LMFDB - Embedded newform 1300.2.a.h.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			$3.37228$ 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)

$f(q)$	$=$	$q+3.37228 q^{3} +2.37228 q^{7} +8.37228 q^{9} +O(q^{10})$	$q+3.37228 q^{3} +2.37228 q^{7} +8.37228 q^{9} -4.37228 q^{11} -1.00000 q^{13} +4.37228 q^{17} -4.00000 q^{19} +8.00000 q^{21} +1.37228 q^{23} +18.1168 q^{27} +5.74456 q^{29} -2.37228 q^{31} -14.7446 q^{33} +10.0000 q^{37} -3.37228 q^{39} -2.74456 q^{41} -9.37228 q^{43} -10.3723 q^{47} -1.37228 q^{49} +14.7446 q^{51} +0.255437 q^{53} -13.4891 q^{57} +4.37228 q^{59} -9.74456 q^{61} +19.8614 q^{63} -9.11684 q^{67} +4.62772 q^{69} +14.7446 q^{71} -4.74456 q^{73} -10.3723 q^{77} +6.62772 q^{79} +35.9783 q^{81} -7.62772 q^{83} +19.3723 q^{87} +12.0000 q^{89} -2.37228 q^{91} -8.00000 q^{93} -19.4891 q^{97} -36.6060 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.37228	1.94699	0.973494	−	0.228714i	$-0.0734519\pi$
$3$	3.37228	1.94699	0.973494	+	0.228714i	$0.0734519\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	8.37228	2.79076
$10$	0	0
$11$	−4.37228	−1.31829	−0.659146	−	0.752015i	$-0.729082\pi$
$11$	−4.37228	−1.31829	−0.659146	+	0.752015i	$0.729082\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.37228	1.06043	0.530217	−	0.847862i	$-0.322110\pi$
$17$	4.37228	1.06043	0.530217	+	0.847862i	$0.322110\pi$
$18$	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
$23$	1.37228	0.286140	0.143070	−	0.989713i	$-0.454303\pi$
$23$	1.37228	0.286140	0.143070	+	0.989713i	$0.454303\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	18.1168	3.48659
$28$	0	0
$29$	5.74456	1.06674	0.533369	−	0.845883i	$-0.320926\pi$
$29$	5.74456	1.06674	0.533369	+	0.845883i	$0.320926\pi$
$30$	0	0
$31$	−2.37228	−0.426074	−0.213037	−	0.977044i	$-0.568336\pi$
$31$	−2.37228	−0.426074	−0.213037	+	0.977044i	$0.568336\pi$
$32$	0	0
$33$	−14.7446	−2.56670
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	−3.37228	−0.539997
$40$	0	0
$41$	−2.74456	−0.428629	−0.214314	−	0.976765i	$-0.568752\pi$
$41$	−2.74456	−0.428629	−0.214314	+	0.976765i	$0.568752\pi$
$42$	0	0
$43$	−9.37228	−1.42926	−0.714630	−	0.699503i	$-0.753405\pi$
$43$	−9.37228	−1.42926	−0.714630	+	0.699503i	$0.753405\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.3723	−1.51295	−0.756476	−	0.654021i	$-0.773081\pi$
$47$	−10.3723	−1.51295	−0.756476	+	0.654021i	$0.773081\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	14.7446	2.06465
$52$	0	0
$53$	0.255437	0.0350870	0.0175435	−	0.999846i	$-0.494415\pi$
$53$	0.255437	0.0350870	0.0175435	+	0.999846i	$0.494415\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.4891	−1.78668
$58$	0	0
$59$	4.37228	0.569223	0.284611	−	0.958643i	$-0.408135\pi$
$59$	4.37228	0.569223	0.284611	+	0.958643i	$0.408135\pi$
$60$	0	0
$61$	−9.74456	−1.24766	−0.623832	−	0.781559i	$-0.714425\pi$
$61$	−9.74456	−1.24766	−0.623832	+	0.781559i	$0.714425\pi$
$62$	0	0
$63$	19.8614	2.50230
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.11684	−1.11380	−0.556900	−	0.830580i	$-0.688009\pi$
$67$	−9.11684	−1.11380	−0.556900	+	0.830580i	$0.688009\pi$
$68$	0	0
$69$	4.62772	0.557112
$70$	0	0
$71$	14.7446	1.74986	0.874929	−	0.484252i	$-0.160908\pi$
$71$	14.7446	1.74986	0.874929	+	0.484252i	$0.160908\pi$
$72$	0	0
$73$	−4.74456	−0.555309	−0.277655	−	0.960681i	$-0.589557\pi$
$73$	−4.74456	−0.555309	−0.277655	+	0.960681i	$0.589557\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.3723	−1.18203
$78$	0	0
$79$	6.62772	0.745677	0.372838	−	0.927896i	$-0.378385\pi$
$79$	6.62772	0.745677	0.372838	+	0.927896i	$0.378385\pi$
$80$	0	0
$81$	35.9783	3.99758
$82$	0	0
$83$	−7.62772	−0.837251	−0.418625	−	0.908159i	$-0.637488\pi$
$83$	−7.62772	−0.837251	−0.418625	+	0.908159i	$0.637488\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.3723	2.07693
$88$	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
$91$	−2.37228	−0.248683
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−19.4891	−1.97882	−0.989410	−	0.145145i	$-0.953635\pi$
$97$	−19.4891	−1.97882	−0.989410	+	0.145145i	$0.953635\pi$
$98$	0	0
$99$	−36.6060	−3.67904
$100$	0	0
$101$	−17.2337	−1.71482	−0.857408	−	0.514637i	$-0.827927\pi$
$101$	−17.2337	−1.71482	−0.857408	+	0.514637i	$0.827927\pi$
$102$	0	0
$103$	−3.37228	−0.332281	−0.166140	−	0.986102i	$-0.553130\pi$
$103$	−3.37228	−0.332281	−0.166140	+	0.986102i	$0.553130\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3723	1.29275	0.646374	−	0.763021i	$-0.276285\pi$
$107$	13.3723	1.29275	0.646374	+	0.763021i	$0.276285\pi$
$108$	0	0
$109$	−1.25544	−0.120249	−0.0601245	−	0.998191i	$-0.519150\pi$
$109$	−1.25544	−0.120249	−0.0601245	+	0.998191i	$0.519150\pi$
$110$	0	0
$111$	33.7228	3.20083
$112$	0	0
$113$	−4.11684	−0.387280	−0.193640	−	0.981073i	$-0.562029\pi$
$113$	−4.11684	−0.387280	−0.193640	+	0.981073i	$0.562029\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.37228	−0.774018
$118$	0	0
$119$	10.3723	0.950825
$120$	0	0
$121$	8.11684	0.737895
$122$	0	0
$123$	−9.25544	−0.834535
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.116844	−0.0103682	−0.00518411	−	0.999987i	$-0.501650\pi$
$127$	−0.116844	−0.0103682	−0.00518411	+	0.999987i	$0.501650\pi$
$128$	0	0
$129$	−31.6060	−2.78275
$130$	0	0
$131$	−1.88316	−0.164532	−0.0822661	−	0.996610i	$-0.526216\pi$
$131$	−1.88316	−0.164532	−0.0822661	+	0.996610i	$0.526216\pi$
$132$	0	0
$133$	−9.48913	−0.822812
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.74456	0.234484	0.117242	−	0.993103i	$-0.462595\pi$
$137$	2.74456	0.234484	0.117242	+	0.993103i	$0.462595\pi$
$138$	0	0
$139$	−17.3723	−1.47350	−0.736749	−	0.676167i	$-0.763640\pi$
$139$	−17.3723	−1.47350	−0.736749	+	0.676167i	$0.763640\pi$
$140$	0	0
$141$	−34.9783	−2.94570
$142$	0	0
$143$	4.37228	0.365629
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.62772	−0.381688
$148$	0	0
$149$	23.4891	1.92430	0.962152	−	0.272513i	$-0.0878548\pi$
$149$	23.4891	1.92430	0.962152	+	0.272513i	$0.0878548\pi$
$150$	0	0
$151$	−5.11684	−0.416403	−0.208201	−	0.978086i	$-0.566761\pi$
$151$	−5.11684	−0.416403	−0.208201	+	0.978086i	$0.566761\pi$
$152$	0	0
$153$	36.6060	2.95942
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.8614	1.10626	0.553130	−	0.833095i	$-0.313433\pi$
$157$	13.8614	1.10626	0.553130	+	0.833095i	$0.313433\pi$
$158$	0	0
$159$	0.861407	0.0683140
$160$	0	0
$161$	3.25544	0.256564
$162$	0	0
$163$	−16.2337	−1.27152	−0.635760	−	0.771887i	$-0.719313\pi$
$163$	−16.2337	−1.27152	−0.635760	+	0.771887i	$0.719313\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.74456	−0.212381	−0.106190	−	0.994346i	$-0.533865\pi$
$167$	−2.74456	−0.212381	−0.106190	+	0.994346i	$0.533865\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−33.4891	−2.56098
$172$	0	0
$173$	−1.11684	−0.0849121	−0.0424560	−	0.999098i	$-0.513518\pi$
$173$	−1.11684	−0.0849121	−0.0424560	+	0.999098i	$0.513518\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.7446	1.10827
$178$	0	0
$179$	−10.1168	−0.756168	−0.378084	−	0.925771i	$-0.623417\pi$
$179$	−10.1168	−0.756168	−0.378084	+	0.925771i	$0.623417\pi$
$180$	0	0
$181$	−5.11684	−0.380332	−0.190166	−	0.981752i	$-0.560903\pi$
$181$	−5.11684	−0.380332	−0.190166	+	0.981752i	$0.560903\pi$
$182$	0	0
$183$	−32.8614	−2.42919
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−19.1168	−1.39796
$188$	0	0
$189$	42.9783	3.12621
$190$	0	0
$191$	7.88316	0.570405	0.285203	−	0.958467i	$-0.407939\pi$
$191$	7.88316	0.570405	0.285203	+	0.958467i	$0.407939\pi$
$192$	0	0
$193$	9.48913	0.683042	0.341521	−	0.939874i	$-0.389058\pi$
$193$	9.48913	0.683042	0.341521	+	0.939874i	$0.389058\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.25544	−0.659423	−0.329711	−	0.944082i	$-0.606951\pi$
$197$	−9.25544	−0.659423	−0.329711	+	0.944082i	$0.606951\pi$
$198$	0	0
$199$	12.1168	0.858940	0.429470	−	0.903081i	$-0.358700\pi$
$199$	12.1168	0.858940	0.429470	+	0.903081i	$0.358700\pi$
$200$	0	0
$201$	−30.7446	−2.16855
$202$	0	0
$203$	13.6277	0.956478
$204$	0	0
$205$	0	0
$206$	0	0
$207$	11.4891	0.798549
$208$	0	0
$209$	17.4891	1.20975
$210$	0	0
$211$	10.2337	0.704516	0.352258	−	0.935903i	$-0.385414\pi$
$211$	10.2337	0.704516	0.352258	+	0.935903i	$0.385414\pi$
$212$	0	0
$213$	49.7228	3.40695
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.62772	−0.382034
$218$	0	0
$219$	−16.0000	−1.08118
$220$	0	0
$221$	−4.37228	−0.294111
$222$	0	0
$223$	12.2337	0.819228	0.409614	−	0.912259i	$-0.365663\pi$
$223$	12.2337	0.819228	0.409614	+	0.912259i	$0.365663\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.62772	−0.108035	−0.0540177	−	0.998540i	$-0.517203\pi$
$227$	−1.62772	−0.108035	−0.0540177	+	0.998540i	$0.517203\pi$
$228$	0	0
$229$	22.2337	1.46924	0.734622	−	0.678477i	$-0.237360\pi$
$229$	22.2337	1.46924	0.734622	+	0.678477i	$0.237360\pi$
$230$	0	0
$231$	−34.9783	−2.30140
$232$	0	0
$233$	−4.11684	−0.269703	−0.134852	−	0.990866i	$-0.543056\pi$
$233$	−4.11684	−0.269703	−0.134852	+	0.990866i	$0.543056\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	22.3505	1.45182
$238$	0	0
$239$	1.62772	0.105288	0.0526442	−	0.998613i	$-0.483235\pi$
$239$	1.62772	0.105288	0.0526442	+	0.998613i	$0.483235\pi$
$240$	0	0
$241$	4.23369	0.272716	0.136358	−	0.990660i	$-0.456460\pi$
$241$	4.23369	0.272716	0.136358	+	0.990660i	$0.456460\pi$
$242$	0	0
$243$	66.9783	4.29666
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−25.7228	−1.63012
$250$	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
$257$	−14.4891	−0.903807	−0.451903	−	0.892067i	$-0.649255\pi$
$257$	−14.4891	−0.903807	−0.451903	+	0.892067i	$0.649255\pi$
$258$	0	0
$259$	23.7228	1.47406
$260$	0	0
$261$	48.0951	2.97701
$262$	0	0
$263$	6.51087	0.401478	0.200739	−	0.979645i	$-0.435666\pi$
$263$	6.51087	0.401478	0.200739	+	0.979645i	$0.435666\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	40.4674	2.47656
$268$	0	0
$269$	−11.7446	−0.716079	−0.358039	−	0.933707i	$-0.616555\pi$
$269$	−11.7446	−0.716079	−0.358039	+	0.933707i	$0.616555\pi$
$270$	0	0
$271$	−19.8614	−1.20649	−0.603247	−	0.797554i	$-0.706127\pi$
$271$	−19.8614	−1.20649	−0.603247	+	0.797554i	$0.706127\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.3505	1.70342	0.851709	−	0.524015i	$-0.175566\pi$
$277$	28.3505	1.70342	0.851709	+	0.524015i	$0.175566\pi$
$278$	0	0
$279$	−19.8614	−1.18907
$280$	0	0
$281$	8.74456	0.521657	0.260828	−	0.965385i	$-0.416004\pi$
$281$	8.74456	0.521657	0.260828	+	0.965385i	$0.416004\pi$
$282$	0	0
$283$	−16.7446	−0.995361	−0.497680	−	0.867360i	$-0.665815\pi$
$283$	−16.7446	−0.995361	−0.497680	+	0.867360i	$0.665815\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.51087	−0.384325
$288$	0	0
$289$	2.11684	0.124520
$290$	0	0
$291$	−65.7228	−3.85274
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−79.2119	−4.59634
$298$	0	0
$299$	−1.37228	−0.0793611
$300$	0	0
$301$	−22.2337	−1.28153
$302$	0	0
$303$	−58.1168	−3.33873
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.2554	0.756528	0.378264	−	0.925698i	$-0.376521\pi$
$307$	13.2554	0.756528	0.378264	+	0.925698i	$0.376521\pi$
$308$	0	0
$309$	−11.3723	−0.646946
$310$	0	0
$311$	−7.88316	−0.447013	−0.223506	−	0.974702i	$-0.571750\pi$
$311$	−7.88316	−0.447013	−0.223506	+	0.974702i	$0.571750\pi$
$312$	0	0
$313$	7.00000	0.395663	0.197832	−	0.980236i	$-0.436610\pi$
$313$	7.00000	0.395663	0.197832	+	0.980236i	$0.436610\pi$
$314$	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
$319$	−25.1168	−1.40627
$320$	0	0
$321$	45.0951	2.51696
$322$	0	0
$323$	−17.4891	−0.973121
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.23369	−0.234123
$328$	0	0
$329$	−24.6060	−1.35657
$330$	0	0
$331$	−21.4891	−1.18115	−0.590575	−	0.806983i	$-0.701099\pi$
$331$	−21.4891	−1.18115	−0.590575	+	0.806983i	$0.701099\pi$
$332$	0	0
$333$	83.7228	4.58798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	25.0000	1.36184	0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000	1.36184	0.680918	+	0.732359i	$0.261581\pi$
$338$	0	0
$339$	−13.8832	−0.754030
$340$	0	0
$341$	10.3723	0.561691
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.8614	−1.65673	−0.828364	−	0.560191i	$-0.810728\pi$
$347$	−30.8614	−1.65673	−0.828364	+	0.560191i	$0.810728\pi$
$348$	0	0
$349$	−35.7228	−1.91220	−0.956099	−	0.293043i	$-0.905332\pi$
$349$	−35.7228	−1.91220	−0.956099	+	0.293043i	$0.905332\pi$
$350$	0	0
$351$	−18.1168	−0.967006
$352$	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
$357$	34.9783	1.85125
$358$	0	0
$359$	15.8614	0.837133	0.418567	−	0.908186i	$-0.362533\pi$
$359$	15.8614	0.837133	0.418567	+	0.908186i	$0.362533\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	27.3723	1.43667
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.11684	−0.319297	−0.159648	−	0.987174i	$-0.551036\pi$
$367$	−6.11684	−0.319297	−0.159648	+	0.987174i	$0.551036\pi$
$368$	0	0
$369$	−22.9783	−1.19620
$370$	0	0
$371$	0.605969	0.0314604
$372$	0	0
$373$	−13.7446	−0.711666	−0.355833	−	0.934549i	$-0.615803\pi$
$373$	−13.7446	−0.711666	−0.355833	+	0.934549i	$0.615803\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.74456	−0.295860
$378$	0	0
$379$	−2.88316	−0.148098	−0.0740489	−	0.997255i	$-0.523592\pi$
$379$	−2.88316	−0.148098	−0.0740489	+	0.997255i	$0.523592\pi$
$380$	0	0
$381$	−0.394031	−0.0201868
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−78.4674	−3.98872
$388$	0	0
$389$	37.3723	1.89485	0.947425	−	0.319978i	$-0.103676\pi$
$389$	37.3723	1.89485	0.947425	+	0.319978i	$0.103676\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	−6.35053	−0.320342
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.2554	−0.564894	−0.282447	−	0.959283i	$-0.591146\pi$
$397$	−11.2554	−0.564894	−0.282447	+	0.959283i	$0.591146\pi$
$398$	0	0
$399$	−32.0000	−1.60200
$400$	0	0
$401$	−2.23369	−0.111545	−0.0557725	−	0.998444i	$-0.517762\pi$
$401$	−2.23369	−0.111545	−0.0557725	+	0.998444i	$0.517762\pi$
$402$	0	0
$403$	2.37228	0.118172
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−43.7228	−2.16726
$408$	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
$411$	9.25544	0.456537
$412$	0	0
$413$	10.3723	0.510387
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−58.5842	−2.86888
$418$	0	0
$419$	31.3723	1.53264	0.766318	−	0.642461i	$-0.222087\pi$
$419$	31.3723	1.53264	0.766318	+	0.642461i	$0.222087\pi$
$420$	0	0
$421$	25.4891	1.24226	0.621132	−	0.783706i	$-0.286673\pi$
$421$	25.4891	1.24226	0.621132	+	0.783706i	$0.286673\pi$
$422$	0	0
$423$	−86.8397	−4.22229
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−23.1168	−1.11870
$428$	0	0
$429$	14.7446	0.711874
$430$	0	0
$431$	34.9783	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$431$	34.9783	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$432$	0	0
$433$	−6.62772	−0.318508	−0.159254	−	0.987238i	$-0.550909\pi$
$433$	−6.62772	−0.318508	−0.159254	+	0.987238i	$0.550909\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.48913	−0.262580
$438$	0	0
$439$	−12.7446	−0.608265	−0.304132	−	0.952630i	$-0.598367\pi$
$439$	−12.7446	−0.608265	−0.304132	+	0.952630i	$0.598367\pi$
$440$	0	0
$441$	−11.4891	−0.547101
$442$	0	0
$443$	−19.7228	−0.937059	−0.468530	−	0.883448i	$-0.655216\pi$
$443$	−19.7228	−0.937059	−0.468530	+	0.883448i	$0.655216\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	79.2119	3.74660
$448$	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
$453$	−17.2554	−0.810731
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	79.2119	3.69730
$460$	0	0
$461$	16.9783	0.790756	0.395378	−	0.918519i	$-0.370614\pi$
$461$	16.9783	0.790756	0.395378	+	0.918519i	$0.370614\pi$
$462$	0	0
$463$	20.3723	0.946780	0.473390	−	0.880853i	$-0.343030\pi$
$463$	20.3723	0.946780	0.473390	+	0.880853i	$0.343030\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.88316	0.0871421	0.0435710	−	0.999050i	$-0.486127\pi$
$467$	1.88316	0.0871421	0.0435710	+	0.999050i	$0.486127\pi$
$468$	0	0
$469$	−21.6277	−0.998675
$470$	0	0
$471$	46.7446	2.15388
$472$	0	0
$473$	40.9783	1.88418
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.13859	0.0979195
$478$	0	0
$479$	−18.6060	−0.850128	−0.425064	−	0.905163i	$-0.639748\pi$
$479$	−18.6060	−0.850128	−0.425064	+	0.905163i	$0.639748\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	10.9783	0.499528
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.11684	−0.141238	−0.0706188	−	0.997503i	$-0.522497\pi$
$487$	−3.11684	−0.141238	−0.0706188	+	0.997503i	$0.522497\pi$
$488$	0	0
$489$	−54.7446	−2.47563
$490$	0	0
$491$	5.13859	0.231901	0.115951	−	0.993255i	$-0.463009\pi$
$491$	5.13859	0.231901	0.115951	+	0.993255i	$0.463009\pi$
$492$	0	0
$493$	25.1168	1.13121
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.9783	1.56899
$498$	0	0
$499$	−13.3505	−0.597652	−0.298826	−	0.954308i	$-0.596595\pi$
$499$	−13.3505	−0.597652	−0.298826	+	0.954308i	$0.596595\pi$
$500$	0	0
$501$	−9.25544	−0.413502
$502$	0	0
$503$	7.37228	0.328714	0.164357	−	0.986401i	$-0.447445\pi$
$503$	7.37228	0.328714	0.164357	+	0.986401i	$0.447445\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.37228	0.149768
$508$	0	0
$509$	−2.74456	−0.121651	−0.0608253	−	0.998148i	$-0.519373\pi$
$509$	−2.74456	−0.121651	−0.0608253	+	0.998148i	$0.519373\pi$
$510$	0	0
$511$	−11.2554	−0.497911
$512$	0	0
$513$	−72.4674	−3.19951
$514$	0	0
$515$	0	0
$516$	0	0
$517$	45.3505	1.99451
$518$	0	0
$519$	−3.76631	−0.165323
$520$	0	0
$521$	39.0951	1.71279	0.856394	−	0.516324i	$-0.172700\pi$
$521$	39.0951	1.71279	0.856394	+	0.516324i	$0.172700\pi$
$522$	0	0
$523$	−22.2337	−0.972211	−0.486106	−	0.873900i	$-0.661583\pi$
$523$	−22.2337	−0.972211	−0.486106	+	0.873900i	$0.661583\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.3723	−0.451824
$528$	0	0
$529$	−21.1168	−0.918124
$530$	0	0
$531$	36.6060	1.58856
$532$	0	0
$533$	2.74456	0.118880
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−34.1168	−1.47225
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−20.9783	−0.901925	−0.450963	−	0.892543i	$-0.648919\pi$
$541$	−20.9783	−0.901925	−0.450963	+	0.892543i	$0.648919\pi$
$542$	0	0
$543$	−17.2554	−0.740502
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.4891	0.918809	0.459404	−	0.888227i	$-0.348063\pi$
$547$	21.4891	0.918809	0.459404	+	0.888227i	$0.348063\pi$
$548$	0	0
$549$	−81.5842	−3.48193
$550$	0	0
$551$	−22.9783	−0.978906
$552$	0	0
$553$	15.7228	0.668602
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.76631	−0.159584	−0.0797919	−	0.996812i	$-0.525426\pi$
$557$	−3.76631	−0.159584	−0.0797919	+	0.996812i	$0.525426\pi$
$558$	0	0
$559$	9.37228	0.396405
$560$	0	0
$561$	−64.4674	−2.72181
$562$	0	0
$563$	41.8397	1.76333	0.881666	−	0.471875i	$-0.156423\pi$
$563$	41.8397	1.76333	0.881666	+	0.471875i	$0.156423\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	85.3505	3.58439
$568$	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
$571$	10.2337	0.428267	0.214133	−	0.976804i	$-0.431307\pi$
$571$	10.2337	0.428267	0.214133	+	0.976804i	$0.431307\pi$
$572$	0	0
$573$	26.5842	1.11057
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	32.0000	1.32987
$580$	0	0
$581$	−18.0951	−0.750711
$582$	0	0
$583$	−1.11684	−0.0462550
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.3723	−1.17105	−0.585525	−	0.810655i	$-0.699111\pi$
$587$	−28.3723	−1.17105	−0.585525	+	0.810655i	$0.699111\pi$
$588$	0	0
$589$	9.48913	0.390993
$590$	0	0
$591$	−31.2119	−1.28389
$592$	0	0
$593$	−8.74456	−0.359096	−0.179548	−	0.983749i	$-0.557464\pi$
$593$	−8.74456	−0.359096	−0.179548	+	0.983749i	$0.557464\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.8614	1.67235
$598$	0	0
$599$	−18.3505	−0.749782	−0.374891	−	0.927069i	$-0.622320\pi$
$599$	−18.3505	−0.749782	−0.374891	+	0.927069i	$0.622320\pi$
$600$	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
$603$	−76.3288	−3.10835
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.2337	1.22715	0.613574	−	0.789637i	$-0.289731\pi$
$607$	30.2337	1.22715	0.613574	+	0.789637i	$0.289731\pi$
$608$	0	0
$609$	45.9565	1.86225
$610$	0	0
$611$	10.3723	0.419618
$612$	0	0
$613$	−28.7446	−1.16098	−0.580491	−	0.814267i	$-0.697139\pi$
$613$	−28.7446	−1.16098	−0.580491	+	0.814267i	$0.697139\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28.9783	1.16662	0.583310	−	0.812249i	$-0.301757\pi$
$617$	28.9783	1.16662	0.583310	+	0.812249i	$0.301757\pi$
$618$	0	0
$619$	−13.2554	−0.532781	−0.266391	−	0.963865i	$-0.585831\pi$
$619$	−13.2554	−0.532781	−0.266391	+	0.963865i	$0.585831\pi$
$620$	0	0
$621$	24.8614	0.997654
$622$	0	0
$623$	28.4674	1.14052
$624$	0	0
$625$	0	0
$626$	0	0
$627$	58.9783	2.35536
$628$	0	0
$629$	43.7228	1.74334
$630$	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
$633$	34.5109	1.37168
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.37228	0.0543718
$638$	0	0
$639$	123.446	4.88343
$640$	0	0
$641$	21.5109	0.849628	0.424814	−	0.905281i	$-0.360339\pi$
$641$	21.5109	0.849628	0.424814	+	0.905281i	$0.360339\pi$
$642$	0	0
$643$	20.4674	0.807155	0.403577	−	0.914946i	$-0.367767\pi$
$643$	20.4674	0.807155	0.403577	+	0.914946i	$0.367767\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.48913	0.215800	0.107900	−	0.994162i	$-0.465587\pi$
$647$	5.48913	0.215800	0.107900	+	0.994162i	$0.465587\pi$
$648$	0	0
$649$	−19.1168	−0.750402
$650$	0	0
$651$	−18.9783	−0.743816
$652$	0	0
$653$	28.3723	1.11029	0.555147	−	0.831753i	$-0.312662\pi$
$653$	28.3723	1.11029	0.555147	+	0.831753i	$0.312662\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−39.7228	−1.54973
$658$	0	0
$659$	−16.1168	−0.627823	−0.313912	−	0.949452i	$-0.601640\pi$
$659$	−16.1168	−0.627823	−0.313912	+	0.949452i	$0.601640\pi$
$660$	0	0
$661$	−6.74456	−0.262333	−0.131167	−	0.991360i	$-0.541872\pi$
$661$	−6.74456	−0.262333	−0.131167	+	0.991360i	$0.541872\pi$
$662$	0	0
$663$	−14.7446	−0.572631
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.88316	0.305237
$668$	0	0
$669$	41.2554	1.59503
$670$	0	0
$671$	42.6060	1.64479
$672$	0	0
$673$	20.3723	0.785294	0.392647	−	0.919689i	$-0.371559\pi$
$673$	20.3723	0.785294	0.392647	+	0.919689i	$0.371559\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.1168	1.08062	0.540309	−	0.841467i	$-0.318307\pi$
$677$	28.1168	1.08062	0.540309	+	0.841467i	$0.318307\pi$
$678$	0	0
$679$	−46.2337	−1.77429
$680$	0	0
$681$	−5.48913	−0.210344
$682$	0	0
$683$	36.6060	1.40069	0.700344	−	0.713805i	$-0.253030\pi$
$683$	36.6060	1.40069	0.700344	+	0.713805i	$0.253030\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	74.9783	2.86060
$688$	0	0
$689$	−0.255437	−0.00973139
$690$	0	0
$691$	35.3505	1.34480	0.672399	−	0.740189i	$-0.265264\pi$
$691$	35.3505	1.34480	0.672399	+	0.740189i	$0.265264\pi$
$692$	0	0
$693$	−86.8397	−3.29877
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	−13.8832	−0.525109
$700$	0	0
$701$	12.7663	0.482177	0.241088	−	0.970503i	$-0.422496\pi$
$701$	12.7663	0.482177	0.241088	+	0.970503i	$0.422496\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−40.8832	−1.53757
$708$	0	0
$709$	18.9783	0.712743	0.356372	−	0.934344i	$-0.384014\pi$
$709$	18.9783	0.712743	0.356372	+	0.934344i	$0.384014\pi$
$710$	0	0
$711$	55.4891	2.08100
$712$	0	0
$713$	−3.25544	−0.121917
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.48913	0.204995
$718$	0	0
$719$	43.3723	1.61751	0.808757	−	0.588144i	$-0.200141\pi$
$719$	43.3723	1.61751	0.808757	+	0.588144i	$0.200141\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	14.2772	0.530974
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35.6060	−1.32055	−0.660276	−	0.751023i	$-0.729561\pi$
$727$	−35.6060	−1.32055	−0.660276	+	0.751023i	$0.729561\pi$
$728$	0	0
$729$	117.935	4.36795
$730$	0	0
$731$	−40.9783	−1.51564
$732$	0	0
$733$	−8.00000	−0.295487	−0.147743	−	0.989026i	$-0.547201\pi$
$733$	−8.00000	−0.295487	−0.147743	+	0.989026i	$0.547201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.8614	1.46831
$738$	0	0
$739$	−13.3505	−0.491107	−0.245554	−	0.969383i	$-0.578970\pi$
$739$	−13.3505	−0.491107	−0.245554	+	0.969383i	$0.578970\pi$
$740$	0	0
$741$	13.4891	0.495535
$742$	0	0
$743$	−25.1168	−0.921448	−0.460724	−	0.887544i	$-0.652410\pi$
$743$	−25.1168	−0.921448	−0.460724	+	0.887544i	$0.652410\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−63.8614	−2.33657
$748$	0	0
$749$	31.7228	1.15913
$750$	0	0
$751$	12.1168	0.442150	0.221075	−	0.975257i	$-0.429043\pi$
$751$	12.1168	0.442150	0.221075	+	0.975257i	$0.429043\pi$
$752$	0	0
$753$	−40.4674	−1.47471
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−47.3505	−1.72098	−0.860492	−	0.509464i	$-0.829844\pi$
$757$	−47.3505	−1.72098	−0.860492	+	0.509464i	$0.829844\pi$
$758$	0	0
$759$	−20.2337	−0.734436
$760$	0	0
$761$	−39.9565	−1.44842	−0.724211	−	0.689578i	$-0.757796\pi$
$761$	−39.9565	−1.44842	−0.724211	+	0.689578i	$0.757796\pi$
$762$	0	0
$763$	−2.97825	−0.107820
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.37228	−0.157874
$768$	0	0
$769$	−20.4674	−0.738072	−0.369036	−	0.929415i	$-0.620312\pi$
$769$	−20.4674	−0.738072	−0.369036	+	0.929415i	$0.620312\pi$
$770$	0	0
$771$	−48.8614	−1.75970
$772$	0	0
$773$	−15.2554	−0.548700	−0.274350	−	0.961630i	$-0.588463\pi$
$773$	−15.2554	−0.548700	−0.274350	+	0.961630i	$0.588463\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	80.0000	2.86998
$778$	0	0
$779$	10.9783	0.393337
$780$	0	0
$781$	−64.4674	−2.30682
$782$	0	0
$783$	104.073	3.71928
$784$	0	0
$785$	0	0
$786$	0	0
$787$	19.8614	0.707983	0.353991	−	0.935249i	$-0.384824\pi$
$787$	19.8614	0.707983	0.353991	+	0.935249i	$0.384824\pi$
$788$	0	0
$789$	21.9565	0.781672
$790$	0	0
$791$	−9.76631	−0.347250
$792$	0	0
$793$	9.74456	0.346040
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.7228	1.44248	0.721238	−	0.692687i	$-0.243573\pi$
$797$	40.7228	1.44248	0.721238	+	0.692687i	$0.243573\pi$
$798$	0	0
$799$	−45.3505	−1.60439
$800$	0	0
$801$	100.467	3.54984
$802$	0	0
$803$	20.7446	0.732060
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−39.6060	−1.39420
$808$	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
$811$	52.8397	1.85545	0.927726	−	0.373263i	$-0.121761\pi$
$811$	52.8397	1.85545	0.927726	+	0.373263i	$0.121761\pi$
$812$	0	0
$813$	−66.9783	−2.34903
$814$	0	0
$815$	0	0
$816$	0	0
$817$	37.4891	1.31158
$818$	0	0
$819$	−19.8614	−0.694014
$820$	0	0
$821$	19.7228	0.688331	0.344165	−	0.938909i	$-0.388162\pi$
$821$	19.7228	0.688331	0.344165	+	0.938909i	$0.388162\pi$
$822$	0	0
$823$	10.3505	0.360797	0.180398	−	0.983594i	$-0.442261\pi$
$823$	10.3505	0.360797	0.180398	+	0.983594i	$0.442261\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.6060	−1.06427	−0.532137	−	0.846658i	$-0.678611\pi$
$827$	−30.6060	−1.06427	−0.532137	+	0.846658i	$0.678611\pi$
$828$	0	0
$829$	25.7446	0.894146	0.447073	−	0.894498i	$-0.352466\pi$
$829$	25.7446	0.894146	0.447073	+	0.894498i	$0.352466\pi$
$830$	0	0
$831$	95.6060	3.31653
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−42.9783	−1.48555
$838$	0	0
$839$	−31.2119	−1.07756	−0.538778	−	0.842448i	$-0.681114\pi$
$839$	−31.2119	−1.07756	−0.538778	+	0.842448i	$0.681114\pi$
$840$	0	0
$841$	4.00000	0.137931
$842$	0	0
$843$	29.4891	1.01566
$844$	0	0
$845$	0	0
$846$	0	0
$847$	19.2554	0.661625
$848$	0	0
$849$	−56.4674	−1.93796
$850$	0	0
$851$	13.7228	0.470412
$852$	0	0
$853$	32.4674	1.11166	0.555831	−	0.831295i	$-0.312400\pi$
$853$	32.4674	1.11166	0.555831	+	0.831295i	$0.312400\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.6277	−0.363036	−0.181518	−	0.983388i	$-0.558101\pi$
$857$	−10.6277	−0.363036	−0.181518	+	0.983388i	$0.558101\pi$
$858$	0	0
$859$	−29.2119	−0.996698	−0.498349	−	0.866976i	$-0.666060\pi$
$859$	−29.2119	−0.996698	−0.498349	+	0.866976i	$0.666060\pi$
$860$	0	0
$861$	−21.9565	−0.748276
$862$	0	0
$863$	13.6277	0.463893	0.231946	−	0.972729i	$-0.425491\pi$
$863$	13.6277	0.463893	0.231946	+	0.972729i	$0.425491\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.13859	0.242439
$868$	0	0
$869$	−28.9783	−0.983020
$870$	0	0
$871$	9.11684	0.308912
$872$	0	0
$873$	−163.168	−5.52241
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−36.4674	−1.23142	−0.615708	−	0.787974i	$-0.711130\pi$
$877$	−36.4674	−1.23142	−0.615708	+	0.787974i	$0.711130\pi$
$878$	0	0
$879$	−60.7011	−2.04740
$880$	0	0
$881$	19.9783	0.673084	0.336542	−	0.941668i	$-0.390743\pi$
$881$	19.9783	0.673084	0.336542	+	0.941668i	$0.390743\pi$
$882$	0	0
$883$	16.8614	0.567432	0.283716	−	0.958908i	$-0.408433\pi$
$883$	16.8614	0.567432	0.283716	+	0.958908i	$0.408433\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.7228	−1.06515	−0.532574	−	0.846383i	$-0.678775\pi$
$887$	−31.7228	−1.06515	−0.532574	+	0.846383i	$0.678775\pi$
$888$	0	0
$889$	−0.277187	−0.00929655
$890$	0	0
$891$	−157.307	−5.26998
$892$	0	0
$893$	41.4891	1.38838
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.62772	−0.154515
$898$	0	0
$899$	−13.6277	−0.454510
$900$	0	0
$901$	1.11684	0.0372075
$902$	0	0
$903$	−74.9783	−2.49512
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.3505	0.343684	0.171842	−	0.985125i	$-0.445028\pi$
$907$	10.3505	0.343684	0.171842	+	0.985125i	$0.445028\pi$
$908$	0	0
$909$	−144.285	−4.78564
$910$	0	0
$911$	−1.37228	−0.0454657	−0.0227329	−	0.999742i	$-0.507237\pi$
$911$	−1.37228	−0.0454657	−0.0227329	+	0.999742i	$0.507237\pi$
$912$	0	0
$913$	33.3505	1.10374
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.46738	−0.147526
$918$	0	0
$919$	46.2337	1.52511	0.762554	−	0.646924i	$-0.223945\pi$
$919$	46.2337	1.52511	0.762554	+	0.646924i	$0.223945\pi$
$920$	0	0
$921$	44.7011	1.47295
$922$	0	0
$923$	−14.7446	−0.485323
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−28.2337	−0.927316
$928$	0	0
$929$	2.23369	0.0732849	0.0366425	−	0.999328i	$-0.488334\pi$
$929$	2.23369	0.0732849	0.0366425	+	0.999328i	$0.488334\pi$
$930$	0	0
$931$	5.48913	0.179899
$932$	0	0
$933$	−26.5842	−0.870328
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.0000	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$937$	25.0000	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$938$	0	0
$939$	23.6060	0.770352
$940$	0	0
$941$	22.4674	0.732416	0.366208	−	0.930533i	$-0.380656\pi$
$941$	22.4674	0.732416	0.366208	+	0.930533i	$0.380656\pi$
$942$	0	0
$943$	−3.76631	−0.122648
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.8397	1.06715	0.533573	−	0.845754i	$-0.320849\pi$
$947$	32.8397	1.06715	0.533573	+	0.845754i	$0.320849\pi$
$948$	0	0
$949$	4.74456	0.154015
$950$	0	0
$951$	−38.7446	−1.25638
$952$	0	0
$953$	−10.8832	−0.352540	−0.176270	−	0.984342i	$-0.556403\pi$
$953$	−10.8832	−0.352540	−0.176270	+	0.984342i	$0.556403\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−84.7011	−2.73800
$958$	0	0
$959$	6.51087	0.210247
$960$	0	0
$961$	−25.3723	−0.818461
$962$	0	0
$963$	111.957	3.60775
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.3505	−0.365008	−0.182504	−	0.983205i	$-0.558420\pi$
$967$	−11.3505	−0.365008	−0.182504	+	0.983205i	$0.558420\pi$
$968$	0	0
$969$	−58.9783	−1.89465
$970$	0	0
$971$	−26.2337	−0.841879	−0.420940	−	0.907089i	$-0.638300\pi$
$971$	−26.2337	−0.841879	−0.420940	+	0.907089i	$0.638300\pi$
$972$	0	0
$973$	−41.2119	−1.32119
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.2119	1.38247	0.691236	−	0.722629i	$-0.257066\pi$
$977$	43.2119	1.38247	0.691236	+	0.722629i	$0.257066\pi$
$978$	0	0
$979$	−52.4674	−1.67686
$980$	0	0
$981$	−10.5109	−0.335586
$982$	0	0
$983$	31.6277	1.00877	0.504384	−	0.863480i	$-0.331720\pi$
$983$	31.6277	1.00877	0.504384	+	0.863480i	$0.331720\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−82.9783	−2.64123
$988$	0	0
$989$	−12.8614	−0.408969
$990$	0	0
$991$	60.1168	1.90967	0.954837	−	0.297129i	$-0.0960293\pi$
$991$	60.1168	1.90967	0.954837	+	0.297129i	$0.0960293\pi$
$992$	0	0
$993$	−72.4674	−2.29968
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.23369	−0.0390713	−0.0195356	−	0.999809i	$-0.506219\pi$
$997$	−1.23369	−0.0390713	−0.0195356	+	0.999809i	$0.506219\pi$
$998$	0	0
$999$	181.168	5.73192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.a.h.1.2	yes	2
4.3	odd	2		5200.2.a.bp.1.1		2
5.2	odd	4		1300.2.c.e.1249.1		4
5.3	odd	4		1300.2.c.e.1249.4		4
5.4	even	2		1300.2.a.g.1.1	✓	2
20.19	odd	2		5200.2.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1300.2.a.g.1.1	✓	2	5.4	even	2
1300.2.a.h.1.2	yes	2	1.1	even	1	trivial
1300.2.c.e.1249.1		4	5.2	odd	4
1300.2.c.e.1249.4		4	5.3	odd	4
5200.2.a.bp.1.1		2	4.3	odd	2
5200.2.a.bx.1.2		2	20.19	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)

\(f(q)\)	\(=\)	\(q+3.37228 q^{3} +2.37228 q^{7} +8.37228 q^{9} +O(q^{10})\)	\(q+3.37228 q^{3} +2.37228 q^{7} +8.37228 q^{9} -4.37228 q^{11} -1.00000 q^{13} +4.37228 q^{17} -4.00000 q^{19} +8.00000 q^{21} +1.37228 q^{23} +18.1168 q^{27} +5.74456 q^{29} -2.37228 q^{31} -14.7446 q^{33} +10.0000 q^{37} -3.37228 q^{39} -2.74456 q^{41} -9.37228 q^{43} -10.3723 q^{47} -1.37228 q^{49} +14.7446 q^{51} +0.255437 q^{53} -13.4891 q^{57} +4.37228 q^{59} -9.74456 q^{61} +19.8614 q^{63} -9.11684 q^{67} +4.62772 q^{69} +14.7446 q^{71} -4.74456 q^{73} -10.3723 q^{77} +6.62772 q^{79} +35.9783 q^{81} -7.62772 q^{83} +19.3723 q^{87} +12.0000 q^{89} -2.37228 q^{91} -8.00000 q^{93} -19.4891 q^{97} -36.6060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{7} + 11 q^{9}+O(q^{10}) \) 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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