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 = [9,-5,0,11,0,0,8,-15,0,0,0,0,6] 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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.68489\) 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)
 
\(f(q)\) \(=\) \(q-2.68489 q^{2} +5.20864 q^{4} -3.48285 q^{7} -8.61484 q^{8} +3.18991 q^{11} -1.81648 q^{13} +9.35107 q^{14} +12.7126 q^{16} -2.82645 q^{17} -5.72714 q^{19} -8.56456 q^{22} +1.43274 q^{23} +4.87704 q^{26} -18.1409 q^{28} +2.30634 q^{29} +6.00714 q^{31} -16.9023 q^{32} +7.58870 q^{34} -1.00000 q^{37} +15.3767 q^{38} -7.82452 q^{41} +11.6967 q^{43} +16.6151 q^{44} -3.84676 q^{46} -2.42814 q^{47} +5.13024 q^{49} -9.46137 q^{52} +9.32573 q^{53} +30.0042 q^{56} -6.19227 q^{58} -2.86539 q^{59} -4.67275 q^{61} -16.1285 q^{62} +19.9557 q^{64} +6.00752 q^{67} -14.7219 q^{68} -3.40566 q^{71} +1.89975 q^{73} +2.68489 q^{74} -29.8306 q^{76} -11.1100 q^{77} -1.60903 q^{79} +21.0080 q^{82} -7.37571 q^{83} -31.4043 q^{86} -27.4806 q^{88} +8.48746 q^{89} +6.32652 q^{91} +7.46264 q^{92} +6.51929 q^{94} +9.00003 q^{97} -13.7741 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} + 11 q^{4} + 8 q^{7} - 15 q^{8} + 6 q^{13} + 4 q^{14} + 11 q^{16} - 18 q^{17} - 4 q^{19} + 6 q^{22} - 16 q^{23} + 6 q^{26} - 20 q^{28} + 2 q^{29} - 6 q^{31} - 35 q^{32} + 6 q^{34} - 9 q^{37}+ \cdots - 21 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.68489 −1.89850 −0.949252 − 0.314516i \(-0.898158\pi\)
−0.949252 + 0.314516i \(0.898158\pi\)
\(3\) 0 0
\(4\) 5.20864 2.60432
\(5\) 0 0
\(6\) 0 0
\(7\) −3.48285 −1.31639 −0.658197 − 0.752846i \(-0.728680\pi\)
−0.658197 + 0.752846i \(0.728680\pi\)
\(8\) −8.61484 −3.04581
\(9\) 0 0
\(10\) 0 0
\(11\) 3.18991 0.961794 0.480897 − 0.876777i \(-0.340311\pi\)
0.480897 + 0.876777i \(0.340311\pi\)
\(12\) 0 0
\(13\) −1.81648 −0.503800 −0.251900 − 0.967753i \(-0.581055\pi\)
−0.251900 + 0.967753i \(0.581055\pi\)
\(14\) 9.35107 2.49918
\(15\) 0 0
\(16\) 12.7126 3.17816
\(17\) −2.82645 −0.685514 −0.342757 − 0.939424i \(-0.611361\pi\)
−0.342757 + 0.939424i \(0.611361\pi\)
\(18\) 0 0
\(19\) −5.72714 −1.31389 −0.656947 − 0.753936i \(-0.728153\pi\)
−0.656947 + 0.753936i \(0.728153\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.56456 −1.82597
\(23\) 1.43274 0.298748 0.149374 − 0.988781i \(-0.452274\pi\)
0.149374 + 0.988781i \(0.452274\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.87704 0.956467
\(27\) 0 0
\(28\) −18.1409 −3.42831
\(29\) 2.30634 0.428276 0.214138 − 0.976803i \(-0.431306\pi\)
0.214138 + 0.976803i \(0.431306\pi\)
\(30\) 0 0
\(31\) 6.00714 1.07891 0.539457 − 0.842013i \(-0.318629\pi\)
0.539457 + 0.842013i \(0.318629\pi\)
\(32\) −16.9023 −2.98794
\(33\) 0 0
\(34\) 7.58870 1.30145
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 15.3767 2.49444
\(39\) 0 0
\(40\) 0 0
\(41\) −7.82452 −1.22198 −0.610992 − 0.791636i \(-0.709229\pi\)
−0.610992 + 0.791636i \(0.709229\pi\)
\(42\) 0 0
\(43\) 11.6967 1.78373 0.891863 − 0.452305i \(-0.149398\pi\)
0.891863 + 0.452305i \(0.149398\pi\)
\(44\) 16.6151 2.50482
\(45\) 0 0
\(46\) −3.84676 −0.567174
\(47\) −2.42814 −0.354181 −0.177090 − 0.984195i \(-0.556668\pi\)
−0.177090 + 0.984195i \(0.556668\pi\)
\(48\) 0 0
\(49\) 5.13024 0.732892
\(50\) 0 0
\(51\) 0 0
\(52\) −9.46137 −1.31206
\(53\) 9.32573 1.28099 0.640493 − 0.767964i \(-0.278730\pi\)
0.640493 + 0.767964i \(0.278730\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 30.0042 4.00948
\(57\) 0 0
\(58\) −6.19227 −0.813084
\(59\) −2.86539 −0.373042 −0.186521 − 0.982451i \(-0.559721\pi\)
−0.186521 + 0.982451i \(0.559721\pi\)
\(60\) 0 0
\(61\) −4.67275 −0.598285 −0.299142 − 0.954208i \(-0.596701\pi\)
−0.299142 + 0.954208i \(0.596701\pi\)
\(62\) −16.1285 −2.04832
\(63\) 0 0
\(64\) 19.9557 2.49446
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00752 0.733936 0.366968 − 0.930234i \(-0.380396\pi\)
0.366968 + 0.930234i \(0.380396\pi\)
\(68\) −14.7219 −1.78530
\(69\) 0 0
\(70\) 0 0
\(71\) −3.40566 −0.404177 −0.202089 − 0.979367i \(-0.564773\pi\)
−0.202089 + 0.979367i \(0.564773\pi\)
\(72\) 0 0
\(73\) 1.89975 0.222349 0.111174 − 0.993801i \(-0.464539\pi\)
0.111174 + 0.993801i \(0.464539\pi\)
\(74\) 2.68489 0.312112
\(75\) 0 0
\(76\) −29.8306 −3.42180
\(77\) −11.1100 −1.26610
\(78\) 0 0
\(79\) −1.60903 −0.181031 −0.0905153 − 0.995895i \(-0.528851\pi\)
−0.0905153 + 0.995895i \(0.528851\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 21.0080 2.31994
\(83\) −7.37571 −0.809589 −0.404795 − 0.914408i \(-0.632657\pi\)
−0.404795 + 0.914408i \(0.632657\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −31.4043 −3.38641
\(87\) 0 0
\(88\) −27.4806 −2.92944
\(89\) 8.48746 0.899669 0.449834 − 0.893112i \(-0.351483\pi\)
0.449834 + 0.893112i \(0.351483\pi\)
\(90\) 0 0
\(91\) 6.32652 0.663199
\(92\) 7.46264 0.778034
\(93\) 0 0
\(94\) 6.51929 0.672413
\(95\) 0 0
\(96\) 0 0
\(97\) 9.00003 0.913814 0.456907 − 0.889514i \(-0.348957\pi\)
0.456907 + 0.889514i \(0.348957\pi\)
\(98\) −13.7741 −1.39140
\(99\) 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.cq.1.1 9
3.2 odd 2 925.2.a.m.1.9 9
5.2 odd 4 1665.2.c.e.334.1 18
5.3 odd 4 1665.2.c.e.334.18 18
5.4 even 2 8325.2.a.cr.1.9 9
15.2 even 4 185.2.b.a.149.18 yes 18
15.8 even 4 185.2.b.a.149.1 ✓ 18
15.14 odd 2 925.2.a.l.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.b.a.149.1 ✓ 18 15.8 even 4
185.2.b.a.149.18 yes 18 15.2 even 4
925.2.a.l.1.1 9 15.14 odd 2
925.2.a.m.1.9 9 3.2 odd 2
1665.2.c.e.334.1 18 5.2 odd 4
1665.2.c.e.334.18 18 5.3 odd 4
8325.2.a.cq.1.1 9 1.1 even 1 trivial
8325.2.a.cr.1.9 9 5.4 even 2