Properties

Label 30.6.c.b
Level $30$
Weight $6$
Character orbit 30.c
Analytic conductor $4.812$
Analytic rank $0$
Dimension $4$
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] = [30,6,Mod(19,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.19");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 30.c (of order \(2\), degree \(1\), 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: \(4.81151459439\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{1249})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 625x^{2} + 97344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - 4 \beta_1 q^{2} + 9 \beta_1 q^{3} - 16 q^{4} + (\beta_{2} + 1) q^{5} + 36 q^{6} + ( - \beta_{3} + 3 \beta_{2} - 58 \beta_1 - 1) q^{7} + 64 \beta_1 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_1 q^{2} + 9 \beta_1 q^{3} - 16 q^{4} + (\beta_{2} + 1) q^{5} + 36 q^{6} + ( - \beta_{3} + 3 \beta_{2} - 58 \beta_1 - 1) q^{7} + 64 \beta_1 q^{8} - 81 q^{9} + ( - 4 \beta_{3} - 4 \beta_1 + 4) q^{10} + ( - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 89) q^{11} - 144 \beta_1 q^{12} + ( - 3 \beta_{3} + 9 \beta_{2} + 318 \beta_1 - 3) q^{13} + ( - 12 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 220) q^{14} + (9 \beta_{3} + 9 \beta_1 - 9) q^{15} + 256 q^{16} + (3 \beta_{3} - 9 \beta_{2} - 754 \beta_1 + 3) q^{17} + 324 \beta_1 q^{18} + (36 \beta_{3} + 12 \beta_{2} - 24 \beta_1 + 36) q^{19} + ( - 16 \beta_{2} - 16) q^{20} + (27 \beta_{3} + 9 \beta_{2} - 18 \beta_1 + 495) q^{21} + (4 \beta_{3} - 12 \beta_{2} - 344 \beta_1 + 4) q^{22} + (12 \beta_{3} - 36 \beta_{2} + 2188 \beta_1 + 12) q^{23} - 576 q^{24} + (\beta_{3} + 3 \beta_{2} - 1874 \beta_1 - 2498) q^{25} + ( - 36 \beta_{3} - 12 \beta_{2} + 24 \beta_1 + 1308) q^{26} - 729 \beta_1 q^{27} + (16 \beta_{3} - 48 \beta_{2} + 928 \beta_1 + 16) q^{28} + ( - 81 \beta_{3} - 27 \beta_{2} + 54 \beta_1 + 1269) q^{29} + (36 \beta_{2} + 36) q^{30} + (18 \beta_{3} + 6 \beta_{2} - 12 \beta_1 + 5830) q^{31} - 1024 \beta_1 q^{32} + ( - 9 \beta_{3} + 27 \beta_{2} + 774 \beta_1 - 9) q^{33} + (36 \beta_{3} + 12 \beta_{2} - 24 \beta_1 - 3052) q^{34} + ( - 57 \beta_{3} + 5 \beta_{2} - 3182 \beta_1 - 9313) q^{35} + 1296 q^{36} + ( - 19 \beta_{3} + 57 \beta_{2} + 8774 \beta_1 - 19) q^{37} + ( - 48 \beta_{3} + 144 \beta_{2} - 288 \beta_1 - 48) q^{38} + (81 \beta_{3} + 27 \beta_{2} - 54 \beta_1 - 2943) q^{39} + (64 \beta_{3} + 64 \beta_1 - 64) q^{40} + ( - 102 \beta_{3} - 34 \beta_{2} + 68 \beta_1 + 12560) q^{41} + ( - 36 \beta_{3} + 108 \beta_{2} - 2088 \beta_1 - 36) q^{42} + ( - 56 \beta_{3} + 168 \beta_{2} - 12140 \beta_1 - 56) q^{43} + (48 \beta_{3} + 16 \beta_{2} - 32 \beta_1 - 1424) q^{44} + ( - 81 \beta_{2} - 81) q^{45} + (144 \beta_{3} + 48 \beta_{2} - 96 \beta_1 + 8608) q^{46} + (6 \beta_{3} - 18 \beta_{2} + 6664 \beta_1 + 6) q^{47} + 2304 \beta_1 q^{48} + ( - 342 \beta_{3} - 114 \beta_{2} + 228 \beta_1 - 17439) q^{49} + ( - 12 \beta_{3} + 4 \beta_{2} + 9988 \beta_1 - 7484) q^{50} + ( - 81 \beta_{3} - 27 \beta_{2} + 54 \beta_1 + 6867) q^{51} + (48 \beta_{3} - 144 \beta_{2} - 5088 \beta_1 + 48) q^{52} + (111 \beta_{3} - 333 \beta_{2} - 6738 \beta_1 + 111) q^{53} - 2916 q^{54} + ( - 5 \beta_{3} + 87 \beta_{2} + 9370 \beta_1 - 3033) q^{55} + (192 \beta_{3} + 64 \beta_{2} - 128 \beta_1 + 3520) q^{56} + (108 \beta_{3} - 324 \beta_{2} + 648 \beta_1 + 108) q^{57} + (108 \beta_{3} - 324 \beta_{2} - 4752 \beta_1 + 108) q^{58} + (555 \beta_{3} + 185 \beta_{2} - 370 \beta_1 + 11495) q^{59} + ( - 144 \beta_{3} - 144 \beta_1 + 144) q^{60} + ( - 288 \beta_{3} - 96 \beta_{2} + 192 \beta_1 + 28754) q^{61} + ( - 24 \beta_{3} + 72 \beta_{2} - 23392 \beta_1 - 24) q^{62} + (81 \beta_{3} - 243 \beta_{2} + 4698 \beta_1 + 81) q^{63} - 4096 q^{64} + (321 \beta_{3} + 15 \beta_{2} - 9054 \beta_1 - 28431) q^{65} + ( - 108 \beta_{3} - 36 \beta_{2} + 72 \beta_1 + 3204) q^{66} + ( - 86 \beta_{3} + 258 \beta_{2} + 19072 \beta_1 - 86) q^{67} + ( - 48 \beta_{3} + 144 \beta_{2} + 12064 \beta_1 - 48) q^{68} + ( - 324 \beta_{3} - 108 \beta_{2} + 216 \beta_1 - 19368) q^{69} + ( - 20 \beta_{3} - 228 \beta_{2} + 37480 \beta_1 - 12708) q^{70} + (882 \beta_{3} + 294 \beta_{2} - 588 \beta_1 - 6126) q^{71} - 5184 \beta_1 q^{72} + ( - 98 \beta_{3} + 294 \beta_{2} - 44372 \beta_1 - 98) q^{73} + ( - 228 \beta_{3} - 76 \beta_{2} + 152 \beta_1 + 35324) q^{74} + (27 \beta_{3} - 9 \beta_{2} - 22473 \beta_1 + 16839) q^{75} + ( - 576 \beta_{3} - 192 \beta_{2} + 384 \beta_1 - 576) q^{76} + ( - 30 \beta_{3} + 90 \beta_{2} + 26236 \beta_1 - 30) q^{77} + ( - 108 \beta_{3} + 324 \beta_{2} + 11448 \beta_1 - 108) q^{78} + (594 \beta_{3} + 198 \beta_{2} - 396 \beta_1 - 7506) q^{79} + (256 \beta_{2} + 256) q^{80} + 6561 q^{81} + (136 \beta_{3} - 408 \beta_{2} - 49832 \beta_1 + 136) q^{82} + ( - 432 \beta_{3} + 1296 \beta_{2} + 24964 \beta_1 - 432) q^{83} + ( - 432 \beta_{3} - 144 \beta_{2} + 288 \beta_1 - 7920) q^{84} + ( - 757 \beta_{3} - 15 \beta_{2} + 8618 \beta_1 + 28867) q^{85} + ( - 672 \beta_{3} - 224 \beta_{2} + 448 \beta_1 - 47888) q^{86} + ( - 243 \beta_{3} + 729 \beta_{2} + 10692 \beta_1 - 243) q^{87} + ( - 64 \beta_{3} + 192 \beta_{2} + 5504 \beta_1 - 64) q^{88} + ( - 600 \beta_{3} - 200 \beta_{2} + 400 \beta_1 + 30610) q^{89} + (324 \beta_{3} + 324 \beta_1 - 324) q^{90} + (450 \beta_{3} + 150 \beta_{2} - 300 \beta_1 - 75678) q^{91} + ( - 192 \beta_{3} + 576 \beta_{2} - 35008 \beta_1 - 192) q^{92} + (54 \beta_{3} - 162 \beta_{2} + 52632 \beta_1 + 54) q^{93} + (72 \beta_{3} + 24 \beta_{2} - 48 \beta_1 + 26584) q^{94} + (60 \beta_{3} + 60 \beta_{2} - 112440 \beta_1 + 37500) q^{95} + 9216 q^{96} + (16 \beta_{3} - 48 \beta_{2} + 85624 \beta_1 + 16) q^{97} + (456 \beta_{3} - 1368 \beta_{2} + 71124 \beta_1 + 456) q^{98} + (243 \beta_{3} + 81 \beta_{2} - 162 \beta_1 - 7209) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{4} + 6 q^{5} + 144 q^{6} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{4} + 6 q^{5} + 144 q^{6} - 324 q^{9} + 8 q^{10} + 348 q^{11} - 912 q^{14} - 18 q^{15} + 1024 q^{16} + 240 q^{19} - 96 q^{20} + 2052 q^{21} - 2304 q^{24} - 9984 q^{25} + 5136 q^{26} + 4860 q^{29} + 216 q^{30} + 23368 q^{31} - 12112 q^{34} - 37356 q^{35} + 5184 q^{36} - 11556 q^{39} - 128 q^{40} + 49968 q^{41} - 5568 q^{44} - 486 q^{45} + 34816 q^{46} - 70668 q^{49} - 29952 q^{50} + 27252 q^{51} - 11664 q^{54} - 11968 q^{55} + 14592 q^{56} + 47460 q^{59} + 288 q^{60} + 114248 q^{61} - 16384 q^{64} - 113052 q^{65} + 12528 q^{66} - 78336 q^{69} - 51328 q^{70} - 22152 q^{71} + 140688 q^{74} + 67392 q^{75} - 3840 q^{76} - 28440 q^{79} + 1536 q^{80} + 26244 q^{81} - 32832 q^{84} + 113924 q^{85} - 193344 q^{86} + 120840 q^{89} - 648 q^{90} - 301512 q^{91} + 106528 q^{94} + 150240 q^{95} + 36864 q^{96} - 28188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 625x^{2} + 97344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 313\nu ) / 312 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 312\nu^{2} - 1249\nu + 97656 ) / 312 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 936\nu^{2} + 625\nu + 292656 ) / 312 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 3\beta_{2} - 4\beta _1 + 1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} + \beta_{2} - 2\beta _1 - 3127 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -313\beta_{3} + 939\beta_{2} + 4372\beta _1 - 313 ) / 10 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
18.1706i
17.1706i
18.1706i
17.1706i
4.00000i 9.00000i −16.0000 −16.1706 + 53.5118i 36.0000 119.706i 64.0000i −81.0000 214.047 + 64.6824i
19.2 4.00000i 9.00000i −16.0000 19.1706 52.5118i 36.0000 233.706i 64.0000i −81.0000 −210.047 76.6824i
19.3 4.00000i 9.00000i −16.0000 −16.1706 53.5118i 36.0000 119.706i 64.0000i −81.0000 214.047 64.6824i
19.4 4.00000i 9.00000i −16.0000 19.1706 + 52.5118i 36.0000 233.706i 64.0000i −81.0000 −210.047 + 76.6824i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.6.c.b 4
3.b odd 2 1 90.6.c.c 4
4.b odd 2 1 240.6.f.b 4
5.b even 2 1 inner 30.6.c.b 4
5.c odd 4 1 150.6.a.n 2
5.c odd 4 1 150.6.a.o 2
12.b even 2 1 720.6.f.i 4
15.d odd 2 1 90.6.c.c 4
15.e even 4 1 450.6.a.bb 2
15.e even 4 1 450.6.a.bc 2
20.d odd 2 1 240.6.f.b 4
60.h even 2 1 720.6.f.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.b 4 1.a even 1 1 trivial
30.6.c.b 4 5.b even 2 1 inner
90.6.c.c 4 3.b odd 2 1
90.6.c.c 4 15.d odd 2 1
150.6.a.n 2 5.c odd 4 1
150.6.a.o 2 5.c odd 4 1
240.6.f.b 4 4.b odd 2 1
240.6.f.b 4 20.d odd 2 1
450.6.a.bb 2 15.e even 4 1
450.6.a.bc 2 15.e even 4 1
720.6.f.i 4 12.b even 2 1
720.6.f.i 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 68948T_{7}^{2} + 782656576 \) acting on \(S_{6}^{\mathrm{new}}(30, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + 5010 T^{2} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{4} + 68948 T^{2} + \cdots + 782656576 \) Copy content Toggle raw display
$11$ \( (T^{2} - 174 T - 23656)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 768132 T^{2} + \cdots + 31678304256 \) Copy content Toggle raw display
$17$ \( T^{4} + 1708148 T^{2} + \cdots + 85278016576 \) Copy content Toggle raw display
$19$ \( (T^{2} - 120 T - 4492800)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18462752 T^{2} + \cdots + 56918507776 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2430 T - 21286800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 11684 T + 33004864)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 177178148 T^{2} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{2} - 24984 T + 119953964)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 487889312 T^{2} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{4} + 90906128 T^{2} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{4} + 863264052 T^{2} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{2} - 23730 T - 927897400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 57124 T + 528018244)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 1195938128 T^{2} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} + 11076 T - 2668294656)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4520143952 T^{2} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{2} + 14220 T - 1173592800)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 12944582432 T^{2} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} - 60420 T - 336355900)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 14673446528 T^{2} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
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