Properties

Label 45.14.a.c
Level $45$
Weight $14$
Character orbit 45.a
Self dual yes
Analytic conductor $48.254$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(48.2539180284\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1609}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 402 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 15)
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 \(\beta = 3\sqrt{1609}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 7) q^{2} + (14 \beta + 6338) q^{4} + 15625 q^{5} + ( - 896 \beta - 14916) q^{7} + (1756 \beta - 189756) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 7) q^{2} + (14 \beta + 6338) q^{4} + 15625 q^{5} + ( - 896 \beta - 14916) q^{7} + (1756 \beta - 189756) q^{8} + ( - 15625 \beta - 109375) q^{10} + (57472 \beta - 1908664) q^{11} + ( - 99584 \beta - 18150062) q^{13} + (21188 \beta + 13079388) q^{14} + (62776 \beta - 76021240) q^{16} + ( - 385280 \beta - 28124038) q^{17} + (1164928 \beta + 34218172) q^{19} + (218750 \beta + 99031250) q^{20} + (1506360 \beta - 818891384) q^{22} + (92288 \beta + 909341688) q^{23} + 244140625 q^{25} + (18847150 \beta + 1569126338) q^{26} + ( - 5887672 \beta - 276187272) q^{28} + (18031104 \beta - 2952305354) q^{29} + (31927296 \beta + 685304136) q^{31} + (61196656 \beta + 1177570576) q^{32} + (30820998 \beta + 5776107946) q^{34} + ( - 14000000 \beta - 233062500) q^{35} + (48481024 \beta - 5652579406) q^{37} + ( - 42372668 \beta - 17108849572) q^{38} + (27437500 \beta - 2964937500) q^{40} + (164849664 \beta - 22332431546) q^{41} + (385691904 \beta + 5811180340) q^{43} + (337536240 \beta - 445583984) q^{44} + ( - 909987704 \beta - 7701814344) q^{46} + (108399232 \beta + 4540641080) q^{47} + (26729472 \beta - 85040944855) q^{49} + ( - 244140625 \beta - 1708984375) q^{50} + ( - 885264260 \beta - 135224155612) q^{52} + ( - 2379215872 \beta + 29873144918) q^{53} + (898000000 \beta - 29822875000) q^{55} + (143828880 \beta - 19953657360) q^{56} + (2826087626 \beta - 240442279546) q^{58} + ( - 1275367552 \beta + 218785088312) q^{59} + ( - 2672183808 \beta - 244050577850) q^{61} + ( - 908795208 \beta - 467136302328) q^{62} + ( - 2120208160 \beta - 271665771488) q^{64} + ( - 1556000000 \beta - 283594718750) q^{65} + ( - 4433811200 \beta - 646020821772) q^{67} + ( - 2835641172 \beta - 256359508364) q^{68} + (331062500 \beta + 204365437500) q^{70} + (8211842560 \beta - 432731086112) q^{71} + ( - 4444331008 \beta - 723819345278) q^{73} + (5313212238 \beta - 662485652702) q^{74} + (7862368072 \beta + 453045287288) q^{76} + (852910592 \beta - 717228188448) q^{77} + (15735834880 \beta - 311846858760) q^{79} + (980875000 \beta - 1187831875000) q^{80} + (21178483898 \beta - 2230860963562) q^{82} + (17069371392 \beta + 1551237122604) q^{83} + ( - 6020000000 \beta - 439438093750) q^{85} + ( - 8511023668 \beta - 5625882724204) q^{86} + ( - 14257270816 \beta + 1823615014176) q^{88} + (13552852992 \beta + 7588798268718) q^{89} + (17747850496 \beta + 1562826334776) q^{91} + (13315704976 \beta + 5782117533936) q^{92} + ( - 5299435704 \beta - 1601513766152) q^{94} + (18202000000 \beta + 534658937500) q^{95} + ( - 36405337600 \beta - 11593467261022) q^{97} + (84853838551 \beta + 208217129953) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{2} + 12676 q^{4} + 31250 q^{5} - 29832 q^{7} - 379512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{2} + 12676 q^{4} + 31250 q^{5} - 29832 q^{7} - 379512 q^{8} - 218750 q^{10} - 3817328 q^{11} - 36300124 q^{13} + 26158776 q^{14} - 152042480 q^{16} - 56248076 q^{17} + 68436344 q^{19} + 198062500 q^{20} - 1637782768 q^{22} + 1818683376 q^{23} + 488281250 q^{25} + 3138252676 q^{26} - 552374544 q^{28} - 5904610708 q^{29} + 1370608272 q^{31} + 2355141152 q^{32} + 11552215892 q^{34} - 466125000 q^{35} - 11305158812 q^{37} - 34217699144 q^{38} - 5929875000 q^{40} - 44664863092 q^{41} + 11622360680 q^{43} - 891167968 q^{44} - 15403628688 q^{46} + 9081282160 q^{47} - 170081889710 q^{49} - 3417968750 q^{50} - 270448311224 q^{52} + 59746289836 q^{53} - 59645750000 q^{55} - 39907314720 q^{56} - 480884559092 q^{58} + 437570176624 q^{59} - 488101155700 q^{61} - 934272604656 q^{62} - 543331542976 q^{64} - 567189437500 q^{65} - 1292041643544 q^{67} - 512719016728 q^{68} + 408730875000 q^{70} - 865462172224 q^{71} - 1447638690556 q^{73} - 1324971305404 q^{74} + 906090574576 q^{76} - 1434456376896 q^{77} - 623693717520 q^{79} - 2375663750000 q^{80} - 4461721927124 q^{82} + 3102474245208 q^{83} - 878876187500 q^{85} - 11251765448408 q^{86} + 3647230028352 q^{88} + 15177596537436 q^{89} + 3125652669552 q^{91} + 11564235067872 q^{92} - 3203027532304 q^{94} + 1069317875000 q^{95} - 23186934522044 q^{97} + 416434259906 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
20.5562
−19.5562
−127.337 0 8022.72 15625.0 0 −122738. 21555.8 0 −1.98964e6
1.2 113.337 0 4653.28 15625.0 0 92906.0 −401068. 0 1.77089e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)

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 45.14.a.c 2
3.b odd 2 1 15.14.a.b 2
15.d odd 2 1 75.14.a.d 2
15.e even 4 2 75.14.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.14.a.b 2 3.b odd 2 1
45.14.a.c 2 1.a even 1 1 trivial
75.14.a.d 2 15.d odd 2 1
75.14.b.d 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 14T_{2} - 14432 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 14T - 14432 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 15625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 11403091440 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 44188190518208 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 185817063779908 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 13\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 18\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 82\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 40\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 20\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 10\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 21\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 14\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 81\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 24\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 43\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 78\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 18\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 54\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
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