LMFDB - Embedded newform 800.2.bg.b.769.1

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			769.1
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	800.769
Dual form			800.2.bg.b.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.98459 + 1.03025i) q^{5} +(0.927051 - 2.85317i) q^{9} +O(q^{10})$	$q+(-1.98459 + 1.03025i) q^{5} +(0.927051 - 2.85317i) q^{9} +(-0.276418 - 0.0898138i) q^{13} +(2.57112 - 3.53884i) q^{17} +(2.87718 - 4.08924i) q^{25} +(5.57768 - 4.05242i) q^{29} +(5.33573 + 1.73368i) q^{37} +(3.95309 - 12.1663i) q^{41} +(1.09966 + 6.61746i) q^{45} +7.00000 q^{49} +(-2.93477 - 4.03936i) q^{53} +(-0.320372 - 0.986005i) q^{61} +(0.641107 - 0.106536i) q^{65} +(-8.95663 + 2.91019i) q^{73} +(-7.28115 - 5.29007i) q^{81} +(-1.45672 + 9.67203i) q^{85} +(-3.74737 - 11.5332i) q^{89} +(11.5154 + 15.8496i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.98459	+	1.03025i	−0.887535	+	0.460741i
$5$	−1.98459	+	1.03025i	−0.887535	+	0.460741i
$6$		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
$13$	−0.276418	−	0.0898138i	−0.0766647	−	0.0249099i	0.270434	−	0.962739i	$-0.412833\pi$
$13$	−0.276418	−	0.0898138i	−0.0766647	−	0.0249099i	−0.347098	+	0.937829i	$0.612833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.57112	−	3.53884i	0.623588	−	0.858295i	−0.374020	−	0.927421i	$-0.622021\pi$
$17$	2.57112	−	3.53884i	0.623588	−	0.858295i	0.997608	+	0.0691254i	$0.0220209\pi$
$18$		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
$25$	2.87718	−	4.08924i	0.575435	−	0.817848i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.57768	−	4.05242i	1.03575	−	0.752516i	0.0662984	−	0.997800i	$-0.478881\pi$
$29$	5.57768	−	4.05242i	1.03575	−	0.752516i	0.969451	+	0.245284i	$0.0788811\pi$
$30$		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
$37$	5.33573	+	1.73368i	0.877188	+	0.285016i	0.712789	−	0.701378i	$-0.247432\pi$
$37$	5.33573	+	1.73368i	0.877188	+	0.285016i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.95309	−	12.1663i	0.617368	−	1.90006i	0.264550	−	0.964372i	$-0.414777\pi$
$41$	3.95309	−	12.1663i	0.617368	−	1.90006i	0.352819	−	0.935692i	$-0.385223\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	1.09966	+	6.61746i	0.163928	+	0.986472i
$46$		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
$53$	−2.93477	−	4.03936i	−0.403121	−	0.554849i	0.558403	−	0.829570i	$-0.311414\pi$
$53$	−2.93477	−	4.03936i	−0.403121	−	0.554849i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		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
$61$	−0.320372	−	0.986005i	−0.0410195	−	0.126245i	0.928450	−	0.371458i	$-0.121142\pi$
$61$	−0.320372	−	0.986005i	−0.0410195	−	0.126245i	−0.969469	+	0.245213i	$0.921142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.641107	−	0.106536i	0.0795195	−	0.0132142i
$66$		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
$73$	−8.95663	+	2.91019i	−1.04829	+	0.340612i	−0.781999	−	0.623280i	$-0.785800\pi$
$73$	−8.95663	+	2.91019i	−1.04829	+	0.340612i	−0.266296	+	0.963891i	$0.585800\pi$
$74$		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
$85$	−1.45672	+	9.67203i	−0.158004	+	1.04908i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.74737	−	11.5332i	−0.397220	−	1.22252i	−0.927219	−	0.374519i	$-0.877808\pi$
$89$	−3.74737	−	11.5332i	−0.397220	−	1.22252i	0.529999	−	0.847998i	$-0.322192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	11.5154	+	15.8496i	1.16921	+	1.60928i	0.668644	+	0.743583i	$0.266875\pi$
$97$	11.5154	+	15.8496i	1.16921	+	1.60928i	0.500567	+	0.865698i	$0.333125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−19.6392			−1.95417			−0.977085	−	0.212850i	$-0.931726\pi$
$101$	−19.6392			−1.95417			−0.977085	+	0.212850i	$0.931726\pi$
$102$		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
$109$	−2.13271	+	6.56381i	−0.204277	+	0.628699i	0.795465	+	0.605999i	$0.207226\pi$
$109$	−2.13271	+	6.56381i	−0.204277	+	0.628699i	−0.999742	+	0.0227005i	$0.992774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.96084	−	2.58663i	−0.748893	−	0.243330i	−0.0903879	−	0.995907i	$-0.528811\pi$
$113$	−7.96084	−	2.58663i	−0.748893	−	0.243330i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.512508	+	0.705407i	−0.0473814	+	0.0652149i
$118$		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
$125$	−1.49707	+	11.0797i	−0.133902	+	0.990995i
$126$		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
$137$	18.4537	+	5.99598i	1.57661	+	0.512271i	0.961180	−	0.275921i	$-0.0889827\pi$
$137$	18.4537	+	5.99598i	1.57661	+	0.512271i	0.615429	+	0.788192i	$0.288983\pi$
$138$		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
$145$	−6.89440	+	13.7888i	−0.572548	+	1.14510i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	23.0819			1.89095			0.945473	−	0.325700i	$-0.105600\pi$
$149$	23.0819			1.89095			0.945473	+	0.325700i	$0.105600\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−7.71336	−	10.6165i	−0.623588	−	0.858295i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			17.2150i			1.37391i	0.726700	+	0.686955i	$0.241053\pi$
$157$			17.2150i			1.37391i	−0.726700	+	0.686955i	$0.758947\pi$
$158$		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
$169$	−10.4489	−	7.59156i	−0.803760	−	0.583966i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−24.6928	+	8.02317i	−1.87736	+	0.609991i	−0.888986	+	0.457933i	$0.848590\pi$
$173$	−24.6928	+	8.02317i	−1.87736	+	0.609991i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		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
$181$	−10.8884	−	7.91090i	−0.809330	−	0.588012i	0.104306	−	0.994545i	$-0.466738\pi$
$181$	−10.8884	−	7.91090i	−0.809330	−	0.588012i	−0.913636	+	0.406533i	$0.866738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.3753	+	2.05648i	−0.909853	+	0.151196i
$186$		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
$193$			27.6454i			1.98996i	0.100076	+	0.994980i	$0.468091\pi$
$193$			27.6454i			1.98996i	−0.100076	+	0.994980i	$0.531909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.0058	−	19.2773i	−0.997870	−	1.37345i	−0.926623	−	0.375992i	$-0.877302\pi$
$197$	−14.0058	−	19.2773i	−0.997870	−	1.37345i	−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
$205$	4.68912	+	28.2178i	0.327502	+	1.97082i
$206$		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
$221$	−1.02854	+	0.747279i	−0.0691872	+	0.0502674i
$222$		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
$229$	21.5374	−	15.6478i	1.42323	−	1.03404i	0.432001	−	0.901873i	$-0.357808\pi$
$229$	21.5374	−	15.6478i	1.42323	−	1.03404i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.5912	+	22.8359i	−1.08693	+	1.49603i	−0.235269	+	0.971930i	$0.575597\pi$
$233$	−16.5912	+	22.8359i	−1.08693	+	1.49603i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		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
$241$	0.513604	−	1.58071i	0.0330841	−	0.101823i	−0.933151	−	0.359485i	$-0.882952\pi$
$241$	0.513604	−	1.58071i	0.0330841	−	0.101823i	0.966235	+	0.257663i	$0.0829523\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.8921	+	7.21174i	−0.887535	+	0.460741i
$246$		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
$257$			27.0641i			1.68821i	0.536175	+	0.844107i	$0.319869\pi$
$257$			27.0641i			1.68821i	−0.536175	+	0.844107i	$0.680131\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.39146	−	19.6709i	−0.395621	−	1.21760i
$262$		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
$265$	9.98585	+	4.99293i	0.613426	+	0.306713i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.50666	+	5.45390i	0.457689	+	0.332530i	0.792624	−	0.609711i	$-0.208714\pi$
$269$	7.50666	+	5.45390i	0.457689	+	0.332530i	−0.334935	+	0.942241i	$0.608714\pi$
$270$		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
$277$	8.05216	−	2.61631i	0.483807	−	0.157199i	−0.0569502	−	0.998377i	$-0.518138\pi$
$277$	8.05216	−	2.61631i	0.483807	−	0.157199i	0.540758	+	0.841178i	$0.318138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.67182	+	6.30045i	0.517317	+	0.375853i	0.815592	−	0.578627i	$-0.196411\pi$
$281$	8.67182	+	6.30045i	0.517317	+	0.375853i	−0.298275	+	0.954480i	$0.596411\pi$
$282$		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
$289$	−0.659459	−	2.02961i	−0.0387917	−	0.119389i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	33.5720i		−	1.96130i	−0.195778	−	0.980648i	$-0.562723\pi$
$293$		−	33.5720i		−	1.96130i	0.195778	−	0.980648i	$-0.437277\pi$
$294$		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
$305$	1.65164	+	1.62675i	0.0945725	+	0.0931474i
$306$		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	−0.418016	−	0.908440i	$-0.637274\pi$
$313$	−22.8254	−	7.41641i	−1.29017	−	0.419200i	−0.872149	+	0.489240i	$0.837274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.4580	−	22.6525i	0.924373	−	1.27229i	−0.0376418	−	0.999291i	$-0.511985\pi$
$317$	16.4580	−	22.6525i	0.924373	−	1.27229i	0.962014	−	0.272999i	$-0.0880154\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.16257	+	0.871931i	−0.0644880	+	0.0483660i
$326$		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
$333$	9.89298	−	13.6165i	0.542132	−	0.746181i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	30.4338	+	9.88854i	1.65784	+	0.538663i	0.980417	−	0.196934i	$-0.0630986\pi$
$337$	30.4338	+	9.88854i	1.65784	+	0.538663i	0.677419	+	0.735598i	$0.263099\pi$
$338$		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
$349$	31.1479			1.66731			0.833653	−	0.552288i	$-0.186245\pi$
$349$	31.1479			1.66731			0.833653	+	0.552288i	$0.186245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.40456	+	12.9443i	0.500554	+	0.688954i	0.982291	−	0.187362i	$-0.0599938\pi$
$353$	9.40456	+	12.9443i	0.500554	+	0.688954i	−0.481736	+	0.876316i	$0.659994\pi$
$354$		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
$365$	14.7770	−	15.0031i	0.773464	−	0.785297i
$366$		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
$369$	−31.0479	−	22.5576i	−1.61629	−	1.17430i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	34.2380	−	11.1246i	1.77278	−	0.576011i	0.774387	−	0.632712i	$-0.218058\pi$
$373$	34.2380	−	11.1246i	1.77278	−	0.576011i	0.998391	+	0.0567016i	$0.0180584\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.90574	+	0.619212i	−0.0981505	+	0.0318910i
$378$		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
$389$	−12.1327	−	37.3406i	−0.615153	−	1.89325i	−0.399300	−	0.916820i	$-0.630747\pi$
$389$	−12.1327	−	37.3406i	−0.615153	−	1.89325i	−0.215852	−	0.976426i	$-0.569253\pi$
$390$		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	0.594464	−	0.804122i	$-0.297364\pi$
$397$	−7.05342	−	9.70820i	−0.354001	−	0.487241i	−0.948465	+	0.316881i	$0.897364\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.8934			1.09330			0.546652	−	0.837360i	$-0.315902\pi$
$401$	21.8934			1.09330			0.546652	+	0.837360i	$0.315902\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	19.9002	+	2.99720i	0.988847	+	0.148932i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−11.1828	+	34.4170i	−0.552952	+	1.70181i	0.148340	+	0.988936i	$0.452607\pi$
$409$	−11.1828	+	34.4170i	−0.552952	+	1.70181i	−0.701292	+	0.712874i	$0.747393\pi$
$410$		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
$421$	−14.0438	+	10.2034i	−0.684452	+	0.497284i	−0.874832	−	0.484427i	$-0.839028\pi$
$421$	−14.0438	+	10.2034i	−0.684452	+	0.497284i	0.190380	+	0.981711i	$0.439028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.07360	−	20.6958i	−0.343120	−	1.00389i
$426$		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
$433$	0.334033	−	0.459757i	0.0160526	−	0.0220945i	−0.800915	−	0.598778i	$-0.795653\pi$
$433$	0.334033	−	0.459757i	0.0160526	−	0.0220945i	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
$445$	19.3191	+	19.0279i	0.915811	+	0.902011i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	42.3685			1.99949			0.999747	−	0.0225137i	$-0.00716692\pi$
$449$	42.3685			1.99949			0.999747	+	0.0225137i	$0.00716692\pi$
$450$		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	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.24918	−	6.92225i	−0.104755	−	0.322401i	0.884918	−	0.465746i	$-0.154214\pi$
$461$	−2.24918	−	6.92225i	−0.104755	−	0.322401i	−0.989673	+	0.143345i	$0.954214\pi$
$462$		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
$477$	−14.2457	+	4.62870i	−0.652264	+	0.211933i
$478$		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
$481$	−1.31919	−	0.958444i	−0.0601497	−	0.0437013i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−39.1823	−	19.5912i	−1.77918	−	0.889588i
$486$		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
$493$		−	30.1578i		−	1.35824i
$494$		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
$505$	38.9756	−	20.2332i	1.73439	−	0.900367i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.8862	+	42.7373i	−0.615495	+	1.89430i	−0.221621	+	0.975133i	$0.571135\pi$
$509$	−13.8862	+	42.7373i	−0.615495	+	1.89430i	−0.393873	+	0.919165i	$0.628865\pi$
$510$		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
$521$	4.62194	−	3.35804i	0.202491	−	0.147118i	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	4.62194	−	3.35804i	0.202491	−	0.147118i	0.684410	+	0.729098i	$0.260060\pi$
$522$		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
$533$	−2.18541	+	3.00796i	−0.0946607	+	0.130289i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.88850	+	30.4337i	−0.425140	+	1.30845i	0.477721	+	0.878512i	$0.341463\pi$
$541$	−9.88850	+	30.4337i	−0.425140	+	1.30845i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.52981	−	15.2237i	−0.108365	−	0.652111i
$546$		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
$549$	−3.11024			−0.132742
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	44.7926i		−	1.89792i	−0.315390	−	0.948962i	$-0.602135\pi$
$557$		−	44.7926i		−	1.89792i	0.315390	−	0.948962i	$-0.397865\pi$
$558$		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
$565$	18.4639	−	3.06824i	0.776780	−	0.129082i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.2768	−	17.6382i	−1.01774	−	0.739430i	−0.0519200	−	0.998651i	$-0.516534\pi$
$569$	−24.2768	−	17.6382i	−1.01774	−	0.739430i	−0.965818	+	0.259221i	$0.916534\pi$
$570$		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	$0.886745\pi$
$577$	−45.6507	+	14.8328i	−1.90046	+	0.617498i	−0.963097	+	0.269156i	$0.913255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	0.290373	−	1.92795i	0.0120054	−	0.0797110i
$586$		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
$593$			38.8043i			1.59350i	0.604307	+	0.796751i	$0.293450\pi$
$593$			38.8043i			1.59350i	−0.604307	+	0.796751i	$0.706550\pi$
$594$		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
$601$	36.3039			1.48087			0.740433	−	0.672130i	$-0.234621\pi$
$601$	36.3039			1.48087			0.740433	+	0.672130i	$0.234621\pi$
$602$		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
$613$	41.3334	+	13.4300i	1.66944	+	0.542434i	0.982818	−	0.184579i	$-0.0590921\pi$
$613$	41.3334	+	13.4300i	1.66944	+	0.542434i	0.686624	+	0.727013i	$0.259092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	28.3456	−	39.0143i	1.14115	−	1.57066i	0.376239	−	0.926523i	$-0.377217\pi$
$617$	28.3456	−	39.0143i	1.14115	−	1.57066i	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
$625$	−8.44373	−	23.5309i	−0.337749	−	0.941236i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	19.8540	−	14.4248i	0.791632	−	0.575154i
$630$		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
$637$	−1.93493	−	0.628697i	−0.0766647	−	0.0249099i
$638$		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
$653$	6.54247	+	9.00494i	0.256027	+	0.352391i	0.917611	−	0.397481i	$-0.130115\pi$
$653$	6.54247	+	9.00494i	0.256027	+	0.352391i	−0.661584	+	0.749871i	$0.730115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			28.2527i			1.10224i
$658$		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
$673$	34.5540	−	11.2273i	1.33196	−	0.432779i	0.445373	−	0.895345i	$-0.353071\pi$
$673$	34.5540	−	11.2273i	1.33196	−	0.432779i	0.886585	+	0.462566i	$0.153071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−49.4549	+	16.0689i	−1.90071	+	0.617577i	−0.938477	+	0.345341i	$0.887763\pi$
$677$	−49.4549	+	16.0689i	−1.90071	+	0.617577i	−0.962230	+	0.272237i	$0.912237\pi$
$678$		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
$685$	−42.8004	+	7.11239i	−1.63532	+	0.271750i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.448434	+	1.38014i	0.0170840	+	0.0525790i
$690$		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
$697$	−32.8909	−	45.2705i	−1.24583	−	1.71474i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.4747			0.848856			0.424428	−	0.905462i	$-0.360475\pi$
$701$	22.4747			0.848856			0.424428	+	0.905462i	$0.360475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−15.4920	+	47.6794i	−0.581813	+	1.79064i	0.0298952	+	0.999553i	$0.490483\pi$
$709$	−15.4920	+	47.6794i	−0.581813	+	1.79064i	−0.611708	+	0.791083i	$0.709517\pi$
$710$		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
$725$	−0.523353	−	34.4680i	−0.0194369	−	1.28011i
$726$		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	−0.763386	−	0.645943i	$-0.776464\pi$
$733$	2.35114	−	3.23607i	0.0868414	−	0.119527i	0.850227	+	0.526416i	$0.176464\pi$
$734$		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
$745$	−45.8081	+	23.7801i	−1.67828	+	0.871237i
$746$		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
$757$			52.6490i			1.91356i	0.290811	+	0.956781i	$0.406075\pi$
$757$			52.6490i			1.91356i	−0.290811	+	0.956781i	$0.593925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.7654	+	51.5987i	0.607746	+	1.87045i	0.476680	+	0.879077i	$0.341840\pi$
$761$	16.7654	+	51.5987i	0.607746	+	1.87045i	0.131066	+	0.991374i	$0.458160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	26.2455	+	13.1227i	0.948908	+	0.474454i
$766$		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
$773$	−52.8611	+	17.1756i	−1.90128	+	0.617764i	−0.941504	+	0.337001i	$0.890588\pi$
$773$	−52.8611	+	17.1756i	−1.90128	+	0.617764i	−0.959777	+	0.280763i	$0.909412\pi$
$774$		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
$785$	−17.7358	−	34.1648i	−0.633017	−	1.21939i
$786$		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
$793$			0.301324i			0.0107003i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−32.3283	−	44.4961i	−1.14513	−	1.57613i	−0.755487	−	0.655164i	$-0.772599\pi$
$797$	−32.3283	−	44.4961i	−1.14513	−	1.57613i	−0.389640	−	0.920967i	$-0.627401\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−36.3802			−1.28543
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.5031	+	47.7135i	−0.545059	+	1.67752i	0.175791	+	0.984428i	$0.443752\pi$
$809$	−15.5031	+	47.7135i	−0.545059	+	1.67752i	−0.720850	+	0.693091i	$0.756248\pi$
$810$		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
$829$	−44.8540	+	32.5884i	−1.55784	+	1.13184i	−0.620096	+	0.784526i	$0.712906\pi$
$829$	−44.8540	+	32.5884i	−1.55784	+	1.13184i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.9978	−	24.7719i	0.623588	−	0.858295i
$834$		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
$841$	5.72692	−	17.6256i	0.197480	−	0.607781i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.5579	+	4.30116i	0.982422	+	0.147964i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	33.0116	+	45.4366i	1.13030	+	1.55572i	0.787505	+	0.616308i	$0.211372\pi$
$853$	33.0116	+	45.4366i	1.13030	+	1.55572i	0.342792	+	0.939411i	$0.388628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	8.00000i		−	0.273275i	−0.990621	−	0.136637i	$-0.956370\pi$
$857$		−	8.00000i		−	0.273275i	0.990621	−	0.136637i	$-0.0436295\pi$
$858$		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
$865$	40.7391	−	41.3624i	1.38517	−	1.40636i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	55.8969	−	18.1620i	1.89182	−	0.614690i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−41.6560	+	13.5349i	−1.40662	+	0.457040i	−0.911327	−	0.411683i	$-0.864941\pi$
$877$	−41.6560	+	13.5349i	−1.40662	+	0.457040i	−0.495297	+	0.868723i	$0.664941\pi$
$878$		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
$901$	−21.8403			−0.727606
$902$		0			0
$903$		0			0
$904$		0			0
$905$	29.7592	+	4.48209i	0.989230	+	0.148990i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−18.2065	+	56.0339i	−0.603872	+	1.85853i
$910$		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
$925$	22.4413	−	16.8310i	0.737864	−	0.553398i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.2768	−	14.0054i	0.632452	−	0.459504i	−0.224797	−	0.974406i	$-0.572172\pi$
$929$	19.2768	−	14.0054i	0.632452	−	0.459504i	0.857249	+	0.514902i	$0.172172\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20.2645	−	6.58433i	−0.662012	−	0.215101i	−0.0413087	−	0.999146i	$-0.513153\pi$
$937$	−20.2645	−	6.58433i	−0.662012	−	0.215101i	−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.339345	−	1.04440i	0.0110623	−	0.0340464i	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	0.339345	−	1.04440i	0.0110623	−	0.0340464i	0.956435	+	0.291944i	$0.0943022\pi$
$942$		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
$949$	2.73715			0.0888517
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.7060	−	34.0049i	−0.800307	−	1.10153i	−0.992747	−	0.120219i	$-0.961640\pi$
$953$	−24.7060	−	34.0049i	−0.800307	−	1.10153i	0.192440	−	0.981309i	$-0.438360\pi$
$954$		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
$965$	−28.4816	−	54.8647i	−0.916856	−	1.76616i
$966$		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
$977$	28.5037	−	9.26141i	0.911914	−	0.296299i	0.184768	−	0.982782i	$-0.440847\pi$
$977$	28.5037	−	9.26141i	0.911914	−	0.296299i	0.727145	+	0.686483i	$0.240847\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	16.7505	+	12.1700i	0.534804	+	0.388558i
$982$		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
$985$	47.6561	+	23.8280i	1.51845	+	0.759225i
$986$		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	0.905969	−	0.423345i	$-0.139144\pi$
$997$	7.05342	+	9.70820i	0.223384	+	0.307462i	−0.682585	+	0.730807i	$0.739144\pi$
$998$		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.769.1	yes	8
4.3	odd	2	CM	800.2.bg.b.769.1	yes	8
25.4	even	10	inner	800.2.bg.b.129.1	✓	8
100.79	odd	10	inner	800.2.bg.b.129.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.bg.b.129.1	✓	8	25.4	even	10	inner
800.2.bg.b.129.1	✓	8	100.79	odd	10	inner
800.2.bg.b.769.1	yes	8	1.1	even	1	trivial
800.2.bg.b.769.1	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)

\(f(q)\)	\(=\)	\(q+(-1.98459 + 1.03025i) q^{5} +(0.927051 - 2.85317i) q^{9} +O(q^{10})\)	\(q+(-1.98459 + 1.03025i) q^{5} +(0.927051 - 2.85317i) q^{9} +(-0.276418 - 0.0898138i) q^{13} +(2.57112 - 3.53884i) q^{17} +(2.87718 - 4.08924i) q^{25} +(5.57768 - 4.05242i) q^{29} +(5.33573 + 1.73368i) q^{37} +(3.95309 - 12.1663i) q^{41} +(1.09966 + 6.61746i) q^{45} +7.00000 q^{49} +(-2.93477 - 4.03936i) q^{53} +(-0.320372 - 0.986005i) q^{61} +(0.641107 - 0.106536i) q^{65} +(-8.95663 + 2.91019i) q^{73} +(-7.28115 - 5.29007i) q^{81} +(-1.45672 + 9.67203i) q^{85} +(-3.74737 - 11.5332i) q^{89} +(11.5154 + 15.8496i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 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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