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Properties

Label 338.4.c.o
Level $338$
Weight $4$
Character orbit 338.c
Analytic conductor $19.943$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(191,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.c (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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 108 x^{10} - 63 x^{9} + 7831 x^{8} - 3348 x^{7} + 317885 x^{6} + 1680 x^{5} + \cdots + 1759886401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_1 - 2) q^{2} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{3} - 4 \beta_1 q^{4} + ( - \beta_{11} - \beta_{9} + \beta_{4} + 3) q^{5} + ( - 2 \beta_{2} - 4 \beta_1) q^{6} + ( - \beta_{8} - 2 \beta_{6} + \cdots - 5 \beta_1) q^{7}+ \cdots + (2 \beta_{8} + 2 \beta_{6} + \cdots - 18 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_1 - 2) q^{2} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{3} - 4 \beta_1 q^{4} + ( - \beta_{11} - \beta_{9} + \beta_{4} + 3) q^{5} + ( - 2 \beta_{2} - 4 \beta_1) q^{6} + ( - \beta_{8} - 2 \beta_{6} + \cdots - 5 \beta_1) q^{7}+ \cdots + (50 \beta_{11} - 80 \beta_{10} + \cdots - 323) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 9 q^{3} - 24 q^{4} + 36 q^{5} - 18 q^{6} - 25 q^{7} + 96 q^{8} - 113 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 9 q^{3} - 24 q^{4} + 36 q^{5} - 18 q^{6} - 25 q^{7} + 96 q^{8} - 113 q^{9} - 36 q^{10} - 37 q^{11} + 72 q^{12} + 100 q^{14} - 118 q^{15} - 96 q^{16} - 99 q^{17} + 452 q^{18} + 81 q^{19} - 72 q^{20} - 52 q^{21} - 74 q^{22} - 267 q^{23} - 72 q^{24} + 736 q^{25} + 1338 q^{27} - 100 q^{28} + 119 q^{29} - 236 q^{30} - 1250 q^{31} - 192 q^{32} - 762 q^{33} + 396 q^{34} - 614 q^{35} - 452 q^{36} - 274 q^{37} - 324 q^{38} + 288 q^{40} - 1140 q^{41} + 52 q^{42} - 428 q^{43} + 296 q^{44} + 1215 q^{45} - 534 q^{46} - 1972 q^{47} - 144 q^{48} - 899 q^{49} - 736 q^{50} + 578 q^{51} + 178 q^{53} - 1338 q^{54} - 1126 q^{55} - 200 q^{56} - 5106 q^{57} + 238 q^{58} - 1088 q^{59} + 944 q^{60} - 1704 q^{61} + 1250 q^{62} + 3222 q^{63} + 768 q^{64} + 3048 q^{66} + 1692 q^{67} - 396 q^{68} - 1168 q^{69} + 2456 q^{70} + 1221 q^{71} - 904 q^{72} - 3108 q^{73} - 548 q^{74} - 1798 q^{75} + 324 q^{76} + 5580 q^{77} - 1750 q^{79} - 288 q^{80} - 3338 q^{81} - 2280 q^{82} - 252 q^{83} + 104 q^{84} + 3721 q^{85} + 1712 q^{86} - 1602 q^{87} - 296 q^{88} + 374 q^{89} - 4860 q^{90} + 2136 q^{92} + 1868 q^{93} + 1972 q^{94} + 4093 q^{95} + 576 q^{96} + 330 q^{97} - 1798 q^{98} - 2688 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 108 x^{10} - 63 x^{9} + 7831 x^{8} - 3348 x^{7} + 317885 x^{6} + 1680 x^{5} + \cdots + 1759886401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 78\!\cdots\!56 \nu^{11} + \cdots - 57\!\cdots\!19 ) / 10\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!33 \nu^{11} + \cdots + 19\!\cdots\!38 ) / 23\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!54 \nu^{11} + \cdots + 37\!\cdots\!22 ) / 23\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!40 \nu^{11} + \cdots + 30\!\cdots\!60 ) / 94\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32\!\cdots\!43 \nu^{11} + \cdots + 93\!\cdots\!08 ) / 23\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!73 \nu^{11} + \cdots + 38\!\cdots\!33 ) / 30\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36\!\cdots\!88 \nu^{11} + \cdots - 21\!\cdots\!67 ) / 72\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!38 \nu^{11} + \cdots - 46\!\cdots\!79 ) / 30\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!21 \nu^{11} + \cdots + 50\!\cdots\!26 ) / 94\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 33\!\cdots\!60 \nu^{11} + \cdots - 10\!\cdots\!16 ) / 94\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 44\!\cdots\!10 \nu^{11} + \cdots + 17\!\cdots\!89 ) / 94\!\cdots\!17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} + 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 4 ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{8} - 5\beta_{6} - 8\beta_{5} + 2\beta_{3} - 3\beta_{2} - 469\beta_1 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -66\beta_{11} - 92\beta_{10} - 40\beta_{9} - 145\beta_{7} - 270\beta_{4} + 91 ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 443 \beta_{11} + 363 \beta_{10} - 625 \beta_{9} + 443 \beta_{8} - 143 \beta_{7} + 363 \beta_{6} + \cdots - 17646 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3566\beta_{8} - 4108\beta_{6} + 91\beta_{5} - 6933\beta_{3} + 9889\beta_{2} + 10806\beta_1 ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -25568\beta_{11} - 20720\beta_{10} + 37260\beta_{9} + 7480\beta_{7} - 2659\beta_{4} + 710376 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 181248 \beta_{11} + 184622 \beta_{10} - 57074 \beta_{9} + 181248 \beta_{8} + 329392 \beta_{7} + \cdots - 494301 ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1359421 \beta_{8} - 1094536 \beta_{6} - 2009604 \beta_{5} + 335375 \beta_{3} + 446732 \beta_{2} - 29887091 \beta_1 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9036326 \beta_{11} - 8466569 \beta_{10} + 4886472 \beta_{9} - 15591752 \beta_{7} - 16312087 \beta_{4} + 35539298 ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 69817796 \beta_{11} + 56028534 \beta_{10} - 103389010 \beta_{9} + 69817796 \beta_{8} + \cdots - 1316074697 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 448638481 \beta_{8} - 397234871 \beta_{6} - 313302931 \beta_{5} - 736608573 \beta_{3} + \cdots - 827715592 \beta_1 ) / 13 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
2.92486 − 5.06601i
2.90876 − 5.03812i
3.32703 − 5.76258i
−2.70234 + 4.68060i
−3.53225 + 6.11803i
−2.42606 + 4.20205i
2.92486 + 5.06601i
2.90876 + 5.03812i
3.32703 + 5.76258i
−2.70234 − 4.68060i
−3.53225 − 6.11803i
−2.42606 − 4.20205i
−1.00000 + 1.73205i −5.00428 + 8.66767i −2.00000 − 3.46410i −13.8136 −10.0086 − 17.3353i 0.131143 + 0.227146i 8.00000 −36.5857 − 63.3682i 13.8136 − 23.9258i
191.2 −1.00000 + 1.73205i −4.47729 + 7.75489i −2.00000 − 3.46410i 17.3267 −8.95458 − 15.5098i 8.37550 + 14.5068i 8.00000 −26.5923 − 46.0591i −17.3267 + 30.0108i
191.3 −1.00000 + 1.73205i −1.67908 + 2.90825i −2.00000 − 3.46410i 6.80407 −3.35815 − 5.81649i −7.76902 − 13.4563i 8.00000 7.86140 + 13.6163i −6.80407 + 11.7850i
191.4 −1.00000 + 1.73205i 0.622927 − 1.07894i −2.00000 − 3.46410i 20.5002 1.24585 + 2.15788i −10.6273 − 18.4070i 8.00000 12.7239 + 22.0385i −20.5002 + 35.5074i
191.5 −1.00000 + 1.73205i 1.96372 − 3.40126i −2.00000 − 3.46410i −0.152827 3.92743 + 6.80251i −16.9956 − 29.4373i 8.00000 5.78763 + 10.0245i 0.152827 − 0.264703i
191.6 −1.00000 + 1.73205i 4.07401 − 7.05638i −2.00000 − 3.46410i −12.6646 8.14801 + 14.1128i 14.3853 + 24.9160i 8.00000 −19.6950 − 34.1128i 12.6646 − 21.9357i
315.1 −1.00000 − 1.73205i −5.00428 − 8.66767i −2.00000 + 3.46410i −13.8136 −10.0086 + 17.3353i 0.131143 − 0.227146i 8.00000 −36.5857 + 63.3682i 13.8136 + 23.9258i
315.2 −1.00000 − 1.73205i −4.47729 − 7.75489i −2.00000 + 3.46410i 17.3267 −8.95458 + 15.5098i 8.37550 − 14.5068i 8.00000 −26.5923 + 46.0591i −17.3267 − 30.0108i
315.3 −1.00000 − 1.73205i −1.67908 − 2.90825i −2.00000 + 3.46410i 6.80407 −3.35815 + 5.81649i −7.76902 + 13.4563i 8.00000 7.86140 − 13.6163i −6.80407 − 11.7850i
315.4 −1.00000 − 1.73205i 0.622927 + 1.07894i −2.00000 + 3.46410i 20.5002 1.24585 − 2.15788i −10.6273 + 18.4070i 8.00000 12.7239 − 22.0385i −20.5002 − 35.5074i
315.5 −1.00000 − 1.73205i 1.96372 + 3.40126i −2.00000 + 3.46410i −0.152827 3.92743 − 6.80251i −16.9956 + 29.4373i 8.00000 5.78763 − 10.0245i 0.152827 + 0.264703i
315.6 −1.00000 − 1.73205i 4.07401 + 7.05638i −2.00000 + 3.46410i −12.6646 8.14801 − 14.1128i 14.3853 − 24.9160i 8.00000 −19.6950 + 34.1128i 12.6646 + 21.9357i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.c.o 12
13.b even 2 1 338.4.c.p 12
13.c even 3 1 338.4.a.o yes 6
13.c even 3 1 inner 338.4.c.o 12
13.d odd 4 2 338.4.e.i 24
13.e even 6 1 338.4.a.n ✓ 6
13.e even 6 1 338.4.c.p 12
13.f odd 12 2 338.4.b.h 12
13.f odd 12 2 338.4.e.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.n ✓ 6 13.e even 6 1
338.4.a.o yes 6 13.c even 3 1
338.4.b.h 12 13.f odd 12 2
338.4.c.o 12 1.a even 1 1 trivial
338.4.c.o 12 13.c even 3 1 inner
338.4.c.p 12 13.b even 2 1
338.4.c.p 12 13.e even 6 1
338.4.e.i 24 13.d odd 4 2
338.4.e.i 24 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{12} + 9 T_{3}^{11} + 178 T_{3}^{10} + 589 T_{3}^{9} + 13675 T_{3}^{8} + 37320 T_{3}^{7} + \cdots + 143976001 \) Copy content Toggle raw display
\( T_{5}^{6} - 18T_{5}^{5} - 397T_{5}^{4} + 5935T_{5}^{3} + 44090T_{5}^{2} - 416208T_{5} - 64616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 143976001 \) Copy content Toggle raw display
$5$ \( (T^{6} - 18 T^{5} + \cdots - 64616)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 2013515592256 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 81\!\cdots\!69 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 18\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 96\!\cdots\!89 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 84\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{6} + 625 T^{5} + \cdots + 254759240632)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 38\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 549423790361944)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 14\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 74\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 52281997669273)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 79\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 99\!\cdots\!91)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 91\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
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