# Properties

 Label 75.22.b.c Level $75$ Weight $22$ Character orbit 75.b Analytic conductor $209.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$209.608008215$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 544 i q^{2} - 59049 i q^{3} + 1801216 q^{4} + 32122656 q^{6} + 1277698380 i q^{7} + 2120712192 i q^{8} - 3486784401 q^{9} +O(q^{10})$$ q + 544*i * q^2 - 59049*i * q^3 + 1801216 * q^4 + 32122656 * q^6 + 1277698380*i * q^7 + 2120712192*i * q^8 - 3486784401 * q^9 $$q + 544 i q^{2} - 59049 i q^{3} + 1801216 q^{4} + 32122656 q^{6} + 1277698380 i q^{7} + 2120712192 i q^{8} - 3486784401 q^{9} - 77585921744 q^{11} - 106360003584 i q^{12} + 434110898702 i q^{13} - 695067918720 q^{14} + 2623756304384 q^{16} + 12848917115782 i q^{17} - 1896810714144 i q^{18} + 28605256159796 q^{19} + 75446811640620 q^{21} - 42206741428736 i q^{22} - 224022192208080 i q^{23} + 125225934225408 q^{24} - 236156328893888 q^{26} + 205891132094649 i q^{27} + 23\!\cdots\!80 i q^{28} + \cdots + 27\!\cdots\!44 q^{99} +O(q^{100})$$ q + 544*i * q^2 - 59049*i * q^3 + 1801216 * q^4 + 32122656 * q^6 + 1277698380*i * q^7 + 2120712192*i * q^8 - 3486784401 * q^9 - 77585921744 * q^11 - 106360003584*i * q^12 + 434110898702*i * q^13 - 695067918720 * q^14 + 2623756304384 * q^16 + 12848917115782*i * q^17 - 1896810714144*i * q^18 + 28605256159796 * q^19 + 75446811640620 * q^21 - 42206741428736*i * q^22 - 224022192208080*i * q^23 + 125225934225408 * q^24 - 236156328893888 * q^26 + 205891132094649*i * q^27 + 2301410765230080*i * q^28 + 51676030833142 * q^29 + 8921108838285000 * q^31 + 5874779244462080*i * q^32 + 4581371093061456*i * q^33 - 6989810910985408 * q^34 - 6280451851631616 * q^36 + 43977154002495890*i * q^37 + 15561259350929024*i * q^38 + 25633814457454398 * q^39 + 58168090830044570 * q^41 + 41043065532497280*i * q^42 + 161437862491900676*i * q^43 - 139749003620040704 * q^44 + 121868072561195520 * q^46 - 160064774442316592*i * q^47 - 154930186017570816*i * q^48 - 1073967286171340393 * q^49 + 758715706769811318 * q^51 + 781927496516421632*i * q^52 - 2299527285858152170*i * q^53 - 112004775859489056 * q^54 - 2709630532164648960 * q^56 - 1689111770979794004*i * q^57 + 28111760773229248*i * q^58 - 5154256088898000016 * q^59 + 1251686105775241798 * q^61 + 4853083208027040000*i * q^62 - 4455058780566970380*i * q^63 + 2306535872264142848 * q^64 - 2492265874625432064 * q^66 - 5407785329527117188*i * q^67 + 23143675091620390912*i * q^68 - 13228286427694915920 * q^69 - 11043230850518282368 * q^71 - 7394466190076116992*i * q^72 + 37701191520217147550*i * q^73 - 23923571777357764160 * q^74 + 51524245079123111936 * q^76 - 99131406523115574720*i * q^77 + 13944795064855192512*i * q^78 - 63155369968366862760 * q^79 + 12157665459056928801 * q^81 + 31643441411544246080*i * q^82 + 145158253921046761428*i * q^83 + 135896004276070993920 * q^84 - 87822197195593967744 * q^86 - 3051417944666201958*i * q^87 - 164537410170058702848*i * q^88 - 137255030734236350514 * q^89 - 554662792011889502760 * q^91 - 403512356960269025280*i * q^92 - 526782555791890965000*i * q^93 + 87075237296620226048 * q^94 + 346899839606241361920 * q^96 - 324306485265230829118*i * q^97 - 584238203677209173792*i * q^98 + 270525381674185915344 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3602432 q^{4} + 64245312 q^{6} - 6973568802 q^{9}+O(q^{10})$$ 2 * q + 3602432 * q^4 + 64245312 * q^6 - 6973568802 * q^9 $$2 q + 3602432 q^{4} + 64245312 q^{6} - 6973568802 q^{9} - 155171843488 q^{11} - 1390135837440 q^{14} + 5247512608768 q^{16} + 57210512319592 q^{19} + 150893623281240 q^{21} + 250451868450816 q^{24} - 472312657787776 q^{26} + 103352061666284 q^{29} + 17\!\cdots\!00 q^{31}+ \cdots + 54\!\cdots\!88 q^{99}+O(q^{100})$$ 2 * q + 3602432 * q^4 + 64245312 * q^6 - 6973568802 * q^9 - 155171843488 * q^11 - 1390135837440 * q^14 + 5247512608768 * q^16 + 57210512319592 * q^19 + 150893623281240 * q^21 + 250451868450816 * q^24 - 472312657787776 * q^26 + 103352061666284 * q^29 + 17842217676570000 * q^31 - 13979621821970816 * q^34 - 12560903703263232 * q^36 + 51267628914908796 * q^39 + 116336181660089140 * q^41 - 279498007240081408 * q^44 + 243736145122391040 * q^46 - 2147934572342680786 * q^49 + 1517431413539622636 * q^51 - 224009551718978112 * q^54 - 5419261064329297920 * q^56 - 10308512177796000032 * q^59 + 2503372211550483596 * q^61 + 4613071744528285696 * q^64 - 4984531749250864128 * q^66 - 26456572855389831840 * q^69 - 22086461701036564736 * q^71 - 47847143554715528320 * q^74 + 103048490158246223872 * q^76 - 126310739936733725520 * q^79 + 24315330918113857602 * q^81 + 271792008552141987840 * q^84 - 175644394391187935488 * q^86 - 274510061468472701028 * q^89 - 1109325584023779005520 * q^91 + 174150474593240452096 * q^94 + 693799679212482723840 * q^96 + 541050763348371830688 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
544.000i 59049.0i 1.80122e6 0 3.21227e7 1.27770e9i 2.12071e9i −3.48678e9 0
49.2 544.000i 59049.0i 1.80122e6 0 3.21227e7 1.27770e9i 2.12071e9i −3.48678e9 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.22.b.c 2
5.b even 2 1 inner 75.22.b.c 2
5.c odd 4 1 15.22.a.a 1
5.c odd 4 1 75.22.a.b 1
15.e even 4 1 45.22.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.a 1 5.c odd 4 1
45.22.a.a 1 15.e even 4 1
75.22.a.b 1 5.c odd 4 1
75.22.b.c 2 1.a even 1 1 trivial
75.22.b.c 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 295936$$ acting on $$S_{22}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 295936$$
$3$ $$T^{2} + 3486784401$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16\!\cdots\!00$$
$11$ $$(T + 77585921744)^{2}$$
$13$ $$T^{2} + 18\!\cdots\!04$$
$17$ $$T^{2} + 16\!\cdots\!24$$
$19$ $$(T - 28605256159796)^{2}$$
$23$ $$T^{2} + 50\!\cdots\!00$$
$29$ $$(T - 51676030833142)^{2}$$
$31$ $$(T - 89\!\cdots\!00)^{2}$$
$37$ $$T^{2} + 19\!\cdots\!00$$
$41$ $$(T - 58\!\cdots\!70)^{2}$$
$43$ $$T^{2} + 26\!\cdots\!76$$
$47$ $$T^{2} + 25\!\cdots\!64$$
$53$ $$T^{2} + 52\!\cdots\!00$$
$59$ $$(T + 51\!\cdots\!16)^{2}$$
$61$ $$(T - 12\!\cdots\!98)^{2}$$
$67$ $$T^{2} + 29\!\cdots\!44$$
$71$ $$(T + 11\!\cdots\!68)^{2}$$
$73$ $$T^{2} + 14\!\cdots\!00$$
$79$ $$(T + 63\!\cdots\!60)^{2}$$
$83$ $$T^{2} + 21\!\cdots\!84$$
$89$ $$(T + 13\!\cdots\!14)^{2}$$
$97$ $$T^{2} + 10\!\cdots\!24$$