Properties

Label 7448.2.a.bx
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7448,2,Mod(1,7448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7448.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 11 x^{12} + 114 x^{11} - 10 x^{10} - 806 x^{9} + 523 x^{8} + 2586 x^{7} - 2226 x^{6} - 3618 x^{5} + 3397 x^{4} + 1570 x^{3} - 1529 x^{2} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{7} q^{5} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{7} q^{5} + (\beta_{2} + \beta_1 + 1) q^{9} + (\beta_{11} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{11} + ( - \beta_{8} + 1) q^{13} + ( - \beta_{12} + \beta_{10} - \beta_{7} - \beta_{3} - \beta_1 + 1) q^{15} + (\beta_{11} + \beta_{6} + \beta_{4}) q^{17} + q^{19} + ( - \beta_{8} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{12} - \beta_{11} - \beta_{6} + \beta_{3} + 2 \beta_1) q^{25} + (\beta_{10} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{2} + \beta_1 + 3) q^{27} + (\beta_{13} + \beta_{12} + \beta_{6} + \beta_1 - 1) q^{29} + (\beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + \beta_{2} + 1) q^{31} + ( - \beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \cdots - 3 \beta_1) q^{33}+ \cdots + ( - \beta_{13} - \beta_{12} + 2 \beta_{10} + 2 \beta_{8} - 3 \beta_{7} + 3 \beta_{4} - 4 \beta_{3} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 6 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{19} - 4 q^{23} + 16 q^{25} + 36 q^{27} - 6 q^{29} + 16 q^{31} - 10 q^{33} + 6 q^{37} + 16 q^{39} - 14 q^{41} - 2 q^{43} + 30 q^{47} + 20 q^{51} - 6 q^{53} + 44 q^{55} + 6 q^{57} + 22 q^{59} + 10 q^{61} - 16 q^{65} + 4 q^{67} + 48 q^{69} + 6 q^{71} + 4 q^{73} + 64 q^{75} + 26 q^{79} + 30 q^{81} + 32 q^{83} - 8 q^{85} + 32 q^{87} - 54 q^{89} - 32 q^{93} + 2 q^{95} + 18 q^{97} - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 11 x^{12} + 114 x^{11} - 10 x^{10} - 806 x^{9} + 523 x^{8} + 2586 x^{7} - 2226 x^{6} - 3618 x^{5} + 3397 x^{4} + 1570 x^{3} - 1529 x^{2} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53 \nu^{13} - 453 \nu^{12} + 161 \nu^{11} + 7954 \nu^{10} - 14859 \nu^{9} - 50427 \nu^{8} + 129821 \nu^{7} + 140670 \nu^{6} - 445797 \nu^{5} - 169935 \nu^{4} + 640171 \nu^{3} + \cdots + 272 ) / 4550 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 140 \nu^{13} + 682 \nu^{12} + 2381 \nu^{11} - 13747 \nu^{10} - 14547 \nu^{9} + 104416 \nu^{8} + 38671 \nu^{7} - 364087 \nu^{6} - 42253 \nu^{5} + 558860 \nu^{4} + \cdots + 7888 ) / 4550 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11 \nu^{13} - 41 \nu^{12} - 243 \nu^{11} + 858 \nu^{10} + 2247 \nu^{9} - 6739 \nu^{8} - 11293 \nu^{7} + 24170 \nu^{6} + 31951 \nu^{5} - 38025 \nu^{4} - 45233 \nu^{3} + 19270 \nu^{2} + \cdots - 556 ) / 350 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 158 \nu^{13} - 841 \nu^{12} - 2213 \nu^{11} + 15947 \nu^{10} + 8424 \nu^{9} - 111891 \nu^{8} + 2047 \nu^{7} + 353893 \nu^{6} - 52340 \nu^{5} - 487225 \nu^{4} + 56171 \nu^{3} + \cdots - 1120 ) / 4550 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 202 \nu^{13} + 1130 \nu^{12} + 2760 \nu^{11} - 22359 \nu^{10} - 7397 \nu^{9} + 165672 \nu^{8} - 45300 \nu^{7} - 559863 \nu^{6} + 276361 \nu^{5} + 824960 \nu^{4} + \cdots - 2216 ) / 4550 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 276 \nu^{13} + 1437 \nu^{12} + 3976 \nu^{11} - 26969 \nu^{10} - 17953 \nu^{9} + 188357 \nu^{8} + 25276 \nu^{7} - 597991 \nu^{6} - 13855 \nu^{5} + 839745 \nu^{4} + \cdots + 16890 ) / 4550 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 296 \nu^{13} - 1436 \nu^{12} - 4903 \nu^{11} + 28138 \nu^{10} + 29432 \nu^{9} - 206284 \nu^{8} - 82403 \nu^{7} + 687800 \nu^{6} + 128326 \nu^{5} - 992930 \nu^{4} + \cdots + 23624 ) / 4550 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 291 \nu^{13} - 1745 \nu^{12} - 3030 \nu^{11} + 31762 \nu^{10} - 2834 \nu^{9} - 212641 \nu^{8} + 123580 \nu^{7} + 637674 \nu^{6} - 453848 \nu^{5} - 832645 \nu^{4} + 579912 \nu^{3} + \cdots - 6232 ) / 4550 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 319 \nu^{13} + 1611 \nu^{12} + 5048 \nu^{11} - 31459 \nu^{10} - 27625 \nu^{9} + 229827 \nu^{8} + 65538 \nu^{7} - 764457 \nu^{6} - 94737 \nu^{5} + 1103225 \nu^{4} + \cdots - 18648 ) / 4550 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 179 \nu^{13} - 942 \nu^{12} - 2641 \nu^{11} + 18375 \nu^{10} + 11141 \nu^{9} - 133770 \nu^{8} + 784 \nu^{7} + 441788 \nu^{6} - 95144 \nu^{5} - 631790 \nu^{4} + 173228 \nu^{3} + \cdots + 939 ) / 2275 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 47 \nu^{13} + 238 \nu^{12} + 744 \nu^{11} - 4687 \nu^{10} - 3935 \nu^{9} + 34536 \nu^{8} + 7534 \nu^{7} - 116221 \nu^{6} - 2481 \nu^{5} + 172470 \nu^{4} - 3464 \nu^{3} + \cdots + 2416 ) / 350 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 9\beta_{2} + 12\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 2 \beta_{12} + \beta_{11} + 11 \beta_{10} + \beta_{9} + 10 \beta_{8} - 4 \beta_{7} - 9 \beta_{6} + 12 \beta_{5} + \beta_{4} - \beta_{3} + 15 \beta_{2} + 58 \beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{13} - 16 \beta_{12} + 4 \beta_{11} + 18 \beta_{10} + 13 \beta_{9} + 16 \beta_{8} - 19 \beta_{7} - 10 \beta_{6} + 19 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 82 \beta_{2} + 129 \beta _1 + 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 19 \beta_{13} - 39 \beta_{12} + 21 \beta_{11} + 110 \beta_{10} + 22 \beta_{9} + 95 \beta_{8} - 69 \beta_{7} - 62 \beta_{6} + 122 \beta_{5} + 4 \beta_{4} - 20 \beta_{3} + 182 \beta_{2} + 524 \beta _1 + 522 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 52 \beta_{13} - 198 \beta_{12} + 77 \beta_{11} + 242 \beta_{10} + 141 \beta_{9} + 207 \beta_{8} - 262 \beta_{7} - 57 \beta_{6} + 260 \beta_{5} + 39 \beta_{4} - 65 \beta_{3} + 787 \beta_{2} + 1353 \beta _1 + 2225 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 283 \beta_{13} - 556 \beta_{12} + 305 \beta_{11} + 1117 \beta_{10} + 315 \beta_{9} + 942 \beta_{8} - 912 \beta_{7} - 319 \beta_{6} + 1234 \beta_{5} - 115 \beta_{4} - 291 \beta_{3} + 2055 \beta_{2} + 4989 \beta _1 + 5798 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 888 \beta_{13} - 2312 \beta_{12} + 1078 \beta_{11} + 2926 \beta_{10} + 1473 \beta_{9} + 2472 \beta_{8} - 3241 \beta_{7} + 44 \beta_{6} + 3169 \beta_{5} - 325 \beta_{4} - 969 \beta_{3} + 7843 \beta_{2} + 14084 \beta _1 + 21886 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3859 \beta_{13} - 7055 \beta_{12} + 3893 \beta_{11} + 11621 \beta_{10} + 3862 \beta_{9} + 9722 \beta_{8} - 11051 \beta_{7} - 221 \beta_{6} + 12781 \beta_{5} - 3258 \beta_{4} - 3724 \beta_{3} + 22487 \beta_{2} + \cdots + 62691 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12832 \beta_{13} - 26531 \beta_{12} + 13513 \beta_{11} + 33641 \beta_{10} + 15380 \beta_{9} + 28348 \beta_{8} - 38187 \beta_{7} + 7644 \beta_{6} + 36701 \beta_{5} - 11582 \beta_{4} - 12455 \beta_{3} + \cdots + 221597 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 50068 \beta_{13} - 84660 \beta_{12} + 47018 \beta_{11} + 122962 \beta_{10} + 44356 \beta_{9} + 102900 \beta_{8} - 128804 \beta_{7} + 27314 \beta_{6} + 135416 \beta_{5} - 57524 \beta_{4} + \cdots + 669808 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
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
−2.48107
−2.41932
−1.76077
−1.66612
−0.870378
−0.0702441
0.0758070
0.884594
0.954523
1.72076
2.12220
3.03680
3.15030
3.32292
0 −2.48107 0 0.551215 0 0 0 3.15571 0
1.2 0 −2.41932 0 −1.56514 0 0 0 2.85310 0
1.3 0 −1.76077 0 2.55769 0 0 0 0.100310 0
1.4 0 −1.66612 0 −2.06660 0 0 0 −0.224056 0
1.5 0 −0.870378 0 −1.04185 0 0 0 −2.24244 0
1.6 0 −0.0702441 0 3.96287 0 0 0 −2.99507 0
1.7 0 0.0758070 0 −2.36208 0 0 0 −2.99425 0
1.8 0 0.884594 0 3.05564 0 0 0 −2.21749 0
1.9 0 0.954523 0 1.32717 0 0 0 −2.08889 0
1.10 0 1.72076 0 −1.55524 0 0 0 −0.0389984 0
1.11 0 2.12220 0 −2.73357 0 0 0 1.50371 0
1.12 0 3.03680 0 −3.65450 0 0 0 6.22218 0
1.13 0 3.15030 0 3.24192 0 0 0 6.92437 0
1.14 0 3.32292 0 2.28248 0 0 0 8.04181 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(19\) \(-1\)

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 7448.2.a.bx yes 14
7.b odd 2 1 7448.2.a.bu 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7448.2.a.bu 14 7.b odd 2 1
7448.2.a.bx yes 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):

\( T_{3}^{14} - 6 T_{3}^{13} - 11 T_{3}^{12} + 114 T_{3}^{11} - 10 T_{3}^{10} - 806 T_{3}^{9} + 523 T_{3}^{8} + 2586 T_{3}^{7} - 2226 T_{3}^{6} - 3618 T_{3}^{5} + 3397 T_{3}^{4} + 1570 T_{3}^{3} - 1529 T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{14} - 2 T_{5}^{13} - 41 T_{5}^{12} + 66 T_{5}^{11} + 673 T_{5}^{10} - 788 T_{5}^{9} - 5716 T_{5}^{8} + 4120 T_{5}^{7} + 26565 T_{5}^{6} - 8214 T_{5}^{5} - 64405 T_{5}^{4} - 514 T_{5}^{3} + 68301 T_{5}^{2} + 10292 T_{5} - 20734 \) Copy content Toggle raw display
\( T_{11}^{14} + 6 T_{11}^{13} - 77 T_{11}^{12} - 486 T_{11}^{11} + 1954 T_{11}^{10} + 13894 T_{11}^{9} - 17761 T_{11}^{8} - 178106 T_{11}^{7} - 5634 T_{11}^{6} + 1029322 T_{11}^{5} + 815663 T_{11}^{4} - 2144578 T_{11}^{3} + \cdots + 609152 \) Copy content Toggle raw display
\( T_{13}^{14} - 16 T_{13}^{13} + 26 T_{13}^{12} + 696 T_{13}^{11} - 2512 T_{13}^{10} - 11944 T_{13}^{9} + 51402 T_{13}^{8} + 107776 T_{13}^{7} - 441329 T_{13}^{6} - 576112 T_{13}^{5} + 1562040 T_{13}^{4} + 1756400 T_{13}^{3} + \cdots - 566264 \) Copy content Toggle raw display
\( T_{17}^{14} + 4 T_{17}^{13} - 88 T_{17}^{12} - 232 T_{17}^{11} + 3007 T_{17}^{10} + 4156 T_{17}^{9} - 47266 T_{17}^{8} - 22520 T_{17}^{7} + 325825 T_{17}^{6} + 42996 T_{17}^{5} - 1020332 T_{17}^{4} - 46048 T_{17}^{3} + \cdots - 648776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 6 T^{13} - 11 T^{12} + 114 T^{11} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} - 41 T^{12} + \cdots - 20734 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 6 T^{13} - 77 T^{12} + \cdots + 609152 \) Copy content Toggle raw display
$13$ \( T^{14} - 16 T^{13} + 26 T^{12} + \cdots - 566264 \) Copy content Toggle raw display
$17$ \( T^{14} + 4 T^{13} - 88 T^{12} + \cdots - 648776 \) Copy content Toggle raw display
$19$ \( (T - 1)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + 4 T^{13} - 144 T^{12} + \cdots + 21044800 \) Copy content Toggle raw display
$29$ \( T^{14} + 6 T^{13} - 173 T^{12} + \cdots - 33222344 \) Copy content Toggle raw display
$31$ \( T^{14} - 16 T^{13} - 152 T^{12} + \cdots - 14605600 \) Copy content Toggle raw display
$37$ \( T^{14} - 6 T^{13} - 205 T^{12} + \cdots - 66460168 \) Copy content Toggle raw display
$41$ \( T^{14} + 14 T^{13} + \cdots - 3537070114 \) Copy content Toggle raw display
$43$ \( T^{14} + 2 T^{13} - 303 T^{12} + \cdots - 1372672 \) Copy content Toggle raw display
$47$ \( T^{14} - 30 T^{13} + \cdots - 1615752568 \) Copy content Toggle raw display
$53$ \( T^{14} + 6 T^{13} - 279 T^{12} + \cdots + 41412196 \) Copy content Toggle raw display
$59$ \( T^{14} - 22 T^{13} + \cdots + 591004400 \) Copy content Toggle raw display
$61$ \( T^{14} - 10 T^{13} + \cdots - 1478120350 \) Copy content Toggle raw display
$67$ \( T^{14} - 4 T^{13} + \cdots - 123038347712 \) Copy content Toggle raw display
$71$ \( T^{14} - 6 T^{13} + \cdots + 687400751644 \) Copy content Toggle raw display
$73$ \( T^{14} - 4 T^{13} - 268 T^{12} + \cdots + 1616608 \) Copy content Toggle raw display
$79$ \( T^{14} - 26 T^{13} - 119 T^{12} + \cdots + 31373248 \) Copy content Toggle raw display
$83$ \( T^{14} - 32 T^{13} + \cdots + 710528800 \) Copy content Toggle raw display
$89$ \( T^{14} + 54 T^{13} + \cdots + 17004293600 \) Copy content Toggle raw display
$97$ \( T^{14} - 18 T^{13} + \cdots - 697028175746 \) Copy content Toggle raw display
show more
show less