LMFDB - Embedded newform 3800.2.a.bd.1.6

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} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2x^5 - 12x^4 + 16x^3 + 33x^2 - 4*x - 7
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			1.6
Root			$3.30105$ 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)

$f(q)$	$=$	$q+3.30105 q^{3} -1.63094 q^{7} +7.89694 q^{9} +O(q^{10})$	$q+3.30105 q^{3} -1.63094 q^{7} +7.89694 q^{9} -4.67473 q^{11} +4.75288 q^{13} +1.41425 q^{17} +1.00000 q^{19} -5.38382 q^{21} +1.96494 q^{23} +16.1650 q^{27} +6.85936 q^{29} +5.08277 q^{31} -15.4315 q^{33} -10.6081 q^{37} +15.6895 q^{39} -4.52536 q^{41} +7.83425 q^{43} +10.9382 q^{47} -4.34003 q^{49} +4.66852 q^{51} +1.55831 q^{53} +3.30105 q^{57} -6.81879 q^{59} -0.109000 q^{61} -12.8794 q^{63} +10.5615 q^{67} +6.48638 q^{69} +12.7070 q^{71} -0.519837 q^{73} +7.62422 q^{77} +0.840485 q^{79} +29.6708 q^{81} -11.9407 q^{83} +22.6431 q^{87} -7.44352 q^{89} -7.75167 q^{91} +16.7785 q^{93} -8.85637 q^{97} -36.9161 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9	$ 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} - 2 q^{17} + 6 q^{19} + 11 q^{21} + 4 q^{23} + 20 q^{27} + 7 q^{29} + 5 q^{31} - 16 q^{33} + 8 q^{39} + 11 q^{41} - 7 q^{43} + 20 q^{47} - 2 q^{49} + 13 q^{51} - 7 q^{53} + 2 q^{57} - 4 q^{59} + 13 q^{61} - q^{63} + 25 q^{67} + 7 q^{69} + 29 q^{71} - 19 q^{73} + 24 q^{77} + 28 q^{79} + 38 q^{81} - 15 q^{83} + 57 q^{87} - 12 q^{89} - 27 q^{93} - 13 q^{97} + 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9 + 3 * q^11 - 3 * q^13 - 2 * q^17 + 6 * q^19 + 11 * q^21 + 4 * q^23 + 20 * q^27 + 7 * q^29 + 5 * q^31 - 16 * q^33 + 8 * q^39 + 11 * q^41 - 7 * q^43 + 20 * q^47 - 2 * q^49 + 13 * q^51 - 7 * q^53 + 2 * q^57 - 4 * q^59 + 13 * q^61 - q^63 + 25 * q^67 + 7 * q^69 + 29 * q^71 - 19 * q^73 + 24 * q^77 + 28 * q^79 + 38 * q^81 - 15 * q^83 + 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 + 10 * 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
$3$	3.30105	1.90586	0.952931	−	0.303186i	$-0.0980505\pi$
$3$	3.30105	1.90586	0.952931	+	0.303186i	$0.0980505\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.63094	−0.616438	−0.308219	−	0.951315i	$-0.599733\pi$
$7$	−1.63094	−0.616438	−0.308219	+	0.951315i	$0.599733\pi$
$8$	0	0
$9$	7.89694	2.63231
$10$	0	0
$11$	−4.67473	−1.40949	−0.704743	−	0.709463i	$-0.748938\pi$
$11$	−4.67473	−1.40949	−0.704743	+	0.709463i	$0.748938\pi$
$12$	0	0
$13$	4.75288	1.31821	0.659106	−	0.752050i	$-0.270935\pi$
$13$	4.75288	1.31821	0.659106	+	0.752050i	$0.270935\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.41425	0.343007	0.171503	−	0.985184i	$-0.445138\pi$
$17$	1.41425	0.343007	0.171503	+	0.985184i	$0.445138\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−5.38382	−1.17485
$22$	0	0
$23$	1.96494	0.409719	0.204860	−	0.978791i	$-0.434326\pi$
$23$	1.96494	0.409719	0.204860	+	0.978791i	$0.434326\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	16.1650	3.11096
$28$	0	0
$29$	6.85936	1.27375	0.636876	−	0.770967i	$-0.280227\pi$
$29$	6.85936	1.27375	0.636876	+	0.770967i	$0.280227\pi$
$30$	0	0
$31$	5.08277	0.912893	0.456446	−	0.889751i	$-0.349122\pi$
$31$	5.08277	0.912893	0.456446	+	0.889751i	$0.349122\pi$
$32$	0	0
$33$	−15.4315	−2.68629
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.6081	−1.74397	−0.871983	−	0.489537i	$-0.837166\pi$
$37$	−10.6081	−1.74397	−0.871983	+	0.489537i	$0.837166\pi$
$38$	0	0
$39$	15.6895	2.51233
$40$	0	0
$41$	−4.52536	−0.706742	−0.353371	−	0.935483i	$-0.614965\pi$
$41$	−4.52536	−0.706742	−0.353371	+	0.935483i	$0.614965\pi$
$42$	0	0
$43$	7.83425	1.19471	0.597356	−	0.801976i	$-0.296218\pi$
$43$	7.83425	1.19471	0.597356	+	0.801976i	$0.296218\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.9382	1.59550	0.797751	−	0.602987i	$-0.206023\pi$
$47$	10.9382	1.59550	0.797751	+	0.602987i	$0.206023\pi$
$48$	0	0
$49$	−4.34003	−0.620004
$50$	0	0
$51$	4.66852	0.653723
$52$	0	0
$53$	1.55831	0.214050	0.107025	−	0.994256i	$-0.465867\pi$
$53$	1.55831	0.214050	0.107025	+	0.994256i	$0.465867\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.30105	0.437235
$58$	0	0
$59$	−6.81879	−0.887731	−0.443865	−	0.896093i	$-0.646393\pi$
$59$	−6.81879	−0.887731	−0.443865	+	0.896093i	$0.646393\pi$
$60$	0	0
$61$	−0.109000	−0.0139561	−0.00697803	−	0.999976i	$-0.502221\pi$
$61$	−0.109000	−0.0139561	−0.00697803	+	0.999976i	$0.502221\pi$
$62$	0	0
$63$	−12.8794	−1.62266
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.5615	1.29030	0.645148	−	0.764057i	$-0.276796\pi$
$67$	10.5615	1.29030	0.645148	+	0.764057i	$0.276796\pi$
$68$	0	0
$69$	6.48638	0.780868
$70$	0	0
$71$	12.7070	1.50804	0.754021	−	0.656850i	$-0.228112\pi$
$71$	12.7070	1.50804	0.754021	+	0.656850i	$0.228112\pi$
$72$	0	0
$73$	−0.519837	−0.0608423	−0.0304211	−	0.999537i	$-0.509685\pi$
$73$	−0.519837	−0.0608423	−0.0304211	+	0.999537i	$0.509685\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.62422	0.868861
$78$	0	0
$79$	0.840485	0.0945620	0.0472810	−	0.998882i	$-0.484944\pi$
$79$	0.840485	0.0945620	0.0472810	+	0.998882i	$0.484944\pi$
$80$	0	0
$81$	29.6708	3.29676
$82$	0	0
$83$	−11.9407	−1.31067	−0.655333	−	0.755340i	$-0.727472\pi$
$83$	−11.9407	−1.31067	−0.655333	+	0.755340i	$0.727472\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	22.6431	2.42759
$88$	0	0
$89$	−7.44352	−0.789011	−0.394506	−	0.918894i	$-0.629084\pi$
$89$	−7.44352	−0.789011	−0.394506	+	0.918894i	$0.629084\pi$
$90$	0	0
$91$	−7.75167	−0.812596
$92$	0	0
$93$	16.7785	1.73985
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.85637	−0.899228	−0.449614	−	0.893223i	$-0.648438\pi$
$97$	−8.85637	−0.899228	−0.449614	+	0.893223i	$0.648438\pi$
$98$	0	0
$99$	−36.9161	−3.71021
$100$	0	0
$101$	−12.9534	−1.28891	−0.644455	−	0.764642i	$-0.722916\pi$
$101$	−12.9534	−1.28891	−0.644455	+	0.764642i	$0.722916\pi$
$102$	0	0
$103$	3.16232	0.311592	0.155796	−	0.987789i	$-0.450206\pi$
$103$	3.16232	0.311592	0.155796	+	0.987789i	$0.450206\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.8458	1.82189	0.910946	−	0.412527i	$-0.135354\pi$
$107$	18.8458	1.82189	0.910946	+	0.412527i	$0.135354\pi$
$108$	0	0
$109$	13.4693	1.29013	0.645063	−	0.764130i	$-0.276831\pi$
$109$	13.4693	1.29013	0.645063	+	0.764130i	$0.276831\pi$
$110$	0	0
$111$	−35.0180	−3.32376
$112$	0	0
$113$	−4.43245	−0.416970	−0.208485	−	0.978026i	$-0.566853\pi$
$113$	−4.43245	−0.416970	−0.208485	+	0.978026i	$0.566853\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	37.5332	3.46994
$118$	0	0
$119$	−2.30656	−0.211442
$120$	0	0
$121$	10.8531	0.986649
$122$	0	0
$123$	−14.9384	−1.34695
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.48637	0.398101	0.199051	−	0.979989i	$-0.436214\pi$
$127$	4.48637	0.398101	0.199051	+	0.979989i	$0.436214\pi$
$128$	0	0
$129$	25.8613	2.27696
$130$	0	0
$131$	−11.1275	−0.972211	−0.486105	−	0.873900i	$-0.661583\pi$
$131$	−11.1275	−0.972211	−0.486105	+	0.873900i	$0.661583\pi$
$132$	0	0
$133$	−1.63094	−0.141421
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.6851	−1.68181	−0.840904	−	0.541184i	$-0.817976\pi$
$137$	−19.6851	−1.68181	−0.840904	+	0.541184i	$0.817976\pi$
$138$	0	0
$139$	9.15699	0.776686	0.388343	−	0.921515i	$-0.373048\pi$
$139$	9.15699	0.776686	0.388343	+	0.921515i	$0.373048\pi$
$140$	0	0
$141$	36.1076	3.04081
$142$	0	0
$143$	−22.2185	−1.85800
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−14.3267	−1.18164
$148$	0	0
$149$	2.68165	0.219689	0.109845	−	0.993949i	$-0.464965\pi$
$149$	2.68165	0.219689	0.109845	+	0.993949i	$0.464965\pi$
$150$	0	0
$151$	18.6183	1.51513	0.757567	−	0.652758i	$-0.226388\pi$
$151$	18.6183	1.51513	0.757567	+	0.652758i	$0.226388\pi$
$152$	0	0
$153$	11.1683	0.902900
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.33933	−0.505934	−0.252967	−	0.967475i	$-0.581406\pi$
$157$	−6.33933	−0.505934	−0.252967	+	0.967475i	$0.581406\pi$
$158$	0	0
$159$	5.14406	0.407950
$160$	0	0
$161$	−3.20471	−0.252566
$162$	0	0
$163$	13.4585	1.05415	0.527074	−	0.849819i	$-0.323289\pi$
$163$	13.4585	1.05415	0.527074	+	0.849819i	$0.323289\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.1325	−1.01622	−0.508112	−	0.861291i	$-0.669656\pi$
$167$	−13.1325	−1.01622	−0.508112	+	0.861291i	$0.669656\pi$
$168$	0	0
$169$	9.58987	0.737682
$170$	0	0
$171$	7.89694	0.603894
$172$	0	0
$173$	0.857255	0.0651759	0.0325879	−	0.999469i	$-0.489625\pi$
$173$	0.857255	0.0651759	0.0325879	+	0.999469i	$0.489625\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−22.5092	−1.69189
$178$	0	0
$179$	8.93269	0.667661	0.333830	−	0.942633i	$-0.391659\pi$
$179$	8.93269	0.667661	0.333830	+	0.942633i	$0.391659\pi$
$180$	0	0
$181$	−6.95157	−0.516706	−0.258353	−	0.966051i	$-0.583180\pi$
$181$	−6.95157	−0.516706	−0.258353	+	0.966051i	$0.583180\pi$
$182$	0	0
$183$	−0.359816	−0.0265983
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.61125	−0.483463
$188$	0	0
$189$	−26.3642	−1.91772
$190$	0	0
$191$	9.36170	0.677389	0.338695	−	0.940896i	$-0.390015\pi$
$191$	9.36170	0.677389	0.338695	+	0.940896i	$0.390015\pi$
$192$	0	0
$193$	−26.7785	−1.92756	−0.963779	−	0.266703i	$-0.914066\pi$
$193$	−26.7785	−1.92756	−0.963779	+	0.266703i	$0.914066\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.92419	0.279587	0.139793	−	0.990181i	$-0.455356\pi$
$197$	3.92419	0.279587	0.139793	+	0.990181i	$0.455356\pi$
$198$	0	0
$199$	−7.52186	−0.533211	−0.266605	−	0.963806i	$-0.585902\pi$
$199$	−7.52186	−0.533211	−0.266605	+	0.963806i	$0.585902\pi$
$200$	0	0
$201$	34.8642	2.45913
$202$	0	0
$203$	−11.1872	−0.785189
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.5170	1.07851
$208$	0	0
$209$	−4.67473	−0.323358
$210$	0	0
$211$	9.97579	0.686761	0.343381	−	0.939196i	$-0.388428\pi$
$211$	9.97579	0.686761	0.343381	+	0.939196i	$0.388428\pi$
$212$	0	0
$213$	41.9464	2.87412
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.28970	−0.562742
$218$	0	0
$219$	−1.71601	−0.115957
$220$	0	0
$221$	6.72177	0.452155
$222$	0	0
$223$	−21.3953	−1.43273	−0.716367	−	0.697724i	$-0.754196\pi$
$223$	−21.3953	−1.43273	−0.716367	+	0.697724i	$0.754196\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.2964	−1.34712	−0.673560	−	0.739132i	$-0.735236\pi$
$227$	−20.2964	−1.34712	−0.673560	+	0.739132i	$0.735236\pi$
$228$	0	0
$229$	19.9525	1.31850	0.659249	−	0.751925i	$-0.270874\pi$
$229$	19.9525	1.31850	0.659249	+	0.751925i	$0.270874\pi$
$230$	0	0
$231$	25.1679	1.65593
$232$	0	0
$233$	13.2102	0.865431	0.432715	−	0.901531i	$-0.357555\pi$
$233$	13.2102	0.865431	0.432715	+	0.901531i	$0.357555\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.77448	0.180222
$238$	0	0
$239$	26.0537	1.68527	0.842636	−	0.538484i	$-0.181003\pi$
$239$	26.0537	1.68527	0.842636	+	0.538484i	$0.181003\pi$
$240$	0	0
$241$	2.12098	0.136624	0.0683122	−	0.997664i	$-0.478239\pi$
$241$	2.12098	0.136624	0.0683122	+	0.997664i	$0.478239\pi$
$242$	0	0
$243$	49.4497	3.17220
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.75288	0.302419
$248$	0	0
$249$	−39.4170	−2.49795
$250$	0	0
$251$	−14.8911	−0.939921	−0.469960	−	0.882688i	$-0.655732\pi$
$251$	−14.8911	−0.939921	−0.469960	+	0.882688i	$0.655732\pi$
$252$	0	0
$253$	−9.18559	−0.577493
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.91006	−0.493416	−0.246708	−	0.969090i	$-0.579349\pi$
$257$	−7.91006	−0.493416	−0.246708	+	0.969090i	$0.579349\pi$
$258$	0	0
$259$	17.3012	1.07505
$260$	0	0
$261$	54.1679	3.35291
$262$	0	0
$263$	−17.1828	−1.05954	−0.529768	−	0.848143i	$-0.677721\pi$
$263$	−17.1828	−1.05954	−0.529768	+	0.848143i	$0.677721\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.5714	−1.50375
$268$	0	0
$269$	−12.9237	−0.787972	−0.393986	−	0.919116i	$-0.628904\pi$
$269$	−12.9237	−0.787972	−0.393986	+	0.919116i	$0.628904\pi$
$270$	0	0
$271$	−20.9897	−1.27503	−0.637515	−	0.770438i	$-0.720038\pi$
$271$	−20.9897	−1.27503	−0.637515	+	0.770438i	$0.720038\pi$
$272$	0	0
$273$	−25.5887	−1.54870
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.27361	−0.376945	−0.188472	−	0.982078i	$-0.560354\pi$
$277$	−6.27361	−0.376945	−0.188472	+	0.982078i	$0.560354\pi$
$278$	0	0
$279$	40.1383	2.40302
$280$	0	0
$281$	−28.7635	−1.71588	−0.857942	−	0.513747i	$-0.828257\pi$
$281$	−28.7635	−1.71588	−0.857942	+	0.513747i	$0.828257\pi$
$282$	0	0
$283$	−28.0145	−1.66529	−0.832644	−	0.553808i	$-0.813174\pi$
$283$	−28.0145	−1.66529	−0.832644	+	0.553808i	$0.813174\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.38059	0.435663
$288$	0	0
$289$	−14.9999	−0.882346
$290$	0	0
$291$	−29.2353	−1.71380
$292$	0	0
$293$	−22.5521	−1.31751	−0.658753	−	0.752359i	$-0.728916\pi$
$293$	−22.5521	−1.31751	−0.658753	+	0.752359i	$0.728916\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−75.5673	−4.38486
$298$	0	0
$299$	9.33914	0.540097
$300$	0	0
$301$	−12.7772	−0.736466
$302$	0	0
$303$	−42.7598	−2.45649
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.5977	−0.833137	−0.416568	−	0.909104i	$-0.636767\pi$
$307$	−14.5977	−0.833137	−0.416568	+	0.909104i	$0.636767\pi$
$308$	0	0
$309$	10.4390	0.593852
$310$	0	0
$311$	27.1608	1.54015	0.770074	−	0.637955i	$-0.220219\pi$
$311$	27.1608	1.54015	0.770074	+	0.637955i	$0.220219\pi$
$312$	0	0
$313$	28.3854	1.60444	0.802219	−	0.597030i	$-0.203653\pi$
$313$	28.3854	1.60444	0.802219	+	0.597030i	$0.203653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.9910	−1.17897	−0.589485	−	0.807779i	$-0.700669\pi$
$317$	−20.9910	−1.17897	−0.589485	+	0.807779i	$0.700669\pi$
$318$	0	0
$319$	−32.0657	−1.79533
$320$	0	0
$321$	62.2109	3.47227
$322$	0	0
$323$	1.41425	0.0786911
$324$	0	0
$325$	0	0
$326$	0	0
$327$	44.4629	2.45880
$328$	0	0
$329$	−17.8396	−0.983528
$330$	0	0
$331$	−1.81631	−0.0998332	−0.0499166	−	0.998753i	$-0.515896\pi$
$331$	−1.81631	−0.0998332	−0.0499166	+	0.998753i	$0.515896\pi$
$332$	0	0
$333$	−83.7717	−4.59066
$334$	0	0
$335$	0	0
$336$	0	0
$337$	25.6967	1.39979	0.699895	−	0.714245i	$-0.253230\pi$
$337$	25.6967	1.39979	0.699895	+	0.714245i	$0.253230\pi$
$338$	0	0
$339$	−14.6317	−0.794687
$340$	0	0
$341$	−23.7606	−1.28671
$342$	0	0
$343$	18.4949	0.998632
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.2851	−0.605814	−0.302907	−	0.953020i	$-0.597957\pi$
$347$	−11.2851	−0.605814	−0.302907	+	0.953020i	$0.597957\pi$
$348$	0	0
$349$	3.62702	0.194150	0.0970750	−	0.995277i	$-0.469051\pi$
$349$	3.62702	0.194150	0.0970750	+	0.995277i	$0.469051\pi$
$350$	0	0
$351$	76.8305	4.10091
$352$	0	0
$353$	−31.6766	−1.68597	−0.842987	−	0.537935i	$-0.819205\pi$
$353$	−31.6766	−1.68597	−0.842987	+	0.537935i	$0.819205\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.61408	−0.402980
$358$	0	0
$359$	−23.4772	−1.23908	−0.619539	−	0.784966i	$-0.712680\pi$
$359$	−23.4772	−1.23908	−0.619539	+	0.784966i	$0.712680\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	35.8268	1.88042
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−32.2584	−1.68387	−0.841937	−	0.539576i	$-0.818584\pi$
$367$	−32.2584	−1.68387	−0.841937	+	0.539576i	$0.818584\pi$
$368$	0	0
$369$	−35.7365	−1.86037
$370$	0	0
$371$	−2.54151	−0.131949
$372$	0	0
$373$	−21.5626	−1.11647	−0.558235	−	0.829683i	$-0.688521\pi$
$373$	−21.5626	−1.11647	−0.558235	+	0.829683i	$0.688521\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.6017	1.67907
$378$	0	0
$379$	12.3881	0.636335	0.318168	−	0.948035i	$-0.396933\pi$
$379$	12.3881	0.636335	0.318168	+	0.948035i	$0.396933\pi$
$380$	0	0
$381$	14.8097	0.758726
$382$	0	0
$383$	6.64117	0.339348	0.169674	−	0.985500i	$-0.445729\pi$
$383$	6.64117	0.339348	0.169674	+	0.985500i	$0.445729\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	61.8666	3.14485
$388$	0	0
$389$	−10.0607	−0.510095	−0.255048	−	0.966928i	$-0.582091\pi$
$389$	−10.0607	−0.510095	−0.255048	+	0.966928i	$0.582091\pi$
$390$	0	0
$391$	2.77893	0.140536
$392$	0	0
$393$	−36.7323	−1.85290
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.64223	−0.333364	−0.166682	−	0.986011i	$-0.553305\pi$
$397$	−6.64223	−0.333364	−0.166682	+	0.986011i	$0.553305\pi$
$398$	0	0
$399$	−5.38382	−0.269528
$400$	0	0
$401$	−16.4296	−0.820456	−0.410228	−	0.911983i	$-0.634551\pi$
$401$	−16.4296	−0.820456	−0.410228	+	0.911983i	$0.634551\pi$
$402$	0	0
$403$	24.1578	1.20339
$404$	0	0
$405$	0	0
$406$	0	0
$407$	49.5902	2.45809
$408$	0	0
$409$	32.6210	1.61300	0.806501	−	0.591232i	$-0.201358\pi$
$409$	32.6210	1.61300	0.806501	+	0.591232i	$0.201358\pi$
$410$	0	0
$411$	−64.9814	−3.20530
$412$	0	0
$413$	11.1211	0.547231
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.2277	1.48026
$418$	0	0
$419$	−17.1286	−0.836786	−0.418393	−	0.908266i	$-0.637406\pi$
$419$	−17.1286	−0.836786	−0.418393	+	0.908266i	$0.637406\pi$
$420$	0	0
$421$	18.5724	0.905162	0.452581	−	0.891723i	$-0.350503\pi$
$421$	18.5724	0.905162	0.452581	+	0.891723i	$0.350503\pi$
$422$	0	0
$423$	86.3783	4.19986
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.177773	0.00860305
$428$	0	0
$429$	−73.3442	−3.54109
$430$	0	0
$431$	2.21969	0.106919	0.0534593	−	0.998570i	$-0.482975\pi$
$431$	2.21969	0.106919	0.0534593	+	0.998570i	$0.482975\pi$
$432$	0	0
$433$	−21.0400	−1.01112	−0.505560	−	0.862791i	$-0.668714\pi$
$433$	−21.0400	−1.01112	−0.505560	+	0.862791i	$0.668714\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.96494	0.0939960
$438$	0	0
$439$	3.27039	0.156087	0.0780435	−	0.996950i	$-0.475133\pi$
$439$	3.27039	0.156087	0.0780435	+	0.996950i	$0.475133\pi$
$440$	0	0
$441$	−34.2729	−1.63204
$442$	0	0
$443$	27.6697	1.31463	0.657314	−	0.753617i	$-0.271693\pi$
$443$	27.6697	1.31463	0.657314	+	0.753617i	$0.271693\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.85226	0.418697
$448$	0	0
$449$	7.69551	0.363174	0.181587	−	0.983375i	$-0.441877\pi$
$449$	7.69551	0.363174	0.181587	+	0.983375i	$0.441877\pi$
$450$	0	0
$451$	21.1548	0.996143
$452$	0	0
$453$	61.4598	2.88764
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.06460	−0.377246	−0.188623	−	0.982050i	$-0.560402\pi$
$457$	−8.06460	−0.377246	−0.188623	+	0.982050i	$0.560402\pi$
$458$	0	0
$459$	22.8614	1.06708
$460$	0	0
$461$	−3.10933	−0.144816	−0.0724079	−	0.997375i	$-0.523068\pi$
$461$	−3.10933	−0.144816	−0.0724079	+	0.997375i	$0.523068\pi$
$462$	0	0
$463$	−12.4068	−0.576592	−0.288296	−	0.957541i	$-0.593089\pi$
$463$	−12.4068	−0.576592	−0.288296	+	0.957541i	$0.593089\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.63093	0.0754706	0.0377353	−	0.999288i	$-0.487986\pi$
$467$	1.63093	0.0754706	0.0377353	+	0.999288i	$0.487986\pi$
$468$	0	0
$469$	−17.2252	−0.795388
$470$	0	0
$471$	−20.9265	−0.964240
$472$	0	0
$473$	−36.6230	−1.68393
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.3059	0.563447
$478$	0	0
$479$	15.4392	0.705433	0.352716	−	0.935730i	$-0.385258\pi$
$479$	15.4392	0.705433	0.352716	+	0.935730i	$0.385258\pi$
$480$	0	0
$481$	−50.4192	−2.29892
$482$	0	0
$483$	−10.5789	−0.481357
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−38.2792	−1.73460	−0.867298	−	0.497789i	$-0.834145\pi$
$487$	−38.2792	−1.73460	−0.867298	+	0.497789i	$0.834145\pi$
$488$	0	0
$489$	44.4271	2.00906
$490$	0	0
$491$	−19.3511	−0.873301	−0.436651	−	0.899631i	$-0.643835\pi$
$491$	−19.3511	−0.873301	−0.436651	+	0.899631i	$0.643835\pi$
$492$	0	0
$493$	9.70086	0.436905
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.7244	−0.929615
$498$	0	0
$499$	18.1775	0.813738	0.406869	−	0.913487i	$-0.366621\pi$
$499$	18.1775	0.813738	0.406869	+	0.913487i	$0.366621\pi$
$500$	0	0
$501$	−43.3511	−1.93678
$502$	0	0
$503$	19.4368	0.866643	0.433321	−	0.901240i	$-0.357342\pi$
$503$	19.4368	0.866643	0.433321	+	0.901240i	$0.357342\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.6566	1.40592
$508$	0	0
$509$	−0.155404	−0.00688816	−0.00344408	−	0.999994i	$-0.501096\pi$
$509$	−0.155404	−0.00688816	−0.00344408	+	0.999994i	$0.501096\pi$
$510$	0	0
$511$	0.847823	0.0375055
$512$	0	0
$513$	16.1650	0.713704
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−51.1332	−2.24884
$518$	0	0
$519$	2.82984	0.124216
$520$	0	0
$521$	−12.7626	−0.559140	−0.279570	−	0.960125i	$-0.590192\pi$
$521$	−12.7626	−0.559140	−0.279570	+	0.960125i	$0.590192\pi$
$522$	0	0
$523$	−9.49811	−0.415323	−0.207662	−	0.978201i	$-0.566585\pi$
$523$	−9.49811	−0.415323	−0.207662	+	0.978201i	$0.566585\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.18832	0.313128
$528$	0	0
$529$	−19.1390	−0.832130
$530$	0	0
$531$	−53.8476	−2.33679
$532$	0	0
$533$	−21.5085	−0.931636
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.4873	1.27247
$538$	0	0
$539$	20.2885	0.873887
$540$	0	0
$541$	−0.840655	−0.0361426	−0.0180713	−	0.999837i	$-0.505753\pi$
$541$	−0.840655	−0.0361426	−0.0180713	+	0.999837i	$0.505753\pi$
$542$	0	0
$543$	−22.9475	−0.984771
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−30.9252	−1.32227	−0.661134	−	0.750268i	$-0.729925\pi$
$547$	−30.9252	−1.32227	−0.661134	+	0.750268i	$0.729925\pi$
$548$	0	0
$549$	−0.860769	−0.0367367
$550$	0	0
$551$	6.85936	0.292219
$552$	0	0
$553$	−1.37078	−0.0582916
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.4896	1.08003	0.540014	−	0.841656i	$-0.318419\pi$
$557$	25.4896	1.08003	0.540014	+	0.841656i	$0.318419\pi$
$558$	0	0
$559$	37.2352	1.57488
$560$	0	0
$561$	−21.8241	−0.921414
$562$	0	0
$563$	−10.2674	−0.432718	−0.216359	−	0.976314i	$-0.569418\pi$
$563$	−10.2674	−0.432718	−0.216359	+	0.976314i	$0.569418\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−48.3913	−2.03225
$568$	0	0
$569$	33.6002	1.40859	0.704296	−	0.709906i	$-0.251263\pi$
$569$	33.6002	1.40859	0.704296	+	0.709906i	$0.251263\pi$
$570$	0	0
$571$	8.89143	0.372095	0.186047	−	0.982541i	$-0.440432\pi$
$571$	8.89143	0.372095	0.186047	+	0.982541i	$0.440432\pi$
$572$	0	0
$573$	30.9035	1.29101
$574$	0	0
$575$	0	0
$576$	0	0
$577$	31.3264	1.30413	0.652067	−	0.758161i	$-0.273902\pi$
$577$	31.3264	1.30413	0.652067	+	0.758161i	$0.273902\pi$
$578$	0	0
$579$	−88.3971	−3.67366
$580$	0	0
$581$	19.4746	0.807944
$582$	0	0
$583$	−7.28468	−0.301701
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−38.7544	−1.59957	−0.799784	−	0.600289i	$-0.795052\pi$
$587$	−38.7544	−1.59957	−0.799784	+	0.600289i	$0.795052\pi$
$588$	0	0
$589$	5.08277	0.209432
$590$	0	0
$591$	12.9539	0.532854
$592$	0	0
$593$	43.7426	1.79629	0.898146	−	0.439696i	$-0.144914\pi$
$593$	43.7426	1.79629	0.898146	+	0.439696i	$0.144914\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.8300	−1.01623
$598$	0	0
$599$	31.2288	1.27597	0.637987	−	0.770047i	$-0.279767\pi$
$599$	31.2288	1.27597	0.637987	+	0.770047i	$0.279767\pi$
$600$	0	0
$601$	−45.1869	−1.84321	−0.921606	−	0.388127i	$-0.873122\pi$
$601$	−45.1869	−1.84321	−0.921606	+	0.388127i	$0.873122\pi$
$602$	0	0
$603$	83.4038	3.39646
$604$	0	0
$605$	0	0
$606$	0	0
$607$	31.3958	1.27432	0.637159	−	0.770733i	$-0.280110\pi$
$607$	31.3958	1.27432	0.637159	+	0.770733i	$0.280110\pi$
$608$	0	0
$609$	−36.9296	−1.49646
$610$	0	0
$611$	51.9880	2.10321
$612$	0	0
$613$	20.7340	0.837439	0.418719	−	0.908116i	$-0.362479\pi$
$613$	20.7340	0.837439	0.418719	+	0.908116i	$0.362479\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.4584	0.421038	0.210519	−	0.977590i	$-0.432485\pi$
$617$	10.4584	0.421038	0.210519	+	0.977590i	$0.432485\pi$
$618$	0	0
$619$	13.9795	0.561883	0.280942	−	0.959725i	$-0.409353\pi$
$619$	13.9795	0.561883	0.280942	+	0.959725i	$0.409353\pi$
$620$	0	0
$621$	31.7634	1.27462
$622$	0	0
$623$	12.1399	0.486377
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−15.4315	−0.616276
$628$	0	0
$629$	−15.0026	−0.598192
$630$	0	0
$631$	13.2899	0.529061	0.264531	−	0.964377i	$-0.414783\pi$
$631$	13.2899	0.529061	0.264531	+	0.964377i	$0.414783\pi$
$632$	0	0
$633$	32.9306	1.30887
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−20.6276	−0.817297
$638$	0	0
$639$	100.346	3.96964
$640$	0	0
$641$	−14.8444	−0.586319	−0.293160	−	0.956064i	$-0.594707\pi$
$641$	−14.8444	−0.586319	−0.293160	+	0.956064i	$0.594707\pi$
$642$	0	0
$643$	−19.8963	−0.784634	−0.392317	−	0.919830i	$-0.628326\pi$
$643$	−19.8963	−0.784634	−0.392317	+	0.919830i	$0.628326\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.90793	−0.389521	−0.194760	−	0.980851i	$-0.562393\pi$
$647$	−9.90793	−0.389521	−0.194760	+	0.980851i	$0.562393\pi$
$648$	0	0
$649$	31.8760	1.25124
$650$	0	0
$651$	−27.3647	−1.07251
$652$	0	0
$653$	−5.31852	−0.208130	−0.104065	−	0.994571i	$-0.533185\pi$
$653$	−5.31852	−0.208130	−0.104065	+	0.994571i	$0.533185\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.10512	−0.160156
$658$	0	0
$659$	−6.86569	−0.267449	−0.133725	−	0.991019i	$-0.542694\pi$
$659$	−6.86569	−0.267449	−0.133725	+	0.991019i	$0.542694\pi$
$660$	0	0
$661$	48.7637	1.89669	0.948344	−	0.317243i	$-0.102757\pi$
$661$	48.7637	1.89669	0.948344	+	0.317243i	$0.102757\pi$
$662$	0	0
$663$	22.1889	0.861746
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.4783	0.521880
$668$	0	0
$669$	−70.6269	−2.73059
$670$	0	0
$671$	0.509548	0.0196709
$672$	0	0
$673$	25.8294	0.995651	0.497825	−	0.867277i	$-0.334132\pi$
$673$	25.8294	0.995651	0.497825	+	0.867277i	$0.334132\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.58131	−0.137641	−0.0688205	−	0.997629i	$-0.521924\pi$
$677$	−3.58131	−0.137641	−0.0688205	+	0.997629i	$0.521924\pi$
$678$	0	0
$679$	14.4442	0.554318
$680$	0	0
$681$	−66.9995	−2.56743
$682$	0	0
$683$	−36.1449	−1.38304	−0.691522	−	0.722355i	$-0.743060\pi$
$683$	−36.1449	−1.38304	−0.691522	+	0.722355i	$0.743060\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	65.8642	2.51288
$688$	0	0
$689$	7.40645	0.282163
$690$	0	0
$691$	−24.0392	−0.914493	−0.457246	−	0.889340i	$-0.651164\pi$
$691$	−24.0392	−0.914493	−0.457246	+	0.889340i	$0.651164\pi$
$692$	0	0
$693$	60.2080	2.28711
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.40000	−0.242417
$698$	0	0
$699$	43.6076	1.64939
$700$	0	0
$701$	−14.5851	−0.550872	−0.275436	−	0.961319i	$-0.588822\pi$
$701$	−14.5851	−0.550872	−0.275436	+	0.961319i	$0.588822\pi$
$702$	0	0
$703$	−10.6081	−0.400093
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.1262	0.794533
$708$	0	0
$709$	46.5636	1.74873	0.874366	−	0.485268i	$-0.161278\pi$
$709$	46.5636	1.74873	0.874366	+	0.485268i	$0.161278\pi$
$710$	0	0
$711$	6.63726	0.248917
$712$	0	0
$713$	9.98736	0.374030
$714$	0	0
$715$	0	0
$716$	0	0
$717$	86.0045	3.21190
$718$	0	0
$719$	4.36765	0.162886	0.0814429	−	0.996678i	$-0.474047\pi$
$719$	4.36765	0.162886	0.0814429	+	0.996678i	$0.474047\pi$
$720$	0	0
$721$	−5.15756	−0.192077
$722$	0	0
$723$	7.00147	0.260387
$724$	0	0
$725$	0	0
$726$	0	0
$727$	49.3720	1.83111	0.915553	−	0.402198i	$-0.131754\pi$
$727$	49.3720	1.83111	0.915553	+	0.402198i	$0.131754\pi$
$728$	0	0
$729$	74.2236	2.74902
$730$	0	0
$731$	11.0796	0.409794
$732$	0	0
$733$	28.3530	1.04724	0.523621	−	0.851952i	$-0.324581\pi$
$733$	28.3530	1.04724	0.523621	+	0.851952i	$0.324581\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−49.3724	−1.81865
$738$	0	0
$739$	25.4558	0.936408	0.468204	−	0.883620i	$-0.344901\pi$
$739$	25.4558	0.936408	0.468204	+	0.883620i	$0.344901\pi$
$740$	0	0
$741$	15.6895	0.576368
$742$	0	0
$743$	7.99036	0.293138	0.146569	−	0.989200i	$-0.453177\pi$
$743$	7.99036	0.293138	0.146569	+	0.989200i	$0.453177\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−94.2952	−3.45008
$748$	0	0
$749$	−30.7364	−1.12308
$750$	0	0
$751$	−50.6911	−1.84974	−0.924872	−	0.380279i	$-0.875828\pi$
$751$	−50.6911	−1.84974	−0.924872	+	0.380279i	$0.875828\pi$
$752$	0	0
$753$	−49.1564	−1.79136
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.449092	0.0163225	0.00816127	−	0.999967i	$-0.497402\pi$
$757$	0.449092	0.0163225	0.00816127	+	0.999967i	$0.497402\pi$
$758$	0	0
$759$	−30.3221	−1.10062
$760$	0	0
$761$	32.9995	1.19623	0.598115	−	0.801410i	$-0.295917\pi$
$761$	32.9995	1.19623	0.598115	+	0.801410i	$0.295917\pi$
$762$	0	0
$763$	−21.9676	−0.795282
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.4089	−1.17022
$768$	0	0
$769$	7.87810	0.284091	0.142046	−	0.989860i	$-0.454632\pi$
$769$	7.87810	0.284091	0.142046	+	0.989860i	$0.454632\pi$
$770$	0	0
$771$	−26.1115	−0.940383
$772$	0	0
$773$	−17.7424	−0.638149	−0.319074	−	0.947730i	$-0.603372\pi$
$773$	−17.7424	−0.638149	−0.319074	+	0.947730i	$0.603372\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	57.1123	2.04889
$778$	0	0
$779$	−4.52536	−0.162138
$780$	0	0
$781$	−59.4018	−2.12556
$782$	0	0
$783$	110.882	3.96259
$784$	0	0
$785$	0	0
$786$	0	0
$787$	47.7349	1.70156	0.850782	−	0.525519i	$-0.176129\pi$
$787$	47.7349	1.70156	0.850782	+	0.525519i	$0.176129\pi$
$788$	0	0
$789$	−56.7213	−2.01933
$790$	0	0
$791$	7.22907	0.257036
$792$	0	0
$793$	−0.518066	−0.0183971
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.5233	−0.620706	−0.310353	−	0.950621i	$-0.600447\pi$
$797$	−17.5233	−0.620706	−0.310353	+	0.950621i	$0.600447\pi$
$798$	0	0
$799$	15.4694	0.547268
$800$	0	0
$801$	−58.7810	−2.07692
$802$	0	0
$803$	2.43010	0.0857563
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−42.6618	−1.50177
$808$	0	0
$809$	41.8643	1.47187	0.735936	−	0.677051i	$-0.236742\pi$
$809$	41.8643	1.47187	0.735936	+	0.677051i	$0.236742\pi$
$810$	0	0
$811$	−29.8960	−1.04979	−0.524895	−	0.851167i	$-0.675895\pi$
$811$	−29.8960	−1.04979	−0.524895	+	0.851167i	$0.675895\pi$
$812$	0	0
$813$	−69.2879	−2.43003
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.83425	0.274086
$818$	0	0
$819$	−61.2144	−2.13901
$820$	0	0
$821$	46.3602	1.61798	0.808992	−	0.587820i	$-0.200014\pi$
$821$	46.3602	1.61798	0.808992	+	0.587820i	$0.200014\pi$
$822$	0	0
$823$	−37.3940	−1.30347	−0.651737	−	0.758445i	$-0.725960\pi$
$823$	−37.3940	−1.30347	−0.651737	+	0.758445i	$0.725960\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.68450	−0.336763	−0.168382	−	0.985722i	$-0.553854\pi$
$827$	−9.68450	−0.336763	−0.168382	+	0.985722i	$0.553854\pi$
$828$	0	0
$829$	16.8732	0.586031	0.293016	−	0.956108i	$-0.405341\pi$
$829$	16.8732	0.586031	0.293016	+	0.956108i	$0.405341\pi$
$830$	0	0
$831$	−20.7095	−0.718405
$832$	0	0
$833$	−6.13790	−0.212665
$834$	0	0
$835$	0	0
$836$	0	0
$837$	82.1632	2.83997
$838$	0	0
$839$	−35.3160	−1.21924	−0.609622	−	0.792692i	$-0.708679\pi$
$839$	−35.3160	−1.21924	−0.609622	+	0.792692i	$0.708679\pi$
$840$	0	0
$841$	18.0508	0.622442
$842$	0	0
$843$	−94.9496	−3.27024
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.7008	−0.608208
$848$	0	0
$849$	−92.4773	−3.17381
$850$	0	0
$851$	−20.8444	−0.714536
$852$	0	0
$853$	−18.3492	−0.628265	−0.314133	−	0.949379i	$-0.601714\pi$
$853$	−18.3492	−0.628265	−0.314133	+	0.949379i	$0.601714\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.4209	0.868361	0.434181	−	0.900826i	$-0.357038\pi$
$857$	25.4209	0.868361	0.434181	+	0.900826i	$0.357038\pi$
$858$	0	0
$859$	27.2820	0.930849	0.465425	−	0.885088i	$-0.345902\pi$
$859$	27.2820	0.930849	0.465425	+	0.885088i	$0.345902\pi$
$860$	0	0
$861$	24.3637	0.830313
$862$	0	0
$863$	−3.63233	−0.123646	−0.0618230	−	0.998087i	$-0.519691\pi$
$863$	−3.63233	−0.123646	−0.0618230	+	0.998087i	$0.519691\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−49.5154	−1.68163
$868$	0	0
$869$	−3.92904	−0.133284
$870$	0	0
$871$	50.1977	1.70088
$872$	0	0
$873$	−69.9382	−2.36705
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−57.1986	−1.93146	−0.965731	−	0.259547i	$-0.916427\pi$
$877$	−57.1986	−1.93146	−0.965731	+	0.259547i	$0.916427\pi$
$878$	0	0
$879$	−74.4455	−2.51099
$880$	0	0
$881$	−48.3846	−1.63012	−0.815059	−	0.579378i	$-0.803296\pi$
$881$	−48.3846	−1.63012	−0.815059	+	0.579378i	$0.803296\pi$
$882$	0	0
$883$	−17.6993	−0.595630	−0.297815	−	0.954624i	$-0.596258\pi$
$883$	−17.6993	−0.595630	−0.297815	+	0.954624i	$0.596258\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.1640	1.34858	0.674288	−	0.738468i	$-0.264451\pi$
$887$	40.1640	1.34858	0.674288	+	0.738468i	$0.264451\pi$
$888$	0	0
$889$	−7.31701	−0.245405
$890$	0	0
$891$	−138.703	−4.64673
$892$	0	0
$893$	10.9382	0.366033
$894$	0	0
$895$	0	0
$896$	0	0
$897$	30.8290	1.02935
$898$	0	0
$899$	34.8646	1.16280
$900$	0	0
$901$	2.20384	0.0734206
$902$	0	0
$903$	−42.1782	−1.40360
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−12.5765	−0.417596	−0.208798	−	0.977959i	$-0.566955\pi$
$907$	−12.5765	−0.417596	−0.208798	+	0.977959i	$0.566955\pi$
$908$	0	0
$909$	−102.292	−3.39281
$910$	0	0
$911$	−29.7278	−0.984927	−0.492464	−	0.870333i	$-0.663904\pi$
$911$	−29.7278	−0.984927	−0.492464	+	0.870333i	$0.663904\pi$
$912$	0	0
$913$	55.8197	1.84736
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.1482	0.599308
$918$	0	0
$919$	22.1146	0.729492	0.364746	−	0.931107i	$-0.381156\pi$
$919$	22.1146	0.729492	0.364746	+	0.931107i	$0.381156\pi$
$920$	0	0
$921$	−48.1879	−1.58784
$922$	0	0
$923$	60.3948	1.98792
$924$	0	0
$925$	0	0
$926$	0	0
$927$	24.9726	0.820208
$928$	0	0
$929$	−23.4746	−0.770175	−0.385088	−	0.922880i	$-0.625829\pi$
$929$	−23.4746	−0.770175	−0.385088	+	0.922880i	$0.625829\pi$
$930$	0	0
$931$	−4.34003	−0.142239
$932$	0	0
$933$	89.6592	2.93531
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.29393	0.238282	0.119141	−	0.992877i	$-0.461986\pi$
$937$	7.29393	0.238282	0.119141	+	0.992877i	$0.461986\pi$
$938$	0	0
$939$	93.7016	3.05784
$940$	0	0
$941$	−36.3844	−1.18610	−0.593048	−	0.805167i	$-0.702076\pi$
$941$	−36.3844	−1.18610	−0.593048	+	0.805167i	$0.702076\pi$
$942$	0	0
$943$	−8.89207	−0.289566
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.77953	0.220305	0.110152	−	0.993915i	$-0.464866\pi$
$947$	6.77953	0.220305	0.110152	+	0.993915i	$0.464866\pi$
$948$	0	0
$949$	−2.47072	−0.0802030
$950$	0	0
$951$	−69.2922	−2.24695
$952$	0	0
$953$	9.00668	0.291755	0.145877	−	0.989303i	$-0.453399\pi$
$953$	9.00668	0.291755	0.145877	+	0.989303i	$0.453399\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−105.850	−3.42166
$958$	0	0
$959$	32.1052	1.03673
$960$	0	0
$961$	−5.16544	−0.166627
$962$	0	0
$963$	148.824	4.79579
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.251168	−0.00807702	−0.00403851	−	0.999992i	$-0.501286\pi$
$967$	−0.251168	−0.00807702	−0.00403851	+	0.999992i	$0.501286\pi$
$968$	0	0
$969$	4.66852	0.149974
$970$	0	0
$971$	−9.76919	−0.313508	−0.156754	−	0.987638i	$-0.550103\pi$
$971$	−9.76919	−0.313508	−0.156754	+	0.987638i	$0.550103\pi$
$972$	0	0
$973$	−14.9345	−0.478779
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.8438	−1.21073	−0.605365	−	0.795948i	$-0.706973\pi$
$977$	−37.8438	−1.21073	−0.605365	+	0.795948i	$0.706973\pi$
$978$	0	0
$979$	34.7965	1.11210
$980$	0	0
$981$	106.366	3.39601
$982$	0	0
$983$	−32.0062	−1.02084	−0.510420	−	0.859925i	$-0.670510\pi$
$983$	−32.0062	−1.02084	−0.510420	+	0.859925i	$0.670510\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−58.8894	−1.87447
$988$	0	0
$989$	15.3939	0.489496
$990$	0	0
$991$	−14.8408	−0.471432	−0.235716	−	0.971822i	$-0.575744\pi$
$991$	−14.8408	−0.471432	−0.235716	+	0.971822i	$0.575744\pi$
$992$	0	0
$993$	−5.99572	−0.190268
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.6666	1.47795	0.738973	−	0.673735i	$-0.235311\pi$
$997$	46.6666	1.47795	0.738973	+	0.673735i	$0.235311\pi$
$998$	0	0
$999$	−171.481	−5.42541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.bd.1.6	yes	6
4.3	odd	2		7600.2.a.ci.1.1		6
5.2	odd	4		3800.2.d.p.3649.1		12
5.3	odd	4		3800.2.d.p.3649.12		12
5.4	even	2		3800.2.a.bb.1.1	✓	6
20.19	odd	2		7600.2.a.cm.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.1	✓	6	5.4	even	2
3800.2.a.bd.1.6	yes	6	1.1	even	1	trivial
3800.2.d.p.3649.1		12	5.2	odd	4
3800.2.d.p.3649.12		12	5.3	odd	4
7600.2.a.ci.1.1		6	4.3	odd	2
7600.2.a.cm.1.6		6	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+3.30105 q^{3} -1.63094 q^{7} +7.89694 q^{9} +O(q^{10})\)	\(q+3.30105 q^{3} -1.63094 q^{7} +7.89694 q^{9} -4.67473 q^{11} +4.75288 q^{13} +1.41425 q^{17} +1.00000 q^{19} -5.38382 q^{21} +1.96494 q^{23} +16.1650 q^{27} +6.85936 q^{29} +5.08277 q^{31} -15.4315 q^{33} -10.6081 q^{37} +15.6895 q^{39} -4.52536 q^{41} +7.83425 q^{43} +10.9382 q^{47} -4.34003 q^{49} +4.66852 q^{51} +1.55831 q^{53} +3.30105 q^{57} -6.81879 q^{59} -0.109000 q^{61} -12.8794 q^{63} +10.5615 q^{67} +6.48638 q^{69} +12.7070 q^{71} -0.519837 q^{73} +7.62422 q^{77} +0.840485 q^{79} +29.6708 q^{81} -11.9407 q^{83} +22.6431 q^{87} -7.44352 q^{89} -7.75167 q^{91} +16.7785 q^{93} -8.85637 q^{97} -36.9161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9	\( 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} - 2 q^{17} + 6 q^{19} + 11 q^{21} + 4 q^{23} + 20 q^{27} + 7 q^{29} + 5 q^{31} - 16 q^{33} + 8 q^{39} + 11 q^{41} - 7 q^{43} + 20 q^{47} - 2 q^{49} + 13 q^{51} - 7 q^{53} + 2 q^{57} - 4 q^{59} + 13 q^{61} - q^{63} + 25 q^{67} + 7 q^{69} + 29 q^{71} - 19 q^{73} + 24 q^{77} + 28 q^{79} + 38 q^{81} - 15 q^{83} + 57 q^{87} - 12 q^{89} - 27 q^{93} - 13 q^{97} + 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 2 * q^7 + 10 * q^9 + 3 * q^11 - 3 * q^13 - 2 * q^17 + 6 * q^19 + 11 * q^21 + 4 * q^23 + 20 * q^27 + 7 * q^29 + 5 * q^31 - 16 * q^33 + 8 * q^39 + 11 * q^41 - 7 * q^43 + 20 * q^47 - 2 * q^49 + 13 * q^51 - 7 * q^53 + 2 * q^57 - 4 * q^59 + 13 * q^61 - q^63 + 25 * q^67 + 7 * q^69 + 29 * q^71 - 19 * q^73 + 24 * q^77 + 28 * q^79 + 38 * q^81 - 15 * q^83 + 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more