LMFDB - Newform orbit 7400.2.a.be

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7400,2,Mod(1,7400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7400 = 2^{3} \cdot 5^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7400.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:	$59.0892974957$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 2 x^{15} - 33 x^{14} + 66 x^{13} + 404 x^{12} - 796 x^{11} - 2273 x^{10} + 4284 x^{9} + \cdots + 16$ x^16 - 2x^15 - 33x^14 + 66x^13 + 404x^12 - 796x^11 - 2273x^10 + 4284x^9 + 6039x^8 - 10158x^7 - 7660x^6 + 10040x^5 + 3914x^4 - 3314x^3 - 232x^2 + 172*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 1480)
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 a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{3} + (\beta_{8} + 1) q^{7} + (\beta_{2} + 1) q^{9} + \beta_{7} q^{11} - \beta_{14} q^{13} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{17} + ( - \beta_{11} + 1) q^{19} + ( - \beta_{15} - \beta_{13} + \cdots + \beta_1) q^{21}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots - 2 \beta_1) q^{99}+O(q^{100})$ q + b1 * q^3 + (b8 + 1) * q^7 + (b2 + 1) * q^9 + b7 * q^11 - b14 * q^13 + (b6 + b5 - b4) * q^17 + (-b11 + 1) * q^19 + (-b15 - b13 + b10 + b7 + b6 + b1) * q^21 + (b14 + b10 + b5 + 1) * q^23 + (-b15 - b14 - b8 + 2b6 + b5 + b3 + b2 + 2b1 - 1) * q^27 + (b13 - b12 - b11 - b10 + b6 + b5 + b3 + b2 + b1) * q^29 + (-b13 - b4 - b1 + 2) * q^31 + (b12 + b11 + b8 - 2b6 - b5 - b3 - b2) q^33 + q^37 + (-b14 + b13 + b11 - b5 - b3 - b1) * q^39 + (b15 + b14 - b11 + b9 + b8 + b7 + b4 + b3 + b1 + 1) * q^41 + (-b12 - b11 + b9 + b8) * q^43 + (b15 - b13 + b8 - b5 + b1 + 2) * q^47 + (b14 + b12 + b11 + b10 - b9 + b8 - 2b6 - b5 - b3 - b2 + 3) q^49 + (-b13 + b12 + b11 - 2b8 - b7 - b6 + b5 - b3 + b2 - b1 + 1) q^51 + (b14 - b13 - b11 - b9 + b5 + b1) * q^53 + (b15 - b14 + 2b13 - b12 - b10 + b8 - 2b5 + b4 - b2 + 2b1 - 2) q^57 + (-2b14 + b13 + b11 - b10 - b9 - b8 + b6 + b3 + 2b2 - b1 + 1) * q^59 + (-b12 - b7 - b6 - b5 + b4 - b3 - b2 + 2) * q^61 + (-2b14 + 2b13 + b12 + 2b11 - 2b10 - b9 + b8 - 2b6 - 2b5 + 2b2 + 3) q^63 + (-b10 - b9 - b7 - b3 + b1 + 1) * q^67 + (b14 + 2b12 + b11 + b10 + b9 - b8 - b6 + b5 + 2b1) * q^69 + (b13 - b12 - b10 + b8 - b6 + b3 - b1 + 2) * q^71 + (b14 - b13 - 2b12 - 2b11 + 2b8 + b3 - b2 + 1) q^73 + (b15 + 2b14 + b12 - 2b11 + b10 + 2b9 + b8 + 3b7 + b4 + b3 - b2 + 4b1 - 3) q^77 + (b13 + b12 + b11 - b8 - b3 - b1) * q^79 + (-b14 - b13 + b12 + 2b11 - b10 - 2b9 - 2b7 - b4 - b3 + 2b2 - 2b1 + 6) q^81 + (-b14 + b13 - b12 + b11 + b8 - b5 - b1 + 1) * q^83 + (-2b13 + b12 + b11 - b9 - 2b8 - 2b7 - b6 - b5 - 2b3 + b2 + b1 + 3) * q^87 + (b14 + b13 - b12 + 2b8 + b7 - b6 - b2 + 1) q^89 + (2b15 - b13 - 2b11 + b9 - b5 + b4 + b3 - b2 + 2b1 + 2) q^91 + (-b10 + b5 + b3 + 2b1 - 2) q^93 + (-b15 + b12 + 2b11 + b10 - b6 - b5 + b4 - b3 + b1 - 2) q^97 + (-b15 + 2b14 - b13 - 2b12 - 2b11 + 3b10 + 2b9 + b8 + b7 + 2b6 + b5 + b3 - 2b2 - 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 2 q^{3} + 9 q^{7} + 22 q^{9} + 3 q^{11} - 4 q^{13} - 9 q^{17} + 14 q^{19} + 8 q^{21} + 18 q^{23} - 4 q^{27} + q^{29} + 23 q^{31} - 10 q^{33} + 16 q^{37} + 23 q^{41} - 11 q^{43} + 32 q^{47} + 45 q^{49}+ \cdots - 15 q^{99}+O(q^{100})$ 16 * q + 2 * q^3 + 9 * q^7 + 22 * q^9 + 3 * q^11 - 4 * q^13 - 9 * q^17 + 14 * q^19 + 8 * q^21 + 18 * q^23 - 4 * q^27 + q^29 + 23 * q^31 - 10 * q^33 + 16 * q^37 + 23 * q^41 - 11 * q^43 + 32 * q^47 + 45 * q^49 + 28 * q^51 + q^53 - 32 * q^57 + 28 * q^59 + 31 * q^61 + 51 * q^63 + 14 * q^67 + 14 * q^69 + 18 * q^71 - 6 * q^73 - 31 * q^77 + 8 * q^79 + 92 * q^81 + 8 * q^83 + 70 * q^87 + 38 * q^91 - 34 * q^93 - 15 * q^97 - 15 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 2 x^{15} - 33 x^{14} + 66 x^{13} + 404 x^{12} - 796 x^{11} - 2273 x^{10} + 4284 x^{9} + \cdots + 16$ x^16 - 2*x^15 - 33*x^14 + 66*x^13 + 404*x^12 - 796*x^11 - 2273*x^10 + 4284*x^9 + 6039*x^8 - 10158*x^7 - 7660*x^6 + 10040*x^5 + 3914*x^4 - 3314*x^3 - 232*x^2 + 172*x + 16 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 4$ v^2 - 4
$\beta_{3}$	$=$	$( 734038251140 \nu^{15} + 22507424628655 \nu^{14} - 66747757257083 \nu^{13} + \cdots - 71\!\cdots\!90 ) / 548023221091718$ (734038251140v^15 + 22507424628655v^14 - 66747757257083v^13 - 753747453690880v^12 + 1690834060452286v^11 + 9453456973954620v^10 - 18289958757068962v^9 - 55400937449692879v^8 + 91900166180257295v^7 + 157496395253269414v^6 - 202951789001367580v^5 - 217200581085329548v^4 + 176593292718165212v^3 + 119825392678794366v^2 - 41358540892964732*v - 7109522945036890) / 548023221091718
$\beta_{4}$	$=$	$( 1978876789657 \nu^{15} - 50267633335272 \nu^{14} + 27826535434397 \nu^{13} + \cdots + 12\!\cdots\!12 ) / 10\!\cdots\!36$ (1978876789657v^15 - 50267633335272v^14 + 27826535434397v^13 + 1638166179029658v^12 - 2226896113595484v^11 - 19646043085266956v^10 + 31073142518088775v^9 + 106755490847094350v^8 - 170489615628055099v^7 - 268048646316985370v^6 + 374874906177819248v^5 + 322328935890044564v^4 - 299825290158092434v^3 - 164520631257408854v^2 + 47185364979144400*v + 12911839673462412) / 1096046442183436
$\beta_{5}$	$=$	$( - 1615550965631 \nu^{15} - 10335920720187 \nu^{14} + 81254903921620 \nu^{13} + \cdots + 48\!\cdots\!56 ) / 548023221091718$ (-1615550965631v^15 - 10335920720187v^14 + 81254903921620v^13 + 334883720206842v^12 - 1556659577336970v^11 - 4000797532422490v^10 + 14239872088854407v^9 + 21784446731766989v^8 - 63585572096982224v^7 - 56066003078678790v^6 + 126707712856910004v^5 + 73971032010600298v^4 - 98513982425359392v^3 - 45091844459243162v^2 + 18848182779095604*v + 4875080359311756) / 548023221091718
$\beta_{6}$	$=$	$( - 7125784976077 \nu^{15} + 1317200650744 \nu^{14} + 256975318051897 \nu^{13} + \cdots + 43\!\cdots\!24 ) / 548023221091718$ (-7125784976077v^15 + 1317200650744v^14 + 256975318051897v^13 - 34729443845434v^12 - 3600259925642186v^11 + 166254895032076v^10 + 24890470692837499v^9 + 2104311197067004v^8 - 89473651809240181v^7 - 21743648845112674v^6 + 161442394152057396v^5 + 59580354829303668v^4 - 121842107998471010v^3 - 49711571397604722v^2 + 20388993851931144*v + 4355863933526824) / 548023221091718
$\beta_{7}$	$=$	$( - 14399954587895 \nu^{15} + 14637956723222 \nu^{14} + 503531838899579 \nu^{13} + \cdots - 22\!\cdots\!72 ) / 10\!\cdots\!36$ (-14399954587895v^15 + 14637956723222v^14 + 503531838899579v^13 - 495127931883014v^12 - 6741618538492636v^11 + 6124401915494172v^10 + 43652654273654399v^9 - 33917732298808208v^8 - 143611510697207149v^7 + 82985220785793298v^6 + 235306793023098540v^5 - 82492530282660552v^4 - 167117303829770526v^3 + 26514942085447614v^2 + 34321541248968664*v - 2233392671333572) / 1096046442183436
$\beta_{8}$	$=$	$( - 30079433014147 \nu^{15} + 40227434856318 \nu^{14} + \cdots + 52\!\cdots\!88 ) / 10\!\cdots\!36$ (-30079433014147v^15 + 40227434856318v^14 + 1021436210242899v^13 - 1318737269896386v^12 - 13097233423011096v^11 + 15592576585648028v^10 + 79570183812617539v^9 - 79893184107833380v^8 - 239489527897224301v^7 + 165471753759831846v^6 + 354451130795592740v^5 - 104402387681372704v^4 - 214103707737635970v^3 - 17453024067854586v^2 + 20074896583106956*v + 5264496219957688) / 1096046442183436
$\beta_{9}$	$=$	$( - 44720099371065 \nu^{15} + 92885284952754 \nu^{14} + \cdots - 54\!\cdots\!52 ) / 10\!\cdots\!36$ (-44720099371065v^15 + 92885284952754v^14 + 1461026975657341v^13 - 3045378917494530v^12 - 17585304196197344v^11 + 36370689884724360v^10 + 95906363110136325v^9 - 192540969214205616v^8 - 239603593799314795v^7 + 442836326917816950v^6 + 271242221387229420v^5 - 417530659704244008v^4 - 101756501875049878v^3 + 129482385525343250v^2 - 17926308641272472*v - 5490238796037352) / 1096046442183436
$\beta_{10}$	$=$	$( - 23630524916430 \nu^{15} + 60147992099071 \nu^{14} + 756186101280111 \nu^{13} + \cdots - 39\!\cdots\!26 ) / 548023221091718$ (-23630524916430v^15 + 60147992099071v^14 + 756186101280111v^13 - 1984725245536492v^12 - 8773341069167930v^11 + 24006620916026330v^10 + 44518394746514092v^9 - 130312665743323439v^8 - 94591125304175999v^7 + 315756586879235562v^6 + 73531683441078738v^5 - 326606362159625032v^4 + 2010916103416386v^3 + 113909836500092334v^2 - 19350467433596384*v - 3957963494328626) / 548023221091718
$\beta_{11}$	$=$	$( 30499877110179 \nu^{15} - 55597545410478 \nu^{14} + \cdots + 887908579904830 ) / 548023221091718$ (30499877110179v^15 - 55597545410478v^14 - 1011604526460943v^13 + 1818425474243972v^12 + 12497821028985104v^11 - 21566588511288424v^10 - 71559853400078137v^9 + 112171886671539842v^8 + 197088910003736697v^7 - 245434071430477934v^6 - 267334377531040690v^5 + 198049206370347286v^4 + 154758246481925750v^3 - 32501347082501478v^2 - 16468372857318534*v + 887908579904830) / 548023221091718
$\beta_{12}$	$=$	$( - 75348438992069 \nu^{15} + 128912803883594 \nu^{14} + \cdots + 17\!\cdots\!56 ) / 10\!\cdots\!36$ (-75348438992069v^15 + 128912803883594v^14 + 2513957479133705v^13 - 4218795805924426v^12 - 31369061307767472v^11 + 50032496395490448v^10 + 182782905578193593v^9 - 259916510823739312v^8 - 519243229098703787v^7 + 566285718949677414v^6 + 735582062365608320v^5 - 450589585464315192v^4 - 447829104053324346v^3 + 65153327308301522v^2 + 49640507829300800*v + 1760470687562256) / 1096046442183436
$\beta_{13}$	$=$	$( 38227715756425 \nu^{15} - 76049503836810 \nu^{14} + \cdots + 34\!\cdots\!40 ) / 548023221091718$ (38227715756425v^15 - 76049503836810v^14 - 1260102866517961v^13 + 2510928039083476v^12 + 15391316821851702v^11 - 30301348714047554v^10 - 86123201170349393v^9 + 163197024375151466v^8 + 225381026708049887v^7 - 387249248097391438v^6 - 273514659182866496v^5 + 382351519319099172v^4 + 120031773342156634v^3 - 123773727417939106v^2 + 5937202806786400*v + 3449321357557140) / 548023221091718
$\beta_{14}$	$=$	$( 66124921197320 \nu^{15} - 135734026778415 \nu^{14} + \cdots + 55\!\cdots\!84 ) / 548023221091718$ (66124921197320v^15 - 135734026778415v^14 - 2174037873134219v^13 + 4476421465532342v^12 + 26442672382438136v^11 - 53947728258350074v^10 - 146929077472990326v^9 + 289983052838541529v^8 + 380326050184539811v^7 - 685526241966348262v^6 - 457929426635359466v^5 + 670569742755182944v^4 + 205752409557317716v^3 - 210096288821877328v^2 + 5571352180336220*v + 5513673784029984) / 548023221091718
$\beta_{15}$	$=$	$( - 132436574713783 \nu^{15} + 260852429120424 \nu^{14} + \cdots - 88\!\cdots\!08 ) / 10\!\cdots\!36$ (-132436574713783v^15 + 260852429120424v^14 + 4383555101562201v^13 - 8610750903518110v^12 - 53920802078203288v^11 + 103873218394244684v^10 + 305749680568283999v^9 - 558888658216833442v^8 - 822427991542265903v^7 + 1321466919141595230v^6 + 1054689146794440624v^5 - 1284874776661237012v^4 - 529706969227455298v^3 + 389362459002476198v^2 + 14086029773674412*v - 8817505436277808) / 1096046442183436

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 4$ b2 + 4
$\nu^{3}$	$=$	$-\beta_{15} - \beta_{14} - \beta_{8} + 2\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 8\beta _1 - 1$ -b15 - b14 - b8 + 2b6 + b5 + b3 + b2 + 8b1 - 1
$\nu^{4}$	$=$	$- \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{7} + \cdots + 33$ -b14 - b13 + b12 + 2b11 - b10 - 2b9 - 2b7 - b4 - b3 + 11b2 - 2*b1 + 33
$\nu^{5}$	$=$	$- 13 \beta_{15} - 12 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 3 \beta_{11} - \beta_{10} - \beta_{9} + \cdots - 16$ -13b15 - 12b14 - 2b13 - 3b12 - 3b11 - b10 - b9 - 13b8 + b7 + 32b6 + 17b5 - 3b4 + 16b3 + 14b2 + 73b1 - 16
$\nu^{6}$	$=$	$2 \beta_{15} - 15 \beta_{14} - 16 \beta_{13} + 20 \beta_{12} + 32 \beta_{11} - 21 \beta_{10} + \cdots + 312$ 2b15 - 15b14 - 16b13 + 20b12 + 32b11 - 21b10 - 32b9 - b8 - 35b7 - 8b6 - 2b5 - 14b4 - 16b3 + 117b2 - 33b1 + 312
$\nu^{7}$	$=$	$- 148 \beta_{15} - 117 \beta_{14} - 44 \beta_{13} - 60 \beta_{12} - 63 \beta_{11} - 7 \beta_{10} + \cdots - 213$ -148b15 - 117b14 - 44b13 - 60b12 - 63b11 - 7b10 - 10b9 - 152b8 + 25b7 + 414b6 + 226b5 - 53b4 + 207b3 + 159b2 + 711*b1 - 213
$\nu^{8}$	$=$	$56 \beta_{15} - 176 \beta_{14} - 189 \beta_{13} + 300 \beta_{12} + 421 \beta_{11} - 310 \beta_{10} + \cdots + 3131$ 56b15 - 176b14 - 189b13 + 300b12 + 421b11 - 310b10 - 405b9 - 478b7 - 203b6 - 50b5 - 161b4 - 215b3 + 1240b2 - 443b1 + 3131
$\nu^{9}$	$=$	$- 1661 \beta_{15} - 1067 \beta_{14} - 701 \beta_{13} - 876 \beta_{12} - 962 \beta_{11} + 65 \beta_{10} + \cdots - 2691$ -1661b15 - 1067b14 - 701b13 - 876b12 - 962b11 + 65b10 - 37b9 - 1768b8 + 447b7 + 4994b6 + 2784b5 - 709b4 + 2486b3 + 1682b2 + 7217*b1 - 2691
$\nu^{10}$	$=$	$1048 \beta_{15} - 1927 \beta_{14} - 1954 \beta_{13} + 4035 \beta_{12} + 5239 \beta_{11} - 4068 \beta_{10} + \cdots + 32571$ 1048b15 - 1927b14 - 1954b13 + 4035b12 + 5239b11 - 4068b10 - 4733b9 + 294b8 - 5977b7 - 3572b6 - 910b5 - 1749b4 - 2754b3 + 13185b2 - 5645*b1 + 32571
$\nu^{11}$	$=$	$- 18718 \beta_{15} - 9349 \beta_{14} - 9865 \beta_{13} - 11457 \beta_{12} - 13063 \beta_{11} + \cdots - 33456$ -18718b15 - 9349b14 - 9865b13 - 11457b12 - 13063b11 + 2789b10 + 683b9 - 20649b8 + 6853b7 + 58448b6 + 33175b5 - 8602b4 + 28867b3 + 17138b2 + 75426*b1 - 33456
$\nu^{12}$	$=$	$16574 \beta_{15} - 20652 \beta_{14} - 18433 \beta_{13} + 51360 \beta_{12} + 63623 \beta_{11} + \cdots + 347406$ 16574b15 - 20652b14 - 18433b13 + 51360b12 + 63623b11 - 50757b10 - 53457b9 + 7619b8 - 71836b7 - 54069b6 - 14431b5 - 18586b4 - 34514b3 + 140961b2 - 70810*b1 + 347406
$\nu^{13}$	$=$	$- 212203 \beta_{15} - 78720 \beta_{14} - 130171 \beta_{13} - 142701 \beta_{12} - 167732 \beta_{11} + \cdots - 413744$ -212203b15 - 78720b14 - 130171b13 - 142701b12 - 167732b11 + 55836b10 + 20695b9 - 241833b8 + 96474b7 + 674508b6 + 388627b5 - 99796b4 + 329826b3 + 169974b2 + 805564*b1 - 413744
$\nu^{14}$	$=$	$239591 \beta_{15} - 220497 \beta_{14} - 158584 \beta_{13} + 633529 \beta_{12} + 762358 \beta_{11} + \cdots + 3774216$ 239591b15 - 220497b14 - 158584b13 + 633529b12 + 762358b11 - 617797b10 - 595012b9 + 139691b8 - 846839b7 - 756217b6 - 211931b5 - 195434b4 - 427072b3 + 1516011b2 - 882822*b1 + 3774216
$\nu^{15}$	$=$	$- 2418060 \beta_{15} - 627694 \beta_{14} - 1651946 \beta_{13} - 1735219 \beta_{12} - 2089599 \beta_{11} + \cdots - 5102286$ -2418060b15 - 627694b14 - 1651946b13 - 1735219b12 - 2089599b11 + 899196b10 + 376346b9 - 2834297b8 + 1289846b7 + 7735480b6 + 4509627b5 - 1131374b4 + 3741409b3 + 1644753b2 + 8747186*b1 - 5102286

Display $\beta_i$ in terms of $\nu^j$

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

−3.43243
−3.04373
−2.70502
−1.47399
−1.16696
−0.902602
−0.189778
−0.100725
0.336056
0.444459
1.18682
1.75598
2.38176
2.53062
3.10845
3.27108

−3.43243

4.96522

8.78154

1.2

−3.04373

−3.50721

6.26427

1.3

−2.70502

0.533645

4.31715

1.4

−1.47399

0.907297

−0.827364

1.5

−1.16696

−2.23151

−1.63819

1.6

−0.902602

−1.32204

−2.18531

1.7

−0.189778

2.89459

−2.96398

1.8

−0.100725

3.98340

−2.98985

1.9

0.336056

3.02156

−2.88707

1.10

0.444459

−4.10506

−2.80246

1.11

1.18682

0.0116605

−1.59145

1.12

1.75598

−4.69529

0.0834815

1.13

2.38176

5.05491

2.67277

1.14

2.53062

−0.665840

3.40405

1.15

3.10845

3.70694

6.66248

1.16

3.27108

0.447728

7.69993

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$37$	$-1$

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	7400.2.a.be		16
5.b	even	2	1		7400.2.a.bd		16
5.c	odd	4	2		1480.2.d.c	✓	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1480.2.d.c	✓	32	5.c	odd	4	2
7400.2.a.bd		16	5.b	even	2	1
7400.2.a.be		16	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7400))$ :

T_{3}^{16} - 2 T_{3}^{15} - 33 T_{3}^{14} + 66 T_{3}^{13} + 404 T_{3}^{12} - 796 T_{3}^{11} - 2273 T_{3}^{10} + \cdots + 16

T_{7}^{16} - 9 T_{7}^{15} - 38 T_{7}^{14} + 522 T_{7}^{13} + 41 T_{7}^{12} - 10867 T_{7}^{11} + \cdots + 1088

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16}$ T^16
$3$	$T^{16} - 2 T^{15} + \cdots + 16$ T^16 - 2T^15 - 33T^14 + 66T^13 + 404T^12 - 796T^11 - 2273T^10 + 4284T^9 + 6039T^8 - 10158T^7 - 7660T^6 + 10040T^5 + 3914T^4 - 3314T^3 - 232T^2 + 172*T + 16
$5$	$T^{16}$ T^16
$7$	$T^{16} - 9 T^{15} + \cdots + 1088$ T^16 - 9T^15 - 38T^14 + 522T^13 + 41T^12 - 10867T^11 + 13692T^10 + 97426T^9 - 184818T^8 - 350232T^7 + 796050T^6 + 368180T^5 - 1036948T^4 + 109920T^3 + 312176T^2 - 96960*T + 1088
$11$	$T^{16} - 3 T^{15} + \cdots - 23763968$ T^16 - 3T^15 - 113T^14 + 349T^13 + 5066T^12 - 15802T^11 - 116365T^10 + 361121T^9 + 1467587T^8 - 4471073T^7 - 10047264T^6 + 29740784T^5 + 33818880T^4 - 98135744T^3 - 37439104T^2 + 124001792*T - 23763968
$13$	$T^{16} + 4 T^{15} + \cdots + 29707264$ T^16 + 4T^15 - 117T^14 - 540T^13 + 4729T^12 + 27318T^11 - 67812T^10 - 620952T^9 - 190400T^8 + 5835396T^7 + 11209048T^6 - 11578784T^5 - 52795968T^4 - 30992848T^3 + 55616192T^2 + 80979072*T + 29707264
$17$	$T^{16} + 9 T^{15} + \cdots + 97800192$ T^16 + 9T^15 - 144T^14 - 1402T^13 + 7500T^12 + 82468T^11 - 177260T^10 - 2337384T^9 + 2141096T^8 + 34366976T^7 - 17263216T^6 - 254827200T^5 + 136502528T^4 + 772051712T^3 - 659290880T^2 - 135661568*T + 97800192
$19$	$T^{16} + \cdots - 538505216$ T^16 - 14T^15 - 94T^14 + 2058T^13 - 442T^12 - 104222T^11 + 245128T^10 + 2289448T^9 - 7597568T^8 - 24172032T^7 + 94585088T^6 + 121244208T^5 - 536251200T^4 - 222264704T^3 + 1223620864T^2 - 86984960*T - 538505216
$23$	$T^{16} + \cdots - 174358528$ T^16 - 18T^15 - 79T^14 + 3246T^13 - 10437T^12 - 170964T^11 + 1241334T^10 + 970196T^9 - 33396340T^8 + 96785360T^7 + 42913760T^6 - 595504928T^5 + 716274432T^4 + 409911104T^3 - 1332043520T^2 + 865875456*T - 174358528
$29$	$T^{16} + \cdots + 49220009984$ T^16 - T^15 - 261T^14 + 411T^13 + 27093T^12 - 53419T^11 - 1438888T^10 + 3140662T^9 + 42332924T^8 - 94225700T^7 - 698132728T^6 + 1464854176T^5 + 6235144384T^4 - 10945479360T^3 - 27823203200T^2 + 30603433984T + 49220009984
$31$	$T^{16} + \cdots - 314427792$ T^16 - 23T^15 + 45T^14 + 2537T^13 - 16439T^12 - 83137T^11 + 926150T^10 + 291822T^9 - 20772228T^8 + 26446396T^7 + 205079998T^6 - 392388892T^5 - 834290432T^4 + 1633607888T^3 + 1070594144T^2 - 758284576*T - 314427792
$37$	$(T - 1)^{16}$ (T - 1)^16
$41$	$T^{16} + \cdots + 31001520832$ T^16 - 23T^15 - 95T^14 + 5301T^13 - 12440T^12 - 426646T^11 + 1603231T^10 + 17929333T^9 - 65782735T^8 - 451526033T^7 + 1169344846T^6 + 6754799716T^5 - 6908846760T^4 - 48832900320T^3 - 19126640464T^2 + 63094264000*T + 31001520832
$43$	$T^{16} + \cdots + 374194176$ T^16 + 11T^15 - 248T^14 - 2942T^13 + 16988T^12 + 230612T^11 - 401120T^10 - 6755440T^9 + 6256640T^8 + 87082432T^7 - 81551168T^6 - 453733888T^5 + 422958848T^4 + 920546304T^3 - 763143168T^2 - 540123136*T + 374194176
$47$	$T^{16} + \cdots - 7578127104$ T^16 - 32T^15 + 128T^14 + 5950T^13 - 70689T^12 - 92332T^11 + 5330174T^10 - 21401812T^9 - 90100134T^8 + 822246218T^7 - 1062722624T^6 - 6285266196T^5 + 22776394960T^4 - 19855233936T^3 - 16013633024T^2 + 29465911360*T - 7578127104
$53$	$T^{16} + \cdots + 442116241408$ T^16 - T^15 - 394T^14 + 492T^13 + 62827T^12 - 95179T^11 - 5191070T^10 + 9323604T^9 + 235848344T^8 - 490045344T^7 - 5734350656T^6 + 13275862912T^5 + 65822813184T^4 - 156442041088T^3 - 255719681536T^2 + 421131531264T + 442116241408
$59$	$T^{16} + \cdots - 63898386432$ T^16 - 28T^15 - 192T^14 + 11032T^13 - 24766T^12 - 1553922T^11 + 7885136T^10 + 98890340T^9 - 628345776T^8 - 2869231408T^7 + 19729908352T^6 + 33022538688T^5 - 214158722304T^4 - 102179938304T^3 + 358871687168T^2 - 23372300288*T - 63898386432
$61$	$T^{16} + \cdots + 990990761984$ T^16 - 31T^15 - 23T^14 + 8981T^13 - 53797T^12 - 886077T^11 + 7586066T^10 + 42875514T^9 - 431139568T^8 - 1248962420T^7 + 11982642896T^6 + 25845830336T^5 - 153554721536T^4 - 346368773120T^3 + 582606843904T^2 + 1753806225408*T + 990990761984
$67$	$T^{16} + \cdots + 147435008$ T^16 - 14T^15 - 367T^14 + 5356T^13 + 46269T^12 - 720538T^11 - 2392816T^10 + 40763444T^9 + 54588718T^8 - 929137406T^7 - 835048184T^6 + 7336091628T^5 + 3973361136T^4 - 17097845040T^3 - 7033230208T^2 + 7604028096*T + 147435008
$71$	$T^{16} + \cdots + 20551437811712$ T^16 - 18T^15 - 358T^14 + 8924T^13 + 16293T^12 - 1474354T^11 + 6076824T^10 + 91389152T^9 - 733391344T^8 - 1653185184T^7 + 30865940992T^6 - 40388899072T^5 - 524274391040T^4 + 1779465951232T^3 + 1986514718720T^2 - 16708297818112*T + 20551437811712
$73$	$T^{16} + \cdots - 313851217482752$ T^16 + 6T^15 - 759T^14 - 2904T^13 + 239236T^12 + 342992T^11 - 40286919T^10 + 37925390T^9 + 3784501021T^8 - 12012503896T^7 - 182100755860T^6 + 995944655760T^5 + 2987382347952T^4 - 30143640830592T^3 + 36577318944704T^2 + 149109331016704*T - 313851217482752
$79$	$T^{16} + \cdots - 2447424256$ T^16 - 8T^15 - 361T^14 + 2608T^13 + 50243T^12 - 309862T^11 - 3526266T^10 + 16945974T^9 + 135225150T^8 - 435047902T^7 - 2758767700T^6 + 4382546540T^5 + 25022650168T^4 - 3541756040T^3 - 43169721648T^2 - 22893682960*T - 2447424256
$83$	$T^{16} + \cdots - 2485440892672$ T^16 - 8T^15 - 424T^14 + 3164T^13 + 71983T^12 - 512884T^11 - 6178482T^10 + 43736714T^9 + 276719626T^8 - 2070132906T^7 - 5807366040T^6 + 51905536132T^5 + 31968248656T^4 - 587993689088T^3 + 330148717952T^2 + 2224977836864*T - 2485440892672
$89$	$T^{16} + \cdots + 205458112512$ T^16 - 636T^14 - 1068T^13 + 139892T^12 + 436952T^11 - 13318256T^10 - 63131616T^9 + 515455296T^8 + 3735248512T^7 - 2254554368T^6 - 73708793344T^5 - 206231160832T^4 - 121281314816T^3 + 311259250688T^2 + 502688088064T + 205458112512
$97$	$T^{16} + \cdots - 1415820529664$ T^16 + 15T^15 - 516T^14 - 8914T^13 + 77956T^12 + 1767668T^11 - 2691448T^10 - 146153520T^9 - 156476512T^8 + 5482203264T^7 + 9846488896T^6 - 99276935680T^5 - 154765252864T^4 + 810400746496T^3 + 663369090048T^2 - 1614542450688*T - 1415820529664
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.16
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more