Properties

Label 1445.2.a.r.1.6
Level $1445$
Weight $2$
Character 1445.1
Self dual yes
Analytic conductor $11.538$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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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 1.6
Root \(-0.0540929\) 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)
 
\(f(q)\) \(=\) \(q-0.0540929 q^{2} -0.605946 q^{3} -1.99707 q^{4} -1.00000 q^{5} +0.0327774 q^{6} +4.08179 q^{7} +0.216213 q^{8} -2.63283 q^{9} +0.0540929 q^{10} -1.50083 q^{11} +1.21012 q^{12} +3.30340 q^{13} -0.220796 q^{14} +0.605946 q^{15} +3.98245 q^{16} +0.142417 q^{18} -4.02443 q^{19} +1.99707 q^{20} -2.47334 q^{21} +0.0811844 q^{22} -2.09556 q^{23} -0.131014 q^{24} +1.00000 q^{25} -0.178691 q^{26} +3.41319 q^{27} -8.15163 q^{28} -10.4424 q^{29} -0.0327774 q^{30} -2.12355 q^{31} -0.647849 q^{32} +0.909424 q^{33} -4.08179 q^{35} +5.25796 q^{36} +5.75990 q^{37} +0.217693 q^{38} -2.00168 q^{39} -0.216213 q^{40} +4.09160 q^{41} +0.133790 q^{42} +11.4355 q^{43} +2.99728 q^{44} +2.63283 q^{45} +0.113355 q^{46} +11.0877 q^{47} -2.41315 q^{48} +9.66099 q^{49} -0.0540929 q^{50} -6.59714 q^{52} +6.78569 q^{53} -0.184629 q^{54} +1.50083 q^{55} +0.882536 q^{56} +2.43859 q^{57} +0.564859 q^{58} +9.05391 q^{59} -1.21012 q^{60} +5.81828 q^{61} +0.114869 q^{62} -10.7466 q^{63} -7.92986 q^{64} -3.30340 q^{65} -0.0491934 q^{66} +6.85741 q^{67} +1.26980 q^{69} +0.220796 q^{70} -12.5305 q^{71} -0.569252 q^{72} +11.5717 q^{73} -0.311569 q^{74} -0.605946 q^{75} +8.03709 q^{76} -6.12608 q^{77} +0.108277 q^{78} -0.181191 q^{79} -3.98245 q^{80} +5.83028 q^{81} -0.221327 q^{82} +1.27674 q^{83} +4.93945 q^{84} -0.618579 q^{86} +6.32753 q^{87} -0.324500 q^{88} -11.3962 q^{89} -0.142417 q^{90} +13.4838 q^{91} +4.18499 q^{92} +1.28676 q^{93} -0.599763 q^{94} +4.02443 q^{95} +0.392561 q^{96} +8.27401 q^{97} -0.522590 q^{98} +3.95144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{3} + 21 q^{4} - 12 q^{5} + 9 q^{6} - 6 q^{7} + 12 q^{8} + 21 q^{9} - 3 q^{10} + 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} + 3 q^{15} + 39 q^{16} - 9 q^{18} + 27 q^{19} - 21 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0540929 −0.0382494 −0.0191247 − 0.999817i \(-0.506088\pi\)
−0.0191247 + 0.999817i \(0.506088\pi\)
\(3\) −0.605946 −0.349843 −0.174922 − 0.984582i \(-0.555967\pi\)
−0.174922 + 0.984582i \(0.555967\pi\)
\(4\) −1.99707 −0.998537
\(5\) −1.00000 −0.447214
\(6\) 0.0327774 0.0133813
\(7\) 4.08179 1.54277 0.771385 − 0.636368i \(-0.219564\pi\)
0.771385 + 0.636368i \(0.219564\pi\)
\(8\) 0.216213 0.0764429
\(9\) −2.63283 −0.877610
\(10\) 0.0540929 0.0171057
\(11\) −1.50083 −0.452518 −0.226259 − 0.974067i \(-0.572650\pi\)
−0.226259 + 0.974067i \(0.572650\pi\)
\(12\) 1.21012 0.349331
\(13\) 3.30340 0.916199 0.458100 − 0.888901i \(-0.348530\pi\)
0.458100 + 0.888901i \(0.348530\pi\)
\(14\) −0.220796 −0.0590101
\(15\) 0.605946 0.156455
\(16\) 3.98245 0.995613
\(17\) 0 0
\(18\) 0.142417 0.0335681
\(19\) −4.02443 −0.923268 −0.461634 − 0.887071i \(-0.652737\pi\)
−0.461634 + 0.887071i \(0.652737\pi\)
\(20\) 1.99707 0.446559
\(21\) −2.47334 −0.539728
\(22\) 0.0811844 0.0173086
\(23\) −2.09556 −0.436955 −0.218477 − 0.975842i \(-0.570109\pi\)
−0.218477 + 0.975842i \(0.570109\pi\)
\(24\) −0.131014 −0.0267430
\(25\) 1.00000 0.200000
\(26\) −0.178691 −0.0350441
\(27\) 3.41319 0.656869
\(28\) −8.15163 −1.54051
\(29\) −10.4424 −1.93910 −0.969552 − 0.244887i \(-0.921249\pi\)
−0.969552 + 0.244887i \(0.921249\pi\)
\(30\) −0.0327774 −0.00598430
\(31\) −2.12355 −0.381401 −0.190701 − 0.981648i \(-0.561076\pi\)
−0.190701 + 0.981648i \(0.561076\pi\)
\(32\) −0.647849 −0.114525
\(33\) 0.909424 0.158310
\(34\) 0 0
\(35\) −4.08179 −0.689948
\(36\) 5.25796 0.876326
\(37\) 5.75990 0.946921 0.473461 − 0.880815i \(-0.343005\pi\)
0.473461 + 0.880815i \(0.343005\pi\)
\(38\) 0.217693 0.0353145
\(39\) −2.00168 −0.320526
\(40\) −0.216213 −0.0341863
\(41\) 4.09160 0.639001 0.319501 − 0.947586i \(-0.396485\pi\)
0.319501 + 0.947586i \(0.396485\pi\)
\(42\) 0.133790 0.0206443
\(43\) 11.4355 1.74390 0.871949 − 0.489597i \(-0.162856\pi\)
0.871949 + 0.489597i \(0.162856\pi\)
\(44\) 2.99728 0.451856
\(45\) 2.63283 0.392479
\(46\) 0.113355 0.0167133
\(47\) 11.0877 1.61730 0.808650 − 0.588290i \(-0.200199\pi\)
0.808650 + 0.588290i \(0.200199\pi\)
\(48\) −2.41315 −0.348308
\(49\) 9.66099 1.38014
\(50\) −0.0540929 −0.00764989
\(51\) 0 0
\(52\) −6.59714 −0.914859
\(53\) 6.78569 0.932086 0.466043 − 0.884762i \(-0.345679\pi\)
0.466043 + 0.884762i \(0.345679\pi\)
\(54\) −0.184629 −0.0251249
\(55\) 1.50083 0.202372
\(56\) 0.882536 0.117934
\(57\) 2.43859 0.322999
\(58\) 0.564859 0.0741696
\(59\) 9.05391 1.17872 0.589359 − 0.807871i \(-0.299380\pi\)
0.589359 + 0.807871i \(0.299380\pi\)
\(60\) −1.21012 −0.156226
\(61\) 5.81828 0.744954 0.372477 − 0.928041i \(-0.378509\pi\)
0.372477 + 0.928041i \(0.378509\pi\)
\(62\) 0.114869 0.0145884
\(63\) −10.7466 −1.35395
\(64\) −7.92986 −0.991233
\(65\) −3.30340 −0.409737
\(66\) −0.0491934 −0.00605528
\(67\) 6.85741 0.837766 0.418883 − 0.908040i \(-0.362422\pi\)
0.418883 + 0.908040i \(0.362422\pi\)
\(68\) 0 0
\(69\) 1.26980 0.152866
\(70\) 0.220796 0.0263901
\(71\) −12.5305 −1.48710 −0.743548 − 0.668683i \(-0.766859\pi\)
−0.743548 + 0.668683i \(0.766859\pi\)
\(72\) −0.569252 −0.0670870
\(73\) 11.5717 1.35437 0.677183 − 0.735815i \(-0.263201\pi\)
0.677183 + 0.735815i \(0.263201\pi\)
\(74\) −0.311569 −0.0362192
\(75\) −0.605946 −0.0699686
\(76\) 8.03709 0.921917
\(77\) −6.12608 −0.698132
\(78\) 0.108277 0.0122599
\(79\) −0.181191 −0.0203856 −0.0101928 − 0.999948i \(-0.503245\pi\)
−0.0101928 + 0.999948i \(0.503245\pi\)
\(80\) −3.98245 −0.445252
\(81\) 5.83028 0.647809
\(82\) −0.221327 −0.0244414
\(83\) 1.27674 0.140140 0.0700701 − 0.997542i \(-0.477678\pi\)
0.0700701 + 0.997542i \(0.477678\pi\)
\(84\) 4.93945 0.538938
\(85\) 0 0
\(86\) −0.618579 −0.0667031
\(87\) 6.32753 0.678382
\(88\) −0.324500 −0.0345918
\(89\) −11.3962 −1.20799 −0.603996 − 0.796987i \(-0.706426\pi\)
−0.603996 + 0.796987i \(0.706426\pi\)
\(90\) −0.142417 −0.0150121
\(91\) 13.4838 1.41348
\(92\) 4.18499 0.436316
\(93\) 1.28676 0.133431
\(94\) −0.599763 −0.0618608
\(95\) 4.02443 0.412898
\(96\) 0.392561 0.0400656
\(97\) 8.27401 0.840098 0.420049 − 0.907501i \(-0.362013\pi\)
0.420049 + 0.907501i \(0.362013\pi\)
\(98\) −0.522590 −0.0527896
\(99\) 3.95144 0.397135
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.1.6 ✓ 12
5.4 even 2 7225.2.a.bo.1.7 12
17.4 even 4 1445.2.d.i.866.13 24
17.13 even 4 1445.2.d.i.866.14 24
17.16 even 2 1445.2.a.s.1.6 yes 12
85.84 even 2 7225.2.a.bn.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.2.a.r.1.6 ✓ 12 1.1 even 1 trivial
1445.2.a.s.1.6 yes 12 17.16 even 2
1445.2.d.i.866.13 24 17.4 even 4
1445.2.d.i.866.14 24 17.13 even 4
7225.2.a.bn.1.7 12 85.84 even 2
7225.2.a.bo.1.7 12 5.4 even 2