149
$470$	0	0
$471$	6039.04	0.590795
$472$	0	0
$473$	−4158.85	−0.404279
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4579.81	−0.439612
$478$	0	0
$479$	−13582.4	−1.29561	−0.647806	−	0.761806i	$-0.724313\pi$
$479$	−13582.4	−1.29561	−0.647806	+	0.761806i	$0.724313\pi$
$480$	0	0
$481$	−664.950	−0.0630335
$482$	0	0
$483$	−7521.27	−0.708550
$484$	0	0
$485$	0	0
$486$	0	0
$487$	830.552	0.0772811	0.0386405	−	0.999253i	$-0.487697\pi$
$487$	830.552	0.0772811	0.0386405	+	0.999253i	$0.487697\pi$
$488$	0	0
$489$	3168.55	0.293020
$490$	0	0
$491$	4954.05	0.455342	0.227671	−	0.973738i	$-0.426889\pi$
$491$	4954.05	0.455342	0.227671	+	0.973738i	$0.426889\pi$
$492$	0	0
$493$	5104.62	0.466329
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6893.66	−0.622179
$498$	0	0
$499$	258.456	0.0231865	0.0115932	−	0.999933i	$-0.496310\pi$
$499$	258.456	0.0231865	0.0115932	+	0.999933i	$0.496310\pi$
$500$	0	0
$501$	−5285.39	−0.471325
$502$	0	0
$503$	−10643.1	−0.943444	−0.471722	−	0.881747i	$-0.656367\pi$
$503$	−10643.1	−0.943444	−0.471722	+	0.881747i	$0.656367\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−363.198	−0.0318149
$508$	0	0
$509$	−3878.37	−0.337732	−0.168866	−	0.985639i	$-0.554011\pi$
$509$	−3878.37	−0.337732	−0.168866	+	0.985639i	$0.554011\pi$
$510$	0	0
$511$	28667.6	2.48176
$512$	0	0
$513$	11138.1	0.958594
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9275.00	−0.789002
$518$	0	0
$519$	3849.23	0.325554
$520$	0	0
$521$	−22966.5	−1.93125	−0.965624	−	0.259943i	$-0.916296\pi$
$521$	−22966.5	−1.93125	−0.965624	+	0.259943i	$0.916296\pi$
$522$	0	0
$523$	4302.19	0.359697	0.179849	−	0.983694i	$-0.442439\pi$
$523$	4302.19	0.359697	0.179849	+	0.983694i	$0.442439\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20689.6	1.71016
$528$	0	0
$529$	5342.64	0.439109
$530$	0	0
$531$	10390.6	0.849180
$532$	0	0
$533$	−1036.32	−0.0842180
$534$	0	0
$535$	0	0
$536$	0	0
$537$	268.122	0.0215462
$538$	0	0
$539$	12412.4	0.991909
$540$	0	0
$541$	−9184.17	−0.729868	−0.364934	−	0.931033i	$-0.618908\pi$
$541$	−9184.17	−0.729868	−0.364934	+	0.931033i	$0.618908\pi$
$542$	0	0
$543$	7556.91	0.597234
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−17191.0	−1.34376	−0.671878	−	0.740662i	$-0.734512\pi$
$547$	−17191.0	−1.34376	−0.671878	+	0.740662i	$0.734512\pi$
$548$	0	0
$549$	6376.77	0.495727
$550$	0	0
$551$	7499.19	0.579811
$552$	0	0
$553$	17880.6	1.37498
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7397.46	−0.562729	−0.281365	−	0.959601i	$-0.590787\pi$
$557$	−7397.46	−0.562729	−0.281365	+	0.959601i	$0.590787\pi$
$558$	0	0
$559$	1552.85	0.117493
$560$	0	0
$561$	−5345.44	−0.402290
$562$	0	0
$563$	−21404.8	−1.60231	−0.801157	−	0.598454i	$-0.795782\pi$
$563$	−21404.8	−1.60231	−0.801157	+	0.598454i	$0.795782\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−9950.42	−0.736999
$568$	0	0
$569$	14062.4	1.03607	0.518036	−	0.855359i	$-0.326663\pi$
$569$	14062.4	1.03607	0.518036	+	0.855359i	$0.326663\pi$
$570$	0	0
$571$	−6904.58	−0.506038	−0.253019	−	0.967461i	$-0.581423\pi$
$571$	−6904.58	−0.506038	−0.253019	+	0.967461i	$0.581423\pi$
$572$	0	0
$573$	−8855.76	−0.645645
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5348.49	0.385893	0.192947	−	0.981209i	$-0.438196\pi$
$577$	5348.49	0.385893	0.192947	+	0.981209i	$0.438196\pi$
$578$	0	0
$579$	2757.56	0.197928
$580$	0	0
$581$	−7055.38	−0.503798
$582$	0	0
$583$	7124.39	0.506110
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3224.74	0.226745	0.113372	−	0.993553i	$-0.463835\pi$
$587$	3224.74	0.226745	0.113372	+	0.993553i	$0.463835\pi$
$588$	0	0
$589$	30395.0	2.12633
$590$	0	0
$591$	8494.21	0.591210
$592$	0	0
$593$	−5352.47	−0.370657	−0.185329	−	0.982677i	$-0.559335\pi$
$593$	−5352.47	−0.370657	−0.185329	+	0.982677i	$0.559335\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1288.57	0.0883378
$598$	0	0
$599$	6921.21	0.472109	0.236054	−	0.971740i	$-0.424146\pi$
$599$	6921.21	0.472109	0.236054	+	0.971740i	$0.424146\pi$
$600$	0	0
$601$	−3114.84	−0.211409	−0.105705	−	0.994398i	$-0.533710\pi$
$601$	−3114.84	−0.211409	−0.105705	+	0.994398i	$0.533710\pi$
$602$	0	0
$603$	13249.2	0.894777
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11491.4	0.768403	0.384201	−	0.923249i	$-0.374477\pi$
$607$	11491.4	0.768403	0.384201	+	0.923249i	$0.374477\pi$
$608$	0	0
$609$	4061.40	0.270240
$610$	0	0
$611$	3463.14	0.229302
$612$	0	0
$613$	−5102.05	−0.336166	−0.168083	−	0.985773i	$-0.553758\pi$
$613$	−5102.05	−0.336166	−0.168083	+	0.985773i	$0.553758\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1713.96	−0.111834	−0.0559169	−	0.998435i	$-0.517808\pi$
$617$	−1713.96	−0.111834	−0.0559169	+	0.998435i	$0.517808\pi$
$618$	0	0
$619$	−6799.73	−0.441525	−0.220763	−	0.975328i	$-0.570855\pi$
$619$	−6799.73	−0.441525	−0.220763	+	0.975328i	$0.570855\pi$
$620$	0	0
$621$	−14042.9	−0.907446
$622$	0	0
$623$	40538.5	2.60696
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−7852.98	−0.500188
$628$	0	0
$629$	3654.14	0.231638
$630$	0	0
$631$	−14897.2	−0.939855	−0.469927	−	0.882705i	$-0.655720\pi$
$631$	−14897.2	−0.939855	−0.469927	+	0.882705i	$0.655720\pi$
$632$	0	0
$633$	−11048.4	−0.693733
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4634.59	−0.288272
$638$	0	0
$639$	−5833.65	−0.361151
$640$	0	0
$641$	−23310.5	−1.43637	−0.718183	−	0.695854i	$-0.755026\pi$
$641$	−23310.5	−1.43637	−0.718183	+	0.695854i	$0.755026\pi$
$642$	0	0
$643$	4945.54	0.303317	0.151659	−	0.988433i	$-0.451539\pi$
$643$	4945.54	0.303317	0.151659	+	0.988433i	$0.451539\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1023.77	0.0622079	0.0311039	−	0.999516i	$-0.490098\pi$
$647$	1023.77	0.0622079	0.0311039	+	0.999516i	$0.490098\pi$
$648$	0	0
$649$	−16163.7	−0.977631
$650$	0	0
$651$	16461.3	0.991043
$652$	0	0
$653$	6339.78	0.379931	0.189965	−	0.981791i	$-0.439162\pi$
$653$	6339.78	0.379931	0.189965	+	0.981791i	$0.439162\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	24259.5	1.44057
$658$	0	0
$659$	25174.6	1.48811	0.744053	−	0.668120i	$-0.232901\pi$
$659$	25174.6	1.48811	0.744053	+	0.668120i	$0.232901\pi$
$660$	0	0
$661$	−3598.02	−0.211720	−0.105860	−	0.994381i	$-0.533759\pi$
$661$	−3598.02	−0.211720	−0.105860	+	0.994381i	$0.533759\pi$
$662$	0	0
$663$	1995.90	0.116915
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9455.00	−0.548874
$668$	0	0
$669$	7020.19	0.405704
$670$	0	0
$671$	−9919.76	−0.570713
$672$	0	0
$673$	−20020.5	−1.14671	−0.573354	−	0.819308i	$-0.694358\pi$
$673$	−20020.5	−1.14671	−0.573354	+	0.819308i	$0.694358\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11360.1	0.644913	0.322456	−	0.946584i	$-0.395491\pi$
$677$	11360.1	0.644913	0.322456	+	0.946584i	$0.395491\pi$
$678$	0	0
$679$	−23046.8	−1.30258
$680$	0	0
$681$	2238.15	0.125941
$682$	0	0
$683$	6125.65	0.343179	0.171590	−	0.985169i	$-0.445110\pi$
$683$	6125.65	0.343179	0.171590	+	0.985169i	$0.445110\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7034.72	0.390672
$688$	0	0
$689$	−2660.14	−0.147087
$690$	0	0
$691$	23867.1	1.31396	0.656980	−	0.753908i	$-0.271834\pi$
$691$	23867.1	1.31396	0.656980	+	0.753908i	$0.271834\pi$
$692$	0	0
$693$	20609.6	1.12972
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5694.98	0.309488
$698$	0	0
$699$	6579.85	0.356041
$700$	0	0
$701$	−20884.7	−1.12526	−0.562628	−	0.826710i	$-0.690210\pi$
$701$	−20884.7	−1.12526	−0.562628	+	0.826710i	$0.690210\pi$
$702$	0	0
$703$	5368.30	0.288007
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5038.22	0.268008
$708$	0	0
$709$	33041.4	1.75021	0.875103	−	0.483937i	$-0.160794\pi$
$709$	33041.4	1.75021	0.875103	+	0.483937i	$0.160794\pi$
$710$	0	0
$711$	15131.2	0.798122
$712$	0	0
$713$	−38322.2	−2.01287
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15499.6	0.807311
$718$	0	0
$719$	4005.93	0.207783	0.103891	−	0.994589i	$-0.466871\pi$
$719$	4005.93	0.207783	0.103891	+	0.994589i	$0.466871\pi$
$720$	0	0
$721$	−14595.1	−0.753883
$722$	0	0
$723$	2615.29	0.134528
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36004.8	1.83679	0.918393	−	0.395670i	$-0.129488\pi$
$727$	36004.8	1.83679	0.918393	+	0.395670i	$0.129488\pi$
$728$	0	0
$729$	−2262.38	−0.114941
$730$	0	0
$731$	−8533.48	−0.431768
$732$	0	0
$733$	−5247.32	−0.264412	−0.132206	−	0.991222i	$-0.542206\pi$
$733$	−5247.32	−0.264412	−0.132206	+	0.991222i	$0.542206\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20610.6	−1.03012
$738$	0	0
$739$	−11474.9	−0.571194	−0.285597	−	0.958350i	$-0.592192\pi$
$739$	−11474.9	−0.571194	−0.285597	+	0.958350i	$0.592192\pi$
$740$	0	0
$741$	2932.18	0.145366
$742$	0	0
$743$	1067.93	0.0527304	0.0263652	−	0.999652i	$-0.491607\pi$
$743$	1067.93	0.0527304	0.0263652	+	0.999652i	$0.491607\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5970.50	−0.292436
$748$	0	0
$749$	−21259.0	−1.03710
$750$	0	0
$751$	−14201.9	−0.690061	−0.345031	−	0.938591i	$-0.612131\pi$
$751$	−14201.9	−0.690061	−0.345031	+	0.938591i	$0.612131\pi$
$752$	0	0
$753$	5454.88	0.263993
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−40376.3	−1.93857	−0.969287	−	0.245932i	$-0.920906\pi$
$757$	−40376.3	−1.93857	−0.969287	+	0.245932i	$0.920906\pi$
$758$	0	0
$759$	9901.07	0.473500
$760$	0	0
$761$	−10127.5	−0.482421	−0.241210	−	0.970473i	$-0.577544\pi$
$761$	−10127.5	−0.482421	−0.241210	+	0.970473i	$0.577544\pi$
$762$	0	0
$763$	31156.8	1.47831
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6035.29	0.284122
$768$	0	0
$769$	22173.1	1.03977	0.519884	−	0.854237i	$-0.325975\pi$
$769$	22173.1	1.03977	0.519884	+	0.854237i	$0.325975\pi$
$770$	0	0
$771$	6913.21	0.322922
$772$	0	0
$773$	−784.687	−0.0365113	−0.0182556	−	0.999833i	$-0.505811\pi$
$773$	−784.687	−0.0365113	−0.0182556	+	0.999833i	$0.505811\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2907.35	0.134235
$778$	0	0
$779$	8366.49	0.384802
$780$	0	0
$781$	9074.88	0.415781
$782$	0	0
$783$	7583.03	0.346099
$784$	0	0
$785$	0	0
$786$	0	0
$787$	40670.0	1.84210	0.921048	−	0.389449i	$-0.127335\pi$
$787$	40670.0	1.84210	0.921048	+	0.389449i	$0.127335\pi$
$788$	0	0
$789$	−8066.05	−0.363953
$790$	0	0
$791$	−34718.5	−1.56062
$792$	0	0
$793$	3703.89	0.165862
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11712.5	0.520548	0.260274	−	0.965535i	$-0.416187\pi$
$797$	11712.5	0.520548	0.260274	+	0.965535i	$0.416187\pi$
$798$	0	0
$799$	−19031.2	−0.842649
$800$	0	0
$801$	34305.0	1.51324
$802$	0	0
$803$	−37738.3	−1.65847
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12872.4	0.561499
$808$	0	0
$809$	15512.2	0.674142	0.337071	−	0.941479i	$-0.390564\pi$
$809$	15512.2	0.674142	0.337071	+	0.941479i	$0.390564\pi$
$810$	0	0
$811$	8903.39	0.385500	0.192750	−	0.981248i	$-0.438259\pi$
$811$	8903.39	0.385500	0.192750	+	0.981248i	$0.438259\pi$
$812$	0	0
$813$	−16418.4	−0.708262
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12536.5	−0.536839
$818$	0	0
$819$	−7695.31	−0.328322
$820$	0	0
$821$	−39364.5	−1.67336	−0.836681	−	0.547691i	$-0.815507\pi$
$821$	−39364.5	−1.67336	−0.836681	+	0.547691i	$0.815507\pi$
$822$	0	0
$823$	3046.59	0.129037	0.0645184	−	0.997917i	$-0.479449\pi$
$823$	3046.59	0.129037	0.0645184	+	0.997917i	$0.479449\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1441.20	0.0605992	0.0302996	−	0.999541i	$-0.490354\pi$
$827$	1441.20	0.0605992	0.0302996	+	0.999541i	$0.490354\pi$
$828$	0	0
$829$	−23069.4	−0.966507	−0.483253	−	0.875481i	$-0.660545\pi$
$829$	−23069.4	−0.966507	−0.483253	+	0.875481i	$0.660545\pi$
$830$	0	0
$831$	18917.4	0.789697
$832$	0	0
$833$	25468.8	1.05935
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30734.9	1.26924
$838$	0	0
$839$	19119.4	0.786742	0.393371	−	0.919380i	$-0.371309\pi$
$839$	19119.4	0.786742	0.393371	+	0.919380i	$0.371309\pi$
$840$	0	0
$841$	−19283.4	−0.790660
$842$	0	0
$843$	5162.51	0.210921
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3142.05	0.127464
$848$	0	0
$849$	9899.84	0.400191
$850$	0	0
$851$	−6768.37	−0.272640
$852$	0	0
$853$	42157.8	1.69221	0.846105	−	0.533016i	$-0.178941\pi$
$853$	42157.8	1.69221	0.846105	+	0.533016i	$0.178941\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44806.3	−1.78595	−0.892973	−	0.450111i	$-0.851384\pi$
$857$	−44806.3	−1.78595	−0.892973	+	0.450111i	$0.851384\pi$
$858$	0	0
$859$	−17101.8	−0.679283	−0.339642	−	0.940555i	$-0.610306\pi$
$859$	−17101.8	−0.679283	−0.339642	+	0.940555i	$0.610306\pi$
$860$	0	0
$861$	4531.11	0.179349
$862$	0	0
$863$	27027.4	1.06608	0.533039	−	0.846091i	$-0.321050\pi$
$863$	27027.4	1.06608	0.533039	+	0.846091i	$0.321050\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−409.703	−0.0160487
$868$	0	0
$869$	−23538.2	−0.918849
$870$	0	0
$871$	7695.68	0.299378
$872$	0	0
$873$	−19503.0	−0.756100
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31970.6	−1.23098	−0.615491	−	0.788144i	$-0.711042\pi$
$877$	−31970.6	−1.23098	−0.615491	+	0.788144i	$0.711042\pi$
$878$	0	0
$879$	−15021.2	−0.576398
$880$	0	0
$881$	24862.9	0.950798	0.475399	−	0.879770i	$-0.342304\pi$
$881$	24862.9	0.950798	0.475399	+	0.879770i	$0.342304\pi$
$882$	0	0
$883$	41043.7	1.56425	0.782124	−	0.623123i	$-0.214136\pi$
$883$	41043.7	1.56425	0.782124	+	0.623123i	$0.214136\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25664.2	−0.971500	−0.485750	−	0.874098i	$-0.661453\pi$
$887$	−25664.2	−0.971500	−0.485750	+	0.874098i	$0.661453\pi$
$888$	0	0
$889$	−28128.2	−1.06118
$890$	0	0
$891$	13098.8	0.492511
$892$	0	0
$893$	−27958.8	−1.04771
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3696.91	−0.137610
$898$	0	0
$899$	20693.5	0.767706
$900$	0	0
$901$	14618.4	0.540522
$902$	0	0
$903$	−6789.51	−0.250211
$904$	0	0
$905$	0	0
$906$	0	0
$907$	980.608	0.0358992	0.0179496	−	0.999839i	$-0.494286\pi$
$907$	980.608	0.0358992	0.0179496	+	0.999839i	$0.494286\pi$
$908$	0	0
$909$	4263.52	0.155569
$910$	0	0
$911$	25008.6	0.909520	0.454760	−	0.890614i	$-0.349725\pi$
$911$	25008.6	0.909520	0.454760	+	0.890614i	$0.349725\pi$
$912$	0	0
$913$	9287.77	0.336671
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−39702.0	−1.42975
$918$	0	0
$919$	−33448.3	−1.20061	−0.600303	−	0.799772i	$-0.704954\pi$
$919$	−33448.3	−1.20061	−0.600303	+	0.799772i	$0.704954\pi$
$920$	0	0
$921$	12580.1	0.450087
$922$	0	0
$923$	−3388.42	−0.120836
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12350.9	−0.437600
$928$	0	0
$929$	43395.1	1.53256	0.766279	−	0.642508i	$-0.222106\pi$
$929$	43395.1	1.53256	0.766279	+	0.642508i	$0.222106\pi$
$930$	0	0
$931$	37416.1	1.31715
$932$	0	0
$933$	17905.3	0.628290
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14907.1	0.519738	0.259869	−	0.965644i	$-0.416321\pi$
$937$	14907.1	0.519738	0.259869	+	0.965644i	$0.416321\pi$
$938$	0	0
$939$	2373.18	0.0824769
$940$	0	0
$941$	−33306.4	−1.15383	−0.576917	−	0.816803i	$-0.695744\pi$
$941$	−33306.4	−1.15383	−0.576917	+	0.816803i	$0.695744\pi$
$942$	0	0
$943$	−10548.5	−0.364270
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33308.2	1.14295	0.571473	−	0.820621i	$-0.306372\pi$
$947$	33308.2	1.14295	0.571473	+	0.820621i	$0.306372\pi$
$948$	0	0
$949$	14090.9	0.481991
$950$	0	0
$951$	−14736.3	−0.502477
$952$	0	0
$953$	−12789.1	−0.434711	−0.217355	−	0.976093i	$-0.569743\pi$
$953$	−12789.1	−0.434711	−0.217355	+	0.976093i	$0.569743\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5346.46	−0.180592
$958$	0	0
$959$	−43336.3	−1.45923
$960$	0	0
$961$	54082.2	1.81539
$962$	0	0
$963$	−17990.1	−0.601997
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−50315.0	−1.67324	−0.836620	−	0.547784i	$-0.815471\pi$
$967$	−50315.0	−1.67324	−0.836620	+	0.547784i	$0.815471\pi$
$968$	0	0
$969$	−16113.4	−0.534197
$970$	0	0
$971$	32991.6	1.09037	0.545185	−	0.838316i	$-0.316459\pi$
$971$	32991.6	1.09037	0.545185	+	0.838316i	$0.316459\pi$
$972$	0	0
$973$	2358.14	0.0776962
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23072.9	−0.755545	−0.377773	−	0.925898i	$-0.623310\pi$
$977$	−23072.9	−0.755545	−0.377773	+	0.925898i	$0.623310\pi$
$978$	0	0
$979$	−53365.2	−1.74215
$980$	0	0
$981$	26366.0	0.858105
$982$	0	0
$983$	−27899.3	−0.905238	−0.452619	−	0.891704i	$-0.649510\pi$
$983$	−27899.3	−0.905238	−0.452619	+	0.891704i	$0.649510\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−15141.9	−0.488319
$988$	0	0
$989$	15806.1	0.508195
$990$	0	0
$991$	−56072.7	−1.79739	−0.898693	−	0.438579i	$-0.855482\pi$
$991$	−56072.7	−1.79739	−0.898693	+	0.438579i	$0.855482\pi$
$992$	0	0
$993$	−23456.3	−0.749611
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10281.6	0.326601	0.163300	−	0.986576i	$-0.447786\pi$
$997$	10281.6	0.326601	0.163300	+	0.986576i	$0.447786\pi$
$998$	0	0
$999$	5428.32	0.171916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.4.a.k.1.2		4
5.2	odd	4		1300.4.c.g.1249.6		8
5.3	odd	4		1300.4.c.g.1249.3		8
5.4	even	2		260.4.a.c.1.3	✓	4
15.14	odd	2		2340.4.a.m.1.4		4
20.19	odd	2		1040.4.a.u.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.4.a.c.1.3	✓	4	5.4	even	2
1040.4.a.u.1.2		4	20.19	odd	2
1300.4.a.k.1.2		4	1.1	even	1	trivial
1300.4.c.g.1249.3		8	5.3	odd	4
1300.4.c.g.1249.6		8	5.2	odd	4
2340.4.a.m.1.4		4	15.14	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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1300.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14910 q^{3} -26.4482 q^{7} -22.3814 q^{9} +O(q^{10})\)	\(q-2.14910 q^{3} -26.4482 q^{7} -22.3814 q^{9} +34.8167 q^{11} -13.0000 q^{13} +71.4398 q^{17} +104.952 q^{19} +56.8398 q^{21} -132.324 q^{23} +106.125 q^{27} +71.4534 q^{29} +289.609 q^{31} -74.8245 q^{33} +51.1500 q^{37} +27.9383 q^{39} +79.7173 q^{41} -119.450 q^{43} -266.396 q^{47} +356.507 q^{49} -153.531 q^{51} +204.626 q^{53} -225.552 q^{57} -464.253 q^{59} -284.914 q^{61} +591.947 q^{63} -591.976 q^{67} +284.377 q^{69} +260.648 q^{71} -1083.91 q^{73} -920.838 q^{77} -676.063 q^{79} +376.223 q^{81} +266.762 q^{83} -153.561 q^{87} -1532.75 q^{89} +343.826 q^{91} -622.398 q^{93} +871.393 q^{97} -779.245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9	\( 4 q - 4 q^{3} + 8 q^{7} + 36 q^{9} + 28 q^{11} - 52 q^{13} - 200 q^{17} + 132 q^{19} + 160 q^{21} - 244 q^{23} - 496 q^{27} - 8 q^{29} + 292 q^{31} - 128 q^{33} - 168 q^{37} + 52 q^{39} + 280 q^{41} - 452 q^{43} + 280 q^{47} + 340 q^{49} + 792 q^{51} - 584 q^{53} - 1200 q^{57} - 708 q^{59} + 1128 q^{61} - 232 q^{63} - 176 q^{67} + 160 q^{69} - 1028 q^{71} - 664 q^{73} + 864 q^{77} - 728 q^{79} + 2244 q^{81} - 552 q^{83} + 2064 q^{87} - 2824 q^{89} - 104 q^{91} - 1392 q^{93} - 1160 q^{97} - 4220 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^7 + 36 * q^9 + 28 * q^11 - 52 * q^13 - 200 * q^17 + 132 * q^19 + 160 * q^21 - 244 * q^23 - 496 * q^27 - 8 * q^29 + 292 * q^31 - 128 * q^33 - 168 * q^37 + 52 * q^39 + 280 * q^41 - 452 * q^43 + 280 * q^47 + 340 * q^49 + 792 * q^51 - 584 * q^53 - 1200 * q^57 - 708 * q^59 + 1128 * q^61 - 232 * q^63 - 176 * q^67 + 160 * q^69 - 1028 * q^71 - 664 * q^73 + 864 * q^77 - 728 * q^79 + 2244 * q^81 - 552 * q^83 + 2064 * q^87 - 2824 * q^89 - 104 * q^91 - 1392 * q^93 - 1160 * q^97 - 4220 * q^99

Introduction

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