Properties

Label 775.2.e.g
Level $775$
Weight $2$
Character orbit 775.e
Analytic conductor $6.188$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 23x^{4} + x^{3} + 16x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{6} + \beta_{4} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{4} + ( - 2 \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + 2) q^{8} + ( - \beta_{5} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{6} + \beta_{4} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{4} + ( - 2 \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + 2) q^{8} + ( - \beta_{5} + \beta_1) q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{11} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{12} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{13} + ( - \beta_{7} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 1) q^{14} + ( - \beta_{7} + 3 \beta_{6} + 1) q^{16} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{17} + (\beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_1 + 3) q^{18} + (2 \beta_{7} - \beta_{6} - 2 \beta_{3} + \beta_1 - 2) q^{19} + (4 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_1 + 2) q^{21} + (\beta_{5} + \beta_{3}) q^{22} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{2}) q^{23} + (4 \beta_{6} + \beta_{4} + \beta_{2} - 4 \beta_1) q^{24} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_1 - 2) q^{26} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{2} - 1) q^{27} + ( - \beta_{7} - 4 \beta_{6} + \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{28} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{2} - 2) q^{29} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} - 3 \beta_1 + 2) q^{31} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{2} + 1) q^{32} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{2} - 3) q^{33} + (2 \beta_{7} + 2 \beta_{6} - 5 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 2) q^{34} + ( - 2 \beta_{4} - \beta_{3} + \beta_1 - 2) q^{36} + (2 \beta_{6} + 4 \beta_{4} - \beta_{2} - 2 \beta_1) q^{37} + ( - 3 \beta_{6} - 3 \beta_{2} + 3 \beta_1) q^{38} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} - 4) q^{39} + (2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 2 \beta_1 - 3) q^{41} + ( - 4 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 6 \beta_1 - 3) q^{42} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{43} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 - 2) q^{44} + (2 \beta_{7} - 5 \beta_{6} - \beta_{5} - \beta_{2} - 2) q^{46} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 3 \beta_{2}) q^{47} + (\beta_{7} + \beta_{6} - 5 \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{48} + ( - 10 \beta_{4} + \beta_{3} + 5 \beta_1 - 10) q^{49} + ( - 5 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2) q^{51} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 3) q^{52} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{53} + ( - 2 \beta_{7} + 4 \beta_{6} + 3) q^{54} + (3 \beta_{7} - 5 \beta_{6} + 6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 3) q^{56} + (3 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{57} + (2 \beta_{7} - 7 \beta_{6} - \beta_{5} - \beta_{2} - 2) q^{58} + ( - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 3) q^{59} + (\beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{2}) q^{61} + (3 \beta_{6} - \beta_{5} - 7 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{62} + ( - \beta_{7} + 4 \beta_{6} - 5 \beta_{5} - 5 \beta_{2} - 1) q^{63} + (\beta_{6} + 4) q^{64} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 2) q^{66} + ( - 4 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_1 + 3) q^{67} + ( - \beta_{7} + 7 \beta_{6} + 3 \beta_{4} + \beta_{3} + 4 \beta_{2} - 7 \beta_1 + 1) q^{68} + (2 \beta_{7} - 4 \beta_{6} - 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 2) q^{69} + (2 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 2 \beta_1 + 4) q^{71} + ( - 5 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 5) q^{72} + ( - 4 \beta_{4} + 2 \beta_{3} - 4) q^{73} + (\beta_{7} - 4 \beta_{6} - 5 \beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 1) q^{74} + ( - \beta_{7} - 7 \beta_{6} + 3 \beta_{4} + \beta_{3} - 3 \beta_{2} + 7 \beta_1 + 1) q^{76} + (4 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{2} - 11) q^{77} + ( - \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{2}) q^{78} + ( - 3 \beta_{7} + 2 \beta_{4} + 3 \beta_{3} + 3) q^{79} + ( - 3 \beta_{7} - \beta_{4} + 3 \beta_{3} + \beta_{2} + 3) q^{81} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_1 + 1) q^{82} + (6 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1 - 2) q^{83} + (2 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} + \beta_1 + 8) q^{84} + ( - 2 \beta_{7} + 3 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 2) q^{86} + (2 \beta_{7} - 6 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 6 \beta_1 - 2) q^{87} + ( - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 2) q^{88} + (3 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} + 5) q^{89} + (3 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{2} - 10) q^{91} + (\beta_{7} - 7 \beta_{6} - 3 \beta_{5} - 3 \beta_{2} - 8) q^{92} + (3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 8) q^{93} + (3 \beta_{7} - 8 \beta_{6} - \beta_{5} - \beta_{2} - 5) q^{94} + ( - 4 \beta_{7} + 5 \beta_{6} - \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 4) q^{96} + ( - 5 \beta_{7} - \beta_{6} + 5) q^{97} + (4 \beta_{5} + 11 \beta_{4} - 6 \beta_1 + 11) q^{98} + ( - \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 3 q^{3} + 6 q^{4} + 10 q^{6} + q^{7} + 18 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 3 q^{3} + 6 q^{4} + 10 q^{6} + q^{7} + 18 q^{8} - q^{9} + 4 q^{11} + 8 q^{12} - q^{13} - 8 q^{14} + 10 q^{16} + 12 q^{17} + 11 q^{18} - 5 q^{19} + 9 q^{21} - 10 q^{23} + 2 q^{24} - 12 q^{26} + 6 q^{27} - 4 q^{28} - 26 q^{29} + 19 q^{31} + 28 q^{32} - 8 q^{33} + 24 q^{34} - 5 q^{36} - 16 q^{37} - 9 q^{38} - 22 q^{39} - 4 q^{41} - 6 q^{42} + q^{43} - 7 q^{44} - 22 q^{46} - 20 q^{47} + 27 q^{48} - 37 q^{49} - 12 q^{51} - 21 q^{52} + q^{53} + 24 q^{54} - 29 q^{56} + 4 q^{57} - 26 q^{58} + 6 q^{59} - 2 q^{61} + 17 q^{62} - 24 q^{63} + 34 q^{64} + 2 q^{66} + 7 q^{67} + 5 q^{68} - 6 q^{69} + 12 q^{71} - 14 q^{72} - 20 q^{73} + 18 q^{74} - 23 q^{76} - 58 q^{77} - 22 q^{78} - 2 q^{79} + 12 q^{81} + 18 q^{82} + 4 q^{83} + 41 q^{84} - 13 q^{86} + 10 q^{88} + 54 q^{89} - 44 q^{91} - 86 q^{92} - 55 q^{93} - 48 q^{94} + 13 q^{96} + 18 q^{97} + 46 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 23x^{4} + x^{3} + 16x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\nu^{7} + 857\nu^{6} - 474\nu^{5} + 4932\nu^{4} + 5726\nu^{3} + 14569\nu^{2} + 4405\nu + 2370 ) / 5817 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -146\nu^{7} - 352\nu^{6} + 792\nu^{5} - 4191\nu^{4} + 1232\nu^{3} - 5984\nu^{2} + 20203\nu - 3960 ) / 5817 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -166\nu^{7} + 556\nu^{6} - 1251\nu^{5} + 1530\nu^{4} - 1946\nu^{3} + 9452\nu^{2} - 1174\nu + 438 ) / 5817 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 197\nu^{7} - 800\nu^{6} + 1800\nu^{5} - 3708\nu^{4} + 2800\nu^{3} - 13600\nu^{2} - 3793\nu - 9000 ) / 5817 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -130\nu^{7} + 85\nu^{6} - 676\nu^{5} - 624\nu^{4} - 3206\nu^{3} - 494\nu^{2} - 312\nu - 498 ) / 1939 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -565\nu^{7} + 444\nu^{6} - 2938\nu^{5} - 2712\nu^{4} - 11249\nu^{3} - 2147\nu^{2} - 1356\nu - 822 ) / 1939 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + 2\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + \beta_{3} - 9\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{7} + 30\beta_{6} - 11\beta_{4} + 6\beta_{3} + 10\beta_{2} - 30\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{7} + 67\beta_{6} + 36\beta_{5} + 36\beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 77\beta_{5} + 88\beta_{4} - 36\beta_{3} + 193\beta _1 + 88 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
501.1
−0.844316 1.46240i
−0.392238 0.679376i
0.434993 + 0.753430i
1.30156 + 2.25437i
−0.844316 + 1.46240i
−0.392238 + 0.679376i
0.434993 0.753430i
1.30156 2.25437i
−1.68863 −1.34432 + 2.32842i 0.851477 0 2.27005 3.93185i 2.52642 4.37588i 1.93943 −2.11437 3.66220i 0
501.2 −0.784476 −0.892238 + 1.54540i −1.38460 0 0.699939 1.21233i −2.44756 + 4.23929i 2.65514 −0.0921773 0.159656i 0
501.3 0.869986 −0.0650072 + 0.112596i −1.24312 0 −0.0565553 + 0.0979567i 1.58680 2.74842i −2.82147 1.49155 + 2.58344i 0
501.4 2.60312 0.801561 1.38834i 4.77625 0 2.08656 3.61403i −1.16566 + 2.01898i 7.22690 0.215000 + 0.372390i 0
676.1 −1.68863 −1.34432 2.32842i 0.851477 0 2.27005 + 3.93185i 2.52642 + 4.37588i 1.93943 −2.11437 + 3.66220i 0
676.2 −0.784476 −0.892238 1.54540i −1.38460 0 0.699939 + 1.21233i −2.44756 4.23929i 2.65514 −0.0921773 + 0.159656i 0
676.3 0.869986 −0.0650072 0.112596i −1.24312 0 −0.0565553 0.0979567i 1.58680 + 2.74842i −2.82147 1.49155 2.58344i 0
676.4 2.60312 0.801561 + 1.38834i 4.77625 0 2.08656 + 3.61403i −1.16566 2.01898i 7.22690 0.215000 0.372390i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 676.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.e.g 8
5.b even 2 1 155.2.e.c 8
5.c odd 4 2 775.2.o.e 16
31.c even 3 1 inner 775.2.e.g 8
155.i odd 6 1 4805.2.a.k 4
155.j even 6 1 155.2.e.c 8
155.j even 6 1 4805.2.a.i 4
155.o odd 12 2 775.2.o.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.e.c 8 5.b even 2 1
155.2.e.c 8 155.j even 6 1
775.2.e.g 8 1.a even 1 1 trivial
775.2.e.g 8 31.c even 3 1 inner
775.2.o.e 16 5.c odd 4 2
775.2.o.e 16 155.o odd 12 2
4805.2.a.i 4 155.j even 6 1
4805.2.a.k 4 155.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} - 5 T^{2} + T + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + 11 T^{6} + 10 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + 33 T^{6} - 12 T^{5} + \cdots + 33489 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + 20 T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + 17 T^{6} + 48 T^{5} + \cdots + 225 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + 137 T^{6} + \cdots + 1067089 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + 52 T^{6} + \cdots + 35721 \) Copy content Toggle raw display
$23$ \( (T^{4} + 5 T^{3} - 25 T^{2} - 57 T + 157)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 13 T^{3} + 29 T^{2} - 65 T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 19 T^{7} + 244 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + 190 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + 77 T^{6} + \cdots + 45369 \) Copy content Toggle raw display
$43$ \( T^{8} - T^{7} + 59 T^{6} - 350 T^{5} + \cdots + 32761 \) Copy content Toggle raw display
$47$ \( (T^{4} + 10 T^{3} - 42 T^{2} - 283 T + 321)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - T^{7} + 90 T^{6} - 533 T^{5} + \cdots + 73441 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + 165 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} - 151 T^{2} - 257 T + 321)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 7 T^{7} + 143 T^{6} + \cdots + 3613801 \) Copy content Toggle raw display
$71$ \( T^{8} - 12 T^{7} + 175 T^{6} + \cdots + 164025 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} + 284 T^{6} + \cdots + 112896 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + 79 T^{6} + \cdots + 80089 \) Copy content Toggle raw display
$83$ \( T^{8} - 4 T^{7} + 271 T^{6} + \cdots + 233998209 \) Copy content Toggle raw display
$89$ \( (T^{4} - 27 T^{3} + 206 T^{2} - 274 T - 459)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 9 T^{3} - 195 T^{2} + 1469 T - 2177)^{2} \) Copy content Toggle raw display
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