Properties

Label 245.6.a.e
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \) 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)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + ( - \beta_{3} + \beta_1 - 4) q^{3} + (3 \beta_{2} - 5 \beta_1 + 15) q^{4} - 25 q^{5} + (11 \beta_{2} + 2 \beta_1 - 29) q^{6} + ( - 6 \beta_{3} + 21 \beta_{2} - 7 \beta_1 + 133) q^{8} + ( - \beta_{3} - 40 \beta_{2} + 33 \beta_1 + 165) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + ( - \beta_{3} + \beta_1 - 4) q^{3} + (3 \beta_{2} - 5 \beta_1 + 15) q^{4} - 25 q^{5} + (11 \beta_{2} + 2 \beta_1 - 29) q^{6} + ( - 6 \beta_{3} + 21 \beta_{2} - 7 \beta_1 + 133) q^{8} + ( - \beta_{3} - 40 \beta_{2} + 33 \beta_1 + 165) q^{9} + (25 \beta_1 - 50) q^{10} + ( - 9 \beta_{3} + 72 \beta_{2} - 7 \beta_1 + 228) q^{11} + (10 \beta_{3} + 16 \beta_{2} - 30 \beta_1 - 192) q^{12} + ( - 39 \beta_{3} - 41 \beta_1 - 14) q^{13} + (25 \beta_{3} - 25 \beta_1 + 100) q^{15} + ( - 42 \beta_{3} + 51 \beta_{2} - 87 \beta_1 - 117) q^{16} + ( - 9 \beta_{3} + 144 \beta_{2} - 55 \beta_1 - 418) q^{17} + (80 \beta_{3} - 165 \beta_{2} + 49 \beta_1 - 427) q^{18} + (18 \beta_{3} - 24 \beta_{2} - 226 \beta_1 - 92) q^{19} + ( - 75 \beta_{2} + 125 \beta_1 - 375) q^{20} + ( - 144 \beta_{3} + 291 \beta_{2} - 510 \beta_1 - 197) q^{22} + ( - 30 \beta_{3} + 72 \beta_{2} - 274 \beta_1 - 1488) q^{23} + ( - 32 \beta_{3} - 370 \beta_{2} + 40 \beta_1 + 1358) q^{24} + 625 q^{25} + (669 \beta_{2} - 304 \beta_1 + 2593) q^{26} + ( - 191 \beta_{3} + 128 \beta_{2} + 559 \beta_1 + 1736) q^{27} + ( - 141 \beta_{3} - 168 \beta_{2} + 333 \beta_1 + 1814) q^{29} + ( - 275 \beta_{2} - 50 \beta_1 + 725) q^{30} + ( - 48 \beta_{3} - 456 \beta_{2} - 576 \beta_1 + 1368) q^{31} + (90 \beta_{3} + 279 \beta_{2} - 283 \beta_1 - 641) q^{32} + (119 \beta_{3} - 1120 \beta_{2} - 519 \beta_1 + 1632) q^{33} + ( - 288 \beta_{3} + 579 \beta_{2} - 224 \beta_1 - 577) q^{34} + (362 \beta_{3} - 317 \beta_{2} + 413 \beta_1 - 7361) q^{36} + ( - 240 \beta_{3} - 264 \beta_{2} - 528 \beta_1 + 7326) q^{37} + (48 \beta_{3} + 378 \beta_{2} - 424 \beta_1 + 9522) q^{38} + ( - 21 \beta_{3} - 680 \beta_{2} + 1429 \beta_1 + 13664) q^{39} + (150 \beta_{3} - 525 \beta_{2} + 175 \beta_1 - 3325) q^{40} + (270 \beta_{3} - 504 \beta_{2} + 1474 \beta_1 - 3842) q^{41} + ( - 210 \beta_{3} - 1464 \beta_{2} + 370 \beta_1 - 2300) q^{43} + ( - 294 \beta_{3} + 1824 \beta_{2} - 2702 \beta_1 + 12752) q^{44} + (25 \beta_{3} + 1000 \beta_{2} - 825 \beta_1 - 4125) q^{45} + ( - 144 \beta_{3} + 1386 \beta_{2} + 300 \beta_1 + 8314) q^{46} + ( - 429 \beta_{3} + 144 \beta_{2} + 1149 \beta_1 - 10304) q^{47} + (420 \beta_{3} - 924 \beta_{2} + 672 \beta_1 + 13764) q^{48} + ( - 625 \beta_1 + 1250) q^{50} + (1221 \beta_{3} - 1528 \beta_{2} - 2245 \beta_1 + 2560) q^{51} + ( - 90 \beta_{3} + 2250 \beta_{2} - 4200 \beta_1 + 8002) q^{52} + ( - 402 \beta_{3} - 960 \beta_{2} - 158 \beta_1 + 4534) q^{53} + ( - 256 \beta_{3} + 1253 \beta_{2} - 1398 \beta_1 - 18411) q^{54} + (225 \beta_{3} - 1800 \beta_{2} + 175 \beta_1 - 5700) q^{55} + (478 \beta_{3} + 3320 \beta_{2} - 398 \beta_1 - 10840) q^{57} + (336 \beta_{3} + 639 \beta_{2} - 1016 \beta_1 - 4901) q^{58} + ( - 288 \beta_{3} - 1344 \beta_{2} + 4128 \beta_1 - 3884) q^{59} + ( - 250 \beta_{3} - 400 \beta_{2} + 750 \beta_1 + 4800) q^{60} + (330 \beta_{3} + 504 \beta_{2} - 186 \beta_1 - 7254) q^{61} + (912 \beta_{3} + 1488 \beta_{2} - 1968 \beta_1 + 35856) q^{62} + (786 \beta_{3} - 1485 \beta_{2} + 2189 \beta_1 + 8187) q^{64} + (975 \beta_{3} + 1025 \beta_1 + 350) q^{65} + (2240 \beta_{3} - 2349 \beta_{2} + 766 \beta_1 + 40883) q^{66} + (372 \beta_{3} - 1968 \beta_{2} + 5404 \beta_1 + 5956) q^{67} + ( - 870 \beta_{3} + 1254 \beta_{2} - 1512 \beta_1 + 18926) q^{68} + (2306 \beta_{3} + 1208 \beta_{2} - 1458 \beta_1 + 10680) q^{69} + ( - 672 \beta_{3} + 4464 \beta_{2} - 1968 \beta_1 - 18424) q^{71} + ( - 1926 \beta_{3} - 1661 \beta_{2} + 9793 \beta_1 - 21709) q^{72} + ( - 1068 \beta_{3} - 768 \beta_{2} + 604 \beta_1 + 18054) q^{73} + (528 \beta_{3} + 4416 \beta_{2} - 9318 \beta_1 + 46860) q^{74} + ( - 625 \beta_{3} + 625 \beta_1 - 2500) q^{75} + ( - 1332 \beta_{3} + 2124 \beta_{2} - 4456 \beta_1 + 33116) q^{76} + (1360 \beta_{3} - 5353 \beta_{2} - 7442 \beta_1 - 22777) q^{78} + ( - 87 \beta_{3} - 2424 \beta_{2} + 295 \beta_1 + 31064) q^{79} + (1050 \beta_{3} - 1275 \beta_{2} + 2175 \beta_1 + 2925) q^{80} + ( - 2544 \beta_{3} - 3632 \beta_{2} - 1136 \beta_1 + 34409) q^{81} + (1008 \beta_{3} - 9210 \beta_{2} + 11126 \beta_1 - 68942) q^{82} + ( - 1392 \beta_{3} + 1536 \beta_{2} - 4704 \beta_1 + 48476) q^{83} + (225 \beta_{3} - 3600 \beta_{2} + 1375 \beta_1 + 10450) q^{85} + (2928 \beta_{3} - 1098 \beta_{2} + 6752 \beta_1 + 7534) q^{86} + ( - 3743 \beta_{3} - 5568 \beta_{2} + 9551 \beta_1 + 53560) q^{87} + (960 \beta_{3} + 6558 \beta_{2} - 11480 \beta_1 + 125278) q^{88} + ( - 510 \beta_{3} + 4872 \beta_{2} - 178 \beta_1 + 9934) q^{89} + ( - 2000 \beta_{3} + 4125 \beta_{2} - 1225 \beta_1 + 10675) q^{90} + ( - 1812 \beta_{3} + 1584 \beta_{2} - 3524 \beta_1 + 32336) q^{92} + ( - 2640 \beta_{3} + 10872 \beta_{2} + 9984 \beta_1 + 4344) q^{93} + ( - 288 \beta_{3} + 2847 \beta_{2} + 11174 \beta_1 - 62881) q^{94} + ( - 450 \beta_{3} + 600 \beta_{2} + 5650 \beta_1 + 2300) q^{95} + (2872 \beta_{3} + 2096 \beta_{2} - 8156 \beta_1 - 39280) q^{96} + (2499 \beta_{3} - 6480 \beta_{2} + 5437 \beta_1 - 49042) q^{97} + ( - 3650 \beta_{3} + 6224 \beta_{2} + 15730 \beta_1 - 106900) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 7 q^{2} - 14 q^{3} + 49 q^{4} - 100 q^{5} - 136 q^{6} + 489 q^{8} + 774 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 7 q^{2} - 14 q^{3} + 49 q^{4} - 100 q^{5} - 136 q^{6} + 489 q^{8} + 774 q^{9} - 175 q^{10} + 770 q^{11} - 840 q^{12} - 58 q^{13} + 350 q^{15} - 615 q^{16} - 2006 q^{17} - 1409 q^{18} - 564 q^{19} - 1225 q^{20} - 1736 q^{22} - 6340 q^{23} + 6244 q^{24} + 2500 q^{25} + 8730 q^{26} + 7438 q^{27} + 8066 q^{29} + 3400 q^{30} + 5856 q^{31} - 3495 q^{32} + 8130 q^{33} - 3402 q^{34} - 28759 q^{36} + 29544 q^{37} + 36860 q^{38} + 57466 q^{39} - 12225 q^{40} - 13156 q^{41} - 5692 q^{43} + 44952 q^{44} - 19350 q^{45} + 30928 q^{46} - 39926 q^{47} + 57156 q^{48} + 4375 q^{50} + 9830 q^{51} + 23398 q^{52} + 20300 q^{53} - 77292 q^{54} - 19250 q^{55} - 50876 q^{57} - 22234 q^{58} - 8432 q^{59} + 21000 q^{60} - 30540 q^{61} + 137568 q^{62} + 37121 q^{64} + 1450 q^{65} + 166756 q^{66} + 32792 q^{67} + 72554 q^{68} + 36540 q^{69} - 83920 q^{71} - 71795 q^{72} + 75424 q^{73} + 168762 q^{74} - 8750 q^{75} + 125092 q^{76} - 89204 q^{78} + 129486 q^{79} + 15375 q^{80} + 146308 q^{81} - 247230 q^{82} + 187520 q^{83} + 50150 q^{85} + 36156 q^{86} + 238670 q^{87} + 475556 q^{88} + 30324 q^{89} + 35225 q^{90} + 124464 q^{92} + 8256 q^{93} - 245756 q^{94} + 14100 q^{95} - 172340 q^{96} - 180270 q^{97} - 420668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + \nu - 43 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} - 52\nu - 48 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} - \beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} - 3\beta_{2} + 53\beta _1 + 5 \) 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
8.05190
4.73688
−3.86448
−7.92431
−6.05190 −15.9755 4.62555 −25.0000 96.6824 0 165.668 12.2177 151.298
1.2 −2.73688 28.3358 −24.5095 −25.0000 −77.5516 0 154.660 559.917 68.4220
1.3 5.86448 −26.2268 2.39209 −25.0000 −153.807 0 −173.635 444.847 −146.612
1.4 9.92431 −0.133419 66.4919 −25.0000 −1.32409 0 342.308 −242.982 −248.108
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 245.6.a.e 4
7.b odd 2 1 35.6.a.d 4
21.c even 2 1 315.6.a.l 4
28.d even 2 1 560.6.a.v 4
35.c odd 2 1 175.6.a.f 4
35.f even 4 2 175.6.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.d 4 7.b odd 2 1
175.6.a.f 4 35.c odd 2 1
175.6.b.f 8 35.f even 4 2
245.6.a.e 4 1.a even 1 1 trivial
315.6.a.l 4 21.c even 2 1
560.6.a.v 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{4} - 7T_{2}^{3} - 64T_{2}^{2} + 250T_{2} + 964 \) Copy content Toggle raw display
\( T_{3}^{4} + 14T_{3}^{3} - 775T_{3}^{2} - 11976T_{3} - 1584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 7 T^{3} - 64 T^{2} + 250 T + 964 \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} - 775 T^{2} + \cdots - 1584 \) Copy content Toggle raw display
$5$ \( (T + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 770 T^{3} + \cdots - 20510223536 \) Copy content Toggle raw display
$13$ \( T^{4} + 58 T^{3} + \cdots + 430417376452 \) Copy content Toggle raw display
$17$ \( T^{4} + 2006 T^{3} + \cdots + 617989800868 \) Copy content Toggle raw display
$19$ \( T^{4} + 564 T^{3} + \cdots + 5221683964480 \) Copy content Toggle raw display
$23$ \( T^{4} + 6340 T^{3} + \cdots - 19307649395456 \) Copy content Toggle raw display
$29$ \( T^{4} - 8066 T^{3} + \cdots + 831691300 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 738407035699200 \) Copy content Toggle raw display
$37$ \( T^{4} - 29544 T^{3} + \cdots - 57\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 225655412958976 \) Copy content Toggle raw display
$43$ \( T^{4} + 5692 T^{3} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{4} + 39926 T^{3} + \cdots - 16\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{4} - 20300 T^{3} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} + 8432 T^{3} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} + 30540 T^{3} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} - 32792 T^{3} + \cdots - 74\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} + 83920 T^{3} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{4} - 75424 T^{3} + \cdots - 32\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{4} - 129486 T^{3} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} - 187520 T^{3} + \cdots - 29\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{4} - 30324 T^{3} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{4} + 180270 T^{3} + \cdots - 75\!\cdots\!24 \) Copy content Toggle raw display
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