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Properties

Label 308.4.i.a
Level $308$
Weight $4$
Character orbit 308.i
Analytic conductor $18.173$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,4,Mod(177,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.177");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 308.i (of order \(3\), degree \(2\), 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: \(18.1725882818\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 194 x^{18} - 432 x^{17} + 24205 x^{16} - 47156 x^{15} + 1632616 x^{14} + \cdots + 7996651776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2} - \beta_1) q^{3} + (\beta_{9} - \beta_{5} + \beta_{3} - 1) q^{5} + ( - \beta_{14} - \beta_{8} - \beta_{3} + \cdots - 1) q^{7} + (\beta_{10} + 11 \beta_{3} + \beta_1 - 11) q^{9}+ \cdots + ( - 11 \beta_{11} - 11 \beta_{10} + \cdots + 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{3} - 10 q^{5} - 20 q^{7} - 104 q^{9} - 110 q^{11} + 16 q^{13} + 108 q^{15} - 166 q^{17} - 342 q^{19} - 42 q^{21} + 54 q^{23} - 198 q^{25} + 612 q^{27} - 160 q^{29} - 492 q^{31} - 66 q^{33} + 310 q^{35}+ \cdots + 2288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 194 x^{18} - 432 x^{17} + 24205 x^{16} - 47156 x^{15} + 1632616 x^{14} + \cdots + 7996651776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!84 \nu^{19} + \cdots - 73\!\cdots\!48 ) / 36\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!63 \nu^{19} + \cdots + 76\!\cdots\!64 ) / 59\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61\!\cdots\!21 \nu^{19} + \cdots - 18\!\cdots\!28 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 40\!\cdots\!72 \nu^{19} + \cdots - 19\!\cdots\!36 ) / 91\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36\!\cdots\!40 \nu^{19} + \cdots + 21\!\cdots\!52 ) / 68\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!25 \nu^{19} + \cdots - 10\!\cdots\!60 ) / 33\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!11 \nu^{19} + \cdots + 11\!\cdots\!60 ) / 22\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49\!\cdots\!59 \nu^{19} + \cdots - 22\!\cdots\!92 ) / 66\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 56\!\cdots\!23 \nu^{19} + \cdots - 62\!\cdots\!16 ) / 59\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 56\!\cdots\!35 \nu^{19} + \cdots + 28\!\cdots\!56 ) / 59\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!79 \nu^{19} + \cdots - 75\!\cdots\!00 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 77\!\cdots\!67 \nu^{19} + \cdots + 49\!\cdots\!12 ) / 66\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!59 \nu^{19} + \cdots + 11\!\cdots\!72 ) / 66\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47\!\cdots\!39 \nu^{19} + \cdots + 44\!\cdots\!36 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 64\!\cdots\!53 \nu^{19} + \cdots + 97\!\cdots\!36 ) / 22\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 78\!\cdots\!84 \nu^{19} + \cdots + 32\!\cdots\!28 ) / 18\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 29\!\cdots\!61 \nu^{19} + \cdots + 77\!\cdots\!24 ) / 66\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 21\!\cdots\!23 \nu^{19} + \cdots + 96\!\cdots\!24 ) / 33\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 37\beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} - 2 \beta_{15} + 2 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} + \beta_{11} + \cdots - 19 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{19} + 5 \beta_{18} + 6 \beta_{17} + 26 \beta_{16} + 16 \beta_{15} + 31 \beta_{14} + \cdots - 2651 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 136 \beta_{19} - 210 \beta_{18} + 336 \beta_{16} - 198 \beta_{15} + 280 \beta_{14} + 466 \beta_{13} + \cdots + 478 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 987 \beta_{17} - 2355 \beta_{16} - 1159 \beta_{15} - 2435 \beta_{14} + 3631 \beta_{13} + \cdots + 217751 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14269 \beta_{19} + 21739 \beta_{18} - 14269 \beta_{17} - 66597 \beta_{16} + 34448 \beta_{15} + \cdots + 460398 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 121599 \beta_{19} - 4518 \beta_{18} - 233769 \beta_{16} - 127371 \beta_{15} - 332337 \beta_{14} + \cdots - 204966 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1443637 \beta_{17} + 4976029 \beta_{16} - 1437284 \beta_{15} - 179278 \beta_{14} - 6234035 \beta_{13} + \cdots - 64005193 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 13573437 \beta_{19} - 2707906 \beta_{18} + 13573437 \beta_{17} + 65049551 \beta_{16} + \cdots - 1679623448 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 146326591 \beta_{19} - 224501367 \beta_{18} + 175070226 \beta_{16} - 116731782 \beta_{15} + \cdots + 676076959 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1451952870 \beta_{17} - 4727356764 \beta_{16} - 1202407774 \beta_{15} - 1833479240 \beta_{14} + \cdots + 158340131897 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14918189656 \beta_{19} + 22390586023 \beta_{18} - 14918189656 \beta_{17} - 76397571852 \beta_{16} + \cdots + 754868225712 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 152093627136 \beta_{19} + 79318393848 \beta_{18} - 233867751861 \beta_{16} - 111296434638 \beta_{15} + \cdots - 412173679878 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1526872041136 \beta_{17} + 6362779623634 \beta_{16} - 715604115908 \beta_{15} - 668564953639 \beta_{14} + \cdots - 89750328272596 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 15755378016579 \beta_{19} - 9980471229007 \beta_{18} + 15755378016579 \beta_{17} + \cdots - 13\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 156570936098419 \beta_{19} - 219863389279494 \beta_{18} + 155429834177496 \beta_{16} + \cdots + 715503907177237 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 16\!\cdots\!02 \beta_{17} + \cdots + 14\!\cdots\!57 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 16\!\cdots\!90 \beta_{19} + \cdots + 94\!\cdots\!60 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−4.54700 + 7.87564i
−3.86353 + 6.69182i
−2.05350 + 3.55676i
−1.14233 + 1.97858i
0.138808 − 0.240422i
0.192948 − 0.334196i
1.43296 − 2.48195i
2.62470 − 4.54611i
4.16059 − 7.20635i
5.05636 − 8.75787i
−4.54700 − 7.87564i
−3.86353 − 6.69182i
−2.05350 − 3.55676i
−1.14233 − 1.97858i
0.138808 + 0.240422i
0.192948 + 0.334196i
1.43296 + 2.48195i
2.62470 + 4.54611i
4.16059 + 7.20635i
5.05636 + 8.75787i
0 −5.04700 − 8.74166i 0 3.77667 − 6.54138i 0 17.9318 + 4.63153i 0 −37.4445 + 64.8557i 0
177.2 0 −4.36353 − 7.55785i 0 −6.34995 + 10.9984i 0 −15.6613 + 9.88547i 0 −24.5807 + 42.5751i 0
177.3 0 −2.55350 − 4.42278i 0 3.75702 − 6.50735i 0 −2.07070 − 18.4041i 0 0.459316 − 0.795558i 0
177.4 0 −1.64233 − 2.84460i 0 −10.0843 + 17.4664i 0 15.8994 − 9.49786i 0 8.10549 − 14.0391i 0
177.5 0 −0.361192 − 0.625604i 0 5.92984 − 10.2708i 0 −5.49821 − 17.6853i 0 13.2391 − 22.9308i 0
177.6 0 −0.307052 − 0.531830i 0 −0.388866 + 0.673536i 0 6.69839 + 17.2665i 0 13.3114 − 23.0561i 0
177.7 0 0.932956 + 1.61593i 0 −7.91970 + 13.7173i 0 −18.5160 + 0.398871i 0 11.7592 − 20.3675i 0
177.8 0 2.12470 + 3.68009i 0 7.32246 − 12.6829i 0 −17.0162 + 7.31095i 0 4.47130 − 7.74452i 0
177.9 0 3.66059 + 6.34032i 0 3.91736 − 6.78507i 0 16.2143 + 8.94974i 0 −13.2998 + 23.0359i 0
177.10 0 4.55636 + 7.89185i 0 −4.96059 + 8.59200i 0 −7.98145 − 16.7122i 0 −28.0208 + 48.5335i 0
221.1 0 −5.04700 + 8.74166i 0 3.77667 + 6.54138i 0 17.9318 − 4.63153i 0 −37.4445 − 64.8557i 0
221.2 0 −4.36353 + 7.55785i 0 −6.34995 − 10.9984i 0 −15.6613 − 9.88547i 0 −24.5807 − 42.5751i 0
221.3 0 −2.55350 + 4.42278i 0 3.75702 + 6.50735i 0 −2.07070 + 18.4041i 0 0.459316 + 0.795558i 0
221.4 0 −1.64233 + 2.84460i 0 −10.0843 − 17.4664i 0 15.8994 + 9.49786i 0 8.10549 + 14.0391i 0
221.5 0 −0.361192 + 0.625604i 0 5.92984 + 10.2708i 0 −5.49821 + 17.6853i 0 13.2391 + 22.9308i 0
221.6 0 −0.307052 + 0.531830i 0 −0.388866 − 0.673536i 0 6.69839 − 17.2665i 0 13.3114 + 23.0561i 0
221.7 0 0.932956 − 1.61593i 0 −7.91970 − 13.7173i 0 −18.5160 − 0.398871i 0 11.7592 + 20.3675i 0
221.8 0 2.12470 − 3.68009i 0 7.32246 + 12.6829i 0 −17.0162 − 7.31095i 0 4.47130 + 7.74452i 0
221.9 0 3.66059 − 6.34032i 0 3.91736 + 6.78507i 0 16.2143 − 8.94974i 0 −13.2998 − 23.0359i 0
221.10 0 4.55636 − 7.89185i 0 −4.96059 − 8.59200i 0 −7.98145 + 16.7122i 0 −28.0208 − 48.5335i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.4.i.a ✓ 20
7.c even 3 1 inner 308.4.i.a ✓ 20
7.c even 3 1 2156.4.a.m 10
7.d odd 6 1 2156.4.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.4.i.a ✓ 20 1.a even 1 1 trivial
308.4.i.a ✓ 20 7.c even 3 1 inner
2156.4.a.j 10 7.d odd 6 1
2156.4.a.m 10 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 6 T_{3}^{19} + 205 T_{3}^{18} + 738 T_{3}^{17} + 25156 T_{3}^{16} + 81030 T_{3}^{15} + \cdots + 120251513529 \) acting on \(S_{4}^{\mathrm{new}}(308, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 120251513529 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 11 T + 121)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 10\!\cdots\!13)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 91\!\cdots\!69)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 76\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 49\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 83\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 75\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 38\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 53\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 18\!\cdots\!79)^{2} \) Copy content Toggle raw display
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