Properties

Label 2205.4.a.ca
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{4} + 5 q^{5} + ( - 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} + 9 \beta_1 + 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{4} + 5 q^{5} + ( - 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} + 9 \beta_1 + 8) q^{8} + 5 \beta_1 q^{10} + (2 \beta_{5} + 5 \beta_{4} + \beta_{3} + 5 \beta_{2} - 5 \beta_1 + 7) q^{11} + ( - 5 \beta_{4} - 8 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 27) q^{13} + ( - 12 \beta_{5} - 3 \beta_{4} - 15 \beta_{3} + \beta_{2} + 27 \beta_1 + 44) q^{16} + (\beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + 16 \beta_1 - 1) q^{17} + ( - 3 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 10 \beta_{2} - 6 \beta_1 + 52) q^{19} + ( - 5 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 10 \beta_{2} + 15 \beta_1 + 10) q^{20} + (22 \beta_{5} + \beta_{4} + 19 \beta_{3} - 17 \beta_{2} - 12 \beta_1 - 48) q^{22} + (6 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} - 3 \beta_1 + 56) q^{23} + 25 q^{25} + ( - 15 \beta_{5} - 5 \beta_{4} - 19 \beta_{3} + 4 \beta_{2} + 58 \beta_1 - 24) q^{26} + (8 \beta_{5} + \beta_{4} - 13 \beta_{3} - 7 \beta_{2} - 39 \beta_1 - 21) q^{29} + (10 \beta_{5} - 17 \beta_{3} + 29 \beta_{2} + 11 \beta_1 + 68) q^{31} + ( - 42 \beta_{5} - 15 \beta_{4} - 31 \beta_{3} + 35 \beta_{2} + 99 \beta_1 + 116) q^{32} + ( - 2 \beta_{5} - 22 \beta_{4} - 11 \beta_{3} + 18 \beta_{2} + 62 \beta_1 + 114) q^{34} + (4 \beta_{5} + 28 \beta_{4} - 9 \beta_{3} - 11 \beta_{2} - 33 \beta_1 - 8) q^{37} + ( - 3 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 16 \beta_1 - 6) q^{38} + ( - 10 \beta_{5} - 15 \beta_{4} - 25 \beta_{3} + 5 \beta_{2} + 45 \beta_1 + 40) q^{40} + ( - 37 \beta_{5} + 15 \beta_{4} + 3 \beta_{3} + 7 \beta_{2} - 11 \beta_1 - 86) q^{41} + (2 \beta_{5} - 34 \beta_{4} - 16 \beta_{3} + 8 \beta_{2} - 74) q^{43} + (\beta_{5} - 10 \beta_{4} + 48 \beta_{3} - 5 \beta_{2} - 122 \beta_1 + 62) q^{44} + (35 \beta_{5} + 20 \beta_{4} + 30 \beta_{3} - 25 \beta_{2} + 42 \beta_1 - 92) q^{46} + (2 \beta_{5} + 9 \beta_{4} - 32 \beta_{3} + 14 \beta_{2} - 50 \beta_1 - 81) q^{47} + 25 \beta_1 q^{50} + ( - 93 \beta_{5} - 32 \beta_{4} - 48 \beta_{3} + 65 \beta_{2} + 196 \beta_1 + 230) q^{52} + ( - 2 \beta_{5} + 60 \beta_{4} + 19 \beta_{3} - 35 \beta_{2} - 73 \beta_1 + 146) q^{53} + (10 \beta_{5} + 25 \beta_{4} + 5 \beta_{3} + 25 \beta_{2} - 25 \beta_1 + 35) q^{55} + (25 \beta_{4} + 15 \beta_{3} - 35 \beta_{2} - 128 \beta_1 - 296) q^{58} + (11 \beta_{5} - 59 \beta_{4} - 47 \beta_{3} + 33 \beta_{2} - 49 \beta_1 - 162) q^{59} + (24 \beta_{5} - 10 \beta_{4} - 4 \beta_{3} + 48 \beta_{2} + 132 \beta_1 + 84) q^{61} + (13 \beta_{5} - 28 \beta_{4} - 6 \beta_{3} + \beta_{2} + 154 \beta_1 - 4) q^{62} + ( - 10 \beta_{5} - 91 \beta_{4} - 63 \beta_{3} + 67 \beta_{2} + 351 \beta_1 + 116) q^{64} + ( - 25 \beta_{4} - 40 \beta_{3} + 20 \beta_{2} + 10 \beta_1 + 135) q^{65} + ( - 12 \beta_{5} + 12 \beta_{4} - 35 \beta_{3} + 31 \beta_{2} - 51 \beta_1 + 330) q^{67} + ( - 78 \beta_{5} - 67 \beta_{4} - 58 \beta_{3} + 15 \beta_{2} + 236 \beta_1 + 420) q^{68} + (14 \beta_{5} + 92 \beta_{3} - 60 \beta_{2} + 116 \beta_1 - 26) q^{71} + ( - 11 \beta_{5} - 85 \beta_{4} + 46 \beta_{3} + 60 \beta_{2} + 64 \beta_1 + 414) q^{73} + (21 \beta_{5} - 4 \beta_{4} + 36 \beta_{3} - 27 \beta_{2} - 132 \beta_1 - 128) q^{74} + (19 \beta_{5} - 43 \beta_{4} - 38 \beta_{3} + 106 \beta_{2} + 76 \beta_1 - 234) q^{76} + (10 \beta_{5} - 55 \beta_{4} + 65 \beta_{3} + 25 \beta_{2} + 67 \beta_1 + 237) q^{79} + ( - 60 \beta_{5} - 15 \beta_{4} - 75 \beta_{3} + 5 \beta_{2} + 135 \beta_1 + 220) q^{80} + (46 \beta_{5} - \beta_{4} + 2 \beta_{3} - 113 \beta_{2} - 96 \beta_1 - 314) q^{82} + (69 \beta_{5} + 41 \beta_{4} + 154 \beta_{3} - 80 \beta_{2} + 32 \beta_1 - 70) q^{83} + (5 \beta_{5} + 10 \beta_{4} - 20 \beta_{3} + 50 \beta_{2} + 80 \beta_1 - 5) q^{85} + ( - 50 \beta_{5} + 18 \beta_{4} - 56 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 172) q^{86} + (22 \beta_{5} + 172 \beta_{4} + 52 \beta_{3} - 144 \beta_{2} - 300 \beta_1 - 840) q^{88} + (33 \beta_{5} + 79 \beta_{4} + 80 \beta_{3} + 62 \beta_{2} - 74 \beta_1 + 378) q^{89} + ( - 60 \beta_{5} + 32 \beta_{4} - 24 \beta_{3} + 86 \beta_{2} - 82 \beta_1 + 412) q^{92} + (23 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} - 92 \beta_{2} - 164 \beta_1 - 536) q^{94} + ( - 15 \beta_{5} + 15 \beta_{4} + 20 \beta_{3} - 50 \beta_{2} - 30 \beta_1 + 260) q^{95} + ( - 77 \beta_{5} - 84 \beta_{4} + 14 \beta_{3} + 72 \beta_{2} + 134 \beta_1 + 63) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31} + 770 q^{32} + 812 q^{34} - 140 q^{37} - 20 q^{38} + 330 q^{40} - 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} - 628 q^{47} + 50 q^{50} + 1520 q^{52} + 676 q^{53} + 80 q^{55} - 2012 q^{58} - 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} + 2940 q^{68} + 224 q^{71} + 2640 q^{73} - 928 q^{74} - 1340 q^{76} + 1636 q^{79} + 1490 q^{80} - 1756 q^{82} - 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} + 1904 q^{89} + 1952 q^{92} - 3332 q^{94} + 1540 q^{95} + 516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 23\nu^{3} + 8\nu^{2} - 86\nu - 64 ) / 26 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 23\nu^{3} + 18\nu^{2} + 60\nu - 144 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 26\nu^{4} + 89\nu^{3} - 298\nu^{2} - 638\nu - 164 ) / 26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 49\nu^{3} + 8\nu^{2} - 476\nu - 428 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{5} - 26\nu^{4} - 155\nu^{3} + 240\nu^{2} + 800\nu + 264 ) / 26 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 56 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{5} - 8\beta_{4} + 15\beta_{3} + 15\beta_{2} + 83\beta _1 + 98 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36\beta_{5} - 29\beta_{4} + 43\beta_{3} + 127\beta_{2} + 265\beta _1 + 784 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 267\beta_{5} - 106\beta_{4} + 267\beta_{3} + 323\beta_{2} + 1315\beta _1 + 2254 ) / 7 \) 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
−3.05323
−0.241849
−2.05886
−1.04490
4.29508
4.10376
−4.46745 0 11.9581 5.00000 0 0 −17.6824 0 −22.3372
1.2 −1.65606 0 −5.25746 5.00000 0 0 21.9552 0 −8.28031
1.3 −0.644648 0 −7.58443 5.00000 0 0 10.0465 0 −3.22324
1.4 0.369315 0 −7.86361 5.00000 0 0 −5.85867 0 1.84657
1.5 2.88087 0 0.299392 5.00000 0 0 −22.1844 0 14.4043
1.6 5.51797 0 22.4480 5.00000 0 0 79.7239 0 27.5899
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 2205.4.a.ca 6
3.b odd 2 1 245.4.a.p yes 6
7.b odd 2 1 2205.4.a.bz 6
15.d odd 2 1 1225.4.a.bi 6
21.c even 2 1 245.4.a.o 6
21.g even 6 2 245.4.e.q 12
21.h odd 6 2 245.4.e.p 12
105.g even 2 1 1225.4.a.bj 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o 6 21.c even 2 1
245.4.a.p yes 6 3.b odd 2 1
245.4.e.p 12 21.h odd 6 2
245.4.e.q 12 21.g even 6 2
1225.4.a.bi 6 15.d odd 2 1
1225.4.a.bj 6 105.g even 2 1
2205.4.a.bz 6 7.b odd 2 1
2205.4.a.ca 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{6} - 2T_{2}^{5} - 29T_{2}^{4} + 28T_{2}^{3} + 134T_{2}^{2} + 24T_{2} - 28 \) Copy content Toggle raw display
\( T_{11}^{6} - 16T_{11}^{5} - 7174T_{11}^{4} + 121780T_{11}^{3} + 14188145T_{11}^{2} - 145959100T_{11} - 9225436100 \) Copy content Toggle raw display
\( T_{13}^{6} - 168T_{13}^{5} + 5774T_{13}^{4} + 335452T_{13}^{3} - 18677143T_{13}^{2} - 142488452T_{13} + 12513937372 \) Copy content Toggle raw display
\( T_{17}^{6} - 4T_{17}^{5} - 15392T_{17}^{4} + 258596T_{17}^{3} + 56343501T_{17}^{2} - 1617286496T_{17} - 3807091762 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} - 29 T^{4} + 28 T^{3} + \cdots - 28 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 16 T^{5} + \cdots - 9225436100 \) Copy content Toggle raw display
$13$ \( T^{6} - 168 T^{5} + \cdots + 12513937372 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots - 3807091762 \) Copy content Toggle raw display
$19$ \( T^{6} - 308 T^{5} + \cdots + 1617325344 \) Copy content Toggle raw display
$23$ \( T^{6} - 336 T^{5} + \cdots + 298897696 \) Copy content Toggle raw display
$29$ \( T^{6} + 176 T^{5} + \cdots - 544215793700 \) Copy content Toggle raw display
$31$ \( T^{6} - 392 T^{5} + \cdots + 16539893268192 \) Copy content Toggle raw display
$37$ \( T^{6} + 140 T^{5} + \cdots + 271258136464 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 144691772208184 \) Copy content Toggle raw display
$43$ \( T^{6} + 388 T^{5} + \cdots + 440374360000 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 251564448569400 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 590408333736048 \) Copy content Toggle raw display
$59$ \( T^{6} + 996 T^{5} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 842000334839552 \) Copy content Toggle raw display
$67$ \( T^{6} - 1768 T^{5} + \cdots + 885207397312 \) Copy content Toggle raw display
$71$ \( T^{6} - 224 T^{5} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{6} - 2640 T^{5} + \cdots - 85\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{6} - 1636 T^{5} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{6} + 140 T^{5} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{6} - 1904 T^{5} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 510868966648482 \) Copy content Toggle raw display
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