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Properties

Label 70.5.f.a
Level $70$
Weight $5$
Character orbit 70.f
Analytic conductor $7.236$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [70,5,Mod(43,70)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(70, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 0])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("70.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 70.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.23589741587\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
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: \( x^{12} + 334x^{10} + 34233x^{8} + 1144512x^{6} + 13607616x^{4} + 38549504x^{2} + 31360000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5^{2}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_1 - 2) q^{2} + ( - \beta_{2} + 2 \beta_1 - 2) q^{3} + 8 \beta_1 q^{4} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{5} + (2 \beta_{3} + 2 \beta_{2} + 8) q^{6} - \beta_{8} q^{7}+ \cdots + (28 \beta_{11} + 54 \beta_{10} + \cdots + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{2} - 20 q^{3} + 8 q^{5} + 80 q^{6} + 192 q^{8} - 144 q^{10} + 4 q^{11} - 160 q^{12} - 180 q^{13} - 736 q^{15} - 768 q^{16} - 236 q^{17} - 464 q^{18} + 512 q^{20} - 196 q^{21} - 8 q^{22} - 1232 q^{23}+ \cdots + 8232 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 334x^{10} + 34233x^{8} + 1144512x^{6} + 13607616x^{4} + 38549504x^{2} + 31360000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 506 \nu^{11} + 168829 \nu^{9} + 17247768 \nu^{7} + 568247977 \nu^{5} + 6262682176 \nu^{3} + 9537109344 \nu ) / 1054636800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7123919 \nu^{11} + 22490860 \nu^{10} + 2359805886 \nu^{9} + 7425952800 \nu^{8} + \cdots + 343603226220800 ) / 3622526745600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7123919 \nu^{11} + 22490860 \nu^{10} - 2359805886 \nu^{9} + 7425952800 \nu^{8} + \cdots + 343603226220800 ) / 3622526745600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1623718639 \nu^{11} - 4777584770 \nu^{10} + 542918989626 \nu^{9} - 1596056740460 \nu^{8} + \cdots - 10\!\cdots\!00 ) / 659299867699200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 429010689 \nu^{11} - 1111871642 \nu^{10} - 143296368358 \nu^{9} - 382519037628 \nu^{8} + \cdots - 28\!\cdots\!00 ) / 131859973539840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 464942395 \nu^{11} + 1090432378 \nu^{10} - 152658933362 \nu^{9} + 355264742924 \nu^{8} + \cdots + 12\!\cdots\!00 ) / 131859973539840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 561098423 \nu^{11} + 580141450 \nu^{10} + 184985811842 \nu^{9} + 192879226540 \nu^{8} + \cdots + 10\!\cdots\!00 ) / 94185695385600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 89885143 \nu^{11} + 25819140 \nu^{10} + 29868150442 \nu^{9} + 8023930200 \nu^{8} + \cdots + 210903248102400 ) / 13455099340800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 89885143 \nu^{11} - 25819140 \nu^{10} + 29868150442 \nu^{9} - 8023930200 \nu^{8} + \cdots - 210903248102400 ) / 13455099340800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 567998817 \nu^{11} - 343022785 \nu^{10} - 189415331313 \nu^{9} - 123873682660 \nu^{8} + \cdots - 16\!\cdots\!00 ) / 82412483462400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5424465777 \nu^{11} - 2014321890 \nu^{10} + 1796640189058 \nu^{9} - 674392880980 \nu^{8} + \cdots - 44\!\cdots\!00 ) / 329649933849600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5 \beta_{11} + \beta_{10} - 7 \beta_{9} + 3 \beta_{8} - 5 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} + \cdots - 5 ) / 35 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 8 \beta_{11} - 16 \beta_{10} - 79 \beta_{9} + 33 \beta_{8} + 16 \beta_{7} - 23 \beta_{6} + \cdots - 1966 ) / 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 589 \beta_{11} - 199 \beta_{10} + 575 \beta_{9} - 799 \beta_{8} + 883 \beta_{7} + 398 \beta_{6} + \cdots + 687 ) / 35 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 974 \beta_{11} + 2998 \beta_{10} + 12917 \beta_{9} - 3379 \beta_{8} - 1948 \beta_{7} + 4769 \beta_{6} + \cdots + 253238 ) / 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 85941 \beta_{11} + 27103 \beta_{10} - 65389 \beta_{9} + 143677 \beta_{8} - 141717 \beta_{7} + \cdots - 104533 ) / 35 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 86624 \beta_{11} - 590448 \beta_{10} - 2117547 \beta_{9} + 333629 \beta_{8} + 173248 \beta_{7} + \cdots - 37426278 ) / 35 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1894239 \beta_{11} - 522621 \beta_{10} + 1187529 \beta_{9} - 3534505 \beta_{8} + 3294573 \beta_{7} + \cdots + 2361017 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4681058 \beta_{11} + 110593946 \beta_{10} + 349703789 \beta_{9} - 29396443 \beta_{8} + \cdots + 5762983726 ) / 35 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2095795537 \beta_{11} + 505607475 \beta_{10} - 1103962541 \beta_{9} + 4215370141 \beta_{8} + \cdots - 2659666341 ) / 35 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 79158964 \beta_{11} - 3988560332 \beta_{10} - 11605685127 \beta_{9} + 364455713 \beta_{8} + \cdots - 181565664374 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 336612779037 \beta_{11} - 71990854455 \beta_{10} + 151439137391 \beta_{9} - 714643009391 \beta_{8} + \cdots + 433041073391 ) / 35 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
− 10.8624i
4.60707i
− 1.27116i
− 4.83398i
12.9547i
1.40575i
10.8624i
− 4.60707i
1.27116i
4.83398i
− 12.9547i
− 1.40575i
−2.00000 + 2.00000i −10.1564 − 10.1564i − 8.00000i 2.02135 − 24.9181i 40.6256 13.0958 − 13.0958i 16.0000 + 16.0000i 125.305i 45.7936 + 53.8790i
43.2 −2.00000 + 2.00000i −7.02600 − 7.02600i − 8.00000i −24.9839 + 0.897502i 28.1040 −13.0958 + 13.0958i 16.0000 + 16.0000i 17.7293i 48.1728 − 51.7628i
43.3 −2.00000 + 2.00000i −6.14353 − 6.14353i − 8.00000i 22.5891 + 10.7113i 24.5741 −13.0958 + 13.0958i 16.0000 + 16.0000i − 5.51396i −66.6008 + 23.7557i
43.4 −2.00000 + 2.00000i 0.137195 + 0.137195i − 8.00000i −19.0601 + 16.1776i −0.548779 13.0958 − 13.0958i 16.0000 + 16.0000i − 80.9624i 5.76501 − 70.4753i
43.5 −2.00000 + 2.00000i 3.14837 + 3.14837i − 8.00000i 24.6512 + 4.16147i −12.5935 13.0958 − 13.0958i 16.0000 + 16.0000i − 61.1755i −57.6254 + 40.9795i
43.6 −2.00000 + 2.00000i 10.0404 + 10.0404i − 8.00000i −1.21772 + 24.9703i −40.1615 −13.0958 + 13.0958i 16.0000 + 16.0000i 120.618i −47.5052 − 52.3761i
57.1 −2.00000 − 2.00000i −10.1564 + 10.1564i 8.00000i 2.02135 + 24.9181i 40.6256 13.0958 + 13.0958i 16.0000 − 16.0000i − 125.305i 45.7936 − 53.8790i
57.2 −2.00000 − 2.00000i −7.02600 + 7.02600i 8.00000i −24.9839 − 0.897502i 28.1040 −13.0958 − 13.0958i 16.0000 − 16.0000i − 17.7293i 48.1728 + 51.7628i
57.3 −2.00000 − 2.00000i −6.14353 + 6.14353i 8.00000i 22.5891 − 10.7113i 24.5741 −13.0958 − 13.0958i 16.0000 − 16.0000i 5.51396i −66.6008 − 23.7557i
57.4 −2.00000 − 2.00000i 0.137195 − 0.137195i 8.00000i −19.0601 − 16.1776i −0.548779 13.0958 + 13.0958i 16.0000 − 16.0000i 80.9624i 5.76501 + 70.4753i
57.5 −2.00000 − 2.00000i 3.14837 − 3.14837i 8.00000i 24.6512 − 4.16147i −12.5935 13.0958 + 13.0958i 16.0000 − 16.0000i 61.1755i −57.6254 − 40.9795i
57.6 −2.00000 − 2.00000i 10.0404 − 10.0404i 8.00000i −1.21772 − 24.9703i −40.1615 −13.0958 − 13.0958i 16.0000 − 16.0000i − 120.618i −47.5052 + 52.3761i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.5.f.a ✓ 12
5.b even 2 1 350.5.f.d 12
5.c odd 4 1 inner 70.5.f.a ✓ 12
5.c odd 4 1 350.5.f.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.5.f.a ✓ 12 1.a even 1 1 trivial
70.5.f.a ✓ 12 5.c odd 4 1 inner
350.5.f.d 12 5.b even 2 1
350.5.f.d 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 20 T_{3}^{11} + 200 T_{3}^{10} + 556 T_{3}^{9} + 40645 T_{3}^{8} + 811144 T_{3}^{7} + \cdots + 231344100 \) acting on \(S_{5}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 8)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 231344100 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + 117649)^{3} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 1780908725200)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 72\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 38\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 69\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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