Properties

Label 1445.2.a.r.1.11
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 1.11
Root \(2.69854\) 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)
 
\(f(q)\) \(=\) \(q+2.69854 q^{2} +2.66942 q^{3} +5.28213 q^{4} -1.00000 q^{5} +7.20354 q^{6} -0.656507 q^{7} +8.85696 q^{8} +4.12579 q^{9} -2.69854 q^{10} -2.89465 q^{11} +14.1002 q^{12} -0.0784138 q^{13} -1.77161 q^{14} -2.66942 q^{15} +13.3366 q^{16} +11.1336 q^{18} -7.45802 q^{19} -5.28213 q^{20} -1.75249 q^{21} -7.81134 q^{22} -5.46884 q^{23} +23.6429 q^{24} +1.00000 q^{25} -0.211603 q^{26} +3.00520 q^{27} -3.46776 q^{28} +2.00368 q^{29} -7.20354 q^{30} -5.25088 q^{31} +18.2755 q^{32} -7.72703 q^{33} +0.656507 q^{35} +21.7930 q^{36} +3.96160 q^{37} -20.1258 q^{38} -0.209319 q^{39} -8.85696 q^{40} -1.20848 q^{41} -4.72917 q^{42} +6.50172 q^{43} -15.2899 q^{44} -4.12579 q^{45} -14.7579 q^{46} +6.11019 q^{47} +35.6010 q^{48} -6.56900 q^{49} +2.69854 q^{50} -0.414192 q^{52} +2.02903 q^{53} +8.10967 q^{54} +2.89465 q^{55} -5.81466 q^{56} -19.9086 q^{57} +5.40701 q^{58} +8.74900 q^{59} -14.1002 q^{60} -4.18013 q^{61} -14.1697 q^{62} -2.70861 q^{63} +22.6440 q^{64} +0.0784138 q^{65} -20.8517 q^{66} +12.9875 q^{67} -14.5986 q^{69} +1.77161 q^{70} +12.6346 q^{71} +36.5420 q^{72} +7.81391 q^{73} +10.6905 q^{74} +2.66942 q^{75} -39.3942 q^{76} +1.90036 q^{77} -0.564856 q^{78} +14.2836 q^{79} -13.3366 q^{80} -4.35523 q^{81} -3.26113 q^{82} +0.946038 q^{83} -9.25689 q^{84} +17.5452 q^{86} +5.34865 q^{87} -25.6378 q^{88} -9.30009 q^{89} -11.1336 q^{90} +0.0514792 q^{91} -28.8871 q^{92} -14.0168 q^{93} +16.4886 q^{94} +7.45802 q^{95} +48.7850 q^{96} +15.7222 q^{97} -17.7267 q^{98} -11.9427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{3} + 21 q^{4} - 12 q^{5} + 9 q^{6} - 6 q^{7} + 12 q^{8} + 21 q^{9} - 3 q^{10} + 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} + 3 q^{15} + 39 q^{16} - 9 q^{18} + 27 q^{19} - 21 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69854 1.90816 0.954079 − 0.299556i \(-0.0968386\pi\)
0.954079 + 0.299556i \(0.0968386\pi\)
\(3\) 2.66942 1.54119 0.770594 − 0.637326i \(-0.219959\pi\)
0.770594 + 0.637326i \(0.219959\pi\)
\(4\) 5.28213 2.64106
\(5\) −1.00000 −0.447214
\(6\) 7.20354 2.94083
\(7\) −0.656507 −0.248137 −0.124068 − 0.992274i \(-0.539594\pi\)
−0.124068 + 0.992274i \(0.539594\pi\)
\(8\) 8.85696 3.13141
\(9\) 4.12579 1.37526
\(10\) −2.69854 −0.853354
\(11\) −2.89465 −0.872770 −0.436385 − 0.899760i \(-0.643741\pi\)
−0.436385 + 0.899760i \(0.643741\pi\)
\(12\) 14.1002 4.07038
\(13\) −0.0784138 −0.0217481 −0.0108740 − 0.999941i \(-0.503461\pi\)
−0.0108740 + 0.999941i \(0.503461\pi\)
\(14\) −1.77161 −0.473483
\(15\) −2.66942 −0.689241
\(16\) 13.3366 3.33416
\(17\) 0 0
\(18\) 11.1336 2.62422
\(19\) −7.45802 −1.71099 −0.855494 − 0.517813i \(-0.826746\pi\)
−0.855494 + 0.517813i \(0.826746\pi\)
\(20\) −5.28213 −1.18112
\(21\) −1.75249 −0.382425
\(22\) −7.81134 −1.66538
\(23\) −5.46884 −1.14033 −0.570166 − 0.821530i \(-0.693121\pi\)
−0.570166 + 0.821530i \(0.693121\pi\)
\(24\) 23.6429 4.82609
\(25\) 1.00000 0.200000
\(26\) −0.211603 −0.0414987
\(27\) 3.00520 0.578352
\(28\) −3.46776 −0.655344
\(29\) 2.00368 0.372073 0.186037 − 0.982543i \(-0.440436\pi\)
0.186037 + 0.982543i \(0.440436\pi\)
\(30\) −7.20354 −1.31518
\(31\) −5.25088 −0.943085 −0.471543 − 0.881843i \(-0.656303\pi\)
−0.471543 + 0.881843i \(0.656303\pi\)
\(32\) 18.2755 3.23069
\(33\) −7.72703 −1.34510
\(34\) 0 0
\(35\) 0.656507 0.110970
\(36\) 21.7930 3.63216
\(37\) 3.96160 0.651283 0.325641 − 0.945493i \(-0.394420\pi\)
0.325641 + 0.945493i \(0.394420\pi\)
\(38\) −20.1258 −3.26483
\(39\) −0.209319 −0.0335179
\(40\) −8.85696 −1.40041
\(41\) −1.20848 −0.188733 −0.0943664 − 0.995538i \(-0.530083\pi\)
−0.0943664 + 0.995538i \(0.530083\pi\)
\(42\) −4.72917 −0.729728
\(43\) 6.50172 0.991503 0.495752 − 0.868464i \(-0.334893\pi\)
0.495752 + 0.868464i \(0.334893\pi\)
\(44\) −15.2899 −2.30504
\(45\) −4.12579 −0.615036
\(46\) −14.7579 −2.17593
\(47\) 6.11019 0.891263 0.445631 − 0.895217i \(-0.352979\pi\)
0.445631 + 0.895217i \(0.352979\pi\)
\(48\) 35.6010 5.13856
\(49\) −6.56900 −0.938428
\(50\) 2.69854 0.381631
\(51\) 0 0
\(52\) −0.414192 −0.0574380
\(53\) 2.02903 0.278708 0.139354 − 0.990243i \(-0.455497\pi\)
0.139354 + 0.990243i \(0.455497\pi\)
\(54\) 8.10967 1.10359
\(55\) 2.89465 0.390315
\(56\) −5.81466 −0.777017
\(57\) −19.9086 −2.63695
\(58\) 5.40701 0.709975
\(59\) 8.74900 1.13902 0.569512 − 0.821983i \(-0.307132\pi\)
0.569512 + 0.821983i \(0.307132\pi\)
\(60\) −14.1002 −1.82033
\(61\) −4.18013 −0.535210 −0.267605 − 0.963529i \(-0.586232\pi\)
−0.267605 + 0.963529i \(0.586232\pi\)
\(62\) −14.1697 −1.79955
\(63\) −2.70861 −0.341253
\(64\) 22.6440 2.83050
\(65\) 0.0784138 0.00972603
\(66\) −20.8517 −2.56667
\(67\) 12.9875 1.58668 0.793341 − 0.608778i \(-0.208340\pi\)
0.793341 + 0.608778i \(0.208340\pi\)
\(68\) 0 0
\(69\) −14.5986 −1.75747
\(70\) 1.77161 0.211748
\(71\) 12.6346 1.49945 0.749727 − 0.661747i \(-0.230185\pi\)
0.749727 + 0.661747i \(0.230185\pi\)
\(72\) 36.5420 4.30651
\(73\) 7.81391 0.914549 0.457274 − 0.889326i \(-0.348826\pi\)
0.457274 + 0.889326i \(0.348826\pi\)
\(74\) 10.6905 1.24275
\(75\) 2.66942 0.308238
\(76\) −39.3942 −4.51883
\(77\) 1.90036 0.216566
\(78\) −0.564856 −0.0639574
\(79\) 14.2836 1.60703 0.803513 − 0.595287i \(-0.202962\pi\)
0.803513 + 0.595287i \(0.202962\pi\)
\(80\) −13.3366 −1.49108
\(81\) −4.35523 −0.483914
\(82\) −3.26113 −0.360132
\(83\) 0.946038 0.103841 0.0519206 − 0.998651i \(-0.483466\pi\)
0.0519206 + 0.998651i \(0.483466\pi\)
\(84\) −9.25689 −1.01001
\(85\) 0 0
\(86\) 17.5452 1.89194
\(87\) 5.34865 0.573435
\(88\) −25.6378 −2.73300
\(89\) −9.30009 −0.985808 −0.492904 − 0.870084i \(-0.664065\pi\)
−0.492904 + 0.870084i \(0.664065\pi\)
\(90\) −11.1336 −1.17359
\(91\) 0.0514792 0.00539649
\(92\) −28.8871 −3.01169
\(93\) −14.0168 −1.45347
\(94\) 16.4886 1.70067
\(95\) 7.45802 0.765177
\(96\) 48.7850 4.97910
\(97\) 15.7222 1.59634 0.798172 − 0.602430i \(-0.205801\pi\)
0.798172 + 0.602430i \(0.205801\pi\)
\(98\) −17.7267 −1.79067
\(99\) −11.9427 −1.20029
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.1.11 ✓ 12
5.4 even 2 7225.2.a.bo.1.2 12
17.4 even 4 1445.2.d.i.866.4 24
17.13 even 4 1445.2.d.i.866.3 24
17.16 even 2 1445.2.a.s.1.11 yes 12
85.84 even 2 7225.2.a.bn.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.2.a.r.1.11 ✓ 12 1.1 even 1 trivial
1445.2.a.s.1.11 yes 12 17.16 even 2
1445.2.d.i.866.3 24 17.13 even 4
1445.2.d.i.866.4 24 17.4 even 4
7225.2.a.bn.1.2 12 85.84 even 2
7225.2.a.bo.1.2 12 5.4 even 2