LMFDB - Embedded newform 8330.2.a.cv.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8330.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,10,-2,10,-10,-2,0,10,12,-10,-2,-2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.5153848837$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 38x^{7} + 100x^{6} - 194x^{5} - 151x^{4} + 282x^{3} + 85x^{2} - 108x - 28 $ x^10 - 2x^9 - 19x^8 + 38x^7 + 100x^6 - 194x^5 - 151x^4 + 282x^3 + 85x^2 - 108*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.255150$ of defining polynomial
Character	$\chi$	$=$	8330.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.255150 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.255150 q^{6} +1.00000 q^{8} -2.93490 q^{9} -1.00000 q^{10} +2.61100 q^{11} +0.255150 q^{12} -5.03829 q^{13} -0.255150 q^{15} +1.00000 q^{16} -1.00000 q^{17} -2.93490 q^{18} +2.49174 q^{19} -1.00000 q^{20} +2.61100 q^{22} -0.0298196 q^{23} +0.255150 q^{24} +1.00000 q^{25} -5.03829 q^{26} -1.51429 q^{27} +4.55631 q^{29} -0.255150 q^{30} -3.07081 q^{31} +1.00000 q^{32} +0.666196 q^{33} -1.00000 q^{34} -2.93490 q^{36} +4.53407 q^{37} +2.49174 q^{38} -1.28552 q^{39} -1.00000 q^{40} -4.29860 q^{41} -5.09136 q^{43} +2.61100 q^{44} +2.93490 q^{45} -0.0298196 q^{46} +6.22417 q^{47} +0.255150 q^{48} +1.00000 q^{50} -0.255150 q^{51} -5.03829 q^{52} -1.36002 q^{53} -1.51429 q^{54} -2.61100 q^{55} +0.635767 q^{57} +4.55631 q^{58} +14.0184 q^{59} -0.255150 q^{60} +4.98656 q^{61} -3.07081 q^{62} +1.00000 q^{64} +5.03829 q^{65} +0.666196 q^{66} +7.02027 q^{67} -1.00000 q^{68} -0.00760847 q^{69} +14.2603 q^{71} -2.93490 q^{72} -12.1613 q^{73} +4.53407 q^{74} +0.255150 q^{75} +2.49174 q^{76} -1.28552 q^{78} +8.95066 q^{79} -1.00000 q^{80} +8.41833 q^{81} -4.29860 q^{82} +10.4460 q^{83} +1.00000 q^{85} -5.09136 q^{86} +1.16254 q^{87} +2.61100 q^{88} +1.30852 q^{89} +2.93490 q^{90} -0.0298196 q^{92} -0.783517 q^{93} +6.22417 q^{94} -2.49174 q^{95} +0.255150 q^{96} -3.38746 q^{97} -7.66302 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} - 2 q^{3} + 10 q^{4} - 10 q^{5} - 2 q^{6} + 10 q^{8} + 12 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 10 q^{16} - 10 q^{17} + 12 q^{18} - 14 q^{19} - 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q + 10 * q^2 - 2 * q^3 + 10 * q^4 - 10 * q^5 - 2 * q^6 + 10 * q^8 + 12 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 10 * q^16 - 10 * q^17 + 12 * q^18 - 14 * q^19 - 10 * q^20 - 2 * q^22 + 10 * q^23 - 2 * q^24 + 10 * q^25 - 4 * q^26 + 4 * q^27 + 10 * q^29 + 2 * q^30 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 10 * q^34 + 12 * q^36 - 14 * q^38 + 48 * q^39 - 10 * q^40 + 22 * q^41 + 14 * q^43 - 2 * q^44 - 12 * q^45 + 10 * q^46 - 18 * q^47 - 2 * q^48 + 10 * q^50 + 2 * q^51 - 4 * q^52 + 22 * q^53 + 4 * q^54 + 2 * q^55 + 32 * q^57 + 10 * q^58 + 6 * q^59 + 2 * q^60 + 16 * q^61 - 2 * q^62 + 10 * q^64 + 4 * q^65 - 4 * q^66 + 14 * q^67 - 10 * q^68 - 10 * q^69 + 20 * q^71 + 12 * q^72 - 12 * q^73 - 2 * q^75 - 14 * q^76 + 48 * q^78 + 44 * q^79 - 10 * q^80 + 50 * q^81 + 22 * q^82 - 20 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^87 - 2 * q^88 - 12 * q^90 + 10 * q^92 + 4 * q^93 - 18 * q^94 + 14 * q^95 - 2 * q^96 + 40 * q^97 - 12 * q^99

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
$3$	0.255150	0.147311	0.0736554	−	0.997284i	$-0.476534\pi$
$3$	0.255150	0.147311	0.0736554	+	0.997284i	$0.476534\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.255150	0.104164
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.93490	−0.978300
$10$	−1.00000	−0.316228
$11$	2.61100	0.787246	0.393623	−	0.919272i	$-0.371222\pi$
$11$	2.61100	0.787246	0.393623	+	0.919272i	$0.371222\pi$
$12$	0.255150	0.0736554
$13$	−5.03829	−1.39737	−0.698685	−	0.715429i	$-0.746231\pi$
$13$	−5.03829	−1.39737	−0.698685	+	0.715429i	$0.746231\pi$
$14$	0	0
$15$	−0.255150	−0.0658794
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.93490	−0.691762
$19$	2.49174	0.571644	0.285822	−	0.958283i	$-0.407733\pi$
$19$	2.49174	0.571644	0.285822	+	0.958283i	$0.407733\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.61100	0.556667
$23$	−0.0298196	−0.00621782	−0.00310891	−	0.999995i	$-0.500990\pi$
$23$	−0.0298196	−0.00621782	−0.00310891	+	0.999995i	$0.500990\pi$
$24$	0.255150	0.0520822
$25$	1.00000	0.200000
$26$	−5.03829	−0.988090
$27$	−1.51429	−0.291425
$28$	0	0
$29$	4.55631	0.846086	0.423043	−	0.906110i	$-0.360962\pi$
$29$	4.55631	0.846086	0.423043	+	0.906110i	$0.360962\pi$
$30$	−0.255150	−0.0465837
$31$	−3.07081	−0.551535	−0.275767	−	0.961224i	$-0.588932\pi$
$31$	−3.07081	−0.551535	−0.275767	+	0.961224i	$0.588932\pi$
$32$	1.00000	0.176777
$33$	0.666196	0.115970
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.93490	−0.489150
$37$	4.53407	0.745397	0.372699	−	0.927952i	$-0.378433\pi$
$37$	4.53407	0.745397	0.372699	+	0.927952i	$0.378433\pi$
$38$	2.49174	0.404214
$39$	−1.28552	−0.205848
$40$	−1.00000	−0.158114
$41$	−4.29860	−0.671328	−0.335664	−	0.941982i	$-0.608961\pi$
$41$	−4.29860	−0.671328	−0.335664	+	0.941982i	$0.608961\pi$
$42$	0	0
$43$	−5.09136	−0.776425	−0.388212	−	0.921570i	$-0.626907\pi$
$43$	−5.09136	−0.776425	−0.388212	+	0.921570i	$0.626907\pi$
$44$	2.61100	0.393623
$45$	2.93490	0.437509
$46$	−0.0298196	−0.00439666
$47$	6.22417	0.907888	0.453944	−	0.891030i	$-0.350017\pi$
$47$	6.22417	0.907888	0.453944	+	0.891030i	$0.350017\pi$
$48$	0.255150	0.0368277
$49$	0	0
$50$	1.00000	0.141421
$51$	−0.255150	−0.0357281
$52$	−5.03829	−0.698685
$53$	−1.36002	−0.186814	−0.0934069	−	0.995628i	$-0.529776\pi$
$53$	−1.36002	−0.186814	−0.0934069	+	0.995628i	$0.529776\pi$
$54$	−1.51429	−0.206068
$55$	−2.61100	−0.352067
$56$	0	0
$57$	0.635767	0.0842094
$58$	4.55631	0.598273
$59$	14.0184	1.82504	0.912519	−	0.409034i	$-0.134134\pi$
$59$	14.0184	1.82504	0.912519	+	0.409034i	$0.134134\pi$
$60$	−0.255150	−0.0329397
$61$	4.98656	0.638463	0.319232	−	0.947677i	$-0.396575\pi$
$61$	4.98656	0.638463	0.319232	+	0.947677i	$0.396575\pi$
$62$	−3.07081	−0.389994
$63$	0	0
$64$	1.00000	0.125000
$65$	5.03829	0.624923
$66$	0.666196	0.0820030
$67$	7.02027	0.857662	0.428831	−	0.903385i	$-0.358925\pi$
$67$	7.02027	0.857662	0.428831	+	0.903385i	$0.358925\pi$
$68$	−1.00000	−0.121268
$69$	−0.00760847	−0.000915952 0
$70$	0	0
$71$	14.2603	1.69239	0.846193	−	0.532876i	$-0.178889\pi$
$71$	14.2603	1.69239	0.846193	+	0.532876i	$0.178889\pi$
$72$	−2.93490	−0.345881
$73$	−12.1613	−1.42337	−0.711685	−	0.702499i	$-0.752068\pi$
$73$	−12.1613	−1.42337	−0.711685	+	0.702499i	$0.752068\pi$
$74$	4.53407	0.527075
$75$	0.255150	0.0294621
$76$	2.49174	0.285822
$77$	0	0
$78$	−1.28552	−0.145556
$79$	8.95066	1.00703	0.503514	−	0.863987i	$-0.332040\pi$
$79$	8.95066	1.00703	0.503514	+	0.863987i	$0.332040\pi$
$80$	−1.00000	−0.111803
$81$	8.41833	0.935370
$82$	−4.29860	−0.474701
$83$	10.4460	1.14660	0.573301	−	0.819345i	$-0.305663\pi$
$83$	10.4460	1.14660	0.573301	+	0.819345i	$0.305663\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	−5.09136	−0.549015
$87$	1.16254	0.124638
$88$	2.61100	0.278333
$89$	1.30852	0.138703	0.0693516	−	0.997592i	$-0.477907\pi$
$89$	1.30852	0.138703	0.0693516	+	0.997592i	$0.477907\pi$
$90$	2.93490	0.309365
$91$	0	0
$92$	−0.0298196	−0.00310891
$93$	−0.783517	−0.0812470
$94$	6.22417	0.641974
$95$	−2.49174	−0.255647
$96$	0.255150	0.0260411
$97$	−3.38746	−0.343945	−0.171972	−	0.985102i	$-0.555014\pi$
$97$	−3.38746	−0.343945	−0.171972	+	0.985102i	$0.555014\pi$
$98$	0	0
$99$	−7.66302	−0.770162
$100$	1.00000	0.100000
$101$	−5.49366	−0.546640	−0.273320	−	0.961923i	$-0.588122\pi$
$101$	−5.49366	−0.546640	−0.273320	+	0.961923i	$0.588122\pi$
$102$	−0.255150	−0.0252636
$103$	3.89469	0.383756	0.191878	−	0.981419i	$-0.438542\pi$
$103$	3.89469	0.383756	0.191878	+	0.981419i	$0.438542\pi$
$104$	−5.03829	−0.494045
$105$	0	0
$106$	−1.36002	−0.132097
$107$	0.837969	0.0810095	0.0405047	−	0.999179i	$-0.487103\pi$
$107$	0.837969	0.0810095	0.0405047	+	0.999179i	$0.487103\pi$
$108$	−1.51429	−0.145712
$109$	−4.09311	−0.392049	−0.196024	−	0.980599i	$-0.562803\pi$
$109$	−4.09311	−0.392049	−0.196024	+	0.980599i	$0.562803\pi$
$110$	−2.61100	−0.248949
$111$	1.15687	0.109805
$112$	0	0
$113$	−6.45057	−0.606819	−0.303409	−	0.952860i	$-0.598125\pi$
$113$	−6.45057	−0.606819	−0.303409	+	0.952860i	$0.598125\pi$
$114$	0.635767	0.0595450
$115$	0.0298196	0.00278069
$116$	4.55631	0.423043
$117$	14.7869	1.36705
$118$	14.0184	1.29050
$119$	0	0
$120$	−0.255150	−0.0232919
$121$	−4.18268	−0.380244
$122$	4.98656	0.451462
$123$	−1.09679	−0.0988938
$124$	−3.07081	−0.275767
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.90849	−0.258087	−0.129043	−	0.991639i	$-0.541191\pi$
$127$	−2.90849	−0.258087	−0.129043	+	0.991639i	$0.541191\pi$
$128$	1.00000	0.0883883
$129$	−1.29906	−0.114376
$130$	5.03829	0.441887
$131$	−3.25071	−0.284016	−0.142008	−	0.989866i	$-0.545356\pi$
$131$	−3.25071	−0.284016	−0.142008	+	0.989866i	$0.545356\pi$
$132$	0.666196	0.0579849
$133$	0	0
$134$	7.02027	0.606459
$135$	1.51429	0.130329
$136$	−1.00000	−0.0857493
$137$	13.4621	1.15014	0.575071	−	0.818103i	$-0.304974\pi$
$137$	13.4621	1.15014	0.575071	+	0.818103i	$0.304974\pi$
$138$	−0.00760847	−0.000647676 0
$139$	6.51919	0.552950	0.276475	−	0.961021i	$-0.410834\pi$
$139$	6.51919	0.552950	0.276475	+	0.961021i	$0.410834\pi$
$140$	0	0
$141$	1.58809	0.133742
$142$	14.2603	1.19670
$143$	−13.1550	−1.10007
$144$	−2.93490	−0.244575
$145$	−4.55631	−0.378381
$146$	−12.1613	−1.00647
$147$	0	0
$148$	4.53407	0.372699
$149$	15.3796	1.25995	0.629974	−	0.776617i	$-0.283066\pi$
$149$	15.3796	1.25995	0.629974	+	0.776617i	$0.283066\pi$
$150$	0.255150	0.0208329
$151$	−4.35620	−0.354503	−0.177251	−	0.984166i	$-0.556721\pi$
$151$	−4.35620	−0.354503	−0.177251	+	0.984166i	$0.556721\pi$
$152$	2.49174	0.202107
$153$	2.93490	0.237272
$154$	0	0
$155$	3.07081	0.246654
$156$	−1.28552	−0.102924
$157$	−18.1056	−1.44499	−0.722493	−	0.691378i	$-0.757004\pi$
$157$	−18.1056	−1.44499	−0.722493	+	0.691378i	$0.757004\pi$
$158$	8.95066	0.712076
$159$	−0.347010	−0.0275197
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	8.41833	0.661406
$163$	23.1691	1.81474	0.907371	−	0.420330i	$-0.138086\pi$
$163$	23.1691	1.81474	0.907371	+	0.420330i	$0.138086\pi$
$164$	−4.29860	−0.335664
$165$	−0.666196	−0.0518633
$166$	10.4460	0.810770
$167$	5.41532	0.419050	0.209525	−	0.977803i	$-0.432808\pi$
$167$	5.41532	0.419050	0.209525	+	0.977803i	$0.432808\pi$
$168$	0	0
$169$	12.3844	0.952645
$170$	1.00000	0.0766965
$171$	−7.31300	−0.559239
$172$	−5.09136	−0.388212
$173$	22.8281	1.73559	0.867796	−	0.496921i	$-0.165536\pi$
$173$	22.8281	1.73559	0.867796	+	0.496921i	$0.165536\pi$
$174$	1.16254	0.0881321
$175$	0	0
$176$	2.61100	0.196812
$177$	3.57679	0.268848
$178$	1.30852	0.0980779
$179$	12.6611	0.946335	0.473167	−	0.880973i	$-0.343111\pi$
$179$	12.6611	0.946335	0.473167	+	0.880973i	$0.343111\pi$
$180$	2.93490	0.218754
$181$	−7.00446	−0.520637	−0.260319	−	0.965523i	$-0.583828\pi$
$181$	−7.00446	−0.520637	−0.260319	+	0.965523i	$0.583828\pi$
$182$	0	0
$183$	1.27232	0.0940525
$184$	−0.0298196	−0.00219833
$185$	−4.53407	−0.333352
$186$	−0.783517	−0.0574503
$187$	−2.61100	−0.190935
$188$	6.22417	0.453944
$189$	0	0
$190$	−2.49174	−0.180770
$191$	−1.13558	−0.0821673	−0.0410837	−	0.999156i	$-0.513081\pi$
$191$	−1.13558	−0.0821673	−0.0410837	+	0.999156i	$0.513081\pi$
$192$	0.255150	0.0184138
$193$	14.0458	1.01104	0.505521	−	0.862815i	$-0.331300\pi$
$193$	14.0458	1.01104	0.505521	+	0.862815i	$0.331300\pi$
$194$	−3.38746	−0.243206
$195$	1.28552	0.0920579
$196$	0	0
$197$	3.04235	0.216759	0.108379	−	0.994110i	$-0.465434\pi$
$197$	3.04235	0.216759	0.108379	+	0.994110i	$0.465434\pi$
$198$	−7.66302	−0.544587
$199$	4.96862	0.352216	0.176108	−	0.984371i	$-0.443649\pi$
$199$	4.96862	0.352216	0.176108	+	0.984371i	$0.443649\pi$
$200$	1.00000	0.0707107
$201$	1.79122	0.126343
$202$	−5.49366	−0.386533
$203$	0	0
$204$	−0.255150	−0.0178641
$205$	4.29860	0.300227
$206$	3.89469	0.271356
$207$	0.0875176	0.00608289
$208$	−5.03829	−0.349343
$209$	6.50593	0.450025
$210$	0	0
$211$	19.6536	1.35301	0.676503	−	0.736440i	$-0.263494\pi$
$211$	19.6536	1.35301	0.676503	+	0.736440i	$0.263494\pi$
$212$	−1.36002	−0.0934069
$213$	3.63851	0.249307
$214$	0.837969	0.0572824
$215$	5.09136	0.347228
$216$	−1.51429	−0.103034
$217$	0	0
$218$	−4.09311	−0.277220
$219$	−3.10295	−0.209678
$220$	−2.61100	−0.176034
$221$	5.03829	0.338912
$222$	1.15687	0.0776439
$223$	6.06689	0.406269	0.203134	−	0.979151i	$-0.434887\pi$
$223$	6.06689	0.406269	0.203134	+	0.979151i	$0.434887\pi$
$224$	0	0
$225$	−2.93490	−0.195660
$226$	−6.45057	−0.429086
$227$	−27.4406	−1.82130	−0.910648	−	0.413183i	$-0.864417\pi$
$227$	−27.4406	−1.82130	−0.910648	+	0.413183i	$0.864417\pi$
$228$	0.635767	0.0421047
$229$	15.8229	1.04560	0.522802	−	0.852454i	$-0.324887\pi$
$229$	15.8229	1.04560	0.522802	+	0.852454i	$0.324887\pi$
$230$	0.0298196	0.00196625
$231$	0	0
$232$	4.55631	0.299137
$233$	−22.3421	−1.46368	−0.731840	−	0.681477i	$-0.761338\pi$
$233$	−22.3421	−1.46368	−0.731840	+	0.681477i	$0.761338\pi$
$234$	14.7869	0.966648
$235$	−6.22417	−0.406020
$236$	14.0184	0.912519
$237$	2.28376	0.148346
$238$	0	0
$239$	17.6457	1.14141	0.570704	−	0.821156i	$-0.306670\pi$
$239$	17.6457	1.14141	0.570704	+	0.821156i	$0.306670\pi$
$240$	−0.255150	−0.0164698
$241$	−0.433723	−0.0279385	−0.0139693	−	0.999902i	$-0.504447\pi$
$241$	−0.433723	−0.0279385	−0.0139693	+	0.999902i	$0.504447\pi$
$242$	−4.18268	−0.268873
$243$	6.69080	0.429215
$244$	4.98656	0.319232
$245$	0	0
$246$	−1.09679	−0.0699285
$247$	−12.5541	−0.798799
$248$	−3.07081	−0.194997
$249$	2.66530	0.168907
$250$	−1.00000	−0.0632456
$251$	22.5819	1.42535	0.712677	−	0.701492i	$-0.247482\pi$
$251$	22.5819	1.42535	0.712677	+	0.701492i	$0.247482\pi$
$252$	0	0
$253$	−0.0778590	−0.00489496
$254$	−2.90849	−0.182495
$255$	0.255150	0.0159781
$256$	1.00000	0.0625000
$257$	−22.2105	−1.38546	−0.692728	−	0.721199i	$-0.743591\pi$
$257$	−22.2105	−1.38546	−0.692728	+	0.721199i	$0.743591\pi$
$258$	−1.29906	−0.0808759
$259$	0	0
$260$	5.03829	0.312462
$261$	−13.3723	−0.827726
$262$	−3.25071	−0.200829
$263$	3.22584	0.198914	0.0994568	−	0.995042i	$-0.468289\pi$
$263$	3.22584	0.198914	0.0994568	+	0.995042i	$0.468289\pi$
$264$	0.666196	0.0410015
$265$	1.36002	0.0835457
$266$	0	0
$267$	0.333869	0.0204325
$268$	7.02027	0.428831
$269$	−12.1343	−0.739842	−0.369921	−	0.929063i	$-0.620615\pi$
$269$	−12.1343	−0.739842	−0.369921	+	0.929063i	$0.620615\pi$
$270$	1.51429	0.0921566
$271$	−15.9209	−0.967123	−0.483562	−	0.875310i	$-0.660657\pi$
$271$	−15.9209	−0.967123	−0.483562	+	0.875310i	$0.660657\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	13.4621	0.813273
$275$	2.61100	0.157449
$276$	−0.00760847	−0.000457976 0
$277$	−23.1982	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$277$	−23.1982	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$278$	6.51919	0.390995
$279$	9.01253	0.539566
$280$	0	0
$281$	26.0349	1.55311	0.776555	−	0.630049i	$-0.216965\pi$
$281$	26.0349	1.55311	0.776555	+	0.630049i	$0.216965\pi$
$282$	1.58809	0.0945696
$283$	15.9654	0.949044	0.474522	−	0.880244i	$-0.342621\pi$
$283$	15.9654	0.949044	0.474522	+	0.880244i	$0.342621\pi$
$284$	14.2603	0.846193
$285$	−0.635767	−0.0376596
$286$	−13.1550	−0.777870
$287$	0	0
$288$	−2.93490	−0.172941
$289$	1.00000	0.0588235
$290$	−4.55631	−0.267556
$291$	−0.864310	−0.0506667
$292$	−12.1613	−0.711685
$293$	−11.6960	−0.683289	−0.341644	−	0.939829i	$-0.610984\pi$
$293$	−11.6960	−0.683289	−0.341644	+	0.939829i	$0.610984\pi$
$294$	0	0
$295$	−14.0184	−0.816182
$296$	4.53407	0.263538
$297$	−3.95380	−0.229423
$298$	15.3796	0.890917
$299$	0.150240	0.00868860
$300$	0.255150	0.0147311
$301$	0	0
$302$	−4.35620	−0.250671
$303$	−1.40171	−0.0805259
$304$	2.49174	0.142911
$305$	−4.98656	−0.285529
$306$	2.93490	0.167777
$307$	21.1653	1.20797	0.603984	−	0.796997i	$-0.293579\pi$
$307$	21.1653	1.20797	0.603984	+	0.796997i	$0.293579\pi$
$308$	0	0
$309$	0.993730	0.0565313
$310$	3.07081	0.174411
$311$	−13.8495	−0.785334	−0.392667	−	0.919681i	$-0.628448\pi$
$311$	−13.8495	−0.785334	−0.392667	+	0.919681i	$0.628448\pi$
$312$	−1.28552	−0.0727782
$313$	17.7135	1.00123	0.500614	−	0.865671i	$-0.333108\pi$
$313$	17.7135	1.00123	0.500614	+	0.865671i	$0.333108\pi$
$314$	−18.1056	−1.02176
$315$	0	0
$316$	8.95066	0.503514
$317$	−2.41294	−0.135524	−0.0677620	−	0.997702i	$-0.521586\pi$
$317$	−2.41294	−0.135524	−0.0677620	+	0.997702i	$0.521586\pi$
$318$	−0.347010	−0.0194594
$319$	11.8965	0.666078
$320$	−1.00000	−0.0559017
$321$	0.213807	0.0119336
$322$	0	0
$323$	−2.49174	−0.138644
$324$	8.41833	0.467685
$325$	−5.03829	−0.279474
$326$	23.1691	1.28322
$327$	−1.04436	−0.0577530
$328$	−4.29860	−0.237350
$329$	0	0
$330$	−0.666196	−0.0366729
$331$	3.55244	0.195260	0.0976298	−	0.995223i	$-0.468874\pi$
$331$	3.55244	0.195260	0.0976298	+	0.995223i	$0.468874\pi$
$332$	10.4460	0.573301
$333$	−13.3070	−0.729222
$334$	5.41532	0.296313
$335$	−7.02027	−0.383558
$336$	0	0
$337$	13.7356	0.748226	0.374113	−	0.927383i	$-0.377947\pi$
$337$	13.7356	0.748226	0.374113	+	0.927383i	$0.377947\pi$
$338$	12.3844	0.673622
$339$	−1.64586	−0.0893909
$340$	1.00000	0.0542326
$341$	−8.01790	−0.434193
$342$	−7.31300	−0.395442
$343$	0	0
$344$	−5.09136	−0.274508
$345$	0.00760847	0.000409626 0
$346$	22.8281	1.22725
$347$	−22.2294	−1.19334	−0.596668	−	0.802489i	$-0.703509\pi$
$347$	−22.2294	−1.19334	−0.596668	+	0.802489i	$0.703509\pi$
$348$	1.16254	0.0623188
$349$	7.02066	0.375807	0.187904	−	0.982187i	$-0.439831\pi$
$349$	7.02066	0.375807	0.187904	+	0.982187i	$0.439831\pi$
$350$	0	0
$351$	7.62942	0.407228
$352$	2.61100	0.139167
$353$	−11.3428	−0.603717	−0.301858	−	0.953353i	$-0.597607\pi$
$353$	−11.3428	−0.603717	−0.301858	+	0.953353i	$0.597607\pi$
$354$	3.57679	0.190104
$355$	−14.2603	−0.756858
$356$	1.30852	0.0693516
$357$	0	0
$358$	12.6611	0.669160
$359$	−11.9550	−0.630961	−0.315481	−	0.948932i	$-0.602166\pi$
$359$	−11.9550	−0.630961	−0.315481	+	0.948932i	$0.602166\pi$
$360$	2.93490	0.154683
$361$	−12.7912	−0.673223
$362$	−7.00446	−0.368146
$363$	−1.06721	−0.0560140
$364$	0	0
$365$	12.1613	0.636551
$366$	1.27232	0.0665052
$367$	9.67804	0.505189	0.252595	−	0.967572i	$-0.418716\pi$
$367$	9.67804	0.505189	0.252595	+	0.967572i	$0.418716\pi$
$368$	−0.0298196	−0.00155446
$369$	12.6159	0.656760
$370$	−4.53407	−0.235715
$371$	0	0
$372$	−0.783517	−0.0406235
$373$	−0.0526126	−0.00272418	−0.00136209	−	0.999999i	$-0.500434\pi$
$373$	−0.0526126	−0.00272418	−0.00136209	+	0.999999i	$0.500434\pi$
$374$	−2.61100	−0.135012
$375$	−0.255150	−0.0131759
$376$	6.22417	0.320987
$377$	−22.9560	−1.18230
$378$	0	0
$379$	23.1498	1.18912	0.594562	−	0.804050i	$-0.297325\pi$
$379$	23.1498	1.18912	0.594562	+	0.804050i	$0.297325\pi$
$380$	−2.49174	−0.127824
$381$	−0.742100	−0.0380190
$382$	−1.13558	−0.0581011
$383$	26.3286	1.34533	0.672664	−	0.739948i	$-0.265150\pi$
$383$	26.3286	1.34533	0.672664	+	0.739948i	$0.265150\pi$
$384$	0.255150	0.0130206
$385$	0	0
$386$	14.0458	0.714914
$387$	14.9426	0.759576
$388$	−3.38746	−0.171972
$389$	1.92283	0.0974913	0.0487456	−	0.998811i	$-0.484478\pi$
$389$	1.92283	0.0974913	0.0487456	+	0.998811i	$0.484478\pi$
$390$	1.28552	0.0650948
$391$	0.0298196	0.00150804
$392$	0	0
$393$	−0.829417	−0.0418386
$394$	3.04235	0.153271
$395$	−8.95066	−0.450357
$396$	−7.66302	−0.385081
$397$	−12.2412	−0.614368	−0.307184	−	0.951650i	$-0.599387\pi$
$397$	−12.2412	−0.614368	−0.307184	+	0.951650i	$0.599387\pi$
$398$	4.96862	0.249055
$399$	0	0
$400$	1.00000	0.0500000
$401$	−27.1504	−1.35583	−0.677913	−	0.735142i	$-0.737115\pi$
$401$	−27.1504	−1.35583	−0.677913	+	0.735142i	$0.737115\pi$
$402$	1.79122	0.0893379
$403$	15.4717	0.770698
$404$	−5.49366	−0.273320
$405$	−8.41833	−0.418310
$406$	0	0
$407$	11.8385	0.586811
$408$	−0.255150	−0.0126318
$409$	−18.5833	−0.918883	−0.459442	−	0.888208i	$-0.651950\pi$
$409$	−18.5833	−0.918883	−0.459442	+	0.888208i	$0.651950\pi$
$410$	4.29860	0.212293
$411$	3.43484	0.169428
$412$	3.89469	0.191878
$413$	0	0
$414$	0.0875176	0.00430125
$415$	−10.4460	−0.512776
$416$	−5.03829	−0.247023
$417$	1.66337	0.0814555
$418$	6.50593	0.318216
$419$	−23.4607	−1.14613	−0.573064	−	0.819510i	$-0.694245\pi$
$419$	−23.4607	−1.14613	−0.573064	+	0.819510i	$0.694245\pi$
$420$	0	0
$421$	−23.0393	−1.12287	−0.561434	−	0.827522i	$-0.689750\pi$
$421$	−23.0393	−1.12287	−0.561434	+	0.827522i	$0.689750\pi$
$422$	19.6536	0.956720
$423$	−18.2673	−0.888186
$424$	−1.36002	−0.0660486
$425$	−1.00000	−0.0485071
$426$	3.63851	0.176286
$427$	0	0
$428$	0.837969	0.0405047
$429$	−3.35649	−0.162053
$430$	5.09136	0.245527
$431$	35.1237	1.69185	0.845923	−	0.533305i	$-0.179050\pi$
$431$	35.1237	1.69185	0.845923	+	0.533305i	$0.179050\pi$
$432$	−1.51429	−0.0728562
$433$	−34.4070	−1.65349	−0.826747	−	0.562573i	$-0.809811\pi$
$433$	−34.4070	−1.65349	−0.826747	+	0.562573i	$0.809811\pi$
$434$	0	0
$435$	−1.16254	−0.0557396
$436$	−4.09311	−0.196024
$437$	−0.0743028	−0.00355438
$438$	−3.10295	−0.148265
$439$	23.4899	1.12111	0.560555	−	0.828117i	$-0.310588\pi$
$439$	23.4899	1.12111	0.560555	+	0.828117i	$0.310588\pi$
$440$	−2.61100	−0.124475
$441$	0	0
$442$	5.03829	0.239647
$443$	19.3910	0.921296	0.460648	−	0.887583i	$-0.347617\pi$
$443$	19.3910	0.921296	0.460648	+	0.887583i	$0.347617\pi$
$444$	1.15687	0.0549025
$445$	−1.30852	−0.0620299
$446$	6.06689	0.287275
$447$	3.92410	0.185604
$448$	0	0
$449$	−27.8525	−1.31444	−0.657220	−	0.753699i	$-0.728268\pi$
$449$	−27.8525	−1.31444	−0.657220	+	0.753699i	$0.728268\pi$
$450$	−2.93490	−0.138352
$451$	−11.2236	−0.528500
$452$	−6.45057	−0.303409
$453$	−1.11148	−0.0522221
$454$	−27.4406	−1.28785
$455$	0	0
$456$	0.635767	0.0297725
$457$	23.0108	1.07640	0.538201	−	0.842817i	$-0.319104\pi$
$457$	23.0108	1.07640	0.538201	+	0.842817i	$0.319104\pi$
$458$	15.8229	0.739354
$459$	1.51429	0.0706809
$460$	0.0298196	0.00139035
$461$	−17.3140	−0.806392	−0.403196	−	0.915114i	$-0.632101\pi$
$461$	−17.3140	−0.806392	−0.403196	+	0.915114i	$0.632101\pi$
$462$	0	0
$463$	4.45573	0.207075	0.103538	−	0.994626i	$-0.466984\pi$
$463$	4.45573	0.207075	0.103538	+	0.994626i	$0.466984\pi$
$464$	4.55631	0.211522
$465$	0.783517	0.0363347
$466$	−22.3421	−1.03498
$467$	36.6836	1.69752	0.848758	−	0.528781i	$-0.177351\pi$
$467$	36.6836	1.69752	0.848758	+	0.528781i	$0.177351\pi$
$468$	14.7869	0.683524
$469$	0	0
$470$	−6.22417	−0.287099
$471$	−4.61965	−0.212862
$472$	14.0184	0.645248
$473$	−13.2935	−0.611237
$474$	2.28376	0.104896
$475$	2.49174	0.114329
$476$	0	0
$477$	3.99154	0.182760
$478$	17.6457	0.807098
$479$	23.0934	1.05517	0.527583	−	0.849503i	$-0.323098\pi$
$479$	23.0934	1.05517	0.527583	+	0.849503i	$0.323098\pi$
$480$	−0.255150	−0.0116459
$481$	−22.8440	−1.04160
$482$	−0.433723	−0.0197555
$483$	0	0
$484$	−4.18268	−0.190122
$485$	3.38746	0.153817
$486$	6.69080	0.303501
$487$	−13.5720	−0.615008	−0.307504	−	0.951547i	$-0.599494\pi$
$487$	−13.5720	−0.615008	−0.307504	+	0.951547i	$0.599494\pi$
$488$	4.98656	0.225731
$489$	5.91159	0.267331
$490$	0	0
$491$	3.49095	0.157544	0.0787722	−	0.996893i	$-0.474900\pi$
$491$	3.49095	0.157544	0.0787722	+	0.996893i	$0.474900\pi$
$492$	−1.09679	−0.0494469
$493$	−4.55631	−0.205206
$494$	−12.5541	−0.564836
$495$	7.66302	0.344427
$496$	−3.07081	−0.137884
$497$	0	0
$498$	2.66530	0.119435
$499$	15.3178	0.685720	0.342860	−	0.939387i	$-0.388604\pi$
$499$	15.3178	0.685720	0.342860	+	0.939387i	$0.388604\pi$
$500$	−1.00000	−0.0447214
$501$	1.38172	0.0617306
$502$	22.5819	1.00788
$503$	−0.341558	−0.0152293	−0.00761466	−	0.999971i	$-0.502424\pi$
$503$	−0.341558	−0.0152293	−0.00761466	+	0.999971i	$0.502424\pi$
$504$	0	0
$505$	5.49366	0.244465
$506$	−0.0778590	−0.00346126
$507$	3.15987	0.140335
$508$	−2.90849	−0.129043
$509$	44.6417	1.97871	0.989355	−	0.145523i	$-0.0464865\pi$
$509$	44.6417	1.97871	0.989355	+	0.145523i	$0.0464865\pi$
$510$	0.255150	0.0112982
$511$	0	0
$512$	1.00000	0.0441942
$513$	−3.77321	−0.166591
$514$	−22.2105	−0.979665
$515$	−3.89469	−0.171621
$516$	−1.29906	−0.0571879
$517$	16.2513	0.714731
$518$	0	0
$519$	5.82459	0.255671
$520$	5.03829	0.220944
$521$	8.97648	0.393267	0.196633	−	0.980477i	$-0.436999\pi$
$521$	8.97648	0.393267	0.196633	+	0.980477i	$0.436999\pi$
$522$	−13.3723	−0.585291
$523$	−22.6832	−0.991867	−0.495933	−	0.868361i	$-0.665174\pi$
$523$	−22.6832	−0.991867	−0.495933	+	0.868361i	$0.665174\pi$
$524$	−3.25071	−0.142008
$525$	0	0
$526$	3.22584	0.140653
$527$	3.07081	0.133767
$528$	0.666196	0.0289924
$529$	−22.9991	−0.999961
$530$	1.36002	0.0590757
$531$	−41.1425	−1.78543
$532$	0	0
$533$	21.6576	0.938094
$534$	0.333869	0.0144479
$535$	−0.837969	−0.0362285
$536$	7.02027	0.303229
$537$	3.23048	0.139405
$538$	−12.1343	−0.523147
$539$	0	0
$540$	1.51429	0.0651646
$541$	8.33431	0.358320	0.179160	−	0.983820i	$-0.442662\pi$
$541$	8.33431	0.358320	0.179160	+	0.983820i	$0.442662\pi$
$542$	−15.9209	−0.683859
$543$	−1.78719	−0.0766955
$544$	−1.00000	−0.0428746
$545$	4.09311	0.175330
$546$	0	0
$547$	−9.74685	−0.416745	−0.208373	−	0.978050i	$-0.566817\pi$
$547$	−9.74685	−0.416745	−0.208373	+	0.978050i	$0.566817\pi$
$548$	13.4621	0.575071
$549$	−14.6350	−0.624608
$550$	2.61100	0.111333
$551$	11.3532	0.483661
$552$	−0.00760847	−0.000323838 0
$553$	0	0
$554$	−23.1982	−0.985599
$555$	−1.15687	−0.0491063
$556$	6.51919	0.276475
$557$	−17.4083	−0.737613	−0.368806	−	0.929506i	$-0.620233\pi$
$557$	−17.4083	−0.737613	−0.368806	+	0.929506i	$0.620233\pi$
$558$	9.01253	0.381531
$559$	25.6518	1.08495
$560$	0	0
$561$	−0.666196	−0.0281268
$562$	26.0349	1.09822
$563$	−2.27680	−0.0959558	−0.0479779	−	0.998848i	$-0.515278\pi$
$563$	−2.27680	−0.0959558	−0.0479779	+	0.998848i	$0.515278\pi$
$564$	1.58809	0.0668708
$565$	6.45057	0.271378
$566$	15.9654	0.671075
$567$	0	0
$568$	14.2603	0.598349
$569$	4.36046	0.182800	0.0914000	−	0.995814i	$-0.470866\pi$
$569$	4.36046	0.182800	0.0914000	+	0.995814i	$0.470866\pi$
$570$	−0.635767	−0.0266293
$571$	9.59804	0.401665	0.200833	−	0.979626i	$-0.435635\pi$
$571$	9.59804	0.401665	0.200833	+	0.979626i	$0.435635\pi$
$572$	−13.1550	−0.550037
$573$	−0.289742	−0.0121041
$574$	0	0
$575$	−0.0298196	−0.00124356
$576$	−2.93490	−0.122287
$577$	−21.0165	−0.874928	−0.437464	−	0.899236i	$-0.644123\pi$
$577$	−21.0165	−0.874928	−0.437464	+	0.899236i	$0.644123\pi$
$578$	1.00000	0.0415945
$579$	3.58379	0.148937
$580$	−4.55631	−0.189191
$581$	0	0
$582$	−0.864310	−0.0358268
$583$	−3.55102	−0.147068
$584$	−12.1613	−0.503237
$585$	−14.7869	−0.611362
$586$	−11.6960	−0.483158
$587$	−5.60652	−0.231406	−0.115703	−	0.993284i	$-0.536912\pi$
$587$	−5.60652	−0.231406	−0.115703	+	0.993284i	$0.536912\pi$
$588$	0	0
$589$	−7.65167	−0.315282
$590$	−14.0184	−0.577128
$591$	0.776255	0.0319309
$592$	4.53407	0.186349
$593$	6.93820	0.284918	0.142459	−	0.989801i	$-0.454499\pi$
$593$	6.93820	0.284918	0.142459	+	0.989801i	$0.454499\pi$
$594$	−3.95380	−0.162227
$595$	0	0
$596$	15.3796	0.629974
$597$	1.26774	0.0518853
$598$	0.150240	0.00614377
$599$	−14.3357	−0.585741	−0.292870	−	0.956152i	$-0.594610\pi$
$599$	−14.3357	−0.585741	−0.292870	+	0.956152i	$0.594610\pi$
$600$	0.255150	0.0104164
$601$	−8.43423	−0.344039	−0.172020	−	0.985094i	$-0.555029\pi$
$601$	−8.43423	−0.344039	−0.172020	+	0.985094i	$0.555029\pi$
$602$	0	0
$603$	−20.6038	−0.839051
$604$	−4.35620	−0.177251
$605$	4.18268	0.170050
$606$	−1.40171	−0.0569404
$607$	4.02725	0.163461	0.0817305	−	0.996654i	$-0.473955\pi$
$607$	4.02725	0.163461	0.0817305	+	0.996654i	$0.473955\pi$
$608$	2.49174	0.101053
$609$	0	0
$610$	−4.98656	−0.201900
$611$	−31.3592	−1.26866
$612$	2.93490	0.118636
$613$	31.3785	1.26737	0.633683	−	0.773593i	$-0.281542\pi$
$613$	31.3785	1.26737	0.633683	+	0.773593i	$0.281542\pi$
$614$	21.1653	0.854162
$615$	1.09679	0.0442267
$616$	0	0
$617$	22.6139	0.910400	0.455200	−	0.890389i	$-0.349568\pi$
$617$	22.6139	0.910400	0.455200	+	0.890389i	$0.349568\pi$
$618$	0.993730	0.0399737
$619$	6.75938	0.271683	0.135841	−	0.990731i	$-0.456626\pi$
$619$	6.75938	0.271683	0.135841	+	0.990731i	$0.456626\pi$
$620$	3.07081	0.123327
$621$	0.0451555	0.00181203
$622$	−13.8495	−0.555315
$623$	0	0
$624$	−1.28552	−0.0514619
$625$	1.00000	0.0400000
$626$	17.7135	0.707975
$627$	1.65999	0.0662935
$628$	−18.1056	−0.722493
$629$	−4.53407	−0.180785
$630$	0	0
$631$	−3.71386	−0.147846	−0.0739232	−	0.997264i	$-0.523552\pi$
$631$	−3.71386	−0.147846	−0.0739232	+	0.997264i	$0.523552\pi$
$632$	8.95066	0.356038
$633$	5.01460	0.199312
$634$	−2.41294	−0.0958299
$635$	2.90849	0.115420
$636$	−0.347010	−0.0137598
$637$	0	0
$638$	11.8965	0.470988
$639$	−41.8525	−1.65566
$640$	−1.00000	−0.0395285
$641$	−47.7676	−1.88671	−0.943354	−	0.331788i	$-0.892348\pi$
$641$	−47.7676	−1.88671	−0.943354	+	0.331788i	$0.892348\pi$
$642$	0.213807	0.00843831
$643$	6.41576	0.253013	0.126506	−	0.991966i	$-0.459624\pi$
$643$	6.41576	0.253013	0.126506	+	0.991966i	$0.459624\pi$
$644$	0	0
$645$	1.29906	0.0511504
$646$	−2.49174	−0.0980362
$647$	0.775811	0.0305003	0.0152501	−	0.999884i	$-0.495146\pi$
$647$	0.775811	0.0305003	0.0152501	+	0.999884i	$0.495146\pi$
$648$	8.41833	0.330703
$649$	36.6020	1.43675
$650$	−5.03829	−0.197618
$651$	0	0
$652$	23.1691	0.907371
$653$	1.32965	0.0520333	0.0260167	−	0.999662i	$-0.491718\pi$
$653$	1.32965	0.0520333	0.0260167	+	0.999662i	$0.491718\pi$
$654$	−1.04436	−0.0408375
$655$	3.25071	0.127016
$656$	−4.29860	−0.167832
$657$	35.6921	1.39248
$658$	0	0
$659$	−42.4940	−1.65533	−0.827665	−	0.561223i	$-0.810331\pi$
$659$	−42.4940	−1.65533	−0.827665	+	0.561223i	$0.810331\pi$
$660$	−0.666196	−0.0259316
$661$	−18.1985	−0.707841	−0.353921	−	0.935275i	$-0.615152\pi$
$661$	−18.1985	−0.707841	−0.353921	+	0.935275i	$0.615152\pi$
$662$	3.55244	0.138069
$663$	1.28552	0.0499254
$664$	10.4460	0.405385
$665$	0	0
$666$	−13.3070	−0.515638
$667$	−0.135868	−0.00526081
$668$	5.41532	0.209525
$669$	1.54796	0.0598478
$670$	−7.02027	−0.271217
$671$	13.0199	0.502628
$672$	0	0
$673$	−9.60707	−0.370326	−0.185163	−	0.982708i	$-0.559281\pi$
$673$	−9.60707	−0.370326	−0.185163	+	0.982708i	$0.559281\pi$
$674$	13.7356	0.529075
$675$	−1.51429	−0.0582850
$676$	12.3844	0.476323
$677$	29.8431	1.14696	0.573482	−	0.819218i	$-0.305592\pi$
$677$	29.8431	1.14696	0.573482	+	0.819218i	$0.305592\pi$
$678$	−1.64586	−0.0632089
$679$	0	0
$680$	1.00000	0.0383482
$681$	−7.00146	−0.268296
$682$	−8.01790	−0.307021
$683$	19.5821	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$683$	19.5821	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$684$	−7.31300	−0.279620
$685$	−13.4621	−0.514359
$686$	0	0
$687$	4.03720	0.154029
$688$	−5.09136	−0.194106
$689$	6.85220	0.261048
$690$	0.00760847	0.000289649 0
$691$	40.8367	1.55350	0.776751	−	0.629808i	$-0.216867\pi$
$691$	40.8367	1.55350	0.776751	+	0.629808i	$0.216867\pi$
$692$	22.8281	0.867796
$693$	0	0
$694$	−22.2294	−0.843815
$695$	−6.51919	−0.247287
$696$	1.16254	0.0440660
$697$	4.29860	0.162821
$698$	7.02066	0.265736
$699$	−5.70058	−0.215616
$700$	0	0
$701$	40.8861	1.54425	0.772123	−	0.635474i	$-0.219195\pi$
$701$	40.8861	1.54425	0.772123	+	0.635474i	$0.219195\pi$
$702$	7.62942	0.287954
$703$	11.2977	0.426102
$704$	2.61100	0.0984058
$705$	−1.58809	−0.0598111
$706$	−11.3428	−0.426892
$707$	0	0
$708$	3.57679	0.134424
$709$	−36.9535	−1.38782	−0.693909	−	0.720062i	$-0.744113\pi$
$709$	−36.9535	−1.38782	−0.693909	+	0.720062i	$0.744113\pi$
$710$	−14.2603	−0.535180
$711$	−26.2693	−0.985175
$712$	1.30852	0.0490390
$713$	0.0915706	0.00342934
$714$	0	0
$715$	13.1550	0.491968
$716$	12.6611	0.473167
$717$	4.50231	0.168142
$718$	−11.9550	−0.446157
$719$	−3.05855	−0.114065	−0.0570323	−	0.998372i	$-0.518164\pi$
$719$	−3.05855	−0.114065	−0.0570323	+	0.998372i	$0.518164\pi$
$720$	2.93490	0.109377
$721$	0	0
$722$	−12.7912	−0.476040
$723$	−0.110664	−0.00411565
$724$	−7.00446	−0.260319
$725$	4.55631	0.169217
$726$	−1.06721	−0.0396079
$727$	26.7938	0.993726	0.496863	−	0.867829i	$-0.334485\pi$
$727$	26.7938	0.993726	0.496863	+	0.867829i	$0.334485\pi$
$728$	0	0
$729$	−23.5478	−0.872142
$730$	12.1613	0.450109
$731$	5.09136	0.188311
$732$	1.27232	0.0470263
$733$	3.22977	0.119294	0.0596472	−	0.998220i	$-0.481002\pi$
$733$	3.22977	0.119294	0.0596472	+	0.998220i	$0.481002\pi$
$734$	9.67804	0.357223
$735$	0	0
$736$	−0.0298196	−0.00109917
$737$	18.3299	0.675191
$738$	12.6159	0.464399
$739$	1.80826	0.0665179	0.0332589	−	0.999447i	$-0.489411\pi$
$739$	1.80826	0.0665179	0.0332589	+	0.999447i	$0.489411\pi$
$740$	−4.53407	−0.166676
$741$	−3.20318	−0.117672
$742$	0	0
$743$	6.11728	0.224421	0.112211	−	0.993684i	$-0.464207\pi$
$743$	6.11728	0.224421	0.112211	+	0.993684i	$0.464207\pi$
$744$	−0.783517	−0.0287251
$745$	−15.3796	−0.563465
$746$	−0.0526126	−0.00192628
$747$	−30.6581	−1.12172
$748$	−2.61100	−0.0954676
$749$	0	0
$750$	−0.255150	−0.00931675
$751$	−4.82306	−0.175996	−0.0879980	−	0.996121i	$-0.528047\pi$
$751$	−4.82306	−0.175996	−0.0879980	+	0.996121i	$0.528047\pi$
$752$	6.22417	0.226972
$753$	5.76175	0.209970
$754$	−22.9560	−0.836010
$755$	4.35620	0.158538
$756$	0	0
$757$	−46.9328	−1.70580	−0.852901	−	0.522073i	$-0.825159\pi$
$757$	−46.9328	−1.70580	−0.852901	+	0.522073i	$0.825159\pi$
$758$	23.1498	0.840838
$759$	−0.0198657	−0.000721080 0
$760$	−2.49174	−0.0903849
$761$	42.4852	1.54009	0.770044	−	0.637991i	$-0.220234\pi$
$761$	42.4852	1.54009	0.770044	+	0.637991i	$0.220234\pi$
$762$	−0.742100	−0.0268835
$763$	0	0
$764$	−1.13558	−0.0410837
$765$	−2.93490	−0.106111
$766$	26.3286	0.951291
$767$	−70.6287	−2.55025
$768$	0.255150	0.00920692
$769$	9.86913	0.355890	0.177945	−	0.984040i	$-0.443055\pi$
$769$	9.86913	0.355890	0.177945	+	0.984040i	$0.443055\pi$
$770$	0	0
$771$	−5.66701	−0.204093
$772$	14.0458	0.505521
$773$	−49.7517	−1.78944	−0.894722	−	0.446623i	$-0.852627\pi$
$773$	−49.7517	−1.78944	−0.894722	+	0.446623i	$0.852627\pi$
$774$	14.9426	0.537101
$775$	−3.07081	−0.110307
$776$	−3.38746	−0.121603
$777$	0	0
$778$	1.92283	0.0689367
$779$	−10.7110	−0.383761
$780$	1.28552	0.0460290
$781$	37.2336	1.33232
$782$	0.0298196	0.00106635
$783$	−6.89957	−0.246571
$784$	0	0
$785$	18.1056	0.646218
$786$	−0.829417	−0.0295843
$787$	−5.73451	−0.204413	−0.102207	−	0.994763i	$-0.532590\pi$
$787$	−5.73451	−0.204413	−0.102207	+	0.994763i	$0.532590\pi$
$788$	3.04235	0.108379
$789$	0.823071	0.0293021
$790$	−8.95066	−0.318450
$791$	0	0
$792$	−7.66302	−0.272294
$793$	−25.1237	−0.892170
$794$	−12.2412	−0.434424
$795$	0.347010	0.0123072
$796$	4.96862	0.176108
$797$	−5.07569	−0.179790	−0.0898951	−	0.995951i	$-0.528653\pi$
$797$	−5.07569	−0.179790	−0.0898951	+	0.995951i	$0.528653\pi$
$798$	0	0
$799$	−6.22417	−0.220195
$800$	1.00000	0.0353553
$801$	−3.84038	−0.135693
$802$	−27.1504	−0.958714
$803$	−31.7531	−1.12054
$804$	1.79122	0.0631714
$805$	0	0
$806$	15.4717	0.544966
$807$	−3.09607	−0.108987
$808$	−5.49366	−0.193266
$809$	−3.44999	−0.121295	−0.0606476	−	0.998159i	$-0.519317\pi$
$809$	−3.44999	−0.121295	−0.0606476	+	0.998159i	$0.519317\pi$
$810$	−8.41833	−0.295790
$811$	−20.4316	−0.717451	−0.358725	−	0.933443i	$-0.616789\pi$
$811$	−20.4316	−0.717451	−0.358725	+	0.933443i	$0.616789\pi$
$812$	0	0
$813$	−4.06220	−0.142468
$814$	11.8385	0.414938
$815$	−23.1691	−0.811578
$816$	−0.255150	−0.00893203
$817$	−12.6863	−0.443839
$818$	−18.5833	−0.649749
$819$	0	0
$820$	4.29860	0.150114
$821$	41.3056	1.44157	0.720787	−	0.693157i	$-0.243781\pi$
$821$	41.3056	1.44157	0.720787	+	0.693157i	$0.243781\pi$
$822$	3.43484	0.119804
$823$	−10.2842	−0.358483	−0.179242	−	0.983805i	$-0.557364\pi$
$823$	−10.2842	−0.358483	−0.179242	+	0.983805i	$0.557364\pi$
$824$	3.89469	0.135678
$825$	0.666196	0.0231940
$826$	0	0
$827$	48.0561	1.67108	0.835538	−	0.549433i	$-0.185156\pi$
$827$	48.0561	1.67108	0.835538	+	0.549433i	$0.185156\pi$
$828$	0.0875176	0.00304145
$829$	3.47412	0.120661	0.0603306	−	0.998178i	$-0.480785\pi$
$829$	3.47412	0.120661	0.0603306	+	0.998178i	$0.480785\pi$
$830$	−10.4460	−0.362587
$831$	−5.91902	−0.205329
$832$	−5.03829	−0.174671
$833$	0	0
$834$	1.66337	0.0575978
$835$	−5.41532	−0.187405
$836$	6.50593	0.225012
$837$	4.65010	0.160731
$838$	−23.4607	−0.810435
$839$	4.17740	0.144220	0.0721099	−	0.997397i	$-0.477027\pi$
$839$	4.17740	0.144220	0.0721099	+	0.997397i	$0.477027\pi$
$840$	0	0
$841$	−8.24000	−0.284138
$842$	−23.0393	−0.793987
$843$	6.64279	0.228790
$844$	19.6536	0.676503
$845$	−12.3844	−0.426036
$846$	−18.2673	−0.628043
$847$	0	0
$848$	−1.36002	−0.0467034
$849$	4.07356	0.139804
$850$	−1.00000	−0.0342997
$851$	−0.135204	−0.00463475
$852$	3.63851	0.124653
$853$	−19.4777	−0.666904	−0.333452	−	0.942767i	$-0.608214\pi$
$853$	−19.4777	−0.666904	−0.333452	+	0.942767i	$0.608214\pi$
$854$	0	0
$855$	7.31300	0.250099
$856$	0.837969	0.0286412
$857$	6.17064	0.210785	0.105393	−	0.994431i	$-0.466390\pi$
$857$	6.17064	0.210785	0.105393	+	0.994431i	$0.466390\pi$
$858$	−3.35649	−0.114589
$859$	6.51129	0.222162	0.111081	−	0.993811i	$-0.464569\pi$
$859$	6.51129	0.222162	0.111081	+	0.993811i	$0.464569\pi$
$860$	5.09136	0.173614
$861$	0	0
$862$	35.1237	1.19632
$863$	−3.42564	−0.116610	−0.0583051	−	0.998299i	$-0.518570\pi$
$863$	−3.42564	−0.116610	−0.0583051	+	0.998299i	$0.518570\pi$
$864$	−1.51429	−0.0515171
$865$	−22.8281	−0.776180
$866$	−34.4070	−1.16920
$867$	0.255150	0.00866534
$868$	0	0
$869$	23.3702	0.792779
$870$	−1.16254	−0.0394139
$871$	−35.3702	−1.19847
$872$	−4.09311	−0.138610
$873$	9.94185	0.336481
$874$	−0.0743028	−0.00251333
$875$	0	0
$876$	−3.10295	−0.104839
$877$	43.6711	1.47467	0.737333	−	0.675529i	$-0.236085\pi$
$877$	43.6711	1.47467	0.737333	+	0.675529i	$0.236085\pi$
$878$	23.4899	0.792744
$879$	−2.98424	−0.100656
$880$	−2.61100	−0.0880168
$881$	−19.3300	−0.651243	−0.325622	−	0.945500i	$-0.605574\pi$
$881$	−19.3300	−0.651243	−0.325622	+	0.945500i	$0.605574\pi$
$882$	0	0
$883$	−52.9606	−1.78226	−0.891132	−	0.453743i	$-0.850088\pi$
$883$	−52.9606	−1.78226	−0.891132	+	0.453743i	$0.850088\pi$
$884$	5.03829	0.169456
$885$	−3.57679	−0.120232
$886$	19.3910	0.651455
$887$	11.7651	0.395032	0.197516	−	0.980300i	$-0.436713\pi$
$887$	11.7651	0.395032	0.197516	+	0.980300i	$0.436713\pi$
$888$	1.15687	0.0388219
$889$	0	0
$890$	−1.30852	−0.0438618
$891$	21.9802	0.736366
$892$	6.06689	0.203134
$893$	15.5090	0.518989
$894$	3.92410	0.131242
$895$	−12.6611	−0.423214
$896$	0	0
$897$	0.0383337	0.00127992
$898$	−27.8525	−0.929450
$899$	−13.9916	−0.466646
$900$	−2.93490	−0.0978300
$901$	1.36002	0.0453090
$902$	−11.2236	−0.373706
$903$	0	0
$904$	−6.45057	−0.214543
$905$	7.00446	0.232836
$906$	−1.11148	−0.0369266
$907$	−10.6873	−0.354866	−0.177433	−	0.984133i	$-0.556779\pi$
$907$	−10.6873	−0.354866	−0.177433	+	0.984133i	$0.556779\pi$
$908$	−27.4406	−0.910648
$909$	16.1233	0.534777
$910$	0	0
$911$	−30.1948	−1.00040	−0.500199	−	0.865911i	$-0.666740\pi$
$911$	−30.1948	−1.00040	−0.500199	+	0.865911i	$0.666740\pi$
$912$	0.635767	0.0210523
$913$	27.2746	0.902658
$914$	23.0108	0.761131
$915$	−1.27232	−0.0420616
$916$	15.8229	0.522802
$917$	0	0
$918$	1.51429	0.0499789
$919$	−33.2984	−1.09841	−0.549206	−	0.835687i	$-0.685070\pi$
$919$	−33.2984	−1.09841	−0.549206	+	0.835687i	$0.685070\pi$
$920$	0.0298196	0.000983124 0
$921$	5.40032	0.177947
$922$	−17.3140	−0.570205
$923$	−71.8476	−2.36489
$924$	0	0
$925$	4.53407	0.149079
$926$	4.45573	0.146424
$927$	−11.4305	−0.375428
$928$	4.55631	0.149568
$929$	46.8081	1.53572	0.767862	−	0.640615i	$-0.221320\pi$
$929$	46.8081	1.53572	0.767862	+	0.640615i	$0.221320\pi$
$930$	0.783517	0.0256925
$931$	0	0
$932$	−22.3421	−0.731840
$933$	−3.53370	−0.115688
$934$	36.6836	1.20033
$935$	2.61100	0.0853888
$936$	14.7869	0.483324
$937$	−18.7599	−0.612860	−0.306430	−	0.951893i	$-0.599134\pi$
$937$	−18.7599	−0.612860	−0.306430	+	0.951893i	$0.599134\pi$
$938$	0	0
$939$	4.51960	0.147492
$940$	−6.22417	−0.203010
$941$	27.9353	0.910663	0.455332	−	0.890322i	$-0.349521\pi$
$941$	27.9353	0.910663	0.455332	+	0.890322i	$0.349521\pi$
$942$	−4.61965	−0.150516
$943$	0.128183	0.00417420
$944$	14.0184	0.456259
$945$	0	0
$946$	−13.2935	−0.432210
$947$	−15.5058	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$947$	−15.5058	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$948$	2.28376	0.0741730
$949$	61.2721	1.98898
$950$	2.49174	0.0808427
$951$	−0.615660	−0.0199641
$952$	0	0
$953$	−27.0793	−0.877185	−0.438592	−	0.898686i	$-0.644523\pi$
$953$	−27.0793	−0.877185	−0.438592	+	0.898686i	$0.644523\pi$
$954$	3.99154	0.129231
$955$	1.13558	0.0367464
$956$	17.6457	0.570704
$957$	3.03540	0.0981205
$958$	23.0934	0.746115
$959$	0	0
$960$	−0.255150	−0.00823492
$961$	−21.5701	−0.695810
$962$	−22.8440	−0.736520
$963$	−2.45935	−0.0792515
$964$	−0.433723	−0.0139693
$965$	−14.0458	−0.452151
$966$	0	0
$967$	14.2320	0.457669	0.228835	−	0.973465i	$-0.426508\pi$
$967$	14.2320	0.457669	0.228835	+	0.973465i	$0.426508\pi$
$968$	−4.18268	−0.134436
$969$	−0.635767	−0.0204238
$970$	3.38746	0.108765
$971$	−49.0281	−1.57339	−0.786693	−	0.617344i	$-0.788208\pi$
$971$	−49.0281	−1.57339	−0.786693	+	0.617344i	$0.788208\pi$
$972$	6.69080	0.214607
$973$	0	0
$974$	−13.5720	−0.434876
$975$	−1.28552	−0.0411695
$976$	4.98656	0.159616
$977$	−24.5909	−0.786733	−0.393366	−	0.919382i	$-0.628690\pi$
$977$	−24.5909	−0.786733	−0.393366	+	0.919382i	$0.628690\pi$
$978$	5.91159	0.189032
$979$	3.41655	0.109194
$980$	0	0
$981$	12.0129	0.383541
$982$	3.49095	0.111401
$983$	−3.31969	−0.105882	−0.0529409	−	0.998598i	$-0.516859\pi$
$983$	−3.31969	−0.105882	−0.0529409	+	0.998598i	$0.516859\pi$
$984$	−1.09679	−0.0349643
$985$	−3.04235	−0.0969374
$986$	−4.55631	−0.145103
$987$	0	0
$988$	−12.5541	−0.399400
$989$	0.151822	0.00482767
$990$	7.66302	0.243547
$991$	26.5670	0.843927	0.421964	−	0.906613i	$-0.361341\pi$
$991$	26.5670	0.843927	0.421964	+	0.906613i	$0.361341\pi$
$992$	−3.07081	−0.0974985
$993$	0.906403	0.0287638
$994$	0	0
$995$	−4.96862	−0.157516
$996$	2.66530	0.0844534
$997$	19.9788	0.632734	0.316367	−	0.948637i	$-0.397537\pi$
$997$	19.9788	0.632734	0.316367	+	0.948637i	$0.397537\pi$
$998$	15.3178	0.484877
$999$	−6.86589	−0.217227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cv.1.6	✓	10
7.6	odd	2		8330.2.a.cw.1.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8330.2.a.cv.1.6	✓	10	1.1	even	1	trivial
8330.2.a.cw.1.5	yes	10	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8330.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.255150 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.255150 q^{6} +1.00000 q^{8} -2.93490 q^{9} -1.00000 q^{10} +2.61100 q^{11} +0.255150 q^{12} -5.03829 q^{13} -0.255150 q^{15} +1.00000 q^{16} -1.00000 q^{17} -2.93490 q^{18} +2.49174 q^{19} -1.00000 q^{20} +2.61100 q^{22} -0.0298196 q^{23} +0.255150 q^{24} +1.00000 q^{25} -5.03829 q^{26} -1.51429 q^{27} +4.55631 q^{29} -0.255150 q^{30} -3.07081 q^{31} +1.00000 q^{32} +0.666196 q^{33} -1.00000 q^{34} -2.93490 q^{36} +4.53407 q^{37} +2.49174 q^{38} -1.28552 q^{39} -1.00000 q^{40} -4.29860 q^{41} -5.09136 q^{43} +2.61100 q^{44} +2.93490 q^{45} -0.0298196 q^{46} +6.22417 q^{47} +0.255150 q^{48} +1.00000 q^{50} -0.255150 q^{51} -5.03829 q^{52} -1.36002 q^{53} -1.51429 q^{54} -2.61100 q^{55} +0.635767 q^{57} +4.55631 q^{58} +14.0184 q^{59} -0.255150 q^{60} +4.98656 q^{61} -3.07081 q^{62} +1.00000 q^{64} +5.03829 q^{65} +0.666196 q^{66} +7.02027 q^{67} -1.00000 q^{68} -0.00760847 q^{69} +14.2603 q^{71} -2.93490 q^{72} -12.1613 q^{73} +4.53407 q^{74} +0.255150 q^{75} +2.49174 q^{76} -1.28552 q^{78} +8.95066 q^{79} -1.00000 q^{80} +8.41833 q^{81} -4.29860 q^{82} +10.4460 q^{83} +1.00000 q^{85} -5.09136 q^{86} +1.16254 q^{87} +2.61100 q^{88} +1.30852 q^{89} +2.93490 q^{90} -0.0298196 q^{92} -0.783517 q^{93} +6.22417 q^{94} -2.49174 q^{95} +0.255150 q^{96} -3.38746 q^{97} -7.66302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} - 2 q^{3} + 10 q^{4} - 10 q^{5} - 2 q^{6} + 10 q^{8} + 12 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 10 q^{16} - 10 q^{17} + 12 q^{18} - 14 q^{19} - 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q + 10 * q^2 - 2 * q^3 + 10 * q^4 - 10 * q^5 - 2 * q^6 + 10 * q^8 + 12 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 10 * q^16 - 10 * q^17 + 12 * q^18 - 14 * q^19 - 10 * q^20 - 2 * q^22 + 10 * q^23 - 2 * q^24 + 10 * q^25 - 4 * q^26 + 4 * q^27 + 10 * q^29 + 2 * q^30 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 10 * q^34 + 12 * q^36 - 14 * q^38 + 48 * q^39 - 10 * q^40 + 22 * q^41 + 14 * q^43 - 2 * q^44 - 12 * q^45 + 10 * q^46 - 18 * q^47 - 2 * q^48 + 10 * q^50 + 2 * q^51 - 4 * q^52 + 22 * q^53 + 4 * q^54 + 2 * q^55 + 32 * q^57 + 10 * q^58 + 6 * q^59 + 2 * q^60 + 16 * q^61 - 2 * q^62 + 10 * q^64 + 4 * q^65 - 4 * q^66 + 14 * q^67 - 10 * q^68 - 10 * q^69 + 20 * q^71 + 12 * q^72 - 12 * q^73 - 2 * q^75 - 14 * q^76 + 48 * q^78 + 44 * q^79 - 10 * q^80 + 50 * q^81 + 22 * q^82 - 20 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^87 - 2 * q^88 - 12 * q^90 + 10 * q^92 + 4 * q^93 - 18 * q^94 + 14 * q^95 - 2 * q^96 + 40 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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