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 = [16,0,0,20,0,0,-20,0,0,0,0,0,-20] 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26x^{14} + 271x^{12} - 1454x^{10} + 4306x^{8} - 7062x^{6} + 6123x^{4} - 2486x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1665)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.571818\) 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)
 
\(f(q)\) \(=\) \(q-0.571818 q^{2} -1.67302 q^{4} -3.31307 q^{7} +2.10030 q^{8} +0.778403 q^{11} -3.69214 q^{13} +1.89447 q^{14} +2.14506 q^{16} -4.20136 q^{17} +5.46957 q^{19} -0.445105 q^{22} +4.59314 q^{23} +2.11123 q^{26} +5.54284 q^{28} -2.33641 q^{29} +5.95031 q^{31} -5.42719 q^{32} +2.40241 q^{34} +1.00000 q^{37} -3.12760 q^{38} -8.98174 q^{41} -8.17900 q^{43} -1.30229 q^{44} -2.62644 q^{46} -2.91979 q^{47} +3.97642 q^{49} +6.17704 q^{52} -1.89150 q^{53} -6.95844 q^{56} +1.33600 q^{58} +11.1106 q^{59} +7.94220 q^{61} -3.40249 q^{62} -1.18675 q^{64} +9.44238 q^{67} +7.02898 q^{68} +1.78726 q^{71} -5.52408 q^{73} -0.571818 q^{74} -9.15072 q^{76} -2.57890 q^{77} +1.93659 q^{79} +5.13592 q^{82} +17.1465 q^{83} +4.67690 q^{86} +1.63488 q^{88} -9.74359 q^{89} +12.2323 q^{91} -7.68444 q^{92} +1.66959 q^{94} +9.88277 q^{97} -2.27379 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} - 20 q^{7} - 20 q^{13} + 20 q^{16} - 8 q^{19} - 12 q^{22} - 72 q^{28} + 8 q^{31} + 12 q^{34} + 16 q^{37} - 24 q^{43} - 4 q^{46} + 40 q^{49} - 52 q^{52} - 64 q^{58} - 8 q^{61} + 40 q^{64}+ \cdots - 28 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.571818 −0.404336 −0.202168 − 0.979351i \(-0.564799\pi\)
−0.202168 + 0.979351i \(0.564799\pi\)
\(3\) 0 0
\(4\) −1.67302 −0.836512
\(5\) 0 0
\(6\) 0 0
\(7\) −3.31307 −1.25222 −0.626111 − 0.779734i \(-0.715354\pi\)
−0.626111 + 0.779734i \(0.715354\pi\)
\(8\) 2.10030 0.742569
\(9\) 0 0
\(10\) 0 0
\(11\) 0.778403 0.234697 0.117349 − 0.993091i \(-0.462560\pi\)
0.117349 + 0.993091i \(0.462560\pi\)
\(12\) 0 0
\(13\) −3.69214 −1.02402 −0.512008 − 0.858981i \(-0.671098\pi\)
−0.512008 + 0.858981i \(0.671098\pi\)
\(14\) 1.89447 0.506319
\(15\) 0 0
\(16\) 2.14506 0.536264
\(17\) −4.20136 −1.01898 −0.509490 − 0.860477i \(-0.670166\pi\)
−0.509490 + 0.860477i \(0.670166\pi\)
\(18\) 0 0
\(19\) 5.46957 1.25481 0.627403 − 0.778695i \(-0.284118\pi\)
0.627403 + 0.778695i \(0.284118\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.445105 −0.0948967
\(23\) 4.59314 0.957737 0.478868 − 0.877887i \(-0.341047\pi\)
0.478868 + 0.877887i \(0.341047\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.11123 0.414047
\(27\) 0 0
\(28\) 5.54284 1.04750
\(29\) −2.33641 −0.433861 −0.216931 − 0.976187i \(-0.569605\pi\)
−0.216931 + 0.976187i \(0.569605\pi\)
\(30\) 0 0
\(31\) 5.95031 1.06871 0.534354 − 0.845261i \(-0.320555\pi\)
0.534354 + 0.845261i \(0.320555\pi\)
\(32\) −5.42719 −0.959400
\(33\) 0 0
\(34\) 2.40241 0.412011
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −3.12760 −0.507364
\(39\) 0 0
\(40\) 0 0
\(41\) −8.98174 −1.40271 −0.701356 − 0.712811i \(-0.747422\pi\)
−0.701356 + 0.712811i \(0.747422\pi\)
\(42\) 0 0
\(43\) −8.17900 −1.24729 −0.623643 − 0.781709i \(-0.714348\pi\)
−0.623643 + 0.781709i \(0.714348\pi\)
\(44\) −1.30229 −0.196327
\(45\) 0 0
\(46\) −2.62644 −0.387248
\(47\) −2.91979 −0.425896 −0.212948 − 0.977064i \(-0.568306\pi\)
−0.212948 + 0.977064i \(0.568306\pi\)
\(48\) 0 0
\(49\) 3.97642 0.568061
\(50\) 0 0
\(51\) 0 0
\(52\) 6.17704 0.856601
\(53\) −1.89150 −0.259817 −0.129909 − 0.991526i \(-0.541468\pi\)
−0.129909 + 0.991526i \(0.541468\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.95844 −0.929861
\(57\) 0 0
\(58\) 1.33600 0.175426
\(59\) 11.1106 1.44647 0.723236 − 0.690601i \(-0.242654\pi\)
0.723236 + 0.690601i \(0.242654\pi\)
\(60\) 0 0
\(61\) 7.94220 1.01689 0.508447 − 0.861093i \(-0.330220\pi\)
0.508447 + 0.861093i \(0.330220\pi\)
\(62\) −3.40249 −0.432117
\(63\) 0 0
\(64\) −1.18675 −0.148344
\(65\) 0 0
\(66\) 0 0
\(67\) 9.44238 1.15357 0.576785 − 0.816896i \(-0.304307\pi\)
0.576785 + 0.816896i \(0.304307\pi\)
\(68\) 7.02898 0.852388
\(69\) 0 0
\(70\) 0 0
\(71\) 1.78726 0.212109 0.106055 − 0.994360i \(-0.466178\pi\)
0.106055 + 0.994360i \(0.466178\pi\)
\(72\) 0 0
\(73\) −5.52408 −0.646544 −0.323272 − 0.946306i \(-0.604783\pi\)
−0.323272 + 0.946306i \(0.604783\pi\)
\(74\) −0.571818 −0.0664725
\(75\) 0 0
\(76\) −9.15072 −1.04966
\(77\) −2.57890 −0.293893
\(78\) 0 0
\(79\) 1.93659 0.217883 0.108941 − 0.994048i \(-0.465254\pi\)
0.108941 + 0.994048i \(0.465254\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 5.13592 0.567167
\(83\) 17.1465 1.88207 0.941036 − 0.338307i \(-0.109854\pi\)
0.941036 + 0.338307i \(0.109854\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.67690 0.504323
\(87\) 0 0
\(88\) 1.63488 0.174279
\(89\) −9.74359 −1.03282 −0.516409 − 0.856342i \(-0.672732\pi\)
−0.516409 + 0.856342i \(0.672732\pi\)
\(90\) 0 0
\(91\) 12.2323 1.28229
\(92\) −7.68444 −0.801158
\(93\) 0 0
\(94\) 1.66959 0.172205
\(95\) 0 0
\(96\) 0 0
\(97\) 9.88277 1.00344 0.501722 − 0.865029i \(-0.332700\pi\)
0.501722 + 0.865029i \(0.332700\pi\)
\(98\) −2.27379 −0.229688
\(99\) 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.cw.1.8 16
3.2 odd 2 inner 8325.2.a.cw.1.9 16
5.2 odd 4 1665.2.c.g.334.16 yes 32
5.3 odd 4 1665.2.c.g.334.18 yes 32
5.4 even 2 8325.2.a.cx.1.9 16
15.2 even 4 1665.2.c.g.334.17 yes 32
15.8 even 4 1665.2.c.g.334.15 ✓ 32
15.14 odd 2 8325.2.a.cx.1.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1665.2.c.g.334.15 ✓ 32 15.8 even 4
1665.2.c.g.334.16 yes 32 5.2 odd 4
1665.2.c.g.334.17 yes 32 15.2 even 4
1665.2.c.g.334.18 yes 32 5.3 odd 4
8325.2.a.cw.1.8 16 1.1 even 1 trivial
8325.2.a.cw.1.9 16 3.2 odd 2 inner
8325.2.a.cx.1.8 16 15.14 odd 2
8325.2.a.cx.1.9 16 5.4 even 2