LMFDB - Embedded newform 800.2.bg.b.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(129,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("800.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.bg (of order $10$, degree $4$, 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:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			769.2
Root			$0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	800.769
Dual form			800.2.bg.b.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.366554 - 2.20582i) q^{5} +(0.927051 - 2.85317i) q^{9} +O(q^{10})$	$q+(0.366554 - 2.20582i) q^{5} +(0.927051 - 2.85317i) q^{9} +(-6.43179 - 2.08981i) q^{13} +(-0.335051 + 0.461158i) q^{17} +(-4.73128 - 1.61710i) q^{25} +(-0.577684 + 0.419712i) q^{29} +(-1.71769 - 0.558113i) q^{37} +(1.04691 - 3.22207i) q^{41} +(-5.95376 - 3.09075i) q^{45} +7.00000 q^{49} +(-6.73899 - 9.27543i) q^{53} +(-4.67963 - 14.4024i) q^{61} +(-6.96735 + 13.4213i) q^{65} +(15.6648 - 5.08981i) q^{73} +(-7.28115 - 5.29007i) q^{81} +(0.894418 + 0.908101i) q^{85} +(5.65720 + 17.4111i) q^{89} +(8.60922 + 11.8496i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 8 * q - 2 * q^5 - 6 * q^9	$ 8 q - 2 q^{5} - 6 q^{9} + 6 q^{25} + 20 q^{29} + 10 q^{37} + 20 q^{41} - 6 q^{45} + 56 q^{49} - 70 q^{53} - 20 q^{61} + 6 q^{65} - 18 q^{81} + 38 q^{85} + 30 q^{89}+O(q^{100}) $ 8 * q - 2 * q^5 - 6 * q^9 + 6 * q^25 + 20 * q^29 + 10 * q^37 + 20 * q^41 - 6 * q^45 + 56 * q^49 - 70 * q^53 - 20 * q^61 + 6 * q^65 - 18 * q^81 + 38 * q^85 + 30 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{9}{10}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$3$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$4$		0			0
$5$	0.366554	−	2.20582i	0.163928	−	0.986472i
$5$	0.366554	−	2.20582i	0.163928	−	0.986472i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	0.927051	−	2.85317i	0.309017	−	0.951057i
$10$		0			0
$11$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$11$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$12$		0			0
$13$	−6.43179	−	2.08981i	−1.78386	−	0.579610i	−0.784669	−	0.619915i	$-0.787167\pi$
$13$	−6.43179	−	2.08981i	−1.78386	−	0.579610i	−0.999187	+	0.0403050i	$0.987167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.335051	+	0.461158i	−0.0812618	+	0.111847i	−0.847713	−	0.530456i	$-0.822021\pi$
$17$	−0.335051	+	0.461158i	−0.0812618	+	0.111847i	0.766451	+	0.642303i	$0.222021\pi$
$18$		0			0
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$23$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$24$		0			0
$25$	−4.73128	−	1.61710i	−0.946255	−	0.323420i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.577684	+	0.419712i	−0.107273	+	0.0779385i	−0.640129	−	0.768268i	$-0.721119\pi$
$29$	−0.577684	+	0.419712i	−0.107273	+	0.0779385i	0.532855	+	0.846206i	$0.321119\pi$
$30$		0			0
$31$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$31$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.71769	−	0.558113i	−0.282387	−	0.0917532i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−1.71769	−	0.558113i	−0.282387	−	0.0917532i	−0.446786	+	0.894641i	$0.647432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.04691	−	3.22207i	0.163501	−	0.503203i	−0.835422	−	0.549609i	$-0.814777\pi$
$41$	1.04691	−	3.22207i	0.163501	−	0.503203i	0.998923	+	0.0464057i	$0.0147767\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−5.95376	−	3.09075i	−0.887535	−	0.460741i
$46$		0			0
$47$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$47$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.73899	−	9.27543i	−0.925672	−	1.27408i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−6.73899	−	9.27543i	−0.925672	−	1.27408i	0.0358519	−	0.999357i	$-0.488586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$59$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$60$		0			0
$61$	−4.67963	−	14.4024i	−0.599165	−	1.84404i	−0.532794	−	0.846245i	$-0.678858\pi$
$61$	−4.67963	−	14.4024i	−0.599165	−	1.84404i	−0.0663709	−	0.997795i	$-0.521142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.96735	+	13.4213i	−0.864193	+	1.66471i
$66$		0			0
$67$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$67$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$71$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$72$		0			0
$73$	15.6648	−	5.08981i	1.83343	−	0.595718i	0.834425	−	0.551121i	$-0.185800\pi$
$73$	15.6648	−	5.08981i	1.83343	−	0.595718i	0.999005	−	0.0445966i	$-0.0142003\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$79$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$80$		0			0
$81$	−7.28115	−	5.29007i	−0.809017	−	0.587785i
$82$		0			0
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$		0			0
$85$	0.894418	+	0.908101i	0.0970132	+	0.0984974i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.65720	+	17.4111i	0.599662	+	1.84557i	0.529999	+	0.847998i	$0.322192\pi$
$89$	5.65720	+	17.4111i	0.599662	+	1.84557i	0.0696627	+	0.997571i	$0.477808\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.60922	+	11.8496i	0.874134	+	1.20314i	0.978011	+	0.208552i	$0.0668751\pi$
$97$	8.60922	+	11.8496i	0.874134	+	1.20314i	−0.103877	+	0.994590i	$0.533125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.4031			1.83118			0.915588	−	0.402117i	$-0.131726\pi$
$101$	18.4031			1.83118			0.915588	+	0.402117i	$0.131726\pi$
$102$		0			0
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	5.13271	−	15.7969i	0.491625	−	1.51307i	−0.330527	−	0.943797i	$-0.607226\pi$
$109$	5.13271	−	15.7969i	0.491625	−	1.51307i	0.822152	−	0.569269i	$-0.192774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.3654	−	5.64236i	−1.63360	−	0.530789i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−17.3654	−	5.64236i	−1.63360	−	0.530789i	−0.975095	+	0.221788i	$0.928811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.9252	+	16.4136i	−1.10248	+	1.51744i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.89919	−	6.46564i	0.809017	−	0.587785i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.30130	+	9.84359i	−0.474163	+	0.880437i
$126$		0			0
$127$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$127$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$131$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.14301	+	1.99598i	0.524832	+	0.170528i	0.559437	−	0.828873i	$-0.311017\pi$
$137$	6.14301	+	1.99598i	0.524832	+	0.170528i	−0.0346048	+	0.999401i	$0.511017\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.714056	+	1.42811i	0.0592991	+	0.118598i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.429467			−0.0351833			−0.0175917	−	0.999845i	$-0.505600\pi$
$149$	−0.429467			−0.0351833			−0.0175917	+	0.999845i	$0.505600\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	1.00515	+	1.38347i	0.0812618	+	0.111847i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			24.6314i			1.96580i	0.184131	+	0.982902i	$0.441053\pi$
$157$			24.6314i			1.96580i	−0.184131	+	0.982902i	$0.558947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$163$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$167$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$168$		0			0
$169$	26.4833	+	19.2413i	2.03718	+	1.48010i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−22.3417	+	7.25924i	−1.69860	+	0.551910i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−22.3417	+	7.25924i	−1.69860	+	0.551910i	−0.710233	+	0.703967i	$0.751410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$179$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$180$		0			0
$181$	19.8884	+	14.4498i	1.47829	+	1.07404i	0.978101	+	0.208130i	$0.0667377\pi$
$181$	19.8884	+	14.4498i	1.47829	+	1.07404i	0.500193	+	0.865914i	$0.333262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.86072	+	3.58435i	−0.136803	+	0.263526i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$191$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$192$		0			0
$193$		−	11.1874i		−	0.805288i	−0.915357	−	0.402644i	$-0.868091\pi$
$193$		−	11.1874i		−	0.805288i	0.915357	−	0.402644i	$-0.131909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.6238	+	17.3752i	0.899409	+	1.23793i	0.970656	+	0.240472i	$0.0773021\pi$
$197$	12.6238	+	17.3752i	0.899409	+	1.23793i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$		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
$205$	−6.72356	−	3.49037i	−0.469594	−	0.243778i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$211$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$212$		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
$221$	3.11871	−	2.26588i	0.209787	−	0.152419i
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$		0			0
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$		0			0
$229$	17.7331	−	12.8839i	1.17184	−	0.851391i	0.180611	−	0.983555i	$-0.442192\pi$
$229$	17.7331	−	12.8839i	1.17184	−	0.851391i	0.991228	+	0.132164i	$0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.37433	+	1.89160i	−0.0900352	+	0.123923i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−1.37433	+	1.89160i	−0.0900352	+	0.123923i	0.761623	+	0.648020i	$0.224403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$239$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$240$		0			0
$241$	5.21589	−	16.0528i	0.335985	−	1.03405i	−0.630250	−	0.776392i	$-0.717048\pi$
$241$	5.21589	−	16.0528i	0.335985	−	1.03405i	0.966235	−	0.257663i	$-0.0829523\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.56587	−	15.4407i	0.163928	−	0.986472i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		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
$256$		0			0
$257$		−	24.7130i		−	1.54155i	−0.637106	−	0.770776i	$-0.719869\pi$
$257$		−	24.7130i		−	1.54155i	0.637106	−	0.770776i	$-0.280131\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.661966	+	2.03732i	0.0409747	+	0.126107i
$262$		0			0
$263$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$263$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$264$		0			0
$265$	−22.9301	+	11.4651i	−1.40859	+	0.704293i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	26.5278	+	19.2736i	1.61743	+	1.17513i	0.824804	+	0.565419i	$0.191286\pi$
$269$	26.5278	+	19.2736i	1.61743	+	1.17513i	0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$271$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	24.5101	−	7.96383i	1.47267	−	0.478500i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	24.5101	−	7.96383i	1.47267	−	0.478500i	0.931914	+	0.362678i	$0.118138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−21.7620	−	15.8110i	−1.29821	−	0.943206i	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−21.7620	−	15.8110i	−1.29821	−	0.943206i	−0.999936	+	0.0112742i	$0.996411\pi$
$282$		0			0
$283$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$283$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.15288	+	15.8589i	0.303111	+	0.932879i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	31.0999i		−	1.81687i	−0.418023	−	0.908436i	$-0.637277\pi$
$293$		−	31.0999i		−	1.81687i	0.418023	−	0.908436i	$-0.362723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−33.4845	+	5.04316i	−1.91731	+	0.288770i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$	22.8254	+	7.41641i	1.29017	+	0.419200i	0.872149	−	0.489240i	$-0.162726\pi$
$313$	22.8254	+	7.41641i	1.29017	+	0.419200i	0.418016	+	0.908440i	$0.362726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.4580	+	22.6525i	−0.924373	+	1.27229i	0.0376418	+	0.999291i	$0.488015\pi$
$317$	−16.4580	+	22.6525i	−0.924373	+	1.27229i	−0.962014	+	0.272999i	$0.911985\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	27.0511	+	20.2883i	1.50053	+	1.12539i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$331$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$332$		0			0
$333$	−3.18478	+	4.38347i	−0.174525	+	0.240213i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−30.4338	−	9.88854i	−1.65784	−	0.538663i	−0.677419	−	0.735598i	$-0.736901\pi$
$337$	−30.4338	−	9.88854i	−1.65784	−	0.538663i	−0.980417	+	0.196934i	$0.936901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$347$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$348$		0			0
$349$	−37.3282			−1.99813			−0.999066	−	0.0431990i	$-0.986245\pi$
$349$	−37.3282			−1.99813			−0.999066	+	0.0431990i	$0.986245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.40456	−	12.9443i	−0.500554	−	0.688954i	0.481736	−	0.876316i	$-0.340006\pi$
$353$	−9.40456	−	12.9443i	−0.500554	−	0.688954i	−0.982291	+	0.187362i	$0.940006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$		0			0
$361$	−5.87132	−	18.0701i	−0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.48521	−	36.4195i	−0.287109	−	1.90628i
$366$		0			0
$367$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$367$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$368$		0			0
$369$	−8.22258	−	5.97405i	−0.428050	−	0.310997i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−34.2380	+	11.1246i	−1.77278	+	0.576011i	−0.998391	−	0.0567016i	$-0.981942\pi$
$373$	−34.2380	+	11.1246i	−1.77278	+	0.576011i	−0.774387	+	0.632712i	$0.781942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.59266	−	1.49224i	0.236534	−	0.0768545i
$378$		0			0
$379$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$379$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$383$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.86729	−	14.9800i	−0.246781	−	0.759515i	−0.995338	−	0.0964443i	$-0.969253\pi$
$389$	−4.86729	−	14.9800i	−0.246781	−	0.759515i	0.748557	−	0.663070i	$-0.230747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.05342	+	9.70820i	0.354001	+	0.487241i	0.948465	−	0.316881i	$-0.102636\pi$
$397$	7.05342	+	9.70820i	0.354001	+	0.487241i	−0.594464	+	0.804122i	$0.702636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−25.1294			−1.25490			−0.627452	−	0.778655i	$-0.715902\pi$
$401$	−25.1294			−1.25490			−0.627452	+	0.778655i	$0.715902\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−14.3379	+	14.1218i	−0.712454	+	0.701719i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	12.3287	−	37.9437i	0.609613	−	1.87619i	0.148340	−	0.988936i	$-0.452607\pi$
$409$	12.3287	−	37.9437i	0.609613	−	1.87619i	0.461272	−	0.887259i	$-0.347393\pi$
$410$		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
$419$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$419$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$420$		0			0
$421$	29.0438	−	21.1015i	1.41551	−	1.02843i	0.423015	−	0.906123i	$-0.360972\pi$
$421$	29.0438	−	21.1015i	1.41551	−	1.02843i	0.992493	−	0.122304i	$-0.0390281\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.33096	−	1.64006i	0.113068	−	0.0795544i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$		0			0
$433$	23.1594	−	31.8762i	1.11297	−	1.53187i	0.296001	−	0.955188i	$-0.404347\pi$
$433$	23.1594	−	31.8762i	1.11297	−	1.53187i	0.816968	−	0.576683i	$-0.195653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$439$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$440$		0			0
$441$	6.48936	−	19.9722i	0.309017	−	0.951057i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	40.4793	−	6.09667i	1.91890	−	0.289010i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.7160			−1.59116			−0.795579	−	0.605850i	$-0.792833\pi$
$449$	−33.7160			−1.59116			−0.795579	+	0.605850i	$0.792833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.50653	+	29.2581i	0.442763	+	1.36269i	0.884918	+	0.465746i	$0.154214\pi$
$461$	9.50653	+	29.2581i	0.442763	+	1.36269i	−0.442155	+	0.896939i	$0.645786\pi$
$462$		0			0
$463$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$463$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$467$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−32.7118	+	10.6287i	−1.49777	+	0.486655i
$478$		0			0
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$	9.88149	+	7.17932i	0.450557	+	0.327349i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.2938	−	14.6469i	1.33016	−	0.665081i
$486$		0			0
$487$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$487$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$491$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$492$		0			0
$493$		−	0.407028i		−	0.0183316i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$503$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$504$		0			0
$505$	6.74572	−	40.5939i	0.300181	−	1.80641i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.9764	−	36.8594i	0.530843	−	1.63377i	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	11.9764	−	36.8594i	0.530843	−	1.63377i	0.752464	−	0.658633i	$-0.228865\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.4203	+	24.2813i	−1.46417	+	1.06378i	−0.481919	+	0.876216i	$0.660060\pi$
$521$	−33.4203	+	24.2813i	−1.46417	+	1.06378i	−0.982252	+	0.187566i	$0.939940\pi$
$522$		0			0
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−18.6074	+	13.5191i	−0.809017	+	0.587785i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13.4671	+	18.5358i	−0.583323	+	0.802876i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.86721	−	5.74668i	0.0802776	−	0.247069i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	1.86721	−	5.74668i	0.0802776	−	0.247069i	0.983138	+	0.182865i	$0.0585370\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−32.9636	−	17.1122i	−1.41201	−	0.733008i
$546$		0			0
$547$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$547$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$548$		0			0
$549$	−45.4308			−1.93894
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	27.4877i		−	1.16469i	−0.812942	−	0.582345i	$-0.802135\pi$
$557$		−	27.4877i		−	1.16469i	0.812942	−	0.582345i	$-0.197865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$		0			0
$565$	−18.8114	+	36.2367i	−0.791401	+	1.52449i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	37.2768	+	27.0832i	1.56273	+	1.13539i	0.933730	+	0.357979i	$0.116534\pi$
$569$	37.2768	+	27.0832i	1.56273	+	1.13539i	0.628997	+	0.777408i	$0.283466\pi$
$570$		0			0
$571$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$571$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	45.6507	−	14.8328i	1.90046	−	0.617498i	0.937367	−	0.348342i	$-0.113255\pi$
$577$	45.6507	−	14.8328i	1.90046	−	0.617498i	0.963097	−	0.269156i	$-0.0867448\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	31.8342	+	32.3213i	1.31618	+	1.33632i
$586$		0			0
$587$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$587$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			48.6929i			1.99958i	0.0205761	+	0.999788i	$0.493450\pi$
$593$			48.6929i			1.99958i	−0.0205761	+	0.999788i	$0.506550\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−20.1235			−0.820856			−0.410428	−	0.911893i	$-0.634621\pi$
$601$	−20.1235			−0.820856			−0.410428	+	0.911893i	$0.634621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.0000	−	22.0000i	−0.447214	−	0.894427i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	20.1732	+	6.55466i	0.814786	+	0.264740i	0.686624	−	0.727013i	$-0.259092\pi$
$613$	20.1732	+	6.55466i	0.814786	+	0.264740i	0.128162	+	0.991753i	$0.459092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.08823	+	2.87420i	−0.0840689	+	0.115711i	−0.848980	−	0.528425i	$-0.822783\pi$
$617$	−2.08823	+	2.87420i	−0.0840689	+	0.115711i	0.764911	+	0.644136i	$0.222783\pi$
$618$		0			0
$619$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$619$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.7700	+	15.3019i	0.790799	+	0.612076i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.832894	−	0.605133i	0.0332096	−	0.0241282i
$630$		0			0
$631$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$631$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−45.0225	−	14.6287i	−1.78386	−	0.579610i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.4508	+	47.5528i	−0.610272	+	1.87822i	−0.154878	+	0.987934i	$0.549498\pi$
$641$	−15.4508	+	47.5528i	−0.610272	+	1.87822i	−0.455394	+	0.890290i	$0.650502\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$647$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.5264	+	31.0049i	0.881527	+	1.21332i	0.975996	+	0.217789i	$0.0698846\pi$
$653$	22.5264	+	31.0049i	0.881527	+	1.21332i	−0.0944693	+	0.995528i	$0.530115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	49.4130i		−	1.92778i
$658$		0			0
$659$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$659$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$660$		0			0
$661$	15.4508	+	47.5528i	0.600968	+	1.84959i	0.522435	+	0.852679i	$0.325024\pi$
$661$	15.4508	+	47.5528i	0.600968	+	1.84959i	0.0785333	+	0.996911i	$0.474976\pi$
$662$		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
$671$		0			0
$672$		0			0
$673$	48.6608	−	15.8109i	1.87574	−	0.609464i	0.886585	−	0.462566i	$-0.153071\pi$
$673$	48.6608	−	15.8109i	1.87574	−	0.609464i	0.989152	−	0.146898i	$-0.0469288\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	49.4549	−	16.0689i	1.90071	−	0.617577i	0.938477	−	0.345341i	$-0.112237\pi$
$677$	49.4549	−	16.0689i	1.90071	−	0.617577i	0.962230	−	0.272237i	$-0.0877633\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$683$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$684$		0			0
$685$	6.65452	−	12.8187i	0.254256	−	0.489778i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	23.9598	+	73.7408i	0.912797	+	2.80930i
$690$		0			0
$691$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$691$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.13512	+	1.56235i	0.0429956	+	0.0591783i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.6550			−1.45998			−0.729990	−	0.683458i	$-0.760475\pi$
$701$	−38.6550			−1.45998			−0.729990	+	0.683458i	$0.760475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.491968	−	1.51412i	0.0184762	−	0.0568640i	−0.941393	−	0.337311i	$-0.890483\pi$
$709$	0.491968	−	1.51412i	0.0184762	−	0.0568640i	0.959870	+	0.280447i	$0.0904826\pi$
$710$		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
$719$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$719$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.41190	−	1.05160i	0.126715	−	0.0390554i
$726$		0			0
$727$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$727$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$728$		0			0
$729$	−21.8435	+	15.8702i	−0.809017	+	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−2.35114	+	3.23607i	−0.0868414	+	0.119527i	−0.850227	−	0.526416i	$-0.823536\pi$
$733$	−2.35114	+	3.23607i	−0.0868414	+	0.119527i	0.763386	+	0.645943i	$0.223536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−0.157423	+	0.947327i	−0.00576752	+	0.0347074i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	31.4887i		−	1.14448i	−0.820087	−	0.572239i	$-0.806075\pi$
$757$		−	31.4887i		−	1.14448i	0.820087	−	0.572239i	$-0.193925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.23457	+	6.87731i	0.0810033	+	0.249302i	0.983354	−	0.181700i	$-0.0581600\pi$
$761$	2.23457	+	6.87731i	0.0810033	+	0.249302i	−0.902351	+	0.431003i	$0.858160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.42014	−	1.71007i	0.123655	−	0.0618277i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−40.4508	−	29.3893i	−1.45869	−	1.05980i	−0.983700	−	0.179815i	$-0.942450\pi$
$769$	−40.4508	−	29.3893i	−1.45869	−	1.05980i	−0.474995	−	0.879989i	$-0.657550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.8479	−	4.82439i	0.534043	−	0.173521i	−0.0295658	−	0.999563i	$-0.509412\pi$
$773$	14.8479	−	4.82439i	0.534043	−	0.173521i	0.563609	+	0.826042i	$0.309412\pi$
$774$		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
$784$		0			0
$785$	54.3325	+	9.02874i	1.93921	+	0.322250i
$786$		0			0
$787$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$787$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			102.413i			3.63678i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.1267	+	23.5728i	0.606657	+	0.834992i	0.996297	−	0.0859751i	$-0.0274006\pi$
$797$	17.1267	+	23.5728i	0.606657	+	0.834992i	−0.389640	+	0.920967i	$0.627401\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	54.9212			1.94055
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.4129	−	53.5914i	0.612205	−	1.88417i	0.175791	−	0.984428i	$-0.443752\pi$
$809$	17.4129	−	53.5914i	0.612205	−	1.88417i	0.436414	−	0.899746i	$-0.356248\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−40.4508	+	29.3893i	−1.41174	+	1.02569i	−0.418678	+	0.908135i	$0.637507\pi$
$821$	−40.4508	+	29.3893i	−1.41174	+	1.02569i	−0.993066	+	0.117558i	$0.962493\pi$
$822$		0			0
$823$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$823$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$827$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$828$		0			0
$829$	−25.8329	+	18.7687i	−0.897213	+	0.651864i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−25.8329	+	18.7687i	−0.897213	+	0.651864i	0.0405353	+	0.999178i	$0.487094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.34536	+	3.22811i	−0.0812618	+	0.111847i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$839$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$840$		0			0
$841$	−8.80393	+	27.0957i	−0.303584	+	0.934335i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	52.1503	−	51.3645i	1.79402	−	1.76699i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.22641	−	1.68801i	−0.0419914	−	0.0577962i	0.787505	−	0.616308i	$-0.211372\pi$
$853$	−1.22641	−	1.68801i	−0.0419914	−	0.0577962i	−0.829496	+	0.558512i	$0.811372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.00000i			0.273275i	0.990621	+	0.136637i	$0.0436295\pi$
$857$			8.00000i			0.273275i	−0.990621	+	0.136637i	$0.956370\pi$
$858$		0			0
$859$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$859$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$863$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$864$		0			0
$865$	7.82317	+	51.9425i	0.265996	+	1.76610i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	41.7900	−	13.5784i	1.41438	−	0.459559i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.1899	+	7.53487i	−0.783068	+	0.254434i	−0.673150	−	0.739506i	$-0.735059\pi$
$877$	−23.1899	+	7.53487i	−0.783068	+	0.254434i	−0.109919	+	0.993941i	$0.535059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	40.4508	+	29.3893i	1.36282	+	0.990149i	0.998260	+	0.0589711i	$0.0187820\pi$
$881$	40.4508	+	29.3893i	1.36282	+	0.990149i	0.364564	+	0.931178i	$0.381218\pi$
$882$		0			0
$883$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$883$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	6.53535			0.217724
$902$		0			0
$903$		0			0
$904$		0			0
$905$	39.1638	−	38.5736i	1.30185	−	1.28223i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	17.0606	−	52.5072i	0.565865	−	1.74155i
$910$		0			0
$911$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$911$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$919$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	7.22436	+	5.41827i	0.237536	+	0.178152i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−42.2768	+	30.7159i	−1.38706	+	1.00776i	−0.390877	+	0.920443i	$0.627828\pi$
$929$	−42.2768	+	30.7159i	−1.38706	+	1.00776i	−0.996181	+	0.0873137i	$0.972172\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−48.4782	−	15.7515i	−1.58371	−	0.514579i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−48.4782	−	15.7515i	−1.58371	−	0.514579i	−0.963010	+	0.269466i	$0.913153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.4164	+	35.1359i	−0.372163	+	1.14540i	0.573210	+	0.819408i	$0.305698\pi$
$941$	−11.4164	+	35.1359i	−0.372163	+	1.14540i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$947$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$948$		0			0
$949$	−111.390			−3.61586
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.36285	−	6.00494i	−0.141326	−	0.194519i	0.732486	−	0.680782i	$-0.238360\pi$
$953$	−4.36285	−	6.00494i	−0.141326	−	0.194519i	−0.873813	+	0.486263i	$0.838360\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−9.57953	−	29.4828i	−0.309017	−	0.951057i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24.6774	−	4.10079i	−0.794394	−	0.132009i
$966$		0			0
$967$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$967$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$971$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	40.8144	−	13.2614i	1.30577	−	0.424270i	0.428184	−	0.903691i	$-0.359153\pi$
$977$	40.8144	−	13.2614i	1.30577	−	0.424270i	0.877585	+	0.479421i	$0.159153\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−40.3128	−	29.2890i	−1.28709	−	0.935126i
$982$		0			0
$983$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$983$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$984$		0			0
$985$	42.9538	−	21.4769i	1.36862	−	0.684311i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$991$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−7.05342	−	9.70820i	−0.223384	−	0.307462i	0.682585	−	0.730807i	$-0.260856\pi$
$997$	−7.05342	−	9.70820i	−0.223384	−	0.307462i	−0.905969	+	0.423345i	$0.860856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.bg.b.769.2	yes	8
4.3	odd	2	CM	800.2.bg.b.769.2	yes	8
25.4	even	10	inner	800.2.bg.b.129.2	✓	8
100.79	odd	10	inner	800.2.bg.b.129.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.bg.b.129.2	✓	8	25.4	even	10	inner
800.2.bg.b.129.2	✓	8	100.79	odd	10	inner
800.2.bg.b.769.2	yes	8	1.1	even	1	trivial
800.2.bg.b.769.2	yes	8	4.3	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.bg (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.366554 - 2.20582i) q^{5} +(0.927051 - 2.85317i) q^{9} +O(q^{10})\)	\(q+(0.366554 - 2.20582i) q^{5} +(0.927051 - 2.85317i) q^{9} +(-6.43179 - 2.08981i) q^{13} +(-0.335051 + 0.461158i) q^{17} +(-4.73128 - 1.61710i) q^{25} +(-0.577684 + 0.419712i) q^{29} +(-1.71769 - 0.558113i) q^{37} +(1.04691 - 3.22207i) q^{41} +(-5.95376 - 3.09075i) q^{45} +7.00000 q^{49} +(-6.73899 - 9.27543i) q^{53} +(-4.67963 - 14.4024i) q^{61} +(-6.96735 + 13.4213i) q^{65} +(15.6648 - 5.08981i) q^{73} +(-7.28115 - 5.29007i) q^{81} +(0.894418 + 0.908101i) q^{85} +(5.65720 + 17.4111i) q^{89} +(8.60922 + 11.8496i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 8 * q - 2 * q^5 - 6 * q^9	\( 8 q - 2 q^{5} - 6 q^{9} + 6 q^{25} + 20 q^{29} + 10 q^{37} + 20 q^{41} - 6 q^{45} + 56 q^{49} - 70 q^{53} - 20 q^{61} + 6 q^{65} - 18 q^{81} + 38 q^{85} + 30 q^{89}+O(q^{100}) \) 8 * q - 2 * q^5 - 6 * q^9 + 6 * q^25 + 20 * q^29 + 10 * q^37 + 20 * q^41 - 6 * q^45 + 56 * q^49 - 70 * q^53 - 20 * q^61 + 6 * q^65 - 18 * q^81 + 38 * q^85 + 30 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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