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Properties

Label 700.2.m.c.657.1
Level $700$
Weight $2$
Character 700.657
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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(293,700)] 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("700.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,2,0,0,0,-12] 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: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.11574317056.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 45x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 657.1
Root \(-1.83051 - 1.83051i\) of defining polynomial
Character \(\chi\) \(=\) 700.657
Dual form 700.2.m.c.293.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.83051 - 1.83051i) q^{3} +(2.12393 + 1.57763i) q^{7} +3.70156i q^{9} -4.70156 q^{11} +(-1.83051 - 1.83051i) q^{13} +(0.737925 - 0.737925i) q^{17} -4.75362 q^{19} +(-1.00000 - 6.77576i) q^{21} +(-3.70156 + 3.70156i) q^{23} +(1.28422 - 1.28422i) q^{27} +0.701562i q^{29} +8.79790i q^{31} +(8.60627 + 8.60627i) q^{33} +(3.70156 + 3.70156i) q^{37} +6.70156i q^{39} -4.75362i q^{41} +(5.00000 - 5.00000i) q^{43} +(-8.05998 + 8.05998i) q^{47} +(2.02214 + 6.70156i) q^{49} -2.70156 q^{51} +(-5.00000 + 5.00000i) q^{53} +(8.70156 + 8.70156i) q^{57} +4.75362 q^{59} +9.50723i q^{61} +(-5.83971 + 7.86185i) q^{63} +(-5.00000 - 5.00000i) q^{67} +13.5515 q^{69} -5.40312 q^{71} +(-3.11473 - 3.11473i) q^{73} +(-9.98578 - 7.41734i) q^{77} -6.70156i q^{79} +6.40312 q^{81} +(-7.86835 - 7.86835i) q^{83} +(1.28422 - 1.28422i) q^{87} -13.5515 q^{89} +(-1.00000 - 6.77576i) q^{91} +(16.1047 - 16.1047i) q^{93} +(5.49154 - 5.49154i) q^{97} -17.4031i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{7} - 12 q^{11} - 8 q^{21} - 4 q^{23} + 4 q^{37} + 40 q^{43} + 4 q^{51} - 40 q^{53} + 44 q^{57} - 42 q^{63} - 40 q^{67} + 8 q^{71} - 44 q^{77} - 8 q^{91} + 52 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) −1.83051 − 1.83051i −1.05685 − 1.05685i −0.998284 − 0.0585640i \(-0.981348\pi\)
−0.0585640 − 0.998284i \(-0.518652\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.12393 + 1.57763i 0.802769 + 0.596289i
\(8\) 0 0
\(9\) 3.70156i 1.23385i
\(10\) 0 0
\(11\) −4.70156 −1.41757 −0.708787 − 0.705422i \(-0.750757\pi\)
−0.708787 + 0.705422i \(0.750757\pi\)
\(12\) 0 0
\(13\) −1.83051 − 1.83051i −0.507693 − 0.507693i 0.406125 − 0.913818i \(-0.366880\pi\)
−0.913818 + 0.406125i \(0.866880\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.737925 − 0.737925i 0.178973 − 0.178973i −0.611935 − 0.790908i \(-0.709609\pi\)
0.790908 + 0.611935i \(0.209609\pi\)
\(18\) 0 0
\(19\) −4.75362 −1.09055 −0.545277 − 0.838256i \(-0.683576\pi\)
−0.545277 + 0.838256i \(0.683576\pi\)
\(20\) 0 0
\(21\) −1.00000 − 6.77576i −0.218218 − 1.47859i
\(22\) 0 0
\(23\) −3.70156 + 3.70156i −0.771829 + 0.771829i −0.978426 − 0.206597i \(-0.933761\pi\)
0.206597 + 0.978426i \(0.433761\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.28422 − 1.28422i 0.247148 − 0.247148i
\(28\) 0 0
\(29\) 0.701562i 0.130277i 0.997876 + 0.0651384i \(0.0207489\pi\)
−0.997876 + 0.0651384i \(0.979251\pi\)
\(30\) 0 0
\(31\) 8.79790i 1.58015i 0.613010 + 0.790075i \(0.289959\pi\)
−0.613010 + 0.790075i \(0.710041\pi\)
\(32\) 0 0
\(33\) 8.60627 + 8.60627i 1.49816 + 1.49816i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.70156 + 3.70156i 0.608533 + 0.608533i 0.942563 − 0.334030i \(-0.108409\pi\)
−0.334030 + 0.942563i \(0.608409\pi\)
\(38\) 0 0
\(39\) 6.70156i 1.07311i
\(40\) 0 0
\(41\) − 4.75362i − 0.742390i −0.928555 − 0.371195i \(-0.878948\pi\)
0.928555 − 0.371195i \(-0.121052\pi\)
\(42\) 0 0
\(43\) 5.00000 − 5.00000i 0.762493 − 0.762493i −0.214280 − 0.976772i \(-0.568740\pi\)
0.976772 + 0.214280i \(0.0687403\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.05998 + 8.05998i −1.17567 + 1.17567i −0.194832 + 0.980837i \(0.562416\pi\)
−0.980837 + 0.194832i \(0.937584\pi\)
\(48\) 0 0
\(49\) 2.02214 + 6.70156i 0.288878 + 0.957366i
\(50\) 0 0
\(51\) −2.70156 −0.378294
\(52\) 0 0
\(53\) −5.00000 + 5.00000i −0.686803 + 0.686803i −0.961524 − 0.274721i \(-0.911414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.70156 + 8.70156i 1.15255 + 1.15255i
\(58\) 0 0
\(59\) 4.75362 0.618868 0.309434 − 0.950921i \(-0.399860\pi\)
0.309434 + 0.950921i \(0.399860\pi\)
\(60\) 0 0
\(61\) 9.50723i 1.21728i 0.793448 + 0.608638i \(0.208284\pi\)
−0.793448 + 0.608638i \(0.791716\pi\)
\(62\) 0 0
\(63\) −5.83971 + 7.86185i −0.735734 + 0.990500i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 − 5.00000i −0.610847 − 0.610847i 0.332320 − 0.943167i \(-0.392169\pi\)
−0.943167 + 0.332320i \(0.892169\pi\)
\(68\) 0 0
\(69\) 13.5515 1.63141
\(70\) 0 0
\(71\) −5.40312 −0.641233 −0.320616 − 0.947209i \(-0.603890\pi\)
−0.320616 + 0.947209i \(0.603890\pi\)
\(72\) 0 0
\(73\) −3.11473 − 3.11473i −0.364552 − 0.364552i 0.500934 − 0.865486i \(-0.332990\pi\)
−0.865486 + 0.500934i \(0.832990\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.98578 − 7.41734i −1.13799 − 0.845285i
\(78\) 0 0
\(79\) − 6.70156i − 0.753985i −0.926216 − 0.376992i \(-0.876958\pi\)
0.926216 − 0.376992i \(-0.123042\pi\)
\(80\) 0 0
\(81\) 6.40312 0.711458
\(82\) 0 0
\(83\) −7.86835 − 7.86835i −0.863664 − 0.863664i 0.128098 − 0.991762i \(-0.459113\pi\)
−0.991762 + 0.128098i \(0.959113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.28422 − 1.28422i 0.137683 − 0.137683i
\(88\) 0 0
\(89\) −13.5515 −1.43646 −0.718229 − 0.695807i \(-0.755047\pi\)
−0.718229 + 0.695807i \(0.755047\pi\)
\(90\) 0 0
\(91\) −1.00000 − 6.77576i −0.104828 − 0.710293i
\(92\) 0 0
\(93\) 16.1047 − 16.1047i 1.66998 − 1.66998i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.49154 − 5.49154i 0.557582 − 0.557582i −0.371037 − 0.928618i \(-0.620998\pi\)
0.928618 + 0.371037i \(0.120998\pi\)
\(98\) 0 0
\(99\) − 17.4031i − 1.74908i
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.m.c.657.1 8
5.2 odd 4 140.2.m.a.13.1 ✓ 8
5.3 odd 4 inner 700.2.m.c.293.4 8
5.4 even 2 140.2.m.a.97.4 yes 8
7.6 odd 2 inner 700.2.m.c.657.4 8
15.2 even 4 1260.2.ba.a.433.4 8
15.14 odd 2 1260.2.ba.a.937.1 8
20.7 even 4 560.2.bj.b.433.4 8
20.19 odd 2 560.2.bj.b.97.1 8
35.2 odd 12 980.2.v.b.913.1 16
35.4 even 6 980.2.v.b.117.4 16
35.9 even 6 980.2.v.b.717.1 16
35.12 even 12 980.2.v.b.913.4 16
35.13 even 4 inner 700.2.m.c.293.1 8
35.17 even 12 980.2.v.b.313.1 16
35.19 odd 6 980.2.v.b.717.4 16
35.24 odd 6 980.2.v.b.117.1 16
35.27 even 4 140.2.m.a.13.4 yes 8
35.32 odd 12 980.2.v.b.313.4 16
35.34 odd 2 140.2.m.a.97.1 yes 8
105.62 odd 4 1260.2.ba.a.433.1 8
105.104 even 2 1260.2.ba.a.937.4 8
140.27 odd 4 560.2.bj.b.433.1 8
140.139 even 2 560.2.bj.b.97.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.m.a.13.1 ✓ 8 5.2 odd 4
140.2.m.a.13.4 yes 8 35.27 even 4
140.2.m.a.97.1 yes 8 35.34 odd 2
140.2.m.a.97.4 yes 8 5.4 even 2
560.2.bj.b.97.1 8 20.19 odd 2
560.2.bj.b.97.4 8 140.139 even 2
560.2.bj.b.433.1 8 140.27 odd 4
560.2.bj.b.433.4 8 20.7 even 4
700.2.m.c.293.1 8 35.13 even 4 inner
700.2.m.c.293.4 8 5.3 odd 4 inner
700.2.m.c.657.1 8 1.1 even 1 trivial
700.2.m.c.657.4 8 7.6 odd 2 inner
980.2.v.b.117.1 16 35.24 odd 6
980.2.v.b.117.4 16 35.4 even 6
980.2.v.b.313.1 16 35.17 even 12
980.2.v.b.313.4 16 35.32 odd 12
980.2.v.b.717.1 16 35.9 even 6
980.2.v.b.717.4 16 35.19 odd 6
980.2.v.b.913.1 16 35.2 odd 12
980.2.v.b.913.4 16 35.12 even 12
1260.2.ba.a.433.1 8 105.62 odd 4
1260.2.ba.a.433.4 8 15.2 even 4
1260.2.ba.a.937.1 8 15.14 odd 2
1260.2.ba.a.937.4 8 105.104 even 2