LMFDB - Embedded newform 435.2.c.e.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(349,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("435.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 435 = 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	435.c (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:	$3.47349248793$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.3899266318336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2 $ x^10 - 2x^9 + 2x^8 + 6x^7 + 19x^6 - 12x^5 + 4x^4 + 2x^3 + 9x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.20964 + 1.20964i$ of defining polynomial
Character	$\chi$	$=$	435.349
Dual form			435.2.c.e.349.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.51908i q^{2} +1.00000i q^{3} -4.34577 q^{4} +(-1.27413 + 1.83755i) q^{5} +2.51908 q^{6} -0.173311i q^{7} +5.90919i q^{8} -1.00000 q^{9} +(4.62893 + 3.20964i) q^{10} +5.08250 q^{11} -4.34577i q^{12} +1.82669i q^{13} -0.436584 q^{14} +(-1.83755 - 1.27413i) q^{15} +6.19418 q^{16} +4.24598i q^{17} +2.51908i q^{18} +8.62093 q^{19} +(5.53709 - 7.98556i) q^{20} +0.173311 q^{21} -12.8032i q^{22} +3.16348i q^{23} -5.90919 q^{24} +(-1.75317 - 4.68256i) q^{25} +4.60158 q^{26} -1.00000i q^{27} +0.753170i q^{28} -1.00000 q^{29} +(-3.20964 + 4.62893i) q^{30} +3.22185 q^{31} -3.78527i q^{32} +5.08250i q^{33} +10.6960 q^{34} +(0.318467 + 0.220821i) q^{35} +4.34577 q^{36} -1.97828i q^{37} -21.7168i q^{38} -1.82669 q^{39} +(-10.8584 - 7.52909i) q^{40} -9.96541 q^{41} -0.436584i q^{42} +7.91070i q^{43} -22.0874 q^{44} +(1.27413 - 1.83755i) q^{45} +7.96907 q^{46} +8.66893i q^{47} +6.19418i q^{48} +6.96996 q^{49} +(-11.7958 + 4.41638i) q^{50} -4.24598 q^{51} -7.93837i q^{52} +5.40285i q^{53} -2.51908 q^{54} +(-6.47578 + 9.33933i) q^{55} +1.02413 q^{56} +8.62093i q^{57} +2.51908i q^{58} -7.66286 q^{59} +(7.98556 + 5.53709i) q^{60} +2.76215 q^{61} -8.11611i q^{62} +0.173311i q^{63} +2.85296 q^{64} +(-3.35663 - 2.32744i) q^{65} +12.8032 q^{66} -8.13872i q^{67} -18.4520i q^{68} -3.16348 q^{69} +(0.556267 - 0.802245i) q^{70} -6.25938 q^{71} -5.90919i q^{72} -6.74356i q^{73} -4.98345 q^{74} +(4.68256 - 1.75317i) q^{75} -37.4646 q^{76} -0.880853i q^{77} +4.60158i q^{78} -4.54763 q^{79} +(-7.89221 + 11.3821i) q^{80} +1.00000 q^{81} +25.1037i q^{82} -11.6224i q^{83} -0.753170 q^{84} +(-7.80219 - 5.40994i) q^{85} +19.9277 q^{86} -1.00000i q^{87} +30.0334i q^{88} +16.4911 q^{89} +(-4.62893 - 3.20964i) q^{90} +0.316585 q^{91} -13.7478i q^{92} +3.22185i q^{93} +21.8377 q^{94} +(-10.9842 + 15.8414i) q^{95} +3.78527 q^{96} -2.74419i q^{97} -17.5579i q^{98} -5.08250 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9} + 4 q^{10} + 24 q^{11} - 12 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{19} + 8 q^{20} + 16 q^{21} - 18 q^{24} + 2 q^{25} - 10 q^{29} - 18 q^{30} + 4 q^{31} - 8 q^{34} + 2 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) $ 10 * q - 10 * q^4 + 6 * q^6 - 10 * q^9 + 4 * q^10 + 24 * q^11 - 12 * q^14 + 2 * q^15 + 2 * q^16 + 4 * q^19 + 8 * q^20 + 16 * q^21 - 18 * q^24 + 2 * q^25 - 10 * q^29 - 18 * q^30 + 4 * q^31 - 8 * q^34 + 2 * q^35 + 10 * q^36 - 4 * q^39 - 14 * q^40 - 28 * q^41 - 40 * q^44 - 12 * q^46 + 14 * q^49 - 12 * q^50 - 6 * q^54 + 2 * q^55 - 4 * q^56 - 8 * q^59 - 2 * q^60 - 24 * q^61 + 18 * q^64 + 6 * q^65 + 28 * q^66 - 16 * q^69 + 20 * q^70 + 60 * q^71 - 4 * q^74 + 12 * q^75 - 88 * q^76 + 36 * q^79 - 30 * q^80 + 10 * q^81 + 12 * q^84 - 14 * q^85 + 60 * q^86 + 44 * q^89 - 4 * q^90 - 24 * q^91 + 100 * q^94 - 36 * q^95 + 2 * q^96 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$.

$n$	$31$	$146$	$262$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	2.51908i		−	1.78126i	−0.454729	−	0.890630i	$-0.650264\pi$
$2$		−	2.51908i		−	1.78126i	0.454729	−	0.890630i	$-0.349736\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−4.34577			−2.17289
$5$	−1.27413	+	1.83755i	−0.569810	+	0.821777i
$5$	−1.27413	+	1.83755i	−0.569810	+	0.821777i
$6$	2.51908			1.02841
$7$		−	0.173311i		−	0.0655054i	−0.999463	−	0.0327527i	$-0.989573\pi$
$7$		−	0.173311i		−	0.0655054i	0.999463	−	0.0327527i	$-0.0104274\pi$
$8$			5.90919i			2.08921i
$9$	−1.00000			−0.333333
$10$	4.62893	+	3.20964i	1.46380	+	1.01498i
$11$	5.08250			1.53243			0.766215	−	0.642584i	$-0.222138\pi$
$11$	5.08250			1.53243			0.766215	+	0.642584i	$0.222138\pi$
$12$		−	4.34577i		−	1.25452i
$13$			1.82669i			0.506632i	0.967384	+	0.253316i	$0.0815213\pi$
$13$			1.82669i			0.506632i	−0.967384	+	0.253316i	$0.918479\pi$
$14$	−0.436584			−0.116682
$15$	−1.83755	−	1.27413i	−0.474453	−	0.328980i
$16$	6.19418			1.54854
$17$			4.24598i			1.02980i	0.857250	+	0.514901i	$0.172171\pi$
$17$			4.24598i			1.02980i	−0.857250	+	0.514901i	$0.827829\pi$
$18$			2.51908i			0.593753i
$19$	8.62093			1.97778			0.988889	−	0.148656i	$-0.0474948\pi$
$19$	8.62093			1.97778			0.988889	+	0.148656i	$0.0474948\pi$
$20$	5.53709	−	7.98556i	1.23813	−	1.78563i
$21$	0.173311			0.0378196
$22$		−	12.8032i		−	2.72966i
$23$			3.16348i			0.659632i	0.944045	+	0.329816i	$0.106987\pi$
$23$			3.16348i			0.659632i	−0.944045	+	0.329816i	$0.893013\pi$
$24$	−5.90919			−1.20621
$25$	−1.75317	−	4.68256i	−0.350634	−	0.936513i
$26$	4.60158			0.902444
$27$		−	1.00000i		−	0.192450i
$28$			0.753170i			0.142336i
$29$	−1.00000			−0.185695
$29$	−1.00000			−0.185695
$30$	−3.20964	+	4.62893i	−0.585998	+	0.845124i
$31$	3.22185			0.578662			0.289331	−	0.957229i	$-0.406567\pi$
$31$	3.22185			0.578662			0.289331	+	0.957229i	$0.406567\pi$
$32$		−	3.78527i		−	0.669147i
$33$			5.08250i			0.884749i
$34$	10.6960			1.83434
$35$	0.318467	+	0.220821i	0.0538308	+	0.0373256i
$36$	4.34577			0.724295
$37$		−	1.97828i		−	0.325227i	−0.986690	−	0.162614i	$-0.948008\pi$
$37$		−	1.97828i		−	0.325227i	0.986690	−	0.162614i	$-0.0519924\pi$
$38$		−	21.7168i		−	3.52294i
$39$	−1.82669			−0.292504
$40$	−10.8584	−	7.52909i	−1.71687	−	1.19045i
$41$	−9.96541			−1.55634			−0.778168	−	0.628056i	$-0.783851\pi$
$41$	−9.96541			−1.55634			−0.778168	+	0.628056i	$0.783851\pi$
$42$		−	0.436584i		−	0.0673664i
$43$			7.91070i			1.20637i	0.797601	+	0.603185i	$0.206102\pi$
$43$			7.91070i			1.20637i	−0.797601	+	0.603185i	$0.793898\pi$
$44$	−22.0874			−3.32980
$45$	1.27413	−	1.83755i	0.189937	−	0.273926i
$46$	7.96907			1.17497
$47$			8.66893i			1.26449i	0.774767	+	0.632246i	$0.217867\pi$
$47$			8.66893i			1.26449i	−0.774767	+	0.632246i	$0.782133\pi$
$48$			6.19418i			0.894053i
$49$	6.96996			0.995709
$50$	−11.7958	+	4.41638i	−1.66817	+	0.624570i
$51$	−4.24598			−0.594556
$52$		−	7.93837i		−	1.10085i
$53$			5.40285i			0.742138i	0.928605	+	0.371069i	$0.121009\pi$
$53$			5.40285i			0.742138i	−0.928605	+	0.371069i	$0.878991\pi$
$54$	−2.51908			−0.342804
$55$	−6.47578	+	9.33933i	−0.873194	+	1.25932i
$56$	1.02413			0.136855
$57$			8.62093i			1.14187i
$58$			2.51908i			0.330772i
$59$	−7.66286			−0.997619			−0.498810	−	0.866712i	$-0.666229\pi$
$59$	−7.66286			−0.997619			−0.498810	+	0.866712i	$0.666229\pi$
$60$	7.98556	+	5.53709i	1.03093	+	0.714835i
$61$	2.76215			0.353657			0.176828	−	0.984242i	$-0.443416\pi$
$61$	2.76215			0.353657			0.176828	+	0.984242i	$0.443416\pi$
$62$		−	8.11611i		−	1.03075i
$63$			0.173311i			0.0218351i
$64$	2.85296			0.356620
$65$	−3.35663	−	2.32744i	−0.416339	−	0.288684i
$66$	12.8032			1.57597
$67$		−	8.13872i		−	0.994303i	−0.867664	−	0.497152i	$-0.834379\pi$
$67$		−	8.13872i		−	0.994303i	0.867664	−	0.497152i	$-0.165621\pi$
$68$		−	18.4520i		−	2.23764i
$69$	−3.16348			−0.380838
$70$	0.556267	−	0.802245i	0.0664866	−	0.0958866i
$71$	−6.25938			−0.742852			−0.371426	−	0.928463i	$-0.621131\pi$
$71$	−6.25938			−0.742852			−0.371426	+	0.928463i	$0.621131\pi$
$72$		−	5.90919i		−	0.696404i
$73$		−	6.74356i		−	0.789274i	−0.918837	−	0.394637i	$-0.870870\pi$
$73$		−	6.74356i		−	0.789274i	0.918837	−	0.394637i	$-0.129130\pi$
$74$	−4.98345			−0.579314
$75$	4.68256	−	1.75317i	0.540696	−	0.202439i
$76$	−37.4646			−4.29748
$77$		−	0.880853i		−	0.100382i
$78$			4.60158i			0.521026i
$79$	−4.54763			−0.511649			−0.255824	−	0.966723i	$-0.582347\pi$
$79$	−4.54763			−0.511649			−0.255824	+	0.966723i	$0.582347\pi$
$80$	−7.89221	+	11.3821i	−0.882376	+	1.27256i
$81$	1.00000			0.111111
$82$			25.1037i			2.77224i
$83$		−	11.6224i		−	1.27573i	−0.770149	−	0.637865i	$-0.779818\pi$
$83$		−	11.6224i		−	1.27573i	0.770149	−	0.637865i	$-0.220182\pi$
$84$	−0.753170			−0.0821776
$85$	−7.80219	−	5.40994i	−0.846267	−	0.586791i
$86$	19.9277			2.14886
$87$		−	1.00000i		−	0.107211i
$88$			30.0334i			3.20157i
$89$	16.4911			1.74805			0.874024	−	0.485882i	$-0.161501\pi$
$89$	16.4911			1.74805			0.874024	+	0.485882i	$0.161501\pi$
$90$	−4.62893	−	3.20964i	−0.487933	−	0.338326i
$91$	0.316585			0.0331872
$92$		−	13.7478i		−	1.43330i
$93$			3.22185i			0.334090i
$94$	21.8377			2.25239
$95$	−10.9842	+	15.8414i	−1.12696	+	1.62529i
$96$	3.78527			0.386332
$97$		−	2.74419i		−	0.278631i	−0.990248	−	0.139315i	$-0.955510\pi$
$97$		−	2.74419i		−	0.278631i	0.990248	−	0.139315i	$-0.0444901\pi$
$98$		−	17.5579i		−	1.77362i
$99$	−5.08250			−0.510810
$100$	7.61887	+	20.3493i	0.761887	+	2.03493i
$101$	6.41474			0.638290			0.319145	−	0.947706i	$-0.396604\pi$
$101$	6.41474			0.638290			0.319145	+	0.947706i	$0.396604\pi$
$102$			10.6960i			1.05906i
$103$		−	7.08642i		−	0.698245i	−0.937077	−	0.349123i	$-0.886480\pi$
$103$		−	7.08642i		−	0.698245i	0.937077	−	0.349123i	$-0.113520\pi$
$104$	−10.7942			−1.05846
$105$	−0.220821	+	0.318467i	−0.0215499	+	0.0310792i
$106$	13.6102			1.32194
$107$		−	0.925187i		−	0.0894412i	−0.999000	−	0.0447206i	$-0.985760\pi$
$107$		−	0.925187i		−	0.0894412i	0.999000	−	0.0447206i	$-0.0142398\pi$
$108$			4.34577i			0.418172i
$109$	−4.11700			−0.394337			−0.197169	−	0.980370i	$-0.563175\pi$
$109$	−4.11700			−0.394337			−0.197169	+	0.980370i	$0.563175\pi$
$110$	23.5265	+	16.3130i	2.24317	+	1.55538i
$111$	1.97828			0.187770
$112$		−	1.07352i		−	0.101438i
$113$			6.84626i			0.644042i	0.946732	+	0.322021i	$0.104362\pi$
$113$			6.84626i			0.644042i	−0.946732	+	0.322021i	$0.895638\pi$
$114$	21.7168			2.03397
$115$	−5.81305	−	4.03070i	−0.542070	−	0.375864i
$116$	4.34577			0.403495
$117$		−	1.82669i		−	0.168877i
$118$			19.3034i			1.77702i
$119$	0.735875			0.0674575
$120$	7.52909	−	10.8584i	0.687309	−	0.991233i
$121$	14.8318			1.34834
$122$		−	6.95807i		−	0.629954i
$123$		−	9.96541i		−	0.898551i
$124$	−14.0014			−1.25737
$125$	10.8382	+	2.74467i	0.969399	+	0.245491i
$126$	0.436584			0.0388940
$127$			5.25850i			0.466617i	0.972403	+	0.233308i	$0.0749551\pi$
$127$			5.25850i			0.466617i	−0.972403	+	0.233308i	$0.925045\pi$
$128$		−	14.7574i		−	1.30438i
$129$	−7.91070			−0.696498
$130$	−5.86302	+	8.45562i	−0.514221	+	0.741607i
$131$	17.7157			1.54783			0.773913	−	0.633293i	$-0.218297\pi$
$131$	17.7157			1.54783			0.773913	+	0.633293i	$0.218297\pi$
$132$		−	22.0874i		−	1.92246i
$133$		−	1.49410i		−	0.129555i
$134$	−20.5021			−1.77111
$135$	1.83755	+	1.27413i	0.158151	+	0.109660i
$136$	−25.0903			−2.15147
$137$			1.14479i			0.0978057i	0.998804	+	0.0489028i	$0.0155725\pi$
$137$			1.14479i			0.0978057i	−0.998804	+	0.0489028i	$0.984428\pi$
$138$			7.96907i			0.678372i
$139$	−4.08670			−0.346630			−0.173315	−	0.984866i	$-0.555448\pi$
$139$	−4.08670			−0.346630			−0.173315	+	0.984866i	$0.555448\pi$
$140$	−1.38399	−	0.959639i	−0.116968	−	0.0811043i
$141$	−8.66893			−0.730055
$142$			15.7679i			1.32321i
$143$			9.28414i			0.776379i
$144$	−6.19418			−0.516182
$145$	1.27413	−	1.83755i	0.105811	−	0.152600i
$146$	−16.9876			−1.40590
$147$			6.96996i			0.574873i
$148$			8.59715i			0.706682i
$149$	−1.50268			−0.123105			−0.0615523	−	0.998104i	$-0.519605\pi$
$149$	−1.50268			−0.123105			−0.0615523	+	0.998104i	$0.519605\pi$
$150$	−4.41638	−	11.7958i	−0.360596	−	0.963119i
$151$	−21.4897			−1.74881			−0.874404	−	0.485199i	$-0.838747\pi$
$151$	−21.4897			−1.74881			−0.874404	+	0.485199i	$0.838747\pi$
$152$			50.9427i			4.13200i
$153$		−	4.24598i		−	0.343267i
$154$	−2.21894			−0.178807
$155$	−4.10507	+	5.92031i	−0.329727	+	0.475531i
$156$	7.93837			0.635578
$157$		−	11.1167i		−	0.887206i	−0.896223	−	0.443603i	$-0.853700\pi$
$157$		−	11.1167i		−	0.887206i	0.896223	−	0.443603i	$-0.146300\pi$
$158$			11.4559i			0.911379i
$159$	−5.40285			−0.428474
$160$	6.95561	+	4.82293i	0.549889	+	0.381286i
$161$	0.548266			0.0432094
$162$		−	2.51908i		−	0.197918i
$163$		−	12.2180i		−	0.956988i	−0.878091	−	0.478494i	$-0.841183\pi$
$163$		−	12.2180i		−	0.956988i	0.878091	−	0.478494i	$-0.158817\pi$
$164$	43.3074			3.38174
$165$	−9.33933	−	6.47578i	−0.727066	−	0.504139i
$166$	−29.2779			−2.27240
$167$		−	21.3634i		−	1.65315i	−0.562826	−	0.826576i	$-0.690286\pi$
$167$		−	21.3634i		−	1.65315i	0.562826	−	0.826576i	$-0.309714\pi$
$168$			1.02413i			0.0790131i
$169$	9.66321			0.743324
$170$	−13.6281	+	19.6544i	−1.04523	+	1.50742i
$171$	−8.62093			−0.659259
$172$		−	34.3781i		−	2.62130i
$173$			13.1686i			1.00119i	0.865682	+	0.500594i	$0.166885\pi$
$173$			13.1686i			1.00119i	−0.865682	+	0.500594i	$0.833115\pi$
$174$	−2.51908			−0.190971
$175$	−0.811540	+	0.303844i	−0.0613466	+	0.0229684i
$176$	31.4819			2.37304
$177$		−	7.66286i		−	0.575976i
$178$		−	41.5423i		−	3.11373i
$179$	−10.6947			−0.799357			−0.399679	−	0.916655i	$-0.630878\pi$
$179$	−10.6947			−0.799357			−0.399679	+	0.916655i	$0.630878\pi$
$180$	−5.53709	+	7.98556i	−0.412710	+	0.595209i
$181$	−15.2511			−1.13361			−0.566804	−	0.823852i	$-0.691820\pi$
$181$	−15.2511			−1.13361			−0.566804	+	0.823852i	$0.691820\pi$
$182$		−	0.797504i		−	0.0591149i
$183$			2.76215i			0.204184i
$184$	−18.6936			−1.37811
$185$	3.63519	+	2.52059i	0.267264	+	0.185318i
$186$	8.11611			0.595102
$187$			21.5802i			1.57810i
$188$		−	37.6732i		−	2.74760i
$189$	−0.173311			−0.0126065
$190$	39.9057	+	27.6701i	2.89507	+	2.00740i
$191$	−25.7131			−1.86053			−0.930266	−	0.366885i	$-0.880424\pi$
$191$	−25.7131			−1.86053			−0.930266	+	0.366885i	$0.880424\pi$
$192$			2.85296i			0.205895i
$193$		−	22.8935i		−	1.64791i	−0.566658	−	0.823953i	$-0.691764\pi$
$193$		−	22.8935i		−	1.64791i	0.566658	−	0.823953i	$-0.308236\pi$
$194$	−6.91284			−0.496313
$195$	2.32744	−	3.35663i	0.166672	−	0.240373i
$196$	−30.2899			−2.16356
$197$		−	0.840810i		−	0.0599052i	−0.999551	−	0.0299526i	$-0.990464\pi$
$197$		−	0.840810i		−	0.0599052i	0.999551	−	0.0299526i	$-0.00953564\pi$
$198$			12.8032i			0.909885i
$199$	26.5996			1.88560			0.942799	−	0.333361i	$-0.108183\pi$
$199$	26.5996			1.88560			0.942799	+	0.333361i	$0.108183\pi$
$200$	27.6701	−	10.3598i	1.95657	−	0.732549i
$201$	8.13872			0.574061
$202$		−	16.1592i		−	1.13696i
$203$			0.173311i			0.0121640i
$204$	18.4520			1.29190
$205$	12.6973	−	18.3119i	0.886815	−	1.27896i
$206$	−17.8513			−1.24376
$207$		−	3.16348i		−	0.219877i
$208$			11.3148i			0.784543i
$209$	43.8159			3.03081
$210$	0.802245	+	0.556267i	0.0553602	+	0.0383861i
$211$	1.86360			0.128296			0.0641478	−	0.997940i	$-0.479567\pi$
$211$	1.86360			0.128296			0.0641478	+	0.997940i	$0.479567\pi$
$212$		−	23.4795i		−	1.61258i
$213$		−	6.25938i		−	0.428886i
$214$	−2.33062			−0.159318
$215$	−14.5363	−	10.0793i	−0.991367	−	0.687401i
$216$	5.90919			0.402069
$217$		−	0.558382i		−	0.0379055i
$218$			10.3711i			0.702417i
$219$	6.74356			0.455687
$220$	28.1422	−	40.5866i	1.89735	−	2.73635i
$221$	−7.75608			−0.521731
$222$		−	4.98345i		−	0.334467i
$223$			25.0759i			1.67920i	0.543202	+	0.839602i	$0.317212\pi$
$223$			25.0759i			1.67920i	−0.543202	+	0.839602i	$0.682788\pi$
$224$	−0.656028			−0.0438327
$225$	1.75317	+	4.68256i	0.116878	+	0.312171i
$226$	17.2463			1.14721
$227$		−	20.3877i		−	1.35318i	−0.736360	−	0.676590i	$-0.763457\pi$
$227$		−	20.3877i		−	1.35318i	0.736360	−	0.676590i	$-0.236543\pi$
$228$		−	37.4646i		−	2.48115i
$229$	5.84179			0.386037			0.193018	−	0.981195i	$-0.438172\pi$
$229$	5.84179			0.386037			0.193018	+	0.981195i	$0.438172\pi$
$230$	−10.1537	+	14.6435i	−0.669512	+	0.965567i
$231$	0.880853			0.0579558
$232$		−	5.90919i		−	0.387957i
$233$			14.1216i			0.925134i	0.886584	+	0.462567i	$0.153072\pi$
$233$			14.1216i			0.925134i	−0.886584	+	0.462567i	$0.846928\pi$
$234$	−4.60158			−0.300815
$235$	−15.9296	−	11.0454i	−1.03913	−	0.720520i
$236$	33.3010			2.16771
$237$		−	4.54763i		−	0.295400i
$238$		−	1.85373i		−	0.120159i
$239$	6.39059			0.413373			0.206686	−	0.978407i	$-0.433732\pi$
$239$	6.39059			0.413373			0.206686	+	0.978407i	$0.433732\pi$
$240$	−11.3821	−	7.89221i	−0.734712	−	0.509440i
$241$	−10.1713			−0.655188			−0.327594	−	0.944819i	$-0.606238\pi$
$241$	−10.1713			−0.655188			−0.327594	+	0.944819i	$0.606238\pi$
$242$		−	37.3624i		−	2.40175i
$243$			1.00000i			0.0641500i
$244$	−12.0037			−0.768455
$245$	−8.88066	+	12.8076i	−0.567365	+	0.818251i
$246$	−25.1037			−1.60055
$247$			15.7478i			1.00201i
$248$			19.0385i			1.20895i
$249$	11.6224			0.736543
$250$	6.91406	−	27.3023i	0.437283	−	1.72675i
$251$	−15.7587			−0.994678			−0.497339	−	0.867556i	$-0.665690\pi$
$251$	−15.7587			−0.994678			−0.497339	+	0.867556i	$0.665690\pi$
$252$		−	0.753170i		−	0.0474452i
$253$			16.0784i			1.01084i
$254$	13.2466			0.831165
$255$	5.40994	−	7.80219i	0.338784	−	0.488592i
$256$	−31.4691			−1.96682
$257$		−	12.3486i		−	0.770283i	−0.922858	−	0.385142i	$-0.874153\pi$
$257$		−	12.3486i		−	0.770283i	0.922858	−	0.385142i	$-0.125847\pi$
$258$			19.9277i			1.24064i
$259$	−0.342858			−0.0213041
$260$	14.5871	+	10.1145i	0.904656	+	0.627277i
$261$	1.00000			0.0618984
$262$		−	44.6272i		−	2.75708i
$263$		−	4.16928i		−	0.257089i	−0.991704	−	0.128545i	$-0.958969\pi$
$263$		−	4.16928i		−	0.257089i	0.991704	−	0.128545i	$-0.0410305\pi$
$264$	−30.0334			−1.84843
$265$	−9.92799	−	6.88395i	−0.609872	−	0.422877i
$266$	−3.76377			−0.230771
$267$			16.4911i			1.00924i
$268$			35.3690i			2.16051i
$269$	1.36011			0.0829273			0.0414636	−	0.999140i	$-0.486798\pi$
$269$	1.36011			0.0829273			0.0414636	+	0.999140i	$0.486798\pi$
$270$	3.20964	−	4.62893i	0.195333	−	0.281708i
$271$	11.1448			0.676998			0.338499	−	0.940967i	$-0.390081\pi$
$271$	11.1448			0.676998			0.338499	+	0.940967i	$0.390081\pi$
$272$			26.3003i			1.59469i
$273$			0.316585i			0.0191606i
$274$	2.88381			0.174217
$275$	−8.91048	−	23.7991i	−0.537322	−	1.43514i
$276$	13.7478			0.827518
$277$		−	0.704673i		−	0.0423397i	−0.999776	−	0.0211699i	$-0.993261\pi$
$277$		−	0.704673i		−	0.0423397i	0.999776	−	0.0211699i	$-0.00673908\pi$
$278$			10.2947i			0.617437i
$279$	−3.22185			−0.192887
$280$	−1.30487	+	1.88188i	−0.0779811	+	0.112464i
$281$	−13.8119			−0.823950			−0.411975	−	0.911195i	$-0.635161\pi$
$281$	−13.8119			−0.823950			−0.411975	+	0.911195i	$0.635161\pi$
$282$			21.8377i			1.30042i
$283$			8.92142i			0.530324i	0.964204	+	0.265162i	$0.0854254\pi$
$283$			8.92142i			0.530324i	−0.964204	+	0.265162i	$0.914575\pi$
$284$	27.2018			1.61413
$285$	−15.8414	−	10.9842i	−0.938363	−	0.650649i
$286$	23.3875			1.38293
$287$			1.72712i			0.101948i
$288$			3.78527i			0.223049i
$289$	−1.02833			−0.0604902
$290$	−4.62893	−	3.20964i	−0.271820	−	0.188477i
$291$	2.74419			0.160867
$292$			29.3060i			1.71500i
$293$		−	31.2817i		−	1.82750i	−0.406278	−	0.913749i	$-0.633174\pi$
$293$		−	31.2817i		−	1.82750i	0.406278	−	0.913749i	$-0.366826\pi$
$294$	17.5579			1.02400
$295$	9.76350	−	14.0809i	0.568453	−	0.819820i
$296$	11.6900			0.679469
$297$		−	5.08250i		−	0.294916i
$298$			3.78538i			0.219281i
$299$	−5.77870			−0.334191
$300$	−20.3493	+	7.61887i	−1.17487	+	0.439876i
$301$	1.37101			0.0790238
$302$			54.1343i			3.11508i
$303$			6.41474i			0.368517i
$304$	53.3996			3.06268
$305$	−3.51934	+	5.07558i	−0.201517	+	0.290627i
$306$	−10.6960			−0.611448
$307$			21.8533i			1.24723i	0.781730	+	0.623617i	$0.214338\pi$
$307$			21.8533i			1.24723i	−0.781730	+	0.623617i	$0.785662\pi$
$308$			3.82798i			0.218120i
$309$	7.08642			0.403132
$310$	14.9137	+	10.3410i	0.847043	+	0.587329i
$311$	28.8838			1.63785			0.818925	−	0.573901i	$-0.194570\pi$
$311$	28.8838			1.63785			0.818925	+	0.573901i	$0.194570\pi$
$312$		−	10.7942i		−	0.611104i
$313$			8.24232i			0.465884i	0.972491	+	0.232942i	$0.0748352\pi$
$313$			8.24232i			0.465884i	−0.972491	+	0.232942i	$0.925165\pi$
$314$	−28.0038			−1.58034
$315$	−0.318467	−	0.220821i	−0.0179436	−	0.0124419i
$316$	19.7630			1.11175
$317$			7.15829i			0.402050i	0.979586	+	0.201025i	$0.0644272\pi$
$317$			7.15829i			0.402050i	−0.979586	+	0.201025i	$0.935573\pi$
$318$			13.6102i			0.763223i
$319$	−5.08250			−0.284565
$320$	−3.63505	+	5.24246i	−0.203206	+	0.293062i
$321$	0.925187			0.0516389
$322$		−	1.38113i		−	0.0769672i
$323$			36.6043i			2.03672i
$324$	−4.34577			−0.241432
$325$	8.55359	−	3.20250i	0.474468	−	0.177643i
$326$	−30.7781			−1.70464
$327$		−	4.11700i		−	0.227671i
$328$		−	58.8875i		−	3.25152i
$329$	1.50242			0.0828311
$330$	−16.3130	+	23.5265i	−0.898002	+	1.29509i
$331$	−24.6936			−1.35728			−0.678642	−	0.734470i	$-0.737431\pi$
$331$	−24.6936			−1.35728			−0.678642	+	0.734470i	$0.737431\pi$
$332$			50.5085i			2.77201i
$333$			1.97828i			0.108409i
$334$	−53.8162			−2.94469
$335$	14.9553	+	10.3698i	0.817095	+	0.566563i
$336$	1.07352			0.0585653
$337$		−	11.7142i		−	0.638116i	−0.947735	−	0.319058i	$-0.896634\pi$
$337$		−	11.7142i		−	0.638116i	0.947735	−	0.319058i	$-0.103366\pi$
$338$		−	24.3424i		−	1.32405i
$339$	−6.84626			−0.371838
$340$	33.9065	+	23.5104i	1.83884	+	1.27503i
$341$	16.3751			0.886759
$342$			21.7168i			1.17431i
$343$		−	2.42115i		−	0.130730i
$344$	−46.7458			−2.52036
$345$	4.03070	−	5.81305i	0.217005	−	0.312964i
$346$	33.1727			1.78337
$347$		−	0.828990i		−	0.0445025i	−0.999752	−	0.0222513i	$-0.992917\pi$
$347$		−	0.828990i		−	0.0445025i	0.999752	−	0.0222513i	$-0.00708338\pi$
$348$			4.34577i			0.232958i
$349$	8.44370			0.451981			0.225991	−	0.974129i	$-0.427438\pi$
$349$	8.44370			0.451981			0.225991	+	0.974129i	$0.427438\pi$
$350$	0.765407	+	2.04433i	0.0409127	+	0.109274i
$351$	1.82669			0.0975014
$352$		−	19.2386i		−	1.02542i
$353$		−	4.90553i		−	0.261095i	−0.991442	−	0.130547i	$-0.958327\pi$
$353$		−	4.90553i		−	0.261095i	0.991442	−	0.130547i	$-0.0416735\pi$
$354$	−19.3034			−1.02596
$355$	7.97528	−	11.5019i	0.423284	−	0.610458i
$356$	−71.6664			−3.79831
$357$			0.735875i			0.0389466i
$358$			26.9407i			1.42386i
$359$	−4.69044			−0.247552			−0.123776	−	0.992310i	$-0.539500\pi$
$359$	−4.69044			−0.247552			−0.123776	+	0.992310i	$0.539500\pi$
$360$	10.8584	+	7.52909i	0.572289	+	0.396818i
$361$	55.3205			2.91161
$362$			38.4189i			2.01925i
$363$			14.8318i			0.778466i
$364$	−1.37581			−0.0721119
$365$	12.3916	+	8.59219i	0.648607	+	0.449736i
$366$	6.95807			0.363704
$367$		−	29.2460i		−	1.52663i	−0.646028	−	0.763314i	$-0.723571\pi$
$367$		−	29.2460i		−	1.52663i	0.646028	−	0.763314i	$-0.276429\pi$
$368$			19.5952i			1.02147i
$369$	9.96541			0.518779
$370$	6.34958	−	9.15733i	0.330099	−	0.476067i
$371$	0.936373			0.0486140
$372$		−	14.0014i		−	0.725940i
$373$			34.5171i			1.78723i	0.448839	+	0.893613i	$0.351838\pi$
$373$			34.5171i			1.78723i	−0.448839	+	0.893613i	$0.648162\pi$
$374$	54.3622			2.81100
$375$	−2.74467	+	10.8382i	−0.141734	+	0.559683i
$376$	−51.2263			−2.64179
$377$		−	1.82669i		−	0.0940793i
$378$			0.436584i			0.0224555i
$379$	3.20131			0.164440			0.0822201	−	0.996614i	$-0.473799\pi$
$379$	3.20131			0.164440			0.0822201	+	0.996614i	$0.473799\pi$
$380$	47.7349	−	68.8430i	2.44875	−	3.53157i
$381$	−5.25850			−0.269401
$382$			64.7733i			3.31409i
$383$			19.4424i			0.993458i	0.867906	+	0.496729i	$0.165466\pi$
$383$			19.4424i			0.993458i	−0.867906	+	0.496729i	$0.834534\pi$
$384$	14.7574			0.753084
$385$	1.61861	+	1.12232i	0.0824920	+	0.0571989i
$386$	−57.6705			−2.93535
$387$		−	7.91070i		−	0.402123i
$388$			11.9256i			0.605432i
$389$	−2.55875			−0.129734			−0.0648669	−	0.997894i	$-0.520662\pi$
$389$	−2.55875			−0.129734			−0.0648669	+	0.997894i	$0.520662\pi$
$390$	−8.45562	−	5.86302i	−0.428167	−	0.296886i
$391$	−13.4321			−0.679289
$392$			41.1868i			2.08025i
$393$			17.7157i			0.893637i
$394$	−2.11807			−0.106707
$395$	5.79429	−	8.35650i	0.291542	−	0.420461i
$396$	22.0874			1.10993
$397$		−	30.6583i		−	1.53869i	−0.638831	−	0.769347i	$-0.720582\pi$
$397$		−	30.6583i		−	1.53869i	0.638831	−	0.769347i	$-0.279418\pi$
$398$		−	67.0067i		−	3.35874i
$399$	1.49410			0.0747987
$400$	−10.8594	−	29.0046i	−0.542972	−	1.45023i
$401$	11.1628			0.557446			0.278723	−	0.960372i	$-0.410089\pi$
$401$	11.1628			0.557446			0.278723	+	0.960372i	$0.410089\pi$
$402$		−	20.5021i		−	1.02255i
$403$			5.88532i			0.293169i
$404$	−27.8770			−1.38693
$405$	−1.27413	+	1.83755i	−0.0633122	+	0.0913085i
$406$	0.436584			0.0216673
$407$		−	10.0546i		−	0.498388i
$408$		−	25.0903i		−	1.24215i
$409$	−13.3880			−0.661994			−0.330997	−	0.943632i	$-0.607385\pi$
$409$	−13.3880			−0.661994			−0.330997	+	0.943632i	$0.607385\pi$
$410$	−46.1292	−	31.9854i	−2.27816	−	1.57965i
$411$	−1.14479			−0.0564681
$412$			30.7959i			1.51721i
$413$			1.32806i			0.0653495i
$414$	−7.96907			−0.391658
$415$	21.3568	+	14.8085i	1.04836	+	0.726923i
$416$	6.91451			0.339012
$417$		−	4.08670i		−	0.200127i
$418$		−	110.376i		−	5.39865i
$419$	−29.2982			−1.43131			−0.715655	−	0.698454i	$-0.753872\pi$
$419$	−29.2982			−1.43131			−0.715655	+	0.698454i	$0.753872\pi$
$420$	0.959639	−	1.38399i	0.0468256	−	0.0675316i
$421$	10.1098			0.492724			0.246362	−	0.969178i	$-0.420765\pi$
$421$	10.1098			0.492724			0.246362	+	0.969178i	$0.420765\pi$
$422$		−	4.69456i		−	0.228528i
$423$		−	8.66893i		−	0.421498i
$424$	−31.9264			−1.55048
$425$	19.8821	−	7.44392i	0.964422	−	0.361083i
$426$	−15.7679			−0.763957
$427$		−	0.478710i		−	0.0231664i
$428$			4.02065i			0.194345i
$429$	−9.28414			−0.448243
$430$	−25.3905	+	36.6181i	−1.22444	+	1.76588i
$431$	30.3330			1.46109			0.730545	−	0.682865i	$-0.239266\pi$
$431$	30.3330			1.46109			0.730545	+	0.682865i	$0.239266\pi$
$432$		−	6.19418i		−	0.298018i
$433$		−	6.50520i		−	0.312620i	−0.987708	−	0.156310i	$-0.950040\pi$
$433$		−	6.50520i		−	0.312620i	0.987708	−	0.156310i	$-0.0499599\pi$
$434$	−1.40661			−0.0675195
$435$	1.83755	+	1.27413i	0.0881037	+	0.0610900i
$436$	17.8915			0.856850
$437$			27.2722i			1.30460i
$438$		−	16.9876i		−	0.811698i
$439$	9.40248			0.448756			0.224378	−	0.974502i	$-0.427965\pi$
$439$	9.40248			0.448756			0.224378	+	0.974502i	$0.427965\pi$
$440$	−55.1879	−	38.2666i	−2.63098	−	1.82429i
$441$	−6.96996			−0.331903
$442$			19.5382i			0.929337i
$443$			28.1657i			1.33819i	0.743175	+	0.669097i	$0.233319\pi$
$443$			28.1657i			1.33819i	−0.743175	+	0.669097i	$0.766681\pi$
$444$	−8.59715			−0.408003
$445$	−21.0118	+	30.3031i	−0.996055	+	1.43651i
$446$	63.1682			2.99110
$447$		−	1.50268i		−	0.0710744i
$448$		−	0.494450i		−	0.0233605i
$449$	32.2625			1.52256			0.761281	−	0.648422i	$-0.224571\pi$
$449$	32.2625			1.52256			0.761281	+	0.648422i	$0.224571\pi$
$450$	11.7958	−	4.41638i	0.556057	−	0.208190i
$451$	−50.6492			−2.38498
$452$		−	29.7523i		−	1.39943i
$453$		−	21.4897i		−	1.00967i
$454$	−51.3583			−2.41037
$455$	−0.403372	+	0.581741i	−0.0189104	+	0.0272724i
$456$	−50.9427			−2.38561
$457$		−	27.1466i		−	1.26986i	−0.772568	−	0.634932i	$-0.781028\pi$
$457$		−	27.1466i		−	1.26986i	0.772568	−	0.634932i	$-0.218972\pi$
$458$		−	14.7160i		−	0.687631i
$459$	4.24598			0.198185
$460$	25.2622	+	17.5165i	1.17786	+	0.816710i
$461$	7.00940			0.326460			0.163230	−	0.986588i	$-0.447809\pi$
$461$	7.00940			0.326460			0.163230	+	0.986588i	$0.447809\pi$
$462$		−	2.21894i		−	0.103234i
$463$		−	21.7590i		−	1.01123i	−0.862760	−	0.505613i	$-0.831266\pi$
$463$		−	21.7590i		−	1.01123i	0.862760	−	0.505613i	$-0.168734\pi$
$464$	−6.19418			−0.287558
$465$	−5.92031	−	4.10507i	−0.274548	−	0.190368i
$466$	35.5733			1.64790
$467$		−	30.7825i		−	1.42445i	−0.701953	−	0.712223i	$-0.747688\pi$
$467$		−	30.7825i		−	1.42445i	0.701953	−	0.712223i	$-0.252312\pi$
$468$			7.93837i			0.366951i
$469$	−1.41053			−0.0651322
$470$	−27.8242	+	40.1279i	−1.28343	+	1.85096i
$471$	11.1167			0.512228
$472$		−	45.2813i		−	2.08424i
$473$			40.2061i			1.84868i
$474$	−11.4559			−0.526185
$475$	−15.1140	−	40.3681i	−0.693476	−	1.85221i
$476$	−3.19794			−0.146577
$477$		−	5.40285i		−	0.247379i
$478$		−	16.0984i		−	0.736324i
$479$	6.29237			0.287506			0.143753	−	0.989614i	$-0.454083\pi$
$479$	6.29237			0.287506			0.143753	+	0.989614i	$0.454083\pi$
$480$	−4.82293	+	6.95561i	−0.220136	+	0.317479i
$481$	3.61370			0.164771
$482$			25.6222i			1.16706i
$483$			0.548266i			0.0249470i
$484$	−64.4555			−2.92979
$485$	5.04259	+	3.49647i	0.228972	+	0.158766i
$486$	2.51908			0.114268
$487$			30.3406i			1.37487i	0.726248	+	0.687433i	$0.241262\pi$
$487$			30.3406i			1.37487i	−0.726248	+	0.687433i	$0.758738\pi$
$488$			16.3220i			0.738864i
$489$	12.2180			0.552517
$490$	32.2635	+	22.3711i	1.45752	+	1.01062i
$491$	17.7822			0.802502			0.401251	−	0.915968i	$-0.368576\pi$
$491$	17.7822			0.802502			0.401251	+	0.915968i	$0.368576\pi$
$492$			43.3074i			1.95245i
$493$		−	4.24598i		−	0.191229i
$494$	39.6699			1.78483
$495$	6.47578	−	9.33933i	0.291065	−	0.419772i
$496$	19.9567			0.896083
$497$			1.08482i			0.0486608i
$498$		−	29.2779i		−	1.31197i
$499$	−0.262248			−0.0117399			−0.00586993	−	0.999983i	$-0.501868\pi$
$499$	−0.262248			−0.0117399			−0.00586993	+	0.999983i	$0.501868\pi$
$500$	−47.1004	−	11.9277i	−2.10639	−	0.533424i
$501$	21.3634			0.954447
$502$			39.6974i			1.77178i
$503$		−	15.0486i		−	0.670986i	−0.942043	−	0.335493i	$-0.891097\pi$
$503$		−	15.0486i		−	0.670986i	0.942043	−	0.335493i	$-0.108903\pi$
$504$	−1.02413			−0.0456182
$505$	−8.17323	+	11.7874i	−0.363704	+	0.524532i
$506$	40.5028			1.80057
$507$			9.66321i			0.429158i
$508$		−	22.8522i		−	1.01390i
$509$	−39.8752			−1.76744			−0.883719	−	0.468019i	$-0.844968\pi$
$509$	−39.8752			−1.76744			−0.883719	+	0.468019i	$0.844968\pi$
$510$	−19.6544	−	13.6281i	−0.870309	−	0.603462i
$511$	−1.16873			−0.0517017
$512$			49.7585i			2.19904i
$513$		−	8.62093i		−	0.380624i
$514$	−31.1071			−1.37207
$515$	13.0216	+	9.02904i	0.573802	+	0.397867i
$516$	34.3781			1.51341
$517$			44.0598i			1.93775i
$518$			0.863687i			0.0379482i
$519$	−13.1686			−0.578036
$520$	13.7533	−	19.8349i	0.603122	−	0.869820i
$521$	18.3187			0.802557			0.401279	−	0.915956i	$-0.368566\pi$
$521$	18.3187			0.802557			0.401279	+	0.915956i	$0.368566\pi$
$522$		−	2.51908i		−	0.110257i
$523$		−	6.04576i		−	0.264363i	−0.991226	−	0.132181i	$-0.957802\pi$
$523$		−	6.04576i		−	0.264363i	0.991226	−	0.132181i	$-0.0421981\pi$
$524$	−76.9882			−3.36325
$525$	−0.303844	−	0.811540i	−0.0132608	−	0.0354185i
$526$	−10.5028			−0.457942
$527$			13.6799i			0.595906i
$528$			31.4819i			1.37007i
$529$	12.9924			0.564886
$530$	−17.3412	+	25.0094i	−0.753254	+	1.08634i
$531$	7.66286			0.332540
$532$			6.49303i			0.281508i
$533$		−	18.2037i		−	0.788490i
$534$	41.5423			1.79771
$535$	1.70008	+	1.17881i	0.0735007	+	0.0509645i
$536$	48.0932			2.07731
$537$		−	10.6947i		−	0.461509i
$538$		−	3.42622i		−	0.147715i
$539$	35.4248			1.52585
$540$	−7.98556	−	5.53709i	−0.343644	−	0.238278i
$541$	−35.6643			−1.53333			−0.766664	−	0.642049i	$-0.778085\pi$
$541$	−35.6643			−1.53333			−0.766664	+	0.642049i	$0.778085\pi$
$542$		−	28.0746i		−	1.20591i
$543$		−	15.2511i		−	0.654489i
$544$	16.0722			0.689088
$545$	5.24561	−	7.56519i	0.224697	−	0.324057i
$546$	0.797504			0.0341300
$547$			5.23087i			0.223656i	0.993728	+	0.111828i	$0.0356706\pi$
$547$			5.23087i			0.223656i	−0.993728	+	0.111828i	$0.964329\pi$
$548$		−	4.97498i		−	0.212520i
$549$	−2.76215			−0.117886
$550$	−59.9519	+	22.4462i	−2.55636	+	0.957110i
$551$	−8.62093			−0.367264
$552$		−	18.6936i		−	0.795653i
$553$			0.788155i			0.0335157i
$554$	−1.77513			−0.0754180
$555$	−2.52059	+	3.63519i	−0.106993	+	0.154305i
$556$	17.7599			0.753186
$557$		−	32.9893i		−	1.39780i	−0.715219	−	0.698900i	$-0.753673\pi$
$557$		−	32.9893i		−	1.39780i	0.715219	−	0.698900i	$-0.246327\pi$
$558$			8.11611i			0.343582i
$559$	−14.4504			−0.611186
$560$	1.97264	+	1.36781i	0.0833594	+	0.0578004i
$561$	−21.5802			−0.911116
$562$			34.7933i			1.46767i
$563$			5.94931i			0.250734i	0.992110	+	0.125367i	$0.0400108\pi$
$563$			5.94931i			0.250734i	−0.992110	+	0.125367i	$0.959989\pi$
$564$	37.6732			1.58633
$565$	−12.5803	−	8.72305i	−0.529259	−	0.366981i
$566$	22.4738			0.944644
$567$		−	0.173311i		−	0.00727838i
$568$		−	36.9878i		−	1.55198i
$569$	14.6940			0.616003			0.308002	−	0.951386i	$-0.400340\pi$
$569$	14.6940			0.616003			0.308002	+	0.951386i	$0.400340\pi$
$570$	−27.6701	+	39.9057i	−1.15897	+	1.67147i
$571$	7.86509			0.329144			0.164572	−	0.986365i	$-0.447376\pi$
$571$	7.86509			0.329144			0.164572	+	0.986365i	$0.447376\pi$
$572$		−	40.3467i		−	1.68698i
$573$		−	25.7131i		−	1.07418i
$574$	4.35074			0.181597
$575$	14.8132	−	5.54612i	0.617753	−	0.231289i
$576$	−2.85296			−0.118873
$577$		−	29.7454i		−	1.23832i	−0.785265	−	0.619159i	$-0.787474\pi$
$577$		−	29.7454i		−	1.23832i	0.785265	−	0.619159i	$-0.212526\pi$
$578$			2.59045i			0.107749i
$579$	22.8935			0.951419
$580$	−5.53709	+	7.98556i	−0.229915	+	0.331582i
$581$	−2.01430			−0.0835671
$582$		−	6.91284i		−	0.286547i
$583$			27.4599i			1.13727i
$584$	39.8489			1.64896
$585$	3.35663	+	2.32744i	0.138780	+	0.0962280i
$586$	−78.8013			−3.25525
$587$			46.1639i			1.90539i	0.303929	+	0.952695i	$0.401701\pi$
$587$			46.1639i			1.90539i	−0.303929	+	0.952695i	$0.598299\pi$
$588$		−	30.2899i		−	1.24913i
$589$	27.7754			1.14446
$590$	−35.4709	−	24.5951i	−1.46031	−	1.01256i
$591$	0.840810			0.0345863
$592$		−	12.2538i		−	0.503629i
$593$		−	22.6806i		−	0.931382i	−0.884948	−	0.465691i	$-0.845806\pi$
$593$		−	22.6806i		−	0.931382i	0.884948	−	0.465691i	$-0.154194\pi$
$594$	−12.8032			−0.525323
$595$	−0.937602	+	1.35221i	−0.0384379	+	0.0554350i
$596$	6.53031			0.267492
$597$			26.5996i			1.08865i
$598$			14.5570i			0.595280i
$599$	−22.8764			−0.934704			−0.467352	−	0.884071i	$-0.654792\pi$
$599$	−22.8764			−0.934704			−0.467352	+	0.884071i	$0.654792\pi$
$600$	10.3598	+	27.6701i	0.422937	+	1.12963i
$601$	14.6918			0.599291			0.299646	−	0.954051i	$-0.403132\pi$
$601$	14.6918			0.599291			0.299646	+	0.954051i	$0.403132\pi$
$602$		−	3.45369i		−	0.140762i
$603$			8.13872i			0.331434i
$604$	93.3893			3.79996
$605$	−18.8977	+	27.2541i	−0.768299	+	1.10804i
$606$	16.1592			0.656424
$607$			20.9484i			0.850270i	0.905130	+	0.425135i	$0.139773\pi$
$607$			20.9484i			0.850270i	−0.905130	+	0.425135i	$0.860227\pi$
$608$		−	32.6325i		−	1.32342i
$609$	−0.173311			−0.00702292
$610$	12.7858	+	8.86551i	0.517682	+	0.358954i
$611$	−15.8354			−0.640633
$612$			18.4520i			0.745880i
$613$		−	42.0373i		−	1.69787i	−0.528496	−	0.848936i	$-0.677244\pi$
$613$		−	42.0373i		−	1.69787i	0.528496	−	0.848936i	$-0.322756\pi$
$614$	55.0503			2.22165
$615$	18.3119	+	12.6973i	0.738408	+	0.512003i
$616$	5.20512			0.209720
$617$		−	44.6853i		−	1.79896i	−0.436958	−	0.899482i	$-0.643944\pi$
$617$		−	44.6853i		−	1.79896i	0.436958	−	0.899482i	$-0.356056\pi$
$618$		−	17.8513i		−	0.718083i
$619$	22.1966			0.892157			0.446079	−	0.894994i	$-0.352820\pi$
$619$	22.1966			0.892157			0.446079	+	0.894994i	$0.352820\pi$
$620$	17.8397	−	25.7283i	0.716459	−	1.03327i
$621$	3.16348			0.126946
$622$		−	72.7606i		−	2.91744i
$623$		−	2.85808i		−	0.114507i
$624$	−11.3148			−0.452956
$625$	−18.8528	+	16.4187i	−0.754112	+	0.656746i
$626$	20.7631			0.829859
$627$			43.8159i			1.74984i
$628$			48.3104i			1.92780i
$629$	8.39974			0.334919
$630$	−0.556267	+	0.802245i	−0.0221622	+	0.0319622i
$631$	−5.29765			−0.210896			−0.105448	−	0.994425i	$-0.533628\pi$
$631$	−5.29765			−0.210896			−0.105448	+	0.994425i	$0.533628\pi$
$632$		−	26.8728i		−	1.06894i
$633$			1.86360i			0.0740715i
$634$	18.0323			0.716155
$635$	−9.66275	−	6.70003i	−0.383455	−	0.265883i
$636$	23.4795			0.931024
$637$			12.7320i			0.504458i
$638$			12.8032i			0.506884i
$639$	6.25938			0.247617
$640$	27.1174	+	18.8029i	1.07191	+	0.743248i
$641$	3.21592			0.127021			0.0635107	−	0.997981i	$-0.479770\pi$
$641$	3.21592			0.127021			0.0635107	+	0.997981i	$0.479770\pi$
$642$		−	2.33062i		−	0.0919823i
$643$			32.0630i			1.26444i	0.774788	+	0.632221i	$0.217857\pi$
$643$			32.0630i			1.26444i	−0.774788	+	0.632221i	$0.782143\pi$
$644$	−2.38264			−0.0938891
$645$	10.0793	−	14.5363i	0.396871	−	0.572366i
$646$	92.2092			3.62792
$647$		−	9.26571i		−	0.364273i	−0.983273	−	0.182136i	$-0.941699\pi$
$647$		−	9.26571i		−	0.364273i	0.983273	−	0.182136i	$-0.0583012\pi$
$648$			5.90919i			0.232135i
$649$	−38.9465			−1.52878
$650$	−8.06735	−	21.5472i	−0.316427	−	0.845150i
$651$	0.558382			0.0218847
$652$			53.0966i			2.07942i
$653$			22.2362i			0.870171i	0.900389	+	0.435085i	$0.143282\pi$
$653$			22.2362i			0.870171i	−0.900389	+	0.435085i	$0.856718\pi$
$654$	−10.3711			−0.405541
$655$	−22.5721	+	32.5534i	−0.881966	+	1.27197i
$656$	−61.7275			−2.41006
$657$			6.74356i			0.263091i
$658$		−	3.78472i		−	0.147544i
$659$	−34.6708			−1.35058			−0.675291	−	0.737552i	$-0.735982\pi$
$659$	−34.6708			−1.35058			−0.675291	+	0.737552i	$0.735982\pi$
$660$	40.5866	+	28.1422i	1.57983	+	1.09544i
$661$	−9.67217			−0.376204			−0.188102	−	0.982150i	$-0.560234\pi$
$661$	−9.67217			−0.376204			−0.188102	+	0.982150i	$0.560234\pi$
$662$			62.2052i			2.41767i
$663$		−	7.75608i		−	0.301221i
$664$	68.6792			2.66527
$665$	2.74549	+	1.90369i	0.106465	+	0.0738218i
$666$	4.98345			0.193105
$667$		−	3.16348i		−	0.122491i
$668$			92.8405i			3.59211i
$669$	−25.0759			−0.969489
$670$	26.1224	−	37.6736i	1.00920	−	1.45546i
$671$	14.0386			0.541954
$672$		−	0.656028i		−	0.0253068i
$673$			24.7689i			0.954770i	0.878694	+	0.477385i	$0.158415\pi$
$673$			24.7689i			0.954770i	−0.878694	+	0.477385i	$0.841585\pi$
$674$	−29.5091			−1.13665
$675$	−4.68256	+	1.75317i	−0.180232	+	0.0674795i
$676$	−41.9941			−1.61516
$677$		−	33.3636i		−	1.28227i	−0.767430	−	0.641133i	$-0.778465\pi$
$677$		−	33.3636i		−	1.28227i	0.767430	−	0.641133i	$-0.221535\pi$
$678$			17.2463i			0.662340i
$679$	−0.475599			−0.0182518
$680$	31.9684	−	46.1046i	1.22593	−	1.76803i
$681$	20.3877			0.781259
$682$		−	41.2501i		−	1.57955i
$683$		−	13.9516i		−	0.533842i	−0.963718	−	0.266921i	$-0.913994\pi$
$683$		−	13.9516i		−	0.533842i	0.963718	−	0.266921i	$-0.0860063\pi$
$684$	37.4646			1.43249
$685$	−2.10360	−	1.45861i	−0.0803744	−	0.0557306i
$686$	−6.09907			−0.232864
$687$			5.84179i			0.222878i
$688$			49.0003i			1.86812i
$689$	−9.86932			−0.375991
$690$	−14.6435	−	10.1537i	−0.557470	−	0.386543i
$691$	50.5611			1.92344			0.961718	−	0.274042i	$-0.0883607\pi$
$691$	50.5611			1.92344			0.961718	+	0.274042i	$0.0883607\pi$
$692$		−	57.2276i		−	2.17547i
$693$			0.880853i			0.0334608i
$694$	−2.08829			−0.0792705
$695$	5.20700	−	7.50952i	0.197513	−	0.284852i
$696$	5.90919			0.223987
$697$		−	42.3129i		−	1.60272i
$698$		−	21.2704i		−	0.805096i
$699$	−14.1216			−0.534126
$700$	3.52676	−	1.32043i	0.133299	−	0.0499077i
$701$	−24.0803			−0.909499			−0.454749	−	0.890620i	$-0.650271\pi$
$701$	−24.0803			−0.909499			−0.454749	+	0.890620i	$0.650271\pi$
$702$		−	4.60158i		−	0.173675i
$703$		−	17.0546i		−	0.643227i
$704$	14.5002			0.546496
$705$	11.0454	−	15.9296i	0.415992	−	0.599942i
$706$	−12.3574			−0.465078
$707$		−	1.11174i		−	0.0418114i
$708$			33.3010i			1.25153i
$709$	−0.769569			−0.0289018			−0.0144509	−	0.999896i	$-0.504600\pi$
$709$	−0.769569			−0.0289018			−0.0144509	+	0.999896i	$0.504600\pi$
$710$	−28.9743	−	20.0904i	−1.08738	−	0.753979i
$711$	4.54763			0.170550
$712$			97.4487i			3.65205i
$713$			10.1923i			0.381703i
$714$	1.85373			0.0693740
$715$	−17.0601	−	11.8292i	−0.638010	−	0.442388i
$716$	46.4766			1.73691
$717$			6.39059i			0.238661i
$718$			11.8156i			0.440955i
$719$	−4.92456			−0.183655			−0.0918275	−	0.995775i	$-0.529271\pi$
$719$	−4.92456			−0.183655			−0.0918275	+	0.995775i	$0.529271\pi$
$720$	7.89221	−	11.3821i	0.294125	−	0.424186i
$721$	−1.22815			−0.0457388
$722$		−	139.357i		−	5.18632i
$723$		−	10.1713i		−	0.378273i
$724$	66.2780			2.46320
$725$	1.75317	+	4.68256i	0.0651111	+	0.173906i
$726$	37.3624			1.38665
$727$			41.8838i			1.55339i	0.629880	+	0.776693i	$0.283104\pi$
$727$			41.8838i			1.55339i	−0.629880	+	0.776693i	$0.716896\pi$
$728$			1.87076i			0.0693350i
$729$	−1.00000			−0.0370370
$730$	21.6444	−	31.2155i	0.801096	−	1.15534i
$731$	−33.5887			−1.24232
$732$		−	12.0037i		−	0.443668i
$733$			27.1372i			1.00234i	0.865350	+	0.501168i	$0.167096\pi$
$733$			27.1372i			1.00234i	−0.865350	+	0.501168i	$0.832904\pi$
$734$	−73.6730			−2.71932
$735$	−12.8076	−	8.88066i	−0.472417	−	0.327568i
$736$	11.9746			0.441390
$737$		−	41.3650i		−	1.52370i
$738$		−	25.1037i		−	0.924079i
$739$	3.41445			0.125603			0.0628013	−	0.998026i	$-0.479997\pi$
$739$	3.41445			0.125603			0.0628013	+	0.998026i	$0.479997\pi$
$740$	−15.7977	−	10.9539i	−0.580735	−	0.402674i
$741$	−15.7478			−0.578509
$742$		−	2.35880i		−	0.0865942i
$743$		−	3.73961i		−	0.137193i	−0.997644	−	0.0685966i	$-0.978148\pi$
$743$		−	3.73961i		−	0.137193i	0.997644	−	0.0685966i	$-0.0218521\pi$
$744$	−19.0385			−0.697986
$745$	1.91462	−	2.76125i	0.0701461	−	0.101164i
$746$	86.9513			3.18351
$747$			11.6224i			0.425243i
$748$		−	93.7825i		−	3.42903i
$749$	−0.160345			−0.00585888
$750$	27.3023	+	6.91406i	0.996940	+	0.252466i
$751$	−24.1900			−0.882707			−0.441354	−	0.897333i	$-0.645502\pi$
$751$	−24.1900			−0.882707			−0.441354	+	0.897333i	$0.645502\pi$
$752$			53.6969i			1.95812i
$753$		−	15.7587i		−	0.574278i
$754$	−4.60158			−0.167580
$755$	27.3807	−	39.4884i	0.996487	−	1.43713i
$756$	0.753170			0.0273925
$757$		−	3.55451i		−	0.129191i	−0.997912	−	0.0645953i	$-0.979424\pi$
$757$		−	3.55451i		−	0.129191i	0.997912	−	0.0645953i	$-0.0205757\pi$
$758$		−	8.06436i		−	0.292911i
$759$	−16.0784			−0.583608
$760$	−93.6097	−	64.9078i	−3.39558	−	2.35445i
$761$	−44.2390			−1.60366			−0.801832	−	0.597550i	$-0.796141\pi$
$761$	−44.2390			−1.60366			−0.801832	+	0.597550i	$0.796141\pi$
$762$			13.2466i			0.479873i
$763$			0.713522i			0.0258312i
$764$	111.743			4.04272
$765$	7.80219	+	5.40994i	0.282089	+	0.195597i
$766$	48.9769			1.76961
$767$		−	13.9977i		−	0.505426i
$768$		−	31.4691i		−	1.13554i
$769$	−6.30558			−0.227385			−0.113692	−	0.993516i	$-0.536268\pi$
$769$	−6.30558			−0.227385			−0.113692	+	0.993516i	$0.536268\pi$
$770$	2.82722	−	4.07741i	0.101886	−	0.146940i
$771$	12.3486			0.444723
$772$			99.4897i			3.58071i
$773$			24.2649i			0.872747i	0.899766	+	0.436374i	$0.143737\pi$
$773$			24.2649i			0.872747i	−0.899766	+	0.436374i	$0.856263\pi$
$774$	−19.9277			−0.716286
$775$	−5.64845	−	15.0865i	−0.202898	−	0.541924i
$776$	16.2159			0.582118
$777$		−	0.342858i		−	0.0123000i
$778$			6.44570i			0.231090i
$779$	−85.9111			−3.07809
$780$	−10.1145	+	14.5871i	−0.362159	+	0.522303i
$781$	−31.8133			−1.13837
$782$			33.8365i			1.20999i
$783$			1.00000i			0.0357371i

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:	\( N \)	\(=\)	\( 435 = 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	435.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.51908i q^{2} +1.00000i q^{3} -4.34577 q^{4} +(-1.27413 + 1.83755i) q^{5} +2.51908 q^{6} -0.173311i q^{7} +5.90919i q^{8} -1.00000 q^{9} +(4.62893 + 3.20964i) q^{10} +5.08250 q^{11} -4.34577i q^{12} +1.82669i q^{13} -0.436584 q^{14} +(-1.83755 - 1.27413i) q^{15} +6.19418 q^{16} +4.24598i q^{17} +2.51908i q^{18} +8.62093 q^{19} +(5.53709 - 7.98556i) q^{20} +0.173311 q^{21} -12.8032i q^{22} +3.16348i q^{23} -5.90919 q^{24} +(-1.75317 - 4.68256i) q^{25} +4.60158 q^{26} -1.00000i q^{27} +0.753170i q^{28} -1.00000 q^{29} +(-3.20964 + 4.62893i) q^{30} +3.22185 q^{31} -3.78527i q^{32} +5.08250i q^{33} +10.6960 q^{34} +(0.318467 + 0.220821i) q^{35} +4.34577 q^{36} -1.97828i q^{37} -21.7168i q^{38} -1.82669 q^{39} +(-10.8584 - 7.52909i) q^{40} -9.96541 q^{41} -0.436584i q^{42} +7.91070i q^{43} -22.0874 q^{44} +(1.27413 - 1.83755i) q^{45} +7.96907 q^{46} +8.66893i q^{47} +6.19418i q^{48} +6.96996 q^{49} +(-11.7958 + 4.41638i) q^{50} -4.24598 q^{51} -7.93837i q^{52} +5.40285i q^{53} -2.51908 q^{54} +(-6.47578 + 9.33933i) q^{55} +1.02413 q^{56} +8.62093i q^{57} +2.51908i q^{58} -7.66286 q^{59} +(7.98556 + 5.53709i) q^{60} +2.76215 q^{61} -8.11611i q^{62} +0.173311i q^{63} +2.85296 q^{64} +(-3.35663 - 2.32744i) q^{65} +12.8032 q^{66} -8.13872i q^{67} -18.4520i q^{68} -3.16348 q^{69} +(0.556267 - 0.802245i) q^{70} -6.25938 q^{71} -5.90919i q^{72} -6.74356i q^{73} -4.98345 q^{74} +(4.68256 - 1.75317i) q^{75} -37.4646 q^{76} -0.880853i q^{77} +4.60158i q^{78} -4.54763 q^{79} +(-7.89221 + 11.3821i) q^{80} +1.00000 q^{81} +25.1037i q^{82} -11.6224i q^{83} -0.753170 q^{84} +(-7.80219 - 5.40994i) q^{85} +19.9277 q^{86} -1.00000i q^{87} +30.0334i q^{88} +16.4911 q^{89} +(-4.62893 - 3.20964i) q^{90} +0.316585 q^{91} -13.7478i q^{92} +3.22185i q^{93} +21.8377 q^{94} +(-10.9842 + 15.8414i) q^{95} +3.78527 q^{96} -2.74419i q^{97} -17.5579i q^{98} -5.08250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9} + 4 q^{10} + 24 q^{11} - 12 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{19} + 8 q^{20} + 16 q^{21} - 18 q^{24} + 2 q^{25} - 10 q^{29} - 18 q^{30} + 4 q^{31} - 8 q^{34} + 2 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) \) 10 * q - 10 * q^4 + 6 * q^6 - 10 * q^9 + 4 * q^10 + 24 * q^11 - 12 * q^14 + 2 * q^15 + 2 * q^16 + 4 * q^19 + 8 * q^20 + 16 * q^21 - 18 * q^24 + 2 * q^25 - 10 * q^29 - 18 * q^30 + 4 * q^31 - 8 * q^34 + 2 * q^35 + 10 * q^36 - 4 * q^39 - 14 * q^40 - 28 * q^41 - 40 * q^44 - 12 * q^46 + 14 * q^49 - 12 * q^50 - 6 * q^54 + 2 * q^55 - 4 * q^56 - 8 * q^59 - 2 * q^60 - 24 * q^61 + 18 * q^64 + 6 * q^65 + 28 * q^66 - 16 * q^69 + 20 * q^70 + 60 * q^71 - 4 * q^74 + 12 * q^75 - 88 * q^76 + 36 * q^79 - 30 * q^80 + 10 * q^81 + 12 * q^84 - 14 * q^85 + 60 * q^86 + 44 * q^89 - 4 * q^90 - 24 * q^91 + 100 * q^94 - 36 * q^95 + 2 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

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