LMFDB - Embedded newform 3800.2.a.ba.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 15x^2 - 14*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.185519$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.185519 q^{3} +4.45651 q^{7} -2.96558 q^{9} +O(q^{10})$	$q+0.185519 q^{3} +4.45651 q^{7} -2.96558 q^{9} +2.64623 q^{11} -1.30142 q^{13} +3.51716 q^{17} -1.00000 q^{19} +0.826767 q^{21} +6.52882 q^{23} -1.10673 q^{27} -5.20946 q^{29} +10.8219 q^{31} +0.490925 q^{33} -2.04607 q^{37} -0.241439 q^{39} -3.80044 q^{41} -4.77089 q^{43} -1.49093 q^{47} +12.8605 q^{49} +0.652501 q^{51} -0.225583 q^{53} -0.185519 q^{57} -2.86056 q^{59} -6.31449 q^{61} -13.2161 q^{63} +13.1831 q^{67} +1.21122 q^{69} +12.3310 q^{71} +5.42276 q^{73} +11.7929 q^{77} +14.9688 q^{79} +8.69143 q^{81} -3.84470 q^{83} -0.966455 q^{87} +1.67666 q^{89} -5.79981 q^{91} +2.00768 q^{93} -9.48523 q^{97} -7.84760 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} + 33 q^{51} - 13 q^{53} + 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} + 19 q^{71} - 3 q^{73} + 36 q^{77} - 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} + 42 q^{91} - 39 q^{93} + 11 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 + 3 * q^11 + q^13 + 14 * q^17 - 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 + 5 * q^31 - 2 * q^33 + 8 * q^37 + 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 + 33 * q^51 - 13 * q^53 + 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 + 19 * q^71 - 3 * q^73 + 36 * q^77 - 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 + 42 * q^91 - 39 * q^93 + 11 * q^97 - 6 * 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
$3$	0.185519	0.107109	0.0535547	−	0.998565i	$-0.482945\pi$
$3$	0.185519	0.107109	0.0535547	+	0.998565i	$0.482945\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.45651	1.68440	0.842201	−	0.539164i	$-0.181260\pi$
$7$	4.45651	1.68440	0.842201	+	0.539164i	$0.181260\pi$
$8$	0	0
$9$	−2.96558	−0.988528
$10$	0	0
$11$	2.64623	0.797867	0.398934	−	0.916980i	$-0.369380\pi$
$11$	2.64623	0.797867	0.398934	+	0.916980i	$0.369380\pi$
$12$	0	0
$13$	−1.30142	−0.360950	−0.180475	−	0.983580i	$-0.557764\pi$
$13$	−1.30142	−0.360950	−0.180475	+	0.983580i	$0.557764\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.51716	0.853037	0.426519	−	0.904479i	$-0.359740\pi$
$17$	3.51716	0.853037	0.426519	+	0.904479i	$0.359740\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.826767	0.180415
$22$	0	0
$23$	6.52882	1.36135	0.680677	−	0.732584i	$-0.261686\pi$
$23$	6.52882	1.36135	0.680677	+	0.732584i	$0.261686\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.10673	−0.212990
$28$	0	0
$29$	−5.20946	−0.967373	−0.483686	−	0.875241i	$-0.660702\pi$
$29$	−5.20946	−0.967373	−0.483686	+	0.875241i	$0.660702\pi$
$30$	0	0
$31$	10.8219	1.94368	0.971839	−	0.235646i	$-0.0757207\pi$
$31$	10.8219	1.94368	0.971839	+	0.235646i	$0.0757207\pi$
$32$	0	0
$33$	0.490925	0.0854591
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.04607	−0.336373	−0.168186	−	0.985755i	$-0.553791\pi$
$37$	−2.04607	−0.336373	−0.168186	+	0.985755i	$0.553791\pi$
$38$	0	0
$39$	−0.241439	−0.0386612
$40$	0	0
$41$	−3.80044	−0.593529	−0.296764	−	0.954951i	$-0.595908\pi$
$41$	−3.80044	−0.593529	−0.296764	+	0.954951i	$0.595908\pi$
$42$	0	0
$43$	−4.77089	−0.727554	−0.363777	−	0.931486i	$-0.618513\pi$
$43$	−4.77089	−0.727554	−0.363777	+	0.931486i	$0.618513\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.49093	−0.217474	−0.108737	−	0.994071i	$-0.534681\pi$
$47$	−1.49093	−0.217474	−0.108737	+	0.994071i	$0.534681\pi$
$48$	0	0
$49$	12.8605	1.83721
$50$	0	0
$51$	0.652501	0.0913684
$52$	0	0
$53$	−0.225583	−0.0309862	−0.0154931	−	0.999880i	$-0.504932\pi$
$53$	−0.225583	−0.0309862	−0.0154931	+	0.999880i	$0.504932\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.185519	−0.0245726
$58$	0	0
$59$	−2.86056	−0.372413	−0.186206	−	0.982511i	$-0.559619\pi$
$59$	−2.86056	−0.372413	−0.186206	+	0.982511i	$0.559619\pi$
$60$	0	0
$61$	−6.31449	−0.808488	−0.404244	−	0.914651i	$-0.632465\pi$
$61$	−6.31449	−0.808488	−0.404244	+	0.914651i	$0.632465\pi$
$62$	0	0
$63$	−13.2161	−1.66508
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1831	1.61058	0.805288	−	0.592884i	$-0.202011\pi$
$67$	13.1831	1.61058	0.805288	+	0.592884i	$0.202011\pi$
$68$	0	0
$69$	1.21122	0.145814
$70$	0	0
$71$	12.3310	1.46342	0.731711	−	0.681615i	$-0.238722\pi$
$71$	12.3310	1.46342	0.731711	+	0.681615i	$0.238722\pi$
$72$	0	0
$73$	5.42276	0.634686	0.317343	−	0.948311i	$-0.397209\pi$
$73$	5.42276	0.634686	0.317343	+	0.948311i	$0.397209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.7929	1.34393
$78$	0	0
$79$	14.9688	1.68413	0.842063	−	0.539379i	$-0.181341\pi$
$79$	14.9688	1.68413	0.842063	+	0.539379i	$0.181341\pi$
$80$	0	0
$81$	8.69143	0.965714
$82$	0	0
$83$	−3.84470	−0.422011	−0.211005	−	0.977485i	$-0.567674\pi$
$83$	−3.84470	−0.422011	−0.211005	+	0.977485i	$0.567674\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.966455	−0.103615
$88$	0	0
$89$	1.67666	0.177726	0.0888629	−	0.996044i	$-0.471677\pi$
$89$	1.67666	0.177726	0.0888629	+	0.996044i	$0.471677\pi$
$90$	0	0
$91$	−5.79981	−0.607985
$92$	0	0
$93$	2.00768	0.208186
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.48523	−0.963079	−0.481539	−	0.876424i	$-0.659922\pi$
$97$	−9.48523	−0.963079	−0.481539	+	0.876424i	$0.659922\pi$
$98$	0	0
$99$	−7.84760	−0.788714
$100$	0	0
$101$	−6.39100	−0.635928	−0.317964	−	0.948103i	$-0.602999\pi$
$101$	−6.39100	−0.635928	−0.317964	+	0.948103i	$0.602999\pi$
$102$	0	0
$103$	−4.14301	−0.408223	−0.204112	−	0.978948i	$-0.565431\pi$
$103$	−4.14301	−0.408223	−0.204112	+	0.978948i	$0.565431\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.27597	−0.703394	−0.351697	−	0.936114i	$-0.614395\pi$
$107$	−7.27597	−0.703394	−0.351697	+	0.936114i	$0.614395\pi$
$108$	0	0
$109$	14.1077	1.35127	0.675637	−	0.737235i	$-0.263869\pi$
$109$	14.1077	1.35127	0.675637	+	0.737235i	$0.263869\pi$
$110$	0	0
$111$	−0.379586	−0.0360287
$112$	0	0
$113$	−7.03290	−0.661600	−0.330800	−	0.943701i	$-0.607319\pi$
$113$	−7.03290	−0.661600	−0.330800	+	0.943701i	$0.607319\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.85948	0.356809
$118$	0	0
$119$	15.6743	1.43686
$120$	0	0
$121$	−3.99749	−0.363408
$122$	0	0
$123$	−0.705053	−0.0635725
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.6240	−1.20893	−0.604467	−	0.796630i	$-0.706614\pi$
$127$	−13.6240	−1.20893	−0.604467	+	0.796630i	$0.706614\pi$
$128$	0	0
$129$	−0.885090	−0.0779279
$130$	0	0
$131$	22.3345	1.95137	0.975685	−	0.219177i	$-0.0703373\pi$
$131$	22.3345	1.95137	0.975685	+	0.219177i	$0.0703373\pi$
$132$	0	0
$133$	−4.45651	−0.386428
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.55578	0.474662	0.237331	−	0.971429i	$-0.423727\pi$
$137$	5.55578	0.474662	0.237331	+	0.971429i	$0.423727\pi$
$138$	0	0
$139$	−11.0452	−0.936841	−0.468420	−	0.883506i	$-0.655177\pi$
$139$	−11.0452	−0.936841	−0.468420	+	0.883506i	$0.655177\pi$
$140$	0	0
$141$	−0.276595	−0.0232935
$142$	0	0
$143$	−3.44386	−0.287990
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.38586	0.196782
$148$	0	0
$149$	17.6003	1.44188	0.720938	−	0.693000i	$-0.243711\pi$
$149$	17.6003	1.44188	0.720938	+	0.693000i	$0.243711\pi$
$150$	0	0
$151$	5.24140	0.426539	0.213269	−	0.976993i	$-0.431589\pi$
$151$	5.24140	0.426539	0.213269	+	0.976993i	$0.431589\pi$
$152$	0	0
$153$	−10.4304	−0.843251
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.7965	1.50012	0.750061	−	0.661368i	$-0.230024\pi$
$157$	18.7965	1.50012	0.750061	+	0.661368i	$0.230024\pi$
$158$	0	0
$159$	−0.0418499	−0.00331891
$160$	0	0
$161$	29.0957	2.29307
$162$	0	0
$163$	−12.1669	−0.952982	−0.476491	−	0.879179i	$-0.658092\pi$
$163$	−12.1669	−0.952982	−0.476491	+	0.879179i	$0.658092\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.46393	−0.577576	−0.288788	−	0.957393i	$-0.593252\pi$
$167$	−7.46393	−0.577576	−0.288788	+	0.957393i	$0.593252\pi$
$168$	0	0
$169$	−11.3063	−0.869715
$170$	0	0
$171$	2.96558	0.226784
$172$	0	0
$173$	−17.8070	−1.35384	−0.676921	−	0.736055i	$-0.736686\pi$
$173$	−17.8070	−1.35384	−0.676921	+	0.736055i	$0.736686\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.530688	−0.0398889
$178$	0	0
$179$	−9.48154	−0.708684	−0.354342	−	0.935116i	$-0.615295\pi$
$179$	−9.48154	−0.708684	−0.354342	+	0.935116i	$0.615295\pi$
$180$	0	0
$181$	4.32832	0.321721	0.160861	−	0.986977i	$-0.448573\pi$
$181$	4.32832	0.321721	0.160861	+	0.986977i	$0.448573\pi$
$182$	0	0
$183$	−1.17146	−0.0865967
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.30720	0.680610
$188$	0	0
$189$	−4.93215	−0.358761
$190$	0	0
$191$	−8.21414	−0.594355	−0.297177	−	0.954822i	$-0.596045\pi$
$191$	−8.21414	−0.594355	−0.297177	+	0.954822i	$0.596045\pi$
$192$	0	0
$193$	−2.89738	−0.208558	−0.104279	−	0.994548i	$-0.533253\pi$
$193$	−2.89738	−0.208558	−0.104279	+	0.994548i	$0.533253\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.4118	1.59678	0.798388	−	0.602143i	$-0.205686\pi$
$197$	22.4118	1.59678	0.798388	+	0.602143i	$0.205686\pi$
$198$	0	0
$199$	2.68542	0.190364	0.0951821	−	0.995460i	$-0.469657\pi$
$199$	2.68542	0.190364	0.0951821	+	0.995460i	$0.469657\pi$
$200$	0	0
$201$	2.44572	0.172508
$202$	0	0
$203$	−23.2160	−1.62944
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−19.3618	−1.34573
$208$	0	0
$209$	−2.64623	−0.183043
$210$	0	0
$211$	7.22869	0.497644	0.248822	−	0.968549i	$-0.419957\pi$
$211$	7.22869	0.497644	0.248822	+	0.968549i	$0.419957\pi$
$212$	0	0
$213$	2.28764	0.156746
$214$	0	0
$215$	0	0
$216$	0	0
$217$	48.2281	3.27393
$218$	0	0
$219$	1.00603	0.0679809
$220$	0	0
$221$	−4.57732	−0.307904
$222$	0	0
$223$	−6.86629	−0.459801	−0.229900	−	0.973214i	$-0.573840\pi$
$223$	−6.86629	−0.459801	−0.229900	+	0.973214i	$0.573840\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.2176	−1.27552	−0.637758	−	0.770237i	$-0.720138\pi$
$227$	−19.2176	−1.27552	−0.637758	+	0.770237i	$0.720138\pi$
$228$	0	0
$229$	25.2205	1.66661	0.833307	−	0.552810i	$-0.186445\pi$
$229$	25.2205	1.66661	0.833307	+	0.552810i	$0.186445\pi$
$230$	0	0
$231$	2.18781	0.143947
$232$	0	0
$233$	24.1123	1.57965	0.789825	−	0.613332i	$-0.210171\pi$
$233$	24.1123	1.57965	0.789825	+	0.613332i	$0.210171\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.77701	0.180386
$238$	0	0
$239$	4.34082	0.280784	0.140392	−	0.990096i	$-0.455164\pi$
$239$	4.34082	0.280784	0.140392	+	0.990096i	$0.455164\pi$
$240$	0	0
$241$	21.9630	1.41476	0.707380	−	0.706834i	$-0.249877\pi$
$241$	21.9630	1.41476	0.707380	+	0.706834i	$0.249877\pi$
$242$	0	0
$243$	4.93261	0.316427
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.30142	0.0828077
$248$	0	0
$249$	−0.713265	−0.0452013
$250$	0	0
$251$	4.41214	0.278492	0.139246	−	0.990258i	$-0.455532\pi$
$251$	4.41214	0.278492	0.139246	+	0.990258i	$0.455532\pi$
$252$	0	0
$253$	17.2767	1.08618
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.9112	0.743000	0.371500	−	0.928433i	$-0.378844\pi$
$257$	11.9112	0.743000	0.371500	+	0.928433i	$0.378844\pi$
$258$	0	0
$259$	−9.11835	−0.566587
$260$	0	0
$261$	15.4491	0.956275
$262$	0	0
$263$	13.7779	0.849581	0.424790	−	0.905292i	$-0.360348\pi$
$263$	13.7779	0.849581	0.424790	+	0.905292i	$0.360348\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.311053	0.0190361
$268$	0	0
$269$	−2.30474	−0.140522	−0.0702612	−	0.997529i	$-0.522383\pi$
$269$	−2.30474	−0.140522	−0.0702612	+	0.997529i	$0.522383\pi$
$270$	0	0
$271$	−20.1878	−1.22632	−0.613161	−	0.789958i	$-0.710102\pi$
$271$	−20.1878	−1.22632	−0.613161	+	0.789958i	$0.710102\pi$
$272$	0	0
$273$	−1.07597	−0.0651210
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.7021	0.643024	0.321512	−	0.946905i	$-0.395809\pi$
$277$	10.7021	0.643024	0.321512	+	0.946905i	$0.395809\pi$
$278$	0	0
$279$	−32.0934	−1.92138
$280$	0	0
$281$	−2.28603	−0.136373	−0.0681865	−	0.997673i	$-0.521721\pi$
$281$	−2.28603	−0.136373	−0.0681865	+	0.997673i	$0.521721\pi$
$282$	0	0
$283$	25.6515	1.52482	0.762411	−	0.647093i	$-0.224016\pi$
$283$	25.6515	1.52482	0.762411	+	0.647093i	$0.224016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16.9367	−0.999741
$288$	0	0
$289$	−4.62957	−0.272328
$290$	0	0
$291$	−1.75969	−0.103155
$292$	0	0
$293$	−19.6027	−1.14520	−0.572602	−	0.819833i	$-0.694066\pi$
$293$	−19.6027	−1.14520	−0.572602	+	0.819833i	$0.694066\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.92866	−0.169938
$298$	0	0
$299$	−8.49677	−0.491381
$300$	0	0
$301$	−21.2615	−1.22549
$302$	0	0
$303$	−1.18565	−0.0681139
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.6679	1.80738	0.903692	−	0.428183i	$-0.140846\pi$
$307$	31.6679	1.80738	0.903692	+	0.428183i	$0.140846\pi$
$308$	0	0
$309$	−0.768608	−0.0437246
$310$	0	0
$311$	14.8957	0.844655	0.422328	−	0.906443i	$-0.361213\pi$
$311$	14.8957	0.844655	0.422328	+	0.906443i	$0.361213\pi$
$312$	0	0
$313$	0.288235	0.0162920	0.00814601	−	0.999967i	$-0.497407\pi$
$313$	0.288235	0.0162920	0.00814601	+	0.999967i	$0.497407\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.9612	−1.23347	−0.616733	−	0.787173i	$-0.711544\pi$
$317$	−21.9612	−1.23347	−0.616733	+	0.787173i	$0.711544\pi$
$318$	0	0
$319$	−13.7854	−0.771835
$320$	0	0
$321$	−1.34983	−0.0753402
$322$	0	0
$323$	−3.51716	−0.195700
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.61725	0.144734
$328$	0	0
$329$	−6.64432	−0.366313
$330$	0	0
$331$	24.0905	1.32413	0.662066	−	0.749445i	$-0.269680\pi$
$331$	24.0905	1.32413	0.662066	+	0.749445i	$0.269680\pi$
$332$	0	0
$333$	6.06780	0.332514
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−17.7090	−0.964673	−0.482337	−	0.875986i	$-0.660212\pi$
$337$	−17.7090	−0.964673	−0.482337	+	0.875986i	$0.660212\pi$
$338$	0	0
$339$	−1.30474	−0.0708636
$340$	0	0
$341$	28.6373	1.55080
$342$	0	0
$343$	26.1172	1.41020
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.158465	−0.00850687	−0.00425344	−	0.999991i	$-0.501354\pi$
$347$	−0.158465	−0.00850687	−0.00425344	+	0.999991i	$0.501354\pi$
$348$	0	0
$349$	27.3776	1.46549	0.732745	−	0.680503i	$-0.238239\pi$
$349$	27.3776	1.46549	0.732745	+	0.680503i	$0.238239\pi$
$350$	0	0
$351$	1.44032	0.0768788
$352$	0	0
$353$	18.3221	0.975185	0.487592	−	0.873071i	$-0.337875\pi$
$353$	18.3221	0.975185	0.487592	+	0.873071i	$0.337875\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.90787	0.153901
$358$	0	0
$359$	35.6265	1.88030	0.940148	−	0.340767i	$-0.110687\pi$
$359$	35.6265	1.88030	0.940148	+	0.340767i	$0.110687\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.741610	−0.0389245
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.03379	−0.158362	−0.0791812	−	0.996860i	$-0.525231\pi$
$367$	−3.03379	−0.158362	−0.0791812	+	0.996860i	$0.525231\pi$
$368$	0	0
$369$	11.2705	0.586719
$370$	0	0
$371$	−1.00531	−0.0521932
$372$	0	0
$373$	−9.43451	−0.488500	−0.244250	−	0.969712i	$-0.578542\pi$
$373$	−9.43451	−0.488500	−0.244250	+	0.969712i	$0.578542\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.77972	0.349173
$378$	0	0
$379$	−10.1522	−0.521486	−0.260743	−	0.965408i	$-0.583967\pi$
$379$	−10.1522	−0.521486	−0.260743	+	0.965408i	$0.583967\pi$
$380$	0	0
$381$	−2.52751	−0.129488
$382$	0	0
$383$	−19.0699	−0.974428	−0.487214	−	0.873283i	$-0.661987\pi$
$383$	−19.0699	−0.974428	−0.487214	+	0.873283i	$0.661987\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14.1485	0.719207
$388$	0	0
$389$	−38.6448	−1.95937	−0.979685	−	0.200544i	$-0.935729\pi$
$389$	−38.6448	−1.95937	−0.979685	+	0.200544i	$0.935729\pi$
$390$	0	0
$391$	22.9629	1.16128
$392$	0	0
$393$	4.14347	0.209010
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.5339	−0.528679	−0.264339	−	0.964430i	$-0.585154\pi$
$397$	−10.5339	−0.528679	−0.264339	+	0.964430i	$0.585154\pi$
$398$	0	0
$399$	−0.826767	−0.0413901
$400$	0	0
$401$	−10.6994	−0.534304	−0.267152	−	0.963654i	$-0.586083\pi$
$401$	−10.6994	−0.534304	−0.267152	+	0.963654i	$0.586083\pi$
$402$	0	0
$403$	−14.0839	−0.701571
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.41438	−0.268381
$408$	0	0
$409$	39.1036	1.93355	0.966775	−	0.255627i	$-0.0822819\pi$
$409$	39.1036	1.93355	0.966775	+	0.255627i	$0.0822819\pi$
$410$	0	0
$411$	1.03070	0.0508408
$412$	0	0
$413$	−12.7481	−0.627292
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.04909	−0.100345
$418$	0	0
$419$	5.00833	0.244673	0.122336	−	0.992489i	$-0.460961\pi$
$419$	5.00833	0.244673	0.122336	+	0.992489i	$0.460961\pi$
$420$	0	0
$421$	−12.7557	−0.621674	−0.310837	−	0.950463i	$-0.600609\pi$
$421$	−12.7557	−0.621674	−0.310837	+	0.950463i	$0.600609\pi$
$422$	0	0
$423$	4.42146	0.214979
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−28.1406	−1.36182
$428$	0	0
$429$	−0.638902	−0.0308465
$430$	0	0
$431$	13.7140	0.660580	0.330290	−	0.943879i	$-0.392853\pi$
$431$	13.7140	0.660580	0.330290	+	0.943879i	$0.392853\pi$
$432$	0	0
$433$	−1.49408	−0.0718007	−0.0359003	−	0.999355i	$-0.511430\pi$
$433$	−1.49408	−0.0718007	−0.0359003	+	0.999355i	$0.511430\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.52882	−0.312316
$438$	0	0
$439$	5.78568	0.276136	0.138068	−	0.990423i	$-0.455911\pi$
$439$	5.78568	0.276136	0.138068	+	0.990423i	$0.455911\pi$
$440$	0	0
$441$	−38.1388	−1.81613
$442$	0	0
$443$	−9.78426	−0.464864	−0.232432	−	0.972613i	$-0.574668\pi$
$443$	−9.78426	−0.464864	−0.232432	+	0.972613i	$0.574668\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.26520	0.154439
$448$	0	0
$449$	−38.0631	−1.79631	−0.898154	−	0.439682i	$-0.855091\pi$
$449$	−38.0631	−1.79631	−0.898154	+	0.439682i	$0.855091\pi$
$450$	0	0
$451$	−10.0568	−0.473557
$452$	0	0
$453$	0.972379	0.0456864
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.3179	−0.856874	−0.428437	−	0.903572i	$-0.640936\pi$
$457$	−18.3179	−0.856874	−0.428437	+	0.903572i	$0.640936\pi$
$458$	0	0
$459$	−3.89255	−0.181688
$460$	0	0
$461$	−35.8641	−1.67036	−0.835179	−	0.549977i	$-0.814636\pi$
$461$	−35.8641	−1.67036	−0.835179	+	0.549977i	$0.814636\pi$
$462$	0	0
$463$	35.3540	1.64304	0.821519	−	0.570181i	$-0.193127\pi$
$463$	35.3540	1.64304	0.821519	+	0.570181i	$0.193127\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.5118	−1.55074	−0.775371	−	0.631506i	$-0.782437\pi$
$467$	−33.5118	−1.55074	−0.775371	+	0.631506i	$0.782437\pi$
$468$	0	0
$469$	58.7507	2.71286
$470$	0	0
$471$	3.48711	0.160677
$472$	0	0
$473$	−12.6248	−0.580491
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.668984	0.0306307
$478$	0	0
$479$	−21.1784	−0.967667	−0.483834	−	0.875160i	$-0.660756\pi$
$479$	−21.1784	−0.967667	−0.483834	+	0.875160i	$0.660756\pi$
$480$	0	0
$481$	2.66281	0.121414
$482$	0	0
$483$	5.39781	0.245609
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−35.2032	−1.59521	−0.797605	−	0.603180i	$-0.793900\pi$
$487$	−35.2032	−1.59521	−0.797605	+	0.603180i	$0.793900\pi$
$488$	0	0
$489$	−2.25719	−0.102073
$490$	0	0
$491$	−42.7908	−1.93112	−0.965561	−	0.260177i	$-0.916219\pi$
$491$	−42.7908	−1.93112	−0.965561	+	0.260177i	$0.916219\pi$
$492$	0	0
$493$	−18.3225	−0.825205
$494$	0	0
$495$	0	0
$496$	0	0
$497$	54.9533	2.46499
$498$	0	0
$499$	18.3564	0.821747	0.410874	−	0.911692i	$-0.365224\pi$
$499$	18.3564	0.821747	0.410874	+	0.911692i	$0.365224\pi$
$500$	0	0
$501$	−1.38470	−0.0618639
$502$	0	0
$503$	−38.3650	−1.71061	−0.855305	−	0.518125i	$-0.826630\pi$
$503$	−38.3650	−1.71061	−0.855305	+	0.518125i	$0.826630\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.09753	−0.0931547
$508$	0	0
$509$	1.73845	0.0770556	0.0385278	−	0.999258i	$-0.487733\pi$
$509$	1.73845	0.0770556	0.0385278	+	0.999258i	$0.487733\pi$
$510$	0	0
$511$	24.1666	1.06907
$512$	0	0
$513$	1.10673	0.0488633
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.94532	−0.173515
$518$	0	0
$519$	−3.30354	−0.145009
$520$	0	0
$521$	−39.3764	−1.72511	−0.862556	−	0.505961i	$-0.831138\pi$
$521$	−39.3764	−1.72511	−0.862556	+	0.505961i	$0.831138\pi$
$522$	0	0
$523$	−43.7296	−1.91216	−0.956080	−	0.293105i	$-0.905311\pi$
$523$	−43.7296	−1.91216	−0.956080	+	0.293105i	$0.905311\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.0625	1.65803
$528$	0	0
$529$	19.6255	0.853282
$530$	0	0
$531$	8.48321	0.368140
$532$	0	0
$533$	4.94598	0.214234
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.75901	−0.0759067
$538$	0	0
$539$	34.0317	1.46585
$540$	0	0
$541$	−17.8936	−0.769305	−0.384652	−	0.923061i	$-0.625679\pi$
$541$	−17.8936	−0.769305	−0.384652	+	0.923061i	$0.625679\pi$
$542$	0	0
$543$	0.802985	0.0344594
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.91310	0.210069	0.105034	−	0.994469i	$-0.466505\pi$
$547$	4.91310	0.210069	0.105034	+	0.994469i	$0.466505\pi$
$548$	0	0
$549$	18.7261	0.799212
$550$	0	0
$551$	5.20946	0.221931
$552$	0	0
$553$	66.7088	2.83675
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.6114	−1.63602	−0.818008	−	0.575206i	$-0.804922\pi$
$557$	−38.6114	−1.63602	−0.818008	+	0.575206i	$0.804922\pi$
$558$	0	0
$559$	6.20895	0.262611
$560$	0	0
$561$	1.72666	0.0728998
$562$	0	0
$563$	−12.2697	−0.517105	−0.258553	−	0.965997i	$-0.583246\pi$
$563$	−12.2697	−0.517105	−0.258553	+	0.965997i	$0.583246\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	38.7334	1.62665
$568$	0	0
$569$	−32.0919	−1.34536	−0.672682	−	0.739932i	$-0.734858\pi$
$569$	−32.0919	−1.34536	−0.672682	+	0.739932i	$0.734858\pi$
$570$	0	0
$571$	6.85246	0.286767	0.143383	−	0.989667i	$-0.454202\pi$
$571$	6.85246	0.286767	0.143383	+	0.989667i	$0.454202\pi$
$572$	0	0
$573$	−1.52388	−0.0636610
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.8983	0.870006	0.435003	−	0.900429i	$-0.356747\pi$
$577$	20.8983	0.870006	0.435003	+	0.900429i	$0.356747\pi$
$578$	0	0
$579$	−0.537519	−0.0223385
$580$	0	0
$581$	−17.1339	−0.710835
$582$	0	0
$583$	−0.596943	−0.0247228
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.5692	−0.683883	−0.341942	−	0.939721i	$-0.611085\pi$
$587$	−16.5692	−0.683883	−0.341942	+	0.939721i	$0.611085\pi$
$588$	0	0
$589$	−10.8219	−0.445910
$590$	0	0
$591$	4.15782	0.171030
$592$	0	0
$593$	−6.96253	−0.285917	−0.142958	−	0.989729i	$-0.545662\pi$
$593$	−6.96253	−0.285917	−0.142958	+	0.989729i	$0.545662\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.498196	0.0203898
$598$	0	0
$599$	−23.8154	−0.973072	−0.486536	−	0.873660i	$-0.661740\pi$
$599$	−23.8154	−0.973072	−0.486536	+	0.873660i	$0.661740\pi$
$600$	0	0
$601$	10.1339	0.413372	0.206686	−	0.978407i	$-0.433732\pi$
$601$	10.1339	0.413372	0.206686	+	0.978407i	$0.433732\pi$
$602$	0	0
$603$	−39.0957	−1.59210
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−37.1179	−1.50657	−0.753286	−	0.657694i	$-0.771532\pi$
$607$	−37.1179	−1.50657	−0.753286	+	0.657694i	$0.771532\pi$
$608$	0	0
$609$	−4.30701	−0.174529
$610$	0	0
$611$	1.94033	0.0784972
$612$	0	0
$613$	−3.21621	−0.129901	−0.0649507	−	0.997888i	$-0.520689\pi$
$613$	−3.21621	−0.129901	−0.0649507	+	0.997888i	$0.520689\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.80395	−0.193400	−0.0967000	−	0.995314i	$-0.530829\pi$
$617$	−4.80395	−0.193400	−0.0967000	+	0.995314i	$0.530829\pi$
$618$	0	0
$619$	−10.3640	−0.416564	−0.208282	−	0.978069i	$-0.566787\pi$
$619$	−10.3640	−0.416564	−0.208282	+	0.978069i	$0.566787\pi$
$620$	0	0
$621$	−7.22564	−0.289955
$622$	0	0
$623$	7.47205	0.299362
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.490925	−0.0196057
$628$	0	0
$629$	−7.19638	−0.286938
$630$	0	0
$631$	−0.213288	−0.00849084	−0.00424542	−	0.999991i	$-0.501351\pi$
$631$	−0.213288	−0.00849084	−0.00424542	+	0.999991i	$0.501351\pi$
$632$	0	0
$633$	1.34106	0.0533024
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−16.7369	−0.663141
$638$	0	0
$639$	−36.5686	−1.44663
$640$	0	0
$641$	4.44567	0.175593	0.0877967	−	0.996138i	$-0.472017\pi$
$641$	4.44567	0.175593	0.0877967	+	0.996138i	$0.472017\pi$
$642$	0	0
$643$	20.0788	0.791830	0.395915	−	0.918287i	$-0.370427\pi$
$643$	20.0788	0.791830	0.395915	+	0.918287i	$0.370427\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.5734	1.04471	0.522353	−	0.852729i	$-0.325054\pi$
$647$	26.5734	1.04471	0.522353	+	0.852729i	$0.325054\pi$
$648$	0	0
$649$	−7.56968	−0.297136
$650$	0	0
$651$	8.94722	0.350669
$652$	0	0
$653$	0.0209494	0.000819814 0	0.000409907	−	1.00000i	$-0.499870\pi$
$653$	0.0209494	0.000819814 0	0.000409907		1.00000i	$0.499870\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−16.0816	−0.627405
$658$	0	0
$659$	−30.9236	−1.20461	−0.602307	−	0.798265i	$-0.705752\pi$
$659$	−30.9236	−1.20461	−0.602307	+	0.798265i	$0.705752\pi$
$660$	0	0
$661$	−26.7194	−1.03927	−0.519633	−	0.854390i	$-0.673931\pi$
$661$	−26.7194	−1.03927	−0.519633	+	0.854390i	$0.673931\pi$
$662$	0	0
$663$	−0.849180	−0.0329794
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34.0116	−1.31694
$668$	0	0
$669$	−1.27383	−0.0492490
$670$	0	0
$671$	−16.7096	−0.645066
$672$	0	0
$673$	17.3676	0.669472	0.334736	−	0.942312i	$-0.391353\pi$
$673$	17.3676	0.669472	0.334736	+	0.942312i	$0.391353\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.7712	1.75913	0.879565	−	0.475779i	$-0.157834\pi$
$677$	45.7712	1.75913	0.879565	+	0.475779i	$0.157834\pi$
$678$	0	0
$679$	−42.2710	−1.62221
$680$	0	0
$681$	−3.56523	−0.136620
$682$	0	0
$683$	36.4020	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$683$	36.4020	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.67887	0.178510
$688$	0	0
$689$	0.293579	0.0111845
$690$	0	0
$691$	−20.3322	−0.773472	−0.386736	−	0.922190i	$-0.626398\pi$
$691$	−20.3322	−0.773472	−0.386736	+	0.922190i	$0.626398\pi$
$692$	0	0
$693$	−34.9729	−1.32851
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13.3668	−0.506302
$698$	0	0
$699$	4.47329	0.169196
$700$	0	0
$701$	−19.4294	−0.733840	−0.366920	−	0.930253i	$-0.619588\pi$
$701$	−19.4294	−0.733840	−0.366920	+	0.930253i	$0.619588\pi$
$702$	0	0
$703$	2.04607	0.0771692
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.4815	−1.07116
$708$	0	0
$709$	−13.0978	−0.491899	−0.245950	−	0.969283i	$-0.579100\pi$
$709$	−13.0978	−0.491899	−0.245950	+	0.969283i	$0.579100\pi$
$710$	0	0
$711$	−44.3913	−1.66481
$712$	0	0
$713$	70.6545	2.64603
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.805305	0.0300747
$718$	0	0
$719$	0.214142	0.00798615	0.00399308	−	0.999992i	$-0.498729\pi$
$719$	0.214142	0.00798615	0.00399308	+	0.999992i	$0.498729\pi$
$720$	0	0
$721$	−18.4634	−0.687612
$722$	0	0
$723$	4.07455	0.151534
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−10.2004	−0.378310	−0.189155	−	0.981947i	$-0.560575\pi$
$727$	−10.2004	−0.378310	−0.189155	+	0.981947i	$0.560575\pi$
$728$	0	0
$729$	−25.1592	−0.931822
$730$	0	0
$731$	−16.7800	−0.620630
$732$	0	0
$733$	31.0210	1.14579	0.572894	−	0.819630i	$-0.305821\pi$
$733$	31.0210	1.14579	0.572894	+	0.819630i	$0.305821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34.8855	1.28503
$738$	0	0
$739$	35.2371	1.29622	0.648108	−	0.761548i	$-0.275560\pi$
$739$	35.2371	1.29622	0.648108	+	0.761548i	$0.275560\pi$
$740$	0	0
$741$	0.241439	0.00886948
$742$	0	0
$743$	−7.82541	−0.287086	−0.143543	−	0.989644i	$-0.545850\pi$
$743$	−7.82541	−0.287086	−0.143543	+	0.989644i	$0.545850\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.4018	0.417169
$748$	0	0
$749$	−32.4254	−1.18480
$750$	0	0
$751$	−24.2499	−0.884892	−0.442446	−	0.896795i	$-0.645889\pi$
$751$	−24.2499	−0.884892	−0.442446	+	0.896795i	$0.645889\pi$
$752$	0	0
$753$	0.818535	0.0298291
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.4717	−1.32559	−0.662793	−	0.748802i	$-0.730629\pi$
$757$	−36.4717	−1.32559	−0.662793	+	0.748802i	$0.730629\pi$
$758$	0	0
$759$	3.20516	0.116340
$760$	0	0
$761$	24.6495	0.893544	0.446772	−	0.894648i	$-0.352573\pi$
$761$	24.6495	0.893544	0.446772	+	0.894648i	$0.352573\pi$
$762$	0	0
$763$	62.8711	2.27609
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.72280	0.134422
$768$	0	0
$769$	14.3625	0.517927	0.258963	−	0.965887i	$-0.416619\pi$
$769$	14.3625	0.517927	0.258963	+	0.965887i	$0.416619\pi$
$770$	0	0
$771$	2.20976	0.0795824
$772$	0	0
$773$	24.1630	0.869081	0.434541	−	0.900652i	$-0.356911\pi$
$773$	24.1630	0.869081	0.434541	+	0.900652i	$0.356911\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.69163	−0.0606868
$778$	0	0
$779$	3.80044	0.136165
$780$	0	0
$781$	32.6306	1.16762
$782$	0	0
$783$	5.76546	0.206041
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.6852	1.05816	0.529082	−	0.848571i	$-0.322537\pi$
$787$	29.6852	1.05816	0.529082	+	0.848571i	$0.322537\pi$
$788$	0	0
$789$	2.55606	0.0909981
$790$	0	0
$791$	−31.3422	−1.11440
$792$	0	0
$793$	8.21783	0.291824
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.0735	−1.49032	−0.745160	−	0.666886i	$-0.767627\pi$
$797$	−42.0735	−1.49032	−0.745160	+	0.666886i	$0.767627\pi$
$798$	0	0
$799$	−5.24383	−0.185513
$800$	0	0
$801$	−4.97228	−0.175687
$802$	0	0
$803$	14.3498	0.506395
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.427573	−0.0150513
$808$	0	0
$809$	27.8348	0.978620	0.489310	−	0.872110i	$-0.337249\pi$
$809$	27.8348	0.978620	0.489310	+	0.872110i	$0.337249\pi$
$810$	0	0
$811$	14.9245	0.524069	0.262035	−	0.965059i	$-0.415607\pi$
$811$	14.9245	0.524069	0.262035	+	0.965059i	$0.415607\pi$
$812$	0	0
$813$	−3.74522	−0.131351
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.77089	0.166912
$818$	0	0
$819$	17.1998	0.601010
$820$	0	0
$821$	0.503399	0.0175688	0.00878438	−	0.999961i	$-0.497204\pi$
$821$	0.503399	0.0175688	0.00878438	+	0.999961i	$0.497204\pi$
$822$	0	0
$823$	20.7905	0.724713	0.362356	−	0.932040i	$-0.381972\pi$
$823$	20.7905	0.724713	0.362356	+	0.932040i	$0.381972\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.3359	−1.29829	−0.649147	−	0.760663i	$-0.724874\pi$
$827$	−37.3359	−1.29829	−0.649147	+	0.760663i	$0.724874\pi$
$828$	0	0
$829$	−20.5586	−0.714029	−0.357014	−	0.934099i	$-0.616205\pi$
$829$	−20.5586	−0.714029	−0.357014	+	0.934099i	$0.616205\pi$
$830$	0	0
$831$	1.98543	0.0688740
$832$	0	0
$833$	45.2323	1.56721
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−11.9770	−0.413984
$838$	0	0
$839$	−8.13209	−0.280751	−0.140375	−	0.990098i	$-0.544831\pi$
$839$	−8.13209	−0.280751	−0.140375	+	0.990098i	$0.544831\pi$
$840$	0	0
$841$	−1.86150	−0.0641896
$842$	0	0
$843$	−0.424102	−0.0146068
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.8148	−0.612125
$848$	0	0
$849$	4.75884	0.163323
$850$	0	0
$851$	−13.3585	−0.457922
$852$	0	0
$853$	−32.7714	−1.12207	−0.561035	−	0.827792i	$-0.689597\pi$
$853$	−32.7714	−1.12207	−0.561035	+	0.827792i	$0.689597\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.21619	−0.212341	−0.106170	−	0.994348i	$-0.533859\pi$
$857$	−6.21619	−0.212341	−0.106170	+	0.994348i	$0.533859\pi$
$858$	0	0
$859$	21.2194	0.723996	0.361998	−	0.932179i	$-0.382095\pi$
$859$	21.2194	0.723996	0.361998	+	0.932179i	$0.382095\pi$
$860$	0	0
$861$	−3.14208	−0.107082
$862$	0	0
$863$	−19.4411	−0.661784	−0.330892	−	0.943669i	$-0.607350\pi$
$863$	−19.4411	−0.661784	−0.330892	+	0.943669i	$0.607350\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−0.858873	−0.0291689
$868$	0	0
$869$	39.6109	1.34371
$870$	0	0
$871$	−17.1569	−0.581338
$872$	0	0
$873$	28.1292	0.952030
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41.4953	−1.40120	−0.700599	−	0.713555i	$-0.747084\pi$
$877$	−41.4953	−1.40120	−0.700599	+	0.713555i	$0.747084\pi$
$878$	0	0
$879$	−3.63668	−0.122662
$880$	0	0
$881$	−15.6473	−0.527171	−0.263585	−	0.964636i	$-0.584905\pi$
$881$	−15.6473	−0.527171	−0.263585	+	0.964636i	$0.584905\pi$
$882$	0	0
$883$	45.4821	1.53059	0.765297	−	0.643677i	$-0.222592\pi$
$883$	45.4821	1.53059	0.765297	+	0.643677i	$0.222592\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.9690	0.435457	0.217729	−	0.976009i	$-0.430135\pi$
$887$	12.9690	0.435457	0.217729	+	0.976009i	$0.430135\pi$
$888$	0	0
$889$	−60.7155	−2.03633
$890$	0	0
$891$	22.9995	0.770512
$892$	0	0
$893$	1.49093	0.0498919
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.57631	−0.0526315
$898$	0	0
$899$	−56.3765	−1.88026
$900$	0	0
$901$	−0.793411	−0.0264324
$902$	0	0
$903$	−3.94441	−0.131262
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.07053	−0.0687508	−0.0343754	−	0.999409i	$-0.510944\pi$
$907$	−2.07053	−0.0687508	−0.0343754	+	0.999409i	$0.510944\pi$
$908$	0	0
$909$	18.9530	0.628633
$910$	0	0
$911$	25.5409	0.846207	0.423104	−	0.906081i	$-0.360941\pi$
$911$	25.5409	0.846207	0.423104	+	0.906081i	$0.360941\pi$
$912$	0	0
$913$	−10.1739	−0.336708
$914$	0	0
$915$	0	0
$916$	0	0
$917$	99.5337	3.28689
$918$	0	0
$919$	25.0300	0.825664	0.412832	−	0.910807i	$-0.364540\pi$
$919$	25.0300	0.825664	0.412832	+	0.910807i	$0.364540\pi$
$920$	0	0
$921$	5.87500	0.193588
$922$	0	0
$923$	−16.0479	−0.528223
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.2865	0.403540
$928$	0	0
$929$	−16.4704	−0.540377	−0.270188	−	0.962807i	$-0.587086\pi$
$929$	−16.4704	−0.540377	−0.270188	+	0.962807i	$0.587086\pi$
$930$	0	0
$931$	−12.8605	−0.421485
$932$	0	0
$933$	2.76343	0.0904706
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.56119	−0.149008	−0.0745039	−	0.997221i	$-0.523737\pi$
$937$	−4.56119	−0.149008	−0.0745039	+	0.997221i	$0.523737\pi$
$938$	0	0
$939$	0.0534732	0.00174503
$940$	0	0
$941$	−10.3721	−0.338119	−0.169060	−	0.985606i	$-0.554073\pi$
$941$	−10.3721	−0.338119	−0.169060	+	0.985606i	$0.554073\pi$
$942$	0	0
$943$	−24.8124	−0.808002
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.70431	−0.0553828	−0.0276914	−	0.999617i	$-0.508816\pi$
$947$	−1.70431	−0.0553828	−0.0276914	+	0.999617i	$0.508816\pi$
$948$	0	0
$949$	−7.05731	−0.229090
$950$	0	0
$951$	−4.07422	−0.132116
$952$	0	0
$953$	−18.5324	−0.600322	−0.300161	−	0.953888i	$-0.597040\pi$
$953$	−18.5324	−0.600322	−0.300161	+	0.953888i	$0.597040\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.55746	−0.0826708
$958$	0	0
$959$	24.7594	0.799522
$960$	0	0
$961$	86.1144	2.77788
$962$	0	0
$963$	21.5775	0.695325
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−43.7127	−1.40570	−0.702852	−	0.711336i	$-0.748091\pi$
$967$	−43.7127	−1.40570	−0.702852	+	0.711336i	$0.748091\pi$
$968$	0	0
$969$	−0.652501	−0.0209613
$970$	0	0
$971$	−32.5273	−1.04385	−0.521925	−	0.852992i	$-0.674786\pi$
$971$	−32.5273	−1.04385	−0.521925	+	0.852992i	$0.674786\pi$
$972$	0	0
$973$	−49.2230	−1.57802
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.03542	−0.0651187	−0.0325594	−	0.999470i	$-0.510366\pi$
$977$	−2.03542	−0.0651187	−0.0325594	+	0.999470i	$0.510366\pi$
$978$	0	0
$979$	4.43682	0.141802
$980$	0	0
$981$	−41.8376	−1.33577
$982$	0	0
$983$	19.2872	0.615166	0.307583	−	0.951521i	$-0.400480\pi$
$983$	19.2872	0.615166	0.307583	+	0.951521i	$0.400480\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.23265	−0.0392356
$988$	0	0
$989$	−31.1483	−0.990457
$990$	0	0
$991$	−0.611835	−0.0194356	−0.00971778	−	0.999953i	$-0.503093\pi$
$991$	−0.611835	−0.0194356	−0.00971778	+	0.999953i	$0.503093\pi$
$992$	0	0
$993$	4.46924	0.141827
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−58.8380	−1.86342	−0.931709	−	0.363207i	$-0.881682\pi$
$997$	−58.8380	−1.86342	−0.931709	+	0.363207i	$0.881682\pi$
$998$	0	0
$999$	2.26445	0.0716441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.ba.1.4	✓	6
4.3	odd	2		7600.2.a.cl.1.3		6
5.2	odd	4		3800.2.d.q.3649.6		12
5.3	odd	4		3800.2.d.q.3649.7		12
5.4	even	2		3800.2.a.bc.1.3	yes	6
20.19	odd	2		7600.2.a.ch.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.4	✓	6	1.1	even	1	trivial
3800.2.a.bc.1.3	yes	6	5.4	even	2
3800.2.d.q.3649.6		12	5.2	odd	4
3800.2.d.q.3649.7		12	5.3	odd	4
7600.2.a.ch.1.4		6	20.19	odd	2
7600.2.a.cl.1.3		6	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.185519 q^{3} +4.45651 q^{7} -2.96558 q^{9} +O(q^{10})\)	\(q+0.185519 q^{3} +4.45651 q^{7} -2.96558 q^{9} +2.64623 q^{11} -1.30142 q^{13} +3.51716 q^{17} -1.00000 q^{19} +0.826767 q^{21} +6.52882 q^{23} -1.10673 q^{27} -5.20946 q^{29} +10.8219 q^{31} +0.490925 q^{33} -2.04607 q^{37} -0.241439 q^{39} -3.80044 q^{41} -4.77089 q^{43} -1.49093 q^{47} +12.8605 q^{49} +0.652501 q^{51} -0.225583 q^{53} -0.185519 q^{57} -2.86056 q^{59} -6.31449 q^{61} -13.2161 q^{63} +13.1831 q^{67} +1.21122 q^{69} +12.3310 q^{71} +5.42276 q^{73} +11.7929 q^{77} +14.9688 q^{79} +8.69143 q^{81} -3.84470 q^{83} -0.966455 q^{87} +1.67666 q^{89} -5.79981 q^{91} +2.00768 q^{93} -9.48523 q^{97} -7.84760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} + 33 q^{51} - 13 q^{53} + 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} + 19 q^{71} - 3 q^{73} + 36 q^{77} - 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} + 42 q^{91} - 39 q^{93} + 11 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 + 3 * q^11 + q^13 + 14 * q^17 - 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 + 5 * q^31 - 2 * q^33 + 8 * q^37 + 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 + 33 * q^51 - 13 * q^53 + 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 + 19 * q^71 - 3 * q^73 + 36 * q^77 - 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 + 42 * q^91 - 39 * q^93 + 11 * q^97 - 6 * q^99

Introduction

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