# Properties

 Label 325.10.a.a Level $325$ Weight $10$ Character orbit 325.a Self dual yes Analytic conductor $167.387$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,10,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 325.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: $$167.386646753$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 1602x^{2} + 1544x + 342272$$ x^4 - x^3 - 1602*x^2 + 1544*x + 342272 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 8) q^{2} + (\beta_{3} - 2 \beta_1 + 41) q^{3} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + ( - 12 \beta_{3} + 14 \beta_{2} + \cdots - 1152) q^{6}+ \cdots + (125 \beta_{3} - 33 \beta_{2} + \cdots - 7400) q^{9}+O(q^{10})$$ q + (b1 + 8) * q^2 + (b3 - 2*b1 + 41) * q^3 + (6*b3 + b2 + 15*b1 + 352) * q^4 + (-12*b3 + 14*b2 + 91*b1 - 1152) * q^6 + (3*b3 - 50*b2 - 126*b1 + 2841) * q^7 + (110*b3 + 129*b2 + 345*b1 + 11272) * q^8 + (125*b3 - 33*b2 - 478*b1 - 7400) * q^9 $$q + (\beta_1 + 8) q^{2} + (\beta_{3} - 2 \beta_1 + 41) q^{3} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + ( - 12 \beta_{3} + 14 \beta_{2} + \cdots - 1152) q^{6}+ \cdots + ( - 3141970 \beta_{3} + \cdots + 536706688) q^{99}+O(q^{100})$$ q + (b1 + 8) * q^2 + (b3 - 2*b1 + 41) * q^3 + (6*b3 + b2 + 15*b1 + 352) * q^4 + (-12*b3 + 14*b2 + 91*b1 - 1152) * q^6 + (3*b3 - 50*b2 - 126*b1 + 2841) * q^7 + (110*b3 + 129*b2 + 345*b1 + 11272) * q^8 + (125*b3 - 33*b2 - 478*b1 - 7400) * q^9 + (436*b3 + 258*b2 - 344*b1 - 10058) * q^11 + (314*b3 + 151*b2 - 35*b1 + 38800) * q^12 + 28561 * q^13 + (-1756*b3 - 978*b2 + 1351*b1 - 69312) * q^14 + (1578*b3 + 3915*b2 + 15111*b1 + 177480) * q^16 + (885*b3 + 2751*b2 + 8794*b1 - 22099) * q^17 + (-3528*b3 + 928*b2 - 3274*b1 - 421056) * q^18 + (-432*b3 + 1906*b2 - 7944*b1 + 54510) * q^19 + (5593*b3 - 329*b2 - 27510*b1 + 290087) * q^21 + (3096*b3 + 11276*b2 + 19566*b1 - 346688) * q^22 + (14740*b3 - 12324*b2 + 6400*b1 + 1059076) * q^23 + (8954*b3 + 539*b2 + 14475*b1 + 884536) * q^24 + (28561*b1 + 228488) * q^26 + (-11957*b3 - 6360*b2 - 24878*b1 + 528659) * q^27 + (-12990*b3 - 18749*b2 - 123375*b1 - 974704) * q^28 + (33728*b3 - 13524*b2 - 29032*b1 - 413158) * q^29 + (-23982*b3 - 29782*b2 + 57276*b1 - 2849824) * q^31 + (112646*b3 + 44781*b2 + 270249*b1 + 7269016) * q^32 + (15490*b3 - 14478*b2 - 55796*b1 + 3613382) * q^33 + (107784*b3 + 72472*b2 + 140115*b1 + 6502440) * q^34 + (-65084*b3 - 26122*b2 - 410182*b1 - 2778112) * q^36 + (-1317*b3 + 39901*b2 - 278562*b1 - 1089253) * q^37 + (-9544*b3 + 19452*b2 + 1750*b1 - 6291168) * q^38 + (28561*b3 - 57122*b1 + 1171001) * q^39 + (51198*b3 + 10134*b2 + 230852*b1 + 3394372) * q^41 + (-171640*b3 + 56056*b2 + 450205*b1 - 18960872) * q^42 + (226419*b3 - 42452*b2 - 316254*b1 + 8364479) * q^43 + (119684*b3 + 139974*b2 + 344962*b1 + 16506144) * q^44 + (-208080*b3 + 20408*b2 + 1850052*b1 + 17431840) * q^46 + (-91721*b3 + 263046*b2 + 637746*b1 + 849245) * q^47 + (-63138*b3 + 90129*b2 + 1585461*b1 - 225384) * q^48 + (178833*b3 - 274077*b2 - 275478*b1 + 5847892) * q^49 + (-164541*b3 + 38912*b2 + 1253242*b1 - 5188293) * q^51 + (171366*b3 + 28561*b2 + 428415*b1 + 10053472) * q^52 + (-192670*b3 - 171078*b2 + 1241756*b1 + 42492560) * q^53 + (-276468*b3 - 330670*b2 - 512495*b1 - 16039488) * q^54 + (-216158*b3 - 167961*b2 - 3661385*b1 - 69418856) * q^56 + (190570*b3 - 164330*b2 - 583068*b1 + 12055382) * q^57 + (-444672*b3 + 267184*b2 + 1325826*b1 - 20211472) * q^58 + (-65820*b3 - 773310*b2 + 1469912*b1 - 16198370) * q^59 + (682194*b3 + 7562*b2 - 1308180*b1 + 19419044) * q^61 + (-251984*b3 - 862512*b2 - 4460252*b1 + 25147744) * q^62 + (1027988*b3 + 580586*b2 - 2112936*b1 + 39195106) * q^63 + (1709178*b3 + 874163*b2 + 9349767*b1 + 189475880) * q^64 + (-624336*b3 - 68560*b2 + 3982522*b1 - 11438640) * q^66 + (1620456*b3 - 256850*b2 - 810648*b1 + 10091066) * q^67 + (1837010*b3 + 1760643*b2 + 11038445*b1 + 176184992) * q^68 + (1604284*b3 + 373296*b2 - 6038576*b1 + 137769212) * q^69 + (1020287*b3 + 380430*b2 - 5253814*b1 + 64173629) * q^71 + (-1177196*b3 - 2396858*b2 - 8556426*b1 - 138211408) * q^72 + (691596*b3 - 305096*b2 + 6481200*b1 - 150396966) * q^73 + (-873352*b3 + 418584*b2 - 2485059*b1 - 238425032) * q^74 + (620724*b3 - 776690*b2 - 2511174*b1 - 81251680) * q^76 + (1722958*b3 + 2013870*b2 - 9461340*b1 - 138553018) * q^77 + (-342732*b3 + 399854*b2 + 2599051*b1 - 32902272) * q^78 + (2407128*b3 + 905092*b2 - 525888*b1 + 28529436) * q^79 + (-1947544*b3 + 545472*b2 + 6873224*b1 + 108219505) * q^81 + (1587792*b3 + 1232432*b2 + 8449152*b1 + 216277824) * q^82 + (-234878*b3 - 1501794*b2 - 10966732*b1 + 22694868) * q^83 + (958734*b3 - 1118579*b2 - 11812381*b1 + 29938272) * q^84 + (-2746564*b3 + 2542314*b2 + 19962285*b1 - 151785152) * q^86 + (1678562*b3 - 182892*b2 - 10528348*b1 + 274094074) * q^87 + (3284100*b3 - 993874*b2 + 18802446*b1 + 576369456) * q^88 + (3984044*b3 + 2129748*b2 + 21923976*b1 - 294537074) * q^89 + (85683*b3 - 1428050*b2 - 3598686*b1 + 81141801) * q^91 + (3961592*b3 + 5198004*b2 + 14114812*b1 + 1048851264) * q^92 + (-4990988*b3 + 1785452*b2 + 11546912*b1 - 406205620) * q^93 + (9087396*b3 + 3905038*b2 + 3652059*b1 + 461792512) * q^94 + (6730898*b3 + 1921607*b2 + 862875*b1 + 790965064) * q^96 + (4723764*b3 - 5763088*b2 + 3356064*b1 - 264294478) * q^97 + (-7134408*b3 - 2347536*b2 + 10979626*b1 - 106094368) * q^98 + (-3141970*b3 - 5516946*b2 - 11503732*b1 + 536706688) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 33 q^{2} + 163 q^{3} + 1429 q^{4} - 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9}+O(q^{10})$$ 4 * q + 33 * q^2 + 163 * q^3 + 1429 * q^4 - 4529 * q^6 + 11241 * q^7 + 45543 * q^8 - 29953 * q^9 $$4 q + 33 q^{2} + 163 q^{3} + 1429 q^{4} - 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9} - 40140 q^{11} + 155479 q^{12} + 114244 q^{13} - 277653 q^{14} + 726609 q^{16} - 78717 q^{17} - 1691026 q^{18} + 209664 q^{19} + 1138431 q^{21} - 1364090 q^{22} + 4257444 q^{23} + 3561573 q^{24} + 942513 q^{26} + 2077801 q^{27} - 4035181 q^{28} - 1647936 q^{29} - 11366002 q^{31} + 29458959 q^{32} + 14413222 q^{33} + 26257659 q^{34} - 11587714 q^{36} - 4636891 q^{37} - 25172466 q^{38} + 4655443 q^{39} + 13859538 q^{41} - 75564923 q^{42} + 33368081 q^{43} + 66489222 q^{44} + 71369332 q^{46} + 3943005 q^{47} + 620787 q^{48} + 23294923 q^{49} - 19664471 q^{51} + 40813669 q^{52} + 171019326 q^{53} - 64946915 q^{54} - 281552967 q^{56} + 47829030 q^{57} - 79964734 q^{58} - 63389388 q^{59} + 77050190 q^{61} + 95878740 q^{62} + 155695476 q^{63} + 768962465 q^{64} - 42396374 q^{66} + 41174072 q^{67} + 717615423 q^{68} + 546642556 q^{69} + 252460989 q^{71} - 562579254 q^{72} - 594415068 q^{73} - 957058539 q^{74} - 326897170 q^{76} - 561950454 q^{77} - 129352769 q^{78} + 115998984 q^{79} + 437803700 q^{81} + 875148240 q^{82} + 79577862 q^{83} + 108899441 q^{84} - 589924887 q^{86} + 1087526510 q^{87} + 2327564370 q^{88} - 1152240276 q^{89} + 321054201 q^{91} + 4213481460 q^{92} - 1618266556 q^{93} + 1859909503 q^{94} + 3171454029 q^{96} - 1049098084 q^{97} - 420532254 q^{98} + 2132181050 q^{99}+O(q^{100})$$ 4 * q + 33 * q^2 + 163 * q^3 + 1429 * q^4 - 4529 * q^6 + 11241 * q^7 + 45543 * q^8 - 29953 * q^9 - 40140 * q^11 + 155479 * q^12 + 114244 * q^13 - 277653 * q^14 + 726609 * q^16 - 78717 * q^17 - 1691026 * q^18 + 209664 * q^19 + 1138431 * q^21 - 1364090 * q^22 + 4257444 * q^23 + 3561573 * q^24 + 942513 * q^26 + 2077801 * q^27 - 4035181 * q^28 - 1647936 * q^29 - 11366002 * q^31 + 29458959 * q^32 + 14413222 * q^33 + 26257659 * q^34 - 11587714 * q^36 - 4636891 * q^37 - 25172466 * q^38 + 4655443 * q^39 + 13859538 * q^41 - 75564923 * q^42 + 33368081 * q^43 + 66489222 * q^44 + 71369332 * q^46 + 3943005 * q^47 + 620787 * q^48 + 23294923 * q^49 - 19664471 * q^51 + 40813669 * q^52 + 171019326 * q^53 - 64946915 * q^54 - 281552967 * q^56 + 47829030 * q^57 - 79964734 * q^58 - 63389388 * q^59 + 77050190 * q^61 + 95878740 * q^62 + 155695476 * q^63 + 768962465 * q^64 - 42396374 * q^66 + 41174072 * q^67 + 717615423 * q^68 + 546642556 * q^69 + 252460989 * q^71 - 562579254 * q^72 - 594415068 * q^73 - 957058539 * q^74 - 326897170 * q^76 - 561950454 * q^77 - 129352769 * q^78 + 115998984 * q^79 + 437803700 * q^81 + 875148240 * q^82 + 79577862 * q^83 + 108899441 * q^84 - 589924887 * q^86 + 1087526510 * q^87 + 2327564370 * q^88 - 1152240276 * q^89 + 321054201 * q^91 + 4213481460 * q^92 - 1618266556 * q^93 + 1859909503 * q^94 + 3171454029 * q^96 - 1049098084 * q^97 - 420532254 * q^98 + 2132181050 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 1602x^{2} + 1544x + 342272$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 3\nu^{3} + 17\nu^{2} - 3586\nu - 12856 ) / 332$$ (3*v^3 + 17*v^2 - 3586*v - 12856) / 332 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 105\nu^{2} + 1306\nu - 84248 ) / 664$$ (-v^3 + 105*v^2 + 1306*v - 84248) / 664
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$6\beta_{3} + \beta_{2} - \beta _1 + 800$$ 6*b3 + b2 - b1 + 800 $$\nu^{3}$$ $$=$$ $$-34\beta_{3} + 105\beta_{2} + 1201\beta _1 - 248$$ -34*b3 + 105*b2 + 1201*b1 - 248

## 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
 −36.8028 −15.3567 16.5360 36.6235
−28.8028 204.594 317.603 0 −5892.88 8862.28 5599.18 22175.6 0
1.2 −7.35673 −42.6243 −457.879 0 313.575 −892.010 7135.13 −17866.2 0
1.3 24.5360 −49.9972 90.0171 0 −1226.73 8707.31 −10353.8 −17183.3 0
1.4 44.6235 51.0278 1479.26 0 2277.04 −5436.58 43162.5 −17079.2 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$13$$ $$-1$$

## 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 325.10.a.a 4
5.b even 2 1 13.10.a.a 4
15.d odd 2 1 117.10.a.c 4
20.d odd 2 1 208.10.a.g 4
65.d even 2 1 169.10.a.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.a.a 4 5.b even 2 1
117.10.a.c 4 15.d odd 2 1
169.10.a.a 4 65.d even 2 1
208.10.a.g 4 20.d odd 2 1
325.10.a.a 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 33T_{2}^{3} - 1194T_{2}^{2} + 24936T_{2} + 232000$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(325))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 33 T^{3} + \cdots + 232000$$
$3$ $$T^{4} - 163 T^{3} + \cdots + 22248576$$
$5$ $$T^{4}$$
$7$ $$T^{4} + \cdots + 374218195104754$$
$11$ $$T^{4} + \cdots + 27\!\cdots\!36$$
$13$ $$(T - 28561)^{4}$$
$17$ $$T^{4} + \cdots + 25\!\cdots\!18$$
$19$ $$T^{4} + \cdots + 50\!\cdots\!08$$
$23$ $$T^{4} + \cdots - 17\!\cdots\!36$$
$29$ $$T^{4} + \cdots + 37\!\cdots\!32$$
$31$ $$T^{4} + \cdots - 58\!\cdots\!60$$
$37$ $$T^{4} + \cdots + 61\!\cdots\!62$$
$41$ $$T^{4} + \cdots + 15\!\cdots\!52$$
$43$ $$T^{4} + \cdots + 26\!\cdots\!40$$
$47$ $$T^{4} + \cdots + 10\!\cdots\!50$$
$53$ $$T^{4} + \cdots - 27\!\cdots\!48$$
$59$ $$T^{4} + \cdots - 26\!\cdots\!68$$
$61$ $$T^{4} + \cdots + 60\!\cdots\!72$$
$67$ $$T^{4} + \cdots + 19\!\cdots\!64$$
$71$ $$T^{4} + \cdots + 45\!\cdots\!54$$
$73$ $$T^{4} + \cdots - 47\!\cdots\!72$$
$79$ $$T^{4} + \cdots + 25\!\cdots\!64$$
$83$ $$T^{4} + \cdots - 12\!\cdots\!88$$
$89$ $$T^{4} + \cdots - 34\!\cdots\!00$$
$97$ $$T^{4} + \cdots + 21\!\cdots\!60$$
show more
show less