LMFDB - Embedded newform 15.4.b.a.4.2

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			4.2
Root			$-2.70156i$ of defining polynomial
Character	$\chi$	$=$	15.4
Dual form			15.4.b.a.4.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} +(-8.10469 + 7.70156i) q^{5} -5.10469 q^{6} +22.2094i q^{7} -22.2984i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} +(-8.10469 + 7.70156i) q^{5} -5.10469 q^{6} +22.2094i q^{7} -22.2984i q^{8} -9.00000 q^{9} +(13.1047 + 13.7906i) q^{10} -1.79063 q^{11} -15.3141i q^{12} -58.2094i q^{13} +37.7906 q^{14} +(23.1047 + 24.3141i) q^{15} +2.89531 q^{16} +18.9844i q^{17} +15.3141i q^{18} -104.837 q^{19} +(-41.3719 + 39.3141i) q^{20} +66.6281 q^{21} +3.04686i q^{22} +49.6125i q^{23} -66.8953 q^{24} +(6.37188 - 124.837i) q^{25} -99.0469 q^{26} +27.0000i q^{27} +113.372i q^{28} +293.466 q^{29} +(41.3719 - 39.3141i) q^{30} +64.4187 q^{31} -183.314i q^{32} +5.37188i q^{33} +32.3031 q^{34} +(-171.047 - 180.000i) q^{35} -45.9422 q^{36} -19.8844i q^{37} +178.388i q^{38} -174.628 q^{39} +(171.733 + 180.722i) q^{40} -165.581 q^{41} -113.372i q^{42} +247.350i q^{43} -9.14059 q^{44} +(72.9422 - 69.3141i) q^{45} +84.4187 q^{46} +384.544i q^{47} -8.68594i q^{48} -150.256 q^{49} +(-212.419 - 10.8422i) q^{50} +56.9531 q^{51} -297.141i q^{52} -463.528i q^{53} +45.9422 q^{54} +(14.5125 - 13.7906i) q^{55} +495.234 q^{56} +314.512i q^{57} -499.350i q^{58} +73.7906 q^{59} +(117.942 + 124.116i) q^{60} -137.350 q^{61} -109.612i q^{62} -199.884i q^{63} -288.758 q^{64} +(448.303 + 471.769i) q^{65} +9.14059 q^{66} +173.906i q^{67} +96.9093i q^{68} +148.837 q^{69} +(-306.281 + 291.047i) q^{70} -594.281 q^{71} +200.686i q^{72} -320.231i q^{73} -33.8345 q^{74} +(-374.512 - 19.1156i) q^{75} -535.163 q^{76} -39.7687i q^{77} +297.141i q^{78} +770.469 q^{79} +(-23.4656 + 22.2984i) q^{80} +81.0000 q^{81} +281.747i q^{82} -173.925i q^{83} +340.116 q^{84} +(-146.209 - 153.862i) q^{85} +420.881 q^{86} -880.397i q^{87} +39.9282i q^{88} -1019.02 q^{89} +(-117.942 - 124.116i) q^{90} +1292.79 q^{91} +253.256i q^{92} -193.256i q^{93} +654.325 q^{94} +(849.675 - 807.412i) q^{95} -549.942 q^{96} +384.375i q^{97} +255.670i q^{98} +16.1156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.70156i		−	0.601593i	−0.953688	−	0.300797i	$-0.902747\pi$
$2$		−	1.70156i		−	0.601593i	0.953688	−	0.300797i	$-0.0972525\pi$
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$	5.10469			0.638086
$5$	−8.10469	+	7.70156i	−0.724905	+	0.688849i
$5$	−8.10469	+	7.70156i	−0.724905	+	0.688849i
$6$	−5.10469			−0.347330
$7$			22.2094i			1.19919i	0.800302	+	0.599597i	$0.204672\pi$
$7$			22.2094i			1.19919i	−0.800302	+	0.599597i	$0.795328\pi$
$8$		−	22.2984i		−	0.985461i
$9$	−9.00000			−0.333333
$10$	13.1047	+	13.7906i	0.414407	+	0.436098i
$11$	−1.79063			−0.0490813			−0.0245407	−	0.999699i	$-0.507812\pi$
$11$	−1.79063			−0.0490813			−0.0245407	+	0.999699i	$0.507812\pi$
$12$		−	15.3141i		−	0.368399i
$13$		−	58.2094i		−	1.24188i	−0.783860	−	0.620938i	$-0.786752\pi$
$13$		−	58.2094i		−	1.24188i	0.783860	−	0.620938i	$-0.213248\pi$
$14$	37.7906			0.721426
$15$	23.1047	+	24.3141i	0.397707	+	0.418524i
$16$	2.89531			0.0452393
$17$			18.9844i			0.270846i	0.990788	+	0.135423i	$0.0432394\pi$
$17$			18.9844i			0.270846i	−0.990788	+	0.135423i	$0.956761\pi$
$18$			15.3141i			0.200531i
$19$	−104.837			−1.26586			−0.632931	−	0.774208i	$-0.718148\pi$
$19$	−104.837			−1.26586			−0.632931	+	0.774208i	$0.718148\pi$
$20$	−41.3719	+	39.3141i	−0.462552	+	0.439545i
$21$	66.6281			0.692355
$22$			3.04686i			0.0295270i
$23$			49.6125i			0.449779i	0.974384	+	0.224890i	$0.0722021\pi$
$23$			49.6125i			0.449779i	−0.974384	+	0.224890i	$0.927798\pi$
$24$	−66.8953			−0.568956
$25$	6.37188	−	124.837i	0.0509751	−	0.998700i
$26$	−99.0469			−0.747103
$27$			27.0000i			0.192450i
$28$			113.372i			0.765188i
$29$	293.466			1.87914			0.939572	−	0.342350i	$-0.111223\pi$
$29$	293.466			1.87914			0.939572	+	0.342350i	$0.111223\pi$
$30$	41.3719	−	39.3141i	0.251781	−	0.239258i
$31$	64.4187			0.373224			0.186612	−	0.982434i	$-0.440249\pi$
$31$	64.4187			0.373224			0.186612	+	0.982434i	$0.440249\pi$
$32$		−	183.314i		−	1.01268i
$33$			5.37188i			0.0283371i
$34$	32.3031			0.162939
$35$	−171.047	−	180.000i	−0.826063	−	0.869302i
$36$	−45.9422			−0.212695
$37$		−	19.8844i		−	0.0883505i	−0.999024	−	0.0441752i	$-0.985934\pi$
$37$		−	19.8844i		−	0.0883505i	0.999024	−	0.0441752i	$-0.0140660\pi$
$38$			178.388i			0.761534i
$39$	−174.628			−0.716997
$40$	171.733	+	180.722i	0.678834	+	0.714366i
$41$	−165.581			−0.630718			−0.315359	−	0.948972i	$-0.602125\pi$
$41$	−165.581			−0.630718			−0.315359	+	0.948972i	$0.602125\pi$
$42$		−	113.372i		−	0.416516i
$43$			247.350i			0.877221i	0.898677	+	0.438611i	$0.144529\pi$
$43$			247.350i			0.877221i	−0.898677	+	0.438611i	$0.855471\pi$
$44$	−9.14059			−0.0313181
$45$	72.9422	−	69.3141i	0.241635	−	0.229616i
$46$	84.4187			0.270584
$47$			384.544i			1.19344i	0.802451	+	0.596718i	$0.203529\pi$
$47$			384.544i			1.19344i	−0.802451	+	0.596718i	$0.796471\pi$
$48$		−	8.68594i		−	0.0261189i
$49$	−150.256			−0.438065
$50$	−212.419	−	10.8422i	−0.600811	−	0.0306662i
$51$	56.9531			0.156373
$52$		−	297.141i		−	0.792423i
$53$		−	463.528i		−	1.20133i	−0.799501	−	0.600665i	$-0.794903\pi$
$53$		−	463.528i		−	1.20133i	0.799501	−	0.600665i	$-0.205097\pi$
$54$	45.9422			0.115777
$55$	14.5125	−	13.7906i	0.0355793	−	0.0338096i
$56$	495.234			1.18176
$57$			314.512i			0.730846i
$58$		−	499.350i		−	1.13048i
$59$	73.7906			0.162826			0.0814129	−	0.996680i	$-0.474057\pi$
$59$	73.7906			0.162826			0.0814129	+	0.996680i	$0.474057\pi$
$60$	117.942	+	124.116i	0.253771	+	0.267054i
$61$	−137.350			−0.288293			−0.144146	−	0.989556i	$-0.546044\pi$
$61$	−137.350			−0.288293			−0.144146	+	0.989556i	$0.546044\pi$
$62$		−	109.612i		−	0.224529i
$63$		−	199.884i		−	0.399731i
$64$	−288.758			−0.563980
$65$	448.303	+	471.769i	0.855464	+	0.900242i
$66$	9.14059			0.0170474
$67$			173.906i			0.317105i	0.987351	+	0.158552i	$0.0506827\pi$
$67$			173.906i			0.317105i	−0.987351	+	0.158552i	$0.949317\pi$
$68$			96.9093i			0.172823i
$69$	148.837			0.259680
$70$	−306.281	+	291.047i	−0.522966	+	0.496954i
$71$	−594.281			−0.993355			−0.496677	−	0.867935i	$-0.665447\pi$
$71$	−594.281			−0.993355			−0.496677	+	0.867935i	$0.665447\pi$
$72$			200.686i			0.328487i
$73$		−	320.231i		−	0.513428i	−0.966487	−	0.256714i	$-0.917360\pi$
$73$		−	320.231i		−	0.513428i	0.966487	−	0.256714i	$-0.0826398\pi$
$74$	−33.8345			−0.0531510
$75$	−374.512	−	19.1156i	−0.576600	−	0.0294305i
$76$	−535.163			−0.807728
$77$		−	39.7687i		−	0.0588580i
$78$			297.141i			0.431340i
$79$	770.469			1.09727			0.548636	−	0.836061i	$-0.315147\pi$
$79$	770.469			1.09727			0.548636	+	0.836061i	$0.315147\pi$
$80$	−23.4656	+	22.2984i	−0.0327942	+	0.0311630i
$81$	81.0000			0.111111
$82$			281.747i			0.379436i
$83$		−	173.925i		−	0.230009i	−0.993365	−	0.115004i	$-0.963312\pi$
$83$		−	173.925i		−	0.230009i	0.993365	−	0.115004i	$-0.0366882\pi$
$84$	340.116			0.441782
$85$	−146.209	−	153.862i	−0.186572	−	0.196338i
$86$	420.881			0.527730
$87$		−	880.397i		−	1.08492i
$88$			39.9282i			0.0483677i
$89$	−1019.02			−1.21367			−0.606834	−	0.794829i	$-0.707561\pi$
$89$	−1019.02			−1.21367			−0.606834	+	0.794829i	$0.707561\pi$
$90$	−117.942	−	124.116i	−0.138136	−	0.145366i
$91$	1292.79			1.48925
$92$			253.256i			0.286998i
$93$		−	193.256i		−	0.215481i
$94$	654.325			0.717962
$95$	849.675	−	807.412i	0.917630	−	0.871987i
$96$	−549.942			−0.584669
$97$			384.375i			0.402344i	0.979556	+	0.201172i	$0.0644750\pi$
$97$			384.375i			0.402344i	−0.979556	+	0.201172i	$0.935525\pi$
$98$			255.670i			0.263537i
$99$	16.1156			0.0163604
$100$	32.5265	−	637.256i	0.0325265	−	0.637256i
$101$	34.4906			0.0339796			0.0169898	−	0.999856i	$-0.494592\pi$
$101$	34.4906			0.0339796			0.0169898	+	0.999856i	$0.494592\pi$
$102$		−	96.9093i		−	0.0940730i
$103$			1756.30i			1.68013i	0.542484	+	0.840066i	$0.317484\pi$
$103$			1756.30i			1.68013i	−0.542484	+	0.840066i	$0.682516\pi$
$104$	−1297.98			−1.22382
$105$	−540.000	+	513.141i	−0.501891	+	0.476928i
$106$	−788.722			−0.722712
$107$		−	1361.74i		−	1.23032i	−0.788403	−	0.615159i	$-0.789092\pi$
$107$		−	1361.74i		−	1.23032i	0.788403	−	0.615159i	$-0.210908\pi$
$108$			137.827i			0.122800i
$109$	−321.119			−0.282180			−0.141090	−	0.989997i	$-0.545061\pi$
$109$	−321.119			−0.282180			−0.141090	+	0.989997i	$0.545061\pi$
$110$	−23.4656	−	24.6939i	−0.0203396	−	0.0214043i
$111$	−59.6531			−0.0510092
$112$			64.3031i			0.0542506i
$113$			1582.25i			1.31721i	0.752487	+	0.658607i	$0.228854\pi$
$113$			1582.25i			1.31721i	−0.752487	+	0.658607i	$0.771146\pi$
$114$	535.163			0.439672
$115$	−382.094	−	402.094i	−0.309830	−	0.326047i
$116$	1498.05			1.19906
$117$			523.884i			0.413958i
$118$		−	125.559i		−	0.0979549i
$119$	−421.631			−0.324797
$120$	542.166	−	515.198i	0.412439	−	0.391925i
$121$	−1327.79			−0.997591
$122$			233.709i			0.173435i
$123$			496.744i			0.364145i
$124$	328.837			0.238149
$125$	909.802	+	1060.84i	0.651001	+	0.759077i
$126$	−340.116			−0.240475
$127$		−	1197.14i		−	0.836449i	−0.908344	−	0.418225i	$-0.862652\pi$
$127$		−	1197.14i		−	0.836449i	0.908344	−	0.418225i	$-0.137348\pi$
$128$		−	975.173i		−	0.673390i
$129$	742.050			0.506464
$130$	802.744	−	762.816i	0.541579	−	0.514641i
$131$	−321.647			−0.214522			−0.107261	−	0.994231i	$-0.534208\pi$
$131$	−321.647			−0.214522			−0.107261	+	0.994231i	$0.534208\pi$
$132$			27.4218i			0.0180815i
$133$		−	2328.37i		−	1.51801i
$134$	295.912			0.190768
$135$	−207.942	−	218.827i	−0.132569	−	0.139508i
$136$	423.322			0.266909
$137$			354.291i			0.220942i	0.993879	+	0.110471i	$0.0352360\pi$
$137$			354.291i			0.220942i	−0.993879	+	0.110471i	$0.964764\pi$
$138$		−	253.256i		−	0.156222i
$139$	−77.2562			−0.0471424			−0.0235712	−	0.999722i	$-0.507504\pi$
$139$	−77.2562			−0.0471424			−0.0235712	+	0.999722i	$0.507504\pi$
$140$	−873.141	−	918.844i	−0.527099	−	0.554689i
$141$	1153.63			0.689030
$142$			1011.21i			0.597595i
$143$			104.231i			0.0609529i
$144$	−26.0578			−0.0150798
$145$	−2378.45	+	2260.14i	−1.36220	+	1.29445i
$146$	−544.893			−0.308875
$147$			450.769i			0.252917i
$148$		−	101.503i		−	0.0563752i
$149$	1705.38			0.937651			0.468826	−	0.883291i	$-0.344677\pi$
$149$	1705.38			0.937651			0.468826	+	0.883291i	$0.344677\pi$
$150$	−32.5265	+	637.256i	−0.0177052	+	0.346878i
$151$	758.281			0.408663			0.204331	−	0.978902i	$-0.434498\pi$
$151$	758.281			0.408663			0.204331	+	0.978902i	$0.434498\pi$
$152$			2337.71i			1.24746i
$153$		−	170.859i		−	0.0902821i
$154$	−67.6689			−0.0354086
$155$	−522.094	+	496.125i	−0.270552	+	0.257095i
$156$	−891.422			−0.457506
$157$		−	1769.05i		−	0.899273i	−0.893212	−	0.449636i	$-0.851554\pi$
$157$		−	1769.05i		−	0.899273i	0.893212	−	0.449636i	$-0.148446\pi$
$158$		−	1311.00i		−	0.660111i
$159$	−1390.58			−0.693588
$160$	1411.80	+	1485.70i	0.697581	+	0.734095i
$161$	−1101.86			−0.539372
$162$		−	137.827i		−	0.0668437i
$163$			881.719i			0.423690i	0.977303	+	0.211845i	$0.0679473\pi$
$163$			881.719i			0.423690i	−0.977303	+	0.211845i	$0.932053\pi$
$164$	−845.240			−0.402452
$165$	−41.3719	−	43.5374i	−0.0195200	−	0.0205417i
$166$	−295.944			−0.138372
$167$		−	216.900i		−	0.100504i	−0.998737	−	0.0502522i	$-0.983997\pi$
$167$		−	216.900i		−	0.100504i	0.998737	−	0.0502522i	$-0.0160025\pi$
$168$		−	1485.70i		−	0.682289i
$169$	−1191.33			−0.542254
$170$	−261.806	+	248.784i	−0.118116	+	0.112241i
$171$	943.537			0.421954
$172$			1262.64i			0.559742i
$173$		−	4125.91i		−	1.81322i	−0.421970	−	0.906610i	$-0.638661\pi$
$173$		−	4125.91i		−	1.81322i	0.421970	−	0.906610i	$-0.361339\pi$
$174$	−1498.05			−0.652683
$175$	2772.56	+	141.515i	1.19763	+	0.0611289i
$176$	−5.18443			−0.00222040
$177$		−	221.372i		−	0.0940075i
$178$			1733.93i			0.730134i
$179$	3213.14			1.34168			0.670842	−	0.741600i	$-0.265933\pi$
$179$	3213.14			1.34168			0.670842	+	0.741600i	$0.265933\pi$
$180$	372.347	−	353.827i	0.154184	−	0.146515i
$181$	3394.42			1.39395			0.696976	−	0.717095i	$-0.254529\pi$
$181$	3394.42			1.39395			0.696976	+	0.717095i	$0.254529\pi$
$182$		−	2199.77i		−	0.895921i
$183$			412.050i			0.166446i
$184$	1106.28			0.443240
$185$	153.141	+	161.156i	0.0608601	+	0.0640457i
$186$	−328.837			−0.129632
$187$		−	33.9939i		−	0.0132935i
$188$			1962.98i			0.761514i
$189$	−599.653			−0.230785
$190$	−1373.86	−	1445.77i	−0.524581	−	0.552040i
$191$	−3467.49			−1.31361			−0.656804	−	0.754062i	$-0.728092\pi$
$191$	−3467.49			−1.31361			−0.656804	+	0.754062i	$0.728092\pi$
$192$			866.273i			0.325614i
$193$		−	1792.14i		−	0.668401i	−0.942502	−	0.334200i	$-0.891534\pi$
$193$		−	1792.14i		−	0.668401i	0.942502	−	0.334200i	$-0.108466\pi$
$194$	654.038			0.242047
$195$	1415.31	−	1344.91i	0.519755	−	0.493902i
$196$	−767.011			−0.279523
$197$			1678.19i			0.606935i	0.952842	+	0.303467i	$0.0981443\pi$
$197$			1678.19i			0.606935i	−0.952842	+	0.303467i	$0.901856\pi$
$198$		−	27.4218i		−	0.00984233i
$199$	−3108.23			−1.10722			−0.553610	−	0.832776i	$-0.686750\pi$
$199$	−3108.23			−1.10722			−0.553610	+	0.832776i	$0.686750\pi$
$200$	−2783.68	−	142.083i	−0.984180	−	0.0502339i
$201$	521.719			0.183081
$202$		−	58.6878i		−	0.0204419i
$203$			6517.69i			2.25346i
$204$	290.728			0.0997795
$205$	1341.98	−	1275.23i	0.457211	−	0.434469i
$206$	2988.46			1.01076
$207$		−	446.512i		−	0.149926i
$208$		−	168.534i		−	0.0561815i
$209$	187.725			0.0621301
$210$	873.141	+	918.844i	0.286916	+	0.301934i
$211$	−4473.27			−1.45949			−0.729745	−	0.683719i	$-0.760361\pi$
$211$	−4473.27			−1.45949			−0.729745	+	0.683719i	$0.760361\pi$
$212$		−	2366.17i		−	0.766551i
$213$			1782.84i			0.573514i
$214$	−2317.08			−0.740151
$215$	−1904.98	−	2004.69i	−0.604273	−	0.635902i
$216$	602.058			0.189652
$217$			1430.70i			0.447568i
$218$			546.403i			0.169757i
$219$	−960.694			−0.296428
$220$	74.0816	−	70.3968i	0.0227026	−	0.0215734i
$221$	1105.07			0.336357
$222$			101.503i			0.0306868i
$223$			1753.42i			0.526535i	0.964723	+	0.263268i	$0.0848003\pi$
$223$			1753.42i			0.526535i	−0.964723	+	0.263268i	$0.915200\pi$
$224$	4071.29			1.21440
$225$	−57.3469	+	1123.54i	−0.0169917	+	0.332900i
$226$	2692.29			0.792427
$227$		−	936.900i		−	0.273939i	−0.990575	−	0.136970i	$-0.956264\pi$
$227$		−	936.900i		−	0.273939i	0.990575	−	0.136970i	$-0.0437363\pi$
$228$			1605.49i			0.466342i
$229$	2582.06			0.745096			0.372548	−	0.928013i	$-0.378484\pi$
$229$	2582.06			0.745096			0.372548	+	0.928013i	$0.378484\pi$
$230$	−684.187	+	650.156i	−0.196148	+	0.186391i
$231$	−119.306			−0.0339817
$232$		−	6543.82i		−	1.85182i
$233$		−	2295.01i		−	0.645284i	−0.946521	−	0.322642i	$-0.895429\pi$
$233$		−	2295.01i		−	0.645284i	0.946521	−	0.322642i	$-0.104571\pi$
$234$	891.422			0.249034
$235$	−2961.59	−	3116.61i	−0.822096	−	0.865128i
$236$	376.678			0.103897
$237$		−	2311.41i		−	0.633510i
$238$			717.432i			0.195396i
$239$	2294.01			0.620866			0.310433	−	0.950595i	$-0.399526\pi$
$239$	2294.01			0.620866			0.310433	+	0.950595i	$0.399526\pi$
$240$	66.8953	+	70.3968i	0.0179920	+	0.0189337i
$241$	382.287			0.102180			0.0510898	−	0.998694i	$-0.483731\pi$
$241$	382.287			0.102180			0.0510898	+	0.998694i	$0.483731\pi$
$242$			2259.32i			0.600144i
$243$		−	243.000i		−	0.0641500i
$244$	−701.128			−0.183956
$245$	1217.78	−	1157.21i	0.317555	−	0.301760i
$246$	845.240			0.219067
$247$			6102.52i			1.57204i
$248$		−	1436.44i		−	0.367798i
$249$	−521.775			−0.132796
$250$	1805.09	−	1548.08i	0.456655	−	0.391638i
$251$	−2259.98			−0.568322			−0.284161	−	0.958777i	$-0.591715\pi$
$251$	−2259.98			−0.568322			−0.284161	+	0.958777i	$0.591715\pi$
$252$		−	1020.35i		−	0.255063i
$253$		−	88.8375i		−	0.0220758i
$254$	−2037.01			−0.503202
$255$	−461.587	+	438.628i	−0.113356	+	0.107717i
$256$	−3969.38			−0.969087
$257$		−	92.7843i		−	0.0225203i	−0.999937	−	0.0112602i	$-0.996416\pi$
$257$		−	92.7843i		−	0.0225203i	0.999937	−	0.0112602i	$-0.00358430\pi$
$258$		−	1262.64i		−	0.304685i
$259$	441.619			0.105949
$260$	2288.45	+	2408.23i	0.545859	+	0.574431i
$261$	−2641.19			−0.626382
$262$			547.302i			0.129055i
$263$			568.312i			0.133246i	0.997778	+	0.0666229i	$0.0212224\pi$
$263$			568.312i			0.133246i	−0.997778	+	0.0666229i	$0.978778\pi$
$264$	119.785			0.0279251
$265$	3569.89	+	3756.75i	0.827534	+	0.870850i
$266$	−3961.87			−0.913226
$267$			3057.07i			0.700711i
$268$			887.737i			0.202340i
$269$	−7582.41			−1.71862			−0.859309	−	0.511458i	$-0.829106\pi$
$269$	−7582.41			−1.71862			−0.859309	+	0.511458i	$0.829106\pi$
$270$	−372.347	+	353.827i	−0.0839271	+	0.0797526i
$271$	7943.69			1.78061			0.890304	−	0.455366i	$-0.150492\pi$
$271$	7943.69			1.78061			0.890304	+	0.455366i	$0.150492\pi$
$272$			54.9657i			0.0122529i
$273$		−	3878.38i		−	0.859818i
$274$	602.847			0.132917
$275$	−11.4097	+	223.537i	−0.00250192	+	0.0490175i
$276$	759.769			0.165698
$277$			6823.00i			1.47998i	0.672618	+	0.739990i	$0.265170\pi$
$277$			6823.00i			1.47998i	−0.672618	+	0.739990i	$0.734830\pi$
$278$			131.456i			0.0283605i
$279$	−579.769			−0.124408
$280$	−4013.72	+	3814.08i	−0.856663	+	0.814053i
$281$	3315.86			0.703942			0.351971	−	0.936011i	$-0.385512\pi$
$281$	3315.86			0.703942			0.351971	+	0.936011i	$0.385512\pi$
$282$		−	1962.98i		−	0.414516i
$283$		−	6602.76i		−	1.38690i	−0.720504	−	0.693451i	$-0.756090\pi$
$283$		−	6602.76i		−	1.38690i	0.720504	−	0.693451i	$-0.243910\pi$
$284$	−3033.62			−0.633846
$285$	−2422.24	−	2549.02i	−0.503442	−	0.529794i
$286$	177.356			0.0366688
$287$		−	3677.46i		−	0.756353i
$288$			1649.83i			0.337559i
$289$	4552.59			0.926642
$290$	3845.77	+	4047.07i	0.778730	+	0.819491i
$291$	1153.12			0.232293
$292$		−	1634.68i		−	0.327611i
$293$		−	5814.14i		−	1.15927i	−0.814877	−	0.579634i	$-0.803195\pi$
$293$		−	5814.14i		−	1.15927i	0.814877	−	0.579634i	$-0.196805\pi$
$294$	767.011			0.152153
$295$	−598.050	+	568.303i	−0.118033	+	0.112162i
$296$	−443.390			−0.0870660
$297$		−	48.3469i		−	0.00944570i
$298$		−	2901.81i		−	0.564084i
$299$	2887.91			0.558570
$300$	−1911.77	−	97.5794i	−0.367920	−	0.0187792i
$301$	−5493.49			−1.05196
$302$		−	1290.26i		−	0.245849i
$303$		−	103.472i		−	0.0196181i
$304$	−303.537			−0.0572667
$305$	1113.18	−	1057.81i	0.208985	−	0.198590i
$306$	−290.728			−0.0543131
$307$		−	8124.86i		−	1.51046i	−0.655462	−	0.755229i	$-0.727526\pi$
$307$		−	8124.86i		−	1.51046i	0.655462	−	0.755229i	$-0.272474\pi$
$308$		−	203.007i		−	0.0375564i
$309$	5268.91			0.970025
$310$	844.187	+	888.375i	0.154667	+	0.162762i
$311$	7336.26			1.33762			0.668812	−	0.743432i	$-0.266803\pi$
$311$	7336.26			1.33762			0.668812	+	0.743432i	$0.266803\pi$
$312$			3893.93i			0.706572i
$313$			2202.66i			0.397768i	0.980023	+	0.198884i	$0.0637318\pi$
$313$			2202.66i			0.397768i	−0.980023	+	0.198884i	$0.936268\pi$
$314$	−3010.15			−0.540996
$315$	1539.42	+	1620.00i	0.275354	+	0.289767i
$316$	3933.00			0.700154
$317$			10008.9i			1.77336i	0.462386	+	0.886679i	$0.346993\pi$
$317$			10008.9i			1.77336i	−0.462386	+	0.886679i	$0.653007\pi$
$318$			2366.17i			0.417258i
$319$	−525.488			−0.0922309
$320$	2340.29	−	2223.89i	0.408832	−	0.388497i
$321$	−4085.21			−0.710325
$322$			1874.89i			0.324483i
$323$		−	1990.27i		−	0.342854i
$324$	413.480			0.0708984
$325$	−7266.71	−	370.903i	−1.24026	−	0.0633046i
$326$	1500.30			0.254889
$327$			963.356i			0.162917i
$328$			3692.20i			0.621548i
$329$	−8540.47			−1.43116
$330$	−74.0816	+	70.3968i	−0.0123578	+	0.0117431i
$331$	−8695.94			−1.44402			−0.722012	−	0.691881i	$-0.756782\pi$
$331$	−8695.94			−1.44402			−0.722012	+	0.691881i	$0.756782\pi$
$332$		−	887.832i		−	0.146765i
$333$			178.959i			0.0294502i
$334$	−369.069			−0.0604627
$335$	−1339.35	−	1409.46i	−0.218437	−	0.229871i
$336$	192.909			0.0313216
$337$		−	7400.61i		−	1.19625i	−0.801402	−	0.598126i	$-0.795912\pi$
$337$		−	7400.61i		−	1.19625i	0.801402	−	0.598126i	$-0.204088\pi$
$338$			2027.12i			0.326216i
$339$	4746.74			0.760494
$340$	−746.353	−	785.419i	−0.119049	−	0.125280i
$341$	−115.350			−0.0183183
$342$		−	1605.49i		−	0.253845i
$343$			4280.72i			0.673869i
$344$	5515.52			0.864467
$345$	−1206.28	+	1146.28i	−0.188243	+	0.178880i
$346$	−7020.49			−1.09082
$347$			7841.44i			1.21311i	0.795040	+	0.606557i	$0.207450\pi$
$347$			7841.44i			1.21311i	−0.795040	+	0.606557i	$0.792550\pi$
$348$		−	4494.15i		−	0.692275i
$349$	4961.26			0.760946			0.380473	−	0.924792i	$-0.375761\pi$
$349$	4961.26			0.760946			0.380473	+	0.924792i	$0.375761\pi$
$350$	240.797	−	4717.69i	0.0367748	−	0.720489i
$351$	1571.65			0.238999
$352$			328.247i			0.0497035i
$353$			12163.0i			1.83392i	0.398981	+	0.916959i	$0.369364\pi$
$353$			12163.0i			1.83392i	−0.398981	+	0.916959i	$0.630636\pi$
$354$	−376.678			−0.0565543
$355$	4816.46	−	4576.89i	0.720088	−	0.684271i
$356$	−5201.80			−0.774424
$357$			1264.89i			0.187522i
$358$		−	5467.36i		−	0.807148i
$359$	−5193.79			−0.763559			−0.381779	−	0.924253i	$-0.624689\pi$
$359$	−5193.79			−0.763559			−0.381779	+	0.924253i	$0.624689\pi$
$360$	−1545.60	−	1626.50i	−0.226278	−	0.238122i
$361$	4131.90			0.602406
$362$		−	5775.81i		−	0.838591i
$363$			3983.38i			0.575959i
$364$	6599.31			0.950268
$365$	2466.28	+	2595.37i	0.353674	+	0.372187i
$366$	701.128			0.100133
$367$		−	6086.09i		−	0.865644i	−0.901479	−	0.432822i	$-0.857518\pi$
$367$		−	6086.09i		−	0.865644i	0.901479	−	0.432822i	$-0.142482\pi$
$368$			143.644i			0.0203477i
$369$	1490.23			0.210239
$370$	274.218	−	260.578i	0.0385295	−	0.0366130i
$371$	10294.7			1.44063
$372$		−	986.512i		−	0.137495i
$373$			10581.9i			1.46893i	0.678646	+	0.734466i	$0.262567\pi$
$373$			10581.9i			1.46893i	−0.678646	+	0.734466i	$0.737433\pi$
$374$	−57.8428			−0.00799727
$375$	3182.53	−	2729.40i	0.438253	−	0.375856i
$376$	8574.72			1.17608
$377$		−	17082.4i		−	2.33366i
$378$			1020.35i			0.138839i
$379$	−11655.2			−1.57964			−0.789822	−	0.613336i	$-0.789827\pi$
$379$	−11655.2			−1.57964			−0.789822	+	0.613336i	$0.789827\pi$
$380$	4337.32	−	4121.59i	0.585526	−	0.556403i
$381$	−3591.42			−0.482924
$382$			5900.16i			0.790257i
$383$		−	6364.97i		−	0.849177i	−0.905387	−	0.424588i	$-0.860419\pi$
$383$		−	6364.97i		−	0.849177i	0.905387	−	0.424588i	$-0.139581\pi$
$384$	−2925.52			−0.388782
$385$	306.281	+	322.313i	0.0405442	+	0.0426665i
$386$	−3049.44			−0.402105
$387$		−	2226.15i		−	0.292407i
$388$			1962.11i			0.256730i
$389$	−6134.33			−0.799545			−0.399773	−	0.916614i	$-0.630911\pi$
$389$	−6134.33			−0.799545			−0.399773	+	0.916614i	$0.630911\pi$
$390$	−2288.45	−	2408.23i	−0.297128	−	0.312681i
$391$	−941.862			−0.121821
$392$			3350.48i			0.431696i
$393$			964.941i			0.123855i
$394$	2855.55			0.365128
$395$	−6244.41	+	5933.81i	−0.795418	+	0.755854i
$396$	82.2653			0.0104394
$397$			9746.46i			1.23214i	0.787690	+	0.616072i	$0.211277\pi$
$397$			9746.46i			1.23214i	−0.787690	+	0.616072i	$0.788723\pi$
$398$			5288.85i			0.666096i
$399$	−6985.12			−0.876425
$400$	18.4486	−	361.444i	0.00230607	−	0.0451805i
$401$	−1306.44			−0.162695			−0.0813474	−	0.996686i	$-0.525922\pi$
$401$	−1306.44			−0.162695			−0.0813474	+	0.996686i	$0.525922\pi$
$402$		−	887.737i		−	0.110140i
$403$		−	3749.77i		−	0.463498i
$404$	176.063			0.0216819
$405$	−656.480	+	623.827i	−0.0805450	+	0.0765387i
$406$	11090.2			1.35566
$407$			35.6055i			0.00433636i
$408$		−	1269.97i		−	0.154100i
$409$	3876.93			0.468709			0.234354	−	0.972151i	$-0.424702\pi$
$409$	3876.93			0.468709			0.234354	+	0.972151i	$0.424702\pi$
$410$	−2169.89	−	2283.47i	−0.261374	−	0.275055i
$411$	1062.87			0.127561
$412$			8965.38i			1.07207i
$413$			1638.84i			0.195260i
$414$	−759.769			−0.0901947
$415$	1339.49	+	1409.61i	0.158441	+	0.166735i
$416$	−10670.6			−1.25762
$417$			231.769i			0.0272177i
$418$		−	319.426i		−	0.0373771i
$419$	16022.5			1.86814			0.934071	−	0.357088i	$-0.116230\pi$
$419$	16022.5			1.86814			0.934071	+	0.357088i	$0.116230\pi$
$420$	−2756.53	+	2619.42i	−0.320250	+	0.304321i
$421$	−8119.73			−0.939980			−0.469990	−	0.882672i	$-0.655742\pi$
$421$	−8119.73			−0.939980			−0.469990	+	0.882672i	$0.655742\pi$
$422$			7611.54i			0.878019i
$423$		−	3460.89i		−	0.397812i
$424$	−10336.0			−1.18386
$425$	2369.96	+	120.966i	0.270494	+	0.0138064i
$426$	3033.62			0.345022
$427$		−	3050.46i		−	0.345719i
$428$		−	6951.24i		−	0.785049i
$429$	312.694			0.0351911
$430$	−3411.11	+	3241.44i	−0.382554	+	0.363526i
$431$	−5713.99			−0.638592			−0.319296	−	0.947655i	$-0.603446\pi$
$431$	−5713.99			−0.638592			−0.319296	+	0.947655i	$0.603446\pi$
$432$			78.1735i			0.00870630i
$433$		−	6251.34i		−	0.693811i	−0.937900	−	0.346906i	$-0.887232\pi$
$433$		−	6251.34i		−	0.693811i	0.937900	−	0.346906i	$-0.112768\pi$
$434$	2434.42			0.269254
$435$	6780.43	+	7135.34i	0.747349	+	0.786468i
$436$	−1639.21			−0.180055
$437$		−	5201.25i		−	0.569358i
$438$			1634.68i			0.178329i
$439$	4230.97			0.459984			0.229992	−	0.973192i	$-0.426130\pi$
$439$	4230.97			0.459984			0.229992	+	0.973192i	$0.426130\pi$
$440$	−307.509	−	323.605i	−0.0333180	−	0.0350620i
$441$	1352.31			0.146022
$442$		−	1880.34i		−	0.202350i
$443$		−	6314.29i		−	0.677203i	−0.940930	−	0.338601i	$-0.890046\pi$
$443$		−	6314.29i		−	0.677203i	0.940930	−	0.338601i	$-0.109954\pi$
$444$	−304.510			−0.0325482
$445$	8258.88	−	7848.08i	0.879794	−	0.836033i
$446$	2983.55			0.316760
$447$		−	5116.13i		−	0.541353i
$448$		−	6413.13i		−	0.676321i
$449$	9349.71			0.982717			0.491358	−	0.870957i	$-0.336501\pi$
$449$	9349.71			0.982717			0.491358	+	0.870957i	$0.336501\pi$
$450$	1911.77	+	97.5794i	0.200270	+	0.0102221i
$451$	296.494			0.0309565
$452$			8076.87i			0.840496i
$453$		−	2274.84i		−	0.235941i
$454$	−1594.19			−0.164800
$455$	−10477.7	+	9956.53i	−1.07956	+	1.02587i
$456$	7013.14			0.720220
$457$		−	9547.46i		−	0.977268i	−0.872489	−	0.488634i	$-0.837495\pi$
$457$		−	9547.46i		−	0.977268i	0.872489	−	0.488634i	$-0.162505\pi$
$458$		−	4393.53i		−	0.448245i
$459$	−512.578			−0.0521244
$460$	−1950.47	−	2052.56i	−0.197698	−	0.208046i
$461$	6237.23			0.630145			0.315073	−	0.949068i	$-0.397971\pi$
$461$	6237.23			0.630145			0.315073	+	0.949068i	$0.397971\pi$
$462$			203.007i			0.0204431i
$463$		−	6469.98i		−	0.649428i	−0.945812	−	0.324714i	$-0.894732\pi$
$463$		−	6469.98i		−	0.649428i	0.945812	−	0.324714i	$-0.105268\pi$
$464$	849.675			0.0850111
$465$	1488.37	+	1566.28i	0.148434	+	0.156203i
$466$	−3905.10			−0.388198
$467$			7206.64i			0.714097i	0.934086	+	0.357049i	$0.116217\pi$
$467$			7206.64i			0.714097i	−0.934086	+	0.357049i	$0.883783\pi$
$468$			2674.27i			0.264141i
$469$	−3862.35			−0.380270
$470$	−5303.10	+	5039.32i	−0.520455	+	0.494567i
$471$	−5307.16			−0.519195
$472$		−	1645.42i		−	0.160458i
$473$		−	442.912i		−	0.0430552i
$474$	−3933.00			−0.381115
$475$	−668.012	+	13087.6i	−0.0645274	+	1.26422i
$476$	−2152.29			−0.207248
$477$			4171.75i			0.400443i
$478$		−	3903.39i		−	0.373509i
$479$	10851.8			1.03514			0.517571	−	0.855640i	$-0.326836\pi$
$479$	10851.8			1.03514			0.517571	+	0.855640i	$0.326836\pi$
$480$	4457.11	−	4235.41i	0.423830	−	0.402749i
$481$	−1157.46			−0.109720
$482$		−	650.485i		−	0.0614705i
$483$			3305.59i			0.311407i
$484$	−6777.97			−0.636549
$485$	−2960.29	−	3115.24i	−0.277154	−	0.291661i
$486$	−413.480			−0.0385922
$487$			12757.1i			1.18702i	0.804827	+	0.593510i	$0.202258\pi$
$487$			12757.1i			1.18702i	−0.804827	+	0.593510i	$0.797742\pi$
$488$			3062.69i			0.284101i
$489$	2645.16			0.244618
$490$	−1969.06	−	2072.13i	−0.181537	−	0.191039i
$491$	−7016.52			−0.644911			−0.322455	−	0.946585i	$-0.604508\pi$
$491$	−7016.52			−0.644911			−0.322455	+	0.946585i	$0.604508\pi$
$492$			2535.72i			0.232356i
$493$			5571.26i			0.508960i
$494$	10383.8			0.945729
$495$	−130.612	+	124.116i	−0.0118598	+	0.0112699i
$496$	186.512			0.0168844
$497$		−	13198.6i		−	1.19122i
$498$			887.832i			0.0798890i
$499$	−11372.3			−1.02023			−0.510113	−	0.860107i	$-0.670396\pi$
$499$	−11372.3			−1.02023			−0.510113	+	0.860107i	$0.670396\pi$
$500$	4644.25	+	5415.27i	0.415395	+	0.484356i
$501$	−650.700			−0.0580262
$502$			3845.50i			0.341899i
$503$			5587.37i			0.495285i	0.968851	+	0.247643i	$0.0796559\pi$
$503$			5587.37i			0.495285i	−0.968851	+	0.247643i	$0.920344\pi$
$504$	−4457.11			−0.393919
$505$	−279.535	+	265.631i	−0.0246320	+	0.0234068i
$506$	−151.163			−0.0132806
$507$			3573.99i			0.313070i
$508$		−	6111.03i		−	0.533726i
$509$	−16256.7			−1.41565			−0.707825	−	0.706388i	$-0.750324\pi$
$509$	−16256.7			−1.41565			−0.707825	+	0.706388i	$0.750324\pi$
$510$	746.353	+	785.419i	0.0648021	+	0.0681940i
$511$	7112.14			0.615699
$512$		−	1047.24i		−	0.0903943i
$513$		−	2830.61i		−	0.243615i
$514$	−157.878			−0.0135481
$515$	−13526.3	−	14234.3i	−1.15736	−	1.21794i
$516$	3787.93			0.323167
$517$		−	688.574i		−	0.0585754i
$518$		−	751.442i		−	0.0637384i
$519$	−12377.7			−1.04686
$520$	10519.7	−	9996.46i	0.887153	−	0.843026i
$521$	19748.4			1.66064			0.830320	−	0.557286i	$-0.188157\pi$
$521$	19748.4			1.66064			0.830320	+	0.557286i	$0.188157\pi$
$522$			4494.15i			0.376827i
$523$			7843.44i			0.655774i	0.944717	+	0.327887i	$0.106337\pi$
$523$			7843.44i			0.655774i	−0.944717	+	0.327887i	$0.893663\pi$
$524$	−1641.91			−0.136884
$525$	424.546	−	8317.69i	0.0352928	−	0.691455i
$526$	967.019			0.0801597
$527$			1222.95i			0.101086i
$528$			15.5533i			0.00128195i
$529$	9705.60			0.797699
$530$	6392.34	−	6074.39i	0.523897	−	0.497839i
$531$	−664.116			−0.0542753
$532$		−	11885.6i		−	0.968622i
$533$			9638.38i			0.783273i
$534$	5201.80			0.421543
$535$	10487.5	+	11036.5i	0.847504	+	0.891865i
$536$	3877.84			0.312495
$537$		−	9639.42i		−	0.774622i
$538$			12902.0i			1.03391i
$539$	269.053			0.0215008
$540$	−1061.48	−	1117.04i	−0.0845904	−	0.0890181i
$541$	7383.29			0.586751			0.293376	−	0.955997i	$-0.405221\pi$
$541$	7383.29			0.586751			0.293376	+	0.955997i	$0.405221\pi$
$542$		−	13516.7i		−	1.07120i
$543$		−	10183.3i		−	0.804798i
$544$	3480.10			0.274280
$545$	2602.57	−	2473.12i	0.204554	−	0.194379i
$546$	−6599.31			−0.517260
$547$		−	3354.90i		−	0.262240i	−0.991367	−	0.131120i	$-0.958143\pi$
$547$		−	3354.90i		−	0.262240i	0.991367	−	0.131120i	$-0.0418573\pi$
$548$			1808.54i			0.140980i
$549$	1236.15			0.0960976
$550$	380.363	+	19.4143i	0.0294886	+	0.00150514i
$551$	−30766.2			−2.37874
$552$		−	3318.84i		−	0.255905i
$553$			17111.6i			1.31584i
$554$	11609.8			0.890346
$555$	483.469	−	459.422i	0.0369768	−	0.0351376i
$556$	−394.369			−0.0300809
$557$			20771.8i			1.58012i	0.613028	+	0.790061i	$0.289951\pi$
$557$			20771.8i			1.58012i	−0.613028	+	0.790061i	$0.710049\pi$
$558$			986.512i			0.0748430i
$559$	14398.1			1.08940
$560$	−495.234	−	521.156i	−0.0373705	−	0.0393266i
$561$	−101.982			−0.00767500
$562$		−	5642.15i		−	0.423487i
$563$		−	7194.86i		−	0.538592i	−0.963057	−	0.269296i	$-0.913209\pi$
$563$		−	7194.86i		−	0.538592i	0.963057	−	0.269296i	$-0.0867910\pi$
$564$	5888.93			0.439660
$565$	−12185.8	−	12823.6i	−0.907361	−	0.954856i
$566$	−11235.0			−0.834350
$567$			1798.96i			0.133244i
$568$			13251.5i			0.978913i
$569$	−11549.5			−0.850931			−0.425466	−	0.904975i	$-0.639890\pi$
$569$	−11549.5			−0.850931			−0.425466	+	0.904975i	$0.639890\pi$
$570$	−4337.32	+	4121.59i	−0.318720	+	0.302867i
$571$	1482.54			0.108655			0.0543277	−	0.998523i	$-0.482698\pi$
$571$	1482.54			0.108655			0.0543277	+	0.998523i	$0.482698\pi$
$572$			532.068i			0.0388932i
$573$			10402.5i			0.758412i
$574$	−6257.42			−0.455017
$575$	6193.50	+	316.125i	0.449194	+	0.0229275i
$576$	2598.82			0.187993
$577$		−	15264.0i		−	1.10130i	−0.834737	−	0.550649i	$-0.814380\pi$
$577$		−	15264.0i		−	1.10130i	0.834737	−	0.550649i	$-0.185620\pi$
$578$		−	7746.52i		−	0.557462i
$579$	−5376.43			−0.385901
$580$	−12141.2	+	11537.3i	−0.869202	+	0.825968i
$581$	3862.76			0.275825
$582$		−	1962.11i		−	0.139746i
$583$			830.006i			0.0589628i
$584$	−7140.66			−0.505963
$585$	−4034.73	−	4245.92i	−0.285155	−	0.300081i
$586$	−9893.12			−0.697408
$587$			1736.89i			0.122128i	0.998134	+	0.0610639i	$0.0194493\pi$
$587$			1736.89i			0.122128i	−0.998134	+	0.0610639i	$0.980551\pi$
$588$			2301.03i			0.161383i
$589$	−6753.50			−0.472450
$590$	967.003	+	1017.62i	0.0674761	+	0.0710080i
$591$	5034.57			0.350414
$592$		−	57.5714i		−	0.00399691i
$593$			11764.8i			0.814707i	0.913271	+	0.407353i	$0.133548\pi$
$593$			11764.8i			0.814707i	−0.913271	+	0.407353i	$0.866452\pi$
$594$	−82.2653			−0.00568247
$595$	3417.19	−	3247.22i	0.235447	−	0.223736i
$596$	8705.42			0.598302
$597$			9324.69i			0.639253i
$598$		−	4913.96i		−	0.336032i
$599$	9451.99			0.644737			0.322369	−	0.946614i	$-0.395521\pi$
$599$	9451.99			0.644737			0.322369	+	0.946614i	$0.395521\pi$
$600$	−426.249	+	8351.04i	−0.0290026	+	0.568217i
$601$	−3131.93			−0.212569			−0.106285	−	0.994336i	$-0.533895\pi$
$601$	−3131.93			−0.212569			−0.106285	+	0.994336i	$0.533895\pi$
$602$			9347.51i			0.632851i
$603$		−	1565.16i		−	0.105702i
$604$	3870.79			0.260762
$605$	10761.4	−	10226.1i	0.723159	−	0.687189i
$606$	−176.063			−0.0118021
$607$			22700.8i			1.51795i	0.651120	+	0.758975i	$0.274300\pi$
$607$			22700.8i			1.51795i	−0.651120	+	0.758975i	$0.725700\pi$
$608$			19218.2i			1.28191i
$609$	19553.1			1.30103
$610$	−1799.93	−	1894.14i	−0.119470	−	0.125724i
$611$	22384.0			1.48210
$612$		−	872.184i		−	0.0576077i
$613$		−	28911.6i		−	1.90494i	−0.304629	−	0.952471i	$-0.598532\pi$
$613$		−	28911.6i		−	1.90494i	0.304629	−	0.952471i	$-0.401468\pi$
$614$	−13825.0			−0.908681
$615$	−3825.70	−	4025.95i	−0.250841	−	0.263971i
$616$	−886.780			−0.0580023
$617$		−	5566.87i		−	0.363231i	−0.983370	−	0.181616i	$-0.941867\pi$
$617$		−	5566.87i		−	0.363231i	0.983370	−	0.181616i	$-0.0581326\pi$
$618$		−	8965.38i		−	0.583560i
$619$	−4150.32			−0.269492			−0.134746	−	0.990880i	$-0.543022\pi$
$619$	−4150.32			−0.269492			−0.134746	+	0.990880i	$0.543022\pi$
$620$	−2665.12	+	2532.56i	−0.172635	+	0.164049i
$621$	−1339.54			−0.0865600
$622$		−	12483.1i		−	0.804705i
$623$		−	22631.9i		−	1.45542i
$624$	−505.603			−0.0324364
$625$	−15543.8	−	1590.90i	−0.994803	−	0.101818i
$626$	3747.96			0.239295
$627$		−	563.175i		−	0.0358709i
$628$		−	9030.46i		−	0.573813i
$629$	377.492			0.0239294
$630$	2756.53	−	2619.42i	0.174322	−	0.165651i
$631$	−4090.09			−0.258041			−0.129021	−	0.991642i	$-0.541183\pi$
$631$	−4090.09			−0.258041			−0.129021	+	0.991642i	$0.541183\pi$
$632$		−	17180.2i		−	1.08132i
$633$			13419.8i			0.842637i
$634$	17030.7			1.06684
$635$	9219.85	+	9702.45i	0.576187	+	0.606346i
$636$	−7098.50			−0.442569
$637$			8746.32i			0.544022i
$638$			894.150i			0.0554855i
$639$	5348.53			0.331118
$640$	7510.36	+	7903.47i	0.463864	+	0.488144i
$641$	3909.35			0.240890			0.120445	−	0.992720i	$-0.461568\pi$
$641$	3909.35			0.240890			0.120445	+	0.992720i	$0.461568\pi$
$642$			6951.24i			0.427327i
$643$		−	30539.5i		−	1.87303i	−0.350624	−	0.936516i	$-0.614031\pi$
$643$		−	30539.5i		−	1.87303i	0.350624	−	0.936516i	$-0.385969\pi$
$644$	−5624.66			−0.344166
$645$	−6014.08	+	5714.94i	−0.367138	+	0.348877i
$646$	−3386.58			−0.206259
$647$		−	12707.7i		−	0.772167i	−0.922464	−	0.386083i	$-0.873828\pi$
$647$		−	12707.7i		−	0.772167i	0.922464	−	0.386083i	$-0.126172\pi$
$648$		−	1806.17i		−	0.109496i
$649$	−132.132			−0.00799170
$650$	−631.115	+	12364.8i	−0.0380836	+	0.746132i
$651$	4292.10			0.258403
$652$			4500.90i			0.270351i
$653$		−	12777.6i		−	0.765737i	−0.923803	−	0.382869i	$-0.874936\pi$
$653$		−	12777.6i		−	0.765737i	0.923803	−	0.382869i	$-0.125064\pi$
$654$	1639.21			0.0980095
$655$	2606.85	−	2477.18i	0.155508	−	0.147773i
$656$	−479.410			−0.0285332
$657$			2882.08i			0.171143i
$658$			14532.1i			0.860976i
$659$	−23563.5			−1.39287			−0.696435	−	0.717620i	$-0.745232\pi$
$659$	−23563.5			−1.39287			−0.696435	+	0.717620i	$0.745232\pi$
$660$	−211.190	−	222.245i	−0.0124554	−	0.0131074i
$661$	−4361.31			−0.256634			−0.128317	−	0.991733i	$-0.540958\pi$
$661$	−4361.31			−0.256634			−0.128317	+	0.991733i	$0.540958\pi$
$662$			14796.7i			0.868715i
$663$		−	3315.21i		−	0.194196i
$664$	−3878.25			−0.226665
$665$	17932.1	+	18870.7i	1.04568	+	1.10042i
$666$	304.510			0.0177170
$667$			14559.6i			0.845200i
$668$		−	1107.21i		−	0.0641304i
$669$	5260.25			0.303995
$670$	−2398.28	+	2278.99i	−0.138289	+	0.131410i
$671$	245.943			0.0141498
$672$		−	12213.9i		−	0.701131i
$673$			8203.52i			0.469870i	0.972011	+	0.234935i	$0.0754877\pi$
$673$			8203.52i			0.469870i	−0.972011	+	0.234935i	$0.924512\pi$
$674$	−12592.6			−0.719657
$675$	3370.61	+	172.041i	0.192200	+	0.00981015i
$676$	−6081.37			−0.346004
$677$			28057.1i			1.59279i	0.604774	+	0.796397i	$0.293263\pi$
$677$			28057.1i			1.59279i	−0.604774	+	0.796397i	$0.706737\pi$
$678$		−	8076.87i		−	0.457508i
$679$	−8536.73			−0.482488
$680$	−3430.89	+	3260.24i	−0.193483	+	0.183860i
$681$	−2810.70			−0.158159
$682$			196.275i			0.0110202i
$683$			3344.62i			0.187377i	0.995602	+	0.0936885i	$0.0298658\pi$
$683$			3344.62i			0.187377i	−0.995602	+	0.0936885i	$0.970134\pi$
$684$	4816.46			0.269243
$685$	−2728.59	−	2871.41i	−0.152196	−	0.160162i
$686$	7283.91			0.405395
$687$		−	7746.17i		−	0.430182i
$688$			716.156i			0.0396849i
$689$	−26981.7			−1.49190
$690$	1950.47	+	2052.56i	0.107613	+	0.113246i
$691$	12964.8			0.713757			0.356879	−	0.934151i	$-0.383841\pi$
$691$	12964.8			0.713757			0.356879	+	0.934151i	$0.383841\pi$
$692$		−	21061.5i		−	1.15699i
$693$			357.918i			0.0196193i
$694$	13342.7			0.729801
$695$	626.138	−	594.994i	0.0341737	−	0.0324740i
$696$	−19631.5			−1.06915
$697$		−	3143.46i		−	0.170828i
$698$		−	8441.90i		−	0.457780i
$699$	−6885.03			−0.372555
$700$	14153.1	+	722.392i	0.764193	+	0.0390055i
$701$	−16162.1			−0.870806			−0.435403	−	0.900236i	$-0.643394\pi$
$701$	−16162.1			−0.870806			−0.435403	+	0.900236i	$0.643394\pi$
$702$		−	2674.27i		−	0.143780i
$703$			2084.63i			0.111839i
$704$	517.058			0.0276809
$705$	−9349.82	+	8884.76i	−0.499482	+	0.474638i
$706$	20696.2			1.10327
$707$			766.014i			0.0407481i
$708$		−	1130.03i		−	0.0599849i
$709$	14244.4			0.754529			0.377265	−	0.926105i	$-0.376865\pi$
$709$	14244.4			0.754529			0.377265	+	0.926105i	$0.376865\pi$
$710$	−7787.87	−	8195.51i	−0.411653	−	0.433200i
$711$	−6934.22			−0.365757
$712$			22722.7i			1.19602i
$713$			3195.97i			0.167868i
$714$	2152.29			0.112812
$715$	−802.744	−	844.762i	−0.0419873	−	0.0441850i
$716$	16402.1			0.856109
$717$		−	6882.02i		−	0.358457i
$718$			8837.55i			0.459352i
$719$	27638.5			1.43358			0.716790	−	0.697289i	$-0.245611\pi$
$719$	27638.5			1.43358			0.716790	+	0.697289i	$0.245611\pi$
$720$	211.190	−	200.686i	0.0109314	−	0.0103877i
$721$	−39006.4			−2.01480
$722$		−	7030.68i		−	0.362403i
$723$		−	1146.86i		−	0.0589934i
$724$	17327.4			0.889460
$725$	1869.93	−	36635.5i	0.0957895	−	1.87670i
$726$	6777.97			0.346493
$727$		−	2525.52i		−	0.128840i	−0.997923	−	0.0644199i	$-0.979480\pi$
$727$		−	2525.52i		−	0.128840i	0.997923	−	0.0644199i	$-0.0205197\pi$
$728$		−	28827.3i		−	1.46760i
$729$	−729.000			−0.0370370
$730$	4416.19	−	4196.53i	0.223905	−	0.212768i
$731$	−4695.79			−0.237592
$732$			2103.39i			0.106207i
$733$			8400.27i			0.423289i	0.977347	+	0.211645i	$0.0678820\pi$
$733$			8400.27i			0.423289i	−0.977347	+	0.211645i	$0.932118\pi$
$734$	−10355.9			−0.520765
$735$	−3471.62	−	3653.34i	−0.174221	−	0.183341i
$736$	9094.67			0.455481
$737$		−	311.401i		−	0.0155639i
$738$		−	2535.72i		−	0.126479i
$739$	19689.1			0.980074			0.490037	−	0.871702i	$-0.336983\pi$
$739$	19689.1			0.980074			0.490037	+	0.871702i	$0.336983\pi$
$740$	781.735	+	822.653i	0.0388340	+	0.0408667i
$741$	18307.6			0.907619
$742$		−	17517.0i		−	0.866671i
$743$			22526.6i			1.11227i	0.831091	+	0.556137i	$0.187717\pi$
$743$			22526.6i			1.11227i	−0.831091	+	0.556137i	$0.812283\pi$
$744$	−4309.31			−0.212348
$745$	−13821.6	+	13134.1i	−0.679708	+	0.645900i
$746$	18005.8			0.883699
$747$			1565.32i			0.0766696i
$748$		−	173.528i		−	0.00848239i
$749$	30243.3			1.47539
$750$	−4644.25	−	5415.27i	−0.226112	−	0.263650i
$751$	34691.1			1.68562			0.842808	−	0.538215i	$-0.180901\pi$
$751$	34691.1			1.68562			0.842808	+	0.538215i	$0.180901\pi$
$752$			1113.37i			0.0539902i
$753$			6779.95i			0.328121i
$754$	−29066.8			−1.40392
$755$	−6145.63	+	5839.95i	−0.296242	+	0.281507i
$756$	−3061.04			−0.147261
$757$			6619.98i			0.317843i	0.987291	+	0.158922i	$0.0508017\pi$
$757$			6619.98i			0.317843i	−0.987291	+	0.158922i	$0.949198\pi$
$758$			19832.0i			0.950303i
$759$	−266.512			−0.0127454
$760$	−18004.0	−	18946.4i	−0.859309	−	0.904288i
$761$	−29368.7			−1.39897			−0.699483	−	0.714649i	$-0.746586\pi$
$761$	−29368.7			−1.39897			−0.699483	+	0.714649i	$0.746586\pi$
$762$			6111.03i			0.290524i
$763$		−	7131.84i		−	0.338388i
$764$	−17700.5			−0.838194
$765$	1315.88	+	1384.76i	0.0621907	+	0.0654460i
$766$	−10830.4			−0.510859
$767$		−	4295.31i		−	0.202209i
$768$			11908.1i			0.559503i
$769$	−32677.4			−1.53235			−0.766174	−	0.642633i	$-0.777842\pi$
$769$	−32677.4			−1.53235			−0.766174	+	0.642633i	$0.777842\pi$
$770$	548.435	−	521.156i	0.0256678	−	0.0243911i
$771$	−278.353			−0.0130021
$772$		−	9148.33i		−	0.426497i
$773$			28047.5i			1.30504i	0.757770	+	0.652522i	$0.226289\pi$
$773$			28047.5i			1.30504i	−0.757770	+	0.652522i	$0.773711\pi$
$774$	−3787.93			−0.175910
$775$	410.469	−	8041.87i	0.0190251	−	0.372739i
$776$	8570.96			0.396494
$777$		−	1324.86i		−	0.0611699i
$778$			10438.0i			0.481001i

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)

\(f(q)\)	\(=\)	\(q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} +(-8.10469 + 7.70156i) q^{5} -5.10469 q^{6} +22.2094i q^{7} -22.2984i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-1.70156i q^{2} -3.00000i q^{3} +5.10469 q^{4} +(-8.10469 + 7.70156i) q^{5} -5.10469 q^{6} +22.2094i q^{7} -22.2984i q^{8} -9.00000 q^{9} +(13.1047 + 13.7906i) q^{10} -1.79063 q^{11} -15.3141i q^{12} -58.2094i q^{13} +37.7906 q^{14} +(23.1047 + 24.3141i) q^{15} +2.89531 q^{16} +18.9844i q^{17} +15.3141i q^{18} -104.837 q^{19} +(-41.3719 + 39.3141i) q^{20} +66.6281 q^{21} +3.04686i q^{22} +49.6125i q^{23} -66.8953 q^{24} +(6.37188 - 124.837i) q^{25} -99.0469 q^{26} +27.0000i q^{27} +113.372i q^{28} +293.466 q^{29} +(41.3719 - 39.3141i) q^{30} +64.4187 q^{31} -183.314i q^{32} +5.37188i q^{33} +32.3031 q^{34} +(-171.047 - 180.000i) q^{35} -45.9422 q^{36} -19.8844i q^{37} +178.388i q^{38} -174.628 q^{39} +(171.733 + 180.722i) q^{40} -165.581 q^{41} -113.372i q^{42} +247.350i q^{43} -9.14059 q^{44} +(72.9422 - 69.3141i) q^{45} +84.4187 q^{46} +384.544i q^{47} -8.68594i q^{48} -150.256 q^{49} +(-212.419 - 10.8422i) q^{50} +56.9531 q^{51} -297.141i q^{52} -463.528i q^{53} +45.9422 q^{54} +(14.5125 - 13.7906i) q^{55} +495.234 q^{56} +314.512i q^{57} -499.350i q^{58} +73.7906 q^{59} +(117.942 + 124.116i) q^{60} -137.350 q^{61} -109.612i q^{62} -199.884i q^{63} -288.758 q^{64} +(448.303 + 471.769i) q^{65} +9.14059 q^{66} +173.906i q^{67} +96.9093i q^{68} +148.837 q^{69} +(-306.281 + 291.047i) q^{70} -594.281 q^{71} +200.686i q^{72} -320.231i q^{73} -33.8345 q^{74} +(-374.512 - 19.1156i) q^{75} -535.163 q^{76} -39.7687i q^{77} +297.141i q^{78} +770.469 q^{79} +(-23.4656 + 22.2984i) q^{80} +81.0000 q^{81} +281.747i q^{82} -173.925i q^{83} +340.116 q^{84} +(-146.209 - 153.862i) q^{85} +420.881 q^{86} -880.397i q^{87} +39.9282i q^{88} -1019.02 q^{89} +(-117.942 - 124.116i) q^{90} +1292.79 q^{91} +253.256i q^{92} -193.256i q^{93} +654.325 q^{94} +(849.675 - 807.412i) q^{95} -549.942 q^{96} +384.375i q^{97} +255.670i q^{98} +16.1156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{4} + 6 q^{5} + 18 q^{6} - 36 q^{9}+O(q^{10}) \) 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

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