LMFDB - Embedded newform 4368.2.h.n.337.3

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:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
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			337.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.n.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +1.43845i q^{11} +(-0.561553 - 3.56155i) q^{13} +0.561553i q^{15} +5.68466 q^{17} +2.56155i q^{19} -1.00000i q^{21} -5.68466 q^{23} +4.68466 q^{25} +1.00000 q^{27} -2.56155 q^{29} +10.2462i q^{31} +1.43845i q^{33} +0.561553 q^{35} +1.68466i q^{37} +(-0.561553 - 3.56155i) q^{39} -4.00000i q^{41} +10.5616 q^{43} +0.561553i q^{45} +6.24621i q^{47} -1.00000 q^{49} +5.68466 q^{51} +13.1231 q^{53} -0.807764 q^{55} +2.56155i q^{57} -12.2462i q^{59} +2.56155 q^{61} -1.00000i q^{63} +(2.00000 - 0.315342i) q^{65} +7.12311i q^{67} -5.68466 q^{69} -15.3693i q^{71} -7.43845i q^{73} +4.68466 q^{75} +1.43845 q^{77} -16.0000 q^{79} +1.00000 q^{81} -2.00000i q^{83} +3.19224i q^{85} -2.56155 q^{87} +8.00000i q^{89} +(-3.56155 + 0.561553i) q^{91} +10.2462i q^{93} -1.43845 q^{95} -10.0000i q^{97} +1.43845i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 6 q^{13} - 2 q^{17} + 2 q^{23} - 6 q^{25} + 4 q^{27} - 2 q^{29} - 6 q^{35} + 6 q^{39} + 34 q^{43} - 4 q^{49} - 2 q^{51} + 36 q^{53} + 38 q^{55} + 2 q^{61} + 8 q^{65} + 2 q^{69} - 6 q^{75} + 14 q^{77} - 64 q^{79} + 4 q^{81} - 2 q^{87} - 6 q^{91} - 14 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 6 * q^13 - 2 * q^17 + 2 * q^23 - 6 * q^25 + 4 * q^27 - 2 * q^29 - 6 * q^35 + 6 * q^39 + 34 * q^43 - 4 * q^49 - 2 * q^51 + 36 * q^53 + 38 * q^55 + 2 * q^61 + 8 * q^65 + 2 * q^69 - 6 * q^75 + 14 * q^77 - 64 * q^79 + 4 * q^81 - 2 * q^87 - 6 * q^91 - 14 * 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
$5$			0.561553i			0.251134i	0.992085	+	0.125567i	$0.0400750\pi$
$5$			0.561553i			0.251134i	−0.992085	+	0.125567i	$0.959925\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.43845i			0.433708i	0.976204	+	0.216854i	$0.0695796\pi$
$11$			1.43845i			0.433708i	−0.976204	+	0.216854i	$0.930420\pi$
$12$		0			0
$13$	−0.561553	−	3.56155i	−0.155747	−	0.987797i
$13$	−0.561553	−	3.56155i	−0.155747	−	0.987797i
$14$		0			0
$15$			0.561553i			0.144992i
$16$		0			0
$17$	5.68466			1.37873			0.689366	−	0.724413i	$-0.257889\pi$
$17$	5.68466			1.37873			0.689366	+	0.724413i	$0.257889\pi$
$18$		0			0
$19$			2.56155i			0.587661i	0.955858	+	0.293830i	$0.0949300\pi$
$19$			2.56155i			0.587661i	−0.955858	+	0.293830i	$0.905070\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	−5.68466			−1.18533			−0.592667	−	0.805448i	$-0.701925\pi$
$23$	−5.68466			−1.18533			−0.592667	+	0.805448i	$0.701925\pi$
$24$		0			0
$25$	4.68466			0.936932
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−2.56155			−0.475668			−0.237834	−	0.971306i	$-0.576437\pi$
$29$	−2.56155			−0.475668			−0.237834	+	0.971306i	$0.576437\pi$
$30$		0			0
$31$			10.2462i			1.84027i	0.391597	+	0.920137i	$0.371923\pi$
$31$			10.2462i			1.84027i	−0.391597	+	0.920137i	$0.628077\pi$
$32$		0			0
$33$			1.43845i			0.250402i
$34$		0			0
$35$	0.561553			0.0949197
$36$		0			0
$37$			1.68466i			0.276956i	0.990366	+	0.138478i	$0.0442210\pi$
$37$			1.68466i			0.276956i	−0.990366	+	0.138478i	$0.955779\pi$
$38$		0			0
$39$	−0.561553	−	3.56155i	−0.0899204	−	0.570305i
$40$		0			0
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$	10.5616			1.61062			0.805311	−	0.592853i	$-0.201998\pi$
$43$	10.5616			1.61062			0.805311	+	0.592853i	$0.201998\pi$
$44$		0			0
$45$			0.561553i			0.0837114i
$46$		0			0
$47$			6.24621i			0.911104i	0.890209	+	0.455552i	$0.150558\pi$
$47$			6.24621i			0.911104i	−0.890209	+	0.455552i	$0.849442\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	5.68466			0.796011
$52$		0			0
$53$	13.1231			1.80260			0.901299	−	0.433198i	$-0.142615\pi$
$53$	13.1231			1.80260			0.901299	+	0.433198i	$0.142615\pi$
$54$		0			0
$55$	−0.807764			−0.108919
$56$		0			0
$57$			2.56155i			0.339286i
$58$		0			0
$59$		−	12.2462i		−	1.59432i	−0.603768	−	0.797160i	$-0.706334\pi$
$59$		−	12.2462i		−	1.59432i	0.603768	−	0.797160i	$-0.293666\pi$
$60$		0			0
$61$	2.56155			0.327973			0.163987	−	0.986463i	$-0.447565\pi$
$61$	2.56155			0.327973			0.163987	+	0.986463i	$0.447565\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	2.00000	−	0.315342i	0.248069	−	0.0391133i
$66$		0			0
$67$			7.12311i			0.870226i	0.900376	+	0.435113i	$0.143292\pi$
$67$			7.12311i			0.870226i	−0.900376	+	0.435113i	$0.856708\pi$
$68$		0			0
$69$	−5.68466			−0.684352
$70$		0			0
$71$		−	15.3693i		−	1.82400i	−0.410188	−	0.912001i	$-0.634537\pi$
$71$		−	15.3693i		−	1.82400i	0.410188	−	0.912001i	$-0.365463\pi$
$72$		0			0
$73$		−	7.43845i		−	0.870604i	−0.900284	−	0.435302i	$-0.856642\pi$
$73$		−	7.43845i		−	0.870604i	0.900284	−	0.435302i	$-0.143358\pi$
$74$		0			0
$75$	4.68466			0.540938
$76$		0			0
$77$	1.43845			0.163926
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	2.00000i		−	0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$		−	2.00000i		−	0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$		0			0
$85$			3.19224i			0.346247i
$86$		0			0
$87$	−2.56155			−0.274627
$88$		0			0
$89$			8.00000i			0.847998i	0.905663	+	0.423999i	$0.139374\pi$
$89$			8.00000i			0.847998i	−0.905663	+	0.423999i	$0.860626\pi$
$90$		0			0
$91$	−3.56155	+	0.561553i	−0.373352	+	0.0588667i
$92$		0			0
$93$			10.2462i			1.06248i
$94$		0			0
$95$	−1.43845			−0.147582
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$			1.43845i			0.144569i
$100$		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
$103$	18.8078			1.85318			0.926592	−	0.376068i	$-0.122724\pi$
$103$	18.8078			1.85318			0.926592	+	0.376068i	$0.122724\pi$
$104$		0			0
$105$	0.561553			0.0548019
$106$		0			0
$107$	13.1231			1.26866			0.634329	−	0.773063i	$-0.281276\pi$
$107$	13.1231			1.26866			0.634329	+	0.773063i	$0.281276\pi$
$108$		0			0
$109$			15.9309i			1.52590i	0.646457	+	0.762950i	$0.276250\pi$
$109$			15.9309i			1.52590i	−0.646457	+	0.762950i	$0.723750\pi$
$110$		0			0
$111$			1.68466i			0.159901i
$112$		0			0
$113$	3.75379			0.353127			0.176563	−	0.984289i	$-0.443502\pi$
$113$	3.75379			0.353127			0.176563	+	0.984289i	$0.443502\pi$
$114$		0			0
$115$		−	3.19224i		−	0.297678i
$116$		0			0
$117$	−0.561553	−	3.56155i	−0.0519156	−	0.329266i
$118$		0			0
$119$		−	5.68466i		−	0.521112i
$120$		0			0
$121$	8.93087			0.811897
$122$		0			0
$123$		−	4.00000i		−	0.360668i
$124$		0			0
$125$			5.43845i			0.486430i
$126$		0			0
$127$	6.24621			0.554262			0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621			0.554262			0.277131	+	0.960832i	$0.410616\pi$
$128$		0			0
$129$	10.5616			0.929893
$130$		0			0
$131$	−2.56155			−0.223804			−0.111902	−	0.993719i	$-0.535694\pi$
$131$	−2.56155			−0.223804			−0.111902	+	0.993719i	$0.535694\pi$
$132$		0			0
$133$	2.56155			0.222115
$134$		0			0
$135$			0.561553i			0.0483308i
$136$		0			0
$137$			5.68466i			0.485673i	0.970067	+	0.242837i	$0.0780779\pi$
$137$			5.68466i			0.485673i	−0.970067	+	0.242837i	$0.921922\pi$
$138$		0			0
$139$	6.24621			0.529797			0.264898	−	0.964276i	$-0.414662\pi$
$139$	6.24621			0.529797			0.264898	+	0.964276i	$0.414662\pi$
$140$		0			0
$141$			6.24621i			0.526026i
$142$		0			0
$143$	5.12311	−	0.807764i	0.428416	−	0.0675486i
$144$		0			0
$145$		−	1.43845i		−	0.119457i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$			4.31534i			0.351178i	0.984464	+	0.175589i	$0.0561829\pi$
$151$			4.31534i			0.351178i	−0.984464	+	0.175589i	$0.943817\pi$
$152$		0			0
$153$	5.68466			0.459577
$154$		0			0
$155$	−5.75379			−0.462155
$156$		0			0
$157$	8.80776			0.702936			0.351468	−	0.936200i	$-0.385683\pi$
$157$	8.80776			0.702936			0.351468	+	0.936200i	$0.385683\pi$
$158$		0			0
$159$	13.1231			1.04073
$160$		0			0
$161$			5.68466i			0.448014i
$162$		0			0
$163$			0.876894i			0.0686837i	0.999410	+	0.0343418i	$0.0109335\pi$
$163$			0.876894i			0.0686837i	−0.999410	+	0.0343418i	$0.989067\pi$
$164$		0			0
$165$	−0.807764			−0.0628843
$166$		0			0
$167$			8.80776i			0.681565i	0.940142	+	0.340783i	$0.110692\pi$
$167$			8.80776i			0.681565i	−0.940142	+	0.340783i	$0.889308\pi$
$168$		0			0
$169$	−12.3693	+	4.00000i	−0.951486	+	0.307692i
$170$		0			0
$171$			2.56155i			0.195887i
$172$		0			0
$173$	−20.2462			−1.53929			−0.769645	−	0.638471i	$-0.779567\pi$
$173$	−20.2462			−1.53929			−0.769645	+	0.638471i	$0.779567\pi$
$174$		0			0
$175$		−	4.68466i		−	0.354127i
$176$		0			0
$177$		−	12.2462i		−	0.920482i
$178$		0			0
$179$	−2.87689			−0.215029			−0.107515	−	0.994204i	$-0.534289\pi$
$179$	−2.87689			−0.215029			−0.107515	+	0.994204i	$0.534289\pi$
$180$		0			0
$181$	11.3693			0.845075			0.422537	−	0.906346i	$-0.361140\pi$
$181$	11.3693			0.845075			0.422537	+	0.906346i	$0.361140\pi$
$182$		0			0
$183$	2.56155			0.189355
$184$		0			0
$185$	−0.946025			−0.0695531
$186$		0			0
$187$			8.17708i			0.597967i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	−13.0540			−0.944553			−0.472276	−	0.881451i	$-0.656568\pi$
$191$	−13.0540			−0.944553			−0.472276	+	0.881451i	$0.656568\pi$
$192$		0			0
$193$			12.0000i			0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$			12.0000i			0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$		0			0
$195$	2.00000	−	0.315342i	0.143223	−	0.0225821i
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	23.9309			1.69641			0.848207	−	0.529665i	$-0.177682\pi$
$199$	23.9309			1.69641			0.848207	+	0.529665i	$0.177682\pi$
$200$		0			0
$201$			7.12311i			0.502425i
$202$		0			0
$203$			2.56155i			0.179786i
$204$		0			0
$205$	2.24621			0.156882
$206$		0			0
$207$	−5.68466			−0.395111
$208$		0			0
$209$	−3.68466			−0.254873
$210$		0			0
$211$	12.8078			0.881723			0.440861	−	0.897575i	$-0.354673\pi$
$211$	12.8078			0.881723			0.440861	+	0.897575i	$0.354673\pi$
$212$		0			0
$213$		−	15.3693i		−	1.05309i
$214$		0			0
$215$			5.93087i			0.404482i
$216$		0			0
$217$	10.2462			0.695558
$218$		0			0
$219$		−	7.43845i		−	0.502644i
$220$		0			0
$221$	−3.19224	−	20.2462i	−0.214733	−	1.36191i
$222$		0			0
$223$		−	18.2462i		−	1.22186i	−0.791686	−	0.610928i	$-0.790796\pi$
$223$		−	18.2462i		−	1.22186i	0.791686	−	0.610928i	$-0.209204\pi$
$224$		0			0
$225$	4.68466			0.312311
$226$		0			0
$227$			23.6155i			1.56742i	0.621128	+	0.783709i	$0.286675\pi$
$227$			23.6155i			1.56742i	−0.621128	+	0.783709i	$0.713325\pi$
$228$		0			0
$229$		−	2.00000i		−	0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$		−	2.00000i		−	0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$		0			0
$231$	1.43845			0.0946429
$232$		0			0
$233$	2.00000			0.131024			0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000			0.131024			0.0655122	+	0.997852i	$0.479132\pi$
$234$		0			0
$235$	−3.50758			−0.228809
$236$		0			0
$237$	−16.0000			−1.03931
$238$		0			0
$239$		−	2.24621i		−	0.145295i	−0.997358	−	0.0726477i	$-0.976855\pi$
$239$		−	2.24621i		−	0.145295i	0.997358	−	0.0726477i	$-0.0231449\pi$
$240$		0			0
$241$		−	6.00000i		−	0.386494i	−0.981150	−	0.193247i	$-0.938098\pi$
$241$		−	6.00000i		−	0.386494i	0.981150	−	0.193247i	$-0.0619019\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	0.561553i		−	0.0358763i
$246$		0			0
$247$	9.12311	−	1.43845i	0.580489	−	0.0915262i
$248$		0			0
$249$		−	2.00000i		−	0.126745i
$250$		0			0
$251$	15.0540			0.950198			0.475099	−	0.879932i	$-0.342412\pi$
$251$	15.0540			0.950198			0.475099	+	0.879932i	$0.342412\pi$
$252$		0			0
$253$		−	8.17708i		−	0.514089i
$254$		0			0
$255$			3.19224i			0.199906i
$256$		0			0
$257$	−26.0000			−1.62184			−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000			−1.62184			−0.810918	+	0.585160i	$0.801032\pi$
$258$		0			0
$259$	1.68466			0.104680
$260$		0			0
$261$	−2.56155			−0.158556
$262$		0			0
$263$	−1.36932			−0.0844357			−0.0422178	−	0.999108i	$-0.513442\pi$
$263$	−1.36932			−0.0844357			−0.0422178	+	0.999108i	$0.513442\pi$
$264$		0			0
$265$			7.36932i			0.452694i
$266$		0			0
$267$			8.00000i			0.489592i
$268$		0			0
$269$	6.63068			0.404280			0.202140	−	0.979357i	$-0.435210\pi$
$269$	6.63068			0.404280			0.202140	+	0.979357i	$0.435210\pi$
$270$		0			0
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$		0			0
$273$	−3.56155	+	0.561553i	−0.215555	+	0.0339867i
$274$		0			0
$275$			6.73863i			0.406355i
$276$		0			0
$277$	−19.1231			−1.14900			−0.574498	−	0.818506i	$-0.694803\pi$
$277$	−19.1231			−1.14900			−0.574498	+	0.818506i	$0.694803\pi$
$278$		0			0
$279$			10.2462i			0.613425i
$280$		0			0
$281$		−	6.00000i		−	0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$		−	6.00000i		−	0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$		0			0
$283$	13.1231			0.780088			0.390044	−	0.920796i	$-0.372460\pi$
$283$	13.1231			0.780088			0.390044	+	0.920796i	$0.372460\pi$
$284$		0			0
$285$	−1.43845			−0.0852063
$286$		0			0
$287$	−4.00000			−0.236113
$288$		0			0
$289$	15.3153			0.900902
$290$		0			0
$291$		−	10.0000i		−	0.586210i
$292$		0			0
$293$			24.2462i			1.41648i	0.705972	+	0.708239i	$0.250510\pi$
$293$			24.2462i			1.41648i	−0.705972	+	0.708239i	$0.749490\pi$
$294$		0			0
$295$	6.87689			0.400388
$296$		0			0
$297$			1.43845i			0.0834672i
$298$		0			0
$299$	3.19224	+	20.2462i	0.184612	+	1.17087i
$300$		0			0
$301$		−	10.5616i		−	0.608758i
$302$		0			0
$303$	6.00000			0.344691
$304$		0			0
$305$			1.43845i			0.0823652i
$306$		0			0
$307$			9.75379i			0.556678i	0.960483	+	0.278339i	$0.0897839\pi$
$307$			9.75379i			0.556678i	−0.960483	+	0.278339i	$0.910216\pi$
$308$		0			0
$309$	18.8078			1.06994
$310$		0			0
$311$	−32.4924			−1.84248			−0.921238	−	0.388999i	$-0.872821\pi$
$311$	−32.4924			−1.84248			−0.921238	+	0.388999i	$0.872821\pi$
$312$		0			0
$313$	15.7538			0.890457			0.445228	−	0.895417i	$-0.353122\pi$
$313$	15.7538			0.890457			0.445228	+	0.895417i	$0.353122\pi$
$314$		0			0
$315$	0.561553			0.0316399
$316$		0			0
$317$		−	21.3693i		−	1.20022i	−0.799917	−	0.600110i	$-0.795123\pi$
$317$		−	21.3693i		−	1.20022i	0.799917	−	0.600110i	$-0.204877\pi$
$318$		0			0
$319$		−	3.68466i		−	0.206301i
$320$		0			0
$321$	13.1231			0.732460
$322$		0			0
$323$			14.5616i			0.810226i
$324$		0			0
$325$	−2.63068	−	16.6847i	−0.145924	−	0.925498i
$326$		0			0
$327$			15.9309i			0.880979i
$328$		0			0
$329$	6.24621			0.344365
$330$		0			0
$331$			7.12311i			0.391521i	0.980652	+	0.195761i	$0.0627176\pi$
$331$			7.12311i			0.391521i	−0.980652	+	0.195761i	$0.937282\pi$
$332$		0			0
$333$			1.68466i			0.0923187i
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−17.0540			−0.928989			−0.464495	−	0.885576i	$-0.653764\pi$
$337$	−17.0540			−0.928989			−0.464495	+	0.885576i	$0.653764\pi$
$338$		0			0
$339$	3.75379			0.203878
$340$		0			0
$341$	−14.7386			−0.798142
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	3.19224i		−	0.171864i
$346$		0			0
$347$	−8.49242			−0.455897			−0.227949	−	0.973673i	$-0.573202\pi$
$347$	−8.49242			−0.455897			−0.227949	+	0.973673i	$0.573202\pi$
$348$		0			0
$349$			21.3693i			1.14387i	0.820298	+	0.571937i	$0.193808\pi$
$349$			21.3693i			1.14387i	−0.820298	+	0.571937i	$0.806192\pi$
$350$		0			0
$351$	−0.561553	−	3.56155i	−0.0299735	−	0.190102i
$352$		0			0
$353$			1.75379i			0.0933448i	0.998910	+	0.0466724i	$0.0148617\pi$
$353$			1.75379i			0.0933448i	−0.998910	+	0.0466724i	$0.985138\pi$
$354$		0			0
$355$	8.63068			0.458069
$356$		0			0
$357$		−	5.68466i		−	0.300864i
$358$		0			0
$359$		−	17.6155i		−	0.929712i	−0.885386	−	0.464856i	$-0.846106\pi$
$359$		−	17.6155i		−	0.929712i	0.885386	−	0.464856i	$-0.153894\pi$
$360$		0			0
$361$	12.4384			0.654655
$362$		0			0
$363$	8.93087			0.468749
$364$		0			0
$365$	4.17708			0.218638
$366$		0			0
$367$	−25.3693			−1.32427			−0.662134	−	0.749386i	$-0.730349\pi$
$367$	−25.3693			−1.32427			−0.662134	+	0.749386i	$0.730349\pi$
$368$		0			0
$369$		−	4.00000i		−	0.208232i
$370$		0			0
$371$		−	13.1231i		−	0.681318i
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$			5.43845i			0.280840i
$376$		0			0
$377$	1.43845	+	9.12311i	0.0740838	+	0.469864i
$378$		0			0
$379$		−	27.1231i		−	1.39322i	−0.717450	−	0.696610i	$-0.754691\pi$
$379$		−	27.1231i		−	1.39322i	0.717450	−	0.696610i	$-0.245309\pi$
$380$		0			0
$381$	6.24621			0.320003
$382$		0			0
$383$			13.4384i			0.686673i	0.939213	+	0.343336i	$0.111557\pi$
$383$			13.4384i			0.686673i	−0.939213	+	0.343336i	$0.888443\pi$
$384$		0			0
$385$			0.807764i			0.0411675i
$386$		0			0
$387$	10.5616			0.536874
$388$		0			0
$389$	18.8769			0.957097			0.478548	−	0.878061i	$-0.341163\pi$
$389$	18.8769			0.957097			0.478548	+	0.878061i	$0.341163\pi$
$390$		0			0
$391$	−32.3153			−1.63426
$392$		0			0
$393$	−2.56155			−0.129213
$394$		0			0
$395$		−	8.98485i		−	0.452077i
$396$		0			0
$397$			9.36932i			0.470233i	0.971967	+	0.235116i	$0.0755471\pi$
$397$			9.36932i			0.470233i	−0.971967	+	0.235116i	$0.924453\pi$
$398$		0			0
$399$	2.56155			0.128238
$400$		0			0
$401$		−	34.9848i		−	1.74706i	−0.486771	−	0.873530i	$-0.661825\pi$
$401$		−	34.9848i		−	1.74706i	0.486771	−	0.873530i	$-0.338175\pi$
$402$		0			0
$403$	36.4924	−	5.75379i	1.81782	−	0.286617i
$404$		0			0
$405$			0.561553i			0.0279038i
$406$		0			0
$407$	−2.42329			−0.120118
$408$		0			0
$409$			6.80776i			0.336622i	0.985734	+	0.168311i	$0.0538313\pi$
$409$			6.80776i			0.336622i	−0.985734	+	0.168311i	$0.946169\pi$
$410$		0			0
$411$			5.68466i			0.280404i
$412$		0			0
$413$	−12.2462			−0.602597
$414$		0			0
$415$	1.12311			0.0551311
$416$		0			0
$417$	6.24621			0.305878
$418$		0			0
$419$	8.31534			0.406231			0.203116	−	0.979155i	$-0.434893\pi$
$419$	8.31534			0.406231			0.203116	+	0.979155i	$0.434893\pi$
$420$		0			0
$421$			10.0000i			0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$			10.0000i			0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$		0			0
$423$			6.24621i			0.303701i
$424$		0			0
$425$	26.6307			1.29178
$426$		0			0
$427$		−	2.56155i		−	0.123962i
$428$		0			0
$429$	5.12311	−	0.807764i	0.247346	−	0.0389992i
$430$		0			0
$431$		−	27.8617i		−	1.34205i	−0.741433	−	0.671026i	$-0.765854\pi$
$431$		−	27.8617i		−	1.34205i	0.741433	−	0.671026i	$-0.234146\pi$
$432$		0			0
$433$	−1.36932			−0.0658052			−0.0329026	−	0.999459i	$-0.510475\pi$
$433$	−1.36932			−0.0658052			−0.0329026	+	0.999459i	$0.510475\pi$
$434$		0			0
$435$		−	1.43845i		−	0.0689683i
$436$		0			0
$437$		−	14.5616i		−	0.696574i
$438$		0			0
$439$	−19.9309			−0.951249			−0.475624	−	0.879649i	$-0.657778\pi$
$439$	−19.9309			−0.951249			−0.475624	+	0.879649i	$0.657778\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−31.3693			−1.49040			−0.745201	−	0.666840i	$-0.767646\pi$
$443$	−31.3693			−1.49040			−0.745201	+	0.666840i	$0.767646\pi$
$444$		0			0
$445$	−4.49242			−0.212961
$446$		0			0
$447$		−	10.0000i		−	0.472984i
$448$		0			0
$449$			21.6847i			1.02336i	0.859175	+	0.511681i	$0.170977\pi$
$449$			21.6847i			1.02336i	−0.859175	+	0.511681i	$0.829023\pi$
$450$		0			0
$451$	5.75379			0.270935
$452$		0			0
$453$			4.31534i			0.202752i
$454$		0			0
$455$	−0.315342	−	2.00000i	−0.0147834	−	0.0937614i
$456$		0			0
$457$			16.0000i			0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$			16.0000i			0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$		0			0
$459$	5.68466			0.265337
$460$		0			0
$461$		−	8.06913i		−	0.375817i	−0.982187	−	0.187908i	$-0.939829\pi$
$461$		−	8.06913i		−	0.375817i	0.982187	−	0.187908i	$-0.0601708\pi$
$462$		0			0
$463$			39.5464i			1.83788i	0.394401	+	0.918938i	$0.370952\pi$
$463$			39.5464i			1.83788i	−0.394401	+	0.918938i	$0.629048\pi$
$464$		0			0
$465$	−5.75379			−0.266826
$466$		0			0
$467$	2.56155			0.118535			0.0592673	−	0.998242i	$-0.481124\pi$
$467$	2.56155			0.118535			0.0592673	+	0.998242i	$0.481124\pi$
$468$		0			0
$469$	7.12311			0.328914
$470$		0			0
$471$	8.80776			0.405840
$472$		0			0
$473$			15.1922i			0.698540i
$474$		0			0
$475$			12.0000i			0.550598i
$476$		0			0
$477$	13.1231			0.600866
$478$		0			0
$479$		−	41.3002i		−	1.88705i	−0.331296	−	0.943527i	$-0.607486\pi$
$479$		−	41.3002i		−	1.88705i	0.331296	−	0.943527i	$-0.392514\pi$
$480$		0			0
$481$	6.00000	−	0.946025i	0.273576	−	0.0431350i
$482$		0			0
$483$			5.68466i			0.258661i
$484$		0			0
$485$	5.61553			0.254988
$486$		0			0
$487$		−	30.2462i		−	1.37059i	−0.728267	−	0.685293i	$-0.759674\pi$
$487$		−	30.2462i		−	1.37059i	0.728267	−	0.685293i	$-0.240326\pi$
$488$		0			0
$489$			0.876894i			0.0396545i
$490$		0			0
$491$	−6.24621			−0.281888			−0.140944	−	0.990018i	$-0.545014\pi$
$491$	−6.24621			−0.281888			−0.140944	+	0.990018i	$0.545014\pi$
$492$		0			0
$493$	−14.5616			−0.655819
$494$		0			0
$495$	−0.807764			−0.0363063
$496$		0			0
$497$	−15.3693			−0.689408
$498$		0			0
$499$		−	34.4924i		−	1.54409i	−0.635566	−	0.772046i	$-0.719233\pi$
$499$		−	34.4924i		−	1.54409i	0.635566	−	0.772046i	$-0.280767\pi$
$500$		0			0
$501$			8.80776i			0.393502i
$502$		0			0
$503$	5.61553			0.250384			0.125192	−	0.992133i	$-0.460045\pi$
$503$	5.61553			0.250384			0.125192	+	0.992133i	$0.460045\pi$
$504$		0			0
$505$			3.36932i			0.149933i
$506$		0			0
$507$	−12.3693	+	4.00000i	−0.549341	+	0.177646i
$508$		0			0
$509$			9.68466i			0.429265i	0.976695	+	0.214632i	$0.0688554\pi$
$509$			9.68466i			0.429265i	−0.976695	+	0.214632i	$0.931145\pi$
$510$		0			0
$511$	−7.43845			−0.329058
$512$		0			0
$513$			2.56155i			0.113095i
$514$		0			0
$515$			10.5616i			0.465398i
$516$		0			0
$517$	−8.98485			−0.395153
$518$		0			0
$519$	−20.2462			−0.888710
$520$		0			0
$521$	−29.0540			−1.27288			−0.636439	−	0.771327i	$-0.719593\pi$
$521$	−29.0540			−1.27288			−0.636439	+	0.771327i	$0.719593\pi$
$522$		0			0
$523$	−24.4924			−1.07098			−0.535489	−	0.844542i	$-0.679873\pi$
$523$	−24.4924			−1.07098			−0.535489	+	0.844542i	$0.679873\pi$
$524$		0			0
$525$		−	4.68466i		−	0.204455i
$526$		0			0
$527$			58.2462i			2.53724i
$528$		0			0
$529$	9.31534			0.405015
$530$		0			0
$531$		−	12.2462i		−	0.531440i
$532$		0			0
$533$	−14.2462	+	2.24621i	−0.617072	+	0.0972942i
$534$		0			0
$535$			7.36932i			0.318603i
$536$		0			0
$537$	−2.87689			−0.124147
$538$		0			0
$539$		−	1.43845i		−	0.0619583i
$540$		0			0
$541$			26.8078i			1.15256i	0.817254	+	0.576278i	$0.195495\pi$
$541$			26.8078i			1.15256i	−0.817254	+	0.576278i	$0.804505\pi$
$542$		0			0
$543$	11.3693			0.487904
$544$		0			0
$545$	−8.94602			−0.383206
$546$		0			0
$547$	36.9848			1.58136			0.790679	−	0.612231i	$-0.209728\pi$
$547$	36.9848			1.58136			0.790679	+	0.612231i	$0.209728\pi$
$548$		0			0
$549$	2.56155			0.109324
$550$		0			0
$551$		−	6.56155i		−	0.279532i
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$	−0.946025			−0.0401565
$556$		0			0
$557$			0.246211i			0.0104323i	0.999986	+	0.00521615i	$0.00166036\pi$
$557$			0.246211i			0.0104323i	−0.999986	+	0.00521615i	$0.998340\pi$
$558$		0			0
$559$	−5.93087	−	37.6155i	−0.250849	−	1.59097i
$560$		0			0
$561$			8.17708i			0.345237i
$562$		0			0
$563$	−0.946025			−0.0398702			−0.0199351	−	0.999801i	$-0.506346\pi$
$563$	−0.946025			−0.0398702			−0.0199351	+	0.999801i	$0.506346\pi$
$564$		0			0
$565$			2.10795i			0.0886821i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$	1.75379			0.0733938			0.0366969	−	0.999326i	$-0.488316\pi$
$571$	1.75379			0.0733938			0.0366969	+	0.999326i	$0.488316\pi$
$572$		0			0
$573$	−13.0540			−0.545338
$574$		0			0
$575$	−26.6307			−1.11058
$576$		0			0
$577$		−	24.2462i		−	1.00938i	−0.863300	−	0.504691i	$-0.831606\pi$
$577$		−	24.2462i		−	1.00938i	0.863300	−	0.504691i	$-0.168394\pi$
$578$		0			0
$579$			12.0000i			0.498703i
$580$		0			0
$581$	−2.00000			−0.0829740
$582$		0			0
$583$			18.8769i			0.781801i
$584$		0			0
$585$	2.00000	−	0.315342i	0.0826898	−	0.0130378i
$586$		0			0
$587$		−	28.8769i		−	1.19188i	−0.803030	−	0.595938i	$-0.796780\pi$
$587$		−	28.8769i		−	1.19188i	0.803030	−	0.595938i	$-0.203220\pi$
$588$		0			0
$589$	−26.2462			−1.08146
$590$		0			0
$591$			6.00000i			0.246807i
$592$		0			0
$593$			28.0000i			1.14982i	0.818216	+	0.574911i	$0.194963\pi$
$593$			28.0000i			1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$		0			0
$595$	3.19224			0.130869
$596$		0			0
$597$	23.9309			0.979425
$598$		0			0
$599$	23.3002			0.952020			0.476010	−	0.879440i	$-0.342083\pi$
$599$	23.3002			0.952020			0.476010	+	0.879440i	$0.342083\pi$
$600$		0			0
$601$	25.8617			1.05492			0.527461	−	0.849579i	$-0.323144\pi$
$601$	25.8617			1.05492			0.527461	+	0.849579i	$0.323144\pi$
$602$		0			0
$603$			7.12311i			0.290075i
$604$		0			0
$605$			5.01515i			0.203895i
$606$		0			0
$607$	−41.5464			−1.68632			−0.843158	−	0.537666i	$-0.819306\pi$
$607$	−41.5464			−1.68632			−0.843158	+	0.537666i	$0.819306\pi$
$608$		0			0
$609$			2.56155i			0.103799i
$610$		0			0
$611$	22.2462	−	3.50758i	0.899985	−	0.141901i
$612$		0			0
$613$		−	18.8078i		−	0.759638i	−0.925061	−	0.379819i	$-0.875986\pi$
$613$		−	18.8078i		−	0.759638i	0.925061	−	0.379819i	$-0.124014\pi$
$614$		0			0
$615$	2.24621			0.0905760
$616$		0			0
$617$		−	26.8078i		−	1.07924i	−0.841909	−	0.539620i	$-0.818568\pi$
$617$		−	26.8078i		−	1.07924i	0.841909	−	0.539620i	$-0.181432\pi$
$618$		0			0
$619$		−	2.06913i		−	0.0831654i	−0.999135	−	0.0415827i	$-0.986760\pi$
$619$		−	2.06913i		−	0.0831654i	0.999135	−	0.0415827i	$-0.0132400\pi$
$620$		0			0
$621$	−5.68466			−0.228117
$622$		0			0
$623$	8.00000			0.320513
$624$		0			0
$625$	20.3693			0.814773
$626$		0			0
$627$	−3.68466			−0.147151
$628$		0			0
$629$			9.57671i			0.381848i
$630$		0			0
$631$			0.315342i			0.0125535i	0.999980	+	0.00627677i	$0.00199797\pi$
$631$			0.315342i			0.0125535i	−0.999980	+	0.00627677i	$0.998002\pi$
$632$		0			0
$633$	12.8078			0.509063
$634$		0			0
$635$			3.50758i			0.139194i
$636$		0			0
$637$	0.561553	+	3.56155i	0.0222495	+	0.141114i
$638$		0			0
$639$		−	15.3693i		−	0.608001i
$640$		0			0
$641$	−44.1080			−1.74216			−0.871080	−	0.491142i	$-0.836580\pi$
$641$	−44.1080			−1.74216			−0.871080	+	0.491142i	$0.836580\pi$
$642$		0			0
$643$		−	20.8078i		−	0.820578i	−0.911956	−	0.410289i	$-0.865428\pi$
$643$		−	20.8078i		−	0.820578i	0.911956	−	0.410289i	$-0.134572\pi$
$644$		0			0
$645$			5.93087i			0.233528i
$646$		0			0
$647$	−35.3693			−1.39051			−0.695256	−	0.718763i	$-0.744709\pi$
$647$	−35.3693			−1.39051			−0.695256	+	0.718763i	$0.744709\pi$
$648$		0			0
$649$	17.6155			0.691470
$650$		0			0
$651$	10.2462			0.401581
$652$		0			0
$653$	−25.4384			−0.995483			−0.497742	−	0.867325i	$-0.665837\pi$
$653$	−25.4384			−0.995483			−0.497742	+	0.867325i	$0.665837\pi$
$654$		0			0
$655$		−	1.43845i		−	0.0562048i
$656$		0			0
$657$		−	7.43845i		−	0.290201i
$658$		0			0
$659$	−21.1231			−0.822839			−0.411420	−	0.911446i	$-0.634967\pi$
$659$	−21.1231			−0.822839			−0.411420	+	0.911446i	$0.634967\pi$
$660$		0			0
$661$		−	4.73863i		−	0.184311i	−0.995745	−	0.0921557i	$-0.970624\pi$
$661$		−	4.73863i		−	0.184311i	0.995745	−	0.0921557i	$-0.0293758\pi$
$662$		0			0
$663$	−3.19224	−	20.2462i	−0.123976	−	0.786298i
$664$		0			0
$665$			1.43845i			0.0557806i
$666$		0			0
$667$	14.5616			0.563826
$668$		0			0
$669$		−	18.2462i		−	0.705439i
$670$		0			0
$671$			3.68466i			0.142245i
$672$		0			0
$673$	17.1922			0.662712			0.331356	−	0.943506i	$-0.392494\pi$
$673$	17.1922			0.662712			0.331356	+	0.943506i	$0.392494\pi$
$674$		0			0
$675$	4.68466			0.180313
$676$		0			0
$677$	6.63068			0.254838			0.127419	−	0.991849i	$-0.459331\pi$
$677$	6.63068			0.254838			0.127419	+	0.991849i	$0.459331\pi$
$678$		0			0
$679$	−10.0000			−0.383765
$680$		0			0
$681$			23.6155i			0.904949i
$682$		0			0
$683$		−	11.6847i		−	0.447101i	−0.974692	−	0.223551i	$-0.928235\pi$
$683$		−	11.6847i		−	0.447101i	0.974692	−	0.223551i	$-0.0717648\pi$
$684$		0			0
$685$	−3.19224			−0.121969
$686$		0			0
$687$		−	2.00000i		−	0.0763048i
$688$		0			0
$689$	−7.36932	−	46.7386i	−0.280749	−	1.78060i
$690$		0			0
$691$			3.50758i			0.133435i	0.997772	+	0.0667173i	$0.0212526\pi$
$691$			3.50758i			0.133435i	−0.997772	+	0.0667173i	$0.978747\pi$
$692$		0			0
$693$	1.43845			0.0546421
$694$		0			0
$695$			3.50758i			0.133050i
$696$		0			0
$697$		−	22.7386i		−	0.861287i
$698$		0			0
$699$	2.00000			0.0756469
$700$		0			0
$701$	37.6155			1.42072			0.710359	−	0.703839i	$-0.248532\pi$
$701$	37.6155			1.42072			0.710359	+	0.703839i	$0.248532\pi$
$702$		0			0
$703$	−4.31534			−0.162756
$704$		0			0
$705$	−3.50758			−0.132103
$706$		0			0
$707$		−	6.00000i		−	0.225653i
$708$		0			0
$709$		−	48.2462i		−	1.81192i	−0.423358	−	0.905962i	$-0.639149\pi$
$709$		−	48.2462i		−	1.81192i	0.423358	−	0.905962i	$-0.360851\pi$
$710$		0			0
$711$	−16.0000			−0.600047
$712$		0			0
$713$		−	58.2462i		−	2.18134i
$714$		0			0
$715$	0.453602	+	2.87689i	0.0169638	+	0.107590i
$716$		0			0
$717$		−	2.24621i		−	0.0838863i
$718$		0			0
$719$	−30.2462			−1.12799			−0.563997	−	0.825777i	$-0.690737\pi$
$719$	−30.2462			−1.12799			−0.563997	+	0.825777i	$0.690737\pi$
$720$		0			0
$721$		−	18.8078i		−	0.700438i
$722$		0			0
$723$		−	6.00000i		−	0.223142i
$724$		0			0
$725$	−12.0000			−0.445669
$726$		0			0
$727$	20.0691			0.744323			0.372161	−	0.928168i	$-0.378617\pi$
$727$	20.0691			0.744323			0.372161	+	0.928168i	$0.378617\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	60.0388			2.22062
$732$		0			0
$733$			31.1231i			1.14956i	0.818309	+	0.574779i	$0.194912\pi$
$733$			31.1231i			1.14956i	−0.818309	+	0.574779i	$0.805088\pi$
$734$		0			0
$735$		−	0.561553i		−	0.0207132i
$736$		0			0
$737$	−10.2462			−0.377424
$738$		0			0
$739$			23.6155i			0.868711i	0.900741	+	0.434356i	$0.143024\pi$
$739$			23.6155i			0.868711i	−0.900741	+	0.434356i	$0.856976\pi$
$740$		0			0
$741$	9.12311	−	1.43845i	0.335146	−	0.0528427i
$742$		0			0
$743$		−	16.0000i		−	0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$		0			0
$745$	5.61553			0.205737
$746$		0			0
$747$		−	2.00000i		−	0.0731762i
$748$		0			0
$749$		−	13.1231i		−	0.479508i
$750$		0			0
$751$	14.2462			0.519852			0.259926	−	0.965629i	$-0.416302\pi$
$751$	14.2462			0.519852			0.259926	+	0.965629i	$0.416302\pi$
$752$		0			0
$753$	15.0540			0.548597
$754$		0			0
$755$	−2.42329			−0.0881926
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		−	8.17708i		−	0.296809i
$760$		0			0
$761$			30.8769i			1.11929i	0.828734	+	0.559643i	$0.189062\pi$
$761$			30.8769i			1.11929i	−0.828734	+	0.559643i	$0.810938\pi$
$762$		0			0
$763$	15.9309			0.576736
$764$		0			0
$765$			3.19224i			0.115416i
$766$		0			0
$767$	−43.6155	+	6.87689i	−1.57487	+	0.248310i
$768$		0			0
$769$			12.5616i			0.452981i	0.974013	+	0.226491i	$0.0727253\pi$
$769$			12.5616i			0.452981i	−0.974013	+	0.226491i	$0.927275\pi$
$770$		0			0
$771$	−26.0000			−0.936367
$772$		0			0
$773$			15.9309i			0.572994i	0.958081	+	0.286497i	$0.0924908\pi$
$773$			15.9309i			0.572994i	−0.958081	+	0.286497i	$0.907509\pi$
$774$		0			0
$775$			48.0000i			1.72421i
$776$		0			0
$777$	1.68466			0.0604368
$778$		0			0
$779$	10.2462			0.367109
$780$		0			0
$781$	22.1080			0.791085
$782$		0			0
$783$	−2.56155			−0.0915424
$784$		0			0
$785$			4.94602i			0.176531i
$786$		0			0
$787$			45.9309i			1.63726i	0.574322	+	0.818629i	$0.305266\pi$
$787$			45.9309i			1.63726i	−0.574322	+	0.818629i	$0.694734\pi$
$788$		0			0
$789$	−1.36932			−0.0487490
$790$		0			0
$791$		−	3.75379i		−	0.133469i
$792$		0			0
$793$	−1.43845	−	9.12311i	−0.0510808	−	0.323971i
$794$		0			0
$795$			7.36932i			0.261363i
$796$		0			0
$797$	20.2462			0.717158			0.358579	−	0.933499i	$-0.383261\pi$
$797$	20.2462			0.717158			0.358579	+	0.933499i	$0.383261\pi$
$798$		0			0
$799$			35.5076i			1.25617i
$800$		0			0
$801$			8.00000i			0.282666i
$802$		0			0
$803$	10.6998			0.377588
$804$		0			0
$805$	−3.19224			−0.112512
$806$		0			0
$807$	6.63068			0.233411
$808$		0			0
$809$	−30.6307			−1.07692			−0.538459	−	0.842652i	$-0.680993\pi$
$809$	−30.6307			−1.07692			−0.538459	+	0.842652i	$0.680993\pi$
$810$		0			0
$811$			35.0540i			1.23091i	0.788171	+	0.615456i	$0.211028\pi$
$811$			35.0540i			1.23091i	−0.788171	+	0.615456i	$0.788972\pi$
$812$		0			0
$813$		−	8.00000i		−	0.280572i
$814$		0			0
$815$	−0.492423			−0.0172488
$816$		0			0
$817$			27.0540i			0.946499i
$818$		0			0
$819$	−3.56155	+	0.561553i	−0.124451	+	0.0196222i
$820$		0			0
$821$			39.1231i			1.36541i	0.730696	+	0.682703i	$0.239196\pi$
$821$			39.1231i			1.36541i	−0.730696	+	0.682703i	$0.760804\pi$
$822$		0			0
$823$	−18.7386			−0.653188			−0.326594	−	0.945165i	$-0.605901\pi$
$823$	−18.7386			−0.653188			−0.326594	+	0.945165i	$0.605901\pi$
$824$		0			0
$825$			6.73863i			0.234609i
$826$		0			0
$827$		−	15.0540i		−	0.523478i	−0.965139	−	0.261739i	$-0.915704\pi$
$827$		−	15.0540i		−	0.523478i	0.965139	−	0.261739i	$-0.0842960\pi$
$828$		0			0
$829$	4.94602			0.171783			0.0858913	−	0.996305i	$-0.472626\pi$
$829$	4.94602			0.171783			0.0858913	+	0.996305i	$0.472626\pi$
$830$		0			0
$831$	−19.1231			−0.663373
$832$		0			0
$833$	−5.68466			−0.196962
$834$		0			0
$835$	−4.94602			−0.171164
$836$		0			0
$837$			10.2462i			0.354161i
$838$		0			0
$839$			42.2462i			1.45850i	0.684247	+	0.729251i	$0.260131\pi$
$839$			42.2462i			1.45850i	−0.684247	+	0.729251i	$0.739869\pi$
$840$		0			0
$841$	−22.4384			−0.773740
$842$		0			0
$843$		−	6.00000i		−	0.206651i
$844$		0			0
$845$	−2.24621	−	6.94602i	−0.0772720	−	0.238951i
$846$		0			0
$847$		−	8.93087i		−	0.306868i
$848$		0			0
$849$	13.1231			0.450384
$850$		0			0
$851$		−	9.57671i		−	0.328285i
$852$		0			0
$853$		−	19.1231i		−	0.654763i	−0.944892	−	0.327381i	$-0.893834\pi$
$853$		−	19.1231i		−	0.654763i	0.944892	−	0.327381i	$-0.106166\pi$
$854$		0			0
$855$	−1.43845			−0.0491939
$856$		0			0
$857$	−38.0000			−1.29806			−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000			−1.29806			−0.649028	+	0.760765i	$0.724824\pi$
$858$		0			0
$859$	−41.6155			−1.41990			−0.709952	−	0.704250i	$-0.751283\pi$
$859$	−41.6155			−1.41990			−0.709952	+	0.704250i	$0.751283\pi$
$860$		0			0
$861$	−4.00000			−0.136320
$862$		0			0
$863$			6.73863i			0.229386i	0.993401	+	0.114693i	$0.0365884\pi$
$863$			6.73863i			0.229386i	−0.993401	+	0.114693i	$0.963412\pi$
$864$		0			0
$865$		−	11.3693i		−	0.386568i
$866$		0			0
$867$	15.3153			0.520136
$868$		0			0
$869$		−	23.0152i		−	0.780736i
$870$		0			0
$871$	25.3693	−	4.00000i	0.859607	−	0.135535i
$872$		0			0
$873$		−	10.0000i		−	0.338449i
$874$		0			0
$875$	5.43845			0.183853
$876$		0			0
$877$		−	42.9848i		−	1.45150i	−0.687961	−	0.725748i	$-0.741494\pi$
$877$		−	42.9848i		−	1.45150i	0.687961	−	0.725748i	$-0.258506\pi$
$878$		0			0
$879$			24.2462i			0.817804i
$880$		0			0
$881$	−39.3002			−1.32406			−0.662028	−	0.749479i	$-0.730304\pi$
$881$	−39.3002			−1.32406			−0.662028	+	0.749479i	$0.730304\pi$
$882$		0			0
$883$	18.5616			0.624646			0.312323	−	0.949976i	$-0.398893\pi$
$883$	18.5616			0.624646			0.312323	+	0.949976i	$0.398893\pi$
$884$		0			0
$885$	6.87689			0.231164
$886$		0			0
$887$	−19.8617			−0.666892			−0.333446	−	0.942769i	$-0.608211\pi$
$887$	−19.8617			−0.666892			−0.333446	+	0.942769i	$0.608211\pi$
$888$		0			0
$889$		−	6.24621i		−	0.209491i
$890$		0			0
$891$			1.43845i			0.0481898i
$892$		0			0
$893$	−16.0000			−0.535420
$894$		0			0
$895$		−	1.61553i		−	0.0540011i
$896$		0			0
$897$	3.19224	+	20.2462i	0.106586	+	0.676001i
$898$		0			0
$899$		−	26.2462i		−	0.875360i
$900$		0			0
$901$	74.6004			2.48530
$902$		0			0
$903$		−	10.5616i		−	0.351466i
$904$		0			0
$905$			6.38447i			0.212227i
$906$		0			0
$907$	−8.49242			−0.281986			−0.140993	−	0.990011i	$-0.545030\pi$
$907$	−8.49242			−0.281986			−0.140993	+	0.990011i	$0.545030\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	8.56155			0.283657			0.141828	−	0.989891i	$-0.454702\pi$
$911$	8.56155			0.283657			0.141828	+	0.989891i	$0.454702\pi$
$912$		0			0
$913$	2.87689			0.0952113
$914$		0			0
$915$			1.43845i			0.0475536i
$916$		0			0
$917$			2.56155i			0.0845899i
$918$		0			0
$919$	−39.8617			−1.31492			−0.657459	−	0.753491i	$-0.728369\pi$
$919$	−39.8617			−1.31492			−0.657459	+	0.753491i	$0.728369\pi$
$920$		0			0
$921$			9.75379i			0.321398i
$922$		0			0
$923$	−54.7386	+	8.63068i	−1.80174	+	0.284082i
$924$		0			0
$925$			7.89205i			0.259489i
$926$		0			0
$927$	18.8078			0.617728
$928$		0			0
$929$			49.1231i			1.61168i	0.592136	+	0.805838i	$0.298285\pi$
$929$			49.1231i			1.61168i	−0.592136	+	0.805838i	$0.701715\pi$
$930$		0			0
$931$		−	2.56155i		−	0.0839515i
$932$		0			0
$933$	−32.4924			−1.06375
$934$		0			0
$935$	−4.59186			−0.150170
$936$		0			0
$937$	37.8617			1.23689			0.618445	−	0.785828i	$-0.287763\pi$
$937$	37.8617			1.23689			0.618445	+	0.785828i	$0.287763\pi$
$938$		0			0
$939$	15.7538			0.514105
$940$		0			0
$941$		−	6.49242i		−	0.211647i	−0.994385	−	0.105823i	$-0.966252\pi$
$941$		−	6.49242i		−	0.211647i	0.994385	−	0.105823i	$-0.0337478\pi$
$942$		0			0
$943$			22.7386i			0.740472i
$944$		0			0
$945$	0.561553			0.0182673
$946$		0			0
$947$			7.19224i			0.233716i	0.993149	+	0.116858i	$0.0372823\pi$
$947$			7.19224i			0.233716i	−0.993149	+	0.116858i	$0.962718\pi$
$948$		0			0
$949$	−26.4924	+	4.17708i	−0.859980	+	0.135594i
$950$		0			0
$951$		−	21.3693i		−	0.692948i
$952$		0			0
$953$	−48.7386			−1.57880			−0.789400	−	0.613880i	$-0.789608\pi$
$953$	−48.7386			−1.57880			−0.789400	+	0.613880i	$0.789608\pi$
$954$		0			0
$955$		−	7.33050i		−	0.237209i
$956$		0			0
$957$		−	3.68466i		−	0.119108i
$958$		0			0
$959$	5.68466			0.183567
$960$		0			0
$961$	−73.9848			−2.38661
$962$		0			0
$963$	13.1231			0.422886
$964$		0			0
$965$	−6.73863			−0.216924
$966$		0			0
$967$		−	25.9309i		−	0.833881i	−0.908934	−	0.416940i	$-0.863102\pi$
$967$		−	25.9309i		−	0.833881i	0.908934	−	0.416940i	$-0.136898\pi$
$968$		0			0
$969$			14.5616i			0.467784i
$970$		0			0
$971$	−4.00000			−0.128366			−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000			−0.128366			−0.0641831	+	0.997938i	$0.520444\pi$
$972$		0			0
$973$		−	6.24621i		−	0.200244i
$974$		0			0
$975$	−2.63068	−	16.6847i	−0.0842493	−	0.534337i
$976$		0			0
$977$		−	39.7926i		−	1.27308i	−0.771244	−	0.636539i	$-0.780365\pi$
$977$		−	39.7926i		−	1.27308i	0.771244	−	0.636539i	$-0.219635\pi$
$978$		0			0
$979$	−11.5076			−0.367784
$980$		0			0
$981$			15.9309i			0.508634i
$982$		0			0
$983$		−	6.56155i		−	0.209281i	−0.994510	−	0.104641i	$-0.966631\pi$
$983$		−	6.56155i		−	0.209281i	0.994510	−	0.104641i	$-0.0333692\pi$
$984$		0			0
$985$	−3.36932			−0.107355
$986$		0			0
$987$	6.24621			0.198819
$988$		0			0
$989$	−60.0388			−1.90912
$990$		0			0
$991$	13.7538			0.436903			0.218452	−	0.975848i	$-0.429899\pi$
$991$	13.7538			0.436903			0.218452	+	0.975848i	$0.429899\pi$
$992$		0			0
$993$			7.12311i			0.226045i
$994$		0			0
$995$			13.4384i			0.426027i
$996$		0			0
$997$	−59.3693			−1.88025			−0.940123	−	0.340837i	$-0.889290\pi$
$997$	−59.3693			−1.88025			−0.940123	+	0.340837i	$0.889290\pi$
$998$		0			0
$999$			1.68466i			0.0533002i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.n.337.3		4
4.3	odd	2		546.2.c.e.337.2	✓	4
12.11	even	2		1638.2.c.h.883.3		4
13.12	even	2	inner	4368.2.h.n.337.2		4
28.27	even	2		3822.2.c.h.883.1		4
52.31	even	4		7098.2.a.bg.1.2		2
52.47	even	4		7098.2.a.bv.1.1		2
52.51	odd	2		546.2.c.e.337.3	yes	4
156.155	even	2		1638.2.c.h.883.2		4
364.363	even	2		3822.2.c.h.883.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.e.337.2	✓	4	4.3	odd	2
546.2.c.e.337.3	yes	4	52.51	odd	2
1638.2.c.h.883.2		4	156.155	even	2
1638.2.c.h.883.3		4	12.11	even	2
3822.2.c.h.883.1		4	28.27	even	2
3822.2.c.h.883.4		4	364.363	even	2
4368.2.h.n.337.2		4	13.12	even	2	inner
4368.2.h.n.337.3		4	1.1	even	1	trivial
7098.2.a.bg.1.2		2	52.31	even	4
7098.2.a.bv.1.1		2	52.47	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +1.43845i q^{11} +(-0.561553 - 3.56155i) q^{13} +0.561553i q^{15} +5.68466 q^{17} +2.56155i q^{19} -1.00000i q^{21} -5.68466 q^{23} +4.68466 q^{25} +1.00000 q^{27} -2.56155 q^{29} +10.2462i q^{31} +1.43845i q^{33} +0.561553 q^{35} +1.68466i q^{37} +(-0.561553 - 3.56155i) q^{39} -4.00000i q^{41} +10.5616 q^{43} +0.561553i q^{45} +6.24621i q^{47} -1.00000 q^{49} +5.68466 q^{51} +13.1231 q^{53} -0.807764 q^{55} +2.56155i q^{57} -12.2462i q^{59} +2.56155 q^{61} -1.00000i q^{63} +(2.00000 - 0.315342i) q^{65} +7.12311i q^{67} -5.68466 q^{69} -15.3693i q^{71} -7.43845i q^{73} +4.68466 q^{75} +1.43845 q^{77} -16.0000 q^{79} +1.00000 q^{81} -2.00000i q^{83} +3.19224i q^{85} -2.56155 q^{87} +8.00000i q^{89} +(-3.56155 + 0.561553i) q^{91} +10.2462i q^{93} -1.43845 q^{95} -10.0000i q^{97} +1.43845i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 6 q^{13} - 2 q^{17} + 2 q^{23} - 6 q^{25} + 4 q^{27} - 2 q^{29} - 6 q^{35} + 6 q^{39} + 34 q^{43} - 4 q^{49} - 2 q^{51} + 36 q^{53} + 38 q^{55} + 2 q^{61} + 8 q^{65} + 2 q^{69} - 6 q^{75} + 14 q^{77} - 64 q^{79} + 4 q^{81} - 2 q^{87} - 6 q^{91} - 14 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 6 * q^13 - 2 * q^17 + 2 * q^23 - 6 * q^25 + 4 * q^27 - 2 * q^29 - 6 * q^35 + 6 * q^39 + 34 * q^43 - 4 * q^49 - 2 * q^51 + 36 * q^53 + 38 * q^55 + 2 * q^61 + 8 * q^65 + 2 * q^69 - 6 * q^75 + 14 * q^77 - 64 * q^79 + 4 * q^81 - 2 * q^87 - 6 * q^91 - 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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