Properties

Label 7.12.a.a
Level $7$
Weight $12$
Character orbit 7.a
Self dual yes
Analytic conductor $5.378$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3369}) \)
Defining polynomial: \( x^{2} - x - 842 \) Copy content Toggle raw display
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 \(\beta = \sqrt{3369}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 27) q^{2} + (6 \beta + 60) q^{3} + (54 \beta + 2050) q^{4} + ( - 10 \beta - 6750) q^{5} + ( - 222 \beta - 21834) q^{6} + 16807 q^{7} + ( - 1460 \beta - 181980) q^{8} + (720 \beta - 52263) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 27) q^{2} + (6 \beta + 60) q^{3} + (54 \beta + 2050) q^{4} + ( - 10 \beta - 6750) q^{5} + ( - 222 \beta - 21834) q^{6} + 16807 q^{7} + ( - 1460 \beta - 181980) q^{8} + (720 \beta - 52263) q^{9} + (7020 \beta + 215940) q^{10} + ( - 4160 \beta - 375408) q^{11} + (15540 \beta + 1214556) q^{12} + ( - 29862 \beta - 4774) q^{13} + ( - 16807 \beta - 453789) q^{14} + ( - 41100 \beta - 607140) q^{15} + (110808 \beta + 5633800) q^{16} + (3100 \beta + 2080026) q^{17} + (32823 \beta - 1014579) q^{18} + (78138 \beta - 8999356) q^{19} + ( - 385000 \beta - 15656760) q^{20} + (100842 \beta + 1008420) q^{21} + (487728 \beta + 24151056) q^{22} + (39500 \beta - 33080508) q^{23} + ( - 1179480 \beta - 40431240) q^{24} + (135000 \beta - 2928725) q^{25} + (811048 \beta + 100733976) q^{26} + ( - 1333260 \beta + 789480) q^{27} + (907578 \beta + 34454350) q^{28} + (1928052 \beta + 30757806) q^{29} + (1716840 \beta + 154858680) q^{30} + (1370844 \beta - 7640776) q^{31} + ( - 5635536 \beta - 152729712) q^{32} + ( - 2502048 \beta - 106614720) q^{33} + ( - 2163726 \beta - 66604602) q^{34} + ( - 168070 \beta - 113447250) q^{35} + ( - 1346202 \beta + 23847570) q^{36} + (5698188 \beta - 263609170) q^{37} + (6889630 \beta - 20264310) q^{38} + ( - 1820364 \beta - 603916908) q^{39} + (11674800 \beta + 1277552400) q^{40} + (1231356 \beta - 89138070) q^{41} + ( - 3731154 \beta - 366964038) q^{42} + ( - 9186912 \beta + 913372616) q^{43} + ( - 28800032 \beta - 1526398560) q^{44} + ( - 4337370 \beta + 328518450) q^{45} + (32014008 \beta + 760098216) q^{46} + ( - 38136388 \beta + 284120352) q^{47} + (40451280 \beta + 2577900912) q^{48} + 282475249 q^{49} + ( - 716275 \beta - 375739425) q^{50} + (12666156 \beta + 187464960) q^{51} + ( - 61474896 \beta - 5442460912) q^{52} + (64945144 \beta - 2092908186) q^{53} + (35208540 \beta + 4470436980) q^{54} + (31834080 \beta + 2674154400) q^{55} + ( - 24538220 \beta - 3058537860) q^{56} + ( - 49307856 \beta + 1039520172) q^{57} + ( - 82815210 \beta - 7326067950) q^{58} + (104471170 \beta + 1555672500) q^{59} + ( - 117040560 \beta - 8721795600) q^{60} + ( - 56906874 \beta + 7521297530) q^{61} + ( - 29372012 \beta - 4412072484) q^{62} + (12101040 \beta - 878384241) q^{63} + (77954400 \beta + 11571800608) q^{64} + (201616240 \beta + 1038275280) q^{65} + (174170016 \beta + 11307997152) q^{66} + ( - 203009004 \beta + 4928261984) q^{67} + (118676404 \beta + 4828023900) q^{68} + ( - 196113048 \beta - 1186377480) q^{69} + (117985140 \beta + 3629303580) q^{70} + ( - 60930912 \beta - 12156005664) q^{71} + ( - 54721620 \beta + 5969327940) q^{72} + ( - 199184616 \beta - 15445000966) q^{73} + (109758094 \beta - 12079747782) q^{74} + ( - 9472350 \beta + 2553166500) q^{75} + ( - 325782324 \beta - 4233346012) q^{76} + ( - 69917120 \beta - 6309482256) q^{77} + (653066736 \beta + 22438562832) q^{78} + (434987496 \beta + 996402128) q^{79} + ( - 804292000 \beta - 41761271520) q^{80} + ( - 202804560 \beta - 17644915179) q^{81} + (55891458 \beta - 1741710474) q^{82} + (334983474 \beta + 2638507284) q^{83} + (261180780 \beta + 20413042692) q^{84} + ( - 41725260 \beta - 14144614500) q^{85} + ( - 665325992 \beta + 6289645896) q^{86} + (300229956 \beta + 40819111488) q^{87} + (1305132480 \beta + 88778706240) q^{88} + (390416072 \beta - 50770656414) q^{89} + ( - 211409460 \beta + 5742601380) q^{90} + ( - 501890634 \beta - 80236618) q^{91} + ( - 1705372432 \beta - 60628964400) q^{92} + (36405984 \beta + 27251794056) q^{93} + (745562124 \beta + 120810241668) q^{94} + ( - 437437940 \beta + 58113183780) q^{95} + ( - 1254510432 \beta - 123080507424) q^{96} + ( - 203366268 \beta - 96114310558) q^{97} + ( - 282475249 \beta - 7626831723) q^{98} + ( - 52879680 \beta + 9529119504) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{2} + 120 q^{3} + 4100 q^{4} - 13500 q^{5} - 43668 q^{6} + 33614 q^{7} - 363960 q^{8} - 104526 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{2} + 120 q^{3} + 4100 q^{4} - 13500 q^{5} - 43668 q^{6} + 33614 q^{7} - 363960 q^{8} - 104526 q^{9} + 431880 q^{10} - 750816 q^{11} + 2429112 q^{12} - 9548 q^{13} - 907578 q^{14} - 1214280 q^{15} + 11267600 q^{16} + 4160052 q^{17} - 2029158 q^{18} - 17998712 q^{19} - 31313520 q^{20} + 2016840 q^{21} + 48302112 q^{22} - 66161016 q^{23} - 80862480 q^{24} - 5857450 q^{25} + 201467952 q^{26} + 1578960 q^{27} + 68908700 q^{28} + 61515612 q^{29} + 309717360 q^{30} - 15281552 q^{31} - 305459424 q^{32} - 213229440 q^{33} - 133209204 q^{34} - 226894500 q^{35} + 47695140 q^{36} - 527218340 q^{37} - 40528620 q^{38} - 1207833816 q^{39} + 2555104800 q^{40} - 178276140 q^{41} - 733928076 q^{42} + 1826745232 q^{43} - 3052797120 q^{44} + 657036900 q^{45} + 1520196432 q^{46} + 568240704 q^{47} + 5155801824 q^{48} + 564950498 q^{49} - 751478850 q^{50} + 374929920 q^{51} - 10884921824 q^{52} - 4185816372 q^{53} + 8940873960 q^{54} + 5348308800 q^{55} - 6117075720 q^{56} + 2079040344 q^{57} - 14652135900 q^{58} + 3111345000 q^{59} - 17443591200 q^{60} + 15042595060 q^{61} - 8824144968 q^{62} - 1756768482 q^{63} + 23143601216 q^{64} + 2076550560 q^{65} + 22615994304 q^{66} + 9856523968 q^{67} + 9656047800 q^{68} - 2372754960 q^{69} + 7258607160 q^{70} - 24312011328 q^{71} + 11938655880 q^{72} - 30890001932 q^{73} - 24159495564 q^{74} + 5106333000 q^{75} - 8466692024 q^{76} - 12618964512 q^{77} + 44877125664 q^{78} + 1992804256 q^{79} - 83522543040 q^{80} - 35289830358 q^{81} - 3483420948 q^{82} + 5277014568 q^{83} + 40826085384 q^{84} - 28289229000 q^{85} + 12579291792 q^{86} + 81638222976 q^{87} + 177557412480 q^{88} - 101541312828 q^{89} + 11485202760 q^{90} - 160473236 q^{91} - 121257928800 q^{92} + 54503588112 q^{93} + 241620483336 q^{94} + 116226367560 q^{95} - 246161014848 q^{96} - 192228621116 q^{97} - 15253663446 q^{98} + 19058239008 q^{99}+O(q^{100}) \) 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
29.5215
−28.5215
−85.0431 408.259 5184.33 −7330.43 −34719.6 16807.0 −266723. −10472.0 623402.
1.2 31.0431 −288.259 −1084.33 −6169.57 −8948.43 16807.0 −97237.1 −94054.0 −191522.
\(n\): 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.a 2
3.b odd 2 1 63.12.a.c 2
4.b odd 2 1 112.12.a.d 2
5.b even 2 1 175.12.a.a 2
5.c odd 4 2 175.12.b.a 4
7.b odd 2 1 49.12.a.c 2
7.c even 3 2 49.12.c.d 4
7.d odd 6 2 49.12.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.12.a.a 2 1.a even 1 1 trivial
49.12.a.c 2 7.b odd 2 1
49.12.c.d 4 7.c even 3 2
49.12.c.e 4 7.d odd 6 2
63.12.a.c 2 3.b odd 2 1
112.12.a.d 2 4.b odd 2 1
175.12.a.a 2 5.b even 2 1
175.12.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 54T_{2} - 2640 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(7))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 54T - 2640 \) Copy content Toggle raw display
$3$ \( T^{2} - 120T - 117684 \) Copy content Toggle raw display
$5$ \( T^{2} + 13500 T + 45225600 \) Copy content Toggle raw display
$7$ \( (T - 16807)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 750816 T + 82628600064 \) Copy content Toggle raw display
$13$ \( T^{2} + 9548 T - 3004246048160 \) Copy content Toggle raw display
$17$ \( T^{2} - 4160052 T + 4294132070676 \) Copy content Toggle raw display
$19$ \( T^{2} + 17998712 T + 60418820423500 \) Copy content Toggle raw display
$23$ \( T^{2} + 66161016 T + 10\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{2} - 61515612 T - 11\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{2} + 15281552 T - 62\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{2} + 527218340 T - 39\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{2} + 178276140 T + 28\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} - 1826745232 T + 54\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{2} - 568240704 T - 48\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{2} + 4185816372 T - 98\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{2} - 3111345000 T - 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} - 15042595060 T + 45\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{2} - 9856523968 T - 11\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{2} + 24312011328 T + 13\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{2} + 30890001932 T + 10\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{2} - 1992804256 T - 63\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{2} - 5277014568 T - 37\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{2} + 101541312828 T + 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + 192228621116 T + 90\!\cdots\!08 \) Copy content Toggle raw display
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