LMFDB - Embedded newform 2320.2.d.g.929.1

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			929.1
Root			$-2.30229i$ of defining polynomial
Character	$\chi$	$=$	2320.929
Dual form			2320.2.d.g.929.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89028i q^{3} +(2.17686 + 0.511167i) q^{5} -3.91261i q^{7} -5.35371 q^{9} -2.65427 q^{11} -5.62692i q^{13} +(1.47742 - 6.29173i) q^{15} -1.86794i q^{17} +1.69944 q^{19} -11.3085 q^{21} +0.691975i q^{23} +(4.47742 + 2.22548i) q^{25} +6.80289i q^{27} +1.00000 q^{29} +0.654273 q^{31} +7.67159i q^{33} +(2.00000 - 8.51720i) q^{35} -3.91261i q^{37} -16.2634 q^{39} +10.7155i q^{43} +(-11.6543 - 2.73664i) q^{45} +4.93495i q^{47} -8.30855 q^{49} -5.39888 q^{51} +7.67159i q^{53} +(-5.77797 - 1.35678i) q^{55} -4.91186i q^{57} +10.0000 q^{59} -4.70743 q^{61} +20.9470i q^{63} +(2.87630 - 12.2490i) q^{65} -6.47253i q^{67} +2.00000 q^{69} -2.00000 q^{71} -10.5619i q^{73} +(6.43225 - 12.9410i) q^{75} +10.3851i q^{77} -2.05316 q^{79} +3.60112 q^{81} -1.86794i q^{83} +(0.954832 - 4.06625i) q^{85} -2.89028i q^{87} -3.30855 q^{89} -22.0160 q^{91} -1.89103i q^{93} +(3.69944 + 0.868699i) q^{95} -0.384703i q^{97} +14.2102 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.89028i		−	1.66870i	−0.551232	−	0.834352i	$-0.685842\pi$
$3$		−	2.89028i		−	1.66870i	0.551232	−	0.834352i	$-0.314158\pi$
$4$		0			0
$5$	2.17686	+	0.511167i	0.973520	+	0.228601i
$5$	2.17686	+	0.511167i	0.973520	+	0.228601i
$6$		0			0
$7$		−	3.91261i		−	1.47883i	−0.673250	−	0.739415i	$-0.735102\pi$
$7$		−	3.91261i		−	1.47883i	0.673250	−	0.739415i	$-0.264898\pi$
$8$		0			0
$9$	−5.35371			−1.78457
$10$		0			0
$11$	−2.65427			−0.800294			−0.400147	−	0.916451i	$-0.631041\pi$
$11$	−2.65427			−0.800294			−0.400147	+	0.916451i	$0.631041\pi$
$12$		0			0
$13$		−	5.62692i		−	1.56063i	−0.625388	−	0.780314i	$-0.715059\pi$
$13$		−	5.62692i		−	1.56063i	0.625388	−	0.780314i	$-0.284941\pi$
$14$		0			0
$15$	1.47742	−	6.29173i	0.381467	−	1.62452i
$16$		0			0
$17$		−	1.86794i		−	0.453043i	−0.974006	−	0.226522i	$-0.927265\pi$
$17$		−	1.86794i		−	0.453043i	0.974006	−	0.226522i	$-0.0727354\pi$
$18$		0			0
$19$	1.69944			0.389879			0.194939	−	0.980815i	$-0.437549\pi$
$19$	1.69944			0.389879			0.194939	+	0.980815i	$0.437549\pi$
$20$		0			0
$21$	−11.3085			−2.46773
$22$		0			0
$23$			0.691975i			0.144287i	0.997394	+	0.0721433i	$0.0229839\pi$
$23$			0.691975i			0.144287i	−0.997394	+	0.0721433i	$0.977016\pi$
$24$		0			0
$25$	4.47742	+	2.22548i	0.895483	+	0.445095i
$26$		0			0
$27$			6.80289i			1.30922i
$28$		0			0
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	0.654273			0.117511			0.0587555	−	0.998272i	$-0.481287\pi$
$31$	0.654273			0.117511			0.0587555	+	0.998272i	$0.481287\pi$
$32$		0			0
$33$			7.67159i			1.33545i
$34$		0			0
$35$	2.00000	−	8.51720i	0.338062	−	1.43967i
$36$		0			0
$37$		−	3.91261i		−	0.643230i	−0.946871	−	0.321615i	$-0.895774\pi$
$37$		−	3.91261i		−	0.643230i	0.946871	−	0.321615i	$-0.104226\pi$
$38$		0			0
$39$	−16.2634			−2.60422
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$			10.7155i			1.63410i	0.576567	+	0.817050i	$0.304392\pi$
$43$			10.7155i			1.63410i	−0.576567	+	0.817050i	$0.695608\pi$
$44$		0			0
$45$	−11.6543	−	2.73664i	−1.73732	−	0.407955i
$46$		0			0
$47$			4.93495i			0.719836i	0.932984	+	0.359918i	$0.117195\pi$
$47$			4.93495i			0.719836i	−0.932984	+	0.359918i	$0.882805\pi$
$48$		0			0
$49$	−8.30855			−1.18694
$50$		0			0
$51$	−5.39888			−0.755995
$52$		0			0
$53$			7.67159i			1.05377i	0.849935	+	0.526887i	$0.176641\pi$
$53$			7.67159i			1.05377i	−0.849935	+	0.526887i	$0.823359\pi$
$54$		0			0
$55$	−5.77797	−	1.35678i	−0.779102	−	0.182948i
$56$		0			0
$57$		−	4.91186i		−	0.650592i
$58$		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
$61$	−4.70743			−0.602725			−0.301362	−	0.953510i	$-0.597441\pi$
$61$	−4.70743			−0.602725			−0.301362	+	0.953510i	$0.597441\pi$
$62$		0			0
$63$			20.9470i			2.63908i
$64$		0			0
$65$	2.87630	−	12.2490i	0.356761	−	1.51930i
$66$		0			0
$67$		−	6.47253i		−	0.790746i	−0.918521	−	0.395373i	$-0.870615\pi$
$67$		−	6.47253i		−	0.790746i	0.918521	−	0.395373i	$-0.129385\pi$
$68$		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
$73$		−	10.5619i		−	1.23617i	−0.786110	−	0.618087i	$-0.787908\pi$
$73$		−	10.5619i		−	1.23617i	0.786110	−	0.618087i	$-0.212092\pi$
$74$		0			0
$75$	6.43225	−	12.9410i	0.742732	−	1.49430i
$76$		0			0
$77$			10.3851i			1.18350i
$78$		0			0
$79$	−2.05316			−0.230998			−0.115499	−	0.993308i	$-0.536847\pi$
$79$	−2.05316			−0.230998			−0.115499	+	0.993308i	$0.536847\pi$
$80$		0			0
$81$	3.60112			0.400124
$82$		0			0
$83$		−	1.86794i		−	0.205034i	−0.994731	−	0.102517i	$-0.967310\pi$
$83$		−	1.86794i		−	0.205034i	0.994731	−	0.102517i	$-0.0326895\pi$
$84$		0			0
$85$	0.954832	−	4.06625i	0.103566	−	0.441047i
$86$		0			0
$87$		−	2.89028i		−	0.309870i
$88$		0			0
$89$	−3.30855			−0.350705			−0.175353	−	0.984506i	$-0.556107\pi$
$89$	−3.30855			−0.350705			−0.175353	+	0.984506i	$0.556107\pi$
$90$		0			0
$91$	−22.0160			−2.30790
$92$		0			0
$93$		−	1.89103i		−	0.196091i
$94$		0			0
$95$	3.69944	+	0.868699i	0.379555	+	0.0891266i
$96$		0			0
$97$		−	0.384703i		−	0.0390607i	−0.999809	−	0.0195303i	$-0.993783\pi$
$97$		−	0.384703i		−	0.0390607i	0.999809	−	0.0195303i	$-0.00621710\pi$
$98$		0			0
$99$	14.2102			1.42818
$100$		0			0
$101$	11.9097			1.18506			0.592528	−	0.805550i	$-0.298130\pi$
$101$	11.9097			1.18506			0.592528	+	0.805550i	$0.298130\pi$
$102$		0			0
$103$		−	14.9585i		−	1.47390i	−0.675946	−	0.736951i	$-0.736265\pi$
$103$		−	14.9585i		−	1.47390i	0.675946	−	0.736951i	$-0.263735\pi$
$104$		0			0
$105$	−24.6171	−	5.78056i	−2.40238	−	0.564125i
$106$		0			0
$107$			3.91261i			0.378247i	0.981953	+	0.189123i	$0.0605646\pi$
$107$			3.91261i			0.378247i	−0.981953	+	0.189123i	$0.939435\pi$
$108$		0			0
$109$	7.55595			0.723729			0.361864	−	0.932231i	$-0.382140\pi$
$109$	7.55595			0.723729			0.361864	+	0.932231i	$0.382140\pi$
$110$		0			0
$111$	−11.3085			−1.07336
$112$		0			0
$113$			18.6944i			1.75862i	0.476251	+	0.879309i	$0.341995\pi$
$113$			18.6944i			1.75862i	−0.476251	+	0.879309i	$0.658005\pi$
$114$		0			0
$115$	−0.353715	+	1.50633i	−0.0329841	+	0.140466i
$116$		0			0
$117$			30.1249i			2.78505i
$118$		0			0
$119$	−7.30855			−0.669973
$120$		0			0
$121$	−3.95483			−0.359530
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.60911	+	7.13325i	0.770022	+	0.638018i
$126$		0			0
$127$		−	4.98929i		−	0.442728i	−0.975191	−	0.221364i	$-0.928949\pi$
$127$		−	4.98929i		−	0.442728i	0.975191	−	0.221364i	$-0.0710509\pi$
$128$		0			0
$129$	30.9708			2.72683
$130$		0			0
$131$	−19.7154			−1.72254			−0.861272	−	0.508144i	$-0.830332\pi$
$131$	−19.7154			−1.72254			−0.861272	+	0.508144i	$0.830332\pi$
$132$		0			0
$133$		−	6.64926i		−	0.576564i
$134$		0			0
$135$	−3.47742	+	14.8089i	−0.299288	+	1.27455i
$136$		0			0
$137$		−	6.26455i		−	0.535217i	−0.963528	−	0.267608i	$-0.913767\pi$
$137$		−	6.26455i		−	0.535217i	0.963528	−	0.267608i	$-0.0862334\pi$
$138$		0			0
$139$	21.9097			1.85835			0.929177	−	0.369636i	$-0.120518\pi$
$139$	21.9097			1.85835			0.929177	+	0.369636i	$0.120518\pi$
$140$		0			0
$141$	14.2634			1.20119
$142$		0			0
$143$			14.9354i			1.24896i
$144$		0			0
$145$	2.17686	+	0.511167i	0.180778	+	0.0424501i
$146$		0			0
$147$			24.0140i			1.98064i
$148$		0			0
$149$	−4.24740			−0.347961			−0.173980	−	0.984749i	$-0.555663\pi$
$149$	−4.24740			−0.347961			−0.173980	+	0.984749i	$0.555663\pi$
$150$		0			0
$151$	11.3085			0.920277			0.460138	−	0.887847i	$-0.347800\pi$
$151$	11.3085			0.920277			0.460138	+	0.887847i	$0.347800\pi$
$152$		0			0
$153$			10.0004i			0.808488i
$154$		0			0
$155$	1.42426	+	0.334443i	0.114399	+	0.0268631i
$156$		0			0
$157$		−	4.12059i		−	0.328859i	−0.986389	−	0.164430i	$-0.947422\pi$
$157$		−	4.12059i		−	0.328859i	0.986389	−	0.164430i	$-0.0525783\pi$
$158$		0			0
$159$	22.1730			1.75844
$160$		0			0
$161$	2.70743			0.213375
$162$		0			0
$163$			3.19755i			0.250452i	0.992128	+	0.125226i	$0.0399655\pi$
$163$			3.19755i			0.250452i	−0.992128	+	0.125226i	$0.960034\pi$
$164$		0			0
$165$	−3.92147	+	16.7000i	−0.305286	+	1.30009i
$166$		0			0
$167$		−	14.6050i		−	1.13017i	−0.825032	−	0.565086i	$-0.808843\pi$
$167$		−	14.6050i		−	1.13017i	0.825032	−	0.565086i	$-0.191157\pi$
$168$		0			0
$169$	−18.6623			−1.43556
$170$		0			0
$171$	−9.09832			−0.695766
$172$		0			0
$173$			9.00120i			0.684348i	0.939637	+	0.342174i	$0.111163\pi$
$173$			9.00120i			0.684348i	−0.939637	+	0.342174i	$0.888837\pi$
$174$		0			0
$175$	8.70743	−	17.5184i	0.658220	−	1.32427i
$176$		0			0
$177$		−	28.9028i		−	2.17247i
$178$		0			0
$179$	−21.3085			−1.59268			−0.796338	−	0.604852i	$-0.793232\pi$
$179$	−21.3085			−1.59268			−0.796338	+	0.604852i	$0.793232\pi$
$180$		0			0
$181$	−16.9708			−1.26143			−0.630715	−	0.776014i	$-0.717238\pi$
$181$	−16.9708			−1.26143			−0.630715	+	0.776014i	$0.717238\pi$
$182$		0			0
$183$			13.6058i			1.00577i
$184$		0			0
$185$	2.00000	−	8.51720i	0.147043	−	0.626197i
$186$		0			0
$187$			4.95804i			0.362568i
$188$		0			0
$189$	26.6171			1.93611
$190$		0			0
$191$	8.40687			0.608300			0.304150	−	0.952624i	$-0.401628\pi$
$191$	8.40687			0.608300			0.304150	+	0.952624i	$0.401628\pi$
$192$		0			0
$193$		−	0.791267i		−	0.0569567i	−0.999594	−	0.0284783i	$-0.990934\pi$
$193$		−	0.791267i		−	0.0569567i	0.999594	−	0.0284783i	$-0.00906616\pi$
$194$		0			0
$195$	−35.4031	−	8.31331i	−2.53527	−	0.595328i
$196$		0			0
$197$			24.5062i			1.74599i	0.487726	+	0.872997i	$0.337826\pi$
$197$			24.5062i			1.74599i	−0.487726	+	0.872997i	$0.662174\pi$
$198$		0			0
$199$	19.3085			1.36875			0.684373	−	0.729132i	$-0.260076\pi$
$199$	19.3085			1.36875			0.684373	+	0.729132i	$0.260076\pi$
$200$		0			0
$201$	−18.7074			−1.31952
$202$		0			0
$203$		−	3.91261i		−	0.274612i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	3.70463i		−	0.257490i
$208$		0			0
$209$	−4.51078			−0.312017
$210$		0			0
$211$	−19.2554			−1.32560			−0.662798	−	0.748798i	$-0.730631\pi$
$211$	−19.2554			−1.32560			−0.662798	+	0.748798i	$0.730631\pi$
$212$		0			0
$213$			5.78056i			0.396077i
$214$		0			0
$215$	−5.47742	+	23.3261i	−0.373557	+	1.59083i
$216$		0			0
$217$		−	2.55992i		−	0.173779i
$218$		0			0
$219$	−30.5268			−2.06281
$220$		0			0
$221$	−10.5108			−0.707032
$222$		0			0
$223$			10.8691i			0.727852i	0.931428	+	0.363926i	$0.118564\pi$
$223$			10.8691i			0.727852i	−0.931428	+	0.363926i	$0.881436\pi$
$224$		0			0
$225$	−23.9708	−	11.9146i	−1.59805	−	0.794304i
$226$		0			0
$227$		−	17.3104i		−	1.14893i	−0.818528	−	0.574467i	$-0.805210\pi$
$227$		−	17.3104i		−	1.14893i	0.818528	−	0.574467i	$-0.194790\pi$
$228$		0			0
$229$	−18.7074			−1.23622			−0.618111	−	0.786091i	$-0.712102\pi$
$229$	−18.7074			−1.23622			−0.618111	+	0.786091i	$0.712102\pi$
$230$		0			0
$231$	30.0160			1.97491
$232$		0			0
$233$		−	14.6281i		−	0.958320i	−0.877728	−	0.479160i	$-0.840941\pi$
$233$		−	14.6281i		−	0.958320i	0.877728	−	0.479160i	$-0.159059\pi$
$234$		0			0
$235$	−2.52258	+	10.7427i	−0.164555	+	0.700775i
$236$		0			0
$237$			5.93419i			0.385467i
$238$		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
$241$	−5.55595			−0.357890			−0.178945	−	0.983859i	$-0.557268\pi$
$241$	−5.55595			−0.357890			−0.178945	+	0.983859i	$0.557268\pi$
$242$		0			0
$243$			10.0004i			0.641529i
$244$		0			0
$245$	−18.0865	−	4.24706i	−1.15551	−	0.271335i
$246$		0			0
$247$		−	9.56263i		−	0.608455i
$248$		0			0
$249$	−5.39888			−0.342140
$250$		0			0
$251$	9.27137			0.585204			0.292602	−	0.956234i	$-0.405479\pi$
$251$	9.27137			0.585204			0.292602	+	0.956234i	$0.405479\pi$
$252$		0			0
$253$		−	1.83669i		−	0.115472i
$254$		0			0
$255$	−11.7526	−	2.75973i	−0.735976	−	0.172821i
$256$		0			0
$257$		−	14.1129i		−	0.880337i	−0.897915	−	0.440168i	$-0.854919\pi$
$257$		−	14.1129i		−	0.880337i	0.897915	−	0.440168i	$-0.145081\pi$
$258$		0			0
$259$	−15.3085			−0.951227
$260$		0			0
$261$	−5.35371			−0.331387
$262$		0			0
$263$			6.97962i			0.430382i	0.976572	+	0.215191i	$0.0690373\pi$
$263$			6.97962i			0.430382i	−0.976572	+	0.215191i	$0.930963\pi$
$264$		0			0
$265$	−3.92147	+	16.7000i	−0.240894	+	1.02587i
$266$		0			0
$267$			9.56263i			0.585223i
$268$		0			0
$269$	7.29257			0.444636			0.222318	−	0.974974i	$-0.428638\pi$
$269$	7.29257			0.444636			0.222318	+	0.974974i	$0.428638\pi$
$270$		0			0
$271$	23.3617			1.41912			0.709561	−	0.704644i	$-0.248893\pi$
$271$	23.3617			1.41912			0.709561	+	0.704644i	$0.248893\pi$
$272$		0			0
$273$			63.6323i			3.85120i
$274$		0			0
$275$	−11.8843	−	5.90702i	−0.716649	−	0.356207i
$276$		0			0
$277$			2.91337i			0.175047i	0.996162	+	0.0875236i	$0.0278953\pi$
$277$			2.91337i			0.175047i	−0.996162	+	0.0875236i	$0.972105\pi$
$278$		0			0
$279$	−3.50279			−0.209707
$280$		0			0
$281$	30.1730			1.79997			0.899986	−	0.435918i	$-0.143576\pi$
$281$	30.1730			1.79997			0.899986	+	0.435918i	$0.143576\pi$
$282$		0			0
$283$			2.01341i			0.119685i	0.998208	+	0.0598425i	$0.0190599\pi$
$283$			2.01341i			0.119685i	−0.998208	+	0.0598425i	$0.980940\pi$
$284$		0			0
$285$	2.51078	−	10.6924i	0.148726	−	0.633364i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.5108			0.794752
$290$		0			0
$291$	−1.11190			−0.0651807
$292$		0			0
$293$		−	10.8691i		−	0.634982i	−0.948261	−	0.317491i	$-0.897160\pi$
$293$		−	10.8691i		−	0.634982i	0.948261	−	0.317491i	$-0.102840\pi$
$294$		0			0
$295$	21.7686	+	5.11167i	1.26742	+	0.297613i
$296$		0			0
$297$		−	18.0567i		−	1.04776i
$298$		0			0
$299$	3.89369			0.225178
$300$		0			0
$301$	41.9256			2.41655
$302$		0			0
$303$		−	34.4223i		−	1.97751i
$304$		0			0
$305$	−10.2474	−	2.40628i	−0.586765	−	0.137783i
$306$		0			0
$307$		−	9.02429i		−	0.515043i	−0.966273	−	0.257522i	$-0.917094\pi$
$307$		−	9.02429i		−	0.515043i	0.966273	−	0.257522i	$-0.0829059\pi$
$308$		0			0
$309$	−43.2342			−2.45951
$310$		0			0
$311$	11.1143			0.630234			0.315117	−	0.949053i	$-0.397956\pi$
$311$	11.1143			0.630234			0.315117	+	0.949053i	$0.397956\pi$
$312$		0			0
$313$		−	25.7365i		−	1.45471i	−0.686260	−	0.727356i	$-0.740749\pi$
$313$		−	25.7365i		−	1.45471i	0.686260	−	0.727356i	$-0.259251\pi$
$314$		0			0
$315$	−10.7074	+	45.5987i	−0.603295	+	2.56919i
$316$		0			0
$317$		−	0.837444i		−	0.0470355i	−0.999723	−	0.0235178i	$-0.992513\pi$
$317$		−	0.837444i		−	0.0470355i	0.999723	−	0.0235178i	$-0.00748663\pi$
$318$		0			0
$319$	−2.65427			−0.148611
$320$		0			0
$321$	11.3085			0.631182
$322$		0			0
$323$		−	3.17446i		−	0.176632i
$324$		0			0
$325$	12.5226	−	25.1941i	0.694628	−	1.39752i
$326$		0			0
$327$		−	21.8388i		−	1.20769i
$328$		0			0
$329$	19.3085			1.06451
$330$		0			0
$331$	−22.0691			−1.21303			−0.606515	−	0.795072i	$-0.707433\pi$
$331$	−22.0691			−1.21303			−0.606515	+	0.795072i	$0.707433\pi$
$332$		0			0
$333$			20.9470i			1.14789i
$334$		0			0
$335$	3.30855	−	14.0898i	0.180765	−	0.769807i
$336$		0			0
$337$		−	7.74780i		−	0.422049i	−0.977481	−	0.211025i	$-0.932320\pi$
$337$		−	7.74780i		−	0.422049i	0.977481	−	0.211025i	$-0.0676800\pi$
$338$		0			0
$339$	54.0320			2.93461
$340$		0			0
$341$	−1.73662			−0.0940433
$342$		0			0
$343$			5.11984i			0.276445i
$344$		0			0
$345$	4.35371	+	1.02233i	0.234396	+	0.0550406i
$346$		0			0
$347$			9.33972i			0.501383i	0.968067	+	0.250691i	$0.0806579\pi$
$347$			9.33972i			0.501383i	−0.968067	+	0.250691i	$0.919342\pi$
$348$		0			0
$349$	21.6623			1.15955			0.579777	−	0.814775i	$-0.303140\pi$
$349$	21.6623			1.15955			0.579777	+	0.814775i	$0.303140\pi$
$350$		0			0
$351$	38.2794			2.04320
$352$		0			0
$353$		−	20.5623i		−	1.09442i	−0.836995	−	0.547211i	$-0.815690\pi$
$353$		−	20.5623i		−	1.09442i	0.836995	−	0.547211i	$-0.184310\pi$
$354$		0			0
$355$	−4.35371	−	1.02233i	−0.231071	−	0.0542599i
$356$		0			0
$357$			21.1237i			1.11799i
$358$		0			0
$359$	30.5639			1.61310			0.806551	−	0.591164i	$-0.201331\pi$
$359$	30.5639			1.61310			0.806551	+	0.591164i	$0.201331\pi$
$360$		0			0
$361$	−16.1119			−0.847995
$362$		0			0
$363$			11.4306i			0.599949i
$364$		0			0
$365$	5.39888	−	22.9917i	0.282590	−	1.20344i
$366$		0			0
$367$			13.7825i			0.719441i	0.933060	+	0.359721i	$0.117128\pi$
$367$			13.7825i			0.719441i	−0.933060	+	0.359721i	$0.882872\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	30.0160			1.55835
$372$		0			0
$373$			27.2659i			1.41178i	0.708324	+	0.705888i	$0.249452\pi$
$373$			27.2659i			1.41178i	−0.708324	+	0.705888i	$0.750548\pi$
$374$		0			0
$375$	20.6171	−	24.8827i	1.06466	−	1.28494i
$376$		0			0
$377$		−	5.62692i		−	0.289801i
$378$		0			0
$379$	16.4069			0.842764			0.421382	−	0.906883i	$-0.361545\pi$
$379$	16.4069			0.842764			0.421382	+	0.906883i	$0.361545\pi$
$380$		0			0
$381$	−14.4204			−0.738782
$382$		0			0
$383$		−	11.7378i		−	0.599776i	−0.953974	−	0.299888i	$-0.903051\pi$
$383$		−	11.7378i		−	0.599776i	0.953974	−	0.299888i	$-0.0969493\pi$
$384$		0			0
$385$	−5.30855	+	22.6070i	−0.270549	+	1.15216i
$386$		0			0
$387$		−	57.3678i		−	2.91617i
$388$		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
$391$	1.29257			0.0653681
$392$		0			0
$393$			56.9831i			2.87442i
$394$		0			0
$395$	−4.46943	−	1.04951i	−0.224881	−	0.0528064i
$396$		0			0
$397$			6.03349i			0.302812i	0.988472	+	0.151406i	$0.0483801\pi$
$397$			6.03349i			0.302812i	−0.988472	+	0.151406i	$0.951620\pi$
$398$		0			0
$399$	−19.2182			−0.962114
$400$		0			0
$401$	−21.4656			−1.07194			−0.535971	−	0.844237i	$-0.680054\pi$
$401$	−21.4656			−1.07194			−0.535971	+	0.844237i	$0.680054\pi$
$402$		0			0
$403$		−	3.68155i		−	0.183391i
$404$		0			0
$405$	7.83912	+	1.84077i	0.389529	+	0.0914688i
$406$		0			0
$407$			10.3851i			0.514773i
$408$		0			0
$409$	4.49481			0.222254			0.111127	−	0.993806i	$-0.464554\pi$
$409$	4.49481			0.222254			0.111127	+	0.993806i	$0.464554\pi$
$410$		0			0
$411$	−18.1063			−0.893119
$412$		0			0
$413$		−	39.1261i		−	1.92527i
$414$		0			0
$415$	0.954832	−	4.06625i	0.0468709	−	0.199604i
$416$		0			0
$417$		−	63.3251i		−	3.10104i
$418$		0			0
$419$	4.19665			0.205020			0.102510	−	0.994732i	$-0.467313\pi$
$419$	4.19665			0.205020			0.102510	+	0.994732i	$0.467313\pi$
$420$		0			0
$421$	−18.5108			−0.902160			−0.451080	−	0.892483i	$-0.648961\pi$
$421$	−18.5108			−0.902160			−0.451080	+	0.892483i	$0.648961\pi$
$422$		0			0
$423$		−	26.4203i		−	1.28460i
$424$		0			0
$425$	4.15707	−	8.36357i	0.201647	−	0.405693i
$426$		0			0
$427$			18.4184i			0.891327i
$428$		0			0
$429$	43.1675			2.08414
$430$		0			0
$431$	−37.1279			−1.78839			−0.894193	−	0.447681i	$-0.852250\pi$
$431$	−37.1279			−1.78839			−0.894193	+	0.447681i	$0.852250\pi$
$432$		0			0
$433$			16.8577i			0.810128i	0.914288	+	0.405064i	$0.132751\pi$
$433$			16.8577i			0.810128i	−0.914288	+	0.405064i	$0.867249\pi$
$434$		0			0
$435$	1.47742	−	6.29173i	0.0708367	−	0.301665i
$436$		0			0
$437$			1.17597i			0.0562543i
$438$		0			0
$439$	−9.21821			−0.439961			−0.219981	−	0.975504i	$-0.570599\pi$
$439$	−9.21821			−0.439961			−0.219981	+	0.975504i	$0.570599\pi$
$440$		0			0
$441$	44.4816			2.11817
$442$		0			0
$443$			18.9023i			0.898078i	0.893512	+	0.449039i	$0.148234\pi$
$443$			18.9023i			0.898078i	−0.893512	+	0.449039i	$0.851766\pi$
$444$		0			0
$445$	−7.20223	−	1.69122i	−0.341419	−	0.0801716i
$446$		0			0
$447$			12.2762i			0.580643i
$448$		0			0
$449$	31.3245			1.47830			0.739148	−	0.673543i	$-0.235228\pi$
$449$	31.3245			1.47830			0.739148	+	0.673543i	$0.235228\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	32.6849i		−	1.53567i
$454$		0			0
$455$	−47.9256	−	11.2538i	−2.24679	−	0.527588i
$456$		0			0
$457$		−	36.6287i		−	1.71342i	−0.515799	−	0.856710i	$-0.672505\pi$
$457$		−	36.6287i		−	1.71342i	0.515799	−	0.856710i	$-0.327495\pi$
$458$		0			0
$459$	12.7074			0.593132
$460$		0			0
$461$	28.8297			1.34273			0.671367	−	0.741125i	$-0.265707\pi$
$461$	28.8297			1.34273			0.671367	+	0.741125i	$0.265707\pi$
$462$		0			0
$463$		−	29.6409i		−	1.37753i	−0.724984	−	0.688766i	$-0.758153\pi$
$463$		−	29.6409i		−	1.37753i	0.724984	−	0.688766i	$-0.241847\pi$
$464$		0			0
$465$	0.966634	−	4.11651i	0.0448266	−	0.190899i
$466$		0			0
$467$		−	26.4746i		−	1.22510i	−0.790432	−	0.612550i	$-0.790144\pi$
$467$		−	26.4746i		−	1.22510i	0.790432	−	0.612550i	$-0.209856\pi$
$468$		0			0
$469$	−25.3245			−1.16938
$470$		0			0
$471$	−11.9097			−0.548768
$472$		0			0
$473$		−	28.4419i		−	1.30776i
$474$		0			0
$475$	7.60911	+	3.78207i	0.349130	+	0.173533i
$476$		0			0
$477$		−	41.0715i		−	1.88054i
$478$		0			0
$479$	27.1810			1.24193			0.620967	−	0.783837i	$-0.286740\pi$
$479$	27.1810			1.24193			0.620967	+	0.783837i	$0.286740\pi$
$480$		0			0
$481$	−22.0160			−1.00384
$482$		0			0
$483$		−	7.82523i		−	0.356060i
$484$		0			0
$485$	0.196648	−	0.837444i	0.00892931	−	0.0380264i
$486$		0			0
$487$		−	32.0922i		−	1.45424i	−0.686513	−	0.727118i	$-0.740859\pi$
$487$		−	32.0922i		−	1.45424i	0.686513	−	0.727118i	$-0.259141\pi$
$488$		0			0
$489$	9.24182			0.417929
$490$		0			0
$491$	10.0691			0.454414			0.227207	−	0.973847i	$-0.427041\pi$
$491$	10.0691			0.454414			0.227207	+	0.973847i	$0.427041\pi$
$492$		0			0
$493$		−	1.86794i		−	0.0841280i
$494$		0			0
$495$	30.9336	+	7.26380i	1.39036	+	0.326484i
$496$		0			0
$497$			7.82523i			0.351009i
$498$		0			0
$499$	−29.9416			−1.34037			−0.670185	−	0.742194i	$-0.733785\pi$
$499$	−29.9416			−1.34037			−0.670185	+	0.742194i	$0.733785\pi$
$500$		0			0
$501$	−42.2126			−1.88592
$502$		0			0
$503$		−	24.0140i		−	1.07073i	−0.844620	−	0.535366i	$-0.820174\pi$
$503$		−	24.0140i		−	1.07073i	0.844620	−	0.535366i	$-0.179826\pi$
$504$		0			0
$505$	25.9256	+	6.08783i	1.15368	+	0.270905i
$506$		0			0
$507$			53.9391i			2.39552i
$508$		0			0
$509$	21.0771			0.934227			0.467113	−	0.884197i	$-0.345294\pi$
$509$	21.0771			0.934227			0.467113	+	0.884197i	$0.345294\pi$
$510$		0			0
$511$	−41.3245			−1.82809
$512$		0			0
$513$			11.5611i			0.510436i
$514$		0			0
$515$	7.64629	−	32.5625i	0.336936	−	1.43487i
$516$		0			0
$517$		−	13.0987i		−	0.576080i
$518$		0			0
$519$	26.0160			1.14197
$520$		0			0
$521$	11.6623			0.510933			0.255466	−	0.966818i	$-0.417771\pi$
$521$	11.6623			0.510933			0.255466	+	0.966818i	$0.417771\pi$
$522$		0			0
$523$		−	36.3895i		−	1.59120i	−0.605821	−	0.795601i	$-0.707155\pi$
$523$		−	36.3895i		−	1.59120i	0.605821	−	0.795601i	$-0.292845\pi$
$524$		0			0
$525$	−50.6331	−	25.1669i	−2.20981	−	1.09837i
$526$		0			0
$527$		−	1.22215i		−	0.0532376i
$528$		0			0
$529$	22.5212			0.979181
$530$		0			0
$531$	−53.5371			−2.32331
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−2.00000	+	8.51720i	−0.0864675	+	0.368231i
$536$		0			0
$537$			61.5877i			2.65770i
$538$		0			0
$539$	22.0532			0.949897
$540$		0			0
$541$	9.92564			0.426737			0.213368	−	0.976972i	$-0.431557\pi$
$541$	9.92564			0.426737			0.213368	+	0.976972i	$0.431557\pi$
$542$		0			0
$543$			49.0504i			2.10495i
$544$		0			0
$545$	16.4482	+	3.86235i	0.704565	+	0.165445i
$546$		0			0
$547$		−	9.03245i		−	0.386200i	−0.981179	−	0.193100i	$-0.938146\pi$
$547$		−	9.03245i		−	0.386200i	0.981179	−	0.193100i	$-0.0618541\pi$
$548$		0			0
$549$	25.2022			1.07561
$550$		0			0
$551$	1.69944			0.0723986
$552$		0			0
$553$			8.03321i			0.341607i
$554$		0			0
$555$	−24.6171	−	5.78056i	−1.04494	−	0.245371i
$556$		0			0
$557$			13.7145i			0.581101i	0.956860	+	0.290550i	$0.0938384\pi$
$557$			13.7145i			0.581101i	−0.956860	+	0.290550i	$0.906162\pi$
$558$		0			0
$559$	60.2953			2.55022
$560$		0			0
$561$	14.3301			0.605018
$562$		0			0
$563$		−	20.2781i		−	0.854621i	−0.904105	−	0.427311i	$-0.859461\pi$
$563$		−	20.2781i		−	0.854621i	0.904105	−	0.427311i	$-0.140539\pi$
$564$		0			0
$565$	−9.55595	+	40.6950i	−0.402022	+	1.71205i
$566$		0			0
$567$		−	14.0898i		−	0.591715i
$568$		0			0
$569$	−13.3085			−0.557923			−0.278962	−	0.960302i	$-0.589990\pi$
$569$	−13.3085			−0.557923			−0.278962	+	0.960302i	$0.589990\pi$
$570$		0			0
$571$	16.4204			0.687174			0.343587	−	0.939121i	$-0.388358\pi$
$571$	16.4204			0.687174			0.343587	+	0.939121i	$0.388358\pi$
$572$		0			0
$573$		−	24.2982i		−	1.01507i
$574$		0			0
$575$	−1.53997	+	3.09826i	−0.0642213	+	0.129206i
$576$		0			0
$577$		−	14.7505i		−	0.614071i	−0.951698	−	0.307036i	$-0.900663\pi$
$577$		−	14.7505i		−	0.614071i	0.951698	−	0.307036i	$-0.0993371\pi$
$578$		0			0
$579$	−2.28698			−0.0950438
$580$		0			0
$581$	−7.30855			−0.303210
$582$		0			0
$583$		−	20.3625i		−	0.843329i
$584$		0			0
$585$	−15.3989	+	65.5777i	−0.636665	+	2.71130i
$586$		0			0
$587$			6.36385i			0.262664i	0.991338	+	0.131332i	$0.0419254\pi$
$587$			6.36385i			0.262664i	−0.991338	+	0.131332i	$0.958075\pi$
$588$		0			0
$589$	1.11190			0.0458150
$590$		0			0
$591$	70.8297			2.91355
$592$		0			0
$593$		−	30.0706i		−	1.23485i	−0.786629	−	0.617426i	$-0.788176\pi$
$593$		−	30.0706i		−	1.23485i	0.786629	−	0.617426i	$-0.211824\pi$
$594$		0			0
$595$	−15.9097	−	3.73589i	−0.652233	−	0.153157i
$596$		0			0
$597$		−	55.8071i		−	2.28403i
$598$		0			0
$599$	−25.0587			−1.02387			−0.511936	−	0.859023i	$-0.671072\pi$
$599$	−25.0587			−1.02387			−0.511936	+	0.859023i	$0.671072\pi$
$600$		0			0
$601$	40.0320			1.63294			0.816469	−	0.577390i	$-0.195929\pi$
$601$	40.0320			1.63294			0.816469	+	0.577390i	$0.195929\pi$
$602$		0			0
$603$			34.6521i			1.41114i
$604$		0			0
$605$	−8.60911	−	2.02158i	−0.350010	−	0.0821889i
$606$		0			0
$607$			41.3557i			1.67858i	0.543687	+	0.839288i	$0.317028\pi$
$607$			41.3557i			1.67858i	−0.543687	+	0.839288i	$0.682972\pi$
$608$		0			0
$609$	−11.3085			−0.458245
$610$		0			0
$611$	27.7686			1.12340
$612$		0			0
$613$		−	40.5183i		−	1.63652i	−0.574851	−	0.818258i	$-0.694940\pi$
$613$		−	40.5183i		−	1.63652i	0.574851	−	0.818258i	$-0.305060\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.76865i			0.0712033i	0.999366	+	0.0356016i	$0.0113347\pi$
$617$			1.76865i			0.0712033i	−0.999366	+	0.0356016i	$0.988665\pi$
$618$		0			0
$619$	−0.249804			−0.0100405			−0.00502023	−	0.999987i	$-0.501598\pi$
$619$	−0.249804			−0.0100405			−0.00502023	+	0.999987i	$0.501598\pi$
$620$		0			0
$621$	−4.70743			−0.188903
$622$		0			0
$623$			12.9451i			0.518633i
$624$		0			0
$625$	15.0945	+	19.9288i	0.603780	+	0.797151i
$626$		0			0
$627$			13.0374i			0.520664i
$628$		0			0
$629$	−7.30855			−0.291411
$630$		0			0
$631$	−9.48922			−0.377760			−0.188880	−	0.982000i	$-0.560486\pi$
$631$	−9.48922			−0.377760			−0.188880	+	0.982000i	$0.560486\pi$
$632$		0			0
$633$			55.6535i			2.21203i
$634$		0			0
$635$	2.55036	−	10.8610i	0.101208	−	0.431005i
$636$		0			0
$637$			46.7516i			1.85236i
$638$		0			0
$639$	10.7074			0.423579
$640$		0			0
$641$	−49.9416			−1.97258			−0.986288	−	0.165035i	$-0.947226\pi$
$641$	−49.9416			−1.97258			−0.986288	+	0.165035i	$0.947226\pi$
$642$		0			0
$643$			25.8589i			1.01977i	0.860241	+	0.509887i	$0.170313\pi$
$643$			25.8589i			1.01977i	−0.860241	+	0.509887i	$0.829687\pi$
$644$		0			0
$645$	67.4190	+	15.8313i	2.65462	+	0.623355i
$646$		0			0
$647$			32.1384i			1.26349i	0.775177	+	0.631745i	$0.217661\pi$
$647$			32.1384i			1.26349i	−0.775177	+	0.631745i	$0.782339\pi$
$648$		0			0
$649$	−26.5427			−1.04189
$650$		0			0
$651$	−7.39888			−0.289985
$652$		0			0
$653$			9.79247i			0.383209i	0.981472	+	0.191604i	$0.0613691\pi$
$653$			9.79247i			0.383209i	−0.981472	+	0.191604i	$0.938631\pi$
$654$		0			0
$655$	−42.9177	−	10.0779i	−1.67693	−	0.393775i
$656$		0			0
$657$			56.5452i			2.20604i
$658$		0			0
$659$	−7.55835			−0.294432			−0.147216	−	0.989104i	$-0.547031\pi$
$659$	−7.55835			−0.294432			−0.147216	+	0.989104i	$0.547031\pi$
$660$		0			0
$661$	−10.9041			−0.424119			−0.212060	−	0.977257i	$-0.568017\pi$
$661$	−10.9041			−0.424119			−0.212060	+	0.977257i	$0.568017\pi$
$662$		0			0
$663$			30.3791i			1.17983i
$664$		0			0
$665$	3.39888	−	14.4745i	0.131803	−	0.561296i
$666$		0			0
$667$			0.691975i			0.0267934i
$668$		0			0
$669$	31.4149			1.21457
$670$		0			0
$671$	12.4948			0.482357
$672$		0			0
$673$		−	12.6296i		−	0.486836i	−0.969921	−	0.243418i	$-0.921731\pi$
$673$		−	12.6296i		−	0.486836i	0.969921	−	0.243418i	$-0.0782687\pi$
$674$		0			0
$675$	−15.1397	+	30.4594i	−0.582726	+	1.17238i
$676$		0			0
$677$		−	8.87065i		−	0.340927i	−0.985364	−	0.170463i	$-0.945474\pi$
$677$		−	8.87065i		−	0.340927i	0.985364	−	0.170463i	$-0.0545265\pi$
$678$		0			0
$679$	−1.50519			−0.0577641
$680$		0			0
$681$	−50.0320			−1.91723
$682$		0			0
$683$			24.4749i			0.936507i	0.883594	+	0.468254i	$0.155117\pi$
$683$			24.4749i			0.936507i	−0.883594	+	0.468254i	$0.844883\pi$
$684$		0			0
$685$	3.20223	−	13.6370i	0.122351	−	0.521045i
$686$		0			0
$687$			54.0697i			2.06289i
$688$		0			0
$689$	43.1675			1.64455
$690$		0			0
$691$	3.30855			0.125863			0.0629315	−	0.998018i	$-0.479955\pi$
$691$	3.30855			0.125863			0.0629315	+	0.998018i	$0.479955\pi$
$692$		0			0
$693$		−	55.5991i		−	2.11204i
$694$		0			0
$695$	47.6942	+	11.1995i	1.80914	+	0.424821i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−42.2794			−1.59915
$700$		0			0
$701$	−10.8485			−0.409743			−0.204871	−	0.978789i	$-0.565678\pi$
$701$	−10.8485			−0.409743			−0.204871	+	0.978789i	$0.565678\pi$
$702$		0			0
$703$		−	6.64926i		−	0.250781i
$704$		0			0
$705$	31.0493	+	7.29097i	1.16939	+	0.274594i
$706$		0			0
$707$		−	46.5979i		−	1.75250i
$708$		0			0
$709$	31.1675			1.17052			0.585259	−	0.810846i	$-0.300993\pi$
$709$	31.1675			1.17052			0.585259	+	0.810846i	$0.300993\pi$
$710$		0			0
$711$	10.9920			0.412233
$712$		0			0
$713$			0.452741i			0.0169553i
$714$		0			0
$715$	−7.63448	+	32.5122i	−0.285513	+	1.21589i
$716$		0			0
$717$			5.78056i			0.215879i
$718$		0			0
$719$	−13.0056			−0.485027			−0.242513	−	0.970148i	$-0.577972\pi$
$719$	−13.0056			−0.485027			−0.242513	+	0.970148i	$0.577972\pi$
$720$		0			0
$721$	−58.5268			−2.17965
$722$		0			0
$723$			16.0582i			0.597213i
$724$		0			0
$725$	4.47742	+	2.22548i	0.166287	+	0.0826521i
$726$		0			0
$727$		−	1.19089i		−	0.0441677i	−0.999756	−	0.0220839i	$-0.992970\pi$
$727$		−	1.19089i		−	0.0441677i	0.999756	−	0.0220839i	$-0.00703008\pi$
$728$		0			0
$729$	39.7074			1.47065
$730$		0			0
$731$	20.0160			0.740318
$732$		0			0
$733$			34.5990i			1.27794i	0.769231	+	0.638971i	$0.220640\pi$
$733$			34.5990i			1.27794i	−0.769231	+	0.638971i	$0.779360\pi$
$734$		0			0
$735$	−12.2752	+	52.2751i	−0.452777	+	1.92820i
$736$		0			0
$737$			17.1799i			0.632829i
$738$		0			0
$739$	−41.7821			−1.53698			−0.768491	−	0.639861i	$-0.778992\pi$
$739$	−41.7821			−1.53698			−0.768491	+	0.639861i	$0.778992\pi$
$740$		0			0
$741$	−27.6387			−1.01533
$742$		0			0
$743$			4.02130i			0.147527i	0.997276	+	0.0737636i	$0.0235010\pi$
$743$			4.02130i			0.147527i	−0.997276	+	0.0737636i	$0.976499\pi$
$744$		0			0
$745$	−9.24599	−	2.17113i	−0.338747	−	0.0795442i
$746$		0			0
$747$			10.0004i			0.365897i
$748$		0			0
$749$	15.3085			0.559362
$750$		0			0
$751$	0.497206			0.0181433			0.00907166	−	0.999959i	$-0.497112\pi$
$751$	0.497206			0.0181433			0.00907166	+	0.999959i	$0.497112\pi$
$752$		0			0
$753$		−	26.7968i		−	0.976531i
$754$		0			0
$755$	24.6171	+	5.78056i	0.895908	+	0.210376i
$756$		0			0
$757$			18.6944i			0.679458i	0.940523	+	0.339729i	$0.110335\pi$
$757$			18.6944i			0.679458i	−0.940523	+	0.339729i	$0.889665\pi$
$758$		0			0
$759$	−5.30855			−0.192688
$760$		0			0
$761$	−3.38291			−0.122630			−0.0613151	−	0.998118i	$-0.519529\pi$
$761$	−3.38291			−0.122630			−0.0613151	+	0.998118i	$0.519529\pi$
$762$		0			0
$763$		−	29.5635i		−	1.07027i
$764$		0			0
$765$	−5.11190	+	21.7695i	−0.184821	+	0.787079i
$766$		0			0
$767$		−	56.2692i		−	2.03176i
$768$		0			0
$769$	7.41486			0.267387			0.133693	−	0.991023i	$-0.457316\pi$
$769$	7.41486			0.267387			0.133693	+	0.991023i	$0.457316\pi$
$770$		0			0
$771$	−40.7901			−1.46902
$772$		0			0
$773$		−	33.2775i		−	1.19691i	−0.801156	−	0.598455i	$-0.795782\pi$
$773$		−	33.2775i		−	1.19691i	0.801156	−	0.598455i	$-0.204218\pi$
$774$		0			0
$775$	2.92945	+	1.45607i	0.105229	+	0.0523036i
$776$		0			0
$777$			44.2460i			1.58732i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	5.30855			0.189955
$782$		0			0
$783$			6.80289i			0.243116i
$784$		0			0
$785$	2.10631	−	8.96994i	0.0751775	−	0.320151i
$786$		0			0
$787$			18.2878i			0.651890i	0.945389	+	0.325945i	$0.105682\pi$
$787$			18.2878i			0.651890i	−0.945389	+	0.325945i	$0.894318\pi$
$788$		0			0
$789$	20.1730			0.718179
$790$		0			0
$791$	73.1439			2.60070
$792$		0			0
$793$			26.4883i			0.940629i
$794$		0			0
$795$	48.2676	+	11.3341i	1.71187	+	0.401980i
$796$		0			0
$797$			44.5845i			1.57926i	0.613581	+	0.789632i	$0.289729\pi$
$797$			44.5845i			1.57926i	−0.613581	+	0.789632i	$0.710271\pi$
$798$		0			0
$799$	9.21821			0.326117
$800$		0			0
$801$	17.7130			0.625859
$802$		0			0
$803$			28.0341i			0.989302i
$804$		0			0
$805$	5.89369	+	1.38395i	0.207725	+	0.0487778i
$806$		0			0
$807$		−	21.0776i		−	0.741965i
$808$		0			0
$809$	−14.1063			−0.495952			−0.247976	−	0.968766i	$-0.579765\pi$
$809$	−14.1063			−0.495952			−0.247976	+	0.968766i	$0.579765\pi$
$810$		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
$813$		−	67.5218i		−	2.36809i
$814$		0			0
$815$	−1.63448	+	6.96061i	−0.0572535	+	0.243820i
$816$		0			0
$817$			18.2104i			0.637100i
$818$		0			0
$819$	117.867			4.11862
$820$		0			0
$821$	18.1571			0.633686			0.316843	−	0.948478i	$-0.397377\pi$
$821$	18.1571			0.633686			0.316843	+	0.948478i	$0.397377\pi$
$822$		0			0
$823$		−	3.25189i		−	0.113354i	−0.998393	−	0.0566770i	$-0.981949\pi$
$823$		−	3.25189i		−	0.113354i	0.998393	−	0.0566770i	$-0.0180505\pi$
$824$		0			0
$825$	−17.0729	+	34.3489i	−0.594404	+	1.19588i
$826$		0			0
$827$			10.7155i			0.372615i	0.982492	+	0.186307i	$0.0596520\pi$
$827$			10.7155i			0.372615i	−0.982492	+	0.186307i	$0.940348\pi$
$828$		0			0
$829$	−52.3461			−1.81805			−0.909027	−	0.416736i	$-0.863174\pi$
$829$	−52.3461			−1.81805			−0.909027	+	0.416736i	$0.863174\pi$
$830$		0			0
$831$	8.42045			0.292102
$832$		0			0
$833$			15.5199i			0.537733i
$834$		0			0
$835$	7.46561	−	31.7931i	0.258358	−	1.10024i
$836$		0			0
$837$			4.45095i			0.153847i
$838$		0			0
$839$	24.6703			0.851712			0.425856	−	0.904791i	$-0.359973\pi$
$839$	24.6703			0.851712			0.425856	+	0.904791i	$0.359973\pi$
$840$		0			0
$841$	1.00000			0.0344828
$842$		0			0
$843$		−	87.2085i		−	3.00362i
$844$		0			0
$845$	−40.6251	−	9.53954i	−1.39755	−	0.328170i
$846$		0			0
$847$			15.4737i			0.531684i
$848$		0			0
$849$	5.81933			0.199719
$850$		0			0
$851$	2.70743			0.0928095
$852$		0			0
$853$		−	28.1115i		−	0.962520i	−0.876578	−	0.481260i	$-0.840179\pi$
$853$		−	28.1115i		−	0.962520i	0.876578	−	0.481260i	$-0.159821\pi$
$854$		0			0
$855$	−19.8058	−	4.65077i	−0.677342	−	0.159053i
$856$		0			0
$857$		−	8.39482i		−	0.286762i	−0.989668	−	0.143381i	$-0.954203\pi$
$857$		−	8.39482i		−	0.286762i	0.989668	−	0.143381i	$-0.0457974\pi$
$858$		0			0
$859$	17.8405			0.608711			0.304356	−	0.952559i	$-0.401559\pi$
$859$	17.8405			0.608711			0.304356	+	0.952559i	$0.401559\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			2.48249i			0.0845049i	0.999107	+	0.0422524i	$0.0134534\pi$
$863$			2.48249i			0.0845049i	−0.999107	+	0.0422524i	$0.986547\pi$
$864$		0			0
$865$	−4.60112	+	19.5943i	−0.156443	+	0.666227i
$866$		0			0
$867$		−	39.0499i		−	1.32621i
$868$		0			0
$869$	5.44964			0.184866
$870$		0			0
$871$	−36.4204			−1.23406
$872$		0			0
$873$			2.05959i			0.0697066i
$874$		0			0
$875$	27.9097	−	33.6841i	0.943519	−	1.13873i
$876$		0			0
$877$			25.9813i			0.877325i	0.898652	+	0.438662i	$0.144548\pi$
$877$			25.9813i			0.877325i	−0.898652	+	0.438662i	$0.855452\pi$
$878$		0			0
$879$	−31.4149			−1.05960
$880$		0			0
$881$	25.1279			0.846580			0.423290	−	0.905994i	$-0.360875\pi$
$881$	25.1279			0.846580			0.423290	+	0.905994i	$0.360875\pi$
$882$		0			0
$883$		−	10.9684i		−	0.369117i	−0.982822	−	0.184559i	$-0.940914\pi$
$883$		−	10.9684i		−	0.369117i	0.982822	−	0.184559i	$-0.0590856\pi$
$884$		0			0
$885$	14.7742	−	62.9173i	0.496628	−	2.11494i
$886$		0			0
$887$			2.99897i			0.100695i	0.998732	+	0.0503477i	$0.0160330\pi$
$887$			2.99897i			0.100695i	−0.998732	+	0.0503477i	$0.983967\pi$
$888$		0			0
$889$	−19.5212			−0.654719
$890$		0			0
$891$	−9.55835			−0.320217
$892$		0			0
$893$			8.38665i			0.280649i
$894$		0			0
$895$	−46.3857	−	10.8922i	−1.55050	−	0.364087i
$896$		0			0
$897$		−	11.2538i		−	0.375755i
$898$		0			0
$899$	0.654273			0.0218212
$900$		0			0
$901$	14.3301			0.477405
$902$		0			0
$903$		−	121.177i		−	4.03251i
$904$		0			0
$905$	−36.9430	−	8.67492i	−1.22803	−	0.288364i
$906$		0			0
$907$			50.5567i			1.67871i	0.543585	+	0.839354i	$0.317066\pi$
$907$			50.5567i			1.67871i	−0.543585	+	0.839354i	$0.682934\pi$
$908$		0			0
$909$	−63.7609			−2.11482
$910$		0			0
$911$	−35.9044			−1.18957			−0.594784	−	0.803886i	$-0.702762\pi$
$911$	−35.9044			−1.18957			−0.594784	+	0.803886i	$0.702762\pi$
$912$		0			0
$913$			4.95804i			0.164087i
$914$		0			0
$915$	−6.95483	+	29.6179i	−0.229920	+	0.979136i
$916$		0			0
$917$			77.1388i			2.54735i
$918$		0			0
$919$	15.2022			0.501475			0.250738	−	0.968055i	$-0.419327\pi$
$919$	15.2022			0.501475			0.250738	+	0.968055i	$0.419327\pi$
$920$		0			0
$921$	−26.0827			−0.859454
$922$		0			0
$923$			11.2538i			0.370425i
$924$		0			0
$925$	8.70743	−	17.5184i	0.286299	−	0.576001i
$926$		0			0
$927$			80.0834i			2.63029i
$928$		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
$931$	−14.1199			−0.462761
$932$		0			0
$933$		−	32.1234i		−	1.05167i
$934$		0			0
$935$	−2.53439	+	10.7929i	−0.0828833	+	0.352967i
$936$		0			0
$937$			32.1859i			1.05147i	0.850649	+	0.525734i	$0.176209\pi$
$937$			32.1859i			1.05147i	−0.850649	+	0.525734i	$0.823791\pi$
$938$		0			0
$939$	−74.3857			−2.42748
$940$		0			0
$941$	22.3697			0.729231			0.364616	−	0.931158i	$-0.381200\pi$
$941$	22.3697			0.729231			0.364616	+	0.931158i	$0.381200\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	57.9416	+	13.6058i	1.88484	+	0.442596i
$946$		0			0
$947$			0.122381i			0.00397684i	0.999998	+	0.00198842i	$0.000632935\pi$
$947$			0.122381i			0.00397684i	−0.999998	+	0.00198842i	$0.999367\pi$
$948$		0			0
$949$	−59.4308			−1.92921
$950$		0			0
$951$	−2.42045			−0.0784884
$952$		0			0
$953$		−	55.4917i		−	1.79755i	−0.438409	−	0.898776i	$-0.644458\pi$
$953$		−	55.4917i		−	1.79755i	0.438409	−	0.898776i	$-0.355542\pi$
$954$		0			0
$955$	18.3006	+	4.29732i	0.592192	+	0.139058i
$956$		0			0
$957$			7.67159i			0.247987i
$958$		0			0
$959$	−24.5108			−0.791494
$960$		0			0
$961$	−30.5719			−0.986191
$962$		0			0
$963$		−	20.9470i		−	0.675008i
$964$		0			0
$965$	0.404470	−	1.72248i	0.0130203	−	0.0554485i
$966$		0			0
$967$			49.0722i			1.57806i	0.614357	+	0.789028i	$0.289416\pi$
$967$			49.0722i			1.57806i	−0.614357	+	0.789028i	$0.710584\pi$
$968$		0			0
$969$	−9.17508			−0.294746
$970$		0			0
$971$	−2.08793			−0.0670050			−0.0335025	−	0.999439i	$-0.510666\pi$
$971$	−2.08793			−0.0670050			−0.0335025	+	0.999439i	$0.510666\pi$
$972$		0			0
$973$		−	85.7241i		−	2.74819i
$974$		0			0
$975$	−72.8179	−	36.1938i	−2.33204	−	1.15913i
$976$		0			0
$977$			36.5119i			1.16812i	0.811711	+	0.584059i	$0.198536\pi$
$977$			36.5119i			1.16812i	−0.811711	+	0.584059i	$0.801464\pi$
$978$		0			0
$979$	8.78179			0.280667
$980$		0			0
$981$	−40.4524			−1.29155
$982$		0			0
$983$		−	32.8698i		−	1.04838i	−0.851601	−	0.524191i	$-0.824368\pi$
$983$		−	32.8698i		−	1.04838i	0.851601	−	0.524191i	$-0.175632\pi$
$984$		0			0
$985$	−12.5268	+	53.3465i	−0.399136	+	1.69976i
$986$		0			0
$987$		−	55.8071i		−	1.77636i
$988$		0			0
$989$	−7.41486			−0.235779
$990$		0			0
$991$	36.0160			1.14409			0.572043	−	0.820224i	$-0.306151\pi$
$991$	36.0160			1.14409			0.572043	+	0.820224i	$0.306151\pi$
$992$		0			0
$993$			63.7860i			2.02419i
$994$		0			0
$995$	42.0320	+	9.86990i	1.33250	+	0.312897i
$996$		0			0
$997$			56.8063i			1.79907i	0.436844	+	0.899537i	$0.356096\pi$
$997$			56.8063i			1.79907i	−0.436844	+	0.899537i	$0.643904\pi$
$998$		0			0
$999$	26.6171			0.842128

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.d.g.929.1		6
4.3	odd	2		145.2.b.c.59.2	✓	6
5.4	even	2	inner	2320.2.d.g.929.6		6
12.11	even	2		1305.2.c.h.784.5		6
20.3	even	4		725.2.a.l.1.2		6
20.7	even	4		725.2.a.l.1.5		6
20.19	odd	2		145.2.b.c.59.5	yes	6
60.23	odd	4		6525.2.a.bt.1.5		6
60.47	odd	4		6525.2.a.bt.1.2		6
60.59	even	2		1305.2.c.h.784.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.2	✓	6	4.3	odd	2
145.2.b.c.59.5	yes	6	20.19	odd	2
725.2.a.l.1.2		6	20.3	even	4
725.2.a.l.1.5		6	20.7	even	4
1305.2.c.h.784.2		6	60.59	even	2
1305.2.c.h.784.5		6	12.11	even	2
2320.2.d.g.929.1		6	1.1	even	1	trivial
2320.2.d.g.929.6		6	5.4	even	2	inner
6525.2.a.bt.1.2		6	60.47	odd	4
6525.2.a.bt.1.5		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)

\(f(q)\)	\(=\)	\(q-2.89028i q^{3} +(2.17686 + 0.511167i) q^{5} -3.91261i q^{7} -5.35371 q^{9} -2.65427 q^{11} -5.62692i q^{13} +(1.47742 - 6.29173i) q^{15} -1.86794i q^{17} +1.69944 q^{19} -11.3085 q^{21} +0.691975i q^{23} +(4.47742 + 2.22548i) q^{25} +6.80289i q^{27} +1.00000 q^{29} +0.654273 q^{31} +7.67159i q^{33} +(2.00000 - 8.51720i) q^{35} -3.91261i q^{37} -16.2634 q^{39} +10.7155i q^{43} +(-11.6543 - 2.73664i) q^{45} +4.93495i q^{47} -8.30855 q^{49} -5.39888 q^{51} +7.67159i q^{53} +(-5.77797 - 1.35678i) q^{55} -4.91186i q^{57} +10.0000 q^{59} -4.70743 q^{61} +20.9470i q^{63} +(2.87630 - 12.2490i) q^{65} -6.47253i q^{67} +2.00000 q^{69} -2.00000 q^{71} -10.5619i q^{73} +(6.43225 - 12.9410i) q^{75} +10.3851i q^{77} -2.05316 q^{79} +3.60112 q^{81} -1.86794i q^{83} +(0.954832 - 4.06625i) q^{85} -2.89028i q^{87} -3.30855 q^{89} -22.0160 q^{91} -1.89103i q^{93} +(3.69944 + 0.868699i) q^{95} -0.384703i q^{97} +14.2102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - 12 q^{9} + 10 q^{11} - 7 q^{15} + 16 q^{19} - 16 q^{21} + 11 q^{25} + 6 q^{29} - 22 q^{31} + 12 q^{35} - 14 q^{39} - 44 q^{45} + 2 q^{49} - 44 q^{51} - 13 q^{55} + 60 q^{59} + 12 q^{61}+ \cdots + 16 q^{99}+O(q^{100}) \) 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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