LMFDB - Embedded newform 75.9.d.b.74.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not 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:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 49 $ x^4 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.4
Root			$1.87083 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.9.d.b.74.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.4499 q^{2} +(67.3498 + 45.0000i) q^{3} +248.000 q^{4} +(1512.00 + 1010.25i) q^{6} +1750.00i q^{7} -179.600 q^{8} +(2511.00 + 6061.48i) q^{9} +O(q^{10})$	\(q+22.4499 q^{2} +(67.3498 + 45.0000i) q^{3} +248.000 q^{4} +(1512.00 + 1010.25i) q^{6} +1750.00i q^{7} -179.600 q^{8} +(2511.00 + 6061.48i) q^{9} +6959.48i q^{11} +(16702.8 + 11160.0i) q^{12} +25730.0i q^{13} +39287.4i q^{14} -67520.0 q^{16} +74893.0 q^{17} +(56371.8 + 136080. i) q^{18} -18938.0 q^{19} +(-78750.0 + 117862. i) q^{21} +156240. i q^{22} +470461. q^{23} +(-12096.0 - 8081.98i) q^{24} +577637. i q^{26} +(-103651. + 521235. i) q^{27} +434000. i q^{28} -460897. i q^{29} -351478. q^{31} -1.46984e6 q^{32} +(-313177. + 468720. i) q^{33} +1.68134e6 q^{34} +(622728. + 1.50325e6i) q^{36} -1.33517e6i q^{37} -425157. q^{38} +(-1.15785e6 + 1.73291e6i) q^{39} +1.87547e6i q^{41} +(-1.76793e6 + 2.64600e6i) q^{42} -3.52615e6i q^{43} +1.72595e6i q^{44} +1.05618e7 q^{46} -4.08104e6 q^{47} +(-4.54746e6 - 3.03840e6i) q^{48} +2.70230e6 q^{49} +(5.04403e6 + 3.37019e6i) q^{51} +6.38104e6i q^{52} +6.60177e6 q^{53} +(-2.32697e6 + 1.17017e7i) q^{54} -314299. i q^{56} +(-1.27547e6 - 852210. i) q^{57} -1.03471e7i q^{58} -1.37149e7i q^{59} +753602. q^{61} -7.89066e6 q^{62} +(-1.06076e7 + 4.39425e6i) q^{63} -1.57128e7 q^{64} +(-7.03080e6 + 1.05227e7i) q^{66} -2.26889e6i q^{67} +1.85735e7 q^{68} +(3.16855e7 + 2.11707e7i) q^{69} +1.70220e7i q^{71} +(-450974. - 1.08864e6i) q^{72} +2.76728e7i q^{73} -2.99745e7i q^{74} -4.69662e6 q^{76} -1.21791e7 q^{77} +(-2.59937e7 + 3.89038e7i) q^{78} +2.29810e7 q^{79} +(-3.04365e7 + 3.04408e7i) q^{81} +4.21042e7i q^{82} +4.63952e7 q^{83} +(-1.95300e7 + 2.92298e7i) q^{84} -7.91619e7i q^{86} +(2.07404e7 - 3.10414e7i) q^{87} -1.24992e6i q^{88} -7.26152e7i q^{89} -4.50275e7 q^{91} +1.16674e8 q^{92} +(-2.36720e7 - 1.58165e7i) q^{93} -9.16191e7 q^{94} +(-9.89937e7 - 6.61429e7i) q^{96} -1.47271e8i q^{97} +6.06665e7 q^{98} +(-4.21848e7 + 1.74753e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9}+O(q^{10}) $ 4 * q + 992 * q^4 + 6048 * q^6 + 10044 * q^9	$ 4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9} - 270080 q^{16} - 75752 q^{19} - 315000 q^{21} - 48384 q^{24} - 1405912 q^{31} + 6725376 q^{34} + 2490912 q^{36} - 4631400 q^{39} + 42247296 q^{46} + 10809204 q^{49} + 20176128 q^{51} - 9307872 q^{54} + 3014408 q^{61} - 62851072 q^{64} - 28123200 q^{66} + 126741888 q^{69} - 18786496 q^{76} + 91923928 q^{79} - 121745916 q^{81} - 78120000 q^{84} - 180110000 q^{91} - 366476544 q^{94} - 395974656 q^{96} - 168739200 q^{99}+O(q^{100}) $ 4 * q + 992 * q^4 + 6048 * q^6 + 10044 * q^9 - 270080 * q^16 - 75752 * q^19 - 315000 * q^21 - 48384 * q^24 - 1405912 * q^31 + 6725376 * q^34 + 2490912 * q^36 - 4631400 * q^39 + 42247296 * q^46 + 10809204 * q^49 + 20176128 * q^51 - 9307872 * q^54 + 3014408 * q^61 - 62851072 * q^64 - 28123200 * q^66 + 126741888 * q^69 - 18786496 * q^76 + 91923928 * q^79 - 121745916 * q^81 - 78120000 * q^84 - 180110000 * q^91 - 366476544 * q^94 - 395974656 * q^96 - 168739200 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

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 $

$2$	22.4499			1.40312			0.701561	−	0.712610i	$-0.252487\pi$
$2$	22.4499			1.40312			0.701561	+	0.712610i	$0.252487\pi$
$3$	67.3498	+	45.0000i	0.831479	+	0.555556i
$3$	67.3498	+	45.0000i	0.831479	+	0.555556i
$4$	248.000			0.968750
$5$		0			0
$5$		0			0
$6$	1512.00	+	1010.25i	1.16667	+	0.779512i
$7$			1750.00i			0.728863i	0.931230	+	0.364431i	$0.118737\pi$
$7$			1750.00i			0.728863i	−0.931230	+	0.364431i	$0.881263\pi$
$8$	−179.600			−0.0438475
$9$	2511.00	+	6061.48i	0.382716	+	0.923866i
$10$		0			0
$11$			6959.48i			0.475342i	0.971346	+	0.237671i	$0.0763840\pi$
$11$			6959.48i			0.475342i	−0.971346	+	0.237671i	$0.923616\pi$
$12$	16702.8	+	11160.0i	0.805496	+	0.538194i
$13$			25730.0i			0.900879i	0.892807	+	0.450439i	$0.148733\pi$
$13$			25730.0i			0.900879i	−0.892807	+	0.450439i	$0.851267\pi$
$14$			39287.4i			1.02268i
$15$		0			0
$16$	−67520.0			−1.03027
$17$	74893.0			0.896697			0.448348	−	0.893859i	$-0.352012\pi$
$17$	74893.0			0.896697			0.448348	+	0.893859i	$0.352012\pi$
$18$	56371.8	+	136080.i	0.536997	+	1.29630i
$19$	−18938.0			−0.145318			−0.0726590	−	0.997357i	$-0.523148\pi$
$19$	−18938.0			−0.145318			−0.0726590	+	0.997357i	$0.523148\pi$
$20$		0			0
$21$	−78750.0	+	117862.i	−0.404924	+	0.606035i
$22$			156240.i			0.666963i
$23$	470461.			1.68117			0.840586	−	0.541678i	$-0.182211\pi$
$23$	470461.			1.68117			0.840586	+	0.541678i	$0.182211\pi$
$24$	−12096.0	−	8081.98i	−0.0364583	−	0.0243597i
$25$		0			0
$26$			577637.i			1.26404i
$27$	−103651.	+	521235.i	−0.195038	+	0.980796i
$28$			434000.i			0.706086i
$29$		−	460897.i		−	0.651647i	−0.945431	−	0.325823i	$-0.894359\pi$
$29$		−	460897.i		−	0.651647i	0.945431	−	0.325823i	$-0.105641\pi$
$30$		0			0
$31$	−351478.			−0.380585			−0.190292	−	0.981727i	$-0.560944\pi$
$31$	−351478.			−0.380585			−0.190292	+	0.981727i	$0.560944\pi$
$32$	−1.46984e6			−1.40175
$33$	−313177.	+	468720.i	−0.264079	+	0.395237i
$34$	1.68134e6			1.25817
$35$		0			0
$36$	622728.	+	1.50325e6i	0.370756	+	0.894995i
$37$		−	1.33517e6i		−	0.712409i	−0.934408	−	0.356205i	$-0.884071\pi$
$37$		−	1.33517e6i		−	0.712409i	0.934408	−	0.356205i	$-0.115929\pi$
$38$	−425157.			−0.203899
$39$	−1.15785e6	+	1.73291e6i	−0.500488	+	0.749062i
$40$		0			0
$41$			1.87547e6i			0.663704i	0.943331	+	0.331852i	$0.107673\pi$
$41$			1.87547e6i			0.663704i	−0.943331	+	0.331852i	$0.892327\pi$
$42$	−1.76793e6	+	2.64600e6i	−0.568157	+	0.850340i
$43$		−	3.52615e6i		−	1.03140i	−0.856769	−	0.515700i	$-0.827532\pi$
$43$		−	3.52615e6i		−	1.03140i	0.856769	−	0.515700i	$-0.172468\pi$
$44$			1.72595e6i			0.460488i
$45$		0			0
$46$	1.05618e7			2.35889
$47$	−4.08104e6			−0.836333			−0.418167	−	0.908370i	$-0.637327\pi$
$47$	−4.08104e6			−0.836333			−0.418167	+	0.908370i	$0.637327\pi$
$48$	−4.54746e6	−	3.03840e6i	−0.856651	−	0.572374i
$49$	2.70230e6			0.468759
$50$		0			0
$51$	5.04403e6	+	3.37019e6i	0.745585	+	0.498165i
$52$			6.38104e6i			0.872726i
$53$	6.60177e6			0.836675			0.418337	−	0.908292i	$-0.362613\pi$
$53$	6.60177e6			0.836675			0.418337	+	0.908292i	$0.362613\pi$
$54$	−2.32697e6	+	1.17017e7i	−0.273663	+	1.37618i
$55$		0			0
$56$		−	314299.i		−	0.0319589i
$57$	−1.27547e6	−	852210.i	−0.120829	−	0.0807323i
$58$		−	1.03471e7i		−	0.914340i
$59$		−	1.37149e7i		−	1.13184i	−0.824461	−	0.565919i	$-0.808521\pi$
$59$		−	1.37149e7i		−	1.13184i	0.824461	−	0.565919i	$-0.191479\pi$
$60$		0			0
$61$	753602.			0.0544280			0.0272140	−	0.999630i	$-0.491336\pi$
$61$	753602.			0.0544280			0.0272140	+	0.999630i	$0.491336\pi$
$62$	−7.89066e6			−0.534007
$63$	−1.06076e7	+	4.39425e6i	−0.673372	+	0.278948i
$64$	−1.57128e7			−0.936554
$65$		0			0
$66$	−7.03080e6	+	1.05227e7i	−0.370535	+	0.554566i
$67$		−	2.26889e6i		−	0.112594i	−0.998414	−	0.0562969i	$-0.982071\pi$
$67$		−	2.26889e6i		−	0.112594i	0.998414	−	0.0562969i	$-0.0179293\pi$
$68$	1.85735e7			0.868675
$69$	3.16855e7	+	2.11707e7i	1.39786	+	0.933985i
$70$		0			0
$71$			1.70220e7i			0.669849i	0.942245	+	0.334925i	$0.108711\pi$
$71$			1.70220e7i			0.669849i	−0.942245	+	0.334925i	$0.891289\pi$
$72$	−450974.	−	1.08864e6i	−0.0167812	−	0.0405093i
$73$			2.76728e7i			0.974454i	0.873275	+	0.487227i	$0.161992\pi$
$73$			2.76728e7i			0.974454i	−0.873275	+	0.487227i	$0.838008\pi$
$74$		−	2.99745e7i		−	0.999597i
$75$		0			0
$76$	−4.69662e6			−0.140777
$77$	−1.21791e7			−0.346459
$78$	−2.59937e7	+	3.89038e7i	−0.702246	+	1.05103i
$79$	2.29810e7			0.590011			0.295006	−	0.955496i	$-0.404678\pi$
$79$	2.29810e7			0.590011			0.295006	+	0.955496i	$0.404678\pi$
$80$		0			0
$81$	−3.04365e7	+	3.04408e7i	−0.707057	+	0.707157i
$82$			4.21042e7i			0.931257i
$83$	4.63952e7			0.977599			0.488799	−	0.872396i	$-0.337435\pi$
$83$	4.63952e7			0.977599			0.488799	+	0.872396i	$0.337435\pi$
$84$	−1.95300e7	+	2.92298e7i	−0.392270	+	0.587096i
$85$		0			0
$86$		−	7.91619e7i		−	1.44718i
$87$	2.07404e7	−	3.10414e7i	0.362026	−	0.541831i
$88$		−	1.24992e6i		−	0.0208426i
$89$		−	7.26152e7i		−	1.15736i	−0.815555	−	0.578679i	$-0.803568\pi$
$89$		−	7.26152e7i		−	1.15736i	0.815555	−	0.578679i	$-0.196432\pi$
$90$		0			0
$91$	−4.50275e7			−0.656617
$92$	1.16674e8			1.62864
$93$	−2.36720e7	−	1.58165e7i	−0.316448	−	0.211436i
$94$	−9.16191e7			−1.17348
$95$		0			0
$96$	−9.89937e7	−	6.61429e7i	−1.16553	−	0.778751i
$97$		−	1.47271e8i		−	1.66353i	−0.555129	−	0.831764i	$-0.687331\pi$
$97$		−	1.47271e8i		−	1.66353i	0.555129	−	0.831764i	$-0.312669\pi$
$98$	6.06665e7			0.657726
$99$	−4.21848e7	+	1.74753e7i	−0.439152	+	0.181921i
$100$		0			0
$101$		−	1.03545e8i		−	0.995045i	−0.867451	−	0.497522i	$-0.834243\pi$
$101$		−	1.03545e8i		−	0.995045i	0.867451	−	0.497522i	$-0.165757\pi$
$102$	1.13238e8	+	7.56605e7i	1.04615	+	0.698986i
$103$		−	1.66064e8i		−	1.47545i	−0.675100	−	0.737726i	$-0.735899\pi$
$103$		−	1.66064e8i		−	1.47545i	0.675100	−	0.737726i	$-0.264101\pi$
$104$		−	4.62110e6i		−	0.0395013i
$105$		0			0
$106$	1.48209e8			1.17396
$107$	−2.25540e7			−0.172063			−0.0860316	−	0.996292i	$-0.527419\pi$
$107$	−2.25540e7			−0.172063			−0.0860316	+	0.996292i	$0.527419\pi$
$108$	−2.57055e7	+	1.29266e8i	−0.188943	+	0.950146i
$109$	1.09975e8			0.779091			0.389546	−	0.921007i	$-0.372632\pi$
$109$	1.09975e8			0.779091			0.389546	+	0.921007i	$0.372632\pi$
$110$		0			0
$111$	6.00826e7	−	8.99235e7i	0.395783	−	0.592354i
$112$		−	1.18160e8i		−	0.750928i
$113$	−2.87748e8			−1.76481			−0.882405	−	0.470490i	$-0.844077\pi$
$113$	−2.87748e8			−1.76481			−0.882405	+	0.470490i	$0.844077\pi$
$114$	−2.86343e7	−	1.91321e7i	−0.169538	−	0.113277i
$115$		0			0
$116$		−	1.14303e8i		−	0.631283i
$117$	−1.55962e8	+	6.46080e7i	−0.832291	+	0.344781i
$118$		−	3.07899e8i		−	1.58811i
$119$			1.31063e8i			0.653569i
$120$		0			0
$121$	1.65924e8			0.774050
$122$	1.69183e7			0.0763692
$123$	−8.43961e7	+	1.26312e8i	−0.368724	+	0.551856i
$124$	−8.71665e7			−0.368691
$125$		0			0
$126$	−2.38140e8	+	9.86507e7i	−0.944822	+	0.391397i
$127$		−	2.75994e8i		−	1.06092i	−0.847708	−	0.530462i	$-0.822018\pi$
$127$		−	2.75994e8i		−	1.06092i	0.847708	−	0.530462i	$-0.177982\pi$
$128$	2.35290e7			0.0876523
$129$	1.58677e8	−	2.37486e8i	0.573000	−	0.857588i
$130$		0			0
$131$			2.89118e8i			0.981725i	0.871237	+	0.490862i	$0.163318\pi$
$131$			2.89118e8i			0.981725i	−0.871237	+	0.490862i	$0.836682\pi$
$132$	−7.76678e7	+	1.16243e8i	−0.255826	+	0.382886i
$133$		−	3.31415e7i		−	0.105917i
$134$		−	5.09365e7i		−	0.157983i
$135$		0			0
$136$	−1.34508e7			−0.0393180
$137$	−2.07562e8			−0.589205			−0.294602	−	0.955620i	$-0.595187\pi$
$137$	−2.07562e8			−0.589205			−0.294602	+	0.955620i	$0.595187\pi$
$138$	7.11337e8	+	4.75282e8i	1.96137	+	1.31049i
$139$	1.42668e8			0.382180			0.191090	−	0.981573i	$-0.438798\pi$
$139$	1.42668e8			0.382180			0.191090	+	0.981573i	$0.438798\pi$
$140$		0			0
$141$	−2.74857e8	−	1.83647e8i	−0.695394	−	0.464630i
$142$			3.82143e8i			0.939880i
$143$	−1.79067e8			−0.428226
$144$	−1.69543e8	−	4.09271e8i	−0.394302	−	0.951835i
$145$		0			0
$146$			6.21252e8i			1.36728i
$147$	1.82000e8	+	1.21604e8i	0.389763	+	0.260422i
$148$		−	3.31122e8i		−	0.690147i
$149$			8.19236e8i			1.66213i	0.556179	+	0.831063i	$0.312267\pi$
$149$			8.19236e8i			1.66213i	−0.556179	+	0.831063i	$0.687733\pi$
$150$		0			0
$151$	4.23861e8			0.815296			0.407648	−	0.913139i	$-0.366349\pi$
$151$	4.23861e8			0.815296			0.407648	+	0.913139i	$0.366349\pi$
$152$	3.40126e6			0.00637184
$153$	1.88056e8	+	4.53963e8i	0.343180	+	0.828428i
$154$	−2.73420e8			−0.486124
$155$		0			0
$156$	−2.87147e8	+	4.29762e8i	−0.484848	+	0.725654i
$157$			7.59851e8i			1.25063i	0.780371	+	0.625316i	$0.215030\pi$
$157$			7.59851e8i			1.25063i	−0.780371	+	0.625316i	$0.784970\pi$
$158$	5.15922e8			0.827857
$159$	4.44628e8	+	2.97079e8i	0.695678	+	0.464819i
$160$		0			0
$161$			8.23307e8i			1.22534i
$162$	−6.83297e8	+	6.83394e8i	−0.992087	+	0.992227i
$163$			6.68160e8i			0.946520i	0.880923	+	0.473260i	$0.156923\pi$
$163$			6.68160e8i			0.946520i	−0.880923	+	0.473260i	$0.843077\pi$
$164$			4.65116e8i			0.642963i
$165$		0			0
$166$	1.04157e9			1.37169
$167$	1.96306e8			0.252387			0.126194	−	0.992006i	$-0.459724\pi$
$167$	1.96306e8			0.252387			0.126194	+	0.992006i	$0.459724\pi$
$168$	1.41435e7	−	2.11680e7i	0.0177549	−	0.0265731i
$169$	1.53698e8			0.188417
$170$		0			0
$171$	−4.75533e7	−	1.14792e8i	−0.0556156	−	0.134254i
$172$		−	8.74485e8i		−	0.999168i
$173$	−1.02319e9			−1.14228			−0.571141	−	0.820852i	$-0.693499\pi$
$173$	−1.02319e9			−1.14228			−0.571141	+	0.820852i	$0.693499\pi$
$174$	4.65620e8	−	6.96877e8i	0.507966	−	0.760255i
$175$		0			0
$176$		−	4.69904e8i		−	0.489732i
$177$	6.17170e8	−	9.23696e8i	0.628799	−	0.941100i
$178$		−	1.63021e9i		−	1.62391i
$179$			1.28895e9i			1.25552i	0.778408	+	0.627759i	$0.216028\pi$
$179$			1.28895e9i			1.25552i	−0.778408	+	0.627759i	$0.783972\pi$
$180$		0			0
$181$	4.71707e8			0.439499			0.219749	−	0.975556i	$-0.429476\pi$
$181$	4.71707e8			0.439499			0.219749	+	0.975556i	$0.429476\pi$
$182$	−1.01086e9			−0.921314
$183$	5.07550e7	+	3.39121e7i	0.0452558	+	0.0302378i
$184$	−8.44946e7			−0.0737153
$185$		0			0
$186$	−5.31435e8	−	3.55080e8i	−0.444016	−	0.296670i
$187$			5.21217e8i			0.426238i
$188$	−1.01210e9			−0.810198
$189$	−9.12161e8	−	1.81390e8i	−0.714866	−	0.142156i
$190$		0			0
$191$		−	1.61787e8i		−	0.121565i	−0.998151	−	0.0607827i	$-0.980640\pi$
$191$		−	1.61787e8i		−	0.121565i	0.998151	−	0.0607827i	$-0.0193597\pi$
$192$	−1.05825e9	−	7.07075e8i	−0.778725	−	0.520308i
$193$		−	1.58840e9i		−	1.14480i	−0.819974	−	0.572401i	$-0.806012\pi$
$193$		−	1.58840e9i		−	1.14480i	0.819974	−	0.572401i	$-0.193988\pi$
$194$		−	3.30623e9i		−	2.33413i
$195$		0			0
$196$	6.70171e8			0.454110
$197$	−5.37769e8			−0.357052			−0.178526	−	0.983935i	$-0.557133\pi$
$197$	−5.37769e8			−0.357052			−0.178526	+	0.983935i	$0.557133\pi$
$198$	−9.47046e8	+	3.92319e8i	−0.616184	+	0.255257i
$199$	−6.47586e8			−0.412938			−0.206469	−	0.978453i	$-0.566197\pi$
$199$	−6.47586e8			−0.412938			−0.206469	+	0.978453i	$0.566197\pi$
$200$		0			0
$201$	1.02100e8	−	1.52809e8i	0.0625521	−	0.0936194i
$202$		−	2.32457e9i		−	1.39617i
$203$	8.06570e8			0.474961
$204$	1.25092e9	+	8.35806e8i	0.722285	+	0.482597i
$205$		0			0
$206$		−	3.72812e9i		−	2.07024i
$207$	1.18133e9	+	2.85169e9i	0.643412	+	1.55318i
$208$		−	1.73729e9i		−	0.928152i
$209$		−	1.31799e8i		−	0.0690758i
$210$		0			0
$211$	5.81104e7			0.0293173			0.0146586	−	0.999893i	$-0.495334\pi$
$211$	5.81104e7			0.0293173			0.0146586	+	0.999893i	$0.495334\pi$
$212$	1.63724e9			0.810529
$213$	−7.65990e8	+	1.14643e9i	−0.372139	+	0.556966i
$214$	−5.06336e8			−0.241426
$215$		0			0
$216$	1.86157e7	−	9.36136e7i	0.00855195	−	0.0430055i
$217$		−	6.15086e8i		−	0.277394i
$218$	2.46893e9			1.09316
$219$	−1.24527e9	+	1.86376e9i	−0.541363	+	0.810238i
$220$		0			0
$221$			1.92700e9i			0.807815i
$222$	1.34885e9	−	2.01878e9i	0.555332	−	0.831144i
$223$			4.40200e9i			1.78004i	0.455920	+	0.890021i	$0.349310\pi$
$223$			4.40200e9i			1.78004i	−0.455920	+	0.890021i	$0.650690\pi$
$224$		−	2.57222e9i		−	1.02168i
$225$		0			0
$226$	−6.45992e9			−2.47624
$227$	3.53592e9			1.33168			0.665839	−	0.746095i	$-0.268074\pi$
$227$	3.53592e9			1.33168			0.665839	+	0.746095i	$0.268074\pi$
$228$	−3.16317e8	−	2.11348e8i	−0.117053	−	0.0782094i
$229$	1.86569e9			0.678420			0.339210	−	0.940711i	$-0.389840\pi$
$229$	1.86569e9			0.678420			0.339210	+	0.940711i	$0.389840\pi$
$230$		0			0
$231$	−8.20260e8	−	5.48059e8i	−0.288074	−	0.192477i
$232$			8.27770e7i			0.0285731i
$233$	2.72132e9			0.923328			0.461664	−	0.887055i	$-0.347253\pi$
$233$	2.72132e9			0.923328			0.461664	+	0.887055i	$0.347253\pi$
$234$	−3.50134e9	+	1.45045e9i	−1.16781	+	0.483769i
$235$		0			0
$236$		−	3.40129e9i		−	1.09647i
$237$	1.54777e9	+	1.03414e9i	0.490582	+	0.327784i
$238$			2.94235e9i			0.917037i
$239$			2.27461e9i			0.697132i	0.937284	+	0.348566i	$0.113331\pi$
$239$			2.27461e9i			0.697132i	−0.937284	+	0.348566i	$0.886669\pi$
$240$		0			0
$241$	−1.74667e9			−0.517778			−0.258889	−	0.965907i	$-0.583356\pi$
$241$	−1.74667e9			−0.517778			−0.258889	+	0.965907i	$0.583356\pi$
$242$	3.72500e9			1.08609
$243$	−3.41973e9	+	6.80540e8i	−0.980768	+	0.195177i
$244$	1.86893e8			0.0527272
$245$		0			0
$246$	−1.89469e9	+	2.83571e9i	−0.517365	+	0.774321i
$247$		−	4.87275e8i		−	0.130914i
$248$	6.31253e7			0.0166877
$249$	3.12471e9	+	2.08778e9i	0.812853	+	0.543110i
$250$		0			0
$251$			1.37549e9i			0.346547i	0.984874	+	0.173274i	$0.0554345\pi$
$251$			1.37549e9i			0.346547i	−0.984874	+	0.173274i	$0.944566\pi$
$252$	−2.63068e9	+	1.08977e9i	−0.652329	+	0.270230i
$253$			3.27417e9i			0.799132i
$254$		−	6.19605e9i		−	1.48861i
$255$		0			0
$256$	4.55069e9			1.05954
$257$	−7.93672e9			−1.81932			−0.909659	−	0.415356i	$-0.863657\pi$
$257$	−7.93672e9			−1.81932			−0.909659	+	0.415356i	$0.863657\pi$
$258$	3.56228e9	−	5.33154e9i	0.803988	−	1.20330i
$259$	2.33655e9			0.519249
$260$		0			0
$261$	2.79372e9	−	1.15731e9i	0.602034	−	0.249396i
$262$			6.49068e9i			1.37748i
$263$	−3.22555e8			−0.0674187			−0.0337093	−	0.999432i	$-0.510732\pi$
$263$	−3.22555e8			−0.0674187			−0.0337093	+	0.999432i	$0.510732\pi$
$264$	5.62464e7	−	8.41819e7i	0.0115792	−	0.0173302i
$265$		0			0
$266$		−	7.44025e8i		−	0.148614i
$267$	3.26769e9	−	4.89062e9i	0.642977	−	0.962319i
$268$		−	5.62685e8i		−	0.109075i
$269$		−	3.47314e9i		−	0.663304i	−0.943402	−	0.331652i	$-0.892394\pi$
$269$		−	3.47314e9i		−	0.663304i	0.943402	−	0.331652i	$-0.107606\pi$
$270$		0			0
$271$	−1.44216e9			−0.267385			−0.133693	−	0.991023i	$-0.542683\pi$
$271$	−1.44216e9			−0.267385			−0.133693	+	0.991023i	$0.542683\pi$
$272$	−5.05678e9			−0.923843
$273$	−3.03259e9	−	2.02624e9i	−0.545964	−	0.364787i
$274$	−4.65976e9			−0.826726
$275$		0			0
$276$	7.85800e9	+	5.25035e9i	1.35418	+	0.904798i
$277$		−	3.38046e9i		−	0.574192i	−0.957902	−	0.287096i	$-0.907310\pi$
$277$		−	3.38046e9i		−	0.574192i	0.957902	−	0.287096i	$-0.0926899\pi$
$278$	3.20289e9			0.536245
$279$	−8.82561e8	−	2.13048e9i	−0.145656	−	0.351609i
$280$		0			0
$281$			4.02262e9i			0.645184i	0.946538	+	0.322592i	$0.104554\pi$
$281$			4.02262e9i			0.645184i	−0.946538	+	0.322592i	$0.895446\pi$
$282$	−6.17053e9	−	4.12286e9i	−0.975722	−	0.651932i
$283$		−	1.04253e10i		−	1.62533i	−0.582730	−	0.812666i	$-0.698016\pi$
$283$		−	1.04253e10i		−	1.62533i	0.582730	−	0.812666i	$-0.301984\pi$
$284$			4.22146e9i			0.648917i
$285$		0			0
$286$	−4.02006e9			−0.600853
$287$	−3.28207e9			−0.483749
$288$	−3.69078e9	−	8.90943e9i	−0.536473	−	1.29503i
$289$	−1.36679e9			−0.195935
$290$		0			0
$291$	6.62720e9	−	9.91868e9i	0.924183	−	1.38319i
$292$			6.86285e9i			0.944002i
$293$	−1.03927e10			−1.41012			−0.705061	−	0.709146i	$-0.749081\pi$
$293$	−1.03927e10			−1.41012			−0.705061	+	0.709146i	$0.749081\pi$
$294$	4.08588e9	+	2.72999e9i	0.546885	+	0.365403i
$295$		0			0
$296$			2.39796e8i			0.0312374i
$297$	−3.62753e9	−	7.21360e8i	−0.466213	−	0.0927099i
$298$			1.83918e10i			2.33216i
$299$			1.21050e10i			1.51453i
$300$		0			0
$301$	6.17076e9			0.751749
$302$	9.51565e9			1.14396
$303$	4.65951e9	−	6.97372e9i	0.552803	−	0.827359i
$304$	1.27869e9			0.149717
$305$		0			0
$306$	4.22185e9	+	1.01914e10i	0.481524	+	1.16238i
$307$			2.99309e9i			0.336951i	0.985706	+	0.168476i	$0.0538844\pi$
$307$			2.99309e9i			0.336951i	−0.985706	+	0.168476i	$0.946116\pi$
$308$	−3.02042e9			−0.335632
$309$	7.47286e9	−	1.11843e10i	0.819696	−	1.22681i
$310$		0			0
$311$			6.44832e9i			0.689295i	0.938732	+	0.344647i	$0.112002\pi$
$311$			6.44832e9i			0.689295i	−0.938732	+	0.344647i	$0.887998\pi$
$312$	2.07949e8	−	3.11230e8i	0.0219452	−	0.0328445i
$313$			3.27737e7i			0.00341467i	0.999999	+	0.00170733i	$0.000543462\pi$
$313$			3.27737e7i			0.00341467i	−0.999999	+	0.00170733i	$0.999457\pi$
$314$			1.70586e10i			1.75479i
$315$		0			0
$316$	5.69928e9			0.571573
$317$	−1.17797e10			−1.16653			−0.583264	−	0.812282i	$-0.698225\pi$
$317$	−1.17797e10			−1.16653			−0.583264	+	0.812282i	$0.698225\pi$
$318$	9.98187e9	+	6.66942e9i	0.976120	+	0.652198i
$319$	3.20761e9			0.309755
$320$		0			0
$321$	−1.51901e9	−	1.01493e9i	−0.143067	−	0.0955907i
$322$			1.84832e10i			1.71931i
$323$	−1.41832e9			−0.130306
$324$	−7.54825e9	+	7.54931e9i	−0.684961	+	0.685058i
$325$		0			0
$326$			1.50002e10i			1.32808i
$327$	7.40680e9	+	4.94888e9i	0.647798	+	0.432829i
$328$		−	3.36833e8i		−	0.0291018i
$329$		−	7.14182e9i		−	0.609573i
$330$		0			0
$331$	−1.20100e10			−1.00053			−0.500265	−	0.865872i	$-0.666764\pi$
$331$	−1.20100e10			−1.00053			−0.500265	+	0.865872i	$0.666764\pi$
$332$	1.15060e10			0.947049
$333$	8.09311e9	−	3.35261e9i	0.658171	−	0.272651i
$334$	4.40705e9			0.354130
$335$		0			0
$336$	5.31720e9	−	7.95806e9i	0.417182	−	0.624381i
$337$		−	1.59214e10i		−	1.23441i	−0.786801	−	0.617207i	$-0.788264\pi$
$337$		−	1.59214e10i		−	1.23441i	0.786801	−	0.617207i	$-0.211736\pi$
$338$	3.45051e9			0.264372
$339$	−1.93798e10	−	1.29486e10i	−1.46740	−	0.980451i
$340$		0			0
$341$		−	2.44611e9i		−	0.180908i
$342$	−1.06757e9	−	2.57708e9i	−0.0780354	−	0.188375i
$343$			1.48174e10i			1.07052i
$344$			6.33295e8i			0.0452243i
$345$		0			0
$346$	−2.29706e10			−1.60276
$347$	−4.94792e9			−0.341275			−0.170638	−	0.985334i	$-0.554583\pi$
$347$	−4.94792e9			−0.341275			−0.170638	+	0.985334i	$0.554583\pi$
$348$	5.14361e9	−	7.69826e9i	0.350713	−	0.524899i
$349$	7.29567e9			0.491772			0.245886	−	0.969299i	$-0.420921\pi$
$349$	7.29567e9			0.491772			0.245886	+	0.969299i	$0.420921\pi$
$350$		0			0
$351$	−1.34114e10	−	2.66695e9i	−0.883578	−	0.175706i
$352$		−	1.02293e10i		−	0.666311i
$353$	6.93875e9			0.446871			0.223436	−	0.974719i	$-0.428273\pi$
$353$	6.93875e9			0.446871			0.223436	+	0.974719i	$0.428273\pi$
$354$	1.38554e10	−	2.07369e10i	0.882282	−	1.32048i
$355$		0			0
$356$		−	1.80086e10i		−	1.12119i
$357$	−5.89782e9	+	8.82706e9i	−0.363094	+	0.543429i
$358$			2.89368e10i			1.76164i
$359$		−	1.60096e10i		−	0.963838i	−0.876216	−	0.481919i	$-0.839940\pi$
$359$		−	1.60096e10i		−	0.963838i	0.876216	−	0.481919i	$-0.160060\pi$
$360$		0			0
$361$	−1.66249e10			−0.978883
$362$	1.05898e10			0.616670
$363$	1.11750e10	+	7.46660e9i	0.643607	+	0.430028i
$364$	−1.11668e10			−0.636098
$365$		0			0
$366$	1.13945e9	+	7.61325e8i	0.0634994	+	0.0424273i
$367$		−	1.36364e10i		−	0.751686i	−0.926683	−	0.375843i	$-0.877353\pi$
$367$		−	1.36364e10i		−	0.751686i	0.926683	−	0.375843i	$-0.122647\pi$
$368$	−3.17655e10			−1.73207
$369$	−1.13681e10	+	4.70930e9i	−0.613173	+	0.254010i
$370$		0			0
$371$			1.15531e10i			0.609821i
$372$	−5.87065e9	−	3.92249e9i	−0.306559	−	0.204829i
$373$		−	2.44062e10i		−	1.26085i	−0.776248	−	0.630427i	$-0.782880\pi$
$373$		−	2.44062e10i		−	1.26085i	0.776248	−	0.630427i	$-0.217120\pi$
$374$			1.17013e10i			0.598063i
$375$		0			0
$376$	7.32953e8			0.0366712
$377$	1.18589e10			0.587055
$378$	−2.04780e10	−	4.07219e9i	−1.00304	−	0.199463i
$379$	1.98392e10			0.961542			0.480771	−	0.876846i	$-0.340357\pi$
$379$	1.98392e10			0.961542			0.480771	+	0.876846i	$0.340357\pi$
$380$		0			0
$381$	1.24197e10	−	1.85881e10i	0.589403	−	0.882137i
$382$		−	3.63211e9i		−	0.170571i
$383$	1.51133e10			0.702366			0.351183	−	0.936307i	$-0.385780\pi$
$383$	1.51133e10			0.702366			0.351183	+	0.936307i	$0.385780\pi$
$384$	1.58467e9	+	1.05880e9i	0.0728811	+	0.0486957i
$385$		0			0
$386$		−	3.56594e10i		−	1.60630i
$387$	2.13737e10	−	8.85416e9i	0.952875	−	0.394733i
$388$		−	3.65232e10i		−	1.61154i
$389$			1.79991e10i			0.786056i	0.919527	+	0.393028i	$0.128572\pi$
$389$			1.79991e10i			0.786056i	−0.919527	+	0.393028i	$0.871428\pi$
$390$		0			0
$391$	3.52342e10			1.50750
$392$	−4.85332e8			−0.0205539
$393$	−1.30103e10	+	1.94720e10i	−0.545403	+	0.816284i
$394$	−1.20729e10			−0.500987
$395$		0			0
$396$	−1.04618e10	+	4.33386e9i	−0.425429	+	0.176236i
$397$			2.35673e10i			0.948739i	0.880326	+	0.474370i	$0.157324\pi$
$397$			2.35673e10i			0.948739i	−0.880326	+	0.474370i	$0.842676\pi$
$398$	−1.45383e10			−0.579403
$399$	1.49137e9	−	2.23207e9i	0.0588428	−	0.0880678i
$400$		0			0
$401$			1.37692e10i			0.532515i	0.963902	+	0.266257i	$0.0857871\pi$
$401$			1.37692e10i			0.532515i	−0.963902	+	0.266257i	$0.914213\pi$
$402$	2.29214e9	−	3.43056e9i	0.0877682	−	0.131359i
$403$		−	9.04353e9i		−	0.342861i
$404$		−	2.56791e10i		−	0.963950i
$405$		0			0
$406$	1.81075e10			0.666428
$407$	9.29209e9			0.338638
$408$	−9.05906e8	−	6.05284e8i	−0.0326921	−	0.0218433i
$409$	−3.58480e10			−1.28107			−0.640533	−	0.767931i	$-0.721286\pi$
$409$	−3.58480e10			−1.28107			−0.640533	+	0.767931i	$0.721286\pi$
$410$		0			0
$411$	−1.39793e10	−	9.34031e9i	−0.489912	−	0.327336i
$412$		−	4.11838e10i		−	1.42934i
$413$	2.40011e10			0.824955
$414$	2.65207e10	+	6.40203e10i	0.902785	+	2.17930i
$415$		0			0
$416$		−	3.78191e10i		−	1.26281i
$417$	9.60868e9	+	6.42007e9i	0.317775	+	0.212322i
$418$		−	2.95887e9i		−	0.0969217i
$419$		−	2.23996e10i		−	0.726750i	−0.931643	−	0.363375i	$-0.881624\pi$
$419$		−	2.23996e10i		−	0.726750i	0.931643	−	0.363375i	$-0.118376\pi$
$420$		0			0
$421$	−1.49535e10			−0.476008			−0.238004	−	0.971264i	$-0.576493\pi$
$421$	−1.49535e10			−0.476008			−0.238004	+	0.971264i	$0.576493\pi$
$422$	1.30457e9			0.0411357
$423$	−1.02475e10	−	2.47372e10i	−0.320078	−	0.772660i
$424$	−1.18567e9			−0.0366861
$425$		0			0
$426$	−1.71964e10	+	2.57373e10i	−0.522156	+	0.781491i
$427$			1.31880e9i			0.0396706i
$428$	−5.59339e9			−0.166686
$429$	−1.20602e10	−	8.05804e9i	−0.356061	−	0.237903i
$430$		0			0
$431$		−	6.40436e10i		−	1.85595i	−0.372640	−	0.927976i	$-0.621547\pi$
$431$		−	6.40436e10i		−	1.85595i	0.372640	−	0.927976i	$-0.378453\pi$
$432$	6.99854e9	−	3.51938e10i	0.200943	−	1.01049i
$433$		−	5.22954e9i		−	0.148769i	−0.997230	−	0.0743843i	$-0.976301\pi$
$433$		−	5.22954e9i		−	0.148769i	0.997230	−	0.0743843i	$-0.0236992\pi$
$434$		−	1.38087e10i		−	0.389218i
$435$		0			0
$436$	2.72738e10			0.754745
$437$	−8.90959e9			−0.244305
$438$	−2.79563e10	+	4.18412e10i	−0.759598	+	1.13686i
$439$	−4.34801e10			−1.17066			−0.585332	−	0.810793i	$-0.699036\pi$
$439$	−4.34801e10			−1.17066			−0.585332	+	0.810793i	$0.699036\pi$
$440$		0			0
$441$	6.78548e9	+	1.63800e10i	0.179402	+	0.433070i
$442$			4.32610e10i			1.13346i
$443$	3.78737e10			0.983383			0.491691	−	0.870770i	$-0.336379\pi$
$443$	3.78737e10			0.983383			0.491691	+	0.870770i	$0.336379\pi$
$444$	1.49005e10	−	2.23010e10i	0.383415	−	0.573843i
$445$		0			0
$446$			9.88246e10i			2.49761i
$447$	−3.68656e10	+	5.51754e10i	−0.923403	+	1.38202i
$448$		−	2.74973e10i		−	0.682620i
$449$		−	2.95505e10i		−	0.727076i	−0.931579	−	0.363538i	$-0.881569\pi$
$449$		−	2.95505e10i		−	0.727076i	0.931579	−	0.363538i	$-0.118431\pi$
$450$		0			0
$451$	−1.30523e10			−0.315486
$452$	−7.13614e10			−1.70966
$453$	2.85469e10	+	1.90737e10i	0.677902	+	0.452942i
$454$	7.93813e10			1.86851
$455$		0			0
$456$	2.29074e8	+	1.53057e8i	0.00529806	+	0.00353991i
$457$			2.02181e10i			0.463529i	0.972772	+	0.231764i	$0.0744498\pi$
$457$			2.02181e10i			0.463529i	−0.972772	+	0.231764i	$0.925550\pi$
$458$	4.18847e10			0.951905
$459$	−7.76277e9	+	3.90369e10i	−0.174890	+	0.879476i
$460$		0			0
$461$			7.01826e10i			1.55391i	0.629556	+	0.776955i	$0.283237\pi$
$461$			7.01826e10i			1.55391i	−0.629556	+	0.776955i	$0.716763\pi$
$462$	−1.84148e10	−	1.23039e10i	−0.404202	−	0.270069i
$463$			4.16009e9i			0.0905271i	0.998975	+	0.0452635i	$0.0144128\pi$
$463$			4.16009e9i			0.0905271i	−0.998975	+	0.0452635i	$0.985587\pi$
$464$			3.11198e10i			0.671374i
$465$		0			0
$466$	6.10935e10			1.29554
$467$	2.88138e10			0.605806			0.302903	−	0.953021i	$-0.402044\pi$
$467$	2.88138e10			0.605806			0.302903	+	0.953021i	$0.402044\pi$
$468$	−3.86786e10	+	1.60228e10i	−0.806282	+	0.334006i
$469$	3.97056e9			0.0820654
$470$		0			0
$471$	−3.41933e10	+	5.11758e10i	−0.694796	+	1.03988i
$472$			2.46319e9i			0.0496283i
$473$	2.45402e10			0.490268
$474$	3.47472e10	+	2.32165e10i	0.688346	+	0.459921i
$475$		0			0
$476$			3.25036e10i			0.633145i
$477$	1.65770e10	+	4.00165e10i	0.320209	+	0.772975i
$478$			5.10649e10i			0.978161i
$479$		−	4.47149e10i		−	0.849395i	−0.905335	−	0.424698i	$-0.860380\pi$
$479$		−	4.47149e10i		−	0.849395i	0.905335	−	0.424698i	$-0.139620\pi$
$480$		0			0
$481$	3.43539e10			0.641795
$482$	−3.92127e10			−0.726505
$483$	−3.70488e10	+	5.54496e10i	−0.680747	+	1.01885i
$484$	4.11493e10			0.749861
$485$		0			0
$486$	−7.67727e10	+	1.52781e10i	−1.37614	+	0.273857i
$487$		−	5.72836e10i		−	1.01839i	−0.860651	−	0.509195i	$-0.829943\pi$
$487$		−	5.72836e10i		−	1.01839i	0.860651	−	0.509195i	$-0.170057\pi$
$488$	−1.35347e8			−0.00238654
$489$	−3.00672e10	+	4.50005e10i	−0.525845	+	0.787012i
$490$		0			0
$491$		−	7.25262e10i		−	1.24787i	−0.781477	−	0.623934i	$-0.785533\pi$
$491$		−	7.25262e10i		−	1.24787i	0.781477	−	0.623934i	$-0.214467\pi$
$492$	−2.09302e10	+	3.13255e10i	−0.357202	+	0.534611i
$493$		−	3.45180e10i		−	0.584330i
$494$		−	1.09393e10i		−	0.183688i
$495$		0			0
$496$	2.37318e10			0.392106
$497$	−2.97885e10			−0.488228
$498$	7.01495e10	+	4.68706e10i	1.14053	+	0.762050i
$499$	2.64368e10			0.426389			0.213195	−	0.977010i	$-0.431613\pi$
$499$	2.64368e10			0.426389			0.213195	+	0.977010i	$0.431613\pi$
$500$		0			0
$501$	1.32212e10	+	8.83376e9i	0.209855	+	0.140215i
$502$			3.08797e10i			0.486248i
$503$	−7.52828e10			−1.17604			−0.588022	−	0.808845i	$-0.700093\pi$
$503$	−7.52828e10			−1.17604			−0.588022	+	0.808845i	$0.700093\pi$
$504$	1.90512e9	−	7.89205e8i	0.0295257	−	0.0122312i
$505$		0			0
$506$			7.35048e10i			1.12128i
$507$	1.03515e10	+	6.91640e9i	0.156665	+	0.104676i
$508$		−	6.84465e10i		−	1.02777i
$509$			6.45184e10i			0.961197i	0.876941	+	0.480599i	$0.159581\pi$
$509$			6.45184e10i			0.961197i	−0.876941	+	0.480599i	$0.840419\pi$
$510$		0			0
$511$	−4.84273e10			−0.710243
$512$	9.61394e10			1.39901
$513$	1.96295e9	−	9.87115e9i	0.0283426	−	0.142527i
$514$	−1.78179e11			−2.55272
$515$		0			0
$516$	3.93518e10	−	5.88964e10i	0.555094	−	0.830788i
$517$		−	2.84019e10i		−	0.397544i
$518$	5.24554e10			0.728569
$519$	−6.89119e10	−	4.60437e10i	−0.949783	−	0.634601i
$520$		0			0
$521$		−	7.65146e10i		−	1.03847i	−0.854632	−	0.519235i	$-0.826217\pi$
$521$		−	7.65146e10i		−	1.03847i	0.854632	−	0.519235i	$-0.173783\pi$
$522$	6.27189e10	−	2.59816e10i	0.844727	−	0.349932i
$523$		−	8.46771e10i		−	1.13177i	−0.824483	−	0.565886i	$-0.808534\pi$
$523$		−	8.46771e10i		−	1.13177i	0.824483	−	0.565886i	$-0.191466\pi$
$524$			7.17012e10i			0.951046i
$525$		0			0
$526$	−7.24133e9			−0.0945966
$527$	−2.63232e10			−0.341269
$528$	2.11457e10	−	3.16480e10i	0.272073	−	0.407202i
$529$	1.43023e11			1.82634
$530$		0			0
$531$	8.31326e10	−	3.44381e10i	1.04567	−	0.433173i
$532$		−	8.21909e9i		−	0.102607i
$533$	−4.82558e10			−0.597917
$534$	7.33594e10	−	1.09794e11i	0.902175	−	1.35025i
$535$		0			0
$536$			4.07492e8i			0.00493696i
$537$	−5.80026e10	+	8.68104e10i	−0.697510	+	1.04394i
$538$		−	7.79717e10i		−	0.930696i
$539$			1.88066e10i			0.222821i
$540$		0			0
$541$	1.43470e11			1.67483			0.837415	−	0.546568i	$-0.184066\pi$
$541$	1.43470e11			1.67483			0.837415	+	0.546568i	$0.184066\pi$
$542$	−3.23765e10			−0.375174
$543$	3.17694e10	+	2.12268e10i	0.365434	+	0.244166i
$544$	−1.10081e11			−1.25695
$545$		0			0
$546$	−6.80816e10	−	4.54889e10i	−0.766053	−	0.511841i
$547$		−	1.64171e11i		−	1.83378i	−0.399134	−	0.916892i	$-0.630689\pi$
$547$		−	1.64171e11i		−	1.83378i	0.399134	−	0.916892i	$-0.369311\pi$
$548$	−5.14755e10			−0.570792
$549$	1.89229e9	+	4.56795e9i	0.0208305	+	0.0502842i
$550$		0			0
$551$			8.72847e9i			0.0946961i
$552$	−5.69070e9	−	3.80226e9i	−0.0612928	−	0.0409529i
$553$			4.02167e10i			0.430037i
$554$		−	7.58912e10i		−	0.805661i
$555$		0			0
$556$	3.53817e10			0.370237
$557$	1.54420e11			1.60429			0.802145	−	0.597130i	$-0.203692\pi$
$557$	1.54420e11			1.60429			0.802145	+	0.597130i	$0.203692\pi$
$558$	−1.98135e10	−	4.78291e10i	−0.204373	−	0.493351i
$559$	9.07278e10			0.929166
$560$		0			0
$561$	−2.34547e10	+	3.51039e10i	−0.236799	+	0.354408i
$562$			9.03075e10i			0.905271i
$563$	−1.54622e11			−1.53900			−0.769500	−	0.638646i	$-0.779495\pi$
$563$	−1.54622e11			−1.53900			−0.769500	+	0.638646i	$0.779495\pi$
$564$	−6.81646e10	−	4.55444e10i	−0.673663	−	0.450110i
$565$		0			0
$566$		−	2.34047e11i		−	2.28054i
$567$	−5.32714e10	−	5.32638e10i	−0.515420	−	0.515348i
$568$		−	3.05714e9i		−	0.0293712i
$569$			1.15380e11i			1.10073i	0.834925	+	0.550364i	$0.185511\pi$
$569$			1.15380e11i			1.10073i	−0.834925	+	0.550364i	$0.814489\pi$
$570$		0			0
$571$	−1.63410e11			−1.53722			−0.768608	−	0.639720i	$-0.779050\pi$
$571$	−1.63410e11			−1.53722			−0.768608	+	0.639720i	$0.779050\pi$
$572$	−4.44087e10			−0.414844
$573$	7.28041e9	−	1.08963e10i	0.0675363	−	0.101079i
$574$	−7.36823e10			−0.678759
$575$		0			0
$576$	−3.94548e10	−	9.52427e10i	−0.358434	−	0.865250i
$577$		−	7.42282e10i		−	0.669678i	−0.942275	−	0.334839i	$-0.891318\pi$
$577$		−	7.42282e10i		−	0.669678i	0.942275	−	0.334839i	$-0.108682\pi$
$578$	−3.06844e10			−0.274920
$579$	7.14779e10	−	1.06978e11i	0.636001	−	0.951879i
$580$		0			0
$581$			8.11916e10i			0.712535i
$582$	1.48780e11	−	2.22674e11i	1.29674	−	1.94078i
$583$			4.59449e10i			0.397707i
$584$		−	4.97002e9i		−	0.0427274i
$585$		0			0
$586$	−2.33315e11			−1.97857
$587$	−6.72877e10			−0.566739			−0.283369	−	0.959011i	$-0.591452\pi$
$587$	−6.72877e10			−0.566739			−0.283369	+	0.959011i	$0.591452\pi$
$588$	4.51359e10	+	3.01577e10i	0.377583	+	0.252283i
$589$	6.65629e9			0.0553059
$590$		0			0
$591$	−3.62187e10	−	2.41996e10i	−0.296881	−	0.198362i
$592$			9.01507e10i			0.733977i
$593$	2.36444e10			0.191210			0.0956048	−	0.995419i	$-0.469521\pi$
$593$	2.36444e10			0.191210			0.0956048	+	0.995419i	$0.469521\pi$
$594$	−8.14378e10	−	1.61945e10i	−0.654154	−	0.130083i
$595$		0			0
$596$			2.03170e11i			1.61018i
$597$	−4.36148e10	−	2.91414e10i	−0.343350	−	0.229410i
$598$			2.71756e11i			2.12507i
$599$		−	3.03370e10i		−	0.235649i	−0.993034	−	0.117825i	$-0.962408\pi$
$599$		−	3.03370e10i		−	0.235649i	0.993034	−	0.117825i	$-0.0375921\pi$
$600$		0			0
$601$	3.37911e10			0.259003			0.129501	−	0.991579i	$-0.458662\pi$
$601$	3.37911e10			0.259003			0.129501	+	0.991579i	$0.458662\pi$
$602$	1.38533e11			1.05480
$603$	1.37528e10	−	5.69718e9i	0.104022	−	0.0430914i
$604$	1.05117e11			0.789818
$605$		0			0
$606$	1.04606e11	−	1.56560e11i	0.775649	−	1.16089i
$607$			3.82366e10i			0.281660i	0.990034	+	0.140830i	$0.0449771\pi$
$607$			3.82366e10i			0.281660i	−0.990034	+	0.140830i	$0.955023\pi$
$608$	2.78359e10			0.203700
$609$	5.43224e10	+	3.62957e10i	0.394920	+	0.263867i
$610$		0			0
$611$		−	1.05005e11i		−	0.753435i
$612$	4.66380e10	+	1.12583e11i	0.332456	+	0.802539i
$613$			1.08066e11i			0.765330i	0.923887	+	0.382665i	$0.124994\pi$
$613$			1.08066e11i			0.765330i	−0.923887	+	0.382665i	$0.875006\pi$
$614$			6.71948e10i			0.472783i
$615$		0			0
$616$	2.18736e9			0.0151914
$617$	−4.72538e9			−0.0326059			−0.0163029	−	0.999867i	$-0.505190\pi$
$617$	−4.72538e9			−0.0326059			−0.0163029	+	0.999867i	$0.505190\pi$
$618$	1.67765e11	−	2.51088e11i	1.15013	−	1.72136i
$619$	−2.29845e10			−0.156557			−0.0782786	−	0.996932i	$-0.524942\pi$
$619$	−2.29845e10			−0.156557			−0.0782786	+	0.996932i	$0.524942\pi$
$620$		0			0
$621$	−4.87639e10	+	2.45221e11i	−0.327893	+	1.64889i
$622$			1.44764e11i			0.967165i
$623$	1.27077e11			0.843555
$624$	7.81780e10	−	1.17006e11i	0.515640	−	0.771739i
$625$		0			0
$626$			7.35768e8i			0.00479119i
$627$	5.93094e9	−	8.87662e9i	0.0383754	−	0.0574351i
$628$			1.88443e11i			1.21155i
$629$		−	9.99949e10i		−	0.638815i
$630$		0			0
$631$	−1.01892e11			−0.642722			−0.321361	−	0.946957i	$-0.604140\pi$
$631$	−1.01892e11			−0.642722			−0.321361	+	0.946957i	$0.604140\pi$
$632$	−4.12737e9			−0.0258705
$633$	3.91372e9	+	2.61497e9i	0.0243767	+	0.0162874i
$634$	−2.64453e11			−1.63678
$635$		0			0
$636$	1.10268e11	+	7.36757e10i	0.673938	+	0.450294i
$637$			6.95302e10i			0.422295i
$638$	7.20106e10			0.434624
$639$	−1.03179e11	+	4.27422e10i	−0.618851	+	0.256362i
$640$		0			0
$641$			1.17803e11i			0.697791i	0.937162	+	0.348896i	$0.113443\pi$
$641$			1.17803e11i			0.697791i	−0.937162	+	0.348896i	$0.886557\pi$
$642$	−3.41016e10	−	2.27851e10i	−0.200740	−	0.134125i
$643$		−	2.62680e10i		−	0.153668i	−0.997044	−	0.0768339i	$-0.975519\pi$
$643$		−	2.62680e10i		−	0.153668i	0.997044	−	0.0768339i	$-0.0244811\pi$
$644$			2.04180e11i			1.18705i
$645$		0			0
$646$	−3.18413e10			−0.182836
$647$	3.10527e11			1.77208			0.886038	−	0.463612i	$-0.153447\pi$
$647$	3.10527e11			1.77208			0.886038	+	0.463612i	$0.153447\pi$
$648$	5.46638e9	−	5.46715e9i	0.0310027	−	0.0310071i
$649$	9.54486e10			0.538010
$650$		0			0
$651$	2.76789e10	−	4.14260e10i	0.154108	−	0.230648i
$652$			1.65704e11i			0.916942i
$653$	−3.48345e10			−0.191583			−0.0957914	−	0.995401i	$-0.530538\pi$
$653$	−3.48345e10			−0.191583			−0.0957914	+	0.995401i	$0.530538\pi$
$654$	1.66282e11	+	1.11102e11i	0.908940	+	0.607311i
$655$		0			0
$656$		−	1.26632e11i		−	0.683796i
$657$	−1.67738e11	+	6.94863e10i	−0.900265	+	0.372939i
$658$		−	1.60333e11i		−	0.855304i
$659$		−	3.77120e10i		−	0.199957i	−0.994990	−	0.0999787i	$-0.968123\pi$
$659$		−	3.77120e10i		−	0.199957i	0.994990	−	0.0999787i	$-0.0318775\pi$
$660$		0			0
$661$	−1.39619e11			−0.731372			−0.365686	−	0.930738i	$-0.619166\pi$
$661$	−1.39619e11			−0.731372			−0.365686	+	0.930738i	$0.619166\pi$
$662$	−2.69623e11			−1.40386
$663$	−8.67149e10	+	1.29783e11i	−0.448786	+	0.671682i
$664$	−8.33256e9			−0.0428653
$665$		0			0
$666$	1.81690e11	−	7.52659e10i	0.923494	−	0.382562i
$667$		−	2.16834e11i		−	1.09553i
$668$	4.86838e10			0.244500
$669$	−1.98090e11	+	2.96474e11i	−0.988912	+	1.48007i
$670$		0			0
$671$			5.24468e9i			0.0258719i
$672$	1.15750e11	−	1.73239e11i	0.567603	−	0.849510i
$673$		−	1.29783e11i		−	0.632642i	−0.948652	−	0.316321i	$-0.897552\pi$
$673$		−	1.29783e11i		−	0.632642i	0.948652	−	0.316321i	$-0.102448\pi$
$674$		−	3.57434e11i		−	1.73203i
$675$		0			0
$676$	3.81171e10			0.182529
$677$	−3.40648e11			−1.62163			−0.810813	−	0.585306i	$-0.800975\pi$
$677$	−3.40648e11			−1.62163			−0.810813	+	0.585306i	$0.800975\pi$
$678$	−4.35075e11	−	2.90696e11i	−2.05895	−	1.37569i
$679$	2.57724e11			1.21248
$680$		0			0
$681$	2.38144e11	+	1.59117e11i	1.10726	+	0.739821i
$682$		−	5.49149e10i		−	0.253836i
$683$	−1.02876e11			−0.472750			−0.236375	−	0.971662i	$-0.575959\pi$
$683$	−1.02876e11			−0.472750			−0.236375	+	0.971662i	$0.575959\pi$
$684$	−1.17932e10	−	2.84685e10i	−0.0538776	−	0.130059i
$685$		0			0
$686$			3.32650e11i			1.50208i
$687$	1.25654e11	+	8.39562e10i	0.564092	+	0.376900i
$688$			2.38086e11i			1.06262i
$689$			1.69863e11i			0.753742i
$690$		0			0
$691$	3.58259e11			1.57140			0.785698	−	0.618611i	$-0.212304\pi$
$691$	3.58259e11			1.57140			0.785698	+	0.618611i	$0.212304\pi$
$692$	−2.53752e11			−1.10659
$693$	−3.05817e10	−	7.38234e10i	−0.132595	−	0.320082i
$694$	−1.11081e11			−0.478851
$695$		0			0
$696$	−3.72496e9	+	5.57501e9i	−0.0158740	+	0.0237580i
$697$			1.40459e11i			0.595141i
$698$	1.63787e11			0.690016
$699$	1.83280e11	+	1.22459e11i	0.767728	+	0.512960i
$700$		0			0
$701$		−	2.49323e11i		−	1.03250i	−0.856438	−	0.516250i	$-0.827327\pi$
$701$		−	2.49323e11i		−	1.03250i	0.856438	−	0.516250i	$-0.172673\pi$
$702$	−3.01085e11	−	5.98729e10i	−1.23977	−	0.246537i
$703$			2.52854e10i			0.103526i
$704$		−	1.09353e11i		−	0.445183i
$705$		0			0
$706$	1.55775e11			0.627015
$707$	1.81203e11			0.725251
$708$	1.53058e11	−	2.29077e11i	0.609149	−	0.911691i
$709$	3.88874e11			1.53895			0.769474	−	0.638678i	$-0.220518\pi$
$709$	3.88874e11			1.53895			0.769474	+	0.638678i	$0.220518\pi$
$710$		0			0
$711$	5.77052e10	+	1.39299e11i	0.225807	+	0.545091i
$712$			1.30417e10i			0.0507473i
$713$	−1.65357e11			−0.639829
$714$	−1.32406e11	+	1.98167e11i	−0.509465	+	0.762497i
$715$		0			0
$716$			3.19659e11i			1.21628i
$717$	−1.02357e11	+	1.53195e11i	−0.387296	+	0.579651i
$718$		−	3.59416e11i		−	1.35238i
$719$		−	3.35735e10i		−	0.125626i	−0.998025	−	0.0628132i	$-0.979993\pi$
$719$		−	3.35735e10i		−	0.125626i	0.998025	−	0.0628132i	$-0.0200072\pi$
$720$		0			0
$721$	2.90611e11			1.07540
$722$	−3.73228e11			−1.37349
$723$	−1.17638e11	−	7.86002e10i	−0.430521	−	0.287654i
$724$	1.16983e11			0.425764
$725$		0			0
$726$	2.50878e11	+	1.67625e11i	0.903058	+	0.603381i
$727$		−	2.53113e11i		−	0.906102i	−0.891485	−	0.453051i	$-0.850336\pi$
$727$		−	2.53113e11i		−	0.906102i	0.891485	−	0.453051i	$-0.149664\pi$
$728$	8.08692e9			0.0287911
$729$	−2.60942e11	−	1.08053e11i	−0.923920	−	0.382586i
$730$		0			0
$731$		−	2.64084e11i		−	0.924853i
$732$	1.25872e10	+	8.41020e9i	0.0438416	+	0.0292929i
$733$			2.07602e11i			0.719144i	0.933117	+	0.359572i	$0.117077\pi$
$733$			2.07602e11i			0.719144i	−0.933117	+	0.359572i	$0.882923\pi$
$734$		−	3.06137e11i		−	1.05471i
$735$		0			0
$736$	−6.91504e11			−2.35659
$737$	1.57903e10			0.0535205
$738$	−2.55214e11	+	1.05724e11i	−0.860357	+	0.356407i
$739$	−4.15014e11			−1.39151			−0.695753	−	0.718281i	$-0.744929\pi$
$739$	−4.15014e11			−1.39151			−0.695753	+	0.718281i	$0.744929\pi$
$740$		0			0
$741$	2.19274e10	−	3.28179e10i	0.0727300	−	0.108852i
$742$			2.59366e11i			0.855653i
$743$	3.30690e11			1.08509			0.542545	−	0.840027i	$-0.317461\pi$
$743$	3.30690e11			1.08509			0.542545	+	0.840027i	$0.317461\pi$
$744$	4.25148e9	+	2.84064e9i	0.0138755	+	0.00927095i
$745$		0			0
$746$		−	5.47918e11i		−	1.76913i
$747$	1.16498e11	+	2.81224e11i	0.374143	+	0.903170i
$748$			1.29262e11i			0.412918i
$749$		−	3.94695e10i		−	0.125411i
$750$		0			0
$751$	−4.77978e11			−1.50262			−0.751308	−	0.659952i	$-0.770577\pi$
$751$	−4.77978e11			−1.50262			−0.751308	+	0.659952i	$0.770577\pi$
$752$	2.75552e11			0.861652
$753$	−6.18970e10	+	9.26390e10i	−0.192526	+	0.288147i
$754$	2.66231e11			0.823709
$755$		0			0
$756$	−2.26216e11	−	4.49847e10i	−0.692526	−	0.137714i
$757$			8.95066e10i			0.272566i	0.990670	+	0.136283i	$0.0435156\pi$
$757$			8.95066e10i			0.272566i	−0.990670	+	0.136283i	$0.956484\pi$
$758$	4.45390e11			1.34916
$759$	−1.47337e11	+	2.20514e11i	−0.443962	+	0.664462i
$760$		0			0
$761$			5.71473e11i			1.70395i	0.523581	+	0.851976i	$0.324596\pi$
$761$			5.71473e11i			1.70395i	−0.523581	+	0.851976i	$0.675404\pi$
$762$	2.78822e11	−	4.17303e11i	0.827004	−	1.23775i
$763$			1.92456e11i			0.567851i
$764$		−	4.01231e10i		−	0.117766i
$765$		0			0
$766$	3.39292e11			0.985504
$767$	3.52884e11			1.01965
$768$	3.06488e11	+	2.04781e11i	0.880986	+	0.588634i
$769$	−2.56194e11			−0.732596			−0.366298	−	0.930498i	$-0.619375\pi$
$769$	−2.56194e11			−0.732596			−0.366298	+	0.930498i	$0.619375\pi$
$770$		0			0
$771$	−5.34537e11	−	3.57152e11i	−1.51273	−	1.01073i
$772$		−	3.93923e11i		−	1.10903i
$773$	1.00523e11			0.281543			0.140772	−	0.990042i	$-0.455042\pi$

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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+22.4499 q^{2} +(67.3498 + 45.0000i) q^{3} +248.000 q^{4} +(1512.00 + 1010.25i) q^{6} +1750.00i q^{7} -179.600 q^{8} +(2511.00 + 6061.48i) q^{9} +O(q^{10})\)	\(q+22.4499 q^{2} +(67.3498 + 45.0000i) q^{3} +248.000 q^{4} +(1512.00 + 1010.25i) q^{6} +1750.00i q^{7} -179.600 q^{8} +(2511.00 + 6061.48i) q^{9} +6959.48i q^{11} +(16702.8 + 11160.0i) q^{12} +25730.0i q^{13} +39287.4i q^{14} -67520.0 q^{16} +74893.0 q^{17} +(56371.8 + 136080. i) q^{18} -18938.0 q^{19} +(-78750.0 + 117862. i) q^{21} +156240. i q^{22} +470461. q^{23} +(-12096.0 - 8081.98i) q^{24} +577637. i q^{26} +(-103651. + 521235. i) q^{27} +434000. i q^{28} -460897. i q^{29} -351478. q^{31} -1.46984e6 q^{32} +(-313177. + 468720. i) q^{33} +1.68134e6 q^{34} +(622728. + 1.50325e6i) q^{36} -1.33517e6i q^{37} -425157. q^{38} +(-1.15785e6 + 1.73291e6i) q^{39} +1.87547e6i q^{41} +(-1.76793e6 + 2.64600e6i) q^{42} -3.52615e6i q^{43} +1.72595e6i q^{44} +1.05618e7 q^{46} -4.08104e6 q^{47} +(-4.54746e6 - 3.03840e6i) q^{48} +2.70230e6 q^{49} +(5.04403e6 + 3.37019e6i) q^{51} +6.38104e6i q^{52} +6.60177e6 q^{53} +(-2.32697e6 + 1.17017e7i) q^{54} -314299. i q^{56} +(-1.27547e6 - 852210. i) q^{57} -1.03471e7i q^{58} -1.37149e7i q^{59} +753602. q^{61} -7.89066e6 q^{62} +(-1.06076e7 + 4.39425e6i) q^{63} -1.57128e7 q^{64} +(-7.03080e6 + 1.05227e7i) q^{66} -2.26889e6i q^{67} +1.85735e7 q^{68} +(3.16855e7 + 2.11707e7i) q^{69} +1.70220e7i q^{71} +(-450974. - 1.08864e6i) q^{72} +2.76728e7i q^{73} -2.99745e7i q^{74} -4.69662e6 q^{76} -1.21791e7 q^{77} +(-2.59937e7 + 3.89038e7i) q^{78} +2.29810e7 q^{79} +(-3.04365e7 + 3.04408e7i) q^{81} +4.21042e7i q^{82} +4.63952e7 q^{83} +(-1.95300e7 + 2.92298e7i) q^{84} -7.91619e7i q^{86} +(2.07404e7 - 3.10414e7i) q^{87} -1.24992e6i q^{88} -7.26152e7i q^{89} -4.50275e7 q^{91} +1.16674e8 q^{92} +(-2.36720e7 - 1.58165e7i) q^{93} -9.16191e7 q^{94} +(-9.89937e7 - 6.61429e7i) q^{96} -1.47271e8i q^{97} +6.06665e7 q^{98} +(-4.21848e7 + 1.74753e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9}+O(q^{10}) \) 4 * q + 992 * q^4 + 6048 * q^6 + 10044 * q^9	\( 4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9} - 270080 q^{16} - 75752 q^{19} - 315000 q^{21} - 48384 q^{24} - 1405912 q^{31} + 6725376 q^{34} + 2490912 q^{36} - 4631400 q^{39} + 42247296 q^{46} + 10809204 q^{49} + 20176128 q^{51} - 9307872 q^{54} + 3014408 q^{61} - 62851072 q^{64} - 28123200 q^{66} + 126741888 q^{69} - 18786496 q^{76} + 91923928 q^{79} - 121745916 q^{81} - 78120000 q^{84} - 180110000 q^{91} - 366476544 q^{94} - 395974656 q^{96} - 168739200 q^{99}+O(q^{100}) \) 4 * q + 992 * q^4 + 6048 * q^6 + 10044 * q^9 - 270080 * q^16 - 75752 * q^19 - 315000 * q^21 - 48384 * q^24 - 1405912 * q^31 + 6725376 * q^34 + 2490912 * q^36 - 4631400 * q^39 + 42247296 * q^46 + 10809204 * q^49 + 20176128 * q^51 - 9307872 * q^54 + 3014408 * q^61 - 62851072 * q^64 - 28123200 * q^66 + 126741888 * q^69 - 18786496 * q^76 + 91923928 * q^79 - 121745916 * q^81 - 78120000 * q^84 - 180110000 * q^91 - 366476544 * q^94 - 395974656 * q^96 - 168739200 * q^99

Introduction

L-functions

Modular forms

Varieties

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