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Properties

Label 1575.4.a.bh.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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 = [4,0,0,10,0,0,-28,0,0,0,0] 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{42 +2 \sqrt{329}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.42371\) 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)
 
\(f(q)\) \(=\) \(q-4.42371 q^{2} +11.5692 q^{4} -7.00000 q^{7} -15.7890 q^{8} -23.3775 q^{11} -3.56918 q^{13} +30.9659 q^{14} -22.7075 q^{16} +57.5082 q^{17} -51.1384 q^{19} +103.415 q^{22} +65.6390 q^{23} +15.7890 q^{26} -80.9843 q^{28} +41.6147 q^{29} -167.538 q^{31} +226.763 q^{32} -254.399 q^{34} +224.538 q^{37} +226.221 q^{38} +196.688 q^{41} -58.9685 q^{43} -270.458 q^{44} -290.368 q^{46} +41.9282 q^{47} +49.0000 q^{49} -41.2925 q^{52} -33.3445 q^{53} +110.523 q^{56} -184.091 q^{58} -229.212 q^{59} -700.613 q^{61} +741.138 q^{62} -821.475 q^{64} +453.890 q^{67} +665.322 q^{68} +930.571 q^{71} -370.182 q^{73} -993.289 q^{74} -591.629 q^{76} +163.642 q^{77} -54.6007 q^{79} -870.091 q^{82} -430.045 q^{83} +260.859 q^{86} +369.107 q^{88} -737.202 q^{89} +24.9843 q^{91} +759.390 q^{92} -185.478 q^{94} -150.371 q^{97} -216.762 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} - 28 q^{7} + 22 q^{13} + 18 q^{16} - 132 q^{19} + 196 q^{22} - 70 q^{28} - 126 q^{31} - 546 q^{34} + 354 q^{37} + 272 q^{43} - 182 q^{46} + 196 q^{49} - 274 q^{52} + 98 q^{58} - 1170 q^{61}+ \cdots - 1980 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.42371 −1.56402 −0.782008 − 0.623268i \(-0.785805\pi\)
−0.782008 + 0.623268i \(0.785805\pi\)
\(3\) 0 0
\(4\) 11.5692 1.44615
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −15.7890 −0.697782
\(9\) 0 0
\(10\) 0 0
\(11\) −23.3775 −0.640779 −0.320390 − 0.947286i \(-0.603814\pi\)
−0.320390 + 0.947286i \(0.603814\pi\)
\(12\) 0 0
\(13\) −3.56918 −0.0761471 −0.0380735 − 0.999275i \(-0.512122\pi\)
−0.0380735 + 0.999275i \(0.512122\pi\)
\(14\) 30.9659 0.591143
\(15\) 0 0
\(16\) −22.7075 −0.354805
\(17\) 57.5082 0.820458 0.410229 − 0.911983i \(-0.365449\pi\)
0.410229 + 0.911983i \(0.365449\pi\)
\(18\) 0 0
\(19\) −51.1384 −0.617471 −0.308735 − 0.951148i \(-0.599906\pi\)
−0.308735 + 0.951148i \(0.599906\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 103.415 1.00219
\(23\) 65.6390 0.595073 0.297537 − 0.954710i \(-0.403835\pi\)
0.297537 + 0.954710i \(0.403835\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 15.7890 0.119095
\(27\) 0 0
\(28\) −80.9843 −0.546592
\(29\) 41.6147 0.266471 0.133235 − 0.991084i \(-0.457463\pi\)
0.133235 + 0.991084i \(0.457463\pi\)
\(30\) 0 0
\(31\) −167.538 −0.970666 −0.485333 − 0.874329i \(-0.661302\pi\)
−0.485333 + 0.874329i \(0.661302\pi\)
\(32\) 226.763 1.25270
\(33\) 0 0
\(34\) −254.399 −1.28321
\(35\) 0 0
\(36\) 0 0
\(37\) 224.538 0.997669 0.498835 − 0.866697i \(-0.333761\pi\)
0.498835 + 0.866697i \(0.333761\pi\)
\(38\) 226.221 0.965734
\(39\) 0 0
\(40\) 0 0
\(41\) 196.688 0.749208 0.374604 − 0.927185i \(-0.377779\pi\)
0.374604 + 0.927185i \(0.377779\pi\)
\(42\) 0 0
\(43\) −58.9685 −0.209131 −0.104565 − 0.994518i \(-0.533345\pi\)
−0.104565 + 0.994518i \(0.533345\pi\)
\(44\) −270.458 −0.926661
\(45\) 0 0
\(46\) −290.368 −0.930704
\(47\) 41.9282 0.130125 0.0650623 − 0.997881i \(-0.479275\pi\)
0.0650623 + 0.997881i \(0.479275\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −41.2925 −0.110120
\(53\) −33.3445 −0.0864192 −0.0432096 − 0.999066i \(-0.513758\pi\)
−0.0432096 + 0.999066i \(0.513758\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 110.523 0.263737
\(57\) 0 0
\(58\) −184.091 −0.416765
\(59\) −229.212 −0.505777 −0.252888 − 0.967495i \(-0.581381\pi\)
−0.252888 + 0.967495i \(0.581381\pi\)
\(60\) 0 0
\(61\) −700.613 −1.47056 −0.735281 − 0.677762i \(-0.762950\pi\)
−0.735281 + 0.677762i \(0.762950\pi\)
\(62\) 741.138 1.51814
\(63\) 0 0
\(64\) −821.475 −1.60444
\(65\) 0 0
\(66\) 0 0
\(67\) 453.890 0.827634 0.413817 − 0.910360i \(-0.364195\pi\)
0.413817 + 0.910360i \(0.364195\pi\)
\(68\) 665.322 1.18650
\(69\) 0 0
\(70\) 0 0
\(71\) 930.571 1.55547 0.777735 − 0.628592i \(-0.216368\pi\)
0.777735 + 0.628592i \(0.216368\pi\)
\(72\) 0 0
\(73\) −370.182 −0.593514 −0.296757 − 0.954953i \(-0.595905\pi\)
−0.296757 + 0.954953i \(0.595905\pi\)
\(74\) −993.289 −1.56037
\(75\) 0 0
\(76\) −591.629 −0.892954
\(77\) 163.642 0.242192
\(78\) 0 0
\(79\) −54.6007 −0.0777602 −0.0388801 − 0.999244i \(-0.512379\pi\)
−0.0388801 + 0.999244i \(0.512379\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −870.091 −1.17177
\(83\) −430.045 −0.568718 −0.284359 − 0.958718i \(-0.591781\pi\)
−0.284359 + 0.958718i \(0.591781\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 260.859 0.327084
\(87\) 0 0
\(88\) 369.107 0.447124
\(89\) −737.202 −0.878014 −0.439007 − 0.898484i \(-0.644670\pi\)
−0.439007 + 0.898484i \(0.644670\pi\)
\(90\) 0 0
\(91\) 24.9843 0.0287809
\(92\) 759.390 0.860564
\(93\) 0 0
\(94\) −185.478 −0.203517
\(95\) 0 0
\(96\) 0 0
\(97\) −150.371 −0.157401 −0.0787004 − 0.996898i \(-0.525077\pi\)
−0.0787004 + 0.996898i \(0.525077\pi\)
\(98\) −216.762 −0.223431
\(99\) 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.bh.1.1 ✓ 4
3.2 odd 2 inner 1575.4.a.bh.1.4 yes 4
5.4 even 2 1575.4.a.bi.1.4 yes 4
15.14 odd 2 1575.4.a.bi.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1575.4.a.bh.1.1 ✓ 4 1.1 even 1 trivial
1575.4.a.bh.1.4 yes 4 3.2 odd 2 inner
1575.4.a.bi.1.1 yes 4 15.14 odd 2
1575.4.a.bi.1.4 yes 4 5.4 even 2