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Properties

Label 7381.2.a.i
Level $7381$
Weight $2$
Character orbit 7381.a
Self dual yes
Analytic conductor $58.938$
Analytic rank $1$
Dimension $19$
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] = [7381,2,Mod(1,7381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7381, 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("7381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7381 = 11^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7381.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: \(58.9375817319\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 \) 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 671)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + ( - \beta_{16} + \beta_{13} + \cdots + \beta_1) q^{6}+ \cdots + (\beta_{17} + \beta_{13} - \beta_{11} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + ( - \beta_{16} + \beta_{13} + \cdots + \beta_1) q^{6}+ \cdots + ( - 2 \beta_{18} + \beta_{17} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9} - 7 q^{10} - 4 q^{12} - 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} - 9 q^{17} - 10 q^{18} - 17 q^{19} - 6 q^{20} - 18 q^{21} - 10 q^{23} - 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} - 36 q^{28} - 27 q^{29} + 30 q^{30} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} - 24 q^{39} - 10 q^{40} - 19 q^{41} + 21 q^{42} - 20 q^{43} - 32 q^{45} - 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} - 36 q^{50} - 47 q^{51} + 28 q^{52} + 3 q^{53} + 33 q^{54} - 44 q^{56} - 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} + 19 q^{61} + 11 q^{62} + 32 q^{63} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} - 19 q^{71} - 34 q^{72} - 20 q^{73} + 22 q^{74} - 50 q^{75} + 25 q^{76} - 94 q^{78} - 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} - q^{83} + 28 q^{84} - 24 q^{85} - 27 q^{86} + 58 q^{87} - 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} - 64 q^{94} + 3 q^{95} + 26 q^{96} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8867201804 \nu^{18} - 28397403169 \nu^{17} - 153963852608 \nu^{16} + 585160018840 \nu^{15} + \cdots + 7406641729547 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9925308116 \nu^{18} + 211585433401 \nu^{17} - 157290256641 \nu^{16} + \cdots + 4643140412416 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16270525372 \nu^{18} - 35753207242 \nu^{17} - 519467802678 \nu^{16} + 1065729788540 \nu^{15} + \cdots + 1737797285981 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37762226457 \nu^{18} - 165903350608 \nu^{17} - 747153495518 \nu^{16} + 3973232345087 \nu^{15} + \cdots - 2002160866471 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41025764486 \nu^{18} + 68688107954 \nu^{17} - 1546829355593 \nu^{16} - 1494342049450 \nu^{15} + \cdots - 165641941058 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 44058942396 \nu^{18} + 100366607650 \nu^{17} + 1163734380318 \nu^{16} + \cdots - 246530814813 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57136613064 \nu^{18} + 46120188886 \nu^{17} + 1573939504037 \nu^{16} + \cdots + 8443491451865 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 81254538506 \nu^{18} + 654428683551 \nu^{17} + 970372146851 \nu^{16} + \cdots - 25082161671395 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 91424960248 \nu^{18} + 290362672773 \nu^{17} + 2163987038842 \nu^{16} + \cdots - 5331850884506 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 110220246484 \nu^{18} - 467343546422 \nu^{17} - 2253974970877 \nu^{16} + \cdots - 8426703234447 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 112916661333 \nu^{18} - 420512816139 \nu^{17} - 2391672731682 \nu^{16} + 10136775407328 \nu^{15} + \cdots - 992900881709 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 134524452259 \nu^{18} - 595936463557 \nu^{17} - 2794615480726 \nu^{16} + \cdots - 6942619491638 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 268063867548 \nu^{18} - 975838766429 \nu^{17} - 5844382212784 \nu^{16} + \cdots + 2625408551097 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 386309035457 \nu^{18} + 1365268186173 \nu^{17} + 8692512067202 \nu^{16} + \cdots - 8519539396534 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 407596389827 \nu^{18} + 996406897672 \nu^{17} + 10303884149056 \nu^{16} + \cdots - 7512666382563 ) / 2584793613763 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 433257329190 \nu^{18} - 1312562946260 \nu^{17} - 10331601737881 \nu^{16} + \cdots + 7601132183693 ) / 2584793613763 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{11} - \beta_{6} + \beta_{5} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{14} + 11 \beta_{13} - \beta_{12} + 7 \beta_{11} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} - 11 \beta_{12} + \beta_{11} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{18} + 24 \beta_{17} - 15 \beta_{16} - 14 \beta_{14} + 94 \beta_{13} - 12 \beta_{12} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{18} - 15 \beta_{17} + 31 \beta_{16} + 14 \beta_{15} + 14 \beta_{14} - 18 \beta_{13} + \cdots + 567 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 78 \beta_{18} + 218 \beta_{17} - 153 \beta_{16} + 2 \beta_{15} - 142 \beta_{14} + 746 \beta_{13} + \cdots + 534 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 47 \beta_{18} - 163 \beta_{17} + 334 \beta_{16} + 143 \beta_{15} + 143 \beta_{14} - 224 \beta_{13} + \cdots + 3874 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 570 \beta_{18} + 1806 \beta_{17} - 1356 \beta_{16} + 33 \beta_{15} - 1270 \beta_{14} + 5763 \beta_{13} + \cdots + 3645 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 492 \beta_{18} - 1566 \beta_{17} + 3118 \beta_{16} + 1289 \beta_{15} + 1305 \beta_{14} - 2393 \beta_{13} + \cdots + 27441 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4103 \beta_{18} + 14404 \beta_{17} - 11290 \beta_{16} + 341 \beta_{15} - 10672 \beta_{14} + 44078 \beta_{13} + \cdots + 24525 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4351 \beta_{18} - 14133 \beta_{17} + 27117 \beta_{16} + 10875 \beta_{15} + 11326 \beta_{14} + \cdots + 198716 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 29563 \beta_{18} + 112951 \beta_{17} - 91244 \beta_{16} + 2777 \beta_{15} - 86679 \beta_{14} + \cdots + 163128 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 35212 \beta_{18} - 122951 \beta_{17} + 226841 \beta_{16} + 88185 \beta_{15} + 95848 \beta_{14} + \cdots + 1458972 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 214159 \beta_{18} + 879024 \beta_{17} - 726921 \beta_{16} + 18877 \beta_{15} - 690523 \beta_{14} + \cdots + 1070635 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 270373 \beta_{18} - 1044853 \beta_{17} + 1855069 \beta_{16} + 697382 \beta_{15} + 799771 \beta_{14} + \cdots + 10807509 \) 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
2.72847
2.69542
2.34443
2.13634
2.04293
1.98166
1.31423
1.04858
0.686089
0.488177
0.0681863
−0.436741
−0.556474
−0.976920
−1.62941
−1.82302
−2.08441
−2.21976
−2.80778
−2.72847 2.09414 5.44452 −3.87909 −5.71379 −0.707409 −9.39826 1.38542 10.5840
1.2 −2.69542 −1.41452 5.26531 1.92020 3.81272 −2.04756 −8.80138 −0.999141 −5.17575
1.3 −2.34443 −3.17986 3.49635 3.40234 7.45497 2.95885 −3.50809 7.11153 −7.97654
1.4 −2.13634 2.50862 2.56396 −0.997990 −5.35927 0.526710 −1.20482 3.29317 2.13205
1.5 −2.04293 0.810410 2.17358 0.258135 −1.65561 −3.70333 −0.354615 −2.34324 −0.527352
1.6 −1.98166 1.84905 1.92698 2.60057 −3.66419 2.18138 0.144707 0.418993 −5.15344
1.7 −1.31423 −1.97692 −0.272811 −1.38757 2.59812 −2.03012 2.98699 0.908205 1.82358
1.8 −1.04858 −3.34590 −0.900472 −4.37157 3.50846 3.70774 3.04139 8.19507 4.58396
1.9 −0.686089 −1.33733 −1.52928 3.77631 0.917525 −4.48234 2.42140 −1.21156 −2.59089
1.10 −0.488177 2.60530 −1.76168 1.92726 −1.27185 −0.259128 1.83637 3.78757 −0.940847
1.11 −0.0681863 0.0544406 −1.99535 −3.91488 −0.00371210 −0.303195 0.272428 −2.99704 0.266941
1.12 0.436741 −0.0103706 −1.80926 3.47884 −0.00452927 1.93516 −1.66366 −2.99989 1.51935
1.13 0.556474 2.53857 −1.69034 0.874867 1.41265 −4.50273 −2.05358 3.44434 0.486841
1.14 0.976920 0.347298 −1.04563 −2.45987 0.339282 4.72156 −2.97533 −2.87938 −2.40309
1.15 1.62941 −2.82994 0.654988 2.04510 −4.61114 3.70972 −2.19158 5.00854 3.33231
1.16 1.82302 3.09920 1.32341 −3.15385 5.64991 0.287436 −1.23344 6.60505 −5.74953
1.17 2.08441 1.49292 2.34475 2.98183 3.11185 −3.78550 0.718591 −0.771187 6.21535
1.18 2.21976 −2.99078 2.92734 −2.17512 −6.63880 −5.05675 2.05847 5.94474 −4.82825
1.19 2.80778 −0.314338 5.88364 −0.925514 −0.882592 −2.15050 10.9044 −2.90119 −2.59864
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(61\) \(-1\)

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 7381.2.a.i 19
11.b odd 2 1 671.2.a.c ✓ 19
33.d even 2 1 6039.2.a.k 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.c ✓ 19 11.b odd 2 1
6039.2.a.k 19 33.d even 2 1
7381.2.a.i 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7381))\):

\( T_{2}^{19} + 5 T_{2}^{18} - 18 T_{2}^{17} - 122 T_{2}^{16} + 78 T_{2}^{15} + 1177 T_{2}^{14} + 387 T_{2}^{13} + \cdots + 43 \) Copy content Toggle raw display
\( T_{7}^{19} + 9 T_{7}^{18} - 47 T_{7}^{17} - 602 T_{7}^{16} + 407 T_{7}^{15} + 15666 T_{7}^{14} + \cdots + 87296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} + 5 T^{18} + \cdots + 43 \) Copy content Toggle raw display
$3$ \( T^{19} - 43 T^{17} + \cdots + 5 \) Copy content Toggle raw display
$5$ \( T^{19} - 70 T^{17} + \cdots + 850564 \) Copy content Toggle raw display
$7$ \( T^{19} + 9 T^{18} + \cdots + 87296 \) Copy content Toggle raw display
$11$ \( T^{19} \) Copy content Toggle raw display
$13$ \( T^{19} + 8 T^{18} + \cdots - 966656 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 16020390437 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots + 7413317632 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots + 76028297216 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 11106421323508 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 7338377216 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots + 88495013888 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 26396783722496 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 25482644176 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 3936280228595 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots - 13303893868544 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 81\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T - 1)^{19} \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 54909964304384 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 7732360953856 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 38776064491520 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 11\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 19\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 21\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 45\!\cdots\!47 \) Copy content Toggle raw display
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