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Properties

Label 927.2.f.c
Level $927$
Weight $2$
Character orbit 927.f
Analytic conductor $7.402$
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] = [927,2,Mod(46,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.f (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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 13 x^{14} - 8 x^{13} + 119 x^{12} - 64 x^{11} + 462 x^{10} + 40 x^{9} + 1089 x^{8} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 309)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + (\beta_{12} + 2 \beta_{6} + \beta_1) q^{4} + ( - \beta_{6} - \beta_{5}) q^{5} + (\beta_{12} - \beta_{7}) q^{7} + ( - 2 \beta_{13} - \beta_{11} + \cdots + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} + (\beta_{12} + 2 \beta_{6} + \beta_1) q^{4} + ( - \beta_{6} - \beta_{5}) q^{5} + (\beta_{12} - \beta_{7}) q^{7} + ( - 2 \beta_{13} - \beta_{11} + \cdots + 2) q^{8}+ \cdots + (3 \beta_{12} + 2 \beta_{10} + \cdots + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - 11 q^{4} + 5 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} - 11 q^{4} + 5 q^{5} + 6 q^{8} + 24 q^{10} - 6 q^{11} + 6 q^{13} + 18 q^{14} - 33 q^{16} + 2 q^{17} - 10 q^{19} + 7 q^{20} + 4 q^{22} - 20 q^{23} - 3 q^{25} - 2 q^{26} - 22 q^{28} + 6 q^{29} + 16 q^{31} + 18 q^{32} - 16 q^{34} - 23 q^{35} - 12 q^{37} - 12 q^{38} - 15 q^{40} + 12 q^{41} - 3 q^{43} - 43 q^{44} - 9 q^{46} - 9 q^{50} + 10 q^{52} + 5 q^{53} + 4 q^{55} - 34 q^{56} - 2 q^{58} + 23 q^{59} - 2 q^{61} + 2 q^{62} + 98 q^{64} + 13 q^{65} - 26 q^{67} - 10 q^{68} - 51 q^{70} + 44 q^{71} - 80 q^{73} + 7 q^{74} + 114 q^{76} - 4 q^{77} - 38 q^{79} + 44 q^{80} - 14 q^{82} - 11 q^{83} + 22 q^{85} - 54 q^{86} + 27 q^{88} + 46 q^{89} - 27 q^{91} + 102 q^{92} + 74 q^{94} - 6 q^{95} + 6 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 13 x^{14} - 8 x^{13} + 119 x^{12} - 64 x^{11} + 462 x^{10} + 40 x^{9} + 1089 x^{8} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 245179874281 \nu^{15} + 30590066672895 \nu^{14} - 9400031540273 \nu^{13} + \cdots + 12\!\cdots\!16 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7410553278337 \nu^{15} + 4320082010367 \nu^{14} - 92093729847565 \nu^{13} + \cdots - 202968589835776 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3809308069300 \nu^{15} - 22474827015235 \nu^{14} + 76773929941837 \nu^{13} + \cdots - 14\!\cdots\!00 ) / 322565464236184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10602401605759 \nu^{15} + 14768672123713 \nu^{14} + 97639040588721 \nu^{13} + \cdots + 13\!\cdots\!20 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12685536864736 \nu^{15} - 20096090143073 \nu^{14} + 169232061251935 \nu^{13} + \cdots - 729950141318440 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13835213923636 \nu^{15} - 7813666849007 \nu^{14} + 168049919856537 \nu^{13} + \cdots - 71298906459640 ) / 322565464236184 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2286190397529 \nu^{15} + 8430180698251 \nu^{14} - 38426631978181 \nu^{13} + \cdots + 500379028092288 ) / 49625456036336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37680737919969 \nu^{15} - 86841398211795 \nu^{14} + 511207393803721 \nu^{13} + \cdots + 31\!\cdots\!12 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 70255183312110 \nu^{15} - 116621795386887 \nu^{14} + 957382780783913 \nu^{13} + \cdots - 50\!\cdots\!08 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 73230812780709 \nu^{15} - 76321284048679 \nu^{14} + 956244028920033 \nu^{13} + \cdots + 10\!\cdots\!60 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 83987373165692 \nu^{15} + 780082947607 \nu^{14} - 932084469999849 \nu^{13} + \cdots - 37\!\cdots\!64 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 90220580655225 \nu^{15} + 137001028650008 \nu^{14} + \cdots + 203847632594360 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 158049360415353 \nu^{15} - 226565344882498 \nu^{14} + \cdots + 714406815648024 ) / 645130928472368 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 173427870873310 \nu^{15} + 296752409804155 \nu^{14} + \cdots + 15\!\cdots\!32 ) / 645130928472368 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{10} + \beta_{9} - 2\beta_{6} - \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - 2\beta_{11} + \beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} - 6\beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{10} + \beta_{8} + 7\beta_{7} + 19\beta_{6} + \beta_{5} + 10\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} + 11 \beta_{13} + 17 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 11 \beta_{6} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13 \beta_{15} + \beta_{14} - 84 \beta_{13} - \beta_{12} - 13 \beta_{11} - 49 \beta_{9} - 13 \beta_{8} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{12} - 31\beta_{10} - 15\beta_{8} - 67\beta_{7} + 26\beta_{6} + 133\beta_{5} + 99\beta_{4} + 303\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 126 \beta_{15} - 15 \beta_{14} + 677 \beta_{13} + 130 \beta_{11} + 354 \beta_{10} + 354 \beta_{9} + \cdots - 654 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 155 \beta_{15} - 126 \beta_{14} - 852 \beta_{13} + 126 \beta_{12} - 1031 \beta_{11} - 344 \beta_{9} + \cdots + 965 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 155 \beta_{12} - 2621 \beta_{10} + 1098 \beta_{8} + 2741 \beta_{7} + 7900 \beta_{6} + 1196 \beta_{5} + \cdots - 1687 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1373 \beta_{15} + 1098 \beta_{14} + 7243 \beta_{13} + 8009 \beta_{11} + 3348 \beta_{10} + 3348 \beta_{9} + \cdots - 8474 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9077 \beta_{15} + 1373 \beta_{14} - 42704 \beta_{13} - 1373 \beta_{12} - 10591 \beta_{11} - 19778 \beta_{9} + \cdots + 38760 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9077 \beta_{12} - 30482 \beta_{10} - 11186 \beta_{8} - 22035 \beta_{7} + 41561 \beta_{6} + \cdots + 122238 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 72780 \beta_{15} - 11186 \beta_{14} + 338060 \beta_{13} + 91746 \beta_{11} + 151499 \beta_{10} + \cdots - 303001 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 86405 \beta_{15} - 72780 \beta_{14} - 516267 \beta_{13} + 72780 \beta_{12} - 489559 \beta_{11} + \cdots + 613451 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\) \(722\) \(829\)
\(\chi(n)\) \(1\) \(-1 - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−0.242953 + 0.420807i
−1.42181 + 2.46265i
1.34500 − 2.32962i
1.19898 − 2.07670i
−0.306705 + 0.531228i
0.318130 − 0.551018i
0.375308 − 0.650053i
−0.765957 + 1.32668i
−0.242953 − 0.420807i
−1.42181 − 2.46265i
1.34500 + 2.32962i
1.19898 + 2.07670i
−0.306705 − 0.531228i
0.318130 + 0.551018i
0.375308 + 0.650053i
−0.765957 − 1.32668i
−1.35346 − 2.34427i 0 −2.66372 + 4.61370i −1.31506 + 2.27775i 0 −2.29707 + 3.97864i 9.00715 0 7.11953
46.2 −1.16677 − 2.02091i 0 −1.72272 + 2.98383i 1.57015 − 2.71958i 0 1.03997 − 1.80127i 3.37298 0 −7.32803
46.3 −0.756270 − 1.30990i 0 −0.143888 + 0.249221i −0.473259 + 0.819708i 0 −0.0165134 + 0.0286020i −2.58981 0 1.43164
46.4 −0.0490316 − 0.0849253i 0 0.995192 − 1.72372i −0.281965 + 0.488377i 0 −1.01813 + 1.76346i −0.391310 0 0.0553007
46.5 0.154206 + 0.267093i 0 0.952441 − 1.64968i −0.823528 + 1.42639i 0 1.18271 − 2.04851i 1.20431 0 −0.507972
46.6 0.327401 + 0.567076i 0 0.785617 − 1.36073i 1.75355 − 3.03724i 0 1.86719 − 3.23407i 2.33845 0 2.29646
46.7 0.943429 + 1.63407i 0 −0.780115 + 1.35120i 1.45693 − 2.52348i 0 −1.22538 + 2.12243i 0.829783 0 5.49804
46.8 1.40050 + 2.42574i 0 −2.92281 + 5.06245i 0.613178 − 1.06206i 0 0.467231 − 0.809267i −10.7716 0 3.43503
262.1 −1.35346 + 2.34427i 0 −2.66372 − 4.61370i −1.31506 − 2.27775i 0 −2.29707 − 3.97864i 9.00715 0 7.11953
262.2 −1.16677 + 2.02091i 0 −1.72272 − 2.98383i 1.57015 + 2.71958i 0 1.03997 + 1.80127i 3.37298 0 −7.32803
262.3 −0.756270 + 1.30990i 0 −0.143888 − 0.249221i −0.473259 − 0.819708i 0 −0.0165134 − 0.0286020i −2.58981 0 1.43164
262.4 −0.0490316 + 0.0849253i 0 0.995192 + 1.72372i −0.281965 − 0.488377i 0 −1.01813 − 1.76346i −0.391310 0 0.0553007
262.5 0.154206 − 0.267093i 0 0.952441 + 1.64968i −0.823528 − 1.42639i 0 1.18271 + 2.04851i 1.20431 0 −0.507972
262.6 0.327401 − 0.567076i 0 0.785617 + 1.36073i 1.75355 + 3.03724i 0 1.86719 + 3.23407i 2.33845 0 2.29646
262.7 0.943429 − 1.63407i 0 −0.780115 − 1.35120i 1.45693 + 2.52348i 0 −1.22538 − 2.12243i 0.829783 0 5.49804
262.8 1.40050 − 2.42574i 0 −2.92281 − 5.06245i 0.613178 + 1.06206i 0 0.467231 + 0.809267i −10.7716 0 3.43503
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.f.c 16
3.b odd 2 1 309.2.e.c ✓ 16
103.c even 3 1 inner 927.2.f.c 16
309.h odd 6 1 309.2.e.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.2.e.c ✓ 16 3.b odd 2 1
309.2.e.c ✓ 16 309.h odd 6 1
927.2.f.c 16 1.a even 1 1 trivial
927.2.f.c 16 103.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + T_{2}^{15} + 14 T_{2}^{14} + 9 T_{2}^{13} + 134 T_{2}^{12} + 74 T_{2}^{11} + 614 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 5 T^{15} + \cdots + 8281 \) Copy content Toggle raw display
$7$ \( T^{16} + 28 T^{14} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{16} + 6 T^{15} + \cdots + 841 \) Copy content Toggle raw display
$13$ \( (T^{8} - 3 T^{7} + \cdots + 448)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{15} + \cdots + 246016 \) Copy content Toggle raw display
$19$ \( T^{16} + 10 T^{15} + \cdots + 10374841 \) Copy content Toggle raw display
$23$ \( (T^{8} + 10 T^{7} + \cdots - 111344)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 35568828409 \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 73936)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 6 T^{7} + \cdots - 60608)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 55756904641 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 34046523289 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1392987702001 \) Copy content Toggle raw display
$53$ \( T^{16} - 5 T^{15} + \cdots + 105625 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5021646255409 \) Copy content Toggle raw display
$61$ \( (T^{8} + T^{7} + \cdots - 33728)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 691953025 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 216779067049561 \) Copy content Toggle raw display
$73$ \( (T^{8} + 40 T^{7} + \cdots - 10496)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 19 T^{7} + \cdots + 10046464)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 46904130098281 \) Copy content Toggle raw display
$89$ \( (T^{8} - 23 T^{7} + \cdots + 12080)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 6 T^{15} + \cdots + 841 \) Copy content Toggle raw display
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