LMFDB - Embedded newform 1344.2.h.e.575.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,2,Mod(575,1344)] 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("1344.575"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,0,0,0,0,0,0,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1344.575
Dual form			1344.2.h.e.575.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.70711 - 0.292893i) q^{3} +0.585786i q^{5} +1.00000i q^{7} +(2.82843 - 1.00000i) q^{9} +2.00000 q^{11} -0.585786 q^{13} +(0.171573 + 1.00000i) q^{15} +0.828427i q^{17} -2.24264i q^{19} +(0.292893 + 1.70711i) q^{21} +4.00000 q^{23} +4.65685 q^{25} +(4.53553 - 2.53553i) q^{27} -0.828427i q^{29} +6.82843i q^{31} +(3.41421 - 0.585786i) q^{33} -0.585786 q^{35} -4.82843 q^{37} +(-1.00000 + 0.171573i) q^{39} +10.0000i q^{41} -6.48528i q^{43} +(0.585786 + 1.65685i) q^{45} +9.65685 q^{47} -1.00000 q^{49} +(0.242641 + 1.41421i) q^{51} -9.31371i q^{53} +1.17157i q^{55} +(-0.656854 - 3.82843i) q^{57} +2.24264 q^{59} -5.75736 q^{61} +(1.00000 + 2.82843i) q^{63} -0.343146i q^{65} +13.3137i q^{67} +(6.82843 - 1.17157i) q^{69} +11.6569 q^{71} -8.82843 q^{73} +(7.94975 - 1.36396i) q^{75} +2.00000i q^{77} +4.00000i q^{79} +(7.00000 - 5.65685i) q^{81} +8.58579 q^{83} -0.485281 q^{85} +(-0.242641 - 1.41421i) q^{87} +3.65685i q^{89} -0.585786i q^{91} +(2.00000 + 11.6569i) q^{93} +1.31371 q^{95} -17.3137 q^{97} +(5.65685 - 2.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 8 q^{11} - 8 q^{13} + 12 q^{15} + 4 q^{21} + 16 q^{23} - 4 q^{25} + 4 q^{27} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 4 q^{39} + 8 q^{45} + 16 q^{47} - 4 q^{49} - 16 q^{51} + 20 q^{57} - 8 q^{59}+ \cdots - 24 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 8 * q^11 - 8 * q^13 + 12 * q^15 + 4 * q^21 + 16 * q^23 - 4 * q^25 + 4 * q^27 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 4 * q^39 + 8 * q^45 + 16 * q^47 - 4 * q^49 - 16 * q^51 + 20 * q^57 - 8 * q^59 - 40 * q^61 + 4 * q^63 + 16 * q^69 + 24 * q^71 - 24 * q^73 + 12 * q^75 + 28 * q^81 + 40 * q^83 + 32 * q^85 + 16 * q^87 + 8 * q^93 - 40 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	1.70711	−	0.292893i	0.985599	−	0.169102i
$3$	1.70711	−	0.292893i	0.985599	−	0.169102i
$4$		0			0
$5$			0.585786i			0.261972i	0.991384	+	0.130986i	$0.0418142\pi$
$5$			0.585786i			0.261972i	−0.991384	+	0.130986i	$0.958186\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.82843	−	1.00000i	0.942809	−	0.333333i
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	−0.585786			−0.162468			−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786			−0.162468			−0.0812340	+	0.996695i	$0.525886\pi$
$14$		0			0
$15$	0.171573	+	1.00000i	0.0442999	+	0.258199i
$16$		0			0
$17$			0.828427i			0.200923i	0.994941	+	0.100462i	$0.0320319\pi$
$17$			0.828427i			0.200923i	−0.994941	+	0.100462i	$0.967968\pi$
$18$		0			0
$19$		−	2.24264i		−	0.514497i	−0.966345	−	0.257249i	$-0.917184\pi$
$19$		−	2.24264i		−	0.514497i	0.966345	−	0.257249i	$-0.0828159\pi$
$20$		0			0
$21$	0.292893	+	1.70711i	0.0639145	+	0.372521i
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$	4.65685			0.931371
$26$		0			0
$27$	4.53553	−	2.53553i	0.872864	−	0.487964i
$28$		0			0
$29$		−	0.828427i		−	0.153835i	−0.997037	−	0.0769175i	$-0.975492\pi$
$29$		−	0.828427i		−	0.153835i	0.997037	−	0.0769175i	$-0.0245078\pi$
$30$		0			0
$31$			6.82843i			1.22642i	0.789919	+	0.613211i	$0.210122\pi$
$31$			6.82843i			1.22642i	−0.789919	+	0.613211i	$0.789878\pi$
$32$		0			0
$33$	3.41421	−	0.585786i	0.594338	−	0.101972i
$34$		0			0
$35$	−0.585786			−0.0990160
$36$		0			0
$37$	−4.82843			−0.793789			−0.396894	−	0.917864i	$-0.629912\pi$
$37$	−4.82843			−0.793789			−0.396894	+	0.917864i	$0.629912\pi$
$38$		0			0
$39$	−1.00000	+	0.171573i	−0.160128	+	0.0274736i
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		−	6.48528i		−	0.988996i	−0.869179	−	0.494498i	$-0.835352\pi$
$43$		−	6.48528i		−	0.988996i	0.869179	−	0.494498i	$-0.164648\pi$
$44$		0			0
$45$	0.585786	+	1.65685i	0.0873239	+	0.246989i
$46$		0			0
$47$	9.65685			1.40860			0.704298	−	0.709904i	$-0.251262\pi$
$47$	9.65685			1.40860			0.704298	+	0.709904i	$0.251262\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.242641	+	1.41421i	0.0339765	+	0.198030i
$52$		0			0
$53$		−	9.31371i		−	1.27934i	−0.768651	−	0.639668i	$-0.779072\pi$
$53$		−	9.31371i		−	1.27934i	0.768651	−	0.639668i	$-0.220928\pi$
$54$		0			0
$55$			1.17157i			0.157975i
$56$		0			0
$57$	−0.656854	−	3.82843i	−0.0870025	−	0.507088i
$58$		0			0
$59$	2.24264			0.291967			0.145983	−	0.989287i	$-0.453365\pi$
$59$	2.24264			0.291967			0.145983	+	0.989287i	$0.453365\pi$
$60$		0			0
$61$	−5.75736			−0.737154			−0.368577	−	0.929597i	$-0.620155\pi$
$61$	−5.75736			−0.737154			−0.368577	+	0.929597i	$0.620155\pi$
$62$		0			0
$63$	1.00000	+	2.82843i	0.125988	+	0.356348i
$64$		0			0
$65$		−	0.343146i		−	0.0425620i
$66$		0			0
$67$			13.3137i			1.62653i	0.581895	+	0.813264i	$0.302312\pi$
$67$			13.3137i			1.62653i	−0.581895	+	0.813264i	$0.697688\pi$
$68$		0			0
$69$	6.82843	−	1.17157i	0.822046	−	0.141041i
$70$		0			0
$71$	11.6569			1.38341			0.691707	−	0.722178i	$-0.256859\pi$
$71$	11.6569			1.38341			0.691707	+	0.722178i	$0.256859\pi$
$72$		0			0
$73$	−8.82843			−1.03329			−0.516645	−	0.856200i	$-0.672819\pi$
$73$	−8.82843			−1.03329			−0.516645	+	0.856200i	$0.672819\pi$
$74$		0			0
$75$	7.94975	−	1.36396i	0.917958	−	0.157497i
$76$		0			0
$77$			2.00000i			0.227921i
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$	8.58579			0.942412			0.471206	−	0.882023i	$-0.343819\pi$
$83$	8.58579			0.942412			0.471206	+	0.882023i	$0.343819\pi$
$84$		0			0
$85$	−0.485281			−0.0526362
$86$		0			0
$87$	−0.242641	−	1.41421i	−0.0260138	−	0.151620i
$88$		0			0
$89$			3.65685i			0.387626i	0.981039	+	0.193813i	$0.0620855\pi$
$89$			3.65685i			0.387626i	−0.981039	+	0.193813i	$0.937915\pi$
$90$		0			0
$91$		−	0.585786i		−	0.0614071i
$92$		0			0
$93$	2.00000	+	11.6569i	0.207390	+	1.20876i
$94$		0			0
$95$	1.31371			0.134784
$96$		0			0
$97$	−17.3137			−1.75794			−0.878970	−	0.476876i	$-0.841769\pi$
$97$	−17.3137			−1.75794			−0.878970	+	0.476876i	$0.841769\pi$
$98$		0			0
$99$	5.65685	−	2.00000i	0.568535	−	0.201008i
$100$		0			0
$101$		−	9.75736i		−	0.970894i	−0.874266	−	0.485447i	$-0.838657\pi$
$101$		−	9.75736i		−	0.970894i	0.874266	−	0.485447i	$-0.161343\pi$
$102$		0			0
$103$		−	16.9706i		−	1.67216i	−0.548608	−	0.836080i	$-0.684842\pi$
$103$		−	16.9706i		−	1.67216i	0.548608	−	0.836080i	$-0.315158\pi$
$104$		0			0
$105$	−1.00000	+	0.171573i	−0.0975900	+	0.0167438i
$106$		0			0
$107$	−2.48528			−0.240261			−0.120131	−	0.992758i	$-0.538331\pi$
$107$	−2.48528			−0.240261			−0.120131	+	0.992758i	$0.538331\pi$
$108$		0			0
$109$	−14.4853			−1.38744			−0.693719	−	0.720246i	$-0.744029\pi$
$109$	−14.4853			−1.38744			−0.693719	+	0.720246i	$0.744029\pi$
$110$		0			0
$111$	−8.24264	+	1.41421i	−0.782357	+	0.134231i
$112$		0			0
$113$		−	17.3137i		−	1.62874i	−0.580348	−	0.814368i	$-0.697084\pi$
$113$		−	17.3137i		−	1.62874i	0.580348	−	0.814368i	$-0.302916\pi$
$114$		0			0
$115$			2.34315i			0.218499i
$116$		0			0
$117$	−1.65685	+	0.585786i	−0.153176	+	0.0541560i
$118$		0			0
$119$	−0.828427			−0.0759418
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	2.92893	+	17.0711i	0.264093	+	1.53925i
$124$		0			0
$125$			5.65685i			0.505964i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$	−1.89949	−	11.0711i	−0.167241	−	0.974753i
$130$		0			0
$131$	−9.55635			−0.834942			−0.417471	−	0.908690i	$-0.637083\pi$
$131$	−9.55635			−0.834942			−0.417471	+	0.908690i	$0.637083\pi$
$132$		0			0
$133$	2.24264			0.194462
$134$		0			0
$135$	1.48528	+	2.65685i	0.127833	+	0.228666i
$136$		0			0
$137$			11.3137i			0.966595i	0.875456	+	0.483298i	$0.160561\pi$
$137$			11.3137i			0.966595i	−0.875456	+	0.483298i	$0.839439\pi$
$138$		0			0
$139$		−	10.2426i		−	0.868769i	−0.900727	−	0.434385i	$-0.856966\pi$
$139$		−	10.2426i		−	0.868769i	0.900727	−	0.434385i	$-0.143034\pi$
$140$		0			0
$141$	16.4853	−	2.82843i	1.38831	−	0.238197i
$142$		0			0
$143$	−1.17157			−0.0979718
$144$		0			0
$145$	0.485281			0.0403004
$146$		0			0
$147$	−1.70711	+	0.292893i	−0.140800	+	0.0241574i
$148$		0			0
$149$		−	3.65685i		−	0.299581i	−0.988718	−	0.149791i	$-0.952140\pi$
$149$		−	3.65685i		−	0.299581i	0.988718	−	0.149791i	$-0.0478599\pi$
$150$		0			0
$151$		−	18.0000i		−	1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$		−	18.0000i		−	1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$		0			0
$153$	0.828427	+	2.34315i	0.0669744	+	0.189432i
$154$		0			0
$155$	−4.00000			−0.321288
$156$		0			0
$157$	6.24264			0.498217			0.249108	−	0.968476i	$-0.419862\pi$
$157$	6.24264			0.498217			0.249108	+	0.968476i	$0.419862\pi$
$158$		0			0
$159$	−2.72792	−	15.8995i	−0.216338	−	1.26091i
$160$		0			0
$161$			4.00000i			0.315244i
$162$		0			0
$163$			20.1421i			1.57765i	0.614615	+	0.788827i	$0.289311\pi$
$163$			20.1421i			1.57765i	−0.614615	+	0.788827i	$0.710689\pi$
$164$		0			0
$165$	0.343146	+	2.00000i	0.0267139	+	0.155700i
$166$		0			0
$167$	5.17157			0.400188			0.200094	−	0.979777i	$-0.435875\pi$
$167$	5.17157			0.400188			0.200094	+	0.979777i	$0.435875\pi$
$168$		0			0
$169$	−12.6569			−0.973604
$170$		0			0
$171$	−2.24264	−	6.34315i	−0.171499	−	0.485072i
$172$		0			0
$173$			21.5563i			1.63890i	0.573151	+	0.819449i	$0.305721\pi$
$173$			21.5563i			1.63890i	−0.573151	+	0.819449i	$0.694279\pi$
$174$		0			0
$175$			4.65685i			0.352025i
$176$		0			0
$177$	3.82843	−	0.656854i	0.287762	−	0.0493722i
$178$		0			0
$179$	−10.4853			−0.783707			−0.391853	−	0.920028i	$-0.628166\pi$
$179$	−10.4853			−0.783707			−0.391853	+	0.920028i	$0.628166\pi$
$180$		0			0
$181$	16.3848			1.21787			0.608935	−	0.793220i	$-0.291597\pi$
$181$	16.3848			1.21787			0.608935	+	0.793220i	$0.291597\pi$
$182$		0			0
$183$	−9.82843	+	1.68629i	−0.726538	+	0.124654i
$184$		0			0
$185$		−	2.82843i		−	0.207950i
$186$		0			0
$187$			1.65685i			0.121161i
$188$		0			0
$189$	2.53553	+	4.53553i	0.184433	+	0.329912i
$190$		0			0
$191$	−7.65685			−0.554031			−0.277015	−	0.960866i	$-0.589345\pi$
$191$	−7.65685			−0.554031			−0.277015	+	0.960866i	$0.589345\pi$
$192$		0			0
$193$	6.34315			0.456590			0.228295	−	0.973592i	$-0.426685\pi$
$193$	6.34315			0.456590			0.228295	+	0.973592i	$0.426685\pi$
$194$		0			0
$195$	−0.100505	−	0.585786i	−0.00719732	−	0.0419490i
$196$		0			0
$197$		−	26.0000i		−	1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$		−	26.0000i		−	1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$		0			0
$199$			15.3137i			1.08556i	0.839875	+	0.542780i	$0.182628\pi$
$199$			15.3137i			1.08556i	−0.839875	+	0.542780i	$0.817372\pi$
$200$		0			0
$201$	3.89949	+	22.7279i	0.275049	+	1.60310i
$202$		0			0
$203$	0.828427			0.0581442
$204$		0			0
$205$	−5.85786			−0.409131
$206$		0			0
$207$	11.3137	−	4.00000i	0.786357	−	0.278019i
$208$		0			0
$209$		−	4.48528i		−	0.310253i
$210$		0			0
$211$		−	11.6569i		−	0.802491i	−0.915971	−	0.401245i	$-0.868577\pi$
$211$		−	11.6569i		−	0.802491i	0.915971	−	0.401245i	$-0.131423\pi$
$212$		0			0
$213$	19.8995	−	3.41421i	1.36349	−	0.233938i
$214$		0			0
$215$	3.79899			0.259089
$216$		0			0
$217$	−6.82843			−0.463544
$218$		0			0
$219$	−15.0711	+	2.58579i	−1.01841	+	0.174731i
$220$		0			0
$221$		−	0.485281i		−	0.0326436i
$222$		0			0
$223$			2.82843i			0.189405i	0.995506	+	0.0947027i	$0.0301901\pi$
$223$			2.82843i			0.189405i	−0.995506	+	0.0947027i	$0.969810\pi$
$224$		0			0
$225$	13.1716	−	4.65685i	0.878105	−	0.310457i
$226$		0			0
$227$	−4.10051			−0.272160			−0.136080	−	0.990698i	$-0.543450\pi$
$227$	−4.10051			−0.272160			−0.136080	+	0.990698i	$0.543450\pi$
$228$		0			0
$229$	−13.0711			−0.863760			−0.431880	−	0.901931i	$-0.642150\pi$
$229$	−13.0711			−0.863760			−0.431880	+	0.901931i	$0.642150\pi$
$230$		0			0
$231$	0.585786	+	3.41421i	0.0385419	+	0.224639i
$232$		0			0
$233$			24.9706i			1.63588i	0.575306	+	0.817938i	$0.304883\pi$
$233$			24.9706i			1.63588i	−0.575306	+	0.817938i	$0.695117\pi$
$234$		0			0
$235$			5.65685i			0.369012i
$236$		0			0
$237$	1.17157	+	6.82843i	0.0761018	+	0.443554i
$238$		0			0
$239$	−11.3137			−0.731823			−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137			−0.731823			−0.365911	+	0.930650i	$0.619243\pi$
$240$		0			0
$241$	−7.65685			−0.493221			−0.246611	−	0.969115i	$-0.579317\pi$
$241$	−7.65685			−0.493221			−0.246611	+	0.969115i	$0.579317\pi$
$242$		0			0
$243$	10.2929	−	11.7071i	0.660289	−	0.751011i
$244$		0			0
$245$		−	0.585786i		−	0.0374245i
$246$		0			0
$247$			1.31371i			0.0835893i
$248$		0			0
$249$	14.6569	−	2.51472i	0.928840	−	0.159364i
$250$		0			0
$251$	−23.4142			−1.47789			−0.738946	−	0.673765i	$-0.764676\pi$
$251$	−23.4142			−1.47789			−0.738946	+	0.673765i	$0.764676\pi$
$252$		0			0
$253$	8.00000			0.502956
$254$		0			0
$255$	−0.828427	+	0.142136i	−0.0518781	+	0.00890088i
$256$		0			0
$257$		−	2.48528i		−	0.155028i	−0.996991	−	0.0775138i	$-0.975302\pi$
$257$		−	2.48528i		−	0.155028i	0.996991	−	0.0775138i	$-0.0246982\pi$
$258$		0			0
$259$		−	4.82843i		−	0.300024i
$260$		0			0
$261$	−0.828427	−	2.34315i	−0.0512784	−	0.145037i
$262$		0			0
$263$	−21.3137			−1.31426			−0.657130	−	0.753777i	$-0.728230\pi$
$263$	−21.3137			−1.31426			−0.657130	+	0.753777i	$0.728230\pi$
$264$		0			0
$265$	5.45584			0.335150
$266$		0			0
$267$	1.07107	+	6.24264i	0.0655483	+	0.382043i
$268$		0			0
$269$		−	6.72792i		−	0.410209i	−0.978740	−	0.205104i	$-0.934247\pi$
$269$		−	6.72792i		−	0.410209i	0.978740	−	0.205104i	$-0.0657534\pi$
$270$		0			0
$271$		−	4.48528i		−	0.272461i	−0.990677	−	0.136231i	$-0.956501\pi$
$271$		−	4.48528i		−	0.272461i	0.990677	−	0.136231i	$-0.0434988\pi$
$272$		0			0
$273$	−0.171573	−	1.00000i	−0.0103841	−	0.0605228i
$274$		0			0
$275$	9.31371			0.561638
$276$		0			0
$277$	−21.3137			−1.28062			−0.640308	−	0.768118i	$-0.721193\pi$
$277$	−21.3137			−1.28062			−0.640308	+	0.768118i	$0.721193\pi$
$278$		0			0
$279$	6.82843	+	19.3137i	0.408807	+	1.15628i
$280$		0			0
$281$		−	24.9706i		−	1.48962i	−0.667277	−	0.744809i	$-0.732540\pi$
$281$		−	24.9706i		−	1.48962i	0.667277	−	0.744809i	$-0.267460\pi$
$282$		0			0
$283$		−	1.75736i		−	0.104464i	−0.998635	−	0.0522321i	$-0.983366\pi$
$283$		−	1.75736i		−	0.104464i	0.998635	−	0.0522321i	$-0.0166336\pi$
$284$		0			0
$285$	2.24264	−	0.384776i	0.132843	−	0.0227922i
$286$		0			0
$287$	−10.0000			−0.590281
$288$		0			0
$289$	16.3137			0.959630
$290$		0			0
$291$	−29.5563	+	5.07107i	−1.73262	+	0.297271i
$292$		0			0
$293$			26.7279i			1.56146i	0.624867	+	0.780731i	$0.285153\pi$
$293$			26.7279i			1.56146i	−0.624867	+	0.780731i	$0.714847\pi$
$294$		0			0
$295$			1.31371i			0.0764871i
$296$		0			0
$297$	9.07107	−	5.07107i	0.526357	−	0.294253i
$298$		0			0
$299$	−2.34315			−0.135508
$300$		0			0
$301$	6.48528			0.373805
$302$		0			0
$303$	−2.85786	−	16.6569i	−0.164180	−	0.956911i
$304$		0			0
$305$		−	3.37258i		−	0.193114i
$306$		0			0
$307$		−	21.5563i		−	1.23029i	−0.788416	−	0.615143i	$-0.789098\pi$
$307$		−	21.5563i		−	1.23029i	0.788416	−	0.615143i	$-0.210902\pi$
$308$		0			0
$309$	−4.97056	−	28.9706i	−0.282765	−	1.64808i
$310$		0			0
$311$	−19.7990			−1.12270			−0.561349	−	0.827579i	$-0.689717\pi$
$311$	−19.7990			−1.12270			−0.561349	+	0.827579i	$0.689717\pi$
$312$		0			0
$313$	−3.17157			−0.179268			−0.0896339	−	0.995975i	$-0.528570\pi$
$313$	−3.17157			−0.179268			−0.0896339	+	0.995975i	$0.528570\pi$
$314$		0			0
$315$	−1.65685	+	0.585786i	−0.0933532	+	0.0330053i
$316$		0			0
$317$			10.0000i			0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$			10.0000i			0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$		0			0
$319$		−	1.65685i		−	0.0927660i
$320$		0			0
$321$	−4.24264	+	0.727922i	−0.236801	+	0.0406286i
$322$		0			0
$323$	1.85786			0.103374
$324$		0			0
$325$	−2.72792			−0.151318
$326$		0			0
$327$	−24.7279	+	4.24264i	−1.36746	+	0.234619i
$328$		0			0
$329$			9.65685i			0.532400i
$330$		0			0
$331$			0.828427i			0.0455345i	0.999741	+	0.0227672i	$0.00724766\pi$
$331$			0.828427i			0.0455345i	−0.999741	+	0.0227672i	$0.992752\pi$
$332$		0			0
$333$	−13.6569	+	4.82843i	−0.748391	+	0.264596i
$334$		0			0
$335$	−7.79899			−0.426104
$336$		0			0
$337$	4.68629			0.255279			0.127639	−	0.991821i	$-0.459260\pi$
$337$	4.68629			0.255279			0.127639	+	0.991821i	$0.459260\pi$
$338$		0			0
$339$	−5.07107	−	29.5563i	−0.275423	−	1.60528i
$340$		0			0
$341$			13.6569i			0.739560i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	0.686292	+	4.00000i	0.0369487	+	0.215353i
$346$		0			0
$347$	11.6569			0.625773			0.312886	−	0.949791i	$-0.398704\pi$
$347$	11.6569			0.625773			0.312886	+	0.949791i	$0.398704\pi$
$348$		0			0
$349$	−29.0711			−1.55614			−0.778069	−	0.628178i	$-0.783801\pi$
$349$	−29.0711			−1.55614			−0.778069	+	0.628178i	$0.783801\pi$
$350$		0			0
$351$	−2.65685	+	1.48528i	−0.141812	+	0.0792785i
$352$		0			0
$353$		−	31.4558i		−	1.67423i	−0.547030	−	0.837113i	$-0.684242\pi$
$353$		−	31.4558i		−	1.67423i	0.547030	−	0.837113i	$-0.315758\pi$
$354$		0			0
$355$			6.82843i			0.362415i
$356$		0			0
$357$	−1.41421	+	0.242641i	−0.0748481	+	0.0128419i
$358$		0			0
$359$	−18.6274			−0.983117			−0.491559	−	0.870844i	$-0.663573\pi$
$359$	−18.6274			−0.983117			−0.491559	+	0.870844i	$0.663573\pi$
$360$		0			0
$361$	13.9706			0.735293
$362$		0			0
$363$	−11.9497	+	2.05025i	−0.627199	+	0.107610i
$364$		0			0
$365$		−	5.17157i		−	0.270692i
$366$		0			0
$367$			18.8284i			0.982836i	0.870924	+	0.491418i	$0.163521\pi$
$367$			18.8284i			0.982836i	−0.870924	+	0.491418i	$0.836479\pi$
$368$		0			0
$369$	10.0000	+	28.2843i	0.520579	+	1.47242i
$370$		0			0
$371$	9.31371			0.483544
$372$		0			0
$373$	−10.9706			−0.568034			−0.284017	−	0.958819i	$-0.591667\pi$
$373$	−10.9706			−0.568034			−0.284017	+	0.958819i	$0.591667\pi$
$374$		0			0
$375$	1.65685	+	9.65685i	0.0855596	+	0.498678i
$376$		0			0
$377$			0.485281i			0.0249933i
$378$		0			0
$379$		−	17.5147i		−	0.899671i	−0.893112	−	0.449835i	$-0.851483\pi$
$379$		−	17.5147i		−	0.899671i	0.893112	−	0.449835i	$-0.148517\pi$
$380$		0			0
$381$	−0.585786	−	3.41421i	−0.0300107	−	0.174915i
$382$		0			0
$383$	5.65685			0.289052			0.144526	−	0.989501i	$-0.453834\pi$
$383$	5.65685			0.289052			0.144526	+	0.989501i	$0.453834\pi$
$384$		0			0
$385$	−1.17157			−0.0597089
$386$		0			0
$387$	−6.48528	−	18.3431i	−0.329665	−	0.932435i
$388$		0			0
$389$		−	10.4853i		−	0.531625i	−0.964025	−	0.265812i	$-0.914360\pi$
$389$		−	10.4853i		−	0.531625i	0.964025	−	0.265812i	$-0.0856402\pi$
$390$		0			0
$391$			3.31371i			0.167581i
$392$		0			0
$393$	−16.3137	+	2.79899i	−0.822918	+	0.141190i
$394$		0			0
$395$	−2.34315			−0.117896
$396$		0			0
$397$	−9.75736			−0.489708			−0.244854	−	0.969560i	$-0.578740\pi$
$397$	−9.75736			−0.489708			−0.244854	+	0.969560i	$0.578740\pi$
$398$		0			0
$399$	3.82843	−	0.656854i	0.191661	−	0.0328838i
$400$		0			0
$401$			20.3431i			1.01589i	0.861390	+	0.507944i	$0.169594\pi$
$401$			20.3431i			1.01589i	−0.861390	+	0.507944i	$0.830406\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$		0			0
$405$	3.31371	+	4.10051i	0.164659	+	0.203756i
$406$		0			0
$407$	−9.65685			−0.478672
$408$		0			0
$409$	9.51472			0.470473			0.235236	−	0.971938i	$-0.424414\pi$
$409$	9.51472			0.470473			0.235236	+	0.971938i	$0.424414\pi$
$410$		0			0
$411$	3.31371	+	19.3137i	0.163453	+	0.952675i
$412$		0			0
$413$			2.24264i			0.110353i
$414$		0			0
$415$			5.02944i			0.246885i
$416$		0			0
$417$	−3.00000	−	17.4853i	−0.146911	−	0.856258i
$418$		0			0
$419$	6.72792			0.328681			0.164340	−	0.986404i	$-0.447450\pi$
$419$	6.72792			0.328681			0.164340	+	0.986404i	$0.447450\pi$
$420$		0			0
$421$	4.34315			0.211672			0.105836	−	0.994384i	$-0.466248\pi$
$421$	4.34315			0.211672			0.105836	+	0.994384i	$0.466248\pi$
$422$		0			0
$423$	27.3137	−	9.65685i	1.32804	−	0.469532i
$424$		0			0
$425$			3.85786i			0.187134i
$426$		0			0
$427$		−	5.75736i		−	0.278618i
$428$		0			0
$429$	−2.00000	+	0.343146i	−0.0965609	+	0.0165672i
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$	−18.9706			−0.911667			−0.455834	−	0.890065i	$-0.650659\pi$
$433$	−18.9706			−0.911667			−0.455834	+	0.890065i	$0.650659\pi$
$434$		0			0
$435$	0.828427	−	0.142136i	0.0397200	−	0.00681488i
$436$		0			0
$437$		−	8.97056i		−	0.429120i
$438$		0			0
$439$			16.2843i			0.777206i	0.921405	+	0.388603i	$0.127042\pi$
$439$			16.2843i			0.777206i	−0.921405	+	0.388603i	$0.872958\pi$
$440$		0			0
$441$	−2.82843	+	1.00000i	−0.134687	+	0.0476190i
$442$		0			0
$443$	28.8284			1.36968			0.684840	−	0.728694i	$-0.259872\pi$
$443$	28.8284			1.36968			0.684840	+	0.728694i	$0.259872\pi$
$444$		0			0
$445$	−2.14214			−0.101547
$446$		0			0
$447$	−1.07107	−	6.24264i	−0.0506598	−	0.295267i
$448$		0			0
$449$		−	29.6569i		−	1.39959i	−0.714342	−	0.699797i	$-0.753274\pi$
$449$		−	29.6569i		−	1.39959i	0.714342	−	0.699797i	$-0.246726\pi$
$450$		0			0
$451$			20.0000i			0.941763i
$452$		0			0
$453$	−5.27208	−	30.7279i	−0.247704	−	1.44372i
$454$		0			0
$455$	0.343146			0.0160869
$456$		0			0
$457$	−24.0000			−1.12267			−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000			−1.12267			−0.561336	+	0.827588i	$0.689713\pi$
$458$		0			0
$459$	2.10051	+	3.75736i	0.0980432	+	0.175379i
$460$		0			0
$461$			26.0416i			1.21288i	0.795129	+	0.606440i	$0.207403\pi$
$461$			26.0416i			1.21288i	−0.795129	+	0.606440i	$0.792597\pi$
$462$		0			0
$463$		−	9.65685i		−	0.448792i	−0.974498	−	0.224396i	$-0.927959\pi$
$463$		−	9.65685i		−	0.448792i	0.974498	−	0.224396i	$-0.0720409\pi$
$464$		0			0
$465$	−6.82843	+	1.17157i	−0.316661	+	0.0543304i
$466$		0			0
$467$	−30.0416			−1.39016			−0.695080	−	0.718932i	$-0.744631\pi$
$467$	−30.0416			−1.39016			−0.695080	+	0.718932i	$0.744631\pi$
$468$		0			0
$469$	−13.3137			−0.614770
$470$		0			0
$471$	10.6569	−	1.82843i	0.491042	−	0.0842495i
$472$		0			0
$473$		−	12.9706i		−	0.596387i
$474$		0			0
$475$		−	10.4437i		−	0.479188i
$476$		0			0
$477$	−9.31371	−	26.3431i	−0.426445	−	1.20617i
$478$		0			0
$479$	−11.3137			−0.516937			−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137			−0.516937			−0.258468	+	0.966020i	$0.583218\pi$
$480$		0			0
$481$	2.82843			0.128965
$482$		0			0
$483$	1.17157	+	6.82843i	0.0533084	+	0.310704i
$484$		0			0
$485$		−	10.1421i		−	0.460531i
$486$		0			0
$487$		−	22.0000i		−	0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$		−	22.0000i		−	0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$		0			0
$489$	5.89949	+	34.3848i	0.266784	+	1.55493i
$490$		0			0
$491$	−12.8284			−0.578939			−0.289469	−	0.957187i	$-0.593479\pi$
$491$	−12.8284			−0.578939			−0.289469	+	0.957187i	$0.593479\pi$
$492$		0			0
$493$	0.686292			0.0309090
$494$		0			0
$495$	1.17157	+	3.31371i	0.0526583	+	0.148940i
$496$		0			0
$497$			11.6569i			0.522881i
$498$		0			0
$499$		−	8.62742i		−	0.386216i	−0.981177	−	0.193108i	$-0.938143\pi$
$499$		−	8.62742i		−	0.386216i	0.981177	−	0.193108i	$-0.0618568\pi$
$500$		0			0
$501$	8.82843	−	1.51472i	0.394425	−	0.0676726i
$502$		0			0
$503$	25.4558			1.13502			0.567510	−	0.823367i	$-0.307907\pi$
$503$	25.4558			1.13502			0.567510	+	0.823367i	$0.307907\pi$
$504$		0			0
$505$	5.71573			0.254347
$506$		0			0
$507$	−21.6066	+	3.70711i	−0.959583	+	0.164638i
$508$		0			0
$509$			32.8701i			1.45694i	0.685078	+	0.728470i	$0.259768\pi$
$509$			32.8701i			1.45694i	−0.685078	+	0.728470i	$0.740232\pi$
$510$		0			0
$511$		−	8.82843i		−	0.390547i
$512$		0			0
$513$	−5.68629	−	10.1716i	−0.251056	−	0.449086i
$514$		0			0
$515$	9.94113			0.438058
$516$		0			0
$517$	19.3137			0.849416
$518$		0			0
$519$	6.31371	+	36.7990i	0.277141	+	1.61530i
$520$		0			0
$521$			23.6569i			1.03643i	0.855252	+	0.518213i	$0.173402\pi$
$521$			23.6569i			1.03643i	−0.855252	+	0.518213i	$0.826598\pi$
$522$		0			0
$523$			20.1005i			0.878934i	0.898259	+	0.439467i	$0.144833\pi$
$523$			20.1005i			0.878934i	−0.898259	+	0.439467i	$0.855167\pi$
$524$		0			0
$525$	1.36396	+	7.94975i	0.0595281	+	0.346955i
$526$		0			0
$527$	−5.65685			−0.246416
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$	6.34315	−	2.24264i	0.275269	−	0.0973223i
$532$		0			0
$533$		−	5.85786i		−	0.253732i
$534$		0			0
$535$		−	1.45584i		−	0.0629416i
$536$		0			0
$537$	−17.8995	+	3.07107i	−0.772420	+	0.132526i
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	32.6274			1.40276			0.701381	−	0.712786i	$-0.252567\pi$
$541$	32.6274			1.40276			0.701381	+	0.712786i	$0.252567\pi$
$542$		0			0
$543$	27.9706	−	4.79899i	1.20033	−	0.205944i
$544$		0			0
$545$		−	8.48528i		−	0.363470i
$546$		0			0
$547$			20.1421i			0.861216i	0.902539	+	0.430608i	$0.141701\pi$
$547$			20.1421i			0.861216i	−0.902539	+	0.430608i	$0.858299\pi$
$548$		0			0
$549$	−16.2843	+	5.75736i	−0.694996	+	0.245718i
$550$		0			0
$551$	−1.85786			−0.0791477
$552$		0			0
$553$	−4.00000			−0.170097
$554$		0			0
$555$	−0.828427	−	4.82843i	−0.0351648	−	0.204955i
$556$		0			0
$557$		−	29.3137i		−	1.24206i	−0.783786	−	0.621031i	$-0.786714\pi$
$557$		−	29.3137i		−	1.24206i	0.783786	−	0.621031i	$-0.213286\pi$
$558$		0			0
$559$			3.79899i			0.160680i
$560$		0			0
$561$	0.485281	+	2.82843i	0.0204886	+	0.119416i
$562$		0			0
$563$	35.2132			1.48406			0.742030	−	0.670367i	$-0.233863\pi$
$563$	35.2132			1.48406			0.742030	+	0.670367i	$0.233863\pi$
$564$		0			0
$565$	10.1421			0.426683
$566$		0			0
$567$	5.65685	+	7.00000i	0.237566	+	0.293972i
$568$		0			0
$569$		−	21.3137i		−	0.893517i	−0.894655	−	0.446759i	$-0.852578\pi$
$569$		−	21.3137i		−	0.893517i	0.894655	−	0.446759i	$-0.147422\pi$
$570$		0			0
$571$		−	20.1421i		−	0.842922i	−0.906846	−	0.421461i	$-0.861517\pi$
$571$		−	20.1421i		−	0.842922i	0.906846	−	0.421461i	$-0.138483\pi$
$572$		0			0
$573$	−13.0711	+	2.24264i	−0.546052	+	0.0936877i
$574$		0			0
$575$	18.6274			0.776817
$576$		0			0
$577$	6.68629			0.278354			0.139177	−	0.990268i	$-0.455554\pi$
$577$	6.68629			0.278354			0.139177	+	0.990268i	$0.455554\pi$
$578$		0			0
$579$	10.8284	−	1.85786i	0.450014	−	0.0772102i
$580$		0			0
$581$			8.58579i			0.356198i
$582$		0			0
$583$		−	18.6274i		−	0.771469i
$584$		0			0
$585$	−0.343146	−	0.970563i	−0.0141873	−	0.0401278i
$586$		0			0
$587$	5.27208			0.217602			0.108801	−	0.994064i	$-0.465299\pi$
$587$	5.27208			0.217602			0.108801	+	0.994064i	$0.465299\pi$
$588$		0			0
$589$	15.3137			0.630990
$590$		0			0
$591$	−7.61522	−	44.3848i	−0.313248	−	1.82575i
$592$		0			0
$593$			5.51472i			0.226462i	0.993569	+	0.113231i	$0.0361201\pi$
$593$			5.51472i			0.226462i	−0.993569	+	0.113231i	$0.963880\pi$
$594$		0			0
$595$		−	0.485281i		−	0.0198946i
$596$		0			0
$597$	4.48528	+	26.1421i	0.183570	+	1.06993i
$598$		0			0
$599$	−28.6274			−1.16968			−0.584842	−	0.811147i	$-0.698844\pi$
$599$	−28.6274			−1.16968			−0.584842	+	0.811147i	$0.698844\pi$
$600$		0			0
$601$	12.1421			0.495288			0.247644	−	0.968851i	$-0.420344\pi$
$601$	12.1421			0.495288			0.247644	+	0.968851i	$0.420344\pi$
$602$		0			0
$603$	13.3137	+	37.6569i	0.542176	+	1.53351i
$604$		0			0
$605$		−	4.10051i		−	0.166709i
$606$		0			0
$607$			26.1421i			1.06108i	0.847661	+	0.530538i	$0.178010\pi$
$607$			26.1421i			1.06108i	−0.847661	+	0.530538i	$0.821990\pi$
$608$		0			0
$609$	1.41421	−	0.242641i	0.0573068	−	0.00983230i
$610$		0			0
$611$	−5.65685			−0.228852
$612$		0			0
$613$	46.7696			1.88900			0.944502	−	0.328505i	$-0.106545\pi$
$613$	46.7696			1.88900			0.944502	+	0.328505i	$0.106545\pi$
$614$		0			0
$615$	−10.0000	+	1.71573i	−0.403239	+	0.0691849i
$616$		0			0
$617$		−	16.6274i		−	0.669395i	−0.942326	−	0.334697i	$-0.891366\pi$
$617$		−	16.6274i		−	0.669395i	0.942326	−	0.334697i	$-0.108634\pi$
$618$		0			0
$619$			9.75736i			0.392181i	0.980586	+	0.196091i	$0.0628247\pi$
$619$			9.75736i			0.392181i	−0.980586	+	0.196091i	$0.937175\pi$
$620$		0			0
$621$	18.1421	−	10.1421i	0.728019	−	0.406990i
$622$		0			0
$623$	−3.65685			−0.146509
$624$		0			0
$625$	19.9706			0.798823
$626$		0			0
$627$	−1.31371	−	7.65685i	−0.0524645	−	0.305785i
$628$		0			0
$629$		−	4.00000i		−	0.159490i
$630$		0			0
$631$		−	24.9706i		−	0.994062i	−0.867733	−	0.497031i	$-0.834423\pi$
$631$		−	24.9706i		−	0.994062i	0.867733	−	0.497031i	$-0.165577\pi$
$632$		0			0
$633$	−3.41421	−	19.8995i	−0.135703	−	0.790934i
$634$		0			0
$635$	1.17157			0.0464925
$636$		0			0
$637$	0.585786			0.0232097
$638$		0			0
$639$	32.9706	−	11.6569i	1.30430	−	0.461138i
$640$		0			0
$641$		−	15.6569i		−	0.618409i	−0.950996	−	0.309204i	$-0.899937\pi$
$641$		−	15.6569i		−	0.618409i	0.950996	−	0.309204i	$-0.100063\pi$
$642$		0			0
$643$			46.2426i			1.82363i	0.410599	+	0.911816i	$0.365319\pi$
$643$			46.2426i			1.82363i	−0.410599	+	0.911816i	$0.634681\pi$
$644$		0			0
$645$	6.48528	−	1.11270i	0.255358	−	0.0438125i
$646$		0			0
$647$	29.4558			1.15803			0.579014	−	0.815317i	$-0.303438\pi$
$647$	29.4558			1.15803			0.579014	+	0.815317i	$0.303438\pi$
$648$		0			0
$649$	4.48528			0.176063
$650$		0			0
$651$	−11.6569	+	2.00000i	−0.456868	+	0.0783862i
$652$		0			0
$653$			14.4853i			0.566853i	0.958994	+	0.283426i	$0.0914712\pi$
$653$			14.4853i			0.566853i	−0.958994	+	0.283426i	$0.908529\pi$
$654$		0			0
$655$		−	5.59798i		−	0.218731i
$656$		0			0
$657$	−24.9706	+	8.82843i	−0.974194	+	0.344430i
$658$		0			0
$659$	18.9706			0.738988			0.369494	−	0.929233i	$-0.379531\pi$
$659$	18.9706			0.738988			0.369494	+	0.929233i	$0.379531\pi$
$660$		0			0
$661$	36.1838			1.40739			0.703693	−	0.710504i	$-0.251533\pi$
$661$	36.1838			1.40739			0.703693	+	0.710504i	$0.251533\pi$
$662$		0			0
$663$	−0.142136	−	0.828427i	−0.00552009	−	0.0321734i
$664$		0			0
$665$			1.31371i			0.0509434i
$666$		0			0
$667$		−	3.31371i		−	0.128307i
$668$		0			0
$669$	0.828427	+	4.82843i	0.0320288	+	0.186678i
$670$		0			0
$671$	−11.5147			−0.444521
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$		0			0
$675$	21.1213	−	11.8076i	0.812960	−	0.454475i
$676$		0			0
$677$			48.8701i			1.87823i	0.343605	+	0.939114i	$0.388352\pi$
$677$			48.8701i			1.87823i	−0.343605	+	0.939114i	$0.611648\pi$
$678$		0			0
$679$		−	17.3137i		−	0.664439i
$680$		0			0
$681$	−7.00000	+	1.20101i	−0.268241	+	0.0460228i
$682$		0			0
$683$	46.4853			1.77871			0.889355	−	0.457217i	$-0.151154\pi$
$683$	46.4853			1.77871			0.889355	+	0.457217i	$0.151154\pi$
$684$		0			0
$685$	−6.62742			−0.253221
$686$		0			0
$687$	−22.3137	+	3.82843i	−0.851321	+	0.146064i
$688$		0			0
$689$			5.45584i			0.207851i
$690$		0			0
$691$			26.0416i			0.990670i	0.868702	+	0.495335i	$0.164955\pi$
$691$			26.0416i			0.990670i	−0.868702	+	0.495335i	$0.835045\pi$
$692$		0			0
$693$	2.00000	+	5.65685i	0.0759737	+	0.214886i
$694$		0			0
$695$	6.00000			0.227593
$696$		0			0
$697$	−8.28427			−0.313789
$698$		0			0
$699$	7.31371	+	42.6274i	0.276630	+	1.61232i
$700$		0			0
$701$		−	0.142136i		−	0.00536839i	−0.999996	−	0.00268419i	$-0.999146\pi$
$701$		−	0.142136i		−	0.00536839i	0.999996	−	0.00268419i	$-0.000854407\pi$
$702$		0			0
$703$			10.8284i			0.408402i
$704$		0			0
$705$	1.65685	+	9.65685i	0.0624007	+	0.363698i
$706$		0			0
$707$	9.75736			0.366963
$708$		0			0
$709$	−22.4853			−0.844452			−0.422226	−	0.906490i	$-0.638751\pi$
$709$	−22.4853			−0.844452			−0.422226	+	0.906490i	$0.638751\pi$
$710$		0			0
$711$	4.00000	+	11.3137i	0.150012	+	0.424297i
$712$		0			0
$713$			27.3137i			1.02291i
$714$		0			0
$715$		−	0.686292i		−	0.0256658i
$716$		0			0
$717$	−19.3137	+	3.31371i	−0.721284	+	0.123753i
$718$		0			0
$719$	43.3137			1.61533			0.807664	−	0.589642i	$-0.200731\pi$
$719$	43.3137			1.61533			0.807664	+	0.589642i	$0.200731\pi$
$720$		0			0
$721$	16.9706			0.632017
$722$		0			0
$723$	−13.0711	+	2.24264i	−0.486118	+	0.0834047i
$724$		0			0
$725$		−	3.85786i		−	0.143277i
$726$		0			0
$727$		−	34.6274i		−	1.28426i	−0.766596	−	0.642130i	$-0.778051\pi$
$727$		−	34.6274i		−	1.28426i	0.766596	−	0.642130i	$-0.221949\pi$
$728$		0			0
$729$	14.1421	−	23.0000i	0.523783	−	0.851852i
$730$		0			0
$731$	5.37258			0.198712
$732$		0			0
$733$	36.3848			1.34390			0.671951	−	0.740595i	$-0.265456\pi$
$733$	36.3848			1.34390			0.671951	+	0.740595i	$0.265456\pi$
$734$		0			0
$735$	−0.171573	−	1.00000i	−0.00632856	−	0.0368856i
$736$		0			0
$737$			26.6274i			0.980834i
$738$		0			0
$739$		−	40.1421i		−	1.47665i	−0.674444	−	0.738326i	$-0.735617\pi$
$739$		−	40.1421i		−	1.47665i	0.674444	−	0.738326i	$-0.264383\pi$
$740$		0			0
$741$	0.384776	+	2.24264i	0.0141351	+	0.0823855i
$742$		0			0
$743$	45.9411			1.68542			0.842708	−	0.538371i	$-0.180960\pi$
$743$	45.9411			1.68542			0.842708	+	0.538371i	$0.180960\pi$
$744$		0			0
$745$	2.14214			0.0784818
$746$		0			0
$747$	24.2843	−	8.58579i	0.888515	−	0.314137i
$748$		0			0
$749$		−	2.48528i		−	0.0908102i
$750$		0			0
$751$			26.2843i			0.959127i	0.877507	+	0.479563i	$0.159205\pi$
$751$			26.2843i			0.959127i	−0.877507	+	0.479563i	$0.840795\pi$
$752$		0			0
$753$	−39.9706	+	6.85786i	−1.45661	+	0.249914i
$754$		0			0
$755$	10.5442			0.383741
$756$		0			0
$757$	−13.1127			−0.476589			−0.238295	−	0.971193i	$-0.576588\pi$
$757$	−13.1127			−0.476589			−0.238295	+	0.971193i	$0.576588\pi$
$758$		0			0
$759$	13.6569	−	2.34315i	0.495712	−	0.0850508i
$760$		0			0
$761$		−	34.9706i		−	1.26768i	−0.773463	−	0.633841i	$-0.781477\pi$
$761$		−	34.9706i		−	1.26768i	0.773463	−	0.633841i	$-0.218523\pi$
$762$		0			0
$763$		−	14.4853i		−	0.524402i
$764$		0			0
$765$	−1.37258	+	0.485281i	−0.0496258	+	0.0175454i
$766$		0			0
$767$	−1.31371			−0.0474353
$768$		0			0
$769$	19.6569			0.708844			0.354422	−	0.935086i	$-0.384678\pi$
$769$	19.6569			0.708844			0.354422	+	0.935086i	$0.384678\pi$
$770$		0			0
$771$	−0.727922	−	4.24264i	−0.0262155	−	0.152795i
$772$		0			0
$773$		−	26.7279i		−	0.961337i	−0.876903	−	0.480668i	$-0.840394\pi$
$773$		−	26.7279i		−	0.961337i	0.876903	−	0.480668i	$-0.159606\pi$
$774$		0			0
$775$			31.7990i			1.14225i
$776$		0			0
$777$	−1.41421	−	8.24264i	−0.0507346	−	0.295703i
$778$		0			0
$779$	22.4264			0.803509
$780$		0			0
$781$	23.3137			0.834230
$782$		0			0
$783$	−2.10051	−	3.75736i	−0.0750659	−	0.134277i
$784$		0			0
$785$			3.65685i			0.130519i
$786$		0			0
$787$			42.2426i			1.50579i	0.658142	+	0.752894i	$0.271343\pi$
$787$			42.2426i			1.50579i	−0.658142	+	0.752894i	$0.728657\pi$
$788$		0			0
$789$	−36.3848	+	6.24264i	−1.29533	+	0.222244i
$790$		0			0
$791$	17.3137			0.615605
$792$		0			0
$793$	3.37258			0.119764
$794$		0			0
$795$	9.31371	−	1.59798i	0.330323	−	0.0566745i
$796$		0			0
$797$		−	31.8995i		−	1.12994i	−0.825112	−	0.564969i	$-0.808888\pi$
$797$		−	31.8995i		−	1.12994i	0.825112	−	0.564969i	$-0.191112\pi$
$798$		0			0
$799$			8.00000i			0.283020i
$800$		0			0
$801$	3.65685	+	10.3431i	0.129209	+	0.365457i
$802$		0			0
$803$	−17.6569			−0.623097
$804$		0			0
$805$	−2.34315			−0.0825850
$806$		0			0
$807$	−1.97056	−	11.4853i	−0.0693671	−	0.404301i
$808$		0			0
$809$			33.3137i			1.17125i	0.810583	+	0.585624i	$0.199150\pi$
$809$			33.3137i			1.17125i	−0.810583	+	0.585624i	$0.800850\pi$
$810$		0			0
$811$			1.55635i			0.0546508i	0.999627	+	0.0273254i	$0.00869903\pi$
$811$			1.55635i			0.0546508i	−0.999627	+	0.0273254i	$0.991301\pi$
$812$		0			0
$813$	−1.31371	−	7.65685i	−0.0460738	−	0.268538i
$814$		0			0
$815$	−11.7990			−0.413301
$816$		0			0
$817$	−14.5442			−0.508836
$818$		0			0
$819$	−0.585786	−	1.65685i	−0.0204690	−	0.0578952i
$820$		0			0
$821$			6.97056i			0.243274i	0.992575	+	0.121637i	$0.0388144\pi$
$821$			6.97056i			0.243274i	−0.992575	+	0.121637i	$0.961186\pi$
$822$		0			0
$823$			0.970563i			0.0338317i	0.999857	+	0.0169158i	$0.00538474\pi$
$823$			0.970563i			0.0338317i	−0.999857	+	0.0169158i	$0.994615\pi$
$824$		0			0
$825$	15.8995	−	2.72792i	0.553549	−	0.0949741i
$826$		0			0
$827$	−47.4558			−1.65020			−0.825101	−	0.564986i	$-0.808882\pi$
$827$	−47.4558			−1.65020			−0.825101	+	0.564986i	$0.808882\pi$
$828$		0			0
$829$	−12.3848			−0.430141			−0.215071	−	0.976599i	$-0.568998\pi$
$829$	−12.3848			−0.430141			−0.215071	+	0.976599i	$0.568998\pi$
$830$		0			0
$831$	−36.3848	+	6.24264i	−1.26217	+	0.216555i
$832$		0			0
$833$		−	0.828427i		−	0.0287033i
$834$		0			0
$835$			3.02944i			0.104838i
$836$		0			0
$837$	17.3137	+	30.9706i	0.598449	+	1.07050i
$838$		0			0
$839$	−6.82843			−0.235743			−0.117872	−	0.993029i	$-0.537607\pi$
$839$	−6.82843			−0.235743			−0.117872	+	0.993029i	$0.537607\pi$
$840$		0			0
$841$	28.3137			0.976335
$842$		0			0
$843$	−7.31371	−	42.6274i	−0.251898	−	1.46817i
$844$		0			0
$845$		−	7.41421i		−	0.255057i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$	−0.514719	−	3.00000i	−0.0176651	−	0.102960i
$850$		0			0
$851$	−19.3137			−0.662065
$852$		0			0
$853$	−7.41421			−0.253858			−0.126929	−	0.991912i	$-0.540512\pi$
$853$	−7.41421			−0.253858			−0.126929	+	0.991912i	$0.540512\pi$
$854$		0			0
$855$	3.71573	−	1.31371i	0.127075	−	0.0449279i
$856$		0			0
$857$		−	38.0000i		−	1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$		−	38.0000i		−	1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$		0			0
$859$		−	27.4142i		−	0.935361i	−0.883898	−	0.467680i	$-0.845090\pi$
$859$		−	27.4142i		−	0.935361i	0.883898	−	0.467680i	$-0.154910\pi$
$860$		0			0
$861$	−17.0711	+	2.92893i	−0.581780	+	0.0998177i
$862$		0			0
$863$	56.6274			1.92762			0.963810	−	0.266591i	$-0.0858972\pi$
$863$	56.6274			1.92762			0.963810	+	0.266591i	$0.0858972\pi$
$864$		0			0
$865$	−12.6274			−0.429345
$866$		0			0
$867$	27.8492	−	4.77817i	0.945810	−	0.162275i
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$		−	7.79899i		−	0.264259i
$872$		0			0
$873$	−48.9706	+	17.3137i	−1.65740	+	0.585980i
$874$		0			0
$875$	−5.65685			−0.191237
$876$		0			0
$877$	18.2010			0.614604			0.307302	−	0.951612i	$-0.400574\pi$
$877$	18.2010			0.614604			0.307302	+	0.951612i	$0.400574\pi$
$878$		0			0
$879$	7.82843	+	45.6274i	0.264046	+	1.53897i
$880$		0			0
$881$			36.1421i			1.21766i	0.793301	+	0.608830i	$0.208361\pi$
$881$			36.1421i			1.21766i	−0.793301	+	0.608830i	$0.791639\pi$
$882$		0			0
$883$			13.3137i			0.448042i	0.974584	+	0.224021i	$0.0719184\pi$
$883$			13.3137i			0.448042i	−0.974584	+	0.224021i	$0.928082\pi$
$884$		0			0
$885$	0.384776	+	2.24264i	0.0129341	+	0.0753855i
$886$		0			0
$887$	−2.82843			−0.0949693			−0.0474846	−	0.998872i	$-0.515121\pi$
$887$	−2.82843			−0.0949693			−0.0474846	+	0.998872i	$0.515121\pi$
$888$		0			0
$889$	2.00000			0.0670778
$890$		0			0
$891$	14.0000	−	11.3137i	0.469018	−	0.379023i
$892$		0			0
$893$		−	21.6569i		−	0.724719i
$894$		0			0
$895$		−	6.14214i		−	0.205309i
$896$		0			0
$897$	−4.00000	+	0.686292i	−0.133556	+	0.0229146i
$898$		0			0
$899$	5.65685			0.188667
$900$		0			0
$901$	7.71573			0.257048
$902$		0			0
$903$	11.0711	−	1.89949i	0.368422	−	0.0632112i
$904$		0			0
$905$			9.59798i			0.319048i
$906$		0			0
$907$		−	41.5980i		−	1.38124i	−0.723219	−	0.690619i	$-0.757338\pi$
$907$		−	41.5980i		−	1.38124i	0.723219	−	0.690619i	$-0.242662\pi$
$908$		0			0
$909$	−9.75736	−	27.5980i	−0.323631	−	0.915367i
$910$		0			0
$911$	−16.9706			−0.562260			−0.281130	−	0.959670i	$-0.590709\pi$
$911$	−16.9706			−0.562260			−0.281130	+	0.959670i	$0.590709\pi$
$912$		0			0
$913$	17.1716			0.568296
$914$		0			0
$915$	−0.987807	−	5.75736i	−0.0326559	−	0.190332i
$916$		0			0
$917$		−	9.55635i		−	0.315578i
$918$		0			0
$919$			18.3431i			0.605085i	0.953136	+	0.302542i	$0.0978353\pi$
$919$			18.3431i			0.605085i	−0.953136	+	0.302542i	$0.902165\pi$
$920$		0			0
$921$	−6.31371	−	36.7990i	−0.208044	−	1.21257i
$922$		0			0
$923$	−6.82843			−0.224760
$924$		0			0
$925$	−22.4853			−0.739311
$926$		0			0
$927$	−16.9706	−	48.0000i	−0.557386	−	1.57653i
$928$		0			0
$929$		−	5.79899i		−	0.190259i	−0.995465	−	0.0951293i	$-0.969674\pi$
$929$		−	5.79899i		−	0.190259i	0.995465	−	0.0951293i	$-0.0303265\pi$
$930$		0			0
$931$			2.24264i			0.0734996i
$932$		0			0
$933$	−33.7990	+	5.79899i	−1.10653	+	0.189850i
$934$		0			0
$935$	−0.970563			−0.0317408
$936$		0			0
$937$	38.4853			1.25726			0.628630	−	0.777705i	$-0.283616\pi$
$937$	38.4853			1.25726			0.628630	+	0.777705i	$0.283616\pi$
$938$		0			0
$939$	−5.41421	+	0.928932i	−0.176686	+	0.0303146i
$940$		0			0
$941$		−	43.6985i		−	1.42453i	−0.701911	−	0.712265i	$-0.747669\pi$
$941$		−	43.6985i		−	1.42453i	0.701911	−	0.712265i	$-0.252331\pi$
$942$		0			0
$943$			40.0000i			1.30258i
$944$		0			0
$945$	−2.65685	+	1.48528i	−0.0864275	+	0.0483162i
$946$		0			0
$947$	−34.6863			−1.12715			−0.563577	−	0.826064i	$-0.690575\pi$
$947$	−34.6863			−1.12715			−0.563577	+	0.826064i	$0.690575\pi$
$948$		0			0
$949$	5.17157			0.167876
$950$		0			0
$951$	2.92893	+	17.0711i	0.0949771	+	0.553567i
$952$		0			0
$953$		−	23.3137i		−	0.755205i	−0.925968	−	0.377603i	$-0.876749\pi$
$953$		−	23.3137i		−	0.755205i	0.925968	−	0.377603i	$-0.123251\pi$
$954$		0			0
$955$		−	4.48528i		−	0.145140i
$956$		0			0
$957$	−0.485281	−	2.82843i	−0.0156869	−	0.0914301i
$958$		0			0
$959$	−11.3137			−0.365339
$960$		0			0
$961$	−15.6274			−0.504110
$962$		0			0
$963$	−7.02944	+	2.48528i	−0.226520	+	0.0800871i
$964$		0			0
$965$			3.71573i			0.119614i
$966$		0			0
$967$		−	16.3431i		−	0.525560i	−0.964856	−	0.262780i	$-0.915361\pi$
$967$		−	16.3431i		−	0.525560i	0.964856	−	0.262780i	$-0.0846394\pi$
$968$		0			0
$969$	3.17157	−	0.544156i	0.101886	−	0.0174808i
$970$		0			0
$971$	−15.6985			−0.503788			−0.251894	−	0.967755i	$-0.581053\pi$
$971$	−15.6985			−0.503788			−0.251894	+	0.967755i	$0.581053\pi$
$972$		0			0
$973$	10.2426			0.328364
$974$		0			0
$975$	−4.65685	+	0.798990i	−0.149139	+	0.0255882i
$976$		0			0
$977$			44.0000i			1.40768i	0.710356	+	0.703842i	$0.248534\pi$
$977$			44.0000i			1.40768i	−0.710356	+	0.703842i	$0.751466\pi$
$978$		0			0
$979$			7.31371i			0.233747i
$980$		0			0
$981$	−40.9706	+	14.4853i	−1.30809	+	0.462479i
$982$		0			0
$983$	33.4558			1.06708			0.533538	−	0.845776i	$-0.320862\pi$
$983$	33.4558			1.06708			0.533538	+	0.845776i	$0.320862\pi$
$984$		0			0
$985$	15.2304			0.485282
$986$		0			0
$987$	2.82843	+	16.4853i	0.0900298	+	0.524732i
$988$		0			0
$989$		−	25.9411i		−	0.824880i
$990$		0			0
$991$			21.9411i			0.696983i	0.937312	+	0.348491i	$0.113306\pi$
$991$			21.9411i			0.696983i	−0.937312	+	0.348491i	$0.886694\pi$
$992$		0			0
$993$	0.242641	+	1.41421i	0.00769997	+	0.0448787i
$994$		0			0
$995$	−8.97056			−0.284386
$996$		0			0
$997$	−58.0416			−1.83820			−0.919098	−	0.394028i	$-0.871081\pi$
$997$	−58.0416			−1.83820			−0.919098	+	0.394028i	$0.871081\pi$
$998$		0			0
$999$	−21.8995	+	12.2426i	−0.692869	+	0.387340i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.h.e.575.3		4
3.2	odd	2		1344.2.h.a.575.1		4
4.3	odd	2		1344.2.h.a.575.2		4
8.3	odd	2		672.2.h.d.575.3	yes	4
8.5	even	2		672.2.h.a.575.2	yes	4
12.11	even	2	inner	1344.2.h.e.575.4		4
24.5	odd	2		672.2.h.d.575.4	yes	4
24.11	even	2		672.2.h.a.575.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.h.a.575.1	✓	4	24.11	even	2
672.2.h.a.575.2	yes	4	8.5	even	2
672.2.h.d.575.3	yes	4	8.3	odd	2
672.2.h.d.575.4	yes	4	24.5	odd	2
1344.2.h.a.575.1		4	3.2	odd	2
1344.2.h.a.575.2		4	4.3	odd	2
1344.2.h.e.575.3		4	1.1	even	1	trivial
1344.2.h.e.575.4		4	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.70711 - 0.292893i) q^{3} +0.585786i q^{5} +1.00000i q^{7} +(2.82843 - 1.00000i) q^{9} +2.00000 q^{11} -0.585786 q^{13} +(0.171573 + 1.00000i) q^{15} +0.828427i q^{17} -2.24264i q^{19} +(0.292893 + 1.70711i) q^{21} +4.00000 q^{23} +4.65685 q^{25} +(4.53553 - 2.53553i) q^{27} -0.828427i q^{29} +6.82843i q^{31} +(3.41421 - 0.585786i) q^{33} -0.585786 q^{35} -4.82843 q^{37} +(-1.00000 + 0.171573i) q^{39} +10.0000i q^{41} -6.48528i q^{43} +(0.585786 + 1.65685i) q^{45} +9.65685 q^{47} -1.00000 q^{49} +(0.242641 + 1.41421i) q^{51} -9.31371i q^{53} +1.17157i q^{55} +(-0.656854 - 3.82843i) q^{57} +2.24264 q^{59} -5.75736 q^{61} +(1.00000 + 2.82843i) q^{63} -0.343146i q^{65} +13.3137i q^{67} +(6.82843 - 1.17157i) q^{69} +11.6569 q^{71} -8.82843 q^{73} +(7.94975 - 1.36396i) q^{75} +2.00000i q^{77} +4.00000i q^{79} +(7.00000 - 5.65685i) q^{81} +8.58579 q^{83} -0.485281 q^{85} +(-0.242641 - 1.41421i) q^{87} +3.65685i q^{89} -0.585786i q^{91} +(2.00000 + 11.6569i) q^{93} +1.31371 q^{95} -17.3137 q^{97} +(5.65685 - 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 8 q^{11} - 8 q^{13} + 12 q^{15} + 4 q^{21} + 16 q^{23} - 4 q^{25} + 4 q^{27} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 4 q^{39} + 8 q^{45} + 16 q^{47} - 4 q^{49} - 16 q^{51} + 20 q^{57} - 8 q^{59}+ \cdots - 24 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 8 * q^11 - 8 * q^13 + 12 * q^15 + 4 * q^21 + 16 * q^23 - 4 * q^25 + 4 * q^27 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 4 * q^39 + 8 * q^45 + 16 * q^47 - 4 * q^49 - 16 * q^51 + 20 * q^57 - 8 * q^59 - 40 * q^61 + 4 * q^63 + 16 * q^69 + 24 * q^71 - 24 * q^73 + 12 * q^75 + 28 * q^81 + 40 * q^83 + 32 * q^85 + 16 * q^87 + 8 * q^93 - 40 * q^95 - 24 * q^97

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