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Properties

Label 414.3.c.b
Level $414$
Weight $3$
Character orbit 414.c
Analytic conductor $11.281$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 414.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.2806829445\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 2 q^{4} - \beta_{7} q^{5} + ( - \beta_{6} - \beta_{2} + 2) q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 2 q^{4} - \beta_{7} q^{5} + ( - \beta_{6} - \beta_{2} + 2) q^{7} - 2 \beta_1 q^{8} + (\beta_{4} - 1) q^{10} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{11} + ( - 2 \beta_{6} - \beta_{4} - 1) q^{13} + ( - \beta_{5} + \beta_{3} + 2 \beta_1) q^{14} + 4 q^{16} + (3 \beta_{7} + 2 \beta_1) q^{17} + ( - \beta_{6} + 3 \beta_{2} - 6) q^{19} + 2 \beta_{7} q^{20} + ( - 2 \beta_{6} - 2 \beta_{2} + 4) q^{22} + \beta_{3} q^{23} + ( - 2 \beta_{6} + \beta_{4}) q^{25} + ( - 2 \beta_{7} + 4 \beta_{3} - 2 \beta_1) q^{26} + (2 \beta_{6} + 2 \beta_{2} - 4) q^{28} + ( - 4 \beta_{7} + 4 \beta_{3} - 3 \beta_1) q^{29} + ( - 2 \beta_{6} + 2 \beta_{2} - 4) q^{31} + 4 \beta_1 q^{32} + ( - 3 \beta_{4} - 1) q^{34} + ( - 2 \beta_{5} + 6 \beta_{3} + 2 \beta_1) q^{35} + ( - 4 \beta_{6} - 2 \beta_{4} + 4) q^{37} + (3 \beta_{5} + 5 \beta_{3} - 6 \beta_1) q^{38} + ( - 2 \beta_{4} + 2) q^{40} + (4 \beta_{7} - 4 \beta_{5} + 4 \beta_{3} + 11 \beta_1) q^{41} + ( - \beta_{6} + 6 \beta_{4} - \beta_{2} + 4) q^{43} + ( - 2 \beta_{5} + 2 \beta_{3} + 4 \beta_1) q^{44} + \beta_{6} q^{46} + ( - 4 \beta_{7} + 4 \beta_{5} - 10 \beta_1) q^{47} + ( - 10 \beta_{6} - 5 \beta_{4} - 8 \beta_{2} + 10) q^{49} + (2 \beta_{7} + 4 \beta_{3} + \beta_1) q^{50} + (4 \beta_{6} + 2 \beta_{4} + 2) q^{52} + (3 \beta_{7} + 2 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{53} + ( - 8 \beta_{6} - 4 \beta_{2} + 4) q^{55} + (2 \beta_{5} - 2 \beta_{3} - 4 \beta_1) q^{56} + (4 \beta_{6} + 4 \beta_{4} + 2) q^{58} + ( - 12 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{59} + (4 \beta_{6} + 4 \beta_{4} - 4 \beta_{2} + 6) q^{61} + (2 \beta_{5} + 6 \beta_{3} - 4 \beta_1) q^{62} - 8 q^{64} + (8 \beta_{7} - 6 \beta_{5} - 2 \beta_{3} + 28 \beta_1) q^{65} + (5 \beta_{6} + 9 \beta_{2} - 2) q^{67} + ( - 6 \beta_{7} - 4 \beta_1) q^{68} + (8 \beta_{6} + 4 \beta_{2} - 4) q^{70} + (8 \beta_{7} + 6 \beta_{5} - 6 \beta_{3} + 16 \beta_1) q^{71} + (10 \beta_{4} - 54) q^{73} + ( - 4 \beta_{7} + 8 \beta_{3} + 2 \beta_1) q^{74} + (2 \beta_{6} - 6 \beta_{2} + 12) q^{76} + (10 \beta_{7} + 8 \beta_{5} - 12 \beta_{3} - 54 \beta_1) q^{77} + (9 \beta_{6} - 6 \beta_{4} + 13 \beta_{2} - 52) q^{79} - 4 \beta_{7} q^{80} + (8 \beta_{6} - 4 \beta_{4} + 8 \beta_{2} - 18) q^{82} + ( - 12 \beta_{7} - \beta_{5} + \beta_{3} - 2 \beta_1) q^{83} + (6 \beta_{6} - \beta_{4} + 73) q^{85} + (12 \beta_{7} - \beta_{5} + \beta_{3} + 10 \beta_1) q^{86} + (4 \beta_{6} + 4 \beta_{2} - 8) q^{88} + ( - \beta_{7} - 6 \beta_{5} + 6 \beta_{3} - 32 \beta_1) q^{89} + ( - 14 \beta_{6} - 10 \beta_{4} - 10 \beta_{2} + 46) q^{91} - 2 \beta_{3} q^{92} + ( - 4 \beta_{6} + 4 \beta_{4} - 8 \beta_{2} + 16) q^{94} + (16 \beta_{7} - 6 \beta_{5} - 14 \beta_{3} + 2 \beta_1) q^{95} + (4 \beta_{6} - 5 \beta_{4} - 4 \beta_{2} - 1) q^{97} + ( - 10 \beta_{7} - 8 \beta_{5} + 12 \beta_{3} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 373 \nu^{7} - 477 \nu^{6} - 4990 \nu^{5} + 11918 \nu^{4} - 51735 \nu^{3} + 94493 \nu^{2} - 436638 \nu + 1216458 ) / 565056 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{7} - 27\nu^{6} + 14\nu^{5} - 142\nu^{4} + 951\nu^{3} + 11\nu^{2} - 4962\nu + 8262 ) / 5184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\nu^{7} + 180\nu^{6} + 2282\nu^{5} - 88\nu^{4} + 18591\nu^{3} - 57976\nu^{2} + 134574\nu - 379080 ) / 70632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 22\nu^{5} - 110\nu^{4} + 411\nu^{3} - 1421\nu^{2} + 2406\nu - 8586 ) / 864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -59\nu^{7} - 363\nu^{6} - 1154\nu^{5} - 3950\nu^{4} + 1383\nu^{3} - 23045\nu^{2} + 156990\nu - 59562 ) / 47088 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 3\nu^{6} - 62\nu^{5} + 238\nu^{4} - 423\nu^{3} + 2269\nu^{2} - 2238\nu + 6858 ) / 1728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1901 \nu^{7} - 1251 \nu^{6} + 40286 \nu^{5} - 105598 \nu^{4} + 460815 \nu^{3} - 1151821 \nu^{2} + 3362766 \nu - 6900714 ) / 565056 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} - 5\beta_{3} - 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 12\beta_{6} - 5\beta_{5} + 13\beta_{4} - 5\beta_{3} + 6\beta_{2} - 30\beta _1 + 79 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -32\beta_{7} - 2\beta_{6} + 23\beta_{5} - 15\beta_{4} + 87\beta_{3} + 36\beta_{2} - 28\beta _1 - 33 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 210\beta_{7} - 24\beta_{6} - 59\beta_{5} - 145\beta_{4} - 71\beta_{3} - 78\beta_{2} + 802\beta _1 - 839 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -212\beta_{7} + 206\beta_{6} - 295\beta_{5} + 531\beta_{4} - 395\beta_{3} - 432\beta_{2} - 1776\beta _1 + 3889 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2270 \beta_{7} - 1164 \beta_{6} + 1171 \beta_{5} - 611 \beta_{4} + 2491 \beta_{3} + 1914 \beta_{2} - 8910 \beta _1 + 1807 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(235\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
2.43130 + 0.475563i
1.41915 + 2.47946i
−0.919152 − 3.46316i
−1.93130 + 3.33657i
−1.93130 − 3.33657i
−0.919152 + 3.46316i
1.41915 − 2.47946i
2.43130 − 0.475563i
− 1.41421i 0 −2.00000 − 6.87676i 0 −5.43723 2.82843i 0 −9.72521
323.2 − 1.41421i 0 −2.00000 − 4.01397i 0 13.7953 2.82843i 0 −5.67661
323.3 − 1.41421i 0 −2.00000 2.59976i 0 −3.01296 2.82843i 0 3.67661
323.4 − 1.41421i 0 −2.00000 5.46255i 0 2.65490 2.82843i 0 7.72521
323.5 1.41421i 0 −2.00000 − 5.46255i 0 2.65490 − 2.82843i 0 7.72521
323.6 1.41421i 0 −2.00000 − 2.59976i 0 −3.01296 − 2.82843i 0 3.67661
323.7 1.41421i 0 −2.00000 4.01397i 0 13.7953 − 2.82843i 0 −5.67661
323.8 1.41421i 0 −2.00000 6.87676i 0 −5.43723 − 2.82843i 0 −9.72521
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.3.c.b ✓ 8
3.b odd 2 1 inner 414.3.c.b ✓ 8
4.b odd 2 1 3312.3.g.b 8
12.b even 2 1 3312.3.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.3.c.b ✓ 8 1.a even 1 1 trivial
414.3.c.b ✓ 8 3.b odd 2 1 inner
3312.3.g.b 8 4.b odd 2 1
3312.3.g.b 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 100T_{5}^{6} + 3284T_{5}^{4} + 40672T_{5}^{2} + 153664 \) acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 100 T^{6} + 3284 T^{4} + \cdots + 153664 \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} - 86 T^{2} + 40 T + 600)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 472 T^{6} + 36944 T^{4} + \cdots + 5760000 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} - 460 T^{2} + 544 T + 18496)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 884 T^{6} + \cdots + 1121982016 \) Copy content Toggle raw display
$19$ \( (T^{4} + 24 T^{3} - 1142 T^{2} + \cdots - 195752)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 3048 T^{6} + \cdots + 676624144 \) Copy content Toggle raw display
$31$ \( (T^{4} + 16 T^{3} - 1080 T^{2} + \cdots - 8064)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} - 1768 T^{2} + \cdots + 203152)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 9000 T^{6} + \cdots + 124832195856 \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} - 3494 T^{2} + \cdots + 1054936)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 7392 T^{6} + \cdots + 5017886724096 \) Copy content Toggle raw display
$53$ \( T^{8} + 6132 T^{6} + \cdots + 1459979223616 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 196109327241216 \) Copy content Toggle raw display
$61$ \( (T^{4} - 24 T^{3} - 5928 T^{2} + \cdots - 421360)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} - 7046 T^{2} + \cdots + 7536600)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 309201557520384 \) Copy content Toggle raw display
$73$ \( (T^{4} + 216 T^{3} + 7696 T^{2} + \cdots - 3423744)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 208 T^{3} - 1958 T^{2} + \cdots + 17983128)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 141567353611264 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 594285713856576 \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} - 7308 T^{2} + \cdots - 1242528)^{2} \) Copy content Toggle raw display
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