Properties

Label 475.2.a.e.1.1
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $1$
Dimension $3$
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] = [475,2,Mod(1,475)] 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("475.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.37720\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.37720 q^{2} -1.27389 q^{3} +3.65109 q^{4} +3.02830 q^{6} -0.726109 q^{7} -3.92498 q^{8} -1.37720 q^{9} -0.273891 q^{11} -4.65109 q^{12} +5.95328 q^{13} +1.72611 q^{14} +2.02830 q^{16} -5.27389 q^{17} +3.27389 q^{18} +1.00000 q^{19} +0.924984 q^{21} +0.651093 q^{22} +3.67939 q^{23} +5.00000 q^{24} -14.1522 q^{26} +5.57608 q^{27} -2.65109 q^{28} -2.27389 q^{29} +3.19887 q^{31} +3.02830 q^{32} +0.348907 q^{33} +12.5371 q^{34} -5.02830 q^{36} -8.12386 q^{37} -2.37720 q^{38} -7.58383 q^{39} -9.43380 q^{41} -2.19887 q^{42} -9.81100 q^{43} -1.00000 q^{44} -8.74666 q^{46} -12.1599 q^{47} -2.58383 q^{48} -6.47277 q^{49} +6.71836 q^{51} +21.7360 q^{52} -5.69781 q^{53} -13.2555 q^{54} +2.84997 q^{56} -1.27389 q^{57} +5.40550 q^{58} -4.20662 q^{59} -0.103312 q^{61} -7.60437 q^{62} +1.00000 q^{63} -11.2555 q^{64} -0.829422 q^{66} +11.7827 q^{67} -19.2555 q^{68} -4.68714 q^{69} +5.75441 q^{71} +5.40550 q^{72} -6.67939 q^{73} +19.3121 q^{74} +3.65109 q^{76} +0.198875 q^{77} +18.0283 q^{78} +3.87826 q^{79} -2.97170 q^{81} +22.4260 q^{82} +0.488265 q^{83} +3.37720 q^{84} +23.3227 q^{86} +2.89669 q^{87} +1.07502 q^{88} -16.4338 q^{89} -4.32273 q^{91} +13.4338 q^{92} -4.07502 q^{93} +28.9066 q^{94} -3.85772 q^{96} +4.44447 q^{97} +15.3871 q^{98} +0.377203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} + 7 q^{14} - 6 q^{16} - 14 q^{17} + 8 q^{18} + 3 q^{19} - 6 q^{21} - 5 q^{22} - 8 q^{23} + 15 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37720 −1.68094 −0.840468 − 0.541861i \(-0.817720\pi\)
−0.840468 + 0.541861i \(0.817720\pi\)
\(3\) −1.27389 −0.735481 −0.367741 − 0.929928i \(-0.619869\pi\)
−0.367741 + 0.929928i \(0.619869\pi\)
\(4\) 3.65109 1.82555
\(5\) 0 0
\(6\) 3.02830 1.23630
\(7\) −0.726109 −0.274444 −0.137222 − 0.990540i \(-0.543817\pi\)
−0.137222 + 0.990540i \(0.543817\pi\)
\(8\) −3.92498 −1.38769
\(9\) −1.37720 −0.459068
\(10\) 0 0
\(11\) −0.273891 −0.0825811 −0.0412906 − 0.999147i \(-0.513147\pi\)
−0.0412906 + 0.999147i \(0.513147\pi\)
\(12\) −4.65109 −1.34266
\(13\) 5.95328 1.65114 0.825571 − 0.564298i \(-0.190853\pi\)
0.825571 + 0.564298i \(0.190853\pi\)
\(14\) 1.72611 0.461322
\(15\) 0 0
\(16\) 2.02830 0.507074
\(17\) −5.27389 −1.27911 −0.639553 − 0.768747i \(-0.720881\pi\)
−0.639553 + 0.768747i \(0.720881\pi\)
\(18\) 3.27389 0.771663
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.924984 0.201848
\(22\) 0.651093 0.138814
\(23\) 3.67939 0.767206 0.383603 − 0.923498i \(-0.374683\pi\)
0.383603 + 0.923498i \(0.374683\pi\)
\(24\) 5.00000 1.02062
\(25\) 0 0
\(26\) −14.1522 −2.77547
\(27\) 5.57608 1.07312
\(28\) −2.65109 −0.501010
\(29\) −2.27389 −0.422251 −0.211125 − 0.977459i \(-0.567713\pi\)
−0.211125 + 0.977459i \(0.567713\pi\)
\(30\) 0 0
\(31\) 3.19887 0.574535 0.287267 − 0.957850i \(-0.407253\pi\)
0.287267 + 0.957850i \(0.407253\pi\)
\(32\) 3.02830 0.535332
\(33\) 0.348907 0.0607368
\(34\) 12.5371 2.15010
\(35\) 0 0
\(36\) −5.02830 −0.838049
\(37\) −8.12386 −1.33555 −0.667777 − 0.744361i \(-0.732754\pi\)
−0.667777 + 0.744361i \(0.732754\pi\)
\(38\) −2.37720 −0.385633
\(39\) −7.58383 −1.21438
\(40\) 0 0
\(41\) −9.43380 −1.47331 −0.736656 − 0.676268i \(-0.763596\pi\)
−0.736656 + 0.676268i \(0.763596\pi\)
\(42\) −2.19887 −0.339294
\(43\) −9.81100 −1.49616 −0.748082 − 0.663607i \(-0.769025\pi\)
−0.748082 + 0.663607i \(0.769025\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −8.74666 −1.28962
\(47\) −12.1599 −1.77370 −0.886852 − 0.462053i \(-0.847113\pi\)
−0.886852 + 0.462053i \(0.847113\pi\)
\(48\) −2.58383 −0.372943
\(49\) −6.47277 −0.924681
\(50\) 0 0
\(51\) 6.71836 0.940758
\(52\) 21.7360 3.01424
\(53\) −5.69781 −0.782655 −0.391327 − 0.920252i \(-0.627984\pi\)
−0.391327 + 0.920252i \(0.627984\pi\)
\(54\) −13.2555 −1.80384
\(55\) 0 0
\(56\) 2.84997 0.380843
\(57\) −1.27389 −0.168731
\(58\) 5.40550 0.709777
\(59\) −4.20662 −0.547656 −0.273828 − 0.961779i \(-0.588290\pi\)
−0.273828 + 0.961779i \(0.588290\pi\)
\(60\) 0 0
\(61\) −0.103312 −0.0132278 −0.00661389 − 0.999978i \(-0.502105\pi\)
−0.00661389 + 0.999978i \(0.502105\pi\)
\(62\) −7.60437 −0.965756
\(63\) 1.00000 0.125988
\(64\) −11.2555 −1.40693
\(65\) 0 0
\(66\) −0.829422 −0.102095
\(67\) 11.7827 1.43949 0.719743 − 0.694241i \(-0.244260\pi\)
0.719743 + 0.694241i \(0.244260\pi\)
\(68\) −19.2555 −2.33507
\(69\) −4.68714 −0.564265
\(70\) 0 0
\(71\) 5.75441 0.682922 0.341461 − 0.939896i \(-0.389078\pi\)
0.341461 + 0.939896i \(0.389078\pi\)
\(72\) 5.40550 0.637044
\(73\) −6.67939 −0.781763 −0.390882 − 0.920441i \(-0.627830\pi\)
−0.390882 + 0.920441i \(0.627830\pi\)
\(74\) 19.3121 2.24498
\(75\) 0 0
\(76\) 3.65109 0.418809
\(77\) 0.198875 0.0226639
\(78\) 18.0283 2.04130
\(79\) 3.87826 0.436339 0.218169 − 0.975911i \(-0.429991\pi\)
0.218169 + 0.975911i \(0.429991\pi\)
\(80\) 0 0
\(81\) −2.97170 −0.330189
\(82\) 22.4260 2.47654
\(83\) 0.488265 0.0535941 0.0267970 − 0.999641i \(-0.491469\pi\)
0.0267970 + 0.999641i \(0.491469\pi\)
\(84\) 3.37720 0.368483
\(85\) 0 0
\(86\) 23.3227 2.51495
\(87\) 2.89669 0.310558
\(88\) 1.07502 0.114597
\(89\) −16.4338 −1.74198 −0.870989 − 0.491302i \(-0.836521\pi\)
−0.870989 + 0.491302i \(0.836521\pi\)
\(90\) 0 0
\(91\) −4.32273 −0.453146
\(92\) 13.4338 1.40057
\(93\) −4.07502 −0.422559
\(94\) 28.9066 2.98148
\(95\) 0 0
\(96\) −3.85772 −0.393727
\(97\) 4.44447 0.451267 0.225634 − 0.974212i \(-0.427555\pi\)
0.225634 + 0.974212i \(0.427555\pi\)
\(98\) 15.3871 1.55433
\(99\) 0.377203 0.0379103
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 475.2.a.e.1.1 ✓ 3
3.2 odd 2 4275.2.a.bm.1.3 3
4.3 odd 2 7600.2.a.cc.1.2 3
5.2 odd 4 475.2.b.b.324.1 6
5.3 odd 4 475.2.b.b.324.6 6
5.4 even 2 475.2.a.g.1.3 yes 3
15.14 odd 2 4275.2.a.ba.1.1 3
19.18 odd 2 9025.2.a.bc.1.3 3
20.19 odd 2 7600.2.a.bh.1.2 3
95.94 odd 2 9025.2.a.y.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.1 ✓ 3 1.1 even 1 trivial
475.2.a.g.1.3 yes 3 5.4 even 2
475.2.b.b.324.1 6 5.2 odd 4
475.2.b.b.324.6 6 5.3 odd 4
4275.2.a.ba.1.1 3 15.14 odd 2
4275.2.a.bm.1.3 3 3.2 odd 2
7600.2.a.bh.1.2 3 20.19 odd 2
7600.2.a.cc.1.2 3 4.3 odd 2
9025.2.a.y.1.1 3 95.94 odd 2
9025.2.a.bc.1.3 3 19.18 odd 2