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Properties

Label 304.5.r.c
Level $304$
Weight $5$
Character orbit 304.r
Analytic conductor $31.424$
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] = [304,5,Mod(65,304)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 304.r (of order \(6\), 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: \(31.4244687775\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 38)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + (\beta_{11} - 2 \beta_{5} - 2) q^{5} + ( - \beta_{4} + \beta_{3} - \beta_1 - 5) q^{7} + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{3} + (\beta_{11} - 2 \beta_{5} - 2) q^{5} + ( - \beta_{4} + \beta_{3} - \beta_1 - 5) q^{7} + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{9}+ \cdots + ( - 38 \beta_{15} + 71 \beta_{14} + \cdots + 201) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{3} - 18 q^{5} - 72 q^{7} + 352 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{3} - 18 q^{5} - 72 q^{7} + 352 q^{9} + 84 q^{11} + 450 q^{13} + 390 q^{15} + 606 q^{17} + 306 q^{19} - 2160 q^{21} + 54 q^{23} - 434 q^{25} - 4914 q^{29} + 7890 q^{33} - 2328 q^{35} - 7620 q^{39} - 1692 q^{41} + 7402 q^{43} - 16720 q^{45} - 3198 q^{47} + 24816 q^{49} - 10710 q^{51} + 3870 q^{53} + 13588 q^{55} + 3702 q^{57} + 18288 q^{59} - 6522 q^{61} + 15676 q^{63} + 30168 q^{67} - 35874 q^{71} - 8080 q^{73} + 34560 q^{77} + 30738 q^{79} - 30920 q^{81} + 1476 q^{83} + 33626 q^{85} - 113100 q^{87} + 19782 q^{89} + 34260 q^{91} - 4272 q^{93} + 23706 q^{95} - 9936 q^{97} - 3848 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 12\!\cdots\!03 \nu^{14} + \cdots + 15\!\cdots\!43 ) / 10\!\cdots\!74 \) 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
\(\beta_{4}\)\(=\) \( ( 57\!\cdots\!39 \nu^{15} + \cdots - 74\!\cdots\!43 ) / 18\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!12 \nu^{15} + \cdots - 13\!\cdots\!23 ) / 10\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!85 \nu^{15} + \cdots - 93\!\cdots\!12 ) / 90\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 46\!\cdots\!30 \nu^{15} + \cdots - 86\!\cdots\!43 ) / 90\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!10 \nu^{15} + \cdots - 64\!\cdots\!84 ) / 30\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!10 \nu^{15} + \cdots - 21\!\cdots\!02 ) / 30\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 37\!\cdots\!62 \nu^{15} + \cdots - 38\!\cdots\!61 ) / 68\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 88\!\cdots\!28 \nu^{15} + \cdots - 76\!\cdots\!98 ) / 68\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!75 \nu^{15} + \cdots + 91\!\cdots\!19 ) / 34\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!75 \nu^{15} + \cdots - 15\!\cdots\!49 ) / 34\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!93 \nu^{15} + \cdots - 10\!\cdots\!88 ) / 34\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 66\!\cdots\!24 \nu^{15} + \cdots - 28\!\cdots\!42 ) / 68\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + 12\beta_{5} - 12\beta_{4} - 12\beta_{3} + 12 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 6 \beta_{7} + 6 \beta_{5} - 6 \beta_{4} - 12 \beta_{3} + \cdots - 738 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 60 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} - 264 \beta_{11} + 18 \beta_{10} - 422 \beta_{9} + \cdots - 3654 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 30 \beta_{14} + 12 \beta_{13} - 18 \beta_{12} - 132 \beta_{11} - 840 \beta_{10} - 1153 \beta_{9} + \cdots + 82500 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6468 \beta_{15} + 14496 \beta_{14} + 3012 \beta_{13} + 2712 \beta_{12} + 73596 \beta_{11} + \cdots + 964944 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9702 \beta_{15} + 21894 \beta_{14} - 15456 \beta_{13} + 24072 \beta_{12} + 111054 \beta_{11} + \cdots - 42277230 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3928842 \beta_{15} - 2704836 \beta_{14} - 1060368 \beta_{13} - 782622 \beta_{12} - 19654350 \beta_{11} + \cdots - 248451186 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3951480 \beta_{15} - 2755992 \beta_{14} + 2966676 \beta_{13} - 4829784 \beta_{12} - 19913784 \beta_{11} + \cdots + 5764477473 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1672538400 \beta_{15} + 346518480 \beta_{14} + 351402816 \beta_{13} + 209398800 \beta_{12} + \cdots + 65360518308 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4240686114 \beta_{15} + 907789638 \beta_{14} - 1903414800 \beta_{13} + 3333425832 \beta_{12} + \cdots - 3263527095432 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 620354842626 \beta_{15} + 12973545168 \beta_{14} - 115293362106 \beta_{13} - 56383353168 \beta_{12} + \cdots - 18099315095292 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 953921343708 \beta_{15} + 14421759546 \beta_{14} + 277991835726 \beta_{13} - 543402805470 \beta_{12} + \cdots + 473549565988053 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 214353280741068 \beta_{15} - 31964056574160 \beta_{14} + 37753450791672 \beta_{13} + \cdots + 53\!\cdots\!00 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 779314160279028 \beta_{15} - 112280971831872 \beta_{14} - 151684356029136 \beta_{13} + \cdots - 27\!\cdots\!78 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 70\!\cdots\!28 \beta_{15} + \cdots - 16\!\cdots\!42 ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 − 15.9560i
0.500000 − 14.3823i
0.500000 − 6.55779i
0.500000 − 0.961202i
0.500000 + 2.12625i
0.500000 + 6.86019i
0.500000 + 11.2515i
0.500000 + 17.6194i
0.500000 + 15.9560i
0.500000 + 14.3823i
0.500000 + 6.55779i
0.500000 + 0.961202i
0.500000 − 2.12625i
0.500000 − 6.86019i
0.500000 − 11.2515i
0.500000 − 17.6194i
0 −14.2931 − 8.25212i 0 −13.5555 + 23.4789i 0 45.5687 0 95.6949 + 165.748i 0
65.2 0 −10.4807 − 6.05105i 0 9.13314 − 15.8191i 0 40.6944 0 32.7305 + 56.6909i 0
65.3 0 −3.70447 − 2.13878i 0 −17.7629 + 30.7663i 0 −57.6374 0 −31.3513 − 54.3020i 0
65.4 0 −1.30717 − 0.754695i 0 1.35492 − 2.34679i 0 −79.4947 0 −39.3609 − 68.1750i 0
65.5 0 1.36664 + 0.789030i 0 11.3097 − 19.5890i 0 73.1169 0 −39.2549 − 67.9914i 0
65.6 0 7.91584 + 4.57021i 0 19.7357 − 34.1832i 0 −91.5785 0 1.27372 + 2.20614i 0
65.7 0 11.7188 + 6.76588i 0 −15.6059 + 27.0302i 0 61.1277 0 51.0542 + 88.4285i 0
65.8 0 14.7841 + 8.53562i 0 −3.60911 + 6.25116i 0 −27.7970 0 105.214 + 182.235i 0
145.1 0 −14.2931 + 8.25212i 0 −13.5555 − 23.4789i 0 45.5687 0 95.6949 − 165.748i 0
145.2 0 −10.4807 + 6.05105i 0 9.13314 + 15.8191i 0 40.6944 0 32.7305 − 56.6909i 0
145.3 0 −3.70447 + 2.13878i 0 −17.7629 − 30.7663i 0 −57.6374 0 −31.3513 + 54.3020i 0
145.4 0 −1.30717 + 0.754695i 0 1.35492 + 2.34679i 0 −79.4947 0 −39.3609 + 68.1750i 0
145.5 0 1.36664 − 0.789030i 0 11.3097 + 19.5890i 0 73.1169 0 −39.2549 + 67.9914i 0
145.6 0 7.91584 − 4.57021i 0 19.7357 + 34.1832i 0 −91.5785 0 1.27372 − 2.20614i 0
145.7 0 11.7188 − 6.76588i 0 −15.6059 − 27.0302i 0 61.1277 0 51.0542 − 88.4285i 0
145.8 0 14.7841 − 8.53562i 0 −3.60911 − 6.25116i 0 −27.7970 0 105.214 − 182.235i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
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 304.5.r.c 16
4.b odd 2 1 38.5.d.a ✓ 16
12.b even 2 1 342.5.m.c 16
19.d odd 6 1 inner 304.5.r.c 16
76.f even 6 1 38.5.d.a ✓ 16
228.n odd 6 1 342.5.m.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.d.a ✓ 16 4.b odd 2 1
38.5.d.a ✓ 16 76.f even 6 1
304.5.r.c 16 1.a even 1 1 trivial
304.5.r.c 16 19.d odd 6 1 inner
342.5.m.c 16 12.b even 2 1
342.5.m.c 16 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 12 T_{3}^{15} - 428 T_{3}^{14} + 5712 T_{3}^{13} + 145222 T_{3}^{12} + \cdots + 18463942586961 \) acting on \(S_{5}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) 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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