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Properties

Label 1600.2.q.e.49.1
Level $1600$
Weight $2$
Character 1600.49
Analytic conductor $12.776$
Analytic rank $0$
Dimension $12$
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] = [1600,2,Mod(49,1600)] 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("1600.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.q (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-2,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.4767670494822400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 49.1
Root \(0.618969 - 1.27156i\) of defining polynomial
Character \(\chi\) \(=\) 1600.49
Dual form 1600.2.q.e.849.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.16859 - 2.16859i) q^{3} +3.30519 q^{7} +6.40553i q^{9} +(-2.01163 - 2.01163i) q^{11} +(0.794042 + 0.794042i) q^{13} -4.61575i q^{17} +(-3.48786 + 3.48786i) q^{19} +(-7.16759 - 7.16759i) q^{21} +7.99801 q^{23} +(7.38518 - 7.38518i) q^{27} +(1.95065 - 1.95065i) q^{29} +5.12695 q^{31} +8.72480i q^{33} +(0.448156 - 0.448156i) q^{37} -3.44390i q^{39} -4.02230i q^{41} +(4.97000 - 4.97000i) q^{43} -5.49112i q^{47} +3.92429 q^{49} +(-10.0096 + 10.0096i) q^{51} +(3.35125 - 3.35125i) q^{53} +15.1274 q^{57} +(2.07673 + 2.07673i) q^{59} +(-0.557208 + 0.557208i) q^{61} +21.1715i q^{63} +(-0.636094 - 0.636094i) q^{67} +(-17.3444 - 17.3444i) q^{69} -6.85258i q^{71} -10.5177 q^{73} +(-6.64883 - 6.64883i) q^{77} -17.3005 q^{79} -12.8142 q^{81} +(-9.48015 - 9.48015i) q^{83} -8.46030 q^{87} +7.62073i q^{89} +(2.62446 + 2.62446i) q^{91} +(-11.1182 - 11.1182i) q^{93} -0.709082i q^{97} +(12.8856 - 12.8856i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 12 q^{7} + 2 q^{11} - 4 q^{13} - 14 q^{19} - 20 q^{21} + 12 q^{23} + 10 q^{27} + 4 q^{31} + 8 q^{37} - 4 q^{49} - 10 q^{51} + 16 q^{53} + 16 q^{57} + 20 q^{59} + 4 q^{61} + 50 q^{67} - 40 q^{73}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −2.16859 − 2.16859i −1.25203 − 1.25203i −0.954807 − 0.297227i \(-0.903938\pi\)
−0.297227 − 0.954807i \(-0.596062\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.30519 1.24924 0.624622 − 0.780927i \(-0.285253\pi\)
0.624622 + 0.780927i \(0.285253\pi\)
\(8\) 0 0
\(9\) 6.40553i 2.13518i
\(10\) 0 0
\(11\) −2.01163 − 2.01163i −0.606530 − 0.606530i 0.335507 − 0.942038i \(-0.391092\pi\)
−0.942038 + 0.335507i \(0.891092\pi\)
\(12\) 0 0
\(13\) 0.794042 + 0.794042i 0.220228 + 0.220228i 0.808594 − 0.588367i \(-0.200229\pi\)
−0.588367 + 0.808594i \(0.700229\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.61575i − 1.11948i −0.828667 − 0.559741i \(-0.810900\pi\)
0.828667 − 0.559741i \(-0.189100\pi\)
\(18\) 0 0
\(19\) −3.48786 + 3.48786i −0.800169 + 0.800169i −0.983122 − 0.182953i \(-0.941434\pi\)
0.182953 + 0.983122i \(0.441434\pi\)
\(20\) 0 0
\(21\) −7.16759 − 7.16759i −1.56410 − 1.56410i
\(22\) 0 0
\(23\) 7.99801 1.66770 0.833850 − 0.551991i \(-0.186132\pi\)
0.833850 + 0.551991i \(0.186132\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 7.38518 − 7.38518i 1.42128 − 1.42128i
\(28\) 0 0
\(29\) 1.95065 − 1.95065i 0.362227 − 0.362227i −0.502406 − 0.864632i \(-0.667552\pi\)
0.864632 + 0.502406i \(0.167552\pi\)
\(30\) 0 0
\(31\) 5.12695 0.920828 0.460414 − 0.887704i \(-0.347701\pi\)
0.460414 + 0.887704i \(0.347701\pi\)
\(32\) 0 0
\(33\) 8.72480i 1.51879i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.448156 − 0.448156i 0.0736764 − 0.0736764i −0.669308 − 0.742985i \(-0.733409\pi\)
0.742985 + 0.669308i \(0.233409\pi\)
\(38\) 0 0
\(39\) − 3.44390i − 0.551465i
\(40\) 0 0
\(41\) − 4.02230i − 0.628177i −0.949394 − 0.314089i \(-0.898301\pi\)
0.949394 − 0.314089i \(-0.101699\pi\)
\(42\) 0 0
\(43\) 4.97000 − 4.97000i 0.757918 − 0.757918i −0.218025 − 0.975943i \(-0.569961\pi\)
0.975943 + 0.218025i \(0.0699615\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 5.49112i − 0.800962i −0.916305 − 0.400481i \(-0.868843\pi\)
0.916305 − 0.400481i \(-0.131157\pi\)
\(48\) 0 0
\(49\) 3.92429 0.560612
\(50\) 0 0
\(51\) −10.0096 + 10.0096i −1.40163 + 1.40163i
\(52\) 0 0
\(53\) 3.35125 − 3.35125i 0.460330 − 0.460330i −0.438434 − 0.898763i \(-0.644467\pi\)
0.898763 + 0.438434i \(0.144467\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 15.1274 2.00368
\(58\) 0 0
\(59\) 2.07673 + 2.07673i 0.270367 + 0.270367i 0.829248 − 0.558881i \(-0.188769\pi\)
−0.558881 + 0.829248i \(0.688769\pi\)
\(60\) 0 0
\(61\) −0.557208 + 0.557208i −0.0713432 + 0.0713432i −0.741878 − 0.670535i \(-0.766065\pi\)
0.670535 + 0.741878i \(0.266065\pi\)
\(62\) 0 0
\(63\) 21.1715i 2.66736i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.636094 − 0.636094i −0.0777112 − 0.0777112i 0.667183 − 0.744894i \(-0.267500\pi\)
−0.744894 + 0.667183i \(0.767500\pi\)
\(68\) 0 0
\(69\) −17.3444 − 17.3444i −2.08802 − 2.08802i
\(70\) 0 0
\(71\) − 6.85258i − 0.813252i −0.913595 − 0.406626i \(-0.866705\pi\)
0.913595 − 0.406626i \(-0.133295\pi\)
\(72\) 0 0
\(73\) −10.5177 −1.23101 −0.615504 − 0.788134i \(-0.711047\pi\)
−0.615504 + 0.788134i \(0.711047\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.64883 − 6.64883i −0.757705 − 0.757705i
\(78\) 0 0
\(79\) −17.3005 −1.94646 −0.973230 − 0.229833i \(-0.926182\pi\)
−0.973230 + 0.229833i \(0.926182\pi\)
\(80\) 0 0
\(81\) −12.8142 −1.42380
\(82\) 0 0
\(83\) −9.48015 − 9.48015i −1.04058 − 1.04058i −0.999141 − 0.0414412i \(-0.986805\pi\)
−0.0414412 − 0.999141i \(-0.513195\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.46030 −0.907040
\(88\) 0 0
\(89\) 7.62073i 0.807796i 0.914804 + 0.403898i \(0.132345\pi\)
−0.914804 + 0.403898i \(0.867655\pi\)
\(90\) 0 0
\(91\) 2.62446 + 2.62446i 0.275118 + 0.275118i
\(92\) 0 0
\(93\) −11.1182 − 11.1182i −1.15291 − 1.15291i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 0.709082i − 0.0719964i −0.999352 − 0.0359982i \(-0.988539\pi\)
0.999352 − 0.0359982i \(-0.0114611\pi\)
\(98\) 0 0
\(99\) 12.8856 − 12.8856i 1.29505 − 1.29505i
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 1600.2.q.e.49.1 12
4.3 odd 2 400.2.q.e.149.6 12
5.2 odd 4 1600.2.l.f.1201.1 12
5.3 odd 4 1600.2.l.g.1201.6 12
5.4 even 2 1600.2.q.f.49.6 12
16.3 odd 4 400.2.q.f.349.1 12
16.13 even 4 1600.2.q.f.849.6 12
20.3 even 4 400.2.l.f.101.4 ✓ 12
20.7 even 4 400.2.l.g.101.3 yes 12
20.19 odd 2 400.2.q.f.149.1 12
80.3 even 4 400.2.l.f.301.4 yes 12
80.13 odd 4 1600.2.l.g.401.6 12
80.19 odd 4 400.2.q.e.349.6 12
80.29 even 4 inner 1600.2.q.e.849.1 12
80.67 even 4 400.2.l.g.301.3 yes 12
80.77 odd 4 1600.2.l.f.401.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.l.f.101.4 ✓ 12 20.3 even 4
400.2.l.f.301.4 yes 12 80.3 even 4
400.2.l.g.101.3 yes 12 20.7 even 4
400.2.l.g.301.3 yes 12 80.67 even 4
400.2.q.e.149.6 12 4.3 odd 2
400.2.q.e.349.6 12 80.19 odd 4
400.2.q.f.149.1 12 20.19 odd 2
400.2.q.f.349.1 12 16.3 odd 4
1600.2.l.f.401.1 12 80.77 odd 4
1600.2.l.f.1201.1 12 5.2 odd 4
1600.2.l.g.401.6 12 80.13 odd 4
1600.2.l.g.1201.6 12 5.3 odd 4
1600.2.q.e.49.1 12 1.1 even 1 trivial
1600.2.q.e.849.1 12 80.29 even 4 inner
1600.2.q.f.49.6 12 5.4 even 2
1600.2.q.f.849.6 12 16.13 even 4