Properties

Label 936.2.m.h
Level $936$
Weight $2$
Character orbit 936.m
Analytic conductor $7.474$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q -\beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} -\beta_{5} q^{7} + ( -\beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} ) q^{8} +O(q^{10})\) \( q -\beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} -\beta_{5} q^{7} + ( -\beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} ) q^{8} + ( 1 - \beta_{2} + \beta_{4} ) q^{10} + ( -\beta_{10} - \beta_{12} - \beta_{14} ) q^{11} + ( \beta_{2} + \beta_{4} + \beta_{15} ) q^{13} + ( -\beta_{1} - \beta_{6} + \beta_{11} ) q^{14} -2 \beta_{2} q^{16} + ( -2 \beta_{1} + \beta_{7} - \beta_{11} ) q^{17} + ( \beta_{13} - \beta_{15} ) q^{19} + ( -2 \beta_{9} - 2 \beta_{12} ) q^{20} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{22} + ( -2 \beta_{6} + \beta_{11} ) q^{23} + ( -\beta_{3} - \beta_{4} ) q^{25} + ( \beta_{1} + 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{26} + ( -\beta_{8} + 2 \beta_{13} - \beta_{15} ) q^{28} -\beta_{7} q^{29} + ( -\beta_{5} + 2 \beta_{8} ) q^{31} -4 \beta_{9} q^{32} + ( \beta_{5} - \beta_{8} - \beta_{13} - 2 \beta_{15} ) q^{34} + ( -2 \beta_{7} - \beta_{11} ) q^{35} + ( -2 \beta_{13} - 2 \beta_{15} ) q^{37} + ( -\beta_{1} - \beta_{6} - \beta_{11} ) q^{38} + ( 6 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{40} + ( -3 \beta_{9} + 3 \beta_{10} - 3 \beta_{12} ) q^{41} + ( 1 - \beta_{3} + \beta_{4} ) q^{43} + ( \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + \beta_{14} ) q^{44} + ( \beta_{5} + 3 \beta_{13} + \beta_{15} ) q^{46} + ( -3 \beta_{9} - 4 \beta_{10} + 4 \beta_{12} ) q^{47} + ( -8 + \beta_{3} + \beta_{4} ) q^{49} + ( \beta_{9} + \beta_{12} - \beta_{14} ) q^{50} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{13} ) q^{52} + ( 2 \beta_{7} - \beta_{11} ) q^{53} + ( -5 + 3 \beta_{3} + 3 \beta_{4} ) q^{55} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{56} + ( -\beta_{8} + \beta_{15} ) q^{58} + ( -3 \beta_{10} - 3 \beta_{12} + \beta_{14} ) q^{59} + ( -3 + 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{61} + ( -3 \beta_{1} - \beta_{6} + 4 \beta_{7} - \beta_{11} ) q^{62} + ( 4 - 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{64} + ( -2 \beta_{1} - 2 \beta_{6} + \beta_{7} + 3 \beta_{9} + \beta_{10} - \beta_{12} ) q^{65} + ( -3 \beta_{13} + \beta_{15} ) q^{67} + ( 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{11} ) q^{68} + ( \beta_{5} - 2 \beta_{8} - \beta_{13} + 3 \beta_{15} ) q^{70} + ( -5 \beta_{9} - 4 \beta_{10} + 4 \beta_{12} ) q^{71} + ( -2 \beta_{5} + 2 \beta_{8} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{6} - 2 \beta_{11} ) q^{74} + ( 2 \beta_{5} - \beta_{8} + \beta_{15} ) q^{76} + ( \beta_{7} + 3 \beta_{11} ) q^{77} + ( 2 - 4 \beta_{3} - 4 \beta_{4} ) q^{79} + ( -2 \beta_{9} - 6 \beta_{10} + 2 \beta_{12} - 2 \beta_{14} ) q^{80} + ( 9 - 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} ) q^{82} + ( \beta_{10} + \beta_{12} + 3 \beta_{14} ) q^{83} + ( -4 \beta_{13} + 2 \beta_{15} ) q^{85} + ( \beta_{9} + \beta_{10} - 3 \beta_{12} - \beta_{14} ) q^{86} + ( -6 + 4 \beta_{3} + 2 \beta_{4} ) q^{88} + ( 7 \beta_{9} - \beta_{10} + \beta_{12} ) q^{89} + ( -5 + 6 \beta_{2} + 5 \beta_{3} + \beta_{4} - 3 \beta_{13} - \beta_{15} ) q^{91} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{92} + ( -5 - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{94} + ( 4 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} + \beta_{11} ) q^{95} + 2 \beta_{8} q^{97} + ( -\beta_{9} + 8 \beta_{10} - \beta_{12} + \beta_{14} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + O(q^{10}) \) \( 16 q + 8 q^{4} - 16 q^{16} + 40 q^{22} + 80 q^{40} - 128 q^{49} + 40 q^{52} - 80 q^{55} + 32 q^{64} + 32 q^{79} + 96 q^{82} - 80 q^{88} - 72 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 310 \nu^{14} - 2816 \nu^{12} - 9222 \nu^{10} + 18376 \nu^{8} + 130292 \nu^{6} - 53906 \nu^{4} + 290050 \nu^{2} + 217000 \)\()/176275\)
\(\beta_{2}\)\(=\)\((\)\( 11992 \nu^{14} + 9614 \nu^{12} - 190612 \nu^{10} - 554504 \nu^{8} + 1001932 \nu^{6} + 2217000 \nu^{4} + 3496000 \nu^{2} + 5574000 \)\()/881375\)
\(\beta_{3}\)\(=\)\((\)\( 16948 \nu^{14} + 5486 \nu^{12} - 299188 \nu^{10} - 779596 \nu^{8} + 1911868 \nu^{6} + 4019270 \nu^{4} + 6074000 \nu^{2} + 6351000 \)\()/881375\)
\(\beta_{4}\)\(=\)\((\)\( -17344 \nu^{14} - 7298 \nu^{12} + 306684 \nu^{10} + 805078 \nu^{8} - 1876924 \nu^{6} - 3981500 \nu^{4} - 6068600 \nu^{2} - 8342875 \)\()/881375\)
\(\beta_{5}\)\(=\)\((\)\( -10661 \nu^{15} - 14207 \nu^{13} + 225956 \nu^{11} + 888177 \nu^{9} - 521116 \nu^{7} - 5727895 \nu^{5} - 13225300 \nu^{3} - 26244000 \nu \)\()/4406875\)
\(\beta_{6}\)\(=\)\((\)\( -25832 \nu^{14} - 21274 \nu^{12} + 450392 \nu^{10} + 1381764 \nu^{8} - 2270262 \nu^{6} - 7392230 \nu^{4} - 12703750 \nu^{2} - 13499250 \)\()/881375\)
\(\beta_{7}\)\(=\)\((\)\( -28806 \nu^{14} - 29832 \nu^{12} + 448156 \nu^{10} + 1496702 \nu^{8} - 2047066 \nu^{6} - 6175530 \nu^{4} - 12354350 \nu^{2} - 13465750 \)\()/881375\)
\(\beta_{8}\)\(=\)\((\)\( 15392 \nu^{15} + 52754 \nu^{13} - 145257 \nu^{11} - 987344 \nu^{9} - 868348 \nu^{7} + 1700315 \nu^{5} + 6288975 \nu^{3} + 15325500 \nu \)\()/4406875\)
\(\beta_{9}\)\(=\)\((\)\( -1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu \)\()/400625\)
\(\beta_{10}\)\(=\)\((\)\( -22194 \nu^{15} - 11523 \nu^{13} + 339409 \nu^{11} + 1024903 \nu^{9} - 1669899 \nu^{7} - 3769000 \nu^{5} - 11127550 \nu^{3} - 9814875 \nu \)\()/4406875\)
\(\beta_{11}\)\(=\)\((\)\( -44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} - 11081230 \nu^{4} - 21249600 \nu^{2} - 22476000 \)\()/881375\)
\(\beta_{12}\)\(=\)\((\)\( -46447 \nu^{15} - 41694 \nu^{13} + 802127 \nu^{11} + 2508384 \nu^{9} - 4035597 \nu^{7} - 13232745 \nu^{5} - 19604750 \nu^{3} - 21119125 \nu \)\()/4406875\)
\(\beta_{13}\)\(=\)\((\)\( 78923 \nu^{15} + 34611 \nu^{13} - 1339113 \nu^{11} - 3553821 \nu^{9} + 8128668 \nu^{7} + 15701920 \nu^{5} + 26400825 \nu^{3} + 32586750 \nu \)\()/4406875\)
\(\beta_{14}\)\(=\)\((\)\( -108813 \nu^{15} - 85331 \nu^{13} + 1829673 \nu^{11} + 5646091 \nu^{9} - 9386828 \nu^{7} - 27713410 \nu^{5} - 49911275 \nu^{3} - 50144500 \nu \)\()/4406875\)
\(\beta_{15}\)\(=\)\((\)\( -128244 \nu^{15} - 47208 \nu^{13} + 2180589 \nu^{11} + 5658588 \nu^{9} - 13459254 \nu^{7} - 25923185 \nu^{5} - 43314975 \nu^{3} - 53041500 \nu \)\()/4406875\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - \beta_{13} + 2 \beta_{12} - 2 \beta_{9} - \beta_{5}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{14} - \beta_{13} + 5 \beta_{12} + 7 \beta_{10} + 5 \beta_{9} + \beta_{8} + \beta_{5}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} - 2 \beta_{6} + 8 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 8\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-7 \beta_{15} - 12 \beta_{13} - 3 \beta_{9}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(11 \beta_{11} - 11 \beta_{7} - 6 \beta_{6} + 11 \beta_{3} - 15 \beta_{2} + 5 \beta_{1} + 30\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-5 \beta_{15} - 55 \beta_{14} - 4 \beta_{13} + 127 \beta_{12} + 55 \beta_{10} - 55 \beta_{9} - 9 \beta_{8} - 4 \beta_{5}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(32 \beta_{11} - 52 \beta_{6} + 53 \beta_{4} + 18 \beta_{3} + 53 \beta_{2} + 32 \beta_{1} + 17\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-137 \beta_{15} - 57 \beta_{14} - 224 \beta_{13} - 57 \beta_{12} + 247 \beta_{10} + 247 \beta_{9} + 87 \beta_{8} + 50 \beta_{5}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(21 \beta_{11} - 36 \beta_{7} + 275 \beta_{4} + 275 \beta_{3} + 607\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-384 \beta_{15} - 341 \beta_{14} - 623 \beta_{13} + 1421 \beta_{12} - 341 \beta_{10} - 1421 \beta_{9} - 239 \beta_{8} - 145 \beta_{5}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(752 \beta_{11} - 462 \beta_{7} - 752 \beta_{6} - 110 \beta_{4} - 177 \beta_{2} + 462 \beta_{1} + 110\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-395 \beta_{15} - 2315 \beta_{14} - 642 \beta_{13} + 2315 \beta_{12} + 5199 \beta_{10} + 2315 \beta_{9} + 1037 \beta_{8} + 642 \beta_{5}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(639 \beta_{7} - 639 \beta_{6} + 5431 \beta_{4} + 3360 \beta_{3} + 3360 \beta_{2} + 400 \beta_{1} + 5431\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-4399 \beta_{15} - 7109 \beta_{13} + 2475 \beta_{12} - 2475 \beta_{10} - 3996 \beta_{9}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−0.556839 1.81878i
1.90184 0.0324487i
1.90184 + 0.0324487i
−0.556839 + 1.81878i
0.752864 0.902863i
0.0783900 + 1.17295i
0.0783900 1.17295i
0.752864 + 0.902863i
−0.752864 0.902863i
−0.0783900 + 1.17295i
−0.0783900 1.17295i
−0.752864 + 0.902863i
0.556839 1.81878i
−1.90184 0.0324487i
−1.90184 + 0.0324487i
0.556839 + 1.81878i
−1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.2 −1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.3 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.4 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.5 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.6 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.7 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.8 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.9 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.10 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.11 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.12 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.13 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.14 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.15 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
181.16 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.m.h 16
3.b odd 2 1 inner 936.2.m.h 16
4.b odd 2 1 3744.2.m.h 16
8.b even 2 1 inner 936.2.m.h 16
8.d odd 2 1 3744.2.m.h 16
12.b even 2 1 3744.2.m.h 16
13.b even 2 1 inner 936.2.m.h 16
24.f even 2 1 3744.2.m.h 16
24.h odd 2 1 inner 936.2.m.h 16
39.d odd 2 1 inner 936.2.m.h 16
52.b odd 2 1 3744.2.m.h 16
104.e even 2 1 inner 936.2.m.h 16
104.h odd 2 1 3744.2.m.h 16
156.h even 2 1 3744.2.m.h 16
312.b odd 2 1 inner 936.2.m.h 16
312.h even 2 1 3744.2.m.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.h 16 1.a even 1 1 trivial
936.2.m.h 16 3.b odd 2 1 inner
936.2.m.h 16 8.b even 2 1 inner
936.2.m.h 16 13.b even 2 1 inner
936.2.m.h 16 24.h odd 2 1 inner
936.2.m.h 16 39.d odd 2 1 inner
936.2.m.h 16 104.e even 2 1 inner
936.2.m.h 16 312.b odd 2 1 inner
3744.2.m.h 16 4.b odd 2 1
3744.2.m.h 16 8.d odd 2 1
3744.2.m.h 16 12.b even 2 1
3744.2.m.h 16 24.f even 2 1
3744.2.m.h 16 52.b odd 2 1
3744.2.m.h 16 104.h odd 2 1
3744.2.m.h 16 156.h even 2 1
3744.2.m.h 16 312.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 10 T_{5}^{2} + 20 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 20 - 10 T^{2} + T^{4} )^{4} \)
$7$ \( ( 220 + 30 T^{2} + T^{4} )^{4} \)
$11$ \( ( 20 - 20 T^{2} + T^{4} )^{4} \)
$13$ \( ( 28561 - 2028 T^{2} + 54 T^{4} - 12 T^{6} + T^{8} )^{2} \)
$17$ \( ( 880 - 60 T^{2} + T^{4} )^{4} \)
$19$ \( ( 44 - 34 T^{2} + T^{4} )^{4} \)
$23$ \( ( 880 - 80 T^{2} + T^{4} )^{4} \)
$29$ \( ( 176 + 28 T^{2} + T^{4} )^{4} \)
$31$ \( ( 5500 + 150 T^{2} + T^{4} )^{4} \)
$37$ \( ( 704 - 104 T^{2} + T^{4} )^{4} \)
$41$ \( ( 324 + 126 T^{2} + T^{4} )^{4} \)
$43$ \( ( 80 + 20 T^{2} + T^{4} )^{4} \)
$47$ \( ( 1444 + 84 T^{2} + T^{4} )^{4} \)
$53$ \( ( 176 + 128 T^{2} + T^{4} )^{4} \)
$59$ \( ( 500 - 100 T^{2} + T^{4} )^{4} \)
$61$ \( ( 6480 + 180 T^{2} + T^{4} )^{4} \)
$67$ \( ( 5324 - 154 T^{2} + T^{4} )^{4} \)
$71$ \( ( 484 + 116 T^{2} + T^{4} )^{4} \)
$73$ \( ( 3520 + 200 T^{2} + T^{4} )^{4} \)
$79$ \( ( -76 - 4 T + T^{2} )^{8} \)
$83$ \( ( 500 - 100 T^{2} + T^{4} )^{4} \)
$89$ \( ( 12100 + 230 T^{2} + T^{4} )^{4} \)
$97$ \( ( 3520 + 160 T^{2} + T^{4} )^{4} \)
show more
show less