LMFDB - Embedded newform 1575.2.m.a.1457.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1268,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

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

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

chi := DirichletCharacter("1575.1268");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.m (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.4
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1575.1457
Dual form			1575.2.m.a.1268.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.366025 - 0.366025i) q^{2} +1.73205i q^{4} +(0.707107 + 0.707107i) q^{7} +(1.36603 + 1.36603i) q^{8} +O(q^{10})$	\(q+(0.366025 - 0.366025i) q^{2} +1.73205i q^{4} +(0.707107 + 0.707107i) q^{7} +(1.36603 + 1.36603i) q^{8} +2.44949i q^{11} +(3.86370 - 3.86370i) q^{13} +0.517638 q^{14} -2.46410 q^{16} +(-2.00000 + 2.00000i) q^{17} -2.00000i q^{19} +(0.896575 + 0.896575i) q^{22} +(6.46410 + 6.46410i) q^{23} -2.82843i q^{26} +(-1.22474 + 1.22474i) q^{28} +6.31319 q^{29} -4.92820 q^{31} +(-3.63397 + 3.63397i) q^{32} +1.46410i q^{34} +(0.378937 + 0.378937i) q^{37} +(-0.732051 - 0.732051i) q^{38} +2.07055i q^{41} +(2.82843 - 2.82843i) q^{43} -4.24264 q^{44} +4.73205 q^{46} +(-7.46410 + 7.46410i) q^{47} +1.00000i q^{49} +(6.69213 + 6.69213i) q^{52} +(2.26795 + 2.26795i) q^{53} +1.93185i q^{56} +(2.31079 - 2.31079i) q^{58} -12.6264 q^{59} -8.92820 q^{61} +(-1.80385 + 1.80385i) q^{62} -2.26795i q^{64} +(6.31319 + 6.31319i) q^{67} +(-3.46410 - 3.46410i) q^{68} +4.52004i q^{71} +(-1.03528 + 1.03528i) q^{73} +0.277401 q^{74} +3.46410 q^{76} +(-1.73205 + 1.73205i) q^{77} +4.00000i q^{79} +(0.757875 + 0.757875i) q^{82} +(4.53590 + 4.53590i) q^{83} -2.07055i q^{86} +(-3.34607 + 3.34607i) q^{88} +14.1421 q^{89} +5.46410 q^{91} +(-11.1962 + 11.1962i) q^{92} +5.46410i q^{94} +(10.8332 + 10.8332i) q^{97} +(0.366025 + 0.366025i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 4 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 4 * q^8	$ 8 q - 4 q^{2} + 4 q^{8} + 8 q^{16} - 16 q^{17} + 24 q^{23} + 16 q^{31} - 36 q^{32} + 8 q^{38} + 24 q^{46} - 32 q^{47} + 32 q^{53} - 16 q^{61} - 56 q^{62} + 64 q^{83} + 16 q^{91} - 48 q^{92} - 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 4 * q^8 + 8 * q^16 - 16 * q^17 + 24 * q^23 + 16 * q^31 - 36 * q^32 + 8 * q^38 + 24 * q^46 - 32 * q^47 + 32 * q^53 - 16 * q^61 - 56 * q^62 + 64 * q^83 + 16 * q^91 - 48 * q^92 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$	0.366025	−	0.366025i	0.258819	−	0.258819i	−0.565755	−	0.824574i	$-0.691415\pi$
$2$	0.366025	−	0.366025i	0.258819	−	0.258819i	0.824574	+	0.565755i	$0.191415\pi$
$3$		0			0
$3$		0			0
$4$			1.73205i			0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.707107	+	0.707107i	0.267261	+	0.267261i
$7$	0.707107	+	0.707107i	0.267261	+	0.267261i
$8$	1.36603	+	1.36603i	0.482963	+	0.482963i
$9$		0			0
$10$		0			0
$11$			2.44949i			0.738549i	0.929320	+	0.369274i	$0.120394\pi$
$11$			2.44949i			0.738549i	−0.929320	+	0.369274i	$0.879606\pi$
$12$		0			0
$13$	3.86370	−	3.86370i	1.07160	−	1.07160i	0.0743676	−	0.997231i	$-0.476306\pi$
$13$	3.86370	−	3.86370i	1.07160	−	1.07160i	0.997231	−	0.0743676i	$-0.0236938\pi$
$14$	0.517638			0.138345
$15$		0			0
$16$	−2.46410			−0.616025
$17$	−2.00000	+	2.00000i	−0.485071	+	0.485071i	−0.906747	−	0.421676i	$-0.861442\pi$
$17$	−2.00000	+	2.00000i	−0.485071	+	0.485071i	0.421676	+	0.906747i	$0.361442\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$	0.896575	+	0.896575i	0.191151	+	0.191151i
$23$	6.46410	+	6.46410i	1.34786	+	1.34786i	0.887984	+	0.459874i	$0.152106\pi$
$23$	6.46410	+	6.46410i	1.34786	+	1.34786i	0.459874	+	0.887984i	$0.347894\pi$
$24$		0			0
$25$		0			0
$26$		−	2.82843i		−	0.554700i
$27$		0			0
$28$	−1.22474	+	1.22474i	−0.231455	+	0.231455i
$29$	6.31319			1.17233			0.586165	−	0.810192i	$-0.300637\pi$
$29$	6.31319			1.17233			0.586165	+	0.810192i	$0.300637\pi$
$30$		0			0
$31$	−4.92820			−0.885131			−0.442566	−	0.896736i	$-0.645932\pi$
$31$	−4.92820			−0.885131			−0.442566	+	0.896736i	$0.645932\pi$
$32$	−3.63397	+	3.63397i	−0.642402	+	0.642402i
$33$		0			0
$34$			1.46410i			0.251091i
$35$		0			0
$36$		0			0
$37$	0.378937	+	0.378937i	0.0622969	+	0.0622969i	0.737569	−	0.675272i	$-0.235974\pi$
$37$	0.378937	+	0.378937i	0.0622969	+	0.0622969i	−0.675272	+	0.737569i	$0.735974\pi$
$38$	−0.732051	−	0.732051i	−0.118754	−	0.118754i
$39$		0			0
$40$		0			0
$41$			2.07055i			0.323366i	0.986843	+	0.161683i	$0.0516922\pi$
$41$			2.07055i			0.323366i	−0.986843	+	0.161683i	$0.948308\pi$
$42$		0			0
$43$	2.82843	−	2.82843i	0.431331	−	0.431331i	−0.457750	−	0.889081i	$-0.651344\pi$
$43$	2.82843	−	2.82843i	0.431331	−	0.431331i	0.889081	+	0.457750i	$0.151344\pi$
$44$	−4.24264			−0.639602
$45$		0			0
$46$	4.73205			0.697703
$47$	−7.46410	+	7.46410i	−1.08875	+	1.08875i	−0.0930938	+	0.995657i	$0.529676\pi$
$47$	−7.46410	+	7.46410i	−1.08875	+	1.08875i	−0.995657	+	0.0930938i	$0.970324\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$		0			0
$52$	6.69213	+	6.69213i	0.928032	+	0.928032i
$53$	2.26795	+	2.26795i	0.311527	+	0.311527i	0.845501	−	0.533974i	$-0.179302\pi$
$53$	2.26795	+	2.26795i	0.311527	+	0.311527i	−0.533974	+	0.845501i	$0.679302\pi$
$54$		0			0
$55$		0			0
$56$			1.93185i			0.258155i
$57$		0			0
$58$	2.31079	−	2.31079i	0.303421	−	0.303421i
$59$	−12.6264			−1.64382			−0.821908	−	0.569621i	$-0.807090\pi$
$59$	−12.6264			−1.64382			−0.821908	+	0.569621i	$0.807090\pi$
$60$		0			0
$61$	−8.92820			−1.14314			−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820			−1.14314			−0.571570	+	0.820554i	$0.693665\pi$
$62$	−1.80385	+	1.80385i	−0.229089	+	0.229089i
$63$		0			0
$64$		−	2.26795i		−	0.283494i
$65$		0			0
$66$		0			0
$67$	6.31319	+	6.31319i	0.771279	+	0.771279i	0.978330	−	0.207051i	$-0.0663866\pi$
$67$	6.31319	+	6.31319i	0.771279	+	0.771279i	−0.207051	+	0.978330i	$0.566387\pi$
$68$	−3.46410	−	3.46410i	−0.420084	−	0.420084i
$69$		0			0
$70$		0			0
$71$			4.52004i			0.536430i	0.963359	+	0.268215i	$0.0864338\pi$
$71$			4.52004i			0.536430i	−0.963359	+	0.268215i	$0.913566\pi$
$72$		0			0
$73$	−1.03528	+	1.03528i	−0.121170	+	0.121170i	−0.765091	−	0.643922i	$-0.777306\pi$
$73$	−1.03528	+	1.03528i	−0.121170	+	0.121170i	0.643922	+	0.765091i	$0.277306\pi$
$74$	0.277401			0.0322473
$75$		0			0
$76$	3.46410			0.397360
$77$	−1.73205	+	1.73205i	−0.197386	+	0.197386i
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$		0			0
$82$	0.757875	+	0.757875i	0.0836933	+	0.0836933i
$83$	4.53590	+	4.53590i	0.497880	+	0.497880i	0.910777	−	0.412898i	$-0.135483\pi$
$83$	4.53590	+	4.53590i	0.497880	+	0.497880i	−0.412898	+	0.910777i	$0.635483\pi$
$84$		0			0
$85$		0			0
$86$		−	2.07055i		−	0.223273i
$87$		0			0
$88$	−3.34607	+	3.34607i	−0.356692	+	0.356692i
$89$	14.1421			1.49906			0.749532	−	0.661968i	$-0.230279\pi$
$89$	14.1421			1.49906			0.749532	+	0.661968i	$0.230279\pi$
$90$		0			0
$91$	5.46410			0.572793
$92$	−11.1962	+	11.1962i	−1.16728	+	1.16728i
$93$		0			0
$94$			5.46410i			0.563579i
$95$		0			0
$96$		0			0
$97$	10.8332	+	10.8332i	1.09995	+	1.09995i	0.994416	+	0.105533i	$0.0336547\pi$
$97$	10.8332	+	10.8332i	1.09995	+	1.09995i	0.105533	+	0.994416i	$0.466345\pi$
$98$	0.366025	+	0.366025i	0.0369741	+	0.0369741i
$99$		0			0
$100$		0			0
$101$		−	11.8685i		−	1.18096i	−0.807052	−	0.590481i	$-0.798938\pi$
$101$		−	11.8685i		−	1.18096i	0.807052	−	0.590481i	$-0.201062\pi$
$102$		0			0
$103$	8.48528	−	8.48528i	0.836080	−	0.836080i	−0.152261	−	0.988340i	$-0.548655\pi$
$103$	8.48528	−	8.48528i	0.836080	−	0.836080i	0.988340	+	0.152261i	$0.0486553\pi$
$104$	10.5558			1.03508
$105$		0			0
$106$	1.66025			0.161258
$107$	−7.92820	+	7.92820i	−0.766448	+	0.766448i	−0.977479	−	0.211031i	$-0.932318\pi$
$107$	−7.92820	+	7.92820i	−0.766448	+	0.766448i	0.211031	+	0.977479i	$0.432318\pi$
$108$		0			0
$109$			1.07180i			0.102660i	0.998682	+	0.0513298i	$0.0163460\pi$
$109$			1.07180i			0.102660i	−0.998682	+	0.0513298i	$0.983654\pi$
$110$		0			0
$111$		0			0
$112$	−1.74238	−	1.74238i	−0.164640	−	0.164640i
$113$	0.267949	+	0.267949i	0.0252065	+	0.0252065i	0.719598	−	0.694391i	$-0.244326\pi$
$113$	0.267949	+	0.267949i	0.0252065	+	0.0252065i	−0.694391	+	0.719598i	$0.744326\pi$
$114$		0			0
$115$		0			0
$116$			10.9348i			1.01527i
$117$		0			0
$118$	−4.62158	+	4.62158i	−0.425451	+	0.425451i
$119$	−2.82843			−0.259281
$120$		0			0
$121$	5.00000			0.454545
$122$	−3.26795	+	3.26795i	−0.295866	+	0.295866i
$123$		0			0
$124$		−	8.53590i		−	0.766546i
$125$		0			0
$126$		0			0
$127$	−5.55532	−	5.55532i	−0.492955	−	0.492955i	0.416281	−	0.909236i	$-0.363333\pi$
$127$	−5.55532	−	5.55532i	−0.492955	−	0.492955i	−0.909236	+	0.416281i	$0.863333\pi$
$128$	−8.09808	−	8.09808i	−0.715776	−	0.715776i
$129$		0			0
$130$		0			0
$131$		−	13.3843i		−	1.16939i	−0.811254	−	0.584694i	$-0.801215\pi$
$131$		−	13.3843i		−	1.16939i	0.811254	−	0.584694i	$-0.198785\pi$
$132$		0			0
$133$	1.41421	−	1.41421i	0.122628	−	0.122628i
$134$	4.62158			0.399244
$135$		0			0
$136$	−5.46410			−0.468543
$137$	7.19615	−	7.19615i	0.614809	−	0.614809i	−0.329386	−	0.944195i	$-0.606842\pi$
$137$	7.19615	−	7.19615i	0.614809	−	0.614809i	0.944195	+	0.329386i	$0.106842\pi$
$138$		0			0
$139$		−	11.8564i		−	1.00565i	−0.864389	−	0.502824i	$-0.832295\pi$
$139$		−	11.8564i		−	1.00565i	0.864389	−	0.502824i	$-0.167705\pi$
$140$		0			0
$141$		0			0
$142$	1.65445	+	1.65445i	0.138838	+	0.138838i
$143$	9.46410	+	9.46410i	0.791428	+	0.791428i
$144$		0			0
$145$		0			0
$146$			0.757875i			0.0627222i
$147$		0			0
$148$	−0.656339	+	0.656339i	−0.0539507	+	0.0539507i
$149$	−18.9396			−1.55159			−0.775795	−	0.630985i	$-0.782651\pi$
$149$	−18.9396			−1.55159			−0.775795	+	0.630985i	$0.782651\pi$
$150$		0			0
$151$	10.3923			0.845714			0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923			0.845714			0.422857	+	0.906196i	$0.361027\pi$
$152$	2.73205	−	2.73205i	0.221599	−	0.221599i
$153$		0			0
$154$			1.26795i			0.102174i
$155$		0			0
$156$		0			0
$157$	−7.45001	−	7.45001i	−0.594575	−	0.594575i	0.344289	−	0.938864i	$-0.388120\pi$
$157$	−7.45001	−	7.45001i	−0.594575	−	0.594575i	−0.938864	+	0.344289i	$0.888120\pi$
$158$	1.46410	+	1.46410i	0.116478	+	0.116478i
$159$		0			0
$160$		0			0
$161$			9.14162i			0.720461i
$162$		0			0
$163$	−8.38375	+	8.38375i	−0.656666	+	0.656666i	−0.954590	−	0.297924i	$-0.903706\pi$
$163$	−8.38375	+	8.38375i	−0.656666	+	0.656666i	0.297924	+	0.954590i	$0.403706\pi$
$164$	−3.58630			−0.280043
$165$		0			0
$166$	3.32051			0.257721
$167$	2.92820	−	2.92820i	0.226591	−	0.226591i	−0.584676	−	0.811267i	$-0.698778\pi$
$167$	2.92820	−	2.92820i	0.226591	−	0.226591i	0.811267	+	0.584676i	$0.198778\pi$
$168$		0			0
$169$		−	16.8564i		−	1.29665i
$170$		0			0
$171$		0			0
$172$	4.89898	+	4.89898i	0.373544	+	0.373544i
$173$	14.3923	+	14.3923i	1.09423	+	1.09423i	0.995072	+	0.0991546i	$0.0316138\pi$
$173$	14.3923	+	14.3923i	1.09423	+	1.09423i	0.0991546	+	0.995072i	$0.468386\pi$
$174$		0			0
$175$		0			0
$176$		−	6.03579i		−	0.454965i
$177$		0			0
$178$	5.17638	−	5.17638i	0.387986	−	0.387986i
$179$	−18.6622			−1.39488			−0.697438	−	0.716645i	$-0.745677\pi$
$179$	−18.6622			−1.39488			−0.697438	+	0.716645i	$0.745677\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$	2.00000	−	2.00000i	0.148250	−	0.148250i
$183$		0			0
$184$			17.6603i			1.30193i
$185$		0			0
$186$		0			0
$187$	−4.89898	−	4.89898i	−0.358249	−	0.358249i
$188$	−12.9282	−	12.9282i	−0.942886	−	0.942886i
$189$		0			0
$190$		0			0
$191$			6.79367i			0.491572i	0.969324	+	0.245786i	$0.0790461\pi$
$191$			6.79367i			0.491572i	−0.969324	+	0.245786i	$0.920954\pi$
$192$		0			0
$193$	17.9043	−	17.9043i	1.28878	−	1.28878i	0.353252	−	0.935528i	$-0.385076\pi$
$193$	17.9043	−	17.9043i	1.28878	−	1.28878i	0.935528	−	0.353252i	$-0.114924\pi$
$194$	7.93048			0.569375
$195$		0			0
$196$	−1.73205			−0.123718
$197$	10.2679	−	10.2679i	0.731561	−	0.731561i	−0.239368	−	0.970929i	$-0.576940\pi$
$197$	10.2679	−	10.2679i	0.731561	−	0.731561i	0.970929	+	0.239368i	$0.0769402\pi$
$198$		0			0
$199$			20.9282i			1.48356i	0.670643	+	0.741780i	$0.266018\pi$
$199$			20.9282i			1.48356i	−0.670643	+	0.741780i	$0.733982\pi$
$200$		0			0
$201$		0			0
$202$	−4.34418	−	4.34418i	−0.305655	−	0.305655i
$203$	4.46410	+	4.46410i	0.313319	+	0.313319i
$204$		0			0
$205$		0			0
$206$		−	6.21166i		−	0.432787i
$207$		0			0
$208$	−9.52056	+	9.52056i	−0.660132	+	0.660132i
$209$	4.89898			0.338869
$210$		0			0
$211$	−0.535898			−0.0368928			−0.0184464	−	0.999830i	$-0.505872\pi$
$211$	−0.535898			−0.0368928			−0.0184464	+	0.999830i	$0.505872\pi$
$212$	−3.92820	+	3.92820i	−0.269790	+	0.269790i
$213$		0			0
$214$			5.80385i			0.396743i
$215$		0			0
$216$		0			0
$217$	−3.48477	−	3.48477i	−0.236561	−	0.236561i
$218$	0.392305	+	0.392305i	0.0265702	+	0.0265702i
$219$		0			0
$220$		0			0
$221$			15.4548i			1.03960i
$222$		0			0
$223$	11.8685	−	11.8685i	0.794774	−	0.794774i	−0.187492	−	0.982266i	$-0.560036\pi$
$223$	11.8685	−	11.8685i	0.794774	−	0.794774i	0.982266	+	0.187492i	$0.0600358\pi$
$224$	−5.13922			−0.343378
$225$		0			0
$226$	0.196152			0.0130479
$227$	16.3923	−	16.3923i	1.08800	−	1.08800i	0.0922606	−	0.995735i	$-0.470591\pi$
$227$	16.3923	−	16.3923i	1.08800	−	1.08800i	0.995735	−	0.0922606i	$-0.0294093\pi$
$228$		0			0
$229$		−	24.9282i		−	1.64730i	−0.567097	−	0.823651i	$-0.691934\pi$
$229$		−	24.9282i		−	1.64730i	0.567097	−	0.823651i	$-0.308066\pi$
$230$		0			0
$231$		0			0
$232$	8.62398	+	8.62398i	0.566192	+	0.566192i
$233$	−0.267949	−	0.267949i	−0.0175539	−	0.0175539i	0.698275	−	0.715829i	$-0.253951\pi$
$233$	−0.267949	−	0.267949i	−0.0175539	−	0.0175539i	−0.715829	+	0.698275i	$0.753951\pi$
$234$		0			0
$235$		0			0
$236$		−	21.8695i		−	1.42359i
$237$		0			0
$238$	−1.03528	+	1.03528i	−0.0671070	+	0.0671070i
$239$	17.3495			1.12225			0.561123	−	0.827732i	$-0.310369\pi$
$239$	17.3495			1.12225			0.561123	+	0.827732i	$0.310369\pi$
$240$		0			0
$241$	12.9282			0.832779			0.416389	−	0.909186i	$-0.363295\pi$
$241$	12.9282			0.832779			0.416389	+	0.909186i	$0.363295\pi$
$242$	1.83013	−	1.83013i	0.117645	−	0.117645i
$243$		0			0
$244$		−	15.4641i		−	0.989988i
$245$		0			0
$246$		0			0
$247$	−7.72741	−	7.72741i	−0.491683	−	0.491683i
$248$	−6.73205	−	6.73205i	−0.427486	−	0.427486i
$249$		0			0
$250$		0			0
$251$		−	14.1421i		−	0.892644i	−0.894873	−	0.446322i	$-0.852734\pi$
$251$		−	14.1421i		−	0.892644i	0.894873	−	0.446322i	$-0.147266\pi$
$252$		0			0
$253$	−15.8338	+	15.8338i	−0.995459	+	0.995459i
$254$	−4.06678			−0.255172
$255$		0			0
$256$	−1.39230			−0.0870191
$257$	14.5359	−	14.5359i	0.906724	−	0.906724i	−0.0892820	−	0.996006i	$-0.528457\pi$
$257$	14.5359	−	14.5359i	0.906724	−	0.906724i	0.996006	+	0.0892820i	$0.0284572\pi$
$258$		0			0
$259$			0.535898i			0.0332991i
$260$		0			0
$261$		0			0
$262$	−4.89898	−	4.89898i	−0.302660	−	0.302660i
$263$	−7.39230	−	7.39230i	−0.455829	−	0.455829i	0.441455	−	0.897284i	$-0.354463\pi$
$263$	−7.39230	−	7.39230i	−0.455829	−	0.455829i	−0.897284	+	0.441455i	$0.854463\pi$
$264$		0			0
$265$		0			0
$266$		−	1.03528i		−	0.0634769i
$267$		0			0
$268$	−10.9348	+	10.9348i	−0.667947	+	0.667947i
$269$	−10.5558			−0.643601			−0.321800	−	0.946808i	$-0.604288\pi$
$269$	−10.5558			−0.643601			−0.321800	+	0.946808i	$0.604288\pi$
$270$		0			0
$271$	−23.8564			−1.44917			−0.724587	−	0.689184i	$-0.757969\pi$
$271$	−23.8564			−1.44917			−0.724587	+	0.689184i	$0.757969\pi$
$272$	4.92820	−	4.92820i	0.298816	−	0.298816i
$273$		0			0
$274$		−	5.26795i		−	0.318248i
$275$		0			0
$276$		0			0
$277$	−21.8695	−	21.8695i	−1.31401	−	1.31401i	−0.918430	−	0.395583i	$-0.870542\pi$
$277$	−21.8695	−	21.8695i	−1.31401	−	1.31401i	−0.395583	−	0.918430i	$-0.629458\pi$
$278$	−4.33975	−	4.33975i	−0.260281	−	0.260281i
$279$		0			0
$280$		0			0
$281$			1.41421i			0.0843649i	0.999110	+	0.0421825i	$0.0134311\pi$
$281$			1.41421i			0.0843649i	−0.999110	+	0.0421825i	$0.986569\pi$
$282$		0			0
$283$	9.24316	−	9.24316i	0.549449	−	0.549449i	−0.376833	−	0.926281i	$-0.622987\pi$
$283$	9.24316	−	9.24316i	0.549449	−	0.549449i	0.926281	+	0.376833i	$0.122987\pi$
$284$	−7.82894			−0.464562
$285$		0			0
$286$	6.92820			0.409673
$287$	−1.46410	+	1.46410i	−0.0864232	+	0.0864232i
$288$		0			0
$289$			9.00000i			0.529412i
$290$		0			0
$291$		0			0
$292$	−1.79315	−	1.79315i	−0.104936	−	0.104936i
$293$	−8.39230	−	8.39230i	−0.490284	−	0.490284i	0.418112	−	0.908396i	$-0.362692\pi$
$293$	−8.39230	−	8.39230i	−0.490284	−	0.490284i	−0.908396	+	0.418112i	$0.862692\pi$
$294$		0			0
$295$		0			0
$296$			1.03528i			0.0601742i
$297$		0			0
$298$	−6.93237	+	6.93237i	−0.401581	+	0.401581i
$299$	49.9507			2.88873
$300$		0			0
$301$	4.00000			0.230556
$302$	3.80385	−	3.80385i	0.218887	−	0.218887i
$303$		0			0
$304$			4.92820i			0.282652i
$305$		0			0
$306$		0			0
$307$	1.31268	+	1.31268i	0.0749185	+	0.0749185i	0.743573	−	0.668655i	$-0.233130\pi$
$307$	1.31268	+	1.31268i	0.0749185	+	0.0749185i	−0.668655	+	0.743573i	$0.733130\pi$
$308$	−3.00000	−	3.00000i	−0.170941	−	0.170941i
$309$		0			0
$310$		0			0
$311$			17.5254i			0.993772i	0.867816	+	0.496886i	$0.165523\pi$
$311$			17.5254i			0.993772i	−0.867816	+	0.496886i	$0.834477\pi$
$312$		0			0
$313$	−19.3185	+	19.3185i	−1.09195	+	1.09195i	−0.0966264	+	0.995321i	$0.530805\pi$
$313$	−19.3185	+	19.3185i	−1.09195	+	1.09195i	−0.995321	+	0.0966264i	$0.969195\pi$
$314$	−5.45378			−0.307775
$315$		0			0
$316$	−6.92820			−0.389742
$317$	14.2679	−	14.2679i	0.801368	−	0.801368i	−0.181941	−	0.983309i	$-0.558238\pi$
$317$	14.2679	−	14.2679i	0.801368	−	0.801368i	0.983309	+	0.181941i	$0.0582381\pi$
$318$		0			0
$319$			15.4641i			0.865823i
$320$		0			0
$321$		0			0
$322$	3.34607	+	3.34607i	0.186469	+	0.186469i
$323$	4.00000	+	4.00000i	0.222566	+	0.222566i
$324$		0			0
$325$		0			0
$326$			6.13733i			0.339915i
$327$		0			0
$328$	−2.82843	+	2.82843i	−0.156174	+	0.156174i
$329$	−10.5558			−0.581962
$330$		0			0
$331$	16.5359			0.908895			0.454448	−	0.890773i	$-0.349837\pi$
$331$	16.5359			0.908895			0.454448	+	0.890773i	$0.349837\pi$
$332$	−7.85641	+	7.85641i	−0.431176	+	0.431176i
$333$		0			0
$334$		−	2.14359i		−	0.117292i
$335$		0			0
$336$		0			0
$337$	−23.5612	−	23.5612i	−1.28346	−	1.28346i	−0.938687	−	0.344771i	$-0.887956\pi$
$337$	−23.5612	−	23.5612i	−1.28346	−	1.28346i	−0.344771	−	0.938687i	$-0.612044\pi$
$338$	−6.16987	−	6.16987i	−0.335597	−	0.335597i
$339$		0			0
$340$		0			0
$341$		−	12.0716i		−	0.653713i
$342$		0			0
$343$	−0.707107	+	0.707107i	−0.0381802	+	0.0381802i
$344$	7.72741			0.416634
$345$		0			0
$346$	10.5359			0.566413
$347$	1.00000	−	1.00000i	0.0536828	−	0.0536828i	−0.679756	−	0.733439i	$-0.737914\pi$
$347$	1.00000	−	1.00000i	0.0536828	−	0.0536828i	0.733439	+	0.679756i	$0.237914\pi$
$348$		0			0
$349$			29.7128i			1.59049i	0.606288	+	0.795245i	$0.292658\pi$
$349$			29.7128i			1.59049i	−0.606288	+	0.795245i	$0.707342\pi$
$350$		0			0
$351$		0			0
$352$	−8.90138	−	8.90138i	−0.474445	−	0.474445i
$353$	−18.9282	−	18.9282i	−1.00745	−	1.00745i	−0.999972	−	0.00747454i	$-0.997621\pi$
$353$	−18.9282	−	18.9282i	−1.00745	−	1.00745i	−0.00747454	−	0.999972i	$-0.502379\pi$
$354$		0			0
$355$		0			0
$356$			24.4949i			1.29823i
$357$		0			0
$358$	−6.83083	+	6.83083i	−0.361021	+	0.361021i
$359$	−11.6926			−0.617114			−0.308557	−	0.951206i	$-0.599846\pi$
$359$	−11.6926			−0.617114			−0.308557	+	0.951206i	$0.599846\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$	−2.19615	+	2.19615i	−0.115427	+	0.115427i
$363$		0			0
$364$			9.46410i			0.496054i
$365$		0			0
$366$		0			0
$367$	14.6969	+	14.6969i	0.767174	+	0.767174i	0.977608	−	0.210434i	$-0.0674877\pi$
$367$	14.6969	+	14.6969i	0.767174	+	0.767174i	−0.210434	+	0.977608i	$0.567488\pi$
$368$	−15.9282	−	15.9282i	−0.830315	−	0.830315i
$369$		0			0
$370$		0			0
$371$			3.20736i			0.166518i
$372$		0			0
$373$	19.7990	−	19.7990i	1.02515	−	1.02515i	0.0254774	−	0.999675i	$-0.491889\pi$
$373$	19.7990	−	19.7990i	1.02515	−	1.02515i	0.999675	−	0.0254774i	$-0.00811060\pi$
$374$	−3.58630			−0.185443
$375$		0			0
$376$	−20.3923			−1.05165
$377$	24.3923	−	24.3923i	1.25627	−	1.25627i
$378$		0			0
$379$		−	13.6077i		−	0.698980i	−0.936940	−	0.349490i	$-0.886355\pi$
$379$		−	13.6077i		−	0.698980i	0.936940	−	0.349490i	$-0.113645\pi$
$380$		0			0
$381$		0			0
$382$	2.48665	+	2.48665i	0.127228	+	0.127228i
$383$	−7.85641	−	7.85641i	−0.401444	−	0.401444i	0.477298	−	0.878742i	$-0.341616\pi$
$383$	−7.85641	−	7.85641i	−0.401444	−	0.401444i	−0.878742	+	0.477298i	$0.841616\pi$
$384$		0			0
$385$		0			0
$386$		−	13.1069i		−	0.667122i
$387$		0			0
$388$	−18.7637	+	18.7637i	−0.952583	+	0.952583i
$389$	−34.5975			−1.75416			−0.877081	−	0.480343i	$-0.840512\pi$
$389$	−34.5975			−1.75416			−0.877081	+	0.480343i	$0.840512\pi$
$390$		0			0
$391$	−25.8564			−1.30761
$392$	−1.36603	+	1.36603i	−0.0689947	+	0.0689947i
$393$		0			0
$394$		−	7.51666i		−	0.378684i
$395$		0			0
$396$		0			0
$397$	1.03528	+	1.03528i	0.0519590	+	0.0519590i	0.732609	−	0.680650i	$-0.238302\pi$
$397$	1.03528	+	1.03528i	0.0519590	+	0.0519590i	−0.680650	+	0.732609i	$0.738302\pi$
$398$	7.66025	+	7.66025i	0.383974	+	0.383974i
$399$		0			0
$400$		0			0
$401$			22.5259i			1.12489i	0.826835	+	0.562444i	$0.190139\pi$
$401$			22.5259i			1.12489i	−0.826835	+	0.562444i	$0.809861\pi$
$402$		0			0
$403$	−19.0411	+	19.0411i	−0.948506	+	0.948506i
$404$	20.5569			1.02274
$405$		0			0
$406$	3.26795			0.162186
$407$	−0.928203	+	0.928203i	−0.0460093	+	0.0460093i
$408$		0			0
$409$		−	8.92820i		−	0.441471i	−0.975334	−	0.220736i	$-0.929154\pi$
$409$		−	8.92820i		−	0.441471i	0.975334	−	0.220736i	$-0.0708458\pi$
$410$		0			0
$411$		0			0
$412$	14.6969	+	14.6969i	0.724066	+	0.724066i
$413$	−8.92820	−	8.92820i	−0.439328	−	0.439328i
$414$		0			0
$415$		0			0
$416$			28.0812i			1.37679i
$417$		0			0
$418$	1.79315	−	1.79315i	0.0877059	−	0.0877059i
$419$	5.65685			0.276355			0.138178	−	0.990407i	$-0.455875\pi$
$419$	5.65685			0.276355			0.138178	+	0.990407i	$0.455875\pi$
$420$		0			0
$421$	−29.7128			−1.44811			−0.724057	−	0.689740i	$-0.757725\pi$
$421$	−29.7128			−1.44811			−0.724057	+	0.689740i	$0.757725\pi$
$422$	−0.196152	+	0.196152i	−0.00954855	+	0.00954855i
$423$		0			0
$424$			6.19615i			0.300912i
$425$		0			0
$426$		0			0
$427$	−6.31319	−	6.31319i	−0.305517	−	0.305517i
$428$	−13.7321	−	13.7321i	−0.663764	−	0.663764i
$429$		0			0
$430$		0			0
$431$		−	5.27792i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$431$		−	5.27792i		−	0.254228i	0.991888	−	0.127114i	$-0.0405715\pi$
$432$		0			0
$433$	−10.2784	+	10.2784i	−0.493950	+	0.493950i	−0.909548	−	0.415598i	$-0.863572\pi$
$433$	−10.2784	+	10.2784i	−0.493950	+	0.493950i	0.415598	+	0.909548i	$0.363572\pi$
$434$	−2.55103			−0.122453
$435$		0			0
$436$	−1.85641			−0.0889057
$437$	12.9282	−	12.9282i	0.618440	−	0.618440i
$438$		0			0
$439$		−	39.8564i		−	1.90224i	−0.308818	−	0.951121i	$-0.599933\pi$
$439$		−	39.8564i		−	1.90224i	0.308818	−	0.951121i	$-0.400067\pi$
$440$		0			0
$441$		0			0
$442$	5.65685	+	5.65685i	0.269069	+	0.269069i
$443$	11.9282	+	11.9282i	0.566726	+	0.566726i	0.931210	−	0.364484i	$-0.118755\pi$
$443$	11.9282	+	11.9282i	0.566726	+	0.566726i	−0.364484	+	0.931210i	$0.618755\pi$
$444$		0			0
$445$		0			0
$446$		−	8.68835i		−	0.411405i
$447$		0			0
$448$	1.60368	−	1.60368i	0.0757669	−	0.0757669i
$449$	22.5259			1.06306			0.531531	−	0.847039i	$-0.321617\pi$
$449$	22.5259			1.06306			0.531531	+	0.847039i	$0.321617\pi$
$450$		0			0
$451$	−5.07180			−0.238822
$452$	−0.464102	+	0.464102i	−0.0218295	+	0.0218295i
$453$		0			0
$454$		−	12.0000i		−	0.563188i
$455$		0			0
$456$		0			0
$457$	−21.4906	−	21.4906i	−1.00529	−	1.00529i	−0.999986	−	0.00530215i	$-0.998312\pi$
$457$	−21.4906	−	21.4906i	−1.00529	−	1.00529i	−0.00530215	−	0.999986i	$-0.501688\pi$
$458$	−9.12436	−	9.12436i	−0.426353	−	0.426353i
$459$		0			0
$460$		0			0
$461$		−	39.3949i		−	1.83480i	−0.397962	−	0.917402i	$-0.630282\pi$
$461$		−	39.3949i		−	1.83480i	0.397962	−	0.917402i	$-0.369718\pi$
$462$		0			0
$463$	25.3543	−	25.3543i	1.17831	−	1.17831i	0.198141	−	0.980174i	$-0.436510\pi$
$463$	25.3543	−	25.3543i	1.17831	−	1.17831i	0.980174	−	0.198141i	$-0.0634904\pi$
$464$	−15.5563			−0.722185
$465$		0			0
$466$	−0.196152			−0.00908659
$467$	−6.53590	+	6.53590i	−0.302445	+	0.302445i	−0.841970	−	0.539525i	$-0.818604\pi$
$467$	−6.53590	+	6.53590i	−0.302445	+	0.302445i	0.539525	+	0.841970i	$0.318604\pi$
$468$		0			0
$469$			8.92820i			0.412266i
$470$		0			0
$471$		0			0
$472$	−17.2480	−	17.2480i	−0.793902	−	0.793902i
$473$	6.92820	+	6.92820i	0.318559	+	0.318559i
$474$		0			0
$475$		0			0
$476$		−	4.89898i		−	0.224544i
$477$		0			0
$478$	6.35036	−	6.35036i	0.290459	−	0.290459i
$479$	−10.7589			−0.491587			−0.245793	−	0.969322i	$-0.579049\pi$
$479$	−10.7589			−0.491587			−0.245793	+	0.969322i	$0.579049\pi$
$480$		0			0
$481$	2.92820			0.133515
$482$	4.73205	−	4.73205i	0.215539	−	0.215539i
$483$		0			0
$484$			8.66025i			0.393648i
$485$		0			0
$486$		0			0
$487$	19.0411	+	19.0411i	0.862835	+	0.862835i	0.991667	−	0.128831i	$-0.0411226\pi$
$487$	19.0411	+	19.0411i	0.862835	+	0.862835i	−0.128831	+	0.991667i	$0.541123\pi$
$488$	−12.1962	−	12.1962i	−0.552094	−	0.552094i
$489$		0			0
$490$		0			0
$491$		−	32.6012i		−	1.47127i	−0.677376	−	0.735637i	$-0.736883\pi$
$491$		−	32.6012i		−	1.47127i	0.677376	−	0.735637i	$-0.263117\pi$
$492$		0			0
$493$	−12.6264	+	12.6264i	−0.568664	+	0.568664i
$494$	−5.65685			−0.254514
$495$		0			0
$496$	12.1436			0.545263
$497$	−3.19615	+	3.19615i	−0.143367	+	0.143367i
$498$		0			0
$499$			17.3205i			0.775372i	0.921791	+	0.387686i	$0.126726\pi$
$499$			17.3205i			0.775372i	−0.921791	+	0.387686i	$0.873274\pi$
$500$		0			0
$501$		0			0
$502$	−5.17638	−	5.17638i	−0.231033	−	0.231033i
$503$	12.9282	+	12.9282i	0.576440	+	0.576440i	0.933921	−	0.357481i	$-0.116364\pi$
$503$	12.9282	+	12.9282i	0.576440	+	0.576440i	−0.357481	+	0.933921i	$0.616364\pi$
$504$		0			0
$505$		0			0
$506$			11.5911i			0.515288i
$507$		0			0
$508$	9.62209	−	9.62209i	0.426911	−	0.426911i
$509$	38.8401			1.72156			0.860779	−	0.508979i	$-0.169977\pi$
$509$	38.8401			1.72156			0.860779	+	0.508979i	$0.169977\pi$
$510$		0			0
$511$	−1.46410			−0.0647680
$512$	15.6865	−	15.6865i	0.693253	−	0.693253i
$513$		0			0
$514$		−	10.6410i		−	0.469355i
$515$		0			0
$516$		0			0
$517$	−18.2832	−	18.2832i	−0.804096	−	0.804096i
$518$	0.196152	+	0.196152i	0.00861844	+	0.00861844i
$519$		0			0
$520$		0			0
$521$			28.6360i			1.25457i	0.778791	+	0.627283i	$0.215833\pi$
$521$			28.6360i			1.25457i	−0.778791	+	0.627283i	$0.784167\pi$
$522$		0			0
$523$	2.82843	−	2.82843i	0.123678	−	0.123678i	−0.642558	−	0.766237i	$-0.722127\pi$
$523$	2.82843	−	2.82843i	0.123678	−	0.123678i	0.766237	+	0.642558i	$0.222127\pi$
$524$	23.1822			1.01272
$525$		0			0
$526$	−5.41154			−0.235954
$527$	9.85641	−	9.85641i	0.429352	−	0.429352i
$528$		0			0
$529$			60.5692i			2.63344i
$530$		0			0
$531$		0			0
$532$	2.44949	+	2.44949i	0.106199	+	0.106199i
$533$	8.00000	+	8.00000i	0.346518	+	0.346518i
$534$		0			0
$535$		0			0
$536$			17.2480i			0.744999i
$537$		0			0
$538$	−3.86370	+	3.86370i	−0.166576	+	0.166576i
$539$	−2.44949			−0.105507
$540$		0			0
$541$	11.8564			0.509747			0.254873	−	0.966974i	$-0.417966\pi$
$541$	11.8564			0.509747			0.254873	+	0.966974i	$0.417966\pi$
$542$	−8.73205	+	8.73205i	−0.375074	+	0.375074i
$543$		0			0
$544$		−	14.5359i		−	0.623222i
$545$		0			0
$546$		0			0
$547$	5.00052	+	5.00052i	0.213807	+	0.213807i	0.805882	−	0.592076i	$-0.201691\pi$
$547$	5.00052	+	5.00052i	0.213807	+	0.213807i	−0.592076	+	0.805882i	$0.701691\pi$
$548$	12.4641	+	12.4641i	0.532440	+	0.532440i
$549$		0			0
$550$		0			0
$551$		−	12.6264i		−	0.537902i
$552$		0			0
$553$	−2.82843	+	2.82843i	−0.120277	+	0.120277i
$554$	−16.0096			−0.680183
$555$		0			0
$556$	20.5359			0.870916
$557$	14.2679	−	14.2679i	0.604552	−	0.604552i	−0.336965	−	0.941517i	$-0.609400\pi$
$557$	14.2679	−	14.2679i	0.604552	−	0.604552i	0.941517	+	0.336965i	$0.109400\pi$
$558$		0			0
$559$		−	21.8564i		−	0.924427i
$560$		0			0
$561$		0			0
$562$	0.517638	+	0.517638i	0.0218352	+	0.0218352i
$563$	22.7846	+	22.7846i	0.960257	+	0.960257i	0.999240	−	0.0389831i	$-0.0124118\pi$
$563$	22.7846	+	22.7846i	0.960257	+	0.960257i	−0.0389831	+	0.999240i	$0.512412\pi$
$564$		0			0
$565$		0			0
$566$		−	6.76646i		−	0.284416i
$567$		0			0
$568$	−6.17449	+	6.17449i	−0.259076	+	0.259076i
$569$	−9.89949			−0.415008			−0.207504	−	0.978234i	$-0.566534\pi$
$569$	−9.89949			−0.415008			−0.207504	+	0.978234i	$0.566534\pi$
$570$		0			0
$571$	−17.8564			−0.747267			−0.373634	−	0.927576i	$-0.621888\pi$
$571$	−17.8564			−0.747267			−0.373634	+	0.927576i	$0.621888\pi$
$572$	−16.3923	+	16.3923i	−0.685397	+	0.685397i
$573$		0			0
$574$			1.07180i			0.0447359i
$575$		0			0
$576$		0			0
$577$	10.8332	+	10.8332i	0.450993	+	0.450993i	0.895684	−	0.444691i	$-0.146686\pi$
$577$	10.8332	+	10.8332i	0.450993	+	0.450993i	−0.444691	+	0.895684i	$0.646686\pi$
$578$	3.29423	+	3.29423i	0.137022	+	0.137022i
$579$		0			0
$580$		0			0
$581$			6.41473i			0.266128i
$582$		0			0
$583$	−5.55532	+	5.55532i	−0.230078	+	0.230078i
$584$	−2.82843			−0.117041
$585$		0			0
$586$	−6.14359			−0.253790
$587$	−14.0000	+	14.0000i	−0.577842	+	0.577842i	−0.934308	−	0.356466i	$-0.883981\pi$
$587$	−14.0000	+	14.0000i	−0.577842	+	0.577842i	0.356466	+	0.934308i	$0.383981\pi$
$588$		0			0
$589$			9.85641i			0.406126i
$590$		0			0
$591$		0			0
$592$	−0.933740	−	0.933740i	−0.0383765	−	0.0383765i
$593$	5.85641	+	5.85641i	0.240494	+	0.240494i	0.817054	−	0.576561i	$-0.195606\pi$
$593$	5.85641	+	5.85641i	0.240494	+	0.240494i	−0.576561	+	0.817054i	$0.695606\pi$
$594$		0			0
$595$		0			0
$596$		−	32.8043i		−	1.34372i
$597$		0			0
$598$	18.2832	−	18.2832i	0.747657	−	0.747657i
$599$	−32.0464			−1.30938			−0.654691	−	0.755897i	$-0.727201\pi$
$599$	−32.0464			−1.30938			−0.654691	+	0.755897i	$0.727201\pi$
$600$		0			0
$601$	−5.21539			−0.212740			−0.106370	−	0.994327i	$-0.533923\pi$
$601$	−5.21539			−0.212740			−0.106370	+	0.994327i	$0.533923\pi$
$602$	1.46410	−	1.46410i	0.0596723	−	0.0596723i
$603$		0			0
$604$			18.0000i			0.732410i
$605$		0			0
$606$		0			0
$607$	−2.82843	−	2.82843i	−0.114802	−	0.114802i	0.647372	−	0.762174i	$-0.275868\pi$
$607$	−2.82843	−	2.82843i	−0.114802	−	0.114802i	−0.762174	+	0.647372i	$0.775868\pi$
$608$	7.26795	+	7.26795i	0.294754	+	0.294754i
$609$		0			0
$610$		0			0
$611$			57.6781i			2.33341i
$612$		0			0
$613$	9.04008	−	9.04008i	0.365126	−	0.365126i	−0.500570	−	0.865696i	$-0.666876\pi$
$613$	9.04008	−	9.04008i	0.365126	−	0.365126i	0.865696	+	0.500570i	$0.166876\pi$
$614$	0.960947			0.0387807
$615$		0			0
$616$	−4.73205			−0.190660
$617$	−29.0526	+	29.0526i	−1.16961	+	1.16961i	−0.187311	+	0.982301i	$0.559977\pi$
$617$	−29.0526	+	29.0526i	−1.16961	+	1.16961i	−0.982301	+	0.187311i	$0.940023\pi$
$618$		0			0
$619$		−	0.928203i		−	0.0373076i	−0.999826	−	0.0186538i	$-0.994062\pi$
$619$		−	0.928203i		−	0.0373076i	0.999826	−	0.0186538i	$-0.00593804\pi$
$620$		0			0
$621$		0			0
$622$	6.41473	+	6.41473i	0.257207	+	0.257207i
$623$	10.0000	+	10.0000i	0.400642	+	0.400642i
$624$		0			0
$625$		0			0
$626$			14.1421i			0.565233i
$627$		0			0
$628$	12.9038	−	12.9038i	0.514917	−	0.514917i
$629$	−1.51575			−0.0604369
$630$		0			0
$631$	25.8564			1.02933			0.514664	−	0.857392i	$-0.327917\pi$
$631$	25.8564			1.02933			0.514664	+	0.857392i	$0.327917\pi$
$632$	−5.46410	+	5.46410i	−0.217350	+	0.217350i
$633$		0			0
$634$		−	10.4449i		−	0.414819i
$635$		0			0
$636$		0			0
$637$	3.86370	+	3.86370i	0.153085	+	0.153085i
$638$	5.66025	+	5.66025i	0.224092	+	0.224092i
$639$		0			0
$640$		0			0
$641$		−	13.8375i		−	0.546549i	−0.961936	−	0.273275i	$-0.911893\pi$
$641$		−	13.8375i		−	0.546549i	0.961936	−	0.273275i	$-0.0881068\pi$
$642$		0			0
$643$	27.5264	−	27.5264i	1.08554	−	1.08554i	0.0895531	−	0.995982i	$-0.471456\pi$
$643$	27.5264	−	27.5264i	1.08554	−	1.08554i	0.995982	−	0.0895531i	$-0.0285439\pi$
$644$	−15.8338			−0.623937
$645$		0			0
$646$	2.92820			0.115209
$647$	−20.0000	+	20.0000i	−0.786281	+	0.786281i	−0.980882	−	0.194601i	$-0.937659\pi$
$647$	−20.0000	+	20.0000i	−0.786281	+	0.786281i	0.194601	+	0.980882i	$0.437659\pi$
$648$		0			0
$649$		−	30.9282i		−	1.21404i
$650$		0			0
$651$		0			0
$652$	−14.5211	−	14.5211i	−0.568689	−	0.568689i
$653$	−3.33975	−	3.33975i	−0.130694	−	0.130694i	0.638734	−	0.769428i	$-0.279459\pi$
$653$	−3.33975	−	3.33975i	−0.130694	−	0.130694i	−0.769428	+	0.638734i	$0.779459\pi$
$654$		0			0
$655$		0			0
$656$		−	5.10205i		−	0.199202i
$657$		0			0
$658$	−3.86370	+	3.86370i	−0.150623	+	0.150623i
$659$	25.0769			0.976858			0.488429	−	0.872604i	$-0.337570\pi$
$659$	25.0769			0.976858			0.488429	+	0.872604i	$0.337570\pi$
$660$		0			0
$661$	18.7846			0.730637			0.365318	−	0.930883i	$-0.380960\pi$
$661$	18.7846			0.730637			0.365318	+	0.930883i	$0.380960\pi$
$662$	6.05256	−	6.05256i	0.235239	−	0.235239i
$663$		0			0
$664$			12.3923i			0.480915i
$665$		0			0
$666$		0			0
$667$	40.8091	+	40.8091i	1.58014	+	1.58014i
$668$	5.07180	+	5.07180i	0.196234	+	0.196234i
$669$		0			0
$670$		0			0
$671$		−	21.8695i		−	0.844264i
$672$		0			0
$673$	−1.89469	+	1.89469i	−0.0730348	+	0.0730348i	−0.742681	−	0.669646i	$-0.766446\pi$
$673$	−1.89469	+	1.89469i	−0.0730348	+	0.0730348i	0.669646	+	0.742681i	$0.266446\pi$
$674$	−17.2480			−0.664367
$675$		0			0
$676$	29.1962			1.12293
$677$	27.3205	−	27.3205i	1.05001	−	1.05001i	0.0513307	−	0.998682i	$-0.483654\pi$
$677$	27.3205	−	27.3205i	1.05001	−	1.05001i	0.998682	−	0.0513307i	$-0.0163463\pi$
$678$		0			0
$679$			15.3205i			0.587947i
$680$		0			0
$681$		0			0
$682$	−4.41851	−	4.41851i	−0.169193	−	0.169193i
$683$	5.14359	+	5.14359i	0.196814	+	0.196814i	0.798633	−	0.601819i	$-0.205557\pi$
$683$	5.14359	+	5.14359i	0.196814	+	0.196814i	−0.601819	+	0.798633i	$0.705557\pi$
$684$		0			0
$685$		0			0
$686$			0.517638i			0.0197635i
$687$		0			0
$688$	−6.96953	+	6.96953i	−0.265711	+	0.265711i
$689$	17.5254			0.667663
$690$		0			0
$691$	10.7846			0.410266			0.205133	−	0.978734i	$-0.434237\pi$
$691$	10.7846			0.410266			0.205133	+	0.978734i	$0.434237\pi$
$692$	−24.9282	+	24.9282i	−0.947628	+	0.947628i
$693$		0			0
$694$		−	0.732051i		−	0.0277883i
$695$		0			0
$696$		0			0
$697$	−4.14110	−	4.14110i	−0.156856	−	0.156856i
$698$	10.8756	+	10.8756i	0.411649	+	0.411649i
$699$		0			0
$700$		0			0
$701$		−	1.96902i		−	0.0743687i	−0.999308	−	0.0371844i	$-0.988161\pi$
$701$		−	1.96902i		−	0.0743687i	0.999308	−	0.0371844i	$-0.0118389\pi$
$702$		0			0
$703$	0.757875	−	0.757875i	0.0285838	−	0.0285838i
$704$	5.55532			0.209374
$705$		0			0
$706$	−13.8564			−0.521493
$707$	8.39230	−	8.39230i	0.315625	−	0.315625i
$708$		0			0
$709$		−	24.7846i		−	0.930806i	−0.885099	−	0.465403i	$-0.845909\pi$
$709$		−	24.7846i		−	0.930806i	0.885099	−	0.465403i	$-0.154091\pi$
$710$		0			0
$711$		0			0
$712$	19.3185	+	19.3185i	0.723992	+	0.723992i
$713$	−31.8564	−	31.8564i	−1.19303	−	1.19303i
$714$		0			0
$715$		0			0
$716$		−	32.3238i		−	1.20800i
$717$		0			0
$718$	−4.27981	+	4.27981i	−0.159721	+	0.159721i
$719$	29.5969			1.10378			0.551890	−	0.833917i	$-0.313907\pi$
$719$	29.5969			1.10378			0.551890	+	0.833917i	$0.313907\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$	5.49038	−	5.49038i	0.204331	−	0.204331i
$723$		0			0
$724$		−	10.3923i		−	0.386227i
$725$		0			0
$726$		0			0
$727$	−8.28221	−	8.28221i	−0.307170	−	0.307170i	0.536641	−	0.843811i	$-0.319693\pi$
$727$	−8.28221	−	8.28221i	−0.307170	−	0.307170i	−0.843811	+	0.536641i	$0.819693\pi$
$728$	7.46410	+	7.46410i	0.276638	+	0.276638i
$729$		0			0
$730$		0			0
$731$			11.3137i			0.418453i
$732$		0			0
$733$	28.5617	−	28.5617i	1.05495	−	1.05495i	0.0565497	−	0.998400i	$-0.481990\pi$
$733$	28.5617	−	28.5617i	1.05495	−	1.05495i	0.998400	−	0.0565497i	$-0.0180099\pi$
$734$	10.7589			0.397118
$735$		0			0
$736$	−46.9808			−1.73173
$737$	−15.4641	+	15.4641i	−0.569628	+	0.569628i
$738$		0			0
$739$		−	9.85641i		−	0.362574i	−0.983430	−	0.181287i	$-0.941974\pi$
$739$		−	9.85641i		−	0.362574i	0.983430	−	0.181287i	$-0.0580263\pi$
$740$		0			0
$741$		0			0
$742$	1.17398	+	1.17398i	0.0430980	+	0.0430980i
$743$	8.60770	+	8.60770i	0.315786	+	0.315786i	0.847146	−	0.531360i	$-0.178319\pi$
$743$	8.60770	+	8.60770i	0.315786	+	0.315786i	−0.531360	+	0.847146i	$0.678319\pi$
$744$		0			0
$745$		0			0
$746$		−	14.4939i		−	0.530658i
$747$		0			0
$748$	8.48528	−	8.48528i	0.310253	−	0.310253i
$749$	−11.2122			−0.409684
$750$		0			0
$751$	50.1051			1.82836			0.914181	−	0.405307i	$-0.132835\pi$
$751$	50.1051			1.82836			0.914181	+	0.405307i	$0.132835\pi$
$752$	18.3923	−	18.3923i	0.670698	−	0.670698i
$753$		0			0
$754$		−	17.8564i		−	0.650292i
$755$		0			0
$756$		0			0
$757$	27.7023	+	27.7023i	1.00686	+	1.00686i	0.999976	+	0.00687950i	$0.00218983\pi$
$757$	27.7023	+	27.7023i	1.00686	+	1.00686i	0.00687950	+	0.999976i	$0.497810\pi$
$758$	−4.98076	−	4.98076i	−0.180909	−	0.180909i
$759$		0			0
$760$		0			0
$761$		−	26.5654i		−	0.962997i	−0.876447	−	0.481498i	$-0.840093\pi$
$761$		−	26.5654i		−	0.962997i	0.876447	−	0.481498i	$-0.159907\pi$
$762$		0			0
$763$	−0.757875	+	0.757875i	−0.0274369	+	0.0274369i
$764$	−11.7670			−0.425714
$765$		0			0
$766$	−5.75129			−0.207803
$767$	−48.7846	+	48.7846i	−1.76151	+	1.76151i
$768$		0			0
$769$		−	10.7846i		−	0.388903i	−0.980912	−	0.194451i	$-0.937707\pi$
$769$		−	10.7846i		−	0.388903i	0.980912	−	0.194451i	$-0.0622927\pi$
$770$		0			0
$771$		0			0
$772$	31.0112	+	31.0112i	1.11612	+	1.11612i
$773$	21.3205	+	21.3205i	0.766845	+	0.766845i	0.977550	−	0.210704i	$-0.0675758\pi$
$773$	21.3205	+	21.3205i	0.766845	+	0.766845i	−0.210704	+	0.977550i	$0.567576\pi$
$774$		0			0
$775$		0			0
$776$			29.5969i			1.06247i
$777$		0			0
$778$	−12.6636	+	12.6636i	−0.454010	+	0.454010i
$779$	4.14110			0.148370
$780$		0			0
$781$	−11.0718			−0.396180
$782$	−9.46410	+	9.46410i	−0.338436	+	0.338436i
$783$		0			0
$784$		−	2.46410i		−	0.0880036i
$785$		0			0
$786$		0			0
$787$	6.41473	+	6.41473i	0.228660	+	0.228660i	0.812133	−	0.583473i	$-0.198306\pi$
$787$	6.41473	+	6.41473i	0.228660	+	0.228660i	−0.583473	+	0.812133i	$0.698306\pi$
$788$	17.7846	+	17.7846i	0.633550	+	0.633550i
$789$		0			0
$790$		0			0
$791$			0.378937i			0.0134735i
$792$		0			0
$793$	−34.4959	+	34.4959i	−1.22499	+	1.22499i
$794$	0.757875			0.0268960
$795$		0			0
$796$	−36.2487			−1.28480
$797$	−27.1769	+	27.1769i	−0.962656	+	0.962656i	−0.999327	−	0.0366717i	$-0.988324\pi$
$797$	−27.1769	+	27.1769i	−0.962656	+	0.962656i	0.0366717	+	0.999327i	$0.488324\pi$
$798$		0			0
$799$		−	29.8564i		−	1.05624i
$800$		0			0
$801$		0			0
$802$	8.24504	+	8.24504i	0.291143	+	0.291143i
$803$	−2.53590	−	2.53590i	−0.0894899	−	0.0894899i
$804$		0			0
$805$		0			0
$806$			13.9391i			0.490983i
$807$		0			0
$808$	16.2127	−	16.2127i	0.570360	−	0.570360i
$809$	−24.0416			−0.845259			−0.422629	−	0.906303i	$-0.638893\pi$
$809$	−24.0416			−0.845259			−0.422629	+	0.906303i	$0.638893\pi$
$810$		0			0
$811$	−24.9282			−0.875348			−0.437674	−	0.899134i	$-0.644198\pi$
$811$	−24.9282			−0.875348			−0.437674	+	0.899134i	$0.644198\pi$
$812$	−7.73205	+	7.73205i	−0.271342	+	0.271342i
$813$		0			0
$814$			0.679492i			0.0238162i
$815$		0			0
$816$		0			0
$817$	−5.65685	−	5.65685i	−0.197908	−	0.197908i
$818$	−3.26795	−	3.26795i	−0.114261	−	0.114261i
$819$		0			0
$820$		0			0
$821$			11.9700i			0.417758i	0.977942	+	0.208879i	$0.0669814\pi$
$821$			11.9700i			0.417758i	−0.977942	+	0.208879i	$0.933019\pi$
$822$		0			0
$823$	15.5563	−	15.5563i	0.542260	−	0.542260i	−0.381931	−	0.924191i	$-0.624741\pi$
$823$	15.5563	−	15.5563i	0.542260	−	0.542260i	0.924191	+	0.381931i	$0.124741\pi$
$824$	23.1822			0.807591
$825$		0			0
$826$	−6.53590			−0.227413
$827$	−21.0000	+	21.0000i	−0.730242	+	0.730242i	−0.970667	−	0.240426i	$-0.922713\pi$
$827$	−21.0000	+	21.0000i	−0.730242	+	0.730242i	0.240426	+	0.970667i	$0.422713\pi$
$828$		0			0
$829$			30.7846i			1.06919i	0.845107	+	0.534597i	$0.179537\pi$
$829$			30.7846i			1.06919i	−0.845107	+	0.534597i	$0.820463\pi$
$830$		0			0
$831$		0			0
$832$	−8.76268	−	8.76268i	−0.303791	−	0.303791i
$833$	−2.00000	−	2.00000i	−0.0692959	−	0.0692959i
$834$		0			0
$835$		0			0
$836$			8.48528i			0.293470i
$837$		0			0
$838$	2.07055	−	2.07055i	0.0715260	−	0.0715260i
$839$	−35.8086			−1.23625			−0.618125	−	0.786080i	$-0.712108\pi$
$839$	−35.8086			−1.23625			−0.618125	+	0.786080i	$0.712108\pi$
$840$		0			0
$841$	10.8564			0.374359
$842$	−10.8756	+	10.8756i	−0.374799	+	0.374799i
$843$		0			0
$844$		−	0.928203i		−	0.0319501i
$845$		0			0
$846$		0			0
$847$	3.53553	+	3.53553i	0.121482	+	0.121482i
$848$	−5.58846	−	5.58846i	−0.191908	−	0.191908i
$849$		0			0
$850$		0			0
$851$			4.89898i			0.167935i
$852$		0			0
$853$	−27.0459	+	27.0459i	−0.926035	+	0.926035i	−0.997447	−	0.0714121i	$-0.977249\pi$
$853$	−27.0459	+	27.0459i	−0.926035	+	0.926035i	0.0714121	+	0.997447i	$0.477249\pi$
$854$	−4.62158			−0.158147
$855$		0			0
$856$	−21.6603			−0.740332
$857$	−10.3923	+	10.3923i	−0.354994	+	0.354994i	−0.861964	−	0.506970i	$-0.830766\pi$
$857$	−10.3923	+	10.3923i	−0.354994	+	0.354994i	0.506970	+	0.861964i	$0.330766\pi$
$858$		0			0
$859$			18.7846i			0.640923i	0.947262	+	0.320461i	$0.103838\pi$
$859$			18.7846i			0.640923i	−0.947262	+	0.320461i	$0.896162\pi$
$860$		0			0
$861$		0			0
$862$	−1.93185	−	1.93185i	−0.0657991	−	0.0657991i
$863$	−23.2487	−	23.2487i	−0.791395	−	0.791395i	0.190326	−	0.981721i	$-0.439046\pi$
$863$	−23.2487	−	23.2487i	−0.791395	−	0.791395i	−0.981721	+	0.190326i	$0.939046\pi$
$864$		0			0
$865$		0			0
$866$			7.52433i			0.255687i
$867$		0			0
$868$	6.03579	−	6.03579i	0.204868	−	0.204868i
$869$	−9.79796			−0.332373
$870$		0			0
$871$	48.7846			1.65300
$872$	−1.46410	+	1.46410i	−0.0495807	+	0.0495807i
$873$		0			0
$874$		−	9.46410i		−	0.320128i
$875$		0			0
$876$		0			0
$877$	−2.27362	−	2.27362i	−0.0767748	−	0.0767748i	0.667677	−	0.744451i	$-0.267289\pi$
$877$	−2.27362	−	2.27362i	−0.0767748	−	0.0767748i	−0.744451	+	0.667677i	$0.767289\pi$
$878$	−14.5885	−	14.5885i	−0.492337	−	0.492337i
$879$		0			0
$880$		0			0
$881$		−	33.9411i		−	1.14351i	−0.820426	−	0.571753i	$-0.806264\pi$
$881$		−	33.9411i		−	1.14351i	0.820426	−	0.571753i	$-0.193736\pi$
$882$		0			0
$883$	24.4949	−	24.4949i	0.824319	−	0.824319i	−0.162405	−	0.986724i	$-0.551925\pi$
$883$	24.4949	−	24.4949i	0.824319	−	0.824319i	0.986724	+	0.162405i	$0.0519252\pi$
$884$	−26.7685			−0.900323
$885$		0			0
$886$	8.73205			0.293359
$887$	−41.3205	+	41.3205i	−1.38741	+	1.38741i	−0.556680	+	0.830727i	$0.687925\pi$
$887$	−41.3205	+	41.3205i	−1.38741	+	1.38741i	−0.830727	+	0.556680i	$0.812075\pi$
$888$		0			0
$889$		−	7.85641i		−	0.263495i
$890$		0			0
$891$		0			0
$892$	20.5569	+	20.5569i	0.688295	+	0.688295i
$893$	14.9282	+	14.9282i	0.499553	+	0.499553i
$894$		0			0
$895$		0			0
$896$		−	11.4524i		−	0.382598i
$897$		0			0
$898$	8.24504	−	8.24504i	0.275141	−	0.275141i
$899$	−31.1127			−1.03767
$900$		0			0
$901$	−9.07180			−0.302225
$902$	−1.85641	+	1.85641i	−0.0618116	+	0.0618116i
$903$		0			0
$904$			0.732051i			0.0243476i
$905$		0			0
$906$		0			0
$907$	12.0716	+	12.0716i	0.400830	+	0.400830i	0.878526	−	0.477695i	$-0.158528\pi$
$907$	12.0716	+	12.0716i	0.400830	+	0.400830i	−0.477695	+	0.878526i	$0.658528\pi$
$908$	28.3923	+	28.3923i	0.942232	+	0.942232i
$909$		0			0
$910$		0			0
$911$			7.55154i			0.250194i	0.992145	+	0.125097i	$0.0399242\pi$
$911$			7.55154i			0.250194i	−0.992145	+	0.125097i	$0.960076\pi$
$912$		0			0
$913$	−11.1106	+	11.1106i	−0.367708	+	0.367708i
$914$	−15.7322			−0.520375
$915$		0			0
$916$	43.1769			1.42661
$917$	9.46410	−	9.46410i	0.312532	−	0.312532i
$918$		0			0
$919$			51.1769i			1.68817i	0.536209	+	0.844085i	$0.319856\pi$
$919$			51.1769i			1.68817i	−0.536209	+	0.844085i	$0.680144\pi$
$920$		0			0
$921$		0			0
$922$	−14.4195	−	14.4195i	−0.474882	−	0.474882i
$923$	17.4641	+	17.4641i	0.574838	+	0.574838i
$924$		0			0
$925$		0			0
$926$		−	18.5606i		−	0.609941i
$927$		0			0
$928$	−22.9420	+	22.9420i	−0.753107	+	0.753107i
$929$	23.9401			0.785449			0.392725	−	0.919656i	$-0.371533\pi$
$929$	23.9401			0.785449			0.392725	+	0.919656i	$0.371533\pi$
$930$		0			0
$931$	2.00000			0.0655474
$932$	0.464102	−	0.464102i	0.0152022	−	0.0152022i
$933$		0			0
$934$			4.78461i			0.156557i
$935$		0			0
$936$		0			0
$937$	22.3500	+	22.3500i	0.730143	+	0.730143i	0.970648	−	0.240505i	$-0.0773130\pi$
$937$	22.3500	+	22.3500i	0.730143	+	0.730143i	−0.240505	+	0.970648i	$0.577313\pi$
$938$	3.26795	+	3.26795i	0.106702	+	0.106702i
$939$		0			0
$940$		0			0
$941$		−	21.1117i		−	0.688221i	−0.938929	−	0.344110i	$-0.888181\pi$
$941$		−	21.1117i		−	0.688221i	0.938929	−	0.344110i	$-0.111819\pi$
$942$		0			0
$943$	−13.3843	+	13.3843i	−0.435852	+	0.435852i
$944$	31.1127			1.01263
$945$		0			0
$946$	5.07180			0.164898
$947$	−31.0000	+	31.0000i	−1.00736	+	1.00736i	−0.00739197	+	0.999973i	$0.502353\pi$
$947$	−31.0000	+	31.0000i	−1.00736	+	1.00736i	−0.999973	+	0.00739197i	$0.997647\pi$
$948$		0			0
$949$			8.00000i			0.259691i
$950$		0			0
$951$		0			0
$952$	−3.86370	−	3.86370i	−0.125223	−	0.125223i
$953$	−38.6603	−	38.6603i	−1.25233	−	1.25233i	−0.954674	−	0.297655i	$-0.903796\pi$
$953$	−38.6603	−	38.6603i	−1.25233	−	1.25233i	−0.297655	−	0.954674i	$-0.596204\pi$
$954$		0			0
$955$		0			0
$956$			30.0502i			0.971893i
$957$		0			0
$958$	−3.93803	+	3.93803i	−0.127232	+	0.127232i
$959$	10.1769			0.328629
$960$		0			0
$961$	−6.71281			−0.216542
$962$	1.07180	−	1.07180i	0.0345561	−	0.0345561i
$963$		0			0
$964$			22.3923i			0.721208i
$965$		0			0
$966$		0			0
$967$	38.7386	+	38.7386i	1.24575	+	1.24575i	0.957580	+	0.288169i	$0.0930464\pi$
$967$	38.7386	+	38.7386i	1.24575	+	1.24575i	0.288169	+	0.957580i	$0.406954\pi$
$968$	6.83013	+	6.83013i	0.219529	+	0.219529i
$969$		0			0
$970$		0			0
$971$		−	10.5558i		−	0.338753i	−0.985551	−	0.169376i	$-0.945825\pi$
$971$		−	10.5558i		−	0.338753i	0.985551	−	0.169376i	$-0.0541754\pi$
$972$		0			0
$973$	8.38375	−	8.38375i	0.268771	−	0.268771i
$974$	13.9391			0.446636
$975$		0			0
$976$	22.0000			0.704203
$977$	15.7321	−	15.7321i	0.503313	−	0.503313i	−0.409153	−	0.912466i	$-0.634176\pi$
$977$	15.7321	−	15.7321i	0.503313	−	0.503313i	0.912466	+	0.409153i	$0.134176\pi$
$978$		0			0
$979$			34.6410i			1.10713i
$980$		0			0
$981$		0			0
$982$	−11.9329	−	11.9329i	−0.380794	−	0.380794i
$983$	23.3205	+	23.3205i	0.743809	+	0.743809i	0.973309	−	0.229500i	$-0.0737091\pi$
$983$	23.3205	+	23.3205i	0.743809	+	0.743809i	−0.229500	+	0.973309i	$0.573709\pi$
$984$		0			0
$985$		0			0
$986$			9.24316i			0.294362i
$987$		0			0
$988$	13.3843	−	13.3843i	0.425810	−	0.425810i
$989$	36.5665			1.16275
$990$		0			0
$991$	−13.3205			−0.423140			−0.211570	−	0.977363i	$-0.567858\pi$
$991$	−13.3205			−0.423140			−0.211570	+	0.977363i	$0.567858\pi$
$992$	17.9090	−	17.9090i	0.568610	−	0.568610i
$993$		0			0
$994$			2.33975i			0.0742122i
$995$		0			0
$996$		0			0
$997$	−13.8647	−	13.8647i	−0.439101	−	0.439101i	0.452609	−	0.891709i	$-0.350493\pi$
$997$	−13.8647	−	13.8647i	−0.439101	−	0.439101i	−0.891709	+	0.452609i	$0.850493\pi$
$998$	6.33975	+	6.33975i	0.200681	+	0.200681i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.m.a.1457.4	yes	8
3.2	odd	2		1575.2.m.b.1457.2	yes	8
5.2	odd	4	inner	1575.2.m.a.1268.3	✓	8
5.3	odd	4		1575.2.m.b.1268.2	yes	8
5.4	even	2		1575.2.m.b.1457.1	yes	8
15.2	even	4		1575.2.m.b.1268.1	yes	8
15.8	even	4	inner	1575.2.m.a.1268.4	yes	8
15.14	odd	2	inner	1575.2.m.a.1457.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.m.a.1268.3	✓	8	5.2	odd	4	inner
1575.2.m.a.1268.4	yes	8	15.8	even	4	inner
1575.2.m.a.1457.3	yes	8	15.14	odd	2	inner
1575.2.m.a.1457.4	yes	8	1.1	even	1	trivial
1575.2.m.b.1268.1	yes	8	15.2	even	4
1575.2.m.b.1268.2	yes	8	5.3	odd	4
1575.2.m.b.1457.1	yes	8	5.4	even	2
1575.2.m.b.1457.2	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.366025 - 0.366025i) q^{2} +1.73205i q^{4} +(0.707107 + 0.707107i) q^{7} +(1.36603 + 1.36603i) q^{8} +O(q^{10})\)	\(q+(0.366025 - 0.366025i) q^{2} +1.73205i q^{4} +(0.707107 + 0.707107i) q^{7} +(1.36603 + 1.36603i) q^{8} +2.44949i q^{11} +(3.86370 - 3.86370i) q^{13} +0.517638 q^{14} -2.46410 q^{16} +(-2.00000 + 2.00000i) q^{17} -2.00000i q^{19} +(0.896575 + 0.896575i) q^{22} +(6.46410 + 6.46410i) q^{23} -2.82843i q^{26} +(-1.22474 + 1.22474i) q^{28} +6.31319 q^{29} -4.92820 q^{31} +(-3.63397 + 3.63397i) q^{32} +1.46410i q^{34} +(0.378937 + 0.378937i) q^{37} +(-0.732051 - 0.732051i) q^{38} +2.07055i q^{41} +(2.82843 - 2.82843i) q^{43} -4.24264 q^{44} +4.73205 q^{46} +(-7.46410 + 7.46410i) q^{47} +1.00000i q^{49} +(6.69213 + 6.69213i) q^{52} +(2.26795 + 2.26795i) q^{53} +1.93185i q^{56} +(2.31079 - 2.31079i) q^{58} -12.6264 q^{59} -8.92820 q^{61} +(-1.80385 + 1.80385i) q^{62} -2.26795i q^{64} +(6.31319 + 6.31319i) q^{67} +(-3.46410 - 3.46410i) q^{68} +4.52004i q^{71} +(-1.03528 + 1.03528i) q^{73} +0.277401 q^{74} +3.46410 q^{76} +(-1.73205 + 1.73205i) q^{77} +4.00000i q^{79} +(0.757875 + 0.757875i) q^{82} +(4.53590 + 4.53590i) q^{83} -2.07055i q^{86} +(-3.34607 + 3.34607i) q^{88} +14.1421 q^{89} +5.46410 q^{91} +(-11.1962 + 11.1962i) q^{92} +5.46410i q^{94} +(10.8332 + 10.8332i) q^{97} +(0.366025 + 0.366025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 4 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 4 * q^8	\( 8 q - 4 q^{2} + 4 q^{8} + 8 q^{16} - 16 q^{17} + 24 q^{23} + 16 q^{31} - 36 q^{32} + 8 q^{38} + 24 q^{46} - 32 q^{47} + 32 q^{53} - 16 q^{61} - 56 q^{62} + 64 q^{83} + 16 q^{91} - 48 q^{92} - 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 4 * q^8 + 8 * q^16 - 16 * q^17 + 24 * q^23 + 16 * q^31 - 36 * q^32 + 8 * q^38 + 24 * q^46 - 32 * q^47 + 32 * q^53 - 16 * q^61 - 56 * q^62 + 64 * q^83 + 16 * q^91 - 48 * q^92 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more