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 = [5,2,0,4,0,0,-4,9,0,0,1,0,-8] 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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.176684.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 6x^{3} - x^{2} + 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2775)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.727897\) 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)
 
\(f(q)\) \(=\) \(q-1.82424 q^{2} +1.32786 q^{4} -2.42958 q^{7} +1.22614 q^{8} +0.0262158 q^{11} -2.72253 q^{13} +4.43214 q^{14} -4.89251 q^{16} +4.47017 q^{17} +0.925960 q^{19} -0.0478240 q^{22} +9.37247 q^{23} +4.96655 q^{26} -3.22614 q^{28} -5.41777 q^{29} -2.25236 q^{31} +6.47283 q^{32} -8.15467 q^{34} +1.00000 q^{37} -1.68918 q^{38} -8.04728 q^{41} -8.72399 q^{43} +0.0348110 q^{44} -17.0977 q^{46} -4.16647 q^{47} -1.09715 q^{49} -3.61514 q^{52} +9.70505 q^{53} -2.97901 q^{56} +9.88333 q^{58} +7.22560 q^{59} -8.71072 q^{61} +4.10885 q^{62} -2.02300 q^{64} +8.78768 q^{67} +5.93576 q^{68} -11.8066 q^{71} +2.40593 q^{73} -1.82424 q^{74} +1.22955 q^{76} -0.0636934 q^{77} +5.27090 q^{79} +14.6802 q^{82} -8.56987 q^{83} +15.9147 q^{86} +0.0321444 q^{88} +15.8983 q^{89} +6.61459 q^{91} +12.4453 q^{92} +7.60066 q^{94} +8.43577 q^{97} +2.00146 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 4 q^{4} - 4 q^{7} + 9 q^{8} + q^{11} - 8 q^{13} - 10 q^{14} + 10 q^{16} + 13 q^{17} - 12 q^{19} - 16 q^{22} - 4 q^{23} + 7 q^{26} - 19 q^{28} + 3 q^{29} - 15 q^{31} + 35 q^{32} - 3 q^{34}+ \cdots - 13 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82424 −1.28993 −0.644967 − 0.764210i \(-0.723129\pi\)
−0.644967 + 0.764210i \(0.723129\pi\)
\(3\) 0 0
\(4\) 1.32786 0.663931
\(5\) 0 0
\(6\) 0 0
\(7\) −2.42958 −0.918294 −0.459147 − 0.888360i \(-0.651845\pi\)
−0.459147 + 0.888360i \(0.651845\pi\)
\(8\) 1.22614 0.433507
\(9\) 0 0
\(10\) 0 0
\(11\) 0.0262158 0.00790437 0.00395218 − 0.999992i \(-0.498742\pi\)
0.00395218 + 0.999992i \(0.498742\pi\)
\(12\) 0 0
\(13\) −2.72253 −0.755093 −0.377546 − 0.925991i \(-0.623232\pi\)
−0.377546 + 0.925991i \(0.623232\pi\)
\(14\) 4.43214 1.18454
\(15\) 0 0
\(16\) −4.89251 −1.22313
\(17\) 4.47017 1.08417 0.542087 − 0.840322i \(-0.317634\pi\)
0.542087 + 0.840322i \(0.317634\pi\)
\(18\) 0 0
\(19\) 0.925960 0.212430 0.106215 − 0.994343i \(-0.466127\pi\)
0.106215 + 0.994343i \(0.466127\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.0478240 −0.0101961
\(23\) 9.37247 1.95430 0.977148 − 0.212561i \(-0.0681804\pi\)
0.977148 + 0.212561i \(0.0681804\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.96655 0.974020
\(27\) 0 0
\(28\) −3.22614 −0.609684
\(29\) −5.41777 −1.00605 −0.503027 − 0.864270i \(-0.667781\pi\)
−0.503027 + 0.864270i \(0.667781\pi\)
\(30\) 0 0
\(31\) −2.25236 −0.404536 −0.202268 − 0.979330i \(-0.564831\pi\)
−0.202268 + 0.979330i \(0.564831\pi\)
\(32\) 6.47283 1.14425
\(33\) 0 0
\(34\) −8.15467 −1.39851
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −1.68918 −0.274021
\(39\) 0 0
\(40\) 0 0
\(41\) −8.04728 −1.25677 −0.628387 − 0.777901i \(-0.716284\pi\)
−0.628387 + 0.777901i \(0.716284\pi\)
\(42\) 0 0
\(43\) −8.72399 −1.33040 −0.665198 − 0.746667i \(-0.731653\pi\)
−0.665198 + 0.746667i \(0.731653\pi\)
\(44\) 0.0348110 0.00524795
\(45\) 0 0
\(46\) −17.0977 −2.52091
\(47\) −4.16647 −0.607743 −0.303871 − 0.952713i \(-0.598279\pi\)
−0.303871 + 0.952713i \(0.598279\pi\)
\(48\) 0 0
\(49\) −1.09715 −0.156735
\(50\) 0 0
\(51\) 0 0
\(52\) −3.61514 −0.501329
\(53\) 9.70505 1.33309 0.666545 − 0.745465i \(-0.267772\pi\)
0.666545 + 0.745465i \(0.267772\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.97901 −0.398087
\(57\) 0 0
\(58\) 9.88333 1.29774
\(59\) 7.22560 0.940693 0.470346 − 0.882482i \(-0.344129\pi\)
0.470346 + 0.882482i \(0.344129\pi\)
\(60\) 0 0
\(61\) −8.71072 −1.11529 −0.557647 − 0.830079i \(-0.688296\pi\)
−0.557647 + 0.830079i \(0.688296\pi\)
\(62\) 4.10885 0.521825
\(63\) 0 0
\(64\) −2.02300 −0.252875
\(65\) 0 0
\(66\) 0 0
\(67\) 8.78768 1.07359 0.536793 − 0.843714i \(-0.319636\pi\)
0.536793 + 0.843714i \(0.319636\pi\)
\(68\) 5.93576 0.719817
\(69\) 0 0
\(70\) 0 0
\(71\) −11.8066 −1.40119 −0.700594 − 0.713560i \(-0.747082\pi\)
−0.700594 + 0.713560i \(0.747082\pi\)
\(72\) 0 0
\(73\) 2.40593 0.281592 0.140796 − 0.990039i \(-0.455034\pi\)
0.140796 + 0.990039i \(0.455034\pi\)
\(74\) −1.82424 −0.212064
\(75\) 0 0
\(76\) 1.22955 0.141039
\(77\) −0.0636934 −0.00725854
\(78\) 0 0
\(79\) 5.27090 0.593022 0.296511 − 0.955029i \(-0.404177\pi\)
0.296511 + 0.955029i \(0.404177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 14.6802 1.62116
\(83\) −8.56987 −0.940666 −0.470333 − 0.882489i \(-0.655866\pi\)
−0.470333 + 0.882489i \(0.655866\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 15.9147 1.71612
\(87\) 0 0
\(88\) 0.0321444 0.00342660
\(89\) 15.8983 1.68521 0.842606 − 0.538531i \(-0.181020\pi\)
0.842606 + 0.538531i \(0.181020\pi\)
\(90\) 0 0
\(91\) 6.61459 0.693397
\(92\) 12.4453 1.29752
\(93\) 0 0
\(94\) 7.60066 0.783948
\(95\) 0 0
\(96\) 0 0
\(97\) 8.43577 0.856522 0.428261 − 0.903655i \(-0.359126\pi\)
0.428261 + 0.903655i \(0.359126\pi\)
\(98\) 2.00146 0.202178
\(99\) 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.ce.1.1 5
3.2 odd 2 2775.2.a.ba.1.5 ✓ 5
5.4 even 2 8325.2.a.bz.1.5 5
15.14 odd 2 2775.2.a.bb.1.1 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.ba.1.5 ✓ 5 3.2 odd 2
2775.2.a.bb.1.1 yes 5 15.14 odd 2
8325.2.a.bz.1.5 5 5.4 even 2
8325.2.a.ce.1.1 5 1.1 even 1 trivial