Properties

Label 1045.2.b.b
Level $1045$
Weight $2$
Character orbit 1045.b
Analytic conductor $8.344$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 19 x^{14} + 144 x^{12} + 552 x^{10} + 1119 x^{8} + 1146 x^{6} + 524 x^{4} + 83 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} + ( \beta_{3} + \beta_{13} ) q^{5} + ( -1 - \beta_{6} - \beta_{8} + \beta_{9} ) q^{6} -\beta_{5} q^{7} + \beta_{3} q^{8} + ( -1 - \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} + ( \beta_{3} + \beta_{13} ) q^{5} + ( -1 - \beta_{6} - \beta_{8} + \beta_{9} ) q^{6} -\beta_{5} q^{7} + \beta_{3} q^{8} + ( -1 - \beta_{8} ) q^{9} + ( -\beta_{2} + \beta_{4} - \beta_{10} ) q^{10} - q^{11} + ( \beta_{5} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{12} + ( 2 \beta_{1} + \beta_{3} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{13} + ( \beta_{4} - \beta_{6} ) q^{14} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{10} - \beta_{12} ) q^{15} + ( -1 + \beta_{2} + \beta_{4} ) q^{16} + ( \beta_{3} + \beta_{5} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{17} + ( -\beta_{1} - \beta_{3} - \beta_{10} - \beta_{11} + \beta_{15} ) q^{18} - q^{19} + ( 2 \beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} ) q^{20} + ( \beta_{2} - \beta_{4} - \beta_{9} ) q^{21} -\beta_{1} q^{22} + ( -\beta_{1} + \beta_{5} - \beta_{10} - 2 \beta_{14} - \beta_{15} ) q^{23} + ( 1 + \beta_{7} + \beta_{9} ) q^{24} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{15} ) q^{25} + ( -1 + \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{26} + ( 2 \beta_{3} + \beta_{5} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{27} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{28} + ( -\beta_{4} - \beta_{7} + \beta_{9} ) q^{29} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{30} + ( -2 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{31} + ( -3 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{32} -\beta_{11} q^{33} + ( -2 - \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} - \beta_{13} ) q^{34} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{35} + ( 2 + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{36} + ( -\beta_{1} - \beta_{3} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{37} -\beta_{1} q^{38} + ( 4 + \beta_{2} - 3 \beta_{4} - 2 \beta_{6} - \beta_{12} - \beta_{13} ) q^{39} + ( -2 + \beta_{4} + \beta_{6} - \beta_{14} ) q^{40} + ( 2 + 2 \beta_{2} - 2 \beta_{4} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{41} + ( -2 \beta_{1} + \beta_{3} - \beta_{10} + \beta_{11} ) q^{42} + ( -\beta_{1} - 2 \beta_{5} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{43} -\beta_{2} q^{44} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{45} + ( -\beta_{2} + \beta_{6} - \beta_{12} - \beta_{13} ) q^{46} + ( -3 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{47} + ( \beta_{1} + 2 \beta_{3} + \beta_{5} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{48} + ( 3 + \beta_{4} + \beta_{7} + \beta_{8} ) q^{49} + ( -2 - \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{11} + 2 \beta_{12} ) q^{50} + ( \beta_{2} - \beta_{4} + 2 \beta_{9} - \beta_{12} - \beta_{13} ) q^{51} + ( 2 \beta_{3} - \beta_{5} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{52} + ( -3 \beta_{3} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{53} + ( -2 \beta_{2} - 2 \beta_{7} + \beta_{9} - \beta_{12} - \beta_{13} ) q^{54} + ( -\beta_{3} - \beta_{13} ) q^{55} + ( 2 + 2 \beta_{2} + \beta_{8} - \beta_{9} ) q^{56} -\beta_{11} q^{57} + ( \beta_{3} - 2 \beta_{5} + 2 \beta_{10} - 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{58} + ( 4 + 3 \beta_{2} - \beta_{4} - \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{59} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{60} + ( -2 + \beta_{2} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{61} + ( -5 \beta_{1} - \beta_{5} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{62} + ( -\beta_{5} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{63} + ( 2 - 3 \beta_{2} + 3 \beta_{4} + \beta_{6} ) q^{64} + ( -2 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{14} ) q^{65} + ( 1 + \beta_{6} + \beta_{8} - \beta_{9} ) q^{66} + ( 3 \beta_{1} + 2 \beta_{3} + 3 \beta_{5} + \beta_{10} - \beta_{11} - 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{67} + ( 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{68} + ( 2 - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{12} - \beta_{13} ) q^{69} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{15} ) q^{70} + ( -4 + 3 \beta_{4} - 2 \beta_{6} + \beta_{7} + 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} ) q^{71} + ( -\beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{72} + ( -3 \beta_{1} - \beta_{3} - 3 \beta_{5} - \beta_{10} + 3 \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{73} + ( -1 - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{12} - \beta_{13} ) q^{74} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{75} -\beta_{2} q^{76} + \beta_{5} q^{77} + ( 4 \beta_{1} + 4 \beta_{3} + \beta_{5} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{78} + ( 6 - 2 \beta_{4} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{12} - 2 \beta_{13} ) q^{79} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{80} + ( -3 - 3 \beta_{4} + \beta_{6} - 2 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} ) q^{81} + ( -2 \beta_{1} + 3 \beta_{3} - \beta_{5} + \beta_{10} + \beta_{11} ) q^{82} + ( 3 \beta_{3} - 3 \beta_{5} + \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{83} + ( 2 - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{84} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{85} + ( -2 - \beta_{2} + \beta_{4} - 3 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{86} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{10} - \beta_{14} + \beta_{15} ) q^{87} -\beta_{3} q^{88} + ( -\beta_{2} + 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{12} + 2 \beta_{13} ) q^{89} + ( 2 \beta_{1} + \beta_{2} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{90} + ( -2 - 2 \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} - \beta_{13} ) q^{91} + ( -\beta_{1} - \beta_{3} + \beta_{11} + \beta_{12} - \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{92} + ( 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - \beta_{10} - 5 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{93} + ( 2 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + \beta_{9} + 2 \beta_{12} + 2 \beta_{13} ) q^{94} + ( -\beta_{3} - \beta_{13} ) q^{95} + ( 5 - \beta_{2} + \beta_{4} + 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{96} + ( 4 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{97} + ( 3 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{11} + \beta_{14} - 2 \beta_{15} ) q^{98} + ( 1 + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + O(q^{10}) \) \( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 19 x^{14} + 144 x^{12} + 552 x^{10} + 1119 x^{8} + 1146 x^{6} + 524 x^{4} + 83 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 5 \nu^{2} + 3 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 6 \nu^{3} + 7 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{6} + 7 \nu^{4} + 12 \nu^{2} + 3 \)
\(\beta_{7}\)\(=\)\( -\nu^{12} - 14 \nu^{10} - 72 \nu^{8} - 165 \nu^{6} - 163 \nu^{4} - 60 \nu^{2} - 6 \)
\(\beta_{8}\)\(=\)\( \nu^{12} + 15 \nu^{10} + 84 \nu^{8} + 215 \nu^{6} + 246 \nu^{4} + 104 \nu^{2} + 7 \)
\(\beta_{9}\)\(=\)\( \nu^{12} + 15 \nu^{10} + 85 \nu^{8} + 225 \nu^{6} + 277 \nu^{4} + 134 \nu^{2} + 13 \)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} - 17 \nu^{13} - 114 \nu^{11} - 382 \nu^{9} - 667 \nu^{7} - 572 \nu^{5} - 198 \nu^{3} - 15 \nu \)\()/2\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{15} - 19 \nu^{13} - 144 \nu^{11} - 552 \nu^{9} - 1117 \nu^{7} - 1128 \nu^{5} - 476 \nu^{3} - 47 \nu \)\()/2\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{15} + 2 \nu^{14} + 19 \nu^{13} + 34 \nu^{12} + 144 \nu^{11} + 228 \nu^{10} + 552 \nu^{9} + 764 \nu^{8} + 1119 \nu^{7} + 1334 \nu^{6} + 1146 \nu^{5} + 1144 \nu^{4} + 524 \nu^{3} + 396 \nu^{2} + 83 \nu + 30 \)\()/4\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} + 2 \nu^{14} - 19 \nu^{13} + 34 \nu^{12} - 144 \nu^{11} + 228 \nu^{10} - 552 \nu^{9} + 764 \nu^{8} - 1119 \nu^{7} + 1334 \nu^{6} - 1146 \nu^{5} + 1144 \nu^{4} - 524 \nu^{3} + 396 \nu^{2} - 83 \nu + 30 \)\()/4\)
\(\beta_{14}\)\(=\)\( -\nu^{15} - 19 \nu^{13} - 143 \nu^{11} - 538 \nu^{9} - 1047 \nu^{7} - 981 \nu^{5} - 362 \nu^{3} - 28 \nu \)
\(\beta_{15}\)\(=\)\( -\nu^{15} - 19 \nu^{13} - 144 \nu^{11} - 551 \nu^{9} - 1107 \nu^{7} - 1096 \nu^{5} - 440 \nu^{3} - 34 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 5 \beta_{2} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 6 \beta_{3} + 17 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{6} - 7 \beta_{4} + 23 \beta_{2} - 28\)
\(\nu^{7}\)\(=\)\(-\beta_{13} + \beta_{12} + \beta_{11} - 9 \beta_{5} + 30 \beta_{3} - 75 \beta_{1}\)
\(\nu^{8}\)\(=\)\(\beta_{9} - \beta_{8} - 10 \beta_{6} + 39 \beta_{4} - 105 \beta_{2} + 117\)
\(\nu^{9}\)\(=\)\(\beta_{15} + 10 \beta_{13} - 10 \beta_{12} - 12 \beta_{11} + 58 \beta_{5} - 144 \beta_{3} + 337 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-12 \beta_{9} + 13 \beta_{8} + \beta_{7} + 70 \beta_{6} - 201 \beta_{4} + 481 \beta_{2} - 498\)
\(\nu^{11}\)\(=\)\(-14 \beta_{15} + \beta_{14} - 70 \beta_{13} + 70 \beta_{12} + 96 \beta_{11} - 329 \beta_{5} + 684 \beta_{3} - 1530 \beta_{1}\)
\(\nu^{12}\)\(=\)\(96 \beta_{9} - 110 \beta_{8} - 15 \beta_{7} - 425 \beta_{6} + 998 \beta_{4} - 2214 \beta_{2} + 2141\)
\(\nu^{13}\)\(=\)\(125 \beta_{15} - 15 \beta_{14} + 425 \beta_{13} - 425 \beta_{12} - 646 \beta_{11} + \beta_{10} + 1752 \beta_{5} - 3241 \beta_{3} + 6994 \beta_{1}\)
\(\nu^{14}\)\(=\)\(\beta_{13} + \beta_{12} - 646 \beta_{9} + 770 \beta_{8} + 141 \beta_{7} + 2398 \beta_{6} - 4853 \beta_{4} + 10235 \beta_{2} - 9266\)
\(\nu^{15}\)\(=\)\(-911 \beta_{15} + 141 \beta_{14} - 2398 \beta_{13} + 2398 \beta_{12} + 3955 \beta_{11} - 19 \beta_{10} - 9003 \beta_{5} + 15353 \beta_{3} - 32134 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
2.18657i
2.10564i
2.05535i
1.78099i
1.19140i
0.853553i
0.387488i
0.301154i
0.301154i
0.387488i
0.853553i
1.19140i
1.78099i
2.05535i
2.10564i
2.18657i
2.18657i 1.58567i −2.78109 −0.558280 + 2.16525i 3.46718 2.56330i 1.70792i 0.485645 4.73448 + 1.22072i
419.2 2.10564i 1.90142i −2.43371 1.75301 + 1.38815i −4.00370 0.116917i 0.913237i −0.615392 2.92295 3.69120i
419.3 2.05535i 2.80376i −2.22445 −2.02523 + 0.947852i −5.76269 1.02917i 0.461316i −4.86106 1.94816 + 4.16256i
419.4 1.78099i 0.213501i −1.17191 2.04104 0.913326i 0.380243 3.50934i 1.47481i 2.95442 −1.62662 3.63506i
419.5 1.19140i 2.62369i 0.580557 0.0644373 2.23514i 3.12588 0.593512i 3.07449i −3.88377 −2.66295 0.0767708i
419.6 0.853553i 1.84364i 1.27145 1.54048 1.62078i −1.57364 2.69678i 2.79235i −0.399003 −1.38342 1.31488i
419.7 0.387488i 0.494325i 1.84985 −1.95299 + 1.08895i −0.191545 2.37207i 1.49177i 2.75564 0.421956 + 0.756762i
419.8 0.301154i 1.85377i 1.90931 0.637551 + 2.14325i 0.558272 1.94668i 1.17730i −0.436480 0.645450 0.192001i
419.9 0.301154i 1.85377i 1.90931 0.637551 2.14325i 0.558272 1.94668i 1.17730i −0.436480 0.645450 + 0.192001i
419.10 0.387488i 0.494325i 1.84985 −1.95299 1.08895i −0.191545 2.37207i 1.49177i 2.75564 0.421956 0.756762i
419.11 0.853553i 1.84364i 1.27145 1.54048 + 1.62078i −1.57364 2.69678i 2.79235i −0.399003 −1.38342 + 1.31488i
419.12 1.19140i 2.62369i 0.580557 0.0644373 + 2.23514i 3.12588 0.593512i 3.07449i −3.88377 −2.66295 + 0.0767708i
419.13 1.78099i 0.213501i −1.17191 2.04104 + 0.913326i 0.380243 3.50934i 1.47481i 2.95442 −1.62662 + 3.63506i
419.14 2.05535i 2.80376i −2.22445 −2.02523 0.947852i −5.76269 1.02917i 0.461316i −4.86106 1.94816 4.16256i
419.15 2.10564i 1.90142i −2.43371 1.75301 1.38815i −4.00370 0.116917i 0.913237i −0.615392 2.92295 + 3.69120i
419.16 2.18657i 1.58567i −2.78109 −0.558280 2.16525i 3.46718 2.56330i 1.70792i 0.485645 4.73448 1.22072i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.b 16
5.b even 2 1 inner 1045.2.b.b 16
5.c odd 4 2 5225.2.a.z 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.b 16 1.a even 1 1 trivial
1045.2.b.b 16 5.b even 2 1 inner
5225.2.a.z 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1045, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 83 T^{2} + 524 T^{4} + 1146 T^{6} + 1119 T^{8} + 552 T^{10} + 144 T^{12} + 19 T^{14} + T^{16} \)
$3$ \( 64 + 1764 T^{2} + 8359 T^{4} + 10408 T^{6} + 5983 T^{8} + 1849 T^{10} + 316 T^{12} + 28 T^{14} + T^{16} \)
$5$ \( 390625 - 234375 T + 125000 T^{2} - 15625 T^{3} + 2500 T^{4} + 3625 T^{5} + 3800 T^{6} - 2705 T^{7} + 1942 T^{8} - 541 T^{9} + 152 T^{10} + 29 T^{11} + 4 T^{12} - 5 T^{13} + 8 T^{14} - 3 T^{15} + T^{16} \)
$7$ \( 64 + 4976 T^{2} + 21900 T^{4} + 28337 T^{6} + 14949 T^{8} + 3908 T^{10} + 537 T^{12} + 37 T^{14} + T^{16} \)
$11$ \( ( 1 + T )^{16} \)
$13$ \( 65536 + 438016 T^{2} + 773284 T^{4} + 560749 T^{6} + 184540 T^{8} + 29479 T^{10} + 2287 T^{12} + 81 T^{14} + T^{16} \)
$17$ \( 6390784 + 24523520 T^{2} + 20835292 T^{4} + 7053981 T^{6} + 1168244 T^{8} + 102859 T^{10} + 4843 T^{12} + 113 T^{14} + T^{16} \)
$19$ \( ( 1 + T )^{16} \)
$23$ \( 1936 + 186672 T^{2} + 2822767 T^{4} + 11409873 T^{6} + 2951461 T^{8} + 276273 T^{10} + 11120 T^{12} + 181 T^{14} + T^{16} \)
$29$ \( ( -784 + 1876 T + 776 T^{2} - 2051 T^{3} + 570 T^{4} + 151 T^{5} - 59 T^{6} - T^{7} + T^{8} )^{2} \)
$31$ \( ( -3104 - 19452 T + 40499 T^{2} + 8843 T^{3} - 3629 T^{4} - 972 T^{5} + 6 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$37$ \( 476636224 + 773955900 T^{2} + 438945343 T^{4} + 107061868 T^{6} + 11462539 T^{8} + 598764 T^{10} + 15893 T^{12} + 204 T^{14} + T^{16} \)
$41$ \( ( -3152 + 13252 T - 8000 T^{2} - 2425 T^{3} + 1391 T^{4} + 157 T^{5} - 68 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$43$ \( 277688896 + 8628416240 T^{2} + 3874432012 T^{4} + 659374457 T^{6} + 52436339 T^{8} + 1996502 T^{10} + 36797 T^{12} + 315 T^{14} + T^{16} \)
$47$ \( 37937690176 + 28902696624 T^{2} + 8210240892 T^{4} + 1110538125 T^{6} + 77749816 T^{8} + 2806125 T^{10} + 48492 T^{12} + 374 T^{14} + T^{16} \)
$53$ \( 32319410176 + 41909201344 T^{2} + 15971706196 T^{4} + 2044023125 T^{6} + 119775800 T^{8} + 3525595 T^{10} + 52063 T^{12} + 369 T^{14} + T^{16} \)
$59$ \( ( 9973312 + 2661196 T - 746833 T^{2} - 148223 T^{3} + 20955 T^{4} + 2498 T^{5} - 250 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$61$ \( ( 269488 + 26300 T - 216380 T^{2} - 135743 T^{3} - 32846 T^{4} - 3213 T^{5} + 3 T^{6} + 21 T^{7} + T^{8} )^{2} \)
$67$ \( 3129530826304 + 1155799732292 T^{2} + 158517530011 T^{4} + 10812950911 T^{6} + 401315556 T^{8} + 8204051 T^{10} + 89127 T^{12} + 479 T^{14} + T^{16} \)
$71$ \( ( 1022368 - 769388 T - 80135 T^{2} + 109333 T^{3} - 1453 T^{4} - 2797 T^{5} - 28 T^{6} + 23 T^{7} + T^{8} )^{2} \)
$73$ \( 3803560873984 + 16011297862400 T^{2} + 1762119306352 T^{4} + 78579467125 T^{6} + 1815314882 T^{8} + 23397804 T^{10} + 169417 T^{12} + 644 T^{14} + T^{16} \)
$79$ \( ( 143552 + 145400 T - 389110 T^{2} + 226251 T^{3} - 48659 T^{4} + 2370 T^{5} + 328 T^{6} - 37 T^{7} + T^{8} )^{2} \)
$83$ \( 28036855560256 + 12138096253104 T^{2} + 1633271181148 T^{4} + 79819688637 T^{6} + 1901499388 T^{8} + 24672307 T^{10} + 177747 T^{12} + 665 T^{14} + T^{16} \)
$89$ \( ( -10084 - 162824 T - 128017 T^{2} - 3533 T^{3} + 9080 T^{4} + 619 T^{5} - 163 T^{6} - 7 T^{7} + T^{8} )^{2} \)
$97$ \( 574639907835904 + 125812288520940 T^{2} + 9942700888443 T^{4} + 364342500805 T^{6} + 6910552192 T^{8} + 70337053 T^{10} + 379133 T^{12} + 997 T^{14} + T^{16} \)
show more
show less