LMFDB - Embedded newform 6930.2.a.ch.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +5.41855 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.26180 q^{17} -1.00000 q^{20} +1.00000 q^{22} +1.26180 q^{23} +1.00000 q^{25} -5.41855 q^{26} +1.00000 q^{28} +6.68035 q^{29} -10.8371 q^{31} -1.00000 q^{32} +3.26180 q^{34} -1.00000 q^{35} -10.0989 q^{37} +1.00000 q^{40} +4.15676 q^{41} -2.15676 q^{43} -1.00000 q^{44} -1.26180 q^{46} -10.8371 q^{47} +1.00000 q^{49} -1.00000 q^{50} +5.41855 q^{52} -0.156755 q^{53} +1.00000 q^{55} -1.00000 q^{56} -6.68035 q^{58} -6.15676 q^{59} -8.52359 q^{61} +10.8371 q^{62} +1.00000 q^{64} -5.41855 q^{65} +9.94214 q^{67} -3.26180 q^{68} +1.00000 q^{70} -8.68035 q^{71} +13.5174 q^{73} +10.0989 q^{74} -1.00000 q^{77} +13.3607 q^{79} -1.00000 q^{80} -4.15676 q^{82} +2.15676 q^{83} +3.26180 q^{85} +2.15676 q^{86} +1.00000 q^{88} +7.57531 q^{89} +5.41855 q^{91} +1.26180 q^{92} +10.8371 q^{94} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{23} + 3 q^{25} - 2 q^{26} + 3 q^{28} - 2 q^{29} - 4 q^{31} - 3 q^{32} + 2 q^{34} - 3 q^{35} + 6 q^{37} + 3 q^{40} + 6 q^{41} - 3 q^{44} + 4 q^{46} - 4 q^{47} + 3 q^{49} - 3 q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{55} - 3 q^{56} + 2 q^{58} - 12 q^{59} - 10 q^{61} + 4 q^{62} + 3 q^{64} - 2 q^{65} - 2 q^{68} + 3 q^{70} - 4 q^{71} - 10 q^{73} - 6 q^{74} - 3 q^{77} - 4 q^{79} - 3 q^{80} - 6 q^{82} + 2 q^{85} + 3 q^{88} + 2 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{94} - 6 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 - 3 * q^8 + 3 * q^10 - 3 * q^11 + 2 * q^13 - 3 * q^14 + 3 * q^16 - 2 * q^17 - 3 * q^20 + 3 * q^22 - 4 * q^23 + 3 * q^25 - 2 * q^26 + 3 * q^28 - 2 * q^29 - 4 * q^31 - 3 * q^32 + 2 * q^34 - 3 * q^35 + 6 * q^37 + 3 * q^40 + 6 * q^41 - 3 * q^44 + 4 * q^46 - 4 * q^47 + 3 * q^49 - 3 * q^50 + 2 * q^52 + 6 * q^53 + 3 * q^55 - 3 * q^56 + 2 * q^58 - 12 * q^59 - 10 * q^61 + 4 * q^62 + 3 * q^64 - 2 * q^65 - 2 * q^68 + 3 * q^70 - 4 * q^71 - 10 * q^73 - 6 * q^74 - 3 * q^77 - 4 * q^79 - 3 * q^80 - 6 * q^82 + 2 * q^85 + 3 * q^88 + 2 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^94 - 6 * q^97 - 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.41855	1.50284	0.751418	−	0.659827i	$-0.229370\pi$
$13$	5.41855	1.50284	0.751418	+	0.659827i	$0.229370\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.26180	−0.791102	−0.395551	−	0.918444i	$-0.629446\pi$
$17$	−3.26180	−0.791102	−0.395551	+	0.918444i	$0.629446\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	1.26180	0.263102	0.131551	−	0.991309i	$-0.458004\pi$
$23$	1.26180	0.263102	0.131551	+	0.991309i	$0.458004\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.41855	−1.06267
$27$	0	0
$28$	1.00000	0.188982
$29$	6.68035	1.24051	0.620255	−	0.784401i	$-0.287029\pi$
$29$	6.68035	1.24051	0.620255	+	0.784401i	$0.287029\pi$
$30$	0	0
$31$	−10.8371	−1.94640	−0.973200	−	0.229958i	$-0.926141\pi$
$31$	−10.8371	−1.94640	−0.973200	+	0.229958i	$0.926141\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.26180	0.559393
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−10.0989	−1.66025	−0.830124	−	0.557579i	$-0.811731\pi$
$37$	−10.0989	−1.66025	−0.830124	+	0.557579i	$0.811731\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	0.158114
$41$	4.15676	0.649176	0.324588	−	0.945855i	$-0.394774\pi$
$41$	4.15676	0.649176	0.324588	+	0.945855i	$0.394774\pi$
$42$	0	0
$43$	−2.15676	−0.328902	−0.164451	−	0.986385i	$-0.552585\pi$
$43$	−2.15676	−0.328902	−0.164451	+	0.986385i	$0.552585\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.26180	−0.186042
$47$	−10.8371	−1.58075	−0.790377	−	0.612621i	$-0.790115\pi$
$47$	−10.8371	−1.58075	−0.790377	+	0.612621i	$0.790115\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.41855	0.751418
$53$	−0.156755	−0.0215320	−0.0107660	−	0.999942i	$-0.503427\pi$
$53$	−0.156755	−0.0215320	−0.0107660	+	0.999942i	$0.503427\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.68035	−0.877172
$59$	−6.15676	−0.801541	−0.400771	−	0.916178i	$-0.631258\pi$
$59$	−6.15676	−0.801541	−0.400771	+	0.916178i	$0.631258\pi$
$60$	0	0
$61$	−8.52359	−1.09133	−0.545667	−	0.838002i	$-0.683724\pi$
$61$	−8.52359	−1.09133	−0.545667	+	0.838002i	$0.683724\pi$
$62$	10.8371	1.37631
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.41855	−0.672089
$66$	0	0
$67$	9.94214	1.21463	0.607313	−	0.794463i	$-0.292247\pi$
$67$	9.94214	1.21463	0.607313	+	0.794463i	$0.292247\pi$
$68$	−3.26180	−0.395551
$69$	0	0
$70$	1.00000	0.119523
$71$	−8.68035	−1.03017	−0.515084	−	0.857140i	$-0.672239\pi$
$71$	−8.68035	−1.03017	−0.515084	+	0.857140i	$0.672239\pi$
$72$	0	0
$73$	13.5174	1.58210	0.791049	−	0.611753i	$-0.209535\pi$
$73$	13.5174	1.58210	0.791049	+	0.611753i	$0.209535\pi$
$74$	10.0989	1.17397
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	13.3607	1.50320	0.751598	−	0.659622i	$-0.229284\pi$
$79$	13.3607	1.50320	0.751598	+	0.659622i	$0.229284\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−4.15676	−0.459037
$83$	2.15676	0.236735	0.118367	−	0.992970i	$-0.462234\pi$
$83$	2.15676	0.236735	0.118367	+	0.992970i	$0.462234\pi$
$84$	0	0
$85$	3.26180	0.353791
$86$	2.15676	0.232569
$87$	0	0
$88$	1.00000	0.106600
$89$	7.57531	0.802981	0.401490	−	0.915863i	$-0.368492\pi$
$89$	7.57531	0.802981	0.401490	+	0.915863i	$0.368492\pi$
$90$	0	0
$91$	5.41855	0.568018
$92$	1.26180	0.131551
$93$	0	0
$94$	10.8371	1.11776
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−12.6803	−1.24943	−0.624716	−	0.780852i	$-0.714785\pi$
$103$	−12.6803	−1.24943	−0.624716	+	0.780852i	$0.714785\pi$
$104$	−5.41855	−0.531333
$105$	0	0
$106$	0.156755	0.0152254
$107$	−10.8371	−1.04766	−0.523831	−	0.851822i	$-0.675498\pi$
$107$	−10.8371	−1.04766	−0.523831	+	0.851822i	$0.675498\pi$
$108$	0	0
$109$	−11.9421	−1.14385	−0.571925	−	0.820306i	$-0.693803\pi$
$109$	−11.9421	−1.14385	−0.571925	+	0.820306i	$0.693803\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	0.738205	0.0694445	0.0347222	−	0.999397i	$-0.488945\pi$
$113$	0.738205	0.0694445	0.0347222	+	0.999397i	$0.488945\pi$
$114$	0	0
$115$	−1.26180	−0.117663
$116$	6.68035	0.620255
$117$	0	0
$118$	6.15676	0.566775
$119$	−3.26180	−0.299008
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.52359	0.771690
$123$	0	0
$124$	−10.8371	−0.973200
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.52359	−0.223932	−0.111966	−	0.993712i	$-0.535715\pi$
$127$	−2.52359	−0.223932	−0.111966	+	0.993712i	$0.535715\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.41855	0.475238
$131$	1.84324	0.161045	0.0805225	−	0.996753i	$-0.474341\pi$
$131$	1.84324	0.161045	0.0805225	+	0.996753i	$0.474341\pi$
$132$	0	0
$133$	0	0
$134$	−9.94214	−0.858870
$135$	0	0
$136$	3.26180	0.279697
$137$	10.0989	0.862807	0.431403	−	0.902159i	$-0.358019\pi$
$137$	10.0989	0.862807	0.431403	+	0.902159i	$0.358019\pi$
$138$	0	0
$139$	−10.5236	−0.892599	−0.446300	−	0.894884i	$-0.647258\pi$
$139$	−10.5236	−0.892599	−0.446300	+	0.894884i	$0.647258\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	8.68035	0.728438
$143$	−5.41855	−0.453122
$144$	0	0
$145$	−6.68035	−0.554773
$146$	−13.5174	−1.11871
$147$	0	0
$148$	−10.0989	−0.830124
$149$	−3.84324	−0.314851	−0.157425	−	0.987531i	$-0.550319\pi$
$149$	−3.84324	−0.314851	−0.157425	+	0.987531i	$0.550319\pi$
$150$	0	0
$151$	5.36069	0.436247	0.218123	−	0.975921i	$-0.430006\pi$
$151$	5.36069	0.436247	0.218123	+	0.975921i	$0.430006\pi$
$152$	0	0
$153$	0	0
$154$	1.00000	0.0805823
$155$	10.8371	0.870457
$156$	0	0
$157$	4.15676	0.331745	0.165873	−	0.986147i	$-0.446956\pi$
$157$	4.15676	0.331745	0.165873	+	0.986147i	$0.446956\pi$
$158$	−13.3607	−1.06292
$159$	0	0
$160$	1.00000	0.0790569
$161$	1.26180	0.0994434
$162$	0	0
$163$	−7.41855	−0.581066	−0.290533	−	0.956865i	$-0.593833\pi$
$163$	−7.41855	−0.581066	−0.290533	+	0.956865i	$0.593833\pi$
$164$	4.15676	0.324588
$165$	0	0
$166$	−2.15676	−0.167397
$167$	7.78539	0.602451	0.301226	−	0.953553i	$-0.402604\pi$
$167$	7.78539	0.602451	0.301226	+	0.953553i	$0.402604\pi$
$168$	0	0
$169$	16.3607	1.25851
$170$	−3.26180	−0.250168
$171$	0	0
$172$	−2.15676	−0.164451
$173$	12.1568	0.924261	0.462131	−	0.886812i	$-0.347085\pi$
$173$	12.1568	0.924261	0.462131	+	0.886812i	$0.347085\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−7.57531	−0.567793
$179$	2.83710	0.212055	0.106027	−	0.994363i	$-0.466187\pi$
$179$	2.83710	0.212055	0.106027	+	0.994363i	$0.466187\pi$
$180$	0	0
$181$	−6.58145	−0.489195	−0.244598	−	0.969625i	$-0.578656\pi$
$181$	−6.58145	−0.489195	−0.244598	+	0.969625i	$0.578656\pi$
$182$	−5.41855	−0.401650
$183$	0	0
$184$	−1.26180	−0.0930208
$185$	10.0989	0.742486
$186$	0	0
$187$	3.26180	0.238526
$188$	−10.8371	−0.790377
$189$	0	0
$190$	0	0
$191$	7.51745	0.543943	0.271972	−	0.962305i	$-0.412324\pi$
$191$	7.51745	0.543943	0.271972	+	0.962305i	$0.412324\pi$
$192$	0	0
$193$	−12.1568	−0.875062	−0.437531	−	0.899203i	$-0.644147\pi$
$193$	−12.1568	−0.875062	−0.437531	+	0.899203i	$0.644147\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	14.3135	1.01980	0.509898	−	0.860235i	$-0.329683\pi$
$197$	14.3135	1.01980	0.509898	+	0.860235i	$0.329683\pi$
$198$	0	0
$199$	−15.1506	−1.07400	−0.536999	−	0.843583i	$-0.680442\pi$
$199$	−15.1506	−1.07400	−0.536999	+	0.843583i	$0.680442\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	2.00000	0.140720
$203$	6.68035	0.468868
$204$	0	0
$205$	−4.15676	−0.290320
$206$	12.6803	0.883482
$207$	0	0
$208$	5.41855	0.375709
$209$	0	0
$210$	0	0
$211$	−9.94214	−0.684445	−0.342223	−	0.939619i	$-0.611180\pi$
$211$	−9.94214	−0.684445	−0.342223	+	0.939619i	$0.611180\pi$
$212$	−0.156755	−0.0107660
$213$	0	0
$214$	10.8371	0.740809
$215$	2.15676	0.147090
$216$	0	0
$217$	−10.8371	−0.735670
$218$	11.9421	0.808824
$219$	0	0
$220$	1.00000	0.0674200
$221$	−17.6742	−1.18890
$222$	0	0
$223$	−2.15676	−0.144427	−0.0722135	−	0.997389i	$-0.523006\pi$
$223$	−2.15676	−0.144427	−0.0722135	+	0.997389i	$0.523006\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−0.738205	−0.0491047
$227$	−15.2039	−1.00912	−0.504560	−	0.863376i	$-0.668345\pi$
$227$	−15.2039	−1.00912	−0.504560	+	0.863376i	$0.668345\pi$
$228$	0	0
$229$	17.6163	1.16412	0.582060	−	0.813146i	$-0.302247\pi$
$229$	17.6163	1.16412	0.582060	+	0.813146i	$0.302247\pi$
$230$	1.26180	0.0832003
$231$	0	0
$232$	−6.68035	−0.438586
$233$	−6.31351	−0.413612	−0.206806	−	0.978382i	$-0.566307\pi$
$233$	−6.31351	−0.413612	−0.206806	+	0.978382i	$0.566307\pi$
$234$	0	0
$235$	10.8371	0.706935
$236$	−6.15676	−0.400771
$237$	0	0
$238$	3.26180	0.211431
$239$	−20.4124	−1.32037	−0.660184	−	0.751104i	$-0.729522\pi$
$239$	−20.4124	−1.32037	−0.660184	+	0.751104i	$0.729522\pi$
$240$	0	0
$241$	−20.8371	−1.34224	−0.671118	−	0.741351i	$-0.734186\pi$
$241$	−20.8371	−1.34224	−0.671118	+	0.741351i	$0.734186\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−8.52359	−0.545667
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	10.8371	0.688157
$249$	0	0
$250$	1.00000	0.0632456
$251$	14.1568	0.893566	0.446783	−	0.894642i	$-0.352570\pi$
$251$	14.1568	0.893566	0.446783	+	0.894642i	$0.352570\pi$
$252$	0	0
$253$	−1.26180	−0.0793284
$254$	2.52359	0.158344
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.52359	−0.531687	−0.265843	−	0.964016i	$-0.585650\pi$
$257$	−8.52359	−0.531687	−0.265843	+	0.964016i	$0.585650\pi$
$258$	0	0
$259$	−10.0989	−0.627515
$260$	−5.41855	−0.336044
$261$	0	0
$262$	−1.84324	−0.113876
$263$	−6.83710	−0.421594	−0.210797	−	0.977530i	$-0.567606\pi$
$263$	−6.83710	−0.421594	−0.210797	+	0.977530i	$0.567606\pi$
$264$	0	0
$265$	0.156755	0.00962941
$266$	0	0
$267$	0	0
$268$	9.94214	0.607313
$269$	−25.1506	−1.53346	−0.766730	−	0.641970i	$-0.778117\pi$
$269$	−25.1506	−1.53346	−0.766730	+	0.641970i	$0.778117\pi$
$270$	0	0
$271$	−20.3135	−1.23396	−0.616979	−	0.786980i	$-0.711644\pi$
$271$	−20.3135	−1.23396	−0.616979	+	0.786980i	$0.711644\pi$
$272$	−3.26180	−0.197775
$273$	0	0
$274$	−10.0989	−0.610097
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	10.5236	0.631163
$279$	0	0
$280$	1.00000	0.0597614
$281$	−2.89496	−0.172699	−0.0863494	−	0.996265i	$-0.527520\pi$
$281$	−2.89496	−0.172699	−0.0863494	+	0.996265i	$0.527520\pi$
$282$	0	0
$283$	−22.3090	−1.32613	−0.663065	−	0.748561i	$-0.730745\pi$
$283$	−22.3090	−1.32613	−0.663065	+	0.748561i	$0.730745\pi$
$284$	−8.68035	−0.515084
$285$	0	0
$286$	5.41855	0.320406
$287$	4.15676	0.245366
$288$	0	0
$289$	−6.36069	−0.374158
$290$	6.68035	0.392283
$291$	0	0
$292$	13.5174	0.791049
$293$	22.6803	1.32500	0.662500	−	0.749062i	$-0.269495\pi$
$293$	22.6803	1.32500	0.662500	+	0.749062i	$0.269495\pi$
$294$	0	0
$295$	6.15676	0.358460
$296$	10.0989	0.586986
$297$	0	0
$298$	3.84324	0.222633
$299$	6.83710	0.395400
$300$	0	0
$301$	−2.15676	−0.124313
$302$	−5.36069	−0.308473
$303$	0	0
$304$	0	0
$305$	8.52359	0.488059
$306$	0	0
$307$	16.0989	0.918813	0.459406	−	0.888226i	$-0.348062\pi$
$307$	16.0989	0.918813	0.459406	+	0.888226i	$0.348062\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−10.8371	−0.615506
$311$	−0.581449	−0.0329710	−0.0164855	−	0.999864i	$-0.505248\pi$
$311$	−0.581449	−0.0329710	−0.0164855	+	0.999864i	$0.505248\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−4.15676	−0.234579
$315$	0	0
$316$	13.3607	0.751598
$317$	−17.5174	−0.983878	−0.491939	−	0.870630i	$-0.663712\pi$
$317$	−17.5174	−0.983878	−0.491939	+	0.870630i	$0.663712\pi$
$318$	0	0
$319$	−6.68035	−0.374028
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−1.26180	−0.0703171
$323$	0	0
$324$	0	0
$325$	5.41855	0.300567
$326$	7.41855	0.410876
$327$	0	0
$328$	−4.15676	−0.229518
$329$	−10.8371	−0.597469
$330$	0	0
$331$	29.9877	1.64827	0.824137	−	0.566391i	$-0.191661\pi$
$331$	29.9877	1.64827	0.824137	+	0.566391i	$0.191661\pi$
$332$	2.15676	0.118367
$333$	0	0
$334$	−7.78539	−0.425997
$335$	−9.94214	−0.543197
$336$	0	0
$337$	5.20394	0.283476	0.141738	−	0.989904i	$-0.454731\pi$
$337$	5.20394	0.283476	0.141738	+	0.989904i	$0.454731\pi$
$338$	−16.3607	−0.889904
$339$	0	0
$340$	3.26180	0.176896
$341$	10.8371	0.586862
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.15676	0.116284
$345$	0	0
$346$	−12.1568	−0.653551
$347$	−5.16290	−0.277159	−0.138579	−	0.990351i	$-0.544254\pi$
$347$	−5.16290	−0.277159	−0.138579	+	0.990351i	$0.544254\pi$
$348$	0	0
$349$	−25.1506	−1.34628	−0.673141	−	0.739514i	$-0.735055\pi$
$349$	−25.1506	−1.34628	−0.673141	+	0.739514i	$0.735055\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	21.1506	1.12573	0.562867	−	0.826548i	$-0.309698\pi$
$353$	21.1506	1.12573	0.562867	+	0.826548i	$0.309698\pi$
$354$	0	0
$355$	8.68035	0.460705
$356$	7.57531	0.401490
$357$	0	0
$358$	−2.83710	−0.149945
$359$	8.09890	0.427443	0.213722	−	0.976895i	$-0.431441\pi$
$359$	8.09890	0.427443	0.213722	+	0.976895i	$0.431441\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	6.58145	0.345913
$363$	0	0
$364$	5.41855	0.284009
$365$	−13.5174	−0.707536
$366$	0	0
$367$	11.3197	0.590881	0.295441	−	0.955361i	$-0.404534\pi$
$367$	11.3197	0.590881	0.295441	+	0.955361i	$0.404534\pi$
$368$	1.26180	0.0657756
$369$	0	0
$370$	−10.0989	−0.525017
$371$	−0.156755	−0.00813834
$372$	0	0
$373$	−3.16290	−0.163769	−0.0818843	−	0.996642i	$-0.526094\pi$
$373$	−3.16290	−0.163769	−0.0818843	+	0.996642i	$0.526094\pi$
$374$	−3.26180	−0.168663
$375$	0	0
$376$	10.8371	0.558881
$377$	36.1978	1.86428
$378$	0	0
$379$	7.68649	0.394828	0.197414	−	0.980320i	$-0.436746\pi$
$379$	7.68649	0.394828	0.197414	+	0.980320i	$0.436746\pi$
$380$	0	0
$381$	0	0
$382$	−7.51745	−0.384626
$383$	18.8371	0.962531	0.481265	−	0.876575i	$-0.340177\pi$
$383$	18.8371	0.962531	0.481265	+	0.876575i	$0.340177\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	12.1568	0.618763
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−27.7275	−1.40584	−0.702921	−	0.711268i	$-0.748121\pi$
$389$	−27.7275	−1.40584	−0.702921	+	0.711268i	$0.748121\pi$
$390$	0	0
$391$	−4.11572	−0.208141
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−14.3135	−0.721104
$395$	−13.3607	−0.672249
$396$	0	0
$397$	−6.36683	−0.319542	−0.159771	−	0.987154i	$-0.551076\pi$
$397$	−6.36683	−0.319542	−0.159771	+	0.987154i	$0.551076\pi$
$398$	15.1506	0.759432
$399$	0	0
$400$	1.00000	0.0500000
$401$	−23.0472	−1.15092	−0.575461	−	0.817829i	$-0.695177\pi$
$401$	−23.0472	−1.15092	−0.575461	+	0.817829i	$0.695177\pi$
$402$	0	0
$403$	−58.7214	−2.92512
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	−6.68035	−0.331540
$407$	10.0989	0.500584
$408$	0	0
$409$	11.7899	0.582974	0.291487	−	0.956575i	$-0.405850\pi$
$409$	11.7899	0.582974	0.291487	+	0.956575i	$0.405850\pi$
$410$	4.15676	0.205288
$411$	0	0
$412$	−12.6803	−0.624716
$413$	−6.15676	−0.302954
$414$	0	0
$415$	−2.15676	−0.105871
$416$	−5.41855	−0.265666
$417$	0	0
$418$	0	0
$419$	−34.6681	−1.69365	−0.846823	−	0.531875i	$-0.821488\pi$
$419$	−34.6681	−1.69365	−0.846823	+	0.531875i	$0.821488\pi$
$420$	0	0
$421$	−23.6742	−1.15381	−0.576905	−	0.816811i	$-0.695740\pi$
$421$	−23.6742	−1.15381	−0.576905	+	0.816811i	$0.695740\pi$
$422$	9.94214	0.483976
$423$	0	0
$424$	0.156755	0.00761272
$425$	−3.26180	−0.158220
$426$	0	0
$427$	−8.52359	−0.412485
$428$	−10.8371	−0.523831
$429$	0	0
$430$	−2.15676	−0.104008
$431$	5.57531	0.268553	0.134277	−	0.990944i	$-0.457129\pi$
$431$	5.57531	0.268553	0.134277	+	0.990944i	$0.457129\pi$
$432$	0	0
$433$	19.6742	0.945482	0.472741	−	0.881201i	$-0.343265\pi$
$433$	19.6742	0.945482	0.472741	+	0.881201i	$0.343265\pi$
$434$	10.8371	0.520198
$435$	0	0
$436$	−11.9421	−0.571925
$437$	0	0
$438$	0	0
$439$	−17.1629	−0.819140	−0.409570	−	0.912279i	$-0.634321\pi$
$439$	−17.1629	−0.819140	−0.409570	+	0.912279i	$0.634321\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	17.6742	0.840676
$443$	−36.8781	−1.75213	−0.876067	−	0.482190i	$-0.839841\pi$
$443$	−36.8781	−1.75213	−0.876067	+	0.482190i	$0.839841\pi$
$444$	0	0
$445$	−7.57531	−0.359104
$446$	2.15676	0.102125
$447$	0	0
$448$	1.00000	0.0472456
$449$	29.0349	1.37024	0.685121	−	0.728430i	$-0.259749\pi$
$449$	29.0349	1.37024	0.685121	+	0.728430i	$0.259749\pi$
$450$	0	0
$451$	−4.15676	−0.195734
$452$	0.738205	0.0347222
$453$	0	0
$454$	15.2039	0.713556
$455$	−5.41855	−0.254026
$456$	0	0
$457$	11.8432	0.554004	0.277002	−	0.960869i	$-0.410659\pi$
$457$	11.8432	0.554004	0.277002	+	0.960869i	$0.410659\pi$
$458$	−17.6163	−0.823158
$459$	0	0
$460$	−1.26180	−0.0588315
$461$	19.0472	0.887116	0.443558	−	0.896246i	$-0.353716\pi$
$461$	19.0472	0.887116	0.443558	+	0.896246i	$0.353716\pi$
$462$	0	0
$463$	36.8781	1.71387	0.856936	−	0.515422i	$-0.172365\pi$
$463$	36.8781	1.71387	0.856936	+	0.515422i	$0.172365\pi$
$464$	6.68035	0.310127
$465$	0	0
$466$	6.31351	0.292468
$467$	15.8843	0.735037	0.367518	−	0.930016i	$-0.380207\pi$
$467$	15.8843	0.735037	0.367518	+	0.930016i	$0.380207\pi$
$468$	0	0
$469$	9.94214	0.459085
$470$	−10.8371	−0.499878
$471$	0	0
$472$	6.15676	0.283388
$473$	2.15676	0.0991677
$474$	0	0
$475$	0	0
$476$	−3.26180	−0.149504
$477$	0	0
$478$	20.4124	0.933642
$479$	−6.63931	−0.303358	−0.151679	−	0.988430i	$-0.548468\pi$
$479$	−6.63931	−0.303358	−0.151679	+	0.988430i	$0.548468\pi$
$480$	0	0
$481$	−54.7214	−2.49508
$482$	20.8371	0.949104
$483$	0	0
$484$	1.00000	0.0454545
$485$	2.00000	0.0908153
$486$	0	0
$487$	27.5174	1.24693	0.623467	−	0.781849i	$-0.285723\pi$
$487$	27.5174	1.24693	0.623467	+	0.781849i	$0.285723\pi$
$488$	8.52359	0.385845
$489$	0	0
$490$	1.00000	0.0451754
$491$	−17.6742	−0.797626	−0.398813	−	0.917032i	$-0.630578\pi$
$491$	−17.6742	−0.797626	−0.398813	+	0.917032i	$0.630578\pi$
$492$	0	0
$493$	−21.7899	−0.981369
$494$	0	0
$495$	0	0
$496$	−10.8371	−0.486600
$497$	−8.68035	−0.389367
$498$	0	0
$499$	−2.83710	−0.127006	−0.0635031	−	0.997982i	$-0.520227\pi$
$499$	−2.83710	−0.127006	−0.0635031	+	0.997982i	$0.520227\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−14.1568	−0.631847
$503$	10.7382	0.478793	0.239396	−	0.970922i	$-0.423050\pi$
$503$	10.7382	0.478793	0.239396	+	0.970922i	$0.423050\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	1.26180	0.0560936
$507$	0	0
$508$	−2.52359	−0.111966
$509$	−25.1506	−1.11478	−0.557391	−	0.830250i	$-0.688197\pi$
$509$	−25.1506	−1.11478	−0.557391	+	0.830250i	$0.688197\pi$
$510$	0	0
$511$	13.5174	0.597977
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	8.52359	0.375959
$515$	12.6803	0.558763
$516$	0	0
$517$	10.8371	0.476615
$518$	10.0989	0.443720
$519$	0	0
$520$	5.41855	0.237619
$521$	−42.4124	−1.85812	−0.929061	−	0.369927i	$-0.879383\pi$
$521$	−42.4124	−1.85812	−0.929061	+	0.369927i	$0.879383\pi$
$522$	0	0
$523$	2.62249	0.114673	0.0573367	−	0.998355i	$-0.481739\pi$
$523$	2.62249	0.114673	0.0573367	+	0.998355i	$0.481739\pi$
$524$	1.84324	0.0805225
$525$	0	0
$526$	6.83710	0.298112
$527$	35.3484	1.53980
$528$	0	0
$529$	−21.4079	−0.930777
$530$	−0.156755	−0.00680902
$531$	0	0
$532$	0	0
$533$	22.5236	0.975605
$534$	0	0
$535$	10.8371	0.468529
$536$	−9.94214	−0.429435
$537$	0	0
$538$	25.1506	1.08432
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−27.4063	−1.17829	−0.589144	−	0.808028i	$-0.700535\pi$
$541$	−27.4063	−1.17829	−0.589144	+	0.808028i	$0.700535\pi$
$542$	20.3135	0.872540
$543$	0	0
$544$	3.26180	0.139848
$545$	11.9421	0.511545
$546$	0	0
$547$	−22.0410	−0.942407	−0.471203	−	0.882025i	$-0.656180\pi$
$547$	−22.0410	−0.942407	−0.471203	+	0.882025i	$0.656180\pi$
$548$	10.0989	0.431403
$549$	0	0
$550$	1.00000	0.0426401
$551$	0	0
$552$	0	0
$553$	13.3607	0.568154
$554$	2.00000	0.0849719
$555$	0	0
$556$	−10.5236	−0.446300
$557$	−15.9877	−0.677421	−0.338711	−	0.940891i	$-0.609991\pi$
$557$	−15.9877	−0.677421	−0.338711	+	0.940891i	$0.609991\pi$
$558$	0	0
$559$	−11.6865	−0.494286
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	2.89496	0.122117
$563$	18.3545	0.773552	0.386776	−	0.922174i	$-0.373589\pi$
$563$	18.3545	0.773552	0.386776	+	0.922174i	$0.373589\pi$
$564$	0	0
$565$	−0.738205	−0.0310565
$566$	22.3090	0.937716
$567$	0	0
$568$	8.68035	0.364219
$569$	−21.4186	−0.897912	−0.448956	−	0.893554i	$-0.648204\pi$
$569$	−21.4186	−0.897912	−0.448956	+	0.893554i	$0.648204\pi$
$570$	0	0
$571$	−24.0456	−1.00628	−0.503138	−	0.864206i	$-0.667821\pi$
$571$	−24.0456	−1.00628	−0.503138	+	0.864206i	$0.667821\pi$
$572$	−5.41855	−0.226561
$573$	0	0
$574$	−4.15676	−0.173500
$575$	1.26180	0.0526205
$576$	0	0
$577$	24.7214	1.02916	0.514582	−	0.857441i	$-0.327947\pi$
$577$	24.7214	1.02916	0.514582	+	0.857441i	$0.327947\pi$
$578$	6.36069	0.264570
$579$	0	0
$580$	−6.68035	−0.277386
$581$	2.15676	0.0894773
$582$	0	0
$583$	0.156755	0.00649215
$584$	−13.5174	−0.559356
$585$	0	0
$586$	−22.6803	−0.936916
$587$	−31.8843	−1.31601	−0.658003	−	0.753016i	$-0.728598\pi$
$587$	−31.8843	−1.31601	−0.658003	+	0.753016i	$0.728598\pi$
$588$	0	0
$589$	0	0
$590$	−6.15676	−0.253470
$591$	0	0
$592$	−10.0989	−0.415062
$593$	31.2618	1.28377	0.641884	−	0.766802i	$-0.278153\pi$
$593$	31.2618	1.28377	0.641884	+	0.766802i	$0.278153\pi$
$594$	0	0
$595$	3.26180	0.133721
$596$	−3.84324	−0.157425
$597$	0	0
$598$	−6.83710	−0.279590
$599$	−33.8432	−1.38280	−0.691399	−	0.722473i	$-0.743005\pi$
$599$	−33.8432	−1.38280	−0.691399	+	0.722473i	$0.743005\pi$
$600$	0	0
$601$	−15.1629	−0.618508	−0.309254	−	0.950980i	$-0.600079\pi$
$601$	−15.1629	−0.618508	−0.309254	+	0.950980i	$0.600079\pi$
$602$	2.15676	0.0879028
$603$	0	0
$604$	5.36069	0.218123
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−16.6270	−0.674870	−0.337435	−	0.941349i	$-0.609559\pi$
$607$	−16.6270	−0.674870	−0.337435	+	0.941349i	$0.609559\pi$
$608$	0	0
$609$	0	0
$610$	−8.52359	−0.345110
$611$	−58.7214	−2.37561
$612$	0	0
$613$	13.0349	0.526474	0.263237	−	0.964731i	$-0.415210\pi$
$613$	13.0349	0.526474	0.263237	+	0.964731i	$0.415210\pi$
$614$	−16.0989	−0.649699
$615$	0	0
$616$	1.00000	0.0402911
$617$	37.7854	1.52118	0.760591	−	0.649231i	$-0.224909\pi$
$617$	37.7854	1.52118	0.760591	+	0.649231i	$0.224909\pi$
$618$	0	0
$619$	44.7792	1.79983	0.899915	−	0.436066i	$-0.143628\pi$
$619$	44.7792	1.79983	0.899915	+	0.436066i	$0.143628\pi$
$620$	10.8371	0.435228
$621$	0	0
$622$	0.581449	0.0233140
$623$	7.57531	0.303498
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	4.15676	0.165873
$629$	32.9405	1.31343
$630$	0	0
$631$	9.16290	0.364769	0.182385	−	0.983227i	$-0.441618\pi$
$631$	9.16290	0.364769	0.182385	+	0.983227i	$0.441618\pi$
$632$	−13.3607	−0.531460
$633$	0	0
$634$	17.5174	0.695707
$635$	2.52359	0.100146
$636$	0	0
$637$	5.41855	0.214691
$638$	6.68035	0.264477
$639$	0	0
$640$	1.00000	0.0395285
$641$	−35.3607	−1.39666	−0.698332	−	0.715774i	$-0.746074\pi$
$641$	−35.3607	−1.39666	−0.698332	+	0.715774i	$0.746074\pi$
$642$	0	0
$643$	−47.3484	−1.86724	−0.933619	−	0.358266i	$-0.883368\pi$
$643$	−47.3484	−1.86724	−0.933619	+	0.358266i	$0.883368\pi$
$644$	1.26180	0.0497217
$645$	0	0
$646$	0	0
$647$	−28.8248	−1.13322	−0.566610	−	0.823986i	$-0.691745\pi$
$647$	−28.8248	−1.13322	−0.566610	+	0.823986i	$0.691745\pi$
$648$	0	0
$649$	6.15676	0.241674
$650$	−5.41855	−0.212533
$651$	0	0
$652$	−7.41855	−0.290533
$653$	16.0410	0.627734	0.313867	−	0.949467i	$-0.398375\pi$
$653$	16.0410	0.627734	0.313867	+	0.949467i	$0.398375\pi$
$654$	0	0
$655$	−1.84324	−0.0720215
$656$	4.15676	0.162294
$657$	0	0
$658$	10.8371	0.422474
$659$	13.3607	0.520459	0.260229	−	0.965547i	$-0.416202\pi$
$659$	13.3607	0.520459	0.260229	+	0.965547i	$0.416202\pi$
$660$	0	0
$661$	32.4534	1.26229	0.631146	−	0.775664i	$-0.282585\pi$
$661$	32.4534	1.26229	0.631146	+	0.775664i	$0.282585\pi$
$662$	−29.9877	−1.16551
$663$	0	0
$664$	−2.15676	−0.0836983
$665$	0	0
$666$	0	0
$667$	8.42923	0.326381
$668$	7.78539	0.301226
$669$	0	0
$670$	9.94214	0.384098
$671$	8.52359	0.329050
$672$	0	0
$673$	0.156755	0.00604248	0.00302124	−	0.999995i	$-0.499038\pi$
$673$	0.156755	0.00604248	0.00302124	+	0.999995i	$0.499038\pi$
$674$	−5.20394	−0.200448
$675$	0	0
$676$	16.3607	0.629257
$677$	−3.10957	−0.119511	−0.0597553	−	0.998213i	$-0.519032\pi$
$677$	−3.10957	−0.119511	−0.0597553	+	0.998213i	$0.519032\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	−3.26180	−0.125084
$681$	0	0
$682$	−10.8371	−0.414974
$683$	14.4703	0.553689	0.276845	−	0.960915i	$-0.410711\pi$
$683$	14.4703	0.553689	0.276845	+	0.960915i	$0.410711\pi$
$684$	0	0
$685$	−10.0989	−0.385859
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−2.15676	−0.0822255
$689$	−0.849388	−0.0323591
$690$	0	0
$691$	−29.9421	−1.13905	−0.569526	−	0.821973i	$-0.692873\pi$
$691$	−29.9421	−1.13905	−0.569526	+	0.821973i	$0.692873\pi$
$692$	12.1568	0.462131
$693$	0	0
$694$	5.16290	0.195981
$695$	10.5236	0.399183
$696$	0	0
$697$	−13.5585	−0.513564
$698$	25.1506	0.951965
$699$	0	0
$700$	1.00000	0.0377964
$701$	39.3074	1.48462	0.742309	−	0.670057i	$-0.233730\pi$
$701$	39.3074	1.48462	0.742309	+	0.670057i	$0.233730\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−21.1506	−0.796014
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	8.95282	0.336230	0.168115	−	0.985767i	$-0.446232\pi$
$709$	8.95282	0.336230	0.168115	+	0.985767i	$0.446232\pi$
$710$	−8.68035	−0.325768
$711$	0	0
$712$	−7.57531	−0.283897
$713$	−13.6742	−0.512103
$714$	0	0
$715$	5.41855	0.202642
$716$	2.83710	0.106027
$717$	0	0
$718$	−8.09890	−0.302248
$719$	−19.1050	−0.712498	−0.356249	−	0.934391i	$-0.615944\pi$
$719$	−19.1050	−0.712498	−0.356249	+	0.934391i	$0.615944\pi$
$720$	0	0
$721$	−12.6803	−0.472241
$722$	19.0000	0.707107
$723$	0	0
$724$	−6.58145	−0.244598
$725$	6.68035	0.248102
$726$	0	0
$727$	−11.5174	−0.427158	−0.213579	−	0.976926i	$-0.568512\pi$
$727$	−11.5174	−0.427158	−0.213579	+	0.976926i	$0.568512\pi$
$728$	−5.41855	−0.200825
$729$	0	0
$730$	13.5174	0.500303
$731$	7.03489	0.260195
$732$	0	0
$733$	−34.5814	−1.27729	−0.638647	−	0.769499i	$-0.720506\pi$
$733$	−34.5814	−1.27729	−0.638647	+	0.769499i	$0.720506\pi$
$734$	−11.3197	−0.417816
$735$	0	0
$736$	−1.26180	−0.0465104
$737$	−9.94214	−0.366223
$738$	0	0
$739$	35.3028	1.29864	0.649318	−	0.760517i	$-0.275054\pi$
$739$	35.3028	1.29864	0.649318	+	0.760517i	$0.275054\pi$
$740$	10.0989	0.371243
$741$	0	0
$742$	0.156755	0.00575468
$743$	−50.7214	−1.86079	−0.930394	−	0.366562i	$-0.880535\pi$
$743$	−50.7214	−1.86079	−0.930394	+	0.366562i	$0.880535\pi$
$744$	0	0
$745$	3.84324	0.140806
$746$	3.16290	0.115802
$747$	0	0
$748$	3.26180	0.119263
$749$	−10.8371	−0.395979
$750$	0	0
$751$	6.10343	0.222717	0.111359	−	0.993780i	$-0.464480\pi$
$751$	6.10343	0.222717	0.111359	+	0.993780i	$0.464480\pi$
$752$	−10.8371	−0.395188
$753$	0	0
$754$	−36.1978	−1.31825
$755$	−5.36069	−0.195096
$756$	0	0
$757$	4.30898	0.156612	0.0783062	−	0.996929i	$-0.475049\pi$
$757$	4.30898	0.156612	0.0783062	+	0.996929i	$0.475049\pi$
$758$	−7.68649	−0.279186
$759$	0	0
$760$	0	0
$761$	−47.1917	−1.71070	−0.855348	−	0.518054i	$-0.826657\pi$
$761$	−47.1917	−1.71070	−0.855348	+	0.518054i	$0.826657\pi$
$762$	0	0
$763$	−11.9421	−0.432335
$764$	7.51745	0.271972
$765$	0	0
$766$	−18.8371	−0.680612
$767$	−33.3607	−1.20458
$768$	0	0
$769$	−37.4641	−1.35099	−0.675495	−	0.737364i	$-0.736070\pi$
$769$	−37.4641	−1.35099	−0.675495	+	0.737364i	$0.736070\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	−12.1568	−0.437531
$773$	46.0821	1.65746	0.828729	−	0.559651i	$-0.189065\pi$
$773$	46.0821	1.65746	0.828729	+	0.559651i	$0.189065\pi$
$774$	0	0
$775$	−10.8371	−0.389280
$776$	2.00000	0.0717958
$777$	0	0
$778$	27.7275	0.994080
$779$	0	0
$780$	0	0
$781$	8.68035	0.310607
$782$	4.11572	0.147178
$783$	0	0
$784$	1.00000	0.0357143
$785$	−4.15676	−0.148361
$786$	0	0
$787$	−16.2967	−0.580914	−0.290457	−	0.956888i	$-0.593807\pi$
$787$	−16.2967	−0.580914	−0.290457	+	0.956888i	$0.593807\pi$
$788$	14.3135	0.509898
$789$	0	0
$790$	13.3607	0.475352
$791$	0.738205	0.0262475
$792$	0	0
$793$	−46.1855	−1.64010
$794$	6.36683	0.225951
$795$	0	0
$796$	−15.1506	−0.536999
$797$	−30.3956	−1.07667	−0.538333	−	0.842732i	$-0.680946\pi$
$797$	−30.3956	−1.07667	−0.538333	+	0.842732i	$0.680946\pi$
$798$	0	0
$799$	35.3484	1.25054
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	23.0472	0.813824
$803$	−13.5174	−0.477020
$804$	0	0
$805$	−1.26180	−0.0444724
$806$	58.7214	2.06837
$807$	0	0
$808$	2.00000	0.0703598
$809$	−54.7792	−1.92594	−0.962968	−	0.269616i	$-0.913103\pi$
$809$	−54.7792	−1.92594	−0.962968	+	0.269616i	$0.913103\pi$
$810$	0	0
$811$	−16.7337	−0.587599	−0.293799	−	0.955867i	$-0.594920\pi$
$811$	−16.7337	−0.587599	−0.293799	+	0.955867i	$0.594920\pi$
$812$	6.68035	0.234434
$813$	0	0
$814$	−10.0989	−0.353966
$815$	7.41855	0.259860
$816$	0	0
$817$	0	0
$818$	−11.7899	−0.412225
$819$	0	0
$820$	−4.15676	−0.145160
$821$	13.7152	0.478665	0.239333	−	0.970938i	$-0.423071\pi$
$821$	13.7152	0.478665	0.239333	+	0.970938i	$0.423071\pi$
$822$	0	0
$823$	−40.3668	−1.40710	−0.703550	−	0.710646i	$-0.748403\pi$
$823$	−40.3668	−1.40710	−0.703550	+	0.710646i	$0.748403\pi$
$824$	12.6803	0.441741
$825$	0	0
$826$	6.15676	0.214221
$827$	22.5236	0.783222	0.391611	−	0.920131i	$-0.371918\pi$
$827$	22.5236	0.783222	0.391611	+	0.920131i	$0.371918\pi$
$828$	0	0
$829$	41.4186	1.43853	0.719263	−	0.694738i	$-0.244480\pi$
$829$	41.4186	1.43853	0.719263	+	0.694738i	$0.244480\pi$
$830$	2.15676	0.0748621
$831$	0	0
$832$	5.41855	0.187854
$833$	−3.26180	−0.113015
$834$	0	0
$835$	−7.78539	−0.269424
$836$	0	0
$837$	0	0
$838$	34.6681	1.19759
$839$	0.152221	0.00525524	0.00262762	−	0.999997i	$-0.499164\pi$
$839$	0.152221	0.00525524	0.00262762	+	0.999997i	$0.499164\pi$
$840$	0	0
$841$	15.6270	0.538863
$842$	23.6742	0.815867
$843$	0	0
$844$	−9.94214	−0.342223
$845$	−16.3607	−0.562825
$846$	0	0
$847$	1.00000	0.0343604
$848$	−0.156755	−0.00538301
$849$	0	0
$850$	3.26180	0.111879
$851$	−12.7427	−0.436815
$852$	0	0
$853$	−54.6635	−1.87164	−0.935822	−	0.352474i	$-0.885341\pi$
$853$	−54.6635	−1.87164	−0.935822	+	0.352474i	$0.885341\pi$
$854$	8.52359	0.291671
$855$	0	0
$856$	10.8371	0.370405
$857$	7.26180	0.248058	0.124029	−	0.992279i	$-0.460418\pi$
$857$	7.26180	0.248058	0.124029	+	0.992279i	$0.460418\pi$
$858$	0	0
$859$	−11.4186	−0.389596	−0.194798	−	0.980843i	$-0.562405\pi$
$859$	−11.4186	−0.389596	−0.194798	+	0.980843i	$0.562405\pi$
$860$	2.15676	0.0735448
$861$	0	0
$862$	−5.57531	−0.189896
$863$	−32.0989	−1.09266	−0.546330	−	0.837570i	$-0.683975\pi$
$863$	−32.0989	−1.09266	−0.546330	+	0.837570i	$0.683975\pi$
$864$	0	0
$865$	−12.1568	−0.413342
$866$	−19.6742	−0.668557
$867$	0	0
$868$	−10.8371	−0.367835
$869$	−13.3607	−0.453230
$870$	0	0
$871$	53.8720	1.82538
$872$	11.9421	0.404412
$873$	0	0
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	45.7686	1.54549	0.772747	−	0.634714i	$-0.218882\pi$
$877$	45.7686	1.54549	0.772747	+	0.634714i	$0.218882\pi$
$878$	17.1629	0.579220
$879$	0	0
$880$	1.00000	0.0337100
$881$	27.8888	0.939598	0.469799	−	0.882773i	$-0.344326\pi$
$881$	27.8888	0.939598	0.469799	+	0.882773i	$0.344326\pi$
$882$	0	0
$883$	−6.45345	−0.217176	−0.108588	−	0.994087i	$-0.534633\pi$
$883$	−6.45345	−0.217176	−0.108588	+	0.994087i	$0.534633\pi$
$884$	−17.6742	−0.594448
$885$	0	0
$886$	36.8781	1.23895
$887$	15.2495	0.512028	0.256014	−	0.966673i	$-0.417591\pi$
$887$	15.2495	0.512028	0.256014	+	0.966673i	$0.417591\pi$
$888$	0	0
$889$	−2.52359	−0.0846385
$890$	7.57531	0.253925
$891$	0	0
$892$	−2.15676	−0.0722135
$893$	0	0
$894$	0	0
$895$	−2.83710	−0.0948338
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−29.0349	−0.968907
$899$	−72.3956	−2.41453
$900$	0	0
$901$	0.511304	0.0170340
$902$	4.15676	0.138405
$903$	0	0
$904$	−0.738205	−0.0245523
$905$	6.58145	0.218775
$906$	0	0
$907$	32.7792	1.08842	0.544208	−	0.838950i	$-0.316830\pi$
$907$	32.7792	1.08842	0.544208	+	0.838950i	$0.316830\pi$
$908$	−15.2039	−0.504560
$909$	0	0
$910$	5.41855	0.179623
$911$	−3.00614	−0.0995980	−0.0497990	−	0.998759i	$-0.515858\pi$
$911$	−3.00614	−0.0995980	−0.0497990	+	0.998759i	$0.515858\pi$
$912$	0	0
$913$	−2.15676	−0.0713782
$914$	−11.8432	−0.391740
$915$	0	0
$916$	17.6163	0.582060
$917$	1.84324	0.0608693
$918$	0	0
$919$	36.1978	1.19406	0.597028	−	0.802221i	$-0.296348\pi$
$919$	36.1978	1.19406	0.597028	+	0.802221i	$0.296348\pi$
$920$	1.26180	0.0416002
$921$	0	0
$922$	−19.0472	−0.627285
$923$	−47.0349	−1.54817
$924$	0	0
$925$	−10.0989	−0.332050
$926$	−36.8781	−1.21189
$927$	0	0
$928$	−6.68035	−0.219293
$929$	−15.2618	−0.500723	−0.250362	−	0.968152i	$-0.580550\pi$
$929$	−15.2618	−0.500723	−0.250362	+	0.968152i	$0.580550\pi$
$930$	0	0
$931$	0	0
$932$	−6.31351	−0.206806
$933$	0	0
$934$	−15.8843	−0.519750
$935$	−3.26180	−0.106672
$936$	0	0
$937$	−18.6803	−0.610260	−0.305130	−	0.952311i	$-0.598700\pi$
$937$	−18.6803	−0.610260	−0.305130	+	0.952311i	$0.598700\pi$
$938$	−9.94214	−0.324622
$939$	0	0
$940$	10.8371	0.353467
$941$	−23.0472	−0.751317	−0.375658	−	0.926758i	$-0.622583\pi$
$941$	−23.0472	−0.751317	−0.375658	+	0.926758i	$0.622583\pi$
$942$	0	0
$943$	5.24497	0.170800
$944$	−6.15676	−0.200385
$945$	0	0
$946$	−2.15676	−0.0701222
$947$	13.3074	0.432431	0.216216	−	0.976346i	$-0.430629\pi$
$947$	13.3074	0.432431	0.216216	+	0.976346i	$0.430629\pi$
$948$	0	0
$949$	73.2450	2.37763
$950$	0	0
$951$	0	0
$952$	3.26180	0.105715
$953$	35.4764	1.14919	0.574597	−	0.818437i	$-0.305159\pi$
$953$	35.4764	1.14919	0.574597	+	0.818437i	$0.305159\pi$
$954$	0	0
$955$	−7.51745	−0.243259
$956$	−20.4124	−0.660184
$957$	0	0
$958$	6.63931	0.214506
$959$	10.0989	0.326110
$960$	0	0
$961$	86.4428	2.78848
$962$	54.7214	1.76429
$963$	0	0
$964$	−20.8371	−0.671118
$965$	12.1568	0.391340
$966$	0	0
$967$	−23.5708	−0.757985	−0.378992	−	0.925400i	$-0.623729\pi$
$967$	−23.5708	−0.757985	−0.378992	+	0.925400i	$0.623729\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−29.7275	−0.954002	−0.477001	−	0.878903i	$-0.658276\pi$
$971$	−29.7275	−0.954002	−0.477001	+	0.878903i	$0.658276\pi$
$972$	0	0
$973$	−10.5236	−0.337371
$974$	−27.5174	−0.881716
$975$	0	0
$976$	−8.52359	−0.272833
$977$	50.9237	1.62919	0.814597	−	0.580027i	$-0.196958\pi$
$977$	50.9237	1.62919	0.814597	+	0.580027i	$0.196958\pi$
$978$	0	0
$979$	−7.57531	−0.242108
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	17.6742	0.564006
$983$	−47.8843	−1.52727	−0.763636	−	0.645647i	$-0.776588\pi$
$983$	−47.8843	−1.52727	−0.763636	+	0.645647i	$0.776588\pi$
$984$	0	0
$985$	−14.3135	−0.456066
$986$	21.7899	0.693932
$987$	0	0
$988$	0	0
$989$	−2.72138	−0.0865350
$990$	0	0
$991$	−56.5934	−1.79775	−0.898874	−	0.438207i	$-0.855614\pi$
$991$	−56.5934	−1.79775	−0.898874	+	0.438207i	$0.855614\pi$
$992$	10.8371	0.344078
$993$	0	0
$994$	8.68035	0.275324
$995$	15.1506	0.480307
$996$	0	0
$997$	−25.8141	−0.817542	−0.408771	−	0.912637i	$-0.634043\pi$
$997$	−25.8141	−0.817542	−0.408771	+	0.912637i	$0.634043\pi$
$998$	2.83710	0.0898069
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.ch.1.3		3
3.2	odd	2		2310.2.a.bd.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.bd.1.3	✓	3	3.2	odd	2
6930.2.a.ch.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +5.41855 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.26180 q^{17} -1.00000 q^{20} +1.00000 q^{22} +1.26180 q^{23} +1.00000 q^{25} -5.41855 q^{26} +1.00000 q^{28} +6.68035 q^{29} -10.8371 q^{31} -1.00000 q^{32} +3.26180 q^{34} -1.00000 q^{35} -10.0989 q^{37} +1.00000 q^{40} +4.15676 q^{41} -2.15676 q^{43} -1.00000 q^{44} -1.26180 q^{46} -10.8371 q^{47} +1.00000 q^{49} -1.00000 q^{50} +5.41855 q^{52} -0.156755 q^{53} +1.00000 q^{55} -1.00000 q^{56} -6.68035 q^{58} -6.15676 q^{59} -8.52359 q^{61} +10.8371 q^{62} +1.00000 q^{64} -5.41855 q^{65} +9.94214 q^{67} -3.26180 q^{68} +1.00000 q^{70} -8.68035 q^{71} +13.5174 q^{73} +10.0989 q^{74} -1.00000 q^{77} +13.3607 q^{79} -1.00000 q^{80} -4.15676 q^{82} +2.15676 q^{83} +3.26180 q^{85} +2.15676 q^{86} +1.00000 q^{88} +7.57531 q^{89} +5.41855 q^{91} +1.26180 q^{92} +10.8371 q^{94} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{10} - 3 q^{11} + 2 q^{13} - 3 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{20} + 3 q^{22} - 4 q^{23} + 3 q^{25} - 2 q^{26} + 3 q^{28} - 2 q^{29} - 4 q^{31} - 3 q^{32} + 2 q^{34} - 3 q^{35} + 6 q^{37} + 3 q^{40} + 6 q^{41} - 3 q^{44} + 4 q^{46} - 4 q^{47} + 3 q^{49} - 3 q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{55} - 3 q^{56} + 2 q^{58} - 12 q^{59} - 10 q^{61} + 4 q^{62} + 3 q^{64} - 2 q^{65} - 2 q^{68} + 3 q^{70} - 4 q^{71} - 10 q^{73} - 6 q^{74} - 3 q^{77} - 4 q^{79} - 3 q^{80} - 6 q^{82} + 2 q^{85} + 3 q^{88} + 2 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{94} - 6 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 - 3 * q^8 + 3 * q^10 - 3 * q^11 + 2 * q^13 - 3 * q^14 + 3 * q^16 - 2 * q^17 - 3 * q^20 + 3 * q^22 - 4 * q^23 + 3 * q^25 - 2 * q^26 + 3 * q^28 - 2 * q^29 - 4 * q^31 - 3 * q^32 + 2 * q^34 - 3 * q^35 + 6 * q^37 + 3 * q^40 + 6 * q^41 - 3 * q^44 + 4 * q^46 - 4 * q^47 + 3 * q^49 - 3 * q^50 + 2 * q^52 + 6 * q^53 + 3 * q^55 - 3 * q^56 + 2 * q^58 - 12 * q^59 - 10 * q^61 + 4 * q^62 + 3 * q^64 - 2 * q^65 - 2 * q^68 + 3 * q^70 - 4 * q^71 - 10 * q^73 - 6 * q^74 - 3 * q^77 - 4 * q^79 - 3 * q^80 - 6 * q^82 + 2 * q^85 + 3 * q^88 + 2 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^94 - 6 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more