Properties

Label 1445.2.a.r.1.12
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 1.12
Root \(2.77092\) 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)
 
\(f(q)\) \(=\) \(q+2.77092 q^{2} -0.449467 q^{3} +5.67798 q^{4} -1.00000 q^{5} -1.24543 q^{6} -1.06235 q^{7} +10.1914 q^{8} -2.79798 q^{9} -2.77092 q^{10} +5.06621 q^{11} -2.55206 q^{12} +1.74590 q^{13} -2.94367 q^{14} +0.449467 q^{15} +16.8835 q^{16} -7.75297 q^{18} +2.82138 q^{19} -5.67798 q^{20} +0.477489 q^{21} +14.0380 q^{22} +1.39195 q^{23} -4.58069 q^{24} +1.00000 q^{25} +4.83775 q^{26} +2.60600 q^{27} -6.03198 q^{28} +4.18716 q^{29} +1.24543 q^{30} -6.69607 q^{31} +26.4001 q^{32} -2.27709 q^{33} +1.06235 q^{35} -15.8869 q^{36} -3.75143 q^{37} +7.81780 q^{38} -0.784724 q^{39} -10.1914 q^{40} +8.29696 q^{41} +1.32308 q^{42} -2.40098 q^{43} +28.7658 q^{44} +2.79798 q^{45} +3.85697 q^{46} -10.6859 q^{47} -7.58858 q^{48} -5.87142 q^{49} +2.77092 q^{50} +9.91320 q^{52} -3.69497 q^{53} +7.22101 q^{54} -5.06621 q^{55} -10.8268 q^{56} -1.26811 q^{57} +11.6023 q^{58} +1.57931 q^{59} +2.55206 q^{60} -10.2515 q^{61} -18.5543 q^{62} +2.97242 q^{63} +39.3854 q^{64} -1.74590 q^{65} -6.30963 q^{66} -6.96485 q^{67} -0.625633 q^{69} +2.94367 q^{70} -3.07563 q^{71} -28.5153 q^{72} +10.6375 q^{73} -10.3949 q^{74} -0.449467 q^{75} +16.0197 q^{76} -5.38206 q^{77} -2.17441 q^{78} -3.78111 q^{79} -16.8835 q^{80} +7.22263 q^{81} +22.9902 q^{82} -10.8621 q^{83} +2.71117 q^{84} -6.65293 q^{86} -1.88199 q^{87} +51.6317 q^{88} +1.80507 q^{89} +7.75297 q^{90} -1.85475 q^{91} +7.90344 q^{92} +3.00966 q^{93} -29.6096 q^{94} -2.82138 q^{95} -11.8660 q^{96} -12.9699 q^{97} -16.2692 q^{98} -14.1751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{3} + 21 q^{4} - 12 q^{5} + 9 q^{6} - 6 q^{7} + 12 q^{8} + 21 q^{9} - 3 q^{10} + 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} + 3 q^{15} + 39 q^{16} - 9 q^{18} + 27 q^{19} - 21 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.77092 1.95933 0.979667 − 0.200629i \(-0.0642986\pi\)
0.979667 + 0.200629i \(0.0642986\pi\)
\(3\) −0.449467 −0.259500 −0.129750 − 0.991547i \(-0.541417\pi\)
−0.129750 + 0.991547i \(0.541417\pi\)
\(4\) 5.67798 2.83899
\(5\) −1.00000 −0.447214
\(6\) −1.24543 −0.508447
\(7\) −1.06235 −0.401529 −0.200764 − 0.979640i \(-0.564343\pi\)
−0.200764 + 0.979640i \(0.564343\pi\)
\(8\) 10.1914 3.60320
\(9\) −2.79798 −0.932660
\(10\) −2.77092 −0.876241
\(11\) 5.06621 1.52752 0.763759 − 0.645501i \(-0.223351\pi\)
0.763759 + 0.645501i \(0.223351\pi\)
\(12\) −2.55206 −0.736717
\(13\) 1.74590 0.484226 0.242113 − 0.970248i \(-0.422160\pi\)
0.242113 + 0.970248i \(0.422160\pi\)
\(14\) −2.94367 −0.786729
\(15\) 0.449467 0.116052
\(16\) 16.8835 4.22088
\(17\) 0 0
\(18\) −7.75297 −1.82739
\(19\) 2.82138 0.647268 0.323634 − 0.946182i \(-0.395095\pi\)
0.323634 + 0.946182i \(0.395095\pi\)
\(20\) −5.67798 −1.26964
\(21\) 0.477489 0.104197
\(22\) 14.0380 2.99292
\(23\) 1.39195 0.290241 0.145120 − 0.989414i \(-0.453643\pi\)
0.145120 + 0.989414i \(0.453643\pi\)
\(24\) −4.58069 −0.935029
\(25\) 1.00000 0.200000
\(26\) 4.83775 0.948761
\(27\) 2.60600 0.501525
\(28\) −6.03198 −1.13994
\(29\) 4.18716 0.777537 0.388768 − 0.921336i \(-0.372901\pi\)
0.388768 + 0.921336i \(0.372901\pi\)
\(30\) 1.24543 0.227384
\(31\) −6.69607 −1.20265 −0.601325 − 0.799004i \(-0.705360\pi\)
−0.601325 + 0.799004i \(0.705360\pi\)
\(32\) 26.4001 4.66692
\(33\) −2.27709 −0.396391
\(34\) 0 0
\(35\) 1.06235 0.179569
\(36\) −15.8869 −2.64781
\(37\) −3.75143 −0.616731 −0.308366 − 0.951268i \(-0.599782\pi\)
−0.308366 + 0.951268i \(0.599782\pi\)
\(38\) 7.81780 1.26821
\(39\) −0.784724 −0.125656
\(40\) −10.1914 −1.61140
\(41\) 8.29696 1.29577 0.647884 − 0.761739i \(-0.275654\pi\)
0.647884 + 0.761739i \(0.275654\pi\)
\(42\) 1.32308 0.204156
\(43\) −2.40098 −0.366147 −0.183073 − 0.983099i \(-0.558605\pi\)
−0.183073 + 0.983099i \(0.558605\pi\)
\(44\) 28.7658 4.33661
\(45\) 2.79798 0.417098
\(46\) 3.85697 0.568679
\(47\) −10.6859 −1.55869 −0.779346 − 0.626594i \(-0.784448\pi\)
−0.779346 + 0.626594i \(0.784448\pi\)
\(48\) −7.58858 −1.09532
\(49\) −5.87142 −0.838775
\(50\) 2.77092 0.391867
\(51\) 0 0
\(52\) 9.91320 1.37471
\(53\) −3.69497 −0.507543 −0.253772 − 0.967264i \(-0.581671\pi\)
−0.253772 + 0.967264i \(0.581671\pi\)
\(54\) 7.22101 0.982655
\(55\) −5.06621 −0.683127
\(56\) −10.8268 −1.44679
\(57\) −1.26811 −0.167966
\(58\) 11.6023 1.52345
\(59\) 1.57931 0.205609 0.102805 − 0.994702i \(-0.467218\pi\)
0.102805 + 0.994702i \(0.467218\pi\)
\(60\) 2.55206 0.329470
\(61\) −10.2515 −1.31257 −0.656284 − 0.754514i \(-0.727873\pi\)
−0.656284 + 0.754514i \(0.727873\pi\)
\(62\) −18.5543 −2.35639
\(63\) 2.97242 0.374490
\(64\) 39.3854 4.92318
\(65\) −1.74590 −0.216552
\(66\) −6.30963 −0.776662
\(67\) −6.96485 −0.850892 −0.425446 − 0.904984i \(-0.639883\pi\)
−0.425446 + 0.904984i \(0.639883\pi\)
\(68\) 0 0
\(69\) −0.625633 −0.0753174
\(70\) 2.94367 0.351836
\(71\) −3.07563 −0.365010 −0.182505 − 0.983205i \(-0.558421\pi\)
−0.182505 + 0.983205i \(0.558421\pi\)
\(72\) −28.5153 −3.36056
\(73\) 10.6375 1.24503 0.622515 − 0.782608i \(-0.286111\pi\)
0.622515 + 0.782608i \(0.286111\pi\)
\(74\) −10.3949 −1.20838
\(75\) −0.449467 −0.0518999
\(76\) 16.0197 1.83759
\(77\) −5.38206 −0.613343
\(78\) −2.17441 −0.246203
\(79\) −3.78111 −0.425408 −0.212704 − 0.977117i \(-0.568227\pi\)
−0.212704 + 0.977117i \(0.568227\pi\)
\(80\) −16.8835 −1.88764
\(81\) 7.22263 0.802514
\(82\) 22.9902 2.53884
\(83\) −10.8621 −1.19227 −0.596133 − 0.802886i \(-0.703297\pi\)
−0.596133 + 0.802886i \(0.703297\pi\)
\(84\) 2.71117 0.295813
\(85\) 0 0
\(86\) −6.65293 −0.717404
\(87\) −1.88199 −0.201771
\(88\) 51.6317 5.50396
\(89\) 1.80507 0.191337 0.0956687 − 0.995413i \(-0.469501\pi\)
0.0956687 + 0.995413i \(0.469501\pi\)
\(90\) 7.75297 0.817235
\(91\) −1.85475 −0.194431
\(92\) 7.90344 0.823991
\(93\) 3.00966 0.312087
\(94\) −29.6096 −3.05400
\(95\) −2.82138 −0.289467
\(96\) −11.8660 −1.21106
\(97\) −12.9699 −1.31690 −0.658448 − 0.752626i \(-0.728787\pi\)
−0.658448 + 0.752626i \(0.728787\pi\)
\(98\) −16.2692 −1.64344
\(99\) −14.1751 −1.42466
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.1.12 ✓ 12
5.4 even 2 7225.2.a.bo.1.1 12
17.4 even 4 1445.2.d.i.866.1 24
17.13 even 4 1445.2.d.i.866.2 24
17.16 even 2 1445.2.a.s.1.12 yes 12
85.84 even 2 7225.2.a.bn.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.2.a.r.1.12 ✓ 12 1.1 even 1 trivial
1445.2.a.s.1.12 yes 12 17.16 even 2
1445.2.d.i.866.1 24 17.4 even 4
1445.2.d.i.866.2 24 17.13 even 4
7225.2.a.bn.1.1 12 85.84 even 2
7225.2.a.bo.1.1 12 5.4 even 2