LMFDB - Embedded newform 2448.2.be.n.1441.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,2,Mod(1441,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2448.1441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.be (of order $4$, 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:	$19.5473784148$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1441.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2448.1441
Dual form			2448.2.be.n.1585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 + 3.00000i) q^{5} +(3.00000 - 3.00000i) q^{7} +(1.00000 - 1.00000i) q^{11} +2.00000 q^{13} +(-1.00000 + 4.00000i) q^{17} +6.00000i q^{19} +(1.00000 - 1.00000i) q^{23} +13.0000i q^{25} +(-1.00000 - 1.00000i) q^{29} +(1.00000 + 1.00000i) q^{31} +18.0000 q^{35} +(-3.00000 - 3.00000i) q^{37} +(-1.00000 + 1.00000i) q^{41} -2.00000i q^{43} -11.0000i q^{49} -4.00000i q^{53} +6.00000 q^{55} -6.00000i q^{59} +(1.00000 - 1.00000i) q^{61} +(6.00000 + 6.00000i) q^{65} +4.00000 q^{67} +(-5.00000 - 5.00000i) q^{71} +(1.00000 + 1.00000i) q^{73} -6.00000i q^{77} +(-9.00000 + 9.00000i) q^{79} -14.0000i q^{83} +(-15.0000 + 9.00000i) q^{85} -6.00000 q^{89} +(6.00000 - 6.00000i) q^{91} +(-18.0000 + 18.0000i) q^{95} +(13.0000 + 13.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 6 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} + 2 q^{23} - 2 q^{29} + 2 q^{31} + 36 q^{35} - 6 q^{37} - 2 q^{41} + 12 q^{55} + 2 q^{61} + 12 q^{65} + 8 q^{67} - 10 q^{71} + 2 q^{73} - 18 q^{79}+ \cdots + 26 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 6 * q^7 + 2 * q^11 + 4 * q^13 - 2 * q^17 + 2 * q^23 - 2 * q^29 + 2 * q^31 + 36 * q^35 - 6 * q^37 - 2 * q^41 + 12 * q^55 + 2 * q^61 + 12 * q^65 + 8 * q^67 - 10 * q^71 + 2 * q^73 - 18 * q^79 - 30 * q^85 - 12 * q^89 + 12 * q^91 - 36 * q^95 + 26 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$.

$n$	$613$	$1361$	$1873$	$2143$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	3.00000	+	3.00000i	1.34164	+	1.34164i	0.894427	+	0.447214i	$0.147584\pi$
$5$	3.00000	+	3.00000i	1.34164	+	1.34164i	0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$	3.00000	−	3.00000i	1.13389	−	1.13389i	0.144370	−	0.989524i	$-0.453885\pi$
$7$	3.00000	−	3.00000i	1.13389	−	1.13389i	0.989524	−	0.144370i	$-0.0461154\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	−0.540094	−	0.841605i	$-0.681611\pi$
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	0.841605	+	0.540094i	$0.181611\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	+	4.00000i	−0.242536	+	0.970143i
$17$	−1.00000	+	4.00000i	−0.242536	+	0.970143i
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	0.803636	+	0.595121i	$0.202896\pi$
$24$		0			0
$25$			13.0000i			2.60000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000	−	1.00000i	−0.185695	−	0.185695i	0.608137	−	0.793832i	$-0.291917\pi$
$29$	−1.00000	−	1.00000i	−0.185695	−	0.185695i	−0.793832	+	0.608137i	$0.791917\pi$
$30$		0			0
$31$	1.00000	+	1.00000i	0.179605	+	0.179605i	0.791184	−	0.611578i	$-0.209465\pi$
$31$	1.00000	+	1.00000i	0.179605	+	0.179605i	−0.611578	+	0.791184i	$0.709465\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	18.0000			3.04256
$36$		0			0
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.00000	+	1.00000i	−0.156174	+	0.156174i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−1.00000	+	1.00000i	−0.156174	+	0.156174i	0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	1.00000	−	1.00000i	0.128037	−	0.128037i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	1.00000	−	1.00000i	0.128037	−	0.128037i	0.768221	+	0.640184i	$0.221142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	+	6.00000i	0.744208	+	0.744208i
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.00000	−	5.00000i	−0.593391	−	0.593391i	0.345155	−	0.938546i	$-0.387826\pi$
$71$	−5.00000	−	5.00000i	−0.593391	−	0.593391i	−0.938546	+	0.345155i	$0.887826\pi$
$72$		0			0
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	6.00000i		−	0.683763i
$78$		0			0
$79$	−9.00000	+	9.00000i	−1.01258	+	1.01258i	−0.0126592	+	0.999920i	$0.504030\pi$
$79$	−9.00000	+	9.00000i	−1.01258	+	1.01258i	−0.999920	+	0.0126592i	$0.995970\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$		0			0
$85$	−15.0000	+	9.00000i	−1.62698	+	0.976187i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	6.00000	−	6.00000i	0.628971	−	0.628971i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−18.0000	+	18.0000i	−1.84676	+	1.84676i
$96$		0			0
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.913812	+	0.406138i	$0.133125\pi$
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.00000	+	7.00000i	0.676716	+	0.676716i	0.959256	−	0.282540i	$-0.0911770\pi$
$107$	7.00000	+	7.00000i	0.676716	+	0.676716i	−0.282540	+	0.959256i	$0.591177\pi$
$108$		0			0
$109$	9.00000	−	9.00000i	0.862044	−	0.862044i	−0.129532	−	0.991575i	$-0.541347\pi$
$109$	9.00000	−	9.00000i	0.862044	−	0.862044i	0.991575	+	0.129532i	$0.0413474\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.00000	+	5.00000i	−0.470360	+	0.470360i	−0.902031	−	0.431671i	$-0.857924\pi$
$113$	−5.00000	+	5.00000i	−0.470360	+	0.470360i	0.431671	+	0.902031i	$0.357924\pi$
$114$		0			0
$115$	6.00000			0.559503
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.00000	+	15.0000i	0.825029	+	1.37505i
$120$		0			0
$121$			9.00000i			0.818182i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−24.0000	+	24.0000i	−2.14663	+	2.14663i
$126$		0			0
$127$		−	14.0000i		−	1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$		−	14.0000i		−	1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.00000	−	9.00000i	−0.786334	−	0.786334i	0.194557	−	0.980891i	$-0.437673\pi$
$131$	−9.00000	−	9.00000i	−0.786334	−	0.786334i	−0.980891	+	0.194557i	$0.937673\pi$
$132$		0			0
$133$	18.0000	+	18.0000i	1.56080	+	1.56080i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	−11.0000	−	11.0000i	−0.933008	−	0.933008i	0.0648849	−	0.997893i	$-0.479332\pi$
$139$	−11.0000	−	11.0000i	−0.933008	−	0.933008i	−0.997893	+	0.0648849i	$0.979332\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000	−	2.00000i	0.167248	−	0.167248i
$144$		0			0
$145$		−	6.00000i		−	0.498273i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$			2.00000i			0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$			2.00000i			0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			6.00000i			0.481932i
$156$		0			0
$157$	−10.0000			−0.798087			−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000			−0.798087			−0.399043	+	0.916932i	$0.630658\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.00000i		−	0.472866i
$162$		0			0
$163$	−1.00000	+	1.00000i	−0.0783260	+	0.0783260i	−0.745184	−	0.666858i	$-0.767639\pi$
$163$	−1.00000	+	1.00000i	−0.0783260	+	0.0783260i	0.666858	+	0.745184i	$0.267639\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.0000	−	13.0000i	−1.00597	−	1.00597i	−0.999982	−	0.00598813i	$-0.998094\pi$
$167$	−13.0000	−	13.0000i	−1.00597	−	1.00597i	−0.00598813	−	0.999982i	$-0.501906\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.00000	+	7.00000i	0.532200	+	0.532200i	0.921227	−	0.389026i	$-0.127189\pi$
$173$	7.00000	+	7.00000i	0.532200	+	0.532200i	−0.389026	+	0.921227i	$0.627189\pi$
$174$		0			0
$175$	39.0000	+	39.0000i	2.94812	+	2.94812i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			18.0000i			1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$			18.0000i			1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$		0			0
$181$	9.00000	−	9.00000i	0.668965	−	0.668965i	−0.288512	−	0.957476i	$-0.593160\pi$
$181$	9.00000	−	9.00000i	0.668965	−	0.668965i	0.957476	+	0.288512i	$0.0931604\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	18.0000i		−	1.32339i
$186$		0			0
$187$	3.00000	+	5.00000i	0.219382	+	0.365636i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$	9.00000	−	9.00000i	0.647834	−	0.647834i	−0.304635	−	0.952469i	$-0.598534\pi$
$193$	9.00000	−	9.00000i	0.647834	−	0.647834i	0.952469	+	0.304635i	$0.0985345\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.0000	−	11.0000i	0.783718	−	0.783718i	−0.196738	−	0.980456i	$-0.563035\pi$
$197$	11.0000	−	11.0000i	0.783718	−	0.783718i	0.980456	+	0.196738i	$0.0630350\pi$
$198$		0			0
$199$	−7.00000	−	7.00000i	−0.496217	−	0.496217i	0.414041	−	0.910258i	$-0.364117\pi$
$199$	−7.00000	−	7.00000i	−0.496217	−	0.496217i	−0.910258	+	0.414041i	$0.864117\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.00000			−0.421117
$204$		0			0
$205$	−6.00000			−0.419058
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.00000	+	6.00000i	0.415029	+	0.415029i
$210$		0			0
$211$	−5.00000	+	5.00000i	−0.344214	+	0.344214i	−0.857949	−	0.513735i	$-0.828262\pi$
$211$	−5.00000	+	5.00000i	−0.344214	+	0.344214i	0.513735	+	0.857949i	$0.328262\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.00000	−	6.00000i	0.409197	−	0.409197i
$216$		0			0
$217$	6.00000			0.407307
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000	+	8.00000i	−0.134535	+	0.538138i
$222$		0			0
$223$			26.0000i			1.74109i	0.492090	+	0.870544i	$0.336233\pi$
$223$			26.0000i			1.74109i	−0.492090	+	0.870544i	$0.663767\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	−0.999990	−	0.00440533i	$-0.998598\pi$
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	0.00440533	+	0.999990i	$0.498598\pi$
$228$		0			0
$229$		−	16.0000i		−	1.05731i	−0.848837	−	0.528655i	$-0.822697\pi$
$229$		−	16.0000i		−	1.05731i	0.848837	−	0.528655i	$-0.177303\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	19.0000	+	19.0000i	1.24473	+	1.24473i	0.958016	+	0.286716i	$0.0925635\pi$
$233$	19.0000	+	19.0000i	1.24473	+	1.24473i	0.286716	+	0.958016i	$0.407437\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	9.00000	+	9.00000i	0.579741	+	0.579741i	0.934832	−	0.355091i	$-0.115550\pi$
$241$	9.00000	+	9.00000i	0.579741	+	0.579741i	−0.355091	+	0.934832i	$0.615550\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	33.0000	−	33.0000i	2.10829	−	2.10829i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.0000i		−	1.24757i	−0.781598	−	0.623783i	$-0.785595\pi$
$257$		−	20.0000i		−	1.24757i	0.781598	−	0.623783i	$-0.214405\pi$
$258$		0			0
$259$	−18.0000			−1.11847
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	18.0000i		−	1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$		−	18.0000i		−	1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$		0			0
$265$	12.0000	−	12.0000i	0.737154	−	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.0000	−	13.0000i	−0.792624	−	0.792624i	0.189296	−	0.981920i	$-0.439379\pi$
$269$	−13.0000	−	13.0000i	−0.792624	−	0.792624i	−0.981920	+	0.189296i	$0.939379\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.0000	+	13.0000i	0.783929	+	0.783929i
$276$		0			0
$277$	9.00000	+	9.00000i	0.540758	+	0.540758i	0.923751	−	0.382993i	$-0.125107\pi$
$277$	9.00000	+	9.00000i	0.540758	+	0.540758i	−0.382993	+	0.923751i	$0.625107\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8.00000i		−	0.477240i	−0.971113	−	0.238620i	$-0.923305\pi$
$281$		−	8.00000i		−	0.477240i	0.971113	−	0.238620i	$-0.0766950\pi$
$282$		0			0
$283$	15.0000	−	15.0000i	0.891657	−	0.891657i	−0.103022	−	0.994679i	$-0.532851\pi$
$283$	15.0000	−	15.0000i	0.891657	−	0.891657i	0.994679	+	0.103022i	$0.0328511\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			6.00000i			0.354169i
$288$		0			0
$289$	−15.0000	−	8.00000i	−0.882353	−	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.00000			0.116841			0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000			0.116841			0.0584206	+	0.998292i	$0.481394\pi$
$294$		0			0
$295$	18.0000	−	18.0000i	1.04800	−	1.04800i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.00000	−	2.00000i	0.115663	−	0.115663i
$300$		0			0
$301$	−6.00000	−	6.00000i	−0.345834	−	0.345834i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000			0.343559
$306$		0			0
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	19.0000	+	19.0000i	1.07739	+	1.07739i	0.996743	+	0.0806486i	$0.0256991\pi$
$311$	19.0000	+	19.0000i	1.07739	+	1.07739i	0.0806486	+	0.996743i	$0.474301\pi$
$312$		0			0
$313$	9.00000	−	9.00000i	0.508710	−	0.508710i	−0.405420	−	0.914130i	$-0.632875\pi$
$313$	9.00000	−	9.00000i	0.508710	−	0.508710i	0.914130	+	0.405420i	$0.132875\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	−0.833439	−	0.552611i	$-0.813631\pi$
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	0.552611	+	0.833439i	$0.313631\pi$
$318$		0			0
$319$	−2.00000			−0.111979
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.0000	−	6.00000i	−1.33540	−	0.333849i
$324$		0			0
$325$			26.0000i			1.44222i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	2.00000i		−	0.109930i	−0.998488	−	0.0549650i	$-0.982495\pi$
$331$		−	2.00000i		−	0.109930i	0.998488	−	0.0549650i	$-0.0175047\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.0000	+	12.0000i	0.655630	+	0.655630i
$336$		0			0
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	0.168585	−	0.985687i	$-0.446080\pi$
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	−0.985687	+	0.168585i	$0.946080\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.00000			0.108306
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.0000	−	17.0000i	0.912608	−	0.912608i	−0.0838690	−	0.996477i	$-0.526728\pi$
$347$	17.0000	−	17.0000i	0.912608	−	0.912608i	0.996477	+	0.0838690i	$0.0267277\pi$
$348$		0			0
$349$			12.0000i			0.642345i	0.947021	+	0.321173i	$0.104077\pi$
$349$			12.0000i			0.642345i	−0.947021	+	0.321173i	$0.895923\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$		−	30.0000i		−	1.59223i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			14.0000i			0.738892i	0.929252	+	0.369446i	$0.120452\pi$
$359$			14.0000i			0.738892i	−0.929252	+	0.369446i	$0.879548\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$			6.00000i			0.314054i
$366$		0			0
$367$	−21.0000	+	21.0000i	−1.09619	+	1.09619i	−0.101339	+	0.994852i	$0.532313\pi$
$367$	−21.0000	+	21.0000i	−1.09619	+	1.09619i	−0.994852	+	0.101339i	$0.967687\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000	−	12.0000i	−0.623009	−	0.623009i
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.00000	−	2.00000i	−0.103005	−	0.103005i
$378$		0			0
$379$	−15.0000	−	15.0000i	−0.770498	−	0.770498i	0.207695	−	0.978194i	$-0.433404\pi$
$379$	−15.0000	−	15.0000i	−0.770498	−	0.770498i	−0.978194	+	0.207695i	$0.933404\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	26.0000i		−	1.32854i	−0.747494	−	0.664269i	$-0.768743\pi$
$383$		−	26.0000i		−	1.32854i	0.747494	−	0.664269i	$-0.231257\pi$
$384$		0			0
$385$	18.0000	−	18.0000i	0.917365	−	0.917365i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			16.0000i			0.811232i	0.914044	+	0.405616i	$0.132943\pi$
$389$			16.0000i			0.811232i	−0.914044	+	0.405616i	$0.867057\pi$
$390$		0			0
$391$	3.00000	+	5.00000i	0.151717	+	0.252861i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−54.0000			−2.71703
$396$		0			0
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.168663	+	0.985674i	$0.553945\pi$
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.985674	+	0.168663i	$0.946055\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	9.00000i	−0.449439	+	0.449439i	−0.895168	−	0.445729i	$-0.852944\pi$
$401$	−9.00000	+	9.00000i	−0.449439	+	0.449439i	0.445729	+	0.895168i	$0.352944\pi$
$402$		0			0
$403$	2.00000	+	2.00000i	0.0996271	+	0.0996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.00000			−0.297409
$408$		0			0
$409$	−38.0000			−1.87898			−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000			−1.87898			−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.0000	−	18.0000i	−0.885722	−	0.885722i
$414$		0			0
$415$	42.0000	−	42.0000i	2.06170	−	2.06170i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.00000	−	5.00000i	0.244266	−	0.244266i	−0.574346	−	0.818612i	$-0.694744\pi$
$419$	5.00000	−	5.00000i	0.244266	−	0.244266i	0.818612	+	0.574346i	$0.194744\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−52.0000	−	13.0000i	−2.52237	−	0.630593i
$426$		0			0
$427$		−	6.00000i		−	0.290360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.00000	+	7.00000i	−0.337178	+	0.337178i	−0.855304	−	0.518126i	$-0.826630\pi$
$431$	−7.00000	+	7.00000i	−0.337178	+	0.337178i	0.518126	+	0.855304i	$0.326630\pi$
$432$		0			0
$433$		−	12.0000i		−	0.576683i	−0.957528	−	0.288342i	$-0.906896\pi$
$433$		−	12.0000i		−	0.576683i	0.957528	−	0.288342i	$-0.0931039\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.00000	+	6.00000i	0.287019	+	0.287019i
$438$		0			0
$439$	−3.00000	−	3.00000i	−0.143182	−	0.143182i	0.631882	−	0.775064i	$-0.282283\pi$
$439$	−3.00000	−	3.00000i	−0.143182	−	0.143182i	−0.775064	+	0.631882i	$0.782283\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$	−18.0000	−	18.0000i	−0.853282	−	0.853282i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.00000	−	3.00000i	0.141579	−	0.141579i	−0.632765	−	0.774344i	$-0.718080\pi$
$449$	3.00000	−	3.00000i	0.141579	−	0.141579i	0.774344	+	0.632765i	$0.218080\pi$
$450$		0			0
$451$			2.00000i			0.0941763i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	36.0000			1.68771
$456$		0			0
$457$		−	20.0000i		−	0.935561i	−0.883845	−	0.467780i	$-0.845054\pi$
$457$		−	20.0000i		−	0.935561i	0.883845	−	0.467780i	$-0.154946\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	28.0000i		−	1.30409i	−0.758180	−	0.652045i	$-0.773911\pi$
$461$		−	28.0000i		−	1.30409i	0.758180	−	0.652045i	$-0.226089\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			18.0000i			0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$			18.0000i			0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$		0			0
$469$	12.0000	−	12.0000i	0.554109	−	0.554109i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.00000	−	2.00000i	−0.0919601	−	0.0919601i
$474$		0			0
$475$	−78.0000			−3.57889
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.00000	−	9.00000i	−0.411220	−	0.411220i	0.470943	−	0.882164i	$-0.343914\pi$
$479$	−9.00000	−	9.00000i	−0.411220	−	0.411220i	−0.882164	+	0.470943i	$0.843914\pi$
$480$		0			0
$481$	−6.00000	−	6.00000i	−0.273576	−	0.273576i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			78.0000i			3.54180i
$486$		0			0
$487$	−13.0000	+	13.0000i	−0.589086	+	0.589086i	−0.937384	−	0.348298i	$-0.886760\pi$
$487$	−13.0000	+	13.0000i	−0.589086	+	0.589086i	0.348298	+	0.937384i	$0.386760\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.00000i			0.0902587i	0.998981	+	0.0451294i	$0.0143700\pi$
$491$			2.00000i			0.0902587i	−0.998981	+	0.0451294i	$0.985630\pi$
$492$		0			0
$493$	5.00000	−	3.00000i	0.225189	−	0.135113i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−30.0000			−1.34568
$498$		0			0
$499$	−13.0000	+	13.0000i	−0.581960	+	0.581960i	−0.935441	−	0.353482i	$-0.884998\pi$
$499$	−13.0000	+	13.0000i	−0.581960	+	0.581960i	0.353482	+	0.935441i	$0.384998\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−19.0000	+	19.0000i	−0.847168	+	0.847168i	−0.989779	−	0.142611i	$-0.954450\pi$
$503$	−19.0000	+	19.0000i	−0.847168	+	0.847168i	0.142611	+	0.989779i	$0.454450\pi$
$504$		0			0
$505$	18.0000	+	18.0000i	0.800989	+	0.800989i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	23.0000	−	23.0000i	1.00765	−	1.00765i	0.00767777	−	0.999971i	$-0.497556\pi$
$521$	23.0000	−	23.0000i	1.00765	−	1.00765i	0.999971	−	0.00767777i	$-0.00244393\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.00000	+	3.00000i	−0.217803	+	0.130682i
$528$		0			0
$529$			21.0000i			0.913043i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.00000	+	2.00000i	−0.0866296	+	0.0866296i
$534$		0			0
$535$			42.0000i			1.81582i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−11.0000	−	11.0000i	−0.473804	−	0.473804i
$540$		0			0
$541$	−19.0000	−	19.0000i	−0.816874	−	0.816874i	0.168780	−	0.985654i	$-0.446017\pi$
$541$	−19.0000	−	19.0000i	−0.816874	−	0.816874i	−0.985654	+	0.168780i	$0.946017\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	54.0000			2.31311
$546$		0			0
$547$	−15.0000	−	15.0000i	−0.641354	−	0.641354i	0.309535	−	0.950888i	$-0.399827\pi$
$547$	−15.0000	−	15.0000i	−0.641354	−	0.641354i	−0.950888	+	0.309535i	$0.899827\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.00000	−	6.00000i	0.255609	−	0.255609i
$552$		0			0
$553$			54.0000i			2.29631i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$		−	4.00000i		−	0.169182i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	22.0000i		−	0.927189i	−0.886047	−	0.463595i	$-0.846559\pi$
$563$		−	22.0000i		−	0.927189i	0.886047	−	0.463595i	$-0.153441\pi$
$564$		0			0
$565$	−30.0000			−1.26211
$566$		0			0
$567$		0			0
$568$		0			0
$569$			16.0000i			0.670755i	0.942084	+	0.335377i	$0.108864\pi$
$569$			16.0000i			0.670755i	−0.942084	+	0.335377i	$0.891136\pi$
$570$		0			0
$571$	−13.0000	+	13.0000i	−0.544033	+	0.544033i	−0.924709	−	0.380676i	$-0.875691\pi$
$571$	−13.0000	+	13.0000i	−0.544033	+	0.544033i	0.380676	+	0.924709i	$0.375691\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.0000	+	13.0000i	0.542137	+	0.542137i
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−42.0000	−	42.0000i	−1.74245	−	1.74245i
$582$		0			0
$583$	−4.00000	−	4.00000i	−0.165663	−	0.165663i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	38.0000i		−	1.56843i	−0.620491	−	0.784214i	$-0.713066\pi$
$587$		−	38.0000i		−	1.56843i	0.620491	−	0.784214i	$-0.286934\pi$
$588$		0			0
$589$	−6.00000	+	6.00000i	−0.247226	+	0.247226i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	24.0000i		−	0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$		−	24.0000i		−	0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$		0			0
$595$	−18.0000	+	72.0000i	−0.737928	+	2.95171i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	1.00000	−	1.00000i	0.0407909	−	0.0407909i	−0.686417	−	0.727208i	$-0.740818\pi$
$601$	1.00000	−	1.00000i	0.0407909	−	0.0407909i	0.727208	+	0.686417i	$0.240818\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−27.0000	+	27.0000i	−1.09771	+	1.09771i
$606$		0			0
$607$	9.00000	+	9.00000i	0.365299	+	0.365299i	0.865759	−	0.500461i	$-0.166836\pi$
$607$	9.00000	+	9.00000i	0.365299	+	0.365299i	−0.500461	+	0.865759i	$0.666836\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.00000	−	5.00000i	−0.201292	−	0.201292i	0.599261	−	0.800554i	$-0.295461\pi$
$617$	−5.00000	−	5.00000i	−0.201292	−	0.201292i	−0.800554	+	0.599261i	$0.795461\pi$
$618$		0			0
$619$	15.0000	−	15.0000i	0.602901	−	0.602901i	−0.338180	−	0.941081i	$-0.609811\pi$
$619$	15.0000	−	15.0000i	0.602901	−	0.602901i	0.941081	+	0.338180i	$0.109811\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.0000	+	18.0000i	−0.721155	+	0.721155i
$624$		0			0
$625$	−79.0000			−3.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15.0000	−	9.00000i	0.598089	−	0.358854i
$630$		0			0
$631$		−	22.0000i		−	0.875806i	−0.899022	−	0.437903i	$-0.855721\pi$
$631$		−	22.0000i		−	0.875806i	0.899022	−	0.437903i	$-0.144279\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	42.0000	−	42.0000i	1.66672	−	1.66672i
$636$		0			0
$637$		−	22.0000i		−	0.871672i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.0000	+	11.0000i	0.434474	+	0.434474i	0.890147	−	0.455673i	$-0.150601\pi$
$641$	11.0000	+	11.0000i	0.434474	+	0.434474i	−0.455673	+	0.890147i	$0.650601\pi$
$642$		0			0
$643$	−31.0000	−	31.0000i	−1.22252	−	1.22252i	−0.966733	−	0.255788i	$-0.917665\pi$
$643$	−31.0000	−	31.0000i	−1.22252	−	1.22252i	−0.255788	−	0.966733i	$-0.582335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.0000			0.629025			0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000			0.629025			0.314512	+	0.949253i	$0.398159\pi$
$648$		0			0
$649$	−6.00000	−	6.00000i	−0.235521	−	0.235521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.0000	−	11.0000i	0.430463	−	0.430463i	−0.458323	−	0.888786i	$-0.651550\pi$
$653$	11.0000	−	11.0000i	0.430463	−	0.430463i	0.888786	+	0.458323i	$0.151550\pi$
$654$		0			0
$655$		−	54.0000i		−	2.10995i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.00000			0.155818			0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000			0.155818			0.0779089	+	0.996960i	$0.475176\pi$
$660$		0			0
$661$		−	36.0000i		−	1.40024i	−0.714026	−	0.700119i	$-0.753130\pi$
$661$		−	36.0000i		−	1.40024i	0.714026	−	0.700119i	$-0.246870\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			108.000i			4.18806i
$666$		0			0
$667$	−2.00000			−0.0774403
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.00000i		−	0.0772091i
$672$		0			0
$673$	−11.0000	+	11.0000i	−0.424019	+	0.424019i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−11.0000	+	11.0000i	−0.424019	+	0.424019i	0.462566	+	0.886585i	$0.346929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.00000	−	1.00000i	−0.0384331	−	0.0384331i	0.687629	−	0.726062i	$-0.258652\pi$
$677$	−1.00000	−	1.00000i	−0.0384331	−	0.0384331i	−0.726062	+	0.687629i	$0.758652\pi$
$678$		0			0
$679$	78.0000			2.99337
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.00000	−	1.00000i	−0.0382639	−	0.0382639i	0.687716	−	0.725980i	$-0.258613\pi$
$683$	−1.00000	−	1.00000i	−0.0382639	−	0.0382639i	−0.725980	+	0.687716i	$0.758613\pi$
$684$		0			0
$685$	−18.0000	−	18.0000i	−0.687745	−	0.687745i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	8.00000i		−	0.304776i
$690$		0			0
$691$	31.0000	−	31.0000i	1.17930	−	1.17930i	0.199372	−	0.979924i	$-0.436110\pi$
$691$	31.0000	−	31.0000i	1.17930	−	1.17930i	0.979924	−	0.199372i	$-0.0638901\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	66.0000i		−	2.50352i
$696$		0			0
$697$	−3.00000	−	5.00000i	−0.113633	−	0.189389i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	18.0000	−	18.0000i	0.678883	−	0.678883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.0000	−	18.0000i	0.676960	−	0.676960i
$708$		0			0
$709$	33.0000	+	33.0000i	1.23934	+	1.23934i	0.960271	+	0.279070i	$0.0900263\pi$
$709$	33.0000	+	33.0000i	1.23934	+	1.23934i	0.279070	+	0.960271i	$0.409974\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.00000			0.0749006
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.0000	−	13.0000i	−0.484818	−	0.484818i	0.421848	−	0.906666i	$-0.361381\pi$
$719$	−13.0000	−	13.0000i	−0.484818	−	0.484818i	−0.906666	+	0.421848i	$0.861381\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.0000	−	13.0000i	0.482808	−	0.482808i
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.00000	+	2.00000i	0.295891	+	0.0739727i
$732$		0			0
$733$		−	28.0000i		−	1.03420i	−0.855924	−	0.517102i	$-0.827011\pi$
$733$		−	28.0000i		−	1.03420i	0.855924	−	0.517102i	$-0.172989\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.00000	−	4.00000i	0.147342	−	0.147342i
$738$		0			0
$739$			14.0000i			0.514998i	0.966279	+	0.257499i	$0.0828985\pi$
$739$			14.0000i			0.514998i	−0.966279	+	0.257499i	$0.917102\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.0000	+	15.0000i	0.550297	+	0.550297i	0.926526	−	0.376230i	$-0.122780\pi$
$743$	15.0000	+	15.0000i	0.550297	+	0.550297i	−0.376230	+	0.926526i	$0.622780\pi$
$744$		0			0
$745$	30.0000	+	30.0000i	1.09911	+	1.09911i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	42.0000			1.53465
$750$		0			0
$751$	−19.0000	−	19.0000i	−0.693320	−	0.693320i	0.269641	−	0.962961i	$-0.413095\pi$
$751$	−19.0000	−	19.0000i	−0.693320	−	0.693320i	−0.962961	+	0.269641i	$0.913095\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.00000	+	6.00000i	−0.218362	+	0.218362i
$756$		0			0
$757$			16.0000i			0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$			16.0000i			0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$		−	54.0000i		−	1.95493i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	12.0000i		−	0.433295i
$768$		0			0
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			24.0000i			0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$			24.0000i			0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$		0			0
$775$	−13.0000	+	13.0000i	−0.466974	+	0.466974i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.00000	−	6.00000i	−0.214972	−	0.214972i
$780$		0			0
$781$	−10.0000			−0.357828
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−30.0000	−	30.0000i	−1.07075	−	1.07075i
$786$		0			0
$787$	5.00000	+	5.00000i	0.178231	+	0.178231i	0.790584	−	0.612353i	$-0.209777\pi$
$787$	5.00000	+	5.00000i	0.178231	+	0.178231i	−0.612353	+	0.790584i	$0.709777\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			30.0000i			1.06668i
$792$		0			0
$793$	2.00000	−	2.00000i	0.0710221	−	0.0710221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	20.0000i		−	0.708436i	−0.935163	−	0.354218i	$-0.884747\pi$
$797$		−	20.0000i		−	0.708436i	0.935163	−	0.354218i	$-0.115253\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.00000			0.0705785
$804$		0			0
$805$	18.0000	−	18.0000i	0.634417	−	0.634417i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.00000	−	3.00000i	0.105474	−	0.105474i	−0.652400	−	0.757875i	$-0.726238\pi$
$809$	3.00000	−	3.00000i	0.105474	−	0.105474i	0.757875	+	0.652400i	$0.226238\pi$
$810$		0			0
$811$	−3.00000	−	3.00000i	−0.105344	−	0.105344i	0.652470	−	0.757814i	$-0.273733\pi$
$811$	−3.00000	−	3.00000i	−0.105344	−	0.105344i	−0.757814	+	0.652470i	$0.773733\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.00000			−0.210171
$816$		0			0
$817$	12.0000			0.419827
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.00000	−	1.00000i	−0.0349002	−	0.0349002i	0.689441	−	0.724342i	$-0.257856\pi$
$821$	−1.00000	−	1.00000i	−0.0349002	−	0.0349002i	−0.724342	+	0.689441i	$0.757856\pi$
$822$		0			0
$823$	−25.0000	+	25.0000i	−0.871445	+	0.871445i	−0.992630	−	0.121185i	$-0.961331\pi$
$823$	−25.0000	+	25.0000i	−0.871445	+	0.871445i	0.121185	+	0.992630i	$0.461331\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.00000	−	5.00000i	0.173867	−	0.173867i	−0.614809	−	0.788676i	$-0.710767\pi$
$827$	5.00000	−	5.00000i	0.173867	−	0.173867i	0.788676	+	0.614809i	$0.210767\pi$
$828$		0			0
$829$	38.0000			1.31979			0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000			1.31979			0.659897	+	0.751356i	$0.270600\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	44.0000	+	11.0000i	1.52451	+	0.381127i
$834$		0			0
$835$		−	78.0000i		−	2.69930i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−27.0000	+	27.0000i	−0.932144	+	0.932144i	−0.997840	−	0.0656962i	$-0.979073\pi$
$839$	−27.0000	+	27.0000i	−0.932144	+	0.932144i	0.0656962	+	0.997840i	$0.479073\pi$
$840$		0			0
$841$		−	27.0000i		−	0.931034i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−27.0000	−	27.0000i	−0.928828	−	0.928828i
$846$		0			0
$847$	27.0000	+	27.0000i	0.927731	+	0.927731i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000			−0.205677
$852$		0			0
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	0.787505	−	0.616308i	$-0.211372\pi$
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	−0.616308	+	0.787505i	$0.711372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−33.0000	+	33.0000i	−1.12726	+	1.12726i	−0.136637	+	0.990621i	$0.543630\pi$
$857$	−33.0000	+	33.0000i	−1.12726	+	1.12726i	−0.990621	+	0.136637i	$0.956370\pi$
$858$		0			0
$859$		−	42.0000i		−	1.43302i	−0.697576	−	0.716511i	$-0.745738\pi$
$859$		−	42.0000i		−	1.43302i	0.697576	−	0.716511i	$-0.254262\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.00000			−0.272323			−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000			−0.272323			−0.136162	+	0.990687i	$0.543477\pi$
$864$		0			0
$865$			42.0000i			1.42804i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			18.0000i			0.610608i
$870$		0			0
$871$	8.00000			0.271070
$872$		0			0
$873$		0			0
$874$		0			0
$875$			144.000i			4.86809i
$876$		0			0
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	0.202606	+	0.979260i	$0.435059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−17.0000	−	17.0000i	−0.572745	−	0.572745i	0.360150	−	0.932894i	$-0.382726\pi$
$881$	−17.0000	−	17.0000i	−0.572745	−	0.572745i	−0.932894	+	0.360150i	$0.882726\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−41.0000	−	41.0000i	−1.37665	−	1.37665i	−0.850224	−	0.526422i	$-0.823533\pi$
$887$	−41.0000	−	41.0000i	−1.37665	−	1.37665i	−0.526422	−	0.850224i	$-0.676467\pi$
$888$		0			0
$889$	−42.0000	−	42.0000i	−1.40863	−	1.40863i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−54.0000	+	54.0000i	−1.80502	+	1.80502i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	2.00000i		−	0.0667037i
$900$		0			0
$901$	16.0000	+	4.00000i	0.533037	+	0.133259i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	54.0000			1.79502
$906$		0			0
$907$	−33.0000	+	33.0000i	−1.09575	+	1.09575i	−0.100845	+	0.994902i	$0.532155\pi$
$907$	−33.0000	+	33.0000i	−1.09575	+	1.09575i	−0.994902	+	0.100845i	$0.967845\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	21.0000	−	21.0000i	0.695761	−	0.695761i	−0.267732	−	0.963493i	$-0.586274\pi$
$911$	21.0000	−	21.0000i	0.695761	−	0.695761i	0.963493	+	0.267732i	$0.0862743\pi$
$912$		0			0
$913$	−14.0000	−	14.0000i	−0.463332	−	0.463332i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−54.0000			−1.78324
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.0000	−	10.0000i	−0.329154	−	0.329154i
$924$		0			0
$925$	39.0000	−	39.0000i	1.28231	−	1.28231i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.00000	−	7.00000i	0.229663	−	0.229663i	−0.582889	−	0.812552i	$-0.698078\pi$
$929$	7.00000	−	7.00000i	0.229663	−	0.229663i	0.812552	+	0.582889i	$0.198078\pi$
$930$		0			0
$931$	66.0000			2.16306
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.00000	+	24.0000i	−0.196221	+	0.784884i
$936$		0			0
$937$			32.0000i			1.04539i	0.852518	+	0.522697i	$0.175074\pi$
$937$			32.0000i			1.04539i	−0.852518	+	0.522697i	$0.824926\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27.0000	−	27.0000i	0.880175	−	0.880175i	−0.113377	−	0.993552i	$-0.536167\pi$
$941$	27.0000	−	27.0000i	0.880175	−	0.880175i	0.993552	+	0.113377i	$0.0361668\pi$
$942$		0			0
$943$			2.00000i			0.0651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.0000	+	27.0000i	0.877382	+	0.877382i	0.993263	−	0.115881i	$-0.0369691\pi$
$947$	27.0000	+	27.0000i	0.877382	+	0.877382i	−0.115881	+	0.993263i	$0.536969\pi$
$948$		0			0
$949$	2.00000	+	2.00000i	0.0649227	+	0.0649227i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.0000			0.323932			0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000			0.323932			0.161966	+	0.986796i	$0.448217\pi$
$954$		0			0
$955$	72.0000	+	72.0000i	2.32987	+	2.32987i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000	+	18.0000i	−0.581250	+	0.581250i
$960$		0			0
$961$		−	29.0000i		−	0.935484i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	54.0000			1.73832
$966$		0			0
$967$			58.0000i			1.86515i	0.360971	+	0.932577i	$0.382445\pi$
$967$			58.0000i			1.86515i	−0.360971	+	0.932577i	$0.617555\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	14.0000i		−	0.449281i	−0.974442	−	0.224641i	$-0.927879\pi$
$971$		−	14.0000i		−	0.449281i	0.974442	−	0.224641i	$-0.0721208\pi$
$972$		0			0
$973$	−66.0000			−2.11586
$974$		0			0
$975$		0			0
$976$		0			0
$977$			8.00000i			0.255943i	0.991778	+	0.127971i	$0.0408466\pi$
$977$			8.00000i			0.255943i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		0			0
$979$	−6.00000	+	6.00000i	−0.191761	+	0.191761i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.00000	+	3.00000i	0.0956851	+	0.0956851i	0.753329	−	0.657644i	$-0.228447\pi$
$983$	3.00000	+	3.00000i	0.0956851	+	0.0956851i	−0.657644	+	0.753329i	$0.728447\pi$
$984$		0			0
$985$	66.0000			2.10293
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.00000	−	2.00000i	−0.0635963	−	0.0635963i
$990$		0			0
$991$	−11.0000	−	11.0000i	−0.349427	−	0.349427i	0.510469	−	0.859896i	$-0.329472\pi$
$991$	−11.0000	−	11.0000i	−0.349427	−	0.349427i	−0.859896	+	0.510469i	$0.829472\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	42.0000i		−	1.33149i
$996$		0			0
$997$	−3.00000	+	3.00000i	−0.0950110	+	0.0950110i	−0.753015	−	0.658004i	$-0.771401\pi$
$997$	−3.00000	+	3.00000i	−0.0950110	+	0.0950110i	0.658004	+	0.753015i	$0.271401\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.be.n.1441.1		2
3.2	odd	2		272.2.o.d.81.1		2
4.3	odd	2		1224.2.w.f.217.1		2
12.11	even	2		136.2.k.a.81.1	✓	2
17.4	even	4	inner	2448.2.be.n.1585.1		2
24.5	odd	2		1088.2.o.g.897.1		2
24.11	even	2		1088.2.o.p.897.1		2
51.2	odd	8		4624.2.a.q.1.2		2
51.32	odd	8		4624.2.a.q.1.1		2
51.38	odd	4		272.2.o.d.225.1		2
68.55	odd	4		1224.2.w.f.361.1		2
204.59	even	8		2312.2.b.f.577.2		2
204.83	even	8		2312.2.a.h.1.2		2
204.155	even	8		2312.2.a.h.1.1		2
204.179	even	8		2312.2.b.f.577.1		2
204.191	even	4		136.2.k.a.89.1	yes	2
408.293	odd	4		1088.2.o.g.769.1		2
408.395	even	4		1088.2.o.p.769.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.k.a.81.1	✓	2	12.11	even	2
136.2.k.a.89.1	yes	2	204.191	even	4
272.2.o.d.81.1		2	3.2	odd	2
272.2.o.d.225.1		2	51.38	odd	4
1088.2.o.g.769.1		2	408.293	odd	4
1088.2.o.g.897.1		2	24.5	odd	2
1088.2.o.p.769.1		2	408.395	even	4
1088.2.o.p.897.1		2	24.11	even	2
1224.2.w.f.217.1		2	4.3	odd	2
1224.2.w.f.361.1		2	68.55	odd	4
2312.2.a.h.1.1		2	204.155	even	8
2312.2.a.h.1.2		2	204.83	even	8
2312.2.b.f.577.1		2	204.179	even	8
2312.2.b.f.577.2		2	204.59	even	8
2448.2.be.n.1441.1		2	1.1	even	1	trivial
2448.2.be.n.1585.1		2	17.4	even	4	inner
4624.2.a.q.1.1		2	51.32	odd	8
4624.2.a.q.1.2		2	51.2	odd	8

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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.be (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.00000 + 3.00000i) q^{5} +(3.00000 - 3.00000i) q^{7} +(1.00000 - 1.00000i) q^{11} +2.00000 q^{13} +(-1.00000 + 4.00000i) q^{17} +6.00000i q^{19} +(1.00000 - 1.00000i) q^{23} +13.0000i q^{25} +(-1.00000 - 1.00000i) q^{29} +(1.00000 + 1.00000i) q^{31} +18.0000 q^{35} +(-3.00000 - 3.00000i) q^{37} +(-1.00000 + 1.00000i) q^{41} -2.00000i q^{43} -11.0000i q^{49} -4.00000i q^{53} +6.00000 q^{55} -6.00000i q^{59} +(1.00000 - 1.00000i) q^{61} +(6.00000 + 6.00000i) q^{65} +4.00000 q^{67} +(-5.00000 - 5.00000i) q^{71} +(1.00000 + 1.00000i) q^{73} -6.00000i q^{77} +(-9.00000 + 9.00000i) q^{79} -14.0000i q^{83} +(-15.0000 + 9.00000i) q^{85} -6.00000 q^{89} +(6.00000 - 6.00000i) q^{91} +(-18.0000 + 18.0000i) q^{95} +(13.0000 + 13.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 6 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} + 2 q^{23} - 2 q^{29} + 2 q^{31} + 36 q^{35} - 6 q^{37} - 2 q^{41} + 12 q^{55} + 2 q^{61} + 12 q^{65} + 8 q^{67} - 10 q^{71} + 2 q^{73} - 18 q^{79}+ \cdots + 26 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 6 * q^7 + 2 * q^11 + 4 * q^13 - 2 * q^17 + 2 * q^23 - 2 * q^29 + 2 * q^31 + 36 * q^35 - 6 * q^37 - 2 * q^41 + 12 * q^55 + 2 * q^61 + 12 * q^65 + 8 * q^67 - 10 * q^71 + 2 * q^73 - 18 * q^79 - 30 * q^85 - 12 * q^89 + 12 * q^91 - 36 * q^95 + 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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