Properties

Label 475.2.e.f
Level $475$
Weight $2$
Character orbit 475.e
Analytic conductor $3.793$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} - 552 x^{3} + 1107 x^{2} + 33 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} - 552 x^{3} + 1107 x^{2} + 33 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-3959208 \nu^{11} - 46922925 \nu^{10} + 38785859 \nu^{9} - 621660414 \nu^{8} - 801763109 \nu^{7} - 3678052623 \nu^{6} - 4177401796 \nu^{5} - 15342668397 \nu^{4} - 38817354653 \nu^{3} - 24355803798 \nu^{2} - 726241735 \nu - 68499350729\)\()/ 66494854340 \)
\(\beta_{3}\)\(=\)\((\)\(690858 \nu^{11} - 11701549 \nu^{10} + 76199247 \nu^{9} - 313859676 \nu^{8} + 833713615 \nu^{7} - 1701731499 \nu^{6} + 2957264154 \nu^{5} - 5508756455 \nu^{4} + 7520604435 \nu^{3} - 6717081048 \nu^{2} - 1547540653 \nu - 6178545\)\()/ 1955731010 \)
\(\beta_{4}\)\(=\)\((\)\(-54841341 \nu^{11} + 153015320 \nu^{10} - 731712017 \nu^{9} + 251450977 \nu^{8} - 3244495858 \nu^{7} + 1416907499 \nu^{6} - 11747270177 \nu^{5} - 19479845384 \nu^{4} + 12276068039 \nu^{3} - 38482449021 \nu^{2} - 1147600790 \nu - 263970961763\)\()/ 66494854340 \)
\(\beta_{5}\)\(=\)\((\)\(134211789 \nu^{11} - 998274384 \nu^{10} + 3072233475 \nu^{9} - 7500316929 \nu^{8} + 1833052570 \nu^{7} - 27189880401 \nu^{6} - 7077581935 \nu^{5} - 102294574896 \nu^{4} - 204345695921 \nu^{3} - 138016601091 \nu^{2} - 4114960306 \nu - 148724948799\)\()/ 66494854340 \)
\(\beta_{6}\)\(=\)\((\)\(-5051595 \nu^{11} + 9282581 \nu^{10} - 64765518 \nu^{9} - 29949491 \nu^{8} - 345971017 \nu^{7} - 412414770 \nu^{6} - 1358444363 \nu^{5} - 2951210507 \nu^{4} - 2448832100 \nu^{3} - 5335902045 \nu^{2} - 159117251 \nu - 4335298296\)\()/ 1955731010 \)
\(\beta_{7}\)\(=\)\((\)\(-6178545 \nu^{11} + 17844777 \nu^{10} - 93333716 \nu^{9} + 35014563 \nu^{8} - 359601729 \nu^{7} - 259108930 \nu^{6} - 1288684281 \nu^{5} - 2049018039 \nu^{4} - 725395450 \nu^{3} - 4110047595 \nu^{2} - 122568267 \nu - 612082342\)\()/ 1955731010 \)
\(\beta_{8}\)\(=\)\((\)\(-469619124 \nu^{11} + 1742271771 \nu^{10} - 9648336987 \nu^{9} + 16279393858 \nu^{8} - 67720087321 \nu^{7} + 86418671699 \nu^{6} - 294644252580 \nu^{5} + 212867920667 \nu^{4} - 647754164847 \nu^{3} + 432662257370 \nu^{2} - 680489253663 \nu + 402309009\)\()/ 66494854340 \)
\(\beta_{9}\)\(=\)\((\)\(-2004496389 \nu^{11} + 6017448375 \nu^{10} - 34029515688 \nu^{9} + 36042149143 \nu^{8} - 217868445987 \nu^{7} + 187219927286 \nu^{6} - 966498199653 \nu^{5} + 298838370979 \nu^{4} - 2007194188104 \nu^{3} + 1145299361381 \nu^{2} - 2194621698825 \nu - 65422139102\)\()/ 66494854340 \)
\(\beta_{10}\)\(=\)\((\)\(-4220156417 \nu^{11} + 12953281828 \nu^{10} - 72904583371 \nu^{9} + 81775705089 \nu^{8} - 469415474758 \nu^{7} + 426177732417 \nu^{6} - 2086126950865 \nu^{5} + 755327449736 \nu^{4} - 4359372249951 \nu^{3} + 2534882432535 \nu^{2} - 4859631701974 \nu + 2372026047\)\()/ 66494854340 \)
\(\beta_{11}\)\(=\)\((\)\(6424307742 \nu^{11} - 19688524501 \nu^{10} + 111043957893 \nu^{9} - 124776050724 \nu^{8} + 717279166315 \nu^{7} - 650339598681 \nu^{6} + 3178214179566 \nu^{5} - 1144205547385 \nu^{4} + 6642795442545 \nu^{3} - 3864903739992 \nu^{2} + 7195985653273 \nu - 3616495515\)\()/ 66494854340 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} + 3 \beta_{9} + \beta_{8} - \beta_{4} - \beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 6 \beta_{2} - 10\)
\(\nu^{4}\)\(=\)\(-10 \beta_{11} - 6 \beta_{10} - 18 \beta_{9} - 6 \beta_{8} + \beta_{3} - 11 \beta_{1} - 18\)
\(\nu^{5}\)\(=\)\(-28 \beta_{11} - 27 \beta_{10} - 31 \beta_{9} - 8 \beta_{8} + 20 \beta_{7} - 31 \beta_{6} + 4 \beta_{5} + 28 \beta_{4} + 4 \beta_{3} + 45 \beta_{2} - 45 \beta_{1} + 73\)
\(\nu^{6}\)\(=\)\(71 \beta_{7} - 107 \beta_{6} + 20 \beta_{5} + 104 \beta_{4} + 112 \beta_{2} + 358\)
\(\nu^{7}\)\(=\)\(323 \beta_{11} + 315 \beta_{10} + 359 \beta_{9} + 52 \beta_{8} - 71 \beta_{3} + 391 \beta_{1} + 359\)
\(\nu^{8}\)\(=\)\(1100 \beta_{11} + 1032 \beta_{10} + 1307 \beta_{9} + 178 \beta_{8} - 922 \beta_{7} + 1303 \beta_{6} - 271 \beta_{5} - 1100 \beta_{4} - 271 \beta_{3} - 1125 \beta_{2} + 1125 \beta_{1} - 2225\)
\(\nu^{9}\)\(=\)\(-3216 \beta_{7} + 4425 \beta_{6} - 922 \beta_{5} - 3528 \beta_{4} - 3710 \beta_{2} - 11087\)
\(\nu^{10}\)\(=\)\(-11663 \beta_{11} - 11481 \beta_{10} - 12949 \beta_{9} - 944 \beta_{8} + 3216 \beta_{3} - 11399 \beta_{1} - 12949\)
\(\nu^{11}\)\(=\)\(-37759 \beta_{11} - 38023 \beta_{10} - 40447 \beta_{9} - 1751 \beta_{8} + 36008 \beta_{7} - 48742 \beta_{6} + 10719 \beta_{5} + 37759 \beta_{4} + 10719 \beta_{3} + 36955 \beta_{2} - 36955 \beta_{1} + 74714\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1 - \beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−0.975939 + 1.69038i
1.20634 2.08945i
−0.0149173 + 0.0258375i
1.62208 2.80952i
−0.928369 + 1.60798i
0.590804 1.02330i
−0.975939 1.69038i
1.20634 + 2.08945i
−0.0149173 0.0258375i
1.62208 + 2.80952i
−0.928369 1.60798i
0.590804 + 1.02330i
−1.08504 1.87935i −1.47594 2.55640i −1.35464 + 2.34630i 0 −3.20292 + 5.54761i 0.591620 1.53919 −2.85679 + 4.94811i 0
26.2 −1.08504 1.87935i 0.706345 + 1.22342i −1.35464 + 2.34630i 0 1.53283 2.65494i −1.76171 1.53919 0.502155 0.869757i 0
26.3 −0.155554 0.269427i −0.514917 0.891863i 0.951606 1.64823i 0 −0.160195 + 0.277466i −3.28038 −1.21432 0.969720 1.67960i 0
26.4 −0.155554 0.269427i 1.12208 + 1.94349i 0.951606 1.64823i 0 0.349087 0.604636i 3.96928 −1.21432 −1.01811 + 1.76343i 0
26.5 0.740597 + 1.28275i −1.42837 2.47401i −0.0969683 + 0.167954i 0 2.11569 3.66449i 3.78541 2.67513 −2.58048 + 4.46952i 0
26.6 0.740597 + 1.28275i 0.0908038 + 0.157277i −0.0969683 + 0.167954i 0 −0.134498 + 0.232958i −1.30422 2.67513 1.48351 2.56951i 0
201.1 −1.08504 + 1.87935i −1.47594 + 2.55640i −1.35464 2.34630i 0 −3.20292 5.54761i 0.591620 1.53919 −2.85679 4.94811i 0
201.2 −1.08504 + 1.87935i 0.706345 1.22342i −1.35464 2.34630i 0 1.53283 + 2.65494i −1.76171 1.53919 0.502155 + 0.869757i 0
201.3 −0.155554 + 0.269427i −0.514917 + 0.891863i 0.951606 + 1.64823i 0 −0.160195 0.277466i −3.28038 −1.21432 0.969720 + 1.67960i 0
201.4 −0.155554 + 0.269427i 1.12208 1.94349i 0.951606 + 1.64823i 0 0.349087 + 0.604636i 3.96928 −1.21432 −1.01811 1.76343i 0
201.5 0.740597 1.28275i −1.42837 + 2.47401i −0.0969683 0.167954i 0 2.11569 + 3.66449i 3.78541 2.67513 −2.58048 4.46952i 0
201.6 0.740597 1.28275i 0.0908038 0.157277i −0.0969683 0.167954i 0 −0.134498 0.232958i −1.30422 2.67513 1.48351 + 2.56951i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.f 12
5.b even 2 1 475.2.e.h yes 12
5.c odd 4 2 475.2.j.d 24
19.c even 3 1 inner 475.2.e.f 12
19.c even 3 1 9025.2.a.bz 6
19.d odd 6 1 9025.2.a.bs 6
95.h odd 6 1 9025.2.a.by 6
95.i even 6 1 475.2.e.h yes 12
95.i even 6 1 9025.2.a.br 6
95.m odd 12 2 475.2.j.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.e.f 12 1.a even 1 1 trivial
475.2.e.f 12 19.c even 3 1 inner
475.2.e.h yes 12 5.b even 2 1
475.2.e.h yes 12 95.i even 6 1
475.2.j.d 24 5.c odd 4 2
475.2.j.d 24 95.m odd 12 2
9025.2.a.br 6 95.i even 6 1
9025.2.a.bs 6 19.d odd 6 1
9025.2.a.by 6 95.h odd 6 1
9025.2.a.bz 6 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} + 4 T_{2}^{4} - T_{2}^{3} + 10 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 3 T + 10 T^{2} - T^{3} + 4 T^{4} + T^{5} + T^{6} )^{2} \)
$3$ \( 25 - 125 T + 715 T^{2} + 240 T^{3} + 809 T^{4} + 7 T^{5} + 500 T^{6} + 85 T^{7} + 109 T^{8} + 18 T^{9} + 17 T^{10} + 3 T^{11} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( -67 + 38 T + 123 T^{2} + 20 T^{3} - 21 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$11$ \( ( 247 - 871 T + 522 T^{2} + 9 T^{3} - 46 T^{4} + T^{5} + T^{6} )^{2} \)
$13$ \( 100489 - 223485 T + 575641 T^{2} + 86714 T^{3} + 148087 T^{4} + 14703 T^{5} + 24090 T^{6} + 3229 T^{7} + 1743 T^{8} + 98 T^{9} + 61 T^{10} + 5 T^{11} + T^{12} \)
$17$ \( 1 + 107 T + 11529 T^{2} - 8454 T^{3} + 12047 T^{4} - 893 T^{5} + 4406 T^{6} - 899 T^{7} + 655 T^{8} - 34 T^{9} + 33 T^{10} - 3 T^{11} + T^{12} \)
$19$ \( 47045881 + 14856594 T + 1954815 T^{2} + 370386 T^{3} + 101802 T^{4} + 16302 T^{5} + 79 T^{6} + 858 T^{7} + 282 T^{8} + 54 T^{9} + 15 T^{10} + 6 T^{11} + T^{12} \)
$23$ \( 1151329 - 2302658 T + 4232985 T^{2} - 1448550 T^{3} + 885458 T^{4} - 137266 T^{5} + 116633 T^{6} - 16678 T^{7} + 4870 T^{8} - 314 T^{9} + 93 T^{10} - 6 T^{11} + T^{12} \)
$29$ \( 163353961 + 34419233 T + 47384589 T^{2} - 10935534 T^{3} + 8218031 T^{4} - 924611 T^{5} + 331046 T^{6} - 11057 T^{7} + 8815 T^{8} - 130 T^{9} + 117 T^{10} + 3 T^{11} + T^{12} \)
$31$ \( ( -631 + 1863 T + 764 T^{2} - 189 T^{3} - 64 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$37$ \( ( -64 + 1056 T + 288 T^{2} - 168 T^{3} - 32 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$41$ \( 1739761 + 1354613 T + 2479249 T^{2} + 286342 T^{3} + 1741339 T^{4} + 635125 T^{5} + 245262 T^{6} + 37483 T^{7} + 7475 T^{8} + 794 T^{9} + 145 T^{10} + 11 T^{11} + T^{12} \)
$43$ \( 829921 + 1518637 T + 1924371 T^{2} + 1314032 T^{3} + 686083 T^{4} + 243355 T^{5} + 74262 T^{6} + 17515 T^{7} + 4163 T^{8} + 768 T^{9} + 131 T^{10} + 13 T^{11} + T^{12} \)
$47$ \( 537266041 - 183809470 T + 121690023 T^{2} - 41259582 T^{3} + 19972138 T^{4} - 5575722 T^{5} + 1419407 T^{6} - 211818 T^{7} + 27642 T^{8} - 1862 T^{9} + 167 T^{10} - 6 T^{11} + T^{12} \)
$53$ \( 116868943321 + 28736151938 T + 9051617913 T^{2} + 1270240062 T^{3} + 273530066 T^{4} + 32694250 T^{5} + 5385041 T^{6} + 470650 T^{7} + 56866 T^{8} + 3902 T^{9} + 393 T^{10} + 18 T^{11} + T^{12} \)
$59$ \( 134397649 - 389385684 T + 978244661 T^{2} - 426814164 T^{3} + 159075790 T^{4} - 20063984 T^{5} + 3327161 T^{6} - 60248 T^{7} + 44534 T^{8} - 300 T^{9} + 253 T^{10} + 4 T^{11} + T^{12} \)
$61$ \( 305935081 + 525132293 T + 662278559 T^{2} + 353918480 T^{3} + 140095873 T^{4} + 27524283 T^{5} + 4663478 T^{6} + 495207 T^{7} + 63649 T^{8} + 5680 T^{9} + 527 T^{10} + 25 T^{11} + T^{12} \)
$67$ \( 145926400 - 158489600 T + 216492160 T^{2} + 30201600 T^{3} + 21263744 T^{4} + 1498912 T^{5} + 1052864 T^{6} + 91072 T^{7} + 27688 T^{8} + 504 T^{9} + 200 T^{10} + 6 T^{11} + T^{12} \)
$71$ \( 135885649 - 349919826 T + 806367199 T^{2} - 323536874 T^{3} + 166936790 T^{4} + 35890178 T^{5} + 9976043 T^{6} + 797342 T^{7} + 88934 T^{8} + 4330 T^{9} + 463 T^{10} + 18 T^{11} + T^{12} \)
$73$ \( 4699788025 - 462403475 T + 567747015 T^{2} + 30131360 T^{3} + 47699269 T^{4} + 989995 T^{5} + 1144758 T^{6} + 17239 T^{7} + 20093 T^{8} + 144 T^{9} + 167 T^{10} + T^{11} + T^{12} \)
$79$ \( 106504975201 + 60624593515 T + 26926848793 T^{2} + 5367195402 T^{3} + 944078261 T^{4} + 83184961 T^{5} + 10171432 T^{6} + 565115 T^{7} + 85285 T^{8} + 2256 T^{9} + 331 T^{10} + 3 T^{11} + T^{12} \)
$83$ \( ( 378053 + 120965 T - 5864 T^{2} - 3729 T^{3} - 74 T^{4} + 23 T^{5} + T^{6} )^{2} \)
$89$ \( 122226452881 - 7908155580 T + 10282187123 T^{2} - 894930972 T^{3} + 723106926 T^{4} - 51342876 T^{5} + 12378927 T^{6} + 22380 T^{7} + 92510 T^{8} + 684 T^{9} + 451 T^{10} + 12 T^{11} + T^{12} \)
$97$ \( 25482056161 + 3286642659 T + 3583004411 T^{2} - 299226492 T^{3} + 347222589 T^{4} - 7029399 T^{5} + 6229806 T^{6} - 248487 T^{7} + 82877 T^{8} - 1644 T^{9} + 331 T^{10} + 3 T^{11} + T^{12} \)
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