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Properties

Label 1815.4.a.bo
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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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: \(0\)
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: \( x^{12} - 3 x^{11} - 69 x^{10} + 157 x^{9} + 1812 x^{8} - 2703 x^{7} - 22379 x^{6} + 16453 x^{5} + \cdots + 196416 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} - 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{10} - 2 \beta_1 + 2) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots + 12) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} - 5 q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{10} - 2 \beta_1 + 2) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots + 12) q^{8}+ \cdots + ( - 19 \beta_{11} + 12 \beta_{10} + \cdots + 553) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} + 27 q^{6} + 21 q^{7} + 123 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} + 27 q^{6} + 21 q^{7} + 123 q^{8} + 108 q^{9} - 45 q^{10} + 171 q^{12} + 78 q^{13} + 262 q^{14} - 180 q^{15} + 225 q^{16} + 313 q^{17} + 81 q^{18} + 51 q^{19} - 285 q^{20} + 63 q^{21} - 34 q^{23} + 369 q^{24} + 300 q^{25} + 28 q^{26} + 324 q^{27} + 376 q^{28} + 31 q^{29} - 135 q^{30} + 655 q^{31} + 1578 q^{32} - 10 q^{34} - 105 q^{35} + 513 q^{36} + 84 q^{37} - 1076 q^{38} + 234 q^{39} - 615 q^{40} + 1463 q^{41} + 786 q^{42} - 111 q^{43} - 540 q^{45} - q^{46} + 278 q^{47} + 675 q^{48} - 325 q^{49} + 225 q^{50} + 939 q^{51} + 1957 q^{52} + 517 q^{53} + 243 q^{54} + 1543 q^{56} + 153 q^{57} + 442 q^{58} - 308 q^{59} - 855 q^{60} + 604 q^{61} + 1773 q^{62} + 189 q^{63} + 4323 q^{64} - 390 q^{65} - 357 q^{67} + 2192 q^{68} - 102 q^{69} - 1310 q^{70} - 620 q^{71} + 1107 q^{72} + 1892 q^{73} + 581 q^{74} + 900 q^{75} + 378 q^{76} + 84 q^{78} + 415 q^{79} - 1125 q^{80} + 972 q^{81} - 2802 q^{82} + 3158 q^{83} + 1128 q^{84} - 1565 q^{85} + 747 q^{86} + 93 q^{87} + 1563 q^{89} - 405 q^{90} + 1434 q^{91} - 3466 q^{92} + 1965 q^{93} - 3 q^{94} - 255 q^{95} + 4734 q^{96} + 714 q^{97} + 6586 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 69 x^{10} + 157 x^{9} + 1812 x^{8} - 2703 x^{7} - 22379 x^{6} + 16453 x^{5} + \cdots + 196416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1039717 \nu^{11} + 7737702 \nu^{10} + 39948432 \nu^{9} - 397439950 \nu^{8} + \cdots - 55129222104 ) / 2424985200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1128971 \nu^{11} - 11624376 \nu^{10} - 36260616 \nu^{9} + 606337100 \nu^{8} + \cdots + 168239621352 ) / 2424985200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 126347 \nu^{11} + 784632 \nu^{10} + 6537162 \nu^{9} - 41205800 \nu^{8} - 113812749 \nu^{7} + \cdots - 1778396664 ) / 220453200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 71618 \nu^{11} - 411933 \nu^{10} - 3846828 \nu^{9} + 22518425 \nu^{8} + 68886756 \nu^{7} + \cdots + 8107808016 ) / 110226600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2130967 \nu^{11} + 12625872 \nu^{10} + 109723002 \nu^{9} - 655832380 \nu^{8} + \cdots - 182963732424 ) / 2424985200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2666033 \nu^{11} + 12236898 \nu^{10} + 149122668 \nu^{9} - 615402950 \nu^{8} + \cdots - 107131619496 ) / 2424985200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4787254 \nu^{11} - 23743509 \nu^{10} - 268640169 \nu^{9} + 1240004515 \nu^{8} + \cdots + 207559016808 ) / 2424985200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12332923 \nu^{11} + 65767533 \nu^{10} + 699719703 \nu^{9} - 3542940955 \nu^{8} + \cdots - 1070926959696 ) / 4849970400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12772667 \nu^{11} - 71145867 \nu^{10} - 702338997 \nu^{9} + 3854140685 \nu^{8} + \cdots + 1147734138144 ) / 4849970400 \) 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
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 20\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{8} + \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \cdots + 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 42 \beta_{11} + 40 \beta_{10} + 27 \beta_{9} - 9 \beta_{8} + 33 \beta_{7} - 37 \beta_{6} + 31 \beta_{5} + \cdots + 366 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 132 \beta_{11} + 124 \beta_{10} + 4 \beta_{9} - 106 \beta_{8} + 72 \beta_{7} - 44 \beta_{6} + \cdots + 5550 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1460 \beta_{11} + 1340 \beta_{10} + 612 \beta_{9} - 534 \beta_{8} + 1006 \beta_{7} - 1174 \beta_{6} + \cdots + 12644 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5853 \beta_{11} + 5305 \beta_{10} + 163 \beta_{9} - 4260 \beta_{8} + 3257 \beta_{7} - 2685 \beta_{6} + \cdots + 141017 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 48564 \beta_{11} + 43472 \beta_{10} + 13216 \beta_{9} - 22762 \beta_{8} + 30811 \beta_{7} + \cdots + 419235 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 223000 \beta_{11} + 197994 \beta_{10} + 4575 \beta_{9} - 155189 \beta_{8} + 124313 \beta_{7} + \cdots + 3788914 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1589828 \beta_{11} + 1400364 \beta_{10} + 279612 \beta_{9} - 854212 \beta_{8} + 955012 \beta_{7} + \cdots + 13666132 \) 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
5.65688
4.72324
4.45803
2.97307
1.76742
0.911819
−1.25962
−1.96095
−2.51077
−2.71454
−4.43454
−4.61005
−4.65688 3.00000 13.6866 −5.00000 −13.9707 −3.38865 −26.4817 9.00000 23.2844
1.2 −3.72324 3.00000 5.86255 −5.00000 −11.1697 −14.1368 7.95826 9.00000 18.6162
1.3 −3.45803 3.00000 3.95794 −5.00000 −10.3741 0.550113 13.9775 9.00000 17.2901
1.4 −1.97307 3.00000 −4.10701 −5.00000 −5.91920 −17.1107 23.8879 9.00000 9.86533
1.5 −0.767423 3.00000 −7.41106 −5.00000 −2.30227 6.29324 11.8268 9.00000 3.83711
1.6 0.0881808 3.00000 −7.99222 −5.00000 0.264542 20.0828 −1.41021 9.00000 −0.440904
1.7 2.25962 3.00000 −2.89411 −5.00000 6.77887 −23.8578 −24.6166 9.00000 −11.2981
1.8 2.96095 3.00000 0.767233 −5.00000 8.88285 25.4365 −21.4159 9.00000 −14.8048
1.9 3.51077 3.00000 4.32548 −5.00000 10.5323 −14.1309 −12.9004 9.00000 −17.5538
1.10 3.71454 3.00000 5.79777 −5.00000 11.1436 21.2047 −8.18025 9.00000 −18.5727
1.11 5.43454 3.00000 21.5342 −5.00000 16.3036 −9.68133 73.5522 9.00000 −27.1727
1.12 5.61005 3.00000 23.4727 −5.00000 16.8301 29.7389 86.8023 9.00000 −28.0502
\(n\): 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.bo 12
11.b odd 2 1 1815.4.a.bg 12
11.d odd 10 2 165.4.m.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.m.d ✓ 24 11.d odd 10 2
1815.4.a.bg 12 11.b odd 2 1
1815.4.a.bo 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{12} - 9 T_{2}^{11} - 36 T_{2}^{10} + 478 T_{2}^{9} + 120 T_{2}^{8} - 8967 T_{2}^{7} + \cdots - 21296 \) Copy content Toggle raw display
\( T_{7}^{12} - 21 T_{7}^{11} - 1675 T_{7}^{10} + 26442 T_{7}^{9} + 1118416 T_{7}^{8} + \cdots + 2983653045684 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 9 T^{11} + \cdots - 21296 \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( (T + 5)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 2983653045684 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 71\!\cdots\!61 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 88\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 83\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 87\!\cdots\!91 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 33\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 83\!\cdots\!59 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 21\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 19\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 77\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 83\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 41\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
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