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Properties

Label 114.4.i.b
Level $114$
Weight $4$
Character orbit 114.i
Analytic conductor $6.726$
Analytic rank $0$
Dimension $12$
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] = [114,4,Mod(25,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("114.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 114.i (of order \(9\), degree \(6\), 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: \(6.72621774065\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 252x^{10} + 25236x^{8} - 1271150x^{6} + 34071732x^{4} - 464617557x^{2} + 2560258801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{7} + 2 \beta_{4}) q^{2} - 3 \beta_{4} q^{3} - 4 \beta_{8} q^{4} + ( - 2 \beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{5}+ \cdots + (9 \beta_{8} - 9 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{7} + 2 \beta_{4}) q^{2} - 3 \beta_{4} q^{3} - 4 \beta_{8} q^{4} + ( - 2 \beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - 54 \beta_{11} - 18 \beta_{10} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{5} + 12 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 9 q^{5} + 12 q^{7} + 48 q^{8} - 18 q^{10} + 93 q^{11} + 72 q^{12} + 171 q^{13} - 108 q^{14} + 27 q^{15} + 249 q^{17} - 216 q^{18} + 168 q^{19} - 24 q^{20} + 162 q^{21} - 162 q^{22} + 24 q^{23} + 81 q^{25} - 30 q^{26} + 162 q^{27} - 216 q^{28} - 513 q^{29} + 18 q^{30} - 345 q^{31} + 438 q^{34} + 330 q^{35} + 510 q^{37} - 270 q^{38} + 90 q^{39} - 72 q^{40} + 489 q^{41} + 324 q^{42} - 1611 q^{43} + 324 q^{44} + 27 q^{45} - 972 q^{46} - 420 q^{47} - 252 q^{49} - 702 q^{50} + 90 q^{51} + 108 q^{52} + 294 q^{53} - 810 q^{55} + 192 q^{56} - 243 q^{57} - 732 q^{58} + 897 q^{59} + 2640 q^{61} - 588 q^{62} - 384 q^{64} - 1023 q^{65} - 486 q^{66} - 39 q^{67} + 996 q^{68} - 1458 q^{69} - 660 q^{70} + 135 q^{71} - 1707 q^{73} - 138 q^{74} + 2106 q^{75} + 1092 q^{76} - 3450 q^{77} - 1188 q^{78} + 1731 q^{79} - 144 q^{80} - 978 q^{82} + 1830 q^{83} - 144 q^{84} + 1371 q^{85} + 2268 q^{86} + 549 q^{87} - 744 q^{88} - 1233 q^{89} + 162 q^{90} - 1338 q^{91} + 780 q^{92} + 711 q^{93} + 5136 q^{94} + 2016 q^{95} + 1152 q^{96} - 3273 q^{97} + 4416 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 252x^{10} + 25236x^{8} - 1271150x^{6} + 34071732x^{4} - 464617557x^{2} + 2560258801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3136 \nu^{10} - 693336 \nu^{8} + 57708585 \nu^{6} - 2212573006 \nu^{4} + 39724323312 \nu^{2} + \cdots - 276215134095 ) / 1126741142 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3136 \nu^{10} + 693336 \nu^{8} - 57708585 \nu^{6} + 2212573006 \nu^{4} - 39724323312 \nu^{2} + \cdots + 276215134095 ) / 1126741142 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61536544858379 \nu^{11} - 234936088277342 \nu^{10} + \cdots + 27\!\cdots\!54 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61536544858379 \nu^{11} - 234936088277342 \nu^{10} + \cdots + 27\!\cdots\!54 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 67457308912972 \nu^{11} + \cdots - 77\!\cdots\!26 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67457308912972 \nu^{11} + \cdots - 77\!\cdots\!26 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 167478779364370 \nu^{11} - 810046411105260 \nu^{10} + \cdots + 80\!\cdots\!53 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 167478779364370 \nu^{11} - 810046411105260 \nu^{10} + \cdots + 80\!\cdots\!53 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 239377408045423 \nu^{11} + \cdots - 58\!\cdots\!58 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 239377408045423 \nu^{11} + \cdots - 58\!\cdots\!58 ) / 35\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5458905 \nu^{11} + 1216965596 \nu^{9} - 102678818316 \nu^{7} + 4019090398335 \nu^{5} + \cdots + 28505987522029 ) / 57011975044058 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + 12\beta_{8} + 12\beta_{7} - \beta_{6} - \beta_{5} - 19\beta_{4} - 19\beta_{3} + 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 60 \beta_{11} - 11 \beta_{10} + 11 \beta_{9} - 69 \beta_{8} + 69 \beta_{7} + 7 \beta_{6} - 7 \beta_{5} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 121 \beta_{10} + 121 \beta_{9} + 958 \beta_{8} + 958 \beta_{7} - 99 \beta_{6} - 99 \beta_{5} + \cdots + 2161 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8896 \beta_{11} - 800 \beta_{10} + 800 \beta_{9} - 9104 \beta_{8} + 9104 \beta_{7} + 779 \beta_{6} + \cdots - 4448 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11966 \beta_{10} + 11966 \beta_{9} + 63932 \beta_{8} + 63932 \beta_{7} - 9418 \beta_{6} - 9418 \beta_{5} + \cdots + 121733 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 941652 \beta_{11} - 46739 \beta_{10} + 46739 \beta_{9} - 921754 \beta_{8} + 921754 \beta_{7} + \cdots - 470826 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1080808 \beta_{10} + 1080808 \beta_{9} + 4061021 \beta_{8} + 4061021 \beta_{7} - 856972 \beta_{6} + \cdots + 6747066 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 86621950 \beta_{11} - 2450270 \beta_{10} + 2450270 \beta_{9} - 83119833 \beta_{8} + 83119833 \beta_{7} + \cdots - 43310975 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 91460197 \beta_{10} + 91460197 \beta_{9} + 245274190 \beta_{8} + 245274190 \beta_{7} - 73338804 \beta_{6} + \cdots + 332304973 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7324291702 \beta_{11} - 106911054 \beta_{10} + 106911054 \beta_{9} - 6959340618 \beta_{8} + \cdots - 3662145851 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
8.59757 − 0.984808i
−8.59757 − 0.984808i
−4.37438 − 0.642788i
4.37438 − 0.642788i
5.86912 − 0.342020i
−5.86912 − 0.342020i
−4.37438 + 0.642788i
4.37438 + 0.642788i
8.59757 + 0.984808i
−8.59757 + 0.984808i
5.86912 + 0.342020i
−5.86912 + 0.342020i
−1.53209 + 1.28558i 2.81908 + 1.02606i 0.694593 − 3.93923i −0.778339 − 4.41418i −5.63816 + 2.05212i −10.3951 + 18.0048i 4.00000 + 6.92820i 6.89440 + 5.78509i 6.86724 + 5.76230i
25.2 −1.53209 + 1.28558i 2.81908 + 1.02606i 0.694593 − 3.93923i 0.810428 + 4.59616i −5.63816 + 2.05212i 15.9494 − 27.6252i 4.00000 + 6.92820i 6.89440 + 5.78509i −7.15036 − 5.99987i
43.1 1.87939 − 0.684040i −0.520945 + 2.95442i 3.06418 − 2.57115i −11.3384 − 9.51407i 1.04189 + 5.90885i −12.3384 − 21.3707i 4.00000 − 6.92820i −8.45723 − 3.07818i −27.8173 − 10.1247i
43.2 1.87939 − 0.684040i −0.520945 + 2.95442i 3.06418 − 2.57115i 7.95904 + 6.67843i 1.04189 + 5.90885i 4.10394 + 7.10823i 4.00000 − 6.92820i −8.45723 − 3.07818i 19.5264 + 7.10704i
55.1 −0.347296 + 1.96962i −2.29813 − 1.92836i −3.75877 − 1.36808i −4.17612 + 1.51998i 4.59627 − 3.85673i 2.30170 + 3.98666i 4.00000 − 6.92820i 1.56283 + 8.86327i −1.54343 − 8.75324i
55.2 −0.347296 + 1.96962i −2.29813 − 1.92836i −3.75877 − 1.36808i 3.02342 − 1.10043i 4.59627 − 3.85673i 6.37835 + 11.0476i 4.00000 − 6.92820i 1.56283 + 8.86327i 1.11741 + 6.33715i
61.1 1.87939 + 0.684040i −0.520945 − 2.95442i 3.06418 + 2.57115i −11.3384 + 9.51407i 1.04189 − 5.90885i −12.3384 + 21.3707i 4.00000 + 6.92820i −8.45723 + 3.07818i −27.8173 + 10.1247i
61.2 1.87939 + 0.684040i −0.520945 − 2.95442i 3.06418 + 2.57115i 7.95904 − 6.67843i 1.04189 − 5.90885i 4.10394 − 7.10823i 4.00000 + 6.92820i −8.45723 + 3.07818i 19.5264 − 7.10704i
73.1 −1.53209 − 1.28558i 2.81908 − 1.02606i 0.694593 + 3.93923i −0.778339 + 4.41418i −5.63816 − 2.05212i −10.3951 − 18.0048i 4.00000 − 6.92820i 6.89440 − 5.78509i 6.86724 − 5.76230i
73.2 −1.53209 − 1.28558i 2.81908 − 1.02606i 0.694593 + 3.93923i 0.810428 − 4.59616i −5.63816 − 2.05212i 15.9494 + 27.6252i 4.00000 − 6.92820i 6.89440 − 5.78509i −7.15036 + 5.99987i
85.1 −0.347296 − 1.96962i −2.29813 + 1.92836i −3.75877 + 1.36808i −4.17612 − 1.51998i 4.59627 + 3.85673i 2.30170 − 3.98666i 4.00000 + 6.92820i 1.56283 − 8.86327i −1.54343 + 8.75324i
85.2 −0.347296 − 1.96962i −2.29813 + 1.92836i −3.75877 + 1.36808i 3.02342 + 1.10043i 4.59627 + 3.85673i 6.37835 − 11.0476i 4.00000 + 6.92820i 1.56283 − 8.86327i 1.11741 − 6.33715i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.i.b ✓ 12
19.e even 9 1 inner 114.4.i.b ✓ 12
19.e even 9 1 2166.4.a.bb 6
19.f odd 18 1 2166.4.a.bg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.i.b ✓ 12 1.a even 1 1 trivial
114.4.i.b ✓ 12 19.e even 9 1 inner
2166.4.a.bb 6 19.e even 9 1
2166.4.a.bg 6 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 9 T_{5}^{11} - 928 T_{5}^{9} + 20934 T_{5}^{8} + 30033 T_{5}^{7} + 383923 T_{5}^{6} + \cdots + 2115908001 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 27 T^{3} + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 2115908001 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 62220433568049 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 88\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 18\!\cdots\!09 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 44\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 28266080172299)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 61\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 21\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 41\!\cdots\!09 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 26\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 34\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 97\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 19\!\cdots\!81 \) Copy content Toggle raw display
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