LMFDB - Embedded newform 4368.2.h.n.337.4

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			337.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.n.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.56155i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.56155i q^{5} +1.00000i q^{7} +1.00000 q^{9} -5.56155i q^{11} +(3.56155 - 0.561553i) q^{13} +3.56155i q^{15} -6.68466 q^{17} +1.56155i q^{19} +1.00000i q^{21} +6.68466 q^{23} -7.68466 q^{25} +1.00000 q^{27} +1.56155 q^{29} +6.24621i q^{31} -5.56155i q^{33} -3.56155 q^{35} +10.6847i q^{37} +(3.56155 - 0.561553i) q^{39} +4.00000i q^{41} +6.43845 q^{43} +3.56155i q^{45} +10.2462i q^{47} -1.00000 q^{49} -6.68466 q^{51} +4.87689 q^{53} +19.8078 q^{55} +1.56155i q^{57} -4.24621i q^{59} -1.56155 q^{61} +1.00000i q^{63} +(2.00000 + 12.6847i) q^{65} +1.12311i q^{67} +6.68466 q^{69} -9.36932i q^{71} +11.5616i q^{73} -7.68466 q^{75} +5.56155 q^{77} -16.0000 q^{79} +1.00000 q^{81} +2.00000i q^{83} -23.8078i q^{85} +1.56155 q^{87} -8.00000i q^{89} +(0.561553 + 3.56155i) q^{91} +6.24621i q^{93} -5.56155 q^{95} +10.0000i q^{97} -5.56155i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			3.56155i			1.59277i	0.604787	+	0.796387i	$0.293258\pi$
$5$			3.56155i			1.59277i	−0.604787	+	0.796387i	$0.706742\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	5.56155i		−	1.67687i	−0.545001	−	0.838436i	$-0.683471\pi$
$11$		−	5.56155i		−	1.67687i	0.545001	−	0.838436i	$-0.316529\pi$
$12$		0			0
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$14$		0			0
$15$			3.56155i			0.919589i
$16$		0			0
$17$	−6.68466			−1.62127			−0.810634	−	0.585553i	$-0.800877\pi$
$17$	−6.68466			−1.62127			−0.810634	+	0.585553i	$0.800877\pi$
$18$		0			0
$19$			1.56155i			0.358245i	0.983827	+	0.179122i	$0.0573258\pi$
$19$			1.56155i			0.358245i	−0.983827	+	0.179122i	$0.942674\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	6.68466			1.39385			0.696924	−	0.717145i	$-0.254552\pi$
$23$	6.68466			1.39385			0.696924	+	0.717145i	$0.254552\pi$
$24$		0			0
$25$	−7.68466			−1.53693
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	1.56155			0.289973			0.144987	−	0.989434i	$-0.453686\pi$
$29$	1.56155			0.289973			0.144987	+	0.989434i	$0.453686\pi$
$30$		0			0
$31$			6.24621i			1.12185i	0.827866	+	0.560926i	$0.189555\pi$
$31$			6.24621i			1.12185i	−0.827866	+	0.560926i	$0.810445\pi$
$32$		0			0
$33$		−	5.56155i		−	0.968142i
$34$		0			0
$35$	−3.56155			−0.602012
$36$		0			0
$37$			10.6847i			1.75655i	0.478159	+	0.878274i	$0.341304\pi$
$37$			10.6847i			1.75655i	−0.478159	+	0.878274i	$0.658696\pi$
$38$		0			0
$39$	3.56155	−	0.561553i	0.570305	−	0.0899204i
$40$		0			0
$41$			4.00000i			0.624695i	0.949968	+	0.312348i	$0.101115\pi$
$41$			4.00000i			0.624695i	−0.949968	+	0.312348i	$0.898885\pi$
$42$		0			0
$43$	6.43845			0.981854			0.490927	−	0.871201i	$-0.336658\pi$
$43$	6.43845			0.981854			0.490927	+	0.871201i	$0.336658\pi$
$44$		0			0
$45$			3.56155i			0.530925i
$46$		0			0
$47$			10.2462i			1.49456i	0.664507	+	0.747282i	$0.268641\pi$
$47$			10.2462i			1.49456i	−0.664507	+	0.747282i	$0.731359\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−6.68466			−0.936039
$52$		0			0
$53$	4.87689			0.669893			0.334946	−	0.942237i	$-0.391282\pi$
$53$	4.87689			0.669893			0.334946	+	0.942237i	$0.391282\pi$
$54$		0			0
$55$	19.8078			2.67088
$56$		0			0
$57$			1.56155i			0.206833i
$58$		0			0
$59$		−	4.24621i		−	0.552810i	−0.961041	−	0.276405i	$-0.910857\pi$
$59$		−	4.24621i		−	0.552810i	0.961041	−	0.276405i	$-0.0891431\pi$
$60$		0			0
$61$	−1.56155			−0.199936			−0.0999682	−	0.994991i	$-0.531874\pi$
$61$	−1.56155			−0.199936			−0.0999682	+	0.994991i	$0.531874\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	2.00000	+	12.6847i	0.248069	+	1.57334i
$66$		0			0
$67$			1.12311i			0.137209i	0.997644	+	0.0686046i	$0.0218547\pi$
$67$			1.12311i			0.137209i	−0.997644	+	0.0686046i	$0.978145\pi$
$68$		0			0
$69$	6.68466			0.804738
$70$		0			0
$71$		−	9.36932i		−	1.11193i	−0.831205	−	0.555967i	$-0.812348\pi$
$71$		−	9.36932i		−	1.11193i	0.831205	−	0.555967i	$-0.187652\pi$
$72$		0			0
$73$			11.5616i			1.35318i	0.736361	+	0.676589i	$0.236542\pi$
$73$			11.5616i			1.35318i	−0.736361	+	0.676589i	$0.763458\pi$
$74$		0			0
$75$	−7.68466			−0.887348
$76$		0			0
$77$	5.56155			0.633798
$78$		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	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$		−	23.8078i		−	2.58231i
$86$		0			0
$87$	1.56155			0.167416
$88$		0			0
$89$		−	8.00000i		−	0.847998i	−0.905663	−	0.423999i	$-0.860626\pi$
$89$		−	8.00000i		−	0.847998i	0.905663	−	0.423999i	$-0.139374\pi$
$90$		0			0
$91$	0.561553	+	3.56155i	0.0588667	+	0.373352i
$92$		0			0
$93$			6.24621i			0.647702i
$94$		0			0
$95$	−5.56155			−0.570603
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$		0			0
$99$		−	5.56155i		−	0.558957i
$100$		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
$103$	−1.80776			−0.178124			−0.0890621	−	0.996026i	$-0.528387\pi$
$103$	−1.80776			−0.178124			−0.0890621	+	0.996026i	$0.528387\pi$
$104$		0			0
$105$	−3.56155			−0.347572
$106$		0			0
$107$	4.87689			0.471467			0.235734	−	0.971818i	$-0.424251\pi$
$107$	4.87689			0.471467			0.235734	+	0.971818i	$0.424251\pi$
$108$		0			0
$109$			12.9309i			1.23855i	0.785173	+	0.619276i	$0.212574\pi$
$109$			12.9309i			1.23855i	−0.785173	+	0.619276i	$0.787426\pi$
$110$		0			0
$111$			10.6847i			1.01414i
$112$		0			0
$113$	20.2462			1.90460			0.952302	−	0.305158i	$-0.0987093\pi$
$113$	20.2462			1.90460			0.952302	+	0.305158i	$0.0987093\pi$
$114$		0			0
$115$			23.8078i			2.22009i
$116$		0			0
$117$	3.56155	−	0.561553i	0.329266	−	0.0519156i
$118$		0			0
$119$		−	6.68466i		−	0.612782i
$120$		0			0
$121$	−19.9309			−1.81190
$122$		0			0
$123$			4.00000i			0.360668i
$124$		0			0
$125$		−	9.56155i		−	0.855211i
$126$		0			0
$127$	−10.2462			−0.909204			−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462			−0.909204			−0.454602	+	0.890695i	$0.650219\pi$
$128$		0			0
$129$	6.43845			0.566874
$130$		0			0
$131$	1.56155			0.136434			0.0682168	−	0.997671i	$-0.478269\pi$
$131$	1.56155			0.136434			0.0682168	+	0.997671i	$0.478269\pi$
$132$		0			0
$133$	−1.56155			−0.135404
$134$		0			0
$135$			3.56155i			0.306530i
$136$		0			0
$137$			6.68466i			0.571109i	0.958362	+	0.285554i	$0.0921777\pi$
$137$			6.68466i			0.571109i	−0.958362	+	0.285554i	$0.907822\pi$
$138$		0			0
$139$	−10.2462			−0.869072			−0.434536	−	0.900654i	$-0.643088\pi$
$139$	−10.2462			−0.869072			−0.434536	+	0.900654i	$0.643088\pi$
$140$		0			0
$141$			10.2462i			0.862887i
$142$		0			0
$143$	−3.12311	−	19.8078i	−0.261167	−	1.65641i
$144$		0			0
$145$			5.56155i			0.461862i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		0			0
$151$		−	16.6847i		−	1.35778i	−0.734241	−	0.678889i	$-0.762462\pi$
$151$		−	16.6847i		−	1.35778i	0.734241	−	0.678889i	$-0.237538\pi$
$152$		0			0
$153$	−6.68466			−0.540423
$154$		0			0
$155$	−22.2462			−1.78686
$156$		0			0
$157$	−11.8078			−0.942362			−0.471181	−	0.882037i	$-0.656172\pi$
$157$	−11.8078			−0.942362			−0.471181	+	0.882037i	$0.656172\pi$
$158$		0			0
$159$	4.87689			0.386763
$160$		0			0
$161$			6.68466i			0.526825i
$162$		0			0
$163$		−	9.12311i		−	0.714577i	−0.933994	−	0.357288i	$-0.883701\pi$
$163$		−	9.12311i		−	0.714577i	0.933994	−	0.357288i	$-0.116299\pi$
$164$		0			0
$165$	19.8078			1.54203
$166$		0			0
$167$			11.8078i			0.913712i	0.889541	+	0.456856i	$0.151025\pi$
$167$			11.8078i			0.913712i	−0.889541	+	0.456856i	$0.848975\pi$
$168$		0			0
$169$	12.3693	−	4.00000i	0.951486	−	0.307692i
$170$		0			0
$171$			1.56155i			0.119415i
$172$		0			0
$173$	−3.75379			−0.285395			−0.142698	−	0.989766i	$-0.545578\pi$
$173$	−3.75379			−0.285395			−0.142698	+	0.989766i	$0.545578\pi$
$174$		0			0
$175$		−	7.68466i		−	0.580906i
$176$		0			0
$177$		−	4.24621i		−	0.319165i
$178$		0			0
$179$	−11.1231			−0.831380			−0.415690	−	0.909506i	$-0.636460\pi$
$179$	−11.1231			−0.831380			−0.415690	+	0.909506i	$0.636460\pi$
$180$		0			0
$181$	−13.3693			−0.993733			−0.496867	−	0.867827i	$-0.665516\pi$
$181$	−13.3693			−0.993733			−0.496867	+	0.867827i	$0.665516\pi$
$182$		0			0
$183$	−1.56155			−0.115433
$184$		0			0
$185$	−38.0540			−2.79778
$186$		0			0
$187$			37.1771i			2.71866i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	24.0540			1.74048			0.870242	−	0.492624i	$-0.163962\pi$
$191$	24.0540			1.74048			0.870242	+	0.492624i	$0.163962\pi$
$192$		0			0
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$		0			0
$195$	2.00000	+	12.6847i	0.143223	+	0.908367i
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	−4.93087			−0.349540			−0.174770	−	0.984609i	$-0.555918\pi$
$199$	−4.93087			−0.349540			−0.174770	+	0.984609i	$0.555918\pi$
$200$		0			0
$201$			1.12311i			0.0792178i
$202$		0			0
$203$			1.56155i			0.109600i
$204$		0			0
$205$	−14.2462			−0.994999
$206$		0			0
$207$	6.68466			0.464616
$208$		0			0
$209$	8.68466			0.600730
$210$		0			0
$211$	−7.80776			−0.537509			−0.268754	−	0.963209i	$-0.586612\pi$
$211$	−7.80776			−0.537509			−0.268754	+	0.963209i	$0.586612\pi$
$212$		0			0
$213$		−	9.36932i		−	0.641975i
$214$		0			0
$215$			22.9309i			1.56387i
$216$		0			0
$217$	−6.24621			−0.424020
$218$		0			0
$219$			11.5616i			0.781257i
$220$		0			0
$221$	−23.8078	+	3.75379i	−1.60148	+	0.252507i
$222$		0			0
$223$			1.75379i			0.117442i	0.998274	+	0.0587212i	$0.0187023\pi$
$223$			1.75379i			0.117442i	−0.998274	+	0.0587212i	$0.981298\pi$
$224$		0			0
$225$	−7.68466			−0.512311
$226$		0			0
$227$			17.6155i			1.16918i	0.811328	+	0.584592i	$0.198745\pi$
$227$			17.6155i			1.16918i	−0.811328	+	0.584592i	$0.801255\pi$
$228$		0			0
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$		0			0
$231$	5.56155			0.365923
$232$		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
$235$	−36.4924			−2.38050
$236$		0			0
$237$	−16.0000			−1.03931
$238$		0			0
$239$		−	14.2462i		−	0.921511i	−0.887527	−	0.460755i	$-0.847579\pi$
$239$		−	14.2462i		−	0.921511i	0.887527	−	0.460755i	$-0.152421\pi$
$240$		0			0
$241$			6.00000i			0.386494i	0.981150	+	0.193247i	$0.0619019\pi$
$241$			6.00000i			0.386494i	−0.981150	+	0.193247i	$0.938098\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	3.56155i		−	0.227539i
$246$		0			0
$247$	0.876894	+	5.56155i	0.0557955	+	0.353873i
$248$		0			0
$249$			2.00000i			0.126745i
$250$		0			0
$251$	−22.0540			−1.39203			−0.696017	−	0.718025i	$-0.745046\pi$
$251$	−22.0540			−1.39203			−0.696017	+	0.718025i	$0.745046\pi$
$252$		0			0
$253$		−	37.1771i		−	2.33730i
$254$		0			0
$255$		−	23.8078i		−	1.49090i
$256$		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
$259$	−10.6847			−0.663912
$260$		0			0
$261$	1.56155			0.0966577
$262$		0			0
$263$	23.3693			1.44101			0.720507	−	0.693448i	$-0.243909\pi$
$263$	23.3693			1.44101			0.720507	+	0.693448i	$0.243909\pi$
$264$		0			0
$265$			17.3693i			1.06699i
$266$		0			0
$267$		−	8.00000i		−	0.489592i
$268$		0			0
$269$	31.3693			1.91262			0.956311	−	0.292353i	$-0.0944382\pi$
$269$	31.3693			1.91262			0.956311	+	0.292353i	$0.0944382\pi$
$270$		0			0
$271$			8.00000i			0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$			8.00000i			0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$		0			0
$273$	0.561553	+	3.56155i	0.0339867	+	0.215555i
$274$		0			0
$275$			42.7386i			2.57724i
$276$		0			0
$277$	−10.8769			−0.653529			−0.326765	−	0.945106i	$-0.605958\pi$
$277$	−10.8769			−0.653529			−0.326765	+	0.945106i	$0.605958\pi$
$278$		0			0
$279$			6.24621i			0.373951i
$280$		0			0
$281$			6.00000i			0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$			6.00000i			0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$		0			0
$283$	4.87689			0.289901			0.144951	−	0.989439i	$-0.453698\pi$
$283$	4.87689			0.289901			0.144951	+	0.989439i	$0.453698\pi$
$284$		0			0
$285$	−5.56155			−0.329438
$286$		0			0
$287$	−4.00000			−0.236113
$288$		0			0
$289$	27.6847			1.62851
$290$		0			0
$291$			10.0000i			0.586210i
$292$		0			0
$293$		−	7.75379i		−	0.452981i	−0.974013	−	0.226491i	$-0.927275\pi$
$293$		−	7.75379i		−	0.452981i	0.974013	−	0.226491i	$-0.0727253\pi$
$294$		0			0
$295$	15.1231			0.880501
$296$		0			0
$297$		−	5.56155i		−	0.322714i
$298$		0			0
$299$	23.8078	−	3.75379i	1.37684	−	0.217087i
$300$		0			0
$301$			6.43845i			0.371106i
$302$		0			0
$303$	6.00000			0.344691
$304$		0			0
$305$		−	5.56155i		−	0.318454i
$306$		0			0
$307$		−	26.2462i		−	1.49795i	−0.662598	−	0.748975i	$-0.730546\pi$
$307$		−	26.2462i		−	1.49795i	0.662598	−	0.748975i	$-0.269454\pi$
$308$		0			0
$309$	−1.80776			−0.102840
$310$		0			0
$311$	0.492423			0.0279227			0.0139614	−	0.999903i	$-0.495556\pi$
$311$	0.492423			0.0279227			0.0139614	+	0.999903i	$0.495556\pi$
$312$		0			0
$313$	32.2462			1.82266			0.911332	−	0.411673i	$-0.135055\pi$
$313$	32.2462			1.82266			0.911332	+	0.411673i	$0.135055\pi$
$314$		0			0
$315$	−3.56155			−0.200671
$316$		0			0
$317$		−	3.36932i		−	0.189240i	−0.995513	−	0.0946198i	$-0.969836\pi$
$317$		−	3.36932i		−	0.189240i	0.995513	−	0.0946198i	$-0.0301636\pi$
$318$		0			0
$319$		−	8.68466i		−	0.486248i
$320$		0			0
$321$	4.87689			0.272202
$322$		0			0
$323$		−	10.4384i		−	0.580811i
$324$		0			0
$325$	−27.3693	+	4.31534i	−1.51818	+	0.239372i
$326$		0			0
$327$			12.9309i			0.715079i
$328$		0			0
$329$	−10.2462			−0.564892
$330$		0			0
$331$			1.12311i			0.0617315i	0.999524	+	0.0308657i	$0.00982643\pi$
$331$			1.12311i			0.0617315i	−0.999524	+	0.0308657i	$0.990174\pi$
$332$		0			0
$333$			10.6847i			0.585516i
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$	20.0540			1.09241			0.546205	−	0.837652i	$-0.316072\pi$
$337$	20.0540			1.09241			0.546205	+	0.837652i	$0.316072\pi$
$338$		0			0
$339$	20.2462			1.09962
$340$		0			0
$341$	34.7386			1.88120
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			23.8078i			1.28177i
$346$		0			0
$347$	24.4924			1.31482			0.657411	−	0.753532i	$-0.271652\pi$
$347$	24.4924			1.31482			0.657411	+	0.753532i	$0.271652\pi$
$348$		0			0
$349$			3.36932i			0.180355i	0.995926	+	0.0901777i	$0.0287435\pi$
$349$			3.36932i			0.180355i	−0.995926	+	0.0901777i	$0.971256\pi$
$350$		0			0
$351$	3.56155	−	0.561553i	0.190102	−	0.0299735i
$352$		0			0
$353$		−	18.2462i		−	0.971148i	−0.874196	−	0.485574i	$-0.838611\pi$
$353$		−	18.2462i		−	0.971148i	0.874196	−	0.485574i	$-0.161389\pi$
$354$		0			0
$355$	33.3693			1.77106
$356$		0			0
$357$		−	6.68466i		−	0.353790i
$358$		0			0
$359$		−	23.6155i		−	1.24638i	−0.782071	−	0.623190i	$-0.785836\pi$
$359$		−	23.6155i		−	1.24638i	0.782071	−	0.623190i	$-0.214164\pi$
$360$		0			0
$361$	16.5616			0.871661
$362$		0			0
$363$	−19.9309			−1.04610
$364$		0			0
$365$	−41.1771			−2.15531
$366$		0			0
$367$	−0.630683			−0.0329214			−0.0164607	−	0.999865i	$-0.505240\pi$
$367$	−0.630683			−0.0329214			−0.0164607	+	0.999865i	$0.505240\pi$
$368$		0			0
$369$			4.00000i			0.208232i
$370$		0			0
$371$			4.87689i			0.253196i
$372$		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
$375$		−	9.56155i		−	0.493756i
$376$		0			0
$377$	5.56155	−	0.876894i	0.286435	−	0.0451624i
$378$		0			0
$379$			18.8769i			0.969641i	0.874614	+	0.484820i	$0.161115\pi$
$379$			18.8769i			0.969641i	−0.874614	+	0.484820i	$0.838885\pi$
$380$		0			0
$381$	−10.2462			−0.524929
$382$		0			0
$383$		−	17.5616i		−	0.897353i	−0.893694	−	0.448677i	$-0.851895\pi$
$383$		−	17.5616i		−	0.897353i	0.893694	−	0.448677i	$-0.148105\pi$
$384$		0			0
$385$			19.8078i			1.00950i
$386$		0			0
$387$	6.43845			0.327285
$388$		0			0
$389$	27.1231			1.37520			0.687598	−	0.726092i	$-0.258665\pi$
$389$	27.1231			1.37520			0.687598	+	0.726092i	$0.258665\pi$
$390$		0			0
$391$	−44.6847			−2.25980
$392$		0			0
$393$	1.56155			0.0787699
$394$		0			0
$395$		−	56.9848i		−	2.86722i
$396$		0			0
$397$			15.3693i			0.771364i	0.922632	+	0.385682i	$0.126034\pi$
$397$			15.3693i			0.771364i	−0.922632	+	0.385682i	$0.873966\pi$
$398$		0			0
$399$	−1.56155			−0.0781754
$400$		0			0
$401$		−	30.9848i		−	1.54731i	−0.633608	−	0.773655i	$-0.718427\pi$
$401$		−	30.9848i		−	1.54731i	0.633608	−	0.773655i	$-0.281573\pi$
$402$		0			0
$403$	3.50758	+	22.2462i	0.174725	+	1.10816i
$404$		0			0
$405$			3.56155i			0.176975i
$406$		0			0
$407$	59.4233			2.94550
$408$		0			0
$409$			13.8078i			0.682750i	0.939927	+	0.341375i	$0.110893\pi$
$409$			13.8078i			0.682750i	−0.939927	+	0.341375i	$0.889107\pi$
$410$		0			0
$411$			6.68466i			0.329730i
$412$		0			0
$413$	4.24621			0.208942
$414$		0			0
$415$	−7.12311			−0.349660
$416$		0			0
$417$	−10.2462			−0.501759
$418$		0			0
$419$	20.6847			1.01051			0.505256	−	0.862970i	$-0.331398\pi$
$419$	20.6847			1.01051			0.505256	+	0.862970i	$0.331398\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$		0			0
$423$			10.2462i			0.498188i
$424$		0			0
$425$	51.3693			2.49178
$426$		0			0
$427$		−	1.56155i		−	0.0755688i
$428$		0			0
$429$	−3.12311	−	19.8078i	−0.150785	−	0.956328i
$430$		0			0
$431$		−	29.8617i		−	1.43839i	−0.694809	−	0.719195i	$-0.744511\pi$
$431$		−	29.8617i		−	1.43839i	0.694809	−	0.719195i	$-0.255489\pi$
$432$		0			0
$433$	23.3693			1.12306			0.561529	−	0.827457i	$-0.310213\pi$
$433$	23.3693			1.12306			0.561529	+	0.827457i	$0.310213\pi$
$434$		0			0
$435$			5.56155i			0.266656i
$436$		0			0
$437$			10.4384i			0.499339i
$438$		0			0
$439$	8.93087			0.426247			0.213124	−	0.977025i	$-0.431636\pi$
$439$	8.93087			0.426247			0.213124	+	0.977025i	$0.431636\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−6.63068			−0.315033			−0.157517	−	0.987516i	$-0.550349\pi$
$443$	−6.63068			−0.315033			−0.157517	+	0.987516i	$0.550349\pi$
$444$		0			0
$445$	28.4924			1.35067
$446$		0			0
$447$			10.0000i			0.472984i
$448$		0			0
$449$		−	9.31534i		−	0.439618i	−0.975543	−	0.219809i	$-0.929457\pi$
$449$		−	9.31534i		−	0.439618i	0.975543	−	0.219809i	$-0.0705434\pi$
$450$		0			0
$451$	22.2462			1.04753
$452$		0			0
$453$		−	16.6847i		−	0.783914i
$454$		0			0
$455$	−12.6847	+	2.00000i	−0.594666	+	0.0937614i
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$		0			0
$459$	−6.68466			−0.312013
$460$		0			0
$461$			36.9309i			1.72004i	0.510259	+	0.860021i	$0.329550\pi$
$461$			36.9309i			1.72004i	−0.510259	+	0.860021i	$0.670450\pi$
$462$		0			0
$463$			30.5464i			1.41961i	0.704397	+	0.709806i	$0.251217\pi$
$463$			30.5464i			1.41961i	−0.704397	+	0.709806i	$0.748783\pi$
$464$		0			0
$465$	−22.2462			−1.03164
$466$		0			0
$467$	−1.56155			−0.0722600			−0.0361300	−	0.999347i	$-0.511503\pi$
$467$	−1.56155			−0.0722600			−0.0361300	+	0.999347i	$0.511503\pi$
$468$		0			0
$469$	−1.12311			−0.0518602
$470$		0			0
$471$	−11.8078			−0.544073
$472$		0			0
$473$		−	35.8078i		−	1.64644i
$474$		0			0
$475$		−	12.0000i		−	0.550598i
$476$		0			0
$477$	4.87689			0.223298
$478$		0			0
$479$		−	12.3002i		−	0.562010i	−0.959706	−	0.281005i	$-0.909332\pi$
$479$		−	12.3002i		−	0.562010i	0.959706	−	0.281005i	$-0.0906677\pi$
$480$		0			0
$481$	6.00000	+	38.0540i	0.273576	+	1.73511i
$482$		0			0
$483$			6.68466i			0.304162i
$484$		0			0
$485$	−35.6155			−1.61722
$486$		0			0
$487$			13.7538i			0.623244i	0.950206	+	0.311622i	$0.100872\pi$
$487$			13.7538i			0.623244i	−0.950206	+	0.311622i	$0.899128\pi$
$488$		0			0
$489$		−	9.12311i		−	0.412561i
$490$		0			0
$491$	10.2462			0.462405			0.231203	−	0.972906i	$-0.425734\pi$
$491$	10.2462			0.462405			0.231203	+	0.972906i	$0.425734\pi$
$492$		0			0
$493$	−10.4384			−0.470124
$494$		0			0
$495$	19.8078			0.890293
$496$		0			0
$497$	9.36932			0.420271
$498$		0			0
$499$			1.50758i			0.0674884i	0.999431	+	0.0337442i	$0.0107432\pi$
$499$			1.50758i			0.0674884i	−0.999431	+	0.0337442i	$0.989257\pi$
$500$		0			0
$501$			11.8078i			0.527532i
$502$		0			0
$503$	−35.6155			−1.58802			−0.794009	−	0.607906i	$-0.792010\pi$
$503$	−35.6155			−1.58802			−0.794009	+	0.607906i	$0.792010\pi$
$504$		0			0
$505$			21.3693i			0.950922i
$506$		0			0
$507$	12.3693	−	4.00000i	0.549341	−	0.177646i
$508$		0			0
$509$			2.68466i			0.118995i	0.998228	+	0.0594977i	$0.0189499\pi$
$509$			2.68466i			0.118995i	−0.998228	+	0.0594977i	$0.981050\pi$
$510$		0			0
$511$	−11.5616			−0.511453
$512$		0			0
$513$			1.56155i			0.0689442i
$514$		0			0
$515$		−	6.43845i		−	0.283712i
$516$		0			0
$517$	56.9848			2.50619
$518$		0			0
$519$	−3.75379			−0.164773
$520$		0			0
$521$	8.05398			0.352851			0.176426	−	0.984314i	$-0.443547\pi$
$521$	8.05398			0.352851			0.176426	+	0.984314i	$0.443547\pi$
$522$		0			0
$523$	8.49242			0.371348			0.185674	−	0.982611i	$-0.440553\pi$
$523$	8.49242			0.371348			0.185674	+	0.982611i	$0.440553\pi$
$524$		0			0
$525$		−	7.68466i		−	0.335386i
$526$		0			0
$527$		−	41.7538i		−	1.81882i
$528$		0			0
$529$	21.6847			0.942811
$530$		0			0
$531$		−	4.24621i		−	0.184270i
$532$		0			0
$533$	2.24621	+	14.2462i	0.0972942	+	0.617072i
$534$		0			0
$535$			17.3693i			0.750941i
$536$		0			0
$537$	−11.1231			−0.479997
$538$		0			0
$539$			5.56155i			0.239553i
$540$		0			0
$541$		−	6.19224i		−	0.266225i	−0.991101	−	0.133113i	$-0.957503\pi$
$541$		−	6.19224i		−	0.266225i	0.991101	−	0.133113i	$-0.0424972\pi$
$542$		0			0
$543$	−13.3693			−0.573732
$544$		0			0
$545$	−46.0540			−1.97274
$546$		0			0
$547$	−28.9848			−1.23930			−0.619651	−	0.784877i	$-0.712726\pi$
$547$	−28.9848			−1.23930			−0.619651	+	0.784877i	$0.712726\pi$
$548$		0			0
$549$	−1.56155			−0.0666455
$550$		0			0
$551$			2.43845i			0.103881i
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$		0			0
$555$	−38.0540			−1.61530
$556$		0			0
$557$			16.2462i			0.688374i	0.938901	+	0.344187i	$0.111845\pi$
$557$			16.2462i			0.688374i	−0.938901	+	0.344187i	$0.888155\pi$
$558$		0			0
$559$	22.9309	−	3.61553i	0.969872	−	0.152921i
$560$		0			0
$561$			37.1771i			1.56962i
$562$		0			0
$563$	−38.0540			−1.60378			−0.801892	−	0.597469i	$-0.796173\pi$
$563$	−38.0540			−1.60378			−0.801892	+	0.597469i	$0.796173\pi$
$564$		0			0
$565$			72.1080i			3.03360i
$566$		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
$571$	18.2462			0.763580			0.381790	−	0.924249i	$-0.375308\pi$
$571$	18.2462			0.763580			0.381790	+	0.924249i	$0.375308\pi$
$572$		0			0
$573$	24.0540			1.00487
$574$		0			0
$575$	−51.3693			−2.14225
$576$		0			0
$577$			7.75379i			0.322794i	0.986890	+	0.161397i	$0.0516000\pi$
$577$			7.75379i			0.322794i	−0.986890	+	0.161397i	$0.948400\pi$
$578$		0			0
$579$		−	12.0000i		−	0.498703i
$580$		0			0
$581$	−2.00000			−0.0829740
$582$		0			0
$583$		−	27.1231i		−	1.12332i
$584$		0			0
$585$	2.00000	+	12.6847i	0.0826898	+	0.524446i
$586$		0			0
$587$			37.1231i			1.53223i	0.642701	+	0.766117i	$0.277814\pi$
$587$			37.1231i			1.53223i	−0.642701	+	0.766117i	$0.722186\pi$
$588$		0			0
$589$	−9.75379			−0.401898
$590$		0			0
$591$		−	6.00000i		−	0.246807i
$592$		0			0
$593$		−	28.0000i		−	1.14982i	−0.818216	−	0.574911i	$-0.805037\pi$
$593$		−	28.0000i		−	1.14982i	0.818216	−	0.574911i	$-0.194963\pi$
$594$		0			0
$595$	23.8078			0.976023
$596$		0			0
$597$	−4.93087			−0.201807
$598$		0			0
$599$	−30.3002			−1.23803			−0.619016	−	0.785378i	$-0.712468\pi$
$599$	−30.3002			−1.23803			−0.619016	+	0.785378i	$0.712468\pi$
$600$		0			0
$601$	−31.8617			−1.29967			−0.649834	−	0.760076i	$-0.725161\pi$
$601$	−31.8617			−1.29967			−0.649834	+	0.760076i	$0.725161\pi$
$602$		0			0
$603$			1.12311i			0.0457364i
$604$		0			0
$605$		−	70.9848i		−	2.88594i
$606$		0			0
$607$	28.5464			1.15866			0.579331	−	0.815092i	$-0.303314\pi$
$607$	28.5464			1.15866			0.579331	+	0.815092i	$0.303314\pi$
$608$		0			0
$609$			1.56155i			0.0632773i
$610$		0			0
$611$	5.75379	+	36.4924i	0.232773	+	1.47633i
$612$		0			0
$613$		−	1.80776i		−	0.0730149i	−0.999333	−	0.0365075i	$-0.988377\pi$
$613$		−	1.80776i		−	0.0730149i	0.999333	−	0.0365075i	$-0.0116233\pi$
$614$		0			0
$615$	−14.2462			−0.574463
$616$		0			0
$617$			6.19224i			0.249290i	0.992201	+	0.124645i	$0.0397792\pi$
$617$			6.19224i			0.249290i	−0.992201	+	0.124645i	$0.960221\pi$
$618$		0			0
$619$			30.9309i			1.24322i	0.783328	+	0.621608i	$0.213520\pi$
$619$			30.9309i			1.24322i	−0.783328	+	0.621608i	$0.786480\pi$
$620$		0			0
$621$	6.68466			0.268246
$622$		0			0
$623$	8.00000			0.320513
$624$		0			0
$625$	−4.36932			−0.174773
$626$		0			0
$627$	8.68466			0.346832
$628$		0			0
$629$		−	71.4233i		−	2.84783i
$630$		0			0
$631$		−	12.6847i		−	0.504968i	−0.967601	−	0.252484i	$-0.918752\pi$
$631$		−	12.6847i		−	0.504968i	0.967601	−	0.252484i	$-0.0812476\pi$
$632$		0			0
$633$	−7.80776			−0.310331
$634$		0			0
$635$		−	36.4924i		−	1.44816i
$636$		0			0
$637$	−3.56155	+	0.561553i	−0.141114	+	0.0222495i
$638$		0			0
$639$		−	9.36932i		−	0.370644i
$640$		0			0
$641$	30.1080			1.18919			0.594596	−	0.804024i	$-0.297312\pi$
$641$	30.1080			1.18919			0.594596	+	0.804024i	$0.297312\pi$
$642$		0			0
$643$			0.192236i			0.00758105i	0.999993	+	0.00379052i	$0.00120656\pi$
$643$			0.192236i			0.00758105i	−0.999993	+	0.00379052i	$0.998793\pi$
$644$		0			0
$645$			22.9309i			0.902902i
$646$		0			0
$647$	−10.6307			−0.417935			−0.208968	−	0.977923i	$-0.567010\pi$
$647$	−10.6307			−0.417935			−0.208968	+	0.977923i	$0.567010\pi$
$648$		0			0
$649$	−23.6155			−0.926991
$650$		0			0
$651$	−6.24621			−0.244808
$652$		0			0
$653$	−29.5616			−1.15683			−0.578416	−	0.815742i	$-0.696329\pi$
$653$	−29.5616			−1.15683			−0.578416	+	0.815742i	$0.696329\pi$
$654$		0			0
$655$			5.56155i			0.217308i
$656$		0			0
$657$			11.5616i			0.451059i
$658$		0			0
$659$	−12.8769			−0.501613			−0.250806	−	0.968037i	$-0.580696\pi$
$659$	−12.8769			−0.501613			−0.250806	+	0.968037i	$0.580696\pi$
$660$		0			0
$661$		−	44.7386i		−	1.74013i	−0.492936	−	0.870066i	$-0.664076\pi$
$661$		−	44.7386i		−	1.74013i	0.492936	−	0.870066i	$-0.335924\pi$
$662$		0			0
$663$	−23.8078	+	3.75379i	−0.924617	+	0.145785i
$664$		0			0
$665$		−	5.56155i		−	0.215668i
$666$		0			0
$667$	10.4384			0.404178
$668$		0			0
$669$			1.75379i			0.0678054i
$670$		0			0
$671$			8.68466i			0.335268i
$672$		0			0
$673$	37.8078			1.45738			0.728691	−	0.684843i	$-0.240129\pi$
$673$	37.8078			1.45738			0.728691	+	0.684843i	$0.240129\pi$
$674$		0			0
$675$	−7.68466			−0.295783
$676$		0			0
$677$	31.3693			1.20562			0.602810	−	0.797884i	$-0.294048\pi$
$677$	31.3693			1.20562			0.602810	+	0.797884i	$0.294048\pi$
$678$		0			0
$679$	−10.0000			−0.383765
$680$		0			0
$681$			17.6155i			0.675029i
$682$		0			0
$683$		−	0.684658i		−	0.0261977i	−0.999914	−	0.0130989i	$-0.995830\pi$
$683$		−	0.684658i		−	0.0261977i	0.999914	−	0.0130989i	$-0.00416962\pi$
$684$		0			0
$685$	−23.8078			−0.909648
$686$		0			0
$687$			2.00000i			0.0763048i
$688$		0			0
$689$	17.3693	−	2.73863i	0.661718	−	0.104334i
$690$		0			0
$691$		−	36.4924i		−	1.38824i	−0.719861	−	0.694119i	$-0.755794\pi$
$691$		−	36.4924i		−	1.38824i	0.719861	−	0.694119i	$-0.244206\pi$
$692$		0			0
$693$	5.56155			0.211266
$694$		0			0
$695$		−	36.4924i		−	1.38424i
$696$		0			0
$697$		−	26.7386i		−	1.01280i
$698$		0			0
$699$	2.00000			0.0756469
$700$		0			0
$701$	−3.61553			−0.136557			−0.0682783	−	0.997666i	$-0.521751\pi$
$701$	−3.61553			−0.136557			−0.0682783	+	0.997666i	$0.521751\pi$
$702$		0			0
$703$	−16.6847			−0.629274
$704$		0			0
$705$	−36.4924			−1.37438
$706$		0			0
$707$			6.00000i			0.225653i
$708$		0			0
$709$			31.7538i			1.19254i	0.802784	+	0.596269i	$0.203351\pi$
$709$			31.7538i			1.19254i	−0.802784	+	0.596269i	$0.796649\pi$
$710$		0			0
$711$	−16.0000			−0.600047
$712$		0			0
$713$			41.7538i			1.56369i
$714$		0			0
$715$	70.5464	−	11.1231i	2.63829	−	0.415981i
$716$		0			0
$717$		−	14.2462i		−	0.532035i
$718$		0			0
$719$	−13.7538			−0.512930			−0.256465	−	0.966554i	$-0.582558\pi$
$719$	−13.7538			−0.512930			−0.256465	+	0.966554i	$0.582558\pi$
$720$		0			0
$721$		−	1.80776i		−	0.0673247i
$722$		0			0
$723$			6.00000i			0.223142i
$724$		0			0
$725$	−12.0000			−0.445669
$726$		0			0
$727$	48.9309			1.81475			0.907373	−	0.420327i	$-0.138085\pi$
$727$	48.9309			1.81475			0.907373	+	0.420327i	$0.138085\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−43.0388			−1.59185
$732$		0			0
$733$		−	22.8769i		−	0.844977i	−0.906368	−	0.422489i	$-0.861157\pi$
$733$		−	22.8769i		−	0.844977i	0.906368	−	0.422489i	$-0.138843\pi$
$734$		0			0
$735$		−	3.56155i		−	0.131370i
$736$		0			0
$737$	6.24621			0.230082
$738$		0			0
$739$			17.6155i			0.647998i	0.946057	+	0.323999i	$0.105027\pi$
$739$			17.6155i			0.647998i	−0.946057	+	0.323999i	$0.894973\pi$
$740$		0			0
$741$	0.876894	+	5.56155i	0.0322135	+	0.204309i
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−35.6155			−1.30485
$746$		0			0
$747$			2.00000i			0.0731762i
$748$		0			0
$749$			4.87689i			0.178198i
$750$		0			0
$751$	−2.24621			−0.0819654			−0.0409827	−	0.999160i	$-0.513049\pi$
$751$	−2.24621			−0.0819654			−0.0409827	+	0.999160i	$0.513049\pi$
$752$		0			0
$753$	−22.0540			−0.803692
$754$		0			0
$755$	59.4233			2.16264
$756$		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
$759$		−	37.1771i		−	1.34944i
$760$		0			0
$761$		−	39.1231i		−	1.41821i	−0.705102	−	0.709106i	$-0.749099\pi$
$761$		−	39.1231i		−	1.41821i	0.705102	−	0.709106i	$-0.250901\pi$
$762$		0			0
$763$	−12.9309			−0.468129
$764$		0			0
$765$		−	23.8078i		−	0.860772i
$766$		0			0
$767$	−2.38447	−	15.1231i	−0.0860983	−	0.546064i
$768$		0			0
$769$		−	8.43845i		−	0.304298i	−0.988358	−	0.152149i	$-0.951381\pi$
$769$		−	8.43845i		−	0.304298i	0.988358	−	0.152149i	$-0.0486194\pi$
$770$		0			0
$771$	−26.0000			−0.936367
$772$		0			0
$773$			12.9309i			0.465091i	0.972586	+	0.232546i	$0.0747055\pi$
$773$			12.9309i			0.465091i	−0.972586	+	0.232546i	$0.925295\pi$
$774$		0			0
$775$		−	48.0000i		−	1.72421i
$776$		0			0
$777$	−10.6847			−0.383310
$778$		0			0
$779$	−6.24621			−0.223794
$780$		0			0
$781$	−52.1080			−1.86457
$782$		0			0
$783$	1.56155			0.0558053
$784$		0			0
$785$		−	42.0540i		−	1.50097i
$786$		0			0
$787$		−	17.0691i		−	0.608449i	−0.952600	−	0.304224i	$-0.901603\pi$
$787$		−	17.0691i		−	0.608449i	0.952600	−	0.304224i	$-0.0983973\pi$
$788$		0			0
$789$	23.3693			0.831970
$790$		0			0
$791$			20.2462i			0.719872i
$792$		0			0
$793$	−5.56155	+	0.876894i	−0.197497	+	0.0311394i
$794$		0			0
$795$			17.3693i			0.616026i
$796$		0			0
$797$	3.75379			0.132966			0.0664830	−	0.997788i	$-0.478822\pi$
$797$	3.75379			0.132966			0.0664830	+	0.997788i	$0.478822\pi$
$798$		0			0
$799$		−	68.4924i		−	2.42309i
$800$		0			0
$801$		−	8.00000i		−	0.282666i
$802$		0			0
$803$	64.3002			2.26910
$804$		0			0
$805$	−23.8078			−0.839113
$806$		0			0
$807$	31.3693			1.10425
$808$		0			0
$809$	−55.3693			−1.94668			−0.973341	−	0.229364i	$-0.926335\pi$
$809$	−55.3693			−1.94668			−0.973341	+	0.229364i	$0.926335\pi$
$810$		0			0
$811$			2.05398i			0.0721248i	0.999350	+	0.0360624i	$0.0114815\pi$
$811$			2.05398i			0.0721248i	−0.999350	+	0.0360624i	$0.988518\pi$
$812$		0			0
$813$			8.00000i			0.280572i
$814$		0			0
$815$	32.4924			1.13816
$816$		0			0
$817$			10.0540i			0.351744i
$818$		0			0
$819$	0.561553	+	3.56155i	0.0196222	+	0.124451i
$820$		0			0
$821$		−	30.8769i		−	1.07761i	−0.842430	−	0.538806i	$-0.818876\pi$
$821$		−	30.8769i		−	1.07761i	0.842430	−	0.538806i	$-0.181124\pi$
$822$		0			0
$823$	30.7386			1.07148			0.535741	−	0.844383i	$-0.320032\pi$
$823$	30.7386			1.07148			0.535741	+	0.844383i	$0.320032\pi$
$824$		0			0
$825$			42.7386i			1.48797i
$826$		0			0
$827$		−	22.0540i		−	0.766892i	−0.923563	−	0.383446i	$-0.874737\pi$
$827$		−	22.0540i		−	0.766892i	0.923563	−	0.383446i	$-0.125263\pi$
$828$		0			0
$829$	42.0540			1.46059			0.730297	−	0.683129i	$-0.239381\pi$
$829$	42.0540			1.46059			0.730297	+	0.683129i	$0.239381\pi$
$830$		0			0
$831$	−10.8769			−0.377315
$832$		0			0
$833$	6.68466			0.231610
$834$		0			0
$835$	−42.0540			−1.45534
$836$		0			0
$837$			6.24621i			0.215901i
$838$		0			0
$839$		−	25.7538i		−	0.889120i	−0.895749	−	0.444560i	$-0.853360\pi$
$839$		−	25.7538i		−	0.889120i	0.895749	−	0.444560i	$-0.146640\pi$
$840$		0			0
$841$	−26.5616			−0.915916
$842$		0			0
$843$			6.00000i			0.206651i
$844$		0			0
$845$	14.2462	+	44.0540i	0.490085	+	1.51550i
$846$		0			0
$847$		−	19.9309i		−	0.684833i
$848$		0			0
$849$	4.87689			0.167375
$850$		0			0
$851$			71.4233i			2.44836i
$852$		0			0
$853$			10.8769i			0.372418i	0.982510	+	0.186209i	$0.0596201\pi$
$853$			10.8769i			0.372418i	−0.982510	+	0.186209i	$0.940380\pi$
$854$		0			0
$855$	−5.56155			−0.190201
$856$		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
$859$	−0.384472			−0.0131180			−0.00655901	−	0.999978i	$-0.502088\pi$
$859$	−0.384472			−0.0131180			−0.00655901	+	0.999978i	$0.502088\pi$
$860$		0			0
$861$	−4.00000			−0.136320
$862$		0			0
$863$			42.7386i			1.45484i	0.686192	+	0.727420i	$0.259281\pi$
$863$			42.7386i			1.45484i	−0.686192	+	0.727420i	$0.740719\pi$
$864$		0			0
$865$		−	13.3693i		−	0.454570i
$866$		0			0
$867$	27.6847			0.940220
$868$		0			0
$869$			88.9848i			3.01860i
$870$		0			0
$871$	0.630683	+	4.00000i	0.0213699	+	0.135535i
$872$		0			0
$873$			10.0000i			0.338449i
$874$		0			0
$875$	9.56155			0.323239
$876$		0			0
$877$		−	22.9848i		−	0.776143i	−0.921629	−	0.388072i	$-0.873141\pi$
$877$		−	22.9848i		−	0.776143i	0.921629	−	0.388072i	$-0.126859\pi$
$878$		0			0
$879$		−	7.75379i		−	0.261529i
$880$		0			0
$881$	14.3002			0.481786			0.240893	−	0.970552i	$-0.422560\pi$
$881$	14.3002			0.481786			0.240893	+	0.970552i	$0.422560\pi$
$882$		0			0
$883$	14.4384			0.485892			0.242946	−	0.970040i	$-0.421886\pi$
$883$	14.4384			0.485892			0.242946	+	0.970040i	$0.421886\pi$
$884$		0			0
$885$	15.1231			0.508358
$886$		0			0
$887$	37.8617			1.27127			0.635636	−	0.771989i	$-0.280738\pi$
$887$	37.8617			1.27127			0.635636	+	0.771989i	$0.280738\pi$
$888$		0			0
$889$		−	10.2462i		−	0.343647i
$890$		0			0
$891$		−	5.56155i		−	0.186319i
$892$		0			0
$893$	−16.0000			−0.535420
$894$		0			0
$895$		−	39.6155i		−	1.32420i
$896$		0			0
$897$	23.8078	−	3.75379i	0.794918	−	0.125335i
$898$		0			0
$899$			9.75379i			0.325307i
$900$		0			0
$901$	−32.6004			−1.08608
$902$		0			0
$903$			6.43845i			0.214258i
$904$		0			0
$905$		−	47.6155i		−	1.58279i
$906$		0			0
$907$	24.4924			0.813258			0.406629	−	0.913593i	$-0.366704\pi$
$907$	24.4924			0.813258			0.406629	+	0.913593i	$0.366704\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	4.43845			0.147052			0.0735262	−	0.997293i	$-0.476575\pi$
$911$	4.43845			0.147052			0.0735262	+	0.997293i	$0.476575\pi$
$912$		0			0
$913$	11.1231			0.368121
$914$		0			0
$915$		−	5.56155i		−	0.183859i
$916$		0			0
$917$			1.56155i			0.0515670i
$918$		0			0
$919$	17.8617			0.589204			0.294602	−	0.955620i	$-0.404813\pi$
$919$	17.8617			0.589204			0.294602	+	0.955620i	$0.404813\pi$
$920$		0			0
$921$		−	26.2462i		−	0.864842i
$922$		0			0
$923$	−5.26137	−	33.3693i	−0.173180	−	1.09836i
$924$		0			0
$925$		−	82.1080i		−	2.69969i
$926$		0			0
$927$	−1.80776			−0.0593748
$928$		0			0
$929$		−	40.8769i		−	1.34113i	−0.741852	−	0.670564i	$-0.766052\pi$
$929$		−	40.8769i		−	1.34113i	0.741852	−	0.670564i	$-0.233948\pi$
$930$		0			0
$931$		−	1.56155i		−	0.0511778i
$932$		0			0
$933$	0.492423			0.0161212
$934$		0			0
$935$	−132.408			−4.33021
$936$		0			0
$937$	−19.8617			−0.648855			−0.324427	−	0.945911i	$-0.605172\pi$
$937$	−19.8617			−0.648855			−0.324427	+	0.945911i	$0.605172\pi$
$938$		0			0
$939$	32.2462			1.05232
$940$		0			0
$941$		−	26.4924i		−	0.863628i	−0.901963	−	0.431814i	$-0.857874\pi$
$941$		−	26.4924i		−	0.863628i	0.901963	−	0.431814i	$-0.142126\pi$
$942$		0			0
$943$			26.7386i			0.870730i
$944$		0			0
$945$	−3.56155			−0.115857
$946$		0			0
$947$		−	27.8078i		−	0.903631i	−0.892111	−	0.451815i	$-0.850777\pi$
$947$		−	27.8078i		−	0.903631i	0.892111	−	0.451815i	$-0.149223\pi$
$948$		0			0
$949$	6.49242	+	41.1771i	0.210753	+	1.33666i
$950$		0			0
$951$		−	3.36932i		−	0.109258i
$952$		0			0
$953$	0.738634			0.0239267			0.0119633	−	0.999928i	$-0.496192\pi$
$953$	0.738634			0.0239267			0.0119633	+	0.999928i	$0.496192\pi$
$954$		0			0
$955$			85.6695i			2.77220i
$956$		0			0
$957$		−	8.68466i		−	0.280735i
$958$		0			0
$959$	−6.68466			−0.215859
$960$		0			0
$961$	−8.01515			−0.258553
$962$		0			0
$963$	4.87689			0.157156
$964$		0			0
$965$	42.7386			1.37581
$966$		0			0
$967$		−	2.93087i		−	0.0942504i	−0.998889	−	0.0471252i	$-0.984994\pi$
$967$		−	2.93087i		−	0.0942504i	0.998889	−	0.0471252i	$-0.0150060\pi$
$968$		0			0
$969$		−	10.4384i		−	0.335331i
$970$		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
$973$		−	10.2462i		−	0.328478i
$974$		0			0
$975$	−27.3693	+	4.31534i	−0.876520	+	0.138202i
$976$		0			0
$977$		−	46.7926i		−	1.49703i	−0.663119	−	0.748514i	$-0.730768\pi$
$977$		−	46.7926i		−	1.49703i	0.663119	−	0.748514i	$-0.269232\pi$
$978$		0			0
$979$	−44.4924			−1.42198
$980$		0			0
$981$			12.9309i			0.412851i
$982$		0			0
$983$			2.43845i			0.0777744i	0.999244	+	0.0388872i	$0.0123813\pi$
$983$			2.43845i			0.0777744i	−0.999244	+	0.0388872i	$0.987619\pi$
$984$		0			0
$985$	21.3693			0.680883
$986$		0			0
$987$	−10.2462			−0.326140
$988$		0			0
$989$	43.0388			1.36855
$990$		0			0
$991$	30.2462			0.960803			0.480401	−	0.877049i	$-0.340491\pi$
$991$	30.2462			0.960803			0.480401	+	0.877049i	$0.340491\pi$
$992$		0			0
$993$			1.12311i			0.0356407i
$994$		0			0
$995$		−	17.5616i		−	0.556739i
$996$		0			0
$997$	−34.6307			−1.09676			−0.548382	−	0.836228i	$-0.684756\pi$
$997$	−34.6307			−1.09676			−0.548382	+	0.836228i	$0.684756\pi$
$998$		0			0
$999$			10.6847i			0.338048i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.n.337.4		4
4.3	odd	2		546.2.c.e.337.4	yes	4
12.11	even	2		1638.2.c.h.883.1		4
13.12	even	2	inner	4368.2.h.n.337.1		4
28.27	even	2		3822.2.c.h.883.3		4
52.31	even	4		7098.2.a.bv.1.2		2
52.47	even	4		7098.2.a.bg.1.1		2
52.51	odd	2		546.2.c.e.337.1	✓	4
156.155	even	2		1638.2.c.h.883.4		4
364.363	even	2		3822.2.c.h.883.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.e.337.1	✓	4	52.51	odd	2
546.2.c.e.337.4	yes	4	4.3	odd	2
1638.2.c.h.883.1		4	12.11	even	2
1638.2.c.h.883.4		4	156.155	even	2
3822.2.c.h.883.2		4	364.363	even	2
3822.2.c.h.883.3		4	28.27	even	2
4368.2.h.n.337.1		4	13.12	even	2	inner
4368.2.h.n.337.4		4	1.1	even	1	trivial
7098.2.a.bg.1.1		2	52.47	even	4
7098.2.a.bv.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.56155i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.56155i q^{5} +1.00000i q^{7} +1.00000 q^{9} -5.56155i q^{11} +(3.56155 - 0.561553i) q^{13} +3.56155i q^{15} -6.68466 q^{17} +1.56155i q^{19} +1.00000i q^{21} +6.68466 q^{23} -7.68466 q^{25} +1.00000 q^{27} +1.56155 q^{29} +6.24621i q^{31} -5.56155i q^{33} -3.56155 q^{35} +10.6847i q^{37} +(3.56155 - 0.561553i) q^{39} +4.00000i q^{41} +6.43845 q^{43} +3.56155i q^{45} +10.2462i q^{47} -1.00000 q^{49} -6.68466 q^{51} +4.87689 q^{53} +19.8078 q^{55} +1.56155i q^{57} -4.24621i q^{59} -1.56155 q^{61} +1.00000i q^{63} +(2.00000 + 12.6847i) q^{65} +1.12311i q^{67} +6.68466 q^{69} -9.36932i q^{71} +11.5616i q^{73} -7.68466 q^{75} +5.56155 q^{77} -16.0000 q^{79} +1.00000 q^{81} +2.00000i q^{83} -23.8078i q^{85} +1.56155 q^{87} -8.00000i q^{89} +(0.561553 + 3.56155i) q^{91} +6.24621i q^{93} -5.56155 q^{95} +10.0000i q^{97} -5.56155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 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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