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Properties

Label 371.2.g.e.76.7
Level $371$
Weight $2$
Character 371.76
Analytic conductor $2.962$
Analytic rank $0$
Dimension $52$
Inner twists $4$

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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] = [371,2,Mod(76,371)] 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("371.76"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [52,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 76.7
Character \(\chi\) \(=\) 371.76
Dual form 371.2.g.e.83.7

$q$-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+(-1.11032 + 1.11032i) q^{2} +(-0.129800 - 0.129800i) q^{3} -0.465639i q^{4} +(-2.62208 + 2.62208i) q^{5} +0.288240 q^{6} +(-1.65924 + 2.06080i) q^{7} +(-1.70364 - 1.70364i) q^{8} -2.96630i q^{9} -5.82272i q^{10} -3.97424i q^{11} +(-0.0604400 + 0.0604400i) q^{12} +3.44772i q^{13} +(-0.445862 - 4.13046i) q^{14} +0.680694 q^{15} +4.71446 q^{16} +3.68013 q^{17} +(3.29356 + 3.29356i) q^{18} +(0.417623 + 0.417623i) q^{19} +(1.22094 + 1.22094i) q^{20} +(0.482863 - 0.0521226i) q^{21} +(4.41270 + 4.41270i) q^{22} +(-2.68877 - 2.68877i) q^{23} +0.442265i q^{24} -8.75064i q^{25} +(-3.82809 - 3.82809i) q^{26} +(-0.774427 + 0.774427i) q^{27} +(0.959590 + 0.772608i) q^{28} +2.59971i q^{29} +(-0.755790 + 0.755790i) q^{30} +(-5.88137 - 5.88137i) q^{31} +(-1.82730 + 1.82730i) q^{32} +(-0.515858 + 0.515858i) q^{33} +(-4.08613 + 4.08613i) q^{34} +(-1.05292 - 9.75427i) q^{35} -1.38123 q^{36} +1.90084i q^{37} -0.927393 q^{38} +(0.447515 - 0.447515i) q^{39} +8.93416 q^{40} +(-7.31617 - 7.31617i) q^{41} +(-0.478261 + 0.594007i) q^{42} +11.6057i q^{43} -1.85056 q^{44} +(7.77789 + 7.77789i) q^{45} +5.97081 q^{46} -4.71906i q^{47} +(-0.611937 - 0.611937i) q^{48} +(-1.49382 - 6.83875i) q^{49} +(9.71604 + 9.71604i) q^{50} +(-0.477681 - 0.477681i) q^{51} +1.60539 q^{52} +(-4.66739 - 5.58708i) q^{53} -1.71973i q^{54} +(10.4208 + 10.4208i) q^{55} +(6.33761 - 0.684114i) q^{56} -0.108415i q^{57} +(-2.88652 - 2.88652i) q^{58} -8.76373 q^{59} -0.316957i q^{60} +(-1.64370 + 1.64370i) q^{61} +13.0604 q^{62} +(6.11297 + 4.92182i) q^{63} +5.37113i q^{64} +(-9.04021 - 9.04021i) q^{65} -1.14554i q^{66} +(-4.77630 - 4.77630i) q^{67} -1.71361i q^{68} +0.698005i q^{69} +(11.9995 + 9.66131i) q^{70} +(-0.604299 - 0.604299i) q^{71} +(-5.05351 + 5.05351i) q^{72} +(2.06628 + 2.06628i) q^{73} +(-2.11054 - 2.11054i) q^{74} +(-1.13583 + 1.13583i) q^{75} +(0.194461 - 0.194461i) q^{76} +(8.19014 + 6.59424i) q^{77} +0.993773i q^{78} +(-5.74084 + 5.74084i) q^{79} +(-12.3617 + 12.3617i) q^{80} -8.69787 q^{81} +16.2466 q^{82} +(-8.66230 - 8.66230i) q^{83} +(-0.0242703 - 0.224840i) q^{84} +(-9.64959 + 9.64959i) q^{85} +(-12.8860 - 12.8860i) q^{86} +(0.337442 - 0.337442i) q^{87} +(-6.77067 + 6.77067i) q^{88} +14.3929i q^{89} -17.2720 q^{90} +(-7.10508 - 5.72061i) q^{91} +(-1.25200 + 1.25200i) q^{92} +1.52680i q^{93} +(5.23969 + 5.23969i) q^{94} -2.19008 q^{95} +0.474368 q^{96} -8.05368i q^{97} +(9.25186 + 5.93460i) q^{98} -11.7888 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52 q - 8 q^{2} + 20 q^{8} - 50 q^{14} - 24 q^{15} - 64 q^{16} - 8 q^{18} - 30 q^{21} + 40 q^{22} - 20 q^{23} + 80 q^{30} + 12 q^{32} - 24 q^{35} - 16 q^{36} + 4 q^{39} - 80 q^{42} + 64 q^{44} + 88 q^{46}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.11032 + 1.11032i −0.785118 + 0.785118i −0.980689 − 0.195572i \(-0.937344\pi\)
0.195572 + 0.980689i \(0.437344\pi\)
\(3\) −0.129800 − 0.129800i −0.0749402 − 0.0749402i 0.668643 − 0.743583i \(-0.266875\pi\)
−0.743583 + 0.668643i \(0.766875\pi\)
\(4\) − 0.465639i − 0.232819i
\(5\) −2.62208 + 2.62208i −1.17263 + 1.17263i −0.191051 + 0.981580i \(0.561190\pi\)
−0.981580 + 0.191051i \(0.938810\pi\)
\(6\) 0.288240 0.117674
\(7\) −1.65924 + 2.06080i −0.627135 + 0.778911i
\(8\) −1.70364 − 1.70364i −0.602327 − 0.602327i
\(9\) − 2.96630i − 0.988768i
\(10\) − 5.82272i − 1.84131i
\(11\) − 3.97424i − 1.19828i −0.800644 − 0.599140i \(-0.795509\pi\)
0.800644 − 0.599140i \(-0.204491\pi\)
\(12\) −0.0604400 + 0.0604400i −0.0174475 + 0.0174475i
\(13\) 3.44772i 0.956226i 0.878298 + 0.478113i \(0.158679\pi\)
−0.878298 + 0.478113i \(0.841321\pi\)
\(14\) −0.445862 − 4.13046i −0.119162 − 1.10391i
\(15\) 0.680694 0.175754
\(16\) 4.71446 1.17861
\(17\) 3.68013 0.892561 0.446281 − 0.894893i \(-0.352748\pi\)
0.446281 + 0.894893i \(0.352748\pi\)
\(18\) 3.29356 + 3.29356i 0.776299 + 0.776299i
\(19\) 0.417623 + 0.417623i 0.0958092 + 0.0958092i 0.753387 − 0.657578i \(-0.228419\pi\)
−0.657578 + 0.753387i \(0.728419\pi\)
\(20\) 1.22094 + 1.22094i 0.273011 + 0.273011i
\(21\) 0.482863 − 0.0521226i 0.105369 − 0.0113741i
\(22\) 4.41270 + 4.41270i 0.940791 + 0.940791i
\(23\) −2.68877 − 2.68877i −0.560647 − 0.560647i 0.368844 − 0.929491i \(-0.379754\pi\)
−0.929491 + 0.368844i \(0.879754\pi\)
\(24\) 0.442265i 0.0902770i
\(25\) − 8.75064i − 1.75013i
\(26\) −3.82809 − 3.82809i −0.750750 − 0.750750i
\(27\) −0.774427 + 0.774427i −0.149039 + 0.149039i
\(28\) 0.959590 + 0.772608i 0.181346 + 0.146009i
\(29\) 2.59971i 0.482753i 0.970432 + 0.241377i \(0.0775989\pi\)
−0.970432 + 0.241377i \(0.922401\pi\)
\(30\) −0.755790 + 0.755790i −0.137988 + 0.137988i
\(31\) −5.88137 − 5.88137i −1.05632 − 1.05632i −0.998316 − 0.0580086i \(-0.981525\pi\)
−0.0580086 − 0.998316i \(-0.518475\pi\)
\(32\) −1.82730 + 1.82730i −0.323024 + 0.323024i
\(33\) −0.515858 + 0.515858i −0.0897993 + 0.0897993i
\(34\) −4.08613 + 4.08613i −0.700766 + 0.700766i
\(35\) −1.05292 − 9.75427i −0.177977 − 1.64877i
\(36\) −1.38123 −0.230204
\(37\) 1.90084i 0.312496i 0.987718 + 0.156248i \(0.0499398\pi\)
−0.987718 + 0.156248i \(0.950060\pi\)
\(38\) −0.927393 −0.150443
\(39\) 0.447515 − 0.447515i 0.0716597 − 0.0716597i
\(40\) 8.93416 1.41261
\(41\) −7.31617 − 7.31617i −1.14259 − 1.14259i −0.987974 − 0.154619i \(-0.950585\pi\)
−0.154619 − 0.987974i \(-0.549415\pi\)
\(42\) −0.478261 + 0.594007i −0.0737973 + 0.0916573i
\(43\) 11.6057i 1.76985i 0.465736 + 0.884924i \(0.345790\pi\)
−0.465736 + 0.884924i \(0.654210\pi\)
\(44\) −1.85056 −0.278983
\(45\) 7.77789 + 7.77789i 1.15946 + 1.15946i
\(46\) 5.97081 0.880348
\(47\) − 4.71906i − 0.688346i −0.938906 − 0.344173i \(-0.888159\pi\)
0.938906 − 0.344173i \(-0.111841\pi\)
\(48\) −0.611937 − 0.611937i −0.0883256 − 0.0883256i
\(49\) −1.49382 − 6.83875i −0.213403 − 0.976964i
\(50\) 9.71604 + 9.71604i 1.37406 + 1.37406i
\(51\) −0.477681 − 0.477681i −0.0668887 − 0.0668887i
\(52\) 1.60539 0.222628
\(53\) −4.66739 − 5.58708i −0.641115 − 0.767444i
\(54\) − 1.71973i − 0.234026i
\(55\) 10.4208 + 10.4208i 1.40514 + 1.40514i
\(56\) 6.33761 − 0.684114i 0.846899 − 0.0914185i
\(57\) − 0.108415i − 0.0143599i
\(58\) −2.88652 − 2.88652i −0.379018 − 0.379018i
\(59\) −8.76373 −1.14094 −0.570471 − 0.821318i \(-0.693239\pi\)
−0.570471 + 0.821318i \(0.693239\pi\)
\(60\) − 0.316957i − 0.0409190i
\(61\) −1.64370 + 1.64370i −0.210455 + 0.210455i −0.804461 − 0.594006i \(-0.797546\pi\)
0.594006 + 0.804461i \(0.297546\pi\)
\(62\) 13.0604 1.65868
\(63\) 6.11297 + 4.92182i 0.770162 + 0.620091i
\(64\) 5.37113i 0.671391i
\(65\) −9.04021 − 9.04021i −1.12130 − 1.12130i
\(66\) − 1.14554i − 0.141006i
\(67\) −4.77630 − 4.77630i −0.583518 − 0.583518i 0.352350 − 0.935868i \(-0.385383\pi\)
−0.935868 + 0.352350i \(0.885383\pi\)
\(68\) − 1.71361i − 0.207806i
\(69\) 0.698005i 0.0840300i
\(70\) 11.9995 + 9.66131i 1.43421 + 1.15475i
\(71\) −0.604299 − 0.604299i −0.0717171 − 0.0717171i 0.670338 − 0.742056i \(-0.266149\pi\)
−0.742056 + 0.670338i \(0.766149\pi\)
\(72\) −5.05351 + 5.05351i −0.595562 + 0.595562i
\(73\) 2.06628 + 2.06628i 0.241839 + 0.241839i 0.817611 − 0.575771i \(-0.195298\pi\)
−0.575771 + 0.817611i \(0.695298\pi\)
\(74\) −2.11054 − 2.11054i −0.245346 − 0.245346i
\(75\) −1.13583 + 1.13583i −0.131155 + 0.131155i
\(76\) 0.194461 − 0.194461i 0.0223062 − 0.0223062i
\(77\) 8.19014 + 6.59424i 0.933353 + 0.751483i
\(78\) 0.993773i 0.112523i
\(79\) −5.74084 + 5.74084i −0.645895 + 0.645895i −0.951998 − 0.306104i \(-0.900975\pi\)
0.306104 + 0.951998i \(0.400975\pi\)
\(80\) −12.3617 + 12.3617i −1.38208 + 1.38208i
\(81\) −8.69787 −0.966430
\(82\) 16.2466 1.79414
\(83\) −8.66230 − 8.66230i −0.950811 − 0.950811i 0.0480348 − 0.998846i \(-0.484704\pi\)
−0.998846 + 0.0480348i \(0.984704\pi\)
\(84\) −0.0242703 − 0.224840i −0.00264811 − 0.0245320i
\(85\) −9.64959 + 9.64959i −1.04665 + 1.04665i
\(86\) −12.8860 − 12.8860i −1.38954 − 1.38954i
\(87\) 0.337442 − 0.337442i 0.0361776 − 0.0361776i
\(88\) −6.77067 + 6.77067i −0.721756 + 0.721756i
\(89\) 14.3929i 1.52564i 0.646608 + 0.762822i \(0.276187\pi\)
−0.646608 + 0.762822i \(0.723813\pi\)
\(90\) −17.2720 −1.82063
\(91\) −7.10508 − 5.72061i −0.744815 − 0.599683i
\(92\) −1.25200 + 1.25200i −0.130530 + 0.130530i
\(93\) 1.52680i 0.158322i
\(94\) 5.23969 + 5.23969i 0.540433 + 0.540433i
\(95\) −2.19008 −0.224698
\(96\) 0.474368 0.0484150
\(97\) − 8.05368i − 0.817728i −0.912595 − 0.408864i \(-0.865925\pi\)
0.912595 − 0.408864i \(-0.134075\pi\)
\(98\) 9.25186 + 5.93460i 0.934579 + 0.599485i
\(99\) −11.7888 −1.18482
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 371.2.g.e.76.7 ✓ 52
7.6 odd 2 inner 371.2.g.e.76.8 yes 52
53.30 odd 4 inner 371.2.g.e.83.8 yes 52
371.83 even 4 inner 371.2.g.e.83.7 yes 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.2.g.e.76.7 ✓ 52 1.1 even 1 trivial
371.2.g.e.76.8 yes 52 7.6 odd 2 inner
371.2.g.e.83.7 yes 52 371.83 even 4 inner
371.2.g.e.83.8 yes 52 53.30 odd 4 inner