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Properties

Label 6845.2.a.f.1.3
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 1.3
Root \(-1.62871\) 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)
 
\(f(q)\) \(=\) \(q-0.728950 q^{2} +2.62871 q^{3} -1.46863 q^{4} +1.00000 q^{5} -1.91620 q^{6} +2.55244 q^{7} +2.52846 q^{8} +3.91009 q^{9} -0.728950 q^{10} +2.46863 q^{11} -3.86060 q^{12} -1.55854 q^{13} -1.86060 q^{14} +2.62871 q^{15} +1.09414 q^{16} +6.83662 q^{17} -2.85026 q^{18} +7.66011 q^{19} -1.46863 q^{20} +6.70960 q^{21} -1.79951 q^{22} +7.50003 q^{23} +6.64658 q^{24} +1.00000 q^{25} +1.13610 q^{26} +2.39236 q^{27} -3.74859 q^{28} -3.25741 q^{29} -1.91620 q^{30} -0.658785 q^{31} -5.85449 q^{32} +6.48930 q^{33} -4.98356 q^{34} +2.55244 q^{35} -5.74248 q^{36} -5.58384 q^{38} -4.09694 q^{39} +2.52846 q^{40} +2.46863 q^{41} -4.89097 q^{42} -10.9579 q^{43} -3.62551 q^{44} +3.91009 q^{45} -5.46715 q^{46} +3.11521 q^{47} +2.87617 q^{48} -0.485072 q^{49} -0.728950 q^{50} +17.9715 q^{51} +2.28892 q^{52} +8.64184 q^{53} -1.74391 q^{54} +2.46863 q^{55} +6.45373 q^{56} +20.1362 q^{57} +2.37449 q^{58} +6.23634 q^{59} -3.86060 q^{60} -3.27808 q^{61} +0.480222 q^{62} +9.98026 q^{63} +2.07935 q^{64} -1.55854 q^{65} -4.73038 q^{66} +1.47764 q^{67} -10.0405 q^{68} +19.7154 q^{69} -1.86060 q^{70} -8.06686 q^{71} +9.88651 q^{72} -4.96199 q^{73} +2.62871 q^{75} -11.2499 q^{76} +6.30102 q^{77} +2.98647 q^{78} -12.8206 q^{79} +1.09414 q^{80} -5.44146 q^{81} -1.79951 q^{82} +1.14934 q^{83} -9.85393 q^{84} +6.83662 q^{85} +7.98779 q^{86} -8.56277 q^{87} +6.24184 q^{88} -11.5207 q^{89} -2.85026 q^{90} -3.97807 q^{91} -11.0148 q^{92} -1.73175 q^{93} -2.27083 q^{94} +7.66011 q^{95} -15.3897 q^{96} -17.2929 q^{97} +0.353594 q^{98} +9.65257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 3 q^{3} + 10 q^{4} + 5 q^{5} + 6 q^{6} + 11 q^{7} - 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} - 4 q^{13} + 8 q^{14} + 3 q^{15} + 16 q^{16} - 2 q^{18} + 4 q^{19} + 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.728950 −0.515446 −0.257723 − 0.966219i \(-0.582972\pi\)
−0.257723 + 0.966219i \(0.582972\pi\)
\(3\) 2.62871 1.51768 0.758842 − 0.651275i \(-0.225766\pi\)
0.758842 + 0.651275i \(0.225766\pi\)
\(4\) −1.46863 −0.734316
\(5\) 1.00000 0.447214
\(6\) −1.91620 −0.782284
\(7\) 2.55244 0.964730 0.482365 − 0.875970i \(-0.339778\pi\)
0.482365 + 0.875970i \(0.339778\pi\)
\(8\) 2.52846 0.893946
\(9\) 3.91009 1.30336
\(10\) −0.728950 −0.230514
\(11\) 2.46863 0.744320 0.372160 − 0.928169i \(-0.378617\pi\)
0.372160 + 0.928169i \(0.378617\pi\)
\(12\) −3.86060 −1.11446
\(13\) −1.55854 −0.432261 −0.216131 − 0.976364i \(-0.569344\pi\)
−0.216131 + 0.976364i \(0.569344\pi\)
\(14\) −1.86060 −0.497266
\(15\) 2.62871 0.678729
\(16\) 1.09414 0.273535
\(17\) 6.83662 1.65812 0.829062 − 0.559156i \(-0.188875\pi\)
0.829062 + 0.559156i \(0.188875\pi\)
\(18\) −2.85026 −0.671813
\(19\) 7.66011 1.75735 0.878675 − 0.477421i \(-0.158428\pi\)
0.878675 + 0.477421i \(0.158428\pi\)
\(20\) −1.46863 −0.328396
\(21\) 6.70960 1.46415
\(22\) −1.79951 −0.383657
\(23\) 7.50003 1.56387 0.781933 − 0.623363i \(-0.214234\pi\)
0.781933 + 0.623363i \(0.214234\pi\)
\(24\) 6.64658 1.35673
\(25\) 1.00000 0.200000
\(26\) 1.13610 0.222807
\(27\) 2.39236 0.460410
\(28\) −3.74859 −0.708416
\(29\) −3.25741 −0.604886 −0.302443 − 0.953167i \(-0.597802\pi\)
−0.302443 + 0.953167i \(0.597802\pi\)
\(30\) −1.91620 −0.349848
\(31\) −0.658785 −0.118321 −0.0591607 − 0.998248i \(-0.518842\pi\)
−0.0591607 + 0.998248i \(0.518842\pi\)
\(32\) −5.85449 −1.03494
\(33\) 6.48930 1.12964
\(34\) −4.98356 −0.854673
\(35\) 2.55244 0.431440
\(36\) −5.74248 −0.957080
\(37\) 0 0
\(38\) −5.58384 −0.905818
\(39\) −4.09694 −0.656036
\(40\) 2.52846 0.399785
\(41\) 2.46863 0.385535 0.192768 − 0.981244i \(-0.438254\pi\)
0.192768 + 0.981244i \(0.438254\pi\)
\(42\) −4.89097 −0.754692
\(43\) −10.9579 −1.67107 −0.835535 − 0.549438i \(-0.814842\pi\)
−0.835535 + 0.549438i \(0.814842\pi\)
\(44\) −3.62551 −0.546566
\(45\) 3.91009 0.582882
\(46\) −5.46715 −0.806088
\(47\) 3.11521 0.454400 0.227200 − 0.973848i \(-0.427043\pi\)
0.227200 + 0.973848i \(0.427043\pi\)
\(48\) 2.87617 0.415140
\(49\) −0.485072 −0.0692960
\(50\) −0.728950 −0.103089
\(51\) 17.9715 2.51651
\(52\) 2.28892 0.317416
\(53\) 8.64184 1.18705 0.593524 − 0.804816i \(-0.297736\pi\)
0.593524 + 0.804816i \(0.297736\pi\)
\(54\) −1.74391 −0.237317
\(55\) 2.46863 0.332870
\(56\) 6.45373 0.862416
\(57\) 20.1362 2.66710
\(58\) 2.37449 0.311786
\(59\) 6.23634 0.811903 0.405951 − 0.913895i \(-0.366940\pi\)
0.405951 + 0.913895i \(0.366940\pi\)
\(60\) −3.86060 −0.498401
\(61\) −3.27808 −0.419716 −0.209858 − 0.977732i \(-0.567300\pi\)
−0.209858 + 0.977732i \(0.567300\pi\)
\(62\) 0.480222 0.0609882
\(63\) 9.98026 1.25739
\(64\) 2.07935 0.259919
\(65\) −1.55854 −0.193313
\(66\) −4.73038 −0.582270
\(67\) 1.47764 0.180523 0.0902615 − 0.995918i \(-0.471230\pi\)
0.0902615 + 0.995918i \(0.471230\pi\)
\(68\) −10.0405 −1.21759
\(69\) 19.7154 2.37345
\(70\) −1.86060 −0.222384
\(71\) −8.06686 −0.957360 −0.478680 − 0.877989i \(-0.658885\pi\)
−0.478680 + 0.877989i \(0.658885\pi\)
\(72\) 9.88651 1.16514
\(73\) −4.96199 −0.580757 −0.290379 − 0.956912i \(-0.593781\pi\)
−0.290379 + 0.956912i \(0.593781\pi\)
\(74\) 0 0
\(75\) 2.62871 0.303537
\(76\) −11.2499 −1.29045
\(77\) 6.30102 0.718068
\(78\) 2.98647 0.338151
\(79\) −12.8206 −1.44243 −0.721214 − 0.692713i \(-0.756415\pi\)
−0.721214 + 0.692713i \(0.756415\pi\)
\(80\) 1.09414 0.122329
\(81\) −5.44146 −0.604607
\(82\) −1.79951 −0.198723
\(83\) 1.14934 0.126157 0.0630784 − 0.998009i \(-0.479908\pi\)
0.0630784 + 0.998009i \(0.479908\pi\)
\(84\) −9.85393 −1.07515
\(85\) 6.83662 0.741536
\(86\) 7.98779 0.861346
\(87\) −8.56277 −0.918026
\(88\) 6.24184 0.665382
\(89\) −11.5207 −1.22119 −0.610596 − 0.791942i \(-0.709070\pi\)
−0.610596 + 0.791942i \(0.709070\pi\)
\(90\) −2.85026 −0.300444
\(91\) −3.97807 −0.417015
\(92\) −11.0148 −1.14837
\(93\) −1.73175 −0.179574
\(94\) −2.27083 −0.234218
\(95\) 7.66011 0.785911
\(96\) −15.3897 −1.57071
\(97\) −17.2929 −1.75583 −0.877916 − 0.478815i \(-0.841067\pi\)
−0.877916 + 0.478815i \(0.841067\pi\)
\(98\) 0.353594 0.0357183
\(99\) 9.65257 0.970120
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.1.3 5
37.36 even 2 185.2.a.e.1.3 ✓ 5
111.110 odd 2 1665.2.a.p.1.3 5
148.147 odd 2 2960.2.a.w.1.1 5
185.73 odd 4 925.2.b.f.149.5 10
185.147 odd 4 925.2.b.f.149.6 10
185.184 even 2 925.2.a.f.1.3 5
259.258 odd 2 9065.2.a.k.1.3 5
555.554 odd 2 8325.2.a.ch.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.3 ✓ 5 37.36 even 2
925.2.a.f.1.3 5 185.184 even 2
925.2.b.f.149.5 10 185.73 odd 4
925.2.b.f.149.6 10 185.147 odd 4
1665.2.a.p.1.3 5 111.110 odd 2
2960.2.a.w.1.1 5 148.147 odd 2
6845.2.a.f.1.3 5 1.1 even 1 trivial
8325.2.a.ch.1.3 5 555.554 odd 2
9065.2.a.k.1.3 5 259.258 odd 2