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Properties

Label 1075.6.a.a
Level $1075$
Weight $6$
Character orbit 1075.a
Self dual yes
Analytic conductor $172.413$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.412606299\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + (\beta_{7} - \beta_{4} + \beta_1 + 2) q^{3} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 15) q^{4} + (4 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 10 \beta_1 - 5) q^{6} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3} + \beta_{2} + 13 \beta_1 + 10) q^{7} + (6 \beta_{7} - 8 \beta_{6} - \beta_{5} - 8 \beta_{4} + \beta_{3} - 5 \beta_{2} - 7 \beta_1 + 82) q^{8} + ( - \beta_{7} + \beta_{6} + 8 \beta_{5} - 15 \beta_{4} - 9 \beta_{3} - 4 \beta_{2} + \cdots + 62) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + (\beta_{7} - \beta_{4} + \beta_1 + 2) q^{3} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 15) q^{4} + (4 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 10 \beta_1 - 5) q^{6} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3} + \beta_{2} + 13 \beta_1 + 10) q^{7} + (6 \beta_{7} - 8 \beta_{6} - \beta_{5} - 8 \beta_{4} + \beta_{3} - 5 \beta_{2} - 7 \beta_1 + 82) q^{8} + ( - \beta_{7} + \beta_{6} + 8 \beta_{5} - 15 \beta_{4} - 9 \beta_{3} - 4 \beta_{2} + \cdots + 62) q^{9}+ \cdots + (1884 \beta_{7} + 2105 \beta_{6} + 1126 \beta_{5} - 3870 \beta_{4} + \cdots + 23821) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8125 \nu^{7} + 510320 \nu^{6} - 2147429 \nu^{5} - 76756386 \nu^{4} + 85006462 \nu^{3} + 2758106176 \nu^{2} - 206135304 \nu - 20471935488 ) / 547265856 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12949 \nu^{7} - 681148 \nu^{6} + 2601727 \nu^{5} + 80176782 \nu^{4} - 378607946 \nu^{3} - 1837549088 \nu^{2} + 7674423864 \nu + 1709840448 ) / 547265856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17485 \nu^{7} + 269956 \nu^{6} + 1611305 \nu^{5} - 37491046 \nu^{4} + 6237658 \nu^{3} + 1274965584 \nu^{2} - 1052986520 \nu - 6710235008 ) / 364843904 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38033 \nu^{7} - 126518 \nu^{6} - 5886547 \nu^{5} + 8711358 \nu^{4} + 252968474 \nu^{3} + 152252096 \nu^{2} - 3214801896 \nu - 7712512416 ) / 273632928 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83131 \nu^{7} + 19780 \nu^{6} + 13177295 \nu^{5} + 14631678 \nu^{4} - 470063386 \nu^{3} - 1263726976 \nu^{2} + 1067316504 \nu + 17642049216 ) / 547265856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 128521 \nu^{7} + 1062904 \nu^{6} + 16607009 \nu^{5} - 129895854 \nu^{4} - 487223974 \nu^{3} + 4067658416 \nu^{2} + 2176112520 \nu - 28238112000 ) / 547265856 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} - 2\beta_{4} + 2\beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{6} + 7\beta_{5} - 4\beta_{4} - \beta_{3} + 5\beta_{2} + 71\beta _1 + 56 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 94\beta_{7} + 28\beta_{6} + 99\beta_{5} - 264\beta_{4} - 23\beta_{3} + 31\beta_{2} + 307\beta _1 + 3114 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 48\beta_{7} + 1092\beta_{6} + 869\beta_{5} - 924\beta_{4} - 71\beta_{3} + 927\beta_{2} + 6567\beta _1 + 9672 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8760 \beta_{7} + 5444 \beta_{6} + 10339 \beta_{5} - 29780 \beta_{4} - 3425 \beta_{3} + 6801 \beta_{2} + 40661 \beta _1 + 285312 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11036 \beta_{7} + 127500 \beta_{6} + 102849 \beta_{5} - 146996 \beta_{4} - 10463 \beta_{3} + 125743 \beta_{2} + 685627 \beta _1 + 1391004 \) Copy content Toggle raw display

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

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
10.9591
7.21373
5.65705
2.58275
−2.08717
−5.06235
−6.09504
−9.16809
−8.95911 25.1057 48.2657 0 −224.924 166.001 −145.726 387.294 0
1.2 −5.21373 11.2683 −4.81697 0 −58.7500 11.1041 191.954 −116.025 0
1.3 −3.65705 −7.84314 −18.6260 0 28.6827 25.5214 185.142 −181.485 0
1.4 −0.582753 −3.05838 −31.6604 0 1.78228 103.690 37.0983 −233.646 0
1.5 4.08717 −25.6605 −15.2950 0 −104.879 184.774 −193.303 415.460 0
1.6 7.06235 9.19190 17.8767 0 64.9164 4.24720 −99.7434 −158.509 0
1.7 8.09504 −11.1803 33.5297 0 −90.5052 −223.489 12.3830 −118.000 0
1.8 11.1681 28.1764 92.7262 0 314.677 −135.849 678.196 550.912 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(43\) \(-1\)

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 1075.6.a.a 8
5.b even 2 1 43.6.a.a ✓ 8
15.d odd 2 1 387.6.a.c 8
20.d odd 2 1 688.6.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.a ✓ 8 5.b even 2 1
387.6.a.c 8 15.d odd 2 1
688.6.a.e 8 20.d odd 2 1
1075.6.a.a 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 12T_{2}^{7} - 117T_{2}^{6} + 1502T_{2}^{5} + 3358T_{2}^{4} - 49104T_{2}^{3} - 39464T_{2}^{2} + 439936T_{2} + 259776 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1075))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 12 T^{7} - 117 T^{6} + \cdots + 259776 \) Copy content Toggle raw display
$3$ \( T^{8} - 26 T^{7} + \cdots + 504223128 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 116222354316288 \) Copy content Toggle raw display
$11$ \( T^{8} + 532 T^{7} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{8} - 2492 T^{7} + \cdots - 20\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{8} - 2534 T^{7} + \cdots + 37\!\cdots\!33 \) Copy content Toggle raw display
$19$ \( T^{8} + 1678 T^{7} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{8} - 2488 T^{7} + \cdots - 18\!\cdots\!83 \) Copy content Toggle raw display
$29$ \( T^{8} + 4360 T^{7} + \cdots - 39\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{8} - 5704 T^{7} + \cdots + 10\!\cdots\!13 \) Copy content Toggle raw display
$37$ \( T^{8} - 3772 T^{7} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{8} + 10698 T^{7} + \cdots + 17\!\cdots\!57 \) Copy content Toggle raw display
$43$ \( (T - 1849)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 77864 T^{7} + \cdots - 19\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{8} - 62352 T^{7} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + 26224 T^{7} + \cdots - 68\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{8} + 82540 T^{7} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{8} + 27784 T^{7} + \cdots + 69\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{8} + 9504 T^{7} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{8} + 14260 T^{7} + \cdots + 80\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{8} - 160248 T^{7} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{8} - 77176 T^{7} + \cdots - 19\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{8} + 265692 T^{7} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + 144742 T^{7} + \cdots + 38\!\cdots\!17 \) Copy content Toggle raw display
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