LMFDB - Embedded newform 105.4.g.a.104.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,4,Mod(104,105)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("105.104");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	105.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$6.19520055060$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 9 $ x^4 - x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			104.2
Root			$1.32288 + 1.11803i$ of defining polynomial
Character	$\chi$	$=$	105.104
Dual form			105.4.g.a.104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +18.5203 q^{7} +(-13.0000 - 23.6643i) q^{9} +O(q^{10})$	\(q+(-2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +18.5203 q^{7} +(-13.0000 - 23.6643i) q^{9} +11.8322i q^{11} +(21.1660 - 35.7771i) q^{12} +84.6640 q^{13} +(50.0000 + 29.5804i) q^{15} +64.0000 q^{16} -102.859i q^{17} +89.4427i q^{20} +(-49.0000 + 82.8251i) q^{21} -125.000 q^{25} +(140.225 + 4.47214i) q^{27} -148.162 q^{28} -307.636i q^{29} +(-52.9150 - 31.3050i) q^{33} -207.063i q^{35} +(104.000 + 189.315i) q^{36} +(-224.000 + 378.629i) q^{39} -94.6573i q^{44} +(-264.575 + 145.344i) q^{45} +178.885i q^{47} +(-169.328 + 286.217i) q^{48} +343.000 q^{49} +(460.000 + 272.140i) q^{51} -677.312 q^{52} +132.288 q^{55} +(-400.000 - 236.643i) q^{60} +(-240.763 - 438.269i) q^{63} -512.000 q^{64} -946.573i q^{65} +822.873i q^{68} +863.748i q^{71} +1132.38 q^{73} +(330.719 - 559.017i) q^{75} +219.135i q^{77} +236.000 q^{79} -715.542i q^{80} +(-391.000 + 615.272i) q^{81} -1511.58i q^{83} +(392.000 - 662.601i) q^{84} -1150.00 q^{85} +(1375.79 + 813.929i) q^{87} +1568.00 q^{91} -963.053 q^{97} +(280.000 - 153.818i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} - 52 q^{9}+O(q^{10}) $ 4 * q - 32 * q^4 - 52 * q^9	$ 4 q - 32 q^{4} - 52 q^{9} + 200 q^{15} + 256 q^{16} - 196 q^{21} - 500 q^{25} + 416 q^{36} - 896 q^{39} + 1372 q^{49} + 1840 q^{51} - 1600 q^{60} - 2048 q^{64} + 944 q^{79} - 1564 q^{81} + 1568 q^{84} - 4600 q^{85} + 6272 q^{91} + 1120 q^{99}+O(q^{100}) $ 4 * q - 32 * q^4 - 52 * q^9 + 200 * q^15 + 256 * q^16 - 196 * q^21 - 500 * q^25 + 416 * q^36 - 896 * q^39 + 1372 * q^49 + 1840 * q^51 - 1600 * q^60 - 2048 * q^64 + 944 * q^79 - 1564 * q^81 + 1568 * q^84 - 4600 * q^85 + 6272 * q^91 + 1120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$.

$n$	$22$	$31$	$71$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	−2.64575	+	4.47214i	−0.509175	+	0.860663i
$3$	−2.64575	+	4.47214i	−0.509175	+	0.860663i
$4$	−8.00000			−1.00000
$5$		−	11.1803i		−	1.00000i
$5$		−	11.1803i		−	1.00000i
$6$		0			0
$7$	18.5203			1.00000
$7$	18.5203			1.00000
$8$		0			0
$9$	−13.0000	−	23.6643i	−0.481481	−	0.876456i
$10$		0			0
$11$			11.8322i			0.324321i	0.986764	+	0.162160i	$0.0518462\pi$
$11$			11.8322i			0.324321i	−0.986764	+	0.162160i	$0.948154\pi$
$12$	21.1660	−	35.7771i	0.509175	−	0.860663i
$13$	84.6640			1.80628			0.903138	−	0.429351i	$-0.141258\pi$
$13$	84.6640			1.80628			0.903138	+	0.429351i	$0.141258\pi$
$14$		0			0
$15$	50.0000	+	29.5804i	0.860663	+	0.509175i
$16$	64.0000			1.00000
$17$		−	102.859i		−	1.46747i	−0.679435	−	0.733735i	$-0.737775\pi$
$17$		−	102.859i		−	1.46747i	0.679435	−	0.733735i	$-0.262225\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$			89.4427i			1.00000i
$21$	−49.0000	+	82.8251i	−0.509175	+	0.860663i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−125.000			−1.00000
$26$		0			0
$27$	140.225	+	4.47214i	0.999492	+	0.0318764i
$28$	−148.162			−1.00000
$29$		−	307.636i		−	1.96988i	−0.172889	−	0.984941i	$-0.555310\pi$
$29$		−	307.636i		−	1.96988i	0.172889	−	0.984941i	$-0.444690\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−52.9150	−	31.3050i	−0.279131	−	0.165136i
$34$		0			0
$35$		−	207.063i		−	1.00000i
$36$	104.000	+	189.315i	0.481481	+	0.876456i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−224.000	+	378.629i	−0.919710	+	1.55459i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		−	94.6573i		−	0.324321i
$45$	−264.575	+	145.344i	−0.876456	+	0.481481i
$46$		0			0
$47$			178.885i			0.555173i	0.960701	+	0.277586i	$0.0895345\pi$
$47$			178.885i			0.555173i	−0.960701	+	0.277586i	$0.910466\pi$
$48$	−169.328	+	286.217i	−0.509175	+	0.860663i
$49$	343.000			1.00000
$50$		0			0
$51$	460.000	+	272.140i	1.26300	+	0.747200i
$52$	−677.312			−1.80628
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	132.288			0.324321
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−400.000	−	236.643i	−0.860663	−	0.509175i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−240.763	−	438.269i	−0.481481	−	0.876456i
$64$	−512.000			−1.00000
$65$		−	946.573i		−	1.80628i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$			822.873i			1.46747i
$69$		0			0
$70$		0			0
$71$			863.748i			1.44377i	0.692011	+	0.721887i	$0.256725\pi$
$71$			863.748i			1.44377i	−0.692011	+	0.721887i	$0.743275\pi$
$72$		0			0
$73$	1132.38			1.81555			0.907776	−	0.419456i	$-0.137779\pi$
$73$	1132.38			1.81555			0.907776	+	0.419456i	$0.137779\pi$
$74$		0			0
$75$	330.719	−	559.017i	0.509175	−	0.860663i
$76$		0			0
$77$			219.135i			0.324321i
$78$		0			0
$79$	236.000			0.336102			0.168051	−	0.985778i	$-0.446253\pi$
$79$	236.000			0.336102			0.168051	+	0.985778i	$0.446253\pi$
$80$		−	715.542i		−	1.00000i
$81$	−391.000	+	615.272i	−0.536351	+	0.843995i
$82$		0			0
$83$		−	1511.58i		−	1.99901i	−0.0314901	−	0.999504i	$-0.510025\pi$
$83$		−	1511.58i		−	1.99901i	0.0314901	−	0.999504i	$-0.489975\pi$
$84$	392.000	−	662.601i	0.509175	−	0.860663i
$85$	−1150.00			−1.46747
$86$		0			0
$87$	1375.79	+	813.929i	1.69541	+	1.00302i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	1568.00			1.80628
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−963.053			−1.00807			−0.504037	−	0.863682i	$-0.668153\pi$
$97$	−963.053			−1.00807			−0.504037	+	0.863682i	$0.668153\pi$
$98$		0			0
$99$	280.000	−	153.818i	0.284253	−	0.156155i
$100$	1000.00			1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	799.017			0.764364			0.382182	−	0.924087i	$-0.375173\pi$
$103$	799.017			0.764364			0.382182	+	0.924087i	$0.375173\pi$
$104$		0			0
$105$	926.013	+	547.837i	0.860663	+	0.509175i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−1121.80	−	35.7771i	−0.999492	−	0.0318764i
$109$	−2266.00			−1.99122			−0.995612	−	0.0935765i	$-0.970170\pi$
$109$	−2266.00			−1.99122			−0.995612	+	0.0935765i	$0.970170\pi$
$110$		0			0
$111$		0			0
$112$	1185.30			1.00000
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$			2461.09i			1.96988i
$117$	−1100.63	−	2003.52i	−0.869688	−	1.58312i
$118$		0			0
$119$		−	1904.98i		−	1.46747i
$120$		0			0
$121$	1191.00			0.894816
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1397.54i			1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	423.320	+	250.440i	0.279131	+	0.165136i
$133$		0			0
$134$		0			0
$135$	50.0000	−	1567.76i	0.0318764	−	0.999492i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$			1656.50i			1.00000i
$141$	−800.000	−	473.286i	−0.477817	−	0.282680i
$142$		0			0
$143$			1001.76i			0.585813i
$144$	−832.000	−	1514.52i	−0.481481	−	0.876456i
$145$	−3439.48			−1.96988
$146$		0			0
$147$	−907.493	+	1533.94i	−0.509175	+	0.860663i
$148$		0			0
$149$			2674.07i			1.47026i	0.677928	+	0.735128i	$0.262878\pi$
$149$			2674.07i			1.47026i	−0.677928	+	0.735128i	$0.737122\pi$
$150$		0			0
$151$	−2788.00			−1.50254			−0.751272	−	0.659992i	$-0.770559\pi$
$151$	−2788.00			−1.50254			−0.751272	+	0.659992i	$0.770559\pi$
$152$		0			0
$153$	−2434.09	+	1337.17i	−1.28617	+	0.706560i
$154$		0			0
$155$		0			0
$156$	1792.00	−	3029.03i	0.919710	−	1.55459i
$157$	−486.818			−0.247467			−0.123734	−	0.992315i	$-0.539487\pi$
$157$	−486.818			−0.247467			−0.123734	+	0.992315i	$0.539487\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	−350.000	+	591.608i	−0.165136	+	0.279131i
$166$		0			0
$167$		−	2558.06i		−	1.18532i	−0.805452	−	0.592661i	$-0.798077\pi$
$167$		−	2558.06i		−	1.18532i	0.805452	−	0.592661i	$-0.201923\pi$
$168$		0			0
$169$	4971.00			2.26263
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4405.05i			1.93590i	0.251150	+	0.967948i	$0.419191\pi$
$173$			4405.05i			1.93590i	−0.251150	+	0.967948i	$0.580809\pi$
$174$		0			0
$175$	−2315.03			−1.00000
$176$			757.258i			0.324321i
$177$		0			0
$178$		0			0
$179$			4697.37i			1.96144i	0.195419	+	0.980720i	$0.437393\pi$
$179$			4697.37i			1.96144i	−0.195419	+	0.980720i	$0.562607\pi$
$180$	2116.60	−	1162.76i	0.876456	−	0.481481i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1217.05			0.475931
$188$		−	1431.08i		−	0.555173i
$189$	2597.00	+	82.8251i	0.999492	+	0.0318764i
$190$		0			0
$191$		−	3182.85i		−	1.20577i	−0.797826	−	0.602887i	$-0.794017\pi$
$191$		−	3182.85i		−	1.20577i	0.797826	−	0.602887i	$-0.205983\pi$
$192$	1354.62	−	2289.73i	0.509175	−	0.860663i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	4233.20	+	2504.40i	1.55459	+	0.919710i
$196$	−2744.00			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	5697.50i		−	1.96988i
$204$	−3680.00	−	2177.12i	−1.26300	−	0.747200i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	5418.50			1.80628
$209$		0			0
$210$		0			0
$211$	−2392.00			−0.780436			−0.390218	−	0.920722i	$-0.627600\pi$
$211$	−2392.00			−0.780436			−0.390218	+	0.920722i	$0.627600\pi$
$212$		0			0
$213$	−3862.80	−	2285.26i	−1.24260	−	0.735134i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2996.00	+	5064.16i	−0.924433	+	1.56258i
$220$	−1058.30			−0.324321
$221$		−	8708.47i		−	2.65066i
$222$		0			0
$223$	5180.38			1.55562			0.777812	−	0.628498i	$-0.216330\pi$
$223$	5180.38			1.55562			0.777812	+	0.628498i	$0.216330\pi$
$224$		0			0
$225$	1625.00	+	2958.04i	0.481481	+	0.876456i
$226$		0			0
$227$			4767.30i			1.39391i	0.717117	+	0.696953i	$0.245461\pi$
$227$			4767.30i			1.39391i	−0.717117	+	0.696953i	$0.754539\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	−980.000	−	579.776i	−0.279131	−	0.165136i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	2000.00			0.555173
$236$		0			0
$237$	−624.397	+	1055.42i	−0.171135	+	0.289271i
$238$		0			0
$239$		−	4673.70i		−	1.26492i	−0.774592	−	0.632462i	$-0.782045\pi$
$239$		−	4673.70i		−	1.26492i	0.774592	−	0.632462i	$-0.217955\pi$
$240$	3200.00	+	1893.15i	0.860663	+	0.509175i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−1717.09	−	3376.46i	−0.453299	−	0.891359i
$244$		0			0
$245$		−	3834.86i		−	1.00000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	6760.00	+	3999.27i	1.72047	+	1.01785i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	1926.11	+	3506.15i	0.481481	+	0.876456i
$253$		0			0
$254$		0			0
$255$	3042.61	−	5142.96i	0.747200	−	1.26300i
$256$	4096.00			1.00000
$257$			3358.57i			0.815183i	0.913164	+	0.407592i	$0.133631\pi$
$257$			3358.57i			0.815183i	−0.913164	+	0.407592i	$0.866369\pi$
$258$		0			0
$259$		0			0
$260$			7572.58i			1.80628i
$261$	−7280.00	+	3999.27i	−1.72652	+	0.948462i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		−	6582.98i		−	1.46747i
$273$	−4148.54	+	7012.31i	−0.919710	+	1.55459i
$274$		0			0
$275$		−	1479.02i		−	0.324321i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4780.19i		−	1.01481i	−0.861707	−	0.507406i	$-0.830604\pi$
$281$		−	4780.19i		−	1.01481i	0.861707	−	0.507406i	$-0.169396\pi$
$282$		0			0
$283$	2227.72			0.467931			0.233965	−	0.972245i	$-0.424830\pi$
$283$	2227.72			0.467931			0.233965	+	0.972245i	$0.424830\pi$
$284$		−	6909.98i		−	1.44377i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5667.00			−1.15347
$290$		0			0
$291$	2548.00	−	4306.91i	0.513287	−	0.867613i
$292$	−9059.05			−1.81555
$293$			6739.51i			1.34378i	0.740653	+	0.671888i	$0.234516\pi$
$293$			6739.51i			1.34378i	−0.740653	+	0.671888i	$0.765484\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−52.9150	+	1659.16i	−0.0103382	+	0.324156i
$298$		0			0
$299$		0			0
$300$	−2645.75	+	4472.14i	−0.509175	+	0.860663i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−9011.43			−1.67527			−0.837637	−	0.546227i	$-0.816064\pi$
$307$	−9011.43			−1.67527			−0.837637	+	0.546227i	$0.816064\pi$
$308$		−	1753.08i		−	0.324321i
$309$	−2114.00	+	3573.31i	−0.389195	+	0.657860i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−2677.50			−0.483518			−0.241759	−	0.970336i	$-0.577724\pi$
$313$	−2677.50			−0.483518			−0.241759	+	0.970336i	$0.577724\pi$
$314$		0			0
$315$	−4900.00	+	2691.82i	−0.876456	+	0.481481i
$316$	−1888.00			−0.336102
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$	3640.00			0.638874
$320$			5724.33i			1.00000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	3128.00	−	4922.18i	0.536351	−	0.843995i
$325$	−10583.0			−1.80628
$326$		0			0
$327$	5995.27	−	10133.9i	1.01388	−	1.71377i
$328$		0			0
$329$			3313.00i			0.555173i
$330$		0			0
$331$	−7432.00			−1.23414			−0.617069	−	0.786909i	$-0.711680\pi$
$331$	−7432.00			−1.23414			−0.617069	+	0.786909i	$0.711680\pi$
$332$			12092.7i			1.99901i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−3136.00	+	5300.81i	−0.509175	+	0.860663i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	9200.00			1.46747
$341$		0			0
$342$		0			0
$343$	6352.45			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−11006.3	−	6511.43i	−1.69541	−	1.00302i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	11872.0	+	378.629i	1.80536	+	0.0575776i
$352$		0			0
$353$		−	6220.74i		−	0.937951i	−0.883211	−	0.468975i	$-0.844623\pi$
$353$		−	6220.74i		−	0.937951i	0.883211	−	0.468975i	$-0.155377\pi$
$354$		0			0
$355$	9656.99			1.44377
$356$		0			0
$357$	8519.32	+	5040.10i	1.26300	+	0.747200i
$358$		0			0
$359$			11086.7i			1.62990i	0.579529	+	0.814952i	$0.303237\pi$
$359$			11086.7i			1.62990i	−0.579529	+	0.814952i	$0.696763\pi$
$360$		0			0
$361$	6859.00			1.00000
$362$		0			0
$363$	−3151.09	+	5326.31i	−0.455618	+	0.770135i
$364$	−12544.0			−1.80628
$365$		−	12660.4i		−	1.81555i
$366$		0			0
$367$	14038.4			1.99672			0.998360	−	0.0572477i	$-0.0182325\pi$
$367$	14038.4			1.99672			0.998360	+	0.0572477i	$0.0182325\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−6250.00	−	3697.55i	−0.860663	−	0.509175i
$376$		0			0
$377$		−	26045.7i		−	3.55815i
$378$		0			0
$379$	14096.0			1.91046			0.955228	−	0.295870i	$-0.0956097\pi$
$379$	14096.0			1.91046			0.955228	+	0.295870i	$0.0956097\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			4686.80i			0.625285i	0.949871	+	0.312643i	$0.101214\pi$
$383$			4686.80i			0.625285i	−0.949871	+	0.312643i	$0.898786\pi$
$384$		0			0
$385$	2450.00			0.324321
$386$		0			0
$387$		0			0
$388$	7704.43			1.00807
$389$		−	6697.00i		−	0.872883i	−0.899733	−	0.436442i	$-0.856239\pi$
$389$		−	6697.00i		−	0.872883i	0.899733	−	0.436442i	$-0.143761\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	2638.56i		−	0.336102i
$396$	−2240.00	+	1230.54i	−0.284253	+	0.156155i
$397$	275.158			0.0347854			0.0173927	−	0.999849i	$-0.494463\pi$
$397$	275.158			0.0347854			0.0173927	+	0.999849i	$0.494463\pi$
$398$		0			0
$399$		0			0
$400$	−8000.00			−1.00000
$401$			9702.37i			1.20826i	0.796885	+	0.604131i	$0.206480\pi$
$401$			9702.37i			1.20826i	−0.796885	+	0.604131i	$0.793520\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	6878.95	+	4371.51i	0.843995	+	0.536351i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−6392.14			−0.764364
$413$		0			0
$414$		0			0
$415$	−16900.0			−1.99901
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−7408.10	−	4382.69i	−0.860663	−	0.509175i
$421$	−3922.00			−0.454030			−0.227015	−	0.973891i	$-0.572897\pi$
$421$	−3922.00			−0.454030			−0.227015	+	0.973891i	$0.572897\pi$
$422$		0			0
$423$	4233.20	−	2325.51i	0.486585	−	0.267305i
$424$		0			0
$425$			12857.4i			1.46747i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−4480.00	−	2650.40i	−0.504188	−	0.298281i
$430$		0			0
$431$			17476.1i			1.95312i	0.215249	+	0.976559i	$0.430944\pi$
$431$			17476.1i			1.95312i	−0.215249	+	0.976559i	$0.569056\pi$
$432$	8974.39	+	286.217i	0.999492	+	0.0318764i
$433$	12562.0			1.39421			0.697105	−	0.716970i	$-0.254471\pi$
$433$	12562.0			1.39421			0.697105	+	0.716970i	$0.254471\pi$
$434$		0			0
$435$	9100.00	−	15381.8i	1.00302	−	1.69541i
$436$	18128.0			1.99122
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−4459.00	−	8116.86i	−0.481481	−	0.876456i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−11958.8	−	7074.92i	−1.26540	−	0.748618i
$448$	−9482.37			−1.00000
$449$			15239.8i			1.60181i	0.598793	+	0.800904i	$0.295647\pi$
$449$			15239.8i			1.60181i	−0.598793	+	0.800904i	$0.704353\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	7376.35	−	12468.3i	0.765058	−	1.29318i
$454$		0			0
$455$		−	17530.8i		−	1.80628i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	460.000	−	14423.4i	0.0467777	−	1.46673i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		−	19688.7i		−	1.96988i
$465$		0			0
$466$		0			0
$467$			17164.1i			1.70077i	0.526164	+	0.850383i	$0.323630\pi$
$467$			17164.1i			1.70077i	−0.526164	+	0.850383i	$0.676370\pi$
$468$	8805.06	+	16028.1i	0.869688	+	1.58312i
$469$		0			0
$470$		0			0
$471$	1288.00	−	2177.12i	0.126004	−	0.212986i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$			15239.8i			1.46747i
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−9528.00			−0.894816
$485$			10767.3i			1.00807i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			17689.1i			1.62586i	0.582362	+	0.812930i	$0.302129\pi$
$491$			17689.1i			1.62586i	−0.582362	+	0.812930i	$0.697871\pi$
$492$		0			0
$493$	−31643.2			−2.89075
$494$		0			0
$495$	−1719.74	−	3130.50i	−0.156155	−	0.284253i
$496$		0			0
$497$			15996.8i			1.44377i
$498$		0			0
$499$	7544.00			0.676785			0.338393	−	0.941005i	$-0.390117\pi$
$499$	7544.00			0.676785			0.338393	+	0.941005i	$0.390117\pi$
$500$		−	11180.3i		−	1.00000i
$501$	11440.0	+	6768.00i	1.02016	+	0.603536i
$502$		0			0
$503$			2432.84i			0.215656i	0.994170	+	0.107828i	$0.0343896\pi$
$503$			2432.84i			0.215656i	−0.994170	+	0.107828i	$0.965610\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13152.0	+	22231.0i	−1.15208	+	1.94736i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	20972.0			1.81555
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	8933.28i		−	0.764364i
$516$		0			0
$517$	−2116.60			−0.180054
$518$		0			0
$519$	−19700.0	−	11654.7i	−1.66615	−	0.985710i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−7296.98			−0.610086			−0.305043	−	0.952339i	$-0.598671\pi$
$523$	−7296.98			−0.610086			−0.305043	+	0.952339i	$0.598671\pi$
$524$		0			0
$525$	6125.00	−	10353.1i	0.509175	−	0.860663i
$526$		0			0
$527$		0			0
$528$	−3386.56	−	2003.52i	−0.279131	−	0.165136i
$529$	−12167.0			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−21007.3	−	12428.1i	−1.68814	−	0.998716i
$538$		0			0
$539$			4058.43i			0.324321i
$540$	−400.000	+	12542.1i	−0.0318764	+	0.999492i
$541$	−9718.00			−0.772291			−0.386146	−	0.922438i	$-0.626194\pi$
$541$	−9718.00			−0.772291			−0.386146	+	0.922438i	$0.626194\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			25334.7i			1.99122i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	4370.78			0.336102
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		−	13252.0i		−	1.00000i
$561$	−3220.00	+	5442.79i	−0.242332	+	0.409617i
$562$		0			0
$563$		−	13908.3i		−	1.04115i	−0.853816	−	0.520574i	$-0.825718\pi$
$563$		−	13908.3i		−	1.04115i	0.853816	−	0.520574i	$-0.174282\pi$
$564$	6400.00	+	3786.29i	0.477817	+	0.282680i
$565$		0			0
$566$		0			0
$567$	−7241.42	+	11395.0i	−0.536351	+	0.843995i
$568$		0			0
$569$		−	25226.2i		−	1.85859i	−0.369343	−	0.929293i	$-0.620417\pi$
$569$		−	25226.2i		−	1.85859i	0.369343	−	0.929293i	$-0.379583\pi$
$570$		0			0
$571$	−22048.0			−1.61590			−0.807951	−	0.589250i	$-0.799423\pi$
$571$	−22048.0			−1.61590			−0.807951	+	0.589250i	$0.799423\pi$
$572$		−	8014.07i		−	0.585813i
$573$	14234.1	+	8421.03i	1.03777	+	0.613951i
$574$		0			0
$575$		0			0
$576$	6656.00	+	12116.1i	0.481481	+	0.876456i
$577$	18848.3			1.35991			0.679954	−	0.733255i	$-0.262000\pi$
$577$	18848.3			1.35991			0.679954	+	0.733255i	$0.262000\pi$
$578$		0			0
$579$		0			0
$580$	27515.8			1.96988
$581$		−	27994.9i		−	1.99901i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−22400.0	+	12305.4i	−1.58312	+	0.869688i
$586$		0			0
$587$		−	15035.3i		−	1.05720i	−0.848872	−	0.528598i	$-0.822718\pi$
$587$		−	15035.3i		−	1.05720i	0.848872	−	0.528598i	$-0.177282\pi$
$588$	7259.94	−	12271.5i	0.509175	−	0.860663i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1104.62i			0.0764944i	0.999268	+	0.0382472i	$0.0121774\pi$
$593$			1104.62i			0.0764944i	−0.999268	+	0.0382472i	$0.987823\pi$
$594$		0			0
$595$	−21298.3			−1.46747
$596$		−	21392.5i		−	1.47026i
$597$		0			0
$598$		0			0
$599$			27273.1i			1.86035i	0.367116	+	0.930175i	$0.380345\pi$
$599$			27273.1i			1.86035i	−0.367116	+	0.930175i	$0.619655\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	22304.0			1.50254
$605$		−	13315.8i		−	0.894816i
$606$		0			0
$607$	9085.51			0.607528			0.303764	−	0.952747i	$-0.401757\pi$
$607$	9085.51			0.607528			0.303764	+	0.952747i	$0.401757\pi$
$608$		0			0
$609$	25480.0	+	15074.2i	1.69541	+	1.00302i
$610$		0			0
$611$			15145.2i			1.00280i
$612$	19472.7	−	10697.3i	1.28617	−	0.706560i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−14336.0	+	24232.3i	−0.919710	+	1.55459i
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$	3894.55			0.247467
$629$		0			0
$630$		0			0
$631$	14852.0			0.937003			0.468501	−	0.883463i	$-0.344794\pi$
$631$	14852.0			0.937003			0.468501	+	0.883463i	$0.344794\pi$
$632$		0			0
$633$	6328.64	−	10697.3i	0.397379	−	0.671693i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	29039.8			1.80628
$638$		0			0
$639$	20440.0	−	11228.7i	1.26541	−	0.695151i
$640$		0			0
$641$		−	15003.2i		−	0.924477i	−0.886756	−	0.462239i	$-0.847046\pi$
$641$		−	15003.2i		−	0.924477i	0.886756	−	0.462239i	$-0.152954\pi$
$642$		0			0
$643$	−30061.0			−1.84369			−0.921844	−	0.387562i	$-0.873317\pi$
$643$	−30061.0			−1.84369			−0.921844	+	0.387562i	$0.873317\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			28997.3i			1.76198i	0.473133	+	0.880991i	$0.343123\pi$
$647$			28997.3i			1.76198i	−0.473133	+	0.880991i	$0.656877\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−14721.0	−	26797.0i	−0.874154	−	1.59125i
$658$		0			0
$659$		−	11489.0i		−	0.679133i	−0.940582	−	0.339567i	$-0.889720\pi$
$659$		−	11489.0i		−	0.679133i	0.940582	−	0.339567i	$-0.110280\pi$
$660$	2800.00	−	4732.86i	0.165136	−	0.279131i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	38945.5	+	23040.4i	2.28132	+	1.34965i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$			20464.5i			1.18532i
$669$	−13706.0	+	23167.4i	−0.792085	+	1.33887i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−17528.1	−	559.017i	−0.999492	−	0.0318764i
$676$	−39768.0			−2.26263
$677$		−	31577.8i		−	1.79266i	−0.443386	−	0.896331i	$-0.646223\pi$
$677$		−	31577.8i		−	1.79266i	0.443386	−	0.896331i	$-0.353777\pi$
$678$		0			0
$679$	−17836.0			−1.00807
$680$		0			0
$681$	−21320.0	−	12613.1i	−1.19968	−	0.709742i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		−	35240.4i		−	1.93590i
$693$	5185.67	−	2848.75i	0.284253	−	0.156155i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	18520.3			1.00000
$701$		−	16068.1i		−	0.865739i	−0.901457	−	0.432869i	$-0.857501\pi$
$701$		−	16068.1i		−	0.865739i	0.901457	−	0.432869i	$-0.142499\pi$
$702$		0			0
$703$		0			0
$704$		−	6058.07i		−	0.324321i
$705$	−5291.50	+	8944.27i	−0.282680	+	0.477817i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2126.00			0.112614			0.0563072	−	0.998413i	$-0.482067\pi$
$709$	2126.00			0.112614			0.0563072	+	0.998413i	$0.482067\pi$
$710$		0			0
$711$	−3068.00	−	5584.78i	−0.161827	−	0.294579i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	11200.0			0.585813
$716$		−	37578.9i		−	1.96144i
$717$	20901.4	+	12365.5i	1.08867	+	0.644068i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−16932.8	+	9302.04i	−0.876456	+	0.481481i
$721$	14798.0			0.764364
$722$		0			0
$723$		0			0
$724$		0			0
$725$			38454.5i			1.96988i
$726$		0			0
$727$	−11392.6			−0.581194			−0.290597	−	0.956845i	$-0.593854\pi$
$727$	−11392.6			−0.581194			−0.290597	+	0.956845i	$0.593854\pi$
$728$		0			0
$729$	19643.0	+	1254.21i	0.997968	+	0.0637204i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−16488.3			−0.830846			−0.415423	−	0.909628i	$-0.636366\pi$
$733$	−16488.3			−0.830846			−0.415423	+	0.909628i	$0.636366\pi$
$734$		0			0
$735$	17150.0	+	10146.1i	0.860663	+	0.509175i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−23056.0			−1.14767			−0.573835	−	0.818971i	$-0.694545\pi$
$739$	−23056.0			−1.14767			−0.573835	+	0.818971i	$0.694545\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	29897.0			1.47026
$746$		0			0
$747$	−35770.6	+	19650.6i	−1.75204	+	0.962485i
$748$	−9736.36			−0.475931
$749$		0			0
$750$		0			0
$751$	27092.0			1.31638			0.658190	−	0.752852i	$-0.271322\pi$
$751$	27092.0			1.31638			0.658190	+	0.752852i	$0.271322\pi$
$752$			11448.7i			0.555173i
$753$		0			0
$754$		0			0
$755$			31170.8i			1.50254i
$756$	−20776.0	−	662.601i	−0.999492	−	0.0318764i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−41966.9			−1.99122
$764$			25462.8i			1.20577i
$765$	14950.0	+	27214.0i	0.706560	+	1.28617i
$766$		0			0
$767$		0			0
$768$	−10837.0	+	18317.9i	−0.509175	+	0.860663i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−15020.0	−	8885.95i	−0.701598	−	0.415071i
$772$		0			0
$773$		−	5174.26i		−	0.240757i	−0.992728	−	0.120379i	$-0.961589\pi$
$773$		−	5174.26i		−	0.240757i	0.992728	−	0.120379i	$-0.0384108\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−33865.6	−	20035.2i	−1.55459	−	0.919710i
$781$	−10220.0			−0.468246
$782$		0			0
$783$	1375.79	−	43138.2i	0.0627928	−	1.96888i
$784$	21952.0			1.00000
$785$			5442.79i			0.247467i
$786$		0			0
$787$	−15202.5			−0.688577			−0.344289	−	0.938864i	$-0.611880\pi$
$787$	−15202.5			−0.688577			−0.344289	+	0.938864i	$0.611880\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			41031.8i			1.82362i	0.410616	+	0.911808i	$0.365314\pi$
$797$			41031.8i			1.82362i	−0.410616	+	0.911808i	$0.634686\pi$
$798$		0			0
$799$	18400.0			0.814700
$800$		0			0
$801$		0			0
$802$		0			0
$803$			13398.5i			0.588821i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	29485.7i		−	1.28141i	−0.767786	−	0.640707i	$-0.778641\pi$
$809$		−	29485.7i		−	1.28141i	0.767786	−	0.640707i	$-0.221359\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$			45580.0i			1.96988i
$813$		0			0
$814$		0			0
$815$		0			0
$816$	29440.0	+	17416.9i	1.26300	+	0.747200i
$817$		0			0
$818$		0			0
$819$	−20384.0	−	37105.7i	−0.869688	−	1.58312i
$820$		0			0
$821$		−	14364.2i		−	0.610616i	−0.952254	−	0.305308i	$-0.901241\pi$
$821$		−	14364.2i		−	0.610616i	0.952254	−	0.305308i	$-0.0987594\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	6614.38	+	3913.12i	0.279131	+	0.165136i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−43348.0			−1.80628
$833$		−	35280.7i		−	1.46747i
$834$		0			0
$835$	−28600.0			−1.18532
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−70251.0			−2.88044
$842$		0			0
$843$	21377.7	+	12647.2i	0.873412	+	0.516717i
$844$	19136.0			0.780436
$845$		−	55577.5i		−	2.26263i
$846$		0			0
$847$	22057.6			0.894816
$848$		0			0
$849$	−5894.00	+	9962.68i	−0.238259	+	0.402731i
$850$		0			0
$851$		0			0
$852$	30902.4	+	18282.1i	1.24260	+	0.735134i
$853$	32468.7			1.30329			0.651645	−	0.758524i	$-0.274079\pi$
$853$	32468.7			1.30329			0.651645	+	0.758524i	$0.274079\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	7428.22i		−	0.296083i	−0.988981	−	0.148041i	$-0.952703\pi$
$857$		−	7428.22i		−	0.296083i	0.988981	−	0.148041i	$-0.0472969\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	49250.0			1.93590
$866$		0			0
$867$	14993.5	−	25343.6i	0.587318	−	0.992749i
$868$		0			0
$869$			2792.39i			0.109005i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	12519.7	+	22790.0i	0.485369	+	0.883534i
$874$		0			0
$875$			25882.8i			1.00000i
$876$	23968.0	−	40513.3i	0.924433	−	1.56258i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	−30140.0	−	17831.1i	−1.15654	−	0.684217i
$880$	8466.40			0.324321
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$			69667.8i			2.65066i
$885$		0			0
$886$		0			0
$887$		−	49408.2i		−	1.87031i	−0.354241	−	0.935154i	$-0.615261\pi$
$887$		−	49408.2i		−	1.87031i	0.354241	−	0.935154i	$-0.384739\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−7280.00	−	4626.37i	−0.273725	−	0.173950i
$892$	−41443.0			−1.55562
$893$		0			0
$894$		0			0
$895$	52518.2			1.96144
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−13000.0	−	23664.3i	−0.481481	−	0.876456i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		−	38138.4i		−	1.39391i
$909$		0			0
$910$		0			0
$911$			45376.3i			1.65026i	0.564944	+	0.825129i	$0.308898\pi$
$911$			45376.3i			1.65026i	−0.564944	+	0.825129i	$0.691102\pi$
$912$		0			0
$913$	17885.3			0.648320
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−55564.0			−1.99444			−0.997218	−	0.0745363i	$-0.976252\pi$
$919$	−55564.0			−1.99444			−0.997218	+	0.0745363i	$0.976252\pi$
$920$		0			0
$921$	23842.0	−	40300.3i	0.853008	−	1.44185i
$922$		0			0
$923$			73128.4i			2.60785i
$924$	7840.00	+	4638.21i	0.279131	+	0.165136i
$925$		0			0
$926$		0			0
$927$	−10387.2	−	18908.2i	−0.368027	−	0.669931i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	13607.0i		−	0.475931i
$936$		0			0
$937$	54280.2			1.89248			0.946242	−	0.323459i	$-0.104846\pi$
$937$	54280.2			1.89248			0.946242	+	0.323459i	$0.104846\pi$
$938$		0			0
$939$	7084.00	−	11974.1i	0.246196	−	0.416146i
$940$	−16000.0			−0.555173
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	926.013	−	29035.3i	0.0318764	−	0.999492i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$	4995.18	−	8443.39i	0.171135	−	0.289271i
$949$	95872.0			3.27939
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−35585.4			−1.20577
$956$			37389.6i			1.26492i
$957$	−9630.53	+	16278.6i	−0.325299	+	0.549855i
$958$		0			0
$959$		0			0
$960$	−25600.0	−	15145.2i	−0.860663	−	0.509175i
$961$	29791.0			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	13736.7	+	27011.7i	0.453299	+	0.891359i
$973$		0			0
$974$		0			0
$975$	28000.0	−	47328.6i	0.919710	−	1.55459i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$			30678.9i			1.00000i
$981$	29458.0	+	53623.3i	0.958738	+	1.74522i
$982$		0			0
$983$		−	61160.9i		−	1.98447i	−0.124390	−	0.992233i	$-0.539698\pi$
$983$		−	61160.9i		−	1.98447i	0.124390	−	0.992233i	$-0.460302\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−14816.2	−	8765.39i	−0.477817	−	0.282680i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	13988.0			0.448379			0.224189	−	0.974546i	$-0.428027\pi$
$991$	13988.0			0.448379			0.224189	+	0.974546i	$0.428027\pi$
$992$		0			0
$993$	19663.2	−	33236.9i	0.628392	−	1.06218i
$994$		0			0
$995$		0			0
$996$	−54080.0	−	31994.2i	−1.72047	−	1.01785i
$997$	49041.6			1.55784			0.778919	−	0.627125i	$-0.215768\pi$
$997$	49041.6			1.55784			0.778919	+	0.627125i	$0.215768\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.g.a.104.2	yes	4
3.2	odd	2	inner	105.4.g.a.104.1	✓	4
5.4	even	2	inner	105.4.g.a.104.3	yes	4
7.6	odd	2	inner	105.4.g.a.104.3	yes	4
15.14	odd	2	inner	105.4.g.a.104.4	yes	4
21.20	even	2	inner	105.4.g.a.104.4	yes	4
35.34	odd	2	CM	105.4.g.a.104.2	yes	4
105.104	even	2	inner	105.4.g.a.104.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.g.a.104.1	✓	4	3.2	odd	2	inner
105.4.g.a.104.1	✓	4	105.104	even	2	inner
105.4.g.a.104.2	yes	4	1.1	even	1	trivial
105.4.g.a.104.2	yes	4	35.34	odd	2	CM
105.4.g.a.104.3	yes	4	5.4	even	2	inner
105.4.g.a.104.3	yes	4	7.6	odd	2	inner
105.4.g.a.104.4	yes	4	15.14	odd	2	inner
105.4.g.a.104.4	yes	4	21.20	even	2	inner

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:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +18.5203 q^{7} +(-13.0000 - 23.6643i) q^{9} +O(q^{10})\)	\(q+(-2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +18.5203 q^{7} +(-13.0000 - 23.6643i) q^{9} +11.8322i q^{11} +(21.1660 - 35.7771i) q^{12} +84.6640 q^{13} +(50.0000 + 29.5804i) q^{15} +64.0000 q^{16} -102.859i q^{17} +89.4427i q^{20} +(-49.0000 + 82.8251i) q^{21} -125.000 q^{25} +(140.225 + 4.47214i) q^{27} -148.162 q^{28} -307.636i q^{29} +(-52.9150 - 31.3050i) q^{33} -207.063i q^{35} +(104.000 + 189.315i) q^{36} +(-224.000 + 378.629i) q^{39} -94.6573i q^{44} +(-264.575 + 145.344i) q^{45} +178.885i q^{47} +(-169.328 + 286.217i) q^{48} +343.000 q^{49} +(460.000 + 272.140i) q^{51} -677.312 q^{52} +132.288 q^{55} +(-400.000 - 236.643i) q^{60} +(-240.763 - 438.269i) q^{63} -512.000 q^{64} -946.573i q^{65} +822.873i q^{68} +863.748i q^{71} +1132.38 q^{73} +(330.719 - 559.017i) q^{75} +219.135i q^{77} +236.000 q^{79} -715.542i q^{80} +(-391.000 + 615.272i) q^{81} -1511.58i q^{83} +(392.000 - 662.601i) q^{84} -1150.00 q^{85} +(1375.79 + 813.929i) q^{87} +1568.00 q^{91} -963.053 q^{97} +(280.000 - 153.818i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} - 52 q^{9}+O(q^{10}) \) 4 * q - 32 * q^4 - 52 * q^9	\( 4 q - 32 q^{4} - 52 q^{9} + 200 q^{15} + 256 q^{16} - 196 q^{21} - 500 q^{25} + 416 q^{36} - 896 q^{39} + 1372 q^{49} + 1840 q^{51} - 1600 q^{60} - 2048 q^{64} + 944 q^{79} - 1564 q^{81} + 1568 q^{84} - 4600 q^{85} + 6272 q^{91} + 1120 q^{99}+O(q^{100}) \) 4 * q - 32 * q^4 - 52 * q^9 + 200 * q^15 + 256 * q^16 - 196 * q^21 - 500 * q^25 + 416 * q^36 - 896 * q^39 + 1372 * q^49 + 1840 * q^51 - 1600 * q^60 - 2048 * q^64 + 944 * q^79 - 1564 * q^81 + 1568 * q^84 - 4600 * q^85 + 6272 * q^91 + 1120 * q^99

Introduction

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