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Properties

Label 700.2.n.d.281.3
Level $700$
Weight $2$
Character 700.281
Analytic conductor $5.590$
Analytic rank $0$
Dimension $28$
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] = [700,2,Mod(141,700)] 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("700.141"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 281.3
Character \(\chi\) \(=\) 700.281
Dual form 700.2.n.d.421.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.743793 + 0.540397i) q^{3} +(2.18245 + 0.486729i) q^{5} -1.00000 q^{7} +(-0.665852 + 2.04928i) q^{9} +(-0.274818 - 0.845803i) q^{11} +(-1.78105 + 5.48151i) q^{13} +(-1.88632 + 0.817365i) q^{15} +(-1.05923 - 0.769575i) q^{17} +(0.0174389 + 0.0126701i) q^{19} +(0.743793 - 0.540397i) q^{21} +(0.850602 + 2.61788i) q^{23} +(4.52619 + 2.12452i) q^{25} +(-1.46448 - 4.50721i) q^{27} +(0.940481 - 0.683300i) q^{29} +(0.203193 + 0.147628i) q^{31} +(0.661477 + 0.480591i) q^{33} +(-2.18245 - 0.486729i) q^{35} +(-3.30416 + 10.1691i) q^{37} +(-1.63746 - 5.03958i) q^{39} +(-1.36094 + 4.18855i) q^{41} +3.47428 q^{43} +(-2.45064 + 4.14837i) q^{45} +(-5.13764 + 3.73272i) q^{47} +1.00000 q^{49} +1.20372 q^{51} +(-0.923230 + 0.670766i) q^{53} +(-0.188100 - 1.97969i) q^{55} -0.0198178 q^{57} +(-0.614391 + 1.89090i) q^{59} +(0.0244181 + 0.0751513i) q^{61} +(0.665852 - 2.04928i) q^{63} +(-6.55506 + 11.0962i) q^{65} +(-4.89952 - 3.55971i) q^{67} +(-2.04737 - 1.48750i) q^{69} +(9.60480 - 6.97830i) q^{71} +(2.76553 + 8.51144i) q^{73} +(-4.51463 + 0.865734i) q^{75} +(0.274818 + 0.845803i) q^{77} +(1.97874 - 1.43764i) q^{79} +(-1.70472 - 1.23855i) q^{81} +(10.7911 + 7.84021i) q^{83} +(-1.93714 - 2.19512i) q^{85} +(-0.330270 + 1.01647i) q^{87} +(-0.143870 - 0.442787i) q^{89} +(1.78105 - 5.48151i) q^{91} -0.230911 q^{93} +(0.0318927 + 0.0361400i) q^{95} +(3.44420 - 2.50236i) q^{97} +1.91628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q - 4 q^{3} + 6 q^{5} - 28 q^{7} - q^{9} - 4 q^{11} - 7 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{21} - 3 q^{23} + 4 q^{25} + 5 q^{27} + 15 q^{29} - 25 q^{31} + 21 q^{33} - 6 q^{35} - 15 q^{37} - 22 q^{39}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.743793 + 0.540397i −0.429429 + 0.311998i −0.781420 − 0.624005i \(-0.785505\pi\)
0.351992 + 0.936003i \(0.385505\pi\)
\(4\) 0 0
\(5\) 2.18245 + 0.486729i 0.976022 + 0.217672i
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −0.665852 + 2.04928i −0.221951 + 0.683094i
\(10\) 0 0
\(11\) −0.274818 − 0.845803i −0.0828608 − 0.255019i 0.901040 − 0.433737i \(-0.142805\pi\)
−0.983900 + 0.178717i \(0.942805\pi\)
\(12\) 0 0
\(13\) −1.78105 + 5.48151i −0.493974 + 1.52030i 0.324575 + 0.945860i \(0.394779\pi\)
−0.818549 + 0.574436i \(0.805221\pi\)
\(14\) 0 0
\(15\) −1.88632 + 0.817365i −0.487045 + 0.211043i
\(16\) 0 0
\(17\) −1.05923 − 0.769575i −0.256901 − 0.186649i 0.451879 − 0.892079i \(-0.350754\pi\)
−0.708780 + 0.705430i \(0.750754\pi\)
\(18\) 0 0
\(19\) 0.0174389 + 0.0126701i 0.00400076 + 0.00290672i 0.589784 − 0.807561i \(-0.299213\pi\)
−0.585783 + 0.810468i \(0.699213\pi\)
\(20\) 0 0
\(21\) 0.743793 − 0.540397i 0.162309 − 0.117924i
\(22\) 0 0
\(23\) 0.850602 + 2.61788i 0.177363 + 0.545866i 0.999733 − 0.0230857i \(-0.00734905\pi\)
−0.822371 + 0.568952i \(0.807349\pi\)
\(24\) 0 0
\(25\) 4.52619 + 2.12452i 0.905238 + 0.424905i
\(26\) 0 0
\(27\) −1.46448 − 4.50721i −0.281839 − 0.867413i
\(28\) 0 0
\(29\) 0.940481 − 0.683300i 0.174643 − 0.126886i −0.497030 − 0.867733i \(-0.665576\pi\)
0.671673 + 0.740848i \(0.265576\pi\)
\(30\) 0 0
\(31\) 0.203193 + 0.147628i 0.0364945 + 0.0265148i 0.605883 − 0.795554i \(-0.292820\pi\)
−0.569388 + 0.822069i \(0.692820\pi\)
\(32\) 0 0
\(33\) 0.661477 + 0.480591i 0.115148 + 0.0836602i
\(34\) 0 0
\(35\) −2.18245 − 0.486729i −0.368902 − 0.0822722i
\(36\) 0 0
\(37\) −3.30416 + 10.1691i −0.543200 + 1.67180i 0.182032 + 0.983293i \(0.441733\pi\)
−0.725232 + 0.688505i \(0.758267\pi\)
\(38\) 0 0
\(39\) −1.63746 − 5.03958i −0.262203 − 0.806978i
\(40\) 0 0
\(41\) −1.36094 + 4.18855i −0.212543 + 0.654141i 0.786776 + 0.617239i \(0.211749\pi\)
−0.999319 + 0.0369018i \(0.988251\pi\)
\(42\) 0 0
\(43\) 3.47428 0.529823 0.264912 − 0.964273i \(-0.414657\pi\)
0.264912 + 0.964273i \(0.414657\pi\)
\(44\) 0 0
\(45\) −2.45064 + 4.14837i −0.365319 + 0.618403i
\(46\) 0 0
\(47\) −5.13764 + 3.73272i −0.749402 + 0.544473i −0.895641 − 0.444777i \(-0.853283\pi\)
0.146239 + 0.989249i \(0.453283\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.20372 0.168555
\(52\) 0 0
\(53\) −0.923230 + 0.670766i −0.126815 + 0.0921368i −0.649385 − 0.760460i \(-0.724974\pi\)
0.522569 + 0.852597i \(0.324974\pi\)
\(54\) 0 0
\(55\) −0.188100 − 1.97969i −0.0253635 − 0.266941i
\(56\) 0 0
\(57\) −0.0198178 −0.00262494
\(58\) 0 0
\(59\) −0.614391 + 1.89090i −0.0799868 + 0.246174i −0.983051 − 0.183331i \(-0.941312\pi\)
0.903064 + 0.429505i \(0.141312\pi\)
\(60\) 0 0
\(61\) 0.0244181 + 0.0751513i 0.00312642 + 0.00962214i 0.952608 − 0.304202i \(-0.0983898\pi\)
−0.949481 + 0.313824i \(0.898390\pi\)
\(62\) 0 0
\(63\) 0.665852 − 2.04928i 0.0838895 − 0.258185i
\(64\) 0 0
\(65\) −6.55506 + 11.0962i −0.813055 + 1.37632i
\(66\) 0 0
\(67\) −4.89952 − 3.55971i −0.598572 − 0.434888i 0.246800 − 0.969066i \(-0.420621\pi\)
−0.845372 + 0.534179i \(0.820621\pi\)
\(68\) 0 0
\(69\) −2.04737 − 1.48750i −0.246474 − 0.179074i
\(70\) 0 0
\(71\) 9.60480 − 6.97830i 1.13988 − 0.828171i 0.152778 − 0.988261i \(-0.451178\pi\)
0.987103 + 0.160089i \(0.0511782\pi\)
\(72\) 0 0
\(73\) 2.76553 + 8.51144i 0.323681 + 0.996189i 0.972032 + 0.234847i \(0.0754590\pi\)
−0.648351 + 0.761342i \(0.724541\pi\)
\(74\) 0 0
\(75\) −4.51463 + 0.865734i −0.521305 + 0.0999664i
\(76\) 0 0
\(77\) 0.274818 + 0.845803i 0.0313184 + 0.0963882i
\(78\) 0 0
\(79\) 1.97874 − 1.43764i 0.222626 − 0.161747i −0.470882 − 0.882196i \(-0.656064\pi\)
0.693508 + 0.720449i \(0.256064\pi\)
\(80\) 0 0
\(81\) −1.70472 − 1.23855i −0.189413 − 0.137617i
\(82\) 0 0
\(83\) 10.7911 + 7.84021i 1.18448 + 0.860575i 0.992670 − 0.120857i \(-0.0385643\pi\)
0.191810 + 0.981432i \(0.438564\pi\)
\(84\) 0 0
\(85\) −1.93714 − 2.19512i −0.210113 − 0.238094i
\(86\) 0 0
\(87\) −0.330270 + 1.01647i −0.0354087 + 0.108977i
\(88\) 0 0
\(89\) −0.143870 − 0.442787i −0.0152502 − 0.0469353i 0.943142 − 0.332391i \(-0.107855\pi\)
−0.958392 + 0.285455i \(0.907855\pi\)
\(90\) 0 0
\(91\) 1.78105 − 5.48151i 0.186705 − 0.574618i
\(92\) 0 0
\(93\) −0.230911 −0.0239443
\(94\) 0 0
\(95\) 0.0318927 + 0.0361400i 0.00327212 + 0.00370788i
\(96\) 0 0
\(97\) 3.44420 − 2.50236i 0.349706 − 0.254076i −0.399040 − 0.916934i \(-0.630657\pi\)
0.748746 + 0.662858i \(0.230657\pi\)
\(98\) 0 0
\(99\) 1.91628 0.192593
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 700.2.n.d.281.3 ✓ 28
25.21 even 5 inner 700.2.n.d.421.3 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.n.d.281.3 ✓ 28 1.1 even 1 trivial
700.2.n.d.421.3 yes 28 25.21 even 5 inner