LMFDB - Embedded newform 792.2.r.e.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [792,2,Mod(289,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("792.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 8])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 792 = 2^{3} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	792.r (of order $5$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,8,0,6,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.32415184009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 264)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			577.1
Root			$-0.309017 + 0.951057i$ of defining polynomial
Character	$\chi$	$=$	792.577
Dual form			792.2.r.e.361.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.881966 - 2.71441i) q^{5} +(1.50000 - 1.08981i) q^{7} +(-1.23607 - 3.07768i) q^{11} +(1.00000 + 3.07768i) q^{13} +(1.23607 - 3.80423i) q^{17} +(0.618034 + 0.449028i) q^{19} -7.23607 q^{23} +(-2.54508 - 1.84911i) q^{25} +(0.309017 - 0.224514i) q^{29} +(-1.80902 - 5.56758i) q^{31} +(-1.63525 - 5.03280i) q^{35} +(3.23607 - 2.35114i) q^{37} +(5.85410 + 4.25325i) q^{41} -9.70820 q^{43} +(7.23607 + 5.25731i) q^{47} +(-1.10081 + 3.38795i) q^{49} +(-3.66312 - 11.2739i) q^{53} +(-9.44427 + 0.640786i) q^{55} +(4.35410 - 3.16344i) q^{59} +(4.38197 - 13.4863i) q^{61} +9.23607 q^{65} +8.00000 q^{67} +(-3.38197 + 10.4086i) q^{71} +(11.6353 - 8.45351i) q^{73} +(-5.20820 - 3.26944i) q^{77} +(-0.118034 - 0.363271i) q^{79} +(-0.500000 + 1.53884i) q^{83} +(-9.23607 - 6.71040i) q^{85} -6.76393 q^{89} +(4.85410 + 3.52671i) q^{91} +(1.76393 - 1.28157i) q^{95} +(5.02786 + 15.4742i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 6 q^{7} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 2 q^{19} - 20 q^{23} + q^{25} - q^{29} - 5 q^{31} + 27 q^{35} + 4 q^{37} + 10 q^{41} - 12 q^{43} + 20 q^{47} - 29 q^{49} + q^{53} - 2 q^{55}+ \cdots + 38 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 6 * q^7 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 2 * q^19 - 20 * q^23 + q^25 - q^29 - 5 * q^31 + 27 * q^35 + 4 * q^37 + 10 * q^41 - 12 * q^43 + 20 * q^47 - 29 * q^49 + q^53 - 2 * q^55 + 4 * q^59 + 22 * q^61 + 28 * q^65 + 32 * q^67 - 18 * q^71 + 13 * q^73 + 6 * q^77 + 4 * q^79 - 2 * q^83 - 28 * q^85 - 36 * q^89 + 6 * q^91 + 16 * q^95 + 38 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/792\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$	$397$
$\chi(n)$	$e\left(\frac{2}{5}\right)$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.881966	−	2.71441i	0.394427	−	1.21392i	−0.534980	−	0.844865i	$-0.679681\pi$
$5$	0.881966	−	2.71441i	0.394427	−	1.21392i	0.929407	−	0.369057i	$-0.120319\pi$
$6$		0			0
$7$	1.50000	−	1.08981i	0.566947	−	0.411911i	−0.267048	−	0.963683i	$-0.586048\pi$
$7$	1.50000	−	1.08981i	0.566947	−	0.411911i	0.833995	+	0.551772i	$0.186048\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.23607	−	3.07768i	−0.372689	−	0.927957i
$11$	−1.23607	−	3.07768i	−0.372689	−	0.927957i
$12$		0			0
$13$	1.00000	+	3.07768i	0.277350	+	0.853596i	0.988588	+	0.150644i	$0.0481349\pi$
$13$	1.00000	+	3.07768i	0.277350	+	0.853596i	−0.711238	+	0.702951i	$0.751865\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.23607	−	3.80423i	0.299791	−	0.922660i	−0.681780	−	0.731558i	$-0.738794\pi$
$17$	1.23607	−	3.80423i	0.299791	−	0.922660i	0.981570	−	0.191103i	$-0.0612063\pi$
$18$		0			0
$19$	0.618034	+	0.449028i	0.141787	+	0.103014i	0.656418	−	0.754398i	$-0.272071\pi$
$19$	0.618034	+	0.449028i	0.141787	+	0.103014i	−0.514631	+	0.857412i	$0.672071\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.23607			−1.50882			−0.754412	−	0.656401i	$-0.772078\pi$
$23$	−7.23607			−1.50882			−0.754412	+	0.656401i	$0.772078\pi$
$24$		0			0
$25$	−2.54508	−	1.84911i	−0.509017	−	0.369822i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.309017	−	0.224514i	0.0573830	−	0.0416912i	−0.558724	−	0.829354i	$-0.688709\pi$
$29$	0.309017	−	0.224514i	0.0573830	−	0.0416912i	0.616107	+	0.787662i	$0.288709\pi$
$30$		0			0
$31$	−1.80902	−	5.56758i	−0.324909	−	0.999967i	−0.971482	−	0.237115i	$-0.923798\pi$
$31$	−1.80902	−	5.56758i	−0.324909	−	0.999967i	0.646573	−	0.762852i	$-0.276202\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.63525	−	5.03280i	−0.276409	−	0.850698i
$36$		0			0
$37$	3.23607	−	2.35114i	0.532006	−	0.386525i	−0.289101	−	0.957298i	$-0.593356\pi$
$37$	3.23607	−	2.35114i	0.532006	−	0.386525i	0.821108	+	0.570773i	$0.193356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.85410	+	4.25325i	0.914257	+	0.664247i	0.942088	−	0.335366i	$-0.108860\pi$
$41$	5.85410	+	4.25325i	0.914257	+	0.664247i	−0.0278308	+	0.999613i	$0.508860\pi$
$42$		0			0
$43$	−9.70820			−1.48049			−0.740244	−	0.672339i	$-0.765290\pi$
$43$	−9.70820			−1.48049			−0.740244	+	0.672339i	$0.765290\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.23607	+	5.25731i	1.05549	+	0.766858i	0.973249	−	0.229755i	$-0.0737924\pi$
$47$	7.23607	+	5.25731i	1.05549	+	0.766858i	0.0822405	+	0.996613i	$0.473792\pi$
$48$		0			0
$49$	−1.10081	+	3.38795i	−0.157259	+	0.483993i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.66312	−	11.2739i	−0.503168	−	1.54859i	−0.803829	−	0.594861i	$-0.797207\pi$
$53$	−3.66312	−	11.2739i	−0.503168	−	1.54859i	0.300661	−	0.953731i	$-0.402793\pi$
$54$		0			0
$55$	−9.44427	+	0.640786i	−1.27347	+	0.0864035i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.35410	−	3.16344i	0.566856	−	0.411845i	−0.267106	−	0.963667i	$-0.586067\pi$
$59$	4.35410	−	3.16344i	0.566856	−	0.411845i	0.833962	+	0.551822i	$0.186067\pi$
$60$		0			0
$61$	4.38197	−	13.4863i	0.561053	−	1.72674i	−0.118343	−	0.992973i	$-0.537758\pi$
$61$	4.38197	−	13.4863i	0.561053	−	1.72674i	0.679396	−	0.733772i	$-0.262242\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.23607			1.14559
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.38197	+	10.4086i	−0.401366	+	1.23528i	0.522527	+	0.852623i	$0.324990\pi$
$71$	−3.38197	+	10.4086i	−0.401366	+	1.23528i	−0.923892	+	0.382653i	$0.875010\pi$
$72$		0			0
$73$	11.6353	−	8.45351i	1.36180	−	0.989408i	0.363476	−	0.931604i	$-0.381590\pi$
$73$	11.6353	−	8.45351i	1.36180	−	0.989408i	0.998328	−	0.0578045i	$-0.0184100\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.20820	−	3.26944i	−0.593530	−	0.372587i
$78$		0			0
$79$	−0.118034	−	0.363271i	−0.0132799	−	0.0408712i	0.944197	−	0.329381i	$-0.106840\pi$
$79$	−0.118034	−	0.363271i	−0.0132799	−	0.0408712i	−0.957477	+	0.288510i	$0.906840\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.500000	+	1.53884i	−0.0548821	+	0.168910i	−0.974740	−	0.223341i	$-0.928304\pi$
$83$	−0.500000	+	1.53884i	−0.0548821	+	0.168910i	0.919858	+	0.392251i	$0.128304\pi$
$84$		0			0
$85$	−9.23607	−	6.71040i	−1.00179	−	0.727845i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.76393			−0.716975			−0.358488	−	0.933534i	$-0.616707\pi$
$89$	−6.76393			−0.716975			−0.358488	+	0.933534i	$0.616707\pi$
$90$		0			0
$91$	4.85410	+	3.52671i	0.508848	+	0.369700i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.76393	−	1.28157i	0.180976	−	0.131486i
$96$		0			0
$97$	5.02786	+	15.4742i	0.510502	+	1.57116i	0.791319	+	0.611403i	$0.209395\pi$
$97$	5.02786	+	15.4742i	0.510502	+	1.57116i	−0.280817	+	0.959761i	$0.590605\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.59017	+	11.0494i	0.357235	+	1.09946i	0.954702	+	0.297563i	$0.0961740\pi$
$101$	3.59017	+	11.0494i	0.357235	+	1.09946i	−0.597467	+	0.801894i	$0.703826\pi$
$102$		0			0
$103$	−6.97214	+	5.06555i	−0.686985	+	0.499124i	−0.875668	−	0.482914i	$-0.839578\pi$
$103$	−6.97214	+	5.06555i	−0.686985	+	0.499124i	0.188683	+	0.982038i	$0.439578\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.11803	−	2.26538i	−0.301432	−	0.219003i	0.426780	−	0.904356i	$-0.359648\pi$
$107$	−3.11803	−	2.26538i	−0.301432	−	0.219003i	−0.728211	+	0.685353i	$0.759648\pi$
$108$		0			0
$109$	14.9443			1.43140			0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443			1.43140			0.715701	+	0.698407i	$0.246107\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.854102	−	0.620541i	−0.0803472	−	0.0583756i	0.546887	−	0.837207i	$-0.315813\pi$
$113$	−0.854102	−	0.620541i	−0.0803472	−	0.0583756i	−0.627234	+	0.778831i	$0.715813\pi$
$114$		0			0
$115$	−6.38197	+	19.6417i	−0.595121	+	1.83160i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.29180	−	7.05342i	−0.210089	−	0.646586i
$120$		0			0
$121$	−7.94427	+	7.60845i	−0.722207	+	0.691677i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.28115	−	3.11044i	0.382918	−	0.278206i
$126$		0			0
$127$	3.70820	−	11.4127i	0.329050	−	1.01271i	−0.640529	−	0.767934i	$-0.721285\pi$
$127$	3.70820	−	11.4127i	0.329050	−	1.01271i	0.969579	−	0.244778i	$-0.0787150\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.09017			0.0952486			0.0476243	−	0.998865i	$-0.484835\pi$
$131$	1.09017			0.0952486			0.0476243	+	0.998865i	$0.484835\pi$
$132$		0			0
$133$	1.41641			0.122818
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.09017	+	18.7436i	−0.520318	+	1.60138i	0.253074	+	0.967447i	$0.418559\pi$
$137$	−6.09017	+	18.7436i	−0.520318	+	1.60138i	−0.773392	+	0.633928i	$0.781441\pi$
$138$		0			0
$139$	−8.23607	+	5.98385i	−0.698574	+	0.507544i	−0.879468	−	0.475959i	$-0.842101\pi$
$139$	−8.23607	+	5.98385i	−0.698574	+	0.507544i	0.180893	+	0.983503i	$0.442101\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.23607	−	6.88191i	0.688735	−	0.575494i
$144$		0			0
$145$	−0.336881	−	1.03681i	−0.0279764	−	0.0861027i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.57295	+	10.9964i	−0.292707	+	0.900861i	0.691275	+	0.722592i	$0.257050\pi$
$149$	−3.57295	+	10.9964i	−0.292707	+	0.900861i	−0.983982	+	0.178268i	$0.942950\pi$
$150$		0			0
$151$	−0.309017	−	0.224514i	−0.0251474	−	0.0182707i	0.575141	−	0.818055i	$-0.304947\pi$
$151$	−0.309017	−	0.224514i	−0.0251474	−	0.0182707i	−0.600288	+	0.799784i	$0.704947\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.7082			−1.34204
$156$		0			0
$157$	1.61803	+	1.17557i	0.129133	+	0.0938207i	0.650477	−	0.759526i	$-0.274569\pi$
$157$	1.61803	+	1.17557i	0.129133	+	0.0938207i	−0.521344	+	0.853347i	$0.674569\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.8541	+	7.88597i	−0.855423	+	0.621501i
$162$		0			0
$163$	3.56231	+	10.9637i	0.279021	+	0.858739i	0.988127	+	0.153637i	$0.0490988\pi$
$163$	3.56231	+	10.9637i	0.279021	+	0.858739i	−0.709106	+	0.705102i	$0.750901\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.527864	−	1.62460i	−0.0408473	−	0.125715i	0.928553	−	0.371199i	$-0.121053\pi$
$167$	−0.527864	−	1.62460i	−0.0408473	−	0.125715i	−0.969401	+	0.245484i	$0.921053\pi$
$168$		0			0
$169$	2.04508	−	1.48584i	0.157314	−	0.114295i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.881966	+	0.640786i	0.0670546	+	0.0487180i	0.620808	−	0.783963i	$-0.286805\pi$
$173$	0.881966	+	0.640786i	0.0670546	+	0.0487180i	−0.553753	+	0.832681i	$0.686805\pi$
$174$		0			0
$175$	−5.83282			−0.440919
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.39919	+	6.10237i	0.627785	+	0.456112i	0.855632	−	0.517585i	$-0.173169\pi$
$179$	8.39919	+	6.10237i	0.627785	+	0.456112i	−0.227847	+	0.973697i	$0.573169\pi$
$180$		0			0
$181$	−1.38197	+	4.25325i	−0.102721	+	0.316142i	−0.989189	−	0.146648i	$-0.953152\pi$
$181$	−1.38197	+	4.25325i	−0.102721	+	0.316142i	0.886468	+	0.462790i	$0.153152\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.52786	−	10.8576i	−0.259374	−	0.798270i
$186$		0			0
$187$	−13.2361	+	0.898056i	−0.967917	+	0.0656724i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.4721	+	7.60845i	−0.757737	+	0.550528i	−0.898215	−	0.439555i	$-0.855136\pi$
$191$	−10.4721	+	7.60845i	−0.757737	+	0.550528i	0.140478	+	0.990084i	$0.455136\pi$
$192$		0			0
$193$	−7.89919	+	24.3112i	−0.568596	+	1.74996i	0.0884218	+	0.996083i	$0.471818\pi$
$193$	−7.89919	+	24.3112i	−0.568596	+	1.74996i	−0.657018	+	0.753875i	$0.728182\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	19.0344			1.35615			0.678074	−	0.734994i	$-0.262815\pi$
$197$	19.0344			1.35615			0.678074	+	0.734994i	$0.262815\pi$
$198$		0			0
$199$	19.0344			1.34932			0.674658	−	0.738131i	$-0.264291\pi$
$199$	19.0344			1.34932			0.674658	+	0.738131i	$0.264291\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.218847	−	0.673542i	0.0153601	−	0.0472734i
$204$		0			0
$205$	16.7082	−	12.1392i	1.16695	−	0.847840i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.618034	−	2.45714i	0.0427503	−	0.169964i
$210$		0			0
$211$	−4.38197	−	13.4863i	−0.301667	−	0.928436i	−0.980900	−	0.194513i	$-0.937687\pi$
$211$	−4.38197	−	13.4863i	−0.301667	−	0.928436i	0.679233	−	0.733923i	$-0.262313\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.56231	+	26.3521i	−0.583944	+	1.79720i
$216$		0			0
$217$	−8.78115	−	6.37988i	−0.596104	−	0.433095i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.9443			0.870726
$222$		0			0
$223$	11.2533	+	8.17599i	0.753576	+	0.547505i	0.896933	−	0.442166i	$-0.145790\pi$
$223$	11.2533	+	8.17599i	0.753576	+	0.547505i	−0.143357	+	0.989671i	$0.545790\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.0172	−	12.3637i	1.12947	−	0.820610i	0.143855	−	0.989599i	$-0.454050\pi$
$227$	17.0172	−	12.3637i	1.12947	−	0.820610i	0.985618	+	0.168989i	$0.0540502\pi$
$228$		0			0
$229$	−3.56231	−	10.9637i	−0.235404	−	0.724498i	−0.997068	−	0.0765260i	$-0.975617\pi$
$229$	−3.56231	−	10.9637i	−0.235404	−	0.724498i	0.761664	−	0.647972i	$-0.224383\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.29180	+	13.2088i	0.281165	+	0.865337i	0.987522	+	0.157481i	$0.0503373\pi$
$233$	4.29180	+	13.2088i	0.281165	+	0.865337i	−0.706357	+	0.707856i	$0.749663\pi$
$234$		0			0
$235$	20.6525	−	15.0049i	1.34722	−	0.978812i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.32624	+	6.77591i	0.603264	+	0.438297i	0.847036	−	0.531536i	$-0.178385\pi$
$239$	9.32624	+	6.77591i	0.603264	+	0.438297i	−0.243772	+	0.969833i	$0.578385\pi$
$240$		0			0
$241$	−12.3262			−0.794003			−0.397001	−	0.917818i	$-0.629949\pi$
$241$	−12.3262			−0.794003			−0.397001	+	0.917818i	$0.629949\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.22542	+	5.97612i	0.525503	+	0.381800i
$246$		0			0
$247$	−0.763932	+	2.35114i	−0.0486078	+	0.149600i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.20820	+	25.2623i	0.518097	+	1.59454i	0.777575	+	0.628790i	$0.216450\pi$
$251$	8.20820	+	25.2623i	0.518097	+	1.59454i	−0.259478	+	0.965749i	$0.583550\pi$
$252$		0			0
$253$	8.94427	+	22.2703i	0.562322	+	1.40012i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−19.1803	+	13.9353i	−1.19644	+	0.869262i	−0.993930	−	0.110019i	$-0.964909\pi$
$257$	−19.1803	+	13.9353i	−1.19644	+	0.869262i	−0.202507	+	0.979281i	$0.564909\pi$
$258$		0			0
$259$	2.29180	−	7.05342i	0.142405	−	0.438278i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.7082			−0.721959			−0.360979	−	0.932574i	$-0.617558\pi$
$263$	−11.7082			−0.721959			−0.360979	+	0.932574i	$0.617558\pi$
$264$		0			0
$265$	−33.8328			−2.07833
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.09017	−	9.51057i	0.188411	−	0.579869i	−0.811579	−	0.584242i	$-0.801392\pi$
$269$	3.09017	−	9.51057i	0.188411	−	0.579869i	0.999990	+	0.00437267i	$0.00139187\pi$
$270$		0			0
$271$	18.9443	−	13.7638i	1.15078	−	0.836092i	0.162198	−	0.986758i	$-0.448142\pi$
$271$	18.9443	−	13.7638i	1.15078	−	0.836092i	0.988585	+	0.150666i	$0.0481417\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.54508	+	10.1186i	−0.153474	+	0.610174i
$276$		0			0
$277$	−4.67376	−	14.3844i	−0.280819	−	0.864272i	−0.987621	−	0.156861i	$-0.949863\pi$
$277$	−4.67376	−	14.3844i	−0.280819	−	0.864272i	0.706802	−	0.707412i	$-0.250137\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.38197	−	19.6417i	0.380716	−	1.17172i	−0.558824	−	0.829286i	$-0.688747\pi$
$281$	6.38197	−	19.6417i	0.380716	−	1.17172i	0.939540	−	0.342438i	$-0.111253\pi$
$282$		0			0
$283$	−21.3262	−	15.4944i	−1.26771	−	0.921048i	−0.268604	−	0.963251i	$-0.586562\pi$
$283$	−21.3262	−	15.4944i	−1.26771	−	0.921048i	−0.999109	+	0.0422030i	$0.986562\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.4164			0.791946
$288$		0			0
$289$	0.809017	+	0.587785i	0.0475892	+	0.0345756i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.118034	+	0.0857567i	−0.00689562	+	0.00500996i	−0.591228	−	0.806505i	$-0.701357\pi$
$293$	−0.118034	+	0.0857567i	−0.00689562	+	0.00500996i	0.584332	+	0.811515i	$0.301357\pi$
$294$		0			0
$295$	−4.74671	−	14.6089i	−0.276364	−	0.850562i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.23607	−	22.2703i	−0.418473	−	1.28793i
$300$		0			0
$301$	−14.5623	+	10.5801i	−0.839357	+	0.609829i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−32.7426	−	23.7889i	−1.87484	−	1.36215i
$306$		0			0
$307$	1.81966			0.103853			0.0519267	−	0.998651i	$-0.483464\pi$
$307$	1.81966			0.103853			0.0519267	+	0.998651i	$0.483464\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.85410	+	4.97980i	0.388660	+	0.282378i	0.764906	−	0.644141i	$-0.222785\pi$
$311$	6.85410	+	4.97980i	0.388660	+	0.282378i	−0.376246	+	0.926520i	$0.622785\pi$
$312$		0			0
$313$	−10.1525	+	31.2461i	−0.573852	+	1.76613i	0.0662041	+	0.997806i	$0.478911\pi$
$313$	−10.1525	+	31.2461i	−0.573852	+	1.76613i	−0.640056	+	0.768328i	$0.721089\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.03444	−	12.4167i	−0.226597	−	0.697393i	−0.998126	−	0.0611989i	$-0.980508\pi$
$317$	−4.03444	−	12.4167i	−0.226597	−	0.697393i	0.771529	−	0.636194i	$-0.219492\pi$
$318$		0			0
$319$	−1.07295	−	0.673542i	−0.0600736	−	0.0377111i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.47214	−	1.79611i	0.137553	−	0.0999383i
$324$		0			0
$325$	3.14590	−	9.68208i	0.174503	−	0.537065i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.5836			0.914283
$330$		0			0
$331$	27.4164			1.50694			0.753471	−	0.657481i	$-0.228378\pi$
$331$	27.4164			1.50694			0.753471	+	0.657481i	$0.228378\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.05573	−	21.7153i	0.385496	−	1.18643i
$336$		0			0
$337$	4.85410	−	3.52671i	0.264420	−	0.192112i	−0.447673	−	0.894197i	$-0.647747\pi$
$337$	4.85410	−	3.52671i	0.264420	−	0.192112i	0.712093	+	0.702085i	$0.247747\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.8992	+	12.4495i	−0.806836	+	0.674178i
$342$		0			0
$343$	6.05166	+	18.6251i	0.326759	+	1.00566i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.28115	+	22.4091i	−0.390873	+	1.20298i	0.541257	+	0.840857i	$0.317949\pi$
$347$	−7.28115	+	22.4091i	−0.390873	+	1.20298i	−0.932129	+	0.362125i	$0.882051\pi$
$348$		0			0
$349$	−17.4721	−	12.6942i	−0.935262	−	0.679508i	0.0120135	−	0.999928i	$-0.496176\pi$
$349$	−17.4721	−	12.6942i	−0.935262	−	0.679508i	−0.947276	+	0.320420i	$0.896176\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.23607			0.0657893			0.0328946	−	0.999459i	$-0.489527\pi$
$353$	1.23607			0.0657893			0.0328946	+	0.999459i	$0.489527\pi$
$354$		0			0
$355$	25.2705	+	18.3601i	1.34122	+	0.974453i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.0344	−	17.4620i	1.26849	−	0.921611i	0.269348	−	0.963043i	$-0.413192\pi$
$359$	24.0344	−	17.4620i	1.26849	−	0.921611i	0.999141	+	0.0414315i	$0.0131918\pi$
$360$		0			0
$361$	−5.69098	−	17.5150i	−0.299525	−	0.921844i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.6844	−	39.0386i	−0.663932	−	2.04337i
$366$		0			0
$367$	−10.7812	+	7.83297i	−0.562772	+	0.408878i	−0.832472	−	0.554067i	$-0.813075\pi$
$367$	−10.7812	+	7.83297i	−0.562772	+	0.408878i	0.269701	+	0.962944i	$0.413075\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−17.7812	−	12.9188i	−0.923151	−	0.670709i
$372$		0			0
$373$	24.0000			1.24267			0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000			1.24267			0.621336	+	0.783544i	$0.286590\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.00000	+	0.726543i	0.0515026	+	0.0374188i
$378$		0			0
$379$	3.14590	−	9.68208i	0.161594	−	0.497335i	−0.837175	−	0.546935i	$-0.815795\pi$
$379$	3.14590	−	9.68208i	0.161594	−	0.497335i	0.998769	+	0.0495998i	$0.0157946\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.50658	−	23.1029i	−0.383568	−	1.18050i	−0.937514	−	0.347949i	$-0.886878\pi$
$383$	−7.50658	−	23.1029i	−0.383568	−	1.18050i	0.553945	−	0.832553i	$-0.313122\pi$
$384$		0			0
$385$	−13.4681	+	11.2537i	−0.686396	+	0.573540i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.38197	−	1.73060i	0.120770	−	0.0877449i	−0.525760	−	0.850633i	$-0.676219\pi$
$389$	2.38197	−	1.73060i	0.120770	−	0.0877449i	0.646531	+	0.762888i	$0.276219\pi$
$390$		0			0
$391$	−8.94427	+	27.5276i	−0.452331	+	1.39213i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.09017			−0.0548524
$396$		0			0
$397$	12.9443			0.649654			0.324827	−	0.945773i	$-0.394694\pi$
$397$	12.9443			0.649654			0.324827	+	0.945773i	$0.394694\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.00000	+	15.3884i	−0.249688	+	0.768461i	0.745142	+	0.666906i	$0.232382\pi$
$401$	−5.00000	+	15.3884i	−0.249688	+	0.768461i	−0.994830	+	0.101555i	$0.967618\pi$
$402$		0			0
$403$	15.3262	−	11.1352i	0.763454	−	0.554682i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11.2361	−	7.05342i	−0.556951	−	0.349625i
$408$		0			0
$409$	−6.20820	−	19.1069i	−0.306976	−	0.944775i	−0.978932	−	0.204184i	$-0.934546\pi$
$409$	−6.20820	−	19.1069i	−0.306976	−	0.944775i	0.671957	−	0.740591i	$-0.265454\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.08359	−	9.49032i	0.151734	−	0.466988i
$414$		0			0
$415$	3.73607	+	2.71441i	0.183396	+	0.133245i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.7984			0.674095			0.337047	−	0.941488i	$-0.390572\pi$
$419$	13.7984			0.674095			0.337047	+	0.941488i	$0.390572\pi$
$420$		0			0
$421$	−5.76393	−	4.18774i	−0.280917	−	0.204098i	0.438400	−	0.898780i	$-0.355545\pi$
$421$	−5.76393	−	4.18774i	−0.280917	−	0.204098i	−0.719317	+	0.694682i	$0.755545\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.1803	+	7.39645i	−0.493819	+	0.358781i
$426$		0			0
$427$	−8.12461	−	25.0050i	−0.393178	−	1.21008i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.94427	−	18.2946i	−0.286326	−	0.881219i	−0.985998	−	0.166756i	$-0.946671\pi$
$431$	−5.94427	−	18.2946i	−0.286326	−	0.881219i	0.699673	−	0.714463i	$-0.253329\pi$
$432$		0			0
$433$	−1.02786	+	0.746787i	−0.0493960	+	0.0358883i	−0.612209	−	0.790696i	$-0.709719\pi$
$433$	−1.02786	+	0.746787i	−0.0493960	+	0.0358883i	0.562813	+	0.826584i	$0.309719\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.47214	−	3.24920i	−0.213931	−	0.155430i
$438$		0			0
$439$	6.50658			0.310542			0.155271	−	0.987872i	$-0.450375\pi$
$439$	6.50658			0.310542			0.155271	+	0.987872i	$0.450375\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.8262	−	9.31881i	−0.609393	−	0.442750i	0.239808	−	0.970820i	$-0.422916\pi$
$443$	−12.8262	−	9.31881i	−0.609393	−	0.442750i	−0.849201	+	0.528071i	$0.822916\pi$
$444$		0			0
$445$	−5.96556	+	18.3601i	−0.282795	+	0.870352i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.09017	−	15.6659i	−0.240220	−	0.739321i	−0.996386	−	0.0849412i	$-0.972930\pi$
$449$	−5.09017	−	15.6659i	−0.240220	−	0.739321i	0.756166	−	0.654380i	$-0.227070\pi$
$450$		0			0
$451$	5.85410	−	23.2744i	0.275659	−	1.09595i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.8541	−	10.0656i	0.649490	−	0.471882i
$456$		0			0
$457$	−3.73607	+	11.4984i	−0.174766	+	0.537874i	−0.999623	−	0.0274676i	$-0.991256\pi$
$457$	−3.73607	+	11.4984i	−0.174766	+	0.537874i	0.824857	+	0.565342i	$0.191256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$	−27.3820			−1.27255			−0.636274	−	0.771463i	$-0.719525\pi$
$463$	−27.3820			−1.27255			−0.636274	+	0.771463i	$0.719525\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.989357	−	3.04493i	0.0457820	−	0.140902i	−0.925553	−	0.378619i	$-0.876399\pi$
$467$	0.989357	−	3.04493i	0.0457820	−	0.140902i	0.971335	+	0.237717i	$0.0763990\pi$
$468$		0			0
$469$	12.0000	−	8.71851i	0.554109	−	0.402583i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.0000	+	29.8788i	0.551761	+	1.37383i
$474$		0			0
$475$	−0.742646	−	2.28563i	−0.0340749	−	0.104872i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.76393	−	23.8949i	0.354743	−	1.09179i	−0.601415	−	0.798937i	$-0.705396\pi$
$479$	7.76393	−	23.8949i	0.354743	−	1.09179i	0.956158	−	0.292850i	$-0.0946038\pi$
$480$		0			0
$481$	10.4721	+	7.60845i	0.477488	+	0.346916i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	46.4377			2.10863
$486$		0			0
$487$	20.4443	+	14.8536i	0.926418	+	0.673082i	0.945113	−	0.326743i	$-0.105951\pi$
$487$	20.4443	+	14.8536i	0.926418	+	0.673082i	−0.0186949	+	0.999825i	$0.505951\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.1803	−	13.2088i	0.820467	−	0.596104i	−0.0963790	−	0.995345i	$-0.530726\pi$
$491$	18.1803	−	13.2088i	0.820467	−	0.596104i	0.916846	+	0.399240i	$0.130726\pi$
$492$		0			0
$493$	−0.472136	−	1.45309i	−0.0212639	−	0.0654437i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.27051	+	19.2986i	0.281271	+	0.865663i
$498$		0			0
$499$	−22.1803	+	16.1150i	−0.992928	+	0.721405i	−0.960560	−	0.278071i	$-0.910305\pi$
$499$	−22.1803	+	16.1150i	−0.992928	+	0.721405i	−0.0323680	+	0.999476i	$0.510305\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−13.2361	−	9.61657i	−0.590167	−	0.428781i	0.252208	−	0.967673i	$-0.418843\pi$
$503$	−13.2361	−	9.61657i	−0.590167	−	0.428781i	−0.842375	+	0.538892i	$0.818843\pi$
$504$		0			0
$505$	33.1591			1.47556
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.5902	−	13.5065i	−0.823995	−	0.598667i	0.0938592	−	0.995585i	$-0.470080\pi$
$509$	−18.5902	−	13.5065i	−0.823995	−	0.598667i	−0.917854	+	0.396918i	$0.870080\pi$
$510$		0			0
$511$	8.24013	−	25.3605i	0.364522	−	1.12188i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.60081	+	23.3929i	0.334932	+	1.03081i
$516$		0			0
$517$	7.23607	−	28.7687i	0.318242	−	1.26525i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.76393	+	6.36737i	−0.383955	+	0.278960i	−0.762974	−	0.646429i	$-0.776261\pi$
$521$	−8.76393	+	6.36737i	−0.383955	+	0.278960i	0.379019	+	0.925389i	$0.376261\pi$
$522$		0			0
$523$	−2.88854	+	8.89002i	−0.126307	+	0.388734i	−0.994137	−	0.108128i	$-0.965514\pi$
$523$	−2.88854	+	8.89002i	−0.126307	+	0.388734i	0.867830	+	0.496862i	$0.165514\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−23.4164			−1.02003
$528$		0			0
$529$	29.3607			1.27655
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.23607	+	22.2703i	−0.313429	+	0.964635i
$534$		0			0
$535$	−8.89919	+	6.46564i	−0.384745	+	0.279534i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.7877	−	0.799788i	0.507734	−	0.0344493i
$540$		0			0
$541$	−7.27051	−	22.3763i	−0.312584	−	0.962033i	−0.976738	−	0.214438i	$-0.931208\pi$
$541$	−7.27051	−	22.3763i	−0.312584	−	0.962033i	0.664154	−	0.747596i	$-0.268792\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13.1803	−	40.5649i	0.564584	−	1.73761i
$546$		0			0
$547$	−24.8885	−	18.0826i	−1.06416	−	0.773156i	−0.0893043	−	0.996004i	$-0.528464\pi$
$547$	−24.8885	−	18.0826i	−1.06416	−	0.773156i	−0.974853	+	0.222849i	$0.928464\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.291796			0.0124309
$552$		0			0
$553$	−0.572949	−	0.416272i	−0.0243643	−	0.0177017i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−27.2984	+	19.8334i	−1.15667	+	0.840369i	−0.989353	−	0.145534i	$-0.953510\pi$
$557$	−27.2984	+	19.8334i	−1.15667	+	0.840369i	−0.167316	+	0.985903i	$0.553510\pi$
$558$		0			0
$559$	−9.70820	−	29.8788i	−0.410613	−	1.26374i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.5279	+	41.6345i	0.570131	+	1.75468i	0.652189	+	0.758057i	$0.273851\pi$
$563$	13.5279	+	41.6345i	0.570131	+	1.75468i	−0.0820572	+	0.996628i	$0.526149\pi$
$564$		0			0
$565$	−2.43769	+	1.77109i	−0.102555	+	0.0745103i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.7984	+	14.3844i	0.829991	+	0.603024i	0.919557	−	0.392957i	$-0.128548\pi$
$569$	19.7984	+	14.3844i	0.829991	+	0.603024i	−0.0895658	+	0.995981i	$0.528548\pi$
$570$		0			0
$571$	37.5967			1.57337			0.786687	−	0.617351i	$-0.211794\pi$
$571$	37.5967			1.57337			0.786687	+	0.617351i	$0.211794\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.4164	+	13.3803i	0.768017	+	0.557997i
$576$		0			0
$577$	−3.33688	+	10.2699i	−0.138916	+	0.427540i	−0.996179	−	0.0873397i	$-0.972163\pi$
$577$	−3.33688	+	10.2699i	−0.138916	+	0.427540i	0.857262	+	0.514880i	$0.172163\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.927051	+	2.85317i	0.0384606	+	0.118369i
$582$		0			0
$583$	−30.1697	+	25.2093i	−1.24950	+	1.04406i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25.1976	−	18.3071i	1.04001	−	0.755615i	0.0697262	−	0.997566i	$-0.477787\pi$
$587$	25.1976	−	18.3071i	1.04001	−	0.755615i	0.970288	+	0.241951i	$0.0777874\pi$
$588$		0			0
$589$	1.38197	−	4.25325i	0.0569429	−	0.175252i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.9443			−0.942208			−0.471104	−	0.882078i	$-0.656144\pi$
$593$	−22.9443			−0.942208			−0.471104	+	0.882078i	$0.656144\pi$
$594$		0			0
$595$	−21.1672			−0.867770
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.29180	−	7.05342i	0.0936402	−	0.288195i	−0.893257	−	0.449547i	$-0.851585\pi$
$599$	2.29180	−	7.05342i	0.0936402	−	0.288195i	0.986897	+	0.161352i	$0.0515854\pi$
$600$		0			0
$601$	−35.6246	+	25.8828i	−1.45316	+	1.05578i	−0.468078	+	0.883687i	$0.655053\pi$
$601$	−35.6246	+	25.8828i	−1.45316	+	1.05578i	−0.985081	+	0.172094i	$0.944947\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.6459	+	28.2744i	0.554785	+	1.14952i
$606$		0			0
$607$	4.65248	+	14.3188i	0.188838	+	0.581184i	0.999993	−	0.00364685i	$-0.00116083\pi$
$607$	4.65248	+	14.3188i	0.188838	+	0.581184i	−0.811155	+	0.584831i	$0.801161\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.94427	+	27.5276i	−0.361847	+	1.11365i
$612$		0			0
$613$	−16.0344	−	11.6497i	−0.647625	−	0.470527i	0.214836	−	0.976650i	$-0.431078\pi$
$613$	−16.0344	−	11.6497i	−0.647625	−	0.470527i	−0.862461	+	0.506123i	$0.831078\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.2361			−0.613381			−0.306691	−	0.951809i	$-0.599222\pi$
$617$	−15.2361			−0.613381			−0.306691	+	0.951809i	$0.599222\pi$
$618$		0			0
$619$	−5.70820	−	4.14725i	−0.229432	−	0.166692i	0.467130	−	0.884189i	$-0.345288\pi$
$619$	−5.70820	−	4.14725i	−0.229432	−	0.166692i	−0.696562	+	0.717496i	$0.745288\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.1459	+	7.37143i	−0.406487	+	0.295330i
$624$		0			0
$625$	−9.52786	−	29.3238i	−0.381115	−	1.17295i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.94427	−	15.2169i	−0.197141	−	0.606738i
$630$		0			0
$631$	−6.30902	+	4.58377i	−0.251158	+	0.182477i	−0.706240	−	0.707973i	$-0.749610\pi$
$631$	−6.30902	+	4.58377i	−0.251158	+	0.182477i	0.455082	+	0.890450i	$0.349610\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.7082	−	20.1312i	−1.09957	−	0.798882i
$636$		0			0
$637$	−11.5279			−0.456751
$638$		0			0
$639$		0			0
$640$		0			0
$641$	16.2361	+	11.7962i	0.641286	+	0.465922i	0.860292	−	0.509802i	$-0.170281\pi$
$641$	16.2361	+	11.7962i	0.641286	+	0.465922i	−0.219006	+	0.975724i	$0.570281\pi$
$642$		0			0
$643$	−9.20163	+	28.3197i	−0.362877	+	1.11682i	0.588423	+	0.808553i	$0.299749\pi$
$643$	−9.20163	+	28.3197i	−0.362877	+	1.11682i	−0.951300	+	0.308267i	$0.900251\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.1246	+	31.1604i	0.398040	+	1.22504i	0.926569	+	0.376125i	$0.122744\pi$
$647$	10.1246	+	31.1604i	0.398040	+	1.22504i	−0.528529	+	0.848915i	$0.677256\pi$
$648$		0			0
$649$	−15.1180	−	9.49032i	−0.593435	−	0.372528i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.69098	−	4.13474i	0.222705	−	0.161805i	−0.470838	−	0.882220i	$-0.656048\pi$
$653$	5.69098	−	4.13474i	0.222705	−	0.161805i	0.693544	+	0.720415i	$0.256048\pi$
$654$		0			0
$655$	0.961493	−	2.95917i	0.0375686	−	0.115624i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.90983			−0.152305			−0.0761527	−	0.997096i	$-0.524264\pi$
$659$	−3.90983			−0.152305			−0.0761527	+	0.997096i	$0.524264\pi$
$660$		0			0
$661$	−32.0689			−1.24734			−0.623668	−	0.781690i	$-0.714358\pi$
$661$	−32.0689			−1.24734			−0.623668	+	0.781690i	$0.714358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.24922	−	3.84471i	0.0484428	−	0.149092i
$666$		0			0
$667$	−2.23607	+	1.62460i	−0.0865809	+	0.0629047i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−46.9230	+	3.18368i	−1.81144	+	0.122905i
$672$		0			0
$673$	0.319660	+	0.983813i	0.0123220	+	0.0379232i	0.957028	−	0.289994i	$-0.0936533\pi$
$673$	0.319660	+	0.983813i	0.0123220	+	0.0379232i	−0.944706	+	0.327917i	$0.893653\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.156541	+	0.481784i	−0.00601637	+	0.0185165i	−0.954020	−	0.299744i	$-0.903099\pi$
$677$	−0.156541	+	0.481784i	−0.00601637	+	0.0185165i	0.948003	+	0.318261i	$0.103099\pi$
$678$		0			0
$679$	24.4058	+	17.7318i	0.936607	+	0.680485i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.437694			0.0167479			0.00837395	−	0.999965i	$-0.497334\pi$
$683$	0.437694			0.0167479			0.00837395	+	0.999965i	$0.497334\pi$
$684$		0			0
$685$	45.5066	+	33.0625i	1.73872	+	1.26325i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	31.0344	−	22.5478i	1.18232	−	0.859004i
$690$		0			0
$691$	10.0902	+	31.0543i	0.383848	+	1.18136i	0.937312	+	0.348490i	$0.113306\pi$
$691$	10.0902	+	31.0543i	0.383848	+	1.18136i	−0.553464	+	0.832873i	$0.686694\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.97871	+	27.6336i	0.340582	+	1.04820i
$696$		0			0
$697$	23.4164	−	17.0130i	0.886960	−	0.644414i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.0901699	+	0.0655123i	0.00340567	+	0.00247437i	0.589487	−	0.807778i	$-0.299330\pi$
$701$	0.0901699	+	0.0655123i	0.00340567	+	0.00247437i	−0.586081	+	0.810252i	$0.699330\pi$
$702$		0			0
$703$	3.05573			0.115249
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.4271	+	12.6615i	0.655412	+	0.476184i
$708$		0			0
$709$	1.00000	−	3.07768i	0.0375558	−	0.115585i	−0.930521	−	0.366238i	$-0.880645\pi$
$709$	1.00000	−	3.07768i	0.0375558	−	0.115585i	0.968077	+	0.250654i	$0.0806455\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.0902	+	40.2874i	0.490231	+	1.50878i
$714$		0			0
$715$	−11.4164	−	28.4257i	−0.426949	−	1.06306i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.0344	+	12.3762i	−0.635277	+	0.461556i	−0.858225	−	0.513274i	$-0.828432\pi$
$719$	−17.0344	+	12.3762i	−0.635277	+	0.461556i	0.222947	+	0.974831i	$0.428432\pi$
$720$		0			0
$721$	−4.93769	+	15.1967i	−0.183889	+	0.565953i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.20163			−0.0446273
$726$		0			0
$727$	−6.47214			−0.240038			−0.120019	−	0.992772i	$-0.538296\pi$
$727$	−6.47214			−0.240038			−0.120019	+	0.992772i	$0.538296\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	+	36.9322i	−0.443836	+	1.36599i
$732$		0			0
$733$	−8.70820	+	6.32688i	−0.321645	+	0.233689i	−0.736877	−	0.676027i	$-0.763700\pi$
$733$	−8.70820	+	6.32688i	−0.321645	+	0.233689i	0.415232	+	0.909715i	$0.363700\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.88854	−	24.6215i	−0.364249	−	0.906943i
$738$		0			0
$739$	1.34752	+	4.14725i	0.0495695	+	0.152559i	0.972777	−	0.231742i	$-0.0744425\pi$
$739$	1.34752	+	4.14725i	0.0495695	+	0.152559i	−0.923208	+	0.384301i	$0.874442\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.20163	+	3.69822i	−0.0440834	+	0.135675i	−0.970676	−	0.240393i	$-0.922724\pi$
$743$	−1.20163	+	3.69822i	−0.0440834	+	0.135675i	0.926592	+	0.376067i	$0.122724\pi$
$744$		0			0
$745$	26.6976	+	19.3969i	0.978123	+	0.710648i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.14590			−0.261105
$750$		0			0
$751$	15.2361	+	11.0697i	0.555972	+	0.403937i	0.829983	−	0.557789i	$-0.188350\pi$
$751$	15.2361	+	11.0697i	0.555972	+	0.403937i	−0.274011	+	0.961727i	$0.588350\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.881966	+	0.640786i	−0.0320980	+	0.0233206i
$756$		0			0
$757$	5.14590	+	15.8374i	0.187031	+	0.575622i	0.999977	−	0.00671649i	$-0.00213794\pi$
$757$	5.14590	+	15.8374i	0.187031	+	0.575622i	−0.812947	+	0.582338i	$0.802138\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.527864	−	1.62460i	−0.0191351	−	0.0588916i	0.941033	−	0.338315i	$-0.109857\pi$
$761$	−0.527864	−	1.62460i	−0.0191351	−	0.0588916i	−0.960168	+	0.279423i	$0.909857\pi$
$762$		0			0
$763$	22.4164	−	16.2865i	0.811528	−	0.589610i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14.0902	+	10.2371i	0.508767	+	0.369641i
$768$		0			0
$769$	29.6180			1.06805			0.534027	−	0.845468i	$-0.320678\pi$
$769$	29.6180			1.06805			0.534027	+	0.845468i	$0.320678\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−38.5238	−	27.9892i	−1.38560	−	1.00670i	−0.996331	−	0.0855785i	$-0.972726\pi$
$773$	−38.5238	−	27.9892i	−1.38560	−	1.00670i	−0.389273	−	0.921122i	$-0.627274\pi$
$774$		0			0
$775$	−5.69098	+	17.5150i	−0.204426	+	0.629159i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.70820	+	5.25731i	0.0612028	+	0.188363i
$780$		0			0
$781$	36.2148	−	2.45714i	1.29587	−	0.0879235i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.61803	−	3.35520i	0.164825	−	0.119752i
$786$		0			0
$787$	−14.8885	+	45.8222i	−0.530719	+	1.63339i	0.222002	+	0.975046i	$0.428741\pi$
$787$	−14.8885	+	45.8222i	−0.530719	+	1.63339i	−0.752721	+	0.658340i	$0.771259\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.95743			−0.0695981
$792$		0			0
$793$	45.8885			1.62955
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.9894	+	33.8218i	−0.389263	+	1.19803i	0.544077	+	0.839035i	$0.316880\pi$
$797$	−10.9894	+	33.8218i	−0.389263	+	1.19803i	−0.933340	+	0.358993i	$0.883120\pi$
$798$		0			0
$799$	28.9443	−	21.0292i	1.02397	−	0.743961i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−40.3992	−	25.3605i	−1.42566	−	0.894953i
$804$		0			0
$805$	11.8328	+	36.4177i	0.417052	+	1.28355i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.3607	−	38.0423i	0.434578	−	1.33749i	−0.458940	−	0.888467i	$-0.651771\pi$
$809$	12.3607	−	38.0423i	0.434578	−	1.33749i	0.893518	−	0.449027i	$-0.148229\pi$
$810$		0			0
$811$	−4.90983	−	3.56720i	−0.172407	−	0.125261i	0.498235	−	0.867042i	$-0.333982\pi$
$811$	−4.90983	−	3.56720i	−0.172407	−	0.125261i	−0.670643	+	0.741781i	$0.733982\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.9017			1.15250
$816$		0			0
$817$	−6.00000	−	4.35926i	−0.209913	−	0.152511i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	32.2426	−	23.4257i	1.12528	−	0.817561i	0.140276	−	0.990112i	$-0.455201\pi$
$821$	32.2426	−	23.4257i	1.12528	−	0.817561i	0.985000	+	0.172552i	$0.0552011\pi$
$822$		0			0
$823$	−15.5344	−	47.8101i	−0.541497	−	1.66656i	−0.729177	−	0.684325i	$-0.760097\pi$
$823$	−15.5344	−	47.8101i	−0.541497	−	1.66656i	0.187680	−	0.982230i	$-0.439903\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.22542	−	13.0045i	−0.146932	−	0.452211i	0.850322	−	0.526263i	$-0.176407\pi$
$827$	−4.22542	−	13.0045i	−0.146932	−	0.452211i	−0.997254	+	0.0740512i	$0.976407\pi$
$828$		0			0
$829$	−25.7984	+	18.7436i	−0.896015	+	0.650993i	−0.937439	−	0.348149i	$-0.886810\pi$
$829$	−25.7984	+	18.7436i	−0.896015	+	0.650993i	0.0414247	+	0.999142i	$0.486810\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11.5279	+	8.37548i	0.399417	+	0.290193i
$834$		0			0
$835$	−4.87539			−0.168720
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.76393	−	1.28157i	−0.0608977	−	0.0442448i	0.556920	−	0.830566i	$-0.311983\pi$
$839$	−1.76393	−	1.28157i	−0.0608977	−	0.0442448i	−0.617818	+	0.786321i	$0.711983\pi$
$840$		0			0
$841$	−8.91641	+	27.4419i	−0.307462	+	0.946272i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.22949	−	6.86167i	−0.0766968	−	0.236048i
$846$		0			0
$847$	−3.62461	+	20.0705i	−0.124543	+	0.689629i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−23.4164	+	17.0130i	−0.802704	+	0.583199i
$852$		0			0
$853$	5.70820	−	17.5680i	0.195445	−	0.601518i	−0.804526	−	0.593918i	$-0.797581\pi$
$853$	5.70820	−	17.5680i	0.195445	−	0.601518i	0.999971	−	0.00760089i	$-0.00241946\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.7639			0.572645			0.286323	−	0.958133i	$-0.407567\pi$
$857$	16.7639			0.572645			0.286323	+	0.958133i	$0.407567\pi$
$858$		0			0
$859$	−39.0132			−1.33111			−0.665556	−	0.746348i	$-0.731805\pi$
$859$	−39.0132			−1.33111			−0.665556	+	0.746348i	$0.731805\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.38197	−	7.33094i	0.0810831	−	0.249548i	−0.902295	−	0.431120i	$-0.858119\pi$
$863$	2.38197	−	7.33094i	0.0810831	−	0.249548i	0.983378	+	0.181572i	$0.0581185\pi$
$864$		0			0
$865$	2.51722	−	1.82887i	0.0855881	−	0.0621834i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.972136	+	0.812299i	−0.0329775	+	0.0275554i
$870$		0			0
$871$	8.00000	+	24.6215i	0.271070	+	0.834267i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.03193	−	9.33132i	0.102498	−	0.315456i
$876$		0			0
$877$	3.90983	+	2.84066i	0.132026	+	0.0959222i	0.651838	−	0.758358i	$-0.273998\pi$
$877$	3.90983	+	2.84066i	0.132026	+	0.0959222i	−0.519812	+	0.854280i	$0.673998\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.0689			0.541374			0.270687	−	0.962667i	$-0.412749\pi$
$881$	16.0689			0.541374			0.270687	+	0.962667i	$0.412749\pi$
$882$		0			0
$883$	−17.0902	−	12.4167i	−0.575130	−	0.417856i	0.261835	−	0.965113i	$-0.415672\pi$
$883$	−17.0902	−	12.4167i	−0.575130	−	0.417856i	−0.836965	+	0.547256i	$0.815672\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.85410	+	1.34708i	−0.0622547	+	0.0452307i	−0.618477	−	0.785803i	$-0.712250\pi$
$887$	−1.85410	+	1.34708i	−0.0622547	+	0.0452307i	0.556223	+	0.831033i	$0.312250\pi$
$888$		0			0
$889$	−6.87539	−	21.1603i	−0.230593	−	0.709693i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.11146	+	6.49839i	0.0706572	+	0.217460i
$894$		0			0
$895$	23.9721	−	17.4168i	0.801300	−	0.582179i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.80902	−	1.31433i	−0.0603341	−	0.0438353i
$900$		0			0
$901$	−47.4164			−1.57967
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.3262	+	7.50245i	0.343256	+	0.249390i
$906$		0			0
$907$	3.47214	−	10.6861i	0.115290	−	0.354827i	−0.876717	−	0.481006i	$-0.840271\pi$
$907$	3.47214	−	10.6861i	0.115290	−	0.354827i	0.992007	+	0.126179i	$0.0402714\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.09017	−	18.7436i	−0.201776	−	0.621004i	−0.999830	−	0.0184178i	$-0.994137\pi$
$911$	−6.09017	−	18.7436i	−0.201776	−	0.621004i	0.798054	−	0.602586i	$-0.205863\pi$
$912$		0			0
$913$	5.35410	−	0.363271i	0.177195	−	0.0120225i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.63525	−	1.18808i	0.0540009	−	0.0392339i
$918$		0			0
$919$	4.40983	−	13.5721i	0.145467	−	0.447701i	−0.851604	−	0.524186i	$-0.824370\pi$
$919$	4.40983	−	13.5721i	0.145467	−	0.447701i	0.997071	+	0.0764849i	$0.0243697\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−35.4164			−1.16575
$924$		0			0
$925$	−12.5836			−0.413746
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.7295	+	36.0997i	−0.384832	+	1.18439i	0.551770	+	0.833996i	$0.313953\pi$
$929$	−11.7295	+	36.0997i	−0.384832	+	1.18439i	−0.936602	+	0.350395i	$0.886047\pi$
$930$		0			0
$931$	−2.20163	+	1.59958i	−0.0721554	+	0.0524240i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.23607	+	36.7202i	−0.302052	+	1.20088i
$936$		0			0
$937$	−8.10081	−	24.9317i	−0.264642	−	0.814484i	−0.991776	−	0.127988i	$-0.959148\pi$
$937$	−8.10081	−	24.9317i	−0.264642	−	0.814484i	0.727134	−	0.686496i	$-0.240852\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.67376	+	11.3067i	−0.119761	+	0.368587i	−0.992910	−	0.118866i	$-0.962074\pi$
$941$	−3.67376	+	11.3067i	−0.119761	+	0.368587i	0.873149	+	0.487453i	$0.162074\pi$
$942$		0			0
$943$	−42.3607	−	30.7768i	−1.37945	−	1.00223i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−58.9787			−1.91655			−0.958275	−	0.285847i	$-0.907725\pi$
$947$	−58.9787			−1.91655			−0.958275	+	0.285847i	$0.907725\pi$
$948$		0			0
$949$	37.6525	+	27.3561i	1.22225	+	0.888017i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−19.7082	+	14.3188i	−0.638411	+	0.463833i	−0.859304	−	0.511465i	$-0.829103\pi$
$953$	−19.7082	+	14.3188i	−0.638411	+	0.463833i	0.220893	+	0.975298i	$0.429103\pi$
$954$		0			0
$955$	11.4164	+	35.1361i	0.369426	+	1.13698i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.2918	+	34.7526i	0.364631	+	1.12222i
$960$		0			0
$961$	−2.64590	+	1.92236i	−0.0853515	+	0.0620115i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	59.0238	+	42.8833i	1.90004	+	1.38046i
$966$		0			0
$967$	−51.9787			−1.67152			−0.835761	−	0.549093i	$-0.814973\pi$
$967$	−51.9787			−1.67152			−0.835761	+	0.549093i	$0.814973\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.47214	−	3.24920i	−0.143518	−	0.104272i	0.513710	−	0.857964i	$-0.328271\pi$
$971$	−4.47214	−	3.24920i	−0.143518	−	0.104272i	−0.657227	+	0.753692i	$0.728271\pi$
$972$		0			0
$973$	−5.83282	+	17.9516i	−0.186991	+	0.575501i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.81966	+	20.9888i	0.218180	+	0.671490i	0.998913	+	0.0466241i	$0.0148463\pi$
$977$	6.81966	+	20.9888i	0.218180	+	0.671490i	−0.780732	+	0.624866i	$0.785154\pi$
$978$		0			0
$979$	8.36068	+	20.8172i	0.267208	+	0.665322i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.14590	−	3.01217i	0.132234	−	0.0960733i	−0.519702	−	0.854347i	$-0.673957\pi$
$983$	4.14590	−	3.01217i	0.132234	−	0.0960733i	0.651936	+	0.758274i	$0.273957\pi$
$984$		0			0
$985$	16.7877	−	51.6673i	0.534902	−	1.64626i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	70.2492			2.23380
$990$		0			0
$991$	−41.6869			−1.32423			−0.662114	−	0.749403i	$-0.730341\pi$
$991$	−41.6869			−1.32423			−0.662114	+	0.749403i	$0.730341\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.7877	−	51.6673i	0.532207	−	1.63796i
$996$		0			0
$997$	2.52786	−	1.83660i	0.0800583	−	0.0581657i	−0.547036	−	0.837109i	$-0.684244\pi$
$997$	2.52786	−	1.83660i	0.0800583	−	0.0581657i	0.627094	+	0.778943i	$0.284244\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	792.2.r.e.577.1		4
3.2	odd	2		264.2.q.a.49.1	✓	4
11.3	even	5		8712.2.a.bg.1.2		2
11.8	odd	10		8712.2.a.bf.1.2		2
11.9	even	5	inner	792.2.r.e.361.1		4
12.11	even	2		528.2.y.e.49.1		4
33.8	even	10		2904.2.a.v.1.1		2
33.14	odd	10		2904.2.a.w.1.1		2
33.20	odd	10		264.2.q.a.97.1	yes	4
132.47	even	10		5808.2.a.br.1.1		2
132.107	odd	10		5808.2.a.bs.1.1		2
132.119	even	10		528.2.y.e.97.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.q.a.49.1	✓	4	3.2	odd	2
264.2.q.a.97.1	yes	4	33.20	odd	10
528.2.y.e.49.1		4	12.11	even	2
528.2.y.e.97.1		4	132.119	even	10
792.2.r.e.361.1		4	11.9	even	5	inner
792.2.r.e.577.1		4	1.1	even	1	trivial
2904.2.a.v.1.1		2	33.8	even	10
2904.2.a.w.1.1		2	33.14	odd	10
5808.2.a.br.1.1		2	132.47	even	10
5808.2.a.bs.1.1		2	132.107	odd	10
8712.2.a.bf.1.2		2	11.8	odd	10
8712.2.a.bg.1.2		2	11.3	even	5

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	792.r (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.881966 - 2.71441i) q^{5} +(1.50000 - 1.08981i) q^{7} +(-1.23607 - 3.07768i) q^{11} +(1.00000 + 3.07768i) q^{13} +(1.23607 - 3.80423i) q^{17} +(0.618034 + 0.449028i) q^{19} -7.23607 q^{23} +(-2.54508 - 1.84911i) q^{25} +(0.309017 - 0.224514i) q^{29} +(-1.80902 - 5.56758i) q^{31} +(-1.63525 - 5.03280i) q^{35} +(3.23607 - 2.35114i) q^{37} +(5.85410 + 4.25325i) q^{41} -9.70820 q^{43} +(7.23607 + 5.25731i) q^{47} +(-1.10081 + 3.38795i) q^{49} +(-3.66312 - 11.2739i) q^{53} +(-9.44427 + 0.640786i) q^{55} +(4.35410 - 3.16344i) q^{59} +(4.38197 - 13.4863i) q^{61} +9.23607 q^{65} +8.00000 q^{67} +(-3.38197 + 10.4086i) q^{71} +(11.6353 - 8.45351i) q^{73} +(-5.20820 - 3.26944i) q^{77} +(-0.118034 - 0.363271i) q^{79} +(-0.500000 + 1.53884i) q^{83} +(-9.23607 - 6.71040i) q^{85} -6.76393 q^{89} +(4.85410 + 3.52671i) q^{91} +(1.76393 - 1.28157i) q^{95} +(5.02786 + 15.4742i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 6 q^{7} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 2 q^{19} - 20 q^{23} + q^{25} - q^{29} - 5 q^{31} + 27 q^{35} + 4 q^{37} + 10 q^{41} - 12 q^{43} + 20 q^{47} - 29 q^{49} + q^{53} - 2 q^{55}+ \cdots + 38 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 6 * q^7 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 2 * q^19 - 20 * q^23 + q^25 - q^29 - 5 * q^31 + 27 * q^35 + 4 * q^37 + 10 * q^41 - 12 * q^43 + 20 * q^47 - 29 * q^49 + q^53 - 2 * q^55 + 4 * q^59 + 22 * q^61 + 28 * q^65 + 32 * q^67 - 18 * q^71 + 13 * q^73 + 6 * q^77 + 4 * q^79 - 2 * q^83 - 28 * q^85 - 36 * q^89 + 6 * q^91 + 16 * q^95 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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