Properties

Label 5.14.a.b
Level $5$
Weight $14$
Character orbit 5.a
Self dual yes
Analytic conductor $5.362$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.36154644760\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 4466x - 18720 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \beta_1 + 47) q^{2} + ( - \beta_{2} - 3 \beta_1 + 138) q^{3} + (6 \beta_{2} - 80 \beta_1 + 5930) q^{4} + 15625 q^{5} + ( - 44 \beta_{2} + 720 \beta_1 + 40380) q^{6} + ( - 99 \beta_{2} + 1879 \beta_1 + 150090) q^{7} + (852 \beta_{2} - 6120 \beta_1 + 857796) q^{8} + ( - 1268 \beta_{2} + 2340 \beta_1 + 429909) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 47) q^{2} + ( - \beta_{2} - 3 \beta_1 + 138) q^{3} + (6 \beta_{2} - 80 \beta_1 + 5930) q^{4} + 15625 q^{5} + ( - 44 \beta_{2} + 720 \beta_1 + 40380) q^{6} + ( - 99 \beta_{2} + 1879 \beta_1 + 150090) q^{7} + (852 \beta_{2} - 6120 \beta_1 + 857796) q^{8} + ( - 1268 \beta_{2} + 2340 \beta_1 + 429909) q^{9} + ( - 15625 \beta_1 + 734375) q^{10} + ( - 330 \beta_{2} + 6050 \beta_1 - 2199208) q^{11} + (1144 \beta_{2} + 50064 \beta_1 - 7891176) q^{12} + (10380 \beta_{2} + 62372 \beta_1 - 11149994) q^{13} + ( - 17412 \beta_{2} + 6660 \beta_1 - 15512952) q^{14} + ( - 15625 \beta_{2} - 46875 \beta_1 + 2156250) q^{15} + (40392 \beta_{2} - 1219760 \beta_1 + 66217352) q^{16} + (7788 \beta_{2} + 957764 \beta_1 + 28026506) q^{17} + ( - 92656 \beta_{2} + 860787 \beta_1 - 10010157) q^{18} + (112404 \beta_{2} + 1339580 \beta_1 + 32906092) q^{19} + (93750 \beta_{2} - 1250000 \beta_1 + 92656250) q^{20} + ( - 268488 \beta_{2} - 2051160 \beta_1 + 145472688) q^{21} + ( - 56760 \beta_{2} + 2714668 \beta_1 - 176045276) q^{22} + (179931 \beta_{2} - 6456303 \beta_1 + 103206534) q^{23} + (130992 \beta_{2} + 2550240 \beta_1 - 1295979984) q^{24} + 244140625 q^{25} + (269328 \beta_{2} + 3274610 \beta_1 - 1247936254) q^{26} + ( - 286858 \beta_{2} + 1695330 \beta_1 + 2173644516) q^{27} + ( - 308496 \beta_{2} + 17003248 \beta_1 - 2070111744) q^{28} + ( - 779784 \beta_{2} - 21088280 \beta_1 + 738621118) q^{29} + ( - 687500 \beta_{2} + 11250000 \beta_1 + 630937500) q^{30} + (1649910 \beta_{2} + 26951650 \beta_1 + 2502765372) q^{31} + (2843280 \beta_{2} - 94989536 \beta_1 + 10690674832) q^{32} + (1805188 \beta_{2} + 1445004 \beta_1 + 119607576) q^{33} + ( - 5263728 \beta_{2} - 3873410 \beta_1 - 10078227890) q^{34} + ( - 1546875 \beta_{2} + 29359375 \beta_1 + 2345156250) q^{35} + ( - 521938 \beta_{2} + 107918640 \beta_1 - 14417797758) q^{36} + ( - 6455208 \beta_{2} + 108165384 \beta_1 + 10520598278) q^{37} + ( - 1068432 \beta_{2} - 96270580 \beta_1 - 14204444836) q^{38} + (22785698 \beta_{2} - 22027290 \beta_1 - 23411128020) q^{39} + (13312500 \beta_{2} - 95625000 \beta_1 + 13403062500) q^{40} + ( - 21811740 \beta_{2} - 180118100 \beta_1 - 3637063598) q^{41} + ( - 4339296 \beta_{2} + 43782048 \beta_1 + 30777325056) q^{42} + ( - 8620077 \beta_{2} - 44724103 \beta_1 + 5631483610) q^{43} + ( - 17103768 \beta_{2} + 270387040 \beta_1 - 22702778120) q^{44} + ( - 19812500 \beta_{2} + 36562500 \beta_1 + 6717328125) q^{45} + (49893540 \beta_{2} - 488458500 \beta_1 + 82096617432) q^{46} + (45223509 \beta_{2} + 246549119 \beta_1 + 10711842434) q^{47} + ( - 16551584 \beta_{2} + 844654272 \beta_1 - 26405874336) q^{48} + ( - 12323100 \beta_{2} + 230960300 \beta_1 - 12904547743) q^{49} + ( - 244140625 \beta_1 + 11474609375) q^{50} + ( - 22029266 \beta_{2} - 910046070 \beta_1 - 43419299916) q^{51} + ( - 87982284 \beta_{2} + 587300064 \beta_1 - 5825771860) q^{52} + (81563364 \beta_{2} - 833062228 \beta_1 - 74021249842) q^{53} + ( - 27957176 \beta_{2} - 1843175520 \beta_1 + 81435572952) q^{54} + ( - 5156250 \beta_{2} + 94531250 \beta_1 - 34362625000) q^{55} + (21492864 \beta_{2} + 2871890880 \beta_1 - 173342017728) q^{56} + (91103324 \beta_{2} - 1271317740 \beta_1 - 254824372248) q^{57} + (78183072 \beta_{2} - 688281070 \beta_1 + 284501170706) q^{58} + ( - 67120488 \beta_{2} + 3593888840 \beta_1 - 17258471524) q^{59} + (17875000 \beta_{2} + 782250000 \beta_1 - 123299625000) q^{60} + ( - 180786000 \beta_{2} + 2927210000 \beta_1 + 166423129582) q^{61} + ( - 59415480 \beta_{2} - 3192324792 \beta_1 - 200400950016) q^{62} + ( - 287289063 \beta_{2} - 1620190773 \beta_1 + 361373827842) q^{63} + (415329312 \beta_{2} - 6554074560 \beta_1 + 1096865363488) q^{64} + (162187500 \beta_{2} + 974562500 \beta_1 - 174218656250) q^{65} + (103251632 \beta_{2} - 1799487360 \beta_1 - 8262204720) q^{66} + ( - 29283327 \beta_{2} - 1675384061 \beta_1 + 152550574686) q^{67} + ( - 366909972 \beta_{2} + 7141790368 \beta_1 - 666837492812) q^{68} + (121103784 \beta_{2} + 5218489080 \beta_1 - 109427302512) q^{69} + ( - 272062500 \beta_{2} + 104062500 \beta_1 - 242389875000) q^{70} + (455730750 \beta_{2} - 6972263750 \beta_1 + 171523294612) q^{71} + (79165956 \beta_{2} + 11427040440 \beta_1 - 1882231022412) q^{72} + (505636956 \beta_{2} - 5146299628 \beta_1 + 833124544834) q^{73} + ( - 1049215200 \beta_{2} - 773506550 \beta_1 - 806015959286) q^{74} + ( - 244140625 \beta_{2} - 732421875 \beta_1 + 33691406250) q^{75} + ( - 409432872 \beta_{2} + 1076165760 \beta_1 + 207724549544) q^{76} + (223452372 \beta_{2} - 4319927172 \beta_1 - 130035279000) q^{77} + (1544877016 \beta_{2} + 878314464 \beta_1 - 795872298360) q^{78} + ( - 593544564 \beta_{2} + 8052774820 \beta_1 + 999761067528) q^{79} + (631125000 \beta_{2} - 19058750000 \beta_1 + 1034646125000) q^{80} + ( - 483860204 \beta_{2} - 11662275780 \beta_1 + 103123545729) q^{81} + ( - 271619280 \beta_{2} + 18567001478 \beta_1 + 1934562275894) q^{82} + ( - 817515513 \beta_{2} - 9286132203 \beta_1 + 1709555617002) q^{83} + (1667725056 \beta_{2} - 8376708480 \beta_1 - 274759521408) q^{84} + (121687500 \beta_{2} + 14965062500 \beta_1 + 437914156250) q^{85} + ( - 266100156 \beta_{2} + 1142034680 \beta_1 + 781573926644) q^{86} + ( - 1563530254 \beta_{2} + 16074488790 \beta_1 + 2301019399308) q^{87} + ( - 2217777936 \beta_{2} + 25755296160 \beta_1 - 2877544930128) q^{88} + (291050568 \beta_{2} + 18339769560 \beta_1 - 6468325746486) q^{89} + ( - 1447750000 \beta_{2} + 13449796875 \beta_1 - 156408703125) q^{90} + (4147432374 \beta_{2} + 5581943330 \beta_1 - 2263695953076) q^{91} + (4550155728 \beta_{2} - 93073831536 \beta_1 + 8924132784576) q^{92} + ( - 704372832 \beta_{2} - 30995801376 \beta_1 - 3708731188224) q^{93} + (1324562844 \beta_{2} - 45854619620 \beta_1 - 2350245686144) q^{94} + (1756312500 \beta_{2} + 20930937500 \beta_1 + 514157687500) q^{95} + ( - 7167210304 \beta_{2} + 49227765120 \beta_1 - 717312079680) q^{96} + ( - 5428709196 \beta_{2} - 12890819556 \beta_1 - 3698595835606) q^{97} + ( - 2149794000 \beta_{2} + 32319444343 \beta_1 - 3380679917321) q^{98} + (2458293134 \beta_{2} - 11557312470 \beta_1 + 37590586968) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 142 q^{2} + 416 q^{3} + 17876 q^{4} + 46875 q^{5} + 120376 q^{6} + 448292 q^{7} + 2580360 q^{8} + 1286119 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 142 q^{2} + 416 q^{3} + 17876 q^{4} + 46875 q^{5} + 120376 q^{6} + 448292 q^{7} + 2580360 q^{8} + 1286119 q^{9} + 2218750 q^{10} - 6604004 q^{11} - 23722448 q^{12} - 33501974 q^{13} - 46562928 q^{14} + 6500000 q^{15} + 199912208 q^{16} + 83129542 q^{17} - 30983914 q^{18} + 97491100 q^{19} + 279312500 q^{20} + 438200736 q^{21} - 530907256 q^{22} + 316255836 q^{23} - 3890359200 q^{24} + 732421875 q^{25} - 3746814044 q^{26} + 6518951360 q^{27} - 6227646976 q^{28} + 2236171850 q^{29} + 1880875000 q^{30} + 7482994376 q^{31} + 32169857312 q^{32} + 359182912 q^{33} - 30236073988 q^{34} + 7004562500 q^{35} - 43361833852 q^{36} + 31447174242 q^{37} - 42518132360 q^{38} - 70188571072 q^{39} + 40318125000 q^{40} - 10752884434 q^{41} + 92283853824 q^{42} + 16930554856 q^{43} - 68395825168 q^{44} + 20095609375 q^{45} + 246828204336 q^{46} + 31934201692 q^{47} - 80078828864 q^{48} - 38956926629 q^{49} + 34667968750 q^{50} - 129369882944 q^{51} - 18152597928 q^{52} - 221149123934 q^{53} + 246121937200 q^{54} - 103187562500 q^{55} - 522876451200 q^{56} - 763110695680 q^{57} + 854269976260 q^{58} - 55436423900 q^{59} - 370663250000 q^{60} + 496161392746 q^{61} - 598069940736 q^{62} + 1085454385236 q^{63} + 3297565494336 q^{64} - 523468343750 q^{65} - 22883875168 q^{66} + 459297824792 q^{67} - 2008021178776 q^{68} - 333379292832 q^{69} - 727545750000 q^{70} + 521997878336 q^{71} - 5658040941720 q^{72} + 2505025571086 q^{73} - 2418323586508 q^{74} + 101562500000 q^{75} + 621688050000 q^{76} - 385562457456 q^{77} - 2386950332528 q^{78} + 2990636883200 q^{79} + 3123628250000 q^{80} + 320549052763 q^{81} + 5784848206924 q^{82} + 5137135467696 q^{83} - 814234130688 q^{84} + 1298899093750 q^{85} + 2343313645096 q^{86} + 6885420178880 q^{87} - 8660607864480 q^{88} - 19423025958450 q^{89} - 484123656250 q^{90} - 6792522370184 q^{91} + 26870022340992 q^{92} - 11095902136128 q^{93} - 7003557875968 q^{94} + 1523298437500 q^{95} - 2208331214464 q^{96} - 11088325396458 q^{97} - 10176508990306 q^{98} + 126787366508 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4466x - 18720 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 16\nu - 5949 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 8\beta _1 + 5957 ) / 2 \) 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
69.3208
−4.21238
−64.1084
−90.6415 −1125.79 23.8902 15625.0 102044. 324482. 740370. −326912. −1.41627e6
1.2 56.4248 2114.98 −5008.25 15625.0 119337. 325303. −744821. 2.87881e6 881637.
1.3 176.217 −573.185 22860.4 15625.0 −101005. −201493. 2.58481e6 −1.26578e6 2.75339e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 5.14.a.b 3
3.b odd 2 1 45.14.a.e 3
4.b odd 2 1 80.14.a.g 3
5.b even 2 1 25.14.a.b 3
5.c odd 4 2 25.14.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.14.a.b 3 1.a even 1 1 trivial
25.14.a.b 3 5.b even 2 1
25.14.b.b 6 5.c odd 4 2
45.14.a.e 3 3.b odd 2 1
80.14.a.g 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 142T_{2}^{2} - 11144T_{2} + 901248 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 142 T^{2} - 11144 T + 901248 \) Copy content Toggle raw display
$3$ \( T^{3} - 416 T^{2} + \cdots - 1364770944 \) Copy content Toggle raw display
$5$ \( (T - 15625)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 448292 T^{2} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
$11$ \( T^{3} + 6604004 T^{2} + \cdots + 88\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{3} + 33501974 T^{2} + \cdots - 15\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{3} - 83129542 T^{2} + \cdots + 55\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{3} - 97491100 T^{2} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} - 316255836 T^{2} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{3} - 2236171850 T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} - 7482994376 T^{2} + \cdots + 61\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{3} - 31447174242 T^{2} + \cdots + 46\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{3} + 10752884434 T^{2} + \cdots - 82\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{3} - 16930554856 T^{2} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{3} - 31934201692 T^{2} + \cdots + 18\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{3} + 221149123934 T^{2} + \cdots - 33\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{3} + 55436423900 T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} - 496161392746 T^{2} + \cdots + 83\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{3} - 459297824792 T^{2} + \cdots + 78\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{3} - 521997878336 T^{2} + \cdots - 40\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{3} - 2505025571086 T^{2} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{3} - 2990636883200 T^{2} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} - 5137135467696 T^{2} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{3} + 19423025958450 T^{2} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + 11088325396458 T^{2} + \cdots - 56\!\cdots\!92 \) Copy content Toggle raw display
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