LMFDB - Embedded newform 1000.2.k.b.307.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1000,2,Mod(307,1000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1000, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1000.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1000 = 2^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1000.k (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:	$7.98504020213$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.1024000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{4} + 25 $ x^8 + 15*x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			307.3
Root			$1.34500 - 1.34500i$ of defining polynomial
Character	$\chi$	$=$	1000.307
Dual form			1000.2.k.b.443.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(0.240706 - 0.240706i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(0.240706 - 0.240706i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} -5.39698 q^{11} +(-4.24357 - 4.24357i) q^{13} -0.481412i q^{14} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{18} -8.57158i q^{19} +(-5.39698 + 5.39698i) q^{22} +(-1.15340 - 1.15340i) q^{23} -8.48715 q^{26} +(-0.481412 - 0.481412i) q^{28} +(-4.00000 + 4.00000i) q^{32} +6.00000 q^{36} +(8.33088 - 8.33088i) q^{37} +(-8.57158 - 8.57158i) q^{38} -9.05299 q^{41} +10.7940i q^{44} -2.30681 q^{46} +(9.64055 - 9.64055i) q^{47} +6.88412i q^{49} +(-8.48715 + 8.48715i) q^{52} +(8.81229 + 8.81229i) q^{53} -0.962825 q^{56} +7.60876i q^{59} +(0.722119 + 0.722119i) q^{63} +8.00000i q^{64} +(6.00000 - 6.00000i) q^{72} -16.6618i q^{74} -17.1432 q^{76} +(-1.29909 + 1.29909i) q^{77} -9.00000 q^{81} +(-9.05299 + 9.05299i) q^{82} +(10.7940 + 10.7940i) q^{88} +7.70378i q^{89} -2.04291 q^{91} +(-2.30681 + 2.30681i) q^{92} -19.2811i q^{94} +(6.88412 + 6.88412i) q^{98} -16.1909i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8	$ 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8} - 4 q^{11} + 8 q^{13} - 32 q^{16} + 24 q^{18} - 4 q^{22} - 12 q^{23} + 16 q^{26} + 8 q^{28} - 32 q^{32} + 48 q^{36} + 16 q^{37} - 12 q^{38} - 4 q^{41} - 24 q^{46} - 4 q^{47} + 16 q^{52} + 8 q^{53} + 16 q^{56} - 12 q^{63} + 48 q^{72} - 24 q^{76} - 8 q^{77} - 72 q^{81} - 4 q^{82} + 8 q^{88} + 32 q^{91} - 24 q^{92} - 68 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8 - 4 * q^11 + 8 * q^13 - 32 * q^16 + 24 * q^18 - 4 * q^22 - 12 * q^23 + 16 * q^26 + 8 * q^28 - 32 * q^32 + 48 * q^36 + 16 * q^37 - 12 * q^38 - 4 * q^41 - 24 * q^46 - 4 * q^47 + 16 * q^52 + 8 * q^53 + 16 * q^56 - 12 * q^63 + 48 * q^72 - 24 * q^76 - 8 * q^77 - 72 * q^81 - 4 * q^82 + 8 * q^88 + 32 * q^91 - 24 * q^92 - 68 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times$.

$n$	$377$	$501$	$751$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.240706	−	0.240706i	0.0909784	−	0.0909784i	−0.660153	−	0.751131i	$-0.729509\pi$
$7$	0.240706	−	0.240706i	0.0909784	−	0.0909784i	0.751131	+	0.660153i	$0.229509\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$			3.00000i			1.00000i
$10$		0			0
$11$	−5.39698			−1.62725			−0.813625	−	0.581390i	$-0.802509\pi$
$11$	−5.39698			−1.62725			−0.813625	+	0.581390i	$0.802509\pi$
$12$		0			0
$13$	−4.24357	−	4.24357i	−1.17696	−	1.17696i	−0.980516	−	0.196439i	$-0.937062\pi$
$13$	−4.24357	−	4.24357i	−1.17696	−	1.17696i	−0.196439	−	0.980516i	$-0.562938\pi$
$14$		−	0.481412i		−	0.128663i
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	3.00000	+	3.00000i	0.707107	+	0.707107i
$19$		−	8.57158i		−	1.96646i	−0.182381	−	0.983228i	$-0.558380\pi$
$19$		−	8.57158i		−	1.96646i	0.182381	−	0.983228i	$-0.441620\pi$
$20$		0			0
$21$		0			0
$22$	−5.39698	+	5.39698i	−1.15064	+	1.15064i
$23$	−1.15340	−	1.15340i	−0.240501	−	0.240501i	0.576556	−	0.817057i	$-0.304396\pi$
$23$	−1.15340	−	1.15340i	−0.240501	−	0.240501i	−0.817057	+	0.576556i	$0.804396\pi$
$24$		0			0
$25$		0			0
$26$	−8.48715			−1.66447
$27$		0			0
$28$	−0.481412	−	0.481412i	−0.0909784	−	0.0909784i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	6.00000			1.00000
$37$	8.33088	−	8.33088i	1.36959	−	1.36959i	0.508563	−	0.861025i	$-0.330177\pi$
$37$	8.33088	−	8.33088i	1.36959	−	1.36959i	0.861025	−	0.508563i	$-0.169823\pi$
$38$	−8.57158	−	8.57158i	−1.39049	−	1.39049i
$39$		0			0
$40$		0			0
$41$	−9.05299			−1.41384			−0.706920	−	0.707293i	$-0.749916\pi$
$41$	−9.05299			−1.41384			−0.706920	+	0.707293i	$0.749916\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$			10.7940i			1.62725i
$45$		0			0
$46$	−2.30681			−0.340120
$47$	9.64055	−	9.64055i	1.40622	−	1.40622i	0.628027	−	0.778192i	$-0.283863\pi$
$47$	9.64055	−	9.64055i	1.40622	−	1.40622i	0.778192	−	0.628027i	$-0.216137\pi$
$48$		0			0
$49$			6.88412i			0.983446i
$50$		0			0
$51$		0			0
$52$	−8.48715	+	8.48715i	−1.17696	+	1.17696i
$53$	8.81229	+	8.81229i	1.21046	+	1.21046i	0.970875	+	0.239586i	$0.0770116\pi$
$53$	8.81229	+	8.81229i	1.21046	+	1.21046i	0.239586	+	0.970875i	$0.422988\pi$
$54$		0			0
$55$		0			0
$56$	−0.962825			−0.128663
$57$		0			0
$58$		0			0
$59$			7.60876i			0.990576i	0.868729	+	0.495288i	$0.164937\pi$
$59$			7.60876i			0.990576i	−0.868729	+	0.495288i	$0.835063\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	0.722119	+	0.722119i	0.0909784	+	0.0909784i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	6.00000	−	6.00000i	0.707107	−	0.707107i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		−	16.6618i		−	1.93689i
$75$		0			0
$76$	−17.1432			−1.96646
$77$	−1.29909	+	1.29909i	−0.148045	+	0.148045i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	−9.05299	+	9.05299i	−0.999736	+	0.999736i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	10.7940	+	10.7940i	1.15064	+	1.15064i
$89$			7.70378i			0.816599i	0.912848	+	0.408300i	$0.133878\pi$
$89$			7.70378i			0.816599i	−0.912848	+	0.408300i	$0.866122\pi$
$90$		0			0
$91$	−2.04291			−0.214155
$92$	−2.30681	+	2.30681i	−0.240501	+	0.240501i
$93$		0			0
$94$		−	19.2811i		−	1.98869i
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	6.88412	+	6.88412i	0.695401	+	0.695401i
$99$		−	16.1909i		−	1.62725i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−7.33374	−	7.33374i	−0.722615	−	0.722615i	0.246522	−	0.969137i	$-0.420712\pi$
$103$	−7.33374	−	7.33374i	−0.722615	−	0.722615i	−0.969137	+	0.246522i	$0.920712\pi$
$104$			16.9743i			1.66447i
$105$		0			0
$106$	17.6246			1.71185
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	−0.962825	+	0.962825i	−0.0909784	+	0.0909784i
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	12.7307	−	12.7307i	1.17696	−	1.17696i
$118$	7.60876	+	7.60876i	0.700443	+	0.700443i
$119$		0			0
$120$		0			0
$121$	18.1273			1.64794
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.44424			0.128663
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	−0.612781	−	0.790253i	$-0.709949\pi$
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	0.790253	+	0.612781i	$0.209949\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$		0			0
$131$	0.783364			0.0684429			0.0342214	−	0.999414i	$-0.489105\pi$
$131$	0.783364			0.0684429			0.0342214	+	0.999414i	$0.489105\pi$
$132$		0			0
$133$	−2.06323	−	2.06323i	−0.178905	−	0.178905i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		−	22.3713i		−	1.89751i	−0.316018	−	0.948753i	$-0.602346\pi$
$139$		−	22.3713i		−	1.89751i	0.316018	−	0.948753i	$-0.397654\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	22.9025	+	22.9025i	1.91520	+	1.91520i
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−16.6618	−	16.6618i	−1.36959	−	1.36959i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−17.1432	+	17.1432i	−1.39049	+	1.39049i
$153$		0			0
$154$			2.59817i			0.209367i
$155$		0			0
$156$		0			0
$157$	−8.00000	+	8.00000i	−0.638470	+	0.638470i	−0.950178	−	0.311708i	$-0.899099\pi$
$157$	−8.00000	+	8.00000i	−0.638470	+	0.638470i	0.311708	+	0.950178i	$0.399099\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.555262			−0.0437608
$162$	−9.00000	+	9.00000i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$			18.1060i			1.41384i
$165$		0			0
$166$		0			0
$167$	16.4210	−	16.4210i	1.27070	−	1.27070i	0.324977	−	0.945722i	$-0.394644\pi$
$167$	16.4210	−	16.4210i	1.27070	−	1.27070i	0.945722	−	0.324977i	$-0.105356\pi$
$168$		0			0
$169$			23.0158i			1.77045i
$170$		0			0
$171$	25.7147			1.96646
$172$		0			0
$173$	−7.36805	−	7.36805i	−0.560183	−	0.560183i	0.369177	−	0.929359i	$-0.379640\pi$
$173$	−7.36805	−	7.36805i	−0.560183	−	0.560183i	−0.929359	+	0.369177i	$0.879640\pi$
$174$		0			0
$175$		0			0
$176$	21.5879			1.62725
$177$		0			0
$178$	7.70378	+	7.70378i	0.577423	+	0.577423i
$179$		−	24.7519i		−	1.85005i	−0.379912	−	0.925023i	$-0.624046\pi$
$179$		−	24.7519i		−	1.85005i	0.379912	−	0.925023i	$-0.375954\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−2.04291	+	2.04291i	−0.151430	+	0.151430i
$183$		0			0
$184$			4.61361i			0.340120i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−19.2811	−	19.2811i	−1.40622	−	1.40622i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		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
$196$	13.7682			0.983446
$197$	12.0000	−	12.0000i	0.854965	−	0.854965i	−0.135775	−	0.990740i	$-0.543352\pi$
$197$	12.0000	−	12.0000i	0.854965	−	0.854965i	0.990740	+	0.135775i	$0.0433525\pi$
$198$	−16.1909	−	16.1909i	−1.15064	−	1.15064i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−14.6675			−1.02193
$207$	3.46021	−	3.46021i	0.240501	−	0.240501i
$208$	16.9743	+	16.9743i	1.17696	+	1.17696i
$209$			46.2606i			3.19991i
$210$		0			0
$211$	−6.64593			−0.457525			−0.228762	−	0.973482i	$-0.573468\pi$
$211$	−6.64593			−0.457525			−0.228762	+	0.973482i	$0.573468\pi$
$212$	17.6246	−	17.6246i	1.21046	−	1.21046i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−14.0000	−	14.0000i	−0.937509	−	0.937509i	0.0606498	−	0.998159i	$-0.480683\pi$
$223$	−14.0000	−	14.0000i	−0.937509	−	0.937509i	−0.998159	+	0.0606498i	$0.980683\pi$
$224$			1.92565i			0.128663i
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		−	25.4614i		−	1.66447i
$235$		0			0
$236$	15.2175			0.990576
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	18.4977			1.19154			0.595772	−	0.803154i	$-0.296846\pi$
$241$	18.4977			1.19154			0.595772	+	0.803154i	$0.296846\pi$
$242$	18.1273	−	18.1273i	1.16527	−	1.16527i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−36.3741	+	36.3741i	−2.31443	+	2.31443i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$	1.44424	−	1.44424i	0.0909784	−	0.0909784i
$253$	6.22489	+	6.22489i	0.391355	+	0.391355i
$254$		−	4.00000i		−	0.250982i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		−	4.01059i		−	0.249206i
$260$		0			0
$261$		0			0
$262$	0.783364	−	0.783364i	0.0483964	−	0.0483964i
$263$	−17.8653	−	17.8653i	−1.10162	−	1.10162i	−0.994215	−	0.107405i	$-0.965746\pi$
$263$	−17.8653	−	17.8653i	−1.10162	−	1.10162i	−0.107405	−	0.994215i	$-0.534254\pi$
$264$		0			0
$265$		0			0
$266$	−4.12647			−0.253010
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−23.5247	+	23.5247i	−1.41346	+	1.41346i	−0.683656	+	0.729805i	$0.739611\pi$
$277$	−23.5247	+	23.5247i	−1.41346	+	1.41346i	−0.729805	+	0.683656i	$0.760389\pi$
$278$	−22.3713	−	22.3713i	−1.34174	−	1.34174i
$279$		0			0
$280$		0			0
$281$	30.8584			1.84086			0.920429	−	0.390909i	$-0.127839\pi$
$281$	30.8584			1.84086			0.920429	+	0.390909i	$0.127839\pi$
$282$		0			0
$283$		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
$286$	45.8049			2.70850
$287$	−2.17911	+	2.17911i	−0.128629	+	0.128629i
$288$	−12.0000	−	12.0000i	−0.707107	−	0.707107i
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.4239	−	10.4239i	−0.608971	−	0.608971i	0.333706	−	0.942677i	$-0.391701\pi$
$293$	−10.4239	−	10.4239i	−0.608971	−	0.608971i	−0.942677	+	0.333706i	$0.891701\pi$
$294$		0			0
$295$		0			0
$296$	−33.3235			−1.93689
$297$		0			0
$298$		0			0
$299$			9.78910i			0.566118i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$			34.2863i			1.96646i
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$	2.59817	+	2.59817i	0.148045	+	0.148045i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$			16.0000i			0.902932i
$315$		0			0
$316$		0			0
$317$	−17.3443	+	17.3443i	−0.974155	+	0.974155i	−0.999674	−	0.0255197i	$-0.991876\pi$
$317$	−17.3443	+	17.3443i	−0.974155	+	0.974155i	0.0255197	+	0.999674i	$0.491876\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−0.555262	+	0.555262i	−0.0309436	+	0.0309436i
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	18.1060	+	18.1060i	0.999736	+	0.999736i
$329$		−	4.64108i		−	0.255871i
$330$		0			0
$331$	−18.0000			−0.989369			−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000			−0.989369			−0.494685	+	0.869072i	$0.664716\pi$
$332$		0			0
$333$	24.9926	+	24.9926i	1.36959	+	1.36959i
$334$		−	32.8421i		−	1.79704i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	23.0158	+	23.0158i	1.25190	+	1.25190i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	25.7147	−	25.7147i	1.39049	−	1.39049i
$343$	3.34199	+	3.34199i	0.180451	+	0.180451i
$344$		0			0
$345$		0			0
$346$	−14.7361			−0.792218
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	21.5879	−	21.5879i	1.15064	−	1.15064i
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$	15.4076			0.816599
$357$		0			0
$358$	−24.7519	−	24.7519i	−1.30818	−	1.30818i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−54.4720			−2.86695
$362$		0			0
$363$		0			0
$364$			4.08582i			0.214155i
$365$		0			0
$366$		0			0
$367$	22.0000	−	22.0000i	1.14839	−	1.14839i	0.161521	−	0.986869i	$-0.448360\pi$
$367$	22.0000	−	22.0000i	1.14839	−	1.14839i	0.986869	−	0.161521i	$-0.0516401\pi$
$368$	4.61361	+	4.61361i	0.240501	+	0.240501i
$369$		−	27.1590i		−	1.41384i
$370$		0			0
$371$	4.24234			0.220252
$372$		0			0
$373$	−25.9555	−	25.9555i	−1.34392	−	1.34392i	−0.892116	−	0.451806i	$-0.850780\pi$
$373$	−25.9555	−	25.9555i	−1.34392	−	1.34392i	−0.451806	−	0.892116i	$-0.649220\pi$
$374$		0			0
$375$		0			0
$376$	−38.5622			−1.98869
$377$		0			0
$378$		0			0
$379$		−	28.5516i		−	1.46660i	−0.679907	−	0.733299i	$-0.737980\pi$
$379$		−	28.5516i		−	1.46660i	0.679907	−	0.733299i	$-0.262020\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.4582	−	15.4582i	−0.789878	−	0.789878i	0.191596	−	0.981474i	$-0.438634\pi$
$383$	−15.4582	−	15.4582i	−0.789878	−	0.789878i	−0.981474	+	0.191596i	$0.938634\pi$
$384$		0			0
$385$		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
$391$		0			0
$392$	13.7682	−	13.7682i	0.695401	−	0.695401i
$393$		0			0
$394$		−	24.0000i		−	1.20910i
$395$		0			0
$396$	−32.3819			−1.62725
$397$	24.5112	−	24.5112i	1.23018	−	1.23018i	0.266290	−	0.963893i	$-0.414202\pi$
$397$	24.5112	−	24.5112i	1.23018	−	1.23018i	0.963893	−	0.266290i	$-0.0857978\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	23.3077			1.16393			0.581965	−	0.813214i	$-0.302284\pi$
$401$	23.3077			1.16393			0.581965	+	0.813214i	$0.302284\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−44.9615	+	44.9615i	−2.22866	+	2.22866i
$408$		0			0
$409$		−	10.9786i		−	0.542859i	−0.962458	−	0.271430i	$-0.912504\pi$
$409$		−	10.9786i		−	0.542859i	0.962458	−	0.271430i	$-0.0874964\pi$
$410$		0			0
$411$		0			0
$412$	−14.6675	+	14.6675i	−0.722615	+	0.722615i
$413$	1.83147	+	1.83147i	0.0901210	+	0.0901210i
$414$		−	6.92042i		−	0.340120i
$415$		0			0
$416$	33.9486			1.66447
$417$		0			0
$418$	46.2606	+	46.2606i	2.26268	+	2.26268i
$419$			26.0000i			1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$419$			26.0000i			1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−6.64593	+	6.64593i	−0.323519	+	0.323519i
$423$	28.9216	+	28.9216i	1.40622	+	1.40622i
$424$		−	35.2492i		−	1.71185i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		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
$436$		0			0
$437$	−9.88649	+	9.88649i	−0.472935	+	0.472935i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−20.6524			−0.983446
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$	−28.0000			−1.32584
$447$		0			0
$448$	1.92565	+	1.92565i	0.0909784	+	0.0909784i
$449$			42.3765i			1.99987i	0.0113733	+	0.999935i	$0.496380\pi$
$449$			42.3765i			1.99987i	−0.0113733	+	0.999935i	$0.503620\pi$
$450$		0			0
$451$	48.8588			2.30067
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	26.0000	+	26.0000i	1.20832	+	1.20832i	0.971570	+	0.236752i	$0.0760830\pi$
$463$	26.0000	+	26.0000i	1.20832	+	1.20832i	0.236752	+	0.971570i	$0.423917\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−25.4614	−	25.4614i	−1.17696	−	1.17696i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	15.2175	−	15.2175i	0.700443	−	0.700443i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−26.4369	+	26.4369i	−1.21046	+	1.21046i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−70.7054			−3.22389
$482$	18.4977	−	18.4977i	0.842549	−	0.842549i
$483$		0			0
$484$		−	36.2547i		−	1.64794i
$485$		0			0
$486$		0			0
$487$	−14.2542	+	14.2542i	−0.645918	+	0.645918i	−0.952004	−	0.306086i	$-0.900980\pi$
$487$	−14.2542	+	14.2542i	−0.645918	+	0.645918i	0.306086	+	0.952004i	$0.400980\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−27.6404			−1.24739			−0.623697	−	0.781666i	$-0.714370\pi$
$491$	−27.6404			−1.24739			−0.623697	+	0.781666i	$0.714370\pi$
$492$		0			0
$493$		0			0
$494$			72.7483i			3.27310i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			43.9592i			1.96788i	0.178492	+	0.983941i	$0.442878\pi$
$499$			43.9592i			1.96788i	−0.178492	+	0.983941i	$0.557122\pi$
$500$		0			0
$501$		0			0
$502$	2.00000	−	2.00000i	0.0892644	−	0.0892644i
$503$	22.7413	+	22.7413i	1.01398	+	1.01398i	0.999901	+	0.0140839i	$0.00448318\pi$
$503$	22.7413	+	22.7413i	1.01398	+	1.01398i	0.0140839	+	0.999901i	$0.495517\pi$
$504$		−	2.88847i		−	0.128663i
$505$		0			0
$506$	12.4498			0.553460
$507$		0			0
$508$	−4.00000	−	4.00000i	−0.177471	−	0.177471i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−52.0298	+	52.0298i	−2.28827	+	2.28827i
$518$	−4.01059	−	4.01059i	−0.176215	−	0.176215i
$519$		0			0
$520$		0			0
$521$	12.3174			0.539635			0.269817	−	0.962912i	$-0.413037\pi$
$521$	12.3174			0.539635			0.269817	+	0.962912i	$0.413037\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		−	1.56673i		−	0.0684429i
$525$		0			0
$526$	−35.7306			−1.55793
$527$		0			0
$528$		0			0
$529$		−	20.3393i		−	0.884318i
$530$		0			0
$531$	−22.8263			−0.990576
$532$	−4.12647	+	4.12647i	−0.178905	+	0.178905i
$533$	38.4170	+	38.4170i	1.66403	+	1.66403i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	37.1534i		−	1.60031i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			47.0493i			1.99893i
$555$		0			0
$556$	−44.7425			−1.89751
$557$	18.9111	−	18.9111i	0.801287	−	0.801287i	−0.182010	−	0.983297i	$-0.558260\pi$
$557$	18.9111	−	18.9111i	0.801287	−	0.801287i	0.983297	+	0.182010i	$0.0582602\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	30.8584	−	30.8584i	1.30168	−	1.30168i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.16636	+	2.16636i	−0.0909784	+	0.0909784i
$568$		0			0
$569$			26.2448i			1.10024i	0.835086	+	0.550120i	$0.185418\pi$
$569$			26.2448i			1.10024i	−0.835086	+	0.550120i	$0.814582\pi$
$570$		0			0
$571$	−11.4601			−0.479588			−0.239794	−	0.970824i	$-0.577080\pi$
$571$	−11.4601			−0.479588			−0.239794	+	0.970824i	$0.577080\pi$
$572$	45.8049	−	45.8049i	1.91520	−	1.91520i
$573$		0			0
$574$			4.35822i			0.181909i
$575$		0			0
$576$	−24.0000			−1.00000
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−17.0000	−	17.0000i	−0.707107	−	0.707107i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−47.5597	−	47.5597i	−1.96972	−	1.96972i
$584$		0			0
$585$		0			0
$586$	−20.8478			−0.861216
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−33.3235	+	33.3235i	−1.36959	+	1.36959i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	9.78910	+	9.78910i	0.400306	+	0.400306i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	37.0388			1.51084			0.755421	−	0.655240i	$-0.227432\pi$
$601$	37.0388			1.51084			0.755421	+	0.655240i	$0.227432\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	18.8281	−	18.8281i	0.764209	−	0.764209i	−0.212871	−	0.977080i	$-0.568281\pi$
$607$	18.8281	−	18.8281i	0.764209	−	0.764209i	0.977080	+	0.212871i	$0.0682814\pi$
$608$	34.2863	+	34.2863i	1.39049	+	1.39049i
$609$		0			0
$610$		0			0
$611$	−81.8207			−3.31011
$612$		0			0
$613$	−34.3186	−	34.3186i	−1.38612	−	1.38612i	−0.833310	−	0.552806i	$-0.813557\pi$
$613$	−34.3186	−	34.3186i	−1.38612	−	1.38612i	−0.552806	−	0.833310i	$-0.686443\pi$
$614$		0			0
$615$		0			0
$616$	5.19634			0.209367
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		−	3.83025i		−	0.153951i	−0.997033	−	0.0769753i	$-0.975474\pi$
$619$		−	3.83025i		−	0.153951i	0.997033	−	0.0769753i	$-0.0245263\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.85435	+	1.85435i	0.0742929	+	0.0742929i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	16.0000	+	16.0000i	0.638470	+	0.638470i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$			34.6887i			1.37766i
$635$		0			0
$636$		0			0
$637$	29.2133	−	29.2133i	1.15747	−	1.15747i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.1218			1.11074			0.555372	−	0.831602i	$-0.312576\pi$
$641$	28.1218			1.11074			0.555372	+	0.831602i	$0.312576\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$			1.11052i			0.0437608i
$645$		0			0
$646$		0			0
$647$	−34.5270	+	34.5270i	−1.35740	+	1.35740i	−0.480286	+	0.877112i	$0.659467\pi$
$647$	−34.5270	+	34.5270i	−1.35740	+	1.35740i	−0.877112	+	0.480286i	$0.840533\pi$
$648$	18.0000	+	18.0000i	0.707107	+	0.707107i
$649$		−	41.0643i		−	1.61191i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8.11711	+	8.11711i	0.317647	+	0.317647i	0.847863	−	0.530216i	$-0.177889\pi$
$653$	8.11711	+	8.11711i	0.317647	+	0.317647i	−0.530216	+	0.847863i	$0.677889\pi$
$654$		0			0
$655$		0			0
$656$	36.2120			1.41384
$657$		0			0
$658$	−4.64108	−	4.64108i	−0.180928	−	0.180928i
$659$			50.1395i			1.95316i	0.215159	+	0.976579i	$0.430973\pi$
$659$			50.1395i			1.95316i	−0.215159	+	0.976579i	$0.569027\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−18.0000	+	18.0000i	−0.699590	+	0.699590i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	49.9853			1.93689
$667$		0			0
$668$	−32.8421	−	32.8421i	−1.27070	−	1.27070i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		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
$676$	46.0316			1.77045
$677$	10.7379	−	10.7379i	0.412692	−	0.412692i	−0.469983	−	0.882675i	$-0.655740\pi$
$677$	10.7379	−	10.7379i	0.412692	−	0.412692i	0.882675	+	0.469983i	$0.155740\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		−	51.4295i		−	1.96646i
$685$		0			0
$686$	6.68399			0.255196
$687$		0			0
$688$		0			0
$689$		−	74.7912i		−	2.84932i
$690$		0			0
$691$	42.0000			1.59776			0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000			1.59776			0.798878	+	0.601494i	$0.205427\pi$
$692$	−14.7361	+	14.7361i	−0.560183	+	0.560183i
$693$	−3.89726	−	3.89726i	−0.148045	−	0.148045i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−71.4088	−	71.4088i	−2.69323	−	2.69323i
$704$		−	43.1758i		−	1.62725i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$	15.4076	−	15.4076i	0.577423	−	0.577423i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−49.5038			−1.85005
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−3.53055			−0.131485
$722$	−54.4720	+	54.4720i	−2.02724	+	2.02724i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	32.6014	−	32.6014i	1.20912	−	1.20912i	0.237806	−	0.971313i	$-0.423572\pi$
$727$	32.6014	−	32.6014i	1.20912	−	1.20912i	0.971313	−	0.237806i	$-0.0764282\pi$
$728$	4.08582	+	4.08582i	0.151430	+	0.151430i
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−28.1383	−	28.1383i	−1.03931	−	1.03931i	−0.999195	−	0.0401161i	$-0.987227\pi$
$733$	−28.1383	−	28.1383i	−1.03931	−	1.03931i	−0.0401161	−	0.999195i	$-0.512773\pi$
$734$		−	44.0000i		−	1.62407i
$735$		0			0
$736$	9.22722			0.340120
$737$		0			0
$738$	−27.1590	−	27.1590i	−0.999736	−	0.999736i
$739$			39.9694i			1.47030i	0.677905	+	0.735150i	$0.262888\pi$
$739$			39.9694i			1.47030i	−0.677905	+	0.735150i	$0.737112\pi$
$740$		0			0
$741$		0			0
$742$	4.24234	−	4.24234i	0.155741	−	0.155741i
$743$	35.1020	+	35.1020i	1.28777	+	1.28777i	0.936140	+	0.351627i	$0.114371\pi$
$743$	35.1020	+	35.1020i	1.28777	+	1.28777i	0.351627	+	0.936140i	$0.385629\pi$
$744$		0			0
$745$		0			0
$746$	−51.9109			−1.90059
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−38.5622	+	38.5622i	−1.40622	+	1.40622i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−11.1640	+	11.1640i	−0.405762	+	0.405762i	−0.880258	−	0.474496i	$-0.842630\pi$
$757$	−11.1640	+	11.1640i	−0.405762	+	0.405762i	0.474496	+	0.880258i	$0.342630\pi$
$758$	−28.5516	−	28.5516i	−1.03704	−	1.03704i
$759$		0			0
$760$		0			0
$761$	11.9415			0.432878			0.216439	−	0.976296i	$-0.430556\pi$
$761$	11.9415			0.432878			0.216439	+	0.976296i	$0.430556\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−30.9164			−1.11706
$767$	32.2883	−	32.2883i	1.16586	−	1.16586i
$768$		0			0
$769$		−	4.65690i		−	0.167932i	−0.996469	−	0.0839660i	$-0.973241\pi$
$769$		−	4.65690i		−	0.167932i	0.996469	−	0.0839660i	$-0.0267587\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	36.0000	+	36.0000i	1.29483	+	1.29483i	0.931763	+	0.363067i	$0.118270\pi$
$773$	36.0000	+	36.0000i	1.29483	+	1.29483i	0.363067	+	0.931763i	$0.381730\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			77.5985i			2.78025i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	27.5365i		−	0.983446i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−24.0000	−	24.0000i	−0.854965	−	0.854965i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−32.3819	+	32.3819i	−1.15064	+	1.15064i
$793$		0			0
$794$		−	49.0224i		−	1.73974i
$795$		0			0
$796$		0			0
$797$	5.92381	−	5.92381i	0.209832	−	0.209832i	−0.594364	−	0.804196i	$-0.702596\pi$
$797$	5.92381	−	5.92381i	0.209832	−	0.209832i	0.804196	+	0.594364i	$0.202596\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−23.1113			−0.816599
$802$	23.3077	−	23.3077i	0.823023	−	0.823023i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	40.0856i		−	1.40934i	−0.709537	−	0.704668i	$-0.751096\pi$
$809$		−	40.0856i		−	1.40934i	0.709537	−	0.704668i	$-0.248904\pi$
$810$		0			0
$811$	54.7531			1.92264			0.961321	−	0.275430i	$-0.0888203\pi$
$811$	54.7531			1.92264			0.961321	+	0.275430i	$0.0888203\pi$
$812$		0			0
$813$		0			0
$814$			89.9231i			3.15180i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−10.9786	−	10.9786i	−0.383859	−	0.383859i
$819$		−	6.12873i		−	0.214155i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	14.4954	+	14.4954i	0.505278	+	0.505278i	0.913073	−	0.407796i	$-0.133702\pi$
$823$	14.4954	+	14.4954i	0.505278	+	0.505278i	−0.407796	+	0.913073i	$0.633702\pi$
$824$			29.3350i			1.02193i
$825$		0			0
$826$	3.66295			0.127450
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$	−6.92042	−	6.92042i	−0.240501	−	0.240501i
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$	33.9486	−	33.9486i	1.17696	−	1.17696i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	92.5212			3.19991
$837$		0			0
$838$	26.0000	+	26.0000i	0.898155	+	0.898155i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$			13.2919i			0.457525i
$845$		0			0
$846$	57.8433			1.98869
$847$	4.36336	−	4.36336i	0.149927	−	0.149927i
$848$	−35.2492	−	35.2492i	−1.21046	−	1.21046i
$849$		0			0
$850$		0			0
$851$	−19.2177			−0.658775
$852$		0			0
$853$	27.3997	+	27.3997i	0.938148	+	0.938148i	0.998195	−	0.0600480i	$-0.0191254\pi$
$853$	27.3997	+	27.3997i	0.938148	+	0.938148i	−0.0600480	+	0.998195i	$0.519125\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$			44.7836i			1.52800i	0.645219	+	0.763998i	$0.276766\pi$
$859$			44.7836i			1.52800i	−0.645219	+	0.763998i	$0.723234\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.3095	+	19.3095i	0.657304	+	0.657304i	0.954741	−	0.297438i	$-0.0961320\pi$
$863$	19.3095	+	19.3095i	0.657304	+	0.657304i	−0.297438	+	0.954741i	$0.596132\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$			19.7730i			0.668831i
$875$		0			0
$876$		0			0
$877$	36.6254	−	36.6254i	1.23675	−	1.23675i	0.275432	−	0.961320i	$-0.411179\pi$
$877$	36.6254	−	36.6254i	1.23675	−	1.23675i	0.961320	−	0.275432i	$-0.0888210\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	39.4880			1.33039			0.665193	−	0.746672i	$-0.268349\pi$
$881$	39.4880			1.33039			0.665193	+	0.746672i	$0.268349\pi$
$882$	−20.6524	+	20.6524i	−0.695401	+	0.695401i
$883$		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
$886$		0			0
$887$	−32.1200	+	32.1200i	−1.07848	+	1.07848i	−0.0818375	+	0.996646i	$0.526079\pi$
$887$	−32.1200	+	32.1200i	−1.07848	+	1.07848i	−0.996646	+	0.0818375i	$0.973921\pi$
$888$		0			0
$889$		−	0.962825i		−	0.0322921i
$890$		0			0
$891$	48.5728			1.62725
$892$	−28.0000	+	28.0000i	−0.937509	+	0.937509i
$893$	−82.6348	−	82.6348i	−2.76527	−	2.76527i
$894$		0			0
$895$		0			0
$896$	3.85130			0.128663
$897$		0			0
$898$	42.3765	+	42.3765i	1.41412	+	1.41412i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	48.8588	−	48.8588i	1.62682	−	1.62682i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.188561	−	0.188561i	0.00622682	−	0.00622682i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	52.0000			1.70883
$927$	22.0012	−	22.0012i	0.722615	−	0.722615i
$928$		0			0
$929$		−	58.6267i		−	1.92348i	−0.273967	−	0.961739i	$-0.588336\pi$
$929$		−	58.6267i		−	1.92348i	0.273967	−	0.961739i	$-0.411664\pi$
$930$		0			0
$931$	59.0078			1.93390
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−50.9229			−1.66447
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	10.4417	+	10.4417i	0.340030	+	0.340030i
$944$		−	30.4350i		−	0.990576i
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$			52.8737i			1.71185i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	−70.7054	+	70.7054i	−2.27963	+	2.27963i
$963$		0			0
$964$		−	36.9955i		−	1.19154i
$965$		0			0
$966$		0			0
$967$	−32.7952	+	32.7952i	−1.05462	+	1.05462i	−0.0562025	+	0.998419i	$0.517899\pi$
$967$	−32.7952	+	32.7952i	−1.05462	+	1.05462i	−0.998419	+	0.0562025i	$0.982101\pi$
$968$	−36.2547	−	36.2547i	−1.16527	−	1.16527i
$969$		0			0
$970$		0			0
$971$	13.1440			0.421812			0.210906	−	0.977506i	$-0.432359\pi$
$971$	13.1440			0.421812			0.210906	+	0.977506i	$0.432359\pi$
$972$		0			0
$973$	−5.38490	−	5.38490i	−0.172632	−	0.172632i
$974$			28.5083i			0.913465i
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		−	41.5771i		−	1.32881i
$980$		0			0
$981$		0			0
$982$	−27.6404	+	27.6404i	−0.882040	+	0.882040i
$983$	16.5610	+	16.5610i	0.528213	+	0.528213i	0.920039	−	0.391826i	$-0.128157\pi$
$983$	16.5610	+	16.5610i	0.528213	+	0.528213i	−0.391826	+	0.920039i	$0.628157\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	72.7483	+	72.7483i	2.31443	+	2.31443i
$989$		0			0
$990$		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
$996$		0			0
$997$	43.0986	−	43.0986i	1.36495	−	1.36495i	0.497460	−	0.867487i	$-0.334266\pi$
$997$	43.0986	−	43.0986i	1.36495	−	1.36495i	0.867487	−	0.497460i	$-0.165734\pi$
$998$	43.9592	+	43.9592i	1.39150	+	1.39150i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1000.2.k.b.307.3	yes	8
5.2	odd	4	inner	1000.2.k.b.443.3	yes	8
5.3	odd	4		1000.2.k.a.443.2	yes	8
5.4	even	2		1000.2.k.a.307.2	✓	8
8.3	odd	2		1000.2.k.a.307.2	✓	8
40.3	even	4	inner	1000.2.k.b.443.3	yes	8
40.19	odd	2	CM	1000.2.k.b.307.3	yes	8
40.27	even	4		1000.2.k.a.443.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.k.a.307.2	✓	8	5.4	even	2
1000.2.k.a.307.2	✓	8	8.3	odd	2
1000.2.k.a.443.2	yes	8	5.3	odd	4
1000.2.k.a.443.2	yes	8	40.27	even	4
1000.2.k.b.307.3	yes	8	1.1	even	1	trivial
1000.2.k.b.307.3	yes	8	40.19	odd	2	CM
1000.2.k.b.443.3	yes	8	5.2	odd	4	inner
1000.2.k.b.443.3	yes	8	40.3	even	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:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1000.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(0.240706 - 0.240706i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(0.240706 - 0.240706i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} -5.39698 q^{11} +(-4.24357 - 4.24357i) q^{13} -0.481412i q^{14} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{18} -8.57158i q^{19} +(-5.39698 + 5.39698i) q^{22} +(-1.15340 - 1.15340i) q^{23} -8.48715 q^{26} +(-0.481412 - 0.481412i) q^{28} +(-4.00000 + 4.00000i) q^{32} +6.00000 q^{36} +(8.33088 - 8.33088i) q^{37} +(-8.57158 - 8.57158i) q^{38} -9.05299 q^{41} +10.7940i q^{44} -2.30681 q^{46} +(9.64055 - 9.64055i) q^{47} +6.88412i q^{49} +(-8.48715 + 8.48715i) q^{52} +(8.81229 + 8.81229i) q^{53} -0.962825 q^{56} +7.60876i q^{59} +(0.722119 + 0.722119i) q^{63} +8.00000i q^{64} +(6.00000 - 6.00000i) q^{72} -16.6618i q^{74} -17.1432 q^{76} +(-1.29909 + 1.29909i) q^{77} -9.00000 q^{81} +(-9.05299 + 9.05299i) q^{82} +(10.7940 + 10.7940i) q^{88} +7.70378i q^{89} -2.04291 q^{91} +(-2.30681 + 2.30681i) q^{92} -19.2811i q^{94} +(6.88412 + 6.88412i) q^{98} -16.1909i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8	\( 8 q + 8 q^{2} - 4 q^{7} - 16 q^{8} - 4 q^{11} + 8 q^{13} - 32 q^{16} + 24 q^{18} - 4 q^{22} - 12 q^{23} + 16 q^{26} + 8 q^{28} - 32 q^{32} + 48 q^{36} + 16 q^{37} - 12 q^{38} - 4 q^{41} - 24 q^{46} - 4 q^{47} + 16 q^{52} + 8 q^{53} + 16 q^{56} - 12 q^{63} + 48 q^{72} - 24 q^{76} - 8 q^{77} - 72 q^{81} - 4 q^{82} + 8 q^{88} + 32 q^{91} - 24 q^{92} - 68 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 4 * q^7 - 16 * q^8 - 4 * q^11 + 8 * q^13 - 32 * q^16 + 24 * q^18 - 4 * q^22 - 12 * q^23 + 16 * q^26 + 8 * q^28 - 32 * q^32 + 48 * q^36 + 16 * q^37 - 12 * q^38 - 4 * q^41 - 24 * q^46 - 4 * q^47 + 16 * q^52 + 8 * q^53 + 16 * q^56 - 12 * q^63 + 48 * q^72 - 24 * q^76 - 8 * q^77 - 72 * q^81 - 4 * q^82 + 8 * q^88 + 32 * q^91 - 24 * q^92 - 68 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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