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Properties

Label 483.6.a.b
Level $483$
Weight $6$
Character orbit 483.a
Self dual yes
Analytic conductor $77.465$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,6,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

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: yes
Analytic conductor: \(77.4653849697\)
Analytic rank: \(1\)
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: \( x^{12} - x^{11} - 268 x^{10} + 83 x^{9} + 25315 x^{8} + 5134 x^{7} - 993368 x^{6} - 511968 x^{5} + \cdots + 102912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + (\beta_{6} - \beta_1 - 13) q^{5} - 9 \beta_1 q^{6} - 49 q^{7} + ( - \beta_{3} - 2 \beta_{2} + \cdots - 40) q^{8}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + (\beta_{6} - \beta_1 - 13) q^{5} - 9 \beta_1 q^{6} - 49 q^{7} + ( - \beta_{3} - 2 \beta_{2} + \cdots - 40) q^{8}+ \cdots + (81 \beta_{9} - 81 \beta_{7} + \cdots - 9639) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 108 q^{3} + 153 q^{4} - 162 q^{5} - 9 q^{6} - 588 q^{7} - 492 q^{8} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 108 q^{3} + 153 q^{4} - 162 q^{5} - 9 q^{6} - 588 q^{7} - 492 q^{8} + 972 q^{9} + 528 q^{10} - 1425 q^{11} + 1377 q^{12} - 70 q^{13} + 49 q^{14} - 1458 q^{15} + 3865 q^{16} - 398 q^{17} - 81 q^{18} - 1293 q^{19} - 8593 q^{20} - 5292 q^{21} + 4961 q^{22} + 6348 q^{23} - 4428 q^{24} + 5830 q^{25} - 5187 q^{26} + 8748 q^{27} - 7497 q^{28} - 5127 q^{29} + 4752 q^{30} + 6498 q^{31} - 28485 q^{32} - 12825 q^{33} - 14527 q^{34} + 7938 q^{35} + 12393 q^{36} - 35545 q^{37} - 32617 q^{38} - 630 q^{39} + 35789 q^{40} - 7806 q^{41} + 441 q^{42} - 66142 q^{43} - 83253 q^{44} - 13122 q^{45} - 529 q^{46} - 16432 q^{47} + 34785 q^{48} + 28812 q^{49} - 177328 q^{50} - 3582 q^{51} - 187010 q^{52} - 67456 q^{53} - 729 q^{54} - 10453 q^{55} + 24108 q^{56} - 11637 q^{57} - 92677 q^{58} - 36346 q^{59} - 77337 q^{60} - 8768 q^{61} - 141813 q^{62} - 47628 q^{63} - 24604 q^{64} + 121875 q^{65} + 44649 q^{66} - 123617 q^{67} + 17217 q^{68} + 57132 q^{69} - 25872 q^{70} - 108667 q^{71} - 39852 q^{72} - 107406 q^{73} - 87825 q^{74} + 52470 q^{75} + 120191 q^{76} + 69825 q^{77} - 46683 q^{78} - 39470 q^{79} - 513682 q^{80} + 78732 q^{81} + 150219 q^{82} - 181838 q^{83} - 67473 q^{84} - 52633 q^{85} + 125713 q^{86} - 46143 q^{87} + 120642 q^{88} - 277361 q^{89} + 42768 q^{90} + 3430 q^{91} + 80937 q^{92} + 58482 q^{93} - 40880 q^{94} - 272491 q^{95} - 256365 q^{96} - 169005 q^{97} - 2401 q^{98} - 115425 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 268 x^{10} + 83 x^{9} + 25315 x^{8} + 5134 x^{7} - 993368 x^{6} - 511968 x^{5} + \cdots + 102912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 78\nu + 50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2031574625 \nu^{11} + 12856961043 \nu^{10} + 615454775966 \nu^{9} + \cdots + 42\!\cdots\!92 ) / 119354720039424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 999513821 \nu^{11} + 1620050505 \nu^{10} - 252615322334 \nu^{9} - 513869754637 \nu^{8} + \cdots + 95\!\cdots\!24 ) / 39784906679808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1842383909 \nu^{11} + 602103219 \nu^{10} - 484909014092 \nu^{9} - 446003791225 \nu^{8} + \cdots - 18\!\cdots\!84 ) / 59677360019712 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1377918191 \nu^{11} - 410055561 \nu^{10} - 365385332414 \nu^{9} - 119037349519 \nu^{8} + \cdots - 39\!\cdots\!92 ) / 39784906679808 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2862015409 \nu^{11} + 829916283 \nu^{10} + 764429966790 \nu^{9} + 280328312169 \nu^{8} + \cdots - 206739256755456 ) / 39784906679808 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10406218379 \nu^{11} - 12352539861 \nu^{10} - 2824835475374 \nu^{9} + \cdots - 17\!\cdots\!88 ) / 119354720039424 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15479062333 \nu^{11} + 23245884543 \nu^{10} + 4182351769054 \nu^{9} + \cdots - 41\!\cdots\!16 ) / 119354720039424 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7048452430 \nu^{11} - 5739172515 \nu^{10} - 1885723002493 \nu^{9} + 242863732396 \nu^{8} + \cdots + 80\!\cdots\!20 ) / 29838680009856 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 80\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - 4 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 7 \beta_{6} + 2 \beta_{5} + \cdots + 3605 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{11} + 16 \beta_{10} - 9 \beta_{9} - \beta_{8} + 8 \beta_{7} - 34 \beta_{6} + 10 \beta_{5} + \cdots + 7485 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 177 \beta_{11} + 196 \beta_{10} - 649 \beta_{9} - 613 \beta_{8} + 468 \beta_{7} - 1186 \beta_{6} + \cdots + 330746 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2786 \beta_{11} + 3437 \beta_{10} - 1997 \beta_{9} - 737 \beta_{8} + 2230 \beta_{7} - 8537 \beta_{6} + \cdots + 1112549 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 26212 \beta_{11} + 31346 \beta_{10} - 83930 \beta_{9} - 76118 \beta_{8} + 76056 \beta_{7} + \cdots + 32843607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 438852 \beta_{11} + 540068 \beta_{10} - 343224 \beta_{9} - 168968 \beta_{8} + 423760 \beta_{7} + \cdots + 150708552 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3670109 \beta_{11} + 4608497 \beta_{10} - 10156288 \beta_{9} - 8959040 \beta_{8} + 10721458 \beta_{7} + \cdots + 3426955569 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 61098725 \beta_{11} + 75134124 \beta_{10} - 52275269 \beta_{9} - 29163405 \beta_{8} + 68091568 \beta_{7} + \cdots + 19484246329 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
10.9835
9.45158
6.14710
6.11405
0.714833
0.238049
−0.0270357
−1.77899
−5.44023
−7.10224
−8.58736
−9.71327
−10.9835 9.00000 88.6377 −108.950 −98.8517 −49.0000 −622.081 81.0000 1196.65
1.2 −9.45158 9.00000 57.3324 43.5642 −85.0642 −49.0000 −239.431 81.0000 −411.751
1.3 −6.14710 9.00000 5.78687 64.1572 −55.3239 −49.0000 161.135 81.0000 −394.381
1.4 −6.11405 9.00000 5.38156 −80.8988 −55.0264 −49.0000 162.746 81.0000 494.619
1.5 −0.714833 9.00000 −31.4890 65.9172 −6.43350 −49.0000 45.3841 81.0000 −47.1198
1.6 −0.238049 9.00000 −31.9433 −89.8479 −2.14244 −49.0000 15.2217 81.0000 21.3882
1.7 0.0270357 9.00000 −31.9993 −35.6847 0.243322 −49.0000 −1.73027 81.0000 −0.964762
1.8 1.77899 9.00000 −28.8352 −4.92580 16.0109 −49.0000 −108.225 81.0000 −8.76294
1.9 5.44023 9.00000 −2.40387 46.9255 48.9621 −49.0000 −187.165 81.0000 255.285
1.10 7.10224 9.00000 18.4419 9.99558 63.9202 −49.0000 −96.2931 81.0000 70.9911
1.11 8.58736 9.00000 41.7428 −47.8265 77.2863 −49.0000 83.6650 81.0000 −410.704
1.12 9.71327 9.00000 62.3475 −24.4259 87.4194 −49.0000 294.774 81.0000 −237.255
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.6.a.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.6.a.b ✓ 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + T_{2}^{11} - 268 T_{2}^{10} - 83 T_{2}^{9} + 25315 T_{2}^{8} - 5134 T_{2}^{7} + \cdots + 102912 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(483))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 102912 \) Copy content Toggle raw display
$3$ \( (T - 9)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$7$ \( (T + 49)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 20\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( (T - 529)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 74\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 27\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 77\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 17\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 20\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 12\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 31\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 37\!\cdots\!80 \) Copy content Toggle raw display
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