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Properties

Label 1815.4.a.bi
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $1$
Dimension $12$
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] = [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: \(1\)
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: \( x^{12} - x^{11} - 72 x^{10} + 68 x^{9} + 1852 x^{8} - 1711 x^{7} - 20848 x^{6} + 20766 x^{5} + \cdots + 56080 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 11^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + ( - \beta_{10} + \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + \cdots - 4 \beta_1) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + ( - \beta_{10} + \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} + \cdots - 4 \beta_1) q^{8}+ \cdots + (20 \beta_{11} - 19 \beta_{10} + \cdots + 95) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} + 3 q^{6} + 5 q^{7} + 3 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} + 3 q^{6} + 5 q^{7} + 3 q^{8} + 108 q^{9} - 5 q^{10} - 147 q^{12} - 34 q^{13} - 134 q^{14} - 180 q^{15} + 265 q^{16} - 169 q^{17} - 9 q^{18} - 203 q^{19} + 245 q^{20} - 15 q^{21} + 116 q^{23} - 9 q^{24} + 300 q^{25} + 216 q^{26} - 324 q^{27} - 80 q^{28} - 409 q^{29} + 15 q^{30} - 645 q^{31} - 532 q^{32} - 1442 q^{34} + 25 q^{35} + 441 q^{36} + 40 q^{37} + 404 q^{38} + 102 q^{39} + 15 q^{40} - 1071 q^{41} + 402 q^{42} + 101 q^{43} + 540 q^{45} - 1539 q^{46} - 572 q^{47} - 795 q^{48} + 1563 q^{49} - 25 q^{50} + 507 q^{51} - 1081 q^{52} + 611 q^{53} + 27 q^{54} - 3221 q^{56} + 609 q^{57} - 354 q^{58} + 958 q^{59} - 735 q^{60} - 2620 q^{61} - 1049 q^{62} + 45 q^{63} + 2931 q^{64} - 170 q^{65} - 1027 q^{67} - 1896 q^{68} - 348 q^{69} - 670 q^{70} + 1020 q^{71} + 27 q^{72} - 2244 q^{73} + 283 q^{74} - 900 q^{75} - 4958 q^{76} - 648 q^{78} - 2199 q^{79} + 1325 q^{80} + 972 q^{81} + 362 q^{82} + 602 q^{83} + 240 q^{84} - 845 q^{85} - 1589 q^{86} + 1227 q^{87} - 2413 q^{89} - 45 q^{90} + 302 q^{91} + 3030 q^{92} + 1935 q^{93} - 2885 q^{94} - 1015 q^{95} + 1596 q^{96} + 886 q^{97} + 1236 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 72 x^{10} + 68 x^{9} + 1852 x^{8} - 1711 x^{7} - 20848 x^{6} + 20766 x^{5} + \cdots + 56080 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6167 \nu^{11} - 6226 \nu^{10} - 407550 \nu^{9} + 306802 \nu^{8} + 9282906 \nu^{7} + \cdots - 89573168 ) / 1933152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10859 \nu^{11} - 13370 \nu^{10} + 809770 \nu^{9} + 892386 \nu^{8} - 21568094 \nu^{7} + \cdots + 584253360 ) / 1933152 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13666 \nu^{11} - 10879 \nu^{10} + 980941 \nu^{9} + 776169 \nu^{8} - 24896139 \nu^{7} + \cdots + 565928176 ) / 1933152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19833 \nu^{11} - 4653 \nu^{10} + 1388491 \nu^{9} + 469367 \nu^{8} - 34179045 \nu^{7} + \cdots + 678699168 ) / 1933152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39837 \nu^{11} - 23050 \nu^{10} + 2855554 \nu^{9} + 1750786 \nu^{8} - 72497590 \nu^{7} + \cdots + 1662422576 ) / 3866304 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1893 \nu^{11} - 890 \nu^{10} + 134058 \nu^{9} + 70090 \nu^{8} - 3346574 \nu^{7} + \cdots + 64425040 ) / 148704 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28787 \nu^{11} - 17174 \nu^{10} + 2052466 \nu^{9} + 1301610 \nu^{8} - 51685838 \nu^{7} + \cdots + 1174335600 ) / 1933152 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 117925 \nu^{11} + 17514 \nu^{10} - 8307546 \nu^{9} - 2099962 \nu^{8} + 206259262 \nu^{7} + \cdots - 4198882416 ) / 7732608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 44689 \nu^{11} + 33838 \nu^{10} - 3220126 \nu^{9} - 2412542 \nu^{8} + 82184834 \nu^{7} + \cdots - 1706830064 ) / 1933152 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 3\beta_{8} - 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{3} + 26\beta_{2} + 7\beta _1 + 244 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{11} + 2 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + 42 \beta_{6} - 42 \beta_{5} + 2 \beta_{4} + \cdots + 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 80 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} + 131 \beta_{8} + 2 \beta_{7} - 104 \beta_{6} + 72 \beta_{5} + \cdots + 5904 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 110 \beta_{11} - 8 \beta_{10} + 146 \beta_{9} - 149 \beta_{8} - 270 \beta_{7} + 1337 \beta_{6} + \cdots + 1863 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2570 \beta_{11} + 244 \beta_{10} + 722 \beta_{9} + 4438 \beta_{8} + 162 \beta_{7} - 4023 \beta_{6} + \cdots + 155160 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4366 \beta_{11} - 464 \beta_{10} + 6696 \beta_{9} - 5483 \beta_{8} - 9312 \beta_{7} + 39465 \beta_{6} + \cdots + 44680 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 77584 \beta_{11} + 9888 \beta_{10} + 33006 \beta_{9} + 139100 \beta_{8} + 7294 \beta_{7} + \cdots + 4255135 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 153156 \beta_{11} - 18068 \beta_{10} + 255740 \beta_{9} - 181981 \beta_{8} - 294772 \beta_{7} + \cdots + 702235 \) 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
5.34844
4.90341
3.04373
2.66682
1.51761
1.48723
−0.374476
−1.17661
−3.41710
−3.69805
−3.83557
−5.46545
−5.34844 −3.00000 20.6059 5.00000 16.0453 17.2500 −67.4218 9.00000 −26.7422
1.2 −4.90341 −3.00000 16.0434 5.00000 14.7102 0.166503 −39.4401 9.00000 −24.5170
1.3 −3.04373 −3.00000 1.26431 5.00000 9.13120 20.6437 20.5016 9.00000 −15.2187
1.4 −2.66682 −3.00000 −0.888052 5.00000 8.00047 −34.9200 23.7029 9.00000 −13.3341
1.5 −1.51761 −3.00000 −5.69685 5.00000 4.55284 −25.9264 20.7865 9.00000 −7.58806
1.6 −1.48723 −3.00000 −5.78815 5.00000 4.46169 25.2790 20.5061 9.00000 −7.43614
1.7 0.374476 −3.00000 −7.85977 5.00000 −1.12343 11.4396 −5.93911 9.00000 1.87238
1.8 1.17661 −3.00000 −6.61558 5.00000 −3.52984 5.87699 −17.1969 9.00000 5.88306
1.9 3.41710 −3.00000 3.67654 5.00000 −10.2513 −25.9117 −14.7737 9.00000 17.0855
1.10 3.69805 −3.00000 5.67556 5.00000 −11.0941 34.0461 −8.59591 9.00000 18.4902
1.11 3.83557 −3.00000 6.71157 5.00000 −11.5067 −2.37609 −4.94185 9.00000 19.1778
1.12 5.46545 −3.00000 21.8712 5.00000 −16.3964 −20.5677 75.8121 9.00000 27.3273
\(n\): 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.bi 12
11.b odd 2 1 1815.4.a.bl 12
11.c even 5 2 165.4.m.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.m.c ✓ 24 11.c even 5 2
1815.4.a.bi 12 1.a even 1 1 trivial
1815.4.a.bl 12 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{12} + T_{2}^{11} - 72 T_{2}^{10} - 68 T_{2}^{9} + 1852 T_{2}^{8} + 1711 T_{2}^{7} - 20848 T_{2}^{6} + \cdots + 56080 \) Copy content Toggle raw display
\( T_{7}^{12} - 5 T_{7}^{11} - 2827 T_{7}^{10} + 19134 T_{7}^{9} + 2818876 T_{7}^{8} - 24860140 T_{7}^{7} + \cdots - 3933260174756 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 56080 \) Copy content Toggle raw display
$3$ \( (T + 3)^{12} \) Copy content Toggle raw display
$5$ \( (T - 5)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 3933260174756 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 19\!\cdots\!99 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 50\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 33\!\cdots\!51 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 61\!\cdots\!29 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 10\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 24\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 11\!\cdots\!45 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 12\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 13\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 66\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
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