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Properties

Label 6845.2.a.f.1.2
Level $6845$
Weight $2$
Character 6845.1
Self dual yes
Analytic conductor $54.658$
Analytic rank $0$
Dimension $5$
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] = [6845,2,Mod(1,6845)] 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("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6845 = 5 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6845.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-2,3,10,5,6,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.6576001836\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.383115\) of defining polynomial
Character \(\chi\) \(=\) 6845.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.15510 q^{2} +1.38311 q^{3} +2.64446 q^{4} +1.00000 q^{5} -2.98075 q^{6} -2.62521 q^{7} -1.38887 q^{8} -1.08699 q^{9} -2.15510 q^{10} -1.64446 q^{11} +3.65759 q^{12} -2.44254 q^{13} +5.65759 q^{14} +1.38311 q^{15} -2.29576 q^{16} +0.578749 q^{17} +2.34258 q^{18} -5.20156 q^{19} +2.64446 q^{20} -3.63096 q^{21} +3.54397 q^{22} -8.22913 q^{23} -1.92097 q^{24} +1.00000 q^{25} +5.26391 q^{26} -5.65278 q^{27} -6.94225 q^{28} -0.766229 q^{29} -2.98075 q^{30} -4.21452 q^{31} +7.72533 q^{32} -2.27447 q^{33} -1.24726 q^{34} -2.62521 q^{35} -2.87451 q^{36} +11.2099 q^{38} -3.37831 q^{39} -1.38887 q^{40} -1.64446 q^{41} +7.82509 q^{42} +1.91893 q^{43} -4.34870 q^{44} -1.08699 q^{45} +17.7346 q^{46} -9.56543 q^{47} -3.17530 q^{48} -0.108279 q^{49} -2.15510 q^{50} +0.800477 q^{51} -6.45918 q^{52} +7.74217 q^{53} +12.1823 q^{54} -1.64446 q^{55} +3.64608 q^{56} -7.19435 q^{57} +1.65130 q^{58} +13.0359 q^{59} +3.65759 q^{60} +3.86379 q^{61} +9.08272 q^{62} +2.85359 q^{63} -12.0574 q^{64} -2.44254 q^{65} +4.90172 q^{66} +11.4566 q^{67} +1.53048 q^{68} -11.3818 q^{69} +5.65759 q^{70} -2.54690 q^{71} +1.50969 q^{72} -9.79732 q^{73} +1.38311 q^{75} -13.7553 q^{76} +4.31704 q^{77} +7.28059 q^{78} +1.81364 q^{79} -2.29576 q^{80} -4.55746 q^{81} +3.54397 q^{82} +10.9822 q^{83} -9.60193 q^{84} +0.578749 q^{85} -4.13549 q^{86} -1.05978 q^{87} +2.28394 q^{88} +8.85915 q^{89} +2.34258 q^{90} +6.41217 q^{91} -21.7616 q^{92} -5.82917 q^{93} +20.6145 q^{94} -5.20156 q^{95} +10.6850 q^{96} +10.5605 q^{97} +0.233352 q^{98} +1.78751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 3 q^{3} + 10 q^{4} + 5 q^{5} + 6 q^{6} + 11 q^{7} - 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} - 4 q^{13} + 8 q^{14} + 3 q^{15} + 16 q^{16} - 2 q^{18} + 4 q^{19} + 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.15510 −1.52389 −0.761943 − 0.647644i \(-0.775754\pi\)
−0.761943 + 0.647644i \(0.775754\pi\)
\(3\) 1.38311 0.798542 0.399271 − 0.916833i \(-0.369263\pi\)
0.399271 + 0.916833i \(0.369263\pi\)
\(4\) 2.64446 1.32223
\(5\) 1.00000 0.447214
\(6\) −2.98075 −1.21689
\(7\) −2.62521 −0.992236 −0.496118 − 0.868255i \(-0.665242\pi\)
−0.496118 + 0.868255i \(0.665242\pi\)
\(8\) −1.38887 −0.491040
\(9\) −1.08699 −0.362331
\(10\) −2.15510 −0.681503
\(11\) −1.64446 −0.495823 −0.247911 − 0.968783i \(-0.579744\pi\)
−0.247911 + 0.968783i \(0.579744\pi\)
\(12\) 3.65759 1.05585
\(13\) −2.44254 −0.677438 −0.338719 − 0.940888i \(-0.609994\pi\)
−0.338719 + 0.940888i \(0.609994\pi\)
\(14\) 5.65759 1.51205
\(15\) 1.38311 0.357119
\(16\) −2.29576 −0.573940
\(17\) 0.578749 0.140367 0.0701837 − 0.997534i \(-0.477641\pi\)
0.0701837 + 0.997534i \(0.477641\pi\)
\(18\) 2.34258 0.552151
\(19\) −5.20156 −1.19332 −0.596660 − 0.802494i \(-0.703506\pi\)
−0.596660 + 0.802494i \(0.703506\pi\)
\(20\) 2.64446 0.591319
\(21\) −3.63096 −0.792341
\(22\) 3.54397 0.755577
\(23\) −8.22913 −1.71589 −0.857946 − 0.513739i \(-0.828260\pi\)
−0.857946 + 0.513739i \(0.828260\pi\)
\(24\) −1.92097 −0.392116
\(25\) 1.00000 0.200000
\(26\) 5.26391 1.03234
\(27\) −5.65278 −1.08788
\(28\) −6.94225 −1.31196
\(29\) −0.766229 −0.142285 −0.0711426 − 0.997466i \(-0.522665\pi\)
−0.0711426 + 0.997466i \(0.522665\pi\)
\(30\) −2.98075 −0.544208
\(31\) −4.21452 −0.756950 −0.378475 − 0.925611i \(-0.623551\pi\)
−0.378475 + 0.925611i \(0.623551\pi\)
\(32\) 7.72533 1.36566
\(33\) −2.27447 −0.395935
\(34\) −1.24726 −0.213904
\(35\) −2.62521 −0.443741
\(36\) −2.87451 −0.479085
\(37\) 0 0
\(38\) 11.2099 1.81848
\(39\) −3.37831 −0.540962
\(40\) −1.38887 −0.219600
\(41\) −1.64446 −0.256821 −0.128411 − 0.991721i \(-0.540988\pi\)
−0.128411 + 0.991721i \(0.540988\pi\)
\(42\) 7.82509 1.20744
\(43\) 1.91893 0.292634 0.146317 − 0.989238i \(-0.453258\pi\)
0.146317 + 0.989238i \(0.453258\pi\)
\(44\) −4.34870 −0.655591
\(45\) −1.08699 −0.162039
\(46\) 17.7346 2.61482
\(47\) −9.56543 −1.39526 −0.697630 − 0.716458i \(-0.745762\pi\)
−0.697630 + 0.716458i \(0.745762\pi\)
\(48\) −3.17530 −0.458315
\(49\) −0.108279 −0.0154684
\(50\) −2.15510 −0.304777
\(51\) 0.800477 0.112089
\(52\) −6.45918 −0.895728
\(53\) 7.74217 1.06347 0.531735 − 0.846911i \(-0.321540\pi\)
0.531735 + 0.846911i \(0.321540\pi\)
\(54\) 12.1823 1.65780
\(55\) −1.64446 −0.221739
\(56\) 3.64608 0.487227
\(57\) −7.19435 −0.952915
\(58\) 1.65130 0.216827
\(59\) 13.0359 1.69713 0.848565 − 0.529092i \(-0.177467\pi\)
0.848565 + 0.529092i \(0.177467\pi\)
\(60\) 3.65759 0.472193
\(61\) 3.86379 0.494707 0.247354 − 0.968925i \(-0.420439\pi\)
0.247354 + 0.968925i \(0.420439\pi\)
\(62\) 9.08272 1.15351
\(63\) 2.85359 0.359518
\(64\) −12.0574 −1.50717
\(65\) −2.44254 −0.302959
\(66\) 4.90172 0.603360
\(67\) 11.4566 1.39965 0.699824 − 0.714315i \(-0.253262\pi\)
0.699824 + 0.714315i \(0.253262\pi\)
\(68\) 1.53048 0.185598
\(69\) −11.3818 −1.37021
\(70\) 5.65759 0.676211
\(71\) −2.54690 −0.302261 −0.151131 − 0.988514i \(-0.548291\pi\)
−0.151131 + 0.988514i \(0.548291\pi\)
\(72\) 1.50969 0.177919
\(73\) −9.79732 −1.14669 −0.573345 − 0.819314i \(-0.694354\pi\)
−0.573345 + 0.819314i \(0.694354\pi\)
\(74\) 0 0
\(75\) 1.38311 0.159708
\(76\) −13.7553 −1.57784
\(77\) 4.31704 0.491973
\(78\) 7.28059 0.824365
\(79\) 1.81364 0.204050 0.102025 − 0.994782i \(-0.467468\pi\)
0.102025 + 0.994782i \(0.467468\pi\)
\(80\) −2.29576 −0.256674
\(81\) −4.55746 −0.506385
\(82\) 3.54397 0.391366
\(83\) 10.9822 1.20546 0.602728 − 0.797947i \(-0.294080\pi\)
0.602728 + 0.797947i \(0.294080\pi\)
\(84\) −9.60193 −1.04766
\(85\) 0.578749 0.0627742
\(86\) −4.13549 −0.445941
\(87\) −1.05978 −0.113621
\(88\) 2.28394 0.243469
\(89\) 8.85915 0.939068 0.469534 − 0.882914i \(-0.344422\pi\)
0.469534 + 0.882914i \(0.344422\pi\)
\(90\) 2.34258 0.246930
\(91\) 6.41217 0.672178
\(92\) −21.7616 −2.26880
\(93\) −5.82917 −0.604456
\(94\) 20.6145 2.12622
\(95\) −5.20156 −0.533669
\(96\) 10.6850 1.09054
\(97\) 10.5605 1.07225 0.536126 − 0.844138i \(-0.319887\pi\)
0.536126 + 0.844138i \(0.319887\pi\)
\(98\) 0.233352 0.0235721
\(99\) 1.78751 0.179652
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 6845.2.a.f.1.2 5
37.36 even 2 185.2.a.e.1.4 ✓ 5
111.110 odd 2 1665.2.a.p.1.2 5
148.147 odd 2 2960.2.a.w.1.3 5
185.73 odd 4 925.2.b.f.149.3 10
185.147 odd 4 925.2.b.f.149.8 10
185.184 even 2 925.2.a.f.1.2 5
259.258 odd 2 9065.2.a.k.1.4 5
555.554 odd 2 8325.2.a.ch.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.4 ✓ 5 37.36 even 2
925.2.a.f.1.2 5 185.184 even 2
925.2.b.f.149.3 10 185.73 odd 4
925.2.b.f.149.8 10 185.147 odd 4
1665.2.a.p.1.2 5 111.110 odd 2
2960.2.a.w.1.3 5 148.147 odd 2
6845.2.a.f.1.2 5 1.1 even 1 trivial
8325.2.a.ch.1.4 5 555.554 odd 2
9065.2.a.k.1.4 5 259.258 odd 2