Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [614,2,Mod(17,614)] 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("614.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(614, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 614 = 2 \cdot 307 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 614.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: no
Analytic conductor: \(4.90281468411\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 17.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 614.17
Dual form 614.2.c.a.289.1

$q$-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+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +(-2.00000 + 3.46410i) q^{5} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +(-2.00000 + 3.46410i) q^{10} +(-1.50000 + 2.59808i) q^{11} -3.00000 q^{12} +(3.00000 - 5.19615i) q^{13} +(6.00000 - 10.3923i) q^{15} +1.00000 q^{16} -7.00000 q^{17} +6.00000 q^{18} -4.00000 q^{19} +(-2.00000 + 3.46410i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(-2.00000 - 3.46410i) q^{23} -3.00000 q^{24} +(-5.50000 - 9.52628i) q^{25} +(3.00000 - 5.19615i) q^{26} -9.00000 q^{27} +(3.00000 + 5.19615i) q^{29} +(6.00000 - 10.3923i) q^{30} +(-5.00000 - 8.66025i) q^{31} +1.00000 q^{32} +(4.50000 - 7.79423i) q^{33} -7.00000 q^{34} +6.00000 q^{36} +(3.00000 - 5.19615i) q^{37} -4.00000 q^{38} +(-9.00000 + 15.5885i) q^{39} +(-2.00000 + 3.46410i) q^{40} +(3.50000 + 6.06218i) q^{41} +(-5.50000 + 9.52628i) q^{43} +(-1.50000 + 2.59808i) q^{44} +(-12.0000 + 20.7846i) q^{45} +(-2.00000 - 3.46410i) q^{46} +(-3.00000 - 5.19615i) q^{47} -3.00000 q^{48} +(3.50000 - 6.06218i) q^{49} +(-5.50000 - 9.52628i) q^{50} +21.0000 q^{51} +(3.00000 - 5.19615i) q^{52} +(2.00000 - 3.46410i) q^{53} -9.00000 q^{54} +(-6.00000 - 10.3923i) q^{55} +12.0000 q^{57} +(3.00000 + 5.19615i) q^{58} +(-0.500000 - 0.866025i) q^{59} +(6.00000 - 10.3923i) q^{60} +(1.00000 + 1.73205i) q^{61} +(-5.00000 - 8.66025i) q^{62} +1.00000 q^{64} +(12.0000 + 20.7846i) q^{65} +(4.50000 - 7.79423i) q^{66} +(-0.500000 + 0.866025i) q^{67} -7.00000 q^{68} +(6.00000 + 10.3923i) q^{69} +(-3.00000 - 5.19615i) q^{71} +6.00000 q^{72} +(-5.50000 + 9.52628i) q^{73} +(3.00000 - 5.19615i) q^{74} +(16.5000 + 28.5788i) q^{75} -4.00000 q^{76} +(-9.00000 + 15.5885i) q^{78} -4.00000 q^{79} +(-2.00000 + 3.46410i) q^{80} +9.00000 q^{81} +(3.50000 + 6.06218i) q^{82} +(-4.00000 - 6.92820i) q^{83} +(14.0000 - 24.2487i) q^{85} +(-5.50000 + 9.52628i) q^{86} +(-9.00000 - 15.5885i) q^{87} +(-1.50000 + 2.59808i) q^{88} +(-3.00000 + 5.19615i) q^{89} +(-12.0000 + 20.7846i) q^{90} +(-2.00000 - 3.46410i) q^{92} +(15.0000 + 25.9808i) q^{93} +(-3.00000 - 5.19615i) q^{94} +(8.00000 - 13.8564i) q^{95} -3.00000 q^{96} +1.00000 q^{97} +(3.50000 - 6.06218i) q^{98} +(-9.00000 + 15.5885i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{3} + 2 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{8} + 12 q^{9} - 4 q^{10} - 3 q^{11} - 6 q^{12} + 6 q^{13} + 12 q^{15} + 2 q^{16} - 14 q^{17} + 12 q^{18} - 8 q^{19} - 4 q^{20} - 3 q^{22} - 4 q^{23}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/614\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 + 3.46410i −0.894427 + 1.54919i −0.0599153 + 0.998203i \(0.519083\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) −3.00000 −1.22474
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) −2.00000 + 3.46410i −0.632456 + 1.09545i
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 − 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) −3.00000 −0.866025
\(13\) 3.00000 − 5.19615i 0.832050 − 1.44115i −0.0643593 − 0.997927i \(-0.520500\pi\)
0.896410 − 0.443227i \(-0.146166\pi\)
\(14\) 0 0
\(15\) 6.00000 − 10.3923i 1.54919 − 2.68328i
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 6.00000 1.41421
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 + 3.46410i −0.447214 + 0.774597i
\(21\) 0 0
\(22\) −1.50000 + 2.59808i −0.319801 + 0.553912i
\(23\) −2.00000 − 3.46410i −0.417029 − 0.722315i 0.578610 − 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) −3.00000 −0.612372
\(25\) −5.50000 − 9.52628i −1.10000 − 1.90526i
\(26\) 3.00000 − 5.19615i 0.588348 − 1.01905i
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 6.00000 − 10.3923i 1.09545 − 1.89737i
\(31\) −5.00000 − 8.66025i −0.898027 − 1.55543i −0.830014 − 0.557743i \(-0.811667\pi\)
−0.0680129 − 0.997684i \(-0.521666\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.50000 − 7.79423i 0.783349 − 1.35680i
\(34\) −7.00000 −1.20049
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 3.00000 − 5.19615i 0.493197 − 0.854242i −0.506772 − 0.862080i \(-0.669162\pi\)
0.999969 + 0.00783774i \(0.00249486\pi\)
\(38\) −4.00000 −0.648886
\(39\) −9.00000 + 15.5885i −1.44115 + 2.49615i
\(40\) −2.00000 + 3.46410i −0.316228 + 0.547723i
\(41\) 3.50000 + 6.06218i 0.546608 + 0.946753i 0.998504 + 0.0546823i \(0.0174146\pi\)
−0.451896 + 0.892071i \(0.649252\pi\)
\(42\) 0 0
\(43\) −5.50000 + 9.52628i −0.838742 + 1.45274i 0.0522047 + 0.998636i \(0.483375\pi\)
−0.890947 + 0.454108i \(0.849958\pi\)
\(44\) −1.50000 + 2.59808i −0.226134 + 0.391675i
\(45\) −12.0000 + 20.7846i −1.78885 + 3.09839i
\(46\) −2.00000 − 3.46410i −0.294884 − 0.510754i
\(47\) −3.00000 − 5.19615i −0.437595 − 0.757937i 0.559908 − 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) −3.00000 −0.433013
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) −5.50000 − 9.52628i −0.777817 − 1.34722i
\(51\) 21.0000 2.94059
\(52\) 3.00000 − 5.19615i 0.416025 − 0.720577i
\(53\) 2.00000 − 3.46410i 0.274721 − 0.475831i −0.695344 − 0.718677i \(-0.744748\pi\)
0.970065 + 0.242846i \(0.0780811\pi\)
\(54\) −9.00000 −1.22474
\(55\) −6.00000 − 10.3923i −0.809040 − 1.40130i
\(56\) 0 0
\(57\) 12.0000 1.58944
\(58\) 3.00000 + 5.19615i 0.393919 + 0.682288i
\(59\) −0.500000 − 0.866025i −0.0650945 − 0.112747i 0.831641 − 0.555313i \(-0.187402\pi\)
−0.896736 + 0.442566i \(0.854068\pi\)
\(60\) 6.00000 − 10.3923i 0.774597 − 1.34164i
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 − 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) −5.00000 − 8.66025i −0.635001 − 1.09985i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.0000 + 20.7846i 1.48842 + 2.57801i
\(66\) 4.50000 − 7.79423i 0.553912 − 0.959403i
\(67\) −0.500000 + 0.866025i −0.0610847 + 0.105802i −0.894951 − 0.446165i \(-0.852789\pi\)
0.833866 + 0.551967i \(0.186123\pi\)
\(68\) −7.00000 −0.848875
\(69\) 6.00000 + 10.3923i 0.722315 + 1.25109i
\(70\) 0 0
\(71\) −3.00000 − 5.19615i −0.356034 − 0.616670i 0.631260 − 0.775571i \(-0.282538\pi\)
−0.987294 + 0.158901i \(0.949205\pi\)
\(72\) 6.00000 0.707107
\(73\) −5.50000 + 9.52628i −0.643726 + 1.11497i 0.340868 + 0.940111i \(0.389279\pi\)
−0.984594 + 0.174855i \(0.944054\pi\)
\(74\) 3.00000 − 5.19615i 0.348743 − 0.604040i
\(75\) 16.5000 + 28.5788i 1.90526 + 3.30000i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −9.00000 + 15.5885i −1.01905 + 1.76505i
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 + 3.46410i −0.223607 + 0.387298i
\(81\) 9.00000 1.00000
\(82\) 3.50000 + 6.06218i 0.386510 + 0.669456i
\(83\) −4.00000 − 6.92820i −0.439057 − 0.760469i 0.558560 − 0.829464i \(-0.311354\pi\)
−0.997617 + 0.0689950i \(0.978021\pi\)
\(84\) 0 0
\(85\) 14.0000 − 24.2487i 1.51851 − 2.63014i
\(86\) −5.50000 + 9.52628i −0.593080 + 1.02725i
\(87\) −9.00000 − 15.5885i −0.964901 − 1.67126i
\(88\) −1.50000 + 2.59808i −0.159901 + 0.276956i
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 − 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) −12.0000 + 20.7846i −1.26491 + 2.19089i
\(91\) 0 0
\(92\) −2.00000 − 3.46410i −0.208514 − 0.361158i
\(93\) 15.0000 + 25.9808i 1.55543 + 2.69408i
\(94\) −3.00000 − 5.19615i −0.309426 − 0.535942i
\(95\) 8.00000 − 13.8564i 0.820783 − 1.42164i
\(96\) −3.00000 −0.306186
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 3.50000 − 6.06218i 0.353553 − 0.612372i
\(99\) −9.00000 + 15.5885i −0.904534 + 1.56670i
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 614.2.c.a.17.1 ✓ 2
307.289 even 3 inner 614.2.c.a.289.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
614.2.c.a.17.1 ✓ 2 1.1 even 1 trivial
614.2.c.a.289.1 yes 2 307.289 even 3 inner