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Properties

Label 605.6.a.p
Level $605$
Weight $6$
Character orbit 605.a
Self dual yes
Analytic conductor $97.032$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,6,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,1] 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: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 523 x^{18} + 521 x^{17} + 115018 x^{16} - 115347 x^{15} - 13821739 x^{14} + \cdots - 32708279373824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{8} \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 20) q^{4} - 25 q^{5} + ( - \beta_{9} + \beta_{4} - \beta_{3} + \cdots - 13) q^{6} + (\beta_{8} + \beta_{4} - \beta_1 - 8) q^{7} + (\beta_{6} + 2 \beta_{4} - \beta_{3} + \cdots - 4) q^{8}+ \cdots + (65 \beta_{19} - 107 \beta_{18} + \cdots + 20168) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{2} + 407 q^{4} - 500 q^{5} - 264 q^{6} - 167 q^{7} - 57 q^{8} + 1598 q^{9} - 25 q^{10} - 253 q^{12} - 769 q^{13} - 1045 q^{14} + 6963 q^{16} + 2989 q^{17} - 3775 q^{18} - 5828 q^{19} - 10175 q^{20}+ \cdots + 420757 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 523 x^{18} + 521 x^{17} + 115018 x^{16} - 115347 x^{15} - 13821739 x^{14} + \cdots - 32708279373824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!77 \nu^{19} + \cdots - 27\!\cdots\!12 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!37 \nu^{19} + \cdots + 66\!\cdots\!72 ) / 19\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 93\!\cdots\!31 \nu^{19} + \cdots - 11\!\cdots\!36 ) / 39\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!17 \nu^{19} + \cdots - 85\!\cdots\!28 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\!\cdots\!31 \nu^{19} + \cdots - 65\!\cdots\!08 ) / 98\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!33 \nu^{19} + \cdots - 28\!\cdots\!28 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 64\!\cdots\!77 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 71\!\cdots\!39 \nu^{19} + \cdots - 50\!\cdots\!68 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 94\!\cdots\!03 \nu^{19} + \cdots - 75\!\cdots\!16 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28\!\cdots\!35 \nu^{19} + \cdots + 86\!\cdots\!00 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!31 \nu^{19} + \cdots + 15\!\cdots\!36 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!85 \nu^{19} + \cdots - 30\!\cdots\!56 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 72\!\cdots\!51 \nu^{19} + \cdots - 66\!\cdots\!12 ) / 39\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 28\!\cdots\!07 \nu^{19} + \cdots - 42\!\cdots\!24 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 68\!\cdots\!59 \nu^{19} + \cdots + 94\!\cdots\!48 ) / 22\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 80\!\cdots\!03 \nu^{19} + \cdots + 11\!\cdots\!12 ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 28\!\cdots\!09 \nu^{19} + \cdots + 20\!\cdots\!92 ) / 49\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 2\beta_{4} - \beta_{3} + 83\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - 2 \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{11} + \beta_{10} + \cdots + 4310 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - 8 \beta_{18} + 2 \beta_{17} + 5 \beta_{16} - 5 \beta_{15} + 9 \beta_{14} + 3 \beta_{13} + \cdots - 157 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 134 \beta_{19} - 361 \beta_{18} + 170 \beta_{17} - 140 \beta_{16} - 147 \beta_{15} + 29 \beta_{14} + \cdots + 401873 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{19} - 1616 \beta_{18} + 536 \beta_{17} + 1130 \beta_{16} - 892 \beta_{15} + 1670 \beta_{14} + \cdots + 7452 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13892 \beta_{19} - 49832 \beta_{18} + 22680 \beta_{17} - 15070 \beta_{16} - 17058 \beta_{15} + \cdots + 39541608 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12481 \beta_{19} - 249102 \beta_{18} + 93275 \beta_{17} + 181555 \beta_{16} - 125929 \beta_{15} + \cdots + 5447998 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1337413 \beta_{19} - 6321964 \beta_{18} + 2797532 \beta_{17} - 1442413 \beta_{16} - 1864295 \beta_{15} + \cdots + 4011058321 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2692252 \beta_{19} - 34661771 \beta_{18} + 13649034 \beta_{17} + 25644280 \beta_{16} + \cdots + 1351708001 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 125800466 \beta_{19} - 774466126 \beta_{18} + 333510036 \beta_{17} - 125448632 \beta_{16} + \cdots + 415378775286 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 405101796 \beta_{19} - 4579359616 \beta_{18} + 1827374616 \beta_{17} + 3397046160 \beta_{16} + \cdots + 253404682644 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11781256913 \beta_{19} - 93275343778 \beta_{18} + 39078781481 \beta_{17} - 9572480593 \beta_{16} + \cdots + 43701921820630 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 52873075427 \beta_{19} - 587133723344 \beta_{18} + 232550480486 \beta_{17} + 433532994865 \beta_{16} + \cdots + 41223854994291 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1107156456910 \beta_{19} - 11130736807385 \beta_{18} + 4536158683702 \beta_{17} - 547555567636 \beta_{16} + \cdots + 46\!\cdots\!89 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 6397086175894 \beta_{19} - 73874705672452 \beta_{18} + 28704694377240 \beta_{17} + \cdots + 61\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 104749406552836 \beta_{19} + \cdots + 50\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 738711912072257 \beta_{19} + \cdots + 86\!\cdots\!62 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.4229
−10.1575
−9.60667
−9.34075
−6.92807
−5.73355
−5.37088
−4.73032
−1.05592
0.238351
1.08038
1.84684
3.12060
5.27089
5.77720
8.03981
8.98229
8.99127
10.1332
10.8658
−10.4229 −11.0208 76.6377 −25.0000 114.870 −4.64894 −465.256 −121.541 260.573
1.2 −10.1575 30.4037 71.1747 −25.0000 −308.825 10.6636 −397.917 681.384 253.937
1.3 −9.60667 21.6000 60.2880 −25.0000 −207.504 −51.1649 −271.754 223.560 240.167
1.4 −9.34075 −29.7904 55.2496 −25.0000 278.265 104.862 −217.169 644.469 233.519
1.5 −6.92807 −8.76999 15.9982 −25.0000 60.7591 −97.5351 110.862 −166.087 173.202
1.6 −5.73355 3.90196 0.873636 −25.0000 −22.3721 −214.940 178.465 −227.775 143.339
1.7 −5.37088 −0.268292 −3.15366 −25.0000 1.44096 221.735 188.806 −242.928 134.272
1.8 −4.73032 8.08839 −9.62406 −25.0000 −38.2607 159.979 196.895 −177.578 118.258
1.9 −1.05592 22.6862 −30.8850 −25.0000 −23.9549 12.2465 66.4016 271.665 26.3980
1.10 0.238351 −19.5755 −31.9432 −25.0000 −4.66584 −208.743 −15.2409 140.200 −5.95878
1.11 1.08038 9.44854 −30.8328 −25.0000 10.2080 −16.3713 −67.8833 −153.725 −27.0095
1.12 1.84684 −21.5804 −28.5892 −25.0000 −39.8555 −85.4988 −111.898 222.716 −46.1709
1.13 3.12060 −6.47958 −22.2618 −25.0000 −20.2202 −46.8054 −169.330 −201.015 −78.0151
1.14 5.27089 −17.5243 −4.21770 −25.0000 −92.3688 132.134 −190.900 64.1015 −131.772
1.15 5.77720 22.4828 1.37599 −25.0000 129.888 −56.5871 −176.921 262.477 −144.430
1.16 8.03981 5.09012 32.6385 −25.0000 40.9236 87.3874 5.13363 −217.091 −200.995
1.17 8.98229 23.5092 48.6815 −25.0000 211.166 −232.955 149.838 309.680 −224.557
1.18 8.99127 10.3783 48.8429 −25.0000 93.3145 132.461 151.440 −135.290 −224.782
1.19 10.1332 −21.6453 70.6821 −25.0000 −219.336 136.893 391.974 225.517 −253.330
1.20 10.8658 −20.9346 86.0646 −25.0000 −227.471 −150.113 587.453 195.259 −271.644
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(11\) \( +1 \)

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 605.6.a.p 20
11.b odd 2 1 605.6.a.o 20
11.c even 5 2 55.6.g.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.g.b ✓ 40 11.c even 5 2
605.6.a.o 20 11.b odd 2 1
605.6.a.p 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - T_{2}^{19} - 523 T_{2}^{18} + 521 T_{2}^{17} + 115018 T_{2}^{16} - 115347 T_{2}^{15} + \cdots - 32708279373824 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots - 32708279373824 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( (T + 25)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 17\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 49\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 35\!\cdots\!59 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 39\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 32\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 27\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 23\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
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