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Properties

Label 441.8.a.z
Level $441$
Weight $8$
Character orbit 441.a
Self dual yes
Analytic conductor $137.762$
Analytic rank $1$
Dimension $6$
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] = [441,8,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(137.761796238\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 553x^{4} + 1386x^{3} + 69169x^{2} - 44864x - 1106944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 147)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + (\beta_{3} + 7 \beta_{2} - 2 \beta_1 + 72) q^{4} + ( - \beta_{5} + 11 \beta_{2} + \cdots + 86) q^{5}+ \cdots + ( - \beta_{5} + 10 \beta_{4} + \cdots + 184) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + (\beta_{3} + 7 \beta_{2} - 2 \beta_1 + 72) q^{4} + ( - \beta_{5} + 11 \beta_{2} + \cdots + 86) q^{5}+ \cdots + ( - 28618 \beta_{5} - 45360 \beta_{4} + \cdots - 2334604) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 14 q^{2} + 438 q^{4} + 500 q^{5} + 1218 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 14 q^{2} + 438 q^{4} + 500 q^{5} + 1218 q^{8} - 6912 q^{10} + 1204 q^{11} - 17576 q^{13} + 15522 q^{16} + 60836 q^{17} + 17280 q^{19} + 43924 q^{20} - 244452 q^{22} - 47708 q^{23} - 126954 q^{25} + 225228 q^{26} + 85880 q^{29} - 344872 q^{31} + 399678 q^{32} + 216304 q^{34} - 535896 q^{37} - 1154872 q^{38} - 817048 q^{40} - 696316 q^{41} - 162720 q^{43} - 1504868 q^{44} + 216540 q^{46} + 2280672 q^{47} - 3654866 q^{50} - 5978752 q^{52} - 5321348 q^{53} - 1716328 q^{55} + 2346000 q^{58} + 2702888 q^{59} - 7891216 q^{61} + 8107760 q^{62} - 4370970 q^{64} - 3217860 q^{65} + 565920 q^{67} + 21007124 q^{68} + 4370788 q^{71} - 11994056 q^{73} + 278200 q^{74} - 3409936 q^{76} - 16816944 q^{79} + 14397828 q^{80} - 29067128 q^{82} + 2431304 q^{83} - 2530620 q^{85} - 16351456 q^{86} - 23615724 q^{88} + 26982380 q^{89} - 22208116 q^{92} + 1880792 q^{94} + 11963336 q^{95} - 14260040 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 553x^{4} + 1386x^{3} + 69169x^{2} - 44864x - 1106944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5981\nu^{5} + 886982\nu^{4} + 762267\nu^{3} - 390236478\nu^{2} - 633536371\nu + 24236526464 ) / 994432992 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5887\nu^{5} - 69426\nu^{4} + 2297911\nu^{3} + 17820810\nu^{2} - 155678495\nu - 300083616 ) / 47353952 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -91481\nu^{5} + 554050\nu^{4} + 41902113\nu^{3} - 325274010\nu^{2} - 1871638457\nu + 16918711744 ) / 142061856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -40655\nu^{5} - 1397952\nu^{4} + 2383671\nu^{3} + 406696140\nu^{2} + 2150909857\nu - 6575704470 ) / 62152062 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 452829 \nu^{5} + 1258022 \nu^{4} - 192240037 \nu^{3} - 173079358 \nu^{2} + 15270363117 \nu + 24745049216 ) / 331477664 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{3} - 13\beta _1 + 4 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -15\beta_{5} - 2\beta_{3} - 164\beta_{2} - 197\beta _1 + 5120 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 218\beta_{5} - 87\beta_{4} + 247\beta_{3} + 1489\beta_{2} - 1754\beta _1 - 2710 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7521\beta_{5} + 660\beta_{4} - 846\beta_{3} - 84396\beta_{2} - 54987\beta _1 + 1537364 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 187031\beta_{5} - 75702\beta_{4} + 143860\beta_{3} + 1436034\beta_{2} - 973403\beta _1 - 6279976 ) / 28 \) 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
−4.11250
4.50858
−20.3630
−10.8782
16.8544
15.9907
−18.5501 0 216.105 422.562 0 0 −1634.36 0 −7838.55
1.2 −11.9730 0 15.3529 −147.782 0 0 1348.72 0 1769.40
1.3 1.54500 0 −125.613 293.726 0 0 −391.831 0 453.805
1.4 6.54757 0 −85.1293 59.7282 0 0 −1395.48 0 391.075
1.5 16.0138 0 128.442 −211.313 0 0 7.07408 0 −3383.93
1.6 20.4167 0 288.842 83.0796 0 0 3283.87 0 1696.21
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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 441.8.a.z 6
3.b odd 2 1 147.8.a.m yes 6
7.b odd 2 1 441.8.a.y 6
21.c even 2 1 147.8.a.l ✓ 6
21.g even 6 2 147.8.e.p 12
21.h odd 6 2 147.8.e.o 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.8.a.l ✓ 6 21.c even 2 1
147.8.a.m yes 6 3.b odd 2 1
147.8.e.o 12 21.h odd 6 2
147.8.e.p 12 21.g even 6 2
441.8.a.y 6 7.b odd 2 1
441.8.a.z 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{6} - 14T_{2}^{5} - 505T_{2}^{4} + 6384T_{2}^{3} + 51640T_{2}^{2} - 568544T_{2} + 734576 \) Copy content Toggle raw display
\( T_{5}^{6} - 500T_{5}^{5} - 45898T_{5}^{4} + 34976880T_{5}^{3} + 199931500T_{5}^{2} - 443351426000T_{5} + 19233338645000 \) Copy content Toggle raw display
\( T_{13}^{6} + 17576 T_{13}^{5} - 30551302 T_{13}^{4} - 1501500447264 T_{13}^{3} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 14 T^{5} + \cdots + 734576 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 19233338645000 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 89\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 99\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 95\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 28\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 83\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 58\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 79\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 60\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 58\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 87\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 39\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 17\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 25\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 16\!\cdots\!04 \) Copy content Toggle raw display
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