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Properties

Label 304.4.n.d
Level $304$
Weight $4$
Character orbit 304.n
Analytic conductor $17.937$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,4,Mod(31,304)]
 
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([3, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.n (of order \(6\), 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: \(17.9365806417\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 175 x^{18} - 24 x^{17} + 20491 x^{16} - 912 x^{15} + 1328590 x^{14} + 339084 x^{13} + \cdots + 729912213801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{5} q^{5} + ( - \beta_{14} + \beta_{6} - 2 \beta_{3} - 1) q^{7} + ( - \beta_{12} + \beta_{9} - \beta_{5} + \cdots - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{5} q^{5} + ( - \beta_{14} + \beta_{6} - 2 \beta_{3} - 1) q^{7} + ( - \beta_{12} + \beta_{9} - \beta_{5} + \cdots - 8) q^{9}+ \cdots + (16 \beta_{18} + 13 \beta_{13} + \cdots + 79) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{5} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{5} - 80 q^{9} + 48 q^{13} + 162 q^{15} - 36 q^{17} - 6 q^{19} + 36 q^{21} - 126 q^{23} - 226 q^{25} - 72 q^{27} - 156 q^{29} - 72 q^{31} - 762 q^{33} - 240 q^{35} - 474 q^{41} - 42 q^{43} - 1552 q^{45} - 2034 q^{47} - 1860 q^{49} - 414 q^{51} + 1584 q^{53} + 1044 q^{55} - 324 q^{57} + 156 q^{59} - 92 q^{61} - 2052 q^{63} + 228 q^{67} - 714 q^{71} + 322 q^{73} + 2316 q^{75} + 1264 q^{77} + 666 q^{79} + 346 q^{81} - 248 q^{85} - 1800 q^{89} + 3540 q^{91} - 1468 q^{93} + 2298 q^{95} + 714 q^{97} + 2232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 175 x^{18} - 24 x^{17} + 20491 x^{16} - 912 x^{15} + 1328590 x^{14} + 339084 x^{13} + \cdots + 729912213801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 45\!\cdots\!90 \nu^{19} + \cdots - 10\!\cdots\!56 ) / 33\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 42\!\cdots\!48 \nu^{19} + \cdots - 11\!\cdots\!77 ) / 94\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!87 \nu^{19} + \cdots + 65\!\cdots\!37 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\!\cdots\!23 \nu^{19} + \cdots - 44\!\cdots\!57 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 28\!\cdots\!93 \nu^{19} + \cdots + 14\!\cdots\!78 ) / 82\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!03 \nu^{19} + \cdots + 19\!\cdots\!31 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!38 \nu^{19} + \cdots - 12\!\cdots\!07 ) / 34\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!74 \nu^{19} + \cdots - 49\!\cdots\!79 ) / 34\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 65\!\cdots\!36 \nu^{19} + \cdots + 32\!\cdots\!87 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 70\!\cdots\!54 \nu^{19} + \cdots + 91\!\cdots\!67 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 77\!\cdots\!23 \nu^{19} + \cdots + 58\!\cdots\!80 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!22 \nu^{19} + \cdots + 96\!\cdots\!72 ) / 16\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!78 \nu^{19} + \cdots + 45\!\cdots\!31 ) / 82\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32\!\cdots\!47 \nu^{19} + \cdots - 30\!\cdots\!10 ) / 16\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 21\!\cdots\!96 \nu^{19} + \cdots + 11\!\cdots\!43 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 34\!\cdots\!97 \nu^{19} + \cdots - 18\!\cdots\!94 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 36\!\cdots\!60 \nu^{19} + \cdots - 72\!\cdots\!07 ) / 98\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 41\!\cdots\!69 \nu^{19} + \cdots + 37\!\cdots\!56 ) / 98\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{8} + \beta_{5} + 35\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{18} + \beta_{17} + \beta_{16} - \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{10} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{15} - 20 \beta_{14} + 4 \beta_{13} - 76 \beta_{12} + 100 \beta_{9} + 10 \beta_{6} - 100 \beta_{5} + \cdots - 2055 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{19} - 2 \beta_{18} + 120 \beta_{17} + 94 \beta_{16} - 80 \beta_{15} - 198 \beta_{14} + \cdots + 230 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 72 \beta_{19} + 72 \beta_{18} + 12 \beta_{17} - 132 \beta_{16} - 500 \beta_{15} + 1286 \beta_{14} + \cdots + 139857 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 596 \beta_{19} + 27089 \beta_{18} - 23218 \beta_{17} - 14894 \beta_{16} + 11754 \beta_{15} + \cdots - 205663 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10716 \beta_{19} - 10716 \beta_{18} + 5376 \beta_{17} - 18912 \beta_{16} - 49248 \beta_{15} + \cdots - 44412 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 36648 \beta_{19} - 2225604 \beta_{18} + 1032988 \beta_{17} + 566392 \beta_{16} - 445180 \beta_{15} + \cdots + 15637268 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2303208 \beta_{19} + 944676 \beta_{18} - 1805472 \beta_{17} + 3938400 \beta_{16} + 8817296 \beta_{15} + \cdots - 746276847 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4052600 \beta_{19} - 4052600 \beta_{18} + 88245117 \beta_{17} + 42520945 \beta_{16} + \cdots + 150544634 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 109352052 \beta_{19} - 13854312 \beta_{18} + 111572832 \beta_{17} - 182308848 \beta_{16} + \cdots + 56762309379 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 826135844 \beta_{19} + 15050493218 \beta_{18} - 14744527768 \beta_{17} - 6362792828 \beta_{16} + \cdots - 106799456560 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 9770080596 \beta_{19} - 9770080596 \beta_{18} + 11898854220 \beta_{17} - 15947153940 \beta_{16} + \cdots - 17820743904 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 39647586138 \beta_{19} - 1174831727139 \beta_{18} + 607706167069 \beta_{17} + 238234800847 \beta_{16} + \cdots + 7235252179463 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1686727404432 \beta_{19} + 1356834896568 \beta_{18} - 2333976213888 \beta_{17} + 2702067617664 \beta_{16} + \cdots - 334291505015529 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3643954288736 \beta_{19} - 3643954288736 \beta_{18} + 49659383919048 \beta_{17} + \cdots + 79307787638192 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 71260814372760 \beta_{19} - 57324044061216 \beta_{18} + 108625916049024 \beta_{17} + \cdots + 26\!\cdots\!07 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 649123213659776 \beta_{19} + \cdots - 49\!\cdots\!99 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(1 + \beta_{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
4.41792 − 7.65207i
3.23642 − 5.60564i
3.12797 − 5.41781i
2.20080 − 3.81191i
−0.177056 + 0.306670i
−0.950956 + 1.64710i
−1.63116 + 2.82526i
−1.70594 + 2.95477i
−4.04104 + 6.99929i
−4.47696 + 7.75433i
4.41792 + 7.65207i
3.23642 + 5.60564i
3.12797 + 5.41781i
2.20080 + 3.81191i
−0.177056 − 0.306670i
−0.950956 − 1.64710i
−1.63116 − 2.82526i
−1.70594 − 2.95477i
−4.04104 − 6.99929i
−4.47696 − 7.75433i
0 −4.41792 − 7.65207i 0 8.39426 + 14.5393i 0 30.5881i 0 −25.5361 + 44.2298i 0
31.2 0 −3.23642 − 5.60564i 0 −2.59895 − 4.50152i 0 9.12666i 0 −7.44878 + 12.9017i 0
31.3 0 −3.12797 − 5.41781i 0 4.69659 + 8.13473i 0 − 34.2033i 0 −6.06843 + 10.5108i 0
31.4 0 −2.20080 − 3.81191i 0 −7.27251 − 12.5964i 0 − 8.36560i 0 3.81292 − 6.60417i 0
31.5 0 0.177056 + 0.306670i 0 1.31812 + 2.28305i 0 13.5579i 0 13.4373 − 23.2741i 0
31.6 0 0.950956 + 1.64710i 0 8.94024 + 15.4850i 0 1.68968i 0 11.6914 − 20.2500i 0
31.7 0 1.63116 + 2.82526i 0 −10.2162 − 17.6950i 0 24.8765i 0 8.17862 − 14.1658i 0
31.8 0 1.70594 + 2.95477i 0 0.584239 + 1.01193i 0 − 17.5578i 0 7.67956 − 13.3014i 0
31.9 0 4.04104 + 6.99929i 0 −4.71455 − 8.16584i 0 − 21.6771i 0 −19.1600 + 33.1861i 0
31.10 0 4.47696 + 7.75433i 0 2.86879 + 4.96888i 0 22.7495i 0 −26.5864 + 46.0490i 0
255.1 0 −4.41792 + 7.65207i 0 8.39426 − 14.5393i 0 − 30.5881i 0 −25.5361 − 44.2298i 0
255.2 0 −3.23642 + 5.60564i 0 −2.59895 + 4.50152i 0 − 9.12666i 0 −7.44878 − 12.9017i 0
255.3 0 −3.12797 + 5.41781i 0 4.69659 − 8.13473i 0 34.2033i 0 −6.06843 − 10.5108i 0
255.4 0 −2.20080 + 3.81191i 0 −7.27251 + 12.5964i 0 8.36560i 0 3.81292 + 6.60417i 0
255.5 0 0.177056 − 0.306670i 0 1.31812 − 2.28305i 0 − 13.5579i 0 13.4373 + 23.2741i 0
255.6 0 0.950956 − 1.64710i 0 8.94024 − 15.4850i 0 − 1.68968i 0 11.6914 + 20.2500i 0
255.7 0 1.63116 − 2.82526i 0 −10.2162 + 17.6950i 0 − 24.8765i 0 8.17862 + 14.1658i 0
255.8 0 1.70594 − 2.95477i 0 0.584239 − 1.01193i 0 17.5578i 0 7.67956 + 13.3014i 0
255.9 0 4.04104 − 6.99929i 0 −4.71455 + 8.16584i 0 21.6771i 0 −19.1600 − 33.1861i 0
255.10 0 4.47696 − 7.75433i 0 2.86879 − 4.96888i 0 − 22.7495i 0 −26.5864 − 46.0490i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.4.n.d yes 20
4.b odd 2 1 304.4.n.c ✓ 20
19.d odd 6 1 304.4.n.c ✓ 20
76.f even 6 1 inner 304.4.n.d yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.4.n.c ✓ 20 4.b odd 2 1
304.4.n.c ✓ 20 19.d odd 6 1
304.4.n.d yes 20 1.a even 1 1 trivial
304.4.n.d yes 20 76.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 175 T_{3}^{18} + 24 T_{3}^{17} + 20491 T_{3}^{16} + 912 T_{3}^{15} + 1328590 T_{3}^{14} + \cdots + 729912213801 \) acting on \(S_{4}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 729912213801 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 52\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 24\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 68\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 96\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 82\!\cdots\!01 \) Copy content Toggle raw display
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