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Properties

Label 304.6.i.d
Level $304$
Weight $6$
Character orbit 304.i
Analytic conductor $48.757$
Analytic rank $0$
Dimension $18$
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] = [304,6,Mod(49,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), degree \(2\), not 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: \(48.7566812231\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} + \cdots + 66\!\cdots\!83 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 76)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{5} + (\beta_{6} + 2 \beta_{2} - 18) q^{7} + (\beta_{12} - \beta_{7} + 3 \beta_{3} + \cdots - 99) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{5} + (\beta_{6} + 2 \beta_{2} - 18) q^{7} + (\beta_{12} - \beta_{7} + 3 \beta_{3} + \cdots - 99) q^{9}+ \cdots + (34 \beta_{17} + 17 \beta_{14} + \cdots + 21036) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9} + 320 q^{11} + 227 q^{13} + 101 q^{15} + 179 q^{17} + 868 q^{19} - 5700 q^{21} + 3425 q^{23} - 7054 q^{25} - 14722 q^{27} - 7349 q^{29} + 9960 q^{31} - 2998 q^{33} - 15888 q^{35} + 26444 q^{37} + 30246 q^{39} - 7311 q^{41} + 8283 q^{43} - 62164 q^{45} - 37603 q^{47} + 124738 q^{49} - 47227 q^{51} - 20337 q^{53} - 716 q^{55} - 57555 q^{57} + 74455 q^{59} - 7569 q^{61} + 52544 q^{63} + 188998 q^{65} + 26177 q^{67} + 116282 q^{69} + 53463 q^{71} - 14103 q^{73} - 120912 q^{75} - 31960 q^{77} - 31825 q^{79} - 21137 q^{81} - 82600 q^{83} - 50787 q^{85} + 339766 q^{87} - 155197 q^{89} + 2800 q^{91} - 46460 q^{93} - 49315 q^{95} + 111241 q^{97} + 193544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} + \cdots + 66\!\cdots\!83 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 96\!\cdots\!86 \nu^{17} + \cdots - 10\!\cdots\!18 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 96\!\cdots\!86 \nu^{17} + \cdots - 10\!\cdots\!18 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!17 \nu^{17} + \cdots + 21\!\cdots\!21 ) / 39\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35\!\cdots\!04 \nu^{17} + \cdots - 67\!\cdots\!27 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33\!\cdots\!28 \nu^{17} + \cdots + 43\!\cdots\!19 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!68 \nu^{17} + \cdots + 26\!\cdots\!59 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!79 \nu^{17} + \cdots - 28\!\cdots\!47 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 85\!\cdots\!71 \nu^{17} + \cdots + 28\!\cdots\!23 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 72\!\cdots\!31 \nu^{17} + \cdots + 82\!\cdots\!03 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!57 \nu^{17} + \cdots - 24\!\cdots\!66 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81\!\cdots\!41 \nu^{17} + \cdots + 18\!\cdots\!93 ) / 96\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!51 \nu^{17} + \cdots - 16\!\cdots\!63 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!04 \nu^{17} + \cdots + 18\!\cdots\!77 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 18\!\cdots\!99 \nu^{17} + \cdots + 19\!\cdots\!87 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 62\!\cdots\!09 \nu^{17} + \cdots + 52\!\cdots\!42 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 23\!\cdots\!43 \nu^{17} + \cdots + 25\!\cdots\!34 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 44\!\cdots\!48 \nu^{17} + \cdots + 63\!\cdots\!99 ) / 48\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 340 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{17} - \beta_{16} + 4 \beta_{15} + \beta_{14} + 2 \beta_{13} + 3 \beta_{12} - \beta_{11} + \cdots - 253 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 32 \beta_{17} + 36 \beta_{16} + 66 \beta_{15} - 40 \beta_{14} - 24 \beta_{13} - 14 \beta_{12} + \cdots + 186850 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1528 \beta_{17} - 882 \beta_{16} + 3510 \beta_{15} + 1072 \beta_{14} + 1380 \beta_{13} + \cdots - 666057 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 34714 \beta_{17} + 44788 \beta_{16} + 80540 \beta_{15} - 50680 \beta_{14} - 27176 \beta_{13} + \cdots + 118830231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 994635 \beta_{17} - 653413 \beta_{16} + 2517178 \beta_{15} + 989546 \beta_{14} + 948458 \beta_{13} + \cdots - 785864070 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 29172564 \beta_{17} + 41601552 \beta_{16} + 70749708 \beta_{15} - 48255960 \beta_{14} + \cdots + 80637457063 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 618658592 \beta_{17} - 470162752 \beta_{16} + 1716048064 \beta_{15} + 880579132 \beta_{14} + \cdots - 771187047865 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 22418713520 \beta_{17} + 34494760608 \beta_{16} + 55338418284 \beta_{15} - 41398565944 \beta_{14} + \cdots + 56891017155247 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 377519170879 \beta_{17} - 336992032644 \beta_{16} + 1155968918484 \beta_{15} + 765725112727 \beta_{14} + \cdots - 701448549462327 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16530979031806 \beta_{17} + 26943465802252 \beta_{16} + 41073808391000 \beta_{15} + \cdots + 41\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 227161506858060 \beta_{17} - 241313300363986 \beta_{16} + 778913741676130 \beta_{15} + \cdots - 61\!\cdots\!86 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11\!\cdots\!14 \beta_{17} + \cdots + 30\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 13\!\cdots\!90 \beta_{17} + \cdots - 52\!\cdots\!01 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 84\!\cdots\!68 \beta_{17} + \cdots + 22\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 77\!\cdots\!08 \beta_{17} + \cdots - 43\!\cdots\!31 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1 - \beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
26.3623 − 0.866025i
22.4078 − 0.866025i
17.4003 − 0.866025i
9.90852 − 0.866025i
−5.10643 − 0.866025i
−6.49256 − 0.866025i
−12.0901 − 0.866025i
−21.8370 − 0.866025i
−28.0529 − 0.866025i
26.3623 + 0.866025i
22.4078 + 0.866025i
17.4003 + 0.866025i
9.90852 + 0.866025i
−5.10643 + 0.866025i
−6.49256 + 0.866025i
−12.0901 + 0.866025i
−21.8370 + 0.866025i
−28.0529 + 0.866025i
0 −12.4311 + 21.5314i 0 −18.0963 + 31.3437i 0 208.885 0 −187.566 − 324.875i 0
49.2 0 −10.4539 + 18.1067i 0 50.5928 − 87.6293i 0 −95.5451 0 −97.0688 − 168.128i 0
49.3 0 −7.95017 + 13.7701i 0 −15.4645 + 26.7852i 0 −132.225 0 −4.91049 − 8.50521i 0
49.4 0 −4.20426 + 7.28199i 0 −18.6530 + 32.3080i 0 −15.8772 0 86.1484 + 149.213i 0
49.5 0 3.30322 − 5.72134i 0 12.6095 − 21.8402i 0 40.7176 0 99.6775 + 172.647i 0
49.6 0 3.99628 − 6.92176i 0 31.6056 − 54.7425i 0 80.2775 0 89.5595 + 155.122i 0
49.7 0 6.79505 − 11.7694i 0 −47.4871 + 82.2500i 0 −189.860 0 29.1546 + 50.4972i 0
49.8 0 11.6685 − 20.2105i 0 −25.2470 + 43.7291i 0 187.942 0 −150.808 − 261.208i 0
49.9 0 14.7764 − 25.5935i 0 35.6401 − 61.7304i 0 −252.315 0 −315.186 − 545.918i 0
273.1 0 −12.4311 − 21.5314i 0 −18.0963 − 31.3437i 0 208.885 0 −187.566 + 324.875i 0
273.2 0 −10.4539 − 18.1067i 0 50.5928 + 87.6293i 0 −95.5451 0 −97.0688 + 168.128i 0
273.3 0 −7.95017 − 13.7701i 0 −15.4645 − 26.7852i 0 −132.225 0 −4.91049 + 8.50521i 0
273.4 0 −4.20426 − 7.28199i 0 −18.6530 − 32.3080i 0 −15.8772 0 86.1484 − 149.213i 0
273.5 0 3.30322 + 5.72134i 0 12.6095 + 21.8402i 0 40.7176 0 99.6775 − 172.647i 0
273.6 0 3.99628 + 6.92176i 0 31.6056 + 54.7425i 0 80.2775 0 89.5595 − 155.122i 0
273.7 0 6.79505 + 11.7694i 0 −47.4871 − 82.2500i 0 −189.860 0 29.1546 − 50.4972i 0
273.8 0 11.6685 + 20.2105i 0 −25.2470 − 43.7291i 0 187.942 0 −150.808 + 261.208i 0
273.9 0 14.7764 + 25.5935i 0 35.6401 + 61.7304i 0 −252.315 0 −315.186 + 545.918i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.9
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 304.6.i.d 18
4.b odd 2 1 76.6.e.a ✓ 18
12.b even 2 1 684.6.k.f 18
19.c even 3 1 inner 304.6.i.d 18
76.g odd 6 1 76.6.e.a ✓ 18
228.m even 6 1 684.6.k.f 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.e.a ✓ 18 4.b odd 2 1
76.6.e.a ✓ 18 76.g odd 6 1
304.6.i.d 18 1.a even 1 1 trivial
304.6.i.d 18 19.c even 3 1 inner
684.6.k.f 18 12.b even 2 1
684.6.k.f 18 228.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 11 T_{3}^{17} + 1605 T_{3}^{16} - 8740 T_{3}^{15} + 1629002 T_{3}^{14} + \cdots + 11\!\cdots\!89 \) acting on \(S_{6}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( (T^{9} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots - 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 34\!\cdots\!99 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 96\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 91\!\cdots\!60)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 21\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 92\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 86\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
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