LMFDB - Embedded newform 15.4.b.a.4.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,4,Mod(4,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("15.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	15.b (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:	$0.885028650086$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4.4
Root			$3.70156i$ of defining polynomial
Character	$\chi$	$=$	15.4
Dual form			15.4.b.a.4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.70156i q^{2} -3.00000i q^{3} -14.1047 q^{4} +(11.1047 + 1.29844i) q^{5} +14.1047 q^{6} -16.2094i q^{7} -28.7016i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+4.70156i q^{2} -3.00000i q^{3} -14.1047 q^{4} +(11.1047 + 1.29844i) q^{5} +14.1047 q^{6} -16.2094i q^{7} -28.7016i q^{8} -9.00000 q^{9} +(-6.10469 + 52.2094i) q^{10} -40.2094 q^{11} +42.3141i q^{12} -19.7906i q^{13} +76.2094 q^{14} +(3.89531 - 33.3141i) q^{15} +22.1047 q^{16} +83.0156i q^{17} -42.3141i q^{18} +48.8375 q^{19} +(-156.628 - 18.3141i) q^{20} -48.6281 q^{21} -189.047i q^{22} -1.61250i q^{23} -86.1047 q^{24} +(121.628 + 28.8375i) q^{25} +93.0469 q^{26} +27.0000i q^{27} +228.628i q^{28} +24.5344 q^{29} +(156.628 + 18.3141i) q^{30} -12.4187 q^{31} -125.686i q^{32} +120.628i q^{33} -390.303 q^{34} +(21.0469 - 180.000i) q^{35} +126.942 q^{36} +325.884i q^{37} +229.612i q^{38} -59.3719 q^{39} +(37.2672 - 318.722i) q^{40} -242.419 q^{41} -228.628i q^{42} -367.350i q^{43} +567.141 q^{44} +(-99.9422 - 11.6859i) q^{45} +7.58125 q^{46} -204.544i q^{47} -66.3141i q^{48} +80.2562 q^{49} +(-135.581 + 571.842i) q^{50} +249.047 q^{51} +279.141i q^{52} +61.5281i q^{53} -126.942 q^{54} +(-446.512 - 52.2094i) q^{55} -465.234 q^{56} -146.512i q^{57} +115.350i q^{58} +112.209 q^{59} +(-54.9422 + 469.884i) q^{60} +477.350 q^{61} -58.3875i q^{62} +145.884i q^{63} +767.758 q^{64} +(25.6969 - 219.769i) q^{65} -567.141 q^{66} +558.094i q^{67} -1170.91i q^{68} -4.83749 q^{69} +(846.281 + 98.9531i) q^{70} +558.281 q^{71} +258.314i q^{72} -1011.77i q^{73} -1532.17 q^{74} +(86.5125 - 364.884i) q^{75} -688.837 q^{76} +651.769i q^{77} -279.141i q^{78} -1150.47 q^{79} +(245.466 + 28.7016i) q^{80} +81.0000 q^{81} -1139.75i q^{82} +1157.92i q^{83} +685.884 q^{84} +(-107.791 + 921.862i) q^{85} +1727.12 q^{86} -73.6032i q^{87} +1154.07i q^{88} -96.9751 q^{89} +(54.9422 - 469.884i) q^{90} -320.794 q^{91} +22.7438i q^{92} +37.2562i q^{93} +961.675 q^{94} +(542.325 + 63.4124i) q^{95} -377.058 q^{96} -1152.37i q^{97} +377.330i q^{98} +361.884 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 18 * q^4 + 6 * q^5 + 18 * q^6 - 36 * q^9	$ 4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9} + 14 q^{10} - 84 q^{11} + 228 q^{14} + 54 q^{15} + 50 q^{16} - 112 q^{19} - 396 q^{20} + 36 q^{21} - 306 q^{24} + 256 q^{25} - 12 q^{26} + 636 q^{29} + 396 q^{30} + 104 q^{31} - 716 q^{34} - 300 q^{35} + 162 q^{36} - 468 q^{39} + 418 q^{40} - 816 q^{41} + 1116 q^{44} - 54 q^{45} + 184 q^{46} - 140 q^{49} - 696 q^{50} + 612 q^{51} - 162 q^{54} - 864 q^{55} + 60 q^{56} + 372 q^{59} + 126 q^{60} + 680 q^{61} + 958 q^{64} + 948 q^{65} - 1116 q^{66} + 288 q^{69} + 1080 q^{70} - 72 q^{71} - 3132 q^{74} - 576 q^{75} - 2448 q^{76} - 760 q^{79} + 444 q^{80} + 324 q^{81} + 2052 q^{84} - 508 q^{85} + 4296 q^{86} - 2232 q^{89} - 126 q^{90} + 1944 q^{91} + 3232 q^{94} + 2784 q^{95} - 1854 q^{96} + 756 q^{99}+O(q^{100}) $ 4 * q - 18 * q^4 + 6 * q^5 + 18 * q^6 - 36 * q^9 + 14 * q^10 - 84 * q^11 + 228 * q^14 + 54 * q^15 + 50 * q^16 - 112 * q^19 - 396 * q^20 + 36 * q^21 - 306 * q^24 + 256 * q^25 - 12 * q^26 + 636 * q^29 + 396 * q^30 + 104 * q^31 - 716 * q^34 - 300 * q^35 + 162 * q^36 - 468 * q^39 + 418 * q^40 - 816 * q^41 + 1116 * q^44 - 54 * q^45 + 184 * q^46 - 140 * q^49 - 696 * q^50 + 612 * q^51 - 162 * q^54 - 864 * q^55 + 60 * q^56 + 372 * q^59 + 126 * q^60 + 680 * q^61 + 958 * q^64 + 948 * q^65 - 1116 * q^66 + 288 * q^69 + 1080 * q^70 - 72 * q^71 - 3132 * q^74 - 576 * q^75 - 2448 * q^76 - 760 * q^79 + 444 * q^80 + 324 * q^81 + 2052 * q^84 - 508 * q^85 + 4296 * q^86 - 2232 * q^89 - 126 * q^90 + 1944 * q^91 + 3232 * q^94 + 2784 * q^95 - 1854 * q^96 + 756 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$1$

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 $

$2$			4.70156i			1.66225i	0.556083	+	0.831127i	$0.312304\pi$
$2$			4.70156i			1.66225i	−0.556083	+	0.831127i	$0.687696\pi$
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$	−14.1047			−1.76309
$5$	11.1047	+	1.29844i	0.993233	+	0.116136i
$5$	11.1047	+	1.29844i	0.993233	+	0.116136i
$6$	14.1047			0.959702
$7$		−	16.2094i		−	0.875224i	−0.899164	−	0.437612i	$-0.855824\pi$
$7$		−	16.2094i		−	0.875224i	0.899164	−	0.437612i	$-0.144176\pi$
$8$		−	28.7016i		−	1.26844i
$9$	−9.00000			−0.333333
$10$	−6.10469	+	52.2094i	−0.193047	+	1.65101i
$11$	−40.2094			−1.10214			−0.551072	−	0.834458i	$-0.685781\pi$
$11$	−40.2094			−1.10214			−0.551072	+	0.834458i	$0.685781\pi$
$12$			42.3141i			1.01792i
$13$		−	19.7906i		−	0.422226i	−0.977462	−	0.211113i	$-0.932291\pi$
$13$		−	19.7906i		−	0.422226i	0.977462	−	0.211113i	$-0.0677087\pi$
$14$	76.2094			1.45484
$15$	3.89531	−	33.3141i	0.0670510	−	0.573444i
$16$	22.1047			0.345386
$17$			83.0156i			1.18437i	0.805803	+	0.592184i	$0.201734\pi$
$17$			83.0156i			1.18437i	−0.805803	+	0.592184i	$0.798266\pi$
$18$		−	42.3141i		−	0.554084i
$19$	48.8375			0.589689			0.294844	−	0.955545i	$-0.404732\pi$
$19$	48.8375			0.589689			0.294844	+	0.955545i	$0.404732\pi$
$20$	−156.628	−	18.3141i	−1.75116	−	0.204757i
$21$	−48.6281			−0.505311
$22$		−	189.047i		−	1.83204i
$23$		−	1.61250i		−	0.0146186i	−0.999973	−	0.00730932i	$-0.997673\pi$
$23$		−	1.61250i		−	0.0146186i	0.999973	−	0.00730932i	$-0.00232665\pi$
$24$	−86.1047			−0.732335
$25$	121.628	+	28.8375i	0.973025	+	0.230700i
$26$	93.0469			0.701846
$27$			27.0000i			0.192450i
$28$			228.628i			1.54309i
$29$	24.5344			0.157101			0.0785504	−	0.996910i	$-0.474971\pi$
$29$	24.5344			0.157101			0.0785504	+	0.996910i	$0.474971\pi$
$30$	156.628	+	18.3141i	0.953208	+	0.111456i
$31$	−12.4187			−0.0719507			−0.0359754	−	0.999353i	$-0.511454\pi$
$31$	−12.4187			−0.0719507			−0.0359754	+	0.999353i	$0.511454\pi$
$32$		−	125.686i		−	0.694323i
$33$			120.628i			0.636323i
$34$	−390.303			−1.96872
$35$	21.0469	−	180.000i	0.101645	−	0.869302i
$36$	126.942			0.587695
$37$			325.884i			1.44797i	0.689813	+	0.723987i	$0.257693\pi$
$37$			325.884i			1.44797i	−0.689813	+	0.723987i	$0.742307\pi$
$38$			229.612i			0.980212i
$39$	−59.3719			−0.243772
$40$	37.2672	−	318.722i	0.147312	−	1.25986i
$41$	−242.419			−0.923401			−0.461701	−	0.887036i	$-0.652761\pi$
$41$	−242.419			−0.923401			−0.461701	+	0.887036i	$0.652761\pi$
$42$		−	228.628i		−	0.839954i
$43$		−	367.350i		−	1.30280i	−0.758735	−	0.651399i	$-0.774182\pi$
$43$		−	367.350i		−	1.30280i	0.758735	−	0.651399i	$-0.225818\pi$
$44$	567.141			1.94317
$45$	−99.9422	−	11.6859i	−0.331078	−	0.0387119i
$46$	7.58125			0.0242999
$47$		−	204.544i		−	0.634804i	−0.948291	−	0.317402i	$-0.897190\pi$
$47$		−	204.544i		−	0.634804i	0.948291	−	0.317402i	$-0.102810\pi$
$48$		−	66.3141i		−	0.199409i
$49$	80.2562			0.233983
$50$	−135.581	+	571.842i	−0.383482	+	1.61741i
$51$	249.047			0.683795
$52$			279.141i			0.744420i
$53$			61.5281i			0.159463i	0.996816	+	0.0797314i	$0.0254063\pi$
$53$			61.5281i			0.159463i	−0.996816	+	0.0797314i	$0.974594\pi$
$54$	−126.942			−0.319901
$55$	−446.512	−	52.2094i	−1.09469	−	0.127998i
$56$	−465.234			−1.11017
$57$		−	146.512i		−	0.340457i
$58$			115.350i			0.261141i
$59$	112.209			0.247600			0.123800	−	0.992307i	$-0.460492\pi$
$59$	112.209			0.247600			0.123800	+	0.992307i	$0.460492\pi$
$60$	−54.9422	+	469.884i	−0.118217	+	1.01103i
$61$	477.350			1.00194			0.500970	−	0.865464i	$-0.332977\pi$
$61$	477.350			1.00194			0.500970	+	0.865464i	$0.332977\pi$
$62$		−	58.3875i		−	0.119600i
$63$			145.884i			0.291741i
$64$	767.758			1.49953
$65$	25.6969	−	219.769i	0.0490355	−	0.419369i
$66$	−567.141			−1.05773
$67$			558.094i			1.01764i	0.860872	+	0.508821i	$0.169918\pi$
$67$			558.094i			1.01764i	−0.860872	+	0.508821i	$0.830082\pi$
$68$		−	1170.91i		−	2.08814i
$69$	−4.83749			−0.00844008
$70$	846.281	+	98.9531i	1.44500	+	0.168959i
$71$	558.281			0.933180			0.466590	−	0.884474i	$-0.345482\pi$
$71$	558.281			0.933180			0.466590	+	0.884474i	$0.345482\pi$
$72$			258.314i			0.422814i
$73$		−	1011.77i		−	1.62217i	−0.584927	−	0.811086i	$-0.698877\pi$
$73$		−	1011.77i		−	1.62217i	0.584927	−	0.811086i	$-0.301123\pi$
$74$	−1532.17			−2.40690
$75$	86.5125	−	364.884i	0.133195	−	0.561776i
$76$	−688.837			−1.03967
$77$			651.769i			0.964623i
$78$		−	279.141i		−	0.405211i
$79$	−1150.47			−1.63845			−0.819227	−	0.573470i	$-0.805597\pi$
$79$	−1150.47			−1.63845			−0.819227	+	0.573470i	$0.805597\pi$
$80$	245.466	+	28.7016i	0.343049	+	0.0401117i
$81$	81.0000			0.111111
$82$		−	1139.75i		−	1.53493i
$83$			1157.92i			1.53131i	0.643251	+	0.765655i	$0.277585\pi$
$83$			1157.92i			1.53131i	−0.643251	+	0.765655i	$0.722415\pi$
$84$	685.884			0.890906
$85$	−107.791	+	921.862i	−0.137547	+	1.17635i
$86$	1727.12			2.16558
$87$		−	73.6032i		−	0.0907022i
$88$			1154.07i			1.39801i
$89$	−96.9751			−0.115498			−0.0577491	−	0.998331i	$-0.518392\pi$
$89$	−96.9751			−0.115498			−0.0577491	+	0.998331i	$0.518392\pi$
$90$	54.9422	−	469.884i	0.0643490	−	0.550335i
$91$	−320.794			−0.369542
$92$			22.7438i			0.0257739i
$93$			37.2562i			0.0415408i
$94$	961.675			1.05520
$95$	542.325	+	63.4124i	0.585699	+	0.0684840i
$96$	−377.058			−0.400868
$97$		−	1152.37i		−	1.20625i	−0.797648	−	0.603123i	$-0.793923\pi$
$97$		−	1152.37i		−	1.20625i	0.797648	−	0.603123i	$-0.206077\pi$
$98$			377.330i			0.388939i
$99$	361.884			0.367381
$100$	−1715.53	−	406.744i	−1.71553	−	0.406744i
$101$	−1156.49			−1.13936			−0.569679	−	0.821867i	$-0.692932\pi$
$101$	−1156.49			−1.13936			−0.569679	+	0.821867i	$0.692932\pi$
$102$			1170.91i			1.13664i
$103$			1333.70i			1.27585i	0.770096	+	0.637927i	$0.220208\pi$
$103$			1333.70i			1.27585i	−0.770096	+	0.637927i	$0.779792\pi$
$104$	−568.022			−0.535569
$105$	−540.000	−	63.1406i	−0.501891	−	0.0586847i
$106$	−289.278			−0.265068
$107$		−	798.263i		−	0.721224i	−0.932716	−	0.360612i	$-0.882568\pi$
$107$		−	798.263i		−	0.721224i	0.932716	−	0.360612i	$-0.117432\pi$
$108$		−	380.827i		−	0.339306i
$109$	985.119			0.865663			0.432831	−	0.901475i	$-0.357515\pi$
$109$	985.119			0.865663			0.432831	+	0.901475i	$0.357515\pi$
$110$	245.466	−	2099.31i	0.212766	−	1.81965i
$111$	977.653			0.835988
$112$		−	358.303i		−	0.302290i
$113$		−	1888.25i		−	1.57196i	−0.618253	−	0.785979i	$-0.712159\pi$
$113$		−	1888.25i		−	1.57196i	0.618253	−	0.785979i	$-0.287841\pi$
$114$	688.837			0.565926
$115$	2.09373	−	17.9063i	0.00169775	−	0.0145197i
$116$	−346.050			−0.276982
$117$			178.116i			0.140742i
$118$			527.559i			0.411574i
$119$	1345.63			1.03659
$120$	−956.166	−	111.802i	−0.727380	−	0.0850503i
$121$	285.794			0.214721
$122$			2244.29i			1.66548i
$123$			727.256i			0.533126i
$124$	175.163			0.126855
$125$	1313.20	+	478.158i	0.939648	+	0.342142i
$126$	−685.884			−0.484948
$127$		−	620.859i		−	0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$127$		−	620.859i		−	0.433798i	0.976194	−	0.216899i	$-0.0695943\pi$
$128$			2604.17i			1.79827i
$129$	−1102.05			−0.752171
$130$	1033.26	+	120.816i	0.697097	+	0.0815094i
$131$	−2588.35			−1.72630			−0.863151	−	0.504947i	$-0.831512\pi$
$131$	−2588.35			−1.72630			−0.863151	+	0.504947i	$0.831512\pi$
$132$		−	1701.42i		−	1.12189i
$133$		−	791.625i		−	0.516110i
$134$	−2623.91			−1.69158
$135$	−35.0578	+	299.827i	−0.0223503	+	0.191148i
$136$	2382.68			1.50230
$137$		−	1656.29i		−	1.03289i	−0.856319	−	0.516447i	$-0.827254\pi$
$137$		−	1656.29i		−	1.03289i	0.856319	−	0.516447i	$-0.172746\pi$
$138$		−	22.7438i		−	0.0140295i
$139$	153.256			0.0935182			0.0467591	−	0.998906i	$-0.485111\pi$
$139$	153.256			0.0935182			0.0467591	+	0.998906i	$0.485111\pi$
$140$	−296.859	+	2538.84i	−0.179209	+	1.53265i
$141$	−613.631			−0.366504
$142$			2624.79i			1.55118i
$143$			795.769i			0.465353i
$144$	−198.942			−0.115129
$145$	272.447	+	31.8564i	0.156038	+	0.0182450i
$146$	4756.89			2.69646
$147$		−	240.769i		−	0.135090i
$148$		−	4596.50i		−	2.55290i
$149$	−1483.38			−0.815591			−0.407795	−	0.913073i	$-0.633702\pi$
$149$	−1483.38			−0.815591			−0.407795	+	0.913073i	$0.633702\pi$
$150$	1715.53	+	406.744i	0.933814	+	0.221403i
$151$	−394.281			−0.212491			−0.106246	−	0.994340i	$-0.533883\pi$
$151$	−394.281			−0.212491			−0.106246	+	0.994340i	$0.533883\pi$
$152$		−	1401.71i		−	0.747986i
$153$		−	747.141i		−	0.394789i
$154$	−3064.33			−1.60345
$155$	−137.906	−	16.1250i	−0.0714639	−	0.00835606i
$156$	837.422			0.429791
$157$			1727.05i			0.877922i	0.898506	+	0.438961i	$0.144653\pi$
$157$			1727.05i			0.877922i	−0.898506	+	0.438961i	$0.855347\pi$
$158$		−	5409.00i		−	2.72352i
$159$	184.584			0.0920659
$160$	163.195	−	1395.70i	0.0806358	−	0.689625i
$161$	−26.1376			−0.0127946
$162$			380.827i			0.184695i
$163$			2034.28i			0.977529i	0.872416	+	0.488764i	$0.162552\pi$
$163$			2034.28i			0.977529i	−0.872416	+	0.488764i	$0.837448\pi$
$164$	3419.24			1.62804
$165$	−156.628	+	1339.54i	−0.0738999	+	0.632017i
$166$	−5444.06			−2.54543
$167$			192.900i			0.0893835i	0.999001	+	0.0446918i	$0.0142306\pi$
$167$			192.900i			0.0893835i	−0.999001	+	0.0446918i	$0.985769\pi$
$168$			1395.70i			0.640957i
$169$	1805.33			0.821726
$170$	−4334.19	−	506.784i	−1.95540	−	0.228639i
$171$	−439.537			−0.196563
$172$			5181.36i			2.29695i
$173$			1239.91i			0.544905i	0.962169	+	0.272452i	$0.0878347\pi$
$173$			1239.91i			0.544905i	−0.962169	+	0.272452i	$0.912165\pi$
$174$	346.050			0.150770
$175$	467.438	−	1971.52i	0.201914	−	0.851615i
$176$	−888.816			−0.380665
$177$		−	336.628i		−	0.142952i
$178$		−	455.934i		−	0.191987i
$179$	2636.86			1.10105			0.550525	−	0.834818i	$-0.314427\pi$
$179$	2636.86			1.10105			0.550525	+	0.834818i	$0.314427\pi$
$180$	1409.65	+	164.827i	0.583719	+	0.0682525i
$181$	3317.58			1.36240			0.681199	−	0.732099i	$-0.261459\pi$
$181$	3317.58			1.36240			0.681199	+	0.732099i	$0.261459\pi$
$182$		−	1508.23i		−	0.614272i
$183$		−	1432.05i		−	0.578471i
$184$	−46.2812			−0.0185429
$185$	−423.141	+	3618.84i	−0.168162	+	1.43818i
$186$	−175.163			−0.0690513
$187$		−	3338.01i		−	1.30534i
$188$			2885.02i			1.11921i
$189$	437.653			0.168437
$190$	−298.138	+	2549.77i	−0.113838	+	0.973579i
$191$	−624.506			−0.236585			−0.118292	−	0.992979i	$-0.537742\pi$
$191$	−624.506			−0.236585			−0.118292	+	0.992979i	$0.537742\pi$
$192$		−	2303.27i		−	0.865752i
$193$			436.144i			0.162665i	0.996687	+	0.0813324i	$0.0259175\pi$
$193$			436.144i			0.162665i	−0.996687	+	0.0813324i	$0.974082\pi$
$194$	5417.96			2.00509
$195$	−659.306	−	77.0907i	−0.242123	−	0.0283107i
$196$	−1131.99			−0.412532
$197$			3355.81i			1.21366i	0.794831	+	0.606831i	$0.207560\pi$
$197$			3355.81i			1.21366i	−0.794831	+	0.606831i	$0.792440\pi$
$198$			1701.42i			0.610681i
$199$	−3799.77			−1.35356			−0.676780	−	0.736185i	$-0.736625\pi$
$199$	−3799.77			−1.35356			−0.676780	+	0.736185i	$0.736625\pi$
$200$	827.681	−	3490.92i	0.292629	−	1.23423i
$201$	1674.28			0.587536
$202$		−	5437.31i		−	1.89390i
$203$		−	397.687i		−	0.137498i
$204$	−3512.73			−1.20559
$205$	−2691.98	−	314.766i	−0.917153	−	0.107240i
$206$	−6270.46			−2.12079
$207$			14.5125i			0.00487288i
$208$		−	437.466i		−	0.145831i
$209$	−1963.72			−0.649922
$210$	296.859	−	2538.84i	0.0975488	−	0.834271i
$211$	2365.27			0.771715			0.385857	−	0.922558i	$-0.373906\pi$
$211$	2365.27			0.771715			0.385857	+	0.922558i	$0.373906\pi$
$212$		−	867.834i		−	0.281147i
$213$		−	1674.84i		−	0.538772i
$214$	3753.08			1.19886
$215$	476.981	−	4079.31i	0.151302	−	1.29398i
$216$	774.942			0.244112
$217$			201.300i			0.0629730i
$218$			4631.60i			1.43895i
$219$	−3035.31			−0.936562
$220$	6297.92	+	736.397i	1.93003	+	0.225672i
$221$	1642.93			0.500070
$222$			4596.50i			1.38962i
$223$			3328.58i			0.999545i	0.866157	+	0.499772i	$0.166583\pi$
$223$			3328.58i			0.999545i	−0.866157	+	0.499772i	$0.833417\pi$
$224$	−2037.29			−0.607688
$225$	−1094.65	−	259.537i	−0.324342	−	0.0769000i
$226$	8877.71			2.61299
$227$		−	527.100i		−	0.154118i	−0.997027	−	0.0770592i	$-0.975447\pi$
$227$		−	527.100i		−	0.154118i	0.997027	−	0.0770592i	$-0.0245530\pi$
$228$			2066.51i			0.600255i
$229$	−2566.06			−0.740479			−0.370240	−	0.928936i	$-0.620724\pi$
$229$	−2566.06			−0.740479			−0.370240	+	0.928936i	$0.620724\pi$
$230$	84.1875	+	9.84379i	0.0241355	+	0.00282209i
$231$	1955.31			0.556925
$232$		−	704.175i		−	0.199273i
$233$		−	5534.99i		−	1.55626i	−0.628101	−	0.778132i	$-0.716168\pi$
$233$		−	5534.99i		−	1.55626i	0.628101	−	0.778132i	$-0.283832\pi$
$234$	−837.422			−0.233949
$235$	265.587	−	2271.39i	0.0737234	−	0.630508i
$236$	−1582.68			−0.436541
$237$			3451.41i			0.945962i
$238$			6326.57i			1.72307i
$239$	−1010.01			−0.273355			−0.136678	−	0.990616i	$-0.543642\pi$
$239$	−1010.01			−0.273355			−0.136678	+	0.990616i	$0.543642\pi$
$240$	86.1047	−	736.397i	0.0231585	−	0.198059i
$241$	−4074.29			−1.08900			−0.544498	−	0.838762i	$-0.683280\pi$
$241$	−4074.29			−1.08900			−0.544498	+	0.838762i	$0.683280\pi$
$242$			1343.68i			0.356921i
$243$		−	243.000i		−	0.0641500i
$244$	−6732.87			−1.76651
$245$	891.220	+	104.208i	0.232400	+	0.0271738i
$246$	−3419.24			−0.886190
$247$		−	966.525i		−	0.248982i
$248$			356.437i			0.0912653i
$249$	3473.77			0.884103
$250$	−2248.09	+	6174.08i	−0.568726	+	1.56193i
$251$	1773.98			0.446107			0.223054	−	0.974806i	$-0.428398\pi$
$251$	1773.98			0.446107			0.223054	+	0.974806i	$0.428398\pi$
$252$		−	2057.65i		−	0.514365i
$253$			64.8375i			0.0161119i
$254$	2919.01			0.721082
$255$	2765.59	+	323.372i	0.679168	+	0.0794131i
$256$	−6101.62			−1.48965
$257$			662.784i			0.160869i	0.996760	+	0.0804345i	$0.0256308\pi$
$257$			662.784i			0.160869i	−0.996760	+	0.0804345i	$0.974369\pi$
$258$		−	5181.36i		−	1.25030i
$259$	5282.38			1.26730
$260$	−362.447	+	3099.77i	−0.0864538	+	0.739383i
$261$	−220.810			−0.0523669
$262$		−	12169.3i		−	2.86955i
$263$		−	712.312i		−	0.167008i	−0.996507	−	0.0835039i	$-0.973389\pi$
$263$		−	712.312i		−	0.167008i	0.996507	−	0.0835039i	$-0.0266111\pi$
$264$	3462.22			0.807139
$265$	−79.8904	+	683.250i	−0.0185194	+	0.158384i
$266$	3721.87			0.857905
$267$			290.925i			0.0666829i
$268$		−	7871.74i		−	1.79419i
$269$	3136.41			0.710894			0.355447	−	0.934696i	$-0.384329\pi$
$269$	3136.41			0.710894			0.355447	+	0.934696i	$0.384329\pi$
$270$	−1409.65	−	164.827i	−0.317736	−	0.0371519i
$271$	−2275.69			−0.510105			−0.255053	−	0.966927i	$-0.582093\pi$
$271$	−2275.69			−0.510105			−0.255053	+	0.966927i	$0.582093\pi$
$272$			1835.03i			0.409064i
$273$			962.381i			0.213355i
$274$	7787.15			1.71693
$275$	−4890.59	−	1159.54i	−1.07241	−	0.254264i
$276$	68.2313			0.0148806
$277$			5171.00i			1.12164i	0.827937	+	0.560821i	$0.189515\pi$
$277$			5171.00i			1.12164i	−0.827937	+	0.560821i	$0.810485\pi$
$278$			720.544i			0.155451i
$279$	111.769			0.0239836
$280$	−5166.28	−	604.078i	−1.10266	−	0.128931i
$281$	2240.14			0.475571			0.237785	−	0.971318i	$-0.423578\pi$
$281$	2240.14			0.475571			0.237785	+	0.971318i	$0.423578\pi$
$282$		−	2885.02i		−	0.609222i
$283$		−	225.244i		−	0.0473123i	−0.999720	−	0.0236561i	$-0.992469\pi$
$283$		−	225.244i		−	0.0473123i	0.999720	−	0.0236561i	$-0.00753068\pi$
$284$	−7874.38			−1.64528
$285$	190.237	−	1626.98i	0.0395393	−	0.338153i
$286$	−3741.36			−0.773535
$287$			3929.46i			0.808183i
$288$			1131.17i			0.231441i
$289$	−1978.59			−0.402726
$290$	−149.775	+	1280.93i	−0.0303279	+	0.259374i
$291$	−3457.12			−0.696427
$292$			14270.7i			2.86003i
$293$		−	1139.86i		−	0.227274i	−0.993522	−	0.113637i	$-0.963750\pi$
$293$		−	1139.86i		−	0.227274i	0.993522	−	0.113637i	$-0.0362501\pi$
$294$	1131.99			0.224554
$295$	1246.05	+	145.697i	0.245925	+	0.0287553i
$296$	9353.39			1.83667
$297$		−	1085.65i		−	0.212108i
$298$		−	6974.19i		−	1.35572i
$299$	−31.9123			−0.00617237
$300$	−1220.23	+	5146.58i	−0.234834	+	0.990460i
$301$	−5954.51			−1.14024
$302$		−	1853.74i		−	0.353214i
$303$			3469.47i			0.657808i
$304$	1079.54			0.203670
$305$	5300.82	+	619.809i	0.995161	+	0.116361i
$306$	3512.73			0.656240
$307$			5244.86i			0.975049i	0.873109	+	0.487525i	$0.162100\pi$
$307$			5244.86i			0.975049i	−0.873109	+	0.487525i	$0.837900\pi$
$308$		−	9192.99i		−	1.70071i
$309$	4001.09			0.736615
$310$	75.8125	−	648.375i	0.0138899	−	0.118791i
$311$	−5188.26			−0.945977			−0.472989	−	0.881068i	$-0.656825\pi$
$311$	−5188.26			−0.945977			−0.472989	+	0.881068i	$0.656825\pi$
$312$			1704.07i			0.309211i
$313$		−	486.656i		−	0.0878832i	−0.999034	−	0.0439416i	$-0.986008\pi$
$313$		−	486.656i		−	0.0878832i	0.999034	−	0.0439416i	$-0.0139915\pi$
$314$	−8119.85			−1.45933
$315$	−189.422	+	1620.00i	−0.0338816	+	0.289767i
$316$	16227.0			2.88873
$317$		−	4218.87i		−	0.747493i	−0.927531	−	0.373747i	$-0.878073\pi$
$317$		−	4218.87i		−	0.747493i	0.927531	−	0.373747i	$-0.121927\pi$
$318$			867.834i			0.153037i
$319$	−986.512			−0.173148
$320$	8525.71	+	996.886i	1.48938	+	0.174149i
$321$	−2394.79			−0.416399
$322$		−	122.887i		−	0.0212678i
$323$			4054.27i			0.698408i
$324$	−1142.48			−0.195898
$325$	570.712	−	2407.10i	0.0974074	−	0.410836i
$326$	−9564.30			−1.62490
$327$		−	2955.36i		−	0.499791i
$328$			6957.80i			1.17128i
$329$	−3315.53			−0.555595
$330$	−6297.92	−	736.397i	−1.05057	−	0.122840i
$331$	7439.94			1.23546			0.617728	−	0.786392i	$-0.288053\pi$
$331$	7439.94			1.23546			0.617728	+	0.786392i	$0.288053\pi$
$332$		−	16332.2i		−	2.69983i
$333$		−	2932.96i		−	0.482658i
$334$	−906.931			−0.148578
$335$	−724.650	+	6197.46i	−0.118185	+	1.01076i
$336$	−1074.91			−0.174527
$337$		−	6555.39i		−	1.05963i	−0.848113	−	0.529815i	$-0.822261\pi$
$337$		−	6555.39i		−	1.05963i	0.848113	−	0.529815i	$-0.177739\pi$
$338$			8487.88i			1.36592i
$339$	−5664.74			−0.907571
$340$	1520.35	−	13002.6i	0.242508	−	2.07401i
$341$	499.350			0.0793000
$342$		−	2066.51i		−	0.326737i
$343$		−	6860.72i		−	1.08001i
$344$	−10543.5			−1.65252
$345$	−53.7188	−	6.28118i	−0.00838297	−	0.000980195i
$346$	−5829.51			−0.905770
$347$			1950.56i			0.301763i	0.988552	+	0.150881i	$0.0482112\pi$
$347$			1950.56i			0.301763i	−0.988552	+	0.150881i	$0.951789\pi$
$348$			1038.15i			0.159916i
$349$	1426.74			0.218830			0.109415	−	0.993996i	$-0.465102\pi$
$349$	1426.74			0.218830			0.109415	+	0.993996i	$0.465102\pi$
$350$	9269.20	+	2197.69i	1.41560	+	0.335632i
$351$	534.347			0.0812573
$352$			5053.75i			0.765244i
$353$			7078.96i			1.06735i	0.845689	+	0.533676i	$0.179190\pi$
$353$			7078.96i			1.06735i	−0.845689	+	0.533676i	$0.820810\pi$
$354$	1582.68			0.237623
$355$	6199.54	+	724.893i	0.926866	+	0.108376i
$356$	1367.80			0.203633
$357$		−	4036.89i		−	0.598474i
$358$			12397.4i			1.83023i
$359$	5409.79			0.795314			0.397657	−	0.917534i	$-0.369823\pi$
$359$	5409.79			0.795314			0.397657	+	0.917534i	$0.369823\pi$
$360$	−335.405	+	2868.50i	−0.0491038	+	0.419953i
$361$	−4473.90			−0.652267
$362$			15597.8i			2.26465i
$363$		−	857.381i		−	0.123969i
$364$	4524.69			0.651534
$365$	1313.72	−	11235.4i	0.188392	−	1.61120i
$366$	6732.87			0.961565
$367$			4940.09i			0.702645i	0.936255	+	0.351322i	$0.114268\pi$
$367$			4940.09i			0.702645i	−0.936255	+	0.351322i	$0.885732\pi$
$368$		−	35.6437i		−	0.00504907i
$369$	2181.77			0.307800
$370$	−17014.2	−	1989.42i	−2.39061	−	0.279527i
$371$	997.332			0.139566
$372$		−	525.488i		−	0.0732399i
$373$		−	12891.9i		−	1.78959i	−0.446473	−	0.894797i	$-0.647320\pi$
$373$		−	12891.9i		−	1.78959i	0.446473	−	0.894797i	$-0.352680\pi$
$374$	15693.8			2.16981
$375$	1434.47	−	3939.60i	0.197536	−	0.542506i
$376$	−5870.72			−0.805211
$377$		−	485.551i		−	0.0663320i
$378$			2057.65i			0.279985i
$379$	9475.15			1.28418			0.642092	−	0.766627i	$-0.278067\pi$
$379$	9475.15			1.28418			0.642092	+	0.766627i	$0.278067\pi$
$380$	−7649.32	−	894.413i	−1.03264	−	0.120743i
$381$	−1862.58			−0.250453
$382$		−	2936.16i		−	0.393264i
$383$			5800.97i			0.773931i	0.922094	+	0.386966i	$0.126477\pi$
$383$			5800.97i			0.773931i	−0.922094	+	0.386966i	$0.873523\pi$
$384$	7812.52			1.03823
$385$	−846.281	+	7237.69i	−0.112027	+	0.958095i
$386$	−2050.56			−0.270390
$387$			3306.15i			0.434266i
$388$			16253.9i			2.12672i
$389$	−13779.7			−1.79603			−0.898016	−	0.439962i	$-0.854992\pi$
$389$	−13779.7			−1.79603			−0.898016	+	0.439962i	$0.854992\pi$
$390$	362.447	−	3099.77i	0.0470595	−	0.402469i
$391$	133.862			0.0173138
$392$		−	2303.48i		−	0.296794i
$393$			7765.06i			0.996680i
$394$	−15777.5			−2.01741
$395$	−12775.6	−	1493.81i	−1.62737	−	0.190283i
$396$	−5104.27			−0.647725
$397$		−	2816.46i		−	0.356056i	−0.984025	−	0.178028i	$-0.943028\pi$
$397$		−	2816.46i		−	0.356056i	0.984025	−	0.178028i	$-0.0569718\pi$
$398$		−	17864.8i		−	2.24996i
$399$	−2374.88			−0.297976
$400$	2688.55	+	637.444i	0.336069	+	0.0796805i
$401$	11986.4			1.49270			0.746352	−	0.665551i	$-0.231804\pi$
$401$	11986.4			1.49270			0.746352	+	0.665551i	$0.231804\pi$
$402$			7871.74i			0.976633i
$403$			245.775i			0.0303794i
$404$	16311.9			2.00879
$405$	899.480	+	105.173i	0.110359	+	0.0129040i
$406$	1869.75			0.228557
$407$		−	13103.6i		−	1.59588i
$408$		−	7148.03i		−	0.867354i
$409$	3339.07			0.403683			0.201841	−	0.979418i	$-0.435307\pi$
$409$	3339.07			0.403683			0.201841	+	0.979418i	$0.435307\pi$
$410$	1479.89	−	12656.5i	0.178260	−	1.52454i
$411$	−4968.87			−0.596342
$412$		−	18811.4i		−	2.24944i
$413$		−	1818.84i		−	0.216706i
$414$	−68.2313			−0.00809996
$415$	−1503.49	+	12858.4i	−0.177840	+	1.52095i
$416$	−2487.40			−0.293161
$417$		−	459.769i		−	0.0539927i
$418$		−	9232.57i		−	1.08033i
$419$	−1688.52			−0.196873			−0.0984363	−	0.995143i	$-0.531384\pi$
$419$	−1688.52			−0.196873			−0.0984363	+	0.995143i	$0.531384\pi$
$420$	7616.53	+	890.578i	0.884878	+	0.103466i
$421$	−2664.27			−0.308429			−0.154214	−	0.988037i	$-0.549285\pi$
$421$	−2664.27			−0.308429			−0.154214	+	0.988037i	$0.549285\pi$
$422$			11120.5i			1.28279i
$423$			1840.89i			0.211601i
$424$	1765.95			0.202269
$425$	−2393.96	+	10097.0i	−0.273233	+	1.15242i
$426$	7874.38			0.895575
$427$		−	7737.54i		−	0.876923i
$428$			11259.2i			1.27158i
$429$	2387.31			0.268672
$430$	19179.1	+	2242.56i	2.15093	+	0.251502i
$431$	12266.0			1.37084			0.685420	−	0.728148i	$-0.259619\pi$
$431$	12266.0			1.37084			0.685420	+	0.728148i	$0.259619\pi$
$432$			596.827i			0.0664695i
$433$			15647.3i			1.73664i	0.496008	+	0.868318i	$0.334799\pi$
$433$			15647.3i			1.73664i	−0.496008	+	0.868318i	$0.665201\pi$
$434$	−946.425			−0.104677
$435$	95.5691	−	817.340i	0.0105338	−	0.0900884i
$436$	−13894.8			−1.52624
$437$		−	78.7503i		−	0.00862045i
$438$		−	14270.7i		−	1.55680i
$439$	−16131.0			−1.75373			−0.876867	−	0.480733i	$-0.840371\pi$
$439$	−16131.0			−1.75373			−0.876867	+	0.480733i	$0.840371\pi$
$440$	−1498.49	+	12815.6i	−0.162358	+	1.38855i
$441$	−722.306			−0.0779944
$442$			7724.34i			0.831243i
$443$		−	10053.7i		−	1.07825i	−0.842225	−	0.539127i	$-0.818754\pi$
$443$		−	10053.7i		−	1.07825i	0.842225	−	0.539127i	$-0.181246\pi$
$444$	−13789.5			−1.47392
$445$	−1076.88	−	125.916i	−0.114717	−	0.0134135i
$446$	−15649.5			−1.66150
$447$			4450.13i			0.470882i
$448$		−	12444.9i		−	1.31242i
$449$	−7477.71			−0.785957			−0.392979	−	0.919548i	$-0.628555\pi$
$449$	−7477.71			−0.785957			−0.392979	+	0.919548i	$0.628555\pi$
$450$	1220.23	−	5146.58i	0.127827	−	0.539138i
$451$	9747.51			1.01772
$452$			26633.1i			2.77150i
$453$			1182.84i			0.122682i
$454$	2478.19			0.256184
$455$	−3562.31	−	416.531i	−0.367041	−	0.0429170i
$456$	−4205.14			−0.431850
$457$			1363.46i			0.139562i	0.997562	+	0.0697812i	$0.0222301\pi$
$457$			1363.46i			0.139562i	−0.997562	+	0.0697812i	$0.977770\pi$
$458$		−	12064.5i		−	1.23086i
$459$	−2241.42			−0.227932
$460$	−29.5314	+	252.562i	−0.00299328	+	0.0255995i
$461$	5276.77			0.533109			0.266555	−	0.963820i	$-0.414115\pi$
$461$	5276.77			0.533109			0.266555	+	0.963820i	$0.414115\pi$
$462$			9192.99i			0.925751i
$463$		−	5740.02i		−	0.576159i	−0.957607	−	0.288079i	$-0.906983\pi$
$463$		−	5740.02i		−	0.576159i	0.957607	−	0.288079i	$-0.0930167\pi$
$464$	542.325			0.0542604
$465$	−48.3749	+	413.719i	−0.00482437	+	0.0412597i
$466$	26023.1			2.58690
$467$			6233.36i			0.617657i	0.951118	+	0.308828i	$0.0999368\pi$
$467$			6233.36i			0.617657i	−0.951118	+	0.308828i	$0.900063\pi$
$468$		−	2512.27i		−	0.248140i
$469$	9046.35			0.890664
$470$	10679.1	+	1248.68i	1.04806	+	0.122547i
$471$	5181.16			0.506869
$472$		−	3220.58i		−	0.314067i
$473$			14770.9i			1.43587i
$474$	−16227.0			−1.57243
$475$	5940.01	+	1408.35i	0.573782	+	0.136041i
$476$	−18979.7			−1.82759
$477$		−	553.753i		−	0.0531543i
$478$		−	4748.61i		−	0.454385i
$479$	19688.2			1.87803			0.939013	−	0.343881i	$-0.111742\pi$
$479$	19688.2			1.87803			0.939013	+	0.343881i	$0.111742\pi$
$480$	−4187.11	−	489.586i	−0.398155	−	0.0465551i
$481$	6449.46			0.611372
$482$		−	19155.5i		−	1.81019i
$483$			78.4127i			0.00738696i
$484$	−4031.03			−0.378572
$485$	1496.29	−	12796.8i	0.140088	−	1.19808i
$486$	1142.48			0.106634
$487$		−	3955.08i		−	0.368012i	−0.982925	−	0.184006i	$-0.941093\pi$
$487$		−	3955.08i		−	0.368012i	0.982925	−	0.184006i	$-0.0589065\pi$
$488$		−	13700.7i		−	1.27090i
$489$	6102.84			0.564377
$490$	−489.939	+	4190.13i	−0.0451698	+	0.386308i
$491$	−13893.5			−1.27699			−0.638497	−	0.769624i	$-0.720443\pi$
$491$	−13893.5			−1.27699			−0.638497	+	0.769624i	$0.720443\pi$
$492$		−	10257.7i		−	0.939947i
$493$			2036.74i			0.186065i
$494$	4544.18			0.413871
$495$	4018.61	+	469.884i	0.364895	+	0.0426661i
$496$	−274.512			−0.0248508
$497$		−	9049.39i		−	0.816741i
$498$			16332.2i			1.46960i
$499$	−13523.7			−1.21324			−0.606618	−	0.794993i	$-0.707474\pi$
$499$	−13523.7			−1.21324			−0.606618	+	0.794993i	$0.707474\pi$
$500$	−18522.3	−	6744.27i	−1.65668	−	0.603226i
$501$	578.700			0.0516056
$502$			8340.50i			0.741543i
$503$		−	13135.4i		−	1.16437i	−0.813057	−	0.582184i	$-0.802198\pi$
$503$		−	13135.4i		−	1.16437i	0.813057	−	0.582184i	$-0.197802\pi$
$504$	4187.11			0.370057
$505$	−12842.5	−	1501.63i	−1.13165	−	0.132320i
$506$	−304.837			−0.0267820
$507$		−	5415.99i		−	0.474423i
$508$			8757.03i			0.764823i
$509$	2222.71			0.193556			0.0967778	−	0.995306i	$-0.469146\pi$
$509$	2222.71			0.193556			0.0967778	+	0.995306i	$0.469146\pi$
$510$	−1520.35	+	13002.6i	−0.132005	+	1.12895i
$511$	−16400.1			−1.41976
$512$		−	7853.76i		−	0.677911i
$513$			1318.61i			0.113486i
$514$	−3116.12			−0.267405
$515$	−1731.72	+	14810.3i	−0.148172	+	1.26722i
$516$	15544.1			1.32614
$517$			8224.57i			0.699645i
$518$			24835.4i			2.10658i
$519$	3719.73			0.314601
$520$	−6307.71	−	737.541i	−0.531945	−	0.0621987i
$521$	−4916.42			−0.413421			−0.206710	−	0.978402i	$-0.566276\pi$
$521$	−4916.42			−0.413421			−0.206710	+	0.978402i	$0.566276\pi$
$522$		−	1038.15i		−	0.0870471i
$523$		−	17743.4i		−	1.48349i	−0.670681	−	0.741746i	$-0.733998\pi$
$523$		−	17743.4i		−	1.48349i	0.670681	−	0.741746i	$-0.266002\pi$
$524$	36507.9			3.04362
$525$	−5914.55	−	1402.31i	−0.491680	−	0.116575i
$526$	3348.98			0.277609
$527$		−	1030.95i		−	0.0852161i
$528$			2666.45i			0.219777i
$529$	12164.4			0.999786
$530$	−3212.34	−	375.610i	−0.263274	−	0.0307839i
$531$	−1009.88			−0.0825334
$532$			11165.6i			0.909946i
$533$			4797.62i			0.389884i
$534$	−1367.80			−0.110844
$535$	1036.49	−	8864.46i	0.0837599	−	0.716344i
$536$	16018.2			1.29082
$537$		−	7910.58i		−	0.635692i
$538$			14746.0i			1.18169i
$539$	−3227.05			−0.257883
$540$	494.480	−	4228.96i	0.0394056	−	0.337010i
$541$	−12671.3			−1.00699			−0.503495	−	0.863998i	$-0.667953\pi$
$541$	−12671.3			−1.00699			−0.503495	+	0.863998i	$0.667953\pi$
$542$		−	10699.3i		−	0.847924i
$543$		−	9952.74i		−	0.786580i
$544$	10433.9			0.822334
$545$	10939.4	+	1279.12i	0.859805	+	0.100534i
$546$	−4524.69			−0.354650
$547$			5250.90i			0.410443i	0.978716	+	0.205221i	$0.0657915\pi$
$547$			5250.90i			0.410443i	−0.978716	+	0.205221i	$0.934209\pi$
$548$			23361.5i			1.82108i
$549$	−4296.15			−0.333980
$550$	5451.64	−	22993.4i	0.422652	−	1.78262i
$551$	1198.20			0.0926406
$552$			138.844i			0.0107057i
$553$			18648.4i			1.43401i
$554$	−24311.8			−1.86445
$555$	10856.5	+	1269.42i	0.830332	+	0.0970882i
$556$	−2161.63			−0.164881
$557$			25830.2i			1.96492i	0.186465	+	0.982462i	$0.440297\pi$
$557$			25830.2i			1.96492i	−0.186465	+	0.982462i	$0.559703\pi$
$558$			525.488i			0.0398668i
$559$	−7270.09			−0.550075
$560$	465.234	−	3978.84i	0.0351067	−	0.300244i
$561$	−10014.0			−0.753640
$562$			10532.1i			0.790519i
$563$		−	2021.14i		−	0.151298i	−0.997135	−	0.0756490i	$-0.975897\pi$
$563$		−	2021.14i		−	0.151298i	0.997135	−	0.0756490i	$-0.0241029\pi$
$564$	8655.07			0.646178
$565$	2451.77	−	20968.4i	0.182561	−	1.56132i
$566$	1059.00			0.0786450
$567$		−	1312.96i		−	0.0972471i
$568$		−	16023.5i		−	1.18368i
$569$	−8706.51			−0.641469			−0.320734	−	0.947169i	$-0.603930\pi$
$569$	−8706.51			−0.641469			−0.320734	+	0.947169i	$0.603930\pi$
$570$	7649.32	+	894.413i	0.562096	+	0.0657243i
$571$	−12194.5			−0.893740			−0.446870	−	0.894599i	$-0.647461\pi$
$571$	−12194.5			−0.893740			−0.446870	+	0.894599i	$0.647461\pi$
$572$		−	11224.1i		−	0.820458i
$573$			1873.52i			0.136592i
$574$	−18474.6			−1.34340
$575$	46.5004	−	196.125i	0.00337252	−	0.0142243i
$576$	−6909.82			−0.499842
$577$		−	15264.0i		−	1.10130i	−0.834737	−	0.550649i	$-0.814380\pi$
$577$		−	15264.0i		−	1.10130i	0.834737	−	0.550649i	$-0.185620\pi$
$578$		−	9302.48i		−	0.669433i
$579$	1308.43			0.0939146
$580$	−3842.78	−	449.324i	−0.275108	−	0.0321675i
$581$	18769.2			1.34024
$582$		−	16253.9i		−	1.15764i
$583$		−	2474.01i		−	0.175751i
$584$	−29039.3			−2.05763
$585$	−231.272	+	1977.92i	−0.0163452	+	0.139790i
$586$	5359.12			0.377787
$587$		−	8456.89i		−	0.594639i	−0.954778	−	0.297319i	$-0.903907\pi$
$587$		−	8456.89i		−	0.594639i	0.954778	−	0.297319i	$-0.0960926\pi$
$588$			3395.97i			0.238176i
$589$	−606.500			−0.0424285
$590$	−685.003	+	5858.38i	−0.0477985	+	0.408789i
$591$	10067.4			0.700708
$592$			7203.57i			0.500110i
$593$			1225.23i			0.0848467i	0.999100	+	0.0424234i	$0.0135078\pi$
$593$			1225.23i			0.0848467i	−0.999100	+	0.0424234i	$0.986492\pi$
$594$	5104.27			0.352577
$595$	14942.8	+	1747.22i	1.02957	+	0.120385i
$596$	20922.6			1.43796
$597$			11399.3i			0.781478i
$598$		−	150.038i		−	0.0102600i
$599$	16060.0			1.09548			0.547741	−	0.836648i	$-0.315488\pi$
$599$	16060.0			1.09548			0.547741	+	0.836648i	$0.315488\pi$
$600$	−10472.8	−	2483.04i	−0.712580	−	0.168950i
$601$	9699.93			0.658350			0.329175	−	0.944269i	$-0.393229\pi$
$601$	9699.93			0.658350			0.329175	+	0.944269i	$0.393229\pi$
$602$		−	27995.5i		−	1.89537i
$603$		−	5022.84i		−	0.339214i
$604$	5561.21			0.374640
$605$	3173.65	+	371.085i	0.213268	+	0.0249368i
$606$	−16311.9			−1.09344
$607$			23661.2i			1.58217i	0.611703	+	0.791087i	$0.290485\pi$
$607$			23661.2i			1.58217i	−0.611703	+	0.791087i	$0.709515\pi$
$608$		−	6138.19i		−	0.409435i
$609$	−1193.06			−0.0793847
$610$	−2914.07	+	24922.1i	−0.193422	+	1.65421i
$611$	−4048.05			−0.268030
$612$			10538.2i			0.696047i
$613$			8085.63i			0.532749i	0.963870	+	0.266375i	$0.0858258\pi$
$613$			8085.63i			0.532749i	−0.963870	+	0.266375i	$0.914174\pi$
$614$	−24659.0			−1.62078
$615$	−944.297	+	8075.95i	−0.0619150	+	0.529518i
$616$	18706.8			1.22357
$617$		−	11035.1i		−	0.720029i	−0.932947	−	0.360014i	$-0.882772\pi$
$617$		−	11035.1i		−	0.720029i	0.932947	−	0.360014i	$-0.117228\pi$
$618$			18811.4i			1.22444i
$619$	16826.3			1.09258			0.546290	−	0.837596i	$-0.316040\pi$
$619$	16826.3			1.09258			0.546290	+	0.837596i	$0.316040\pi$
$620$	1945.12	+	227.438i	0.125997	+	0.0147324i
$621$	43.5374			0.00281336
$622$		−	24392.9i		−	1.57245i
$623$			1571.90i			0.101087i
$624$	−1312.40			−0.0841954
$625$	13961.8	+	7014.90i	0.893555	+	0.448954i
$626$	2288.04			0.146084
$627$			5891.17i			0.375233i
$628$		−	24359.5i		−	1.54785i
$629$	−27053.5			−1.71493
$630$	−7616.53	−	890.578i	−0.481666	−	0.0563198i
$631$	−3705.91			−0.233803			−0.116902	−	0.993144i	$-0.537296\pi$
$631$	−3705.91			−0.233803			−0.116902	+	0.993144i	$0.537296\pi$
$632$			33020.2i			2.07828i
$633$		−	7095.81i		−	0.445550i
$634$	19835.3			1.24252
$635$	806.147	−	6894.45i	0.0503795	−	0.430863i
$636$	−2603.50			−0.162320
$637$		−	1588.32i		−	0.0987937i
$638$		−	4638.15i		−	0.287815i
$639$	−5024.53			−0.311060
$640$	−3381.36	+	28918.5i	−0.208844	+	1.78610i
$641$	−24597.4			−1.51566			−0.757829	−	0.652453i	$-0.773740\pi$
$641$	−24597.4			−1.51566			−0.757829	+	0.652453i	$0.773740\pi$
$642$		−	11259.2i		−	0.692160i
$643$			21479.5i			1.31737i	0.752419	+	0.658685i	$0.228887\pi$
$643$			21479.5i			1.31737i	−0.752419	+	0.658685i	$0.771113\pi$
$644$	368.662			0.0225580
$645$	−12237.9	−	1430.94i	−0.747081	−	0.0873540i
$646$	−19061.4			−1.16093
$647$			27119.7i			1.64789i	0.566668	+	0.823946i	$0.308232\pi$
$647$			27119.7i			1.64789i	−0.566668	+	0.823946i	$0.691768\pi$
$648$		−	2324.83i		−	0.140938i
$649$	−4511.87			−0.272891
$650$	11317.1	+	2683.24i	0.682913	+	0.161916i
$651$	603.900			0.0363575
$652$		−	28692.9i		−	1.72347i
$653$		−	18476.4i		−	1.10725i	−0.832765	−	0.553627i	$-0.813243\pi$
$653$		−	18476.4i		−	1.10725i	0.832765	−	0.553627i	$-0.186757\pi$
$654$	13894.8			0.830779
$655$	−28742.8	−	3360.82i	−1.71462	−	0.200485i
$656$	−5358.59			−0.318930
$657$			9105.92i			0.540724i
$658$		−	15588.1i		−	0.923540i
$659$	19273.5			1.13928			0.569641	−	0.821894i	$-0.307082\pi$
$659$	19273.5			1.13928			0.569641	+	0.821894i	$0.307082\pi$
$660$	2209.19	−	18893.8i	0.130292	−	1.11430i
$661$	25605.3			1.50670			0.753352	−	0.657618i	$-0.228436\pi$
$661$	25605.3			1.50670			0.753352	+	0.657618i	$0.228436\pi$
$662$			34979.3i			2.05364i
$663$		−	4928.79i		−	0.288716i
$664$	33234.3			1.94238
$665$	1027.88	−	8790.75i	0.0599388	−	0.512617i
$666$	13789.5			0.802300
$667$		−	39.5616i		−	0.00229660i
$668$		−	2720.79i		−	0.157591i
$669$	9985.75			0.577087
$670$	−29137.7	−	3406.99i	−1.68013	−	0.196453i
$671$	−19193.9			−1.10428
$672$			6111.87i			0.350849i
$673$		−	7855.52i		−	0.449938i	−0.974366	−	0.224969i	$-0.927772\pi$
$673$		−	7855.52i		−	0.449938i	0.974366	−	0.224969i	$-0.0722280\pi$
$674$	30820.6			1.76137
$675$	−778.612	+	3283.96i	−0.0443982	+	0.187259i
$676$	−25463.6			−1.44877
$677$		−	6763.09i		−	0.383939i	−0.981401	−	0.191970i	$-0.938512\pi$
$677$		−	6763.09i		−	0.383939i	0.981401	−	0.191970i	$-0.0614875\pi$
$678$		−	26633.1i		−	1.50861i
$679$	−18679.3			−1.05574
$680$	26458.9	+	3093.76i	1.49214	+	0.174471i
$681$	−1581.30			−0.0889803
$682$			2347.72i			0.131817i
$683$		−	15608.6i		−	0.874447i	−0.899353	−	0.437224i	$-0.855962\pi$
$683$		−	15608.6i		−	0.874447i	0.899353	−	0.437224i	$-0.144038\pi$
$684$	6199.54			0.346557
$685$	2150.59	−	18392.6i	0.119956	−	1.02590i
$686$	32256.1			1.79525
$687$			7698.17i			0.427516i
$688$		−	8120.16i		−	0.449968i
$689$	1217.68			0.0673293
$690$	29.5314	−	252.562i	0.00162933	−	0.0139346i
$691$	6203.15			0.341504			0.170752	−	0.985314i	$-0.445380\pi$
$691$	6203.15			0.341504			0.170752	+	0.985314i	$0.445380\pi$
$692$		−	17488.5i		−	0.960714i
$693$		−	5865.92i		−	0.321541i
$694$	−9170.69			−0.501606
$695$	1701.86	+	198.994i	0.0928854	+	0.0108608i
$696$	−2112.53			−0.115050
$697$		−	20124.5i		−	1.09365i
$698$			6707.90i			0.363750i
$699$	−16605.0			−0.898509
$700$	−6593.06	+	27807.6i	−0.355992	+	1.50147i
$701$	−16507.9			−0.889435			−0.444718	−	0.895671i	$-0.646696\pi$
$701$	−16507.9			−0.889435			−0.444718	+	0.895671i	$0.646696\pi$
$702$			2512.27i			0.135070i
$703$			15915.4i			0.853854i
$704$	−30871.1			−1.65269
$705$	−6814.18	−	796.762i	−0.364024	−	0.0425642i
$706$	−33282.2			−1.77421
$707$			18746.0i			0.997193i
$708$			4748.03i			0.252037i
$709$	25539.6			1.35283			0.676416	−	0.736520i	$-0.263532\pi$
$709$	25539.6			1.35283			0.676416	+	0.736520i	$0.263532\pi$
$710$	−3408.13	+	29147.5i	−0.180148	+	1.54069i
$711$	10354.2			0.546151
$712$			2783.34i			0.146503i
$713$			20.0252i			0.00105182i
$714$	18979.7			0.994815
$715$	−1033.26	+	8836.76i	−0.0540442	+	0.462204i
$716$	−37192.1			−1.94125
$717$			3030.02i			0.157822i
$718$			25434.4i			1.32201i
$719$	7353.45			0.381415			0.190708	−	0.981647i	$-0.438922\pi$
$719$	7353.45			0.381415			0.190708	+	0.981647i	$0.438922\pi$
$720$	−2209.19	−	258.314i	−0.114350	−	0.0133706i
$721$	21618.4			1.11666
$722$		−	21034.3i		−	1.08423i
$723$			12222.9i			0.628732i
$724$	−46793.4			−2.40202
$725$	2984.07	+	707.510i	0.152863	+	0.0362431i
$726$	4031.03			0.206068
$727$		−	21696.5i		−	1.10685i	−0.832900	−	0.553424i	$-0.813321\pi$
$727$		−	21696.5i		−	1.10685i	0.832900	−	0.553424i	$-0.186679\pi$
$728$			9207.28i			0.468742i
$729$	−729.000			−0.0370370
$730$	52823.8	+	6176.53i	2.67821	+	0.313156i
$731$	30495.8			1.54299
$732$			20198.6i			1.01989i
$733$		−	90.2714i		−	0.00454877i	−0.999997	−	0.00227439i	$-0.999276\pi$
$733$		−	90.2714i		−	0.00454877i	0.999997	−	0.00227439i	$-0.000723960\pi$
$734$	−23226.1			−1.16797
$735$	312.623	−	2673.66i	0.0156888	−	0.134176i
$736$	−202.668			−0.0101501
$737$		−	22440.6i		−	1.12159i
$738$			10257.7i			0.511642i
$739$	−14273.1			−0.710479			−0.355239	−	0.934775i	$-0.615601\pi$
$739$	−14273.1			−0.710479			−0.355239	+	0.934775i	$0.615601\pi$
$740$	5968.27	−	51042.7i	0.296484	−	2.53563i
$741$	−2899.57			−0.143750
$742$			4689.02i			0.231994i
$743$		−	15866.6i		−	0.783429i	−0.920087	−	0.391715i	$-0.871882\pi$
$743$		−	15866.6i		−	0.783429i	0.920087	−	0.391715i	$-0.128118\pi$
$744$	1069.31			0.0526921
$745$	−16472.4	−	1926.07i	−0.810072	−	0.0947193i
$746$	60612.2			2.97476
$747$		−	10421.3i		−	0.510437i
$748$			47081.5i			2.30143i
$749$	−12939.3			−0.631232
$750$	18522.3	+	6744.27i	0.901783	+	0.328354i
$751$	26776.9			1.30107			0.650534	−	0.759477i	$-0.274545\pi$
$751$	26776.9			1.30107			0.650534	+	0.759477i	$0.274545\pi$
$752$		−	4521.37i		−	0.219252i
$753$		−	5321.95i		−	0.257560i
$754$	2282.85			0.110261
$755$	−4378.37	−	511.950i	−0.211053	−	0.0246778i
$756$	−6172.96			−0.296969
$757$			30478.0i			1.46333i	0.681663	+	0.731666i	$0.261257\pi$
$757$			30478.0i			1.46333i	−0.681663	+	0.731666i	$0.738743\pi$
$758$			44548.0i			2.13464i
$759$	194.512			0.00930218
$760$	1820.04	−	15565.6i	0.0868680	−	0.742925i
$761$	29104.7			1.38639			0.693195	−	0.720750i	$-0.256202\pi$
$761$	29104.7			1.38639			0.693195	+	0.720750i	$0.256202\pi$
$762$		−	8757.03i		−	0.416317i
$763$		−	15968.2i		−	0.757649i
$764$	8808.47			0.417119
$765$	970.116	−	8296.76i	0.0458492	−	0.392118i
$766$	−27273.6			−1.28647
$767$		−	2220.69i		−	0.104543i
$768$			18304.9i			0.860052i
$769$	−4170.65			−0.195575			−0.0977876	−	0.995207i	$-0.531177\pi$
$769$	−4170.65			−0.195575			−0.0977876	+	0.995207i	$0.531177\pi$
$770$	−34028.4	−	3978.84i	−1.59260	−	0.186218i
$771$	1988.35			0.0928778
$772$		−	6151.67i		−	0.286792i
$773$			17738.5i			0.825367i	0.910874	+	0.412684i	$0.135409\pi$
$773$			17738.5i			0.825367i	−0.910874	+	0.412684i	$0.864591\pi$
$774$	−15544.1			−0.721860
$775$	−1510.47	−	358.125i	−0.0700099	−	0.0165990i
$776$	−33075.0			−1.53005
$777$		−	15847.1i		−	0.731677i
$778$		−	64786.0i		−	2.98546i

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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	15.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.70156i q^{2} -3.00000i q^{3} -14.1047 q^{4} +(11.1047 + 1.29844i) q^{5} +14.1047 q^{6} -16.2094i q^{7} -28.7016i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+4.70156i q^{2} -3.00000i q^{3} -14.1047 q^{4} +(11.1047 + 1.29844i) q^{5} +14.1047 q^{6} -16.2094i q^{7} -28.7016i q^{8} -9.00000 q^{9} +(-6.10469 + 52.2094i) q^{10} -40.2094 q^{11} +42.3141i q^{12} -19.7906i q^{13} +76.2094 q^{14} +(3.89531 - 33.3141i) q^{15} +22.1047 q^{16} +83.0156i q^{17} -42.3141i q^{18} +48.8375 q^{19} +(-156.628 - 18.3141i) q^{20} -48.6281 q^{21} -189.047i q^{22} -1.61250i q^{23} -86.1047 q^{24} +(121.628 + 28.8375i) q^{25} +93.0469 q^{26} +27.0000i q^{27} +228.628i q^{28} +24.5344 q^{29} +(156.628 + 18.3141i) q^{30} -12.4187 q^{31} -125.686i q^{32} +120.628i q^{33} -390.303 q^{34} +(21.0469 - 180.000i) q^{35} +126.942 q^{36} +325.884i q^{37} +229.612i q^{38} -59.3719 q^{39} +(37.2672 - 318.722i) q^{40} -242.419 q^{41} -228.628i q^{42} -367.350i q^{43} +567.141 q^{44} +(-99.9422 - 11.6859i) q^{45} +7.58125 q^{46} -204.544i q^{47} -66.3141i q^{48} +80.2562 q^{49} +(-135.581 + 571.842i) q^{50} +249.047 q^{51} +279.141i q^{52} +61.5281i q^{53} -126.942 q^{54} +(-446.512 - 52.2094i) q^{55} -465.234 q^{56} -146.512i q^{57} +115.350i q^{58} +112.209 q^{59} +(-54.9422 + 469.884i) q^{60} +477.350 q^{61} -58.3875i q^{62} +145.884i q^{63} +767.758 q^{64} +(25.6969 - 219.769i) q^{65} -567.141 q^{66} +558.094i q^{67} -1170.91i q^{68} -4.83749 q^{69} +(846.281 + 98.9531i) q^{70} +558.281 q^{71} +258.314i q^{72} -1011.77i q^{73} -1532.17 q^{74} +(86.5125 - 364.884i) q^{75} -688.837 q^{76} +651.769i q^{77} -279.141i q^{78} -1150.47 q^{79} +(245.466 + 28.7016i) q^{80} +81.0000 q^{81} -1139.75i q^{82} +1157.92i q^{83} +685.884 q^{84} +(-107.791 + 921.862i) q^{85} +1727.12 q^{86} -73.6032i q^{87} +1154.07i q^{88} -96.9751 q^{89} +(54.9422 - 469.884i) q^{90} -320.794 q^{91} +22.7438i q^{92} +37.2562i q^{93} +961.675 q^{94} +(542.325 + 63.4124i) q^{95} -377.058 q^{96} -1152.37i q^{97} +377.330i q^{98} +361.884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 18 * q^4 + 6 * q^5 + 18 * q^6 - 36 * q^9	\( 4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9} + 14 q^{10} - 84 q^{11} + 228 q^{14} + 54 q^{15} + 50 q^{16} - 112 q^{19} - 396 q^{20} + 36 q^{21} - 306 q^{24} + 256 q^{25} - 12 q^{26} + 636 q^{29} + 396 q^{30} + 104 q^{31} - 716 q^{34} - 300 q^{35} + 162 q^{36} - 468 q^{39} + 418 q^{40} - 816 q^{41} + 1116 q^{44} - 54 q^{45} + 184 q^{46} - 140 q^{49} - 696 q^{50} + 612 q^{51} - 162 q^{54} - 864 q^{55} + 60 q^{56} + 372 q^{59} + 126 q^{60} + 680 q^{61} + 958 q^{64} + 948 q^{65} - 1116 q^{66} + 288 q^{69} + 1080 q^{70} - 72 q^{71} - 3132 q^{74} - 576 q^{75} - 2448 q^{76} - 760 q^{79} + 444 q^{80} + 324 q^{81} + 2052 q^{84} - 508 q^{85} + 4296 q^{86} - 2232 q^{89} - 126 q^{90} + 1944 q^{91} + 3232 q^{94} + 2784 q^{95} - 1854 q^{96} + 756 q^{99}+O(q^{100}) \) 4 * q - 18 * q^4 + 6 * q^5 + 18 * q^6 - 36 * q^9 + 14 * q^10 - 84 * q^11 + 228 * q^14 + 54 * q^15 + 50 * q^16 - 112 * q^19 - 396 * q^20 + 36 * q^21 - 306 * q^24 + 256 * q^25 - 12 * q^26 + 636 * q^29 + 396 * q^30 + 104 * q^31 - 716 * q^34 - 300 * q^35 + 162 * q^36 - 468 * q^39 + 418 * q^40 - 816 * q^41 + 1116 * q^44 - 54 * q^45 + 184 * q^46 - 140 * q^49 - 696 * q^50 + 612 * q^51 - 162 * q^54 - 864 * q^55 + 60 * q^56 + 372 * q^59 + 126 * q^60 + 680 * q^61 + 958 * q^64 + 948 * q^65 - 1116 * q^66 + 288 * q^69 + 1080 * q^70 - 72 * q^71 - 3132 * q^74 - 576 * q^75 - 2448 * q^76 - 760 * q^79 + 444 * q^80 + 324 * q^81 + 2052 * q^84 - 508 * q^85 + 4296 * q^86 - 2232 * q^89 - 126 * q^90 + 1944 * q^91 + 3232 * q^94 + 2784 * q^95 - 1854 * q^96 + 756 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more