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Properties

Label 1700.2.o.e.1101.1
Level $1700$
Weight $2$
Character 1700.1101
Analytic conductor $13.575$
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] = [1700,2,Mod(701,1700)] 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("1700.701"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1700.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-2] 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: \(13.5745683436\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 2 x^{9} + 55 x^{8} - 106 x^{7} + 104 x^{6} + 102 x^{5} + 187 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1101.1
Root \(1.89391 - 1.89391i\) of defining polynomial
Character \(\chi\) \(=\) 1700.1101
Dual form 1700.2.o.e.701.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.89391 - 1.89391i) q^{3} +(-0.735996 + 0.735996i) q^{7} +4.17377i q^{9} +(3.03089 - 3.03089i) q^{11} -2.63086 q^{13} +(-1.48702 + 3.84562i) q^{17} -1.90121i q^{19} +2.78782 q^{21} +(-4.03089 + 4.03089i) q^{23} +(2.22302 - 2.22302i) q^{27} +(-1.32816 - 1.32816i) q^{29} +(3.52476 + 3.52476i) q^{31} -11.4805 q^{33} +(-1.41419 - 1.41419i) q^{37} +(4.98260 + 4.98260i) q^{39} +(-6.81775 + 6.81775i) q^{41} +2.04497i q^{43} +12.7936 q^{47} +5.91662i q^{49} +(10.0995 - 4.46696i) q^{51} -2.67442i q^{53} +(-3.60072 + 3.60072i) q^{57} +2.17237i q^{59} +(-2.54482 + 2.54482i) q^{61} +(-3.07188 - 3.07188i) q^{63} +1.34369 q^{67} +15.2683 q^{69} +(2.23664 + 2.23664i) q^{71} +(-1.57165 - 1.57165i) q^{73} +4.46144i q^{77} +(-2.42275 + 2.42275i) q^{79} +4.10094 q^{81} -6.62151i q^{83} +5.03081i q^{87} +9.17647 q^{89} +(1.93630 - 1.93630i) q^{91} -13.3512i q^{93} +(13.8371 + 13.8371i) q^{97} +(12.6502 + 12.6502i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 4 q^{13} + 2 q^{17} - 8 q^{21} - 12 q^{23} - 2 q^{27} - 14 q^{29} - 14 q^{31} + 12 q^{33} - 2 q^{37} - 10 q^{41} + 4 q^{47} + 12 q^{51} - 30 q^{57} - 8 q^{61} - 38 q^{63} + 20 q^{67} - 8 q^{69}+ \cdots + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(851\) \(1601\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.89391 − 1.89391i −1.09345 − 1.09345i −0.995158 − 0.0982903i \(-0.968663\pi\)
−0.0982903 − 0.995158i \(-0.531337\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.735996 + 0.735996i −0.278180 + 0.278180i −0.832382 − 0.554202i \(-0.813024\pi\)
0.554202 + 0.832382i \(0.313024\pi\)
\(8\) 0 0
\(9\) 4.17377i 1.39126i
\(10\) 0 0
\(11\) 3.03089 − 3.03089i 0.913848 − 0.913848i −0.0827247 − 0.996572i \(-0.526362\pi\)
0.996572 + 0.0827247i \(0.0263622\pi\)
\(12\) 0 0
\(13\) −2.63086 −0.729668 −0.364834 − 0.931073i \(-0.618874\pi\)
−0.364834 + 0.931073i \(0.618874\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.48702 + 3.84562i −0.360656 + 0.932699i
\(18\) 0 0
\(19\) − 1.90121i − 0.436168i −0.975930 − 0.218084i \(-0.930019\pi\)
0.975930 − 0.218084i \(-0.0699806\pi\)
\(20\) 0 0
\(21\) 2.78782 0.608351
\(22\) 0 0
\(23\) −4.03089 + 4.03089i −0.840499 + 0.840499i −0.988924 − 0.148425i \(-0.952580\pi\)
0.148425 + 0.988924i \(0.452580\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.22302 − 2.22302i 0.427820 − 0.427820i
\(28\) 0 0
\(29\) −1.32816 − 1.32816i −0.246632 − 0.246632i 0.572955 − 0.819587i \(-0.305797\pi\)
−0.819587 + 0.572955i \(0.805797\pi\)
\(30\) 0 0
\(31\) 3.52476 + 3.52476i 0.633066 + 0.633066i 0.948836 − 0.315770i \(-0.102263\pi\)
−0.315770 + 0.948836i \(0.602263\pi\)
\(32\) 0 0
\(33\) −11.4805 −1.99849
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.41419 − 1.41419i −0.232491 − 0.232491i 0.581240 − 0.813732i \(-0.302568\pi\)
−0.813732 + 0.581240i \(0.802568\pi\)
\(38\) 0 0
\(39\) 4.98260 + 4.98260i 0.797854 + 0.797854i
\(40\) 0 0
\(41\) −6.81775 + 6.81775i −1.06475 + 1.06475i −0.0670011 + 0.997753i \(0.521343\pi\)
−0.997753 + 0.0670011i \(0.978657\pi\)
\(42\) 0 0
\(43\) 2.04497i 0.311854i 0.987769 + 0.155927i \(0.0498365\pi\)
−0.987769 + 0.155927i \(0.950163\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.7936 1.86614 0.933068 − 0.359701i \(-0.117121\pi\)
0.933068 + 0.359701i \(0.117121\pi\)
\(48\) 0 0
\(49\) 5.91662i 0.845232i
\(50\) 0 0
\(51\) 10.0995 − 4.46696i 1.41422 − 0.625500i
\(52\) 0 0
\(53\) − 2.67442i − 0.367360i −0.982986 − 0.183680i \(-0.941199\pi\)
0.982986 − 0.183680i \(-0.0588010\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.60072 + 3.60072i −0.476927 + 0.476927i
\(58\) 0 0
\(59\) 2.17237i 0.282818i 0.989951 + 0.141409i \(0.0451633\pi\)
−0.989951 + 0.141409i \(0.954837\pi\)
\(60\) 0 0
\(61\) −2.54482 + 2.54482i −0.325831 + 0.325831i −0.850999 − 0.525168i \(-0.824003\pi\)
0.525168 + 0.850999i \(0.324003\pi\)
\(62\) 0 0
\(63\) −3.07188 − 3.07188i −0.387020 − 0.387020i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.34369 0.164158 0.0820789 − 0.996626i \(-0.473844\pi\)
0.0820789 + 0.996626i \(0.473844\pi\)
\(68\) 0 0
\(69\) 15.2683 1.83808
\(70\) 0 0
\(71\) 2.23664 + 2.23664i 0.265441 + 0.265441i 0.827260 − 0.561819i \(-0.189898\pi\)
−0.561819 + 0.827260i \(0.689898\pi\)
\(72\) 0 0
\(73\) −1.57165 − 1.57165i −0.183948 − 0.183948i 0.609126 − 0.793074i \(-0.291520\pi\)
−0.793074 + 0.609126i \(0.791520\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.46144i 0.508429i
\(78\) 0 0
\(79\) −2.42275 + 2.42275i −0.272580 + 0.272580i −0.830138 − 0.557558i \(-0.811738\pi\)
0.557558 + 0.830138i \(0.311738\pi\)
\(80\) 0 0
\(81\) 4.10094 0.455660
\(82\) 0 0
\(83\) − 6.62151i − 0.726805i −0.931632 − 0.363403i \(-0.881615\pi\)
0.931632 − 0.363403i \(-0.118385\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.03081i 0.539359i
\(88\) 0 0
\(89\) 9.17647 0.972704 0.486352 − 0.873763i \(-0.338327\pi\)
0.486352 + 0.873763i \(0.338327\pi\)
\(90\) 0 0
\(91\) 1.93630 − 1.93630i 0.202979 − 0.202979i
\(92\) 0 0
\(93\) − 13.3512i − 1.38445i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8371 + 13.8371i 1.40495 + 1.40495i 0.783281 + 0.621668i \(0.213545\pi\)
0.621668 + 0.783281i \(0.286455\pi\)
\(98\) 0 0
\(99\) 12.6502 + 12.6502i 1.27140 + 1.27140i
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 1700.2.o.e.1101.1 yes 12
5.2 odd 4 1700.2.m.d.149.1 12
5.3 odd 4 1700.2.m.e.149.6 12
5.4 even 2 1700.2.o.g.1101.6 yes 12
17.4 even 4 inner 1700.2.o.e.701.1 ✓ 12
85.4 even 4 1700.2.o.g.701.6 yes 12
85.38 odd 4 1700.2.m.d.1449.1 12
85.72 odd 4 1700.2.m.e.1449.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1700.2.m.d.149.1 12 5.2 odd 4
1700.2.m.d.1449.1 12 85.38 odd 4
1700.2.m.e.149.6 12 5.3 odd 4
1700.2.m.e.1449.6 12 85.72 odd 4
1700.2.o.e.701.1 ✓ 12 17.4 even 4 inner
1700.2.o.e.1101.1 yes 12 1.1 even 1 trivial
1700.2.o.g.701.6 yes 12 85.4 even 4
1700.2.o.g.1101.6 yes 12 5.4 even 2