Properties

Label 1445.2.a.r.1.10
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 1.10
Root \(2.27524\) 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)
 
\(f(q)\) \(=\) \(q+2.27524 q^{2} +1.70271 q^{3} +3.17670 q^{4} -1.00000 q^{5} +3.87408 q^{6} +2.70724 q^{7} +2.67727 q^{8} -0.100765 q^{9} -2.27524 q^{10} -2.17632 q^{11} +5.40901 q^{12} +5.37970 q^{13} +6.15962 q^{14} -1.70271 q^{15} -0.261974 q^{16} -0.229265 q^{18} +8.53588 q^{19} -3.17670 q^{20} +4.60966 q^{21} -4.95164 q^{22} -5.32570 q^{23} +4.55863 q^{24} +1.00000 q^{25} +12.2401 q^{26} -5.27972 q^{27} +8.60011 q^{28} +1.31247 q^{29} -3.87408 q^{30} +0.757667 q^{31} -5.95060 q^{32} -3.70565 q^{33} -2.70724 q^{35} -0.320102 q^{36} -8.20380 q^{37} +19.4212 q^{38} +9.16010 q^{39} -2.67727 q^{40} +0.0130696 q^{41} +10.4881 q^{42} -0.330596 q^{43} -6.91352 q^{44} +0.100765 q^{45} -12.1172 q^{46} +11.3395 q^{47} -0.446067 q^{48} +0.329174 q^{49} +2.27524 q^{50} +17.0897 q^{52} -8.04151 q^{53} -12.0126 q^{54} +2.17632 q^{55} +7.24803 q^{56} +14.5342 q^{57} +2.98619 q^{58} -13.1546 q^{59} -5.40901 q^{60} +7.20639 q^{61} +1.72387 q^{62} -0.272797 q^{63} -13.0151 q^{64} -5.37970 q^{65} -8.43123 q^{66} +2.67143 q^{67} -9.06814 q^{69} -6.15962 q^{70} -1.87106 q^{71} -0.269777 q^{72} +11.9780 q^{73} -18.6656 q^{74} +1.70271 q^{75} +27.1160 q^{76} -5.89183 q^{77} +20.8414 q^{78} -17.0477 q^{79} +0.261974 q^{80} -8.68755 q^{81} +0.0297365 q^{82} -3.35585 q^{83} +14.6435 q^{84} -0.752183 q^{86} +2.23477 q^{87} -5.82660 q^{88} -10.8909 q^{89} +0.229265 q^{90} +14.5642 q^{91} -16.9182 q^{92} +1.29009 q^{93} +25.8001 q^{94} -8.53588 q^{95} -10.1322 q^{96} -10.9119 q^{97} +0.748948 q^{98} +0.219298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{3} + 21 q^{4} - 12 q^{5} + 9 q^{6} - 6 q^{7} + 12 q^{8} + 21 q^{9} - 3 q^{10} + 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} + 3 q^{15} + 39 q^{16} - 9 q^{18} + 27 q^{19} - 21 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.27524 1.60884 0.804418 − 0.594064i \(-0.202478\pi\)
0.804418 + 0.594064i \(0.202478\pi\)
\(3\) 1.70271 0.983062 0.491531 − 0.870860i \(-0.336437\pi\)
0.491531 + 0.870860i \(0.336437\pi\)
\(4\) 3.17670 1.58835
\(5\) −1.00000 −0.447214
\(6\) 3.87408 1.58159
\(7\) 2.70724 1.02324 0.511621 − 0.859211i \(-0.329045\pi\)
0.511621 + 0.859211i \(0.329045\pi\)
\(8\) 2.67727 0.946559
\(9\) −0.100765 −0.0335885
\(10\) −2.27524 −0.719493
\(11\) −2.17632 −0.656185 −0.328093 − 0.944646i \(-0.606406\pi\)
−0.328093 + 0.944646i \(0.606406\pi\)
\(12\) 5.40901 1.56145
\(13\) 5.37970 1.49206 0.746031 − 0.665912i \(-0.231957\pi\)
0.746031 + 0.665912i \(0.231957\pi\)
\(14\) 6.15962 1.64623
\(15\) −1.70271 −0.439639
\(16\) −0.261974 −0.0654935
\(17\) 0 0
\(18\) −0.229265 −0.0540384
\(19\) 8.53588 1.95827 0.979133 − 0.203220i \(-0.0651406\pi\)
0.979133 + 0.203220i \(0.0651406\pi\)
\(20\) −3.17670 −0.710332
\(21\) 4.60966 1.00591
\(22\) −4.95164 −1.05569
\(23\) −5.32570 −1.11049 −0.555243 − 0.831688i \(-0.687375\pi\)
−0.555243 + 0.831688i \(0.687375\pi\)
\(24\) 4.55863 0.930526
\(25\) 1.00000 0.200000
\(26\) 12.2401 2.40048
\(27\) −5.27972 −1.01608
\(28\) 8.60011 1.62527
\(29\) 1.31247 0.243720 0.121860 − 0.992547i \(-0.461114\pi\)
0.121860 + 0.992547i \(0.461114\pi\)
\(30\) −3.87408 −0.707306
\(31\) 0.757667 0.136081 0.0680405 − 0.997683i \(-0.478325\pi\)
0.0680405 + 0.997683i \(0.478325\pi\)
\(32\) −5.95060 −1.05193
\(33\) −3.70565 −0.645071
\(34\) 0 0
\(35\) −2.70724 −0.457608
\(36\) −0.320102 −0.0533503
\(37\) −8.20380 −1.34870 −0.674348 − 0.738414i \(-0.735575\pi\)
−0.674348 + 0.738414i \(0.735575\pi\)
\(38\) 19.4212 3.15053
\(39\) 9.16010 1.46679
\(40\) −2.67727 −0.423314
\(41\) 0.0130696 0.00204113 0.00102057 − 0.999999i \(-0.499675\pi\)
0.00102057 + 0.999999i \(0.499675\pi\)
\(42\) 10.4881 1.61834
\(43\) −0.330596 −0.0504154 −0.0252077 − 0.999682i \(-0.508025\pi\)
−0.0252077 + 0.999682i \(0.508025\pi\)
\(44\) −6.91352 −1.04225
\(45\) 0.100765 0.0150212
\(46\) −12.1172 −1.78659
\(47\) 11.3395 1.65404 0.827019 − 0.562175i \(-0.190035\pi\)
0.827019 + 0.562175i \(0.190035\pi\)
\(48\) −0.446067 −0.0643842
\(49\) 0.329174 0.0470248
\(50\) 2.27524 0.321767
\(51\) 0 0
\(52\) 17.0897 2.36992
\(53\) −8.04151 −1.10459 −0.552293 − 0.833650i \(-0.686247\pi\)
−0.552293 + 0.833650i \(0.686247\pi\)
\(54\) −12.0126 −1.63471
\(55\) 2.17632 0.293455
\(56\) 7.24803 0.968559
\(57\) 14.5342 1.92510
\(58\) 2.98619 0.392106
\(59\) −13.1546 −1.71259 −0.856293 − 0.516490i \(-0.827238\pi\)
−0.856293 + 0.516490i \(0.827238\pi\)
\(60\) −5.40901 −0.698300
\(61\) 7.20639 0.922684 0.461342 − 0.887222i \(-0.347368\pi\)
0.461342 + 0.887222i \(0.347368\pi\)
\(62\) 1.72387 0.218932
\(63\) −0.272797 −0.0343692
\(64\) −13.0151 −1.62688
\(65\) −5.37970 −0.667270
\(66\) −8.43123 −1.03781
\(67\) 2.67143 0.326367 0.163184 − 0.986596i \(-0.447824\pi\)
0.163184 + 0.986596i \(0.447824\pi\)
\(68\) 0 0
\(69\) −9.06814 −1.09168
\(70\) −6.15962 −0.736216
\(71\) −1.87106 −0.222054 −0.111027 − 0.993817i \(-0.535414\pi\)
−0.111027 + 0.993817i \(0.535414\pi\)
\(72\) −0.269777 −0.0317935
\(73\) 11.9780 1.40192 0.700962 − 0.713199i \(-0.252754\pi\)
0.700962 + 0.713199i \(0.252754\pi\)
\(74\) −18.6656 −2.16983
\(75\) 1.70271 0.196612
\(76\) 27.1160 3.11041
\(77\) −5.89183 −0.671437
\(78\) 20.8414 2.35982
\(79\) −17.0477 −1.91802 −0.959009 − 0.283375i \(-0.908546\pi\)
−0.959009 + 0.283375i \(0.908546\pi\)
\(80\) 0.261974 0.0292896
\(81\) −8.68755 −0.965283
\(82\) 0.0297365 0.00328385
\(83\) −3.35585 −0.368352 −0.184176 − 0.982893i \(-0.558962\pi\)
−0.184176 + 0.982893i \(0.558962\pi\)
\(84\) 14.6435 1.59774
\(85\) 0 0
\(86\) −0.752183 −0.0811100
\(87\) 2.23477 0.239592
\(88\) −5.82660 −0.621118
\(89\) −10.8909 −1.15444 −0.577218 − 0.816590i \(-0.695862\pi\)
−0.577218 + 0.816590i \(0.695862\pi\)
\(90\) 0.229265 0.0241667
\(91\) 14.5642 1.52674
\(92\) −16.9182 −1.76384
\(93\) 1.29009 0.133776
\(94\) 25.8001 2.66107
\(95\) −8.53588 −0.875763
\(96\) −10.1322 −1.03411
\(97\) −10.9119 −1.10793 −0.553965 − 0.832540i \(-0.686886\pi\)
−0.553965 + 0.832540i \(0.686886\pi\)
\(98\) 0.748948 0.0756552
\(99\) 0.219298 0.0220403
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.1.10 ✓ 12
5.4 even 2 7225.2.a.bo.1.3 12
17.4 even 4 1445.2.d.i.866.5 24
17.13 even 4 1445.2.d.i.866.6 24
17.16 even 2 1445.2.a.s.1.10 yes 12
85.84 even 2 7225.2.a.bn.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.2.a.r.1.10 ✓ 12 1.1 even 1 trivial
1445.2.a.s.1.10 yes 12 17.16 even 2
1445.2.d.i.866.5 24 17.4 even 4
1445.2.d.i.866.6 24 17.13 even 4
7225.2.a.bn.1.3 12 85.84 even 2
7225.2.a.bo.1.3 12 5.4 even 2