Properties

Label 448.2.j.c.111.1
Level 448448
Weight 22
Character 448.111
Analytic conductor 3.5773.577
Analytic rank 00
Dimension 44
Inner twists 44

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(111,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 448=267 448 = 2^{6} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 448.j (of order 44, degree 22, not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 3.577298010553.57729801055
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(ζ12)\Q(\zeta_{12})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x2+1 x^{4} - x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a25]\Z[a_1, \ldots, a_{25}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

Embedding invariants

Embedding label 111.1
Root 0.866025+0.500000i0.866025 + 0.500000i of defining polynomial
Character χ\chi == 448.111
Dual form 448.2.j.c.335.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+(1.732051.73205i)q3+(1.732051.73205i)q5+(2.000001.73205i)q7+3.00000iq9+(3.000003.00000i)q11+(1.73205+1.73205i)q13+6.00000iq15+(5.19615+5.19615i)q19+(6.464100.464102i)q214.00000q23+1.00000iq25+(1.000001.00000i)q296.92820q31+10.3923iq33+(6.464100.464102i)q35+(5.00000+5.00000i)q37+6.00000q39+3.46410q41+(1.00000+1.00000i)q43+(5.196155.19615i)q456.92820q47+(1.000006.92820i)q49+(3.000003.00000i)q53+10.3923iq5518.0000iq57+(8.660258.66025i)q59+(5.196155.19615i)q61+(5.19615+6.00000i)q63+6.00000q65+(7.000007.00000i)q67+(6.92820+6.92820i)q694.00000q7110.3923q73+(1.732051.73205i)q75+(11.19620.803848i)q77+2.00000iq79+9.00000q81+(1.732051.73205i)q83+3.46410iq87+3.46410q89+(0.464102+6.46410i)q91+(12.0000+12.0000i)q9318.0000iq95+(9.000009.00000i)q99+O(q100)q+(-1.73205 - 1.73205i) q^{3} +(-1.73205 - 1.73205i) q^{5} +(2.00000 - 1.73205i) q^{7} +3.00000i q^{9} +(-3.00000 - 3.00000i) q^{11} +(-1.73205 + 1.73205i) q^{13} +6.00000i q^{15} +(5.19615 + 5.19615i) q^{19} +(-6.46410 - 0.464102i) q^{21} -4.00000 q^{23} +1.00000i q^{25} +(-1.00000 - 1.00000i) q^{29} -6.92820 q^{31} +10.3923i q^{33} +(-6.46410 - 0.464102i) q^{35} +(-5.00000 + 5.00000i) q^{37} +6.00000 q^{39} +3.46410 q^{41} +(1.00000 + 1.00000i) q^{43} +(5.19615 - 5.19615i) q^{45} -6.92820 q^{47} +(1.00000 - 6.92820i) q^{49} +(3.00000 - 3.00000i) q^{53} +10.3923i q^{55} -18.0000i q^{57} +(8.66025 - 8.66025i) q^{59} +(5.19615 - 5.19615i) q^{61} +(5.19615 + 6.00000i) q^{63} +6.00000 q^{65} +(7.00000 - 7.00000i) q^{67} +(6.92820 + 6.92820i) q^{69} -4.00000 q^{71} -10.3923 q^{73} +(1.73205 - 1.73205i) q^{75} +(-11.1962 - 0.803848i) q^{77} +2.00000i q^{79} +9.00000 q^{81} +(-1.73205 - 1.73205i) q^{83} +3.46410i q^{87} +3.46410 q^{89} +(-0.464102 + 6.46410i) q^{91} +(12.0000 + 12.0000i) q^{93} -18.0000i q^{95} +(9.00000 - 9.00000i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+8q712q1112q2116q234q2912q3520q37+24q39+4q43+4q49+12q53+24q65+28q6716q7124q77+36q81+12q91++36q99+O(q100) 4 q + 8 q^{7} - 12 q^{11} - 12 q^{21} - 16 q^{23} - 4 q^{29} - 12 q^{35} - 20 q^{37} + 24 q^{39} + 4 q^{43} + 4 q^{49} + 12 q^{53} + 24 q^{65} + 28 q^{67} - 16 q^{71} - 24 q^{77} + 36 q^{81} + 12 q^{91}+ \cdots + 36 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/448Z)×\left(\mathbb{Z}/448\mathbb{Z}\right)^\times.

nn 127127 129129 197197
χ(n)\chi(n) 1-1 1-1 e(34)e\left(\frac{3}{4}\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 −1.73205 1.73205i −1.00000 1.00000i 1.00000i 0.5π-0.5\pi
−1.00000 π\pi
44 0 0
55 −1.73205 1.73205i −0.774597 0.774597i 0.204310 0.978906i 0.434505π-0.434505\pi
−0.978906 + 0.204310i 0.934505π0.934505\pi
66 0 0
77 2.00000 1.73205i 0.755929 0.654654i
88 0 0
99 3.00000i 1.00000i
1010 0 0
1111 −3.00000 3.00000i −0.904534 0.904534i 0.0912903 0.995824i 0.470901π-0.470901\pi
−0.995824 + 0.0912903i 0.970901π0.970901\pi
1212 0 0
1313 −1.73205 + 1.73205i −0.480384 + 0.480384i −0.905254 0.424870i 0.860320π-0.860320\pi
0.424870 + 0.905254i 0.360320π0.360320\pi
1414 0 0
1515 6.00000i 1.54919i
1616 0 0
1717 0 0 1.00000 00
−1.00000 π\pi
1818 0 0
1919 5.19615 + 5.19615i 1.19208 + 1.19208i 0.976483 + 0.215597i 0.0691696π0.0691696\pi
0.215597 + 0.976483i 0.430830π0.430830\pi
2020 0 0
2121 −6.46410 0.464102i −1.41058 0.101275i
2222 0 0
2323 −4.00000 −0.834058 −0.417029 0.908893i 0.636929π-0.636929\pi
−0.417029 + 0.908893i 0.636929π0.636929\pi
2424 0 0
2525 1.00000i 0.200000i
2626 0 0
2727 0 0
2828 0 0
2929 −1.00000 1.00000i −0.185695 0.185695i 0.608137 0.793832i 0.291917π-0.291917\pi
−0.793832 + 0.608137i 0.791917π0.791917\pi
3030 0 0
3131 −6.92820 −1.24434 −0.622171 0.782881i 0.713749π-0.713749\pi
−0.622171 + 0.782881i 0.713749π0.713749\pi
3232 0 0
3333 10.3923i 1.80907i
3434 0 0
3535 −6.46410 0.464102i −1.09263 0.0784475i
3636 0 0
3737 −5.00000 + 5.00000i −0.821995 + 0.821995i −0.986394 0.164399i 0.947432π-0.947432\pi
0.164399 + 0.986394i 0.447432π0.447432\pi
3838 0 0
3939 6.00000 0.960769
4040 0 0
4141 3.46410 0.541002 0.270501 0.962720i 0.412811π-0.412811\pi
0.270501 + 0.962720i 0.412811π0.412811\pi
4242 0 0
4343 1.00000 + 1.00000i 0.152499 + 0.152499i 0.779233 0.626734i 0.215609π-0.215609\pi
−0.626734 + 0.779233i 0.715609π0.715609\pi
4444 0 0
4545 5.19615 5.19615i 0.774597 0.774597i
4646 0 0
4747 −6.92820 −1.01058 −0.505291 0.862949i 0.668615π-0.668615\pi
−0.505291 + 0.862949i 0.668615π0.668615\pi
4848 0 0
4949 1.00000 6.92820i 0.142857 0.989743i
5050 0 0
5151 0 0
5252 0 0
5353 3.00000 3.00000i 0.412082 0.412082i −0.470381 0.882463i 0.655884π-0.655884\pi
0.882463 + 0.470381i 0.155884π0.155884\pi
5454 0 0
5555 10.3923i 1.40130i
5656 0 0
5757 18.0000i 2.38416i
5858 0 0
5959 8.66025 8.66025i 1.12747 1.12747i 0.136882 0.990587i 0.456292π-0.456292\pi
0.990587 0.136882i 0.0437080π-0.0437080\pi
6060 0 0
6161 5.19615 5.19615i 0.665299 0.665299i −0.291325 0.956624i 0.594096π-0.594096\pi
0.956624 + 0.291325i 0.0940961π0.0940961\pi
6262 0 0
6363 5.19615 + 6.00000i 0.654654 + 0.755929i
6464 0 0
6565 6.00000 0.744208
6666 0 0
6767 7.00000 7.00000i 0.855186 0.855186i −0.135580 0.990766i 0.543290π-0.543290\pi
0.990766 + 0.135580i 0.0432899π0.0432899\pi
6868 0 0
6969 6.92820 + 6.92820i 0.834058 + 0.834058i
7070 0 0
7171 −4.00000 −0.474713 −0.237356 0.971423i 0.576281π-0.576281\pi
−0.237356 + 0.971423i 0.576281π0.576281\pi
7272 0 0
7373 −10.3923 −1.21633 −0.608164 0.793812i 0.708094π-0.708094\pi
−0.608164 + 0.793812i 0.708094π0.708094\pi
7474 0 0
7575 1.73205 1.73205i 0.200000 0.200000i
7676 0 0
7777 −11.1962 0.803848i −1.27592 0.0916069i
7878 0 0
7979 2.00000i 0.225018i 0.993651 + 0.112509i 0.0358886π0.0358886\pi
−0.993651 + 0.112509i 0.964111π0.964111\pi
8080 0 0
8181 9.00000 1.00000
8282 0 0
8383 −1.73205 1.73205i −0.190117 0.190117i 0.605629 0.795747i 0.292921π-0.292921\pi
−0.795747 + 0.605629i 0.792921π0.792921\pi
8484 0 0
8585 0 0
8686 0 0
8787 3.46410i 0.371391i
8888 0 0
8989 3.46410 0.367194 0.183597 0.983002i 0.441226π-0.441226\pi
0.183597 + 0.983002i 0.441226π0.441226\pi
9090 0 0
9191 −0.464102 + 6.46410i −0.0486511 + 0.677622i
9292 0 0
9393 12.0000 + 12.0000i 1.24434 + 1.24434i
9494 0 0
9595 18.0000i 1.84676i
9696 0 0
9797 0 0 1.00000 00
−1.00000 π\pi
9898 0 0
9999 9.00000 9.00000i 0.904534 0.904534i
100100 0 0
101101 −1.73205 1.73205i −0.172345 0.172345i 0.615664 0.788009i 0.288888π-0.288888\pi
−0.788009 + 0.615664i 0.788888π0.788888\pi
102102 0 0
103103 10.3923i 1.02398i −0.858990 0.511992i 0.828908π-0.828908\pi
0.858990 0.511992i 0.171092π-0.171092\pi
104104 0 0
105105 10.3923 + 12.0000i 1.01419 + 1.17108i
106106 0 0
107107 −7.00000 7.00000i −0.676716 0.676716i 0.282540 0.959256i 0.408823π-0.408823\pi
−0.959256 + 0.282540i 0.908823π0.908823\pi
108108 0 0
109109 −13.0000 13.0000i −1.24517 1.24517i −0.957826 0.287348i 0.907226π-0.907226\pi
−0.287348 0.957826i 0.592774π-0.592774\pi
110110 0 0
111111 17.3205 1.64399
112112 0 0
113113 −10.0000 −0.940721 −0.470360 0.882474i 0.655876π-0.655876\pi
−0.470360 + 0.882474i 0.655876π0.655876\pi
114114 0 0
115115 6.92820 + 6.92820i 0.646058 + 0.646058i
116116 0 0
117117 −5.19615 5.19615i −0.480384 0.480384i
118118 0 0
119119 0 0
120120 0 0
121121 7.00000i 0.636364i
122122 0 0
123123 −6.00000 6.00000i −0.541002 0.541002i
124124 0 0
125125 −6.92820 + 6.92820i −0.619677 + 0.619677i
126126 0 0
127127 6.00000i 0.532414i −0.963916 0.266207i 0.914230π-0.914230\pi
0.963916 0.266207i 0.0857705π-0.0857705\pi
128128 0 0
129129 3.46410i 0.304997i
130130 0 0
131131 5.19615 + 5.19615i 0.453990 + 0.453990i 0.896676 0.442687i 0.145975π-0.145975\pi
−0.442687 + 0.896676i 0.645975π0.645975\pi
132132 0 0
133133 19.3923 + 1.39230i 1.68153 + 0.120728i
134134 0 0
135135 0 0
136136 0 0
137137 8.00000i 0.683486i −0.939793 0.341743i 0.888983π-0.888983\pi
0.939793 0.341743i 0.111017π-0.111017\pi
138138 0 0
139139 −5.19615 + 5.19615i −0.440732 + 0.440732i −0.892258 0.451526i 0.850880π-0.850880\pi
0.451526 + 0.892258i 0.350880π0.350880\pi
140140 0 0
141141 12.0000 + 12.0000i 1.01058 + 1.01058i
142142 0 0
143143 10.3923 0.869048
144144 0 0
145145 3.46410i 0.287678i
146146 0 0
147147 −13.7321 + 10.2679i −1.13260 + 0.846886i
148148 0 0
149149 −5.00000 + 5.00000i −0.409616 + 0.409616i −0.881605 0.471989i 0.843536π-0.843536\pi
0.471989 + 0.881605i 0.343536π0.343536\pi
150150 0 0
151151 12.0000 0.976546 0.488273 0.872691i 0.337627π-0.337627\pi
0.488273 + 0.872691i 0.337627π0.337627\pi
152152 0 0
153153 0 0
154154 0 0
155155 12.0000 + 12.0000i 0.963863 + 0.963863i
156156 0 0
157157 −8.66025 + 8.66025i −0.691164 + 0.691164i −0.962488 0.271324i 0.912538π-0.912538\pi
0.271324 + 0.962488i 0.412538π0.412538\pi
158158 0 0
159159 −10.3923 −0.824163
160160 0 0
161161 −8.00000 + 6.92820i −0.630488 + 0.546019i
162162 0 0
163163 −5.00000 + 5.00000i −0.391630 + 0.391630i −0.875268 0.483638i 0.839315π-0.839315\pi
0.483638 + 0.875268i 0.339315π0.339315\pi
164164 0 0
165165 18.0000 18.0000i 1.40130 1.40130i
166166 0 0
167167 10.3923i 0.804181i −0.915600 0.402090i 0.868284π-0.868284\pi
0.915600 0.402090i 0.131716π-0.131716\pi
168168 0 0
169169 7.00000i 0.538462i
170170 0 0
171171 −15.5885 + 15.5885i −1.19208 + 1.19208i
172172 0 0
173173 12.1244 12.1244i 0.921798 0.921798i −0.0753588 0.997156i 0.524010π-0.524010\pi
0.997156 + 0.0753588i 0.0240102π0.0240102\pi
174174 0 0
175175 1.73205 + 2.00000i 0.130931 + 0.151186i
176176 0 0
177177 −30.0000 −2.25494
178178 0 0
179179 −17.0000 + 17.0000i −1.27064 + 1.27064i −0.324887 + 0.945753i 0.605326π0.605326\pi
−0.945753 + 0.324887i 0.894674π0.894674\pi
180180 0 0
181181 −8.66025 8.66025i −0.643712 0.643712i 0.307754 0.951466i 0.400423π-0.400423\pi
−0.951466 + 0.307754i 0.900423π0.900423\pi
182182 0 0
183183 −18.0000 −1.33060
184184 0 0
185185 17.3205 1.27343
186186 0 0
187187 0 0
188188 0 0
189189 0 0
190190 0 0
191191 10.0000i 0.723575i 0.932261 + 0.361787i 0.117833π0.117833\pi
−0.932261 + 0.361787i 0.882167π0.882167\pi
192192 0 0
193193 10.0000 0.719816 0.359908 0.932988i 0.382808π-0.382808\pi
0.359908 + 0.932988i 0.382808π0.382808\pi
194194 0 0
195195 −10.3923 10.3923i −0.744208 0.744208i
196196 0 0
197197 −1.00000 + 1.00000i −0.0712470 + 0.0712470i −0.741832 0.670585i 0.766043π-0.766043\pi
0.670585 + 0.741832i 0.266043π0.266043\pi
198198 0 0
199199 10.3923i 0.736691i −0.929689 0.368345i 0.879924π-0.879924\pi
0.929689 0.368345i 0.120076π-0.120076\pi
200200 0 0
201201 −24.2487 −1.71037
202202 0 0
203203 −3.73205 0.267949i −0.261939 0.0188063i
204204 0 0
205205 −6.00000 6.00000i −0.419058 0.419058i
206206 0 0
207207 12.0000i 0.834058i
208208 0 0
209209 31.1769i 2.15655i
210210 0 0
211211 19.0000 19.0000i 1.30801 1.30801i 0.385167 0.922847i 0.374144π-0.374144\pi
0.922847 0.385167i 0.125856π-0.125856\pi
212212 0 0
213213 6.92820 + 6.92820i 0.474713 + 0.474713i
214214 0 0
215215 3.46410i 0.236250i
216216 0 0
217217 −13.8564 + 12.0000i −0.940634 + 0.814613i
218218 0 0
219219 18.0000 + 18.0000i 1.21633 + 1.21633i
220220 0 0
221221 0 0
222222 0 0
223223 −13.8564 −0.927894 −0.463947 0.885863i 0.653567π-0.653567\pi
−0.463947 + 0.885863i 0.653567π0.653567\pi
224224 0 0
225225 −3.00000 −0.200000
226226 0 0
227227 −8.66025 8.66025i −0.574801 0.574801i 0.358665 0.933466i 0.383232π-0.383232\pi
−0.933466 + 0.358665i 0.883232π0.883232\pi
228228 0 0
229229 5.19615 + 5.19615i 0.343371 + 0.343371i 0.857633 0.514262i 0.171934π-0.171934\pi
−0.514262 + 0.857633i 0.671934π0.671934\pi
230230 0 0
231231 18.0000 + 20.7846i 1.18431 + 1.36753i
232232 0 0
233233 4.00000i 0.262049i 0.991379 + 0.131024i 0.0418266π0.0418266\pi
−0.991379 + 0.131024i 0.958173π0.958173\pi
234234 0 0
235235 12.0000 + 12.0000i 0.782794 + 0.782794i
236236 0 0
237237 3.46410 3.46410i 0.225018 0.225018i
238238 0 0
239239 2.00000i 0.129369i 0.997906 + 0.0646846i 0.0206041π0.0206041\pi
−0.997906 + 0.0646846i 0.979396π0.979396\pi
240240 0 0
241241 13.8564i 0.892570i 0.894891 + 0.446285i 0.147253π0.147253\pi
−0.894891 + 0.446285i 0.852747π0.852747\pi
242242 0 0
243243 −15.5885 15.5885i −1.00000 1.00000i
244244 0 0
245245 −13.7321 + 10.2679i −0.877309 + 0.655995i
246246 0 0
247247 −18.0000 −1.14531
248248 0 0
249249 6.00000i 0.380235i
250250 0 0
251251 1.73205 1.73205i 0.109326 0.109326i −0.650328 0.759654i 0.725368π-0.725368\pi
0.759654 + 0.650328i 0.225368π0.225368\pi
252252 0 0
253253 12.0000 + 12.0000i 0.754434 + 0.754434i
254254 0 0
255255 0 0
256256 0 0
257257 27.7128i 1.72868i −0.502910 0.864339i 0.667737π-0.667737\pi
0.502910 0.864339i 0.332263π-0.332263\pi
258258 0 0
259259 −1.33975 + 18.6603i −0.0832478 + 1.15949i
260260 0 0
261261 3.00000 3.00000i 0.185695 0.185695i
262262 0 0
263263 4.00000 0.246651 0.123325 0.992366i 0.460644π-0.460644\pi
0.123325 + 0.992366i 0.460644π0.460644\pi
264264 0 0
265265 −10.3923 −0.638394
266266 0 0
267267 −6.00000 6.00000i −0.367194 0.367194i
268268 0 0
269269 5.19615 5.19615i 0.316815 0.316815i −0.530728 0.847543i 0.678081π-0.678081\pi
0.847543 + 0.530728i 0.178081π0.178081\pi
270270 0 0
271271 13.8564 0.841717 0.420858 0.907126i 0.361729π-0.361729\pi
0.420858 + 0.907126i 0.361729π0.361729\pi
272272 0 0
273273 12.0000 10.3923i 0.726273 0.628971i
274274 0 0
275275 3.00000 3.00000i 0.180907 0.180907i
276276 0 0
277277 −1.00000 + 1.00000i −0.0600842 + 0.0600842i −0.736510 0.676426i 0.763528π-0.763528\pi
0.676426 + 0.736510i 0.263528π0.263528\pi
278278 0 0
279279 20.7846i 1.24434i
280280 0 0
281281 28.0000i 1.67034i −0.549992 0.835170i 0.685369π-0.685369\pi
0.549992 0.835170i 0.314631π-0.314631\pi
282282 0 0
283283 −5.19615 + 5.19615i −0.308879 + 0.308879i −0.844475 0.535595i 0.820087π-0.820087\pi
0.535595 + 0.844475i 0.320087π0.320087\pi
284284 0 0
285285 −31.1769 + 31.1769i −1.84676 + 1.84676i
286286 0 0
287287 6.92820 6.00000i 0.408959 0.354169i
288288 0 0
289289 17.0000 1.00000
290290 0 0
291291 0 0
292292 0 0
293293 −1.73205 1.73205i −0.101187 0.101187i 0.654701 0.755888i 0.272795π-0.272795\pi
−0.755888 + 0.654701i 0.772795π0.772795\pi
294294 0 0
295295 −30.0000 −1.74667
296296 0 0
297297 0 0
298298 0 0
299299 6.92820 6.92820i 0.400668 0.400668i
300300 0 0
301301 3.73205 + 0.267949i 0.215112 + 0.0154443i
302302 0 0
303303 6.00000i 0.344691i
304304 0 0
305305 −18.0000 −1.03068
306306 0 0
307307 −1.73205 1.73205i −0.0988534 0.0988534i 0.655951 0.754804i 0.272268π-0.272268\pi
−0.754804 + 0.655951i 0.772268π0.772268\pi
308308 0 0
309309 −18.0000 + 18.0000i −1.02398 + 1.02398i
310310 0 0
311311 10.3923i 0.589294i 0.955606 + 0.294647i 0.0952020π0.0952020\pi
−0.955606 + 0.294647i 0.904798π0.904798\pi
312312 0 0
313313 17.3205 0.979013 0.489506 0.872000i 0.337177π-0.337177\pi
0.489506 + 0.872000i 0.337177π0.337177\pi
314314 0 0
315315 1.39230 19.3923i 0.0784475 1.09263i
316316 0 0
317317 3.00000 + 3.00000i 0.168497 + 0.168497i 0.786318 0.617822i 0.211985π-0.211985\pi
−0.617822 + 0.786318i 0.711985π0.711985\pi
318318 0 0
319319 6.00000i 0.335936i
320320 0 0
321321 24.2487i 1.35343i
322322 0 0
323323 0 0
324324 0 0
325325 −1.73205 1.73205i −0.0960769 0.0960769i
326326 0 0
327327 45.0333i 2.49035i
328328 0 0
329329 −13.8564 + 12.0000i −0.763928 + 0.661581i
330330 0 0
331331 17.0000 + 17.0000i 0.934405 + 0.934405i 0.997977 0.0635727i 0.0202495π-0.0202495\pi
−0.0635727 + 0.997977i 0.520249π0.520249\pi
332332 0 0
333333 −15.0000 15.0000i −0.821995 0.821995i
334334 0 0
335335 −24.2487 −1.32485
336336 0 0
337337 −10.0000 −0.544735 −0.272367 0.962193i 0.587807π-0.587807\pi
−0.272367 + 0.962193i 0.587807π0.587807\pi
338338 0 0
339339 17.3205 + 17.3205i 0.940721 + 0.940721i
340340 0 0
341341 20.7846 + 20.7846i 1.12555 + 1.12555i
342342 0 0
343343 −10.0000 15.5885i −0.539949 0.841698i
344344 0 0
345345 24.0000i 1.29212i
346346 0 0
347347 −23.0000 23.0000i −1.23470 1.23470i −0.962136 0.272568i 0.912127π-0.912127\pi
−0.272568 0.962136i 0.587873π-0.587873\pi
348348 0 0
349349 19.0526 19.0526i 1.01986 1.01986i 0.0200613 0.999799i 0.493614π-0.493614\pi
0.999799 0.0200613i 0.00638615π-0.00638615\pi
350350 0 0
351351 0 0
352352 0 0
353353 13.8564i 0.737502i 0.929528 + 0.368751i 0.120215π0.120215\pi
−0.929528 + 0.368751i 0.879785π0.879785\pi
354354 0 0
355355 6.92820 + 6.92820i 0.367711 + 0.367711i
356356 0 0
357357 0 0
358358 0 0
359359 4.00000 0.211112 0.105556 0.994413i 0.466338π-0.466338\pi
0.105556 + 0.994413i 0.466338π0.466338\pi
360360 0 0
361361 35.0000i 1.84211i
362362 0 0
363363 12.1244 12.1244i 0.636364 0.636364i
364364 0 0
365365 18.0000 + 18.0000i 0.942163 + 0.942163i
366366 0 0
367367 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
368368 0 0
369369 10.3923i 0.541002i
370370 0 0
371371 0.803848 11.1962i 0.0417337 0.581275i
372372 0 0
373373 7.00000 7.00000i 0.362446 0.362446i −0.502267 0.864713i 0.667500π-0.667500\pi
0.864713 + 0.502267i 0.167500π0.167500\pi
374374 0 0
375375 24.0000 1.23935
376376 0 0
377377 3.46410 0.178410
378378 0 0
379379 −11.0000 11.0000i −0.565032 0.565032i 0.365701 0.930733i 0.380829π-0.380829\pi
−0.930733 + 0.365701i 0.880829π0.880829\pi
380380 0 0
381381 −10.3923 + 10.3923i −0.532414 + 0.532414i
382382 0 0
383383 34.6410 1.77007 0.885037 0.465521i 0.154133π-0.154133\pi
0.885037 + 0.465521i 0.154133π0.154133\pi
384384 0 0
385385 18.0000 + 20.7846i 0.917365 + 1.05928i
386386 0 0
387387 −3.00000 + 3.00000i −0.152499 + 0.152499i
388388 0 0
389389 3.00000 3.00000i 0.152106 0.152106i −0.626952 0.779058i 0.715698π-0.715698\pi
0.779058 + 0.626952i 0.215698π0.215698\pi
390390 0 0
391391 0 0
392392 0 0
393393 18.0000i 0.907980i
394394 0 0
395395 3.46410 3.46410i 0.174298 0.174298i
396396 0 0
397397 19.0526 19.0526i 0.956221 0.956221i −0.0428605 0.999081i 0.513647π-0.513647\pi
0.999081 + 0.0428605i 0.0136471π0.0136471\pi
398398 0 0
399399 −31.1769 36.0000i −1.56080 1.80225i
400400 0 0
401401 2.00000 0.0998752 0.0499376 0.998752i 0.484098π-0.484098\pi
0.0499376 + 0.998752i 0.484098π0.484098\pi
402402 0 0
403403 12.0000 12.0000i 0.597763 0.597763i
404404 0 0
405405 −15.5885 15.5885i −0.774597 0.774597i
406406 0 0
407407 30.0000 1.48704
408408 0 0
409409 17.3205 0.856444 0.428222 0.903674i 0.359140π-0.359140\pi
0.428222 + 0.903674i 0.359140π0.359140\pi
410410 0 0
411411 −13.8564 + 13.8564i −0.683486 + 0.683486i
412412 0 0
413413 2.32051 32.3205i 0.114185 1.59039i
414414 0 0
415415 6.00000i 0.294528i
416416 0 0
417417 18.0000 0.881464
418418 0 0
419419 25.9808 + 25.9808i 1.26924 + 1.26924i 0.946480 + 0.322764i 0.104612π0.104612\pi
0.322764 + 0.946480i 0.395388π0.395388\pi
420420 0 0
421421 11.0000 11.0000i 0.536107 0.536107i −0.386276 0.922383i 0.626239π-0.626239\pi
0.922383 + 0.386276i 0.126239π0.126239\pi
422422 0 0
423423 20.7846i 1.01058i
424424 0 0
425425 0 0
426426 0 0
427427 1.39230 19.3923i 0.0673784 0.938459i
428428 0 0
429429 −18.0000 18.0000i −0.869048 0.869048i
430430 0 0
431431 10.0000i 0.481683i 0.970564 + 0.240842i 0.0774234π0.0774234\pi
−0.970564 + 0.240842i 0.922577π0.922577\pi
432432 0 0
433433 27.7128i 1.33179i 0.746044 + 0.665896i 0.231951π0.231951\pi
−0.746044 + 0.665896i 0.768049π0.768049\pi
434434 0 0
435435 6.00000 6.00000i 0.287678 0.287678i
436436 0 0
437437 −20.7846 20.7846i −0.994263 0.994263i
438438 0 0
439439 17.3205i 0.826663i −0.910581 0.413331i 0.864365π-0.864365\pi
0.910581 0.413331i 0.135635π-0.135635\pi
440440 0 0
441441 20.7846 + 3.00000i 0.989743 + 0.142857i
442442 0 0
443443 −15.0000 15.0000i −0.712672 0.712672i 0.254422 0.967093i 0.418115π-0.418115\pi
−0.967093 + 0.254422i 0.918115π0.918115\pi
444444 0 0
445445 −6.00000 6.00000i −0.284427 0.284427i
446446 0 0
447447 17.3205 0.819232
448448 0 0
449449 22.0000 1.03824 0.519122 0.854700i 0.326259π-0.326259\pi
0.519122 + 0.854700i 0.326259π0.326259\pi
450450 0 0
451451 −10.3923 10.3923i −0.489355 0.489355i
452452 0 0
453453 −20.7846 20.7846i −0.976546 0.976546i
454454 0 0
455455 12.0000 10.3923i 0.562569 0.487199i
456456 0 0
457457 36.0000i 1.68401i −0.539471 0.842004i 0.681376π-0.681376\pi
0.539471 0.842004i 0.318624π-0.318624\pi
458458 0 0
459459 0 0
460460 0 0
461461 −8.66025 + 8.66025i −0.403348 + 0.403348i −0.879411 0.476063i 0.842063π-0.842063\pi
0.476063 + 0.879411i 0.342063π0.342063\pi
462462 0 0
463463 34.0000i 1.58011i 0.613033 + 0.790057i 0.289949π0.289949\pi
−0.613033 + 0.790057i 0.710051π0.710051\pi
464464 0 0
465465 41.5692i 1.92773i
466466 0 0
467467 −15.5885 15.5885i −0.721348 0.721348i 0.247532 0.968880i 0.420380π-0.420380\pi
−0.968880 + 0.247532i 0.920380π0.920380\pi
468468 0 0
469469 1.87564 26.1244i 0.0866092 1.20631i
470470 0 0
471471 30.0000 1.38233
472472 0 0
473473 6.00000i 0.275880i
474474 0 0
475475 −5.19615 + 5.19615i −0.238416 + 0.238416i
476476 0 0
477477 9.00000 + 9.00000i 0.412082 + 0.412082i
478478 0 0
479479 6.92820 0.316558 0.158279 0.987394i 0.449406π-0.449406\pi
0.158279 + 0.987394i 0.449406π0.449406\pi
480480 0 0
481481 17.3205i 0.789747i
482482 0 0
483483 25.8564 + 1.85641i 1.17651 + 0.0844694i
484484 0 0
485485 0 0
486486 0 0
487487 12.0000 0.543772 0.271886 0.962329i 0.412353π-0.412353\pi
0.271886 + 0.962329i 0.412353π0.412353\pi
488488 0 0
489489 17.3205 0.783260
490490 0 0
491491 9.00000 + 9.00000i 0.406164 + 0.406164i 0.880399 0.474234i 0.157275π-0.157275\pi
−0.474234 + 0.880399i 0.657275π0.657275\pi
492492 0 0
493493 0 0
494494 0 0
495495 −31.1769 −1.40130
496496 0 0
497497 −8.00000 + 6.92820i −0.358849 + 0.310772i
498498 0 0
499499 11.0000 11.0000i 0.492428 0.492428i −0.416643 0.909070i 0.636793π-0.636793\pi
0.909070 + 0.416643i 0.136793π0.136793\pi
500500 0 0
501501 −18.0000 + 18.0000i −0.804181 + 0.804181i
502502 0 0
503503 38.1051i 1.69902i 0.527570 + 0.849512i 0.323103π0.323103\pi
−0.527570 + 0.849512i 0.676897π0.676897\pi
504504 0 0
505505 6.00000i 0.266996i
506506 0 0
507507 12.1244 12.1244i 0.538462 0.538462i
508508 0 0
509509 −29.4449 + 29.4449i −1.30512 + 1.30512i −0.380228 + 0.924893i 0.624154π0.624154\pi
−0.924893 + 0.380228i 0.875846π0.875846\pi
510510 0 0
511511 −20.7846 + 18.0000i −0.919457 + 0.796273i
512512 0 0
513513 0 0
514514 0 0
515515 −18.0000 + 18.0000i −0.793175 + 0.793175i
516516 0 0
517517 20.7846 + 20.7846i 0.914106 + 0.914106i
518518 0 0
519519 −42.0000 −1.84360
520520 0 0
521521 −10.3923 −0.455295 −0.227648 0.973744i 0.573103π-0.573103\pi
−0.227648 + 0.973744i 0.573103π0.573103\pi
522522 0 0
523523 1.73205 1.73205i 0.0757373 0.0757373i −0.668223 0.743961i 0.732945π-0.732945\pi
0.743961 + 0.668223i 0.232945π0.232945\pi
524524 0 0
525525 0.464102 6.46410i 0.0202551 0.282117i
526526 0 0
527527 0 0
528528 0 0
529529 −7.00000 −0.304348
530530 0 0
531531 25.9808 + 25.9808i 1.12747 + 1.12747i
532532 0 0
533533 −6.00000 + 6.00000i −0.259889 + 0.259889i
534534 0 0
535535 24.2487i 1.04836i
536536 0 0
537537 58.8897 2.54128
538538 0 0
539539 −23.7846 + 17.7846i −1.02448 + 0.766037i
540540 0 0
541541 −5.00000 5.00000i −0.214967 0.214967i 0.591407 0.806373i 0.298573π-0.298573\pi
−0.806373 + 0.591407i 0.798573π0.798573\pi
542542 0 0
543543 30.0000i 1.28742i
544544 0 0
545545 45.0333i 1.92902i
546546 0 0
547547 −1.00000 + 1.00000i −0.0427569 + 0.0427569i −0.728162 0.685405i 0.759625π-0.759625\pi
0.685405 + 0.728162i 0.259625π0.259625\pi
548548 0 0
549549 15.5885 + 15.5885i 0.665299 + 0.665299i
550550 0 0
551551 10.3923i 0.442727i
552552 0 0
553553 3.46410 + 4.00000i 0.147309 + 0.170097i
554554 0 0
555555 −30.0000 30.0000i −1.27343 1.27343i
556556 0 0
557557 7.00000 + 7.00000i 0.296600 + 0.296600i 0.839680 0.543081i 0.182742π-0.182742\pi
−0.543081 + 0.839680i 0.682742π0.682742\pi
558558 0 0
559559 −3.46410 −0.146516
560560 0 0
561561 0 0
562562 0 0
563563 −22.5167 22.5167i −0.948964 0.948964i 0.0497953 0.998759i 0.484143π-0.484143\pi
−0.998759 + 0.0497953i 0.984143π0.984143\pi
564564 0 0
565565 17.3205 + 17.3205i 0.728679 + 0.728679i
566566 0 0
567567 18.0000 15.5885i 0.755929 0.654654i
568568 0 0
569569 4.00000i 0.167689i −0.996479 0.0838444i 0.973280π-0.973280\pi
0.996479 0.0838444i 0.0267199π-0.0267199\pi
570570 0 0
571571 5.00000 + 5.00000i 0.209243 + 0.209243i 0.803946 0.594702i 0.202730π-0.202730\pi
−0.594702 + 0.803946i 0.702730π0.702730\pi
572572 0 0
573573 17.3205 17.3205i 0.723575 0.723575i
574574 0 0
575575 4.00000i 0.166812i
576576 0 0
577577 13.8564i 0.576850i −0.957503 0.288425i 0.906868π-0.906868\pi
0.957503 0.288425i 0.0931316π-0.0931316\pi
578578 0 0
579579 −17.3205 17.3205i −0.719816 0.719816i
580580 0 0
581581 −6.46410 0.464102i −0.268176 0.0192542i
582582 0 0
583583 −18.0000 −0.745484
584584 0 0
585585 18.0000i 0.744208i
586586 0 0
587587 1.73205 1.73205i 0.0714894 0.0714894i −0.670458 0.741947i 0.733902π-0.733902\pi
0.741947 + 0.670458i 0.233902π0.233902\pi
588588 0 0
589589 −36.0000 36.0000i −1.48335 1.48335i
590590 0 0
591591 3.46410 0.142494
592592 0 0
593593 13.8564i 0.569014i 0.958674 + 0.284507i 0.0918300π0.0918300\pi
−0.958674 + 0.284507i 0.908170π0.908170\pi
594594 0 0
595595 0 0
596596 0 0
597597 −18.0000 + 18.0000i −0.736691 + 0.736691i
598598 0 0
599599 −44.0000 −1.79779 −0.898896 0.438163i 0.855629π-0.855629\pi
−0.898896 + 0.438163i 0.855629π0.855629\pi
600600 0 0
601601 −38.1051 −1.55434 −0.777170 0.629291i 0.783346π-0.783346\pi
−0.777170 + 0.629291i 0.783346π0.783346\pi
602602 0 0
603603 21.0000 + 21.0000i 0.855186 + 0.855186i
604604 0 0
605605 12.1244 12.1244i 0.492925 0.492925i
606606 0 0
607607 27.7128 1.12483 0.562414 0.826856i 0.309873π-0.309873\pi
0.562414 + 0.826856i 0.309873π0.309873\pi
608608 0 0
609609 6.00000 + 6.92820i 0.243132 + 0.280745i
610610 0 0
611611 12.0000 12.0000i 0.485468 0.485468i
612612 0 0
613613 −29.0000 + 29.0000i −1.17130 + 1.17130i −0.189399 + 0.981900i 0.560654π0.560654\pi
−0.981900 + 0.189399i 0.939346π0.939346\pi
614614 0 0
615615 20.7846i 0.838116i
616616 0 0
617617 16.0000i 0.644136i 0.946717 + 0.322068i 0.104378π0.104378\pi
−0.946717 + 0.322068i 0.895622π0.895622\pi
618618 0 0
619619 22.5167 22.5167i 0.905021 0.905021i −0.0908441 0.995865i 0.528957π-0.528957\pi
0.995865 + 0.0908441i 0.0289565π0.0289565\pi
620620 0 0
621621 0 0
622622 0 0
623623 6.92820 6.00000i 0.277573 0.240385i
624624 0 0
625625 29.0000 1.16000
626626 0 0
627627 −54.0000 + 54.0000i −2.15655 + 2.15655i
628628 0 0
629629 0 0
630630 0 0
631631 −12.0000 −0.477712 −0.238856 0.971055i 0.576772π-0.576772\pi
−0.238856 + 0.971055i 0.576772π0.576772\pi
632632 0 0
633633 −65.8179 −2.61603
634634 0 0
635635 −10.3923 + 10.3923i −0.412406 + 0.412406i
636636 0 0
637637 10.2679 + 13.7321i 0.406831 + 0.544084i
638638 0 0
639639 12.0000i 0.474713i
640640 0 0
641641 10.0000 0.394976 0.197488 0.980305i 0.436722π-0.436722\pi
0.197488 + 0.980305i 0.436722π0.436722\pi
642642 0 0
643643 −1.73205 1.73205i −0.0683054 0.0683054i 0.672129 0.740434i 0.265380π-0.265380\pi
−0.740434 + 0.672129i 0.765380π0.765380\pi
644644 0 0
645645 −6.00000 + 6.00000i −0.236250 + 0.236250i
646646 0 0
647647 17.3205i 0.680939i 0.940255 + 0.340470i 0.110586π0.110586\pi
−0.940255 + 0.340470i 0.889414π0.889414\pi
648648 0 0
649649 −51.9615 −2.03967
650650 0 0
651651 44.7846 + 3.21539i 1.75525 + 0.126021i
652652 0 0
653653 31.0000 + 31.0000i 1.21312 + 1.21312i 0.969994 + 0.243130i 0.0781742π0.0781742\pi
0.243130 + 0.969994i 0.421826π0.421826\pi
654654 0 0
655655 18.0000i 0.703318i
656656 0 0
657657 31.1769i 1.21633i
658658 0 0
659659 23.0000 23.0000i 0.895953 0.895953i −0.0991224 0.995075i 0.531604π-0.531604\pi
0.995075 + 0.0991224i 0.0316036π0.0316036\pi
660660 0 0
661661 −8.66025 8.66025i −0.336845 0.336845i 0.518334 0.855178i 0.326553π-0.326553\pi
−0.855178 + 0.518334i 0.826553π0.826553\pi
662662 0 0
663663 0 0
664664 0 0
665665 −31.1769 36.0000i −1.20899 1.39602i
666666 0 0
667667 4.00000 + 4.00000i 0.154881 + 0.154881i
668668 0 0
669669 24.0000 + 24.0000i 0.927894 + 0.927894i
670670 0 0
671671 −31.1769 −1.20357
672672 0 0
673673 −42.0000 −1.61898 −0.809491 0.587133i 0.800257π-0.800257\pi
−0.809491 + 0.587133i 0.800257π0.800257\pi
674674 0 0
675675 0 0
676676 0 0
677677 −22.5167 22.5167i −0.865386 0.865386i 0.126572 0.991957i 0.459603π-0.459603\pi
−0.991957 + 0.126572i 0.959603π0.959603\pi
678678 0 0
679679 0 0
680680 0 0
681681 30.0000i 1.14960i
682682 0 0
683683 21.0000 + 21.0000i 0.803543 + 0.803543i 0.983647 0.180105i 0.0576437π-0.0576437\pi
−0.180105 + 0.983647i 0.557644π0.557644\pi
684684 0 0
685685 −13.8564 + 13.8564i −0.529426 + 0.529426i
686686 0 0
687687 18.0000i 0.686743i
688688 0 0
689689 10.3923i 0.395915i
690690 0 0
691691 −1.73205 1.73205i −0.0658903 0.0658903i 0.673394 0.739284i 0.264836π-0.264836\pi
−0.739284 + 0.673394i 0.764836π0.764836\pi
692692 0 0
693693 2.41154 33.5885i 0.0916069 1.27592i
694694 0 0
695695 18.0000 0.682779
696696 0 0
697697 0 0
698698 0 0
699699 6.92820 6.92820i 0.262049 0.262049i
700700 0 0
701701 27.0000 + 27.0000i 1.01978 + 1.01978i 0.999800 + 0.0199755i 0.00635881π0.00635881\pi
0.0199755 + 0.999800i 0.493641π0.493641\pi
702702 0 0
703703 −51.9615 −1.95977
704704 0 0
705705 41.5692i 1.56559i
706706 0 0
707707 −6.46410 0.464102i −0.243108 0.0174543i
708708 0 0
709709 19.0000 19.0000i 0.713560 0.713560i −0.253718 0.967278i 0.581654π-0.581654\pi
0.967278 + 0.253718i 0.0816536π0.0816536\pi
710710 0 0
711711 −6.00000 −0.225018
712712 0 0
713713 27.7128 1.03785
714714 0 0
715715 −18.0000 18.0000i −0.673162 0.673162i
716716 0 0
717717 3.46410 3.46410i 0.129369 0.129369i
718718 0 0
719719 6.92820 0.258378 0.129189 0.991620i 0.458763π-0.458763\pi
0.129189 + 0.991620i 0.458763π0.458763\pi
720720 0 0
721721 −18.0000 20.7846i −0.670355 0.774059i
722722 0 0
723723 24.0000 24.0000i 0.892570 0.892570i
724724 0 0
725725 1.00000 1.00000i 0.0371391 0.0371391i
726726 0 0
727727 38.1051i 1.41324i −0.707593 0.706620i 0.750219π-0.750219\pi
0.707593 0.706620i 0.249781π-0.249781\pi
728728 0 0
729729 27.0000i 1.00000i
730730 0 0
731731 0 0
732732 0 0
733733 −1.73205 + 1.73205i −0.0639748 + 0.0639748i −0.738370 0.674396i 0.764404π-0.764404\pi
0.674396 + 0.738370i 0.264404π0.264404\pi
734734 0 0
735735 41.5692 + 6.00000i 1.53330 + 0.221313i
736736 0 0
737737 −42.0000 −1.54709
738738 0 0
739739 11.0000 11.0000i 0.404642 0.404642i −0.475224 0.879865i 0.657633π-0.657633\pi
0.879865 + 0.475224i 0.157633π0.157633\pi
740740 0 0
741741 31.1769 + 31.1769i 1.14531 + 1.14531i
742742 0 0
743743 20.0000 0.733729 0.366864 0.930274i 0.380431π-0.380431\pi
0.366864 + 0.930274i 0.380431π0.380431\pi
744744 0 0
745745 17.3205 0.634574
746746 0 0
747747 5.19615 5.19615i 0.190117 0.190117i
748748 0 0
749749 −26.1244 1.87564i −0.954563 0.0685346i
750750 0 0
751751 46.0000i 1.67856i −0.543696 0.839282i 0.682976π-0.682976\pi
0.543696 0.839282i 0.317024π-0.317024\pi
752752 0 0
753753 −6.00000 −0.218652
754754 0 0
755755 −20.7846 20.7846i −0.756429 0.756429i
756756 0 0
757757 −13.0000 + 13.0000i −0.472493 + 0.472493i −0.902721 0.430227i 0.858433π-0.858433\pi
0.430227 + 0.902721i 0.358433π0.358433\pi
758758 0 0
759759 41.5692i 1.50887i
760760 0 0
761761 3.46410 0.125574 0.0627868 0.998027i 0.480001π-0.480001\pi
0.0627868 + 0.998027i 0.480001π0.480001\pi
762762 0 0
763763 −48.5167 3.48334i −1.75642 0.126105i
764764 0 0
765765 0 0
766766 0 0
767767 30.0000i 1.08324i
768768 0 0
769769 0 0 1.00000 00
−1.00000 π\pi
770770 0 0
771771 −48.0000 + 48.0000i −1.72868 + 1.72868i
772772 0 0
773773 12.1244 + 12.1244i 0.436083 + 0.436083i 0.890691 0.454609i 0.150221π-0.150221\pi
−0.454609 + 0.890691i 0.650221π0.650221\pi
774774 0 0
775775 6.92820i 0.248868i
776776 0 0
777777 34.6410 30.0000i 1.24274 1.07624i
778778 0 0
779779 18.0000 + 18.0000i 0.644917 + 0.644917i
780780 0 0
781781 12.0000 + 12.0000i 0.429394 + 0.429394i
782782 0 0
783783 0 0
784784 0 0
785785 30.0000 1.07075
786786 0 0
787787 −1.73205 1.73205i −0.0617409 0.0617409i 0.675562 0.737303i 0.263901π-0.263901\pi
−0.737303 + 0.675562i 0.763901π0.763901\pi
788788 0 0
789789 −6.92820 6.92820i −0.246651 0.246651i
790790 0 0
791791 −20.0000 + 17.3205i −0.711118 + 0.615846i
792792 0 0
793793 18.0000i 0.639199i
794794 0 0
795795 18.0000 + 18.0000i 0.638394 + 0.638394i
796796 0 0
797797 −15.5885 + 15.5885i −0.552171 + 0.552171i −0.927067 0.374896i 0.877679π-0.877679\pi
0.374896 + 0.927067i 0.377679π0.377679\pi
798798 0 0
799799 0 0
800800 0 0
801801 10.3923i 0.367194i
802802 0 0
803803 31.1769 + 31.1769i 1.10021 + 1.10021i
804804 0 0
805805 25.8564 + 1.85641i 0.911319 + 0.0654297i
806806 0 0
807807 −18.0000 −0.633630
808808 0 0
809809 20.0000i 0.703163i 0.936157 + 0.351581i 0.114356π0.114356\pi
−0.936157 + 0.351581i 0.885644π0.885644\pi
810810 0 0
811811 −19.0526 + 19.0526i −0.669026 + 0.669026i −0.957491 0.288465i 0.906855π-0.906855\pi
0.288465 + 0.957491i 0.406855π0.406855\pi
812812 0 0
813813 −24.0000 24.0000i −0.841717 0.841717i
814814 0 0
815815 17.3205 0.606711
816816 0 0
817817 10.3923i 0.363581i
818818 0 0
819819 −19.3923 1.39230i −0.677622 0.0486511i
820820 0 0
821821 39.0000 39.0000i 1.36111 1.36111i 0.488603 0.872506i 0.337507π-0.337507\pi
0.872506 0.488603i 0.162493π-0.162493\pi
822822 0 0
823823 −12.0000 −0.418294 −0.209147 0.977884i 0.567069π-0.567069\pi
−0.209147 + 0.977884i 0.567069π0.567069\pi
824824 0 0
825825 −10.3923 −0.361814
826826 0 0
827827 29.0000 + 29.0000i 1.00843 + 1.00843i 0.999964 + 0.00846463i 0.00269441π0.00269441\pi
0.00846463 + 0.999964i 0.497306π0.497306\pi
828828 0 0
829829 12.1244 12.1244i 0.421096 0.421096i −0.464485 0.885581i 0.653760π-0.653760\pi
0.885581 + 0.464485i 0.153760π0.153760\pi
830830 0 0
831831 3.46410 0.120168
832832 0 0
833833 0 0
834834 0 0
835835 −18.0000 + 18.0000i −0.622916 + 0.622916i
836836 0 0
837837 0 0
838838 0 0
839839 10.3923i 0.358782i −0.983778 0.179391i 0.942587π-0.942587\pi
0.983778 0.179391i 0.0574128π-0.0574128\pi
840840 0 0
841841 27.0000i 0.931034i
842842 0 0
843843 −48.4974 + 48.4974i −1.67034 + 1.67034i
844844 0 0
845845 12.1244 12.1244i 0.417091 0.417091i
846846 0 0
847847 12.1244 + 14.0000i 0.416598 + 0.481046i
848848 0 0
849849 18.0000 0.617758
850850 0 0
851851 20.0000 20.0000i 0.685591 0.685591i
852852 0 0
853853 −29.4449 29.4449i −1.00817 1.00817i −0.999966 0.00820661i 0.997388π-0.997388\pi
−0.00820661 0.999966i 0.502612π-0.502612\pi
854854 0 0
855855 54.0000 1.84676
856856 0 0
857857 45.0333 1.53831 0.769154 0.639063i 0.220678π-0.220678\pi
0.769154 + 0.639063i 0.220678π0.220678\pi
858858 0 0
859859 −19.0526 + 19.0526i −0.650065 + 0.650065i −0.953008 0.302944i 0.902031π-0.902031\pi
0.302944 + 0.953008i 0.402031π0.402031\pi
860860 0 0
861861 −22.3923 1.60770i −0.763128 0.0547901i
862862 0 0
863863 38.0000i 1.29354i −0.762687 0.646768i 0.776120π-0.776120\pi
0.762687 0.646768i 0.223880π-0.223880\pi
864864 0 0
865865 −42.0000 −1.42804
866866 0 0
867867 −29.4449 29.4449i −1.00000 1.00000i
868868 0 0
869869 6.00000 6.00000i 0.203536 0.203536i
870870 0 0
871871 24.2487i 0.821636i
872872 0 0
873873 0 0
874874 0 0
875875 −1.85641 + 25.8564i −0.0627580 + 0.874106i
876876 0 0
877877 −5.00000 5.00000i −0.168838 0.168838i 0.617630 0.786468i 0.288093π-0.288093\pi
−0.786468 + 0.617630i 0.788093π0.788093\pi
878878 0 0
879879 6.00000i 0.202375i
880880 0 0
881881 55.4256i 1.86734i −0.358139 0.933668i 0.616589π-0.616589\pi
0.358139 0.933668i 0.383411π-0.383411\pi
882882 0 0
883883 −13.0000 + 13.0000i −0.437485 + 0.437485i −0.891165 0.453680i 0.850111π-0.850111\pi
0.453680 + 0.891165i 0.350111π0.350111\pi
884884 0 0
885885 51.9615 + 51.9615i 1.74667 + 1.74667i
886886 0 0
887887 10.3923i 0.348939i −0.984663 0.174470i 0.944179π-0.944179\pi
0.984663 0.174470i 0.0558211π-0.0558211\pi
888888 0 0
889889 −10.3923 12.0000i −0.348547 0.402467i
890890 0 0
891891 −27.0000 27.0000i −0.904534 0.904534i
892892 0 0
893893 −36.0000 36.0000i −1.20469 1.20469i
894894 0 0
895895 58.8897 1.96847
896896 0 0
897897 −24.0000 −0.801337
898898 0 0
899899 6.92820 + 6.92820i 0.231069 + 0.231069i
900900 0 0
901901 0 0
902902 0 0
903903 −6.00000 6.92820i −0.199667 0.230556i
904904 0 0
905905 30.0000i 0.997234i
906906 0 0
907907 5.00000 + 5.00000i 0.166022 + 0.166022i 0.785228 0.619206i 0.212545π-0.212545\pi
−0.619206 + 0.785228i 0.712545π0.712545\pi
908908 0 0
909909 5.19615 5.19615i 0.172345 0.172345i
910910 0 0
911911 34.0000i 1.12647i 0.826297 + 0.563235i 0.190443π0.190443\pi
−0.826297 + 0.563235i 0.809557π0.809557\pi
912912 0 0
913913 10.3923i 0.343935i
914914 0 0
915915 31.1769 + 31.1769i 1.03068 + 1.03068i
916916 0 0
917917 19.3923 + 1.39230i 0.640390 + 0.0459780i
918918 0 0
919919 −28.0000 −0.923635 −0.461817 0.886975i 0.652802π-0.652802\pi
−0.461817 + 0.886975i 0.652802π0.652802\pi
920920 0 0
921921 6.00000i 0.197707i
922922 0 0
923923 6.92820 6.92820i 0.228045 0.228045i
924924 0 0
925925 −5.00000 5.00000i −0.164399 0.164399i
926926 0 0
927927 31.1769 1.02398
928928 0 0
929929 41.5692i 1.36384i 0.731426 + 0.681921i 0.238855π0.238855\pi
−0.731426 + 0.681921i 0.761145π0.761145\pi
930930 0 0
931931 41.1962 30.8038i 1.35015 1.00956i
932932 0 0
933933 18.0000 18.0000i 0.589294 0.589294i
934934 0 0
935935 0 0
936936 0 0
937937 17.3205 0.565836 0.282918 0.959144i 0.408698π-0.408698\pi
0.282918 + 0.959144i 0.408698π0.408698\pi
938938 0 0
939939 −30.0000 30.0000i −0.979013 0.979013i
940940 0 0
941941 −22.5167 + 22.5167i −0.734022 + 0.734022i −0.971414 0.237392i 0.923707π-0.923707\pi
0.237392 + 0.971414i 0.423707π0.423707\pi
942942 0 0
943943 −13.8564 −0.451227
944944 0 0
945945 0 0
946946 0 0
947947 −5.00000 + 5.00000i −0.162478 + 0.162478i −0.783664 0.621185i 0.786651π-0.786651\pi
0.621185 + 0.783664i 0.286651π0.286651\pi
948948 0 0
949949 18.0000 18.0000i 0.584305 0.584305i
950950 0 0
951951 10.3923i 0.336994i
952952 0 0
953953 32.0000i 1.03658i 0.855204 + 0.518291i 0.173432π0.173432\pi
−0.855204 + 0.518291i 0.826568π0.826568\pi
954954 0 0
955955 17.3205 17.3205i 0.560478 0.560478i
956956 0 0
957957 10.3923 10.3923i 0.335936 0.335936i
958958 0 0
959959 −13.8564 16.0000i −0.447447 0.516667i
960960 0 0
961961 17.0000 0.548387
962962 0 0
963963 21.0000 21.0000i 0.676716 0.676716i
964964 0 0
965965 −17.3205 17.3205i −0.557567 0.557567i
966966 0 0
967967 44.0000 1.41494 0.707472 0.706741i 0.249835π-0.249835\pi
0.707472 + 0.706741i 0.249835π0.249835\pi
968968 0 0
969969 0 0
970970 0 0
971971 1.73205 1.73205i 0.0555842 0.0555842i −0.678768 0.734352i 0.737486π-0.737486\pi
0.734352 + 0.678768i 0.237486π0.237486\pi
972972 0 0
973973 −1.39230 + 19.3923i −0.0446352 + 0.621689i
974974 0 0
975975 6.00000i 0.192154i
976976 0 0
977977 −10.0000 −0.319928 −0.159964 0.987123i 0.551138π-0.551138\pi
−0.159964 + 0.987123i 0.551138π0.551138\pi
978978 0 0
979979 −10.3923 10.3923i −0.332140 0.332140i
980980 0 0
981981 39.0000 39.0000i 1.24517 1.24517i
982982 0 0
983983 45.0333i 1.43634i 0.695868 + 0.718170i 0.255020π0.255020\pi
−0.695868 + 0.718170i 0.744980π0.744980\pi
984984 0 0
985985 3.46410 0.110375
986986 0 0
987987 44.7846 + 3.21539i 1.42551 + 0.102347i
988988 0 0
989989 −4.00000 4.00000i −0.127193 0.127193i
990990 0 0
991991 6.00000i 0.190596i −0.995449 0.0952981i 0.969620π-0.969620\pi
0.995449 0.0952981i 0.0303804π-0.0303804\pi
992992 0 0
993993 58.8897i 1.86881i
994994 0 0
995995 −18.0000 + 18.0000i −0.570638 + 0.570638i
996996 0 0
997997 −22.5167 22.5167i −0.713110 0.713110i 0.254075 0.967185i 0.418229π-0.418229\pi
−0.967185 + 0.254075i 0.918229π0.918229\pi
998998 0 0
999999 0 0
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 448.2.j.c.111.1 4
4.3 odd 2 112.2.j.c.83.2 yes 4
7.6 odd 2 inner 448.2.j.c.111.2 4
8.3 odd 2 896.2.j.a.223.1 4
8.5 even 2 896.2.j.f.223.2 4
16.3 odd 4 896.2.j.f.671.1 4
16.5 even 4 112.2.j.c.27.1 4
16.11 odd 4 inner 448.2.j.c.335.2 4
16.13 even 4 896.2.j.a.671.2 4
28.3 even 6 784.2.w.a.19.1 4
28.11 odd 6 784.2.w.b.19.1 4
28.19 even 6 784.2.w.b.227.1 4
28.23 odd 6 784.2.w.a.227.1 4
28.27 even 2 112.2.j.c.83.1 yes 4
56.13 odd 2 896.2.j.f.223.1 4
56.27 even 2 896.2.j.a.223.2 4
112.5 odd 12 784.2.w.b.619.1 4
112.13 odd 4 896.2.j.a.671.1 4
112.27 even 4 inner 448.2.j.c.335.1 4
112.37 even 12 784.2.w.a.619.1 4
112.53 even 12 784.2.w.b.411.1 4
112.69 odd 4 112.2.j.c.27.2 yes 4
112.83 even 4 896.2.j.f.671.2 4
112.101 odd 12 784.2.w.a.411.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.j.c.27.1 4 16.5 even 4
112.2.j.c.27.2 yes 4 112.69 odd 4
112.2.j.c.83.1 yes 4 28.27 even 2
112.2.j.c.83.2 yes 4 4.3 odd 2
448.2.j.c.111.1 4 1.1 even 1 trivial
448.2.j.c.111.2 4 7.6 odd 2 inner
448.2.j.c.335.1 4 112.27 even 4 inner
448.2.j.c.335.2 4 16.11 odd 4 inner
784.2.w.a.19.1 4 28.3 even 6
784.2.w.a.227.1 4 28.23 odd 6
784.2.w.a.411.1 4 112.101 odd 12
784.2.w.a.619.1 4 112.37 even 12
784.2.w.b.19.1 4 28.11 odd 6
784.2.w.b.227.1 4 28.19 even 6
784.2.w.b.411.1 4 112.53 even 12
784.2.w.b.619.1 4 112.5 odd 12
896.2.j.a.223.1 4 8.3 odd 2
896.2.j.a.223.2 4 56.27 even 2
896.2.j.a.671.1 4 112.13 odd 4
896.2.j.a.671.2 4 16.13 even 4
896.2.j.f.223.1 4 56.13 odd 2
896.2.j.f.223.2 4 8.5 even 2
896.2.j.f.671.1 4 16.3 odd 4
896.2.j.f.671.2 4 112.83 even 4