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Properties

Label 700.2.w.a.391.59
Level $700$
Weight $2$
Character 700.391
Analytic conductor $5.590$
Analytic rank $0$
Dimension $464$
Inner twists $8$

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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(111,700)] 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("700.111"); 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([5, 8, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(464\)
Relative dimension: \(116\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 391.59
Character \(\chi\) \(=\) 700.391
Dual form 700.2.w.a.111.59

$q$-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+(0.0439545 - 1.41353i) q^{2} +(-0.723930 - 2.22803i) q^{3} +(-1.99614 - 0.124262i) q^{4} +(1.71454 + 1.43539i) q^{5} +(-3.18120 + 0.925365i) q^{6} +(2.60491 - 0.463059i) q^{7} +(-0.263387 + 2.81614i) q^{8} +(-2.01298 + 1.46251i) q^{9} +(2.10433 - 2.36047i) q^{10} +(2.35442 - 3.24058i) q^{11} +(1.16820 + 4.53740i) q^{12} +(-3.58757 - 4.93786i) q^{13} +(-0.540050 - 3.70248i) q^{14} +(1.95688 - 4.85917i) q^{15} +(3.96912 + 0.496088i) q^{16} +(-6.90843 - 2.24469i) q^{17} +(1.97883 + 2.90969i) q^{18} +(-1.77353 + 5.45836i) q^{19} +(-3.24410 - 3.07829i) q^{20} +(-2.91748 - 5.46859i) q^{21} +(-4.47717 - 3.47048i) q^{22} +(3.21641 - 4.42701i) q^{23} +(6.46510 - 1.45185i) q^{24} +(0.879313 + 4.92207i) q^{25} +(-7.13751 + 4.85409i) q^{26} +(-0.970047 - 0.704781i) q^{27} +(-5.25730 + 0.600637i) q^{28} +(0.208511 + 0.641730i) q^{29} +(-6.78257 - 2.97969i) q^{30} +(2.41883 - 7.44438i) q^{31} +(0.875696 - 5.58866i) q^{32} +(-8.92454 - 2.89976i) q^{33} +(-3.47659 + 9.66661i) q^{34} +(5.13091 + 2.94513i) q^{35} +(4.19991 - 2.66924i) q^{36} +(0.226384 - 0.164478i) q^{37} +(7.63760 + 2.74685i) q^{38} +(-8.40454 + 11.5679i) q^{39} +(-4.49384 + 4.45032i) q^{40} +(3.40534 + 4.68705i) q^{41} +(-7.85826 + 3.88358i) q^{42} -4.29038i q^{43} +(-5.10243 + 6.17608i) q^{44} +(-5.55061 - 0.381865i) q^{45} +(-6.11633 - 4.74108i) q^{46} +(0.315322 + 0.970462i) q^{47} +(-1.76807 - 9.20243i) q^{48} +(6.57115 - 2.41246i) q^{49} +(6.99615 - 1.02659i) q^{50} +17.0172i q^{51} +(6.54768 + 10.3024i) q^{52} +(2.25631 + 6.94421i) q^{53} +(-1.03887 + 1.34021i) q^{54} +(8.68825 - 2.17661i) q^{55} +(0.617936 + 7.45776i) q^{56} +13.4453 q^{57} +(0.916270 - 0.266529i) q^{58} +(-4.88971 + 3.55258i) q^{59} +(-4.51000 + 9.45639i) q^{60} +(-0.920124 + 1.26644i) q^{61} +(-10.4165 - 3.74630i) q^{62} +(-4.56640 + 4.74185i) q^{63} +(-7.86125 - 1.48347i) q^{64} +(0.936720 - 13.6157i) q^{65} +(-4.49117 + 12.4876i) q^{66} +(-3.72351 - 1.20984i) q^{67} +(13.5112 + 5.33915i) q^{68} +(-12.1919 - 3.96140i) q^{69} +(4.38856 - 7.12324i) q^{70} +(1.74050 - 0.565522i) q^{71} +(-3.58844 - 6.05402i) q^{72} +(-2.82328 + 3.88591i) q^{73} +(-0.222543 - 0.327230i) q^{74} +(10.3299 - 5.52237i) q^{75} +(4.21847 - 10.6752i) q^{76} +(4.63248 - 9.53167i) q^{77} +(15.9821 + 12.3885i) q^{78} +(6.90000 - 2.24195i) q^{79} +(6.09314 + 6.54779i) q^{80} +(-3.17469 + 9.77069i) q^{81} +(6.77496 - 4.60753i) q^{82} +(1.78182 - 5.48387i) q^{83} +(5.14415 + 11.2786i) q^{84} +(-8.62280 - 13.7649i) q^{85} +(-6.06459 - 0.188582i) q^{86} +(1.27884 - 0.929135i) q^{87} +(8.50580 + 7.48390i) q^{88} +(-0.789327 + 1.08642i) q^{89} +(-0.783752 + 7.82917i) q^{90} +(-11.6318 - 11.2015i) q^{91} +(-6.97050 + 8.43723i) q^{92} -18.3373 q^{93} +(1.38564 - 0.403062i) q^{94} +(-10.8757 + 6.81288i) q^{95} +(-13.0856 + 2.09473i) q^{96} +(5.45196 - 1.77145i) q^{97} +(-3.12125 - 9.39456i) q^{98} +9.96659i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 464 q - 6 q^{2} - 6 q^{4} + 12 q^{8} - 120 q^{9} + 9 q^{14} - 6 q^{16} + 4 q^{18} - 24 q^{21} - 14 q^{22} - 24 q^{25} + 31 q^{28} - 12 q^{29} - 90 q^{30} + 4 q^{32} + 52 q^{36} - 12 q^{37} - 31 q^{42} - 22 q^{44}+ \cdots + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0439545 − 1.41353i 0.0310805 − 0.999517i
\(3\) −0.723930 − 2.22803i −0.417961 − 1.28635i −0.909576 − 0.415539i \(-0.863593\pi\)
0.491615 − 0.870813i \(-0.336407\pi\)
\(4\) −1.99614 − 0.124262i −0.998068 − 0.0621310i
\(5\) 1.71454 + 1.43539i 0.766767 + 0.641926i
\(6\) −3.18120 + 0.925365i −1.29872 + 0.377779i
\(7\) 2.60491 − 0.463059i 0.984565 − 0.175020i
\(8\) −0.263387 + 2.81614i −0.0931215 + 0.995655i
\(9\) −2.01298 + 1.46251i −0.670992 + 0.487504i
\(10\) 2.10433 − 2.36047i 0.665447 − 0.746445i
\(11\) 2.35442 − 3.24058i 0.709885 − 0.977072i −0.289915 − 0.957052i \(-0.593627\pi\)
0.999800 − 0.0200200i \(-0.00637300\pi\)
\(12\) 1.16820 + 4.53740i 0.337231 + 1.30983i
\(13\) −3.58757 − 4.93786i −0.995012 − 1.36952i −0.928336 − 0.371742i \(-0.878761\pi\)
−0.0666758 − 0.997775i \(-0.521239\pi\)
\(14\) −0.540050 − 3.70248i −0.144334 − 0.989529i
\(15\) 1.95688 − 4.85917i 0.505264 − 1.25463i
\(16\) 3.96912 + 0.496088i 0.992279 + 0.124022i
\(17\) −6.90843 − 2.24469i −1.67554 − 0.544416i −0.691503 − 0.722374i \(-0.743051\pi\)
−0.984038 + 0.177958i \(0.943051\pi\)
\(18\) 1.97883 + 2.90969i 0.466414 + 0.685820i
\(19\) −1.77353 + 5.45836i −0.406875 + 1.25223i 0.512444 + 0.858720i \(0.328740\pi\)
−0.919319 + 0.393512i \(0.871260\pi\)
\(20\) −3.24410 − 3.07829i −0.725402 − 0.688326i
\(21\) −2.91748 − 5.46859i −0.636647 − 1.19335i
\(22\) −4.47717 − 3.47048i −0.954537 − 0.739910i
\(23\) 3.21641 − 4.42701i 0.670667 − 0.923094i −0.329108 − 0.944292i \(-0.606748\pi\)
0.999775 + 0.0211979i \(0.00674801\pi\)
\(24\) 6.46510 − 1.45185i 1.31968 − 0.296358i
\(25\) 0.879313 + 4.92207i 0.175863 + 0.984415i
\(26\) −7.13751 + 4.85409i −1.39978 + 0.951966i
\(27\) −0.970047 − 0.704781i −0.186686 − 0.135635i
\(28\) −5.25730 + 0.600637i −0.993537 + 0.113510i
\(29\) 0.208511 + 0.641730i 0.0387195 + 0.119166i 0.968548 − 0.248827i \(-0.0800450\pi\)
−0.929829 + 0.367993i \(0.880045\pi\)
\(30\) −6.78257 − 2.97969i −1.23832 − 0.544014i
\(31\) 2.41883 − 7.44438i 0.434434 − 1.33705i −0.459231 − 0.888317i \(-0.651875\pi\)
0.893665 − 0.448734i \(-0.148125\pi\)
\(32\) 0.875696 − 5.58866i 0.154803 − 0.987945i
\(33\) −8.92454 − 2.89976i −1.55356 − 0.504783i
\(34\) −3.47659 + 9.66661i −0.596230 + 1.65781i
\(35\) 5.13091 + 2.94513i 0.867281 + 0.497818i
\(36\) 4.19991 − 2.66924i 0.699985 − 0.444873i
\(37\) 0.226384 − 0.164478i 0.0372173 − 0.0270399i −0.569021 − 0.822323i \(-0.692678\pi\)
0.606238 + 0.795283i \(0.292678\pi\)
\(38\) 7.63760 + 2.74685i 1.23898 + 0.445599i
\(39\) −8.40454 + 11.5679i −1.34580 + 1.85234i
\(40\) −4.49384 + 4.45032i −0.710539 + 0.703658i
\(41\) 3.40534 + 4.68705i 0.531824 + 0.731993i 0.987407 − 0.158200i \(-0.0505691\pi\)
−0.455583 + 0.890193i \(0.650569\pi\)
\(42\) −7.85826 + 3.88358i −1.21256 + 0.599249i
\(43\) − 4.29038i − 0.654277i −0.944976 − 0.327139i \(-0.893916\pi\)
0.944976 − 0.327139i \(-0.106084\pi\)
\(44\) −5.10243 + 6.17608i −0.769220 + 0.931079i
\(45\) −5.55061 − 0.381865i −0.827436 − 0.0569250i
\(46\) −6.11633 − 4.74108i −0.901804 − 0.699034i
\(47\) 0.315322 + 0.970462i 0.0459945 + 0.141556i 0.971416 − 0.237382i \(-0.0762892\pi\)
−0.925422 + 0.378938i \(0.876289\pi\)
\(48\) −1.76807 − 9.20243i −0.255198 − 1.32826i
\(49\) 6.57115 − 2.41246i 0.938736 − 0.344637i
\(50\) 6.99615 − 1.02659i 0.989405 − 0.145181i
\(51\) 17.0172i 2.38288i
\(52\) 6.54768 + 10.3024i 0.908000 + 1.42869i
\(53\) 2.25631 + 6.94421i 0.309928 + 0.953860i 0.977792 + 0.209577i \(0.0672088\pi\)
−0.667864 + 0.744283i \(0.732791\pi\)
\(54\) −1.03887 + 1.34021i −0.141372 + 0.182380i
\(55\) 8.68825 − 2.17661i 1.17152 − 0.293493i
\(56\) 0.617936 + 7.45776i 0.0825752 + 0.996585i
\(57\) 13.4453 1.78087
\(58\) 0.916270 − 0.266529i 0.120312 − 0.0349970i
\(59\) −4.88971 + 3.55258i −0.636586 + 0.462507i −0.858676 − 0.512520i \(-0.828712\pi\)
0.222090 + 0.975026i \(0.428712\pi\)
\(60\) −4.51000 + 9.45639i −0.582239 + 1.22082i
\(61\) −0.920124 + 1.26644i −0.117810 + 0.162151i −0.863849 − 0.503751i \(-0.831953\pi\)
0.746039 + 0.665902i \(0.231953\pi\)
\(62\) −10.4165 − 3.74630i −1.32290 − 0.475781i
\(63\) −4.56640 + 4.74185i −0.575312 + 0.597416i
\(64\) −7.86125 − 1.48347i −0.982657 − 0.185434i
\(65\) 0.936720 − 13.6157i 0.116186 − 1.68882i
\(66\) −4.49117 + 12.4876i −0.552825 + 1.53712i
\(67\) −3.72351 − 1.20984i −0.454899 − 0.147806i 0.0726002 − 0.997361i \(-0.476870\pi\)
−0.527499 + 0.849555i \(0.676870\pi\)
\(68\) 13.5112 + 5.33915i 1.63848 + 0.647467i
\(69\) −12.1919 − 3.96140i −1.46774 − 0.476897i
\(70\) 4.38856 − 7.12324i 0.524533 − 0.851390i
\(71\) 1.74050 − 0.565522i 0.206559 − 0.0671151i −0.203910 − 0.978990i \(-0.565365\pi\)
0.410469 + 0.911875i \(0.365365\pi\)
\(72\) −3.58844 − 6.05402i −0.422902 − 0.713473i
\(73\) −2.82328 + 3.88591i −0.330440 + 0.454811i −0.941619 − 0.336681i \(-0.890696\pi\)
0.611179 + 0.791492i \(0.290696\pi\)
\(74\) −0.222543 − 0.327230i −0.0258701 − 0.0380397i
\(75\) 10.3299 − 5.52237i 1.19280 − 0.637668i
\(76\) 4.21847 − 10.6752i 0.483891 − 1.22453i
\(77\) 4.63248 − 9.53167i 0.527921 − 1.08624i
\(78\) 15.9821 + 12.3885i 1.80962 + 1.40272i
\(79\) 6.90000 − 2.24195i 0.776310 − 0.252239i 0.106046 − 0.994361i \(-0.466181\pi\)
0.670264 + 0.742123i \(0.266181\pi\)
\(80\) 6.09314 + 6.54779i 0.681234 + 0.732066i
\(81\) −3.17469 + 9.77069i −0.352743 + 1.08563i
\(82\) 6.77496 − 4.60753i 0.748169 − 0.508817i
\(83\) 1.78182 − 5.48387i 0.195580 − 0.601933i −0.804389 − 0.594102i \(-0.797507\pi\)
0.999969 − 0.00783071i \(-0.00249262\pi\)
\(84\) 5.14415 + 11.2786i 0.561273 + 1.23060i
\(85\) −8.62280 − 13.7649i −0.935274 − 1.49301i
\(86\) −6.06459 − 0.188582i −0.653961 − 0.0203353i
\(87\) 1.27884 − 0.929135i 0.137107 − 0.0996137i
\(88\) 8.50580 + 7.48390i 0.906721 + 0.797787i
\(89\) −0.789327 + 1.08642i −0.0836685 + 0.115160i −0.848800 − 0.528715i \(-0.822674\pi\)
0.765131 + 0.643875i \(0.222674\pi\)
\(90\) −0.783752 + 7.82917i −0.0826147 + 0.825267i
\(91\) −11.6318 − 11.2015i −1.21935 − 1.17423i
\(92\) −6.97050 + 8.43723i −0.726724 + 0.879642i
\(93\) −18.3373 −1.90149
\(94\) 1.38564 − 0.403062i 0.142918 − 0.0415726i
\(95\) −10.8757 + 6.81288i −1.11582 + 0.698987i
\(96\) −13.0856 + 2.09473i −1.33555 + 0.213792i
\(97\) 5.45196 − 1.77145i 0.553562 − 0.179863i −0.0188603 − 0.999822i \(-0.506004\pi\)
0.572422 + 0.819959i \(0.306004\pi\)
\(98\) −3.12125 − 9.39456i −0.315294 − 0.948994i
\(99\) 9.96659i 1.00168i
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.w.a.391.59 yes 464
4.3 odd 2 inner 700.2.w.a.391.14 yes 464
7.6 odd 2 inner 700.2.w.a.391.60 yes 464
25.11 even 5 inner 700.2.w.a.111.13 ✓ 464
28.27 even 2 inner 700.2.w.a.391.13 yes 464
100.11 odd 10 inner 700.2.w.a.111.60 yes 464
175.111 odd 10 inner 700.2.w.a.111.14 yes 464
700.111 even 10 inner 700.2.w.a.111.59 yes 464
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.w.a.111.13 ✓ 464 25.11 even 5 inner
700.2.w.a.111.14 yes 464 175.111 odd 10 inner
700.2.w.a.111.59 yes 464 700.111 even 10 inner
700.2.w.a.111.60 yes 464 100.11 odd 10 inner
700.2.w.a.391.13 yes 464 28.27 even 2 inner
700.2.w.a.391.14 yes 464 4.3 odd 2 inner
700.2.w.a.391.59 yes 464 1.1 even 1 trivial
700.2.w.a.391.60 yes 464 7.6 odd 2 inner