Properties

Label 245.4.e.h
Level $245$
Weight $4$
Character orbit 245.e
Analytic conductor $14.455$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 4 \beta_1 - 1) q^{3} + ( - 10 \beta_{2} + 8 \beta_1 - 10) q^{4} - 5 \beta_{2} q^{5} + ( - 15 \beta_{3} - 4) q^{6} + (34 \beta_{3} + 24) q^{8} + (8 \beta_{3} + 6 \beta_{2} + 8 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 4 \beta_1 - 1) q^{3} + ( - 10 \beta_{2} + 8 \beta_1 - 10) q^{4} - 5 \beta_{2} q^{5} + ( - 15 \beta_{3} - 4) q^{6} + (34 \beta_{3} + 24) q^{8} + (8 \beta_{3} + 6 \beta_{2} + 8 \beta_1) q^{9} + (20 \beta_{2} - 5 \beta_1 + 20) q^{10} + (7 \beta_{2} - 32 \beta_1 + 7) q^{11} + (32 \beta_{3} - 54 \beta_{2} + 32 \beta_1) q^{12} + (4 \beta_{3} + 25) q^{13} + (20 \beta_{3} - 5) q^{15} + ( - 96 \beta_{3} + 84 \beta_{2} - 96 \beta_1) q^{16} + (25 \beta_{2} + 44 \beta_1 + 25) q^{17} + ( - 8 \beta_{2} - 26 \beta_1 - 8) q^{18} + ( - 44 \beta_{3} + 18 \beta_{2} - 44 \beta_1) q^{19} + ( - 40 \beta_{3} - 50) q^{20} + ( - 135 \beta_{3} - 92) q^{22} + (68 \beta_{3} + 122 \beta_{2} + 68 \beta_1) q^{23} + (248 \beta_{2} - 62 \beta_1 + 248) q^{24} + ( - 25 \beta_{2} - 25) q^{25} + ( - 41 \beta_{3} + 108 \beta_{2} - 41 \beta_1) q^{26} + (76 \beta_{3} + 43) q^{27} + (24 \beta_{3} - 13) q^{29} + ( - 75 \beta_{3} + 20 \beta_{2} - 75 \beta_1) q^{30} + (60 \beta_{2} - 180 \beta_1 + 60) q^{31} + ( - 336 \beta_{2} + 196 \beta_1 - 336) q^{32} + (4 \beta_{3} + 249 \beta_{2} + 4 \beta_1) q^{33} + (151 \beta_{3} - 12) q^{34} + ( - 32 \beta_{3} - 68) q^{36} + (60 \beta_{3} + 282 \beta_{2} + 60 \beta_1) q^{37} + ( - 160 \beta_{2} + 194 \beta_1 - 160) q^{38} + (7 \beta_{2} - 96 \beta_1 + 7) q^{39} + (170 \beta_{3} - 120 \beta_{2} + 170 \beta_1) q^{40} + (124 \beta_{3} - 164) q^{41} + (68 \beta_{3} - 130) q^{43} + (376 \beta_{3} - 582 \beta_{2} + 376 \beta_1) q^{44} + (30 \beta_{2} + 40 \beta_1 + 30) q^{45} + ( - 352 \beta_{2} - 150 \beta_1 - 352) q^{46} + (132 \beta_{3} - 175 \beta_{2} + 132 \beta_1) q^{47} + ( - 240 \beta_{3} - 684) q^{48} + (25 \beta_{3} + 100) q^{50} + ( - 144 \beta_{3} - 377 \beta_{2} - 144 \beta_1) q^{51} + ( - 314 \beta_{2} + 240 \beta_1 - 314) q^{52} + (28 \beta_{2} + 128 \beta_1 + 28) q^{53} + ( - 347 \beta_{3} + 324 \beta_{2} - 347 \beta_1) q^{54} + (160 \beta_{3} + 35) q^{55} + ( - 28 \beta_{3} - 334) q^{57} + ( - 83 \beta_{3} - 4 \beta_{2} - 83 \beta_1) q^{58} + (616 \beta_{2} + 616) q^{59} + ( - 270 \beta_{2} + 160 \beta_1 - 270) q^{60} + (108 \beta_{3} + 168 \beta_{2} + 108 \beta_1) q^{61} + ( - 780 \beta_{3} - 600) q^{62} + (352 \beta_{3} + 1064) q^{64} + (20 \beta_{3} - 125 \beta_{2} + 20 \beta_1) q^{65} + ( - 988 \beta_{2} + 233 \beta_1 - 988) q^{66} + (76 \beta_{2} - 64 \beta_1 + 76) q^{67} + ( - 240 \beta_{3} + 454 \beta_{2} - 240 \beta_1) q^{68} + ( - 556 \beta_{3} + 666) q^{69} - 952 q^{71} + ( - 12 \beta_{3} - 400 \beta_{2} - 12 \beta_1) q^{72} + ( - 338 \beta_{2} - 344 \beta_1 - 338) q^{73} + ( - 1008 \beta_{2} + 42 \beta_1 - 1008) q^{74} + (100 \beta_{3} + 25 \beta_{2} + 100 \beta_1) q^{75} + (584 \beta_{3} + 884) q^{76} + ( - 391 \beta_{3} - 220) q^{78} + ( - 248 \beta_{3} + 507 \beta_{2} - 248 \beta_1) q^{79} + (420 \beta_{2} - 480 \beta_1 + 420) q^{80} + (727 \beta_{2} + 120 \beta_1 + 727) q^{81} + ( - 332 \beta_{3} - 408 \beta_{2} - 332 \beta_1) q^{82} + (600 \beta_{3} - 188) q^{83} + ( - 220 \beta_{3} + 125) q^{85} + ( - 142 \beta_{3} - 384 \beta_{2} - 142 \beta_1) q^{86} + (205 \beta_{2} + 76 \beta_1 + 205) q^{87} + (2344 \beta_{2} - 1006 \beta_1 + 2344) q^{88} + ( - 44 \beta_{3} - 108 \beta_{2} - 44 \beta_1) q^{89} + (130 \beta_{3} - 40) q^{90} + (296 \beta_{3} + 132) q^{92} + ( - 60 \beta_{3} + 1380 \beta_{2} - 60 \beta_1) q^{93} + (964 \beta_{2} - 703 \beta_1 + 964) q^{94} + (90 \beta_{2} - 220 \beta_1 + 90) q^{95} + (1148 \beta_{3} - 1232 \beta_{2} + 1148 \beta_1) q^{96} + (220 \beta_{3} + 1371) q^{97} + ( - 136 \beta_{3} + 470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 2 q^{3} - 20 q^{4} + 10 q^{5} - 16 q^{6} + 96 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} - 2 q^{3} - 20 q^{4} + 10 q^{5} - 16 q^{6} + 96 q^{8} - 12 q^{9} + 40 q^{10} + 14 q^{11} + 108 q^{12} + 100 q^{13} - 20 q^{15} - 168 q^{16} + 50 q^{17} - 16 q^{18} - 36 q^{19} - 200 q^{20} - 368 q^{22} - 244 q^{23} + 496 q^{24} - 50 q^{25} - 216 q^{26} + 172 q^{27} - 52 q^{29} - 40 q^{30} + 120 q^{31} - 672 q^{32} - 498 q^{33} - 48 q^{34} - 272 q^{36} - 564 q^{37} - 320 q^{38} + 14 q^{39} + 240 q^{40} - 656 q^{41} - 520 q^{43} + 1164 q^{44} + 60 q^{45} - 704 q^{46} + 350 q^{47} - 2736 q^{48} + 400 q^{50} + 754 q^{51} - 628 q^{52} + 56 q^{53} - 648 q^{54} + 140 q^{55} - 1336 q^{57} + 8 q^{58} + 1232 q^{59} - 540 q^{60} - 336 q^{61} - 2400 q^{62} + 4256 q^{64} + 250 q^{65} - 1976 q^{66} + 152 q^{67} - 908 q^{68} + 2664 q^{69} - 3808 q^{71} + 800 q^{72} - 676 q^{73} - 2016 q^{74} - 50 q^{75} + 3536 q^{76} - 880 q^{78} - 1014 q^{79} + 840 q^{80} + 1454 q^{81} + 816 q^{82} - 752 q^{83} + 500 q^{85} + 768 q^{86} + 410 q^{87} + 4688 q^{88} + 216 q^{89} - 160 q^{90} + 528 q^{92} - 2760 q^{93} + 1928 q^{94} + 180 q^{95} + 2464 q^{96} + 5484 q^{97} + 1880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−2.70711 4.68885i 2.32843 4.03295i −10.6569 + 18.4582i 2.50000 + 4.33013i −25.2132 0 72.0833 2.65685 + 4.60181i 13.5355 23.4442i
116.2 −1.29289 2.23936i −3.32843 + 5.76500i 0.656854 1.13770i 2.50000 + 4.33013i 17.2132 0 −24.0833 −8.65685 14.9941i 6.46447 11.1968i
226.1 −2.70711 + 4.68885i 2.32843 + 4.03295i −10.6569 18.4582i 2.50000 4.33013i −25.2132 0 72.0833 2.65685 4.60181i 13.5355 + 23.4442i
226.2 −1.29289 + 2.23936i −3.32843 5.76500i 0.656854 + 1.13770i 2.50000 4.33013i 17.2132 0 −24.0833 −8.65685 + 14.9941i 6.46447 + 11.1968i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.e.h 4
7.b odd 2 1 245.4.e.i 4
7.c even 3 1 35.4.a.b 2
7.c even 3 1 inner 245.4.e.h 4
7.d odd 6 1 245.4.a.k 2
7.d odd 6 1 245.4.e.i 4
21.g even 6 1 2205.4.a.u 2
21.h odd 6 1 315.4.a.f 2
28.g odd 6 1 560.4.a.r 2
35.i odd 6 1 1225.4.a.m 2
35.j even 6 1 175.4.a.c 2
35.l odd 12 2 175.4.b.c 4
56.k odd 6 1 2240.4.a.bo 2
56.p even 6 1 2240.4.a.bn 2
105.o odd 6 1 1575.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 7.c even 3 1
175.4.a.c 2 35.j even 6 1
175.4.b.c 4 35.l odd 12 2
245.4.a.k 2 7.d odd 6 1
245.4.e.h 4 1.a even 1 1 trivial
245.4.e.h 4 7.c even 3 1 inner
245.4.e.i 4 7.b odd 2 1
245.4.e.i 4 7.d odd 6 1
315.4.a.f 2 21.h odd 6 1
560.4.a.r 2 28.g odd 6 1
1225.4.a.m 2 35.i odd 6 1
1575.4.a.z 2 105.o odd 6 1
2205.4.a.u 2 21.g even 6 1
2240.4.a.bn 2 56.p even 6 1
2240.4.a.bo 2 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} + 8T_{2}^{3} + 50T_{2}^{2} + 112T_{2} + 196 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} + 35T_{3}^{2} - 62T_{3} + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 8 T^{3} + 50 T^{2} + 112 T + 196 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 35 T^{2} - 62 T + 961 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 14 T^{3} + 2195 T^{2} + \cdots + 3996001 \) Copy content Toggle raw display
$13$ \( (T^{2} - 50 T + 593)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 50 T^{3} + 5747 T^{2} + \cdots + 10543009 \) Copy content Toggle raw display
$19$ \( T^{4} + 36 T^{3} + 4844 T^{2} + \cdots + 12588304 \) Copy content Toggle raw display
$23$ \( T^{4} + 244 T^{3} + \cdots + 31764496 \) Copy content Toggle raw display
$29$ \( (T^{2} + 26 T - 983)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 120 T^{3} + \cdots + 3745440000 \) Copy content Toggle raw display
$37$ \( T^{4} + 564 T^{3} + \cdots + 5230760976 \) Copy content Toggle raw display
$41$ \( (T^{2} + 328 T - 3856)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 260 T + 7652)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 350 T^{3} + \cdots + 17833729 \) Copy content Toggle raw display
$53$ \( T^{4} - 56 T^{3} + \cdots + 1022976256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 616 T + 379456)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 336 T^{3} + \cdots + 23970816 \) Copy content Toggle raw display
$67$ \( T^{4} - 152 T^{3} + 25520 T^{2} + \cdots + 5837056 \) Copy content Toggle raw display
$71$ \( (T + 952)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 676 T^{3} + \cdots + 14988615184 \) Copy content Toggle raw display
$79$ \( T^{4} + 1014 T^{3} + \cdots + 17966989681 \) Copy content Toggle raw display
$83$ \( (T^{2} + 376 T - 684656)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 216 T^{3} + \cdots + 60715264 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2742 T + 1782841)^{2} \) Copy content Toggle raw display
show more
show less