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Properties

Label 114.4.i.c
Level $114$
Weight $4$
Character orbit 114.i
Analytic conductor $6.726$
Analytic rank $0$
Dimension $18$
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: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 1488 x^{16} + 4798 x^{15} + 939930 x^{14} - 421443 x^{13} - 319752461 x^{12} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{6} - 2 \beta_{4}) q^{2} - 3 \beta_{4} q^{3} + (4 \beta_{7} + 4 \beta_{5}) q^{4} + \beta_{9} q^{5} + 6 \beta_{5} q^{6} + ( - \beta_{17} - \beta_{16} + \cdots - 3 \beta_{2}) q^{7}+ \cdots - 9 \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{6} - 2 \beta_{4}) q^{2} - 3 \beta_{4} q^{3} + (4 \beta_{7} + 4 \beta_{5}) q^{4} + \beta_{9} q^{5} + 6 \beta_{5} q^{6} + ( - \beta_{17} - \beta_{16} + \cdots - 3 \beta_{2}) q^{7}+ \cdots + (27 \beta_{17} + 9 \beta_{15} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{5} - 30 q^{7} + 72 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{5} - 30 q^{7} + 72 q^{8} + 18 q^{10} + 57 q^{11} - 108 q^{12} - 228 q^{13} + 6 q^{14} + 27 q^{15} + 27 q^{17} - 324 q^{18} + 105 q^{19} - 24 q^{20} + 45 q^{21} + 222 q^{22} - 576 q^{23} - 243 q^{25} - 186 q^{26} - 243 q^{27} + 60 q^{28} - 339 q^{29} - 18 q^{30} + 429 q^{31} + 288 q^{33} - 450 q^{34} + 1680 q^{35} - 318 q^{37} + 114 q^{38} - 558 q^{39} + 72 q^{40} + 531 q^{41} + 18 q^{42} + 2172 q^{43} - 444 q^{44} + 27 q^{45} - 48 q^{46} - 816 q^{47} - 249 q^{49} + 780 q^{50} - 756 q^{51} + 276 q^{52} - 744 q^{53} + 2862 q^{55} - 480 q^{56} - 171 q^{57} + 900 q^{58} - 2307 q^{59} + 168 q^{61} + 1770 q^{62} - 108 q^{63} - 576 q^{64} + 1209 q^{65} - 90 q^{66} + 1944 q^{67} + 588 q^{68} + 72 q^{69} - 3360 q^{70} - 1011 q^{71} + 4152 q^{73} + 1854 q^{74} + 2340 q^{75} - 276 q^{76} - 7686 q^{77} - 954 q^{78} - 4929 q^{79} + 144 q^{80} - 1062 q^{82} + 156 q^{83} - 360 q^{84} - 2049 q^{85} + 2064 q^{86} + 675 q^{87} - 456 q^{88} - 4785 q^{89} - 162 q^{90} + 5166 q^{91} + 3852 q^{92} + 315 q^{93} + 1368 q^{94} - 1350 q^{95} - 1728 q^{96} + 2247 q^{97} - 2538 q^{98} + 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 1488 x^{16} + 4798 x^{15} + 939930 x^{14} - 421443 x^{13} - 319752461 x^{12} + \cdots + 20\!\cdots\!49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\!\cdots\!13 \nu^{17} + \cdots + 12\!\cdots\!14 ) / 74\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!27 \nu^{17} + \cdots - 10\!\cdots\!40 ) / 39\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!66 \nu^{17} + \cdots + 17\!\cdots\!63 ) / 74\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!98 \nu^{17} + \cdots + 22\!\cdots\!91 ) / 74\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!98 \nu^{17} + \cdots - 71\!\cdots\!29 ) / 39\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29\!\cdots\!20 \nu^{17} + \cdots + 16\!\cdots\!76 ) / 74\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!22 \nu^{17} + \cdots + 14\!\cdots\!42 ) / 74\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!06 \nu^{17} + \cdots - 38\!\cdots\!45 ) / 74\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 80\!\cdots\!28 \nu^{17} + \cdots + 45\!\cdots\!17 ) / 74\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\!\cdots\!36 \nu^{17} + \cdots + 21\!\cdots\!01 ) / 15\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!95 \nu^{17} + \cdots + 84\!\cdots\!99 ) / 74\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40\!\cdots\!10 \nu^{17} + \cdots + 17\!\cdots\!96 ) / 15\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 62\!\cdots\!66 \nu^{17} + \cdots - 57\!\cdots\!08 ) / 15\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 81\!\cdots\!86 \nu^{17} + \cdots + 54\!\cdots\!86 ) / 15\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!15 \nu^{17} + \cdots - 84\!\cdots\!02 ) / 15\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 21\!\cdots\!18 \nu^{17} + \cdots + 12\!\cdots\!61 ) / 15\!\cdots\!21 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 2 \beta_{16} - \beta_{15} + 6 \beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + \cdots + 170 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6 \beta_{17} + 10 \beta_{16} - 10 \beta_{15} + 6 \beta_{14} + 7 \beta_{13} - 33 \beta_{12} + 23 \beta_{11} + \cdots + 598 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 26 \beta_{17} - 328 \beta_{16} - 302 \beta_{15} + 1745 \beta_{14} + 256 \beta_{13} - 55 \beta_{12} + \cdots + 42502 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1870 \beta_{17} + 4662 \beta_{16} - 2715 \beta_{15} + 6775 \beta_{14} + 1195 \beta_{13} - 14333 \beta_{12} + \cdots + 273460 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11088 \beta_{17} - 22748 \beta_{16} - 85141 \beta_{15} + 509355 \beta_{14} + 67351 \beta_{13} + \cdots + 11159019 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 426965 \beta_{17} + 1898582 \beta_{16} - 835586 \beta_{15} + 3524804 \beta_{14} + 208204 \beta_{13} + \cdots + 101030540 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2387602 \beta_{17} + 13345002 \beta_{16} - 22357110 \beta_{15} + 152738044 \beta_{14} + \cdots + 3039351162 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 49425714 \beta_{17} + 729878092 \beta_{16} - 261632473 \beta_{15} + 1494534276 \beta_{14} + \cdots + 34347183360 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 296286550 \beta_{17} + 9359643809 \beta_{16} - 5681615897 \beta_{15} + 47201568086 \beta_{14} + \cdots + 855008089708 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 19869659447 \beta_{17} + 269983955781 \beta_{16} - 79734716553 \beta_{15} + 578536379041 \beta_{14} + \cdots + 11211865615895 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 639041443272 \beta_{17} + 4243787181307 \beta_{16} - 1427954751301 \beta_{15} + \cdots + 247157525895675 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 19929459292561 \beta_{17} + 96981626134376 \beta_{16} - 23539342066601 \beta_{15} + \cdots + 35\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 436938517812377 \beta_{17} + \cdots + 73\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 11\!\cdots\!01 \beta_{17} + \cdots + 11\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 23\!\cdots\!80 \beta_{17} + \cdots + 21\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 54\!\cdots\!02 \beta_{17} + \cdots + 35\!\cdots\!74 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(\beta_{5} + \beta_{7}\)

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
18.4957 − 0.342020i
−3.03906 − 0.342020i
−15.0491 − 0.342020i
16.8282 − 0.984808i
0.476726 − 0.984808i
−14.5992 − 0.984808i
−13.1496 + 0.642788i
−2.92272 + 0.642788i
15.9590 + 0.642788i
16.8282 + 0.984808i
0.476726 + 0.984808i
−14.5992 + 0.984808i
18.4957 + 0.342020i
−3.03906 + 0.342020i
−15.0491 + 0.342020i
−13.1496 − 0.642788i
−2.92272 − 0.642788i
15.9590 − 0.642788i
−1.53209 + 1.28558i −2.81908 − 1.02606i 0.694593 − 3.93923i −3.31462 − 18.7981i 5.63816 − 2.05212i −9.15421 + 15.8556i 4.00000 + 6.92820i 6.89440 + 5.78509i 29.2447 + 24.5392i
25.2 −1.53209 + 1.28558i −2.81908 − 1.02606i 0.694593 − 3.93923i 0.424858 + 2.40949i 5.63816 − 2.05212i −6.23258 + 10.7951i 4.00000 + 6.92820i 6.89440 + 5.78509i −3.74850 − 3.14536i
25.3 −1.53209 + 1.28558i −2.81908 − 1.02606i 0.694593 − 3.93923i 2.51037 + 14.2370i 5.63816 − 2.05212i 7.91501 − 13.7092i 4.00000 + 6.92820i 6.89440 + 5.78509i −22.1489 − 18.5851i
43.1 1.87939 − 0.684040i 0.520945 − 2.95442i 3.06418 − 2.57115i −11.5845 − 9.72055i −1.04189 − 5.90885i −2.24091 − 3.88137i 4.00000 − 6.92820i −8.45723 − 3.07818i −28.4210 − 10.3444i
43.2 1.87939 − 0.684040i 0.520945 − 2.95442i 3.06418 − 2.57115i 0.941477 + 0.789993i −1.04189 − 5.90885i −12.0128 − 20.8069i 4.00000 − 6.92820i −8.45723 − 3.07818i 2.30978 + 0.840693i
43.3 1.87939 − 0.684040i 0.520945 − 2.95442i 3.06418 − 2.57115i 12.4903 + 10.4806i −1.04189 − 5.90885i 11.3068 + 19.5839i 4.00000 − 6.92820i −8.45723 − 3.07818i 30.6433 + 11.1532i
55.1 −0.347296 + 1.96962i 2.29813 + 1.92836i −3.75877 − 1.36808i −11.3104 + 4.11665i −4.59627 + 3.85673i −16.9274 − 29.3191i 4.00000 − 6.92820i 1.56283 + 8.86327i −4.18016 − 23.7068i
55.2 −0.347296 + 1.96962i 2.29813 + 1.92836i −3.75877 − 1.36808i −1.70026 + 0.618843i −4.59627 + 3.85673i 8.26871 + 14.3218i 4.00000 − 6.92820i 1.56283 + 8.86327i −0.628389 − 3.56377i
55.3 −0.347296 + 1.96962i 2.29813 + 1.92836i −3.75877 − 1.36808i 16.0427 − 5.83908i −4.59627 + 3.85673i 4.07744 + 7.06234i 4.00000 − 6.92820i 1.56283 + 8.86327i 5.92916 + 33.6259i
61.1 1.87939 + 0.684040i 0.520945 + 2.95442i 3.06418 + 2.57115i −11.5845 + 9.72055i −1.04189 + 5.90885i −2.24091 + 3.88137i 4.00000 + 6.92820i −8.45723 + 3.07818i −28.4210 + 10.3444i
61.2 1.87939 + 0.684040i 0.520945 + 2.95442i 3.06418 + 2.57115i 0.941477 − 0.789993i −1.04189 + 5.90885i −12.0128 + 20.8069i 4.00000 + 6.92820i −8.45723 + 3.07818i 2.30978 − 0.840693i
61.3 1.87939 + 0.684040i 0.520945 + 2.95442i 3.06418 + 2.57115i 12.4903 − 10.4806i −1.04189 + 5.90885i 11.3068 − 19.5839i 4.00000 + 6.92820i −8.45723 + 3.07818i 30.6433 − 11.1532i
73.1 −1.53209 − 1.28558i −2.81908 + 1.02606i 0.694593 + 3.93923i −3.31462 + 18.7981i 5.63816 + 2.05212i −9.15421 − 15.8556i 4.00000 − 6.92820i 6.89440 − 5.78509i 29.2447 − 24.5392i
73.2 −1.53209 − 1.28558i −2.81908 + 1.02606i 0.694593 + 3.93923i 0.424858 − 2.40949i 5.63816 + 2.05212i −6.23258 − 10.7951i 4.00000 − 6.92820i 6.89440 − 5.78509i −3.74850 + 3.14536i
73.3 −1.53209 − 1.28558i −2.81908 + 1.02606i 0.694593 + 3.93923i 2.51037 − 14.2370i 5.63816 + 2.05212i 7.91501 + 13.7092i 4.00000 − 6.92820i 6.89440 − 5.78509i −22.1489 + 18.5851i
85.1 −0.347296 − 1.96962i 2.29813 − 1.92836i −3.75877 + 1.36808i −11.3104 − 4.11665i −4.59627 − 3.85673i −16.9274 + 29.3191i 4.00000 + 6.92820i 1.56283 − 8.86327i −4.18016 + 23.7068i
85.2 −0.347296 − 1.96962i 2.29813 − 1.92836i −3.75877 + 1.36808i −1.70026 − 0.618843i −4.59627 − 3.85673i 8.26871 − 14.3218i 4.00000 + 6.92820i 1.56283 − 8.86327i −0.628389 + 3.56377i
85.3 −0.347296 − 1.96962i 2.29813 − 1.92836i −3.75877 + 1.36808i 16.0427 + 5.83908i −4.59627 − 3.85673i 4.07744 − 7.06234i 4.00000 + 6.92820i 1.56283 − 8.86327i 5.92916 − 33.6259i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
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.c ✓ 18
19.e even 9 1 inner 114.4.i.c ✓ 18
19.e even 9 1 2166.4.a.bk 9
19.f odd 18 1 2166.4.a.bl 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.i.c ✓ 18 1.a even 1 1 trivial
114.4.i.c ✓ 18 19.e even 9 1 inner
2166.4.a.bk 9 19.e even 9 1
2166.4.a.bl 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 9 T_{5}^{17} + 162 T_{5}^{16} - 3304 T_{5}^{15} + 12366 T_{5}^{14} + 553023 T_{5}^{13} + \cdots + 57\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 8 T^{3} + 64)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} + 27 T^{3} + 729)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 57\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 16\!\cdots\!81 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 70\!\cdots\!69 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 94\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 33\!\cdots\!39 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 24\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots + 18\!\cdots\!13)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 16\!\cdots\!09 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 89\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 52\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 46\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 92\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 25\!\cdots\!21 \) Copy content Toggle raw display
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