LMFDB - Embedded newform 300.3.l.e.143.4

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:	8.0.3317760000.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 17x^{4} + 256 $ x^8 + 17*x^4 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			143.4
Root			$1.72286 + 1.01575i$ of defining polynomial
Character	$\chi$	$=$	300.143
Dual form			300.3.l.e.107.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72286 + 1.01575i) q^{2} +(2.12132 + 2.12132i) q^{3} +(1.93649 + 3.50000i) q^{4} +(1.50000 + 5.80948i) q^{6} +(-0.218832 + 7.99701i) q^{8} +9.00000i q^{9} +O(q^{10})$	\(q+(1.72286 + 1.01575i) q^{2} +(2.12132 + 2.12132i) q^{3} +(1.93649 + 3.50000i) q^{4} +(1.50000 + 5.80948i) q^{6} +(-0.218832 + 7.99701i) q^{8} +9.00000i q^{9} +(-3.31670 + 11.5325i) q^{12} +(-8.50000 + 13.5554i) q^{16} +(-21.9089 - 21.9089i) q^{17} +(-9.14178 + 15.5057i) q^{18} +30.9839 q^{19} +(24.0416 + 24.0416i) q^{23} +(-17.4284 + 16.5000i) q^{24} +(-19.0919 + 19.0919i) q^{27} -61.9677i q^{31} +(-28.4133 + 14.7202i) q^{32} +(-15.4919 - 60.0000i) q^{34} +(-31.5000 + 17.4284i) q^{36} +(53.3809 + 31.4720i) q^{38} +(17.0000 + 65.8407i) q^{46} +(9.89949 - 9.89949i) q^{47} +(-46.7867 + 10.7242i) q^{48} -49.0000i q^{49} -92.9516i q^{51} +(-43.8178 + 43.8178i) q^{53} +(-52.2853 + 13.5000i) q^{54} +(65.7267 + 65.7267i) q^{57} +118.000 q^{61} +(62.9439 - 106.762i) q^{62} +(-63.9042 - 3.50000i) q^{64} +(34.2548 - 119.108i) q^{68} +102.000i q^{69} +(-71.9731 - 1.96949i) q^{72} +(60.0000 + 108.444i) q^{76} +123.935 q^{79} -81.0000 q^{81} +(-108.894 - 108.894i) q^{83} +(-37.5893 + 130.702i) q^{92} +(131.453 - 131.453i) q^{93} +(27.1109 - 7.00000i) q^{94} +(-91.5000 - 29.0474i) q^{96} +(49.7719 - 84.4201i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{6}+O(q^{10}) $ 8 * q + 12 * q^6	$ 8 q + 12 q^{6} - 68 q^{16} - 252 q^{36} + 136 q^{46} + 944 q^{61} + 480 q^{76} - 648 q^{81} - 732 q^{96}+O(q^{100}) $ 8 * q + 12 * q^6 - 68 * q^16 - 252 * q^36 + 136 * q^46 + 944 * q^61 + 480 * q^76 - 648 * q^81 - 732 * q^96

Display 100 coefficients 10 coefficients

Character values

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

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	1.72286	+	1.01575i	0.861430	+	0.507877i
$2$	1.72286	+	1.01575i	0.861430	+	0.507877i
$3$	2.12132	+	2.12132i	0.707107	+	0.707107i
$3$	2.12132	+	2.12132i	0.707107	+	0.707107i
$4$	1.93649	+	3.50000i	0.484123	+	0.875000i
$5$		0			0
$5$		0			0
$6$	1.50000	+	5.80948i	0.250000	+	0.968246i
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−0.218832	+	7.99701i	−0.0273540	+	0.999626i
$9$			9.00000i			1.00000i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−3.31670	+	11.5325i	−0.276392	+	0.961045i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	−8.50000	+	13.5554i	−0.531250	+	0.847215i
$17$	−21.9089	−	21.9089i	−1.28876	−	1.28876i	−0.935541	−	0.353218i	$-0.885087\pi$
$17$	−21.9089	−	21.9089i	−1.28876	−	1.28876i	−0.353218	−	0.935541i	$-0.614913\pi$
$18$	−9.14178	+	15.5057i	−0.507877	+	0.861430i
$19$	30.9839			1.63073			0.815365	−	0.578947i	$-0.196536\pi$
$19$	30.9839			1.63073			0.815365	+	0.578947i	$0.196536\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	24.0416	+	24.0416i	1.04529	+	1.04529i	0.998925	+	0.0463637i	$0.0147633\pi$
$23$	24.0416	+	24.0416i	1.04529	+	1.04529i	0.0463637	+	0.998925i	$0.485237\pi$
$24$	−17.4284	+	16.5000i	−0.726184	+	0.687500i
$25$		0			0
$26$		0			0
$27$	−19.0919	+	19.0919i	−0.707107	+	0.707107i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		−	61.9677i		−	1.99896i	−0.0322581	−	0.999480i	$-0.510270\pi$
$31$		−	61.9677i		−	1.99896i	0.0322581	−	0.999480i	$-0.489730\pi$
$32$	−28.4133	+	14.7202i	−0.887915	+	0.460007i
$33$		0			0
$34$	−15.4919	−	60.0000i	−0.455645	−	1.76471i
$35$		0			0
$36$	−31.5000	+	17.4284i	−0.875000	+	0.484123i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	53.3809	+	31.4720i	1.40476	+	0.828209i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	17.0000	+	65.8407i	0.369565	+	1.43132i
$47$	9.89949	−	9.89949i	0.210628	−	0.210628i	−0.593907	−	0.804534i	$-0.702415\pi$
$47$	9.89949	−	9.89949i	0.210628	−	0.210628i	0.804534	+	0.593907i	$0.202415\pi$
$48$	−46.7867	+	10.7242i	−0.974722	+	0.223421i
$49$		−	49.0000i		−	1.00000i
$50$		0			0
$51$		−	92.9516i		−	1.82258i
$52$		0			0
$53$	−43.8178	+	43.8178i	−0.826751	+	0.826751i	−0.987066	−	0.160315i	$-0.948749\pi$
$53$	−43.8178	+	43.8178i	−0.826751	+	0.826751i	0.160315	+	0.987066i	$0.448749\pi$
$54$	−52.2853	+	13.5000i	−0.968246	+	0.250000i
$55$		0			0
$56$		0			0
$57$	65.7267	+	65.7267i	1.15310	+	1.15310i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	118.000			1.93443			0.967213	−	0.253966i	$-0.0817352\pi$
$61$	118.000			1.93443			0.967213	+	0.253966i	$0.0817352\pi$
$62$	62.9439	−	106.762i	1.01522	−	1.72196i
$63$		0			0
$64$	−63.9042	−	3.50000i	−0.998504	−	0.0546875i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	34.2548	−	119.108i	0.503746	−	1.75158i
$69$			102.000i			1.47826i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−71.9731	−	1.96949i	−0.999626	−	0.0273540i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$	60.0000	+	108.444i	0.789474	+	1.42689i
$77$		0			0
$78$		0			0
$79$	123.935			1.56880			0.784402	−	0.620253i	$-0.212970\pi$
$79$	123.935			1.56880			0.784402	+	0.620253i	$0.212970\pi$
$80$		0			0
$81$	−81.0000			−1.00000
$82$		0			0
$83$	−108.894	−	108.894i	−1.31198	−	1.31198i	−0.919953	−	0.392028i	$-0.871774\pi$
$83$	−108.894	−	108.894i	−1.31198	−	1.31198i	−0.392028	−	0.919953i	$-0.628226\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−37.5893	+	130.702i	−0.408579	+	1.42068i
$93$	131.453	−	131.453i	1.41348	−	1.41348i
$94$	27.1109	−	7.00000i	0.288414	−	0.0744681i
$95$		0			0
$96$	−91.5000	−	29.0474i	−0.953125	−	0.302577i
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	49.7719	−	84.4201i	0.507877	−	0.861430i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	94.4159	−	160.143i	0.925646	−	1.57003i
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$	−120.000	+	30.9839i	−1.13208	+	0.292301i
$107$	74.9533	−	74.9533i	0.700498	−	0.700498i	−0.264019	−	0.964517i	$-0.585048\pi$
$107$	74.9533	−	74.9533i	0.700498	−	0.700498i	0.964517	+	0.264019i	$0.0850482\pi$
$108$	−103.793	−	29.8503i	−0.961045	−	0.276392i
$109$		−	22.0000i		−	0.201835i	−0.994895	−	0.100917i	$-0.967822\pi$
$109$		−	22.0000i		−	0.201835i	0.994895	−	0.100917i	$-0.0321778\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−65.7267	+	65.7267i	−0.581652	+	0.581652i	−0.935357	−	0.353705i	$-0.884922\pi$
$113$	−65.7267	+	65.7267i	−0.581652	+	0.581652i	0.353705	+	0.935357i	$0.384922\pi$
$114$	46.4758	+	180.000i	0.407682	+	1.57895i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−121.000			−1.00000
$122$	203.297	+	119.859i	1.66637	+	0.982450i
$123$		0			0
$124$	216.887	−	120.000i	1.74909	−	0.967742i
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−106.543	−	70.9409i	−0.832366	−	0.554226i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	180.000	−	170.411i	1.32353	−	1.25302i
$137$	−109.545	−	109.545i	−0.799595	−	0.799595i	0.183437	−	0.983032i	$-0.441278\pi$
$137$	−109.545	−	109.545i	−0.799595	−	0.799595i	−0.983032	+	0.183437i	$0.941278\pi$
$138$	−103.607	+	175.732i	−0.750774	+	1.27342i
$139$	−92.9516			−0.668717			−0.334358	−	0.942446i	$-0.608520\pi$
$139$	−92.9516			−0.668717			−0.334358	+	0.942446i	$0.608520\pi$
$140$		0			0
$141$	42.0000			0.297872
$142$		0			0
$143$		0			0
$144$	−121.999	−	76.5000i	−0.847215	−	0.531250i
$145$		0			0
$146$		0			0
$147$	103.945	−	103.945i	0.707107	−	0.707107i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$			185.903i			1.23115i	0.788079	+	0.615574i	$0.211076\pi$
$151$			185.903i			1.23115i	−0.788079	+	0.615574i	$0.788924\pi$
$152$	−6.78026	+	247.778i	−0.0446070	+	1.63012i
$153$	197.180	−	197.180i	1.28876	−	1.28876i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	213.523	+	125.888i	1.35141	+	0.796758i
$159$	−185.903			−1.16920
$160$		0			0
$161$		0			0
$162$	−139.552	−	82.2760i	−0.861430	−	0.507877i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	−77.0000	−	298.220i	−0.463855	−	1.79650i
$167$	−179.605	+	179.605i	−1.07548	+	1.07548i	−0.0785713	+	0.996908i	$0.525036\pi$
$167$	−179.605	+	179.605i	−1.07548	+	1.07548i	−0.996908	+	0.0785713i	$0.974964\pi$
$168$		0			0
$169$		−	169.000i		−	1.00000i
$170$		0			0
$171$			278.855i			1.63073i
$172$		0			0
$173$	−219.089	+	219.089i	−1.26641	+	1.26641i	−0.318481	+	0.947929i	$0.603173\pi$
$173$	−219.089	+	219.089i	−1.26641	+	1.26641i	−0.947929	+	0.318481i	$0.896827\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	122.000			0.674033			0.337017	−	0.941499i	$-0.390582\pi$
$181$	122.000			0.674033			0.337017	+	0.941499i	$0.390582\pi$
$182$		0			0
$183$	250.316	+	250.316i	1.36785	+	1.36785i
$184$	−197.522	+	187.000i	−1.07349	+	1.01630i
$185$		0			0
$186$	360.000	−	92.9516i	1.93548	−	0.499740i
$187$		0			0
$188$	53.8185	+	15.4779i	0.286269	+	0.0823295i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−128.137	−	142.986i	−0.667379	−	0.744719i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	171.500	−	94.8881i	0.875000	−	0.484123i
$197$	−87.6356	−	87.6356i	−0.444851	−	0.444851i	0.448788	−	0.893638i	$-0.351856\pi$
$197$	−87.6356	−	87.6356i	−0.444851	−	0.444851i	−0.893638	+	0.448788i	$0.851856\pi$
$198$		0			0
$199$	−371.806			−1.86837			−0.934187	−	0.356784i	$-0.883873\pi$
$199$	−371.806			−1.86837			−0.934187	+	0.356784i	$0.883873\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	325.331	−	180.000i	1.59476	−	0.882353i
$205$		0			0
$206$		0			0
$207$	−216.375	+	216.375i	−1.04529	+	1.04529i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			216.887i			1.02790i	0.857820	+	0.513950i	$0.171818\pi$
$211$			216.887i			1.02790i	−0.857820	+	0.513950i	$0.828182\pi$
$212$	−238.215	−	68.5095i	−1.12366	−	0.323158i
$213$		0			0
$214$	205.268	−	53.0000i	0.959197	−	0.247664i
$215$		0			0
$216$	−148.500	−	156.856i	−0.687500	−	0.726184i
$217$		0			0
$218$	22.3466	−	37.9029i	0.102507	−	0.173867i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$	−180.000	+	46.4758i	−0.796460	+	0.205645i
$227$	94.7523	−	94.7523i	0.417411	−	0.417411i	−0.466899	−	0.884310i	$-0.654629\pi$
$227$	94.7523	−	94.7523i	0.417411	−	0.417411i	0.884310	+	0.466899i	$0.154629\pi$
$228$	−102.764	+	357.323i	−0.450720	+	1.56720i
$229$			218.000i			0.951965i	0.879455	+	0.475983i	$0.157907\pi$
$229$			218.000i			0.951965i	−0.879455	+	0.475983i	$0.842093\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−328.634	+	328.634i	−1.41044	+	1.41044i	−0.653631	+	0.756814i	$0.726755\pi$
$233$	−328.634	+	328.634i	−1.41044	+	1.41044i	−0.756814	+	0.653631i	$0.773245\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	262.907	+	262.907i	1.10931	+	1.10931i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−478.000			−1.98340			−0.991701	−	0.128564i	$-0.958963\pi$
$241$	−478.000			−1.98340			−0.991701	+	0.128564i	$0.958963\pi$
$242$	−208.466	−	122.906i	−0.861430	−	0.507877i
$243$	−171.827	−	171.827i	−0.707107	−	0.707107i
$244$	228.506	+	413.000i	0.936500	+	1.69262i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	495.556	+	13.5605i	1.99821	+	0.0546795i
$249$		−	462.000i		−	1.85542i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−111.500	−	230.443i	−0.435547	−	0.900166i
$257$	153.362	+	153.362i	0.596741	+	0.596741i	0.939444	−	0.342703i	$-0.111343\pi$
$257$	153.362	+	153.362i	0.596741	+	0.596741i	−0.342703	+	0.939444i	$0.611343\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	315.370	+	315.370i	1.19912	+	1.19912i	0.974429	+	0.224695i	$0.0721385\pi$
$263$	315.370	+	315.370i	1.19912	+	1.19912i	0.224695	+	0.974429i	$0.427861\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			247.871i			0.914653i	0.889299	+	0.457326i	$0.151193\pi$
$271$			247.871i			0.914653i	−0.889299	+	0.457326i	$0.848807\pi$
$272$	483.211	−	110.759i	1.77651	−	0.407203i
$273$		0			0
$274$	−77.4597	−	300.000i	−0.282700	−	1.09489i
$275$		0			0
$276$	−357.000	+	197.522i	−1.29348	+	0.715660i
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	−160.143	−	94.4159i	−0.576052	−	0.339625i
$279$	557.710			1.99896
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	72.3601	+	42.6616i	0.256596	+	0.151282i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−132.482	−	255.720i	−0.460007	−	0.887915i
$289$			671.000i			2.32180i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	306.725	−	306.725i	1.04684	−	1.04684i	0.0479941	−	0.998848i	$-0.484717\pi$
$293$	306.725	−	306.725i	1.04684	−	1.04684i	0.998848	−	0.0479941i	$-0.0152829\pi$
$294$	284.664	−	73.5000i	0.968246	−	0.250000i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−188.832	+	320.285i	−0.625271	+	1.06055i
$303$		0			0
$304$	−263.363	+	420.000i	−0.866325	+	1.38158i
$305$		0			0
$306$	540.000	−	139.427i	1.76471	−	0.455645i
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		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	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$	240.000	+	433.774i	0.759494	+	1.37270i
$317$	−438.178	−	438.178i	−1.38227	−	1.38227i	−0.840584	−	0.541681i	$-0.817788\pi$
$317$	−438.178	−	438.178i	−1.38227	−	1.38227i	−0.541681	−	0.840584i	$-0.682212\pi$
$318$	−320.285	−	188.832i	−1.00719	−	0.593810i
$319$		0			0
$320$		0			0
$321$	318.000			0.990654
$322$		0			0
$323$	−678.823	−	678.823i	−2.10162	−	2.10162i
$324$	−156.856	−	283.500i	−0.484123	−	0.875000i
$325$		0			0
$326$		0			0
$327$	46.6690	−	46.6690i	0.142719	−	0.142719i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	650.661i		−	1.96574i	−0.184290	−	0.982872i	$-0.558999\pi$
$331$		−	650.661i		−	1.96574i	0.184290	−	0.982872i	$-0.441001\pi$
$332$	170.257	−	592.004i	0.512823	−	1.78314i
$333$		0			0
$334$	−491.869	+	127.000i	−1.47266	+	0.380240i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	171.662	−	291.163i	0.507877	−	0.861430i
$339$	−278.855			−0.822581
$340$		0			0
$341$		0			0
$342$	−283.248	+	480.428i	−0.828209	+	1.40476i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−600.000	+	154.919i	−1.73410	+	0.447744i
$347$	414.365	−	414.365i	1.19413	−	1.19413i	0.218239	−	0.975895i	$-0.429969\pi$
$347$	414.365	−	414.365i	1.19413	−	1.19413i	0.975895	−	0.218239i	$-0.0700312\pi$
$348$		0			0
$349$		−	458.000i		−	1.31232i	−0.754621	−	0.656160i	$-0.772179\pi$
$349$		−	458.000i		−	1.31232i	0.754621	−	0.656160i	$-0.227821\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	460.087	−	460.087i	1.30336	−	1.30336i	0.377252	−	0.926111i	$-0.376869\pi$
$353$	460.087	−	460.087i	1.30336	−	1.30336i	0.926111	−	0.377252i	$-0.123131\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	599.000			1.65928
$362$	210.189	+	123.922i	0.580632	+	0.342326i
$363$	−256.680	−	256.680i	−0.707107	−	0.707107i
$364$		0			0
$365$		0			0
$366$	177.000	+	685.518i	0.483607	+	1.87300i
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	−530.249	+	121.541i	−1.44089	+	0.330275i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	714.645	+	205.529i	1.92109	+	0.552496i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	77.0000	+	81.3327i	0.204787	+	0.216310i
$377$		0			0
$378$		0			0
$379$	154.919			0.408758			0.204379	−	0.978892i	$-0.434482\pi$
$379$	154.919			0.408758			0.204379	+	0.978892i	$0.434482\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	485.075	+	485.075i	1.26652	+	1.26652i	0.947876	+	0.318639i	$0.103226\pi$
$383$	485.075	+	485.075i	1.26652	+	1.26652i	0.318639	+	0.947876i	$0.396774\pi$
$384$	−75.5232	−	376.500i	−0.196675	−	0.980469i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		−	1053.45i		−	2.69425i
$392$	391.853	+	10.7228i	0.999626	+	0.0273540i
$393$		0			0
$394$	−61.9677	−	240.000i	−0.157279	−	0.609137i
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	−640.570	−	377.663i	−1.60947	−	0.948903i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	743.335	+	20.3408i	1.82190	+	0.0498548i
$409$			142.000i			0.347188i	0.984817	+	0.173594i	$0.0555381\pi$
$409$			142.000i			0.347188i	−0.984817	+	0.173594i	$0.944462\pi$
$410$		0			0
$411$		−	464.758i		−	1.13080i
$412$		0			0
$413$		0			0
$414$	−592.566	+	153.000i	−1.43132	+	0.369565i
$415$		0			0
$416$		0			0
$417$	−197.180	−	197.180i	−0.472854	−	0.472854i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−602.000			−1.42993			−0.714964	−	0.699161i	$-0.753557\pi$
$421$	−602.000			−1.42993			−0.714964	+	0.699161i	$0.753557\pi$
$422$	−220.304	+	373.666i	−0.522047	+	0.885464i
$423$	89.0955	+	89.0955i	0.210628	+	0.210628i
$424$	−340.823	−	360.000i	−0.803827	−	0.849057i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	407.483	+	117.190i	0.952063	+	0.273809i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−96.5179	−	421.080i	−0.223421	−	0.974722i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	77.0000	−	42.6028i	0.176606	−	0.0977129i
$437$	744.903	+	744.903i	1.70458	+	1.70458i
$438$		0			0
$439$	619.677			1.41157			0.705783	−	0.708428i	$-0.250595\pi$
$439$	619.677			1.41157			0.705783	+	0.708428i	$0.250595\pi$
$440$		0			0
$441$	441.000			1.00000
$442$		0			0
$443$	−400.222	−	400.222i	−0.903437	−	0.903437i	0.0922950	−	0.995732i	$-0.470580\pi$
$443$	−400.222	−	400.222i	−0.903437	−	0.903437i	−0.995732	+	0.0922950i	$0.970580\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−357.323	−	102.764i	−0.790537	−	0.227355i
$453$	−394.360	+	394.360i	−0.870552	+	0.870552i
$454$	259.490	−	67.0000i	0.571564	−	0.147577i
$455$		0			0
$456$	−540.000	+	511.234i	−1.18421	+	1.12113i
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	−221.434	+	375.583i	−0.483481	+	0.820051i
$459$	836.564			1.82258
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	−900.000	+	232.379i	−1.93133	+	0.498667i
$467$	−244.659	+	244.659i	−0.523895	+	0.523895i	−0.918745	−	0.394850i	$-0.870796\pi$
$467$	−244.659	+	244.659i	−0.523895	+	0.523895i	0.394850	+	0.918745i	$0.370796\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$	185.903	+	720.000i	0.392201	+	1.51899i
$475$		0			0
$476$		0			0
$477$	−394.360	−	394.360i	−0.826751	−	0.826751i
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	−823.527	−	485.530i	−1.70856	−	1.00732i
$483$		0			0
$484$	−234.315	−	423.500i	−0.484123	−	0.875000i
$485$		0			0
$486$	−121.500	−	470.567i	−0.250000	−	0.968246i
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	−25.8222	+	943.647i	−0.0529143	+	1.93370i
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	840.000	+	526.726i	1.69355	+	1.06195i
$497$		0			0
$498$	469.278	−	795.961i	0.942325	−	1.59832i
$499$	340.823			0.683011			0.341506	−	0.939880i	$-0.389063\pi$
$499$	340.823			0.683011			0.341506	+	0.939880i	$0.389063\pi$
$500$		0			0
$501$	−762.000			−1.52096
$502$		0			0
$503$	702.864	+	702.864i	1.39734	+	1.39734i	0.807563	+	0.589781i	$0.200786\pi$
$503$	702.864	+	702.864i	1.39734	+	1.39734i	0.589781	+	0.807563i	$0.299214\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	358.503	−	358.503i	0.707107	−	0.707107i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	41.9738	−	510.277i	0.0819801	−	0.996634i
$513$	−591.540	+	591.540i	−1.15310	+	1.15310i
$514$	108.444	+	420.000i	0.210980	+	0.817121i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−929.516			−1.79097
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$	223.000	+	863.675i	0.423954	+	1.64197i
$527$	−1357.65	+	1357.65i	−2.57618	+	2.57618i
$528$		0			0
$529$			627.000i			1.18526i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1078.00			1.99261			0.996303	−	0.0859072i	$-0.0273789\pi$
$541$	1078.00			1.99261			0.996303	+	0.0859072i	$0.0273789\pi$
$542$	−251.776	+	427.047i	−0.464531	+	0.787909i
$543$	258.801	+	258.801i	0.476613	+	0.476613i
$544$	945.008	+	300.000i	1.73715	+	0.551471i
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	171.274	−	595.538i	0.312543	−	1.08675i
$549$			1062.00i			1.93443i
$550$		0			0
$551$		0			0
$552$	−815.695	−	22.3209i	−1.47771	−	0.0404363i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−180.000	−	325.331i	−0.323741	−	0.585127i
$557$	657.267	+	657.267i	1.18001	+	1.18001i	0.979740	+	0.200272i	$0.0641827\pi$
$557$	657.267	+	657.267i	1.18001	+	1.18001i	0.200272	+	0.979740i	$0.435817\pi$
$558$	960.855	+	566.495i	1.72196	+	1.01522i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	108.894	+	108.894i	0.193418	+	0.193418i	0.797171	−	0.603753i	$-0.206329\pi$
$563$	108.894	+	108.894i	0.193418	+	0.193418i	−0.603753	+	0.797171i	$0.706329\pi$
$564$	81.3327	+	147.000i	0.144207	+	0.260638i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			1084.44i			1.89919i	0.313485	+	0.949593i	$0.398503\pi$
$571$			1084.44i			1.89919i	−0.313485	+	0.949593i	$0.601497\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	31.5000	−	575.138i	0.0546875	−	0.998504i
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−681.570	+	1156.04i	−1.17919	+	2.00007i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	840.000	−	216.887i	1.43345	−	0.370114i
$587$	603.869	−	603.869i	1.02874	−	1.02874i	0.0291633	−	0.999575i	$-0.490716\pi$
$587$	603.869	−	603.869i	1.02874	−	1.02874i	0.999575	−	0.0291633i	$-0.00928429\pi$
$588$	565.094	+	162.518i	0.961045	+	0.276392i
$589$		−	1920.00i		−	3.25976i
$590$		0			0
$591$		−	371.806i		−	0.629114i
$592$		0			0
$593$	153.362	−	153.362i	0.258621	−	0.258621i	−0.565872	−	0.824493i	$-0.691460\pi$
$593$	153.362	−	153.362i	0.258621	−	0.258621i	0.824493	+	0.565872i	$0.191460\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−788.720	−	788.720i	−1.32114	−	1.32114i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	242.000			0.402662			0.201331	−	0.979523i	$-0.435473\pi$
$601$	242.000			0.402662			0.201331	+	0.979523i	$0.435473\pi$
$602$		0			0
$603$		0			0
$604$	−650.661	+	360.000i	−1.07725	+	0.596026i
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−880.354	+	456.089i	−1.44795	+	0.750147i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1071.97	+	308.293i	1.75158	+	0.503746i
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	240.998	+	240.998i	0.390596	+	0.390596i	0.874900	−	0.484304i	$-0.160927\pi$
$617$	240.998	+	240.998i	0.390596	+	0.390596i	−0.484304	+	0.874900i	$0.660927\pi$
$618$		0			0
$619$	−1022.47			−1.65181			−0.825903	−	0.563813i	$-0.809334\pi$
$619$	−1022.47			−1.65181			−0.825903	+	0.563813i	$0.809334\pi$
$620$		0			0
$621$	−918.000			−1.47826
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			1239.35i			1.96411i	0.188590	+	0.982056i	$0.439608\pi$
$631$			1239.35i			1.96411i	−0.188590	+	0.982056i	$0.560392\pi$
$632$	−27.1210	+	991.113i	−0.0429130	+	1.56822i
$633$	−460.087	+	460.087i	−0.726836	+	0.726836i
$634$	−309.839	−	1200.00i	−0.488705	−	1.89274i
$635$		0			0
$636$	−360.000	−	650.661i	−0.566038	−	1.02305i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	547.869	+	323.009i	0.853379	+	0.503130i
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	−480.000	−	1859.03i	−0.743034	−	2.87776i
$647$	499.217	−	499.217i	0.771588	−	0.771588i	−0.206796	−	0.978384i	$-0.566304\pi$
$647$	499.217	−	499.217i	0.771588	−	0.771588i	0.978384	+	0.206796i	$0.0663037\pi$
$648$	17.7254	−	647.758i	0.0273540	−	0.999626i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	481.996	−	481.996i	0.738125	−	0.738125i	−0.234090	−	0.972215i	$-0.575211\pi$
$653$	481.996	−	481.996i	0.738125	−	0.738125i	0.972215	+	0.234090i	$0.0752109\pi$
$654$	127.808	−	33.0000i	0.195426	−	0.0504587i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−838.000			−1.26778			−0.633888	−	0.773425i	$-0.718542\pi$
$661$	−838.000			−1.26778			−0.633888	+	0.773425i	$0.718542\pi$
$662$	660.911	−	1121.00i	0.998355	−	1.69335i
$663$		0			0
$664$	894.659	−	847.000i	1.34738	−	1.27560i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−976.422	−	280.814i	−1.46171	−	0.420380i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	591.500	−	327.267i	0.875000	−	0.484123i
$677$	−920.174	−	920.174i	−1.35919	−	1.35919i	−0.874913	−	0.484281i	$-0.839081\pi$
$677$	−920.174	−	920.174i	−1.35919	−	1.35919i	−0.484281	−	0.874913i	$-0.660919\pi$
$678$	−480.428	−	283.248i	−0.708595	−	0.417769i
$679$		0			0
$680$		0			0
$681$	402.000			0.590308
$682$		0			0
$683$	−60.8112	−	60.8112i	−0.0890354	−	0.0890354i	0.661186	−	0.750222i	$-0.270053\pi$
$683$	−60.8112	−	60.8112i	−0.0890354	−	0.0890354i	−0.750222	+	0.661186i	$0.770053\pi$
$684$	−975.992	+	540.000i	−1.42689	+	0.789474i
$685$		0			0
$686$		0			0
$687$	−462.448	+	462.448i	−0.673141	+	0.673141i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	402.790i		−	0.582909i	−0.956585	−	0.291455i	$-0.905861\pi$
$691$		−	402.790i		−	0.582909i	0.956585	−	0.291455i	$-0.0941392\pi$
$692$	−1191.08	−	342.548i	−1.72121	−	0.495011i
$693$		0			0
$694$	1134.78	−	293.000i	1.63514	−	0.422190i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	465.215	−	789.070i	0.666497	−	1.13047i
$699$	−1394.27			−1.99467
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1260.00	−	325.331i	1.78470	−	0.460808i
$707$		0			0
$708$		0			0
$709$		−	742.000i		−	1.04654i	−0.852166	−	0.523272i	$-0.824711\pi$
$709$		−	742.000i		−	1.04654i	0.852166	−	0.523272i	$-0.175289\pi$
$710$		0			0
$711$			1115.42i			1.56880i
$712$		0			0
$713$	1489.81	−	1489.81i	2.08949	−	2.08949i
$714$		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
$721$		0			0
$722$	1031.99	+	608.436i	1.42935	+	0.842709i
$723$	−1013.99	−	1013.99i	−1.40248	−	1.40248i
$724$	236.252	+	427.000i	0.326315	+	0.589779i
$725$		0			0
$726$	−181.500	−	702.946i	−0.250000	−	0.968246i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	729.000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	−391.371	+	1360.84i	−0.534660	+	1.85907i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	−1037.00	−	329.204i	−1.40897	−	0.447287i
$737$		0			0
$738$		0			0
$739$	−216.887			−0.293487			−0.146744	−	0.989175i	$-0.546879\pi$
$739$	−216.887			−0.293487			−0.146744	+	0.989175i	$0.546879\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−363.453	−	363.453i	−0.489169	−	0.489169i	0.418875	−	0.908044i	$-0.362425\pi$
$743$	−363.453	−	363.453i	−0.489169	−	0.489169i	−0.908044	+	0.418875i	$0.862425\pi$
$744$	1022.47	+	1080.00i	1.37428	+	1.45161i
$745$		0			0
$746$		0			0
$747$	980.050	−	980.050i	1.31198	−	1.31198i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			433.774i			0.577595i	0.957390	+	0.288798i	$0.0932555\pi$
$751$			433.774i			0.577595i	−0.957390	+	0.288798i	$0.906745\pi$
$752$	50.0463	+	218.338i	0.0665510	+	0.290343i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	266.904	+	157.360i	0.352116	+	0.207599i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	343.000	+	1328.43i	0.447781	+	1.73425i
$767$		0			0
$768$	252.315	−	725.370i	0.328535	−	0.944492i
$769$			578.000i			0.751625i	0.926696	+	0.375813i	$0.122636\pi$
$769$			578.000i			0.751625i	−0.926696	+	0.375813i	$0.877364\pi$
$770$		0			0
$771$			650.661i			0.843919i
$772$		0			0
$773$	175.271	−	175.271i	0.226742	−	0.226742i	−0.584588	−	0.811330i	$-0.698744\pi$
$773$	175.271	−	175.271i	0.226742	−	0.226742i	0.811330	+	0.584588i	$0.198744\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	1070.05	−	1814.95i	1.36835	−	2.32091i
$783$		0			0
$784$	664.217	+	416.500i	0.847215	+	0.531250i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	137.019	−	476.430i	0.173882	−	0.604607i
$789$			1338.00i			1.69582i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−720.000	−	1301.32i	−0.904523	−	1.63483i
$797$	963.992	+	963.992i	1.20953	+	1.20953i	0.971181	+	0.238345i	$0.0766048\pi$
$797$	963.992	+	963.992i	1.20953	+	1.20953i	0.238345	+	0.971181i	$0.423395\pi$
$798$		0			0
$799$	−433.774			−0.542896
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$			1208.37i			1.48998i	0.667078	+	0.744988i	$0.267545\pi$
$811$			1208.37i			1.48998i	−0.667078	+	0.744988i	$0.732455\pi$
$812$		0			0
$813$	−525.814	+	525.814i	−0.646757	+	0.646757i
$814$		0			0
$815$		0			0
$816$	1260.00	+	790.089i	1.54412	+	0.968246i
$817$		0			0
$818$	−144.237	+	244.646i	−0.176329	+	0.299078i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	472.079	−	800.713i	0.574306	−	0.974103i
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	264.458	−	264.458i	0.319780	−	0.319780i	−0.528903	−	0.848683i	$-0.677396\pi$
$827$	264.458	−	264.458i	0.319780	−	0.319780i	0.848683	+	0.528903i	$0.177396\pi$
$828$	−1176.32	−	338.304i	−1.42068	−	0.408579i
$829$			502.000i			0.605549i	0.953062	+	0.302774i	$0.0979129\pi$
$829$			502.000i			0.605549i	−0.953062	+	0.302774i	$0.902087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1073.54	+	1073.54i	−1.28876	+	1.28876i
$834$	−139.427	−	540.000i	−0.167179	−	0.647482i
$835$		0			0
$836$		0			0
$837$	1183.08	+	1183.08i	1.41348	+	1.41348i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−841.000			−1.00000
$842$	−1037.16	−	611.483i	−1.23178	−	0.726227i
$843$		0			0
$844$	−759.105	+	420.000i	−0.899413	+	0.497630i
$845$		0			0
$846$	63.0000	+	243.998i	0.0744681	+	0.288414i
$847$		0			0
$848$	−221.518	−	966.421i	−0.261224	−	1.13965i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	583.000	+	615.804i	0.681075	+	0.719398i
$857$	−284.816	−	284.816i	−0.332340	−	0.332340i	0.521134	−	0.853475i	$-0.325509\pi$
$857$	−284.816	−	284.816i	−0.332340	−	0.332340i	−0.853475	+	0.521134i	$0.825509\pi$
$858$		0			0
$859$	1704.11			1.98383			0.991917	−	0.126892i	$-0.0405001\pi$
$859$	1704.11			1.98383			0.991917	+	0.126892i	$0.0405001\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−193.747	−	193.747i	−0.224504	−	0.224504i	0.585888	−	0.810392i	$-0.300746\pi$
$863$	−193.747	−	193.747i	−0.224504	−	0.224504i	−0.810392	+	0.585888i	$0.800746\pi$
$864$	261.426	−	823.500i	0.302577	−	0.953125i
$865$		0			0
$866$		0			0
$867$	−1423.41	+	1423.41i	−1.64176	+	1.64176i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	175.934	+	4.81430i	0.201759	+	0.00552099i
$873$		0			0
$874$	526.726	+	2040.00i	0.602661	+	2.33410i
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	1067.62	+	629.439i	1.21596	+	0.716901i
$879$	1301.32			1.48046
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	759.781	+	447.947i	0.861430	+	0.507877i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$	−283.000	−	1096.05i	−0.319413	−	1.23708i
$887$	1197.84	−	1197.84i	1.35044	−	1.35044i	0.465269	−	0.885169i	$-0.345958\pi$
$887$	1197.84	−	1197.84i	1.35044	−	1.35044i	0.885169	−	0.465269i	$-0.154042\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	306.725	−	306.725i	0.343477	−	0.343477i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1920.00			2.13097
$902$		0			0
$903$		0			0
$904$	−511.234	−	540.000i	−0.565524	−	0.597345i
$905$		0			0
$906$	−1080.00	+	278.855i	−1.19205	+	0.307787i
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	515.120	+	148.146i	0.567313	+	0.163156i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−1449.63	+	332.278i	−1.58951	+	0.364339i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−763.000	+	422.155i	−0.832969	+	0.460868i
$917$		0			0
$918$	1441.28	+	849.743i	1.57003	+	0.925646i
$919$	−1301.32			−1.41602			−0.708010	−	0.706202i	$-0.750407\pi$
$919$	−1301.32			−1.41602			−0.708010	+	0.706202i	$0.750407\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		−	1518.21i		−	1.63073i
$932$	−1786.61	−	513.821i	−1.91697	−	0.551310i
$933$		0			0
$934$	−670.026	+	173.000i	−0.717373	+	0.185225i
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1112.99	+	1112.99i	−1.17528	+	1.17528i	−0.194342	+	0.980934i	$0.562257\pi$
$947$	−1112.99	+	1112.99i	−1.17528	+	1.17528i	−0.980934	+	0.194342i	$0.937743\pi$
$948$	−411.057	+	1429.29i	−0.433604	+	1.50769i
$949$		0			0
$950$		0			0
$951$		−	1859.03i		−	1.95482i
$952$		0			0
$953$	−854.447	+	854.447i	−0.896587	+	0.896587i	−0.995133	−	0.0985458i	$-0.968581\pi$
$953$	−854.447	+	854.447i	−0.896587	+	0.896587i	0.0985458	+	0.995133i	$0.468581\pi$
$954$	−278.855	−	1080.00i	−0.292301	−	1.13208i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2879.00			−2.99584
$962$		0			0
$963$	674.580	+	674.580i	0.700498	+	0.700498i
$964$	−925.643	−	1673.00i	−0.960211	−	1.73548i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	26.4787	−	967.638i	0.0273540	−	0.999626i
$969$		−	2880.00i		−	2.97214i
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	268.653	−	934.136i	0.276392	−	0.961045i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1003.00	+	1599.54i	−1.02766	+	1.63888i
$977$	197.180	+	197.180i	0.201822	+	0.201822i	0.800780	−	0.598958i	$-0.204418\pi$
$977$	197.180	+	197.180i	0.201822	+	0.201822i	−0.598958	+	0.800780i	$0.704418\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	198.000			0.201835
$982$		0			0
$983$	1381.69	+	1381.69i	1.40558	+	1.40558i	0.780798	+	0.624783i	$0.214813\pi$
$983$	1381.69	+	1381.69i	1.40558	+	1.40558i	0.624783	+	0.780798i	$0.285187\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	1735.10i		−	1.75085i	−0.483350	−	0.875427i	$-0.660580\pi$
$991$		−	1735.10i		−	1.75085i	0.483350	−	0.875427i	$-0.339420\pi$
$992$	912.179	+	1760.71i	0.919535	+	1.77491i
$993$	1380.26	−	1380.26i	1.38999	−	1.38999i
$994$		0			0
$995$		0			0
$996$	1617.00	−	894.659i	1.62349	−	0.898252i
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	587.189	+	346.192i	0.588366	+	0.346885i
$999$		0			0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+(1.72286 + 1.01575i) q^{2} +(2.12132 + 2.12132i) q^{3} +(1.93649 + 3.50000i) q^{4} +(1.50000 + 5.80948i) q^{6} +(-0.218832 + 7.99701i) q^{8} +9.00000i q^{9} +O(q^{10})\)	\(q+(1.72286 + 1.01575i) q^{2} +(2.12132 + 2.12132i) q^{3} +(1.93649 + 3.50000i) q^{4} +(1.50000 + 5.80948i) q^{6} +(-0.218832 + 7.99701i) q^{8} +9.00000i q^{9} +(-3.31670 + 11.5325i) q^{12} +(-8.50000 + 13.5554i) q^{16} +(-21.9089 - 21.9089i) q^{17} +(-9.14178 + 15.5057i) q^{18} +30.9839 q^{19} +(24.0416 + 24.0416i) q^{23} +(-17.4284 + 16.5000i) q^{24} +(-19.0919 + 19.0919i) q^{27} -61.9677i q^{31} +(-28.4133 + 14.7202i) q^{32} +(-15.4919 - 60.0000i) q^{34} +(-31.5000 + 17.4284i) q^{36} +(53.3809 + 31.4720i) q^{38} +(17.0000 + 65.8407i) q^{46} +(9.89949 - 9.89949i) q^{47} +(-46.7867 + 10.7242i) q^{48} -49.0000i q^{49} -92.9516i q^{51} +(-43.8178 + 43.8178i) q^{53} +(-52.2853 + 13.5000i) q^{54} +(65.7267 + 65.7267i) q^{57} +118.000 q^{61} +(62.9439 - 106.762i) q^{62} +(-63.9042 - 3.50000i) q^{64} +(34.2548 - 119.108i) q^{68} +102.000i q^{69} +(-71.9731 - 1.96949i) q^{72} +(60.0000 + 108.444i) q^{76} +123.935 q^{79} -81.0000 q^{81} +(-108.894 - 108.894i) q^{83} +(-37.5893 + 130.702i) q^{92} +(131.453 - 131.453i) q^{93} +(27.1109 - 7.00000i) q^{94} +(-91.5000 - 29.0474i) q^{96} +(49.7719 - 84.4201i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{6}+O(q^{10}) \) 8 * q + 12 * q^6	\( 8 q + 12 q^{6} - 68 q^{16} - 252 q^{36} + 136 q^{46} + 944 q^{61} + 480 q^{76} - 648 q^{81} - 732 q^{96}+O(q^{100}) \) 8 * q + 12 * q^6 - 68 * q^16 - 252 * q^36 + 136 * q^46 + 944 * q^61 + 480 * q^76 - 648 * q^81 - 732 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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