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Properties

Label 1014.4.b.q
Level $1014$
Weight $4$
Character orbit 1014.b
Analytic conductor $59.828$
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] = [1014,4,Mod(337,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), 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: \(59.8279367458\)
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: \( x^{12} + 845x^{10} + 287958x^{8} + 50362537x^{6} + 4731667920x^{4} + 224458698240x^{2} + 4178851762176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + 3 q^{3} - 4 q^{4} + \beta_{8} q^{5} - 6 \beta_{3} q^{6} + ( - \beta_{10} + \beta_{9} - \beta_{8}) q^{7} + 8 \beta_{3} q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + 3 q^{3} - 4 q^{4} + \beta_{8} q^{5} - 6 \beta_{3} q^{6} + ( - \beta_{10} + \beta_{9} - \beta_{8}) q^{7} + 8 \beta_{3} q^{8} + 9 q^{9} + ( - 2 \beta_{5} + 2) q^{10} + ( - \beta_{10} + 2 \beta_{9} + 10 \beta_{3}) q^{11} - 12 q^{12} + (2 \beta_{5} - 2 \beta_{2} + 2 \beta_1 - 2) q^{14} + 3 \beta_{8} q^{15} + 16 q^{16} + (2 \beta_{7} - \beta_{5} + 3 \beta_{4} + \cdots - 5) q^{17}+ \cdots + ( - 9 \beta_{10} + 18 \beta_{9} + 90 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} + 12 q^{10} - 144 q^{12} - 4 q^{14} + 192 q^{16} - 102 q^{17} + 256 q^{22} - 444 q^{23} - 370 q^{25} + 324 q^{27} + 658 q^{29} + 36 q^{30} + 1688 q^{35} - 432 q^{36} - 852 q^{38} - 48 q^{40} - 12 q^{42} - 982 q^{43} + 576 q^{48} - 2266 q^{49} - 306 q^{51} + 4604 q^{53} - 658 q^{55} + 16 q^{56} + 690 q^{61} - 1156 q^{62} - 768 q^{64} + 768 q^{66} + 408 q^{68} - 1332 q^{69} + 1880 q^{74} - 1110 q^{75} - 6582 q^{77} + 6200 q^{79} + 972 q^{81} + 1284 q^{82} + 1974 q^{87} - 1024 q^{88} + 108 q^{90} + 1776 q^{92} - 2564 q^{94} + 1330 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 845x^{10} + 287958x^{8} + 50362537x^{6} + 4731667920x^{4} + 224458698240x^{2} + 4178851762176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 254679227 \nu^{10} + 142769968975 \nu^{8} + 28618044849186 \nu^{6} + \cdots + 76\!\cdots\!00 ) / 42\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19352151011 \nu^{10} - 13104030855607 \nu^{8} + \cdots - 48\!\cdots\!80 ) / 42\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 447775 \nu^{11} - 302892707 \nu^{9} - 77891726586 \nu^{7} - 9426147340375 \nu^{5} + \cdots - 11\!\cdots\!08 \nu ) / 10261971772416 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2175408533 \nu^{10} + 1472552490625 \nu^{8} + 378924665061246 \nu^{6} + \cdots + 54\!\cdots\!16 ) / 329774636282112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2160967510 \nu^{10} + 1461602528585 \nu^{8} + 375794682802131 \nu^{6} + \cdots + 53\!\cdots\!44 ) / 267941891979216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45124973671 \nu^{11} - 30899064670331 \nu^{9} + \cdots - 11\!\cdots\!60 \nu ) / 66\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 489398321 \nu^{10} - 331233973639 \nu^{8} - 85220941965720 \nu^{6} + \cdots - 12\!\cdots\!56 ) / 41221829535264 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 27881483581 \nu^{11} - 18852044969069 \nu^{9} + \cdots - 68\!\cdots\!88 \nu ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 115005844379 \nu^{11} + 77904138878215 \nu^{9} + \cdots + 28\!\cdots\!64 \nu ) / 33\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2953462321 \nu^{11} + 2001783658997 \nu^{9} + 515830116892878 \nu^{7} + \cdots + 73\!\cdots\!84 \nu ) / 85\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10348550369 \nu^{11} - 6984074142643 \nu^{9} + \cdots - 25\!\cdots\!96 \nu ) / 27\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -4\beta_{11} + 13\beta_{8} - 3\beta_{6} - 11\beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -41\beta_{7} - 13\beta_{5} - 49\beta_{4} + 13\beta_{2} - 1850 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 744\beta_{11} - 143\beta_{10} + 481\beta_{9} - 2210\beta_{8} + 740\beta_{6} + 3632\beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11895\beta_{7} + 5343\beta_{5} + 12082\beta_{4} - 4095\beta_{2} - 949\beta _1 + 303192 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -134733\beta_{11} + 18213\beta_{10} - 136708\beta_{9} + 399750\beta_{8} - 169382\beta_{6} - 1052872\beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2834904\beta_{7} - 1570114\beta_{5} - 2425901\beta_{4} + 1110837\beta_{2} + 394420\beta _1 - 53606212 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24931777 \beta_{11} + 471705 \beta_{10} + 31886270 \beta_{9} - 74978995 \beta_{8} + \cdots + 270236179 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 641104415 \beta_{7} + 405901509 \beta_{5} + 464155478 \beta_{4} - 284207404 \beta_{2} + \cdots + 9892622830 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4739190743 \beta_{11} - 977265484 \beta_{10} - 7072568087 \beta_{9} + 14428825333 \beta_{8} + \cdots - 64966644871 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 142321549570 \beta_{7} - 98503171260 \beta_{5} - 88225042034 \beta_{4} + 69808002485 \beta_{2} + \cdots - 1878795357878 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 922342605226 \beta_{11} + 365144410553 \beta_{10} + 1545034066667 \beta_{9} + \cdots + 15068511104193 \beta_{3} ) / 13 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 7.49280i
− 12.9910i
− 14.7697i
8.73978i
12.5460i
12.9677i
− 12.9677i
− 12.5460i
− 8.73978i
14.7697i
12.9910i
7.49280i
− 2.00000i 3.00000 −4.00000 − 13.0357i − 6.00000i − 14.8081i 8.00000i 9.00000 −26.0714
337.2 − 2.00000i 3.00000 −4.00000 − 10.4089i − 6.00000i 18.5089i 8.00000i 9.00000 −20.8178
337.3 − 2.00000i 3.00000 −4.00000 − 9.80890i − 6.00000i 11.4763i 8.00000i 9.00000 −19.6178
337.4 − 2.00000i 3.00000 −4.00000 3.19691i − 6.00000i 34.6837i 8.00000i 9.00000 6.39381
337.5 − 2.00000i 3.00000 −4.00000 15.1281i − 6.00000i − 24.7003i 8.00000i 9.00000 30.2562
337.6 − 2.00000i 3.00000 −4.00000 17.9285i − 6.00000i − 26.1605i 8.00000i 9.00000 35.8570
337.7 2.00000i 3.00000 −4.00000 − 17.9285i 6.00000i 26.1605i − 8.00000i 9.00000 35.8570
337.8 2.00000i 3.00000 −4.00000 − 15.1281i 6.00000i 24.7003i − 8.00000i 9.00000 30.2562
337.9 2.00000i 3.00000 −4.00000 − 3.19691i 6.00000i − 34.6837i − 8.00000i 9.00000 6.39381
337.10 2.00000i 3.00000 −4.00000 9.80890i 6.00000i − 11.4763i − 8.00000i 9.00000 −19.6178
337.11 2.00000i 3.00000 −4.00000 10.4089i 6.00000i − 18.5089i − 8.00000i 9.00000 −20.8178
337.12 2.00000i 3.00000 −4.00000 13.0357i 6.00000i 14.8081i − 8.00000i 9.00000 −26.0714
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.4.b.q 12
13.b even 2 1 inner 1014.4.b.q 12
13.d odd 4 1 1014.4.a.bc ✓ 6
13.d odd 4 1 1014.4.a.be yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.4.a.bc ✓ 6 13.d odd 4 1
1014.4.a.be yes 6 13.d odd 4 1
1014.4.b.q 12 1.a even 1 1 trivial
1014.4.b.q 12 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{12} + 935 T_{5}^{10} + 334277 T_{5}^{8} + 57504531 T_{5}^{6} + 4852504985 T_{5}^{4} + \cdots + 1331792165089 \) Copy content Toggle raw display
\( T_{7}^{12} + 3191 T_{7}^{10} + 3855993 T_{7}^{8} + 2254203111 T_{7}^{6} + 667547496841 T_{7}^{4} + \cdots + 49\!\cdots\!09 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1331792165089 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 49\!\cdots\!09 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 26\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 51 T^{5} + \cdots + 3061293976)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{6} + 222 T^{5} + \cdots - 934743636008)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 15597636448127)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 40\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 514860237502472)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 11\!\cdots\!43)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 90\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 68\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 63\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 867165206982607)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 47\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 28\!\cdots\!61 \) Copy content Toggle raw display
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