LMFDB - Embedded newform 4368.2.h.b.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, 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:	$34.8786556029$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.b.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +5.00000i q^{11} +(-3.00000 - 2.00000i) q^{13} +3.00000i q^{15} +3.00000 q^{17} +1.00000i q^{19} +1.00000i q^{21} +1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} -5.00000i q^{33} -3.00000 q^{35} -7.00000i q^{37} +(3.00000 + 2.00000i) q^{39} +1.00000 q^{43} -3.00000i q^{45} -8.00000i q^{47} -1.00000 q^{49} -3.00000 q^{51} +14.0000 q^{53} +15.0000 q^{55} -1.00000i q^{57} -14.0000i q^{59} -3.00000 q^{61} -1.00000i q^{63} +(-6.00000 + 9.00000i) q^{65} -8.00000i q^{67} -1.00000 q^{69} +10.0000i q^{71} +11.0000i q^{73} +4.00000 q^{75} +5.00000 q^{77} +1.00000 q^{81} -6.00000i q^{83} -9.00000i q^{85} -5.00000 q^{87} -16.0000i q^{89} +(-2.00000 + 3.00000i) q^{91} +3.00000 q^{95} -2.00000i q^{97} +5.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} + 6 q^{17} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} - 6 q^{35} + 6 q^{39} + 2 q^{43} - 2 q^{49} - 6 q^{51} + 28 q^{53} + 30 q^{55} - 6 q^{61} - 12 q^{65} - 2 q^{69} + 8 q^{75} + 10 q^{77} + 2 q^{81} - 10 q^{87} - 4 q^{91} + 6 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 + 6 * q^17 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 10 * q^29 - 6 * q^35 + 6 * q^39 + 2 * q^43 - 2 * q^49 - 6 * q^51 + 28 * q^53 + 30 * q^55 - 6 * q^61 - 12 * q^65 - 2 * q^69 + 8 * q^75 + 10 * q^77 + 2 * q^81 - 10 * q^87 - 4 * q^91 + 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	3.00000i		−	1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			5.00000i			1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$			5.00000i			1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$		0			0
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$		0			0
$15$			3.00000i			0.774597i
$16$		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
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	1.00000			0.208514			0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000			0.208514			0.104257	+	0.994550i	$0.466753\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		−	5.00000i		−	0.870388i
$34$		0			0
$35$	−3.00000			−0.507093
$36$		0			0
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$		0			0
$39$	3.00000	+	2.00000i	0.480384	+	0.320256i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$		0			0
$45$		−	3.00000i		−	0.447214i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.00000			−0.420084
$52$		0			0
$53$	14.0000			1.92305			0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000			1.92305			0.961524	+	0.274721i	$0.0885855\pi$
$54$		0			0
$55$	15.0000			2.02260
$56$		0			0
$57$		−	1.00000i		−	0.132453i
$58$		0			0
$59$		−	14.0000i		−	1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$		−	14.0000i		−	1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$		0			0
$61$	−3.00000			−0.384111			−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000			−0.384111			−0.192055	+	0.981384i	$0.561515\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−6.00000	+	9.00000i	−0.744208	+	1.11631i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$	−1.00000			−0.120386
$70$		0			0
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$		0			0
$73$			11.0000i			1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$			11.0000i			1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$		0			0
$75$	4.00000			0.461880
$76$		0			0
$77$	5.00000			0.569803
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$		−	9.00000i		−	0.976187i
$86$		0			0
$87$	−5.00000			−0.536056
$88$		0			0
$89$		−	16.0000i		−	1.69600i	−0.529999	−	0.847998i	$-0.677808\pi$
$89$		−	16.0000i		−	1.69600i	0.529999	−	0.847998i	$-0.322192\pi$
$90$		0			0
$91$	−2.00000	+	3.00000i	−0.209657	+	0.314485i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.00000			0.307794
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$		0			0
$99$			5.00000i			0.502519i
$100$		0			0
$101$	−18.0000			−1.79107			−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000			−1.79107			−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$	11.0000			1.08386			0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000			1.08386			0.541931	+	0.840423i	$0.317693\pi$
$104$		0			0
$105$	3.00000			0.292770
$106$		0			0
$107$	−18.0000			−1.74013			−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000			−1.74013			−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$			9.00000i			0.862044i	0.902342	+	0.431022i	$0.141847\pi$
$109$			9.00000i			0.862044i	−0.902342	+	0.431022i	$0.858153\pi$
$110$		0			0
$111$			7.00000i			0.664411i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		−	3.00000i		−	0.279751i
$116$		0			0
$117$	−3.00000	−	2.00000i	−0.277350	−	0.184900i
$118$		0			0
$119$		−	3.00000i		−	0.275010i
$120$		0			0
$121$	−14.0000			−1.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3.00000i		−	0.268328i
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$	−1.00000			−0.0880451
$130$		0			0
$131$	−7.00000			−0.611593			−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000			−0.611593			−0.305796	+	0.952097i	$0.598923\pi$
$132$		0			0
$133$	1.00000			0.0867110
$134$		0			0
$135$			3.00000i			0.258199i
$136$		0			0
$137$			3.00000i			0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$			3.00000i			0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$		0			0
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$			8.00000i			0.673722i
$142$		0			0
$143$	10.0000	−	15.0000i	0.836242	−	1.25436i
$144$		0			0
$145$		−	15.0000i		−	1.24568i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$		−	15.0000i		−	1.22068i	−0.792139	−	0.610341i	$-0.791032\pi$
$151$		−	15.0000i		−	1.22068i	0.792139	−	0.610341i	$-0.208968\pi$
$152$		0			0
$153$	3.00000			0.242536
$154$		0			0
$155$		0			0
$156$		0			0
$157$	13.0000			1.03751			0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000			1.03751			0.518756	+	0.854922i	$0.326395\pi$
$158$		0			0
$159$	−14.0000			−1.11027
$160$		0			0
$161$		−	1.00000i		−	0.0788110i
$162$		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
$165$	−15.0000			−1.16775
$166$		0			0
$167$			7.00000i			0.541676i	0.962625	+	0.270838i	$0.0873008\pi$
$167$			7.00000i			0.541676i	−0.962625	+	0.270838i	$0.912699\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$			1.00000i			0.0764719i
$172$		0			0
$173$	−26.0000			−1.97674			−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−26.0000			−1.97674			−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$		0			0
$177$			14.0000i			1.05230i
$178$		0			0
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$	3.00000			0.221766
$184$		0			0
$185$	−21.0000			−1.54395
$186$		0			0
$187$			15.0000i			1.09691i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	−17.0000			−1.23008			−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000			−1.23008			−0.615038	+	0.788497i	$0.710860\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	6.00000	−	9.00000i	0.429669	−	0.644503i
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	5.00000			0.354441			0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000			0.354441			0.177220	+	0.984171i	$0.443289\pi$
$200$		0			0
$201$			8.00000i			0.564276i
$202$		0			0
$203$		−	5.00000i		−	0.350931i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.00000			0.0695048
$208$		0			0
$209$	−5.00000			−0.345857
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		−	10.0000i		−	0.685189i
$214$		0			0
$215$		−	3.00000i		−	0.204598i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	11.0000i		−	0.743311i
$220$		0			0
$221$	−9.00000	−	6.00000i	−0.605406	−	0.403604i
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	−4.00000			−0.266667
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$		−	26.0000i		−	1.71813i	−0.511868	−	0.859064i	$-0.671046\pi$
$229$		−	26.0000i		−	1.71813i	0.511868	−	0.859064i	$-0.328954\pi$
$230$		0			0
$231$	−5.00000			−0.328976
$232$		0			0
$233$	14.0000			0.917170			0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000			0.917170			0.458585	+	0.888650i	$0.348356\pi$
$234$		0			0
$235$	−24.0000			−1.56559
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.00000i		−	0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$		−	4.00000i		−	0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$		0			0
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			3.00000i			0.191663i
$246$		0			0
$247$	2.00000	−	3.00000i	0.127257	−	0.190885i
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	−17.0000			−1.07303			−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000			−1.07303			−0.536515	+	0.843891i	$0.680260\pi$
$252$		0			0
$253$			5.00000i			0.314347i
$254$		0			0
$255$			9.00000i			0.563602i
$256$		0			0
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$	−7.00000			−0.434959
$260$		0			0
$261$	5.00000			0.309492
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$		−	42.0000i		−	2.58004i
$266$		0			0
$267$			16.0000i			0.979184i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	2.00000	−	3.00000i	0.121046	−	0.181568i
$274$		0			0
$275$		−	20.0000i		−	1.20605i
$276$		0			0
$277$	28.0000			1.68236			0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000			1.68236			0.841178	+	0.540758i	$0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.0000i		−	0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$		0			0
$283$	−14.0000			−0.832214			−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000			−0.832214			−0.416107	+	0.909316i	$0.636606\pi$
$284$		0			0
$285$	−3.00000			−0.177705
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$			2.00000i			0.117242i
$292$		0			0
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$	−42.0000			−2.44533
$296$		0			0
$297$		−	5.00000i		−	0.290129i
$298$		0			0
$299$	−3.00000	−	2.00000i	−0.173494	−	0.115663i
$300$		0			0
$301$		−	1.00000i		−	0.0576390i
$302$		0			0
$303$	18.0000			1.03407
$304$		0			0
$305$			9.00000i			0.515339i
$306$		0			0
$307$		−	8.00000i		−	0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$		−	8.00000i		−	0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$		0			0
$309$	−11.0000			−0.625768
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$		0			0
$315$	−3.00000			−0.169031
$316$		0			0
$317$			8.00000i			0.449325i	0.974437	+	0.224662i	$0.0721279\pi$
$317$			8.00000i			0.449325i	−0.974437	+	0.224662i	$0.927872\pi$
$318$		0			0
$319$			25.0000i			1.39973i
$320$		0			0
$321$	18.0000			1.00466
$322$		0			0
$323$			3.00000i			0.166924i
$324$		0			0
$325$	12.0000	+	8.00000i	0.665640	+	0.443760i
$326$		0			0
$327$		−	9.00000i		−	0.497701i
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$		−	20.0000i		−	1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$		−	20.0000i		−	1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$		0			0
$333$		−	7.00000i		−	0.383598i
$334$		0			0
$335$	−24.0000			−1.31126
$336$		0			0
$337$	33.0000			1.79762			0.898812	−	0.438334i	$-0.144431\pi$
$337$	33.0000			1.79762			0.898812	+	0.438334i	$0.144431\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			3.00000i			0.161515i
$346$		0			0
$347$	−28.0000			−1.50312			−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000			−1.50312			−0.751559	+	0.659665i	$0.770698\pi$
$348$		0			0
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$		0			0
$351$	3.00000	+	2.00000i	0.160128	+	0.106752i
$352$		0			0
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$		0			0
$355$	30.0000			1.59223
$356$		0			0
$357$			3.00000i			0.158777i
$358$		0			0
$359$		−	34.0000i		−	1.79445i	−0.441572	−	0.897226i	$-0.645579\pi$
$359$		−	34.0000i		−	1.79445i	0.441572	−	0.897226i	$-0.354421\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$	14.0000			0.734809
$364$		0			0
$365$	33.0000			1.72730
$366$		0			0
$367$	32.0000			1.67039			0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000			1.67039			0.835193	+	0.549957i	$0.185356\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	14.0000i		−	0.726844i
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$			3.00000i			0.154919i
$376$		0			0
$377$	−15.0000	−	10.0000i	−0.772539	−	0.515026i
$378$		0			0
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$		−	31.0000i		−	1.58403i	−0.610504	−	0.792013i	$-0.709033\pi$
$383$		−	31.0000i		−	1.58403i	0.610504	−	0.792013i	$-0.290967\pi$
$384$		0			0
$385$		−	15.0000i		−	0.764471i
$386$		0			0
$387$	1.00000			0.0508329
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	3.00000			0.151717
$392$		0			0
$393$	7.00000			0.353103
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	12.0000i		−	0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$		−	12.0000i		−	0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$		0			0
$399$	−1.00000			−0.0500626
$400$		0			0
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	3.00000i		−	0.149071i
$406$		0			0
$407$	35.0000			1.73489
$408$		0			0
$409$			19.0000i			0.939490i	0.882802	+	0.469745i	$0.155654\pi$
$409$			19.0000i			0.939490i	−0.882802	+	0.469745i	$0.844346\pi$
$410$		0			0
$411$		−	3.00000i		−	0.147979i
$412$		0			0
$413$	−14.0000			−0.688895
$414$		0			0
$415$	−18.0000			−0.883585
$416$		0			0
$417$	20.0000			0.979404
$418$		0			0
$419$	35.0000			1.70986			0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000			1.70986			0.854931	+	0.518742i	$0.173599\pi$
$420$		0			0
$421$		−	30.0000i		−	1.46211i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$		−	30.0000i		−	1.46211i	0.682318	−	0.731055i	$-0.260972\pi$
$422$		0			0
$423$		−	8.00000i		−	0.388973i
$424$		0			0
$425$	−12.0000			−0.582086
$426$		0			0
$427$			3.00000i			0.145180i
$428$		0			0
$429$	−10.0000	+	15.0000i	−0.482805	+	0.724207i
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		0			0
$433$	4.00000			0.192228			0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000			0.192228			0.0961139	+	0.995370i	$0.469359\pi$
$434$		0			0
$435$			15.0000i			0.719195i
$436$		0			0
$437$			1.00000i			0.0478365i
$438$		0			0
$439$	15.0000			0.715911			0.357955	−	0.933739i	$-0.383474\pi$
$439$	15.0000			0.715911			0.357955	+	0.933739i	$0.383474\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$		0			0
$445$	−48.0000			−2.27542
$446$		0			0
$447$			6.00000i			0.283790i
$448$		0			0
$449$		−	21.0000i		−	0.991051i	−0.868593	−	0.495526i	$-0.834975\pi$
$449$		−	21.0000i		−	0.991051i	0.868593	−	0.495526i	$-0.165025\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			15.0000i			0.704761i
$454$		0			0
$455$	9.00000	+	6.00000i	0.421927	+	0.281284i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$	−3.00000			−0.140028
$460$		0			0
$461$			15.0000i			0.698620i	0.937007	+	0.349310i	$0.113584\pi$
$461$			15.0000i			0.698620i	−0.937007	+	0.349310i	$0.886416\pi$
$462$		0			0
$463$		−	11.0000i		−	0.511213i	−0.966781	−	0.255607i	$-0.917725\pi$
$463$		−	11.0000i		−	0.511213i	0.966781	−	0.255607i	$-0.0822752\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.00000			0.323921			0.161961	−	0.986797i	$-0.448218\pi$
$467$	7.00000			0.323921			0.161961	+	0.986797i	$0.448218\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$	−13.0000			−0.599008
$472$		0			0
$473$			5.00000i			0.229900i
$474$		0			0
$475$		−	4.00000i		−	0.183533i
$476$		0			0
$477$	14.0000			0.641016
$478$		0			0
$479$			21.0000i			0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$			21.0000i			0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$		0			0
$481$	−14.0000	+	21.0000i	−0.638345	+	0.957518i
$482$		0			0
$483$			1.00000i			0.0455016i
$484$		0			0
$485$	−6.00000			−0.272446
$486$		0			0
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		0			0
$489$		−	4.00000i		−	0.180886i
$490$		0			0
$491$	28.0000			1.26362			0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000			1.26362			0.631811	+	0.775122i	$0.282312\pi$
$492$		0			0
$493$	15.0000			0.675566
$494$		0			0
$495$	15.0000			0.674200
$496$		0			0
$497$	10.0000			0.448561
$498$		0			0
$499$		−	14.0000i		−	0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$		−	14.0000i		−	0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$		−	7.00000i		−	0.312737i
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$			54.0000i			2.40297i
$506$		0			0
$507$	−5.00000	−	12.0000i	−0.222058	−	0.532939i
$508$		0			0
$509$		−	1.00000i		−	0.0443242i	−0.999754	−	0.0221621i	$-0.992945\pi$
$509$		−	1.00000i		−	0.0443242i	0.999754	−	0.0221621i	$-0.00705500\pi$
$510$		0			0
$511$	11.0000			0.486611
$512$		0			0
$513$		−	1.00000i		−	0.0441511i
$514$		0			0
$515$		−	33.0000i		−	1.45415i
$516$		0			0
$517$	40.0000			1.75920
$518$		0			0
$519$	26.0000			1.14127
$520$		0			0
$521$	27.0000			1.18289			0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000			1.18289			0.591446	+	0.806345i	$0.298557\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$		0			0
$525$		−	4.00000i		−	0.174574i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.0000			−0.956522
$530$		0			0
$531$		−	14.0000i		−	0.607548i
$532$		0			0
$533$		0			0
$534$		0			0
$535$			54.0000i			2.33462i
$536$		0			0
$537$	10.0000			0.431532
$538$		0			0
$539$		−	5.00000i		−	0.215365i
$540$		0			0
$541$			15.0000i			0.644900i	0.946586	+	0.322450i	$0.104506\pi$
$541$			15.0000i			0.644900i	−0.946586	+	0.322450i	$0.895494\pi$
$542$		0			0
$543$	−2.00000			−0.0858282
$544$		0			0
$545$	27.0000			1.15655
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$	−3.00000			−0.128037
$550$		0			0
$551$			5.00000i			0.213007i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	21.0000			0.891400
$556$		0			0
$557$		−	22.0000i		−	0.932170i	−0.884740	−	0.466085i	$-0.845664\pi$
$557$		−	22.0000i		−	0.932170i	0.884740	−	0.466085i	$-0.154336\pi$
$558$		0			0
$559$	−3.00000	−	2.00000i	−0.126886	−	0.0845910i
$560$		0			0
$561$		−	15.0000i		−	0.633300i
$562$		0			0
$563$	−9.00000			−0.379305			−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000			−0.379305			−0.189652	+	0.981851i	$0.560736\pi$
$564$		0			0
$565$			18.0000i			0.757266i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$	17.0000			0.710185
$574$		0			0
$575$	−4.00000			−0.166812
$576$		0			0
$577$			38.0000i			1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$			38.0000i			1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$		0			0
$579$		−	16.0000i		−	0.664937i
$580$		0			0
$581$	−6.00000			−0.248922
$582$		0			0
$583$			70.0000i			2.89910i
$584$		0			0
$585$	−6.00000	+	9.00000i	−0.248069	+	0.372104i
$586$		0			0
$587$		−	8.00000i		−	0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$		−	8.00000i		−	0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			2.00000i			0.0822690i
$592$		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
$595$	−9.00000			−0.368964
$596$		0			0
$597$	−5.00000			−0.204636
$598$		0			0
$599$	45.0000			1.83865			0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000			1.83865			0.919325	+	0.393499i	$0.128735\pi$
$600$		0			0
$601$	−48.0000			−1.95796			−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−48.0000			−1.95796			−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		−	8.00000i		−	0.325785i
$604$		0			0
$605$			42.0000i			1.70754i
$606$		0			0
$607$	−23.0000			−0.933541			−0.466771	−	0.884378i	$-0.654583\pi$
$607$	−23.0000			−0.933541			−0.466771	+	0.884378i	$0.654583\pi$
$608$		0			0
$609$			5.00000i			0.202610i
$610$		0			0
$611$	−16.0000	+	24.0000i	−0.647291	+	0.970936i
$612$		0			0
$613$		−	39.0000i		−	1.57520i	−0.616190	−	0.787598i	$-0.711325\pi$
$613$		−	39.0000i		−	1.57520i	0.616190	−	0.787598i	$-0.288675\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	17.0000i		−	0.684394i	−0.939628	−	0.342197i	$-0.888829\pi$
$617$		−	17.0000i		−	0.684394i	0.939628	−	0.342197i	$-0.111171\pi$
$618$		0			0
$619$			11.0000i			0.442127i	0.975259	+	0.221064i	$0.0709529\pi$
$619$			11.0000i			0.442127i	−0.975259	+	0.221064i	$0.929047\pi$
$620$		0			0
$621$	−1.00000			−0.0401286
$622$		0			0
$623$	−16.0000			−0.641026
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$	5.00000			0.199681
$628$		0			0
$629$		−	21.0000i		−	0.837325i
$630$		0			0
$631$		−	35.0000i		−	1.39333i	−0.717398	−	0.696664i	$-0.754667\pi$
$631$		−	35.0000i		−	1.39333i	0.717398	−	0.696664i	$-0.245333\pi$
$632$		0			0
$633$	−13.0000			−0.516704
$634$		0			0
$635$		−	36.0000i		−	1.42862i
$636$		0			0
$637$	3.00000	+	2.00000i	0.118864	+	0.0792429i
$638$		0			0
$639$			10.0000i			0.395594i
$640$		0			0
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$			19.0000i			0.749287i	0.927169	+	0.374643i	$0.122235\pi$
$643$			19.0000i			0.749287i	−0.927169	+	0.374643i	$0.877765\pi$
$644$		0			0
$645$			3.00000i			0.118125i
$646$		0			0
$647$	2.00000			0.0786281			0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000			0.0786281			0.0393141	+	0.999227i	$0.487483\pi$
$648$		0			0
$649$	70.0000			2.74774
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.00000			−0.0391330			−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000			−0.0391330			−0.0195665	+	0.999809i	$0.506229\pi$
$654$		0			0
$655$			21.0000i			0.820538i
$656$		0			0
$657$			11.0000i			0.429151i
$658$		0			0
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$			50.0000i			1.94477i	0.233373	+	0.972387i	$0.425024\pi$
$661$			50.0000i			1.94477i	−0.233373	+	0.972387i	$0.574976\pi$
$662$		0			0
$663$	9.00000	+	6.00000i	0.349531	+	0.233021i
$664$		0			0
$665$		−	3.00000i		−	0.116335i
$666$		0			0
$667$	5.00000			0.193601
$668$		0			0
$669$			16.0000i			0.618596i
$670$		0			0
$671$		−	15.0000i		−	0.579069i
$672$		0			0
$673$	−11.0000			−0.424019			−0.212009	−	0.977268i	$-0.568001\pi$
$673$	−11.0000			−0.424019			−0.212009	+	0.977268i	$0.568001\pi$
$674$		0			0
$675$	4.00000			0.153960
$676$		0			0
$677$	8.00000			0.307465			0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000			0.307465			0.153732	+	0.988113i	$0.450871\pi$
$678$		0			0
$679$	−2.00000			−0.0767530
$680$		0			0
$681$			8.00000i			0.306561i
$682$		0			0
$683$		−	1.00000i		−	0.0382639i	−0.999817	−	0.0191320i	$-0.993910\pi$
$683$		−	1.00000i		−	0.0382639i	0.999817	−	0.0191320i	$-0.00609027\pi$
$684$		0			0
$685$	9.00000			0.343872
$686$		0			0
$687$			26.0000i			0.991962i
$688$		0			0
$689$	−42.0000	−	28.0000i	−1.60007	−	1.06672i
$690$		0			0
$691$			40.0000i			1.52167i	0.648944	+	0.760836i	$0.275211\pi$
$691$			40.0000i			1.52167i	−0.648944	+	0.760836i	$0.724789\pi$
$692$		0			0
$693$	5.00000			0.189934
$694$		0			0
$695$			60.0000i			2.27593i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−14.0000			−0.529529
$700$		0			0
$701$	2.00000			0.0755390			0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000			0.0755390			0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$	7.00000			0.264010
$704$		0			0
$705$	24.0000			0.903892
$706$		0			0
$707$			18.0000i			0.676960i
$708$		0			0
$709$		−	26.0000i		−	0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$		−	26.0000i		−	0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−45.0000	−	30.0000i	−1.68290	−	1.12194i
$716$		0			0
$717$			4.00000i			0.149383i
$718$		0			0
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		−	11.0000i		−	0.409661i
$722$		0			0
$723$		−	10.0000i		−	0.371904i
$724$		0			0
$725$	−20.0000			−0.742781
$726$		0			0
$727$	7.00000			0.259616			0.129808	−	0.991539i	$-0.458564\pi$
$727$	7.00000			0.259616			0.129808	+	0.991539i	$0.458564\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	3.00000			0.110959
$732$		0			0
$733$			36.0000i			1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$			36.0000i			1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$		0			0
$735$		−	3.00000i		−	0.110657i
$736$		0			0
$737$	40.0000			1.47342
$738$		0			0
$739$		−	24.0000i		−	0.882854i	−0.897297	−	0.441427i	$-0.854472\pi$
$739$		−	24.0000i		−	0.882854i	0.897297	−	0.441427i	$-0.145528\pi$
$740$		0			0
$741$	−2.00000	+	3.00000i	−0.0734718	+	0.110208i
$742$		0			0
$743$			4.00000i			0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$			4.00000i			0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$		0			0
$747$		−	6.00000i		−	0.219529i
$748$		0			0
$749$			18.0000i			0.657706i
$750$		0			0
$751$	48.0000			1.75154			0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000			1.75154			0.875772	+	0.482724i	$0.160353\pi$
$752$		0			0
$753$	17.0000			0.619514
$754$		0			0
$755$	−45.0000			−1.63772
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		−	5.00000i		−	0.181489i
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	9.00000			0.325822
$764$		0			0
$765$		−	9.00000i		−	0.325396i
$766$		0			0
$767$	−28.0000	+	42.0000i	−1.01102	+	1.51653i
$768$		0			0
$769$		−	1.00000i		−	0.0360609i	−0.999837	−	0.0180305i	$-0.994260\pi$
$769$		−	1.00000i		−	0.0360609i	0.999837	−	0.0180305i	$-0.00573959\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$		−	9.00000i		−	0.323708i	−0.986815	−	0.161854i	$-0.948253\pi$
$773$		−	9.00000i		−	0.323708i	0.986815	−	0.161854i	$-0.0517473\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	7.00000			0.251124
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−50.0000			−1.78914
$782$		0			0
$783$	−5.00000			−0.178685
$784$		0			0
$785$		−	39.0000i		−	1.39197i
$786$		0			0
$787$		−	13.0000i		−	0.463400i	−0.972787	−	0.231700i	$-0.925571\pi$
$787$		−	13.0000i		−	0.463400i	0.972787	−	0.231700i	$-0.0744288\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$			6.00000i			0.213335i
$792$		0			0
$793$	9.00000	+	6.00000i	0.319599	+	0.213066i
$794$		0			0
$795$			42.0000i			1.48959i
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$		−	24.0000i		−	0.849059i
$800$		0			0
$801$		−	16.0000i		−	0.565332i
$802$		0			0
$803$	−55.0000			−1.94091
$804$		0			0
$805$	−3.00000			−0.105736
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		−	35.0000i		−	1.22902i	−0.788911	−	0.614508i	$-0.789355\pi$
$811$		−	35.0000i		−	1.22902i	0.788911	−	0.614508i	$-0.210645\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000			0.420342
$816$		0			0
$817$			1.00000i			0.0349856i
$818$		0			0
$819$	−2.00000	+	3.00000i	−0.0698857	+	0.104828i
$820$		0			0
$821$		−	20.0000i		−	0.698005i	−0.937122	−	0.349002i	$-0.886521\pi$
$821$		−	20.0000i		−	0.698005i	0.937122	−	0.349002i	$-0.113479\pi$
$822$		0			0
$823$	−44.0000			−1.53374			−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000			−1.53374			−0.766872	+	0.641800i	$0.778188\pi$
$824$		0			0
$825$			20.0000i			0.696311i
$826$		0			0
$827$		−	23.0000i		−	0.799788i	−0.916561	−	0.399894i	$-0.869047\pi$
$827$		−	23.0000i		−	0.799788i	0.916561	−	0.399894i	$-0.130953\pi$
$828$		0			0
$829$	−25.0000			−0.868286			−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000			−0.868286			−0.434143	+	0.900844i	$0.642949\pi$
$830$		0			0
$831$	−28.0000			−0.971309
$832$		0			0
$833$	−3.00000			−0.103944
$834$		0			0
$835$	21.0000			0.726735
$836$		0			0
$837$		0			0
$838$		0			0
$839$			36.0000i			1.24286i	0.783470	+	0.621429i	$0.213448\pi$
$839$			36.0000i			1.24286i	−0.783470	+	0.621429i	$0.786552\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$			10.0000i			0.344418i
$844$		0			0
$845$	36.0000	−	15.0000i	1.23844	−	0.516016i
$846$		0			0
$847$			14.0000i			0.481046i
$848$		0			0
$849$	14.0000			0.480479
$850$		0			0
$851$		−	7.00000i		−	0.239957i
$852$		0			0
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$		0			0
$855$	3.00000			0.102598
$856$		0			0
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$		0			0
$859$	50.0000			1.70598			0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000			1.70598			0.852989	+	0.521929i	$0.174787\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$			78.0000i			2.65208i
$866$		0			0
$867$	8.00000			0.271694
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−16.0000	+	24.0000i	−0.542139	+	0.813209i
$872$		0			0
$873$		−	2.00000i		−	0.0676897i
$874$		0			0
$875$	−3.00000			−0.101419
$876$		0			0
$877$			38.0000i			1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$			38.0000i			1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$		0			0
$879$		−	6.00000i		−	0.202375i
$880$		0			0
$881$	7.00000			0.235836			0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000			0.235836			0.117918	+	0.993023i	$0.462378\pi$
$882$		0			0
$883$	41.0000			1.37976			0.689880	−	0.723924i	$-0.257663\pi$
$883$	41.0000			1.37976			0.689880	+	0.723924i	$0.257663\pi$
$884$		0			0
$885$	42.0000			1.41181
$886$		0			0
$887$	2.00000			0.0671534			0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000			0.0671534			0.0335767	+	0.999436i	$0.489310\pi$
$888$		0			0
$889$		−	12.0000i		−	0.402467i
$890$		0			0
$891$			5.00000i			0.167506i
$892$		0			0
$893$	8.00000			0.267710
$894$		0			0
$895$			30.0000i			1.00279i
$896$		0			0
$897$	3.00000	+	2.00000i	0.100167	+	0.0667781i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	42.0000			1.39922
$902$		0			0
$903$			1.00000i			0.0332779i
$904$		0			0
$905$		−	6.00000i		−	0.199447i
$906$		0			0
$907$	12.0000			0.398453			0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000			0.398453			0.199227	+	0.979953i	$0.436157\pi$
$908$		0			0
$909$	−18.0000			−0.597022
$910$		0			0
$911$	−47.0000			−1.55718			−0.778590	−	0.627533i	$-0.784065\pi$
$911$	−47.0000			−1.55718			−0.778590	+	0.627533i	$0.784065\pi$
$912$		0			0
$913$	30.0000			0.992855
$914$		0			0
$915$		−	9.00000i		−	0.297531i
$916$		0			0
$917$			7.00000i			0.231160i
$918$		0			0
$919$	−10.0000			−0.329870			−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000			−0.329870			−0.164935	+	0.986304i	$0.552741\pi$
$920$		0			0
$921$			8.00000i			0.263609i
$922$		0			0
$923$	20.0000	−	30.0000i	0.658308	−	0.987462i
$924$		0			0
$925$			28.0000i			0.920634i
$926$		0			0
$927$	11.0000			0.361287
$928$		0			0
$929$			54.0000i			1.77168i	0.463988	+	0.885841i	$0.346418\pi$
$929$			54.0000i			1.77168i	−0.463988	+	0.885841i	$0.653582\pi$
$930$		0			0
$931$		−	1.00000i		−	0.0327737i
$932$		0			0
$933$	−8.00000			−0.261908
$934$		0			0
$935$	45.0000			1.47166
$936$		0			0
$937$	−32.0000			−1.04539			−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000			−1.04539			−0.522697	+	0.852518i	$0.675074\pi$
$938$		0			0
$939$	6.00000			0.195803
$940$		0			0
$941$		−	30.0000i		−	0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$		−	30.0000i		−	0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	3.00000			0.0975900
$946$		0			0
$947$			57.0000i			1.85225i	0.377215	+	0.926126i	$0.376882\pi$
$947$			57.0000i			1.85225i	−0.377215	+	0.926126i	$0.623118\pi$
$948$		0			0
$949$	22.0000	−	33.0000i	0.714150	−	1.07123i
$950$		0			0
$951$		−	8.00000i		−	0.259418i
$952$		0			0
$953$	54.0000			1.74923			0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000			1.74923			0.874616	+	0.484817i	$0.161114\pi$
$954$		0			0
$955$			51.0000i			1.65032i
$956$		0			0
$957$		−	25.0000i		−	0.808135i
$958$		0			0
$959$	3.00000			0.0968751
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−18.0000			−0.580042
$964$		0			0
$965$	48.0000			1.54517
$966$		0			0
$967$			17.0000i			0.546683i	0.961917	+	0.273342i	$0.0881289\pi$
$967$			17.0000i			0.546683i	−0.961917	+	0.273342i	$0.911871\pi$
$968$		0			0
$969$		−	3.00000i		−	0.0963739i
$970$		0			0
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$			20.0000i			0.641171i
$974$		0			0
$975$	−12.0000	−	8.00000i	−0.384308	−	0.256205i
$976$		0			0
$977$			3.00000i			0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$			3.00000i			0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$		0			0
$979$	80.0000			2.55681
$980$		0			0
$981$			9.00000i			0.287348i
$982$		0			0
$983$		−	11.0000i		−	0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$983$		−	11.0000i		−	0.350846i	0.984493	−	0.175423i	$-0.0561292\pi$
$984$		0			0
$985$	−6.00000			−0.191176
$986$		0			0
$987$	8.00000			0.254643
$988$		0			0
$989$	1.00000			0.0317982
$990$		0			0
$991$	28.0000			0.889449			0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000			0.889449			0.444725	+	0.895667i	$0.353302\pi$
$992$		0			0
$993$			20.0000i			0.634681i
$994$		0			0
$995$		−	15.0000i		−	0.475532i
$996$		0			0
$997$	38.0000			1.20347			0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000			1.20347			0.601736	+	0.798695i	$0.294476\pi$
$998$		0			0
$999$			7.00000i			0.221470i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.b.337.1		2
4.3	odd	2		546.2.c.d.337.2	yes	2
12.11	even	2		1638.2.c.g.883.1		2
13.12	even	2	inner	4368.2.h.b.337.2		2
28.27	even	2		3822.2.c.a.883.2		2
52.31	even	4		7098.2.a.x.1.1		1
52.47	even	4		7098.2.a.p.1.1		1
52.51	odd	2		546.2.c.d.337.1	✓	2
156.155	even	2		1638.2.c.g.883.2		2
364.363	even	2		3822.2.c.a.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.d.337.1	✓	2	52.51	odd	2
546.2.c.d.337.2	yes	2	4.3	odd	2
1638.2.c.g.883.1		2	12.11	even	2
1638.2.c.g.883.2		2	156.155	even	2
3822.2.c.a.883.1		2	364.363	even	2
3822.2.c.a.883.2		2	28.27	even	2
4368.2.h.b.337.1		2	1.1	even	1	trivial
4368.2.h.b.337.2		2	13.12	even	2	inner
7098.2.a.p.1.1		1	52.47	even	4
7098.2.a.x.1.1		1	52.31	even	4

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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +5.00000i q^{11} +(-3.00000 - 2.00000i) q^{13} +3.00000i q^{15} +3.00000 q^{17} +1.00000i q^{19} +1.00000i q^{21} +1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} -5.00000i q^{33} -3.00000 q^{35} -7.00000i q^{37} +(3.00000 + 2.00000i) q^{39} +1.00000 q^{43} -3.00000i q^{45} -8.00000i q^{47} -1.00000 q^{49} -3.00000 q^{51} +14.0000 q^{53} +15.0000 q^{55} -1.00000i q^{57} -14.0000i q^{59} -3.00000 q^{61} -1.00000i q^{63} +(-6.00000 + 9.00000i) q^{65} -8.00000i q^{67} -1.00000 q^{69} +10.0000i q^{71} +11.0000i q^{73} +4.00000 q^{75} +5.00000 q^{77} +1.00000 q^{81} -6.00000i q^{83} -9.00000i q^{85} -5.00000 q^{87} -16.0000i q^{89} +(-2.00000 + 3.00000i) q^{91} +3.00000 q^{95} -2.00000i q^{97} +5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} + 6 q^{17} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} - 6 q^{35} + 6 q^{39} + 2 q^{43} - 2 q^{49} - 6 q^{51} + 28 q^{53} + 30 q^{55} - 6 q^{61} - 12 q^{65} - 2 q^{69} + 8 q^{75} + 10 q^{77} + 2 q^{81} - 10 q^{87} - 4 q^{91} + 6 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 + 6 * q^17 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 10 * q^29 - 6 * q^35 + 6 * q^39 + 2 * q^43 - 2 * q^49 - 6 * q^51 + 28 * q^53 + 30 * q^55 - 6 * q^61 - 12 * q^65 - 2 * q^69 + 8 * q^75 + 10 * q^77 + 2 * q^81 - 10 * q^87 - 4 * q^91 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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