LMFDB - Embedded newform 2496.2.d.b.1535.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(1535,2496)] 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("2496.1535"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,0,0,-2,0,0] 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:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	2496.1535
Dual form			2496.2.d.b.1535.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.41421i) q^{3} +1.41421i q^{5} +4.24264i q^{7} +(-1.00000 + 2.82843i) q^{9} -1.00000 q^{13} +(2.00000 - 1.41421i) q^{15} -5.65685i q^{17} +4.24264i q^{19} +(6.00000 - 4.24264i) q^{21} +6.00000 q^{23} +3.00000 q^{25} +(5.00000 - 1.41421i) q^{27} -2.82843i q^{29} +4.24264i q^{31} -6.00000 q^{35} -2.00000 q^{37} +(1.00000 + 1.41421i) q^{39} -1.41421i q^{41} +8.48528i q^{43} +(-4.00000 - 1.41421i) q^{45} -12.0000 q^{47} -11.0000 q^{49} +(-8.00000 + 5.65685i) q^{51} +5.65685i q^{53} +(6.00000 - 4.24264i) q^{57} -8.00000 q^{61} +(-12.0000 - 4.24264i) q^{63} -1.41421i q^{65} +4.24264i q^{67} +(-6.00000 - 8.48528i) q^{69} -12.0000 q^{71} +2.00000 q^{73} +(-3.00000 - 4.24264i) q^{75} +8.48528i q^{79} +(-7.00000 - 5.65685i) q^{81} +12.0000 q^{83} +8.00000 q^{85} +(-4.00000 + 2.82843i) q^{87} +7.07107i q^{89} -4.24264i q^{91} +(6.00000 - 4.24264i) q^{93} -6.00000 q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{9} - 2 q^{13} + 4 q^{15} + 12 q^{21} + 12 q^{23} + 6 q^{25} + 10 q^{27} - 12 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{45} - 24 q^{47} - 22 q^{49} - 16 q^{51} + 12 q^{57} - 16 q^{61} - 24 q^{63}+ \cdots - 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^9 - 2 * q^13 + 4 * q^15 + 12 * q^21 + 12 * q^23 + 6 * q^25 + 10 * q^27 - 12 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^45 - 24 * q^47 - 22 * q^49 - 16 * q^51 + 12 * q^57 - 16 * q^61 - 24 * q^63 - 12 * q^69 - 24 * q^71 + 4 * q^73 - 6 * q^75 - 14 * q^81 + 24 * q^83 + 16 * q^85 - 8 * q^87 + 12 * q^93 - 12 * q^95 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$4$		0			0
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$			4.24264i			1.60357i	0.597614	+	0.801784i	$0.296115\pi$
$7$			4.24264i			1.60357i	−0.597614	+	0.801784i	$0.703885\pi$
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	2.00000	−	1.41421i	0.516398	−	0.365148i
$16$		0			0
$17$		−	5.65685i		−	1.37199i	−0.727607	−	0.685994i	$-0.759367\pi$
$17$		−	5.65685i		−	1.37199i	0.727607	−	0.685994i	$-0.240633\pi$
$18$		0			0
$19$			4.24264i			0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$			4.24264i			0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$		0			0
$21$	6.00000	−	4.24264i	1.30931	−	0.925820i
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	5.00000	−	1.41421i	0.962250	−	0.272166i
$28$		0			0
$29$		−	2.82843i		−	0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$		−	2.82843i		−	0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$		0			0
$31$			4.24264i			0.762001i	0.924575	+	0.381000i	$0.124420\pi$
$31$			4.24264i			0.762001i	−0.924575	+	0.381000i	$0.875580\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000			−1.01419
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	1.00000	+	1.41421i	0.160128	+	0.226455i
$40$		0			0
$41$		−	1.41421i		−	0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$		−	1.41421i		−	0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$		0			0
$43$			8.48528i			1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$			8.48528i			1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$	−4.00000	−	1.41421i	−0.596285	−	0.210819i
$46$		0			0
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$		0			0
$49$	−11.0000			−1.57143
$50$		0			0
$51$	−8.00000	+	5.65685i	−1.12022	+	0.792118i
$52$		0			0
$53$			5.65685i			0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$			5.65685i			0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.00000	−	4.24264i	0.794719	−	0.561951i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	−12.0000	−	4.24264i	−1.51186	−	0.534522i
$64$		0			0
$65$		−	1.41421i		−	0.175412i
$66$		0			0
$67$			4.24264i			0.518321i	0.965834	+	0.259161i	$0.0834459\pi$
$67$			4.24264i			0.518321i	−0.965834	+	0.259161i	$0.916554\pi$
$68$		0			0
$69$	−6.00000	−	8.48528i	−0.722315	−	1.02151i
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	−3.00000	−	4.24264i	−0.346410	−	0.489898i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			8.48528i			0.954669i	0.878722	+	0.477334i	$0.158397\pi$
$79$			8.48528i			0.954669i	−0.878722	+	0.477334i	$0.841603\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$	−4.00000	+	2.82843i	−0.428845	+	0.303239i
$88$		0			0
$89$			7.07107i			0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$			7.07107i			0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$		0			0
$91$		−	4.24264i		−	0.444750i
$92$		0			0
$93$	6.00000	−	4.24264i	0.622171	−	0.439941i
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.82843i		−	0.281439i	−0.990050	−	0.140720i	$-0.955058\pi$
$101$		−	2.82843i		−	0.281439i	0.990050	−	0.140720i	$-0.0449416\pi$
$102$		0			0
$103$		−	8.48528i		−	0.836080i	−0.908429	−	0.418040i	$-0.862717\pi$
$103$		−	8.48528i		−	0.836080i	0.908429	−	0.418040i	$-0.137283\pi$
$104$		0			0
$105$	6.00000	+	8.48528i	0.585540	+	0.828079i
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	2.00000	+	2.82843i	0.189832	+	0.268462i
$112$		0			0
$113$		−	14.1421i		−	1.33038i	−0.746674	−	0.665190i	$-0.768350\pi$
$113$		−	14.1421i		−	1.33038i	0.746674	−	0.665190i	$-0.231650\pi$
$114$		0			0
$115$			8.48528i			0.791257i
$116$		0			0
$117$	1.00000	−	2.82843i	0.0924500	−	0.261488i
$118$		0			0
$119$	24.0000			2.20008
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−2.00000	+	1.41421i	−0.180334	+	0.127515i
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$			16.9706i			1.50589i	0.658081	+	0.752947i	$0.271368\pi$
$127$			16.9706i			1.50589i	−0.658081	+	0.752947i	$0.728632\pi$
$128$		0			0
$129$	12.0000	−	8.48528i	1.05654	−	0.747087i
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−18.0000			−1.56080
$134$		0			0
$135$	2.00000	+	7.07107i	0.172133	+	0.608581i
$136$		0			0
$137$			15.5563i			1.32907i	0.747258	+	0.664534i	$0.231370\pi$
$137$			15.5563i			1.32907i	−0.747258	+	0.664534i	$0.768630\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	12.0000	+	16.9706i	1.01058	+	1.42918i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000			0.332182
$146$		0			0
$147$	11.0000	+	15.5563i	0.907265	+	1.28307i
$148$		0			0
$149$			1.41421i			0.115857i	0.998321	+	0.0579284i	$0.0184495\pi$
$149$			1.41421i			0.115857i	−0.998321	+	0.0579284i	$0.981550\pi$
$150$		0			0
$151$		−	4.24264i		−	0.345261i	−0.984987	−	0.172631i	$-0.944773\pi$
$151$		−	4.24264i		−	0.345261i	0.984987	−	0.172631i	$-0.0552267\pi$
$152$		0			0
$153$	16.0000	+	5.65685i	1.29352	+	0.457330i
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$		0			0
$159$	8.00000	−	5.65685i	0.634441	−	0.448618i
$160$		0			0
$161$			25.4558i			2.00620i
$162$		0			0
$163$			4.24264i			0.332309i	0.986100	+	0.166155i	$0.0531351\pi$
$163$			4.24264i			0.332309i	−0.986100	+	0.166155i	$0.946865\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−12.0000	−	4.24264i	−0.917663	−	0.324443i
$172$		0			0
$173$		−	2.82843i		−	0.215041i	−0.994203	−	0.107521i	$-0.965709\pi$
$173$		−	2.82843i		−	0.215041i	0.994203	−	0.107521i	$-0.0342912\pi$
$174$		0			0
$175$			12.7279i			0.962140i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	16.0000			1.18927			0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000			1.18927			0.594635	+	0.803996i	$0.297296\pi$
$182$		0			0
$183$	8.00000	+	11.3137i	0.591377	+	0.836333i
$184$		0			0
$185$		−	2.82843i		−	0.207950i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	6.00000	+	21.2132i	0.436436	+	1.54303i
$190$		0			0
$191$	−6.00000			−0.434145			−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000			−0.434145			−0.217072	+	0.976156i	$0.569651\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$	−2.00000	+	1.41421i	−0.143223	+	0.101274i
$196$		0			0
$197$		−	7.07107i		−	0.503793i	−0.967754	−	0.251896i	$-0.918946\pi$
$197$		−	7.07107i		−	0.503793i	0.967754	−	0.251896i	$-0.0810542\pi$
$198$		0			0
$199$		−	16.9706i		−	1.20301i	−0.798869	−	0.601506i	$-0.794568\pi$
$199$		−	16.9706i		−	1.20301i	0.798869	−	0.601506i	$-0.205432\pi$
$200$		0			0
$201$	6.00000	−	4.24264i	0.423207	−	0.299253i
$202$		0			0
$203$	12.0000			0.842235
$204$		0			0
$205$	2.00000			0.139686
$206$		0			0
$207$	−6.00000	+	16.9706i	−0.417029	+	1.17954i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	8.48528i		−	0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$		−	8.48528i		−	0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$		0			0
$213$	12.0000	+	16.9706i	0.822226	+	1.16280i
$214$		0			0
$215$	−12.0000			−0.818393
$216$		0			0
$217$	−18.0000			−1.22192
$218$		0			0
$219$	−2.00000	−	2.82843i	−0.135147	−	0.191127i
$220$		0			0
$221$			5.65685i			0.380521i
$222$		0			0
$223$			12.7279i			0.852325i	0.904647	+	0.426162i	$0.140135\pi$
$223$			12.7279i			0.852325i	−0.904647	+	0.426162i	$0.859865\pi$
$224$		0			0
$225$	−3.00000	+	8.48528i	−0.200000	+	0.565685i
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			19.7990i			1.29707i	0.761183	+	0.648537i	$0.224619\pi$
$233$			19.7990i			1.29707i	−0.761183	+	0.648537i	$0.775381\pi$
$234$		0			0
$235$		−	16.9706i		−	1.10704i
$236$		0			0
$237$	12.0000	−	8.48528i	0.779484	−	0.551178i
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$		0			0
$243$	−1.00000	+	15.5563i	−0.0641500	+	0.997940i
$244$		0			0
$245$		−	15.5563i		−	0.993859i
$246$		0			0
$247$		−	4.24264i		−	0.269953i
$248$		0			0
$249$	−12.0000	−	16.9706i	−0.760469	−	1.07547i
$250$		0			0
$251$	30.0000			1.89358			0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000			1.89358			0.946792	+	0.321847i	$0.104304\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−8.00000	−	11.3137i	−0.500979	−	0.708492i
$256$		0			0
$257$		−	14.1421i		−	0.882162i	−0.897467	−	0.441081i	$-0.854595\pi$
$257$		−	14.1421i		−	0.882162i	0.897467	−	0.441081i	$-0.145405\pi$
$258$		0			0
$259$		−	8.48528i		−	0.527250i
$260$		0			0
$261$	8.00000	+	2.82843i	0.495188	+	0.175075i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	−8.00000			−0.491436
$266$		0			0
$267$	10.0000	−	7.07107i	0.611990	−	0.432742i
$268$		0			0
$269$		−	19.7990i		−	1.20717i	−0.797300	−	0.603583i	$-0.793739\pi$
$269$		−	19.7990i		−	1.20717i	0.797300	−	0.603583i	$-0.206261\pi$
$270$		0			0
$271$		−	21.2132i		−	1.28861i	−0.764768	−	0.644305i	$-0.777147\pi$
$271$		−	21.2132i		−	1.28861i	0.764768	−	0.644305i	$-0.222853\pi$
$272$		0			0
$273$	−6.00000	+	4.24264i	−0.363137	+	0.256776i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−26.0000			−1.56219			−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000			−1.56219			−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	−12.0000	−	4.24264i	−0.718421	−	0.254000i
$280$		0			0
$281$			15.5563i			0.928014i	0.885832	+	0.464007i	$0.153589\pi$
$281$			15.5563i			0.928014i	−0.885832	+	0.464007i	$0.846411\pi$
$282$		0			0
$283$			16.9706i			1.00880i	0.863472	+	0.504398i	$0.168285\pi$
$283$			16.9706i			1.00880i	−0.863472	+	0.504398i	$0.831715\pi$
$284$		0			0
$285$	6.00000	+	8.48528i	0.355409	+	0.502625i
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$	10.0000	+	14.1421i	0.586210	+	0.829027i
$292$		0			0
$293$			1.41421i			0.0826192i	0.999146	+	0.0413096i	$0.0131530\pi$
$293$			1.41421i			0.0826192i	−0.999146	+	0.0413096i	$0.986847\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	−36.0000			−2.07501
$302$		0			0
$303$	−4.00000	+	2.82843i	−0.229794	+	0.162489i
$304$		0			0
$305$		−	11.3137i		−	0.647821i
$306$		0			0
$307$			12.7279i			0.726421i	0.931707	+	0.363210i	$0.118319\pi$
$307$			12.7279i			0.726421i	−0.931707	+	0.363210i	$0.881681\pi$
$308$		0			0
$309$	−12.0000	+	8.48528i	−0.682656	+	0.482711i
$310$		0			0
$311$	−6.00000			−0.340229			−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000			−0.340229			−0.170114	+	0.985424i	$0.554414\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$	6.00000	−	16.9706i	0.338062	−	0.956183i
$316$		0			0
$317$			9.89949i			0.556011i	0.960579	+	0.278006i	$0.0896734\pi$
$317$			9.89949i			0.556011i	−0.960579	+	0.278006i	$0.910327\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	12.0000	+	16.9706i	0.669775	+	0.947204i
$322$		0			0
$323$	24.0000			1.33540
$324$		0			0
$325$	−3.00000			−0.166410
$326$		0			0
$327$	2.00000	+	2.82843i	0.110600	+	0.156412i
$328$		0			0
$329$		−	50.9117i		−	2.80685i
$330$		0			0
$331$			21.2132i			1.16598i	0.812478	+	0.582992i	$0.198118\pi$
$331$			21.2132i			1.16598i	−0.812478	+	0.582992i	$0.801882\pi$
$332$		0			0
$333$	2.00000	−	5.65685i	0.109599	−	0.309994i
$334$		0			0
$335$	−6.00000			−0.327815
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	−20.0000	+	14.1421i	−1.08625	+	0.768095i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$	12.0000	−	8.48528i	0.646058	−	0.456832i
$346$		0			0
$347$	18.0000			0.966291			0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000			0.966291			0.483145	+	0.875540i	$0.339494\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	−5.00000	+	1.41421i	−0.266880	+	0.0754851i
$352$		0			0
$353$		−	9.89949i		−	0.526897i	−0.964673	−	0.263448i	$-0.915140\pi$
$353$		−	9.89949i		−	0.526897i	0.964673	−	0.263448i	$-0.0848599\pi$
$354$		0			0
$355$		−	16.9706i		−	0.900704i
$356$		0			0
$357$	−24.0000	−	33.9411i	−1.27021	−	1.79635i
$358$		0			0
$359$	36.0000			1.90001			0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000			1.90001			0.950004	+	0.312239i	$0.101079\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$	11.0000	+	15.5563i	0.577350	+	0.816497i
$364$		0			0
$365$			2.82843i			0.148047i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	4.00000	+	1.41421i	0.208232	+	0.0736210i
$370$		0			0
$371$	−24.0000			−1.24602
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$	16.0000	−	11.3137i	0.826236	−	0.584237i
$376$		0			0
$377$			2.82843i			0.145671i
$378$		0			0
$379$		−	29.6985i		−	1.52551i	−0.646688	−	0.762754i	$-0.723847\pi$
$379$		−	29.6985i		−	1.52551i	0.646688	−	0.762754i	$-0.276153\pi$
$380$		0			0
$381$	24.0000	−	16.9706i	1.22956	−	0.869428i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−24.0000	−	8.48528i	−1.21999	−	0.431331i
$388$		0			0
$389$		−	28.2843i		−	1.43407i	−0.697037	−	0.717035i	$-0.745499\pi$
$389$		−	28.2843i		−	1.43407i	0.697037	−	0.717035i	$-0.254501\pi$
$390$		0			0
$391$		−	33.9411i		−	1.71648i
$392$		0			0
$393$	12.0000	+	16.9706i	0.605320	+	0.856052i
$394$		0			0
$395$	−12.0000			−0.603786
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$	18.0000	+	25.4558i	0.901127	+	1.27439i
$400$		0			0
$401$		−	1.41421i		−	0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$		−	1.41421i		−	0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$		0			0
$403$		−	4.24264i		−	0.211341i
$404$		0			0
$405$	8.00000	−	9.89949i	0.397523	−	0.491910i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−22.0000			−1.08783			−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000			−1.08783			−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	22.0000	−	15.5563i	1.08518	−	0.767338i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			16.9706i			0.833052i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	12.0000	−	33.9411i	0.583460	−	1.65027i
$424$		0			0
$425$		−	16.9706i		−	0.823193i
$426$		0			0
$427$		−	33.9411i		−	1.64253i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.0000			−1.15604			−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000			−1.15604			−0.578020	+	0.816023i	$0.696174\pi$
$432$		0			0
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$		0			0
$435$	−4.00000	−	5.65685i	−0.191785	−	0.271225i
$436$		0			0
$437$			25.4558i			1.21772i
$438$		0			0
$439$			25.4558i			1.21494i	0.794342	+	0.607471i	$0.207816\pi$
$439$			25.4558i			1.21494i	−0.794342	+	0.607471i	$0.792184\pi$
$440$		0			0
$441$	11.0000	−	31.1127i	0.523810	−	1.48156i
$442$		0			0
$443$	12.0000			0.570137			0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000			0.570137			0.285069	+	0.958507i	$0.407984\pi$
$444$		0			0
$445$	−10.0000			−0.474045
$446$		0			0
$447$	2.00000	−	1.41421i	0.0945968	−	0.0668900i
$448$		0			0
$449$			32.5269i			1.53504i	0.641025	+	0.767520i	$0.278509\pi$
$449$			32.5269i			1.53504i	−0.641025	+	0.767520i	$0.721491\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−6.00000	+	4.24264i	−0.281905	+	0.199337i
$454$		0			0
$455$	6.00000			0.281284
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$	−8.00000	−	28.2843i	−0.373408	−	1.32020i
$460$		0			0
$461$			35.3553i			1.64666i	0.567561	+	0.823331i	$0.307887\pi$
$461$			35.3553i			1.64666i	−0.567561	+	0.823331i	$0.692113\pi$
$462$		0			0
$463$			4.24264i			0.197172i	0.995129	+	0.0985861i	$0.0314320\pi$
$463$			4.24264i			0.197172i	−0.995129	+	0.0985861i	$0.968568\pi$
$464$		0			0
$465$	6.00000	+	8.48528i	0.278243	+	0.393496i
$466$		0			0
$467$	−18.0000			−0.832941			−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000			−0.832941			−0.416470	+	0.909149i	$0.636733\pi$
$468$		0			0
$469$	−18.0000			−0.831163
$470$		0			0
$471$	−4.00000	−	5.65685i	−0.184310	−	0.260654i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			12.7279i			0.583997i
$476$		0			0
$477$	−16.0000	−	5.65685i	−0.732590	−	0.259010i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$		0			0
$483$	36.0000	−	25.4558i	1.63806	−	1.15828i
$484$		0			0
$485$		−	14.1421i		−	0.642161i
$486$		0			0
$487$		−	4.24264i		−	0.192252i	−0.995369	−	0.0961262i	$-0.969355\pi$
$487$		−	4.24264i		−	0.192252i	0.995369	−	0.0961262i	$-0.0306452\pi$
$488$		0			0
$489$	6.00000	−	4.24264i	0.271329	−	0.191859i
$490$		0			0
$491$	−6.00000			−0.270776			−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000			−0.270776			−0.135388	+	0.990793i	$0.543228\pi$
$492$		0			0
$493$	−16.0000			−0.720604
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	50.9117i		−	2.28370i
$498$		0			0
$499$		−	4.24264i		−	0.189927i	−0.995481	−	0.0949633i	$-0.969727\pi$
$499$		−	4.24264i		−	0.189927i	0.995481	−	0.0949633i	$-0.0302734\pi$
$500$		0			0
$501$	12.0000	+	16.9706i	0.536120	+	0.758189i
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	4.00000			0.177998
$506$		0			0
$507$	−1.00000	−	1.41421i	−0.0444116	−	0.0628074i
$508$		0			0
$509$		−	24.0416i		−	1.06563i	−0.846233	−	0.532813i	$-0.821135\pi$
$509$		−	24.0416i		−	1.06563i	0.846233	−	0.532813i	$-0.178865\pi$
$510$		0			0
$511$			8.48528i			0.375367i
$512$		0			0
$513$	6.00000	+	21.2132i	0.264906	+	0.936586i
$514$		0			0
$515$	12.0000			0.528783
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−4.00000	+	2.82843i	−0.175581	+	0.124154i
$520$		0			0
$521$			19.7990i			0.867409i	0.901055	+	0.433705i	$0.142794\pi$
$521$			19.7990i			0.867409i	−0.901055	+	0.433705i	$0.857206\pi$
$522$		0			0
$523$		−	16.9706i		−	0.742071i	−0.928619	−	0.371035i	$-0.879003\pi$
$523$		−	16.9706i		−	0.742071i	0.928619	−	0.371035i	$-0.120997\pi$
$524$		0			0
$525$	18.0000	−	12.7279i	0.785584	−	0.555492i
$526$		0			0
$527$	24.0000			1.04546
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$			1.41421i			0.0612564i
$534$		0			0
$535$		−	16.9706i		−	0.733701i
$536$		0			0
$537$	12.0000	+	16.9706i	0.517838	+	0.732334i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	−16.0000	−	22.6274i	−0.686626	−	0.971035i
$544$		0			0
$545$		−	2.82843i		−	0.121157i
$546$		0			0
$547$			16.9706i			0.725609i	0.931865	+	0.362804i	$0.118181\pi$
$547$			16.9706i			0.725609i	−0.931865	+	0.362804i	$0.881819\pi$
$548$		0			0
$549$	8.00000	−	22.6274i	0.341432	−	0.965715i
$550$		0			0
$551$	12.0000			0.511217
$552$		0			0
$553$	−36.0000			−1.53088
$554$		0			0
$555$	−4.00000	+	2.82843i	−0.169791	+	0.120060i
$556$		0			0
$557$		−	7.07107i		−	0.299611i	−0.988716	−	0.149805i	$-0.952135\pi$
$557$		−	7.07107i		−	0.299611i	0.988716	−	0.149805i	$-0.0478647\pi$
$558$		0			0
$559$		−	8.48528i		−	0.358889i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$	20.0000			0.841406
$566$		0			0
$567$	24.0000	−	29.6985i	1.00791	−	1.24722i
$568$		0			0
$569$			2.82843i			0.118574i	0.998241	+	0.0592869i	$0.0188827\pi$
$569$			2.82843i			0.118574i	−0.998241	+	0.0592869i	$0.981117\pi$
$570$		0			0
$571$		−	16.9706i		−	0.710196i	−0.934829	−	0.355098i	$-0.884448\pi$
$571$		−	16.9706i		−	0.710196i	0.934829	−	0.355098i	$-0.115552\pi$
$572$		0			0
$573$	6.00000	+	8.48528i	0.250654	+	0.354478i
$574$		0			0
$575$	18.0000			0.750652
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$	−14.0000	−	19.7990i	−0.581820	−	0.822818i
$580$		0			0
$581$			50.9117i			2.11217i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	4.00000	+	1.41421i	0.165380	+	0.0584705i
$586$		0			0
$587$	36.0000			1.48588			0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000			1.48588			0.742940	+	0.669359i	$0.233431\pi$
$588$		0			0
$589$	−18.0000			−0.741677
$590$		0			0
$591$	−10.0000	+	7.07107i	−0.411345	+	0.290865i
$592$		0			0
$593$			7.07107i			0.290374i	0.989404	+	0.145187i	$0.0463784\pi$
$593$			7.07107i			0.290374i	−0.989404	+	0.145187i	$0.953622\pi$
$594$		0			0
$595$			33.9411i			1.39145i
$596$		0			0
$597$	−24.0000	+	16.9706i	−0.982255	+	0.694559i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$		0			0
$603$	−12.0000	−	4.24264i	−0.488678	−	0.172774i
$604$		0			0
$605$		−	15.5563i		−	0.632456i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$	−12.0000	−	16.9706i	−0.486265	−	0.687682i
$610$		0			0
$611$	12.0000			0.485468
$612$		0			0
$613$	34.0000			1.37325			0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000			1.37325			0.686624	+	0.727013i	$0.259092\pi$
$614$		0			0
$615$	−2.00000	−	2.82843i	−0.0806478	−	0.114053i
$616$		0			0
$617$		−	1.41421i		−	0.0569341i	−0.999595	−	0.0284670i	$-0.990937\pi$
$617$		−	1.41421i		−	0.0569341i	0.999595	−	0.0284670i	$-0.00906257\pi$
$618$		0			0
$619$			4.24264i			0.170526i	0.996358	+	0.0852631i	$0.0271731\pi$
$619$			4.24264i			0.170526i	−0.996358	+	0.0852631i	$0.972827\pi$
$620$		0			0
$621$	30.0000	−	8.48528i	1.20386	−	0.340503i
$622$		0			0
$623$	−30.0000			−1.20192
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			11.3137i			0.451107i
$630$		0			0
$631$			21.2132i			0.844484i	0.906483	+	0.422242i	$0.138757\pi$
$631$			21.2132i			0.844484i	−0.906483	+	0.422242i	$0.861243\pi$
$632$		0			0
$633$	−12.0000	+	8.48528i	−0.476957	+	0.337260i
$634$		0			0
$635$	−24.0000			−0.952411
$636$		0			0
$637$	11.0000			0.435836
$638$		0			0
$639$	12.0000	−	33.9411i	0.474713	−	1.34269i
$640$		0			0
$641$			28.2843i			1.11716i	0.829450	+	0.558581i	$0.188654\pi$
$641$			28.2843i			1.11716i	−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$			12.7279i			0.501940i	0.967995	+	0.250970i	$0.0807496\pi$
$643$			12.7279i			0.501940i	−0.967995	+	0.250970i	$0.919250\pi$
$644$		0			0
$645$	12.0000	+	16.9706i	0.472500	+	0.668215i
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	18.0000	+	25.4558i	0.705476	+	0.997693i
$652$		0			0
$653$			14.1421i			0.553425i	0.960953	+	0.276712i	$0.0892449\pi$
$653$			14.1421i			0.553425i	−0.960953	+	0.276712i	$0.910755\pi$
$654$		0			0
$655$		−	16.9706i		−	0.663095i
$656$		0			0
$657$	−2.00000	+	5.65685i	−0.0780274	+	0.220695i
$658$		0			0
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$		0			0
$663$	8.00000	−	5.65685i	0.310694	−	0.219694i
$664$		0			0
$665$		−	25.4558i		−	0.987135i
$666$		0			0
$667$		−	16.9706i		−	0.657103i
$668$		0			0
$669$	18.0000	−	12.7279i	0.695920	−	0.492090i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−28.0000			−1.07932			−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000			−1.07932			−0.539660	+	0.841883i	$0.681447\pi$
$674$		0			0
$675$	15.0000	−	4.24264i	0.577350	−	0.163299i
$676$		0			0
$677$			22.6274i			0.869642i	0.900517	+	0.434821i	$0.143188\pi$
$677$			22.6274i			0.869642i	−0.900517	+	0.434821i	$0.856812\pi$
$678$		0			0
$679$		−	42.4264i		−	1.62818i
$680$		0			0
$681$	12.0000	+	16.9706i	0.459841	+	0.650313i
$682$		0			0
$683$	48.0000			1.83667			0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000			1.83667			0.918334	+	0.395805i	$0.129534\pi$
$684$		0			0
$685$	−22.0000			−0.840577
$686$		0			0
$687$	14.0000	+	19.7990i	0.534133	+	0.755379i
$688$		0			0
$689$		−	5.65685i		−	0.215509i
$690$		0			0
$691$			12.7279i			0.484193i	0.970252	+	0.242096i	$0.0778351\pi$
$691$			12.7279i			0.484193i	−0.970252	+	0.242096i	$0.922165\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−8.00000			−0.303022
$698$		0			0
$699$	28.0000	−	19.7990i	1.05906	−	0.748867i
$700$		0			0
$701$		−	45.2548i		−	1.70925i	−0.519244	−	0.854626i	$-0.673787\pi$
$701$		−	45.2548i		−	1.70925i	0.519244	−	0.854626i	$-0.326213\pi$
$702$		0			0
$703$		−	8.48528i		−	0.320028i
$704$		0			0
$705$	−24.0000	+	16.9706i	−0.903892	+	0.639148i
$706$		0			0
$707$	12.0000			0.451306
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$	−24.0000	−	8.48528i	−0.900070	−	0.318223i
$712$		0			0
$713$			25.4558i			0.953329i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−12.0000	−	16.9706i	−0.448148	−	0.633777i
$718$		0			0
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$	36.0000			1.34071
$722$		0			0
$723$	−26.0000	−	36.7696i	−0.966950	−	1.36747i
$724$		0			0
$725$		−	8.48528i		−	0.315135i
$726$		0			0
$727$		−	16.9706i		−	0.629403i	−0.949191	−	0.314702i	$-0.898096\pi$
$727$		−	16.9706i		−	0.629403i	0.949191	−	0.314702i	$-0.101904\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$	48.0000			1.77534
$732$		0			0
$733$	−50.0000			−1.84679			−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000			−1.84679			−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$	−22.0000	+	15.5563i	−0.811482	+	0.573805i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	4.24264i		−	0.156068i	−0.996951	−	0.0780340i	$-0.975136\pi$
$739$		−	4.24264i		−	0.156068i	0.996951	−	0.0780340i	$-0.0248643\pi$
$740$		0			0
$741$	−6.00000	+	4.24264i	−0.220416	+	0.155857i
$742$		0			0
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$	−2.00000			−0.0732743
$746$		0			0
$747$	−12.0000	+	33.9411i	−0.439057	+	1.24184i
$748$		0			0
$749$		−	50.9117i		−	1.86027i
$750$		0			0
$751$			8.48528i			0.309632i	0.987943	+	0.154816i	$0.0494785\pi$
$751$			8.48528i			0.309632i	−0.987943	+	0.154816i	$0.950521\pi$
$752$		0			0
$753$	−30.0000	−	42.4264i	−1.09326	−	1.54610i
$754$		0			0
$755$	6.00000			0.218362
$756$		0			0
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	26.8701i		−	0.974039i	−0.873391	−	0.487019i	$-0.838084\pi$
$761$		−	26.8701i		−	0.974039i	0.873391	−	0.487019i	$-0.161916\pi$
$762$		0			0
$763$		−	8.48528i		−	0.307188i
$764$		0			0
$765$	−8.00000	+	22.6274i	−0.289241	+	0.818096i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	−20.0000	+	14.1421i	−0.720282	+	0.509317i
$772$		0			0
$773$			43.8406i			1.57684i	0.615139	+	0.788419i	$0.289100\pi$
$773$			43.8406i			1.57684i	−0.615139	+	0.788419i	$0.710900\pi$
$774$		0			0
$775$			12.7279i			0.457200i
$776$		0			0
$777$	−12.0000	+	8.48528i	−0.430498	+	0.304408i
$778$		0			0
$779$	6.00000			0.214972
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−4.00000	−	14.1421i	−0.142948	−	0.505399i
$784$		0			0
$785$			5.65685i			0.201902i
$786$		0			0
$787$			12.7279i			0.453701i	0.973930	+	0.226851i	$0.0728429\pi$
$787$			12.7279i			0.453701i	−0.973930	+	0.226851i	$0.927157\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	60.0000			2.13335
$792$		0			0
$793$	8.00000			0.284088
$794$		0			0
$795$	8.00000	+	11.3137i	0.283731	+	0.401256i
$796$		0			0
$797$			39.5980i			1.40263i	0.712850	+	0.701316i	$0.247404\pi$
$797$			39.5980i			1.40263i	−0.712850	+	0.701316i	$0.752596\pi$
$798$		0			0
$799$			67.8823i			2.40150i
$800$		0			0
$801$	−20.0000	−	7.07107i	−0.706665	−	0.249844i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−36.0000			−1.26883
$806$		0			0
$807$	−28.0000	+	19.7990i	−0.985647	+	0.696957i
$808$		0			0
$809$			19.7990i			0.696095i	0.937477	+	0.348048i	$0.113155\pi$
$809$			19.7990i			0.696095i	−0.937477	+	0.348048i	$0.886845\pi$
$810$		0			0
$811$		−	38.1838i		−	1.34081i	−0.741994	−	0.670407i	$-0.766120\pi$
$811$		−	38.1838i		−	1.34081i	0.741994	−	0.670407i	$-0.233880\pi$
$812$		0			0
$813$	−30.0000	+	21.2132i	−1.05215	+	0.743980i
$814$		0			0
$815$	−6.00000			−0.210171
$816$		0			0
$817$	−36.0000			−1.25948
$818$		0			0
$819$	12.0000	+	4.24264i	0.419314	+	0.148250i
$820$		0			0
$821$			9.89949i			0.345495i	0.984966	+	0.172747i	$0.0552644\pi$
$821$			9.89949i			0.345495i	−0.984966	+	0.172747i	$0.944736\pi$
$822$		0			0
$823$			25.4558i			0.887335i	0.896191	+	0.443667i	$0.146323\pi$
$823$			25.4558i			0.887335i	−0.896191	+	0.443667i	$0.853677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$	26.0000	+	36.7696i	0.901930	+	1.27552i
$832$		0			0
$833$			62.2254i			2.15598i
$834$		0			0
$835$		−	16.9706i		−	0.587291i
$836$		0			0
$837$	6.00000	+	21.2132i	0.207390	+	0.733236i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	22.0000	−	15.5563i	0.757720	−	0.535789i
$844$		0			0
$845$			1.41421i			0.0486504i
$846$		0			0
$847$		−	46.6690i		−	1.60357i
$848$		0			0
$849$	24.0000	−	16.9706i	0.823678	−	0.582428i
$850$		0			0
$851$	−12.0000			−0.411355
$852$		0			0
$853$	46.0000			1.57501			0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000			1.57501			0.787505	+	0.616308i	$0.211372\pi$
$854$		0			0
$855$	6.00000	−	16.9706i	0.205196	−	0.580381i
$856$		0			0
$857$			2.82843i			0.0966172i	0.998832	+	0.0483086i	$0.0153831\pi$
$857$			2.82843i			0.0966172i	−0.998832	+	0.0483086i	$0.984617\pi$
$858$		0			0
$859$		−	42.4264i		−	1.44757i	−0.690025	−	0.723785i	$-0.742401\pi$
$859$		−	42.4264i		−	1.44757i	0.690025	−	0.723785i	$-0.257599\pi$
$860$		0			0
$861$	−6.00000	−	8.48528i	−0.204479	−	0.289178i
$862$		0			0
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$	4.00000			0.136004
$866$		0			0
$867$	15.0000	+	21.2132i	0.509427	+	0.720438i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	4.24264i		−	0.143756i
$872$		0			0
$873$	10.0000	−	28.2843i	0.338449	−	0.957278i
$874$		0			0
$875$	−48.0000			−1.62270
$876$		0			0
$877$	22.0000			0.742887			0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000			0.742887			0.371444	+	0.928456i	$0.378863\pi$
$878$		0			0
$879$	2.00000	−	1.41421i	0.0674583	−	0.0477002i
$880$		0			0
$881$			19.7990i			0.667045i	0.942742	+	0.333522i	$0.108237\pi$
$881$			19.7990i			0.667045i	−0.942742	+	0.333522i	$0.891763\pi$
$882$		0			0
$883$			50.9117i			1.71331i	0.515886	+	0.856657i	$0.327463\pi$
$883$			50.9117i			1.71331i	−0.515886	+	0.856657i	$0.672537\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.00000			0.201460			0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000			0.201460			0.100730	+	0.994914i	$0.467882\pi$
$888$		0			0
$889$	−72.0000			−2.41480
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	50.9117i		−	1.70369i
$894$		0			0
$895$		−	16.9706i		−	0.567263i
$896$		0			0
$897$	6.00000	+	8.48528i	0.200334	+	0.283315i
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$	32.0000			1.06607
$902$		0			0
$903$	36.0000	+	50.9117i	1.19800	+	1.69423i
$904$		0			0
$905$			22.6274i			0.752161i
$906$		0			0
$907$			50.9117i			1.69049i	0.534375	+	0.845247i	$0.320547\pi$
$907$			50.9117i			1.69049i	−0.534375	+	0.845247i	$0.679453\pi$
$908$		0			0
$909$	8.00000	+	2.82843i	0.265343	+	0.0938130i
$910$		0			0
$911$	42.0000			1.39152			0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000			1.39152			0.695761	+	0.718273i	$0.255067\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−16.0000	+	11.3137i	−0.528944	+	0.374020i
$916$		0			0
$917$		−	50.9117i		−	1.68125i
$918$		0			0
$919$			8.48528i			0.279904i	0.990158	+	0.139952i	$0.0446948\pi$
$919$			8.48528i			0.279904i	−0.990158	+	0.139952i	$0.955305\pi$
$920$		0			0
$921$	18.0000	−	12.7279i	0.593120	−	0.419399i
$922$		0			0
$923$	12.0000			0.394985
$924$		0			0
$925$	−6.00000			−0.197279
$926$		0			0
$927$	24.0000	+	8.48528i	0.788263	+	0.278693i
$928$		0			0
$929$			15.5563i			0.510387i	0.966890	+	0.255194i	$0.0821392\pi$
$929$			15.5563i			0.510387i	−0.966890	+	0.255194i	$0.917861\pi$
$930$		0			0
$931$		−	46.6690i		−	1.52952i
$932$		0			0
$933$	6.00000	+	8.48528i	0.196431	+	0.277796i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	20.0000			0.653372			0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000			0.653372			0.326686	+	0.945133i	$0.394068\pi$
$938$		0			0
$939$	10.0000	+	14.1421i	0.326338	+	0.461511i
$940$		0			0
$941$			1.41421i			0.0461020i	0.999734	+	0.0230510i	$0.00733802\pi$
$941$			1.41421i			0.0461020i	−0.999734	+	0.0230510i	$0.992662\pi$
$942$		0			0
$943$		−	8.48528i		−	0.276319i
$944$		0			0
$945$	−30.0000	+	8.48528i	−0.975900	+	0.276026i
$946$		0			0
$947$	−36.0000			−1.16984			−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000			−1.16984			−0.584921	+	0.811090i	$0.698875\pi$
$948$		0			0
$949$	−2.00000			−0.0649227
$950$		0			0
$951$	14.0000	−	9.89949i	0.453981	−	0.321013i
$952$		0			0
$953$		−	31.1127i		−	1.00784i	−0.863751	−	0.503920i	$-0.831891\pi$
$953$		−	31.1127i		−	1.00784i	0.863751	−	0.503920i	$-0.168109\pi$
$954$		0			0
$955$		−	8.48528i		−	0.274577i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−66.0000			−2.13125
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	12.0000	−	33.9411i	0.386695	−	1.09374i
$964$		0			0
$965$			19.7990i			0.637352i
$966$		0			0
$967$		−	21.2132i		−	0.682171i	−0.940032	−	0.341085i	$-0.889205\pi$
$967$		−	21.2132i		−	0.682171i	0.940032	−	0.341085i	$-0.110795\pi$
$968$		0			0
$969$	−24.0000	−	33.9411i	−0.770991	−	1.09035i
$970$		0			0
$971$	−42.0000			−1.34784			−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000			−1.34784			−0.673922	+	0.738802i	$0.735392\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	3.00000	+	4.24264i	0.0960769	+	0.135873i
$976$		0			0
$977$			49.4975i			1.58356i	0.610803	+	0.791782i	$0.290847\pi$
$977$			49.4975i			1.58356i	−0.610803	+	0.791782i	$0.709153\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.00000	−	5.65685i	0.0638551	−	0.180609i
$982$		0			0
$983$	−12.0000			−0.382741			−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000			−0.382741			−0.191370	+	0.981518i	$0.561293\pi$
$984$		0			0
$985$	10.0000			0.318626
$986$		0			0
$987$	−72.0000	+	50.9117i	−2.29179	+	1.62054i
$988$		0			0
$989$			50.9117i			1.61890i
$990$		0			0
$991$		−	33.9411i		−	1.07818i	−0.842250	−	0.539088i	$-0.818769\pi$
$991$		−	33.9411i		−	1.07818i	0.842250	−	0.539088i	$-0.181231\pi$
$992$		0			0
$993$	30.0000	−	21.2132i	0.952021	−	0.673181i
$994$		0			0
$995$	24.0000			0.760851
$996$		0			0
$997$	46.0000			1.45683			0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000			1.45683			0.728417	+	0.685134i	$0.240256\pi$
$998$		0			0
$999$	−10.0000	+	2.82843i	−0.316386	+	0.0894875i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.d.b.1535.1		2
3.2	odd	2		2496.2.d.g.1535.1		2
4.3	odd	2		2496.2.d.g.1535.2		2
8.3	odd	2		156.2.c.a.131.2	yes	2
8.5	even	2		156.2.c.b.131.2	yes	2
12.11	even	2	inner	2496.2.d.b.1535.2		2
24.5	odd	2		156.2.c.a.131.1	✓	2
24.11	even	2		156.2.c.b.131.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.c.a.131.1	✓	2	24.5	odd	2
156.2.c.a.131.2	yes	2	8.3	odd	2
156.2.c.b.131.1	yes	2	24.11	even	2
156.2.c.b.131.2	yes	2	8.5	even	2
2496.2.d.b.1535.1		2	1.1	even	1	trivial
2496.2.d.b.1535.2		2	12.11	even	2	inner
2496.2.d.g.1535.1		2	3.2	odd	2
2496.2.d.g.1535.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.41421i) q^{3} +1.41421i q^{5} +4.24264i q^{7} +(-1.00000 + 2.82843i) q^{9} -1.00000 q^{13} +(2.00000 - 1.41421i) q^{15} -5.65685i q^{17} +4.24264i q^{19} +(6.00000 - 4.24264i) q^{21} +6.00000 q^{23} +3.00000 q^{25} +(5.00000 - 1.41421i) q^{27} -2.82843i q^{29} +4.24264i q^{31} -6.00000 q^{35} -2.00000 q^{37} +(1.00000 + 1.41421i) q^{39} -1.41421i q^{41} +8.48528i q^{43} +(-4.00000 - 1.41421i) q^{45} -12.0000 q^{47} -11.0000 q^{49} +(-8.00000 + 5.65685i) q^{51} +5.65685i q^{53} +(6.00000 - 4.24264i) q^{57} -8.00000 q^{61} +(-12.0000 - 4.24264i) q^{63} -1.41421i q^{65} +4.24264i q^{67} +(-6.00000 - 8.48528i) q^{69} -12.0000 q^{71} +2.00000 q^{73} +(-3.00000 - 4.24264i) q^{75} +8.48528i q^{79} +(-7.00000 - 5.65685i) q^{81} +12.0000 q^{83} +8.00000 q^{85} +(-4.00000 + 2.82843i) q^{87} +7.07107i q^{89} -4.24264i q^{91} +(6.00000 - 4.24264i) q^{93} -6.00000 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{9} - 2 q^{13} + 4 q^{15} + 12 q^{21} + 12 q^{23} + 6 q^{25} + 10 q^{27} - 12 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{45} - 24 q^{47} - 22 q^{49} - 16 q^{51} + 12 q^{57} - 16 q^{61} - 24 q^{63}+ \cdots - 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^9 - 2 * q^13 + 4 * q^15 + 12 * q^21 + 12 * q^23 + 6 * q^25 + 10 * q^27 - 12 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^45 - 24 * q^47 - 22 * q^49 - 16 * q^51 + 12 * q^57 - 16 * q^61 - 24 * q^63 - 12 * q^69 - 24 * q^71 + 4 * q^73 - 6 * q^75 - 14 * q^81 + 24 * q^83 + 16 * q^85 - 8 * q^87 + 12 * q^93 - 12 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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