LMFDB - Embedded newform 475.2.g.a.18.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(18,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 2]))

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

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

chi := DirichletCharacter("475.18");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.g (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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			18.2
Root			$2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	475.18
Dual form			475.2.g.a.132.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{4} +(0.679449 - 0.679449i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q-2.00000i q^{4} +(0.679449 - 0.679449i) q^{7} +3.00000i q^{9} +4.35890 q^{11} -4.00000 q^{16} +(5.67945 - 5.67945i) q^{17} -4.35890i q^{19} +(-6.35890 - 6.35890i) q^{23} +(-1.35890 - 1.35890i) q^{28} +6.00000 q^{36} +(7.03835 + 7.03835i) q^{43} -8.71780i q^{44} +(-4.32055 + 4.32055i) q^{47} +6.07670i q^{49} +4.35890 q^{61} +(2.03835 + 2.03835i) q^{63} +8.00000i q^{64} +(-11.3589 - 11.3589i) q^{68} +(12.0383 + 12.0383i) q^{73} -8.71780 q^{76} +(2.96165 - 2.96165i) q^{77} -9.00000 q^{81} +(3.64110 + 3.64110i) q^{83} +(-12.7178 + 12.7178i) q^{92} +13.0767i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7}+O(q^{10}) $ 4 * q - 6 * q^7	$ 4 q - 6 q^{7} - 16 q^{16} + 14 q^{17} - 8 q^{23} + 12 q^{28} + 24 q^{36} + 2 q^{43} - 26 q^{47} - 18 q^{63} - 28 q^{68} + 22 q^{73} + 38 q^{77} - 36 q^{81} + 32 q^{83} - 16 q^{92}+O(q^{100}) $ 4 * q - 6 * q^7 - 16 * q^16 + 14 * q^17 - 8 * q^23 + 12 * q^28 + 24 * q^36 + 2 * q^43 - 26 * q^47 - 18 * q^63 - 28 * q^68 + 22 * q^73 + 38 * q^77 - 36 * q^81 + 32 * q^83 - 16 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$

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 $

$2$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$2$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$3$		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
$7$	0.679449	−	0.679449i	0.256808	−	0.256808i	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	0.679449	−	0.679449i	0.256808	−	0.256808i	0.823754	+	0.566947i	$0.191875\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	4.35890			1.31426			0.657129	−	0.753778i	$-0.271771\pi$
$11$	4.35890			1.31426			0.657129	+	0.753778i	$0.271771\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	5.67945	−	5.67945i	1.37747	−	1.37747i	0.528594	−	0.848875i	$-0.322719\pi$
$17$	5.67945	−	5.67945i	1.37747	−	1.37747i	0.848875	−	0.528594i	$-0.177281\pi$
$18$		0			0
$19$		−	4.35890i		−	1.00000i
$19$		−	4.35890i		−	1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.35890	−	6.35890i	−1.32592	−	1.32592i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	−6.35890	−	6.35890i	−1.32592	−	1.32592i	−0.417029	−	0.908893i	$-0.636929\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	−1.35890	−	1.35890i	−0.256808	−	0.256808i
$29$		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			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	6.00000			1.00000
$37$		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
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	7.03835	+	7.03835i	1.07334	+	1.07334i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	7.03835	+	7.03835i	1.07334	+	1.07334i	0.0762493	+	0.997089i	$0.475706\pi$
$44$		−	8.71780i		−	1.31426i
$45$		0			0
$46$		0			0
$47$	−4.32055	+	4.32055i	−0.630217	+	0.630217i	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−4.32055	+	4.32055i	−0.630217	+	0.630217i	0.317905	+	0.948122i	$0.397021\pi$
$48$		0			0
$49$			6.07670i			0.868100i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	4.35890			0.558100			0.279050	−	0.960277i	$-0.409981\pi$
$61$	4.35890			0.558100			0.279050	+	0.960277i	$0.409981\pi$
$62$		0			0
$63$	2.03835	+	2.03835i	0.256808	+	0.256808i
$64$			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	$0.750000\pi$
$68$	−11.3589	−	11.3589i	−1.37747	−	1.37747i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	12.0383	+	12.0383i	1.40898	+	1.40898i	0.765256	+	0.643726i	$0.222612\pi$
$73$	12.0383	+	12.0383i	1.40898	+	1.40898i	0.643726	+	0.765256i	$0.277388\pi$
$74$		0			0
$75$		0			0
$76$	−8.71780			−1.00000
$77$	2.96165	−	2.96165i	0.337512	−	0.337512i
$78$		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
$82$		0			0
$83$	3.64110	+	3.64110i	0.399663	+	0.399663i	0.878114	−	0.478451i	$-0.158802\pi$
$83$	3.64110	+	3.64110i	0.399663	+	0.399663i	−0.478451	+	0.878114i	$0.658802\pi$
$84$		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
$92$	−12.7178	+	12.7178i	−1.32592	+	1.32592i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$			13.0767i			1.31426i
$100$		0			0
$101$	−17.4356			−1.73491			−0.867453	−	0.497519i	$-0.834245\pi$
$101$	−17.4356			−1.73491			−0.867453	+	0.497519i	$0.834245\pi$
$102$		0			0
$103$		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
$106$		0			0
$107$		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
$112$	−2.71780	+	2.71780i	−0.256808	+	0.256808i
$113$		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
$117$		0			0
$118$		0			0
$119$		−	7.71780i		−	0.707489i
$120$		0			0
$121$	8.00000			0.727273
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.00000			0.611593			0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000			0.611593			0.305796	+	0.952097i	$0.401077\pi$
$132$		0			0
$133$	−2.96165	−	2.96165i	−0.256808	−	0.256808i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.32055	+	9.32055i	−0.796308	+	0.796308i	−0.982511	−	0.186203i	$-0.940382\pi$
$137$	−9.32055	+	9.32055i	−0.796308	+	0.796308i	0.186203	+	0.982511i	$0.440382\pi$
$138$		0			0
$139$		−	9.00000i		−	0.763370i	−0.924292	−	0.381685i	$-0.875344\pi$
$139$		−	9.00000i		−	0.763370i	0.924292	−	0.381685i	$-0.124656\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			11.0000i			0.901155i	0.892737	+	0.450578i	$0.148782\pi$
$149$			11.0000i			0.901155i	−0.892737	+	0.450578i	$0.851218\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	17.0383	+	17.0383i	1.37747	+	1.37747i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.7178	+	17.7178i	−1.41403	+	1.41403i	−0.695756	+	0.718278i	$0.744931\pi$
$157$	−17.7178	+	17.7178i	−1.41403	+	1.41403i	−0.718278	+	0.695756i	$0.755069\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.64110			−0.681014
$162$		0			0
$163$	−16.3589	−	16.3589i	−1.28133	−	1.28133i	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−16.3589	−	16.3589i	−1.28133	−	1.28133i	−0.341415	−	0.939913i	$-0.610906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	13.0767			1.00000
$172$	14.0767	−	14.0767i	1.07334	−	1.07334i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$	−17.4356			−1.31426
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	24.7561	−	24.7561i	1.81035	−	1.81035i
$188$	8.64110	+	8.64110i	0.630217	+	0.630217i
$189$		0			0
$190$		0			0
$191$	17.0000			1.23008			0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000			1.23008			0.615038	+	0.788497i	$0.289140\pi$
$192$		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
$196$	12.1534			0.868100
$197$	2.28220	−	2.28220i	0.162600	−	0.162600i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	2.28220	−	2.28220i	0.162600	−	0.162600i	0.783718	+	0.621117i	$0.213321\pi$
$198$		0			0
$199$			13.0767i			0.926982i	0.886102	+	0.463491i	$0.153403\pi$
$199$			13.0767i			0.926982i	−0.886102	+	0.463491i	$0.846597\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	19.0767	−	19.0767i	1.32592	−	1.32592i
$208$		0			0
$209$		−	19.0000i		−	1.31426i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		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
$223$		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
$226$		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
$229$			21.0000i			1.38772i	0.720110	+	0.693860i	$0.244091\pi$
$229$			21.0000i			1.38772i	−0.720110	+	0.693860i	$0.755909\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.7561	−	14.7561i	−0.966707	−	0.966707i	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	−14.7561	−	14.7561i	−0.966707	−	0.966707i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	30.5123i		−	1.97368i	−0.161712	−	0.986838i	$-0.551701\pi$
$239$		−	30.5123i		−	1.97368i	0.161712	−	0.986838i	$-0.448299\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		−	8.71780i		−	0.558100i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−23.0000			−1.45175			−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000			−1.45175			−0.725874	+	0.687828i	$0.758564\pi$
$252$	4.07670	−	4.07670i	0.256808	−	0.256808i
$253$	−27.7178	−	27.7178i	−1.74260	−	1.74260i
$254$		0			0
$255$		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
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.75615	−	9.75615i	−0.601590	−	0.601590i	0.339145	−	0.940734i	$-0.389862\pi$
$263$	−9.75615	−	9.75615i	−0.601590	−	0.601590i	−0.940734	+	0.339145i	$0.889862\pi$
$264$		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
$271$	26.1534			1.58871			0.794353	−	0.607457i	$-0.207810\pi$
$271$	26.1534			1.58871			0.794353	+	0.607457i	$0.207810\pi$
$272$	−22.7178	+	22.7178i	−1.37747	+	1.37747i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−14.3206	+	14.3206i	−0.860438	+	0.860438i	−0.991389	−	0.130950i	$-0.958197\pi$
$277$	−14.3206	+	14.3206i	−0.860438	+	0.860438i	0.130950	+	0.991389i	$0.458197\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	22.0383	+	22.0383i	1.31004	+	1.31004i	0.921379	+	0.388664i	$0.127063\pi$
$283$	22.0383	+	22.0383i	1.31004	+	1.31004i	0.388664	+	0.921379i	$0.372937\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	47.5123i		−	2.79484i
$290$		0			0
$291$		0			0
$292$	24.0767	−	24.0767i	1.40898	−	1.40898i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	9.56440			0.551283
$302$		0			0
$303$		0			0
$304$			17.4356i			1.00000i
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$	−5.92330	−	5.92330i	−0.337512	−	0.337512i
$309$		0			0
$310$		0			0
$311$	4.35890			0.247170			0.123585	−	0.992334i	$-0.460561\pi$
$311$	4.35890			0.247170			0.123585	+	0.992334i	$0.460561\pi$
$312$		0			0
$313$	20.4356	+	20.4356i	1.15509	+	1.15509i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	20.4356	+	20.4356i	1.15509	+	1.15509i	0.169570	+	0.985518i	$0.445762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.7561	−	24.7561i	−1.37747	−	1.37747i
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			5.87119i			0.323689i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	7.28220	−	7.28220i	0.399663	−	0.399663i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	8.88495	+	8.88495i	0.479742	+	0.479742i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.6794	−	20.6794i	1.11013	−	1.11013i	0.116999	−	0.993132i	$-0.462673\pi$
$347$	20.6794	−	20.6794i	1.11013	−	1.11013i	0.993132	−	0.116999i	$-0.0373274\pi$
$348$		0			0
$349$			13.0767i			0.699980i	0.936754	+	0.349990i	$0.113815\pi$
$349$			13.0767i			0.699980i	−0.936754	+	0.349990i	$0.886185\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.4356	+	10.4356i	0.555431	+	0.555431i	0.928003	−	0.372572i	$-0.121524\pi$
$353$	10.4356	+	10.4356i	0.555431	+	0.555431i	−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			31.0000i			1.63612i	0.575135	+	0.818059i	$0.304950\pi$
$359$			31.0000i			1.63612i	−0.575135	+	0.818059i	$0.695050\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.923303	+	0.923303i	−0.0481960	+	0.0481960i	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−0.923303	+	0.923303i	−0.0481960	+	0.0481960i	0.682598	+	0.730794i	$0.260850\pi$
$368$	25.4356	+	25.4356i	1.32592	+	1.32592i
$369$		0			0
$370$		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
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−21.1150	+	21.1150i	−1.07334	+	1.07334i
$388$		0			0
$389$		−	30.5123i		−	1.54703i	−0.633775	−	0.773517i	$-0.718496\pi$
$389$		−	30.5123i		−	1.54703i	0.633775	−	0.773517i	$-0.281504\pi$
$390$		0			0
$391$	−72.2301			−3.65283
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	26.1534			1.31426
$397$	−16.1150	+	16.1150i	−0.808791	+	0.808791i	−0.984451	−	0.175660i	$-0.943794\pi$
$397$	−16.1150	+	16.1150i	−0.808791	+	0.808791i	0.175660	+	0.984451i	$0.443794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$			34.8712i			1.73491i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	8.71780i		−	0.425892i	−0.977064	−	0.212946i	$-0.931694\pi$
$419$		−	8.71780i		−	0.425892i	0.977064	−	0.212946i	$-0.0683059\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	−12.9617	−	12.9617i	−0.630217	−	0.630217i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.96165	−	2.96165i	0.143324	−	0.143324i
$428$		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
$437$	−27.7178	+	27.7178i	−1.32592	+	1.32592i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−18.2301			−0.868100
$442$		0			0
$443$	−29.7561	−	29.7561i	−1.41376	−	1.41376i	−0.724841	−	0.688916i	$-0.758087\pi$
$443$	−29.7561	−	29.7561i	−1.41376	−	1.41376i	−0.688916	−	0.724841i	$-0.741913\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	5.43560	+	5.43560i	0.256808	+	0.256808i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.1150	+	11.1150i	−0.519940	+	0.519940i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	−11.1150	+	11.1150i	−0.519940	+	0.519940i	0.397613	+	0.917553i	$0.369839\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	37.0000			1.72326			0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000			1.72326			0.861631	+	0.507535i	$0.169443\pi$
$462$		0			0
$463$	27.0383	+	27.0383i	1.25658	+	1.25658i	0.952716	+	0.303863i	$0.0982765\pi$
$463$	27.0383	+	27.0383i	1.25658	+	1.25658i	0.303863	+	0.952716i	$0.401724\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.3206	+	19.3206i	−0.894048	+	0.894048i	−0.994901	−	0.100853i	$-0.967843\pi$
$467$	−19.3206	+	19.3206i	−0.894048	+	0.894048i	0.100853	+	0.994901i	$0.467843\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	30.6794	+	30.6794i	1.41064	+	1.41064i
$474$		0			0
$475$		0			0
$476$	−15.4356			−0.707489
$477$		0			0
$478$		0			0
$479$		−	4.00000i		−	0.182765i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$		−	4.00000i		−	0.182765i	0.995816	−	0.0913823i	$-0.0291285\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		−	16.0000i		−	0.727273i
$485$		0			0
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	39.0000i		−	1.74588i	−0.487828	−	0.872940i	$-0.662211\pi$
$499$		−	39.0000i		−	1.74588i	0.487828	−	0.872940i	$-0.337789\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.3589	−	26.3589i	−1.17529	−	1.17529i	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−26.3589	−	26.3589i	−1.17529	−	1.17529i	−0.194354	−	0.980932i	$-0.562261\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	16.3589			0.723675
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−18.8328	+	18.8328i	−0.828267	+	0.828267i
$518$		0			0
$519$		0			0
$520$		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	$0.250000\pi$
$524$		−	14.0000i		−	0.611593i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			57.8712i			2.51614i
$530$		0			0
$531$		0			0
$532$	−5.92330	+	5.92330i	−0.256808	+	0.256808i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			26.4877i			1.14091i
$540$		0			0
$541$	−39.2301			−1.68663			−0.843317	−	0.537417i	$-0.819400\pi$
$541$	−39.2301			−1.68663			−0.843317	+	0.537417i	$0.819400\pi$
$542$		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	$0.750000\pi$
$548$	18.6411	+	18.6411i	0.796308	+	0.796308i
$549$			13.0767i			0.558100i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−18.0000			−0.763370
$557$	25.6794	−	25.6794i	1.08807	−	1.08807i	0.0923462	−	0.995727i	$-0.470563\pi$
$557$	25.6794	−	25.6794i	1.08807	−	1.08807i	0.995727	−	0.0923462i	$-0.0294367\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		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
$567$	−6.11505	+	6.11505i	−0.256808	+	0.256808i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	26.1534			1.09449			0.547243	−	0.836974i	$-0.315677\pi$
$571$	26.1534			1.09449			0.547243	+	0.836974i	$0.315677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	22.4739	−	22.4739i	0.935603	−	0.935603i	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	22.4739	−	22.4739i	0.935603	−	0.935603i	0.998048	+	0.0624458i	$0.0198901\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.94789			0.205273
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.4739	−	27.4739i	1.13397	−	1.13397i	0.144460	−	0.989511i	$-0.453855\pi$
$587$	27.4739	−	27.4739i	1.13397	−	1.13397i	0.989511	−	0.144460i	$-0.0461446\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.435596	+	0.435596i	0.0178878	+	0.0178878i	0.715994	−	0.698106i	$-0.245974\pi$
$593$	0.435596	+	0.435596i	0.0178878	+	0.0178878i	−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$		0			0
$596$	22.0000			0.901155
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	34.0767	−	34.0767i	1.37747	−	1.37747i
$613$	−34.7561	−	34.7561i	−1.40379	−	1.40379i	−0.787598	−	0.616190i	$-0.788675\pi$
$613$	−34.7561	−	34.7561i	−1.40379	−	1.40379i	−0.616190	−	0.787598i	$-0.711325\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.4739	−	17.4739i	0.703475	−	0.703475i	−0.261680	−	0.965155i	$-0.584277\pi$
$617$	17.4739	−	17.4739i	0.703475	−	0.703475i	0.965155	+	0.261680i	$0.0842766\pi$
$618$		0			0
$619$		−	24.0000i		−	0.964641i	−0.875995	−	0.482321i	$-0.839794\pi$
$619$		−	24.0000i		−	0.964641i	0.875995	−	0.482321i	$-0.160206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	35.4356	+	35.4356i	1.41403	+	1.41403i
$629$		0			0
$630$		0			0
$631$	47.9479			1.90878			0.954388	−	0.298570i	$-0.0965097\pi$
$631$	47.9479			1.90878			0.954388	+	0.298570i	$0.0965097\pi$
$632$		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
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	−17.9617	−	17.9617i	−0.708338	−	0.708338i	0.257847	−	0.966186i	$-0.416987\pi$
$643$	−17.9617	−	17.9617i	−0.708338	−	0.708338i	−0.966186	+	0.257847i	$0.916987\pi$
$644$			17.2822i			0.681014i
$645$		0			0
$646$		0			0
$647$	32.4739	−	32.4739i	1.27668	−	1.27668i	0.334169	−	0.942513i	$-0.391544\pi$
$647$	32.4739	−	32.4739i	1.27668	−	1.27668i	0.942513	−	0.334169i	$-0.108456\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−32.7178	+	32.7178i	−1.28133	+	1.28133i
$653$	5.24385	+	5.24385i	0.205208	+	0.205208i	0.802227	−	0.597019i	$-0.203648\pi$
$653$	5.24385	+	5.24385i	0.205208	+	0.205208i	−0.597019	+	0.802227i	$0.703648\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−36.1150	+	36.1150i	−1.40898	+	1.40898i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	19.0000			0.733487
$672$		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
$676$	26.0000			1.00000
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		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	$0.250000\pi$
$684$		−	26.1534i		−	1.00000i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−28.1534	−	28.1534i	−1.07334	−	1.07334i
$689$		0			0
$690$		0			0
$691$	−39.2301			−1.49238			−0.746191	−	0.665731i	$-0.768120\pi$
$691$	−39.2301			−1.49238			−0.746191	+	0.665731i	$0.768120\pi$
$692$		0			0
$693$	8.88495	+	8.88495i	0.337512	+	0.337512i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.4356			−0.658533			−0.329267	−	0.944237i	$-0.606802\pi$
$701$	−17.4356			−0.658533			−0.329267	+	0.944237i	$0.606802\pi$
$702$		0			0
$703$		0			0
$704$			34.8712i			1.31426i
$705$		0			0
$706$		0			0
$707$	−11.8466	+	11.8466i	−0.445537	+	0.445537i
$708$		0			0
$709$		−	52.3068i		−	1.96442i	−0.187779	−	0.982211i	$-0.560129\pi$
$709$		−	52.3068i		−	1.96442i	0.187779	−	0.982211i	$-0.439871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	49.0000i		−	1.82739i	−0.406399	−	0.913696i	$-0.633216\pi$
$719$		−	49.0000i		−	1.82739i	0.406399	−	0.913696i	$-0.366784\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.11505	+	1.11505i	−0.0413547	+	0.0413547i	−0.727482	−	0.686127i	$-0.759309\pi$
$727$	−1.11505	+	1.11505i	−0.0413547	+	0.0413547i	0.686127	+	0.727482i	$0.259309\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	79.9479			2.95698
$732$		0			0
$733$	−33.1534	−	33.1534i	−1.22455	−	1.22455i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	−33.1534	−	33.1534i	−1.22455	−	1.22455i	−0.258551	−	0.965998i	$-0.583245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	30.5123i		−	1.12241i	−0.827676	−	0.561206i	$-0.810337\pi$
$739$		−	30.5123i		−	1.12241i	0.827676	−	0.561206i	$-0.189663\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−10.9233	+	10.9233i	−0.399663	+	0.399663i
$748$	−49.5123	−	49.5123i	−1.81035	−	1.81035i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	17.2822	−	17.2822i	0.630217	−	0.630217i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	37.4739	−	37.4739i	1.36201	−	1.36201i	0.490666	−	0.871348i	$-0.336754\pi$
$757$	37.4739	−	37.4739i	1.36201	−	1.36201i	0.871348	−	0.490666i	$-0.163246\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.35890			0.158010			0.0790050	−	0.996874i	$-0.474826\pi$
$761$	4.35890			0.158010			0.0790050	+	0.996874i	$0.474826\pi$
$762$		0			0
$763$		0			0
$764$		−	34.0000i		−	1.23008i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			51.0000i			1.83911i	0.392965	+	0.919554i	$0.371449\pi$
$769$			51.0000i			1.83911i	−0.392965	+	0.919554i	$0.628551\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	24.3068i		−	0.868100i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−4.56440	−	4.56440i	−0.162600	−	0.162600i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	26.1534			0.926982
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$			49.0767i			1.73621i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	52.4739	+	52.4739i	1.85177	+	1.85177i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			56.6657i			1.99226i	0.0878953	+	0.996130i	$0.471986\pi$
$809$			56.6657i			1.99226i	−0.0878953	+	0.996130i	$0.528014\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	30.6794	−	30.6794i	1.07334	−	1.07334i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−53.0000			−1.84971			−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000			−1.84971			−0.924856	+	0.380317i	$0.875815\pi$
$822$		0			0
$823$	23.8328	+	23.8328i	0.830761	+	0.830761i	0.987621	−	0.156860i	$-0.0501372\pi$
$823$	23.8328	+	23.8328i	0.830761	+	0.830761i	−0.156860	+	0.987621i	$0.550137\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$	−38.1534	−	38.1534i	−1.32592	−	1.32592i
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.5123	+	34.5123i	1.19578	+	1.19578i
$834$		0			0
$835$		0			0
$836$	−38.0000			−1.31426
$837$		0			0
$838$		0			0
$839$		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
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.43560	−	5.43560i	0.186769	−	0.186769i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−13.1534	−	13.1534i	−0.450364	−	0.450364i	0.445112	−	0.895475i	$-0.353164\pi$
$853$	−13.1534	−	13.1534i	−0.450364	−	0.450364i	−0.895475	+	0.445112i	$0.853164\pi$
$854$		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
$859$			56.6657i			1.93341i	0.255897	+	0.966704i	$0.417629\pi$
$859$			56.6657i			1.93341i	−0.255897	+	0.966704i	$0.582371\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		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
$874$		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
$881$	47.9479			1.61541			0.807703	−	0.589590i	$-0.200711\pi$
$881$	47.9479			1.61541			0.807703	+	0.589590i	$0.200711\pi$
$882$		0			0
$883$	10.2439	+	10.2439i	0.344733	+	0.344733i	0.858143	−	0.513410i	$-0.171618\pi$
$883$	10.2439	+	10.2439i	0.344733	+	0.344733i	−0.513410	+	0.858143i	$0.671618\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−39.2301			−1.31426
$892$		0			0
$893$	18.8328	+	18.8328i	0.630217	+	0.630217i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		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
$909$		−	52.3068i		−	1.73491i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	15.8712	+	15.8712i	0.525260	+	0.525260i
$914$		0			0
$915$		0			0
$916$	42.0000			1.38772
$917$	4.75615	−	4.75615i	0.157062	−	0.157062i
$918$		0			0
$919$		−	8.71780i		−	0.287574i	−0.989609	−	0.143787i	$-0.954072\pi$
$919$		−	8.71780i		−	0.287574i	0.989609	−	0.143787i	$-0.0459280\pi$
$920$		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
$929$			34.8712i			1.14409i	0.820223	+	0.572043i	$0.193849\pi$
$929$			34.8712i			1.14409i	−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$	26.4877			0.868100
$932$	−29.5123	+	29.5123i	−0.966707	+	0.966707i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	3.88495	−	3.88495i	0.126916	−	0.126916i	−0.640796	−	0.767712i	$-0.721395\pi$
$937$	3.88495	−	3.88495i	0.126916	−	0.126916i	0.767712	+	0.640796i	$0.221395\pi$
$938$		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
$947$	−34.5123	+	34.5123i	−1.12150	+	1.12150i	−0.129983	+	0.991516i	$0.541492\pi$
$947$	−34.5123	+	34.5123i	−1.12150	+	1.12150i	−0.991516	+	0.129983i	$0.958508\pi$
$948$		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	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$	−61.0246			−1.97368
$957$		0			0
$958$		0			0
$959$			12.6657i			0.408996i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−24.5123	+	24.5123i	−0.788262	+	0.788262i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	−24.5123	+	24.5123i	−0.788262	+	0.788262i	0.192947	+	0.981209i	$0.438195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−6.11505	−	6.11505i	−0.196039	−	0.196039i
$974$		0			0
$975$		0			0
$976$	−17.4356			−0.558100
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	89.5123i		−	2.84633i
$990$		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
$997$	−29.3206	+	29.3206i	−0.928591	+	0.928591i	−0.997615	−	0.0690239i	$-0.978012\pi$
$997$	−29.3206	+	29.3206i	−0.928591	+	0.928591i	0.0690239	+	0.997615i	$0.478012\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.g.a.18.2		4
5.2	odd	4	inner	475.2.g.a.132.2		4
5.3	odd	4		95.2.g.a.37.2	yes	4
5.4	even	2		95.2.g.a.18.1	✓	4
15.8	even	4		855.2.p.a.37.1		4
15.14	odd	2		855.2.p.a.208.2		4
19.18	odd	2	CM	475.2.g.a.18.2		4
95.18	even	4		95.2.g.a.37.2	yes	4
95.37	even	4	inner	475.2.g.a.132.2		4
95.94	odd	2		95.2.g.a.18.1	✓	4
285.113	odd	4		855.2.p.a.37.1		4
285.284	even	2		855.2.p.a.208.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.g.a.18.1	✓	4	5.4	even	2
95.2.g.a.18.1	✓	4	95.94	odd	2
95.2.g.a.37.2	yes	4	5.3	odd	4
95.2.g.a.37.2	yes	4	95.18	even	4
475.2.g.a.18.2		4	1.1	even	1	trivial
475.2.g.a.18.2		4	19.18	odd	2	CM
475.2.g.a.132.2		4	5.2	odd	4	inner
475.2.g.a.132.2		4	95.37	even	4	inner
855.2.p.a.37.1		4	15.8	even	4
855.2.p.a.37.1		4	285.113	odd	4
855.2.p.a.208.2		4	15.14	odd	2
855.2.p.a.208.2		4	285.284	even	2

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{4} +(0.679449 - 0.679449i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q-2.00000i q^{4} +(0.679449 - 0.679449i) q^{7} +3.00000i q^{9} +4.35890 q^{11} -4.00000 q^{16} +(5.67945 - 5.67945i) q^{17} -4.35890i q^{19} +(-6.35890 - 6.35890i) q^{23} +(-1.35890 - 1.35890i) q^{28} +6.00000 q^{36} +(7.03835 + 7.03835i) q^{43} -8.71780i q^{44} +(-4.32055 + 4.32055i) q^{47} +6.07670i q^{49} +4.35890 q^{61} +(2.03835 + 2.03835i) q^{63} +8.00000i q^{64} +(-11.3589 - 11.3589i) q^{68} +(12.0383 + 12.0383i) q^{73} -8.71780 q^{76} +(2.96165 - 2.96165i) q^{77} -9.00000 q^{81} +(3.64110 + 3.64110i) q^{83} +(-12.7178 + 12.7178i) q^{92} +13.0767i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7}+O(q^{10}) \) 4 * q - 6 * q^7	\( 4 q - 6 q^{7} - 16 q^{16} + 14 q^{17} - 8 q^{23} + 12 q^{28} + 24 q^{36} + 2 q^{43} - 26 q^{47} - 18 q^{63} - 28 q^{68} + 22 q^{73} + 38 q^{77} - 36 q^{81} + 32 q^{83} - 16 q^{92}+O(q^{100}) \) 4 * q - 6 * q^7 - 16 * q^16 + 14 * q^17 - 8 * q^23 + 12 * q^28 + 24 * q^36 + 2 * q^43 - 26 * q^47 - 18 * q^63 - 28 * q^68 + 22 * q^73 + 38 * q^77 - 36 * q^81 + 32 * q^83 - 16 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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