LMFDB - Embedded newform 6930.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, 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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{20} -1.00000 q^{22} +1.46410 q^{23} +1.00000 q^{25} -3.46410 q^{26} -1.00000 q^{28} -8.92820 q^{29} +10.9282 q^{31} +1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} +3.46410 q^{37} -1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} +1.46410 q^{46} -6.92820 q^{47} +1.00000 q^{49} +1.00000 q^{50} -3.46410 q^{52} -6.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} -8.92820 q^{58} -8.00000 q^{59} -4.92820 q^{61} +10.9282 q^{62} +1.00000 q^{64} +3.46410 q^{65} -5.46410 q^{67} +3.46410 q^{68} +1.00000 q^{70} -1.07180 q^{71} +15.8564 q^{73} +3.46410 q^{74} +1.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} -13.8564 q^{83} -3.46410 q^{85} -1.00000 q^{88} -8.53590 q^{89} +3.46410 q^{91} +1.46410 q^{92} -6.92820 q^{94} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{35} - 2 q^{40} - 4 q^{41} - 2 q^{44} - 4 q^{46} + 2 q^{49} + 2 q^{50} - 12 q^{53} + 2 q^{55} - 2 q^{56} - 4 q^{58} - 16 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{70} - 16 q^{71} + 4 q^{73} + 2 q^{77} - 8 q^{79} - 2 q^{80} - 4 q^{82} - 2 q^{88} - 24 q^{89} - 4 q^{92} - 20 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^14 + 2 * q^16 - 2 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^25 - 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^35 - 2 * q^40 - 4 * q^41 - 2 * q^44 - 4 * q^46 + 2 * q^49 + 2 * q^50 - 12 * q^53 + 2 * q^55 - 2 * q^56 - 4 * q^58 - 16 * q^59 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^67 + 2 * q^70 - 16 * q^71 + 4 * q^73 + 2 * q^77 - 8 * q^79 - 2 * q^80 - 4 * q^82 - 2 * q^88 - 24 * q^89 - 4 * q^92 - 20 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	1.46410	0.305286	0.152643	−	0.988281i	$-0.451221\pi$
$23$	1.46410	0.305286	0.152643	+	0.988281i	$0.451221\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.46410	−0.679366
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−8.92820	−1.65793	−0.828963	−	0.559304i	$-0.811069\pi$
$29$	−8.92820	−1.65793	−0.828963	+	0.559304i	$0.811069\pi$
$30$	0	0
$31$	10.9282	1.96276	0.981382	−	0.192068i	$-0.0615194\pi$
$31$	10.9282	1.96276	0.981382	+	0.192068i	$0.0615194\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.46410	0.594089
$35$	1.00000	0.169031
$36$	0	0
$37$	3.46410	0.569495	0.284747	−	0.958603i	$-0.408090\pi$
$37$	3.46410	0.569495	0.284747	+	0.958603i	$0.408090\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	1.46410	0.215870
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−3.46410	−0.480384
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−8.92820	−1.17233
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	10.9282	1.38788
$63$	0	0
$64$	1.00000	0.125000
$65$	3.46410	0.429669
$66$	0	0
$67$	−5.46410	−0.667546	−0.333773	−	0.942653i	$-0.608322\pi$
$67$	−5.46410	−0.667546	−0.333773	+	0.942653i	$0.608322\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	1.00000	0.119523
$71$	−1.07180	−0.127199	−0.0635994	−	0.997976i	$-0.520258\pi$
$71$	−1.07180	−0.127199	−0.0635994	+	0.997976i	$0.520258\pi$
$72$	0	0
$73$	15.8564	1.85585	0.927926	−	0.372764i	$-0.121590\pi$
$73$	15.8564	1.85585	0.927926	+	0.372764i	$0.121590\pi$
$74$	3.46410	0.402694
$75$	0	0
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−13.8564	−1.52094	−0.760469	−	0.649374i	$-0.775031\pi$
$83$	−13.8564	−1.52094	−0.760469	+	0.649374i	$0.775031\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	0	0
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−8.53590	−0.904803	−0.452402	−	0.891814i	$-0.649433\pi$
$89$	−8.53590	−0.904803	−0.452402	+	0.891814i	$0.649433\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	1.46410	0.152643
$93$	0	0
$94$	−6.92820	−0.714590
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	7.85641	0.781742	0.390871	−	0.920446i	$-0.372174\pi$
$101$	7.85641	0.781742	0.390871	+	0.920446i	$0.372174\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	−3.46410	−0.331801	−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410	−0.331801	−0.165900	+	0.986143i	$0.553053\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	2.39230	0.225049	0.112525	−	0.993649i	$-0.464106\pi$
$113$	2.39230	0.225049	0.112525	+	0.993649i	$0.464106\pi$
$114$	0	0
$115$	−1.46410	−0.136528
$116$	−8.92820	−0.828963
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.92820	−0.446179
$123$	0	0
$124$	10.9282	0.981382
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.9282	−0.969721	−0.484861	−	0.874591i	$-0.661130\pi$
$127$	−10.9282	−0.969721	−0.484861	+	0.874591i	$0.661130\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.46410	0.303822
$131$	−17.8564	−1.56012	−0.780061	−	0.625704i	$-0.784812\pi$
$131$	−17.8564	−1.56012	−0.780061	+	0.625704i	$0.784812\pi$
$132$	0	0
$133$	0	0
$134$	−5.46410	−0.472026
$135$	0	0
$136$	3.46410	0.297044
$137$	−3.46410	−0.295958	−0.147979	−	0.988990i	$-0.547277\pi$
$137$	−3.46410	−0.295958	−0.147979	+	0.988990i	$0.547277\pi$
$138$	0	0
$139$	−10.9282	−0.926918	−0.463459	−	0.886118i	$-0.653392\pi$
$139$	−10.9282	−0.926918	−0.463459	+	0.886118i	$0.653392\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−1.07180	−0.0899432
$143$	3.46410	0.289683
$144$	0	0
$145$	8.92820	0.741447
$146$	15.8564	1.31229
$147$	0	0
$148$	3.46410	0.284747
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0	0
$154$	1.00000	0.0805823
$155$	−10.9282	−0.877774
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−1.46410	−0.115387
$162$	0	0
$163$	−19.3205	−1.51330	−0.756649	−	0.653821i	$-0.773165\pi$
$163$	−19.3205	−1.51330	−0.756649	+	0.653821i	$0.773165\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−13.8564	−1.07547
$167$	17.4641	1.35141	0.675706	−	0.737171i	$-0.263839\pi$
$167$	17.4641	1.35141	0.675706	+	0.737171i	$0.263839\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	−3.46410	−0.265684
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−8.53590	−0.639793
$179$	6.92820	0.517838	0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820	0.517838	0.258919	+	0.965899i	$0.416634\pi$
$180$	0	0
$181$	3.46410	0.257485	0.128742	−	0.991678i	$-0.458906\pi$
$181$	3.46410	0.257485	0.128742	+	0.991678i	$0.458906\pi$
$182$	3.46410	0.256776
$183$	0	0
$184$	1.46410	0.107935
$185$	−3.46410	−0.254686
$186$	0	0
$187$	−3.46410	−0.253320
$188$	−6.92820	−0.505291
$189$	0	0
$190$	0	0
$191$	9.85641	0.713185	0.356592	−	0.934260i	$-0.383939\pi$
$191$	9.85641	0.713185	0.356592	+	0.934260i	$0.383939\pi$
$192$	0	0
$193$	15.8564	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$193$	15.8564	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	18.9282	1.34178	0.670892	−	0.741555i	$-0.265911\pi$
$199$	18.9282	1.34178	0.670892	+	0.741555i	$0.265911\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	7.85641	0.552775
$203$	8.92820	0.626637
$204$	0	0
$205$	2.00000	0.139686
$206$	−2.92820	−0.204018
$207$	0	0
$208$	−3.46410	−0.240192
$209$	0	0
$210$	0	0
$211$	4.39230	0.302379	0.151189	−	0.988505i	$-0.451690\pi$
$211$	4.39230	0.302379	0.151189	+	0.988505i	$0.451690\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−6.92820	−0.473602
$215$	0	0
$216$	0	0
$217$	−10.9282	−0.741855
$218$	−3.46410	−0.234619
$219$	0	0
$220$	1.00000	0.0674200
$221$	−12.0000	−0.807207
$222$	0	0
$223$	27.7128	1.85579	0.927894	−	0.372845i	$-0.121618\pi$
$223$	27.7128	1.85579	0.927894	+	0.372845i	$0.121618\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	2.39230	0.159134
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	16.5359	1.09272	0.546361	−	0.837549i	$-0.316012\pi$
$229$	16.5359	1.09272	0.546361	+	0.837549i	$0.316012\pi$
$230$	−1.46410	−0.0965400
$231$	0	0
$232$	−8.92820	−0.586165
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	0	0
$235$	6.92820	0.451946
$236$	−8.00000	−0.520756
$237$	0	0
$238$	−3.46410	−0.224544
$239$	−21.4641	−1.38840	−0.694199	−	0.719783i	$-0.744241\pi$
$239$	−21.4641	−1.38840	−0.694199	+	0.719783i	$0.744241\pi$
$240$	0	0
$241$	−18.7846	−1.21002	−0.605012	−	0.796217i	$-0.706832\pi$
$241$	−18.7846	−1.21002	−0.605012	+	0.796217i	$0.706832\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−4.92820	−0.315496
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	10.9282	0.693942
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	−1.46410	−0.0920473
$254$	−10.9282	−0.685696
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.07180	−0.191613	−0.0958067	−	0.995400i	$-0.530543\pi$
$257$	−3.07180	−0.191613	−0.0958067	+	0.995400i	$0.530543\pi$
$258$	0	0
$259$	−3.46410	−0.215249
$260$	3.46410	0.214834
$261$	0	0
$262$	−17.8564	−1.10317
$263$	−5.07180	−0.312740	−0.156370	−	0.987699i	$-0.549979\pi$
$263$	−5.07180	−0.312740	−0.156370	+	0.987699i	$0.549979\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	−5.46410	−0.333773
$269$	−16.9282	−1.03213	−0.516065	−	0.856549i	$-0.672604\pi$
$269$	−16.9282	−1.03213	−0.516065	+	0.856549i	$0.672604\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	−3.46410	−0.209274
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	−10.9282	−0.655430
$279$	0	0
$280$	1.00000	0.0597614
$281$	−21.3205	−1.27187	−0.635937	−	0.771741i	$-0.719386\pi$
$281$	−21.3205	−1.27187	−0.635937	+	0.771741i	$0.719386\pi$
$282$	0	0
$283$	−24.3923	−1.44997	−0.724986	−	0.688764i	$-0.758154\pi$
$283$	−24.3923	−1.44997	−0.724986	+	0.688764i	$0.758154\pi$
$284$	−1.07180	−0.0635994
$285$	0	0
$286$	3.46410	0.204837
$287$	2.00000	0.118056
$288$	0	0
$289$	−5.00000	−0.294118
$290$	8.92820	0.524282
$291$	0	0
$292$	15.8564	0.927926
$293$	18.7846	1.09741	0.548704	−	0.836016i	$-0.315121\pi$
$293$	18.7846	1.09741	0.548704	+	0.836016i	$0.315121\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	3.46410	0.201347
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−5.07180	−0.293310
$300$	0	0
$301$	0	0
$302$	−20.0000	−1.15087
$303$	0	0
$304$	0	0
$305$	4.92820	0.282188
$306$	0	0
$307$	−24.3923	−1.39214	−0.696071	−	0.717973i	$-0.745070\pi$
$307$	−24.3923	−1.39214	−0.696071	+	0.717973i	$0.745070\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−10.9282	−0.620680
$311$	21.4641	1.21712	0.608559	−	0.793509i	$-0.291748\pi$
$311$	21.4641	1.21712	0.608559	+	0.793509i	$0.291748\pi$
$312$	0	0
$313$	27.8564	1.57454	0.787269	−	0.616610i	$-0.211495\pi$
$313$	27.8564	1.57454	0.787269	+	0.616610i	$0.211495\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−11.8564	−0.665922	−0.332961	−	0.942941i	$-0.608048\pi$
$317$	−11.8564	−0.665922	−0.332961	+	0.942941i	$0.608048\pi$
$318$	0	0
$319$	8.92820	0.499883
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−1.46410	−0.0815912
$323$	0	0
$324$	0	0
$325$	−3.46410	−0.192154
$326$	−19.3205	−1.07006
$327$	0	0
$328$	−2.00000	−0.110432
$329$	6.92820	0.381964
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−13.8564	−0.760469
$333$	0	0
$334$	17.4641	0.955593
$335$	5.46410	0.298536
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−3.46410	−0.187867
$341$	−10.9282	−0.591795
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	−6.00000	−0.322562
$347$	14.9282	0.801388	0.400694	−	0.916212i	$-0.368769\pi$
$347$	14.9282	0.801388	0.400694	+	0.916212i	$0.368769\pi$
$348$	0	0
$349$	−7.07180	−0.378545	−0.189272	−	0.981925i	$-0.560613\pi$
$349$	−7.07180	−0.378545	−0.189272	+	0.981925i	$0.560613\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	12.9282	0.688099	0.344049	−	0.938952i	$-0.388201\pi$
$353$	12.9282	0.688099	0.344049	+	0.938952i	$0.388201\pi$
$354$	0	0
$355$	1.07180	0.0568851
$356$	−8.53590	−0.452402
$357$	0	0
$358$	6.92820	0.366167
$359$	5.46410	0.288384	0.144192	−	0.989550i	$-0.453942\pi$
$359$	5.46410	0.288384	0.144192	+	0.989550i	$0.453942\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	3.46410	0.182069
$363$	0	0
$364$	3.46410	0.181568
$365$	−15.8564	−0.829962
$366$	0	0
$367$	16.7846	0.876149	0.438075	−	0.898939i	$-0.355661\pi$
$367$	16.7846	0.876149	0.438075	+	0.898939i	$0.355661\pi$
$368$	1.46410	0.0763216
$369$	0	0
$370$	−3.46410	−0.180090
$371$	6.00000	0.311504
$372$	0	0
$373$	34.7846	1.80108	0.900539	−	0.434774i	$-0.143172\pi$
$373$	34.7846	1.80108	0.900539	+	0.434774i	$0.143172\pi$
$374$	−3.46410	−0.179124
$375$	0	0
$376$	−6.92820	−0.357295
$377$	30.9282	1.59288
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	9.85641	0.504298
$383$	1.07180	0.0547663	0.0273831	−	0.999625i	$-0.491283\pi$
$383$	1.07180	0.0547663	0.0273831	+	0.999625i	$0.491283\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	15.8564	0.807070
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−28.9282	−1.46672	−0.733359	−	0.679842i	$-0.762049\pi$
$389$	−28.9282	−1.46672	−0.733359	+	0.679842i	$0.762049\pi$
$390$	0	0
$391$	5.07180	0.256492
$392$	1.00000	0.0505076
$393$	0	0
$394$	−2.00000	−0.100759
$395$	4.00000	0.201262
$396$	0	0
$397$	34.7846	1.74579	0.872895	−	0.487909i	$-0.162240\pi$
$397$	34.7846	1.74579	0.872895	+	0.487909i	$0.162240\pi$
$398$	18.9282	0.948785
$399$	0	0
$400$	1.00000	0.0500000
$401$	19.8564	0.991582	0.495791	−	0.868442i	$-0.334878\pi$
$401$	19.8564	0.991582	0.495791	+	0.868442i	$0.334878\pi$
$402$	0	0
$403$	−37.8564	−1.88576
$404$	7.85641	0.390871
$405$	0	0
$406$	8.92820	0.443099
$407$	−3.46410	−0.171709
$408$	0	0
$409$	−7.07180	−0.349678	−0.174839	−	0.984597i	$-0.555940\pi$
$409$	−7.07180	−0.349678	−0.174839	+	0.984597i	$0.555940\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	−2.92820	−0.144262
$413$	8.00000	0.393654
$414$	0	0
$415$	13.8564	0.680184
$416$	−3.46410	−0.169842
$417$	0	0
$418$	0	0
$419$	2.92820	0.143052	0.0715260	−	0.997439i	$-0.477213\pi$
$419$	2.92820	0.143052	0.0715260	+	0.997439i	$0.477213\pi$
$420$	0	0
$421$	−31.8564	−1.55259	−0.776293	−	0.630372i	$-0.782902\pi$
$421$	−31.8564	−1.55259	−0.776293	+	0.630372i	$0.782902\pi$
$422$	4.39230	0.213814
$423$	0	0
$424$	−6.00000	−0.291386
$425$	3.46410	0.168034
$426$	0	0
$427$	4.92820	0.238492
$428$	−6.92820	−0.334887
$429$	0	0
$430$	0	0
$431$	32.3923	1.56028	0.780141	−	0.625603i	$-0.215147\pi$
$431$	32.3923	1.56028	0.780141	+	0.625603i	$0.215147\pi$
$432$	0	0
$433$	−23.8564	−1.14647	−0.573233	−	0.819393i	$-0.694311\pi$
$433$	−23.8564	−1.14647	−0.573233	+	0.819393i	$0.694311\pi$
$434$	−10.9282	−0.524571
$435$	0	0
$436$	−3.46410	−0.165900
$437$	0	0
$438$	0	0
$439$	34.9282	1.66703	0.833516	−	0.552495i	$-0.186324\pi$
$439$	34.9282	1.66703	0.833516	+	0.552495i	$0.186324\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−12.0000	−0.570782
$443$	6.14359	0.291891	0.145945	−	0.989293i	$-0.453378\pi$
$443$	6.14359	0.291891	0.145945	+	0.989293i	$0.453378\pi$
$444$	0	0
$445$	8.53590	0.404640
$446$	27.7128	1.31224
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	11.8564	0.559538	0.279769	−	0.960067i	$-0.409742\pi$
$449$	11.8564	0.559538	0.279769	+	0.960067i	$0.409742\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	2.39230	0.112525
$453$	0	0
$454$	8.00000	0.375459
$455$	−3.46410	−0.162400
$456$	0	0
$457$	−11.8564	−0.554619	−0.277310	−	0.960781i	$-0.589443\pi$
$457$	−11.8564	−0.554619	−0.277310	+	0.960781i	$0.589443\pi$
$458$	16.5359	0.772672
$459$	0	0
$460$	−1.46410	−0.0682641
$461$	−25.7128	−1.19757	−0.598783	−	0.800912i	$-0.704349\pi$
$461$	−25.7128	−1.19757	−0.598783	+	0.800912i	$0.704349\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−8.92820	−0.414481
$465$	0	0
$466$	−19.8564	−0.919830
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	0	0
$469$	5.46410	0.252309
$470$	6.92820	0.319574
$471$	0	0
$472$	−8.00000	−0.368230
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−3.46410	−0.158777
$477$	0	0
$478$	−21.4641	−0.981745
$479$	−21.8564	−0.998645	−0.499322	−	0.866416i	$-0.666418\pi$
$479$	−21.8564	−0.998645	−0.499322	+	0.866416i	$0.666418\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−18.7846	−0.855616
$483$	0	0
$484$	1.00000	0.0454545
$485$	10.0000	0.454077
$486$	0	0
$487$	−41.8564	−1.89669	−0.948347	−	0.317234i	$-0.897246\pi$
$487$	−41.8564	−1.89669	−0.948347	+	0.317234i	$0.897246\pi$
$488$	−4.92820	−0.223089
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−1.85641	−0.0837785	−0.0418892	−	0.999122i	$-0.513338\pi$
$491$	−1.85641	−0.0837785	−0.0418892	+	0.999122i	$0.513338\pi$
$492$	0	0
$493$	−30.9282	−1.39294
$494$	0	0
$495$	0	0
$496$	10.9282	0.490691
$497$	1.07180	0.0480767
$498$	0	0
$499$	−12.7846	−0.572318	−0.286159	−	0.958182i	$-0.592379\pi$
$499$	−12.7846	−0.572318	−0.286159	+	0.958182i	$0.592379\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	8.00000	0.357057
$503$	19.6077	0.874264	0.437132	−	0.899397i	$-0.355994\pi$
$503$	19.6077	0.874264	0.437132	+	0.899397i	$0.355994\pi$
$504$	0	0
$505$	−7.85641	−0.349605
$506$	−1.46410	−0.0650873
$507$	0	0
$508$	−10.9282	−0.484861
$509$	−0.928203	−0.0411419	−0.0205709	−	0.999788i	$-0.506548\pi$
$509$	−0.928203	−0.0411419	−0.0205709	+	0.999788i	$0.506548\pi$
$510$	0	0
$511$	−15.8564	−0.701446
$512$	1.00000	0.0441942
$513$	0	0
$514$	−3.07180	−0.135491
$515$	2.92820	0.129032
$516$	0	0
$517$	6.92820	0.304702
$518$	−3.46410	−0.152204
$519$	0	0
$520$	3.46410	0.151911
$521$	−34.1051	−1.49417	−0.747086	−	0.664727i	$-0.768548\pi$
$521$	−34.1051	−1.49417	−0.747086	+	0.664727i	$0.768548\pi$
$522$	0	0
$523$	−19.3205	−0.844827	−0.422413	−	0.906403i	$-0.638817\pi$
$523$	−19.3205	−0.844827	−0.422413	+	0.906403i	$0.638817\pi$
$524$	−17.8564	−0.780061
$525$	0	0
$526$	−5.07180	−0.221141
$527$	37.8564	1.64905
$528$	0	0
$529$	−20.8564	−0.906800
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	6.92820	0.299532
$536$	−5.46410	−0.236013
$537$	0	0
$538$	−16.9282	−0.729827
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−0.535898	−0.0230401	−0.0115200	−	0.999934i	$-0.503667\pi$
$541$	−0.535898	−0.0230401	−0.0115200	+	0.999934i	$0.503667\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	3.46410	0.148522
$545$	3.46410	0.148386
$546$	0	0
$547$	38.6410	1.65217	0.826085	−	0.563545i	$-0.190563\pi$
$547$	38.6410	1.65217	0.826085	+	0.563545i	$0.190563\pi$
$548$	−3.46410	−0.147979
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	0	0
$553$	4.00000	0.170097
$554$	−14.0000	−0.594803
$555$	0	0
$556$	−10.9282	−0.463459
$557$	16.1436	0.684026	0.342013	−	0.939695i	$-0.388891\pi$
$557$	16.1436	0.684026	0.342013	+	0.939695i	$0.388891\pi$
$558$	0	0
$559$	0	0
$560$	1.00000	0.0422577
$561$	0	0
$562$	−21.3205	−0.899351
$563$	21.0718	0.888070	0.444035	−	0.896009i	$-0.353547\pi$
$563$	21.0718	0.888070	0.444035	+	0.896009i	$0.353547\pi$
$564$	0	0
$565$	−2.39230	−0.100645
$566$	−24.3923	−1.02529
$567$	0	0
$568$	−1.07180	−0.0449716
$569$	−6.67949	−0.280019	−0.140009	−	0.990150i	$-0.544713\pi$
$569$	−6.67949	−0.280019	−0.140009	+	0.990150i	$0.544713\pi$
$570$	0	0
$571$	21.1769	0.886226	0.443113	−	0.896466i	$-0.353874\pi$
$571$	21.1769	0.886226	0.443113	+	0.896466i	$0.353874\pi$
$572$	3.46410	0.144841
$573$	0	0
$574$	2.00000	0.0834784
$575$	1.46410	0.0610573
$576$	0	0
$577$	−15.8564	−0.660111	−0.330055	−	0.943962i	$-0.607067\pi$
$577$	−15.8564	−0.660111	−0.330055	+	0.943962i	$0.607067\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	8.92820	0.370723
$581$	13.8564	0.574861
$582$	0	0
$583$	6.00000	0.248495
$584$	15.8564	0.656143
$585$	0	0
$586$	18.7846	0.775985
$587$	22.9282	0.946348	0.473174	−	0.880969i	$-0.343108\pi$
$587$	22.9282	0.946348	0.473174	+	0.880969i	$0.343108\pi$
$588$	0	0
$589$	0	0
$590$	8.00000	0.329355
$591$	0	0
$592$	3.46410	0.142374
$593$	12.2487	0.502994	0.251497	−	0.967858i	$-0.419077\pi$
$593$	12.2487	0.502994	0.251497	+	0.967858i	$0.419077\pi$
$594$	0	0
$595$	3.46410	0.142014
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−5.07180	−0.207401
$599$	−23.7128	−0.968879	−0.484440	−	0.874825i	$-0.660976\pi$
$599$	−23.7128	−0.968879	−0.484440	+	0.874825i	$0.660976\pi$
$600$	0	0
$601$	3.07180	0.125301	0.0626506	−	0.998036i	$-0.480045\pi$
$601$	3.07180	0.125301	0.0626506	+	0.998036i	$0.480045\pi$
$602$	0	0
$603$	0	0
$604$	−20.0000	−0.813788
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	4.92820	0.199537
$611$	24.0000	0.970936
$612$	0	0
$613$	−43.8564	−1.77134	−0.885672	−	0.464312i	$-0.846302\pi$
$613$	−43.8564	−1.77134	−0.885672	+	0.464312i	$0.846302\pi$
$614$	−24.3923	−0.984393
$615$	0	0
$616$	1.00000	0.0402911
$617$	−3.46410	−0.139459	−0.0697297	−	0.997566i	$-0.522214\pi$
$617$	−3.46410	−0.139459	−0.0697297	+	0.997566i	$0.522214\pi$
$618$	0	0
$619$	−30.5359	−1.22734	−0.613671	−	0.789562i	$-0.710308\pi$
$619$	−30.5359	−1.22734	−0.613671	+	0.789562i	$0.710308\pi$
$620$	−10.9282	−0.438887
$621$	0	0
$622$	21.4641	0.860632
$623$	8.53590	0.341984
$624$	0	0
$625$	1.00000	0.0400000
$626$	27.8564	1.11337
$627$	0	0
$628$	−6.00000	−0.239426
$629$	12.0000	0.478471
$630$	0	0
$631$	−2.92820	−0.116570	−0.0582850	−	0.998300i	$-0.518563\pi$
$631$	−2.92820	−0.116570	−0.0582850	+	0.998300i	$0.518563\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	−11.8564	−0.470878
$635$	10.9282	0.433673
$636$	0	0
$637$	−3.46410	−0.137253
$638$	8.92820	0.353471
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	0	0
$643$	−33.8564	−1.33517	−0.667583	−	0.744535i	$-0.732671\pi$
$643$	−33.8564	−1.33517	−0.667583	+	0.744535i	$0.732671\pi$
$644$	−1.46410	−0.0576937
$645$	0	0
$646$	0	0
$647$	30.9282	1.21591	0.607957	−	0.793970i	$-0.291989\pi$
$647$	30.9282	1.21591	0.607957	+	0.793970i	$0.291989\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	−3.46410	−0.135873
$651$	0	0
$652$	−19.3205	−0.756649
$653$	−16.9282	−0.662452	−0.331226	−	0.943551i	$-0.607462\pi$
$653$	−16.9282	−0.662452	−0.331226	+	0.943551i	$0.607462\pi$
$654$	0	0
$655$	17.8564	0.697708
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	6.92820	0.270089
$659$	9.85641	0.383951	0.191976	−	0.981400i	$-0.438511\pi$
$659$	9.85641	0.383951	0.191976	+	0.981400i	$0.438511\pi$
$660$	0	0
$661$	43.4641	1.69056	0.845279	−	0.534325i	$-0.179434\pi$
$661$	43.4641	1.69056	0.845279	+	0.534325i	$0.179434\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−13.8564	−0.537733
$665$	0	0
$666$	0	0
$667$	−13.0718	−0.506142
$668$	17.4641	0.675706
$669$	0	0
$670$	5.46410	0.211097
$671$	4.92820	0.190251
$672$	0	0
$673$	−35.8564	−1.38216	−0.691081	−	0.722777i	$-0.742865\pi$
$673$	−35.8564	−1.38216	−0.691081	+	0.722777i	$0.742865\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−19.8564	−0.763144	−0.381572	−	0.924339i	$-0.624617\pi$
$677$	−19.8564	−0.763144	−0.381572	+	0.924339i	$0.624617\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	−3.46410	−0.132842
$681$	0	0
$682$	−10.9282	−0.418463
$683$	−37.5692	−1.43755	−0.718773	−	0.695245i	$-0.755296\pi$
$683$	−37.5692	−1.43755	−0.718773	+	0.695245i	$0.755296\pi$
$684$	0	0
$685$	3.46410	0.132357
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	0	0
$689$	20.7846	0.791831
$690$	0	0
$691$	−14.5359	−0.552972	−0.276486	−	0.961018i	$-0.589170\pi$
$691$	−14.5359	−0.552972	−0.276486	+	0.961018i	$0.589170\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	14.9282	0.566667
$695$	10.9282	0.414530
$696$	0	0
$697$	−6.92820	−0.262424
$698$	−7.07180	−0.267671
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	10.7846	0.407329	0.203665	−	0.979041i	$-0.434715\pi$
$701$	10.7846	0.407329	0.203665	+	0.979041i	$0.434715\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	12.9282	0.486559
$707$	−7.85641	−0.295471
$708$	0	0
$709$	9.71281	0.364772	0.182386	−	0.983227i	$-0.441618\pi$
$709$	9.71281	0.364772	0.182386	+	0.983227i	$0.441618\pi$
$710$	1.07180	0.0402238
$711$	0	0
$712$	−8.53590	−0.319896
$713$	16.0000	0.599205
$714$	0	0
$715$	−3.46410	−0.129550
$716$	6.92820	0.258919
$717$	0	0
$718$	5.46410	0.203918
$719$	−29.4641	−1.09883	−0.549413	−	0.835551i	$-0.685149\pi$
$719$	−29.4641	−1.09883	−0.549413	+	0.835551i	$0.685149\pi$
$720$	0	0
$721$	2.92820	0.109052
$722$	−19.0000	−0.707107
$723$	0	0
$724$	3.46410	0.128742
$725$	−8.92820	−0.331585
$726$	0	0
$727$	18.1436	0.672909	0.336454	−	0.941700i	$-0.390772\pi$
$727$	18.1436	0.672909	0.336454	+	0.941700i	$0.390772\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	−15.8564	−0.586872
$731$	0	0
$732$	0	0
$733$	36.5359	1.34948	0.674742	−	0.738054i	$-0.264255\pi$
$733$	36.5359	1.34948	0.674742	+	0.738054i	$0.264255\pi$
$734$	16.7846	0.619531
$735$	0	0
$736$	1.46410	0.0539675
$737$	5.46410	0.201273
$738$	0	0
$739$	−14.5359	−0.534712	−0.267356	−	0.963598i	$-0.586150\pi$
$739$	−14.5359	−0.534712	−0.267356	+	0.963598i	$0.586150\pi$
$740$	−3.46410	−0.127343
$741$	0	0
$742$	6.00000	0.220267
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	34.7846	1.27356
$747$	0	0
$748$	−3.46410	−0.126660
$749$	6.92820	0.253151
$750$	0	0
$751$	−26.9282	−0.982624	−0.491312	−	0.870984i	$-0.663483\pi$
$751$	−26.9282	−0.982624	−0.491312	+	0.870984i	$0.663483\pi$
$752$	−6.92820	−0.252646
$753$	0	0
$754$	30.9282	1.12634
$755$	20.0000	0.727875
$756$	0	0
$757$	−4.53590	−0.164860	−0.0824300	−	0.996597i	$-0.526268\pi$
$757$	−4.53590	−0.164860	−0.0824300	+	0.996597i	$0.526268\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	−21.7128	−0.787089	−0.393544	−	0.919306i	$-0.628751\pi$
$761$	−21.7128	−0.787089	−0.393544	+	0.919306i	$0.628751\pi$
$762$	0	0
$763$	3.46410	0.125409
$764$	9.85641	0.356592
$765$	0	0
$766$	1.07180	0.0387256
$767$	27.7128	1.00065
$768$	0	0
$769$	28.6410	1.03282	0.516411	−	0.856341i	$-0.327268\pi$
$769$	28.6410	1.03282	0.516411	+	0.856341i	$0.327268\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	15.8564	0.570685
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	0	0
$775$	10.9282	0.392553
$776$	−10.0000	−0.358979
$777$	0	0
$778$	−28.9282	−1.03713
$779$	0	0
$780$	0	0
$781$	1.07180	0.0383519
$782$	5.07180	0.181367
$783$	0	0
$784$	1.00000	0.0357143
$785$	6.00000	0.214149
$786$	0	0
$787$	21.4641	0.765113	0.382556	−	0.923932i	$-0.375044\pi$
$787$	21.4641	0.765113	0.382556	+	0.923932i	$0.375044\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	4.00000	0.142314
$791$	−2.39230	−0.0850606
$792$	0	0
$793$	17.0718	0.606237
$794$	34.7846	1.23446
$795$	0	0
$796$	18.9282	0.670892
$797$	−33.7128	−1.19417	−0.597085	−	0.802178i	$-0.703674\pi$
$797$	−33.7128	−1.19417	−0.597085	+	0.802178i	$0.703674\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	0	0
$802$	19.8564	0.701154
$803$	−15.8564	−0.559560
$804$	0	0
$805$	1.46410	0.0516028
$806$	−37.8564	−1.33344
$807$	0	0
$808$	7.85641	0.276387
$809$	7.17691	0.252327	0.126163	−	0.992009i	$-0.459734\pi$
$809$	7.17691	0.252327	0.126163	+	0.992009i	$0.459734\pi$
$810$	0	0
$811$	−35.7128	−1.25405	−0.627023	−	0.779001i	$-0.715727\pi$
$811$	−35.7128	−1.25405	−0.627023	+	0.779001i	$0.715727\pi$
$812$	8.92820	0.313319
$813$	0	0
$814$	−3.46410	−0.121417
$815$	19.3205	0.676768
$816$	0	0
$817$	0	0
$818$	−7.07180	−0.247260
$819$	0	0
$820$	2.00000	0.0698430
$821$	42.7846	1.49319	0.746597	−	0.665277i	$-0.231687\pi$
$821$	42.7846	1.49319	0.746597	+	0.665277i	$0.231687\pi$
$822$	0	0
$823$	28.7846	1.00337	0.501684	−	0.865051i	$-0.332714\pi$
$823$	28.7846	1.00337	0.501684	+	0.865051i	$0.332714\pi$
$824$	−2.92820	−0.102009
$825$	0	0
$826$	8.00000	0.278356
$827$	30.9282	1.07548	0.537740	−	0.843111i	$-0.319278\pi$
$827$	30.9282	1.07548	0.537740	+	0.843111i	$0.319278\pi$
$828$	0	0
$829$	−16.2487	−0.564341	−0.282171	−	0.959364i	$-0.591054\pi$
$829$	−16.2487	−0.564341	−0.282171	+	0.959364i	$0.591054\pi$
$830$	13.8564	0.480963
$831$	0	0
$832$	−3.46410	−0.120096
$833$	3.46410	0.120024
$834$	0	0
$835$	−17.4641	−0.604370
$836$	0	0
$837$	0	0
$838$	2.92820	0.101153
$839$	27.3205	0.943209	0.471604	−	0.881810i	$-0.343675\pi$
$839$	27.3205	0.943209	0.471604	+	0.881810i	$0.343675\pi$
$840$	0	0
$841$	50.7128	1.74872
$842$	−31.8564	−1.09784
$843$	0	0
$844$	4.39230	0.151189
$845$	1.00000	0.0344010
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−6.00000	−0.206041
$849$	0	0
$850$	3.46410	0.118818
$851$	5.07180	0.173859
$852$	0	0
$853$	26.3923	0.903655	0.451828	−	0.892105i	$-0.350772\pi$
$853$	26.3923	0.903655	0.451828	+	0.892105i	$0.350772\pi$
$854$	4.92820	0.168640
$855$	0	0
$856$	−6.92820	−0.236801
$857$	8.53590	0.291581	0.145790	−	0.989316i	$-0.453427\pi$
$857$	8.53590	0.291581	0.145790	+	0.989316i	$0.453427\pi$
$858$	0	0
$859$	36.3923	1.24169	0.620845	−	0.783934i	$-0.286790\pi$
$859$	36.3923	1.24169	0.620845	+	0.783934i	$0.286790\pi$
$860$	0	0
$861$	0	0
$862$	32.3923	1.10329
$863$	−24.1051	−0.820548	−0.410274	−	0.911962i	$-0.634567\pi$
$863$	−24.1051	−0.820548	−0.410274	+	0.911962i	$0.634567\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	−23.8564	−0.810674
$867$	0	0
$868$	−10.9282	−0.370927
$869$	4.00000	0.135691
$870$	0	0
$871$	18.9282	0.641358
$872$	−3.46410	−0.117309
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	20.1436	0.680201	0.340100	−	0.940389i	$-0.389539\pi$
$877$	20.1436	0.680201	0.340100	+	0.940389i	$0.389539\pi$
$878$	34.9282	1.17877
$879$	0	0
$880$	1.00000	0.0337100
$881$	−0.535898	−0.0180549	−0.00902744	−	0.999959i	$-0.502874\pi$
$881$	−0.535898	−0.0180549	−0.00902744	+	0.999959i	$0.502874\pi$
$882$	0	0
$883$	8.39230	0.282424	0.141212	−	0.989979i	$-0.454900\pi$
$883$	8.39230	0.282424	0.141212	+	0.989979i	$0.454900\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	6.14359	0.206398
$887$	6.53590	0.219454	0.109727	−	0.993962i	$-0.465002\pi$
$887$	6.53590	0.219454	0.109727	+	0.993962i	$0.465002\pi$
$888$	0	0
$889$	10.9282	0.366520
$890$	8.53590	0.286124
$891$	0	0
$892$	27.7128	0.927894
$893$	0	0
$894$	0	0
$895$	−6.92820	−0.231584
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	11.8564	0.395653
$899$	−97.5692	−3.25412
$900$	0	0
$901$	−20.7846	−0.692436
$902$	2.00000	0.0665927
$903$	0	0
$904$	2.39230	0.0795669
$905$	−3.46410	−0.115151
$906$	0	0
$907$	−26.5359	−0.881110	−0.440555	−	0.897726i	$-0.645218\pi$
$907$	−26.5359	−0.881110	−0.440555	+	0.897726i	$0.645218\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	−3.46410	−0.114834
$911$	42.6410	1.41276	0.706380	−	0.707833i	$-0.250327\pi$
$911$	42.6410	1.41276	0.706380	+	0.707833i	$0.250327\pi$
$912$	0	0
$913$	13.8564	0.458580
$914$	−11.8564	−0.392175
$915$	0	0
$916$	16.5359	0.546361
$917$	17.8564	0.589670
$918$	0	0
$919$	12.7846	0.421725	0.210863	−	0.977516i	$-0.432373\pi$
$919$	12.7846	0.421725	0.210863	+	0.977516i	$0.432373\pi$
$920$	−1.46410	−0.0482700
$921$	0	0
$922$	−25.7128	−0.846806
$923$	3.71281	0.122209
$924$	0	0
$925$	3.46410	0.113899
$926$	4.00000	0.131448
$927$	0	0
$928$	−8.92820	−0.293083
$929$	−27.4641	−0.901068	−0.450534	−	0.892759i	$-0.648766\pi$
$929$	−27.4641	−0.901068	−0.450534	+	0.892759i	$0.648766\pi$
$930$	0	0
$931$	0	0
$932$	−19.8564	−0.650418
$933$	0	0
$934$	20.7846	0.680093
$935$	3.46410	0.113288
$936$	0	0
$937$	−14.7846	−0.482992	−0.241496	−	0.970402i	$-0.577638\pi$
$937$	−14.7846	−0.482992	−0.241496	+	0.970402i	$0.577638\pi$
$938$	5.46410	0.178409
$939$	0	0
$940$	6.92820	0.225973
$941$	21.7128	0.707817	0.353909	−	0.935280i	$-0.384852\pi$
$941$	21.7128	0.707817	0.353909	+	0.935280i	$0.384852\pi$
$942$	0	0
$943$	−2.92820	−0.0953554
$944$	−8.00000	−0.260378
$945$	0	0
$946$	0	0
$947$	44.7846	1.45530	0.727652	−	0.685946i	$-0.240612\pi$
$947$	44.7846	1.45530	0.727652	+	0.685946i	$0.240612\pi$
$948$	0	0
$949$	−54.9282	−1.78304
$950$	0	0
$951$	0	0
$952$	−3.46410	−0.112272
$953$	23.0718	0.747369	0.373684	−	0.927556i	$-0.378094\pi$
$953$	23.0718	0.747369	0.373684	+	0.927556i	$0.378094\pi$
$954$	0	0
$955$	−9.85641	−0.318946
$956$	−21.4641	−0.694199
$957$	0	0
$958$	−21.8564	−0.706148
$959$	3.46410	0.111862
$960$	0	0
$961$	88.4256	2.85244
$962$	−12.0000	−0.386896
$963$	0	0
$964$	−18.7846	−0.605012
$965$	−15.8564	−0.510436
$966$	0	0
$967$	−8.78461	−0.282494	−0.141247	−	0.989974i	$-0.545111\pi$
$967$	−8.78461	−0.282494	−0.141247	+	0.989974i	$0.545111\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	10.0000	0.321081
$971$	58.9282	1.89110	0.945548	−	0.325483i	$-0.105527\pi$
$971$	58.9282	1.89110	0.945548	+	0.325483i	$0.105527\pi$
$972$	0	0
$973$	10.9282	0.350342
$974$	−41.8564	−1.34117
$975$	0	0
$976$	−4.92820	−0.157748
$977$	1.60770	0.0514347	0.0257174	−	0.999669i	$-0.491813\pi$
$977$	1.60770	0.0514347	0.0257174	+	0.999669i	$0.491813\pi$
$978$	0	0
$979$	8.53590	0.272808
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−1.85641	−0.0592403
$983$	26.6410	0.849716	0.424858	−	0.905260i	$-0.360324\pi$
$983$	26.6410	0.849716	0.424858	+	0.905260i	$0.360324\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	−30.9282	−0.984955
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−10.9282	−0.347146	−0.173573	−	0.984821i	$-0.555531\pi$
$991$	−10.9282	−0.347146	−0.173573	+	0.984821i	$0.555531\pi$
$992$	10.9282	0.346971
$993$	0	0
$994$	1.07180	0.0339953
$995$	−18.9282	−0.600064
$996$	0	0
$997$	1.60770	0.0509162	0.0254581	−	0.999676i	$-0.491896\pi$
$997$	1.60770	0.0509162	0.0254581	+	0.999676i	$0.491896\pi$
$998$	−12.7846	−0.404690
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.bw.1.1		2
3.2	odd	2		2310.2.a.z.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.z.1.1	✓	2	3.2	odd	2
6930.2.a.bw.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{20} -1.00000 q^{22} +1.46410 q^{23} +1.00000 q^{25} -3.46410 q^{26} -1.00000 q^{28} -8.92820 q^{29} +10.9282 q^{31} +1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} +3.46410 q^{37} -1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} +1.46410 q^{46} -6.92820 q^{47} +1.00000 q^{49} +1.00000 q^{50} -3.46410 q^{52} -6.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} -8.92820 q^{58} -8.00000 q^{59} -4.92820 q^{61} +10.9282 q^{62} +1.00000 q^{64} +3.46410 q^{65} -5.46410 q^{67} +3.46410 q^{68} +1.00000 q^{70} -1.07180 q^{71} +15.8564 q^{73} +3.46410 q^{74} +1.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} -13.8564 q^{83} -3.46410 q^{85} -1.00000 q^{88} -8.53590 q^{89} +3.46410 q^{91} +1.46410 q^{92} -6.92820 q^{94} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{35} - 2 q^{40} - 4 q^{41} - 2 q^{44} - 4 q^{46} + 2 q^{49} + 2 q^{50} - 12 q^{53} + 2 q^{55} - 2 q^{56} - 4 q^{58} - 16 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{70} - 16 q^{71} + 4 q^{73} + 2 q^{77} - 8 q^{79} - 2 q^{80} - 4 q^{82} - 2 q^{88} - 24 q^{89} - 4 q^{92} - 20 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^14 + 2 * q^16 - 2 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^25 - 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^35 - 2 * q^40 - 4 * q^41 - 2 * q^44 - 4 * q^46 + 2 * q^49 + 2 * q^50 - 12 * q^53 + 2 * q^55 - 2 * q^56 - 4 * q^58 - 16 * q^59 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^67 + 2 * q^70 - 16 * q^71 + 4 * q^73 + 2 * q^77 - 8 * q^79 - 2 * q^80 - 4 * q^82 - 2 * q^88 - 24 * q^89 - 4 * q^92 - 20 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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