LMFDB - Embedded newform 4368.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.138892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 2x + 12 $ x^4 - x^3 - 10x^2 + 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.44214$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.69903 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.69903 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.84830 q^{11} +1.00000 q^{13} +1.69903 q^{15} +0.964026 q^{17} -6.58330 q^{19} +1.00000 q^{21} +7.18524 q^{23} -2.11329 q^{25} +1.00000 q^{27} +8.58330 q^{29} +5.54733 q^{31} -5.84830 q^{33} +1.69903 q^{35} +0.964026 q^{37} +1.00000 q^{39} -4.88427 q^{41} +7.18524 q^{43} +1.69903 q^{45} -4.73501 q^{47} +1.00000 q^{49} +0.964026 q^{51} +13.0173 q^{53} -9.93644 q^{55} -6.58330 q^{57} +4.81232 q^{59} +13.2626 q^{61} +1.00000 q^{63} +1.69903 q^{65} +10.3705 q^{67} +7.18524 q^{69} -4.81232 q^{71} +13.1852 q^{73} -2.11329 q^{75} -5.84830 q^{77} +7.61928 q^{79} +1.00000 q^{81} +0.133069 q^{83} +1.63791 q^{85} +8.58330 q^{87} +2.73501 q^{89} +1.00000 q^{91} +5.54733 q^{93} -11.1852 q^{95} -5.01734 q^{97} -5.84830 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.69903	0.759830	0.379915	−	0.925021i	$-0.375953\pi$
$5$	1.69903	0.759830	0.379915	+	0.925021i	$0.375953\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.84830	−1.76333	−0.881664	−	0.471878i	$-0.843576\pi$
$11$	−5.84830	−1.76333	−0.881664	+	0.471878i	$0.843576\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.69903	0.438688
$16$	0	0
$17$	0.964026	0.233811	0.116905	−	0.993143i	$-0.462703\pi$
$17$	0.964026	0.233811	0.116905	+	0.993143i	$0.462703\pi$
$18$	0	0
$19$	−6.58330	−1.51031	−0.755157	−	0.655544i	$-0.772439\pi$
$19$	−6.58330	−1.51031	−0.755157	+	0.655544i	$0.772439\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	7.18524	1.49823	0.749113	−	0.662442i	$-0.230480\pi$
$23$	7.18524	1.49823	0.749113	+	0.662442i	$0.230480\pi$
$24$	0	0
$25$	−2.11329	−0.422658
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.58330	1.59388	0.796940	−	0.604059i	$-0.206451\pi$
$29$	8.58330	1.59388	0.796940	+	0.604059i	$0.206451\pi$
$30$	0	0
$31$	5.54733	0.996330	0.498165	−	0.867082i	$-0.334007\pi$
$31$	5.54733	0.996330	0.498165	+	0.867082i	$0.334007\pi$
$32$	0	0
$33$	−5.84830	−1.01806
$34$	0	0
$35$	1.69903	0.287189
$36$	0	0
$37$	0.964026	0.158485	0.0792425	−	0.996855i	$-0.474750\pi$
$37$	0.964026	0.158485	0.0792425	+	0.996855i	$0.474750\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−4.88427	−0.762795	−0.381397	−	0.924411i	$-0.624557\pi$
$41$	−4.88427	−0.762795	−0.381397	+	0.924411i	$0.624557\pi$
$42$	0	0
$43$	7.18524	1.09574	0.547869	−	0.836564i	$-0.315439\pi$
$43$	7.18524	1.09574	0.547869	+	0.836564i	$0.315439\pi$
$44$	0	0
$45$	1.69903	0.253277
$46$	0	0
$47$	−4.73501	−0.690672	−0.345336	−	0.938479i	$-0.612235\pi$
$47$	−4.73501	−0.690672	−0.345336	+	0.938479i	$0.612235\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.964026	0.134991
$52$	0	0
$53$	13.0173	1.78807	0.894035	−	0.447998i	$-0.147863\pi$
$53$	13.0173	1.78807	0.894035	+	0.447998i	$0.147863\pi$
$54$	0	0
$55$	−9.93644	−1.33983
$56$	0	0
$57$	−6.58330	−0.871980
$58$	0	0
$59$	4.81232	0.626511	0.313256	−	0.949669i	$-0.398580\pi$
$59$	4.81232	0.626511	0.313256	+	0.949669i	$0.398580\pi$
$60$	0	0
$61$	13.2626	1.69810	0.849048	−	0.528315i	$-0.177176\pi$
$61$	13.2626	1.69810	0.849048	+	0.528315i	$0.177176\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.69903	0.210739
$66$	0	0
$67$	10.3705	1.26696	0.633478	−	0.773761i	$-0.281627\pi$
$67$	10.3705	1.26696	0.633478	+	0.773761i	$0.281627\pi$
$68$	0	0
$69$	7.18524	0.865001
$70$	0	0
$71$	−4.81232	−0.571118	−0.285559	−	0.958361i	$-0.592179\pi$
$71$	−4.81232	−0.571118	−0.285559	+	0.958361i	$0.592179\pi$
$72$	0	0
$73$	13.1852	1.54322	0.771608	−	0.636099i	$-0.219453\pi$
$73$	13.1852	1.54322	0.771608	+	0.636099i	$0.219453\pi$
$74$	0	0
$75$	−2.11329	−0.244022
$76$	0	0
$77$	−5.84830	−0.666475
$78$	0	0
$79$	7.61928	0.857236	0.428618	−	0.903486i	$-0.359001\pi$
$79$	7.61928	0.857236	0.428618	+	0.903486i	$0.359001\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.133069	0.0146062	0.00730310	−	0.999973i	$-0.497675\pi$
$83$	0.133069	0.0146062	0.00730310	+	0.999973i	$0.497675\pi$
$84$	0	0
$85$	1.63791	0.177656
$86$	0	0
$87$	8.58330	0.920227
$88$	0	0
$89$	2.73501	0.289910	0.144955	−	0.989438i	$-0.453696\pi$
$89$	2.73501	0.289910	0.144955	+	0.989438i	$0.453696\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	5.54733	0.575231
$94$	0	0
$95$	−11.1852	−1.14758
$96$	0	0
$97$	−5.01734	−0.509434	−0.254717	−	0.967016i	$-0.581982\pi$
$97$	−5.01734	−0.509434	−0.254717	+	0.967016i	$0.581982\pi$
$98$	0	0
$99$	−5.84830	−0.587776
$100$	0	0
$101$	14.9005	1.48265	0.741326	−	0.671145i	$-0.234197\pi$
$101$	14.9005	1.48265	0.741326	+	0.671145i	$0.234197\pi$
$102$	0	0
$103$	−2.36209	−0.232744	−0.116372	−	0.993206i	$-0.537126\pi$
$103$	−2.36209	−0.232744	−0.116372	+	0.993206i	$0.537126\pi$
$104$	0	0
$105$	1.69903	0.165809
$106$	0	0
$107$	5.92805	0.573086	0.286543	−	0.958067i	$-0.407494\pi$
$107$	5.92805	0.573086	0.286543	+	0.958067i	$0.407494\pi$
$108$	0	0
$109$	9.56596	0.916253	0.458127	−	0.888887i	$-0.348521\pi$
$109$	9.56596	0.916253	0.458127	+	0.888887i	$0.348521\pi$
$110$	0	0
$111$	0.964026	0.0915013
$112$	0	0
$113$	0.751202	0.0706672	0.0353336	−	0.999376i	$-0.488751\pi$
$113$	0.751202	0.0706672	0.0353336	+	0.999376i	$0.488751\pi$
$114$	0	0
$115$	12.2079	1.13840
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0.964026	0.0883721
$120$	0	0
$121$	23.2026	2.10933
$122$	0	0
$123$	−4.88427	−0.440400
$124$	0	0
$125$	−12.0857	−1.08098
$126$	0	0
$127$	−15.8405	−1.40562	−0.702808	−	0.711379i	$-0.748071\pi$
$127$	−15.8405	−1.40562	−0.702808	+	0.711379i	$0.748071\pi$
$128$	0	0
$129$	7.18524	0.632625
$130$	0	0
$131$	−6.36209	−0.555858	−0.277929	−	0.960602i	$-0.589648\pi$
$131$	−6.36209	−0.555858	−0.277929	+	0.960602i	$0.589648\pi$
$132$	0	0
$133$	−6.58330	−0.570845
$134$	0	0
$135$	1.69903	0.146229
$136$	0	0
$137$	−19.6888	−1.68213	−0.841063	−	0.540937i	$-0.818070\pi$
$137$	−19.6888	−1.68213	−0.841063	+	0.540937i	$0.818070\pi$
$138$	0	0
$139$	−4.26614	−0.361849	−0.180925	−	0.983497i	$-0.557909\pi$
$139$	−4.26614	−0.361849	−0.180925	+	0.983497i	$0.557909\pi$
$140$	0	0
$141$	−4.73501	−0.398759
$142$	0	0
$143$	−5.84830	−0.489059
$144$	0	0
$145$	14.5833	1.21108
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−22.5089	−1.84400	−0.922001	−	0.387187i	$-0.873447\pi$
$149$	−22.5089	−1.84400	−0.922001	+	0.387187i	$0.873447\pi$
$150$	0	0
$151$	−18.3621	−1.49429	−0.747143	−	0.664664i	$-0.768575\pi$
$151$	−18.3621	−1.49429	−0.747143	+	0.664664i	$0.768575\pi$
$152$	0	0
$153$	0.964026	0.0779369
$154$	0	0
$155$	9.42509	0.757041
$156$	0	0
$157$	8.96403	0.715407	0.357704	−	0.933835i	$-0.383560\pi$
$157$	8.96403	0.715407	0.357704	+	0.933835i	$0.383560\pi$
$158$	0	0
$159$	13.0173	1.03234
$160$	0	0
$161$	7.18524	0.566276
$162$	0	0
$163$	−7.69659	−0.602844	−0.301422	−	0.953491i	$-0.597461\pi$
$163$	−7.69659	−0.602844	−0.301422	+	0.953491i	$0.597461\pi$
$164$	0	0
$165$	−9.93644	−0.773551
$166$	0	0
$167$	1.16904	0.0904632	0.0452316	−	0.998977i	$-0.485597\pi$
$167$	1.16904	0.0904632	0.0452316	+	0.998977i	$0.485597\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.58330	−0.503438
$172$	0	0
$173$	−0.673885	−0.0512345	−0.0256173	−	0.999672i	$-0.508155\pi$
$173$	−0.673885	−0.0512345	−0.0256173	+	0.999672i	$0.508155\pi$
$174$	0	0
$175$	−2.11329	−0.159750
$176$	0	0
$177$	4.81232	0.361716
$178$	0	0
$179$	−1.24880	−0.0933395	−0.0466698	−	0.998910i	$-0.514861\pi$
$179$	−1.24880	−0.0933395	−0.0466698	+	0.998910i	$0.514861\pi$
$180$	0	0
$181$	−0.796126	−0.0591756	−0.0295878	−	0.999562i	$-0.509419\pi$
$181$	−0.796126	−0.0591756	−0.0295878	+	0.999562i	$0.509419\pi$
$182$	0	0
$183$	13.2626	0.980396
$184$	0	0
$185$	1.63791	0.120422
$186$	0	0
$187$	−5.63791	−0.412285
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	4.29014	0.310424	0.155212	−	0.987881i	$-0.450394\pi$
$191$	4.29014	0.310424	0.155212	+	0.987881i	$0.450394\pi$
$192$	0	0
$193$	12.0719	0.868958	0.434479	−	0.900682i	$-0.356933\pi$
$193$	12.0719	0.868958	0.434479	+	0.900682i	$0.356933\pi$
$194$	0	0
$195$	1.69903	0.121670
$196$	0	0
$197$	−19.6804	−1.40217	−0.701085	−	0.713078i	$-0.747301\pi$
$197$	−19.6804	−1.40217	−0.701085	+	0.713078i	$0.747301\pi$
$198$	0	0
$199$	−14.0587	−0.996594	−0.498297	−	0.867007i	$-0.666041\pi$
$199$	−14.0587	−0.996594	−0.498297	+	0.867007i	$0.666041\pi$
$200$	0	0
$201$	10.3705	0.731477
$202$	0	0
$203$	8.58330	0.602430
$204$	0	0
$205$	−8.29853	−0.579595
$206$	0	0
$207$	7.18524	0.499409
$208$	0	0
$209$	38.5011	2.66318
$210$	0	0
$211$	−4.51135	−0.310574	−0.155287	−	0.987869i	$-0.549630\pi$
$211$	−4.51135	−0.310574	−0.155287	+	0.987869i	$0.549630\pi$
$212$	0	0
$213$	−4.81232	−0.329735
$214$	0	0
$215$	12.2079	0.832575
$216$	0	0
$217$	5.54733	0.376577
$218$	0	0
$219$	13.1852	0.890976
$220$	0	0
$221$	0.964026	0.0648474
$222$	0	0
$223$	−7.61928	−0.510225	−0.255112	−	0.966911i	$-0.582112\pi$
$223$	−7.61928	−0.510225	−0.255112	+	0.966911i	$0.582112\pi$
$224$	0	0
$225$	−2.11329	−0.140886
$226$	0	0
$227$	−16.5089	−1.09574	−0.547868	−	0.836565i	$-0.684560\pi$
$227$	−16.5089	−1.09574	−0.547868	+	0.836565i	$0.684560\pi$
$228$	0	0
$229$	4.79613	0.316937	0.158468	−	0.987364i	$-0.449344\pi$
$229$	4.79613	0.316937	0.158468	+	0.987364i	$0.449344\pi$
$230$	0	0
$231$	−5.84830	−0.384790
$232$	0	0
$233$	−21.6193	−1.41633	−0.708163	−	0.706049i	$-0.750476\pi$
$233$	−21.6193	−1.41633	−0.708163	+	0.706049i	$0.750476\pi$
$234$	0	0
$235$	−8.04492	−0.524793
$236$	0	0
$237$	7.61928	0.494925
$238$	0	0
$239$	−6.01620	−0.389155	−0.194578	−	0.980887i	$-0.562334\pi$
$239$	−6.01620	−0.389155	−0.194578	+	0.980887i	$0.562334\pi$
$240$	0	0
$241$	8.57491	0.552359	0.276179	−	0.961106i	$-0.410932\pi$
$241$	8.57491	0.552359	0.276179	+	0.961106i	$0.410932\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.69903	0.108547
$246$	0	0
$247$	−6.58330	−0.418886
$248$	0	0
$249$	0.133069	0.00843289
$250$	0	0
$251$	5.93644	0.374705	0.187352	−	0.982293i	$-0.440009\pi$
$251$	5.93644	0.374705	0.187352	+	0.982293i	$0.440009\pi$
$252$	0	0
$253$	−42.0214	−2.64186
$254$	0	0
$255$	1.63791	0.102570
$256$	0	0
$257$	−20.8285	−1.29925	−0.649624	−	0.760256i	$-0.725074\pi$
$257$	−20.8285	−1.29925	−0.649624	+	0.760256i	$0.725074\pi$
$258$	0	0
$259$	0.964026	0.0599017
$260$	0	0
$261$	8.58330	0.531293
$262$	0	0
$263$	11.0173	0.679358	0.339679	−	0.940541i	$-0.389682\pi$
$263$	11.0173	0.679358	0.339679	+	0.940541i	$0.389682\pi$
$264$	0	0
$265$	22.1169	1.35863
$266$	0	0
$267$	2.73501	0.167380
$268$	0	0
$269$	−6.26614	−0.382053	−0.191027	−	0.981585i	$-0.561182\pi$
$269$	−6.26614	−0.382053	−0.191027	+	0.981585i	$0.561182\pi$
$270$	0	0
$271$	24.2613	1.47377	0.736883	−	0.676020i	$-0.236297\pi$
$271$	24.2613	1.47377	0.736883	+	0.676020i	$0.236297\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	12.3592	0.745285
$276$	0	0
$277$	25.3159	1.52108	0.760542	−	0.649289i	$-0.224933\pi$
$277$	25.3159	1.52108	0.760542	+	0.649289i	$0.224933\pi$
$278$	0	0
$279$	5.54733	0.332110
$280$	0	0
$281$	5.75235	0.343156	0.171578	−	0.985171i	$-0.445113\pi$
$281$	5.75235	0.343156	0.171578	+	0.985171i	$0.445113\pi$
$282$	0	0
$283$	8.14390	0.484104	0.242052	−	0.970263i	$-0.422179\pi$
$283$	8.14390	0.484104	0.242052	+	0.970263i	$0.422179\pi$
$284$	0	0
$285$	−11.1852	−0.662556
$286$	0	0
$287$	−4.88427	−0.288309
$288$	0	0
$289$	−16.0707	−0.945333
$290$	0	0
$291$	−5.01734	−0.294122
$292$	0	0
$293$	18.0060	1.05192	0.525959	−	0.850510i	$-0.323706\pi$
$293$	18.0060	1.05192	0.525959	+	0.850510i	$0.323706\pi$
$294$	0	0
$295$	8.17629	0.476042
$296$	0	0
$297$	−5.84830	−0.339353
$298$	0	0
$299$	7.18524	0.415533
$300$	0	0
$301$	7.18524	0.414150
$302$	0	0
$303$	14.9005	0.856010
$304$	0	0
$305$	22.5335	1.29026
$306$	0	0
$307$	−20.3758	−1.16291	−0.581456	−	0.813578i	$-0.697517\pi$
$307$	−20.3758	−1.16291	−0.581456	+	0.813578i	$0.697517\pi$
$308$	0	0
$309$	−2.36209	−0.134375
$310$	0	0
$311$	−25.7313	−1.45909	−0.729543	−	0.683935i	$-0.760267\pi$
$311$	−25.7313	−1.45909	−0.729543	+	0.683935i	$0.760267\pi$
$312$	0	0
$313$	−15.4651	−0.874141	−0.437071	−	0.899427i	$-0.643984\pi$
$313$	−15.4651	−0.874141	−0.437071	+	0.899427i	$0.643984\pi$
$314$	0	0
$315$	1.69903	0.0957296
$316$	0	0
$317$	26.9513	1.51374	0.756869	−	0.653566i	$-0.226728\pi$
$317$	26.9513	1.51374	0.756869	+	0.653566i	$0.226728\pi$
$318$	0	0
$319$	−50.1977	−2.81053
$320$	0	0
$321$	5.92805	0.330872
$322$	0	0
$323$	−6.34648	−0.353127
$324$	0	0
$325$	−2.11329	−0.117224
$326$	0	0
$327$	9.56596	0.528999
$328$	0	0
$329$	−4.73501	−0.261049
$330$	0	0
$331$	22.2266	1.22168	0.610842	−	0.791753i	$-0.290831\pi$
$331$	22.2266	1.22168	0.610842	+	0.791753i	$0.290831\pi$
$332$	0	0
$333$	0.964026	0.0528283
$334$	0	0
$335$	17.6198	0.962671
$336$	0	0
$337$	−30.3518	−1.65337	−0.826685	−	0.562665i	$-0.809776\pi$
$337$	−30.3518	−1.65337	−0.826685	+	0.562665i	$0.809776\pi$
$338$	0	0
$339$	0.751202	0.0407997
$340$	0	0
$341$	−32.4424	−1.75686
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	12.2079	0.657254
$346$	0	0
$347$	2.23146	0.119791	0.0598955	−	0.998205i	$-0.480923\pi$
$347$	2.23146	0.119791	0.0598955	+	0.998205i	$0.480923\pi$
$348$	0	0
$349$	3.54733	0.189884	0.0949421	−	0.995483i	$-0.469733\pi$
$349$	3.54733	0.189884	0.0949421	+	0.995483i	$0.469733\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−10.8123	−0.575482	−0.287741	−	0.957708i	$-0.592904\pi$
$353$	−10.8123	−0.575482	−0.287741	+	0.957708i	$0.592904\pi$
$354$	0	0
$355$	−8.17629	−0.433952
$356$	0	0
$357$	0.964026	0.0510217
$358$	0	0
$359$	−6.01620	−0.317523	−0.158761	−	0.987317i	$-0.550750\pi$
$359$	−6.01620	−0.317523	−0.158761	+	0.987317i	$0.550750\pi$
$360$	0	0
$361$	24.3399	1.28105
$362$	0	0
$363$	23.2026	1.21782
$364$	0	0
$365$	22.4021	1.17258
$366$	0	0
$367$	12.1222	0.632776	0.316388	−	0.948630i	$-0.397530\pi$
$367$	12.1222	0.632776	0.316388	+	0.948630i	$0.397530\pi$
$368$	0	0
$369$	−4.88427	−0.254265
$370$	0	0
$371$	13.0173	0.675827
$372$	0	0
$373$	−32.5251	−1.68409	−0.842043	−	0.539410i	$-0.818647\pi$
$373$	−32.5251	−1.68409	−0.842043	+	0.539410i	$0.818647\pi$
$374$	0	0
$375$	−12.0857	−0.624103
$376$	0	0
$377$	8.58330	0.442063
$378$	0	0
$379$	24.1222	1.23908	0.619538	−	0.784967i	$-0.287320\pi$
$379$	24.1222	1.23908	0.619538	+	0.784967i	$0.287320\pi$
$380$	0	0
$381$	−15.8405	−0.811533
$382$	0	0
$383$	12.2188	0.624350	0.312175	−	0.950025i	$-0.398942\pi$
$383$	12.2188	0.624350	0.312175	+	0.950025i	$0.398942\pi$
$384$	0	0
$385$	−9.93644	−0.506408
$386$	0	0
$387$	7.18524	0.365246
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	6.92676	0.350301
$392$	0	0
$393$	−6.36209	−0.320925
$394$	0	0
$395$	12.9454	0.651353
$396$	0	0
$397$	7.69122	0.386011	0.193006	−	0.981198i	$-0.438176\pi$
$397$	7.69122	0.386011	0.193006	+	0.981198i	$0.438176\pi$
$398$	0	0
$399$	−6.58330	−0.329577
$400$	0	0
$401$	−14.6577	−0.731970	−0.365985	−	0.930621i	$-0.619268\pi$
$401$	−14.6577	−0.731970	−0.365985	+	0.930621i	$0.619268\pi$
$402$	0	0
$403$	5.54733	0.276332
$404$	0	0
$405$	1.69903	0.0844256
$406$	0	0
$407$	−5.63791	−0.279461
$408$	0	0
$409$	29.4465	1.45604	0.728018	−	0.685558i	$-0.240442\pi$
$409$	29.4465	1.45604	0.728018	+	0.685558i	$0.240442\pi$
$410$	0	0
$411$	−19.6888	−0.971176
$412$	0	0
$413$	4.81232	0.236799
$414$	0	0
$415$	0.226088	0.0110982
$416$	0	0
$417$	−4.26614	−0.208914
$418$	0	0
$419$	21.6006	1.05526	0.527630	−	0.849474i	$-0.323081\pi$
$419$	21.6006	1.05526	0.527630	+	0.849474i	$0.323081\pi$
$420$	0	0
$421$	−2.26614	−0.110445	−0.0552224	−	0.998474i	$-0.517587\pi$
$421$	−2.26614	−0.110445	−0.0552224	+	0.998474i	$0.517587\pi$
$422$	0	0
$423$	−4.73501	−0.230224
$424$	0	0
$425$	−2.03727	−0.0988220
$426$	0	0
$427$	13.2626	0.641820
$428$	0	0
$429$	−5.84830	−0.282358
$430$	0	0
$431$	40.7751	1.96407	0.982033	−	0.188711i	$-0.0604310\pi$
$431$	40.7751	1.96407	0.982033	+	0.188711i	$0.0604310\pi$
$432$	0	0
$433$	−35.7637	−1.71869	−0.859346	−	0.511394i	$-0.829129\pi$
$433$	−35.7637	−1.71869	−0.859346	+	0.511394i	$0.829129\pi$
$434$	0	0
$435$	14.5833	0.699216
$436$	0	0
$437$	−47.3026	−2.26279
$438$	0	0
$439$	12.5887	0.600825	0.300412	−	0.953809i	$-0.402876\pi$
$439$	12.5887	0.600825	0.300412	+	0.953809i	$0.402876\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	32.0400	1.52227	0.761134	−	0.648594i	$-0.224643\pi$
$443$	32.0400	1.52227	0.761134	+	0.648594i	$0.224643\pi$
$444$	0	0
$445$	4.64686	0.220282
$446$	0	0
$447$	−22.5089	−1.06464
$448$	0	0
$449$	−13.9202	−0.656937	−0.328468	−	0.944515i	$-0.606532\pi$
$449$	−13.9202	−0.656937	−0.328468	+	0.944515i	$0.606532\pi$
$450$	0	0
$451$	28.5647	1.34506
$452$	0	0
$453$	−18.3621	−0.862726
$454$	0	0
$455$	1.69903	0.0796518
$456$	0	0
$457$	34.5274	1.61512	0.807562	−	0.589783i	$-0.200787\pi$
$457$	34.5274	1.61512	0.807562	+	0.589783i	$0.200787\pi$
$458$	0	0
$459$	0.964026	0.0449969
$460$	0	0
$461$	−19.8434	−0.924200	−0.462100	−	0.886828i	$-0.652904\pi$
$461$	−19.8434	−0.924200	−0.462100	+	0.886828i	$0.652904\pi$
$462$	0	0
$463$	15.7050	0.729873	0.364936	−	0.931033i	$-0.381091\pi$
$463$	15.7050	0.729873	0.364936	+	0.931033i	$0.381091\pi$
$464$	0	0
$465$	9.42509	0.437078
$466$	0	0
$467$	−38.1977	−1.76758	−0.883789	−	0.467885i	$-0.845016\pi$
$467$	−38.1977	−1.76758	−0.883789	+	0.467885i	$0.845016\pi$
$468$	0	0
$469$	10.3705	0.478864
$470$	0	0
$471$	8.96403	0.413041
$472$	0	0
$473$	−42.0214	−1.93215
$474$	0	0
$475$	13.9124	0.638346
$476$	0	0
$477$	13.0173	0.596023
$478$	0	0
$479$	33.6066	1.53552	0.767762	−	0.640735i	$-0.221370\pi$
$479$	33.6066	1.53552	0.767762	+	0.640735i	$0.221370\pi$
$480$	0	0
$481$	0.964026	0.0439558
$482$	0	0
$483$	7.18524	0.326940
$484$	0	0
$485$	−8.52462	−0.387083
$486$	0	0
$487$	26.0671	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$487$	26.0671	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$488$	0	0
$489$	−7.69659	−0.348052
$490$	0	0
$491$	2.07195	0.0935057	0.0467529	−	0.998906i	$-0.485113\pi$
$491$	2.07195	0.0935057	0.0467529	+	0.998906i	$0.485113\pi$
$492$	0	0
$493$	8.27453	0.372666
$494$	0	0
$495$	−9.93644	−0.446610
$496$	0	0
$497$	−4.81232	−0.215862
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	1.16904	0.0522290
$502$	0	0
$503$	−31.9951	−1.42659	−0.713296	−	0.700863i	$-0.752798\pi$
$503$	−31.9951	−1.42659	−0.713296	+	0.700863i	$0.752798\pi$
$504$	0	0
$505$	25.3164	1.12656
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−36.8261	−1.63229	−0.816144	−	0.577849i	$-0.803892\pi$
$509$	−36.8261	−1.63229	−0.816144	+	0.577849i	$0.803892\pi$
$510$	0	0
$511$	13.1852	0.583281
$512$	0	0
$513$	−6.58330	−0.290660
$514$	0	0
$515$	−4.01326	−0.176846
$516$	0	0
$517$	27.6917	1.21788
$518$	0	0
$519$	−0.673885	−0.0295803
$520$	0	0
$521$	−29.0658	−1.27339	−0.636697	−	0.771114i	$-0.719700\pi$
$521$	−29.0658	−1.27339	−0.636697	+	0.771114i	$0.719700\pi$
$522$	0	0
$523$	24.1222	1.05479	0.527396	−	0.849620i	$-0.323168\pi$
$523$	24.1222	1.05479	0.527396	+	0.849620i	$0.323168\pi$
$524$	0	0
$525$	−2.11329	−0.0922316
$526$	0	0
$527$	5.34777	0.232952
$528$	0	0
$529$	28.6277	1.24468
$530$	0	0
$531$	4.81232	0.208837
$532$	0	0
$533$	−4.88427	−0.211561
$534$	0	0
$535$	10.0719	0.435448
$536$	0	0
$537$	−1.24880	−0.0538896
$538$	0	0
$539$	−5.84830	−0.251904
$540$	0	0
$541$	23.6006	1.01467	0.507335	−	0.861749i	$-0.330631\pi$
$541$	23.6006	1.01467	0.507335	+	0.861749i	$0.330631\pi$
$542$	0	0
$543$	−0.796126	−0.0341651
$544$	0	0
$545$	16.2529	0.696197
$546$	0	0
$547$	−38.7091	−1.65508	−0.827540	−	0.561407i	$-0.810260\pi$
$547$	−38.7091	−1.65508	−0.827540	+	0.561407i	$0.810260\pi$
$548$	0	0
$549$	13.2626	0.566032
$550$	0	0
$551$	−56.5065	−2.40726
$552$	0	0
$553$	7.61928	0.324005
$554$	0	0
$555$	1.63791	0.0695254
$556$	0	0
$557$	−0.474238	−0.0200941	−0.0100470	−	0.999950i	$-0.503198\pi$
$557$	−0.474238	−0.0200941	−0.0100470	+	0.999950i	$0.503198\pi$
$558$	0	0
$559$	7.18524	0.303903
$560$	0	0
$561$	−5.63791	−0.238033
$562$	0	0
$563$	−21.8668	−0.921575	−0.460787	−	0.887511i	$-0.652433\pi$
$563$	−21.8668	−0.921575	−0.460787	+	0.887511i	$0.652433\pi$
$564$	0	0
$565$	1.27632	0.0536950
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−33.7366	−1.41431	−0.707157	−	0.707057i	$-0.750023\pi$
$569$	−33.7366	−1.41431	−0.707157	+	0.707057i	$0.750023\pi$
$570$	0	0
$571$	−2.59657	−0.108663	−0.0543315	−	0.998523i	$-0.517303\pi$
$571$	−2.59657	−0.108663	−0.0543315	+	0.998523i	$0.517303\pi$
$572$	0	0
$573$	4.29014	0.179223
$574$	0	0
$575$	−15.1845	−0.633238
$576$	0	0
$577$	46.5647	1.93851	0.969256	−	0.246053i	$-0.0791339\pi$
$577$	46.5647	1.93851	0.969256	+	0.246053i	$0.0791339\pi$
$578$	0	0
$579$	12.0719	0.501693
$580$	0	0
$581$	0.133069	0.00552062
$582$	0	0
$583$	−76.1293	−3.15295
$584$	0	0
$585$	1.69903	0.0702463
$586$	0	0
$587$	−38.9411	−1.60727	−0.803636	−	0.595122i	$-0.797104\pi$
$587$	−38.9411	−1.60727	−0.803636	+	0.595122i	$0.797104\pi$
$588$	0	0
$589$	−36.5197	−1.50477
$590$	0	0
$591$	−19.6804	−0.809543
$592$	0	0
$593$	−15.7626	−0.647292	−0.323646	−	0.946178i	$-0.604909\pi$
$593$	−15.7626	−0.647292	−0.323646	+	0.946178i	$0.604909\pi$
$594$	0	0
$595$	1.63791	0.0671478
$596$	0	0
$597$	−14.0587	−0.575384
$598$	0	0
$599$	−15.3399	−0.626770	−0.313385	−	0.949626i	$-0.601463\pi$
$599$	−15.3399	−0.626770	−0.313385	+	0.949626i	$0.601463\pi$
$600$	0	0
$601$	26.8681	1.09597	0.547986	−	0.836488i	$-0.315395\pi$
$601$	26.8681	1.09597	0.547986	+	0.836488i	$0.315395\pi$
$602$	0	0
$603$	10.3705	0.422319
$604$	0	0
$605$	39.4219	1.60273
$606$	0	0
$607$	−11.7206	−0.475724	−0.237862	−	0.971299i	$-0.576447\pi$
$607$	−11.7206	−0.475724	−0.237862	+	0.971299i	$0.576447\pi$
$608$	0	0
$609$	8.58330	0.347813
$610$	0	0
$611$	−4.73501	−0.191558
$612$	0	0
$613$	−35.4411	−1.43145	−0.715727	−	0.698380i	$-0.753904\pi$
$613$	−35.4411	−1.43145	−0.715727	+	0.698380i	$0.753904\pi$
$614$	0	0
$615$	−8.29853	−0.334629
$616$	0	0
$617$	−34.9753	−1.40805	−0.704027	−	0.710173i	$-0.748617\pi$
$617$	−34.9753	−1.40805	−0.704027	+	0.710173i	$0.748617\pi$
$618$	0	0
$619$	−6.53838	−0.262800	−0.131400	−	0.991329i	$-0.541947\pi$
$619$	−6.53838	−0.262800	−0.131400	+	0.991329i	$0.541947\pi$
$620$	0	0
$621$	7.18524	0.288334
$622$	0	0
$623$	2.73501	0.109576
$624$	0	0
$625$	−9.96754	−0.398702
$626$	0	0
$627$	38.5011	1.53759
$628$	0	0
$629$	0.929346	0.0370555
$630$	0	0
$631$	−14.0636	−0.559861	−0.279931	−	0.960020i	$-0.590311\pi$
$631$	−14.0636	−0.559861	−0.279931	+	0.960020i	$0.590311\pi$
$632$	0	0
$633$	−4.51135	−0.179310
$634$	0	0
$635$	−26.9135	−1.06803
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−4.81232	−0.190373
$640$	0	0
$641$	−23.6696	−0.934892	−0.467446	−	0.884022i	$-0.654826\pi$
$641$	−23.6696	−0.934892	−0.467446	+	0.884022i	$0.654826\pi$
$642$	0	0
$643$	9.19061	0.362442	0.181221	−	0.983442i	$-0.441995\pi$
$643$	9.19061	0.362442	0.181221	+	0.983442i	$0.441995\pi$
$644$	0	0
$645$	12.2079	0.480688
$646$	0	0
$647$	−41.8908	−1.64690	−0.823448	−	0.567391i	$-0.807953\pi$
$647$	−41.8908	−1.64690	−0.823448	+	0.567391i	$0.807953\pi$
$648$	0	0
$649$	−28.1439	−1.10474
$650$	0	0
$651$	5.54733	0.217417
$652$	0	0
$653$	45.3643	1.77524	0.887621	−	0.460574i	$-0.152356\pi$
$653$	45.3643	1.77524	0.887621	+	0.460574i	$0.152356\pi$
$654$	0	0
$655$	−10.8094	−0.422358
$656$	0	0
$657$	13.1852	0.514405
$658$	0	0
$659$	39.4549	1.53694	0.768472	−	0.639883i	$-0.221017\pi$
$659$	39.4549	1.53694	0.768472	+	0.639883i	$0.221017\pi$
$660$	0	0
$661$	−7.39269	−0.287542	−0.143771	−	0.989611i	$-0.545923\pi$
$661$	−7.39269	−0.287542	−0.143771	+	0.989611i	$0.545923\pi$
$662$	0	0
$663$	0.964026	0.0374397
$664$	0	0
$665$	−11.1852	−0.433745
$666$	0	0
$667$	61.6731	2.38799
$668$	0	0
$669$	−7.61928	−0.294578
$670$	0	0
$671$	−77.5634	−2.99430
$672$	0	0
$673$	−6.67086	−0.257143	−0.128571	−	0.991700i	$-0.541039\pi$
$673$	−6.67086	−0.257143	−0.128571	+	0.991700i	$0.541039\pi$
$674$	0	0
$675$	−2.11329	−0.0813406
$676$	0	0
$677$	−0.940022	−0.0361280	−0.0180640	−	0.999837i	$-0.505750\pi$
$677$	−0.940022	−0.0361280	−0.0180640	+	0.999837i	$0.505750\pi$
$678$	0	0
$679$	−5.01734	−0.192548
$680$	0	0
$681$	−16.5089	−0.632623
$682$	0	0
$683$	22.1144	0.846185	0.423093	−	0.906086i	$-0.360944\pi$
$683$	22.1144	0.846185	0.423093	+	0.906086i	$0.360944\pi$
$684$	0	0
$685$	−33.4519	−1.27813
$686$	0	0
$687$	4.79613	0.182984
$688$	0	0
$689$	13.0173	0.495921
$690$	0	0
$691$	47.1515	1.79373	0.896864	−	0.442307i	$-0.145840\pi$
$691$	47.1515	1.79373	0.896864	+	0.442307i	$0.145840\pi$
$692$	0	0
$693$	−5.84830	−0.222158
$694$	0	0
$695$	−7.24830	−0.274944
$696$	0	0
$697$	−4.70856	−0.178350
$698$	0	0
$699$	−21.6193	−0.817716
$700$	0	0
$701$	49.9129	1.88519	0.942593	−	0.333945i	$-0.108380\pi$
$701$	49.9129	1.88519	0.942593	+	0.333945i	$0.108380\pi$
$702$	0	0
$703$	−6.34648	−0.239362
$704$	0	0
$705$	−8.04492	−0.302989
$706$	0	0
$707$	14.9005	0.560390
$708$	0	0
$709$	−40.8979	−1.53595	−0.767976	−	0.640479i	$-0.778736\pi$
$709$	−40.8979	−1.53595	−0.767976	+	0.640479i	$0.778736\pi$
$710$	0	0
$711$	7.61928	0.285745
$712$	0	0
$713$	39.8589	1.49273
$714$	0	0
$715$	−9.93644	−0.371602
$716$	0	0
$717$	−6.01620	−0.224679
$718$	0	0
$719$	13.3429	0.497606	0.248803	−	0.968554i	$-0.419963\pi$
$719$	13.3429	0.497606	0.248803	+	0.968554i	$0.419963\pi$
$720$	0	0
$721$	−2.36209	−0.0879688
$722$	0	0
$723$	8.57491	0.318904
$724$	0	0
$725$	−18.1390	−0.673666
$726$	0	0
$727$	24.9089	0.923818	0.461909	−	0.886927i	$-0.347165\pi$
$727$	24.9089	0.923818	0.461909	+	0.886927i	$0.347165\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.92676	0.256195
$732$	0	0
$733$	10.5030	0.387936	0.193968	−	0.981008i	$-0.437864\pi$
$733$	10.5030	0.387936	0.193968	+	0.981008i	$0.437864\pi$
$734$	0	0
$735$	1.69903	0.0626697
$736$	0	0
$737$	−60.6496	−2.23406
$738$	0	0
$739$	−15.4305	−0.567619	−0.283809	−	0.958881i	$-0.591598\pi$
$739$	−15.4305	−0.567619	−0.283809	+	0.958881i	$0.591598\pi$
$740$	0	0
$741$	−6.58330	−0.241844
$742$	0	0
$743$	15.9119	0.583749	0.291875	−	0.956457i	$-0.405721\pi$
$743$	15.9119	0.583749	0.291875	+	0.956457i	$0.405721\pi$
$744$	0	0
$745$	−38.2434	−1.40113
$746$	0	0
$747$	0.133069	0.00486873
$748$	0	0
$749$	5.92805	0.216606
$750$	0	0
$751$	19.3159	0.704846	0.352423	−	0.935841i	$-0.385358\pi$
$751$	19.3159	0.704846	0.352423	+	0.935841i	$0.385358\pi$
$752$	0	0
$753$	5.93644	0.216336
$754$	0	0
$755$	−31.1978	−1.13540
$756$	0	0
$757$	−1.93342	−0.0702714	−0.0351357	−	0.999383i	$-0.511186\pi$
$757$	−1.93342	−0.0702714	−0.0351357	+	0.999383i	$0.511186\pi$
$758$	0	0
$759$	−42.0214	−1.52528
$760$	0	0
$761$	0.0779003	0.00282388	0.00141194	−	0.999999i	$-0.499551\pi$
$761$	0.0779003	0.00282388	0.00141194	+	0.999999i	$0.499551\pi$
$762$	0	0
$763$	9.56596	0.346311
$764$	0	0
$765$	1.63791	0.0592188
$766$	0	0
$767$	4.81232	0.173763
$768$	0	0
$769$	49.0365	1.76830	0.884150	−	0.467203i	$-0.154738\pi$
$769$	49.0365	1.76830	0.884150	+	0.467203i	$0.154738\pi$
$770$	0	0
$771$	−20.8285	−0.750121
$772$	0	0
$773$	−1.62171	−0.0583290	−0.0291645	−	0.999575i	$-0.509285\pi$
$773$	−1.62171	−0.0583290	−0.0291645	+	0.999575i	$0.509285\pi$
$774$	0	0
$775$	−11.7231	−0.421107
$776$	0	0
$777$	0.964026	0.0345842
$778$	0	0
$779$	32.1546	1.15206
$780$	0	0
$781$	28.1439	1.00707
$782$	0	0
$783$	8.58330	0.306742
$784$	0	0
$785$	15.2302	0.543588
$786$	0	0
$787$	−38.0880	−1.35769	−0.678845	−	0.734281i	$-0.737519\pi$
$787$	−38.0880	−1.35769	−0.678845	+	0.734281i	$0.737519\pi$
$788$	0	0
$789$	11.0173	0.392228
$790$	0	0
$791$	0.751202	0.0267097
$792$	0	0
$793$	13.2626	0.470967
$794$	0	0
$795$	22.1169	0.784405
$796$	0	0
$797$	−25.5203	−0.903976	−0.451988	−	0.892024i	$-0.649285\pi$
$797$	−25.5203	−0.903976	−0.451988	+	0.892024i	$0.649285\pi$
$798$	0	0
$799$	−4.56467	−0.161486
$800$	0	0
$801$	2.73501	0.0966367
$802$	0	0
$803$	−77.1112	−2.72120
$804$	0	0
$805$	12.2079	0.430274
$806$	0	0
$807$	−6.26614	−0.220578
$808$	0	0
$809$	9.28348	0.326390	0.163195	−	0.986594i	$-0.447820\pi$
$809$	9.28348	0.326390	0.163195	+	0.986594i	$0.447820\pi$
$810$	0	0
$811$	−28.3069	−0.993990	−0.496995	−	0.867753i	$-0.665563\pi$
$811$	−28.3069	−0.993990	−0.496995	+	0.867753i	$0.665563\pi$
$812$	0	0
$813$	24.2613	0.850880
$814$	0	0
$815$	−13.0768	−0.458059
$816$	0	0
$817$	−47.3026	−1.65491
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−3.25962	−0.113762	−0.0568808	−	0.998381i	$-0.518116\pi$
$821$	−3.25962	−0.113762	−0.0568808	+	0.998381i	$0.518116\pi$
$822$	0	0
$823$	1.73615	0.0605183	0.0302592	−	0.999542i	$-0.490367\pi$
$823$	1.73615	0.0605183	0.0302592	+	0.999542i	$0.490367\pi$
$824$	0	0
$825$	12.3592	0.430291
$826$	0	0
$827$	32.7834	1.13999	0.569996	−	0.821647i	$-0.306945\pi$
$827$	32.7834	1.13999	0.569996	+	0.821647i	$0.306945\pi$
$828$	0	0
$829$	−3.33450	−0.115812	−0.0579061	−	0.998322i	$-0.518442\pi$
$829$	−3.33450	−0.115812	−0.0579061	+	0.998322i	$0.518442\pi$
$830$	0	0
$831$	25.3159	0.878198
$832$	0	0
$833$	0.964026	0.0334015
$834$	0	0
$835$	1.98624	0.0687367
$836$	0	0
$837$	5.54733	0.191744
$838$	0	0
$839$	−16.3916	−0.565899	−0.282950	−	0.959135i	$-0.591313\pi$
$839$	−16.3916	−0.565899	−0.282950	+	0.959135i	$0.591313\pi$
$840$	0	0
$841$	44.6731	1.54045
$842$	0	0
$843$	5.75235	0.198121
$844$	0	0
$845$	1.69903	0.0584485
$846$	0	0
$847$	23.2026	0.797250
$848$	0	0
$849$	8.14390	0.279498
$850$	0	0
$851$	6.92676	0.237446
$852$	0	0
$853$	25.3110	0.866632	0.433316	−	0.901242i	$-0.357343\pi$
$853$	25.3110	0.866632	0.433316	+	0.901242i	$0.357343\pi$
$854$	0	0
$855$	−11.1852	−0.382527
$856$	0	0
$857$	10.0324	0.342700	0.171350	−	0.985210i	$-0.445187\pi$
$857$	10.0324	0.342700	0.171350	+	0.985210i	$0.445187\pi$
$858$	0	0
$859$	−47.6917	−1.62722	−0.813610	−	0.581411i	$-0.802501\pi$
$859$	−47.6917	−1.62722	−0.813610	+	0.581411i	$0.802501\pi$
$860$	0	0
$861$	−4.88427	−0.166456
$862$	0	0
$863$	−22.7080	−0.772989	−0.386494	−	0.922292i	$-0.626314\pi$
$863$	−22.7080	−0.772989	−0.386494	+	0.922292i	$0.626314\pi$
$864$	0	0
$865$	−1.14495	−0.0389295
$866$	0	0
$867$	−16.0707	−0.545788
$868$	0	0
$869$	−44.5598	−1.51159
$870$	0	0
$871$	10.3705	0.351390
$872$	0	0
$873$	−5.01734	−0.169811
$874$	0	0
$875$	−12.0857	−0.408571
$876$	0	0
$877$	−14.2985	−0.482827	−0.241414	−	0.970422i	$-0.577611\pi$
$877$	−14.2985	−0.482827	−0.241414	+	0.970422i	$0.577611\pi$
$878$	0	0
$879$	18.0060	0.607326
$880$	0	0
$881$	−8.81526	−0.296994	−0.148497	−	0.988913i	$-0.547443\pi$
$881$	−8.81526	−0.296994	−0.148497	+	0.988913i	$0.547443\pi$
$882$	0	0
$883$	39.2302	1.32020	0.660100	−	0.751178i	$-0.270514\pi$
$883$	39.2302	1.32020	0.660100	+	0.751178i	$0.270514\pi$
$884$	0	0
$885$	8.17629	0.274843
$886$	0	0
$887$	−7.08392	−0.237855	−0.118927	−	0.992903i	$-0.537946\pi$
$887$	−7.08392	−0.237855	−0.118927	+	0.992903i	$0.537946\pi$
$888$	0	0
$889$	−15.8405	−0.531273
$890$	0	0
$891$	−5.84830	−0.195925
$892$	0	0
$893$	31.1720	1.04313
$894$	0	0
$895$	−2.12175	−0.0709222
$896$	0	0
$897$	7.18524	0.239908
$898$	0	0
$899$	47.6144	1.58803
$900$	0	0
$901$	12.5491	0.418070
$902$	0	0
$903$	7.18524	0.239110
$904$	0	0
$905$	−1.35264	−0.0449634
$906$	0	0
$907$	−12.1097	−0.402096	−0.201048	−	0.979581i	$-0.564435\pi$
$907$	−12.1097	−0.402096	−0.201048	+	0.979581i	$0.564435\pi$
$908$	0	0
$909$	14.9005	0.494217
$910$	0	0
$911$	5.37943	0.178228	0.0891142	−	0.996021i	$-0.471596\pi$
$911$	5.37943	0.178228	0.0891142	+	0.996021i	$0.471596\pi$
$912$	0	0
$913$	−0.778226	−0.0257555
$914$	0	0
$915$	22.5335	0.744935
$916$	0	0
$917$	−6.36209	−0.210095
$918$	0	0
$919$	54.0298	1.78228	0.891139	−	0.453730i	$-0.149907\pi$
$919$	54.0298	1.78228	0.891139	+	0.453730i	$0.149907\pi$
$920$	0	0
$921$	−20.3758	−0.671407
$922$	0	0
$923$	−4.81232	−0.158400
$924$	0	0
$925$	−2.03727	−0.0669850
$926$	0	0
$927$	−2.36209	−0.0775812
$928$	0	0
$929$	−1.41376	−0.0463841	−0.0231921	−	0.999731i	$-0.507383\pi$
$929$	−1.41376	−0.0463841	−0.0231921	+	0.999731i	$0.507383\pi$
$930$	0	0
$931$	−6.58330	−0.215759
$932$	0	0
$933$	−25.7313	−0.842404
$934$	0	0
$935$	−9.57899	−0.313266
$936$	0	0
$937$	34.4424	1.12519	0.562593	−	0.826734i	$-0.309804\pi$
$937$	34.4424	1.12519	0.562593	+	0.826734i	$0.309804\pi$
$938$	0	0
$939$	−15.4651	−0.504686
$940$	0	0
$941$	37.7074	1.22923	0.614613	−	0.788828i	$-0.289312\pi$
$941$	37.7074	1.22923	0.614613	+	0.788828i	$0.289312\pi$
$942$	0	0
$943$	−35.0947	−1.14284
$944$	0	0
$945$	1.69903	0.0552695
$946$	0	0
$947$	19.4573	0.632278	0.316139	−	0.948713i	$-0.397613\pi$
$947$	19.4573	0.632278	0.316139	+	0.948713i	$0.397613\pi$
$948$	0	0
$949$	13.1852	0.428011
$950$	0	0
$951$	26.9513	0.873957
$952$	0	0
$953$	−45.1768	−1.46342	−0.731711	−	0.681615i	$-0.761278\pi$
$953$	−45.1768	−1.46342	−0.731711	+	0.681615i	$0.761278\pi$
$954$	0	0
$955$	7.28909	0.235869
$956$	0	0
$957$	−50.1977	−1.62266
$958$	0	0
$959$	−19.6888	−0.635784
$960$	0	0
$961$	−0.227144	−0.00732722
$962$	0	0
$963$	5.92805	0.191029
$964$	0	0
$965$	20.5106	0.660260
$966$	0	0
$967$	2.19770	0.0706734	0.0353367	−	0.999375i	$-0.488750\pi$
$967$	2.19770	0.0706734	0.0353367	+	0.999375i	$0.488750\pi$
$968$	0	0
$969$	−6.34648	−0.203878
$970$	0	0
$971$	17.4279	0.559287	0.279643	−	0.960104i	$-0.409784\pi$
$971$	17.4279	0.559287	0.279643	+	0.960104i	$0.409784\pi$
$972$	0	0
$973$	−4.26614	−0.136766
$974$	0	0
$975$	−2.11329	−0.0676795
$976$	0	0
$977$	−15.5498	−0.497481	−0.248741	−	0.968570i	$-0.580017\pi$
$977$	−15.5498	−0.497481	−0.248741	+	0.968570i	$0.580017\pi$
$978$	0	0
$979$	−15.9951	−0.511206
$980$	0	0
$981$	9.56596	0.305418
$982$	0	0
$983$	22.6018	0.720885	0.360443	−	0.932781i	$-0.382626\pi$
$983$	22.6018	0.720885	0.360443	+	0.932781i	$0.382626\pi$
$984$	0	0
$985$	−33.4376	−1.06541
$986$	0	0
$987$	−4.73501	−0.150717
$988$	0	0
$989$	51.6277	1.64166
$990$	0	0
$991$	−2.67876	−0.0850936	−0.0425468	−	0.999094i	$-0.513547\pi$
$991$	−2.67876	−0.0850936	−0.0425468	+	0.999094i	$0.513547\pi$
$992$	0	0
$993$	22.2266	0.705339
$994$	0	0
$995$	−23.8861	−0.757242
$996$	0	0
$997$	−38.8308	−1.22978	−0.614892	−	0.788611i	$-0.710800\pi$
$997$	−38.8308	−1.22978	−0.614892	+	0.788611i	$0.710800\pi$
$998$	0	0
$999$	0.964026	0.0305004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bs.1.3		4
4.3	odd	2		2184.2.a.v.1.3	✓	4
12.11	even	2		6552.2.a.bs.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.v.1.3	✓	4	4.3	odd	2
4368.2.a.bs.1.3		4	1.1	even	1	trivial
6552.2.a.bs.1.2		4	12.11	even	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.69903 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.69903 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.84830 q^{11} +1.00000 q^{13} +1.69903 q^{15} +0.964026 q^{17} -6.58330 q^{19} +1.00000 q^{21} +7.18524 q^{23} -2.11329 q^{25} +1.00000 q^{27} +8.58330 q^{29} +5.54733 q^{31} -5.84830 q^{33} +1.69903 q^{35} +0.964026 q^{37} +1.00000 q^{39} -4.88427 q^{41} +7.18524 q^{43} +1.69903 q^{45} -4.73501 q^{47} +1.00000 q^{49} +0.964026 q^{51} +13.0173 q^{53} -9.93644 q^{55} -6.58330 q^{57} +4.81232 q^{59} +13.2626 q^{61} +1.00000 q^{63} +1.69903 q^{65} +10.3705 q^{67} +7.18524 q^{69} -4.81232 q^{71} +13.1852 q^{73} -2.11329 q^{75} -5.84830 q^{77} +7.61928 q^{79} +1.00000 q^{81} +0.133069 q^{83} +1.63791 q^{85} +8.58330 q^{87} +2.73501 q^{89} +1.00000 q^{91} +5.54733 q^{93} -11.1852 q^{95} -5.01734 q^{97} -5.84830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * q^99

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