Properties

Label 7.10.a.a
Level 7
Weight 10
Character orbit 7.a
Self dual yes
Analytic conductor 3.605
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.60525085315\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - \beta ) q^{2} + ( -43 + 11 \beta ) q^{3} + ( -310 + 6 \beta ) q^{4} + ( -1119 - 95 \beta ) q^{5} + ( -1994 + 10 \beta ) q^{6} -2401 q^{7} + ( 1308 + 804 \beta ) q^{8} + ( 5519 - 946 \beta ) q^{9} +O(q^{10})\) \( q + ( -3 - \beta ) q^{2} + ( -43 + 11 \beta ) q^{3} + ( -310 + 6 \beta ) q^{4} + ( -1119 - 95 \beta ) q^{5} + ( -1994 + 10 \beta ) q^{6} -2401 q^{7} + ( 1308 + 804 \beta ) q^{8} + ( 5519 - 946 \beta ) q^{9} + ( 21692 + 1404 \beta ) q^{10} + ( 17658 - 3326 \beta ) q^{11} + ( 26068 - 3668 \beta ) q^{12} + ( -13265 + 10899 \beta ) q^{13} + ( 7203 + 2401 \beta ) q^{14} + ( -153568 - 8224 \beta ) q^{15} + ( -376 - 6792 \beta ) q^{16} + ( -231960 + 9426 \beta ) q^{17} + ( 166021 - 2681 \beta ) q^{18} + ( -462713 - 1887 \beta ) q^{19} + ( 236880 + 22736 \beta ) q^{20} + ( 103243 - 26411 \beta ) q^{21} + ( 588944 - 7680 \beta ) q^{22} + ( 389064 - 38088 \beta ) q^{23} + ( 1650648 - 20184 \beta ) q^{24} + ( 1040861 + 212610 \beta ) q^{25} + ( -2063712 - 19432 \beta ) q^{26} + ( -1399306 - 115126 \beta ) q^{27} + ( 744310 - 14406 \beta ) q^{28} + ( -5001792 - 94682 \beta ) q^{29} + ( 2047936 + 178240 \beta ) q^{30} + ( 1233630 - 161430 \beta ) q^{31} + ( 642288 - 390896 \beta ) q^{32} + ( -7820392 + 337256 \beta ) q^{33} + ( -1123338 + 203682 \beta ) q^{34} + ( 2686719 + 228095 \beta ) q^{35} + ( -2806358 + 326374 \beta ) q^{36} + ( 15367776 - 248130 \beta ) q^{37} + ( 1752330 + 468374 \beta ) q^{38} + ( 23708972 - 614572 \beta ) q^{39} + ( -16204992 - 1023936 \beta ) q^{40} + ( -9551724 - 860818 \beta ) q^{41} + ( 4787594 - 24010 \beta ) q^{42} + ( 2032550 + 1048278 \beta ) q^{43} + ( -9325488 + 1137008 \beta ) q^{44} + ( 11169149 + 534269 \beta ) q^{45} + ( 6183792 - 274800 \beta ) q^{46} + ( -41097510 + 1033182 \beta ) q^{47} + ( -14403248 + 287920 \beta ) q^{48} + 5764801 q^{49} + ( -44156313 - 1678691 \beta ) q^{50} + ( 29985678 - 2956878 \beta ) q^{51} + ( 16733192 - 3458280 \beta ) q^{52} + ( -27594906 + 4685568 \beta ) q^{53} + ( 26417236 + 1744684 \beta ) q^{54} + ( 41222908 + 2044284 \beta ) q^{55} + ( -3140508 - 1930404 \beta ) q^{56} + ( 15890558 - 5008702 \beta ) q^{57} + ( 33279002 + 5285838 \beta ) q^{58} + ( -3534609 + 1563825 \beta ) q^{59} + ( 38082688 + 1628032 \beta ) q^{60} + ( 22158193 - 3395319 \beta ) q^{61} + ( 27455100 - 749340 \beta ) q^{62} + ( -13251119 + 2271346 \beta ) q^{63} + ( 73708576 + 4007904 \beta ) q^{64} + ( -184989630 - 10935806 \beta ) q^{65} + ( -41629232 + 6808624 \beta ) q^{66} + ( -120960668 - 7026216 \beta ) q^{67} + ( 82822908 - 4313820 \beta ) q^{68} + ( -97590576 + 5917488 \beta ) q^{69} + ( -52082492 - 3371004 \beta ) q^{70} + ( 103246908 + 15075900 \beta ) q^{71} + ( -139573860 + 3199908 \beta ) q^{72} + ( -249576594 - 2840484 \beta ) q^{73} + ( 1785762 - 14623386 \beta ) q^{74} + ( 406614007 + 2307241 \beta ) q^{75} + ( 141255884 - 2191308 \beta ) q^{76} + ( -42396858 + 7985726 \beta ) q^{77} + ( 47485480 - 21865256 \beta ) q^{78} + ( 234267548 + 16873716 \beta ) q^{79} + ( 124952064 + 7635968 \beta ) q^{80} + ( -292872817 + 8178170 \beta ) q^{81} + ( 194793046 + 12134178 \beta ) q^{82} + ( 222011979 - 21562275 \beta ) q^{83} + ( -62589268 + 8806868 \beta ) q^{84} + ( 86737530 + 11488506 \beta ) q^{85} + ( -208415304 - 5177384 \beta ) q^{86} + ( 14067170 - 50948386 \beta ) q^{87} + ( -493005408 + 9846624 \beta ) q^{88} + ( 318133698 + 1406968 \beta ) q^{89} + ( -136621364 - 12771956 \beta ) q^{90} + ( 31849265 - 26168499 \beta ) q^{91} + ( -164715744 + 14141664 \beta ) q^{92} + ( -395761980 + 20511420 \beta ) q^{93} + ( -76111596 + 37997964 \beta ) q^{94} + ( 552373992 + 46069288 \beta ) q^{95} + ( -857490592 + 23873696 \beta ) q^{96} + ( -816358032 + 5731530 \beta ) q^{97} + ( -17294403 - 5764801 \beta ) q^{98} + ( 704708930 - 35060662 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{2} - 86q^{3} - 620q^{4} - 2238q^{5} - 3988q^{6} - 4802q^{7} + 2616q^{8} + 11038q^{9} + O(q^{10}) \) \( 2q - 6q^{2} - 86q^{3} - 620q^{4} - 2238q^{5} - 3988q^{6} - 4802q^{7} + 2616q^{8} + 11038q^{9} + 43384q^{10} + 35316q^{11} + 52136q^{12} - 26530q^{13} + 14406q^{14} - 307136q^{15} - 752q^{16} - 463920q^{17} + 332042q^{18} - 925426q^{19} + 473760q^{20} + 206486q^{21} + 1177888q^{22} + 778128q^{23} + 3301296q^{24} + 2081722q^{25} - 4127424q^{26} - 2798612q^{27} + 1488620q^{28} - 10003584q^{29} + 4095872q^{30} + 2467260q^{31} + 1284576q^{32} - 15640784q^{33} - 2246676q^{34} + 5373438q^{35} - 5612716q^{36} + 30735552q^{37} + 3504660q^{38} + 47417944q^{39} - 32409984q^{40} - 19103448q^{41} + 9575188q^{42} + 4065100q^{43} - 18650976q^{44} + 22338298q^{45} + 12367584q^{46} - 82195020q^{47} - 28806496q^{48} + 11529602q^{49} - 88312626q^{50} + 59971356q^{51} + 33466384q^{52} - 55189812q^{53} + 52834472q^{54} + 82445816q^{55} - 6281016q^{56} + 31781116q^{57} + 66558004q^{58} - 7069218q^{59} + 76165376q^{60} + 44316386q^{61} + 54910200q^{62} - 26502238q^{63} + 147417152q^{64} - 369979260q^{65} - 83258464q^{66} - 241921336q^{67} + 165645816q^{68} - 195181152q^{69} - 104164984q^{70} + 206493816q^{71} - 279147720q^{72} - 499153188q^{73} + 3571524q^{74} + 813228014q^{75} + 282511768q^{76} - 84793716q^{77} + 94970960q^{78} + 468535096q^{79} + 249904128q^{80} - 585745634q^{81} + 389586092q^{82} + 444023958q^{83} - 125178536q^{84} + 173475060q^{85} - 416830608q^{86} + 28134340q^{87} - 986010816q^{88} + 636267396q^{89} - 273242728q^{90} + 63698530q^{91} - 329431488q^{92} - 791523960q^{93} - 152223192q^{94} + 1104747984q^{95} - 1714981184q^{96} - 1632716064q^{97} - 34588806q^{98} + 1409417860q^{99} + O(q^{100}) \)

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
7.44622
−6.44622
−16.8924 109.817 −226.645 −2438.78 −1855.08 −2401.00 12477.5 −7623.25 41197.0
1.2 10.8924 −195.817 −393.355 200.782 −2132.92 −2401.00 −9861.52 18661.3 2187.01
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 7.10.a.a 2
3.b odd 2 1 63.10.a.d 2
4.b odd 2 1 112.10.a.e 2
5.b even 2 1 175.10.a.b 2
5.c odd 4 2 175.10.b.b 4
7.b odd 2 1 49.10.a.b 2
7.c even 3 2 49.10.c.c 4
7.d odd 6 2 49.10.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.a 2 1.a even 1 1 trivial
49.10.a.b 2 7.b odd 2 1
49.10.c.b 4 7.d odd 6 2
49.10.c.c 4 7.c even 3 2
63.10.a.d 2 3.b odd 2 1
112.10.a.e 2 4.b odd 2 1
175.10.a.b 2 5.b even 2 1
175.10.b.b 4 5.c odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 6 T_{2} - 184 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 840 T^{2} + 3072 T^{3} + 262144 T^{4} \)
$3$ \( 1 + 86 T + 17862 T^{2} + 1692738 T^{3} + 387420489 T^{4} \)
$5$ \( 1 + 2238 T + 3416586 T^{2} + 4371093750 T^{3} + 3814697265625 T^{4} \)
$7$ \( ( 1 + 2401 T )^{2} \)
$11$ \( 1 - 35316 T + 2892681078 T^{2} - 83273280655356 T^{3} + 5559917313492231481 T^{4} \)
$13$ \( 1 + 26530 T - 1541163822 T^{2} + 281337368365690 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 + 463920 T + 273833245726 T^{2} + 55015287664488240 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \)
$19$ \( 1 + 925426 T + 858791487510 T^{2} + 298623585404828854 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \)
$23$ \( 1 - 778128 T + 3473691840430 T^{2} - 1401527318158881264 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \)
$29$ \( 1 + 10003584 T + 52302031706070 T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \)
$31$ \( 1 - 2467260 T + 49371575832542 T^{2} - 65233422172137131460 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \)
$37$ \( 1 - 30735552 T + 484209298874630 T^{2} - \)\(39\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \)
$41$ \( 1 + 19103448 T + 602984827739166 T^{2} + \)\(62\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \)
$43$ \( 1 - 4065100 T + 797231337676374 T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + 82195020 T + 3721245520696702 T^{2} + \)\(91\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 + 55189812 T + 3123778356606670 T^{2} + \)\(18\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \)
$59$ \( 1 + 7069218 T + 16866494212382134 T^{2} + \)\(61\!\cdots\!02\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$61$ \( 1 - 44316386 T + 21654336818123658 T^{2} - \)\(51\!\cdots\!26\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \)
$67$ \( 1 + 241921336 T + 59516583718815510 T^{2} + \)\(65\!\cdots\!92\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \)
$71$ \( 1 - 206493816 T + 58491352612128526 T^{2} - \)\(94\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \)
$73$ \( 1 + 499153188 T + 178474458263805254 T^{2} + \)\(29\!\cdots\!44\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \)
$79$ \( 1 - 468535096 T + 239633073722978334 T^{2} - \)\(56\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \)
$83$ \( 1 - 444023958 T + 333438010641681622 T^{2} - \)\(83\!\cdots\!74\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \)
$89$ \( 1 - 636267396 T + 801539802340191990 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \)
$97$ \( 1 + 1632716064 T + 2180562419544849758 T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \)
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