Properties

Label 1445.2.a.r.1.2
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 1.2
Root \(-2.54480\) 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)
 
\(f(q)\) \(=\) \(q-2.54480 q^{2} -3.06453 q^{3} +4.47602 q^{4} -1.00000 q^{5} +7.79863 q^{6} -0.261390 q^{7} -6.30099 q^{8} +6.39135 q^{9} +2.54480 q^{10} +0.545043 q^{11} -13.7169 q^{12} +5.48254 q^{13} +0.665187 q^{14} +3.06453 q^{15} +7.08273 q^{16} -16.2647 q^{18} +4.34857 q^{19} -4.47602 q^{20} +0.801039 q^{21} -1.38703 q^{22} +3.75127 q^{23} +19.3096 q^{24} +1.00000 q^{25} -13.9520 q^{26} -10.3929 q^{27} -1.16999 q^{28} +2.69600 q^{29} -7.79863 q^{30} +8.95269 q^{31} -5.42218 q^{32} -1.67030 q^{33} +0.261390 q^{35} +28.6078 q^{36} -4.86119 q^{37} -11.0663 q^{38} -16.8014 q^{39} +6.30099 q^{40} -7.76968 q^{41} -2.03849 q^{42} +2.35019 q^{43} +2.43963 q^{44} -6.39135 q^{45} -9.54625 q^{46} +2.35604 q^{47} -21.7052 q^{48} -6.93168 q^{49} -2.54480 q^{50} +24.5400 q^{52} -10.8975 q^{53} +26.4479 q^{54} -0.545043 q^{55} +1.64702 q^{56} -13.3263 q^{57} -6.86078 q^{58} +6.23938 q^{59} +13.7169 q^{60} +0.633641 q^{61} -22.7828 q^{62} -1.67064 q^{63} -0.367091 q^{64} -5.48254 q^{65} +4.25059 q^{66} +8.44378 q^{67} -11.4959 q^{69} -0.665187 q^{70} +5.40706 q^{71} -40.2718 q^{72} -8.12625 q^{73} +12.3708 q^{74} -3.06453 q^{75} +19.4643 q^{76} -0.142469 q^{77} +42.7563 q^{78} -1.73147 q^{79} -7.08273 q^{80} +12.6753 q^{81} +19.7723 q^{82} -0.107006 q^{83} +3.58547 q^{84} -5.98077 q^{86} -8.26196 q^{87} -3.43431 q^{88} -16.8773 q^{89} +16.2647 q^{90} -1.43308 q^{91} +16.7908 q^{92} -27.4358 q^{93} -5.99566 q^{94} -4.34857 q^{95} +16.6164 q^{96} +7.14812 q^{97} +17.6397 q^{98} +3.48356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{3} + 21 q^{4} - 12 q^{5} + 9 q^{6} - 6 q^{7} + 12 q^{8} + 21 q^{9} - 3 q^{10} + 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} + 3 q^{15} + 39 q^{16} - 9 q^{18} + 27 q^{19} - 21 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.54480 −1.79945 −0.899724 − 0.436460i \(-0.856232\pi\)
−0.899724 + 0.436460i \(0.856232\pi\)
\(3\) −3.06453 −1.76931 −0.884654 − 0.466248i \(-0.845605\pi\)
−0.884654 + 0.466248i \(0.845605\pi\)
\(4\) 4.47602 2.23801
\(5\) −1.00000 −0.447214
\(6\) 7.79863 3.18378
\(7\) −0.261390 −0.0987963 −0.0493981 − 0.998779i \(-0.515730\pi\)
−0.0493981 + 0.998779i \(0.515730\pi\)
\(8\) −6.30099 −2.22774
\(9\) 6.39135 2.13045
\(10\) 2.54480 0.804737
\(11\) 0.545043 0.164337 0.0821683 − 0.996618i \(-0.473815\pi\)
0.0821683 + 0.996618i \(0.473815\pi\)
\(12\) −13.7169 −3.95973
\(13\) 5.48254 1.52058 0.760291 − 0.649582i \(-0.225056\pi\)
0.760291 + 0.649582i \(0.225056\pi\)
\(14\) 0.665187 0.177779
\(15\) 3.06453 0.791259
\(16\) 7.08273 1.77068
\(17\) 0 0
\(18\) −16.2647 −3.83363
\(19\) 4.34857 0.997631 0.498816 − 0.866708i \(-0.333768\pi\)
0.498816 + 0.866708i \(0.333768\pi\)
\(20\) −4.47602 −1.00087
\(21\) 0.801039 0.174801
\(22\) −1.38703 −0.295715
\(23\) 3.75127 0.782194 0.391097 − 0.920349i \(-0.372096\pi\)
0.391097 + 0.920349i \(0.372096\pi\)
\(24\) 19.3096 3.94155
\(25\) 1.00000 0.200000
\(26\) −13.9520 −2.73621
\(27\) −10.3929 −2.00012
\(28\) −1.16999 −0.221107
\(29\) 2.69600 0.500634 0.250317 − 0.968164i \(-0.419465\pi\)
0.250317 + 0.968164i \(0.419465\pi\)
\(30\) −7.79863 −1.42383
\(31\) 8.95269 1.60795 0.803975 − 0.594663i \(-0.202714\pi\)
0.803975 + 0.594663i \(0.202714\pi\)
\(32\) −5.42218 −0.958514
\(33\) −1.67030 −0.290762
\(34\) 0 0
\(35\) 0.261390 0.0441830
\(36\) 28.6078 4.76797
\(37\) −4.86119 −0.799175 −0.399588 − 0.916695i \(-0.630847\pi\)
−0.399588 + 0.916695i \(0.630847\pi\)
\(38\) −11.0663 −1.79519
\(39\) −16.8014 −2.69038
\(40\) 6.30099 0.996274
\(41\) −7.76968 −1.21342 −0.606710 − 0.794923i \(-0.707511\pi\)
−0.606710 + 0.794923i \(0.707511\pi\)
\(42\) −2.03849 −0.314545
\(43\) 2.35019 0.358401 0.179200 − 0.983813i \(-0.442649\pi\)
0.179200 + 0.983813i \(0.442649\pi\)
\(44\) 2.43963 0.367787
\(45\) −6.39135 −0.952766
\(46\) −9.54625 −1.40752
\(47\) 2.35604 0.343664 0.171832 − 0.985126i \(-0.445031\pi\)
0.171832 + 0.985126i \(0.445031\pi\)
\(48\) −21.7052 −3.13288
\(49\) −6.93168 −0.990239
\(50\) −2.54480 −0.359889
\(51\) 0 0
\(52\) 24.5400 3.40308
\(53\) −10.8975 −1.49689 −0.748445 − 0.663197i \(-0.769199\pi\)
−0.748445 + 0.663197i \(0.769199\pi\)
\(54\) 26.4479 3.59910
\(55\) −0.545043 −0.0734936
\(56\) 1.64702 0.220092
\(57\) −13.3263 −1.76512
\(58\) −6.86078 −0.900864
\(59\) 6.23938 0.812298 0.406149 − 0.913807i \(-0.366871\pi\)
0.406149 + 0.913807i \(0.366871\pi\)
\(60\) 13.7169 1.77085
\(61\) 0.633641 0.0811294 0.0405647 − 0.999177i \(-0.487084\pi\)
0.0405647 + 0.999177i \(0.487084\pi\)
\(62\) −22.7828 −2.89342
\(63\) −1.67064 −0.210481
\(64\) −0.367091 −0.0458863
\(65\) −5.48254 −0.680025
\(66\) 4.25059 0.523211
\(67\) 8.44378 1.03157 0.515786 − 0.856717i \(-0.327500\pi\)
0.515786 + 0.856717i \(0.327500\pi\)
\(68\) 0 0
\(69\) −11.4959 −1.38394
\(70\) −0.665187 −0.0795051
\(71\) 5.40706 0.641700 0.320850 − 0.947130i \(-0.396031\pi\)
0.320850 + 0.947130i \(0.396031\pi\)
\(72\) −40.2718 −4.74608
\(73\) −8.12625 −0.951105 −0.475553 − 0.879687i \(-0.657752\pi\)
−0.475553 + 0.879687i \(0.657752\pi\)
\(74\) 12.3708 1.43807
\(75\) −3.06453 −0.353862
\(76\) 19.4643 2.23271
\(77\) −0.142469 −0.0162359
\(78\) 42.7563 4.84120
\(79\) −1.73147 −0.194805 −0.0974026 − 0.995245i \(-0.531053\pi\)
−0.0974026 + 0.995245i \(0.531053\pi\)
\(80\) −7.08273 −0.791873
\(81\) 12.6753 1.40837
\(82\) 19.7723 2.18349
\(83\) −0.107006 −0.0117455 −0.00587273 − 0.999983i \(-0.501869\pi\)
−0.00587273 + 0.999983i \(0.501869\pi\)
\(84\) 3.58547 0.391207
\(85\) 0 0
\(86\) −5.98077 −0.644923
\(87\) −8.26196 −0.885775
\(88\) −3.43431 −0.366099
\(89\) −16.8773 −1.78899 −0.894497 − 0.447075i \(-0.852466\pi\)
−0.894497 + 0.447075i \(0.852466\pi\)
\(90\) 16.2647 1.71445
\(91\) −1.43308 −0.150228
\(92\) 16.7908 1.75056
\(93\) −27.4358 −2.84496
\(94\) −5.99566 −0.618405
\(95\) −4.34857 −0.446154
\(96\) 16.6164 1.69591
\(97\) 7.14812 0.725782 0.362891 − 0.931832i \(-0.381790\pi\)
0.362891 + 0.931832i \(0.381790\pi\)
\(98\) 17.6397 1.78188
\(99\) 3.48356 0.350111
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.1.2 ✓ 12
5.4 even 2 7225.2.a.bo.1.11 12
17.4 even 4 1445.2.d.i.866.21 24
17.13 even 4 1445.2.d.i.866.22 24
17.16 even 2 1445.2.a.s.1.2 yes 12
85.84 even 2 7225.2.a.bn.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.2.a.r.1.2 ✓ 12 1.1 even 1 trivial
1445.2.a.s.1.2 yes 12 17.16 even 2
1445.2.d.i.866.21 24 17.4 even 4
1445.2.d.i.866.22 24 17.13 even 4
7225.2.a.bn.1.11 12 85.84 even 2
7225.2.a.bo.1.11 12 5.4 even 2