LMFDB - Embedded newform 2320.2.d.g.929.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(929,2320)] 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("2320.929"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,3,0,0,0,-12,0,10] 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:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.84345856.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 13x^{4} + 41x^{2} + 1 $ x^6 + 13x^4 + 41x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.3
Root			$-2.77035i$ of defining polynomial
Character	$\chi$	$=$	2320.929
Dual form			2320.2.d.g.929.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.269894i q^{3} +(-1.96358 - 1.06975i) q^{5} +1.86960i q^{7} +2.92716 q^{9} +3.25230 q^{11} -3.40121i q^{13} +(-0.288719 + 0.529959i) q^{15} -2.40939i q^{17} -0.674860 q^{19} +0.504595 q^{21} +7.41031i q^{23} +(2.71128 + 4.20107i) q^{25} -1.59971i q^{27} +1.00000 q^{29} -5.25230 q^{31} -0.877777i q^{33} +(2.00000 - 3.67111i) q^{35} +1.86960i q^{37} -0.917968 q^{39} -3.46931i q^{43} +(-5.74770 - 3.13132i) q^{45} -4.00910i q^{47} +3.50459 q^{49} -0.650280 q^{51} -0.877777i q^{53} +(-6.38614 - 3.47914i) q^{55} +0.182141i q^{57} +10.0000 q^{59} +11.8543 q^{61} +5.47261i q^{63} +(-3.63844 + 6.67855i) q^{65} -7.95010i q^{67} +2.00000 q^{69} -2.00000 q^{71} +0.607882i q^{73} +(1.13384 - 0.731759i) q^{75} +6.08050i q^{77} +8.60202 q^{79} +8.34972 q^{81} -2.40939i q^{83} +(-2.57744 + 4.73102i) q^{85} -0.269894i q^{87} +8.50459 q^{89} +6.35891 q^{91} +1.41757i q^{93} +(1.32514 + 0.721929i) q^{95} -13.1332i q^{97} +9.51998 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} - 12 q^{9} + 10 q^{11} - 7 q^{15} + 16 q^{19} - 16 q^{21} + 11 q^{25} + 6 q^{29} - 22 q^{31} + 12 q^{35} - 14 q^{39} - 44 q^{45} + 2 q^{49} - 44 q^{51} - 13 q^{55} + 60 q^{59} + 12 q^{61}+ \cdots + 16 q^{99}+O(q^{100}) $ 6 * q + 3 * q^5 - 12 * q^9 + 10 * q^11 - 7 * q^15 + 16 * q^19 - 16 * q^21 + 11 * q^25 + 6 * q^29 - 22 * q^31 + 12 * q^35 - 14 * q^39 - 44 * q^45 + 2 * q^49 - 44 * q^51 - 13 * q^55 + 60 * q^59 + 12 * q^61 + 13 * q^65 + 12 * q^69 - 12 * q^71 - 9 * q^75 + 2 * q^79 + 10 * q^81 - 26 * q^85 + 32 * q^89 - 40 * q^91 + 28 * q^95 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		−	0.269894i		−	0.155824i	−0.996960	−	0.0779118i	$-0.975175\pi$
$3$		−	0.269894i		−	0.155824i	0.996960	−	0.0779118i	$-0.0248252\pi$
$4$		0			0
$5$	−1.96358	−	1.06975i	−0.878139	−	0.478406i
$5$	−1.96358	−	1.06975i	−0.878139	−	0.478406i
$6$		0			0
$7$			1.86960i			0.706643i	0.935502	+	0.353321i	$0.114948\pi$
$7$			1.86960i			0.706643i	−0.935502	+	0.353321i	$0.885052\pi$
$8$		0			0
$9$	2.92716			0.975719
$10$		0			0
$11$	3.25230			0.980605			0.490302	−	0.871552i	$-0.336886\pi$
$11$	3.25230			0.980605			0.490302	+	0.871552i	$0.336886\pi$
$12$		0			0
$13$		−	3.40121i		−	0.943327i	−0.881779	−	0.471663i	$-0.843654\pi$
$13$		−	3.40121i		−	0.943327i	0.881779	−	0.471663i	$-0.156346\pi$
$14$		0			0
$15$	−0.288719	+	0.529959i	−0.0745469	+	0.136835i
$16$		0			0
$17$		−	2.40939i		−	0.584363i	−0.956363	−	0.292181i	$-0.905619\pi$
$17$		−	2.40939i		−	0.584363i	0.956363	−	0.292181i	$-0.0943811\pi$
$18$		0			0
$19$	−0.674860			−0.154823			−0.0774117	−	0.996999i	$-0.524666\pi$
$19$	−0.674860			−0.154823			−0.0774117	+	0.996999i	$0.524666\pi$
$20$		0			0
$21$	0.504595			0.110112
$22$		0			0
$23$			7.41031i			1.54516i	0.634920	+	0.772578i	$0.281033\pi$
$23$			7.41031i			1.54516i	−0.634920	+	0.772578i	$0.718967\pi$
$24$		0			0
$25$	2.71128	+	4.20107i	0.542256	+	0.840213i
$26$		0			0
$27$		−	1.59971i		−	0.307864i
$28$		0			0
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	−5.25230			−0.943340			−0.471670	−	0.881775i	$-0.656349\pi$
$31$	−5.25230			−0.943340			−0.471670	+	0.881775i	$0.656349\pi$
$32$		0			0
$33$		−	0.877777i		−	0.152801i
$34$		0			0
$35$	2.00000	−	3.67111i	0.338062	−	0.620530i
$36$		0			0
$37$			1.86960i			0.307360i	0.988121	+	0.153680i	$0.0491126\pi$
$37$			1.86960i			0.307360i	−0.988121	+	0.153680i	$0.950887\pi$
$38$		0			0
$39$	−0.917968			−0.146993
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		−	3.46931i		−	0.529064i	−0.964377	−	0.264532i	$-0.914782\pi$
$43$		−	3.46931i		−	0.529064i	0.964377	−	0.264532i	$-0.0852175\pi$
$44$		0			0
$45$	−5.74770	−	3.13132i	−0.856817	−	0.466789i
$46$		0			0
$47$		−	4.00910i		−	0.584787i	−0.956298	−	0.292393i	$-0.905548\pi$
$47$		−	4.00910i		−	0.584787i	0.956298	−	0.292393i	$-0.0944516\pi$
$48$		0			0
$49$	3.50459			0.500656
$50$		0			0
$51$	−0.650280			−0.0910575
$52$		0			0
$53$		−	0.877777i		−	0.120572i	−0.998181	−	0.0602859i	$-0.980799\pi$
$53$		−	0.877777i		−	0.120572i	0.998181	−	0.0602859i	$-0.0192013\pi$
$54$		0			0
$55$	−6.38614	−	3.47914i	−0.861107	−	0.469127i
$56$		0			0
$57$			0.182141i			0.0241251i
$58$		0			0
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	11.8543			1.51779			0.758895	−	0.651213i	$-0.225740\pi$
$61$	11.8543			1.51779			0.758895	+	0.651213i	$0.225740\pi$
$62$		0			0
$63$			5.47261i			0.689485i
$64$		0			0
$65$	−3.63844	+	6.67855i	−0.451293	+	0.828372i
$66$		0			0
$67$		−	7.95010i		−	0.971259i	−0.874165	−	0.485629i	$-0.838590\pi$
$67$		−	7.95010i		−	0.971259i	0.874165	−	0.485629i	$-0.161410\pi$
$68$		0			0
$69$	2.00000			0.240772
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$			0.607882i			0.0711472i	0.999367	+	0.0355736i	$0.0113258\pi$
$73$			0.607882i			0.0711472i	−0.999367	+	0.0355736i	$0.988674\pi$
$74$		0			0
$75$	1.13384	−	0.731759i	0.130925	−	0.0844963i
$76$		0			0
$77$			6.08050i			0.692937i
$78$		0			0
$79$	8.60202			0.967803			0.483901	−	0.875123i	$-0.339219\pi$
$79$	8.60202			0.967803			0.483901	+	0.875123i	$0.339219\pi$
$80$		0			0
$81$	8.34972			0.927747
$82$		0			0
$83$		−	2.40939i		−	0.264465i	−0.991219	−	0.132232i	$-0.957785\pi$
$83$		−	2.40939i		−	0.264465i	0.991219	−	0.132232i	$-0.0422145\pi$
$84$		0			0
$85$	−2.57744	+	4.73102i	−0.279562	+	0.513152i
$86$		0			0
$87$		−	0.269894i		−	0.0289357i
$88$		0			0
$89$	8.50459			0.901485			0.450743	−	0.892654i	$-0.351159\pi$
$89$	8.50459			0.901485			0.450743	+	0.892654i	$0.351159\pi$
$90$		0			0
$91$	6.35891			0.666595
$92$		0			0
$93$			1.41757i			0.146995i
$94$		0			0
$95$	1.32514	+	0.721929i	0.135957	+	0.0740684i
$96$		0			0
$97$		−	13.1332i		−	1.33347i	−0.745295	−	0.666735i	$-0.767691\pi$
$97$		−	13.1332i		−	1.33347i	0.745295	−	0.666735i	$-0.232309\pi$
$98$		0			0
$99$	9.51998			0.956794
$100$		0			0
$101$	4.84513			0.482108			0.241054	−	0.970512i	$-0.422507\pi$
$101$	4.84513			0.482108			0.241054	+	0.970512i	$0.422507\pi$
$102$		0			0
$103$			14.8887i			1.46703i	0.679674	+	0.733514i	$0.262121\pi$
$103$			14.8887i			1.46703i	−0.679674	+	0.733514i	$0.737879\pi$
$104$		0			0
$105$	−0.990811	−	0.539789i	−0.0966932	−	0.0526780i
$106$		0			0
$107$		−	1.86960i		−	0.180741i	−0.995908	−	0.0903705i	$-0.971195\pi$
$107$		−	1.86960i		−	0.180741i	0.995908	−	0.0903705i	$-0.0288051\pi$
$108$		0			0
$109$	8.77228			0.840232			0.420116	−	0.907470i	$-0.361989\pi$
$109$	8.77228			0.840232			0.420116	+	0.907470i	$0.361989\pi$
$110$		0			0
$111$	0.504595			0.0478940
$112$		0			0
$113$		−	10.0699i		−	0.947299i	−0.880713	−	0.473650i	$-0.842936\pi$
$113$		−	10.0699i		−	0.947299i	0.880713	−	0.473650i	$-0.157064\pi$
$114$		0			0
$115$	7.92716	−	14.5507i	0.739211	−	1.35686i
$116$		0			0
$117$		−	9.95588i		−	0.920422i
$118$		0			0
$119$	4.50459			0.412936
$120$		0			0
$121$	−0.422563			−0.0384148
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.829735	−	11.1495i	−0.0742137	−	0.997242i
$126$		0			0
$127$		−	18.6739i		−	1.65704i	−0.559961	−	0.828519i	$-0.689184\pi$
$127$		−	18.6739i		−	1.65704i	0.559961	−	0.828519i	$-0.310816\pi$
$128$		0			0
$129$	−0.936346			−0.0824407
$130$		0			0
$131$	11.0338			0.964025			0.482012	−	0.876164i	$-0.339906\pi$
$131$	11.0338			0.964025			0.482012	+	0.876164i	$0.339906\pi$
$132$		0			0
$133$		−	1.26172i		−	0.109405i
$134$		0			0
$135$	−1.71128	+	3.14115i	−0.147284	+	0.270347i
$136$		0			0
$137$			11.8714i			1.01425i	0.861874	+	0.507123i	$0.169291\pi$
$137$			11.8714i			1.01425i	−0.861874	+	0.507123i	$0.830709\pi$
$138$		0			0
$139$	14.8451			1.25915			0.629574	−	0.776941i	$-0.283230\pi$
$139$	14.8451			1.25915			0.629574	+	0.776941i	$0.283230\pi$
$140$		0			0
$141$	−1.08203			−0.0911235
$142$		0			0
$143$		−	11.0618i		−	0.925030i
$144$		0			0
$145$	−1.96358	−	1.06975i	−0.163066	−	0.0888377i
$146$		0			0
$147$		−	0.945870i		−	0.0780141i
$148$		0			0
$149$	−17.2769			−1.41538			−0.707688	−	0.706525i	$-0.750262\pi$
$149$	−17.2769			−1.41538			−0.707688	+	0.706525i	$0.750262\pi$
$150$		0			0
$151$	−0.504595			−0.0410633			−0.0205317	−	0.999789i	$-0.506536\pi$
$151$	−0.504595			−0.0410633			−0.0205317	+	0.999789i	$0.506536\pi$
$152$		0			0
$153$		−	7.05266i		−	0.570174i
$154$		0			0
$155$	10.3133	+	5.61863i	0.828384	+	0.451299i
$156$		0			0
$157$		−	17.9519i		−	1.43272i	−0.697731	−	0.716360i	$-0.745807\pi$
$157$		−	17.9519i		−	1.43272i	0.697731	−	0.716360i	$-0.254193\pi$
$158$		0			0
$159$	−0.236907			−0.0187879
$160$		0			0
$161$	−13.8543			−1.09187
$162$		0			0
$163$		−	5.45295i		−	0.427108i	−0.976931	−	0.213554i	$-0.931496\pi$
$163$		−	5.45295i		−	0.427108i	0.976931	−	0.213554i	$-0.0685040\pi$
$164$		0			0
$165$	−0.938999	+	1.72358i	−0.0731010	+	0.134181i
$166$		0			0
$167$			1.51195i			0.116998i	0.998287	+	0.0584992i	$0.0186315\pi$
$167$			1.51195i			0.116998i	−0.998287	+	0.0584992i	$0.981368\pi$
$168$		0			0
$169$	1.43175			0.110135
$170$		0			0
$171$	−1.97542			−0.151064
$172$		0			0
$173$		−	8.74012i		−	0.664499i	−0.943192	−	0.332249i	$-0.892192\pi$
$173$		−	8.74012i		−	0.664499i	0.943192	−	0.332249i	$-0.107808\pi$
$174$		0			0
$175$	−7.85431	+	5.06901i	−0.593730	+	0.383181i
$176$		0			0
$177$		−	2.69894i		−	0.202865i
$178$		0			0
$179$	−9.49541			−0.709720			−0.354860	−	0.934919i	$-0.615471\pi$
$179$	−9.49541			−0.709720			−0.354860	+	0.934919i	$0.615471\pi$
$180$		0			0
$181$	14.9363			1.11021			0.555105	−	0.831780i	$-0.312678\pi$
$181$	14.9363			1.11021			0.555105	+	0.831780i	$0.312678\pi$
$182$		0			0
$183$		−	3.19941i		−	0.236507i
$184$		0			0
$185$	2.00000	−	3.67111i	0.147043	−	0.269905i
$186$		0			0
$187$		−	7.83605i		−	0.573029i
$188$		0			0
$189$	2.99081			0.217549
$190$		0			0
$191$	−10.5292			−0.761864			−0.380932	−	0.924603i	$-0.624397\pi$
$191$	−10.5292			−0.761864			−0.380932	+	0.924603i	$0.624397\pi$
$192$		0			0
$193$			18.1341i			1.30532i	0.757651	+	0.652660i	$0.226347\pi$
$193$			18.1341i			1.30532i	−0.757651	+	0.652660i	$0.773653\pi$
$194$		0			0
$195$	1.80250	+	0.981994i	0.129080	+	0.0703220i
$196$		0			0
$197$			16.9798i			1.20976i	0.796317	+	0.604879i	$0.206779\pi$
$197$			16.9798i			1.20976i	−0.796317	+	0.604879i	$0.793221\pi$
$198$		0			0
$199$	7.49541			0.531335			0.265668	−	0.964065i	$-0.414408\pi$
$199$	7.49541			0.531335			0.265668	+	0.964065i	$0.414408\pi$
$200$		0			0
$201$	−2.14569			−0.151345
$202$		0			0
$203$			1.86960i			0.131220i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			21.6911i			1.50764i
$208$		0			0
$209$	−2.19484			−0.151821
$210$		0			0
$211$	−18.0974			−1.24588			−0.622939	−	0.782270i	$-0.714062\pi$
$211$	−18.0974			−1.24588			−0.622939	+	0.782270i	$0.714062\pi$
$212$		0			0
$213$			0.539789i			0.0369857i
$214$		0			0
$215$	−3.71128	+	6.81226i	−0.253107	+	0.464592i
$216$		0			0
$217$		−	9.81970i		−	0.666604i
$218$		0			0
$219$	0.164064			0.0110864
$220$		0			0
$221$	−8.19484			−0.551245
$222$		0			0
$223$		−	6.33073i		−	0.423937i	−0.977276	−	0.211969i	$-0.932013\pi$
$223$		−	6.33073i		−	0.423937i	0.977276	−	0.211969i	$-0.0679874\pi$
$224$		0			0
$225$	7.93635	+	12.2972i	0.529090	+	0.819812i
$226$		0			0
$227$			24.8905i			1.65204i	0.563638	+	0.826022i	$0.309401\pi$
$227$			24.8905i			1.65204i	−0.563638	+	0.826022i	$0.690599\pi$
$228$		0			0
$229$	−2.14569			−0.141791			−0.0708955	−	0.997484i	$-0.522586\pi$
$229$	−2.14569			−0.141791			−0.0708955	+	0.997484i	$0.522586\pi$
$230$		0			0
$231$	1.64109			0.107976
$232$		0			0
$233$			5.33891i			0.349763i	0.984589	+	0.174882i	$0.0559543\pi$
$233$			5.33891i			0.349763i	−0.984589	+	0.174882i	$0.944046\pi$
$234$		0			0
$235$	−4.28872	+	7.87217i	−0.279765	+	0.513524i
$236$		0			0
$237$		−	2.32164i		−	0.150806i
$238$		0			0
$239$	−2.00000			−0.129369			−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000			−0.129369			−0.0646846	+	0.997906i	$0.520604\pi$
$240$		0			0
$241$	−6.77228			−0.436241			−0.218121	−	0.975922i	$-0.569993\pi$
$241$	−6.77228			−0.436241			−0.218121	+	0.975922i	$0.569993\pi$
$242$		0			0
$243$		−	7.05266i		−	0.452428i
$244$		0			0
$245$	−6.88155	−	3.74903i	−0.439646	−	0.239517i
$246$		0			0
$247$			2.29534i			0.146049i
$248$		0			0
$249$	−0.650280			−0.0412098
$250$		0			0
$251$	−20.2615			−1.27889			−0.639447	−	0.768835i	$-0.720837\pi$
$251$	−20.2615			−1.27889			−0.639447	+	0.768835i	$0.720837\pi$
$252$		0			0
$253$			24.1005i			1.51519i
$254$		0			0
$255$	1.27688	+	0.695636i	0.0799611	+	0.0435624i
$256$		0			0
$257$			19.4376i			1.21248i	0.795280	+	0.606242i	$0.207324\pi$
$257$			19.4376i			1.21248i	−0.795280	+	0.606242i	$0.792676\pi$
$258$		0			0
$259$	−3.49541			−0.217194
$260$		0			0
$261$	2.92716			0.181186
$262$		0			0
$263$		−	8.28808i		−	0.511065i	−0.966800	−	0.255533i	$-0.917749\pi$
$263$		−	8.28808i		−	0.511065i	0.966800	−	0.255533i	$-0.0822508\pi$
$264$		0			0
$265$	−0.938999	+	1.72358i	−0.0576823	+	0.105879i
$266$		0			0
$267$		−	2.29534i		−	0.140473i
$268$		0			0
$269$	23.8543			1.45442			0.727212	−	0.686413i	$-0.240816\pi$
$269$	23.8543			1.45442			0.727212	+	0.686413i	$0.240816\pi$
$270$		0			0
$271$	0.893389			0.0542695			0.0271347	−	0.999632i	$-0.491362\pi$
$271$	0.893389			0.0542695			0.0271347	+	0.999632i	$0.491362\pi$
$272$		0			0
$273$		−	1.71623i		−	0.103871i
$274$		0			0
$275$	8.81789	+	13.6631i	0.531739	+	0.823917i
$276$		0			0
$277$		−	3.55706i		−	0.213723i	−0.994274	−	0.106862i	$-0.965920\pi$
$277$		−	3.55706i		−	0.213723i	0.994274	−	0.106862i	$-0.0340801\pi$
$278$		0			0
$279$	−15.3743			−0.920435
$280$		0			0
$281$	7.76309			0.463107			0.231554	−	0.972822i	$-0.425619\pi$
$281$	7.76309			0.463107			0.231554	+	0.972822i	$0.425619\pi$
$282$		0			0
$283$		−	30.7889i		−	1.83021i	−0.403215	−	0.915105i	$-0.632107\pi$
$283$		−	30.7889i		−	1.83021i	0.403215	−	0.915105i	$-0.367893\pi$
$284$		0			0
$285$	0.194845	−	0.357648i	0.0115416	−	0.0211852i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	11.1948			0.658520
$290$		0			0
$291$	−3.54456			−0.207786
$292$		0			0
$293$			6.33073i			0.369845i	0.982753	+	0.184923i	$0.0592034\pi$
$293$			6.33073i			0.369845i	−0.982753	+	0.184923i	$0.940797\pi$
$294$		0			0
$295$	−19.6358	−	10.6975i	−1.14324	−	0.622831i
$296$		0			0
$297$		−	5.20272i		−	0.301892i
$298$		0			0
$299$	25.2040			1.45759
$300$		0			0
$301$	6.48622			0.373859
$302$		0			0
$303$		−	1.30767i		−	0.0751238i
$304$		0			0
$305$	−23.2769	−	12.6811i	−1.33283	−	0.726119i
$306$		0			0
$307$			12.5671i			0.717241i	0.933483	+	0.358620i	$0.116753\pi$
$307$			12.5671i			0.717241i	−0.933483	+	0.358620i	$0.883247\pi$
$308$		0			0
$309$	4.01838			0.228598
$310$		0			0
$311$	−24.3835			−1.38266			−0.691330	−	0.722539i	$-0.742975\pi$
$311$	−24.3835			−1.38266			−0.691330	+	0.722539i	$0.742975\pi$
$312$		0			0
$313$		−	34.6618i		−	1.95920i	−0.200954	−	0.979601i	$-0.564404\pi$
$313$		−	34.6618i		−	1.95920i	0.200954	−	0.979601i	$-0.435596\pi$
$314$		0			0
$315$	5.85431	−	10.7459i	0.329853	−	0.605463i
$316$		0			0
$317$			25.7880i			1.44840i	0.689591	+	0.724199i	$0.257790\pi$
$317$			25.7880i			1.44840i	−0.689591	+	0.724199i	$0.742210\pi$
$318$		0			0
$319$	3.25230			0.182094
$320$		0			0
$321$	−0.504595			−0.0281637
$322$		0			0
$323$			1.62600i			0.0904730i
$324$		0			0
$325$	14.2887	−	9.22164i	0.792596	−	0.511525i
$326$		0			0
$327$		−	2.36759i		−	0.130928i
$328$		0			0
$329$	7.49541			0.413235
$330$		0			0
$331$	16.9609			0.932257			0.466128	−	0.884717i	$-0.345648\pi$
$331$	16.9609			0.932257			0.466128	+	0.884717i	$0.345648\pi$
$332$		0			0
$333$			5.47261i			0.299897i
$334$		0			0
$335$	−8.50459	+	15.6106i	−0.464656	+	0.852900i
$336$		0			0
$337$			22.5952i			1.23084i	0.788200	+	0.615420i	$0.211013\pi$
$337$			22.5952i			1.23084i	−0.788200	+	0.615420i	$0.788987\pi$
$338$		0			0
$339$	−2.71782			−0.147612
$340$		0			0
$341$	−17.0820			−0.925044
$342$		0			0
$343$			19.6394i			1.06043i
$344$		0			0
$345$	−3.92716	−	2.13949i	−0.211431	−	0.115187i
$346$		0			0
$347$			12.0469i			0.646714i	0.946277	+	0.323357i	$0.104811\pi$
$347$			12.0469i			0.646714i	−0.946277	+	0.323357i	$0.895189\pi$
$348$		0			0
$349$	1.56825			0.0839464			0.0419732	−	0.999119i	$-0.486636\pi$
$349$	1.56825			0.0839464			0.0419732	+	0.999119i	$0.486636\pi$
$350$		0			0
$351$	−5.44094			−0.290416
$352$		0			0
$353$			7.66054i			0.407730i	0.978999	+	0.203865i	$0.0653503\pi$
$353$			7.66054i			0.407730i	−0.978999	+	0.203865i	$0.934650\pi$
$354$		0			0
$355$	3.92716	+	2.13949i	0.208432	+	0.113553i
$356$		0			0
$357$		−	1.21576i		−	0.0643451i
$358$		0			0
$359$	17.5928			0.928514			0.464257	−	0.885701i	$-0.346321\pi$
$359$	17.5928			0.928514			0.464257	+	0.885701i	$0.346321\pi$
$360$		0			0
$361$	−18.5446			−0.976030
$362$		0			0
$363$			0.114047i			0.00598593i
$364$		0			0
$365$	0.650280	−	1.19362i	0.0340372	−	0.0624772i
$366$		0			0
$367$		−	9.88779i		−	0.516138i	−0.966126	−	0.258069i	$-0.916914\pi$
$367$		−	9.88779i		−	0.516138i	0.966126	−	0.258069i	$-0.0830863\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.64109			0.0852012
$372$		0			0
$373$			16.2841i			0.843161i	0.906791	+	0.421580i	$0.138524\pi$
$373$			16.2841i			0.843161i	−0.906791	+	0.421580i	$0.861476\pi$
$374$		0			0
$375$	−3.00919	+	0.223941i	−0.155394	+	0.0115642i
$376$		0			0
$377$		−	3.40121i		−	0.175171i
$378$		0			0
$379$	−2.52917			−0.129915			−0.0649575	−	0.997888i	$-0.520691\pi$
$379$	−2.52917			−0.129915			−0.0649575	+	0.997888i	$0.520691\pi$
$380$		0			0
$381$	−5.03997			−0.258205
$382$		0			0
$383$			5.60880i			0.286596i	0.989680	+	0.143298i	$0.0457708\pi$
$383$			5.60880i			0.286596i	−0.989680	+	0.143298i	$0.954229\pi$
$384$		0			0
$385$	6.50459	−	11.9395i	0.331505	−	0.608495i
$386$		0			0
$387$		−	10.1552i		−	0.516218i
$388$		0			0
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$	17.8543			0.902931
$392$		0			0
$393$		−	2.97795i		−	0.150218i
$394$		0			0
$395$	−16.8907	−	9.20198i	−0.849865	−	0.463002i
$396$		0			0
$397$		−	27.8660i		−	1.39856i	−0.714850	−	0.699278i	$-0.753505\pi$
$397$		−	27.8660i		−	1.39856i	0.714850	−	0.699278i	$-0.246495\pi$
$398$		0			0
$399$	−0.340531			−0.0170479
$400$		0			0
$401$	−15.6174			−0.779896			−0.389948	−	0.920837i	$-0.627507\pi$
$401$	−15.6174			−0.779896			−0.389948	+	0.920837i	$0.627507\pi$
$402$		0			0
$403$			17.8642i			0.889878i
$404$		0			0
$405$	−16.3953	−	8.93209i	−0.814691	−	0.443839i
$406$		0			0
$407$			6.08050i			0.301399i
$408$		0			0
$409$	30.5538			1.51079			0.755393	−	0.655272i	$-0.227446\pi$
$409$	30.5538			1.51079			0.755393	+	0.655272i	$0.227446\pi$
$410$		0			0
$411$	3.20403			0.158043
$412$		0			0
$413$			18.6960i			0.919970i
$414$		0			0
$415$	−2.57744	+	4.73102i	−0.126521	+	0.232237i
$416$		0			0
$417$		−	4.00661i		−	0.196205i
$418$		0			0
$419$	−10.0492			−0.490934			−0.245467	−	0.969405i	$-0.578941\pi$
$419$	−10.0492			−0.490934			−0.245467	+	0.969405i	$0.578941\pi$
$420$		0			0
$421$	−16.1948			−0.789288			−0.394644	−	0.918834i	$-0.629132\pi$
$421$	−16.1948			−0.789288			−0.394644	+	0.918834i	$0.629132\pi$
$422$		0			0
$423$		−	11.7353i		−	0.570587i
$424$		0			0
$425$	10.1220	−	6.53253i	0.490989	−	0.316874i
$426$		0			0
$427$			22.1628i			1.07253i
$428$		0			0
$429$	−2.98550			−0.144142
$430$		0			0
$431$	−11.1857			−0.538794			−0.269397	−	0.963029i	$-0.586824\pi$
$431$	−11.1857			−0.538794			−0.269397	+	0.963029i	$0.586824\pi$
$432$		0			0
$433$			14.0306i			0.674267i	0.941457	+	0.337134i	$0.109457\pi$
$433$			14.0306i			0.674267i	−0.941457	+	0.337134i	$0.890543\pi$
$434$		0			0
$435$	−0.288719	+	0.529959i	−0.0138430	+	0.0254096i
$436$		0			0
$437$		−	5.00092i		−	0.239226i
$438$		0			0
$439$	9.65947			0.461021			0.230511	−	0.973070i	$-0.425960\pi$
$439$	9.65947			0.461021			0.230511	+	0.973070i	$0.425960\pi$
$440$		0			0
$441$	10.2585			0.488500
$442$		0			0
$443$			9.75160i			0.463313i	0.972798	+	0.231656i	$0.0744145\pi$
$443$			9.75160i			0.463313i	−0.972798	+	0.231656i	$0.925586\pi$
$444$		0			0
$445$	−16.6994	−	9.09777i	−0.791629	−	0.431276i
$446$		0			0
$447$			4.66293i			0.220549i
$448$		0			0
$449$	−8.86350			−0.418295			−0.209147	−	0.977884i	$-0.567069\pi$
$449$	−8.86350			−0.418295			−0.209147	+	0.977884i	$0.567069\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			0.136187i			0.00639863i
$454$		0			0
$455$	−12.4862	−	6.80243i	−0.585363	−	0.318903i
$456$		0			0
$457$		−	24.5041i		−	1.14625i	−0.819466	−	0.573127i	$-0.805730\pi$
$457$		−	24.5041i		−	1.14625i	0.819466	−	0.573127i	$-0.194270\pi$
$458$		0			0
$459$	−3.85431			−0.179904
$460$		0			0
$461$	−37.4173			−1.74270			−0.871348	−	0.490666i	$-0.836753\pi$
$461$	−37.4173			−1.74270			−0.871348	+	0.490666i	$0.836753\pi$
$462$		0			0
$463$		−	2.45534i		−	0.114109i	−0.998371	−	0.0570547i	$-0.981829\pi$
$463$		−	2.45534i		−	0.114109i	0.998371	−	0.0570547i	$-0.0181710\pi$
$464$		0			0
$465$	1.51644	−	2.78350i	0.0703231	−	0.129082i
$466$		0			0
$467$		−	34.4182i		−	1.59268i	−0.604846	−	0.796342i	$-0.706765\pi$
$467$		−	34.4182i		−	1.59268i	0.604846	−	0.796342i	$-0.293235\pi$
$468$		0			0
$469$	14.8635			0.686333
$470$		0			0
$471$	−4.84513			−0.223252
$472$		0			0
$473$		−	11.2832i		−	0.518803i
$474$		0			0
$475$	−1.82973	−	2.83513i	−0.0839540	−	0.130085i
$476$		0			0
$477$		−	2.56939i		−	0.117644i
$478$		0			0
$479$	−9.41636			−0.430245			−0.215122	−	0.976587i	$-0.569015\pi$
$479$	−9.41636			−0.430245			−0.215122	+	0.976587i	$0.569015\pi$
$480$		0			0
$481$	6.35891			0.289941
$482$		0			0
$483$			3.73920i			0.170140i
$484$		0			0
$485$	−14.0492	+	25.7880i	−0.637939	+	1.17097i
$486$		0			0
$487$			33.0909i			1.49949i	0.661726	+	0.749745i	$0.269824\pi$
$487$			33.0909i			1.49949i	−0.661726	+	0.749745i	$0.730176\pi$
$488$		0			0
$489$	−1.47172			−0.0665535
$490$		0			0
$491$	−28.9609			−1.30699			−0.653494	−	0.756932i	$-0.726698\pi$
$491$	−28.9609			−1.30699			−0.653494	+	0.756932i	$0.726698\pi$
$492$		0			0
$493$		−	2.40939i		−	0.108513i
$494$		0			0
$495$	−18.6932	−	10.1840i	−0.840199	−	0.457736i
$496$		0			0
$497$		−	3.73920i		−	0.167726i
$498$		0			0
$499$	33.8727			1.51635			0.758175	−	0.652051i	$-0.226091\pi$
$499$	33.8727			1.51635			0.758175	+	0.652051i	$0.226091\pi$
$500$		0			0
$501$	0.408067			0.0182311
$502$		0			0
$503$			0.945870i			0.0421743i	0.999778	+	0.0210871i	$0.00671274\pi$
$503$			0.945870i			0.0421743i	−0.999778	+	0.0210871i	$0.993287\pi$
$504$		0			0
$505$	−9.51378	−	5.18306i	−0.423358	−	0.230643i
$506$		0			0
$507$		−	0.386422i		−	0.0171616i
$508$		0			0
$509$	−32.1404			−1.42460			−0.712299	−	0.701877i	$-0.752346\pi$
$509$	−32.1404			−1.42460			−0.712299	+	0.701877i	$0.752346\pi$
$510$		0			0
$511$	−1.13650			−0.0502757
$512$		0			0
$513$			1.07958i			0.0476645i
$514$		0			0
$515$	15.9272	−	29.2351i	0.701834	−	1.28825i
$516$		0			0
$517$		−	13.0388i		−	0.573444i
$518$		0			0
$519$	−2.35891			−0.103545
$520$		0			0
$521$	−8.43175			−0.369402			−0.184701	−	0.982795i	$-0.559132\pi$
$521$	−8.43175			−0.369402			−0.184701	+	0.982795i	$0.559132\pi$
$522$		0			0
$523$			21.8273i			0.954442i	0.878783	+	0.477221i	$0.158356\pi$
$523$			21.8273i			0.954442i	−0.878783	+	0.477221i	$0.841644\pi$
$524$		0			0
$525$	1.36810	+	2.11983i	0.0597087	+	0.0925172i
$526$		0			0
$527$			12.6548i			0.551253i
$528$		0			0
$529$	−31.9127			−1.38751
$530$		0			0
$531$	29.2716			1.27028
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−2.00000	+	3.67111i	−0.0864675	+	0.158716i
$536$		0			0
$537$			2.56276i			0.110591i
$538$		0			0
$539$	11.3980			0.490946
$540$		0			0
$541$	−25.5138			−1.09692			−0.548462	−	0.836176i	$-0.684786\pi$
$541$	−25.5138			−1.09692			−0.548462	+	0.836176i	$0.684786\pi$
$542$		0			0
$543$		−	4.03123i		−	0.172997i
$544$		0			0
$545$	−17.2251	−	9.38413i	−0.737841	−	0.401972i
$546$		0			0
$547$		−	17.7698i		−	0.759781i	−0.925031	−	0.379891i	$-0.875962\pi$
$547$		−	17.7698i		−	0.759781i	0.925031	−	0.379891i	$-0.124038\pi$
$548$		0			0
$549$	34.6994			1.48094
$550$		0			0
$551$	−0.674860			−0.0287500
$552$		0			0
$553$			16.0823i			0.683890i
$554$		0			0
$555$	−0.990811	−	0.539789i	−0.0420576	−	0.0229128i
$556$		0			0
$557$			42.1665i			1.78665i	0.449409	+	0.893326i	$0.351635\pi$
$557$			42.1665i			1.78665i	−0.449409	+	0.893326i	$0.648365\pi$
$558$		0			0
$559$	−11.7998			−0.499080
$560$		0			0
$561$	−2.11491			−0.0892914
$562$		0			0
$563$			5.76465i			0.242951i	0.992594	+	0.121475i	$0.0387626\pi$
$563$			5.76465i			0.242951i	−0.992594	+	0.121475i	$0.961237\pi$
$564$		0			0
$565$	−10.7723	+	19.7731i	−0.453193	+	0.831861i
$566$		0			0
$567$			15.6106i			0.655585i
$568$		0			0
$569$	−1.49541			−0.0626907			−0.0313453	−	0.999509i	$-0.509979\pi$
$569$	−1.49541			−0.0626907			−0.0313453	+	0.999509i	$0.509979\pi$
$570$		0			0
$571$	7.03997			0.294614			0.147307	−	0.989091i	$-0.452940\pi$
$571$	7.03997			0.294614			0.147307	+	0.989091i	$0.452940\pi$
$572$		0			0
$573$			2.84176i			0.118716i
$574$		0			0
$575$	−31.1312	+	20.0914i	−1.29826	+	0.837870i
$576$		0			0
$577$			34.7102i			1.44501i	0.691368	+	0.722503i	$0.257009\pi$
$577$			34.7102i			1.44501i	−0.691368	+	0.722503i	$0.742991\pi$
$578$		0			0
$579$	4.89428			0.203399
$580$		0			0
$581$	4.50459			0.186882
$582$		0			0
$583$		−	2.85479i		−	0.118233i
$584$		0			0
$585$	−10.6503	+	19.5492i	−0.440335	+	0.808258i
$586$		0			0
$587$		−	37.4158i		−	1.54432i	−0.635430	−	0.772158i	$-0.719177\pi$
$587$		−	37.4158i		−	1.54432i	0.635430	−	0.772158i	$-0.280823\pi$
$588$		0			0
$589$	3.54456			0.146051
$590$		0			0
$591$	4.58274			0.188509
$592$		0			0
$593$			32.6388i			1.34032i	0.742218	+	0.670158i	$0.233774\pi$
$593$			32.6388i			1.34032i	−0.742218	+	0.670158i	$0.766226\pi$
$594$		0			0
$595$	−8.84513	−	4.81878i	−0.362615	−	0.197551i
$596$		0			0
$597$		−	2.02297i		−	0.0827945i
$598$		0			0
$599$	−38.1466			−1.55863			−0.779314	−	0.626634i	$-0.784432\pi$
$599$	−38.1466			−1.55863			−0.779314	+	0.626634i	$0.784432\pi$
$600$		0			0
$601$	−16.7178			−0.681934			−0.340967	−	0.940075i	$-0.610754\pi$
$601$	−16.7178			−0.681934			−0.340967	+	0.940075i	$0.610754\pi$
$602$		0			0
$603$		−	23.2712i		−	0.947676i
$604$		0			0
$605$	0.829735	+	0.452035i	0.0337335	+	0.0183778i
$606$		0			0
$607$			0.673496i			0.0273364i	0.999907	+	0.0136682i	$0.00435085\pi$
$607$			0.673496i			0.0273364i	−0.999907	+	0.0136682i	$0.995649\pi$
$608$		0			0
$609$	0.504595			0.0204472
$610$		0			0
$611$	−13.6358			−0.551645
$612$		0			0
$613$		−	26.4615i		−	1.06877i	−0.845241	−	0.534385i	$-0.820543\pi$
$613$		−	26.4615i		−	1.06877i	0.845241	−	0.534385i	$-0.179457\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			27.9538i			1.12538i	0.826669	+	0.562688i	$0.190233\pi$
$617$			27.9538i			1.12538i	−0.826669	+	0.562688i	$0.809767\pi$
$618$		0			0
$619$	24.6512			0.990814			0.495407	−	0.868661i	$-0.335019\pi$
$619$	24.6512			0.990814			0.495407	+	0.868661i	$0.335019\pi$
$620$		0			0
$621$	11.8543			0.475697
$622$		0			0
$623$			15.9002i			0.637028i
$624$		0			0
$625$	−10.2979	+	22.7805i	−0.411916	+	0.911222i
$626$		0			0
$627$			0.592376i			0.0236572i
$628$		0			0
$629$	4.50459			0.179610
$630$		0			0
$631$	−11.8052			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$631$	−11.8052			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$632$		0			0
$633$			4.88439i			0.194137i
$634$		0			0
$635$	−19.9763	+	36.6676i	−0.792736	+	1.45511i
$636$		0			0
$637$		−	11.9199i		−	0.472283i
$638$		0			0
$639$	−5.85431			−0.231593
$640$		0			0
$641$	13.8727			0.547938			0.273969	−	0.961738i	$-0.411663\pi$
$641$	13.8727			0.547938			0.273969	+	0.961738i	$0.411663\pi$
$642$		0			0
$643$			5.29047i			0.208636i	0.994544	+	0.104318i	$0.0332660\pi$
$643$			5.29047i			0.208636i	−0.994544	+	0.104318i	$0.966734\pi$
$644$		0			0
$645$	1.83859	+	1.00165i	0.0723944	+	0.0394401i
$646$		0			0
$647$		−	40.7448i		−	1.60184i	−0.598769	−	0.800921i	$-0.704343\pi$
$647$		−	40.7448i		−	1.60184i	0.598769	−	0.800921i	$-0.295657\pi$
$648$		0			0
$649$	32.5230			1.27664
$650$		0			0
$651$	−2.65028			−0.103873
$652$		0			0
$653$		−	26.8742i		−	1.05167i	−0.850587	−	0.525834i	$-0.823753\pi$
$653$		−	26.8742i		−	1.05167i	0.850587	−	0.525834i	$-0.176247\pi$
$654$		0			0
$655$	−21.6657	−	11.8033i	−0.846548	−	0.461195i
$656$		0			0
$657$			1.77937i			0.0694197i
$658$		0			0
$659$	29.1558			1.13575			0.567874	−	0.823116i	$-0.307766\pi$
$659$	29.1558			1.13575			0.567874	+	0.823116i	$0.307766\pi$
$660$		0			0
$661$	19.9035			0.774155			0.387078	−	0.922047i	$-0.373485\pi$
$661$	19.9035			0.774155			0.387078	+	0.922047i	$0.373485\pi$
$662$		0			0
$663$			2.21174i			0.0858969i
$664$		0			0
$665$	−1.34972	+	2.47748i	−0.0523399	+	0.0960727i
$666$		0			0
$667$			7.41031i			0.286928i
$668$		0			0
$669$	−1.70863			−0.0660594
$670$		0			0
$671$	38.5538			1.48835
$672$		0			0
$673$			8.71383i			0.335893i	0.985796	+	0.167947i	$0.0537136\pi$
$673$			8.71383i			0.335893i	−0.985796	+	0.167947i	$0.946286\pi$
$674$		0			0
$675$	6.72047	−	4.33725i	0.258671	−	0.166941i
$676$		0			0
$677$			9.70565i			0.373018i	0.982453	+	0.186509i	$0.0597174\pi$
$677$			9.70565i			0.373018i	−0.982453	+	0.186509i	$0.940283\pi$
$678$		0			0
$679$	24.5538			0.942287
$680$		0			0
$681$	6.71782			0.257427
$682$		0			0
$683$		−	9.53014i		−	0.364661i	−0.983237	−	0.182330i	$-0.941636\pi$
$683$		−	9.53014i		−	0.364661i	0.983237	−	0.182330i	$-0.0583640\pi$
$684$		0			0
$685$	12.6994	−	23.3105i	0.485221	−	0.890648i
$686$		0			0
$687$			0.579108i			0.0220944i
$688$		0			0
$689$	−2.98550			−0.113739
$690$		0			0
$691$	−8.50459			−0.323530			−0.161765	−	0.986829i	$-0.551719\pi$
$691$	−8.50459			−0.323530			−0.161765	+	0.986829i	$0.551719\pi$
$692$		0			0
$693$			17.7986i			0.676112i
$694$		0			0
$695$	−29.1496	−	15.8805i	−1.10571	−	0.602383i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	1.44094			0.0545014
$700$		0			0
$701$	−28.6266			−1.08121			−0.540606	−	0.841276i	$-0.681805\pi$
$701$	−28.6266			−1.08121			−0.540606	+	0.841276i	$0.681805\pi$
$702$		0			0
$703$		−	1.26172i		−	0.0475866i
$704$		0			0
$705$	2.12465	+	1.15750i	0.0800191	+	0.0435940i
$706$		0			0
$707$			9.05845i			0.340678i
$708$		0			0
$709$	−14.9855			−0.562792			−0.281396	−	0.959592i	$-0.590798\pi$
$709$	−14.9855			−0.562792			−0.281396	+	0.959592i	$0.590798\pi$
$710$		0			0
$711$	25.1795			0.944303
$712$		0			0
$713$		−	38.9211i		−	1.45761i
$714$		0			0
$715$	−11.8333	+	21.7206i	−0.442540	+	0.812305i
$716$		0			0
$717$			0.539789i			0.0201588i
$718$		0			0
$719$	−36.7486			−1.37049			−0.685246	−	0.728312i	$-0.740305\pi$
$719$	−36.7486			−1.37049			−0.685246	+	0.728312i	$0.740305\pi$
$720$		0			0
$721$	−27.8359			−1.03666
$722$		0			0
$723$			1.82780i			0.0679766i
$724$		0			0
$725$	2.71128	+	4.20107i	0.100694	+	0.156024i
$726$		0			0
$727$			39.1647i			1.45254i	0.687410	+	0.726270i	$0.258748\pi$
$727$			39.1647i			1.45254i	−0.687410	+	0.726270i	$0.741252\pi$
$728$		0			0
$729$	23.1457			0.857248
$730$		0			0
$731$	−8.35891			−0.309165
$732$		0			0
$733$		−	5.38071i		−	0.198741i	−0.995051	−	0.0993705i	$-0.968317\pi$
$733$		−	5.38071i		−	0.198741i	0.995051	−	0.0993705i	$-0.0316829\pi$
$734$		0			0
$735$	−1.01184	+	1.85729i	−0.0373224	+	0.0685072i
$736$		0			0
$737$		−	25.8561i		−	0.952421i
$738$		0			0
$739$	−9.93336			−0.365404			−0.182702	−	0.983168i	$-0.558484\pi$
$739$	−9.93336			−0.365404			−0.182702	+	0.983168i	$0.558484\pi$
$740$		0			0
$741$	0.619500			0.0227579
$742$		0			0
$743$			43.4963i			1.59573i	0.602839	+	0.797863i	$0.294036\pi$
$743$			43.4963i			1.59573i	−0.602839	+	0.797863i	$0.705964\pi$
$744$		0			0
$745$	33.9245	+	18.4819i	1.24290	+	0.677124i
$746$		0			0
$747$		−	7.05266i		−	0.258043i
$748$		0			0
$749$	3.49541			0.127719
$750$		0			0
$751$	−11.3743			−0.415054			−0.207527	−	0.978229i	$-0.566541\pi$
$751$	−11.3743			−0.415054			−0.207527	+	0.978229i	$0.566541\pi$
$752$		0			0
$753$			5.46846i			0.199282i
$754$		0			0
$755$	0.990811	+	0.539789i	0.0360593	+	0.0196449i
$756$		0			0
$757$		−	10.0699i		−	0.365998i	−0.983113	−	0.182999i	$-0.941420\pi$
$757$		−	10.0699i		−	0.365998i	0.983113	−	0.182999i	$-0.0585805\pi$
$758$		0			0
$759$	6.50459			0.236102
$760$		0			0
$761$	−27.0092			−0.979082			−0.489541	−	0.871980i	$-0.662836\pi$
$761$	−27.0092			−0.979082			−0.489541	+	0.871980i	$0.662836\pi$
$762$		0			0
$763$			16.4007i			0.593744i
$764$		0			0
$765$	−7.54456	+	13.8485i	−0.272774	+	0.500692i
$766$		0			0
$767$		−	34.0121i		−	1.22811i
$768$		0			0
$769$	−25.7086			−0.927077			−0.463538	−	0.886077i	$-0.653420\pi$
$769$	−25.7086			−0.927077			−0.463538	+	0.886077i	$0.653420\pi$
$770$		0			0
$771$	5.24610			0.188934
$772$		0			0
$773$		−	32.8185i		−	1.18040i	−0.807257	−	0.590200i	$-0.799049\pi$
$773$		−	32.8185i		−	1.18040i	0.807257	−	0.590200i	$-0.200951\pi$
$774$		0			0
$775$	−14.2405	−	22.0652i	−0.511532	−	0.792607i
$776$		0			0
$777$			0.943390i			0.0338439i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−6.50459			−0.232753
$782$		0			0
$783$		−	1.59971i		−	0.0571688i
$784$		0			0
$785$	−19.2040	+	35.2500i	−0.685421	+	1.25813i
$786$		0			0
$787$			21.1973i			0.755602i	0.925887	+	0.377801i	$0.123320\pi$
$787$			21.1973i			0.755602i	−0.925887	+	0.377801i	$0.876680\pi$
$788$		0			0
$789$	−2.23691			−0.0796360
$790$		0			0
$791$	18.8267			0.669402
$792$		0			0
$793$		−	40.3190i		−	1.43177i
$794$		0			0
$795$	0.465185	+	0.253431i	0.0164984	+	0.00898825i
$796$		0			0
$797$			21.7305i			0.769732i	0.922972	+	0.384866i	$0.125752\pi$
$797$			21.7305i			0.769732i	−0.922972	+	0.384866i	$0.874248\pi$
$798$		0			0
$799$	−9.65947			−0.341727
$800$		0			0
$801$	24.8943			0.879596
$802$		0			0
$803$			1.97701i			0.0697673i
$804$		0			0
$805$	27.2040	+	14.8206i	0.958816	+	0.522358i
$806$		0			0
$807$		−	6.43814i		−	0.226633i
$808$		0			0
$809$	7.20403			0.253280			0.126640	−	0.991949i	$-0.459581\pi$
$809$	7.20403			0.253280			0.126640	+	0.991949i	$0.459581\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		−	0.241121i		−	0.00845647i
$814$		0			0
$815$	−5.83328	+	10.7073i	−0.204331	+	0.375060i
$816$		0			0
$817$			2.34130i			0.0819116i
$818$		0			0
$819$	18.6135			0.650409
$820$		0			0
$821$	24.1220			0.841864			0.420932	−	0.907092i	$-0.361703\pi$
$821$	24.1220			0.841864			0.420932	+	0.907092i	$0.361703\pi$
$822$		0			0
$823$		−	17.2300i		−	0.600600i	−0.953845	−	0.300300i	$-0.902913\pi$
$823$		−	17.2300i		−	0.600600i	0.953845	−	0.300300i	$-0.0970868\pi$
$824$		0			0
$825$	3.68760	−	2.37990i	0.128386	−	0.0828575i
$826$		0			0
$827$		−	3.46931i		−	0.120640i	−0.998179	−	0.0603198i	$-0.980788\pi$
$827$		−	3.46931i		−	0.120640i	0.998179	−	0.0603198i	$-0.0192121\pi$
$828$		0			0
$829$	−7.52619			−0.261395			−0.130698	−	0.991422i	$-0.541722\pi$
$829$	−7.52619			−0.261395			−0.130698	+	0.991422i	$0.541722\pi$
$830$		0			0
$831$	−0.960030			−0.0333031
$832$		0			0
$833$		−	8.44393i		−	0.292565i
$834$		0			0
$835$	1.61741	−	2.96884i	0.0559727	−	0.102741i
$836$		0			0
$837$			8.40213i			0.290420i
$838$		0			0
$839$	−9.61121			−0.331816			−0.165908	−	0.986141i	$-0.553055\pi$
$839$	−9.61121			−0.331816			−0.165908	+	0.986141i	$0.553055\pi$
$840$		0			0
$841$	1.00000			0.0344828
$842$		0			0
$843$		−	2.09521i		−	0.0721630i
$844$		0			0
$845$	−2.81136	−	1.53161i	−0.0967136	−	0.0526891i
$846$		0			0
$847$		−	0.790023i		−	0.0271455i
$848$		0			0
$849$	−8.30975			−0.285190
$850$		0			0
$851$	−13.8543			−0.474920
$852$		0			0
$853$		−	20.8330i		−	0.713309i	−0.934236	−	0.356654i	$-0.883917\pi$
$853$		−	20.8330i		−	0.713309i	0.934236	−	0.356654i	$-0.116083\pi$
$854$		0			0
$855$	3.87889	+	2.11320i	0.132655	+	0.0722699i
$856$		0			0
$857$		−	33.0424i		−	1.12871i	−0.825533	−	0.564354i	$-0.809125\pi$
$857$		−	33.0424i		−	1.12871i	0.825533	−	0.564354i	$-0.190875\pi$
$858$		0			0
$859$	49.8061			1.69936			0.849680	−	0.527298i	$-0.176795\pi$
$859$	49.8061			1.69936			0.849680	+	0.527298i	$0.176795\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	9.03631i		−	0.307599i	−0.988102	−	0.153800i	$-0.950849\pi$
$863$		−	9.03631i		−	0.307599i	0.988102	−	0.153800i	$-0.0491511\pi$
$864$		0			0
$865$	−9.34972	+	17.1619i	−0.317900	+	0.583522i
$866$		0			0
$867$		−	3.02143i		−	0.102613i
$868$		0			0
$869$	27.9763			0.949032
$870$		0			0
$871$	−27.0400			−0.916214
$872$		0			0
$873$		−	38.4428i		−	1.30109i
$874$		0			0
$875$	20.8451	−	1.55127i	0.704694	−	0.0524426i
$876$		0			0
$877$		−	24.0809i		−	0.813153i	−0.913617	−	0.406576i	$-0.866722\pi$
$877$		−	24.0809i		−	0.813153i	0.913617	−	0.406576i	$-0.133278\pi$
$878$		0			0
$879$	1.70863			0.0576306
$880$		0			0
$881$	−0.814344			−0.0274360			−0.0137180	−	0.999906i	$-0.504367\pi$
$881$	−0.814344			−0.0274360			−0.0137180	+	0.999906i	$0.504367\pi$
$882$		0			0
$883$			31.8751i			1.07268i	0.844001	+	0.536341i	$0.180194\pi$
$883$			31.8751i			1.07268i	−0.844001	+	0.536341i	$0.819806\pi$
$884$		0			0
$885$	−2.88719	+	5.29959i	−0.0970517	+	0.178144i
$886$		0			0
$887$			45.6358i			1.53230i	0.642661	+	0.766150i	$0.277830\pi$
$887$			45.6358i			1.53230i	−0.642661	+	0.766150i	$0.722170\pi$
$888$		0			0
$889$	34.9127			1.17093
$890$		0			0
$891$	27.1558			0.909753
$892$		0			0
$893$			2.70558i			0.0905387i
$894$		0			0
$895$	18.6450	+	10.1577i	0.623233	+	0.339534i
$896$		0			0
$897$		−	6.80243i		−	0.227126i
$898$		0			0
$899$	−5.25230			−0.175174
$900$		0			0
$901$	−2.11491			−0.0704577
$902$		0			0
$903$		−	1.75059i		−	0.0582561i
$904$		0			0
$905$	−29.3287	−	15.9781i	−0.974919	−	0.531131i
$906$		0			0
$907$		−	18.5820i		−	0.617004i	−0.951224	−	0.308502i	$-0.900172\pi$
$907$		−	18.5820i		−	0.617004i	0.951224	−	0.308502i	$-0.0998276\pi$
$908$		0			0
$909$	14.1824			0.470402
$910$		0			0
$911$	45.6296			1.51178			0.755888	−	0.654701i	$-0.227206\pi$
$911$	45.6296			1.51178			0.755888	+	0.654701i	$0.227206\pi$
$912$		0			0
$913$		−	7.83605i		−	0.259335i
$914$		0			0
$915$	−3.42256	+	6.28230i	−0.113146	+	0.207686i
$916$		0			0
$917$			20.6287i			0.681221i
$918$		0			0
$919$	24.6994			0.814759			0.407380	−	0.913259i	$-0.366443\pi$
$919$	24.6994			0.814759			0.407380	+	0.913259i	$0.366443\pi$
$920$		0			0
$921$	3.39178			0.111763
$922$		0			0
$923$			6.80243i			0.223905i
$924$		0			0
$925$	−7.85431	+	5.06901i	−0.258248	+	0.166668i
$926$		0			0
$927$			43.5816i			1.43141i
$928$		0			0
$929$	6.00000			0.196854			0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000			0.196854			0.0984268	+	0.995144i	$0.468619\pi$
$930$		0			0
$931$	−2.36511			−0.0775133
$932$		0			0
$933$			6.58096i			0.215451i
$934$		0			0
$935$	−8.38259	+	15.3867i	−0.274140	+	0.503199i
$936$		0			0
$937$			46.4389i			1.51709i	0.651620	+	0.758546i	$0.274090\pi$
$937$			46.4389i			1.51709i	−0.651620	+	0.758546i	$0.725910\pi$
$938$		0			0
$939$	−9.35503			−0.305290
$940$		0			0
$941$	−14.2861			−0.465712			−0.232856	−	0.972511i	$-0.574807\pi$
$941$	−14.2861			−0.465712			−0.232856	+	0.972511i	$0.574807\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−5.87269	−	3.19941i	−0.191039	−	0.104077i
$946$		0			0
$947$		−	29.3713i		−	0.954440i	−0.878784	−	0.477220i	$-0.841644\pi$
$947$		−	29.3713i		−	0.954440i	0.878784	−	0.477220i	$-0.158356\pi$
$948$		0			0
$949$	2.06754			0.0671151
$950$		0			0
$951$	6.96003			0.225694
$952$		0			0
$953$			22.5911i			0.731796i	0.930655	+	0.365898i	$0.119238\pi$
$953$			22.5911i			0.731796i	−0.930655	+	0.365898i	$0.880762\pi$
$954$		0			0
$955$	20.6749	+	11.2636i	0.669023	+	0.364480i
$956$		0			0
$957$		−	0.877777i		−	0.0283745i
$958$		0			0
$959$	−22.1948			−0.716709
$960$		0			0
$961$	−3.41337			−0.110109
$962$		0			0
$963$		−	5.47261i		−	0.176353i
$964$		0			0
$965$	19.3989	−	35.6077i	0.624472	−	1.14625i
$966$		0			0
$967$		−	48.4316i		−	1.55746i	−0.627362	−	0.778728i	$-0.715865\pi$
$967$		−	48.4316i		−	1.55746i	0.627362	−	0.778728i	$-0.284135\pi$
$968$		0			0
$969$	0.438848			0.0140978
$970$		0			0
$971$	−47.0829			−1.51096			−0.755482	−	0.655170i	$-0.772597\pi$
$971$	−47.0829			−1.51096			−0.755482	+	0.655170i	$0.772597\pi$
$972$		0			0
$973$			27.7545i			0.889767i
$974$		0			0
$975$	−2.48887	−	3.85644i	−0.0797076	−	0.123505i
$976$		0			0
$977$		−	51.1987i		−	1.63799i	−0.573800	−	0.818995i	$-0.694532\pi$
$977$		−	51.1987i		−	1.63799i	0.573800	−	0.818995i	$-0.305468\pi$
$978$		0			0
$979$	27.6595			0.884000
$980$		0			0
$981$	25.6778			0.819831
$982$		0			0
$983$		−	23.5123i		−	0.749926i	−0.927040	−	0.374963i	$-0.877655\pi$
$983$		−	23.5123i		−	0.749926i	0.927040	−	0.374963i	$-0.122345\pi$
$984$		0			0
$985$	18.1641	−	33.3411i	0.578755	−	1.06234i
$986$		0			0
$987$		−	2.02297i		−	0.0643918i
$988$		0			0
$989$	25.7086			0.817487
$990$		0			0
$991$	7.64109			0.242727			0.121364	−	0.992608i	$-0.461273\pi$
$991$	7.64109			0.242727			0.121364	+	0.992608i	$0.461273\pi$
$992$		0			0
$993$		−	4.57766i		−	0.145268i
$994$		0			0
$995$	−14.7178	−	8.01819i	−0.466586	−	0.254194i
$996$		0			0
$997$			3.71043i			0.117510i	0.998272	+	0.0587552i	$0.0187131\pi$
$997$			3.71043i			0.117510i	−0.998272	+	0.0587552i	$0.981287\pi$
$998$		0			0
$999$	2.99081			0.0946251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.d.g.929.3		6
4.3	odd	2		145.2.b.c.59.1	✓	6
5.4	even	2	inner	2320.2.d.g.929.4		6
12.11	even	2		1305.2.c.h.784.6		6
20.3	even	4		725.2.a.l.1.1		6
20.7	even	4		725.2.a.l.1.6		6
20.19	odd	2		145.2.b.c.59.6	yes	6
60.23	odd	4		6525.2.a.bt.1.6		6
60.47	odd	4		6525.2.a.bt.1.1		6
60.59	even	2		1305.2.c.h.784.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.1	✓	6	4.3	odd	2
145.2.b.c.59.6	yes	6	20.19	odd	2
725.2.a.l.1.1		6	20.3	even	4
725.2.a.l.1.6		6	20.7	even	4
1305.2.c.h.784.1		6	60.59	even	2
1305.2.c.h.784.6		6	12.11	even	2
2320.2.d.g.929.3		6	1.1	even	1	trivial
2320.2.d.g.929.4		6	5.4	even	2	inner
6525.2.a.bt.1.1		6	60.47	odd	4
6525.2.a.bt.1.6		6	60.23	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.269894i q^{3} +(-1.96358 - 1.06975i) q^{5} +1.86960i q^{7} +2.92716 q^{9} +3.25230 q^{11} -3.40121i q^{13} +(-0.288719 + 0.529959i) q^{15} -2.40939i q^{17} -0.674860 q^{19} +0.504595 q^{21} +7.41031i q^{23} +(2.71128 + 4.20107i) q^{25} -1.59971i q^{27} +1.00000 q^{29} -5.25230 q^{31} -0.877777i q^{33} +(2.00000 - 3.67111i) q^{35} +1.86960i q^{37} -0.917968 q^{39} -3.46931i q^{43} +(-5.74770 - 3.13132i) q^{45} -4.00910i q^{47} +3.50459 q^{49} -0.650280 q^{51} -0.877777i q^{53} +(-6.38614 - 3.47914i) q^{55} +0.182141i q^{57} +10.0000 q^{59} +11.8543 q^{61} +5.47261i q^{63} +(-3.63844 + 6.67855i) q^{65} -7.95010i q^{67} +2.00000 q^{69} -2.00000 q^{71} +0.607882i q^{73} +(1.13384 - 0.731759i) q^{75} +6.08050i q^{77} +8.60202 q^{79} +8.34972 q^{81} -2.40939i q^{83} +(-2.57744 + 4.73102i) q^{85} -0.269894i q^{87} +8.50459 q^{89} +6.35891 q^{91} +1.41757i q^{93} +(1.32514 + 0.721929i) q^{95} -13.1332i q^{97} +9.51998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - 12 q^{9} + 10 q^{11} - 7 q^{15} + 16 q^{19} - 16 q^{21} + 11 q^{25} + 6 q^{29} - 22 q^{31} + 12 q^{35} - 14 q^{39} - 44 q^{45} + 2 q^{49} - 44 q^{51} - 13 q^{55} + 60 q^{59} + 12 q^{61}+ \cdots + 16 q^{99}+O(q^{100}) \) 6 * q + 3 * q^5 - 12 * q^9 + 10 * q^11 - 7 * q^15 + 16 * q^19 - 16 * q^21 + 11 * q^25 + 6 * q^29 - 22 * q^31 + 12 * q^35 - 14 * q^39 - 44 * q^45 + 2 * q^49 - 44 * q^51 - 13 * q^55 + 60 * q^59 + 12 * q^61 + 13 * q^65 + 12 * q^69 - 12 * q^71 - 9 * q^75 + 2 * q^79 + 10 * q^81 - 26 * q^85 + 32 * q^89 - 40 * q^91 + 28 * q^95 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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