Properties

Label 425.4.b.f
Level $425$
Weight $4$
Character orbit 425.b
Analytic conductor $25.076$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27793984.1
Defining polynomial: \( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{3} + ( - 3 \beta_{2} - \beta_1 - 8) q^{4} + ( - 6 \beta_{2} - 2 \beta_1 - 24) q^{6} + ( - 4 \beta_{5} - 6 \beta_{4} + \beta_{3}) q^{7} + (9 \beta_{5} - 16 \beta_{4} + 5 \beta_{3}) q^{8} + ( - 8 \beta_{2} - 2 \beta_1 - 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{3} + ( - 3 \beta_{2} - \beta_1 - 8) q^{4} + ( - 6 \beta_{2} - 2 \beta_1 - 24) q^{6} + ( - 4 \beta_{5} - 6 \beta_{4} + \beta_{3}) q^{7} + (9 \beta_{5} - 16 \beta_{4} + 5 \beta_{3}) q^{8} + ( - 8 \beta_{2} - 2 \beta_1 - 19) q^{9} + ( - 11 \beta_{2} + 2 \beta_1 - 10) q^{11} + (26 \beta_{5} - 16 \beta_{4} + 26 \beta_{3}) q^{12} + ( - 6 \beta_{5} + 12 \beta_{4} + 8 \beta_{3}) q^{13} + (4 \beta_{2} - 20 \beta_1 - 24) q^{14} + (11 \beta_{2} - 7 \beta_1 + 48) q^{16} + 17 \beta_{4} q^{17} + (41 \beta_{5} - 48 \beta_{4} + 33 \beta_{3}) q^{18} + (22 \beta_{2} - 8 \beta_1 - 24) q^{19} + (12 \beta_{2} - 30 \beta_1 - 54) q^{21} + (26 \beta_{5} - 104 \beta_{4} + 34 \beta_{3}) q^{22} + (4 \beta_{5} + 46 \beta_{4} - 39 \beta_{3}) q^{23} + (46 \beta_{2} - 6 \beta_1 + 224) q^{24} + ( - 2 \beta_{2} - 22 \beta_1 + 16) q^{26} + (8 \beta_{5} + 4 \beta_{4} + 40 \beta_{3}) q^{27} + (44 \beta_{5} + 144 \beta_{4} + 4 \beta_{3}) q^{28} + ( - 16 \beta_{2} + 30 \beta_1 + 142) q^{29} + ( - 39 \beta_{2} - 16 \beta_1 + 82) q^{31} + (23 \beta_{5} + 16 \beta_{4} - 37 \beta_{3}) q^{32} + (34 \beta_{5} - 122 \beta_{4} + 76 \beta_{3}) q^{33} + ( - 17 \beta_{2} + 17 \beta_1) q^{34} + (91 \beta_{2} - 7 \beta_1 + 440) q^{36} + ( - 50 \beta_{5} - 102 \beta_{4} + 28 \beta_{3}) q^{37} + (4 \beta_{5} + 240 \beta_{4} - 28 \beta_{3}) q^{38} + (36 \beta_{2} - 16 \beta_1 - 84) q^{39} + (52 \beta_{2} + 60 \beta_1 - 118) q^{41} + (120 \beta_{5} + 336 \beta_{4}) q^{42} + ( - 56 \beta_{5} + 204 \beta_{4} + 2 \beta_{3}) q^{43} + (110 \beta_{2} - 78 \beta_1 + 400) q^{44} + ( - 120 \beta_{2} + 136 \beta_1 - 280) q^{46} + (44 \beta_{5} - 228 \beta_{4} + 48 \beta_{3}) q^{47} + ( - 90 \beta_{5} + 288 \beta_{4} - 114 \beta_{3}) q^{48} + (20 \beta_{2} - 94 \beta_1 + 121) q^{49} + ( - 17 \beta_{2} + 34 \beta_1 - 34) q^{51} + (6 \beta_{5} + 256 \beta_{4} + 30 \beta_{3}) q^{52} + (8 \beta_{5} + 98 \beta_{4} + 116 \beta_{3}) q^{53} + (84 \beta_{2} - 52 \beta_1 + 384) q^{54} + ( - 60 \beta_{2} + 108 \beta_1 + 192) q^{56} + (36 \beta_{5} + 228 \beta_{4} - 108 \beta_{3}) q^{57} + ( - 200 \beta_{5} - 368 \beta_{4} - 80 \beta_{3}) q^{58} + ( - 130 \beta_{2} - 212) q^{59} + (64 \beta_{2} - 78 \beta_1 - 2) q^{61} + (44 \beta_{5} - 184 \beta_{4} - 20 \beta_{3}) q^{62} + (96 \beta_{5} + 342 \beta_{4} + 9 \beta_{3}) q^{63} + (21 \beta_{2} + 103 \beta_1 + 272) q^{64} + (308 \beta_{2} - 172 \beta_1 + 880) q^{66} + ( - 132 \beta_{5} - 292 \beta_{4} - 24 \beta_{3}) q^{67} + ( - 17 \beta_{5} - 136 \beta_{4} + 51 \beta_{3}) q^{68} + ( - 280 \beta_{2} + 186 \beta_1 - 254) q^{69} + ( - 185 \beta_{2} - 72 \beta_1 - 110) q^{71} + ( - 273 \beta_{5} + 400 \beta_{4} - 365 \beta_{3}) q^{72} + (16 \beta_{5} + 274 \beta_{4} + 16 \beta_{3}) q^{73} + (108 \beta_{2} - 308 \beta_1 - 176) q^{74} + ( - 116 \beta_{2} + 244 \beta_1 - 384) q^{76} + ( - 18 \beta_{5} + 174 \beta_{4}) q^{77} + (60 \beta_{5} + 416 \beta_{4} - 4 \beta_{3}) q^{78} + ( - 267 \beta_{2} + 180 \beta_1 + 138) q^{79} + (20 \beta_{2} - 94 \beta_1 - 137) q^{81} + ( - 166 \beta_{5} - 64 \beta_{4} + 74 \beta_{3}) q^{82} + ( - 128 \beta_{5} - 756 \beta_{4} + 82 \beta_{3}) q^{83} + ( - 120 \beta_{2} + 456 \beta_1 + 528) q^{84} + ( - 256 \beta_{2} + 32 \beta_1 - 432) q^{86} + ( - 372 \beta_{5} - 352 \beta_{4} - 46 \beta_{3}) q^{87} + ( - 178 \beta_{5} + 672 \beta_{4} - 426 \beta_{3}) q^{88} + ( - 276 \beta_{2} + 110 \beta_1 + 20) q^{89} + (64 \beta_{2} - 44 \beta_1 - 324) q^{91} + (144 \beta_{5} - 1680 \beta_{4} + 344 \beta_{3}) q^{92} + ( - 22 \beta_{5} + 218 \beta_{4} + 152 \beta_{3}) q^{93} + (368 \beta_{2} - 192 \beta_1 + 736) q^{94} + ( - 238 \beta_{2} + 198 \beta_1 + 160) q^{96} + (120 \beta_{5} + 50 \beta_{4} - 140 \beta_{3}) q^{97} + (121 \beta_{5} + 912 \beta_{4} - 255 \beta_{3}) q^{98} + (281 \beta_{2} - 206 \beta_1 + 1042) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9} - 56 q^{11} - 184 q^{14} + 274 q^{16} - 160 q^{19} - 384 q^{21} + 1332 q^{24} + 52 q^{26} + 912 q^{29} + 460 q^{31} + 34 q^{34} + 2626 q^{36} - 536 q^{39} - 588 q^{41} + 2244 q^{44} - 1408 q^{46} + 538 q^{49} - 136 q^{51} + 2200 q^{54} + 1368 q^{56} - 1272 q^{59} - 168 q^{61} + 1838 q^{64} + 4936 q^{66} - 1152 q^{69} - 804 q^{71} - 1672 q^{74} - 1816 q^{76} + 1188 q^{79} - 1010 q^{81} + 4080 q^{84} - 2528 q^{86} + 340 q^{89} - 2032 q^{91} + 4032 q^{94} + 1356 q^{96} + 5840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 49\nu^{4} + 7\nu^{3} - \nu^{2} + 1696 ) / 342 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{5} - \nu^{4} - 49\nu^{3} + 7\nu^{2} + 98 ) / 342 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} + \nu^{4} + 49\nu^{3} - 7\nu^{2} + 684\nu - 98 ) / 342 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -49\nu^{5} - 7\nu^{4} - \nu^{3} + 49\nu^{2} - 2394\nu + 344 ) / 342 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -245\nu^{5} - 35\nu^{4} - 5\nu^{3} + 587\nu^{2} - 11970\nu + 1720 ) / 342 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{4} + 7\beta_{3} - 7\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} + 7\beta _1 - 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} - 24\beta_{4} - 49\beta_{3} - 49\beta_{2} - 2\beta _1 + 24 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.79483 + 1.79483i
0.143705 0.143705i
−1.93854 + 1.93854i
−1.93854 1.93854i
0.143705 + 0.143705i
1.79483 1.79483i
5.03251i 8.47535i −17.3261 0 −42.6523 3.81828i 46.9339i −44.8316 0
324.2 4.67129i 7.62999i −13.8209 0 −35.6419 26.1222i 27.1912i −31.2167 0
324.3 1.36122i 3.15463i 6.14708 0 4.29415 7.94049i 19.2573i 17.0483 0
324.4 1.36122i 3.15463i 6.14708 0 4.29415 7.94049i 19.2573i 17.0483 0
324.5 4.67129i 7.62999i −13.8209 0 −35.6419 26.1222i 27.1912i −31.2167 0
324.6 5.03251i 8.47535i −17.3261 0 −42.6523 3.81828i 46.9339i −44.8316 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.6
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 425.4.b.f 6
5.b even 2 1 inner 425.4.b.f 6
5.c odd 4 1 17.4.a.b 3
5.c odd 4 1 425.4.a.g 3
15.e even 4 1 153.4.a.g 3
20.e even 4 1 272.4.a.h 3
35.f even 4 1 833.4.a.d 3
40.i odd 4 1 1088.4.a.v 3
40.k even 4 1 1088.4.a.x 3
55.e even 4 1 2057.4.a.e 3
60.l odd 4 1 2448.4.a.bi 3
85.f odd 4 1 289.4.b.b 6
85.g odd 4 1 289.4.a.b 3
85.i odd 4 1 289.4.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 5.c odd 4 1
153.4.a.g 3 15.e even 4 1
272.4.a.h 3 20.e even 4 1
289.4.a.b 3 85.g odd 4 1
289.4.b.b 6 85.f odd 4 1
289.4.b.b 6 85.i odd 4 1
425.4.a.g 3 5.c odd 4 1
425.4.b.f 6 1.a even 1 1 trivial
425.4.b.f 6 5.b even 2 1 inner
833.4.a.d 3 35.f even 4 1
1088.4.a.v 3 40.i odd 4 1
1088.4.a.x 3 40.k even 4 1
2057.4.a.e 3 55.e even 4 1
2448.4.a.bi 3 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{6} + 49T_{2}^{4} + 640T_{2}^{2} + 1024 \) Copy content Toggle raw display
\( T_{3}^{6} + 140T_{3}^{4} + 5476T_{3}^{2} + 41616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 49 T^{4} + 640 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( T^{6} + 140 T^{4} + 5476 T^{2} + \cdots + 41616 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 760 T^{4} + 53892 T^{2} + \cdots + 627264 \) Copy content Toggle raw display
$11$ \( (T^{3} + 28 T^{2} - 1366 T - 4692)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 3844 T^{4} + \cdots + 88209664 \) Copy content Toggle raw display
$17$ \( (T^{2} + 289)^{3} \) Copy content Toggle raw display
$19$ \( (T^{3} + 80 T^{2} - 4632 T - 340128)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 51704 T^{4} + \cdots + 2561741095936 \) Copy content Toggle raw display
$29$ \( (T^{3} - 456 T^{2} + 53908 T - 1518624)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 230 T^{2} - 11586 T - 81608)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 162664 T^{4} + \cdots + 38152265269504 \) Copy content Toggle raw display
$41$ \( (T^{3} + 294 T^{2} - 86564 T - 1638744)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 206944 T^{4} + \cdots + 52856854953984 \) Copy content Toggle raw display
$47$ \( T^{6} + 238944 T^{4} + \cdots + 2792802484224 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 329860859333184 \) Copy content Toggle raw display
$59$ \( (T^{3} + 636 T^{2} - 101768 T - 49419072)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 84 T^{2} - 124412 T - 6792784)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 885984 T^{4} + \cdots + 586682466304 \) Copy content Toggle raw display
$71$ \( (T^{3} + 402 T^{2} - 589874 T - 274866016)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 398302285230144 \) Copy content Toggle raw display
$79$ \( (T^{3} - 594 T^{2} - 1121274 T + 742135824)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 2763040 T^{4} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{3} - 170 T^{2} - 1072304 T + 446571376)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 1245100 T^{4} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
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