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Properties

Label 2738.2.a.x
Level $2738$
Weight $2$
Character orbit 2738.a
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
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] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.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: \(21.8630400734\)
Analytic rank: \(0\)
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: \( x^{18} - 8 x^{17} - 8 x^{16} + 209 x^{15} - 253 x^{14} - 1996 x^{13} + 4196 x^{12} + 8667 x^{11} + \cdots + 113 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{12} - \beta_1 + 1) q^{3} + q^{4} + (\beta_{15} + \beta_{14} - \beta_{12} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{14} + \beta_{12} - \beta_{11} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{12} - \beta_1 + 1) q^{3} + q^{4} + (\beta_{15} + \beta_{14} - \beta_{12} + \cdots + 1) q^{5}+ \cdots + (2 \beta_{17} - 3 \beta_{15} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{2} + 8 q^{3} + 18 q^{4} + 9 q^{5} + 8 q^{6} + 18 q^{7} + 18 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{2} + 8 q^{3} + 18 q^{4} + 9 q^{5} + 8 q^{6} + 18 q^{7} + 18 q^{8} + 26 q^{9} + 9 q^{10} + 10 q^{11} + 8 q^{12} + 9 q^{13} + 18 q^{14} - 4 q^{15} + 18 q^{16} + 13 q^{17} + 26 q^{18} + 2 q^{19} + 9 q^{20} + 24 q^{21} + 10 q^{22} - 11 q^{23} + 8 q^{24} + 49 q^{25} + 9 q^{26} + 29 q^{27} + 18 q^{28} + 30 q^{29} - 4 q^{30} - 8 q^{31} + 18 q^{32} + 42 q^{33} + 13 q^{34} + 25 q^{35} + 26 q^{36} + 2 q^{38} - 45 q^{39} + 9 q^{40} + 5 q^{41} + 24 q^{42} - 3 q^{43} + 10 q^{44} - 30 q^{45} - 11 q^{46} + 37 q^{47} + 8 q^{48} + 50 q^{49} + 49 q^{50} + 10 q^{51} + 9 q^{52} + 25 q^{53} + 29 q^{54} - 44 q^{55} + 18 q^{56} + 22 q^{57} + 30 q^{58} + 26 q^{59} - 4 q^{60} + 27 q^{61} - 8 q^{62} + 74 q^{63} + 18 q^{64} - 16 q^{65} + 42 q^{66} + 23 q^{67} + 13 q^{68} - 2 q^{69} + 25 q^{70} - 25 q^{71} + 26 q^{72} + 77 q^{73} - q^{75} + 2 q^{76} - 6 q^{77} - 45 q^{78} - 13 q^{79} + 9 q^{80} + 38 q^{81} + 5 q^{82} - 10 q^{83} + 24 q^{84} + 16 q^{85} - 3 q^{86} - 55 q^{87} + 10 q^{88} + 55 q^{89} - 30 q^{90} + 12 q^{91} - 11 q^{92} - 58 q^{93} + 37 q^{94} - 18 q^{95} + 8 q^{96} - 59 q^{97} + 50 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 8 x^{17} - 8 x^{16} + 209 x^{15} - 253 x^{14} - 1996 x^{13} + 4196 x^{12} + 8667 x^{11} + \cdots + 113 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 288951249349 \nu^{17} - 3591379024255 \nu^{16} + 5588056150692 \nu^{15} + \cdots + 481706958614797 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 339789756367 \nu^{17} - 4967748483248 \nu^{16} + 52736315932460 \nu^{15} + \cdots + 569976441271657 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1076022812465 \nu^{17} + 6107715698223 \nu^{16} + 21640413863764 \nu^{15} + \cdots - 13\!\cdots\!79 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1542780927592 \nu^{17} - 9170574598705 \nu^{16} - 28764108899129 \nu^{15} + \cdots + 12\!\cdots\!17 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1549894163847 \nu^{17} + 9333877080572 \nu^{16} + 29760010628922 \nu^{15} + \cdots - 32933534425465 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1997441834796 \nu^{17} + 11105151266670 \nu^{16} + 43686727585558 \nu^{15} + \cdots + 429365903773692 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2063781192451 \nu^{17} - 6051607290075 \nu^{16} - 81391785790979 \nu^{15} + \cdots + 16075387819939 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2977590344659 \nu^{17} + 17346377645528 \nu^{16} + 60178134110193 \nu^{15} + \cdots - 321722815924100 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5193825773890 \nu^{17} + 33161215679707 \nu^{16} + 86575370956769 \nu^{15} + \cdots - 239751347303384 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6655598780904 \nu^{17} - 31157708048006 \nu^{16} - 185835614582087 \nu^{15} + \cdots - 502765674810731 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6762901520895 \nu^{17} - 42145279389393 \nu^{16} - 120457688091949 \nu^{15} + \cdots - 671196427105821 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11661402651090 \nu^{17} - 79767685631079 \nu^{16} - 160932521842093 \nu^{15} + \cdots + 554147111971995 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11957932777767 \nu^{17} - 66354475924789 \nu^{16} - 265735968553660 \nu^{15} + \cdots - 10\!\cdots\!86 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 16593280437543 \nu^{17} + 100278150348575 \nu^{16} + 315461018420552 \nu^{15} + \cdots - 11\!\cdots\!58 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 19304209555584 \nu^{17} - 121340222289561 \nu^{16} - 336951345493012 \nu^{15} + \cdots - 532122377771700 ) / 299494704712751 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 23081286448329 \nu^{17} - 134279987934766 \nu^{16} - 472778247498487 \nu^{15} + \cdots - 260913772800456 ) / 299494704712751 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - 2\beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} - \beta_{3} - \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - 2 \beta_{16} - 2 \beta_{15} + 9 \beta_{14} + 3 \beta_{12} - 16 \beta_{11} - 10 \beta_{10} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{17} + 11 \beta_{16} + 11 \beta_{15} - 13 \beta_{14} - \beta_{13} + 28 \beta_{12} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 20 \beta_{17} - 24 \beta_{16} - 29 \beta_{15} + 75 \beta_{14} - 2 \beta_{13} + 50 \beta_{12} + \cdots + 294 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 134 \beta_{17} + 109 \beta_{16} + 98 \beta_{15} - 142 \beta_{14} - 17 \beta_{13} + 317 \beta_{12} + \cdots + 183 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 273 \beta_{17} - 216 \beta_{16} - 317 \beta_{15} + 614 \beta_{14} - 32 \beta_{13} + 616 \beta_{12} + \cdots + 2549 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1451 \beta_{17} + 1080 \beta_{16} + 843 \beta_{15} - 1478 \beta_{14} - 192 \beta_{13} + 3338 \beta_{12} + \cdots + 1717 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3282 \beta_{17} - 1707 \beta_{16} - 3156 \beta_{15} + 4942 \beta_{14} - 345 \beta_{13} + 6902 \beta_{12} + \cdots + 22832 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 15402 \beta_{17} + 10783 \beta_{16} + 7257 \beta_{15} - 15083 \beta_{14} - 1805 \beta_{13} + \cdots + 16778 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 37278 \beta_{17} - 12128 \beta_{16} - 30258 \beta_{15} + 38910 \beta_{14} - 3016 \beta_{13} + \cdots + 208575 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 161516 \beta_{17} + 107998 \beta_{16} + 62841 \beta_{15} - 152320 \beta_{14} - 14923 \beta_{13} + \cdots + 169512 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 410845 \beta_{17} - 74224 \beta_{16} - 285391 \beta_{15} + 297077 \beta_{14} - 21476 \beta_{13} + \cdots + 1931025 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1681533 \beta_{17} + 1081255 \beta_{16} + 546312 \beta_{15} - 1528015 \beta_{14} - 107233 \beta_{13} + \cdots + 1750513 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 4446713 \beta_{17} - 316333 \beta_{16} - 2672210 \beta_{15} + 2165884 \beta_{14} - 102250 \beta_{13} + \cdots + 18060389 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 17428676 \beta_{17} + 10805515 \beta_{16} + 4755694 \beta_{15} - 15256403 \beta_{14} + \cdots + 18301860 \) 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
2.24542
3.18423
2.72365
1.39715
0.0277653
2.93806
2.26200
3.21182
−1.00049
2.01728
−1.75172
0.561002
−3.00437
−0.771034
−1.88881
−0.844166
−2.98762
−0.320182
1.00000 −3.00437 1.00000 4.06246 −3.00437 0.951094 1.00000 6.02625 4.06246
1.2 1.00000 −2.98762 1.00000 −2.67223 −2.98762 2.38780 1.00000 5.92589 −2.67223
1.3 1.00000 −1.88881 1.00000 −0.545033 −1.88881 −1.80620 1.00000 0.567610 −0.545033
1.4 1.00000 −1.75172 1.00000 −2.09131 −1.75172 1.21116 1.00000 0.0685155 −2.09131
1.5 1.00000 −1.00049 1.00000 2.36317 −1.00049 3.55615 1.00000 −1.99902 2.36317
1.6 1.00000 −0.844166 1.00000 2.97364 −0.844166 4.24887 1.00000 −2.28738 2.97364
1.7 1.00000 −0.771034 1.00000 −0.437041 −0.771034 −4.61128 1.00000 −2.40551 −0.437041
1.8 1.00000 −0.320182 1.00000 3.73221 −0.320182 2.93372 1.00000 −2.89748 3.73221
1.9 1.00000 0.0277653 1.00000 −3.81163 0.0277653 −3.33489 1.00000 −2.99923 −3.81163
1.10 1.00000 0.561002 1.00000 3.29795 0.561002 −3.86349 1.00000 −2.68528 3.29795
1.11 1.00000 1.39715 1.00000 3.69423 1.39715 −0.398050 1.00000 −1.04796 3.69423
1.12 1.00000 2.01728 1.00000 −2.91611 2.01728 5.00881 1.00000 1.06941 −2.91611
1.13 1.00000 2.24542 1.00000 0.488539 2.24542 3.12577 1.00000 2.04193 0.488539
1.14 1.00000 2.26200 1.00000 3.49383 2.26200 4.18290 1.00000 2.11666 3.49383
1.15 1.00000 2.72365 1.00000 2.50059 2.72365 −0.328765 1.00000 4.41829 2.50059
1.16 1.00000 2.93806 1.00000 −3.37430 2.93806 2.74077 1.00000 5.63221 −3.37430
1.17 1.00000 3.18423 1.00000 0.335004 3.18423 3.81934 1.00000 7.13933 0.335004
1.18 1.00000 3.21182 1.00000 −2.09398 3.21182 −1.82370 1.00000 7.31576 −2.09398
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(37\) \(1\)

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 2738.2.a.x yes 18
37.b even 2 1 2738.2.a.w ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2738.2.a.w ✓ 18 37.b even 2 1
2738.2.a.x yes 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\):

\( T_{3}^{18} - 8 T_{3}^{17} - 8 T_{3}^{16} + 209 T_{3}^{15} - 253 T_{3}^{14} - 1996 T_{3}^{13} + \cdots + 113 \) Copy content Toggle raw display
\( T_{5}^{18} - 9 T_{5}^{17} - 29 T_{5}^{16} + 455 T_{5}^{15} - 78 T_{5}^{14} - 9261 T_{5}^{13} + \cdots + 194048 \) Copy content Toggle raw display
\( T_{7}^{18} - 18 T_{7}^{17} + 74 T_{7}^{16} + 524 T_{7}^{15} - 5105 T_{7}^{14} + 4893 T_{7}^{13} + \cdots - 2140672 \) Copy content Toggle raw display
\( T_{13}^{18} - 9 T_{13}^{17} - 105 T_{13}^{16} + 1082 T_{13}^{15} + 3874 T_{13}^{14} - 50966 T_{13}^{13} + \cdots + 66558464 \) Copy content Toggle raw display
\( T_{17}^{18} - 13 T_{17}^{17} - 78 T_{17}^{16} + 1565 T_{17}^{15} + 593 T_{17}^{14} - 73468 T_{17}^{13} + \cdots + 278237 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 8 T^{17} + \cdots + 113 \) Copy content Toggle raw display
$5$ \( T^{18} - 9 T^{17} + \cdots + 194048 \) Copy content Toggle raw display
$7$ \( T^{18} - 18 T^{17} + \cdots - 2140672 \) Copy content Toggle raw display
$11$ \( T^{18} - 10 T^{17} + \cdots - 1678687 \) Copy content Toggle raw display
$13$ \( T^{18} - 9 T^{17} + \cdots + 66558464 \) Copy content Toggle raw display
$17$ \( T^{18} - 13 T^{17} + \cdots + 278237 \) Copy content Toggle raw display
$19$ \( T^{18} - 2 T^{17} + \cdots + 153521 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 702069248 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 755457536 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 12374814208 \) Copy content Toggle raw display
$37$ \( T^{18} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 11940934483 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 605849481581 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 35338905088 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 53823866757632 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 5283151765979 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 147836571136 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 26881668076057 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 42725861888 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 13545811777 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 862427513344 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 40118279748961 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 21059632979023 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 160408755241271 \) Copy content Toggle raw display
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