LMFDB - Embedded newform 2100.2.x.d.1357.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1357,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("2100.1357");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.x (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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 $ x^16 - 8x^14 + 8x^12 - 8x^10 + 212x^8 + 248x^6 + 368x^4 + 32*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1357.3
Root			$-0.522506 - 1.01508i$ of defining polynomial
Character	$\chi$	$=$	2100.1357
Dual form			2100.2.x.d.1693.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{3} +(0.235858 + 2.63522i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{3} +(0.235858 + 2.63522i) q^{7} +1.00000i q^{9} +4.99261 q^{11} +(2.63111 + 2.63111i) q^{13} +(-4.14626 + 4.14626i) q^{17} -4.66126 q^{19} +(1.69660 - 2.03016i) q^{21} +(-4.41755 + 4.41755i) q^{23} +(0.707107 - 0.707107i) q^{27} -2.34895i q^{29} -2.57124i q^{31} +(-3.53031 - 3.53031i) q^{33} +(-7.06989 - 7.06989i) q^{37} -3.72095i q^{39} +7.87389i q^{41} +(2.59202 - 2.59202i) q^{43} +(-0.813264 + 0.813264i) q^{47} +(-6.88874 + 1.24308i) q^{49} +5.86369 q^{51} +(0.0317294 - 0.0317294i) q^{53} +(3.29601 + 3.29601i) q^{57} +0.858136 q^{59} -1.59654i q^{61} +(-2.63522 + 0.235858i) q^{63} +(10.5074 + 10.5074i) q^{67} +6.24736 q^{69} -12.5404 q^{71} +(-7.92331 - 7.92331i) q^{73} +(1.17755 + 13.1566i) q^{77} +8.69789i q^{79} -1.00000 q^{81} +(-5.50896 - 5.50896i) q^{83} +(-1.66096 + 1.66096i) q^{87} +1.62052 q^{89} +(-6.31297 + 7.55411i) q^{91} +(-1.81814 + 1.81814i) q^{93} +(-3.51923 + 3.51923i) q^{97} +4.99261i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{11} + 8 q^{21} - 8 q^{23} - 16 q^{37} - 48 q^{43} - 16 q^{51} + 40 q^{53} + 8 q^{57} + 48 q^{67} - 32 q^{71} + 24 q^{77} - 16 q^{81} + 32 q^{91} + 8 q^{93}+O(q^{100}) $ 16 * q + 16 * q^11 + 8 * q^21 - 8 * q^23 - 16 * q^37 - 48 * q^43 - 16 * q^51 + 40 * q^53 + 8 * q^57 + 48 * q^67 - 32 * q^71 + 24 * q^77 - 16 * q^81 + 32 * q^91 + 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.235858	+	2.63522i	0.0891461	+	0.996019i
$7$	0.235858	+	2.63522i	0.0891461	+	0.996019i
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	4.99261			1.50533			0.752665	−	0.658404i	$-0.228768\pi$
$11$	4.99261			1.50533			0.752665	+	0.658404i	$0.228768\pi$
$12$		0			0
$13$	2.63111	+	2.63111i	0.729738	+	0.729738i	0.970567	−	0.240830i	$-0.0774195\pi$
$13$	2.63111	+	2.63111i	0.729738	+	0.729738i	−0.240830	+	0.970567i	$0.577420\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.14626	+	4.14626i	−1.00561	+	1.00561i	−0.00563041	+	0.999984i	$0.501792\pi$
$17$	−4.14626	+	4.14626i	−1.00561	+	1.00561i	−0.999984	+	0.00563041i	$0.998208\pi$
$18$		0			0
$19$	−4.66126			−1.06937			−0.534684	−	0.845052i	$-0.679569\pi$
$19$	−4.66126			−1.06937			−0.534684	+	0.845052i	$0.679569\pi$
$20$		0			0
$21$	1.69660	−	2.03016i	0.370229	−	0.443017i
$22$		0			0
$23$	−4.41755	+	4.41755i	−0.921123	+	0.921123i	−0.997109	−	0.0759861i	$-0.975790\pi$
$23$	−4.41755	+	4.41755i	−0.921123	+	0.921123i	0.0759861	+	0.997109i	$0.475790\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.707107	−	0.707107i	0.136083	−	0.136083i
$28$		0			0
$29$		−	2.34895i		−	0.436188i	−0.975928	−	0.218094i	$-0.930016\pi$
$29$		−	2.34895i		−	0.436188i	0.975928	−	0.218094i	$-0.0699840\pi$
$30$		0			0
$31$		−	2.57124i		−	0.461808i	−0.972976	−	0.230904i	$-0.925832\pi$
$31$		−	2.57124i		−	0.461808i	0.972976	−	0.230904i	$-0.0741684\pi$
$32$		0			0
$33$	−3.53031	−	3.53031i	−0.614548	−	0.614548i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.06989	−	7.06989i	−1.16228	−	1.16228i	−0.983975	−	0.178309i	$-0.942937\pi$
$37$	−7.06989	−	7.06989i	−1.16228	−	1.16228i	−0.178309	−	0.983975i	$-0.557063\pi$
$38$		0			0
$39$		−	3.72095i		−	0.595828i
$40$		0			0
$41$			7.87389i			1.22969i	0.788646	+	0.614847i	$0.210782\pi$
$41$			7.87389i			1.22969i	−0.788646	+	0.614847i	$0.789218\pi$
$42$		0			0
$43$	2.59202	−	2.59202i	0.395280	−	0.395280i	−0.481285	−	0.876564i	$-0.659830\pi$
$43$	2.59202	−	2.59202i	0.395280	−	0.395280i	0.876564	+	0.481285i	$0.159830\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.813264	+	0.813264i	−0.118627	+	0.118627i	−0.763928	−	0.645301i	$-0.776732\pi$
$47$	−0.813264	+	0.813264i	−0.118627	+	0.118627i	0.645301	+	0.763928i	$0.276732\pi$
$48$		0			0
$49$	−6.88874	+	1.24308i	−0.984106	+	0.177582i
$50$		0			0
$51$	5.86369			0.821081
$52$		0			0
$53$	0.0317294	−	0.0317294i	0.00435837	−	0.00435837i	−0.704924	−	0.709283i	$-0.749019\pi$
$53$	0.0317294	−	0.0317294i	0.00435837	−	0.00435837i	0.709283	+	0.704924i	$0.249019\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.29601	+	3.29601i	0.436567	+	0.436567i
$58$		0			0
$59$	0.858136			0.111720			0.0558599	−	0.998439i	$-0.482210\pi$
$59$	0.858136			0.111720			0.0558599	+	0.998439i	$0.482210\pi$
$60$		0			0
$61$		−	1.59654i		−	0.204416i	−0.994763	−	0.102208i	$-0.967409\pi$
$61$		−	1.59654i		−	0.204416i	0.994763	−	0.102208i	$-0.0325907\pi$
$62$		0			0
$63$	−2.63522	+	0.235858i	−0.332006	+	0.0297154i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	10.5074	+	10.5074i	1.28368	+	1.28368i	0.938559	+	0.345119i	$0.112162\pi$
$67$	10.5074	+	10.5074i	1.28368	+	1.28368i	0.345119	+	0.938559i	$0.387838\pi$
$68$		0			0
$69$	6.24736			0.752094
$70$		0			0
$71$	−12.5404			−1.48827			−0.744134	−	0.668030i	$-0.767138\pi$
$71$	−12.5404			−1.48827			−0.744134	+	0.668030i	$0.767138\pi$
$72$		0			0
$73$	−7.92331	−	7.92331i	−0.927353	−	0.927353i	0.0701812	−	0.997534i	$-0.477642\pi$
$73$	−7.92331	−	7.92331i	−0.927353	−	0.927353i	−0.997534	+	0.0701812i	$0.977642\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.17755	+	13.1566i	0.134194	+	1.49934i
$78$		0			0
$79$			8.69789i			0.978589i	0.872119	+	0.489295i	$0.162746\pi$
$79$			8.69789i			0.978589i	−0.872119	+	0.489295i	$0.837254\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−5.50896	−	5.50896i	−0.604686	−	0.604686i	0.336866	−	0.941553i	$-0.390633\pi$
$83$	−5.50896	−	5.50896i	−0.604686	−	0.604686i	−0.941553	+	0.336866i	$0.890633\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.66096	+	1.66096i	−0.178073	+	0.178073i
$88$		0			0
$89$	1.62052			0.171775			0.0858875	−	0.996305i	$-0.472627\pi$
$89$	1.62052			0.171775			0.0858875	+	0.996305i	$0.472627\pi$
$90$		0			0
$91$	−6.31297	+	7.55411i	−0.661779	+	0.791886i
$92$		0			0
$93$	−1.81814	+	1.81814i	−0.188532	+	0.188532i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.51923	+	3.51923i	−0.357324	+	0.357324i	−0.862826	−	0.505502i	$-0.831307\pi$
$97$	−3.51923	+	3.51923i	−0.357324	+	0.357324i	0.505502	+	0.862826i	$0.331307\pi$
$98$		0			0
$99$			4.99261i			0.501777i
$100$		0			0
$101$			15.6729i			1.55951i	0.626083	+	0.779757i	$0.284657\pi$
$101$			15.6729i			1.55951i	−0.626083	+	0.779757i	$0.715343\pi$
$102$		0			0
$103$	1.83284	+	1.83284i	0.180595	+	0.180595i	0.791615	−	0.611020i	$-0.209241\pi$
$103$	1.83284	+	1.83284i	0.180595	+	0.180595i	−0.611020	+	0.791615i	$0.709241\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.869787	+	0.869787i	0.0840855	+	0.0840855i	0.747899	−	0.663813i	$-0.231063\pi$
$107$	0.869787	+	0.869787i	0.0840855	+	0.0840855i	−0.663813	+	0.747899i	$0.731063\pi$
$108$		0			0
$109$			11.7274i			1.12328i	0.827382	+	0.561640i	$0.189829\pi$
$109$			11.7274i			1.12328i	−0.827382	+	0.561640i	$0.810171\pi$
$110$		0			0
$111$			9.99834i			0.949000i
$112$		0			0
$113$	−10.3172	+	10.3172i	−0.970562	+	0.970562i	−0.999579	−	0.0290168i	$-0.990762\pi$
$113$	−10.3172	+	10.3172i	−0.970562	+	0.970562i	0.0290168	+	0.999579i	$0.490762\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.63111	+	2.63111i	−0.243246	+	0.243246i
$118$		0			0
$119$	−11.9042	−	9.94835i	−1.09126	−	0.911964i
$120$		0			0
$121$	13.9262			1.26602
$122$		0			0
$123$	5.56768	−	5.56768i	0.502021	−	0.502021i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	11.4843	+	11.4843i	1.01907	+	1.01907i	0.999815	+	0.0192525i	$0.00612864\pi$
$127$	11.4843	+	11.4843i	1.01907	+	1.01907i	0.0192525	+	0.999815i	$0.493871\pi$
$128$		0			0
$129$	−3.66567			−0.322745
$130$		0			0
$131$			11.2928i			0.986658i	0.869843	+	0.493329i	$0.164220\pi$
$131$			11.2928i			0.986658i	−0.869843	+	0.493329i	$0.835780\pi$
$132$		0			0
$133$	−1.09940	−	12.2834i	−0.0953299	−	1.06511i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.75911	−	9.75911i	−0.833777	−	0.833777i	0.154254	−	0.988031i	$-0.450703\pi$
$137$	−9.75911	−	9.75911i	−0.833777	−	0.833777i	−0.988031	+	0.154254i	$0.950703\pi$
$138$		0			0
$139$	7.88433			0.668740			0.334370	−	0.942442i	$-0.391476\pi$
$139$	7.88433			0.668740			0.334370	+	0.942442i	$0.391476\pi$
$140$		0			0
$141$	1.15013			0.0968584
$142$		0			0
$143$	13.1361	+	13.1361i	1.09850	+	1.09850i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.75006	+	3.99209i	0.474257	+	0.329262i
$148$		0			0
$149$			11.4991i			0.942041i	0.882122	+	0.471020i	$0.156114\pi$
$149$			11.4991i			0.942041i	−0.882122	+	0.471020i	$0.843886\pi$
$150$		0			0
$151$	19.7274			1.60539			0.802696	−	0.596389i	$-0.203398\pi$
$151$	19.7274			1.60539			0.802696	+	0.596389i	$0.203398\pi$
$152$		0			0
$153$	−4.14626	−	4.14626i	−0.335205	−	0.335205i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.1764	−	12.1764i	0.971782	−	0.971782i	−0.0278306	−	0.999613i	$-0.508860\pi$
$157$	12.1764	−	12.1764i	0.971782	−	0.971782i	0.999613	+	0.0278306i	$0.00885990\pi$
$158$		0			0
$159$	−0.0448722			−0.00355859
$160$		0			0
$161$	−12.6831	−	10.5993i	−0.999570	−	0.835341i
$162$		0			0
$163$	1.75692	−	1.75692i	0.137613	−	0.137613i	−0.634945	−	0.772558i	$-0.718977\pi$
$163$	1.75692	−	1.75692i	0.137613	−	0.137613i	0.772558	+	0.634945i	$0.218977\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.14217	+	2.14217i	−0.165766	+	0.165766i	−0.785116	−	0.619349i	$-0.787396\pi$
$167$	−2.14217	+	2.14217i	−0.165766	+	0.165766i	0.619349	+	0.785116i	$0.287396\pi$
$168$		0			0
$169$			0.845444i			0.0650341i
$170$		0			0
$171$		−	4.66126i		−	0.356456i
$172$		0			0
$173$	−6.06440	−	6.06440i	−0.461068	−	0.461068i	0.437938	−	0.899005i	$-0.355709\pi$
$173$	−6.06440	−	6.06440i	−0.461068	−	0.461068i	−0.899005	+	0.437938i	$0.855709\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.606794	−	0.606794i	−0.0456094	−	0.0456094i
$178$		0			0
$179$		−	12.1280i		−	0.906487i	−0.891387	−	0.453244i	$-0.850267\pi$
$179$		−	12.1280i		−	0.906487i	0.891387	−	0.453244i	$-0.149733\pi$
$180$		0			0
$181$		−	17.9366i		−	1.33322i	−0.745407	−	0.666610i	$-0.767745\pi$
$181$		−	17.9366i		−	1.33322i	0.745407	−	0.666610i	$-0.232255\pi$
$182$		0			0
$183$	−1.12892	+	1.12892i	−0.0834525	+	0.0834525i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−20.7007	+	20.7007i	−1.51378	+	1.51378i
$188$		0			0
$189$	2.03016	+	1.69660i	0.147672	+	0.123410i
$190$		0			0
$191$	19.8764			1.43821			0.719103	−	0.694904i	$-0.244553\pi$
$191$	19.8764			1.43821			0.719103	+	0.694904i	$0.244553\pi$
$192$		0			0
$193$	−4.35723	+	4.35723i	−0.313640	+	0.313640i	−0.846318	−	0.532678i	$-0.821186\pi$
$193$	−4.35723	+	4.35723i	−0.313640	+	0.313640i	0.532678	+	0.846318i	$0.321186\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.75911	+	1.75911i	0.125331	+	0.125331i	0.766990	−	0.641659i	$-0.221753\pi$
$197$	1.75911	+	1.75911i	0.125331	+	0.125331i	−0.641659	+	0.766990i	$0.721753\pi$
$198$		0			0
$199$	25.9162			1.83715			0.918574	−	0.395248i	$-0.129341\pi$
$199$	25.9162			1.83715			0.918574	+	0.395248i	$0.129341\pi$
$200$		0			0
$201$		−	14.8596i		−	1.04812i
$202$		0			0
$203$	6.18998	−	0.554019i	0.434452	−	0.0388845i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−4.41755	−	4.41755i	−0.307041	−	0.307041i
$208$		0			0
$209$	−23.2719			−1.60975
$210$		0			0
$211$	−0.412405			−0.0283911			−0.0141956	−	0.999899i	$-0.504519\pi$
$211$	−0.412405			−0.0283911			−0.0141956	+	0.999899i	$0.504519\pi$
$212$		0			0
$213$	8.86739	+	8.86739i	0.607583	+	0.607583i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.77578	−	0.606449i	0.459970	−	0.0411684i
$218$		0			0
$219$			11.2053i			0.757181i
$220$		0			0
$221$	−21.8185			−1.46767
$222$		0			0
$223$	12.0779	+	12.0779i	0.808797	+	0.808797i	0.984452	−	0.175655i	$-0.0562042\pi$
$223$	12.0779	+	12.0779i	0.808797	+	0.808797i	−0.175655	+	0.984452i	$0.556204\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.84390	−	7.84390i	0.520618	−	0.520618i	−0.397140	−	0.917758i	$-0.629997\pi$
$227$	7.84390	−	7.84390i	0.520618	−	0.520618i	0.917758	+	0.397140i	$0.129997\pi$
$228$		0			0
$229$	−13.7475			−0.908460			−0.454230	−	0.890884i	$-0.650086\pi$
$229$	−13.7475			−0.908460			−0.454230	+	0.890884i	$0.650086\pi$
$230$		0			0
$231$	8.47048	−	10.1358i	0.557317	−	0.666886i
$232$		0			0
$233$	−13.7104	+	13.7104i	−0.898200	+	0.898200i	−0.995277	−	0.0970772i	$-0.969051\pi$
$233$	−13.7104	+	13.7104i	−0.898200	+	0.898200i	0.0970772	+	0.995277i	$0.469051\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6.15034	−	6.15034i	0.399507	−	0.399507i
$238$		0			0
$239$			2.69236i			0.174154i	0.996202	+	0.0870770i	$0.0277526\pi$
$239$			2.69236i			0.174154i	−0.996202	+	0.0870770i	$0.972247\pi$
$240$		0			0
$241$			27.2501i			1.75533i	0.479273	+	0.877666i	$0.340900\pi$
$241$			27.2501i			1.75533i	−0.479273	+	0.877666i	$0.659100\pi$
$242$		0			0
$243$	0.707107	+	0.707107i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−12.2643	−	12.2643i	−0.780358	−	0.780358i
$248$		0			0
$249$			7.79084i			0.493724i
$250$		0			0
$251$			6.91873i			0.436706i	0.975870	+	0.218353i	$0.0700685\pi$
$251$			6.91873i			0.436706i	−0.975870	+	0.218353i	$0.929932\pi$
$252$		0			0
$253$	−22.0551	+	22.0551i	−1.38659	+	1.38659i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.405721	+	0.405721i	−0.0253082	+	0.0253082i	−0.719648	−	0.694339i	$-0.755697\pi$
$257$	−0.405721	+	0.405721i	−0.0253082	+	0.0253082i	0.694339	+	0.719648i	$0.255697\pi$
$258$		0			0
$259$	16.9632	−	20.2982i	1.05404	−	1.26127i
$260$		0			0
$261$	2.34895			0.145396
$262$		0			0
$263$	−5.57949	+	5.57949i	−0.344046	+	0.344046i	−0.857886	−	0.513840i	$-0.828223\pi$
$263$	−5.57949	+	5.57949i	−0.344046	+	0.344046i	0.513840	+	0.857886i	$0.328223\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.14588	−	1.14588i	−0.0701268	−	0.0701268i
$268$		0			0
$269$	8.43131			0.514066			0.257033	−	0.966403i	$-0.417255\pi$
$269$	8.43131			0.514066			0.257033	+	0.966403i	$0.417255\pi$
$270$		0			0
$271$		−	15.2157i		−	0.924287i	−0.886805	−	0.462143i	$-0.847081\pi$
$271$		−	15.2157i		−	0.924287i	0.886805	−	0.462143i	$-0.152919\pi$
$272$		0			0
$273$	9.80550	−	0.877617i	0.593456	−	0.0531158i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.76521	−	1.76521i	−0.106061	−	0.106061i	0.652085	−	0.758146i	$-0.273895\pi$
$277$	−1.76521	−	1.76521i	−0.106061	−	0.106061i	−0.758146	+	0.652085i	$0.773895\pi$
$278$		0			0
$279$	2.57124			0.153936
$280$		0			0
$281$	−16.7230			−0.997608			−0.498804	−	0.866715i	$-0.666227\pi$
$281$	−16.7230			−0.997608			−0.498804	+	0.866715i	$0.666227\pi$
$282$		0			0
$283$	−7.00974	−	7.00974i	−0.416686	−	0.416686i	0.467374	−	0.884060i	$-0.345200\pi$
$283$	−7.00974	−	7.00974i	−0.416686	−	0.416686i	−0.884060	+	0.467374i	$0.845200\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−20.7494	+	1.85712i	−1.22480	+	0.109622i
$288$		0			0
$289$		−	17.3829i		−	1.02252i
$290$		0			0
$291$	4.97694			0.291754
$292$		0			0
$293$	0.405721	+	0.405721i	0.0237025	+	0.0237025i	0.718859	−	0.695156i	$-0.244665\pi$
$293$	0.405721	+	0.405721i	0.0237025	+	0.0237025i	−0.695156	+	0.718859i	$0.744665\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.53031	−	3.53031i	0.204849	−	0.204849i
$298$		0			0
$299$	−23.2461			−1.34436
$300$		0			0
$301$	7.44189	+	6.21919i	0.428944	+	0.358468i
$302$		0			0
$303$	11.0824	−	11.0824i	0.636669	−	0.636669i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−3.01258	+	3.01258i	−0.171937	+	0.171937i	−0.787830	−	0.615893i	$-0.788795\pi$
$307$	−3.01258	+	3.01258i	−0.171937	+	0.171937i	0.615893	+	0.787830i	$0.288795\pi$
$308$		0			0
$309$		−	2.59202i		−	0.147455i
$310$		0			0
$311$			1.49770i			0.0849266i	0.999098	+	0.0424633i	$0.0135206\pi$
$311$			1.49770i			0.0849266i	−0.999098	+	0.0424633i	$0.986479\pi$
$312$		0			0
$313$	−16.0870	−	16.0870i	−0.909290	−	0.909290i	0.0869246	−	0.996215i	$-0.472296\pi$
$313$	−16.0870	−	16.0870i	−0.909290	−	0.909290i	−0.996215	+	0.0869246i	$0.972296\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.95350	−	1.95350i	−0.109719	−	0.109719i	0.650116	−	0.759835i	$-0.274720\pi$
$317$	−1.95350	−	1.95350i	−0.109719	−	0.109719i	−0.759835	+	0.650116i	$0.774720\pi$
$318$		0			0
$319$		−	11.7274i		−	0.656607i
$320$		0			0
$321$		−	1.23006i		−	0.0686555i
$322$		0			0
$323$	19.3268	−	19.3268i	1.07537	−	1.07537i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	8.29251	−	8.29251i	0.458577	−	0.458577i
$328$		0			0
$329$	−2.33494	−	1.95131i	−0.128730	−	0.107579i
$330$		0			0
$331$	−25.4271			−1.39760			−0.698801	−	0.715317i	$-0.746282\pi$
$331$	−25.4271			−1.39760			−0.698801	+	0.715317i	$0.746282\pi$
$332$		0			0
$333$	7.06989	−	7.06989i	0.387428	−	0.387428i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−8.43562	−	8.43562i	−0.459517	−	0.459517i	0.438980	−	0.898497i	$-0.355340\pi$
$337$	−8.43562	−	8.43562i	−0.459517	−	0.459517i	−0.898497	+	0.438980i	$0.855340\pi$
$338$		0			0
$339$	14.5907			0.792461
$340$		0			0
$341$		−	12.8372i		−	0.695174i
$342$		0			0
$343$	−4.90055	−	17.8601i	−0.264605	−	0.964357i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.66801	+	7.66801i	0.411640	+	0.411640i	0.882310	−	0.470669i	$-0.155987\pi$
$347$	7.66801	+	7.66801i	0.411640	+	0.411640i	−0.470669	+	0.882310i	$0.655987\pi$
$348$		0			0
$349$	−13.2122			−0.707234			−0.353617	−	0.935390i	$-0.615048\pi$
$349$	−13.2122			−0.707234			−0.353617	+	0.935390i	$0.615048\pi$
$350$		0			0
$351$	3.72095			0.198609
$352$		0			0
$353$	18.4154	+	18.4154i	0.980153	+	0.980153i	0.999807	−	0.0196540i	$-0.00625647\pi$
$353$	18.4154	+	18.4154i	0.980153	+	0.980153i	−0.0196540	+	0.999807i	$0.506256\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.38300	+	15.4521i	0.0731962	+	0.817812i
$358$		0			0
$359$		−	17.5636i		−	0.926971i	−0.886105	−	0.463485i	$-0.846599\pi$
$359$		−	17.5636i		−	0.926971i	0.886105	−	0.463485i	$-0.153401\pi$
$360$		0			0
$361$	2.72738			0.143546
$362$		0			0
$363$	−9.84731	−	9.84731i	−0.516850	−	0.516850i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	18.5241	−	18.5241i	0.966948	−	0.966948i	−0.0325233	−	0.999471i	$-0.510354\pi$
$367$	18.5241	−	18.5241i	0.966948	−	0.966948i	0.999471	+	0.0325233i	$0.0103543\pi$
$368$		0			0
$369$	−7.87389			−0.409898
$370$		0			0
$371$	0.0910975	+	0.0761302i	0.00472955	+	0.00395249i
$372$		0			0
$373$	14.7274	−	14.7274i	0.762555	−	0.762555i	−0.214229	−	0.976784i	$-0.568724\pi$
$373$	14.7274	−	14.7274i	0.762555	−	0.762555i	0.976784	+	0.214229i	$0.0687238\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.18033	−	6.18033i	0.318303	−	0.318303i
$378$		0			0
$379$			22.2943i			1.14518i	0.819841	+	0.572592i	$0.194062\pi$
$379$			22.2943i			1.14518i	−0.819841	+	0.572592i	$0.805938\pi$
$380$		0			0
$381$		−	16.2413i		−	0.832065i
$382$		0			0
$383$	18.8978	+	18.8978i	0.965633	+	0.965633i	0.999429	−	0.0337961i	$-0.0107597\pi$
$383$	18.8978	+	18.8978i	0.965633	+	0.965633i	−0.0337961	+	0.999429i	$0.510760\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.59202	+	2.59202i	0.131760	+	0.131760i
$388$		0			0
$389$			2.77164i			0.140528i	0.997528	+	0.0702639i	$0.0223841\pi$
$389$			2.77164i			0.140528i	−0.997528	+	0.0702639i	$0.977616\pi$
$390$		0			0
$391$		−	36.6326i		−	1.85259i
$392$		0			0
$393$	7.98523	−	7.98523i	0.402801	−	0.402801i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	20.6199	−	20.6199i	1.03488	−	1.03488i	0.0355137	−	0.999369i	$-0.488693\pi$
$397$	20.6199	−	20.6199i	1.03488	−	1.03488i	0.999369	−	0.0355137i	$-0.0113067\pi$
$398$		0			0
$399$	−7.90831	+	9.46310i	−0.395911	+	0.473747i
$400$		0			0
$401$	−28.3385			−1.41516			−0.707580	−	0.706633i	$-0.750213\pi$
$401$	−28.3385			−1.41516			−0.707580	+	0.706633i	$0.750213\pi$
$402$		0			0
$403$	6.76521	−	6.76521i	0.336999	−	0.336999i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−35.2972	−	35.2972i	−1.74962	−	1.74962i
$408$		0			0
$409$	38.1691			1.88734			0.943671	−	0.330885i	$-0.107347\pi$
$409$	38.1691			1.88734			0.943671	+	0.330885i	$0.107347\pi$
$410$		0			0
$411$			13.8015i			0.680776i
$412$		0			0
$413$	0.202399	+	2.26138i	0.00995939	+	0.111275i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.57506	−	5.57506i	−0.273012	−	0.273012i
$418$		0			0
$419$	33.3495			1.62923			0.814616	−	0.580001i	$-0.196948\pi$
$419$	33.3495			1.62923			0.814616	+	0.580001i	$0.196948\pi$
$420$		0			0
$421$	7.41055			0.361168			0.180584	−	0.983560i	$-0.442201\pi$
$421$	7.41055			0.361168			0.180584	+	0.983560i	$0.442201\pi$
$422$		0			0
$423$	−0.813264	−	0.813264i	−0.0395423	−	0.0395423i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.20723	−	0.376557i	0.203602	−	0.0182229i
$428$		0			0
$429$		−	18.5773i		−	0.896918i
$430$		0			0
$431$	−15.4005			−0.741818			−0.370909	−	0.928669i	$-0.620954\pi$
$431$	−15.4005			−0.741818			−0.370909	+	0.928669i	$0.620954\pi$
$432$		0			0
$433$	6.12490	+	6.12490i	0.294344	+	0.294344i	0.838794	−	0.544450i	$-0.183261\pi$
$433$	6.12490	+	6.12490i	0.294344	+	0.294344i	−0.544450	+	0.838794i	$0.683261\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	20.5914	−	20.5914i	0.985019	−	0.985019i
$438$		0			0
$439$	−4.18867			−0.199914			−0.0999572	−	0.994992i	$-0.531871\pi$
$439$	−4.18867			−0.199914			−0.0999572	+	0.994992i	$0.531871\pi$
$440$		0			0
$441$	−1.24308	−	6.88874i	−0.0591941	−	0.328035i
$442$		0			0
$443$	2.45442	−	2.45442i	0.116613	−	0.116613i	−0.646392	−	0.763005i	$-0.723723\pi$
$443$	2.45442	−	2.45442i	0.116613	−	0.116613i	0.763005	+	0.646392i	$0.223723\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	8.13107	−	8.13107i	0.384587	−	0.384587i
$448$		0			0
$449$		−	31.6407i		−	1.49321i	−0.665265	−	0.746607i	$-0.731681\pi$
$449$		−	31.6407i		−	1.49321i	0.665265	−	0.746607i	$-0.268319\pi$
$450$		0			0
$451$			39.3113i			1.85110i
$452$		0			0
$453$	−13.9494	−	13.9494i	−0.655398	−	0.655398i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.4887	+	11.4887i	0.537420	+	0.537420i	0.922770	−	0.385350i	$-0.125919\pi$
$457$	11.4887	+	11.4887i	0.537420	+	0.537420i	−0.385350	+	0.922770i	$0.625919\pi$
$458$		0			0
$459$			5.86369i			0.273694i
$460$		0			0
$461$			37.4080i			1.74227i	0.491048	+	0.871133i	$0.336614\pi$
$461$			37.4080i			1.74227i	−0.491048	+	0.871133i	$0.663386\pi$
$462$		0			0
$463$	21.4695	−	21.4695i	0.997774	−	0.997774i	−0.00222376	−	0.999998i	$-0.500708\pi$
$463$	21.4695	−	21.4695i	0.997774	−	0.997774i	0.999998	+	0.00222376i	$0.000707846\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.1279	+	21.1279i	−0.977683	+	0.977683i	−0.999756	−	0.0220738i	$-0.992973\pi$
$467$	−21.1279	+	21.1279i	−0.977683	+	0.977683i	0.0220738	+	0.999756i	$0.492973\pi$
$468$		0			0
$469$	−25.2109	+	30.1674i	−1.16413	+	1.39300i
$470$		0			0
$471$	−17.2200			−0.793457
$472$		0			0
$473$	12.9410	−	12.9410i	0.595026	−	0.595026i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.0317294	+	0.0317294i	0.00145279	+	0.00145279i
$478$		0			0
$479$	36.6835			1.67611			0.838055	−	0.545586i	$-0.183693\pi$
$479$	36.6835			1.67611			0.838055	+	0.545586i	$0.183693\pi$
$480$		0			0
$481$		−	37.2033i		−	1.69632i
$482$		0			0
$483$	1.47349	+	16.4631i	0.0670462	+	0.749099i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.19439	−	2.19439i	−0.0994373	−	0.0994373i	0.655638	−	0.755075i	$-0.272400\pi$
$487$	−2.19439	−	2.19439i	−0.0994373	−	0.0994373i	−0.755075	+	0.655638i	$0.772400\pi$
$488$		0			0
$489$	−2.48466			−0.112360
$490$		0			0
$491$	−13.6566			−0.616313			−0.308157	−	0.951336i	$-0.599712\pi$
$491$	−13.6566			−0.616313			−0.308157	+	0.951336i	$0.599712\pi$
$492$		0			0
$493$	9.73933	+	9.73933i	0.438637	+	0.438637i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.95775	−	33.0466i	−0.132673	−	1.48234i
$498$		0			0
$499$		−	3.27077i		−	0.146420i	−0.997317	−	0.0732099i	$-0.976676\pi$
$499$		−	3.27077i		−	0.146420i	0.997317	−	0.0732099i	$-0.0233243\pi$
$500$		0			0
$501$	3.02949			0.135348
$502$		0			0
$503$	9.91904	+	9.91904i	0.442268	+	0.442268i	0.892774	−	0.450506i	$-0.148756\pi$
$503$	9.91904	+	9.91904i	0.442268	+	0.442268i	−0.450506	+	0.892774i	$0.648756\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.597819	−	0.597819i	0.0265501	−	0.0265501i
$508$		0			0
$509$	−20.6081			−0.913437			−0.456718	−	0.889611i	$-0.650975\pi$
$509$	−20.6081			−0.913437			−0.456718	+	0.889611i	$0.650975\pi$
$510$		0			0
$511$	19.0109	−	22.7484i	0.840991	−	1.00633i
$512$		0			0
$513$	−3.29601	+	3.29601i	−0.145522	+	0.145522i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−4.06031	+	4.06031i	−0.178572	+	0.178572i
$518$		0			0
$519$			8.57635i			0.376460i
$520$		0			0
$521$		−	14.3938i		−	0.630603i	−0.948991	−	0.315302i	$-0.897894\pi$
$521$		−	14.3938i		−	0.630603i	0.948991	−	0.315302i	$-0.102106\pi$
$522$		0			0
$523$	14.7222	+	14.7222i	0.643757	+	0.643757i	0.951477	−	0.307720i	$-0.0995661\pi$
$523$	14.7222	+	14.7222i	0.643757	+	0.643757i	−0.307720	+	0.951477i	$0.599566\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.6610	+	10.6610i	0.464401	+	0.464401i
$528$		0			0
$529$		−	16.0295i		−	0.696934i
$530$		0			0
$531$			0.858136i			0.0372399i
$532$		0			0
$533$	−20.7170	+	20.7170i	−0.897354	+	0.897354i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−8.57577	+	8.57577i	−0.370072	+	0.370072i
$538$		0			0
$539$	−34.3928	+	6.20620i	−1.48140	+	0.267320i
$540$		0			0
$541$	−12.5138			−0.538010			−0.269005	−	0.963139i	$-0.586695\pi$
$541$	−12.5138			−0.538010			−0.269005	+	0.963139i	$0.586695\pi$
$542$		0			0
$543$	−12.6831	+	12.6831i	−0.544285	+	0.544285i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.66191	−	4.66191i	−0.199329	−	0.199329i	0.600383	−	0.799712i	$-0.295015\pi$
$547$	−4.66191	−	4.66191i	−0.199329	−	0.199329i	−0.799712	+	0.600383i	$0.795015\pi$
$548$		0			0
$549$	1.59654			0.0681387
$550$		0			0
$551$			10.9491i			0.466445i
$552$		0			0
$553$	−22.9208	+	2.05147i	−0.974693	+	0.0872374i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.2626	+	18.2626i	0.773812	+	0.773812i	0.978771	−	0.204959i	$-0.0657061\pi$
$557$	18.2626	+	18.2626i	0.773812	+	0.773812i	−0.204959	+	0.978771i	$0.565706\pi$
$558$		0			0
$559$	13.6398			0.576901
$560$		0			0
$561$	29.2751			1.23600
$562$		0			0
$563$	0.858136	+	0.858136i	0.0361661	+	0.0361661i	0.724959	−	0.688792i	$-0.241859\pi$
$563$	0.858136	+	0.858136i	0.0361661	+	0.0361661i	−0.688792	+	0.724959i	$0.741859\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.235858	−	2.63522i	−0.00990512	−	0.110669i
$568$		0			0
$569$			24.5451i			1.02899i	0.857495	+	0.514493i	$0.172020\pi$
$569$			24.5451i			1.02899i	−0.857495	+	0.514493i	$0.827980\pi$
$570$		0			0
$571$	−0.528564			−0.0221197			−0.0110599	−	0.999939i	$-0.503521\pi$
$571$	−0.528564			−0.0221197			−0.0110599	+	0.999939i	$0.503521\pi$
$572$		0			0
$573$	−14.0547	−	14.0547i	−0.587145	−	0.587145i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.90536	−	8.90536i	0.370735	−	0.370735i	−0.497010	−	0.867745i	$-0.665569\pi$
$577$	8.90536	−	8.90536i	0.370735	−	0.370735i	0.867745	+	0.497010i	$0.165569\pi$
$578$		0			0
$579$	6.16205			0.256086
$580$		0			0
$581$	13.2180	−	15.8166i	0.548373	−	0.656184i
$582$		0			0
$583$	0.158413	−	0.158413i	0.00656078	−	0.00656078i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.0394	−	17.0394i	0.703292	−	0.703292i	−0.261824	−	0.965116i	$-0.584324\pi$
$587$	17.0394	−	17.0394i	0.703292	−	0.703292i	0.965116	+	0.261824i	$0.0843239\pi$
$588$		0			0
$589$			11.9852i			0.493843i
$590$		0			0
$591$		−	2.48776i		−	0.102333i
$592$		0			0
$593$	−9.25748	−	9.25748i	−0.380159	−	0.380159i	0.491000	−	0.871159i	$-0.336631\pi$
$593$	−9.25748	−	9.25748i	−0.380159	−	0.380159i	−0.871159	+	0.491000i	$0.836631\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−18.3255	−	18.3255i	−0.750013	−	0.750013i
$598$		0			0
$599$		−	27.6462i		−	1.12959i	−0.825230	−	0.564796i	$-0.808955\pi$
$599$		−	27.6462i		−	1.12959i	0.825230	−	0.564796i	$-0.191045\pi$
$600$		0			0
$601$		−	20.7142i		−	0.844949i	−0.906375	−	0.422475i	$-0.861162\pi$
$601$		−	20.7142i		−	0.844949i	0.906375	−	0.422475i	$-0.138838\pi$
$602$		0			0
$603$	−10.5074	+	10.5074i	−0.427893	+	0.427893i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.59685	+	4.59685i	−0.186580	+	0.186580i	−0.794216	−	0.607636i	$-0.792118\pi$
$607$	−4.59685	+	4.59685i	−0.186580	+	0.186580i	0.607636	+	0.794216i	$0.292118\pi$
$608$		0			0
$609$	−4.76873	−	3.98523i	−0.193239	−	0.161490i
$610$		0			0
$611$	−4.27957			−0.173133
$612$		0			0
$613$	29.5964	−	29.5964i	1.19539	−	1.19539i	0.219854	−	0.975533i	$-0.429442\pi$
$613$	29.5964	−	29.5964i	1.19539	−	1.19539i	0.975533	−	0.219854i	$-0.0705582\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23.2730	−	23.2730i	−0.936934	−	0.936934i	0.0611921	−	0.998126i	$-0.480510\pi$
$617$	−23.2730	−	23.2730i	−0.936934	−	0.936934i	−0.998126	+	0.0611921i	$0.980510\pi$
$618$		0			0
$619$	−19.8683			−0.798575			−0.399288	−	0.916826i	$-0.630743\pi$
$619$	−19.8683			−0.798575			−0.399288	+	0.916826i	$0.630743\pi$
$620$		0			0
$621$			6.24736i			0.250698i
$622$		0			0
$623$	0.382214	+	4.27043i	0.0153131	+	0.171091i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	16.4557	+	16.4557i	0.657178	+	0.657178i
$628$		0			0
$629$	58.6272			2.33762
$630$		0			0
$631$	8.05496			0.320663			0.160332	−	0.987063i	$-0.448744\pi$
$631$	8.05496			0.320663			0.160332	+	0.987063i	$0.448744\pi$
$632$		0			0
$633$	0.291614	+	0.291614i	0.0115906	+	0.0115906i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−21.3957	−	14.8543i	−0.847728	−	0.588551i
$638$		0			0
$639$		−	12.5404i		−	0.496090i
$640$		0			0
$641$	10.5201			0.415518			0.207759	−	0.978180i	$-0.433383\pi$
$641$	10.5201			0.415518			0.207759	+	0.978180i	$0.433383\pi$
$642$		0			0
$643$	24.2466	+	24.2466i	0.956192	+	0.956192i	0.999080	−	0.0428876i	$-0.0136558\pi$
$643$	24.2466	+	24.2466i	0.956192	+	0.956192i	−0.0428876	+	0.999080i	$0.513656\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.1304	+	15.1304i	−0.594836	+	0.594836i	−0.938934	−	0.344098i	$-0.888185\pi$
$647$	−15.1304	+	15.1304i	−0.594836	+	0.594836i	0.344098	+	0.938934i	$0.388185\pi$
$648$		0			0
$649$	4.28434			0.168175
$650$		0			0
$651$	−5.22002	−	4.36237i	−0.204589	−	0.170975i
$652$		0			0
$653$	−27.7591	+	27.7591i	−1.08630	+	1.08630i	−0.0903916	+	0.995906i	$0.528812\pi$
$653$	−27.7591	+	27.7591i	−1.08630	+	1.08630i	−0.995906	+	0.0903916i	$0.971188\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	7.92331	−	7.92331i	0.309118	−	0.309118i
$658$		0			0
$659$		−	5.15936i		−	0.200980i	−0.994938	−	0.100490i	$-0.967959\pi$
$659$		−	5.15936i		−	0.200980i	0.994938	−	0.100490i	$-0.0320411\pi$
$660$		0			0
$661$		−	8.62844i		−	0.335608i	−0.985820	−	0.167804i	$-0.946332\pi$
$661$		−	8.62844i		−	0.335608i	0.985820	−	0.167804i	$-0.0536675\pi$
$662$		0			0
$663$	15.4280	+	15.4280i	0.599174	+	0.599174i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.3766	+	10.3766i	0.401783	+	0.401783i
$668$		0			0
$669$		−	17.0808i		−	0.660380i
$670$		0			0
$671$		−	7.97091i		−	0.307713i
$672$		0			0
$673$	1.88585	−	1.88585i	0.0726941	−	0.0726941i	−0.669825	−	0.742519i	$-0.733631\pi$
$673$	1.88585	−	1.88585i	0.0726941	−	0.0726941i	0.742519	+	0.669825i	$0.233631\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.56673	+	5.56673i	−0.213947	+	0.213947i	−0.805942	−	0.591995i	$-0.798341\pi$
$677$	−5.56673	+	5.56673i	−0.213947	+	0.213947i	0.591995	+	0.805942i	$0.298341\pi$
$678$		0			0
$679$	−10.1040	−	8.44390i	−0.387755	−	0.324047i
$680$		0			0
$681$	−11.0929			−0.425083
$682$		0			0
$683$	−25.4526	+	25.4526i	−0.973916	+	0.973916i	−0.999668	−	0.0257527i	$-0.991802\pi$
$683$	−25.4526	+	25.4526i	−0.973916	+	0.973916i	0.0257527	+	0.999668i	$0.491802\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	9.72095	+	9.72095i	0.370877	+	0.370877i
$688$		0			0
$689$	0.166967			0.00636093
$690$		0			0
$691$		−	15.5594i		−	0.591909i	−0.955202	−	0.295955i	$-0.904362\pi$
$691$		−	15.5594i		−	0.591909i	0.955202	−	0.295955i	$-0.0956377\pi$
$692$		0			0
$693$	−13.1566	+	1.17755i	−0.499779	+	0.0447314i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−32.6471	−	32.6471i	−1.23660	−	1.23660i
$698$		0			0
$699$	19.3895			0.733377
$700$		0			0
$701$	8.61966			0.325560			0.162780	−	0.986662i	$-0.447954\pi$
$701$	8.61966			0.325560			0.162780	+	0.986662i	$0.447954\pi$
$702$		0			0
$703$	32.9546	+	32.9546i	1.24291	+	1.24291i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−41.3015	+	3.69659i	−1.55330	+	0.139025i
$708$		0			0
$709$			8.58353i			0.322361i	0.986925	+	0.161181i	$0.0515302\pi$
$709$			8.58353i			0.322361i	−0.986925	+	0.161181i	$0.948470\pi$
$710$		0			0
$711$	−8.69789			−0.326196
$712$		0			0
$713$	11.3586	+	11.3586i	0.425382	+	0.425382i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.90378	−	1.90378i	0.0710981	−	0.0710981i
$718$		0			0
$719$	−17.8287			−0.664898			−0.332449	−	0.943121i	$-0.607875\pi$
$719$	−17.8287			−0.664898			−0.332449	+	0.943121i	$0.607875\pi$
$720$		0			0
$721$	−4.39763	+	5.26221i	−0.163776	+	0.195975i
$722$		0			0
$723$	19.2687	−	19.2687i	0.716611	−	0.716611i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	18.5253	−	18.5253i	0.687066	−	0.687066i	−0.274516	−	0.961582i	$-0.588518\pi$
$727$	18.5253	−	18.5253i	0.687066	−	0.687066i	0.961582	+	0.274516i	$0.0885177\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$			21.4944i			0.794998i
$732$		0			0
$733$	10.8206	+	10.8206i	0.399668	+	0.399668i	0.878116	−	0.478448i	$-0.158801\pi$
$733$	10.8206	+	10.8206i	0.399668	+	0.399668i	−0.478448	+	0.878116i	$0.658801\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	52.4592	+	52.4592i	1.93236	+	1.93236i
$738$		0			0
$739$			26.7993i			0.985827i	0.870078	+	0.492914i	$0.164068\pi$
$739$			26.7993i			0.985827i	−0.870078	+	0.492914i	$0.835932\pi$
$740$		0			0
$741$			17.3443i			0.637159i
$742$		0			0
$743$	13.5456	−	13.5456i	0.496939	−	0.496939i	−0.413545	−	0.910484i	$-0.635709\pi$
$743$	13.5456	−	13.5456i	0.496939	−	0.496939i	0.910484	+	0.413545i	$0.135709\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5.50896	−	5.50896i	0.201562	−	0.201562i
$748$		0			0
$749$	−2.08693	+	2.49723i	−0.0762548	+	0.0912466i
$750$		0			0
$751$	−27.3386			−0.997600			−0.498800	−	0.866717i	$-0.666226\pi$
$751$	−27.3386			−0.997600			−0.498800	+	0.866717i	$0.666226\pi$
$752$		0			0
$753$	4.89228	−	4.89228i	0.178285	−	0.178285i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	9.47988	+	9.47988i	0.344552	+	0.344552i	0.858076	−	0.513524i	$-0.171660\pi$
$757$	9.47988	+	9.47988i	0.344552	+	0.344552i	−0.513524	+	0.858076i	$0.671660\pi$
$758$		0			0
$759$	31.1907			1.13215
$760$		0			0
$761$		−	21.2685i		−	0.770983i	−0.922711	−	0.385492i	$-0.874032\pi$
$761$		−	21.2685i		−	0.770983i	0.922711	−	0.385492i	$-0.125968\pi$
$762$		0			0
$763$	−30.9042	+	2.76600i	−1.11881	+	0.100136i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.25785	+	2.25785i	0.0815262	+	0.0815262i
$768$		0			0
$769$	−18.1127			−0.653161			−0.326580	−	0.945169i	$-0.605896\pi$
$769$	−18.1127			−0.653161			−0.326580	+	0.945169i	$0.605896\pi$
$770$		0			0
$771$	0.573776			0.0206640
$772$		0			0
$773$	−22.5967	−	22.5967i	−0.812747	−	0.812747i	0.172298	−	0.985045i	$-0.444881\pi$
$773$	−22.5967	−	22.5967i	−0.812747	−	0.812747i	−0.985045	+	0.172298i	$0.944881\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−26.3478	+	2.35819i	−0.945222	+	0.0845997i
$778$		0			0
$779$		−	36.7023i		−	1.31500i
$780$		0			0
$781$	−62.6093			−2.24034
$782$		0			0
$783$	−1.66096	−	1.66096i	−0.0593577	−	0.0593577i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	18.4776	−	18.4776i	0.658656	−	0.658656i	−0.296406	−	0.955062i	$-0.595788\pi$
$787$	18.4776	−	18.4776i	0.658656	−	0.658656i	0.955062	+	0.296406i	$0.0957882\pi$
$788$		0			0
$789$	7.89059			0.280913
$790$		0			0
$791$	−29.6215	−	24.7547i	−1.05322	−	0.880176i
$792$		0			0
$793$	4.20067	−	4.20067i	0.149170	−	0.149170i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.01770	−	6.01770i	0.213158	−	0.213158i	−0.592450	−	0.805608i	$-0.701839\pi$
$797$	6.01770	−	6.01770i	0.213158	−	0.213158i	0.805608	+	0.592450i	$0.201839\pi$
$798$		0			0
$799$		−	6.74400i		−	0.238586i
$800$		0			0
$801$			1.62052i			0.0572583i
$802$		0			0
$803$	−39.5580	−	39.5580i	−1.39597	−	1.39597i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5.96184	−	5.96184i	−0.209867	−	0.209867i
$808$		0			0
$809$			55.4002i			1.94777i	0.227051	+	0.973883i	$0.427092\pi$
$809$			55.4002i			1.94777i	−0.227051	+	0.973883i	$0.572908\pi$
$810$		0			0
$811$			54.3175i			1.90734i	0.300847	+	0.953672i	$0.402731\pi$
$811$			54.3175i			1.90734i	−0.300847	+	0.953672i	$0.597269\pi$
$812$		0			0
$813$	−10.7591	+	10.7591i	−0.377338	+	0.377338i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−12.0821	+	12.0821i	−0.422699	+	0.422699i
$818$		0			0
$819$	−7.55411	−	6.31297i	−0.263962	−	0.220593i
$820$		0			0
$821$	33.2327			1.15983			0.579915	−	0.814677i	$-0.303086\pi$
$821$	33.2327			1.15983			0.579915	+	0.814677i	$0.303086\pi$
$822$		0			0
$823$	−14.3976	+	14.3976i	−0.501870	+	0.501870i	−0.912019	−	0.410149i	$-0.865477\pi$
$823$	−14.3976	+	14.3976i	−0.501870	+	0.501870i	0.410149	+	0.912019i	$0.365477\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.5824	+	18.5824i	0.646173	+	0.646173i	0.952066	−	0.305893i	$-0.0989549\pi$
$827$	18.5824	+	18.5824i	0.646173	+	0.646173i	−0.305893	+	0.952066i	$0.598955\pi$
$828$		0			0
$829$	36.0402			1.25173			0.625865	−	0.779932i	$-0.284746\pi$
$829$	36.0402			1.25173			0.625865	+	0.779932i	$0.284746\pi$
$830$		0			0
$831$			2.49638i			0.0865984i
$832$		0			0
$833$	23.4084	−	33.7166i	0.811052	−	1.16821i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.81814	−	1.81814i	−0.0628442	−	0.0628442i
$838$		0			0
$839$	−37.2603			−1.28637			−0.643185	−	0.765711i	$-0.722387\pi$
$839$	−37.2603			−1.28637			−0.643185	+	0.765711i	$0.722387\pi$
$840$		0			0
$841$	23.4825			0.809740
$842$		0			0
$843$	11.8249	+	11.8249i	0.407272	+	0.407272i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	3.28461	+	36.6986i	0.112861	+	1.26098i
$848$		0			0
$849$			9.91328i			0.340223i
$850$		0			0
$851$	62.4632			2.14121
$852$		0			0
$853$	5.65231	+	5.65231i	0.193531	+	0.193531i	0.797220	−	0.603689i	$-0.206303\pi$
$853$	5.65231	+	5.65231i	0.193531	+	0.193531i	−0.603689	+	0.797220i	$0.706303\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	31.5964	−	31.5964i	1.07931	−	1.07931i	0.0827404	−	0.996571i	$-0.473633\pi$
$857$	31.5964	−	31.5964i	1.07931	−	1.07931i	0.996571	−	0.0827404i	$-0.0263672\pi$
$858$		0			0
$859$	−23.7273			−0.809565			−0.404783	−	0.914413i	$-0.632653\pi$
$859$	−23.7273			−0.809565			−0.404783	+	0.914413i	$0.632653\pi$
$860$		0			0
$861$	15.9852	+	13.3589i	0.544775	+	0.455269i
$862$		0			0
$863$	36.0121	−	36.0121i	1.22587	−	1.22587i	0.260352	−	0.965514i	$-0.416161\pi$
$863$	36.0121	−	36.0121i	1.22587	−	1.22587i	0.965514	−	0.260352i	$-0.0838387\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−12.2915	+	12.2915i	−0.417443	+	0.417443i
$868$		0			0
$869$			43.4252i			1.47310i
$870$		0			0
$871$			55.2920i			1.87350i
$872$		0			0
$873$	−3.51923	−	3.51923i	−0.119108	−	0.119108i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	9.97231	+	9.97231i	0.336741	+	0.336741i	0.855139	−	0.518398i	$-0.173472\pi$
$877$	9.97231	+	9.97231i	0.336741	+	0.336741i	−0.518398	+	0.855139i	$0.673472\pi$
$878$		0			0
$879$		−	0.573776i		−	0.0193530i
$880$		0			0
$881$			35.5199i			1.19669i	0.801237	+	0.598347i	$0.204176\pi$
$881$			35.5199i			1.19669i	−0.801237	+	0.598347i	$0.795824\pi$
$882$		0			0
$883$	−15.9006	+	15.9006i	−0.535097	+	0.535097i	−0.922085	−	0.386988i	$-0.873515\pi$
$883$	−15.9006	+	15.9006i	−0.535097	+	0.535097i	0.386988	+	0.922085i	$0.373515\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.98186	−	8.98186i	0.301581	−	0.301581i	−0.540051	−	0.841632i	$-0.681595\pi$
$887$	8.98186	−	8.98186i	0.301581	−	0.301581i	0.841632	+	0.540051i	$0.181595\pi$
$888$		0			0
$889$	−27.5550	+	32.9723i	−0.924164	+	1.10586i
$890$		0			0
$891$	−4.99261			−0.167259
$892$		0			0
$893$	3.79084	−	3.79084i	0.126856	−	0.126856i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	16.4375	+	16.4375i	0.548831	+	0.548831i
$898$		0			0
$899$	−6.03970			−0.201435
$900$		0			0
$901$			0.263116i			0.00876568i
$902$		0			0
$903$	−0.864580	−	9.65985i	−0.0287714	−	0.321460i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	9.91534	+	9.91534i	0.329233	+	0.329233i	0.852295	−	0.523062i	$-0.175210\pi$
$907$	9.91534	+	9.91534i	0.329233	+	0.329233i	−0.523062	+	0.852295i	$0.675210\pi$
$908$		0			0
$909$	−15.6729			−0.519838
$910$		0			0
$911$	26.7476			0.886189			0.443094	−	0.896475i	$-0.353881\pi$
$911$	26.7476			0.886189			0.443094	+	0.896475i	$0.353881\pi$
$912$		0			0
$913$	−27.5041	−	27.5041i	−0.910252	−	0.910252i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29.7590	+	2.66351i	−0.982730	+	0.0879567i
$918$		0			0
$919$			4.81405i			0.158801i	0.996843	+	0.0794004i	$0.0253006\pi$
$919$			4.81405i			0.158801i	−0.996843	+	0.0794004i	$0.974699\pi$
$920$		0			0
$921$	4.26043			0.140386
$922$		0			0
$923$	−32.9951	−	32.9951i	−1.08605	−	1.08605i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.83284	+	1.83284i	−0.0601983	+	0.0601983i
$928$		0			0
$929$	19.1224			0.627385			0.313693	−	0.949525i	$-0.398434\pi$
$929$	19.1224			0.627385			0.313693	+	0.949525i	$0.398434\pi$
$930$		0			0
$931$	32.1102	−	5.79431i	1.05237	−	0.189901i
$932$		0			0
$933$	1.05903	−	1.05903i	0.0346711	−	0.0346711i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−18.2759	+	18.2759i	−0.597046	+	0.597046i	−0.939525	−	0.342479i	$-0.888733\pi$
$937$	−18.2759	+	18.2759i	−0.597046	+	0.597046i	0.342479	+	0.939525i	$0.388733\pi$
$938$		0			0
$939$			22.7504i			0.742432i
$940$		0			0
$941$		−	3.37097i		−	0.109890i	−0.998489	−	0.0549452i	$-0.982502\pi$
$941$		−	3.37097i		−	0.109890i	0.998489	−	0.0549452i	$-0.0174984\pi$
$942$		0			0
$943$	−34.7833	−	34.7833i	−1.13270	−	1.13270i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	35.6046	+	35.6046i	1.15699	+	1.15699i	0.985119	+	0.171873i	$0.0549820\pi$
$947$	35.6046	+	35.6046i	1.15699	+	1.15699i	0.171873	+	0.985119i	$0.445018\pi$
$948$		0			0
$949$		−	41.6941i		−	1.35345i
$950$		0			0
$951$			2.76266i			0.0895855i
$952$		0			0
$953$	24.2600	−	24.2600i	0.785860	−	0.785860i	−0.194953	−	0.980813i	$-0.562455\pi$
$953$	24.2600	−	24.2600i	0.785860	−	0.785860i	0.980813	+	0.194953i	$0.0624554\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−8.29251	+	8.29251i	−0.268059	+	0.268059i
$958$		0			0
$959$	23.4156	−	28.0191i	0.756129	−	0.904785i
$960$		0			0
$961$	24.3887			0.786733
$962$		0			0
$963$	−0.869787	+	0.869787i	−0.0280285	+	0.0280285i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.00833799	+	0.00833799i	0.000268132	+	0.000268132i	0.707241	−	0.706973i	$-0.249940\pi$
$967$	0.00833799	+	0.00833799i	0.000268132	+	0.000268132i	−0.706973	+	0.707241i	$0.749940\pi$
$968$		0			0
$969$	−27.3322			−0.878037
$970$		0			0
$971$		−	23.9127i		−	0.767396i	−0.923459	−	0.383698i	$-0.874650\pi$
$971$		−	23.9127i		−	0.767396i	0.923459	−	0.383698i	$-0.125350\pi$
$972$		0			0
$973$	1.85959	+	20.7769i	0.0596156	+	0.666078i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.23497	−	4.23497i	−0.135489	−	0.135489i	0.636110	−	0.771599i	$-0.280543\pi$
$977$	−4.23497	−	4.23497i	−0.135489	−	0.135489i	−0.771599	+	0.636110i	$0.780543\pi$
$978$		0			0
$979$	8.09064			0.258578
$980$		0			0
$981$	−11.7274			−0.374426
$982$		0			0
$983$	8.01578	+	8.01578i	0.255664	+	0.255664i	0.823288	−	0.567624i	$-0.192137\pi$
$983$	8.01578	+	8.01578i	0.255664	+	0.255664i	−0.567624	+	0.823288i	$0.692137\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.271268	+	3.03084i	0.00863455	+	0.0964727i
$988$		0			0
$989$			22.9008i			0.728202i
$990$		0			0
$991$	−55.1397			−1.75157			−0.875786	−	0.482700i	$-0.839656\pi$
$991$	−55.1397			−1.75157			−0.875786	+	0.482700i	$0.839656\pi$
$992$		0			0
$993$	17.9797	+	17.9797i	0.570568	+	0.570568i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−37.6893	+	37.6893i	−1.19363	+	1.19363i	−0.217592	+	0.976040i	$0.569820\pi$
$997$	−37.6893	+	37.6893i	−1.19363	+	1.19363i	−0.976040	+	0.217592i	$0.930180\pi$
$998$		0			0
$999$	−9.99834			−0.316333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.x.d.1357.3		16
5.2	odd	4		420.2.x.a.13.3	✓	16
5.3	odd	4	inner	2100.2.x.d.1693.8		16
5.4	even	2		420.2.x.a.97.6	yes	16
7.6	odd	2	inner	2100.2.x.d.1357.8		16
15.2	even	4		1260.2.ba.b.433.3		16
15.14	odd	2		1260.2.ba.b.937.6		16
20.7	even	4		1680.2.cz.c.433.7		16
20.19	odd	2		1680.2.cz.c.97.2		16
35.13	even	4	inner	2100.2.x.d.1693.3		16
35.27	even	4		420.2.x.a.13.6	yes	16
35.34	odd	2		420.2.x.a.97.3	yes	16
105.62	odd	4		1260.2.ba.b.433.6		16
105.104	even	2		1260.2.ba.b.937.3		16
140.27	odd	4		1680.2.cz.c.433.2		16
140.139	even	2		1680.2.cz.c.97.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.x.a.13.3	✓	16	5.2	odd	4
420.2.x.a.13.6	yes	16	35.27	even	4
420.2.x.a.97.3	yes	16	35.34	odd	2
420.2.x.a.97.6	yes	16	5.4	even	2
1260.2.ba.b.433.3		16	15.2	even	4
1260.2.ba.b.433.6		16	105.62	odd	4
1260.2.ba.b.937.3		16	105.104	even	2
1260.2.ba.b.937.6		16	15.14	odd	2
1680.2.cz.c.97.2		16	20.19	odd	2
1680.2.cz.c.97.7		16	140.139	even	2
1680.2.cz.c.433.2		16	140.27	odd	4
1680.2.cz.c.433.7		16	20.7	even	4
2100.2.x.d.1357.3		16	1.1	even	1	trivial
2100.2.x.d.1357.8		16	7.6	odd	2	inner
2100.2.x.d.1693.3		16	35.13	even	4	inner
2100.2.x.d.1693.8		16	5.3	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{3} +(0.235858 + 2.63522i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{3} +(0.235858 + 2.63522i) q^{7} +1.00000i q^{9} +4.99261 q^{11} +(2.63111 + 2.63111i) q^{13} +(-4.14626 + 4.14626i) q^{17} -4.66126 q^{19} +(1.69660 - 2.03016i) q^{21} +(-4.41755 + 4.41755i) q^{23} +(0.707107 - 0.707107i) q^{27} -2.34895i q^{29} -2.57124i q^{31} +(-3.53031 - 3.53031i) q^{33} +(-7.06989 - 7.06989i) q^{37} -3.72095i q^{39} +7.87389i q^{41} +(2.59202 - 2.59202i) q^{43} +(-0.813264 + 0.813264i) q^{47} +(-6.88874 + 1.24308i) q^{49} +5.86369 q^{51} +(0.0317294 - 0.0317294i) q^{53} +(3.29601 + 3.29601i) q^{57} +0.858136 q^{59} -1.59654i q^{61} +(-2.63522 + 0.235858i) q^{63} +(10.5074 + 10.5074i) q^{67} +6.24736 q^{69} -12.5404 q^{71} +(-7.92331 - 7.92331i) q^{73} +(1.17755 + 13.1566i) q^{77} +8.69789i q^{79} -1.00000 q^{81} +(-5.50896 - 5.50896i) q^{83} +(-1.66096 + 1.66096i) q^{87} +1.62052 q^{89} +(-6.31297 + 7.55411i) q^{91} +(-1.81814 + 1.81814i) q^{93} +(-3.51923 + 3.51923i) q^{97} +4.99261i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{11} + 8 q^{21} - 8 q^{23} - 16 q^{37} - 48 q^{43} - 16 q^{51} + 40 q^{53} + 8 q^{57} + 48 q^{67} - 32 q^{71} + 24 q^{77} - 16 q^{81} + 32 q^{91} + 8 q^{93}+O(q^{100}) \) 16 * q + 16 * q^11 + 8 * q^21 - 8 * q^23 - 16 * q^37 - 48 * q^43 - 16 * q^51 + 40 * q^53 + 8 * q^57 + 48 * q^67 - 32 * q^71 + 24 * q^77 - 16 * q^81 + 32 * q^91 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more