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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [725,2,Mod(157,725)] 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("725.157"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 568.2
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 725.568
Dual form 725.2.e.a.157.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.23607i q^{2} -3.00000 q^{4} +(2.23607 - 2.23607i) q^{7} -2.23607i q^{8} -3.00000 q^{9} +(4.00000 - 4.00000i) q^{11} +(4.47214 - 4.47214i) q^{13} +(5.00000 + 5.00000i) q^{14} -1.00000 q^{16} -4.47214i q^{17} -6.70820i q^{18} +(-2.00000 - 2.00000i) q^{19} +(8.94427 + 8.94427i) q^{22} +(-2.23607 - 2.23607i) q^{23} +(10.0000 + 10.0000i) q^{26} +(-6.70820 + 6.70820i) q^{28} +(2.00000 + 5.00000i) q^{29} +(-4.00000 + 4.00000i) q^{31} -6.70820i q^{32} +10.0000 q^{34} +9.00000 q^{36} -4.47214 q^{37} +(4.47214 - 4.47214i) q^{38} +(1.00000 + 1.00000i) q^{41} +4.47214 q^{43} +(-12.0000 + 12.0000i) q^{44} +(5.00000 - 5.00000i) q^{46} +4.47214 q^{47} -3.00000i q^{49} +(-13.4164 + 13.4164i) q^{52} +(8.94427 + 8.94427i) q^{53} +(-5.00000 - 5.00000i) q^{56} +(-11.1803 + 4.47214i) q^{58} +(-9.00000 + 9.00000i) q^{61} +(-8.94427 - 8.94427i) q^{62} +(-6.70820 + 6.70820i) q^{63} +13.0000 q^{64} +(-2.23607 - 2.23607i) q^{67} +13.4164i q^{68} -12.0000i q^{71} +6.70820i q^{72} +4.47214i q^{73} -10.0000i q^{74} +(6.00000 + 6.00000i) q^{76} -17.8885i q^{77} +(-2.00000 - 2.00000i) q^{79} +9.00000 q^{81} +(-2.23607 + 2.23607i) q^{82} +(6.70820 + 6.70820i) q^{83} +10.0000i q^{86} +(-8.94427 - 8.94427i) q^{88} +(-7.00000 - 7.00000i) q^{89} -20.0000i q^{91} +(6.70820 + 6.70820i) q^{92} +10.0000i q^{94} +4.47214 q^{97} +6.70820 q^{98} +(-12.0000 + 12.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 12 q^{9} + 16 q^{11} + 20 q^{14} - 4 q^{16} - 8 q^{19} + 40 q^{26} + 8 q^{29} - 16 q^{31} + 40 q^{34} + 36 q^{36} + 4 q^{41} - 48 q^{44} + 20 q^{46} - 20 q^{56} - 36 q^{61} + 52 q^{64}+ \cdots - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607i 1.58114i 0.612372 + 0.790569i \(0.290215\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −3.00000 −1.50000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.23607 − 2.23607i 0.845154 − 0.845154i −0.144370 − 0.989524i \(-0.546115\pi\)
0.989524 + 0.144370i \(0.0461154\pi\)
\(8\) − 2.23607i − 0.790569i
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.00000 − 4.00000i 1.20605 − 1.20605i 0.233748 − 0.972297i \(-0.424901\pi\)
0.972297 − 0.233748i \(-0.0750991\pi\)
\(12\) 0 0
\(13\) 4.47214 − 4.47214i 1.24035 − 1.24035i 0.280491 − 0.959857i \(-0.409503\pi\)
0.959857 − 0.280491i \(-0.0904971\pi\)
\(14\) 5.00000 + 5.00000i 1.33631 + 1.33631i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 4.47214i − 1.08465i −0.840168 − 0.542326i \(-0.817544\pi\)
0.840168 − 0.542326i \(-0.182456\pi\)
\(18\) − 6.70820i − 1.58114i
\(19\) −2.00000 − 2.00000i −0.458831 − 0.458831i 0.439440 − 0.898272i \(-0.355177\pi\)
−0.898272 + 0.439440i \(0.855177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8.94427 + 8.94427i 1.90693 + 1.90693i
\(23\) −2.23607 − 2.23607i −0.466252 − 0.466252i 0.434446 − 0.900698i \(-0.356944\pi\)
−0.900698 + 0.434446i \(0.856944\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 10.0000 + 10.0000i 1.96116 + 1.96116i
\(27\) 0 0
\(28\) −6.70820 + 6.70820i −1.26773 + 1.26773i
\(29\) 2.00000 + 5.00000i 0.371391 + 0.928477i
\(30\) 0 0
\(31\) −4.00000 + 4.00000i −0.718421 + 0.718421i −0.968282 − 0.249861i \(-0.919615\pi\)
0.249861 + 0.968282i \(0.419615\pi\)
\(32\) − 6.70820i − 1.18585i
\(33\) 0 0
\(34\) 10.0000 1.71499
\(35\) 0 0
\(36\) 9.00000 1.50000
\(37\) −4.47214 −0.735215 −0.367607 − 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 4.47214 − 4.47214i 0.725476 − 0.725476i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 + 1.00000i 0.156174 + 0.156174i 0.780869 − 0.624695i \(-0.214777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 4.47214 0.681994 0.340997 − 0.940064i \(-0.389235\pi\)
0.340997 + 0.940064i \(0.389235\pi\)
\(44\) −12.0000 + 12.0000i −1.80907 + 1.80907i
\(45\) 0 0
\(46\) 5.00000 − 5.00000i 0.737210 − 0.737210i
\(47\) 4.47214 0.652328 0.326164 − 0.945313i \(-0.394244\pi\)
0.326164 + 0.945313i \(0.394244\pi\)
\(48\) 0 0
\(49\) − 3.00000i − 0.428571i
\(50\) 0 0
\(51\) 0 0
\(52\) −13.4164 + 13.4164i −1.86052 + 1.86052i
\(53\) 8.94427 + 8.94427i 1.22859 + 1.22859i 0.964497 + 0.264093i \(0.0850726\pi\)
0.264093 + 0.964497i \(0.414927\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.00000 − 5.00000i −0.668153 − 0.668153i
\(57\) 0 0
\(58\) −11.1803 + 4.47214i −1.46805 + 0.587220i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −9.00000 + 9.00000i −1.15233 + 1.15233i −0.166248 + 0.986084i \(0.553165\pi\)
−0.986084 + 0.166248i \(0.946835\pi\)
\(62\) −8.94427 − 8.94427i −1.13592 − 1.13592i
\(63\) −6.70820 + 6.70820i −0.845154 + 0.845154i
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) −2.23607 − 2.23607i −0.273179 − 0.273179i 0.557199 − 0.830379i \(-0.311876\pi\)
−0.830379 + 0.557199i \(0.811876\pi\)
\(68\) 13.4164i 1.62698i
\(69\) 0 0
\(70\) 0 0
\(71\) − 12.0000i − 1.42414i −0.702109 − 0.712069i \(-0.747758\pi\)
0.702109 − 0.712069i \(-0.252242\pi\)
\(72\) 6.70820i 0.790569i
\(73\) 4.47214i 0.523424i 0.965146 + 0.261712i \(0.0842870\pi\)
−0.965146 + 0.261712i \(0.915713\pi\)
\(74\) − 10.0000i − 1.16248i
\(75\) 0 0
\(76\) 6.00000 + 6.00000i 0.688247 + 0.688247i
\(77\) − 17.8885i − 2.03859i
\(78\) 0 0
\(79\) −2.00000 − 2.00000i −0.225018 − 0.225018i 0.585590 − 0.810607i \(-0.300863\pi\)
−0.810607 + 0.585590i \(0.800863\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −2.23607 + 2.23607i −0.246932 + 0.246932i
\(83\) 6.70820 + 6.70820i 0.736321 + 0.736321i 0.971864 − 0.235543i \(-0.0756868\pi\)
−0.235543 + 0.971864i \(0.575687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000i 1.07833i
\(87\) 0 0
\(88\) −8.94427 − 8.94427i −0.953463 − 0.953463i
\(89\) −7.00000 − 7.00000i −0.741999 − 0.741999i 0.230964 − 0.972962i \(-0.425812\pi\)
−0.972962 + 0.230964i \(0.925812\pi\)
\(90\) 0 0
\(91\) − 20.0000i − 2.09657i
\(92\) 6.70820 + 6.70820i 0.699379 + 0.699379i
\(93\) 0 0
\(94\) 10.0000i 1.03142i
\(95\) 0 0
\(96\) 0 0
\(97\) 4.47214 0.454077 0.227038 − 0.973886i \(-0.427096\pi\)
0.227038 + 0.973886i \(0.427096\pi\)
\(98\) 6.70820 0.677631
\(99\) −12.0000 + 12.0000i −1.20605 + 1.20605i
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 725.2.e.a.568.2 yes 4
5.2 odd 4 725.2.j.a.307.1 yes 4
5.3 odd 4 725.2.j.a.307.2 yes 4
5.4 even 2 inner 725.2.e.a.568.1 yes 4
29.12 odd 4 725.2.j.a.418.1 yes 4
145.12 even 4 inner 725.2.e.a.157.1 ✓ 4
145.99 odd 4 725.2.j.a.418.2 yes 4
145.128 even 4 inner 725.2.e.a.157.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
725.2.e.a.157.1 ✓ 4 145.12 even 4 inner
725.2.e.a.157.2 yes 4 145.128 even 4 inner
725.2.e.a.568.1 yes 4 5.4 even 2 inner
725.2.e.a.568.2 yes 4 1.1 even 1 trivial
725.2.j.a.307.1 yes 4 5.2 odd 4
725.2.j.a.307.2 yes 4 5.3 odd 4
725.2.j.a.418.1 yes 4 29.12 odd 4
725.2.j.a.418.2 yes 4 145.99 odd 4