LMFDB - Embedded newform 2800.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	2800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} -4.89898 q^{11} +0.449490 q^{13} -2.00000 q^{17} -6.44949 q^{19} -2.44949 q^{21} -6.89898 q^{23} -2.89898 q^{29} +0.898979 q^{31} -12.0000 q^{33} -2.00000 q^{37} +1.10102 q^{39} -10.8990 q^{41} +8.89898 q^{43} -0.898979 q^{47} +1.00000 q^{49} -4.89898 q^{51} +1.10102 q^{53} -15.7980 q^{57} +6.44949 q^{59} +8.44949 q^{61} -3.00000 q^{63} -8.00000 q^{67} -16.8990 q^{69} +10.8990 q^{71} +6.89898 q^{73} +4.89898 q^{77} +2.89898 q^{79} -9.00000 q^{81} +2.44949 q^{83} -7.10102 q^{87} -10.0000 q^{89} -0.449490 q^{91} +2.20204 q^{93} +3.79796 q^{97} -14.6969 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^7 + 6 * q^9	$ 2 q - 2 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} - 8 q^{19} - 4 q^{23} + 4 q^{29} - 8 q^{31} - 24 q^{33} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 8 q^{43} + 8 q^{47} + 2 q^{49} + 12 q^{53} - 12 q^{57} + 8 q^{59} + 12 q^{61} - 6 q^{63} - 16 q^{67} - 24 q^{69} + 12 q^{71} + 4 q^{73} - 4 q^{79} - 18 q^{81} - 24 q^{87} - 20 q^{89} + 4 q^{91} + 24 q^{93} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 - 8 * q^19 - 4 * q^23 + 4 * q^29 - 8 * q^31 - 24 * q^33 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 8 * q^43 + 8 * q^47 + 2 * q^49 + 12 * q^53 - 12 * q^57 + 8 * q^59 + 12 * q^61 - 6 * q^63 - 16 * q^67 - 24 * q^69 + 12 * q^71 + 4 * q^73 - 4 * q^79 - 18 * q^81 - 24 * q^87 - 20 * q^89 + 4 * q^91 + 24 * q^93 - 12 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	0.449490	0.124666	0.0623330	−	0.998055i	$-0.480146\pi$
$13$	0.449490	0.124666	0.0623330	+	0.998055i	$0.480146\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−6.44949	−1.47961	−0.739807	−	0.672819i	$-0.765083\pi$
$19$	−6.44949	−1.47961	−0.739807	+	0.672819i	$0.765083\pi$
$20$	0	0
$21$	−2.44949	−0.534522
$22$	0	0
$23$	−6.89898	−1.43854	−0.719268	−	0.694732i	$-0.755523\pi$
$23$	−6.89898	−1.43854	−0.719268	+	0.694732i	$0.755523\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.89898	−0.538327	−0.269163	−	0.963095i	$-0.586747\pi$
$29$	−2.89898	−0.538327	−0.269163	+	0.963095i	$0.586747\pi$
$30$	0	0
$31$	0.898979	0.161461	0.0807307	−	0.996736i	$-0.474275\pi$
$31$	0.898979	0.161461	0.0807307	+	0.996736i	$0.474275\pi$
$32$	0	0
$33$	−12.0000	−2.08893
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	1.10102	0.176304
$40$	0	0
$41$	−10.8990	−1.70213	−0.851067	−	0.525057i	$-0.824044\pi$
$41$	−10.8990	−1.70213	−0.851067	+	0.525057i	$0.824044\pi$
$42$	0	0
$43$	8.89898	1.35708	0.678541	−	0.734563i	$-0.262613\pi$
$43$	8.89898	1.35708	0.678541	+	0.734563i	$0.262613\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.898979	−0.131130	−0.0655648	−	0.997848i	$-0.520885\pi$
$47$	−0.898979	−0.131130	−0.0655648	+	0.997848i	$0.520885\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.89898	−0.685994
$52$	0	0
$53$	1.10102	0.151237	0.0756184	−	0.997137i	$-0.475907\pi$
$53$	1.10102	0.151237	0.0756184	+	0.997137i	$0.475907\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−15.7980	−2.09249
$58$	0	0
$59$	6.44949	0.839652	0.419826	−	0.907605i	$-0.362091\pi$
$59$	6.44949	0.839652	0.419826	+	0.907605i	$0.362091\pi$
$60$	0	0
$61$	8.44949	1.08185	0.540923	−	0.841072i	$-0.318075\pi$
$61$	8.44949	1.08185	0.540923	+	0.841072i	$0.318075\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−16.8990	−2.03440
$70$	0	0
$71$	10.8990	1.29347	0.646735	−	0.762714i	$-0.276134\pi$
$71$	10.8990	1.29347	0.646735	+	0.762714i	$0.276134\pi$
$72$	0	0
$73$	6.89898	0.807464	0.403732	−	0.914877i	$-0.367713\pi$
$73$	6.89898	0.807464	0.403732	+	0.914877i	$0.367713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.89898	0.558291
$78$	0	0
$79$	2.89898	0.326161	0.163080	−	0.986613i	$-0.447857\pi$
$79$	2.89898	0.326161	0.163080	+	0.986613i	$0.447857\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	2.44949	0.268866	0.134433	−	0.990923i	$-0.457079\pi$
$83$	2.44949	0.268866	0.134433	+	0.990923i	$0.457079\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.10102	−0.761309
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−0.449490	−0.0471193
$92$	0	0
$93$	2.20204	0.228341
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.79796	0.385624	0.192812	−	0.981236i	$-0.438239\pi$
$97$	3.79796	0.385624	0.192812	+	0.981236i	$0.438239\pi$
$98$	0	0
$99$	−14.6969	−1.47710
$100$	0	0
$101$	8.44949	0.840756	0.420378	−	0.907349i	$-0.361898\pi$
$101$	8.44949	0.840756	0.420378	+	0.907349i	$0.361898\pi$
$102$	0	0
$103$	3.10102	0.305553	0.152776	−	0.988261i	$-0.451179\pi$
$103$	3.10102	0.305553	0.152776	+	0.988261i	$0.451179\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−2.89898	−0.277672	−0.138836	−	0.990315i	$-0.544336\pi$
$109$	−2.89898	−0.277672	−0.138836	+	0.990315i	$0.544336\pi$
$110$	0	0
$111$	−4.89898	−0.464991
$112$	0	0
$113$	−0.202041	−0.0190064	−0.00950321	−	0.999955i	$-0.503025\pi$
$113$	−0.202041	−0.0190064	−0.00950321	+	0.999955i	$0.503025\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.34847	0.124666
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	13.0000	1.18182
$122$	0	0
$123$	−26.6969	−2.40718
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.10102	−0.452642	−0.226321	−	0.974053i	$-0.572670\pi$
$127$	−5.10102	−0.452642	−0.226321	+	0.974053i	$0.572670\pi$
$128$	0	0
$129$	21.7980	1.91920
$130$	0	0
$131$	1.55051	0.135469	0.0677344	−	0.997703i	$-0.478423\pi$
$131$	1.55051	0.135469	0.0677344	+	0.997703i	$0.478423\pi$
$132$	0	0
$133$	6.44949	0.559242
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.7980	−1.52058	−0.760291	−	0.649582i	$-0.774944\pi$
$137$	−17.7980	−1.52058	−0.760291	+	0.649582i	$0.774944\pi$
$138$	0	0
$139$	−6.44949	−0.547039	−0.273519	−	0.961867i	$-0.588188\pi$
$139$	−6.44949	−0.547039	−0.273519	+	0.961867i	$0.588188\pi$
$140$	0	0
$141$	−2.20204	−0.185445
$142$	0	0
$143$	−2.20204	−0.184144
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.44949	0.202031
$148$	0	0
$149$	15.7980	1.29422	0.647110	−	0.762397i	$-0.275978\pi$
$149$	15.7980	1.29422	0.647110	+	0.762397i	$0.275978\pi$
$150$	0	0
$151$	19.5959	1.59469	0.797347	−	0.603522i	$-0.206236\pi$
$151$	19.5959	1.59469	0.797347	+	0.603522i	$0.206236\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.44949	−0.674343	−0.337171	−	0.941443i	$-0.609470\pi$
$157$	−8.44949	−0.674343	−0.337171	+	0.941443i	$0.609470\pi$
$158$	0	0
$159$	2.69694	0.213881
$160$	0	0
$161$	6.89898	0.543716
$162$	0	0
$163$	−16.8990	−1.32363	−0.661815	−	0.749667i	$-0.730214\pi$
$163$	−16.8990	−1.32363	−0.661815	+	0.749667i	$0.730214\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.89898	0.379094	0.189547	−	0.981872i	$-0.439298\pi$
$167$	4.89898	0.379094	0.189547	+	0.981872i	$0.439298\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	−19.3485	−1.47961
$172$	0	0
$173$	−18.2474	−1.38733	−0.693664	−	0.720299i	$-0.744005\pi$
$173$	−18.2474	−1.38733	−0.693664	+	0.720299i	$0.744005\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.7980	1.18745
$178$	0	0
$179$	5.79796	0.433360	0.216680	−	0.976243i	$-0.430477\pi$
$179$	5.79796	0.433360	0.216680	+	0.976243i	$0.430477\pi$
$180$	0	0
$181$	14.2474	1.05900	0.529502	−	0.848309i	$-0.322379\pi$
$181$	14.2474	1.05900	0.529502	+	0.848309i	$0.322379\pi$
$182$	0	0
$183$	20.6969	1.52996
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.79796	0.716498
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.6969	1.20815	0.604074	−	0.796928i	$-0.293543\pi$
$191$	16.6969	1.20815	0.604074	+	0.796928i	$0.293543\pi$
$192$	0	0
$193$	−17.5959	−1.26658	−0.633291	−	0.773914i	$-0.718296\pi$
$193$	−17.5959	−1.26658	−0.633291	+	0.773914i	$0.718296\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.10102	−0.648421	−0.324210	−	0.945985i	$-0.605099\pi$
$197$	−9.10102	−0.648421	−0.324210	+	0.945985i	$0.605099\pi$
$198$	0	0
$199$	−7.10102	−0.503378	−0.251689	−	0.967808i	$-0.580986\pi$
$199$	−7.10102	−0.503378	−0.251689	+	0.967808i	$0.580986\pi$
$200$	0	0
$201$	−19.5959	−1.38219
$202$	0	0
$203$	2.89898	0.203468
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−20.6969	−1.43854
$208$	0	0
$209$	31.5959	2.18554
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	26.6969	1.82924
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.898979	−0.0610267
$218$	0	0
$219$	16.8990	1.14193
$220$	0	0
$221$	−0.898979	−0.0604719
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.34847	−0.487735	−0.243868	−	0.969809i	$-0.578416\pi$
$227$	−7.34847	−0.487735	−0.243868	+	0.969809i	$0.578416\pi$
$228$	0	0
$229$	15.1464	1.00090	0.500452	−	0.865764i	$-0.333167\pi$
$229$	15.1464	1.00090	0.500452	+	0.865764i	$0.333167\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	−10.2020	−0.668358	−0.334179	−	0.942510i	$-0.608459\pi$
$233$	−10.2020	−0.668358	−0.334179	+	0.942510i	$0.608459\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.10102	0.461261
$238$	0	0
$239$	25.7980	1.66873	0.834366	−	0.551211i	$-0.185834\pi$
$239$	25.7980	1.66873	0.834366	+	0.551211i	$0.185834\pi$
$240$	0	0
$241$	20.6969	1.33321	0.666604	−	0.745412i	$-0.267747\pi$
$241$	20.6969	1.33321	0.666604	+	0.745412i	$0.267747\pi$
$242$	0	0
$243$	−22.0454	−1.41421
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.89898	−0.184458
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	1.55051	0.0978673	0.0489337	−	0.998802i	$-0.484418\pi$
$251$	1.55051	0.0978673	0.0489337	+	0.998802i	$0.484418\pi$
$252$	0	0
$253$	33.7980	2.12486
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.6969	−1.29104	−0.645520	−	0.763744i	$-0.723359\pi$
$257$	−20.6969	−1.29104	−0.645520	+	0.763744i	$0.723359\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−8.69694	−0.538327
$262$	0	0
$263$	−9.79796	−0.604168	−0.302084	−	0.953281i	$-0.597682\pi$
$263$	−9.79796	−0.604168	−0.302084	+	0.953281i	$0.597682\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.4949	−1.49906
$268$	0	0
$269$	−15.1464	−0.923494	−0.461747	−	0.887012i	$-0.652777\pi$
$269$	−15.1464	−0.923494	−0.461747	+	0.887012i	$0.652777\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	−1.10102	−0.0666368
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.10102	0.306491	0.153245	−	0.988188i	$-0.451028\pi$
$277$	5.10102	0.306491	0.153245	+	0.988188i	$0.451028\pi$
$278$	0	0
$279$	2.69694	0.161461
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	28.2474	1.67914	0.839568	−	0.543254i	$-0.182808\pi$
$283$	28.2474	1.67914	0.839568	+	0.543254i	$0.182808\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.8990	0.643346
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	9.30306	0.545355
$292$	0	0
$293$	6.24745	0.364980	0.182490	−	0.983208i	$-0.441584\pi$
$293$	6.24745	0.364980	0.182490	+	0.983208i	$0.441584\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.10102	−0.179337
$300$	0	0
$301$	−8.89898	−0.512929
$302$	0	0
$303$	20.6969	1.18901
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.24745	0.242415	0.121207	−	0.992627i	$-0.461323\pi$
$307$	4.24745	0.242415	0.121207	+	0.992627i	$0.461323\pi$
$308$	0	0
$309$	7.59592	0.432117
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−17.5959	−0.994580	−0.497290	−	0.867584i	$-0.665672\pi$
$313$	−17.5959	−0.994580	−0.497290	+	0.867584i	$0.665672\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.4949	−1.48810	−0.744051	−	0.668123i	$-0.767098\pi$
$317$	−26.4949	−1.48810	−0.744051	+	0.668123i	$0.767098\pi$
$318$	0	0
$319$	14.2020	0.795162
$320$	0	0
$321$	−19.5959	−1.09374
$322$	0	0
$323$	12.8990	0.717718
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.10102	−0.392687
$328$	0	0
$329$	0.898979	0.0495623
$330$	0	0
$331$	−10.6969	−0.587957	−0.293978	−	0.955812i	$-0.594979\pi$
$331$	−10.6969	−0.587957	−0.293978	+	0.955812i	$0.594979\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.5959	1.61219	0.806096	−	0.591785i	$-0.201576\pi$
$337$	29.5959	1.61219	0.806096	+	0.591785i	$0.201576\pi$
$338$	0	0
$339$	−0.494897	−0.0268791
$340$	0	0
$341$	−4.40408	−0.238494
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.1010	1.02540	0.512698	−	0.858569i	$-0.328646\pi$
$347$	19.1010	1.02540	0.512698	+	0.858569i	$0.328646\pi$
$348$	0	0
$349$	−3.55051	−0.190054	−0.0950272	−	0.995475i	$-0.530294\pi$
$349$	−3.55051	−0.190054	−0.0950272	+	0.995475i	$0.530294\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.1010	−0.697297	−0.348648	−	0.937254i	$-0.613359\pi$
$353$	−13.1010	−0.697297	−0.348648	+	0.937254i	$0.613359\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.89898	0.259281
$358$	0	0
$359$	−11.5959	−0.612009	−0.306005	−	0.952030i	$-0.598992\pi$
$359$	−11.5959	−0.612009	−0.306005	+	0.952030i	$0.598992\pi$
$360$	0	0
$361$	22.5959	1.18926
$362$	0	0
$363$	31.8434	1.67134
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−32.6969	−1.70213
$370$	0	0
$371$	−1.10102	−0.0571621
$372$	0	0
$373$	−24.6969	−1.27876	−0.639380	−	0.768891i	$-0.720809\pi$
$373$	−24.6969	−1.27876	−0.639380	+	0.768891i	$0.720809\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.30306	−0.0671111
$378$	0	0
$379$	−1.30306	−0.0669338	−0.0334669	−	0.999440i	$-0.510655\pi$
$379$	−1.30306	−0.0669338	−0.0334669	+	0.999440i	$0.510655\pi$
$380$	0	0
$381$	−12.4949	−0.640133
$382$	0	0
$383$	−16.8990	−0.863498	−0.431749	−	0.901994i	$-0.642103\pi$
$383$	−16.8990	−0.863498	−0.431749	+	0.901994i	$0.642103\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.6969	1.35708
$388$	0	0
$389$	22.8990	1.16102	0.580512	−	0.814252i	$-0.302852\pi$
$389$	22.8990	1.16102	0.580512	+	0.814252i	$0.302852\pi$
$390$	0	0
$391$	13.7980	0.697793
$392$	0	0
$393$	3.79796	0.191582
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.3485	0.870695	0.435347	−	0.900263i	$-0.356626\pi$
$397$	17.3485	0.870695	0.435347	+	0.900263i	$0.356626\pi$
$398$	0	0
$399$	15.7980	0.790887
$400$	0	0
$401$	29.3939	1.46786	0.733930	−	0.679225i	$-0.237684\pi$
$401$	29.3939	1.46786	0.733930	+	0.679225i	$0.237684\pi$
$402$	0	0
$403$	0.404082	0.0201288
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.79796	0.485667
$408$	0	0
$409$	−14.4949	−0.716727	−0.358363	−	0.933582i	$-0.616665\pi$
$409$	−14.4949	−0.716727	−0.358363	+	0.933582i	$0.616665\pi$
$410$	0	0
$411$	−43.5959	−2.15043
$412$	0	0
$413$	−6.44949	−0.317359
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−15.7980	−0.773629
$418$	0	0
$419$	6.44949	0.315078	0.157539	−	0.987513i	$-0.449644\pi$
$419$	6.44949	0.315078	0.157539	+	0.987513i	$0.449644\pi$
$420$	0	0
$421$	−23.7980	−1.15984	−0.579921	−	0.814673i	$-0.696917\pi$
$421$	−23.7980	−1.15984	−0.579921	+	0.814673i	$0.696917\pi$
$422$	0	0
$423$	−2.69694	−0.131130
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.44949	−0.408899
$428$	0	0
$429$	−5.39388	−0.260419
$430$	0	0
$431$	−17.7980	−0.857298	−0.428649	−	0.903471i	$-0.641010\pi$
$431$	−17.7980	−0.857298	−0.428649	+	0.903471i	$0.641010\pi$
$432$	0	0
$433$	19.7980	0.951429	0.475715	−	0.879600i	$-0.342190\pi$
$433$	19.7980	0.951429	0.475715	+	0.879600i	$0.342190\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	44.4949	2.12848
$438$	0	0
$439$	−37.3939	−1.78471	−0.892356	−	0.451332i	$-0.850949\pi$
$439$	−37.3939	−1.78471	−0.892356	+	0.451332i	$0.850949\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	−9.79796	−0.465515	−0.232758	−	0.972535i	$-0.574775\pi$
$443$	−9.79796	−0.465515	−0.232758	+	0.972535i	$0.574775\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	38.6969	1.83030
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	53.3939	2.51422
$452$	0	0
$453$	48.0000	2.25524
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.59592	0.448878	0.224439	−	0.974488i	$-0.427945\pi$
$457$	9.59592	0.448878	0.224439	+	0.974488i	$0.427945\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.65153	0.123494	0.0617470	−	0.998092i	$-0.480333\pi$
$461$	2.65153	0.123494	0.0617470	+	0.998092i	$0.480333\pi$
$462$	0	0
$463$	−35.5959	−1.65428	−0.827141	−	0.561994i	$-0.810034\pi$
$463$	−35.5959	−1.65428	−0.827141	+	0.561994i	$0.810034\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.55051	0.256847	0.128423	−	0.991719i	$-0.459008\pi$
$467$	5.55051	0.256847	0.128423	+	0.991719i	$0.459008\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−20.6969	−0.953665
$472$	0	0
$473$	−43.5959	−2.00454
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.30306	0.151237
$478$	0	0
$479$	−38.6969	−1.76811	−0.884054	−	0.467385i	$-0.845196\pi$
$479$	−38.6969	−1.76811	−0.884054	+	0.467385i	$0.845196\pi$
$480$	0	0
$481$	−0.898979	−0.0409899
$482$	0	0
$483$	16.8990	0.768930
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.6969	−1.66290	−0.831449	−	0.555602i	$-0.812488\pi$
$487$	−36.6969	−1.66290	−0.831449	+	0.555602i	$0.812488\pi$
$488$	0	0
$489$	−41.3939	−1.87190
$490$	0	0
$491$	19.5959	0.884351	0.442176	−	0.896928i	$-0.354207\pi$
$491$	19.5959	0.884351	0.442176	+	0.896928i	$0.354207\pi$
$492$	0	0
$493$	5.79796	0.261127
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.8990	−0.488886
$498$	0	0
$499$	−25.7980	−1.15488	−0.577438	−	0.816435i	$-0.695947\pi$
$499$	−25.7980	−1.15488	−0.577438	+	0.816435i	$0.695947\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.3485	−1.39223
$508$	0	0
$509$	−36.4495	−1.61560	−0.807798	−	0.589460i	$-0.799341\pi$
$509$	−36.4495	−1.61560	−0.807798	+	0.589460i	$0.799341\pi$
$510$	0	0
$511$	−6.89898	−0.305193
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.40408	0.193691
$518$	0	0
$519$	−44.6969	−1.96198
$520$	0	0
$521$	3.30306	0.144710	0.0723549	−	0.997379i	$-0.476949\pi$
$521$	3.30306	0.144710	0.0723549	+	0.997379i	$0.476949\pi$
$522$	0	0
$523$	1.14643	0.0501298	0.0250649	−	0.999686i	$-0.492021\pi$
$523$	1.14643	0.0501298	0.0250649	+	0.999686i	$0.492021\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.79796	−0.0783203
$528$	0	0
$529$	24.5959	1.06939
$530$	0	0
$531$	19.3485	0.839652
$532$	0	0
$533$	−4.89898	−0.212198
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.2020	0.612863
$538$	0	0
$539$	−4.89898	−0.211014
$540$	0	0
$541$	−29.5959	−1.27243	−0.636214	−	0.771513i	$-0.719500\pi$
$541$	−29.5959	−1.27243	−0.636214	+	0.771513i	$0.719500\pi$
$542$	0	0
$543$	34.8990	1.49766
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.6969	0.457368	0.228684	−	0.973501i	$-0.426558\pi$
$547$	10.6969	0.457368	0.228684	+	0.973501i	$0.426558\pi$
$548$	0	0
$549$	25.3485	1.08185
$550$	0	0
$551$	18.6969	0.796516
$552$	0	0
$553$	−2.89898	−0.123277
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.6969	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$557$	16.6969	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	14.0454	0.591943	0.295972	−	0.955197i	$-0.404357\pi$
$563$	14.0454	0.591943	0.295972	+	0.955197i	$0.404357\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	14.2020	0.595381	0.297690	−	0.954663i	$-0.403784\pi$
$569$	14.2020	0.595381	0.297690	+	0.954663i	$0.403784\pi$
$570$	0	0
$571$	20.8990	0.874595	0.437298	−	0.899317i	$-0.355936\pi$
$571$	20.8990	0.874595	0.437298	+	0.899317i	$0.355936\pi$
$572$	0	0
$573$	40.8990	1.70858
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−46.4949	−1.93561	−0.967804	−	0.251705i	$-0.919009\pi$
$577$	−46.4949	−1.93561	−0.967804	+	0.251705i	$0.919009\pi$
$578$	0	0
$579$	−43.1010	−1.79122
$580$	0	0
$581$	−2.44949	−0.101622
$582$	0	0
$583$	−5.39388	−0.223392
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.1464	−1.36810	−0.684050	−	0.729435i	$-0.739783\pi$
$587$	−33.1464	−1.36810	−0.684050	+	0.729435i	$0.739783\pi$
$588$	0	0
$589$	−5.79796	−0.238901
$590$	0	0
$591$	−22.2929	−0.917006
$592$	0	0
$593$	1.10102	0.0452135	0.0226067	−	0.999744i	$-0.492803\pi$
$593$	1.10102	0.0452135	0.0226067	+	0.999744i	$0.492803\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.3939	−0.711884
$598$	0	0
$599$	22.8990	0.935627	0.467813	−	0.883827i	$-0.345042\pi$
$599$	22.8990	0.935627	0.467813	+	0.883827i	$0.345042\pi$
$600$	0	0
$601$	19.3939	0.791093	0.395546	−	0.918446i	$-0.370555\pi$
$601$	19.3939	0.791093	0.395546	+	0.918446i	$0.370555\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−25.3939	−1.03071	−0.515353	−	0.856978i	$-0.672339\pi$
$607$	−25.3939	−1.03071	−0.515353	+	0.856978i	$0.672339\pi$
$608$	0	0
$609$	7.10102	0.287748
$610$	0	0
$611$	−0.404082	−0.0163474
$612$	0	0
$613$	8.20204	0.331277	0.165639	−	0.986187i	$-0.447031\pi$
$613$	8.20204	0.331277	0.165639	+	0.986187i	$0.447031\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.59592	0.386317	0.193159	−	0.981168i	$-0.438127\pi$
$617$	9.59592	0.386317	0.193159	+	0.981168i	$0.438127\pi$
$618$	0	0
$619$	46.4495	1.86696	0.933481	−	0.358626i	$-0.116755\pi$
$619$	46.4495	1.86696	0.933481	+	0.358626i	$0.116755\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.0000	0.400642
$624$	0	0
$625$	0	0
$626$	0	0
$627$	77.3939	3.09081
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−6.49490	−0.258558	−0.129279	−	0.991608i	$-0.541266\pi$
$631$	−6.49490	−0.258558	−0.129279	+	0.991608i	$0.541266\pi$
$632$	0	0
$633$	−29.3939	−1.16830
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.449490	0.0178094
$638$	0	0
$639$	32.6969	1.29347
$640$	0	0
$641$	6.20204	0.244966	0.122483	−	0.992471i	$-0.460914\pi$
$641$	6.20204	0.244966	0.122483	+	0.992471i	$0.460914\pi$
$642$	0	0
$643$	−9.14643	−0.360700	−0.180350	−	0.983603i	$-0.557723\pi$
$643$	−9.14643	−0.360700	−0.180350	+	0.983603i	$0.557723\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.2929	0.876423	0.438211	−	0.898872i	$-0.355612\pi$
$647$	22.2929	0.876423	0.438211	+	0.898872i	$0.355612\pi$
$648$	0	0
$649$	−31.5959	−1.24025
$650$	0	0
$651$	−2.20204	−0.0863048
$652$	0	0
$653$	39.7980	1.55741	0.778707	−	0.627387i	$-0.215876\pi$
$653$	39.7980	1.55741	0.778707	+	0.627387i	$0.215876\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	20.6969	0.807464
$658$	0	0
$659$	−7.10102	−0.276616	−0.138308	−	0.990389i	$-0.544166\pi$
$659$	−7.10102	−0.276616	−0.138308	+	0.990389i	$0.544166\pi$
$660$	0	0
$661$	12.9444	0.503478	0.251739	−	0.967795i	$-0.418997\pi$
$661$	12.9444	0.503478	0.251739	+	0.967795i	$0.418997\pi$
$662$	0	0
$663$	−2.20204	−0.0855202
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.0000	0.774403
$668$	0	0
$669$	−9.79796	−0.378811
$670$	0	0
$671$	−41.3939	−1.59799
$672$	0	0
$673$	−1.79796	−0.0693062	−0.0346531	−	0.999399i	$-0.511033\pi$
$673$	−1.79796	−0.0693062	−0.0346531	+	0.999399i	$0.511033\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.5505	1.21258	0.606292	−	0.795242i	$-0.292656\pi$
$677$	31.5505	1.21258	0.606292	+	0.795242i	$0.292656\pi$
$678$	0	0
$679$	−3.79796	−0.145752
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	−35.5959	−1.36204	−0.681020	−	0.732265i	$-0.738463\pi$
$683$	−35.5959	−1.36204	−0.681020	+	0.732265i	$0.738463\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	37.1010	1.41549
$688$	0	0
$689$	0.494897	0.0188541
$690$	0	0
$691$	13.1464	0.500114	0.250057	−	0.968231i	$-0.419551\pi$
$691$	13.1464	0.500114	0.250057	+	0.968231i	$0.419551\pi$
$692$	0	0
$693$	14.6969	0.558291
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21.7980	0.825657
$698$	0	0
$699$	−24.9898	−0.945201
$700$	0	0
$701$	40.6969	1.53710	0.768551	−	0.639788i	$-0.220978\pi$
$701$	40.6969	1.53710	0.768551	+	0.639788i	$0.220978\pi$
$702$	0	0
$703$	12.8990	0.486494
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.44949	−0.317776
$708$	0	0
$709$	40.2929	1.51323	0.756615	−	0.653861i	$-0.226852\pi$
$709$	40.2929	1.51323	0.756615	+	0.653861i	$0.226852\pi$
$710$	0	0
$711$	8.69694	0.326161
$712$	0	0
$713$	−6.20204	−0.232268
$714$	0	0
$715$	0	0
$716$	0	0
$717$	63.1918	2.35994
$718$	0	0
$719$	−44.4949	−1.65938	−0.829690	−	0.558225i	$-0.811483\pi$
$719$	−44.4949	−1.65938	−0.829690	+	0.558225i	$0.811483\pi$
$720$	0	0
$721$	−3.10102	−0.115488
$722$	0	0
$723$	50.6969	1.88544
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−6.69694	−0.248376	−0.124188	−	0.992259i	$-0.539633\pi$
$727$	−6.69694	−0.248376	−0.124188	+	0.992259i	$0.539633\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−17.7980	−0.658281
$732$	0	0
$733$	43.6413	1.61193	0.805965	−	0.591964i	$-0.201647\pi$
$733$	43.6413	1.61193	0.805965	+	0.591964i	$0.201647\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.1918	1.44365
$738$	0	0
$739$	44.4949	1.63677	0.818386	−	0.574669i	$-0.194869\pi$
$739$	44.4949	1.63677	0.818386	+	0.574669i	$0.194869\pi$
$740$	0	0
$741$	−7.10102	−0.260863
$742$	0	0
$743$	−15.3031	−0.561415	−0.280707	−	0.959793i	$-0.590569\pi$
$743$	−15.3031	−0.561415	−0.280707	+	0.959793i	$0.590569\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.34847	0.268866
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	22.2020	0.810164	0.405082	−	0.914280i	$-0.367243\pi$
$751$	22.2020	0.810164	0.405082	+	0.914280i	$0.367243\pi$
$752$	0	0
$753$	3.79796	0.138405
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.2020	1.17040	0.585202	−	0.810888i	$-0.301015\pi$
$757$	32.2020	1.17040	0.585202	+	0.810888i	$0.301015\pi$
$758$	0	0
$759$	82.7878	3.00501
$760$	0	0
$761$	−30.8990	−1.12009	−0.560044	−	0.828463i	$-0.689216\pi$
$761$	−30.8990	−1.12009	−0.560044	+	0.828463i	$0.689216\pi$
$762$	0	0
$763$	2.89898	0.104950
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.89898	0.104676
$768$	0	0
$769$	−11.3031	−0.407599	−0.203799	−	0.979013i	$-0.565329\pi$
$769$	−11.3031	−0.407599	−0.203799	+	0.979013i	$0.565329\pi$
$770$	0	0
$771$	−50.6969	−1.82581
$772$	0	0
$773$	13.3485	0.480111	0.240056	−	0.970759i	$-0.422834\pi$
$773$	13.3485	0.480111	0.240056	+	0.970759i	$0.422834\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.89898	0.175750
$778$	0	0
$779$	70.2929	2.51850
$780$	0	0
$781$	−53.3939	−1.91058
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	45.5505	1.62370	0.811850	−	0.583866i	$-0.198461\pi$
$787$	45.5505	1.62370	0.811850	+	0.583866i	$0.198461\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0.202041	0.00718375
$792$	0	0
$793$	3.79796	0.134869
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−52.9444	−1.87539	−0.937693	−	0.347464i	$-0.887043\pi$
$797$	−52.9444	−1.87539	−0.937693	+	0.347464i	$0.887043\pi$
$798$	0	0
$799$	1.79796	0.0636072
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	−33.7980	−1.19270
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−37.1010	−1.30602
$808$	0	0
$809$	8.40408	0.295472	0.147736	−	0.989027i	$-0.452801\pi$
$809$	8.40408	0.295472	0.147736	+	0.989027i	$0.452801\pi$
$810$	0	0
$811$	38.9444	1.36752	0.683761	−	0.729706i	$-0.260343\pi$
$811$	38.9444	1.36752	0.683761	+	0.729706i	$0.260343\pi$
$812$	0	0
$813$	−29.3939	−1.03089
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−57.3939	−2.00796
$818$	0	0
$819$	−1.34847	−0.0471193
$820$	0	0
$821$	27.7980	0.970155	0.485078	−	0.874471i	$-0.338791\pi$
$821$	27.7980	0.970155	0.485078	+	0.874471i	$0.338791\pi$
$822$	0	0
$823$	39.1918	1.36614	0.683071	−	0.730352i	$-0.260644\pi$
$823$	39.1918	1.36614	0.683071	+	0.730352i	$0.260644\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.5959	0.820510	0.410255	−	0.911971i	$-0.365440\pi$
$827$	23.5959	0.820510	0.410255	+	0.911971i	$0.365440\pi$
$828$	0	0
$829$	39.6413	1.37680	0.688400	−	0.725331i	$-0.258313\pi$
$829$	39.6413	1.37680	0.688400	+	0.725331i	$0.258313\pi$
$830$	0	0
$831$	12.4949	0.433443
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27.1010	−0.935631	−0.467816	−	0.883826i	$-0.654959\pi$
$839$	−27.1010	−0.935631	−0.467816	+	0.883826i	$0.654959\pi$
$840$	0	0
$841$	−20.5959	−0.710204
$842$	0	0
$843$	−44.0908	−1.51857
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.0000	−0.446685
$848$	0	0
$849$	69.1918	2.37466
$850$	0	0
$851$	13.7980	0.472988
$852$	0	0
$853$	−29.8434	−1.02182	−0.510909	−	0.859635i	$-0.670691\pi$
$853$	−29.8434	−1.02182	−0.510909	+	0.859635i	$0.670691\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.1918	−0.860537	−0.430268	−	0.902701i	$-0.641581\pi$
$857$	−25.1918	−0.860537	−0.430268	+	0.902701i	$0.641581\pi$
$858$	0	0
$859$	29.6413	1.01135	0.505674	−	0.862724i	$-0.331244\pi$
$859$	29.6413	1.01135	0.505674	+	0.862724i	$0.331244\pi$
$860$	0	0
$861$	26.6969	0.909829
$862$	0	0
$863$	13.3939	0.455933	0.227966	−	0.973669i	$-0.426792\pi$
$863$	13.3939	0.455933	0.227966	+	0.973669i	$0.426792\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−31.8434	−1.08146
$868$	0	0
$869$	−14.2020	−0.481771
$870$	0	0
$871$	−3.59592	−0.121843
$872$	0	0
$873$	11.3939	0.385624
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.3939	−0.654885	−0.327442	−	0.944871i	$-0.606187\pi$
$877$	−19.3939	−0.654885	−0.327442	+	0.944871i	$0.606187\pi$
$878$	0	0
$879$	15.3031	0.516159
$880$	0	0
$881$	27.7980	0.936537	0.468269	−	0.883586i	$-0.344878\pi$
$881$	27.7980	0.936537	0.468269	+	0.883586i	$0.344878\pi$
$882$	0	0
$883$	41.7980	1.40661	0.703307	−	0.710887i	$-0.251706\pi$
$883$	41.7980	1.40661	0.703307	+	0.710887i	$0.251706\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.6969	−0.896395	−0.448198	−	0.893934i	$-0.647934\pi$
$887$	−26.6969	−0.896395	−0.448198	+	0.893934i	$0.647934\pi$
$888$	0	0
$889$	5.10102	0.171083
$890$	0	0
$891$	44.0908	1.47710
$892$	0	0
$893$	5.79796	0.194021
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.59592	−0.253620
$898$	0	0
$899$	−2.60612	−0.0869191
$900$	0	0
$901$	−2.20204	−0.0733606
$902$	0	0
$903$	−21.7980	−0.725391
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22.2020	−0.737207	−0.368603	−	0.929587i	$-0.620164\pi$
$907$	−22.2020	−0.737207	−0.368603	+	0.929587i	$0.620164\pi$
$908$	0	0
$909$	25.3485	0.840756
$910$	0	0
$911$	−3.59592	−0.119138	−0.0595690	−	0.998224i	$-0.518973\pi$
$911$	−3.59592	−0.119138	−0.0595690	+	0.998224i	$0.518973\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.55051	−0.0512024
$918$	0	0
$919$	17.1010	0.564111	0.282055	−	0.959398i	$-0.408984\pi$
$919$	17.1010	0.564111	0.282055	+	0.959398i	$0.408984\pi$
$920$	0	0
$921$	10.4041	0.342826
$922$	0	0
$923$	4.89898	0.161252
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.30306	0.305553
$928$	0	0
$929$	40.2929	1.32197	0.660983	−	0.750401i	$-0.270140\pi$
$929$	40.2929	1.32197	0.660983	+	0.750401i	$0.270140\pi$
$930$	0	0
$931$	−6.44949	−0.211373
$932$	0	0
$933$	−29.3939	−0.962312
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50.8990	1.66280	0.831399	−	0.555677i	$-0.187541\pi$
$937$	50.8990	1.66280	0.831399	+	0.555677i	$0.187541\pi$
$938$	0	0
$939$	−43.1010	−1.40655
$940$	0	0
$941$	−24.4495	−0.797031	−0.398515	−	0.917162i	$-0.630474\pi$
$941$	−24.4495	−0.797031	−0.398515	+	0.917162i	$0.630474\pi$
$942$	0	0
$943$	75.1918	2.44858
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.0908	−1.43276	−0.716379	−	0.697711i	$-0.754202\pi$
$947$	−44.0908	−1.43276	−0.716379	+	0.697711i	$0.754202\pi$
$948$	0	0
$949$	3.10102	0.100663
$950$	0	0
$951$	−64.8990	−2.10449
$952$	0	0
$953$	−21.7980	−0.706105	−0.353053	−	0.935603i	$-0.614856\pi$
$953$	−21.7980	−0.706105	−0.353053	+	0.935603i	$0.614856\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	34.7878	1.12453
$958$	0	0
$959$	17.7980	0.574726
$960$	0	0
$961$	−30.1918	−0.973930
$962$	0	0
$963$	−24.0000	−0.773389
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.2929	1.03847	0.519234	−	0.854632i	$-0.326217\pi$
$967$	32.2929	1.03847	0.519234	+	0.854632i	$0.326217\pi$
$968$	0	0
$969$	31.5959	1.01501
$970$	0	0
$971$	14.4495	0.463706	0.231853	−	0.972751i	$-0.425521\pi$
$971$	14.4495	0.463706	0.231853	+	0.972751i	$0.425521\pi$
$972$	0	0
$973$	6.44949	0.206761
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.3939	−0.940393	−0.470197	−	0.882562i	$-0.655817\pi$
$977$	−29.3939	−0.940393	−0.470197	+	0.882562i	$0.655817\pi$
$978$	0	0
$979$	48.9898	1.56572
$980$	0	0
$981$	−8.69694	−0.277672
$982$	0	0
$983$	−42.6969	−1.36182	−0.680910	−	0.732367i	$-0.738416\pi$
$983$	−42.6969	−1.36182	−0.680910	+	0.732367i	$0.738416\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.20204	0.0700917
$988$	0	0
$989$	−61.3939	−1.95221
$990$	0	0
$991$	−60.6969	−1.92810	−0.964051	−	0.265718i	$-0.914391\pi$
$991$	−60.6969	−1.92810	−0.964051	+	0.265718i	$0.914391\pi$
$992$	0	0
$993$	−26.2020	−0.831497
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−42.6515	−1.35079	−0.675394	−	0.737457i	$-0.736026\pi$
$997$	−42.6515	−1.35079	−0.675394	+	0.737457i	$0.736026\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bl.1.2		2
4.3	odd	2		350.2.a.h.1.1		2
5.2	odd	4		560.2.g.e.449.1		4
5.3	odd	4		560.2.g.e.449.3		4
5.4	even	2		2800.2.a.bm.1.1		2
12.11	even	2		3150.2.a.bs.1.1		2
15.2	even	4		5040.2.t.t.1009.4		4
15.8	even	4		5040.2.t.t.1009.3		4
20.3	even	4		70.2.c.a.29.1	✓	4
20.7	even	4		70.2.c.a.29.4	yes	4
20.19	odd	2		350.2.a.g.1.2		2
28.27	even	2		2450.2.a.bq.1.2		2
40.3	even	4		2240.2.g.j.449.4		4
40.13	odd	4		2240.2.g.i.449.2		4
40.27	even	4		2240.2.g.j.449.2		4
40.37	odd	4		2240.2.g.i.449.4		4
60.23	odd	4		630.2.g.g.379.4		4
60.47	odd	4		630.2.g.g.379.2		4
60.59	even	2		3150.2.a.bt.1.1		2
140.3	odd	12		490.2.i.f.79.4		8
140.23	even	12		490.2.i.c.459.2		8
140.27	odd	4		490.2.c.e.99.3		4
140.47	odd	12		490.2.i.f.459.4		8
140.67	even	12		490.2.i.c.79.2		8
140.83	odd	4		490.2.c.e.99.2		4
140.87	odd	12		490.2.i.f.79.1		8
140.103	odd	12		490.2.i.f.459.1		8
140.107	even	12		490.2.i.c.459.3		8
140.123	even	12		490.2.i.c.79.3		8
140.139	even	2		2450.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.c.a.29.1	✓	4	20.3	even	4
70.2.c.a.29.4	yes	4	20.7	even	4
350.2.a.g.1.2		2	20.19	odd	2
350.2.a.h.1.1		2	4.3	odd	2
490.2.c.e.99.2		4	140.83	odd	4
490.2.c.e.99.3		4	140.27	odd	4
490.2.i.c.79.2		8	140.67	even	12
490.2.i.c.79.3		8	140.123	even	12
490.2.i.c.459.2		8	140.23	even	12
490.2.i.c.459.3		8	140.107	even	12
490.2.i.f.79.1		8	140.87	odd	12
490.2.i.f.79.4		8	140.3	odd	12
490.2.i.f.459.1		8	140.103	odd	12
490.2.i.f.459.4		8	140.47	odd	12
560.2.g.e.449.1		4	5.2	odd	4
560.2.g.e.449.3		4	5.3	odd	4
630.2.g.g.379.2		4	60.47	odd	4
630.2.g.g.379.4		4	60.23	odd	4
2240.2.g.i.449.2		4	40.13	odd	4
2240.2.g.i.449.4		4	40.37	odd	4
2240.2.g.j.449.2		4	40.27	even	4
2240.2.g.j.449.4		4	40.3	even	4
2450.2.a.bl.1.1		2	140.139	even	2
2450.2.a.bq.1.2		2	28.27	even	2
2800.2.a.bl.1.2		2	1.1	even	1	trivial
2800.2.a.bm.1.1		2	5.4	even	2
3150.2.a.bs.1.1		2	12.11	even	2
3150.2.a.bt.1.1		2	60.59	even	2
5040.2.t.t.1009.3		4	15.8	even	4
5040.2.t.t.1009.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{3} -1.00000 q^{7} +3.00000 q^{9} -4.89898 q^{11} +0.449490 q^{13} -2.00000 q^{17} -6.44949 q^{19} -2.44949 q^{21} -6.89898 q^{23} -2.89898 q^{29} +0.898979 q^{31} -12.0000 q^{33} -2.00000 q^{37} +1.10102 q^{39} -10.8990 q^{41} +8.89898 q^{43} -0.898979 q^{47} +1.00000 q^{49} -4.89898 q^{51} +1.10102 q^{53} -15.7980 q^{57} +6.44949 q^{59} +8.44949 q^{61} -3.00000 q^{63} -8.00000 q^{67} -16.8990 q^{69} +10.8990 q^{71} +6.89898 q^{73} +4.89898 q^{77} +2.89898 q^{79} -9.00000 q^{81} +2.44949 q^{83} -7.10102 q^{87} -10.0000 q^{89} -0.449490 q^{91} +2.20204 q^{93} +3.79796 q^{97} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 + 6 * q^9	\( 2 q - 2 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} - 8 q^{19} - 4 q^{23} + 4 q^{29} - 8 q^{31} - 24 q^{33} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 8 q^{43} + 8 q^{47} + 2 q^{49} + 12 q^{53} - 12 q^{57} + 8 q^{59} + 12 q^{61} - 6 q^{63} - 16 q^{67} - 24 q^{69} + 12 q^{71} + 4 q^{73} - 4 q^{79} - 18 q^{81} - 24 q^{87} - 20 q^{89} + 4 q^{91} + 24 q^{93} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 - 8 * q^19 - 4 * q^23 + 4 * q^29 - 8 * q^31 - 24 * q^33 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 8 * q^43 + 8 * q^47 + 2 * q^49 + 12 * q^53 - 12 * q^57 + 8 * q^59 + 12 * q^61 - 6 * q^63 - 16 * q^67 - 24 * q^69 + 12 * q^71 + 4 * q^73 - 4 * q^79 - 18 * q^81 - 24 * q^87 - 20 * q^89 + 4 * q^91 + 24 * q^93 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more