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Properties

Label 927.3.d.c
Level $927$
Weight $3$
Character orbit 927.d
Analytic conductor $25.259$
Analytic rank $0$
Dimension $12$
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] = [927,3,Mod(514,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.514");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 927.d (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: \(25.2589205062\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 92x^{10} + 3344x^{8} + 60552x^{6} + 561888x^{4} + 2410584x^{2} + 3274879 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 103)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + (\beta_{10} + 1) q^{4} - \beta_{3} q^{5} + (\beta_{10} + \beta_{8} - \beta_{6}) q^{7} + (\beta_{8} - \beta_{6} + \beta_{5} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + (\beta_{10} + 1) q^{4} - \beta_{3} q^{5} + (\beta_{10} + \beta_{8} - \beta_{6}) q^{7} + (\beta_{8} - \beta_{6} + \beta_{5} + 2) q^{8} + (\beta_{7} - \beta_{3}) q^{10} + \beta_{4} q^{11} + ( - 3 \beta_{11} + 2 \beta_{10} + \cdots + 4) q^{13}+ \cdots + ( - 15 \beta_{11} + 21 \beta_{10} + \cdots + 94) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 14 q^{4} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 14 q^{4} + 16 q^{8} + 30 q^{13} + 50 q^{14} - 50 q^{16} + 14 q^{17} - 10 q^{19} - 48 q^{23} - 242 q^{25} - 176 q^{26} + 202 q^{28} - 96 q^{29} - 74 q^{32} + 104 q^{34} + 46 q^{38} + 40 q^{41} - 48 q^{46} - 92 q^{49} - 276 q^{50} + 254 q^{52} + 128 q^{55} + 280 q^{56} - 408 q^{58} - 194 q^{59} - 150 q^{61} - 396 q^{64} + 230 q^{68} + 284 q^{76} + 158 q^{79} - 440 q^{82} - 162 q^{83} + 238 q^{91} - 548 q^{92} + 192 q^{97} + 1104 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 92x^{10} + 3344x^{8} + 60552x^{6} + 561888x^{4} + 2410584x^{2} + 3274879 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1773 \nu^{11} - 133388 \nu^{9} - 10421356 \nu^{7} - 205154332 \nu^{5} - 1441031403 \nu^{3} - 2257379672 \nu ) / 95725001 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7411 \nu^{11} + 792209 \nu^{9} + 29272621 \nu^{7} + 467341819 \nu^{5} + 3122794327 \nu^{3} + 6320623191 \nu ) / 191450002 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10957\nu^{11} - 525433\nu^{9} - 8429909\nu^{7} - 57033155\nu^{5} - 49281519\nu^{3} + 2023136193\nu ) / 191450002 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7411\nu^{10} + 792209\nu^{8} + 29272621\nu^{6} + 467341819\nu^{4} + 3122794327\nu^{2} + 6129173189 ) / 191450002 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9184\nu^{10} - 658821\nu^{8} - 18851265\nu^{6} - 262187487\nu^{4} - 1586037923\nu^{2} - 2340193501 ) / 95725001 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68807 \nu^{11} + 4487731 \nu^{9} + 109392325 \nu^{7} + 1248708953 \nu^{5} + \cdots + 13608260439 \nu ) / 191450002 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53985\nu^{10} + 2903313\nu^{8} + 50847083\nu^{6} + 314025315\nu^{4} + 476141839\nu^{2} + 584114053 ) / 191450002 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 187972 \nu^{11} - 13609386 \nu^{9} - 365781259 \nu^{7} - 4428415801 \nu^{5} + \cdots - 33487533459 \nu ) / 95725001 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 361441 \nu^{10} + 26960115 \nu^{8} + 743975321 \nu^{6} + 9210107111 \nu^{4} + 48121836605 \nu^{2} + 70832662427 ) / 574350006 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 372079 \nu^{10} + 26159787 \nu^{8} + 681447185 \nu^{6} + 7979181119 \nu^{4} + 39475648187 \nu^{2} + 56139684383 ) / 574350006 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{6} + 2\beta_{5} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{4} + 3\beta_{3} + 3\beta_{2} - 20\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -40\beta_{11} + 37\beta_{10} + 7\beta_{8} - 27\beta_{6} - 50\beta_{5} + 266 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{9} + 21\beta_{7} - 81\beta_{4} - 108\beta_{3} - 114\beta_{2} + 430\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1180\beta_{11} - 1072\beta_{10} - 277\beta_{8} + 601\beta_{6} + 1187\beta_{5} - 5596 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -108\beta_{9} - 831\beta_{7} + 1803\beta_{4} + 3312\beta_{3} + 3324\beta_{2} - 9575\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -31483\beta_{11} + 28600\beta_{10} + 8076\beta_{8} - 12704\beta_{6} - 28383\beta_{5} + 123607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2883\beta_{9} + 24228\beta_{7} - 38112\beta_{4} - 92610\beta_{3} - 88683\beta_{2} + 218446\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 805600\beta_{11} - 734824\beta_{10} - 210600\beta_{8} + 265391\beta_{6} + 681638\beta_{5} - 2811492 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -70776\beta_{9} - 631800\beta_{7} + 796173\beta_{4} + 2441565\beta_{3} + 2275248\beta_{2} - 5072360\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(722\) \(829\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
514.1
− 4.94529i
4.94529i
1.54521i
− 1.54521i
4.18237i
− 4.18237i
− 3.02467i
3.02467i
3.83937i
− 3.83937i
− 4.87595i
4.87595i
−3.04667 0 5.28219 − 3.95241i 0 0.175006 −3.90641 0 12.0417i
514.2 −3.04667 0 5.28219 3.95241i 0 0.175006 −3.90641 0 − 12.0417i
514.3 −1.88143 0 −0.460236 − 9.32562i 0 −0.103817 8.39160 0 17.5455i
514.4 −1.88143 0 −0.460236 9.32562i 0 −0.103817 8.39160 0 − 17.5455i
514.5 −0.671610 0 −3.54894 − 1.09651i 0 −0.741172 5.06994 0 0.736424i
514.6 −0.671610 0 −3.54894 1.09651i 0 −0.741172 5.06994 0 − 0.736424i
514.7 0.771370 0 −3.40499 − 7.47643i 0 −8.64515 −5.71198 0 − 5.76709i
514.8 0.771370 0 −3.40499 7.47643i 0 −8.64515 −5.71198 0 5.76709i
514.9 2.64285 0 2.98465 − 6.46444i 0 −3.38248 −2.68341 0 − 17.0845i
514.10 2.64285 0 2.98465 6.46444i 0 −3.38248 −2.68341 0 17.0845i
514.11 3.18549 0 6.14732 − 8.33804i 0 12.6976 6.84026 0 − 26.5607i
514.12 3.18549 0 6.14732 8.33804i 0 12.6976 6.84026 0 26.5607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 514.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.3.d.c 12
3.b odd 2 1 103.3.b.b ✓ 12
103.b odd 2 1 inner 927.3.d.c 12
309.c even 2 1 103.3.b.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.3.b.b ✓ 12 3.b odd 2 1
103.3.b.b ✓ 12 309.c even 2 1
927.3.d.c 12 1.a even 1 1 trivial
927.3.d.c 12 103.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} - 15T_{2}^{4} + 10T_{2}^{3} + 55T_{2}^{2} - 9T_{2} - 25 \) acting on \(S_{3}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} - 15 T^{4} + \cdots - 25)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 265265199 \) Copy content Toggle raw display
$7$ \( (T^{6} - 124 T^{4} + \cdots + 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 1697697273600 \) Copy content Toggle raw display
$13$ \( (T^{6} - 15 T^{5} + \cdots - 262480)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 7 T^{5} + \cdots + 226885)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 5 T^{5} + \cdots - 2887376)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 24 T^{5} + \cdots - 2308060)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 48 T^{5} + \cdots + 56410112)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 67\!\cdots\!91 \) Copy content Toggle raw display
$41$ \( (T^{6} - 20 T^{5} + \cdots + 3565468)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!79 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14\!\cdots\!39 \) Copy content Toggle raw display
$59$ \( (T^{6} + 97 T^{5} + \cdots - 18095168848)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 75 T^{5} + \cdots - 19383425104)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 48\!\cdots\!71 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{6} - 79 T^{5} + \cdots - 205105099109)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 81 T^{5} + \cdots - 167129829040)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} - 96 T^{5} + \cdots - 28741698880)^{2} \) Copy content Toggle raw display
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