Properties

Label 156.2.h.b
Level $156$
Weight $2$
Character orbit 156.h
Analytic conductor $1.246$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 43x^{12} + 517x^{8} + 1804x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + \beta_{6} q^{3} - \beta_{12} q^{4} - \beta_{4} q^{5} + ( - \beta_{11} + \beta_{5} - \beta_1) q^{6} + ( - \beta_{13} + \beta_{11} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8} + (\beta_{15} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + \beta_{6} q^{3} - \beta_{12} q^{4} - \beta_{4} q^{5} + ( - \beta_{11} + \beta_{5} - \beta_1) q^{6} + ( - \beta_{13} + \beta_{11} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8} + (\beta_{15} + 1) q^{9} + ( - \beta_{14} - \beta_{7} - \beta_{6}) q^{10} - \beta_1 q^{11} + ( - \beta_{14} - \beta_{10} + 1) q^{12} + (\beta_{14} + \beta_{12} + \beta_{2} - 1) q^{13} + (\beta_{15} - \beta_{14} - \beta_{12} + \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6}) q^{14} + (\beta_{13} + \beta_{5}) q^{15} + (\beta_{12} - 2 \beta_{7} - 2 \beta_{6} - 2) q^{16} + ( - \beta_{15} + \beta_{14} + \beta_{10} + \beta_{8}) q^{17} + (\beta_{13} - \beta_{9} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{18} + ( - \beta_{13} - \beta_{11} + \beta_{9} + \beta_{4} + \beta_{3} - \beta_1) q^{19} + (\beta_{5} + \beta_{4} + \beta_1) q^{20} + ( - \beta_{11} + \beta_{9} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{21} + ( - \beta_{14} - \beta_{12} + \beta_{7} + \beta_{6}) q^{22} + (\beta_{14} + \beta_{12} - 2 \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6}) q^{23} + (\beta_{13} - 2 \beta_{9} + 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{24} + (\beta_{14} + \beta_{12} - 3) q^{25} + ( - \beta_{15} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - \beta_1) q^{26} + ( - \beta_{12} + \beta_{10} - \beta_{8} + 3 \beta_{7} + 2 \beta_{6}) q^{27} + (2 \beta_{11} - \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_1) q^{28} + ( - \beta_{15} + \beta_{12} - \beta_{10} - \beta_{8}) q^{29} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{8} + 1) q^{30} + (\beta_{13} + \beta_{11} - \beta_{9} - \beta_{4} - \beta_{3} + \beta_1) q^{31} + (2 \beta_{9} - \beta_{5} + 3 \beta_{4} + \beta_1) q^{32} + (\beta_{9} - \beta_{5} - \beta_{2}) q^{33} + ( - 2 \beta_{13} + 2 \beta_{9} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_1) q^{34} + (\beta_{7} - \beta_{6}) q^{35} + (\beta_{14} - \beta_{12} - \beta_{10} + 2 \beta_{8} + 2 \beta_{6} - 1) q^{36} + (2 \beta_{11} + \beta_{9} - \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{37} + (\beta_{15} - \beta_{14} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6}) q^{38} + ( - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{39} + (2 \beta_{14} + \beta_{12} + 2) q^{40} + ( - 3 \beta_{9} + 3 \beta_{5} + \beta_{4}) q^{41} + ( - \beta_{15} + 2 \beta_{14} + 2 \beta_{12} - \beta_{8} + 3 \beta_{7} + \beta_{6} - 3) q^{42} + (\beta_{14} - \beta_{12}) q^{43} + (2 \beta_{5} - 2 \beta_{4}) q^{44} + (\beta_{11} + \beta_{9} - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{45} + (2 \beta_{13} - 2 \beta_{11} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{46} + (3 \beta_{9} + 3 \beta_{5} - \beta_1) q^{47} + ( - 2 \beta_{15} + \beta_{14} + \beta_{10} - 2 \beta_{6} + 3) q^{48} + ( - 3 \beta_{14} - 3 \beta_{12} + 3) q^{49} + (3 \beta_{9} - \beta_{5} + \beta_{4} - \beta_1) q^{50} + ( - 3 \beta_{14} + 2 \beta_{12} + \beta_{10} - \beta_{8} - 3 \beta_{7} - 3 \beta_{6}) q^{51} + ( - 2 \beta_{13} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{52} + ( - \beta_{13} + \beta_{11} - 4 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{54} + ( - \beta_{14} + \beta_{12} - \beta_{7} - \beta_{6}) q^{55} + (2 \beta_{15} - \beta_{12} + 2 \beta_{8}) q^{56} + ( - 2 \beta_{9} + 2 \beta_{5} + 3 \beta_{4} - \beta_{2}) q^{57} + ( - 2 \beta_{9} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{58} + ( - 2 \beta_{9} - 2 \beta_{5} - 3 \beta_1) q^{59} + ( - \beta_{13} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{60} + ( - 2 \beta_{14} - 2 \beta_{12}) q^{61} + ( - \beta_{15} + \beta_{14} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6}) q^{62} + ( - \beta_{13} - 3 \beta_{9} - 4 \beta_{5} + 3 \beta_1) q^{63} + (4 \beta_{14} + 3 \beta_{12} + 2 \beta_{7} + 2 \beta_{6} - 2) q^{64} + (\beta_{15} - \beta_{14} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} - \beta_{4}) q^{65} + (\beta_{15} + \beta_{12} - \beta_{10} + \beta_{8} - 2) q^{66} + (3 \beta_{13} - \beta_{11} - \beta_{9} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{67} + (2 \beta_{15} - 2 \beta_{14} - \beta_{12} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6}) q^{68} + (\beta_{15} - 4 \beta_{14} - 5 \beta_{12} + \beta_{10} + \beta_{8} + 2) q^{69} + (2 \beta_{11} - \beta_{5} - \beta_{4} + \beta_1) q^{70} + ( - 3 \beta_{9} - 3 \beta_{5} + 5 \beta_1) q^{71} + (\beta_{13} - 2 \beta_{11} + 2 \beta_{9} + \beta_{5} + 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{72} + ( - 4 \beta_{11} - 2 \beta_{9} + 2 \beta_{5} - 4 \beta_{3} + 2 \beta_{2}) q^{73} + (2 \beta_{15} - \beta_{12} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6}) q^{74} + ( - \beta_{12} + \beta_{10} - \beta_{8} - 3 \beta_{6}) q^{75} + (2 \beta_{13} - 2 \beta_{11} - 2 \beta_{9} + 2 \beta_{5} - 2 \beta_{3}) q^{76} + ( - \beta_{15} + \beta_{12} - \beta_{10} - \beta_{8}) q^{77} + (2 \beta_{14} - \beta_{13} + 3 \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{7} - \beta_{6} - 3 \beta_{5} + \cdots + 1) q^{78}+ \cdots + (2 \beta_{11} - 2 \beta_{9} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} + 20 q^{9} - 4 q^{10} + 10 q^{12} - 8 q^{13} - 28 q^{16} - 8 q^{22} - 40 q^{25} + 18 q^{30} - 22 q^{36} + 44 q^{40} - 34 q^{42} + 46 q^{48} + 24 q^{49} - 32 q^{52} - 16 q^{61} - 4 q^{64} - 28 q^{66} + 34 q^{78} - 60 q^{81} + 88 q^{82} + 56 q^{88} - 22 q^{90} + 100 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 43x^{12} + 517x^{8} + 1804x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -383\nu^{14} - 16513\nu^{10} - 200087\nu^{6} - 728592\nu^{2} ) / 34268 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 85\nu^{14} + 3651\nu^{10} + 43277\nu^{6} + 142248\nu^{2} ) / 5272 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2069 \nu^{15} - 3460 \nu^{14} + 62 \nu^{13} + 89831 \nu^{11} - 150788 \nu^{10} - 190 \nu^{9} + 1097977 \nu^{7} - 1871100 \nu^{6} - 65314 \nu^{5} + 3861388 \nu^{3} + \cdots - 363776 \nu ) / 274144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 667 \nu^{15} - 185 \nu^{13} + 28355 \nu^{11} - 7171 \nu^{9} + 332573 \nu^{7} - 64885 \nu^{5} + 1111158 \nu^{3} - 105076 \nu ) / 68536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2199 \nu^{15} + 797 \nu^{14} + 229 \nu^{13} - 94407 \nu^{11} + 32931 \nu^{10} + 9247 \nu^{9} - 1132921 \nu^{7} + 367517 \nu^{6} + 102545 \nu^{5} - 3956990 \nu^{3} + \cdots + 373092 \nu ) / 137072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2199 \nu^{15} + 229 \nu^{13} - 220 \nu^{12} + 94407 \nu^{11} + 9247 \nu^{9} - 10380 \nu^{8} + 1132921 \nu^{7} + 102545 \nu^{5} - 119764 \nu^{4} + 3956990 \nu^{3} + \cdots - 174968 ) / 137072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2199 \nu^{15} + 229 \nu^{13} + 220 \nu^{12} + 94407 \nu^{11} + 9247 \nu^{9} + 10380 \nu^{8} + 1132921 \nu^{7} + 102545 \nu^{5} + 119764 \nu^{4} + 3956990 \nu^{3} + \cdots + 174968 ) / 137072 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4873 \nu^{15} + 644 \nu^{13} + 1604 \nu^{12} - 212783 \nu^{11} + 17924 \nu^{9} + 56988 \nu^{8} - 2628785 \nu^{7} + 9148 \nu^{5} + 388452 \nu^{4} - 9498920 \nu^{3} + \cdots - 161088 ) / 274144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2199 \nu^{15} + 797 \nu^{14} - 229 \nu^{13} + 94407 \nu^{11} + 32931 \nu^{10} - 9247 \nu^{9} + 1132921 \nu^{7} + 367517 \nu^{6} - 102545 \nu^{5} + 3956990 \nu^{3} + \cdots - 373092 \nu ) / 137072 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4873 \nu^{15} + 644 \nu^{13} - 1604 \nu^{12} - 212783 \nu^{11} + 17924 \nu^{9} - 56988 \nu^{8} - 2628785 \nu^{7} + 9148 \nu^{5} - 388452 \nu^{4} - 9498920 \nu^{3} + \cdots + 161088 ) / 274144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6467 \nu^{15} + 1198 \nu^{14} + 396 \nu^{13} - 278645 \nu^{11} + 47178 \nu^{10} + 18684 \nu^{9} - 3363819 \nu^{7} + 464630 \nu^{6} + 270404 \nu^{5} + \cdots + 1109960 \nu ) / 274144 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3533 \nu^{15} + 599 \nu^{13} - 520 \nu^{12} + 151117 \nu^{11} + 23589 \nu^{9} - 18304 \nu^{8} + 1798067 \nu^{7} + 232315 \nu^{5} - 139776 \nu^{4} + 6179306 \nu^{3} + \cdots - 108264 ) / 137072 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3601 \nu^{15} + 62 \nu^{13} + 155883 \nu^{11} - 190 \nu^{9} + 1898325 \nu^{7} - 65314 \nu^{5} + 6775756 \nu^{3} - 637920 \nu ) / 137072 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3533 \nu^{15} - 599 \nu^{13} - 520 \nu^{12} - 151117 \nu^{11} - 23589 \nu^{9} - 18304 \nu^{8} - 1798067 \nu^{7} - 232315 \nu^{5} - 139776 \nu^{4} - 6179306 \nu^{3} + \cdots - 108264 ) / 137072 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5602 \nu^{15} - 537 \nu^{13} - 520 \nu^{12} - 240948 \nu^{11} - 23779 \nu^{9} - 18304 \nu^{8} - 2896044 \nu^{7} - 297629 \nu^{5} - 139776 \nu^{4} - 10040694 \nu^{3} + \cdots - 108264 ) / 137072 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} - \beta_{9} - 3\beta_{5} + 2\beta_{3} - \beta_{2} - 5\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{15} - 7 \beta_{14} + 4 \beta_{13} + \beta_{12} + 4 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 4 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{14} - 15\beta_{12} + 10\beta_{10} - 10\beta_{8} + \beta_{7} - \beta_{6} - 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 49 \beta_{15} + 67 \beta_{14} + 38 \beta_{13} - 18 \beta_{12} + 30 \beta_{11} + 19 \beta_{10} - 28 \beta_{9} + 19 \beta_{8} + 27 \beta_{7} + 27 \beta_{6} - 2 \beta_{5} + 36 \beta_{4} - 30 \beta_{3} + 30 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -24\beta_{11} + 14\beta_{9} + 38\beta_{5} - 24\beta_{3} - 4\beta_{2} + 39\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 213 \beta_{15} + 325 \beta_{14} - 190 \beta_{13} - 112 \beta_{12} - 118 \beta_{11} + 95 \beta_{10} + 168 \beta_{9} + 95 \beta_{8} + 185 \beta_{7} + 185 \beta_{6} - 50 \beta_{5} - 224 \beta_{4} + 118 \beta_{3} - 118 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 389\beta_{14} + 389\beta_{12} - 230\beta_{10} + 230\beta_{8} + 81\beta_{7} - 81\beta_{6} + 678 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 486 \beta_{15} - 797 \beta_{14} - 484 \beta_{13} + 311 \beta_{12} - 244 \beta_{11} - 242 \beta_{10} + 470 \beta_{9} - 242 \beta_{8} - 539 \beta_{7} - 539 \beta_{6} - 226 \beta_{5} - 622 \beta_{4} + 244 \beta_{3} - 244 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1110\beta_{11} - 883\beta_{9} - 1993\beta_{5} + 1110\beta_{3} + 537\beta_{2} - 1541\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4591 \beta_{15} - 7885 \beta_{14} + 4954 \beta_{13} + 3294 \beta_{12} + 2114 \beta_{11} - 2477 \beta_{10} - 5060 \beta_{9} - 2477 \beta_{8} - 5877 \beta_{7} - 5877 \beta_{6} + 2946 \beta_{5} + 6588 \beta_{4} + \cdots + 2114 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -2547\beta_{14} - 2547\beta_{12} + 1352\beta_{10} - 1352\beta_{8} - 780\beta_{7} + 780\beta_{6} - 3621 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22195 \beta_{15} + 39251 \beta_{14} + 25330 \beta_{13} - 17056 \beta_{12} + 9530 \beta_{11} + 12665 \beta_{10} - 26616 \beta_{9} + 12665 \beta_{8} + 31007 \beta_{7} + 31007 \beta_{6} + 17086 \beta_{5} + \cdots + 9530 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -26586\beta_{11} + 25345\beta_{9} + 51931\beta_{5} - 26586\beta_{3} - 17071\beta_{2} + 34845\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 54478 \beta_{15} + 98135 \beta_{14} - 64596 \beta_{13} - 43657 \beta_{12} - 22180 \beta_{11} + 32298 \beta_{10} + 69002 \beta_{9} + 32298 \beta_{8} + 80361 \beta_{7} + 80361 \beta_{6} + \cdots - 22180 \beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
−0.217136 0.217136i
−1.29309 1.29309i
−0.217136 + 0.217136i
−1.29309 + 1.29309i
1.58894 1.58894i
−1.12073 + 1.12073i
1.58894 + 1.58894i
−1.12073 1.12073i
−1.58894 + 1.58894i
1.12073 1.12073i
−1.58894 1.58894i
1.12073 + 1.12073i
0.217136 + 0.217136i
1.29309 + 1.29309i
0.217136 0.217136i
1.29309 1.29309i
−1.17915 0.780776i −1.26870 1.17915i 0.780776 + 1.84130i −0.662153 0.575339 + 2.38096i −3.83206 0.516994 2.78078i 0.219224 + 2.99198i 0.780776 + 0.516994i
155.2 −1.17915 0.780776i 1.26870 1.17915i 0.780776 + 1.84130i −0.662153 −2.41664 + 0.399813i 3.83206 0.516994 2.78078i 0.219224 2.99198i 0.780776 + 0.516994i
155.3 −1.17915 + 0.780776i −1.26870 + 1.17915i 0.780776 1.84130i −0.662153 0.575339 2.38096i −3.83206 0.516994 + 2.78078i 0.219224 2.99198i 0.780776 0.516994i
155.4 −1.17915 + 0.780776i 1.26870 + 1.17915i 0.780776 1.84130i −0.662153 −2.41664 0.399813i 3.83206 0.516994 + 2.78078i 0.219224 + 2.99198i 0.780776 0.516994i
155.5 −0.599676 1.28078i −1.62493 + 0.599676i −1.28078 + 1.53610i 2.13578 1.74248 + 1.72156i 1.52162 2.73546 + 0.719224i 2.28078 1.94886i −1.28078 2.73546i
155.6 −0.599676 1.28078i 1.62493 + 0.599676i −1.28078 + 1.53610i 2.13578 −0.206379 2.44078i −1.52162 2.73546 + 0.719224i 2.28078 + 1.94886i −1.28078 2.73546i
155.7 −0.599676 + 1.28078i −1.62493 0.599676i −1.28078 1.53610i 2.13578 1.74248 1.72156i 1.52162 2.73546 0.719224i 2.28078 + 1.94886i −1.28078 + 2.73546i
155.8 −0.599676 + 1.28078i 1.62493 0.599676i −1.28078 1.53610i 2.13578 −0.206379 + 2.44078i −1.52162 2.73546 0.719224i 2.28078 1.94886i −1.28078 + 2.73546i
155.9 0.599676 1.28078i −1.62493 0.599676i −1.28078 1.53610i −2.13578 −1.74248 + 1.72156i −1.52162 −2.73546 + 0.719224i 2.28078 + 1.94886i −1.28078 + 2.73546i
155.10 0.599676 1.28078i 1.62493 0.599676i −1.28078 1.53610i −2.13578 0.206379 2.44078i 1.52162 −2.73546 + 0.719224i 2.28078 1.94886i −1.28078 + 2.73546i
155.11 0.599676 + 1.28078i −1.62493 + 0.599676i −1.28078 + 1.53610i −2.13578 −1.74248 1.72156i −1.52162 −2.73546 0.719224i 2.28078 1.94886i −1.28078 2.73546i
155.12 0.599676 + 1.28078i 1.62493 + 0.599676i −1.28078 + 1.53610i −2.13578 0.206379 + 2.44078i 1.52162 −2.73546 0.719224i 2.28078 + 1.94886i −1.28078 2.73546i
155.13 1.17915 0.780776i −1.26870 + 1.17915i 0.780776 1.84130i 0.662153 −0.575339 + 2.38096i 3.83206 −0.516994 2.78078i 0.219224 2.99198i 0.780776 0.516994i
155.14 1.17915 0.780776i 1.26870 + 1.17915i 0.780776 1.84130i 0.662153 2.41664 + 0.399813i −3.83206 −0.516994 2.78078i 0.219224 + 2.99198i 0.780776 0.516994i
155.15 1.17915 + 0.780776i −1.26870 1.17915i 0.780776 + 1.84130i 0.662153 −0.575339 2.38096i 3.83206 −0.516994 + 2.78078i 0.219224 + 2.99198i 0.780776 + 0.516994i
155.16 1.17915 + 0.780776i 1.26870 1.17915i 0.780776 + 1.84130i 0.662153 2.41664 0.399813i −3.83206 −0.516994 + 2.78078i 0.219224 2.99198i 0.780776 + 0.516994i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner
156.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.h.b 16
3.b odd 2 1 inner 156.2.h.b 16
4.b odd 2 1 inner 156.2.h.b 16
12.b even 2 1 inner 156.2.h.b 16
13.b even 2 1 inner 156.2.h.b 16
39.d odd 2 1 inner 156.2.h.b 16
52.b odd 2 1 inner 156.2.h.b 16
156.h even 2 1 inner 156.2.h.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.h.b 16 1.a even 1 1 trivial
156.2.h.b 16 3.b odd 2 1 inner
156.2.h.b 16 4.b odd 2 1 inner
156.2.h.b 16 12.b even 2 1 inner
156.2.h.b 16 13.b even 2 1 inner
156.2.h.b 16 39.d odd 2 1 inner
156.2.h.b 16 52.b odd 2 1 inner
156.2.h.b 16 156.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 5T_{5}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + T^{6} + 4 T^{4} + 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 5 T^{6} + 20 T^{4} - 45 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 17 T^{2} + 34)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + 10 T^{2} + 26 T + 169)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 51 T^{2} + 136)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 34 T^{2} + 136)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 68 T^{2} + 1088)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 68 T^{2} + 544)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 34 T^{2} + 136)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 85 T^{2} + 272)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 74 T^{2} + 1352)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 23 T^{2} + 128)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 77 T^{2} + 1444)^{4} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{4} + 132 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 16)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 102 T^{2} + 2176)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 221 T^{2} + 1156)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 272 T^{2} + 17408)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 148 T^{2} + 5408)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - 266 T^{2} + 17672)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 340 T^{2} + 17408)^{4} \) Copy content Toggle raw display
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