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Properties

Label 80.2.l.a.61.7
Level $80$
Weight $2$
Character 80.61
Analytic conductor $0.639$
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] = [80,2,Mod(21,80)] 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("80.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.638803216170\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 61.7
Root \(-0.296075 + 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 80.61
Dual form 80.2.l.a.21.7

$q$-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} +(-0.120009 - 0.120009i) q^{3} +(0.418713 - 1.95568i) q^{4} +(-0.707107 + 0.707107i) q^{5} +(-0.238684 - 0.0252650i) q^{6} +2.66881i q^{7} +(-1.27849 - 2.52299i) q^{8} -2.97120i q^{9} +(-0.148864 + 1.40636i) q^{10} +(-3.49714 + 3.49714i) q^{11} +(-0.284948 + 0.184450i) q^{12} +(2.94072 + 2.94072i) q^{13} +(2.37306 + 2.93491i) q^{14} +0.169718 q^{15} +(-3.64936 - 1.63774i) q^{16} +1.85116 q^{17} +(-2.64193 - 3.26745i) q^{18} +(-3.44856 - 3.44856i) q^{19} +(1.08680 + 1.67895i) q^{20} +(0.320281 - 0.320281i) q^{21} +(-0.736240 + 6.95543i) q^{22} -0.707288i q^{23} +(-0.149351 + 0.456211i) q^{24} -1.00000i q^{25} +(5.84877 + 0.619099i) q^{26} +(-0.716597 + 0.716597i) q^{27} +(5.21934 + 1.11747i) q^{28} +(-3.49909 - 3.49909i) q^{29} +(0.186640 - 0.150910i) q^{30} +6.84272 q^{31} +(-5.46947 + 1.44391i) q^{32} +0.839377 q^{33} +(2.03573 - 1.64601i) q^{34} +(-1.88714 - 1.88714i) q^{35} +(-5.81070 - 1.24408i) q^{36} +(-0.0975060 + 0.0975060i) q^{37} +(-6.85881 - 0.726013i) q^{38} -0.705826i q^{39} +(2.68805 + 0.879991i) q^{40} -10.2052i q^{41} +(0.0674276 - 0.637004i) q^{42} +(4.43844 - 4.43844i) q^{43} +(5.37499 + 8.30359i) q^{44} +(2.10095 + 2.10095i) q^{45} +(-0.628908 - 0.777810i) q^{46} -1.89428 q^{47} +(0.241413 + 0.634498i) q^{48} -0.122561 q^{49} +(-0.889181 - 1.09971i) q^{50} +(-0.222155 - 0.222155i) q^{51} +(6.98243 - 4.51979i) q^{52} +(-7.43897 + 7.43897i) q^{53} +(-0.150862 + 1.42523i) q^{54} -4.94571i q^{55} +(6.73338 - 3.41205i) q^{56} +0.827717i q^{57} +(-6.95931 - 0.736651i) q^{58} +(0.959574 - 0.959574i) q^{59} +(0.0710632 - 0.331914i) q^{60} +(6.49825 + 6.49825i) q^{61} +(7.52499 - 6.08442i) q^{62} +7.92956 q^{63} +(-4.73092 + 6.45123i) q^{64} -4.15881 q^{65} +(0.923069 - 0.746358i) q^{66} +(3.49691 + 3.49691i) q^{67} +(0.775103 - 3.62027i) q^{68} +(-0.0848809 + 0.0848809i) q^{69} +(-3.75330 - 0.397291i) q^{70} +7.86777i q^{71} +(-7.49629 + 3.79865i) q^{72} +15.6564i q^{73} +(-0.0205276 + 0.193929i) q^{74} +(-0.120009 + 0.120009i) q^{75} +(-8.18824 + 5.30033i) q^{76} +(-9.33322 - 9.33322i) q^{77} +(-0.627607 - 0.776202i) q^{78} -6.70212 q^{79} +(3.73854 - 1.42243i) q^{80} -8.74159 q^{81} +(-9.07431 - 11.2228i) q^{82} +(-3.87327 - 3.87327i) q^{83} +(-0.492261 - 0.760473i) q^{84} +(-1.30896 + 1.30896i) q^{85} +(0.934407 - 8.82755i) q^{86} +0.839845i q^{87} +(13.2943 + 4.35218i) q^{88} +10.5055i q^{89} +(4.17856 + 0.442305i) q^{90} +(-7.84824 + 7.84824i) q^{91} +(-1.38323 - 0.296151i) q^{92} +(-0.821187 - 0.821187i) q^{93} +(-2.08316 + 1.68436i) q^{94} +4.87701 q^{95} +(0.829667 + 0.483103i) q^{96} +4.79937 q^{97} +(-0.134781 + 0.108979i) q^{98} +(10.3907 + 10.3907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 12 q^{6} + 4 q^{10} - 8 q^{11} - 12 q^{12} + 4 q^{14} - 8 q^{15} + 16 q^{16} - 8 q^{19} + 8 q^{20} - 20 q^{22} + 8 q^{24} - 16 q^{26} + 24 q^{27} - 4 q^{28} - 16 q^{29} + 16 q^{34} - 4 q^{36}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −0.120009 − 0.120009i −0.0692872 − 0.0692872i 0.671614 − 0.740901i \(-0.265601\pi\)
−0.740901 + 0.671614i \(0.765601\pi\)
\(4\) 0.418713 − 1.95568i 0.209357 − 0.977839i
\(5\) −0.707107 + 0.707107i −0.316228 + 0.316228i
\(6\) −0.238684 − 0.0252650i −0.0974425 − 0.0103144i
\(7\) 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
\(9\) − 2.97120i − 0.990399i
\(10\) −0.148864 + 1.40636i −0.0470751 + 0.444729i
\(11\) −3.49714 + 3.49714i −1.05443 + 1.05443i −0.0559977 + 0.998431i \(0.517834\pi\)
−0.998431 + 0.0559977i \(0.982166\pi\)
\(12\) −0.284948 + 0.184450i −0.0822574 + 0.0532460i
\(13\) 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
\(15\) 0.169718 0.0438211
\(16\) −3.64936 − 1.63774i −0.912340 − 0.409434i
\(17\) 1.85116 0.448971 0.224486 − 0.974477i \(-0.427930\pi\)
0.224486 + 0.974477i \(0.427930\pi\)
\(18\) −2.64193 − 3.26745i −0.622709 − 0.770144i
\(19\) −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
\(21\) 0.320281 − 0.320281i 0.0698911 − 0.0698911i
\(22\) −0.736240 + 6.95543i −0.156967 + 1.48290i
\(23\) − 0.707288i − 0.147480i −0.997278 − 0.0737399i \(-0.976507\pi\)
0.997278 − 0.0737399i \(-0.0234935\pi\)
\(24\) −0.149351 + 0.456211i −0.0304860 + 0.0931237i
\(25\) − 1.00000i − 0.200000i
\(26\) 5.84877 + 0.619099i 1.14704 + 0.121415i
\(27\) −0.716597 + 0.716597i −0.137909 + 0.137909i
\(28\) 5.21934 + 1.11747i 0.986363 + 0.211181i
\(29\) −3.49909 − 3.49909i −0.649766 − 0.649766i 0.303171 − 0.952936i \(-0.401955\pi\)
−0.952936 + 0.303171i \(0.901955\pi\)
\(30\) 0.186640 − 0.150910i 0.0340757 − 0.0275523i
\(31\) 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
\(33\) 0.839377 0.146117
\(34\) 2.03573 − 1.64601i 0.349125 − 0.282289i
\(35\) −1.88714 − 1.88714i −0.318984 − 0.318984i
\(36\) −5.81070 − 1.24408i −0.968451 − 0.207346i
\(37\) −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
\(39\) − 0.705826i − 0.113023i
\(40\) 2.68805 + 0.879991i 0.425018 + 0.139139i
\(41\) − 10.2052i − 1.59379i −0.604117 − 0.796896i \(-0.706474\pi\)
0.604117 − 0.796896i \(-0.293526\pi\)
\(42\) 0.0674276 − 0.637004i 0.0104043 − 0.0982918i
\(43\) 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
\(45\) 2.10095 + 2.10095i 0.313192 + 0.313192i
\(46\) −0.628908 − 0.777810i −0.0927274 − 0.114682i
\(47\) −1.89428 −0.276310 −0.138155 − 0.990411i \(-0.544117\pi\)
−0.138155 + 0.990411i \(0.544117\pi\)
\(48\) 0.241413 + 0.634498i 0.0348449 + 0.0915820i
\(49\) −0.122561 −0.0175087
\(50\) −0.889181 − 1.09971i −0.125749 − 0.155522i
\(51\) −0.222155 − 0.222155i −0.0311079 − 0.0311079i
\(52\) 6.98243 − 4.51979i 0.968289 − 0.626782i
\(53\) −7.43897 + 7.43897i −1.02182 + 1.02182i −0.0220650 + 0.999757i \(0.507024\pi\)
−0.999757 + 0.0220650i \(0.992976\pi\)
\(54\) −0.150862 + 1.42523i −0.0205298 + 0.193949i
\(55\) − 4.94571i − 0.666879i
\(56\) 6.73338 − 3.41205i 0.899786 − 0.455955i
\(57\) 0.827717i 0.109634i
\(58\) −6.95931 − 0.736651i −0.913802 − 0.0967270i
\(59\) 0.959574 − 0.959574i 0.124926 − 0.124926i −0.641880 − 0.766805i \(-0.721845\pi\)
0.766805 + 0.641880i \(0.221845\pi\)
\(60\) 0.0710632 − 0.331914i 0.00917422 − 0.0428500i
\(61\) 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
\(63\) 7.92956 0.999031
\(64\) −4.73092 + 6.45123i −0.591365 + 0.806404i
\(65\) −4.15881 −0.515837
\(66\) 0.923069 − 0.746358i 0.113622 − 0.0918703i
\(67\) 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
\(69\) −0.0848809 + 0.0848809i −0.0102185 + 0.0102185i
\(70\) −3.75330 − 0.397291i −0.448605 − 0.0474854i
\(71\) 7.86777i 0.933733i 0.884328 + 0.466866i \(0.154617\pi\)
−0.884328 + 0.466866i \(0.845383\pi\)
\(72\) −7.49629 + 3.79865i −0.883446 + 0.447675i
\(73\) 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
\(75\) −0.120009 + 0.120009i −0.0138574 + 0.0138574i
\(76\) −8.18824 + 5.30033i −0.939256 + 0.607989i
\(77\) −9.33322 − 9.33322i −1.06362 − 1.06362i
\(78\) −0.627607 − 0.776202i −0.0710625 − 0.0878876i
\(79\) −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
\(81\) −8.74159 −0.971288
\(82\) −9.07431 − 11.2228i −1.00209 − 1.23935i
\(83\) −3.87327 − 3.87327i −0.425147 − 0.425147i 0.461825 − 0.886971i \(-0.347195\pi\)
−0.886971 + 0.461825i \(0.847195\pi\)
\(84\) −0.492261 − 0.760473i −0.0537101 − 0.0829744i
\(85\) −1.30896 + 1.30896i −0.141977 + 0.141977i
\(86\) 0.934407 − 8.82755i 0.100760 − 0.951900i
\(87\) 0.839845i 0.0900408i
\(88\) 13.2943 + 4.35218i 1.41718 + 0.463944i
\(89\) 10.5055i 1.11358i 0.830653 + 0.556790i \(0.187967\pi\)
−0.830653 + 0.556790i \(0.812033\pi\)
\(90\) 4.17856 + 0.442305i 0.440459 + 0.0466231i
\(91\) −7.84824 + 7.84824i −0.822719 + 0.822719i
\(92\) −1.38323 − 0.296151i −0.144212 − 0.0308759i
\(93\) −0.821187 − 0.821187i −0.0851531 − 0.0851531i
\(94\) −2.08316 + 1.68436i −0.214861 + 0.173729i
\(95\) 4.87701 0.500370
\(96\) 0.829667 + 0.483103i 0.0846776 + 0.0493065i
\(97\) 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
\(99\) 10.3907 + 10.3907i 1.04430 + 1.04430i
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 80.2.l.a.61.7 yes 16
3.2 odd 2 720.2.t.c.541.2 16
4.3 odd 2 320.2.l.a.81.5 16
5.2 odd 4 400.2.q.h.349.6 16
5.3 odd 4 400.2.q.g.349.3 16
5.4 even 2 400.2.l.h.301.2 16
8.3 odd 2 640.2.l.a.161.4 16
8.5 even 2 640.2.l.b.161.5 16
12.11 even 2 2880.2.t.c.721.6 16
16.3 odd 4 640.2.l.a.481.4 16
16.5 even 4 inner 80.2.l.a.21.7 ✓ 16
16.11 odd 4 320.2.l.a.241.5 16
16.13 even 4 640.2.l.b.481.5 16
20.3 even 4 1600.2.q.h.849.4 16
20.7 even 4 1600.2.q.g.849.5 16
20.19 odd 2 1600.2.l.i.401.4 16
32.5 even 8 5120.2.a.s.1.5 8
32.11 odd 8 5120.2.a.t.1.5 8
32.21 even 8 5120.2.a.v.1.4 8
32.27 odd 8 5120.2.a.u.1.4 8
48.5 odd 4 720.2.t.c.181.2 16
48.11 even 4 2880.2.t.c.2161.7 16
80.27 even 4 1600.2.q.h.49.4 16
80.37 odd 4 400.2.q.g.149.3 16
80.43 even 4 1600.2.q.g.49.5 16
80.53 odd 4 400.2.q.h.149.6 16
80.59 odd 4 1600.2.l.i.1201.4 16
80.69 even 4 400.2.l.h.101.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 16.5 even 4 inner
80.2.l.a.61.7 yes 16 1.1 even 1 trivial
320.2.l.a.81.5 16 4.3 odd 2
320.2.l.a.241.5 16 16.11 odd 4
400.2.l.h.101.2 16 80.69 even 4
400.2.l.h.301.2 16 5.4 even 2
400.2.q.g.149.3 16 80.37 odd 4
400.2.q.g.349.3 16 5.3 odd 4
400.2.q.h.149.6 16 80.53 odd 4
400.2.q.h.349.6 16 5.2 odd 4
640.2.l.a.161.4 16 8.3 odd 2
640.2.l.a.481.4 16 16.3 odd 4
640.2.l.b.161.5 16 8.5 even 2
640.2.l.b.481.5 16 16.13 even 4
720.2.t.c.181.2 16 48.5 odd 4
720.2.t.c.541.2 16 3.2 odd 2
1600.2.l.i.401.4 16 20.19 odd 2
1600.2.l.i.1201.4 16 80.59 odd 4
1600.2.q.g.49.5 16 80.43 even 4
1600.2.q.g.849.5 16 20.7 even 4
1600.2.q.h.49.4 16 80.27 even 4
1600.2.q.h.849.4 16 20.3 even 4
2880.2.t.c.721.6 16 12.11 even 2
2880.2.t.c.2161.7 16 48.11 even 4
5120.2.a.s.1.5 8 32.5 even 8
5120.2.a.t.1.5 8 32.11 odd 8
5120.2.a.u.1.4 8 32.27 odd 8
5120.2.a.v.1.4 8 32.21 even 8