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Properties

Label 242.4.c.r
Level $242$
Weight $4$
Character orbit 242.c
Analytic conductor $14.278$
Analytic rank $0$
Dimension $8$
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] = [242,4,Mod(3,242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(242, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.c (of order \(5\), degree \(4\), 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: \(14.2784622214\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 71x^{6} - 141x^{5} + 2911x^{4} + 2710x^{3} + 75340x^{2} + 169400x + 5856400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{4} q^{2} + (\beta_{7} + \beta_{4} - 1) q^{3} + 4 \beta_{3} q^{4} + (\beta_{5} - 4 \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{6} - 3 \beta_{5} - 21 \beta_{4} + \cdots - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{4} q^{2} + (\beta_{7} + \beta_{4} - 1) q^{3} + 4 \beta_{3} q^{4} + (\beta_{5} - 4 \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + ( - 10 \beta_{7} + 4 \beta_{6} + \cdots - 474) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 7 q^{3} - 8 q^{4} - 30 q^{5} - 6 q^{6} + 4 q^{7} + 16 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 7 q^{3} - 8 q^{4} - 30 q^{5} - 6 q^{6} + 4 q^{7} + 16 q^{8} - 81 q^{9} - 100 q^{10} + 32 q^{12} - 48 q^{13} - 8 q^{14} - 279 q^{15} - 32 q^{16} - 109 q^{17} + 42 q^{18} + 288 q^{19} - 120 q^{20} - 50 q^{21} + 628 q^{23} - 24 q^{24} + 38 q^{25} - 14 q^{26} + 242 q^{27} - 4 q^{28} + 528 q^{29} + 558 q^{30} - 522 q^{31} - 256 q^{32} + 208 q^{34} - 17 q^{35} - 84 q^{36} - 406 q^{37} + 544 q^{38} - 1429 q^{39} - 40 q^{40} - 329 q^{41} - 1480 q^{42} + 1442 q^{43} + 2652 q^{45} + 1044 q^{46} + 666 q^{47} - 112 q^{48} - 114 q^{49} + 34 q^{50} + 1158 q^{51} + 28 q^{52} + 414 q^{53} - 1144 q^{54} + 48 q^{56} - 593 q^{57} - 1056 q^{58} - 888 q^{59} + 844 q^{60} - 302 q^{61} - 646 q^{62} - 2061 q^{63} - 128 q^{64} - 138 q^{65} + 578 q^{67} - 436 q^{68} + 1930 q^{69} + 1394 q^{70} + 1090 q^{71} + 648 q^{72} + 253 q^{73} + 812 q^{74} + 2763 q^{75} - 128 q^{76} - 4152 q^{78} - 674 q^{79} + 80 q^{80} - 230 q^{81} - 722 q^{82} - 428 q^{83} - 2860 q^{84} + 1046 q^{85} - 984 q^{86} + 2122 q^{87} - 2202 q^{89} + 1366 q^{90} - 2217 q^{91} + 832 q^{92} - 3721 q^{93} + 2138 q^{94} - 973 q^{95} + 224 q^{96} + 3012 q^{97} - 3292 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 71x^{6} - 141x^{5} + 2911x^{4} + 2710x^{3} + 75340x^{2} + 169400x + 5856400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22554025143 \nu^{7} + 151013926876 \nu^{6} - 1356924294556 \nu^{5} + 28230146638036 \nu^{4} + \cdots + 48\!\cdots\!00 ) / 34\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14112299722 \nu^{7} + 955320989289 \nu^{6} + 1256420838616 \nu^{5} - 5250112809736 \nu^{4} + \cdots - 66\!\cdots\!20 ) / 17\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 123788106187 \nu^{7} + 5732748190805 \nu^{6} - 11662655347575 \nu^{5} + \cdots + 42\!\cdots\!20 ) / 68\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15778904729 \nu^{7} - 268932734519 \nu^{6} + 2855478562109 \nu^{5} - 16065997834439 \nu^{4} + \cdots - 12\!\cdots\!00 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15728307097 \nu^{7} - 23011705883 \nu^{6} - 3489045673392 \nu^{5} + \cdots - 22\!\cdots\!05 ) / 77\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 254952731119 \nu^{7} - 130622718559 \nu^{6} + 16218883337689 \nu^{5} + \cdots + 32\!\cdots\!00 ) / 31\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{5} - 8\beta_{4} + 8\beta_{3} + 54\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{6} + 47\beta_{5} + 46\beta_{4} + 8\beta_{3} - 8\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 109\beta_{7} + 16\beta_{6} - 16\beta_{5} + 2226\beta_{4} - 3034\beta_{3} - 3034\beta_{2} - 2210 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -824\beta_{7} - 3143\beta_{6} - 1608\beta_{4} + 1608\beta_{2} + 824\beta _1 - 7549 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2432\beta_{7} + 2432\beta_{5} + 176314\beta_{3} + 63048\beta_{2} - 6725\beta _1 + 63048 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 185471 \beta_{7} + 185471 \beta_{6} - 119991 \beta_{5} + 185128 \beta_{4} - 185128 \beta_{3} + \cdots + 185471 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\) \(123\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.22300 − 6.84169i
−2.53202 + 7.79275i
5.60402 + 4.07156i
−4.79501 − 3.48378i
5.60402 − 4.07156i
−4.79501 + 3.48378i
2.22300 + 6.84169i
−2.53202 − 7.79275i
1.61803 − 1.17557i −2.41398 − 7.42948i 1.23607 − 3.80423i −12.0690 − 8.76866i −12.6398 − 9.18334i 6.72664 − 20.7025i −2.47214 − 7.60845i −27.5264 + 19.9991i −29.8363
3.2 1.61803 − 1.17557i 2.34103 + 7.20496i 1.23607 − 3.80423i −4.37525 − 3.17880i 12.2578 + 8.90583i −6.84467 + 21.0657i −2.47214 − 7.60845i −24.5876 + 17.8639i −10.8162
9.1 −0.618034 + 1.90211i −6.91304 + 5.02262i −3.23607 − 2.35114i 3.93561 + 12.1126i −5.28109 − 16.2535i −18.9804 − 13.7901i 6.47214 − 4.70228i 14.2200 − 43.7646i −25.4718
9.2 −0.618034 + 1.90211i 3.48599 − 2.53272i −3.23607 − 2.35114i −2.49134 − 7.66756i 2.66306 + 8.19605i 21.0985 + 15.3289i 6.47214 − 4.70228i −2.60601 + 8.02046i 16.1243
27.1 −0.618034 − 1.90211i −6.91304 − 5.02262i −3.23607 + 2.35114i 3.93561 − 12.1126i −5.28109 + 16.2535i −18.9804 + 13.7901i 6.47214 + 4.70228i 14.2200 + 43.7646i −25.4718
27.2 −0.618034 − 1.90211i 3.48599 + 2.53272i −3.23607 + 2.35114i −2.49134 + 7.66756i 2.66306 − 8.19605i 21.0985 − 15.3289i 6.47214 + 4.70228i −2.60601 − 8.02046i 16.1243
81.1 1.61803 + 1.17557i −2.41398 + 7.42948i 1.23607 + 3.80423i −12.0690 + 8.76866i −12.6398 + 9.18334i 6.72664 + 20.7025i −2.47214 + 7.60845i −27.5264 − 19.9991i −29.8363
81.2 1.61803 + 1.17557i 2.34103 − 7.20496i 1.23607 + 3.80423i −4.37525 + 3.17880i 12.2578 − 8.90583i −6.84467 − 21.0657i −2.47214 + 7.60845i −24.5876 − 17.8639i −10.8162
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.4.c.r 8
11.b odd 2 1 242.4.c.n 8
11.c even 5 2 22.4.c.b ✓ 8
11.c even 5 1 242.4.a.n 4
11.c even 5 1 inner 242.4.c.r 8
11.d odd 10 1 242.4.a.o 4
11.d odd 10 1 242.4.c.n 8
11.d odd 10 2 242.4.c.q 8
33.f even 10 1 2178.4.a.bt 4
33.h odd 10 2 198.4.f.d 8
33.h odd 10 1 2178.4.a.by 4
44.g even 10 1 1936.4.a.bm 4
44.h odd 10 2 176.4.m.b 8
44.h odd 10 1 1936.4.a.bn 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b ✓ 8 11.c even 5 2
176.4.m.b 8 44.h odd 10 2
198.4.f.d 8 33.h odd 10 2
242.4.a.n 4 11.c even 5 1
242.4.a.o 4 11.d odd 10 1
242.4.c.n 8 11.b odd 2 1
242.4.c.n 8 11.d odd 10 1
242.4.c.q 8 11.d odd 10 2
242.4.c.r 8 1.a even 1 1 trivial
242.4.c.r 8 11.c even 5 1 inner
1936.4.a.bm 4 44.g even 10 1
1936.4.a.bn 4 44.h odd 10 1
2178.4.a.bt 4 33.f even 10 1
2178.4.a.by 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(242, [\chi])\):

\( T_{3}^{8} + 7T_{3}^{7} + 92T_{3}^{6} + 395T_{3}^{5} + 4301T_{3}^{4} + 65T_{3}^{3} + 115218T_{3}^{2} - 895569T_{3} + 4748041 \) Copy content Toggle raw display
\( T_{5}^{8} + 30 T_{5}^{7} + 556 T_{5}^{6} + 7795 T_{5}^{5} + 106181 T_{5}^{4} + 964060 T_{5}^{3} + \cdots + 68624656 \) Copy content Toggle raw display
\( T_{7}^{8} - 4 T_{7}^{7} + 408 T_{7}^{6} - 915 T_{7}^{5} + 318211 T_{7}^{4} + 1168230 T_{7}^{3} + \cdots + 87027360016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 7 T^{7} + \cdots + 4748041 \) Copy content Toggle raw display
$5$ \( T^{8} + 30 T^{7} + \cdots + 68624656 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 87027360016 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 16022496400 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 4078198497025 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3649418019025 \) Copy content Toggle raw display
$23$ \( (T^{4} - 314 T^{3} + \cdots - 257882816)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 60\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{4} - 721 T^{3} + \cdots - 2713875120)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} - 289 T^{3} + \cdots + 35027256944)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 32\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1101 T^{3} + \cdots - 64943655580)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 46\!\cdots\!81 \) Copy content Toggle raw display
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