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Properties

Label 720.2.t.c.541.2
Level $720$
Weight $2$
Character 720.541
Analytic conductor $5.749$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [720,2,Mod(181,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.181"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 541.2
Root \(-0.296075 + 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 720.541
Dual form 720.2.t.c.181.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.09971 + 0.889181i) q^{2} +(0.418713 - 1.95568i) q^{4} +(0.707107 - 0.707107i) q^{5} +2.66881i q^{7} +(1.27849 + 2.52299i) q^{8} +(-0.148864 + 1.40636i) q^{10} +(3.49714 - 3.49714i) q^{11} +(2.94072 + 2.94072i) q^{13} +(-2.37306 - 2.93491i) q^{14} +(-3.64936 - 1.63774i) q^{16} -1.85116 q^{17} +(-3.44856 - 3.44856i) q^{19} +(-1.08680 - 1.67895i) q^{20} +(-0.736240 + 6.95543i) q^{22} +0.707288i q^{23} -1.00000i q^{25} +(-5.84877 - 0.619099i) q^{26} +(5.21934 + 1.11747i) q^{28} +(3.49909 + 3.49909i) q^{29} +6.84272 q^{31} +(5.46947 - 1.44391i) q^{32} +(2.03573 - 1.64601i) q^{34} +(1.88714 + 1.88714i) q^{35} +(-0.0975060 + 0.0975060i) q^{37} +(6.85881 + 0.726013i) q^{38} +(2.68805 + 0.879991i) q^{40} +10.2052i q^{41} +(4.43844 - 4.43844i) q^{43} +(-5.37499 - 8.30359i) q^{44} +(-0.628908 - 0.777810i) q^{46} +1.89428 q^{47} -0.122561 q^{49} +(0.889181 + 1.09971i) q^{50} +(6.98243 - 4.51979i) q^{52} +(7.43897 - 7.43897i) q^{53} -4.94571i q^{55} +(-6.73338 + 3.41205i) q^{56} +(-6.95931 - 0.736651i) q^{58} +(-0.959574 + 0.959574i) q^{59} +(6.49825 + 6.49825i) q^{61} +(-7.52499 + 6.08442i) q^{62} +(-4.73092 + 6.45123i) q^{64} +4.15881 q^{65} +(3.49691 + 3.49691i) q^{67} +(-0.775103 + 3.62027i) q^{68} +(-3.75330 - 0.397291i) q^{70} -7.86777i q^{71} +15.6564i q^{73} +(0.0205276 - 0.193929i) q^{74} +(-8.18824 + 5.30033i) q^{76} +(9.33322 + 9.33322i) q^{77} -6.70212 q^{79} +(-3.73854 + 1.42243i) q^{80} +(-9.07431 - 11.2228i) q^{82} +(3.87327 + 3.87327i) q^{83} +(-1.30896 + 1.30896i) q^{85} +(-0.934407 + 8.82755i) q^{86} +(13.2943 + 4.35218i) q^{88} -10.5055i q^{89} +(-7.84824 + 7.84824i) q^{91} +(1.38323 + 0.296151i) q^{92} +(-2.08316 + 1.68436i) q^{94} -4.87701 q^{95} +4.79937 q^{97} +(0.134781 - 0.108979i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} + 4 q^{10} + 8 q^{11} - 4 q^{14} + 16 q^{16} - 8 q^{19} - 8 q^{20} - 20 q^{22} + 16 q^{26} - 4 q^{28} + 16 q^{29} + 16 q^{34} - 16 q^{37} - 20 q^{38} + 8 q^{43} - 40 q^{44} - 4 q^{46} + 40 q^{47}+ \cdots - 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\) \(1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.09971 + 0.889181i −0.777611 + 0.628746i
\(3\) 0 0
\(4\) 0.418713 − 1.95568i 0.209357 − 0.977839i
\(5\) 0.707107 − 0.707107i 0.316228 − 0.316228i
\(6\) 0 0
\(7\) 2.66881i 1.00872i 0.863495 + 0.504358i \(0.168271\pi\)
−0.863495 + 0.504358i \(0.831729\pi\)
\(8\) 1.27849 + 2.52299i 0.452015 + 0.892010i
\(9\) 0 0
\(10\) −0.148864 + 1.40636i −0.0470751 + 0.444729i
\(11\) 3.49714 − 3.49714i 1.05443 − 1.05443i 0.0559977 − 0.998431i \(-0.482166\pi\)
0.998431 − 0.0559977i \(-0.0178339\pi\)
\(12\) 0 0
\(13\) 2.94072 + 2.94072i 0.815610 + 0.815610i 0.985468 − 0.169858i \(-0.0543310\pi\)
−0.169858 + 0.985468i \(0.554331\pi\)
\(14\) −2.37306 − 2.93491i −0.634227 − 0.784389i
\(15\) 0 0
\(16\) −3.64936 − 1.63774i −0.912340 − 0.409434i
\(17\) −1.85116 −0.448971 −0.224486 − 0.974477i \(-0.572070\pi\)
−0.224486 + 0.974477i \(0.572070\pi\)
\(18\) 0 0
\(19\) −3.44856 − 3.44856i −0.791155 − 0.791155i 0.190527 − 0.981682i \(-0.438980\pi\)
−0.981682 + 0.190527i \(0.938980\pi\)
\(20\) −1.08680 − 1.67895i −0.243016 − 0.375424i
\(21\) 0 0
\(22\) −0.736240 + 6.95543i −0.156967 + 1.48290i
\(23\) 0.707288i 0.147480i 0.997278 + 0.0737399i \(0.0234935\pi\)
−0.997278 + 0.0737399i \(0.976507\pi\)
\(24\) 0 0
\(25\) − 1.00000i − 0.200000i
\(26\) −5.84877 − 0.619099i −1.14704 − 0.121415i
\(27\) 0 0
\(28\) 5.21934 + 1.11747i 0.986363 + 0.211181i
\(29\) 3.49909 + 3.49909i 0.649766 + 0.649766i 0.952936 − 0.303171i \(-0.0980452\pi\)
−0.303171 + 0.952936i \(0.598045\pi\)
\(30\) 0 0
\(31\) 6.84272 1.22899 0.614494 − 0.788921i \(-0.289360\pi\)
0.614494 + 0.788921i \(0.289360\pi\)
\(32\) 5.46947 − 1.44391i 0.966875 − 0.255250i
\(33\) 0 0
\(34\) 2.03573 − 1.64601i 0.349125 − 0.282289i
\(35\) 1.88714 + 1.88714i 0.318984 + 0.318984i
\(36\) 0 0
\(37\) −0.0975060 + 0.0975060i −0.0160299 + 0.0160299i −0.715076 − 0.699046i \(-0.753608\pi\)
0.699046 + 0.715076i \(0.253608\pi\)
\(38\) 6.85881 + 0.726013i 1.11265 + 0.117775i
\(39\) 0 0
\(40\) 2.68805 + 0.879991i 0.425018 + 0.139139i
\(41\) 10.2052i 1.59379i 0.604117 + 0.796896i \(0.293526\pi\)
−0.604117 + 0.796896i \(0.706474\pi\)
\(42\) 0 0
\(43\) 4.43844 − 4.43844i 0.676855 − 0.676855i −0.282432 − 0.959287i \(-0.591141\pi\)
0.959287 + 0.282432i \(0.0911412\pi\)
\(44\) −5.37499 − 8.30359i −0.810310 − 1.25181i
\(45\) 0 0
\(46\) −0.628908 − 0.777810i −0.0927274 − 0.114682i
\(47\) 1.89428 0.276310 0.138155 − 0.990411i \(-0.455883\pi\)
0.138155 + 0.990411i \(0.455883\pi\)
\(48\) 0 0
\(49\) −0.122561 −0.0175087
\(50\) 0.889181 + 1.09971i 0.125749 + 0.155522i
\(51\) 0 0
\(52\) 6.98243 − 4.51979i 0.968289 − 0.626782i
\(53\) 7.43897 − 7.43897i 1.02182 − 1.02182i 0.0220650 − 0.999757i \(-0.492976\pi\)
0.999757 − 0.0220650i \(-0.00702407\pi\)
\(54\) 0 0
\(55\) − 4.94571i − 0.666879i
\(56\) −6.73338 + 3.41205i −0.899786 + 0.455955i
\(57\) 0 0
\(58\) −6.95931 − 0.736651i −0.913802 − 0.0967270i
\(59\) −0.959574 + 0.959574i −0.124926 + 0.124926i −0.766805 − 0.641880i \(-0.778155\pi\)
0.641880 + 0.766805i \(0.278155\pi\)
\(60\) 0 0
\(61\) 6.49825 + 6.49825i 0.832015 + 0.832015i 0.987792 − 0.155777i \(-0.0497881\pi\)
−0.155777 + 0.987792i \(0.549788\pi\)
\(62\) −7.52499 + 6.08442i −0.955674 + 0.772722i
\(63\) 0 0
\(64\) −4.73092 + 6.45123i −0.591365 + 0.806404i
\(65\) 4.15881 0.515837
\(66\) 0 0
\(67\) 3.49691 + 3.49691i 0.427216 + 0.427216i 0.887679 − 0.460463i \(-0.152317\pi\)
−0.460463 + 0.887679i \(0.652317\pi\)
\(68\) −0.775103 + 3.62027i −0.0939951 + 0.439022i
\(69\) 0 0
\(70\) −3.75330 − 0.397291i −0.448605 − 0.0474854i
\(71\) − 7.86777i − 0.933733i −0.884328 − 0.466866i \(-0.845383\pi\)
0.884328 − 0.466866i \(-0.154617\pi\)
\(72\) 0 0
\(73\) 15.6564i 1.83244i 0.400675 + 0.916220i \(0.368776\pi\)
−0.400675 + 0.916220i \(0.631224\pi\)
\(74\) 0.0205276 − 0.193929i 0.00238628 − 0.0225437i
\(75\) 0 0
\(76\) −8.18824 + 5.30033i −0.939256 + 0.607989i
\(77\) 9.33322 + 9.33322i 1.06362 + 1.06362i
\(78\) 0 0
\(79\) −6.70212 −0.754047 −0.377024 − 0.926204i \(-0.623052\pi\)
−0.377024 + 0.926204i \(0.623052\pi\)
\(80\) −3.73854 + 1.42243i −0.417982 + 0.159033i
\(81\) 0 0
\(82\) −9.07431 − 11.2228i −1.00209 − 1.23935i
\(83\) 3.87327 + 3.87327i 0.425147 + 0.425147i 0.886971 − 0.461825i \(-0.152805\pi\)
−0.461825 + 0.886971i \(0.652805\pi\)
\(84\) 0 0
\(85\) −1.30896 + 1.30896i −0.141977 + 0.141977i
\(86\) −0.934407 + 8.82755i −0.100760 + 0.951900i
\(87\) 0 0
\(88\) 13.2943 + 4.35218i 1.41718 + 0.463944i
\(89\) − 10.5055i − 1.11358i −0.830653 − 0.556790i \(-0.812033\pi\)
0.830653 − 0.556790i \(-0.187967\pi\)
\(90\) 0 0
\(91\) −7.84824 + 7.84824i −0.822719 + 0.822719i
\(92\) 1.38323 + 0.296151i 0.144212 + 0.0308759i
\(93\) 0 0
\(94\) −2.08316 + 1.68436i −0.214861 + 0.173729i
\(95\) −4.87701 −0.500370
\(96\) 0 0
\(97\) 4.79937 0.487303 0.243651 − 0.969863i \(-0.421655\pi\)
0.243651 + 0.969863i \(0.421655\pi\)
\(98\) 0.134781 − 0.108979i 0.0136150 − 0.0110085i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 720.2.t.c.541.2 16
3.2 odd 2 80.2.l.a.61.7 yes 16
4.3 odd 2 2880.2.t.c.721.6 16
12.11 even 2 320.2.l.a.81.5 16
15.2 even 4 400.2.q.h.349.6 16
15.8 even 4 400.2.q.g.349.3 16
15.14 odd 2 400.2.l.h.301.2 16
16.5 even 4 inner 720.2.t.c.181.2 16
16.11 odd 4 2880.2.t.c.2161.7 16
24.5 odd 2 640.2.l.b.161.5 16
24.11 even 2 640.2.l.a.161.4 16
48.5 odd 4 80.2.l.a.21.7 ✓ 16
48.11 even 4 320.2.l.a.241.5 16
48.29 odd 4 640.2.l.b.481.5 16
48.35 even 4 640.2.l.a.481.4 16
60.23 odd 4 1600.2.q.h.849.4 16
60.47 odd 4 1600.2.q.g.849.5 16
60.59 even 2 1600.2.l.i.401.4 16
96.5 odd 8 5120.2.a.s.1.5 8
96.11 even 8 5120.2.a.t.1.5 8
96.53 odd 8 5120.2.a.v.1.4 8
96.59 even 8 5120.2.a.u.1.4 8
240.53 even 4 400.2.q.h.149.6 16
240.59 even 4 1600.2.l.i.1201.4 16
240.107 odd 4 1600.2.q.h.49.4 16
240.149 odd 4 400.2.l.h.101.2 16
240.197 even 4 400.2.q.g.149.3 16
240.203 odd 4 1600.2.q.g.49.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 48.5 odd 4
80.2.l.a.61.7 yes 16 3.2 odd 2
320.2.l.a.81.5 16 12.11 even 2
320.2.l.a.241.5 16 48.11 even 4
400.2.l.h.101.2 16 240.149 odd 4
400.2.l.h.301.2 16 15.14 odd 2
400.2.q.g.149.3 16 240.197 even 4
400.2.q.g.349.3 16 15.8 even 4
400.2.q.h.149.6 16 240.53 even 4
400.2.q.h.349.6 16 15.2 even 4
640.2.l.a.161.4 16 24.11 even 2
640.2.l.a.481.4 16 48.35 even 4
640.2.l.b.161.5 16 24.5 odd 2
640.2.l.b.481.5 16 48.29 odd 4
720.2.t.c.181.2 16 16.5 even 4 inner
720.2.t.c.541.2 16 1.1 even 1 trivial
1600.2.l.i.401.4 16 60.59 even 2
1600.2.l.i.1201.4 16 240.59 even 4
1600.2.q.g.49.5 16 240.203 odd 4
1600.2.q.g.849.5 16 60.47 odd 4
1600.2.q.h.49.4 16 240.107 odd 4
1600.2.q.h.849.4 16 60.23 odd 4
2880.2.t.c.721.6 16 4.3 odd 2
2880.2.t.c.2161.7 16 16.11 odd 4
5120.2.a.s.1.5 8 96.5 odd 8
5120.2.a.t.1.5 8 96.11 even 8
5120.2.a.u.1.4 8 96.59 even 8
5120.2.a.v.1.4 8 96.53 odd 8