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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: \(0\)
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: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{8} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{9} + \beta_{6} + \cdots + \beta_1) q^{11} + (\beta_{12} + 1) q^{13} + (\beta_{11} + \beta_{5} + \cdots + 2 \beta_1) q^{14}+ \cdots + (\beta_{13} + 3 \beta_{11} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 20 q^{7} + 20 q^{13} + 20 q^{16} - 8 q^{19} + 12 q^{22} + 72 q^{28} + 8 q^{31} + 12 q^{34} - 16 q^{37} + 24 q^{43} - 4 q^{46} + 40 q^{49} + 52 q^{52} + 64 q^{58} - 8 q^{61} + 40 q^{64}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26x^{14} + 271x^{12} - 1454x^{10} + 4306x^{8} - 7062x^{6} + 6123x^{4} - 2486x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{14} + 115\nu^{12} - 801\nu^{10} + 2475\nu^{8} - 3166\nu^{6} + 371\nu^{4} + 1901\nu^{2} - 615 ) / 94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{15} - 196\nu^{13} + 1413\nu^{11} - 2655\nu^{9} - 10855\nu^{7} + 47877\nu^{5} - 51426\nu^{3} + 11427\nu ) / 893 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17 \nu^{15} - 271 \nu^{13} + 883 \nu^{11} + 6594 \nu^{9} - 55827 \nu^{7} + 150563 \nu^{5} + \cdots + 39742 \nu ) / 1786 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{14} + 81\nu^{12} - 847\nu^{10} + 4316\nu^{8} - 11077\nu^{6} + 13369\nu^{4} - 6217\nu^{2} + 844 ) / 94 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\nu^{14} - 223\nu^{12} + 1899\nu^{10} - 7697\nu^{8} + 14896\nu^{6} - 11491\nu^{4} + 1093\nu^{2} + 649 ) / 94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 119 \nu^{15} - 2790 \nu^{13} + 24934 \nu^{11} - 104759 \nu^{9} + 202163 \nu^{7} - 134642 \nu^{5} + \cdots + 14759 \nu ) / 1786 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{12} - 19\nu^{10} + 131\nu^{8} - 406\nu^{6} + 583\nu^{4} - 365\nu^{2} + 75 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 172 \nu^{15} - 3845 \nu^{13} + 33255 \nu^{11} - 140933 \nu^{9} + 304734 \nu^{7} - 314083 \nu^{5} + \cdots - 3379 \nu ) / 1786 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 9\nu^{14} - 243\nu^{12} + 2541\nu^{10} - 12948\nu^{8} + 33325\nu^{6} - 41047\nu^{4} + 20813\nu^{2} - 3190 ) / 94 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 239 \nu^{15} - 5701 \nu^{13} + 53597 \nu^{11} - 253570 \nu^{9} + 640261 \nu^{7} - 847391 \nu^{5} + \cdots - 121320 \nu ) / 1786 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 225 \nu^{15} - 5793 \nu^{13} + 58543 \nu^{11} - 294090 \nu^{9} + 769863 \nu^{7} - 1015929 \nu^{5} + \cdots - 115282 \nu ) / 1786 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 15\nu^{14} - 358\nu^{12} + 3342\nu^{10} - 15470\nu^{8} + 37196\nu^{6} - 44755\nu^{4} + 24129\nu^{2} - 4455 ) / 47 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{10} + \beta_{7} + \beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} + \beta_{11} + \beta_{9} - 2\beta_{6} - \beta_{5} + 9\beta_{3} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{12} + 10\beta_{10} + 13\beta_{7} + 10\beta_{4} + 47\beta_{2} + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{14} - 11\beta_{13} + 10\beta_{11} + 18\beta_{9} - 27\beta_{6} - 18\beta_{5} + 68\beta_{3} + 165\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -\beta_{15} + 96\beta_{12} + 79\beta_{10} + 124\beta_{7} + 77\beta_{4} + 319\beta_{2} + 548 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 31 \beta_{14} - 95 \beta_{13} + 78 \beta_{11} + 206 \beta_{9} - 264 \beta_{6} - 209 \beta_{5} + \cdots + 1000 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -17\beta_{15} + 779\beta_{12} + 587\beta_{10} - 3\beta_{8} + 1057\beta_{7} + 550\beta_{4} + 2193\beta_{2} + 3520 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 332 \beta_{14} - 759 \beta_{13} + 567 \beta_{11} + 1965 \beta_{9} - 2286 \beta_{6} + \cdots + 6184 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 192 \beta_{15} + 6108 \beta_{12} + 4283 \beta_{10} - 57 \beta_{8} + 8534 \beta_{7} + 3840 \beta_{4} + \cdots + 23095 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3061 \beta_{14} - 5857 \beta_{13} + 4034 \beta_{11} + 17070 \beta_{9} - 18642 \beta_{6} + \cdots + 38916 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1823 \beta_{15} + 46931 \beta_{12} + 31099 \beta_{10} - 692 \beta_{8} + 66811 \beta_{7} + \cdots + 154125 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 26095 \beta_{14} - 44362 \beta_{13} + 28584 \beta_{11} + 140402 \beta_{9} - 146885 \beta_{6} + \cdots + 248974 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.70149
−2.39280
−2.38341
−1.77531
−1.32131
−1.22977
−0.747627
−0.571818
0.571818
0.747627
1.22977
1.32131
1.77531
2.38341
2.39280
2.70149
−2.70149 0 5.29803 0 0 4.46991 −8.90958 0 0
1.2 −2.39280 0 3.72549 0 0 0.617585 −4.12875 0 0
1.3 −2.38341 0 3.68064 0 0 3.82708 −4.00566 0 0
1.4 −1.77531 0 1.15172 0 0 −2.10808 1.50596 0 0
1.5 −1.32131 0 −0.254137 0 0 3.90085 2.97842 0 0
1.6 −1.22977 0 −0.487664 0 0 −0.944151 3.05926 0 0
1.7 −0.747627 0 −1.44105 0 0 −3.07627 2.57262 0 0
1.8 −0.571818 0 −1.67302 0 0 3.31307 2.10030 0 0
1.9 0.571818 0 −1.67302 0 0 3.31307 −2.10030 0 0
1.10 0.747627 0 −1.44105 0 0 −3.07627 −2.57262 0 0
1.11 1.22977 0 −0.487664 0 0 −0.944151 −3.05926 0 0
1.12 1.32131 0 −0.254137 0 0 3.90085 −2.97842 0 0
1.13 1.77531 0 1.15172 0 0 −2.10808 −1.50596 0 0
1.14 2.38341 0 3.68064 0 0 3.82708 4.00566 0 0
1.15 2.39280 0 3.72549 0 0 0.617585 4.12875 0 0
1.16 2.70149 0 5.29803 0 0 4.46991 8.90958 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(37\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8325.2.a.cx 16
3.b odd 2 1 inner 8325.2.a.cx 16
5.b even 2 1 8325.2.a.cw 16
5.c odd 4 2 1665.2.c.g ✓ 32
15.d odd 2 1 8325.2.a.cw 16
15.e even 4 2 1665.2.c.g ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1665.2.c.g ✓ 32 5.c odd 4 2
1665.2.c.g ✓ 32 15.e even 4 2
8325.2.a.cw 16 5.b even 2 1
8325.2.a.cw 16 15.d odd 2 1
8325.2.a.cx 16 1.a even 1 1 trivial
8325.2.a.cx 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):

\( T_{2}^{16} - 26T_{2}^{14} + 271T_{2}^{12} - 1454T_{2}^{10} + 4306T_{2}^{8} - 7062T_{2}^{6} + 6123T_{2}^{4} - 2486T_{2}^{2} + 361 \) Copy content Toggle raw display
\( T_{7}^{8} - 10T_{7}^{7} + 12T_{7}^{6} + 146T_{7}^{5} - 357T_{7}^{4} - 556T_{7}^{3} + 1548T_{7}^{2} + 672T_{7} - 836 \) Copy content Toggle raw display
\( T_{11}^{16} - 90 T_{11}^{14} + 3155 T_{11}^{12} - 54734 T_{11}^{10} + 496469 T_{11}^{8} - 2323136 T_{11}^{6} + \cdots + 1478656 \) Copy content Toggle raw display
\( T_{13}^{8} - 10T_{13}^{7} - 18T_{13}^{6} + 462T_{13}^{5} - 1191T_{13}^{4} - 1832T_{13}^{3} + 9064T_{13}^{2} - 4672T_{13} - 5168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 26 T^{14} + \cdots + 361 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 10 T^{7} + \cdots - 836)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 90 T^{14} + \cdots + 1478656 \) Copy content Toggle raw display
$13$ \( (T^{8} - 10 T^{7} + \cdots - 5168)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 134 T^{14} + \cdots + 4884100 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots + 38000)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 22760549956 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 24190825156 \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} + \cdots - 464)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{16} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7559259136 \) Copy content Toggle raw display
$43$ \( (T^{8} - 12 T^{7} + \cdots - 38000)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 650454016 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 655360000 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 95890554244 \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{7} + \cdots - 1754000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 38 T^{7} + \cdots + 1739564)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 7772185600 \) Copy content Toggle raw display
$73$ \( (T^{8} - 48 T^{7} + \cdots + 7476044)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 12 T^{7} + \cdots - 197200)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 658 T^{14} + \cdots + 640000 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 277446654292900 \) Copy content Toggle raw display
$97$ \( (T^{8} - 14 T^{7} + \cdots + 12062416)^{2} \) Copy content Toggle raw display
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