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Properties

Label 455.2.c.c
Level $455$
Weight $2$
Character orbit 455.c
Analytic conductor $3.633$
Analytic rank $0$
Dimension $20$
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] = [455,2,Mod(274,455)] 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("455.274"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(455, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 455 = 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 455.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] 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: \(3.63319329197\)
Analytic rank: \(0\)
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: \( x^{20} + 33 x^{18} + 456 x^{16} + 3416 x^{14} + 14988 x^{12} + 38773 x^{10} + 56577 x^{8} + 42596 x^{6} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{19} q^{3} + (\beta_{2} - 1) q^{4} + \beta_{17} q^{5} + ( - \beta_{19} + \beta_{18} - \beta_{17} + \cdots + 1) q^{6} + \beta_{4} q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{12} - \beta_{10} - 2) q^{9}+ \cdots + (2 \beta_{17} + 2 \beta_{16} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 26 q^{4} + 6 q^{5} + 4 q^{6} - 40 q^{9} - 2 q^{10} + 16 q^{11} + 2 q^{14} - 10 q^{15} + 38 q^{16} - 4 q^{19} + 10 q^{20} + 8 q^{21} - 54 q^{24} - 20 q^{25} - 2 q^{26} - 60 q^{29} - 28 q^{30} + 4 q^{31}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 33 x^{18} + 456 x^{16} + 3416 x^{14} + 14988 x^{12} + 38773 x^{10} + 56577 x^{8} + 42596 x^{6} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 82 \nu^{19} + 2043 \nu^{17} + 18315 \nu^{15} + 58622 \nu^{13} - 91752 \nu^{11} - 1063346 \nu^{9} + \cdots - 118148 \nu ) / 23640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 171 \nu^{18} - 4849 \nu^{16} - 54470 \nu^{14} - 303176 \nu^{12} - 837484 \nu^{10} - 919207 \nu^{8} + \cdots + 12144 ) / 7880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 221 \nu^{18} + 6359 \nu^{16} + 73830 \nu^{14} + 440256 \nu^{12} + 1414244 \nu^{10} + 2335097 \nu^{8} + \cdots + 3936 ) / 7880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 139 \nu^{19} + 4316 \nu^{17} + 55515 \nu^{15} + 381634 \nu^{13} + 1505996 \nu^{11} + \cdots + 122084 \nu ) / 7880 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1205 \nu^{19} - 552 \nu^{18} + 37770 \nu^{17} - 24708 \nu^{16} + 491595 \nu^{15} - 439260 \nu^{14} + \cdots - 719352 ) / 47280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 267 \nu^{18} - 8615 \nu^{16} - 115754 \nu^{14} - 836496 \nu^{12} - 3496908 \nu^{10} + \cdots - 104464 ) / 1576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1331 \nu^{19} + 3 \nu^{18} + 43624 \nu^{17} + 327 \nu^{16} + 596015 \nu^{15} + 6520 \nu^{14} + \cdots - 80672 ) / 15760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1331 \nu^{19} - 1163 \nu^{18} + 43624 \nu^{17} - 40087 \nu^{16} + 596015 \nu^{15} - 578600 \nu^{14} + \cdots - 913328 ) / 15760 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1331 \nu^{19} + 1163 \nu^{18} + 43624 \nu^{17} + 40087 \nu^{16} + 596015 \nu^{15} + 578600 \nu^{14} + \cdots + 913328 ) / 15760 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1691 \nu^{19} + 1128 \nu^{18} + 54299 \nu^{17} + 36272 \nu^{16} + 725360 \nu^{15} + 485460 \nu^{14} + \cdots + 336288 ) / 15760 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5306 \nu^{19} + 4101 \nu^{18} - 171549 \nu^{17} + 133779 \nu^{16} - 2310465 \nu^{15} + \cdots + 1577976 ) / 47280 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5306 \nu^{19} - 4101 \nu^{18} - 171549 \nu^{17} - 133779 \nu^{16} - 2310465 \nu^{15} + \cdots - 1577976 ) / 47280 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 7403 \nu^{19} - 2412 \nu^{18} + 240552 \nu^{17} - 79698 \nu^{16} + 3256935 \nu^{15} - 1099170 \nu^{14} + \cdots - 1284432 ) / 47280 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 7403 \nu^{19} - 2412 \nu^{18} - 240552 \nu^{17} - 79698 \nu^{16} - 3256935 \nu^{15} + \cdots - 1284432 ) / 47280 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 9629 \nu^{19} + 552 \nu^{18} + 317706 \nu^{17} + 24708 \nu^{16} + 4379355 \nu^{15} + 439260 \nu^{14} + \cdots + 719352 ) / 47280 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 3034 \nu^{19} + 99231 \nu^{17} + 1354350 \nu^{15} + 9964304 \nu^{13} + 42502236 \nu^{11} + \cdots + 1349404 \nu ) / 11820 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} - \beta_{16} + \beta_{11} - \beta_{10} - \beta_{6} - 6\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} - \beta_{15} + \beta_{13} - \beta_{11} + \beta_{9} - \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} - \beta_{18} + 10 \beta_{17} + 9 \beta_{16} - \beta_{15} - \beta_{13} - 11 \beta_{11} + \cdots - 105 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{19} + 11 \beta_{18} - 15 \beta_{17} + 13 \beta_{15} - 2 \beta_{14} - 15 \beta_{13} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{19} + 14 \beta_{18} - 78 \beta_{17} - 64 \beta_{16} + 18 \beta_{15} - 4 \beta_{14} + \cdots + 672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 89 \beta_{19} - 90 \beta_{18} + 158 \beta_{17} + 3 \beta_{16} - 125 \beta_{15} + 36 \beta_{14} + \cdots - 161 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 146 \beta_{19} - 146 \beta_{18} + 565 \beta_{17} + 419 \beta_{16} - 217 \beta_{15} + 71 \beta_{14} + \cdots - 4415 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 646 \beta_{19} + 664 \beta_{18} - 1437 \beta_{17} - 65 \beta_{16} + 1068 \beta_{15} - 434 \beta_{14} + \cdots + 1502 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1359 \beta_{19} + 1359 \beta_{18} - 4005 \beta_{17} - 2646 \beta_{16} + 2197 \beta_{15} + \cdots + 29626 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4466 \beta_{19} - 4682 \beta_{18} + 12088 \beta_{17} + 898 \beta_{16} - 8592 \beta_{15} + 4394 \beta_{14} + \cdots - 12986 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11890 \beta_{19} - 11890 \beta_{18} + 28351 \beta_{17} + 16461 \beta_{16} - 20194 \beta_{15} + \cdots - 202283 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 30145 \beta_{19} + 32333 \beta_{18} - 97077 \beta_{17} - 10096 \beta_{16} + 66785 \beta_{15} + \cdots + 107173 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 99925 \beta_{19} + 99925 \beta_{18} - 201970 \beta_{17} - 102045 \beta_{16} + 174765 \beta_{15} + \cdots + 1401149 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 201103 \beta_{19} - 221359 \beta_{18} + 757159 \beta_{17} + 100792 \beta_{16} - 508421 \beta_{15} + \cdots - 857951 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 816094 \beta_{19} - 816094 \beta_{18} + 1450866 \beta_{17} + 634772 \beta_{16} - 1452686 \beta_{15} + \cdots - 9821388 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1334933 \beta_{19} + 1512498 \beta_{18} - 5794062 \beta_{17} - 932027 \beta_{16} + 3820717 \beta_{15} + \cdots + 6726089 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/455\mathbb{Z}\right)^\times\).

\(n\) \(66\) \(92\) \(106\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 2.71750i
− 2.45474i
− 2.38987i
− 2.27890i
− 1.90093i
− 1.88895i
− 0.858932i
− 0.590489i
− 0.557283i
− 0.325442i
0.325442i
0.557283i
0.590489i
0.858932i
1.88895i
1.90093i
2.27890i
2.38987i
2.45474i
2.71750i
− 2.71750i 2.22684i −5.38482 −1.90467 − 1.17142i 6.05144 1.00000i 9.19827i −1.95880 −3.18335 + 5.17595i
274.2 − 2.45474i 1.52977i −4.02573 1.58985 − 1.57238i 3.75517 − 1.00000i 4.97264i 0.659817 −3.85977 − 3.90267i
274.3 − 2.38987i − 3.28504i −3.71148 −1.22385 + 1.87142i −7.85083 1.00000i 4.09022i −7.79152 4.47245 + 2.92483i
274.4 − 2.27890i 2.83679i −3.19338 −0.381651 + 2.20326i 6.46476 − 1.00000i 2.71958i −5.04739 5.02100 + 0.869743i
274.5 − 1.90093i − 0.527099i −1.61353 1.66682 − 1.49054i −1.00198 1.00000i − 0.734648i 2.72217 −2.83341 − 3.16851i
274.6 − 1.88895i − 2.55412i −1.56814 2.13016 + 0.680019i −4.82460 − 1.00000i − 0.815771i −3.52350 1.28452 − 4.02376i
274.7 − 0.858932i − 0.269566i 1.26224 0.625454 − 2.14681i −0.231539 − 1.00000i − 2.80204i 2.92733 −1.84397 − 0.537223i
274.8 − 0.590489i 2.53915i 1.65132 −0.200698 + 2.22704i 1.49934 1.00000i − 2.15607i −3.44730 1.31504 + 0.118510i
274.9 − 0.557283i − 3.24228i 1.68944 1.80012 − 1.32649i −1.80686 1.00000i − 2.05606i −7.51235 −0.739232 − 1.00317i
274.10 − 0.325442i − 0.168695i 1.89409 −1.10154 − 1.94592i −0.0549004 1.00000i − 1.26730i 2.97154 −0.633286 + 0.358487i
274.11 0.325442i 0.168695i 1.89409 −1.10154 + 1.94592i −0.0549004 − 1.00000i 1.26730i 2.97154 −0.633286 − 0.358487i
274.12 0.557283i 3.24228i 1.68944 1.80012 + 1.32649i −1.80686 − 1.00000i 2.05606i −7.51235 −0.739232 + 1.00317i
274.13 0.590489i − 2.53915i 1.65132 −0.200698 − 2.22704i 1.49934 − 1.00000i 2.15607i −3.44730 1.31504 − 0.118510i
274.14 0.858932i 0.269566i 1.26224 0.625454 + 2.14681i −0.231539 1.00000i 2.80204i 2.92733 −1.84397 + 0.537223i
274.15 1.88895i 2.55412i −1.56814 2.13016 − 0.680019i −4.82460 1.00000i 0.815771i −3.52350 1.28452 + 4.02376i
274.16 1.90093i 0.527099i −1.61353 1.66682 + 1.49054i −1.00198 − 1.00000i 0.734648i 2.72217 −2.83341 + 3.16851i
274.17 2.27890i − 2.83679i −3.19338 −0.381651 − 2.20326i 6.46476 1.00000i − 2.71958i −5.04739 5.02100 − 0.869743i
274.18 2.38987i 3.28504i −3.71148 −1.22385 − 1.87142i −7.85083 − 1.00000i − 4.09022i −7.79152 4.47245 − 2.92483i
274.19 2.45474i − 1.52977i −4.02573 1.58985 + 1.57238i 3.75517 1.00000i − 4.97264i 0.659817 −3.85977 + 3.90267i
274.20 2.71750i − 2.22684i −5.38482 −1.90467 + 1.17142i 6.05144 − 1.00000i − 9.19827i −1.95880 −3.18335 − 5.17595i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 455.2.c.c ✓ 20
5.b even 2 1 inner 455.2.c.c ✓ 20
5.c odd 4 1 2275.2.a.ba 10
5.c odd 4 1 2275.2.a.bb 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.c.c ✓ 20 1.a even 1 1 trivial
455.2.c.c ✓ 20 5.b even 2 1 inner
2275.2.a.ba 10 5.c odd 4 1
2275.2.a.bb 10 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 33 T_{2}^{18} + 456 T_{2}^{16} + 3416 T_{2}^{14} + 14988 T_{2}^{12} + 38773 T_{2}^{10} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(455, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 33 T^{18} + \cdots + 144 \) Copy content Toggle raw display
$3$ \( T^{20} + 50 T^{18} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{20} - 6 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$11$ \( (T^{10} - 8 T^{9} + \cdots - 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{20} + 206 T^{18} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( (T^{10} + 2 T^{9} + \cdots + 6860)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 106172416 \) Copy content Toggle raw display
$29$ \( (T^{10} + 30 T^{9} + \cdots + 235740)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 2 T^{9} + \cdots + 265372)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 29776953600 \) Copy content Toggle raw display
$41$ \( (T^{10} - 32 T^{9} + \cdots + 25968640)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 716609654784 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 55389794074624 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 98344960000 \) Copy content Toggle raw display
$59$ \( (T^{10} + 4 T^{9} + \cdots - 10829760)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 40 T^{9} + \cdots - 263961600)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 846988902400 \) Copy content Toggle raw display
$71$ \( (T^{10} - 36 T^{9} + \cdots + 37056512)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 216525809913856 \) Copy content Toggle raw display
$79$ \( (T^{10} - 24 T^{9} + \cdots - 15997760)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + 152 T^{18} + \cdots + 589824 \) Copy content Toggle raw display
$89$ \( (T^{10} + 14 T^{9} + \cdots + 48683600)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 25334452224 \) Copy content Toggle raw display
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