↑

Properties

Label 1815.4.a.bn
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $12$
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] = [1815,4,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(107.088466660\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 59 x^{10} + 269 x^{9} + 1318 x^{8} - 5253 x^{7} - 13369 x^{6} + 44853 x^{5} + \cdots + 17600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 165)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{10} + \beta_1 + 6) q^{7} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 11) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{10} + \beta_1 + 6) q^{7} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 11) q^{8} + 9 q^{9} + ( - 5 \beta_1 + 5) q^{10} + (3 \beta_{2} - 3 \beta_1 + 15) q^{12} + ( - \beta_{11} - 2 \beta_{10} + \cdots + 14) q^{13}+ \cdots + ( - 22 \beta_{11} - 43 \beta_{10} + \cdots + 54) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} + 21 q^{6} + 77 q^{7} + 111 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} + 21 q^{6} + 77 q^{7} + 111 q^{8} + 108 q^{9} + 35 q^{10} + 147 q^{12} + 172 q^{13} - 30 q^{14} + 180 q^{15} + 161 q^{16} + 317 q^{17} + 63 q^{18} + 237 q^{19} + 245 q^{20} + 231 q^{21} + 210 q^{23} + 333 q^{24} + 300 q^{25} + 8 q^{26} + 324 q^{27} + 542 q^{28} + 759 q^{29} + 105 q^{30} - 193 q^{31} + 410 q^{32} - 78 q^{34} + 385 q^{35} + 441 q^{36} + 286 q^{37} - 168 q^{38} + 516 q^{39} + 555 q^{40} + 1189 q^{41} - 90 q^{42} + 775 q^{43} + 540 q^{45} + 529 q^{46} + 382 q^{47} + 483 q^{48} + 1195 q^{49} + 175 q^{50} + 951 q^{51} + 1741 q^{52} + 275 q^{53} + 189 q^{54} - 419 q^{56} + 711 q^{57} - 418 q^{58} + 646 q^{59} + 735 q^{60} + 1340 q^{61} + 983 q^{62} + 693 q^{63} - 1489 q^{64} + 860 q^{65} - 185 q^{67} + 3322 q^{68} + 630 q^{69} - 150 q^{70} - 932 q^{71} + 999 q^{72} + 2860 q^{73} + 4187 q^{74} + 900 q^{75} + 1594 q^{76} + 24 q^{78} + 1429 q^{79} + 805 q^{80} + 972 q^{81} - 30 q^{82} + 2590 q^{83} + 1626 q^{84} + 1585 q^{85} - 3195 q^{86} + 2277 q^{87} - 473 q^{89} + 315 q^{90} - 4302 q^{91} + 5462 q^{92} - 579 q^{93} + 2875 q^{94} + 1185 q^{95} + 1230 q^{96} + 318 q^{97} - 194 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 59 x^{10} + 269 x^{9} + 1318 x^{8} - 5253 x^{7} - 13369 x^{6} + 44853 x^{5} + \cdots + 17600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 17\nu + 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 96262 \nu^{11} - 678861 \nu^{10} - 3890483 \nu^{9} + 30748837 \nu^{8} + 48361337 \nu^{7} + \cdots + 19569545720 ) / 576954640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 111517 \nu^{11} - 717891 \nu^{10} - 5417673 \nu^{9} + 33853987 \nu^{8} + 111044592 \nu^{7} + \cdots + 6277569280 ) / 576954640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 127626 \nu^{11} - 815837 \nu^{10} - 7380199 \nu^{9} + 46635417 \nu^{8} + 157872765 \nu^{7} + \cdots + 6403216920 ) / 576954640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 344997 \nu^{11} + 4286939 \nu^{10} - 54344143 \nu^{9} - 171854403 \nu^{8} + 1852765582 \nu^{7} + \cdots + 996822640 ) / 1153909280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 578917 \nu^{11} + 467095 \nu^{10} + 48096453 \nu^{9} - 45343183 \nu^{8} - 1361867036 \nu^{7} + \cdots - 22018596800 ) / 1153909280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 593265 \nu^{11} - 2943239 \nu^{10} - 30920335 \nu^{9} + 139543951 \nu^{8} + 583355114 \nu^{7} + \cdots + 12399713520 ) / 576954640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1717069 \nu^{11} - 12271653 \nu^{10} - 60616831 \nu^{9} + 504434173 \nu^{8} + 575213110 \nu^{7} + \cdots - 1046240240 ) / 1153909280 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1909593 \nu^{11} + 13629375 \nu^{10} + 68397797 \nu^{9} - 565931847 \nu^{8} + \cdots + 19602612800 ) / 1153909280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 19\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{4} + 3\beta_{3} + 27\beta_{2} + 42\beta _1 + 232 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} + 4 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + \cdots + 476 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 44 \beta_{11} + 46 \beta_{10} + 12 \beta_{8} + 6 \beta_{7} + 12 \beta_{6} + 10 \beta_{5} + 24 \beta_{4} + \cdots + 5382 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 158 \beta_{11} + 160 \beta_{10} + 12 \beta_{9} + 170 \beta_{8} + 80 \beta_{7} + 84 \beta_{6} + \cdots + 17350 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1686 \beta_{11} + 1752 \beta_{10} + 124 \beta_{9} + 662 \beta_{8} + 284 \beta_{7} + 560 \beta_{6} + \cdots + 139556 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7831 \beta_{11} + 7889 \beta_{10} + 1188 \beta_{9} + 5818 \beta_{8} + 2448 \beta_{7} + 2620 \beta_{6} + \cdots + 572862 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 61990 \beta_{11} + 63400 \beta_{10} + 9348 \beta_{9} + 26434 \beta_{8} + 9894 \beta_{7} + 17714 \beta_{6} + \cdots + 3865846 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 323938 \beta_{11} + 324366 \beta_{10} + 68300 \beta_{9} + 188926 \beta_{8} + 69922 \beta_{7} + \cdots + 18092328 \) 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
5.58387
5.28037
3.59117
3.07517
2.35682
0.843186
−0.161453
−1.01648
−2.69379
−3.77011
−3.84968
−4.23908
−4.58387 3.00000 13.0119 5.00000 −13.7516 29.4297 −22.9739 9.00000 −22.9194
1.2 −4.28037 3.00000 10.3215 5.00000 −12.8411 9.81276 −9.93705 9.00000 −21.4018
1.3 −2.59117 3.00000 −1.28583 5.00000 −7.77352 −12.5999 24.0612 9.00000 −12.9559
1.4 −2.07517 3.00000 −3.69366 5.00000 −6.22551 −24.2415 24.2664 9.00000 −10.3759
1.5 −1.35682 3.00000 −6.15904 5.00000 −4.07045 36.6735 19.2112 9.00000 −6.78409
1.6 0.156814 3.00000 −7.97541 5.00000 0.470441 3.41884 −2.50516 9.00000 0.784068
1.7 1.16145 3.00000 −6.65103 5.00000 3.48436 4.01638 −17.0165 9.00000 5.80727
1.8 2.01648 3.00000 −3.93383 5.00000 6.04943 17.1804 −24.0643 9.00000 10.0824
1.9 3.69379 3.00000 5.64411 5.00000 11.0814 0.0404379 −8.70216 9.00000 18.4690
1.10 4.77011 3.00000 14.7539 5.00000 14.3103 −30.6111 32.2168 9.00000 23.8505
1.11 4.84968 3.00000 15.5194 5.00000 14.5491 26.0919 36.4669 9.00000 24.2484
1.12 5.23908 3.00000 19.4479 5.00000 15.7172 17.7886 59.9765 9.00000 26.1954
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 1815.4.a.bn 12
11.b odd 2 1 1815.4.a.bh 12
11.c even 5 2 165.4.m.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.m.a ✓ 24 11.c even 5 2
1815.4.a.bh 12 11.b odd 2 1
1815.4.a.bn 12 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{12} - 7 T_{2}^{11} - 48 T_{2}^{10} + 376 T_{2}^{9} + 754 T_{2}^{8} - 7037 T_{2}^{7} + \cdots - 23536 \) Copy content Toggle raw display
\( T_{7}^{12} - 77 T_{7}^{11} + 309 T_{7}^{10} + 108586 T_{7}^{9} - 2474180 T_{7}^{8} - 25763424 T_{7}^{7} + \cdots - 438449674204 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 7 T^{11} + \cdots - 23536 \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( (T - 5)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 438449674204 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 10\!\cdots\!39 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 69\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 16\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 52\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 36\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 12\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 13\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 26\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
show more
show less