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Properties

Label 7774.2.a.bo
Level $7774$
Weight $2$
Character orbit 7774.a
Self dual yes
Analytic conductor $62.076$
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] = [7774,2,Mod(1,7774)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7774, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7774.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7774 = 2 \cdot 13^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7774.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: \(62.0757025313\)
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: \( x^{18} - 3 x^{17} - 30 x^{16} + 86 x^{15} + 356 x^{14} - 952 x^{13} - 2153 x^{12} + 5148 x^{11} + \cdots - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{11} q^{5} + \beta_1 q^{6} + ( - \beta_{15} + 1) q^{7} + q^{8} + (\beta_{17} + \beta_{16} + \cdots + \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{11} q^{5} + \beta_1 q^{6} + ( - \beta_{15} + 1) q^{7} + q^{8} + (\beta_{17} + \beta_{16} + \cdots + \beta_{7}) q^{9}+ \cdots + ( - 2 \beta_{17} + 2 \beta_{16} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{2} + 3 q^{3} + 18 q^{4} + 2 q^{5} + 3 q^{6} + 15 q^{7} + 18 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{2} + 3 q^{3} + 18 q^{4} + 2 q^{5} + 3 q^{6} + 15 q^{7} + 18 q^{8} + 15 q^{9} + 2 q^{10} + 2 q^{11} + 3 q^{12} + 15 q^{14} + 18 q^{15} + 18 q^{16} + 23 q^{17} + 15 q^{18} + 21 q^{19} + 2 q^{20} + 18 q^{21} + 2 q^{22} - 18 q^{23} + 3 q^{24} + 26 q^{25} + 21 q^{27} + 15 q^{28} + 33 q^{29} + 18 q^{30} + 19 q^{31} + 18 q^{32} - 11 q^{33} + 23 q^{34} - 12 q^{35} + 15 q^{36} + 6 q^{37} + 21 q^{38} + 2 q^{40} - 13 q^{41} + 18 q^{42} + 4 q^{43} + 2 q^{44} + 32 q^{45} - 18 q^{46} + 23 q^{47} + 3 q^{48} + 23 q^{49} + 26 q^{50} + 32 q^{51} + 26 q^{53} + 21 q^{54} - 17 q^{55} + 15 q^{56} - 55 q^{57} + 33 q^{58} + 4 q^{59} + 18 q^{60} + 18 q^{61} + 19 q^{62} + 55 q^{63} + 18 q^{64} - 11 q^{66} + 28 q^{67} + 23 q^{68} - 3 q^{69} - 12 q^{70} + 35 q^{71} + 15 q^{72} - 18 q^{73} + 6 q^{74} + 30 q^{75} + 21 q^{76} + 52 q^{77} + 28 q^{79} + 2 q^{80} + 46 q^{81} - 13 q^{82} - 18 q^{83} + 18 q^{84} - 11 q^{85} + 4 q^{86} - 42 q^{87} + 2 q^{88} - 23 q^{89} + 32 q^{90} - 18 q^{92} - q^{93} + 23 q^{94} + 42 q^{95} + 3 q^{96} - 5 q^{97} + 23 q^{98} + 47 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} - 30 x^{16} + 86 x^{15} + 356 x^{14} - 952 x^{13} - 2153 x^{12} + 5148 x^{11} + \cdots - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 748 \nu^{17} - 12220 \nu^{16} - 2641 \nu^{15} + 380053 \nu^{14} - 266640 \nu^{13} - 4683340 \nu^{12} + \cdots - 1358368 ) / 92456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3831 \nu^{17} + 24934 \nu^{16} - 193333 \nu^{15} - 800394 \nu^{14} + 3460914 \nu^{13} + \cdots + 3331936 ) / 184912 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17345 \nu^{17} - 187941 \nu^{16} - 84772 \nu^{15} + 5260334 \nu^{14} - 5652720 \nu^{13} + \cdots - 5703296 ) / 369824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35116 \nu^{17} + 145795 \nu^{16} + 884311 \nu^{15} - 4030846 \nu^{14} - 7836678 \nu^{13} + \cdots + 2089296 ) / 184912 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35953 \nu^{17} - 157456 \nu^{16} - 867015 \nu^{15} + 4315292 \nu^{14} + 6971922 \nu^{13} + \cdots - 1377136 ) / 184912 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40447 \nu^{17} + 169169 \nu^{16} + 1010870 \nu^{15} - 4664618 \nu^{14} - 8822864 \nu^{13} + \cdots + 2247424 ) / 184912 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49597 \nu^{17} + 211575 \nu^{16} + 1223334 \nu^{15} - 5827346 \nu^{14} - 10357216 \nu^{13} + \cdots + 2300992 ) / 184912 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63939 \nu^{17} - 228865 \nu^{16} - 1690514 \nu^{15} + 6209742 \nu^{14} + 16416352 \nu^{13} + \cdots + 24720 ) / 184912 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31817 \nu^{17} - 153790 \nu^{16} - 736162 \nu^{15} + 4268929 \nu^{14} + 5327254 \nu^{13} + \cdots - 2372112 ) / 92456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 153579 \nu^{17} - 777803 \nu^{16} - 3445976 \nu^{15} + 21602450 \nu^{14} + 22837784 \nu^{13} + \cdots - 12533728 ) / 369824 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 101760 \nu^{17} - 414843 \nu^{16} - 2584475 \nu^{15} + 11469162 \nu^{14} + 23263138 \nu^{13} + \cdots - 5801344 ) / 184912 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 111209 \nu^{17} - 499577 \nu^{16} - 2665886 \nu^{15} + 13770196 \nu^{14} + 21158348 \nu^{13} + \cdots - 5169552 ) / 184912 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 167827 \nu^{17} + 671710 \nu^{16} + 4290625 \nu^{15} - 18523634 \nu^{14} - 39166434 \nu^{13} + \cdots + 8822176 ) / 184912 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 95655 \nu^{17} - 379301 \nu^{16} - 2456406 \nu^{15} + 10460654 \nu^{14} + 22605498 \nu^{13} + \cdots - 4647080 ) / 92456 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 213401 \nu^{17} - 921557 \nu^{16} - 5262684 \nu^{15} + 25499502 \nu^{14} + 44557484 \nu^{13} + \cdots - 13102624 ) / 184912 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 229572 \nu^{17} + 924521 \nu^{16} + 5876553 \nu^{15} - 25588322 \nu^{14} - 53742706 \nu^{13} + \cdots + 13953088 ) / 184912 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} + \beta_{16} - \beta_{14} - \beta_{13} + \beta_{7} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{15} - \beta_{11} - 2\beta_{6} + \beta_{5} + \beta_{4} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{17} + 11 \beta_{16} - \beta_{15} - 8 \beta_{14} - 8 \beta_{13} - 2 \beta_{11} + 8 \beta_{7} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{17} + 15 \beta_{16} + 9 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 11 \beta_{11} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 80 \beta_{17} + 106 \beta_{16} - 10 \beta_{15} - 65 \beta_{14} - 69 \beta_{13} - 3 \beta_{12} + \cdots + 179 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 123 \beta_{17} + 176 \beta_{16} + 71 \beta_{15} - 36 \beta_{14} - 42 \beta_{13} - 39 \beta_{12} + \cdots + 201 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 718 \beta_{17} + 1002 \beta_{16} - 84 \beta_{15} - 546 \beta_{14} - 624 \beta_{13} - 68 \beta_{12} + \cdots + 1636 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1223 \beta_{17} + 1917 \beta_{16} + 544 \beta_{15} - 481 \beta_{14} - 617 \beta_{13} - 546 \beta_{12} + \cdots + 2471 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6489 \beta_{17} + 9499 \beta_{16} - 713 \beta_{15} - 4720 \beta_{14} - 5812 \beta_{13} - 1086 \beta_{12} + \cdots + 15495 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12009 \beta_{17} + 20229 \beta_{16} + 4053 \beta_{15} - 5672 \beta_{14} - 7844 \beta_{13} - 6784 \beta_{12} + \cdots + 28683 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 58920 \beta_{17} + 90765 \beta_{16} - 6441 \beta_{15} - 41736 \beta_{14} - 55304 \beta_{13} + \cdots + 150159 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 116832 \beta_{17} + 210014 \beta_{16} + 28718 \beta_{15} - 62487 \beta_{14} - 92603 \beta_{13} + \cdots + 320925 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 536667 \beta_{17} + 874792 \beta_{16} - 62675 \beta_{15} - 375636 \beta_{14} - 535198 \beta_{13} + \cdots + 1479187 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1127500 \beta_{17} + 2160272 \beta_{16} + 184032 \beta_{15} - 660686 \beta_{14} - 1047544 \beta_{13} + \cdots + 3506424 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 4898033 \beta_{17} + 8500301 \beta_{16} - 650728 \beta_{15} - 3428705 \beta_{14} - 5250833 \beta_{13} + \cdots + 14750299 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 10803073 \beta_{17} + 22101337 \beta_{16} + 918433 \beta_{15} - 6804936 \beta_{14} - 11540488 \beta_{13} + \cdots + 37717149 \) 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
−2.85659
−2.85573
−2.61224
−1.31604
−1.20697
−1.20246
−0.684421
−0.495396
−0.392866
0.0783325
0.827224
1.34160
1.35017
1.64672
2.00224
3.05084
3.09154
3.23403
1.00000 −2.85659 1.00000 −2.50619 −2.85659 1.50974 1.00000 5.16009 −2.50619
1.2 1.00000 −2.85573 1.00000 1.15363 −2.85573 4.78157 1.00000 5.15518 1.15363
1.3 1.00000 −2.61224 1.00000 3.42025 −2.61224 −1.00089 1.00000 3.82378 3.42025
1.4 1.00000 −1.31604 1.00000 −1.99628 −1.31604 −2.30119 1.00000 −1.26805 −1.99628
1.5 1.00000 −1.20697 1.00000 −3.56035 −1.20697 2.17256 1.00000 −1.54323 −3.56035
1.6 1.00000 −1.20246 1.00000 0.486453 −1.20246 −4.33778 1.00000 −1.55409 0.486453
1.7 1.00000 −0.684421 1.00000 −3.50148 −0.684421 1.47336 1.00000 −2.53157 −3.50148
1.8 1.00000 −0.495396 1.00000 −0.0910187 −0.495396 −3.08752 1.00000 −2.75458 −0.0910187
1.9 1.00000 −0.392866 1.00000 2.39330 −0.392866 −0.944953 1.00000 −2.84566 2.39330
1.10 1.00000 0.0783325 1.00000 −0.293525 0.0783325 2.47596 1.00000 −2.99386 −0.293525
1.11 1.00000 0.827224 1.00000 −1.34757 0.827224 4.91605 1.00000 −2.31570 −1.34757
1.12 1.00000 1.34160 1.00000 −3.10649 1.34160 −1.98484 1.00000 −1.20012 −3.10649
1.13 1.00000 1.35017 1.00000 3.61712 1.35017 2.17416 1.00000 −1.17705 3.61712
1.14 1.00000 1.64672 1.00000 1.91242 1.64672 2.54630 1.00000 −0.288299 1.91242
1.15 1.00000 2.00224 1.00000 3.56010 2.00224 2.70614 1.00000 1.00895 3.56010
1.16 1.00000 3.05084 1.00000 0.548833 3.05084 1.20151 1.00000 6.30764 0.548833
1.17 1.00000 3.09154 1.00000 3.79130 3.09154 −2.08284 1.00000 6.55762 3.79130
1.18 1.00000 3.23403 1.00000 −2.48050 3.23403 4.78268 1.00000 7.45895 −2.48050
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( +1 \)
\(23\) \( +1 \)

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 7774.2.a.bo yes 18
13.b even 2 1 7774.2.a.bn ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7774.2.a.bn ✓ 18 13.b even 2 1
7774.2.a.bo yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7774))\):

\( T_{3}^{18} - 3 T_{3}^{17} - 30 T_{3}^{16} + 86 T_{3}^{15} + 356 T_{3}^{14} - 952 T_{3}^{13} - 2153 T_{3}^{12} + \cdots - 64 \) Copy content Toggle raw display
\( T_{5}^{18} - 2 T_{5}^{17} - 56 T_{5}^{16} + 98 T_{5}^{15} + 1294 T_{5}^{14} - 1928 T_{5}^{13} + \cdots - 4073 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{17} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{18} - 2 T^{17} + \cdots - 4073 \) Copy content Toggle raw display
$7$ \( T^{18} - 15 T^{17} + \cdots - 2917888 \) Copy content Toggle raw display
$11$ \( T^{18} - 2 T^{17} + \cdots - 2796032 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} - 23 T^{17} + \cdots + 33754624 \) Copy content Toggle raw display
$19$ \( T^{18} - 21 T^{17} + \cdots - 19726848 \) Copy content Toggle raw display
$23$ \( (T + 1)^{18} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 1536940544 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 354663251048 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 40738332713833 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 14447795823 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 2501529123904 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 396525725128 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 739485853 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 856558394048 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 557880657109 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 29184778420288 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 12\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 2879393965337 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 1454775463232 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 1346925664328 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
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