Properties

Label 275.2.z.a
Level $275$
Weight $2$
Character orbit 275.z
Analytic conductor $2.196$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.z (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{14} + 15 x^{12} - 59 x^{10} + 104 x^{8} - 59 x^{6} + 15 x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( -\beta_{1} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{3} + ( -\beta_{2} - \beta_{6} - \beta_{10} + \beta_{12} ) q^{4} + ( -1 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{6} + ( \beta_{1} - 2 \beta_{8} + \beta_{9} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{13} - \beta_{15} ) q^{8} + ( 1 - 3 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{10} + \beta_{12} ) q^{9} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( -\beta_{1} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{3} + ( -\beta_{2} - \beta_{6} - \beta_{10} + \beta_{12} ) q^{4} + ( -1 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{6} + ( \beta_{1} - 2 \beta_{8} + \beta_{9} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{13} - \beta_{15} ) q^{8} + ( 1 - 3 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{10} + \beta_{12} ) q^{9} + ( -\beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{12} ) q^{11} + ( -2 \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{12} + ( -3 \beta_{4} - 2 \beta_{9} - 2 \beta_{11} ) q^{13} + ( 2 \beta_{3} + \beta_{7} - 2 \beta_{10} - 2 \beta_{12} ) q^{14} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{16} + ( -\beta_{1} - \beta_{4} + \beta_{11} + 3 \beta_{14} - 3 \beta_{15} ) q^{17} + ( -\beta_{9} - \beta_{14} - 2 \beta_{15} ) q^{18} + ( -\beta_{5} - \beta_{6} - 5 \beta_{7} + 2 \beta_{10} ) q^{19} + ( -2 + \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{12} ) q^{21} + ( \beta_{1} - \beta_{4} + \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{14} ) q^{22} + ( \beta_{4} - \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{23} + ( -4 + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{10} - 4 \beta_{12} ) q^{24} + ( -\beta_{2} - \beta_{6} - \beta_{10} - 3 \beta_{12} ) q^{26} + ( -4 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{27} + ( 2 \beta_{11} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{28} + ( 2 - \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - \beta_{10} + 5 \beta_{12} ) q^{29} -5 \beta_{2} q^{31} + ( 2 \beta_{4} - 3 \beta_{9} + \beta_{11} + 2 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{32} + ( -3 \beta_{1} + \beta_{4} + 3 \beta_{9} + 2 \beta_{11} + 4 \beta_{14} + \beta_{15} ) q^{33} + ( -2 \beta_{2} - 2 \beta_{5} + \beta_{7} - \beta_{12} ) q^{34} + ( -4 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{10} + 4 \beta_{12} ) q^{36} + ( 3 \beta_{8} - 7 \beta_{9} - 3 \beta_{13} - 7 \beta_{14} + 4 \beta_{15} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{4} - 3 \beta_{11} - 4 \beta_{13} - \beta_{14} + \beta_{15} ) q^{38} + ( -3 + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{39} + ( \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{12} ) q^{41} + ( \beta_{1} + 4 \beta_{4} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} ) q^{42} + ( -2 \beta_{4} + \beta_{8} - 6 \beta_{9} - 6 \beta_{11} - 2 \beta_{13} - 6 \beta_{14} + 6 \beta_{15} ) q^{43} + ( 5 - 2 \beta_{2} - 5 \beta_{3} - \beta_{5} - 4 \beta_{7} + 2 \beta_{12} ) q^{44} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{46} + ( -2 \beta_{1} + \beta_{4} - \beta_{8} + 3 \beta_{9} - 2 \beta_{13} ) q^{47} + ( \beta_{1} + \beta_{8} - \beta_{9} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{48} + ( -1 + 3 \beta_{2} + 4 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{49} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{6} + 2 \beta_{10} - 3 \beta_{12} ) q^{51} + ( 2 \beta_{1} + \beta_{4} - \beta_{8} + 5 \beta_{9} + 5 \beta_{11} + 2 \beta_{13} - 5 \beta_{15} ) q^{52} + ( -5 \beta_{1} - \beta_{4} + 5 \beta_{8} + \beta_{9} + \beta_{11} + 6 \beta_{14} ) q^{53} + ( -1 + 4 \beta_{2} + 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{10} ) q^{54} + ( 3 + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} - 4 \beta_{7} - \beta_{10} + 4 \beta_{12} ) q^{56} + ( -7 \beta_{1} - \beta_{4} + 7 \beta_{8} + 8 \beta_{9} + 8 \beta_{11} + 9 \beta_{14} ) q^{57} + ( 4 \beta_{1} + 2 \beta_{9} + \beta_{11} + 4 \beta_{13} - \beta_{15} ) q^{58} + ( 5 + 4 \beta_{2} - 5 \beta_{3} + \beta_{6} + 4 \beta_{10} - 2 \beta_{12} ) q^{59} + ( -1 + 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{61} + ( 5 \beta_{8} - 5 \beta_{13} - 5 \beta_{15} ) q^{62} + ( -\beta_{1} + \beta_{4} - \beta_{8} + 3 \beta_{9} + 6 \beta_{11} - \beta_{13} - 6 \beta_{15} ) q^{63} + ( 2 + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} + 2 \beta_{12} ) q^{64} + ( -\beta_{2} - 2 \beta_{5} + 3 \beta_{7} - 5 \beta_{10} + \beta_{12} ) q^{66} + ( -4 \beta_{4} + 3 \beta_{8} - \beta_{9} + 5 \beta_{11} - 4 \beta_{13} + 5 \beta_{14} + \beta_{15} ) q^{67} + ( \beta_{1} + 3 \beta_{4} - \beta_{8} + 4 \beta_{9} + 4 \beta_{11} ) q^{68} + ( 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{10} - 2 \beta_{12} ) q^{69} + ( -2 - \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{4} + 9 \beta_{11} + \beta_{13} + 3 \beta_{14} - 3 \beta_{15} ) q^{72} + ( \beta_{1} + 2 \beta_{8} - 2 \beta_{13} - \beta_{15} ) q^{73} + ( -3 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{10} + 3 \beta_{12} ) q^{74} + ( 6 - \beta_{2} - \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 6 \beta_{10} + 2 \beta_{12} ) q^{76} + ( 2 \beta_{1} + \beta_{4} + \beta_{8} - \beta_{9} - 2 \beta_{11} + 3 \beta_{13} - 8 \beta_{14} + 6 \beta_{15} ) q^{77} + ( 4 \beta_{4} - 2 \beta_{8} - \beta_{9} + 2 \beta_{11} + 4 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{78} + ( -9 + \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{10} - 9 \beta_{12} ) q^{79} + ( -1 - 2 \beta_{2} + \beta_{3} - 6 \beta_{6} - 2 \beta_{10} + 3 \beta_{12} ) q^{81} + ( -4 \beta_{11} + 4 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{82} + ( -3 \beta_{1} - 3 \beta_{4} - \beta_{11} - 4 \beta_{13} + 6 \beta_{14} - 6 \beta_{15} ) q^{83} + ( -3 + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{6} + 3 \beta_{10} - \beta_{12} ) q^{84} + ( -1 + 4 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{86} + ( -3 \beta_{4} - 5 \beta_{8} - 3 \beta_{9} - 7 \beta_{11} - 3 \beta_{13} - 7 \beta_{14} + 3 \beta_{15} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{8} - 7 \beta_{9} - 5 \beta_{11} + \beta_{13} - 7 \beta_{14} + 3 \beta_{15} ) q^{88} + ( -1 + 6 \beta_{2} + 6 \beta_{5} ) q^{89} + ( 8 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} - 8 \beta_{12} ) q^{91} + ( -\beta_{1} + \beta_{8} - \beta_{9} - \beta_{13} - \beta_{14} + 4 \beta_{15} ) q^{92} + ( -5 \beta_{1} - 5 \beta_{4} - 5 \beta_{11} - 5 \beta_{14} + 5 \beta_{15} ) q^{93} + ( -3 + \beta_{2} + \beta_{3} + \beta_{6} + 3 \beta_{7} ) q^{94} + ( 3 \beta_{3} - \beta_{5} - \beta_{6} - 6 \beta_{7} + 6 \beta_{10} - 3 \beta_{12} ) q^{96} + ( -2 \beta_{1} + 2 \beta_{8} - 8 \beta_{9} - 8 \beta_{11} + 5 \beta_{14} ) q^{97} + ( -2 \beta_{8} + \beta_{9} + 4 \beta_{11} + 4 \beta_{14} - \beta_{15} ) q^{98} + ( 7 - 4 \beta_{2} - \beta_{3} - 5 \beta_{5} - 2 \beta_{6} - 12 \beta_{7} - 4 \beta_{10} + 5 \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{4} - 14q^{6} + 10q^{9} + O(q^{10}) \) \( 16q + 4q^{4} - 14q^{6} + 10q^{9} + 6q^{11} + 32q^{14} + 8q^{16} - 30q^{19} - 40q^{21} - 26q^{24} + 20q^{26} + 18q^{29} - 20q^{31} - 8q^{34} - 30q^{36} - 42q^{39} + 16q^{41} + 24q^{44} + 6q^{46} - 2q^{49} + 2q^{51} - 32q^{54} + 44q^{56} + 54q^{59} + 12q^{61} + 52q^{64} + 26q^{66} + 2q^{69} - 40q^{71} - 40q^{74} - 74q^{79} + 16q^{81} - 56q^{84} - 6q^{86} + 32q^{89} + 88q^{91} - 34q^{94} - 34q^{96} + 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{14} + 15 x^{12} - 59 x^{10} + 104 x^{8} - 59 x^{6} + 15 x^{4} - x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 21 \nu^{14} + 2 \nu^{12} + 289 \nu^{10} - 908 \nu^{8} + 772 \nu^{6} + 1045 \nu^{4} - 622 \nu^{2} - 63 \)\()/384\)
\(\beta_{3}\)\(=\)\((\)\( 21 \nu^{14} - 22 \nu^{12} + 329 \nu^{10} - 1260 \nu^{8} + 2436 \nu^{6} - 1995 \nu^{4} + 1498 \nu^{2} - 119 \)\()/384\)
\(\beta_{4}\)\(=\)\((\)\( -21 \nu^{15} + 22 \nu^{13} - 329 \nu^{11} + 1260 \nu^{9} - 2436 \nu^{7} + 1995 \nu^{5} - 1498 \nu^{3} + 119 \nu \)\()/384\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{14} - 11 \nu^{12} + 76 \nu^{10} - 384 \nu^{8} + 804 \nu^{6} - 671 \nu^{4} + 137 \nu^{2} - 40 \)\()/96\)
\(\beta_{6}\)\(=\)\((\)\( -21 \nu^{14} + 10 \nu^{12} - 309 \nu^{10} + 1084 \nu^{8} - 1604 \nu^{6} + 475 \nu^{4} - 246 \nu^{2} + 91 \)\()/192\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{14} - 17 \nu^{12} + 138 \nu^{10} - 648 \nu^{8} + 1332 \nu^{6} - 1107 \nu^{4} + 155 \nu^{2} + 30 \)\()/96\)
\(\beta_{8}\)\(=\)\((\)\( 9 \nu^{15} - 17 \nu^{13} + 138 \nu^{11} - 648 \nu^{9} + 1332 \nu^{7} - 1107 \nu^{5} + 155 \nu^{3} + 30 \nu \)\()/96\)
\(\beta_{9}\)\(=\)\((\)\( -9 \nu^{15} + 23 \nu^{13} - 148 \nu^{11} + 736 \nu^{9} - 1748 \nu^{7} + 1867 \nu^{5} - 685 \nu^{3} - 16 \nu \)\()/96\)
\(\beta_{10}\)\(=\)\((\)\( 57 \nu^{14} - 30 \nu^{12} + 845 \nu^{10} - 2972 \nu^{8} + 4564 \nu^{6} - 1511 \nu^{4} + 466 \nu^{2} + 77 \)\()/384\)
\(\beta_{11}\)\(=\)\((\)\( -63 \nu^{15} + 42 \nu^{13} - 947 \nu^{11} + 3428 \nu^{9} - 5644 \nu^{7} + 2945 \nu^{5} - 1990 \nu^{3} + 685 \nu \)\()/384\)
\(\beta_{12}\)\(=\)\((\)\( -17 \nu^{14} + 5 \nu^{12} - 246 \nu^{10} + 824 \nu^{8} - 1108 \nu^{6} - 101 \nu^{4} + 201 \nu^{2} - 58 \)\()/96\)
\(\beta_{13}\)\(=\)\((\)\( -17 \nu^{15} + 5 \nu^{13} - 246 \nu^{11} + 824 \nu^{9} - 1108 \nu^{7} - 101 \nu^{5} + 201 \nu^{3} - 58 \nu \)\()/96\)
\(\beta_{14}\)\(=\)\((\)\( 77 \nu^{15} - 134 \nu^{13} + 1185 \nu^{11} - 5388 \nu^{9} + 10980 \nu^{7} - 9107 \nu^{5} + 2666 \nu^{3} - 543 \nu \)\()/384\)
\(\beta_{15}\)\(=\)\((\)\( 40 \nu^{15} - 35 \nu^{13} + 589 \nu^{11} - 2284 \nu^{9} + 3776 \nu^{7} - 1556 \nu^{5} - 71 \nu^{3} + 97 \nu \)\()/96\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{11} + \beta_{9} + \beta_{8} - 3 \beta_{4} - \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12} + 5 \beta_{10} - 3 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - \beta_{3}\)
\(\nu^{5}\)\(=\)\(5 \beta_{15} - \beta_{14} + 6 \beta_{13} - \beta_{11} - 5 \beta_{9} - 12 \beta_{8} + 6 \beta_{4}\)
\(\nu^{6}\)\(=\)\(-6 \beta_{12} - 16 \beta_{10} - 23 \beta_{6} - 6 \beta_{3} - 16 \beta_{2} + 6\)
\(\nu^{7}\)\(=\)\(-16 \beta_{15} + 16 \beta_{14} - 22 \beta_{13} - 7 \beta_{11} + 29 \beta_{4} + 29 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-22 \beta_{12} - 67 \beta_{10} + 51 \beta_{7} - 67 \beta_{5} + 51 \beta_{3} + 36 \beta_{2} - 22\)
\(\nu^{9}\)\(=\)\(-67 \beta_{15} - 89 \beta_{13} + 67 \beta_{11} + 103 \beta_{9} + 221 \beta_{8} - 221 \beta_{4} - 89 \beta_{1}\)
\(\nu^{10}\)\(=\)\(132 \beta_{12} + 456 \beta_{10} - 132 \beta_{7} + 456 \beta_{6} + 168 \beta_{5} + 168 \beta_{2} - 89\)
\(\nu^{11}\)\(=\)\(456 \beta_{15} - 288 \beta_{14} + 588 \beta_{13} - 288 \beta_{9} - 588 \beta_{8} - 377 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-588 \beta_{7} - 1253 \beta_{6} + 756 \beta_{5} - 965 \beta_{3} - 1253 \beta_{2} + 588\)
\(\nu^{13}\)\(=\)\(756 \beta_{14} - 1253 \beta_{11} - 1253 \beta_{9} - 2597 \beta_{8} + 4227 \beta_{4} + 2597 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-2597 \beta_{12} - 8833 \beta_{10} + 4227 \beta_{7} - 5480 \beta_{6} - 5480 \beta_{5} + 2597 \beta_{3}\)
\(\nu^{15}\)\(=\)\(-8833 \beta_{15} + 3353 \beta_{14} - 11430 \beta_{13} + 3353 \beta_{11} + 8833 \beta_{9} + 18540 \beta_{8} - 11430 \beta_{4}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(\beta_{12}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.23158 + 1.69513i
0.280526 + 0.386111i
−0.280526 0.386111i
−1.23158 1.69513i
−1.28932 0.418926i
−0.701538 0.227943i
0.701538 + 0.227943i
1.28932 + 0.418926i
1.23158 1.69513i
0.280526 0.386111i
−0.280526 + 0.386111i
−1.23158 + 1.69513i
−1.28932 + 0.418926i
−0.701538 + 0.227943i
0.701538 0.227943i
1.28932 0.418926i
−1.99274 + 0.647481i 1.12443 1.54765i 1.93376 1.40496i 0 −1.23863 + 3.81211i −1.80285 2.48141i −0.480634 + 0.661536i −0.203814 0.627276i 0
49.2 −0.453901 + 0.147481i −0.189896 + 0.261370i −1.43376 + 1.04169i 0 0.0476470 0.146642i −1.57833 2.17239i 1.05821 1.45650i 0.894797 + 2.75390i 0
49.3 0.453901 0.147481i 0.189896 0.261370i −1.43376 + 1.04169i 0 0.0476470 0.146642i 1.57833 + 2.17239i −1.05821 + 1.45650i 0.894797 + 2.75390i 0
49.4 1.99274 0.647481i −1.12443 + 1.54765i 1.93376 1.40496i 0 −1.23863 + 3.81211i 1.80285 + 2.48141i 0.480634 0.661536i −0.203814 0.627276i 0
124.1 −0.796845 1.09676i 0.547326 0.177837i 0.0501062 0.154211i 0 −0.631180 0.458579i −3.47080 1.12773i −2.78771 + 0.905781i −2.15911 + 1.56869i 0
124.2 −0.433574 0.596764i 2.67395 0.868820i 0.449894 1.38463i 0 −1.67784 1.21902i −0.980901 0.318714i −2.42443 + 0.787747i 3.96813 2.88301i 0
124.3 0.433574 + 0.596764i −2.67395 + 0.868820i 0.449894 1.38463i 0 −1.67784 1.21902i 0.980901 + 0.318714i 2.42443 0.787747i 3.96813 2.88301i 0
124.4 0.796845 + 1.09676i −0.547326 + 0.177837i 0.0501062 0.154211i 0 −0.631180 0.458579i 3.47080 + 1.12773i 2.78771 0.905781i −2.15911 + 1.56869i 0
174.1 −1.99274 0.647481i 1.12443 + 1.54765i 1.93376 + 1.40496i 0 −1.23863 3.81211i −1.80285 + 2.48141i −0.480634 0.661536i −0.203814 + 0.627276i 0
174.2 −0.453901 0.147481i −0.189896 0.261370i −1.43376 1.04169i 0 0.0476470 + 0.146642i −1.57833 + 2.17239i 1.05821 + 1.45650i 0.894797 2.75390i 0
174.3 0.453901 + 0.147481i 0.189896 + 0.261370i −1.43376 1.04169i 0 0.0476470 + 0.146642i 1.57833 2.17239i −1.05821 1.45650i 0.894797 2.75390i 0
174.4 1.99274 + 0.647481i −1.12443 1.54765i 1.93376 + 1.40496i 0 −1.23863 3.81211i 1.80285 2.48141i 0.480634 + 0.661536i −0.203814 + 0.627276i 0
224.1 −0.796845 + 1.09676i 0.547326 + 0.177837i 0.0501062 + 0.154211i 0 −0.631180 + 0.458579i −3.47080 + 1.12773i −2.78771 0.905781i −2.15911 1.56869i 0
224.2 −0.433574 + 0.596764i 2.67395 + 0.868820i 0.449894 + 1.38463i 0 −1.67784 + 1.21902i −0.980901 + 0.318714i −2.42443 0.787747i 3.96813 + 2.88301i 0
224.3 0.433574 0.596764i −2.67395 0.868820i 0.449894 + 1.38463i 0 −1.67784 + 1.21902i 0.980901 0.318714i 2.42443 + 0.787747i 3.96813 + 2.88301i 0
224.4 0.796845 1.09676i −0.547326 0.177837i 0.0501062 + 0.154211i 0 −0.631180 + 0.458579i 3.47080 1.12773i 2.78771 + 0.905781i −2.15911 1.56869i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 224.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.z.a 16
5.b even 2 1 inner 275.2.z.a 16
5.c odd 4 1 55.2.g.b 8
5.c odd 4 1 275.2.h.a 8
11.c even 5 1 inner 275.2.z.a 16
15.e even 4 1 495.2.n.e 8
20.e even 4 1 880.2.bo.h 8
55.e even 4 1 605.2.g.k 8
55.j even 10 1 inner 275.2.z.a 16
55.k odd 20 1 55.2.g.b 8
55.k odd 20 1 275.2.h.a 8
55.k odd 20 1 605.2.a.j 4
55.k odd 20 2 605.2.g.m 8
55.k odd 20 1 3025.2.a.bd 4
55.l even 20 1 605.2.a.k 4
55.l even 20 2 605.2.g.e 8
55.l even 20 1 605.2.g.k 8
55.l even 20 1 3025.2.a.w 4
165.u odd 20 1 5445.2.a.bi 4
165.v even 20 1 495.2.n.e 8
165.v even 20 1 5445.2.a.bp 4
220.v even 20 1 880.2.bo.h 8
220.v even 20 1 9680.2.a.cn 4
220.w odd 20 1 9680.2.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b 8 5.c odd 4 1
55.2.g.b 8 55.k odd 20 1
275.2.h.a 8 5.c odd 4 1
275.2.h.a 8 55.k odd 20 1
275.2.z.a 16 1.a even 1 1 trivial
275.2.z.a 16 5.b even 2 1 inner
275.2.z.a 16 11.c even 5 1 inner
275.2.z.a 16 55.j even 10 1 inner
495.2.n.e 8 15.e even 4 1
495.2.n.e 8 165.v even 20 1
605.2.a.j 4 55.k odd 20 1
605.2.a.k 4 55.l even 20 1
605.2.g.e 8 55.l even 20 2
605.2.g.k 8 55.e even 4 1
605.2.g.k 8 55.l even 20 1
605.2.g.m 8 55.k odd 20 2
880.2.bo.h 8 20.e even 4 1
880.2.bo.h 8 220.v even 20 1
3025.2.a.w 4 55.l even 20 1
3025.2.a.bd 4 55.k odd 20 1
5445.2.a.bi 4 165.u odd 20 1
5445.2.a.bp 4 165.v even 20 1
9680.2.a.cm 4 220.w odd 20 1
9680.2.a.cn 4 220.v even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T^{2} + 15 T^{4} - 4 T^{6} + 69 T^{8} - 4 T^{10} + 15 T^{12} - 6 T^{14} + T^{16} \)
$3$ \( 1 + T^{2} + 72 T^{4} - 397 T^{6} + 855 T^{8} - 53 T^{10} + 52 T^{12} - 11 T^{14} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( 923521 - 1377113 T^{2} + 795743 T^{4} - 9671 T^{6} + 21080 T^{8} - 1291 T^{10} + 143 T^{12} - 13 T^{14} + T^{16} \)
$11$ \( ( 14641 - 3993 T + 2178 T^{2} + 99 T^{3} + 75 T^{4} + 9 T^{5} + 18 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$13$ \( 373301041 - 44476942 T^{2} + 7909555 T^{4} - 1139672 T^{6} + 130089 T^{8} - 7768 T^{10} + 715 T^{12} - 38 T^{14} + T^{16} \)
$17$ \( 130321 - 970007 T^{2} + 2743680 T^{4} + 186743 T^{6} + 88379 T^{8} - 2293 T^{10} + 300 T^{12} - 23 T^{14} + T^{16} \)
$19$ \( ( 625 - 3750 T + 67125 T^{2} + 33875 T^{3} + 7950 T^{4} + 1025 T^{5} + 135 T^{6} + 15 T^{7} + T^{8} )^{2} \)
$23$ \( ( 121 + 188 T^{2} + 94 T^{4} + 17 T^{6} + T^{8} )^{2} \)
$29$ \( ( 203401 - 18942 T - 2329 T^{2} + 429 T^{3} + 1384 T^{4} - 363 T^{5} + 99 T^{6} - 9 T^{7} + T^{8} )^{2} \)
$31$ \( ( 390625 - 156250 T + 78125 T^{2} - 6250 T^{3} - 625 T^{4} + 250 T^{5} + 125 T^{6} + 10 T^{7} + T^{8} )^{2} \)
$37$ \( 1755097689601 + 219078366967 T^{2} + 135958661335 T^{4} - 8194385503 T^{6} + 207982794 T^{8} - 2073497 T^{10} + 17185 T^{12} - 142 T^{14} + T^{16} \)
$41$ \( ( 101761 - 120263 T + 69513 T^{2} - 23171 T^{3} + 5430 T^{4} - 831 T^{5} + 93 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$43$ \( ( 44521 + 32459 T^{2} + 4081 T^{4} + 119 T^{6} + T^{8} )^{2} \)
$47$ \( 815730721 - 222033214 T^{2} + 29903367 T^{4} - 1958372 T^{6} + 513605 T^{8} - 11588 T^{10} + 1047 T^{12} - 46 T^{14} + T^{16} \)
$53$ \( 784076601361 - 46435509121 T^{2} + 8364167295 T^{4} - 995972579 T^{6} + 61378464 T^{8} - 1493299 T^{10} + 15415 T^{12} - T^{14} + T^{16} \)
$59$ \( ( 687241 - 395433 T + 157155 T^{2} - 54243 T^{3} + 18994 T^{4} - 3177 T^{5} + 385 T^{6} - 27 T^{7} + T^{8} )^{2} \)
$61$ \( ( 28561 + 26364 T + 17745 T^{2} + 6396 T^{3} + 1504 T^{4} + 96 T^{5} + 10 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$67$ \( ( 16638241 + 1207672 T^{2} + 30858 T^{4} + 317 T^{6} + T^{8} )^{2} \)
$71$ \( ( 17161 + 9825 T + 11477 T^{2} + 10445 T^{3} + 5634 T^{4} + 1325 T^{5} + 213 T^{6} + 20 T^{7} + T^{8} )^{2} \)
$73$ \( 14641 - 39567 T^{2} + 53120 T^{4} - 44097 T^{6} + 48919 T^{8} - 30873 T^{10} + 9320 T^{12} + 57 T^{14} + T^{16} \)
$79$ \( ( 45954841 + 17171207 T + 6346418 T^{2} + 1002329 T^{3} + 104405 T^{4} + 8839 T^{5} + 698 T^{6} + 37 T^{7} + T^{8} )^{2} \)
$83$ \( 8332465333201 + 1753503303263 T^{2} + 1187703883823 T^{4} - 33962438979 T^{6} + 471603680 T^{8} - 4076979 T^{10} + 36743 T^{12} - 257 T^{14} + T^{16} \)
$89$ \( ( 1861 + 472 T - 102 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$97$ \( 82194549986641 + 10269576033782 T^{2} + 2697321461235 T^{4} + 68217625252 T^{6} + 5563040429 T^{8} - 70415372 T^{10} + 371715 T^{12} - 922 T^{14} + T^{16} \)
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