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Properties

Label 700.2.w.a.391.60
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 391.60
Character \(\chi\) \(=\) 700.391
Dual form 700.2.w.a.111.60

$q$-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} +(-0.769045 + 3.66177i) 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} +(-0.854067 - 6.13904i) 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} +(5.14222 + 1.24802i) 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} +(3.80157 + 4.53300i) 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} +(-8.71526 + 0.937411i) 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} +(1.99014 - 7.21383i) 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} +(5.92086 - 2.87759i) 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} +(6.57463 - 5.17438i) 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} +(-7.63365 + 7.35120i) 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} +(0.941985 + 12.3605i) 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} +(-7.05878 - 14.5240i) 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} +(3.69891 - 9.18249i) 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 \(0.136407\pi\)
−0.491615 + 0.870813i \(0.663593\pi\)
\(4\) −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 \(0.121239\pi\)
0.0666758 + 0.997775i \(0.478761\pi\)
\(14\) −0.769045 + 3.66177i −0.205536 + 0.978650i
\(15\) 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 0.984038 − 0.177958i \(-0.0569490\pi\)
0.691503 + 0.722374i \(0.256949\pi\)
\(18\) 1.97883 + 2.90969i 0.466414 + 0.685820i
\(19\) 1.77353 − 5.45836i 0.406875 − 1.25223i −0.512444 − 0.858720i \(-0.671260\pi\)
0.919319 − 0.393512i \(-0.128740\pi\)
\(20\) 3.24410 + 3.07829i 0.725402 + 0.688326i
\(21\) −0.854067 − 6.13904i −0.186373 − 1.33965i
\(22\) −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
\(28\) 5.14222 + 1.24802i 0.971789 + 0.235854i
\(29\) 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 \(0.348125\pi\)
−0.893665 + 0.448734i \(0.851875\pi\)
\(32\) 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
\(35\) 3.80157 + 4.53300i 0.642582 + 0.766217i
\(36\) 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 0.455583 − 0.890193i \(-0.349431\pi\)
−0.987407 + 0.158200i \(0.949431\pi\)
\(42\) −8.71526 + 0.937411i −1.34479 + 0.144646i
\(43\) − 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 0.925422 − 0.378938i \(-0.123711\pi\)
−0.971416 + 0.237382i \(0.923711\pi\)
\(48\) 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
\(56\) 1.99014 − 7.21383i 0.265943 − 0.963989i
\(57\) 13.4453 1.78087
\(58\) 0.916270 − 0.266529i 0.120312 − 0.0349970i
\(59\) 4.88971 − 3.55258i 0.636586 − 0.462507i −0.222090 − 0.975026i \(-0.571288\pi\)
0.858676 + 0.512520i \(0.171288\pi\)
\(60\) −4.51000 + 9.45639i −0.582239 + 1.22082i
\(61\) 0.920124 − 1.26644i 0.117810 − 0.162151i −0.746039 − 0.665902i \(-0.768047\pi\)
0.863849 + 0.503751i \(0.168047\pi\)
\(62\) 10.4165 + 3.74630i 1.32290 + 0.475781i
\(63\) 5.92086 − 2.87759i 0.745958 − 0.362543i
\(64\) −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
\(70\) 6.57463 − 5.17438i 0.785819 − 0.618457i
\(71\) 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 −0.611179 − 0.791492i \(-0.709304\pi\)
0.941619 + 0.336681i \(0.109304\pi\)
\(74\) −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
\(77\) −7.63365 + 7.35120i −0.869935 + 0.837747i
\(78\) 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 \(0.202493\pi\)
−0.999969 + 0.00783071i \(0.997507\pi\)
\(84\) 0.941985 + 12.3605i 0.102779 + 1.34864i
\(85\) −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 −0.765131 − 0.643875i \(-0.777326\pi\)
0.848800 + 0.528715i \(0.177326\pi\)
\(90\) 0.783752 − 7.82917i 0.0826147 − 0.825267i
\(91\) −7.05878 − 14.5240i −0.739961 − 1.52252i
\(92\) −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 −0.572422 − 0.819959i \(-0.693996\pi\)
0.0188603 + 0.999822i \(0.493996\pi\)
\(98\) 3.69891 − 9.18249i 0.373647 − 0.927571i
\(99\) 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.391.60 yes 464
4.3 odd 2 inner 700.2.w.a.391.13 yes 464
7.6 odd 2 inner 700.2.w.a.391.59 yes 464
25.11 even 5 inner 700.2.w.a.111.14 yes 464
28.27 even 2 inner 700.2.w.a.391.14 yes 464
100.11 odd 10 inner 700.2.w.a.111.59 yes 464
175.111 odd 10 inner 700.2.w.a.111.13 ✓ 464
700.111 even 10 inner 700.2.w.a.111.60 yes 464
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.w.a.111.13 ✓ 464 175.111 odd 10 inner
700.2.w.a.111.14 yes 464 25.11 even 5 inner
700.2.w.a.111.59 yes 464 100.11 odd 10 inner
700.2.w.a.111.60 yes 464 700.111 even 10 inner
700.2.w.a.391.13 yes 464 4.3 odd 2 inner
700.2.w.a.391.14 yes 464 28.27 even 2 inner
700.2.w.a.391.59 yes 464 7.6 odd 2 inner
700.2.w.a.391.60 yes 464 1.1 even 1 trivial