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Properties

Label 605.6.a.n
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,-17] 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: \( x^{20} - 3 x^{19} - 436 x^{18} + 1032 x^{17} + 79722 x^{16} - 137625 x^{15} - 7955280 x^{14} + \cdots + 80139762021616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5\cdot 11^{8} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{2} - \beta_1 + 13) q^{4} + 25 q^{5} + ( - \beta_{8} - \beta_{4} + \beta_{3} + \cdots - 14) q^{6} + (\beta_{12} - \beta_{2} + \beta_1 - 28) q^{7}+ \cdots + ( - 108 \beta_{19} + 19 \beta_{18} + \cdots - 11611) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 17 q^{2} + 22 q^{3} + 255 q^{4} + 500 q^{5} - 256 q^{6} - 559 q^{7} - 711 q^{8} + 1598 q^{9} - 425 q^{10} + 1309 q^{12} - 2323 q^{13} + 1471 q^{14} + 550 q^{15} + 1131 q^{16} - 4947 q^{17} - 5553 q^{18}+ \cdots - 209377 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 436 x^{18} + 1032 x^{17} + 79722 x^{16} - 137625 x^{15} - 7955280 x^{14} + \cdots + 80139762021616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33\!\cdots\!91 \nu^{19} + \cdots - 91\!\cdots\!16 ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!97 \nu^{19} + \cdots - 22\!\cdots\!12 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\!\cdots\!99 \nu^{19} + \cdots - 26\!\cdots\!96 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 68\!\cdots\!61 \nu^{19} + \cdots - 60\!\cdots\!56 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38\!\cdots\!37 \nu^{19} + \cdots + 88\!\cdots\!32 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!13 \nu^{19} + \cdots + 27\!\cdots\!88 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22\!\cdots\!36 \nu^{19} + \cdots - 12\!\cdots\!76 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!93 \nu^{19} + \cdots + 46\!\cdots\!88 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!17 \nu^{19} + \cdots - 22\!\cdots\!32 ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21\!\cdots\!51 \nu^{19} + \cdots + 51\!\cdots\!36 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!79 \nu^{19} + \cdots - 46\!\cdots\!36 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26\!\cdots\!19 \nu^{19} + \cdots + 61\!\cdots\!04 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 28\!\cdots\!81 \nu^{19} + \cdots - 69\!\cdots\!76 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 22\!\cdots\!79 \nu^{19} + \cdots + 36\!\cdots\!64 ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 53\!\cdots\!47 \nu^{19} + \cdots + 12\!\cdots\!52 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 19\!\cdots\!89 \nu^{19} + \cdots - 51\!\cdots\!24 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 14\!\cdots\!63 \nu^{19} + \cdots + 30\!\cdots\!08 ) / 30\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{10} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} + 73\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + 4 \beta_{7} + \cdots + 3235 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{19} + 6 \beta_{18} + 2 \beta_{16} - 9 \beta_{15} + 7 \beta_{14} + 3 \beta_{13} - 126 \beta_{12} + \cdots + 4456 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 61 \beta_{19} + 52 \beta_{18} + 16 \beta_{17} + 40 \beta_{16} + 99 \beta_{15} + 345 \beta_{14} + \cdots + 278502 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 592 \beta_{19} + 1446 \beta_{18} - 132 \beta_{17} + 224 \beta_{16} - 1925 \beta_{15} + 1410 \beta_{14} + \cdots + 524677 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13436 \beta_{19} + 11564 \beta_{18} + 3160 \beta_{17} + 9032 \beta_{16} + 5714 \beta_{15} + \cdots + 25599852 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 87004 \beta_{19} + 235008 \beta_{18} - 26464 \beta_{17} + 18528 \beta_{16} - 288465 \beta_{15} + \cdots + 58438297 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2054309 \beta_{19} + 1832474 \beta_{18} + 463392 \beta_{17} + 1383526 \beta_{16} - 2842 \beta_{15} + \cdots + 2436032367 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 11539774 \beta_{19} + 32636570 \beta_{18} - 3583776 \beta_{17} + 1366890 \beta_{16} + \cdots + 6362725497 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 271575108 \beta_{19} + 253905572 \beta_{18} + 60875140 \beta_{17} + 180656874 \beta_{16} + \cdots + 237068131781 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1454884697 \beta_{19} + 4182431214 \beta_{18} - 406788188 \beta_{17} + 95948840 \beta_{16} + \cdots + 687241069259 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 33342049256 \beta_{19} + 32804771242 \beta_{18} + 7560877620 \beta_{17} + 21704440984 \beta_{16} + \cdots + 23459616118208 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 177778368331 \beta_{19} + 511447348986 \beta_{18} - 41317194632 \beta_{17} + 6711433594 \beta_{16} + \cdots + 74150322302858 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3923113674559 \beta_{19} + 4063815496656 \beta_{18} + 908031506784 \beta_{17} + 2482937401780 \beta_{16} + \cdots + 23\!\cdots\!15 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 21255554557483 \beta_{19} + 60708015231800 \beta_{18} - 3829896076292 \beta_{17} + \cdots + 80\!\cdots\!03 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 449610493190765 \beta_{19} + 489670124139272 \beta_{18} + 106672111798256 \beta_{17} + \cdots + 23\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 25\!\cdots\!09 \beta_{19} + \cdots + 86\!\cdots\!64 \) 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
−10.1153
−8.94562
−8.70460
−6.97082
−5.90598
−5.47490
−3.62368
−3.45406
−2.66765
−1.91082
0.888819
1.62818
3.74430
4.82252
5.51773
5.90749
8.89249
9.11748
9.79354
10.4609
−11.1153 3.26137 91.5500 25.0000 −36.2511 27.4041 −661.917 −232.363 −277.883
1.2 −9.94562 23.0590 66.9153 25.0000 −229.336 80.2641 −347.254 288.718 −248.640
1.3 −9.70460 −7.56086 62.1793 25.0000 73.3751 −199.276 −292.879 −185.833 −242.615
1.4 −7.97082 27.8505 31.5340 25.0000 −221.991 −220.809 3.71413 532.648 −199.271
1.5 −6.90598 −1.02322 15.6925 25.0000 7.06633 3.68045 112.619 −241.953 −172.649
1.6 −6.47490 −29.5208 9.92436 25.0000 191.144 16.3316 142.938 628.477 −161.873
1.7 −4.62368 −27.1017 −10.6216 25.0000 125.310 −225.648 197.069 491.501 −115.592
1.8 −4.45406 −2.47240 −12.1614 25.0000 11.0122 82.4631 196.697 −236.887 −111.351
1.9 −3.66765 24.7362 −18.5483 25.0000 −90.7237 −169.967 185.394 368.878 −91.6913
1.10 −2.91082 18.7218 −23.5272 25.0000 −54.4957 208.489 161.629 107.505 −72.7704
1.11 −0.111181 4.60961 −31.9876 25.0000 −0.512503 −61.3924 7.11423 −221.751 −2.77953
1.12 0.628180 −4.58825 −31.6054 25.0000 −2.88225 −116.292 −39.9557 −221.948 15.7045
1.13 2.74430 −12.8364 −24.4688 25.0000 −35.2269 161.649 −154.967 −78.2269 68.6075
1.14 3.82252 −23.1708 −17.3883 25.0000 −88.5706 62.4720 −188.788 293.884 95.5630
1.15 4.51773 22.9237 −11.5902 25.0000 103.563 −56.4114 −196.928 282.498 112.943
1.16 4.90749 11.7032 −7.91656 25.0000 57.4334 217.582 −195.890 −106.035 122.687
1.17 7.89249 20.7410 30.2914 25.0000 163.698 −181.733 −13.4853 187.189 197.312
1.18 8.11748 −11.1242 33.8935 25.0000 −90.3005 −93.8217 15.3706 −119.252 202.937
1.19 8.79354 −22.5193 45.3263 25.0000 −198.024 38.0979 117.185 264.118 219.838
1.20 9.46090 6.31148 57.5086 25.0000 59.7123 −132.083 241.334 −203.165 236.522
\(n\): 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.n 20
11.b odd 2 1 605.6.a.q 20
11.d odd 10 2 55.6.g.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.g.a ✓ 40 11.d odd 10 2
605.6.a.n 20 1.a even 1 1 trivial
605.6.a.q 20 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 17 T_{2}^{19} - 303 T_{2}^{18} - 6189 T_{2}^{17} + 32496 T_{2}^{16} + 926379 T_{2}^{15} + \cdots - 7278582132736 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots - 7278582132736 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 82\!\cdots\!80 \) Copy content Toggle raw display
$5$ \( (T - 25)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 29\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 24\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 82\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 50\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 58\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 56\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 56\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 22\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 18\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 15\!\cdots\!75 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
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