LMFDB - Embedded newform 1440.2.x.q.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(127,1440)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1440.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.x (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.1698758656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 $ x^8 + 18x^6 + 97x^4 + 176*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			127.1
Root			$3.16053i$ of defining polynomial
Character	$\chi$	$=$	1440.127
Dual form			1440.2.x.q.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23483 + 0.0743018i) q^{5} +(-3.16053 + 3.16053i) q^{7} +O(q^{10})$	$q+(-2.23483 + 0.0743018i) q^{5} +(-3.16053 + 3.16053i) q^{7} -4.46967i q^{11} +(-2.51929 + 2.51929i) q^{13} +(2.30913 + 2.30913i) q^{17} +2.61827 q^{19} +(-1.64124 - 1.64124i) q^{23} +(4.98896 - 0.332104i) q^{25} -8.17246i q^{29} -4.00000i q^{31} +(6.82843 - 7.29809i) q^{35} +(5.80177 + 5.80177i) q^{37} +2.61827 q^{41} +(-5.14949 - 5.14949i) q^{43} +(-0.679824 + 0.679824i) q^{47} -12.9779i q^{49} +(7.81739 - 7.81739i) q^{53} +(0.332104 + 9.98896i) q^{55} -4.88998 q^{59} +12.2751 q^{61} +(5.44301 - 5.81739i) q^{65} +(9.44670 - 9.44670i) q^{67} -5.65685i q^{71} +(-5.61827 + 5.61827i) q^{73} +(14.1265 + 14.1265i) q^{77} -3.57969 q^{79} +(-1.34403 - 1.34403i) q^{83} +(-5.33210 - 4.98896i) q^{85} +17.6348i q^{89} -15.9246i q^{91} +(-5.85140 + 0.194542i) q^{95} +(-1.32106 - 1.32106i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} - 12 q^{13} + 4 q^{17} - 8 q^{19} - 4 q^{25} + 32 q^{35} + 12 q^{37} - 8 q^{41} + 24 q^{43} + 24 q^{47} - 4 q^{53} + 4 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{67} - 16 q^{73} + 32 q^{77} - 16 q^{79} + 16 q^{83} - 44 q^{85} - 40 q^{95} + 32 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 12 * q^13 + 4 * q^17 - 8 * q^19 - 4 * q^25 + 32 * q^35 + 12 * q^37 - 8 * q^41 + 24 * q^43 + 24 * q^47 - 4 * q^53 + 4 * q^55 - 16 * q^59 + 24 * q^61 - 4 * q^65 + 24 * q^67 - 16 * q^73 + 32 * q^77 - 16 * q^79 + 16 * q^83 - 44 * q^85 - 40 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.23483	+	0.0743018i	−0.999448	+	0.0332288i
$5$	−2.23483	+	0.0743018i	−0.999448	+	0.0332288i
$6$		0			0
$7$	−3.16053	+	3.16053i	−1.19457	+	1.19457i	−0.218799	+	0.975770i	$0.570214\pi$
$7$	−3.16053	+	3.16053i	−1.19457	+	1.19457i	−0.975770	+	0.218799i	$0.929786\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.46967i		−	1.34766i	−0.738889	−	0.673828i	$-0.764649\pi$
$11$		−	4.46967i		−	1.34766i	0.738889	−	0.673828i	$-0.235351\pi$
$12$		0			0
$13$	−2.51929	+	2.51929i	−0.698726	+	0.698726i	−0.964136	−	0.265410i	$-0.914493\pi$
$13$	−2.51929	+	2.51929i	−0.698726	+	0.698726i	0.265410	+	0.964136i	$0.414493\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.30913	+	2.30913i	0.560047	+	0.560047i	0.929321	−	0.369273i	$-0.120393\pi$
$17$	2.30913	+	2.30913i	0.560047	+	0.560047i	−0.369273	+	0.929321i	$0.620393\pi$
$18$		0			0
$19$	2.61827			0.600672			0.300336	−	0.953833i	$-0.402901\pi$
$19$	2.61827			0.600672			0.300336	+	0.953833i	$0.402901\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.64124	−	1.64124i	−0.342222	−	0.342222i	0.514980	−	0.857202i	$-0.327799\pi$
$23$	−1.64124	−	1.64124i	−0.342222	−	0.342222i	−0.857202	+	0.514980i	$0.827799\pi$
$24$		0			0
$25$	4.98896	−	0.332104i	0.997792	−	0.0664208i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.17246i		−	1.51759i	−0.651331	−	0.758794i	$-0.725789\pi$
$29$		−	8.17246i		−	1.51759i	0.651331	−	0.758794i	$-0.274211\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.82843	−	7.29809i	1.15421	−	1.23360i
$36$		0			0
$37$	5.80177	+	5.80177i	0.953805	+	0.953805i	0.998979	−	0.0451739i	$-0.0143842\pi$
$37$	5.80177	+	5.80177i	0.953805	+	0.953805i	−0.0451739	+	0.998979i	$0.514384\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.61827			0.408905			0.204453	−	0.978876i	$-0.434459\pi$
$41$	2.61827			0.408905			0.204453	+	0.978876i	$0.434459\pi$
$42$		0			0
$43$	−5.14949	−	5.14949i	−0.785290	−	0.785290i	0.195428	−	0.980718i	$-0.437390\pi$
$43$	−5.14949	−	5.14949i	−0.785290	−	0.785290i	−0.980718	+	0.195428i	$0.937390\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.679824	+	0.679824i	−0.0991625	+	0.0991625i	−0.754948	−	0.655785i	$-0.772338\pi$
$47$	−0.679824	+	0.679824i	−0.0991625	+	0.0991625i	0.655785	+	0.754948i	$0.272338\pi$
$48$		0			0
$49$		−	12.9779i		−	1.85399i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.81739	−	7.81739i	1.07380	−	1.07380i	0.0767501	−	0.997050i	$-0.475546\pi$
$53$	7.81739	−	7.81739i	1.07380	−	1.07380i	0.997050	−	0.0767501i	$-0.0244544\pi$
$54$		0			0
$55$	0.332104	+	9.98896i	0.0447809	+	1.34691i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.88998			−0.636621			−0.318311	−	0.947986i	$-0.603115\pi$
$59$	−4.88998			−0.636621			−0.318311	+	0.947986i	$0.603115\pi$
$60$		0			0
$61$	12.2751			1.57167			0.785834	−	0.618437i	$-0.212234\pi$
$61$	12.2751			1.57167			0.785834	+	0.618437i	$0.212234\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.44301	−	5.81739i	0.675122	−	0.721558i
$66$		0			0
$67$	9.44670	−	9.44670i	1.15410	−	1.15410i	0.168375	−	0.985723i	$-0.446148\pi$
$67$	9.44670	−	9.44670i	1.15410	−	1.15410i	0.985723	−	0.168375i	$-0.0538518\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		0			0
$73$	−5.61827	+	5.61827i	−0.657569	+	0.657569i	−0.954804	−	0.297235i	$-0.903935\pi$
$73$	−5.61827	+	5.61827i	−0.657569	+	0.657569i	0.297235	+	0.954804i	$0.403935\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	14.1265	+	14.1265i	1.60987	+	1.60987i
$78$		0			0
$79$	−3.57969			−0.402746			−0.201373	−	0.979515i	$-0.564540\pi$
$79$	−3.57969			−0.402746			−0.201373	+	0.979515i	$0.564540\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.34403	−	1.34403i	−0.147527	−	0.147527i	0.629486	−	0.777012i	$-0.283266\pi$
$83$	−1.34403	−	1.34403i	−0.147527	−	0.147527i	−0.777012	+	0.629486i	$0.783266\pi$
$84$		0			0
$85$	−5.33210	−	4.98896i	−0.578348	−	0.541129i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			17.6348i			1.86928i	0.355593	+	0.934641i	$0.384279\pi$
$89$			17.6348i			1.86928i	−0.355593	+	0.934641i	$0.615721\pi$
$90$		0			0
$91$		−	15.9246i		−	1.66935i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.85140	+	0.194542i	−0.600341	+	0.0199596i
$96$		0			0
$97$	−1.32106	−	1.32106i	−0.134134	−	0.134134i	0.636852	−	0.770986i	$-0.280236\pi$
$97$	−1.32106	−	1.32106i	−0.134134	−	0.134134i	−0.770986	+	0.636852i	$0.780236\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.47702			0.146969			0.0734845	−	0.997296i	$-0.476588\pi$
$101$	1.47702			0.146969			0.0734845	+	0.997296i	$0.476588\pi$
$102$		0			0
$103$	−9.11459	−	9.11459i	−0.898088	−	0.898088i	0.0971794	−	0.995267i	$-0.469018\pi$
$103$	−9.11459	−	9.11459i	−0.898088	−	0.898088i	−0.995267	+	0.0971794i	$0.969018\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.06155	+	2.06155i	−0.199298	+	0.199298i	−0.799699	−	0.600401i	$-0.795008\pi$
$107$	−2.06155	+	2.06155i	−0.199298	+	0.199298i	0.600401	+	0.799699i	$0.295008\pi$
$108$		0			0
$109$		−	9.70279i		−	0.929359i	−0.885479	−	0.464679i	$-0.846170\pi$
$109$		−	9.70279i		−	0.929359i	0.885479	−	0.464679i	$-0.153830\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.63020	−	4.63020i	0.435572	−	0.435572i	−0.454946	−	0.890519i	$-0.650342\pi$
$113$	4.63020	−	4.63020i	0.435572	−	0.435572i	0.890519	+	0.454946i	$0.150342\pi$
$114$		0			0
$115$	3.78984	+	3.54595i	0.353405	+	0.330661i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.5962			−1.33803
$120$		0			0
$121$	−8.97792			−0.816174
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1248	+	1.11289i	−0.995034	+	0.0995396i
$126$		0			0
$127$	10.1991	−	10.1991i	0.905025	−	0.905025i	−0.0908403	−	0.995865i	$-0.528955\pi$
$127$	10.1991	−	10.1991i	0.905025	−	0.905025i	0.995865	+	0.0908403i	$0.0289553\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.4697i		−	1.08948i	−0.838605	−	0.544740i	$-0.816628\pi$
$131$		−	12.4697i		−	1.08948i	0.838605	−	0.544740i	$-0.183372\pi$
$132$		0			0
$133$	−8.27512	+	8.27512i	−0.717544	+	0.717544i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.63020	−	4.63020i	−0.395584	−	0.395584i	0.481088	−	0.876672i	$-0.340242\pi$
$137$	−4.63020	−	4.63020i	−0.395584	−	0.395584i	−0.876672	+	0.481088i	$0.840242\pi$
$138$		0			0
$139$	−3.00735			−0.255080			−0.127540	−	0.991833i	$-0.540708\pi$
$139$	−3.00735			−0.255080			−0.127540	+	0.991833i	$0.540708\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.2604	+	11.2604i	0.941642	+	0.941642i
$144$		0			0
$145$	0.607228	+	18.2641i	0.0504276	+	1.51675i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	0.148604i		−	0.0121741i	−0.999981	−	0.00608704i	$-0.998062\pi$
$149$		−	0.148604i		−	0.0121741i	0.999981	−	0.00608704i	$-0.00193758\pi$
$150$		0			0
$151$			20.6200i			1.67804i	0.544104	+	0.839018i	$0.316870\pi$
$151$			20.6200i			1.67804i	−0.544104	+	0.839018i	$0.683130\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.297207	+	8.93933i	0.0238723	+	0.718024i
$156$		0			0
$157$	−9.77969	−	9.77969i	−0.780504	−	0.780504i	0.199412	−	0.979916i	$-0.436097\pi$
$157$	−9.77969	−	9.77969i	−0.780504	−	0.780504i	−0.979916	+	0.199412i	$0.936097\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.3744			0.817615
$162$		0			0
$163$	10.4081	+	10.4081i	0.815226	+	0.815226i	0.985412	−	0.170186i	$-0.0544368\pi$
$163$	10.4081	+	10.4081i	0.815226	+	0.815226i	−0.170186	+	0.985412i	$0.554437\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.06155	−	6.06155i	0.469057	−	0.469057i	−0.432552	−	0.901609i	$-0.642387\pi$
$167$	6.06155	−	6.06155i	0.469057	−	0.469057i	0.901609	+	0.432552i	$0.142387\pi$
$168$		0			0
$169$			0.306334i			0.0235641i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.355074	+	0.355074i	−0.0269957	+	0.0269957i	−0.720476	−	0.693480i	$-0.756077\pi$
$173$	−0.355074	+	0.355074i	−0.0269957	+	0.0269957i	0.693480	+	0.720476i	$0.256077\pi$
$174$		0			0
$175$	−14.7181	+	16.8174i	−1.11259	+	1.27127i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.7521			−1.17737			−0.588685	−	0.808362i	$-0.700354\pi$
$179$	−15.7521			−1.17737			−0.588685	+	0.808362i	$0.700354\pi$
$180$		0			0
$181$	7.77996			0.578280			0.289140	−	0.957287i	$-0.406631\pi$
$181$	7.77996			0.578280			0.289140	+	0.957287i	$0.406631\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.3971	−	12.5349i	−0.984972	−	0.921585i
$186$		0			0
$187$	10.3211	−	10.3211i	0.754751	−	0.754751i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			9.35965i			0.677240i	0.940923	+	0.338620i	$0.109960\pi$
$191$			9.35965i			0.677240i	−0.940923	+	0.338620i	$0.890040\pi$
$192$		0			0
$193$	17.1759	−	17.1759i	1.23635	−	1.23635i	0.274863	−	0.961483i	$-0.411368\pi$
$193$	17.1759	−	17.1759i	1.23635	−	1.23635i	0.961483	−	0.274863i	$-0.0886325\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.83947	−	1.83947i	−0.131057	−	0.131057i	0.638536	−	0.769592i	$-0.279541\pi$
$197$	−1.83947	−	1.83947i	−0.131057	−	0.131057i	−0.769592	+	0.638536i	$0.779541\pi$
$198$		0			0
$199$	−0.442397			−0.0313607			−0.0156804	−	0.999877i	$-0.504991\pi$
$199$	−0.442397			−0.0313607			−0.0156804	+	0.999877i	$0.504991\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	25.8293	+	25.8293i	1.81286	+	1.81286i
$204$		0			0
$205$	−5.85140	+	0.194542i	−0.408679	+	0.0135874i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	11.7028i		−	0.809499i
$210$		0			0
$211$			5.30633i			0.365303i	0.983178	+	0.182652i	$0.0584680\pi$
$211$			5.30633i			0.365303i	−0.983178	+	0.182652i	$0.941532\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.8909	+	11.1256i	0.810950	+	0.758762i
$216$		0			0
$217$	12.6421	+	12.6421i	0.858203	+	0.858203i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−11.6348			−0.782639
$222$		0			0
$223$	5.48159	+	5.48159i	0.367075	+	0.367075i	0.866409	−	0.499335i	$-0.166422\pi$
$223$	5.48159	+	5.48159i	0.367075	+	0.367075i	−0.499335	+	0.866409i	$0.666422\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.51649	−	9.51649i	0.631632	−	0.631632i	−0.316845	−	0.948477i	$-0.602624\pi$
$227$	9.51649	−	9.51649i	0.631632	−	0.631632i	0.948477	+	0.316845i	$0.102624\pi$
$228$		0			0
$229$		−	5.25862i		−	0.347500i	−0.984790	−	0.173750i	$-0.944412\pi$
$229$		−	5.25862i		−	0.347500i	0.984790	−	0.173750i	$-0.0555884\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.92740	+	2.92740i	−0.191781	+	0.191781i	−0.796465	−	0.604684i	$-0.793299\pi$
$233$	−2.92740	+	2.92740i	−0.191781	+	0.191781i	0.604684	+	0.796465i	$0.293299\pi$
$234$		0			0
$235$	1.46878	−	1.56980i	0.0958126	−	0.102403i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.6421			0.817751			0.408876	−	0.912590i	$-0.365921\pi$
$239$	12.6421			0.817751			0.408876	+	0.912590i	$0.365921\pi$
$240$		0			0
$241$	14.5650			0.938211			0.469106	−	0.883142i	$-0.344576\pi$
$241$	14.5650			0.938211			0.469106	+	0.883142i	$0.344576\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.964283	+	29.0035i	0.0616058	+	1.85296i
$246$		0			0
$247$	−6.59619	+	6.59619i	−0.419705	+	0.419705i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	18.5468i		−	1.17067i	−0.810793	−	0.585333i	$-0.800964\pi$
$251$		−	18.5468i		−	1.17067i	0.810793	−	0.585333i	$-0.199036\pi$
$252$		0			0
$253$	−7.33579	+	7.33579i	−0.461197	+	0.461197i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.2485	−	13.2485i	−0.826417	−	0.826417i	0.160602	−	0.987019i	$-0.448656\pi$
$257$	−13.2485	−	13.2485i	−0.826417	−	0.826417i	−0.987019	+	0.160602i	$0.948656\pi$
$258$		0			0
$259$	−36.6734			−2.27877
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.5806	−	22.5806i	−1.39238	−	1.39238i	−0.819967	−	0.572411i	$-0.806008\pi$
$263$	−22.5806	−	22.5806i	−1.39238	−	1.39238i	−0.572411	−	0.819967i	$-0.693992\pi$
$264$		0			0
$265$	−16.8897	+	18.0514i	−1.03753	+	1.10889i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6.82019i		−	0.415834i	−0.978146	−	0.207917i	$-0.933332\pi$
$269$		−	6.82019i		−	0.415834i	0.978146	−	0.207917i	$-0.0666684\pi$
$270$		0			0
$271$		−	2.74138i		−	0.166527i	−0.996528	−	0.0832634i	$-0.973466\pi$
$271$		−	2.74138i		−	0.166527i	0.996528	−	0.0832634i	$-0.0265343\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.48440	−	22.2990i	−0.0895124	−	1.34468i
$276$		0			0
$277$	−7.75583	−	7.75583i	−0.466003	−	0.466003i	0.434614	−	0.900617i	$-0.356885\pi$
$277$	−7.75583	−	7.75583i	−0.466003	−	0.466003i	−0.900617	+	0.434614i	$0.856885\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.54110			−0.270899			−0.135450	−	0.990784i	$-0.543248\pi$
$281$	−4.54110			−0.270899			−0.135450	+	0.990784i	$0.543248\pi$
$282$		0			0
$283$	7.91295	+	7.91295i	0.470376	+	0.470376i	0.902036	−	0.431660i	$-0.142072\pi$
$283$	7.91295	+	7.91295i	0.470376	+	0.470376i	−0.431660	+	0.902036i	$0.642072\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.27512	+	8.27512i	−0.488465	+	0.488465i
$288$		0			0
$289$		−	6.33579i		−	0.372694i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.0999	+	15.0999i	−0.882143	+	0.882143i	−0.993752	−	0.111609i	$-0.964400\pi$
$293$	−15.0999	+	15.0999i	−0.882143	+	0.882143i	0.111609	+	0.993752i	$0.464400\pi$
$294$		0			0
$295$	10.9283	−	0.363334i	0.636270	−	0.0211541i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.26952			0.478239
$300$		0			0
$301$	32.5502			1.87617
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−27.4329	+	0.912064i	−1.57080	+	0.0522246i
$306$		0			0
$307$	4.43197	−	4.43197i	0.252946	−	0.252946i	−0.569232	−	0.822177i	$-0.692759\pi$
$307$	4.43197	−	4.43197i	0.252946	−	0.252946i	0.822177	+	0.569232i	$0.192759\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.5355i			1.33458i	0.744799	+	0.667288i	$0.232545\pi$
$311$			23.5355i			1.33458i	−0.744799	+	0.667288i	$0.767455\pi$
$312$		0			0
$313$	−18.6035	+	18.6035i	−1.05153	+	1.05153i	−0.0529364	+	0.998598i	$0.516858\pi$
$313$	−18.6035	+	18.6035i	−1.05153	+	1.05153i	−0.998598	+	0.0529364i	$0.983142\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.56953	−	7.56953i	−0.425147	−	0.425147i	0.461824	−	0.886971i	$-0.347195\pi$
$317$	−7.56953	−	7.56953i	−0.425147	−	0.425147i	−0.886971	+	0.461824i	$0.847195\pi$
$318$		0			0
$319$	−36.5282			−2.04518
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.04594	+	6.04594i	0.336405	+	0.336405i
$324$		0			0
$325$	−11.7320	+	13.4053i	−0.650773	+	0.743593i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	4.29721i		−	0.236913i
$330$		0			0
$331$		−	13.8020i		−	0.758629i	−0.925268	−	0.379314i	$-0.876160\pi$
$331$		−	13.8020i		−	0.758629i	0.925268	−	0.379314i	$-0.123840\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−20.4099	+	21.8137i	−1.11511	+	1.19181i
$336$		0			0
$337$	10.9541	+	10.9541i	0.596706	+	0.596706i	0.939434	−	0.342729i	$-0.111351\pi$
$337$	10.9541	+	10.9541i	0.596706	+	0.596706i	−0.342729	+	0.939434i	$0.611351\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.8787			−0.968184
$342$		0			0
$343$	18.8934	+	18.8934i	1.02015	+	1.02015i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.3934	+	13.3934i	−0.718995	+	0.718995i	−0.968399	−	0.249405i	$-0.919765\pi$
$347$	−13.3934	+	13.3934i	−0.718995	+	0.718995i	0.249405	+	0.968399i	$0.419765\pi$
$348$		0			0
$349$		−	21.6035i		−	1.15641i	−0.815891	−	0.578206i	$-0.803753\pi$
$349$		−	21.6035i		−	1.15641i	0.815891	−	0.578206i	$-0.196247\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.9678	+	25.9678i	−1.38212	+	1.38212i	−0.541286	+	0.840839i	$0.682062\pi$
$353$	−25.9678	+	25.9678i	−1.38212	+	1.38212i	−0.840839	+	0.541286i	$0.817938\pi$
$354$		0			0
$355$	0.420314	+	12.6421i	0.0223080	+	0.670974i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.93933			0.471800			0.235900	−	0.971777i	$-0.424196\pi$
$359$	8.93933			0.471800			0.235900	+	0.971777i	$0.424196\pi$
$360$		0			0
$361$	−12.1447			−0.639193
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.1384	−	12.9733i	0.635355	−	0.679056i
$366$		0			0
$367$	13.7788	−	13.7788i	0.719248	−	0.719248i	−0.249204	−	0.968451i	$-0.580169\pi$
$367$	13.7788	−	13.7788i	0.719248	−	0.719248i	0.968451	+	0.249204i	$0.0801688\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			49.4142i			2.56546i
$372$		0			0
$373$	−12.4439	+	12.4439i	−0.644321	+	0.644321i	−0.951615	−	0.307294i	$-0.900576\pi$
$373$	−12.4439	+	12.4439i	−0.644321	+	0.644321i	0.307294	+	0.951615i	$0.400576\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.5888	+	20.5888i	1.06038	+	1.06038i
$378$		0			0
$379$	34.9190			1.79367			0.896835	−	0.442366i	$-0.145861\pi$
$379$	34.9190			1.79367			0.896835	+	0.442366i	$0.145861\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.69455	−	5.69455i	−0.290978	−	0.290978i	0.546489	−	0.837467i	$-0.315964\pi$
$383$	−5.69455	−	5.69455i	−0.290978	−	0.290978i	−0.837467	+	0.546489i	$0.815964\pi$
$384$		0			0
$385$	−32.6200	−	30.5208i	−1.66247	−	1.55548i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	8.44581i		−	0.428220i	−0.976810	−	0.214110i	$-0.931315\pi$
$389$		−	8.44581i		−	0.428220i	0.976810	−	0.214110i	$-0.0686850\pi$
$390$		0			0
$391$		−	7.57969i		−	0.383321i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.00000	−	0.265977i	0.402524	−	0.0133828i
$396$		0			0
$397$	18.0769	+	18.0769i	0.907253	+	0.907253i	0.996050	−	0.0887965i	$-0.0283021\pi$
$397$	18.0769	+	18.0769i	0.907253	+	0.907253i	−0.0887965	+	0.996050i	$0.528302\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	31.8787			1.59194			0.795972	−	0.605333i	$-0.206960\pi$
$401$	31.8787			1.59194			0.795972	+	0.605333i	$0.206960\pi$
$402$		0			0
$403$	10.0772	+	10.0772i	0.501980	+	0.501980i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25.9320	−	25.9320i	1.28540	−	1.28540i
$408$		0			0
$409$		−	27.3063i		−	1.35021i	−0.737721	−	0.675106i	$-0.764098\pi$
$409$		−	27.3063i		−	1.35021i	0.737721	−	0.675106i	$-0.235902\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	15.4549	−	15.4549i	0.760488	−	0.760488i
$414$		0			0
$415$	3.10355	+	2.90382i	0.152347	+	0.142543i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.1412			−0.788551			−0.394275	−	0.918992i	$-0.629004\pi$
$419$	−16.1412			−0.788551			−0.394275	+	0.918992i	$0.629004\pi$
$420$		0			0
$421$	−20.6200			−1.00496			−0.502480	−	0.864589i	$-0.667579\pi$
$421$	−20.6200			−1.00496			−0.502480	+	0.864589i	$0.667579\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.2871	+	10.7533i	0.596010	+	0.521612i
$426$		0			0
$427$	−38.7959	+	38.7959i	−1.87747	+	1.87747i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2.54848i			0.122756i	0.998115	+	0.0613779i	$0.0195495\pi$
$431$			2.54848i			0.122756i	−0.998115	+	0.0613779i	$0.980451\pi$
$432$		0			0
$433$	6.77996	−	6.77996i	0.325824	−	0.325824i	−0.525172	−	0.850996i	$-0.675999\pi$
$433$	6.77996	−	6.77996i	0.325824	−	0.325824i	0.850996	+	0.525172i	$0.175999\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.29721	−	4.29721i	−0.205563	−	0.205563i
$438$		0			0
$439$	15.0698			0.719243			0.359622	−	0.933098i	$-0.382906\pi$
$439$	15.0698			0.719243			0.359622	+	0.933098i	$0.382906\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.69059	+	9.69059i	0.460414	+	0.460414i	0.898791	−	0.438377i	$-0.144447\pi$
$443$	9.69059	+	9.69059i	0.460414	+	0.460414i	−0.438377	+	0.898791i	$0.644447\pi$
$444$		0			0
$445$	−1.31030	−	39.4108i	−0.0621139	−	1.86825i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	20.1759i		−	0.952158i	−0.879402	−	0.476079i	$-0.842058\pi$
$449$		−	20.1759i		−	0.952158i	0.879402	−	0.476079i	$-0.157942\pi$
$450$		0			0
$451$		−	11.7028i		−	0.551063i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.18323	+	35.5888i	0.0554705	+	1.66843i
$456$		0			0
$457$	−0.579686	−	0.579686i	−0.0271165	−	0.0271165i	0.693419	−	0.720535i	$-0.256104\pi$
$457$	−0.579686	−	0.579686i	−0.0271165	−	0.0271165i	−0.720535	+	0.693419i	$0.756104\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.40164			0.205005			0.102503	−	0.994733i	$-0.467315\pi$
$461$	4.40164			0.205005			0.102503	+	0.994733i	$0.467315\pi$
$462$		0			0
$463$	−0.786155	−	0.786155i	−0.0365357	−	0.0365357i	0.688603	−	0.725139i	$-0.258224\pi$
$463$	−0.786155	−	0.786155i	−0.0365357	−	0.0365357i	−0.725139	+	0.688603i	$0.758224\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.6430	+	19.6430i	−0.908970	+	0.908970i	−0.996189	−	0.0872190i	$-0.972202\pi$
$467$	−19.6430	+	19.6430i	−0.908970	+	0.908970i	0.0872190	+	0.996189i	$0.472202\pi$
$468$		0			0
$469$			59.7132i			2.75730i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−23.0165	+	23.0165i	−1.05830	+	1.05830i
$474$		0			0
$475$	13.0624	−	0.869539i	0.599346	−	0.0398972i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.6274			1.39940			0.699701	−	0.714436i	$-0.253316\pi$
$479$	30.6274			1.39940			0.699701	+	0.714436i	$0.253316\pi$
$480$		0			0
$481$	−29.2327			−1.33290
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.05051	+	2.85420i	0.138517	+	0.129602i
$486$		0			0
$487$	24.7641	−	24.7641i	1.12217	−	1.12217i	0.130752	−	0.991415i	$-0.458261\pi$
$487$	24.7641	−	24.7641i	1.12217	−	1.12217i	0.991415	−	0.130752i	$-0.0417392\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	23.3318i		−	1.05295i	−0.850190	−	0.526475i	$-0.823513\pi$
$491$		−	23.3318i		−	1.05295i	0.850190	−	0.526475i	$-0.176487\pi$
$492$		0			0
$493$	18.8713	−	18.8713i	0.849921	−	0.849921i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	17.8787	+	17.8787i	0.801968	+	0.801968i
$498$		0			0
$499$	−40.4051			−1.80878			−0.904389	−	0.426708i	$-0.859673\pi$
$499$	−40.4051			−1.80878			−0.904389	+	0.426708i	$0.859673\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.160805	−	0.160805i	−0.00716996	−	0.00716996i	0.703513	−	0.710683i	$-0.251614\pi$
$503$	−0.160805	−	0.160805i	−0.00716996	−	0.00716996i	−0.710683	+	0.703513i	$0.751614\pi$
$504$		0			0
$505$	−3.30089	+	0.109745i	−0.146888	+	0.00488360i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	11.4862i		−	0.509115i	−0.967058	−	0.254558i	$-0.918070\pi$
$509$		−	11.4862i		−	0.509115i	0.967058	−	0.254558i	$-0.0819299\pi$
$510$		0			0
$511$		−	35.5134i		−	1.57102i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	21.0468	+	19.6924i	0.927434	+	0.867749i
$516$		0			0
$517$	3.03858	+	3.03858i	0.133637	+	0.133637i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.3744			−0.717374			−0.358687	−	0.933458i	$-0.616776\pi$
$521$	−16.3744			−0.717374			−0.358687	+	0.933458i	$0.616776\pi$
$522$		0			0
$523$	5.02638	+	5.02638i	0.219788	+	0.219788i	0.808409	−	0.588621i	$-0.200329\pi$
$523$	5.02638	+	5.02638i	0.219788	+	0.219788i	−0.588621	+	0.808409i	$0.700329\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.23654	−	9.23654i	0.402350	−	0.402350i
$528$		0			0
$529$		−	17.6127i		−	0.765768i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.59619	+	6.59619i	−0.285713	+	0.285713i
$534$		0			0
$535$	4.45405	−	4.76041i	0.192565	−	0.205810i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−58.0070			−2.49854
$540$		0			0
$541$	−32.3302			−1.38998			−0.694992	−	0.719017i	$-0.744592\pi$
$541$	−32.3302			−1.38998			−0.694992	+	0.719017i	$0.744592\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.720935	+	21.6841i	0.0308815	+	0.928846i
$546$		0			0
$547$	−8.01165	+	8.01165i	−0.342554	+	0.342554i	−0.857327	−	0.514773i	$-0.827876\pi$
$547$	−8.01165	+	8.01165i	−0.342554	+	0.342554i	0.514773	+	0.857327i	$0.327876\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	21.3977i		−	0.911573i
$552$		0			0
$553$	11.3137	−	11.3137i	0.481108	−	0.481108i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.138448	−	0.138448i	−0.00586624	−	0.00586624i	0.704168	−	0.710034i	$-0.251320\pi$
$557$	−0.138448	−	0.138448i	−0.00586624	−	0.00586624i	−0.710034	+	0.704168i	$0.751320\pi$
$558$		0			0
$559$	25.9461			1.09740
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.7843	−	14.7843i	−0.623082	−	0.623082i	0.323236	−	0.946318i	$-0.395229\pi$
$563$	−14.7843	−	14.7843i	−0.623082	−	0.623082i	−0.946318	+	0.323236i	$0.895229\pi$
$564$		0			0
$565$	−10.0037	+	10.6918i	−0.420858	+	0.449805i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	29.9558i		−	1.25581i	−0.778288	−	0.627907i	$-0.783912\pi$
$569$		−	29.9558i		−	1.25581i	0.778288	−	0.627907i	$-0.216088\pi$
$570$		0			0
$571$			8.52134i			0.356607i	0.983976	+	0.178303i	$0.0570609\pi$
$571$			8.52134i			0.356607i	−0.983976	+	0.178303i	$0.942939\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.73314	−	7.64301i	−0.364197	−	0.318736i
$576$		0			0
$577$	10.9779	+	10.9779i	0.457017	+	0.457017i	0.897675	−	0.440658i	$-0.145255\pi$
$577$	10.9779	+	10.9779i	0.457017	+	0.457017i	−0.440658	+	0.897675i	$0.645255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.49571			0.352461
$582$		0			0
$583$	−34.9411	−	34.9411i	−1.44711	−	1.44711i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0174	−	18.0174i	0.743657	−	0.743657i	−0.229623	−	0.973280i	$-0.573749\pi$
$587$	18.0174	−	18.0174i	0.743657	−	0.743657i	0.973280	+	0.229623i	$0.0737492\pi$
$588$		0			0
$589$		−	10.4731i		−	0.431536i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	25.8906	−	25.8906i	1.06320	−	1.06320i	0.0653359	−	0.997863i	$-0.479188\pi$
$593$	25.8906	−	25.8906i	1.06320	−	1.06320i	0.997863	−	0.0653359i	$-0.0208119\pi$
$594$		0			0
$595$	32.6200	−	1.08452i	1.33729	−	0.0444611i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.89517			−0.363447			−0.181723	−	0.983350i	$-0.558168\pi$
$599$	−8.89517			−0.363447			−0.181723	+	0.983350i	$0.558168\pi$
$600$		0			0
$601$	35.3393			1.44152			0.720761	−	0.693184i	$-0.243793\pi$
$601$	35.3393			1.44152			0.720761	+	0.693184i	$0.243793\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	20.0641	−	0.667075i	0.815724	−	0.0271205i
$606$		0			0
$607$	−4.90926	+	4.90926i	−0.199261	+	0.199261i	−0.799683	−	0.600422i	$-0.794999\pi$
$607$	−4.90926	+	4.90926i	−0.199261	+	0.199261i	0.600422	+	0.799683i	$0.294999\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	3.42535i		−	0.138575i
$612$		0			0
$613$	1.87894	−	1.87894i	0.0758896	−	0.0758896i	−0.668143	−	0.744033i	$-0.732911\pi$
$613$	1.87894	−	1.87894i	0.0758896	−	0.0758896i	0.744033	+	0.668143i	$0.232911\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.07260	+	5.07260i	0.204215	+	0.204215i	0.801803	−	0.597588i	$-0.203874\pi$
$617$	5.07260	+	5.07260i	0.204215	+	0.204215i	−0.597588	+	0.801803i	$0.703874\pi$
$618$		0			0
$619$	−46.2769			−1.86003			−0.930013	−	0.367527i	$-0.880204\pi$
$619$	−46.2769			−1.86003			−0.930013	+	0.367527i	$0.880204\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−55.7352	−	55.7352i	−2.23299	−	2.23299i
$624$		0			0
$625$	24.7794	−	3.31371i	0.991177	−	0.132548i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			26.7941i			1.06835i
$630$		0			0
$631$		−	36.6457i		−	1.45884i	−0.684066	−	0.729421i	$-0.739790\pi$
$631$		−	36.6457i		−	1.45884i	0.684066	−	0.729421i	$-0.260210\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−22.0355	+	23.5511i	−0.874453	+	0.934598i
$636$		0			0
$637$	32.6952	+	32.6952i	1.29543	+	1.29543i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.1520			0.479976			0.239988	−	0.970776i	$-0.422857\pi$
$641$	12.1520			0.479976			0.239988	+	0.970776i	$0.422857\pi$
$642$		0			0
$643$	6.53122	+	6.53122i	0.257566	+	0.257566i	0.824064	−	0.566497i	$-0.191702\pi$
$643$	6.53122	+	6.53122i	0.257566	+	0.257566i	−0.566497	+	0.824064i	$0.691702\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.6192	−	17.6192i	0.692681	−	0.692681i	−0.270140	−	0.962821i	$-0.587070\pi$
$647$	17.6192	−	17.6192i	0.692681	−	0.692681i	0.962821	+	0.270140i	$0.0870701\pi$
$648$		0			0
$649$			21.8566i			0.857946i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.727677	+	0.727677i	−0.0284762	+	0.0284762i	−0.721202	−	0.692725i	$-0.756410\pi$
$653$	−0.727677	+	0.727677i	−0.0284762	+	0.0284762i	0.692725	+	0.721202i	$0.256410\pi$
$654$		0			0
$655$	0.926519	+	27.8676i	0.0362021	+	1.08888i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	44.9004			1.74907			0.874535	−	0.484963i	$-0.161167\pi$
$659$	44.9004			1.74907			0.874535	+	0.484963i	$0.161167\pi$
$660$		0			0
$661$	14.9320			0.580786			0.290393	−	0.956907i	$-0.406214\pi$
$661$	14.9320			0.580786			0.290393	+	0.956907i	$0.406214\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	17.8787	−	19.1084i	0.693305	−	0.740991i
$666$		0			0
$667$	−13.4130	+	13.4130i	−0.519352	+	0.519352i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	54.8657i		−	2.11807i
$672$		0			0
$673$	2.55583	−	2.55583i	0.0985200	−	0.0985200i	−0.656129	−	0.754649i	$-0.727807\pi$
$673$	2.55583	−	2.55583i	0.0985200	−	0.0985200i	0.754649	+	0.656129i	$0.227807\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.7147	−	17.7147i	−0.680832	−	0.680832i	0.279356	−	0.960188i	$-0.409879\pi$
$677$	−17.7147	−	17.7147i	−0.680832	−	0.680832i	−0.960188	+	0.279356i	$0.909879\pi$
$678$		0			0
$679$	8.35052			0.320464
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.3968	−	11.3968i	−0.436086	−	0.436086i	0.454606	−	0.890693i	$-0.349780\pi$
$683$	−11.3968	−	11.3968i	−0.436086	−	0.436086i	−0.890693	+	0.454606i	$0.849780\pi$
$684$		0			0
$685$	10.6918	+	10.0037i	0.408511	+	0.382221i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			39.3886i			1.50058i
$690$		0			0
$691$		−	6.94671i		−	0.264265i	−0.991232	−	0.132133i	$-0.957818\pi$
$691$		−	6.94671i		−	0.264265i	0.991232	−	0.132133i	$-0.0421825\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.72093	−	0.223452i	0.254940	−	0.00847601i
$696$		0			0
$697$	6.04594	+	6.04594i	0.229006	+	0.229006i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.70621			0.366598			0.183299	−	0.983057i	$-0.441322\pi$
$701$	9.70621			0.366598			0.183299	+	0.983057i	$0.441322\pi$
$702$		0			0
$703$	15.1906	+	15.1906i	0.572924	+	0.572924i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.66817	+	4.66817i	−0.175565	+	0.175565i
$708$		0			0
$709$		−	10.7414i		−	0.403401i	−0.979447	−	0.201700i	$-0.935353\pi$
$709$		−	10.7414i		−	0.403401i	0.979447	−	0.201700i	$-0.0646467\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.56496	+	6.56496i	−0.245860	+	0.245860i
$714$		0			0
$715$	−26.0018	−	24.3284i	−0.972411	−	0.909832i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	43.8640			1.63585			0.817925	−	0.575325i	$-0.195124\pi$
$719$	43.8640			1.63585			0.817925	+	0.575325i	$0.195124\pi$
$720$		0			0
$721$	57.6139			2.14565
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.71411	−	40.7721i	−0.100799	−	1.51424i
$726$		0			0
$727$	−9.95522	+	9.95522i	−0.369219	+	0.369219i	−0.867192	−	0.497974i	$-0.834078\pi$
$727$	−9.95522	+	9.95522i	−0.369219	+	0.369219i	0.497974	+	0.867192i	$0.334078\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	23.7817i		−	0.879599i
$732$		0			0
$733$	16.9924	−	16.9924i	0.627628	−	0.627628i	−0.319843	−	0.947471i	$-0.603630\pi$
$733$	16.9924	−	16.9924i	0.627628	−	0.627628i	0.947471	+	0.319843i	$0.103630\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−42.2236	−	42.2236i	−1.55533	−	1.55533i
$738$		0			0
$739$	−27.9025			−1.02641			−0.513205	−	0.858266i	$-0.671542\pi$
$739$	−27.9025			−1.02641			−0.513205	+	0.858266i	$0.671542\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20.2907	+	20.2907i	0.744395	+	0.744395i	0.973420	−	0.229025i	$-0.0735539\pi$
$743$	20.2907	+	20.2907i	0.744395	+	0.744395i	−0.229025	+	0.973420i	$0.573554\pi$
$744$		0			0
$745$	0.0110415	+	0.332104i	0.000404530	+	0.0121674i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	13.0312i		−	0.476150i
$750$		0			0
$751$			1.97054i			0.0719061i	0.999353	+	0.0359531i	$0.0114467\pi$
$751$			1.97054i			0.0719061i	−0.999353	+	0.0359531i	$0.988553\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.53211	−	46.0824i	−0.0557591	−	1.67711i
$756$		0			0
$757$	9.13933	+	9.13933i	0.332175	+	0.332175i	0.853412	−	0.521237i	$-0.174529\pi$
$757$	9.13933	+	9.13933i	0.332175	+	0.332175i	−0.521237	+	0.853412i	$0.674529\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.7800			0.862023			0.431011	−	0.902346i	$-0.358157\pi$
$761$	23.7800			0.862023			0.431011	+	0.902346i	$0.358157\pi$
$762$		0			0
$763$	30.6660	+	30.6660i	1.11018	+	1.11018i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.3193	−	12.3193i	0.444824	−	0.444824i
$768$		0			0
$769$		−	17.7022i		−	0.638359i	−0.947694	−	0.319180i	$-0.896593\pi$
$769$		−	17.7022i		−	0.638359i	0.947694	−	0.319180i	$-0.103407\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.30178	+	9.30178i	−0.334562	+	0.334562i	−0.854316	−	0.519754i	$-0.826024\pi$
$773$	−9.30178	+	9.30178i	−0.334562	+	0.334562i	0.519754	+	0.854316i	$0.326024\pi$
$774$		0			0
$775$	−1.32842	−	19.9558i	−0.0477181	−	0.716835i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.85534			0.245618
$780$		0			0
$781$	−25.2843			−0.904742
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.5826	+	21.1293i	0.806008	+	0.754138i
$786$		0			0
$787$	18.1642	−	18.1642i	0.647484	−	0.647484i	−0.304900	−	0.952384i	$-0.598623\pi$
$787$	18.1642	−	18.1642i	0.647484	−	0.647484i	0.952384	+	0.304900i	$0.0986230\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			29.2678i			1.04064i
$792$		0			0
$793$	−30.9246	+	30.9246i	−1.09817	+	1.09817i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.75098	+	4.75098i	0.168289	+	0.168289i	0.786227	−	0.617938i	$-0.212032\pi$
$797$	4.75098	+	4.75098i	0.168289	+	0.168289i	−0.617938	+	0.786227i	$0.712032\pi$
$798$		0			0
$799$	−3.13961			−0.111071
$800$		0			0
$801$		0			0
$802$		0			0
$803$	25.1118	+	25.1118i	0.886176	+	0.886176i
$804$		0			0
$805$	−23.1850	+	0.770835i	−0.817164	+	0.0271684i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	37.7891i		−	1.32859i	−0.747469	−	0.664297i	$-0.768731\pi$
$809$		−	37.7891i		−	1.32859i	0.747469	−	0.664297i	$-0.231269\pi$
$810$		0			0
$811$			44.4215i			1.55985i	0.625872	+	0.779926i	$0.284743\pi$
$811$			44.4215i			1.55985i	−0.625872	+	0.779926i	$0.715257\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−24.0337	−	22.4871i	−0.841865	−	0.787687i
$816$		0			0
$817$	−13.4828	−	13.4828i	−0.471702	−	0.471702i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	29.3648			1.02484			0.512420	−	0.858735i	$-0.328749\pi$
$821$	29.3648			1.02484			0.512420	+	0.858735i	$0.328749\pi$
$822$		0			0
$823$	−22.7402	−	22.7402i	−0.792674	−	0.792674i	0.189254	−	0.981928i	$-0.439393\pi$
$823$	−22.7402	−	22.7402i	−0.792674	−	0.792674i	−0.981928	+	0.189254i	$0.939393\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.50001	−	1.50001i	0.0521605	−	0.0521605i	−0.680545	−	0.732706i	$-0.738257\pi$
$827$	1.50001	−	1.50001i	0.0521605	−	0.0521605i	0.732706	+	0.680545i	$0.238257\pi$
$828$		0			0
$829$		−	29.0642i		−	1.00944i	−0.863283	−	0.504721i	$-0.831595\pi$
$829$		−	29.0642i		−	1.00944i	0.863283	−	0.504721i	$-0.168405\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	29.9678	−	29.9678i	1.03832	−	1.03832i
$834$		0			0
$835$	−13.0962	+	13.9969i	−0.453212	+	0.484384i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−38.3779			−1.32495			−0.662476	−	0.749083i	$-0.730495\pi$
$839$	−38.3779			−1.32495			−0.662476	+	0.749083i	$0.730495\pi$
$840$		0			0
$841$	−37.7891			−1.30307
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.0227612	−	0.684605i	−0.000783008	−	0.0235511i
$846$		0			0
$847$	28.3750	−	28.3750i	0.974976	−	0.974976i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	19.0442i		−	0.652826i
$852$		0			0
$853$	28.1467	−	28.1467i	0.963724	−	0.963724i	−0.0356404	−	0.999365i	$-0.511347\pi$
$853$	28.1467	−	28.1467i	0.963724	−	0.963724i	0.999365	+	0.0356404i	$0.0113471\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13.5990	+	13.5990i	0.464533	+	0.464533i	0.900138	−	0.435605i	$-0.143466\pi$
$857$	13.5990	+	13.5990i	0.464533	+	0.464533i	−0.435605	+	0.900138i	$0.643466\pi$
$858$		0			0
$859$	−14.9155			−0.508910			−0.254455	−	0.967085i	$-0.581896\pi$
$859$	−14.9155			−0.508910			−0.254455	+	0.967085i	$0.581896\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.95495	+	8.95495i	0.304830	+	0.304830i	0.842900	−	0.538070i	$-0.180846\pi$
$863$	8.95495	+	8.95495i	0.304830	+	0.304830i	−0.538070	+	0.842900i	$0.680846\pi$
$864$		0			0
$865$	0.767147	−	0.819913i	0.0260838	−	0.0278779i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			16.0000i			0.542763i
$870$		0			0
$871$			47.5980i			1.61280i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.6430	−	38.6776i	1.06973	−	1.30754i
$876$		0			0
$877$	−37.8274	−	37.8274i	−1.27734	−	1.27734i	−0.942151	−	0.335190i	$-0.891200\pi$
$877$	−37.8274	−	37.8274i	−1.27734	−	1.27734i	−0.335190	−	0.942151i	$-0.608800\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23.7119			−0.798876			−0.399438	−	0.916760i	$-0.630795\pi$
$881$	−23.7119			−0.798876			−0.399438	+	0.916760i	$0.630795\pi$
$882$		0			0
$883$	−9.39338	−	9.39338i	−0.316113	−	0.316113i	0.531159	−	0.847272i	$-0.321757\pi$
$883$	−9.39338	−	9.39338i	−0.316113	−	0.316113i	−0.847272	+	0.531159i	$0.821757\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.825014	+	0.825014i	−0.0277013	+	0.0277013i	−0.720822	−	0.693120i	$-0.756235\pi$
$887$	−0.825014	+	0.825014i	−0.0277013	+	0.0277013i	0.693120	+	0.720822i	$0.256235\pi$
$888$		0			0
$889$			64.4693i			2.16223i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.77996	+	1.77996i	−0.0595641	+	0.0595641i
$894$		0			0
$895$	35.2034	−	1.17041i	1.17672	−	0.0391226i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−32.6898			−1.09027
$900$		0			0
$901$	36.1028			1.20276
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17.3869	+	0.578065i	−0.577961	+	0.0192155i
$906$		0			0
$907$	−1.59189	+	1.59189i	−0.0528578	+	0.0528578i	−0.733042	−	0.680184i	$-0.761900\pi$
$907$	−1.59189	+	1.59189i	−0.0528578	+	0.0528578i	0.680184	+	0.733042i	$0.261900\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			55.3926i			1.83524i	0.397458	+	0.917620i	$0.369892\pi$
$911$			55.3926i			1.83524i	−0.397458	+	0.917620i	$0.630108\pi$
$912$		0			0
$913$	−6.00737	+	6.00737i	−0.198815	+	0.198815i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	39.4108	+	39.4108i	1.30146	+	1.30146i
$918$		0			0
$919$	−4.31369			−0.142295			−0.0711477	−	0.997466i	$-0.522666\pi$
$919$	−4.31369			−0.142295			−0.0711477	+	0.997466i	$0.522666\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	14.2513	+	14.2513i	0.469086	+	0.469086i
$924$		0			0
$925$	30.8716	+	27.0180i	1.01505	+	0.888346i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	19.9774i		−	0.655436i	−0.944776	−	0.327718i	$-0.893720\pi$
$929$		−	19.9774i		−	0.655436i	0.944776	−	0.327718i	$-0.106280\pi$
$930$		0			0
$931$		−	33.9797i		−	1.11364i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−22.2990	+	23.8327i	−0.729255	+	0.779414i
$936$		0			0
$937$	−8.97792	−	8.97792i	−0.293296	−	0.293296i	0.545085	−	0.838381i	$-0.316497\pi$
$937$	−8.97792	−	8.97792i	−0.293296	−	0.293296i	−0.838381	+	0.545085i	$0.816497\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	37.9831			1.23821			0.619107	−	0.785307i	$-0.287495\pi$
$941$	37.9831			1.23821			0.619107	+	0.785307i	$0.287495\pi$
$942$		0			0
$943$	−4.29721	−	4.29721i	−0.139936	−	0.139936i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−39.0815	+	39.0815i	−1.26998	+	1.26998i	−0.323878	+	0.946099i	$0.604987\pi$
$947$	−39.0815	+	39.0815i	−1.26998	+	1.26998i	−0.946099	+	0.323878i	$0.895013\pi$
$948$		0			0
$949$		−	28.3081i		−	0.918921i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.27232	−	1.27232i	0.0412146	−	0.0412146i	−0.686199	−	0.727414i	$-0.740722\pi$
$953$	1.27232	−	1.27232i	0.0412146	−	0.0412146i	0.727414	+	0.686199i	$0.240722\pi$
$954$		0			0
$955$	−0.695439	−	20.9172i	−0.0225039	−	0.676866i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	29.2678			0.945106
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−37.1090	+	39.6614i	−1.19458	+	1.27675i
$966$		0			0
$967$	−1.67778	+	1.67778i	−0.0539537	+	0.0539537i	−0.733569	−	0.679615i	$-0.762147\pi$
$967$	−1.67778	+	1.67778i	−0.0539537	+	0.0539537i	0.679615	+	0.733569i	$0.262147\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	15.9887i		−	0.513102i	−0.966531	−	0.256551i	$-0.917414\pi$
$971$		−	15.9887i		−	0.513102i	0.966531	−	0.256551i	$-0.0825862\pi$
$972$		0			0
$973$	9.50484	−	9.50484i	0.304711	−	0.304711i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−16.7754	−	16.7754i	−0.536692	−	0.536692i	0.385864	−	0.922556i	$-0.373904\pi$
$977$	−16.7754	−	16.7754i	−0.536692	−	0.536692i	−0.922556	+	0.385864i	$0.873904\pi$
$978$		0			0
$979$	78.8215			2.51915
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−34.3367	−	34.3367i	−1.09517	−	1.09517i	−0.994967	−	0.100203i	$-0.968051\pi$
$983$	−34.3367	−	34.3367i	−1.09517	−	1.09517i	−0.100203	−	0.994967i	$-0.531949\pi$
$984$		0			0
$985$	4.24758	+	3.97423i	0.135339	+	0.126629i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			16.9031i			0.537487i
$990$		0			0
$991$			26.7414i			0.849468i	0.905318	+	0.424734i	$0.139632\pi$
$991$			26.7414i			0.849468i	−0.905318	+	0.424734i	$0.860368\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.988685	−	0.0328709i	0.0313434	−	0.00104208i
$996$		0			0
$997$	−9.80177	−	9.80177i	−0.310425	−	0.310425i	0.534649	−	0.845074i	$-0.320444\pi$
$997$	−9.80177	−	9.80177i	−0.310425	−	0.310425i	−0.845074	+	0.534649i	$0.820444\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.x.q.127.1		8
3.2	odd	2		480.2.w.c.127.2	✓	8
4.3	odd	2		1440.2.x.r.127.1		8
5.3	odd	4		1440.2.x.r.703.1		8
12.11	even	2		480.2.w.d.127.4	yes	8
15.2	even	4		2400.2.w.i.2143.1		8
15.8	even	4		480.2.w.d.223.4	yes	8
15.14	odd	2		2400.2.w.j.607.4		8
20.3	even	4	inner	1440.2.x.q.703.1		8
24.5	odd	2		960.2.w.e.127.3		8
24.11	even	2		960.2.w.f.127.1		8
60.23	odd	4		480.2.w.c.223.2	yes	8
60.47	odd	4		2400.2.w.j.2143.4		8
60.59	even	2		2400.2.w.i.607.1		8
120.53	even	4		960.2.w.f.703.1		8
120.83	odd	4		960.2.w.e.703.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.w.c.127.2	✓	8	3.2	odd	2
480.2.w.c.223.2	yes	8	60.23	odd	4
480.2.w.d.127.4	yes	8	12.11	even	2
480.2.w.d.223.4	yes	8	15.8	even	4
960.2.w.e.127.3		8	24.5	odd	2
960.2.w.e.703.3		8	120.83	odd	4
960.2.w.f.127.1		8	24.11	even	2
960.2.w.f.703.1		8	120.53	even	4
1440.2.x.q.127.1		8	1.1	even	1	trivial
1440.2.x.q.703.1		8	20.3	even	4	inner
1440.2.x.r.127.1		8	4.3	odd	2
1440.2.x.r.703.1		8	5.3	odd	4
2400.2.w.i.607.1		8	60.59	even	2
2400.2.w.i.2143.1		8	15.2	even	4
2400.2.w.j.607.4		8	15.14	odd	2
2400.2.w.j.2143.4		8	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.23483 + 0.0743018i) q^{5} +(-3.16053 + 3.16053i) q^{7} +O(q^{10})\)	\(q+(-2.23483 + 0.0743018i) q^{5} +(-3.16053 + 3.16053i) q^{7} -4.46967i q^{11} +(-2.51929 + 2.51929i) q^{13} +(2.30913 + 2.30913i) q^{17} +2.61827 q^{19} +(-1.64124 - 1.64124i) q^{23} +(4.98896 - 0.332104i) q^{25} -8.17246i q^{29} -4.00000i q^{31} +(6.82843 - 7.29809i) q^{35} +(5.80177 + 5.80177i) q^{37} +2.61827 q^{41} +(-5.14949 - 5.14949i) q^{43} +(-0.679824 + 0.679824i) q^{47} -12.9779i q^{49} +(7.81739 - 7.81739i) q^{53} +(0.332104 + 9.98896i) q^{55} -4.88998 q^{59} +12.2751 q^{61} +(5.44301 - 5.81739i) q^{65} +(9.44670 - 9.44670i) q^{67} -5.65685i q^{71} +(-5.61827 + 5.61827i) q^{73} +(14.1265 + 14.1265i) q^{77} -3.57969 q^{79} +(-1.34403 - 1.34403i) q^{83} +(-5.33210 - 4.98896i) q^{85} +17.6348i q^{89} -15.9246i q^{91} +(-5.85140 + 0.194542i) q^{95} +(-1.32106 - 1.32106i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} - 12 q^{13} + 4 q^{17} - 8 q^{19} - 4 q^{25} + 32 q^{35} + 12 q^{37} - 8 q^{41} + 24 q^{43} + 24 q^{47} - 4 q^{53} + 4 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{67} - 16 q^{73} + 32 q^{77} - 16 q^{79} + 16 q^{83} - 44 q^{85} - 40 q^{95} + 32 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 12 * q^13 + 4 * q^17 - 8 * q^19 - 4 * q^25 + 32 * q^35 + 12 * q^37 - 8 * q^41 + 24 * q^43 + 24 * q^47 - 4 * q^53 + 4 * q^55 - 16 * q^59 + 24 * q^61 - 4 * q^65 + 24 * q^67 - 16 * q^73 + 32 * q^77 - 16 * q^79 + 16 * q^83 - 44 * q^85 - 40 * q^95 + 32 * q^97

Introduction

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