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Properties

Label 1255.2.a.d
Level $1255$
Weight $2$
Character orbit 1255.a
Self dual yes
Analytic conductor $10.021$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1255,2,Mod(1,1255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1255, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1255.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1255 = 5 \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1255.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.0212254537\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} + 14 x^{16} + 88 x^{15} - 267 x^{14} - 279 x^{13} + 1508 x^{12} + 130 x^{11} + \cdots + 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + \beta_{7} q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{17} + \beta_{15} - 2 \beta_{14} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{7} + \beta_{6} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + \beta_{7} q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{17} + \beta_{15} - 2 \beta_{14} + \cdots - 1) q^{6}+ \cdots + ( - 4 \beta_{17} + \beta_{16} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 9 q^{2} - 7 q^{3} + 17 q^{4} + 18 q^{5} - 3 q^{6} - 13 q^{7} - 27 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 9 q^{2} - 7 q^{3} + 17 q^{4} + 18 q^{5} - 3 q^{6} - 13 q^{7} - 27 q^{8} + 9 q^{9} - 9 q^{10} - 9 q^{11} - 9 q^{12} - 31 q^{13} - 6 q^{14} - 7 q^{15} + 23 q^{16} - 46 q^{17} - 27 q^{18} + 17 q^{20} - 24 q^{21} - 9 q^{22} - 13 q^{23} + 2 q^{24} + 18 q^{25} + 7 q^{26} - 25 q^{27} - 26 q^{28} - 9 q^{29} - 3 q^{30} - 7 q^{31} - 53 q^{32} - 36 q^{33} + 18 q^{34} - 13 q^{35} + 38 q^{36} - 39 q^{37} + 9 q^{39} - 27 q^{40} - 64 q^{41} + 42 q^{42} - 15 q^{43} + 6 q^{44} + 9 q^{45} + 23 q^{46} - 15 q^{47} + 18 q^{48} + 25 q^{49} - 9 q^{50} + 20 q^{51} - 21 q^{52} - 32 q^{53} + 23 q^{54} - 9 q^{55} + 9 q^{56} - 20 q^{57} + 20 q^{58} - 3 q^{59} - 9 q^{60} - 4 q^{61} + 13 q^{62} - 4 q^{63} + 39 q^{64} - 31 q^{65} + 49 q^{66} - 25 q^{67} - 75 q^{68} - 3 q^{69} - 6 q^{70} - 5 q^{71} - 75 q^{72} - 50 q^{73} + 19 q^{74} - 7 q^{75} + 14 q^{76} - 32 q^{77} + 43 q^{78} + 8 q^{79} + 23 q^{80} + 14 q^{81} + 13 q^{82} - 16 q^{83} - 53 q^{84} - 46 q^{85} - 32 q^{86} - 21 q^{87} + 24 q^{88} - 69 q^{89} - 27 q^{90} + 29 q^{91} - 70 q^{92} - 23 q^{93} + 36 q^{94} - q^{96} - 64 q^{97} + 36 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 9 x^{17} + 14 x^{16} + 88 x^{15} - 267 x^{14} - 279 x^{13} + 1508 x^{12} + 130 x^{11} + \cdots + 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 303003 \nu^{17} - 2684483 \nu^{16} + 4024133 \nu^{15} + 25881802 \nu^{14} - 75675346 \nu^{13} + \cdots + 4118559 ) / 512638 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 525458 \nu^{17} + 5058751 \nu^{16} - 10462378 \nu^{15} - 40236741 \nu^{14} + 166048191 \nu^{13} + \cdots - 48445241 ) / 512638 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 628223 \nu^{17} + 5717758 \nu^{16} - 9576085 \nu^{15} - 52575007 \nu^{14} + 170839947 \nu^{13} + \cdots - 33257858 ) / 512638 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 691097 \nu^{17} + 6637565 \nu^{16} - 13540241 \nu^{15} - 53915416 \nu^{14} + 218865862 \nu^{13} + \cdots - 87792485 ) / 512638 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1160195 \nu^{17} + 10861988 \nu^{16} - 20339551 \nu^{15} - 93311115 \nu^{14} + 341608321 \nu^{13} + \cdots - 86670812 ) / 512638 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 639151 \nu^{17} - 6067741 \nu^{16} + 11922962 \nu^{15} + 50514783 \nu^{14} - 195687467 \nu^{13} + \cdots + 52203481 ) / 256319 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2837529 \nu^{17} + 26915204 \nu^{16} - 52828841 \nu^{15} - 223743783 \nu^{14} + 865876161 \nu^{13} + \cdots - 252976592 ) / 512638 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3243397 \nu^{17} - 30762807 \nu^{16} + 60167789 \nu^{15} + 257556802 \nu^{14} - 992442732 \nu^{13} + \cdots + 327601895 ) / 512638 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4194523 \nu^{17} + 39775444 \nu^{16} - 77777025 \nu^{15} - 332802331 \nu^{14} + 1281900619 \nu^{13} + \cdots - 411600254 ) / 512638 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2259192 \nu^{17} + 21430247 \nu^{16} - 41938468 \nu^{15} - 179333962 \nu^{14} + 691724058 \nu^{13} + \cdots - 222930217 ) / 256319 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5209937 \nu^{17} + 49459185 \nu^{16} - 97114491 \nu^{15} - 412418562 \nu^{14} + 1596563420 \nu^{13} + \cdots - 489578081 ) / 512638 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5342396 \nu^{17} - 50534447 \nu^{16} + 97974362 \nu^{15} + 425338507 \nu^{14} - 1621936205 \nu^{13} + \cdots + 504832879 ) / 512638 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5973160 \nu^{17} - 56484143 \nu^{16} + 109435372 \nu^{15} + 475406807 \nu^{14} - 1811520185 \nu^{13} + \cdots + 560825141 ) / 512638 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 7145507 \nu^{17} - 67591768 \nu^{16} + 131079989 \nu^{15} + 568603213 \nu^{14} - 2168786407 \nu^{13} + \cdots + 665475572 ) / 512638 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 9691723 \nu^{17} + 91723033 \nu^{16} - 178144049 \nu^{15} - 771070466 \nu^{14} + \cdots - 917670617 ) / 512638 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + \beta_{12} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 2\beta_{2} + 4\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{15} + \beta_{14} - 4 \beta_{13} + 3 \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{17} + 5 \beta_{15} + 3 \beta_{14} - 17 \beta_{13} + 14 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20 \beta_{17} + \beta_{16} + 25 \beta_{15} + 14 \beta_{14} - 61 \beta_{13} + 47 \beta_{12} - 10 \beta_{11} + \cdots + 115 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 74 \beta_{17} + 7 \beta_{16} + 101 \beta_{15} + 40 \beta_{14} - 214 \beta_{13} + 172 \beta_{12} + \cdots + 340 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 282 \beta_{17} + 38 \beta_{16} + 395 \beta_{15} + 132 \beta_{14} - 724 \beta_{13} + 579 \beta_{12} + \cdots + 1083 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 988 \beta_{17} + 173 \beta_{16} + 1454 \beta_{15} + 352 \beta_{14} - 2407 \beta_{13} + 1979 \beta_{12} + \cdots + 3315 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3451 \beta_{17} + 718 \beta_{16} + 5232 \beta_{15} + 964 \beta_{14} - 7908 \beta_{13} + 6587 \beta_{12} + \cdots + 10402 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11699 \beta_{17} + 2803 \beta_{16} + 18362 \beta_{15} + 2237 \beta_{14} - 25740 \beta_{13} + \cdots + 32271 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 39396 \beta_{17} + 10475 \beta_{16} + 63535 \beta_{15} + 4654 \beta_{14} - 83362 \beta_{13} + \cdots + 101087 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 131001 \beta_{17} + 37988 \beta_{16} + 217024 \beta_{15} + 5781 \beta_{14} - 268770 \beta_{13} + \cdots + 315949 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 433349 \beta_{17} + 134567 \beta_{16} + 734719 \beta_{15} - 9501 \beta_{14} - 864577 \beta_{13} + \cdots + 992151 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1424812 \beta_{17} + 468620 \beta_{16} + 2468490 \beta_{15} - 121747 \beta_{14} - 2775766 \beta_{13} + \cdots + 3117881 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 4668672 \beta_{17} + 1609669 \beta_{16} + 8245500 \beta_{15} - 670354 \beta_{14} - 8903744 \beta_{13} + \cdots + 9827163 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 15248640 \beta_{17} + 5471551 \beta_{16} + 27408831 \beta_{15} - 3022475 \beta_{14} - 28542083 \beta_{13} + \cdots + 31023424 \) 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.

comment: embeddings in the coefficient field
 
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
−1.80283
−1.68311
−1.52539
−1.29458
−1.17927
−0.497182
−0.459765
−0.451856
0.482844
0.698372
0.895270
1.03751
1.49537
2.19617
2.31223
2.63525
2.90311
3.23785
−2.80283 −0.457270 5.85583 1.00000 1.28165 −0.513654 −10.8072 −2.79090 −2.80283
1.2 −2.68311 2.97098 5.19910 1.00000 −7.97147 −2.84209 −8.58356 5.82671 −2.68311
1.3 −2.52539 −3.10263 4.37757 1.00000 7.83533 4.59385 −6.00428 6.62630 −2.52539
1.4 −2.29458 −2.38728 3.26511 1.00000 5.47782 −3.94924 −2.90291 2.69913 −2.29458
1.5 −2.17927 1.50435 2.74921 1.00000 −3.27838 −2.26053 −1.63273 −0.736932 −2.17927
1.6 −1.49718 −1.86072 0.241554 1.00000 2.78584 0.283683 2.63271 0.462279 −1.49718
1.7 −1.45976 0.861325 0.130914 1.00000 −1.25733 −0.688319 2.72843 −2.25812 −1.45976
1.8 −1.45186 0.497037 0.107885 1.00000 −0.721625 3.61036 2.74708 −2.75295 −1.45186
1.9 −0.517156 0.397485 −1.73255 1.00000 −0.205562 −0.865641 1.93031 −2.84201 −0.517156
1.10 −0.301628 −3.05452 −1.90902 1.00000 0.921331 −1.85015 1.17907 6.33011 −0.301628
1.11 −0.104730 2.21811 −1.98903 1.00000 −0.232303 −2.08752 0.417773 1.92000 −0.104730
1.12 0.0375113 0.746333 −1.99859 1.00000 0.0279960 −1.82492 −0.149993 −2.44299 0.0375113
1.13 0.495370 −2.38368 −1.75461 1.00000 −1.18081 2.66611 −1.85992 2.68195 0.495370
1.14 1.19617 0.136633 −0.569188 1.00000 0.163436 3.61962 −3.07317 −2.98133 1.19617
1.15 1.31223 1.51451 −0.278042 1.00000 1.98739 −5.15418 −2.98932 −0.706258 1.31223
1.16 1.63525 −1.31301 0.674052 1.00000 −2.14710 0.759931 −2.16826 −1.27600 1.63525
1.17 1.90311 −2.60213 1.62183 1.00000 −4.95214 −1.38600 −0.719706 3.77109 1.90311
1.18 2.23785 −0.685508 3.00798 1.00000 −1.53406 −5.11130 2.25570 −2.53008 2.23785
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(251\) \( -1 \)

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 1255.2.a.d ✓ 18
5.b even 2 1 6275.2.a.f 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1255.2.a.d ✓ 18 1.a even 1 1 trivial
6275.2.a.f 18 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 9 T_{2}^{17} + 14 T_{2}^{16} - 96 T_{2}^{15} - 327 T_{2}^{14} + 211 T_{2}^{13} + 1976 T_{2}^{12} + \cdots - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1255))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 9 T^{17} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{18} + 7 T^{17} + \cdots - 28 \) Copy content Toggle raw display
$5$ \( (T - 1)^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + 13 T^{17} + \cdots + 68956 \) Copy content Toggle raw display
$11$ \( T^{18} + 9 T^{17} + \cdots + 72892 \) Copy content Toggle raw display
$13$ \( T^{18} + 31 T^{17} + \cdots - 356 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 138760432 \) Copy content Toggle raw display
$19$ \( T^{18} - 177 T^{16} + \cdots + 13814960 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 6338147188 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 307940562700 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 181558408769 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 228090935659 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 6078805432 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 1272556496 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 523237140499 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 6535070445748 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 68170300478140 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 17\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 36\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 159647039664212 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 140160626817892 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 2413735405585 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 26451116427952 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 182844731899465 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 29\!\cdots\!96 \) Copy content Toggle raw display
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