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Properties

Label 4805.2.a.x
Level $4805$
Weight $2$
Character orbit 4805.a
Self dual yes
Analytic conductor $38.368$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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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] = [4805,2,Mod(1,4805)] 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("4805.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4805 = 5 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4805.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,-1,-4,15,-20,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.3681181712\)
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: \( x^{20} - x^{19} - 27 x^{18} + 29 x^{17} + 298 x^{16} - 343 x^{15} - 1729 x^{14} + 2127 x^{13} + 5634 x^{12} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{19} q^{3} + (\beta_{10} + \beta_{9} + 1) q^{4} - q^{5} + (\beta_{17} - \beta_{13} - \beta_{10} + \cdots - 1) q^{6} + ( - 2 \beta_{19} - \beta_{18} + \cdots - \beta_1) q^{7}+ \cdots + (2 \beta_{19} - \beta_{18} + 2 \beta_{17} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 4 q^{3} + 15 q^{4} - 20 q^{5} - 4 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + q^{10} - 17 q^{11} + 13 q^{12} - 9 q^{13} + 6 q^{14} + 4 q^{15} + 9 q^{16} + 20 q^{17} - 28 q^{18} - 7 q^{19} - 15 q^{20}+ \cdots - 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 27 x^{18} + 29 x^{17} + 298 x^{16} - 343 x^{15} - 1729 x^{14} + 2127 x^{13} + 5634 x^{12} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 912719283 \nu^{19} + 329340740 \nu^{18} - 23761982370 \nu^{17} - 5803980404 \nu^{16} + \cdots - 1622319754 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1622319754 \nu^{19} + 709600471 \nu^{18} + 43473292618 \nu^{17} - 23285290496 \nu^{16} + \cdots - 21302456477 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5561967652 \nu^{19} - 12035399474 \nu^{18} - 151093454251 \nu^{17} + 334125829341 \nu^{16} + \cdots + 142755367207 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8521139329 \nu^{19} - 4947676590 \nu^{18} - 225440707423 \nu^{17} + 160233405499 \nu^{16} + \cdots + 79621375617 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3283653558 \nu^{19} - 1180143605 \nu^{18} - 87338297126 \nu^{17} + 40993919906 \nu^{16} + \cdots + 47620046823 ) / 14362102591 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11467232016 \nu^{19} + 19623982447 \nu^{18} - 291520148094 \nu^{17} - 485882516278 \nu^{16} + \cdots - 147976341746 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12477578635 \nu^{19} - 14453757800 \nu^{18} - 337224783355 \nu^{17} + 410702585118 \nu^{16} + \cdots + 70330104088 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13257473408 \nu^{19} - 12309656997 \nu^{18} + 353491157840 \nu^{17} + 287751495235 \nu^{16} + \cdots + 12634401405 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13257473408 \nu^{19} + 12309656997 \nu^{18} - 353491157840 \nu^{17} - 287751495235 \nu^{16} + \cdots - 141893324724 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16338170846 \nu^{19} + 6312885976 \nu^{18} - 425322139109 \nu^{17} - 123595519214 \nu^{16} + \cdots + 97249642891 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20182916744 \nu^{19} + 5062488184 \nu^{18} - 542505357662 \nu^{17} - 72251104796 \nu^{16} + \cdots + 312389352427 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 20950593296 \nu^{19} - 32093645432 \nu^{18} - 558892956296 \nu^{17} + 894747445084 \nu^{16} + \cdots + 285687607432 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22046283734 \nu^{19} + 8206856295 \nu^{18} + 585491372639 \nu^{17} - 279346103441 \nu^{16} + \cdots - 223947408915 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26479849688 \nu^{19} + 4789964916 \nu^{18} - 695980206437 \nu^{17} - 57698621068 \nu^{16} + \cdots - 14879793162 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 30112458980 \nu^{19} - 2549224641 \nu^{18} + 796713669230 \nu^{17} - 14683445513 \nu^{16} + \cdots - 248694700059 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 31091214463 \nu^{19} - 18095116338 \nu^{18} + 818432244742 \nu^{17} + 390011390237 \nu^{16} + \cdots + 11467232016 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 32979038084 \nu^{19} + 4326876616 \nu^{18} + 877585981907 \nu^{17} - 194606993232 \nu^{16} + \cdots - 151177168766 ) / 43086307773 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 37776886092 \nu^{19} + 5193855766 \nu^{18} - 1003670607966 \nu^{17} - 36046987477 \nu^{16} + \cdots + 103644876166 ) / 43086307773 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{8} - \beta_{7} + \beta_{3} - \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} + 2 \beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{14} - \beta_{12} + 2 \beta_{11} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{19} - \beta_{18} - 2 \beta_{17} + \beta_{16} + 8 \beta_{15} - \beta_{14} + \beta_{13} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 21 \beta_{19} + 20 \beta_{18} + 21 \beta_{17} - 9 \beta_{16} - 10 \beta_{14} - 11 \beta_{12} + 20 \beta_{11} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 27 \beta_{19} - 11 \beta_{18} - 26 \beta_{17} + 9 \beta_{16} + 57 \beta_{15} - 12 \beta_{14} + \cdots - 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 173 \beta_{19} + 157 \beta_{18} + 174 \beta_{17} - 67 \beta_{16} - 83 \beta_{14} - 4 \beta_{13} + \cdots + 528 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 256 \beta_{19} - 90 \beta_{18} - 241 \beta_{17} + 61 \beta_{16} + 395 \beta_{15} - 110 \beta_{14} + \cdots - 275 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1315 \beta_{19} + 1138 \beta_{18} + 1329 \beta_{17} - 472 \beta_{16} - 645 \beta_{14} - 66 \beta_{13} + \cdots + 3364 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2105 \beta_{19} - 650 \beta_{18} - 1961 \beta_{17} + 373 \beta_{16} + 2716 \beta_{15} - 910 \beta_{14} + \cdots - 1472 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9637 \beta_{19} + 7977 \beta_{18} + 9779 \beta_{17} - 3255 \beta_{16} + 9 \beta_{15} - 4839 \beta_{14} + \cdots + 21917 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 16105 \beta_{19} - 4376 \beta_{18} - 14961 \beta_{17} + 2166 \beta_{16} + 18658 \beta_{15} + \cdots - 7526 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 69236 \beta_{19} + 55065 \beta_{18} + 70529 \beta_{17} - 22249 \beta_{16} + 216 \beta_{15} + \cdots + 144961 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 118305 \beta_{19} - 28115 \beta_{18} - 110081 \beta_{17} + 12145 \beta_{16} + 128390 \beta_{15} + \cdots - 35945 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 491468 \beta_{19} + 377477 \beta_{18} + 502653 \beta_{17} - 151577 \beta_{16} + 3206 \beta_{15} + \cdots + 969418 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 848014 \beta_{19} - 174248 \beta_{18} - 792269 \beta_{17} + 65832 \beta_{16} + 885699 \beta_{15} + \cdots - 150558 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 3461713 \beta_{19} + 2580671 \beta_{18} + 3555246 \beta_{17} - 1032094 \beta_{16} + 38031 \beta_{15} + \cdots + 6538105 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 5985666 \beta_{19} - 1045826 \beta_{18} - 5621438 \beta_{17} + 341413 \beta_{16} + 6125668 \beta_{15} + \cdots - 438642 \) 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
2.65915
2.22127
2.14600
1.75464
1.54509
1.47391
1.43916
0.842160
0.428361
0.331395
−0.0405845
−0.105871
−0.470058
−1.03335
−1.04508
−1.77100
−2.09217
−2.20707
−2.44834
−2.62761
−2.65915 3.01427 5.07110 −1.00000 −8.01540 −1.08231 −8.16653 6.08581 2.65915
1.2 −2.22127 −1.93838 2.93405 −1.00000 4.30566 0.482944 −2.07478 0.757308 2.22127
1.3 −2.14600 0.339468 2.60531 −1.00000 −0.728498 −2.47106 −1.29898 −2.88476 2.14600
1.4 −1.75464 0.106823 1.07875 −1.00000 −0.187436 3.37688 1.61646 −2.98859 1.75464
1.5 −1.54509 −1.42833 0.387308 −1.00000 2.20690 0.359248 2.49176 −0.959876 1.54509
1.6 −1.47391 −2.90581 0.172406 −1.00000 4.28290 3.45832 2.69371 5.44372 1.47391
1.7 −1.43916 2.55788 0.0711934 −1.00000 −3.68121 0.465673 2.77587 3.54274 1.43916
1.8 −0.842160 1.62007 −1.29077 −1.00000 −1.36435 −4.34754 2.77135 −0.375389 0.842160
1.9 −0.428361 −1.27631 −1.81651 −1.00000 0.546723 −2.32283 1.63484 −1.37103 0.428361
1.10 −0.331395 −3.34630 −1.89018 −1.00000 1.10895 2.06578 1.28919 8.19769 0.331395
1.11 0.0405845 2.11646 −1.99835 −1.00000 0.0858953 0.968366 −0.162271 1.47939 −0.0405845
1.12 0.105871 −0.544916 −1.98879 −1.00000 −0.0576909 −2.50109 −0.422298 −2.70307 −0.105871
1.13 0.470058 −1.65534 −1.77905 −1.00000 −0.778104 −4.13795 −1.77637 −0.259863 −0.470058
1.14 1.03335 0.650984 −0.932183 −1.00000 0.672695 2.64987 −3.02998 −2.57622 −1.03335
1.15 1.04508 −0.909816 −0.907815 −1.00000 −0.950828 −0.553397 −3.03889 −2.17223 −1.04508
1.16 1.77100 1.10522 1.13645 −1.00000 1.95735 1.77335 −1.52935 −1.77849 −1.77100
1.17 2.09217 1.52922 2.37719 −1.00000 3.19940 3.03261 0.789144 −0.661478 −2.09217
1.18 2.20707 −2.64497 2.87114 −1.00000 −5.83763 3.89513 1.92267 3.99588 −2.20707
1.19 2.44834 −1.45056 3.99439 −1.00000 −3.55147 0.552093 4.88296 −0.895871 −2.44834
1.20 2.62761 1.06034 4.90435 −1.00000 2.78616 −4.66409 7.63151 −1.87568 −2.62761
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(31\) \( +1 \)

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 4805.2.a.x 20
31.b odd 2 1 4805.2.a.y 20
31.g even 15 2 155.2.q.a ✓ 40
155.u even 30 2 775.2.bl.c 40
155.w odd 60 4 775.2.ck.c 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.q.a ✓ 40 31.g even 15 2
775.2.bl.c 40 155.u even 30 2
775.2.ck.c 80 155.w odd 60 4
4805.2.a.x 20 1.a even 1 1 trivial
4805.2.a.y 20 31.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4805))\):

\( T_{2}^{20} + T_{2}^{19} - 27 T_{2}^{18} - 29 T_{2}^{17} + 298 T_{2}^{16} + 343 T_{2}^{15} - 1729 T_{2}^{14} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{20} + 4 T_{3}^{19} - 26 T_{3}^{18} - 113 T_{3}^{17} + 256 T_{3}^{16} + 1275 T_{3}^{15} + \cdots + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( (T + 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - T^{19} + \cdots + 41731 \) Copy content Toggle raw display
$11$ \( T^{20} + 17 T^{19} + \cdots + 17557261 \) Copy content Toggle raw display
$13$ \( T^{20} + 9 T^{19} + \cdots - 232955 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 116661299 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 333141779 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 3730919279 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 1571297513911 \) Copy content Toggle raw display
$31$ \( T^{20} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 4733133599 \) Copy content Toggle raw display
$41$ \( T^{20} - 9 T^{19} + \cdots + 72361 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 12378561336639 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 59737958814631 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 152307459 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 3924090903105 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 26526875211 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 445085894793359 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 33592479000445 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 5813719257344 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 679538552539289 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 89\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 603092821593231 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
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