LMFDB - Embedded newform 6930.2.a.cl.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.31955$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -5.87847 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.63910 q^{17} -5.57834 q^{19} +1.00000 q^{20} -1.00000 q^{22} -5.57834 q^{23} +1.00000 q^{25} -5.87847 q^{26} -1.00000 q^{28} +9.45681 q^{29} +6.00000 q^{31} +1.00000 q^{32} +4.63910 q^{34} -1.00000 q^{35} +2.30013 q^{37} -5.57834 q^{38} +1.00000 q^{40} +3.06077 q^{41} +10.5176 q^{43} -1.00000 q^{44} -5.57834 q^{46} -8.51757 q^{47} +1.00000 q^{49} +1.00000 q^{50} -5.87847 q^{52} +9.45681 q^{53} -1.00000 q^{55} -1.00000 q^{56} +9.45681 q^{58} -7.23937 q^{59} +14.5176 q^{61} +6.00000 q^{62} +1.00000 q^{64} -5.87847 q^{65} +8.00000 q^{67} +4.63910 q^{68} -1.00000 q^{70} +7.15667 q^{71} +12.3960 q^{73} +2.30013 q^{74} -5.57834 q^{76} +1.00000 q^{77} -10.8565 q^{79} +1.00000 q^{80} +3.06077 q^{82} +13.8785 q^{83} +4.63910 q^{85} +10.5176 q^{86} -1.00000 q^{88} -3.87847 q^{89} +5.87847 q^{91} -5.57834 q^{92} -8.51757 q^{94} -5.57834 q^{95} +7.57834 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} + 8 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} - 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 4 q^{37} - 2 q^{38} + 3 q^{40} + 18 q^{41} + 8 q^{43} - 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} + 3 q^{50} - 4 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 18 q^{62} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 3 q^{70} - 8 q^{71} - 4 q^{73} + 4 q^{74} - 2 q^{76} + 3 q^{77} - 6 q^{79} + 3 q^{80} + 18 q^{82} + 24 q^{83} + 8 q^{85} + 8 q^{86} - 3 q^{88} + 6 q^{89} - 2 q^{92} - 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8 + 3 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 + 8 * q^17 - 2 * q^19 + 3 * q^20 - 3 * q^22 - 2 * q^23 + 3 * q^25 - 3 * q^28 - 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 - 3 * q^35 + 4 * q^37 - 2 * q^38 + 3 * q^40 + 18 * q^41 + 8 * q^43 - 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 + 3 * q^50 - 4 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 10 * q^59 + 20 * q^61 + 18 * q^62 + 3 * q^64 + 24 * q^67 + 8 * q^68 - 3 * q^70 - 8 * q^71 - 4 * q^73 + 4 * q^74 - 2 * q^76 + 3 * q^77 - 6 * q^79 + 3 * q^80 + 18 * q^82 + 24 * q^83 + 8 * q^85 + 8 * q^86 - 3 * q^88 + 6 * q^89 - 2 * q^92 - 2 * q^94 - 2 * q^95 + 8 * q^97 + 3 * 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
$13$	−5.87847	−1.63039	−0.815197	−	0.579184i	$-0.803371\pi$
$13$	−5.87847	−1.63039	−0.815197	+	0.579184i	$0.803371\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.63910	1.12515	0.562574	−	0.826747i	$-0.309811\pi$
$17$	4.63910	1.12515	0.562574	+	0.826747i	$0.309811\pi$
$18$	0	0
$19$	−5.57834	−1.27976	−0.639879	−	0.768476i	$-0.721016\pi$
$19$	−5.57834	−1.27976	−0.639879	+	0.768476i	$0.721016\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−5.57834	−1.16316	−0.581582	−	0.813488i	$-0.697566\pi$
$23$	−5.57834	−1.16316	−0.581582	+	0.813488i	$0.697566\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.87847	−1.15286
$27$	0	0
$28$	−1.00000	−0.188982
$29$	9.45681	1.75608	0.878042	−	0.478583i	$-0.158849\pi$
$29$	9.45681	1.75608	0.878042	+	0.478583i	$0.158849\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.63910	0.795599
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.30013	0.378140	0.189070	−	0.981964i	$-0.439453\pi$
$37$	2.30013	0.378140	0.189070	+	0.981964i	$0.439453\pi$
$38$	−5.57834	−0.904926
$39$	0	0
$40$	1.00000	0.158114
$41$	3.06077	0.478011	0.239006	−	0.971018i	$-0.423179\pi$
$41$	3.06077	0.478011	0.239006	+	0.971018i	$0.423179\pi$
$42$	0	0
$43$	10.5176	1.60391	0.801957	−	0.597381i	$-0.203792\pi$
$43$	10.5176	1.60391	0.801957	+	0.597381i	$0.203792\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−5.57834	−0.822481
$47$	−8.51757	−1.24242	−0.621208	−	0.783646i	$-0.713358\pi$
$47$	−8.51757	−1.24242	−0.621208	+	0.783646i	$0.713358\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.87847	−0.815197
$53$	9.45681	1.29899	0.649496	−	0.760365i	$-0.274980\pi$
$53$	9.45681	1.29899	0.649496	+	0.760365i	$0.274980\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	9.45681	1.24174
$59$	−7.23937	−0.942485	−0.471243	−	0.882004i	$-0.656194\pi$
$59$	−7.23937	−0.942485	−0.471243	+	0.882004i	$0.656194\pi$
$60$	0	0
$61$	14.5176	1.85878	0.929392	−	0.369093i	$-0.120332\pi$
$61$	14.5176	1.85878	0.929392	+	0.369093i	$0.120332\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.87847	−0.729134
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	4.63910	0.562574
$69$	0	0
$70$	−1.00000	−0.119523
$71$	7.15667	0.849341	0.424670	−	0.905348i	$-0.360390\pi$
$71$	7.15667	0.849341	0.424670	+	0.905348i	$0.360390\pi$
$72$	0	0
$73$	12.3960	1.45085	0.725423	−	0.688303i	$-0.241644\pi$
$73$	12.3960	1.45085	0.725423	+	0.688303i	$0.241644\pi$
$74$	2.30013	0.267385
$75$	0	0
$76$	−5.57834	−0.639879
$77$	1.00000	0.113961
$78$	0	0
$79$	−10.8565	−1.22146	−0.610728	−	0.791840i	$-0.709123\pi$
$79$	−10.8565	−1.22146	−0.610728	+	0.791840i	$0.709123\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	3.06077	0.338005
$83$	13.8785	1.52336	0.761680	−	0.647953i	$-0.224375\pi$
$83$	13.8785	1.52336	0.761680	+	0.647953i	$0.224375\pi$
$84$	0	0
$85$	4.63910	0.503181
$86$	10.5176	1.13414
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−3.87847	−0.411117	−0.205558	−	0.978645i	$-0.565901\pi$
$89$	−3.87847	−0.411117	−0.205558	+	0.978645i	$0.565901\pi$
$90$	0	0
$91$	5.87847	0.616231
$92$	−5.57834	−0.581582
$93$	0	0
$94$	−8.51757	−0.878520
$95$	−5.57834	−0.572325
$96$	0	0
$97$	7.57834	0.769463	0.384732	−	0.923028i	$-0.374294\pi$
$97$	7.57834	0.769463	0.384732	+	0.923028i	$0.374294\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−11.7958	−1.17372	−0.586862	−	0.809687i	$-0.699637\pi$
$101$	−11.7958	−1.17372	−0.586862	+	0.809687i	$0.699637\pi$
$102$	0	0
$103$	17.1178	1.68667	0.843335	−	0.537388i	$-0.180589\pi$
$103$	17.1178	1.68667	0.843335	+	0.537388i	$0.180589\pi$
$104$	−5.87847	−0.576431
$105$	0	0
$106$	9.45681	0.918526
$107$	−2.51757	−0.243383	−0.121691	−	0.992568i	$-0.538832\pi$
$107$	−2.51757	−0.243383	−0.121691	+	0.992568i	$0.538832\pi$
$108$	0	0
$109$	10.3001	0.986574	0.493287	−	0.869867i	$-0.335795\pi$
$109$	10.3001	0.986574	0.493287	+	0.869867i	$0.335795\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−5.57834	−0.520183
$116$	9.45681	0.878042
$117$	0	0
$118$	−7.23937	−0.666438
$119$	−4.63910	−0.425266
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.5176	1.31436
$123$	0	0
$124$	6.00000	0.538816
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4787	0.929837	0.464919	−	0.885353i	$-0.346084\pi$
$127$	10.4787	0.929837	0.464919	+	0.885353i	$0.346084\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.87847	−0.515576
$131$	1.57834	0.137900	0.0689499	−	0.997620i	$-0.478035\pi$
$131$	1.57834	0.137900	0.0689499	+	0.997620i	$0.478035\pi$
$132$	0	0
$133$	5.57834	0.483703
$134$	8.00000	0.691095
$135$	0	0
$136$	4.63910	0.397800
$137$	3.27820	0.280076	0.140038	−	0.990146i	$-0.455278\pi$
$137$	3.27820	0.280076	0.140038	+	0.990146i	$0.455278\pi$
$138$	0	0
$139$	−16.0571	−1.36194	−0.680972	−	0.732310i	$-0.738442\pi$
$139$	−16.0571	−1.36194	−0.680972	+	0.732310i	$0.738442\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	7.15667	0.600575
$143$	5.87847	0.491582
$144$	0	0
$145$	9.45681	0.785345
$146$	12.3960	1.02590
$147$	0	0
$148$	2.30013	0.189070
$149$	−10.7350	−0.879446	−0.439723	−	0.898133i	$-0.644923\pi$
$149$	−10.7350	−0.879446	−0.439723	+	0.898133i	$0.644923\pi$
$150$	0	0
$151$	−2.17860	−0.177292	−0.0886461	−	0.996063i	$-0.528254\pi$
$151$	−2.17860	−0.177292	−0.0886461	+	0.996063i	$0.528254\pi$
$152$	−5.57834	−0.452463
$153$	0	0
$154$	1.00000	0.0805823
$155$	6.00000	0.481932
$156$	0	0
$157$	−17.1567	−1.36925	−0.684626	−	0.728895i	$-0.740034\pi$
$157$	−17.1567	−1.36925	−0.684626	+	0.728895i	$0.740034\pi$
$158$	−10.8565	−0.863700
$159$	0	0
$160$	1.00000	0.0790569
$161$	5.57834	0.439635
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	3.06077	0.239006
$165$	0	0
$166$	13.8785	1.07718
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	21.5564	1.65819
$170$	4.63910	0.355803
$171$	0	0
$172$	10.5176	0.801957
$173$	5.48243	0.416821	0.208411	−	0.978041i	$-0.433171\pi$
$173$	5.48243	0.416821	0.208411	+	0.978041i	$0.433171\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−3.87847	−0.290704
$179$	−1.23937	−0.0926347	−0.0463174	−	0.998927i	$-0.514749\pi$
$179$	−1.23937	−0.0926347	−0.0463174	+	0.998927i	$0.514749\pi$
$180$	0	0
$181$	8.55641	0.635993	0.317996	−	0.948092i	$-0.396990\pi$
$181$	8.55641	0.635993	0.317996	+	0.948092i	$0.396990\pi$
$182$	5.87847	0.435741
$183$	0	0
$184$	−5.57834	−0.411240
$185$	2.30013	0.169109
$186$	0	0
$187$	−4.63910	−0.339245
$188$	−8.51757	−0.621208
$189$	0	0
$190$	−5.57834	−0.404695
$191$	1.27820	0.0924875	0.0462438	−	0.998930i	$-0.485275\pi$
$191$	1.27820	0.0924875	0.0462438	+	0.998930i	$0.485275\pi$
$192$	0	0
$193$	−10.6391	−0.765819	−0.382910	−	0.923786i	$-0.625078\pi$
$193$	−10.6391	−0.765819	−0.382910	+	0.923786i	$0.625078\pi$
$194$	7.57834	0.544093
$195$	0	0
$196$	1.00000	0.0714286
$197$	−7.87847	−0.561318	−0.280659	−	0.959808i	$-0.590553\pi$
$197$	−7.87847	−0.561318	−0.280659	+	0.959808i	$0.590553\pi$
$198$	0	0
$199$	0.517571	0.0366897	0.0183448	−	0.999832i	$-0.494160\pi$
$199$	0.517571	0.0366897	0.0183448	+	0.999832i	$0.494160\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−11.7958	−0.829948
$203$	−9.45681	−0.663738
$204$	0	0
$205$	3.06077	0.213773
$206$	17.1178	1.19266
$207$	0	0
$208$	−5.87847	−0.407599
$209$	5.57834	0.385862
$210$	0	0
$211$	2.72180	0.187376	0.0936881	−	0.995602i	$-0.470134\pi$
$211$	2.72180	0.187376	0.0936881	+	0.995602i	$0.470134\pi$
$212$	9.45681	0.649496
$213$	0	0
$214$	−2.51757	−0.172098
$215$	10.5176	0.717292
$216$	0	0
$217$	−6.00000	−0.407307
$218$	10.3001	0.697613
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	−27.2708	−1.83443
$222$	0	0
$223$	−22.3960	−1.49975	−0.749875	−	0.661580i	$-0.769886\pi$
$223$	−22.3960	−1.49975	−0.749875	+	0.661580i	$0.769886\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	−13.0351	−0.865173	−0.432586	−	0.901592i	$-0.642399\pi$
$227$	−13.0351	−0.865173	−0.432586	+	0.901592i	$0.642399\pi$
$228$	0	0
$229$	−6.63910	−0.438724	−0.219362	−	0.975644i	$-0.570398\pi$
$229$	−6.63910	−0.438724	−0.219362	+	0.975644i	$0.570398\pi$
$230$	−5.57834	−0.367825
$231$	0	0
$232$	9.45681	0.620870
$233$	−19.0351	−1.24703	−0.623517	−	0.781810i	$-0.714297\pi$
$233$	−19.0351	−1.24703	−0.623517	+	0.781810i	$0.714297\pi$
$234$	0	0
$235$	−8.51757	−0.555625
$236$	−7.23937	−0.471243
$237$	0	0
$238$	−4.63910	−0.300708
$239$	−22.6135	−1.46274	−0.731372	−	0.681979i	$-0.761120\pi$
$239$	−22.6135	−1.46274	−0.731372	+	0.681979i	$0.761120\pi$
$240$	0	0
$241$	−17.9744	−1.15783	−0.578916	−	0.815387i	$-0.696524\pi$
$241$	−17.9744	−1.15783	−0.578916	+	0.815387i	$0.696524\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	14.5176	0.929392
$245$	1.00000	0.0638877
$246$	0	0
$247$	32.7921	2.08651
$248$	6.00000	0.381000
$249$	0	0
$250$	1.00000	0.0632456
$251$	−8.76063	−0.552966	−0.276483	−	0.961019i	$-0.589169\pi$
$251$	−8.76063	−0.552966	−0.276483	+	0.961019i	$0.589169\pi$
$252$	0	0
$253$	5.57834	0.350707
$254$	10.4787	0.657494
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.09960	0.318104	0.159052	−	0.987270i	$-0.449156\pi$
$257$	5.09960	0.318104	0.159052	+	0.987270i	$0.449156\pi$
$258$	0	0
$259$	−2.30013	−0.142923
$260$	−5.87847	−0.364567
$261$	0	0
$262$	1.57834	0.0975100
$263$	11.7569	0.724964	0.362482	−	0.931991i	$-0.381929\pi$
$263$	11.7569	0.724964	0.362482	+	0.931991i	$0.381929\pi$
$264$	0	0
$265$	9.45681	0.580927
$266$	5.57834	0.342030
$267$	0	0
$268$	8.00000	0.488678
$269$	11.6742	0.711791	0.355896	−	0.934526i	$-0.384176\pi$
$269$	11.6742	0.711791	0.355896	+	0.934526i	$0.384176\pi$
$270$	0	0
$271$	17.6354	1.07127	0.535637	−	0.844448i	$-0.320071\pi$
$271$	17.6354	1.07127	0.535637	+	0.844448i	$0.320071\pi$
$272$	4.63910	0.281287
$273$	0	0
$274$	3.27820	0.198043
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−16.0571	−0.963039
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−1.15667	−0.0690013	−0.0345007	−	0.999405i	$-0.510984\pi$
$281$	−1.15667	−0.0690013	−0.0345007	+	0.999405i	$0.510984\pi$
$282$	0	0
$283$	17.2782	1.02708	0.513541	−	0.858065i	$-0.328333\pi$
$283$	17.2782	1.02708	0.513541	+	0.858065i	$0.328333\pi$
$284$	7.15667	0.424670
$285$	0	0
$286$	5.87847	0.347601
$287$	−3.06077	−0.180671
$288$	0	0
$289$	4.52126	0.265957
$290$	9.45681	0.555323
$291$	0	0
$292$	12.3960	0.725423
$293$	−1.87847	−0.109741	−0.0548707	−	0.998493i	$-0.517475\pi$
$293$	−1.87847	−0.109741	−0.0548707	+	0.998493i	$0.517475\pi$
$294$	0	0
$295$	−7.23937	−0.421492
$296$	2.30013	0.133693
$297$	0	0
$298$	−10.7350	−0.621862
$299$	32.7921	1.89642
$300$	0	0
$301$	−10.5176	−0.606223
$302$	−2.17860	−0.125365
$303$	0	0
$304$	−5.57834	−0.319940
$305$	14.5176	0.831274
$306$	0	0
$307$	8.60027	0.490843	0.245422	−	0.969416i	$-0.421074\pi$
$307$	8.60027	0.490843	0.245422	+	0.969416i	$0.421074\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	6.00000	0.340777
$311$	−13.7569	−0.780084	−0.390042	−	0.920797i	$-0.627540\pi$
$311$	−13.7569	−0.780084	−0.390042	+	0.920797i	$0.627540\pi$
$312$	0	0
$313$	2.90040	0.163940	0.0819701	−	0.996635i	$-0.473879\pi$
$313$	2.90040	0.163940	0.0819701	+	0.996635i	$0.473879\pi$
$314$	−17.1567	−0.968207
$315$	0	0
$316$	−10.8565	−0.610728
$317$	−19.3353	−1.08598	−0.542989	−	0.839740i	$-0.682707\pi$
$317$	−19.3353	−1.08598	−0.542989	+	0.839740i	$0.682707\pi$
$318$	0	0
$319$	−9.45681	−0.529479
$320$	1.00000	0.0559017
$321$	0	0
$322$	5.57834	0.310869
$323$	−25.8785	−1.43992
$324$	0	0
$325$	−5.87847	−0.326079
$326$	−8.00000	−0.443079
$327$	0	0
$328$	3.06077	0.169002
$329$	8.51757	0.469589
$330$	0	0
$331$	10.5176	0.578098	0.289049	−	0.957314i	$-0.406661\pi$
$331$	10.5176	0.578098	0.289049	+	0.957314i	$0.406661\pi$
$332$	13.8785	0.761680
$333$	0	0
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	7.27820	0.396469	0.198234	−	0.980155i	$-0.436479\pi$
$337$	7.27820	0.396469	0.198234	+	0.980155i	$0.436479\pi$
$338$	21.5564	1.17251
$339$	0	0
$340$	4.63910	0.251591
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	10.5176	0.567069
$345$	0	0
$346$	5.48243	0.294737
$347$	14.2745	0.766296	0.383148	−	0.923687i	$-0.374840\pi$
$347$	14.2745	0.766296	0.383148	+	0.923687i	$0.374840\pi$
$348$	0	0
$349$	0.396041	0.0211996	0.0105998	−	0.999944i	$-0.496626\pi$
$349$	0.396041	0.0211996	0.0105998	+	0.999944i	$0.496626\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	5.09960	0.271424	0.135712	−	0.990748i	$-0.456668\pi$
$353$	5.09960	0.271424	0.135712	+	0.990748i	$0.456668\pi$
$354$	0	0
$355$	7.15667	0.379837
$356$	−3.87847	−0.205558
$357$	0	0
$358$	−1.23937	−0.0655026
$359$	18.3704	0.969554	0.484777	−	0.874638i	$-0.338901\pi$
$359$	18.3704	0.969554	0.484777	+	0.874638i	$0.338901\pi$
$360$	0	0
$361$	12.1178	0.637781
$362$	8.55641	0.449715
$363$	0	0
$364$	5.87847	0.308116
$365$	12.3960	0.648838
$366$	0	0
$367$	17.1178	0.893544	0.446772	−	0.894648i	$-0.352574\pi$
$367$	17.1178	0.893544	0.446772	+	0.894648i	$0.352574\pi$
$368$	−5.57834	−0.290791
$369$	0	0
$370$	2.30013	0.119578
$371$	−9.45681	−0.490973
$372$	0	0
$373$	−6.35721	−0.329164	−0.164582	−	0.986363i	$-0.552627\pi$
$373$	−6.35721	−0.329164	−0.164582	+	0.986363i	$0.552627\pi$
$374$	−4.63910	−0.239882
$375$	0	0
$376$	−8.51757	−0.439260
$377$	−55.5915	−2.86311
$378$	0	0
$379$	−5.03514	−0.258638	−0.129319	−	0.991603i	$-0.541279\pi$
$379$	−5.03514	−0.258638	−0.129319	+	0.991603i	$0.541279\pi$
$380$	−5.57834	−0.286163
$381$	0	0
$382$	1.27820	0.0653986
$383$	7.43118	0.379716	0.189858	−	0.981812i	$-0.439197\pi$
$383$	7.43118	0.379716	0.189858	+	0.981812i	$0.439197\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	−10.6391	−0.541516
$387$	0	0
$388$	7.57834	0.384732
$389$	5.15667	0.261454	0.130727	−	0.991418i	$-0.458269\pi$
$389$	5.15667	0.261454	0.130727	+	0.991418i	$0.458269\pi$
$390$	0	0
$391$	−25.8785	−1.30873
$392$	1.00000	0.0505076
$393$	0	0
$394$	−7.87847	−0.396912
$395$	−10.8565	−0.546252
$396$	0	0
$397$	−19.6354	−0.985473	−0.492736	−	0.870179i	$-0.664003\pi$
$397$	−19.6354	−0.985473	−0.492736	+	0.870179i	$0.664003\pi$
$398$	0.517571	0.0259435
$399$	0	0
$400$	1.00000	0.0500000
$401$	5.11784	0.255573	0.127786	−	0.991802i	$-0.459213\pi$
$401$	5.11784	0.255573	0.127786	+	0.991802i	$0.459213\pi$
$402$	0	0
$403$	−35.2708	−1.75696
$404$	−11.7958	−0.586862
$405$	0	0
$406$	−9.45681	−0.469333
$407$	−2.30013	−0.114013
$408$	0	0
$409$	2.21744	0.109645	0.0548226	−	0.998496i	$-0.482541\pi$
$409$	2.21744	0.109645	0.0548226	+	0.998496i	$0.482541\pi$
$410$	3.06077	0.151160
$411$	0	0
$412$	17.1178	0.843335
$413$	7.23937	0.356226
$414$	0	0
$415$	13.8785	0.681267
$416$	−5.87847	−0.288216
$417$	0	0
$418$	5.57834	0.272845
$419$	4.51757	0.220698	0.110349	−	0.993893i	$-0.464803\pi$
$419$	4.51757	0.220698	0.110349	+	0.993893i	$0.464803\pi$
$420$	0	0
$421$	−12.3133	−0.600116	−0.300058	−	0.953921i	$-0.597006\pi$
$421$	−12.3133	−0.600116	−0.300058	+	0.953921i	$0.597006\pi$
$422$	2.72180	0.132495
$423$	0	0
$424$	9.45681	0.459263
$425$	4.63910	0.225029
$426$	0	0
$427$	−14.5176	−0.702555
$428$	−2.51757	−0.121691
$429$	0	0
$430$	10.5176	0.507202
$431$	2.17860	0.104940	0.0524698	−	0.998623i	$-0.483291\pi$
$431$	2.17860	0.104940	0.0524698	+	0.998623i	$0.483291\pi$
$432$	0	0
$433$	5.02193	0.241339	0.120669	−	0.992693i	$-0.461496\pi$
$433$	5.02193	0.241339	0.120669	+	0.992693i	$0.461496\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	10.3001	0.493287
$437$	31.1178	1.48857
$438$	0	0
$439$	7.15667	0.341569	0.170785	−	0.985308i	$-0.445370\pi$
$439$	7.15667	0.341569	0.170785	+	0.985308i	$0.445370\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−27.2708	−1.29714
$443$	19.5139	0.927132	0.463566	−	0.886062i	$-0.346570\pi$
$443$	19.5139	0.927132	0.463566	+	0.886062i	$0.346570\pi$
$444$	0	0
$445$	−3.87847	−0.183857
$446$	−22.3960	−1.06048
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−1.36090	−0.0642248	−0.0321124	−	0.999484i	$-0.510223\pi$
$449$	−1.36090	−0.0642248	−0.0321124	+	0.999484i	$0.510223\pi$
$450$	0	0
$451$	−3.06077	−0.144126
$452$	14.0000	0.658505
$453$	0	0
$454$	−13.0351	−0.611770
$455$	5.87847	0.275587
$456$	0	0
$457$	4.47874	0.209506	0.104753	−	0.994498i	$-0.466595\pi$
$457$	4.47874	0.209506	0.104753	+	0.994498i	$0.466595\pi$
$458$	−6.63910	−0.310225
$459$	0	0
$460$	−5.57834	−0.260091
$461$	36.1530	1.68381	0.841906	−	0.539624i	$-0.181434\pi$
$461$	36.1530	1.68381	0.841906	+	0.539624i	$0.181434\pi$
$462$	0	0
$463$	21.0922	0.980238	0.490119	−	0.871655i	$-0.336953\pi$
$463$	21.0922	0.980238	0.490119	+	0.871655i	$0.336953\pi$
$464$	9.45681	0.439021
$465$	0	0
$466$	−19.0351	−0.881786
$467$	−19.1311	−0.885279	−0.442640	−	0.896700i	$-0.645958\pi$
$467$	−19.1311	−0.885279	−0.442640	+	0.896700i	$0.645958\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	−8.51757	−0.392886
$471$	0	0
$472$	−7.23937	−0.333219
$473$	−10.5176	−0.483598
$474$	0	0
$475$	−5.57834	−0.255952
$476$	−4.63910	−0.212633
$477$	0	0
$478$	−22.6135	−1.03432
$479$	−5.27820	−0.241167	−0.120584	−	0.992703i	$-0.538477\pi$
$479$	−5.27820	−0.241167	−0.120584	+	0.992703i	$0.538477\pi$
$480$	0	0
$481$	−13.5213	−0.616517
$482$	−17.9744	−0.818710
$483$	0	0
$484$	1.00000	0.0454545
$485$	7.57834	0.344115
$486$	0	0
$487$	−19.8140	−0.897859	−0.448929	−	0.893567i	$-0.648194\pi$
$487$	−19.8140	−0.897859	−0.448929	+	0.893567i	$0.648194\pi$
$488$	14.5176	0.657180
$489$	0	0
$490$	1.00000	0.0451754
$491$	6.47874	0.292381	0.146191	−	0.989256i	$-0.453299\pi$
$491$	6.47874	0.292381	0.146191	+	0.989256i	$0.453299\pi$
$492$	0	0
$493$	43.8711	1.97585
$494$	32.7921	1.47539
$495$	0	0
$496$	6.00000	0.269408
$497$	−7.15667	−0.321021
$498$	0	0
$499$	−21.2394	−0.950805	−0.475402	−	0.879768i	$-0.657698\pi$
$499$	−21.2394	−0.950805	−0.475402	+	0.879768i	$0.657698\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−8.76063	−0.391006
$503$	−42.2357	−1.88320	−0.941598	−	0.336739i	$-0.890676\pi$
$503$	−42.2357	−1.88320	−0.941598	+	0.336739i	$0.890676\pi$
$504$	0	0
$505$	−11.7958	−0.524905
$506$	5.57834	0.247987
$507$	0	0
$508$	10.4787	0.464919
$509$	−38.8698	−1.72287	−0.861436	−	0.507867i	$-0.830434\pi$
$509$	−38.8698	−1.72287	−0.861436	+	0.507867i	$0.830434\pi$
$510$	0	0
$511$	−12.3960	−0.548369
$512$	1.00000	0.0441942
$513$	0	0
$514$	5.09960	0.224934
$515$	17.1178	0.754302
$516$	0	0
$517$	8.51757	0.374602
$518$	−2.30013	−0.101062
$519$	0	0
$520$	−5.87847	−0.257788
$521$	−25.9488	−1.13684	−0.568418	−	0.822740i	$-0.692444\pi$
$521$	−25.9488	−1.13684	−0.568418	+	0.822740i	$0.692444\pi$
$522$	0	0
$523$	42.7482	1.86925	0.934625	−	0.355636i	$-0.115736\pi$
$523$	42.7482	1.86925	0.934625	+	0.355636i	$0.115736\pi$
$524$	1.57834	0.0689499
$525$	0	0
$526$	11.7569	0.512627
$527$	27.8346	1.21249
$528$	0	0
$529$	8.11784	0.352949
$530$	9.45681	0.410777
$531$	0	0
$532$	5.57834	0.241852
$533$	−17.9926	−0.779347
$534$	0	0
$535$	−2.51757	−0.108844
$536$	8.00000	0.345547
$537$	0	0
$538$	11.6742	0.503312
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−6.73501	−0.289561	−0.144780	−	0.989464i	$-0.546248\pi$
$541$	−6.73501	−0.289561	−0.144780	+	0.989464i	$0.546248\pi$
$542$	17.6354	0.757506
$543$	0	0
$544$	4.63910	0.198900
$545$	10.3001	0.441209
$546$	0	0
$547$	23.7569	1.01577	0.507887	−	0.861424i	$-0.330427\pi$
$547$	23.7569	1.01577	0.507887	+	0.861424i	$0.330427\pi$
$548$	3.27820	0.140038
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−52.7532	−2.24736
$552$	0	0
$553$	10.8565	0.461667
$554$	22.0000	0.934690
$555$	0	0
$556$	−16.0571	−0.680972
$557$	34.1918	1.44875	0.724377	−	0.689404i	$-0.242128\pi$
$557$	34.1918	1.44875	0.724377	+	0.689404i	$0.242128\pi$
$558$	0	0
$559$	−61.8272	−2.61501
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−1.15667	−0.0487913
$563$	31.3485	1.32118	0.660591	−	0.750746i	$-0.270306\pi$
$563$	31.3485	1.32118	0.660591	+	0.750746i	$0.270306\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	17.2782	0.726257
$567$	0	0
$568$	7.15667	0.300287
$569$	38.5490	1.61606	0.808030	−	0.589142i	$-0.200534\pi$
$569$	38.5490	1.61606	0.808030	+	0.589142i	$0.200534\pi$
$570$	0	0
$571$	26.9136	1.12630	0.563150	−	0.826355i	$-0.309589\pi$
$571$	26.9136	1.12630	0.563150	+	0.826355i	$0.309589\pi$
$572$	5.87847	0.245791
$573$	0	0
$574$	−3.06077	−0.127754
$575$	−5.57834	−0.232633
$576$	0	0
$577$	−15.3353	−0.638416	−0.319208	−	0.947685i	$-0.603417\pi$
$577$	−15.3353	−0.638416	−0.319208	+	0.947685i	$0.603417\pi$
$578$	4.52126	0.188060
$579$	0	0
$580$	9.45681	0.392673
$581$	−13.8785	−0.575776
$582$	0	0
$583$	−9.45681	−0.391661
$584$	12.3960	0.512952
$585$	0	0
$586$	−1.87847	−0.0775989
$587$	21.2526	0.877188	0.438594	−	0.898685i	$-0.355477\pi$
$587$	21.2526	0.877188	0.438594	+	0.898685i	$0.355477\pi$
$588$	0	0
$589$	−33.4700	−1.37911
$590$	−7.23937	−0.298040
$591$	0	0
$592$	2.30013	0.0945349
$593$	−19.1955	−0.788265	−0.394133	−	0.919054i	$-0.628955\pi$
$593$	−19.1955	−0.788265	−0.394133	+	0.919054i	$0.628955\pi$
$594$	0	0
$595$	−4.63910	−0.190185
$596$	−10.7350	−0.439723
$597$	0	0
$598$	32.7921	1.34097
$599$	−3.39973	−0.138909	−0.0694547	−	0.997585i	$-0.522126\pi$
$599$	−3.39973	−0.138909	−0.0694547	+	0.997585i	$0.522126\pi$
$600$	0	0
$601$	−7.90409	−0.322415	−0.161207	−	0.986921i	$-0.551539\pi$
$601$	−7.90409	−0.322415	−0.161207	+	0.986921i	$0.551539\pi$
$602$	−10.5176	−0.428664
$603$	0	0
$604$	−2.17860	−0.0886461
$605$	1.00000	0.0406558
$606$	0	0
$607$	−23.7181	−0.962688	−0.481344	−	0.876532i	$-0.659851\pi$
$607$	−23.7181	−0.962688	−0.481344	+	0.876532i	$0.659851\pi$
$608$	−5.57834	−0.226231
$609$	0	0
$610$	14.5176	0.587799
$611$	50.0703	2.02563
$612$	0	0
$613$	40.9136	1.65249	0.826243	−	0.563314i	$-0.190474\pi$
$613$	40.9136	1.65249	0.826243	+	0.563314i	$0.190474\pi$
$614$	8.60027	0.347079
$615$	0	0
$616$	1.00000	0.0402911
$617$	30.3572	1.22214	0.611068	−	0.791578i	$-0.290740\pi$
$617$	30.3572	1.22214	0.611068	+	0.791578i	$0.290740\pi$
$618$	0	0
$619$	−34.9963	−1.40662	−0.703310	−	0.710883i	$-0.748295\pi$
$619$	−34.9963	−1.40662	−0.703310	+	0.710883i	$0.748295\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	−13.7569	−0.551603
$623$	3.87847	0.155388
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.90040	0.115923
$627$	0	0
$628$	−17.1567	−0.684626
$629$	10.6706	0.425463
$630$	0	0
$631$	38.2357	1.52214	0.761069	−	0.648671i	$-0.224675\pi$
$631$	38.2357	1.52214	0.761069	+	0.648671i	$0.224675\pi$
$632$	−10.8565	−0.431850
$633$	0	0
$634$	−19.3353	−0.767902
$635$	10.4787	0.415836
$636$	0	0
$637$	−5.87847	−0.232913
$638$	−9.45681	−0.374399
$639$	0	0
$640$	1.00000	0.0395285
$641$	25.5139	1.00774	0.503869	−	0.863780i	$-0.331909\pi$
$641$	25.5139	1.00774	0.503869	+	0.863780i	$0.331909\pi$
$642$	0	0
$643$	−37.8528	−1.49277	−0.746385	−	0.665514i	$-0.768212\pi$
$643$	−37.8528	−1.49277	−0.746385	+	0.665514i	$0.768212\pi$
$644$	5.57834	0.219817
$645$	0	0
$646$	−25.8785	−1.01817
$647$	24.7094	0.971426	0.485713	−	0.874118i	$-0.338560\pi$
$647$	24.7094	0.971426	0.485713	+	0.874118i	$0.338560\pi$
$648$	0	0
$649$	7.23937	0.284170
$650$	−5.87847	−0.230573
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−16.7789	−0.656608	−0.328304	−	0.944572i	$-0.606477\pi$
$653$	−16.7789	−0.656608	−0.328304	+	0.944572i	$0.606477\pi$
$654$	0	0
$655$	1.57834	0.0616707
$656$	3.06077	0.119503
$657$	0	0
$658$	8.51757	0.332050
$659$	−10.1215	−0.394279	−0.197139	−	0.980375i	$-0.563165\pi$
$659$	−10.1215	−0.394279	−0.197139	+	0.980375i	$0.563165\pi$
$660$	0	0
$661$	23.1128	0.898984	0.449492	−	0.893284i	$-0.351605\pi$
$661$	23.1128	0.898984	0.449492	+	0.893284i	$0.351605\pi$
$662$	10.5176	0.408777
$663$	0	0
$664$	13.8785	0.538589
$665$	5.57834	0.216319
$666$	0	0
$667$	−52.7532	−2.04261
$668$	−8.00000	−0.309529
$669$	0	0
$670$	8.00000	0.309067
$671$	−14.5176	−0.560445
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	7.27820	0.280346
$675$	0	0
$676$	21.5564	0.829093
$677$	−8.83092	−0.339400	−0.169700	−	0.985496i	$-0.554280\pi$
$677$	−8.83092	−0.339400	−0.169700	+	0.985496i	$0.554280\pi$
$678$	0	0
$679$	−7.57834	−0.290830
$680$	4.63910	0.177901
$681$	0	0
$682$	−6.00000	−0.229752
$683$	−11.3997	−0.436199	−0.218099	−	0.975927i	$-0.569986\pi$
$683$	−11.3997	−0.436199	−0.218099	+	0.975927i	$0.569986\pi$
$684$	0	0
$685$	3.27820	0.125254
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	10.5176	0.400979
$689$	−55.5915	−2.11787
$690$	0	0
$691$	−0.0826952	−0.00314587	−0.00157294	−	0.999999i	$-0.500501\pi$
$691$	−0.0826952	−0.00314587	−0.00157294	+	0.999999i	$0.500501\pi$
$692$	5.48243	0.208411
$693$	0	0
$694$	14.2745	0.541853
$695$	−16.0571	−0.609079
$696$	0	0
$697$	14.1992	0.537833
$698$	0.396041	0.0149904
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−38.0571	−1.43740	−0.718698	−	0.695322i	$-0.755262\pi$
$701$	−38.0571	−1.43740	−0.718698	+	0.695322i	$0.755262\pi$
$702$	0	0
$703$	−12.8309	−0.483927
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	5.09960	0.191926
$707$	11.7958	0.443626
$708$	0	0
$709$	36.4275	1.36806	0.684032	−	0.729452i	$-0.260225\pi$
$709$	36.4275	1.36806	0.684032	+	0.729452i	$0.260225\pi$
$710$	7.15667	0.268585
$711$	0	0
$712$	−3.87847	−0.145352
$713$	−33.4700	−1.25346
$714$	0	0
$715$	5.87847	0.219842
$716$	−1.23937	−0.0463174
$717$	0	0
$718$	18.3704	0.685578
$719$	−37.3097	−1.39142	−0.695708	−	0.718325i	$-0.744909\pi$
$719$	−37.3097	−1.39142	−0.695708	+	0.718325i	$0.744909\pi$
$720$	0	0
$721$	−17.1178	−0.637502
$722$	12.1178	0.450979
$723$	0	0
$724$	8.55641	0.317996
$725$	9.45681	0.351217
$726$	0	0
$727$	3.23937	0.120142	0.0600708	−	0.998194i	$-0.480867\pi$
$727$	3.23937	0.120142	0.0600708	+	0.998194i	$0.480867\pi$
$728$	5.87847	0.217871
$729$	0	0
$730$	12.3960	0.458798
$731$	48.7921	1.80464
$732$	0	0
$733$	−26.3522	−0.973340	−0.486670	−	0.873586i	$-0.661789\pi$
$733$	−26.3522	−0.973340	−0.486670	+	0.873586i	$0.661789\pi$
$734$	17.1178	0.631831
$735$	0	0
$736$	−5.57834	−0.205620
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−11.9223	−0.438570	−0.219285	−	0.975661i	$-0.570372\pi$
$739$	−11.9223	−0.438570	−0.219285	+	0.975661i	$0.570372\pi$
$740$	2.30013	0.0845546
$741$	0	0
$742$	−9.45681	−0.347170
$743$	26.0703	0.956426	0.478213	−	0.878244i	$-0.341285\pi$
$743$	26.0703	0.956426	0.478213	+	0.878244i	$0.341285\pi$
$744$	0	0
$745$	−10.7350	−0.393300
$746$	−6.35721	−0.232754
$747$	0	0
$748$	−4.63910	−0.169622
$749$	2.51757	0.0919901
$750$	0	0
$751$	36.6003	1.33556	0.667781	−	0.744357i	$-0.267244\pi$
$751$	36.6003	1.33556	0.667781	+	0.744357i	$0.267244\pi$
$752$	−8.51757	−0.310604
$753$	0	0
$754$	−55.5915	−2.02452
$755$	−2.17860	−0.0792875
$756$	0	0
$757$	−43.0922	−1.56621	−0.783107	−	0.621887i	$-0.786366\pi$
$757$	−43.0922	−1.56621	−0.783107	+	0.621887i	$0.786366\pi$
$758$	−5.03514	−0.182885
$759$	0	0
$760$	−5.57834	−0.202348
$761$	44.0959	1.59848	0.799238	−	0.601015i	$-0.205237\pi$
$761$	44.0959	1.59848	0.799238	+	0.601015i	$0.205237\pi$
$762$	0	0
$763$	−10.3001	−0.372890
$764$	1.27820	0.0462438
$765$	0	0
$766$	7.43118	0.268500
$767$	42.5564	1.53662
$768$	0	0
$769$	−8.33897	−0.300711	−0.150355	−	0.988632i	$-0.548042\pi$
$769$	−8.33897	−0.300711	−0.150355	+	0.988632i	$0.548042\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−10.6391	−0.382910
$773$	19.2782	0.693389	0.346694	−	0.937978i	$-0.387304\pi$
$773$	19.2782	0.693389	0.346694	+	0.937978i	$0.387304\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	7.57834	0.272046
$777$	0	0
$778$	5.15667	0.184876
$779$	−17.0740	−0.611739
$780$	0	0
$781$	−7.15667	−0.256086
$782$	−25.8785	−0.925412
$783$	0	0
$784$	1.00000	0.0357143
$785$	−17.1567	−0.612348
$786$	0	0
$787$	43.5915	1.55387	0.776935	−	0.629580i	$-0.216773\pi$
$787$	43.5915	1.55387	0.776935	+	0.629580i	$0.216773\pi$
$788$	−7.87847	−0.280659
$789$	0	0
$790$	−10.8565	−0.386258
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−85.3411	−3.03055
$794$	−19.6354	−0.696835
$795$	0	0
$796$	0.517571	0.0183448
$797$	51.1493	1.81180	0.905900	−	0.423491i	$-0.139195\pi$
$797$	51.1493	1.81180	0.905900	+	0.423491i	$0.139195\pi$
$798$	0	0
$799$	−39.5139	−1.39790
$800$	1.00000	0.0353553
$801$	0	0
$802$	5.11784	0.180717
$803$	−12.3960	−0.437447
$804$	0	0
$805$	5.57834	0.196611
$806$	−35.2708	−1.24236
$807$	0	0
$808$	−11.7958	−0.414974
$809$	−45.1567	−1.58762	−0.793812	−	0.608163i	$-0.791907\pi$
$809$	−45.1567	−1.58762	−0.793812	+	0.608163i	$0.791907\pi$
$810$	0	0
$811$	39.2914	1.37971	0.689854	−	0.723948i	$-0.257675\pi$
$811$	39.2914	1.37971	0.689854	+	0.723948i	$0.257675\pi$
$812$	−9.45681	−0.331869
$813$	0	0
$814$	−2.30013	−0.0806196
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−58.6706	−2.05262
$818$	2.21744	0.0775309
$819$	0	0
$820$	3.06077	0.106887
$821$	−25.6486	−0.895143	−0.447572	−	0.894248i	$-0.647711\pi$
$821$	−25.6486	−0.895143	−0.447572	+	0.894248i	$0.647711\pi$
$822$	0	0
$823$	−51.6486	−1.80036	−0.900179	−	0.435520i	$-0.856564\pi$
$823$	−51.6486	−1.80036	−0.900179	+	0.435520i	$0.856564\pi$
$824$	17.1178	0.596328
$825$	0	0
$826$	7.23937	0.251890
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	5.36090	0.186192	0.0930958	−	0.995657i	$-0.470324\pi$
$829$	5.36090	0.186192	0.0930958	+	0.995657i	$0.470324\pi$
$830$	13.8785	0.481729
$831$	0	0
$832$	−5.87847	−0.203799
$833$	4.63910	0.160735
$834$	0	0
$835$	−8.00000	−0.276851
$836$	5.57834	0.192931
$837$	0	0
$838$	4.51757	0.156057
$839$	−16.9649	−0.585692	−0.292846	−	0.956160i	$-0.594602\pi$
$839$	−16.9649	−0.585692	−0.292846	+	0.956160i	$0.594602\pi$
$840$	0	0
$841$	60.4312	2.08383
$842$	−12.3133	−0.424346
$843$	0	0
$844$	2.72180	0.0936881
$845$	21.5564	0.741563
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	9.45681	0.324748
$849$	0	0
$850$	4.63910	0.159120
$851$	−12.8309	−0.439838
$852$	0	0
$853$	31.0666	1.06370	0.531850	−	0.846839i	$-0.321497\pi$
$853$	31.0666	1.06370	0.531850	+	0.846839i	$0.321497\pi$
$854$	−14.5176	−0.496781
$855$	0	0
$856$	−2.51757	−0.0860488
$857$	−37.5966	−1.28427	−0.642137	−	0.766590i	$-0.721952\pi$
$857$	−37.5966	−1.28427	−0.642137	+	0.766590i	$0.721952\pi$
$858$	0	0
$859$	−14.0388	−0.478999	−0.239499	−	0.970897i	$-0.576983\pi$
$859$	−14.0388	−0.478999	−0.239499	+	0.970897i	$0.576983\pi$
$860$	10.5176	0.358646
$861$	0	0
$862$	2.17860	0.0742035
$863$	−38.8053	−1.32095	−0.660474	−	0.750849i	$-0.729645\pi$
$863$	−38.8053	−1.32095	−0.660474	+	0.750849i	$0.729645\pi$
$864$	0	0
$865$	5.48243	0.186408
$866$	5.02193	0.170652
$867$	0	0
$868$	−6.00000	−0.203653
$869$	10.8565	0.368283
$870$	0	0
$871$	−47.0278	−1.59347
$872$	10.3001	0.348807
$873$	0	0
$874$	31.1178	1.05258
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	31.7131	1.07087	0.535437	−	0.844575i	$-0.320147\pi$
$877$	31.7131	1.07087	0.535437	+	0.844575i	$0.320147\pi$
$878$	7.15667	0.241526
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−2.24306	−0.0755706	−0.0377853	−	0.999286i	$-0.512030\pi$
$881$	−2.24306	−0.0755706	−0.0377853	+	0.999286i	$0.512030\pi$
$882$	0	0
$883$	24.4349	0.822299	0.411150	−	0.911568i	$-0.365127\pi$
$883$	24.4349	0.822299	0.411150	+	0.911568i	$0.365127\pi$
$884$	−27.2708	−0.917217
$885$	0	0
$886$	19.5139	0.655582
$887$	−35.5915	−1.19505	−0.597524	−	0.801851i	$-0.703849\pi$
$887$	−35.5915	−1.19505	−0.597524	+	0.801851i	$0.703849\pi$
$888$	0	0
$889$	−10.4787	−0.351446
$890$	−3.87847	−0.130007
$891$	0	0
$892$	−22.3960	−0.749875
$893$	47.5139	1.58999
$894$	0	0
$895$	−1.23937	−0.0414275
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−1.36090	−0.0454138
$899$	56.7408	1.89241
$900$	0	0
$901$	43.8711	1.46156
$902$	−3.06077	−0.101912
$903$	0	0
$904$	14.0000	0.465633
$905$	8.55641	0.284425
$906$	0	0
$907$	17.3923	0.577503	0.288752	−	0.957404i	$-0.406760\pi$
$907$	17.3923	0.577503	0.288752	+	0.957404i	$0.406760\pi$
$908$	−13.0351	−0.432586
$909$	0	0
$910$	5.87847	0.194869
$911$	−52.1141	−1.72662	−0.863309	−	0.504675i	$-0.831612\pi$
$911$	−52.1141	−1.72662	−0.863309	+	0.504675i	$0.831612\pi$
$912$	0	0
$913$	−13.8785	−0.459310
$914$	4.47874	0.148143
$915$	0	0
$916$	−6.63910	−0.219362
$917$	−1.57834	−0.0521213
$918$	0	0
$919$	12.1347	0.400288	0.200144	−	0.979766i	$-0.435859\pi$
$919$	12.1347	0.400288	0.200144	+	0.979766i	$0.435859\pi$
$920$	−5.57834	−0.183912
$921$	0	0
$922$	36.1530	1.19064
$923$	−42.0703	−1.38476
$924$	0	0
$925$	2.30013	0.0756279
$926$	21.0922	0.693133
$927$	0	0
$928$	9.45681	0.310435
$929$	−23.8008	−0.780879	−0.390439	−	0.920629i	$-0.627677\pi$
$929$	−23.8008	−0.780879	−0.390439	+	0.920629i	$0.627677\pi$
$930$	0	0
$931$	−5.57834	−0.182823
$932$	−19.0351	−0.623517
$933$	0	0
$934$	−19.1311	−0.625987
$935$	−4.63910	−0.151715
$936$	0	0
$937$	9.16170	0.299300	0.149650	−	0.988739i	$-0.452185\pi$
$937$	9.16170	0.299300	0.149650	+	0.988739i	$0.452185\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	−8.51757	−0.277813
$941$	−0.996308	−0.0324787	−0.0162393	−	0.999868i	$-0.505169\pi$
$941$	−0.996308	−0.0324787	−0.0162393	+	0.999868i	$0.505169\pi$
$942$	0	0
$943$	−17.0740	−0.556005
$944$	−7.23937	−0.235621
$945$	0	0
$946$	−10.5176	−0.341956
$947$	18.8359	0.612086	0.306043	−	0.952018i	$-0.400995\pi$
$947$	18.8359	0.612086	0.306043	+	0.952018i	$0.400995\pi$
$948$	0	0
$949$	−72.8698	−2.36545
$950$	−5.57834	−0.180985
$951$	0	0
$952$	−4.63910	−0.150354
$953$	−21.4386	−0.694463	−0.347232	−	0.937779i	$-0.612878\pi$
$953$	−21.4386	−0.694463	−0.347232	+	0.937779i	$0.612878\pi$
$954$	0	0
$955$	1.27820	0.0413617
$956$	−22.6135	−0.731372
$957$	0	0
$958$	−5.27820	−0.170531
$959$	−3.27820	−0.105859
$960$	0	0
$961$	5.00000	0.161290
$962$	−13.5213	−0.435943
$963$	0	0
$964$	−17.9744	−0.578916
$965$	−10.6391	−0.342485
$966$	0	0
$967$	31.5139	1.01342	0.506709	−	0.862117i	$-0.330862\pi$
$967$	31.5139	1.01342	0.506709	+	0.862117i	$0.330862\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	7.57834	0.243326
$971$	−16.1968	−0.519781	−0.259891	−	0.965638i	$-0.583687\pi$
$971$	−16.1968	−0.519781	−0.259891	+	0.965638i	$0.583687\pi$
$972$	0	0
$973$	16.0571	0.514766
$974$	−19.8140	−0.634882
$975$	0	0
$976$	14.5176	0.464696
$977$	−17.5139	−0.560319	−0.280159	−	0.959954i	$-0.590387\pi$
$977$	−17.5139	−0.560319	−0.280159	+	0.959954i	$0.590387\pi$
$978$	0	0
$979$	3.87847	0.123956
$980$	1.00000	0.0319438
$981$	0	0
$982$	6.47874	0.206745
$983$	−41.1178	−1.31146	−0.655728	−	0.754997i	$-0.727638\pi$
$983$	−41.1178	−1.31146	−0.655728	+	0.754997i	$0.727638\pi$
$984$	0	0
$985$	−7.87847	−0.251029
$986$	43.8711	1.39714
$987$	0	0
$988$	32.7921	1.04326
$989$	−58.6706	−1.86562
$990$	0	0
$991$	−2.79947	−0.0889280	−0.0444640	−	0.999011i	$-0.514158\pi$
$991$	−2.79947	−0.0889280	−0.0444640	+	0.999011i	$0.514158\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	−7.15667	−0.226996
$995$	0.517571	0.0164081
$996$	0	0
$997$	11.7181	0.371116	0.185558	−	0.982633i	$-0.440591\pi$
$997$	11.7181	0.371116	0.185558	+	0.982633i	$0.440591\pi$
$998$	−21.2394	−0.672320
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.cl.1.1		3
3.2	odd	2		770.2.a.l.1.1	✓	3
12.11	even	2		6160.2.a.bi.1.3		3
15.2	even	4		3850.2.c.z.1849.3		6
15.8	even	4		3850.2.c.z.1849.4		6
15.14	odd	2		3850.2.a.bu.1.3		3
21.20	even	2		5390.2.a.bz.1.3		3
33.32	even	2		8470.2.a.cl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.l.1.1	✓	3	3.2	odd	2
3850.2.a.bu.1.3		3	15.14	odd	2
3850.2.c.z.1849.3		6	15.2	even	4
3850.2.c.z.1849.4		6	15.8	even	4
5390.2.a.bz.1.3		3	21.20	even	2
6160.2.a.bi.1.3		3	12.11	even	2
6930.2.a.cl.1.1		3	1.1	even	1	trivial
8470.2.a.cl.1.1		3	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -5.87847 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.63910 q^{17} -5.57834 q^{19} +1.00000 q^{20} -1.00000 q^{22} -5.57834 q^{23} +1.00000 q^{25} -5.87847 q^{26} -1.00000 q^{28} +9.45681 q^{29} +6.00000 q^{31} +1.00000 q^{32} +4.63910 q^{34} -1.00000 q^{35} +2.30013 q^{37} -5.57834 q^{38} +1.00000 q^{40} +3.06077 q^{41} +10.5176 q^{43} -1.00000 q^{44} -5.57834 q^{46} -8.51757 q^{47} +1.00000 q^{49} +1.00000 q^{50} -5.87847 q^{52} +9.45681 q^{53} -1.00000 q^{55} -1.00000 q^{56} +9.45681 q^{58} -7.23937 q^{59} +14.5176 q^{61} +6.00000 q^{62} +1.00000 q^{64} -5.87847 q^{65} +8.00000 q^{67} +4.63910 q^{68} -1.00000 q^{70} +7.15667 q^{71} +12.3960 q^{73} +2.30013 q^{74} -5.57834 q^{76} +1.00000 q^{77} -10.8565 q^{79} +1.00000 q^{80} +3.06077 q^{82} +13.8785 q^{83} +4.63910 q^{85} +10.5176 q^{86} -1.00000 q^{88} -3.87847 q^{89} +5.87847 q^{91} -5.57834 q^{92} -8.51757 q^{94} -5.57834 q^{95} +7.57834 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} + 8 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} - 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 4 q^{37} - 2 q^{38} + 3 q^{40} + 18 q^{41} + 8 q^{43} - 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} + 3 q^{50} - 4 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 18 q^{62} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 3 q^{70} - 8 q^{71} - 4 q^{73} + 4 q^{74} - 2 q^{76} + 3 q^{77} - 6 q^{79} + 3 q^{80} + 18 q^{82} + 24 q^{83} + 8 q^{85} + 8 q^{86} - 3 q^{88} + 6 q^{89} - 2 q^{92} - 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8 + 3 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 + 8 * q^17 - 2 * q^19 + 3 * q^20 - 3 * q^22 - 2 * q^23 + 3 * q^25 - 3 * q^28 - 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 - 3 * q^35 + 4 * q^37 - 2 * q^38 + 3 * q^40 + 18 * q^41 + 8 * q^43 - 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 + 3 * q^50 - 4 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 10 * q^59 + 20 * q^61 + 18 * q^62 + 3 * q^64 + 24 * q^67 + 8 * q^68 - 3 * q^70 - 8 * q^71 - 4 * q^73 + 4 * q^74 - 2 * q^76 + 3 * q^77 - 6 * q^79 + 3 * q^80 + 18 * q^82 + 24 * q^83 + 8 * q^85 + 8 * q^86 - 3 * q^88 + 6 * q^89 - 2 * q^92 - 2 * q^94 - 2 * q^95 + 8 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more