Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13,0,0,16,0,0,8,0,0,0,-10,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 21x^{11} + 165x^{9} - 2x^{8} - 600x^{7} + 8x^{6} + 1005x^{5} + 32x^{4} - 657x^{3} - 80x^{2} + 73x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(2.36117\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.36117 q^{2} +3.57513 q^{4} -3.86072 q^{7} +3.71916 q^{8} +2.15745 q^{11} -0.289584 q^{13} -9.11583 q^{14} +1.63132 q^{16} +6.14007 q^{17} +2.80806 q^{19} +5.09411 q^{22} +0.0934537 q^{23} -0.683759 q^{26} -13.8026 q^{28} +0.794700 q^{29} -2.43423 q^{31} -3.58651 q^{32} +14.4978 q^{34} +1.00000 q^{37} +6.63032 q^{38} -7.71321 q^{41} +1.72469 q^{43} +7.71317 q^{44} +0.220660 q^{46} +11.9611 q^{47} +7.90517 q^{49} -1.03530 q^{52} +4.34574 q^{53} -14.3587 q^{56} +1.87642 q^{58} +13.2726 q^{59} +7.85176 q^{61} -5.74763 q^{62} -11.7310 q^{64} +9.27887 q^{67} +21.9516 q^{68} -6.28396 q^{71} +7.25206 q^{73} +2.36117 q^{74} +10.0392 q^{76} -8.32932 q^{77} +10.5780 q^{79} -18.2122 q^{82} -17.4100 q^{83} +4.07229 q^{86} +8.02391 q^{88} -8.47792 q^{89} +1.11800 q^{91} +0.334110 q^{92} +28.2422 q^{94} +16.2108 q^{97} +18.6655 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 16 q^{4} + 8 q^{7} - 10 q^{11} + 11 q^{13} - 8 q^{14} + 22 q^{16} - 2 q^{17} + 14 q^{19} + 14 q^{22} + 2 q^{23} - 16 q^{26} + 4 q^{28} - 23 q^{29} + 16 q^{31} + 10 q^{32} - 10 q^{34} + 13 q^{37}+ \cdots + 42 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.36117 1.66960 0.834800 − 0.550553i \(-0.185583\pi\)
0.834800 + 0.550553i \(0.185583\pi\)
\(3\) 0 0
\(4\) 3.57513 1.78757
\(5\) 0 0
\(6\) 0 0
\(7\) −3.86072 −1.45922 −0.729608 − 0.683866i \(-0.760297\pi\)
−0.729608 + 0.683866i \(0.760297\pi\)
\(8\) 3.71916 1.31492
\(9\) 0 0
\(10\) 0 0
\(11\) 2.15745 0.650496 0.325248 − 0.945629i \(-0.394552\pi\)
0.325248 + 0.945629i \(0.394552\pi\)
\(12\) 0 0
\(13\) −0.289584 −0.0803163 −0.0401581 − 0.999193i \(-0.512786\pi\)
−0.0401581 + 0.999193i \(0.512786\pi\)
\(14\) −9.11583 −2.43631
\(15\) 0 0
\(16\) 1.63132 0.407829
\(17\) 6.14007 1.48919 0.744593 − 0.667519i \(-0.232644\pi\)
0.744593 + 0.667519i \(0.232644\pi\)
\(18\) 0 0
\(19\) 2.80806 0.644214 0.322107 − 0.946703i \(-0.395609\pi\)
0.322107 + 0.946703i \(0.395609\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.09411 1.08607
\(23\) 0.0934537 0.0194865 0.00974323 − 0.999953i \(-0.496899\pi\)
0.00974323 + 0.999953i \(0.496899\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.683759 −0.134096
\(27\) 0 0
\(28\) −13.8026 −2.60845
\(29\) 0.794700 0.147572 0.0737860 − 0.997274i \(-0.476492\pi\)
0.0737860 + 0.997274i \(0.476492\pi\)
\(30\) 0 0
\(31\) −2.43423 −0.437200 −0.218600 − 0.975815i \(-0.570149\pi\)
−0.218600 + 0.975815i \(0.570149\pi\)
\(32\) −3.58651 −0.634011
\(33\) 0 0
\(34\) 14.4978 2.48635
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 6.63032 1.07558
\(39\) 0 0
\(40\) 0 0
\(41\) −7.71321 −1.20460 −0.602300 − 0.798269i \(-0.705749\pi\)
−0.602300 + 0.798269i \(0.705749\pi\)
\(42\) 0 0
\(43\) 1.72469 0.263013 0.131506 − 0.991315i \(-0.458019\pi\)
0.131506 + 0.991315i \(0.458019\pi\)
\(44\) 7.71317 1.16280
\(45\) 0 0
\(46\) 0.220660 0.0325346
\(47\) 11.9611 1.74471 0.872353 − 0.488877i \(-0.162593\pi\)
0.872353 + 0.488877i \(0.162593\pi\)
\(48\) 0 0
\(49\) 7.90517 1.12931
\(50\) 0 0
\(51\) 0 0
\(52\) −1.03530 −0.143571
\(53\) 4.34574 0.596933 0.298467 − 0.954420i \(-0.403525\pi\)
0.298467 + 0.954420i \(0.403525\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −14.3587 −1.91876
\(57\) 0 0
\(58\) 1.87642 0.246386
\(59\) 13.2726 1.72794 0.863971 − 0.503542i \(-0.167970\pi\)
0.863971 + 0.503542i \(0.167970\pi\)
\(60\) 0 0
\(61\) 7.85176 1.00532 0.502658 − 0.864486i \(-0.332356\pi\)
0.502658 + 0.864486i \(0.332356\pi\)
\(62\) −5.74763 −0.729950
\(63\) 0 0
\(64\) −11.7310 −1.46637
\(65\) 0 0
\(66\) 0 0
\(67\) 9.27887 1.13359 0.566797 − 0.823857i \(-0.308182\pi\)
0.566797 + 0.823857i \(0.308182\pi\)
\(68\) 21.9516 2.66202
\(69\) 0 0
\(70\) 0 0
\(71\) −6.28396 −0.745768 −0.372884 − 0.927878i \(-0.621631\pi\)
−0.372884 + 0.927878i \(0.621631\pi\)
\(72\) 0 0
\(73\) 7.25206 0.848790 0.424395 − 0.905477i \(-0.360487\pi\)
0.424395 + 0.905477i \(0.360487\pi\)
\(74\) 2.36117 0.274481
\(75\) 0 0
\(76\) 10.0392 1.15158
\(77\) −8.32932 −0.949214
\(78\) 0 0
\(79\) 10.5780 1.19011 0.595057 − 0.803684i \(-0.297129\pi\)
0.595057 + 0.803684i \(0.297129\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −18.2122 −2.01120
\(83\) −17.4100 −1.91100 −0.955499 − 0.294995i \(-0.904682\pi\)
−0.955499 + 0.294995i \(0.904682\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.07229 0.439126
\(87\) 0 0
\(88\) 8.02391 0.855352
\(89\) −8.47792 −0.898658 −0.449329 − 0.893366i \(-0.648337\pi\)
−0.449329 + 0.893366i \(0.648337\pi\)
\(90\) 0 0
\(91\) 1.11800 0.117199
\(92\) 0.334110 0.0348333
\(93\) 0 0
\(94\) 28.2422 2.91296
\(95\) 0 0
\(96\) 0 0
\(97\) 16.2108 1.64595 0.822977 − 0.568074i \(-0.192311\pi\)
0.822977 + 0.568074i \(0.192311\pi\)
\(98\) 18.6655 1.88550
\(99\) 0 0
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 8325.2.a.cv.1.12 13
3.2 odd 2 2775.2.a.bh.1.2 13
5.2 odd 4 1665.2.c.f.334.23 26
5.3 odd 4 1665.2.c.f.334.4 26
5.4 even 2 8325.2.a.cu.1.2 13
15.2 even 4 555.2.c.c.334.4 ✓ 26
15.8 even 4 555.2.c.c.334.23 yes 26
15.14 odd 2 2775.2.a.bg.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.c.334.4 ✓ 26 15.2 even 4
555.2.c.c.334.23 yes 26 15.8 even 4
1665.2.c.f.334.4 26 5.3 odd 4
1665.2.c.f.334.23 26 5.2 odd 4
2775.2.a.bg.1.12 13 15.14 odd 2
2775.2.a.bh.1.2 13 3.2 odd 2
8325.2.a.cu.1.2 13 5.4 even 2
8325.2.a.cv.1.12 13 1.1 even 1 trivial