LMFDB - Embedded newform 1728.2.r.b.1441.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(289,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("1728.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.r (of order $6$, 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1441.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.1441
Dual form			1728.2.r.b.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 + 1.73205i) q^{5} +O(q^{10})$	$q+(3.00000 + 1.73205i) q^{5} +(-2.59808 + 1.50000i) q^{11} +(-3.00000 - 1.73205i) q^{13} +3.00000 q^{17} +7.00000i q^{19} +(-1.73205 + 3.00000i) q^{23} +(3.50000 + 6.06218i) q^{25} +(6.00000 - 3.46410i) q^{29} +(-3.46410 + 6.00000i) q^{31} +10.3923i q^{37} +(1.50000 - 2.59808i) q^{41} +(4.33013 - 2.50000i) q^{43} +(-1.73205 - 3.00000i) q^{47} +(3.50000 - 6.06218i) q^{49} +13.8564i q^{53} -10.3923 q^{55} +(7.79423 + 4.50000i) q^{59} +(-6.00000 + 3.46410i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(-4.33013 - 2.50000i) q^{67} -3.46410 q^{71} -7.00000 q^{73} +(-8.66025 - 15.0000i) q^{79} +(10.3923 - 6.00000i) q^{83} +(9.00000 + 5.19615i) q^{85} +6.00000 q^{89} +(-12.1244 + 21.0000i) q^{95} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5}+O(q^{10}) $ 4 * q + 12 * q^5	$ 4 q + 12 q^{5} - 12 q^{13} + 12 q^{17} + 14 q^{25} + 24 q^{29} + 6 q^{41} + 14 q^{49} - 24 q^{61} - 24 q^{65} - 28 q^{73} + 36 q^{85} + 24 q^{89} - 2 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 - 12 * q^13 + 12 * q^17 + 14 * q^25 + 24 * q^29 + 6 * q^41 + 14 * q^49 - 24 * q^61 - 24 * q^65 - 28 * q^73 + 36 * q^85 + 24 * q^89 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.00000	+	1.73205i	1.34164	+	0.774597i	0.987048	−	0.160424i	$-0.0512862\pi$
$5$	3.00000	+	1.73205i	1.34164	+	0.774597i	0.354593	+	0.935021i	$0.384620\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.59808	+	1.50000i	−0.783349	+	0.452267i	−0.837616	−	0.546259i	$-0.816051\pi$
$11$	−2.59808	+	1.50000i	−0.783349	+	0.452267i	0.0542666	+	0.998526i	$0.482718\pi$
$12$		0			0
$13$	−3.00000	−	1.73205i	−0.832050	−	0.480384i	0.0225039	−	0.999747i	$-0.492836\pi$
$13$	−3.00000	−	1.73205i	−0.832050	−	0.480384i	−0.854554	+	0.519362i	$0.826170\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$			7.00000i			1.60591i	0.596040	+	0.802955i	$0.296740\pi$
$19$			7.00000i			1.60591i	−0.596040	+	0.802955i	$0.703260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.73205	+	3.00000i	−0.361158	+	0.625543i	−0.988152	−	0.153481i	$-0.950952\pi$
$23$	−1.73205	+	3.00000i	−0.361158	+	0.625543i	0.626994	+	0.779024i	$0.284285\pi$
$24$		0			0
$25$	3.50000	+	6.06218i	0.700000	+	1.21244i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000	−	3.46410i	1.11417	−	0.643268i	0.174265	−	0.984699i	$-0.444245\pi$
$29$	6.00000	−	3.46410i	1.11417	−	0.643268i	0.939907	+	0.341431i	$0.110912\pi$
$30$		0			0
$31$	−3.46410	+	6.00000i	−0.622171	+	1.07763i	0.366910	+	0.930257i	$0.380416\pi$
$31$	−3.46410	+	6.00000i	−0.622171	+	1.07763i	−0.989081	+	0.147375i	$0.952918\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			10.3923i			1.70848i	0.519875	+	0.854242i	$0.325978\pi$
$37$			10.3923i			1.70848i	−0.519875	+	0.854242i	$0.674022\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	−0.724797	−	0.688963i	$-0.758066\pi$
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	0.959058	+	0.283211i	$0.0913998\pi$
$42$		0			0
$43$	4.33013	−	2.50000i	0.660338	−	0.381246i	−0.132068	−	0.991241i	$-0.542162\pi$
$43$	4.33013	−	2.50000i	0.660338	−	0.381246i	0.792406	+	0.609994i	$0.208828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	0.711608	−	0.702577i	$-0.247967\pi$
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	−0.964253	+	0.264982i	$0.914634\pi$
$48$		0			0
$49$	3.50000	−	6.06218i	0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			13.8564i			1.90332i	0.307148	+	0.951662i	$0.400625\pi$
$53$			13.8564i			1.90332i	−0.307148	+	0.951662i	$0.599375\pi$
$54$		0			0
$55$	−10.3923			−1.40130
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.79423	+	4.50000i	1.01472	+	0.585850i	0.912571	−	0.408919i	$-0.134094\pi$
$59$	7.79423	+	4.50000i	1.01472	+	0.585850i	0.102151	+	0.994769i	$0.467427\pi$
$60$		0			0
$61$	−6.00000	+	3.46410i	−0.768221	+	0.443533i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−6.00000	+	3.46410i	−0.768221	+	0.443533i	0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		0			0
$67$	−4.33013	−	2.50000i	−0.529009	−	0.305424i	0.211604	−	0.977356i	$-0.432131\pi$
$67$	−4.33013	−	2.50000i	−0.529009	−	0.305424i	−0.740613	+	0.671932i	$0.765465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.46410			−0.411113			−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410			−0.411113			−0.205557	+	0.978645i	$0.565900\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.66025	−	15.0000i	−0.974355	−	1.68763i	−0.682048	−	0.731307i	$-0.738911\pi$
$79$	−8.66025	−	15.0000i	−0.974355	−	1.68763i	−0.292306	−	0.956325i	$-0.594423\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.3923	−	6.00000i	1.14070	−	0.658586i	0.194099	−	0.980982i	$-0.437822\pi$
$83$	10.3923	−	6.00000i	1.14070	−	0.658586i	0.946605	+	0.322396i	$0.104488\pi$
$84$		0			0
$85$	9.00000	+	5.19615i	0.976187	+	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−12.1244	+	21.0000i	−1.24393	+	2.15455i
$96$		0			0
$97$	−0.500000	−	0.866025i	−0.0507673	−	0.0879316i	0.839525	−	0.543321i	$-0.182833\pi$
$97$	−0.500000	−	0.866025i	−0.0507673	−	0.0879316i	−0.890292	+	0.455389i	$0.849500\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.00000	−	1.73205i	0.298511	−	0.172345i	−0.343263	−	0.939239i	$-0.611532\pi$
$101$	3.00000	−	1.73205i	0.298511	−	0.172345i	0.641774	+	0.766894i	$0.278199\pi$
$102$		0			0
$103$	1.73205	−	3.00000i	0.170664	−	0.295599i	−0.767988	−	0.640464i	$-0.778742\pi$
$103$	1.73205	−	3.00000i	0.170664	−	0.295599i	0.938652	+	0.344865i	$0.112075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$			6.92820i			0.663602i	0.943349	+	0.331801i	$0.107656\pi$
$109$			6.92820i			0.663602i	−0.943349	+	0.331801i	$0.892344\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	−10.3923	+	6.00000i	−0.969087	+	0.559503i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000	+	1.73205i	−0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			6.92820i			0.619677i
$126$		0			0
$127$	3.46410			0.307389			0.153695	−	0.988118i	$-0.450883\pi$
$127$	3.46410			0.307389			0.153695	+	0.988118i	$0.450883\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.3923	+	6.00000i	0.907980	+	0.524222i	0.879781	−	0.475380i	$-0.157689\pi$
$131$	10.3923	+	6.00000i	0.907980	+	0.524222i	0.0281993	+	0.999602i	$0.491023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.50000	+	12.9904i	0.640768	+	1.10984i	0.985262	+	0.171054i	$0.0547174\pi$
$137$	7.50000	+	12.9904i	0.640768	+	1.10984i	−0.344493	+	0.938789i	$0.611949\pi$
$138$		0			0
$139$	4.33013	+	2.50000i	0.367277	+	0.212047i	0.672268	−	0.740308i	$-0.265320\pi$
$139$	4.33013	+	2.50000i	0.367277	+	0.212047i	−0.304991	+	0.952355i	$0.598654\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.3923			0.869048
$144$		0			0
$145$	24.0000			1.99309
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.00000	−	5.19615i	−0.737309	−	0.425685i	0.0837813	−	0.996484i	$-0.473300\pi$
$149$	−9.00000	−	5.19615i	−0.737309	−	0.425685i	−0.821090	+	0.570799i	$0.806634\pi$
$150$		0			0
$151$	1.73205	+	3.00000i	0.140952	+	0.244137i	0.927855	−	0.372940i	$-0.121650\pi$
$151$	1.73205	+	3.00000i	0.140952	+	0.244137i	−0.786903	+	0.617076i	$0.788317\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−20.7846	+	12.0000i	−1.66946	+	0.963863i
$156$		0			0
$157$	6.00000	+	3.46410i	0.478852	+	0.276465i	0.719938	−	0.694038i	$-0.244170\pi$
$157$	6.00000	+	3.46410i	0.478852	+	0.276465i	−0.241086	+	0.970504i	$0.577504\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.66025	+	15.0000i	−0.670151	+	1.16073i	0.307711	+	0.951480i	$0.400437\pi$
$167$	−8.66025	+	15.0000i	−0.670151	+	1.16073i	−0.977861	+	0.209255i	$0.932896\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.0384615	−	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.00000	−	3.46410i	0.456172	−	0.263371i	−0.254262	−	0.967135i	$-0.581832\pi$
$173$	6.00000	−	3.46410i	0.456172	−	0.263371i	0.710433	+	0.703765i	$0.248499\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.0000i		−	0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$		−	12.0000i		−	0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$		0			0
$181$		−	10.3923i		−	0.772454i	−0.922404	−	0.386227i	$-0.873778\pi$
$181$		−	10.3923i		−	0.772454i	0.922404	−	0.386227i	$-0.126222\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−18.0000	+	31.1769i	−1.32339	+	2.29217i
$186$		0			0
$187$	−7.79423	+	4.50000i	−0.569970	+	0.329073i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.46410	−	6.00000i	−0.250654	−	0.434145i	0.713052	−	0.701111i	$-0.247312\pi$
$191$	−3.46410	−	6.00000i	−0.250654	−	0.434145i	−0.963706	+	0.266966i	$0.913979\pi$
$192$		0			0
$193$	3.50000	−	6.06218i	0.251936	−	0.436365i	−0.712123	−	0.702055i	$-0.752266\pi$
$193$	3.50000	−	6.06218i	0.251936	−	0.436365i	0.964059	+	0.265689i	$0.0855996\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−6.92820			−0.491127			−0.245564	−	0.969380i	$-0.578973\pi$
$199$	−6.92820			−0.491127			−0.245564	+	0.969380i	$0.578973\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	9.00000	−	5.19615i	0.628587	−	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.5000	−	18.1865i	−0.726300	−	1.25799i
$210$		0			0
$211$	13.8564	+	8.00000i	0.953914	+	0.550743i	0.894295	−	0.447478i	$-0.147678\pi$
$211$	13.8564	+	8.00000i	0.953914	+	0.550743i	0.0596196	+	0.998221i	$0.481011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	17.3205			1.18125
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.00000	−	5.19615i	−0.605406	−	0.349531i
$222$		0			0
$223$	8.66025	+	15.0000i	0.579934	+	1.00447i	0.995486	+	0.0949052i	$0.0302548\pi$
$223$	8.66025	+	15.0000i	0.579934	+	1.00447i	−0.415553	+	0.909569i	$0.636412\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.79423	+	4.50000i	−0.517321	+	0.298675i	−0.735838	−	0.677158i	$-0.763211\pi$
$227$	−7.79423	+	4.50000i	−0.517321	+	0.298675i	0.218517	+	0.975833i	$0.429878\pi$
$228$		0			0
$229$	−12.0000	−	6.92820i	−0.792982	−	0.457829i	0.0480291	−	0.998846i	$-0.484706\pi$
$229$	−12.0000	−	6.92820i	−0.792982	−	0.457829i	−0.841011	+	0.541017i	$0.818039\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	27.0000			1.76883			0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000			1.76883			0.884414	+	0.466702i	$0.154558\pi$
$234$		0			0
$235$		−	12.0000i		−	0.782794i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.1244	−	21.0000i	0.784259	−	1.35838i	−0.145181	−	0.989405i	$-0.546376\pi$
$239$	12.1244	−	21.0000i	0.784259	−	1.35838i	0.929441	−	0.368972i	$-0.120290\pi$
$240$		0			0
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	0.450910	−	0.892570i	$-0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	21.0000	−	12.1244i	1.34164	−	0.774597i
$246$		0			0
$247$	12.1244	−	21.0000i	0.771454	−	1.33620i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.0000i		−	0.946792i	−0.880850	−	0.473396i	$-0.843028\pi$
$251$		−	15.0000i		−	0.946792i	0.880850	−	0.473396i	$-0.156972\pi$
$252$		0			0
$253$		−	10.3923i		−	0.653359i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	−0.531487	−	0.847066i	$-0.678367\pi$
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	0.999325	+	0.0367485i	$0.0117000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.3923	−	18.0000i	−0.640817	−	1.10993i	−0.985251	−	0.171117i	$-0.945262\pi$
$263$	−10.3923	−	18.0000i	−0.640817	−	1.10993i	0.344434	−	0.938811i	$-0.388071\pi$
$264$		0			0
$265$	−24.0000	+	41.5692i	−1.47431	+	2.55358i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			20.7846i			1.26726i	0.773636	+	0.633630i	$0.218436\pi$
$269$			20.7846i			1.26726i	−0.773636	+	0.633630i	$0.781564\pi$
$270$		0			0
$271$	6.92820			0.420858			0.210429	−	0.977609i	$-0.432514\pi$
$271$	6.92820			0.420858			0.210429	+	0.977609i	$0.432514\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−18.1865	−	10.5000i	−1.09669	−	0.633174i
$276$		0			0
$277$	−15.0000	+	8.66025i	−0.901263	+	0.520344i	−0.877610	−	0.479376i	$-0.840863\pi$
$277$	−15.0000	+	8.66025i	−0.901263	+	0.520344i	−0.0236530	+	0.999720i	$0.507530\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	0.762561	−	0.646916i	$-0.223942\pi$
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	−0.941526	+	0.336939i	$0.890608\pi$
$282$		0			0
$283$	−3.46410	−	2.00000i	−0.205919	−	0.118888i	0.393494	−	0.919327i	$-0.371266\pi$
$283$	−3.46410	−	2.00000i	−0.205919	−	0.118888i	−0.599414	+	0.800439i	$0.704600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.0000	−	6.92820i	−0.701047	−	0.404750i	0.106690	−	0.994292i	$-0.465975\pi$
$293$	−12.0000	−	6.92820i	−0.701047	−	0.404750i	−0.807737	+	0.589542i	$0.799308\pi$
$294$		0			0
$295$	15.5885	+	27.0000i	0.907595	+	1.57200i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.3923	−	6.00000i	0.601003	−	0.346989i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−24.0000			−1.37424
$306$		0			0
$307$		−	31.0000i		−	1.76926i	−0.466290	−	0.884632i	$-0.654410\pi$
$307$		−	31.0000i		−	1.76926i	0.466290	−	0.884632i	$-0.345590\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.92820	−	12.0000i	0.392862	−	0.680458i	−0.599963	−	0.800027i	$-0.704818\pi$
$311$	6.92820	−	12.0000i	0.392862	−	0.680458i	0.992826	+	0.119570i	$0.0381515\pi$
$312$		0			0
$313$	−3.50000	−	6.06218i	−0.197832	−	0.342655i	0.749993	−	0.661445i	$-0.230057\pi$
$313$	−3.50000	−	6.06218i	−0.197832	−	0.342655i	−0.947825	+	0.318791i	$0.896723\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	5.19615i	−0.505490	+	0.291845i	−0.730978	−	0.682401i	$-0.760936\pi$
$317$	−9.00000	+	5.19615i	−0.505490	+	0.291845i	0.225488	+	0.974246i	$0.427602\pi$
$318$		0			0
$319$	−10.3923	+	18.0000i	−0.581857	+	1.00781i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			21.0000i			1.16847i
$324$		0			0
$325$		−	24.2487i		−	1.34508i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	24.2487	−	14.0000i	1.33283	−	0.769510i	0.347097	−	0.937829i	$-0.387167\pi$
$331$	24.2487	−	14.0000i	1.33283	−	0.769510i	0.985732	+	0.168320i	$0.0538340\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.66025	−	15.0000i	−0.473160	−	0.819538i
$336$		0			0
$337$	8.50000	−	14.7224i	0.463025	−	0.801982i	−0.536085	−	0.844164i	$-0.680098\pi$
$337$	8.50000	−	14.7224i	0.463025	−	0.801982i	0.999110	+	0.0421818i	$0.0134309\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	20.7846i		−	1.12555i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	23.3827	+	13.5000i	1.25525	+	0.724718i	0.972147	−	0.234372i	$-0.0753034\pi$
$347$	23.3827	+	13.5000i	1.25525	+	0.724718i	0.283101	+	0.959090i	$0.408637\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.50000	−	7.79423i	−0.239511	−	0.414845i	0.721063	−	0.692869i	$-0.243654\pi$
$353$	−4.50000	−	7.79423i	−0.239511	−	0.414845i	−0.960574	+	0.278024i	$0.910320\pi$
$354$		0			0
$355$	−10.3923	−	6.00000i	−0.551566	−	0.318447i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	27.7128			1.46263			0.731313	−	0.682042i	$-0.238908\pi$
$359$	27.7128			1.46263			0.731313	+	0.682042i	$0.238908\pi$
$360$		0			0
$361$	−30.0000			−1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−21.0000	−	12.1244i	−1.09919	−	0.634618i
$366$		0			0
$367$	−8.66025	−	15.0000i	−0.452062	−	0.782994i	0.546452	−	0.837490i	$-0.315978\pi$
$367$	−8.66025	−	15.0000i	−0.452062	−	0.782994i	−0.998514	+	0.0544966i	$0.982645\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−21.0000	−	12.1244i	−1.08734	−	0.627775i	−0.154472	−	0.987997i	$-0.549368\pi$
$373$	−21.0000	−	12.1244i	−1.08734	−	0.627775i	−0.932867	+	0.360222i	$0.882701\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.0000			−1.23606
$378$		0			0
$379$			5.00000i			0.256833i	0.991720	+	0.128416i	$0.0409894\pi$
$379$			5.00000i			0.256833i	−0.991720	+	0.128416i	$0.959011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.73205	−	3.00000i	0.0885037	−	0.153293i	−0.818375	−	0.574684i	$-0.805125\pi$
$383$	1.73205	−	3.00000i	0.0885037	−	0.153293i	0.906879	+	0.421392i	$0.138458\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.00000	+	3.46410i	−0.304212	+	0.175637i	−0.644334	−	0.764745i	$-0.722865\pi$
$389$	−6.00000	+	3.46410i	−0.304212	+	0.175637i	0.340121	+	0.940382i	$0.389532\pi$
$390$		0			0
$391$	−5.19615	+	9.00000i	−0.262781	+	0.455150i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	60.0000i		−	3.01893i
$396$		0			0
$397$			3.46410i			0.173858i	0.996214	+	0.0869291i	$0.0277054\pi$
$397$			3.46410i			0.173858i	−0.996214	+	0.0869291i	$0.972295\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	−0.990257	−	0.139253i	$-0.955530\pi$
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	0.615725	+	0.787961i	$0.288863\pi$
$402$		0			0
$403$	20.7846	−	12.0000i	1.03536	−	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−15.5885	−	27.0000i	−0.772691	−	1.33834i
$408$		0			0
$409$	0.500000	−	0.866025i	0.0247234	−	0.0428222i	−0.853399	−	0.521258i	$-0.825463\pi$
$409$	0.500000	−	0.866025i	0.0247234	−	0.0428222i	0.878122	+	0.478436i	$0.158796\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	41.5692			2.04055
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−31.1769	−	18.0000i	−1.52309	−	0.879358i	−0.999627	−	0.0273103i	$-0.991306\pi$
$419$	−31.1769	−	18.0000i	−1.52309	−	0.879358i	−0.523465	−	0.852047i	$-0.675361\pi$
$420$		0			0
$421$	27.0000	−	15.5885i	1.31590	−	0.759735i	0.332833	−	0.942986i	$-0.391995\pi$
$421$	27.0000	−	15.5885i	1.31590	−	0.759735i	0.983066	+	0.183251i	$0.0586620\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.5000	+	18.1865i	0.509325	+	0.882176i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	27.7128			1.33488			0.667440	−	0.744664i	$-0.267390\pi$
$431$	27.7128			1.33488			0.667440	+	0.744664i	$0.267390\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.0000	−	12.1244i	−1.00457	−	0.579987i
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−28.5788	+	16.5000i	−1.35782	+	0.783939i	−0.989330	−	0.145692i	$-0.953459\pi$
$443$	−28.5788	+	16.5000i	−1.35782	+	0.783939i	−0.368492	+	0.929631i	$0.620126\pi$
$444$		0			0
$445$	18.0000	+	10.3923i	0.853282	+	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.0000			1.27421			0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000			1.27421			0.637104	+	0.770778i	$0.280132\pi$
$450$		0			0
$451$			9.00000i			0.423793i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.50000	+	11.2583i	0.304057	+	0.526642i	0.977051	−	0.213006i	$-0.0683253\pi$
$457$	6.50000	+	11.2583i	0.304057	+	0.526642i	−0.672994	+	0.739648i	$0.734992\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.00000	−	5.19615i	0.419172	−	0.242009i	−0.275551	−	0.961286i	$-0.588860\pi$
$461$	9.00000	−	5.19615i	0.419172	−	0.242009i	0.694723	+	0.719277i	$0.255527\pi$
$462$		0			0
$463$	12.1244	−	21.0000i	0.563467	−	0.975953i	−0.433724	−	0.901046i	$-0.642801\pi$
$463$	12.1244	−	21.0000i	0.563467	−	0.975953i	0.997191	−	0.0749070i	$-0.0238660\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			27.0000i			1.24941i	0.780860	+	0.624705i	$0.214781\pi$
$467$			27.0000i			1.24941i	−0.780860	+	0.624705i	$0.785219\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.50000	+	12.9904i	−0.344850	+	0.597298i
$474$		0			0
$475$	−42.4352	+	24.5000i	−1.94706	+	1.12414i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.3923	+	18.0000i	0.474837	+	0.822441i	0.999585	−	0.0288165i	$-0.00917385\pi$
$479$	10.3923	+	18.0000i	0.474837	+	0.822441i	−0.524748	+	0.851258i	$0.675841\pi$
$480$		0			0
$481$	18.0000	−	31.1769i	0.820729	−	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	3.46410i		−	0.157297i
$486$		0			0
$487$	27.7128			1.25579			0.627894	−	0.778299i	$-0.283917\pi$
$487$	27.7128			1.25579			0.627894	+	0.778299i	$0.283917\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.59808	+	1.50000i	0.117250	+	0.0676941i	0.557478	−	0.830192i	$-0.311769\pi$
$491$	2.59808	+	1.50000i	0.117250	+	0.0676941i	−0.440228	+	0.897886i	$0.645102\pi$
$492$		0			0
$493$	18.0000	−	10.3923i	0.810679	−	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.866025	+	0.500000i	0.0387686	+	0.0223831i	0.519259	−	0.854617i	$-0.326208\pi$
$499$	0.866025	+	0.500000i	0.0387686	+	0.0223831i	−0.480490	+	0.877000i	$0.659541\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6.92820			−0.308913			−0.154457	−	0.988000i	$-0.549363\pi$
$503$	−6.92820			−0.308913			−0.154457	+	0.988000i	$0.549363\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000	+	3.46410i	0.265945	+	0.153544i	0.627044	−	0.778984i	$-0.284265\pi$
$509$	6.00000	+	3.46410i	0.265945	+	0.153544i	−0.361098	+	0.932528i	$0.617598\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.3923	−	6.00000i	0.457940	−	0.264392i
$516$		0			0
$517$	9.00000	+	5.19615i	0.395820	+	0.228527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.00000			0.394297			0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000			0.394297			0.197149	+	0.980374i	$0.436832\pi$
$522$		0			0
$523$			20.0000i			0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$			20.0000i			0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.3923	+	18.0000i	−0.452696	+	0.784092i
$528$		0			0
$529$	5.50000	+	9.52628i	0.239130	+	0.414186i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.00000	+	5.19615i	−0.389833	+	0.225070i
$534$		0			0
$535$	−5.19615	+	9.00000i	−0.224649	+	0.389104i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			21.0000i			0.904534i
$540$		0			0
$541$		−	31.1769i		−	1.34040i	−0.742180	−	0.670200i	$-0.766208\pi$
$541$		−	31.1769i		−	1.34040i	0.742180	−	0.670200i	$-0.233792\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000	+	20.7846i	−0.514024	+	0.890315i
$546$		0			0
$547$	16.4545	−	9.50000i	0.703543	−	0.406191i	−0.105123	−	0.994459i	$-0.533524\pi$
$547$	16.4545	−	9.50000i	0.703543	−	0.406191i	0.808666	+	0.588269i	$0.200190\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	24.2487	+	42.0000i	1.03303	+	1.78926i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	20.7846i		−	0.880672i	−0.897833	−	0.440336i	$-0.854859\pi$
$557$		−	20.7846i		−	0.880672i	0.897833	−	0.440336i	$-0.145141\pi$
$558$		0			0
$559$	−17.3205			−0.732579
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.79423	+	4.50000i	0.328488	+	0.189652i	0.655169	−	0.755482i	$-0.272597\pi$
$563$	7.79423	+	4.50000i	0.328488	+	0.189652i	−0.326682	+	0.945134i	$0.605931\pi$
$564$		0			0
$565$	18.0000	−	10.3923i	0.757266	−	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.5000	+	33.7750i	0.817483	+	1.41592i	0.907532	+	0.419984i	$0.137964\pi$
$569$	19.5000	+	33.7750i	0.817483	+	1.41592i	−0.0900490	+	0.995937i	$0.528702\pi$
$570$		0			0
$571$	−32.0429	−	18.5000i	−1.34096	−	0.774201i	−0.354008	−	0.935243i	$-0.615181\pi$
$571$	−32.0429	−	18.5000i	−1.34096	−	0.774201i	−0.986948	+	0.161042i	$0.948515\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.2487			−1.01124
$576$		0			0
$577$	7.00000			0.291414			0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000			0.291414			0.145707	+	0.989328i	$0.453454\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−20.7846	−	36.0000i	−0.860811	−	1.49097i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.59808	−	1.50000i	0.107234	−	0.0619116i	−0.445424	−	0.895320i	$-0.646947\pi$
$587$	2.59808	−	1.50000i	0.107234	−	0.0619116i	0.552658	+	0.833408i	$0.313614\pi$
$588$		0			0
$589$	−42.0000	−	24.2487i	−1.73058	−	0.999151i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.73205	+	3.00000i	−0.0707697	+	0.122577i	−0.899239	−	0.437458i	$-0.855879\pi$
$599$	−1.73205	+	3.00000i	−0.0707697	+	0.122577i	0.828469	+	0.560035i	$0.189212\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.00000	+	3.46410i	−0.243935	+	0.140836i
$606$		0			0
$607$	10.3923	−	18.0000i	0.421811	−	0.730597i	−0.574306	−	0.818641i	$-0.694728\pi$
$607$	10.3923	−	18.0000i	0.421811	−	0.730597i	0.996117	+	0.0880432i	$0.0280614\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			12.0000i			0.485468i
$612$		0			0
$613$			24.2487i			0.979396i	0.871892	+	0.489698i	$0.162893\pi$
$613$			24.2487i			0.979396i	−0.871892	+	0.489698i	$0.837107\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	−0.894639	−	0.446790i	$-0.852567\pi$
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	0.834251	+	0.551385i	$0.185900\pi$
$618$		0			0
$619$	11.2583	−	6.50000i	0.452510	−	0.261257i	−0.256379	−	0.966576i	$-0.582530\pi$
$619$	11.2583	−	6.50000i	0.452510	−	0.261257i	0.708890	+	0.705319i	$0.249196\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			31.1769i			1.24310i
$630$		0			0
$631$	31.1769			1.24113			0.620567	−	0.784154i	$-0.286903\pi$
$631$	31.1769			1.24113			0.620567	+	0.784154i	$0.286903\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.3923	+	6.00000i	0.412406	+	0.238103i
$636$		0			0
$637$	−21.0000	+	12.1244i	−0.832050	+	0.480384i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	−0.937451	−	0.348117i	$-0.886821\pi$
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	0.167247	−	0.985915i	$-0.446512\pi$
$642$		0			0
$643$	−16.4545	−	9.50000i	−0.648901	−	0.374643i	0.139134	−	0.990274i	$-0.455568\pi$
$643$	−16.4545	−	9.50000i	−0.648901	−	0.374643i	−0.788035	+	0.615630i	$0.788902\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−34.6410			−1.36188			−0.680939	−	0.732340i	$-0.738428\pi$
$647$	−34.6410			−1.36188			−0.680939	+	0.732340i	$0.738428\pi$
$648$		0			0
$649$	−27.0000			−1.05984
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$	20.7846	+	36.0000i	0.812122	+	1.40664i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.3923	−	6.00000i	0.404827	−	0.233727i	−0.283738	−	0.958902i	$-0.591575\pi$
$659$	10.3923	−	6.00000i	0.404827	−	0.233727i	0.688565	+	0.725175i	$0.258241\pi$
$660$		0			0
$661$	33.0000	+	19.0526i	1.28355	+	0.741059i	0.977496	−	0.210955i	$-0.0676574\pi$
$661$	33.0000	+	19.0526i	1.28355	+	0.741059i	0.306055	+	0.952014i	$0.400991\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			24.0000i			0.929284i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.3923	−	18.0000i	0.401190	−	0.694882i
$672$		0			0
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	0.753420	−	0.657539i	$-0.228403\pi$
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	−0.946156	+	0.323711i	$0.895069\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.0000	+	6.92820i	−0.461197	+	0.266272i	−0.712548	−	0.701624i	$-0.752459\pi$
$677$	−12.0000	+	6.92820i	−0.461197	+	0.266272i	0.251350	+	0.967896i	$0.419125\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	45.0000i		−	1.72188i	−0.508709	−	0.860939i	$-0.669877\pi$
$683$		−	45.0000i		−	1.72188i	0.508709	−	0.860939i	$-0.330123\pi$
$684$		0			0
$685$			51.9615i			1.98535i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.0000	−	41.5692i	0.914327	−	1.58366i
$690$		0			0
$691$	24.2487	−	14.0000i	0.922464	−	0.532585i	0.0380440	−	0.999276i	$-0.487887\pi$
$691$	24.2487	−	14.0000i	0.922464	−	0.532585i	0.884420	+	0.466691i	$0.154554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.66025	+	15.0000i	0.328502	+	0.568982i
$696$		0			0
$697$	4.50000	−	7.79423i	0.170450	−	0.295227i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.1769i		−	1.17754i	−0.808302	−	0.588768i	$-0.799613\pi$
$701$		−	31.1769i		−	1.17754i	0.808302	−	0.588768i	$-0.200387\pi$
$702$		0			0
$703$	−72.7461			−2.74367
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−24.0000	+	13.8564i	−0.901339	+	0.520388i	−0.877634	−	0.479331i	$-0.840879\pi$
$709$	−24.0000	+	13.8564i	−0.901339	+	0.520388i	−0.0237046	+	0.999719i	$0.507546\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.0000	−	20.7846i	−0.449404	−	0.778390i
$714$		0			0
$715$	31.1769	+	18.0000i	1.16595	+	0.673162i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.3923			−0.387568			−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923			−0.387568			−0.193784	+	0.981044i	$0.562076\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	42.0000	+	24.2487i	1.55984	+	0.900575i
$726$		0			0
$727$	−15.5885	−	27.0000i	−0.578144	−	1.00137i	−0.995692	−	0.0927199i	$-0.970444\pi$
$727$	−15.5885	−	27.0000i	−0.578144	−	1.00137i	0.417548	−	0.908655i	$-0.362889\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.9904	−	7.50000i	0.480467	−	0.277398i
$732$		0			0
$733$	12.0000	+	6.92820i	0.443230	+	0.255899i	0.704967	−	0.709240i	$-0.250962\pi$
$733$	12.0000	+	6.92820i	0.443230	+	0.255899i	−0.261737	+	0.965139i	$0.584295\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.0000			0.552532
$738$		0			0
$739$		−	1.00000i		−	0.0367856i	−0.999831	−	0.0183928i	$-0.994145\pi$
$739$		−	1.00000i		−	0.0367856i	0.999831	−	0.0183928i	$-0.00585494\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.92820	−	12.0000i	0.254171	−	0.440237i	−0.710499	−	0.703698i	$-0.751531\pi$
$743$	6.92820	−	12.0000i	0.254171	−	0.440237i	0.964670	+	0.263461i	$0.0848640\pi$
$744$		0			0
$745$	−18.0000	−	31.1769i	−0.659469	−	1.14223i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.5885	+	27.0000i	−0.568831	+	0.985244i	0.427851	+	0.903849i	$0.359271\pi$
$751$	−15.5885	+	27.0000i	−0.568831	+	0.985244i	−0.996682	+	0.0813947i	$0.974063\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			12.0000i			0.436725i
$756$		0			0
$757$			27.7128i			1.00724i	0.863925	+	0.503620i	$0.167999\pi$
$757$			27.7128i			1.00724i	−0.863925	+	0.503620i	$0.832001\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.5885	−	27.0000i	−0.562867	−	0.974913i
$768$		0			0
$769$	−25.0000	+	43.3013i	−0.901523	+	1.56148i	−0.0760054	+	0.997107i	$0.524217\pi$
$769$	−25.0000	+	43.3013i	−0.901523	+	1.56148i	−0.825518	+	0.564376i	$0.809117\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			24.2487i			0.872166i	0.899907	+	0.436083i	$0.143635\pi$
$773$			24.2487i			0.872166i	−0.899907	+	0.436083i	$0.856365\pi$
$774$		0			0
$775$	−48.4974			−1.74208
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18.1865	+	10.5000i	0.651600	+	0.376202i
$780$		0			0
$781$	9.00000	−	5.19615i	0.322045	−	0.185933i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.0000	+	20.7846i	0.428298	+	0.741835i
$786$		0			0
$787$	−3.46410	−	2.00000i	−0.123482	−	0.0712923i	0.436987	−	0.899468i	$-0.356046\pi$
$787$	−3.46410	−	2.00000i	−0.123482	−	0.0712923i	−0.560469	+	0.828176i	$0.689379\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	24.0000			0.852265
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.0000	+	12.1244i	0.743858	+	0.429467i	0.823470	−	0.567359i	$-0.192035\pi$
$797$	21.0000	+	12.1244i	0.743858	+	0.429467i	−0.0796123	+	0.996826i	$0.525368\pi$
$798$		0			0
$799$	−5.19615	−	9.00000i	−0.183827	−	0.318397i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18.1865	−	10.5000i	0.641789	−	0.370537i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.0000			−0.527372			−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000			−0.527372			−0.263686	+	0.964609i	$0.584938\pi$
$810$		0			0
$811$		−	25.0000i		−	0.877869i	−0.898519	−	0.438934i	$-0.855356\pi$
$811$		−	25.0000i		−	0.877869i	0.898519	−	0.438934i	$-0.144644\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.92820	+	12.0000i	−0.242684	+	0.420342i
$816$		0			0
$817$	17.5000	+	30.3109i	0.612247	+	1.06044i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.0000	+	6.92820i	−0.418803	+	0.241796i	−0.694565	−	0.719430i	$-0.744403\pi$
$821$	−12.0000	+	6.92820i	−0.418803	+	0.241796i	0.275762	+	0.961226i	$0.411070\pi$
$822$		0			0
$823$	−22.5167	+	39.0000i	−0.784881	+	1.35945i	0.144188	+	0.989550i	$0.453943\pi$
$823$	−22.5167	+	39.0000i	−0.784881	+	1.35945i	−0.929070	+	0.369904i	$0.879390\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			48.0000i			1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$			48.0000i			1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$		0			0
$829$		−	34.6410i		−	1.20313i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$		−	34.6410i		−	1.20313i	0.798823	−	0.601566i	$-0.205456\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.5000	−	18.1865i	0.363803	−	0.630126i
$834$		0			0
$835$	−51.9615	+	30.0000i	−1.79820	+	1.03819i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.3205	−	30.0000i	−0.597970	−	1.03572i	−0.993120	−	0.117098i	$-0.962641\pi$
$839$	−17.3205	−	30.0000i	−0.597970	−	1.03572i	0.395150	−	0.918617i	$-0.370693\pi$
$840$		0			0
$841$	9.50000	−	16.4545i	0.327586	−	0.567396i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	3.46410i		−	0.119169i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−31.1769	−	18.0000i	−1.06873	−	0.617032i
$852$		0			0
$853$	−21.0000	+	12.1244i	−0.719026	+	0.415130i	−0.814394	−	0.580312i	$-0.802931\pi$
$853$	−21.0000	+	12.1244i	−0.719026	+	0.415130i	0.0953679	+	0.995442i	$0.469597\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	0.670366	−	0.742030i	$-0.266137\pi$
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	−0.977800	+	0.209539i	$0.932804\pi$
$858$		0			0
$859$	16.4545	+	9.50000i	0.561420	+	0.324136i	0.753715	−	0.657201i	$-0.228260\pi$
$859$	16.4545	+	9.50000i	0.561420	+	0.324136i	−0.192295	+	0.981337i	$0.561593\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.3205			0.589597			0.294798	−	0.955559i	$-0.404747\pi$
$863$	17.3205			0.589597			0.294798	+	0.955559i	$0.404747\pi$
$864$		0			0
$865$	24.0000			0.816024
$866$		0			0
$867$		0			0
$868$		0			0
$869$	45.0000	+	25.9808i	1.52652	+	0.881337i
$870$		0			0
$871$	8.66025	+	15.0000i	0.293442	+	0.508256i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−36.0000	−	20.7846i	−1.21563	−	0.701846i	−0.251653	−	0.967818i	$-0.580974\pi$
$877$	−36.0000	−	20.7846i	−1.21563	−	0.701846i	−0.963981	+	0.265971i	$0.914307\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$			1.00000i			0.0336527i	0.999858	+	0.0168263i	$0.00535624\pi$
$883$			1.00000i			0.0336527i	−0.999858	+	0.0168263i	$0.994644\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.5885	+	27.0000i	−0.523409	+	0.906571i	0.476220	+	0.879326i	$0.342007\pi$
$887$	−15.5885	+	27.0000i	−0.523409	+	0.906571i	−0.999629	+	0.0272449i	$0.991327\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	21.0000	−	12.1244i	0.702738	−	0.405726i
$894$		0			0
$895$	20.7846	−	36.0000i	0.694753	−	1.20335i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			48.0000i			1.60089i
$900$		0			0
$901$			41.5692i			1.38487i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.0000	−	31.1769i	0.598340	−	1.03636i
$906$		0			0
$907$	−0.866025	+	0.500000i	−0.0287559	+	0.0166022i	−0.514309	−	0.857605i	$-0.671952\pi$
$907$	−0.866025	+	0.500000i	−0.0287559	+	0.0166022i	0.485553	+	0.874207i	$0.338618\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.2487	+	42.0000i	0.803396	+	1.39152i	0.917369	+	0.398038i	$0.130309\pi$
$911$	24.2487	+	42.0000i	0.803396	+	1.39152i	−0.113973	+	0.993484i	$0.536358\pi$
$912$		0			0
$913$	−18.0000	+	31.1769i	−0.595713	+	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−10.3923			−0.342811			−0.171405	−	0.985201i	$-0.554831\pi$
$919$	−10.3923			−0.342811			−0.171405	+	0.985201i	$0.554831\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.3923	+	6.00000i	0.342067	+	0.197492i
$924$		0			0
$925$	−63.0000	+	36.3731i	−2.07143	+	1.19594i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.0000	+	25.9808i	0.492134	+	0.852401i	0.999959	−	0.00905914i	$-0.00288365\pi$
$929$	15.0000	+	25.9808i	0.492134	+	0.852401i	−0.507825	+	0.861460i	$0.669550\pi$
$930$		0			0
$931$	42.4352	+	24.5000i	1.39076	+	0.802955i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.1769			−1.01959
$936$		0			0
$937$	34.0000			1.11073			0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000			1.11073			0.555366	+	0.831606i	$0.312578\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	48.0000	+	27.7128i	1.56476	+	0.903412i	0.996765	+	0.0803769i	$0.0256124\pi$
$941$	48.0000	+	27.7128i	1.56476	+	0.903412i	0.567991	+	0.823035i	$0.307721\pi$
$942$		0			0
$943$	5.19615	+	9.00000i	0.169210	+	0.293080i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.1865	−	10.5000i	0.590983	−	0.341204i	−0.174503	−	0.984657i	$-0.555832\pi$
$947$	18.1865	−	10.5000i	0.590983	−	0.341204i	0.765486	+	0.643452i	$0.222499\pi$
$948$		0			0
$949$	21.0000	+	12.1244i	0.681689	+	0.393573i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.0000			0.680257			0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000			0.680257			0.340128	+	0.940379i	$0.389529\pi$
$954$		0			0
$955$		−	24.0000i		−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−8.50000	−	14.7224i	−0.274194	−	0.474917i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	21.0000	−	12.1244i	0.676014	−	0.390297i
$966$		0			0
$967$	−6.92820	+	12.0000i	−0.222796	+	0.385894i	−0.955656	−	0.294486i	$-0.904852\pi$
$967$	−6.92820	+	12.0000i	−0.222796	+	0.385894i	0.732860	+	0.680380i	$0.238185\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	36.0000i		−	1.15529i	−0.816286	−	0.577647i	$-0.803971\pi$
$971$		−	36.0000i		−	1.15529i	0.816286	−	0.577647i	$-0.196029\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.50000	−	7.79423i	0.143968	−	0.249359i	−0.785020	−	0.619471i	$-0.787347\pi$
$977$	4.50000	−	7.79423i	0.143968	−	0.249359i	0.928987	+	0.370111i	$0.120681\pi$
$978$		0			0
$979$	−15.5885	+	9.00000i	−0.498209	+	0.287641i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25.9808	+	45.0000i	0.828658	+	1.43528i	0.899092	+	0.437761i	$0.144228\pi$
$983$	25.9808	+	45.0000i	0.828658	+	1.43528i	−0.0704339	+	0.997516i	$0.522438\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			17.3205i			0.550760i
$990$		0			0
$991$	−13.8564			−0.440163			−0.220082	−	0.975481i	$-0.570632\pi$
$991$	−13.8564			−0.440163			−0.220082	+	0.975481i	$0.570632\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.7846	−	12.0000i	−0.658916	−	0.380426i
$996$		0			0
$997$	−18.0000	+	10.3923i	−0.570066	+	0.329128i	−0.757176	−	0.653211i	$-0.773421\pi$
$997$	−18.0000	+	10.3923i	−0.570066	+	0.329128i	0.187110	+	0.982339i	$0.440088\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.r.b.1441.1		4
3.2	odd	2		576.2.r.a.481.1	yes	4
4.3	odd	2	inner	1728.2.r.b.1441.2		4
8.3	odd	2		1728.2.r.a.1441.1		4
8.5	even	2		1728.2.r.a.1441.2		4
9.2	odd	6		576.2.r.b.97.2	yes	4
9.4	even	3		5184.2.d.j.2593.1		4
9.5	odd	6		5184.2.d.c.2593.4		4
9.7	even	3		1728.2.r.a.289.2		4
12.11	even	2		576.2.r.a.481.2	yes	4
24.5	odd	2		576.2.r.b.481.2	yes	4
24.11	even	2		576.2.r.b.481.1	yes	4
36.7	odd	6		1728.2.r.a.289.1		4
36.11	even	6		576.2.r.b.97.1	yes	4
36.23	even	6		5184.2.d.c.2593.3		4
36.31	odd	6		5184.2.d.j.2593.2		4
72.5	odd	6		5184.2.d.c.2593.1		4
72.11	even	6		576.2.r.a.97.2	yes	4
72.13	even	6		5184.2.d.j.2593.4		4
72.29	odd	6		576.2.r.a.97.1	✓	4
72.43	odd	6	inner	1728.2.r.b.289.2		4
72.59	even	6		5184.2.d.c.2593.2		4
72.61	even	6	inner	1728.2.r.b.289.1		4
72.67	odd	6		5184.2.d.j.2593.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.a.97.1	✓	4	72.29	odd	6
576.2.r.a.97.2	yes	4	72.11	even	6
576.2.r.a.481.1	yes	4	3.2	odd	2
576.2.r.a.481.2	yes	4	12.11	even	2
576.2.r.b.97.1	yes	4	36.11	even	6
576.2.r.b.97.2	yes	4	9.2	odd	6
576.2.r.b.481.1	yes	4	24.11	even	2
576.2.r.b.481.2	yes	4	24.5	odd	2
1728.2.r.a.289.1		4	36.7	odd	6
1728.2.r.a.289.2		4	9.7	even	3
1728.2.r.a.1441.1		4	8.3	odd	2
1728.2.r.a.1441.2		4	8.5	even	2
1728.2.r.b.289.1		4	72.61	even	6	inner
1728.2.r.b.289.2		4	72.43	odd	6	inner
1728.2.r.b.1441.1		4	1.1	even	1	trivial
1728.2.r.b.1441.2		4	4.3	odd	2	inner
5184.2.d.c.2593.1		4	72.5	odd	6
5184.2.d.c.2593.2		4	72.59	even	6
5184.2.d.c.2593.3		4	36.23	even	6
5184.2.d.c.2593.4		4	9.5	odd	6
5184.2.d.j.2593.1		4	9.4	even	3
5184.2.d.j.2593.2		4	36.31	odd	6
5184.2.d.j.2593.3		4	72.67	odd	6
5184.2.d.j.2593.4		4	72.13	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.00000 + 1.73205i) q^{5} +O(q^{10})\)	\(q+(3.00000 + 1.73205i) q^{5} +(-2.59808 + 1.50000i) q^{11} +(-3.00000 - 1.73205i) q^{13} +3.00000 q^{17} +7.00000i q^{19} +(-1.73205 + 3.00000i) q^{23} +(3.50000 + 6.06218i) q^{25} +(6.00000 - 3.46410i) q^{29} +(-3.46410 + 6.00000i) q^{31} +10.3923i q^{37} +(1.50000 - 2.59808i) q^{41} +(4.33013 - 2.50000i) q^{43} +(-1.73205 - 3.00000i) q^{47} +(3.50000 - 6.06218i) q^{49} +13.8564i q^{53} -10.3923 q^{55} +(7.79423 + 4.50000i) q^{59} +(-6.00000 + 3.46410i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(-4.33013 - 2.50000i) q^{67} -3.46410 q^{71} -7.00000 q^{73} +(-8.66025 - 15.0000i) q^{79} +(10.3923 - 6.00000i) q^{83} +(9.00000 + 5.19615i) q^{85} +6.00000 q^{89} +(-12.1244 + 21.0000i) q^{95} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5}+O(q^{10}) \) 4 * q + 12 * q^5	\( 4 q + 12 q^{5} - 12 q^{13} + 12 q^{17} + 14 q^{25} + 24 q^{29} + 6 q^{41} + 14 q^{49} - 24 q^{61} - 24 q^{65} - 28 q^{73} + 36 q^{85} + 24 q^{89} - 2 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 - 12 * q^13 + 12 * q^17 + 14 * q^25 + 24 * q^29 + 6 * q^41 + 14 * q^49 - 24 * q^61 - 24 * q^65 - 28 * q^73 + 36 * q^85 + 24 * q^89 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more