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Properties

Label 78.5.c.a
Level $78$
Weight $5$
Character orbit 78.c
Analytic conductor $8.063$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,5,Mod(53,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 78.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(8.06285712054\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 154 x^{14} - 938 x^{13} + 8635 x^{12} - 39980 x^{11} + 231013 x^{10} + \cdots + 81960012 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{9}\cdot 13^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{4} + \beta_{3} + 1) q^{3} - 8 q^{4} - \beta_{12} q^{5} + ( - \beta_{9} + \beta_{3} - 6) q^{6} + ( - \beta_{2} + 5) q^{7} - 8 \beta_{3} q^{8} + ( - \beta_{9} + \beta_{6} - \beta_{4} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{4} + \beta_{3} + 1) q^{3} - 8 q^{4} - \beta_{12} q^{5} + ( - \beta_{9} + \beta_{3} - 6) q^{6} + ( - \beta_{2} + 5) q^{7} - 8 \beta_{3} q^{8} + ( - \beta_{9} + \beta_{6} - \beta_{4} + \cdots + 7) q^{9}+ \cdots + (63 \beta_{15} + 46 \beta_{14} + \cdots - 3137) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{3} - 128 q^{4} - 96 q^{6} + 80 q^{7} + 116 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{3} - 128 q^{4} - 96 q^{6} + 80 q^{7} + 116 q^{9} - 64 q^{10} - 96 q^{12} + 40 q^{15} + 1024 q^{16} + 448 q^{18} + 816 q^{19} - 128 q^{21} - 1344 q^{22} + 768 q^{24} - 3896 q^{25} - 2520 q^{27} - 640 q^{28} - 544 q^{30} + 5472 q^{31} + 2920 q^{33} + 1408 q^{34} - 928 q^{36} - 3456 q^{37} + 512 q^{40} - 3520 q^{42} - 3112 q^{43} + 3632 q^{45} + 4864 q^{46} + 768 q^{48} + 10776 q^{49} + 4132 q^{51} + 10080 q^{54} - 13792 q^{55} - 17264 q^{57} - 10688 q^{58} - 320 q^{60} + 11504 q^{61} - 2904 q^{63} - 8192 q^{64} - 8512 q^{66} - 10080 q^{67} + 16320 q^{69} + 8576 q^{70} - 3584 q^{72} + 19568 q^{73} - 4840 q^{75} - 6528 q^{76} - 7216 q^{79} - 6364 q^{81} + 6592 q^{82} + 1024 q^{84} + 8320 q^{85} + 14664 q^{87} + 10752 q^{88} + 7360 q^{90} - 1352 q^{91} + 49392 q^{93} + 4224 q^{94} - 6144 q^{96} - 45056 q^{97} - 51016 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 154 x^{14} - 938 x^{13} + 8635 x^{12} - 39980 x^{11} + 231013 x^{10} + \cdots + 81960012 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 725829 \nu^{14} + 5080803 \nu^{13} - 106280473 \nu^{12} + 571632399 \nu^{11} + \cdots - 26460003342708 ) / 6782149224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19843467 \nu^{14} + 138904269 \nu^{13} - 2834969947 \nu^{12} + 15204064185 \nu^{11} + \cdots - 395017954027740 ) / 61039343016 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1000871278 \nu^{15} - 7506534585 \nu^{14} + 147848077311 \nu^{13} - 847163394649 \nu^{12} + \cdots - 14\!\cdots\!92 ) / 180601435416258 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29242537618866 \nu^{15} - 177475327735381 \nu^{14} + \cdots + 63\!\cdots\!56 ) / 72\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 44362144778172 \nu^{15} - 140139560066833 \nu^{14} + \cdots + 39\!\cdots\!04 ) / 72\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16197018273035 \nu^{15} + 85231115306994 \nu^{14} + \cdots - 62\!\cdots\!40 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 73494756215868 \nu^{15} - 358634145849553 \nu^{14} + \cdots + 37\!\cdots\!88 ) / 72\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 36802341198519 \nu^{15} - 540799499102764 \nu^{14} + \cdots - 66\!\cdots\!92 ) / 36\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19454217066726 \nu^{15} + 193260510358495 \nu^{14} + \cdots + 13\!\cdots\!70 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38611947950213 \nu^{15} - 374482882756922 \nu^{14} + \cdots - 23\!\cdots\!14 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 155185645760002 \nu^{15} + \cdots - 48\!\cdots\!00 ) / 72\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3820035630176 \nu^{15} + 28650267226320 \nu^{14} - 560902125996024 \nu^{13} + \cdots + 42\!\cdots\!95 ) / 15\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 73346745664148 \nu^{15} + 510701263860951 \nu^{14} + \cdots - 59\!\cdots\!16 ) / 24\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 211382248621218 \nu^{15} + \cdots + 52\!\cdots\!68 ) / 36\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 453290148005514 \nu^{15} + \cdots + 28\!\cdots\!96 ) / 72\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 23 \beta_{15} - 12 \beta_{14} - 84 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} + 50 \beta_{9} + \cdots + 503 ) / 936 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11 \beta_{15} - 4 \beta_{14} + 42 \beta_{13} - 28 \beta_{12} + 10 \beta_{11} + 43 \beta_{10} + \cdots - 4677 ) / 312 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 352 \beta_{15} + 108 \beta_{14} + 63 \beta_{13} + 834 \beta_{12} - 77 \beta_{11} - 50 \beta_{10} + \cdots - 7393 ) / 312 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 167 \beta_{15} + 220 \beta_{14} - 1968 \beta_{13} + 1696 \beta_{12} - 948 \beta_{11} + \cdots + 143031 ) / 312 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18612 \beta_{15} - 6436 \beta_{14} - 5025 \beta_{13} - 31480 \beta_{12} + 5751 \beta_{11} + \cdots + 379919 ) / 312 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11425 \beta_{15} - 19860 \beta_{14} + 92232 \beta_{13} - 98694 \beta_{12} + 61662 \beta_{11} + \cdots - 5933099 ) / 312 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 987133 \beta_{15} + 366286 \beta_{14} + 340473 \beta_{13} + 1355314 \beta_{12} - 308436 \beta_{11} + \cdots - 22523328 ) / 312 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1626753 \beta_{15} + 1558340 \beta_{14} - 4430568 \beta_{13} + 5885804 \beta_{12} - 3713676 \beta_{11} + \cdots + 271849739 ) / 312 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 51400141 \beta_{15} - 19152240 \beta_{14} - 22001625 \beta_{13} - 61786746 \beta_{12} + \cdots + 1380058396 ) / 312 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 69518659 \beta_{15} - 53794438 \beta_{14} + 105911781 \beta_{13} - 176888320 \beta_{12} + \cdots - 6361864290 ) / 156 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1308011256 \beta_{15} + 470617726 \beta_{14} + 685255989 \beta_{13} + 1425009115 \beta_{12} + \cdots - 41953554686 ) / 156 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 5033061013 \beta_{15} + 3428521764 \beta_{14} - 4945743144 \beta_{13} + 10545483789 \beta_{12} + \cdots + 293945737607 ) / 156 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 9991670819 \beta_{15} - 3377790314 \beta_{14} - 6357678180 \beta_{13} - 9944081810 \beta_{12} + \cdots + 384655750569 ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 334108374018 \beta_{15} - 207540360158 \beta_{14} + 222322236825 \beta_{13} - 618513155189 \beta_{12} + \cdots - 13129362730883 ) / 156 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 6273086330534 \beta_{15} + 1937300289294 \beta_{14} + 4846259675538 \beta_{13} + \cdots - 291416929657202 ) / 312 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 − 7.28865i
0.500000 + 2.71385i
0.500000 + 1.90592i
0.500000 − 5.07409i
0.500000 − 1.58453i
0.500000 + 3.94477i
0.500000 + 1.80454i
0.500000 + 3.57818i
0.500000 + 7.28865i
0.500000 − 2.71385i
0.500000 − 1.90592i
0.500000 + 5.07409i
0.500000 + 1.58453i
0.500000 − 3.94477i
0.500000 − 1.80454i
0.500000 − 3.57818i
− 2.82843i −8.72175 − 2.22059i −8.00000 − 21.1459i −6.28079 + 24.6688i −44.0341 22.6274i 71.1379 + 38.7349i −59.8095
53.2 − 2.82843i −8.51219 + 2.92276i −8.00000 15.0297i 8.26682 + 24.0761i 35.8465 22.6274i 63.9149 − 49.7583i 42.5104
53.3 − 2.82843i −3.84426 − 8.13767i −8.00000 39.9993i −23.0168 + 10.8732i 25.3326 22.6274i −51.4434 + 62.5666i 113.135
53.4 − 2.82843i 0.217281 − 8.99738i −8.00000 − 47.4886i −25.4484 − 0.614563i 76.3806 22.6274i −80.9056 − 3.90991i −134.318
53.5 − 2.82843i 4.20657 − 7.95643i −8.00000 6.32497i −22.5042 − 11.8980i −79.1829 22.6274i −45.6095 − 66.9386i 17.8897
53.6 − 2.82843i 7.08834 + 5.54576i −8.00000 20.4771i 15.6858 − 20.0489i −5.83303 22.6274i 19.4891 + 78.6204i 57.9180
53.7 − 2.82843i 7.25482 + 5.32612i −8.00000 − 41.0662i 15.0646 − 20.5197i −47.8807 22.6274i 24.2648 + 77.2801i −116.153
53.8 − 2.82843i 8.31119 − 3.45314i −8.00000 16.5559i −9.76695 − 23.5076i 79.3711 22.6274i 57.1517 − 57.3994i 46.8271
53.9 2.82843i −8.72175 + 2.22059i −8.00000 21.1459i −6.28079 − 24.6688i −44.0341 − 22.6274i 71.1379 − 38.7349i −59.8095
53.10 2.82843i −8.51219 − 2.92276i −8.00000 − 15.0297i 8.26682 − 24.0761i 35.8465 − 22.6274i 63.9149 + 49.7583i 42.5104
53.11 2.82843i −3.84426 + 8.13767i −8.00000 − 39.9993i −23.0168 − 10.8732i 25.3326 − 22.6274i −51.4434 − 62.5666i 113.135
53.12 2.82843i 0.217281 + 8.99738i −8.00000 47.4886i −25.4484 + 0.614563i 76.3806 − 22.6274i −80.9056 + 3.90991i −134.318
53.13 2.82843i 4.20657 + 7.95643i −8.00000 − 6.32497i −22.5042 + 11.8980i −79.1829 − 22.6274i −45.6095 + 66.9386i 17.8897
53.14 2.82843i 7.08834 − 5.54576i −8.00000 − 20.4771i 15.6858 + 20.0489i −5.83303 − 22.6274i 19.4891 − 78.6204i 57.9180
53.15 2.82843i 7.25482 − 5.32612i −8.00000 41.0662i 15.0646 + 20.5197i −47.8807 − 22.6274i 24.2648 − 77.2801i −116.153
53.16 2.82843i 8.31119 + 3.45314i −8.00000 − 16.5559i −9.76695 + 23.5076i 79.3711 − 22.6274i 57.1517 + 57.3994i 46.8271
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.5.c.a ✓ 16
3.b odd 2 1 inner 78.5.c.a ✓ 16
4.b odd 2 1 624.5.f.a 16
12.b even 2 1 624.5.f.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.5.c.a ✓ 16 1.a even 1 1 trivial
78.5.c.a ✓ 16 3.b odd 2 1 inner
624.5.f.a 16 4.b odd 2 1
624.5.f.a 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(78, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 5361013811728)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2197)^{8} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 43\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 28\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 51\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 24\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 53\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 65\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 68\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 15\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
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