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Properties

Label 38.5.d.a
Level $38$
Weight $5$
Character orbit 38.d
Analytic conductor $3.928$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,5,Mod(27,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.27");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 38.d (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(3.92805859719\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 1024 x^{14} - 7028 x^{13} + 404698 x^{12} - 2337188 x^{11} + 77836288 x^{10} + \cdots + 23840536514409 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{5}) q^{2} + (\beta_{3} - 1) q^{3} + 8 \beta_{7} q^{4} + ( - \beta_{12} + 2 \beta_{7} + \cdots - 2) q^{5}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{5}) q^{2} + (\beta_{3} - 1) q^{3} + 8 \beta_{7} q^{4} + ( - \beta_{12} + 2 \beta_{7} + \cdots - 2) q^{5}+ \cdots + ( - 71 \beta_{15} - 114 \beta_{14} + \cdots - 71) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{3} + 64 q^{4} - 18 q^{5} - 16 q^{6} + 72 q^{7} + 352 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{3} + 64 q^{4} - 18 q^{5} - 16 q^{6} + 72 q^{7} + 352 q^{9} - 84 q^{11} + 450 q^{13} + 288 q^{14} - 390 q^{15} - 512 q^{16} + 606 q^{17} - 306 q^{19} - 288 q^{20} - 2160 q^{21} - 1680 q^{22} - 54 q^{23} + 128 q^{24} - 434 q^{25} + 1344 q^{26} + 288 q^{28} - 4914 q^{29} + 2752 q^{30} + 7890 q^{33} - 1536 q^{34} + 2328 q^{35} - 2816 q^{36} + 1344 q^{38} + 7620 q^{39} - 1692 q^{41} + 2080 q^{42} - 7402 q^{43} - 336 q^{44} - 16720 q^{45} + 3198 q^{47} + 768 q^{48} + 24816 q^{49} + 10710 q^{51} + 3600 q^{52} + 3870 q^{53} - 16 q^{54} - 13588 q^{55} + 3702 q^{57} - 1728 q^{58} - 18288 q^{59} - 3120 q^{60} - 6522 q^{61} - 6144 q^{62} - 15676 q^{63} - 8192 q^{64} + 4960 q^{66} - 30168 q^{67} + 9696 q^{68} + 15360 q^{70} + 35874 q^{71} + 5376 q^{72} - 8080 q^{73} - 9120 q^{74} + 480 q^{76} + 34560 q^{77} - 46560 q^{78} - 30738 q^{79} - 1152 q^{80} - 30920 q^{81} + 6720 q^{82} - 1476 q^{83} + 33626 q^{85} + 288 q^{86} + 113100 q^{87} + 19782 q^{89} + 44256 q^{90} - 34260 q^{91} + 432 q^{92} - 4272 q^{93} - 23706 q^{95} + 2048 q^{96} - 9936 q^{97} + 12672 q^{98} + 3848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 1024 x^{14} - 7028 x^{13} + 404698 x^{12} - 2337188 x^{11} + 77836288 x^{10} + \cdots + 23840536514409 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 35\!\cdots\!67 \nu^{14} + \cdots + 49\!\cdots\!15 ) / 32\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!90 \nu^{14} + \cdots - 75\!\cdots\!67 ) / 64\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\!\cdots\!39 \nu^{15} + \cdots + 91\!\cdots\!54 ) / 18\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 89\!\cdots\!10 \nu^{15} + \cdots + 12\!\cdots\!61 ) / 10\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 76\!\cdots\!05 \nu^{15} + \cdots + 10\!\cdots\!51 ) / 90\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44\!\cdots\!55 \nu^{15} + \cdots - 19\!\cdots\!91 ) / 52\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!12 \nu^{15} + \cdots + 13\!\cdots\!23 ) / 10\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!75 \nu^{15} + \cdots - 13\!\cdots\!65 ) / 90\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 89\!\cdots\!10 \nu^{15} + \cdots - 12\!\cdots\!61 ) / 52\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 64\!\cdots\!72 \nu^{15} + \cdots + 19\!\cdots\!55 ) / 68\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64\!\cdots\!72 \nu^{15} + \cdots + 31\!\cdots\!52 ) / 68\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 88\!\cdots\!28 \nu^{15} + \cdots - 40\!\cdots\!25 ) / 68\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 21\!\cdots\!74 \nu^{15} + \cdots - 11\!\cdots\!49 ) / 68\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!74 \nu^{15} + \cdots + 98\!\cdots\!46 ) / 68\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!84 \nu^{15} + \cdots - 88\!\cdots\!42 ) / 34\!\cdots\!18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{13} + 2\beta_{9} - \beta_{7} - 2\beta_{6} - 7\beta_{5} - 2\beta_{3} + \beta_{2} + \beta _1 - 122 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 44 \beta_{12} + \beta_{11} - \beta_{10} + 335 \beta_{9} + \cdots - 175 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 10 \beta_{15} - 284 \beta_{14} - 286 \beta_{13} + 44 \beta_{12} - 4 \beta_{11} - 6 \beta_{10} + \cdots + 27478 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2416 \beta_{15} - 3524 \beta_{14} + 664 \beta_{13} - 12266 \beta_{12} - 502 \beta_{11} + 452 \beta_{10} + \cdots + 54280 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3649 \beta_{15} + 76808 \beta_{14} + 83095 \beta_{13} - 18509 \beta_{12} + 2576 \beta_{11} + \cdots - 7129316 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 450806 \beta_{15} + 1700462 \beta_{14} - 571124 \beta_{13} + 3275725 \beta_{12} + 176728 \beta_{11} + \cdots - 13866106 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 918664 \beta_{15} - 20280184 \beta_{14} - 24852700 \beta_{13} + 6637928 \beta_{12} + \cdots + 1970744963 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 57753080 \beta_{15} - 671360072 \beta_{14} + 258376064 \beta_{13} - 921077488 \beta_{12} + \cdots + 3579331694 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 151298273 \beta_{15} + 5242407157 \beta_{14} + 7601085386 \beta_{13} - 2352607963 \beta_{12} + \cdots - 565056048674 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2162257528 \beta_{15} + 241933383058 \beta_{14} - 96856824134 \beta_{13} + 270375367717 \beta_{12} + \cdots - 1043554554971 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4807253182 \beta_{15} - 1320323145370 \beta_{14} - 2362714812326 \beta_{13} + 837114724480 \beta_{12} + \cdots + 165852806135551 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 5327342762360 \beta_{15} - 82924460318752 \beta_{14} + 33149082552820 \beta_{13} + \cdots + 354690150531692 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 18713495305312 \beta_{15} + 320402143333631 \beta_{14} + 742548179127581 \beta_{13} + \cdots - 49\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 26\!\cdots\!98 \beta_{15} + \cdots - 13\!\cdots\!01 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
0.500000 + 17.6194i
0.500000 + 2.12625i
0.500000 − 0.961202i
0.500000 − 15.9560i
0.500000 + 11.2515i
0.500000 + 6.86019i
0.500000 − 6.55779i
0.500000 − 14.3823i
0.500000 − 17.6194i
0.500000 − 2.12625i
0.500000 + 0.961202i
0.500000 + 15.9560i
0.500000 − 11.2515i
0.500000 − 6.86019i
0.500000 + 6.55779i
0.500000 + 14.3823i
−2.44949 − 1.41421i −14.7841 − 8.53562i 4.00000 + 6.92820i −3.60911 + 6.25116i 24.1424 + 41.8158i 27.7970 − 22.6274i 105.214 + 182.235i 17.6809 − 10.2081i
27.2 −2.44949 − 1.41421i −1.36664 − 0.789030i 4.00000 + 6.92820i 11.3097 − 19.5890i 2.23171 + 3.86544i −73.1169 − 22.6274i −39.2549 − 67.9914i −55.4061 + 31.9887i
27.3 −2.44949 − 1.41421i 1.30717 + 0.754695i 4.00000 + 6.92820i 1.35492 − 2.34679i −2.13460 − 3.69723i 79.4947 − 22.6274i −39.3609 − 68.1750i −6.63772 + 3.83229i
27.4 −2.44949 − 1.41421i 14.2931 + 8.25212i 4.00000 + 6.92820i −13.5555 + 23.4789i −23.3405 − 40.4270i −45.5687 − 22.6274i 95.6949 + 165.748i 66.4083 − 38.3408i
27.5 2.44949 + 1.41421i −11.7188 − 6.76588i 4.00000 + 6.92820i −15.6059 + 27.0302i −19.1368 − 33.1459i −61.1277 22.6274i 51.0542 + 88.4285i −76.4529 + 44.1401i
27.6 2.44949 + 1.41421i −7.91584 − 4.57021i 4.00000 + 6.92820i 19.7357 − 34.1832i −12.9265 − 22.3894i 91.5785 22.6274i 1.27372 + 2.20614i 96.6847 − 55.8209i
27.7 2.44949 + 1.41421i 3.70447 + 2.13878i 4.00000 + 6.92820i −17.7629 + 30.7663i 6.04937 + 10.4778i 57.6374 22.6274i −31.3513 − 54.3020i −87.0203 + 50.2412i
27.8 2.44949 + 1.41421i 10.4807 + 6.05105i 4.00000 + 6.92820i 9.13314 − 15.8191i 17.1150 + 29.6440i −40.6944 22.6274i 32.7305 + 56.6909i 44.7431 − 25.8324i
31.1 −2.44949 + 1.41421i −14.7841 + 8.53562i 4.00000 − 6.92820i −3.60911 − 6.25116i 24.1424 − 41.8158i 27.7970 22.6274i 105.214 − 182.235i 17.6809 + 10.2081i
31.2 −2.44949 + 1.41421i −1.36664 + 0.789030i 4.00000 − 6.92820i 11.3097 + 19.5890i 2.23171 − 3.86544i −73.1169 22.6274i −39.2549 + 67.9914i −55.4061 − 31.9887i
31.3 −2.44949 + 1.41421i 1.30717 − 0.754695i 4.00000 − 6.92820i 1.35492 + 2.34679i −2.13460 + 3.69723i 79.4947 22.6274i −39.3609 + 68.1750i −6.63772 − 3.83229i
31.4 −2.44949 + 1.41421i 14.2931 − 8.25212i 4.00000 − 6.92820i −13.5555 − 23.4789i −23.3405 + 40.4270i −45.5687 22.6274i 95.6949 − 165.748i 66.4083 + 38.3408i
31.5 2.44949 − 1.41421i −11.7188 + 6.76588i 4.00000 − 6.92820i −15.6059 − 27.0302i −19.1368 + 33.1459i −61.1277 − 22.6274i 51.0542 − 88.4285i −76.4529 − 44.1401i
31.6 2.44949 − 1.41421i −7.91584 + 4.57021i 4.00000 − 6.92820i 19.7357 + 34.1832i −12.9265 + 22.3894i 91.5785 − 22.6274i 1.27372 − 2.20614i 96.6847 + 55.8209i
31.7 2.44949 − 1.41421i 3.70447 − 2.13878i 4.00000 − 6.92820i −17.7629 − 30.7663i 6.04937 − 10.4778i 57.6374 − 22.6274i −31.3513 + 54.3020i −87.0203 − 50.2412i
31.8 2.44949 − 1.41421i 10.4807 − 6.05105i 4.00000 − 6.92820i 9.13314 + 15.8191i 17.1150 − 29.6440i −40.6944 − 22.6274i 32.7305 − 56.6909i 44.7431 + 25.8324i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.5.d.a ✓ 16
3.b odd 2 1 342.5.m.c 16
4.b odd 2 1 304.5.r.c 16
19.d odd 6 1 inner 38.5.d.a ✓ 16
57.f even 6 1 342.5.m.c 16
76.f even 6 1 304.5.r.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.d.a ✓ 16 1.a even 1 1 trivial
38.5.d.a ✓ 16 19.d odd 6 1 inner
304.5.r.c 16 4.b odd 2 1
304.5.r.c 16 76.f even 6 1
342.5.m.c 16 3.b odd 2 1
342.5.m.c 16 57.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 64)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18463942586961 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 96669830323984)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 86\!\cdots\!40)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 84\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 83\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 81\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 22\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 65\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 97\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 65\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
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