Properties

Label 175.6.b.d
Level $175$
Weight $6$
Character orbit 175.b
Analytic conductor $28.067$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{65})\)
Defining polynomial: \( x^{4} + 33x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (3 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 15) q^{4} - 48 q^{6} + 49 \beta_{2} q^{7} + (16 \beta_{2} + 47 \beta_1) q^{8} + ( - 9 \beta_{3} + 99) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (3 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 15) q^{4} - 48 q^{6} + 49 \beta_{2} q^{7} + (16 \beta_{2} + 47 \beta_1) q^{8} + ( - 9 \beta_{3} + 99) q^{9} + ( - 97 \beta_{3} - 252) q^{11} + (96 \beta_{2} + 48 \beta_1) q^{12} + ( - 315 \beta_{2} - 53 \beta_1) q^{13} + ( - 49 \beta_{3} + 49) q^{14} + (63 \beta_{3} - 303) q^{16} + (105 \beta_{2} + 251 \beta_1) q^{17} + ( - 144 \beta_{2} + 99 \beta_1) q^{18} + ( - 86 \beta_{3} - 272) q^{19} - 147 \beta_{3} q^{21} + ( - 1552 \beta_{2} - 252 \beta_1) q^{22} + (230 \beta_{2} + 902 \beta_1) q^{23} + ( - 48 \beta_{3} - 2256) q^{24} + (262 \beta_{3} + 586) q^{26} + (567 \beta_{2} + 999 \beta_1) q^{27} + (784 \beta_{2} + 49 \beta_1) q^{28} + ( - 945 \beta_{3} - 2470) q^{29} + ( - 924 \beta_{3} + 264) q^{31} + (1520 \beta_{2} + 1201 \beta_1) q^{32} + ( - 5703 \beta_{2} - 1047 \beta_1) q^{33} + (146 \beta_{3} - 4162) q^{34} + ( - 45 \beta_{3} + 1341) q^{36} + (3822 \beta_{2} - 1260 \beta_1) q^{37} + ( - 1376 \beta_{2} - 272 \beta_1) q^{38} + (945 \beta_{3} + 2544) q^{39} + (3818 \beta_{3} - 1022) q^{41} - 2352 \beta_{2} q^{42} + ( - 14022 \beta_{2} - 922 \beta_1) q^{43} + ( - 1804 \beta_{3} - 5332) q^{44} + (672 \beta_{3} - 15104) q^{46} + (9857 \beta_{2} - 1575 \beta_1) q^{47} + (2304 \beta_{2} - 720 \beta_1) q^{48} - 2401 q^{49} + ( - 315 \beta_{3} - 12048) q^{51} + ( - 5888 \beta_{2} - 1110 \beta_1) q^{52} + ( - 27564 \beta_{2} + 454 \beta_1) q^{53} + (432 \beta_{3} - 16416) q^{54} + ( - 2303 \beta_{3} + 1519) q^{56} + ( - 5202 \beta_{2} - 1074 \beta_1) q^{57} + ( - 15120 \beta_{2} - 2470 \beta_1) q^{58} + (5184 \beta_{3} - 32392) q^{59} + ( - 5706 \beta_{3} - 23070) q^{61} + ( - 14784 \beta_{2} + 264 \beta_1) q^{62} + (4410 \beta_{2} - 441 \beta_1) q^{63} + (1697 \beta_{3} - 28593) q^{64} + (4656 \beta_{3} + 12096) q^{66} + (20388 \beta_{2} - 4568 \beta_1) q^{67} + (5696 \beta_{2} + 3870 \beta_1) q^{68} + ( - 690 \beta_{3} - 43296) q^{69} + ( - 5304 \beta_{3} + 43024) q^{71} + ( - 5328 \beta_{2} + 4509 \beta_1) q^{72} + ( - 4670 \beta_{2} + 4192 \beta_1) q^{73} + ( - 5082 \beta_{3} + 25242) q^{74} + ( - 1648 \beta_{3} - 5456) q^{76} + ( - 17101 \beta_{2} - 4753 \beta_1) q^{77} + (15120 \beta_{2} + 2544 \beta_1) q^{78} + (17635 \beta_{3} + 17080) q^{79} + ( - 3888 \beta_{3} - 23895) q^{81} + (61088 \beta_{2} - 1022 \beta_1) q^{82} + (56876 \beta_{2} + 3924 \beta_1) q^{83} + ( - 2352 \beta_{3} - 2352) q^{84} + (13100 \beta_{3} + 1652) q^{86} + ( - 55605 \beta_{2} - 10245 \beta_1) q^{87} + ( - 78528 \beta_{2} - 13396 \beta_1) q^{88} + ( - 5722 \beta_{3} + 21686) q^{89} + (2597 \beta_{3} + 12838) q^{91} + (18112 \beta_{2} + 13760 \beta_1) q^{92} + ( - 46332 \beta_{2} - 1980 \beta_1) q^{93} + ( - 11432 \beta_{3} + 36632) q^{94} + ( - 4560 \beta_{3} - 57648) q^{96} + (55141 \beta_{2} + 13943 \beta_1) q^{97} - 2401 \beta_1 q^{98} + ( - 6462 \beta_{3} - 10980) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9} - 1202 q^{11} + 98 q^{14} - 1086 q^{16} - 1260 q^{19} - 294 q^{21} - 9120 q^{24} + 2868 q^{26} - 11770 q^{29} - 792 q^{31} - 16356 q^{34} + 5274 q^{36} + 12066 q^{39} + 3548 q^{41} - 24936 q^{44} - 59072 q^{46} - 9604 q^{49} - 48822 q^{51} - 64800 q^{54} + 1470 q^{56} - 119200 q^{59} - 103692 q^{61} - 110978 q^{64} + 57696 q^{66} - 174564 q^{69} + 161488 q^{71} + 90804 q^{74} - 25120 q^{76} + 103590 q^{79} - 103356 q^{81} - 14112 q^{84} + 32808 q^{86} + 75300 q^{89} + 56546 q^{91} + 123664 q^{94} - 239712 q^{96} - 56844 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 33x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 17\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 17 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16\beta_{2} - 17\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
4.53113i
3.53113i
3.53113i
4.53113i
4.53113i 10.5934i 11.4689 0 −48.0000 49.0000i 196.963i 130.780 0
99.2 3.53113i 13.5934i 19.5311 0 −48.0000 49.0000i 181.963i 58.2198 0
99.3 3.53113i 13.5934i 19.5311 0 −48.0000 49.0000i 181.963i 58.2198 0
99.4 4.53113i 10.5934i 11.4689 0 −48.0000 49.0000i 196.963i 130.780 0
\(n\): e.g. 2-40 or 990-1000
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 175.6.b.d 4
5.b even 2 1 inner 175.6.b.d 4
5.c odd 4 1 35.6.a.b 2
5.c odd 4 1 175.6.a.d 2
15.e even 4 1 315.6.a.c 2
20.e even 4 1 560.6.a.l 2
35.f even 4 1 245.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 5.c odd 4 1
175.6.a.d 2 5.c odd 4 1
175.6.b.d 4 1.a even 1 1 trivial
175.6.b.d 4 5.b even 2 1 inner
245.6.a.c 2 35.f even 4 1
315.6.a.c 2 15.e even 4 1
560.6.a.l 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 33T_{2}^{2} + 256 \) acting on \(S_{6}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 33T^{2} + 256 \) Copy content Toggle raw display
$3$ \( T^{4} + 297 T^{2} + 20736 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 601 T - 62596)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 257757 T^{2} + \cdots + 1412707396 \) Copy content Toggle raw display
$17$ \( T^{4} + 2048373 T^{2} + \cdots + 1047237035716 \) Copy content Toggle raw display
$19$ \( (T^{2} + 630 T - 20960)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 173507485106176 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5885 T - 5853350)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 396 T - 13834656)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 91237608 T^{2} + \cdots + 35738827414416 \) Copy content Toggle raw display
$41$ \( (T^{2} - 1774 T - 236091496)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 395429172 T^{2} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{4} + 307231073 T^{2} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + 1551378132 T^{2} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 59600 T + 451339840)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 51846 T + 142927344)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 1706204448 T^{2} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{2} - 80744 T + 1172746624)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 662675592 T^{2} + \cdots + 57\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{2} - 51795 T - 4382959400)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 6531522512 T^{2} + \cdots + 76\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} - 37650 T - 177665240)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 10958837053 T^{2} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
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