LMFDB - Embedded newform 6930.2.a.ce.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:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
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			$-1.81361$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -3.62721 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.20555 q^{17} -8.15165 q^{19} -1.00000 q^{20} +1.00000 q^{22} -0.897225 q^{23} +1.00000 q^{25} +3.62721 q^{26} -1.00000 q^{28} +7.30833 q^{29} -3.42166 q^{31} -1.00000 q^{32} -4.20555 q^{34} +1.00000 q^{35} -1.10278 q^{37} +8.15165 q^{38} +1.00000 q^{40} -12.3572 q^{41} +10.6761 q^{43} -1.00000 q^{44} +0.897225 q^{46} +1.15667 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.62721 q^{52} -11.3083 q^{53} +1.00000 q^{55} +1.00000 q^{56} -7.30833 q^{58} -7.25443 q^{59} -3.15667 q^{61} +3.42166 q^{62} +1.00000 q^{64} +3.62721 q^{65} -8.41110 q^{67} +4.20555 q^{68} -1.00000 q^{70} +13.0489 q^{71} +0.205550 q^{73} +1.10278 q^{74} -8.15165 q^{76} +1.00000 q^{77} +12.1517 q^{79} -1.00000 q^{80} +12.3572 q^{82} -2.20555 q^{83} -4.20555 q^{85} -10.6761 q^{86} +1.00000 q^{88} +7.45998 q^{89} +3.62721 q^{91} -0.897225 q^{92} -1.15667 q^{94} +8.15165 q^{95} +2.25945 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{13} + 3 q^{14} + 3 q^{16} - 2 q^{17} - 6 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 3 q^{25} - 2 q^{26} - 3 q^{28} - 12 q^{31} - 3 q^{32} + 2 q^{34} + 3 q^{35} + 4 q^{37} + 6 q^{38} + 3 q^{40} - 4 q^{41} + 8 q^{43} - 3 q^{44} + 10 q^{46} + 3 q^{49} - 3 q^{50} + 2 q^{52} - 12 q^{53} + 3 q^{55} + 3 q^{56} + 4 q^{59} - 6 q^{61} + 12 q^{62} + 3 q^{64} - 2 q^{65} + 4 q^{67} - 2 q^{68} - 3 q^{70} + 28 q^{71} - 14 q^{73} - 4 q^{74} - 6 q^{76} + 3 q^{77} + 18 q^{79} - 3 q^{80} + 4 q^{82} + 8 q^{83} + 2 q^{85} - 8 q^{86} + 3 q^{88} - 18 q^{89} - 2 q^{91} - 10 q^{92} + 6 q^{95} - 4 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 + 2 * q^13 + 3 * q^14 + 3 * q^16 - 2 * q^17 - 6 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 3 * q^25 - 2 * q^26 - 3 * q^28 - 12 * q^31 - 3 * q^32 + 2 * q^34 + 3 * q^35 + 4 * q^37 + 6 * q^38 + 3 * q^40 - 4 * q^41 + 8 * q^43 - 3 * q^44 + 10 * q^46 + 3 * q^49 - 3 * q^50 + 2 * q^52 - 12 * q^53 + 3 * q^55 + 3 * q^56 + 4 * q^59 - 6 * q^61 + 12 * q^62 + 3 * q^64 - 2 * q^65 + 4 * q^67 - 2 * q^68 - 3 * q^70 + 28 * q^71 - 14 * q^73 - 4 * q^74 - 6 * q^76 + 3 * q^77 + 18 * q^79 - 3 * q^80 + 4 * q^82 + 8 * q^83 + 2 * q^85 - 8 * q^86 + 3 * q^88 - 18 * q^89 - 2 * q^91 - 10 * q^92 + 6 * q^95 - 4 * 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
$13$	−3.62721	−1.00601	−0.503004	−	0.864284i	$-0.667772\pi$
$13$	−3.62721	−1.00601	−0.503004	+	0.864284i	$0.667772\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.20555	1.02000	0.509998	−	0.860176i	$-0.329646\pi$
$17$	4.20555	1.02000	0.509998	+	0.860176i	$0.329646\pi$
$18$	0	0
$19$	−8.15165	−1.87012	−0.935058	−	0.354493i	$-0.884653\pi$
$19$	−8.15165	−1.87012	−0.935058	+	0.354493i	$0.884653\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−0.897225	−0.187084	−0.0935422	−	0.995615i	$-0.529819\pi$
$23$	−0.897225	−0.187084	−0.0935422	+	0.995615i	$0.529819\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.62721	0.711355
$27$	0	0
$28$	−1.00000	−0.188982
$29$	7.30833	1.35712	0.678561	−	0.734544i	$-0.262604\pi$
$29$	7.30833	1.35712	0.678561	+	0.734544i	$0.262604\pi$
$30$	0	0
$31$	−3.42166	−0.614549	−0.307274	−	0.951621i	$-0.599417\pi$
$31$	−3.42166	−0.614549	−0.307274	+	0.951621i	$0.599417\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.20555	−0.721246
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.10278	−0.181295	−0.0906476	−	0.995883i	$-0.528894\pi$
$37$	−1.10278	−0.181295	−0.0906476	+	0.995883i	$0.528894\pi$
$38$	8.15165	1.32237
$39$	0	0
$40$	1.00000	0.158114
$41$	−12.3572	−1.92987	−0.964935	−	0.262488i	$-0.915457\pi$
$41$	−12.3572	−1.92987	−0.964935	+	0.262488i	$0.915457\pi$
$42$	0	0
$43$	10.6761	1.62809	0.814044	−	0.580803i	$-0.197261\pi$
$43$	10.6761	1.62809	0.814044	+	0.580803i	$0.197261\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0.897225	0.132289
$47$	1.15667	0.168718	0.0843591	−	0.996435i	$-0.473116\pi$
$47$	1.15667	0.168718	0.0843591	+	0.996435i	$0.473116\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.62721	−0.503004
$53$	−11.3083	−1.55332	−0.776659	−	0.629921i	$-0.783087\pi$
$53$	−11.3083	−1.55332	−0.776659	+	0.629921i	$0.783087\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	−7.30833	−0.959630
$59$	−7.25443	−0.944446	−0.472223	−	0.881479i	$-0.656548\pi$
$59$	−7.25443	−0.944446	−0.472223	+	0.881479i	$0.656548\pi$
$60$	0	0
$61$	−3.15667	−0.404171	−0.202085	−	0.979368i	$-0.564772\pi$
$61$	−3.15667	−0.404171	−0.202085	+	0.979368i	$0.564772\pi$
$62$	3.42166	0.434552
$63$	0	0
$64$	1.00000	0.125000
$65$	3.62721	0.449900
$66$	0	0
$67$	−8.41110	−1.02758	−0.513790	−	0.857916i	$-0.671759\pi$
$67$	−8.41110	−1.02758	−0.513790	+	0.857916i	$0.671759\pi$
$68$	4.20555	0.509998
$69$	0	0
$70$	−1.00000	−0.119523
$71$	13.0489	1.54862	0.774308	−	0.632809i	$-0.218098\pi$
$71$	13.0489	1.54862	0.774308	+	0.632809i	$0.218098\pi$
$72$	0	0
$73$	0.205550	0.0240578	0.0120289	−	0.999928i	$-0.496171\pi$
$73$	0.205550	0.0240578	0.0120289	+	0.999928i	$0.496171\pi$
$74$	1.10278	0.128195
$75$	0	0
$76$	−8.15165	−0.935058
$77$	1.00000	0.113961
$78$	0	0
$79$	12.1517	1.36717	0.683584	−	0.729872i	$-0.260420\pi$
$79$	12.1517	1.36717	0.683584	+	0.729872i	$0.260420\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	12.3572	1.36462
$83$	−2.20555	−0.242091	−0.121045	−	0.992647i	$-0.538625\pi$
$83$	−2.20555	−0.242091	−0.121045	+	0.992647i	$0.538625\pi$
$84$	0	0
$85$	−4.20555	−0.456156
$86$	−10.6761	−1.15123
$87$	0	0
$88$	1.00000	0.106600
$89$	7.45998	0.790756	0.395378	−	0.918519i	$-0.370614\pi$
$89$	7.45998	0.790756	0.395378	+	0.918519i	$0.370614\pi$
$90$	0	0
$91$	3.62721	0.380235
$92$	−0.897225	−0.0935422
$93$	0	0
$94$	−1.15667	−0.119302
$95$	8.15165	0.836342
$96$	0	0
$97$	2.25945	0.229412	0.114706	−	0.993399i	$-0.463407\pi$
$97$	2.25945	0.229412	0.114706	+	0.993399i	$0.463407\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.6655	−1.35977	−0.679885	−	0.733318i	$-0.737970\pi$
$101$	−13.6655	−1.35977	−0.679885	+	0.733318i	$0.737970\pi$
$102$	0	0
$103$	−5.04888	−0.497481	−0.248740	−	0.968570i	$-0.580017\pi$
$103$	−5.04888	−0.497481	−0.248740	+	0.968570i	$0.580017\pi$
$104$	3.62721	0.355677
$105$	0	0
$106$	11.3083	1.09836
$107$	−12.9894	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$107$	−12.9894	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$108$	0	0
$109$	5.10278	0.488757	0.244379	−	0.969680i	$-0.421416\pi$
$109$	5.10278	0.488757	0.244379	+	0.969680i	$0.421416\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	10.4111	0.979394	0.489697	−	0.871893i	$-0.337107\pi$
$113$	10.4111	0.979394	0.489697	+	0.871893i	$0.337107\pi$
$114$	0	0
$115$	0.897225	0.0836667
$116$	7.30833	0.678561
$117$	0	0
$118$	7.25443	0.667824
$119$	−4.20555	−0.385522
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.15667	0.285792
$123$	0	0
$124$	−3.42166	−0.307274
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.25443	0.288784	0.144392	−	0.989521i	$-0.453877\pi$
$127$	3.25443	0.288784	0.144392	+	0.989521i	$0.453877\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.62721	−0.318128
$131$	17.3083	1.51224	0.756118	−	0.654436i	$-0.227094\pi$
$131$	17.3083	1.51224	0.756118	+	0.654436i	$0.227094\pi$
$132$	0	0
$133$	8.15165	0.706838
$134$	8.41110	0.726608
$135$	0	0
$136$	−4.20555	−0.360623
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	13.2005	1.11965	0.559827	−	0.828609i	$-0.310868\pi$
$139$	13.2005	1.11965	0.559827	+	0.828609i	$0.310868\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−13.0489	−1.09504
$143$	3.62721	0.303323
$144$	0	0
$145$	−7.30833	−0.606923
$146$	−0.205550	−0.0170114
$147$	0	0
$148$	−1.10278	−0.0906476
$149$	23.6116	1.93434	0.967170	−	0.254131i	$-0.0817893\pi$
$149$	23.6116	1.93434	0.967170	+	0.254131i	$0.0817893\pi$
$150$	0	0
$151$	11.5139	0.936986	0.468493	−	0.883467i	$-0.344797\pi$
$151$	11.5139	0.936986	0.468493	+	0.883467i	$0.344797\pi$
$152$	8.15165	0.661186
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	3.42166	0.274835
$156$	0	0
$157$	−16.2056	−1.29334	−0.646672	−	0.762768i	$-0.723840\pi$
$157$	−16.2056	−1.29334	−0.646672	+	0.762768i	$0.723840\pi$
$158$	−12.1517	−0.966733
$159$	0	0
$160$	1.00000	0.0790569
$161$	0.897225	0.0707112
$162$	0	0
$163$	6.31335	0.494500	0.247250	−	0.968952i	$-0.420473\pi$
$163$	6.31335	0.494500	0.247250	+	0.968952i	$0.420473\pi$
$164$	−12.3572	−0.964935
$165$	0	0
$166$	2.20555	0.171184
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	0.156674	0.0120519
$170$	4.20555	0.322551
$171$	0	0
$172$	10.6761	0.814044
$173$	−9.25443	−0.703601	−0.351800	−	0.936075i	$-0.614430\pi$
$173$	−9.25443	−0.703601	−0.351800	+	0.936075i	$0.614430\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−7.45998	−0.559149
$179$	−4.98944	−0.372928	−0.186464	−	0.982462i	$-0.559703\pi$
$179$	−4.98944	−0.372928	−0.186464	+	0.982462i	$0.559703\pi$
$180$	0	0
$181$	−9.66553	−0.718433	−0.359216	−	0.933254i	$-0.616956\pi$
$181$	−9.66553	−0.718433	−0.359216	+	0.933254i	$0.616956\pi$
$182$	−3.62721	−0.268867
$183$	0	0
$184$	0.897225	0.0661443
$185$	1.10278	0.0810776
$186$	0	0
$187$	−4.20555	−0.307540
$188$	1.15667	0.0843591
$189$	0	0
$190$	−8.15165	−0.591383
$191$	9.15667	0.662554	0.331277	−	0.943534i	$-0.392521\pi$
$191$	9.15667	0.662554	0.331277	+	0.943534i	$0.392521\pi$
$192$	0	0
$193$	1.52946	0.110093	0.0550465	−	0.998484i	$-0.482469\pi$
$193$	1.52946	0.110093	0.0550465	+	0.998484i	$0.482469\pi$
$194$	−2.25945	−0.162219
$195$	0	0
$196$	1.00000	0.0714286
$197$	−14.3033	−1.01907	−0.509534	−	0.860451i	$-0.670182\pi$
$197$	−14.3033	−1.01907	−0.509534	+	0.860451i	$0.670182\pi$
$198$	0	0
$199$	−10.0978	−0.715811	−0.357905	−	0.933758i	$-0.616509\pi$
$199$	−10.0978	−0.715811	−0.357905	+	0.933758i	$0.616509\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	13.6655	0.961503
$203$	−7.30833	−0.512944
$204$	0	0
$205$	12.3572	0.863064
$206$	5.04888	0.351772
$207$	0	0
$208$	−3.62721	−0.251502
$209$	8.15165	0.563861
$210$	0	0
$211$	10.8433	0.746485	0.373243	−	0.927734i	$-0.378246\pi$
$211$	10.8433	0.746485	0.373243	+	0.927734i	$0.378246\pi$
$212$	−11.3083	−0.776659
$213$	0	0
$214$	12.9894	0.887940
$215$	−10.6761	−0.728103
$216$	0	0
$217$	3.42166	0.232278
$218$	−5.10278	−0.345604
$219$	0	0
$220$	1.00000	0.0674200
$221$	−15.2544	−1.02612
$222$	0	0
$223$	−9.45998	−0.633487	−0.316743	−	0.948511i	$-0.602589\pi$
$223$	−9.45998	−0.633487	−0.316743	+	0.948511i	$0.602589\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−10.4111	−0.692536
$227$	17.5678	1.16601	0.583007	−	0.812467i	$-0.301876\pi$
$227$	17.5678	1.16601	0.583007	+	0.812467i	$0.301876\pi$
$228$	0	0
$229$	25.7250	1.69995	0.849977	−	0.526820i	$-0.176616\pi$
$229$	25.7250	1.69995	0.849977	+	0.526820i	$0.176616\pi$
$230$	−0.897225	−0.0591613
$231$	0	0
$232$	−7.30833	−0.479815
$233$	−19.1567	−1.25500	−0.627498	−	0.778618i	$-0.715921\pi$
$233$	−19.1567	−1.25500	−0.627498	+	0.778618i	$0.715921\pi$
$234$	0	0
$235$	−1.15667	−0.0754531
$236$	−7.25443	−0.472223
$237$	0	0
$238$	4.20555	0.272605
$239$	18.0539	1.16781	0.583905	−	0.811822i	$-0.301524\pi$
$239$	18.0539	1.16781	0.583905	+	0.811822i	$0.301524\pi$
$240$	0	0
$241$	6.15165	0.396263	0.198131	−	0.980175i	$-0.436513\pi$
$241$	6.15165	0.396263	0.198131	+	0.980175i	$0.436513\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−3.15667	−0.202085
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	29.5678	1.88135
$248$	3.42166	0.217276
$249$	0	0
$250$	1.00000	0.0632456
$251$	29.9789	1.89225	0.946125	−	0.323802i	$-0.104961\pi$
$251$	29.9789	1.89225	0.946125	+	0.323802i	$0.104961\pi$
$252$	0	0
$253$	0.897225	0.0564080
$254$	−3.25443	−0.204201
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.05390	−0.252875	−0.126438	−	0.991975i	$-0.540354\pi$
$257$	−4.05390	−0.252875	−0.126438	+	0.991975i	$0.540354\pi$
$258$	0	0
$259$	1.10278	0.0685231
$260$	3.62721	0.224950
$261$	0	0
$262$	−17.3083	−1.06931
$263$	14.5089	0.894654	0.447327	−	0.894370i	$-0.352376\pi$
$263$	14.5089	0.894654	0.447327	+	0.894370i	$0.352376\pi$
$264$	0	0
$265$	11.3083	0.694665
$266$	−8.15165	−0.499810
$267$	0	0
$268$	−8.41110	−0.513790
$269$	−4.37279	−0.266614	−0.133307	−	0.991075i	$-0.542560\pi$
$269$	−4.37279	−0.266614	−0.133307	+	0.991075i	$0.542560\pi$
$270$	0	0
$271$	−10.6167	−0.644916	−0.322458	−	0.946584i	$-0.604509\pi$
$271$	−10.6167	−0.644916	−0.322458	+	0.946584i	$0.604509\pi$
$272$	4.20555	0.254999
$273$	0	0
$274$	10.0000	0.604122
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−19.7633	−1.18746	−0.593730	−	0.804664i	$-0.702345\pi$
$277$	−19.7633	−1.18746	−0.593730	+	0.804664i	$0.702345\pi$
$278$	−13.2005	−0.791715
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	27.0489	1.61360	0.806800	−	0.590824i	$-0.201197\pi$
$281$	27.0489	1.61360	0.806800	+	0.590824i	$0.201197\pi$
$282$	0	0
$283$	27.6655	1.64454	0.822272	−	0.569094i	$-0.192706\pi$
$283$	27.6655	1.64454	0.822272	+	0.569094i	$0.192706\pi$
$284$	13.0489	0.774308
$285$	0	0
$286$	−3.62721	−0.214482
$287$	12.3572	0.729423
$288$	0	0
$289$	0.686652	0.0403913
$290$	7.30833	0.429160
$291$	0	0
$292$	0.205550	0.0120289
$293$	8.37279	0.489143	0.244572	−	0.969631i	$-0.421353\pi$
$293$	8.37279	0.489143	0.244572	+	0.969631i	$0.421353\pi$
$294$	0	0
$295$	7.25443	0.422369
$296$	1.10278	0.0640975
$297$	0	0
$298$	−23.6116	−1.36778
$299$	3.25443	0.188208
$300$	0	0
$301$	−10.6761	−0.615360
$302$	−11.5139	−0.662549
$303$	0	0
$304$	−8.15165	−0.467529
$305$	3.15667	0.180751
$306$	0	0
$307$	−10.2056	−0.582462	−0.291231	−	0.956653i	$-0.594065\pi$
$307$	−10.2056	−0.582462	−0.291231	+	0.956653i	$0.594065\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−3.42166	−0.194337
$311$	8.16724	0.463122	0.231561	−	0.972820i	$-0.425617\pi$
$311$	8.16724	0.463122	0.231561	+	0.972820i	$0.425617\pi$
$312$	0	0
$313$	34.5628	1.95360	0.976801	−	0.214149i	$-0.0686977\pi$
$313$	34.5628	1.95360	0.976801	+	0.214149i	$0.0686977\pi$
$314$	16.2056	0.914532
$315$	0	0
$316$	12.1517	0.683584
$317$	−9.21057	−0.517317	−0.258659	−	0.965969i	$-0.583280\pi$
$317$	−9.21057	−0.517317	−0.258659	+	0.965969i	$0.583280\pi$
$318$	0	0
$319$	−7.30833	−0.409188
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−0.897225	−0.0500004
$323$	−34.2822	−1.90751
$324$	0	0
$325$	−3.62721	−0.201202
$326$	−6.31335	−0.349664
$327$	0	0
$328$	12.3572	0.682312
$329$	−1.15667	−0.0637695
$330$	0	0
$331$	16.2439	0.892843	0.446422	−	0.894823i	$-0.352698\pi$
$331$	16.2439	0.892843	0.446422	+	0.894823i	$0.352698\pi$
$332$	−2.20555	−0.121045
$333$	0	0
$334$	−8.00000	−0.437741
$335$	8.41110	0.459547
$336$	0	0
$337$	−1.25443	−0.0683329	−0.0341665	−	0.999416i	$-0.510878\pi$
$337$	−1.25443	−0.0683329	−0.0341665	+	0.999416i	$0.510878\pi$
$338$	−0.156674	−0.00852195
$339$	0	0
$340$	−4.20555	−0.228078
$341$	3.42166	0.185293
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−10.6761	−0.575616
$345$	0	0
$346$	9.25443	0.497521
$347$	−7.42166	−0.398416	−0.199208	−	0.979957i	$-0.563837\pi$
$347$	−7.42166	−0.398416	−0.199208	+	0.979957i	$0.563837\pi$
$348$	0	0
$349$	−18.3033	−0.979753	−0.489877	−	0.871792i	$-0.662958\pi$
$349$	−18.3033	−0.979753	−0.489877	+	0.871792i	$0.662958\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	−28.8761	−1.53692	−0.768460	−	0.639898i	$-0.778977\pi$
$353$	−28.8761	−1.53692	−0.768460	+	0.639898i	$0.778977\pi$
$354$	0	0
$355$	−13.0489	−0.692562
$356$	7.45998	0.395378
$357$	0	0
$358$	4.98944	0.263700
$359$	−4.15165	−0.219116	−0.109558	−	0.993980i	$-0.534943\pi$
$359$	−4.15165	−0.219116	−0.109558	+	0.993980i	$0.534943\pi$
$360$	0	0
$361$	47.4494	2.49734
$362$	9.66553	0.508009
$363$	0	0
$364$	3.62721	0.190118
$365$	−0.205550	−0.0107590
$366$	0	0
$367$	−5.26447	−0.274803	−0.137402	−	0.990515i	$-0.543875\pi$
$367$	−5.26447	−0.274803	−0.137402	+	0.990515i	$0.543875\pi$
$368$	−0.897225	−0.0467711
$369$	0	0
$370$	−1.10278	−0.0573305
$371$	11.3083	0.587099
$372$	0	0
$373$	35.8711	1.85733	0.928667	−	0.370915i	$-0.120956\pi$
$373$	35.8711	1.85733	0.928667	+	0.370915i	$0.120956\pi$
$374$	4.20555	0.217464
$375$	0	0
$376$	−1.15667	−0.0596509
$377$	−26.5089	−1.36528
$378$	0	0
$379$	−33.0177	−1.69601	−0.848003	−	0.529992i	$-0.822195\pi$
$379$	−33.0177	−1.69601	−0.848003	+	0.529992i	$0.822195\pi$
$380$	8.15165	0.418171
$381$	0	0
$382$	−9.15667	−0.468496
$383$	−2.95112	−0.150795	−0.0753977	−	0.997154i	$-0.524023\pi$
$383$	−2.95112	−0.150795	−0.0753977	+	0.997154i	$0.524023\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−1.52946	−0.0778475
$387$	0	0
$388$	2.25945	0.114706
$389$	−1.89220	−0.0959384	−0.0479692	−	0.998849i	$-0.515275\pi$
$389$	−1.89220	−0.0959384	−0.0479692	+	0.998849i	$0.515275\pi$
$390$	0	0
$391$	−3.77332	−0.190825
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	14.3033	0.720590
$395$	−12.1517	−0.611416
$396$	0	0
$397$	5.36222	0.269122	0.134561	−	0.990905i	$-0.457038\pi$
$397$	5.36222	0.269122	0.134561	+	0.990905i	$0.457038\pi$
$398$	10.0978	0.506155
$399$	0	0
$400$	1.00000	0.0500000
$401$	−23.7038	−1.18371	−0.591857	−	0.806043i	$-0.701605\pi$
$401$	−23.7038	−1.18371	−0.591857	+	0.806043i	$0.701605\pi$
$402$	0	0
$403$	12.4111	0.618241
$404$	−13.6655	−0.679885
$405$	0	0
$406$	7.30833	0.362706
$407$	1.10278	0.0546625
$408$	0	0
$409$	27.0816	1.33910	0.669551	−	0.742766i	$-0.266487\pi$
$409$	27.0816	1.33910	0.669551	+	0.742766i	$0.266487\pi$
$410$	−12.3572	−0.610279
$411$	0	0
$412$	−5.04888	−0.248740
$413$	7.25443	0.356967
$414$	0	0
$415$	2.20555	0.108266
$416$	3.62721	0.177839
$417$	0	0
$418$	−8.15165	−0.398710
$419$	3.05892	0.149438	0.0747191	−	0.997205i	$-0.476194\pi$
$419$	3.05892	0.149438	0.0747191	+	0.997205i	$0.476194\pi$
$420$	0	0
$421$	21.6655	1.05591	0.527957	−	0.849271i	$-0.322958\pi$
$421$	21.6655	1.05591	0.527957	+	0.849271i	$0.322958\pi$
$422$	−10.8433	−0.527845
$423$	0	0
$424$	11.3083	0.549181
$425$	4.20555	0.203999
$426$	0	0
$427$	3.15667	0.152762
$428$	−12.9894	−0.627868
$429$	0	0
$430$	10.6761	0.514847
$431$	34.3260	1.65343	0.826713	−	0.562623i	$-0.190208\pi$
$431$	34.3260	1.65343	0.826713	+	0.562623i	$0.190208\pi$
$432$	0	0
$433$	−0.0538991	−0.00259023	−0.00129511	−	0.999999i	$-0.500412\pi$
$433$	−0.0538991	−0.00259023	−0.00129511	+	0.999999i	$0.500412\pi$
$434$	−3.42166	−0.164245
$435$	0	0
$436$	5.10278	0.244379
$437$	7.31386	0.349870
$438$	0	0
$439$	−7.14663	−0.341090	−0.170545	−	0.985350i	$-0.554553\pi$
$439$	−7.14663	−0.341090	−0.170545	+	0.985350i	$0.554553\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	15.2544	0.725579
$443$	−8.74557	−0.415515	−0.207757	−	0.978180i	$-0.566616\pi$
$443$	−8.74557	−0.415515	−0.207757	+	0.978180i	$0.566616\pi$
$444$	0	0
$445$	−7.45998	−0.353637
$446$	9.45998	0.447943
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−39.7038	−1.87374	−0.936870	−	0.349678i	$-0.886291\pi$
$449$	−39.7038	−1.87374	−0.936870	+	0.349678i	$0.886291\pi$
$450$	0	0
$451$	12.3572	0.581878
$452$	10.4111	0.489697
$453$	0	0
$454$	−17.5678	−0.824497
$455$	−3.62721	−0.170046
$456$	0	0
$457$	12.3133	0.575994	0.287997	−	0.957631i	$-0.407011\pi$
$457$	12.3133	0.575994	0.287997	+	0.957631i	$0.407011\pi$
$458$	−25.7250	−1.20205
$459$	0	0
$460$	0.897225	0.0418333
$461$	15.0489	0.700896	0.350448	−	0.936582i	$-0.386029\pi$
$461$	15.0489	0.700896	0.350448	+	0.936582i	$0.386029\pi$
$462$	0	0
$463$	31.4061	1.45956	0.729782	−	0.683680i	$-0.239622\pi$
$463$	31.4061	1.45956	0.729782	+	0.683680i	$0.239622\pi$
$464$	7.30833	0.339280
$465$	0	0
$466$	19.1567	0.887416
$467$	19.0716	0.882529	0.441264	−	0.897377i	$-0.354530\pi$
$467$	19.0716	0.882529	0.441264	+	0.897377i	$0.354530\pi$
$468$	0	0
$469$	8.41110	0.388389
$470$	1.15667	0.0533534
$471$	0	0
$472$	7.25443	0.333912
$473$	−10.6761	−0.490887
$474$	0	0
$475$	−8.15165	−0.374023
$476$	−4.20555	−0.192761
$477$	0	0
$478$	−18.0539	−0.825766
$479$	9.49115	0.433662	0.216831	−	0.976209i	$-0.430428\pi$
$479$	9.49115	0.433662	0.216831	+	0.976209i	$0.430428\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−6.15165	−0.280200
$483$	0	0
$484$	1.00000	0.0454545
$485$	−2.25945	−0.102596
$486$	0	0
$487$	0.0438527	0.00198715	0.000993577	−	1.00000i	$-0.499684\pi$
$487$	0.0438527	0.00198715	0.000993577		1.00000i	$0.499684\pi$
$488$	3.15667	0.142896
$489$	0	0
$490$	1.00000	0.0451754
$491$	−1.56777	−0.0707527	−0.0353763	−	0.999374i	$-0.511263\pi$
$491$	−1.56777	−0.0707527	−0.0353763	+	0.999374i	$0.511263\pi$
$492$	0	0
$493$	30.7355	1.38426
$494$	−29.5678	−1.33032
$495$	0	0
$496$	−3.42166	−0.153637
$497$	−13.0489	−0.585322
$498$	0	0
$499$	−42.3416	−1.89547	−0.947736	−	0.319057i	$-0.896634\pi$
$499$	−42.3416	−1.89547	−0.947736	+	0.319057i	$0.896634\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−29.9789	−1.33802
$503$	36.4111	1.62349	0.811745	−	0.584012i	$-0.198518\pi$
$503$	36.4111	1.62349	0.811745	+	0.584012i	$0.198518\pi$
$504$	0	0
$505$	13.6655	0.608108
$506$	−0.897225	−0.0398865
$507$	0	0
$508$	3.25443	0.144392
$509$	30.0766	1.33312	0.666562	−	0.745450i	$-0.267765\pi$
$509$	30.0766	1.33312	0.666562	+	0.745450i	$0.267765\pi$
$510$	0	0
$511$	−0.205550	−0.00909300
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.05390	0.178810
$515$	5.04888	0.222480
$516$	0	0
$517$	−1.15667	−0.0508705
$518$	−1.10278	−0.0484532
$519$	0	0
$520$	−3.62721	−0.159064
$521$	33.2233	1.45554	0.727769	−	0.685823i	$-0.240557\pi$
$521$	33.2233	1.45554	0.727769	+	0.685823i	$0.240557\pi$
$522$	0	0
$523$	−15.5577	−0.680292	−0.340146	−	0.940373i	$-0.610476\pi$
$523$	−15.5577	−0.680292	−0.340146	+	0.940373i	$0.610476\pi$
$524$	17.3083	0.756118
$525$	0	0
$526$	−14.5089	−0.632616
$527$	−14.3900	−0.626837
$528$	0	0
$529$	−22.1950	−0.964999
$530$	−11.3083	−0.491202
$531$	0	0
$532$	8.15165	0.353419
$533$	44.8222	1.94147
$534$	0	0
$535$	12.9894	0.561582
$536$	8.41110	0.363304
$537$	0	0
$538$	4.37279	0.188524
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	23.4161	1.00674	0.503369	−	0.864072i	$-0.332094\pi$
$541$	23.4161	1.00674	0.503369	+	0.864072i	$0.332094\pi$
$542$	10.6167	0.456024
$543$	0	0
$544$	−4.20555	−0.180311
$545$	−5.10278	−0.218579
$546$	0	0
$547$	−8.41110	−0.359633	−0.179816	−	0.983700i	$-0.557550\pi$
$547$	−8.41110	−0.359633	−0.179816	+	0.983700i	$0.557550\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	1.00000	0.0426401
$551$	−59.5749	−2.53798
$552$	0	0
$553$	−12.1517	−0.516741
$554$	19.7633	0.839661
$555$	0	0
$556$	13.2005	0.559827
$557$	−37.0278	−1.56892	−0.784458	−	0.620182i	$-0.787059\pi$
$557$	−37.0278	−1.56892	−0.784458	+	0.620182i	$0.787059\pi$
$558$	0	0
$559$	−38.7244	−1.63787
$560$	1.00000	0.0422577
$561$	0	0
$562$	−27.0489	−1.14099
$563$	−22.9200	−0.965961	−0.482980	−	0.875631i	$-0.660446\pi$
$563$	−22.9200	−0.965961	−0.482980	+	0.875631i	$0.660446\pi$
$564$	0	0
$565$	−10.4111	−0.437998
$566$	−27.6655	−1.16287
$567$	0	0
$568$	−13.0489	−0.547519
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	20.3033	0.849667	0.424833	−	0.905272i	$-0.360333\pi$
$571$	20.3033	0.849667	0.424833	+	0.905272i	$0.360333\pi$
$572$	3.62721	0.151661
$573$	0	0
$574$	−12.3572	−0.515780
$575$	−0.897225	−0.0374169
$576$	0	0
$577$	16.4650	0.685447	0.342723	−	0.939436i	$-0.388651\pi$
$577$	16.4650	0.685447	0.342723	+	0.939436i	$0.388651\pi$
$578$	−0.686652	−0.0285609
$579$	0	0
$580$	−7.30833	−0.303462
$581$	2.20555	0.0915016
$582$	0	0
$583$	11.3083	0.468343
$584$	−0.205550	−0.00850572
$585$	0	0
$586$	−8.37279	−0.345877
$587$	30.6605	1.26549	0.632747	−	0.774358i	$-0.281927\pi$
$587$	30.6605	1.26549	0.632747	+	0.774358i	$0.281927\pi$
$588$	0	0
$589$	27.8922	1.14928
$590$	−7.25443	−0.298660
$591$	0	0
$592$	−1.10278	−0.0453238
$593$	−18.3033	−0.751627	−0.375813	−	0.926695i	$-0.622637\pi$
$593$	−18.3033	−0.751627	−0.375813	+	0.926695i	$0.622637\pi$
$594$	0	0
$595$	4.20555	0.172411
$596$	23.6116	0.967170
$597$	0	0
$598$	−3.25443	−0.133083
$599$	−29.6555	−1.21169	−0.605845	−	0.795583i	$-0.707165\pi$
$599$	−29.6555	−1.21169	−0.605845	+	0.795583i	$0.707165\pi$
$600$	0	0
$601$	18.7783	0.765985	0.382992	−	0.923751i	$-0.374894\pi$
$601$	18.7783	0.765985	0.382992	+	0.923751i	$0.374894\pi$
$602$	10.6761	0.435125
$603$	0	0
$604$	11.5139	0.468493
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	14.6761	0.595684	0.297842	−	0.954615i	$-0.403733\pi$
$607$	14.6761	0.595684	0.297842	+	0.954615i	$0.403733\pi$
$608$	8.15165	0.330593
$609$	0	0
$610$	−3.15667	−0.127810
$611$	−4.19550	−0.169732
$612$	0	0
$613$	−5.28560	−0.213483	−0.106742	−	0.994287i	$-0.534042\pi$
$613$	−5.28560	−0.213483	−0.106742	+	0.994287i	$0.534042\pi$
$614$	10.2056	0.411862
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	44.6933	1.79928	0.899642	−	0.436629i	$-0.143828\pi$
$617$	44.6933	1.79928	0.899642	+	0.436629i	$0.143828\pi$
$618$	0	0
$619$	−17.5678	−0.706108	−0.353054	−	0.935603i	$-0.614857\pi$
$619$	−17.5678	−0.706108	−0.353054	+	0.935603i	$0.614857\pi$
$620$	3.42166	0.137417
$621$	0	0
$622$	−8.16724	−0.327476
$623$	−7.45998	−0.298878
$624$	0	0
$625$	1.00000	0.0400000
$626$	−34.5628	−1.38141
$627$	0	0
$628$	−16.2056	−0.646672
$629$	−4.63778	−0.184920
$630$	0	0
$631$	4.74557	0.188918	0.0944592	−	0.995529i	$-0.469888\pi$
$631$	4.74557	0.188918	0.0944592	+	0.995529i	$0.469888\pi$
$632$	−12.1517	−0.483367
$633$	0	0
$634$	9.21057	0.365799
$635$	−3.25443	−0.129148
$636$	0	0
$637$	−3.62721	−0.143715
$638$	7.30833	0.289339
$639$	0	0
$640$	1.00000	0.0395285
$641$	2.52998	0.0999281	0.0499641	−	0.998751i	$-0.484089\pi$
$641$	2.52998	0.0999281	0.0499641	+	0.998751i	$0.484089\pi$
$642$	0	0
$643$	−0.670549	−0.0264439	−0.0132219	−	0.999913i	$-0.504209\pi$
$643$	−0.670549	−0.0264439	−0.0132219	+	0.999913i	$0.504209\pi$
$644$	0.897225	0.0353556
$645$	0	0
$646$	34.2822	1.34881
$647$	−11.8922	−0.467531	−0.233765	−	0.972293i	$-0.575105\pi$
$647$	−11.8922	−0.467531	−0.233765	+	0.972293i	$0.575105\pi$
$648$	0	0
$649$	7.25443	0.284761
$650$	3.62721	0.142271
$651$	0	0
$652$	6.31335	0.247250
$653$	11.4161	0.446747	0.223374	−	0.974733i	$-0.428293\pi$
$653$	11.4161	0.446747	0.223374	+	0.974733i	$0.428293\pi$
$654$	0	0
$655$	−17.3083	−0.676292
$656$	−12.3572	−0.482468
$657$	0	0
$658$	1.15667	0.0450919
$659$	−1.87108	−0.0728868	−0.0364434	−	0.999336i	$-0.511603\pi$
$659$	−1.87108	−0.0728868	−0.0364434	+	0.999336i	$0.511603\pi$
$660$	0	0
$661$	33.1355	1.28882	0.644412	−	0.764679i	$-0.277102\pi$
$661$	33.1355	1.28882	0.644412	+	0.764679i	$0.277102\pi$
$662$	−16.2439	−0.631336
$663$	0	0
$664$	2.20555	0.0855919
$665$	−8.15165	−0.316107
$666$	0	0
$667$	−6.55721	−0.253896
$668$	8.00000	0.309529
$669$	0	0
$670$	−8.41110	−0.324949
$671$	3.15667	0.121862
$672$	0	0
$673$	−18.1955	−0.701385	−0.350693	−	0.936491i	$-0.614054\pi$
$673$	−18.1955	−0.701385	−0.350693	+	0.936491i	$0.614054\pi$
$674$	1.25443	0.0483187
$675$	0	0
$676$	0.156674	0.00602593
$677$	−15.9789	−0.614118	−0.307059	−	0.951690i	$-0.599345\pi$
$677$	−15.9789	−0.614118	−0.307059	+	0.951690i	$0.599345\pi$
$678$	0	0
$679$	−2.25945	−0.0867097
$680$	4.20555	0.161275
$681$	0	0
$682$	−3.42166	−0.131022
$683$	17.8711	0.683818	0.341909	−	0.939733i	$-0.388927\pi$
$683$	17.8711	0.683818	0.341909	+	0.939733i	$0.388927\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	1.00000	0.0381802
$687$	0	0
$688$	10.6761	0.407022
$689$	41.0177	1.56265
$690$	0	0
$691$	−2.20555	−0.0839031	−0.0419515	−	0.999120i	$-0.513358\pi$
$691$	−2.20555	−0.0839031	−0.0419515	+	0.999120i	$0.513358\pi$
$692$	−9.25443	−0.351800
$693$	0	0
$694$	7.42166	0.281722
$695$	−13.2005	−0.500725
$696$	0	0
$697$	−51.9688	−1.96846
$698$	18.3033	0.692790
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	25.6217	0.967717	0.483859	−	0.875146i	$-0.339235\pi$
$701$	25.6217	0.967717	0.483859	+	0.875146i	$0.339235\pi$
$702$	0	0
$703$	8.98944	0.339043
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	28.8761	1.08677
$707$	13.6655	0.513945
$708$	0	0
$709$	5.48110	0.205847	0.102924	−	0.994689i	$-0.467180\pi$
$709$	5.48110	0.205847	0.102924	+	0.994689i	$0.467180\pi$
$710$	13.0489	0.489716
$711$	0	0
$712$	−7.45998	−0.279574
$713$	3.07000	0.114972
$714$	0	0
$715$	−3.62721	−0.135650
$716$	−4.98944	−0.186464
$717$	0	0
$718$	4.15165	0.154938
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	5.04888	0.188030
$722$	−47.4494	−1.76588
$723$	0	0
$724$	−9.66553	−0.359216
$725$	7.30833	0.271424
$726$	0	0
$727$	18.5855	0.689297	0.344649	−	0.938732i	$-0.387998\pi$
$727$	18.5855	0.689297	0.344649	+	0.938732i	$0.387998\pi$
$728$	−3.62721	−0.134433
$729$	0	0
$730$	0.205550	0.00760775
$731$	44.8988	1.66064
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	5.26447	0.194315
$735$	0	0
$736$	0.897225	0.0330722
$737$	8.41110	0.309827
$738$	0	0
$739$	5.97887	0.219936	0.109968	−	0.993935i	$-0.464925\pi$
$739$	5.97887	0.219936	0.109968	+	0.993935i	$0.464925\pi$
$740$	1.10278	0.0405388
$741$	0	0
$742$	−11.3083	−0.415142
$743$	50.0978	1.83791	0.918954	−	0.394364i	$-0.129035\pi$
$743$	50.0978	1.83791	0.918954	+	0.394364i	$0.129035\pi$
$744$	0	0
$745$	−23.6116	−0.865063
$746$	−35.8711	−1.31333
$747$	0	0
$748$	−4.20555	−0.153770
$749$	12.9894	0.474624
$750$	0	0
$751$	−8.63778	−0.315197	−0.157598	−	0.987503i	$-0.550375\pi$
$751$	−8.63778	−0.315197	−0.157598	+	0.987503i	$0.550375\pi$
$752$	1.15667	0.0421796
$753$	0	0
$754$	26.5089	0.965395
$755$	−11.5139	−0.419033
$756$	0	0
$757$	25.5139	0.927318	0.463659	−	0.886014i	$-0.346536\pi$
$757$	25.5139	0.927318	0.463659	+	0.886014i	$0.346536\pi$
$758$	33.0177	1.19926
$759$	0	0
$760$	−8.15165	−0.295691
$761$	32.6605	1.18394	0.591971	−	0.805959i	$-0.298350\pi$
$761$	32.6605	1.18394	0.591971	+	0.805959i	$0.298350\pi$
$762$	0	0
$763$	−5.10278	−0.184733
$764$	9.15667	0.331277
$765$	0	0
$766$	2.95112	0.106628
$767$	26.3133	0.950120
$768$	0	0
$769$	16.7683	0.604680	0.302340	−	0.953200i	$-0.402232\pi$
$769$	16.7683	0.604680	0.302340	+	0.953200i	$0.402232\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	1.52946	0.0550465
$773$	2.82220	0.101507	0.0507537	−	0.998711i	$-0.483838\pi$
$773$	2.82220	0.101507	0.0507537	+	0.998711i	$0.483838\pi$
$774$	0	0
$775$	−3.42166	−0.122910
$776$	−2.25945	−0.0811095
$777$	0	0
$778$	1.89220	0.0678387
$779$	100.732	3.60908
$780$	0	0
$781$	−13.0489	−0.466925
$782$	3.77332	0.134934
$783$	0	0
$784$	1.00000	0.0357143
$785$	16.2056	0.578401
$786$	0	0
$787$	−4.82220	−0.171893	−0.0859464	−	0.996300i	$-0.527391\pi$
$787$	−4.82220	−0.171893	−0.0859464	+	0.996300i	$0.527391\pi$
$788$	−14.3033	−0.509534
$789$	0	0
$790$	12.1517	0.432336
$791$	−10.4111	−0.370176
$792$	0	0
$793$	11.4499	0.406599
$794$	−5.36222	−0.190298
$795$	0	0
$796$	−10.0978	−0.357905
$797$	0.540024	0.0191286	0.00956431	−	0.999954i	$-0.496956\pi$
$797$	0.540024	0.0191286	0.00956431	+	0.999954i	$0.496956\pi$
$798$	0	0
$799$	4.86445	0.172092
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	23.7038	0.837012
$803$	−0.205550	−0.00725371
$804$	0	0
$805$	−0.897225	−0.0316230
$806$	−12.4111	−0.437162
$807$	0	0
$808$	13.6655	0.480752
$809$	17.8922	0.629056	0.314528	−	0.949248i	$-0.398154\pi$
$809$	17.8922	0.629056	0.314528	+	0.949248i	$0.398154\pi$
$810$	0	0
$811$	−12.2283	−0.429393	−0.214696	−	0.976681i	$-0.568876\pi$
$811$	−12.2283	−0.429393	−0.214696	+	0.976681i	$0.568876\pi$
$812$	−7.30833	−0.256472
$813$	0	0
$814$	−1.10278	−0.0386522
$815$	−6.31335	−0.221147
$816$	0	0
$817$	−87.0278	−3.04472
$818$	−27.0816	−0.946888
$819$	0	0
$820$	12.3572	0.431532
$821$	−38.5316	−1.34476	−0.672381	−	0.740206i	$-0.734728\pi$
$821$	−38.5316	−1.34476	−0.672381	+	0.740206i	$0.734728\pi$
$822$	0	0
$823$	21.9461	0.764993	0.382496	−	0.923957i	$-0.375064\pi$
$823$	21.9461	0.764993	0.382496	+	0.923957i	$0.375064\pi$
$824$	5.04888	0.175886
$825$	0	0
$826$	−7.25443	−0.252414
$827$	−9.68665	−0.336838	−0.168419	−	0.985716i	$-0.553866\pi$
$827$	−9.68665	−0.336838	−0.168419	+	0.985716i	$0.553866\pi$
$828$	0	0
$829$	31.0771	1.07935	0.539677	−	0.841872i	$-0.318546\pi$
$829$	31.0771	1.07935	0.539677	+	0.841872i	$0.318546\pi$
$830$	−2.20555	−0.0765558
$831$	0	0
$832$	−3.62721	−0.125751
$833$	4.20555	0.145714
$834$	0	0
$835$	−8.00000	−0.276851
$836$	8.15165	0.281931
$837$	0	0
$838$	−3.05892	−0.105669
$839$	24.7738	0.855288	0.427644	−	0.903947i	$-0.359344\pi$
$839$	24.7738	0.855288	0.427644	+	0.903947i	$0.359344\pi$
$840$	0	0
$841$	24.4116	0.841780
$842$	−21.6655	−0.746643
$843$	0	0
$844$	10.8433	0.373243
$845$	−0.156674	−0.00538976
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−11.3083	−0.388329
$849$	0	0
$850$	−4.20555	−0.144249
$851$	0.989437	0.0339175
$852$	0	0
$853$	1.58890	0.0544029	0.0272014	−	0.999630i	$-0.491340\pi$
$853$	1.58890	0.0544029	0.0272014	+	0.999630i	$0.491340\pi$
$854$	−3.15667	−0.108019
$855$	0	0
$856$	12.9894	0.443970
$857$	−19.1255	−0.653315	−0.326657	−	0.945143i	$-0.605922\pi$
$857$	−19.1255	−0.653315	−0.326657	+	0.945143i	$0.605922\pi$
$858$	0	0
$859$	4.94108	0.168587	0.0842937	−	0.996441i	$-0.473137\pi$
$859$	4.94108	0.168587	0.0842937	+	0.996441i	$0.473137\pi$
$860$	−10.6761	−0.364052
$861$	0	0
$862$	−34.3260	−1.16915
$863$	19.3295	0.657982	0.328991	−	0.944333i	$-0.393291\pi$
$863$	19.3295	0.657982	0.328991	+	0.944333i	$0.393291\pi$
$864$	0	0
$865$	9.25443	0.314660
$866$	0.0538991	0.00183157
$867$	0	0
$868$	3.42166	0.116139
$869$	−12.1517	−0.412217
$870$	0	0
$871$	30.5089	1.03375
$872$	−5.10278	−0.172802
$873$	0	0
$874$	−7.31386	−0.247395
$875$	1.00000	0.0338062
$876$	0	0
$877$	16.4011	0.553824	0.276912	−	0.960895i	$-0.410689\pi$
$877$	16.4011	0.553824	0.276912	+	0.960895i	$0.410689\pi$
$878$	7.14663	0.241187
$879$	0	0
$880$	1.00000	0.0337100
$881$	−39.5678	−1.33307	−0.666536	−	0.745473i	$-0.732224\pi$
$881$	−39.5678	−1.33307	−0.666536	+	0.745473i	$0.732224\pi$
$882$	0	0
$883$	55.4389	1.86567	0.932833	−	0.360309i	$-0.117329\pi$
$883$	55.4389	1.86567	0.932833	+	0.360309i	$0.117329\pi$
$884$	−15.2544	−0.513062
$885$	0	0
$886$	8.74557	0.293813
$887$	−44.1955	−1.48394	−0.741970	−	0.670433i	$-0.766108\pi$
$887$	−44.1955	−1.48394	−0.741970	+	0.670433i	$0.766108\pi$
$888$	0	0
$889$	−3.25443	−0.109150
$890$	7.45998	0.250059
$891$	0	0
$892$	−9.45998	−0.316743
$893$	−9.42880	−0.315523
$894$	0	0
$895$	4.98944	0.166778
$896$	1.00000	0.0334077
$897$	0	0
$898$	39.7038	1.32493
$899$	−25.0066	−0.834018
$900$	0	0
$901$	−47.5577	−1.58438
$902$	−12.3572	−0.411450
$903$	0	0
$904$	−10.4111	−0.346268
$905$	9.66553	0.321293
$906$	0	0
$907$	9.26447	0.307622	0.153811	−	0.988100i	$-0.450845\pi$
$907$	9.26447	0.307622	0.153811	+	0.988100i	$0.450845\pi$
$908$	17.5678	0.583007
$909$	0	0
$910$	3.62721	0.120241
$911$	37.0489	1.22748	0.613742	−	0.789507i	$-0.289664\pi$
$911$	37.0489	1.22748	0.613742	+	0.789507i	$0.289664\pi$
$912$	0	0
$913$	2.20555	0.0729931
$914$	−12.3133	−0.407289
$915$	0	0
$916$	25.7250	0.849977
$917$	−17.3083	−0.571571
$918$	0	0
$919$	−13.9149	−0.459011	−0.229506	−	0.973307i	$-0.573711\pi$
$919$	−13.9149	−0.459011	−0.229506	+	0.973307i	$0.573711\pi$
$920$	−0.897225	−0.0295806
$921$	0	0
$922$	−15.0489	−0.495608
$923$	−47.3311	−1.55792
$924$	0	0
$925$	−1.10278	−0.0362590
$926$	−31.4061	−1.03207
$927$	0	0
$928$	−7.30833	−0.239908
$929$	32.0666	1.05207	0.526035	−	0.850463i	$-0.323678\pi$
$929$	32.0666	1.05207	0.526035	+	0.850463i	$0.323678\pi$
$930$	0	0
$931$	−8.15165	−0.267160
$932$	−19.1567	−0.627498
$933$	0	0
$934$	−19.0716	−0.624042
$935$	4.20555	0.137536
$936$	0	0
$937$	25.4499	0.831413	0.415706	−	0.909499i	$-0.363534\pi$
$937$	25.4499	0.831413	0.415706	+	0.909499i	$0.363534\pi$
$938$	−8.41110	−0.274632
$939$	0	0
$940$	−1.15667	−0.0377266
$941$	3.49115	0.113808	0.0569041	−	0.998380i	$-0.481877\pi$
$941$	3.49115	0.113808	0.0569041	+	0.998380i	$0.481877\pi$
$942$	0	0
$943$	11.0872	0.361049
$944$	−7.25443	−0.236111
$945$	0	0
$946$	10.6761	0.347110
$947$	−30.6167	−0.994907	−0.497454	−	0.867491i	$-0.665732\pi$
$947$	−30.6167	−0.994907	−0.497454	+	0.867491i	$0.665732\pi$
$948$	0	0
$949$	−0.745574	−0.0242024
$950$	8.15165	0.264474
$951$	0	0
$952$	4.20555	0.136303
$953$	33.9406	1.09944	0.549721	−	0.835348i	$-0.314734\pi$
$953$	33.9406	1.09944	0.549721	+	0.835348i	$0.314734\pi$
$954$	0	0
$955$	−9.15667	−0.296303
$956$	18.0539	0.583905
$957$	0	0
$958$	−9.49115	−0.306645
$959$	10.0000	0.322917
$960$	0	0
$961$	−19.2922	−0.622330
$962$	−4.00000	−0.128965
$963$	0	0
$964$	6.15165	0.198131
$965$	−1.52946	−0.0492351
$966$	0	0
$967$	−60.8021	−1.95526	−0.977632	−	0.210323i	$-0.932548\pi$
$967$	−60.8021	−1.95526	−0.977632	+	0.210323i	$0.932548\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	2.25945	0.0725465
$971$	−50.2933	−1.61399	−0.806994	−	0.590560i	$-0.798907\pi$
$971$	−50.2933	−1.61399	−0.806994	+	0.590560i	$0.798907\pi$
$972$	0	0
$973$	−13.2005	−0.423189
$974$	−0.0438527	−0.00140513
$975$	0	0
$976$	−3.15667	−0.101043
$977$	−28.0978	−0.898927	−0.449463	−	0.893299i	$-0.648385\pi$
$977$	−28.0978	−0.898927	−0.449463	+	0.893299i	$0.648385\pi$
$978$	0	0
$979$	−7.45998	−0.238422
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	1.56777	0.0500297
$983$	21.0489	0.671355	0.335677	−	0.941977i	$-0.391035\pi$
$983$	21.0489	0.671355	0.335677	+	0.941977i	$0.391035\pi$
$984$	0	0
$985$	14.3033	0.455741
$986$	−30.7355	−0.978819
$987$	0	0
$988$	29.5678	0.940676
$989$	−9.57885	−0.304590
$990$	0	0
$991$	−44.6832	−1.41941	−0.709705	−	0.704499i	$-0.751172\pi$
$991$	−44.6832	−1.41941	−0.709705	+	0.704499i	$0.751172\pi$
$992$	3.42166	0.108638
$993$	0	0
$994$	13.0489	0.413885
$995$	10.0978	0.320120
$996$	0	0
$997$	9.47002	0.299919	0.149959	−	0.988692i	$-0.452086\pi$
$997$	9.47002	0.299919	0.149959	+	0.988692i	$0.452086\pi$
$998$	42.3416	1.34030
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.ce.1.1		3
3.2	odd	2		770.2.a.m.1.3	✓	3
12.11	even	2		6160.2.a.bf.1.1		3
15.2	even	4		3850.2.c.ba.1849.4		6
15.8	even	4		3850.2.c.ba.1849.3		6
15.14	odd	2		3850.2.a.bt.1.1		3
21.20	even	2		5390.2.a.ca.1.1		3
33.32	even	2		8470.2.a.ci.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.m.1.3	✓	3	3.2	odd	2
3850.2.a.bt.1.1		3	15.14	odd	2
3850.2.c.ba.1849.3		6	15.8	even	4
3850.2.c.ba.1849.4		6	15.2	even	4
5390.2.a.ca.1.1		3	21.20	even	2
6160.2.a.bf.1.1		3	12.11	even	2
6930.2.a.ce.1.1		3	1.1	even	1	trivial
8470.2.a.ci.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -3.62721 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.20555 q^{17} -8.15165 q^{19} -1.00000 q^{20} +1.00000 q^{22} -0.897225 q^{23} +1.00000 q^{25} +3.62721 q^{26} -1.00000 q^{28} +7.30833 q^{29} -3.42166 q^{31} -1.00000 q^{32} -4.20555 q^{34} +1.00000 q^{35} -1.10278 q^{37} +8.15165 q^{38} +1.00000 q^{40} -12.3572 q^{41} +10.6761 q^{43} -1.00000 q^{44} +0.897225 q^{46} +1.15667 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.62721 q^{52} -11.3083 q^{53} +1.00000 q^{55} +1.00000 q^{56} -7.30833 q^{58} -7.25443 q^{59} -3.15667 q^{61} +3.42166 q^{62} +1.00000 q^{64} +3.62721 q^{65} -8.41110 q^{67} +4.20555 q^{68} -1.00000 q^{70} +13.0489 q^{71} +0.205550 q^{73} +1.10278 q^{74} -8.15165 q^{76} +1.00000 q^{77} +12.1517 q^{79} -1.00000 q^{80} +12.3572 q^{82} -2.20555 q^{83} -4.20555 q^{85} -10.6761 q^{86} +1.00000 q^{88} +7.45998 q^{89} +3.62721 q^{91} -0.897225 q^{92} -1.15667 q^{94} +8.15165 q^{95} +2.25945 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8} + 3 q^{10} - 3 q^{11} + 2 q^{13} + 3 q^{14} + 3 q^{16} - 2 q^{17} - 6 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 3 q^{25} - 2 q^{26} - 3 q^{28} - 12 q^{31} - 3 q^{32} + 2 q^{34} + 3 q^{35} + 4 q^{37} + 6 q^{38} + 3 q^{40} - 4 q^{41} + 8 q^{43} - 3 q^{44} + 10 q^{46} + 3 q^{49} - 3 q^{50} + 2 q^{52} - 12 q^{53} + 3 q^{55} + 3 q^{56} + 4 q^{59} - 6 q^{61} + 12 q^{62} + 3 q^{64} - 2 q^{65} + 4 q^{67} - 2 q^{68} - 3 q^{70} + 28 q^{71} - 14 q^{73} - 4 q^{74} - 6 q^{76} + 3 q^{77} + 18 q^{79} - 3 q^{80} + 4 q^{82} + 8 q^{83} + 2 q^{85} - 8 q^{86} + 3 q^{88} - 18 q^{89} - 2 q^{91} - 10 q^{92} + 6 q^{95} - 4 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 3 * q^8 + 3 * q^10 - 3 * q^11 + 2 * q^13 + 3 * q^14 + 3 * q^16 - 2 * q^17 - 6 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 3 * q^25 - 2 * q^26 - 3 * q^28 - 12 * q^31 - 3 * q^32 + 2 * q^34 + 3 * q^35 + 4 * q^37 + 6 * q^38 + 3 * q^40 - 4 * q^41 + 8 * q^43 - 3 * q^44 + 10 * q^46 + 3 * q^49 - 3 * q^50 + 2 * q^52 - 12 * q^53 + 3 * q^55 + 3 * q^56 + 4 * q^59 - 6 * q^61 + 12 * q^62 + 3 * q^64 - 2 * q^65 + 4 * q^67 - 2 * q^68 - 3 * q^70 + 28 * q^71 - 14 * q^73 - 4 * q^74 - 6 * q^76 + 3 * q^77 + 18 * q^79 - 3 * q^80 + 4 * q^82 + 8 * q^83 + 2 * q^85 - 8 * q^86 + 3 * q^88 - 18 * q^89 - 2 * q^91 - 10 * q^92 + 6 * q^95 - 4 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

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Database

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