LMFDB - Embedded newform 6930.2.a.cm.1.3

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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +4.68585 q^{13} -1.00000 q^{14} +1.00000 q^{16} -0.292731 q^{17} -4.97858 q^{19} +1.00000 q^{20} +1.00000 q^{22} +0.292731 q^{23} +1.00000 q^{25} +4.68585 q^{26} -1.00000 q^{28} +2.00000 q^{29} +6.97858 q^{31} +1.00000 q^{32} -0.292731 q^{34} -1.00000 q^{35} +4.29273 q^{37} -4.97858 q^{38} +1.00000 q^{40} +4.39312 q^{41} -10.3503 q^{43} +1.00000 q^{44} +0.292731 q^{46} +4.97858 q^{47} +1.00000 q^{49} +1.00000 q^{50} +4.68585 q^{52} +2.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} -9.37169 q^{59} +2.97858 q^{61} +6.97858 q^{62} +1.00000 q^{64} +4.68585 q^{65} +6.29273 q^{67} -0.292731 q^{68} -1.00000 q^{70} +3.60688 q^{71} +1.41454 q^{73} +4.29273 q^{74} -4.97858 q^{76} -1.00000 q^{77} +2.97858 q^{79} +1.00000 q^{80} +4.39312 q^{82} +12.3503 q^{83} -0.292731 q^{85} -10.3503 q^{86} +1.00000 q^{88} -0.100384 q^{89} -4.68585 q^{91} +0.292731 q^{92} +4.97858 q^{94} -4.97858 q^{95} +3.60688 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 3 q^{20} + 3 q^{22} - 2 q^{23} + 3 q^{25} + 2 q^{26} - 3 q^{28} + 6 q^{29} + 6 q^{31} + 3 q^{32} + 2 q^{34} - 3 q^{35} + 10 q^{37} + 3 q^{40} + 4 q^{41} + 8 q^{43} + 3 q^{44} - 2 q^{46} + 3 q^{49} + 3 q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{55} - 3 q^{56} + 6 q^{58} - 4 q^{59} - 6 q^{61} + 6 q^{62} + 3 q^{64} + 2 q^{65} + 16 q^{67} + 2 q^{68} - 3 q^{70} + 20 q^{71} + 10 q^{73} + 10 q^{74} - 3 q^{77} - 6 q^{79} + 3 q^{80} + 4 q^{82} - 2 q^{83} + 2 q^{85} + 8 q^{86} + 3 q^{88} + 6 q^{89} - 2 q^{91} - 2 q^{92} + 20 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 + 3 * q^20 + 3 * q^22 - 2 * q^23 + 3 * q^25 + 2 * q^26 - 3 * q^28 + 6 * q^29 + 6 * q^31 + 3 * q^32 + 2 * q^34 - 3 * q^35 + 10 * q^37 + 3 * q^40 + 4 * q^41 + 8 * q^43 + 3 * q^44 - 2 * q^46 + 3 * q^49 + 3 * q^50 + 2 * q^52 + 6 * q^53 + 3 * q^55 - 3 * q^56 + 6 * q^58 - 4 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 + 2 * q^65 + 16 * q^67 + 2 * q^68 - 3 * q^70 + 20 * q^71 + 10 * q^73 + 10 * q^74 - 3 * q^77 - 6 * q^79 + 3 * q^80 + 4 * q^82 - 2 * q^83 + 2 * q^85 + 8 * q^86 + 3 * q^88 + 6 * q^89 - 2 * q^91 - 2 * q^92 + 20 * 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
$13$	4.68585	1.29962	0.649810	−	0.760097i	$-0.274848\pi$
$13$	4.68585	1.29962	0.649810	+	0.760097i	$0.274848\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.292731	−0.0709977	−0.0354988	−	0.999370i	$-0.511302\pi$
$17$	−0.292731	−0.0709977	−0.0354988	+	0.999370i	$0.511302\pi$
$18$	0	0
$19$	−4.97858	−1.14216	−0.571082	−	0.820893i	$-0.693476\pi$
$19$	−4.97858	−1.14216	−0.571082	+	0.820893i	$0.693476\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	0.292731	0.0610386	0.0305193	−	0.999534i	$-0.490284\pi$
$23$	0.292731	0.0610386	0.0305193	+	0.999534i	$0.490284\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.68585	0.918970
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	6.97858	1.25339	0.626695	−	0.779265i	$-0.284407\pi$
$31$	6.97858	1.25339	0.626695	+	0.779265i	$0.284407\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.292731	−0.0502029
$35$	−1.00000	−0.169031
$36$	0	0
$37$	4.29273	0.705721	0.352860	−	0.935676i	$-0.385209\pi$
$37$	4.29273	0.705721	0.352860	+	0.935676i	$0.385209\pi$
$38$	−4.97858	−0.807632
$39$	0	0
$40$	1.00000	0.158114
$41$	4.39312	0.686089	0.343045	−	0.939319i	$-0.388542\pi$
$41$	4.39312	0.686089	0.343045	+	0.939319i	$0.388542\pi$
$42$	0	0
$43$	−10.3503	−1.57840	−0.789201	−	0.614135i	$-0.789505\pi$
$43$	−10.3503	−1.57840	−0.789201	+	0.614135i	$0.789505\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0.292731	0.0431608
$47$	4.97858	0.726200	0.363100	−	0.931750i	$-0.381718\pi$
$47$	4.97858	0.726200	0.363100	+	0.931750i	$0.381718\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	4.68585	0.649810
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	−9.37169	−1.22009	−0.610045	−	0.792367i	$-0.708849\pi$
$59$	−9.37169	−1.22009	−0.610045	+	0.792367i	$0.708849\pi$
$60$	0	0
$61$	2.97858	0.381368	0.190684	−	0.981651i	$-0.438929\pi$
$61$	2.97858	0.381368	0.190684	+	0.981651i	$0.438929\pi$
$62$	6.97858	0.886280
$63$	0	0
$64$	1.00000	0.125000
$65$	4.68585	0.581208
$66$	0	0
$67$	6.29273	0.768779	0.384390	−	0.923171i	$-0.374412\pi$
$67$	6.29273	0.768779	0.384390	+	0.923171i	$0.374412\pi$
$68$	−0.292731	−0.0354988
$69$	0	0
$70$	−1.00000	−0.119523
$71$	3.60688	0.428058	0.214029	−	0.976827i	$-0.431341\pi$
$71$	3.60688	0.428058	0.214029	+	0.976827i	$0.431341\pi$
$72$	0	0
$73$	1.41454	0.165559	0.0827796	−	0.996568i	$-0.473620\pi$
$73$	1.41454	0.165559	0.0827796	+	0.996568i	$0.473620\pi$
$74$	4.29273	0.499020
$75$	0	0
$76$	−4.97858	−0.571082
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.97858	0.335116	0.167558	−	0.985862i	$-0.446412\pi$
$79$	2.97858	0.335116	0.167558	+	0.985862i	$0.446412\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	4.39312	0.485138
$83$	12.3503	1.35562	0.677809	−	0.735238i	$-0.262930\pi$
$83$	12.3503	1.35562	0.677809	+	0.735238i	$0.262930\pi$
$84$	0	0
$85$	−0.292731	−0.0317511
$86$	−10.3503	−1.11610
$87$	0	0
$88$	1.00000	0.106600
$89$	−0.100384	−0.0106407	−0.00532035	−	0.999986i	$-0.501694\pi$
$89$	−0.100384	−0.0106407	−0.00532035	+	0.999986i	$0.501694\pi$
$90$	0	0
$91$	−4.68585	−0.491210
$92$	0.292731	0.0305193
$93$	0	0
$94$	4.97858	0.513501
$95$	−4.97858	−0.510791
$96$	0	0
$97$	3.60688	0.366224	0.183112	−	0.983092i	$-0.441383\pi$
$97$	3.60688	0.366224	0.183112	+	0.983092i	$0.441383\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	8.39312	0.835146	0.417573	−	0.908643i	$-0.362881\pi$
$101$	8.39312	0.835146	0.417573	+	0.908643i	$0.362881\pi$
$102$	0	0
$103$	−15.3288	−1.51040	−0.755198	−	0.655497i	$-0.772459\pi$
$103$	−15.3288	−1.51040	−0.755198	+	0.655497i	$0.772459\pi$
$104$	4.68585	0.459485
$105$	0	0
$106$	2.00000	0.194257
$107$	−5.37169	−0.519301	−0.259651	−	0.965703i	$-0.583607\pi$
$107$	−5.37169	−0.519301	−0.259651	+	0.965703i	$0.583607\pi$
$108$	0	0
$109$	14.6430	1.40255	0.701273	−	0.712893i	$-0.252616\pi$
$109$	14.6430	1.40255	0.701273	+	0.712893i	$0.252616\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−10.6430	−1.00121	−0.500605	−	0.865676i	$-0.666889\pi$
$113$	−10.6430	−1.00121	−0.500605	+	0.865676i	$0.666889\pi$
$114$	0	0
$115$	0.292731	0.0272973
$116$	2.00000	0.185695
$117$	0	0
$118$	−9.37169	−0.862734
$119$	0.292731	0.0268346
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.97858	0.269668
$123$	0	0
$124$	6.97858	0.626695
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.7434	1.30826	0.654132	−	0.756380i	$-0.273034\pi$
$127$	14.7434	1.30826	0.654132	+	0.756380i	$0.273034\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.68585	0.410976
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	4.97858	0.431697
$134$	6.29273	0.543609
$135$	0	0
$136$	−0.292731	−0.0251015
$137$	18.0575	1.54276	0.771380	−	0.636375i	$-0.219567\pi$
$137$	18.0575	1.54276	0.771380	+	0.636375i	$0.219567\pi$
$138$	0	0
$139$	−9.56404	−0.811211	−0.405606	−	0.914048i	$-0.632939\pi$
$139$	−9.56404	−0.811211	−0.405606	+	0.914048i	$0.632939\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	3.60688	0.302683
$143$	4.68585	0.391850
$144$	0	0
$145$	2.00000	0.166091
$146$	1.41454	0.117068
$147$	0	0
$148$	4.29273	0.352860
$149$	−7.95715	−0.651875	−0.325938	−	0.945391i	$-0.605680\pi$
$149$	−7.95715	−0.651875	−0.325938	+	0.945391i	$0.605680\pi$
$150$	0	0
$151$	−15.7648	−1.28292	−0.641461	−	0.767156i	$-0.721671\pi$
$151$	−15.7648	−1.28292	−0.641461	+	0.767156i	$0.721671\pi$
$152$	−4.97858	−0.403816
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	6.97858	0.560533
$156$	0	0
$157$	−2.35027	−0.187572	−0.0937860	−	0.995592i	$-0.529897\pi$
$157$	−2.35027	−0.187572	−0.0937860	+	0.995592i	$0.529897\pi$
$158$	2.97858	0.236963
$159$	0	0
$160$	1.00000	0.0790569
$161$	−0.292731	−0.0230704
$162$	0	0
$163$	17.0361	1.33437	0.667186	−	0.744891i	$-0.267499\pi$
$163$	17.0361	1.33437	0.667186	+	0.744891i	$0.267499\pi$
$164$	4.39312	0.343045
$165$	0	0
$166$	12.3503	0.958567
$167$	−10.0575	−0.778276	−0.389138	−	0.921180i	$-0.627227\pi$
$167$	−10.0575	−0.778276	−0.389138	+	0.921180i	$0.627227\pi$
$168$	0	0
$169$	8.95715	0.689012
$170$	−0.292731	−0.0224514
$171$	0	0
$172$	−10.3503	−0.789201
$173$	−7.37169	−0.560459	−0.280230	−	0.959933i	$-0.590411\pi$
$173$	−7.37169	−0.560459	−0.280230	+	0.959933i	$0.590411\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−0.100384	−0.00752411
$179$	14.7434	1.10197	0.550986	−	0.834514i	$-0.314252\pi$
$179$	14.7434	1.10197	0.550986	+	0.834514i	$0.314252\pi$
$180$	0	0
$181$	−2.33558	−0.173602	−0.0868010	−	0.996226i	$-0.527664\pi$
$181$	−2.33558	−0.173602	−0.0868010	+	0.996226i	$0.527664\pi$
$182$	−4.68585	−0.347338
$183$	0	0
$184$	0.292731	0.0215804
$185$	4.29273	0.315608
$186$	0	0
$187$	−0.292731	−0.0214066
$188$	4.97858	0.363100
$189$	0	0
$190$	−4.97858	−0.361184
$191$	8.97858	0.649667	0.324834	−	0.945771i	$-0.394692\pi$
$191$	8.97858	0.649667	0.324834	+	0.945771i	$0.394692\pi$
$192$	0	0
$193$	−10.3503	−0.745029	−0.372514	−	0.928026i	$-0.621504\pi$
$193$	−10.3503	−0.745029	−0.372514	+	0.928026i	$0.621504\pi$
$194$	3.60688	0.258959
$195$	0	0
$196$	1.00000	0.0714286
$197$	19.9572	1.42189	0.710944	−	0.703248i	$-0.248268\pi$
$197$	19.9572	1.42189	0.710944	+	0.703248i	$0.248268\pi$
$198$	0	0
$199$	−5.60688	−0.397462	−0.198731	−	0.980054i	$-0.563682\pi$
$199$	−5.60688	−0.397462	−0.198731	+	0.980054i	$0.563682\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	8.39312	0.590538
$203$	−2.00000	−0.140372
$204$	0	0
$205$	4.39312	0.306828
$206$	−15.3288	−1.06801
$207$	0	0
$208$	4.68585	0.324905
$209$	−4.97858	−0.344375
$210$	0	0
$211$	−22.8353	−1.57205	−0.786025	−	0.618195i	$-0.787864\pi$
$211$	−22.8353	−1.57205	−0.786025	+	0.618195i	$0.787864\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−5.37169	−0.367201
$215$	−10.3503	−0.705883
$216$	0	0
$217$	−6.97858	−0.473737
$218$	14.6430	0.991749
$219$	0	0
$220$	1.00000	0.0674200
$221$	−1.37169	−0.0922700
$222$	0	0
$223$	5.95715	0.398921	0.199460	−	0.979906i	$-0.436081\pi$
$223$	5.95715	0.398921	0.199460	+	0.979906i	$0.436081\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−10.6430	−0.707962
$227$	−13.7220	−0.910759	−0.455379	−	0.890298i	$-0.650496\pi$
$227$	−13.7220	−0.910759	−0.455379	+	0.890298i	$0.650496\pi$
$228$	0	0
$229$	23.0361	1.52227	0.761135	−	0.648594i	$-0.224643\pi$
$229$	23.0361	1.52227	0.761135	+	0.648594i	$0.224643\pi$
$230$	0.292731	0.0193021
$231$	0	0
$232$	2.00000	0.131306
$233$	−13.9143	−0.911557	−0.455778	−	0.890093i	$-0.650639\pi$
$233$	−13.9143	−0.911557	−0.455778	+	0.890093i	$0.650639\pi$
$234$	0	0
$235$	4.97858	0.324767
$236$	−9.37169	−0.610045
$237$	0	0
$238$	0.292731	0.0189749
$239$	19.8652	1.28497	0.642486	−	0.766297i	$-0.277903\pi$
$239$	19.8652	1.28497	0.642486	+	0.766297i	$0.277903\pi$
$240$	0	0
$241$	−11.9572	−0.770228	−0.385114	−	0.922869i	$-0.625838\pi$
$241$	−11.9572	−0.770228	−0.385114	+	0.922869i	$0.625838\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.97858	0.190684
$245$	1.00000	0.0638877
$246$	0	0
$247$	−23.3288	−1.48438
$248$	6.97858	0.443140
$249$	0	0
$250$	1.00000	0.0632456
$251$	30.7434	1.94051	0.970253	−	0.242095i	$-0.0778345\pi$
$251$	30.7434	1.94051	0.970253	+	0.242095i	$0.0778345\pi$
$252$	0	0
$253$	0.292731	0.0184038
$254$	14.7434	0.925082
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.5640	−0.970858	−0.485429	−	0.874276i	$-0.661337\pi$
$257$	−15.5640	−0.970858	−0.485429	+	0.874276i	$0.661337\pi$
$258$	0	0
$259$	−4.29273	−0.266737
$260$	4.68585	0.290604
$261$	0	0
$262$	4.00000	0.247121
$263$	−9.37169	−0.577883	−0.288942	−	0.957347i	$-0.593303\pi$
$263$	−9.37169	−0.577883	−0.288942	+	0.957347i	$0.593303\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	4.97858	0.305256
$267$	0	0
$268$	6.29273	0.384390
$269$	20.7434	1.26475	0.632373	−	0.774664i	$-0.282081\pi$
$269$	20.7434	1.26475	0.632373	+	0.774664i	$0.282081\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	−0.292731	−0.0177494
$273$	0	0
$274$	18.0575	1.09090
$275$	1.00000	0.0603023
$276$	0	0
$277$	24.1151	1.44893	0.724467	−	0.689309i	$-0.242086\pi$
$277$	24.1151	1.44893	0.724467	+	0.689309i	$0.242086\pi$
$278$	−9.56404	−0.573613
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	21.6644	1.29239	0.646196	−	0.763172i	$-0.276359\pi$
$281$	21.6644	1.29239	0.646196	+	0.763172i	$0.276359\pi$
$282$	0	0
$283$	−0.728692	−0.0433162	−0.0216581	−	0.999765i	$-0.506895\pi$
$283$	−0.728692	−0.0433162	−0.0216581	+	0.999765i	$0.506895\pi$
$284$	3.60688	0.214029
$285$	0	0
$286$	4.68585	0.277080
$287$	−4.39312	−0.259317
$288$	0	0
$289$	−16.9143	−0.994959
$290$	2.00000	0.117444
$291$	0	0
$292$	1.41454	0.0827796
$293$	17.1281	1.00063	0.500317	−	0.865843i	$-0.333217\pi$
$293$	17.1281	1.00063	0.500317	+	0.865843i	$0.333217\pi$
$294$	0	0
$295$	−9.37169	−0.545641
$296$	4.29273	0.249510
$297$	0	0
$298$	−7.95715	−0.460946
$299$	1.37169	0.0793270
$300$	0	0
$301$	10.3503	0.596580
$302$	−15.7648	−0.907163
$303$	0	0
$304$	−4.97858	−0.285541
$305$	2.97858	0.170553
$306$	0	0
$307$	−19.4721	−1.11133	−0.555665	−	0.831406i	$-0.687536\pi$
$307$	−19.4721	−1.11133	−0.555665	+	0.831406i	$0.687536\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	6.97858	0.396357
$311$	26.6002	1.50836	0.754178	−	0.656670i	$-0.228035\pi$
$311$	26.6002	1.50836	0.754178	+	0.656670i	$0.228035\pi$
$312$	0	0
$313$	−24.3074	−1.37394	−0.686968	−	0.726687i	$-0.741059\pi$
$313$	−24.3074	−1.37394	−0.686968	+	0.726687i	$0.741059\pi$
$314$	−2.35027	−0.132633
$315$	0	0
$316$	2.97858	0.167558
$317$	25.3288	1.42261	0.711305	−	0.702884i	$-0.248105\pi$
$317$	25.3288	1.42261	0.711305	+	0.702884i	$0.248105\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	1.00000	0.0559017
$321$	0	0
$322$	−0.292731	−0.0163133
$323$	1.45738	0.0810910
$324$	0	0
$325$	4.68585	0.259924
$326$	17.0361	0.943543
$327$	0	0
$328$	4.39312	0.242569
$329$	−4.97858	−0.274478
$330$	0	0
$331$	−18.7434	−1.03023	−0.515115	−	0.857121i	$-0.672251\pi$
$331$	−18.7434	−1.03023	−0.515115	+	0.857121i	$0.672251\pi$
$332$	12.3503	0.677809
$333$	0	0
$334$	−10.0575	−0.550324
$335$	6.29273	0.343809
$336$	0	0
$337$	8.19235	0.446266	0.223133	−	0.974788i	$-0.428372\pi$
$337$	8.19235	0.446266	0.223133	+	0.974788i	$0.428372\pi$
$338$	8.95715	0.487205
$339$	0	0
$340$	−0.292731	−0.0158756
$341$	6.97858	0.377911
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−10.3503	−0.558049
$345$	0	0
$346$	−7.37169	−0.396305
$347$	4.20077	0.225509	0.112755	−	0.993623i	$-0.464033\pi$
$347$	4.20077	0.225509	0.112755	+	0.993623i	$0.464033\pi$
$348$	0	0
$349$	−9.02142	−0.482906	−0.241453	−	0.970413i	$-0.577624\pi$
$349$	−9.02142	−0.482906	−0.241453	+	0.970413i	$0.577624\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	−6.39312	−0.340271	−0.170136	−	0.985421i	$-0.554421\pi$
$353$	−6.39312	−0.340271	−0.170136	+	0.985421i	$0.554421\pi$
$354$	0	0
$355$	3.60688	0.191434
$356$	−0.100384	−0.00532035
$357$	0	0
$358$	14.7434	0.779212
$359$	4.92104	0.259722	0.129861	−	0.991532i	$-0.458547\pi$
$359$	4.92104	0.259722	0.129861	+	0.991532i	$0.458547\pi$
$360$	0	0
$361$	5.78623	0.304538
$362$	−2.33558	−0.122755
$363$	0	0
$364$	−4.68585	−0.245605
$365$	1.41454	0.0740403
$366$	0	0
$367$	29.2860	1.52872	0.764358	−	0.644792i	$-0.223056\pi$
$367$	29.2860	1.52872	0.764358	+	0.644792i	$0.223056\pi$
$368$	0.292731	0.0152597
$369$	0	0
$370$	4.29273	0.223168
$371$	−2.00000	−0.103835
$372$	0	0
$373$	22.7434	1.17761	0.588804	−	0.808276i	$-0.299599\pi$
$373$	22.7434	1.17761	0.588804	+	0.808276i	$0.299599\pi$
$374$	−0.292731	−0.0151368
$375$	0	0
$376$	4.97858	0.256751
$377$	9.37169	0.482667
$378$	0	0
$379$	−34.7434	−1.78465	−0.892324	−	0.451396i	$-0.850926\pi$
$379$	−34.7434	−1.78465	−0.892324	+	0.451396i	$0.850926\pi$
$380$	−4.97858	−0.255396
$381$	0	0
$382$	8.97858	0.459384
$383$	23.7220	1.21214	0.606068	−	0.795413i	$-0.292746\pi$
$383$	23.7220	1.21214	0.606068	+	0.795413i	$0.292746\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−10.3503	−0.526815
$387$	0	0
$388$	3.60688	0.183112
$389$	−15.8077	−0.801480	−0.400740	−	0.916192i	$-0.631247\pi$
$389$	−15.8077	−0.801480	−0.400740	+	0.916192i	$0.631247\pi$
$390$	0	0
$391$	−0.0856914	−0.00433360
$392$	1.00000	0.0505076
$393$	0	0
$394$	19.9572	1.00543
$395$	2.97858	0.149868
$396$	0	0
$397$	8.19235	0.411162	0.205581	−	0.978640i	$-0.434092\pi$
$397$	8.19235	0.411162	0.205581	+	0.978640i	$0.434092\pi$
$398$	−5.60688	−0.281048
$399$	0	0
$400$	1.00000	0.0500000
$401$	9.17092	0.457974	0.228987	−	0.973429i	$-0.426459\pi$
$401$	9.17092	0.457974	0.228987	+	0.973429i	$0.426459\pi$
$402$	0	0
$403$	32.7005	1.62893
$404$	8.39312	0.417573
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	4.29273	0.212783
$408$	0	0
$409$	−20.1579	−0.996745	−0.498373	−	0.866963i	$-0.666069\pi$
$409$	−20.1579	−0.996745	−0.498373	+	0.866963i	$0.666069\pi$
$410$	4.39312	0.216960
$411$	0	0
$412$	−15.3288	−0.755198
$413$	9.37169	0.461151
$414$	0	0
$415$	12.3503	0.606251
$416$	4.68585	0.229743
$417$	0	0
$418$	−4.97858	−0.243510
$419$	−21.1709	−1.03427	−0.517134	−	0.855905i	$-0.673001\pi$
$419$	−21.1709	−1.03427	−0.517134	+	0.855905i	$0.673001\pi$
$420$	0	0
$421$	−14.7862	−0.720637	−0.360318	−	0.932829i	$-0.617332\pi$
$421$	−14.7862	−0.720637	−0.360318	+	0.932829i	$0.617332\pi$
$422$	−22.8353	−1.11161
$423$	0	0
$424$	2.00000	0.0971286
$425$	−0.292731	−0.0141995
$426$	0	0
$427$	−2.97858	−0.144143
$428$	−5.37169	−0.259651
$429$	0	0
$430$	−10.3503	−0.499134
$431$	20.2499	0.975403	0.487701	−	0.873011i	$-0.337836\pi$
$431$	20.2499	0.975403	0.487701	+	0.873011i	$0.337836\pi$
$432$	0	0
$433$	1.56404	0.0751629	0.0375815	−	0.999294i	$-0.488035\pi$
$433$	1.56404	0.0751629	0.0375815	+	0.999294i	$0.488035\pi$
$434$	−6.97858	−0.334982
$435$	0	0
$436$	14.6430	0.701273
$437$	−1.45738	−0.0697161
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−1.37169	−0.0652448
$443$	−5.80765	−0.275930	−0.137965	−	0.990437i	$-0.544056\pi$
$443$	−5.80765	−0.275930	−0.137965	+	0.990437i	$0.544056\pi$
$444$	0	0
$445$	−0.100384	−0.00475867
$446$	5.95715	0.282079
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−4.70054	−0.221832	−0.110916	−	0.993830i	$-0.535378\pi$
$449$	−4.70054	−0.221832	−0.110916	+	0.993830i	$0.535378\pi$
$450$	0	0
$451$	4.39312	0.206864
$452$	−10.6430	−0.500605
$453$	0	0
$454$	−13.7220	−0.644004
$455$	−4.68585	−0.219676
$456$	0	0
$457$	−4.77781	−0.223496	−0.111748	−	0.993737i	$-0.535645\pi$
$457$	−4.77781	−0.223496	−0.111748	+	0.993737i	$0.535645\pi$
$458$	23.0361	1.07641
$459$	0	0
$460$	0.292731	0.0136487
$461$	−34.2646	−1.59586	−0.797930	−	0.602750i	$-0.794072\pi$
$461$	−34.2646	−1.59586	−0.797930	+	0.602750i	$0.794072\pi$
$462$	0	0
$463$	30.6577	1.42478	0.712392	−	0.701782i	$-0.247612\pi$
$463$	30.6577	1.42478	0.712392	+	0.701782i	$0.247612\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−13.9143	−0.644568
$467$	−29.8715	−1.38229	−0.691143	−	0.722718i	$-0.742893\pi$
$467$	−29.8715	−1.38229	−0.691143	+	0.722718i	$0.742893\pi$
$468$	0	0
$469$	−6.29273	−0.290571
$470$	4.97858	0.229645
$471$	0	0
$472$	−9.37169	−0.431367
$473$	−10.3503	−0.475906
$474$	0	0
$475$	−4.97858	−0.228433
$476$	0.292731	0.0134173
$477$	0	0
$478$	19.8652	0.908613
$479$	−6.74338	−0.308113	−0.154057	−	0.988062i	$-0.549234\pi$
$479$	−6.74338	−0.308113	−0.154057	+	0.988062i	$0.549234\pi$
$480$	0	0
$481$	20.1151	0.917169
$482$	−11.9572	−0.544633
$483$	0	0
$484$	1.00000	0.0454545
$485$	3.60688	0.163780
$486$	0	0
$487$	19.4145	0.879757	0.439878	−	0.898057i	$-0.355022\pi$
$487$	19.4145	0.879757	0.439878	+	0.898057i	$0.355022\pi$
$488$	2.97858	0.134834
$489$	0	0
$490$	1.00000	0.0451754
$491$	−3.32885	−0.150229	−0.0751144	−	0.997175i	$-0.523932\pi$
$491$	−3.32885	−0.150229	−0.0751144	+	0.997175i	$0.523932\pi$
$492$	0	0
$493$	−0.585462	−0.0263679
$494$	−23.3288	−1.04961
$495$	0	0
$496$	6.97858	0.313347
$497$	−3.60688	−0.161791
$498$	0	0
$499$	−26.1579	−1.17099	−0.585495	−	0.810676i	$-0.699100\pi$
$499$	−26.1579	−1.17099	−0.585495	+	0.810676i	$0.699100\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	30.7434	1.37214
$503$	−34.7581	−1.54979	−0.774893	−	0.632092i	$-0.782196\pi$
$503$	−34.7581	−1.54979	−0.774893	+	0.632092i	$0.782196\pi$
$504$	0	0
$505$	8.39312	0.373489
$506$	0.292731	0.0130135
$507$	0	0
$508$	14.7434	0.654132
$509$	−36.1579	−1.60267	−0.801336	−	0.598215i	$-0.795877\pi$
$509$	−36.1579	−1.60267	−0.801336	+	0.598215i	$0.795877\pi$
$510$	0	0
$511$	−1.41454	−0.0625755
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.5640	−0.686500
$515$	−15.3288	−0.675470
$516$	0	0
$517$	4.97858	0.218958
$518$	−4.29273	−0.188612
$519$	0	0
$520$	4.68585	0.205488
$521$	7.89962	0.346088	0.173044	−	0.984914i	$-0.444640\pi$
$521$	7.89962	0.346088	0.173044	+	0.984914i	$0.444640\pi$
$522$	0	0
$523$	−22.8866	−1.00076	−0.500381	−	0.865805i	$-0.666807\pi$
$523$	−22.8866	−1.00076	−0.500381	+	0.865805i	$0.666807\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−9.37169	−0.408625
$527$	−2.04285	−0.0889878
$528$	0	0
$529$	−22.9143	−0.996274
$530$	2.00000	0.0868744
$531$	0	0
$532$	4.97858	0.215849
$533$	20.5855	0.891655
$534$	0	0
$535$	−5.37169	−0.232239
$536$	6.29273	0.271805
$537$	0	0
$538$	20.7434	0.894311
$539$	1.00000	0.0430730
$540$	0	0
$541$	−19.4292	−0.835328	−0.417664	−	0.908602i	$-0.637151\pi$
$541$	−19.4292	−0.835328	−0.417664	+	0.908602i	$0.637151\pi$
$542$	−4.00000	−0.171815
$543$	0	0
$544$	−0.292731	−0.0125507
$545$	14.6430	0.627237
$546$	0	0
$547$	−25.7648	−1.10162	−0.550812	−	0.834629i	$-0.685682\pi$
$547$	−25.7648	−1.10162	−0.550812	+	0.834629i	$0.685682\pi$
$548$	18.0575	0.771380
$549$	0	0
$550$	1.00000	0.0426401
$551$	−9.95715	−0.424189
$552$	0	0
$553$	−2.97858	−0.126662
$554$	24.1151	1.02455
$555$	0	0
$556$	−9.56404	−0.405606
$557$	8.82908	0.374100	0.187050	−	0.982350i	$-0.440107\pi$
$557$	8.82908	0.374100	0.187050	+	0.982350i	$0.440107\pi$
$558$	0	0
$559$	−48.4998	−2.05132
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	21.6644	0.913859
$563$	−10.9786	−0.462692	−0.231346	−	0.972872i	$-0.574313\pi$
$563$	−10.9786	−0.462692	−0.231346	+	0.972872i	$0.574313\pi$
$564$	0	0
$565$	−10.6430	−0.447754
$566$	−0.728692	−0.0306292
$567$	0	0
$568$	3.60688	0.151342
$569$	24.9933	1.04777	0.523886	−	0.851788i	$-0.324482\pi$
$569$	24.9933	1.04777	0.523886	+	0.851788i	$0.324482\pi$
$570$	0	0
$571$	−0.677425	−0.0283493	−0.0141747	−	0.999900i	$-0.504512\pi$
$571$	−0.677425	−0.0283493	−0.0141747	+	0.999900i	$0.504512\pi$
$572$	4.68585	0.195925
$573$	0	0
$574$	−4.39312	−0.183365
$575$	0.292731	0.0122077
$576$	0	0
$577$	11.1365	0.463619	0.231809	−	0.972761i	$-0.425535\pi$
$577$	11.1365	0.463619	0.231809	+	0.972761i	$0.425535\pi$
$578$	−16.9143	−0.703542
$579$	0	0
$580$	2.00000	0.0830455
$581$	−12.3503	−0.512376
$582$	0	0
$583$	2.00000	0.0828315
$584$	1.41454	0.0585340
$585$	0	0
$586$	17.1281	0.707554
$587$	8.70054	0.359110	0.179555	−	0.983748i	$-0.442534\pi$
$587$	8.70054	0.359110	0.179555	+	0.983748i	$0.442534\pi$
$588$	0	0
$589$	−34.7434	−1.43158
$590$	−9.37169	−0.385826
$591$	0	0
$592$	4.29273	0.176430
$593$	46.9504	1.92802	0.964011	−	0.265861i	$-0.0856562\pi$
$593$	46.9504	1.92802	0.964011	+	0.265861i	$0.0856562\pi$
$594$	0	0
$595$	0.292731	0.0120008
$596$	−7.95715	−0.325938
$597$	0	0
$598$	1.37169	0.0560927
$599$	−3.80765	−0.155576	−0.0777882	−	0.996970i	$-0.524786\pi$
$599$	−3.80765	−0.155576	−0.0777882	+	0.996970i	$0.524786\pi$
$600$	0	0
$601$	−34.1151	−1.39158	−0.695792	−	0.718244i	$-0.744946\pi$
$601$	−34.1151	−1.39158	−0.695792	+	0.718244i	$0.744946\pi$
$602$	10.3503	0.421845
$603$	0	0
$604$	−15.7648	−0.641461
$605$	1.00000	0.0406558
$606$	0	0
$607$	−33.2860	−1.35104	−0.675519	−	0.737343i	$-0.736080\pi$
$607$	−33.2860	−1.35104	−0.675519	+	0.737343i	$0.736080\pi$
$608$	−4.97858	−0.201908
$609$	0	0
$610$	2.97858	0.120599
$611$	23.3288	0.943784
$612$	0	0
$613$	−31.3288	−1.26536	−0.632680	−	0.774413i	$-0.718045\pi$
$613$	−31.3288	−1.26536	−0.632680	+	0.774413i	$0.718045\pi$
$614$	−19.4721	−0.785829
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−13.8568	−0.557853	−0.278926	−	0.960313i	$-0.589979\pi$
$617$	−13.8568	−0.557853	−0.278926	+	0.960313i	$0.589979\pi$
$618$	0	0
$619$	25.3864	1.02036	0.510182	−	0.860066i	$-0.329578\pi$
$619$	25.3864	1.02036	0.510182	+	0.860066i	$0.329578\pi$
$620$	6.97858	0.280266
$621$	0	0
$622$	26.6002	1.06657
$623$	0.100384	0.00402181
$624$	0	0
$625$	1.00000	0.0400000
$626$	−24.3074	−0.971520
$627$	0	0
$628$	−2.35027	−0.0937860
$629$	−1.25662	−0.0501045
$630$	0	0
$631$	−32.4998	−1.29380	−0.646898	−	0.762577i	$-0.723934\pi$
$631$	−32.4998	−1.29380	−0.646898	+	0.762577i	$0.723934\pi$
$632$	2.97858	0.118481
$633$	0	0
$634$	25.3288	1.00594
$635$	14.7434	0.585073
$636$	0	0
$637$	4.68585	0.185660
$638$	2.00000	0.0791808
$639$	0	0
$640$	1.00000	0.0395285
$641$	−29.8715	−1.17985	−0.589926	−	0.807457i	$-0.700843\pi$
$641$	−29.8715	−1.17985	−0.589926	+	0.807457i	$0.700843\pi$
$642$	0	0
$643$	−39.1281	−1.54306	−0.771530	−	0.636192i	$-0.780508\pi$
$643$	−39.1281	−1.54306	−0.771530	+	0.636192i	$0.780508\pi$
$644$	−0.292731	−0.0115352
$645$	0	0
$646$	1.45738	0.0573400
$647$	24.1923	0.951099	0.475550	−	0.879689i	$-0.342249\pi$
$647$	24.1923	0.951099	0.475550	+	0.879689i	$0.342249\pi$
$648$	0	0
$649$	−9.37169	−0.367871
$650$	4.68585	0.183794
$651$	0	0
$652$	17.0361	0.667186
$653$	9.61531	0.376276	0.188138	−	0.982143i	$-0.439755\pi$
$653$	9.61531	0.376276	0.188138	+	0.982143i	$0.439755\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	4.39312	0.171522
$657$	0	0
$658$	−4.97858	−0.194085
$659$	6.74338	0.262685	0.131342	−	0.991337i	$-0.458071\pi$
$659$	6.74338	0.262685	0.131342	+	0.991337i	$0.458071\pi$
$660$	0	0
$661$	12.0920	0.470323	0.235161	−	0.971956i	$-0.424438\pi$
$661$	12.0920	0.470323	0.235161	+	0.971956i	$0.424438\pi$
$662$	−18.7434	−0.728482
$663$	0	0
$664$	12.3503	0.479283
$665$	4.97858	0.193061
$666$	0	0
$667$	0.585462	0.0226692
$668$	−10.0575	−0.389138
$669$	0	0
$670$	6.29273	0.243109
$671$	2.97858	0.114987
$672$	0	0
$673$	−18.1495	−0.699612	−0.349806	−	0.936822i	$-0.613752\pi$
$673$	−18.1495	−0.699612	−0.349806	+	0.936822i	$0.613752\pi$
$674$	8.19235	0.315557
$675$	0	0
$676$	8.95715	0.344506
$677$	−2.87192	−0.110377	−0.0551885	−	0.998476i	$-0.517576\pi$
$677$	−2.87192	−0.110377	−0.0551885	+	0.998476i	$0.517576\pi$
$678$	0	0
$679$	−3.60688	−0.138420
$680$	−0.292731	−0.0112257
$681$	0	0
$682$	6.97858	0.267224
$683$	26.3074	1.00663	0.503313	−	0.864104i	$-0.332114\pi$
$683$	26.3074	1.00663	0.503313	+	0.864104i	$0.332114\pi$
$684$	0	0
$685$	18.0575	0.689943
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−10.3503	−0.394600
$689$	9.37169	0.357033
$690$	0	0
$691$	22.6430	0.861380	0.430690	−	0.902500i	$-0.358270\pi$
$691$	22.6430	0.861380	0.430690	+	0.902500i	$0.358270\pi$
$692$	−7.37169	−0.280230
$693$	0	0
$694$	4.20077	0.159459
$695$	−9.56404	−0.362785
$696$	0	0
$697$	−1.28600	−0.0487108
$698$	−9.02142	−0.341466
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	31.9572	1.20701	0.603503	−	0.797361i	$-0.293771\pi$
$701$	31.9572	1.20701	0.603503	+	0.797361i	$0.293771\pi$
$702$	0	0
$703$	−21.3717	−0.806049
$704$	1.00000	0.0376889
$705$	0	0
$706$	−6.39312	−0.240608
$707$	−8.39312	−0.315656
$708$	0	0
$709$	8.65769	0.325146	0.162573	−	0.986696i	$-0.448021\pi$
$709$	8.65769	0.325146	0.162573	+	0.986696i	$0.448021\pi$
$710$	3.60688	0.135364
$711$	0	0
$712$	−0.100384	−0.00376206
$713$	2.04285	0.0765052
$714$	0	0
$715$	4.68585	0.175241
$716$	14.7434	0.550986
$717$	0	0
$718$	4.92104	0.183652
$719$	−11.9425	−0.445379	−0.222689	−	0.974889i	$-0.571484\pi$
$719$	−11.9425	−0.445379	−0.222689	+	0.974889i	$0.571484\pi$
$720$	0	0
$721$	15.3288	0.570876
$722$	5.78623	0.215341
$723$	0	0
$724$	−2.33558	−0.0868010
$725$	2.00000	0.0742781
$726$	0	0
$727$	−10.3418	−0.383558	−0.191779	−	0.981438i	$-0.561426\pi$
$727$	−10.3418	−0.383558	−0.191779	+	0.981438i	$0.561426\pi$
$728$	−4.68585	−0.173669
$729$	0	0
$730$	1.41454	0.0523544
$731$	3.02984	0.112063
$732$	0	0
$733$	27.8139	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$733$	27.8139	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$734$	29.2860	1.08097
$735$	0	0
$736$	0.292731	0.0107902
$737$	6.29273	0.231796
$738$	0	0
$739$	14.3650	0.528424	0.264212	−	0.964465i	$-0.414888\pi$
$739$	14.3650	0.528424	0.264212	+	0.964465i	$0.414888\pi$
$740$	4.29273	0.157804
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	19.2138	0.704885	0.352442	−	0.935833i	$-0.385351\pi$
$743$	19.2138	0.704885	0.352442	+	0.935833i	$0.385351\pi$
$744$	0	0
$745$	−7.95715	−0.291528
$746$	22.7434	0.832694
$747$	0	0
$748$	−0.292731	−0.0107033
$749$	5.37169	0.196277
$750$	0	0
$751$	1.37169	0.0500538	0.0250269	−	0.999687i	$-0.492033\pi$
$751$	1.37169	0.0500538	0.0250269	+	0.999687i	$0.492033\pi$
$752$	4.97858	0.181550
$753$	0	0
$754$	9.37169	0.341297
$755$	−15.7648	−0.573740
$756$	0	0
$757$	12.2927	0.446787	0.223393	−	0.974728i	$-0.428287\pi$
$757$	12.2927	0.446787	0.223393	+	0.974728i	$0.428287\pi$
$758$	−34.7434	−1.26194
$759$	0	0
$760$	−4.97858	−0.180592
$761$	23.5212	0.852643	0.426321	−	0.904572i	$-0.359809\pi$
$761$	23.5212	0.852643	0.426321	+	0.904572i	$0.359809\pi$
$762$	0	0
$763$	−14.6430	−0.530112
$764$	8.97858	0.324834
$765$	0	0
$766$	23.7220	0.857109
$767$	−43.9143	−1.58565
$768$	0	0
$769$	−18.2008	−0.656336	−0.328168	−	0.944619i	$-0.606431\pi$
$769$	−18.2008	−0.656336	−0.328168	+	0.944619i	$0.606431\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	−10.3503	−0.372514
$773$	−27.2860	−0.981409	−0.490705	−	0.871326i	$-0.663261\pi$
$773$	−27.2860	−0.981409	−0.490705	+	0.871326i	$0.663261\pi$
$774$	0	0
$775$	6.97858	0.250678
$776$	3.60688	0.129480
$777$	0	0
$778$	−15.8077	−0.566732
$779$	−21.8715	−0.783626
$780$	0	0
$781$	3.60688	0.129064
$782$	−0.0856914	−0.00306432
$783$	0	0
$784$	1.00000	0.0357143
$785$	−2.35027	−0.0838847
$786$	0	0
$787$	−19.4721	−0.694105	−0.347052	−	0.937846i	$-0.612817\pi$
$787$	−19.4721	−0.694105	−0.347052	+	0.937846i	$0.612817\pi$
$788$	19.9572	0.710944
$789$	0	0
$790$	2.97858	0.105973
$791$	10.6430	0.378422
$792$	0	0
$793$	13.9572	0.495633
$794$	8.19235	0.290736
$795$	0	0
$796$	−5.60688	−0.198731
$797$	−2.20077	−0.0779552	−0.0389776	−	0.999240i	$-0.512410\pi$
$797$	−2.20077	−0.0779552	−0.0389776	+	0.999240i	$0.512410\pi$
$798$	0	0
$799$	−1.45738	−0.0515585
$800$	1.00000	0.0353553
$801$	0	0
$802$	9.17092	0.323837
$803$	1.41454	0.0499180
$804$	0	0
$805$	−0.292731	−0.0103174
$806$	32.7005	1.15183
$807$	0	0
$808$	8.39312	0.295269
$809$	29.3780	1.03287	0.516437	−	0.856325i	$-0.327258\pi$
$809$	29.3780	1.03287	0.516437	+	0.856325i	$0.327258\pi$
$810$	0	0
$811$	−11.0214	−0.387015	−0.193507	−	0.981099i	$-0.561986\pi$
$811$	−11.0214	−0.387015	−0.193507	+	0.981099i	$0.561986\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	4.29273	0.150460
$815$	17.0361	0.596749
$816$	0	0
$817$	51.5296	1.80279
$818$	−20.1579	−0.704805
$819$	0	0
$820$	4.39312	0.153414
$821$	20.1579	0.703516	0.351758	−	0.936091i	$-0.385584\pi$
$821$	20.1579	0.703516	0.351758	+	0.936091i	$0.385584\pi$
$822$	0	0
$823$	−37.4868	−1.30671	−0.653353	−	0.757053i	$-0.726638\pi$
$823$	−37.4868	−1.30671	−0.653353	+	0.757053i	$0.726638\pi$
$824$	−15.3288	−0.534006
$825$	0	0
$826$	9.37169	0.326083
$827$	23.7135	0.824601	0.412300	−	0.911048i	$-0.364725\pi$
$827$	23.7135	0.824601	0.412300	+	0.911048i	$0.364725\pi$
$828$	0	0
$829$	−41.7795	−1.45106	−0.725531	−	0.688189i	$-0.758406\pi$
$829$	−41.7795	−1.45106	−0.725531	+	0.688189i	$0.758406\pi$
$830$	12.3503	0.428684
$831$	0	0
$832$	4.68585	0.162452
$833$	−0.292731	−0.0101425
$834$	0	0
$835$	−10.0575	−0.348055
$836$	−4.97858	−0.172188
$837$	0	0
$838$	−21.1709	−0.731337
$839$	−55.1856	−1.90522	−0.952610	−	0.304196i	$-0.901612\pi$
$839$	−55.1856	−1.90522	−0.952610	+	0.304196i	$0.901612\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−14.7862	−0.509567
$843$	0	0
$844$	−22.8353	−0.786025
$845$	8.95715	0.308135
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	2.00000	0.0686803
$849$	0	0
$850$	−0.292731	−0.0100406
$851$	1.25662	0.0430762
$852$	0	0
$853$	−6.52792	−0.223512	−0.111756	−	0.993736i	$-0.535647\pi$
$853$	−6.52792	−0.223512	−0.111756	+	0.993736i	$0.535647\pi$
$854$	−2.97858	−0.101925
$855$	0	0
$856$	−5.37169	−0.183601
$857$	−15.6216	−0.533623	−0.266811	−	0.963749i	$-0.585970\pi$
$857$	−15.6216	−0.533623	−0.266811	+	0.963749i	$0.585970\pi$
$858$	0	0
$859$	−1.47208	−0.0502266	−0.0251133	−	0.999685i	$-0.507995\pi$
$859$	−1.47208	−0.0502266	−0.0251133	+	0.999685i	$0.507995\pi$
$860$	−10.3503	−0.352941
$861$	0	0
$862$	20.2499	0.689714
$863$	32.6086	1.11001	0.555004	−	0.831847i	$-0.312717\pi$
$863$	32.6086	1.11001	0.555004	+	0.831847i	$0.312717\pi$
$864$	0	0
$865$	−7.37169	−0.250645
$866$	1.56404	0.0531482
$867$	0	0
$868$	−6.97858	−0.236868
$869$	2.97858	0.101041
$870$	0	0
$871$	29.4868	0.999121
$872$	14.6430	0.495875
$873$	0	0
$874$	−1.45738	−0.0492967
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−21.3717	−0.721671	−0.360835	−	0.932630i	$-0.617508\pi$
$877$	−21.3717	−0.721671	−0.360835	+	0.932630i	$0.617508\pi$
$878$	−16.0000	−0.539974
$879$	0	0
$880$	1.00000	0.0337100
$881$	−23.1428	−0.779700	−0.389850	−	0.920878i	$-0.627473\pi$
$881$	−23.1428	−0.779700	−0.389850	+	0.920878i	$0.627473\pi$
$882$	0	0
$883$	−9.70727	−0.326676	−0.163338	−	0.986570i	$-0.552226\pi$
$883$	−9.70727	−0.326676	−0.163338	+	0.986570i	$0.552226\pi$
$884$	−1.37169	−0.0461350
$885$	0	0
$886$	−5.80765	−0.195112
$887$	15.6300	0.524804	0.262402	−	0.964959i	$-0.415485\pi$
$887$	15.6300	0.524804	0.262402	+	0.964959i	$0.415485\pi$
$888$	0	0
$889$	−14.7434	−0.494477
$890$	−0.100384	−0.00336489
$891$	0	0
$892$	5.95715	0.199460
$893$	−24.7862	−0.829440
$894$	0	0
$895$	14.7434	0.492817
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−4.70054	−0.156859
$899$	13.9572	0.465497
$900$	0	0
$901$	−0.585462	−0.0195046
$902$	4.39312	0.146275
$903$	0	0
$904$	−10.6430	−0.353981
$905$	−2.33558	−0.0776372
$906$	0	0
$907$	39.5787	1.31419	0.657095	−	0.753808i	$-0.271785\pi$
$907$	39.5787	1.31419	0.657095	+	0.753808i	$0.271785\pi$
$908$	−13.7220	−0.455379
$909$	0	0
$910$	−4.68585	−0.155334
$911$	27.6069	0.914657	0.457328	−	0.889298i	$-0.348806\pi$
$911$	27.6069	0.914657	0.457328	+	0.889298i	$0.348806\pi$
$912$	0	0
$913$	12.3503	0.408734
$914$	−4.77781	−0.158036
$915$	0	0
$916$	23.0361	0.761135
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−42.9786	−1.41773	−0.708866	−	0.705343i	$-0.750793\pi$
$919$	−42.9786	−1.41773	−0.708866	+	0.705343i	$0.750793\pi$
$920$	0.292731	0.00965105
$921$	0	0
$922$	−34.2646	−1.12844
$923$	16.9013	0.556313
$924$	0	0
$925$	4.29273	0.141144
$926$	30.6577	1.00747
$927$	0	0
$928$	2.00000	0.0656532
$929$	−7.89962	−0.259178	−0.129589	−	0.991568i	$-0.541366\pi$
$929$	−7.89962	−0.259178	−0.129589	+	0.991568i	$0.541366\pi$
$930$	0	0
$931$	−4.97858	−0.163166
$932$	−13.9143	−0.455778
$933$	0	0
$934$	−29.8715	−0.977424
$935$	−0.292731	−0.00957333
$936$	0	0
$937$	−12.1579	−0.397182	−0.198591	−	0.980082i	$-0.563637\pi$
$937$	−12.1579	−0.397182	−0.198591	+	0.980082i	$0.563637\pi$
$938$	−6.29273	−0.205465
$939$	0	0
$940$	4.97858	0.162383
$941$	18.4360	0.600995	0.300498	−	0.953783i	$-0.402847\pi$
$941$	18.4360	0.600995	0.300498	+	0.953783i	$0.402847\pi$
$942$	0	0
$943$	1.28600	0.0418780
$944$	−9.37169	−0.305023
$945$	0	0
$946$	−10.3503	−0.336516
$947$	−20.9357	−0.680320	−0.340160	−	0.940368i	$-0.610481\pi$
$947$	−20.9357	−0.680320	−0.340160	+	0.940368i	$0.610481\pi$
$948$	0	0
$949$	6.62831	0.215164
$950$	−4.97858	−0.161526
$951$	0	0
$952$	0.292731	0.00948747
$953$	52.9442	1.71503	0.857515	−	0.514460i	$-0.172007\pi$
$953$	52.9442	1.71503	0.857515	+	0.514460i	$0.172007\pi$
$954$	0	0
$955$	8.97858	0.290540
$956$	19.8652	0.642486
$957$	0	0
$958$	−6.74338	−0.217869
$959$	−18.0575	−0.583108
$960$	0	0
$961$	17.7005	0.570985
$962$	20.1151	0.648536
$963$	0	0
$964$	−11.9572	−0.385114
$965$	−10.3503	−0.333187
$966$	0	0
$967$	56.6148	1.82061	0.910305	−	0.413937i	$-0.135847\pi$
$967$	56.6148	1.82061	0.910305	+	0.413937i	$0.135847\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	3.60688	0.115810
$971$	−44.1151	−1.41572	−0.707860	−	0.706353i	$-0.750339\pi$
$971$	−44.1151	−1.41572	−0.707860	+	0.706353i	$0.750339\pi$
$972$	0	0
$973$	9.56404	0.306609
$974$	19.4145	0.622082
$975$	0	0
$976$	2.97858	0.0953419
$977$	28.6002	0.915000	0.457500	−	0.889210i	$-0.348745\pi$
$977$	28.6002	0.915000	0.457500	+	0.889210i	$0.348745\pi$
$978$	0	0
$979$	−0.100384	−0.00320829
$980$	1.00000	0.0319438
$981$	0	0
$982$	−3.32885	−0.106228
$983$	41.5934	1.32662	0.663312	−	0.748343i	$-0.269150\pi$
$983$	41.5934	1.32662	0.663312	+	0.748343i	$0.269150\pi$
$984$	0	0
$985$	19.9572	0.635888
$986$	−0.585462	−0.0186449
$987$	0	0
$988$	−23.3288	−0.742189
$989$	−3.02984	−0.0963435
$990$	0	0
$991$	30.1579	0.957998	0.478999	−	0.877815i	$-0.341000\pi$
$991$	30.1579	0.957998	0.478999	+	0.877815i	$0.341000\pi$
$992$	6.97858	0.221570
$993$	0	0
$994$	−3.60688	−0.114403
$995$	−5.60688	−0.177750
$996$	0	0
$997$	−15.8139	−0.500832	−0.250416	−	0.968138i	$-0.580567\pi$
$997$	−15.8139	−0.500832	−0.250416	+	0.968138i	$0.580567\pi$
$998$	−26.1579	−0.828015
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.cm.1.3	yes	3
3.2	odd	2		6930.2.a.cf.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6930.2.a.cf.1.3	✓	3	3.2	odd	2
6930.2.a.cm.1.3	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +4.68585 q^{13} -1.00000 q^{14} +1.00000 q^{16} -0.292731 q^{17} -4.97858 q^{19} +1.00000 q^{20} +1.00000 q^{22} +0.292731 q^{23} +1.00000 q^{25} +4.68585 q^{26} -1.00000 q^{28} +2.00000 q^{29} +6.97858 q^{31} +1.00000 q^{32} -0.292731 q^{34} -1.00000 q^{35} +4.29273 q^{37} -4.97858 q^{38} +1.00000 q^{40} +4.39312 q^{41} -10.3503 q^{43} +1.00000 q^{44} +0.292731 q^{46} +4.97858 q^{47} +1.00000 q^{49} +1.00000 q^{50} +4.68585 q^{52} +2.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} -9.37169 q^{59} +2.97858 q^{61} +6.97858 q^{62} +1.00000 q^{64} +4.68585 q^{65} +6.29273 q^{67} -0.292731 q^{68} -1.00000 q^{70} +3.60688 q^{71} +1.41454 q^{73} +4.29273 q^{74} -4.97858 q^{76} -1.00000 q^{77} +2.97858 q^{79} +1.00000 q^{80} +4.39312 q^{82} +12.3503 q^{83} -0.292731 q^{85} -10.3503 q^{86} +1.00000 q^{88} -0.100384 q^{89} -4.68585 q^{91} +0.292731 q^{92} +4.97858 q^{94} -4.97858 q^{95} +3.60688 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{20} + 3 q^{22} - 2 q^{23} + 3 q^{25} + 2 q^{26} - 3 q^{28} + 6 q^{29} + 6 q^{31} + 3 q^{32} + 2 q^{34} - 3 q^{35} + 10 q^{37} + 3 q^{40} + 4 q^{41} + 8 q^{43} + 3 q^{44} - 2 q^{46} + 3 q^{49} + 3 q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{55} - 3 q^{56} + 6 q^{58} - 4 q^{59} - 6 q^{61} + 6 q^{62} + 3 q^{64} + 2 q^{65} + 16 q^{67} + 2 q^{68} - 3 q^{70} + 20 q^{71} + 10 q^{73} + 10 q^{74} - 3 q^{77} - 6 q^{79} + 3 q^{80} + 4 q^{82} - 2 q^{83} + 2 q^{85} + 8 q^{86} + 3 q^{88} + 6 q^{89} - 2 q^{91} - 2 q^{92} + 20 q^{97} + 3 q^{98}+O(q^{100}) \) 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 + 3 * q^20 + 3 * q^22 - 2 * q^23 + 3 * q^25 + 2 * q^26 - 3 * q^28 + 6 * q^29 + 6 * q^31 + 3 * q^32 + 2 * q^34 - 3 * q^35 + 10 * q^37 + 3 * q^40 + 4 * q^41 + 8 * q^43 + 3 * q^44 - 2 * q^46 + 3 * q^49 + 3 * q^50 + 2 * q^52 + 6 * q^53 + 3 * q^55 - 3 * q^56 + 6 * q^58 - 4 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 + 2 * q^65 + 16 * q^67 + 2 * q^68 - 3 * q^70 + 20 * q^71 + 10 * q^73 + 10 * q^74 - 3 * q^77 - 6 * q^79 + 3 * q^80 + 4 * q^82 - 2 * q^83 + 2 * q^85 + 8 * q^86 + 3 * q^88 + 6 * q^89 - 2 * q^91 - 2 * q^92 + 20 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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