Properties

Label 414.8.a.i
Level $414$
Weight $8$
Character orbit 414.a
Self dual yes
Analytic conductor $129.327$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(129.327400550\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{2} + 64 q^{4} + ( - \beta_{2} + \beta_1 - 68) q^{5} + (\beta_{3} + \beta_1 + 506) q^{7} - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 64 q^{4} + ( - \beta_{2} + \beta_1 - 68) q^{5} + (\beta_{3} + \beta_1 + 506) q^{7} - 512 q^{8} + (8 \beta_{2} - 8 \beta_1 + 544) q^{10} + ( - 3 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 1028) q^{11} + ( - 6 \beta_{3} + 20 \beta_{2} + 7 \beta_1 + 2016) q^{13} + ( - 8 \beta_{3} - 8 \beta_1 - 4048) q^{14} + 4096 q^{16} + (7 \beta_{3} + 80 \beta_{2} + 19 \beta_1 - 9252) q^{17} + ( - 32 \beta_{3} + 113 \beta_{2} + 31 \beta_1 + 1466) q^{19} + ( - 64 \beta_{2} + 64 \beta_1 - 4352) q^{20} + (24 \beta_{3} - 56 \beta_{2} - 32 \beta_1 + 8224) q^{22} + 12167 q^{23} + ( - 20 \beta_{3} - 230 \beta_{2} - 115 \beta_1 + 30245) q^{25} + (48 \beta_{3} - 160 \beta_{2} - 56 \beta_1 - 16128) q^{26} + (64 \beta_{3} + 64 \beta_1 + 32384) q^{28} + (198 \beta_{3} + 304 \beta_{2} - 214 \beta_1 - 54178) q^{29} + (118 \beta_{3} - 200 \beta_{2} + 648 \beta_1 + 55672) q^{31} - 32768 q^{32} + ( - 56 \beta_{3} - 640 \beta_{2} - 152 \beta_1 + 74016) q^{34} + ( - 340 \beta_{3} - 92 \beta_{2} + 247 \beta_1 - 17326) q^{35} + (287 \beta_{3} - 561 \beta_{2} + 1464 \beta_1 + 121470) q^{37} + (256 \beta_{3} - 904 \beta_{2} - 248 \beta_1 - 11728) q^{38} + (512 \beta_{2} - 512 \beta_1 + 34816) q^{40} + (206 \beta_{3} - 786 \beta_{2} + 448 \beta_1 - 84874) q^{41} + ( - 490 \beta_{3} - 1201 \beta_{2} + 599 \beta_1 + 181898) q^{43} + ( - 192 \beta_{3} + 448 \beta_{2} + 256 \beta_1 - 65792) q^{44} - 97336 q^{46} + (60 \beta_{3} + 492 \beta_{2} + 3671 \beta_1 - 84286) q^{47} + (716 \beta_{3} + 990 \beta_{2} - 439 \beta_1 - 76785) q^{49} + (160 \beta_{3} + 1840 \beta_{2} + 920 \beta_1 - 241960) q^{50} + ( - 384 \beta_{3} + 1280 \beta_{2} + 448 \beta_1 + 129024) q^{52} + (806 \beta_{3} - 4709 \beta_{2} + 381 \beta_1 + 91924) q^{53} + (1160 \beta_{3} + 1340 \beta_{2} - 1630 \beta_1 + 107460) q^{55} + ( - 512 \beta_{3} - 512 \beta_1 - 259072) q^{56} + ( - 1584 \beta_{3} - 2432 \beta_{2} + 1712 \beta_1 + 433424) q^{58} + ( - 2338 \beta_{3} - 5346 \beta_{2} - 3229 \beta_1 - 21690) q^{59} + ( - 2379 \beta_{3} - 4903 \beta_{2} - 3478 \beta_1 + 521650) q^{61} + ( - 944 \beta_{3} + 1600 \beta_{2} - 5184 \beta_1 - 445376) q^{62} + 262144 q^{64} + (2440 \beta_{3} + 206 \beta_{2} + 484 \beta_1 - 393372) q^{65} + ( - 306 \beta_{3} - 5799 \beta_{2} + 10509 \beta_1 + 1081238) q^{67} + (448 \beta_{3} + 5120 \beta_{2} + 1216 \beta_1 - 592128) q^{68} + (2720 \beta_{3} + 736 \beta_{2} - 1976 \beta_1 + 138608) q^{70} + ( - 2430 \beta_{3} + 12294 \beta_{2} + 2388 \beta_1 - 567072) q^{71} + ( - 212 \beta_{3} + 7240 \beta_{2} + 2072 \beta_1 - 273318) q^{73} + ( - 2296 \beta_{3} + 4488 \beta_{2} - 11712 \beta_1 - 971760) q^{74} + ( - 2048 \beta_{3} + 7232 \beta_{2} + 1984 \beta_1 + 93824) q^{76} + ( - 2862 \beta_{3} + 3218 \beta_{2} + 7946 \beta_1 - 1456372) q^{77} + (4019 \beta_{3} - 9176 \beta_{2} - 7373 \beta_1 + 12574) q^{79} + ( - 4096 \beta_{2} + 4096 \beta_1 - 278528) q^{80} + ( - 1648 \beta_{3} + 6288 \beta_{2} - 3584 \beta_1 + 678992) q^{82} + (2903 \beta_{3} - 13259 \beta_{2} - 11034 \beta_1 - 376544) q^{83} + ( - 780 \beta_{3} + 32494 \beta_{2} - 18529 \beta_1 - 2195098) q^{85} + (3920 \beta_{3} + 9608 \beta_{2} - 4792 \beta_1 - 1455184) q^{86} + (1536 \beta_{3} - 3584 \beta_{2} - 2048 \beta_1 + 526336) q^{88} + (4239 \beta_{3} - 1350 \beta_{2} - 657 \beta_1 - 369828) q^{89} + ( - 1130 \beta_{3} + 8968 \beta_{2} + 21054 \beta_1 - 720684) q^{91} + 778688 q^{92} + ( - 480 \beta_{3} - 3936 \beta_{2} - 29368 \beta_1 + 674288) q^{94} + (13140 \beta_{3} + 12272 \beta_{2} - 6407 \beta_1 - 2123594) q^{95} + (9616 \beta_{3} - 33518 \beta_{2} - 6966 \beta_1 + 471910) q^{97} + ( - 5728 \beta_{3} - 7920 \beta_{2} + 3512 \beta_1 + 614280) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{2} + 256 q^{4} - 270 q^{5} + 2022 q^{7} - 2048 q^{8} + 2160 q^{10} - 4120 q^{11} + 8036 q^{13} - 16176 q^{14} + 16384 q^{16} - 37182 q^{17} + 5702 q^{19} - 17280 q^{20} + 32960 q^{22} + 48668 q^{23} + 121480 q^{25} - 64288 q^{26} + 129408 q^{28} - 217716 q^{29} + 222852 q^{31} - 131072 q^{32} + 297456 q^{34} - 68440 q^{35} + 486428 q^{37} - 45616 q^{38} + 138240 q^{40} - 338336 q^{41} + 730974 q^{43} - 263680 q^{44} - 389344 q^{46} - 338248 q^{47} - 310552 q^{49} - 971840 q^{50} + 514304 q^{52} + 375502 q^{53} + 424840 q^{55} - 1035264 q^{56} + 1741728 q^{58} - 71392 q^{59} + 2101164 q^{61} - 1782816 q^{62} + 1048576 q^{64} - 1578780 q^{65} + 4337162 q^{67} - 2379648 q^{68} + 547520 q^{70} - 2288016 q^{71} - 1107328 q^{73} - 3891424 q^{74} + 364928 q^{76} - 5826200 q^{77} + 60610 q^{79} - 1105920 q^{80} + 2706688 q^{82} - 1485464 q^{83} - 8843820 q^{85} - 5847792 q^{86} + 2109440 q^{88} - 1485090 q^{89} - 2898412 q^{91} + 3114752 q^{92} + 2705984 q^{94} - 8545200 q^{95} + 1935444 q^{97} + 2484416 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 8\nu^{2} - 5551\nu - 110450 ) / 684 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 68\nu^{2} + 4411\nu - 207382 ) / 76 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + 36\beta_{2} + 15\beta _1 + 16758 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -32\beta_{3} + 2448\beta_{2} + 5431\beta _1 + 318838 ) / 4 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−56.1267
−65.5856
90.4389
33.2734
−8.00000 0 64.0000 −366.876 0 −561.721 −512.000 0 2935.00
1.2 −8.00000 0 64.0000 −340.987 0 1267.12 −512.000 0 2727.89
1.3 −8.00000 0 64.0000 10.0669 0 971.172 −512.000 0 −80.5353
1.4 −8.00000 0 64.0000 427.795 0 345.429 −512.000 0 −3422.36
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.8.a.i 4
3.b odd 2 1 138.8.a.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.h 4 3.b odd 2 1
414.8.a.i 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 270T_{5}^{3} - 180540T_{5}^{2} - 51728000T_{5} + 538752000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(414))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 270 T^{3} + \cdots + 538752000 \) Copy content Toggle raw display
$7$ \( T^{4} - 2022 T^{3} + \cdots - 238777204176 \) Copy content Toggle raw display
$11$ \( T^{4} + 4120 T^{3} + \cdots + 25210679155200 \) Copy content Toggle raw display
$13$ \( T^{4} - 8036 T^{3} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{4} + 37182 T^{3} + \cdots - 69\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{4} - 5702 T^{3} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T - 12167)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 217716 T^{3} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{4} - 222852 T^{3} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} - 486428 T^{3} + \cdots - 26\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{4} + 338336 T^{3} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{4} - 730974 T^{3} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + 338248 T^{3} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} - 375502 T^{3} + \cdots + 12\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{4} + 71392 T^{3} + \cdots + 23\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{4} - 2101164 T^{3} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{4} - 4337162 T^{3} + \cdots + 44\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{4} + 2288016 T^{3} + \cdots + 92\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{4} + 1107328 T^{3} + \cdots - 25\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{4} - 60610 T^{3} + \cdots + 83\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{4} + 1485464 T^{3} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{4} + 1485090 T^{3} + \cdots - 10\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{4} - 1935444 T^{3} + \cdots + 94\!\cdots\!60 \) Copy content Toggle raw display
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