Properties

Label 138.6.a.i
Level $138$
Weight $6$
Character orbit 138.a
Self dual yes
Analytic conductor $22.133$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,6,Mod(1,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 138.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: \(22.1329671342\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8369x^{2} - 182616x - 370980 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( - \beta_1 + 13) q^{5} + 36 q^{6} + ( - \beta_{2} + 87) q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( - \beta_1 + 13) q^{5} + 36 q^{6} + ( - \beta_{2} + 87) q^{7} + 64 q^{8} + 81 q^{9} + ( - 4 \beta_1 + 52) q^{10} + (2 \beta_{3} + 3 \beta_{2} + \beta_1) q^{11} + 144 q^{12} + ( - 5 \beta_{3} - \beta_{2} + 5 \beta_1 + 312) q^{13} + ( - 4 \beta_{2} + 348) q^{14} + ( - 9 \beta_1 + 117) q^{15} + 256 q^{16} + (11 \beta_{2} + 10 \beta_1 - 9) q^{17} + 324 q^{18} + (10 \beta_{3} + 2 \beta_{2} + 19 \beta_1 + 397) q^{19} + ( - 16 \beta_1 + 208) q^{20} + ( - 9 \beta_{2} + 783) q^{21} + (8 \beta_{3} + 12 \beta_{2} + 4 \beta_1) q^{22} + 529 q^{23} + 576 q^{24} + ( - 19 \beta_{3} - 31 \beta_{2} + 27 \beta_1 + 2217) q^{25} + ( - 20 \beta_{3} - 4 \beta_{2} + 20 \beta_1 + 1248) q^{26} + 729 q^{27} + ( - 16 \beta_{2} + 1392) q^{28} + ( - 2 \beta_{3} - 6 \beta_{2} - 40 \beta_1 + 552) q^{29} + ( - 36 \beta_1 + 468) q^{30} + (28 \beta_{3} + 12 \beta_{2} + 38 \beta_1 + 422) q^{31} + 1024 q^{32} + (18 \beta_{3} + 27 \beta_{2} + 9 \beta_1) q^{33} + (44 \beta_{2} + 40 \beta_1 - 36) q^{34} + ( - 5 \beta_{3} - 51 \beta_{2} - 143 \beta_1 + 1348) q^{35} + 1296 q^{36} + ( - 10 \beta_{3} + 63 \beta_{2} - 61 \beta_1 - 2432) q^{37} + (40 \beta_{3} + 8 \beta_{2} + 76 \beta_1 + 1588) q^{38} + ( - 45 \beta_{3} - 9 \beta_{2} + 45 \beta_1 + 2808) q^{39} + ( - 64 \beta_1 + 832) q^{40} + (8 \beta_{3} + 80 \beta_{2} - 80 \beta_1 - 2978) q^{41} + ( - 36 \beta_{2} + 3132) q^{42} + ( - 28 \beta_{3} - 76 \beta_{2} - 21 \beta_1 + 3415) q^{43} + (32 \beta_{3} + 48 \beta_{2} + 16 \beta_1) q^{44} + ( - 81 \beta_1 + 1053) q^{45} + 2116 q^{46} + (47 \beta_{3} - 25 \beta_{2} + 115 \beta_1 + 2740) q^{47} + 2304 q^{48} + ( - 45 \beta_{3} - 107 \beta_{2} - 85 \beta_1 + 2775) q^{49} + ( - 76 \beta_{3} - 124 \beta_{2} + 108 \beta_1 + 8868) q^{50} + (99 \beta_{2} + 90 \beta_1 - 81) q^{51} + ( - 80 \beta_{3} - 16 \beta_{2} + 80 \beta_1 + 4992) q^{52} + (20 \beta_{3} - 136 \beta_{2} + 159 \beta_1 + 6761) q^{53} + 2916 q^{54} + (94 \beta_{3} + 196 \beta_{2} + 244 \beta_1 - 7056) q^{55} + ( - 64 \beta_{2} + 5568) q^{56} + (90 \beta_{3} + 18 \beta_{2} + 171 \beta_1 + 3573) q^{57} + ( - 8 \beta_{3} - 24 \beta_{2} - 160 \beta_1 + 2208) q^{58} + ( - 163 \beta_{3} + 153 \beta_{2} + 325 \beta_1 - 2196) q^{59} + ( - 144 \beta_1 + 1872) q^{60} + (100 \beta_{3} + 71 \beta_{2} - 121 \beta_1 - 9480) q^{61} + (112 \beta_{3} + 48 \beta_{2} + 152 \beta_1 + 1688) q^{62} + ( - 81 \beta_{2} + 7047) q^{63} + 4096 q^{64} + ( - 60 \beta_{3} + 74 \beta_{2} - 858 \beta_1 - 18512) q^{65} + (72 \beta_{3} + 108 \beta_{2} + 36 \beta_1) q^{66} + (202 \beta_{3} - 180 \beta_{2} - 369 \beta_1 - 5909) q^{67} + (176 \beta_{2} + 160 \beta_1 - 144) q^{68} + 4761 q^{69} + ( - 20 \beta_{3} - 204 \beta_{2} - 572 \beta_1 + 5392) q^{70} + ( - 230 \beta_{3} - 294 \beta_{2} + 398 \beta_1 + 8016) q^{71} + 5184 q^{72} + ( - 308 \beta_{3} - 86 \beta_{2} - 158 \beta_1 - 7318) q^{73} + ( - 40 \beta_{3} + 252 \beta_{2} - 244 \beta_1 - 9728) q^{74} + ( - 171 \beta_{3} - 279 \beta_{2} + 243 \beta_1 + 19953) q^{75} + (160 \beta_{3} + 32 \beta_{2} + 304 \beta_1 + 6352) q^{76} + (336 \beta_{3} + 322 \beta_{2} + 398 \beta_1 - 29952) q^{77} + ( - 180 \beta_{3} - 36 \beta_{2} + 180 \beta_1 + 11232) q^{78} + ( - 210 \beta_{3} - 27 \beta_{2} + 238 \beta_1 - 6653) q^{79} + ( - 256 \beta_1 + 3328) q^{80} + 6561 q^{81} + (32 \beta_{3} + 320 \beta_{2} - 320 \beta_1 - 11912) q^{82} + (296 \beta_{3} - 77 \beta_{2} - 71 \beta_1 - 29644) q^{83} + ( - 144 \beta_{2} + 12528) q^{84} + (245 \beta_{3} + 871 \beta_{2} + 225 \beta_1 - 54234) q^{85} + ( - 112 \beta_{3} - 304 \beta_{2} - 84 \beta_1 + 13660) q^{86} + ( - 18 \beta_{3} - 54 \beta_{2} - 360 \beta_1 + 4968) q^{87} + (128 \beta_{3} + 192 \beta_{2} + 64 \beta_1) q^{88} + ( - 152 \beta_{3} - 303 \beta_{2} - 892 \beta_1 - 37581) q^{89} + ( - 324 \beta_1 + 4212) q^{90} + ( - 510 \beta_{3} - 702 \beta_{2} + 630 \beta_1 + 22312) q^{91} + 8464 q^{92} + (252 \beta_{3} + 108 \beta_{2} + 342 \beta_1 + 3798) q^{93} + (188 \beta_{3} - 100 \beta_{2} + 460 \beta_1 + 10960) q^{94} + (671 \beta_{3} + 751 \beta_{2} - 465 \beta_1 - 99720) q^{95} + 9216 q^{96} + (132 \beta_{3} - 424 \beta_{2} + 530 \beta_1 - 3692) q^{97} + ( - 180 \beta_{3} - 428 \beta_{2} - 340 \beta_1 + 11100) q^{98} + (162 \beta_{3} + 243 \beta_{2} + 81 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} + 36 q^{3} + 64 q^{4} + 54 q^{5} + 144 q^{6} + 348 q^{7} + 256 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{2} + 36 q^{3} + 64 q^{4} + 54 q^{5} + 144 q^{6} + 348 q^{7} + 256 q^{8} + 324 q^{9} + 216 q^{10} - 6 q^{11} + 576 q^{12} + 1248 q^{13} + 1392 q^{14} + 486 q^{15} + 1024 q^{16} - 56 q^{17} + 1296 q^{18} + 1530 q^{19} + 864 q^{20} + 3132 q^{21} - 24 q^{22} + 2116 q^{23} + 2304 q^{24} + 8852 q^{25} + 4992 q^{26} + 2916 q^{27} + 5568 q^{28} + 2292 q^{29} + 1944 q^{30} + 1556 q^{31} + 4096 q^{32} - 54 q^{33} - 224 q^{34} + 5688 q^{35} + 5184 q^{36} - 9586 q^{37} + 6120 q^{38} + 11232 q^{39} + 3456 q^{40} - 11768 q^{41} + 12528 q^{42} + 13758 q^{43} - 96 q^{44} + 4374 q^{45} + 8464 q^{46} + 10636 q^{47} + 9216 q^{48} + 11360 q^{49} + 35408 q^{50} - 504 q^{51} + 19968 q^{52} + 26686 q^{53} + 11664 q^{54} - 28900 q^{55} + 22272 q^{56} + 13770 q^{57} + 9168 q^{58} - 9108 q^{59} + 7776 q^{60} - 37878 q^{61} + 6224 q^{62} + 28188 q^{63} + 16384 q^{64} - 72212 q^{65} - 216 q^{66} - 23302 q^{67} - 896 q^{68} + 19044 q^{69} + 22752 q^{70} + 31728 q^{71} + 20736 q^{72} - 28340 q^{73} - 38344 q^{74} + 79668 q^{75} + 24480 q^{76} - 121276 q^{77} + 44928 q^{78} - 26668 q^{79} + 13824 q^{80} + 26244 q^{81} - 47072 q^{82} - 119026 q^{83} + 50112 q^{84} - 217876 q^{85} + 55032 q^{86} + 20628 q^{87} - 384 q^{88} - 148236 q^{89} + 17496 q^{90} + 89008 q^{91} + 33856 q^{92} + 14004 q^{93} + 42544 q^{94} - 399292 q^{95} + 36864 q^{96} - 16092 q^{97} + 45440 q^{98} - 486 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8369x^{2} - 182616x - 370980 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{3} + 307\nu^{2} + 58579\nu - 16554 ) / 12684 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{3} + 307\nu^{2} + 83947\nu - 16554 ) / 12684 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{3} + 38\nu^{2} + 65224\nu + 969000 ) / 3171 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -27\beta_{3} + 56\beta_{2} + 40\beta _1 + 8376 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -921\beta_{3} + 8419\beta_{2} - 7963\beta _1 + 282036 ) / 2 \) Copy content Toggle raw display

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
−77.4267
−2.26674
101.516
−20.8222
4.00000 9.00000 16.0000 −102.561 36.0000 126.292 64.0000 81.0000 −410.245
1.2 4.00000 9.00000 16.0000 24.6410 36.0000 103.175 64.0000 81.0000 98.5642
1.3 4.00000 9.00000 16.0000 38.3508 36.0000 −90.6804 64.0000 81.0000 153.403
1.4 4.00000 9.00000 16.0000 93.5694 36.0000 209.214 64.0000 81.0000 374.278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)

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 138.6.a.i 4
3.b odd 2 1 414.6.a.o 4
4.b odd 2 1 1104.6.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.i 4 1.a even 1 1 trivial
414.6.a.o 4 3.b odd 2 1
1104.6.a.m 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 54T_{5}^{3} - 9218T_{5}^{2} + 613004T_{5} - 9068808 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{4} \) Copy content Toggle raw display
$3$ \( (T - 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 54 T^{3} - 9218 T^{2} + \cdots - 9068808 \) Copy content Toggle raw display
$7$ \( T^{4} - 348 T^{3} + \cdots - 247202152 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 6955214976 \) Copy content Toggle raw display
$13$ \( T^{4} - 1248 T^{3} + \cdots - 368203080832 \) Copy content Toggle raw display
$17$ \( T^{4} + 56 T^{3} + \cdots + 791612460000 \) Copy content Toggle raw display
$19$ \( T^{4} - 1530 T^{3} + \cdots + 12533606427240 \) Copy content Toggle raw display
$23$ \( (T - 529)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 2292 T^{3} + \cdots - 7814693873520 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 270426164868224 \) Copy content Toggle raw display
$37$ \( T^{4} + 9586 T^{3} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{4} + 11768 T^{3} + \cdots - 17\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{4} - 13758 T^{3} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{4} - 10636 T^{3} + \cdots + 17\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} - 26686 T^{3} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + 9108 T^{3} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{4} + 37878 T^{3} + \cdots - 55\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{4} + 23302 T^{3} + \cdots + 38\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{4} - 31728 T^{3} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + 28340 T^{3} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + 26668 T^{3} + \cdots + 42\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{4} + 119026 T^{3} + \cdots - 77\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} + 148236 T^{3} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + 16092 T^{3} + \cdots + 16\!\cdots\!20 \) Copy content Toggle raw display
show more
show less