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Properties

Label 1127.4.a.g
Level $1127$
Weight $4$
Character orbit 1127.a
Self dual yes
Analytic conductor $66.495$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1127,4,Mod(1,1127)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1127.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1127, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1127 = 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1127.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4951525765\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 109x^{10} + 3303x^{8} - 34203x^{6} + 83010x^{4} - 42334x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{9} q^{3} + ( - \beta_{4} - \beta_{2} + 4) q^{4} + (\beta_{9} + \beta_{6}) q^{5} + (\beta_{8} - 2 \beta_{7}) q^{6} + (\beta_{5} + \beta_{4} + 4 \beta_{2} + \cdots - 5) q^{8}+ \cdots + (2 \beta_{5} - 27 \beta_{4} + \cdots + 133) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 56 q^{4} - 84 q^{8} + 84 q^{9} + 60 q^{11} - 344 q^{15} + 232 q^{16} - 244 q^{18} - 572 q^{22} - 276 q^{23} - 540 q^{25} + 424 q^{29} - 72 q^{30} - 1636 q^{32} + 824 q^{36} - 604 q^{37}+ \cdots + 2148 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 109x^{10} + 3303x^{8} - 34203x^{6} + 83010x^{4} - 42334x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5516812 \nu^{10} + 627854346 \nu^{8} - 20982945429 \nu^{6} + 264549253872 \nu^{4} + \cdots + 919604744854 ) / 64704814191 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34957267 \nu^{10} + 3795641685 \nu^{8} - 113954051745 \nu^{6} + 1154735150205 \nu^{4} + \cdots + 478870476970 ) / 129409628382 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29338859 \nu^{10} + 3158523774 \nu^{8} - 92861111583 \nu^{6} + 897596991312 \nu^{4} + \cdots + 1230022716503 ) / 64704814191 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34676464 \nu^{10} + 3757937166 \nu^{8} - 111864850938 \nu^{6} + 1088773976226 \nu^{4} + \cdots - 55416221432 ) / 64704814191 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50770901 \nu^{10} + 5476578927 \nu^{8} - 161799499359 \nu^{6} + 1581707705151 \nu^{4} + \cdots + 203601697616 ) / 43136542794 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40474079 \nu^{11} + 4423496031 \nu^{9} - 134936997174 \nu^{7} + 1419284404077 \nu^{5} + \cdots + 1527884850206 \nu ) / 64704814191 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55601513 \nu^{11} + 6118291293 \nu^{9} - 189908850339 \nu^{7} + 2088106637547 \nu^{5} + \cdots + 5452644241514 \nu ) / 86273085588 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 388392559 \nu^{11} - 42281307603 \nu^{9} + 1276985539629 \nu^{7} - 13106207616609 \nu^{5} + \cdots - 14276200991170 \nu ) / 258819256764 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 152769031 \nu^{11} + 16624196991 \nu^{9} - 501631214373 \nu^{7} + 5138559305673 \nu^{5} + \cdots + 4991707562782 \nu ) / 86273085588 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 537321245 \nu^{11} + 58671890037 \nu^{9} - 1784842046955 \nu^{7} + \cdots + 22429090423286 \nu ) / 258819256764 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 246632180 \nu^{11} + 26834116329 \nu^{9} - 809423847432 \nu^{7} + 8285206533612 \nu^{5} + \cdots + 8179930281995 \nu ) / 64704814191 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 3\beta_{9} - \beta_{8} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{3} - \beta_{2} + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 31\beta_{11} - 12\beta_{10} - 137\beta_{9} - 83\beta_{8} - 12\beta_{7} + 52\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 163\beta_{5} + 9\beta_{4} - 224\beta_{3} - 383\beta_{2} + 98\beta _1 + 3521 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1419\beta_{11} - 960\beta_{10} - 8081\beta_{9} - 6099\beta_{8} - 792\beta_{7} + 3600\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5690\beta_{5} + 689\beta_{4} - 7034\beta_{3} - 15694\beta_{2} + 4252\beta _1 + 105092 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 82605\beta_{11} - 65524\beta_{10} - 522811\beta_{9} - 420505\beta_{8} - 53652\beta_{7} + 240596\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 769435\beta_{5} + 117609\beta_{4} - 926116\beta_{3} - 2224975\beta_{2} + 613594\beta _1 + 13587045 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5295605 \beta_{11} - 4381392 \beta_{10} - 34738535 \beta_{9} - 28450573 \beta_{8} + \cdots + 16095248 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 25844613\beta_{5} + 4287600\beta_{4} - 30905436\beta_{3} - 75895365\beta_{2} + 21069750\beta _1 + 450587231 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 350379367 \beta_{11} - 292984956 \beta_{10} - 2324807001 \beta_{9} - 1913940331 \beta_{8} + \cdots + 1078507836 \beta_{6} ) / 8 \) Copy content Toggle raw display

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.811872
0.811872
−8.19132
8.19132
1.57414
−1.57414
−4.06778
4.06778
0.159514
−0.159514
−4.71095
4.71095
−5.42902 −6.85728 21.4742 10.5247 37.2283 0 −73.1518 20.0223 −57.1389
1.2 −5.42902 6.85728 21.4742 −10.5247 −37.2283 0 −73.1518 20.0223 57.1389
1.3 −3.36700 −7.12288 3.33669 0.205677 23.9827 0 15.7014 23.7354 −0.692513
1.4 −3.36700 7.12288 3.33669 −0.205677 −23.9827 0 15.7014 23.7354 0.692513
1.5 −1.79537 −2.80763 −4.77666 −13.4005 5.04073 0 22.9388 −19.1172 24.0588
1.6 −1.79537 2.80763 −4.77666 13.4005 −5.04073 0 22.9388 −19.1172 −24.0588
1.7 0.794389 −4.69173 −7.36895 6.34776 −3.72706 0 −12.2089 −4.98766 5.04259
1.8 0.794389 4.69173 −7.36895 −6.34776 3.72706 0 −12.2089 −4.98766 −5.04259
1.9 3.21363 −8.67793 2.32740 12.2185 −27.8876 0 −18.2296 48.3065 39.2658
1.10 3.21363 8.67793 2.32740 −12.2185 27.8876 0 −18.2296 48.3065 −39.2658
1.11 4.58337 −1.02017 13.0073 0.170678 −4.67582 0 22.9502 −25.9592 0.782279
1.12 4.58337 1.02017 13.0073 −0.170678 4.67582 0 22.9502 −25.9592 −0.782279
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(23\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1127.4.a.g ✓ 12
7.b odd 2 1 inner 1127.4.a.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1127.4.a.g ✓ 12 1.a even 1 1 trivial
1127.4.a.g ✓ 12 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1127))\):

\( T_{2}^{6} + 2T_{2}^{5} - 36T_{2}^{4} - 50T_{2}^{3} + 307T_{2}^{2} + 288T_{2} - 384 \) Copy content Toggle raw display
\( T_{3}^{12} - 204T_{3}^{10} + 15306T_{3}^{8} - 516800T_{3}^{6} + 7583784T_{3}^{4} - 38524128T_{3}^{2} + 32444416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 2 T^{5} + \cdots - 384)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 204 T^{10} + \cdots + 32444416 \) Copy content Toggle raw display
$5$ \( T^{12} - 480 T^{10} + \cdots + 147456 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 30 T^{5} + \cdots - 46346816)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T + 23)^{12} \) Copy content Toggle raw display
$29$ \( (T^{6} - 212 T^{5} + \cdots - 994497977408)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{6} + 302 T^{5} + \cdots + 470248793344)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 43184303523328)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 19260956496000)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 169250197941568)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 97\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 76\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
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