LMFDB - Newform orbit 174.2.g.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [174,2,Mod(7,174)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("174.7"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(174, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 174 = 2 \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	174.g (of order $7$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$1.38939699517$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{7})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 21 x^{10} - 48 x^{9} + 157 x^{8} - 126 x^{7} + 99 x^{6} + 434 x^{5} - 345 x^{4} + \cdots + 9409 $ x^12 - 4x^11 + 21x^10 - 48x^9 + 157x^8 - 126x^7 + 99x^6 + 434x^5 - 345x^4 - 986x^3 + 6699x^2 + 11058*x + 9409
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{8} q^{2} + \beta_{6} q^{3} + \beta_{4} q^{4} + ( - \beta_{11} - 2 \beta_{8} + \cdots + 1) q^{5} - \beta_{5} q^{6} + ( - \beta_{10} - \beta_{8} + \cdots + \beta_1) q^{7} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 1) q^{8}+ \cdots + (\beta_{11} + 2 \beta_{6} + \cdots + \beta_{2}) q^{99}+O(q^{100}) $ q + b8 * q^2 + b6 * q^3 + b4 * q^4 + (-b11 - 2b8 + b6 + b4 + b3 + b1 + 1) q^5 - b5 * q^6 + (-b10 - b8 + b6 + b4 + b1) * q^7 + (-b8 + b6 + b5 + b4 + b3 + 1) * q^8 + (b8 - b6 - b5 - b4 - b3 - 1) * q^9 + (b10 + b9 - b7 - b6 - b5 - 2b4 + b3 - b2 - b1 + 1) q^10 + (-b11 + b5 + b4) * q^11 + q^12 + (-b11 + b10 - b9 + 2b8 - b7 - b5 - b4 - b3 - 2) q^13 + (b9 - b8 - b7 + b6 + b3 + 1) * q^14 + (b8 - b6 - b4 - b3 - b2 - b1) * q^15 + b3 * q^16 + (b11 + b9 - 2b6 + b5 - 2b4 + b2 - b1) * q^17 - b3 * q^18 + (-2b4 + 2b3 - 2b2 + 2) q^19 + (b11 + 2b8 + b7 - b6 - b3 + b2 - 1) q^20 + (b11 + b9 + b8 + b7 - b6 - b4 - b3 - b1) * q^21 + (b10 - b8 - b7 + b3 - b2 - b1) * q^22 + (-4b8 + 2b6 + 2b5 + 4b3 + 2) * q^23 + b8 * q^24 + (-2b10 - 2b9 + 2b7 - b6 - b5 + 3b4 - 5b3 + b2 + 2b1 - 5) * q^25 + (-b11 + b10 - b9 - b8 - b5 + b4 - b3) * q^26 - b8 * q^27 + (b11 + b9 + b8 + b2 - b1) * q^28 + (-b11 - 2b8 + b5 + b3 + b2 + 2b1 - 2) * q^29 + (-b9 + b6 + b4 + b1 - 1) * q^30 + (2b11 - b10 + b8 + b7 + 2b6 + b5 + 2b4 + b3 + b2 - 2b1 + 1) * q^31 - b6 * q^32 + (-b7 - b6 - b5 - b4 + b3 - b2 - b1 + 1) * q^33 + (b11 - b10 + 3b8 + b7 - 2b6 + b5 - 2b4 - 3b3 + b2 - b1 - 3) * q^34 + (b11 - b10 - b9 + 6b8 + 2b7 - 5b6 + 2b5 + b4 - 6b3 + b2 - 6) q^35 + b6 * q^36 + (b11 - b9 - 2b8 + b7 + 4b6 + b5 + 3b4 + b3 + 2b2 + 2b1 - 1) q^37 + (2b8 - 2b6 - 2b5 + 2b1 - 2) * q^38 + (-b10 - b9 - b6 - b5 - b3 + b2 - 1) * q^39 + (-b10 + b7 + b5 + b4 - b3) * q^40 + (b11 + b10 + b9 - 2b8 + 2b6 + 3b5 + 2b4 + b2 - b1 + 5) * q^41 + (b11 - b10 + b8 + b7 - b3 - 1) * q^42 + (-2b11 + 2b10 - 2b8 - 2b7 + 2b6 - 2b5 - 2b4 + 2b3 - 2b2 + 4) q^43 + (-b9 - b8 + b7 + b6 + b5 + b4 + b3 + b2 + b1) * q^44 + (b10 - b7 - b5 - b4 + b3) * q^45 + (2b8 - 4b6 - 2b5 - 4b4 - 2) * q^46 + (-2b9 + 2b8 + 2b5 + 2b4 + 2b3 - 2) q^47 + b4 * q^48 + (b11 + b9 + 4b8 + b7 - 4b6 - 6b4 - 6b3) * q^49 + (-2b11 - 7b8 - 2b7 + 5b6 + 2b5 + 2b3 - 2b2 + b1 + 4) q^50 + (-b10 - 3b8 + 3b6 + 3b5 + 3b4 + 3b3 + b1) q^51 + (-b11 + b10 - b9 - b8 + b6 - b4 + b3 - b2 + 2) * q^52 + (2b10 - 4b8 - 3b7 + b6 - 2b5 - 3b4 + 3b3 - 2b2 - 1) q^53 - b4 * q^54 + (b11 + b9 - b8 + b7 - 5b6 - 4b5 + 2b4 - 5b3 + b2 - b1) * q^55 + (b11 - b10 + b8 + b7 + b5 + b4 - b3 + b2 - b1) * q^56 + (2b10 + 2b8 - 2b5 - 2) q^57 + (b10 + 2b9 - b8 - b7 - 3b6 - b5 - 4b4 - b3 - b2 - 2b1 - 1) * q^58 + (-b11 - 2b10 + 2b8 - 3b5 - b2) q^59 + (-b11 - 2b8 + b6 + b4 + b3 + b1 + 1) q^60 + (-b10 + 3b8 + 2b7 + b6 + b5 - b4 - b3 + 2b2 + b1) q^61 + (-2b10 - 2b9 + b7 + b6 + b5 + 3b4 + b3 + b2 + b1 + 1) q^62 + (-b11 + b10 - b8 - b7 - b5 - b4 + b3 - b2 + b1) * q^63 + b5 * q^64 + (b11 + 2b10 + b9 + 4b8 - 2b7 + b6 - 4b5 - 6b4 - 2b3 - 2b2 - 3b1) * q^65 + (-b9 + b8 + b5 + b4 + b2 + b1) * q^66 + (4b8 - 4b6 + 2b4 + 2b3) * q^67 + (-b10 - b9 - 3b3 - 3) q^68 + (2b8 + 2b5 + 2b4 - 2) q^69 + (-b11 - b10 - b9 - 6b8 + 5b6 + 5b5 + 5b4 - b2 + b1 - 1) * q^70 + (2b11 - 2b10 + 2b9 + 4b8 + 2b7 - 2b5 - 2b4 + 2b3 - 4) * q^71 - b5 * q^72 + (-b11 + b10 + 2b9 + 3b8 - 3b7 - b6 - 6b5 - 3b4 - 3b3 - 3b2 - 2b1 + 2) * q^73 + (-b11 - b10 + b9 - 2b8 + b7 - b5 - b4 + b3 + 2) q^74 + (b10 + 2b9 - 2b8 - 3b6 + 2b5 - 3b4 - 2b1 + 4) * q^75 + (2b9 - 2b8 + 2b5 + 2b4 + 2) * q^76 + (b10 + b9 - b7 + 2b6 + 2b5 + 7b4 + 3b3 + 2b2 - b1 + 3) q^77 + (-b11 - b9 - b7 + b5 - b4 - b3 + 1) * q^78 + (-2b9 + 2b6 + 2b5 - 2b3 + 2b2) q^79 + (-b8 - b7 + b6 + b3 - b2) * q^80 + b5 * q^81 + (b11 - b10 + 4b8 + 2b7 + 2b6 + b5 + b3 + b2 - b1 - 1) q^82 + (2b10 + 2b9 - b7 - 7b6 - 7b5 - b4 - b3 - b2 - b1 - 1) * q^83 + (-b10 - b8 + b6 + b4 + b1) * q^84 + (-3b11 - 13b8 - 3b7 + 9b6 + 2b4 + 9b3 + 3b1 + 2) q^85 + (2b10 + 4b8 - 2b6 - 4b5 - 2b4) q^86 + (-b10 - b8 + b7 - 2b6 + 2b5 + b4 + b3 - b2 - b1) * q^87 + (-b11 - 2b6 - b5 - 2b4 - b2) * q^88 + (b10 - 5b8 + b7 + 3b6 - b5 + 4b4 + 4b3 - b2 + 5) * q^89 + (b8 + b7 - b6 - b3 + b2) * q^90 + (2b10 + 2b9 - 2b7 + 6b6 + 6b5 + 12b4 + 10b3 - 2b1 + 10) * q^91 + (2b8 - 4b6 - 2b4 - 4b3 - 2) * q^92 + (b9 + b8 - b7 + b6 - b5 + b4 - b3 + b2 + b1) * q^93 + (-2b11 - 2b9 - 4b8 + 2b5 + 4b4 + 2b3 + 2b1) q^94 + (-2b11 + 2b9 - 4b8 - 2b7 + 8b5 + 4b3 - 4b2 - 2b1 + 12) * q^95 + (-b8 + b6 + b5 + b4 + b3 + 1) * q^96 + (-b10 - b9 + 3b7 - 3b4 + 2b2 + 3b1) * q^97 + (b11 - b10 + b9 + 6b8 + b7 - 2b5 - 2b4 - 6b3 - 6) * q^98 + (b11 + 2b6 + b5 + 2b4 + b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} - 2 q^{9} + 6 q^{10} - 9 q^{11} + 12 q^{12} - 16 q^{13} + 5 q^{14} + q^{15} - 2 q^{16} + 6 q^{17} + 2 q^{18} + 18 q^{19} + q^{20}+ \cdots - 2 q^{99}+O(q^{100}) $ 12 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 - 2 * q^9 + 6 * q^10 - 9 * q^11 + 12 * q^12 - 16 * q^13 + 5 * q^14 + q^15 - 2 * q^16 + 6 * q^17 + 2 * q^18 + 18 * q^19 + q^20 + 2 * q^21 - 12 * q^22 + 2 * q^24 - 31 * q^25 - 5 * q^26 - 2 * q^27 + 2 * q^28 - 26 * q^29 - 8 * q^30 + 8 * q^31 + 2 * q^32 + 12 * q^33 - 13 * q^34 - 34 * q^35 - 2 * q^36 - 14 * q^37 - 4 * q^38 + 5 * q^39 - q^40 + 38 * q^41 - 2 * q^42 + 26 * q^43 - 2 * q^44 + q^45 - 24 * q^47 - 2 * q^48 + 38 * q^49 + 10 * q^50 - 22 * q^51 + 12 * q^52 - 23 * q^53 + 2 * q^54 + 19 * q^55 + 5 * q^56 - 24 * q^57 - 16 * q^58 + 10 * q^59 + q^60 + 14 * q^61 + 20 * q^62 - 5 * q^63 - 2 * q^64 + 11 * q^65 + 9 * q^66 + 8 * q^67 - 22 * q^68 - 28 * q^69 - 50 * q^70 - 32 * q^71 + 2 * q^72 + 31 * q^73 + 14 * q^74 + 32 * q^75 + 4 * q^76 + 5 * q^77 + 16 * q^78 + 10 * q^79 - 6 * q^80 - 2 * q^81 - 10 * q^82 + 2 * q^84 - 36 * q^85 + 16 * q^86 - 12 * q^87 + 2 * q^88 + 20 * q^89 + 6 * q^90 + 34 * q^91 + 8 * q^93 - 18 * q^94 + 80 * q^95 + 2 * q^96 + 23 * q^97 - 38 * q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4 x^{11} + 21 x^{10} - 48 x^{9} + 157 x^{8} - 126 x^{7} + 99 x^{6} + 434 x^{5} - 345 x^{4} + \cdots + 9409 $ x^12 - 4*x^11 + 21*x^10 - 48*x^9 + 157*x^8 - 126*x^7 + 99*x^6 + 434*x^5 - 345*x^4 - 986*x^3 + 6699*x^2 + 11058*x + 9409 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 12544269397937 \nu^{11} + \cdots + 11\!\cdots\!83 ) / 47\!\cdots\!86 $ (12544269397937v^11 + 1265819008041312v^10 - 7294572711592952v^9 + 36540756699209664v^8 - 100425319089866968v^7 + 284548755096930233v^6 - 347648531572510539v^5 - 9586293944699250v^4 + 2200352243420612539v^3 - 3253935648173631237v^2 + 1230898644156134971*v + 11287461980847792283) / 4756385226169908686
$\beta_{3}$	$=$	$ ( 108436957618889 \nu^{11} + 455837426093101 \nu^{10} + \cdots + 43\!\cdots\!03 ) / 47\!\cdots\!86 $ (108436957618889v^11 + 455837426093101v^10 - 2284423556425730v^9 + 18888656481963664v^8 - 53856597958959693v^7 + 214615623340466630v^6 - 350352783729515018v^5 + 512523861245810431v^4 - 624707075374529v^3 - 943855174878683484v^2 + 1794113974898869816*v + 4365374933946904803) / 4756385226169908686
$\beta_{4}$	$=$	$ ( - 209685379633323 \nu^{11} + 437488181441742 \nu^{10} + \cdots - 61\!\cdots\!19 ) / 47\!\cdots\!86 $ (-209685379633323v^11 + 437488181441742v^10 - 1386205284461405v^9 - 5955535204491530v^8 + 19627177263755316v^7 - 136497070532915151v^6 + 301707683597213253v^5 - 606464860987967326v^4 + 221485094633076505v^3 + 17810629308702892v^2 - 1365800105190560397*v - 6158708705193737919) / 4756385226169908686
$\beta_{5}$	$=$	$ ( - 277320117783728 \nu^{11} + \cdots - 27\!\cdots\!85 ) / 47\!\cdots\!86 $ (-277320117783728v^11 + 1029914318340416v^10 - 5155798839880026v^9 + 9739732902691722v^8 - 30263149582053585v^7 + 455625335895458v^6 + 43923586392120318v^5 - 147838573027678808v^4 - 64125049035223288v^3 + 1732372063025426574v^2 - 2819610744942380328*v - 2709868583301746385) / 4756385226169908686
$\beta_{6}$	$=$	$ ( 395095075224618 \nu^{11} + \cdots + 19\!\cdots\!73 ) / 47\!\cdots\!86 $ (395095075224618v^11 - 2065297874245208v^10 + 10264944085007161v^9 - 27048845487418145v^8 + 76574581903310967v^7 - 77129031847201150v^6 - 25464380326363888v^5 + 604089238289735936v^4 - 898777862127794990v^3 + 541909679930315315v^2 + 3311249485127523593*v + 190420295874788773) / 4756385226169908686
$\beta_{7}$	$=$	$ ( - 4999150240688 \nu^{11} + 20288118611239 \nu^{10} - 83343112130273 \nu^{9} + \cdots - 38\!\cdots\!46 ) / 49\!\cdots\!38 $ (-4999150240688v^11 + 20288118611239v^10 - 83343112130273v^9 + 149944897866453v^8 - 281928374937106v^7 - 665760750243310v^6 + 4459979130332492v^5 - 7860516094590740v^4 + 9602818805173079v^3 + 6850593569049563v^2 - 43077742741845743*v - 38324222296787946) / 49034899238865038
$\beta_{8}$	$=$	$ ( - 811835915228413 \nu^{11} + \cdots - 37\!\cdots\!08 ) / 47\!\cdots\!86 $ (-811835915228413v^11 + 4112180798339979v^10 - 20183544786790921v^9 + 57101856129148745v^8 - 168781312390444813v^7 + 253271139965707323v^6 - 224323696635524503v^5 - 88771221347747020v^4 + 445530032375795476v^3 + 574070269340282820v^2 - 5513044135942270856*v - 3727406218167765408) / 4756385226169908686
$\beta_{9}$	$=$	$ ( 8915846777591 \nu^{11} - 32319490381384 \nu^{10} + 186945692764793 \nu^{9} + \cdots + 78\!\cdots\!61 ) / 49\!\cdots\!38 $ (8915846777591v^11 - 32319490381384v^10 + 186945692764793v^9 - 426011069067876v^8 + 1556492934504405v^7 - 1484040629153728v^6 + 2717170782076126v^5 + 1705635480639103v^4 - 2334020031700334v^3 - 768611750795177v^2 + 54122426107505418*v + 78748083777156061) / 49034899238865038
$\beta_{10}$	$=$	$ ( - 9170982026481 \nu^{11} + 47026800684767 \nu^{10} - 248387942759488 \nu^{9} + \cdots + 10\!\cdots\!33 ) / 49\!\cdots\!38 $ (-9170982026481v^11 + 47026800684767v^10 - 248387942759488v^9 + 730734023764178v^8 - 2353388453612852v^7 + 3722557139523557v^6 - 4798579604527965v^5 - 379237559826208v^4 + 8628209635736690v^3 - 11007162843401365v^2 - 32642052129868353*v + 10518384889032233) / 49034899238865038
$\beta_{11}$	$=$	$ ( - 13052479118741 \nu^{11} + 63424518091058 \nu^{10} - 352104800258515 \nu^{9} + \cdots - 58\!\cdots\!30 ) / 49\!\cdots\!38 $ (-13052479118741v^11 + 63424518091058v^10 - 352104800258515v^9 + 967740779028567v^8 - 3236054051687022v^7 + 4808437909369318v^6 - 8031205897819478v^5 - 168072195488793v^4 + 386193691393596v^3 + 1169459719589717v^2 - 44675235912026485*v - 58408601952723730) / 49034899238865038

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} + 5\beta_{5} + \beta_{3} + \beta _1 - 1 $ -b11 + b10 - b8 - b7 - b6 + 5*b5 + b3 + b1 - 1
$\nu^{3}$	$=$	$ \beta_{10} + \beta_{9} - 2\beta_{7} + 10\beta_{6} + 10\beta_{5} + 11\beta_{4} + \beta_{3} + 6\beta_{2} - 2\beta _1 + 1 $ b10 + b9 - 2b7 + 10b6 + 10b5 + 11b4 + b3 + 6b2 - 2b1 + 1
$\nu^{4}$	$=$	$ 14 \beta_{11} + 14 \beta_{9} + 5 \beta_{8} + 11 \beta_{7} + 40 \beta_{6} - 9 \beta_{5} + 6 \beta_{4} + \cdots - 14 \beta_1 $ 14b11 + 14b9 + 5b8 + 11b7 + 40b6 - 9b5 + 6b4 - 20b3 + 11b2 - 14b1
$\nu^{5}$	$=$	$ 20 \beta_{11} + 9 \beta_{10} - 56 \beta_{8} + 65 \beta_{7} + 21 \beta_{6} - 9 \beta_{5} - 26 \beta_{4} + \cdots - 17 $ 20b11 + 9b10 - 56b8 + 65b7 + 21b6 - 9b5 - 26b4 + 30b3 - 9b2 - 20b1 - 17
$\nu^{6}$	$=$	$ 3 \beta_{11} - 115 \beta_{10} - 167 \beta_{9} - 170 \beta_{8} + 115 \beta_{7} + 173 \beta_{5} + \cdots + 170 $ 3b11 - 115b10 - 167b9 - 170b8 + 115b7 + 173b5 + 173b4 + 442b3 + 170
$\nu^{7}$	$=$	$ 58 \beta_{11} - 724 \beta_{10} - 282 \beta_{9} + 500 \beta_{8} - 238 \beta_{6} - 442 \beta_{5} + \cdots + 381 $ 58b11 - 724b10 - 282b9 + 500b8 - 238b6 - 442b5 - 238b4 + 58b2 + 282*b1 + 381
$\nu^{8}$	$=$	$ - 1244 \beta_{11} - 78 \beta_{9} + 2722 \beta_{8} - 1244 \beta_{7} - 3888 \beta_{6} - 3542 \beta_{5} + \cdots - 2376 $ -1244b11 - 78b9 + 2722b8 - 1244b7 - 3888b6 - 3542b5 - 5344b4 - 4178b3 - 1166b2 + 1965b1 - 2376
$\nu^{9}$	$=$	$ - 7309 \beta_{11} + 7309 \beta_{10} - 212 \beta_{9} + 3167 \beta_{8} - 7097 \beta_{7} - 8849 \beta_{6} + \cdots - 5307 $ -7309b11 + 7309b10 - 212b9 + 3167b8 - 7097b7 - 8849b6 + 1073b5 - 3542b4 - 3167b3 - 3542b2 + 3767*b1 - 5307
$\nu^{10}$	$=$	$ 13819 \beta_{10} + 13819 \beta_{9} - 12404 \beta_{7} + 27204 \beta_{6} + 27204 \beta_{5} + 49105 \beta_{4} + \cdots + 12123 $ 13819b10 + 13819b9 - 12404b7 + 27204b6 + 27204b5 + 49105b4 + 12123b3 + 8382b2 - 12404*b1 + 12123
$\nu^{11}$	$=$	$ 75328 \beta_{11} + 1696 \beta_{10} + 75328 \beta_{9} - 35905 \beta_{8} + 41023 \beta_{7} + \cdots - 77024 \beta_1 $ 75328b11 + 1696b10 + 75328b9 - 35905b8 + 41023b7 + 128412b6 + 20345b5 + 76928b4 - 20678b3 + 41023b2 - 77024*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/174\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$\beta_{4}$	$1$

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} $

7.1

−0.680420	−	0.853220i
2.08139	+	2.60998i
−0.680420	+	0.853220i
2.08139	−	2.60998i
−0.696063	+	3.04965i
0.572573	−	2.50861i
−0.696063	−	3.04965i
0.572573	+	2.50861i
2.03928	+	0.982065i
−1.31676	−	0.634117i
2.03928	−	0.982065i
−1.31676	+	0.634117i

0.222521

0.974928i

−0.900969

−

0.433884i

−0.900969

0.433884i

−0.926305

−

4.05841i

0.222521

−

0.974928i

−3.79619

−

1.82815i

−0.623490

−

0.781831i

0.623490

0.781831i

3.75053

−

1.80616i

7.2

0.222521

0.974928i

−0.900969

−

0.433884i

−0.900969

0.433884i

0.302815

1.32672i

0.222521

−

0.974928i

3.39522

1.63505i

−0.623490

−

0.781831i

0.623490

0.781831i

−1.22607

0.590446i

25.1

0.222521

−

0.974928i

−0.900969

0.433884i

−0.900969

−

0.433884i

−0.926305

4.05841i

0.222521

0.974928i

−3.79619

1.82815i

−0.623490

0.781831i

0.623490

−

0.781831i

3.75053

1.80616i

25.2

0.222521

−

0.974928i

−0.900969

0.433884i

−0.900969

−

0.433884i

0.302815

−

1.32672i

0.222521

0.974928i

3.39522

−

1.63505i

−0.623490

0.781831i

0.623490

−

0.781831i

−1.22607

−

0.590446i

49.1

0.900969

−

0.433884i

0.623490

0.781831i

0.623490

−

0.781831i

−1.03174

0.496860i

0.900969

0.433884i

2.77803

3.48353i

0.222521

−

0.974928i

−0.222521

0.974928i

−0.713987

0.895311i

49.2

0.900969

−

0.433884i

0.623490

0.781831i

0.623490

−

0.781831i

1.25426

−

0.604021i

0.900969

0.433884i

−1.65454

−

2.07472i

0.222521

−

0.974928i

−0.222521

0.974928i

0.867976

−

1.08841i

103.1

0.900969

0.433884i

0.623490

−

0.781831i

0.623490

0.781831i

−1.03174

−

0.496860i

0.900969

−

0.433884i

2.77803

−

3.48353i

0.222521

0.974928i

−0.222521

−

0.974928i

−0.713987

−

0.895311i

103.2

0.900969

0.433884i

0.623490

−

0.781831i

0.623490

0.781831i

1.25426

0.604021i

0.900969

−

0.433884i

−1.65454

2.07472i

0.222521

0.974928i

−0.222521

−

0.974928i

0.867976

1.08841i

139.1

−0.623490

0.781831i

−0.222521

0.974928i

−0.222521

−

0.974928i

−1.64197

2.05896i

−0.623490

−

0.781831i

−0.0457019

0.200233i

0.900969

0.433884i

−0.900969

−

0.433884i

−0.586012

−

2.56749i

139.2

−0.623490

0.781831i

−0.222521

0.974928i

−0.222521

−

0.974928i

2.54294

−

3.18874i

−0.623490

−

0.781831i

0.323181

−

1.41595i

0.900969

0.433884i

−0.900969

−

0.433884i

0.907564

3.97630i

169.1

−0.623490

−

0.781831i

−0.222521

−

0.974928i

−0.222521

0.974928i

−1.64197

−

2.05896i

−0.623490

0.781831i

−0.0457019

−

0.200233i

0.900969

−

0.433884i

−0.900969

0.433884i

−0.586012

2.56749i

169.2

−0.623490

−

0.781831i

−0.222521

−

0.974928i

−0.222521

0.974928i

2.54294

3.18874i

−0.623490

0.781831i

0.323181

1.41595i

0.900969

−

0.433884i

−0.900969

0.433884i

0.907564

−

3.97630i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	174.2.g.c	✓	12
3.b	odd	2	1		522.2.k.g		12
29.d	even	7	1	inner	174.2.g.c	✓	12
29.d	even	7	1		5046.2.a.bo		6
29.e	even	14	1		5046.2.a.bq		6
87.j	odd	14	1		522.2.k.g		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
174.2.g.c	✓	12	1.a	even	1	1	trivial
174.2.g.c	✓	12	29.d	even	7	1	inner
522.2.k.g		12	3.b	odd	2	1
522.2.k.g		12	87.j	odd	14	1
5046.2.a.bo		6	29.d	even	7	1
5046.2.a.bq		6	29.e	even	14	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} - T_{5}^{11} + 21 T_{5}^{10} - 20 T_{5}^{9} + 272 T_{5}^{8} + 287 T_{5}^{7} + 1450 T_{5}^{6} + \cdots + 9409 $ T5^12 - T5^11 + 21*T5^10 - 20*T5^9 + 272*T5^8 + 287*T5^7 + 1450*T5^6 - 1715*T5^5 + 671*T5^4 + 1247*T5^3 - 1764*T5^2 + 2134*T5 + 9409 acting on $S_{2}^{\mathrm{new}}(174, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	$ (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} $ (T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$5$	$ T^{12} - T^{11} + \cdots + 9409 $ T^12 - T^11 + 21T^10 - 20T^9 + 272T^8 + 287T^7 + 1450T^6 - 1715T^5 + 671T^4 + 1247T^3 - 1764T^2 + 2134T + 9409
$7$	$ T^{12} - 2 T^{11} + \cdots + 3136 $ T^12 - 2T^11 - 10T^10 + 69T^9 + 212T^8 - 1103T^7 + 253T^6 + 3220T^5 + 30408T^4 - 9520T^3 + 74480T^2 + 6272*T + 3136
$11$	$ T^{12} + 9 T^{11} + \cdots + 53824 $ T^12 + 9T^11 + 35T^10 + 27T^9 - 247T^8 - 931T^7 + 3025T^6 + 24136T^5 + 85508T^4 + 97040T^3 + 101472T^2 + 46400*T + 53824
$13$	$ T^{12} + 16 T^{11} + \cdots + 1000000 $ T^12 + 16T^11 + 196T^10 + 1805T^9 + 13354T^8 + 78883T^7 + 367109T^6 + 1290380T^5 + 3250600T^4 + 5419000T^3 + 5600000T^2 + 3300000*T + 1000000
$17$	$ (T^{6} - 3 T^{5} - 50 T^{4} + \cdots + 8)^{2} $ (T^6 - 3T^5 - 50T^4 + 133T^3 - 8T^2 - 52*T + 8)^2
$19$	$ T^{12} - 18 T^{11} + \cdots + 4096 $ T^12 - 18T^11 + 188T^10 - 1216T^9 + 5504T^8 - 15904T^7 + 91904T^6 - 445312T^5 + 2121984T^4 + 5600768T^3 + 3868672T^2 + 61440*T + 4096
$23$	$ (T^{6} + 28 T^{4} + \cdots + 3136)^{2} $ (T^6 + 28T^4 + 112T^3 + 3136)^2
$29$	$ (T^{6} + 13 T^{5} + \cdots + 24389)^{2} $ (T^6 + 13T^5 + 113T^4 + 755T^3 + 3277T^2 + 10933*T + 24389)^2
$31$	$ T^{12} - 8 T^{11} + \cdots + 3182656 $ T^12 - 8T^11 + 90T^10 - 1061T^9 + 5326T^8 + 1843T^7 + 101143T^6 + 87602T^5 + 1128536T^4 + 340056T^3 + 168992T^2 + 2754496*T + 3182656
$37$	$ T^{12} + 14 T^{11} + \cdots + 11999296 $ T^12 + 14T^11 + 8T^10 - 959T^9 - 3044T^8 + 48027T^7 + 662775T^6 + 3599050T^5 + 13057528T^4 + 26912088T^3 + 41751200T^2 - 2521792*T + 11999296
$41$	$ (T^{6} - 19 T^{5} + \cdots - 13208)^{2} $ (T^6 - 19T^5 + 68T^4 + 637T^3 - 5604T^2 + 14988*T - 13208)^2
$43$	$ T^{12} - 26 T^{11} + \cdots + 200704 $ T^12 - 26T^11 + 340T^10 - 2736T^9 + 14464T^8 - 34912T^7 + 2304T^6 + 425600T^5 + 1198848T^4 + 1788416T^3 + 1906688T^2 + 802816*T + 200704
$47$	$ T^{12} + 24 T^{11} + \cdots + 4096 $ T^12 + 24T^11 + 384T^10 + 3880T^9 + 23984T^8 + 77280T^7 + 105344T^6 + 18816T^5 + 2073600T^4 + 916480T^3 + 263168T^2 + 47104*T + 4096
$53$	$ T^{12} + \cdots + 537451489 $ T^12 + 23T^11 + 381T^10 + 4099T^9 + 41676T^8 + 400252T^7 + 3902955T^6 + 31458888T^5 + 167624476T^4 + 497770634T^3 + 800500578T^2 + 711231257*T + 537451489
$59$	$ (T^{6} - 5 T^{5} + \cdots + 17576)^{2} $ (T^6 - 5T^5 - 146T^4 + 1075T^3 + 598T^2 - 13520*T + 17576)^2
$61$	$ T^{12} - 14 T^{11} + \cdots + 1827904 $ T^12 - 14T^11 + 226T^10 - 3017T^9 + 26002T^8 - 148827T^7 + 607951T^6 - 1823374T^5 + 4071692T^4 - 6596072T^3 + 7350032T^2 - 5110560*T + 1827904
$67$	$ (T^{6} - 4 T^{5} + \cdots + 53824)^{2} $ (T^6 - 4T^5 + 72T^4 + 160T^3 - 80T^2 + 9280*T + 53824)^2
$71$	$ T^{12} + \cdots + 14109638656 $ T^12 + 32T^11 + 672T^10 + 11080T^9 + 141536T^8 + 1401568T^7 + 11710784T^6 + 63103488T^5 + 248262656T^4 + 1904705536T^3 + 13452443648T^2 + 17576230912*T + 14109638656
$73$	$ T^{12} + \cdots + 5545334089 $ T^12 - 31T^11 + 453T^10 - 5267T^9 + 73860T^8 - 937934T^7 + 9356655T^6 - 77055456T^5 + 570414172T^4 - 3222185620T^3 + 10336944498T^2 - 4986384787*T + 5545334089
$79$	$ T^{12} - 10 T^{11} + \cdots + 12845056 $ T^12 - 10T^11 + 212T^10 - 2568T^9 + 20304T^8 - 106272T^7 + 366528T^6 + 976640T^5 + 2214912T^4 + 258048T^3 + 5619712T^2 + 16056320*T + 12845056
$83$	$ T^{12} + \cdots + 14482678336 $ T^12 + 210T^10 - 1568T^9 + 7987T^8 - 72128T^7 + 662921T^6 + 5682138T^5 + 13312516T^4 + 33095384T^3 + 1048411056T^2 + 5590219488T + 14482678336
$89$	$ T^{12} + \cdots + 2562789376 $ T^12 - 20T^11 + 274T^10 - 3975T^9 + 42624T^8 - 192233T^7 + 1369845T^6 - 3612028T^5 + 10324048T^4 + 58120384T^3 - 43740928T^2 - 96388096*T + 2562789376
$97$	$ T^{12} + \cdots + 8851234561 $ T^12 - 23T^11 + 385T^10 - 4927T^9 + 51476T^8 - 578662T^7 + 6981507T^6 - 51973600T^5 + 306535568T^4 - 1738335892T^3 + 6963378646T^2 - 10258874483*T + 8851234561
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 174 = 2 \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	174.g (of order \(7\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{2} + \beta_{6} q^{3} + \beta_{4} q^{4} + ( - \beta_{11} - 2 \beta_{8} + \cdots + 1) q^{5} - \beta_{5} q^{6} + ( - \beta_{10} - \beta_{8} + \cdots + \beta_1) q^{7} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 1) q^{8}+ \cdots + (\beta_{11} + 2 \beta_{6} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) q + b8 * q^2 + b6 * q^3 + b4 * q^4 + (-b11 - 2b8 + b6 + b4 + b3 + b1 + 1) q^5 - b5 * q^6 + (-b10 - b8 + b6 + b4 + b1) * q^7 + (-b8 + b6 + b5 + b4 + b3 + 1) * q^8 + (b8 - b6 - b5 - b4 - b3 - 1) * q^9 + (b10 + b9 - b7 - b6 - b5 - 2b4 + b3 - b2 - b1 + 1) q^10 + (-b11 + b5 + b4) * q^11 + q^12 + (-b11 + b10 - b9 + 2b8 - b7 - b5 - b4 - b3 - 2) q^13 + (b9 - b8 - b7 + b6 + b3 + 1) * q^14 + (b8 - b6 - b4 - b3 - b2 - b1) * q^15 + b3 * q^16 + (b11 + b9 - 2b6 + b5 - 2b4 + b2 - b1) * q^17 - b3 * q^18 + (-2b4 + 2b3 - 2b2 + 2) q^19 + (b11 + 2b8 + b7 - b6 - b3 + b2 - 1) q^20 + (b11 + b9 + b8 + b7 - b6 - b4 - b3 - b1) * q^21 + (b10 - b8 - b7 + b3 - b2 - b1) * q^22 + (-4b8 + 2b6 + 2b5 + 4b3 + 2) * q^23 + b8 * q^24 + (-2b10 - 2b9 + 2b7 - b6 - b5 + 3b4 - 5b3 + b2 + 2b1 - 5) * q^25 + (-b11 + b10 - b9 - b8 - b5 + b4 - b3) * q^26 - b8 * q^27 + (b11 + b9 + b8 + b2 - b1) * q^28 + (-b11 - 2b8 + b5 + b3 + b2 + 2b1 - 2) * q^29 + (-b9 + b6 + b4 + b1 - 1) * q^30 + (2b11 - b10 + b8 + b7 + 2b6 + b5 + 2b4 + b3 + b2 - 2b1 + 1) * q^31 - b6 * q^32 + (-b7 - b6 - b5 - b4 + b3 - b2 - b1 + 1) * q^33 + (b11 - b10 + 3b8 + b7 - 2b6 + b5 - 2b4 - 3b3 + b2 - b1 - 3) * q^34 + (b11 - b10 - b9 + 6b8 + 2b7 - 5b6 + 2b5 + b4 - 6b3 + b2 - 6) q^35 + b6 * q^36 + (b11 - b9 - 2b8 + b7 + 4b6 + b5 + 3b4 + b3 + 2b2 + 2b1 - 1) q^37 + (2b8 - 2b6 - 2b5 + 2b1 - 2) * q^38 + (-b10 - b9 - b6 - b5 - b3 + b2 - 1) * q^39 + (-b10 + b7 + b5 + b4 - b3) * q^40 + (b11 + b10 + b9 - 2b8 + 2b6 + 3b5 + 2b4 + b2 - b1 + 5) * q^41 + (b11 - b10 + b8 + b7 - b3 - 1) * q^42 + (-2b11 + 2b10 - 2b8 - 2b7 + 2b6 - 2b5 - 2b4 + 2b3 - 2b2 + 4) q^43 + (-b9 - b8 + b7 + b6 + b5 + b4 + b3 + b2 + b1) * q^44 + (b10 - b7 - b5 - b4 + b3) * q^45 + (2b8 - 4b6 - 2b5 - 4b4 - 2) * q^46 + (-2b9 + 2b8 + 2b5 + 2b4 + 2b3 - 2) q^47 + b4 * q^48 + (b11 + b9 + 4b8 + b7 - 4b6 - 6b4 - 6b3) * q^49 + (-2b11 - 7b8 - 2b7 + 5b6 + 2b5 + 2b3 - 2b2 + b1 + 4) q^50 + (-b10 - 3b8 + 3b6 + 3b5 + 3b4 + 3b3 + b1) q^51 + (-b11 + b10 - b9 - b8 + b6 - b4 + b3 - b2 + 2) * q^52 + (2b10 - 4b8 - 3b7 + b6 - 2b5 - 3b4 + 3b3 - 2b2 - 1) q^53 - b4 * q^54 + (b11 + b9 - b8 + b7 - 5b6 - 4b5 + 2b4 - 5b3 + b2 - b1) * q^55 + (b11 - b10 + b8 + b7 + b5 + b4 - b3 + b2 - b1) * q^56 + (2b10 + 2b8 - 2b5 - 2) q^57 + (b10 + 2b9 - b8 - b7 - 3b6 - b5 - 4b4 - b3 - b2 - 2b1 - 1) * q^58 + (-b11 - 2b10 + 2b8 - 3b5 - b2) q^59 + (-b11 - 2b8 + b6 + b4 + b3 + b1 + 1) q^60 + (-b10 + 3b8 + 2b7 + b6 + b5 - b4 - b3 + 2b2 + b1) q^61 + (-2b10 - 2b9 + b7 + b6 + b5 + 3b4 + b3 + b2 + b1 + 1) q^62 + (-b11 + b10 - b8 - b7 - b5 - b4 + b3 - b2 + b1) * q^63 + b5 * q^64 + (b11 + 2b10 + b9 + 4b8 - 2b7 + b6 - 4b5 - 6b4 - 2b3 - 2b2 - 3b1) * q^65 + (-b9 + b8 + b5 + b4 + b2 + b1) * q^66 + (4b8 - 4b6 + 2b4 + 2b3) * q^67 + (-b10 - b9 - 3b3 - 3) q^68 + (2b8 + 2b5 + 2b4 - 2) q^69 + (-b11 - b10 - b9 - 6b8 + 5b6 + 5b5 + 5b4 - b2 + b1 - 1) * q^70 + (2b11 - 2b10 + 2b9 + 4b8 + 2b7 - 2b5 - 2b4 + 2b3 - 4) * q^71 - b5 * q^72 + (-b11 + b10 + 2b9 + 3b8 - 3b7 - b6 - 6b5 - 3b4 - 3b3 - 3b2 - 2b1 + 2) * q^73 + (-b11 - b10 + b9 - 2b8 + b7 - b5 - b4 + b3 + 2) q^74 + (b10 + 2b9 - 2b8 - 3b6 + 2b5 - 3b4 - 2b1 + 4) * q^75 + (2b9 - 2b8 + 2b5 + 2b4 + 2) * q^76 + (b10 + b9 - b7 + 2b6 + 2b5 + 7b4 + 3b3 + 2b2 - b1 + 3) q^77 + (-b11 - b9 - b7 + b5 - b4 - b3 + 1) * q^78 + (-2b9 + 2b6 + 2b5 - 2b3 + 2b2) q^79 + (-b8 - b7 + b6 + b3 - b2) * q^80 + b5 * q^81 + (b11 - b10 + 4b8 + 2b7 + 2b6 + b5 + b3 + b2 - b1 - 1) q^82 + (2b10 + 2b9 - b7 - 7b6 - 7b5 - b4 - b3 - b2 - b1 - 1) * q^83 + (-b10 - b8 + b6 + b4 + b1) * q^84 + (-3b11 - 13b8 - 3b7 + 9b6 + 2b4 + 9b3 + 3b1 + 2) q^85 + (2b10 + 4b8 - 2b6 - 4b5 - 2b4) q^86 + (-b10 - b8 + b7 - 2b6 + 2b5 + b4 + b3 - b2 - b1) * q^87 + (-b11 - 2b6 - b5 - 2b4 - b2) * q^88 + (b10 - 5b8 + b7 + 3b6 - b5 + 4b4 + 4b3 - b2 + 5) * q^89 + (b8 + b7 - b6 - b3 + b2) * q^90 + (2b10 + 2b9 - 2b7 + 6b6 + 6b5 + 12b4 + 10b3 - 2b1 + 10) * q^91 + (2b8 - 4b6 - 2b4 - 4b3 - 2) * q^92 + (b9 + b8 - b7 + b6 - b5 + b4 - b3 + b2 + b1) * q^93 + (-2b11 - 2b9 - 4b8 + 2b5 + 4b4 + 2b3 + 2b1) q^94 + (-2b11 + 2b9 - 4b8 - 2b7 + 8b5 + 4b3 - 4b2 - 2b1 + 12) * q^95 + (-b8 + b6 + b5 + b4 + b3 + 1) * q^96 + (-b10 - b9 + 3b7 - 3b4 + 2b2 + 3b1) * q^97 + (b11 - b10 + b9 + 6b8 + b7 - 2b5 - 2b4 - 6b3 - 6) * q^98 + (b11 + 2b6 + b5 + 2b4 + b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} - 2 q^{9} + 6 q^{10} - 9 q^{11} + 12 q^{12} - 16 q^{13} + 5 q^{14} + q^{15} - 2 q^{16} + 6 q^{17} + 2 q^{18} + 18 q^{19} + q^{20}+ \cdots - 2 q^{99}+O(q^{100}) \) 12 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 - 2 * q^9 + 6 * q^10 - 9 * q^11 + 12 * q^12 - 16 * q^13 + 5 * q^14 + q^15 - 2 * q^16 + 6 * q^17 + 2 * q^18 + 18 * q^19 + q^20 + 2 * q^21 - 12 * q^22 + 2 * q^24 - 31 * q^25 - 5 * q^26 - 2 * q^27 + 2 * q^28 - 26 * q^29 - 8 * q^30 + 8 * q^31 + 2 * q^32 + 12 * q^33 - 13 * q^34 - 34 * q^35 - 2 * q^36 - 14 * q^37 - 4 * q^38 + 5 * q^39 - q^40 + 38 * q^41 - 2 * q^42 + 26 * q^43 - 2 * q^44 + q^45 - 24 * q^47 - 2 * q^48 + 38 * q^49 + 10 * q^50 - 22 * q^51 + 12 * q^52 - 23 * q^53 + 2 * q^54 + 19 * q^55 + 5 * q^56 - 24 * q^57 - 16 * q^58 + 10 * q^59 + q^60 + 14 * q^61 + 20 * q^62 - 5 * q^63 - 2 * q^64 + 11 * q^65 + 9 * q^66 + 8 * q^67 - 22 * q^68 - 28 * q^69 - 50 * q^70 - 32 * q^71 + 2 * q^72 + 31 * q^73 + 14 * q^74 + 32 * q^75 + 4 * q^76 + 5 * q^77 + 16 * q^78 + 10 * q^79 - 6 * q^80 - 2 * q^81 - 10 * q^82 + 2 * q^84 - 36 * q^85 + 16 * q^86 - 12 * q^87 + 2 * q^88 + 20 * q^89 + 6 * q^90 + 34 * q^91 + 8 * q^93 - 18 * q^94 + 80 * q^95 + 2 * q^96 + 23 * q^97 - 38 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	174.2.g.c

Level	$174$
Weight	$2$
Character orbit	174.g
Analytic conductor	$1.389$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(1.38939699517\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{7})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4 x^{11} + 21 x^{10} - 48 x^{9} + 157 x^{8} - 126 x^{7} + 99 x^{6} + 434 x^{5} - 345 x^{4} + \cdots + 9409 \) x^12 - 4x^11 + 21x^10 - 48x^9 + 157x^8 - 126x^7 + 99x^6 + 434x^5 - 345x^4 - 986x^3 + 6699x^2 + 11058*x + 9409
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 12544269397937 \nu^{11} + \cdots + 11\!\cdots\!83 ) / 47\!\cdots\!86 \) (12544269397937v^11 + 1265819008041312v^10 - 7294572711592952v^9 + 36540756699209664v^8 - 100425319089866968v^7 + 284548755096930233v^6 - 347648531572510539v^5 - 9586293944699250v^4 + 2200352243420612539v^3 - 3253935648173631237v^2 + 1230898644156134971*v + 11287461980847792283) / 4756385226169908686
\(\beta_{3}\)	\(=\)	\( ( 108436957618889 \nu^{11} + 455837426093101 \nu^{10} + \cdots + 43\!\cdots\!03 ) / 47\!\cdots\!86 \) (108436957618889v^11 + 455837426093101v^10 - 2284423556425730v^9 + 18888656481963664v^8 - 53856597958959693v^7 + 214615623340466630v^6 - 350352783729515018v^5 + 512523861245810431v^4 - 624707075374529v^3 - 943855174878683484v^2 + 1794113974898869816*v + 4365374933946904803) / 4756385226169908686
\(\beta_{4}\)	\(=\)	\( ( - 209685379633323 \nu^{11} + 437488181441742 \nu^{10} + \cdots - 61\!\cdots\!19 ) / 47\!\cdots\!86 \) (-209685379633323v^11 + 437488181441742v^10 - 1386205284461405v^9 - 5955535204491530v^8 + 19627177263755316v^7 - 136497070532915151v^6 + 301707683597213253v^5 - 606464860987967326v^4 + 221485094633076505v^3 + 17810629308702892v^2 - 1365800105190560397*v - 6158708705193737919) / 4756385226169908686
\(\beta_{5}\)	\(=\)	\( ( - 277320117783728 \nu^{11} + \cdots - 27\!\cdots\!85 ) / 47\!\cdots\!86 \) (-277320117783728v^11 + 1029914318340416v^10 - 5155798839880026v^9 + 9739732902691722v^8 - 30263149582053585v^7 + 455625335895458v^6 + 43923586392120318v^5 - 147838573027678808v^4 - 64125049035223288v^3 + 1732372063025426574v^2 - 2819610744942380328*v - 2709868583301746385) / 4756385226169908686
\(\beta_{6}\)	\(=\)	\( ( 395095075224618 \nu^{11} + \cdots + 19\!\cdots\!73 ) / 47\!\cdots\!86 \) (395095075224618v^11 - 2065297874245208v^10 + 10264944085007161v^9 - 27048845487418145v^8 + 76574581903310967v^7 - 77129031847201150v^6 - 25464380326363888v^5 + 604089238289735936v^4 - 898777862127794990v^3 + 541909679930315315v^2 + 3311249485127523593*v + 190420295874788773) / 4756385226169908686
\(\beta_{7}\)	\(=\)	\( ( - 4999150240688 \nu^{11} + 20288118611239 \nu^{10} - 83343112130273 \nu^{9} + \cdots - 38\!\cdots\!46 ) / 49\!\cdots\!38 \) (-4999150240688v^11 + 20288118611239v^10 - 83343112130273v^9 + 149944897866453v^8 - 281928374937106v^7 - 665760750243310v^6 + 4459979130332492v^5 - 7860516094590740v^4 + 9602818805173079v^3 + 6850593569049563v^2 - 43077742741845743*v - 38324222296787946) / 49034899238865038
\(\beta_{8}\)	\(=\)	\( ( - 811835915228413 \nu^{11} + \cdots - 37\!\cdots\!08 ) / 47\!\cdots\!86 \) (-811835915228413v^11 + 4112180798339979v^10 - 20183544786790921v^9 + 57101856129148745v^8 - 168781312390444813v^7 + 253271139965707323v^6 - 224323696635524503v^5 - 88771221347747020v^4 + 445530032375795476v^3 + 574070269340282820v^2 - 5513044135942270856*v - 3727406218167765408) / 4756385226169908686
\(\beta_{9}\)	\(=\)	\( ( 8915846777591 \nu^{11} - 32319490381384 \nu^{10} + 186945692764793 \nu^{9} + \cdots + 78\!\cdots\!61 ) / 49\!\cdots\!38 \) (8915846777591v^11 - 32319490381384v^10 + 186945692764793v^9 - 426011069067876v^8 + 1556492934504405v^7 - 1484040629153728v^6 + 2717170782076126v^5 + 1705635480639103v^4 - 2334020031700334v^3 - 768611750795177v^2 + 54122426107505418*v + 78748083777156061) / 49034899238865038
\(\beta_{10}\)	\(=\)	\( ( - 9170982026481 \nu^{11} + 47026800684767 \nu^{10} - 248387942759488 \nu^{9} + \cdots + 10\!\cdots\!33 ) / 49\!\cdots\!38 \) (-9170982026481v^11 + 47026800684767v^10 - 248387942759488v^9 + 730734023764178v^8 - 2353388453612852v^7 + 3722557139523557v^6 - 4798579604527965v^5 - 379237559826208v^4 + 8628209635736690v^3 - 11007162843401365v^2 - 32642052129868353*v + 10518384889032233) / 49034899238865038
\(\beta_{11}\)	\(=\)	\( ( - 13052479118741 \nu^{11} + 63424518091058 \nu^{10} - 352104800258515 \nu^{9} + \cdots - 58\!\cdots\!30 ) / 49\!\cdots\!38 \) (-13052479118741v^11 + 63424518091058v^10 - 352104800258515v^9 + 967740779028567v^8 - 3236054051687022v^7 + 4808437909369318v^6 - 8031205897819478v^5 - 168072195488793v^4 + 386193691393596v^3 + 1169459719589717v^2 - 44675235912026485*v - 58408601952723730) / 49034899238865038

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} + 5\beta_{5} + \beta_{3} + \beta _1 - 1 \) -b11 + b10 - b8 - b7 - b6 + 5*b5 + b3 + b1 - 1
\(\nu^{3}\)	\(=\)	\( \beta_{10} + \beta_{9} - 2\beta_{7} + 10\beta_{6} + 10\beta_{5} + 11\beta_{4} + \beta_{3} + 6\beta_{2} - 2\beta _1 + 1 \) b10 + b9 - 2b7 + 10b6 + 10b5 + 11b4 + b3 + 6b2 - 2b1 + 1
\(\nu^{4}\)	\(=\)	\( 14 \beta_{11} + 14 \beta_{9} + 5 \beta_{8} + 11 \beta_{7} + 40 \beta_{6} - 9 \beta_{5} + 6 \beta_{4} + \cdots - 14 \beta_1 \) 14b11 + 14b9 + 5b8 + 11b7 + 40b6 - 9b5 + 6b4 - 20b3 + 11b2 - 14b1
\(\nu^{5}\)	\(=\)	\( 20 \beta_{11} + 9 \beta_{10} - 56 \beta_{8} + 65 \beta_{7} + 21 \beta_{6} - 9 \beta_{5} - 26 \beta_{4} + \cdots - 17 \) 20b11 + 9b10 - 56b8 + 65b7 + 21b6 - 9b5 - 26b4 + 30b3 - 9b2 - 20b1 - 17
\(\nu^{6}\)	\(=\)	\( 3 \beta_{11} - 115 \beta_{10} - 167 \beta_{9} - 170 \beta_{8} + 115 \beta_{7} + 173 \beta_{5} + \cdots + 170 \) 3b11 - 115b10 - 167b9 - 170b8 + 115b7 + 173b5 + 173b4 + 442b3 + 170
\(\nu^{7}\)	\(=\)	\( 58 \beta_{11} - 724 \beta_{10} - 282 \beta_{9} + 500 \beta_{8} - 238 \beta_{6} - 442 \beta_{5} + \cdots + 381 \) 58b11 - 724b10 - 282b9 + 500b8 - 238b6 - 442b5 - 238b4 + 58b2 + 282*b1 + 381
\(\nu^{8}\)	\(=\)	\( - 1244 \beta_{11} - 78 \beta_{9} + 2722 \beta_{8} - 1244 \beta_{7} - 3888 \beta_{6} - 3542 \beta_{5} + \cdots - 2376 \) -1244b11 - 78b9 + 2722b8 - 1244b7 - 3888b6 - 3542b5 - 5344b4 - 4178b3 - 1166b2 + 1965b1 - 2376
\(\nu^{9}\)	\(=\)	\( - 7309 \beta_{11} + 7309 \beta_{10} - 212 \beta_{9} + 3167 \beta_{8} - 7097 \beta_{7} - 8849 \beta_{6} + \cdots - 5307 \) -7309b11 + 7309b10 - 212b9 + 3167b8 - 7097b7 - 8849b6 + 1073b5 - 3542b4 - 3167b3 - 3542b2 + 3767*b1 - 5307
\(\nu^{10}\)	\(=\)	\( 13819 \beta_{10} + 13819 \beta_{9} - 12404 \beta_{7} + 27204 \beta_{6} + 27204 \beta_{5} + 49105 \beta_{4} + \cdots + 12123 \) 13819b10 + 13819b9 - 12404b7 + 27204b6 + 27204b5 + 49105b4 + 12123b3 + 8382b2 - 12404*b1 + 12123
\(\nu^{11}\)	\(=\)	\( 75328 \beta_{11} + 1696 \beta_{10} + 75328 \beta_{9} - 35905 \beta_{8} + 41023 \beta_{7} + \cdots - 77024 \beta_1 \) 75328b11 + 1696b10 + 75328b9 - 35905b8 + 41023b7 + 128412b6 + 20345b5 + 76928b4 - 20678b3 + 41023b2 - 77024*b1

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 7.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	\( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) (T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$5$	\( T^{12} - T^{11} + \cdots + 9409 \) T^12 - T^11 + 21T^10 - 20T^9 + 272T^8 + 287T^7 + 1450T^6 - 1715T^5 + 671T^4 + 1247T^3 - 1764T^2 + 2134T + 9409
$7$	\( T^{12} - 2 T^{11} + \cdots + 3136 \) T^12 - 2T^11 - 10T^10 + 69T^9 + 212T^8 - 1103T^7 + 253T^6 + 3220T^5 + 30408T^4 - 9520T^3 + 74480T^2 + 6272*T + 3136
$11$	\( T^{12} + 9 T^{11} + \cdots + 53824 \) T^12 + 9T^11 + 35T^10 + 27T^9 - 247T^8 - 931T^7 + 3025T^6 + 24136T^5 + 85508T^4 + 97040T^3 + 101472T^2 + 46400*T + 53824
$13$	\( T^{12} + 16 T^{11} + \cdots + 1000000 \) T^12 + 16T^11 + 196T^10 + 1805T^9 + 13354T^8 + 78883T^7 + 367109T^6 + 1290380T^5 + 3250600T^4 + 5419000T^3 + 5600000T^2 + 3300000*T + 1000000
$17$	\( (T^{6} - 3 T^{5} - 50 T^{4} + \cdots + 8)^{2} \) (T^6 - 3T^5 - 50T^4 + 133T^3 - 8T^2 - 52*T + 8)^2
$19$	\( T^{12} - 18 T^{11} + \cdots + 4096 \) T^12 - 18T^11 + 188T^10 - 1216T^9 + 5504T^8 - 15904T^7 + 91904T^6 - 445312T^5 + 2121984T^4 + 5600768T^3 + 3868672T^2 + 61440*T + 4096
$23$	\( (T^{6} + 28 T^{4} + \cdots + 3136)^{2} \) (T^6 + 28T^4 + 112T^3 + 3136)^2
$29$	\( (T^{6} + 13 T^{5} + \cdots + 24389)^{2} \) (T^6 + 13T^5 + 113T^4 + 755T^3 + 3277T^2 + 10933*T + 24389)^2
$31$	\( T^{12} - 8 T^{11} + \cdots + 3182656 \) T^12 - 8T^11 + 90T^10 - 1061T^9 + 5326T^8 + 1843T^7 + 101143T^6 + 87602T^5 + 1128536T^4 + 340056T^3 + 168992T^2 + 2754496*T + 3182656
$37$	\( T^{12} + 14 T^{11} + \cdots + 11999296 \) T^12 + 14T^11 + 8T^10 - 959T^9 - 3044T^8 + 48027T^7 + 662775T^6 + 3599050T^5 + 13057528T^4 + 26912088T^3 + 41751200T^2 - 2521792*T + 11999296
$41$	\( (T^{6} - 19 T^{5} + \cdots - 13208)^{2} \) (T^6 - 19T^5 + 68T^4 + 637T^3 - 5604T^2 + 14988*T - 13208)^2
$43$	\( T^{12} - 26 T^{11} + \cdots + 200704 \) T^12 - 26T^11 + 340T^10 - 2736T^9 + 14464T^8 - 34912T^7 + 2304T^6 + 425600T^5 + 1198848T^4 + 1788416T^3 + 1906688T^2 + 802816*T + 200704
$47$	\( T^{12} + 24 T^{11} + \cdots + 4096 \) T^12 + 24T^11 + 384T^10 + 3880T^9 + 23984T^8 + 77280T^7 + 105344T^6 + 18816T^5 + 2073600T^4 + 916480T^3 + 263168T^2 + 47104*T + 4096
$53$	\( T^{12} + \cdots + 537451489 \) T^12 + 23T^11 + 381T^10 + 4099T^9 + 41676T^8 + 400252T^7 + 3902955T^6 + 31458888T^5 + 167624476T^4 + 497770634T^3 + 800500578T^2 + 711231257*T + 537451489
$59$	\( (T^{6} - 5 T^{5} + \cdots + 17576)^{2} \) (T^6 - 5T^5 - 146T^4 + 1075T^3 + 598T^2 - 13520*T + 17576)^2
$61$	\( T^{12} - 14 T^{11} + \cdots + 1827904 \) T^12 - 14T^11 + 226T^10 - 3017T^9 + 26002T^8 - 148827T^7 + 607951T^6 - 1823374T^5 + 4071692T^4 - 6596072T^3 + 7350032T^2 - 5110560*T + 1827904
$67$	\( (T^{6} - 4 T^{5} + \cdots + 53824)^{2} \) (T^6 - 4T^5 + 72T^4 + 160T^3 - 80T^2 + 9280*T + 53824)^2
$71$	\( T^{12} + \cdots + 14109638656 \) T^12 + 32T^11 + 672T^10 + 11080T^9 + 141536T^8 + 1401568T^7 + 11710784T^6 + 63103488T^5 + 248262656T^4 + 1904705536T^3 + 13452443648T^2 + 17576230912*T + 14109638656
$73$	\( T^{12} + \cdots + 5545334089 \) T^12 - 31T^11 + 453T^10 - 5267T^9 + 73860T^8 - 937934T^7 + 9356655T^6 - 77055456T^5 + 570414172T^4 - 3222185620T^3 + 10336944498T^2 - 4986384787*T + 5545334089
$79$	\( T^{12} - 10 T^{11} + \cdots + 12845056 \) T^12 - 10T^11 + 212T^10 - 2568T^9 + 20304T^8 - 106272T^7 + 366528T^6 + 976640T^5 + 2214912T^4 + 258048T^3 + 5619712T^2 + 16056320*T + 12845056
$83$	\( T^{12} + \cdots + 14482678336 \) T^12 + 210T^10 - 1568T^9 + 7987T^8 - 72128T^7 + 662921T^6 + 5682138T^5 + 13312516T^4 + 33095384T^3 + 1048411056T^2 + 5590219488T + 14482678336
$89$	\( T^{12} + \cdots + 2562789376 \) T^12 - 20T^11 + 274T^10 - 3975T^9 + 42624T^8 - 192233T^7 + 1369845T^6 - 3612028T^5 + 10324048T^4 + 58120384T^3 - 43740928T^2 - 96388096*T + 2562789376
$97$	\( T^{12} + \cdots + 8851234561 \) T^12 - 23T^11 + 385T^10 - 4927T^9 + 51476T^8 - 578662T^7 + 6981507T^6 - 51973600T^5 + 306535568T^4 - 1738335892T^3 + 6963378646T^2 - 10258874483*T + 8851234561
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\(n\)	\(31\)	\(59\)
\(\chi(n)\)	\(\beta_{4}\)	\(1\)