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Properties

Label 8013.2.a.c.1.20
Level $8013$
Weight $2$
Character 8013.1
Self dual yes
Analytic conductor $63.984$
Analytic rank $1$
Dimension $116$
CM no
Inner twists $1$

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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] = [8013,2,Mod(1,8013)] 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("8013.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8013, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8013 = 3 \cdot 2671 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8013.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [116] 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: yes
Analytic conductor: \(63.9841271397\)
Analytic rank: \(1\)
Dimension: \(116\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 8013.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.15908 q^{2} -1.00000 q^{3} +2.66162 q^{4} -0.295164 q^{5} +2.15908 q^{6} +3.97405 q^{7} -1.42849 q^{8} +1.00000 q^{9} +0.637282 q^{10} -3.95151 q^{11} -2.66162 q^{12} +3.40735 q^{13} -8.58029 q^{14} +0.295164 q^{15} -2.23902 q^{16} +3.81766 q^{17} -2.15908 q^{18} +2.91295 q^{19} -0.785614 q^{20} -3.97405 q^{21} +8.53161 q^{22} -5.02941 q^{23} +1.42849 q^{24} -4.91288 q^{25} -7.35673 q^{26} -1.00000 q^{27} +10.5774 q^{28} -6.89560 q^{29} -0.637282 q^{30} +6.77252 q^{31} +7.69120 q^{32} +3.95151 q^{33} -8.24262 q^{34} -1.17300 q^{35} +2.66162 q^{36} +3.10390 q^{37} -6.28928 q^{38} -3.40735 q^{39} +0.421638 q^{40} -5.07139 q^{41} +8.58029 q^{42} +5.52948 q^{43} -10.5174 q^{44} -0.295164 q^{45} +10.8589 q^{46} +12.2391 q^{47} +2.23902 q^{48} +8.79310 q^{49} +10.6073 q^{50} -3.81766 q^{51} +9.06905 q^{52} -2.58965 q^{53} +2.15908 q^{54} +1.16634 q^{55} -5.67688 q^{56} -2.91295 q^{57} +14.8881 q^{58} -4.77837 q^{59} +0.785614 q^{60} -9.95217 q^{61} -14.6224 q^{62} +3.97405 q^{63} -12.1279 q^{64} -1.00573 q^{65} -8.53161 q^{66} -7.46330 q^{67} +10.1611 q^{68} +5.02941 q^{69} +2.53259 q^{70} -9.75586 q^{71} -1.42849 q^{72} -10.5593 q^{73} -6.70157 q^{74} +4.91288 q^{75} +7.75316 q^{76} -15.7035 q^{77} +7.35673 q^{78} -11.0357 q^{79} +0.660879 q^{80} +1.00000 q^{81} +10.9495 q^{82} +6.56685 q^{83} -10.5774 q^{84} -1.12683 q^{85} -11.9386 q^{86} +6.89560 q^{87} +5.64467 q^{88} -9.87316 q^{89} +0.637282 q^{90} +13.5410 q^{91} -13.3864 q^{92} -6.77252 q^{93} -26.4252 q^{94} -0.859797 q^{95} -7.69120 q^{96} +3.28209 q^{97} -18.9850 q^{98} -3.95151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18}+ \cdots - 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.15908 −1.52670 −0.763349 − 0.645986i \(-0.776447\pi\)
−0.763349 + 0.645986i \(0.776447\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.66162 1.33081
\(5\) −0.295164 −0.132001 −0.0660006 − 0.997820i \(-0.521024\pi\)
−0.0660006 + 0.997820i \(0.521024\pi\)
\(6\) 2.15908 0.881440
\(7\) 3.97405 1.50205 0.751025 − 0.660273i \(-0.229560\pi\)
0.751025 + 0.660273i \(0.229560\pi\)
\(8\) −1.42849 −0.505046
\(9\) 1.00000 0.333333
\(10\) 0.637282 0.201526
\(11\) −3.95151 −1.19142 −0.595712 − 0.803198i \(-0.703130\pi\)
−0.595712 + 0.803198i \(0.703130\pi\)
\(12\) −2.66162 −0.768343
\(13\) 3.40735 0.945028 0.472514 − 0.881323i \(-0.343347\pi\)
0.472514 + 0.881323i \(0.343347\pi\)
\(14\) −8.58029 −2.29318
\(15\) 0.295164 0.0762110
\(16\) −2.23902 −0.559756
\(17\) 3.81766 0.925918 0.462959 − 0.886380i \(-0.346788\pi\)
0.462959 + 0.886380i \(0.346788\pi\)
\(18\) −2.15908 −0.508900
\(19\) 2.91295 0.668276 0.334138 − 0.942524i \(-0.391555\pi\)
0.334138 + 0.942524i \(0.391555\pi\)
\(20\) −0.785614 −0.175669
\(21\) −3.97405 −0.867209
\(22\) 8.53161 1.81895
\(23\) −5.02941 −1.04870 −0.524352 − 0.851502i \(-0.675692\pi\)
−0.524352 + 0.851502i \(0.675692\pi\)
\(24\) 1.42849 0.291589
\(25\) −4.91288 −0.982576
\(26\) −7.35673 −1.44277
\(27\) −1.00000 −0.192450
\(28\) 10.5774 1.99894
\(29\) −6.89560 −1.28048 −0.640240 − 0.768175i \(-0.721165\pi\)
−0.640240 + 0.768175i \(0.721165\pi\)
\(30\) −0.637282 −0.116351
\(31\) 6.77252 1.21638 0.608190 − 0.793792i \(-0.291896\pi\)
0.608190 + 0.793792i \(0.291896\pi\)
\(32\) 7.69120 1.35962
\(33\) 3.95151 0.687869
\(34\) −8.24262 −1.41360
\(35\) −1.17300 −0.198273
\(36\) 2.66162 0.443603
\(37\) 3.10390 0.510278 0.255139 − 0.966904i \(-0.417879\pi\)
0.255139 + 0.966904i \(0.417879\pi\)
\(38\) −6.28928 −1.02026
\(39\) −3.40735 −0.545612
\(40\) 0.421638 0.0666668
\(41\) −5.07139 −0.792017 −0.396009 − 0.918247i \(-0.629605\pi\)
−0.396009 + 0.918247i \(0.629605\pi\)
\(42\) 8.58029 1.32397
\(43\) 5.52948 0.843237 0.421619 − 0.906773i \(-0.361462\pi\)
0.421619 + 0.906773i \(0.361462\pi\)
\(44\) −10.5174 −1.58556
\(45\) −0.295164 −0.0440004
\(46\) 10.8589 1.60105
\(47\) 12.2391 1.78526 0.892630 − 0.450790i \(-0.148858\pi\)
0.892630 + 0.450790i \(0.148858\pi\)
\(48\) 2.23902 0.323175
\(49\) 8.79310 1.25616
\(50\) 10.6073 1.50010
\(51\) −3.81766 −0.534579
\(52\) 9.06905 1.25765
\(53\) −2.58965 −0.355716 −0.177858 − 0.984056i \(-0.556917\pi\)
−0.177858 + 0.984056i \(0.556917\pi\)
\(54\) 2.15908 0.293813
\(55\) 1.16634 0.157270
\(56\) −5.67688 −0.758605
\(57\) −2.91295 −0.385829
\(58\) 14.8881 1.95491
\(59\) −4.77837 −0.622090 −0.311045 − 0.950395i \(-0.600679\pi\)
−0.311045 + 0.950395i \(0.600679\pi\)
\(60\) 0.785614 0.101422
\(61\) −9.95217 −1.27424 −0.637122 − 0.770763i \(-0.719875\pi\)
−0.637122 + 0.770763i \(0.719875\pi\)
\(62\) −14.6224 −1.85705
\(63\) 3.97405 0.500684
\(64\) −12.1279 −1.51598
\(65\) −1.00573 −0.124745
\(66\) −8.53161 −1.05017
\(67\) −7.46330 −0.911787 −0.455893 − 0.890034i \(-0.650680\pi\)
−0.455893 + 0.890034i \(0.650680\pi\)
\(68\) 10.1611 1.23222
\(69\) 5.02941 0.605469
\(70\) 2.53259 0.302703
\(71\) −9.75586 −1.15781 −0.578904 − 0.815396i \(-0.696519\pi\)
−0.578904 + 0.815396i \(0.696519\pi\)
\(72\) −1.42849 −0.168349
\(73\) −10.5593 −1.23588 −0.617938 − 0.786227i \(-0.712032\pi\)
−0.617938 + 0.786227i \(0.712032\pi\)
\(74\) −6.70157 −0.779041
\(75\) 4.91288 0.567290
\(76\) 7.75316 0.889348
\(77\) −15.7035 −1.78958
\(78\) 7.35673 0.832985
\(79\) −11.0357 −1.24162 −0.620809 − 0.783962i \(-0.713196\pi\)
−0.620809 + 0.783962i \(0.713196\pi\)
\(80\) 0.660879 0.0738885
\(81\) 1.00000 0.111111
\(82\) 10.9495 1.20917
\(83\) 6.56685 0.720805 0.360403 − 0.932797i \(-0.382639\pi\)
0.360403 + 0.932797i \(0.382639\pi\)
\(84\) −10.5774 −1.15409
\(85\) −1.12683 −0.122222
\(86\) −11.9386 −1.28737
\(87\) 6.89560 0.739286
\(88\) 5.64467 0.601724
\(89\) −9.87316 −1.04655 −0.523276 − 0.852163i \(-0.675290\pi\)
−0.523276 + 0.852163i \(0.675290\pi\)
\(90\) 0.637282 0.0671754
\(91\) 13.5410 1.41948
\(92\) −13.3864 −1.39562
\(93\) −6.77252 −0.702277
\(94\) −26.4252 −2.72555
\(95\) −0.859797 −0.0882133
\(96\) −7.69120 −0.784980
\(97\) 3.28209 0.333245 0.166623 − 0.986021i \(-0.446714\pi\)
0.166623 + 0.986021i \(0.446714\pi\)
\(98\) −18.9850 −1.91777
\(99\) −3.95151 −0.397141
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 8013.2.a.c.1.20 ✓ 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8013.2.a.c.1.20 ✓ 116 1.1 even 1 trivial