Properties

Label 7935.2.a.bf.1.2
Level $7935$
Weight $2$
Character 7935.1
Self dual yes
Analytic conductor $63.361$
Analytic rank $0$
Dimension $6$
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] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,6,4,-6,0,2] 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: \(63.3612940039\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.35100\) of defining polynomial
Character \(\chi\) \(=\) 7935.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.17619 q^{2} +1.00000 q^{3} -0.616586 q^{4} -1.00000 q^{5} -1.17619 q^{6} +4.49962 q^{7} +3.07759 q^{8} +1.00000 q^{9} +1.17619 q^{10} +5.81200 q^{11} -0.616586 q^{12} +2.32343 q^{13} -5.29239 q^{14} -1.00000 q^{15} -2.38665 q^{16} -1.46396 q^{17} -1.17619 q^{18} -3.05141 q^{19} +0.616586 q^{20} +4.49962 q^{21} -6.83600 q^{22} +3.07759 q^{24} +1.00000 q^{25} -2.73279 q^{26} +1.00000 q^{27} -2.77440 q^{28} +7.50582 q^{29} +1.17619 q^{30} +0.278383 q^{31} -3.34804 q^{32} +5.81200 q^{33} +1.72190 q^{34} -4.49962 q^{35} -0.616586 q^{36} +8.24841 q^{37} +3.58902 q^{38} +2.32343 q^{39} -3.07759 q^{40} -6.03831 q^{41} -5.29239 q^{42} -2.09697 q^{43} -3.58360 q^{44} -1.00000 q^{45} +11.7754 q^{47} -2.38665 q^{48} +13.2466 q^{49} -1.17619 q^{50} -1.46396 q^{51} -1.43260 q^{52} +9.33724 q^{53} -1.17619 q^{54} -5.81200 q^{55} +13.8480 q^{56} -3.05141 q^{57} -8.82824 q^{58} -2.63301 q^{59} +0.616586 q^{60} +11.7760 q^{61} -0.327430 q^{62} +4.49962 q^{63} +8.71122 q^{64} -2.32343 q^{65} -6.83600 q^{66} -6.13444 q^{67} +0.902660 q^{68} +5.29239 q^{70} +1.67610 q^{71} +3.07759 q^{72} -1.23216 q^{73} -9.70166 q^{74} +1.00000 q^{75} +1.88145 q^{76} +26.1518 q^{77} -2.73279 q^{78} -14.0972 q^{79} +2.38665 q^{80} +1.00000 q^{81} +7.10217 q^{82} -15.1366 q^{83} -2.77440 q^{84} +1.46396 q^{85} +2.46643 q^{86} +7.50582 q^{87} +17.8870 q^{88} +0.840086 q^{89} +1.17619 q^{90} +10.4546 q^{91} +0.278383 q^{93} -13.8500 q^{94} +3.05141 q^{95} -3.34804 q^{96} +15.3893 q^{97} -15.5804 q^{98} +5.81200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 4 q^{4} - 6 q^{5} + 2 q^{7} + 6 q^{9} + 12 q^{11} + 4 q^{12} - 4 q^{13} + 8 q^{14} - 6 q^{15} + 14 q^{17} + 8 q^{19} - 4 q^{20} + 2 q^{21} + 12 q^{22} + 6 q^{25} + 24 q^{26} + 6 q^{27} + 4 q^{28}+ \cdots + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17619 −0.831689 −0.415845 − 0.909436i \(-0.636514\pi\)
−0.415845 + 0.909436i \(0.636514\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.616586 −0.308293
\(5\) −1.00000 −0.447214
\(6\) −1.17619 −0.480176
\(7\) 4.49962 1.70070 0.850348 − 0.526220i \(-0.176391\pi\)
0.850348 + 0.526220i \(0.176391\pi\)
\(8\) 3.07759 1.08809
\(9\) 1.00000 0.333333
\(10\) 1.17619 0.371943
\(11\) 5.81200 1.75238 0.876192 − 0.481961i \(-0.160075\pi\)
0.876192 + 0.481961i \(0.160075\pi\)
\(12\) −0.616586 −0.177993
\(13\) 2.32343 0.644405 0.322202 − 0.946671i \(-0.395577\pi\)
0.322202 + 0.946671i \(0.395577\pi\)
\(14\) −5.29239 −1.41445
\(15\) −1.00000 −0.258199
\(16\) −2.38665 −0.596663
\(17\) −1.46396 −0.355064 −0.177532 − 0.984115i \(-0.556811\pi\)
−0.177532 + 0.984115i \(0.556811\pi\)
\(18\) −1.17619 −0.277230
\(19\) −3.05141 −0.700041 −0.350020 − 0.936742i \(-0.613825\pi\)
−0.350020 + 0.936742i \(0.613825\pi\)
\(20\) 0.616586 0.137873
\(21\) 4.49962 0.981898
\(22\) −6.83600 −1.45744
\(23\) 0 0
\(24\) 3.07759 0.628211
\(25\) 1.00000 0.200000
\(26\) −2.73279 −0.535944
\(27\) 1.00000 0.192450
\(28\) −2.77440 −0.524312
\(29\) 7.50582 1.39379 0.696897 − 0.717171i \(-0.254563\pi\)
0.696897 + 0.717171i \(0.254563\pi\)
\(30\) 1.17619 0.214741
\(31\) 0.278383 0.0499990 0.0249995 − 0.999687i \(-0.492042\pi\)
0.0249995 + 0.999687i \(0.492042\pi\)
\(32\) −3.34804 −0.591855
\(33\) 5.81200 1.01174
\(34\) 1.72190 0.295303
\(35\) −4.49962 −0.760575
\(36\) −0.616586 −0.102764
\(37\) 8.24841 1.35603 0.678015 − 0.735048i \(-0.262841\pi\)
0.678015 + 0.735048i \(0.262841\pi\)
\(38\) 3.58902 0.582216
\(39\) 2.32343 0.372047
\(40\) −3.07759 −0.486610
\(41\) −6.03831 −0.943025 −0.471513 − 0.881859i \(-0.656292\pi\)
−0.471513 + 0.881859i \(0.656292\pi\)
\(42\) −5.29239 −0.816634
\(43\) −2.09697 −0.319786 −0.159893 − 0.987134i \(-0.551115\pi\)
−0.159893 + 0.987134i \(0.551115\pi\)
\(44\) −3.58360 −0.540248
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 11.7754 1.71762 0.858808 − 0.512297i \(-0.171205\pi\)
0.858808 + 0.512297i \(0.171205\pi\)
\(48\) −2.38665 −0.344483
\(49\) 13.2466 1.89237
\(50\) −1.17619 −0.166338
\(51\) −1.46396 −0.204996
\(52\) −1.43260 −0.198665
\(53\) 9.33724 1.28257 0.641284 − 0.767303i \(-0.278402\pi\)
0.641284 + 0.767303i \(0.278402\pi\)
\(54\) −1.17619 −0.160059
\(55\) −5.81200 −0.783690
\(56\) 13.8480 1.85052
\(57\) −3.05141 −0.404169
\(58\) −8.82824 −1.15920
\(59\) −2.63301 −0.342789 −0.171394 − 0.985203i \(-0.554827\pi\)
−0.171394 + 0.985203i \(0.554827\pi\)
\(60\) 0.616586 0.0796008
\(61\) 11.7760 1.50776 0.753882 − 0.657009i \(-0.228179\pi\)
0.753882 + 0.657009i \(0.228179\pi\)
\(62\) −0.327430 −0.0415837
\(63\) 4.49962 0.566899
\(64\) 8.71122 1.08890
\(65\) −2.32343 −0.288186
\(66\) −6.83600 −0.841453
\(67\) −6.13444 −0.749441 −0.374720 − 0.927138i \(-0.622261\pi\)
−0.374720 + 0.927138i \(0.622261\pi\)
\(68\) 0.902660 0.109464
\(69\) 0 0
\(70\) 5.29239 0.632562
\(71\) 1.67610 0.198916 0.0994581 − 0.995042i \(-0.468289\pi\)
0.0994581 + 0.995042i \(0.468289\pi\)
\(72\) 3.07759 0.362698
\(73\) −1.23216 −0.144213 −0.0721067 − 0.997397i \(-0.522972\pi\)
−0.0721067 + 0.997397i \(0.522972\pi\)
\(74\) −9.70166 −1.12780
\(75\) 1.00000 0.115470
\(76\) 1.88145 0.215817
\(77\) 26.1518 2.98027
\(78\) −2.73279 −0.309428
\(79\) −14.0972 −1.58605 −0.793027 − 0.609186i \(-0.791496\pi\)
−0.793027 + 0.609186i \(0.791496\pi\)
\(80\) 2.38665 0.266836
\(81\) 1.00000 0.111111
\(82\) 7.10217 0.784304
\(83\) −15.1366 −1.66145 −0.830727 − 0.556680i \(-0.812075\pi\)
−0.830727 + 0.556680i \(0.812075\pi\)
\(84\) −2.77440 −0.302712
\(85\) 1.46396 0.158789
\(86\) 2.46643 0.265962
\(87\) 7.50582 0.804708
\(88\) 17.8870 1.90676
\(89\) 0.840086 0.0890490 0.0445245 − 0.999008i \(-0.485823\pi\)
0.0445245 + 0.999008i \(0.485823\pi\)
\(90\) 1.17619 0.123981
\(91\) 10.4546 1.09594
\(92\) 0 0
\(93\) 0.278383 0.0288670
\(94\) −13.8500 −1.42852
\(95\) 3.05141 0.313068
\(96\) −3.34804 −0.341708
\(97\) 15.3893 1.56254 0.781272 − 0.624190i \(-0.214571\pi\)
0.781272 + 0.624190i \(0.214571\pi\)
\(98\) −15.5804 −1.57386
\(99\) 5.81200 0.584128
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 7935.2.a.bf.1.2 ✓ 6
23.22 odd 2 7935.2.a.bg.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.bf.1.2 ✓ 6 1.1 even 1 trivial
7935.2.a.bg.1.2 yes 6 23.22 odd 2