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Properties

Label 7581.2.a.bs
Level $7581$
Weight $2$
Character orbit 7581.a
Self dual yes
Analytic conductor $60.535$
Analytic rank $0$
Dimension $18$
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] = [7581,2,Mod(1,7581)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7581, 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("7581.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7581 = 3 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7581.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: \(60.5345897723\)
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: \( x^{18} - 30 x^{16} - x^{15} + 366 x^{14} + 21 x^{13} - 2326 x^{12} - 150 x^{11} + 8172 x^{10} + 379 x^{9} + \cdots - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 399)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{12} q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{12} q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9} + (\beta_{16} - \beta_{12} + \beta_{8} + \cdots + 1) q^{10}+ \cdots + ( - \beta_{15} + \beta_{11} - \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{3} + 24 q^{4} - 18 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{3} + 24 q^{4} - 18 q^{7} + 3 q^{8} + 18 q^{9} + 18 q^{10} + 18 q^{11} + 24 q^{12} + 15 q^{13} + 48 q^{16} - 6 q^{17} + 30 q^{20} - 18 q^{21} + 9 q^{22} + 30 q^{23} + 3 q^{24} + 24 q^{25} - 27 q^{26} + 18 q^{27} - 24 q^{28} + 21 q^{29} + 18 q^{30} + 18 q^{31} + 21 q^{32} + 18 q^{33} - 9 q^{34} + 24 q^{36} + 24 q^{37} + 15 q^{39} - 24 q^{40} - 21 q^{41} + 27 q^{43} + 72 q^{44} + 6 q^{46} - 15 q^{47} + 48 q^{48} + 18 q^{49} - 12 q^{50} - 6 q^{51} + 45 q^{52} - 3 q^{53} + 33 q^{55} - 3 q^{56} + 51 q^{58} + 3 q^{59} + 30 q^{60} - 3 q^{61} - 15 q^{62} - 18 q^{63} + 15 q^{64} - 48 q^{65} + 9 q^{66} + 36 q^{67} - 12 q^{68} + 30 q^{69} - 18 q^{70} + 3 q^{72} + 21 q^{73} + 15 q^{74} + 24 q^{75} - 18 q^{77} - 27 q^{78} - 18 q^{79} + 42 q^{80} + 18 q^{81} - 75 q^{82} + 30 q^{83} - 24 q^{84} + 9 q^{85} + 54 q^{86} + 21 q^{87} + 24 q^{88} + 3 q^{89} + 18 q^{90} - 15 q^{91} + 114 q^{92} + 18 q^{93} - 9 q^{94} + 21 q^{96} + 75 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 30 x^{16} - x^{15} + 366 x^{14} + 21 x^{13} - 2326 x^{12} - 150 x^{11} + 8172 x^{10} + 379 x^{9} + \cdots - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 149356 \nu^{17} - 79739 \nu^{16} - 4531298 \nu^{15} + 2247607 \nu^{14} + 56137267 \nu^{13} + \cdots - 2226045 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 213320 \nu^{17} - 61384 \nu^{16} - 6488167 \nu^{15} + 1641749 \nu^{14} + 80337608 \nu^{13} + \cdots - 3881283 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 282142 \nu^{17} + 24799 \nu^{16} + 8565491 \nu^{15} - 821724 \nu^{14} - 105841407 \nu^{13} + \cdots + 20283108 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 551746 \nu^{17} - 63964 \nu^{16} - 16570735 \nu^{15} + 1405123 \nu^{14} + 202544894 \nu^{13} + \cdots - 283383 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 674308 \nu^{17} - 14497 \nu^{16} + 20423750 \nu^{15} + 816099 \nu^{14} - 251664911 \nu^{13} + \cdots - 9135423 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 675256 \nu^{17} - 91648 \nu^{16} - 19991045 \nu^{15} + 2049147 \nu^{14} + 239734734 \nu^{13} + \cdots + 3705687 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 742015 \nu^{17} + 149356 \nu^{16} + 22180711 \nu^{15} - 3789283 \nu^{14} - 269329883 \nu^{13} + \cdots + 14669928 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 808669 \nu^{17} - 212754 \nu^{16} - 24238022 \nu^{15} + 5526104 \nu^{14} + 295609115 \nu^{13} + \cdots - 27196221 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 826232 \nu^{17} - 241116 \nu^{16} - 24569169 \nu^{15} + 6080593 \nu^{14} + 296783758 \nu^{13} + \cdots - 7645491 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1646745 \nu^{17} + 179579 \nu^{16} + 49402736 \nu^{15} - 3894727 \nu^{14} - 602342196 \nu^{13} + \cdots + 3080088 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1720905 \nu^{17} + 308776 \nu^{16} + 51628002 \nu^{15} - 7758734 \nu^{14} - 630236361 \nu^{13} + \cdots + 954063 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1778433 \nu^{17} - 236646 \nu^{16} - 53470532 \nu^{15} + 5584530 \nu^{14} + 653988903 \nu^{13} + \cdots - 5776065 ) / 4410492 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1362450 \nu^{17} - 306456 \nu^{16} - 40783634 \nu^{15} + 7678051 \nu^{14} + 496437278 \nu^{13} + \cdots - 20711634 ) / 2205246 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3027921 \nu^{17} - 993934 \nu^{16} - 90583350 \nu^{15} + 26340566 \nu^{14} + 1102094487 \nu^{13} + \cdots - 51838779 ) / 4410492 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + 6\beta_{2} - \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} + \beta_{16} - 23 \beta_{15} - 11 \beta_{14} - 11 \beta_{13} + 11 \beta_{11} + 9 \beta_{10} + \cdots + 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{15} + 13 \beta_{14} + 2 \beta_{13} - 10 \beta_{12} + 10 \beta_{11} + 18 \beta_{10} + 15 \beta_{9} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15 \beta_{17} + 15 \beta_{16} - 204 \beta_{15} - 97 \beta_{14} - 95 \beta_{13} + 2 \beta_{12} + \cdots + 724 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{17} - 5 \beta_{16} + 163 \beta_{15} + 127 \beta_{14} + 36 \beta_{13} - 71 \beta_{12} + 75 \beta_{11} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 158 \beta_{17} + 158 \beta_{16} - 1657 \beta_{15} - 792 \beta_{14} - 760 \beta_{13} + 42 \beta_{12} + \cdots + 5007 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 23 \beta_{17} - 107 \beta_{16} + 1566 \beta_{15} + 1128 \beta_{14} + 436 \beta_{13} - 425 \beta_{12} + \cdots - 172 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1441 \beta_{17} + 1443 \beta_{16} - 12959 \beta_{15} - 6243 \beta_{14} - 5899 \beta_{13} + \cdots + 35328 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 334 \beta_{17} - 1466 \beta_{16} + 14139 \beta_{15} + 9603 \beta_{14} + 4482 \beta_{13} - 2178 \beta_{12} + \cdots - 4704 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 12207 \beta_{17} + 12255 \beta_{16} - 99602 \beta_{15} - 48372 \beta_{14} - 45190 \beta_{13} + \cdots + 252828 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3939 \beta_{17} - 16429 \beta_{16} + 123076 \beta_{15} + 79910 \beta_{14} + 42286 \beta_{13} + \cdots - 63037 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 99185 \beta_{17} + 99927 \beta_{16} - 759718 \beta_{15} - 371782 \beta_{14} - 344184 \beta_{13} + \cdots + 1829270 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 41323 \beta_{17} - 164401 \beta_{16} + 1046076 \beta_{15} + 655630 \beta_{14} + 379206 \beta_{13} + \cdots - 689509 \) 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
−2.79826
−2.41092
−2.34858
−2.01029
−1.79990
−1.73451
−0.487733
−0.243863
−0.203351
0.0646193
0.320918
0.908052
0.982896
1.70831
2.20697
2.54264
2.63538
2.66762
−2.79826 1.00000 5.83027 2.25314 −2.79826 −1.00000 −10.7181 1.00000 −6.30488
1.2 −2.41092 1.00000 3.81254 −2.28250 −2.41092 −1.00000 −4.36988 1.00000 5.50292
1.3 −2.34858 1.00000 3.51581 2.54884 −2.34858 −1.00000 −3.55998 1.00000 −5.98613
1.4 −2.01029 1.00000 2.04125 −4.24775 −2.01029 −1.00000 −0.0829219 1.00000 8.53919
1.5 −1.79990 1.00000 1.23965 2.50427 −1.79990 −1.00000 1.36855 1.00000 −4.50744
1.6 −1.73451 1.00000 1.00851 −2.60928 −1.73451 −1.00000 1.71974 1.00000 4.52581
1.7 −0.487733 1.00000 −1.76212 −3.52472 −0.487733 −1.00000 1.83491 1.00000 1.71912
1.8 −0.243863 1.00000 −1.94053 −1.39605 −0.243863 −1.00000 0.960951 1.00000 0.340445
1.9 −0.203351 1.00000 −1.95865 0.904849 −0.203351 −1.00000 0.804994 1.00000 −0.184002
1.10 0.0646193 1.00000 −1.99582 −0.998703 0.0646193 −1.00000 −0.258207 1.00000 −0.0645355
1.11 0.320918 1.00000 −1.89701 −2.14822 0.320918 −1.00000 −1.25062 1.00000 −0.689405
1.12 0.908052 1.00000 −1.17544 0.893947 0.908052 −1.00000 −2.88347 1.00000 0.811750
1.13 0.982896 1.00000 −1.03392 2.72885 0.982896 −1.00000 −2.98202 1.00000 2.68218
1.14 1.70831 1.00000 0.918308 2.98533 1.70831 −1.00000 −1.84786 1.00000 5.09986
1.15 2.20697 1.00000 2.87071 −1.62484 2.20697 −1.00000 1.92162 1.00000 −3.58597
1.16 2.54264 1.00000 4.46501 4.45863 2.54264 −1.00000 6.26764 1.00000 11.3367
1.17 2.63538 1.00000 4.94522 1.43744 2.63538 −1.00000 7.76176 1.00000 3.78821
1.18 2.66762 1.00000 5.11622 −1.88325 2.66762 −1.00000 8.31290 1.00000 −5.02379
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(19\) \(1\)

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 7581.2.a.bs 18
19.b odd 2 1 7581.2.a.bq 18
19.e even 9 2 399.2.bo.e ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.bo.e ✓ 36 19.e even 9 2
7581.2.a.bq 18 19.b odd 2 1
7581.2.a.bs 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):

\( T_{2}^{18} - 30 T_{2}^{16} - T_{2}^{15} + 366 T_{2}^{14} + 21 T_{2}^{13} - 2326 T_{2}^{12} - 150 T_{2}^{11} + \cdots - 3 \) Copy content Toggle raw display
\( T_{5}^{18} - 57 T_{5}^{16} - 3 T_{5}^{15} + 1314 T_{5}^{14} + 126 T_{5}^{13} - 16172 T_{5}^{12} + \cdots - 496376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 30 T^{16} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( (T - 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 57 T^{16} + \cdots - 496376 \) Copy content Toggle raw display
$7$ \( (T + 1)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 164736339 \) Copy content Toggle raw display
$13$ \( T^{18} - 15 T^{17} + \cdots + 44908352 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 349854009 \) Copy content Toggle raw display
$19$ \( T^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 32082645129 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 1288009664 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 2334717160569 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 132508466667 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 1409820686193 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 21021596352 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 734624721408 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 16107102912 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 4950969333312 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 105917569984 \) Copy content Toggle raw display
$67$ \( T^{18} - 36 T^{17} + \cdots + 35172864 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 17434175949447 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 26670421813824 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 1926558134272 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 1371311520441 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 150775396772544 \) Copy content Toggle raw display
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