LMFDB - Embedded newform 1300.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,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, 2, names="a")

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

chi := DirichletCharacter("1300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q+2.37228 q^{3} +3.37228 q^{7} +2.62772 q^{9} +O(q^{10})$	$q+2.37228 q^{3} +3.37228 q^{7} +2.62772 q^{9} +1.37228 q^{11} +1.00000 q^{13} +1.37228 q^{17} -4.00000 q^{19} +8.00000 q^{21} +4.37228 q^{23} -0.883156 q^{27} -5.74456 q^{29} +3.37228 q^{31} +3.25544 q^{33} -10.0000 q^{37} +2.37228 q^{39} +8.74456 q^{41} +3.62772 q^{43} +4.62772 q^{47} +4.37228 q^{49} +3.25544 q^{51} -11.7446 q^{53} -9.48913 q^{57} -1.37228 q^{59} +1.74456 q^{61} +8.86141 q^{63} -8.11684 q^{67} +10.3723 q^{69} +3.25544 q^{71} -6.74456 q^{73} +4.62772 q^{77} +12.3723 q^{79} -9.97825 q^{81} +13.3723 q^{83} -13.6277 q^{87} +12.0000 q^{89} +3.37228 q^{91} +8.00000 q^{93} -3.48913 q^{97} +3.60597 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + q^{7} + 11 q^{9}+O(q^{10}) $ 2 * q - q^3 + q^7 + 11 * q^9	$ 2 q - q^{3} + q^{7} + 11 q^{9} - 3 q^{11} + 2 q^{13} - 3 q^{17} - 8 q^{19} + 16 q^{21} + 3 q^{23} - 19 q^{27} + q^{31} + 18 q^{33} - 20 q^{37} - q^{39} + 6 q^{41} + 13 q^{43} + 15 q^{47} + 3 q^{49} + 18 q^{51} - 12 q^{53} + 4 q^{57} + 3 q^{59} - 8 q^{61} - 11 q^{63} + q^{67} + 15 q^{69} + 18 q^{71} - 2 q^{73} + 15 q^{77} + 19 q^{79} + 26 q^{81} + 21 q^{83} - 33 q^{87} + 24 q^{89} + q^{91} + 16 q^{93} + 16 q^{97} - 33 q^{99}+O(q^{100}) $ 2 * q - q^3 + q^7 + 11 * q^9 - 3 * q^11 + 2 * q^13 - 3 * q^17 - 8 * q^19 + 16 * q^21 + 3 * q^23 - 19 * q^27 + q^31 + 18 * q^33 - 20 * q^37 - q^39 + 6 * q^41 + 13 * q^43 + 15 * q^47 + 3 * q^49 + 18 * q^51 - 12 * q^53 + 4 * q^57 + 3 * q^59 - 8 * q^61 - 11 * q^63 + q^67 + 15 * q^69 + 18 * q^71 - 2 * q^73 + 15 * q^77 + 19 * q^79 + 26 * q^81 + 21 * q^83 - 33 * q^87 + 24 * q^89 + q^91 + 16 * q^93 + 16 * q^97 - 33 * 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
$3$	2.37228	1.36964	0.684819	−	0.728714i	$-0.259881\pi$
$3$	2.37228	1.36964	0.684819	+	0.728714i	$0.259881\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	0	0
$9$	2.62772	0.875906
$10$	0	0
$11$	1.37228	0.413758	0.206879	−	0.978366i	$-0.433669\pi$
$11$	1.37228	0.413758	0.206879	+	0.978366i	$0.433669\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	0	0
$23$	4.37228	0.911684	0.455842	−	0.890061i	$-0.349338\pi$
$23$	4.37228	0.911684	0.455842	+	0.890061i	$0.349338\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.883156	−0.169963
$28$	0	0
$29$	−5.74456	−1.06674	−0.533369	−	0.845883i	$-0.679074\pi$
$29$	−5.74456	−1.06674	−0.533369	+	0.845883i	$0.679074\pi$
$30$	0	0
$31$	3.37228	0.605680	0.302840	−	0.953041i	$-0.402065\pi$
$31$	3.37228	0.605680	0.302840	+	0.953041i	$0.402065\pi$
$32$	0	0
$33$	3.25544	0.566699
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	2.37228	0.379869
$40$	0	0
$41$	8.74456	1.36567	0.682836	−	0.730572i	$-0.260747\pi$
$41$	8.74456	1.36567	0.682836	+	0.730572i	$0.260747\pi$
$42$	0	0
$43$	3.62772	0.553222	0.276611	−	0.960982i	$-0.410789\pi$
$43$	3.62772	0.553222	0.276611	+	0.960982i	$0.410789\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.62772	0.675022	0.337511	−	0.941322i	$-0.390415\pi$
$47$	4.62772	0.675022	0.337511	+	0.941322i	$0.390415\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	3.25544	0.455852
$52$	0	0
$53$	−11.7446	−1.61324	−0.806620	−	0.591071i	$-0.798705\pi$
$53$	−11.7446	−1.61324	−0.806620	+	0.591071i	$0.798705\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.48913	−1.25687
$58$	0	0
$59$	−1.37228	−0.178656	−0.0893279	−	0.996002i	$-0.528472\pi$
$59$	−1.37228	−0.178656	−0.0893279	+	0.996002i	$0.528472\pi$
$60$	0	0
$61$	1.74456	0.223368	0.111684	−	0.993744i	$-0.464375\pi$
$61$	1.74456	0.223368	0.111684	+	0.993744i	$0.464375\pi$
$62$	0	0
$63$	8.86141	1.11643
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.11684	−0.991630	−0.495815	−	0.868428i	$-0.665131\pi$
$67$	−8.11684	−0.991630	−0.495815	+	0.868428i	$0.665131\pi$
$68$	0	0
$69$	10.3723	1.24868
$70$	0	0
$71$	3.25544	0.386349	0.193175	−	0.981164i	$-0.438122\pi$
$71$	3.25544	0.386349	0.193175	+	0.981164i	$0.438122\pi$
$72$	0	0
$73$	−6.74456	−0.789391	−0.394696	−	0.918812i	$-0.629150\pi$
$73$	−6.74456	−0.789391	−0.394696	+	0.918812i	$0.629150\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.62772	0.527377
$78$	0	0
$79$	12.3723	1.39199	0.695995	−	0.718046i	$-0.254963\pi$
$79$	12.3723	1.39199	0.695995	+	0.718046i	$0.254963\pi$
$80$	0	0
$81$	−9.97825	−1.10869
$82$	0	0
$83$	13.3723	1.46780	0.733899	−	0.679258i	$-0.237698\pi$
$83$	13.3723	1.46780	0.733899	+	0.679258i	$0.237698\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−13.6277	−1.46104
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	3.37228	0.353511
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.48913	−0.354267	−0.177133	−	0.984187i	$-0.556682\pi$
$97$	−3.48913	−0.354267	−0.177133	+	0.984187i	$0.556682\pi$
$98$	0	0
$99$	3.60597	0.362414
$100$	0	0
$101$	17.2337	1.71482	0.857408	−	0.514637i	$-0.172073\pi$
$101$	17.2337	1.71482	0.857408	+	0.514637i	$0.172073\pi$
$102$	0	0
$103$	−2.37228	−0.233748	−0.116874	−	0.993147i	$-0.537287\pi$
$103$	−2.37228	−0.233748	−0.116874	+	0.993147i	$0.537287\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.62772	−0.737399	−0.368700	−	0.929549i	$-0.620197\pi$
$107$	−7.62772	−0.737399	−0.368700	+	0.929549i	$0.620197\pi$
$108$	0	0
$109$	−12.7446	−1.22071	−0.610354	−	0.792129i	$-0.708973\pi$
$109$	−12.7446	−1.22071	−0.610354	+	0.792129i	$0.708973\pi$
$110$	0	0
$111$	−23.7228	−2.25167
$112$	0	0
$113$	−13.1168	−1.23393	−0.616964	−	0.786991i	$-0.711638\pi$
$113$	−13.1168	−1.23393	−0.616964	+	0.786991i	$0.711638\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.62772	0.242933
$118$	0	0
$119$	4.62772	0.424222
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	20.7446	1.87047
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.1168	−1.51887	−0.759437	−	0.650581i	$-0.774526\pi$
$127$	−17.1168	−1.51887	−0.759437	+	0.650581i	$0.774526\pi$
$128$	0	0
$129$	8.60597	0.757713
$130$	0	0
$131$	−19.1168	−1.67025	−0.835123	−	0.550063i	$-0.814604\pi$
$131$	−19.1168	−1.67025	−0.835123	+	0.550063i	$0.814604\pi$
$132$	0	0
$133$	−13.4891	−1.16966
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.74456	0.747098	0.373549	−	0.927610i	$-0.378141\pi$
$137$	8.74456	0.747098	0.373549	+	0.927610i	$0.378141\pi$
$138$	0	0
$139$	−11.6277	−0.986250	−0.493125	−	0.869958i	$-0.664146\pi$
$139$	−11.6277	−0.986250	−0.493125	+	0.869958i	$0.664146\pi$
$140$	0	0
$141$	10.9783	0.924535
$142$	0	0
$143$	1.37228	0.114756
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.3723	0.855491
$148$	0	0
$149$	0.510875	0.0418525	0.0209262	−	0.999781i	$-0.493338\pi$
$149$	0.510875	0.0418525	0.0209262	+	0.999781i	$0.493338\pi$
$150$	0	0
$151$	12.1168	0.986055	0.493027	−	0.870014i	$-0.335890\pi$
$151$	12.1168	0.986055	0.493027	+	0.870014i	$0.335890\pi$
$152$	0	0
$153$	3.60597	0.291525
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.8614	1.18607	0.593035	−	0.805177i	$-0.297930\pi$
$157$	14.8614	1.18607	0.593035	+	0.805177i	$0.297930\pi$
$158$	0	0
$159$	−27.8614	−2.20955
$160$	0	0
$161$	14.7446	1.16203
$162$	0	0
$163$	−18.2337	−1.42817	−0.714086	−	0.700058i	$-0.753158\pi$
$163$	−18.2337	−1.42817	−0.714086	+	0.700058i	$0.753158\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.74456	−0.676675	−0.338337	−	0.941025i	$-0.609864\pi$
$167$	−8.74456	−0.676675	−0.338337	+	0.941025i	$0.609864\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.5109	−0.803787
$172$	0	0
$173$	−16.1168	−1.22534	−0.612670	−	0.790338i	$-0.709905\pi$
$173$	−16.1168	−1.22534	−0.612670	+	0.790338i	$0.709905\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.25544	−0.244694
$178$	0	0
$179$	7.11684	0.531938	0.265969	−	0.963982i	$-0.414308\pi$
$179$	7.11684	0.531938	0.265969	+	0.963982i	$0.414308\pi$
$180$	0	0
$181$	12.1168	0.900638	0.450319	−	0.892868i	$-0.351310\pi$
$181$	12.1168	0.900638	0.450319	+	0.892868i	$0.351310\pi$
$182$	0	0
$183$	4.13859	0.305934
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.88316	0.137710
$188$	0	0
$189$	−2.97825	−0.216636
$190$	0	0
$191$	25.1168	1.81739	0.908696	−	0.417460i	$-0.137079\pi$
$191$	25.1168	1.81739	0.908696	+	0.417460i	$0.137079\pi$
$192$	0	0
$193$	13.4891	0.970968	0.485484	−	0.874245i	$-0.338643\pi$
$193$	13.4891	0.970968	0.485484	+	0.874245i	$0.338643\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.7446	1.47799	0.738994	−	0.673712i	$-0.235301\pi$
$197$	20.7446	1.47799	0.738994	+	0.673712i	$0.235301\pi$
$198$	0	0
$199$	−5.11684	−0.362723	−0.181362	−	0.983416i	$-0.558050\pi$
$199$	−5.11684	−0.362723	−0.181362	+	0.983416i	$0.558050\pi$
$200$	0	0
$201$	−19.2554	−1.35817
$202$	0	0
$203$	−19.3723	−1.35967
$204$	0	0
$205$	0	0
$206$	0	0
$207$	11.4891	0.798549
$208$	0	0
$209$	−5.48913	−0.379691
$210$	0	0
$211$	−24.2337	−1.66832	−0.834158	−	0.551526i	$-0.814046\pi$
$211$	−24.2337	−1.66832	−0.834158	+	0.551526i	$0.814046\pi$
$212$	0	0
$213$	7.72281	0.529158
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.3723	0.772001
$218$	0	0
$219$	−16.0000	−1.08118
$220$	0	0
$221$	1.37228	0.0923096
$222$	0	0
$223$	22.2337	1.48888	0.744439	−	0.667691i	$-0.232717\pi$
$223$	22.2337	1.48888	0.744439	+	0.667691i	$0.232717\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.37228	0.489315	0.244658	−	0.969610i	$-0.421324\pi$
$227$	7.37228	0.489315	0.244658	+	0.969610i	$0.421324\pi$
$228$	0	0
$229$	−12.2337	−0.808425	−0.404212	−	0.914665i	$-0.632454\pi$
$229$	−12.2337	−0.808425	−0.404212	+	0.914665i	$0.632454\pi$
$230$	0	0
$231$	10.9783	0.722316
$232$	0	0
$233$	−13.1168	−0.859313	−0.429657	−	0.902992i	$-0.641365\pi$
$233$	−13.1168	−0.859313	−0.429657	+	0.902992i	$0.641365\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	29.3505	1.90652
$238$	0	0
$239$	7.37228	0.476873	0.238437	−	0.971158i	$-0.423365\pi$
$239$	7.37228	0.476873	0.238437	+	0.971158i	$0.423365\pi$
$240$	0	0
$241$	−30.2337	−1.94752	−0.973762	−	0.227571i	$-0.926922\pi$
$241$	−30.2337	−1.94752	−0.973762	+	0.227571i	$0.926922\pi$
$242$	0	0
$243$	−21.0217	−1.34855
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	31.7228	2.01035
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.48913	−0.529537	−0.264769	−	0.964312i	$-0.585296\pi$
$257$	−8.48913	−0.529537	−0.264769	+	0.964312i	$0.585296\pi$
$258$	0	0
$259$	−33.7228	−2.09543
$260$	0	0
$261$	−15.0951	−0.934363
$262$	0	0
$263$	−29.4891	−1.81838	−0.909189	−	0.416384i	$-0.863297\pi$
$263$	−29.4891	−1.81838	−0.909189	+	0.416384i	$0.863297\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	28.4674	1.74218
$268$	0	0
$269$	−0.255437	−0.0155743	−0.00778715	−	0.999970i	$-0.502479\pi$
$269$	−0.255437	−0.0155743	−0.00778715	+	0.999970i	$0.502479\pi$
$270$	0	0
$271$	8.86141	0.538292	0.269146	−	0.963099i	$-0.413259\pi$
$271$	8.86141	0.538292	0.269146	+	0.963099i	$0.413259\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.3505	1.40300	0.701499	−	0.712671i	$-0.252515\pi$
$277$	23.3505	1.40300	0.701499	+	0.712671i	$0.252515\pi$
$278$	0	0
$279$	8.86141	0.530519
$280$	0	0
$281$	−2.74456	−0.163727	−0.0818634	−	0.996644i	$-0.526087\pi$
$281$	−2.74456	−0.163727	−0.0818634	+	0.996644i	$0.526087\pi$
$282$	0	0
$283$	5.25544	0.312403	0.156202	−	0.987725i	$-0.450075\pi$
$283$	5.25544	0.312403	0.156202	+	0.987725i	$0.450075\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	29.4891	1.74069
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	−8.27719	−0.485217
$292$	0	0
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.21194	−0.0703238
$298$	0	0
$299$	4.37228	0.252856
$300$	0	0
$301$	12.2337	0.705138
$302$	0	0
$303$	40.8832	2.34868
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.7446	−1.41225	−0.706123	−	0.708089i	$-0.749558\pi$
$307$	−24.7446	−1.41225	−0.706123	+	0.708089i	$0.749558\pi$
$308$	0	0
$309$	−5.62772	−0.320150
$310$	0	0
$311$	−25.1168	−1.42425	−0.712123	−	0.702055i	$-0.752266\pi$
$311$	−25.1168	−1.42425	−0.712123	+	0.702055i	$0.752266\pi$
$312$	0	0
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.4891	−0.645294	−0.322647	−	0.946519i	$-0.604573\pi$
$317$	−11.4891	−0.645294	−0.322647	+	0.946519i	$0.604573\pi$
$318$	0	0
$319$	−7.88316	−0.441372
$320$	0	0
$321$	−18.0951	−1.00997
$322$	0	0
$323$	−5.48913	−0.305423
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−30.2337	−1.67193
$328$	0	0
$329$	15.6060	0.860385
$330$	0	0
$331$	1.48913	0.0818497	0.0409249	−	0.999162i	$-0.486970\pi$
$331$	1.48913	0.0818497	0.0409249	+	0.999162i	$0.486970\pi$
$332$	0	0
$333$	−26.2772	−1.43998
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	0	0
$339$	−31.1168	−1.69003
$340$	0	0
$341$	4.62772	0.250605
$342$	0	0
$343$	−8.86141	−0.478471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.13859	0.114806	0.0574029	−	0.998351i	$-0.481718\pi$
$347$	2.13859	0.114806	0.0574029	+	0.998351i	$0.481718\pi$
$348$	0	0
$349$	21.7228	1.16280	0.581398	−	0.813619i	$-0.302506\pi$
$349$	21.7228	1.16280	0.581398	+	0.813619i	$0.302506\pi$
$350$	0	0
$351$	−0.883156	−0.0471394
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.9783	0.581031
$358$	0	0
$359$	−12.8614	−0.678799	−0.339400	−	0.940642i	$-0.610224\pi$
$359$	−12.8614	−0.678799	−0.339400	+	0.940642i	$0.610224\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−21.6277	−1.13516
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11.1168	−0.580295	−0.290147	−	0.956982i	$-0.593704\pi$
$367$	−11.1168	−0.580295	−0.290147	+	0.956982i	$0.593704\pi$
$368$	0	0
$369$	22.9783	1.19620
$370$	0	0
$371$	−39.6060	−2.05624
$372$	0	0
$373$	2.25544	0.116782	0.0583911	−	0.998294i	$-0.481403\pi$
$373$	2.25544	0.116782	0.0583911	+	0.998294i	$0.481403\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.74456	−0.295860
$378$	0	0
$379$	−20.1168	−1.03333	−0.516666	−	0.856187i	$-0.672827\pi$
$379$	−20.1168	−1.03333	−0.516666	+	0.856187i	$0.672827\pi$
$380$	0	0
$381$	−40.6060	−2.08031
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.53262	0.484571
$388$	0	0
$389$	31.6277	1.60359	0.801794	−	0.597600i	$-0.203879\pi$
$389$	31.6277	1.60359	0.801794	+	0.597600i	$0.203879\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	−45.3505	−2.28763
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.7446	1.14152	0.570758	−	0.821118i	$-0.306649\pi$
$397$	22.7446	1.14152	0.570758	+	0.821118i	$0.306649\pi$
$398$	0	0
$399$	−32.0000	−1.60200
$400$	0	0
$401$	32.2337	1.60967	0.804837	−	0.593496i	$-0.202253\pi$
$401$	32.2337	1.60967	0.804837	+	0.593496i	$0.202253\pi$
$402$	0	0
$403$	3.37228	0.167985
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.7228	−0.680215
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	20.7446	1.02325
$412$	0	0
$413$	−4.62772	−0.227715
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−27.5842	−1.35081
$418$	0	0
$419$	25.6277	1.25200	0.625998	−	0.779825i	$-0.284692\pi$
$419$	25.6277	1.25200	0.625998	+	0.779825i	$0.284692\pi$
$420$	0	0
$421$	2.51087	0.122373	0.0611863	−	0.998126i	$-0.480512\pi$
$421$	2.51087	0.122373	0.0611863	+	0.998126i	$0.480512\pi$
$422$	0	0
$423$	12.1603	0.591256
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.88316	0.284706
$428$	0	0
$429$	3.25544	0.157174
$430$	0	0
$431$	−10.9783	−0.528804	−0.264402	−	0.964413i	$-0.585175\pi$
$431$	−10.9783	−0.528804	−0.264402	+	0.964413i	$0.585175\pi$
$432$	0	0
$433$	12.3723	0.594574	0.297287	−	0.954788i	$-0.403918\pi$
$433$	12.3723	0.594574	0.297287	+	0.954788i	$0.403918\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.4891	−0.836618
$438$	0	0
$439$	−1.25544	−0.0599188	−0.0299594	−	0.999551i	$-0.509538\pi$
$439$	−1.25544	−0.0599188	−0.0299594	+	0.999551i	$0.509538\pi$
$440$	0	0
$441$	11.4891	0.547101
$442$	0	0
$443$	−37.7228	−1.79226	−0.896132	−	0.443787i	$-0.853635\pi$
$443$	−37.7228	−1.79226	−0.896132	+	0.443787i	$0.853635\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.21194	0.0573227
$448$	0	0
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	28.7446	1.35054
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	−1.21194	−0.0565684
$460$	0	0
$461$	−28.9783	−1.34965	−0.674826	−	0.737977i	$-0.735781\pi$
$461$	−28.9783	−1.34965	−0.674826	+	0.737977i	$0.735781\pi$
$462$	0	0
$463$	−14.6277	−0.679808	−0.339904	−	0.940460i	$-0.610395\pi$
$463$	−14.6277	−0.679808	−0.339904	+	0.940460i	$0.610395\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.1168	−0.884622	−0.442311	−	0.896862i	$-0.645841\pi$
$467$	−19.1168	−0.884622	−0.442311	+	0.896862i	$0.645841\pi$
$468$	0	0
$469$	−27.3723	−1.26393
$470$	0	0
$471$	35.2554	1.62448
$472$	0	0
$473$	4.97825	0.228900
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−30.8614	−1.41305
$478$	0	0
$479$	21.6060	0.987202	0.493601	−	0.869688i	$-0.335680\pi$
$479$	21.6060	0.987202	0.493601	+	0.869688i	$0.335680\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	34.9783	1.59157
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−14.1168	−0.639695	−0.319848	−	0.947469i	$-0.603632\pi$
$487$	−14.1168	−0.639695	−0.319848	+	0.947469i	$0.603632\pi$
$488$	0	0
$489$	−43.2554	−1.95608
$490$	0	0
$491$	33.8614	1.52814	0.764072	−	0.645131i	$-0.223197\pi$
$491$	33.8614	1.52814	0.764072	+	0.645131i	$0.223197\pi$
$492$	0	0
$493$	−7.88316	−0.355039
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.9783	0.492442
$498$	0	0
$499$	38.3505	1.71681	0.858403	−	0.512976i	$-0.171457\pi$
$499$	38.3505	1.71681	0.858403	+	0.512976i	$0.171457\pi$
$500$	0	0
$501$	−20.7446	−0.926799
$502$	0	0
$503$	−1.62772	−0.0725764	−0.0362882	−	0.999341i	$-0.511553\pi$
$503$	−1.62772	−0.0725764	−0.0362882	+	0.999341i	$0.511553\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.37228	0.105357
$508$	0	0
$509$	8.74456	0.387596	0.193798	−	0.981041i	$-0.437919\pi$
$509$	8.74456	0.387596	0.193798	+	0.981041i	$0.437919\pi$
$510$	0	0
$511$	−22.7446	−1.00616
$512$	0	0
$513$	3.53262	0.155969
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.35053	0.279296
$518$	0	0
$519$	−38.2337	−1.67827
$520$	0	0
$521$	−24.0951	−1.05563	−0.527813	−	0.849361i	$-0.676988\pi$
$521$	−24.0951	−1.05563	−0.527813	+	0.849361i	$0.676988\pi$
$522$	0	0
$523$	−12.2337	−0.534942	−0.267471	−	0.963566i	$-0.586188\pi$
$523$	−12.2337	−0.534942	−0.267471	+	0.963566i	$0.586188\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.62772	0.201587
$528$	0	0
$529$	−3.88316	−0.168833
$530$	0	0
$531$	−3.60597	−0.156486
$532$	0	0
$533$	8.74456	0.378769
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.8832	0.728562
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	24.9783	1.07390	0.536949	−	0.843614i	$-0.319577\pi$
$541$	24.9783	1.07390	0.536949	+	0.843614i	$0.319577\pi$
$542$	0	0
$543$	28.7446	1.23355
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.48913	0.0636704	0.0318352	−	0.999493i	$-0.489865\pi$
$547$	1.48913	0.0636704	0.0318352	+	0.999493i	$0.489865\pi$
$548$	0	0
$549$	4.58422	0.195650
$550$	0	0
$551$	22.9783	0.978906
$552$	0	0
$553$	41.7228	1.77423
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.2337	1.62001	0.810007	−	0.586421i	$-0.199463\pi$
$557$	38.2337	1.62001	0.810007	+	0.586421i	$0.199463\pi$
$558$	0	0
$559$	3.62772	0.153436
$560$	0	0
$561$	4.46738	0.188613
$562$	0	0
$563$	32.8397	1.38403	0.692013	−	0.721885i	$-0.256724\pi$
$563$	32.8397	1.38403	0.692013	+	0.721885i	$0.256724\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−33.6495	−1.41314
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	−24.2337	−1.01415	−0.507074	−	0.861902i	$-0.669273\pi$
$571$	−24.2337	−1.01415	−0.507074	+	0.861902i	$0.669273\pi$
$572$	0	0
$573$	59.5842	2.48917
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	32.0000	1.32987
$580$	0	0
$581$	45.0951	1.87086
$582$	0	0
$583$	−16.1168	−0.667491
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.6277	0.933946	0.466973	−	0.884272i	$-0.345345\pi$
$587$	22.6277	0.933946	0.466973	+	0.884272i	$0.345345\pi$
$588$	0	0
$589$	−13.4891	−0.555810
$590$	0	0
$591$	49.2119	2.02431
$592$	0	0
$593$	−2.74456	−0.112706	−0.0563528	−	0.998411i	$-0.517947\pi$
$593$	−2.74456	−0.112706	−0.0563528	+	0.998411i	$0.517947\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.1386	−0.496800
$598$	0	0
$599$	33.3505	1.36267	0.681333	−	0.731974i	$-0.261401\pi$
$599$	33.3505	1.36267	0.681333	+	0.731974i	$0.261401\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	−21.3288	−0.868575
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.23369	0.171840	0.0859200	−	0.996302i	$-0.472617\pi$
$607$	4.23369	0.171840	0.0859200	+	0.996302i	$0.472617\pi$
$608$	0	0
$609$	−45.9565	−1.86225
$610$	0	0
$611$	4.62772	0.187217
$612$	0	0
$613$	17.2554	0.696941	0.348470	−	0.937320i	$-0.386701\pi$
$613$	17.2554	0.696941	0.348470	+	0.937320i	$0.386701\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.9783	0.683519	0.341759	−	0.939788i	$-0.388977\pi$
$617$	16.9783	0.683519	0.341759	+	0.939788i	$0.388977\pi$
$618$	0	0
$619$	−24.7446	−0.994568	−0.497284	−	0.867588i	$-0.665669\pi$
$619$	−24.7446	−0.994568	−0.497284	+	0.867588i	$0.665669\pi$
$620$	0	0
$621$	−3.86141	−0.154953
$622$	0	0
$623$	40.4674	1.62129
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13.0217	−0.520039
$628$	0	0
$629$	−13.7228	−0.547164
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	−57.4891	−2.28499
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.37228	0.173236
$638$	0	0
$639$	8.55437	0.338406
$640$	0	0
$641$	44.4891	1.75721	0.878607	−	0.477545i	$-0.158473\pi$
$641$	44.4891	1.75721	0.878607	+	0.477545i	$0.158473\pi$
$642$	0	0
$643$	48.4674	1.91137	0.955683	−	0.294397i	$-0.0951186\pi$
$643$	48.4674	1.91137	0.955683	+	0.294397i	$0.0951186\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.4891	0.687568	0.343784	−	0.939049i	$-0.388291\pi$
$647$	17.4891	0.687568	0.343784	+	0.939049i	$0.388291\pi$
$648$	0	0
$649$	−1.88316	−0.0739203
$650$	0	0
$651$	26.9783	1.05736
$652$	0	0
$653$	−22.6277	−0.885491	−0.442746	−	0.896647i	$-0.645995\pi$
$653$	−22.6277	−0.885491	−0.442746	+	0.896647i	$0.645995\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−17.7228	−0.691433
$658$	0	0
$659$	1.11684	0.0435061	0.0217530	−	0.999763i	$-0.493075\pi$
$659$	1.11684	0.0435061	0.0217530	+	0.999763i	$0.493075\pi$
$660$	0	0
$661$	4.74456	0.184542	0.0922710	−	0.995734i	$-0.470587\pi$
$661$	4.74456	0.184542	0.0922710	+	0.995734i	$0.470587\pi$
$662$	0	0
$663$	3.25544	0.126431
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−25.1168	−0.972528
$668$	0	0
$669$	52.7446	2.03922
$670$	0	0
$671$	2.39403	0.0924205
$672$	0	0
$673$	−14.6277	−0.563857	−0.281929	−	0.959435i	$-0.590974\pi$
$673$	−14.6277	−0.563857	−0.281929	+	0.959435i	$0.590974\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.8832	−0.418274	−0.209137	−	0.977886i	$-0.567065\pi$
$677$	−10.8832	−0.418274	−0.209137	+	0.977886i	$0.567065\pi$
$678$	0	0
$679$	−11.7663	−0.451550
$680$	0	0
$681$	17.4891	0.670185
$682$	0	0
$683$	3.60597	0.137979	0.0689893	−	0.997617i	$-0.478023\pi$
$683$	3.60597	0.137979	0.0689893	+	0.997617i	$0.478023\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−29.0217	−1.10725
$688$	0	0
$689$	−11.7446	−0.447432
$690$	0	0
$691$	−16.3505	−0.622004	−0.311002	−	0.950409i	$-0.600665\pi$
$691$	−16.3505	−0.622004	−0.311002	+	0.950409i	$0.600665\pi$
$692$	0	0
$693$	12.1603	0.461933
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	−31.1168	−1.17695
$700$	0	0
$701$	47.2337	1.78399	0.891996	−	0.452044i	$-0.149305\pi$
$701$	47.2337	1.78399	0.891996	+	0.452044i	$0.149305\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	58.1168	2.18571
$708$	0	0
$709$	−26.9783	−1.01319	−0.506595	−	0.862184i	$-0.669096\pi$
$709$	−26.9783	−1.01319	−0.506595	+	0.862184i	$0.669096\pi$
$710$	0	0
$711$	32.5109	1.21925
$712$	0	0
$713$	14.7446	0.552188
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.4891	0.653143
$718$	0	0
$719$	37.6277	1.40328	0.701639	−	0.712533i	$-0.252452\pi$
$719$	37.6277	1.40328	0.701639	+	0.712533i	$0.252452\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−71.7228	−2.66740
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.60597	−0.170826	−0.0854130	−	0.996346i	$-0.527221\pi$
$727$	−4.60597	−0.170826	−0.0854130	+	0.996346i	$0.527221\pi$
$728$	0	0
$729$	−19.9348	−0.738324
$730$	0	0
$731$	4.97825	0.184127
$732$	0	0
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.1386	−0.410295
$738$	0	0
$739$	38.3505	1.41075	0.705374	−	0.708836i	$-0.250779\pi$
$739$	38.3505	1.41075	0.705374	+	0.708836i	$0.250779\pi$
$740$	0	0
$741$	−9.48913	−0.348592
$742$	0	0
$743$	7.88316	0.289205	0.144602	−	0.989490i	$-0.453810\pi$
$743$	7.88316	0.289205	0.144602	+	0.989490i	$0.453810\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	35.1386	1.28565
$748$	0	0
$749$	−25.7228	−0.939891
$750$	0	0
$751$	−5.11684	−0.186716	−0.0933581	−	0.995633i	$-0.529760\pi$
$751$	−5.11684	−0.186716	−0.0933581	+	0.995633i	$0.529760\pi$
$752$	0	0
$753$	−28.4674	−1.03741
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.35053	−0.158123	−0.0790614	−	0.996870i	$-0.525192\pi$
$757$	−4.35053	−0.158123	−0.0790614	+	0.996870i	$0.525192\pi$
$758$	0	0
$759$	14.2337	0.516650
$760$	0	0
$761$	51.9565	1.88342	0.941711	−	0.336423i	$-0.109217\pi$
$761$	51.9565	1.88342	0.941711	+	0.336423i	$0.109217\pi$
$762$	0	0
$763$	−42.9783	−1.55592
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.37228	−0.0495502
$768$	0	0
$769$	48.4674	1.74778	0.873889	−	0.486125i	$-0.161590\pi$
$769$	48.4674	1.74778	0.873889	+	0.486125i	$0.161590\pi$
$770$	0	0
$771$	−20.1386	−0.725274
$772$	0	0
$773$	26.7446	0.961935	0.480968	−	0.876738i	$-0.340285\pi$
$773$	26.7446	0.961935	0.480968	+	0.876738i	$0.340285\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−80.0000	−2.86998
$778$	0	0
$779$	−34.9783	−1.25323
$780$	0	0
$781$	4.46738	0.159855
$782$	0	0
$783$	5.07335	0.181307
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.86141	0.315875	0.157938	−	0.987449i	$-0.449516\pi$
$787$	8.86141	0.315875	0.157938	+	0.987449i	$0.449516\pi$
$788$	0	0
$789$	−69.9565	−2.49052
$790$	0	0
$791$	−44.2337	−1.57277
$792$	0	0
$793$	1.74456	0.0619512
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.7228	0.592352	0.296176	−	0.955133i	$-0.404288\pi$
$797$	16.7228	0.592352	0.296176	+	0.955133i	$0.404288\pi$
$798$	0	0
$799$	6.35053	0.224666
$800$	0	0
$801$	31.5326	1.11415
$802$	0	0
$803$	−9.25544	−0.326617
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.605969	−0.0213311
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−21.8397	−0.766894	−0.383447	−	0.923563i	$-0.625263\pi$
$811$	−21.8397	−0.766894	−0.383447	+	0.923563i	$0.625263\pi$
$812$	0	0
$813$	21.0217	0.737265
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.5109	−0.507671
$818$	0	0
$819$	8.86141	0.309643
$820$	0	0
$821$	−37.7228	−1.31654	−0.658268	−	0.752784i	$-0.728710\pi$
$821$	−37.7228	−1.31654	−0.658268	+	0.752784i	$0.728710\pi$
$822$	0	0
$823$	41.3505	1.44139	0.720694	−	0.693253i	$-0.243823\pi$
$823$	41.3505	1.44139	0.720694	+	0.693253i	$0.243823\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.60597	−0.334032	−0.167016	−	0.985954i	$-0.553413\pi$
$827$	−9.60597	−0.334032	−0.167016	+	0.985954i	$0.553413\pi$
$828$	0	0
$829$	14.2554	0.495112	0.247556	−	0.968874i	$-0.420373\pi$
$829$	14.2554	0.495112	0.247556	+	0.968874i	$0.420373\pi$
$830$	0	0
$831$	55.3940	1.92160
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.97825	−0.102943
$838$	0	0
$839$	49.2119	1.69898	0.849492	−	0.527601i	$-0.176908\pi$
$839$	49.2119	1.69898	0.849492	+	0.527601i	$0.176908\pi$
$840$	0	0
$841$	4.00000	0.137931
$842$	0	0
$843$	−6.51087	−0.224246
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−30.7446	−1.05640
$848$	0	0
$849$	12.4674	0.427879
$850$	0	0
$851$	−43.7228	−1.49880
$852$	0	0
$853$	36.4674	1.24862	0.624310	−	0.781177i	$-0.285380\pi$
$853$	36.4674	1.24862	0.624310	+	0.781177i	$0.285380\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.3723	0.559266	0.279633	−	0.960107i	$-0.409787\pi$
$857$	16.3723	0.559266	0.279633	+	0.960107i	$0.409787\pi$
$858$	0	0
$859$	51.2119	1.74733	0.873664	−	0.486529i	$-0.161737\pi$
$859$	51.2119	1.74733	0.873664	+	0.486529i	$0.161737\pi$
$860$	0	0
$861$	69.9565	2.38411
$862$	0	0
$863$	−19.3723	−0.659440	−0.329720	−	0.944079i	$-0.606954\pi$
$863$	−19.3723	−0.659440	−0.329720	+	0.944079i	$0.606954\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−35.8614	−1.21792
$868$	0	0
$869$	16.9783	0.575948
$870$	0	0
$871$	−8.11684	−0.275029
$872$	0	0
$873$	−9.16844	−0.310305
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.4674	−1.09635	−0.548173	−	0.836365i	$-0.684676\pi$
$877$	−32.4674	−1.09635	−0.548173	+	0.836365i	$0.684676\pi$
$878$	0	0
$879$	42.7011	1.44027
$880$	0	0
$881$	−25.9783	−0.875230	−0.437615	−	0.899163i	$-0.644177\pi$
$881$	−25.9783	−0.875230	−0.437615	+	0.899163i	$0.644177\pi$
$882$	0	0
$883$	11.8614	0.399168	0.199584	−	0.979881i	$-0.436041\pi$
$883$	11.8614	0.399168	0.199584	+	0.979881i	$0.436041\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.7228	−0.863688	−0.431844	−	0.901948i	$-0.642137\pi$
$887$	−25.7228	−0.863688	−0.431844	+	0.901948i	$0.642137\pi$
$888$	0	0
$889$	−57.7228	−1.93596
$890$	0	0
$891$	−13.6930	−0.458732
$892$	0	0
$893$	−18.5109	−0.619443
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.3723	0.346320
$898$	0	0
$899$	−19.3723	−0.646102
$900$	0	0
$901$	−16.1168	−0.536930
$902$	0	0
$903$	29.0217	0.965783
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41.3505	1.37302	0.686511	−	0.727119i	$-0.259141\pi$
$907$	41.3505	1.37302	0.686511	+	0.727119i	$0.259141\pi$
$908$	0	0
$909$	45.2853	1.50202
$910$	0	0
$911$	4.37228	0.144860	0.0724301	−	0.997373i	$-0.476925\pi$
$911$	4.37228	0.144860	0.0724301	+	0.997373i	$0.476925\pi$
$912$	0	0
$913$	18.3505	0.607314
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−64.4674	−2.12890
$918$	0	0
$919$	11.7663	0.388135	0.194067	−	0.980988i	$-0.437832\pi$
$919$	11.7663	0.388135	0.194067	+	0.980988i	$0.437832\pi$
$920$	0	0
$921$	−58.7011	−1.93427
$922$	0	0
$923$	3.25544	0.107154
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6.23369	−0.204741
$928$	0	0
$929$	−32.2337	−1.05755	−0.528776	−	0.848761i	$-0.677349\pi$
$929$	−32.2337	−1.05755	−0.528776	+	0.848761i	$0.677349\pi$
$930$	0	0
$931$	−17.4891	−0.573183
$932$	0	0
$933$	−59.5842	−1.95070
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	0	0
$939$	−16.6060	−0.541915
$940$	0	0
$941$	−46.4674	−1.51479	−0.757397	−	0.652955i	$-0.773529\pi$
$941$	−46.4674	−1.51479	−0.757397	+	0.652955i	$0.773529\pi$
$942$	0	0
$943$	38.2337	1.24506
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.8397	1.35961	0.679803	−	0.733395i	$-0.262065\pi$
$947$	41.8397	1.35961	0.679803	+	0.733395i	$0.262065\pi$
$948$	0	0
$949$	−6.74456	−0.218938
$950$	0	0
$951$	−27.2554	−0.883818
$952$	0	0
$953$	28.1168	0.910794	0.455397	−	0.890288i	$-0.349497\pi$
$953$	28.1168	0.910794	0.455397	+	0.890288i	$0.349497\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−18.7011	−0.604520
$958$	0	0
$959$	29.4891	0.952254
$960$	0	0
$961$	−19.6277	−0.633152
$962$	0	0
$963$	−20.0435	−0.645893
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40.3505	−1.29759	−0.648793	−	0.760965i	$-0.724726\pi$
$967$	−40.3505	−1.29759	−0.648793	+	0.760965i	$0.724726\pi$
$968$	0	0
$969$	−13.0217	−0.418319
$970$	0	0
$971$	8.23369	0.264232	0.132116	−	0.991234i	$-0.457823\pi$
$971$	8.23369	0.264232	0.132116	+	0.991234i	$0.457823\pi$
$972$	0	0
$973$	−39.2119	−1.25708
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.2119	1.19052	0.595258	−	0.803535i	$-0.297050\pi$
$977$	37.2119	1.19052	0.595258	+	0.803535i	$0.297050\pi$
$978$	0	0
$979$	16.4674	0.526300
$980$	0	0
$981$	−33.4891	−1.06923
$982$	0	0
$983$	−37.3723	−1.19199	−0.595995	−	0.802988i	$-0.703242\pi$
$983$	−37.3723	−1.19199	−0.595995	+	0.802988i	$0.703242\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	37.0217	1.17842
$988$	0	0
$989$	15.8614	0.504363
$990$	0	0
$991$	42.8832	1.36223	0.681114	−	0.732177i	$-0.261496\pi$
$991$	42.8832	1.36223	0.681114	+	0.732177i	$0.261496\pi$
$992$	0	0
$993$	3.53262	0.112104
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−33.2337	−1.05252	−0.526261	−	0.850323i	$-0.676406\pi$
$997$	−33.2337	−1.05252	−0.526261	+	0.850323i	$0.676406\pi$
$998$	0	0
$999$	8.83156	0.279418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.a.g.1.2	✓	2
4.3	odd	2		5200.2.a.bx.1.1		2
5.2	odd	4		1300.2.c.e.1249.2		4
5.3	odd	4		1300.2.c.e.1249.3		4
5.4	even	2		1300.2.a.h.1.1	yes	2
20.19	odd	2		5200.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1300.2.a.g.1.2	✓	2	1.1	even	1	trivial
1300.2.a.h.1.1	yes	2	5.4	even	2
1300.2.c.e.1249.2		4	5.2	odd	4
1300.2.c.e.1249.3		4	5.3	odd	4
5200.2.a.bp.1.2		2	20.19	odd	2
5200.2.a.bx.1.1		2	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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37228 q^{3} +3.37228 q^{7} +2.62772 q^{9} +O(q^{10})\)	\(q+2.37228 q^{3} +3.37228 q^{7} +2.62772 q^{9} +1.37228 q^{11} +1.00000 q^{13} +1.37228 q^{17} -4.00000 q^{19} +8.00000 q^{21} +4.37228 q^{23} -0.883156 q^{27} -5.74456 q^{29} +3.37228 q^{31} +3.25544 q^{33} -10.0000 q^{37} +2.37228 q^{39} +8.74456 q^{41} +3.62772 q^{43} +4.62772 q^{47} +4.37228 q^{49} +3.25544 q^{51} -11.7446 q^{53} -9.48913 q^{57} -1.37228 q^{59} +1.74456 q^{61} +8.86141 q^{63} -8.11684 q^{67} +10.3723 q^{69} +3.25544 q^{71} -6.74456 q^{73} +4.62772 q^{77} +12.3723 q^{79} -9.97825 q^{81} +13.3723 q^{83} -13.6277 q^{87} +12.0000 q^{89} +3.37228 q^{91} +8.00000 q^{93} -3.48913 q^{97} +3.60597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{7} + 11 q^{9}+O(q^{10}) \) 2 * q - q^3 + q^7 + 11 * q^9	\( 2 q - q^{3} + q^{7} + 11 q^{9} - 3 q^{11} + 2 q^{13} - 3 q^{17} - 8 q^{19} + 16 q^{21} + 3 q^{23} - 19 q^{27} + q^{31} + 18 q^{33} - 20 q^{37} - q^{39} + 6 q^{41} + 13 q^{43} + 15 q^{47} + 3 q^{49} + 18 q^{51} - 12 q^{53} + 4 q^{57} + 3 q^{59} - 8 q^{61} - 11 q^{63} + q^{67} + 15 q^{69} + 18 q^{71} - 2 q^{73} + 15 q^{77} + 19 q^{79} + 26 q^{81} + 21 q^{83} - 33 q^{87} + 24 q^{89} + q^{91} + 16 q^{93} + 16 q^{97} - 33 q^{99}+O(q^{100}) \) 2 * q - q^3 + q^7 + 11 * q^9 - 3 * q^11 + 2 * q^13 - 3 * q^17 - 8 * q^19 + 16 * q^21 + 3 * q^23 - 19 * q^27 + q^31 + 18 * q^33 - 20 * q^37 - q^39 + 6 * q^41 + 13 * q^43 + 15 * q^47 + 3 * q^49 + 18 * q^51 - 12 * q^53 + 4 * q^57 + 3 * q^59 - 8 * q^61 - 11 * q^63 + q^67 + 15 * q^69 + 18 * q^71 - 2 * q^73 + 15 * q^77 + 19 * q^79 + 26 * q^81 + 21 * q^83 - 33 * q^87 + 24 * q^89 + q^91 + 16 * q^93 + 16 * q^97 - 33 * q^99

Introduction

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