Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1449,2,Mod(1,1449)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1449.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1449, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1449.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,0,3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.5703232529\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1449.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.381966 q^{2} -1.85410 q^{4} +1.38197 q^{5} +1.00000 q^{7} -1.47214 q^{8} +0.527864 q^{10} +5.47214 q^{11} -2.38197 q^{13} +0.381966 q^{14} +3.14590 q^{16} +1.00000 q^{17} -3.00000 q^{19} -2.56231 q^{20} +2.09017 q^{22} +1.00000 q^{23} -3.09017 q^{25} -0.909830 q^{26} -1.85410 q^{28} +7.47214 q^{29} -3.76393 q^{31} +4.14590 q^{32} +0.381966 q^{34} +1.38197 q^{35} +1.47214 q^{37} -1.14590 q^{38} -2.03444 q^{40} +4.70820 q^{41} +8.09017 q^{43} -10.1459 q^{44} +0.381966 q^{46} -1.70820 q^{47} +1.00000 q^{49} -1.18034 q^{50} +4.41641 q^{52} +3.38197 q^{53} +7.56231 q^{55} -1.47214 q^{56} +2.85410 q^{58} +6.14590 q^{59} +13.7984 q^{61} -1.43769 q^{62} -4.70820 q^{64} -3.29180 q^{65} +4.14590 q^{67} -1.85410 q^{68} +0.527864 q^{70} +3.90983 q^{71} +2.70820 q^{73} +0.562306 q^{74} +5.56231 q^{76} +5.47214 q^{77} +0.527864 q^{79} +4.34752 q^{80} +1.79837 q^{82} +3.00000 q^{83} +1.38197 q^{85} +3.09017 q^{86} -8.05573 q^{88} -3.14590 q^{89} -2.38197 q^{91} -1.85410 q^{92} -0.652476 q^{94} -4.14590 q^{95} +5.00000 q^{97} +0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} + 5 q^{5} + 2 q^{7} + 6 q^{8} + 10 q^{10} + 2 q^{11} - 7 q^{13} + 3 q^{14} + 13 q^{16} + 2 q^{17} - 6 q^{19} + 15 q^{20} - 7 q^{22} + 2 q^{23} + 5 q^{25} - 13 q^{26} + 3 q^{28} + 6 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.381966 0.270091 0.135045 − 0.990839i \(-0.456882\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) 0 0
\(4\) −1.85410 −0.927051
\(5\) 1.38197 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.47214 −0.520479
\(9\) 0 0
\(10\) 0.527864 0.166925
\(11\) 5.47214 1.64991 0.824956 − 0.565198i \(-0.191200\pi\)
0.824956 + 0.565198i \(0.191200\pi\)
\(12\) 0 0
\(13\) −2.38197 −0.660639 −0.330319 − 0.943869i \(-0.607156\pi\)
−0.330319 + 0.943869i \(0.607156\pi\)
\(14\) 0.381966 0.102085
\(15\) 0 0
\(16\) 3.14590 0.786475
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −2.56231 −0.572949
\(21\) 0 0
\(22\) 2.09017 0.445626
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.09017 −0.618034
\(26\) −0.909830 −0.178432
\(27\) 0 0
\(28\) −1.85410 −0.350392
\(29\) 7.47214 1.38754 0.693770 − 0.720196i \(-0.255948\pi\)
0.693770 + 0.720196i \(0.255948\pi\)
\(30\) 0 0
\(31\) −3.76393 −0.676022 −0.338011 − 0.941142i \(-0.609754\pi\)
−0.338011 + 0.941142i \(0.609754\pi\)
\(32\) 4.14590 0.732898
\(33\) 0 0
\(34\) 0.381966 0.0655066
\(35\) 1.38197 0.233595
\(36\) 0 0
\(37\) 1.47214 0.242018 0.121009 − 0.992651i \(-0.461387\pi\)
0.121009 + 0.992651i \(0.461387\pi\)
\(38\) −1.14590 −0.185889
\(39\) 0 0
\(40\) −2.03444 −0.321674
\(41\) 4.70820 0.735298 0.367649 − 0.929965i \(-0.380163\pi\)
0.367649 + 0.929965i \(0.380163\pi\)
\(42\) 0 0
\(43\) 8.09017 1.23374 0.616870 − 0.787065i \(-0.288401\pi\)
0.616870 + 0.787065i \(0.288401\pi\)
\(44\) −10.1459 −1.52955
\(45\) 0 0
\(46\) 0.381966 0.0563178
\(47\) −1.70820 −0.249167 −0.124584 − 0.992209i \(-0.539759\pi\)
−0.124584 + 0.992209i \(0.539759\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.18034 −0.166925
\(51\) 0 0
\(52\) 4.41641 0.612446
\(53\) 3.38197 0.464549 0.232274 − 0.972650i \(-0.425383\pi\)
0.232274 + 0.972650i \(0.425383\pi\)
\(54\) 0 0
\(55\) 7.56231 1.01970
\(56\) −1.47214 −0.196722
\(57\) 0 0
\(58\) 2.85410 0.374762
\(59\) 6.14590 0.800128 0.400064 − 0.916487i \(-0.368988\pi\)
0.400064 + 0.916487i \(0.368988\pi\)
\(60\) 0 0
\(61\) 13.7984 1.76670 0.883350 − 0.468713i \(-0.155282\pi\)
0.883350 + 0.468713i \(0.155282\pi\)
\(62\) −1.43769 −0.182587
\(63\) 0 0
\(64\) −4.70820 −0.588525
\(65\) −3.29180 −0.408297
\(66\) 0 0
\(67\) 4.14590 0.506502 0.253251 − 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(68\) −1.85410 −0.224843
\(69\) 0 0
\(70\) 0.527864 0.0630918
\(71\) 3.90983 0.464011 0.232006 − 0.972714i \(-0.425471\pi\)
0.232006 + 0.972714i \(0.425471\pi\)
\(72\) 0 0
\(73\) 2.70820 0.316971 0.158486 − 0.987361i \(-0.449339\pi\)
0.158486 + 0.987361i \(0.449339\pi\)
\(74\) 0.562306 0.0653667
\(75\) 0 0
\(76\) 5.56231 0.638040
\(77\) 5.47214 0.623608
\(78\) 0 0
\(79\) 0.527864 0.0593893 0.0296947 − 0.999559i \(-0.490547\pi\)
0.0296947 + 0.999559i \(0.490547\pi\)
\(80\) 4.34752 0.486068
\(81\) 0 0
\(82\) 1.79837 0.198597
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 1.38197 0.149895
\(86\) 3.09017 0.333222
\(87\) 0 0
\(88\) −8.05573 −0.858743
\(89\) −3.14590 −0.333465 −0.166732 − 0.986002i \(-0.553322\pi\)
−0.166732 + 0.986002i \(0.553322\pi\)
\(90\) 0 0
\(91\) −2.38197 −0.249698
\(92\) −1.85410 −0.193303
\(93\) 0 0
\(94\) −0.652476 −0.0672977
\(95\) −4.14590 −0.425360
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0.381966 0.0385844
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1449.2.a.k.1.1 2
3.2 odd 2 483.2.a.c.1.2 ✓ 2
12.11 even 2 7728.2.a.v.1.2 2
21.20 even 2 3381.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.c.1.2 ✓ 2 3.2 odd 2
1449.2.a.k.1.1 2 1.1 even 1 trivial
3381.2.a.n.1.2 2 21.20 even 2
7728.2.a.v.1.2 2 12.11 even 2