↑

Properties

Label 961.4.a.i
Level $961$
Weight $4$
Character orbit 961.a
Self dual yes
Analytic conductor $56.701$
Analytic rank $0$
Dimension $14$
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] = [961,4,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("961.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 961.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,1,-1] 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: \(56.7008355155\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 77 x^{12} + 54 x^{11} + 2250 x^{10} - 1046 x^{9} - 31002 x^{8} + 8912 x^{7} + \cdots - 79056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 31)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{4} + \beta_{3} + 4) q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{10} - \beta_{8} - \beta_{6} + \cdots + 3) q^{6} + ( - \beta_{13} + \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + ( - 8 \beta_{13} + 5 \beta_{12} + \cdots + 21) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + q^{2} - q^{3} + 43 q^{4} + 19 q^{6} + 5 q^{7} + 54 q^{8} + 107 q^{9} + 57 q^{10} + 79 q^{11} - 5 q^{12} + 47 q^{13} - 129 q^{14} + 228 q^{15} + 127 q^{16} + 143 q^{17} - 392 q^{18} + 47 q^{19}+ \cdots + 2002 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 77 x^{12} + 54 x^{11} + 2250 x^{10} - 1046 x^{9} - 31002 x^{8} + 8912 x^{7} + \cdots - 79056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12128635 \nu^{13} + 232340546 \nu^{12} - 1461095393 \nu^{11} - 16667944077 \nu^{10} + \cdots + 18202718369808 ) / 1178279728128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7548800 \nu^{13} + 18435563 \nu^{12} + 547517641 \nu^{11} - 1146257832 \nu^{10} + \cdots + 225707208624 ) / 294569932032 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7548800 \nu^{13} - 18435563 \nu^{12} - 547517641 \nu^{11} + 1146257832 \nu^{10} + \cdots - 3760546393008 ) / 294569932032 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10850156 \nu^{13} + 17159841 \nu^{12} - 984034493 \nu^{11} - 852624704 \nu^{10} + \cdots - 7813776276336 ) / 392759909376 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3628921 \nu^{13} - 11246653 \nu^{12} - 246207544 \nu^{11} + 658312209 \nu^{10} + \cdots - 198925977600 ) / 98189977344 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43614928 \nu^{13} + 21542591 \nu^{12} - 3403438007 \nu^{11} - 1542232812 \nu^{10} + \cdots + 3151910335536 ) / 1178279728128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30305307 \nu^{13} + 67736507 \nu^{12} + 2260074708 \nu^{11} - 4270240295 \nu^{10} + \cdots + 2313200785920 ) / 392759909376 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 169162847 \nu^{13} + 171030464 \nu^{12} + 11681018059 \nu^{11} - 6209442423 \nu^{10} + \cdots - 9507393327024 ) / 1178279728128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5990501 \nu^{13} - 14420036 \nu^{12} - 414058381 \nu^{11} + 820115757 \nu^{10} + \cdots - 315519772464 ) / 38009023488 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 192163607 \nu^{13} + 133584107 \nu^{12} + 14702321056 \nu^{11} - 5743737483 \nu^{10} + \cdots + 21244729415616 ) / 1178279728128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30091802 \nu^{13} - 51566537 \nu^{12} - 2317821313 \nu^{11} + 3448363542 \nu^{10} + \cdots - 958342645584 ) / 147284966016 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 199645376 \nu^{13} + 310346631 \nu^{12} + 15271001577 \nu^{11} - 20047481908 \nu^{10} + \cdots + 21171162553008 ) / 392759909376 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 19\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + 3 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 239 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{13} + 11 \beta_{12} + 30 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 14 \beta_{8} + 2 \beta_{7} + \cdots + 225 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37 \beta_{13} + 137 \beta_{12} + 83 \beta_{11} + 74 \beta_{10} - 6 \beta_{9} - 58 \beta_{8} + \cdots + 5539 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 152 \beta_{13} + 558 \beta_{12} + 816 \beta_{11} + 96 \beta_{10} + 86 \beta_{9} + 512 \beta_{8} + \cdots + 9180 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1192 \beta_{13} + 4916 \beta_{12} + 2764 \beta_{11} + 2092 \beta_{10} - 332 \beta_{9} - 1308 \beta_{8} + \cdots + 138240 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5920 \beta_{13} + 21464 \beta_{12} + 22129 \beta_{11} + 3312 \beta_{10} + 2512 \beta_{9} + 14032 \beta_{8} + \cdots + 322868 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 37769 \beta_{13} + 161947 \beta_{12} + 86018 \beta_{11} + 54082 \beta_{10} - 13024 \beta_{9} + \cdots + 3609767 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 206907 \beta_{13} + 743035 \beta_{12} + 608302 \beta_{11} + 100434 \beta_{10} + 59426 \beta_{9} + \cdots + 10558721 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1191477 \beta_{13} + 5116841 \beta_{12} + 2600723 \beta_{11} + 1350842 \beta_{10} - 445174 \beta_{9} + \cdots + 97193619 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6811896 \beta_{13} + 24361222 \beta_{12} + 16972744 \beta_{11} + 2865792 \beta_{10} + 1148390 \beta_{9} + \cdots + 332204980 \) 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
−4.79880
−4.14269
−4.05875
−3.42862
−1.60866
−1.03879
−0.708460
0.291376
1.25419
2.30989
2.82057
3.93746
4.74510
5.42617
−4.79880 −8.90345 15.0285 −17.4274 42.7259 −10.9339 −33.7285 52.2714 83.6307
1.2 −4.14269 −1.31757 9.16185 −4.51879 5.45827 −15.4338 −4.81316 −25.2640 18.7199
1.3 −4.05875 4.61106 8.47346 2.02857 −18.7151 10.8094 −1.92165 −5.73813 −8.23348
1.4 −3.42862 10.0046 3.75542 10.5489 −34.3019 25.5036 14.5531 73.0920 −36.1681
1.5 −1.60866 −5.64879 −5.41223 18.5767 9.08696 5.73537 21.5757 4.90882 −29.8836
1.6 −1.03879 −7.34597 −6.92092 −8.60252 7.63091 34.9288 15.4997 26.9633 8.93620
1.7 −0.708460 2.78489 −7.49808 −9.52038 −1.97298 −25.4772 10.9798 −19.2444 6.74481
1.8 0.291376 −0.959449 −7.91510 19.3340 −0.279561 −9.07633 −4.63728 −26.0795 5.63346
1.9 1.25419 5.46788 −6.42700 −14.2300 6.85779 −7.64482 −18.0943 2.89775 −17.8472
1.10 2.30989 7.03386 −2.66440 3.56481 16.2475 26.8071 −24.6336 22.4752 8.23432
1.11 2.82057 −5.67988 −0.0443646 −5.41022 −16.0205 0.305009 −22.6897 5.26103 −15.2599
1.12 3.93746 −7.33613 7.50358 −3.86557 −28.8857 −27.6218 −1.95465 26.8188 −15.2205
1.13 4.74510 4.34075 14.5159 5.80623 20.5973 18.6149 30.9188 −8.15793 27.5511
1.14 5.42617 1.94821 21.4433 3.71575 10.5713 −21.5163 72.9459 −23.2045 20.1623
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

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

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 961.4.a.i 14
31.b odd 2 1 961.4.a.j 14
31.d even 5 2 31.4.d.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.4.d.a ✓ 28 31.d even 5 2
961.4.a.i 14 1.a even 1 1 trivial
961.4.a.j 14 31.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(961))\):

\( T_{2}^{14} - T_{2}^{13} - 77 T_{2}^{12} + 54 T_{2}^{11} + 2250 T_{2}^{10} - 1046 T_{2}^{9} + \cdots - 79056 \) Copy content Toggle raw display
\( T_{3}^{14} + T_{3}^{13} - 242 T_{3}^{12} - 195 T_{3}^{11} + 22230 T_{3}^{10} + 7629 T_{3}^{9} + \cdots - 813177461 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - T^{13} + \cdots - 79056 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots - 813177461 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 1134574950636 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 14\!\cdots\!19 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 24\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 80\!\cdots\!51 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 23\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 14\!\cdots\!79 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 22\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{14} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 98\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 34\!\cdots\!51 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 74\!\cdots\!59 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 11\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 19\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 18\!\cdots\!51 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 53\!\cdots\!31 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 46\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 17\!\cdots\!31 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 51\!\cdots\!91 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 46\!\cdots\!01 \) Copy content Toggle raw display
show more
show less