# 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 $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7,12,Mod(1,7)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$5.37840226392$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3369})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 842$$ x^2 - x - 842 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3369}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{12}^{\mathrm{new}}(\Gamma_0(7))$$.

## Hecke characteristic polynomials

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