Properties

Label 676.2.f.h
Level $676$
Weight $2$
Character orbit 676.f
Analytic conductor $5.398$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [676,2,Mod(99,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.f (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.102930383934669717504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + 64 x^{7} - 48 x^{6} - 64 x^{5} + 80 x^{4} - 64 x^{3} + 320 x^{2} - 512 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + (\beta_{14} - \beta_{13} + \beta_{10} - \beta_{8} - \beta_{5}) q^{4} + ( - \beta_{13} + \beta_{4} + \beta_{2} - 1) q^{5} + ( - \beta_{15} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{5} + \beta_{2} + 1) q^{6} + ( - \beta_{14} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{13} - \beta_{12} + \beta_{8} - \beta_{6} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + (\beta_{14} - \beta_{13} + \beta_{10} - \beta_{8} - \beta_{5}) q^{4} + ( - \beta_{13} + \beta_{4} + \beta_{2} - 1) q^{5} + ( - \beta_{15} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{5} + \beta_{2} + 1) q^{6} + ( - \beta_{14} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{7} - 2 \beta_{3} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 12 q^{5} + 4 q^{6} + 10 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} - 12 q^{5} + 4 q^{6} + 10 q^{8} - 8 q^{9} + 8 q^{14} + 4 q^{16} - 6 q^{18} - 22 q^{20} - 28 q^{21} + 4 q^{24} - 36 q^{28} + 16 q^{29} - 2 q^{32} + 28 q^{33} + 14 q^{34} + 8 q^{37} - 40 q^{40} - 24 q^{41} + 56 q^{42} - 8 q^{44} + 20 q^{45} + 56 q^{46} + 20 q^{48} - 32 q^{50} - 32 q^{53} + 44 q^{54} + 12 q^{57} + 30 q^{58} - 24 q^{60} - 8 q^{61} + 56 q^{66} - 32 q^{68} + 28 q^{70} - 46 q^{72} + 20 q^{73} - 8 q^{74} - 8 q^{76} - 22 q^{80} - 96 q^{81} - 48 q^{84} - 52 q^{85} + 16 q^{86} + 44 q^{89} - 12 q^{92} + 112 q^{93} + 76 q^{94} - 72 q^{96} - 52 q^{97} + 70 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + 64 x^{7} - 48 x^{6} - 64 x^{5} + 80 x^{4} - 64 x^{3} + 320 x^{2} - 512 x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4 \nu^{15} + 9 \nu^{14} + 14 \nu^{13} - 39 \nu^{12} - 56 \nu^{11} + 29 \nu^{10} + 134 \nu^{9} + 128 \nu^{8} - 184 \nu^{7} - 324 \nu^{6} - 344 \nu^{5} + 448 \nu^{4} + 992 \nu^{3} - 240 \nu^{2} - 1056 \nu - 832 ) / 704 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19 \nu^{15} - 26 \nu^{14} - 49 \nu^{13} - 12 \nu^{12} + 163 \nu^{11} + 146 \nu^{10} - 172 \nu^{9} - 184 \nu^{8} - 412 \nu^{7} + 408 \nu^{6} + 1072 \nu^{5} + 544 \nu^{4} - 1360 \nu^{3} - 2592 \nu^{2} + \cdots + 1152 ) / 1408 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10 \nu^{15} + 37 \nu^{14} + 12 \nu^{13} - 7 \nu^{12} - 106 \nu^{11} - 87 \nu^{10} + 266 \nu^{9} + 80 \nu^{8} + 264 \nu^{7} - 612 \nu^{6} - 872 \nu^{5} + 480 \nu^{4} + 800 \nu^{3} + 1744 \nu^{2} + \cdots - 1216 ) / 704 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25 \nu^{15} + 39 \nu^{14} + 9 \nu^{13} + 21 \nu^{12} - 105 \nu^{11} - 105 \nu^{10} + 198 \nu^{9} - 48 \nu^{8} + 492 \nu^{7} - 420 \nu^{6} - 792 \nu^{5} + 384 \nu^{4} + 336 \nu^{3} + 2160 \nu^{2} + \cdots + 1024 ) / 704 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7 \nu^{15} + 28 \nu^{14} - 21 \nu^{13} - 6 \nu^{12} - 57 \nu^{11} + 24 \nu^{10} + 166 \nu^{9} - 124 \nu^{8} + 152 \nu^{7} - 520 \nu^{6} - 40 \nu^{5} + 672 \nu^{4} + 96 \nu^{3} + 640 \nu^{2} - 2240 \nu + 1600 ) / 352 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{15} - 28 \nu^{14} + 21 \nu^{13} + 6 \nu^{12} + 57 \nu^{11} - 24 \nu^{10} - 166 \nu^{9} + 124 \nu^{8} - 152 \nu^{7} + 520 \nu^{6} + 40 \nu^{5} - 672 \nu^{4} - 96 \nu^{3} - 640 \nu^{2} + 2944 \nu - 1600 ) / 352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59 \nu^{15} + 100 \nu^{14} + 77 \nu^{13} + 6 \nu^{12} - 355 \nu^{11} - 368 \nu^{10} + 808 \nu^{9} + 360 \nu^{8} + 1132 \nu^{7} - 1472 \nu^{6} - 3104 \nu^{5} + 1152 \nu^{4} + 3344 \nu^{3} + \cdots - 4224 ) / 1408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47 \nu^{15} + 78 \nu^{14} + 13 \nu^{13} + 32 \nu^{12} - 191 \nu^{11} - 150 \nu^{10} + 420 \nu^{9} - 160 \nu^{8} + 876 \nu^{7} - 952 \nu^{6} - 1136 \nu^{5} + 992 \nu^{4} + 368 \nu^{3} + 3744 \nu^{2} + \cdots + 3200 ) / 704 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 81 \nu^{15} + 206 \nu^{14} - 109 \nu^{13} + 28 \nu^{12} - 393 \nu^{11} + 146 \nu^{10} + 1100 \nu^{9} - 1120 \nu^{8} + 1492 \nu^{7} - 3288 \nu^{6} - 176 \nu^{5} + 4384 \nu^{4} + \cdots + 16384 ) / 1408 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51 \nu^{15} - 109 \nu^{14} + 33 \nu^{13} - 27 \nu^{12} + 239 \nu^{11} + 39 \nu^{10} - 622 \nu^{9} + 492 \nu^{8} - 892 \nu^{7} + 1564 \nu^{6} + 680 \nu^{5} - 1840 \nu^{4} - 176 \nu^{3} - 3984 \nu^{2} + \cdots - 6336 ) / 704 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81 \nu^{15} - 272 \nu^{14} + 153 \nu^{13} + 82 \nu^{12} + 481 \nu^{11} - 300 \nu^{10} - 1496 \nu^{9} + 1296 \nu^{8} - 1316 \nu^{7} + 4432 \nu^{6} + 352 \nu^{5} - 6496 \nu^{4} - 624 \nu^{3} + \cdots - 19200 ) / 1408 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 147 \nu^{15} - 376 \nu^{14} + 179 \nu^{13} - 50 \nu^{12} + 747 \nu^{11} - 92 \nu^{10} - 2000 \nu^{9} + 1712 \nu^{8} - 2780 \nu^{7} + 5616 \nu^{6} + 1504 \nu^{5} - 7296 \nu^{4} + 112 \nu^{3} + \cdots - 24064 ) / 1408 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 73 \nu^{15} + 204 \nu^{14} - 109 \nu^{13} - 6 \nu^{12} - 365 \nu^{11} + 112 \nu^{10} + 1068 \nu^{9} - 960 \nu^{8} + 1252 \nu^{7} - 3072 \nu^{6} - 480 \nu^{5} + 4544 \nu^{4} - 80 \nu^{3} + \cdots + 13568 ) / 704 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 113 \nu^{15} + 295 \nu^{14} - 143 \nu^{13} + 21 \nu^{12} - 577 \nu^{11} + 87 \nu^{10} + 1574 \nu^{9} - 1424 \nu^{8} + 2092 \nu^{7} - 4404 \nu^{6} - 920 \nu^{5} + 6144 \nu^{4} - 176 \nu^{3} + \cdots + 19712 ) / 704 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 247 \nu^{15} + 726 \nu^{14} - 483 \nu^{13} - 64 \nu^{12} - 1183 \nu^{11} + 714 \nu^{10} + 3788 \nu^{9} - 4088 \nu^{8} + 4140 \nu^{7} - 10872 \nu^{6} + 368 \nu^{5} + 17344 \nu^{4} + \cdots + 55808 ) / 1408 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} + 3 \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} + \beta_{14} + 3 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{6} - 9 \beta_{4} - \beta_{3} - 6 \beta_{2} - 3 \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5 \beta_{15} + \beta_{14} - 15 \beta_{13} - 4 \beta_{12} - \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 5 \beta_{8} + \beta_{7} - 5 \beta_{6} - 4 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3 \beta_{15} - 7 \beta_{14} + 5 \beta_{13} - 14 \beta_{12} + 9 \beta_{11} + 10 \beta_{10} - 7 \beta_{9} + 11 \beta_{8} + 5 \beta_{7} - 7 \beta_{6} - 4 \beta_{5} - 21 \beta_{4} - 5 \beta_{3} - 6 \beta_{2} + \beta _1 + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7 \beta_{15} - 5 \beta_{14} - 12 \beta_{13} - 7 \beta_{12} + 5 \beta_{11} - 2 \beta_{10} + 7 \beta_{8} - 2 \beta_{6} - 8 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} + 4 \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - \beta_{15} - 5 \beta_{14} - 19 \beta_{13} - 38 \beta_{12} - 11 \beta_{11} + 14 \beta_{10} + \beta_{9} - \beta_{8} + 5 \beta_{7} + 9 \beta_{6} - 12 \beta_{5} - 3 \beta_{4} - 9 \beta_{3} + 10 \beta_{2} + 9 \beta _1 - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9 \beta_{15} - 17 \beta_{14} + 21 \beta_{13} + 6 \beta_{12} - 27 \beta_{11} - 10 \beta_{10} + 17 \beta_{9} - 9 \beta_{8} + 13 \beta_{7} + 23 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} + 13 \beta_{3} + 14 \beta_{2} - 33 \beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7 \beta_{15} + 27 \beta_{13} - 4 \beta_{12} + 23 \beta_{11} + 23 \beta_{9} - 27 \beta_{8} + 7 \beta_{7} + 27 \beta_{6} + 9 \beta_{3} + 4 \beta_{2} + 3 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 27 \beta_{15} + 53 \beta_{14} + \beta_{13} + 36 \beta_{12} - 33 \beta_{11} + 18 \beta_{10} + 37 \beta_{9} + 5 \beta_{8} + \beta_{7} + 75 \beta_{6} - 12 \beta_{5} - 51 \beta_{4} + 27 \beta_{3} + 8 \beta_{2} - 9 \beta _1 - 57 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 15 \beta_{15} + 15 \beta_{14} + 171 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} + 54 \beta_{10} - 57 \beta_{9} - 135 \beta_{8} + 75 \beta_{7} + 171 \beta_{6} - 60 \beta_{5} - 111 \beta_{4} + 45 \beta_{3} - 166 \beta_{2} - 133 \beta _1 + 73 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 78 \beta_{14} - 45 \beta_{13} + 15 \beta_{12} + 60 \beta_{10} + 27 \beta_{9} - 72 \beta_{8} + 15 \beta_{7} + 33 \beta_{6} - 60 \beta_{5} + \beta_{4} + 78 \beta_{3} - 15 \beta_{2} - 33 \beta _1 + 67 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−1.39427 + 0.236640i
1.41121 + 0.0921725i
−0.00757716 + 1.41419i
1.17605 0.785427i
−0.873468 1.11223i
1.08916 + 0.902074i
−0.713659 1.22094i
1.31256 0.526485i
−1.39427 0.236640i
1.41121 0.0921725i
−0.00757716 1.41419i
1.17605 + 0.785427i
−0.873468 + 1.11223i
1.08916 0.902074i
−0.713659 + 1.22094i
1.31256 + 0.526485i
−1.32580 + 0.492201i 0.850043i 1.51548 1.30512i −0.166404 0.166404i 0.418392 + 1.12698i −1.86977 1.86977i −1.36683 + 2.47624i 2.27743 0.302522 + 0.138714i
99.2 −1.26823 0.625780i 2.09440i 1.21680 + 1.58726i 0.894007 + 0.894007i −1.31063 + 2.65617i −3.20020 3.20020i −0.549903 2.77446i −1.38651 −0.574352 1.69325i
99.3 −0.700535 + 1.22852i 1.61663i −1.01850 1.72124i −1.52798 1.52798i 1.98605 + 1.13250i 1.44528 + 1.44528i 2.82806 0.0454612i 0.386509 2.94755 0.806745i
99.4 −0.625780 1.26823i 2.09440i −1.21680 + 1.58726i 0.894007 + 0.894007i 2.65617 1.31063i 3.20020 + 3.20020i 2.77446 + 0.549903i −1.38651 0.574352 1.69325i
99.5 −0.200331 + 1.39995i 2.50548i −1.91973 0.560908i −2.19962 2.19962i −3.50755 0.501925i 0.416921 + 0.416921i 1.16983 2.57517i −3.27743 3.52002 2.63871i
99.6 0.492201 1.32580i 0.850043i −1.51548 1.30512i −0.166404 0.166404i 1.12698 + 0.418392i 1.86977 + 1.86977i −2.47624 + 1.36683i 2.27743 −0.302522 + 0.138714i
99.7 1.22852 0.700535i 1.61663i 1.01850 1.72124i −1.52798 1.52798i 1.13250 + 1.98605i −1.44528 1.44528i 0.0454612 2.82806i 0.386509 −2.94755 0.806745i
99.8 1.39995 0.200331i 2.50548i 1.91973 0.560908i −2.19962 2.19962i −0.501925 3.50755i −0.416921 0.416921i 2.57517 1.16983i −3.27743 −3.52002 2.63871i
239.1 −1.32580 0.492201i 0.850043i 1.51548 + 1.30512i −0.166404 + 0.166404i 0.418392 1.12698i −1.86977 + 1.86977i −1.36683 2.47624i 2.27743 0.302522 0.138714i
239.2 −1.26823 + 0.625780i 2.09440i 1.21680 1.58726i 0.894007 0.894007i −1.31063 2.65617i −3.20020 + 3.20020i −0.549903 + 2.77446i −1.38651 −0.574352 + 1.69325i
239.3 −0.700535 1.22852i 1.61663i −1.01850 + 1.72124i −1.52798 + 1.52798i 1.98605 1.13250i 1.44528 1.44528i 2.82806 + 0.0454612i 0.386509 2.94755 + 0.806745i
239.4 −0.625780 + 1.26823i 2.09440i −1.21680 1.58726i 0.894007 0.894007i 2.65617 + 1.31063i 3.20020 3.20020i 2.77446 0.549903i −1.38651 0.574352 + 1.69325i
239.5 −0.200331 1.39995i 2.50548i −1.91973 + 0.560908i −2.19962 + 2.19962i −3.50755 + 0.501925i 0.416921 0.416921i 1.16983 + 2.57517i −3.27743 3.52002 + 2.63871i
239.6 0.492201 + 1.32580i 0.850043i −1.51548 + 1.30512i −0.166404 + 0.166404i 1.12698 0.418392i 1.86977 1.86977i −2.47624 1.36683i 2.27743 −0.302522 0.138714i
239.7 1.22852 + 0.700535i 1.61663i 1.01850 + 1.72124i −1.52798 + 1.52798i 1.13250 1.98605i −1.44528 + 1.44528i 0.0454612 + 2.82806i 0.386509 −2.94755 + 0.806745i
239.8 1.39995 + 0.200331i 2.50548i 1.91973 + 0.560908i −2.19962 + 2.19962i −0.501925 + 3.50755i −0.416921 + 0.416921i 2.57517 + 1.16983i −3.27743 −3.52002 + 2.63871i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.d odd 4 1 inner
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.2.f.h 16
4.b odd 2 1 inner 676.2.f.h 16
13.b even 2 1 676.2.f.i 16
13.c even 3 1 52.2.l.b 16
13.c even 3 1 676.2.l.m 16
13.d odd 4 1 inner 676.2.f.h 16
13.d odd 4 1 676.2.f.i 16
13.e even 6 1 676.2.l.i 16
13.e even 6 1 676.2.l.k 16
13.f odd 12 1 52.2.l.b 16
13.f odd 12 1 676.2.l.i 16
13.f odd 12 1 676.2.l.k 16
13.f odd 12 1 676.2.l.m 16
39.i odd 6 1 468.2.cb.f 16
39.k even 12 1 468.2.cb.f 16
52.b odd 2 1 676.2.f.i 16
52.f even 4 1 inner 676.2.f.h 16
52.f even 4 1 676.2.f.i 16
52.i odd 6 1 676.2.l.i 16
52.i odd 6 1 676.2.l.k 16
52.j odd 6 1 52.2.l.b 16
52.j odd 6 1 676.2.l.m 16
52.l even 12 1 52.2.l.b 16
52.l even 12 1 676.2.l.i 16
52.l even 12 1 676.2.l.k 16
52.l even 12 1 676.2.l.m 16
104.n odd 6 1 832.2.bu.n 16
104.r even 6 1 832.2.bu.n 16
104.u even 12 1 832.2.bu.n 16
104.x odd 12 1 832.2.bu.n 16
156.p even 6 1 468.2.cb.f 16
156.v odd 12 1 468.2.cb.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.b 16 13.c even 3 1
52.2.l.b 16 13.f odd 12 1
52.2.l.b 16 52.j odd 6 1
52.2.l.b 16 52.l even 12 1
468.2.cb.f 16 39.i odd 6 1
468.2.cb.f 16 39.k even 12 1
468.2.cb.f 16 156.p even 6 1
468.2.cb.f 16 156.v odd 12 1
676.2.f.h 16 1.a even 1 1 trivial
676.2.f.h 16 4.b odd 2 1 inner
676.2.f.h 16 13.d odd 4 1 inner
676.2.f.h 16 52.f even 4 1 inner
676.2.f.i 16 13.b even 2 1
676.2.f.i 16 13.d odd 4 1
676.2.f.i 16 52.b odd 2 1
676.2.f.i 16 52.f even 4 1
676.2.l.i 16 13.e even 6 1
676.2.l.i 16 13.f odd 12 1
676.2.l.i 16 52.i odd 6 1
676.2.l.i 16 52.l even 12 1
676.2.l.k 16 13.e even 6 1
676.2.l.k 16 13.f odd 12 1
676.2.l.k 16 52.i odd 6 1
676.2.l.k 16 52.l even 12 1
676.2.l.m 16 13.c even 3 1
676.2.l.m 16 13.f odd 12 1
676.2.l.m 16 52.j odd 6 1
676.2.l.m 16 52.l even 12 1
832.2.bu.n 16 104.n odd 6 1
832.2.bu.n 16 104.r even 6 1
832.2.bu.n 16 104.u even 12 1
832.2.bu.n 16 104.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(676, [\chi])\):

\( T_{3}^{8} + 14T_{3}^{6} + 65T_{3}^{4} + 112T_{3}^{2} + 52 \) Copy content Toggle raw display
\( T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} + 18T_{5}^{5} + 5T_{5}^{4} + 72T_{5}^{2} + 24T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{16} + 486T_{7}^{12} + 28745T_{7}^{8} + 361440T_{7}^{4} + 43264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + 2 T^{14} - 2 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{8} + 14 T^{6} + 65 T^{4} + 112 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 6 T^{7} + 18 T^{6} + 18 T^{5} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 486 T^{12} + 28745 T^{8} + \cdots + 43264 \) Copy content Toggle raw display
$11$ \( T^{16} + 438 T^{12} + 31193 T^{8} + \cdots + 692224 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} + 32 T^{6} + 222 T^{4} + 320 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 1134 T^{12} + \cdots + 77228944 \) Copy content Toggle raw display
$23$ \( (T^{8} - 106 T^{6} + 2393 T^{4} + \cdots + 8788)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} - 46 T^{2} + 244 T - 227)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + 9072 T^{12} + \cdots + 1235663104 \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} + 18 T^{2} - 54 T + 81)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 266 T^{6} + 22613 T^{4} + \cdots + 2381392)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 20976 T^{12} + \cdots + 5671027857664 \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} - 3 T^{2} - 112 T - 128)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 171720267307264 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} - 58 T^{2} - 158 T - 83)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + 12702 T^{12} + \cdots + 2205735869584 \) Copy content Toggle raw display
$71$ \( T^{16} + 16158 T^{12} + \cdots + 5067731344 \) Copy content Toggle raw display
$73$ \( (T^{8} - 10 T^{7} + 50 T^{6} - 6 T^{5} + \cdots + 676)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 160 T^{6} + 7040 T^{4} + \cdots + 53248)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 538409280507904 \) Copy content Toggle raw display
$89$ \( (T^{8} - 22 T^{7} + 242 T^{6} + \cdots + 2896804)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 26 T^{7} + 338 T^{6} + 1134 T^{5} + \cdots + 2116)^{2} \) Copy content Toggle raw display
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