Properties

Label 136.4.k.a
Level $136$
Weight $4$
Character orbit 136.k
Analytic conductor $8.024$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \( x^{14} + 142x^{12} + 7785x^{10} + 208489x^{8} + 2822686x^{6} + 17977041x^{4} + 45020416x^{2} + 31719424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{7} - \beta_{5} + 1) q^{5} + (\beta_{9} - \beta_{5} - 1) q^{7} + (\beta_{12} - 13 \beta_{5} + \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{7} - \beta_{5} + 1) q^{5} + (\beta_{9} - \beta_{5} - 1) q^{7} + (\beta_{12} - 13 \beta_{5} + \beta_{2} - \beta_1) q^{9} + (\beta_{11} + \beta_{6} + \beta_{5} - 2 \beta_{2} + 1) q^{11} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 + 5) q^{13} + (\beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots + 2 \beta_1) q^{15}+ \cdots + ( - 3 \beta_{13} - \beta_{12} - 22 \beta_{10} - 24 \beta_{8} + 82 \beta_{7} + \cdots + 187) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{3} + 8 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{3} + 8 q^{5} - 10 q^{7} + 16 q^{11} + 72 q^{13} - 40 q^{17} - 60 q^{21} + 246 q^{23} + 572 q^{27} + 260 q^{29} + 566 q^{31} - 904 q^{33} - 804 q^{35} - 28 q^{37} + 476 q^{39} - 786 q^{41} - 604 q^{45} + 448 q^{47} - 380 q^{51} + 1612 q^{55} + 1004 q^{57} + 660 q^{61} - 2250 q^{63} - 796 q^{65} - 284 q^{67} - 2532 q^{69} + 326 q^{71} + 730 q^{73} - 864 q^{75} + 342 q^{79} - 950 q^{81} + 4148 q^{85} - 4516 q^{89} - 3112 q^{91} + 3356 q^{95} - 1194 q^{97} + 2876 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 142x^{12} + 7785x^{10} + 208489x^{8} + 2822686x^{6} + 17977041x^{4} + 45020416x^{2} + 31719424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 95 \nu^{12} - 10342 \nu^{10} - 385555 \nu^{8} - 5793399 \nu^{6} - 30510566 \nu^{4} - 44912651 \nu^{2} - 49770288 \nu - 72523264 ) / 49770288 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 95 \nu^{12} - 10342 \nu^{10} - 385555 \nu^{8} - 5793399 \nu^{6} - 30510566 \nu^{4} - 44912651 \nu^{2} + 49770288 \nu - 72523264 ) / 49770288 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 95 \nu^{12} + 10342 \nu^{10} + 385555 \nu^{8} + 5793399 \nu^{6} + 30510566 \nu^{4} + 94682939 \nu^{2} + 1067929024 ) / 24885144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1007 \nu^{12} + 137938 \nu^{10} + 7222375 \nu^{8} + 181591239 \nu^{6} + 2215243778 \nu^{4} + 11287190879 \nu^{2} + 14732140048 ) / 49770288 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12877 \nu^{13} - 1293494 \nu^{11} - 42001301 \nu^{9} - 513267093 \nu^{7} - 3719304454 \nu^{5} - 59654849245 \nu^{3} + \cdots - 326779846400 \nu ) / 280306262016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37343 \nu^{13} - 735680 \nu^{12} + 4283314 \nu^{11} - 90690688 \nu^{10} + 189294199 \nu^{9} - 4086250432 \nu^{8} + 4496161911 \nu^{7} + \cdots - 2300930351104 ) / 140153131008 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37343 \nu^{13} - 735680 \nu^{12} - 4283314 \nu^{11} - 90690688 \nu^{10} - 189294199 \nu^{9} - 4086250432 \nu^{8} - 4496161911 \nu^{7} + \cdots - 2300930351104 ) / 140153131008 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 92817 \nu^{13} + 461824 \nu^{12} - 13906542 \nu^{11} + 48728064 \nu^{10} - 800065401 \nu^{9} + 1710021632 \nu^{8} - 21974038905 \nu^{7} + \cdots - 1927881990144 ) / 93435420672 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 92817 \nu^{13} + 461824 \nu^{12} + 13906542 \nu^{11} + 48728064 \nu^{10} + 800065401 \nu^{9} + 1710021632 \nu^{8} + 21974038905 \nu^{7} + \cdots - 1927881990144 ) / 93435420672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42065 \nu^{13} - 219264 \nu^{12} + 4674286 \nu^{11} - 29660928 \nu^{10} + 176604985 \nu^{9} - 1512193152 \nu^{8} + 2499780921 \nu^{7} + \cdots - 1040149929984 ) / 25482387456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 42065 \nu^{13} + 219264 \nu^{12} + 4674286 \nu^{11} + 29660928 \nu^{10} + 176604985 \nu^{9} + 1512193152 \nu^{8} + 2499780921 \nu^{7} + \cdots + 1040149929984 ) / 25482387456 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23125 \nu^{13} + 2698022 \nu^{11} + 110951645 \nu^{9} + 1863590109 \nu^{7} + 8120784886 \nu^{5} - 78345744539 \nu^{3} + \cdots - 533954347264 \nu ) / 11679427584 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1072661 \nu^{13} + 152053158 \nu^{11} + 8248400029 \nu^{9} + 214633695901 \nu^{7} + 2716254447222 \nu^{5} + 14628546330917 \nu^{3} + \cdots + 21378896606976 \nu ) / 93435420672 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{12} + 3\beta_{11} + 3\beta_{10} - 3\beta_{7} + 3\beta_{6} + 78\beta_{5} - 65\beta_{2} + 65\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{11} - 6 \beta_{10} - 24 \beta_{9} - 24 \beta_{8} + 12 \beta_{7} + 12 \beta_{6} - 3 \beta_{4} - 75 \beta_{3} - 159 \beta_{2} - 159 \beta _1 + 2557 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 15 \beta_{13} + 213 \beta_{12} - 168 \beta_{11} - 168 \beta_{10} + 84 \beta_{9} - 84 \beta_{8} + 270 \beta_{7} - 270 \beta_{6} - 6141 \beta_{5} + 2453 \beta_{2} - 2453 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 597 \beta_{11} + 597 \beta_{10} + 1704 \beta_{9} + 1704 \beta_{8} - 1095 \beta_{7} - 1095 \beta_{6} + 285 \beta_{4} + 2801 \beta_{3} + 9044 \beta_{2} + 9044 \beta _1 - 95105 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 690 \beta_{13} - 5276 \beta_{12} + 4125 \beta_{11} + 4125 \beta_{10} - 3702 \beta_{9} + 3702 \beta_{8} - 7917 \beta_{7} + 7917 \beta_{6} + 173220 \beta_{5} - 50236 \beta_{2} + 50236 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 19455 \beta_{11} - 19455 \beta_{10} - 46698 \beta_{9} - 46698 \beta_{8} + 36309 \beta_{7} + 36309 \beta_{6} - 8607 \beta_{4} - 55578 \beta_{3} - 234267 \beta_{2} - 234267 \beta _1 + 1926935 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 89832 \beta_{13} + 513192 \beta_{12} - 395841 \beta_{11} - 395841 \beta_{10} + 471372 \beta_{9} - 471372 \beta_{8} + 814257 \beta_{7} - 814257 \beta_{6} - 17837898 \beta_{5} + \cdots - 4342823 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2175864 \beta_{11} + 2175864 \beta_{10} + 4713684 \beta_{9} + 4713684 \beta_{8} - 4195866 \beta_{7} - 4195866 \beta_{6} + 904089 \beta_{4} + 4668077 \beta_{3} + \cdots - 165378299 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5099955 \beta_{13} - 24777805 \beta_{12} + 18901530 \beta_{11} + 18901530 \beta_{10} - 26372424 \beta_{9} + 26372424 \beta_{8} - 39876168 \beta_{7} + 39876168 \beta_{6} + \cdots + 194596091 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 113436171 \beta_{11} - 113436171 \beta_{10} - 230308668 \beta_{9} - 230308668 \beta_{8} + 224979561 \beta_{7} + 224979561 \beta_{6} - 44976123 \beta_{4} + \cdots + 7376127727 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 134977842 \beta_{13} + 596319498 \beta_{12} - 450871848 \beta_{11} - 450871848 \beta_{10} + 691237980 \beta_{9} - 691237980 \beta_{8} + 958358088 \beta_{7} + \cdots - 4464323335 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
6.91794i
5.35029i
1.84168i
1.06483i
3.07347i
4.53742i
5.56386i
6.91794i
5.35029i
1.84168i
1.06483i
3.07347i
4.53742i
5.56386i
0 −6.91794 6.91794i 0 8.19992 + 8.19992i 0 −18.1123 + 18.1123i 0 68.7157i 0
81.2 0 −5.35029 5.35029i 0 −11.0962 11.0962i 0 19.2933 19.2933i 0 30.2512i 0
81.3 0 −1.84168 1.84168i 0 4.90158 + 4.90158i 0 6.14401 6.14401i 0 20.2164i 0
81.4 0 −1.06483 1.06483i 0 −0.550686 0.550686i 0 −11.3140 + 11.3140i 0 24.7323i 0
81.5 0 3.07347 + 3.07347i 0 −8.67626 8.67626i 0 9.02315 9.02315i 0 8.10760i 0
81.6 0 4.53742 + 4.53742i 0 14.5849 + 14.5849i 0 9.46047 9.46047i 0 14.1763i 0
81.7 0 5.56386 + 5.56386i 0 −3.36317 3.36317i 0 −19.4947 + 19.4947i 0 34.9130i 0
89.1 0 −6.91794 + 6.91794i 0 8.19992 8.19992i 0 −18.1123 18.1123i 0 68.7157i 0
89.2 0 −5.35029 + 5.35029i 0 −11.0962 + 11.0962i 0 19.2933 + 19.2933i 0 30.2512i 0
89.3 0 −1.84168 + 1.84168i 0 4.90158 4.90158i 0 6.14401 + 6.14401i 0 20.2164i 0
89.4 0 −1.06483 + 1.06483i 0 −0.550686 + 0.550686i 0 −11.3140 11.3140i 0 24.7323i 0
89.5 0 3.07347 3.07347i 0 −8.67626 + 8.67626i 0 9.02315 + 9.02315i 0 8.10760i 0
89.6 0 4.53742 4.53742i 0 14.5849 14.5849i 0 9.46047 + 9.46047i 0 14.1763i 0
89.7 0 5.56386 5.56386i 0 −3.36317 + 3.36317i 0 −19.4947 19.4947i 0 34.9130i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.k.a 14
4.b odd 2 1 272.4.o.g 14
17.c even 4 1 inner 136.4.k.a 14
17.d even 8 2 2312.4.a.m 14
68.f odd 4 1 272.4.o.g 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.k.a 14 1.a even 1 1 trivial
136.4.k.a 14 17.c even 4 1 inner
272.4.o.g 14 4.b odd 2 1
272.4.o.g 14 68.f odd 4 1
2312.4.a.m 14 17.d even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 4 T_{3}^{13} + 8 T_{3}^{12} - 252 T_{3}^{11} + 8148 T_{3}^{10} + 14304 T_{3}^{9} + 23784 T_{3}^{8} - 796464 T_{3}^{7} + 16150080 T_{3}^{6} + 1753536 T_{3}^{5} - 12967392 T_{3}^{4} + \cdots + 4060086272 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 4 T^{13} + \cdots + 4060086272 \) Copy content Toggle raw display
$5$ \( T^{14} - 8 T^{13} + \cdots + 1398380605952 \) Copy content Toggle raw display
$7$ \( T^{14} + 10 T^{13} + \cdots + 20\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( T^{14} - 16 T^{13} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T^{7} - 36 T^{6} - 8876 T^{5} + \cdots - 2332419584)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + 40 T^{13} + \cdots + 69\!\cdots\!17 \) Copy content Toggle raw display
$19$ \( T^{14} + 67636 T^{12} + \cdots + 90\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{14} - 246 T^{13} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} - 260 T^{13} + \cdots + 54\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{14} - 566 T^{13} + \cdots + 44\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{14} + 28 T^{13} + \cdots + 65\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{14} + 786 T^{13} + \cdots + 45\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{14} + 427000 T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{7} - 224 T^{6} + \cdots + 65\!\cdots\!32)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + 524980 T^{12} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{14} + 873928 T^{12} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{14} - 660 T^{13} + \cdots + 39\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( (T^{7} + 142 T^{6} + \cdots - 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} - 326 T^{13} + \cdots + 43\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{14} - 730 T^{13} + \cdots + 31\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{14} - 342 T^{13} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{14} + 4438952 T^{12} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{7} + 2258 T^{6} + \cdots + 98\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 1194 T^{13} + \cdots + 53\!\cdots\!92 \) Copy content Toggle raw display
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