[N,k,chi] = [45,22,Mod(1,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
p p p
Sign
3 3 3
− 1 -1 − 1
5 5 5
+ 1 +1 + 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 2 4 − 897 T 2 3 − 3210438 T 2 2 + 1891862624 T 2 + 2065963121664 T_{2}^{4} - 897T_{2}^{3} - 3210438T_{2}^{2} + 1891862624T_{2} + 2065963121664 T 2 4 − 8 9 7 T 2 3 − 3 2 1 0 4 3 8 T 2 2 + 1 8 9 1 8 6 2 6 2 4 T 2 + 2 0 6 5 9 6 3 1 2 1 6 6 4
T2^4 - 897*T2^3 - 3210438*T2^2 + 1891862624*T2 + 2065963121664
acting on S 22 n e w ( Γ 0 ( 45 ) ) S_{22}^{\mathrm{new}}(\Gamma_0(45)) S 2 2 n e w ( Γ 0 ( 4 5 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 + ⋯ + 2065963121664 T^{4} + \cdots + 2065963121664 T 4 + ⋯ + 2 0 6 5 9 6 3 1 2 1 6 6 4
T^4 - 897*T^3 - 3210438*T^2 + 1891862624*T + 2065963121664
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
( T + 9765625 ) 4 (T + 9765625)^{4} ( T + 9 7 6 5 6 2 5 ) 4
(T + 9765625)^4
7 7 7
T 4 + ⋯ + 38 ⋯ 00 T^{4} + \cdots + 38\!\cdots\!00 T 4 + ⋯ + 3 8 ⋯ 0 0
T^4 + 234577504*T^3 - 1862168801977510656*T^2 - 396328883227766325337374720*T + 383743482120644300925415661666304000
11 11 1 1
T 4 + ⋯ + 28 ⋯ 64 T^{4} + \cdots + 28\!\cdots\!64 T 4 + ⋯ + 2 8 ⋯ 6 4
T^4 + 31491830256*T^3 - 19312068120074551638432*T^2 - 135828530855307080234681438933248*T + 28381277872317888219670139675616823821099264
13 13 1 3
T 4 + ⋯ + 26 ⋯ 84 T^{4} + \cdots + 26\!\cdots\!84 T 4 + ⋯ + 2 6 ⋯ 8 4
T^4 + 27017977768*T^3 - 1108036936737950777531688*T^2 + 36800716070593616924938915126169504*T + 268873426021349807386537001480915737460523405584
17 17 1 7
T 4 + ⋯ + 31 ⋯ 56 T^{4} + \cdots + 31\!\cdots\!56 T 4 + ⋯ + 3 1 ⋯ 5 6
T^4 - 2946095028888*T^3 - 161949279185939054113955496*T^2 - 134270127188204789858228137516735511392*T + 3126117540636703998211126081393748221064885507075856
19 19 1 9
T 4 + ⋯ + 34 ⋯ 00 T^{4} + \cdots + 34\!\cdots\!00 T 4 + ⋯ + 3 4 ⋯ 0 0
T^4 + 24270353300752*T^3 - 1122016742390773364602642464*T^2 - 26784438194108588296787353606124744323840*T + 34500498078945485655609309571828525112069536990726400
23 23 2 3
T 4 + ⋯ − 97 ⋯ 00 T^{4} + \cdots - 97\!\cdots\!00 T 4 + ⋯ − 9 7 ⋯ 0 0
T^4 + 10350924920928*T^3 - 87375522042917234112129801984*T^2 + 5857681076185688239010378768097020725985280*T - 97882467896375776778600843562402968945696606150590464000
29 29 2 9
T 4 + ⋯ − 51 ⋯ 00 T^{4} + \cdots - 51\!\cdots\!00 T 4 + ⋯ − 5 1 ⋯ 0 0
T^4 + 4728924677079096*T^3 - 8615954128680003089181792109416*T^2 - 55419366035739131036274669582211845253426230560*T - 51698111869114919407013904642078816008357499442382589806742000
31 31 3 1
T 4 + ⋯ + 15 ⋯ 00 T^{4} + \cdots + 15\!\cdots\!00 T 4 + ⋯ + 1 5 ⋯ 0 0
T^4 + 1094923910405536*T^3 - 15156177263813620949946948216576*T^2 - 23497228991071117292317787916848512360253030400*T + 15093410036755809776644877250045788748316048818091917312000000
37 37 3 7
T 4 + ⋯ + 50 ⋯ 00 T^{4} + \cdots + 50\!\cdots\!00 T 4 + ⋯ + 5 0 ⋯ 0 0
T^4 - 4813435696247096*T^3 - 1705834780620370056883044361244136*T^2 + 8983718392327521355605213392645332306489573484320*T + 50902738028276392754835826318461524106289535057504308221626256400
41 41 4 1
T 4 + ⋯ − 70 ⋯ 00 T^{4} + \cdots - 70\!\cdots\!00 T 4 + ⋯ − 7 0 ⋯ 0 0
T^4 + 289731591445930344*T^3 + 20040100471128790395212720627129304*T^2 - 626717668457751634185028633974815287340950462916960*T - 70988148097466694306937224635755838526356528447560110781120580790000
43 43 4 3
T 4 + ⋯ − 16 ⋯ 04 T^{4} + \cdots - 16\!\cdots\!04 T 4 + ⋯ − 1 6 ⋯ 0 4
T^4 - 451091912658458000*T^3 + 53628003549075469769577221428204128*T^2 + 170723039919168078381360773034697421295228688889600*T - 168142645200656821462271400373249337797668046106286833261808276315904
47 47 4 7
T 4 + ⋯ − 21 ⋯ 44 T^{4} + \cdots - 21\!\cdots\!44 T 4 + ⋯ − 2 1 ⋯ 4 4
T^4 + 813883435638492480*T^3 + 70996736532884308065355717627189632*T^2 - 41469855076302193875730133323223232087146163091640320*T - 2138282441670918163149287458126602544875689169627082198304261357727744
53 53 5 3
T 4 + ⋯ + 34 ⋯ 00 T^{4} + \cdots + 34\!\cdots\!00 T 4 + ⋯ + 3 4 ⋯ 0 0
T^4 + 697278335404085208*T^3 - 1751504316097265064122009922806505384*T^2 - 1112410904110422387734151233417074967993524228690487200*T + 343049512911392603732127080919361151128523776416223802671027584756080400
59 59 5 9
T 4 + ⋯ + 97 ⋯ 00 T^{4} + \cdots + 97\!\cdots\!00 T 4 + ⋯ + 9 7 ⋯ 0 0
T^4 + 6622888614569598192*T^3 - 20793368156049157056622924058857704864*T^2 - 106209756365166093460538215589859685076290107897579700480*T + 97735591953825442742647737025190871064295690763197115617191783742714323200
61 61 6 1
T 4 + ⋯ + 27 ⋯ 76 T^{4} + \cdots + 27\!\cdots\!76 T 4 + ⋯ + 2 7 ⋯ 7 6
T^4 - 7390887218011683320*T^3 - 94130994221938032607154340302683356648*T^2 + 408657984562613119556592010625197705791750721801388802080*T + 2772748853140256623478100799493790410058598971685572177396824249132296284176
67 67 6 7
T 4 + ⋯ + 28 ⋯ 56 T^{4} + \cdots + 28\!\cdots\!56 T 4 + ⋯ + 2 8 ⋯ 5 6
T^4 + 24188188449376788688*T^3 - 229042213596636635406215328262853083296*T^2 - 3775330820854699506521328410729408975665271817224760945408*T + 28101773869313131338352518536798403955571804051674653865985795157035971481856
71 71 7 1
T 4 + ⋯ − 17 ⋯ 44 T^{4} + \cdots - 17\!\cdots\!44 T 4 + ⋯ − 1 7 ⋯ 4 4
T^4 - 37390337803999713312*T^3 - 1139463428933347426505646390231682375296*T^2 + 34885138177175458229586900690027461748786696708493950318592*T - 174155452652628671764487922142823029352266186382196214441059763328791533318144
73 73 7 3
T 4 + ⋯ − 99 ⋯ 00 T^{4} + \cdots - 99\!\cdots\!00 T 4 + ⋯ − 9 9 ⋯ 0 0
T^4 + 37253672904265201432*T^3 - 34326670508947858402927007441557140264*T^2 - 12178398494082618130220355106486656860561566991319385743520*T - 99224270188047384046262956583035411054218726917645555817471227781947804162800
79 79 7 9
T 4 + ⋯ − 12 ⋯ 00 T^{4} + \cdots - 12\!\cdots\!00 T 4 + ⋯ − 1 2 ⋯ 0 0
T^4 + 109751291042259570400*T^3 - 7989976050687852877434024218805467769600*T^2 - 979320490199289162418845200807165522419367125829420083200000*T - 12491659494476605469195544050095629894496474702724019536037320574822150144000000
83 83 8 3
T 4 + ⋯ − 79 ⋯ 56 T^{4} + \cdots - 79\!\cdots\!56 T 4 + ⋯ − 7 9 ⋯ 5 6
T^4 - 149023592088638482896*T^3 - 30149560645279164228205996073477325581472*T^2 + 3510562983549102883166340450603451397844516592423171290594048*T - 79906271600397952499194113081421659552297186737552301904332541122481503618445056
89 89 8 9
T 4 + ⋯ + 85 ⋯ 00 T^{4} + \cdots + 85\!\cdots\!00 T 4 + ⋯ + 8 5 ⋯ 0 0
T^4 + 558502237151273959848*T^3 + 88379878473011091764935528702921534539096*T^2 + 4988745067107050271415050446682710227600024951734344794507680*T + 85928687855151514423903558307575585638097404912023250992369418199007126089866000
97 97 9 7
T 4 + ⋯ − 16 ⋯ 24 T^{4} + \cdots - 16\!\cdots\!24 T 4 + ⋯ − 1 6 ⋯ 2 4
T^4 + 236244923748673777528*T^3 - 800778384912282913045222904652702838836456*T^2 + 236981001326529697888318286476571490238100529544085777828411872*T - 16986617770411665031361304084824596665919882978220925885493241279225356388816010224
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