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Properties

Label 35.8.e.a
Level $35$
Weight $8$
Character orbit 35.e
Analytic conductor $10.933$
Analytic rank $0$
Dimension $16$
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] = [35,8,Mod(11,35)] 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("35.11"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 35.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9334758919\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 586 x^{14} + 203 x^{13} + 238710 x^{12} + 138005 x^{11} + 49781891 x^{10} + \cdots + 1458259117056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{5} - 4 \beta_{4}) q^{3} + (\beta_{10} + 18 \beta_{4} + \cdots + \beta_{2}) q^{4} + ( - 125 \beta_{4} - 125) q^{5} + (\beta_{15} + \beta_{12} + \beta_{11} + \cdots - 44) q^{6}+ \cdots + ( - 27775 \beta_{15} - 6663 \beta_{13} + \cdots - 10071522) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} + 34 q^{3} - 147 q^{4} - 1000 q^{5} - 672 q^{6} - 1106 q^{7} - 1854 q^{8} - 6762 q^{9} + 125 q^{10} + 6100 q^{11} + 9152 q^{12} + 7716 q^{13} - 37555 q^{14} - 8500 q^{15} + 47389 q^{16} + 16832 q^{17}+ \cdots - 162635968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 586 x^{14} + 203 x^{13} + 238710 x^{12} + 138005 x^{11} + 49781891 x^{10} + \cdots + 1458259117056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69\!\cdots\!13 \nu^{15} + \cdots - 75\!\cdots\!00 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 61\!\cdots\!21 \nu^{15} + \cdots + 29\!\cdots\!24 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 93\!\cdots\!00 \nu^{15} + \cdots - 10\!\cdots\!52 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 97\!\cdots\!99 \nu^{15} + \cdots + 47\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83\!\cdots\!50 \nu^{15} + \cdots - 74\!\cdots\!08 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 97\!\cdots\!77 \nu^{15} + \cdots + 43\!\cdots\!20 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!15 \nu^{15} + \cdots + 10\!\cdots\!04 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!64 \nu^{15} + \cdots + 22\!\cdots\!16 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!25 \nu^{15} + \cdots - 85\!\cdots\!28 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!79 \nu^{15} + \cdots - 19\!\cdots\!12 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!29 \nu^{15} + \cdots - 19\!\cdots\!56 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25\!\cdots\!85 \nu^{15} + \cdots + 55\!\cdots\!52 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 88\!\cdots\!02 \nu^{15} + \cdots - 78\!\cdots\!92 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!11 \nu^{15} + \cdots - 11\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 146\beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{12} + \beta_{11} + \beta_{8} - \beta_{7} + 10\beta_{5} + 2\beta_{3} + 230\beta_{2} - 229\beta _1 - 123 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 22 \beta_{15} + 10 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 24 \beta_{11} - 317 \beta_{10} + \cdots - 33808 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 527 \beta_{15} - 393 \beta_{14} + 527 \beta_{13} - 10 \beta_{12} + 10 \beta_{11} + \cdots - 58993 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1572 \beta_{15} + 9272 \beta_{13} - 10844 \beta_{12} - 1572 \beta_{11} + 4852 \beta_{8} + \cdots + 8759346 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6744 \beta_{15} + 129185 \beta_{14} - 195873 \beta_{13} - 195873 \beta_{12} - 202617 \beta_{11} + \cdots + 18878051 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3914428 \beta_{15} - 1789566 \beta_{14} - 3914428 \beta_{13} + 3147422 \beta_{12} + \cdots + 48133617 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 66834265 \beta_{15} - 3353434 \beta_{13} + 70187699 \beta_{12} + 66834265 \beta_{11} + 39859677 \beta_{8} + \cdots - 7469162163 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1001936096 \beta_{15} + 590926896 \beta_{14} + 313238456 \beta_{13} + 313238456 \beta_{12} + \cdots - 696643808438 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 23261255925 \beta_{15} - 11939688829 \beta_{14} + 23261255925 \beta_{13} + \cdots - 1372650354245 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 117682528666 \beta_{15} + 311222428134 \beta_{13} - 428904956800 \beta_{12} - 117682528666 \beta_{11} + \cdots + 205381567277272 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 651887241962 \beta_{15} + 3519182595969 \beta_{14} - 6897549969149 \beta_{13} - 6897549969149 \beta_{12} + \cdots + 10\!\cdots\!11 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 137937251396916 \beta_{15} - 55600033180396 \beta_{14} - 137937251396916 \beta_{13} + \cdots + 23\!\cdots\!49 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 21\!\cdots\!53 \beta_{15} - 263944054504312 \beta_{13} + \cdots - 36\!\cdots\!95 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−8.45260 − 14.6403i
−6.95790 − 12.0514i
−4.27009 − 7.39600i
−0.931966 − 1.61421i
0.0619863 + 0.107363i
5.46541 + 9.46637i
6.68219 + 11.5739i
8.90296 + 15.4204i
−8.45260 + 14.6403i
−6.95790 + 12.0514i
−4.27009 + 7.39600i
−0.931966 + 1.61421i
0.0619863 − 0.107363i
5.46541 − 9.46637i
6.68219 − 11.5739i
8.90296 − 15.4204i
−8.45260 − 14.6403i 5.52621 − 9.57167i −78.8929 + 136.647i −62.5000 − 108.253i −186.843 52.7706 + 905.957i 503.535 1032.42 + 1788.21i −1056.58 + 1830.04i
11.2 −6.95790 − 12.0514i 36.7309 − 63.6198i −32.8247 + 56.8541i −62.5000 − 108.253i −1022.28 501.799 − 756.135i −867.657 −1604.82 − 2779.63i −869.738 + 1506.43i
11.3 −4.27009 − 7.39600i −26.1566 + 45.3046i 27.5327 − 47.6881i −62.5000 − 108.253i 446.764 804.156 − 420.567i −1563.41 −274.837 − 476.032i −533.761 + 924.501i
11.4 −0.931966 − 1.61421i −34.7675 + 60.2191i 62.2629 − 107.842i −62.5000 − 108.253i 129.609 −642.672 + 640.715i −470.691 −1324.06 − 2293.33i −116.496 + 201.777i
11.5 0.0619863 + 0.107363i 16.9602 − 29.3760i 63.9923 − 110.838i −62.5000 − 108.253i 4.20521 −624.098 − 658.820i 31.7351 518.200 + 897.549i 7.74829 − 13.4204i
11.6 5.46541 + 9.46637i −1.18361 + 2.05007i 4.25854 − 7.37601i −62.5000 − 108.253i −25.8757 823.721 + 380.825i 1492.24 1090.70 + 1889.14i 683.177 − 1183.30i
11.7 6.68219 + 11.5739i 43.8936 − 76.0259i −25.3033 + 43.8267i −62.5000 − 108.253i 1173.22 −609.320 + 672.511i 1034.31 −2759.79 − 4780.10i 835.274 − 1446.74i
11.8 8.90296 + 15.4204i −24.0032 + 41.5748i −94.5255 + 163.723i −62.5000 − 108.253i −854.800 −859.355 − 291.636i −1087.07 −58.8113 − 101.864i 1112.87 − 1927.55i
16.1 −8.45260 + 14.6403i 5.52621 + 9.57167i −78.8929 − 136.647i −62.5000 + 108.253i −186.843 52.7706 − 905.957i 503.535 1032.42 − 1788.21i −1056.58 − 1830.04i
16.2 −6.95790 + 12.0514i 36.7309 + 63.6198i −32.8247 − 56.8541i −62.5000 + 108.253i −1022.28 501.799 + 756.135i −867.657 −1604.82 + 2779.63i −869.738 − 1506.43i
16.3 −4.27009 + 7.39600i −26.1566 − 45.3046i 27.5327 + 47.6881i −62.5000 + 108.253i 446.764 804.156 + 420.567i −1563.41 −274.837 + 476.032i −533.761 − 924.501i
16.4 −0.931966 + 1.61421i −34.7675 − 60.2191i 62.2629 + 107.842i −62.5000 + 108.253i 129.609 −642.672 − 640.715i −470.691 −1324.06 + 2293.33i −116.496 − 201.777i
16.5 0.0619863 − 0.107363i 16.9602 + 29.3760i 63.9923 + 110.838i −62.5000 + 108.253i 4.20521 −624.098 + 658.820i 31.7351 518.200 − 897.549i 7.74829 + 13.4204i
16.6 5.46541 − 9.46637i −1.18361 − 2.05007i 4.25854 + 7.37601i −62.5000 + 108.253i −25.8757 823.721 − 380.825i 1492.24 1090.70 − 1889.14i 683.177 + 1183.30i
16.7 6.68219 − 11.5739i 43.8936 + 76.0259i −25.3033 − 43.8267i −62.5000 + 108.253i 1173.22 −609.320 − 672.511i 1034.31 −2759.79 + 4780.10i 835.274 + 1446.74i
16.8 8.90296 − 15.4204i −24.0032 − 41.5748i −94.5255 − 163.723i −62.5000 + 108.253i −854.800 −859.355 + 291.636i −1087.07 −58.8113 + 101.864i 1112.87 + 1927.55i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.8.e.a ✓ 16
7.c even 3 1 inner 35.8.e.a ✓ 16
7.c even 3 1 245.8.a.i 8
7.d odd 6 1 245.8.a.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.e.a ✓ 16 1.a even 1 1 trivial
35.8.e.a ✓ 16 7.c even 3 1 inner
245.8.a.i 8 7.c even 3 1
245.8.a.j 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - T_{2}^{15} + 586 T_{2}^{14} + 203 T_{2}^{13} + 238710 T_{2}^{12} + 138005 T_{2}^{11} + \cdots + 1458259117056 \) acting on \(S_{8}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 1458259117056 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 99\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( (T^{2} + 125 T + 15625)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 32\!\cdots\!25)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 36\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 85\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 59\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 72\!\cdots\!76)^{2} \) Copy content Toggle raw display
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