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Properties

Label 7105.2.a.bd
Level $7105$
Weight $2$
Character orbit 7105.a
Self dual yes
Analytic conductor $56.734$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7105,2,Mod(1,7105)] 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("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7105.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [19,2,-5,30,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.7337106361\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 2 x^{18} - 32 x^{17} + 59 x^{16} + 431 x^{15} - 711 x^{14} - 3185 x^{13} + 4489 x^{12} + \cdots - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1015)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{16} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_{18} - \beta_{15} + \cdots - \beta_1) q^{6} + (\beta_{17} + \beta_{16} + \beta_{13} + \cdots + 1) q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{18} + \beta_{15} - \beta_{13} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 2 q^{2} - 5 q^{3} + 30 q^{4} - 19 q^{5} + 15 q^{8} + 22 q^{9} - 2 q^{10} + 17 q^{11} - 22 q^{12} - 15 q^{13} + 5 q^{15} + 48 q^{16} - 8 q^{17} + 9 q^{18} + 9 q^{19} - 30 q^{20} + 19 q^{22} + 45 q^{23}+ \cdots + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 2 x^{18} - 32 x^{17} + 59 x^{16} + 431 x^{15} - 711 x^{14} - 3185 x^{13} + 4489 x^{12} + \cdots - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 41868343147 \nu^{18} + 658290687175 \nu^{17} - 392352354517 \nu^{16} + \cdots - 31476326346351 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57532142929 \nu^{18} - 280668534517 \nu^{17} + 2959893627187 \nu^{16} + \cdots + 109028131100091 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42541736694 \nu^{18} - 151321126084 \nu^{17} - 725006281194 \nu^{16} + 3696653715408 \nu^{15} + \cdots + 53879190006555 ) / 7207735751817 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 112482125217 \nu^{18} - 256538843699 \nu^{17} - 3097022726709 \nu^{16} + \cdots - 3202255505895 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114770415623 \nu^{18} - 287914853033 \nu^{17} - 3873019472389 \nu^{16} + \cdots - 48662889276231 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 152453289309 \nu^{18} - 15230556069 \nu^{17} - 5343194409799 \nu^{16} + \cdots - 41497260711009 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 217028294565 \nu^{18} + 220111793399 \nu^{17} + 6217522601247 \nu^{16} + \cdots - 40078548247755 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 143059048354 \nu^{18} + 759080528916 \nu^{17} + 3746756633511 \nu^{16} + \cdots + 5692156973130 ) / 7207735751817 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 146570714563 \nu^{18} - 201665620534 \nu^{17} - 4554678344467 \nu^{16} + \cdots - 7552108679907 ) / 7207735751817 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 300276327295 \nu^{18} + 425248216631 \nu^{17} + 9779353318889 \nu^{16} + \cdots + 74235098849517 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 212183042712 \nu^{18} - 245834387138 \nu^{17} - 6544850830941 \nu^{16} + \cdots - 1635370698378 ) / 7207735751817 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 218045278875 \nu^{18} + 604320915372 \nu^{17} + 6494377080200 \nu^{16} + \cdots + 23123492406852 ) / 7207735751817 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 231523051664 \nu^{18} + 755062437016 \nu^{17} + 6494597332453 \nu^{16} + \cdots + 235847649564 ) / 7207735751817 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 243660445482 \nu^{18} - 629479712749 \nu^{17} - 7122353445320 \nu^{16} + \cdots + 5801551876596 ) / 7207735751817 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 832135759555 \nu^{18} + 1365021809563 \nu^{17} + 26256585969629 \nu^{16} + \cdots + 73556007066039 ) / 14415471503634 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 428974346530 \nu^{18} - 1108313181567 \nu^{17} - 13108458800426 \nu^{16} + \cdots - 38915585187582 ) / 7207735751817 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{13} - \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} - \beta_{17} + \beta_{12} - \beta_{10} + \beta_{8} - \beta_{6} + 6\beta_{2} + \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{18} + 9 \beta_{17} + 11 \beta_{16} + \beta_{15} + 9 \beta_{13} + \beta_{12} + \beta_{11} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{18} - 9 \beta_{17} + \beta_{16} - \beta_{15} + \beta_{13} + 10 \beta_{12} + \beta_{11} + \cdots + 137 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 17 \beta_{18} + 69 \beta_{17} + 97 \beta_{16} + 11 \beta_{15} - 3 \beta_{14} + 69 \beta_{13} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 98 \beta_{18} - 66 \beta_{17} + 18 \beta_{16} - 17 \beta_{15} + 16 \beta_{13} + 83 \beta_{12} + \cdots + 913 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 196 \beta_{18} + 507 \beta_{17} + 792 \beta_{16} + 89 \beta_{15} - 49 \beta_{14} + 508 \beta_{13} + \cdots + 641 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 817 \beta_{18} - 454 \beta_{17} + 231 \beta_{16} - 186 \beta_{15} - 3 \beta_{14} + 178 \beta_{13} + \cdots + 6340 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1932 \beta_{18} + 3665 \beta_{17} + 6232 \beta_{16} + 639 \beta_{15} - 545 \beta_{14} + 3697 \beta_{13} + \cdots + 5372 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6617 \beta_{18} - 3032 \beta_{17} + 2540 \beta_{16} - 1690 \beta_{15} - 66 \beta_{14} + 1725 \beta_{13} + \cdots + 45223 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 17561 \beta_{18} + 26289 \beta_{17} + 48053 \beta_{16} + 4311 \beta_{15} - 5186 \beta_{14} + \cdots + 44586 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 52799 \beta_{18} - 19874 \beta_{17} + 25447 \beta_{16} - 13912 \beta_{15} - 951 \beta_{14} + \cdots + 328591 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 152191 \beta_{18} + 187775 \beta_{17} + 366179 \beta_{16} + 28010 \beta_{15} - 45576 \beta_{14} + \cdots + 366571 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 417813 \beta_{18} - 128319 \beta_{17} + 239535 \beta_{16} - 107911 \beta_{15} - 11421 \beta_{14} + \cdots + 2419410 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1279471 \beta_{18} + 1337836 \beta_{17} + 2771105 \beta_{16} + 177602 \beta_{15} - 382857 \beta_{14} + \cdots + 2989875 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 3290885 \beta_{18} - 816920 \beta_{17} + 2157607 \beta_{16} - 804794 \beta_{15} - 124064 \beta_{14} + \cdots + 17989436 \) 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
−2.67821
−2.57181
−2.29213
−2.21398
−1.58921
−1.23028
−1.19647
−0.811060
−0.169175
−0.0409760
0.0869586
0.807724
1.73757
1.76782
2.13874
2.23815
2.42770
2.77866
2.80998
−2.67821 −0.716469 5.17283 −1.00000 1.91886 0 −8.49753 −2.48667 2.67821
1.2 −2.57181 −3.32381 4.61420 −1.00000 8.54821 0 −6.72321 8.04772 2.57181
1.3 −2.29213 1.87844 3.25387 −1.00000 −4.30564 0 −2.87403 0.528551 2.29213
1.4 −2.21398 1.45224 2.90172 −1.00000 −3.21523 0 −1.99640 −0.891000 2.21398
1.5 −1.58921 −1.70282 0.525587 −1.00000 2.70614 0 2.34315 −0.100409 1.58921
1.6 −1.23028 −2.56429 −0.486408 −1.00000 3.15480 0 3.05898 3.57559 1.23028
1.7 −1.19647 2.37529 −0.568458 −1.00000 −2.84196 0 3.07308 2.64200 1.19647
1.8 −0.811060 −2.04083 −1.34218 −1.00000 1.65523 0 2.71071 1.16498 0.811060
1.9 −0.169175 0.597648 −1.97138 −1.00000 −0.101107 0 0.671858 −2.64282 0.169175
1.10 −0.0409760 0.533327 −1.99832 −1.00000 −0.0218536 0 0.163835 −2.71556 0.0409760
1.11 0.0869586 2.73757 −1.99244 −1.00000 0.238055 0 −0.347177 4.49427 −0.0869586
1.12 0.807724 −1.13759 −1.34758 −1.00000 −0.918856 0 −2.70392 −1.70590 −0.807724
1.13 1.73757 −0.172597 1.01915 −1.00000 −0.299899 0 −1.70430 −2.97021 −1.73757
1.14 1.76782 −0.647858 1.12520 −1.00000 −1.14530 0 −1.54649 −2.58028 −1.76782
1.15 2.13874 2.69429 2.57419 −1.00000 5.76237 0 1.22805 4.25919 −2.13874
1.16 2.23815 −3.01974 3.00934 −1.00000 −6.75864 0 2.25905 6.11880 −2.23815
1.17 2.42770 −3.02809 3.89374 −1.00000 −7.35131 0 4.59745 6.16933 −2.42770
1.18 2.77866 2.34590 5.72093 −1.00000 6.51846 0 10.3392 2.50327 −2.77866
1.19 2.80998 −1.26062 5.89601 −1.00000 −3.54232 0 10.9477 −1.41084 −2.80998
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)
\(29\) \( +1 \)

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 7105.2.a.bd 19
7.b odd 2 1 7105.2.a.be 19
7.d odd 6 2 1015.2.i.d ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1015.2.i.d ✓ 38 7.d odd 6 2
7105.2.a.bd 19 1.a even 1 1 trivial
7105.2.a.be 19 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7105))\):

\( T_{2}^{19} - 2 T_{2}^{18} - 32 T_{2}^{17} + 59 T_{2}^{16} + 431 T_{2}^{15} - 711 T_{2}^{14} - 3185 T_{2}^{13} + \cdots - 9 \) Copy content Toggle raw display
\( T_{3}^{19} + 5 T_{3}^{18} - 27 T_{3}^{17} - 160 T_{3}^{16} + 258 T_{3}^{15} + 2086 T_{3}^{14} + \cdots + 1112 \) Copy content Toggle raw display
\( T_{17}^{19} + 8 T_{17}^{18} - 127 T_{17}^{17} - 957 T_{17}^{16} + 6770 T_{17}^{15} + 45436 T_{17}^{14} + \cdots - 141949521 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - 2 T^{18} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{19} + 5 T^{18} + \cdots + 1112 \) Copy content Toggle raw display
$5$ \( (T + 1)^{19} \) Copy content Toggle raw display
$7$ \( T^{19} \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots + 242305887 \) Copy content Toggle raw display
$13$ \( T^{19} + 15 T^{18} + \cdots + 42211736 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots - 141949521 \) Copy content Toggle raw display
$19$ \( T^{19} - 9 T^{18} + \cdots - 22444324 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots - 264454317 \) Copy content Toggle raw display
$29$ \( (T + 1)^{19} \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 237997686467 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 75711194041 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 202886949324 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots - 12414551792536 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots + 12538246584 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 497948131836288 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 1064491604736 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots - 4292337300336 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 6737183923212 \) Copy content Toggle raw display
$73$ \( T^{19} + 47 T^{18} + \cdots - 5748867 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 96372030925156 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 21\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 1615199874972 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 72\!\cdots\!32 \) Copy content Toggle raw display
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