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Properties

Label 700.2.p.c.451.4
Level $700$
Weight $2$
Character 700.451
Analytic conductor $5.590$
Analytic rank $0$
Dimension $32$
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(451,700)] 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("700.451"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 451.4
Character \(\chi\) \(=\) 700.451
Dual form 700.2.p.c.551.4

$q$-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+(-1.05472 - 0.942109i) q^{2} +(-0.450639 + 0.780530i) q^{3} +(0.224860 + 1.98732i) q^{4} +(1.21064 - 0.398687i) q^{6} +(-2.29962 + 1.30833i) q^{7} +(1.63511 - 2.30790i) q^{8} +(1.09385 + 1.89460i) q^{9} +(-3.24107 - 1.87123i) q^{11} +(-1.65249 - 0.720054i) q^{12} +2.41990i q^{13} +(3.65805 + 0.786573i) q^{14} +(-3.89888 + 0.893735i) q^{16} +(0.505515 + 0.291859i) q^{17} +(0.631220 - 3.02880i) q^{18} +(-3.07977 - 5.33433i) q^{19} +(0.0151060 - 2.38451i) q^{21} +(1.65551 + 5.02706i) q^{22} +(3.73439 - 2.15605i) q^{23} +(1.06454 + 2.31628i) q^{24} +(2.27981 - 2.55232i) q^{26} -4.67556 q^{27} +(-3.11717 - 4.27589i) q^{28} -0.435463 q^{29} +(-1.26933 + 2.19854i) q^{31} +(4.95421 + 2.73053i) q^{32} +(2.92110 - 1.68650i) q^{33} +(-0.258212 - 0.784080i) q^{34} +(-3.51922 + 2.59985i) q^{36} +(-5.65039 - 9.78676i) q^{37} +(-1.77723 + 8.52769i) q^{38} +(-1.88881 - 1.09050i) q^{39} -7.35068i q^{41} +(-2.26240 + 2.50075i) q^{42} +5.80096i q^{43} +(2.98995 - 6.86180i) q^{44} +(-5.96996 - 1.24418i) q^{46} +(-5.78826 - 10.0256i) q^{47} +(1.05940 - 3.44594i) q^{48} +(3.57652 - 6.01735i) q^{49} +(-0.455610 + 0.263046i) q^{51} +(-4.80912 + 0.544138i) q^{52} +(-1.55746 + 2.69759i) q^{53} +(4.93139 + 4.40489i) q^{54} +(-0.740624 + 7.44657i) q^{56} +5.55147 q^{57} +(0.459290 + 0.410254i) q^{58} +(-1.73534 + 3.00569i) q^{59} +(-8.99597 + 5.19383i) q^{61} +(3.41004 - 1.12299i) q^{62} +(-4.99421 - 2.92575i) q^{63} +(-2.65284 - 7.54735i) q^{64} +(-4.66981 - 0.973217i) q^{66} +(-8.52602 - 4.92250i) q^{67} +(-0.466348 + 1.07025i) q^{68} +3.88640i q^{69} +9.96771i q^{71} +(6.16112 + 0.573383i) q^{72} +(-8.48612 - 4.89946i) q^{73} +(-3.26063 + 15.6456i) q^{74} +(9.90849 - 7.31997i) q^{76} +(9.90142 + 0.0627260i) q^{77} +(0.964785 + 2.92964i) q^{78} +(-0.397549 + 0.229525i) q^{79} +(-1.17456 + 2.03439i) q^{81} +(-6.92515 + 7.75290i) q^{82} +2.59747 q^{83} +(4.74218 - 0.506159i) q^{84} +(5.46514 - 6.11837i) q^{86} +(0.196236 - 0.339892i) q^{87} +(-9.61812 + 4.42040i) q^{88} +(-8.55647 + 4.94008i) q^{89} +(-3.16604 - 5.56486i) q^{91} +(5.12447 + 6.93662i) q^{92} +(-1.14402 - 1.98149i) q^{93} +(-3.34020 + 16.0273i) q^{94} +(-4.36382 + 2.63643i) q^{96} +4.54044i q^{97} +(-9.44122 + 2.97713i) q^{98} -8.18738i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 16 q^{9} + 30 q^{12} + 2 q^{14} - 14 q^{16} - 12 q^{21} + 8 q^{22} + 36 q^{24} + 30 q^{26} - 2 q^{28} - 40 q^{29} - 2 q^{32} + 60 q^{36} - 8 q^{37} + 60 q^{38} + 62 q^{42}+ \cdots - 78 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.05472 − 0.942109i −0.745798 − 0.666172i
\(3\) −0.450639 + 0.780530i −0.260177 + 0.450639i −0.966289 − 0.257461i \(-0.917114\pi\)
0.706112 + 0.708100i \(0.250447\pi\)
\(4\) 0.224860 + 1.98732i 0.112430 + 0.993660i
\(5\) 0 0
\(6\) 1.21064 − 0.398687i 0.494242 − 0.162763i
\(7\) −2.29962 + 1.30833i −0.869175 + 0.494504i
\(8\) 1.63511 − 2.30790i 0.578098 − 0.815967i
\(9\) 1.09385 + 1.89460i 0.364616 + 0.631534i
\(10\) 0 0
\(11\) −3.24107 − 1.87123i −0.977218 − 0.564197i −0.0757892 − 0.997124i \(-0.524148\pi\)
−0.901429 + 0.432927i \(0.857481\pi\)
\(12\) −1.65249 − 0.720054i −0.477033 − 0.207862i
\(13\) 2.41990i 0.671161i 0.942012 + 0.335580i \(0.108932\pi\)
−0.942012 + 0.335580i \(0.891068\pi\)
\(14\) 3.65805 + 0.786573i 0.977654 + 0.210220i
\(15\) 0 0
\(16\) −3.89888 + 0.893735i −0.974719 + 0.223434i
\(17\) 0.505515 + 0.291859i 0.122605 + 0.0707863i 0.560048 − 0.828460i \(-0.310782\pi\)
−0.437443 + 0.899246i \(0.644116\pi\)
\(18\) 0.631220 − 3.02880i 0.148780 − 0.713894i
\(19\) −3.07977 − 5.33433i −0.706549 − 1.22378i −0.966130 − 0.258057i \(-0.916918\pi\)
0.259581 − 0.965721i \(-0.416416\pi\)
\(20\) 0 0
\(21\) 0.0151060 − 2.38451i 0.00329639 − 0.520343i
\(22\) 1.65551 + 5.02706i 0.352955 + 1.07177i
\(23\) 3.73439 − 2.15605i 0.778674 − 0.449568i −0.0572861 − 0.998358i \(-0.518245\pi\)
0.835960 + 0.548790i \(0.184911\pi\)
\(24\) 1.06454 + 2.31628i 0.217299 + 0.472809i
\(25\) 0 0
\(26\) 2.27981 − 2.55232i 0.447108 − 0.500550i
\(27\) −4.67556 −0.899812
\(28\) −3.11717 − 4.27589i −0.589090 − 0.808068i
\(29\) −0.435463 −0.0808634 −0.0404317 − 0.999182i \(-0.512873\pi\)
−0.0404317 + 0.999182i \(0.512873\pi\)
\(30\) 0 0
\(31\) −1.26933 + 2.19854i −0.227978 + 0.394869i −0.957209 − 0.289399i \(-0.906545\pi\)
0.729231 + 0.684268i \(0.239878\pi\)
\(32\) 4.95421 + 2.73053i 0.875789 + 0.482694i
\(33\) 2.92110 − 1.68650i 0.508499 − 0.293582i
\(34\) −0.258212 − 0.784080i −0.0442831 − 0.134469i
\(35\) 0 0
\(36\) −3.51922 + 2.59985i −0.586536 + 0.433308i
\(37\) −5.65039 − 9.78676i −0.928918 − 1.60893i −0.785136 − 0.619324i \(-0.787407\pi\)
−0.143782 − 0.989609i \(-0.545927\pi\)
\(38\) −1.77723 + 8.52769i −0.288304 + 1.38337i
\(39\) −1.88881 − 1.09050i −0.302451 − 0.174620i
\(40\) 0 0
\(41\) − 7.35068i − 1.14798i −0.818861 − 0.573992i \(-0.805394\pi\)
0.818861 − 0.573992i \(-0.194606\pi\)
\(42\) −2.26240 + 2.50075i −0.349096 + 0.385875i
\(43\) 5.80096i 0.884637i 0.896858 + 0.442319i \(0.145844\pi\)
−0.896858 + 0.442319i \(0.854156\pi\)
\(44\) 2.98995 − 6.86180i 0.450752 − 1.03446i
\(45\) 0 0
\(46\) −5.96996 − 1.24418i −0.880223 − 0.183444i
\(47\) −5.78826 − 10.0256i −0.844305 − 1.46238i −0.886223 − 0.463258i \(-0.846680\pi\)
0.0419181 − 0.999121i \(-0.486653\pi\)
\(48\) 1.05940 − 3.44594i 0.152911 − 0.497379i
\(49\) 3.57652 − 6.01735i 0.510932 − 0.859621i
\(50\) 0 0
\(51\) −0.455610 + 0.263046i −0.0637981 + 0.0368339i
\(52\) −4.80912 + 0.544138i −0.666905 + 0.0754584i
\(53\) −1.55746 + 2.69759i −0.213933 + 0.370543i −0.952942 − 0.303153i \(-0.901961\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(54\) 4.93139 + 4.40489i 0.671078 + 0.599429i
\(55\) 0 0
\(56\) −0.740624 + 7.44657i −0.0989701 + 0.995090i
\(57\) 5.55147 0.735310
\(58\) 0.459290 + 0.410254i 0.0603078 + 0.0538689i
\(59\) −1.73534 + 3.00569i −0.225922 + 0.391308i −0.956596 − 0.291419i \(-0.905873\pi\)
0.730674 + 0.682727i \(0.239206\pi\)
\(60\) 0 0
\(61\) −8.99597 + 5.19383i −1.15182 + 0.665001i −0.949329 − 0.314284i \(-0.898236\pi\)
−0.202487 + 0.979285i \(0.564902\pi\)
\(62\) 3.41004 − 1.12299i 0.433076 − 0.142620i
\(63\) −4.99421 − 2.92575i −0.629211 − 0.368610i
\(64\) −2.65284 − 7.54735i −0.331605 − 0.943418i
\(65\) 0 0
\(66\) −4.66981 − 0.973217i −0.574813 − 0.119795i
\(67\) −8.52602 − 4.92250i −1.04162 − 0.601379i −0.121327 − 0.992613i \(-0.538715\pi\)
−0.920291 + 0.391234i \(0.872048\pi\)
\(68\) −0.466348 + 1.07025i −0.0565530 + 0.129787i
\(69\) 3.88640i 0.467868i
\(70\) 0 0
\(71\) 9.96771i 1.18295i 0.806324 + 0.591475i \(0.201454\pi\)
−0.806324 + 0.591475i \(0.798546\pi\)
\(72\) 6.16112 + 0.573383i 0.726095 + 0.0675738i
\(73\) −8.48612 − 4.89946i −0.993225 − 0.573439i −0.0869881 − 0.996209i \(-0.527724\pi\)
−0.906237 + 0.422771i \(0.861058\pi\)
\(74\) −3.26063 + 15.6456i −0.379041 + 1.81876i
\(75\) 0 0
\(76\) 9.90849 − 7.31997i 1.13658 − 0.839658i
\(77\) 9.90142 + 0.0627260i 1.12837 + 0.00714829i
\(78\) 0.964785 + 2.92964i 0.109240 + 0.331716i
\(79\) −0.397549 + 0.229525i −0.0447278 + 0.0258236i −0.522197 − 0.852825i \(-0.674887\pi\)
0.477469 + 0.878648i \(0.341554\pi\)
\(80\) 0 0
\(81\) −1.17456 + 2.03439i −0.130506 + 0.226044i
\(82\) −6.92515 + 7.75290i −0.764755 + 0.856164i
\(83\) 2.59747 0.285109 0.142554 − 0.989787i \(-0.454468\pi\)
0.142554 + 0.989787i \(0.454468\pi\)
\(84\) 4.74218 − 0.506159i 0.517414 − 0.0552265i
\(85\) 0 0
\(86\) 5.46514 − 6.11837i 0.589321 − 0.659761i
\(87\) 0.196236 − 0.339892i 0.0210388 − 0.0364402i
\(88\) −9.61812 + 4.42040i −1.02529 + 0.471217i
\(89\) −8.55647 + 4.94008i −0.906984 + 0.523648i −0.879460 − 0.475973i \(-0.842096\pi\)
−0.0275247 + 0.999621i \(0.508763\pi\)
\(90\) 0 0
\(91\) −3.16604 − 5.56486i −0.331891 − 0.583356i
\(92\) 5.12447 + 6.93662i 0.534263 + 0.723192i
\(93\) −1.14402 − 1.98149i −0.118629 − 0.205471i
\(94\) −3.34020 + 16.0273i −0.344515 + 1.65309i
\(95\) 0 0
\(96\) −4.36382 + 2.63643i −0.445381 + 0.269079i
\(97\) 4.54044i 0.461011i 0.973071 + 0.230506i \(0.0740380\pi\)
−0.973071 + 0.230506i \(0.925962\pi\)
\(98\) −9.44122 + 2.97713i −0.953708 + 0.300735i
\(99\) − 8.18738i − 0.822862i
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.p.c.451.4 32
4.3 odd 2 inner 700.2.p.c.451.8 32
5.2 odd 4 700.2.t.d.199.11 32
5.3 odd 4 700.2.t.c.199.6 32
5.4 even 2 140.2.o.a.31.13 yes 32
7.5 odd 6 inner 700.2.p.c.551.8 32
20.3 even 4 700.2.t.c.199.1 32
20.7 even 4 700.2.t.d.199.16 32
20.19 odd 2 140.2.o.a.31.9 ✓ 32
28.19 even 6 inner 700.2.p.c.551.4 32
35.4 even 6 980.2.g.a.391.7 32
35.9 even 6 980.2.o.f.411.9 32
35.12 even 12 700.2.t.c.299.1 32
35.19 odd 6 140.2.o.a.131.9 yes 32
35.24 odd 6 980.2.g.a.391.8 32
35.33 even 12 700.2.t.d.299.16 32
35.34 odd 2 980.2.o.f.31.13 32
140.19 even 6 140.2.o.a.131.13 yes 32
140.39 odd 6 980.2.g.a.391.6 32
140.47 odd 12 700.2.t.c.299.6 32
140.59 even 6 980.2.g.a.391.5 32
140.79 odd 6 980.2.o.f.411.13 32
140.103 odd 12 700.2.t.d.299.11 32
140.139 even 2 980.2.o.f.31.9 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.o.a.31.9 ✓ 32 20.19 odd 2
140.2.o.a.31.13 yes 32 5.4 even 2
140.2.o.a.131.9 yes 32 35.19 odd 6
140.2.o.a.131.13 yes 32 140.19 even 6
700.2.p.c.451.4 32 1.1 even 1 trivial
700.2.p.c.451.8 32 4.3 odd 2 inner
700.2.p.c.551.4 32 28.19 even 6 inner
700.2.p.c.551.8 32 7.5 odd 6 inner
700.2.t.c.199.1 32 20.3 even 4
700.2.t.c.199.6 32 5.3 odd 4
700.2.t.c.299.1 32 35.12 even 12
700.2.t.c.299.6 32 140.47 odd 12
700.2.t.d.199.11 32 5.2 odd 4
700.2.t.d.199.16 32 20.7 even 4
700.2.t.d.299.11 32 140.103 odd 12
700.2.t.d.299.16 32 35.33 even 12
980.2.g.a.391.5 32 140.59 even 6
980.2.g.a.391.6 32 140.39 odd 6
980.2.g.a.391.7 32 35.4 even 6
980.2.g.a.391.8 32 35.24 odd 6
980.2.o.f.31.9 32 140.139 even 2
980.2.o.f.31.13 32 35.34 odd 2
980.2.o.f.411.9 32 35.9 even 6
980.2.o.f.411.13 32 140.79 odd 6