LMFDB - Embedded newform 6930.2.a.ci.1.2

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:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -0.694593 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.69459 q^{17} +5.51754 q^{19} +1.00000 q^{20} -1.00000 q^{22} +7.43376 q^{23} +1.00000 q^{25} +0.694593 q^{26} +1.00000 q^{28} +0.610815 q^{29} -7.51754 q^{31} -1.00000 q^{32} -2.69459 q^{34} +1.00000 q^{35} -1.30541 q^{37} -5.51754 q^{38} -1.00000 q^{40} +9.51754 q^{41} +5.51754 q^{43} +1.00000 q^{44} -7.43376 q^{46} +6.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -0.694593 q^{52} -4.12836 q^{53} +1.00000 q^{55} -1.00000 q^{56} -0.610815 q^{58} -3.51754 q^{59} -4.00000 q^{61} +7.51754 q^{62} +1.00000 q^{64} -0.694593 q^{65} -8.95130 q^{67} +2.69459 q^{68} -1.00000 q^{70} +4.77837 q^{71} +5.51754 q^{73} +1.30541 q^{74} +5.51754 q^{76} +1.00000 q^{77} -2.77837 q^{79} +1.00000 q^{80} -9.51754 q^{82} -4.73917 q^{83} +2.69459 q^{85} -5.51754 q^{86} -1.00000 q^{88} +8.04458 q^{89} -0.694593 q^{91} +7.43376 q^{92} -6.00000 q^{94} +5.51754 q^{95} -6.90673 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{17} - 6 q^{19} + 3 q^{20} - 3 q^{22} + 6 q^{23} + 3 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} - 6 q^{34} + 3 q^{35} - 6 q^{37} + 6 q^{38} - 3 q^{40} + 6 q^{41} - 6 q^{43} + 3 q^{44} - 6 q^{46} + 18 q^{47} + 3 q^{49} - 3 q^{50} + 6 q^{53} + 3 q^{55} - 3 q^{56} - 6 q^{58} + 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} + 6 q^{68} - 3 q^{70} + 6 q^{71} - 6 q^{73} + 6 q^{74} - 6 q^{76} + 3 q^{77} + 3 q^{80} - 6 q^{82} + 6 q^{85} + 6 q^{86} - 3 q^{88} + 12 q^{89} + 6 q^{92} - 18 q^{94} - 6 q^{95} + 6 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 + 6 * q^17 - 6 * q^19 + 3 * q^20 - 3 * q^22 + 6 * q^23 + 3 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 - 6 * q^34 + 3 * q^35 - 6 * q^37 + 6 * q^38 - 3 * q^40 + 6 * q^41 - 6 * q^43 + 3 * q^44 - 6 * q^46 + 18 * q^47 + 3 * q^49 - 3 * q^50 + 6 * q^53 + 3 * q^55 - 3 * q^56 - 6 * q^58 + 12 * q^59 - 12 * q^61 + 3 * q^64 + 12 * q^67 + 6 * q^68 - 3 * q^70 + 6 * q^71 - 6 * q^73 + 6 * q^74 - 6 * q^76 + 3 * q^77 + 3 * q^80 - 6 * q^82 + 6 * q^85 + 6 * q^86 - 3 * q^88 + 12 * q^89 + 6 * q^92 - 18 * q^94 - 6 * q^95 + 6 * 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
$13$	−0.694593	−0.192645	−0.0963227	−	0.995350i	$-0.530708\pi$
$13$	−0.694593	−0.192645	−0.0963227	+	0.995350i	$0.530708\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.69459	0.653535	0.326767	−	0.945105i	$-0.394041\pi$
$17$	2.69459	0.653535	0.326767	+	0.945105i	$0.394041\pi$
$18$	0	0
$19$	5.51754	1.26581	0.632905	−	0.774229i	$-0.281862\pi$
$19$	5.51754	1.26581	0.632905	+	0.774229i	$0.281862\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	7.43376	1.55005	0.775023	−	0.631933i	$-0.217738\pi$
$23$	7.43376	1.55005	0.775023	+	0.631933i	$0.217738\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.694593	0.136221
$27$	0	0
$28$	1.00000	0.188982
$29$	0.610815	0.113425	0.0567127	−	0.998391i	$-0.481938\pi$
$29$	0.610815	0.113425	0.0567127	+	0.998391i	$0.481938\pi$
$30$	0	0
$31$	−7.51754	−1.35019	−0.675095	−	0.737731i	$-0.735897\pi$
$31$	−7.51754	−1.35019	−0.675095	+	0.737731i	$0.735897\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.69459	−0.462119
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.30541	−0.214608	−0.107304	−	0.994226i	$-0.534222\pi$
$37$	−1.30541	−0.214608	−0.107304	+	0.994226i	$0.534222\pi$
$38$	−5.51754	−0.895063
$39$	0	0
$40$	−1.00000	−0.158114
$41$	9.51754	1.48639	0.743195	−	0.669075i	$-0.233309\pi$
$41$	9.51754	1.48639	0.743195	+	0.669075i	$0.233309\pi$
$42$	0	0
$43$	5.51754	0.841417	0.420709	−	0.907196i	$-0.361782\pi$
$43$	5.51754	0.841417	0.420709	+	0.907196i	$0.361782\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−7.43376	−1.09605
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−0.694593	−0.0963227
$53$	−4.12836	−0.567073	−0.283537	−	0.958961i	$-0.591508\pi$
$53$	−4.12836	−0.567073	−0.283537	+	0.958961i	$0.591508\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−0.610815	−0.0802039
$59$	−3.51754	−0.457945	−0.228972	−	0.973433i	$-0.573537\pi$
$59$	−3.51754	−0.457945	−0.228972	+	0.973433i	$0.573537\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	7.51754	0.954729
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.694593	−0.0861536
$66$	0	0
$67$	−8.95130	−1.09358	−0.546788	−	0.837271i	$-0.684150\pi$
$67$	−8.95130	−1.09358	−0.546788	+	0.837271i	$0.684150\pi$
$68$	2.69459	0.326767
$69$	0	0
$70$	−1.00000	−0.119523
$71$	4.77837	0.567088	0.283544	−	0.958959i	$-0.408490\pi$
$71$	4.77837	0.567088	0.283544	+	0.958959i	$0.408490\pi$
$72$	0	0
$73$	5.51754	0.645779	0.322890	−	0.946437i	$-0.395346\pi$
$73$	5.51754	0.645779	0.322890	+	0.946437i	$0.395346\pi$
$74$	1.30541	0.151751
$75$	0	0
$76$	5.51754	0.632905
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.77837	−0.312591	−0.156296	−	0.987710i	$-0.549955\pi$
$79$	−2.77837	−0.312591	−0.156296	+	0.987710i	$0.549955\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−9.51754	−1.05104
$83$	−4.73917	−0.520192	−0.260096	−	0.965583i	$-0.583754\pi$
$83$	−4.73917	−0.520192	−0.260096	+	0.965583i	$0.583754\pi$
$84$	0	0
$85$	2.69459	0.292270
$86$	−5.51754	−0.594972
$87$	0	0
$88$	−1.00000	−0.106600
$89$	8.04458	0.852724	0.426362	−	0.904553i	$-0.359795\pi$
$89$	8.04458	0.852724	0.426362	+	0.904553i	$0.359795\pi$
$90$	0	0
$91$	−0.694593	−0.0728131
$92$	7.43376	0.775023
$93$	0	0
$94$	−6.00000	−0.618853
$95$	5.51754	0.566088
$96$	0	0
$97$	−6.90673	−0.701272	−0.350636	−	0.936512i	$-0.614035\pi$
$97$	−6.90673	−0.701272	−0.350636	+	0.936512i	$0.614035\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−7.64590	−0.760795	−0.380398	−	0.924823i	$-0.624213\pi$
$101$	−7.64590	−0.760795	−0.380398	+	0.924823i	$0.624213\pi$
$102$	0	0
$103$	−7.51754	−0.740725	−0.370363	−	0.928887i	$-0.620767\pi$
$103$	−7.51754	−0.740725	−0.370363	+	0.928887i	$0.620767\pi$
$104$	0.694593	0.0681104
$105$	0	0
$106$	4.12836	0.400981
$107$	−17.1634	−1.65925	−0.829626	−	0.558319i	$-0.811446\pi$
$107$	−17.1634	−1.65925	−0.829626	+	0.558319i	$0.811446\pi$
$108$	0	0
$109$	−1.91622	−0.183541	−0.0917704	−	0.995780i	$-0.529253\pi$
$109$	−1.91622	−0.183541	−0.0917704	+	0.995780i	$0.529253\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	12.2121	1.14882	0.574410	−	0.818567i	$-0.305231\pi$
$113$	12.2121	1.14882	0.574410	+	0.818567i	$0.305231\pi$
$114$	0	0
$115$	7.43376	0.693202
$116$	0.610815	0.0567127
$117$	0	0
$118$	3.51754	0.323816
$119$	2.69459	0.247013
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.00000	0.362143
$123$	0	0
$124$	−7.51754	−0.675095
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.5175	1.02202	0.511008	−	0.859576i	$-0.329272\pi$
$127$	11.5175	1.02202	0.511008	+	0.859576i	$0.329272\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.694593	0.0609198
$131$	5.38919	0.470855	0.235428	−	0.971892i	$-0.424351\pi$
$131$	5.38919	0.470855	0.235428	+	0.971892i	$0.424351\pi$
$132$	0	0
$133$	5.51754	0.478431
$134$	8.95130	0.773275
$135$	0	0
$136$	−2.69459	−0.231059
$137$	13.4338	1.14772	0.573862	−	0.818952i	$-0.305445\pi$
$137$	13.4338	1.14772	0.573862	+	0.818952i	$0.305445\pi$
$138$	0	0
$139$	−18.9067	−1.60365	−0.801824	−	0.597561i	$-0.796137\pi$
$139$	−18.9067	−1.60365	−0.801824	+	0.597561i	$0.796137\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−4.77837	−0.400992
$143$	−0.694593	−0.0580848
$144$	0	0
$145$	0.610815	0.0507254
$146$	−5.51754	−0.456635
$147$	0	0
$148$	−1.30541	−0.107304
$149$	−0.610815	−0.0500399	−0.0250199	−	0.999687i	$-0.507965\pi$
$149$	−0.610815	−0.0500399	−0.0250199	+	0.999687i	$0.507965\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−5.51754	−0.447532
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−7.51754	−0.603823
$156$	0	0
$157$	24.5526	1.95951	0.979756	−	0.200194i	$-0.0641572\pi$
$157$	24.5526	1.95951	0.979756	+	0.200194i	$0.0641572\pi$
$158$	2.77837	0.221035
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	7.43376	0.585863
$162$	0	0
$163$	−1.91622	−0.150090	−0.0750450	−	0.997180i	$-0.523910\pi$
$163$	−1.91622	−0.150090	−0.0750450	+	0.997180i	$0.523910\pi$
$164$	9.51754	0.743195
$165$	0	0
$166$	4.73917	0.367831
$167$	24.2121	1.87359	0.936796	−	0.349877i	$-0.113777\pi$
$167$	24.2121	1.87359	0.936796	+	0.349877i	$0.113777\pi$
$168$	0	0
$169$	−12.5175	−0.962888
$170$	−2.69459	−0.206666
$171$	0	0
$172$	5.51754	0.420709
$173$	7.22163	0.549050	0.274525	−	0.961580i	$-0.411479\pi$
$173$	7.22163	0.549050	0.274525	+	0.961580i	$0.411479\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−8.04458	−0.602967
$179$	14.2959	1.06853	0.534263	−	0.845318i	$-0.320589\pi$
$179$	14.2959	1.06853	0.534263	+	0.845318i	$0.320589\pi$
$180$	0	0
$181$	−2.52704	−0.187833	−0.0939166	−	0.995580i	$-0.529939\pi$
$181$	−2.52704	−0.187833	−0.0939166	+	0.995580i	$0.529939\pi$
$182$	0.694593	0.0514866
$183$	0	0
$184$	−7.43376	−0.548024
$185$	−1.30541	−0.0959755
$186$	0	0
$187$	2.69459	0.197048
$188$	6.00000	0.437595
$189$	0	0
$190$	−5.51754	−0.400284
$191$	−0.610815	−0.0441970	−0.0220985	−	0.999756i	$-0.507035\pi$
$191$	−0.610815	−0.0441970	−0.0220985	+	0.999756i	$0.507035\pi$
$192$	0	0
$193$	−1.51754	−0.109235	−0.0546175	−	0.998507i	$-0.517394\pi$
$193$	−1.51754	−0.109235	−0.0546175	+	0.998507i	$0.517394\pi$
$194$	6.90673	0.495874
$195$	0	0
$196$	1.00000	0.0714286
$197$	−10.7392	−0.765134	−0.382567	−	0.923928i	$-0.624960\pi$
$197$	−10.7392	−0.765134	−0.382567	+	0.923928i	$0.624960\pi$
$198$	0	0
$199$	−0.906726	−0.0642761	−0.0321381	−	0.999483i	$-0.510232\pi$
$199$	−0.906726	−0.0642761	−0.0321381	+	0.999483i	$0.510232\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	7.64590	0.537963
$203$	0.610815	0.0428708
$204$	0	0
$205$	9.51754	0.664734
$206$	7.51754	0.523772
$207$	0	0
$208$	−0.694593	−0.0481613
$209$	5.51754	0.381656
$210$	0	0
$211$	−27.3756	−1.88461	−0.942306	−	0.334753i	$-0.891347\pi$
$211$	−27.3756	−1.88461	−0.942306	+	0.334753i	$0.891347\pi$
$212$	−4.12836	−0.283537
$213$	0	0
$214$	17.1634	1.17327
$215$	5.51754	0.376293
$216$	0	0
$217$	−7.51754	−0.510324
$218$	1.91622	0.129783
$219$	0	0
$220$	1.00000	0.0674200
$221$	−1.87164	−0.125900
$222$	0	0
$223$	21.6459	1.44952	0.724758	−	0.689003i	$-0.241951\pi$
$223$	21.6459	1.44952	0.724758	+	0.689003i	$0.241951\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−12.2121	−0.812339
$227$	11.3500	0.753325	0.376662	−	0.926351i	$-0.377072\pi$
$227$	11.3500	0.753325	0.376662	+	0.926351i	$0.377072\pi$
$228$	0	0
$229$	−6.69459	−0.442391	−0.221196	−	0.975229i	$-0.570996\pi$
$229$	−6.69459	−0.442391	−0.221196	+	0.975229i	$0.570996\pi$
$230$	−7.43376	−0.490168
$231$	0	0
$232$	−0.610815	−0.0401019
$233$	14.2567	0.933988	0.466994	−	0.884260i	$-0.345337\pi$
$233$	14.2567	0.933988	0.466994	+	0.884260i	$0.345337\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	−3.51754	−0.228972
$237$	0	0
$238$	−2.69459	−0.174665
$239$	−3.72967	−0.241253	−0.120626	−	0.992698i	$-0.538490\pi$
$239$	−3.72967	−0.241253	−0.120626	+	0.992698i	$0.538490\pi$
$240$	0	0
$241$	−14.7392	−0.949433	−0.474717	−	0.880139i	$-0.657449\pi$
$241$	−14.7392	−0.949433	−0.474717	+	0.880139i	$0.657449\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−4.00000	−0.256074
$245$	1.00000	0.0638877
$246$	0	0
$247$	−3.83244	−0.243853
$248$	7.51754	0.477364
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	18.3851	1.16046	0.580228	−	0.814454i	$-0.302964\pi$
$251$	18.3851	1.16046	0.580228	+	0.814454i	$0.302964\pi$
$252$	0	0
$253$	7.43376	0.467357
$254$	−11.5175	−0.722675
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.09327	0.192953	0.0964766	−	0.995335i	$-0.469243\pi$
$257$	3.09327	0.192953	0.0964766	+	0.995335i	$0.469243\pi$
$258$	0	0
$259$	−1.30541	−0.0811141
$260$	−0.694593	−0.0430768
$261$	0	0
$262$	−5.38919	−0.332945
$263$	−6.61081	−0.407640	−0.203820	−	0.979008i	$-0.565336\pi$
$263$	−6.61081	−0.407640	−0.203820	+	0.979008i	$0.565336\pi$
$264$	0	0
$265$	−4.12836	−0.253603
$266$	−5.51754	−0.338302
$267$	0	0
$268$	−8.95130	−0.546788
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	13.8135	0.839107	0.419554	−	0.907730i	$-0.362187\pi$
$271$	13.8135	0.839107	0.419554	+	0.907730i	$0.362187\pi$
$272$	2.69459	0.163384
$273$	0	0
$274$	−13.4338	−0.811563
$275$	1.00000	0.0603023
$276$	0	0
$277$	25.1634	1.51192	0.755962	−	0.654615i	$-0.227169\pi$
$277$	25.1634	1.51192	0.755962	+	0.654615i	$0.227169\pi$
$278$	18.9067	1.13395
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−15.9162	−0.949482	−0.474741	−	0.880125i	$-0.657458\pi$
$281$	−15.9162	−0.949482	−0.474741	+	0.880125i	$0.657458\pi$
$282$	0	0
$283$	−10.2121	−0.607048	−0.303524	−	0.952824i	$-0.598163\pi$
$283$	−10.2121	−0.607048	−0.303524	+	0.952824i	$0.598163\pi$
$284$	4.77837	0.283544
$285$	0	0
$286$	0.694593	0.0410721
$287$	9.51754	0.561803
$288$	0	0
$289$	−9.73917	−0.572892
$290$	−0.610815	−0.0358683
$291$	0	0
$292$	5.51754	0.322890
$293$	−26.6810	−1.55872	−0.779360	−	0.626577i	$-0.784455\pi$
$293$	−26.6810	−1.55872	−0.779360	+	0.626577i	$0.784455\pi$
$294$	0	0
$295$	−3.51754	−0.204799
$296$	1.30541	0.0758753
$297$	0	0
$298$	0.610815	0.0353835
$299$	−5.16344	−0.298609
$300$	0	0
$301$	5.51754	0.318026
$302$	−8.00000	−0.460348
$303$	0	0
$304$	5.51754	0.316453
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−17.2472	−0.984351	−0.492175	−	0.870496i	$-0.663798\pi$
$307$	−17.2472	−0.984351	−0.492175	+	0.870496i	$0.663798\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	7.51754	0.426968
$311$	12.1729	0.690264	0.345132	−	0.938554i	$-0.387834\pi$
$311$	12.1729	0.690264	0.345132	+	0.938554i	$0.387834\pi$
$312$	0	0
$313$	−18.6810	−1.05591	−0.527956	−	0.849272i	$-0.677041\pi$
$313$	−18.6810	−1.05591	−0.527956	+	0.849272i	$0.677041\pi$
$314$	−24.5526	−1.38558
$315$	0	0
$316$	−2.77837	−0.156296
$317$	29.7743	1.67229	0.836144	−	0.548510i	$-0.184805\pi$
$317$	29.7743	1.67229	0.836144	+	0.548510i	$0.184805\pi$
$318$	0	0
$319$	0.610815	0.0341991
$320$	1.00000	0.0559017
$321$	0	0
$322$	−7.43376	−0.414267
$323$	14.8675	0.827251
$324$	0	0
$325$	−0.694593	−0.0385291
$326$	1.91622	0.106130
$327$	0	0
$328$	−9.51754	−0.525518
$329$	6.00000	0.330791
$330$	0	0
$331$	4.25671	0.233970	0.116985	−	0.993134i	$-0.462677\pi$
$331$	4.25671	0.233970	0.116985	+	0.993134i	$0.462677\pi$
$332$	−4.73917	−0.260096
$333$	0	0
$334$	−24.2121	−1.32483
$335$	−8.95130	−0.489062
$336$	0	0
$337$	3.64590	0.198605	0.0993023	−	0.995057i	$-0.468339\pi$
$337$	3.64590	0.198605	0.0993023	+	0.995057i	$0.468339\pi$
$338$	12.5175	0.680864
$339$	0	0
$340$	2.69459	0.146135
$341$	−7.51754	−0.407098
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.51754	−0.297486
$345$	0	0
$346$	−7.22163	−0.388237
$347$	−27.9418	−1.49999	−0.749997	−	0.661441i	$-0.769945\pi$
$347$	−27.9418	−1.49999	−0.749997	+	0.661441i	$0.769945\pi$
$348$	0	0
$349$	−25.4783	−1.36382	−0.681912	−	0.731434i	$-0.738851\pi$
$349$	−25.4783	−1.36382	−0.681912	+	0.731434i	$0.738851\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−17.1634	−0.913518	−0.456759	−	0.889591i	$-0.650990\pi$
$353$	−17.1634	−0.913518	−0.456759	+	0.889591i	$0.650990\pi$
$354$	0	0
$355$	4.77837	0.253610
$356$	8.04458	0.426362
$357$	0	0
$358$	−14.2959	−0.755562
$359$	−31.8188	−1.67933	−0.839667	−	0.543102i	$-0.817250\pi$
$359$	−31.8188	−1.67933	−0.839667	+	0.543102i	$0.817250\pi$
$360$	0	0
$361$	11.4433	0.602277
$362$	2.52704	0.132818
$363$	0	0
$364$	−0.694593	−0.0364066
$365$	5.51754	0.288801
$366$	0	0
$367$	−27.7743	−1.44980	−0.724902	−	0.688852i	$-0.758115\pi$
$367$	−27.7743	−1.44980	−0.724902	+	0.688852i	$0.758115\pi$
$368$	7.43376	0.387512
$369$	0	0
$370$	1.30541	0.0678649
$371$	−4.12836	−0.214334
$372$	0	0
$373$	0.739170	0.0382728	0.0191364	−	0.999817i	$-0.493908\pi$
$373$	0.739170	0.0382728	0.0191364	+	0.999817i	$0.493908\pi$
$374$	−2.69459	−0.139334
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−0.424267	−0.0218509
$378$	0	0
$379$	24.5134	1.25917	0.629585	−	0.776932i	$-0.283225\pi$
$379$	24.5134	1.25917	0.629585	+	0.776932i	$0.283225\pi$
$380$	5.51754	0.283044
$381$	0	0
$382$	0.610815	0.0312520
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	1.51754	0.0772408
$387$	0	0
$388$	−6.90673	−0.350636
$389$	6.65002	0.337169	0.168585	−	0.985687i	$-0.446080\pi$
$389$	6.65002	0.337169	0.168585	+	0.985687i	$0.446080\pi$
$390$	0	0
$391$	20.0310	1.01301
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	10.7392	0.541032
$395$	−2.77837	−0.139795
$396$	0	0
$397$	34.9067	1.75192	0.875959	−	0.482385i	$-0.160229\pi$
$397$	34.9067	1.75192	0.875959	+	0.482385i	$0.160229\pi$
$398$	0.906726	0.0454501
$399$	0	0
$400$	1.00000	0.0500000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	5.22163	0.260108
$404$	−7.64590	−0.380398
$405$	0	0
$406$	−0.610815	−0.0303142
$407$	−1.30541	−0.0647066
$408$	0	0
$409$	3.64590	0.180278	0.0901390	−	0.995929i	$-0.471269\pi$
$409$	3.64590	0.180278	0.0901390	+	0.995929i	$0.471269\pi$
$410$	−9.51754	−0.470038
$411$	0	0
$412$	−7.51754	−0.370363
$413$	−3.51754	−0.173087
$414$	0	0
$415$	−4.73917	−0.232637
$416$	0.694593	0.0340552
$417$	0	0
$418$	−5.51754	−0.269872
$419$	18.3851	0.898169	0.449085	−	0.893489i	$-0.351750\pi$
$419$	18.3851	0.898169	0.449085	+	0.893489i	$0.351750\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	27.3756	1.33262
$423$	0	0
$424$	4.12836	0.200491
$425$	2.69459	0.130707
$426$	0	0
$427$	−4.00000	−0.193574
$428$	−17.1634	−0.829626
$429$	0	0
$430$	−5.51754	−0.266079
$431$	15.3054	0.737236	0.368618	−	0.929581i	$-0.379831\pi$
$431$	15.3054	0.737236	0.368618	+	0.929581i	$0.379831\pi$
$432$	0	0
$433$	−9.34998	−0.449332	−0.224666	−	0.974436i	$-0.572129\pi$
$433$	−9.34998	−0.449332	−0.224666	+	0.974436i	$0.572129\pi$
$434$	7.51754	0.360854
$435$	0	0
$436$	−1.91622	−0.0917704
$437$	41.0161	1.96207
$438$	0	0
$439$	30.5526	1.45820	0.729099	−	0.684409i	$-0.239940\pi$
$439$	30.5526	1.45820	0.729099	+	0.684409i	$0.239940\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	1.87164	0.0890250
$443$	36.1985	1.71984	0.859922	−	0.510426i	$-0.170512\pi$
$443$	36.1985	1.71984	0.859922	+	0.510426i	$0.170512\pi$
$444$	0	0
$445$	8.04458	0.381350
$446$	−21.6459	−1.02496
$447$	0	0
$448$	1.00000	0.0472456
$449$	−1.22163	−0.0576522	−0.0288261	−	0.999584i	$-0.509177\pi$
$449$	−1.22163	−0.0576522	−0.0288261	+	0.999584i	$0.509177\pi$
$450$	0	0
$451$	9.51754	0.448164
$452$	12.2121	0.574410
$453$	0	0
$454$	−11.3500	−0.532681
$455$	−0.694593	−0.0325630
$456$	0	0
$457$	−11.0743	−0.518033	−0.259017	−	0.965873i	$-0.583398\pi$
$457$	−11.0743	−0.518033	−0.259017	+	0.965873i	$0.583398\pi$
$458$	6.69459	0.312818
$459$	0	0
$460$	7.43376	0.346601
$461$	−7.87164	−0.366619	−0.183310	−	0.983055i	$-0.558681\pi$
$461$	−7.87164	−0.366619	−0.183310	+	0.983055i	$0.558681\pi$
$462$	0	0
$463$	24.1676	1.12316	0.561581	−	0.827422i	$-0.310193\pi$
$463$	24.1676	1.12316	0.561581	+	0.827422i	$0.310193\pi$
$464$	0.610815	0.0283564
$465$	0	0
$466$	−14.2567	−0.660429
$467$	27.5175	1.27336	0.636680	−	0.771128i	$-0.280307\pi$
$467$	27.5175	1.27336	0.636680	+	0.771128i	$0.280307\pi$
$468$	0	0
$469$	−8.95130	−0.413333
$470$	−6.00000	−0.276759
$471$	0	0
$472$	3.51754	0.161908
$473$	5.51754	0.253697
$474$	0	0
$475$	5.51754	0.253162
$476$	2.69459	0.123506
$477$	0	0
$478$	3.72967	0.170591
$479$	26.0702	1.19118	0.595588	−	0.803290i	$-0.296919\pi$
$479$	26.0702	1.19118	0.595588	+	0.803290i	$0.296919\pi$
$480$	0	0
$481$	0.906726	0.0413432
$482$	14.7392	0.671351
$483$	0	0
$484$	1.00000	0.0454545
$485$	−6.90673	−0.313618
$486$	0	0
$487$	12.9649	0.587497	0.293748	−	0.955883i	$-0.405097\pi$
$487$	12.9649	0.587497	0.293748	+	0.955883i	$0.405097\pi$
$488$	4.00000	0.181071
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−36.4243	−1.64380	−0.821902	−	0.569629i	$-0.807087\pi$
$491$	−36.4243	−1.64380	−0.821902	+	0.569629i	$0.807087\pi$
$492$	0	0
$493$	1.64590	0.0741275
$494$	3.83244	0.172430
$495$	0	0
$496$	−7.51754	−0.337548
$497$	4.77837	0.214339
$498$	0	0
$499$	19.1242	0.856118	0.428059	−	0.903751i	$-0.359197\pi$
$499$	19.1242	0.856118	0.428059	+	0.903751i	$0.359197\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−18.3851	−0.820566
$503$	21.7689	0.970626	0.485313	−	0.874340i	$-0.338706\pi$
$503$	21.7689	0.970626	0.485313	+	0.874340i	$0.338706\pi$
$504$	0	0
$505$	−7.64590	−0.340238
$506$	−7.43376	−0.330471
$507$	0	0
$508$	11.5175	0.511008
$509$	22.1676	0.982560	0.491280	−	0.871002i	$-0.336529\pi$
$509$	22.1676	0.982560	0.491280	+	0.871002i	$0.336529\pi$
$510$	0	0
$511$	5.51754	0.244082
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.09327	−0.136438
$515$	−7.51754	−0.331262
$516$	0	0
$517$	6.00000	0.263880
$518$	1.30541	0.0573563
$519$	0	0
$520$	0.694593	0.0304599
$521$	−32.4688	−1.42249	−0.711243	−	0.702946i	$-0.751867\pi$
$521$	−32.4688	−1.42249	−0.711243	+	0.702946i	$0.751867\pi$
$522$	0	0
$523$	−3.60132	−0.157475	−0.0787373	−	0.996895i	$-0.525089\pi$
$523$	−3.60132	−0.157475	−0.0787373	+	0.996895i	$0.525089\pi$
$524$	5.38919	0.235428
$525$	0	0
$526$	6.61081	0.288245
$527$	−20.2567	−0.882396
$528$	0	0
$529$	32.2608	1.40264
$530$	4.12836	0.179324
$531$	0	0
$532$	5.51754	0.239216
$533$	−6.61081	−0.286346
$534$	0	0
$535$	−17.1634	−0.742040
$536$	8.95130	0.386637
$537$	0	0
$538$	6.00000	0.258678
$539$	1.00000	0.0430730
$540$	0	0
$541$	0.951304	0.0408997	0.0204499	−	0.999791i	$-0.493490\pi$
$541$	0.951304	0.0408997	0.0204499	+	0.999791i	$0.493490\pi$
$542$	−13.8135	−0.593339
$543$	0	0
$544$	−2.69459	−0.115530
$545$	−1.91622	−0.0820819
$546$	0	0
$547$	38.1985	1.63325	0.816625	−	0.577168i	$-0.195842\pi$
$547$	38.1985	1.63325	0.816625	+	0.577168i	$0.195842\pi$
$548$	13.4338	0.573862
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	3.37019	0.143575
$552$	0	0
$553$	−2.77837	−0.118148
$554$	−25.1634	−1.06909
$555$	0	0
$556$	−18.9067	−0.801824
$557$	8.29591	0.351509	0.175755	−	0.984434i	$-0.443763\pi$
$557$	8.29591	0.351509	0.175755	+	0.984434i	$0.443763\pi$
$558$	0	0
$559$	−3.83244	−0.162095
$560$	1.00000	0.0422577
$561$	0	0
$562$	15.9162	0.671385
$563$	−15.0933	−0.636106	−0.318053	−	0.948073i	$-0.603029\pi$
$563$	−15.0933	−0.636106	−0.318053	+	0.948073i	$0.603029\pi$
$564$	0	0
$565$	12.2121	0.513768
$566$	10.2121	0.429248
$567$	0	0
$568$	−4.77837	−0.200496
$569$	43.2080	1.81137	0.905687	−	0.423947i	$-0.139356\pi$
$569$	43.2080	1.81137	0.905687	+	0.423947i	$0.139356\pi$
$570$	0	0
$571$	13.5621	0.567557	0.283778	−	0.958890i	$-0.408412\pi$
$571$	13.5621	0.567557	0.283778	+	0.958890i	$0.408412\pi$
$572$	−0.694593	−0.0290424
$573$	0	0
$574$	−9.51754	−0.397254
$575$	7.43376	0.310009
$576$	0	0
$577$	19.3892	0.807182	0.403591	−	0.914939i	$-0.367762\pi$
$577$	19.3892	0.807182	0.403591	+	0.914939i	$0.367762\pi$
$578$	9.73917	0.405096
$579$	0	0
$580$	0.610815	0.0253627
$581$	−4.73917	−0.196614
$582$	0	0
$583$	−4.12836	−0.170979
$584$	−5.51754	−0.228317
$585$	0	0
$586$	26.6810	1.10218
$587$	28.7392	1.18619	0.593096	−	0.805132i	$-0.297905\pi$
$587$	28.7392	1.18619	0.593096	+	0.805132i	$0.297905\pi$
$588$	0	0
$589$	−41.4783	−1.70909
$590$	3.51754	0.144815
$591$	0	0
$592$	−1.30541	−0.0536519
$593$	30.7837	1.26414	0.632068	−	0.774913i	$-0.282206\pi$
$593$	30.7837	1.26414	0.632068	+	0.774913i	$0.282206\pi$
$594$	0	0
$595$	2.69459	0.110468
$596$	−0.610815	−0.0250199
$597$	0	0
$598$	5.16344	0.211149
$599$	−5.20264	−0.212574	−0.106287	−	0.994335i	$-0.533896\pi$
$599$	−5.20264	−0.212574	−0.106287	+	0.994335i	$0.533896\pi$
$600$	0	0
$601$	−6.68098	−0.272523	−0.136261	−	0.990673i	$-0.543509\pi$
$601$	−6.68098	−0.272523	−0.136261	+	0.990673i	$0.543509\pi$
$602$	−5.51754	−0.224878
$603$	0	0
$604$	8.00000	0.325515
$605$	1.00000	0.0406558
$606$	0	0
$607$	15.8324	0.642619	0.321310	−	0.946974i	$-0.395877\pi$
$607$	15.8324	0.642619	0.321310	+	0.946974i	$0.395877\pi$
$608$	−5.51754	−0.223766
$609$	0	0
$610$	4.00000	0.161955
$611$	−4.16756	−0.168601
$612$	0	0
$613$	−18.7202	−0.756101	−0.378050	−	0.925785i	$-0.623405\pi$
$613$	−18.7202	−0.756101	−0.378050	+	0.925785i	$0.623405\pi$
$614$	17.2472	0.696041
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−15.0797	−0.607084	−0.303542	−	0.952818i	$-0.598169\pi$
$617$	−15.0797	−0.607084	−0.303542	+	0.952818i	$0.598169\pi$
$618$	0	0
$619$	−45.1498	−1.81472	−0.907362	−	0.420349i	$-0.861907\pi$
$619$	−45.1498	−1.81472	−0.907362	+	0.420349i	$0.861907\pi$
$620$	−7.51754	−0.301912
$621$	0	0
$622$	−12.1729	−0.488090
$623$	8.04458	0.322299
$624$	0	0
$625$	1.00000	0.0400000
$626$	18.6810	0.746642
$627$	0	0
$628$	24.5526	0.979756
$629$	−3.51754	−0.140254
$630$	0	0
$631$	7.42839	0.295719	0.147860	−	0.989008i	$-0.452762\pi$
$631$	7.42839	0.295719	0.147860	+	0.989008i	$0.452762\pi$
$632$	2.77837	0.110518
$633$	0	0
$634$	−29.7743	−1.18249
$635$	11.5175	0.457060
$636$	0	0
$637$	−0.694593	−0.0275208
$638$	−0.610815	−0.0241824
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−36.8485	−1.45543	−0.727715	−	0.685880i	$-0.759418\pi$
$641$	−36.8485	−1.45543	−0.727715	+	0.685880i	$0.759418\pi$
$642$	0	0
$643$	−32.5134	−1.28220	−0.641102	−	0.767456i	$-0.721522\pi$
$643$	−32.5134	−1.28220	−0.641102	+	0.767456i	$0.721522\pi$
$644$	7.43376	0.292931
$645$	0	0
$646$	−14.8675	−0.584955
$647$	−3.47834	−0.136748	−0.0683738	−	0.997660i	$-0.521781\pi$
$647$	−3.47834	−0.136748	−0.0683738	+	0.997660i	$0.521781\pi$
$648$	0	0
$649$	−3.51754	−0.138076
$650$	0.694593	0.0272442
$651$	0	0
$652$	−1.91622	−0.0750450
$653$	−20.7202	−0.810843	−0.405422	−	0.914130i	$-0.632875\pi$
$653$	−20.7202	−0.810843	−0.405422	+	0.914130i	$0.632875\pi$
$654$	0	0
$655$	5.38919	0.210573
$656$	9.51754	0.371598
$657$	0	0
$658$	−6.00000	−0.233904
$659$	8.25671	0.321636	0.160818	−	0.986984i	$-0.448587\pi$
$659$	8.25671	0.321636	0.160818	+	0.986984i	$0.448587\pi$
$660$	0	0
$661$	−40.5972	−1.57905	−0.789524	−	0.613720i	$-0.789673\pi$
$661$	−40.5972	−1.57905	−0.789524	+	0.613720i	$0.789673\pi$
$662$	−4.25671	−0.165442
$663$	0	0
$664$	4.73917	0.183915
$665$	5.51754	0.213961
$666$	0	0
$667$	4.54065	0.175815
$668$	24.2121	0.936796
$669$	0	0
$670$	8.95130	0.345819
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−22.3459	−0.861370	−0.430685	−	0.902502i	$-0.641728\pi$
$673$	−22.3459	−0.861370	−0.430685	+	0.902502i	$0.641728\pi$
$674$	−3.64590	−0.140435
$675$	0	0
$676$	−12.5175	−0.481444
$677$	15.9026	0.611187	0.305593	−	0.952162i	$-0.401145\pi$
$677$	15.9026	0.611187	0.305593	+	0.952162i	$0.401145\pi$
$678$	0	0
$679$	−6.90673	−0.265056
$680$	−2.69459	−0.103333
$681$	0	0
$682$	7.51754	0.287862
$683$	−33.2526	−1.27238	−0.636188	−	0.771534i	$-0.719490\pi$
$683$	−33.2526	−1.27238	−0.636188	+	0.771534i	$0.719490\pi$
$684$	0	0
$685$	13.4338	0.513278
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	5.51754	0.210354
$689$	2.86753	0.109244
$690$	0	0
$691$	10.7338	0.408333	0.204166	−	0.978936i	$-0.434552\pi$
$691$	10.7338	0.408333	0.204166	+	0.978936i	$0.434552\pi$
$692$	7.22163	0.274525
$693$	0	0
$694$	27.9418	1.06066
$695$	−18.9067	−0.717173
$696$	0	0
$697$	25.6459	0.971408
$698$	25.4783	0.964369
$699$	0	0
$700$	1.00000	0.0377964
$701$	−37.4593	−1.41482	−0.707410	−	0.706803i	$-0.750137\pi$
$701$	−37.4593	−1.41482	−0.707410	+	0.706803i	$0.750137\pi$
$702$	0	0
$703$	−7.20264	−0.271653
$704$	1.00000	0.0376889
$705$	0	0
$706$	17.1634	0.645954
$707$	−7.64590	−0.287554
$708$	0	0
$709$	19.8135	0.744110	0.372055	−	0.928211i	$-0.378653\pi$
$709$	19.8135	0.744110	0.372055	+	0.928211i	$0.378653\pi$
$710$	−4.77837	−0.179329
$711$	0	0
$712$	−8.04458	−0.301483
$713$	−55.8836	−2.09286
$714$	0	0
$715$	−0.694593	−0.0259763
$716$	14.2959	0.534263
$717$	0	0
$718$	31.8188	1.18747
$719$	32.0838	1.19652	0.598262	−	0.801301i	$-0.295858\pi$
$719$	32.0838	1.19652	0.598262	+	0.801301i	$0.295858\pi$
$720$	0	0
$721$	−7.51754	−0.279968
$722$	−11.4433	−0.425874
$723$	0	0
$724$	−2.52704	−0.0939166
$725$	0.610815	0.0226851
$726$	0	0
$727$	34.8675	1.29316	0.646582	−	0.762844i	$-0.276198\pi$
$727$	34.8675	1.29316	0.646582	+	0.762844i	$0.276198\pi$
$728$	0.694593	0.0257433
$729$	0	0
$730$	−5.51754	−0.204213
$731$	14.8675	0.549895
$732$	0	0
$733$	29.0405	1.07263	0.536317	−	0.844017i	$-0.319815\pi$
$733$	29.0405	1.07263	0.536317	+	0.844017i	$0.319815\pi$
$734$	27.7743	1.02517
$735$	0	0
$736$	−7.43376	−0.274012
$737$	−8.95130	−0.329726
$738$	0	0
$739$	50.4107	1.85439	0.927193	−	0.374584i	$-0.122215\pi$
$739$	50.4107	1.85439	0.927193	+	0.374584i	$0.122215\pi$
$740$	−1.30541	−0.0479877
$741$	0	0
$742$	4.12836	0.151557
$743$	−10.7784	−0.395420	−0.197710	−	0.980261i	$-0.563350\pi$
$743$	−10.7784	−0.395420	−0.197710	+	0.980261i	$0.563350\pi$
$744$	0	0
$745$	−0.610815	−0.0223785
$746$	−0.739170	−0.0270629
$747$	0	0
$748$	2.69459	0.0985241
$749$	−17.1634	−0.627138
$750$	0	0
$751$	−7.94181	−0.289801	−0.144900	−	0.989446i	$-0.546286\pi$
$751$	−7.94181	−0.289801	−0.144900	+	0.989446i	$0.546286\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	0.424267	0.0154509
$755$	8.00000	0.291150
$756$	0	0
$757$	21.3946	0.777599	0.388799	−	0.921322i	$-0.372890\pi$
$757$	21.3946	0.777599	0.388799	+	0.921322i	$0.372890\pi$
$758$	−24.5134	−0.890368
$759$	0	0
$760$	−5.51754	−0.200142
$761$	−29.7743	−1.07932	−0.539658	−	0.841884i	$-0.681446\pi$
$761$	−29.7743	−1.07932	−0.539658	+	0.841884i	$0.681446\pi$
$762$	0	0
$763$	−1.91622	−0.0693719
$764$	−0.610815	−0.0220985
$765$	0	0
$766$	−6.00000	−0.216789
$767$	2.44326	0.0882209
$768$	0	0
$769$	−12.9459	−0.466842	−0.233421	−	0.972376i	$-0.574992\pi$
$769$	−12.9459	−0.466842	−0.233421	+	0.972376i	$0.574992\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	−1.51754	−0.0546175
$773$	2.90673	0.104548	0.0522738	−	0.998633i	$-0.483353\pi$
$773$	2.90673	0.104548	0.0522738	+	0.998633i	$0.483353\pi$
$774$	0	0
$775$	−7.51754	−0.270038
$776$	6.90673	0.247937
$777$	0	0
$778$	−6.65002	−0.238415
$779$	52.5134	1.88149
$780$	0	0
$781$	4.77837	0.170984
$782$	−20.0310	−0.716306
$783$	0	0
$784$	1.00000	0.0357143
$785$	24.5526	0.876321
$786$	0	0
$787$	29.1581	1.03937	0.519686	−	0.854357i	$-0.326049\pi$
$787$	29.1581	1.03937	0.519686	+	0.854357i	$0.326049\pi$
$788$	−10.7392	−0.382567
$789$	0	0
$790$	2.77837	0.0988500
$791$	12.2121	0.434213
$792$	0	0
$793$	2.77837	0.0986628
$794$	−34.9067	−1.23879
$795$	0	0
$796$	−0.906726	−0.0321381
$797$	25.6851	0.909813	0.454906	−	0.890539i	$-0.349673\pi$
$797$	25.6851	0.909813	0.454906	+	0.890539i	$0.349673\pi$
$798$	0	0
$799$	16.1676	0.571967
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	0	0
$803$	5.51754	0.194710
$804$	0	0
$805$	7.43376	0.262006
$806$	−5.22163	−0.183924
$807$	0	0
$808$	7.64590	0.268982
$809$	32.4296	1.14017	0.570083	−	0.821587i	$-0.306911\pi$
$809$	32.4296	1.14017	0.570083	+	0.821587i	$0.306911\pi$
$810$	0	0
$811$	16.2175	0.569474	0.284737	−	0.958606i	$-0.408094\pi$
$811$	16.2175	0.569474	0.284737	+	0.958606i	$0.408094\pi$
$812$	0.610815	0.0214354
$813$	0	0
$814$	1.30541	0.0457545
$815$	−1.91622	−0.0671223
$816$	0	0
$817$	30.4433	1.06507
$818$	−3.64590	−0.127476
$819$	0	0
$820$	9.51754	0.332367
$821$	−16.7000	−0.582833	−0.291416	−	0.956596i	$-0.594127\pi$
$821$	−16.7000	−0.582833	−0.291416	+	0.956596i	$0.594127\pi$
$822$	0	0
$823$	17.9810	0.626779	0.313389	−	0.949625i	$-0.398536\pi$
$823$	17.9810	0.626779	0.313389	+	0.949625i	$0.398536\pi$
$824$	7.51754	0.261886
$825$	0	0
$826$	3.51754	0.122391
$827$	−11.4284	−0.397404	−0.198702	−	0.980060i	$-0.563673\pi$
$827$	−11.4284	−0.397404	−0.198702	+	0.980060i	$0.563673\pi$
$828$	0	0
$829$	22.2431	0.772535	0.386267	−	0.922387i	$-0.373764\pi$
$829$	22.2431	0.772535	0.386267	+	0.922387i	$0.373764\pi$
$830$	4.73917	0.164499
$831$	0	0
$832$	−0.694593	−0.0240807
$833$	2.69459	0.0933621
$834$	0	0
$835$	24.2121	0.837895
$836$	5.51754	0.190828
$837$	0	0
$838$	−18.3851	−0.635102
$839$	−13.8972	−0.479786	−0.239893	−	0.970799i	$-0.577112\pi$
$839$	−13.8972	−0.479786	−0.239893	+	0.970799i	$0.577112\pi$
$840$	0	0
$841$	−28.6269	−0.987135
$842$	−14.0000	−0.482472
$843$	0	0
$844$	−27.3756	−0.942306
$845$	−12.5175	−0.430617
$846$	0	0
$847$	1.00000	0.0343604
$848$	−4.12836	−0.141768
$849$	0	0
$850$	−2.69459	−0.0924238
$851$	−9.70409	−0.332652
$852$	0	0
$853$	26.5972	0.910671	0.455335	−	0.890320i	$-0.349519\pi$
$853$	26.5972	0.910671	0.455335	+	0.890320i	$0.349519\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	17.1634	0.586634
$857$	−3.91622	−0.133776	−0.0668878	−	0.997761i	$-0.521307\pi$
$857$	−3.91622	−0.133776	−0.0668878	+	0.997761i	$0.521307\pi$
$858$	0	0
$859$	−12.4688	−0.425431	−0.212716	−	0.977114i	$-0.568231\pi$
$859$	−12.4688	−0.425431	−0.212716	+	0.977114i	$0.568231\pi$
$860$	5.51754	0.188147
$861$	0	0
$862$	−15.3054	−0.521304
$863$	−37.5931	−1.27968	−0.639842	−	0.768507i	$-0.721000\pi$
$863$	−37.5931	−1.27968	−0.639842	+	0.768507i	$0.721000\pi$
$864$	0	0
$865$	7.22163	0.245543
$866$	9.34998	0.317725
$867$	0	0
$868$	−7.51754	−0.255162
$869$	−2.77837	−0.0942498
$870$	0	0
$871$	6.21751	0.210672
$872$	1.91622	0.0648915
$873$	0	0
$874$	−41.0161	−1.38739
$875$	1.00000	0.0338062
$876$	0	0
$877$	−30.7202	−1.03735	−0.518673	−	0.854972i	$-0.673574\pi$
$877$	−30.7202	−1.03735	−0.518673	+	0.854972i	$0.673574\pi$
$878$	−30.5526	−1.03110
$879$	0	0
$880$	1.00000	0.0337100
$881$	−3.95542	−0.133262	−0.0666308	−	0.997778i	$-0.521225\pi$
$881$	−3.95542	−0.133262	−0.0666308	+	0.997778i	$0.521225\pi$
$882$	0	0
$883$	49.3756	1.66162	0.830810	−	0.556556i	$-0.187878\pi$
$883$	49.3756	1.66162	0.830810	+	0.556556i	$0.187878\pi$
$884$	−1.87164	−0.0629502
$885$	0	0
$886$	−36.1985	−1.21611
$887$	4.30129	0.144423	0.0722116	−	0.997389i	$-0.476994\pi$
$887$	4.30129	0.144423	0.0722116	+	0.997389i	$0.476994\pi$
$888$	0	0
$889$	11.5175	0.386286
$890$	−8.04458	−0.269655
$891$	0	0
$892$	21.6459	0.724758
$893$	33.1052	1.10782
$894$	0	0
$895$	14.2959	0.477860
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	1.22163	0.0407663
$899$	−4.59182	−0.153146
$900$	0	0
$901$	−11.1242	−0.370602
$902$	−9.51754	−0.316899
$903$	0	0
$904$	−12.2121	−0.406170
$905$	−2.52704	−0.0840015
$906$	0	0
$907$	16.6946	0.554335	0.277167	−	0.960822i	$-0.410604\pi$
$907$	16.6946	0.554335	0.277167	+	0.960822i	$0.410604\pi$
$908$	11.3500	0.376662
$909$	0	0
$910$	0.694593	0.0230255
$911$	−22.9649	−0.760862	−0.380431	−	0.924809i	$-0.624224\pi$
$911$	−22.9649	−0.760862	−0.380431	+	0.924809i	$0.624224\pi$
$912$	0	0
$913$	−4.73917	−0.156844
$914$	11.0743	0.366305
$915$	0	0
$916$	−6.69459	−0.221196
$917$	5.38919	0.177967
$918$	0	0
$919$	−46.5836	−1.53665	−0.768325	−	0.640059i	$-0.778910\pi$
$919$	−46.5836	−1.53665	−0.768325	+	0.640059i	$0.778910\pi$
$920$	−7.43376	−0.245084
$921$	0	0
$922$	7.87164	0.259239
$923$	−3.31902	−0.109247
$924$	0	0
$925$	−1.30541	−0.0429215
$926$	−24.1676	−0.794195
$927$	0	0
$928$	−0.610815	−0.0200510
$929$	39.5039	1.29608	0.648041	−	0.761606i	$-0.275589\pi$
$929$	39.5039	1.29608	0.648041	+	0.761606i	$0.275589\pi$
$930$	0	0
$931$	5.51754	0.180830
$932$	14.2567	0.466994
$933$	0	0
$934$	−27.5175	−0.900401
$935$	2.69459	0.0881226
$936$	0	0
$937$	15.0743	0.492455	0.246228	−	0.969212i	$-0.420809\pi$
$937$	15.0743	0.492455	0.246228	+	0.969212i	$0.420809\pi$
$938$	8.95130	0.292270
$939$	0	0
$940$	6.00000	0.195698
$941$	−22.1676	−0.722642	−0.361321	−	0.932442i	$-0.617674\pi$
$941$	−22.1676	−0.722642	−0.361321	+	0.932442i	$0.617674\pi$
$942$	0	0
$943$	70.7511	2.30397
$944$	−3.51754	−0.114486
$945$	0	0
$946$	−5.51754	−0.179391
$947$	−5.61493	−0.182461	−0.0912304	−	0.995830i	$-0.529080\pi$
$947$	−5.61493	−0.182461	−0.0912304	+	0.995830i	$0.529080\pi$
$948$	0	0
$949$	−3.83244	−0.124406
$950$	−5.51754	−0.179013
$951$	0	0
$952$	−2.69459	−0.0873323
$953$	−17.5757	−0.569334	−0.284667	−	0.958626i	$-0.591883\pi$
$953$	−17.5757	−0.569334	−0.284667	+	0.958626i	$0.591883\pi$
$954$	0	0
$955$	−0.610815	−0.0197655
$956$	−3.72967	−0.120626
$957$	0	0
$958$	−26.0702	−0.842289
$959$	13.4338	0.433799
$960$	0	0
$961$	25.5134	0.823014
$962$	−0.906726	−0.0292340
$963$	0	0
$964$	−14.7392	−0.474717
$965$	−1.51754	−0.0488514
$966$	0	0
$967$	11.5175	0.370379	0.185190	−	0.982703i	$-0.440710\pi$
$967$	11.5175	0.370379	0.185190	+	0.982703i	$0.440710\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	6.90673	0.221762
$971$	24.2257	0.777441	0.388721	−	0.921356i	$-0.372917\pi$
$971$	24.2257	0.777441	0.388721	+	0.921356i	$0.372917\pi$
$972$	0	0
$973$	−18.9067	−0.606122
$974$	−12.9649	−0.415423
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−14.7338	−0.471376	−0.235688	−	0.971829i	$-0.575734\pi$
$977$	−14.7338	−0.471376	−0.235688	+	0.971829i	$0.575734\pi$
$978$	0	0
$979$	8.04458	0.257106
$980$	1.00000	0.0319438
$981$	0	0
$982$	36.4243	1.16235
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	−10.7392	−0.342178
$986$	−1.64590	−0.0524160
$987$	0	0
$988$	−3.83244	−0.121926
$989$	41.0161	1.30424
$990$	0	0
$991$	43.3500	1.37706	0.688529	−	0.725209i	$-0.258257\pi$
$991$	43.3500	1.37706	0.688529	+	0.725209i	$0.258257\pi$
$992$	7.51754	0.238682
$993$	0	0
$994$	−4.77837	−0.151561
$995$	−0.906726	−0.0287452
$996$	0	0
$997$	−2.34049	−0.0741240	−0.0370620	−	0.999313i	$-0.511800\pi$
$997$	−2.34049	−0.0741240	−0.0370620	+	0.999313i	$0.511800\pi$
$998$	−19.1242	−0.605367
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.ci.1.2	✓	3
3.2	odd	2		6930.2.a.cj.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6930.2.a.ci.1.2	✓	3	1.1	even	1	trivial
6930.2.a.cj.1.2	yes	3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -0.694593 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.69459 q^{17} +5.51754 q^{19} +1.00000 q^{20} -1.00000 q^{22} +7.43376 q^{23} +1.00000 q^{25} +0.694593 q^{26} +1.00000 q^{28} +0.610815 q^{29} -7.51754 q^{31} -1.00000 q^{32} -2.69459 q^{34} +1.00000 q^{35} -1.30541 q^{37} -5.51754 q^{38} -1.00000 q^{40} +9.51754 q^{41} +5.51754 q^{43} +1.00000 q^{44} -7.43376 q^{46} +6.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -0.694593 q^{52} -4.12836 q^{53} +1.00000 q^{55} -1.00000 q^{56} -0.610815 q^{58} -3.51754 q^{59} -4.00000 q^{61} +7.51754 q^{62} +1.00000 q^{64} -0.694593 q^{65} -8.95130 q^{67} +2.69459 q^{68} -1.00000 q^{70} +4.77837 q^{71} +5.51754 q^{73} +1.30541 q^{74} +5.51754 q^{76} +1.00000 q^{77} -2.77837 q^{79} +1.00000 q^{80} -9.51754 q^{82} -4.73917 q^{83} +2.69459 q^{85} -5.51754 q^{86} -1.00000 q^{88} +8.04458 q^{89} -0.694593 q^{91} +7.43376 q^{92} -6.00000 q^{94} +5.51754 q^{95} -6.90673 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} + 3 q^{11} - 3 q^{14} + 3 q^{16} + 6 q^{17} - 6 q^{19} + 3 q^{20} - 3 q^{22} + 6 q^{23} + 3 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} - 6 q^{34} + 3 q^{35} - 6 q^{37} + 6 q^{38} - 3 q^{40} + 6 q^{41} - 6 q^{43} + 3 q^{44} - 6 q^{46} + 18 q^{47} + 3 q^{49} - 3 q^{50} + 6 q^{53} + 3 q^{55} - 3 q^{56} - 6 q^{58} + 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} + 6 q^{68} - 3 q^{70} + 6 q^{71} - 6 q^{73} + 6 q^{74} - 6 q^{76} + 3 q^{77} + 3 q^{80} - 6 q^{82} + 6 q^{85} + 6 q^{86} - 3 q^{88} + 12 q^{89} + 6 q^{92} - 18 q^{94} - 6 q^{95} + 6 q^{97} - 3 q^{98}+O(q^{100}) \) 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 + 6 * q^17 - 6 * q^19 + 3 * q^20 - 3 * q^22 + 6 * q^23 + 3 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 - 6 * q^34 + 3 * q^35 - 6 * q^37 + 6 * q^38 - 3 * q^40 + 6 * q^41 - 6 * q^43 + 3 * q^44 - 6 * q^46 + 18 * q^47 + 3 * q^49 - 3 * q^50 + 6 * q^53 + 3 * q^55 - 3 * q^56 - 6 * q^58 + 12 * q^59 - 12 * q^61 + 3 * q^64 + 12 * q^67 + 6 * q^68 - 3 * q^70 + 6 * q^71 - 6 * q^73 + 6 * q^74 - 6 * q^76 + 3 * q^77 + 3 * q^80 - 6 * q^82 + 6 * q^85 + 6 * q^86 - 3 * q^88 + 12 * q^89 + 6 * q^92 - 18 * q^94 - 6 * q^95 + 6 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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