LMFDB - Embedded newform 3150.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(1,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3150.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.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:	$25.1528766367$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{13}]$
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			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	3150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} -4.89898 q^{11} -0.449490 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.44949 q^{19} -4.89898 q^{22} +6.89898 q^{23} -0.449490 q^{26} -1.00000 q^{28} +2.89898 q^{29} -0.898979 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{37} +6.44949 q^{38} +10.8990 q^{41} +8.89898 q^{43} -4.89898 q^{44} +6.89898 q^{46} +0.898979 q^{47} +1.00000 q^{49} -0.449490 q^{52} +1.10102 q^{53} -1.00000 q^{56} +2.89898 q^{58} +6.44949 q^{59} +8.44949 q^{61} -0.898979 q^{62} +1.00000 q^{64} -8.00000 q^{67} -2.00000 q^{68} +10.8990 q^{71} -6.89898 q^{73} +2.00000 q^{74} +6.44949 q^{76} +4.89898 q^{77} -2.89898 q^{79} +10.8990 q^{82} -2.44949 q^{83} +8.89898 q^{86} -4.89898 q^{88} +10.0000 q^{89} +0.449490 q^{91} +6.89898 q^{92} +0.898979 q^{94} -3.79796 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{26} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{37} + 8 q^{38} + 12 q^{41} + 8 q^{43} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 4 q^{52} + 12 q^{53} - 2 q^{56} - 4 q^{58} + 8 q^{59} + 12 q^{61} + 8 q^{62} + 2 q^{64} - 16 q^{67} - 4 q^{68} + 12 q^{71} - 4 q^{73} + 4 q^{74} + 8 q^{76} + 4 q^{79} + 12 q^{82} + 8 q^{86} + 20 q^{89} - 4 q^{91} + 4 q^{92} - 8 q^{94} + 12 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^19 + 4 * q^23 + 4 * q^26 - 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 4 * q^34 + 4 * q^37 + 8 * q^38 + 12 * q^41 + 8 * q^43 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 4 * q^52 + 12 * q^53 - 2 * q^56 - 4 * q^58 + 8 * q^59 + 12 * q^61 + 8 * q^62 + 2 * q^64 - 16 * q^67 - 4 * q^68 + 12 * q^71 - 4 * q^73 + 4 * q^74 + 8 * q^76 + 4 * q^79 + 12 * q^82 + 8 * q^86 + 20 * q^89 - 4 * q^91 + 4 * q^92 - 8 * q^94 + 12 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	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	$-0.519854\pi$
$13$	−0.449490	−0.124666	−0.0623330	+	0.998055i	$0.519854\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−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	$-0.234917\pi$
$19$	6.44949	1.47961	0.739807	+	0.672819i	$0.234917\pi$
$20$	0	0
$21$	0	0
$22$	−4.89898	−1.04447
$23$	6.89898	1.43854	0.719268	−	0.694732i	$-0.244477\pi$
$23$	6.89898	1.43854	0.719268	+	0.694732i	$0.244477\pi$
$24$	0	0
$25$	0	0
$26$	−0.449490	−0.0881522
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.89898	0.538327	0.269163	−	0.963095i	$-0.413253\pi$
$29$	2.89898	0.538327	0.269163	+	0.963095i	$0.413253\pi$
$30$	0	0
$31$	−0.898979	−0.161461	−0.0807307	−	0.996736i	$-0.525725\pi$
$31$	−0.898979	−0.161461	−0.0807307	+	0.996736i	$0.525725\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	6.44949	1.04625
$39$	0	0
$40$	0	0
$41$	10.8990	1.70213	0.851067	−	0.525057i	$-0.175956\pi$
$41$	10.8990	1.70213	0.851067	+	0.525057i	$0.175956\pi$
$42$	0	0
$43$	8.89898	1.35708	0.678541	−	0.734563i	$-0.262613\pi$
$43$	8.89898	1.35708	0.678541	+	0.734563i	$0.262613\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	6.89898	1.01720
$47$	0.898979	0.131130	0.0655648	−	0.997848i	$-0.479115\pi$
$47$	0.898979	0.131130	0.0655648	+	0.997848i	$0.479115\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−0.449490	−0.0623330
$53$	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
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.89898	0.380655
$59$	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	$0.318075\pi$
$62$	−0.898979	−0.114171
$63$	0	0
$64$	1.00000	0.125000
$65$	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	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	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	$-0.632287\pi$
$73$	−6.89898	−0.807464	−0.403732	+	0.914877i	$0.632287\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	6.44949	0.739807
$77$	4.89898	0.558291
$78$	0	0
$79$	−2.89898	−0.326161	−0.163080	−	0.986613i	$-0.552143\pi$
$79$	−2.89898	−0.326161	−0.163080	+	0.986613i	$0.552143\pi$
$80$	0	0
$81$	0	0
$82$	10.8990	1.20359
$83$	−2.44949	−0.268866	−0.134433	−	0.990923i	$-0.542921\pi$
$83$	−2.44949	−0.268866	−0.134433	+	0.990923i	$0.542921\pi$
$84$	0	0
$85$	0	0
$86$	8.89898	0.959602
$87$	0	0
$88$	−4.89898	−0.522233
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0.449490	0.0471193
$92$	6.89898	0.719268
$93$	0	0
$94$	0.898979	0.0927227
$95$	0	0
$96$	0	0
$97$	−3.79796	−0.385624	−0.192812	−	0.981236i	$-0.561761\pi$
$97$	−3.79796	−0.385624	−0.192812	+	0.981236i	$0.561761\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	0	0
$101$	−8.44949	−0.840756	−0.420378	−	0.907349i	$-0.638102\pi$
$101$	−8.44949	−0.840756	−0.420378	+	0.907349i	$0.638102\pi$
$102$	0	0
$103$	3.10102	0.305553	0.152776	−	0.988261i	$-0.451179\pi$
$103$	3.10102	0.305553	0.152776	+	0.988261i	$0.451179\pi$
$104$	−0.449490	−0.0440761
$105$	0	0
$106$	1.10102	0.106941
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	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
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−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
$116$	2.89898	0.269163
$117$	0	0
$118$	6.44949	0.593724
$119$	2.00000	0.183340
$120$	0	0
$121$	13.0000	1.18182
$122$	8.44949	0.764981
$123$	0	0
$124$	−0.898979	−0.0807307
$125$	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	$0.572670\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	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
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−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	$-0.411812\pi$
$139$	6.44949	0.547039	0.273519	+	0.961867i	$0.411812\pi$
$140$	0	0
$141$	0	0
$142$	10.8990	0.914622
$143$	2.20204	0.184144
$144$	0	0
$145$	0	0
$146$	−6.89898	−0.570964
$147$	0	0
$148$	2.00000	0.164399
$149$	−15.7980	−1.29422	−0.647110	−	0.762397i	$-0.724022\pi$
$149$	−15.7980	−1.29422	−0.647110	+	0.762397i	$0.724022\pi$
$150$	0	0
$151$	−19.5959	−1.59469	−0.797347	−	0.603522i	$-0.793764\pi$
$151$	−19.5959	−1.59469	−0.797347	+	0.603522i	$0.793764\pi$
$152$	6.44949	0.523123
$153$	0	0
$154$	4.89898	0.394771
$155$	0	0
$156$	0	0
$157$	8.44949	0.674343	0.337171	−	0.941443i	$-0.390530\pi$
$157$	8.44949	0.674343	0.337171	+	0.941443i	$0.390530\pi$
$158$	−2.89898	−0.230630
$159$	0	0
$160$	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	$0.730214\pi$
$164$	10.8990	0.851067
$165$	0	0
$166$	−2.44949	−0.190117
$167$	−4.89898	−0.379094	−0.189547	−	0.981872i	$-0.560702\pi$
$167$	−4.89898	−0.379094	−0.189547	+	0.981872i	$0.560702\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	0	0
$172$	8.89898	0.678541
$173$	−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
$176$	−4.89898	−0.369274
$177$	0	0
$178$	10.0000	0.749532
$179$	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	$0.322379\pi$
$182$	0.449490	0.0333184
$183$	0	0
$184$	6.89898	0.508600
$185$	0	0
$186$	0	0
$187$	9.79796	0.716498
$188$	0.898979	0.0655648
$189$	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	$-0.281704\pi$
$193$	17.5959	1.26658	0.633291	+	0.773914i	$0.281704\pi$
$194$	−3.79796	−0.272678
$195$	0	0
$196$	1.00000	0.0714286
$197$	−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	$-0.419014\pi$
$199$	7.10102	0.503378	0.251689	+	0.967808i	$0.419014\pi$
$200$	0	0
$201$	0	0
$202$	−8.44949	−0.594504
$203$	−2.89898	−0.203468
$204$	0	0
$205$	0	0
$206$	3.10102	0.216058
$207$	0	0
$208$	−0.449490	−0.0311665
$209$	−31.5959	−2.18554
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	1.10102	0.0756184
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	0.898979	0.0610267
$218$	−2.89898	−0.196344
$219$	0	0
$220$	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	$0.542760\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−0.202041	−0.0134396
$227$	7.34847	0.487735	0.243868	−	0.969809i	$-0.421584\pi$
$227$	7.34847	0.487735	0.243868	+	0.969809i	$0.421584\pi$
$228$	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
$231$	0	0
$232$	2.89898	0.190327
$233$	−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
$236$	6.44949	0.419826
$237$	0	0
$238$	2.00000	0.129641
$239$	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	$0.267747\pi$
$242$	13.0000	0.835672
$243$	0	0
$244$	8.44949	0.540923
$245$	0	0
$246$	0	0
$247$	−2.89898	−0.184458
$248$	−0.898979	−0.0570853
$249$	0	0
$250$	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
$254$	−5.10102	−0.320066
$255$	0	0
$256$	1.00000	0.0625000
$257$	−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
$261$	0	0
$262$	1.55051	0.0957908
$263$	9.79796	0.604168	0.302084	−	0.953281i	$-0.402318\pi$
$263$	9.79796	0.604168	0.302084	+	0.953281i	$0.402318\pi$
$264$	0	0
$265$	0	0
$266$	−6.44949	−0.395444
$267$	0	0
$268$	−8.00000	−0.488678
$269$	15.1464	0.923494	0.461747	−	0.887012i	$-0.347223\pi$
$269$	15.1464	0.923494	0.461747	+	0.887012i	$0.347223\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−17.7980	−1.07521
$275$	0	0
$276$	0	0
$277$	−5.10102	−0.306491	−0.153245	−	0.988188i	$-0.548972\pi$
$277$	−5.10102	−0.306491	−0.153245	+	0.988188i	$0.548972\pi$
$278$	6.44949	0.386815
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	28.2474	1.67914	0.839568	−	0.543254i	$-0.182808\pi$
$283$	28.2474	1.67914	0.839568	+	0.543254i	$0.182808\pi$
$284$	10.8990	0.646735
$285$	0	0
$286$	2.20204	0.130209
$287$	−10.8990	−0.643346
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	−6.89898	−0.403732
$293$	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
$296$	2.00000	0.116248
$297$	0	0
$298$	−15.7980	−0.915151
$299$	−3.10102	−0.179337
$300$	0	0
$301$	−8.89898	−0.512929
$302$	−19.5959	−1.12762
$303$	0	0
$304$	6.44949	0.369904
$305$	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	$0.461323\pi$
$308$	4.89898	0.279145
$309$	0	0
$310$	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	$-0.334328\pi$
$313$	17.5959	0.994580	0.497290	+	0.867584i	$0.334328\pi$
$314$	8.44949	0.476832
$315$	0	0
$316$	−2.89898	−0.163080
$317$	−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
$321$	0	0
$322$	−6.89898	−0.384465
$323$	−12.8990	−0.717718
$324$	0	0
$325$	0	0
$326$	−16.8990	−0.935948
$327$	0	0
$328$	10.8990	0.601795
$329$	−0.898979	−0.0495623
$330$	0	0
$331$	10.6969	0.587957	0.293978	−	0.955812i	$-0.405021\pi$
$331$	10.6969	0.587957	0.293978	+	0.955812i	$0.405021\pi$
$332$	−2.44949	−0.134433
$333$	0	0
$334$	−4.89898	−0.268060
$335$	0	0
$336$	0	0
$337$	−29.5959	−1.61219	−0.806096	−	0.591785i	$-0.798424\pi$
$337$	−29.5959	−1.61219	−0.806096	+	0.591785i	$0.798424\pi$
$338$	−12.7980	−0.696117
$339$	0	0
$340$	0	0
$341$	4.40408	0.238494
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.89898	0.479801
$345$	0	0
$346$	−18.2474	−0.980989
$347$	−19.1010	−1.02540	−0.512698	−	0.858569i	$-0.671354\pi$
$347$	−19.1010	−1.02540	−0.512698	+	0.858569i	$0.671354\pi$
$348$	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
$352$	−4.89898	−0.261116
$353$	−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
$356$	10.0000	0.529999
$357$	0	0
$358$	5.79796	0.306432
$359$	−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
$362$	14.2474	0.748829
$363$	0	0
$364$	0.449490	0.0235597
$365$	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	$0.185356\pi$
$368$	6.89898	0.359634
$369$	0	0
$370$	0	0
$371$	−1.10102	−0.0571621
$372$	0	0
$373$	24.6969	1.27876	0.639380	−	0.768891i	$-0.279191\pi$
$373$	24.6969	1.27876	0.639380	+	0.768891i	$0.279191\pi$
$374$	9.79796	0.506640
$375$	0	0
$376$	0.898979	0.0463613
$377$	−1.30306	−0.0671111
$378$	0	0
$379$	1.30306	0.0669338	0.0334669	−	0.999440i	$-0.489345\pi$
$379$	1.30306	0.0669338	0.0334669	+	0.999440i	$0.489345\pi$
$380$	0	0
$381$	0	0
$382$	16.6969	0.854290
$383$	16.8990	0.863498	0.431749	−	0.901994i	$-0.357897\pi$
$383$	16.8990	0.863498	0.431749	+	0.901994i	$0.357897\pi$
$384$	0	0
$385$	0	0
$386$	17.5959	0.895609
$387$	0	0
$388$	−3.79796	−0.192812
$389$	−22.8990	−1.16102	−0.580512	−	0.814252i	$-0.697148\pi$
$389$	−22.8990	−1.16102	−0.580512	+	0.814252i	$0.697148\pi$
$390$	0	0
$391$	−13.7980	−0.697793
$392$	1.00000	0.0505076
$393$	0	0
$394$	−9.10102	−0.458503
$395$	0	0
$396$	0	0
$397$	−17.3485	−0.870695	−0.435347	−	0.900263i	$-0.643374\pi$
$397$	−17.3485	−0.870695	−0.435347	+	0.900263i	$0.643374\pi$
$398$	7.10102	0.355942
$399$	0	0
$400$	0	0
$401$	−29.3939	−1.46786	−0.733930	−	0.679225i	$-0.762316\pi$
$401$	−29.3939	−1.46786	−0.733930	+	0.679225i	$0.762316\pi$
$402$	0	0
$403$	0.404082	0.0201288
$404$	−8.44949	−0.420378
$405$	0	0
$406$	−2.89898	−0.143874
$407$	−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
$411$	0	0
$412$	3.10102	0.152776
$413$	−6.44949	−0.317359
$414$	0	0
$415$	0	0
$416$	−0.449490	−0.0220380
$417$	0	0
$418$	−31.5959	−1.54541
$419$	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	$0.696917\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	1.10102	0.0534703
$425$	0	0
$426$	0	0
$427$	−8.44949	−0.408899
$428$	8.00000	0.386695
$429$	0	0
$430$	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	$-0.657810\pi$
$433$	−19.7980	−0.951429	−0.475715	+	0.879600i	$0.657810\pi$
$434$	0.898979	0.0431524
$435$	0	0
$436$	−2.89898	−0.138836
$437$	44.4949	2.12848
$438$	0	0
$439$	37.3939	1.78471	0.892356	−	0.451332i	$-0.149051\pi$
$439$	37.3939	1.78471	0.892356	+	0.451332i	$0.149051\pi$
$440$	0	0
$441$	0	0
$442$	0.898979	0.0427601
$443$	9.79796	0.465515	0.232758	−	0.972535i	$-0.425225\pi$
$443$	9.79796	0.465515	0.232758	+	0.972535i	$0.425225\pi$
$444$	0	0
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−53.3939	−2.51422
$452$	−0.202041	−0.00950321
$453$	0	0
$454$	7.34847	0.344881
$455$	0	0
$456$	0	0
$457$	−9.59592	−0.448878	−0.224439	−	0.974488i	$-0.572055\pi$
$457$	−9.59592	−0.448878	−0.224439	+	0.974488i	$0.572055\pi$
$458$	15.1464	0.707746
$459$	0	0
$460$	0	0
$461$	−2.65153	−0.123494	−0.0617470	−	0.998092i	$-0.519667\pi$
$461$	−2.65153	−0.123494	−0.0617470	+	0.998092i	$0.519667\pi$
$462$	0	0
$463$	−35.5959	−1.65428	−0.827141	−	0.561994i	$-0.810034\pi$
$463$	−35.5959	−1.65428	−0.827141	+	0.561994i	$0.810034\pi$
$464$	2.89898	0.134582
$465$	0	0
$466$	−10.2020	−0.472600
$467$	−5.55051	−0.256847	−0.128423	−	0.991719i	$-0.540992\pi$
$467$	−5.55051	−0.256847	−0.128423	+	0.991719i	$0.540992\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	6.44949	0.296862
$473$	−43.5959	−2.00454
$474$	0	0
$475$	0	0
$476$	2.00000	0.0916698
$477$	0	0
$478$	25.7980	1.17997
$479$	−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
$482$	20.6969	0.942720
$483$	0	0
$484$	13.0000	0.590909
$485$	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	$0.812488\pi$
$488$	8.44949	0.382490
$489$	0	0
$490$	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
$494$	−2.89898	−0.130431
$495$	0	0
$496$	−0.898979	−0.0403654
$497$	−10.8990	−0.488886
$498$	0	0
$499$	25.7980	1.15488	0.577438	−	0.816435i	$-0.304053\pi$
$499$	25.7980	1.15488	0.577438	+	0.816435i	$0.304053\pi$
$500$	0	0
$501$	0	0
$502$	1.55051	0.0692027
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	0	0
$506$	−33.7980	−1.50250
$507$	0	0
$508$	−5.10102	−0.226321
$509$	36.4495	1.61560	0.807798	−	0.589460i	$-0.200659\pi$
$509$	36.4495	1.61560	0.807798	+	0.589460i	$0.200659\pi$
$510$	0	0
$511$	6.89898	0.305193
$512$	1.00000	0.0441942
$513$	0	0
$514$	−20.6969	−0.912903
$515$	0	0
$516$	0	0
$517$	−4.40408	−0.193691
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	0	0
$521$	−3.30306	−0.144710	−0.0723549	−	0.997379i	$-0.523051\pi$
$521$	−3.30306	−0.144710	−0.0723549	+	0.997379i	$0.523051\pi$
$522$	0	0
$523$	1.14643	0.0501298	0.0250649	−	0.999686i	$-0.492021\pi$
$523$	1.14643	0.0501298	0.0250649	+	0.999686i	$0.492021\pi$
$524$	1.55051	0.0677344
$525$	0	0
$526$	9.79796	0.427211
$527$	1.79796	0.0783203
$528$	0	0
$529$	24.5959	1.06939
$530$	0	0
$531$	0	0
$532$	−6.44949	−0.279621
$533$	−4.89898	−0.212198
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	15.1464	0.653009
$539$	−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	$0.719500\pi$
$542$	12.0000	0.515444
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	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	$0.426558\pi$
$548$	−17.7980	−0.760291
$549$	0	0
$550$	0	0
$551$	18.6969	0.796516
$552$	0	0
$553$	2.89898	0.123277
$554$	−5.10102	−0.216722
$555$	0	0
$556$	6.44949	0.273519
$557$	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
$561$	0	0
$562$	18.0000	0.759284
$563$	−14.0454	−0.591943	−0.295972	−	0.955197i	$-0.595643\pi$
$563$	−14.0454	−0.591943	−0.295972	+	0.955197i	$0.595643\pi$
$564$	0	0
$565$	0	0
$566$	28.2474	1.18733
$567$	0	0
$568$	10.8990	0.457311
$569$	−14.2020	−0.595381	−0.297690	−	0.954663i	$-0.596216\pi$
$569$	−14.2020	−0.595381	−0.297690	+	0.954663i	$0.596216\pi$
$570$	0	0
$571$	−20.8990	−0.874595	−0.437298	−	0.899317i	$-0.644064\pi$
$571$	−20.8990	−0.874595	−0.437298	+	0.899317i	$0.644064\pi$
$572$	2.20204	0.0920720
$573$	0	0
$574$	−10.8990	−0.454915
$575$	0	0
$576$	0	0
$577$	46.4949	1.93561	0.967804	−	0.251705i	$-0.0809913\pi$
$577$	46.4949	1.93561	0.967804	+	0.251705i	$0.0809913\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	0	0
$581$	2.44949	0.101622
$582$	0	0
$583$	−5.39388	−0.223392
$584$	−6.89898	−0.285482
$585$	0	0
$586$	6.24745	0.258080
$587$	33.1464	1.36810	0.684050	−	0.729435i	$-0.260217\pi$
$587$	33.1464	1.36810	0.684050	+	0.729435i	$0.260217\pi$
$588$	0	0
$589$	−5.79796	−0.238901
$590$	0	0
$591$	0	0
$592$	2.00000	0.0821995
$593$	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
$596$	−15.7980	−0.647110
$597$	0	0
$598$	−3.10102	−0.126810
$599$	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	$0.370555\pi$
$602$	−8.89898	−0.362695
$603$	0	0
$604$	−19.5959	−0.797347
$605$	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	$0.672339\pi$
$608$	6.44949	0.261561
$609$	0	0
$610$	0	0
$611$	−0.404082	−0.0163474
$612$	0	0
$613$	−8.20204	−0.331277	−0.165639	−	0.986187i	$-0.552969\pi$
$613$	−8.20204	−0.331277	−0.165639	+	0.986187i	$0.552969\pi$
$614$	4.24745	0.171413
$615$	0	0
$616$	4.89898	0.197386
$617$	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	$-0.883245\pi$
$619$	−46.4495	−1.86696	−0.933481	+	0.358626i	$0.883245\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−10.0000	−0.400642
$624$	0	0
$625$	0	0
$626$	17.5959	0.703274
$627$	0	0
$628$	8.44949	0.337171
$629$	−4.00000	−0.159490
$630$	0	0
$631$	6.49490	0.258558	0.129279	−	0.991608i	$-0.458734\pi$
$631$	6.49490	0.258558	0.129279	+	0.991608i	$0.458734\pi$
$632$	−2.89898	−0.115315
$633$	0	0
$634$	−26.4949	−1.05225
$635$	0	0
$636$	0	0
$637$	−0.449490	−0.0178094
$638$	−14.2020	−0.562264
$639$	0	0
$640$	0	0
$641$	−6.20204	−0.244966	−0.122483	−	0.992471i	$-0.539086\pi$
$641$	−6.20204	−0.244966	−0.122483	+	0.992471i	$0.539086\pi$
$642$	0	0
$643$	−9.14643	−0.360700	−0.180350	−	0.983603i	$-0.557723\pi$
$643$	−9.14643	−0.360700	−0.180350	+	0.983603i	$0.557723\pi$
$644$	−6.89898	−0.271858
$645$	0	0
$646$	−12.8990	−0.507504
$647$	−22.2929	−0.876423	−0.438211	−	0.898872i	$-0.644388\pi$
$647$	−22.2929	−0.876423	−0.438211	+	0.898872i	$0.644388\pi$
$648$	0	0
$649$	−31.5959	−1.24025
$650$	0	0
$651$	0	0
$652$	−16.8990	−0.661815
$653$	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
$656$	10.8990	0.425534
$657$	0	0
$658$	−0.898979	−0.0350459
$659$	−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	$0.418997\pi$
$662$	10.6969	0.415748
$663$	0	0
$664$	−2.44949	−0.0950586
$665$	0	0
$666$	0	0
$667$	20.0000	0.774403
$668$	−4.89898	−0.189547
$669$	0	0
$670$	0	0
$671$	−41.3939	−1.59799
$672$	0	0
$673$	1.79796	0.0693062	0.0346531	−	0.999399i	$-0.488967\pi$
$673$	1.79796	0.0693062	0.0346531	+	0.999399i	$0.488967\pi$
$674$	−29.5959	−1.13999
$675$	0	0
$676$	−12.7980	−0.492229
$677$	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
$681$	0	0
$682$	4.40408	0.168641
$683$	35.5959	1.36204	0.681020	−	0.732265i	$-0.261537\pi$
$683$	35.5959	1.36204	0.681020	+	0.732265i	$0.261537\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	8.89898	0.339270
$689$	−0.494897	−0.0188541
$690$	0	0
$691$	−13.1464	−0.500114	−0.250057	−	0.968231i	$-0.580449\pi$
$691$	−13.1464	−0.500114	−0.250057	+	0.968231i	$0.580449\pi$
$692$	−18.2474	−0.693664
$693$	0	0
$694$	−19.1010	−0.725065
$695$	0	0
$696$	0	0
$697$	−21.7980	−0.825657
$698$	−3.55051	−0.134389
$699$	0	0
$700$	0	0
$701$	−40.6969	−1.53710	−0.768551	−	0.639788i	$-0.779022\pi$
$701$	−40.6969	−1.53710	−0.768551	+	0.639788i	$0.779022\pi$
$702$	0	0
$703$	12.8990	0.486494
$704$	−4.89898	−0.184637
$705$	0	0
$706$	−13.1010	−0.493063
$707$	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
$711$	0	0
$712$	10.0000	0.374766
$713$	−6.20204	−0.232268
$714$	0	0
$715$	0	0
$716$	5.79796	0.216680
$717$	0	0
$718$	−11.5959	−0.432756
$719$	−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
$722$	22.5959	0.840933
$723$	0	0
$724$	14.2474	0.529502
$725$	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	$0.539633\pi$
$728$	0.449490	0.0166592
$729$	0	0
$730$	0	0
$731$	−17.7980	−0.658281
$732$	0	0
$733$	−43.6413	−1.61193	−0.805965	−	0.591964i	$-0.798353\pi$
$733$	−43.6413	−1.61193	−0.805965	+	0.591964i	$0.798353\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	6.89898	0.254300
$737$	39.1918	1.44365
$738$	0	0
$739$	−44.4949	−1.63677	−0.818386	−	0.574669i	$-0.805131\pi$
$739$	−44.4949	−1.63677	−0.818386	+	0.574669i	$0.805131\pi$
$740$	0	0
$741$	0	0
$742$	−1.10102	−0.0404197
$743$	15.3031	0.561415	0.280707	−	0.959793i	$-0.409431\pi$
$743$	15.3031	0.561415	0.280707	+	0.959793i	$0.409431\pi$
$744$	0	0
$745$	0	0
$746$	24.6969	0.904219
$747$	0	0
$748$	9.79796	0.358249
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−22.2020	−0.810164	−0.405082	−	0.914280i	$-0.632757\pi$
$751$	−22.2020	−0.810164	−0.405082	+	0.914280i	$0.632757\pi$
$752$	0.898979	0.0327824
$753$	0	0
$754$	−1.30306	−0.0474547
$755$	0	0
$756$	0	0
$757$	−32.2020	−1.17040	−0.585202	−	0.810888i	$-0.698985\pi$
$757$	−32.2020	−1.17040	−0.585202	+	0.810888i	$0.698985\pi$
$758$	1.30306	0.0473293
$759$	0	0
$760$	0	0
$761$	30.8990	1.12009	0.560044	−	0.828463i	$-0.310784\pi$
$761$	30.8990	1.12009	0.560044	+	0.828463i	$0.310784\pi$
$762$	0	0
$763$	2.89898	0.104950
$764$	16.6969	0.604074
$765$	0	0
$766$	16.8990	0.610585
$767$	−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
$771$	0	0
$772$	17.5959	0.633291
$773$	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
$776$	−3.79796	−0.136339
$777$	0	0
$778$	−22.8990	−0.820968
$779$	70.2929	2.51850
$780$	0	0
$781$	−53.3939	−1.91058
$782$	−13.7980	−0.493414
$783$	0	0
$784$	1.00000	0.0357143
$785$	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	$0.198461\pi$
$788$	−9.10102	−0.324210
$789$	0	0
$790$	0	0
$791$	0.202041	0.00718375
$792$	0	0
$793$	−3.79796	−0.134869
$794$	−17.3485	−0.615674
$795$	0	0
$796$	7.10102	0.251689
$797$	−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
$801$	0	0
$802$	−29.3939	−1.03793
$803$	33.7980	1.19270
$804$	0	0
$805$	0	0
$806$	0.404082	0.0142332
$807$	0	0
$808$	−8.44949	−0.297252
$809$	−8.40408	−0.295472	−0.147736	−	0.989027i	$-0.547199\pi$
$809$	−8.40408	−0.295472	−0.147736	+	0.989027i	$0.547199\pi$
$810$	0	0
$811$	−38.9444	−1.36752	−0.683761	−	0.729706i	$-0.739657\pi$
$811$	−38.9444	−1.36752	−0.683761	+	0.729706i	$0.739657\pi$
$812$	−2.89898	−0.101734
$813$	0	0
$814$	−9.79796	−0.343418
$815$	0	0
$816$	0	0
$817$	57.3939	2.00796
$818$	−14.4949	−0.506802
$819$	0	0
$820$	0	0
$821$	−27.7980	−0.970155	−0.485078	−	0.874471i	$-0.661209\pi$
$821$	−27.7980	−0.970155	−0.485078	+	0.874471i	$0.661209\pi$
$822$	0	0
$823$	39.1918	1.36614	0.683071	−	0.730352i	$-0.260644\pi$
$823$	39.1918	1.36614	0.683071	+	0.730352i	$0.260644\pi$
$824$	3.10102	0.108029
$825$	0	0
$826$	−6.44949	−0.224406
$827$	−23.5959	−0.820510	−0.410255	−	0.911971i	$-0.634560\pi$
$827$	−23.5959	−0.820510	−0.410255	+	0.911971i	$0.634560\pi$
$828$	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
$831$	0	0
$832$	−0.449490	−0.0155833
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	−31.5959	−1.09277
$837$	0	0
$838$	6.44949	0.222794
$839$	−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
$842$	−23.7980	−0.820132
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	−13.0000	−0.446685
$848$	1.10102	0.0378092
$849$	0	0
$850$	0	0
$851$	13.7980	0.472988
$852$	0	0
$853$	29.8434	1.02182	0.510909	−	0.859635i	$-0.329309\pi$
$853$	29.8434	1.02182	0.510909	+	0.859635i	$0.329309\pi$
$854$	−8.44949	−0.289136
$855$	0	0
$856$	8.00000	0.273434
$857$	−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	$-0.668756\pi$
$859$	−29.6413	−1.01135	−0.505674	+	0.862724i	$0.668756\pi$
$860$	0	0
$861$	0	0
$862$	−17.7980	−0.606201
$863$	−13.3939	−0.455933	−0.227966	−	0.973669i	$-0.573208\pi$
$863$	−13.3939	−0.455933	−0.227966	+	0.973669i	$0.573208\pi$
$864$	0	0
$865$	0	0
$866$	−19.7980	−0.672762
$867$	0	0
$868$	0.898979	0.0305134
$869$	14.2020	0.481771
$870$	0	0
$871$	3.59592	0.121843
$872$	−2.89898	−0.0981718
$873$	0	0
$874$	44.4949	1.50506
$875$	0	0
$876$	0	0
$877$	19.3939	0.654885	0.327442	−	0.944871i	$-0.393813\pi$
$877$	19.3939	0.654885	0.327442	+	0.944871i	$0.393813\pi$
$878$	37.3939	1.26198
$879$	0	0
$880$	0	0
$881$	−27.7980	−0.936537	−0.468269	−	0.883586i	$-0.655122\pi$
$881$	−27.7980	−0.936537	−0.468269	+	0.883586i	$0.655122\pi$
$882$	0	0
$883$	41.7980	1.40661	0.703307	−	0.710887i	$-0.251706\pi$
$883$	41.7980	1.40661	0.703307	+	0.710887i	$0.251706\pi$
$884$	0.898979	0.0302360
$885$	0	0
$886$	9.79796	0.329169
$887$	26.6969	0.896395	0.448198	−	0.893934i	$-0.352066\pi$
$887$	26.6969	0.896395	0.448198	+	0.893934i	$0.352066\pi$
$888$	0	0
$889$	5.10102	0.171083
$890$	0	0
$891$	0	0
$892$	−4.00000	−0.133930
$893$	5.79796	0.194021
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	−2.60612	−0.0869191
$900$	0	0
$901$	−2.20204	−0.0733606
$902$	−53.3939	−1.77782
$903$	0	0
$904$	−0.202041	−0.00671978
$905$	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	$0.620164\pi$
$908$	7.34847	0.243868
$909$	0	0
$910$	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
$914$	−9.59592	−0.317405
$915$	0	0
$916$	15.1464	0.500452
$917$	−1.55051	−0.0512024
$918$	0	0
$919$	−17.1010	−0.564111	−0.282055	−	0.959398i	$-0.591016\pi$
$919$	−17.1010	−0.564111	−0.282055	+	0.959398i	$0.591016\pi$
$920$	0	0
$921$	0	0
$922$	−2.65153	−0.0873235
$923$	−4.89898	−0.161252
$924$	0	0
$925$	0	0
$926$	−35.5959	−1.16975
$927$	0	0
$928$	2.89898	0.0951637
$929$	−40.2929	−1.32197	−0.660983	−	0.750401i	$-0.729860\pi$
$929$	−40.2929	−1.32197	−0.660983	+	0.750401i	$0.729860\pi$
$930$	0	0
$931$	6.44949	0.211373
$932$	−10.2020	−0.334179
$933$	0	0
$934$	−5.55051	−0.181618
$935$	0	0
$936$	0	0
$937$	−50.8990	−1.66280	−0.831399	−	0.555677i	$-0.812459\pi$
$937$	−50.8990	−1.66280	−0.831399	+	0.555677i	$0.812459\pi$
$938$	8.00000	0.261209
$939$	0	0
$940$	0	0
$941$	24.4495	0.797031	0.398515	−	0.917162i	$-0.369526\pi$
$941$	24.4495	0.797031	0.398515	+	0.917162i	$0.369526\pi$
$942$	0	0
$943$	75.1918	2.44858
$944$	6.44949	0.209913
$945$	0	0
$946$	−43.5959	−1.41743
$947$	44.0908	1.43276	0.716379	−	0.697711i	$-0.245798\pi$
$947$	44.0908	1.43276	0.716379	+	0.697711i	$0.245798\pi$
$948$	0	0
$949$	3.10102	0.100663
$950$	0	0
$951$	0	0
$952$	2.00000	0.0648204
$953$	−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
$956$	25.7980	0.834366
$957$	0	0
$958$	−38.6969	−1.25024
$959$	17.7980	0.574726
$960$	0	0
$961$	−30.1918	−0.973930
$962$	−0.898979	−0.0289843
$963$	0	0
$964$	20.6969	0.666604
$965$	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	$0.326217\pi$
$968$	13.0000	0.417836
$969$	0	0
$970$	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
$974$	−36.6969	−1.17585
$975$	0	0
$976$	8.44949	0.270462
$977$	−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
$981$	0	0
$982$	19.5959	0.625331
$983$	42.6969	1.36182	0.680910	−	0.732367i	$-0.261584\pi$
$983$	42.6969	1.36182	0.680910	+	0.732367i	$0.261584\pi$
$984$	0	0
$985$	0	0
$986$	−5.79796	−0.184645
$987$	0	0
$988$	−2.89898	−0.0922288
$989$	61.3939	1.95221
$990$	0	0
$991$	60.6969	1.92810	0.964051	−	0.265718i	$-0.0856089\pi$
$991$	60.6969	1.92810	0.964051	+	0.265718i	$0.0856089\pi$
$992$	−0.898979	−0.0285426
$993$	0	0
$994$	−10.8990	−0.345695
$995$	0	0
$996$	0	0
$997$	42.6515	1.35079	0.675394	−	0.737457i	$-0.263974\pi$
$997$	42.6515	1.35079	0.675394	+	0.737457i	$0.263974\pi$
$998$	25.7980	0.816620
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.a.bt.1.1		2
3.2	odd	2		350.2.a.g.1.2		2
5.2	odd	4		630.2.g.g.379.4		4
5.3	odd	4		630.2.g.g.379.2		4
5.4	even	2		3150.2.a.bs.1.1		2
12.11	even	2		2800.2.a.bm.1.1		2
15.2	even	4		70.2.c.a.29.1	✓	4
15.8	even	4		70.2.c.a.29.4	yes	4
15.14	odd	2		350.2.a.h.1.1		2
20.3	even	4		5040.2.t.t.1009.4		4
20.7	even	4		5040.2.t.t.1009.3		4
21.20	even	2		2450.2.a.bl.1.1		2
60.23	odd	4		560.2.g.e.449.1		4
60.47	odd	4		560.2.g.e.449.3		4
60.59	even	2		2800.2.a.bl.1.2		2
105.2	even	12		490.2.i.c.459.2		8
105.17	odd	12		490.2.i.f.79.4		8
105.23	even	12		490.2.i.c.459.3		8
105.32	even	12		490.2.i.c.79.3		8
105.38	odd	12		490.2.i.f.79.1		8
105.47	odd	12		490.2.i.f.459.1		8
105.53	even	12		490.2.i.c.79.2		8
105.62	odd	4		490.2.c.e.99.2		4
105.68	odd	12		490.2.i.f.459.4		8
105.83	odd	4		490.2.c.e.99.3		4
105.104	even	2		2450.2.a.bq.1.2		2
120.53	even	4		2240.2.g.j.449.2		4
120.77	even	4		2240.2.g.j.449.4		4
120.83	odd	4		2240.2.g.i.449.4		4
120.107	odd	4		2240.2.g.i.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.c.a.29.1	✓	4	15.2	even	4
70.2.c.a.29.4	yes	4	15.8	even	4
350.2.a.g.1.2		2	3.2	odd	2
350.2.a.h.1.1		2	15.14	odd	2
490.2.c.e.99.2		4	105.62	odd	4
490.2.c.e.99.3		4	105.83	odd	4
490.2.i.c.79.2		8	105.53	even	12
490.2.i.c.79.3		8	105.32	even	12
490.2.i.c.459.2		8	105.2	even	12
490.2.i.c.459.3		8	105.23	even	12
490.2.i.f.79.1		8	105.38	odd	12
490.2.i.f.79.4		8	105.17	odd	12
490.2.i.f.459.1		8	105.47	odd	12
490.2.i.f.459.4		8	105.68	odd	12
560.2.g.e.449.1		4	60.23	odd	4
560.2.g.e.449.3		4	60.47	odd	4
630.2.g.g.379.2		4	5.3	odd	4
630.2.g.g.379.4		4	5.2	odd	4
2240.2.g.i.449.2		4	120.107	odd	4
2240.2.g.i.449.4		4	120.83	odd	4
2240.2.g.j.449.2		4	120.53	even	4
2240.2.g.j.449.4		4	120.77	even	4
2450.2.a.bl.1.1		2	21.20	even	2
2450.2.a.bq.1.2		2	105.104	even	2
2800.2.a.bl.1.2		2	60.59	even	2
2800.2.a.bm.1.1		2	12.11	even	2
3150.2.a.bs.1.1		2	5.4	even	2
3150.2.a.bt.1.1		2	1.1	even	1	trivial
5040.2.t.t.1009.3		4	20.7	even	4
5040.2.t.t.1009.4		4	20.3	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} -4.89898 q^{11} -0.449490 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.44949 q^{19} -4.89898 q^{22} +6.89898 q^{23} -0.449490 q^{26} -1.00000 q^{28} +2.89898 q^{29} -0.898979 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{37} +6.44949 q^{38} +10.8990 q^{41} +8.89898 q^{43} -4.89898 q^{44} +6.89898 q^{46} +0.898979 q^{47} +1.00000 q^{49} -0.449490 q^{52} +1.10102 q^{53} -1.00000 q^{56} +2.89898 q^{58} +6.44949 q^{59} +8.44949 q^{61} -0.898979 q^{62} +1.00000 q^{64} -8.00000 q^{67} -2.00000 q^{68} +10.8990 q^{71} -6.89898 q^{73} +2.00000 q^{74} +6.44949 q^{76} +4.89898 q^{77} -2.89898 q^{79} +10.8990 q^{82} -2.44949 q^{83} +8.89898 q^{86} -4.89898 q^{88} +10.0000 q^{89} +0.449490 q^{91} +6.89898 q^{92} +0.898979 q^{94} -3.79796 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{26} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{37} + 8 q^{38} + 12 q^{41} + 8 q^{43} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 4 q^{52} + 12 q^{53} - 2 q^{56} - 4 q^{58} + 8 q^{59} + 12 q^{61} + 8 q^{62} + 2 q^{64} - 16 q^{67} - 4 q^{68} + 12 q^{71} - 4 q^{73} + 4 q^{74} + 8 q^{76} + 4 q^{79} + 12 q^{82} + 8 q^{86} + 20 q^{89} - 4 q^{91} + 4 q^{92} - 8 q^{94} + 12 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^19 + 4 * q^23 + 4 * q^26 - 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 4 * q^34 + 4 * q^37 + 8 * q^38 + 12 * q^41 + 8 * q^43 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 4 * q^52 + 12 * q^53 - 2 * q^56 - 4 * q^58 + 8 * q^59 + 12 * q^61 + 8 * q^62 + 2 * q^64 - 16 * q^67 - 4 * q^68 + 12 * q^71 - 4 * q^73 + 4 * q^74 + 8 * q^76 + 4 * q^79 + 12 * q^82 + 8 * q^86 + 20 * q^89 - 4 * q^91 + 4 * q^92 - 8 * q^94 + 12 * q^97 + 2 * q^98

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