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Properties

Label 8649.2.a.bs
Level $8649$
Weight $2$
Character orbit 8649.a
Self dual yes
Analytic conductor $69.063$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8649,2,Mod(1,8649)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8649, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8649.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 8649 = 3^{2} \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8649.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,8,0,8,16,0,-16,0,0,8,0,0,0] 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: \(69.0626127082\)
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: \( x^{16} - 24x^{14} + 220x^{12} - 992x^{10} + 2366x^{8} - 2944x^{6} + 1688x^{4} - 288x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 961)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{8}) q^{2} + (\beta_{15} - \beta_{4}) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{13} - \beta_{10} + \beta_{4} + \cdots - 1) q^{7} + (\beta_{15} - \beta_{13} + \cdots - \beta_{8}) q^{8}+ \cdots + (\beta_{15} - \beta_{13} + \beta_{12} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7} + 8 q^{10} + 8 q^{14} - 8 q^{16} - 32 q^{19} + 24 q^{20} - 8 q^{28} + 8 q^{32} + 16 q^{35} + 24 q^{38} + 32 q^{41} + 32 q^{47} - 16 q^{49} + 32 q^{50} + 48 q^{56}+ \cdots - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24x^{14} + 220x^{12} - 992x^{10} + 2366x^{8} - 2944x^{6} + 1688x^{4} - 288x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{14} + 276 \nu^{12} - 7520 \nu^{10} + 39462 \nu^{8} + 3707 \nu^{6} - 356455 \nu^{4} + \cdots - 32184 ) / 24769 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 55 \nu^{15} - 3073 \nu^{13} + 85164 \nu^{11} - 732831 \nu^{9} + 2798456 \nu^{7} + \cdots - 725036 \nu ) / 49538 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 104 \nu^{14} + 2163 \nu^{12} - 15604 \nu^{10} + 45640 \nu^{8} - 41312 \nu^{6} - 27392 \nu^{4} + \cdots - 4788 ) / 2914 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1797 \nu^{15} + 34250 \nu^{13} - 195144 \nu^{11} + 121973 \nu^{9} + 2103566 \nu^{7} + \cdots - 870800 \nu ) / 49538 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4312 \nu^{15} - 100889 \nu^{13} + 887595 \nu^{11} - 3740341 \nu^{9} + 7970562 \nu^{7} + \cdots - 710426 \nu ) / 49538 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6080 \nu^{15} + 137660 \nu^{13} - 1152863 \nu^{11} + 4516221 \nu^{9} - 8672866 \nu^{7} + \cdots + 629030 \nu ) / 49538 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6717 \nu^{14} + 148905 \nu^{12} - 1205456 \nu^{10} + 4465027 \nu^{8} - 7789442 \nu^{6} + \cdots + 174988 ) / 49538 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6717 \nu^{15} + 148905 \nu^{13} - 1205456 \nu^{11} + 4465027 \nu^{9} - 7789442 \nu^{7} + \cdots + 174988 \nu ) / 49538 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8267 \nu^{14} - 188399 \nu^{12} + 1593018 \nu^{10} - 6327538 \nu^{8} + 12377380 \nu^{6} + \cdots - 277908 ) / 49538 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8322 \nu^{15} + 185326 \nu^{13} - 1507854 \nu^{11} + 5594707 \nu^{9} - 9578924 \nu^{7} + \cdots - 447128 \nu ) / 49538 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 8904 \nu^{14} + 199644 \nu^{12} - 1645611 \nu^{10} + 6276344 \nu^{8} - 11493956 \nu^{6} + \cdots + 71556 ) / 49538 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14402 \nu^{14} + 322986 \nu^{12} - 2660717 \nu^{10} + 10110928 \nu^{8} - 18251790 \nu^{6} + \cdots + 132364 ) / 49538 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19741 \nu^{15} - 451609 \nu^{13} + 3845916 \nu^{11} - 15489419 \nu^{9} + 31147008 \nu^{7} + \cdots - 1205654 \nu ) / 49538 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21756 \nu^{14} + 483136 \nu^{12} - 3918766 \nu^{10} + 14524761 \nu^{8} - 25157808 \nu^{6} + \cdots - 47614 ) / 49538 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{13} - \beta_{12} - \beta_{10} - \beta_{8} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + 2\beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{15} - 10\beta_{13} - 10\beta_{12} - 9\beta_{10} - 5\beta_{8} - 2\beta_{4} + \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{14} + 18\beta_{11} + 13\beta_{9} - 13\beta_{7} - 13\beta_{6} - 9\beta_{5} - 10\beta_{3} + 34\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 78\beta_{15} - 93\beta_{13} - 81\beta_{12} - 72\beta_{10} - 26\beta_{8} - 32\beta_{4} + 14\beta_{2} + 94 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 81\beta_{14} + 138\beta_{11} + 130\beta_{9} - 139\beta_{7} - 127\beta_{6} - 67\beta_{5} - 86\beta_{3} + 265\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 672\beta_{15} - 852\beta_{13} - 632\beta_{12} - 582\beta_{10} - 134\beta_{8} - 356\beta_{4} + 158\beta_{2} + 664 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 632 \beta_{14} + 1034 \beta_{11} + 1210 \beta_{9} - 1366 \beta_{7} - 1146 \beta_{6} - 474 \beta_{5} + \cdots + 2188 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5798 \beta_{15} - 7722 \beta_{13} - 4960 \beta_{12} - 4814 \beta_{10} - 646 \beta_{8} - 3496 \beta_{4} + \cdots + 5022 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4960 \beta_{14} + 7850 \beta_{11} + 10950 \beta_{9} - 12850 \beta_{7} - 10088 \beta_{6} + \cdots + 18542 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 50192 \beta_{15} - 69360 \beta_{13} - 39640 \beta_{12} - 40560 \beta_{10} - 2592 \beta_{8} + \cdots + 39676 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 39640 \beta_{14} + 61032 \beta_{11} + 97792 \beta_{9} - 117898 \beta_{7} - 88178 \beta_{6} + \cdots + 159228 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 435950 \beta_{15} - 618704 \beta_{13} - 323212 \beta_{12} - 346148 \beta_{10} - 4570 \beta_{8} + \cdots + 323026 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 323212 \beta_{14} + 486606 \beta_{11} + 867330 \beta_{9} - 1065336 \beta_{7} - 769844 \beta_{6} + \cdots + 1377680 \beta_1 \) 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
−1.23447
1.23447
−2.46287
2.46287
−1.86943
1.86943
−1.29905
1.29905
−0.123325
0.123325
0.501299
−0.501299
1.47925
−1.47925
−2.96199
2.96199
−2.16383 0 2.68217 2.65612 0 −3.05284 −1.47609 0 −5.74741
1.2 −2.16383 0 2.68217 2.65612 0 −3.05284 −1.47609 0 −5.74741
1.3 −1.21652 0 −0.520070 −1.24784 0 −1.96096 3.06572 0 1.51803
1.4 −1.21652 0 −0.520070 −1.24784 0 −1.96096 3.06572 0 1.51803
1.5 −0.124175 0 −1.98458 1.22944 0 2.66703 0.494784 0 −0.152666
1.6 −0.124175 0 −1.98458 1.22944 0 2.66703 0.494784 0 −0.152666
1.7 0.337752 0 −1.88592 1.25850 0 −2.44342 −1.31248 0 0.425062
1.8 0.337752 0 −1.88592 1.25850 0 −2.44342 −1.31248 0 0.425062
1.9 1.31607 0 −0.267953 −0.00727825 0 −0.794917 −2.98479 0 −0.00957870
1.10 1.31607 0 −0.267953 −0.00727825 0 −0.794917 −2.98479 0 −0.00957870
1.11 1.45116 0 0.105856 3.68139 0 −0.804403 −2.74870 0 5.34227
1.12 1.45116 0 0.105856 3.68139 0 −0.804403 −2.74870 0 5.34227
1.13 1.88954 0 1.57037 −2.49142 0 −3.90166 −0.811809 0 −4.70764
1.14 1.88954 0 1.57037 −2.49142 0 −3.90166 −0.811809 0 −4.70764
1.15 2.51001 0 4.30014 2.92108 0 2.29118 5.77336 0 7.33192
1.16 2.51001 0 4.30014 2.92108 0 2.29118 5.77336 0 7.33192
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(31\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8649.2.a.bs 16
3.b odd 2 1 961.2.a.l ✓ 16
31.b odd 2 1 inner 8649.2.a.bs 16
93.c even 2 1 961.2.a.l ✓ 16
93.g even 6 2 961.2.c.l 32
93.h odd 6 2 961.2.c.l 32
93.k even 10 4 961.2.d.s 64
93.l odd 10 4 961.2.d.s 64
93.o odd 30 8 961.2.g.w 128
93.p even 30 8 961.2.g.w 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.a.l ✓ 16 3.b odd 2 1
961.2.a.l ✓ 16 93.c even 2 1
961.2.c.l 32 93.g even 6 2
961.2.c.l 32 93.h odd 6 2
961.2.d.s 64 93.k even 10 4
961.2.d.s 64 93.l odd 10 4
961.2.g.w 128 93.o odd 30 8
961.2.g.w 128 93.p even 30 8
8649.2.a.bs 16 1.a even 1 1 trivial
8649.2.a.bs 16 31.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8649))\):

\( T_{2}^{8} - 4T_{2}^{7} - 2T_{2}^{6} + 24T_{2}^{5} - 19T_{2}^{4} - 24T_{2}^{3} + 30T_{2}^{2} - 4T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 8T_{5}^{7} + 12T_{5}^{6} + 48T_{5}^{5} - 144T_{5}^{4} + 32T_{5}^{3} + 192T_{5}^{2} - 136T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 8T_{7}^{7} + 8T_{7}^{6} - 80T_{7}^{5} - 208T_{7}^{4} + 64T_{7}^{3} + 684T_{7}^{2} + 712T_{7} + 223 \) Copy content Toggle raw display
\( T_{11}^{16} - 80 T_{11}^{14} + 2376 T_{11}^{12} - 33376 T_{11}^{10} + 232404 T_{11}^{8} - 779040 T_{11}^{6} + \cdots + 1156 \) Copy content Toggle raw display
\( T_{13}^{16} - 120 T_{13}^{14} + 5564 T_{13}^{12} - 125664 T_{13}^{10} + 1418334 T_{13}^{8} - 7265728 T_{13}^{6} + \cdots + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{7} - 2 T^{6} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 8 T^{7} + 12 T^{6} + \cdots - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 8 T^{7} + \cdots + 223)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 80 T^{14} + \cdots + 1156 \) Copy content Toggle raw display
$13$ \( T^{16} - 120 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{16} - 120 T^{14} + \cdots + 3844 \) Copy content Toggle raw display
$19$ \( (T^{8} + 16 T^{7} + \cdots - 10433)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 184 T^{14} + \cdots + 1110916 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 41380510084 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 478996996 \) Copy content Toggle raw display
$41$ \( (T^{8} - 16 T^{7} + \cdots - 50111)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 184 T^{14} + \cdots + 8282884 \) Copy content Toggle raw display
$47$ \( (T^{8} - 16 T^{7} + \cdots - 35972)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 235465621504 \) Copy content Toggle raw display
$59$ \( (T^{8} - 32 T^{7} + \cdots + 37519)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1025954255236 \) Copy content Toggle raw display
$67$ \( (T^{8} - 8 T^{7} + \cdots - 511996)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 24 T^{7} + \cdots + 498767)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 645641176324 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 85674460804 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1891032196 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 7783827076 \) Copy content Toggle raw display
$97$ \( (T^{8} - 244 T^{6} + \cdots + 658321)^{2} \) Copy content Toggle raw display
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