↑

Properties

Label 950.6.a.v
Level $950$
Weight $6$
Character orbit 950.a
Self dual yes
Analytic conductor $152.365$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,6,Mod(1,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

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: yes
Analytic conductor: \(152.364628822\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4 x^{10} - 1757 x^{9} + 3226 x^{8} + 1058248 x^{7} - 656072 x^{6} - 254660592 x^{5} + \cdots + 2758188563136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + \beta_1 q^{3} + 16 q^{4} - 4 \beta_1 q^{6} + (\beta_{3} - \beta_1 + 22) q^{7} - 64 q^{8} + (\beta_{6} + \beta_{4} + \beta_{3} + \cdots + 77) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + \beta_1 q^{3} + 16 q^{4} - 4 \beta_1 q^{6} + (\beta_{3} - \beta_1 + 22) q^{7} - 64 q^{8} + (\beta_{6} + \beta_{4} + \beta_{3} + \cdots + 77) q^{9}+ \cdots + (147 \beta_{10} - 81 \beta_{9} + \cdots - 40634) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 44 q^{2} + 4 q^{3} + 176 q^{4} - 16 q^{6} + 240 q^{7} - 704 q^{8} + 857 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 44 q^{2} + 4 q^{3} + 176 q^{4} - 16 q^{6} + 240 q^{7} - 704 q^{8} + 857 q^{9} - 1098 q^{11} + 64 q^{12} + 554 q^{13} - 960 q^{14} + 2816 q^{16} + 506 q^{17} - 3428 q^{18} - 3971 q^{19} - 2590 q^{21} + 4392 q^{22} + 2520 q^{23} - 256 q^{24} - 2216 q^{26} + 9526 q^{27} + 3840 q^{28} + 588 q^{29} - 8356 q^{31} - 11264 q^{32} + 468 q^{33} - 2024 q^{34} + 13712 q^{36} + 21636 q^{37} + 15884 q^{38} - 17946 q^{39} - 10086 q^{41} + 10360 q^{42} + 47230 q^{43} - 17568 q^{44} - 10080 q^{46} + 26570 q^{47} + 1024 q^{48} + 32667 q^{49} - 40890 q^{51} + 8864 q^{52} + 18542 q^{53} - 38104 q^{54} - 15360 q^{56} - 1444 q^{57} - 2352 q^{58} + 786 q^{59} + 51504 q^{61} + 33424 q^{62} + 149170 q^{63} + 45056 q^{64} - 1872 q^{66} + 58456 q^{67} + 8096 q^{68} - 151694 q^{69} - 73924 q^{71} - 54848 q^{72} + 149950 q^{73} - 86544 q^{74} - 63536 q^{76} + 125388 q^{77} + 71784 q^{78} - 122220 q^{79} + 78363 q^{81} + 40344 q^{82} + 106590 q^{83} - 41440 q^{84} - 188920 q^{86} + 165182 q^{87} + 70272 q^{88} - 144726 q^{89} + 53650 q^{91} + 40320 q^{92} + 110432 q^{93} - 106280 q^{94} - 4096 q^{96} + 76508 q^{97} - 130668 q^{98} - 445522 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 4 x^{10} - 1757 x^{9} + 3226 x^{8} + 1058248 x^{7} - 656072 x^{6} - 254660592 x^{5} + \cdots + 2758188563136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 80\!\cdots\!23 \nu^{10} + \cdots + 77\!\cdots\!96 ) / 50\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 78\!\cdots\!15 \nu^{10} + \cdots + 29\!\cdots\!52 ) / 50\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31\!\cdots\!13 \nu^{10} + \cdots + 31\!\cdots\!28 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 58\!\cdots\!67 \nu^{10} + \cdots - 23\!\cdots\!92 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!67 \nu^{10} + \cdots - 44\!\cdots\!64 ) / 50\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 70\!\cdots\!15 \nu^{10} + \cdots - 22\!\cdots\!52 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!35 \nu^{10} + \cdots - 37\!\cdots\!52 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 51\!\cdots\!57 \nu^{10} + \cdots - 16\!\cdots\!40 ) / 50\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!83 \nu^{10} + \cdots - 98\!\cdots\!64 ) / 26\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + \beta_{3} + 2\beta _1 + 320 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16 \beta_{10} - 12 \beta_{9} - 4 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{4} + 10 \beta_{3} + \cdots + 838 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{10} + 76 \beta_{9} - 71 \beta_{8} - 44 \beta_{7} + 774 \beta_{6} - 84 \beta_{5} + 785 \beta_{4} + \cdots + 180564 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13456 \beta_{10} - 9835 \beta_{9} - 1926 \beta_{8} - 1464 \beta_{7} + 662 \beta_{6} - 1098 \beta_{5} + \cdots + 1301464 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19937 \beta_{10} + 63930 \beta_{9} - 76871 \beta_{8} - 49862 \beta_{7} + 566918 \beta_{6} + \cdots + 120242488 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9895860 \beta_{10} - 7122043 \beta_{9} - 666020 \beta_{8} - 1439954 \beta_{7} + 1917336 \beta_{6} + \cdots + 1337189040 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 26826565 \beta_{10} + 43558898 \beta_{9} - 60598215 \beta_{8} - 46337076 \beta_{7} + 416997288 \beta_{6} + \cdots + 85009753410 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7091293710 \beta_{10} - 4988825667 \beta_{9} - 99639810 \beta_{8} - 1303258038 \beta_{7} + \cdots + 1203267317112 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 28973546793 \beta_{10} + 27371880396 \beta_{9} - 41712204915 \beta_{8} - 40940873136 \beta_{7} + \cdots + 61935850399682 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−25.6125
−20.1054
−20.0050
−10.0929
−9.69248
3.59939
8.70979
8.79040
12.8957
27.2428
28.2702
−4.00000 −25.6125 16.0000 0 102.450 −21.8301 −64.0000 413.001 0
1.2 −4.00000 −20.1054 16.0000 0 80.4215 189.825 −64.0000 161.227 0
1.3 −4.00000 −20.0050 16.0000 0 80.0200 200.402 −64.0000 157.200 0
1.4 −4.00000 −10.0929 16.0000 0 40.3716 −181.357 −64.0000 −141.133 0
1.5 −4.00000 −9.69248 16.0000 0 38.7699 −36.6998 −64.0000 −149.056 0
1.6 −4.00000 3.59939 16.0000 0 −14.3975 110.999 −64.0000 −230.044 0
1.7 −4.00000 8.70979 16.0000 0 −34.8392 −80.6272 −64.0000 −167.140 0
1.8 −4.00000 8.79040 16.0000 0 −35.1616 22.9562 −64.0000 −165.729 0
1.9 −4.00000 12.8957 16.0000 0 −51.5827 −118.602 −64.0000 −76.7013 0
1.10 −4.00000 27.2428 16.0000 0 −108.971 252.422 −64.0000 499.170 0
1.11 −4.00000 28.2702 16.0000 0 −113.081 −97.4878 −64.0000 556.206 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.6.a.v 11
5.b even 2 1 950.6.a.w 11
5.c odd 4 2 190.6.b.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.6.b.a ✓ 22 5.c odd 4 2
950.6.a.v 11 1.a even 1 1 trivial
950.6.a.w 11 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} - 4 T_{3}^{10} - 1757 T_{3}^{9} + 3226 T_{3}^{8} + 1058248 T_{3}^{7} - 656072 T_{3}^{6} + \cdots + 2758188563136 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{11} \) Copy content Toggle raw display
$3$ \( T^{11} + \cdots + 2758188563136 \) Copy content Toggle raw display
$5$ \( T^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 33\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots - 87\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 37\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( (T + 361)^{11} \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 47\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 65\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 52\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots + 45\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots - 51\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 33\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 73\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 13\!\cdots\!08 \) Copy content Toggle raw display
show more
show less