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Properties

Label 128.5.f.a
Level $128$
Weight $5$
Character orbit 128.f
Analytic conductor $13.231$
Analytic rank $0$
Dimension $14$
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] = [128,5,Mod(31,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.f (of order \(4\), degree \(2\), not 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: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{42} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{8} q^{5} + \beta_{9} q^{7} + ( - \beta_{12} + \beta_{5} + \cdots - 19 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + \beta_{8} q^{5} + \beta_{9} q^{7} + ( - \beta_{12} + \beta_{5} + \cdots - 19 \beta_1) q^{9}+ \cdots + ( - 2 \beta_{13} - 34 \beta_{12} + \cdots + 3450) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{3} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{3} + 2 q^{5} + 4 q^{7} + 94 q^{11} + 2 q^{13} - 4 q^{17} - 706 q^{19} + 164 q^{21} - 1148 q^{23} - 1664 q^{27} - 862 q^{29} - 4 q^{33} + 1340 q^{35} + 1826 q^{37} - 2684 q^{39} + 1694 q^{43} - 1410 q^{45} + 682 q^{49} - 3012 q^{51} + 482 q^{53} + 11780 q^{55} - 2786 q^{59} + 3778 q^{61} - 2020 q^{65} + 7998 q^{67} - 9628 q^{69} - 19964 q^{71} + 17570 q^{75} + 9508 q^{77} + 1454 q^{81} - 17282 q^{83} - 9948 q^{85} + 49284 q^{87} - 28036 q^{91} - 8896 q^{93} - 4 q^{97} + 49214 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2875 \nu^{13} + 13444 \nu^{12} - 26581 \nu^{11} + 16062 \nu^{10} - 24954 \nu^{9} + \cdots + 1279524864 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11609 \nu^{13} + 81068 \nu^{12} - 216535 \nu^{11} + 92170 \nu^{10} - 445086 \nu^{9} + \cdots + 7904428032 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23279 \nu^{13} + 167756 \nu^{12} - 482591 \nu^{11} + 959162 \nu^{10} - 197678 \nu^{9} + \cdots + 45595492352 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3499 \nu^{13} + 6652 \nu^{12} + 13019 \nu^{11} - 10778 \nu^{10} + 122070 \nu^{9} + \cdots - 3419275264 ) / 84869120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30299 \nu^{13} + 175428 \nu^{12} - 396085 \nu^{11} + 537790 \nu^{10} - 1159226 \nu^{9} + \cdots + 20694433792 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 187 \nu^{13} + 588 \nu^{12} - 3179 \nu^{11} + 8306 \nu^{10} - 18854 \nu^{9} + 936 \nu^{8} + \cdots + 359268352 ) / 2424832 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54457 \nu^{13} - 666028 \nu^{12} + 2372471 \nu^{11} - 2827082 \nu^{10} - 5318178 \nu^{9} + \cdots - 123104133120 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 63941 \nu^{13} + 248956 \nu^{12} - 325931 \nu^{11} + 303682 \nu^{10} - 1485958 \nu^{9} + \cdots + 4344512512 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1373 \nu^{13} - 1028 \nu^{12} + 16237 \nu^{11} - 30894 \nu^{10} + 21354 \nu^{9} + \cdots - 1804861440 ) / 9699328 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 60737 \nu^{13} - 163572 \nu^{12} + 819089 \nu^{11} - 871398 \nu^{10} - 2291022 \nu^{9} + \cdots - 98412789760 ) / 339476480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 191167 \nu^{13} - 198708 \nu^{12} - 780399 \nu^{11} + 1949018 \nu^{10} + 2939442 \nu^{9} + \cdots + 139540561920 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 107577 \nu^{13} + 484524 \nu^{12} - 1270135 \nu^{11} + 2145610 \nu^{10} - 3926878 \nu^{9} + \cdots + 33873985536 ) / 339476480 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 108593 \nu^{13} - 103372 \nu^{12} - 236161 \nu^{11} + 473862 \nu^{10} + 1410158 \nu^{9} + \cdots + 31120424960 ) / 339476480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{2} - 9\beta _1 + 17 ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{12} + 2 \beta_{11} + \beta_{10} - 4 \beta_{9} + 7 \beta_{8} + \beta_{7} - 3 \beta_{6} + \cdots - 63 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{12} - 2 \beta_{11} + \beta_{10} - 12 \beta_{9} + 31 \beta_{8} + \beta_{7} - 7 \beta_{6} + \cdots - 67 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{13} + 11 \beta_{12} + 12 \beta_{11} - 5 \beta_{10} + 12 \beta_{9} + 45 \beta_{8} + \cdots + 121 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 44 \beta_{13} + 15 \beta_{12} - 34 \beta_{11} - 23 \beta_{10} - 36 \beta_{9} + 107 \beta_{8} + \cdots + 5561 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 38 \beta_{13} - 189 \beta_{12} - 108 \beta_{11} - 5 \beta_{10} - 132 \beta_{9} + 445 \beta_{8} + \cdots - 12447 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 280 \beta_{13} + 7 \beta_{12} + 242 \beta_{11} - 291 \beta_{10} - 84 \beta_{9} + 1143 \beta_{8} + \cdots - 10215 ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 858 \beta_{13} + 611 \beta_{12} - 1136 \beta_{11} - 153 \beta_{10} - 1156 \beta_{9} + 409 \beta_{8} + \cdots + 66329 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2564 \beta_{13} - 153 \beta_{12} + 1158 \beta_{11} - 127 \beta_{10} + 3660 \beta_{9} - 6677 \beta_{8} + \cdots + 89825 ) / 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3434 \beta_{13} + 6691 \beta_{12} + 548 \beta_{11} + 2587 \beta_{10} - 4804 \beta_{9} + 3021 \beta_{8} + \cdots + 628385 ) / 64 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15088 \beta_{13} - 31849 \beta_{12} - 5174 \beta_{11} + 8437 \beta_{10} + 2892 \beta_{9} + \cdots - 1111399 ) / 64 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 9886 \beta_{13} + 1059 \beta_{12} + 65352 \beta_{11} - 12833 \beta_{10} + 144956 \beta_{9} + \cdots + 1768521 ) / 64 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 124644 \beta_{13} + 4807 \beta_{12} - 241810 \beta_{11} + 41769 \beta_{10} - 273012 \beta_{9} + \cdots - 514479 ) / 64 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.153862 − 2.82424i
0.336831 + 2.80830i
2.79265 + 0.448449i
−2.40693 + 1.48549i
−2.15805 − 1.82834i
2.24452 − 1.72109i
1.03712 + 2.63142i
0.153862 + 2.82424i
0.336831 − 2.80830i
2.79265 − 0.448449i
−2.40693 − 1.48549i
−2.15805 + 1.82834i
2.24452 + 1.72109i
1.03712 − 2.63142i
0 −9.42589 − 9.42589i 0 2.84710 + 2.84710i 0 76.7794 0 96.6949i 0
31.2 0 −7.86839 − 7.86839i 0 −27.2309 − 27.2309i 0 −50.3097 0 42.8233i 0
31.3 0 −4.63552 − 4.63552i 0 29.2002 + 29.2002i 0 −59.6196 0 − 38.0239i 0
31.4 0 −0.0461995 − 0.0461995i 0 8.04297 + 8.04297i 0 49.8797 0 − 80.9957i 0
31.5 0 3.91498 + 3.91498i 0 −4.72348 − 4.72348i 0 −45.3712 0 − 50.3458i 0
31.6 0 5.54016 + 5.54016i 0 −21.7374 − 21.7374i 0 6.62054 0 − 19.6133i 0
31.7 0 11.5209 + 11.5209i 0 14.6016 + 14.6016i 0 24.0210 0 184.461i 0
95.1 0 −9.42589 + 9.42589i 0 2.84710 − 2.84710i 0 76.7794 0 − 96.6949i 0
95.2 0 −7.86839 + 7.86839i 0 −27.2309 + 27.2309i 0 −50.3097 0 − 42.8233i 0
95.3 0 −4.63552 + 4.63552i 0 29.2002 − 29.2002i 0 −59.6196 0 38.0239i 0
95.4 0 −0.0461995 + 0.0461995i 0 8.04297 − 8.04297i 0 49.8797 0 80.9957i 0
95.5 0 3.91498 − 3.91498i 0 −4.72348 + 4.72348i 0 −45.3712 0 50.3458i 0
95.6 0 5.54016 − 5.54016i 0 −21.7374 + 21.7374i 0 6.62054 0 19.6133i 0
95.7 0 11.5209 − 11.5209i 0 14.6016 − 14.6016i 0 24.0210 0 − 184.461i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.5.f.a 14
4.b odd 2 1 128.5.f.b 14
8.b even 2 1 64.5.f.a 14
8.d odd 2 1 16.5.f.a ✓ 14
16.e even 4 1 16.5.f.a ✓ 14
16.e even 4 1 128.5.f.b 14
16.f odd 4 1 64.5.f.a 14
16.f odd 4 1 inner 128.5.f.a 14
24.f even 2 1 144.5.m.a 14
24.h odd 2 1 576.5.m.a 14
48.i odd 4 1 144.5.m.a 14
48.k even 4 1 576.5.m.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a ✓ 14 8.d odd 2 1
16.5.f.a ✓ 14 16.e even 4 1
64.5.f.a 14 8.b even 2 1
64.5.f.a 14 16.f odd 4 1
128.5.f.a 14 1.a even 1 1 trivial
128.5.f.a 14 16.f odd 4 1 inner
128.5.f.b 14 4.b odd 2 1
128.5.f.b 14 16.e even 4 1
144.5.m.a 14 24.f even 2 1
144.5.m.a 14 48.i odd 4 1
576.5.m.a 14 24.h odd 2 1
576.5.m.a 14 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 2 T_{3}^{13} + 2 T_{3}^{12} + 448 T_{3}^{11} + 62860 T_{3}^{10} + 288664 T_{3}^{9} + \cdots + 2016379008 \) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 2016379008 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 95\!\cdots\!08 \) Copy content Toggle raw display
$7$ \( (T^{7} - 2 T^{6} + \cdots + 82884464768)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 47\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 24\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 37\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 54\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 27\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 51\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 35\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 40\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 38\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 67\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 30\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 50\!\cdots\!96)^{2} \) Copy content Toggle raw display
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