Properties

Label 136.2.n.c.121.3
Level 136136
Weight 22
Character 136.121
Analytic conductor 1.0861.086
Analytic rank 00
Dimension 1212
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [136,2,Mod(9,136)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(136, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("136.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 136=2317 136 = 2^{3} \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 136.n (of order 88, degree 44, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0] 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: 1.085965467491.08596546749
Analytic rank: 00
Dimension: 1212
Relative dimension: 33 over Q(ζ8)\Q(\zeta_{8})
Coefficient field: Q[x]/(x12+)\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: x12+28x10+258x8+880x6+1033x4+132x2+4 x^{12} + 28x^{10} + 258x^{8} + 880x^{6} + 1033x^{4} + 132x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C8]\mathrm{SU}(2)[C_{8}]

Embedding invariants

Embedding label 121.3
Root 3.49562i3.49562i of defining polynomial
Character χ\chi == 136.121
Dual form 136.2.n.c.9.3

qq-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)f(q) == q+(2.471781.02384i)q3+(0.3925310.947653i)q5+(0.893542+2.15720i)q7+(2.940102.94010i)q9+(4.811961.99318i)q11+4.35777iq13+(1.940501.94050i)q15+(3.77648+1.65476i)q17+(1.892811.89281i)q19+6.24696iq21+(3.79373+1.57141i)q23+(2.791572.79157i)q25+(1.185542.86215i)q27+(1.469623.54797i)q29+(10.1411+4.20057i)q3113.9348q33+2.39502q35+(1.021680.423195i)q37+(4.46167+10.7714i)q39+(1.00448+2.42504i)q41+(2.86647+2.86647i)q43+(3.940281.63212i)q4510.4047iq47+(1.09465+1.09465i)q49+(11.0288+0.223677i)q51+(5.498675.49867i)q53+5.34245iq55+(6.616542.74066i)q57+(7.694897.69489i)q59+(3.045767.35311i)q61+(3.71528+8.96949i)q63+(4.129651.71056i)q65+6.44882q67+10.9861q69+(1.735610.718913i)q71+(5.05217+12.1970i)q73+(4.042009.75826i)q75+(8.599378.59937i)q77+(2.31129+0.957366i)q79+4.18540iq81+(0.0232879+0.0232879i)q83+(0.08575564.22833i)q85+(7.265127.26512i)q87+7.34936iq89+(9.400573.89384i)q91+(20.7657+20.7657i)q93+(1.05074+2.53671i)q95+(6.9877916.8700i)q97+(20.0078+8.28750i)q99+O(q100)q+(2.47178 - 1.02384i) q^{3} +(-0.392531 - 0.947653i) q^{5} +(-0.893542 + 2.15720i) q^{7} +(2.94010 - 2.94010i) q^{9} +(-4.81196 - 1.99318i) q^{11} +4.35777i q^{13} +(-1.94050 - 1.94050i) q^{15} +(3.77648 + 1.65476i) q^{17} +(-1.89281 - 1.89281i) q^{19} +6.24696i q^{21} +(3.79373 + 1.57141i) q^{23} +(2.79157 - 2.79157i) q^{25} +(1.18554 - 2.86215i) q^{27} +(-1.46962 - 3.54797i) q^{29} +(-10.1411 + 4.20057i) q^{31} -13.9348 q^{33} +2.39502 q^{35} +(1.02168 - 0.423195i) q^{37} +(4.46167 + 10.7714i) q^{39} +(-1.00448 + 2.42504i) q^{41} +(-2.86647 + 2.86647i) q^{43} +(-3.94028 - 1.63212i) q^{45} -10.4047i q^{47} +(1.09465 + 1.09465i) q^{49} +(11.0288 + 0.223677i) q^{51} +(-5.49867 - 5.49867i) q^{53} +5.34245i q^{55} +(-6.61654 - 2.74066i) q^{57} +(7.69489 - 7.69489i) q^{59} +(3.04576 - 7.35311i) q^{61} +(3.71528 + 8.96949i) q^{63} +(4.12965 - 1.71056i) q^{65} +6.44882 q^{67} +10.9861 q^{69} +(1.73561 - 0.718913i) q^{71} +(5.05217 + 12.1970i) q^{73} +(4.04200 - 9.75826i) q^{75} +(8.59937 - 8.59937i) q^{77} +(2.31129 + 0.957366i) q^{79} +4.18540i q^{81} +(0.0232879 + 0.0232879i) q^{83} +(0.0857556 - 4.22833i) q^{85} +(-7.26512 - 7.26512i) q^{87} +7.34936i q^{89} +(-9.40057 - 3.89384i) q^{91} +(-20.7657 + 20.7657i) q^{93} +(-1.05074 + 2.53671i) q^{95} +(-6.98779 - 16.8700i) q^{97} +(-20.0078 + 8.28750i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q+4q58q912q11+20q15+4q174q198q2316q25+24q27+8q2932q3124q3332q35+4q378q39+16q41+8q4364q45+68q99+O(q100) 12 q + 4 q^{5} - 8 q^{9} - 12 q^{11} + 20 q^{15} + 4 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{25} + 24 q^{27} + 8 q^{29} - 32 q^{31} - 24 q^{33} - 32 q^{35} + 4 q^{37} - 8 q^{39} + 16 q^{41} + 8 q^{43} - 64 q^{45}+ \cdots - 68 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/136Z)×\left(\mathbb{Z}/136\mathbb{Z}\right)^\times.

nn 6969 103103 105105
χ(n)\chi(n) 11 11 e(78)e\left(\frac{7}{8}\right)

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0 0
33 2.47178 1.02384i 1.42708 0.591116i 0.470452 0.882426i 0.344091π-0.344091\pi
0.956629 + 0.291310i 0.0940910π0.0940910\pi
44 0 0
55 −0.392531 0.947653i −0.175545 0.423803i 0.811478 0.584383i 0.198664π-0.198664\pi
−0.987023 + 0.160580i 0.948664π0.948664\pi
66 0 0
77 −0.893542 + 2.15720i −0.337727 + 0.815345i 0.660206 + 0.751084i 0.270469π0.270469\pi
−0.997933 + 0.0642606i 0.979531π0.979531\pi
88 0 0
99 2.94010 2.94010i 0.980034 0.980034i
1010 0 0
1111 −4.81196 1.99318i −1.45086 0.600966i −0.488457 0.872588i 0.662440π-0.662440\pi
−0.962404 + 0.271622i 0.912440π0.912440\pi
1212 0 0
1313 4.35777i 1.20863i 0.796747 + 0.604313i 0.206552π0.206552\pi
−0.796747 + 0.604313i 0.793448π0.793448\pi
1414 0 0
1515 −1.94050 1.94050i −0.501034 0.501034i
1616 0 0
1717 3.77648 + 1.65476i 0.915930 + 0.401338i
1818 0 0
1919 −1.89281 1.89281i −0.434240 0.434240i 0.455828 0.890068i 0.349343π-0.349343\pi
−0.890068 + 0.455828i 0.849343π0.849343\pi
2020 0 0
2121 6.24696i 1.36320i
2222 0 0
2323 3.79373 + 1.57141i 0.791047 + 0.327662i 0.741364 0.671103i 0.234179π-0.234179\pi
0.0496826 + 0.998765i 0.484179π0.484179\pi
2424 0 0
2525 2.79157 2.79157i 0.558314 0.558314i
2626 0 0
2727 1.18554 2.86215i 0.228157 0.550820i
2828 0 0
2929 −1.46962 3.54797i −0.272901 0.658841i 0.726704 0.686951i 0.241051π-0.241051\pi
−0.999605 + 0.0281099i 0.991051π0.991051\pi
3030 0 0
3131 −10.1411 + 4.20057i −1.82139 + 0.754444i −0.846272 + 0.532751i 0.821158π0.821158\pi
−0.975117 + 0.221693i 0.928842π0.928842\pi
3232 0 0
3333 −13.9348 −2.42574
3434 0 0
3535 2.39502 0.404832
3636 0 0
3737 1.02168 0.423195i 0.167964 0.0695728i −0.297117 0.954841i 0.596025π-0.596025\pi
0.465081 + 0.885268i 0.346025π0.346025\pi
3838 0 0
3939 4.46167 + 10.7714i 0.714439 + 1.72481i
4040 0 0
4141 −1.00448 + 2.42504i −0.156874 + 0.378728i −0.982702 0.185195i 0.940708π-0.940708\pi
0.825828 + 0.563923i 0.190708π0.190708\pi
4242 0 0
4343 −2.86647 + 2.86647i −0.437132 + 0.437132i −0.891046 0.453914i 0.850027π-0.850027\pi
0.453914 + 0.891046i 0.350027π0.350027\pi
4444 0 0
4545 −3.94028 1.63212i −0.587382 0.243302i
4646 0 0
4747 10.4047i 1.51768i −0.651280 0.758838i 0.725767π-0.725767\pi
0.651280 0.758838i 0.274233π-0.274233\pi
4848 0 0
4949 1.09465 + 1.09465i 0.156379 + 0.156379i
5050 0 0
5151 11.0288 + 0.223677i 1.54434 + 0.0313211i
5252 0 0
5353 −5.49867 5.49867i −0.755301 0.755301i 0.220162 0.975463i 0.429341π-0.429341\pi
−0.975463 + 0.220162i 0.929341π0.929341\pi
5454 0 0
5555 5.34245i 0.720376i
5656 0 0
5757 −6.61654 2.74066i −0.876383 0.363010i
5858 0 0
5959 7.69489 7.69489i 1.00179 1.00179i 0.00179152 0.999998i 0.499430π-0.499430\pi
0.999998 0.00179152i 0.000570259π-0.000570259\pi
6060 0 0
6161 3.04576 7.35311i 0.389969 0.941469i −0.599976 0.800018i 0.704823π-0.704823\pi
0.989945 0.141451i 0.0451767π-0.0451767\pi
6262 0 0
6363 3.71528 + 8.96949i 0.468082 + 1.13005i
6464 0 0
6565 4.12965 1.71056i 0.512220 0.212169i
6666 0 0
6767 6.44882 0.787848 0.393924 0.919143i 0.371117π-0.371117\pi
0.393924 + 0.919143i 0.371117π0.371117\pi
6868 0 0
6969 10.9861 1.32257
7070 0 0
7171 1.73561 0.718913i 0.205979 0.0853193i −0.277309 0.960781i 0.589442π-0.589442\pi
0.483288 + 0.875462i 0.339442π0.339442\pi
7272 0 0
7373 5.05217 + 12.1970i 0.591312 + 1.42755i 0.882236 + 0.470807i 0.156037π0.156037\pi
−0.290924 + 0.956746i 0.593963π0.593963\pi
7474 0 0
7575 4.04200 9.75826i 0.466730 1.12679i
7676 0 0
7777 8.59937 8.59937i 0.979989 0.979989i
7878 0 0
7979 2.31129 + 0.957366i 0.260040 + 0.107712i 0.508895 0.860829i 0.330054π-0.330054\pi
−0.248855 + 0.968541i 0.580054π0.580054\pi
8080 0 0
8181 4.18540i 0.465045i
8282 0 0
8383 0.0232879 + 0.0232879i 0.00255618 + 0.00255618i 0.708384 0.705828i 0.249425π-0.249425\pi
−0.705828 + 0.708384i 0.749425π0.749425\pi
8484 0 0
8585 0.0857556 4.22833i 0.00930150 0.458627i
8686 0 0
8787 −7.26512 7.26512i −0.778903 0.778903i
8888 0 0
8989 7.34936i 0.779030i 0.921020 + 0.389515i 0.127357π0.127357\pi
−0.921020 + 0.389515i 0.872643π0.872643\pi
9090 0 0
9191 −9.40057 3.89384i −0.985448 0.408186i
9292 0 0
9393 −20.7657 + 20.7657i −2.15330 + 2.15330i
9494 0 0
9595 −1.05074 + 2.53671i −0.107804 + 0.260261i
9696 0 0
9797 −6.98779 16.8700i −0.709502 1.71289i −0.701241 0.712924i 0.747370π-0.747370\pi
−0.00826130 0.999966i 0.502630π-0.502630\pi
9898 0 0
9999 −20.0078 + 8.28750i −2.01086 + 0.832925i
100100 0 0
101101 9.20094 0.915528 0.457764 0.889074i 0.348650π-0.348650\pi
0.457764 + 0.889074i 0.348650π0.348650\pi
102102 0 0
103103 7.88599 0.777029 0.388515 0.921443i 0.372988π-0.372988\pi
0.388515 + 0.921443i 0.372988π0.372988\pi
104104 0 0
105105 5.91995 2.45213i 0.577728 0.239303i
106106 0 0
107107 −1.67810 4.05130i −0.162228 0.391654i 0.821773 0.569815i 0.192985π-0.192985\pi
−0.984001 + 0.178161i 0.942985π0.942985\pi
108108 0 0
109109 2.32974 5.62449i 0.223149 0.538729i −0.772166 0.635421i 0.780826π-0.780826\pi
0.995314 + 0.0966928i 0.0308264π0.0308264\pi
110110 0 0
111111 2.09209 2.09209i 0.198572 0.198572i
112112 0 0
113113 −16.8902 6.99617i −1.58890 0.658144i −0.599109 0.800667i 0.704478π-0.704478\pi
−0.989791 + 0.142523i 0.954478π0.954478\pi
114114 0 0
115115 4.21197i 0.392768i
116116 0 0
117117 12.8123 + 12.8123i 1.18449 + 1.18449i
118118 0 0
119119 −6.94409 + 6.66802i −0.636563 + 0.611256i
120120 0 0
121121 11.4040 + 11.4040i 1.03673 + 1.03673i
122122 0 0
123123 7.02259i 0.633206i
124124 0 0
125125 −8.47948 3.51232i −0.758428 0.314151i
126126 0 0
127127 −6.83989 + 6.83989i −0.606942 + 0.606942i −0.942146 0.335203i 0.891195π-0.891195\pi
0.335203 + 0.942146i 0.391195π0.391195\pi
128128 0 0
129129 −4.15045 + 10.0201i −0.365427 + 0.882219i
130130 0 0
131131 3.79494 + 9.16179i 0.331565 + 0.800469i 0.998468 + 0.0553249i 0.0176195π0.0176195\pi
−0.666903 + 0.745144i 0.732381π0.732381\pi
132132 0 0
133133 5.77448 2.39187i 0.500711 0.207401i
134134 0 0
135135 −3.17768 −0.273491
136136 0 0
137137 2.40322 0.205321 0.102661 0.994716i 0.467264π-0.467264\pi
0.102661 + 0.994716i 0.467264π0.467264\pi
138138 0 0
139139 −8.03405 + 3.32781i −0.681439 + 0.282261i −0.696428 0.717626i 0.745229π-0.745229\pi
0.0149893 + 0.999888i 0.495229π0.495229\pi
140140 0 0
141141 −10.6527 25.7180i −0.897123 2.16585i
142142 0 0
143143 8.68581 20.9694i 0.726344 1.75355i
144144 0 0
145145 −2.78537 + 2.78537i −0.231313 + 0.231313i
146146 0 0
147147 3.82649 + 1.58498i 0.315603 + 0.130727i
148148 0 0
149149 3.90029i 0.319524i 0.987156 + 0.159762i 0.0510727π0.0510727\pi
−0.987156 + 0.159762i 0.948927π0.948927\pi
150150 0 0
151151 2.53060 + 2.53060i 0.205938 + 0.205938i 0.802538 0.596601i 0.203482π-0.203482\pi
−0.596601 + 0.802538i 0.703482π0.703482\pi
152152 0 0
153153 15.9684 6.23806i 1.29097 0.504317i
154154 0 0
155155 7.96136 + 7.96136i 0.639472 + 0.639472i
156156 0 0
157157 13.5186i 1.07890i 0.842018 + 0.539449i 0.181367π0.181367\pi
−0.842018 + 0.539449i 0.818633π0.818633\pi
158158 0 0
159159 −19.2213 7.96171i −1.52435 0.631405i
160160 0 0
161161 −6.77971 + 6.77971i −0.534316 + 0.534316i
162162 0 0
163163 0.517312 1.24890i 0.0405190 0.0978216i −0.902324 0.431058i 0.858141π-0.858141\pi
0.942843 + 0.333236i 0.108141π0.108141\pi
164164 0 0
165165 5.46984 + 13.2053i 0.425826 + 1.02804i
166166 0 0
167167 2.52757 1.04696i 0.195589 0.0810158i −0.282739 0.959197i 0.591243π-0.591243\pi
0.478328 + 0.878181i 0.341243π0.341243\pi
168168 0 0
169169 −5.99012 −0.460778
170170 0 0
171171 −11.1301 −0.851141
172172 0 0
173173 −3.58906 + 1.48664i −0.272871 + 0.113027i −0.514923 0.857236i 0.672180π-0.672180\pi
0.242052 + 0.970263i 0.422180π0.422180\pi
174174 0 0
175175 3.52759 + 8.51635i 0.266661 + 0.643776i
176176 0 0
177177 11.1417 26.8984i 0.837461 2.02181i
178178 0 0
179179 −5.91821 + 5.91821i −0.442348 + 0.442348i −0.892800 0.450452i 0.851263π-0.851263\pi
0.450452 + 0.892800i 0.351263π0.351263\pi
180180 0 0
181181 −6.48711 2.68705i −0.482183 0.199727i 0.128332 0.991731i 0.459038π-0.459038\pi
−0.610515 + 0.792004i 0.709038π0.709038\pi
182182 0 0
183183 21.2936i 1.57407i
184184 0 0
185185 −0.802084 0.802084i −0.0589704 0.0589704i
186186 0 0
187187 −14.8740 15.4898i −1.08770 1.13273i
188188 0 0
189189 5.11489 + 5.11489i 0.372054 + 0.372054i
190190 0 0
191191 12.6238i 0.913427i 0.889614 + 0.456713i 0.150973π0.150973\pi
−0.889614 + 0.456713i 0.849027π0.849027\pi
192192 0 0
193193 17.3393 + 7.18217i 1.24811 + 0.516984i 0.906240 0.422764i 0.138940π-0.138940\pi
0.341869 + 0.939748i 0.388940π0.388940\pi
194194 0 0
195195 8.45623 8.45623i 0.605563 0.605563i
196196 0 0
197197 −5.51876 + 13.3235i −0.393196 + 0.949258i 0.596044 + 0.802952i 0.296739π0.296739\pi
−0.989239 + 0.146306i 0.953261π0.953261\pi
198198 0 0
199199 −3.66350 8.84447i −0.259699 0.626968i 0.739220 0.673464i 0.235194π-0.235194\pi
−0.998918 + 0.0464963i 0.985194π0.985194\pi
200200 0 0
201201 15.9400 6.60258i 1.12432 0.465710i
202202 0 0
203203 8.96683 0.629348
204204 0 0
205205 2.69239 0.188045
206206 0 0
207207 15.7741 6.53383i 1.09637 0.454133i
208208 0 0
209209 5.33542 + 12.8808i 0.369059 + 0.890986i
210210 0 0
211211 0.615646 1.48630i 0.0423828 0.102321i −0.901270 0.433257i 0.857364π-0.857364\pi
0.943653 + 0.330936i 0.107364π0.107364\pi
212212 0 0
213213 3.55399 3.55399i 0.243515 0.243515i
214214 0 0
215215 3.84159 + 1.59124i 0.261995 + 0.108522i
216216 0 0
217217 25.6297i 1.73986i
218218 0 0
219219 24.9757 + 24.9757i 1.68770 + 1.68770i
220220 0 0
221221 −7.21106 + 16.4570i −0.485068 + 1.10702i
222222 0 0
223223 6.70407 + 6.70407i 0.448937 + 0.448937i 0.895001 0.446064i 0.147175π-0.147175\pi
−0.446064 + 0.895001i 0.647175π0.647175\pi
224224 0 0
225225 16.4150i 1.09433i
226226 0 0
227227 2.41865 + 1.00184i 0.160531 + 0.0664943i 0.461502 0.887139i 0.347311π-0.347311\pi
−0.300971 + 0.953633i 0.597311π0.597311\pi
228228 0 0
229229 3.48993 3.48993i 0.230621 0.230621i −0.582331 0.812952i 0.697859π-0.697859\pi
0.812952 + 0.582331i 0.197859π0.197859\pi
230230 0 0
231231 12.4513 30.0601i 0.819236 1.97781i
232232 0 0
233233 −0.873453 2.10870i −0.0572218 0.138146i 0.892683 0.450685i 0.148820π-0.148820\pi
−0.949905 + 0.312540i 0.898820π0.898820\pi
234234 0 0
235235 −9.86001 + 4.08415i −0.643196 + 0.266421i
236236 0 0
237237 6.69317 0.434768
238238 0 0
239239 −12.1821 −0.787995 −0.393997 0.919112i 0.628908π-0.628908\pi
−0.393997 + 0.919112i 0.628908π0.628908\pi
240240 0 0
241241 −13.3105 + 5.51338i −0.857404 + 0.355148i −0.767692 0.640820i 0.778595π-0.778595\pi
−0.0897121 + 0.995968i 0.528595π0.528595\pi
242242 0 0
243243 7.84181 + 18.9318i 0.503053 + 1.21448i
244244 0 0
245245 0.607666 1.46704i 0.0388223 0.0937254i
246246 0 0
247247 8.24842 8.24842i 0.524835 0.524835i
248248 0 0
249249 0.0814055 + 0.0337193i 0.00515887 + 0.00213687i
250250 0 0
251251 17.8445i 1.12634i 0.826342 + 0.563168i 0.190418π0.190418\pi
−0.826342 + 0.563168i 0.809582π0.809582\pi
252252 0 0
253253 −15.1232 15.1232i −0.950785 0.950785i
254254 0 0
255255 −4.11718 10.5393i −0.257828 0.659996i
256256 0 0
257257 −3.82080 3.82080i −0.238335 0.238335i 0.577825 0.816160i 0.303901π-0.303901\pi
−0.816160 + 0.577825i 0.803901π0.803901\pi
258258 0 0
259259 2.58212i 0.160445i
260260 0 0
261261 −14.7522 6.11056i −0.913138 0.378234i
262262 0 0
263263 6.98795 6.98795i 0.430896 0.430896i −0.458037 0.888933i 0.651447π-0.651447\pi
0.888933 + 0.458037i 0.151447π0.151447\pi
264264 0 0
265265 −3.05244 + 7.36924i −0.187510 + 0.452689i
266266 0 0
267267 7.52459 + 18.1660i 0.460497 + 1.11174i
268268 0 0
269269 −24.7296 + 10.2433i −1.50779 + 0.624546i −0.975100 0.221765i 0.928818π-0.928818\pi
−0.532688 + 0.846312i 0.678818π0.678818\pi
270270 0 0
271271 26.6483 1.61877 0.809384 0.587280i 0.199801π-0.199801\pi
0.809384 + 0.587280i 0.199801π0.199801\pi
272272 0 0
273273 −27.2228 −1.64760
274274 0 0
275275 −18.9970 + 7.86882i −1.14556 + 0.474508i
276276 0 0
277277 −10.7366 25.9204i −0.645097 1.55740i −0.819718 0.572767i 0.805870π-0.805870\pi
0.174621 0.984636i 0.444130π-0.444130\pi
278278 0 0
279279 −17.4657 + 42.1658i −1.04564 + 2.52440i
280280 0 0
281281 −14.9649 + 14.9649i −0.892731 + 0.892731i −0.994779 0.102049i 0.967460π-0.967460\pi
0.102049 + 0.994779i 0.467460π0.467460\pi
282282 0 0
283283 4.05293 + 1.67878i 0.240922 + 0.0997931i 0.499877 0.866096i 0.333378π-0.333378\pi
−0.258956 + 0.965889i 0.583378π0.583378\pi
284284 0 0
285285 7.34598i 0.435139i
286286 0 0
287287 −4.33375 4.33375i −0.255813 0.255813i
288288 0 0
289289 11.5235 + 12.4983i 0.677855 + 0.735195i
290290 0 0
291291 −34.5445 34.5445i −2.02503 2.02503i
292292 0 0
293293 11.8892i 0.694573i −0.937759 0.347286i 0.887103π-0.887103\pi
0.937759 0.347286i 0.112897π-0.112897\pi
294294 0 0
295295 −10.3126 4.27161i −0.600421 0.248703i
296296 0 0
297297 −11.4095 + 11.4095i −0.662049 + 0.662049i
298298 0 0
299299 −6.84785 + 16.5322i −0.396022 + 0.956080i
300300 0 0
301301 −3.62224 8.74485i −0.208782 0.504045i
302302 0 0
303303 22.7427 9.42032i 1.30653 0.541183i
304304 0 0
305305 −8.16375 −0.467455
306306 0 0
307307 20.9185 1.19388 0.596941 0.802285i 0.296383π-0.296383\pi
0.596941 + 0.802285i 0.296383π0.296383\pi
308308 0 0
309309 19.4924 8.07401i 1.10888 0.459315i
310310 0 0
311311 2.34082 + 5.65124i 0.132736 + 0.320453i 0.976248 0.216658i 0.0695157π-0.0695157\pi
−0.843512 + 0.537111i 0.819516π0.819516\pi
312312 0 0
313313 −10.9736 + 26.4926i −0.620264 + 1.49745i 0.231130 + 0.972923i 0.425758π0.425758\pi
−0.851394 + 0.524527i 0.824242π0.824242\pi
314314 0 0
315315 7.04160 7.04160i 0.396749 0.396749i
316316 0 0
317317 5.79187 + 2.39907i 0.325304 + 0.134745i 0.539358 0.842077i 0.318667π-0.318667\pi
−0.214054 + 0.976822i 0.568667π0.568667\pi
318318 0 0
319319 20.0019i 1.11989i
320320 0 0
321321 −8.29578 8.29578i −0.463026 0.463026i
322322 0 0
323323 −4.01601 10.2803i −0.223457 0.572011i
324324 0 0
325325 12.1650 + 12.1650i 0.674793 + 0.674793i
326326 0 0
327327 16.2878i 0.900716i
328328 0 0
329329 22.4449 + 9.29700i 1.23743 + 0.512560i
330330 0 0
331331 6.76610 6.76610i 0.371899 0.371899i −0.496270 0.868168i 0.665297π-0.665297\pi
0.868168 + 0.496270i 0.165297π0.165297\pi
332332 0 0
333333 1.75961 4.24809i 0.0964263 0.232794i
334334 0 0
335335 −2.53136 6.11124i −0.138303 0.333893i
336336 0 0
337337 −2.11552 + 0.876277i −0.115240 + 0.0477339i −0.439558 0.898214i 0.644865π-0.644865\pi
0.324319 + 0.945948i 0.394865π0.394865\pi
338338 0 0
339339 −48.9119 −2.65653
340340 0 0
341341 57.1709 3.09598
342342 0 0
343343 −18.4399 + 7.63806i −0.995661 + 0.412416i
344344 0 0
345345 −4.31239 10.4110i −0.232171 0.560512i
346346 0 0
347347 −1.53048 + 3.69490i −0.0821603 + 0.198353i −0.959621 0.281296i 0.909236π-0.909236\pi
0.877461 + 0.479648i 0.159236π0.159236\pi
348348 0 0
349349 18.7738 18.7738i 1.00494 1.00494i 0.00495064 0.999988i 0.498424π-0.498424\pi
0.999988 0.00495064i 0.00157584π-0.00157584\pi
350350 0 0
351351 12.4726 + 5.16630i 0.665736 + 0.275757i
352352 0 0
353353 1.75320i 0.0933134i −0.998911 0.0466567i 0.985143π-0.985143\pi
0.998911 0.0466567i 0.0148567π-0.0148567\pi
354354 0 0
355355 −1.36256 1.36256i −0.0723172 0.0723172i
356356 0 0
357357 −10.3372 + 23.5915i −0.547104 + 1.24859i
358358 0 0
359359 −14.0466 14.0466i −0.741353 0.741353i 0.231486 0.972838i 0.425641π-0.425641\pi
−0.972838 + 0.231486i 0.925641π0.925641\pi
360360 0 0
361361 11.8345i 0.622870i
362362 0 0
363363 39.8641 + 16.5123i 2.09232 + 0.866669i
364364 0 0
365365 9.57541 9.57541i 0.501200 0.501200i
366366 0 0
367367 6.63148 16.0098i 0.346160 0.835705i −0.650906 0.759159i 0.725611π-0.725611\pi
0.997066 0.0765462i 0.0243893π-0.0243893\pi
368368 0 0
369369 4.17658 + 10.0832i 0.217424 + 0.524908i
370370 0 0
371371 16.7750 6.94845i 0.870916 0.360745i
372372 0 0
373373 0.854460 0.0442423 0.0221211 0.999755i 0.492958π-0.492958\pi
0.0221211 + 0.999755i 0.492958π0.492958\pi
374374 0 0
375375 −24.5554 −1.26804
376376 0 0
377377 15.4612 6.40424i 0.796292 0.329835i
378378 0 0
379379 −11.4441 27.6284i −0.587843 1.41918i −0.885561 0.464524i 0.846226π-0.846226\pi
0.297718 0.954654i 0.403774π-0.403774\pi
380380 0 0
381381 −9.90370 + 23.9097i −0.507382 + 1.22493i
382382 0 0
383383 −19.5522 + 19.5522i −0.999073 + 0.999073i −1.00000 0.000926910i 0.999705π-0.999705\pi
0.000926910 1.00000i 0.499705π0.499705\pi
384384 0 0
385385 −11.5247 4.77370i −0.587355 0.243291i
386386 0 0
387387 16.8554i 0.856809i
388388 0 0
389389 −20.7260 20.7260i −1.05085 1.05085i −0.998636 0.0522161i 0.983372π-0.983372\pi
−0.0522161 0.998636i 0.516628π-0.516628\pi
390390 0 0
391391 11.7266 + 12.2121i 0.593040 + 0.617593i
392392 0 0
393393 18.7605 + 18.7605i 0.946341 + 0.946341i
394394 0 0
395395 2.56609i 0.129114i
396396 0 0
397397 17.1844 + 7.11800i 0.862458 + 0.357242i 0.769668 0.638444i 0.220422π-0.220422\pi
0.0927899 + 0.995686i 0.470422π0.470422\pi
398398 0 0
399399 11.8243 11.8243i 0.591956 0.591956i
400400 0 0
401401 3.03684 7.33159i 0.151653 0.366122i −0.829735 0.558157i 0.811509π-0.811509\pi
0.981388 + 0.192035i 0.0615087π0.0615087\pi
402402 0 0
403403 −18.3051 44.1924i −0.911841 2.20138i
404404 0 0
405405 3.96631 1.64290i 0.197088 0.0816363i
406406 0 0
407407 −5.75980 −0.285503
408408 0 0
409409 −6.93345 −0.342837 −0.171418 0.985198i 0.554835π-0.554835\pi
−0.171418 + 0.985198i 0.554835π0.554835\pi
410410 0 0
411411 5.94023 2.46052i 0.293010 0.121369i
412412 0 0
413413 9.72372 + 23.4751i 0.478473 + 1.15514i
414414 0 0
415415 0.0129276 0.0312100i 0.000634592 0.00153204i
416416 0 0
417417 −16.4512 + 16.4512i −0.805619 + 0.805619i
418418 0 0
419419 23.6912 + 9.81323i 1.15739 + 0.479407i 0.877006 0.480480i 0.159537π-0.159537\pi
0.280386 + 0.959887i 0.409537π0.409537\pi
420420 0 0
421421 34.5518i 1.68395i 0.539513 + 0.841977i 0.318608π0.318608\pi
−0.539513 + 0.841977i 0.681392π0.681392\pi
422422 0 0
423423 −30.5908 30.5908i −1.48737 1.48737i
424424 0 0
425425 15.1617 5.92291i 0.735449 0.287303i
426426 0 0
427427 13.1406 + 13.1406i 0.635919 + 0.635919i
428428 0 0
429429 60.7245i 2.93181i
430430 0 0
431431 −2.65090 1.09804i −0.127689 0.0528907i 0.317924 0.948116i 0.397014π-0.397014\pi
−0.445613 + 0.895226i 0.647014π0.647014\pi
432432 0 0
433433 11.9564 11.9564i 0.574586 0.574586i −0.358820 0.933407i 0.616821π-0.616821\pi
0.933407 + 0.358820i 0.116821π0.116821\pi
434434 0 0
435435 −4.03303 + 9.73660i −0.193369 + 0.466834i
436436 0 0
437437 −4.20642 10.1552i −0.201220 0.485789i
438438 0 0
439439 19.2330 7.96657i 0.917941 0.380223i 0.126850 0.991922i 0.459513π-0.459513\pi
0.791091 + 0.611698i 0.209513π0.209513\pi
440440 0 0
441441 6.43678 0.306513
442442 0 0
443443 −23.5157 −1.11726 −0.558632 0.829416i 0.688673π-0.688673\pi
−0.558632 + 0.829416i 0.688673π0.688673\pi
444444 0 0
445445 6.96464 2.88485i 0.330156 0.136755i
446446 0 0
447447 3.99328 + 9.64063i 0.188876 + 0.455986i
448448 0 0
449449 3.67748 8.87821i 0.173551 0.418989i −0.813039 0.582210i 0.802188π-0.802188\pi
0.986590 + 0.163221i 0.0521883π0.0521883\pi
450450 0 0
451451 9.66708 9.66708i 0.455205 0.455205i
452452 0 0
453453 8.84603 + 3.66414i 0.415623 + 0.172157i
454454 0 0
455455 10.4369i 0.489291i
456456 0 0
457457 −12.1391 12.1391i −0.567843 0.567843i 0.363681 0.931524i 0.381520π-0.381520\pi
−0.931524 + 0.363681i 0.881520π0.881520\pi
458458 0 0
459459 9.21333 8.84704i 0.430041 0.412945i
460460 0 0
461461 −21.3878 21.3878i −0.996131 0.996131i 0.00386124 0.999993i 0.498771π-0.498771\pi
−0.999993 + 0.00386124i 0.998771π0.998771\pi
462462 0 0
463463 33.1695i 1.54152i 0.637128 + 0.770758i 0.280122π0.280122\pi
−0.637128 + 0.770758i 0.719878π0.719878\pi
464464 0 0
465465 27.8299 + 11.5275i 1.29058 + 0.534576i
466466 0 0
467467 3.94231 3.94231i 0.182428 0.182428i −0.609985 0.792413i 0.708824π-0.708824\pi
0.792413 + 0.609985i 0.208824π0.208824\pi
468468 0 0
469469 −5.76229 + 13.9114i −0.266078 + 0.642368i
470470 0 0
471471 13.8409 + 33.4148i 0.637754 + 1.53967i
472472 0 0
473473 19.5067 8.07994i 0.896919 0.371516i
474474 0 0
475475 −10.5678 −0.484885
476476 0 0
477477 −32.3333 −1.48044
478478 0 0
479479 21.3489 8.84301i 0.975457 0.404047i 0.162716 0.986673i 0.447975π-0.447975\pi
0.812741 + 0.582626i 0.197975π0.197975\pi
480480 0 0
481481 1.84418 + 4.45225i 0.0840875 + 0.203005i
482482 0 0
483483 −9.81656 + 23.6993i −0.446669 + 1.07835i
484484 0 0
485485 −13.2440 + 13.2440i −0.601379 + 0.601379i
486486 0 0
487487 13.6778 + 5.66554i 0.619801 + 0.256730i 0.670413 0.741988i 0.266117π-0.266117\pi
−0.0506118 + 0.998718i 0.516117π0.516117\pi
488488 0 0
489489 3.61665i 0.163551i
490490 0 0
491491 −0.370833 0.370833i −0.0167355 0.0167355i 0.698690 0.715425i 0.253767π-0.253767\pi
−0.715425 + 0.698690i 0.753767π0.753767\pi
492492 0 0
493493 0.321065 15.8307i 0.0144600 0.712977i
494494 0 0
495495 15.7074 + 15.7074i 0.705993 + 0.705993i
496496 0 0
497497 4.38644i 0.196759i
498498 0 0
499499 14.9111 + 6.17637i 0.667512 + 0.276492i 0.690596 0.723241i 0.257348π-0.257348\pi
−0.0230839 + 0.999734i 0.507348π0.507348\pi
500500 0 0
501501 5.17568 5.17568i 0.231232 0.231232i
502502 0 0
503503 4.92666 11.8940i 0.219669 0.530328i −0.775175 0.631747i 0.782338π-0.782338\pi
0.994844 + 0.101419i 0.0323382π0.0323382\pi
504504 0 0
505505 −3.61165 8.71931i −0.160717 0.388004i
506506 0 0
507507 −14.8062 + 6.13294i −0.657568 + 0.272374i
508508 0 0
509509 0.684018 0.0303185 0.0151593 0.999885i 0.495174π-0.495174\pi
0.0151593 + 0.999885i 0.495174π0.495174\pi
510510 0 0
511511 −30.8257 −1.36365
512512 0 0
513513 −7.66150 + 3.17350i −0.338264 + 0.140113i
514514 0 0
515515 −3.09549 7.47318i −0.136404 0.329308i
516516 0 0
517517 −20.7384 + 50.0668i −0.912072 + 2.20194i
518518 0 0
519519 −7.34927 + 7.34927i −0.322597 + 0.322597i
520520 0 0
521521 −4.38938 1.81814i −0.192302 0.0796543i 0.284454 0.958690i 0.408188π-0.408188\pi
−0.476757 + 0.879035i 0.658188π0.658188\pi
522522 0 0
523523 40.6707i 1.77840i −0.457515 0.889202i 0.651260π-0.651260\pi
0.457515 0.889202i 0.348740π-0.348740\pi
524524 0 0
525525 17.4388 + 17.4388i 0.761092 + 0.761092i
526526 0 0
527527 −45.2484 0.917691i −1.97105 0.0399753i
528528 0 0
529529 −4.34042 4.34042i −0.188714 0.188714i
530530 0 0
531531 45.2475i 1.96358i
532532 0 0
533533 −10.5678 4.37731i −0.457740 0.189602i
534534 0 0
535535 −3.18052 + 3.18052i −0.137506 + 0.137506i
536536 0 0
537537 −8.56918 + 20.6878i −0.369787 + 0.892745i
538538 0 0
539539 −3.08558 7.44926i −0.132906 0.320862i
540540 0 0
541541 −31.9376 + 13.2290i −1.37310 + 0.568759i −0.942629 0.333843i 0.891654π-0.891654\pi
−0.430476 + 0.902602i 0.641654π0.641654\pi
542542 0 0
543543 −18.7858 −0.806176
544544 0 0
545545 −6.24456 −0.267488
546546 0 0
547547 −24.4795 + 10.1397i −1.04667 + 0.433544i −0.838701 0.544592i 0.816685π-0.816685\pi
−0.207967 + 0.978136i 0.566685π0.566685\pi
548548 0 0
549549 −12.6640 30.5737i −0.540488 1.30485i
550550 0 0
551551 −3.93392 + 9.49733i −0.167591 + 0.404600i
552552 0 0
553553 −4.13046 + 4.13046i −0.175645 + 0.175645i
554554 0 0
555555 −2.80378 1.16136i −0.119014 0.0492971i
556556 0 0
557557 24.1708i 1.02415i 0.858941 + 0.512075i 0.171123π0.171123\pi
−0.858941 + 0.512075i 0.828877π0.828877\pi
558558 0 0
559559 −12.4914 12.4914i −0.528330 0.528330i
560560 0 0
561561 −52.6244 23.0587i −2.22180 0.973541i
562562 0 0
563563 2.50308 + 2.50308i 0.105492 + 0.105492i 0.757883 0.652391i 0.226234π-0.226234\pi
−0.652391 + 0.757883i 0.726234π0.726234\pi
564564 0 0
565565 18.7523i 0.788916i
566566 0 0
567567 −9.02875 3.73983i −0.379172 0.157058i
568568 0 0
569569 11.2348 11.2348i 0.470987 0.470987i −0.431247 0.902234i 0.641926π-0.641926\pi
0.902234 + 0.431247i 0.141926π0.141926\pi
570570 0 0
571571 −6.56569 + 15.8510i −0.274765 + 0.663342i −0.999675 0.0255019i 0.991882π-0.991882\pi
0.724909 + 0.688844i 0.241882π0.241882\pi
572572 0 0
573573 12.9248 + 31.2032i 0.539941 + 1.30353i
574574 0 0
575575 14.9772 6.20374i 0.624591 0.258714i
576576 0 0
577577 39.0319 1.62492 0.812460 0.583017i 0.198128π-0.198128\pi
0.812460 + 0.583017i 0.198128π0.198128\pi
578578 0 0
579579 50.2122 2.08675
580580 0 0
581581 −0.0710453 + 0.0294279i −0.00294745 + 0.00122088i
582582 0 0
583583 15.4996 + 37.4192i 0.641926 + 1.54975i
584584 0 0
585585 7.11238 17.1708i 0.294061 0.709925i
586586 0 0
587587 −11.2227 + 11.2227i −0.463212 + 0.463212i −0.899707 0.436495i 0.856220π-0.856220\pi
0.436495 + 0.899707i 0.356220π0.356220\pi
588588 0 0
589589 27.1460 + 11.2442i 1.11853 + 0.463311i
590590 0 0
591591 38.5830i 1.58709i
592592 0 0
593593 7.35714 + 7.35714i 0.302121 + 0.302121i 0.841843 0.539722i 0.181471π-0.181471\pi
−0.539722 + 0.841843i 0.681471π0.681471\pi
594594 0 0
595595 9.04474 + 3.96318i 0.370798 + 0.162475i
596596 0 0
597597 −18.1107 18.1107i −0.741222 0.741222i
598598 0 0
599599 35.7297i 1.45988i −0.683513 0.729938i 0.739549π-0.739549\pi
0.683513 0.729938i 0.260451π-0.260451\pi
600600 0 0
601601 −25.7150 10.6515i −1.04894 0.434484i −0.209425 0.977825i 0.567159π-0.567159\pi
−0.839512 + 0.543341i 0.817159π0.817159\pi
602602 0 0
603603 18.9602 18.9602i 0.772118 0.772118i
604604 0 0
605605 6.33063 15.2835i 0.257377 0.621362i
606606 0 0
607607 6.58165 + 15.8895i 0.267141 + 0.644935i 0.999346 0.0361486i 0.0115090π-0.0115090\pi
−0.732206 + 0.681084i 0.761509π0.761509\pi
608608 0 0
609609 22.1640 9.18063i 0.898131 0.372018i
610610 0 0
611611 45.3411 1.83430
612612 0 0
613613 −44.0031 −1.77727 −0.888634 0.458618i 0.848345π-0.848345\pi
−0.888634 + 0.458618i 0.848345π0.848345\pi
614614 0 0
615615 6.65498 2.75658i 0.268355 0.111156i
616616 0 0
617617 −11.3310 27.3555i −0.456169 1.10129i −0.969936 0.243360i 0.921750π-0.921750\pi
0.513767 0.857930i 0.328250π-0.328250\pi
618618 0 0
619619 10.0162 24.1812i 0.402584 0.971923i −0.584453 0.811428i 0.698691π-0.698691\pi
0.987037 0.160495i 0.0513091π-0.0513091\pi
620620 0 0
621621 8.99523 8.99523i 0.360966 0.360966i
622622 0 0
623623 −15.8540 6.56695i −0.635178 0.263099i
624624 0 0
625625 10.3251i 0.413003i
626626 0 0
627627 26.3759 + 26.3759i 1.05335 + 1.05335i
628628 0 0
629629 4.55865 + 0.0924547i 0.181765 + 0.00368641i
630630 0 0
631631 28.4881 + 28.4881i 1.13409 + 1.13409i 0.989490 + 0.144604i 0.0461908π0.0461908\pi
0.144604 + 0.989490i 0.453809π0.453809\pi
632632 0 0
633633 4.30413i 0.171074i
634634 0 0
635635 9.16672 + 3.79698i 0.363770 + 0.150678i
636636 0 0
637637 −4.77024 + 4.77024i −0.189004 + 0.189004i
638638 0 0
639639 2.98919 7.21655i 0.118251 0.285482i
640640 0 0
641641 3.02621 + 7.30593i 0.119528 + 0.288567i 0.972308 0.233705i 0.0750848π-0.0750848\pi
−0.852779 + 0.522271i 0.825085π0.825085\pi
642642 0 0
643643 −5.12127 + 2.12130i −0.201963 + 0.0836559i −0.481372 0.876516i 0.659861π-0.659861\pi
0.279409 + 0.960172i 0.409861π0.409861\pi
644644 0 0
645645 11.1247 0.438036
646646 0 0
647647 18.3835 0.722731 0.361366 0.932424i 0.382311π-0.382311\pi
0.361366 + 0.932424i 0.382311π0.382311\pi
648648 0 0
649649 −52.3648 + 21.6902i −2.05550 + 0.851416i
650650 0 0
651651 −26.2408 63.3508i −1.02846 2.48291i
652652 0 0
653653 13.5240 32.6499i 0.529236 1.27769i −0.402789 0.915293i 0.631959π-0.631959\pi
0.932024 0.362395i 0.118041π-0.118041\pi
654654 0 0
655655 7.19257 7.19257i 0.281037 0.281037i
656656 0 0
657657 50.7144 + 21.0066i 1.97856 + 0.819545i
658658 0 0
659659 0.950773i 0.0370369i −0.999829 0.0185184i 0.994105π-0.994105\pi
0.999829 0.0185184i 0.00589494π-0.00589494\pi
660660 0 0
661661 −8.16636 8.16636i −0.317635 0.317635i 0.530223 0.847858i 0.322108π-0.322108\pi
−0.847858 + 0.530223i 0.822108π0.822108\pi
662662 0 0
663663 −0.974734 + 48.0610i −0.0378555 + 1.86653i
664664 0 0
665665 −4.53332 4.53332i −0.175795 0.175795i
666666 0 0
667667 15.7694i 0.610593i
668668 0 0
669669 23.4349 + 9.70704i 0.906044 + 0.375296i
670670 0 0
671671 −29.3121 + 29.3121i −1.13158 + 1.13158i
672672 0 0
673673 4.59155 11.0850i 0.176991 0.427295i −0.810341 0.585958i 0.800718π-0.800718\pi
0.987333 + 0.158663i 0.0507183π0.0507183\pi
674674 0 0
675675 −4.68036 11.2994i −0.180147 0.434914i
676676 0 0
677677 38.5010 15.9476i 1.47971 0.612917i 0.510663 0.859781i 0.329400π-0.329400\pi
0.969050 + 0.246864i 0.0794001π0.0794001\pi
678678 0 0
679679 42.6359 1.63621
680680 0 0
681681 7.00409 0.268397
682682 0 0
683683 10.4611 4.33312i 0.400282 0.165802i −0.173455 0.984842i 0.555493π-0.555493\pi
0.573737 + 0.819039i 0.305493π0.305493\pi
684684 0 0
685685 −0.943339 2.27742i −0.0360431 0.0870158i
686686 0 0
687687 5.05319 12.1995i 0.192791 0.465439i
688688 0 0
689689 23.9619 23.9619i 0.912877 0.912877i
690690 0 0
691691 22.8129 + 9.44941i 0.867843 + 0.359472i 0.771770 0.635902i 0.219372π-0.219372\pi
0.0960732 + 0.995374i 0.469372π0.469372\pi
692692 0 0
693693 50.5661i 1.92085i
694694 0 0
695695 6.30722 + 6.30722i 0.239247 + 0.239247i
696696 0 0
697697 −7.80627 + 7.49592i −0.295684 + 0.283928i
698698 0 0
699699 −4.31796 4.31796i −0.163320 0.163320i
700700 0 0
701701 38.0011i 1.43528i 0.696413 + 0.717642i 0.254778π0.254778\pi
−0.696413 + 0.717642i 0.745222π0.745222\pi
702702 0 0
703703 −2.73488 1.13282i −0.103148 0.0427253i
704704 0 0
705705 −20.1902 + 20.1902i −0.760407 + 0.760407i
706706 0 0
707707 −8.22143 + 19.8483i −0.309199 + 0.746471i
708708 0 0
709709 −7.48697 18.0752i −0.281179 0.678827i 0.718685 0.695336i 0.244745π-0.244745\pi
−0.999864 + 0.0165097i 0.994745π0.994745\pi
710710 0 0
711711 9.61016 3.98066i 0.360409 0.149286i
712712 0 0
713713 −45.0733 −1.68801
714714 0 0
715715 −23.2812 −0.870666
716716 0 0
717717 −30.1114 + 12.4726i −1.12453 + 0.465796i
718718 0 0
719719 −15.0002 36.2137i −0.559414 1.35054i −0.910231 0.414100i 0.864096π-0.864096\pi
0.350818 0.936444i 0.385904π-0.385904\pi
720720 0 0
721721 −7.04646 + 17.0116i −0.262424 + 0.633547i
722722 0 0
723723 −27.2557 + 27.2557i −1.01365 + 1.01365i
724724 0 0
725725 −14.0069 5.80186i −0.520204 0.215475i
726726 0 0
727727 49.8981i 1.85062i −0.379212 0.925310i 0.623805π-0.623805\pi
0.379212 0.925310i 0.376195π-0.376195\pi
728728 0 0
729729 29.8878 + 29.8878i 1.10696 + 1.10696i
730730 0 0
731731 −15.5685 + 6.08183i −0.575820 + 0.224945i
732732 0 0
733733 19.7537 + 19.7537i 0.729620 + 0.729620i 0.970544 0.240924i 0.0774505π-0.0774505\pi
−0.240924 + 0.970544i 0.577451π0.577451\pi
734734 0 0
735735 4.24834i 0.156702i
736736 0 0
737737 −31.0315 12.8536i −1.14306 0.473470i
738738 0 0
739739 −21.3501 + 21.3501i −0.785375 + 0.785375i −0.980732 0.195357i 0.937413π-0.937413\pi
0.195357 + 0.980732i 0.437413π0.437413\pi
740740 0 0
741741 11.9432 28.8333i 0.438743 1.05922i
742742 0 0
743743 −2.00765 4.84689i −0.0736534 0.177815i 0.882765 0.469815i 0.155679π-0.155679\pi
−0.956418 + 0.292000i 0.905679π0.905679\pi
744744 0 0
745745 3.69612 1.53098i 0.135415 0.0560909i
746746 0 0
747747 0.136937 0.00501028
748748 0 0
749749 10.2389 0.374122
750750 0 0
751751 13.3168 5.51601i 0.485938 0.201282i −0.126244 0.991999i 0.540292π-0.540292\pi
0.612182 + 0.790717i 0.290292π0.290292\pi
752752 0 0
753753 18.2700 + 44.1077i 0.665796 + 1.60737i
754754 0 0
755755 1.40480 3.39148i 0.0511257 0.123428i
756756 0 0
757757 −4.45055 + 4.45055i −0.161758 + 0.161758i −0.783345 0.621587i 0.786488π-0.786488\pi
0.621587 + 0.783345i 0.286488π0.286488\pi
758758 0 0
759759 −52.8648 21.8973i −1.91887 0.794822i
760760 0 0
761761 6.45544i 0.234010i −0.993131 0.117005i 0.962671π-0.962671\pi
0.993131 0.117005i 0.0373293π-0.0373293\pi
762762 0 0
763763 10.0514 + 10.0514i 0.363886 + 0.363886i
764764 0 0
765765 −12.1796 12.6839i −0.440354 0.458586i
766766 0 0
767767 33.5325 + 33.5325i 1.21079 + 1.21079i
768768 0 0
769769 0.665285i 0.0239908i −0.999928 0.0119954i 0.996182π-0.996182\pi
0.999928 0.0119954i 0.00381835π-0.00381835\pi
770770 0 0
771771 −13.3561 5.53227i −0.481007 0.199240i
772772 0 0
773773 3.09332 3.09332i 0.111259 0.111259i −0.649286 0.760545i 0.724932π-0.724932\pi
0.760545 + 0.649286i 0.224932π0.224932\pi
774774 0 0
775775 −16.5833 + 40.0356i −0.595690 + 1.43812i
776776 0 0
777777 2.64368 + 6.38241i 0.0948416 + 0.228968i
778778 0 0
779779 6.49144 2.68884i 0.232580 0.0963378i
780780 0 0
781781 −9.78461 −0.350121
782782 0 0
783783 −11.8971 −0.425167
784784 0 0
785785 12.8109 5.30645i 0.457241 0.189395i
786786 0 0
787787 −2.82833 6.82819i −0.100819 0.243399i 0.865419 0.501048i 0.167052π-0.167052\pi
−0.966238 + 0.257649i 0.917052π0.917052\pi
788788 0 0
789789 10.1181 24.4272i 0.360213 0.869632i
790790 0 0
791791 30.1843 30.1843i 1.07323 1.07323i
792792 0 0
793793 32.0431 + 13.2727i 1.13788 + 0.471327i
794794 0 0
795795 21.3403i 0.756863i
796796 0 0
797797 15.9635 + 15.9635i 0.565458 + 0.565458i 0.930853 0.365395i 0.119066π-0.119066\pi
−0.365395 + 0.930853i 0.619066π0.619066\pi
798798 0 0
799799 17.2172 39.2930i 0.609101 1.39008i
800800 0 0
801801 21.6078 + 21.6078i 0.763476 + 0.763476i
802802 0 0
803803 68.7615i 2.42654i
804804 0 0
805805 9.08606 + 3.76357i 0.320241 + 0.132648i
806806 0 0
807807 −50.6384 + 50.6384i −1.78256 + 1.78256i
808808 0 0
809809 1.68152 4.05955i 0.0591191 0.142726i −0.891560 0.452903i 0.850388π-0.850388\pi
0.950679 + 0.310177i 0.100388π0.100388\pi
810810 0 0
811811 4.35027 + 10.5025i 0.152759 + 0.368792i 0.981670 0.190588i 0.0610394π-0.0610394\pi
−0.828912 + 0.559380i 0.811039π0.811039\pi
812812 0 0
813813 65.8686 27.2837i 2.31011 0.956880i
814814 0 0
815815 −1.38659 −0.0485700
816816 0 0
817817 10.8514 0.379641
818818 0 0
819819 −39.0869 + 16.1903i −1.36581 + 0.565736i
820820 0 0
821821 17.2551 + 41.6575i 0.602207 + 1.45386i 0.871304 + 0.490743i 0.163275π0.163275\pi
−0.269097 + 0.963113i 0.586725π0.586725\pi
822822 0 0
823823 4.09403 9.88385i 0.142709 0.344529i −0.836323 0.548237i 0.815299π-0.815299\pi
0.979032 + 0.203708i 0.0652992π0.0652992\pi
824824 0 0
825825 −38.8999 + 38.8999i −1.35432 + 1.35432i
826826 0 0
827827 −46.9906 19.4641i −1.63402 0.676834i −0.638348 0.769748i 0.720382π-0.720382\pi
−0.995674 + 0.0929138i 0.970382π0.970382\pi
828828 0 0
829829 6.52156i 0.226503i 0.993566 + 0.113252i 0.0361266π0.0361266\pi
−0.993566 + 0.113252i 0.963873π0.963873\pi
830830 0 0
831831 −53.0767 53.0767i −1.84121 1.84121i
832832 0 0
833833 2.32254 + 5.94531i 0.0804712 + 0.205993i
834834 0 0
835835 −1.98430 1.98430i −0.0686696 0.0686696i
836836 0 0
837837 34.0051i 1.17539i
838838 0 0
839839 7.72808 + 3.20108i 0.266803 + 0.110513i 0.512075 0.858941i 0.328877π-0.328877\pi
−0.245272 + 0.969454i 0.578877π0.578877\pi
840840 0 0
841841 10.0778 10.0778i 0.347511 0.347511i
842842 0 0
843843 −21.6682 + 52.3116i −0.746291 + 1.80171i
844844 0 0
845845 2.35131 + 5.67656i 0.0808874 + 0.195279i
846846 0 0
847847 −34.7907 + 14.4108i −1.19542 + 0.495161i
848848 0 0
849849 11.7367 0.402804
850850 0 0
851851 4.54100 0.155663
852852 0 0
853853 −2.86110 + 1.18511i −0.0979622 + 0.0405773i −0.431126 0.902292i 0.641884π-0.641884\pi
0.333164 + 0.942869i 0.391884π0.391884\pi
854854 0 0
855855 4.36891 + 10.5475i 0.149414 + 0.360716i
856856 0 0
857857 3.14145 7.58414i 0.107310 0.259069i −0.861099 0.508438i 0.830223π-0.830223\pi
0.968409 + 0.249369i 0.0802231π0.0802231\pi
858858 0 0
859859 −7.93014 + 7.93014i −0.270573 + 0.270573i −0.829331 0.558758i 0.811278π-0.811278\pi
0.558758 + 0.829331i 0.311278π0.311278\pi
860860 0 0
861861 −15.1491 6.27498i −0.516281 0.213851i
862862 0 0
863863 34.8722i 1.18706i −0.804811 0.593532i 0.797733π-0.797733\pi
0.804811 0.593532i 0.202267π-0.202267\pi
864864 0 0
865865 2.81763 + 2.81763i 0.0958024 + 0.0958024i
866866 0 0
867867 41.2799 + 19.0948i 1.40194 + 0.648492i
868868 0 0
869869 −9.21361 9.21361i −0.312550 0.312550i
870870 0 0
871871 28.1024i 0.952215i
872872 0 0
873873 −70.1443 29.0547i −2.37403 0.983354i
874874 0 0
875875 15.1535 15.1535i 0.512283 0.512283i
876876 0 0
877877 4.71367 11.3798i 0.159169 0.384269i −0.824095 0.566451i 0.808316π-0.808316\pi
0.983265 + 0.182182i 0.0583160π0.0583160\pi
878878 0 0
879879 −12.1726 29.3874i −0.410573 0.991211i
880880 0 0
881881 −38.7338 + 16.0441i −1.30498 + 0.540538i −0.923414 0.383805i 0.874613π-0.874613\pi
−0.381561 + 0.924344i 0.624613π0.624613\pi
882882 0 0
883883 14.6148 0.491828 0.245914 0.969292i 0.420912π-0.420912\pi
0.245914 + 0.969292i 0.420912π0.420912\pi
884884 0 0
885885 −29.8638 −1.00386
886886 0 0
887887 −5.37765 + 2.22750i −0.180564 + 0.0747920i −0.471134 0.882062i 0.656155π-0.656155\pi
0.290570 + 0.956854i 0.406155π0.406155\pi
888888 0 0
889889 −8.64329 20.8667i −0.289887 0.699848i
890890 0 0
891891 8.34226 20.1400i 0.279476 0.674715i
892892 0 0
893893 −19.6941 + 19.6941i −0.659036 + 0.659036i
894894 0 0
895895 7.93150 + 3.28533i 0.265121 + 0.109817i
896896 0 0
897897 47.8750i 1.59850i
898898 0 0
899899 29.8069 + 29.8069i 0.994117 + 0.994117i
900900 0 0
901901 −11.6666 29.8646i −0.388672 0.994934i
902902 0 0
903903 −17.9067 17.9067i −0.595898 0.595898i
904904 0 0
905905 7.20228i 0.239412i
906906 0 0
907907 −2.22827 0.922978i −0.0739884 0.0306470i 0.345382 0.938462i 0.387749π-0.387749\pi
−0.419371 + 0.907815i 0.637749π0.637749\pi
908908 0 0
909909 27.0517 27.0517i 0.897249 0.897249i
910910 0 0
911911 −7.59570 + 18.3377i −0.251657 + 0.607553i −0.998338 0.0576286i 0.981646π-0.981646\pi
0.746681 + 0.665182i 0.231646π0.231646\pi
912912 0 0
913913 −0.0656434 0.158477i −0.00217248 0.00524483i
914914 0 0
915915 −20.1790 + 8.35840i −0.667096 + 0.276320i
916916 0 0
917917 −23.1547 −0.764637
918918 0 0
919919 16.7830 0.553619 0.276810 0.960925i 0.410723π-0.410723\pi
0.276810 + 0.960925i 0.410723π0.410723\pi
920920 0 0
921921 51.7058 21.4173i 1.70377 0.705723i
922922 0 0
923923 3.13286 + 7.56338i 0.103119 + 0.248952i
924924 0 0
925925 1.67072 4.03347i 0.0549329 0.132620i
926926 0 0
927927 23.1856 23.1856i 0.761515 0.761515i
928928 0 0
929929 −21.5034 8.90700i −0.705504 0.292229i 0.000938752 1.00000i 0.499701π-0.499701\pi
−0.706443 + 0.707770i 0.749701π0.749701\pi
930930 0 0
931931 4.14394i 0.135812i
932932 0 0
933933 11.5720 + 11.5720i 0.378849 + 0.378849i
934934 0 0
935935 −8.84048 + 20.1756i −0.289115 + 0.659814i
936936 0 0
937937 −28.1329 28.1329i −0.919062 0.919062i 0.0778997 0.996961i 0.475179π-0.475179\pi
−0.996961 + 0.0778997i 0.975179π0.975179\pi
938938 0 0
939939 76.7190i 2.50363i
940940 0 0
941941 26.4136 + 10.9409i 0.861059 + 0.356662i 0.769122 0.639102i 0.220694π-0.220694\pi
0.0919374 + 0.995765i 0.470694π0.470694\pi
942942 0 0
943943 −7.62148 + 7.62148i −0.248190 + 0.248190i
944944 0 0
945945 2.83939 6.85490i 0.0923654 0.222990i
946946 0 0
947947 16.3787 + 39.5418i 0.532238 + 1.28494i 0.930038 + 0.367463i 0.119774π0.119774\pi
−0.397801 + 0.917472i 0.630226π0.630226\pi
948948 0 0
949949 −53.1517 + 22.0162i −1.72538 + 0.714675i
950950 0 0
951951 16.7725 0.543885
952952 0 0
953953 −41.2093 −1.33490 −0.667451 0.744654i 0.732614π-0.732614\pi
−0.667451 + 0.744654i 0.732614π0.732614\pi
954954 0 0
955955 11.9630 4.95523i 0.387113 0.160348i
956956 0 0
957957 20.4788 + 49.4402i 0.661985 + 1.59817i
958958 0 0
959959 −2.14738 + 5.18423i −0.0693425 + 0.167408i
960960 0 0
961961 63.2761 63.2761i 2.04116 2.04116i
962962 0 0
963963 −16.8450 6.97743i −0.542823 0.224845i
964964 0 0
965965 19.2509i 0.619707i
966966 0 0
967967 −21.3856 21.3856i −0.687714 0.687714i 0.274013 0.961726i 0.411649π-0.411649\pi
−0.961726 + 0.274013i 0.911649π0.911649\pi
968968 0 0
969969 −20.4521 21.2988i −0.657015 0.684217i
970970 0 0
971971 −26.4835 26.4835i −0.849896 0.849896i 0.140224 0.990120i 0.455218π-0.455218\pi
−0.990120 + 0.140224i 0.955218π0.955218\pi
972972 0 0
973973 20.3046i 0.650935i
974974 0 0
975975 42.5242 + 17.6141i 1.36186 + 0.564103i
976976 0 0
977977 26.9082 26.9082i 0.860868 0.860868i −0.130571 0.991439i 0.541681π-0.541681\pi
0.991439 + 0.130571i 0.0416810π0.0416810\pi
978978 0 0
979979 14.6486 35.3648i 0.468171 1.13026i
980980 0 0
981981 −9.68690 23.3862i −0.309279 0.746665i
982982 0 0
983983 45.3329 18.7775i 1.44590 0.598909i 0.484676 0.874694i 0.338938π-0.338938\pi
0.961219 + 0.275785i 0.0889376π0.0889376\pi
984984 0 0
985985 14.7923 0.471323
986986 0 0
987987 64.9975 2.06889
988988 0 0
989989 −15.3790 + 6.37019i −0.489024 + 0.202560i
990990 0 0
991991 −12.8773 31.0886i −0.409062 0.987563i −0.985385 0.170341i 0.945513π-0.945513\pi
0.576323 0.817222i 0.304487π-0.304487\pi
992992 0 0
993993 9.79686 23.6517i 0.310894 0.750565i
994994 0 0
995995 −6.94345 + 6.94345i −0.220122 + 0.220122i
996996 0 0
997997 35.8385 + 14.8448i 1.13502 + 0.470139i 0.869483 0.493963i 0.164452π-0.164452\pi
0.265533 + 0.964102i 0.414452π0.414452\pi
998998 0 0
999999 3.42592i 0.108391i
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 136.2.n.c.121.3 yes 12
3.2 odd 2 1224.2.bq.c.937.2 12
4.3 odd 2 272.2.v.f.257.1 12
17.3 odd 16 2312.2.a.w.1.10 12
17.5 odd 16 2312.2.b.n.577.10 12
17.9 even 8 inner 136.2.n.c.9.3 12
17.12 odd 16 2312.2.b.n.577.3 12
17.14 odd 16 2312.2.a.w.1.3 12
51.26 odd 8 1224.2.bq.c.145.2 12
68.3 even 16 4624.2.a.bt.1.3 12
68.31 even 16 4624.2.a.bt.1.10 12
68.43 odd 8 272.2.v.f.145.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.n.c.9.3 12 17.9 even 8 inner
136.2.n.c.121.3 yes 12 1.1 even 1 trivial
272.2.v.f.145.1 12 68.43 odd 8
272.2.v.f.257.1 12 4.3 odd 2
1224.2.bq.c.145.2 12 51.26 odd 8
1224.2.bq.c.937.2 12 3.2 odd 2
2312.2.a.w.1.3 12 17.14 odd 16
2312.2.a.w.1.10 12 17.3 odd 16
2312.2.b.n.577.3 12 17.12 odd 16
2312.2.b.n.577.10 12 17.5 odd 16
4624.2.a.bt.1.3 12 68.3 even 16
4624.2.a.bt.1.10 12 68.31 even 16