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Properties

Label 38.6.c.b
Level $38$
Weight $6$
Character orbit 38.c
Analytic conductor $6.095$
Analytic rank $0$
Dimension $8$
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] = [38,6,Mod(7,38)]
 
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([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), 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: \(6.09458515289\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 386x^{6} + 3436x^{5} + 128708x^{4} + 568528x^{3} + 7340704x^{2} - 19430784x + 211527936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \beta_{2} - 4) q^{2} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{3} - 16 \beta_{2} q^{4} + (\beta_{5} + \beta_{4} + 9 \beta_{2} - 9) q^{5} + ( - 12 \beta_{2} - 4 \beta_1) q^{6} + ( - \beta_{7} - \beta_{6} + \beta_{3} + \cdots + 9) q^{7}+ \cdots + ( - 2 \beta_{5} + 46 \beta_{2} - 14 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \beta_{2} - 4) q^{2} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{3} - 16 \beta_{2} q^{4} + (\beta_{5} + \beta_{4} + 9 \beta_{2} - 9) q^{5} + ( - 12 \beta_{2} - 4 \beta_1) q^{6} + ( - \beta_{7} - \beta_{6} + \beta_{3} + \cdots + 9) q^{7}+ \cdots + ( - 364 \beta_{7} - 1108 \beta_{5} + \cdots - 804 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{2} - 14 q^{3} - 64 q^{4} - 36 q^{5} - 56 q^{6} + 76 q^{7} + 512 q^{8} + 156 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{2} - 14 q^{3} - 64 q^{4} - 36 q^{5} - 56 q^{6} + 76 q^{7} + 512 q^{8} + 156 q^{9} - 144 q^{10} + 144 q^{11} + 448 q^{12} - 674 q^{13} - 152 q^{14} - 20 q^{15} - 1024 q^{16} + 522 q^{17} - 1248 q^{18} + 320 q^{19} + 1152 q^{20} - 770 q^{21} - 288 q^{22} - 204 q^{23} - 896 q^{24} - 2158 q^{25} + 5392 q^{26} + 13156 q^{27} - 608 q^{28} - 5712 q^{29} + 160 q^{30} - 2324 q^{31} - 4096 q^{32} - 15650 q^{33} + 2088 q^{34} - 4188 q^{35} + 2496 q^{36} - 25508 q^{37} - 2440 q^{38} + 44156 q^{39} - 2304 q^{40} - 3480 q^{41} - 3080 q^{42} + 4066 q^{43} - 1152 q^{44} + 51560 q^{45} + 1632 q^{46} - 45768 q^{47} - 3584 q^{48} + 1416 q^{49} + 17264 q^{50} + 1506 q^{51} - 10784 q^{52} + 45654 q^{53} - 26312 q^{54} + 36570 q^{55} + 4864 q^{56} - 66702 q^{57} + 45696 q^{58} - 84006 q^{59} - 320 q^{60} - 14012 q^{61} + 4648 q^{62} - 15128 q^{63} + 32768 q^{64} - 4020 q^{65} - 62600 q^{66} - 19046 q^{67} - 16704 q^{68} + 111592 q^{69} - 16752 q^{70} - 53274 q^{71} + 9984 q^{72} - 41084 q^{73} + 51016 q^{74} + 153216 q^{75} + 4640 q^{76} + 271524 q^{77} - 88312 q^{78} - 54170 q^{79} - 9216 q^{80} - 9528 q^{81} - 13920 q^{82} + 52392 q^{83} + 24640 q^{84} - 221850 q^{85} + 16264 q^{86} - 457120 q^{87} + 9216 q^{88} - 137862 q^{89} - 103120 q^{90} + 308516 q^{91} - 3264 q^{92} + 112746 q^{93} + 366144 q^{94} + 153078 q^{95} + 28672 q^{96} - 292388 q^{97} - 2832 q^{98} - 138328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 386x^{6} + 3436x^{5} + 128708x^{4} + 568528x^{3} + 7340704x^{2} - 19430784x + 211527936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 390471533 \nu^{7} + 3142267604 \nu^{6} + 128794858006 \nu^{5} + 2734091869688 \nu^{4} + \cdots + 21\!\cdots\!64 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6473945 \nu^{7} + 36183422 \nu^{6} - 2297742050 \nu^{5} - 13997685940 \nu^{4} + \cdots + 136296431422848 ) / 44885936230176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18756773 \nu^{7} + 22066664 \nu^{6} - 6657181370 \nu^{5} - 40555089316 \nu^{4} + \cdots + 20\!\cdots\!88 ) / 22442968115088 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14831978953 \nu^{7} - 117204668308 \nu^{6} - 4803969780662 \nu^{5} + \cdots - 79\!\cdots\!28 ) / 13\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35647275847 \nu^{7} - 1347014053248 \nu^{6} + 26507480732490 \nu^{5} + \cdots - 12\!\cdots\!44 ) / 90\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71425529135 \nu^{7} + 280447752060 \nu^{6} + 11494956177090 \nu^{5} + 561804417935112 \nu^{4} + \cdots + 19\!\cdots\!60 ) / 90\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{5} + 2\beta_{4} - 8\beta_{3} + 188\beta_{2} - 188 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 2\beta_{6} + 28\beta_{4} - 326\beta_{3} - 326\beta _1 - 1414 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} - 820\beta_{5} - 60100\beta_{2} - 5044\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 772\beta_{6} - 15008\beta_{5} - 15008\beta_{4} + 130764\beta_{3} - 911516\beta_{2} + 911516 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5744\beta_{7} + 5744\beta_{6} - 351576\beta_{4} + 2507520\beta_{3} + 2507520\beta _1 + 23936064 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 282648\beta_{7} + 7124496\beta_{5} + 455799624\beta_{2} + 56964328\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(21\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
10.6694 − 18.4800i
2.36148 − 4.09020i
−5.68703 + 9.85023i
−6.34386 + 10.9879i
10.6694 + 18.4800i
2.36148 + 4.09020i
−5.68703 − 9.85023i
−6.34386 − 10.9879i
−2.00000 − 3.46410i −12.1694 − 21.0781i −8.00000 + 13.8564i −28.6588 − 49.6385i −48.6777 + 84.3122i 33.7394 64.0000 −174.689 + 302.571i −114.635 + 198.554i
7.2 −2.00000 − 3.46410i −3.86148 − 6.68827i −8.00000 + 13.8564i 46.3693 + 80.3140i −15.4459 + 26.7531i 13.5472 64.0000 91.6780 − 158.791i 185.477 − 321.256i
7.3 −2.00000 − 3.46410i 4.18703 + 7.25215i −8.00000 + 13.8564i −12.5904 − 21.8073i 16.7481 − 29.0086i −187.086 64.0000 86.4375 − 149.714i −50.3618 + 87.2292i
7.4 −2.00000 − 3.46410i 4.84386 + 8.38982i −8.00000 + 13.8564i −23.1201 − 40.0451i 19.3755 − 33.5593i 177.800 64.0000 74.5740 − 129.166i −92.4803 + 160.181i
11.1 −2.00000 + 3.46410i −12.1694 + 21.0781i −8.00000 − 13.8564i −28.6588 + 49.6385i −48.6777 − 84.3122i 33.7394 64.0000 −174.689 − 302.571i −114.635 − 198.554i
11.2 −2.00000 + 3.46410i −3.86148 + 6.68827i −8.00000 − 13.8564i 46.3693 − 80.3140i −15.4459 − 26.7531i 13.5472 64.0000 91.6780 + 158.791i 185.477 + 321.256i
11.3 −2.00000 + 3.46410i 4.18703 − 7.25215i −8.00000 − 13.8564i −12.5904 + 21.8073i 16.7481 + 29.0086i −187.086 64.0000 86.4375 + 149.714i −50.3618 − 87.2292i
11.4 −2.00000 + 3.46410i 4.84386 − 8.38982i −8.00000 − 13.8564i −23.1201 + 40.0451i 19.3755 + 33.5593i 177.800 64.0000 74.5740 + 129.166i −92.4803 − 160.181i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.6.c.b ✓ 8
3.b odd 2 1 342.6.g.d 8
4.b odd 2 1 304.6.i.b 8
19.c even 3 1 inner 38.6.c.b ✓ 8
19.c even 3 1 722.6.a.j 4
19.d odd 6 1 722.6.a.g 4
57.h odd 6 1 342.6.g.d 8
76.g odd 6 1 304.6.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.c.b ✓ 8 1.a even 1 1 trivial
38.6.c.b ✓ 8 19.c even 3 1 inner
304.6.i.b 8 4.b odd 2 1
304.6.i.b 8 76.g odd 6 1
342.6.g.d 8 3.b odd 2 1
342.6.g.d 8 57.h odd 6 1
722.6.a.g 4 19.d odd 6 1
722.6.a.j 4 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 14 T_{3}^{7} + 506 T_{3}^{6} - 2752 T_{3}^{5} + 91967 T_{3}^{4} - 180832 T_{3}^{3} + \cdots + 232532001 \) acting on \(S_{6}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 14 T^{7} + \cdots + 232532001 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 38307063132900 \) Copy content Toggle raw display
$7$ \( (T^{4} - 38 T^{3} + \cdots - 15204096)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 72 T^{3} + \cdots - 423612144)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 9213535294816)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 27\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 28\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 61\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 61\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
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