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Properties

Label 9196.2.a.x
Level $9196$
Weight $2$
Character orbit 9196.a
Self dual yes
Analytic conductor $73.430$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9196,2,Mod(1,9196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9196, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9196.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9196.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: \(73.4304296988\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} - 26 x^{18} + 208 x^{17} + 185 x^{16} - 2910 x^{15} + 687 x^{14} + 21067 x^{13} + \cdots - 3520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 836)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{14} q^{5} - \beta_{10} q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{14} q^{5} - \beta_{10} q^{7} + (\beta_{2} + 1) q^{9} + \beta_{15} q^{13} + ( - \beta_{19} - \beta_{18} + \beta_{17} + \cdots + 1) q^{15}+ \cdots + (\beta_{19} + 2 \beta_{18} - \beta_{17} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41} - 8 q^{43} - 15 q^{45} + 34 q^{47} + 42 q^{49} + 14 q^{51} - 10 q^{53} - 6 q^{57} + 52 q^{59} - 2 q^{61} + 59 q^{63} + 25 q^{65} + 14 q^{67} + 15 q^{69} + 88 q^{71} - 13 q^{73} + 15 q^{75} - 2 q^{79} + 48 q^{81} - 2 q^{83} + 11 q^{85} + 37 q^{87} + 46 q^{89} - 4 q^{91} + 2 q^{93} - 2 q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} - 26 x^{18} + 208 x^{17} + 185 x^{16} - 2910 x^{15} + 687 x^{14} + 21067 x^{13} + \cdots - 3520 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!04 \nu^{19} + \cdots - 23\!\cdots\!60 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!87 \nu^{19} + \cdots + 13\!\cdots\!60 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!49 \nu^{19} + \cdots - 46\!\cdots\!20 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14357728478791 \nu^{19} - 58897586862604 \nu^{18} - 516407111138704 \nu^{17} + \cdots + 12\!\cdots\!00 ) / 401907441152880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13624392005071 \nu^{19} + 71553085345069 \nu^{18} + 402657550953634 \nu^{17} + \cdots - 25\!\cdots\!60 ) / 200953720576440 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 44\!\cdots\!53 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 60\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1234812684815 \nu^{19} - 4624787660228 \nu^{18} - 46875255232586 \nu^{17} + \cdots + 12\!\cdots\!52 ) / 13396914705096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!37 \nu^{19} + \cdots + 42\!\cdots\!40 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 44\!\cdots\!21 \nu^{19} + \cdots - 28\!\cdots\!40 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!29 \nu^{19} + \cdots + 22\!\cdots\!40 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 73\!\cdots\!28 \nu^{19} + \cdots + 44\!\cdots\!20 ) / 50\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!57 \nu^{19} + \cdots + 12\!\cdots\!60 ) / 90\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 69\!\cdots\!27 \nu^{19} + \cdots + 53\!\cdots\!40 ) / 40\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 16\!\cdots\!42 \nu^{19} + \cdots - 80\!\cdots\!20 ) / 60\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 48\!\cdots\!29 \nu^{19} + \cdots - 55\!\cdots\!60 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 10\!\cdots\!95 \nu^{19} + \cdots - 27\!\cdots\!80 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 55\!\cdots\!41 \nu^{19} + \cdots - 11\!\cdots\!68 ) / 18\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} + \beta_{14} + \beta_{10} - \beta_{6} + \beta_{5} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} - \beta_{17} + \beta_{16} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} - 13 \beta_{17} - \beta_{16} - 3 \beta_{15} + 11 \beta_{14} - 6 \beta_{13} - 3 \beta_{11} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 33 \beta_{19} - \beta_{18} - 18 \beta_{17} + 18 \beta_{16} - 6 \beta_{15} + 18 \beta_{14} + \cdots + 244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8 \beta_{19} + 8 \beta_{18} - 143 \beta_{17} - 18 \beta_{16} - 47 \beta_{15} + 112 \beta_{14} + \cdots + 258 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 429 \beta_{19} - 30 \beta_{18} - 242 \beta_{17} + 227 \beta_{16} - 113 \beta_{15} + 247 \beta_{14} + \cdots + 2233 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 219 \beta_{19} - 17 \beta_{18} - 1517 \beta_{17} - 225 \beta_{16} - 572 \beta_{15} + 1170 \beta_{14} + \cdots + 3330 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5144 \beta_{19} - 613 \beta_{18} - 2937 \beta_{17} + 2518 \beta_{16} - 1551 \beta_{15} + 3085 \beta_{14} + \cdots + 21733 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4081 \beta_{19} - 1613 \beta_{18} - 16008 \beta_{17} - 2411 \beta_{16} - 6503 \beta_{15} + \cdots + 40860 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 59624 \beta_{19} - 10383 \beta_{18} - 34163 \beta_{17} + 26401 \beta_{16} - 19141 \beta_{15} + \cdots + 221394 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 64470 \beta_{19} - 34010 \beta_{18} - 169712 \beta_{17} - 23612 \beta_{16} - 72744 \beta_{15} + \cdots + 488316 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 681406 \beta_{19} - 157374 \beta_{18} - 390480 \beta_{17} + 269741 \beta_{16} - 226494 \beta_{15} + \cdots + 2331867 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 929673 \beta_{19} - 549643 \beta_{18} - 1813986 \beta_{17} - 216078 \beta_{16} - 815087 \beta_{15} + \cdots + 5752559 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 7750702 \beta_{19} - 2220876 \beta_{18} - 4436064 \beta_{17} + 2726182 \beta_{16} - 2635856 \beta_{15} + \cdots + 25164468 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 12664139 \beta_{19} - 7906988 \beta_{18} - 19569278 \beta_{17} - 1849672 \beta_{16} - 9199609 \beta_{15} + \cdots + 67232467 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 88147628 \beta_{19} - 29860551 \beta_{18} - 50359549 \beta_{17} + 27477832 \beta_{16} + \cdots + 276426763 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 166040567 \beta_{19} - 106653455 \beta_{18} - 213088207 \beta_{17} - 14517125 \beta_{16} + \cdots + 782409475 \) 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
−3.09432
−2.76219
−2.45218
−2.33451
−1.88355
−1.24316
−0.600910
−0.593598
−0.493868
0.301667
0.907309
1.01428
1.02691
1.29453
2.21297
2.49726
2.74725
2.92279
3.12040
3.41291
0 −3.09432 0 −4.18501 0 3.94873 0 6.57481 0
1.2 0 −2.76219 0 1.14665 0 3.79035 0 4.62969 0
1.3 0 −2.45218 0 0.520700 0 0.586094 0 3.01318 0
1.4 0 −2.33451 0 −3.18058 0 −2.36116 0 2.44994 0
1.5 0 −1.88355 0 3.33948 0 −0.742056 0 0.547760 0
1.6 0 −1.24316 0 1.72864 0 −3.40076 0 −1.45456 0
1.7 0 −0.600910 0 3.64728 0 −3.75931 0 −2.63891 0
1.8 0 −0.593598 0 −3.09847 0 1.91364 0 −2.64764 0
1.9 0 −0.493868 0 −0.708542 0 −2.31217 0 −2.75609 0
1.10 0 0.301667 0 2.84384 0 4.03114 0 −2.90900 0
1.11 0 0.907309 0 −1.52006 0 2.67549 0 −2.17679 0
1.12 0 1.01428 0 −2.65495 0 −2.23891 0 −1.97123 0
1.13 0 1.02691 0 1.23436 0 −4.96370 0 −1.94545 0
1.14 0 1.29453 0 −0.783903 0 2.58491 0 −1.32419 0
1.15 0 2.21297 0 3.85083 0 1.71998 0 1.89723 0
1.16 0 2.49726 0 −4.07266 0 0.853391 0 3.23632 0
1.17 0 2.74725 0 3.77127 0 2.22761 0 4.54740 0
1.18 0 2.92279 0 0.920955 0 −2.92346 0 5.54270 0
1.19 0 3.12040 0 −2.72877 0 5.12972 0 6.73690 0
1.20 0 3.41291 0 1.92896 0 −2.75953 0 8.64792 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 9196.2.a.x 20
11.b odd 2 1 9196.2.a.w 20
11.c even 5 2 836.2.j.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.j.c ✓ 40 11.c even 5 2
9196.2.a.w 20 11.b odd 2 1
9196.2.a.x 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\):

\( T_{3}^{20} - 6 T_{3}^{19} - 26 T_{3}^{18} + 208 T_{3}^{17} + 185 T_{3}^{16} - 2910 T_{3}^{15} + \cdots - 3520 \) Copy content Toggle raw display
\( T_{5}^{20} - 2 T_{5}^{19} - 70 T_{5}^{18} + 146 T_{5}^{17} + 2016 T_{5}^{16} - 4389 T_{5}^{15} + \cdots - 1169671 \) Copy content Toggle raw display
\( T_{7}^{20} - 4 T_{7}^{19} - 83 T_{7}^{18} + 327 T_{7}^{17} + 2852 T_{7}^{16} - 11092 T_{7}^{15} + \cdots - 36449281 \) Copy content Toggle raw display
\( T_{13}^{20} + 5 T_{13}^{19} - 146 T_{13}^{18} - 912 T_{13}^{17} + 7726 T_{13}^{16} + 64163 T_{13}^{15} + \cdots - 1788189120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 6 T^{19} + \cdots - 3520 \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + \cdots - 1169671 \) Copy content Toggle raw display
$7$ \( T^{20} - 4 T^{19} + \cdots - 36449281 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 1788189120 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 131553424 \) Copy content Toggle raw display
$19$ \( (T + 1)^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 7030043681 \) Copy content Toggle raw display
$29$ \( T^{20} - T^{19} + \cdots - 70570816 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 5350643858624 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 850382080 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 1990567494080 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 13245095572795 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 4986323380841 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 34\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 148831597769536 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 12\!\cdots\!19 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 92\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 91862584153856 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 25\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 4403854411979 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
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