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Properties

Label 190.4.b.b
Level $190$
Weight $4$
Character orbit 190.b
Analytic conductor $11.210$
Analytic rank $0$
Dimension $14$
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] = [190,4,Mod(39,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("190.39");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 190.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: \(11.2103629011\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 301 x^{12} + 34918 x^{10} + 2019538 x^{8} + 62069257 x^{6} + 985407997 x^{4} + \cdots + 13853760804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + \beta_{3} q^{3} - 4 q^{4} + ( - \beta_{8} + \beta_{5} - 1) q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{9} - 2 \beta_{8}) q^{7} + 4 \beta_{8} q^{8} + (\beta_{12} - \beta_{10} - \beta_{8} + \cdots - 17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + \beta_{3} q^{3} - 4 q^{4} + ( - \beta_{8} + \beta_{5} - 1) q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{9} - 2 \beta_{8}) q^{7} + 4 \beta_{8} q^{8} + (\beta_{12} - \beta_{10} - \beta_{8} + \cdots - 17) q^{9}+ \cdots + (7 \beta_{13} + 26 \beta_{12} + \cdots - 424) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 56 q^{4} - 8 q^{5} + 16 q^{6} - 226 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 56 q^{4} - 8 q^{5} + 16 q^{6} - 226 q^{9} - 40 q^{10} + 204 q^{11} - 96 q^{14} - 16 q^{15} + 224 q^{16} + 266 q^{19} + 32 q^{20} - 188 q^{21} - 64 q^{24} + 444 q^{25} + 392 q^{26} - 120 q^{29} + 544 q^{30} + 552 q^{31} - 1000 q^{34} + 98 q^{35} + 904 q^{36} + 36 q^{39} + 160 q^{40} - 252 q^{41} - 816 q^{44} + 1768 q^{45} + 640 q^{46} - 2638 q^{49} - 72 q^{50} + 372 q^{51} - 520 q^{54} + 310 q^{55} + 384 q^{56} - 3324 q^{59} + 64 q^{60} - 2072 q^{61} - 896 q^{64} + 1456 q^{65} - 48 q^{66} - 596 q^{69} + 288 q^{70} + 2008 q^{71} - 528 q^{74} + 1672 q^{75} - 1064 q^{76} - 1256 q^{79} - 128 q^{80} + 1974 q^{81} + 752 q^{84} + 3026 q^{85} + 3208 q^{86} - 5412 q^{89} - 2312 q^{90} + 1116 q^{91} + 520 q^{94} - 152 q^{95} + 256 q^{96} - 5356 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 301 x^{12} + 34918 x^{10} + 2019538 x^{8} + 62069257 x^{6} + 985407997 x^{4} + \cdots + 13853760804 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 429375 \nu^{12} - 116287564 \nu^{10} - 11525287502 \nu^{8} - 529631475432 \nu^{6} + \cdots - 217079588959260 ) / 2388023974896 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2783177891 \nu^{12} + 729710546036 \nu^{10} + 69554767935422 \nu^{8} + \cdots + 13\!\cdots\!92 ) / 10\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 912013241 \nu^{13} + 249246837416 \nu^{11} + 25002038920274 \nu^{9} + \cdots + 58\!\cdots\!76 \nu ) / 28\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 59286546502447 \nu^{13} + \cdots - 36\!\cdots\!64 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59286546502447 \nu^{13} - 178864427893749 \nu^{12} + \cdots - 41\!\cdots\!36 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 59286546502447 \nu^{13} - 546095302111359 \nu^{12} + \cdots + 10\!\cdots\!64 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30590498233549 \nu^{13} + \cdots - 83\!\cdots\!08 ) / 71\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 912013241 \nu^{13} + 249246837416 \nu^{11} + 25002038920274 \nu^{9} + \cdots + 30\!\cdots\!84 \nu ) / 14\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 92794791607 \nu^{13} + 26000080706620 \nu^{11} + \cdots + 22\!\cdots\!56 \nu ) / 52\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 126153046544389 \nu^{13} + 59621475964583 \nu^{12} + \cdots + 13\!\cdots\!12 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 247563856191551 \nu^{13} + \cdots - 10\!\cdots\!28 ) / 71\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 98585711138172 \nu^{13} + 348296741994149 \nu^{12} + \cdots + 27\!\cdots\!36 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 337440900927547 \nu^{13} - 628708596883926 \nu^{12} + \cdots - 54\!\cdots\!24 ) / 71\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{2} - 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 13 \beta_{10} + 12 \beta_{9} + 115 \beta_{8} + 6 \beta_{7} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{13} - 110 \beta_{12} + 11 \beta_{11} + 110 \beta_{10} + 110 \beta_{8} + 99 \beta_{7} + \cdots + 3089 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 810 \beta_{13} + 564 \beta_{12} + 270 \beta_{11} + 1704 \beta_{10} - 1716 \beta_{9} - 15937 \beta_{8} + \cdots + 810 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 966 \beta_{13} + 11029 \beta_{12} - 966 \beta_{11} - 11029 \beta_{10} - 11029 \beta_{8} + \cdots - 271034 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 116272 \beta_{13} - 81146 \beta_{12} - 32126 \beta_{11} - 193801 \beta_{10} + 206460 \beta_{9} + \cdots - 106325 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 65807 \beta_{13} - 1126958 \beta_{12} + 65807 \beta_{11} + 1126958 \beta_{10} + 1126958 \beta_{8} + \cdots + 26721491 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14627250 \beta_{13} + 9946692 \beta_{12} + 3719022 \beta_{11} + 21588636 \beta_{10} - 23563692 \beta_{9} + \cdots + 12892158 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4068006 \beta_{13} + 118539637 \beta_{12} - 4068006 \beta_{11} - 118539637 \beta_{10} + \cdots - 2796235208 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1732003360 \beta_{13} - 1153642250 \beta_{12} - 425733398 \beta_{11} - 2398951165 \beta_{10} + \cdots - 1504601777 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 231497879 \beta_{13} - 12746106854 \beta_{12} + 231497879 \beta_{11} + 12746106854 \beta_{10} + \cdots + 301537882349 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 199066307922 \beta_{13} + 130881905508 \beta_{12} + 48335070486 \beta_{11} + 266647653552 \beta_{10} + \cdots + 172035759690 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
− 10.5490i
− 6.18013i
− 4.03438i
− 1.74481i
4.69006i
6.95165i
7.86657i
− 7.86657i
− 6.95165i
− 4.69006i
1.74481i
4.03438i
6.18013i
10.5490i
− 2.00000i − 9.54896i −4.00000 −11.0664 + 1.59220i −19.0979 − 34.2294i 8.00000i −64.1826 3.18439 + 22.1328i
39.2 − 2.00000i − 5.18013i −4.00000 −10.3419 − 4.24790i −10.3603 13.1250i 8.00000i 0.166258 −8.49581 + 20.6838i
39.3 − 2.00000i − 3.03438i −4.00000 10.4701 + 3.92123i −6.06875 − 10.1105i 8.00000i 17.7926 7.84247 − 20.9403i
39.4 − 2.00000i − 0.744814i −4.00000 7.17931 − 8.57074i −1.48963 31.9425i 8.00000i 26.4453 −17.1415 − 14.3586i
39.5 − 2.00000i 5.69006i −4.00000 −7.81282 − 7.99749i 11.3801 − 5.63584i 8.00000i −5.37675 −15.9950 + 15.6256i
39.6 − 2.00000i 7.95165i −4.00000 9.62570 − 5.68734i 15.9033 − 32.3442i 8.00000i −36.2288 −11.3747 − 19.2514i
39.7 − 2.00000i 8.86657i −4.00000 −2.05403 + 10.9900i 17.7331 13.2525i 8.00000i −51.6160 21.9801 + 4.10806i
39.8 2.00000i − 8.86657i −4.00000 −2.05403 − 10.9900i 17.7331 − 13.2525i − 8.00000i −51.6160 21.9801 − 4.10806i
39.9 2.00000i − 7.95165i −4.00000 9.62570 + 5.68734i 15.9033 32.3442i − 8.00000i −36.2288 −11.3747 + 19.2514i
39.10 2.00000i − 5.69006i −4.00000 −7.81282 + 7.99749i 11.3801 5.63584i − 8.00000i −5.37675 −15.9950 − 15.6256i
39.11 2.00000i 0.744814i −4.00000 7.17931 + 8.57074i −1.48963 − 31.9425i − 8.00000i 26.4453 −17.1415 + 14.3586i
39.12 2.00000i 3.03438i −4.00000 10.4701 − 3.92123i −6.06875 10.1105i − 8.00000i 17.7926 7.84247 + 20.9403i
39.13 2.00000i 5.18013i −4.00000 −10.3419 + 4.24790i −10.3603 − 13.1250i − 8.00000i 0.166258 −8.49581 − 20.6838i
39.14 2.00000i 9.54896i −4.00000 −11.0664 − 1.59220i −19.0979 34.2294i − 8.00000i −64.1826 3.18439 − 22.1328i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.14
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 190.4.b.b ✓ 14
5.b even 2 1 inner 190.4.b.b ✓ 14
5.c odd 4 1 950.4.a.w 7
5.c odd 4 1 950.4.a.x 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.4.b.b ✓ 14 1.a even 1 1 trivial
190.4.b.b ✓ 14 5.b even 2 1 inner
950.4.a.w 7 5.c odd 4 1
950.4.a.x 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 302 T_{3}^{12} + 35429 T_{3}^{10} + 2035296 T_{3}^{8} + 59309124 T_{3}^{6} + \cdots + 2011343104 \) acting on \(S_{4}^{\mathrm{new}}(190, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 2011343104 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 476837158203125 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{7} - 102 T^{6} + \cdots + 1713841120)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 3751070285824 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T - 19)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 4350225899520)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 43\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 79\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 88\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 60\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
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