LMFDB - Embedded newform 6930.2.a.br.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 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			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.65685 q^{17} +2.82843 q^{19} +1.00000 q^{20} -1.00000 q^{22} -2.82843 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -8.82843 q^{29} +5.65685 q^{31} -1.00000 q^{32} -3.65685 q^{34} -1.00000 q^{35} +8.82843 q^{37} -2.82843 q^{38} -1.00000 q^{40} -4.82843 q^{41} +1.65685 q^{43} +1.00000 q^{44} +2.82843 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -6.48528 q^{53} +1.00000 q^{55} +1.00000 q^{56} +8.82843 q^{58} +8.00000 q^{59} +13.3137 q^{61} -5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} -7.31371 q^{67} +3.65685 q^{68} +1.00000 q^{70} -5.65685 q^{71} -3.65685 q^{73} -8.82843 q^{74} +2.82843 q^{76} -1.00000 q^{77} -10.8284 q^{79} +1.00000 q^{80} +4.82843 q^{82} +13.6569 q^{83} +3.65685 q^{85} -1.65685 q^{86} -1.00000 q^{88} +0.343146 q^{89} -2.00000 q^{91} -2.82843 q^{92} -8.00000 q^{94} +2.82843 q^{95} +4.82843 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{20} - 2 q^{22} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 12 q^{37} - 2 q^{40} - 4 q^{41} - 8 q^{43} + 2 q^{44} + 16 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} + 2 q^{55} + 2 q^{56} + 12 q^{58} + 16 q^{59} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} + 2 q^{70} + 4 q^{73} - 12 q^{74} - 2 q^{77} - 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} - 4 q^{85} + 8 q^{86} - 2 q^{88} + 12 q^{89} - 4 q^{91} - 16 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^20 - 2 * q^22 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 12 * q^29 - 2 * q^32 + 4 * q^34 - 2 * q^35 + 12 * q^37 - 2 * q^40 - 4 * q^41 - 8 * q^43 + 2 * q^44 + 16 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 2 * q^55 + 2 * q^56 + 12 * q^58 + 16 * q^59 + 4 * q^61 + 2 * q^64 + 4 * q^65 + 8 * q^67 - 4 * q^68 + 2 * q^70 + 4 * q^73 - 12 * q^74 - 2 * q^77 - 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 - 4 * q^85 + 8 * q^86 - 2 * q^88 + 12 * q^89 - 4 * q^91 - 16 * q^94 + 4 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−8.82843	−1.63940	−0.819699	−	0.572795i	$-0.805859\pi$
$29$	−8.82843	−1.63940	−0.819699	+	0.572795i	$0.805859\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.65685	−0.627145
$35$	−1.00000	−0.169031
$36$	0	0
$37$	8.82843	1.45138	0.725692	−	0.688019i	$-0.241520\pi$
$37$	8.82843	1.45138	0.725692	+	0.688019i	$0.241520\pi$
$38$	−2.82843	−0.458831
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−4.82843	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$41$	−4.82843	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	2.82843	0.417029
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.48528	−0.890822	−0.445411	−	0.895326i	$-0.646942\pi$
$53$	−6.48528	−0.890822	−0.445411	+	0.895326i	$0.646942\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	8.82843	1.15923
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	−5.65685	−0.718421
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−7.31371	−0.893512	−0.446756	−	0.894656i	$-0.647421\pi$
$67$	−7.31371	−0.893512	−0.446756	+	0.894656i	$0.647421\pi$
$68$	3.65685	0.443459
$69$	0	0
$70$	1.00000	0.119523
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	−3.65685	−0.428002	−0.214001	−	0.976833i	$-0.568650\pi$
$73$	−3.65685	−0.428002	−0.214001	+	0.976833i	$0.568650\pi$
$74$	−8.82843	−1.02628
$75$	0	0
$76$	2.82843	0.324443
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.8284	−1.21829	−0.609147	−	0.793058i	$-0.708488\pi$
$79$	−10.8284	−1.21829	−0.609147	+	0.793058i	$0.708488\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	4.82843	0.533211
$83$	13.6569	1.49903	0.749517	−	0.661985i	$-0.230286\pi$
$83$	13.6569	1.49903	0.749517	+	0.661985i	$0.230286\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	−1.65685	−0.178663
$87$	0	0
$88$	−1.00000	−0.106600
$89$	0.343146	0.0363734	0.0181867	−	0.999835i	$-0.494211\pi$
$89$	0.343146	0.0363734	0.0181867	+	0.999835i	$0.494211\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−2.82843	−0.294884
$93$	0	0
$94$	−8.00000	−0.825137
$95$	2.82843	0.290191
$96$	0	0
$97$	4.82843	0.490252	0.245126	−	0.969491i	$-0.421171\pi$
$97$	4.82843	0.490252	0.245126	+	0.969491i	$0.421171\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	5.65685	0.557386	0.278693	−	0.960380i	$-0.410099\pi$
$103$	5.65685	0.557386	0.278693	+	0.960380i	$0.410099\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.48528	0.629906
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	0	0
$109$	−2.48528	−0.238047	−0.119023	−	0.992891i	$-0.537976\pi$
$109$	−2.48528	−0.238047	−0.119023	+	0.992891i	$0.537976\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−2.82843	−0.263752
$116$	−8.82843	−0.819699
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−3.65685	−0.335223
$120$	0	0
$121$	1.00000	0.0909091
$122$	−13.3137	−1.20537
$123$	0	0
$124$	5.65685	0.508001
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.31371	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$127$	−3.31371	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−13.1716	−1.15081	−0.575403	−	0.817870i	$-0.695155\pi$
$131$	−13.1716	−1.15081	−0.575403	+	0.817870i	$0.695155\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	7.31371	0.631808
$135$	0	0
$136$	−3.65685	−0.313573
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	0	0
$139$	13.1716	1.11720	0.558599	−	0.829438i	$-0.311339\pi$
$139$	13.1716	1.11720	0.558599	+	0.829438i	$0.311339\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	5.65685	0.474713
$143$	2.00000	0.167248
$144$	0	0
$145$	−8.82843	−0.733161
$146$	3.65685	0.302643
$147$	0	0
$148$	8.82843	0.725692
$149$	18.4853	1.51437	0.757187	−	0.653199i	$-0.226573\pi$
$149$	18.4853	1.51437	0.757187	+	0.653199i	$0.226573\pi$
$150$	0	0
$151$	5.17157	0.420857	0.210428	−	0.977609i	$-0.432514\pi$
$151$	5.17157	0.420857	0.210428	+	0.977609i	$0.432514\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	1.00000	0.0805823
$155$	5.65685	0.454369
$156$	0	0
$157$	4.34315	0.346621	0.173310	−	0.984867i	$-0.444554\pi$
$157$	4.34315	0.346621	0.173310	+	0.984867i	$0.444554\pi$
$158$	10.8284	0.861463
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	2.82843	0.222911
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−4.82843	−0.377037
$165$	0	0
$166$	−13.6569	−1.05998
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−3.65685	−0.280468
$171$	0	0
$172$	1.65685	0.126334
$173$	−21.3137	−1.62045	−0.810226	−	0.586118i	$-0.800655\pi$
$173$	−21.3137	−1.62045	−0.810226	+	0.586118i	$0.800655\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−0.343146	−0.0257199
$179$	−17.6569	−1.31974	−0.659868	−	0.751382i	$-0.729388\pi$
$179$	−17.6569	−1.31974	−0.659868	+	0.751382i	$0.729388\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	2.82843	0.208514
$185$	8.82843	0.649079
$186$	0	0
$187$	3.65685	0.267416
$188$	8.00000	0.583460
$189$	0	0
$190$	−2.82843	−0.205196
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	0	0
$193$	−1.31371	−0.0945628	−0.0472814	−	0.998882i	$-0.515056\pi$
$193$	−1.31371	−0.0945628	−0.0472814	+	0.998882i	$0.515056\pi$
$194$	−4.82843	−0.346661
$195$	0	0
$196$	1.00000	0.0714286
$197$	23.6569	1.68548	0.842741	−	0.538320i	$-0.180941\pi$
$197$	23.6569	1.68548	0.842741	+	0.538320i	$0.180941\pi$
$198$	0	0
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	10.0000	0.703598
$203$	8.82843	0.619634
$204$	0	0
$205$	−4.82843	−0.337232
$206$	−5.65685	−0.394132
$207$	0	0
$208$	2.00000	0.138675
$209$	2.82843	0.195646
$210$	0	0
$211$	−7.31371	−0.503496	−0.251748	−	0.967793i	$-0.581005\pi$
$211$	−7.31371	−0.503496	−0.251748	+	0.967793i	$0.581005\pi$
$212$	−6.48528	−0.445411
$213$	0	0
$214$	9.65685	0.660129
$215$	1.65685	0.112997
$216$	0	0
$217$	−5.65685	−0.384012
$218$	2.48528	0.168324
$219$	0	0
$220$	1.00000	0.0674200
$221$	7.31371	0.491973
$222$	0	0
$223$	28.2843	1.89405	0.947027	−	0.321153i	$-0.104070\pi$
$223$	28.2843	1.89405	0.947027	+	0.321153i	$0.104070\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	2.82843	0.186501
$231$	0	0
$232$	8.82843	0.579615
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	8.00000	0.520756
$237$	0	0
$238$	3.65685	0.237039
$239$	24.4853	1.58382	0.791911	−	0.610637i	$-0.209087\pi$
$239$	24.4853	1.58382	0.791911	+	0.610637i	$0.209087\pi$
$240$	0	0
$241$	−3.17157	−0.204299	−0.102149	−	0.994769i	$-0.532572\pi$
$241$	−3.17157	−0.204299	−0.102149	+	0.994769i	$0.532572\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	13.3137	0.852323
$245$	1.00000	0.0638877
$246$	0	0
$247$	5.65685	0.359937
$248$	−5.65685	−0.359211
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−11.3137	−0.714115	−0.357057	−	0.934082i	$-0.616220\pi$
$251$	−11.3137	−0.714115	−0.357057	+	0.934082i	$0.616220\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	3.31371	0.207921
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.8284	−1.79827	−0.899134	−	0.437674i	$-0.855803\pi$
$257$	−28.8284	−1.79827	−0.899134	+	0.437674i	$0.855803\pi$
$258$	0	0
$259$	−8.82843	−0.548572
$260$	2.00000	0.124035
$261$	0	0
$262$	13.1716	0.813742
$263$	−30.6274	−1.88857	−0.944284	−	0.329133i	$-0.893244\pi$
$263$	−30.6274	−1.88857	−0.944284	+	0.329133i	$0.893244\pi$
$264$	0	0
$265$	−6.48528	−0.398388
$266$	2.82843	0.173422
$267$	0	0
$268$	−7.31371	−0.446756
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	3.65685	0.221729
$273$	0	0
$274$	−17.3137	−1.04596
$275$	1.00000	0.0603023
$276$	0	0
$277$	9.31371	0.559607	0.279803	−	0.960057i	$-0.409731\pi$
$277$	9.31371	0.559607	0.279803	+	0.960057i	$0.409731\pi$
$278$	−13.1716	−0.789978
$279$	0	0
$280$	1.00000	0.0597614
$281$	−26.9706	−1.60893	−0.804464	−	0.594001i	$-0.797548\pi$
$281$	−26.9706	−1.60893	−0.804464	+	0.594001i	$0.797548\pi$
$282$	0	0
$283$	22.6274	1.34506	0.672530	−	0.740070i	$-0.265208\pi$
$283$	22.6274	1.34506	0.672530	+	0.740070i	$0.265208\pi$
$284$	−5.65685	−0.335673
$285$	0	0
$286$	−2.00000	−0.118262
$287$	4.82843	0.285013
$288$	0	0
$289$	−3.62742	−0.213377
$290$	8.82843	0.518423
$291$	0	0
$292$	−3.65685	−0.214001
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−8.82843	−0.513142
$297$	0	0
$298$	−18.4853	−1.07082
$299$	−5.65685	−0.327144
$300$	0	0
$301$	−1.65685	−0.0954995
$302$	−5.17157	−0.297591
$303$	0	0
$304$	2.82843	0.162221
$305$	13.3137	0.762341
$306$	0	0
$307$	24.9706	1.42515	0.712573	−	0.701598i	$-0.247530\pi$
$307$	24.9706	1.42515	0.712573	+	0.701598i	$0.247530\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−5.65685	−0.321288
$311$	32.9706	1.86959	0.934795	−	0.355189i	$-0.115583\pi$
$311$	32.9706	1.86959	0.934795	+	0.355189i	$0.115583\pi$
$312$	0	0
$313$	−3.17157	−0.179268	−0.0896339	−	0.995975i	$-0.528570\pi$
$313$	−3.17157	−0.179268	−0.0896339	+	0.995975i	$0.528570\pi$
$314$	−4.34315	−0.245098
$315$	0	0
$316$	−10.8284	−0.609147
$317$	−8.82843	−0.495854	−0.247927	−	0.968779i	$-0.579749\pi$
$317$	−8.82843	−0.495854	−0.247927	+	0.968779i	$0.579749\pi$
$318$	0	0
$319$	−8.82843	−0.494297
$320$	1.00000	0.0559017
$321$	0	0
$322$	−2.82843	−0.157622
$323$	10.3431	0.575508
$324$	0	0
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	0	0
$328$	4.82843	0.266605
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−4.97056	−0.273207	−0.136603	−	0.990626i	$-0.543619\pi$
$331$	−4.97056	−0.273207	−0.136603	+	0.990626i	$0.543619\pi$
$332$	13.6569	0.749517
$333$	0	0
$334$	−16.0000	−0.875481
$335$	−7.31371	−0.399591
$336$	0	0
$337$	6.68629	0.364226	0.182113	−	0.983278i	$-0.441706\pi$
$337$	6.68629	0.364226	0.182113	+	0.983278i	$0.441706\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	3.65685	0.198321
$341$	5.65685	0.306336
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.65685	−0.0893316
$345$	0	0
$346$	21.3137	1.14583
$347$	4.97056	0.266834	0.133417	−	0.991060i	$-0.457405\pi$
$347$	4.97056	0.266834	0.133417	+	0.991060i	$0.457405\pi$
$348$	0	0
$349$	−6.97056	−0.373126	−0.186563	−	0.982443i	$-0.559735\pi$
$349$	−6.97056	−0.373126	−0.186563	+	0.982443i	$0.559735\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	19.1716	1.02040	0.510200	−	0.860056i	$-0.329571\pi$
$353$	19.1716	1.02040	0.510200	+	0.860056i	$0.329571\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	0.343146	0.0181867
$357$	0	0
$358$	17.6569	0.933194
$359$	17.4558	0.921284	0.460642	−	0.887586i	$-0.347619\pi$
$359$	17.4558	0.921284	0.460642	+	0.887586i	$0.347619\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	14.0000	0.735824
$363$	0	0
$364$	−2.00000	−0.104828
$365$	−3.65685	−0.191408
$366$	0	0
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	−2.82843	−0.147442
$369$	0	0
$370$	−8.82843	−0.458968
$371$	6.48528	0.336699
$372$	0	0
$373$	−7.65685	−0.396457	−0.198228	−	0.980156i	$-0.563519\pi$
$373$	−7.65685	−0.396457	−0.198228	+	0.980156i	$0.563519\pi$
$374$	−3.65685	−0.189091
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−17.6569	−0.909374
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	2.82843	0.145095
$381$	0	0
$382$	−3.31371	−0.169544
$383$	18.3431	0.937291	0.468645	−	0.883386i	$-0.344742\pi$
$383$	18.3431	0.937291	0.468645	+	0.883386i	$0.344742\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	1.31371	0.0668660
$387$	0	0
$388$	4.82843	0.245126
$389$	2.97056	0.150614	0.0753068	−	0.997160i	$-0.476006\pi$
$389$	2.97056	0.150614	0.0753068	+	0.997160i	$0.476006\pi$
$390$	0	0
$391$	−10.3431	−0.523075
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−23.6569	−1.19182
$395$	−10.8284	−0.544837
$396$	0	0
$397$	10.9706	0.550597	0.275298	−	0.961359i	$-0.411223\pi$
$397$	10.9706	0.550597	0.275298	+	0.961359i	$0.411223\pi$
$398$	−11.3137	−0.567105
$399$	0	0
$400$	1.00000	0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	11.3137	0.563576
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−8.82843	−0.438147
$407$	8.82843	0.437609
$408$	0	0
$409$	16.1421	0.798177	0.399089	−	0.916912i	$-0.369327\pi$
$409$	16.1421	0.798177	0.399089	+	0.916912i	$0.369327\pi$
$410$	4.82843	0.238459
$411$	0	0
$412$	5.65685	0.278693
$413$	−8.00000	−0.393654
$414$	0	0
$415$	13.6569	0.670389
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	−2.82843	−0.138343
$419$	−14.6274	−0.714596	−0.357298	−	0.933990i	$-0.616302\pi$
$419$	−14.6274	−0.714596	−0.357298	+	0.933990i	$0.616302\pi$
$420$	0	0
$421$	−16.6274	−0.810371	−0.405185	−	0.914235i	$-0.632793\pi$
$421$	−16.6274	−0.810371	−0.405185	+	0.914235i	$0.632793\pi$
$422$	7.31371	0.356026
$423$	0	0
$424$	6.48528	0.314953
$425$	3.65685	0.177383
$426$	0	0
$427$	−13.3137	−0.644296
$428$	−9.65685	−0.466782
$429$	0	0
$430$	−1.65685	−0.0799006
$431$	−19.7990	−0.953684	−0.476842	−	0.878989i	$-0.658219\pi$
$431$	−19.7990	−0.953684	−0.476842	+	0.878989i	$0.658219\pi$
$432$	0	0
$433$	15.1716	0.729099	0.364550	−	0.931184i	$-0.381223\pi$
$433$	15.1716	0.729099	0.364550	+	0.931184i	$0.381223\pi$
$434$	5.65685	0.271538
$435$	0	0
$436$	−2.48528	−0.119023
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−8.97056	−0.428142	−0.214071	−	0.976818i	$-0.568672\pi$
$439$	−8.97056	−0.428142	−0.214071	+	0.976818i	$0.568672\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−7.31371	−0.347878
$443$	31.3137	1.48776	0.743880	−	0.668314i	$-0.232984\pi$
$443$	31.3137	1.48776	0.743880	+	0.668314i	$0.232984\pi$
$444$	0	0
$445$	0.343146	0.0162667
$446$	−28.2843	−1.33930
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−4.82843	−0.227362
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	−24.0000	−1.12638
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−21.3137	−0.995924
$459$	0	0
$460$	−2.82843	−0.131876
$461$	0.343146	0.0159819	0.00799095	−	0.999968i	$-0.497456\pi$
$461$	0.343146	0.0159819	0.00799095	+	0.999968i	$0.497456\pi$
$462$	0	0
$463$	41.4558	1.92662	0.963308	−	0.268398i	$-0.0864941\pi$
$463$	41.4558	1.92662	0.963308	+	0.268398i	$0.0864941\pi$
$464$	−8.82843	−0.409849
$465$	0	0
$466$	−6.00000	−0.277945
$467$	27.7990	1.28638	0.643192	−	0.765705i	$-0.277610\pi$
$467$	27.7990	1.28638	0.643192	+	0.765705i	$0.277610\pi$
$468$	0	0
$469$	7.31371	0.337716
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−8.00000	−0.368230
$473$	1.65685	0.0761822
$474$	0	0
$475$	2.82843	0.129777
$476$	−3.65685	−0.167612
$477$	0	0
$478$	−24.4853	−1.11993
$479$	−12.6863	−0.579651	−0.289826	−	0.957079i	$-0.593597\pi$
$479$	−12.6863	−0.579651	−0.289826	+	0.957079i	$0.593597\pi$
$480$	0	0
$481$	17.6569	0.805083
$482$	3.17157	0.144461
$483$	0	0
$484$	1.00000	0.0454545
$485$	4.82843	0.219248
$486$	0	0
$487$	36.7696	1.66619	0.833094	−	0.553132i	$-0.186567\pi$
$487$	36.7696	1.66619	0.833094	+	0.553132i	$0.186567\pi$
$488$	−13.3137	−0.602683
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	31.3137	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$491$	31.3137	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$492$	0	0
$493$	−32.2843	−1.45401
$494$	−5.65685	−0.254514
$495$	0	0
$496$	5.65685	0.254000
$497$	5.65685	0.253745
$498$	0	0
$499$	−4.97056	−0.222513	−0.111256	−	0.993792i	$-0.535488\pi$
$499$	−4.97056	−0.222513	−0.111256	+	0.993792i	$0.535488\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	11.3137	0.504956
$503$	−19.3137	−0.861156	−0.430578	−	0.902553i	$-0.641690\pi$
$503$	−19.3137	−0.861156	−0.430578	+	0.902553i	$0.641690\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	2.82843	0.125739
$507$	0	0
$508$	−3.31371	−0.147022
$509$	33.3137	1.47660	0.738302	−	0.674470i	$-0.235628\pi$
$509$	33.3137	1.47660	0.738302	+	0.674470i	$0.235628\pi$
$510$	0	0
$511$	3.65685	0.161770
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	28.8284	1.27157
$515$	5.65685	0.249271
$516$	0	0
$517$	8.00000	0.351840
$518$	8.82843	0.387899
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	37.5980	1.64720	0.823599	−	0.567173i	$-0.191963\pi$
$521$	37.5980	1.64720	0.823599	+	0.567173i	$0.191963\pi$
$522$	0	0
$523$	−21.6569	−0.946988	−0.473494	−	0.880797i	$-0.657007\pi$
$523$	−21.6569	−0.946988	−0.473494	+	0.880797i	$0.657007\pi$
$524$	−13.1716	−0.575403
$525$	0	0
$526$	30.6274	1.33542
$527$	20.6863	0.901109
$528$	0	0
$529$	−15.0000	−0.652174
$530$	6.48528	0.281703
$531$	0	0
$532$	−2.82843	−0.122628
$533$	−9.65685	−0.418285
$534$	0	0
$535$	−9.65685	−0.417502
$536$	7.31371	0.315904
$537$	0	0
$538$	2.00000	0.0862261
$539$	1.00000	0.0430730
$540$	0	0
$541$	29.1127	1.25165	0.625826	−	0.779962i	$-0.284762\pi$
$541$	29.1127	1.25165	0.625826	+	0.779962i	$0.284762\pi$
$542$	16.9706	0.728948
$543$	0	0
$544$	−3.65685	−0.156786
$545$	−2.48528	−0.106458
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	17.3137	0.739605
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−24.9706	−1.06378
$552$	0	0
$553$	10.8284	0.460472
$554$	−9.31371	−0.395702
$555$	0	0
$556$	13.1716	0.558599
$557$	−6.97056	−0.295352	−0.147676	−	0.989036i	$-0.547179\pi$
$557$	−6.97056	−0.295352	−0.147676	+	0.989036i	$0.547179\pi$
$558$	0	0
$559$	3.31371	0.140155
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	26.9706	1.13768
$563$	−22.6274	−0.953632	−0.476816	−	0.879003i	$-0.658209\pi$
$563$	−22.6274	−0.953632	−0.476816	+	0.879003i	$0.658209\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	−22.6274	−0.951101
$567$	0	0
$568$	5.65685	0.237356
$569$	28.6274	1.20012	0.600062	−	0.799954i	$-0.295143\pi$
$569$	28.6274	1.20012	0.600062	+	0.799954i	$0.295143\pi$
$570$	0	0
$571$	43.5980	1.82452	0.912259	−	0.409613i	$-0.134336\pi$
$571$	43.5980	1.82452	0.912259	+	0.409613i	$0.134336\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−4.82843	−0.201535
$575$	−2.82843	−0.117954
$576$	0	0
$577$	−37.1127	−1.54502	−0.772511	−	0.635001i	$-0.780999\pi$
$577$	−37.1127	−1.54502	−0.772511	+	0.635001i	$0.780999\pi$
$578$	3.62742	0.150881
$579$	0	0
$580$	−8.82843	−0.366580
$581$	−13.6569	−0.566582
$582$	0	0
$583$	−6.48528	−0.268593
$584$	3.65685	0.151322
$585$	0	0
$586$	2.00000	0.0826192
$587$	9.85786	0.406878	0.203439	−	0.979088i	$-0.434788\pi$
$587$	9.85786	0.406878	0.203439	+	0.979088i	$0.434788\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−8.00000	−0.329355
$591$	0	0
$592$	8.82843	0.362846
$593$	−31.6569	−1.29999	−0.649996	−	0.759938i	$-0.725229\pi$
$593$	−31.6569	−1.29999	−0.649996	+	0.759938i	$0.725229\pi$
$594$	0	0
$595$	−3.65685	−0.149916
$596$	18.4853	0.757187
$597$	0	0
$598$	5.65685	0.231326
$599$	−10.3431	−0.422609	−0.211305	−	0.977420i	$-0.567771\pi$
$599$	−10.3431	−0.422609	−0.211305	+	0.977420i	$0.567771\pi$
$600$	0	0
$601$	7.17157	0.292535	0.146267	−	0.989245i	$-0.453274\pi$
$601$	7.17157	0.292535	0.146267	+	0.989245i	$0.453274\pi$
$602$	1.65685	0.0675283
$603$	0	0
$604$	5.17157	0.210428
$605$	1.00000	0.0406558
$606$	0	0
$607$	18.3431	0.744525	0.372263	−	0.928127i	$-0.378582\pi$
$607$	18.3431	0.744525	0.372263	+	0.928127i	$0.378582\pi$
$608$	−2.82843	−0.114708
$609$	0	0
$610$	−13.3137	−0.539056
$611$	16.0000	0.647291
$612$	0	0
$613$	−10.9706	−0.443097	−0.221548	−	0.975149i	$-0.571111\pi$
$613$	−10.9706	−0.443097	−0.221548	+	0.975149i	$0.571111\pi$
$614$	−24.9706	−1.00773
$615$	0	0
$616$	1.00000	0.0402911
$617$	43.6569	1.75756	0.878779	−	0.477228i	$-0.158358\pi$
$617$	43.6569	1.75756	0.878779	+	0.477228i	$0.158358\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	5.65685	0.227185
$621$	0	0
$622$	−32.9706	−1.32200
$623$	−0.343146	−0.0137478
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.17157	0.126762
$627$	0	0
$628$	4.34315	0.173310
$629$	32.2843	1.28726
$630$	0	0
$631$	22.6274	0.900783	0.450392	−	0.892831i	$-0.351284\pi$
$631$	22.6274	0.900783	0.450392	+	0.892831i	$0.351284\pi$
$632$	10.8284	0.430732
$633$	0	0
$634$	8.82843	0.350622
$635$	−3.31371	−0.131501
$636$	0	0
$637$	2.00000	0.0792429
$638$	8.82843	0.349521
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	−14.1421	−0.557711	−0.278856	−	0.960333i	$-0.589955\pi$
$643$	−14.1421	−0.557711	−0.278856	+	0.960333i	$0.589955\pi$
$644$	2.82843	0.111456
$645$	0	0
$646$	−10.3431	−0.406946
$647$	44.2843	1.74099	0.870497	−	0.492173i	$-0.163797\pi$
$647$	44.2843	1.74099	0.870497	+	0.492173i	$0.163797\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	12.0000	0.469956
$653$	43.4558	1.70056	0.850279	−	0.526332i	$-0.176433\pi$
$653$	43.4558	1.70056	0.850279	+	0.526332i	$0.176433\pi$
$654$	0	0
$655$	−13.1716	−0.514656
$656$	−4.82843	−0.188518
$657$	0	0
$658$	8.00000	0.311872
$659$	−1.65685	−0.0645419	−0.0322709	−	0.999479i	$-0.510274\pi$
$659$	−1.65685	−0.0645419	−0.0322709	+	0.999479i	$0.510274\pi$
$660$	0	0
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	4.97056	0.193186
$663$	0	0
$664$	−13.6569	−0.529989
$665$	−2.82843	−0.109682
$666$	0	0
$667$	24.9706	0.966864
$668$	16.0000	0.619059
$669$	0	0
$670$	7.31371	0.282553
$671$	13.3137	0.513970
$672$	0	0
$673$	8.62742	0.332562	0.166281	−	0.986078i	$-0.446824\pi$
$673$	8.62742	0.332562	0.166281	+	0.986078i	$0.446824\pi$
$674$	−6.68629	−0.257546
$675$	0	0
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	−4.82843	−0.185298
$680$	−3.65685	−0.140234
$681$	0	0
$682$	−5.65685	−0.216612
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	0	0
$685$	17.3137	0.661523
$686$	1.00000	0.0381802
$687$	0	0
$688$	1.65685	0.0631670
$689$	−12.9706	−0.494139
$690$	0	0
$691$	−32.9706	−1.25426	−0.627130	−	0.778915i	$-0.715770\pi$
$691$	−32.9706	−1.25426	−0.627130	+	0.778915i	$0.715770\pi$
$692$	−21.3137	−0.810226
$693$	0	0
$694$	−4.97056	−0.188680
$695$	13.1716	0.499626
$696$	0	0
$697$	−17.6569	−0.668801
$698$	6.97056	0.263840
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−35.1716	−1.32841	−0.664206	−	0.747550i	$-0.731230\pi$
$701$	−35.1716	−1.32841	−0.664206	+	0.747550i	$0.731230\pi$
$702$	0	0
$703$	24.9706	0.941783
$704$	1.00000	0.0376889
$705$	0	0
$706$	−19.1716	−0.721532
$707$	10.0000	0.376089
$708$	0	0
$709$	−30.2843	−1.13735	−0.568675	−	0.822562i	$-0.692544\pi$
$709$	−30.2843	−1.13735	−0.568675	+	0.822562i	$0.692544\pi$
$710$	5.65685	0.212298
$711$	0	0
$712$	−0.343146	−0.0128599
$713$	−16.0000	−0.599205
$714$	0	0
$715$	2.00000	0.0747958
$716$	−17.6569	−0.659868
$717$	0	0
$718$	−17.4558	−0.651446
$719$	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
$721$	−5.65685	−0.210672
$722$	11.0000	0.409378
$723$	0	0
$724$	−14.0000	−0.520306
$725$	−8.82843	−0.327880
$726$	0	0
$727$	27.3137	1.01301	0.506505	−	0.862237i	$-0.330937\pi$
$727$	27.3137	1.01301	0.506505	+	0.862237i	$0.330937\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	3.65685	0.135346
$731$	6.05887	0.224096
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	28.2843	1.04399
$735$	0	0
$736$	2.82843	0.104257
$737$	−7.31371	−0.269404
$738$	0	0
$739$	−49.2548	−1.81187	−0.905934	−	0.423419i	$-0.860830\pi$
$739$	−49.2548	−1.81187	−0.905934	+	0.423419i	$0.860830\pi$
$740$	8.82843	0.324539
$741$	0	0
$742$	−6.48528	−0.238082
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	18.4853	0.677248
$746$	7.65685	0.280337
$747$	0	0
$748$	3.65685	0.133708
$749$	9.65685	0.352854
$750$	0	0
$751$	−39.5980	−1.44495	−0.722475	−	0.691397i	$-0.756996\pi$
$751$	−39.5980	−1.44495	−0.722475	+	0.691397i	$0.756996\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	17.6569	0.643025
$755$	5.17157	0.188213
$756$	0	0
$757$	37.1127	1.34888	0.674442	−	0.738328i	$-0.264384\pi$
$757$	37.1127	1.34888	0.674442	+	0.738328i	$0.264384\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	−2.82843	−0.102598
$761$	−43.4558	−1.57527	−0.787637	−	0.616140i	$-0.788695\pi$
$761$	−43.4558	−1.57527	−0.787637	+	0.616140i	$0.788695\pi$
$762$	0	0
$763$	2.48528	0.0899732
$764$	3.31371	0.119886
$765$	0	0
$766$	−18.3431	−0.662765
$767$	16.0000	0.577727
$768$	0	0
$769$	2.48528	0.0896215	0.0448108	−	0.998995i	$-0.485732\pi$
$769$	2.48528	0.0896215	0.0448108	+	0.998995i	$0.485732\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−1.31371	−0.0472814
$773$	33.3137	1.19821	0.599105	−	0.800670i	$-0.295523\pi$
$773$	33.3137	1.19821	0.599105	+	0.800670i	$0.295523\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	−4.82843	−0.173330
$777$	0	0
$778$	−2.97056	−0.106500
$779$	−13.6569	−0.489308
$780$	0	0
$781$	−5.65685	−0.202418
$782$	10.3431	0.369870
$783$	0	0
$784$	1.00000	0.0357143
$785$	4.34315	0.155014
$786$	0	0
$787$	−14.6274	−0.521411	−0.260706	−	0.965418i	$-0.583955\pi$
$787$	−14.6274	−0.521411	−0.260706	+	0.965418i	$0.583955\pi$
$788$	23.6569	0.842741
$789$	0	0
$790$	10.8284	0.385258
$791$	2.00000	0.0711118
$792$	0	0
$793$	26.6274	0.945567
$794$	−10.9706	−0.389331
$795$	0	0
$796$	11.3137	0.401004
$797$	50.2843	1.78116	0.890580	−	0.454826i	$-0.150299\pi$
$797$	50.2843	1.78116	0.890580	+	0.454826i	$0.150299\pi$
$798$	0	0
$799$	29.2548	1.03496
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	10.0000	0.353112
$803$	−3.65685	−0.129048
$804$	0	0
$805$	2.82843	0.0996890
$806$	−11.3137	−0.398508
$807$	0	0
$808$	10.0000	0.351799
$809$	34.2843	1.20537	0.602685	−	0.797979i	$-0.294097\pi$
$809$	34.2843	1.20537	0.602685	+	0.797979i	$0.294097\pi$
$810$	0	0
$811$	−6.14214	−0.215680	−0.107840	−	0.994168i	$-0.534393\pi$
$811$	−6.14214	−0.215680	−0.107840	+	0.994168i	$0.534393\pi$
$812$	8.82843	0.309817
$813$	0	0
$814$	−8.82843	−0.309436
$815$	12.0000	0.420342
$816$	0	0
$817$	4.68629	0.163953
$818$	−16.1421	−0.564397
$819$	0	0
$820$	−4.82843	−0.168616
$821$	−9.79899	−0.341987	−0.170994	−	0.985272i	$-0.554698\pi$
$821$	−9.79899	−0.341987	−0.170994	+	0.985272i	$0.554698\pi$
$822$	0	0
$823$	26.8284	0.935180	0.467590	−	0.883945i	$-0.345122\pi$
$823$	26.8284	0.935180	0.467590	+	0.883945i	$0.345122\pi$
$824$	−5.65685	−0.197066
$825$	0	0
$826$	8.00000	0.278356
$827$	−41.2548	−1.43457	−0.717286	−	0.696779i	$-0.754616\pi$
$827$	−41.2548	−1.43457	−0.717286	+	0.696779i	$0.754616\pi$
$828$	0	0
$829$	−51.2548	−1.78015	−0.890077	−	0.455810i	$-0.849350\pi$
$829$	−51.2548	−1.78015	−0.890077	+	0.455810i	$0.849350\pi$
$830$	−13.6569	−0.474036
$831$	0	0
$832$	2.00000	0.0693375
$833$	3.65685	0.126702
$834$	0	0
$835$	16.0000	0.553703
$836$	2.82843	0.0978232
$837$	0	0
$838$	14.6274	0.505296
$839$	45.6569	1.57625	0.788125	−	0.615515i	$-0.211052\pi$
$839$	45.6569	1.57625	0.788125	+	0.615515i	$0.211052\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	16.6274	0.573019
$843$	0	0
$844$	−7.31371	−0.251748
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−6.48528	−0.222705
$849$	0	0
$850$	−3.65685	−0.125429
$851$	−24.9706	−0.855980
$852$	0	0
$853$	−2.68629	−0.0919769	−0.0459884	−	0.998942i	$-0.514644\pi$
$853$	−2.68629	−0.0919769	−0.0459884	+	0.998942i	$0.514644\pi$
$854$	13.3137	0.455586
$855$	0	0
$856$	9.65685	0.330064
$857$	−26.9706	−0.921297	−0.460648	−	0.887583i	$-0.652383\pi$
$857$	−26.9706	−0.921297	−0.460648	+	0.887583i	$0.652383\pi$
$858$	0	0
$859$	27.3137	0.931932	0.465966	−	0.884803i	$-0.345707\pi$
$859$	27.3137	0.931932	0.465966	+	0.884803i	$0.345707\pi$
$860$	1.65685	0.0564983
$861$	0	0
$862$	19.7990	0.674356
$863$	−36.7696	−1.25165	−0.625825	−	0.779963i	$-0.715238\pi$
$863$	−36.7696	−1.25165	−0.625825	+	0.779963i	$0.715238\pi$
$864$	0	0
$865$	−21.3137	−0.724688
$866$	−15.1716	−0.515551
$867$	0	0
$868$	−5.65685	−0.192006
$869$	−10.8284	−0.367329
$870$	0	0
$871$	−14.6274	−0.495631
$872$	2.48528	0.0841622
$873$	0	0
$874$	8.00000	0.270604
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	30.9706	1.04580	0.522901	−	0.852394i	$-0.324850\pi$
$877$	30.9706	1.04580	0.522901	+	0.852394i	$0.324850\pi$
$878$	8.97056	0.302742
$879$	0	0
$880$	1.00000	0.0337100
$881$	25.3137	0.852841	0.426420	−	0.904525i	$-0.359774\pi$
$881$	25.3137	0.852841	0.426420	+	0.904525i	$0.359774\pi$
$882$	0	0
$883$	30.3431	1.02113	0.510564	−	0.859840i	$-0.329437\pi$
$883$	30.3431	1.02113	0.510564	+	0.859840i	$0.329437\pi$
$884$	7.31371	0.245987
$885$	0	0
$886$	−31.3137	−1.05200
$887$	−22.6274	−0.759754	−0.379877	−	0.925037i	$-0.624034\pi$
$887$	−22.6274	−0.759754	−0.379877	+	0.925037i	$0.624034\pi$
$888$	0	0
$889$	3.31371	0.111138
$890$	−0.343146	−0.0115023
$891$	0	0
$892$	28.2843	0.947027
$893$	22.6274	0.757198
$894$	0	0
$895$	−17.6569	−0.590204
$896$	1.00000	0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−49.9411	−1.66563
$900$	0	0
$901$	−23.7157	−0.790085
$902$	4.82843	0.160769
$903$	0	0
$904$	2.00000	0.0665190
$905$	−14.0000	−0.465376
$906$	0	0
$907$	6.34315	0.210621	0.105310	−	0.994439i	$-0.466416\pi$
$907$	6.34315	0.210621	0.105310	+	0.994439i	$0.466416\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	2.00000	0.0662994
$911$	10.3431	0.342684	0.171342	−	0.985212i	$-0.445190\pi$
$911$	10.3431	0.342684	0.171342	+	0.985212i	$0.445190\pi$
$912$	0	0
$913$	13.6569	0.451976
$914$	22.0000	0.727695
$915$	0	0
$916$	21.3137	0.704225
$917$	13.1716	0.434964
$918$	0	0
$919$	−52.7696	−1.74071	−0.870353	−	0.492428i	$-0.836110\pi$
$919$	−52.7696	−1.74071	−0.870353	+	0.492428i	$0.836110\pi$
$920$	2.82843	0.0932505
$921$	0	0
$922$	−0.343146	−0.0113009
$923$	−11.3137	−0.372395
$924$	0	0
$925$	8.82843	0.290277
$926$	−41.4558	−1.36232
$927$	0	0
$928$	8.82843	0.289807
$929$	5.02944	0.165010	0.0825052	−	0.996591i	$-0.473708\pi$
$929$	5.02944	0.165010	0.0825052	+	0.996591i	$0.473708\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	6.00000	0.196537
$933$	0	0
$934$	−27.7990	−0.909611
$935$	3.65685	0.119592
$936$	0	0
$937$	−20.6274	−0.673868	−0.336934	−	0.941528i	$-0.609390\pi$
$937$	−20.6274	−0.673868	−0.336934	+	0.941528i	$0.609390\pi$
$938$	−7.31371	−0.238801
$939$	0	0
$940$	8.00000	0.260931
$941$	9.31371	0.303618	0.151809	−	0.988410i	$-0.451490\pi$
$941$	9.31371	0.303618	0.151809	+	0.988410i	$0.451490\pi$
$942$	0	0
$943$	13.6569	0.444728
$944$	8.00000	0.260378
$945$	0	0
$946$	−1.65685	−0.0538690
$947$	43.5980	1.41674	0.708372	−	0.705839i	$-0.249430\pi$
$947$	43.5980	1.41674	0.708372	+	0.705839i	$0.249430\pi$
$948$	0	0
$949$	−7.31371	−0.237413
$950$	−2.82843	−0.0917663
$951$	0	0
$952$	3.65685	0.118519
$953$	−47.2548	−1.53073	−0.765367	−	0.643594i	$-0.777443\pi$
$953$	−47.2548	−1.53073	−0.765367	+	0.643594i	$0.777443\pi$
$954$	0	0
$955$	3.31371	0.107229
$956$	24.4853	0.791911
$957$	0	0
$958$	12.6863	0.409875
$959$	−17.3137	−0.559089
$960$	0	0
$961$	1.00000	0.0322581
$962$	−17.6569	−0.569280
$963$	0	0
$964$	−3.17157	−0.102149
$965$	−1.31371	−0.0422898
$966$	0	0
$967$	36.6863	1.17975	0.589876	−	0.807494i	$-0.299177\pi$
$967$	36.6863	1.17975	0.589876	+	0.807494i	$0.299177\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−4.82843	−0.155031
$971$	−25.9411	−0.832490	−0.416245	−	0.909252i	$-0.636654\pi$
$971$	−25.9411	−0.832490	−0.416245	+	0.909252i	$0.636654\pi$
$972$	0	0
$973$	−13.1716	−0.422261
$974$	−36.7696	−1.17817
$975$	0	0
$976$	13.3137	0.426161
$977$	−24.6274	−0.787901	−0.393950	−	0.919132i	$-0.628892\pi$
$977$	−24.6274	−0.787901	−0.393950	+	0.919132i	$0.628892\pi$
$978$	0	0
$979$	0.343146	0.0109670
$980$	1.00000	0.0319438
$981$	0	0
$982$	−31.3137	−0.999261
$983$	39.5980	1.26298	0.631490	−	0.775384i	$-0.282444\pi$
$983$	39.5980	1.26298	0.631490	+	0.775384i	$0.282444\pi$
$984$	0	0
$985$	23.6569	0.753770
$986$	32.2843	1.02814
$987$	0	0
$988$	5.65685	0.179969
$989$	−4.68629	−0.149015
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−5.65685	−0.179605
$993$	0	0
$994$	−5.65685	−0.179425
$995$	11.3137	0.358669
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	4.97056	0.157340
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.br.1.2		2
3.2	odd	2		770.2.a.i.1.1	✓	2
12.11	even	2		6160.2.a.ba.1.2		2
15.2	even	4		3850.2.c.u.1849.4		4
15.8	even	4		3850.2.c.u.1849.1		4
15.14	odd	2		3850.2.a.bi.1.2		2
21.20	even	2		5390.2.a.bt.1.2		2
33.32	even	2		8470.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.i.1.1	✓	2	3.2	odd	2
3850.2.a.bi.1.2		2	15.14	odd	2
3850.2.c.u.1849.1		4	15.8	even	4
3850.2.c.u.1849.4		4	15.2	even	4
5390.2.a.bt.1.2		2	21.20	even	2
6160.2.a.ba.1.2		2	12.11	even	2
6930.2.a.br.1.2		2	1.1	even	1	trivial
8470.2.a.bo.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.65685 q^{17} +2.82843 q^{19} +1.00000 q^{20} -1.00000 q^{22} -2.82843 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -8.82843 q^{29} +5.65685 q^{31} -1.00000 q^{32} -3.65685 q^{34} -1.00000 q^{35} +8.82843 q^{37} -2.82843 q^{38} -1.00000 q^{40} -4.82843 q^{41} +1.65685 q^{43} +1.00000 q^{44} +2.82843 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -6.48528 q^{53} +1.00000 q^{55} +1.00000 q^{56} +8.82843 q^{58} +8.00000 q^{59} +13.3137 q^{61} -5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} -7.31371 q^{67} +3.65685 q^{68} +1.00000 q^{70} -5.65685 q^{71} -3.65685 q^{73} -8.82843 q^{74} +2.82843 q^{76} -1.00000 q^{77} -10.8284 q^{79} +1.00000 q^{80} +4.82843 q^{82} +13.6569 q^{83} +3.65685 q^{85} -1.65685 q^{86} -1.00000 q^{88} +0.343146 q^{89} -2.00000 q^{91} -2.82843 q^{92} -8.00000 q^{94} +2.82843 q^{95} +4.82843 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{20} - 2 q^{22} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 12 q^{37} - 2 q^{40} - 4 q^{41} - 8 q^{43} + 2 q^{44} + 16 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} + 2 q^{55} + 2 q^{56} + 12 q^{58} + 16 q^{59} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} + 2 q^{70} + 4 q^{73} - 12 q^{74} - 2 q^{77} - 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} - 4 q^{85} + 8 q^{86} - 2 q^{88} + 12 q^{89} - 4 q^{91} - 16 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^20 - 2 * q^22 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 12 * q^29 - 2 * q^32 + 4 * q^34 - 2 * q^35 + 12 * q^37 - 2 * q^40 - 4 * q^41 - 8 * q^43 + 2 * q^44 + 16 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 2 * q^55 + 2 * q^56 + 12 * q^58 + 16 * q^59 + 4 * q^61 + 2 * q^64 + 4 * q^65 + 8 * q^67 - 4 * q^68 + 2 * q^70 + 4 * q^73 - 12 * q^74 - 2 * q^77 - 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 - 4 * q^85 + 8 * q^86 - 2 * q^88 + 12 * q^89 - 4 * q^91 - 16 * q^94 + 4 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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