Properties

Label 475.2.e.h
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: \( x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} - 552 x^{3} + 1107 x^{2} + 33 x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_{7}) q^{2} + ( - \beta_{9} - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{10} - \beta_{8}) q^{4} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{7} - \beta_{4} - 1) q^{7} + ( - \beta_{6} - 1) q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_{7}) q^{2} + ( - \beta_{9} - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{10} - \beta_{8}) q^{4} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{7} - \beta_{4} - 1) q^{7} + ( - \beta_{6} - 1) q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_1 - 1) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2}) q^{11} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{12} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{3} - \beta_1 + 1) q^{13} + ( - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{3}) q^{14} + (2 \beta_{10} - \beta_{9} + 2 \beta_{6}) q^{16} + (\beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{17} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{2}) q^{18} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - 1) q^{19} + ( - 2 \beta_{10} + \beta_{9} - 3 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{21} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{22} + ( - 3 \beta_{10} - \beta_{9} - \beta_{3} - 1) q^{23} + (\beta_{11} + 2 \beta_{9} - \beta_{7} - \beta_{4} + \beta_{2} - \beta_1) q^{24} + ( - \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{2} + 2) q^{26} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - 3) q^{27} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_1 + 1) q^{28} + ( - 2 \beta_{11} - \beta_{10} - \beta_{8} + \beta_{3} + \beta_1) q^{29} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{31} + (2 \beta_{10} - 3 \beta_{8}) q^{32} + (\beta_{10} + 4 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{33} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{3} - 3 \beta_1 + 1) q^{34} + (3 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{36} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{37} + (\beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{38}+ \cdots + (\beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 6 \beta_{8} + 2 \beta_{3} + 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 3 q^{3} - 2 q^{4} + q^{6} - 4 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 3 q^{3} - 2 q^{4} + q^{6} - 4 q^{7} - 12 q^{8} - 7 q^{9} - 2 q^{11} - 14 q^{12} + 5 q^{13} + 6 q^{14} + 6 q^{16} - 3 q^{17} - 14 q^{18} - 6 q^{19} - 3 q^{21} + 9 q^{22} - 6 q^{23} - 11 q^{24} + 38 q^{26} - 36 q^{27} - 4 q^{28} - 3 q^{29} - 6 q^{31} - 6 q^{32} - 18 q^{33} + q^{34} - 13 q^{36} + 12 q^{37} + 18 q^{38} + 16 q^{39} - 11 q^{41} - 11 q^{42} + 13 q^{43} - 21 q^{44} - 24 q^{46} - 6 q^{47} - 19 q^{48} + 8 q^{49} + 17 q^{51} - q^{52} + 18 q^{53} - 18 q^{54} + 8 q^{56} + 20 q^{57} - 10 q^{58} - 4 q^{59} - 25 q^{61} - 21 q^{62} + 43 q^{63} - 44 q^{64} - 34 q^{66} + 6 q^{67} + 2 q^{68} + 26 q^{69} - 18 q^{71} + 13 q^{72} + q^{73} + 6 q^{74} + 24 q^{76} + 22 q^{77} + 72 q^{78} - 3 q^{79} - 2 q^{81} + 31 q^{82} + 46 q^{83} + 74 q^{84} - 9 q^{86} - 22 q^{87} - 22 q^{88} - 12 q^{89} + 11 q^{91} + 28 q^{92} - 13 q^{93} + 16 q^{94} - 26 q^{96} + 3 q^{97} - 22 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3959208 \nu^{11} - 46922925 \nu^{10} + 38785859 \nu^{9} - 621660414 \nu^{8} - 801763109 \nu^{7} - 3678052623 \nu^{6} + \cdots - 68499350729 ) / 66494854340 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 690858 \nu^{11} - 11701549 \nu^{10} + 76199247 \nu^{9} - 313859676 \nu^{8} + 833713615 \nu^{7} - 1701731499 \nu^{6} + 2957264154 \nu^{5} + \cdots - 6178545 ) / 1955731010 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54841341 \nu^{11} + 153015320 \nu^{10} - 731712017 \nu^{9} + 251450977 \nu^{8} - 3244495858 \nu^{7} + 1416907499 \nu^{6} + \cdots - 263970961763 ) / 66494854340 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 134211789 \nu^{11} - 998274384 \nu^{10} + 3072233475 \nu^{9} - 7500316929 \nu^{8} + 1833052570 \nu^{7} - 27189880401 \nu^{6} + \cdots - 148724948799 ) / 66494854340 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5051595 \nu^{11} + 9282581 \nu^{10} - 64765518 \nu^{9} - 29949491 \nu^{8} - 345971017 \nu^{7} - 412414770 \nu^{6} - 1358444363 \nu^{5} + \cdots - 4335298296 ) / 1955731010 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6178545 \nu^{11} + 17844777 \nu^{10} - 93333716 \nu^{9} + 35014563 \nu^{8} - 359601729 \nu^{7} - 259108930 \nu^{6} - 1288684281 \nu^{5} + \cdots - 612082342 ) / 1955731010 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 469619124 \nu^{11} + 1742271771 \nu^{10} - 9648336987 \nu^{9} + 16279393858 \nu^{8} - 67720087321 \nu^{7} + 86418671699 \nu^{6} + \cdots + 402309009 ) / 66494854340 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2004496389 \nu^{11} + 6017448375 \nu^{10} - 34029515688 \nu^{9} + 36042149143 \nu^{8} - 217868445987 \nu^{7} + \cdots - 65422139102 ) / 66494854340 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4220156417 \nu^{11} + 12953281828 \nu^{10} - 72904583371 \nu^{9} + 81775705089 \nu^{8} - 469415474758 \nu^{7} + \cdots + 2372026047 ) / 66494854340 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6424307742 \nu^{11} - 19688524501 \nu^{10} + 111043957893 \nu^{9} - 124776050724 \nu^{8} + 717279166315 \nu^{7} - 650339598681 \nu^{6} + \cdots - 3616495515 ) / 66494854340 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 3\beta_{9} + \beta_{8} - \beta_{4} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 2\beta_{6} - 2\beta_{4} - 6\beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{11} - 6\beta_{10} - 18\beta_{9} - 6\beta_{8} + \beta_{3} - 11\beta _1 - 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 28 \beta_{11} - 27 \beta_{10} - 31 \beta_{9} - 8 \beta_{8} + 20 \beta_{7} - 31 \beta_{6} + 4 \beta_{5} + 28 \beta_{4} + 4 \beta_{3} + 45 \beta_{2} - 45 \beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 71\beta_{7} - 107\beta_{6} + 20\beta_{5} + 104\beta_{4} + 112\beta_{2} + 358 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 323\beta_{11} + 315\beta_{10} + 359\beta_{9} + 52\beta_{8} - 71\beta_{3} + 391\beta _1 + 359 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1100 \beta_{11} + 1032 \beta_{10} + 1307 \beta_{9} + 178 \beta_{8} - 922 \beta_{7} + 1303 \beta_{6} - 271 \beta_{5} - 1100 \beta_{4} - 271 \beta_{3} - 1125 \beta_{2} + 1125 \beta _1 - 2225 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3216\beta_{7} + 4425\beta_{6} - 922\beta_{5} - 3528\beta_{4} - 3710\beta_{2} - 11087 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -11663\beta_{11} - 11481\beta_{10} - 12949\beta_{9} - 944\beta_{8} + 3216\beta_{3} - 11399\beta _1 - 12949 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 37759 \beta_{11} - 38023 \beta_{10} - 40447 \beta_{9} - 1751 \beta_{8} + 36008 \beta_{7} - 48742 \beta_{6} + 10719 \beta_{5} + 37759 \beta_{4} + 10719 \beta_{3} + 36955 \beta_{2} - 36955 \beta _1 + 74714 \) Copy content Toggle raw display

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
0.590804 1.02330i
−0.928369 + 1.60798i
1.62208 2.80952i
−0.0149173 + 0.0258375i
1.20634 2.08945i
−0.975939 + 1.69038i
0.590804 + 1.02330i
−0.928369 1.60798i
1.62208 + 2.80952i
−0.0149173 0.0258375i
1.20634 + 2.08945i
−0.975939 1.69038i
−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
26.2 −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
26.3 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
26.4 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
26.5 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
26.6 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
201.1 −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
201.2 −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
201.3 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
201.4 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
201.5 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
201.6 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
\(n\): 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.h yes 12
5.b even 2 1 475.2.e.f 12
5.c odd 4 2 475.2.j.d 24
19.c even 3 1 inner 475.2.e.h yes 12
19.c even 3 1 9025.2.a.br 6
19.d odd 6 1 9025.2.a.by 6
95.h odd 6 1 9025.2.a.bs 6
95.i even 6 1 475.2.e.f 12
95.i even 6 1 9025.2.a.bz 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 5.b even 2 1
475.2.e.f 12 95.i even 6 1
475.2.e.h yes 12 1.a even 1 1 trivial
475.2.e.h yes 12 19.c even 3 1 inner
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 19.c even 3 1
9025.2.a.bs 6 95.h odd 6 1
9025.2.a.by 6 19.d odd 6 1
9025.2.a.bz 6 95.i even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} + 4 T^{4} + T^{3} + 10 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + 17 T^{10} - 18 T^{9} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} - 21 T^{4} - 20 T^{3} + \cdots - 67)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + T^{5} - 46 T^{4} + 9 T^{3} + 522 T^{2} + \cdots + 247)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 5 T^{11} + 61 T^{10} + \cdots + 100489 \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} + 33 T^{10} + 34 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + 15 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + 6 T^{11} + 93 T^{10} + \cdots + 1151329 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + 117 T^{10} + \cdots + 163353961 \) Copy content Toggle raw display
$31$ \( (T^{6} + 3 T^{5} - 64 T^{4} - 189 T^{3} + \cdots - 631)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} - 32 T^{4} + 168 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 11 T^{11} + 145 T^{10} + \cdots + 1739761 \) Copy content Toggle raw display
$43$ \( T^{12} - 13 T^{11} + 131 T^{10} + \cdots + 829921 \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} + 167 T^{10} + \cdots + 537266041 \) Copy content Toggle raw display
$53$ \( T^{12} - 18 T^{11} + \cdots + 116868943321 \) Copy content Toggle raw display
$59$ \( T^{12} + 4 T^{11} + 253 T^{10} + \cdots + 134397649 \) Copy content Toggle raw display
$61$ \( T^{12} + 25 T^{11} + \cdots + 305935081 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + 200 T^{10} + \cdots + 145926400 \) Copy content Toggle raw display
$71$ \( T^{12} + 18 T^{11} + \cdots + 135885649 \) Copy content Toggle raw display
$73$ \( T^{12} - T^{11} + 167 T^{10} + \cdots + 4699788025 \) Copy content Toggle raw display
$79$ \( T^{12} + 3 T^{11} + \cdots + 106504975201 \) Copy content Toggle raw display
$83$ \( (T^{6} - 23 T^{5} - 74 T^{4} + \cdots + 378053)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 12 T^{11} + \cdots + 122226452881 \) Copy content Toggle raw display
$97$ \( T^{12} - 3 T^{11} + \cdots + 25482056161 \) Copy content Toggle raw display
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