Properties

Label 7.12.a.b
Level $7$
Weight $12$
Character orbit 7.a
Self dual yes
Analytic conductor $5.378$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.37840226392\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 818x - 4704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 26) q^{2} + (10 \beta_{2} - 11 \beta_1 - 47) q^{3} + (9 \beta_{2} + 21 \beta_1 + 1841) q^{4} + (2 \beta_{2} + 177 \beta_1 + 1735) q^{5} + ( - 118 \beta_{2} - 538 \beta_1 + 21206) q^{6} - 16807 q^{7} + ( - 549 \beta_{2} + 1617 \beta_1 + 42057) q^{8} + ( - 356 \beta_{2} - 5138 \beta_1 + 217675) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 26) q^{2} + (10 \beta_{2} - 11 \beta_1 - 47) q^{3} + (9 \beta_{2} + 21 \beta_1 + 1841) q^{4} + (2 \beta_{2} + 177 \beta_1 + 1735) q^{5} + ( - 118 \beta_{2} - 538 \beta_1 + 21206) q^{6} - 16807 q^{7} + ( - 549 \beta_{2} + 1617 \beta_1 + 42057) q^{8} + ( - 356 \beta_{2} - 5138 \beta_1 + 217675) q^{9} + (108 \beta_{2} + 12078 \beta_1 + 207650) q^{10} + (7436 \beta_{2} - 5082 \beta_1 - 345566) q^{11} + (7574 \beta_{2} - 16534 \beta_1 - 206038) q^{12} + ( - 9450 \beta_{2} + 2331 \beta_1 - 629447) q^{13} + ( - 16807 \beta_{2} - 436982) q^{14} + ( - 46064 \beta_{2} + 30856 \beta_1 - 2745320) q^{15} + (18405 \beta_{2} + 55419 \beta_1 - 3014629) q^{16} + ( - 61548 \beta_{2} + 41898 \beta_1 + 5259228) q^{17} + (269969 \beta_{2} - 356860 \beta_1 - 15994) q^{18} + (32958 \beta_{2} - 48081 \beta_1 + 4108411) q^{19} + (93016 \beta_{2} + 461076 \beta_1 + 12845420) q^{20} + ( - 168070 \beta_{2} + 184877 \beta_1 + 789929) q^{21} + ( - 426240 \beta_{2} - 189420 \beta_1 + 10424828) q^{22} + ( - 508416 \beta_{2} - 553728 \beta_1 + 2071536) q^{23} + (55674 \beta_{2} + 136566 \beta_1 - 39034602) q^{24} + (752292 \beta_{2} + 894642 \beta_1 + 23523905) q^{25} + ( - 489776 \beta_{2} - 39942 \beta_1 - 44672530) q^{26} + (2358700 \beta_{2} - 1820426 \beta_1 + 68737630) q^{27} + ( - 151263 \beta_{2} - 352947 \beta_1 - 30941687) q^{28} + ( - 487060 \beta_{2} + 1978326 \beta_1 - 41787380) q^{29} + ( - 2239936 \beta_{2} + 1130864 \beta_1 - 192166960) q^{30} + (2285244 \beta_{2} - 1586946 \beta_1 + 54294270) q^{31} + ( - 2701933 \beta_{2} + 843381 \beta_1 - 56498267) q^{32} + ( - 4467440 \beta_{2} + 478984 \beta_1 + 260786056) q^{33} + (5928462 \beta_{2} + 1556556 \beta_1 - 24059760) q^{34} + ( - 33614 \beta_{2} - 2974839 \beta_1 - 29160145) q^{35} + ( - 664639 \beta_{2} - 8074507 \beta_1 + 106445633) q^{36} + (290844 \beta_{2} + 4496814 \beta_1 + 190636332) q^{37} + (3980854 \beta_{2} - 2577390 \beta_1 + 170305298) q^{38} + ( - 3547544 \beta_{2} + 9969652 \beta_1 - 218031884) q^{39} + (6893280 \beta_{2} + 8570760 \beta_1 + 614243160) q^{40} + ( - 8852564 \beta_{2} - 10082058 \beta_1 + 291582704) q^{41} + (1983226 \beta_{2} + 9042166 \beta_1 - 356409242) q^{42} + (6422724 \beta_{2} - 5533374 \beta_1 + 153695726) q^{43} + (4146760 \beta_{2} - 11423664 \beta_1 - 557812864) q^{44} + ( - 21318326 \beta_{2} + 19245589 \beta_1 - 1683632765) q^{45} + (15698160 \beta_{2} - 48330240 \beta_1 - 2068068768) q^{46} + (10946196 \beta_{2} - 16537302 \beta_1 + 906078186) q^{47} + ( - 56721706 \beta_{2} + 44317274 \beta_1 - 293602054) q^{48} + 282475249 q^{49} + (2683163 \beta_{2} + 76633788 \beta_1 + 3817809970) q^{50} + (61025532 \beta_{2} - 30400482 \beta_1 - 2268747354) q^{51} + ( - 16633260 \beta_{2} - 17775240 \beta_1 - 1481257456) q^{52} + (5811360 \beta_{2} + 35890512 \beta_1 + 45042798) q^{53} + (45023564 \beta_{2} - 74256268 \beta_1 + 7760065748) q^{54} + ( - 21771576 \beta_{2} - 22013916 \beta_1 - 1372508420) q^{55} + (9227043 \beta_{2} - 27176919 \beta_1 - 706851999) q^{56} + (45646660 \beta_{2} - 67201022 \beta_1 + 1281870250) q^{57} + ( - 51312294 \beta_{2} + 124297908 \beta_1 - 906512128) q^{58} + ( - 38340978 \beta_{2} + 18634647 \beta_1 + 1959822699) q^{59} + ( - 69926752 \beta_{2} - 33332992 \beta_1 - 5573417920) q^{60} + ( - 86546430 \beta_{2} + 161957553 \beta_1 + 727406287) q^{61} + (29727636 \beta_{2} - 59922204 \beta_1 + 7354453620) q^{62} + (5983292 \beta_{2} + 86354366 \beta_1 - 3658463725) q^{63} + ( - 55849275 \beta_{2} - 112888797 \beta_1 - 3232443437) q^{64} + (8072036 \beta_{2} - 174519534 \beta_1 - 1719178370) q^{65} + (332421680 \beta_{2} - 61245328 \beta_1 - 7150983376) q^{66} + ( - 93370464 \beta_{2} + 167269200 \beta_1 - 8984462204) q^{67} + ( - 12802314 \beta_{2} + 144536406 \beta_1 + 9024578094) q^{68} + (408916320 \beta_{2} - 52478544 \beta_1 - 3026416656) q^{69} + ( - 1815156 \beta_{2} - 202994946 \beta_1 - 3489973550) q^{70} + ( - 42991032 \beta_{2} - 137467932 \beta_1 + 5461196988) q^{71} + ( - 362481453 \beta_{2} + 167825385 \beta_1 - 6456858111) q^{72} + ( - 270287784 \beta_{2} - 21412500 \beta_1 + 1915034550) q^{73} + (145220658 \beta_{2} + 311891076 \beta_1 + 9857216352) q^{74} + ( - 365830594 \beta_{2} - 193367569 \beta_1 + 2076074915) q^{75} + (58329306 \beta_{2} + 6805302 \beta_1 + 6531137942) q^{76} + ( - 124976852 \beta_{2} + 85413174 \beta_1 + 5807927762) q^{77} + ( - 247450504 \beta_{2} + 603437912 \beta_1 - 8273854792) q^{78} + (157585464 \beta_{2} + 83144796 \beta_1 - 18218095492) q^{79} + (229423792 \beta_{2} - 216713088 \beta_1 + 19370420960) q^{80} + (503282836 \beta_{2} - 959146454 \beta_1 + 38967760843) q^{81} + (532814814 \beta_{2} - 871483788 \beta_1 - 29754512984) q^{82} + (172922022 \beta_{2} + 178290147 \beta_1 + 41304138687) q^{83} + ( - 127296218 \beta_{2} + 277886938 \beta_1 + 3462880666) q^{84} + (184300308 \beta_{2} + 606290538 \beta_1 + 15457826310) q^{85} + (94309784 \beta_{2} - 241392228 \beta_1 + 19751865220) q^{86} + ( - 934092524 \beta_{2} + 1146223210 \beta_1 - 39234097598) q^{87} + (347444712 \beta_{2} - 301795032 \beta_1 - 32605313976) q^{88} + (167810960 \beta_{2} + 773412936 \beta_1 - 30235729070) q^{89} + ( - 1494431524 \beta_{2} + 861015206 \beta_1 - 95295623830) q^{90} + (158826150 \beta_{2} - 39177117 \beta_1 + 10579115729) q^{91} + ( - 858729360 \beta_{2} - 1822760016 \beta_1 - 50201377296) q^{92} + (240899352 \beta_{2} - 1625398068 \beta_1 + 72986656380) q^{93} + (868828572 \beta_{2} - 894666420 \beta_1 + 44142260220) q^{94} + ( - 193320160 \beta_{2} + 818040720 \beta_1 - 6262897520) q^{95} + (157791130 \beta_{2} + 1542731638 \beta_1 - 70849794218) q^{96} + ( - 599814684 \beta_{2} - 661659054 \beta_1 - 24166196124) q^{97} + (282475249 \beta_{2} + 7344356474) q^{98} + (2809727452 \beta_{2} - 706080242 \beta_1 - 58566600902) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 77 q^{2} - 140 q^{3} + 5493 q^{4} + 5026 q^{5} + 64274 q^{6} - 50421 q^{7} + 125103 q^{8} + 658519 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 77 q^{2} - 140 q^{3} + 5493 q^{4} + 5026 q^{5} + 64274 q^{6} - 50421 q^{7} + 125103 q^{8} + 658519 q^{9} + 610764 q^{10} - 1039052 q^{11} - 609154 q^{12} - 1881222 q^{13} - 1294139 q^{14} - 8220752 q^{15} - 9117711 q^{16} + 15797334 q^{17} + 38909 q^{18} + 12340356 q^{19} + 37982168 q^{20} + 2352980 q^{21} + 31890144 q^{22} + 7276752 q^{23} - 117296046 q^{24} + 68924781 q^{25} - 133487872 q^{26} + 205674616 q^{27} - 92320851 q^{28} - 126853406 q^{29} - 575391808 q^{30} + 162184512 q^{31} - 167636249 q^{32} + 786346624 q^{33} - 79664298 q^{34} - 84471982 q^{35} + 328076045 q^{36} + 567121338 q^{37} + 509512430 q^{38} - 660517760 q^{39} + 1827265440 q^{40} + 893682734 q^{41} - 1080253118 q^{42} + 460197828 q^{43} - 1666161688 q^{44} - 5048825558 q^{45} - 6171574224 q^{46} + 2723825664 q^{47} - 868401730 q^{48} + 847425747 q^{49} + 11374112959 q^{50} - 6836867112 q^{51} - 4409363868 q^{52} + 93426522 q^{53} + 23309429948 q^{54} - 4073739768 q^{55} - 2102606121 q^{56} + 3867165112 q^{57} - 2792521998 q^{58} + 5899174428 q^{59} - 16616994016 q^{60} + 2106807738 q^{61} + 22093555428 q^{62} - 11067728833 q^{63} - 9528592239 q^{64} - 4991087612 q^{65} - 21724126480 q^{66} - 27027285348 q^{67} + 26942000190 q^{68} - 9435687744 q^{69} - 10265110548 q^{70} + 16564049928 q^{71} - 19175918265 q^{72} + 6036803934 q^{73} + 29114537322 q^{74} + 6787422908 q^{75} + 19528279218 q^{76} + 17463346964 q^{77} - 25177551784 q^{78} - 54895016736 q^{79} + 58098552176 q^{80} + 117359146147 q^{81} - 88924869978 q^{82} + 123561203892 q^{83} + 10238051278 q^{84} + 45582888084 q^{85} + 59402678104 q^{86} - 117914423480 q^{87} - 97861591608 q^{88} - 91648411106 q^{89} - 285253455172 q^{90} + 31617698154 q^{91} - 147922642512 q^{92} + 220344467856 q^{93} + 132452618508 q^{94} - 19413413120 q^{95} - 214249905422 q^{96} - 71237114634 q^{97} + 21750594173 q^{98} - 177803449916 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 818x - 4704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 5\nu - 546 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12\beta_{2} + 5\beta _1 + 1097 ) / 2 \) 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
−6.06890
−24.5296
31.5985
−53.8040 −700.524 846.870 −749.998 37691.0 −16807.0 64625.6 313587. 40352.9
1.2 55.7251 800.902 1057.28 −7066.04 44630.3 −16807.0 −55207.8 464297. −393755.
1.3 75.0789 −240.378 3588.85 12842.0 −18047.3 −16807.0 115685. −119365. 964166.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 7.12.a.b 3
3.b odd 2 1 63.12.a.d 3
4.b odd 2 1 112.12.a.h 3
5.b even 2 1 175.12.a.b 3
5.c odd 4 2 175.12.b.b 6
7.b odd 2 1 49.12.a.d 3
7.c even 3 2 49.12.c.g 6
7.d odd 6 2 49.12.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.12.a.b 3 1.a even 1 1 trivial
49.12.a.d 3 7.b odd 2 1
49.12.c.f 6 7.d odd 6 2
49.12.c.g 6 7.c even 3 2
63.12.a.d 3 3.b odd 2 1
112.12.a.h 3 4.b odd 2 1
175.12.a.b 3 5.b even 2 1
175.12.b.b 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 77T_{2}^{2} - 2854T_{2} + 225104 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(7))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 77 T^{2} - 2854 T + 225104 \) Copy content Toggle raw display
$3$ \( T^{3} + 140 T^{2} + \cdots - 134864352 \) Copy content Toggle raw display
$5$ \( T^{3} - 5026 T^{2} + \cdots - 68056486400 \) Copy content Toggle raw display
$7$ \( (T + 16807)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 1039052 T^{2} + \cdots - 33\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{3} + 1881222 T^{2} + \cdots - 91\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{3} - 15797334 T^{2} + \cdots - 62\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{3} - 12340356 T^{2} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} - 7276752 T^{2} + \cdots + 42\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{3} + 126853406 T^{2} + \cdots - 25\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{3} - 162184512 T^{2} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} - 567121338 T^{2} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{3} - 893682734 T^{2} + \cdots + 46\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{3} - 460197828 T^{2} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{3} - 2723825664 T^{2} + \cdots - 21\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{3} - 93426522 T^{2} + \cdots - 36\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{3} - 5899174428 T^{2} + \cdots + 66\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{3} - 2106807738 T^{2} + \cdots + 35\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{3} + 27027285348 T^{2} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{3} - 16564049928 T^{2} + \cdots + 61\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{3} - 6036803934 T^{2} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{3} + 54895016736 T^{2} + \cdots + 29\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{3} - 123561203892 T^{2} + \cdots - 57\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{3} + 91648411106 T^{2} + \cdots - 89\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + 71237114634 T^{2} + \cdots - 27\!\cdots\!84 \) Copy content Toggle raw display
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