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Properties

Label 441.7.d.c
Level $441$
Weight $7$
Character orbit 441.d
Analytic conductor $101.454$
Analytic rank $0$
Dimension $8$
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] = [441,7,Mod(244,441)]
 
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, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.244");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(101.453850876\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} + ( - \beta_{4} - 4 \beta_{3} + 42) q^{4} + (\beta_{5} - 3 \beta_{2} + 25 \beta_1) q^{5} + ( - 4 \beta_{7} + 5 \beta_{4} + \cdots - 404) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{2} + ( - \beta_{4} - 4 \beta_{3} + 42) q^{4} + (\beta_{5} - 3 \beta_{2} + 25 \beta_1) q^{5} + ( - 4 \beta_{7} + 5 \beta_{4} + \cdots - 404) q^{8}+ \cdots + (4 \beta_{6} - 4027 \beta_{5} + \cdots - 114285 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8} - 628 q^{11} + 25442 q^{16} + 86106 q^{22} + 7856 q^{23} + 34076 q^{25} + 8300 q^{29} - 372414 q^{32} - 129412 q^{37} + 45740 q^{43} + 185058 q^{44} + 223008 q^{46} - 967216 q^{50} - 1081948 q^{53} - 1079598 q^{58} + 2378626 q^{64} - 828408 q^{65} + 2317804 q^{67} - 1442344 q^{71} - 865880 q^{74} - 1222904 q^{79} - 275112 q^{85} + 1632448 q^{86} + 732882 q^{88} - 678720 q^{92} - 1183584 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 405518177 \nu^{7} + 12595814413 \nu^{6} + 65484675874 \nu^{5} + 2673339327091 \nu^{4} + \cdots + 67\!\cdots\!35 ) / 57\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + \cdots - 5583985297290 ) / 7542500798721 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + \cdots - 5583985297290 ) / 7542500798721 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7295536 \nu^{7} - 50933099 \nu^{6} - 1392479248 \nu^{5} - 2649247696 \nu^{4} + \cdots + 208034068231995 ) / 2514166932907 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 65881610383 \nu^{7} - 648926656693 \nu^{6} + 19199653600241 \nu^{5} - 98352920361781 \nu^{4} + \cdots - 27\!\cdots\!60 ) / 76\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 135000035243 \nu^{7} - 904004937787 \nu^{6} - 30723933002041 \nu^{5} + \cdots - 47\!\cdots\!40 ) / 76\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1392321691 \nu^{7} - 2568340439 \nu^{6} + 265748679913 \nu^{5} + 505597536901 \nu^{4} + \cdots - 13\!\cdots\!76 ) / 30170003194884 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} + 105\beta _1 - 105 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 2\beta_{4} + 169\beta_{3} - 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} - 217\beta_{6} - 205\beta_{5} + 213\beta_{4} + 722\beta_{3} - 722\beta_{2} - 17781\beta _1 - 17781 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 848 \beta_{7} - 1614 \beta_{6} + 930 \beta_{5} - 766 \beta_{4} - 32053 \beta_{3} - 32053 \beta_{2} + \cdots + 77112 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2216\beta_{7} - 41381\beta_{4} - 197354\beta_{3} + 3375249 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 163308 \beta_{7} + 385038 \beta_{6} - 104886 \beta_{5} - 221730 \beta_{4} - 6279409 \beta_{3} + \cdots + 20983752 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
7.29767 + 12.6399i
7.29767 − 12.6399i
2.26350 − 3.92050i
2.26350 + 3.92050i
−2.75320 − 4.76869i
−2.75320 + 4.76869i
−6.30797 + 10.9257i
−6.30797 − 10.9257i
−15.5953 0 179.215 − 25.8331i 0 0 −1796.81 0 402.876i
244.2 −15.5953 0 179.215 25.8331i 0 0 −1796.81 0 − 402.876i
244.3 −5.52700 0 −33.4522 − 66.9661i 0 0 538.619 0 370.122i
244.4 −5.52700 0 −33.4522 66.9661i 0 0 538.619 0 − 370.122i
244.5 4.50641 0 −43.6923 − 62.2665i 0 0 −485.305 0 − 280.598i
244.6 4.50641 0 −43.6923 62.2665i 0 0 −485.305 0 280.598i
244.7 11.6159 0 70.9299 − 190.875i 0 0 80.4975 0 − 2217.19i
244.8 11.6159 0 70.9299 190.875i 0 0 80.4975 0 2217.19i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.8
Significant digits:
Format:

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 441.7.d.c 8
3.b odd 2 1 147.7.d.b 8
7.b odd 2 1 inner 441.7.d.c 8
7.c even 3 1 63.7.m.d 8
7.d odd 6 1 63.7.m.d 8
21.c even 2 1 147.7.d.b 8
21.g even 6 1 21.7.f.a ✓ 8
21.g even 6 1 147.7.f.d 8
21.h odd 6 1 21.7.f.a ✓ 8
21.h odd 6 1 147.7.f.d 8
84.j odd 6 1 336.7.bh.d 8
84.n even 6 1 336.7.bh.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a ✓ 8 21.g even 6 1
21.7.f.a ✓ 8 21.h odd 6 1
63.7.m.d 8 7.c even 3 1
63.7.m.d 8 7.d odd 6 1
147.7.d.b 8 3.b odd 2 1
147.7.d.b 8 21.c even 2 1
147.7.f.d 8 21.g even 6 1
147.7.f.d 8 21.h odd 6 1
336.7.bh.d 8 84.j odd 6 1
336.7.bh.d 8 84.n even 6 1
441.7.d.c 8 1.a even 1 1 trivial
441.7.d.c 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5T_{2}^{3} - 202T_{2}^{2} - 284T_{2} + 4512 \) acting on \(S_{7}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 5 T^{3} + \cdots + 4512)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 422734160250000 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 314 T^{3} + \cdots - 43002975612)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 3970225373184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 48\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 85\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 17\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 19\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 77\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 99\!\cdots\!51)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
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