Properties

Label 775.2.b.e
Level $775$
Weight $2$
Character orbit 775.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{3} - 2) q^{4} + (\beta_{3} + 2) q^{6} - \beta_{2} q^{7} + (3 \beta_{7} + 2 \beta_{4} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{6} - 2 \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{3} - 2) q^{4} + (\beta_{3} + 2) q^{6} - \beta_{2} q^{7} + (3 \beta_{7} + 2 \beta_{4} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{6} - 2 \beta_{3} - 2) q^{9} + ( - 2 \beta_{6} - \beta_{3} - 2) q^{11} + \beta_1 q^{12} + ( - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{6} - \beta_{5} - 2) q^{14} + ( - 4 \beta_{6} - \beta_{5} - 3 \beta_{3} + 2) q^{16} + (2 \beta_{7} + \beta_{4} - \beta_{2}) q^{17} + ( - 3 \beta_{7} - 4 \beta_{4} + \beta_{2}) q^{18} + ( - \beta_{5} - 1) q^{19} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{21} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{22} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_{2}) q^{23} + (2 \beta_{6} + 2 \beta_{3} + 4) q^{24} + (3 \beta_{6} - \beta_{5} - 2) q^{26} + (3 \beta_{4} - \beta_{2} - \beta_1) q^{27} + (2 \beta_{2} + 2 \beta_1) q^{28} + ( - \beta_{6} + \beta_{5} + \beta_{3} - 2) q^{29} + q^{31} + ( - 7 \beta_{7} - 2 \beta_{4} + 5 \beta_{2} + 3 \beta_1) q^{32} + (4 \beta_{4} + \beta_{2} - 2 \beta_1) q^{33} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{3} + 4) q^{34} + (2 \beta_{6} + \beta_{5} + 3 \beta_{3} - 6) q^{36} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{37} + 3 \beta_{2} q^{38} + ( - 3 \beta_{6} + 3 \beta_{5} + \beta_{3}) q^{39} + (\beta_{5} - \beta_{3} + 3) q^{41} + (2 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{42} + (2 \beta_{7} + \beta_{4} - \beta_{2} + \beta_1) q^{43} + (6 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 4) q^{44} + (\beta_{6} - \beta_{5} + 4 \beta_{3} - 6) q^{46} + ( - 2 \beta_{4} + 2 \beta_1) q^{47} + (2 \beta_{7} + 4 \beta_{4} - 2 \beta_{2}) q^{48} + ( - \beta_{6} - \beta_{5} + 1) q^{49} + (\beta_{5} + \beta_{3} + 1) q^{51} + (4 \beta_{7} - 2 \beta_{2} - 2 \beta_1) q^{52} + (3 \beta_{4} + 2 \beta_{2} + \beta_1) q^{53} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{3} - 8) q^{54} + 4 \beta_{6} q^{56} + (2 \beta_{7} + \beta_{4} - 3 \beta_{2} + 4 \beta_1) q^{57} + (4 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{58} + ( - \beta_{6} - 2 \beta_{5} - \beta_{3} - 3) q^{59} + (\beta_{6} - \beta_{5} + 6) q^{61} - \beta_{7} q^{62} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2} - 5 \beta_1) q^{63} + (10 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} - 10) q^{64} + ( - 3 \beta_{6} + \beta_{5} - 4 \beta_{3} - 6) q^{66} + ( - 2 \beta_{7} + \beta_{2} + \beta_1) q^{67} + ( - 10 \beta_{7} - 4 \beta_{4} + 4 \beta_{2} + 3 \beta_1) q^{68} + ( - 3 \beta_{6} + \beta_{5} - \beta_{3} - 6) q^{69} + (\beta_{6} + 2 \beta_{3} + 1) q^{71} + (9 \beta_{7} - 2 \beta_{4} - 3 \beta_{2} - \beta_1) q^{72} + ( - 4 \beta_{7} - 3 \beta_{4} + \beta_1) q^{73} + ( - \beta_{6} + \beta_{5} + \beta_{3} + 4) q^{74} + (3 \beta_{6} + \beta_{5} + 4) q^{76} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2} - \beta_1) q^{77} + (2 \beta_{7} + 2 \beta_{4} - 6 \beta_{2} + 7 \beta_1) q^{78} + ( - 3 \beta_{3} + 4) q^{79} + (\beta_{3} + 9) q^{81} + ( - 4 \beta_{7} - 2 \beta_{4} - 3 \beta_{2} - \beta_1) q^{82} + ( - \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{83} - 2 \beta_{3} q^{84} + ( - \beta_{6} - \beta_{5} - 3 \beta_{3} + 4) q^{86} + ( - 2 \beta_{7} + 5 \beta_{2} - 5 \beta_1) q^{87} + (12 \beta_{7} - 8 \beta_{2} - 4 \beta_1) q^{88} + ( - 4 \beta_{6} + \beta_{3} + 2) q^{89} + ( - \beta_{6} - \beta_{5} - 4 \beta_{3} - 6) q^{91} + (10 \beta_{7} + 4 \beta_{4} + 2 \beta_1) q^{92} - \beta_{4} q^{93} + (4 \beta_{6} + 2 \beta_{3} + 4) q^{94} + ( - 2 \beta_{5} - 2 \beta_{3} + 4) q^{96} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{97} + ( - 3 \beta_{7} + 4 \beta_{2} + 2 \beta_1) q^{98} + (\beta_{6} - 3 \beta_{5} + 4 \beta_{3} + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9} - 12 q^{11} - 16 q^{14} + 22 q^{16} - 10 q^{19} + 4 q^{21} + 28 q^{24} - 24 q^{26} - 12 q^{29} + 8 q^{31} + 36 q^{34} - 50 q^{36} + 12 q^{39} + 26 q^{41} - 40 q^{44} - 52 q^{46} + 8 q^{49} + 10 q^{51} - 60 q^{54} - 8 q^{56} - 26 q^{59} + 44 q^{61} - 94 q^{64} - 40 q^{66} - 40 q^{69} + 6 q^{71} + 36 q^{74} + 28 q^{76} + 32 q^{79} + 72 q^{81} + 32 q^{86} + 24 q^{89} - 48 q^{91} + 24 q^{94} + 28 q^{96} + 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -215\nu^{7} + 864\nu^{6} + 144\nu^{5} + 454\nu^{4} - 7744\nu^{3} + 30084\nu^{2} - 6214\nu - 956 ) / 10798 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -216\nu^{7} - 36\nu^{6} - 6\nu^{5} + 431\nu^{4} - 6876\nu^{3} + 1446\nu^{2} - 5590\nu - 860 ) / 5399 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -216\nu^{7} - 36\nu^{6} - 6\nu^{5} + 431\nu^{4} - 6876\nu^{3} + 1446\nu^{2} + 5208\nu - 860 ) / 5399 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -655\nu^{7} - 1009\nu^{6} - 1068\nu^{5} + 1132\nu^{4} - 14552\nu^{3} - 12562\nu^{2} - 12402\nu - 1908 ) / 10798 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 714\nu^{7} + 119\nu^{6} - 880\nu^{5} - 5174\nu^{4} + 17330\nu^{3} - 3880\nu^{2} - 13616\nu - 36150 ) / 10798 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -858\nu^{7} - 143\nu^{6} + 876\nu^{5} + 1862\nu^{4} - 21914\nu^{3} + 4844\nu^{2} + 17088\nu - 14814 ) / 10798 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1955\nu^{7} - 574\nu^{6} + 1704\nu^{5} - 3626\nu^{4} + 52336\nu^{3} - 43532\nu^{2} + 43446\nu + 6684 ) / 21596 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{4} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - 4\beta_{3} - 4\beta_{2} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{6} - 3\beta_{5} + \beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14\beta_{7} + 13\beta_{6} + \beta_{5} + 2\beta_{4} - 20\beta_{3} + 20\beta_{2} + 9\beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{7} - 32\beta_{4} - 10\beta_{2} - 37\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 44\beta_{7} - 38\beta_{6} - 6\beta_{5} + 12\beta_{4} + 53\beta_{3} + 53\beta_{2} + 32\beta _1 - 64 \) Copy content Toggle raw display

Character values

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

\(n\) \(251\) \(652\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
−0.520627 + 0.520627i
−1.71822 1.71822i
1.48716 1.48716i
0.751690 + 0.751690i
0.751690 0.751690i
1.48716 + 1.48716i
−1.71822 + 1.71822i
−0.520627 0.520627i
2.80027i 0.342376i −5.84153 0 0.958747 1.04125i 10.7573i 2.88278 0
249.2 2.27244i 0.632112i −3.16400 0 −1.43644 3.43644i 2.64511i 2.60043 0
249.3 1.62946i 3.05273i −0.655151 0 4.97431 2.97431i 2.19138i −6.31916 0
249.4 1.15729i 3.02722i 0.660672 0 3.50338 1.50338i 3.07918i −6.16405 0
249.5 1.15729i 3.02722i 0.660672 0 3.50338 1.50338i 3.07918i −6.16405 0
249.6 1.62946i 3.05273i −0.655151 0 4.97431 2.97431i 2.19138i −6.31916 0
249.7 2.27244i 0.632112i −3.16400 0 −1.43644 3.43644i 2.64511i 2.60043 0
249.8 2.80027i 0.342376i −5.84153 0 0.958747 1.04125i 10.7573i 2.88278 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.8
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 775.2.b.e 8
5.b even 2 1 inner 775.2.b.e 8
5.c odd 4 1 155.2.a.d 4
5.c odd 4 1 775.2.a.g 4
15.e even 4 1 1395.2.a.m 4
15.e even 4 1 6975.2.a.bj 4
20.e even 4 1 2480.2.a.z 4
35.f even 4 1 7595.2.a.q 4
40.i odd 4 1 9920.2.a.ch 4
40.k even 4 1 9920.2.a.cd 4
155.f even 4 1 4805.2.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.d 4 5.c odd 4 1
775.2.a.g 4 5.c odd 4 1
775.2.b.e 8 1.a even 1 1 trivial
775.2.b.e 8 5.b even 2 1 inner
1395.2.a.m 4 15.e even 4 1
2480.2.a.z 4 20.e even 4 1
4805.2.a.j 4 155.f even 4 1
6975.2.a.bj 4 15.e even 4 1
7595.2.a.q 4 35.f even 4 1
9920.2.a.cd 4 40.k even 4 1
9920.2.a.ch 4 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 17T_{2}^{6} + 96T_{2}^{4} + 208T_{2}^{2} + 144 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 17 T^{6} + 96 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$3$ \( T^{8} + 19 T^{6} + 95 T^{4} + 45 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 24 T^{6} + 176 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} - 16 T^{2} - 124 T - 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 88 T^{6} + 2192 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} + 51 T^{6} + 479 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( (T^{4} + 5 T^{3} - 21 T^{2} - 81 T + 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 128 T^{6} + 4048 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} - 40 T^{2} - 308 T - 456)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 67 T^{6} + 167 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( (T^{4} - 13 T^{3} + 17 T^{2} + 161 T - 294)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 63 T^{6} + 1427 T^{4} + \cdots + 45796 \) Copy content Toggle raw display
$47$ \( T^{8} + 124 T^{6} + 2480 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( T^{8} + 271 T^{6} + 24107 T^{4} + \cdots + 8363664 \) Copy content Toggle raw display
$59$ \( (T^{4} + 13 T^{3} - 65 T^{2} - 625 T + 2484)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 22 T^{3} + 144 T^{2} - 288 T - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 84 T^{6} + 720 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( (T^{4} - 3 T^{3} - 37 T^{2} + 59 T + 384)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 271 T^{6} + 21611 T^{4} + \cdots + 204304 \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{3} - 12 T^{2} + 500 T + 256)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 295 T^{6} + 16955 T^{4} + \cdots + 544644 \) Copy content Toggle raw display
$89$ \( (T^{4} - 12 T^{3} - 124 T^{2} + 1348 T + 1656)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 144 T^{6} + 4640 T^{4} + \cdots + 256 \) Copy content Toggle raw display
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