LMFDB - Embedded newform 300.3.l.f.107.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,3,Mod(107,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("300.107");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	300.l (of order $4$, degree $2$, 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:	$8.17440793081$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			107.4
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	300.107
Dual form			300.3.l.f.143.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 + 1.41421i) q^{2} +(2.99535 + 0.166925i) q^{3} +4.00000i q^{4} +(4.00000 + 4.47214i) q^{6} +(9.48683 - 9.48683i) q^{7} +(-5.65685 + 5.65685i) q^{8} +(8.94427 + 1.00000i) q^{9} +O(q^{10})$	\(q+(1.41421 + 1.41421i) q^{2} +(2.99535 + 0.166925i) q^{3} +4.00000i q^{4} +(4.00000 + 4.47214i) q^{6} +(9.48683 - 9.48683i) q^{7} +(-5.65685 + 5.65685i) q^{8} +(8.94427 + 1.00000i) q^{9} +(-0.667701 + 11.9814i) q^{12} +26.8328 q^{14} -16.0000 q^{16} +(11.2349 + 14.0633i) q^{18} +(30.0000 - 26.8328i) q^{21} +(-31.1127 + 31.1127i) q^{23} +(-17.8885 + 16.0000i) q^{24} +(26.6243 + 4.48838i) q^{27} +(37.9473 + 37.9473i) q^{28} -53.6656 q^{29} +(-22.6274 - 22.6274i) q^{32} +(-4.00000 + 35.7771i) q^{36} -53.6656i q^{41} +(80.3737 + 4.47907i) q^{42} +(28.4605 + 28.4605i) q^{43} -88.0000 q^{46} +(-2.82843 - 2.82843i) q^{47} +(-47.9256 - 2.67080i) q^{48} -131.000i q^{49} +(31.3050 + 44.0000i) q^{54} +107.331i q^{56} +(-75.8947 - 75.8947i) q^{58} +58.0000 q^{61} +(94.3396 - 75.3660i) q^{63} -64.0000i q^{64} +(-47.4342 + 47.4342i) q^{67} +(-98.3870 + 88.0000i) q^{69} +(-56.2533 + 44.9396i) q^{72} +(79.0000 + 17.8885i) q^{81} +(75.8947 - 75.8947i) q^{82} +(-53.7401 + 53.7401i) q^{83} +(107.331 + 120.000i) q^{84} +80.4984i q^{86} +(-160.747 - 8.95815i) q^{87} +107.331 q^{89} +(-124.451 - 124.451i) q^{92} -8.00000i q^{94} +(-64.0000 - 71.5542i) q^{96} +(185.262 - 185.262i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{6}+O(q^{10}) $ 8 * q + 32 * q^6	$ 8 q + 32 q^{6} - 128 q^{16} + 240 q^{21} - 32 q^{36} - 704 q^{46} + 464 q^{61} + 632 q^{81} - 512 q^{96}+O(q^{100}) $ 8 * q + 32 * q^6 - 128 * q^16 + 240 * q^21 - 32 * q^36 - 704 * q^46 + 464 * q^61 + 632 * q^81 - 512 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	1.41421	+	1.41421i	0.707107	+	0.707107i
$2$	1.41421	+	1.41421i	0.707107	+	0.707107i
$3$	2.99535	+	0.166925i	0.998451	+	0.0556418i
$3$	2.99535	+	0.166925i	0.998451	+	0.0556418i
$4$			4.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$	4.00000	+	4.47214i	0.666667	+	0.745356i
$7$	9.48683	−	9.48683i	1.35526	−	1.35526i	0.475600	−	0.879661i	$-0.342231\pi$
$7$	9.48683	−	9.48683i	1.35526	−	1.35526i	0.879661	−	0.475600i	$-0.157769\pi$
$8$	−5.65685	+	5.65685i	−0.707107	+	0.707107i
$9$	8.94427	+	1.00000i	0.993808	+	0.111111i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.667701	+	11.9814i	−0.0556418	+	0.998451i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	26.8328			1.91663
$15$		0			0
$16$	−16.0000			−1.00000
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	11.2349	+	14.0633i	0.624161	+	0.781296i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	30.0000	−	26.8328i	1.42857	−	1.27775i
$22$		0			0
$23$	−31.1127	+	31.1127i	−1.35273	+	1.35273i	−0.470128	+	0.882599i	$0.655792\pi$
$23$	−31.1127	+	31.1127i	−1.35273	+	1.35273i	−0.882599	+	0.470128i	$0.844208\pi$
$24$	−17.8885	+	16.0000i	−0.745356	+	0.666667i
$25$		0			0
$26$		0			0
$27$	26.6243	+	4.48838i	0.986086	+	0.166236i
$28$	37.9473	+	37.9473i	1.35526	+	1.35526i
$29$	−53.6656			−1.85054			−0.925270	−	0.379310i	$-0.876161\pi$
$29$	−53.6656			−1.85054			−0.925270	+	0.379310i	$0.876161\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−22.6274	−	22.6274i	−0.707107	−	0.707107i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−4.00000	+	35.7771i	−0.111111	+	0.993808i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	53.6656i		−	1.30892i	−0.756098	−	0.654459i	$-0.772896\pi$
$41$		−	53.6656i		−	1.30892i	0.756098	−	0.654459i	$-0.227104\pi$
$42$	80.3737	+	4.47907i	1.91366	+	0.106645i
$43$	28.4605	+	28.4605i	0.661872	+	0.661872i	0.955821	−	0.293949i	$-0.0949696\pi$
$43$	28.4605	+	28.4605i	0.661872	+	0.661872i	−0.293949	+	0.955821i	$0.594970\pi$
$44$		0			0
$45$		0			0
$46$	−88.0000			−1.91304
$47$	−2.82843	−	2.82843i	−0.0601793	−	0.0601793i	0.676377	−	0.736556i	$-0.263549\pi$
$47$	−2.82843	−	2.82843i	−0.0601793	−	0.0601793i	−0.736556	+	0.676377i	$0.763549\pi$
$48$	−47.9256	−	2.67080i	−0.998451	−	0.0556418i
$49$		−	131.000i		−	2.67347i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	31.3050	+	44.0000i	0.579721	+	0.814815i
$55$		0			0
$56$			107.331i			1.91663i
$57$		0			0
$58$	−75.8947	−	75.8947i	−1.30853	−	1.30853i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	58.0000			0.950820			0.475410	−	0.879764i	$-0.342300\pi$
$61$	58.0000			0.950820			0.475410	+	0.879764i	$0.342300\pi$
$62$		0			0
$63$	94.3396	−	75.3660i	1.49745	−	1.19629i
$64$		−	64.0000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−47.4342	+	47.4342i	−0.707973	+	0.707973i	−0.966109	−	0.258136i	$-0.916892\pi$
$67$	−47.4342	+	47.4342i	−0.707973	+	0.707973i	0.258136	+	0.966109i	$0.416892\pi$
$68$		0			0
$69$	−98.3870	+	88.0000i	−1.42590	+	1.27536i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−56.2533	+	44.9396i	−0.781296	+	0.624161i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	79.0000	+	17.8885i	0.975309	+	0.220846i
$82$	75.8947	−	75.8947i	0.925545	−	0.925545i
$83$	−53.7401	+	53.7401i	−0.647471	+	0.647471i	−0.952381	−	0.304910i	$-0.901374\pi$
$83$	−53.7401	+	53.7401i	−0.647471	+	0.647471i	0.304910	+	0.952381i	$0.401374\pi$
$84$	107.331	+	120.000i	1.27775	+	1.42857i
$85$		0			0
$86$			80.4984i			0.936028i
$87$	−160.747	−	8.95815i	−1.84767	−	0.102967i
$88$		0			0
$89$	107.331			1.20597			0.602985	−	0.797753i	$-0.293978\pi$
$89$	107.331			1.20597			0.602985	+	0.797753i	$0.293978\pi$
$90$		0			0
$91$		0			0
$92$	−124.451	−	124.451i	−1.35273	−	1.35273i
$93$		0			0
$94$		−	8.00000i		−	0.0851064i
$95$		0			0
$96$	−64.0000	−	71.5542i	−0.666667	−	0.745356i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	185.262	−	185.262i	1.89043	−	1.89043i
$99$		0			0
$100$		0			0
$101$		−	160.997i		−	1.59403i	−0.603960	−	0.797014i	$-0.706411\pi$
$101$		−	160.997i		−	1.59403i	0.603960	−	0.797014i	$-0.293589\pi$
$102$		0			0
$103$	−142.302	−	142.302i	−1.38158	−	1.38158i	−0.841821	−	0.539756i	$-0.818516\pi$
$103$	−142.302	−	142.302i	−1.38158	−	1.38158i	−0.539756	−	0.841821i	$-0.681484\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	87.6812	+	87.6812i	0.819451	+	0.819451i	0.986028	−	0.166578i	$-0.0532716\pi$
$107$	87.6812	+	87.6812i	0.819451	+	0.819451i	−0.166578	+	0.986028i	$0.553272\pi$
$108$	−17.9535	+	106.497i	−0.166236	+	0.986086i
$109$		−	38.0000i		−	0.348624i	−0.984690	−	0.174312i	$-0.944230\pi$
$109$		−	38.0000i		−	0.348624i	0.984690	−	0.174312i	$-0.0557701\pi$
$110$		0			0
$111$		0			0
$112$	−151.789	+	151.789i	−1.35526	+	1.35526i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		−	214.663i		−	1.85054i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−121.000			−1.00000
$122$	82.0244	+	82.0244i	0.672331	+	0.672331i
$123$	8.95815	−	160.747i	0.0728305	−	1.30689i
$124$		0			0
$125$		0			0
$126$	240.000	+	26.8328i	1.90476	+	0.212959i
$127$	66.4078	−	66.4078i	0.522896	−	0.522896i	−0.395549	−	0.918445i	$-0.629446\pi$
$127$	66.4078	−	66.4078i	0.522896	−	0.522896i	0.918445	+	0.395549i	$0.129446\pi$
$128$	90.5097	−	90.5097i	0.707107	−	0.707107i
$129$	80.4984	+	90.0000i	0.624019	+	0.697674i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	−134.164			−1.00122
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$	−263.591	−	14.6894i	−1.91008	−	0.106445i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−8.00000	−	8.94427i	−0.0567376	−	0.0634346i
$142$		0			0
$143$		0			0
$144$	−143.108	−	16.0000i	−0.993808	−	0.111111i
$145$		0			0
$146$		0			0
$147$	21.8672	−	392.391i	0.148757	−	2.66933i
$148$		0			0
$149$	−107.331			−0.720344			−0.360172	−	0.932886i	$-0.617282\pi$
$149$	−107.331			−0.720344			−0.360172	+	0.932886i	$0.617282\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			590.322i			3.66660i
$162$	86.4247	+	137.021i	0.533485	+	0.845809i
$163$	199.223	+	199.223i	1.22223	+	1.22223i	0.966837	+	0.255393i	$0.0822047\pi$
$163$	199.223	+	199.223i	1.22223	+	1.22223i	0.255393	+	0.966837i	$0.417795\pi$
$164$	214.663			1.30892
$165$		0			0
$166$	−152.000			−0.915663
$167$	172.534	+	172.534i	1.03314	+	1.03314i	0.999432	+	0.0337063i	$0.0107311\pi$
$167$	172.534	+	172.534i	1.03314	+	1.03314i	0.0337063	+	0.999432i	$0.489269\pi$
$168$	−17.9163	+	321.495i	−0.106645	+	1.91366i
$169$			169.000i			1.00000i
$170$		0			0
$171$		0			0
$172$	−113.842	+	113.842i	−0.661872	+	0.661872i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$	−214.663	−	240.000i	−1.23369	−	1.37931i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	151.789	+	151.789i	0.852749	+	0.852749i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−358.000			−1.97790			−0.988950	−	0.148248i	$-0.952637\pi$
$181$	−358.000			−1.97790			−0.988950	+	0.148248i	$0.952637\pi$
$182$		0			0
$183$	173.730	+	9.68167i	0.949347	+	0.0529053i
$184$		−	352.000i		−	1.91304i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	11.3137	−	11.3137i	0.0601793	−	0.0601793i
$189$	295.161	−	210.000i	1.56170	−	1.11111i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	10.6832	−	191.703i	0.0556418	−	0.998451i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	524.000			2.67347
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−150.000	+	134.164i	−0.746269	+	0.667483i
$202$	227.684	−	227.684i	1.12715	−	1.12715i
$203$	−509.117	+	509.117i	−2.50796	+	2.50796i
$204$		0			0
$205$		0			0
$206$		−	402.492i		−	1.95385i
$207$	−309.393	+	247.168i	−1.49465	+	1.19405i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$			248.000i			1.15888i
$215$		0			0
$216$	−176.000	+	125.220i	−0.814815	+	0.579721i
$217$		0			0
$218$	53.7401	−	53.7401i	0.246514	−	0.246514i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−66.4078	−	66.4078i	−0.297793	−	0.297793i	0.542356	−	0.840149i	$-0.317532\pi$
$223$	−66.4078	−	66.4078i	−0.297793	−	0.297793i	−0.840149	+	0.542356i	$0.817532\pi$
$224$	−429.325			−1.91663
$225$		0			0
$226$		0			0
$227$	251.730	+	251.730i	1.10894	+	1.10894i	0.993290	+	0.115653i	$0.0368961\pi$
$227$	251.730	+	251.730i	1.10894	+	1.10894i	0.115653	+	0.993290i	$0.463104\pi$
$228$		0			0
$229$			262.000i			1.14410i	0.820217	+	0.572052i	$0.193853\pi$
$229$			262.000i			1.14410i	−0.820217	+	0.572052i	$0.806147\pi$
$230$		0			0
$231$		0			0
$232$	303.579	−	303.579i	1.30853	−	1.30853i
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	302.000			1.25311			0.626556	−	0.779376i	$-0.284464\pi$
$241$	302.000			1.25311			0.626556	+	0.779376i	$0.284464\pi$
$242$	−171.120	−	171.120i	−0.707107	−	0.707107i
$243$	233.647	+	66.7696i	0.961509	+	0.274772i
$244$			232.000i			0.950820i
$245$		0			0
$246$	240.000	−	214.663i	0.975610	−	0.872612i
$247$		0			0
$248$		0			0
$249$	−169.941	+	152.000i	−0.682495	+	0.610442i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	301.464	+	377.359i	1.19629	+	1.49745i
$253$		0			0
$254$	187.830			0.739487
$255$		0			0
$256$	256.000			1.00000
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$	−13.4372	+	241.121i	−0.0520823	+	0.934578i
$259$		0			0
$260$		0			0
$261$	−480.000	−	53.6656i	−1.83908	−	0.205615i
$262$		0			0
$263$	200.818	−	200.818i	0.763568	−	0.763568i	−0.213398	−	0.976965i	$-0.568453\pi$
$263$	200.818	−	200.818i	0.763568	−	0.763568i	0.976965	+	0.213398i	$0.0684530\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	321.495	+	17.9163i	1.20410	+	0.0671022i
$268$	−189.737	−	189.737i	−0.707973	−	0.707973i
$269$	536.656			1.99500			0.997502	−	0.0706320i	$-0.0225016\pi$
$269$	536.656			1.99500			0.997502	+	0.0706320i	$0.0225016\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−352.000	−	393.548i	−1.27536	−	1.42590i
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	375.659i		−	1.33687i	−0.743772	−	0.668433i	$-0.766965\pi$
$281$		−	375.659i		−	1.33687i	0.743772	−	0.668433i	$-0.233035\pi$
$282$	1.33540	−	23.9628i	0.00473547	−	0.0849745i
$283$	332.039	+	332.039i	1.17328	+	1.17328i	0.981422	+	0.191861i	$0.0614523\pi$
$283$	332.039	+	332.039i	1.17328	+	1.17328i	0.191861	+	0.981422i	$0.438548\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−509.117	−	509.117i	−1.77393	−	1.77393i
$288$	−179.758	−	225.013i	−0.624161	−	0.781296i
$289$			289.000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	585.850	−	524.000i	1.99269	−	1.78231i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−151.789	−	151.789i	−0.509360	−	0.509360i
$299$		0			0
$300$		0			0
$301$	540.000			1.79402
$302$		0			0
$303$	26.8744	−	482.242i	0.0886946	−	1.59156i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−104.355	+	104.355i	−0.339919	+	0.339919i	−0.856337	−	0.516418i	$-0.827265\pi$
$307$	−104.355	+	104.355i	−0.339919	+	0.339919i	0.516418	+	0.856337i	$0.327265\pi$
$308$		0			0
$309$	−402.492	−	450.000i	−1.30256	−	1.45631i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	248.000	+	277.272i	0.772586	+	0.863777i
$322$	−834.841	+	834.841i	−2.59267	+	2.59267i
$323$		0			0
$324$	−71.5542	+	316.000i	−0.220846	+	0.975309i
$325$		0			0
$326$			563.489i			1.72849i
$327$	6.34316	−	113.823i	0.0193980	−	0.348084i
$328$	303.579	+	303.579i	0.925545	+	0.925545i
$329$	−53.6656			−0.163117
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−214.960	−	214.960i	−0.647471	−	0.647471i
$333$		0			0
$334$			488.000i			1.46108i
$335$		0			0
$336$	−480.000	+	429.325i	−1.42857	+	1.27775i
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−239.002	+	239.002i	−0.707107	+	0.707107i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−777.920	−	777.920i	−2.26799	−	2.26799i
$344$	−321.994			−0.936028
$345$		0			0
$346$		0			0
$347$	−82.0244	−	82.0244i	−0.236382	−	0.236382i	0.578968	−	0.815350i	$-0.303455\pi$
$347$	−82.0244	−	82.0244i	−0.236382	−	0.236382i	−0.815350	+	0.578968i	$0.803455\pi$
$348$	35.8326	−	642.990i	0.102967	−	1.84767i
$349$		−	22.0000i		−	0.0630372i	−0.999503	−	0.0315186i	$-0.989966\pi$
$349$		−	22.0000i		−	0.0630372i	0.999503	−	0.0315186i	$-0.0100344\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$			429.325i			1.20597i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−361.000			−1.00000
$362$	−506.288	−	506.288i	−1.39859	−	1.39859i
$363$	−362.438	−	20.1980i	−0.998451	−	0.0556418i
$364$		0			0
$365$		0			0
$366$	232.000	+	259.384i	0.633880	+	0.708699i
$367$	−85.3815	+	85.3815i	−0.232647	+	0.232647i	−0.813797	−	0.581150i	$-0.802603\pi$
$367$	−85.3815	+	85.3815i	−0.232647	+	0.232647i	0.581150	+	0.813797i	$0.302603\pi$
$368$	497.803	−	497.803i	1.35273	−	1.35273i
$369$	53.6656	−	480.000i	0.145435	−	1.30081i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	32.0000			0.0851064
$377$		0			0
$378$	714.405	+	120.436i	1.88996	+	0.318613i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	210.000	−	187.830i	0.551181	−	0.492991i
$382$		0			0
$383$	31.1127	−	31.1127i	0.0812342	−	0.0812342i	−0.665322	−	0.746556i	$-0.731706\pi$
$383$	31.1127	−	31.1127i	0.0812342	−	0.0812342i	0.746556	+	0.665322i	$0.231706\pi$
$384$	286.217	−	256.000i	0.745356	−	0.666667i
$385$		0			0
$386$		0			0
$387$	226.098	+	283.019i	0.584232	+	0.731315i
$388$		0			0
$389$	−751.319			−1.93141			−0.965705	−	0.259640i	$-0.916396\pi$
$389$	−751.319			−1.93141			−0.965705	+	0.259640i	$0.916396\pi$
$390$		0			0
$391$		0			0
$392$	741.048	+	741.048i	1.89043	+	1.89043i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	643.988i		−	1.60595i	−0.596010	−	0.802977i	$-0.703248\pi$
$401$		−	643.988i		−	1.60595i	0.596010	−	0.802977i	$-0.296752\pi$
$402$	−401.869	−	22.3954i	−0.999673	−	0.0557099i
$403$		0			0
$404$	643.988			1.59403
$405$		0			0
$406$	−1440.00			−3.54680
$407$		0			0
$408$		0			0
$409$		−	802.000i		−	1.96088i	−0.196818	−	0.980440i	$-0.563061\pi$
$409$		−	802.000i		−	1.96088i	0.196818	−	0.980440i	$-0.436939\pi$
$410$		0			0
$411$		0			0
$412$	569.210	−	569.210i	1.38158	−	1.38158i
$413$		0			0
$414$	−787.096	−	88.0000i	−1.90120	−	0.212560i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	778.000			1.84798			0.923990	−	0.382415i	$-0.124908\pi$
$421$	778.000			1.84798			0.923990	+	0.382415i	$0.124908\pi$
$422$		0			0
$423$	−22.4698	−	28.1266i	−0.0531201	−	0.0664933i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	550.236	−	550.236i	1.28861	−	1.28861i
$428$	−350.725	+	350.725i	−0.819451	+	0.819451i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−425.989	−	71.8140i	−0.986086	−	0.166236i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$	152.000			0.348624
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	131.000	−	1171.70i	0.297052	−	2.65692i
$442$		0			0
$443$	562.857	−	562.857i	1.27056	−	1.27056i	0.324762	−	0.945796i	$-0.394716\pi$
$443$	562.857	−	562.857i	1.27056	−	1.27056i	0.945796	−	0.324762i	$-0.105284\pi$
$444$		0			0
$445$		0			0
$446$		−	187.830i		−	0.421143i
$447$	−321.495	−	17.9163i	−0.719228	−	0.0400812i
$448$	−607.157	−	607.157i	−1.35526	−	1.35526i
$449$	−804.984			−1.79284			−0.896419	−	0.443207i	$-0.853841\pi$
$449$	−804.984			−1.79284			−0.896419	+	0.443207i	$0.853841\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$			712.000i			1.56828i
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$	−370.524	+	370.524i	−0.809004	+	0.809004i
$459$		0			0
$460$		0			0
$461$			375.659i			0.814879i	0.913232	+	0.407440i	$0.133578\pi$
$461$			375.659i			0.814879i	−0.913232	+	0.407440i	$0.866422\pi$
$462$		0			0
$463$	369.986	+	369.986i	0.799107	+	0.799107i	0.982955	−	0.183848i	$-0.0588554\pi$
$463$	369.986	+	369.986i	0.799107	+	0.799107i	−0.183848	+	0.982955i	$0.558855\pi$
$464$	858.650			1.85054
$465$		0			0
$466$		0			0
$467$	−87.6812	−	87.6812i	−0.187754	−	0.187754i	0.606970	−	0.794725i	$-0.292385\pi$
$467$	−87.6812	−	87.6812i	−0.187754	−	0.187754i	−0.794725	+	0.606970i	$0.792385\pi$
$468$		0			0
$469$			900.000i			1.91898i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	427.092	+	427.092i	0.886084	+	0.886084i
$483$	−98.5396	+	1768.22i	−0.204016	+	3.66092i
$484$		−	484.000i		−	1.00000i
$485$		0			0
$486$	236.000	+	424.853i	0.485597	+	0.874183i
$487$	−597.670	+	597.670i	−1.22725	+	1.22725i	−0.262249	+	0.965000i	$0.584464\pi$
$487$	−597.670	+	597.670i	−1.22725	+	1.22725i	−0.965000	+	0.262249i	$0.915536\pi$
$488$	−328.098	+	328.098i	−0.672331	+	0.672331i
$489$	563.489	+	630.000i	1.15233	+	1.28834i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	642.990	+	35.8326i	1.30689	+	0.0728305i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−455.294	−	25.3726i	−0.914244	−	0.0509491i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	488.000	+	545.601i	0.974052	+	1.08902i
$502$		0			0
$503$	647.710	−	647.710i	1.28769	−	1.28769i	0.351509	−	0.936185i	$-0.385669\pi$
$503$	647.710	−	647.710i	1.28769	−	1.28769i	0.936185	−	0.351509i	$-0.114331\pi$
$504$	−107.331	+	960.000i	−0.212959	+	1.90476i
$505$		0			0
$506$		0			0
$507$	−28.2104	+	506.215i	−0.0556418	+	0.998451i
$508$	265.631	+	265.631i	0.522896	+	0.522896i
$509$	268.328			0.527167			0.263584	−	0.964637i	$-0.415095\pi$
$509$	268.328			0.527167			0.263584	+	0.964637i	$0.415095\pi$
$510$		0			0
$511$		0			0
$512$	362.039	+	362.039i	0.707107	+	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−360.000	+	321.994i	−0.697674	+	0.624019i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	751.319i		−	1.44207i	−0.692898	−	0.721035i	$-0.743667\pi$
$521$		−	751.319i		−	1.44207i	0.692898	−	0.721035i	$-0.256333\pi$
$522$	−602.928	−	754.717i	−1.15503	−	1.44582i
$523$	730.486	+	730.486i	1.39672	+	1.39672i	0.809227	+	0.587496i	$0.199886\pi$
$523$	730.486	+	730.486i	1.39672	+	1.39672i	0.587496	+	0.809227i	$0.300114\pi$
$524$		0			0
$525$		0			0
$526$	568.000			1.07985
$527$		0			0
$528$		0			0
$529$		−	1407.00i		−	2.65974i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	429.325	+	480.000i	0.803979	+	0.898876i
$535$		0			0
$536$		−	536.656i		−	1.00122i
$537$		0			0
$538$	758.947	+	758.947i	1.41068	+	1.41068i
$539$		0			0
$540$		0			0
$541$	−362.000			−0.669131			−0.334566	−	0.942372i	$-0.608590\pi$
$541$	−362.000			−0.669131			−0.334566	+	0.942372i	$0.608590\pi$
$542$		0			0
$543$	−1072.34	−	59.7592i	−1.97484	−	0.110054i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	104.355	−	104.355i	0.190777	−	0.190777i	−0.605255	−	0.796032i	$-0.706929\pi$
$547$	104.355	−	104.355i	0.190777	−	0.190777i	0.796032	+	0.605255i	$0.206929\pi$
$548$		0			0
$549$	518.768	+	58.0000i	0.944932	+	0.105647i
$550$		0			0
$551$		0			0
$552$	58.7577	−	1054.36i	0.106445	−	1.91008i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	531.263	−	531.263i	0.945307	−	0.945307i
$563$	−794.788	+	794.788i	−1.41170	+	1.41170i	−0.663725	+	0.747977i	$0.731025\pi$
$563$	−794.788	+	794.788i	−1.41170	+	1.41170i	−0.747977	+	0.663725i	$0.768975\pi$
$564$	35.7771	−	32.0000i	0.0634346	−	0.0567376i
$565$		0			0
$566$			939.149i			1.65927i
$567$	919.165	−	579.754i	1.62110	−	1.02249i
$568$		0			0
$569$	1126.98			1.98063			0.990315	−	0.138840i	$-0.0443374\pi$
$569$	1126.98			1.98063			0.990315	+	0.138840i	$0.0443374\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		−	1440.00i		−	2.50871i
$575$		0			0
$576$	64.0000	−	572.433i	0.111111	−	0.993808i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−408.708	+	408.708i	−0.707107	+	0.707107i
$579$		0			0
$580$		0			0
$581$			1019.65i			1.75499i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	760.847	+	760.847i	1.29616	+	1.29616i	0.930908	+	0.365254i	$0.119018\pi$
$587$	760.847	+	760.847i	1.29616	+	1.29616i	0.365254	+	0.930908i	$0.380982\pi$
$588$	1569.56	+	87.4688i	2.66933	+	0.148757i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		−	429.325i		−	0.720344i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−418.000			−0.695507			−0.347754	−	0.937586i	$-0.613055\pi$
$601$	−418.000			−0.695507			−0.347754	+	0.937586i	$0.613055\pi$
$602$	763.675	+	763.675i	1.26856	+	1.26856i
$603$	−471.698	+	376.830i	−0.782252	+	0.624925i
$604$		0			0
$605$		0			0
$606$	720.000	−	643.988i	1.18812	−	1.06269i
$607$	−521.776	+	521.776i	−0.859598	+	0.859598i	−0.991291	−	0.131693i	$-0.957959\pi$
$607$	−521.776	+	521.776i	−0.859598	+	0.859598i	0.131693	+	0.991291i	$0.457959\pi$
$608$		0			0
$609$	−1609.97	+	1440.00i	−2.64363	+	2.36453i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$	−295.161			−0.480718
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$	67.1861	−	1205.61i	0.108715	−	1.95082i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−968.000	+	688.709i	−1.55878	+	1.10903i
$622$		0			0
$623$	1018.23	−	1018.23i	1.63440	−	1.63440i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			590.322i			0.920939i	0.887676	+	0.460470i	$0.152319\pi$
$641$			590.322i			0.920939i	−0.887676	+	0.460470i	$0.847681\pi$
$642$	−41.3975	+	742.847i	−0.0644820	+	1.15708i
$643$	−863.302	−	863.302i	−1.34262	−	1.34262i	−0.893447	−	0.449168i	$-0.851720\pi$
$643$	−863.302	−	863.302i	−1.34262	−	1.34262i	−0.449168	−	0.893447i	$-0.648280\pi$
$644$	−2361.29			−3.66660
$645$		0			0
$646$		0			0
$647$	−675.994	−	675.994i	−1.04481	−	1.04481i	−0.998948	−	0.0458655i	$-0.985395\pi$
$647$	−675.994	−	675.994i	−1.04481	−	1.04481i	−0.0458655	−	0.998948i	$-0.514605\pi$
$648$	−548.084	+	345.699i	−0.845809	+	0.533485i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−796.894	+	796.894i	−1.22223	+	1.22223i
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$	169.941	−	152.000i	0.259849	−	0.232416i
$655$		0			0
$656$			858.650i			1.30892i
$657$		0			0
$658$	−75.8947	−	75.8947i	−0.115341	−	0.115341i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−298.000			−0.450832			−0.225416	−	0.974263i	$-0.572374\pi$
$661$	−298.000			−0.450832			−0.225416	+	0.974263i	$0.572374\pi$
$662$		0			0
$663$		0			0
$664$		−	608.000i		−	0.915663i
$665$		0			0
$666$		0			0
$667$	1669.68	−	1669.68i	2.50327	−	2.50327i
$668$	−690.136	+	690.136i	−1.03314	+	1.03314i
$669$	−187.830	−	210.000i	−0.280762	−	0.313901i
$670$		0			0
$671$		0			0
$672$	−1285.98	−	71.6652i	−1.91366	−	0.106645i
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	−676.000			−1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	712.000	+	796.040i	1.04552	+	1.16893i
$682$		0			0
$683$	393.151	−	393.151i	0.575624	−	0.575624i	−0.358070	−	0.933695i	$-0.616565\pi$
$683$	393.151	−	393.151i	0.575624	−	0.575624i	0.933695	+	0.358070i	$0.116565\pi$
$684$		0			0
$685$		0			0
$686$		−	2200.29i		−	3.20742i
$687$	−43.7344	+	784.782i	−0.0636600	+	1.14233i
$688$	−455.368	−	455.368i	−0.661872	−	0.661872i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		−	232.000i		−	0.334294i
$695$		0			0
$696$	960.000	−	858.650i	1.37931	−	1.23369i
$697$		0			0
$698$	31.1127	−	31.1127i	0.0445741	−	0.0445741i
$699$		0			0
$700$		0			0
$701$		−	1073.31i		−	1.53112i	−0.643367	−	0.765558i	$-0.722463\pi$
$701$		−	1073.31i		−	1.53112i	0.643367	−	0.765558i	$-0.277537\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1527.35	−	1527.35i	−2.16033	−	2.16033i
$708$		0			0
$709$		−	698.000i		−	0.984485i	−0.870458	−	0.492243i	$-0.836177\pi$
$709$		−	698.000i		−	0.984485i	0.870458	−	0.492243i	$-0.163823\pi$
$710$		0			0
$711$		0			0
$712$	−607.157	+	607.157i	−0.852749	+	0.852749i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−2700.00			−3.74480
$722$	−510.531	−	510.531i	−0.707107	−	0.707107i
$723$	904.596	+	50.4114i	1.25117	+	0.0697254i
$724$		−	1432.00i		−	1.97790i
$725$		0			0
$726$	−484.000	−	541.128i	−0.666667	−	0.745356i
$727$	−161.276	+	161.276i	−0.221838	+	0.221838i	−0.809272	−	0.587434i	$-0.800138\pi$
$727$	−161.276	+	161.276i	−0.221838	+	0.221838i	0.587434	+	0.809272i	$0.300138\pi$
$728$		0			0
$729$	688.709	+	239.000i	0.944731	+	0.327846i
$730$		0			0
$731$		0			0
$732$	−38.7267	+	694.922i	−0.0529053	+	0.949347i
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$	−241.495			−0.329013
$735$		0			0
$736$	1408.00			1.91304
$737$		0			0
$738$	754.717	−	602.928i	1.02265	−	0.816975i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	540.230	−	540.230i	0.727092	−	0.727092i	−0.242947	−	0.970040i	$-0.578114\pi$
$743$	540.230	−	540.230i	0.727092	−	0.727092i	0.970040	+	0.242947i	$0.0781142\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−534.406	+	426.926i	−0.715403	+	0.571521i
$748$		0			0
$749$	1663.63			2.22114
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	45.2548	+	45.2548i	0.0601793	+	0.0601793i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	840.000	+	1180.64i	1.11111	+	1.56170i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1502.64i			1.97456i	0.159001	+	0.987278i	$0.449173\pi$
$761$			1502.64i			1.97456i	−0.159001	+	0.987278i	$0.550827\pi$
$762$	562.616	+	31.3535i	0.738341	+	0.0411464i
$763$	−360.500	−	360.500i	−0.472477	−	0.472477i
$764$		0			0
$765$		0			0
$766$	88.0000			0.114883
$767$		0			0
$768$	766.810	+	42.7329i	0.998451	+	0.0556418i
$769$			1342.00i			1.74512i	0.488504	+	0.872562i	$0.337543\pi$
$769$			1342.00i			1.74512i	−0.488504	+	0.872562i	$0.662457\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$	−80.4984	+	720.000i	−0.104003	+	0.930233i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1062.53	−	1062.53i	−1.36571	−	1.36571i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1428.81	−	240.872i	−1.82479	−	0.307627i
$784$			2096.00i			2.67347i
$785$		0			0
$786$		0			0
$787$	−1109.96	+	1109.96i	−1.41037	+	1.41037i	−0.653072	+	0.757296i	$0.726520\pi$
$787$	−1109.96	+	1109.96i	−1.41037	+	1.41037i	−0.757296	+	0.653072i	$0.773480\pi$
$788$		0			0
$789$	635.043	−	568.000i	0.804871	−	0.719899i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	960.000	+	107.331i	1.19850	+	0.133997i
$802$	910.736	−	910.736i	1.13558	−	1.13558i
$803$		0			0
$804$	−536.656	−	600.000i	−0.667483	−	0.746269i
$805$		0			0
$806$		0			0
$807$	1607.47	+	89.5815i	1.99191	+	0.111006i
$808$	910.736	+	910.736i	1.12715	+	1.12715i
$809$	−965.981			−1.19404			−0.597022	−	0.802225i	$-0.703649\pi$
$809$	−965.981			−1.19404			−0.597022	+	0.802225i	$0.703649\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−2036.47	−	2036.47i	−2.50796	−	2.50796i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1134.20	−	1134.20i	1.38655	−	1.38655i
$819$		0			0
$820$		0			0
$821$			1502.64i			1.83025i	0.403167	+	0.915126i	$0.367909\pi$
$821$			1502.64i			1.83025i	−0.403167	+	0.915126i	$0.632091\pi$
$822$		0			0
$823$	−616.644	−	616.644i	−0.749264	−	0.749264i	0.225077	−	0.974341i	$-0.427737\pi$
$823$	−616.644	−	616.644i	−0.749264	−	0.749264i	−0.974341	+	0.225077i	$0.927737\pi$
$824$	1609.97			1.95385
$825$		0			0
$826$		0			0
$827$	421.436	+	421.436i	0.509596	+	0.509596i	0.914402	−	0.404807i	$-0.132661\pi$
$827$	421.436	+	421.436i	0.509596	+	0.509596i	−0.404807	+	0.914402i	$0.632661\pi$
$828$	−988.671	−	1237.57i	−1.19405	−	1.49465i
$829$			1478.00i			1.78287i	0.453148	+	0.891435i	$0.350301\pi$
$829$			1478.00i			1.78287i	−0.453148	+	0.891435i	$0.649699\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	2039.00			2.42449
$842$	1100.26	+	1100.26i	1.30672	+	1.30672i
$843$	62.7070	−	1125.23i	0.0743856	−	1.33480i
$844$		0			0
$845$		0			0
$846$	8.00000	−	71.5542i	0.00945626	−	0.0845794i
$847$	−1147.91	+	1147.91i	−1.35526	+	1.35526i
$848$		0			0
$849$	939.149	+	1050.00i	1.10618	+	1.23675i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$	1556.30			1.82237
$855$		0			0
$856$	−992.000			−1.15888
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$	−1440.00	−	1609.97i	−1.67247	−	1.86988i
$862$		0			0
$863$	−1156.83	+	1156.83i	−1.34047	+	1.34047i	−0.444882	+	0.895589i	$0.646754\pi$
$863$	−1156.83	+	1156.83i	−1.34047	+	1.34047i	−0.895589	+	0.444882i	$0.853246\pi$
$864$	−500.879	−	704.000i	−0.579721	−	0.814815i
$865$		0			0
$866$		0			0
$867$	−48.2414	+	865.657i	−0.0556418	+	0.998451i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	214.960	+	214.960i	0.246514	+	0.246514i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	697.653i		−	0.791888i	−0.918275	−	0.395944i	$-0.870417\pi$
$881$		−	697.653i		−	0.791888i	0.918275	−	0.395944i	$-0.129583\pi$
$882$	1842.30	−	1471.77i	2.08877	−	1.66868i
$883$	863.302	+	863.302i	0.977692	+	0.977692i	0.999757	−	0.0220648i	$-0.00702402\pi$
$883$	863.302	+	863.302i	0.977692	+	0.977692i	−0.0220648	+	0.999757i	$0.507024\pi$
$884$		0			0
$885$		0			0
$886$	1592.00			1.79684
$887$	−681.651	−	681.651i	−0.768490	−	0.768490i	0.209350	−	0.977841i	$-0.432865\pi$
$887$	−681.651	−	681.651i	−0.768490	−	0.768490i	−0.977841	+	0.209350i	$0.932865\pi$
$888$		0			0
$889$		−	1260.00i		−	1.41732i
$890$		0			0
$891$		0			0
$892$	265.631	−	265.631i	0.297793	−	0.297793i
$893$		0			0
$894$	−429.325	−	480.000i	−0.480229	−	0.536913i
$895$		0			0
$896$		−	1717.30i		−	1.91663i
$897$		0			0
$898$	−1138.42	−	1138.42i	−1.26773	−	1.26773i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	1617.49	+	90.1396i	1.79124	+	0.0998224i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	180.250	−	180.250i	0.198732	−	0.198732i	−0.600724	−	0.799456i	$-0.705121\pi$
$907$	180.250	−	180.250i	0.198732	−	0.198732i	0.799456	+	0.600724i	$0.205121\pi$
$908$	−1006.92	+	1006.92i	−1.10894	+	1.10894i
$909$	160.997	−	1440.00i	0.177114	−	1.58416i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1048.00			−1.14410
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−330.000	+	295.161i	−0.358306	+	0.320479i
$922$	−531.263	+	531.263i	−0.576207	+	0.576207i
$923$		0			0
$924$		0			0
$925$		0			0
$926$			1046.48i			1.13011i
$927$	−1130.49	−	1415.09i	−1.21951	−	1.52653i
$928$	1214.31	+	1214.31i	1.30853	+	1.30853i
$929$	−1770.97			−1.90631			−0.953157	−	0.302476i	$-0.902187\pi$
$929$	−1770.97			−1.90631			−0.953157	+	0.302476i	$0.902187\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		−	248.000i		−	0.265525i
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$	−1272.79	+	1272.79i	−1.35692	+	1.35692i
$939$		0			0
$940$		0			0
$941$		−	1878.30i		−	1.99606i	−0.0626993	−	0.998032i	$-0.519971\pi$
$941$		−	1878.30i		−	1.99606i	0.0626993	−	0.998032i	$-0.480029\pi$
$942$		0			0
$943$	1669.68	+	1669.68i	1.77061	+	1.77061i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1275.62	+	1275.62i	1.34701	+	1.34701i	0.888888	+	0.458124i	$0.151478\pi$
$947$	1275.62	+	1275.62i	1.34701	+	1.34701i	0.458124	+	0.888888i	$0.348522\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$	696.564	+	871.926i	0.723327	+	0.905427i
$964$			1208.00i			1.25311i
$965$		0			0
$966$	−2640.00	+	2361.29i	−2.73292	+	2.44440i
$967$	1356.62	−	1356.62i	1.40291	−	1.40291i	0.612246	−	0.790668i	$-0.290266\pi$
$967$	1356.62	−	1356.62i	1.40291	−	1.40291i	0.790668	−	0.612246i	$-0.209734\pi$
$968$	684.479	−	684.479i	0.707107	−	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−267.078	+	934.587i	−0.274772	+	0.961509i
$973$		0			0
$974$	−1690.47			−1.73559
$975$		0			0
$976$	−928.000			−0.950820
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$	−94.0606	+	1687.85i	−0.0961765	+	1.72582i
$979$		0			0
$980$		0			0
$981$	38.0000	−	339.882i	0.0387360	−	0.346465i
$982$		0			0
$983$	−200.818	+	200.818i	−0.204291	+	0.204291i	−0.801836	−	0.597544i	$-0.796143\pi$
$983$	−200.818	+	200.818i	−0.204291	+	0.204291i	0.597544	+	0.801836i	$0.296143\pi$
$984$	858.650	+	960.000i	0.872612	+	0.975610i
$985$		0			0
$986$		0			0
$987$	−160.747	−	8.95815i	−0.162865	−	0.00907614i
$988$		0			0
$989$	−1770.97			−1.79066
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−608.000	−	679.765i	−0.610442	−	0.682495i
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.3.l.f.107.4	yes	8
3.2	odd	2	inner	300.3.l.f.107.2	yes	8
4.3	odd	2	inner	300.3.l.f.107.1	✓	8
5.2	odd	4	inner	300.3.l.f.143.2	yes	8
5.3	odd	4	inner	300.3.l.f.143.3	yes	8
5.4	even	2	inner	300.3.l.f.107.1	✓	8
12.11	even	2	inner	300.3.l.f.107.3	yes	8
15.2	even	4	inner	300.3.l.f.143.4	yes	8
15.8	even	4	inner	300.3.l.f.143.1	yes	8
15.14	odd	2	inner	300.3.l.f.107.3	yes	8
20.3	even	4	inner	300.3.l.f.143.2	yes	8
20.7	even	4	inner	300.3.l.f.143.3	yes	8
20.19	odd	2	CM	300.3.l.f.107.4	yes	8
60.23	odd	4	inner	300.3.l.f.143.4	yes	8
60.47	odd	4	inner	300.3.l.f.143.1	yes	8
60.59	even	2	inner	300.3.l.f.107.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.l.f.107.1	✓	8	4.3	odd	2	inner
300.3.l.f.107.1	✓	8	5.4	even	2	inner
300.3.l.f.107.2	yes	8	3.2	odd	2	inner
300.3.l.f.107.2	yes	8	60.59	even	2	inner
300.3.l.f.107.3	yes	8	12.11	even	2	inner
300.3.l.f.107.3	yes	8	15.14	odd	2	inner
300.3.l.f.107.4	yes	8	1.1	even	1	trivial
300.3.l.f.107.4	yes	8	20.19	odd	2	CM
300.3.l.f.143.1	yes	8	15.8	even	4	inner
300.3.l.f.143.1	yes	8	60.47	odd	4	inner
300.3.l.f.143.2	yes	8	5.2	odd	4	inner
300.3.l.f.143.2	yes	8	20.3	even	4	inner
300.3.l.f.143.3	yes	8	5.3	odd	4	inner
300.3.l.f.143.3	yes	8	20.7	even	4	inner
300.3.l.f.143.4	yes	8	15.2	even	4	inner
300.3.l.f.143.4	yes	8	60.23	odd	4	inner

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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	300.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 + 1.41421i) q^{2} +(2.99535 + 0.166925i) q^{3} +4.00000i q^{4} +(4.00000 + 4.47214i) q^{6} +(9.48683 - 9.48683i) q^{7} +(-5.65685 + 5.65685i) q^{8} +(8.94427 + 1.00000i) q^{9} +O(q^{10})\)	\(q+(1.41421 + 1.41421i) q^{2} +(2.99535 + 0.166925i) q^{3} +4.00000i q^{4} +(4.00000 + 4.47214i) q^{6} +(9.48683 - 9.48683i) q^{7} +(-5.65685 + 5.65685i) q^{8} +(8.94427 + 1.00000i) q^{9} +(-0.667701 + 11.9814i) q^{12} +26.8328 q^{14} -16.0000 q^{16} +(11.2349 + 14.0633i) q^{18} +(30.0000 - 26.8328i) q^{21} +(-31.1127 + 31.1127i) q^{23} +(-17.8885 + 16.0000i) q^{24} +(26.6243 + 4.48838i) q^{27} +(37.9473 + 37.9473i) q^{28} -53.6656 q^{29} +(-22.6274 - 22.6274i) q^{32} +(-4.00000 + 35.7771i) q^{36} -53.6656i q^{41} +(80.3737 + 4.47907i) q^{42} +(28.4605 + 28.4605i) q^{43} -88.0000 q^{46} +(-2.82843 - 2.82843i) q^{47} +(-47.9256 - 2.67080i) q^{48} -131.000i q^{49} +(31.3050 + 44.0000i) q^{54} +107.331i q^{56} +(-75.8947 - 75.8947i) q^{58} +58.0000 q^{61} +(94.3396 - 75.3660i) q^{63} -64.0000i q^{64} +(-47.4342 + 47.4342i) q^{67} +(-98.3870 + 88.0000i) q^{69} +(-56.2533 + 44.9396i) q^{72} +(79.0000 + 17.8885i) q^{81} +(75.8947 - 75.8947i) q^{82} +(-53.7401 + 53.7401i) q^{83} +(107.331 + 120.000i) q^{84} +80.4984i q^{86} +(-160.747 - 8.95815i) q^{87} +107.331 q^{89} +(-124.451 - 124.451i) q^{92} -8.00000i q^{94} +(-64.0000 - 71.5542i) q^{96} +(185.262 - 185.262i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{6}+O(q^{10}) \) 8 * q + 32 * q^6	\( 8 q + 32 q^{6} - 128 q^{16} + 240 q^{21} - 32 q^{36} - 704 q^{46} + 464 q^{61} + 632 q^{81} - 512 q^{96}+O(q^{100}) \) 8 * q + 32 * q^6 - 128 * q^16 + 240 * q^21 - 32 * q^36 - 704 * q^46 + 464 * q^61 + 632 * q^81 - 512 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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