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Properties

Label 334.4.a.c
Level $334$
Weight $4$
Character orbit 334.a
Self dual yes
Analytic conductor $19.707$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [334,4,Mod(1,334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("334.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 334 = 2 \cdot 167 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 334.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: \(19.7066379419\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 245 x^{10} + 220 x^{9} + 20500 x^{8} - 15289 x^{7} - 682899 x^{6} + 405442 x^{5} + \cdots - 4940224 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + (\beta_{8} + 1) q^{5} - 2 \beta_1 q^{6} + (\beta_{10} + \beta_{6} - \beta_{2} + \cdots - 3) q^{7}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{9} + \cdots + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + (\beta_{8} + 1) q^{5} - 2 \beta_1 q^{6} + (\beta_{10} + \beta_{6} - \beta_{2} + \cdots - 3) q^{7}+ \cdots + ( - 40 \beta_{11} + 3 \beta_{10} + \cdots + 37) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{2} + q^{3} + 48 q^{4} + 10 q^{5} - 2 q^{6} - 28 q^{7} - 96 q^{8} + 167 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{2} + q^{3} + 48 q^{4} + 10 q^{5} - 2 q^{6} - 28 q^{7} - 96 q^{8} + 167 q^{9} - 20 q^{10} + 60 q^{11} + 4 q^{12} - 57 q^{13} + 56 q^{14} - 18 q^{15} + 192 q^{16} + 138 q^{17} - 334 q^{18} + 27 q^{19} + 40 q^{20} + 177 q^{21} - 120 q^{22} + 35 q^{23} - 8 q^{24} + 514 q^{25} + 114 q^{26} + 22 q^{27} - 112 q^{28} + 709 q^{29} + 36 q^{30} - 267 q^{31} - 384 q^{32} - 80 q^{33} - 276 q^{34} + 303 q^{35} + 668 q^{36} - 101 q^{37} - 54 q^{38} + 408 q^{39} - 80 q^{40} + 772 q^{41} - 354 q^{42} + 357 q^{43} + 240 q^{44} + 930 q^{45} - 70 q^{46} + 253 q^{47} + 16 q^{48} + 1052 q^{49} - 1028 q^{50} + 4016 q^{51} - 228 q^{52} + 3183 q^{53} - 44 q^{54} + 1479 q^{55} + 224 q^{56} + 2299 q^{57} - 1418 q^{58} + 1726 q^{59} - 72 q^{60} + 914 q^{61} + 534 q^{62} + 2312 q^{63} + 768 q^{64} + 4701 q^{65} + 160 q^{66} - 559 q^{67} + 552 q^{68} + 3891 q^{69} - 606 q^{70} + 3975 q^{71} - 1336 q^{72} + 2592 q^{73} + 202 q^{74} + 4437 q^{75} + 108 q^{76} + 3647 q^{77} - 816 q^{78} + 2092 q^{79} + 160 q^{80} + 7128 q^{81} - 1544 q^{82} + 1167 q^{83} + 708 q^{84} - 468 q^{85} - 714 q^{86} - 3080 q^{87} - 480 q^{88} + 1887 q^{89} - 1860 q^{90} - 1866 q^{91} + 140 q^{92} - 1323 q^{93} - 506 q^{94} - 478 q^{95} - 32 q^{96} - 2786 q^{97} - 2104 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 245 x^{10} + 220 x^{9} + 20500 x^{8} - 15289 x^{7} - 682899 x^{6} + 405442 x^{5} + \cdots - 4940224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43336369156439 \nu^{11} - 80345916584907 \nu^{10} + \cdots - 40\!\cdots\!88 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 692314591404539 \nu^{11} + 198573687818751 \nu^{10} + \cdots - 32\!\cdots\!28 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 78\!\cdots\!01 \nu^{11} + \cdots + 11\!\cdots\!08 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!13 \nu^{11} + \cdots + 20\!\cdots\!56 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 75\!\cdots\!93 \nu^{11} + \cdots - 45\!\cdots\!16 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\!\cdots\!87 \nu^{11} + \cdots + 14\!\cdots\!04 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 74\!\cdots\!97 \nu^{11} + \cdots - 63\!\cdots\!04 ) / 81\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 713567803297003 \nu^{11} + 206692095790047 \nu^{10} + \cdots - 71\!\cdots\!36 ) / 75\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42\!\cdots\!17 \nu^{11} + \cdots - 41\!\cdots\!64 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 61\!\cdots\!93 \nu^{11} + \cdots - 46\!\cdots\!76 ) / 40\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{11} + 2\beta_{9} + \beta_{6} - \beta_{5} - 2\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{11} + 2 \beta_{10} - \beta_{9} - 9 \beta_{8} - 3 \beta_{7} + 5 \beta_{6} - 13 \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 168 \beta_{11} - 33 \beta_{10} + 213 \beta_{9} - 24 \beta_{8} + 15 \beta_{7} + 123 \beta_{6} + \cdots + 3073 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 384 \beta_{11} + 165 \beta_{10} - 150 \beta_{9} - 1023 \beta_{8} - 357 \beta_{7} + 762 \beta_{6} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14441 \beta_{11} - 4452 \beta_{10} + 21461 \beta_{9} - 3363 \beta_{8} + 2985 \beta_{7} + \cdots + 270664 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 43641 \beta_{11} + 13754 \beta_{10} - 21610 \beta_{9} - 99645 \beta_{8} - 44343 \beta_{7} + \cdots - 24356 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1269723 \beta_{11} - 477372 \beta_{10} + 2139771 \beta_{9} - 382827 \beta_{8} + 418089 \beta_{7} + \cdots + 25078381 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4703466 \beta_{11} + 1254273 \beta_{10} - 2881821 \beta_{9} - 9438690 \beta_{8} - 5409111 \beta_{7} + \cdots - 6384801 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 113136335 \beta_{11} - 48006615 \beta_{10} + 213100073 \beta_{9} - 40931976 \beta_{8} + \cdots + 2375486947 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 491088327 \beta_{11} + 123544973 \beta_{10} - 363330442 \beta_{9} - 891663123 \beta_{8} + \cdots - 1112854343 \) Copy content Toggle raw display

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

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
−10.1965
−8.95170
−5.51344
−4.88823
−1.97028
0.412906
0.757412
1.22741
4.56533
6.22866
9.52532
9.80316
−2.00000 −10.1965 4.00000 −4.66805 20.3931 −6.05585 −8.00000 76.9695 9.33610
1.2 −2.00000 −8.95170 4.00000 19.3252 17.9034 1.21847 −8.00000 53.1329 −38.6504
1.3 −2.00000 −5.51344 4.00000 −4.18134 11.0269 32.6862 −8.00000 3.39805 8.36267
1.4 −2.00000 −4.88823 4.00000 1.93486 9.77646 −34.5834 −8.00000 −3.10518 −3.86972
1.5 −2.00000 −1.97028 4.00000 −7.17718 3.94056 −1.57921 −8.00000 −23.1180 14.3544
1.6 −2.00000 0.412906 4.00000 −16.2363 −0.825812 −30.6190 −8.00000 −26.8295 32.4726
1.7 −2.00000 0.757412 4.00000 10.1821 −1.51482 19.7514 −8.00000 −26.4263 −20.3642
1.8 −2.00000 1.22741 4.00000 −11.9190 −2.45482 −10.3056 −8.00000 −25.4935 23.8380
1.9 −2.00000 4.56533 4.00000 19.2752 −9.13065 −31.3699 −8.00000 −6.15779 −38.5503
1.10 −2.00000 6.22866 4.00000 6.79165 −12.4573 8.17804 −8.00000 11.7962 −13.5833
1.11 −2.00000 9.52532 4.00000 15.8501 −19.0506 18.4068 −8.00000 63.7317 −31.7003
1.12 −2.00000 9.80316 4.00000 −19.1773 −19.6063 6.27204 −8.00000 69.1019 38.3545
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(167\) \( +1 \)

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 334.4.a.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
334.4.a.c ✓ 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - T_{3}^{11} - 245 T_{3}^{10} + 220 T_{3}^{9} + 20500 T_{3}^{8} - 15289 T_{3}^{7} + \cdots - 4940224 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(334))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots - 4940224 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 410709341888 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 2431512209088 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 28\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 14\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 95\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 17\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 61\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 32\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 14\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 78\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 60\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 24\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 39\!\cdots\!44 \) Copy content Toggle raw display
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