Properties

Label 1445.2.a.r.1.4
Level $1445$
Weight $2$
Character 1445.1
Self dual yes
Analytic conductor $11.538$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,2,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,3,-3,21,-12] 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: \(11.5383830921\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 18 x^{10} + 55 x^{9} + 114 x^{8} - 354 x^{7} - 309 x^{6} + 936 x^{5} + 396 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.02921\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.02921 q^{2} -1.23410 q^{3} -0.940729 q^{4} -1.00000 q^{5} +1.27015 q^{6} -0.979236 q^{7} +3.02662 q^{8} -1.47699 q^{9} +1.02921 q^{10} -6.35575 q^{11} +1.16096 q^{12} -2.29653 q^{13} +1.00784 q^{14} +1.23410 q^{15} -1.23357 q^{16} +1.52013 q^{18} -1.91355 q^{19} +0.940729 q^{20} +1.20848 q^{21} +6.54140 q^{22} -8.96509 q^{23} -3.73517 q^{24} +1.00000 q^{25} +2.36361 q^{26} +5.52507 q^{27} +0.921196 q^{28} -2.21557 q^{29} -1.27015 q^{30} +5.74037 q^{31} -4.78365 q^{32} +7.84366 q^{33} +0.979236 q^{35} +1.38945 q^{36} -5.49260 q^{37} +1.96944 q^{38} +2.83416 q^{39} -3.02662 q^{40} -9.14368 q^{41} -1.24378 q^{42} -4.57764 q^{43} +5.97904 q^{44} +1.47699 q^{45} +9.22695 q^{46} -1.95008 q^{47} +1.52235 q^{48} -6.04110 q^{49} -1.02921 q^{50} +2.16041 q^{52} -4.00762 q^{53} -5.68645 q^{54} +6.35575 q^{55} -2.96378 q^{56} +2.36152 q^{57} +2.28028 q^{58} +12.6197 q^{59} -1.16096 q^{60} -5.87984 q^{61} -5.90804 q^{62} +1.44632 q^{63} +7.39051 q^{64} +2.29653 q^{65} -8.07276 q^{66} +6.72054 q^{67} +11.0638 q^{69} -1.00784 q^{70} +5.08034 q^{71} -4.47029 q^{72} -3.68032 q^{73} +5.65303 q^{74} -1.23410 q^{75} +1.80013 q^{76} +6.22378 q^{77} -2.91694 q^{78} +1.08356 q^{79} +1.23357 q^{80} -2.38754 q^{81} +9.41075 q^{82} +3.74449 q^{83} -1.13685 q^{84} +4.71135 q^{86} +2.73424 q^{87} -19.2365 q^{88} +9.03669 q^{89} -1.52013 q^{90} +2.24885 q^{91} +8.43372 q^{92} -7.08421 q^{93} +2.00704 q^{94} +1.91355 q^{95} +5.90351 q^{96} -2.17633 q^{97} +6.21755 q^{98} +9.38737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{3} + 21 q^{4} - 12 q^{5} + 9 q^{6} - 6 q^{7} + 12 q^{8} + 21 q^{9} - 3 q^{10} + 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} + 3 q^{15} + 39 q^{16} - 9 q^{18} + 27 q^{19} - 21 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.02921 −0.727761 −0.363880 − 0.931446i \(-0.618548\pi\)
−0.363880 + 0.931446i \(0.618548\pi\)
\(3\) −1.23410 −0.712510 −0.356255 − 0.934389i \(-0.615947\pi\)
−0.356255 + 0.934389i \(0.615947\pi\)
\(4\) −0.940729 −0.470364
\(5\) −1.00000 −0.447214
\(6\) 1.27015 0.518537
\(7\) −0.979236 −0.370116 −0.185058 − 0.982728i \(-0.559247\pi\)
−0.185058 + 0.982728i \(0.559247\pi\)
\(8\) 3.02662 1.07007
\(9\) −1.47699 −0.492330
\(10\) 1.02921 0.325464
\(11\) −6.35575 −1.91633 −0.958166 − 0.286214i \(-0.907603\pi\)
−0.958166 + 0.286214i \(0.907603\pi\)
\(12\) 1.16096 0.335139
\(13\) −2.29653 −0.636943 −0.318472 − 0.947932i \(-0.603170\pi\)
−0.318472 + 0.947932i \(0.603170\pi\)
\(14\) 1.00784 0.269356
\(15\) 1.23410 0.318644
\(16\) −1.23357 −0.308393
\(17\) 0 0
\(18\) 1.52013 0.358298
\(19\) −1.91355 −0.438999 −0.219499 − 0.975613i \(-0.570442\pi\)
−0.219499 + 0.975613i \(0.570442\pi\)
\(20\) 0.940729 0.210353
\(21\) 1.20848 0.263712
\(22\) 6.54140 1.39463
\(23\) −8.96509 −1.86935 −0.934675 − 0.355503i \(-0.884310\pi\)
−0.934675 + 0.355503i \(0.884310\pi\)
\(24\) −3.73517 −0.762438
\(25\) 1.00000 0.200000
\(26\) 2.36361 0.463542
\(27\) 5.52507 1.06330
\(28\) 0.921196 0.174090
\(29\) −2.21557 −0.411421 −0.205710 − 0.978613i \(-0.565950\pi\)
−0.205710 + 0.978613i \(0.565950\pi\)
\(30\) −1.27015 −0.231897
\(31\) 5.74037 1.03100 0.515501 − 0.856889i \(-0.327606\pi\)
0.515501 + 0.856889i \(0.327606\pi\)
\(32\) −4.78365 −0.845637
\(33\) 7.84366 1.36541
\(34\) 0 0
\(35\) 0.979236 0.165521
\(36\) 1.38945 0.231574
\(37\) −5.49260 −0.902977 −0.451489 − 0.892277i \(-0.649107\pi\)
−0.451489 + 0.892277i \(0.649107\pi\)
\(38\) 1.96944 0.319486
\(39\) 2.83416 0.453828
\(40\) −3.02662 −0.478551
\(41\) −9.14368 −1.42800 −0.714001 − 0.700144i \(-0.753119\pi\)
−0.714001 + 0.700144i \(0.753119\pi\)
\(42\) −1.24378 −0.191919
\(43\) −4.57764 −0.698083 −0.349042 − 0.937107i \(-0.613493\pi\)
−0.349042 + 0.937107i \(0.613493\pi\)
\(44\) 5.97904 0.901374
\(45\) 1.47699 0.220176
\(46\) 9.22695 1.36044
\(47\) −1.95008 −0.284449 −0.142224 − 0.989834i \(-0.545425\pi\)
−0.142224 + 0.989834i \(0.545425\pi\)
\(48\) 1.52235 0.219733
\(49\) −6.04110 −0.863014
\(50\) −1.02921 −0.145552
\(51\) 0 0
\(52\) 2.16041 0.299595
\(53\) −4.00762 −0.550488 −0.275244 − 0.961374i \(-0.588759\pi\)
−0.275244 + 0.961374i \(0.588759\pi\)
\(54\) −5.68645 −0.773828
\(55\) 6.35575 0.857010
\(56\) −2.96378 −0.396052
\(57\) 2.36152 0.312791
\(58\) 2.28028 0.299416
\(59\) 12.6197 1.64294 0.821471 − 0.570250i \(-0.193154\pi\)
0.821471 + 0.570250i \(0.193154\pi\)
\(60\) −1.16096 −0.149879
\(61\) −5.87984 −0.752837 −0.376418 − 0.926450i \(-0.622844\pi\)
−0.376418 + 0.926450i \(0.622844\pi\)
\(62\) −5.90804 −0.750322
\(63\) 1.44632 0.182219
\(64\) 7.39051 0.923814
\(65\) 2.29653 0.284850
\(66\) −8.07276 −0.993688
\(67\) 6.72054 0.821044 0.410522 − 0.911851i \(-0.365346\pi\)
0.410522 + 0.911851i \(0.365346\pi\)
\(68\) 0 0
\(69\) 11.0638 1.33193
\(70\) −1.00784 −0.120460
\(71\) 5.08034 0.602926 0.301463 − 0.953478i \(-0.402525\pi\)
0.301463 + 0.953478i \(0.402525\pi\)
\(72\) −4.47029 −0.526829
\(73\) −3.68032 −0.430749 −0.215374 − 0.976532i \(-0.569097\pi\)
−0.215374 + 0.976532i \(0.569097\pi\)
\(74\) 5.65303 0.657151
\(75\) −1.23410 −0.142502
\(76\) 1.80013 0.206489
\(77\) 6.22378 0.709266
\(78\) −2.91694 −0.330278
\(79\) 1.08356 0.121910 0.0609548 − 0.998141i \(-0.480585\pi\)
0.0609548 + 0.998141i \(0.480585\pi\)
\(80\) 1.23357 0.137917
\(81\) −2.38754 −0.265282
\(82\) 9.41075 1.03924
\(83\) 3.74449 0.411011 0.205506 − 0.978656i \(-0.434116\pi\)
0.205506 + 0.978656i \(0.434116\pi\)
\(84\) −1.13685 −0.124041
\(85\) 0 0
\(86\) 4.71135 0.508038
\(87\) 2.73424 0.293141
\(88\) −19.2365 −2.05062
\(89\) 9.03669 0.957888 0.478944 − 0.877846i \(-0.341020\pi\)
0.478944 + 0.877846i \(0.341020\pi\)
\(90\) −1.52013 −0.160236
\(91\) 2.24885 0.235743
\(92\) 8.43372 0.879276
\(93\) −7.08421 −0.734599
\(94\) 2.00704 0.207011
\(95\) 1.91355 0.196326
\(96\) 5.90351 0.602525
\(97\) −2.17633 −0.220973 −0.110487 − 0.993878i \(-0.535241\pi\)
−0.110487 + 0.993878i \(0.535241\pi\)
\(98\) 6.21755 0.628067
\(99\) 9.38737 0.943467
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 1445.2.a.r.1.4 ✓ 12
5.4 even 2 7225.2.a.bo.1.9 12
17.4 even 4 1445.2.d.i.866.17 24
17.13 even 4 1445.2.d.i.866.18 24
17.16 even 2 1445.2.a.s.1.4 yes 12
85.84 even 2 7225.2.a.bn.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.2.a.r.1.4 ✓ 12 1.1 even 1 trivial
1445.2.a.s.1.4 yes 12 17.16 even 2
1445.2.d.i.866.17 24 17.4 even 4
1445.2.d.i.866.18 24 17.13 even 4
7225.2.a.bn.1.9 12 85.84 even 2
7225.2.a.bo.1.9 12 5.4 even 2