LMFDB - Embedded newform 90.4.c.b.19.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,4,Mod(19,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, 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("90.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	90.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.31017190052$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	90.19
Dual form			90.4.c.b.19.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +(5.00000 - 10.0000i) q^{5} -26.0000i q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+2.00000i q^{2} -4.00000 q^{4} +(5.00000 - 10.0000i) q^{5} -26.0000i q^{7} -8.00000i q^{8} +(20.0000 + 10.0000i) q^{10} +28.0000 q^{11} +12.0000i q^{13} +52.0000 q^{14} +16.0000 q^{16} -64.0000i q^{17} +60.0000 q^{19} +(-20.0000 + 40.0000i) q^{20} +56.0000i q^{22} +58.0000i q^{23} +(-75.0000 - 100.000i) q^{25} -24.0000 q^{26} +104.000i q^{28} +90.0000 q^{29} -128.000 q^{31} +32.0000i q^{32} +128.000 q^{34} +(-260.000 - 130.000i) q^{35} -236.000i q^{37} +120.000i q^{38} +(-80.0000 - 40.0000i) q^{40} -242.000 q^{41} +362.000i q^{43} -112.000 q^{44} -116.000 q^{46} +226.000i q^{47} -333.000 q^{49} +(200.000 - 150.000i) q^{50} -48.0000i q^{52} +108.000i q^{53} +(140.000 - 280.000i) q^{55} -208.000 q^{56} +180.000i q^{58} -20.0000 q^{59} +542.000 q^{61} -256.000i q^{62} -64.0000 q^{64} +(120.000 + 60.0000i) q^{65} +434.000i q^{67} +256.000i q^{68} +(260.000 - 520.000i) q^{70} +1128.00 q^{71} +632.000i q^{73} +472.000 q^{74} -240.000 q^{76} -728.000i q^{77} +720.000 q^{79} +(80.0000 - 160.000i) q^{80} -484.000i q^{82} +478.000i q^{83} +(-640.000 - 320.000i) q^{85} -724.000 q^{86} -224.000i q^{88} -490.000 q^{89} +312.000 q^{91} -232.000i q^{92} -452.000 q^{94} +(300.000 - 600.000i) q^{95} -1456.00i q^{97} -666.000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 10 q^{5}+O(q^{10}) $ 2 * q - 8 * q^4 + 10 * q^5	$ 2 q - 8 q^{4} + 10 q^{5} + 40 q^{10} + 56 q^{11} + 104 q^{14} + 32 q^{16} + 120 q^{19} - 40 q^{20} - 150 q^{25} - 48 q^{26} + 180 q^{29} - 256 q^{31} + 256 q^{34} - 520 q^{35} - 160 q^{40} - 484 q^{41} - 224 q^{44} - 232 q^{46} - 666 q^{49} + 400 q^{50} + 280 q^{55} - 416 q^{56} - 40 q^{59} + 1084 q^{61} - 128 q^{64} + 240 q^{65} + 520 q^{70} + 2256 q^{71} + 944 q^{74} - 480 q^{76} + 1440 q^{79} + 160 q^{80} - 1280 q^{85} - 1448 q^{86} - 980 q^{89} + 624 q^{91} - 904 q^{94} + 600 q^{95}+O(q^{100}) $ 2 * q - 8 * q^4 + 10 * q^5 + 40 * q^10 + 56 * q^11 + 104 * q^14 + 32 * q^16 + 120 * q^19 - 40 * q^20 - 150 * q^25 - 48 * q^26 + 180 * q^29 - 256 * q^31 + 256 * q^34 - 520 * q^35 - 160 * q^40 - 484 * q^41 - 224 * q^44 - 232 * q^46 - 666 * q^49 + 400 * q^50 + 280 * q^55 - 416 * q^56 - 40 * q^59 + 1084 * q^61 - 128 * q^64 + 240 * q^65 + 520 * q^70 + 2256 * q^71 + 944 * q^74 - 480 * q^76 + 1440 * q^79 + 160 * q^80 - 1280 * q^85 - 1448 * q^86 - 980 * q^89 + 624 * q^91 - 904 * q^94 + 600 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$			2.00000i			0.707107i
$2$			2.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−4.00000			−0.500000
$5$	5.00000	−	10.0000i	0.447214	−	0.894427i
$5$	5.00000	−	10.0000i	0.447214	−	0.894427i
$6$		0			0
$7$		−	26.0000i		−	1.40387i	−0.712242	−	0.701934i	$-0.752320\pi$
$7$		−	26.0000i		−	1.40387i	0.712242	−	0.701934i	$-0.247680\pi$
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	20.0000	+	10.0000i	0.632456	+	0.316228i
$11$	28.0000			0.767483			0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000			0.767483			0.383742	+	0.923440i	$0.374635\pi$
$12$		0			0
$13$			12.0000i			0.256015i	0.991773	+	0.128008i	$0.0408582\pi$
$13$			12.0000i			0.256015i	−0.991773	+	0.128008i	$0.959142\pi$
$14$	52.0000			0.992685
$15$		0			0
$16$	16.0000			0.250000
$17$		−	64.0000i		−	0.913075i	−0.889704	−	0.456538i	$-0.849089\pi$
$17$		−	64.0000i		−	0.913075i	0.889704	−	0.456538i	$-0.150911\pi$
$18$		0			0
$19$	60.0000			0.724471			0.362235	−	0.932087i	$-0.382014\pi$
$19$	60.0000			0.724471			0.362235	+	0.932087i	$0.382014\pi$
$20$	−20.0000	+	40.0000i	−0.223607	+	0.447214i
$21$		0			0
$22$			56.0000i			0.542693i
$23$			58.0000i			0.525819i	0.964821	+	0.262909i	$0.0846821\pi$
$23$			58.0000i			0.525819i	−0.964821	+	0.262909i	$0.915318\pi$
$24$		0			0
$25$	−75.0000	−	100.000i	−0.600000	−	0.800000i
$26$	−24.0000			−0.181030
$27$		0			0
$28$			104.000i			0.701934i
$29$	90.0000			0.576296			0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000			0.576296			0.288148	+	0.957586i	$0.406961\pi$
$30$		0			0
$31$	−128.000			−0.741596			−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000			−0.741596			−0.370798	+	0.928714i	$0.620916\pi$
$32$			32.0000i			0.176777i
$33$		0			0
$34$	128.000			0.645642
$35$	−260.000	−	130.000i	−1.25566	−	0.627829i
$36$		0			0
$37$		−	236.000i		−	1.04860i	−0.851534	−	0.524299i	$-0.824327\pi$
$37$		−	236.000i		−	1.04860i	0.851534	−	0.524299i	$-0.175673\pi$
$38$			120.000i			0.512278i
$39$		0			0
$40$	−80.0000	−	40.0000i	−0.316228	−	0.158114i
$41$	−242.000			−0.921806			−0.460903	−	0.887450i	$-0.652474\pi$
$41$	−242.000			−0.921806			−0.460903	+	0.887450i	$0.652474\pi$
$42$		0			0
$43$			362.000i			1.28383i	0.766778	+	0.641913i	$0.221859\pi$
$43$			362.000i			1.28383i	−0.766778	+	0.641913i	$0.778141\pi$
$44$	−112.000			−0.383742
$45$		0			0
$46$	−116.000			−0.371810
$47$			226.000i			0.701393i	0.936489	+	0.350697i	$0.114055\pi$
$47$			226.000i			0.701393i	−0.936489	+	0.350697i	$0.885945\pi$
$48$		0			0
$49$	−333.000			−0.970845
$50$	200.000	−	150.000i	0.565685	−	0.424264i
$51$		0			0
$52$		−	48.0000i		−	0.128008i
$53$			108.000i			0.279905i	0.990158	+	0.139952i	$0.0446949\pi$
$53$			108.000i			0.279905i	−0.990158	+	0.139952i	$0.955305\pi$
$54$		0			0
$55$	140.000	−	280.000i	0.343229	−	0.686458i
$56$	−208.000			−0.496342
$57$		0			0
$58$			180.000i			0.407503i
$59$	−20.0000			−0.0441318			−0.0220659	−	0.999757i	$-0.507024\pi$
$59$	−20.0000			−0.0441318			−0.0220659	+	0.999757i	$0.507024\pi$
$60$		0			0
$61$	542.000			1.13764			0.568820	−	0.822462i	$-0.307400\pi$
$61$	542.000			1.13764			0.568820	+	0.822462i	$0.307400\pi$
$62$		−	256.000i		−	0.524388i
$63$		0			0
$64$	−64.0000			−0.125000
$65$	120.000	+	60.0000i	0.228987	+	0.114494i
$66$		0			0
$67$			434.000i			0.791366i	0.918387	+	0.395683i	$0.129492\pi$
$67$			434.000i			0.791366i	−0.918387	+	0.395683i	$0.870508\pi$
$68$			256.000i			0.456538i
$69$		0			0
$70$	260.000	−	520.000i	0.443942	−	0.887884i
$71$	1128.00			1.88548			0.942739	−	0.333531i	$-0.108240\pi$
$71$	1128.00			1.88548			0.942739	+	0.333531i	$0.108240\pi$
$72$		0			0
$73$			632.000i			1.01329i	0.862155	+	0.506644i	$0.169114\pi$
$73$			632.000i			1.01329i	−0.862155	+	0.506644i	$0.830886\pi$
$74$	472.000			0.741471
$75$		0			0
$76$	−240.000			−0.362235
$77$		−	728.000i		−	1.07745i
$78$		0			0
$79$	720.000			1.02540			0.512698	−	0.858569i	$-0.328646\pi$
$79$	720.000			1.02540			0.512698	+	0.858569i	$0.328646\pi$
$80$	80.0000	−	160.000i	0.111803	−	0.223607i
$81$		0			0
$82$		−	484.000i		−	0.651815i
$83$			478.000i			0.632136i	0.948736	+	0.316068i	$0.102363\pi$
$83$			478.000i			0.632136i	−0.948736	+	0.316068i	$0.897637\pi$
$84$		0			0
$85$	−640.000	−	320.000i	−0.816679	−	0.408340i
$86$	−724.000			−0.907801
$87$		0			0
$88$		−	224.000i		−	0.271346i
$89$	−490.000			−0.583594			−0.291797	−	0.956480i	$-0.594253\pi$
$89$	−490.000			−0.583594			−0.291797	+	0.956480i	$0.594253\pi$
$90$		0			0
$91$	312.000			0.359412
$92$		−	232.000i		−	0.262909i
$93$		0			0
$94$	−452.000			−0.495960
$95$	300.000	−	600.000i	0.323993	−	0.647986i
$96$		0			0
$97$		−	1456.00i		−	1.52407i	−0.647538	−	0.762033i	$-0.724201\pi$
$97$		−	1456.00i		−	1.52407i	0.647538	−	0.762033i	$-0.275799\pi$
$98$		−	666.000i		−	0.686491i
$99$		0			0
$100$	300.000	+	400.000i	0.300000	+	0.400000i
$101$	578.000			0.569437			0.284719	−	0.958611i	$-0.408100\pi$
$101$	578.000			0.569437			0.284719	+	0.958611i	$0.408100\pi$
$102$		0			0
$103$			1462.00i			1.39859i	0.714831	+	0.699297i	$0.246503\pi$
$103$			1462.00i			1.39859i	−0.714831	+	0.699297i	$0.753497\pi$
$104$	96.0000			0.0905151
$105$		0			0
$106$	−216.000			−0.197922
$107$			966.000i			0.872773i	0.899759	+	0.436387i	$0.143742\pi$
$107$			966.000i			0.872773i	−0.899759	+	0.436387i	$0.856258\pi$
$108$		0			0
$109$	−370.000			−0.325134			−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000			−0.325134			−0.162567	+	0.986698i	$0.551977\pi$
$110$	560.000	+	280.000i	0.485399	+	0.242700i
$111$		0			0
$112$		−	416.000i		−	0.350967i
$113$			528.000i			0.439558i	0.975550	+	0.219779i	$0.0705336\pi$
$113$			528.000i			0.439558i	−0.975550	+	0.219779i	$0.929466\pi$
$114$		0			0
$115$	580.000	+	290.000i	0.470307	+	0.235153i
$116$	−360.000			−0.288148
$117$		0			0
$118$		−	40.0000i		−	0.0312059i
$119$	−1664.00			−1.28184
$120$		0			0
$121$	−547.000			−0.410969
$122$			1084.00i			0.804432i
$123$		0			0
$124$	512.000			0.370798
$125$	−1375.00	+	250.000i	−0.983870	+	0.178885i
$126$		0			0
$127$			1534.00i			1.07181i	0.844277	+	0.535907i	$0.180030\pi$
$127$			1534.00i			1.07181i	−0.844277	+	0.535907i	$0.819970\pi$
$128$		−	128.000i		−	0.0883883i
$129$		0			0
$130$	−120.000	+	240.000i	−0.0809592	+	0.161918i
$131$	−12.0000			−0.00800340			−0.00400170	−	0.999992i	$-0.501274\pi$
$131$	−12.0000			−0.00800340			−0.00400170	+	0.999992i	$0.501274\pi$
$132$		0			0
$133$		−	1560.00i		−	1.01706i
$134$	−868.000			−0.559580
$135$		0			0
$136$	−512.000			−0.322821
$137$		−	1224.00i		−	0.763309i	−0.924305	−	0.381655i	$-0.875354\pi$
$137$		−	1224.00i		−	0.763309i	0.924305	−	0.381655i	$-0.124646\pi$
$138$		0			0
$139$	−3100.00			−1.89164			−0.945822	−	0.324685i	$-0.894742\pi$
$139$	−3100.00			−1.89164			−0.945822	+	0.324685i	$0.894742\pi$
$140$	1040.00	+	520.000i	0.627829	+	0.313914i
$141$		0			0
$142$			2256.00i			1.33323i
$143$			336.000i			0.196488i
$144$		0			0
$145$	450.000	−	900.000i	0.257727	−	0.515455i
$146$	−1264.00			−0.716503
$147$		0			0
$148$			944.000i			0.524299i
$149$	250.000			0.137455			0.0687275	−	0.997635i	$-0.478106\pi$
$149$	250.000			0.137455			0.0687275	+	0.997635i	$0.478106\pi$
$150$		0			0
$151$	2152.00			1.15978			0.579892	−	0.814694i	$-0.303095\pi$
$151$	2152.00			1.15978			0.579892	+	0.814694i	$0.303095\pi$
$152$		−	480.000i		−	0.256139i
$153$		0			0
$154$	1456.00			0.761869
$155$	−640.000	+	1280.00i	−0.331652	+	0.663304i
$156$		0			0
$157$			524.000i			0.266368i	0.991091	+	0.133184i	$0.0425201\pi$
$157$			524.000i			0.266368i	−0.991091	+	0.133184i	$0.957480\pi$
$158$			1440.00i			0.725065i
$159$		0			0
$160$	320.000	+	160.000i	0.158114	+	0.0790569i
$161$	1508.00			0.738180
$162$		0			0
$163$		−	3518.00i		−	1.69050i	−0.534373	−	0.845249i	$-0.679452\pi$
$163$		−	3518.00i		−	1.69050i	0.534373	−	0.845249i	$-0.320548\pi$
$164$	968.000			0.460903
$165$		0			0
$166$	−956.000			−0.446988
$167$		−	534.000i		−	0.247438i	−0.992317	−	0.123719i	$-0.960518\pi$
$167$		−	534.000i		−	0.247438i	0.992317	−	0.123719i	$-0.0394822\pi$
$168$		0			0
$169$	2053.00			0.934456
$170$	640.000	−	1280.00i	0.288740	−	0.577480i
$171$		0			0
$172$		−	1448.00i		−	0.641913i
$173$		−	4252.00i		−	1.86863i	−0.356444	−	0.934317i	$-0.616011\pi$
$173$		−	4252.00i		−	1.86863i	0.356444	−	0.934317i	$-0.383989\pi$
$174$		0			0
$175$	−2600.00	+	1950.00i	−1.12309	+	0.842321i
$176$	448.000			0.191871
$177$		0			0
$178$		−	980.000i		−	0.412664i
$179$	2500.00			1.04390			0.521952	−	0.852975i	$-0.325204\pi$
$179$	2500.00			1.04390			0.521952	+	0.852975i	$0.325204\pi$
$180$		0			0
$181$	−2578.00			−1.05868			−0.529340	−	0.848410i	$-0.677561\pi$
$181$	−2578.00			−1.05868			−0.529340	+	0.848410i	$0.677561\pi$
$182$			624.000i			0.254143i
$183$		0			0
$184$	464.000			0.185905
$185$	−2360.00	−	1180.00i	−0.937895	−	0.468948i
$186$		0			0
$187$		−	1792.00i		−	0.700770i
$188$		−	904.000i		−	0.350697i
$189$		0			0
$190$	1200.00	+	600.000i	0.458196	+	0.229098i
$191$	768.000			0.290945			0.145473	−	0.989362i	$-0.453530\pi$
$191$	768.000			0.290945			0.145473	+	0.989362i	$0.453530\pi$
$192$		0			0
$193$		−	2608.00i		−	0.972684i	−0.873769	−	0.486342i	$-0.838331\pi$
$193$		−	2608.00i		−	0.972684i	0.873769	−	0.486342i	$-0.161669\pi$
$194$	2912.00			1.07768
$195$		0			0
$196$	1332.00			0.485423
$197$			5116.00i			1.85025i	0.379659	+	0.925127i	$0.376041\pi$
$197$			5116.00i			1.85025i	−0.379659	+	0.925127i	$0.623959\pi$
$198$		0			0
$199$	3480.00			1.23965			0.619826	−	0.784739i	$-0.287203\pi$
$199$	3480.00			1.23965			0.619826	+	0.784739i	$0.287203\pi$
$200$	−800.000	+	600.000i	−0.282843	+	0.212132i
$201$		0			0
$202$			1156.00i			0.402653i
$203$		−	2340.00i		−	0.809043i
$204$		0			0
$205$	−1210.00	+	2420.00i	−0.412244	+	0.824488i
$206$	−2924.00			−0.988955
$207$		0			0
$208$			192.000i			0.0640039i
$209$	1680.00			0.556019
$210$		0			0
$211$	3132.00			1.02188			0.510938	−	0.859618i	$-0.329298\pi$
$211$	3132.00			1.02188			0.510938	+	0.859618i	$0.329298\pi$
$212$		−	432.000i		−	0.139952i
$213$		0			0
$214$	−1932.00			−0.617144
$215$	3620.00	+	1810.00i	1.14829	+	0.574144i
$216$		0			0
$217$			3328.00i			1.04110i
$218$		−	740.000i		−	0.229904i
$219$		0			0
$220$	−560.000	+	1120.00i	−0.171615	+	0.343229i
$221$	768.000			0.233761
$222$		0			0
$223$			62.0000i			0.0186181i	0.999957	+	0.00930903i	$0.00296320\pi$
$223$			62.0000i			0.0186181i	−0.999957	+	0.00930903i	$0.997037\pi$
$224$	832.000			0.248171
$225$		0			0
$226$	−1056.00			−0.310814
$227$		−	5314.00i		−	1.55376i	−0.629651	−	0.776878i	$-0.716802\pi$
$227$		−	5314.00i		−	1.55376i	0.629651	−	0.776878i	$-0.283198\pi$
$228$		0			0
$229$	190.000			0.0548277			0.0274139	−	0.999624i	$-0.491273\pi$
$229$	190.000			0.0548277			0.0274139	+	0.999624i	$0.491273\pi$
$230$	−580.000	+	1160.00i	−0.166279	+	0.332557i
$231$		0			0
$232$		−	720.000i		−	0.203751i
$233$			2408.00i			0.677053i	0.940957	+	0.338526i	$0.109928\pi$
$233$			2408.00i			0.677053i	−0.940957	+	0.338526i	$0.890072\pi$
$234$		0			0
$235$	2260.00	+	1130.00i	0.627345	+	0.313673i
$236$	80.0000			0.0220659
$237$		0			0
$238$		−	3328.00i		−	0.906396i
$239$	−5680.00			−1.53727			−0.768637	−	0.639685i	$-0.779065\pi$
$239$	−5680.00			−1.53727			−0.768637	+	0.639685i	$0.779065\pi$
$240$		0			0
$241$	−278.000			−0.0743052			−0.0371526	−	0.999310i	$-0.511829\pi$
$241$	−278.000			−0.0743052			−0.0371526	+	0.999310i	$0.511829\pi$
$242$		−	1094.00i		−	0.290599i
$243$		0			0
$244$	−2168.00			−0.568820
$245$	−1665.00	+	3330.00i	−0.434175	+	0.868351i
$246$		0			0
$247$			720.000i			0.185476i
$248$			1024.00i			0.262194i
$249$		0			0
$250$	−500.000	−	2750.00i	−0.126491	−	0.695701i
$251$	−3252.00			−0.817787			−0.408893	−	0.912582i	$-0.634085\pi$
$251$	−3252.00			−0.817787			−0.408893	+	0.912582i	$0.634085\pi$
$252$		0			0
$253$			1624.00i			0.403557i
$254$	−3068.00			−0.757888
$255$		0			0
$256$	256.000			0.0625000
$257$			1536.00i			0.372813i	0.982473	+	0.186407i	$0.0596842\pi$
$257$			1536.00i			0.372813i	−0.982473	+	0.186407i	$0.940316\pi$
$258$		0			0
$259$	−6136.00			−1.47209
$260$	−480.000	−	240.000i	−0.114494	−	0.0572468i
$261$		0			0
$262$		−	24.0000i		−	0.00565926i
$263$			4858.00i			1.13900i	0.821991	+	0.569500i	$0.192863\pi$
$263$			4858.00i			1.13900i	−0.821991	+	0.569500i	$0.807137\pi$
$264$		0			0
$265$	1080.00	+	540.000i	0.250354	+	0.125177i
$266$	3120.00			0.719171
$267$		0			0
$268$		−	1736.00i		−	0.395683i
$269$	2610.00			0.591578			0.295789	−	0.955253i	$-0.404417\pi$
$269$	2610.00			0.591578			0.295789	+	0.955253i	$0.404417\pi$
$270$		0			0
$271$	−5168.00			−1.15843			−0.579213	−	0.815176i	$-0.696640\pi$
$271$	−5168.00			−1.15843			−0.579213	+	0.815176i	$0.696640\pi$
$272$		−	1024.00i		−	0.228269i
$273$		0			0
$274$	2448.00			0.539741
$275$	−2100.00	−	2800.00i	−0.460490	−	0.613987i
$276$		0			0
$277$			1924.00i			0.417336i	0.977987	+	0.208668i	$0.0669127\pi$
$277$			1924.00i			0.417336i	−0.977987	+	0.208668i	$0.933087\pi$
$278$		−	6200.00i		−	1.33759i
$279$		0			0
$280$	−1040.00	+	2080.00i	−0.221971	+	0.443942i
$281$	−3042.00			−0.645803			−0.322901	−	0.946433i	$-0.604658\pi$
$281$	−3042.00			−0.645803			−0.322901	+	0.946433i	$0.604658\pi$
$282$		0			0
$283$		−	1718.00i		−	0.360864i	−0.983587	−	0.180432i	$-0.942250\pi$
$283$		−	1718.00i		−	0.360864i	0.983587	−	0.180432i	$-0.0577496\pi$
$284$	−4512.00			−0.942739
$285$		0			0
$286$	−672.000			−0.138938
$287$			6292.00i			1.29409i
$288$		0			0
$289$	817.000			0.166294
$290$	1800.00	+	900.000i	0.364482	+	0.182241i
$291$		0			0
$292$		−	2528.00i		−	0.506644i
$293$		−	2292.00i		−	0.456997i	−0.973544	−	0.228498i	$-0.926618\pi$
$293$		−	2292.00i		−	0.456997i	0.973544	−	0.228498i	$-0.0733816\pi$
$294$		0			0
$295$	−100.000	+	200.000i	−0.0197364	+	0.0394727i
$296$	−1888.00			−0.370736
$297$		0			0
$298$			500.000i			0.0971954i
$299$	−696.000			−0.134618
$300$		0			0
$301$	9412.00			1.80232
$302$			4304.00i			0.820091i
$303$		0			0
$304$	960.000			0.181118
$305$	2710.00	−	5420.00i	0.508768	−	1.01754i
$306$		0			0
$307$		−	5406.00i		−	1.00501i	−0.864576	−	0.502503i	$-0.832413\pi$
$307$		−	5406.00i		−	1.00501i	0.864576	−	0.502503i	$-0.167587\pi$
$308$			2912.00i			0.538723i
$309$		0			0
$310$	−2560.00	−	1280.00i	−0.469027	−	0.234513i
$311$	5688.00			1.03710			0.518548	−	0.855048i	$-0.326473\pi$
$311$	5688.00			1.03710			0.518548	+	0.855048i	$0.326473\pi$
$312$		0			0
$313$			7352.00i			1.32767i	0.747881	+	0.663833i	$0.231072\pi$
$313$			7352.00i			1.32767i	−0.747881	+	0.663833i	$0.768928\pi$
$314$	−1048.00			−0.188351
$315$		0			0
$316$	−2880.00			−0.512698
$317$		−	3484.00i		−	0.617290i	−0.951177	−	0.308645i	$-0.900124\pi$
$317$		−	3484.00i		−	0.617290i	0.951177	−	0.308645i	$-0.0998755\pi$
$318$		0			0
$319$	2520.00			0.442298
$320$	−320.000	+	640.000i	−0.0559017	+	0.111803i
$321$		0			0
$322$			3016.00i			0.521972i
$323$		−	3840.00i		−	0.661496i
$324$		0			0
$325$	1200.00	−	900.000i	0.204812	−	0.153609i
$326$	7036.00			1.19536
$327$		0			0
$328$			1936.00i			0.325908i
$329$	5876.00			0.984664
$330$		0			0
$331$	−7868.00			−1.30654			−0.653269	−	0.757125i	$-0.726603\pi$
$331$	−7868.00			−1.30654			−0.653269	+	0.757125i	$0.726603\pi$
$332$		−	1912.00i		−	0.316068i
$333$		0			0
$334$	1068.00			0.174965
$335$	4340.00	+	2170.00i	0.707819	+	0.353910i
$336$		0			0
$337$		−	656.000i		−	0.106037i	−0.998594	−	0.0530187i	$-0.983116\pi$
$337$		−	656.000i		−	0.106037i	0.998594	−	0.0530187i	$-0.0168843\pi$
$338$			4106.00i			0.660760i
$339$		0			0
$340$	2560.00	+	1280.00i	0.408340	+	0.204170i
$341$	−3584.00			−0.569163
$342$		0			0
$343$		−	260.000i		−	0.0409291i
$344$	2896.00			0.453901
$345$		0			0
$346$	8504.00			1.32132
$347$		−	5754.00i		−	0.890176i	−0.895487	−	0.445088i	$-0.853172\pi$
$347$		−	5754.00i		−	0.890176i	0.895487	−	0.445088i	$-0.146828\pi$
$348$		0			0
$349$	3110.00			0.477004			0.238502	−	0.971142i	$-0.423344\pi$
$349$	3110.00			0.477004			0.238502	+	0.971142i	$0.423344\pi$
$350$	−3900.00	−	5200.00i	−0.595611	−	0.794148i
$351$		0			0
$352$			896.000i			0.135673i
$353$			7808.00i			1.17727i	0.808397	+	0.588637i	$0.200335\pi$
$353$			7808.00i			1.17727i	−0.808397	+	0.588637i	$0.799665\pi$
$354$		0			0
$355$	5640.00	−	11280.0i	0.843212	−	1.68642i
$356$	1960.00			0.291797
$357$		0			0
$358$			5000.00i			0.738151i
$359$	−9240.00			−1.35841			−0.679204	−	0.733949i	$-0.737675\pi$
$359$	−9240.00			−1.35841			−0.679204	+	0.733949i	$0.737675\pi$
$360$		0			0
$361$	−3259.00			−0.475142
$362$		−	5156.00i		−	0.748600i
$363$		0			0
$364$	−1248.00			−0.179706
$365$	6320.00	+	3160.00i	0.906312	+	0.453156i
$366$		0			0
$367$			3214.00i			0.457137i	0.973528	+	0.228569i	$0.0734046\pi$
$367$			3214.00i			0.457137i	−0.973528	+	0.228569i	$0.926595\pi$
$368$			928.000i			0.131455i
$369$		0			0
$370$	2360.00	−	4720.00i	0.331596	−	0.663192i
$371$	2808.00			0.392949
$372$		0			0
$373$		−	348.000i		−	0.0483077i	−0.999708	−	0.0241538i	$-0.992311\pi$
$373$		−	348.000i		−	0.0483077i	0.999708	−	0.0241538i	$-0.00768915\pi$
$374$	3584.00			0.495519
$375$		0			0
$376$	1808.00			0.247980
$377$			1080.00i			0.147541i
$378$		0			0
$379$	−4940.00			−0.669527			−0.334764	−	0.942302i	$-0.608656\pi$
$379$	−4940.00			−0.669527			−0.334764	+	0.942302i	$0.608656\pi$
$380$	−1200.00	+	2400.00i	−0.161997	+	0.323993i
$381$		0			0
$382$			1536.00i			0.205729i
$383$		−	6142.00i		−	0.819430i	−0.912214	−	0.409715i	$-0.865628\pi$
$383$		−	6142.00i		−	0.819430i	0.912214	−	0.409715i	$-0.134372\pi$
$384$		0			0
$385$	−7280.00	−	3640.00i	−0.963697	−	0.481848i
$386$	5216.00			0.687791
$387$		0			0
$388$			5824.00i			0.762033i
$389$	3050.00			0.397535			0.198768	−	0.980047i	$-0.436306\pi$
$389$	3050.00			0.397535			0.198768	+	0.980047i	$0.436306\pi$
$390$		0			0
$391$	3712.00			0.480112
$392$			2664.00i			0.343246i
$393$		0			0
$394$	−10232.0			−1.30833
$395$	3600.00	−	7200.00i	0.458571	−	0.917143i
$396$		0			0
$397$		−	5396.00i		−	0.682160i	−0.940034	−	0.341080i	$-0.889207\pi$
$397$		−	5396.00i		−	0.682160i	0.940034	−	0.341080i	$-0.110793\pi$
$398$			6960.00i			0.876566i
$399$		0			0
$400$	−1200.00	−	1600.00i	−0.150000	−	0.200000i
$401$	−14482.0			−1.80348			−0.901741	−	0.432276i	$-0.857711\pi$
$401$	−14482.0			−1.80348			−0.901741	+	0.432276i	$0.857711\pi$
$402$		0			0
$403$		−	1536.00i		−	0.189860i
$404$	−2312.00			−0.284719
$405$		0			0
$406$	4680.00			0.572080
$407$		−	6608.00i		−	0.804782i
$408$		0			0
$409$	1090.00			0.131778			0.0658888	−	0.997827i	$-0.479012\pi$
$409$	1090.00			0.131778			0.0658888	+	0.997827i	$0.479012\pi$
$410$	−4840.00	−	2420.00i	−0.583001	−	0.291501i
$411$		0			0
$412$		−	5848.00i		−	0.699297i
$413$			520.000i			0.0619553i
$414$		0			0
$415$	4780.00	+	2390.00i	0.565400	+	0.282700i
$416$	−384.000			−0.0452576
$417$		0			0
$418$			3360.00i			0.393165i
$419$	−7180.00			−0.837150			−0.418575	−	0.908182i	$-0.637470\pi$
$419$	−7180.00			−0.837150			−0.418575	+	0.908182i	$0.637470\pi$
$420$		0			0
$421$	−8138.00			−0.942095			−0.471047	−	0.882108i	$-0.656124\pi$
$421$	−8138.00			−0.942095			−0.471047	+	0.882108i	$0.656124\pi$
$422$			6264.00i			0.722575i
$423$		0			0
$424$	864.000			0.0989612
$425$	−6400.00	+	4800.00i	−0.730460	+	0.547845i
$426$		0			0
$427$		−	14092.0i		−	1.59710i
$428$		−	3864.00i		−	0.436387i
$429$		0			0
$430$	−3620.00	+	7240.00i	−0.405981	+	0.811962i
$431$	208.000			0.0232460			0.0116230	−	0.999932i	$-0.496300\pi$
$431$	208.000			0.0232460			0.0116230	+	0.999932i	$0.496300\pi$
$432$		0			0
$433$			12992.0i			1.44193i	0.692971	+	0.720965i	$0.256301\pi$
$433$			12992.0i			1.44193i	−0.692971	+	0.720965i	$0.743699\pi$
$434$	−6656.00			−0.736171
$435$		0			0
$436$	1480.00			0.162567
$437$			3480.00i			0.380940i
$438$		0			0
$439$	−1080.00			−0.117416			−0.0587080	−	0.998275i	$-0.518698\pi$
$439$	−1080.00			−0.117416			−0.0587080	+	0.998275i	$0.518698\pi$
$440$	−2240.00	−	1120.00i	−0.242700	−	0.121350i
$441$		0			0
$442$			1536.00i			0.165294i
$443$			9078.00i			0.973609i	0.873511	+	0.486805i	$0.161838\pi$
$443$			9078.00i			0.973609i	−0.873511	+	0.486805i	$0.838162\pi$
$444$		0			0
$445$	−2450.00	+	4900.00i	−0.260991	+	0.521983i
$446$	−124.000			−0.0131650
$447$		0			0
$448$			1664.00i			0.175484i
$449$	14310.0			1.50408			0.752039	−	0.659119i	$-0.229071\pi$
$449$	14310.0			1.50408			0.752039	+	0.659119i	$0.229071\pi$
$450$		0			0
$451$	−6776.00			−0.707471
$452$		−	2112.00i		−	0.219779i
$453$		0			0
$454$	10628.0			1.09867
$455$	1560.00	−	3120.00i	0.160734	−	0.321468i
$456$		0			0
$457$			2344.00i			0.239929i	0.992778	+	0.119965i	$0.0382781\pi$
$457$			2344.00i			0.239929i	−0.992778	+	0.119965i	$0.961722\pi$
$458$			380.000i			0.0387691i
$459$		0			0
$460$	−2320.00	−	1160.00i	−0.235153	−	0.117577i
$461$	−11382.0			−1.14992			−0.574959	−	0.818182i	$-0.694982\pi$
$461$	−11382.0			−1.14992			−0.574959	+	0.818182i	$0.694982\pi$
$462$		0			0
$463$			16062.0i			1.61223i	0.591756	+	0.806117i	$0.298435\pi$
$463$			16062.0i			1.61223i	−0.591756	+	0.806117i	$0.701565\pi$
$464$	1440.00			0.144074
$465$		0			0
$466$	−4816.00			−0.478749
$467$			17166.0i			1.70096i	0.526008	+	0.850479i	$0.323688\pi$
$467$			17166.0i			1.70096i	−0.526008	+	0.850479i	$0.676312\pi$
$468$		0			0
$469$	11284.0			1.11097
$470$	−2260.00	+	4520.00i	−0.221800	+	0.443600i
$471$		0			0
$472$			160.000i			0.0156030i
$473$			10136.0i			0.985315i
$474$		0			0
$475$	−4500.00	−	6000.00i	−0.434682	−	0.579577i
$476$	6656.00			0.640919
$477$		0			0
$478$		−	11360.0i		−	1.08702i
$479$	7520.00			0.717323			0.358661	−	0.933468i	$-0.383233\pi$
$479$	7520.00			0.717323			0.358661	+	0.933468i	$0.383233\pi$
$480$		0			0
$481$	2832.00			0.268458
$482$		−	556.000i		−	0.0525417i
$483$		0			0
$484$	2188.00			0.205485
$485$	−14560.0	−	7280.00i	−1.36317	−	0.681583i
$486$		0			0
$487$			11814.0i			1.09927i	0.835406	+	0.549634i	$0.185233\pi$
$487$			11814.0i			1.09927i	−0.835406	+	0.549634i	$0.814767\pi$
$488$		−	4336.00i		−	0.402216i
$489$		0			0
$490$	−6660.00	−	3330.00i	−0.614017	−	0.307008i
$491$	−14052.0			−1.29156			−0.645782	−	0.763522i	$-0.723468\pi$
$491$	−14052.0			−1.29156			−0.645782	+	0.763522i	$0.723468\pi$
$492$		0			0
$493$		−	5760.00i		−	0.526202i
$494$	−1440.00			−0.131151
$495$		0			0
$496$	−2048.00			−0.185399
$497$		−	29328.0i		−	2.64696i
$498$		0			0
$499$	−7620.00			−0.683603			−0.341802	−	0.939772i	$-0.611037\pi$
$499$	−7620.00			−0.683603			−0.341802	+	0.939772i	$0.611037\pi$
$500$	5500.00	−	1000.00i	0.491935	−	0.0894427i
$501$		0			0
$502$		−	6504.00i		−	0.578262i
$503$			1818.00i			0.161154i	0.996748	+	0.0805772i	$0.0256763\pi$
$503$			1818.00i			0.161154i	−0.996748	+	0.0805772i	$0.974324\pi$
$504$		0			0
$505$	2890.00	−	5780.00i	0.254660	−	0.509320i
$506$	−3248.00			−0.285358
$507$		0			0
$508$		−	6136.00i		−	0.535907i
$509$	17850.0			1.55440			0.777198	−	0.629256i	$-0.216640\pi$
$509$	17850.0			1.55440			0.777198	+	0.629256i	$0.216640\pi$
$510$		0			0
$511$	16432.0			1.42252
$512$			512.000i			0.0441942i
$513$		0			0
$514$	−3072.00			−0.263619
$515$	14620.0	+	7310.00i	1.25094	+	0.625470i
$516$		0			0
$517$			6328.00i			0.538308i
$518$		−	12272.0i		−	1.04093i
$519$		0			0
$520$	480.000	−	960.000i	0.0404796	−	0.0809592i
$521$	19238.0			1.61772			0.808860	−	0.588001i	$-0.200085\pi$
$521$	19238.0			1.61772			0.808860	+	0.588001i	$0.200085\pi$
$522$		0			0
$523$		−	6278.00i		−	0.524891i	−0.964947	−	0.262445i	$-0.915471\pi$
$523$		−	6278.00i		−	0.524891i	0.964947	−	0.262445i	$-0.0845289\pi$
$524$	48.0000			0.00400170
$525$		0			0
$526$	−9716.00			−0.805395
$527$			8192.00i			0.677133i
$528$		0			0
$529$	8803.00			0.723514
$530$	−1080.00	+	2160.00i	−0.0885136	+	0.177027i
$531$		0			0
$532$			6240.00i			0.508531i
$533$		−	2904.00i		−	0.235997i
$534$		0			0
$535$	9660.00	+	4830.00i	0.780632	+	0.390316i
$536$	3472.00			0.279790
$537$		0			0
$538$			5220.00i			0.418309i
$539$	−9324.00			−0.745108
$540$		0			0
$541$	−9818.00			−0.780238			−0.390119	−	0.920764i	$-0.627566\pi$
$541$	−9818.00			−0.780238			−0.390119	+	0.920764i	$0.627566\pi$
$542$		−	10336.0i		−	0.819131i
$543$		0			0
$544$	2048.00			0.161410
$545$	−1850.00	+	3700.00i	−0.145404	+	0.290808i
$546$		0			0
$547$			12514.0i			0.978172i	0.872236	+	0.489086i	$0.162670\pi$
$547$			12514.0i			0.978172i	−0.872236	+	0.489086i	$0.837330\pi$
$548$			4896.00i			0.381655i
$549$		0			0
$550$	5600.00	−	4200.00i	0.434154	−	0.325616i
$551$	5400.00			0.417509
$552$		0			0
$553$		−	18720.0i		−	1.43952i
$554$	−3848.00			−0.295101
$555$		0			0
$556$	12400.0			0.945822
$557$			10596.0i			0.806045i	0.915190	+	0.403022i	$0.132040\pi$
$557$			10596.0i			0.806045i	−0.915190	+	0.403022i	$0.867960\pi$
$558$		0			0
$559$	−4344.00			−0.328679
$560$	−4160.00	−	2080.00i	−0.313914	−	0.156957i
$561$		0			0
$562$		−	6084.00i		−	0.456651i
$563$		−	14002.0i		−	1.04816i	−0.851669	−	0.524080i	$-0.824409\pi$
$563$		−	14002.0i		−	1.04816i	0.851669	−	0.524080i	$-0.175591\pi$
$564$		0			0
$565$	5280.00	+	2640.00i	0.393153	+	0.196576i
$566$	3436.00			0.255169
$567$		0			0
$568$		−	9024.00i		−	0.666617i
$569$	−7330.00			−0.540052			−0.270026	−	0.962853i	$-0.587032\pi$
$569$	−7330.00			−0.540052			−0.270026	+	0.962853i	$0.587032\pi$
$570$		0			0
$571$	5812.00			0.425963			0.212981	−	0.977056i	$-0.431683\pi$
$571$	5812.00			0.425963			0.212981	+	0.977056i	$0.431683\pi$
$572$		−	1344.00i		−	0.0982438i
$573$		0			0
$574$	−12584.0			−0.915063
$575$	5800.00	−	4350.00i	0.420655	−	0.315491i
$576$		0			0
$577$		−	16736.0i		−	1.20750i	−0.797173	−	0.603751i	$-0.793672\pi$
$577$		−	16736.0i		−	1.20750i	0.797173	−	0.603751i	$-0.206328\pi$
$578$			1634.00i			0.117587i
$579$		0			0
$580$	−1800.00	+	3600.00i	−0.128864	+	0.257727i
$581$	12428.0			0.887436
$582$		0			0
$583$			3024.00i			0.214822i
$584$	5056.00			0.358251
$585$		0			0
$586$	4584.00			0.323146
$587$		−	7434.00i		−	0.522716i	−0.965242	−	0.261358i	$-0.915830\pi$
$587$		−	7434.00i		−	0.522716i	0.965242	−	0.261358i	$-0.0841702\pi$
$588$		0			0
$589$	−7680.00			−0.537265
$590$	−400.000	−	200.000i	−0.0279114	−	0.0139557i
$591$		0			0
$592$		−	3776.00i		−	0.262150i
$593$		−	25872.0i		−	1.79163i	−0.444429	−	0.895814i	$-0.646593\pi$
$593$		−	25872.0i		−	1.79163i	0.444429	−	0.895814i	$-0.353407\pi$
$594$		0			0
$595$	−8320.00	+	16640.0i	−0.573255	+	1.14651i
$596$	−1000.00			−0.0687275
$597$		0			0
$598$		−	1392.00i		−	0.0951892i
$599$	−3720.00			−0.253748			−0.126874	−	0.991919i	$-0.540494\pi$
$599$	−3720.00			−0.253748			−0.126874	+	0.991919i	$0.540494\pi$
$600$		0			0
$601$	−12958.0			−0.879481			−0.439740	−	0.898125i	$-0.644930\pi$
$601$	−12958.0			−0.879481			−0.439740	+	0.898125i	$0.644930\pi$
$602$			18824.0i			1.27443i
$603$		0			0
$604$	−8608.00			−0.579892
$605$	−2735.00	+	5470.00i	−0.183791	+	0.367582i
$606$		0			0
$607$			7214.00i			0.482384i	0.970477	+	0.241192i	$0.0775384\pi$
$607$			7214.00i			0.482384i	−0.970477	+	0.241192i	$0.922462\pi$
$608$			1920.00i			0.128070i
$609$		0			0
$610$	10840.0	+	5420.00i	0.719506	+	0.359753i
$611$	−2712.00			−0.179568
$612$		0			0
$613$		−	4828.00i		−	0.318109i	−0.987270	−	0.159055i	$-0.949155\pi$
$613$		−	4828.00i		−	0.318109i	0.987270	−	0.159055i	$-0.0508446\pi$
$614$	10812.0			0.710646
$615$		0			0
$616$	−5824.00			−0.380934
$617$			27656.0i			1.80452i	0.431193	+	0.902260i	$0.358093\pi$
$617$			27656.0i			1.80452i	−0.431193	+	0.902260i	$0.641907\pi$
$618$		0			0
$619$	21220.0			1.37787			0.688937	−	0.724821i	$-0.258078\pi$
$619$	21220.0			1.37787			0.688937	+	0.724821i	$0.258078\pi$
$620$	2560.00	−	5120.00i	0.165826	−	0.331652i
$621$		0			0
$622$			11376.0i			0.733338i
$623$			12740.0i			0.819289i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$	−14704.0			−0.938802
$627$		0			0
$628$		−	2096.00i		−	0.133184i
$629$	−15104.0			−0.957450
$630$		0			0
$631$	17672.0			1.11491			0.557457	−	0.830206i	$-0.311777\pi$
$631$	17672.0			1.11491			0.557457	+	0.830206i	$0.311777\pi$
$632$		−	5760.00i		−	0.362532i
$633$		0			0
$634$	6968.00			0.436490
$635$	15340.0	+	7670.00i	0.958660	+	0.479330i
$636$		0			0
$637$		−	3996.00i		−	0.248551i
$638$			5040.00i			0.312752i
$639$		0			0
$640$	−1280.00	−	640.000i	−0.0790569	−	0.0395285i
$641$	−7322.00			−0.451173			−0.225586	−	0.974223i	$-0.572430\pi$
$641$	−7322.00			−0.451173			−0.225586	+	0.974223i	$0.572430\pi$
$642$		0			0
$643$		−	8238.00i		−	0.505249i	−0.967564	−	0.252624i	$-0.918706\pi$
$643$		−	8238.00i		−	0.505249i	0.967564	−	0.252624i	$-0.0812937\pi$
$644$	−6032.00			−0.369090
$645$		0			0
$646$	7680.00			0.467749
$647$			6426.00i			0.390467i	0.980757	+	0.195233i	$0.0625465\pi$
$647$			6426.00i			0.390467i	−0.980757	+	0.195233i	$0.937454\pi$
$648$		0			0
$649$	−560.000			−0.0338705
$650$	1800.00	+	2400.00i	0.108618	+	0.144824i
$651$		0			0
$652$			14072.0i			0.845249i
$653$			5908.00i			0.354055i	0.984206	+	0.177027i	$0.0566482\pi$
$653$			5908.00i			0.354055i	−0.984206	+	0.177027i	$0.943352\pi$
$654$		0			0
$655$	−60.0000	+	120.000i	−0.00357923	+	0.00715845i
$656$	−3872.00			−0.230452
$657$		0			0
$658$			11752.0i			0.696262i
$659$	−26780.0			−1.58301			−0.791503	−	0.611166i	$-0.790701\pi$
$659$	−26780.0			−1.58301			−0.791503	+	0.611166i	$0.790701\pi$
$660$		0			0
$661$	−24538.0			−1.44390			−0.721950	−	0.691945i	$-0.756754\pi$
$661$	−24538.0			−1.44390			−0.721950	+	0.691945i	$0.756754\pi$
$662$		−	15736.0i		−	0.923863i
$663$		0			0
$664$	3824.00			0.223494
$665$	−15600.0	−	7800.00i	−0.909687	−	0.454844i
$666$		0			0
$667$			5220.00i			0.303027i
$668$			2136.00i			0.123719i
$669$		0			0
$670$	−4340.00	+	8680.00i	−0.250252	+	0.500504i
$671$	15176.0			0.873119
$672$		0			0
$673$		−	28848.0i		−	1.65232i	−0.563439	−	0.826158i	$-0.690522\pi$
$673$		−	28848.0i		−	1.65232i	0.563439	−	0.826158i	$-0.309478\pi$
$674$	1312.00			0.0749798
$675$		0			0
$676$	−8212.00			−0.467228
$677$		−	26884.0i		−	1.52620i	−0.646282	−	0.763099i	$-0.723677\pi$
$677$		−	26884.0i		−	1.52620i	0.646282	−	0.763099i	$-0.276323\pi$
$678$		0			0
$679$	−37856.0			−2.13959
$680$	−2560.00	+	5120.00i	−0.144370	+	0.288740i
$681$		0			0
$682$		−	7168.00i		−	0.402459i
$683$		−	14282.0i		−	0.800125i	−0.916488	−	0.400063i	$-0.868988\pi$
$683$		−	14282.0i		−	0.800125i	0.916488	−	0.400063i	$-0.131012\pi$
$684$		0			0
$685$	−12240.0	−	6120.00i	−0.682725	−	0.341362i
$686$	520.000			0.0289412
$687$		0			0
$688$			5792.00i			0.320956i
$689$	−1296.00			−0.0716599
$690$		0			0
$691$	−3428.00			−0.188723			−0.0943613	−	0.995538i	$-0.530081\pi$
$691$	−3428.00			−0.188723			−0.0943613	+	0.995538i	$0.530081\pi$
$692$			17008.0i			0.934317i
$693$		0			0
$694$	11508.0			0.629449
$695$	−15500.0	+	31000.0i	−0.845969	+	1.69194i
$696$		0			0
$697$			15488.0i			0.841678i
$698$			6220.00i			0.337293i
$699$		0			0
$700$	10400.0	−	7800.00i	0.561547	−	0.421160i
$701$	−26942.0			−1.45162			−0.725810	−	0.687895i	$-0.758535\pi$
$701$	−26942.0			−1.45162			−0.725810	+	0.687895i	$0.758535\pi$
$702$		0			0
$703$		−	14160.0i		−	0.759679i
$704$	−1792.00			−0.0959354
$705$		0			0
$706$	−15616.0			−0.832459
$707$		−	15028.0i		−	0.799415i
$708$		0			0
$709$	1950.00			0.103292			0.0516458	−	0.998665i	$-0.483553\pi$
$709$	1950.00			0.103292			0.0516458	+	0.998665i	$0.483553\pi$
$710$	22560.0	+	11280.0i	1.19248	+	0.596241i
$711$		0			0
$712$			3920.00i			0.206332i
$713$		−	7424.00i		−	0.389945i
$714$		0			0
$715$	3360.00	+	1680.00i	0.175744	+	0.0878719i
$716$	−10000.0			−0.521952
$717$		0			0
$718$		−	18480.0i		−	0.960540i
$719$	12080.0			0.626576			0.313288	−	0.949658i	$-0.398570\pi$
$719$	12080.0			0.626576			0.313288	+	0.949658i	$0.398570\pi$
$720$		0			0
$721$	38012.0			1.96344
$722$		−	6518.00i		−	0.335976i
$723$		0			0
$724$	10312.0			0.529340
$725$	−6750.00	−	9000.00i	−0.345778	−	0.461037i
$726$		0			0
$727$		−	17226.0i		−	0.878785i	−0.898295	−	0.439393i	$-0.855194\pi$
$727$		−	17226.0i		−	0.878785i	0.898295	−	0.439393i	$-0.144806\pi$
$728$		−	2496.00i		−	0.127071i
$729$		0			0
$730$	−6320.00	+	12640.0i	−0.320430	+	0.640859i
$731$	23168.0			1.17223
$732$		0			0
$733$		−	788.000i		−	0.0397073i	−0.999803	−	0.0198536i	$-0.993680\pi$
$733$		−	788.000i		−	0.0397073i	0.999803	−	0.0198536i	$-0.00632003\pi$
$734$	−6428.00			−0.323245
$735$		0			0
$736$	−1856.00			−0.0929525
$737$			12152.0i			0.607360i
$738$		0			0
$739$	2060.00			0.102542			0.0512709	−	0.998685i	$-0.483673\pi$
$739$	2060.00			0.102542			0.0512709	+	0.998685i	$0.483673\pi$
$740$	9440.00	+	4720.00i	0.468948	+	0.234474i
$741$		0			0
$742$			5616.00i			0.277857i
$743$			3258.00i			0.160867i	0.996760	+	0.0804337i	$0.0256305\pi$
$743$			3258.00i			0.160867i	−0.996760	+	0.0804337i	$0.974369\pi$
$744$		0			0
$745$	1250.00	−	2500.00i	0.0614718	−	0.122944i
$746$	696.000			0.0341587
$747$		0			0
$748$			7168.00i			0.350385i
$749$	25116.0			1.22526
$750$		0			0
$751$	−4528.00			−0.220012			−0.110006	−	0.993931i	$-0.535087\pi$
$751$	−4528.00			−0.220012			−0.110006	+	0.993931i	$0.535087\pi$
$752$			3616.00i			0.175348i
$753$		0			0
$754$	−2160.00			−0.104327
$755$	10760.0	−	21520.0i	0.518671	−	1.03734i
$756$		0			0
$757$		−	18236.0i		−	0.875560i	−0.899082	−	0.437780i	$-0.855765\pi$
$757$		−	18236.0i		−	0.875560i	0.899082	−	0.437780i	$-0.144235\pi$
$758$		−	9880.00i		−	0.473427i
$759$		0			0
$760$	−4800.00	−	2400.00i	−0.229098	−	0.114549i
$761$	18678.0			0.889720			0.444860	−	0.895600i	$-0.353253\pi$
$761$	18678.0			0.889720			0.444860	+	0.895600i	$0.353253\pi$
$762$		0			0
$763$			9620.00i			0.456445i
$764$	−3072.00			−0.145473
$765$		0			0
$766$	12284.0			0.579424
$767$		−	240.000i		−	0.0112984i
$768$		0			0
$769$	−27390.0			−1.28441			−0.642203	−	0.766534i	$-0.721980\pi$
$769$	−27390.0			−1.28441			−0.642203	+	0.766534i	$0.721980\pi$
$770$	7280.00	−	14560.0i	0.340718	−	0.681436i
$771$		0			0
$772$			10432.0i			0.486342i
$773$		−	9252.00i		−	0.430493i	−0.976560	−	0.215247i	$-0.930944\pi$
$773$		−	9252.00i		−	0.430493i	0.976560	−	0.215247i	$-0.0690555\pi$
$774$		0			0
$775$	9600.00	+	12800.0i	0.444958	+	0.593277i
$776$	−11648.0			−0.538839
$777$		0			0
$778$			6100.00i			0.281100i
$779$	−14520.0			−0.667822
$780$		0			0
$781$	31584.0			1.44707
$782$			7424.00i			0.339491i
$783$		0			0
$784$	−5328.00			−0.242711
$785$	5240.00	+	2620.00i	0.238247	+	0.119123i
$786$		0			0
$787$		−	5726.00i

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:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +(5.00000 - 10.0000i) q^{5} -26.0000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +(5.00000 - 10.0000i) q^{5} -26.0000i q^{7} -8.00000i q^{8} +(20.0000 + 10.0000i) q^{10} +28.0000 q^{11} +12.0000i q^{13} +52.0000 q^{14} +16.0000 q^{16} -64.0000i q^{17} +60.0000 q^{19} +(-20.0000 + 40.0000i) q^{20} +56.0000i q^{22} +58.0000i q^{23} +(-75.0000 - 100.000i) q^{25} -24.0000 q^{26} +104.000i q^{28} +90.0000 q^{29} -128.000 q^{31} +32.0000i q^{32} +128.000 q^{34} +(-260.000 - 130.000i) q^{35} -236.000i q^{37} +120.000i q^{38} +(-80.0000 - 40.0000i) q^{40} -242.000 q^{41} +362.000i q^{43} -112.000 q^{44} -116.000 q^{46} +226.000i q^{47} -333.000 q^{49} +(200.000 - 150.000i) q^{50} -48.0000i q^{52} +108.000i q^{53} +(140.000 - 280.000i) q^{55} -208.000 q^{56} +180.000i q^{58} -20.0000 q^{59} +542.000 q^{61} -256.000i q^{62} -64.0000 q^{64} +(120.000 + 60.0000i) q^{65} +434.000i q^{67} +256.000i q^{68} +(260.000 - 520.000i) q^{70} +1128.00 q^{71} +632.000i q^{73} +472.000 q^{74} -240.000 q^{76} -728.000i q^{77} +720.000 q^{79} +(80.0000 - 160.000i) q^{80} -484.000i q^{82} +478.000i q^{83} +(-640.000 - 320.000i) q^{85} -724.000 q^{86} -224.000i q^{88} -490.000 q^{89} +312.000 q^{91} -232.000i q^{92} -452.000 q^{94} +(300.000 - 600.000i) q^{95} -1456.00i q^{97} -666.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 10 q^{5}+O(q^{10}) \) 2 * q - 8 * q^4 + 10 * q^5	\( 2 q - 8 q^{4} + 10 q^{5} + 40 q^{10} + 56 q^{11} + 104 q^{14} + 32 q^{16} + 120 q^{19} - 40 q^{20} - 150 q^{25} - 48 q^{26} + 180 q^{29} - 256 q^{31} + 256 q^{34} - 520 q^{35} - 160 q^{40} - 484 q^{41} - 224 q^{44} - 232 q^{46} - 666 q^{49} + 400 q^{50} + 280 q^{55} - 416 q^{56} - 40 q^{59} + 1084 q^{61} - 128 q^{64} + 240 q^{65} + 520 q^{70} + 2256 q^{71} + 944 q^{74} - 480 q^{76} + 1440 q^{79} + 160 q^{80} - 1280 q^{85} - 1448 q^{86} - 980 q^{89} + 624 q^{91} - 904 q^{94} + 600 q^{95}+O(q^{100}) \) 2 * q - 8 * q^4 + 10 * q^5 + 40 * q^10 + 56 * q^11 + 104 * q^14 + 32 * q^16 + 120 * q^19 - 40 * q^20 - 150 * q^25 - 48 * q^26 + 180 * q^29 - 256 * q^31 + 256 * q^34 - 520 * q^35 - 160 * q^40 - 484 * q^41 - 224 * q^44 - 232 * q^46 - 666 * q^49 + 400 * q^50 + 280 * q^55 - 416 * q^56 - 40 * q^59 + 1084 * q^61 - 128 * q^64 + 240 * q^65 + 520 * q^70 + 2256 * q^71 + 944 * q^74 - 480 * q^76 + 1440 * q^79 + 160 * q^80 - 1280 * q^85 - 1448 * q^86 - 980 * q^89 + 624 * q^91 - 904 * q^94 + 600 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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