LMFDB - Embedded newform 4732.2.g.k.337.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(337,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4732.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.g (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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704 $ x^16 + 38x^14 + 587x^12 + 4762x^10 + 21849x^8 + 56552x^6 + 76456x^4 + 42624*x^2 + 2704
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.14
Root			$-1.77673i$ of defining polynomial
Character	$\chi$	$=$	4732.337
Dual form			4732.2.g.k.337.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77673 q^{3} +3.82804i q^{5} -1.00000i q^{7} +0.156761 q^{9} +4.32707i q^{11} +6.80139i q^{15} -2.91488 q^{17} +5.45810i q^{19} -1.77673i q^{21} +0.614224 q^{23} -9.65391 q^{25} -5.05166 q^{27} +9.25744 q^{29} +5.30753i q^{31} +7.68802i q^{33} +3.82804 q^{35} +1.12470i q^{37} -12.0239i q^{41} -1.28310 q^{43} +0.600087i q^{45} -5.68074i q^{47} -1.00000 q^{49} -5.17895 q^{51} -3.96013 q^{53} -16.5642 q^{55} +9.69757i q^{57} -7.71754i q^{59} -14.3462 q^{61} -0.156761i q^{63} -0.541525i q^{67} +1.09131 q^{69} +9.20202i q^{71} -7.36798i q^{73} -17.1524 q^{75} +4.32707 q^{77} +0.331686 q^{79} -9.44571 q^{81} +16.6777i q^{83} -11.1583i q^{85} +16.4479 q^{87} -2.95414i q^{89} +9.43004i q^{93} -20.8939 q^{95} +9.65583i q^{97} +0.678314i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 28 q^{9} - 4 q^{17} - 44 q^{25} - 12 q^{27} + 44 q^{29} + 12 q^{35} - 12 q^{43} - 16 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{61} + 104 q^{69} + 20 q^{75} - 24 q^{77} + 8 q^{79} - 52 q^{87}+ \cdots + 60 q^{95}+O(q^{100}) $ 16 * q + 28 * q^9 - 4 * q^17 - 44 * q^25 - 12 * q^27 + 44 * q^29 + 12 * q^35 - 12 * q^43 - 16 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 - 8 * q^61 + 104 * q^69 + 20 * q^75 - 24 * q^77 + 8 * q^79 - 52 * q^87 + 60 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times$.

$n$	$2367$	$2705$	$4565$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.77673	1.02579	0.512897	−	0.858450i	$-0.328572\pi$
$3$	1.77673	1.02579	0.512897	+	0.858450i	$0.328572\pi$
$4$	0	0
$5$	3.82804i	1.71195i	0.517015	+	0.855976i	$0.327043\pi$
$5$	3.82804i	1.71195i	−0.517015	+	0.855976i	$0.672957\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0.156761	0.0522536
$10$	0	0
$11$	4.32707i	1.30466i	0.757935	+	0.652330i	$0.226208\pi$
$11$	4.32707i	1.30466i	−0.757935	+	0.652330i	$0.773792\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	6.80139i	1.75611i
$16$	0	0
$17$	−2.91488	−0.706962	−0.353481	−	0.935442i	$-0.615002\pi$
$17$	−2.91488	−0.706962	−0.353481	+	0.935442i	$0.615002\pi$
$18$	0	0
$19$	5.45810i	1.25218i	0.779753	+	0.626088i	$0.215345\pi$
$19$	5.45810i	1.25218i	−0.779753	+	0.626088i	$0.784655\pi$
$20$	0	0
$21$	− 1.77673i	− 0.387714i
$22$	0	0
$23$	0.614224	0.128075	0.0640373	−	0.997948i	$-0.479602\pi$
$23$	0.614224	0.128075	0.0640373	+	0.997948i	$0.479602\pi$
$24$	0	0
$25$	−9.65391	−1.93078
$26$	0	0
$27$	−5.05166	−0.972193
$28$	0	0
$29$	9.25744	1.71906	0.859531	−	0.511083i	$-0.170756\pi$
$29$	9.25744	1.71906	0.859531	+	0.511083i	$0.170756\pi$
$30$	0	0
$31$	5.30753i	0.953261i	0.879104	+	0.476630i	$0.158142\pi$
$31$	5.30753i	0.953261i	−0.879104	+	0.476630i	$0.841858\pi$
$32$	0	0
$33$	7.68802i	1.33831i
$34$	0	0
$35$	3.82804	0.647057
$36$	0	0
$37$	1.12470i	0.184900i	0.995717	+	0.0924501i	$0.0294699\pi$
$37$	1.12470i	0.184900i	−0.995717	+	0.0924501i	$0.970530\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 12.0239i	− 1.87782i	−0.344165	−	0.938909i	$-0.611838\pi$
$41$	− 12.0239i	− 1.87782i	0.344165	−	0.938909i	$-0.388162\pi$
$42$	0	0
$43$	−1.28310	−0.195672	−0.0978358	−	0.995203i	$-0.531192\pi$
$43$	−1.28310	−0.195672	−0.0978358	+	0.995203i	$0.531192\pi$
$44$	0	0
$45$	0.600087i	0.0894557i
$46$	0	0
$47$	− 5.68074i	− 0.828620i	−0.910136	−	0.414310i	$-0.864023\pi$
$47$	− 5.68074i	− 0.828620i	0.910136	−	0.414310i	$-0.135977\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−5.17895	−0.725197
$52$	0	0
$53$	−3.96013	−0.543966	−0.271983	−	0.962302i	$-0.587679\pi$
$53$	−3.96013	−0.543966	−0.271983	+	0.962302i	$0.587679\pi$
$54$	0	0
$55$	−16.5642	−2.23351
$56$	0	0
$57$	9.69757i	1.28447i
$58$	0	0
$59$	− 7.71754i	− 1.00474i	−0.864653	−	0.502369i	$-0.832462\pi$
$59$	− 7.71754i	− 1.00474i	0.864653	−	0.502369i	$-0.167538\pi$
$60$	0	0
$61$	−14.3462	−1.83684	−0.918419	−	0.395610i	$-0.870533\pi$
$61$	−14.3462	−1.83684	−0.918419	+	0.395610i	$0.870533\pi$
$62$	0	0
$63$	− 0.156761i	− 0.0197500i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.541525i	− 0.0661578i	−0.999453	−	0.0330789i	$-0.989469\pi$
$67$	− 0.541525i	− 0.0661578i	0.999453	−	0.0330789i	$-0.0105313\pi$
$68$	0	0
$69$	1.09131	0.131378
$70$	0	0
$71$	9.20202i	1.09208i	0.837759	+	0.546040i	$0.183865\pi$
$71$	9.20202i	1.09208i	−0.837759	+	0.546040i	$0.816135\pi$
$72$	0	0
$73$	− 7.36798i	− 0.862357i	−0.902267	−	0.431179i	$-0.858098\pi$
$73$	− 7.36798i	− 0.862357i	0.902267	−	0.431179i	$-0.141902\pi$
$74$	0	0
$75$	−17.1524	−1.98059
$76$	0	0
$77$	4.32707	0.493115
$78$	0	0
$79$	0.331686	0.0373176	0.0186588	−	0.999826i	$-0.494060\pi$
$79$	0.331686	0.0373176	0.0186588	+	0.999826i	$0.494060\pi$
$80$	0	0
$81$	−9.44571	−1.04952
$82$	0	0
$83$	16.6777i	1.83061i	0.402758	+	0.915306i	$0.368052\pi$
$83$	16.6777i	1.83061i	−0.402758	+	0.915306i	$0.631948\pi$
$84$	0	0
$85$	− 11.1583i	− 1.21029i
$86$	0	0
$87$	16.4479	1.76340
$88$	0	0
$89$	− 2.95414i	− 0.313138i	−0.987667	−	0.156569i	$-0.949957\pi$
$89$	− 2.95414i	− 0.313138i	0.987667	−	0.156569i	$-0.0500433\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.43004i	0.977849i
$94$	0	0
$95$	−20.8939	−2.14366
$96$	0	0
$97$	9.65583i	0.980401i	0.871610	+	0.490201i	$0.163076\pi$
$97$	9.65583i	0.980401i	−0.871610	+	0.490201i	$0.836924\pi$
$98$	0	0
$99$	0.678314i	0.0681731i
$100$	0	0
$101$	6.35056	0.631905	0.315952	−	0.948775i	$-0.397676\pi$
$101$	6.35056	0.631905	0.315952	+	0.948775i	$0.397676\pi$
$102$	0	0
$103$	−5.07117	−0.499678	−0.249839	−	0.968287i	$-0.580378\pi$
$103$	−5.07117	−0.499678	−0.249839	+	0.968287i	$0.580378\pi$
$104$	0	0
$105$	6.80139	0.663748
$106$	0	0
$107$	−5.48579	−0.530331	−0.265166	−	0.964203i	$-0.585427\pi$
$107$	−5.48579	−0.530331	−0.265166	+	0.964203i	$0.585427\pi$
$108$	0	0
$109$	− 6.80685i	− 0.651978i	−0.945374	−	0.325989i	$-0.894303\pi$
$109$	− 6.80685i	− 0.651978i	0.945374	−	0.325989i	$-0.105697\pi$
$110$	0	0
$111$	1.99829i	0.189669i
$112$	0	0
$113$	−6.56768	−0.617835	−0.308918	−	0.951089i	$-0.599967\pi$
$113$	−6.56768	−0.617835	−0.308918	+	0.951089i	$0.599967\pi$
$114$	0	0
$115$	2.35128i	0.219258i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.91488i	0.267207i
$120$	0	0
$121$	−7.72349	−0.702136
$122$	0	0
$123$	− 21.3632i	− 1.92626i
$124$	0	0
$125$	− 17.8154i	− 1.59346i
$126$	0	0
$127$	20.2850	1.80001	0.900003	−	0.435884i	$-0.143564\pi$
$127$	20.2850	1.80001	0.900003	+	0.435884i	$0.143564\pi$
$128$	0	0
$129$	−2.27973	−0.200719
$130$	0	0
$131$	4.92442	0.430249	0.215124	−	0.976587i	$-0.430984\pi$
$131$	4.92442	0.430249	0.215124	+	0.976587i	$0.430984\pi$
$132$	0	0
$133$	5.45810	0.473278
$134$	0	0
$135$	− 19.3380i	− 1.66435i
$136$	0	0
$137$	12.9318i	1.10484i	0.833567	+	0.552419i	$0.186295\pi$
$137$	12.9318i	1.10484i	−0.833567	+	0.552419i	$0.813705\pi$
$138$	0	0
$139$	15.8093	1.34093	0.670464	−	0.741942i	$-0.266095\pi$
$139$	15.8093	1.34093	0.670464	+	0.741942i	$0.266095\pi$
$140$	0	0
$141$	− 10.0931i	− 0.849994i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	35.4379i	2.94295i
$146$	0	0
$147$	−1.77673	−0.146542
$148$	0	0
$149$	15.9310i	1.30512i	0.757737	+	0.652561i	$0.226305\pi$
$149$	15.9310i	1.30512i	−0.757737	+	0.652561i	$0.773695\pi$
$150$	0	0
$151$	6.37358i	0.518675i	0.965787	+	0.259337i	$0.0835041\pi$
$151$	6.37358i	0.518675i	−0.965787	+	0.259337i	$0.916496\pi$
$152$	0	0
$153$	−0.456939	−0.0369413
$154$	0	0
$155$	−20.3175	−1.63194
$156$	0	0
$157$	−7.38800	−0.589627	−0.294813	−	0.955555i	$-0.595257\pi$
$157$	−7.38800	−0.589627	−0.294813	+	0.955555i	$0.595257\pi$
$158$	0	0
$159$	−7.03608	−0.557997
$160$	0	0
$161$	− 0.614224i	− 0.0484076i
$162$	0	0
$163$	− 0.0875638i	− 0.00685852i	−0.999994	−	0.00342926i	$-0.998908\pi$
$163$	− 0.0875638i	− 0.00685852i	0.999994	−	0.00342926i	$-0.00109157\pi$
$164$	0	0
$165$	−29.4301	−2.29113
$166$	0	0
$167$	− 18.5108i	− 1.43241i	−0.697889	−	0.716206i	$-0.745877\pi$
$167$	− 18.5108i	− 1.43241i	0.697889	−	0.716206i	$-0.254123\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.855617i	0.0654307i
$172$	0	0
$173$	−0.410903	−0.0312404	−0.0156202	−	0.999878i	$-0.504972\pi$
$173$	−0.410903	−0.0312404	−0.0156202	+	0.999878i	$0.504972\pi$
$174$	0	0
$175$	9.65391i	0.729767i
$176$	0	0
$177$	− 13.7120i	− 1.03065i
$178$	0	0
$179$	−2.15475	−0.161054	−0.0805268	−	0.996752i	$-0.525660\pi$
$179$	−2.15475	−0.161054	−0.0805268	+	0.996752i	$0.525660\pi$
$180$	0	0
$181$	6.50124	0.483233	0.241617	−	0.970372i	$-0.422322\pi$
$181$	6.50124	0.483233	0.241617	+	0.970372i	$0.422322\pi$
$182$	0	0
$183$	−25.4892	−1.88422
$184$	0	0
$185$	−4.30541	−0.316540
$186$	0	0
$187$	− 12.6129i	− 0.922344i
$188$	0	0
$189$	5.05166i	0.367454i
$190$	0	0
$191$	−18.7750	−1.35851	−0.679256	−	0.733901i	$-0.737698\pi$
$191$	−18.7750	−1.35851	−0.679256	+	0.733901i	$0.737698\pi$
$192$	0	0
$193$	25.6173i	1.84397i	0.387224	+	0.921986i	$0.373434\pi$
$193$	25.6173i	1.84397i	−0.387224	+	0.921986i	$0.626566\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.589556i	0.0420041i	0.999779	+	0.0210021i	$0.00668565\pi$
$197$	0.589556i	0.0420041i	−0.999779	+	0.0210021i	$0.993314\pi$
$198$	0	0
$199$	−16.0903	−1.14061	−0.570304	−	0.821433i	$-0.693175\pi$
$199$	−16.0903	−1.14061	−0.570304	+	0.821433i	$0.693175\pi$
$200$	0	0
$201$	− 0.962142i	− 0.0678643i
$202$	0	0
$203$	− 9.25744i	− 0.649745i
$204$	0	0
$205$	46.0280	3.21474
$206$	0	0
$207$	0.0962862	0.00669236
$208$	0	0
$209$	−23.6176	−1.63366
$210$	0	0
$211$	−11.3849	−0.783772	−0.391886	−	0.920014i	$-0.628177\pi$
$211$	−11.3849	−0.783772	−0.391886	+	0.920014i	$0.628177\pi$
$212$	0	0
$213$	16.3495i	1.12025i
$214$	0	0
$215$	− 4.91178i	− 0.334981i
$216$	0	0
$217$	5.30753	0.360299
$218$	0	0
$219$	− 13.0909i	− 0.884601i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.34614i	0.157109i	0.996910	+	0.0785545i	$0.0250305\pi$
$223$	2.34614i	0.157109i	−0.996910	+	0.0785545i	$0.974970\pi$
$224$	0	0
$225$	−1.51335	−0.100890
$226$	0	0
$227$	− 8.25134i	− 0.547660i	−0.961778	−	0.273830i	$-0.911709\pi$
$227$	− 8.25134i	− 0.547660i	0.961778	−	0.273830i	$-0.0882906\pi$
$228$	0	0
$229$	17.2750i	1.14156i	0.821103	+	0.570780i	$0.193359\pi$
$229$	17.2750i	1.14156i	−0.821103	+	0.570780i	$0.806641\pi$
$230$	0	0
$231$	7.68802	0.505834
$232$	0	0
$233$	−4.13778	−0.271075	−0.135537	−	0.990772i	$-0.543276\pi$
$233$	−4.13778	−0.271075	−0.135537	+	0.990772i	$0.543276\pi$
$234$	0	0
$235$	21.7461	1.41856
$236$	0	0
$237$	0.589316	0.0382802
$238$	0	0
$239$	− 5.83879i	− 0.377680i	−0.982008	−	0.188840i	$-0.939527\pi$
$239$	− 5.83879i	− 0.377680i	0.982008	−	0.188840i	$-0.0604727\pi$
$240$	0	0
$241$	28.4467i	1.83241i	0.400704	+	0.916207i	$0.368766\pi$
$241$	28.4467i	1.83241i	−0.400704	+	0.916207i	$0.631234\pi$
$242$	0	0
$243$	−1.62747	−0.104402
$244$	0	0
$245$	− 3.82804i	− 0.244565i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	29.6317i	1.87783i
$250$	0	0
$251$	12.7800	0.806667	0.403334	−	0.915053i	$-0.367851\pi$
$251$	12.7800	0.806667	0.403334	+	0.915053i	$0.367851\pi$
$252$	0	0
$253$	2.65779i	0.167094i
$254$	0	0
$255$	− 19.8252i	− 1.24150i
$256$	0	0
$257$	−0.671390	−0.0418802	−0.0209401	−	0.999781i	$-0.506666\pi$
$257$	−0.671390	−0.0418802	−0.0209401	+	0.999781i	$0.506666\pi$
$258$	0	0
$259$	1.12470	0.0698857
$260$	0	0
$261$	1.45120	0.0898272
$262$	0	0
$263$	−18.9702	−1.16976	−0.584878	−	0.811122i	$-0.698857\pi$
$263$	−18.9702	−1.16976	−0.584878	+	0.811122i	$0.698857\pi$
$264$	0	0
$265$	− 15.1596i	− 0.931244i
$266$	0	0
$267$	− 5.24870i	− 0.321215i
$268$	0	0
$269$	−9.58482	−0.584397	−0.292198	−	0.956358i	$-0.594387\pi$
$269$	−9.58482	−0.584397	−0.292198	+	0.956358i	$0.594387\pi$
$270$	0	0
$271$	− 10.4715i	− 0.636097i	−0.948075	−	0.318048i	$-0.896973\pi$
$271$	− 10.4715i	− 0.636097i	0.948075	−	0.318048i	$-0.103027\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 41.7731i	− 2.51901i
$276$	0	0
$277$	19.3371	1.16185	0.580927	−	0.813956i	$-0.302690\pi$
$277$	19.3371	1.16185	0.580927	+	0.813956i	$0.302690\pi$
$278$	0	0
$279$	0.832013i	0.0498113i
$280$	0	0
$281$	− 18.7039i	− 1.11578i	−0.829915	−	0.557890i	$-0.811611\pi$
$281$	− 18.7039i	− 1.11578i	0.829915	−	0.557890i	$-0.188389\pi$
$282$	0	0
$283$	−1.17591	−0.0699007	−0.0349504	−	0.999389i	$-0.511127\pi$
$283$	−1.17591	−0.0699007	−0.0349504	+	0.999389i	$0.511127\pi$
$284$	0	0
$285$	−37.1227	−2.19896
$286$	0	0
$287$	−12.0239	−0.709749
$288$	0	0
$289$	−8.50348	−0.500205
$290$	0	0
$291$	17.1558i	1.00569i
$292$	0	0
$293$	− 6.50698i	− 0.380142i	−0.981770	−	0.190071i	$-0.939128\pi$
$293$	− 6.50698i	− 0.380142i	0.981770	−	0.190071i	$-0.0608718\pi$
$294$	0	0
$295$	29.5431	1.72007
$296$	0	0
$297$	− 21.8589i	− 1.26838i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.28310i	0.0739569i
$302$	0	0
$303$	11.2832	0.648204
$304$	0	0
$305$	− 54.9177i	− 3.14458i
$306$	0	0
$307$	− 27.0573i	− 1.54424i	−0.635475	−	0.772121i	$-0.719196\pi$
$307$	− 27.0573i	− 1.54424i	0.635475	−	0.772121i	$-0.280804\pi$
$308$	0	0
$309$	−9.01009	−0.512566
$310$	0	0
$311$	30.9061	1.75253	0.876263	−	0.481834i	$-0.160029\pi$
$311$	30.9061	1.75253	0.876263	+	0.481834i	$0.160029\pi$
$312$	0	0
$313$	30.0816	1.70031	0.850157	−	0.526530i	$-0.176507\pi$
$313$	30.0816	1.70031	0.850157	+	0.526530i	$0.176507\pi$
$314$	0	0
$315$	0.600087	0.0338111
$316$	0	0
$317$	− 3.78351i	− 0.212503i	−0.994339	−	0.106251i	$-0.966115\pi$
$317$	− 3.78351i	− 0.212503i	0.994339	−	0.106251i	$-0.0338848\pi$
$318$	0	0
$319$	40.0575i	2.24279i
$320$	0	0
$321$	−9.74675	−0.544011
$322$	0	0
$323$	− 15.9097i	− 0.885240i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 12.0939i	− 0.668795i
$328$	0	0
$329$	−5.68074	−0.313189
$330$	0	0
$331$	23.2552i	1.27822i	0.769115	+	0.639110i	$0.220697\pi$
$331$	23.2552i	1.27822i	−0.769115	+	0.639110i	$0.779303\pi$
$332$	0	0
$333$	0.176309i	0.00966170i
$334$	0	0
$335$	2.07298	0.113259
$336$	0	0
$337$	6.51910	0.355118	0.177559	−	0.984110i	$-0.443180\pi$
$337$	6.51910	0.355118	0.177559	+	0.984110i	$0.443180\pi$
$338$	0	0
$339$	−11.6690	−0.633772
$340$	0	0
$341$	−22.9660	−1.24368
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	4.17758i	0.224913i
$346$	0	0
$347$	−13.1139	−0.703992	−0.351996	−	0.936001i	$-0.614497\pi$
$347$	−13.1139	−0.703992	−0.351996	+	0.936001i	$0.614497\pi$
$348$	0	0
$349$	− 9.71447i	− 0.520004i	−0.965608	−	0.260002i	$-0.916277\pi$
$349$	− 9.71447i	− 0.520004i	0.965608	−	0.260002i	$-0.0837232\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.3262i	1.61410i	0.590484	+	0.807049i	$0.298937\pi$
$353$	30.3262i	1.61410i	−0.590484	+	0.807049i	$0.701063\pi$
$354$	0	0
$355$	−35.2257	−1.86959
$356$	0	0
$357$	5.17895i	0.274099i
$358$	0	0
$359$	10.5854i	0.558677i	0.960193	+	0.279339i	$0.0901152\pi$
$359$	10.5854i	0.558677i	−0.960193	+	0.279339i	$0.909885\pi$
$360$	0	0
$361$	−10.7909	−0.567943
$362$	0	0
$363$	−13.7225	−0.720247
$364$	0	0
$365$	28.2050	1.47631
$366$	0	0
$367$	26.6376	1.39047	0.695234	−	0.718783i	$-0.255301\pi$
$367$	26.6376	1.39047	0.695234	+	0.718783i	$0.255301\pi$
$368$	0	0
$369$	− 1.88488i	− 0.0981228i
$370$	0	0
$371$	3.96013i	0.205600i
$372$	0	0
$373$	−6.14128	−0.317984	−0.158992	−	0.987280i	$-0.550824\pi$
$373$	−6.14128	−0.317984	−0.158992	+	0.987280i	$0.550824\pi$
$374$	0	0
$375$	− 31.6531i	− 1.63456i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 23.5761i	− 1.21102i	−0.795837	−	0.605510i	$-0.792969\pi$
$379$	− 23.5761i	− 1.21102i	0.795837	−	0.605510i	$-0.207031\pi$
$380$	0	0
$381$	36.0410	1.84643
$382$	0	0
$383$	− 20.1768i	− 1.03099i	−0.856894	−	0.515493i	$-0.827609\pi$
$383$	− 20.1768i	− 1.03099i	0.856894	−	0.515493i	$-0.172391\pi$
$384$	0	0
$385$	16.5642i	0.844189i
$386$	0	0
$387$	−0.201141	−0.0102245
$388$	0	0
$389$	−32.8506	−1.66559	−0.832795	−	0.553581i	$-0.813261\pi$
$389$	−32.8506	−1.66559	−0.832795	+	0.553581i	$0.813261\pi$
$390$	0	0
$391$	−1.79039	−0.0905438
$392$	0	0
$393$	8.74936	0.441347
$394$	0	0
$395$	1.26971i	0.0638860i
$396$	0	0
$397$	25.2635i	1.26794i	0.773358	+	0.633970i	$0.218576\pi$
$397$	25.2635i	1.26794i	−0.773358	+	0.633970i	$0.781424\pi$
$398$	0	0
$399$	9.69757	0.485486
$400$	0	0
$401$	29.2208i	1.45922i	0.683866	+	0.729608i	$0.260297\pi$
$401$	29.2208i	1.45922i	−0.683866	+	0.729608i	$0.739703\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 36.1586i	− 1.79673i
$406$	0	0
$407$	−4.86667	−0.241232
$408$	0	0
$409$	27.3315i	1.35145i	0.737152	+	0.675727i	$0.236170\pi$
$409$	27.3315i	1.35145i	−0.737152	+	0.675727i	$0.763830\pi$
$410$	0	0
$411$	22.9763i	1.13334i
$412$	0	0
$413$	−7.71754	−0.379755
$414$	0	0
$415$	−63.8429	−3.13392
$416$	0	0
$417$	28.0889	1.37552
$418$	0	0
$419$	20.3386	0.993604	0.496802	−	0.867864i	$-0.334508\pi$
$419$	20.3386	0.993604	0.496802	+	0.867864i	$0.334508\pi$
$420$	0	0
$421$	− 21.0712i	− 1.02695i	−0.858106	−	0.513473i	$-0.828359\pi$
$421$	− 21.0712i	− 1.02695i	0.858106	−	0.513473i	$-0.171641\pi$
$422$	0	0
$423$	− 0.890517i	− 0.0432984i
$424$	0	0
$425$	28.1400	1.36499
$426$	0	0
$427$	14.3462i	0.694259i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.6333i	1.42739i	0.700458	+	0.713694i	$0.252979\pi$
$431$	29.6333i	1.42739i	−0.700458	+	0.713694i	$0.747021\pi$
$432$	0	0
$433$	28.5965	1.37426	0.687130	−	0.726534i	$-0.258870\pi$
$433$	28.5965	1.37426	0.687130	+	0.726534i	$0.258870\pi$
$434$	0	0
$435$	62.9634i	3.01887i
$436$	0	0
$437$	3.35250i	0.160372i
$438$	0	0
$439$	21.8248	1.04164	0.520821	−	0.853666i	$-0.325626\pi$
$439$	21.8248	1.04164	0.520821	+	0.853666i	$0.325626\pi$
$440$	0	0
$441$	−0.156761	−0.00746480
$442$	0	0
$443$	−6.32956	−0.300726	−0.150363	−	0.988631i	$-0.548044\pi$
$443$	−6.32956	−0.300726	−0.150363	+	0.988631i	$0.548044\pi$
$444$	0	0
$445$	11.3086	0.536077
$446$	0	0
$447$	28.3051i	1.33879i
$448$	0	0
$449$	24.5693i	1.15950i	0.814795	+	0.579749i	$0.196849\pi$
$449$	24.5693i	1.15950i	−0.814795	+	0.579749i	$0.803151\pi$
$450$	0	0
$451$	52.0282	2.44991
$452$	0	0
$453$	11.3241i	0.532053i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.50611i	0.210787i	0.994431	+	0.105393i	$0.0336102\pi$
$457$	4.50611i	0.210787i	−0.994431	+	0.105393i	$0.966390\pi$
$458$	0	0
$459$	14.7250	0.687303
$460$	0	0
$461$	0.300252i	0.0139841i	0.999976	+	0.00699206i	$0.00222566\pi$
$461$	0.300252i	0.0139841i	−0.999976	+	0.00699206i	$0.997774\pi$
$462$	0	0
$463$	19.2907i	0.896514i	0.893905	+	0.448257i	$0.147955\pi$
$463$	19.2907i	0.896514i	−0.893905	+	0.448257i	$0.852045\pi$
$464$	0	0
$465$	−36.0986	−1.67403
$466$	0	0
$467$	13.9509	0.645570	0.322785	−	0.946472i	$-0.395381\pi$
$467$	13.9509	0.645570	0.322785	+	0.946472i	$0.395381\pi$
$468$	0	0
$469$	−0.541525	−0.0250053
$470$	0	0
$471$	−13.1265	−0.604836
$472$	0	0
$473$	− 5.55208i	− 0.255285i
$474$	0	0
$475$	− 52.6921i	− 2.41768i
$476$	0	0
$477$	−0.620794	−0.0284242
$478$	0	0
$479$	27.4521i	1.25432i	0.778890	+	0.627160i	$0.215783\pi$
$479$	27.4521i	1.25432i	−0.778890	+	0.627160i	$0.784217\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 1.09131i	− 0.0496563i
$484$	0	0
$485$	−36.9629	−1.67840
$486$	0	0
$487$	13.9957i	0.634205i	0.948391	+	0.317103i	$0.102710\pi$
$487$	13.9957i	0.634205i	−0.948391	+	0.317103i	$0.897290\pi$
$488$	0	0
$489$	− 0.155577i	− 0.00703543i
$490$	0	0
$491$	7.52661	0.339671	0.169836	−	0.985472i	$-0.445676\pi$
$491$	7.52661	0.339671	0.169836	+	0.985472i	$0.445676\pi$
$492$	0	0
$493$	−26.9843	−1.21531
$494$	0	0
$495$	−2.59662	−0.116709
$496$	0	0
$497$	9.20202	0.412767
$498$	0	0
$499$	− 1.16848i	− 0.0523083i	−0.999658	−	0.0261541i	$-0.991674\pi$
$499$	− 1.16848i	− 0.0523083i	0.999658	−	0.0261541i	$-0.00832607\pi$
$500$	0	0
$501$	− 32.8887i	− 1.46936i
$502$	0	0
$503$	−10.6175	−0.473412	−0.236706	−	0.971581i	$-0.576068\pi$
$503$	−10.6175	−0.473412	−0.236706	+	0.971581i	$0.576068\pi$
$504$	0	0
$505$	24.3102i	1.08179i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.3007i	1.07711i	0.842591	+	0.538554i	$0.181029\pi$
$509$	24.3007i	1.07711i	−0.842591	+	0.538554i	$0.818971\pi$
$510$	0	0
$511$	−7.36798	−0.325940
$512$	0	0
$513$	− 27.5725i	− 1.21736i
$514$	0	0
$515$	− 19.4127i	− 0.855424i
$516$	0	0
$517$	24.5809	1.08107
$518$	0	0
$519$	−0.730063	−0.0320462
$520$	0	0
$521$	11.3280	0.496289	0.248144	−	0.968723i	$-0.420179\pi$
$521$	11.3280	0.496289	0.248144	+	0.968723i	$0.420179\pi$
$522$	0	0
$523$	30.2656	1.32342	0.661711	−	0.749759i	$-0.269830\pi$
$523$	30.2656	1.32342	0.661711	+	0.749759i	$0.269830\pi$
$524$	0	0
$525$	17.1524i	0.748591i
$526$	0	0
$527$	− 15.4708i	− 0.673919i
$528$	0	0
$529$	−22.6227	−0.983597
$530$	0	0
$531$	− 1.20981i	− 0.0525012i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 20.9998i	− 0.907902i
$536$	0	0
$537$	−3.82840	−0.165208
$538$	0	0
$539$	− 4.32707i	− 0.186380i
$540$	0	0
$541$	11.1529i	0.479500i	0.970835	+	0.239750i	$0.0770654\pi$
$541$	11.1529i	0.479500i	−0.970835	+	0.239750i	$0.922935\pi$
$542$	0	0
$543$	11.5509	0.495698
$544$	0	0
$545$	26.0569	1.11616
$546$	0	0
$547$	4.87804	0.208570	0.104285	−	0.994547i	$-0.466745\pi$
$547$	4.87804	0.208570	0.104285	+	0.994547i	$0.466745\pi$
$548$	0	0
$549$	−2.24892	−0.0959814
$550$	0	0
$551$	50.5281i	2.15257i
$552$	0	0
$553$	− 0.331686i	− 0.0141047i
$554$	0	0
$555$	−7.64955	−0.324705
$556$	0	0
$557$	− 3.02585i	− 0.128209i	−0.997943	−	0.0641047i	$-0.979581\pi$
$557$	− 3.02585i	− 0.128209i	0.997943	−	0.0641047i	$-0.0204192\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 22.4096i	− 0.946136i
$562$	0	0
$563$	27.3183	1.15133	0.575664	−	0.817686i	$-0.304744\pi$
$563$	27.3183	1.15133	0.575664	+	0.817686i	$0.304744\pi$
$564$	0	0
$565$	− 25.1414i	− 1.05770i
$566$	0	0
$567$	9.44571i	0.396682i
$568$	0	0
$569$	12.9374	0.542365	0.271183	−	0.962528i	$-0.412585\pi$
$569$	12.9374	0.542365	0.271183	+	0.962528i	$0.412585\pi$
$570$	0	0
$571$	10.6779	0.446855	0.223427	−	0.974721i	$-0.428275\pi$
$571$	10.6779	0.446855	0.223427	+	0.974721i	$0.428275\pi$
$572$	0	0
$573$	−33.3581	−1.39355
$574$	0	0
$575$	−5.92966	−0.247284
$576$	0	0
$577$	4.76467i	0.198356i	0.995070	+	0.0991778i	$0.0316213\pi$
$577$	4.76467i	0.198356i	−0.995070	+	0.0991778i	$0.968379\pi$
$578$	0	0
$579$	45.5149i	1.89153i
$580$	0	0
$581$	16.6777	0.691907
$582$	0	0
$583$	− 17.1358i	− 0.709690i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 31.9269i	− 1.31776i	−0.752247	−	0.658881i	$-0.771030\pi$
$587$	− 31.9269i	− 1.31776i	0.752247	−	0.658881i	$-0.228970\pi$
$588$	0	0
$589$	−28.9691	−1.19365
$590$	0	0
$591$	1.04748i	0.0430876i
$592$	0	0
$593$	1.27362i	0.0523015i	0.999658	+	0.0261508i	$0.00832499\pi$
$593$	1.27362i	0.0523015i	−0.999658	+	0.0261508i	$0.991675\pi$
$594$	0	0
$595$	−11.1583	−0.457445
$596$	0	0
$597$	−28.5880	−1.17003
$598$	0	0
$599$	−15.1462	−0.618857	−0.309428	−	0.950923i	$-0.600138\pi$
$599$	−15.1462	−0.618857	−0.309428	+	0.950923i	$0.600138\pi$
$600$	0	0
$601$	34.2850	1.39852	0.699258	−	0.714869i	$-0.253514\pi$
$601$	34.2850	1.39852	0.699258	+	0.714869i	$0.253514\pi$
$602$	0	0
$603$	− 0.0848899i	− 0.00345698i
$604$	0	0
$605$	− 29.5659i	− 1.20202i
$606$	0	0
$607$	9.47474	0.384568	0.192284	−	0.981339i	$-0.438411\pi$
$607$	9.47474	0.384568	0.192284	+	0.981339i	$0.438411\pi$
$608$	0	0
$609$	− 16.4479i	− 0.666504i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 11.6437i	− 0.470283i	−0.971961	−	0.235141i	$-0.924445\pi$
$613$	− 11.6437i	− 0.470283i	0.971961	−	0.235141i	$-0.0755553\pi$
$614$	0	0
$615$	81.7792	3.29766
$616$	0	0
$617$	− 6.40045i	− 0.257672i	−0.991666	−	0.128836i	$-0.958876\pi$
$617$	− 6.40045i	− 0.257672i	0.991666	−	0.128836i	$-0.0411242\pi$
$618$	0	0
$619$	11.2065i	0.450425i	0.974310	+	0.225213i	$0.0723077\pi$
$619$	11.2065i	0.450425i	−0.974310	+	0.225213i	$0.927692\pi$
$620$	0	0
$621$	−3.10285	−0.124513
$622$	0	0
$623$	−2.95414	−0.118355
$624$	0	0
$625$	19.9284	0.797138
$626$	0	0
$627$	−41.9620	−1.67580
$628$	0	0
$629$	− 3.27837i	− 0.130717i
$630$	0	0
$631$	45.1659i	1.79803i	0.437922	+	0.899013i	$0.355715\pi$
$631$	45.1659i	1.79803i	−0.437922	+	0.899013i	$0.644285\pi$
$632$	0	0
$633$	−20.2279	−0.803988
$634$	0	0
$635$	77.6520i	3.08152i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.44252i	0.0570651i
$640$	0	0
$641$	7.15994	0.282800	0.141400	−	0.989953i	$-0.454840\pi$
$641$	7.15994	0.282800	0.141400	+	0.989953i	$0.454840\pi$
$642$	0	0
$643$	− 41.4555i	− 1.63485i	−0.576038	−	0.817423i	$-0.695402\pi$
$643$	− 41.4555i	− 1.63485i	0.576038	−	0.817423i	$-0.304598\pi$
$644$	0	0
$645$	− 8.72690i	− 0.343621i
$646$	0	0
$647$	−15.5852	−0.612716	−0.306358	−	0.951916i	$-0.599110\pi$
$647$	−15.5852	−0.612716	−0.306358	+	0.951916i	$0.599110\pi$
$648$	0	0
$649$	33.3943	1.31084
$650$	0	0
$651$	9.43004	0.369592
$652$	0	0
$653$	−23.8242	−0.932312	−0.466156	−	0.884702i	$-0.654362\pi$
$653$	−23.8242	−0.932312	−0.466156	+	0.884702i	$0.654362\pi$
$654$	0	0
$655$	18.8509i	0.736565i
$656$	0	0
$657$	− 1.15501i	− 0.0450613i
$658$	0	0
$659$	4.03723	0.157268	0.0786340	−	0.996904i	$-0.474944\pi$
$659$	4.03723	0.157268	0.0786340	+	0.996904i	$0.474944\pi$
$660$	0	0
$661$	− 37.1979i	− 1.44683i	−0.690412	−	0.723416i	$-0.742571\pi$
$661$	− 37.1979i	− 1.44683i	0.690412	−	0.723416i	$-0.257429\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20.8939i	0.810229i
$666$	0	0
$667$	5.68614	0.220168
$668$	0	0
$669$	4.16845i	0.161162i
$670$	0	0
$671$	− 62.0768i	− 2.39645i
$672$	0	0
$673$	10.0573	0.387679	0.193839	−	0.981033i	$-0.437906\pi$
$673$	10.0573	0.387679	0.193839	+	0.981033i	$0.437906\pi$
$674$	0	0
$675$	48.7683	1.87709
$676$	0	0
$677$	−19.8807	−0.764076	−0.382038	−	0.924147i	$-0.624778\pi$
$677$	−19.8807	−0.764076	−0.382038	+	0.924147i	$0.624778\pi$
$678$	0	0
$679$	9.65583	0.370557
$680$	0	0
$681$	− 14.6604i	− 0.561787i
$682$	0	0
$683$	− 19.8832i	− 0.760810i	−0.924820	−	0.380405i	$-0.875785\pi$
$683$	− 19.8832i	− 0.760810i	0.924820	−	0.380405i	$-0.124215\pi$
$684$	0	0
$685$	−49.5034	−1.89143
$686$	0	0
$687$	30.6929i	1.17101i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	19.1644i	0.729047i	0.931194	+	0.364524i	$0.118768\pi$
$691$	19.1644i	0.729047i	−0.931194	+	0.364524i	$0.881232\pi$
$692$	0	0
$693$	0.678314	0.0257670
$694$	0	0
$695$	60.5187i	2.29561i
$696$	0	0
$697$	35.0482i	1.32755i
$698$	0	0
$699$	−7.35170	−0.278067
$700$	0	0
$701$	18.5012	0.698780	0.349390	−	0.936977i	$-0.386389\pi$
$701$	18.5012	0.698780	0.349390	+	0.936977i	$0.386389\pi$
$702$	0	0
$703$	−6.13875	−0.231527
$704$	0	0
$705$	38.6369	1.45515
$706$	0	0
$707$	− 6.35056i	− 0.238838i
$708$	0	0
$709$	− 24.0214i	− 0.902144i	−0.892488	−	0.451072i	$-0.851042\pi$
$709$	− 24.0214i	− 0.902144i	0.892488	−	0.451072i	$-0.148958\pi$
$710$	0	0
$711$	0.0519954	0.00194998
$712$	0	0
$713$	3.26001i	0.122088i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10.3739i	− 0.387422i
$718$	0	0
$719$	−24.4297	−0.911076	−0.455538	−	0.890216i	$-0.650553\pi$
$719$	−24.4297	−0.911076	−0.455538	+	0.890216i	$0.650553\pi$
$720$	0	0
$721$	5.07117i	0.188860i
$722$	0	0
$723$	50.5421i	1.87968i
$724$	0	0
$725$	−89.3705	−3.31914
$726$	0	0
$727$	20.9000	0.775138	0.387569	−	0.921841i	$-0.373315\pi$
$727$	20.9000	0.775138	0.387569	+	0.921841i	$0.373315\pi$
$728$	0	0
$729$	25.4456	0.942428
$730$	0	0
$731$	3.74010	0.138332
$732$	0	0
$733$	− 7.08118i	− 0.261549i	−0.991412	−	0.130775i	$-0.958254\pi$
$733$	− 7.08118i	− 0.261549i	0.991412	−	0.130775i	$-0.0417465\pi$
$734$	0	0
$735$	− 6.80139i	− 0.250873i
$736$	0	0
$737$	2.34321	0.0863134
$738$	0	0
$739$	23.7150i	0.872371i	0.899857	+	0.436186i	$0.143671\pi$
$739$	23.7150i	0.872371i	−0.899857	+	0.436186i	$0.856329\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.2404i	1.54965i	0.632175	+	0.774825i	$0.282162\pi$
$743$	42.2404i	1.54965i	−0.632175	+	0.774825i	$0.717838\pi$
$744$	0	0
$745$	−60.9847	−2.23431
$746$	0	0
$747$	2.61441i	0.0956561i
$748$	0	0
$749$	5.48579i	0.200446i
$750$	0	0
$751$	−24.1421	−0.880956	−0.440478	−	0.897763i	$-0.645191\pi$
$751$	−24.1421	−0.880956	−0.440478	+	0.897763i	$0.645191\pi$
$752$	0	0
$753$	22.7066	0.827475
$754$	0	0
$755$	−24.3983	−0.887947
$756$	0	0
$757$	−31.9432	−1.16099	−0.580497	−	0.814262i	$-0.697142\pi$
$757$	−31.9432	−1.16099	−0.580497	+	0.814262i	$0.697142\pi$
$758$	0	0
$759$	4.72216i	0.171404i
$760$	0	0
$761$	− 42.2716i	− 1.53234i	−0.642635	−	0.766172i	$-0.722159\pi$
$761$	− 42.2716i	− 1.53234i	0.642635	−	0.766172i	$-0.277841\pi$
$762$	0	0
$763$	−6.80685	−0.246425
$764$	0	0
$765$	− 1.74918i	− 0.0632418i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.73396i	0.0985892i	0.998784	+	0.0492946i	$0.0156973\pi$
$769$	2.73396i	0.0985892i	−0.998784	+	0.0492946i	$0.984303\pi$
$770$	0	0
$771$	−1.19288	−0.0429604
$772$	0	0
$773$	− 19.5541i	− 0.703310i	−0.936130	−	0.351655i	$-0.885619\pi$
$773$	− 19.5541i	− 0.703310i	0.936130	−	0.351655i	$-0.114381\pi$
$774$	0	0
$775$	− 51.2384i	− 1.84054i
$776$	0	0
$777$	1.99829	0.0716883
$778$	0	0
$779$	65.6277	2.35136
$780$	0	0
$781$	−39.8177	−1.42479
$782$	0	0
$783$	−46.7654	−1.67126
$784$	0	0
$785$	− 28.2816i	− 1.00941i
$786$	0	0
$787$	32.4390i	1.15633i	0.815922	+	0.578163i	$0.196230\pi$
$787$	32.4390i	1.15633i	−0.815922	+	0.578163i	$0.803770\pi$
$788$	0	0
$789$	−33.7050	−1.19993
$790$	0	0
$791$	6.56768i	0.233520i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 26.9344i	− 0.955265i
$796$	0	0
$797$	−25.8795	−0.916697	−0.458349	−	0.888772i	$-0.651559\pi$
$797$	−25.8795	−0.916697	−0.458349	+	0.888772i	$0.651559\pi$
$798$	0	0
$799$	16.5587i	0.585803i
$800$	0	0
$801$	− 0.463093i	− 0.0163626i
$802$	0	0
$803$	31.8817	1.12508
$804$	0	0
$805$	2.35128	0.0828716
$806$	0	0
$807$	−17.0296	−0.599471
$808$	0	0
$809$	22.4133	0.788008	0.394004	−	0.919109i	$-0.371090\pi$
$809$	22.4133	0.788008	0.394004	+	0.919109i	$0.371090\pi$
$810$	0	0
$811$	− 5.44486i	− 0.191195i	−0.995420	−	0.0955974i	$-0.969524\pi$
$811$	− 5.44486i	− 0.191195i	0.995420	−	0.0955974i	$-0.0304761\pi$
$812$	0	0
$813$	− 18.6049i	− 0.652504i
$814$	0	0
$815$	0.335198	0.0117415
$816$	0	0
$817$	− 7.00332i	− 0.245015i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 30.8123i	− 1.07536i	−0.843150	−	0.537679i	$-0.819301\pi$
$821$	− 30.8123i	− 1.07536i	0.843150	−	0.537679i	$-0.180699\pi$
$822$	0	0
$823$	−0.205700	−0.00717025	−0.00358513	−	0.999994i	$-0.501141\pi$
$823$	−0.205700	−0.00717025	−0.00358513	+	0.999994i	$0.501141\pi$
$824$	0	0
$825$	− 74.2194i	− 2.58399i
$826$	0	0
$827$	9.66411i	0.336054i	0.985782	+	0.168027i	$0.0537396\pi$
$827$	9.66411i	0.336054i	−0.985782	+	0.168027i	$0.946260\pi$
$828$	0	0
$829$	−30.5334	−1.06047	−0.530234	−	0.847851i	$-0.677896\pi$
$829$	−30.5334	−1.06047	−0.530234	+	0.847851i	$0.677896\pi$
$830$	0	0
$831$	34.3568	1.19182
$832$	0	0
$833$	2.91488	0.100995
$834$	0	0
$835$	70.8603	2.45222
$836$	0	0
$837$	− 26.8119i	− 0.926753i
$838$	0	0
$839$	− 38.1885i	− 1.31841i	−0.751962	−	0.659206i	$-0.770892\pi$
$839$	− 38.1885i	− 1.31841i	0.751962	−	0.659206i	$-0.229108\pi$
$840$	0	0
$841$	56.7001	1.95518
$842$	0	0
$843$	− 33.2317i	− 1.14456i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.72349i	0.265382i
$848$	0	0
$849$	−2.08928	−0.0717038
$850$	0	0
$851$	0.690820i	0.0236810i
$852$	0	0
$853$	28.3480i	0.970618i	0.874343	+	0.485309i	$0.161293\pi$
$853$	28.3480i	0.970618i	−0.874343	+	0.485309i	$0.838707\pi$
$854$	0	0
$855$	−3.27534	−0.112014
$856$	0	0
$857$	25.8488	0.882978	0.441489	−	0.897267i	$-0.354450\pi$
$857$	25.8488	0.882978	0.441489	+	0.897267i	$0.354450\pi$
$858$	0	0
$859$	19.6084	0.669029	0.334514	−	0.942391i	$-0.391428\pi$
$859$	19.6084	0.669029	0.334514	+	0.942391i	$0.391428\pi$
$860$	0	0
$861$	−21.3632	−0.728056
$862$	0	0
$863$	− 13.4368i	− 0.457395i	−0.973498	−	0.228697i	$-0.926553\pi$
$863$	− 13.4368i	− 0.457395i	0.973498	−	0.228697i	$-0.0734466\pi$
$864$	0	0
$865$	− 1.57295i	− 0.0534821i
$866$	0	0
$867$	−15.1084	−0.513107
$868$	0	0
$869$	1.43523i	0.0486868i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.51366i	0.0512295i
$874$	0	0
$875$	−17.8154	−0.602269
$876$	0	0
$877$	30.1421i	1.01783i	0.860818	+	0.508914i	$0.169953\pi$
$877$	30.1421i	1.01783i	−0.860818	+	0.508914i	$0.830047\pi$
$878$	0	0
$879$	− 11.5611i	− 0.389947i
$880$	0	0
$881$	−9.04954	−0.304887	−0.152443	−	0.988312i	$-0.548714\pi$
$881$	−9.04954	−0.304887	−0.152443	+	0.988312i	$0.548714\pi$
$882$	0	0
$883$	47.3475	1.59337	0.796685	−	0.604395i	$-0.206585\pi$
$883$	47.3475	1.59337	0.796685	+	0.604395i	$0.206585\pi$
$884$	0	0
$885$	52.4900	1.76443
$886$	0	0
$887$	37.9790	1.27521	0.637605	−	0.770363i	$-0.279925\pi$
$887$	37.9790	1.27521	0.637605	+	0.770363i	$0.279925\pi$
$888$	0	0
$889$	− 20.2850i	− 0.680338i
$890$	0	0
$891$	− 40.8722i	− 1.36927i
$892$	0	0
$893$	31.0061	1.03758
$894$	0	0
$895$	− 8.24848i	− 0.275716i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	49.1341i	1.63872i
$900$	0	0
$901$	11.5433	0.384563
$902$	0	0
$903$	2.27973i	0.0758646i
$904$	0	0
$905$	24.8870i	0.827272i
$906$	0	0
$907$	−27.8867	−0.925962	−0.462981	−	0.886368i	$-0.653220\pi$
$907$	−27.8867	−0.925962	−0.462981	+	0.886368i	$0.653220\pi$
$908$	0	0
$909$	0.995519	0.0330193
$910$	0	0
$911$	−26.1435	−0.866172	−0.433086	−	0.901353i	$-0.642575\pi$
$911$	−26.1435	−0.866172	−0.433086	+	0.901353i	$0.642575\pi$
$912$	0	0
$913$	−72.1654	−2.38833
$914$	0	0
$915$	− 97.5738i	− 3.22569i
$916$	0	0
$917$	− 4.92442i	− 0.162619i
$918$	0	0
$919$	−3.08812	−0.101868	−0.0509338	−	0.998702i	$-0.516220\pi$
$919$	−3.08812	−0.101868	−0.0509338	+	0.998702i	$0.516220\pi$
$920$	0	0
$921$	− 48.0735i	− 1.58408i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 10.8578i	− 0.357002i
$926$	0	0
$927$	−0.794961	−0.0261100
$928$	0	0
$929$	38.5771i	1.26567i	0.774286	+	0.632836i	$0.218109\pi$
$929$	38.5771i	1.26567i	−0.774286	+	0.632836i	$0.781891\pi$
$930$	0	0
$931$	− 5.45810i	− 0.178882i
$932$	0	0
$933$	54.9118	1.79773
$934$	0	0
$935$	48.2826	1.57901
$936$	0	0
$937$	8.28836	0.270769	0.135384	−	0.990793i	$-0.456773\pi$
$937$	8.28836	0.270769	0.135384	+	0.990793i	$0.456773\pi$
$938$	0	0
$939$	53.4468	1.74417
$940$	0	0
$941$	7.72406i	0.251797i	0.992043	+	0.125899i	$0.0401814\pi$
$941$	7.72406i	0.251797i	−0.992043	+	0.125899i	$0.959819\pi$
$942$	0	0
$943$	− 7.38537i	− 0.240501i
$944$	0	0
$945$	−19.3380	−0.629064
$946$	0	0
$947$	− 51.7803i	− 1.68263i	−0.540543	−	0.841316i	$-0.681781\pi$
$947$	− 51.7803i	− 1.68263i	0.540543	−	0.841316i	$-0.318219\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 6.72226i	− 0.217984i
$952$	0	0
$953$	−48.7824	−1.58022	−0.790109	−	0.612967i	$-0.789976\pi$
$953$	−48.7824	−1.58022	−0.790109	+	0.612967i	$0.789976\pi$
$954$	0	0
$955$	− 71.8716i	− 2.32571i
$956$	0	0
$957$	71.1713i	2.30064i
$958$	0	0
$959$	12.9318	0.417589
$960$	0	0
$961$	2.83011	0.0912938
$962$	0	0
$963$	−0.859956	−0.0277117
$964$	0	0
$965$	−98.0640	−3.15679
$966$	0	0
$967$	− 3.09186i	− 0.0994274i	−0.998764	−	0.0497137i	$-0.984169\pi$
$967$	− 3.09186i	− 0.0994274i	0.998764	−	0.0497137i	$-0.0158309\pi$
$968$	0	0
$969$	− 28.2672i	− 0.908074i
$970$	0	0
$971$	54.4761	1.74822	0.874111	−	0.485727i	$-0.161445\pi$
$971$	54.4761	1.74822	0.874111	+	0.485727i	$0.161445\pi$
$972$	0	0
$973$	− 15.8093i	− 0.506823i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 19.3385i	− 0.618693i	−0.950949	−	0.309347i	$-0.899890\pi$
$977$	− 19.3385i	− 0.618693i	0.950949	−	0.309347i	$-0.100110\pi$
$978$	0	0
$979$	12.7827	0.408538
$980$	0	0
$981$	− 1.06705i	− 0.0340682i
$982$	0	0
$983$	4.93770i	0.157488i	0.996895	+	0.0787440i	$0.0250910\pi$
$983$	4.93770i	0.157488i	−0.996895	+	0.0787440i	$0.974909\pi$
$984$	0	0
$985$	−2.25684	−0.0719091
$986$	0	0
$987$	−10.0931	−0.321268
$988$	0	0
$989$	−0.788114	−0.0250606
$990$	0	0
$991$	48.3434	1.53568	0.767839	−	0.640643i	$-0.221332\pi$
$991$	48.3434	1.53568	0.767839	+	0.640643i	$0.221332\pi$
$992$	0	0
$993$	41.3181i	1.31119i
$994$	0	0
$995$	− 61.5942i	− 1.95267i
$996$	0	0
$997$	−43.3269	−1.37218	−0.686088	−	0.727518i	$-0.740674\pi$
$997$	−43.3269	−1.37218	−0.686088	+	0.727518i	$0.740674\pi$
$998$	0	0
$999$	− 5.68162i	− 0.179759i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.g.k.337.14		16
13.3	even	3		364.2.u.a.225.2	✓	16
13.4	even	6		364.2.u.a.309.2	yes	16
13.5	odd	4		4732.2.a.t.1.7		8
13.8	odd	4		4732.2.a.s.1.7		8
13.12	even	2	inner	4732.2.g.k.337.13		16
39.17	odd	6		3276.2.cf.c.1765.7		16
39.29	odd	6		3276.2.cf.c.2773.2		16
52.3	odd	6		1456.2.cc.f.225.7		16
52.43	odd	6		1456.2.cc.f.673.7		16
91.3	odd	6		2548.2.bq.c.1941.2		16
91.4	even	6		2548.2.bb.d.569.2		16
91.16	even	3		2548.2.bb.d.1733.2		16
91.17	odd	6		2548.2.bb.c.569.7		16
91.30	even	6		2548.2.bq.e.361.7		16
91.55	odd	6		2548.2.u.c.589.7		16
91.68	odd	6		2548.2.bb.c.1733.7		16
91.69	odd	6		2548.2.u.c.1765.7		16
91.81	even	3		2548.2.bq.e.1941.7		16
91.82	odd	6		2548.2.bq.c.361.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.u.a.225.2	✓	16	13.3	even	3
364.2.u.a.309.2	yes	16	13.4	even	6
1456.2.cc.f.225.7		16	52.3	odd	6
1456.2.cc.f.673.7		16	52.43	odd	6
2548.2.u.c.589.7		16	91.55	odd	6
2548.2.u.c.1765.7		16	91.69	odd	6
2548.2.bb.c.569.7		16	91.17	odd	6
2548.2.bb.c.1733.7		16	91.68	odd	6
2548.2.bb.d.569.2		16	91.4	even	6
2548.2.bb.d.1733.2		16	91.16	even	3
2548.2.bq.c.361.2		16	91.82	odd	6
2548.2.bq.c.1941.2		16	91.3	odd	6
2548.2.bq.e.361.7		16	91.30	even	6
2548.2.bq.e.1941.7		16	91.81	even	3
3276.2.cf.c.1765.7		16	39.17	odd	6
3276.2.cf.c.2773.2		16	39.29	odd	6
4732.2.a.s.1.7		8	13.8	odd	4
4732.2.a.t.1.7		8	13.5	odd	4
4732.2.g.k.337.13		16	13.12	even	2	inner
4732.2.g.k.337.14		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.77673 q^{3} +3.82804i q^{5} -1.00000i q^{7} +0.156761 q^{9} +4.32707i q^{11} +6.80139i q^{15} -2.91488 q^{17} +5.45810i q^{19} -1.77673i q^{21} +0.614224 q^{23} -9.65391 q^{25} -5.05166 q^{27} +9.25744 q^{29} +5.30753i q^{31} +7.68802i q^{33} +3.82804 q^{35} +1.12470i q^{37} -12.0239i q^{41} -1.28310 q^{43} +0.600087i q^{45} -5.68074i q^{47} -1.00000 q^{49} -5.17895 q^{51} -3.96013 q^{53} -16.5642 q^{55} +9.69757i q^{57} -7.71754i q^{59} -14.3462 q^{61} -0.156761i q^{63} -0.541525i q^{67} +1.09131 q^{69} +9.20202i q^{71} -7.36798i q^{73} -17.1524 q^{75} +4.32707 q^{77} +0.331686 q^{79} -9.44571 q^{81} +16.6777i q^{83} -11.1583i q^{85} +16.4479 q^{87} -2.95414i q^{89} +9.43004i q^{93} -20.8939 q^{95} +9.65583i q^{97} +0.678314i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 28 q^{9} - 4 q^{17} - 44 q^{25} - 12 q^{27} + 44 q^{29} + 12 q^{35} - 12 q^{43} - 16 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{61} + 104 q^{69} + 20 q^{75} - 24 q^{77} + 8 q^{79} - 52 q^{87}+ \cdots + 60 q^{95}+O(q^{100}) \) 16 * q + 28 * q^9 - 4 * q^17 - 44 * q^25 - 12 * q^27 + 44 * q^29 + 12 * q^35 - 12 * q^43 - 16 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 - 8 * q^61 + 104 * q^69 + 20 * q^75 - 24 * q^77 + 8 * q^79 - 52 * q^87 + 60 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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