LMFDB - Embedded newform 3528.2.s.bc.3313.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$, degree $2$, not 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:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3528.3313
Dual form			3528.2.s.bc.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70711 + 2.95680i) q^{5} +(0.414214 + 0.717439i) q^{11} +4.24264 q^{13} +(-3.70711 - 6.42090i) q^{17} +(3.41421 - 5.91359i) q^{19} +(-2.41421 + 4.18154i) q^{23} +(-3.32843 - 5.76500i) q^{25} -2.82843 q^{29} +(-1.41421 - 2.44949i) q^{31} +(0.828427 - 1.43488i) q^{37} +10.2426 q^{41} -11.3137 q^{43} +(2.24264 - 3.88437i) q^{47} +(-1.00000 - 1.73205i) q^{53} -2.82843 q^{55} +(-4.24264 - 7.34847i) q^{59} +(5.53553 - 9.58783i) q^{61} +(-7.24264 + 12.5446i) q^{65} +(-5.65685 - 9.79796i) q^{67} +10.4853 q^{71} +(3.87868 + 6.71807i) q^{73} +(-6.82843 + 11.8272i) q^{79} -4.00000 q^{83} +25.3137 q^{85} +(-2.87868 + 4.98602i) q^{89} +(11.6569 + 20.1903i) q^{95} +0.242641 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{11} - 12 q^{17} + 8 q^{19} - 4 q^{23} - 2 q^{25} - 8 q^{37} + 24 q^{41} - 8 q^{47} - 4 q^{53} + 8 q^{61} - 12 q^{65} + 8 q^{71} + 24 q^{73} - 16 q^{79} - 16 q^{83} + 56 q^{85} - 20 q^{89}+ \cdots - 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^11 - 12 * q^17 + 8 * q^19 - 4 * q^23 - 2 * q^25 - 8 * q^37 + 24 * q^41 - 8 * q^47 - 4 * q^53 + 8 * q^61 - 12 * q^65 + 8 * q^71 + 24 * q^73 - 16 * q^79 - 16 * q^83 + 56 * q^85 - 20 * q^89 + 24 * q^95 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.70711	+	2.95680i	−0.763441	+	1.32232i	0.177625	+	0.984098i	$0.443158\pi$
$5$	−1.70711	+	2.95680i	−0.763441	+	1.32232i	−0.941067	+	0.338221i	$0.890175\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.414214	+	0.717439i	0.124890	+	0.216316i	0.921690	−	0.387927i	$-0.126809\pi$
$11$	0.414214	+	0.717439i	0.124890	+	0.216316i	−0.796800	+	0.604243i	$0.793476\pi$
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.70711	−	6.42090i	−0.899105	−	1.55730i	−0.828640	−	0.559782i	$-0.810885\pi$
$17$	−3.70711	−	6.42090i	−0.899105	−	1.55730i	−0.0704656	−	0.997514i	$-0.522449\pi$
$18$		0			0
$19$	3.41421	−	5.91359i	0.783274	−	1.35667i	−0.146750	−	0.989174i	$-0.546881\pi$
$19$	3.41421	−	5.91359i	0.783274	−	1.35667i	0.930025	−	0.367497i	$-0.119785\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.41421	+	4.18154i	−0.503398	+	0.871911i	0.496594	+	0.867983i	$0.334584\pi$
$23$	−2.41421	+	4.18154i	−0.503398	+	0.871911i	−0.999992	+	0.00392850i	$0.998750\pi$
$24$		0			0
$25$	−3.32843	−	5.76500i	−0.665685	−	1.15300i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.82843			−0.525226			−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843			−0.525226			−0.262613	+	0.964901i	$0.584584\pi$
$30$		0			0
$31$	−1.41421	−	2.44949i	−0.254000	−	0.439941i	0.710623	−	0.703573i	$-0.248413\pi$
$31$	−1.41421	−	2.44949i	−0.254000	−	0.439941i	−0.964623	+	0.263631i	$0.915080\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.828427	−	1.43488i	0.136193	−	0.235892i	−0.789860	−	0.613287i	$-0.789847\pi$
$37$	0.828427	−	1.43488i	0.136193	−	0.235892i	0.926052	+	0.377395i	$0.123180\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.2426			1.59963			0.799816	−	0.600245i	$-0.204930\pi$
$41$	10.2426			1.59963			0.799816	+	0.600245i	$0.204930\pi$
$42$		0			0
$43$	−11.3137			−1.72532			−0.862662	−	0.505781i	$-0.831205\pi$
$43$	−11.3137			−1.72532			−0.862662	+	0.505781i	$0.831205\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.24264	−	3.88437i	0.327123	−	0.566593i	−0.654817	−	0.755788i	$-0.727254\pi$
$47$	2.24264	−	3.88437i	0.327123	−	0.566593i	0.981940	+	0.189194i	$0.0605876\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.00000	−	1.73205i	−0.137361	−	0.237915i	0.789136	−	0.614218i	$-0.210529\pi$
$53$	−1.00000	−	1.73205i	−0.137361	−	0.237915i	−0.926497	+	0.376303i	$0.877195\pi$
$54$		0			0
$55$	−2.82843			−0.381385
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.24264	−	7.34847i	−0.552345	−	0.956689i	−0.998105	−	0.0615367i	$-0.980400\pi$
$59$	−4.24264	−	7.34847i	−0.552345	−	0.956689i	0.445760	−	0.895152i	$-0.352933\pi$
$60$		0			0
$61$	5.53553	−	9.58783i	0.708752	−	1.22760i	−0.256568	−	0.966526i	$-0.582592\pi$
$61$	5.53553	−	9.58783i	0.708752	−	1.22760i	0.965320	−	0.261069i	$-0.0840750\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.24264	+	12.5446i	−0.898339	+	1.55597i
$66$		0			0
$67$	−5.65685	−	9.79796i	−0.691095	−	1.19701i	−0.971480	−	0.237124i	$-0.923795\pi$
$67$	−5.65685	−	9.79796i	−0.691095	−	1.19701i	0.280385	−	0.959888i	$-0.409538\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.4853			1.24437			0.622187	−	0.782869i	$-0.286244\pi$
$71$	10.4853			1.24437			0.622187	+	0.782869i	$0.286244\pi$
$72$		0			0
$73$	3.87868	+	6.71807i	0.453965	+	0.786291i	0.998628	−	0.0523644i	$-0.0166757\pi$
$73$	3.87868	+	6.71807i	0.453965	+	0.786291i	−0.544663	+	0.838655i	$0.683342\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.82843	+	11.8272i	−0.768258	+	1.33066i	0.170249	+	0.985401i	$0.445543\pi$
$79$	−6.82843	+	11.8272i	−0.768258	+	1.33066i	−0.938507	+	0.345261i	$0.887790\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	25.3137			2.74566
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.87868	+	4.98602i	−0.305139	+	0.528517i	−0.977292	−	0.211895i	$-0.932036\pi$
$89$	−2.87868	+	4.98602i	−0.305139	+	0.528517i	0.672153	+	0.740412i	$0.265370\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.6569	+	20.1903i	1.19597	+	2.07148i
$96$		0			0
$97$	0.242641			0.0246364			0.0123182	−	0.999924i	$-0.496079\pi$
$97$	0.242641			0.0246364			0.0123182	+	0.999924i	$0.496079\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.36396	+	9.29065i	0.533734	+	0.924455i	0.999223	+	0.0394011i	$0.0125450\pi$
$101$	5.36396	+	9.29065i	0.533734	+	0.924455i	−0.465489	+	0.885053i	$0.654122\pi$
$102$		0			0
$103$	7.07107	−	12.2474i	0.696733	−	1.20678i	−0.272860	−	0.962054i	$-0.587970\pi$
$103$	7.07107	−	12.2474i	0.696733	−	1.20678i	0.969593	−	0.244723i	$-0.0786971\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.41421	−	11.1097i	0.620085	−	1.07402i	−0.369384	−	0.929277i	$-0.620431\pi$
$107$	6.41421	−	11.1097i	0.620085	−	1.07402i	0.989469	−	0.144743i	$-0.0462354\pi$
$108$		0			0
$109$	1.65685	+	2.86976i	0.158698	+	0.274873i	0.934399	−	0.356227i	$-0.115937\pi$
$109$	1.65685	+	2.86976i	0.158698	+	0.274873i	−0.775702	+	0.631100i	$0.782604\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.0000			0.940721			0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000			0.940721			0.470360	+	0.882474i	$0.344124\pi$
$114$		0			0
$115$	−8.24264	−	14.2767i	−0.768630	−	1.33131i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.15685	−	8.93193i	0.468805	−	0.811994i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.00000	−	3.46410i	0.174741	−	0.302660i	−0.765331	−	0.643637i	$-0.777425\pi$
$131$	2.00000	−	3.46410i	0.174741	−	0.302660i	0.940072	+	0.340977i	$0.110758\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.58579	−	7.94282i	−0.391790	−	0.678600i	0.600896	−	0.799328i	$-0.294811\pi$
$137$	−4.58579	−	7.94282i	−0.391790	−	0.678600i	−0.992686	+	0.120727i	$0.961477\pi$
$138$		0			0
$139$	20.9706			1.77870			0.889350	−	0.457227i	$-0.151157\pi$
$139$	20.9706			1.77870			0.889350	+	0.457227i	$0.151157\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.75736	+	3.04384i	0.146958	+	0.254538i
$144$		0			0
$145$	4.82843	−	8.36308i	0.400979	−	0.694516i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.656854	−	1.13770i	0.0538116	−	0.0932044i	−0.837865	−	0.545878i	$-0.816196\pi$
$149$	0.656854	−	1.13770i	0.0538116	−	0.0932044i	0.891676	+	0.452673i	$0.149530\pi$
$150$		0			0
$151$	4.82843	+	8.36308i	0.392932	+	0.680578i	0.992835	−	0.119495i	$-0.0381275\pi$
$151$	4.82843	+	8.36308i	0.392932	+	0.680578i	−0.599903	+	0.800073i	$0.704794\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.65685			0.775657
$156$		0			0
$157$	0.121320	+	0.210133i	0.00968242	+	0.0167704i	0.870826	−	0.491591i	$-0.163585\pi$
$157$	0.121320	+	0.210133i	0.00968242	+	0.0167704i	−0.861144	+	0.508362i	$0.830251\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.82843	+	4.89898i	−0.221540	+	0.383718i	−0.955276	−	0.295717i	$-0.904442\pi$
$163$	−2.82843	+	4.89898i	−0.221540	+	0.383718i	0.733736	+	0.679435i	$0.237775\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.8284			−1.14746			−0.573729	−	0.819045i	$-0.694504\pi$
$167$	−14.8284			−1.14746			−0.573729	+	0.819045i	$0.694504\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.29289	−	14.3637i	0.630497	−	1.09205i	−0.356953	−	0.934122i	$-0.616184\pi$
$173$	8.29289	−	14.3637i	0.630497	−	1.09205i	0.987450	−	0.157931i	$-0.0504822\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.24264	−	5.61642i	−0.242366	−	0.419791i	0.719022	−	0.694988i	$-0.244590\pi$
$179$	−3.24264	−	5.61642i	−0.242366	−	0.419791i	−0.961388	+	0.275197i	$0.911257\pi$
$180$		0			0
$181$	7.07107			0.525588			0.262794	−	0.964852i	$-0.415356\pi$
$181$	7.07107			0.525588			0.262794	+	0.964852i	$0.415356\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.82843	+	4.89898i	0.207950	+	0.360180i
$186$		0			0
$187$	3.07107	−	5.31925i	0.224579	−	0.388982i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.07107	−	10.5154i	0.439287	−	0.760867i	−0.558348	−	0.829607i	$-0.688564\pi$
$191$	6.07107	−	10.5154i	0.439287	−	0.760867i	0.997635	+	0.0687396i	$0.0218978\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.68629			0.191390			0.0956952	−	0.995411i	$-0.469493\pi$
$197$	2.68629			0.191390			0.0956952	+	0.995411i	$0.469493\pi$
$198$		0			0
$199$	−2.82843	−	4.89898i	−0.200502	−	0.347279i	0.748188	−	0.663486i	$-0.230924\pi$
$199$	−2.82843	−	4.89898i	−0.200502	−	0.347279i	−0.948690	+	0.316207i	$0.897591\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−17.4853	+	30.2854i	−1.22123	+	2.11522i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.65685			0.391293
$210$		0			0
$211$	1.65685			0.114063			0.0570313	−	0.998372i	$-0.481837\pi$
$211$	1.65685			0.114063			0.0570313	+	0.998372i	$0.481837\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.3137	−	33.4523i	1.31718	−	2.28143i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.7279	−	27.2416i	−1.05797	−	1.83247i
$222$		0			0
$223$	−13.6569			−0.914531			−0.457265	−	0.889330i	$-0.651171\pi$
$223$	−13.6569			−0.914531			−0.457265	+	0.889330i	$0.651171\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.58579	+	4.47871i	0.171625	+	0.297263i	0.938988	−	0.343950i	$-0.111765\pi$
$227$	2.58579	+	4.47871i	0.171625	+	0.297263i	−0.767363	+	0.641213i	$0.778432\pi$
$228$		0			0
$229$	−12.7071	+	22.0094i	−0.839709	+	1.45442i	0.0504286	+	0.998728i	$0.483941\pi$
$229$	−12.7071	+	22.0094i	−0.839709	+	1.45442i	−0.890138	+	0.455691i	$0.849392\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.41421	+	12.8418i	−0.485721	+	0.841294i	−0.999865	−	0.0164099i	$-0.994776\pi$
$233$	−7.41421	+	12.8418i	−0.485721	+	0.841294i	0.514144	+	0.857704i	$0.328110\pi$
$234$		0			0
$235$	7.65685	+	13.2621i	0.499478	+	0.865121i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.8284			0.829802			0.414901	−	0.909867i	$-0.363816\pi$
$239$	12.8284			0.829802			0.414901	+	0.909867i	$0.363816\pi$
$240$		0			0
$241$	6.94975	+	12.0373i	0.447673	+	0.775392i	0.998234	−	0.0594029i	$-0.0189197\pi$
$241$	6.94975	+	12.0373i	0.447673	+	0.775392i	−0.550561	+	0.834795i	$0.685586\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	14.4853	−	25.0892i	0.921676	−	1.59639i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.14214			−0.387688			−0.193844	−	0.981032i	$-0.562096\pi$
$251$	−6.14214			−0.387688			−0.193844	+	0.981032i	$0.562096\pi$
$252$		0			0
$253$	−4.00000			−0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.70711	−	2.95680i	0.106486	−	0.184440i	−0.807858	−	0.589377i	$-0.799373\pi$
$257$	1.70711	−	2.95680i	0.106486	−	0.184440i	0.914345	+	0.404937i	$0.132707\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.0711	−	17.4436i	−0.621009	−	1.07562i	−0.989298	−	0.145907i	$-0.953390\pi$
$263$	−10.0711	−	17.4436i	−0.621009	−	1.07562i	0.368290	−	0.929711i	$-0.379943\pi$
$264$		0			0
$265$	6.82843			0.419467
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.0502525	+	0.0870399i	0.00306395	+	0.00530692i	0.867553	−	0.497344i	$-0.165691\pi$
$269$	0.0502525	+	0.0870399i	0.00306395	+	0.00530692i	−0.864489	+	0.502651i	$0.832358\pi$
$270$		0			0
$271$	3.07107	−	5.31925i	0.186554	−	0.323121i	−0.757545	−	0.652783i	$-0.773601\pi$
$271$	3.07107	−	5.31925i	0.186554	−	0.323121i	0.944099	+	0.329662i	$0.106935\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.75736	−	4.77589i	0.166275	−	0.287997i
$276$		0			0
$277$	14.3137	+	24.7921i	0.860027	+	1.48961i	0.871901	+	0.489682i	$0.162887\pi$
$277$	14.3137	+	24.7921i	0.860027	+	1.48961i	−0.0118739	+	0.999930i	$0.503780\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.17157			−0.0698902			−0.0349451	−	0.999389i	$-0.511126\pi$
$281$	−1.17157			−0.0698902			−0.0349451	+	0.999389i	$0.511126\pi$
$282$		0			0
$283$	−13.0711	−	22.6398i	−0.776994	−	1.34579i	−0.933667	−	0.358143i	$-0.883410\pi$
$283$	−13.0711	−	22.6398i	−0.776994	−	1.34579i	0.156672	−	0.987651i	$-0.449923\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−18.9853	+	32.8835i	−1.11678	+	1.93432i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.75736			−0.102666			−0.0513330	−	0.998682i	$-0.516347\pi$
$293$	−1.75736			−0.102666			−0.0513330	+	0.998682i	$0.516347\pi$
$294$		0			0
$295$	28.9706			1.68673
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.2426	+	17.7408i	−0.592347	+	1.02598i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.8995	+	32.7349i	1.08218	+	1.87439i
$306$		0			0
$307$	11.5147			0.657180			0.328590	−	0.944473i	$-0.393427\pi$
$307$	11.5147			0.657180			0.328590	+	0.944473i	$0.393427\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.8995	−	20.6105i	−0.674758	−	1.16872i	−0.976539	−	0.215339i	$-0.930914\pi$
$311$	−11.8995	−	20.6105i	−0.674758	−	1.16872i	0.301781	−	0.953377i	$-0.402419\pi$
$312$		0			0
$313$	14.3640	−	24.8791i	0.811899	−	1.40625i	−0.0996342	−	0.995024i	$-0.531767\pi$
$313$	14.3640	−	24.8791i	0.811899	−	1.40625i	0.911533	−	0.411226i	$-0.134899\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.0000	+	19.0526i	−0.617822	+	1.07010i	0.372061	+	0.928208i	$0.378651\pi$
$317$	−11.0000	+	19.0526i	−0.617822	+	1.07010i	−0.989882	+	0.141890i	$0.954682\pi$
$318$		0			0
$319$	−1.17157	−	2.02922i	−0.0655955	−	0.113615i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−50.6274			−2.81698
$324$		0			0
$325$	−14.1213	−	24.4588i	−0.783310	−	1.35673i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.65685	+	6.33386i	−0.200999	+	0.348140i	−0.948851	−	0.315726i	$-0.897752\pi$
$331$	−3.65685	+	6.33386i	−0.200999	+	0.348140i	0.747852	+	0.663866i	$0.231085\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	38.6274			2.11044
$336$		0			0
$337$	10.3431			0.563427			0.281714	−	0.959499i	$-0.409097\pi$
$337$	10.3431			0.563427			0.281714	+	0.959499i	$0.409097\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.17157	−	2.02922i	0.0634442	−	0.109889i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.24264	−	2.15232i	−0.0667084	−	0.115542i	0.830742	−	0.556657i	$-0.187916\pi$
$347$	−1.24264	−	2.15232i	−0.0667084	−	0.115542i	−0.897451	+	0.441115i	$0.854583\pi$
$348$		0			0
$349$	−10.5858			−0.566644			−0.283322	−	0.959025i	$-0.591437\pi$
$349$	−10.5858			−0.566644			−0.283322	+	0.959025i	$0.591437\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.8492	+	23.9876i	0.737121	+	1.27673i	0.953787	+	0.300485i	$0.0971485\pi$
$353$	13.8492	+	23.9876i	0.737121	+	1.27673i	−0.216666	+	0.976246i	$0.569518\pi$
$354$		0			0
$355$	−17.8995	+	31.0028i	−0.950007	+	1.64546i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.4142	−	24.9662i	0.760753	−	1.31766i	−0.181710	−	0.983352i	$-0.558163\pi$
$359$	14.4142	−	24.9662i	0.760753	−	1.31766i	0.942463	−	0.334311i	$-0.108504\pi$
$360$		0			0
$361$	−13.8137	−	23.9260i	−0.727037	−	1.25927i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.4853			−1.38630
$366$		0			0
$367$	2.34315	+	4.05845i	0.122311	+	0.211849i	0.920679	−	0.390321i	$-0.127636\pi$
$367$	2.34315	+	4.05845i	0.122311	+	0.211849i	−0.798368	+	0.602171i	$0.794303\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.65685	+	4.60181i	−0.137567	+	0.238273i	−0.926575	−	0.376110i	$-0.877261\pi$
$373$	−2.65685	+	4.60181i	−0.137567	+	0.238273i	0.789008	+	0.614383i	$0.210595\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	−23.3137			−1.19754			−0.598772	−	0.800919i	$-0.704345\pi$
$379$	−23.3137			−1.19754			−0.598772	+	0.800919i	$0.704345\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.48528	−	7.76874i	0.229187	−	0.396964i	−0.728380	−	0.685173i	$-0.759727\pi$
$383$	4.48528	−	7.76874i	0.229187	−	0.396964i	0.957567	+	0.288209i	$0.0930599\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.8995	−	31.0028i	−0.907540	−	1.57191i	−0.817470	−	0.575971i	$-0.804624\pi$
$389$	−17.8995	−	31.0028i	−0.907540	−	1.57191i	−0.0900701	−	0.995935i	$-0.528709\pi$
$390$		0			0
$391$	35.7990			1.81043
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−23.3137	−	40.3805i	−1.17304	−	2.03176i
$396$		0			0
$397$	−8.12132	+	14.0665i	−0.407597	+	0.705979i	−0.994620	−	0.103591i	$-0.966967\pi$
$397$	−8.12132	+	14.0665i	−0.407597	+	0.705979i	0.587023	+	0.809571i	$0.300300\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.585786	−	1.01461i	0.0292528	−	0.0506673i	−0.851028	−	0.525120i	$-0.824021\pi$
$401$	0.585786	−	1.01461i	0.0292528	−	0.0506673i	0.880281	+	0.474452i	$0.157354\pi$
$402$		0			0
$403$	−6.00000	−	10.3923i	−0.298881	−	0.517678i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.37258			0.0680364
$408$		0			0
$409$	−4.12132	−	7.13834i	−0.203786	−	0.352968i	0.745959	−	0.665992i	$-0.231991\pi$
$409$	−4.12132	−	7.13834i	−0.203786	−	0.352968i	−0.949745	+	0.313024i	$0.898658\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6.82843	−	11.8272i	0.335194	−	0.580574i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.51472			0.367118			0.183559	−	0.983009i	$-0.441238\pi$
$419$	7.51472			0.367118			0.183559	+	0.983009i	$0.441238\pi$
$420$		0			0
$421$	−25.3137			−1.23371			−0.616857	−	0.787075i	$-0.711594\pi$
$421$	−25.3137			−1.23371			−0.616857	+	0.787075i	$0.711594\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.6777	+	42.7430i	−1.19704	+	2.07334i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.07107	−	7.05130i	−0.196096	−	0.339649i	0.751163	−	0.660117i	$-0.229493\pi$
$431$	−4.07107	−	7.05130i	−0.196096	−	0.339649i	−0.947259	+	0.320468i	$0.896160\pi$
$432$		0			0
$433$	−0.928932			−0.0446416			−0.0223208	−	0.999751i	$-0.507106\pi$
$433$	−0.928932			−0.0446416			−0.0223208	+	0.999751i	$0.507106\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.4853	+	28.5533i	0.788598	+	1.36589i
$438$		0			0
$439$	6.34315	−	10.9867i	0.302742	−	0.524364i	−0.674014	−	0.738718i	$-0.735431\pi$
$439$	6.34315	−	10.9867i	0.302742	−	0.524364i	0.976756	+	0.214354i	$0.0687647\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.58579	+	9.67487i	−0.265389	+	0.459667i	−0.967665	−	0.252237i	$-0.918834\pi$
$443$	−5.58579	+	9.67487i	−0.265389	+	0.459667i	0.702277	+	0.711904i	$0.252167\pi$
$444$		0			0
$445$	−9.82843	−	17.0233i	−0.465912	−	0.806983i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16.6274			0.784696			0.392348	−	0.919817i	$-0.371663\pi$
$449$	16.6274			0.784696			0.392348	+	0.919817i	$0.371663\pi$
$450$		0			0
$451$	4.24264	+	7.34847i	0.199778	+	0.346026i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	12.3137	−	21.3280i	0.576011	−	0.997680i	−0.419920	−	0.907561i	$-0.637942\pi$
$457$	12.3137	−	21.3280i	0.576011	−	0.997680i	0.995931	−	0.0901192i	$-0.0287248\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.3848			0.763115			0.381558	−	0.924345i	$-0.375388\pi$
$461$	16.3848			0.763115			0.381558	+	0.924345i	$0.375388\pi$
$462$		0			0
$463$	1.65685			0.0770005			0.0385003	−	0.999259i	$-0.487742\pi$
$463$	1.65685			0.0770005			0.0385003	+	0.999259i	$0.487742\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.41421	−	16.3059i	0.435638	−	0.754547i	−0.561710	−	0.827334i	$-0.689856\pi$
$467$	9.41421	−	16.3059i	0.435638	−	0.754547i	0.997347	+	0.0727876i	$0.0231895\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.68629	−	8.11689i	−0.215476	−	0.373215i
$474$		0			0
$475$	−45.4558			−2.08566
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.4142	−	19.7700i	−0.521529	−	0.903314i	−0.999686	−	0.0250403i	$-0.992029\pi$
$479$	−11.4142	−	19.7700i	−0.521529	−	0.903314i	0.478158	−	0.878274i	$-0.341305\pi$
$480$		0			0
$481$	3.51472	−	6.08767i	0.160257	−	0.277574i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.414214	+	0.717439i	−0.0188085	+	0.0325772i
$486$		0			0
$487$	−2.48528	−	4.30463i	−0.112619	−	0.195062i	0.804207	−	0.594350i	$-0.202591\pi$
$487$	−2.48528	−	4.30463i	−0.112619	−	0.195062i	−0.916825	+	0.399288i	$0.869257\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−19.1716			−0.865201			−0.432600	−	0.901586i	$-0.642404\pi$
$491$	−19.1716			−0.865201			−0.432600	+	0.901586i	$0.642404\pi$
$492$		0			0
$493$	10.4853	+	18.1610i	0.472233	+	0.817932i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.4853	−	21.6251i	0.558918	−	0.968074i	−0.438669	−	0.898649i	$-0.644550\pi$
$499$	12.4853	−	21.6251i	0.558918	−	0.968074i	0.997587	−	0.0694257i	$-0.0221167\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−19.3137			−0.861156			−0.430578	−	0.902553i	$-0.641690\pi$
$503$	−19.3137			−0.861156			−0.430578	+	0.902553i	$0.641690\pi$
$504$		0			0
$505$	−36.6274			−1.62990
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.1924	−	31.5101i	0.806363	−	1.39666i	−0.109003	−	0.994041i	$-0.534766\pi$
$509$	18.1924	−	31.5101i	0.806363	−	1.39666i	0.915367	−	0.402621i	$-0.131901\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.1421	+	41.8154i	1.06383	+	1.84261i
$516$		0			0
$517$	3.71573			0.163418
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.12132	−	12.3345i	−0.311991	−	0.540384i	0.666803	−	0.745234i	$-0.267662\pi$
$521$	−7.12132	−	12.3345i	−0.311991	−	0.540384i	−0.978793	+	0.204851i	$0.934329\pi$
$522$		0			0
$523$	−2.48528	+	4.30463i	−0.108674	+	0.188228i	−0.915233	−	0.402924i	$-0.867994\pi$
$523$	−2.48528	+	4.30463i	−0.108674	+	0.188228i	0.806559	+	0.591153i	$0.201327\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.4853	+	18.1610i	−0.456746	+	0.791107i
$528$		0			0
$529$	−0.156854	−	0.271680i	−0.00681975	−	0.0118122i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	43.4558			1.88228
$534$		0			0
$535$	21.8995	+	37.9310i	0.946798	+	1.63990i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9.00000	−	15.5885i	0.386940	−	0.670200i	−0.605096	−	0.796152i	$-0.706865\pi$
$541$	9.00000	−	15.5885i	0.386940	−	0.670200i	0.992036	+	0.125952i	$0.0401986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.3137			−0.484626
$546$		0			0
$547$	−3.02944			−0.129529			−0.0647647	−	0.997901i	$-0.520630\pi$
$547$	−3.02944			−0.129529			−0.0647647	+	0.997901i	$0.520630\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.65685	+	16.7262i	−0.411396	+	0.712558i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.3137	+	28.2562i	0.691234	+	1.19725i	0.971434	+	0.237311i	$0.0762660\pi$
$557$	16.3137	+	28.2562i	0.691234	+	1.19725i	−0.280200	+	0.959942i	$0.590401\pi$
$558$		0			0
$559$	−48.0000			−2.03018
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.928932	−	1.60896i	−0.0391498	−	0.0678095i	0.845787	−	0.533521i	$-0.179132\pi$
$563$	−0.928932	−	1.60896i	−0.0391498	−	0.0678095i	−0.884936	+	0.465712i	$0.845798\pi$
$564$		0			0
$565$	−17.0711	+	29.5680i	−0.718185	+	1.24393i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.92893	+	5.07306i	−0.122787	+	0.212674i	−0.920866	−	0.389880i	$-0.872517\pi$
$569$	−2.92893	+	5.07306i	−0.122787	+	0.212674i	0.798079	+	0.602553i	$0.205850\pi$
$570$		0			0
$571$	20.8284	+	36.0759i	0.871643	+	1.50973i	0.860297	+	0.509794i	$0.170278\pi$
$571$	20.8284	+	36.0759i	0.871643	+	1.50973i	0.0113458	+	0.999936i	$0.496388\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	32.1421			1.34042
$576$		0			0
$577$	6.60660	+	11.4430i	0.275036	+	0.476377i	0.970144	−	0.242528i	$-0.0779767\pi$
$577$	6.60660	+	11.4430i	0.275036	+	0.476377i	−0.695108	+	0.718905i	$0.744643\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.828427	−	1.43488i	0.0343099	−	0.0594266i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−35.7990			−1.47758			−0.738791	−	0.673934i	$-0.764603\pi$
$587$	−35.7990			−1.47758			−0.738791	+	0.673934i	$0.764603\pi$
$588$		0			0
$589$	−19.3137			−0.795807
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−23.7071	+	41.0619i	−0.973534	+	1.68621i	−0.288844	+	0.957376i	$0.593271\pi$
$593$	−23.7071	+	41.0619i	−0.973534	+	1.68621i	−0.684690	+	0.728835i	$0.740062\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.7279	+	20.3134i	0.479190	+	0.829981i	0.999715	−	0.0238650i	$-0.00759718\pi$
$599$	11.7279	+	20.3134i	0.479190	+	0.829981i	−0.520525	+	0.853846i	$0.674264\pi$
$600$		0			0
$601$	12.2426			0.499388			0.249694	−	0.968325i	$-0.419670\pi$
$601$	12.2426			0.499388			0.249694	+	0.968325i	$0.419670\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	17.6066	+	30.4955i	0.715810	+	1.23982i
$606$		0			0
$607$	12.4853	−	21.6251i	0.506762	−	0.877737i	−0.493207	−	0.869912i	$-0.664176\pi$
$607$	12.4853	−	21.6251i	0.506762	−	0.877737i	0.999969	−	0.00782569i	$-0.00249102\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.51472	−	16.4800i	0.384924	−	0.666708i
$612$		0			0
$613$	5.17157	+	8.95743i	0.208878	+	0.361787i	0.951361	−	0.308077i	$-0.0996856\pi$
$613$	5.17157	+	8.95743i	0.208878	+	0.361787i	−0.742483	+	0.669864i	$0.766352\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.48528			0.180571			0.0902853	−	0.995916i	$-0.471222\pi$
$617$	4.48528			0.180571			0.0902853	+	0.995916i	$0.471222\pi$
$618$		0			0
$619$	16.8284	+	29.1477i	0.676392	+	1.17154i	0.976060	+	0.217501i	$0.0697906\pi$
$619$	16.8284	+	29.1477i	0.676392	+	1.17154i	−0.299668	+	0.954043i	$0.596876\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	6.98528	−	12.0989i	0.279411	−	0.483954i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.2843			−0.489806
$630$		0			0
$631$	−3.02944			−0.120600			−0.0603000	−	0.998180i	$-0.519206\pi$
$631$	−3.02944			−0.120600			−0.0603000	+	0.998180i	$0.519206\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−34.1421	+	59.1359i	−1.35489	+	2.34674i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.5563	+	30.4085i	0.693434	+	1.20106i	0.970706	+	0.240272i	$0.0772365\pi$
$641$	17.5563	+	30.4085i	0.693434	+	1.20106i	−0.277272	+	0.960792i	$0.589430\pi$
$642$		0			0
$643$	−31.7990			−1.25403			−0.627015	−	0.779007i	$-0.715723\pi$
$643$	−31.7990			−1.25403			−0.627015	+	0.779007i	$0.715723\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.100505	+	0.174080i	0.00395126	+	0.00684379i	0.867994	−	0.496574i	$-0.165409\pi$
$647$	0.100505	+	0.174080i	0.00395126	+	0.00684379i	−0.864043	+	0.503418i	$0.832076\pi$
$648$		0			0
$649$	3.51472	−	6.08767i	0.137965	−	0.238962i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.0711	−	39.9603i	0.902841	−	1.56377i	0.0790508	−	0.996871i	$-0.474811\pi$
$653$	23.0711	−	39.9603i	0.902841	−	1.56377i	0.823790	−	0.566895i	$-0.191856\pi$
$654$		0			0
$655$	6.82843	+	11.8272i	0.266809	+	0.462126i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−21.5147			−0.838094			−0.419047	−	0.907964i	$-0.637636\pi$
$659$	−21.5147			−0.838094			−0.419047	+	0.907964i	$0.637636\pi$
$660$		0			0
$661$	2.46447	+	4.26858i	0.0958566	+	0.166029i	0.909966	−	0.414683i	$-0.136108\pi$
$661$	2.46447	+	4.26858i	0.0958566	+	0.166029i	−0.814109	+	0.580712i	$0.802774\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6.82843	−	11.8272i	0.264398	−	0.457950i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9.17157			0.354065
$672$		0			0
$673$	−23.3137			−0.898677			−0.449339	−	0.893361i	$-0.648340\pi$
$673$	−23.3137			−0.898677			−0.449339	+	0.893361i	$0.648340\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.8492	+	23.9876i	−0.532270	+	0.921918i	0.467021	+	0.884246i	$0.345327\pi$
$677$	−13.8492	+	23.9876i	−0.532270	+	0.921918i	−0.999290	+	0.0376716i	$0.988006\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.899495	−	1.55797i	−0.0344182	−	0.0596141i	0.848303	−	0.529511i	$-0.177624\pi$
$683$	−0.899495	−	1.55797i	−0.0344182	−	0.0596141i	−0.882721	+	0.469897i	$0.844291\pi$
$684$		0			0
$685$	31.3137			1.19644
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.24264	−	7.34847i	−0.161632	−	0.279954i
$690$		0			0
$691$	4.34315	−	7.52255i	0.165221	−	0.286171i	−0.771513	−	0.636214i	$-0.780500\pi$
$691$	4.34315	−	7.52255i	0.165221	−	0.286171i	0.936734	+	0.350043i	$0.113833\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−35.7990	+	62.0057i	−1.35793	+	2.35201i
$696$		0			0
$697$	−37.9706	−	65.7669i	−1.43824	−	2.49110i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.14214			0.231985			0.115993	−	0.993250i	$-0.462995\pi$
$701$	6.14214			0.231985			0.115993	+	0.993250i	$0.462995\pi$
$702$		0			0
$703$	−5.65685	−	9.79796i	−0.213352	−	0.369537i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.31371	+	5.73951i	−0.124449	+	0.215552i	−0.921517	−	0.388337i	$-0.873050\pi$
$709$	−3.31371	+	5.73951i	−0.124449	+	0.215552i	0.797068	+	0.603889i	$0.206383\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.6569			0.511453
$714$		0			0
$715$	−12.0000			−0.448775
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.31371	−	5.73951i	0.123580	−	0.214048i	−0.797597	−	0.603191i	$-0.793896\pi$
$719$	3.31371	−	5.73951i	0.123580	−	0.214048i	0.921177	+	0.389143i	$0.127229\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.41421	+	16.3059i	0.349635	+	0.605586i
$726$		0			0
$727$	9.85786			0.365608			0.182804	−	0.983149i	$-0.441483\pi$
$727$	9.85786			0.365608			0.182804	+	0.983149i	$0.441483\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	41.9411	+	72.6442i	1.55125	+	2.68684i
$732$		0			0
$733$	−4.02082	+	6.96426i	−0.148512	+	0.257231i	−0.930678	−	0.365840i	$-0.880782\pi$
$733$	−4.02082	+	6.96426i	−0.148512	+	0.257231i	0.782166	+	0.623071i	$0.214115\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.68629	−	8.11689i	0.172622	−	0.298990i
$738$		0			0
$739$	−16.4853	−	28.5533i	−0.606421	−	1.05035i	−0.991825	−	0.127604i	$-0.959271\pi$
$739$	−16.4853	−	28.5533i	−0.606421	−	1.05035i	0.385404	−	0.922748i	$-0.374062\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.82843			0.177138			0.0885689	−	0.996070i	$-0.471771\pi$
$743$	4.82843			0.177138			0.0885689	+	0.996070i	$0.471771\pi$
$744$		0			0
$745$	2.24264	+	3.88437i	0.0821640	+	0.142312i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.1421	+	17.5667i	−0.370092	+	0.641018i	−0.989579	−	0.143989i	$-0.954007\pi$
$751$	−10.1421	+	17.5667i	−0.370092	+	0.641018i	0.619488	+	0.785006i	$0.287340\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−32.9706			−1.19992
$756$		0			0
$757$	41.9411			1.52438			0.762188	−	0.647356i	$-0.224125\pi$
$757$	41.9411			1.52438			0.762188	+	0.647356i	$0.224125\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.9497	−	27.6258i	0.578178	−	1.00143i	−0.417510	−	0.908672i	$-0.637097\pi$
$761$	15.9497	−	27.6258i	0.578178	−	1.00143i	0.995688	−	0.0927614i	$-0.0295694\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000	−	31.1769i	−0.649942	−	1.12573i
$768$		0			0
$769$	−34.8701			−1.25745			−0.628723	−	0.777629i	$-0.716422\pi$
$769$	−34.8701			−1.25745			−0.628723	+	0.777629i	$0.716422\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.6777	−	28.8866i	−0.599854	−	1.03898i	−0.992842	−	0.119434i	$-0.961892\pi$
$773$	−16.6777	−	28.8866i	−0.599854	−	1.03898i	0.392988	−	0.919544i	$-0.371441\pi$
$774$		0			0
$775$	−9.41421	+	16.3059i	−0.338169	+	0.585725i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	34.9706	−	60.5708i	1.25295	−	2.17017i
$780$		0			0
$781$	4.34315	+	7.52255i	0.155410	+	0.269178i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.828427			−0.0295678
$786$		0			0
$787$	7.65685	+	13.2621i	0.272937	+	0.472741i	0.969613	−	0.244645i	$-0.0786715\pi$
$787$	7.65685	+	13.2621i	0.272937	+	0.472741i	−0.696675	+	0.717387i	$0.745338\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	23.4853	−	40.6777i	0.833987	−	1.44451i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−39.4142			−1.39612			−0.698062	−	0.716038i	$-0.745954\pi$
$797$	−39.4142			−1.39612			−0.698062	+	0.716038i	$0.745954\pi$
$798$		0			0
$799$	−33.2548			−1.17647
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.21320	+	5.56543i	−0.113391	+	0.196400i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.31371	−	4.00746i	−0.0813457	−	0.140895i	0.822483	−	0.568790i	$-0.192588\pi$
$809$	−2.31371	−	4.00746i	−0.0813457	−	0.140895i	−0.903828	+	0.427896i	$0.859255\pi$
$810$		0			0
$811$	−25.6569			−0.900934			−0.450467	−	0.892793i	$-0.648743\pi$
$811$	−25.6569			−0.900934			−0.450467	+	0.892793i	$0.648743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.65685	−	16.7262i	−0.338265	−	0.585892i
$816$		0			0
$817$	−38.6274	+	66.9046i	−1.35140	+	2.34070i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.34315	+	9.25460i	−0.186477	+	0.322988i	−0.944073	−	0.329736i	$-0.893040\pi$
$821$	−5.34315	+	9.25460i	−0.186477	+	0.322988i	0.757596	+	0.652724i	$0.226374\pi$
$822$		0			0
$823$	12.4853	+	21.6251i	0.435210	+	0.753805i	0.997313	−	0.0732621i	$-0.0233410\pi$
$823$	12.4853	+	21.6251i	0.435210	+	0.753805i	−0.562103	+	0.827067i	$0.690008\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−40.1421			−1.39588			−0.697939	−	0.716157i	$-0.745900\pi$
$827$	−40.1421			−1.39588			−0.697939	+	0.716157i	$0.745900\pi$
$828$		0			0
$829$	17.1924	+	29.7781i	0.597116	+	1.03424i	0.993244	+	0.116041i	$0.0370203\pi$
$829$	17.1924	+	29.7781i	0.597116	+	1.03424i	−0.396128	+	0.918195i	$0.629646\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	25.3137	−	43.8446i	0.876017	−	1.51731i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.7990			0.821632			0.410816	−	0.911718i	$-0.365244\pi$
$839$	23.7990			0.821632			0.410816	+	0.911718i	$0.365244\pi$
$840$		0			0
$841$	−21.0000			−0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.53553	+	14.7840i	−0.293631	+	0.508584i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.00000	+	6.92820i	0.137118	+	0.237496i
$852$		0			0
$853$	−4.92893			−0.168763			−0.0843817	−	0.996434i	$-0.526892\pi$
$853$	−4.92893			−0.168763			−0.0843817	+	0.996434i	$0.526892\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.60660	+	9.71092i	0.191518	+	0.331719i	0.945753	−	0.324885i	$-0.105326\pi$
$857$	5.60660	+	9.71092i	0.191518	+	0.331719i	−0.754236	+	0.656604i	$0.771992\pi$
$858$		0			0
$859$	1.27208	−	2.20330i	0.0434027	−	0.0751757i	−0.843508	−	0.537117i	$-0.819513\pi$
$859$	1.27208	−	2.20330i	0.0434027	−	0.0751757i	0.886911	+	0.461941i	$0.152847\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.0416	−	46.8375i	0.920508	−	1.59437i	0.121877	−	0.992545i	$-0.461109\pi$
$863$	27.0416	−	46.8375i	0.920508	−	1.59437i	0.798631	−	0.601821i	$-0.205558\pi$
$864$		0			0
$865$	28.3137	+	49.0408i	0.962695	+	1.66744i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.3137			−0.383791
$870$		0			0
$871$	−24.0000	−	41.5692i	−0.813209	−	1.40852i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14.1421	−	24.4949i	0.477546	−	0.827134i	−0.522123	−	0.852870i	$-0.674860\pi$
$877$	14.1421	−	24.4949i	0.477546	−	0.827134i	0.999669	+	0.0257364i	$0.00819306\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0416			1.01213			0.506064	−	0.862496i	$-0.331100\pi$
$881$	30.0416			1.01213			0.506064	+	0.862496i	$0.331100\pi$
$882$		0			0
$883$	−10.3431			−0.348075			−0.174037	−	0.984739i	$-0.555681\pi$
$883$	−10.3431			−0.348075			−0.174037	+	0.984739i	$0.555681\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19.8995	−	34.4669i	0.668160	−	1.15729i	−0.310259	−	0.950652i	$-0.600416\pi$
$887$	19.8995	−	34.4669i	0.668160	−	1.15729i	0.978418	−	0.206634i	$-0.0662510\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.3137	−	26.5241i	−0.512454	−	0.887596i
$894$		0			0
$895$	22.1421			0.740130
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.00000	+	6.92820i	0.133407	+	0.231069i
$900$		0			0
$901$	−7.41421	+	12.8418i	−0.247003	+	0.427822i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.0711	+	20.9077i	−0.401256	+	0.694996i
$906$		0			0
$907$	−26.8284	−	46.4682i	−0.890823	−	1.54295i	−0.838890	−	0.544301i	$-0.816795\pi$
$907$	−26.8284	−	46.4682i	−0.890823	−	1.54295i	−0.0519331	−	0.998651i	$-0.516538\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.48528			−0.214867			−0.107433	−	0.994212i	$-0.534263\pi$
$911$	−6.48528			−0.214867			−0.107433	+	0.994212i	$0.534263\pi$
$912$		0			0
$913$	−1.65685	−	2.86976i	−0.0548339	−	0.0949751i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−15.6569	+	27.1185i	−0.516472	+	0.894556i	0.483345	+	0.875430i	$0.339422\pi$
$919$	−15.6569	+	27.1185i	−0.516472	+	0.894556i	−0.999817	+	0.0191258i	$0.993912\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	44.4853			1.46425
$924$		0			0
$925$	−11.0294			−0.362646
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.94975	−	13.7694i	0.260823	−	0.451758i	−0.705638	−	0.708572i	$-0.749340\pi$
$929$	7.94975	−	13.7694i	0.260823	−	0.451758i	0.966461	+	0.256814i	$0.0826728\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	10.4853	+	18.1610i	0.342905	+	0.593930i
$936$		0			0
$937$	−51.3553			−1.67771			−0.838853	−	0.544358i	$-0.816773\pi$
$937$	−51.3553			−1.67771			−0.838853	+	0.544358i	$0.816773\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7.36396	+	12.7548i	0.240058	+	0.415793i	0.960731	−	0.277483i	$-0.0895001\pi$
$941$	7.36396	+	12.7548i	0.240058	+	0.415793i	−0.720672	+	0.693276i	$0.756167\pi$
$942$		0			0
$943$	−24.7279	+	42.8300i	−0.805252	+	1.39474i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−29.3848	+	50.8959i	−0.954877	+	1.65390i	−0.220227	+	0.975449i	$0.570680\pi$
$947$	−29.3848	+	50.8959i	−0.954877	+	1.65390i	−0.734650	+	0.678447i	$0.762653\pi$
$948$		0			0
$949$	16.4558	+	28.5024i	0.534179	+	0.925226i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.00000			−0.0647864			−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000			−0.0647864			−0.0323932	+	0.999475i	$0.510313\pi$
$954$		0			0
$955$	20.7279	+	35.9018i	0.670740	+	1.16176i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.5000	−	19.9186i	0.370968	−	0.642535i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.82843			0.219815
$966$		0			0
$967$	−31.3137			−1.00698			−0.503490	−	0.864001i	$-0.667951\pi$
$967$	−31.3137			−1.00698			−0.503490	+	0.864001i	$0.667951\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.0000	−	17.3205i	0.320915	−	0.555842i	−0.659762	−	0.751475i	$-0.729343\pi$
$971$	10.0000	−	17.3205i	0.320915	−	0.555842i	0.980677	+	0.195633i	$0.0626762\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.89949	−	6.75412i	−0.124756	−	0.216084i	0.796882	−	0.604136i	$-0.206481\pi$
$977$	−3.89949	−	6.75412i	−0.124756	−	0.216084i	−0.921637	+	0.388052i	$0.873148\pi$
$978$		0			0
$979$	−4.76955			−0.152436
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.686292	−	1.18869i	−0.0218893	−	0.0379134i	0.854873	−	0.518837i	$-0.173635\pi$
$983$	−0.686292	−	1.18869i	−0.0218893	−	0.0379134i	−0.876763	+	0.480924i	$0.840301\pi$
$984$		0			0
$985$	−4.58579	+	7.94282i	−0.146115	+	0.253079i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	27.3137	−	47.3087i	0.868525	−	1.50433i
$990$		0			0
$991$	9.31371	+	16.1318i	0.295860	+	0.512444i	0.975185	−	0.221394i	$-0.0710606\pi$
$991$	9.31371	+	16.1318i	0.295860	+	0.512444i	−0.679325	+	0.733838i	$0.737727\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	19.3137			0.612286
$996$		0			0
$997$	−27.1924	−	47.0986i	−0.861192	−	1.49163i	−0.870780	−	0.491674i	$-0.836385\pi$
$997$	−27.1924	−	47.0986i	−0.861192	−	1.49163i	0.00958804	−	0.999954i	$-0.496948\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.bc.3313.1		4
3.2	odd	2		1176.2.q.m.961.2		4
7.2	even	3		3528.2.a.bm.1.2		2
7.3	odd	6		3528.2.s.bl.361.2		4
7.4	even	3	inner	3528.2.s.bc.361.1		4
7.5	odd	6		3528.2.a.bc.1.1		2
7.6	odd	2		3528.2.s.bl.3313.2		4
12.11	even	2		2352.2.q.bg.961.2		4
21.2	odd	6		1176.2.a.m.1.1	yes	2
21.5	even	6		1176.2.a.l.1.2	✓	2
21.11	odd	6		1176.2.q.m.361.2		4
21.17	even	6		1176.2.q.n.361.1		4
21.20	even	2		1176.2.q.n.961.1		4
28.19	even	6		7056.2.a.ce.1.1		2
28.23	odd	6		7056.2.a.cw.1.2		2
84.11	even	6		2352.2.q.bg.1537.2		4
84.23	even	6		2352.2.a.z.1.1		2
84.47	odd	6		2352.2.a.bg.1.2		2
84.59	odd	6		2352.2.q.ba.1537.1		4
84.83	odd	2		2352.2.q.ba.961.1		4
168.5	even	6		9408.2.a.dv.1.1		2
168.107	even	6		9408.2.a.ed.1.2		2
168.131	odd	6		9408.2.a.dh.1.1		2
168.149	odd	6		9408.2.a.dr.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.a.l.1.2	✓	2	21.5	even	6
1176.2.a.m.1.1	yes	2	21.2	odd	6
1176.2.q.m.361.2		4	21.11	odd	6
1176.2.q.m.961.2		4	3.2	odd	2
1176.2.q.n.361.1		4	21.17	even	6
1176.2.q.n.961.1		4	21.20	even	2
2352.2.a.z.1.1		2	84.23	even	6
2352.2.a.bg.1.2		2	84.47	odd	6
2352.2.q.ba.961.1		4	84.83	odd	2
2352.2.q.ba.1537.1		4	84.59	odd	6
2352.2.q.bg.961.2		4	12.11	even	2
2352.2.q.bg.1537.2		4	84.11	even	6
3528.2.a.bc.1.1		2	7.5	odd	6
3528.2.a.bm.1.2		2	7.2	even	3
3528.2.s.bc.361.1		4	7.4	even	3	inner
3528.2.s.bc.3313.1		4	1.1	even	1	trivial
3528.2.s.bl.361.2		4	7.3	odd	6
3528.2.s.bl.3313.2		4	7.6	odd	2
7056.2.a.ce.1.1		2	28.19	even	6
7056.2.a.cw.1.2		2	28.23	odd	6
9408.2.a.dh.1.1		2	168.131	odd	6
9408.2.a.dr.1.2		2	168.149	odd	6
9408.2.a.dv.1.1		2	168.5	even	6
9408.2.a.ed.1.2		2	168.107	even	6

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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.70711 + 2.95680i) q^{5} +(0.414214 + 0.717439i) q^{11} +4.24264 q^{13} +(-3.70711 - 6.42090i) q^{17} +(3.41421 - 5.91359i) q^{19} +(-2.41421 + 4.18154i) q^{23} +(-3.32843 - 5.76500i) q^{25} -2.82843 q^{29} +(-1.41421 - 2.44949i) q^{31} +(0.828427 - 1.43488i) q^{37} +10.2426 q^{41} -11.3137 q^{43} +(2.24264 - 3.88437i) q^{47} +(-1.00000 - 1.73205i) q^{53} -2.82843 q^{55} +(-4.24264 - 7.34847i) q^{59} +(5.53553 - 9.58783i) q^{61} +(-7.24264 + 12.5446i) q^{65} +(-5.65685 - 9.79796i) q^{67} +10.4853 q^{71} +(3.87868 + 6.71807i) q^{73} +(-6.82843 + 11.8272i) q^{79} -4.00000 q^{83} +25.3137 q^{85} +(-2.87868 + 4.98602i) q^{89} +(11.6569 + 20.1903i) q^{95} +0.242641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{11} - 12 q^{17} + 8 q^{19} - 4 q^{23} - 2 q^{25} - 8 q^{37} + 24 q^{41} - 8 q^{47} - 4 q^{53} + 8 q^{61} - 12 q^{65} + 8 q^{71} + 24 q^{73} - 16 q^{79} - 16 q^{83} + 56 q^{85} - 20 q^{89}+ \cdots - 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^11 - 12 * q^17 + 8 * q^19 - 4 * q^23 - 2 * q^25 - 8 * q^37 + 24 * q^41 - 8 * q^47 - 4 * q^53 + 8 * q^61 - 12 * q^65 + 8 * q^71 + 24 * q^73 - 16 * q^79 - 16 * q^83 + 56 * q^85 - 20 * q^89 + 24 * q^95 - 16 * q^97

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