↑

Properties

Label 185.2.b.a
Level $185$
Weight $2$
Character orbit 185.b
Analytic conductor $1.477$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [185,2,Mod(149,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.b (of order \(2\), degree \(1\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 28x^{16} + 306x^{14} + 1684x^{12} + 5049x^{10} + 8280x^{8} + 7004x^{6} + 2672x^{4} + 368x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{13} + \beta_1) q^{2} + ( - \beta_{13} + \beta_{12}) q^{3} + (\beta_{4} + \beta_{3} - 2) q^{4} + \beta_{11} q^{5} + ( - \beta_{17} + \beta_{15} - \beta_{8} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{17} - \beta_{14} - \beta_{8} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} + \beta_1) q^{2} + ( - \beta_{13} + \beta_{12}) q^{3} + (\beta_{4} + \beta_{3} - 2) q^{4} + \beta_{11} q^{5} + ( - \beta_{17} + \beta_{15} - \beta_{8} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{17} + 2 \beta_{15} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 22 q^{4} + 2 q^{5} + 4 q^{6} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 22 q^{4} + 2 q^{5} + 4 q^{6} - 26 q^{9} + 6 q^{10} + 8 q^{14} - 2 q^{15} + 22 q^{16} + 8 q^{19} - 4 q^{20} + 8 q^{21} - 12 q^{24} - 18 q^{25} - 12 q^{26} + 4 q^{29} - 12 q^{31} - 12 q^{34} - 2 q^{35} + 22 q^{36} + 32 q^{39} - 6 q^{40} + 4 q^{41} - 8 q^{44} - 16 q^{45} + 32 q^{46} + 14 q^{49} + 8 q^{50} + 16 q^{51} + 36 q^{54} - 34 q^{55} - 48 q^{56} - 20 q^{59} + 28 q^{60} + 16 q^{61} - 30 q^{64} + 20 q^{65} - 68 q^{66} + 20 q^{69} + 40 q^{70} + 16 q^{71} + 10 q^{74} - 56 q^{75} - 96 q^{76} + 52 q^{79} + 36 q^{80} + 26 q^{81} + 48 q^{84} - 8 q^{85} - 32 q^{86} + 16 q^{89} - 86 q^{90} - 36 q^{91} + 52 q^{94} - 44 q^{95} + 104 q^{96} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 28x^{16} + 306x^{14} + 1684x^{12} + 5049x^{10} + 8280x^{8} + 7004x^{6} + 2672x^{4} + 368x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 177 \nu^{16} - 6496 \nu^{14} - 93358 \nu^{12} - 669460 \nu^{10} - 2540885 \nu^{8} - 5036892 \nu^{6} + \cdots - 132264 ) / 19592 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1077 \nu^{16} + 29689 \nu^{14} + 317307 \nu^{12} + 1691115 \nu^{10} + 4838164 \nu^{8} + \cdots + 232640 ) / 39184 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1077 \nu^{16} + 29689 \nu^{14} + 317307 \nu^{12} + 1691115 \nu^{10} + 4838164 \nu^{8} + \cdots + 115088 ) / 39184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1977 \nu^{16} + 55331 \nu^{14} + 602481 \nu^{12} + 3278489 \nu^{10} + 9562402 \nu^{8} + \cdots + 352608 ) / 39184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3059 \nu^{16} - 83557 \nu^{14} - 882923 \nu^{12} - 4649903 \nu^{10} - 13225414 \nu^{8} + \cdots - 489696 ) / 39184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 123 \nu^{17} - 1799 \nu^{16} - 3684 \nu^{15} - 50597 \nu^{14} - 43333 \nu^{13} - 555680 \nu^{12} + \cdots - 235536 ) / 19592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1253 \nu^{16} - 35498 \nu^{14} - 393548 \nu^{12} - 2199697 \nu^{10} - 6674643 \nu^{8} + \cdots - 223360 ) / 9796 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 685 \nu^{17} - 19979 \nu^{15} - 229539 \nu^{13} - 1336033 \nu^{11} - 4218614 \nu^{9} + \cdots - 75200 \nu ) / 19592 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1911 \nu^{17} + 3598 \nu^{16} - 54071 \nu^{15} + 101194 \nu^{14} - 599299 \nu^{13} + \cdots + 471072 ) / 39184 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9783 \nu^{17} - 649 \nu^{16} + 270255 \nu^{15} - 19737 \nu^{14} + 2897505 \nu^{13} - 240271 \nu^{12} + \cdots - 504560 ) / 78368 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7055 \nu^{17} - 198583 \nu^{15} - 2184753 \nu^{13} - 12115545 \nu^{11} - 36568528 \nu^{9} + \cdots - 1536448 \nu ) / 39184 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7193 \nu^{17} - 200327 \nu^{15} - 2171369 \nu^{13} - 11795705 \nu^{11} - 34626342 \nu^{9} + \cdots - 1046496 \nu ) / 39184 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17917 \nu^{17} + 8701 \nu^{16} + 499901 \nu^{15} + 242029 \nu^{14} + 5433799 \nu^{13} + \cdots + 824496 ) / 78368 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17917 \nu^{17} - 7137 \nu^{16} - 499901 \nu^{15} - 196141 \nu^{14} - 5433799 \nu^{13} + \cdots - 917648 ) / 78368 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 17775 \nu^{17} + 649 \nu^{16} + 500307 \nu^{15} + 19737 \nu^{14} + 5501165 \nu^{13} + 240271 \nu^{12} + \cdots + 504560 ) / 78368 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17917 \nu^{17} + 7137 \nu^{16} - 499901 \nu^{15} + 196141 \nu^{14} - 5433799 \nu^{13} + \cdots + 917648 ) / 78368 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} - \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - 3\beta_{15} - 2\beta_{14} + \beta_{8} + 3\beta_{6} + 2\beta_{5} + 10\beta_{4} - 6\beta_{3} - \beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{17} + 4 \beta_{16} + 13 \beta_{15} + 24 \beta_{14} - 20 \beta_{13} + 13 \beta_{12} - 22 \beta_{11} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 17 \beta_{17} + 41 \beta_{15} + 24 \beta_{14} - 19 \beta_{8} - 41 \beta_{6} - 28 \beta_{5} + \cdots - 106 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 95 \beta_{17} - 70 \beta_{16} - 135 \beta_{15} - 236 \beta_{14} + 164 \beta_{13} - 139 \beta_{12} + \cdots - 236 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 209 \beta_{17} - 445 \beta_{15} - 236 \beta_{14} + 259 \beta_{8} + 435 \beta_{6} + 320 \beta_{5} + \cdots + 774 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 771 \beta_{17} + 896 \beta_{16} + 1333 \beta_{15} + 2204 \beta_{14} - 1292 \beta_{13} + 1381 \beta_{12} + \cdots + 2204 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2285 \beta_{17} + 4489 \beta_{15} + 2204 \beta_{14} - 3055 \beta_{8} - 4281 \beta_{6} - 3384 \beta_{5} + \cdots - 5914 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6179 \beta_{17} - 10150 \beta_{16} - 12939 \beta_{15} - 20288 \beta_{14} + 10148 \beta_{13} + \cdots - 20288 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 23609 \beta_{17} - 43897 \beta_{15} - 20288 \beta_{14} + 33303 \beta_{8} + 40963 \beta_{6} + \cdots + 46338 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 49747 \beta_{17} + 107844 \beta_{16} + 124413 \beta_{15} + 186232 \beta_{14} - 80356 \beta_{13} + \cdots + 186232 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 236269 \beta_{17} + 422501 \beta_{15} + 186232 \beta_{14} - 346147 \beta_{8} - 387289 \beta_{6} + \cdots - 369662 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 404923 \beta_{17} - 1102274 \beta_{16} - 1188003 \beta_{15} - 1711068 \beta_{14} + 643996 \beta_{13} + \cdots - 1711068 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2317633 \beta_{17} - 4028701 \beta_{15} - 1711068 \beta_{14} + 3488643 \beta_{8} + 3640771 \beta_{6} + \cdots + 2995654 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3340147 \beta_{17} + 10983688 \beta_{16} + 11282085 \beta_{15} + 15751236 \beta_{14} - 5232060 \beta_{13} + \cdots + 15751236 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
− 1.68489i
− 1.58855i
− 1.15179i
− 3.05104i
− 2.70607i
− 0.356037i
− 1.97415i
0.296889i
0.753109i
− 0.753109i
− 0.296889i
1.97415i
0.356037i
2.70607i
3.05104i
1.15179i
1.58855i
1.68489i
− 2.68489i 2.42453i −5.20864 0.587164 + 2.15760i 6.50959 3.48285i 8.61484i −2.87832 5.79292 − 1.57647i
149.2 − 2.58855i − 1.45668i −4.70058 0.873361 − 2.05846i −3.77067 0.648627i 6.99059i 0.878097 −5.32841 − 2.26074i
149.3 − 2.15179i 1.93821i −2.63020 −1.56968 − 1.59252i 4.17062 − 4.19034i 1.35605i −0.756662 −3.42676 + 3.37761i
149.4 − 2.05104i − 3.14116i −2.20678 −0.760195 + 2.10288i −6.44266 0.536422i 0.424122i −6.86691 4.31310 + 1.55919i
149.5 − 1.70607i − 0.742055i −0.910675 2.11017 + 0.739704i −1.26600 − 0.663740i − 1.85846i 2.44935 1.26199 − 3.60011i
149.6 − 1.35604i 1.13547i 0.161165 −0.722164 + 2.11624i 1.53974 − 0.748140i − 2.93062i 1.71070 2.86970 + 0.979281i
149.7 − 0.974151i 1.62921i 1.05103 0.497118 − 2.18011i 1.58709 4.15169i − 2.97216i 0.345685 −2.12376 − 0.484268i
149.8 − 0.703111i − 1.64070i 1.50564 −2.08504 − 0.807843i −1.15359 − 0.501390i − 2.46485i 0.308108 −0.568003 + 1.46601i
149.9 − 0.246891i 3.34515i 1.93904 2.06926 + 0.847451i 0.825888 − 2.66723i − 0.972515i −8.19006 0.209228 − 0.510881i
149.10 0.246891i − 3.34515i 1.93904 2.06926 − 0.847451i 0.825888 2.66723i 0.972515i −8.19006 0.209228 + 0.510881i
149.11 0.703111i 1.64070i 1.50564 −2.08504 + 0.807843i −1.15359 0.501390i 2.46485i 0.308108 −0.568003 − 1.46601i
149.12 0.974151i − 1.62921i 1.05103 0.497118 + 2.18011i 1.58709 − 4.15169i 2.97216i 0.345685 −2.12376 + 0.484268i
149.13 1.35604i − 1.13547i 0.161165 −0.722164 − 2.11624i 1.53974 0.748140i 2.93062i 1.71070 2.86970 − 0.979281i
149.14 1.70607i 0.742055i −0.910675 2.11017 − 0.739704i −1.26600 0.663740i 1.85846i 2.44935 1.26199 + 3.60011i
149.15 2.05104i 3.14116i −2.20678 −0.760195 − 2.10288i −6.44266 − 0.536422i − 0.424122i −6.86691 4.31310 − 1.55919i
149.16 2.15179i − 1.93821i −2.63020 −1.56968 + 1.59252i 4.17062 4.19034i − 1.35605i −0.756662 −3.42676 − 3.37761i
149.17 2.58855i 1.45668i −4.70058 0.873361 + 2.05846i −3.77067 − 0.648627i − 6.99059i 0.878097 −5.32841 + 2.26074i
149.18 2.68489i − 2.42453i −5.20864 0.587164 − 2.15760i 6.50959 − 3.48285i − 8.61484i −2.87832 5.79292 + 1.57647i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.b.a ✓ 18
3.b odd 2 1 1665.2.c.e 18
5.b even 2 1 inner 185.2.b.a ✓ 18
5.c odd 4 1 925.2.a.l 9
5.c odd 4 1 925.2.a.m 9
15.d odd 2 1 1665.2.c.e 18
15.e even 4 1 8325.2.a.cq 9
15.e even 4 1 8325.2.a.cr 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.b.a ✓ 18 1.a even 1 1 trivial
185.2.b.a ✓ 18 5.b even 2 1 inner
925.2.a.l 9 5.c odd 4 1
925.2.a.m 9 5.c odd 4 1
1665.2.c.e 18 3.b odd 2 1
1665.2.c.e 18 15.d odd 2 1
8325.2.a.cq 9 15.e even 4 1
8325.2.a.cr 9 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 29 T^{16} + \cdots + 144 \) Copy content Toggle raw display
$3$ \( T^{18} + 40 T^{16} + \cdots + 26244 \) Copy content Toggle raw display
$5$ \( T^{18} - 2 T^{17} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} + 56 T^{16} + \cdots + 196 \) Copy content Toggle raw display
$11$ \( (T^{9} - 42 T^{7} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + 116 T^{16} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{18} + 128 T^{16} + \cdots + 59721984 \) Copy content Toggle raw display
$19$ \( (T^{9} - 4 T^{8} + \cdots - 63768)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 108 T^{16} + \cdots + 15241216 \) Copy content Toggle raw display
$29$ \( (T^{9} - 2 T^{8} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + 6 T^{8} + \cdots + 2168344)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$41$ \( (T^{9} - 2 T^{8} + \cdots + 8368)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 3432630685696 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 138917480578276 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 169893914960896 \) Copy content Toggle raw display
$59$ \( (T^{9} + 10 T^{8} + \cdots - 689664)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 8 T^{8} + \cdots + 6156288)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 42279957308416 \) Copy content Toggle raw display
$71$ \( (T^{9} - 8 T^{8} + \cdots - 5516928)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 1074489803776 \) Copy content Toggle raw display
$79$ \( (T^{9} - 26 T^{8} + \cdots + 70088)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{9} - 8 T^{8} + \cdots + 27504768)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
show more
show less