LMFDB - Embedded newform 400.2.bi.a.303.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,2,Mod(47,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 0, 17]))

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

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

chi := DirichletCharacter("400.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	400.bi (of order $20$, degree $8$, 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:	$3.19401608085$
Analytic rank:	$0$
Dimension:	$8$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			303.1
Root			$-0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	400.303
Dual form			400.2.bi.a.367.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.03025 + 1.98459i) q^{5} +(-2.85317 + 0.927051i) q^{9} +O(q^{10})$	$q+(-1.03025 + 1.98459i) q^{5} +(-2.85317 + 0.927051i) q^{9} +(-2.21972 + 4.35645i) q^{13} +(-0.289390 + 1.82714i) q^{17} +(-2.87718 - 4.08924i) q^{25} +(-4.86363 + 6.69421i) q^{29} +(10.3357 + 5.26632i) q^{37} +(-0.183644 - 0.565197i) q^{41} +(1.09966 - 6.61746i) q^{45} -7.00000i q^{49} +(2.06523 + 13.0394i) q^{53} +(4.81636 - 14.8232i) q^{61} +(-6.35889 - 8.89346i) q^{65} +(3.05007 - 1.55409i) q^{73} +(7.28115 - 5.29007i) q^{81} +(-3.32797 - 2.45672i) q^{85} +(-13.7474 - 4.46679i) q^{89} +(15.1115 - 2.39343i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5}+O(q^{10}) $ 8 * q - 4 * q^5	$ 8 q - 4 q^{5} + 10 q^{13} + 10 q^{17} - 6 q^{25} + 50 q^{37} - 16 q^{41} - 6 q^{45} - 30 q^{53} + 24 q^{61} - 50 q^{65} - 10 q^{73} + 18 q^{81} - 60 q^{85} - 50 q^{89} - 10 q^{97}+O(q^{100}) $ 8 * q - 4 * q^5 + 10 * q^13 + 10 * q^17 - 6 * q^25 + 50 * q^37 - 16 * q^41 - 6 * q^45 - 30 * q^53 + 24 * q^61 - 50 * q^65 - 10 * q^73 + 18 * q^81 - 60 * q^85 - 50 * q^89 - 10 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{7}{20}\right)$	$-1$

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
$2$		0			0
$3$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$3$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$4$		0			0
$5$	−1.03025	+	1.98459i	−0.460741	+	0.887535i
$5$	−1.03025	+	1.98459i	−0.460741	+	0.887535i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$	−2.85317	+	0.927051i	−0.951057	+	0.309017i
$10$		0			0
$11$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$11$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$12$		0			0
$13$	−2.21972	+	4.35645i	−0.615640	+	1.20826i	0.347098	+	0.937829i	$0.387167\pi$
$13$	−2.21972	+	4.35645i	−0.615640	+	1.20826i	−0.962739	+	0.270434i	$0.912833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.289390	+	1.82714i	−0.0701873	+	0.443145i	0.927421	+	0.374020i	$0.122021\pi$
$17$	−0.289390	+	1.82714i	−0.0701873	+	0.443145i	−0.997608	+	0.0691254i	$0.977979\pi$
$18$		0			0
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$23$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$24$		0			0
$25$	−2.87718	−	4.08924i	−0.575435	−	0.817848i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.86363	+	6.69421i	−0.903153	+	1.24308i	0.0662984	+	0.997800i	$0.478881\pi$
$29$	−4.86363	+	6.69421i	−0.903153	+	1.24308i	−0.969451	+	0.245284i	$0.921119\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.3357	+	5.26632i	1.69918	+	0.865777i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	10.3357	+	5.26632i	1.69918	+	0.865777i	0.712789	+	0.701378i	$0.247432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.183644	−	0.565197i	−0.0286803	−	0.0882690i	0.935692	−	0.352819i	$-0.114777\pi$
$41$	−0.183644	−	0.565197i	−0.0286803	−	0.0882690i	−0.964372	+	0.264550i	$0.914777\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	1.09966	−	6.61746i	0.163928	−	0.986472i
$46$		0			0
$47$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$47$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.06523	+	13.0394i	0.283681	+	1.79109i	0.558403	+	0.829570i	$0.311414\pi$
$53$	2.06523	+	13.0394i	0.283681	+	1.79109i	−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$59$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$60$		0			0
$61$	4.81636	−	14.8232i	0.616671	−	1.89792i	0.245213	−	0.969469i	$-0.421142\pi$
$61$	4.81636	−	14.8232i	0.616671	−	1.89792i	0.371458	−	0.928450i	$-0.378858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.35889	−	8.89346i	−0.788724	−	1.10310i
$66$		0			0
$67$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$67$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$		0			0
$73$	3.05007	−	1.55409i	0.356984	−	0.181893i	−0.266296	−	0.963891i	$-0.585800\pi$
$73$	3.05007	−	1.55409i	0.356984	−	0.181893i	0.623280	+	0.781999i	$0.285800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$79$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$80$		0			0
$81$	7.28115	−	5.29007i	0.809017	−	0.587785i
$82$		0			0
$83$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$83$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$84$		0			0
$85$	−3.32797	−	2.45672i	−0.360969	−	0.266469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.7474	−	4.46679i	−1.45722	−	0.473479i	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−13.7474	−	4.46679i	−1.45722	−	0.473479i	−0.927219	+	0.374519i	$0.877808\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	15.1115	−	2.39343i	1.53434	−	0.243016i	0.668644	−	0.743583i	$-0.266875\pi$
$97$	15.1115	−	2.39343i	1.53434	−	0.243016i	0.865698	+	0.500567i	$0.166875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	19.6392			1.95417			0.977085	−	0.212850i	$-0.0682745\pi$
$101$	19.6392			1.95417			0.977085	+	0.212850i	$0.0682745\pi$
$102$		0			0
$103$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$103$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$104$		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
$109$	−6.56381	+	2.13271i	−0.628699	+	0.204277i	−0.605999	−	0.795465i	$-0.707226\pi$
$109$	−6.56381	+	2.13271i	−0.628699	+	0.204277i	−0.0227005	+	0.999742i	$0.507226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.58663	+	7.03916i	−0.337402	+	0.662189i	−0.995907	−	0.0903879i	$-0.971189\pi$
$113$	−3.58663	+	7.03916i	−0.337402	+	0.662189i	0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.29459	−	14.4875i	0.212135	−	1.33937i
$118$		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
$125$	11.0797	−	1.49707i	0.990995	−	0.133902i
$126$		0			0
$127$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$127$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$131$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	20.4759	+	10.4330i	1.74937	+	0.891350i	0.961180	+	0.275921i	$0.0889827\pi$
$137$	20.4759	+	10.4330i	1.74937	+	0.891350i	0.788192	+	0.615429i	$0.211017\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−8.27450	−	16.5490i	−0.687160	−	1.37432i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			23.0819i			1.89095i	0.325700	+	0.945473i	$0.394400\pi$
$149$			23.0819i			1.89095i	−0.325700	+	0.945473i	$0.605600\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−0.868169	−	5.48141i	−0.0701873	−	0.443145i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.7130	+	17.7130i	−1.41365	+	1.41365i	−0.686955	+	0.726700i	$0.741053\pi$
$157$	−17.7130	+	17.7130i	−1.41365	+	1.41365i	−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		0.453990	−	0.891007i	$-0.350000\pi$
$163$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$167$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$168$		0			0
$169$	−6.41029	−	8.82301i	−0.493100	−	0.678693i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−19.0232	+	9.69279i	−1.44631	+	0.736929i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−19.0232	+	9.69279i	−1.44631	+	0.736929i	−0.457933	+	0.888986i	$0.651410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$179$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$180$		0			0
$181$	−10.8884	+	7.91090i	−0.809330	+	0.588012i	−0.913636	−	0.406533i	$-0.866738\pi$
$181$	−10.8884	+	7.91090i	−0.809330	+	0.588012i	0.104306	+	0.994545i	$0.466738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−21.0998	+	15.0865i	−1.55129	+	1.10918i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$		0			0
$193$	15.2130	+	15.2130i	1.09506	+	1.09506i	0.994980	+	0.100076i	$0.0319087\pi$
$193$	15.2130	+	15.2130i	1.09506	+	1.09506i	0.100076	+	0.994980i	$0.468091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.27729	+	0.994226i	−0.447239	+	0.0708357i	−0.375992	−	0.926623i	$-0.622698\pi$
$197$	−6.27729	+	0.994226i	−0.447239	+	0.0708357i	−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
$205$	1.31088	+	0.217837i	0.0915560	+	0.0152144i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$		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
$221$	−7.31746	−	5.31645i	−0.492226	−	0.357623i
$222$		0			0
$223$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$223$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$224$		0			0
$225$	12.0000	+	9.00000i	0.800000	+	0.600000i
$226$		0			0
$227$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$227$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$228$		0			0
$229$	8.46263	−	11.6478i	0.559226	−	0.769709i	−0.432001	−	0.901873i	$-0.642192\pi$
$229$	8.46263	−	11.6478i	0.559226	−	0.769709i	0.991228	+	0.132164i	$0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.5912	+	1.83587i	0.759367	+	0.120272i	0.524097	−	0.851658i	$-0.324403\pi$
$233$	11.5912	+	1.83587i	0.759367	+	0.120272i	0.235269	+	0.971930i	$0.424403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$	9.58071	+	29.4864i	0.617148	+	1.89939i	0.359485	+	0.933151i	$0.382952\pi$
$241$	9.58071	+	29.4864i	0.617148	+	1.89939i	0.257663	+	0.966235i	$0.417048\pi$
$242$		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		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.1276	+	22.1276i	−1.38028	+	1.38028i	−0.536175	+	0.844107i	$0.680131\pi$
$257$	−22.1276	+	22.1276i	−1.38028	+	1.38028i	−0.844107	+	0.536175i	$0.819869\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7.67088	−	23.6085i	0.474816	−	1.46133i
$262$		0			0
$263$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$263$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$264$		0			0
$265$	−28.0055	−	9.33515i	−1.72036	−	0.573454i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.45390	+	7.50666i	0.332530	+	0.457689i	0.942241	−	0.334935i	$-0.108714\pi$
$269$	5.45390	+	7.50666i	0.332530	+	0.457689i	−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.0522	−	25.6163i	−0.784228	−	1.53913i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−13.0522	−	25.6163i	−0.784228	−	1.53913i	0.0569502	−	0.998377i	$-0.481862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−25.6996	+	18.6718i	−1.53311	+	1.11387i	−0.578627	+	0.815592i	$0.696411\pi$
$281$	−25.6996	+	18.6718i	−1.53311	+	1.11387i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$283$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	12.9133	+	4.19578i	0.759605	+	0.246811i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.4348	−	13.4348i	−0.784871	−	0.784871i	0.195778	−	0.980648i	$-0.437277\pi$
$293$	−13.4348	−	13.4348i	−0.784871	−	0.784871i	−0.980648	+	0.195778i	$0.937277\pi$
$294$		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
$305$	24.4559	+	24.8301i	1.40034	+	1.42177i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	16.0510	−	31.5018i	0.907255	−	1.78059i	0.418016	−	0.908440i	$-0.362726\pi$
$313$	16.0510	−	31.5018i	0.907255	−	1.78059i	0.489240	−	0.872149i	$-0.337274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.53079	−	34.9201i	0.310640	−	1.96131i	0.0376418	−	0.999291i	$-0.488015\pi$
$317$	5.53079	−	34.9201i	0.310640	−	1.96131i	0.272999	−	0.962014i	$-0.411985\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	24.2011	−	3.45730i	1.34244	−	0.191776i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$331$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$332$		0			0
$333$	−34.3717	−	5.44395i	−1.88356	−	0.298326i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	31.5018	+	16.0510i	1.71601	+	0.874353i	0.980417	+	0.196934i	$0.0630986\pi$
$337$	31.5018	+	16.0510i	1.71601	+	0.874353i	0.735598	+	0.677419i	$0.236901\pi$
$338$		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		0.987688	−	0.156434i	$-0.0500000\pi$
$347$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$348$		0			0
$349$		−	20.6352i		−	1.10458i	−0.833653	−	0.552288i	$-0.813755\pi$
$349$		−	20.6352i		−	1.10458i	0.833653	−	0.552288i	$-0.186245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.53079	−	34.9201i	−0.294374	−	1.85861i	−0.481736	−	0.876316i	$-0.659994\pi$
$353$	−5.53079	−	34.9201i	−0.294374	−	1.85861i	0.187362	−	0.982291i	$-0.440006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$		0			0
$361$	−5.87132	+	18.0701i	−0.309017	+	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.0581067	+	7.65424i	−0.00304144	+	0.400641i
$366$		0			0
$367$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$367$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$368$		0			0
$369$	1.04793	+	1.44236i	0.0545532	+	0.0750861i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	31.5018	−	16.0510i	1.63110	−	0.831089i	0.632712	−	0.774387i	$-0.281942\pi$
$373$	31.5018	−	16.0510i	1.63110	−	0.831089i	0.998391	−	0.0567016i	$-0.0180584\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.3671	−	36.0474i	−0.945953	−	1.85654i
$378$		0			0
$379$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$379$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$383$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	37.3406	+	12.1327i	1.89325	+	0.615153i	0.976426	+	0.215852i	$0.0692530\pi$
$389$	37.3406	+	12.1327i	1.89325	+	0.615153i	0.916820	+	0.399300i	$0.130747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−34.9201	+	5.53079i	−1.75259	+	0.277583i	−0.948465	−	0.316881i	$-0.897364\pi$
$397$	−34.9201	+	5.53079i	−1.75259	+	0.277583i	−0.804122	+	0.594464i	$0.797364\pi$
$398$		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
$405$	2.99720	+	19.9002i	0.148932	+	0.988847i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	34.4170	−	11.1828i	1.70181	−	0.552952i	0.712874	−	0.701292i	$-0.247393\pi$
$409$	34.4170	−	11.1828i	1.70181	−	0.552952i	0.988936	+	0.148340i	$0.0473931\pi$
$410$		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	$-0.700000\pi$
$419$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$420$		0			0
$421$	−30.0826	−	21.8563i	−1.46614	−	1.06521i	−0.981711	−	0.190380i	$-0.939028\pi$
$421$	−30.0826	−	21.8563i	−1.46614	−	1.06521i	−0.484427	−	0.874832i	$-0.660972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.30422	−	4.07360i	0.402814	−	0.197599i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$		0			0
$433$	29.4598	+	4.66597i	1.41575	+	0.224232i	0.816968	−	0.576683i	$-0.195653\pi$
$433$	29.4598	+	4.66597i	1.41575	+	0.224232i	0.598778	+	0.800915i	$0.295653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$439$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$440$		0			0
$441$	6.48936	+	19.9722i	0.309017	+	0.951057i
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	23.0279	−	22.6809i	1.09163	−	1.07518i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	42.3685i		−	1.99949i	−0.0225137	−	0.999747i	$-0.507167\pi$
$449$		−	42.3685i		−	1.99949i	0.0225137	−	0.999747i	$-0.492833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.0000	+	25.0000i	−1.16945	+	1.16945i	−0.187112	+	0.982339i	$0.559913\pi$
$457$	−25.0000	+	25.0000i	−1.16945	+	1.16945i	−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	$-0.845786\pi$
$461$	2.24918	−	6.92225i	0.104755	−	0.322401i	0.989673	+	0.143345i	$0.0457859\pi$
$462$		0			0
$463$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$463$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$467$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$468$		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
$477$	−17.9806	−	35.2889i	−0.823276	−	1.61577i
$478$		0			0
$479$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$	−45.8849	+	33.3373i	−2.09217	+	1.52005i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.8186	+	32.4559i	−0.491250	+	1.47375i
$486$		0			0
$487$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$487$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$491$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$	−10.8237	−	10.8237i	−0.487477	−	0.487477i
$494$		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		−0.156434	−	0.987688i	$-0.550000\pi$
$503$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$504$		0			0
$505$	−20.2332	+	38.9756i	−0.900367	+	1.73439i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.88619	+	1.26270i	−0.172252	+	0.0559682i	−0.393873	−	0.919165i	$-0.628865\pi$
$509$	−3.88619	+	1.26270i	−0.172252	+	0.0559682i	0.221621	+	0.975133i	$0.428865\pi$
$510$		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	$-0.339940\pi$
$521$	−4.62194	−	3.35804i	−0.202491	−	0.147118i	−0.684410	+	0.729098i	$0.739940\pi$
$522$		0			0
$523$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$523$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.5191	−	18.6074i	0.587785	−	0.809017i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.86989	+	0.454546i	0.124309	+	0.0196886i
$534$		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	0.902861	+	0.429934i	$0.141463\pi$
$541$	9.88850	+	30.4337i	0.425140	+	1.30845i	−0.477721	+	0.878512i	$0.658537\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.52981	−	15.2237i	0.108365	−	0.652111i
$546$		0			0
$547$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$547$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$548$		0			0
$549$			46.7582i			1.99559i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−29.8398	+	29.8398i	−1.26435	+	1.26435i	−0.315390	+	0.948962i	$0.602135\pi$
$557$	−29.8398	+	29.8398i	−1.26435	+	1.26435i	−0.948962	+	0.315390i	$0.897865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$563$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$564$		0			0
$565$	−10.2747	−	14.3701i	−0.432260	−	0.604554i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.6382	+	24.2768i	0.739430	+	1.01774i	0.998651	+	0.0519200i	$0.0165341\pi$
$569$	17.6382	+	24.2768i	0.739430	+	1.01774i	−0.259221	+	0.965818i	$0.583466\pi$
$570$		0			0
$571$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$571$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	16.0510	+	31.5018i	0.668211	+	1.31144i	0.937367	+	0.348342i	$0.113255\pi$
$577$	16.0510	+	31.5018i	0.668211	+	1.31144i	−0.269156	+	0.963097i	$0.586745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	26.3877	+	19.4795i	1.09100	+	0.805380i
$586$		0			0
$587$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$587$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−34.1180	−	34.1180i	−1.40106	−	1.40106i	−0.796751	−	0.604307i	$-0.793450\pi$
$593$	−34.1180	−	34.1180i	−1.40106	−	1.40106i	−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
$601$	−32.9550			−1.34426			−0.672130	−	0.740433i	$-0.734621\pi$
$601$	−32.9550			−1.34426			−0.672130	+	0.740433i	$0.734621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.0000	−	11.0000i	0.894427	−	0.447214i
$606$		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
$612$		0			0
$613$	−6.33342	+	12.4300i	−0.255805	+	0.502045i	−0.982818	−	0.184579i	$-0.940908\pi$
$613$	−6.33342	+	12.4300i	−0.255805	+	0.502045i	0.727013	+	0.686624i	$0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.65442	−	42.0143i	0.267897	−	1.69143i	−0.376239	−	0.926523i	$-0.622783\pi$
$617$	6.65442	−	42.0143i	0.267897	−	1.69143i	0.644136	−	0.764911i	$-0.277217\pi$
$618$		0			0
$619$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$619$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$620$		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
$629$	−12.6133	+	17.3608i	−0.502926	+	0.692219i
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	30.4952	+	15.5381i	1.20826	+	0.615640i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.47214	+	7.60845i	0.0976435	+	0.300516i	0.987934	−	0.154878i	$-0.0494985\pi$
$641$	2.47214	+	7.60845i	0.0976435	+	0.300516i	−0.890290	+	0.455394i	$0.849498\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$647$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.28639	+	27.0632i	0.167739	+	1.05906i	0.917611	+	0.397481i	$0.130115\pi$
$653$	4.28639	+	27.0632i	0.167739	+	1.05906i	−0.749871	+	0.661584i	$0.769885\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7.26166	+	7.26166i	−0.283304	+	0.283304i
$658$		0			0
$659$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$659$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$660$		0			0
$661$	−3.70820	+	11.4127i	−0.144232	+	0.443902i	−0.996911	−	0.0785333i	$-0.974976\pi$
$661$	−3.70820	+	11.4127i	−0.144232	+	0.443902i	0.852679	+	0.522435i	$0.174976\pi$
$662$		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
$673$	−0.446032	+	0.227265i	−0.0171933	+	0.00876041i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−0.446032	+	0.227265i	−0.0171933	+	0.00876041i	0.445373	+	0.895345i	$0.353071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.0510	−	31.5018i	−0.616890	−	1.21071i	−0.962230	−	0.272237i	$-0.912237\pi$
$677$	−16.0510	−	31.5018i	−0.616890	−	1.21071i	0.345341	−	0.938477i	$-0.387763\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$683$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$684$		0			0
$685$	−41.8004	+	29.8876i	−1.59711	+	1.14195i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−61.3896	−	19.9467i	−2.33876	−	0.759908i
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.08584	−	0.171980i	0.0411290	−	0.00651419i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	47.9467			1.81092			0.905462	−	0.424428i	$-0.139525\pi$
$701$	47.9467			1.81092			0.905462	+	0.424428i	$0.139525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.0840	−	5.55093i	0.641603	−	0.208470i	0.0298952	−	0.999553i	$-0.490483\pi$
$709$	17.0840	−	5.55093i	0.641603	−	0.208470i	0.611708	+	0.791083i	$0.290483\pi$
$710$		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	$-0.700000\pi$
$719$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	41.3677	+	0.628117i	1.53636	+	0.0233277i
$726$		0			0
$727$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$727$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$728$		0			0
$729$	−15.8702	+	21.8435i	−0.587785	+	0.809017i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−34.9201	−	5.53079i	−1.28980	−	0.204285i	−0.526416	−	0.850227i	$-0.676464\pi$
$733$	−34.9201	−	5.53079i	−1.28980	−	0.204285i	−0.763386	+	0.645943i	$0.776464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$		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
$745$	−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		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−18.3232	+	18.3232i	−0.665970	+	0.665970i	−0.956781	−	0.290811i	$-0.906075\pi$
$757$	−18.3232	+	18.3232i	−0.665970	+	0.665970i	0.290811	+	0.956781i	$0.406075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.7654	−	51.5987i	0.607746	−	1.87045i	0.131066	−	0.991374i	$-0.458160\pi$
$761$	16.7654	−	51.5987i	0.607746	−	1.87045i	0.476680	−	0.879077i	$-0.341840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	11.7728	+	3.92425i	0.425645	+	0.141882i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.1068	−	19.4164i	−0.508706	−	0.700174i	0.474995	−	0.879989i	$-0.342450\pi$
$769$	−14.1068	−	19.4164i	−0.508706	−	0.700174i	−0.983700	+	0.179815i	$0.942450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	33.9826	−	17.3150i	1.22227	−	0.622776i	0.280763	−	0.959777i	$-0.409412\pi$
$773$	33.9826	−	17.3150i	1.22227	−	0.622776i	0.941504	+	0.337001i	$0.109412\pi$
$774$		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
$785$	−16.9042	−	53.4019i	−0.603338	−	1.90600i
$786$		0			0
$787$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$787$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	53.8857	+	53.8857i	1.91354	+	1.91354i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	47.3283	−	7.49606i	1.67645	−	0.265524i	0.755487	−	0.655164i	$-0.227401\pi$
$797$	47.3283	−	7.49606i	1.67645	−	0.265524i	0.920967	+	0.389640i	$0.127401\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	43.3645			1.53221
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.5031	−	8.28645i	0.896640	−	0.291336i	0.175791	−	0.984428i	$-0.443752\pi$
$809$	25.5031	−	8.28645i	0.896640	−	0.291336i	0.720850	+	0.693091i	$0.243752\pi$
$810$		0			0
$811$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$811$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$812$		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
$821$	−22.6525	−	16.4580i	−0.790577	−	0.574388i	0.117558	−	0.993066i	$-0.462493\pi$
$821$	−22.6525	−	16.4580i	−0.790577	−	0.574388i	−0.908135	+	0.418678i	$0.862493\pi$
$822$		0			0
$823$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$823$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$827$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$828$		0			0
$829$	32.5884	−	44.8540i	1.13184	−	1.55784i	0.347314	−	0.937749i	$-0.387094\pi$
$829$	32.5884	−	44.8540i	1.13184	−	1.55784i	0.784526	−	0.620096i	$-0.212906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.7899	+	2.02573i	0.443145	+	0.0701873i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$839$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	−12.1961	−	37.5356i	−0.420554	−	1.29433i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.1142	−	3.63189i	0.829555	−	0.124941i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	4.43661	+	28.0116i	0.151906	+	0.959100i	0.939411	+	0.342792i	$0.111372\pi$
$853$	4.43661	+	28.0116i	0.151906	+	0.959100i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	0.990621	+	0.136637i	$0.0436295\pi$
$858$		0			0
$859$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$859$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$863$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$864$		0			0
$865$	0.362409	−	47.7391i	0.0123223	−	1.62318i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−40.8969	+	20.8380i	−1.38415	+	0.705260i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	26.8595	+	52.7147i	0.906981	+	1.78005i	0.495297	+	0.868723i	$0.335059\pi$
$877$	26.8595	+	52.7147i	0.906981	+	1.78005i	0.411683	+	0.911327i	$0.364941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25.8885	−	18.8091i	0.872207	−	0.633696i	−0.0589711	−	0.998260i	$-0.518782\pi$
$881$	25.8885	−	18.8091i	0.872207	−	0.633696i	0.931178	+	0.364564i	$0.118782\pi$
$882$		0			0
$883$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$883$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$887$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$888$		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
$901$	−24.4223			−0.813626
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.48209	−	29.7592i	−0.148990	−	0.989230i
$906$		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
$909$	−56.0339	+	18.2065i	−1.85853	+	0.603872i
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$		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	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−8.20248	−	57.4174i	−0.269696	−	1.88787i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−14.0054	+	19.2768i	−0.459504	+	0.632452i	−0.974406	−	0.224797i	$-0.927828\pi$
$929$	−14.0054	+	19.2768i	−0.459504	+	0.632452i	0.514902	+	0.857249i	$0.327828\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	49.5843	+	25.2645i	1.61985	+	0.825354i	0.999146	+	0.0413087i	$0.0131527\pi$
$937$	49.5843	+	25.2645i	1.61985	+	0.825354i	0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.339345	+	1.04440i	0.0110623	+	0.0340464i	0.956435	−	0.291944i	$-0.0943022\pi$
$941$	0.339345	+	1.04440i	0.0110623	+	0.0340464i	−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		0.987688	−	0.156434i	$-0.0500000\pi$
$947$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$948$		0			0
$949$			16.7372i			0.543311i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.65203	−	60.9405i	−0.312660	−	1.97406i	−0.192440	−	0.981309i	$-0.561640\pi$
$953$	−9.65203	−	60.9405i	−0.312660	−	1.97406i	−0.120219	−	0.992747i	$-0.538360\pi$
$954$		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
$965$	−45.8647	+	14.5184i	−1.47644	+	0.467363i
$966$		0			0
$967$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$967$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$971$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.99043	−	15.6821i	−0.255637	−	0.501715i	0.727145	−	0.686483i	$-0.240847\pi$
$977$	−7.99043	−	15.6821i	−0.255637	−	0.501715i	−0.982782	+	0.184768i	$0.940847\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	16.7505	−	12.1700i	0.534804	−	0.388558i
$982$		0			0
$983$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$983$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$984$		0			0
$985$	4.49405	−	13.4821i	0.143192	−	0.429577i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$991$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	34.9201	−	5.53079i	1.10593	−	0.175162i	0.423345	−	0.905969i	$-0.360856\pi$
$997$	34.9201	−	5.53079i	1.10593	−	0.175162i	0.682585	+	0.730807i	$0.260856\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bi.a.303.1	✓	8
4.3	odd	2	CM	400.2.bi.a.303.1	✓	8
25.17	odd	20	inner	400.2.bi.a.367.1	yes	8
100.67	even	20	inner	400.2.bi.a.367.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bi.a.303.1	✓	8	1.1	even	1	trivial
400.2.bi.a.303.1	✓	8	4.3	odd	2	CM
400.2.bi.a.367.1	yes	8	25.17	odd	20	inner
400.2.bi.a.367.1	yes	8	100.67	even	20	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bi (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.03025 + 1.98459i) q^{5} +(-2.85317 + 0.927051i) q^{9} +O(q^{10})\)	\(q+(-1.03025 + 1.98459i) q^{5} +(-2.85317 + 0.927051i) q^{9} +(-2.21972 + 4.35645i) q^{13} +(-0.289390 + 1.82714i) q^{17} +(-2.87718 - 4.08924i) q^{25} +(-4.86363 + 6.69421i) q^{29} +(10.3357 + 5.26632i) q^{37} +(-0.183644 - 0.565197i) q^{41} +(1.09966 - 6.61746i) q^{45} -7.00000i q^{49} +(2.06523 + 13.0394i) q^{53} +(4.81636 - 14.8232i) q^{61} +(-6.35889 - 8.89346i) q^{65} +(3.05007 - 1.55409i) q^{73} +(7.28115 - 5.29007i) q^{81} +(-3.32797 - 2.45672i) q^{85} +(-13.7474 - 4.46679i) q^{89} +(15.1115 - 2.39343i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5}+O(q^{10}) \) 8 * q - 4 * q^5	\( 8 q - 4 q^{5} + 10 q^{13} + 10 q^{17} - 6 q^{25} + 50 q^{37} - 16 q^{41} - 6 q^{45} - 30 q^{53} + 24 q^{61} - 50 q^{65} - 10 q^{73} + 18 q^{81} - 60 q^{85} - 50 q^{89} - 10 q^{97}+O(q^{100}) \) 8 * q - 4 * q^5 + 10 * q^13 + 10 * q^17 - 6 * q^25 + 50 * q^37 - 16 * q^41 - 6 * q^45 - 30 * q^53 + 24 * q^61 - 50 * q^65 - 10 * q^73 + 18 * q^81 - 60 * q^85 - 50 * q^89 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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