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Properties

Label 1575.4.a.bf.1.4
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $4$
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] = [1575,4,Mod(1,1575)] 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("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-6,0,16,0,0,-28,-93,0,0,-57] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.26729725.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 18x^{2} + 19x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.56826\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.56826 q^{2} -1.40404 q^{4} -7.00000 q^{7} -24.1520 q^{8} -66.4181 q^{11} +34.0329 q^{13} -17.9778 q^{14} -50.7963 q^{16} +6.12934 q^{17} -163.424 q^{19} -170.579 q^{22} +35.8799 q^{23} +87.4053 q^{26} +9.82830 q^{28} -27.7256 q^{29} +74.3417 q^{31} +62.7581 q^{32} +15.7417 q^{34} +260.966 q^{37} -419.716 q^{38} +445.485 q^{41} +474.985 q^{43} +93.2538 q^{44} +92.1489 q^{46} -51.0436 q^{47} +49.0000 q^{49} -47.7837 q^{52} -676.667 q^{53} +169.064 q^{56} -71.2066 q^{58} +115.979 q^{59} -390.356 q^{61} +190.929 q^{62} +567.550 q^{64} +713.721 q^{67} -8.60586 q^{68} -810.524 q^{71} -350.196 q^{73} +670.227 q^{74} +229.455 q^{76} +464.927 q^{77} -50.9307 q^{79} +1144.12 q^{82} -84.8904 q^{83} +1219.88 q^{86} +1604.13 q^{88} -1521.82 q^{89} -238.230 q^{91} -50.3769 q^{92} -131.093 q^{94} +1232.08 q^{97} +125.845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56826 0.908017 0.454008 − 0.890997i \(-0.349994\pi\)
0.454008 + 0.890997i \(0.349994\pi\)
\(3\) 0 0
\(4\) −1.40404 −0.175505
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −24.1520 −1.06738
\(9\) 0 0
\(10\) 0 0
\(11\) −66.4181 −1.82053 −0.910264 − 0.414029i \(-0.864121\pi\)
−0.910264 + 0.414029i \(0.864121\pi\)
\(12\) 0 0
\(13\) 34.0329 0.726079 0.363040 − 0.931774i \(-0.381739\pi\)
0.363040 + 0.931774i \(0.381739\pi\)
\(14\) −17.9778 −0.343198
\(15\) 0 0
\(16\) −50.7963 −0.793692
\(17\) 6.12934 0.0874461 0.0437231 − 0.999044i \(-0.486078\pi\)
0.0437231 + 0.999044i \(0.486078\pi\)
\(18\) 0 0
\(19\) −163.424 −1.97327 −0.986634 − 0.162951i \(-0.947899\pi\)
−0.986634 + 0.162951i \(0.947899\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −170.579 −1.65307
\(23\) 35.8799 0.325282 0.162641 − 0.986685i \(-0.447999\pi\)
0.162641 + 0.986685i \(0.447999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 87.4053 0.659292
\(27\) 0 0
\(28\) 9.82830 0.0663348
\(29\) −27.7256 −0.177535 −0.0887676 − 0.996052i \(-0.528293\pi\)
−0.0887676 + 0.996052i \(0.528293\pi\)
\(30\) 0 0
\(31\) 74.3417 0.430715 0.215358 − 0.976535i \(-0.430908\pi\)
0.215358 + 0.976535i \(0.430908\pi\)
\(32\) 62.7581 0.346693
\(33\) 0 0
\(34\) 15.7417 0.0794026
\(35\) 0 0
\(36\) 0 0
\(37\) 260.966 1.15953 0.579763 − 0.814785i \(-0.303145\pi\)
0.579763 + 0.814785i \(0.303145\pi\)
\(38\) −419.716 −1.79176
\(39\) 0 0
\(40\) 0 0
\(41\) 445.485 1.69690 0.848452 − 0.529272i \(-0.177535\pi\)
0.848452 + 0.529272i \(0.177535\pi\)
\(42\) 0 0
\(43\) 474.985 1.68452 0.842261 − 0.539070i \(-0.181224\pi\)
0.842261 + 0.539070i \(0.181224\pi\)
\(44\) 93.2538 0.319512
\(45\) 0 0
\(46\) 92.1489 0.295361
\(47\) −51.0436 −0.158414 −0.0792072 − 0.996858i \(-0.525239\pi\)
−0.0792072 + 0.996858i \(0.525239\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −47.7837 −0.127431
\(53\) −676.667 −1.75372 −0.876862 − 0.480742i \(-0.840367\pi\)
−0.876862 + 0.480742i \(0.840367\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 169.064 0.403431
\(57\) 0 0
\(58\) −71.2066 −0.161205
\(59\) 115.979 0.255917 0.127959 − 0.991779i \(-0.459158\pi\)
0.127959 + 0.991779i \(0.459158\pi\)
\(60\) 0 0
\(61\) −390.356 −0.819344 −0.409672 − 0.912233i \(-0.634357\pi\)
−0.409672 + 0.912233i \(0.634357\pi\)
\(62\) 190.929 0.391097
\(63\) 0 0
\(64\) 567.550 1.10850
\(65\) 0 0
\(66\) 0 0
\(67\) 713.721 1.30142 0.650708 − 0.759328i \(-0.274472\pi\)
0.650708 + 0.759328i \(0.274472\pi\)
\(68\) −8.60586 −0.0153473
\(69\) 0 0
\(70\) 0 0
\(71\) −810.524 −1.35481 −0.677405 − 0.735610i \(-0.736896\pi\)
−0.677405 + 0.735610i \(0.736896\pi\)
\(72\) 0 0
\(73\) −350.196 −0.561470 −0.280735 − 0.959785i \(-0.590578\pi\)
−0.280735 + 0.959785i \(0.590578\pi\)
\(74\) 670.227 1.05287
\(75\) 0 0
\(76\) 229.455 0.346319
\(77\) 464.927 0.688095
\(78\) 0 0
\(79\) −50.9307 −0.0725336 −0.0362668 − 0.999342i \(-0.511547\pi\)
−0.0362668 + 0.999342i \(0.511547\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1144.12 1.54082
\(83\) −84.8904 −0.112264 −0.0561321 − 0.998423i \(-0.517877\pi\)
−0.0561321 + 0.998423i \(0.517877\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1219.88 1.52957
\(87\) 0 0
\(88\) 1604.13 1.94319
\(89\) −1521.82 −1.81251 −0.906253 − 0.422735i \(-0.861070\pi\)
−0.906253 + 0.422735i \(0.861070\pi\)
\(90\) 0 0
\(91\) −238.230 −0.274432
\(92\) −50.3769 −0.0570887
\(93\) 0 0
\(94\) −131.093 −0.143843
\(95\) 0 0
\(96\) 0 0
\(97\) 1232.08 1.28968 0.644840 − 0.764317i \(-0.276924\pi\)
0.644840 + 0.764317i \(0.276924\pi\)
\(98\) 125.845 0.129717
\(99\) 0 0
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 1575.4.a.bf.1.4 4
3.2 odd 2 525.4.a.v.1.1 yes 4
5.4 even 2 1575.4.a.bm.1.1 4
15.2 even 4 525.4.d.o.274.3 8
15.8 even 4 525.4.d.o.274.6 8
15.14 odd 2 525.4.a.s.1.4 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.s.1.4 ✓ 4 15.14 odd 2
525.4.a.v.1.1 yes 4 3.2 odd 2
525.4.d.o.274.3 8 15.2 even 4
525.4.d.o.274.6 8 15.8 even 4
1575.4.a.bf.1.4 4 1.1 even 1 trivial
1575.4.a.bm.1.1 4 5.4 even 2