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 = [5,-1,0,1,0,0,0,0,0,0,16,0,3] 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: \(5\)
Coefficient field: 5.5.65657.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 5x^{3} + 2x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 925)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.233963\) 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)
 
\(f(q)\) \(=\) \(q-1.32892 q^{2} -0.233963 q^{4} +3.11718 q^{7} +2.96877 q^{8} +4.02350 q^{11} +6.45341 q^{13} -4.14249 q^{14} -3.47734 q^{16} -0.628844 q^{17} -8.46282 q^{19} -5.34693 q^{22} +4.61042 q^{23} -8.57609 q^{26} -0.729303 q^{28} +1.58162 q^{29} -0.511666 q^{31} -1.31642 q^{32} +0.835685 q^{34} -1.00000 q^{37} +11.2464 q^{38} +9.85929 q^{41} -5.71479 q^{43} -0.941350 q^{44} -6.12689 q^{46} +2.47605 q^{47} +2.71680 q^{49} -1.50986 q^{52} +11.0294 q^{53} +9.25417 q^{56} -2.10185 q^{58} +10.9761 q^{59} +3.99227 q^{61} +0.679965 q^{62} +8.70409 q^{64} +2.41138 q^{67} +0.147126 q^{68} -1.93074 q^{71} -2.43974 q^{73} +1.32892 q^{74} +1.97998 q^{76} +12.5420 q^{77} +9.87782 q^{79} -13.1022 q^{82} -4.70778 q^{83} +7.59451 q^{86} +11.9448 q^{88} +6.33464 q^{89} +20.1164 q^{91} -1.07867 q^{92} -3.29048 q^{94} -2.78521 q^{97} -3.61042 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + q^{4} + 16 q^{11} + 3 q^{13} + 5 q^{14} - 11 q^{16} + 2 q^{17} - 11 q^{19} - 19 q^{22} - 7 q^{23} + 4 q^{26} + 7 q^{28} + 15 q^{29} - 13 q^{31} - q^{32} + 13 q^{34} - 5 q^{37} - 2 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32892 −0.939691 −0.469845 − 0.882749i \(-0.655690\pi\)
−0.469845 + 0.882749i \(0.655690\pi\)
\(3\) 0 0
\(4\) −0.233963 −0.116981
\(5\) 0 0
\(6\) 0 0
\(7\) 3.11718 1.17818 0.589091 − 0.808067i \(-0.299486\pi\)
0.589091 + 0.808067i \(0.299486\pi\)
\(8\) 2.96877 1.04962
\(9\) 0 0
\(10\) 0 0
\(11\) 4.02350 1.21313 0.606566 − 0.795033i \(-0.292547\pi\)
0.606566 + 0.795033i \(0.292547\pi\)
\(12\) 0 0
\(13\) 6.45341 1.78985 0.894927 − 0.446213i \(-0.147228\pi\)
0.894927 + 0.446213i \(0.147228\pi\)
\(14\) −4.14249 −1.10713
\(15\) 0 0
\(16\) −3.47734 −0.869334
\(17\) −0.628844 −0.152517 −0.0762585 − 0.997088i \(-0.524297\pi\)
−0.0762585 + 0.997088i \(0.524297\pi\)
\(18\) 0 0
\(19\) −8.46282 −1.94150 −0.970752 − 0.240085i \(-0.922825\pi\)
−0.970752 + 0.240085i \(0.922825\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.34693 −1.13997
\(23\) 4.61042 0.961338 0.480669 − 0.876902i \(-0.340394\pi\)
0.480669 + 0.876902i \(0.340394\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.57609 −1.68191
\(27\) 0 0
\(28\) −0.729303 −0.137825
\(29\) 1.58162 0.293699 0.146849 − 0.989159i \(-0.453087\pi\)
0.146849 + 0.989159i \(0.453087\pi\)
\(30\) 0 0
\(31\) −0.511666 −0.0918979 −0.0459490 − 0.998944i \(-0.514631\pi\)
−0.0459490 + 0.998944i \(0.514631\pi\)
\(32\) −1.31642 −0.232712
\(33\) 0 0
\(34\) 0.835685 0.143319
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 11.2464 1.82441
\(39\) 0 0
\(40\) 0 0
\(41\) 9.85929 1.53976 0.769881 − 0.638187i \(-0.220316\pi\)
0.769881 + 0.638187i \(0.220316\pi\)
\(42\) 0 0
\(43\) −5.71479 −0.871497 −0.435748 − 0.900069i \(-0.643516\pi\)
−0.435748 + 0.900069i \(0.643516\pi\)
\(44\) −0.941350 −0.141914
\(45\) 0 0
\(46\) −6.12689 −0.903361
\(47\) 2.47605 0.361169 0.180584 − 0.983559i \(-0.442201\pi\)
0.180584 + 0.983559i \(0.442201\pi\)
\(48\) 0 0
\(49\) 2.71680 0.388114
\(50\) 0 0
\(51\) 0 0
\(52\) −1.50986 −0.209380
\(53\) 11.0294 1.51500 0.757502 − 0.652833i \(-0.226420\pi\)
0.757502 + 0.652833i \(0.226420\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.25417 1.23664
\(57\) 0 0
\(58\) −2.10185 −0.275986
\(59\) 10.9761 1.42896 0.714481 − 0.699654i \(-0.246663\pi\)
0.714481 + 0.699654i \(0.246663\pi\)
\(60\) 0 0
\(61\) 3.99227 0.511158 0.255579 − 0.966788i \(-0.417734\pi\)
0.255579 + 0.966788i \(0.417734\pi\)
\(62\) 0.679965 0.0863556
\(63\) 0 0
\(64\) 8.70409 1.08801
\(65\) 0 0
\(66\) 0 0
\(67\) 2.41138 0.294597 0.147298 − 0.989092i \(-0.452942\pi\)
0.147298 + 0.989092i \(0.452942\pi\)
\(68\) 0.147126 0.0178417
\(69\) 0 0
\(70\) 0 0
\(71\) −1.93074 −0.229137 −0.114569 − 0.993415i \(-0.536549\pi\)
−0.114569 + 0.993415i \(0.536549\pi\)
\(72\) 0 0
\(73\) −2.43974 −0.285550 −0.142775 − 0.989755i \(-0.545603\pi\)
−0.142775 + 0.989755i \(0.545603\pi\)
\(74\) 1.32892 0.154484
\(75\) 0 0
\(76\) 1.97998 0.227120
\(77\) 12.5420 1.42929
\(78\) 0 0
\(79\) 9.87782 1.11134 0.555671 − 0.831403i \(-0.312462\pi\)
0.555671 + 0.831403i \(0.312462\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −13.1022 −1.44690
\(83\) −4.70778 −0.516746 −0.258373 − 0.966045i \(-0.583186\pi\)
−0.258373 + 0.966045i \(0.583186\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.59451 0.818937
\(87\) 0 0
\(88\) 11.9448 1.27332
\(89\) 6.33464 0.671471 0.335735 − 0.941956i \(-0.391015\pi\)
0.335735 + 0.941956i \(0.391015\pi\)
\(90\) 0 0
\(91\) 20.1164 2.10877
\(92\) −1.07867 −0.112459
\(93\) 0 0
\(94\) −3.29048 −0.339387
\(95\) 0 0
\(96\) 0 0
\(97\) −2.78521 −0.282795 −0.141397 − 0.989953i \(-0.545160\pi\)
−0.141397 + 0.989953i \(0.545160\pi\)
\(98\) −3.61042 −0.364707
\(99\) 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.cb.1.2 5
3.2 odd 2 925.2.a.i.1.4 yes 5
5.4 even 2 8325.2.a.cd.1.4 5
15.2 even 4 925.2.b.h.149.8 10
15.8 even 4 925.2.b.h.149.3 10
15.14 odd 2 925.2.a.g.1.2 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.g.1.2 ✓ 5 15.14 odd 2
925.2.a.i.1.4 yes 5 3.2 odd 2
925.2.b.h.149.3 10 15.8 even 4
925.2.b.h.149.8 10 15.2 even 4
8325.2.a.cb.1.2 5 1.1 even 1 trivial
8325.2.a.cd.1.4 5 5.4 even 2