↑

Properties

Label 1875.4.a.i
Level $1875$
Weight $4$
Character orbit 1875.a
Self dual yes
Analytic conductor $110.629$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1875,4,Mod(1,1875)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1875.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-1,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(110.628581261\)
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: \( x^{16} - x^{15} - 82 x^{14} + 58 x^{13} + 2691 x^{12} - 1196 x^{11} - 45020 x^{10} + 10424 x^{9} + \cdots + 1284336 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{6} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 2) q^{4} - 3 \beta_1 q^{6} + ( - \beta_{13} - \beta_{10} + \beta_{2} - 3) q^{7} + (\beta_{15} + 2 \beta_{10} + \beta_{8} + \cdots - 4) q^{8} + 9 q^{9} + ( - \beta_{15} + \beta_{12} + \beta_{9} + \cdots - 5) q^{11}+ \cdots + ( - 9 \beta_{15} + 9 \beta_{12} + \cdots - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} + 48 q^{3} + 37 q^{4} - 3 q^{6} - 52 q^{7} - 57 q^{8} + 144 q^{9} - 96 q^{11} + 111 q^{12} - 86 q^{13} - 62 q^{14} - 155 q^{16} + 46 q^{17} - 9 q^{18} - 192 q^{19} - 156 q^{21} - 158 q^{22}+ \cdots - 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 82 x^{14} + 58 x^{13} + 2691 x^{12} - 1196 x^{11} - 45020 x^{10} + 10424 x^{9} + \cdots + 1284336 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 772705 \nu^{15} + 2571284 \nu^{14} - 65979148 \nu^{13} - 198935036 \nu^{12} + \cdots - 2951952508368 ) / 562286689920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 92296829 \nu^{15} + 473840558 \nu^{14} + 6493964468 \nu^{13} - 34471780286 \nu^{12} + \cdots - 188848349764848 ) / 6747440279040 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 36369239 \nu^{15} + 57284378 \nu^{14} + 2873567540 \nu^{13} - 3663400442 \nu^{12} + \cdots + 16827020746416 ) / 2249146759680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31226671 \nu^{15} + 46907134 \nu^{14} + 2518021984 \nu^{13} - 2917773406 \nu^{12} + \cdots + 27493540344432 ) / 1124573379840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 205877135 \nu^{15} + 447219239 \nu^{14} + 16289148023 \nu^{13} - 33576283127 \nu^{12} + \cdots - 390946026735360 ) / 6747440279040 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 131534021 \nu^{15} + 198878649 \nu^{14} - 11051639755 \nu^{13} - 18266978577 \nu^{12} + \cdots - 221102154590112 ) / 2249146759680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 203638258 \nu^{15} - 149703455 \nu^{14} + 16828427803 \nu^{13} + 15961038155 \nu^{12} + \cdots + 176218566198192 ) / 3373720139520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 687855200 \nu^{15} + 389954785 \nu^{14} - 56231947553 \nu^{13} - 47746328497 \nu^{12} + \cdots - 628192536278256 ) / 6747440279040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 704384116 \nu^{15} - 443292989 \nu^{14} + 57230005141 \nu^{13} + 54737561837 \nu^{12} + \cdots + 989502035971824 ) / 6747440279040 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 941839693 \nu^{15} - 67535596 \nu^{14} - 75768416662 \nu^{13} - 20094316124 \nu^{12} + \cdots - 751089716695152 ) / 6747440279040 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 420516865 \nu^{15} - 129424201 \nu^{14} - 34033564921 \nu^{13} + 171709345 \nu^{12} + \cdots - 290377920249216 ) / 2249146759680 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1612523330 \nu^{15} + 741422125 \nu^{14} - 130041142433 \nu^{13} - 103538250661 \nu^{12} + \cdots - 17\!\cdots\!24 ) / 6747440279040 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 206956588 \nu^{15} - 100300551 \nu^{14} + 16901419871 \nu^{13} + 12869090271 \nu^{12} + \cdots + 187817616098640 ) / 749715586560 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - 2\beta_{10} - \beta_{8} + \beta_{4} + \beta_{2} + 16\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} + 3 \beta_{11} + \beta_{10} + 3 \beta_{9} + \cdots + 168 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 33 \beta_{15} - 3 \beta_{13} - 3 \beta_{12} - 59 \beta_{10} + 4 \beta_{9} - 29 \beta_{8} - 9 \beta_{6} + \cdots + 155 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 41 \beta_{15} + 36 \beta_{14} + 36 \beta_{13} - 76 \beta_{12} + 97 \beta_{11} + 13 \beta_{10} + \cdots + 3257 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 878 \beta_{15} + 5 \beta_{14} - 92 \beta_{13} - 133 \beta_{12} - 1495 \beta_{10} + 172 \beta_{9} + \cdots + 4508 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1327 \beta_{15} + 970 \beta_{14} + 970 \beta_{13} - 2214 \beta_{12} + 2454 \beta_{11} - 142 \beta_{10} + \cdots + 67212 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21768 \beta_{15} + 357 \beta_{14} - 2097 \beta_{13} - 4500 \beta_{12} + 51 \beta_{11} - 36095 \beta_{10} + \cdots + 118976 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 38471 \beta_{15} + 23865 \beta_{14} + 23814 \beta_{13} - 58741 \beta_{12} + 57505 \beta_{11} + \cdots + 1437793 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 523567 \beta_{15} + 14657 \beta_{14} - 42848 \beta_{13} - 134947 \beta_{12} + 2814 \beta_{11} + \cdots + 3005284 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1046226 \beta_{15} + 566415 \beta_{14} + 563601 \beta_{13} - 1489846 \beta_{12} + 1307115 \beta_{11} + \cdots + 31480986 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12405222 \beta_{15} + 482625 \beta_{14} - 824490 \beta_{13} - 3767853 \beta_{12} + 104823 \beta_{11} + \cdots + 74173101 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 27340749 \beta_{15} + 13229712 \beta_{14} + 13124889 \beta_{13} - 36831531 \beta_{12} + \cdots + 700319216 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 291550766 \beta_{15} + 14215860 \beta_{14} - 15099261 \beta_{13} - 100469631 \beta_{12} + \cdots + 1805940941 \) 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
4.86419
4.81450
4.18272
3.60005
2.40820
1.41452
1.35690
0.805223
−0.580363
−1.44160
−1.73218
−2.82472
−3.20828
−3.95401
−4.03397
−4.67120
−4.86419 3.00000 15.6604 0 −14.5926 8.50814 −37.2616 9.00000 0
1.2 −4.81450 3.00000 15.1795 0 −14.4435 9.51252 −34.5655 9.00000 0
1.3 −4.18272 3.00000 9.49515 0 −12.5482 11.3434 −6.25378 9.00000 0
1.4 −3.60005 3.00000 4.96039 0 −10.8002 −14.7522 10.9428 9.00000 0
1.5 −2.40820 3.00000 −2.20058 0 −7.22460 −17.0733 24.5650 9.00000 0
1.6 −1.41452 3.00000 −5.99913 0 −4.24356 0.413195 19.8020 9.00000 0
1.7 −1.35690 3.00000 −6.15882 0 −4.07070 −32.5572 19.2121 9.00000 0
1.8 −0.805223 3.00000 −7.35162 0 −2.41567 24.2423 12.3615 9.00000 0
1.9 0.580363 3.00000 −7.66318 0 1.74109 −30.2933 −9.09033 9.00000 0
1.10 1.44160 3.00000 −5.92179 0 4.32480 1.72551 −20.0697 9.00000 0
1.11 1.73218 3.00000 −4.99957 0 5.19653 −26.1229 −22.5175 9.00000 0
1.12 2.82472 3.00000 −0.0209768 0 8.47415 5.77741 −22.6570 9.00000 0
1.13 3.20828 3.00000 2.29304 0 9.62483 21.7328 −18.3095 9.00000 0
1.14 3.95401 3.00000 7.63421 0 11.8620 7.28822 −1.44634 9.00000 0
1.15 4.03397 3.00000 8.27293 0 12.1019 −2.17756 1.10098 9.00000 0
1.16 4.67120 3.00000 13.8201 0 14.0136 −19.5671 27.1869 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1875.4.a.i ✓ 16
5.b even 2 1 1875.4.a.j yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.4.a.i ✓ 16 1.a even 1 1 trivial
1875.4.a.j yes 16 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + T_{2}^{15} - 82 T_{2}^{14} - 58 T_{2}^{13} + 2691 T_{2}^{12} + 1196 T_{2}^{11} + \cdots + 1284336 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 1284336 \) Copy content Toggle raw display
$3$ \( (T - 3)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots - 40\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 52\!\cdots\!91 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots - 17\!\cdots\!19 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 49\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 27\!\cdots\!75 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 15\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 84\!\cdots\!75 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 79\!\cdots\!75 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 58\!\cdots\!75 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 33\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 14\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 45\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 26\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 54\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 19\!\cdots\!71 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 13\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 16\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 38\!\cdots\!71 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 24\!\cdots\!75 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 87\!\cdots\!19 \) Copy content Toggle raw display
show more
show less