Properties

Label 105.2.b.a.41.1
Level 105105
Weight 22
Character 105.41
Analytic conductor 0.8380.838
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,2,Mod(41,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 105=357 105 = 3 \cdot 5 \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 105.b (of order 22, degree 11, 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: 0.8384292212230.838429221223
Analytic rank: 00
Dimension: 22
Coefficient field: Q(ζ6)\Q(\zeta_{6})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 2 2
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 41.1
Root 0.500000+0.866025i0.500000 + 0.866025i of defining polynomial
Character χ\chi == 105.41
Dual form 105.2.b.a.41.2

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) == q1.73205iq21.73205iq31.00000q41.00000q53.00000q6+(2.00000+1.73205i)q71.73205iq83.00000q9+1.73205iq10+3.46410iq11+1.73205iq12+(3.000003.46410i)q14+1.73205iq155.00000q16+6.00000q17+5.19615iq183.46410iq19+1.00000q20+(3.000003.46410i)q21+6.00000q223.46410iq233.00000q24+1.00000q25+5.19615iq27+(2.000001.73205i)q28+6.92820iq29+3.00000q30+3.46410iq31+5.19615iq32+6.00000q3310.3923iq34+(2.000001.73205i)q35+3.00000q362.00000q376.00000q38+1.73205iq40+6.00000q41+(6.000005.19615i)q428.00000q433.46410iq44+3.00000q456.00000q4612.0000q47+8.66025iq48+(1.00000+6.92820i)q491.73205iq5010.3923iq51+9.00000q543.46410iq55+(3.000003.46410i)q566.00000q57+12.0000q5812.0000q591.73205iq60+6.92820iq61+6.00000q62+(6.000005.19615i)q631.00000q6410.3923iq66+8.00000q676.00000q686.00000q69+(3.00000+3.46410i)q703.46410iq71+5.19615iq726.92820iq73+3.46410iq741.73205iq75+3.46410iq76+(6.00000+6.92820i)q77+8.00000q79+5.00000q80+9.00000q8110.3923iq82+(3.00000+3.46410i)q846.00000q85+13.8564iq86+12.0000q87+6.00000q88+6.00000q895.19615iq90+3.46410iq92+6.00000q93+20.7846iq94+3.46410iq95+9.00000q966.92820iq97+(12.00001.73205i)q9810.3923iq99+O(q100)q-1.73205i q^{2} -1.73205i q^{3} -1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +(2.00000 + 1.73205i) q^{7} -1.73205i q^{8} -3.00000 q^{9} +1.73205i q^{10} +3.46410i q^{11} +1.73205i q^{12} +(3.00000 - 3.46410i) q^{14} +1.73205i q^{15} -5.00000 q^{16} +6.00000 q^{17} +5.19615i q^{18} -3.46410i q^{19} +1.00000 q^{20} +(3.00000 - 3.46410i) q^{21} +6.00000 q^{22} -3.46410i q^{23} -3.00000 q^{24} +1.00000 q^{25} +5.19615i q^{27} +(-2.00000 - 1.73205i) q^{28} +6.92820i q^{29} +3.00000 q^{30} +3.46410i q^{31} +5.19615i q^{32} +6.00000 q^{33} -10.3923i q^{34} +(-2.00000 - 1.73205i) q^{35} +3.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +1.73205i q^{40} +6.00000 q^{41} +(-6.00000 - 5.19615i) q^{42} -8.00000 q^{43} -3.46410i q^{44} +3.00000 q^{45} -6.00000 q^{46} -12.0000 q^{47} +8.66025i q^{48} +(1.00000 + 6.92820i) q^{49} -1.73205i q^{50} -10.3923i q^{51} +9.00000 q^{54} -3.46410i q^{55} +(3.00000 - 3.46410i) q^{56} -6.00000 q^{57} +12.0000 q^{58} -12.0000 q^{59} -1.73205i q^{60} +6.92820i q^{61} +6.00000 q^{62} +(-6.00000 - 5.19615i) q^{63} -1.00000 q^{64} -10.3923i q^{66} +8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +(-3.00000 + 3.46410i) q^{70} -3.46410i q^{71} +5.19615i q^{72} -6.92820i q^{73} +3.46410i q^{74} -1.73205i q^{75} +3.46410i q^{76} +(-6.00000 + 6.92820i) q^{77} +8.00000 q^{79} +5.00000 q^{80} +9.00000 q^{81} -10.3923i q^{82} +(-3.00000 + 3.46410i) q^{84} -6.00000 q^{85} +13.8564i q^{86} +12.0000 q^{87} +6.00000 q^{88} +6.00000 q^{89} -5.19615i q^{90} +3.46410i q^{92} +6.00000 q^{93} +20.7846i q^{94} +3.46410i q^{95} +9.00000 q^{96} -6.92820i q^{97} +(12.0000 - 1.73205i) q^{98} -10.3923i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q42q56q6+4q76q9+6q1410q16+12q17+2q20+6q21+12q226q24+2q254q28+6q30+12q334q35+6q364q37++24q98+O(q100) 2 q - 2 q^{4} - 2 q^{5} - 6 q^{6} + 4 q^{7} - 6 q^{9} + 6 q^{14} - 10 q^{16} + 12 q^{17} + 2 q^{20} + 6 q^{21} + 12 q^{22} - 6 q^{24} + 2 q^{25} - 4 q^{28} + 6 q^{30} + 12 q^{33} - 4 q^{35} + 6 q^{36} - 4 q^{37}+ \cdots + 24 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/105Z)×\left(\mathbb{Z}/105\mathbb{Z}\right)^\times.

nn 2222 3131 7171
χ(n)\chi(n) 11 1-1 1-1

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 1.73205i 1.22474i −0.790569 0.612372i 0.790215π-0.790215\pi
0.790569 0.612372i 0.209785π-0.209785\pi
33 1.73205i 1.00000i
44 −1.00000 −0.500000
55 −1.00000 −0.447214
66 −3.00000 −1.22474
77 2.00000 + 1.73205i 0.755929 + 0.654654i
88 1.73205i 0.612372i
99 −3.00000 −1.00000
1010 1.73205i 0.547723i
1111 3.46410i 1.04447i 0.852803 + 0.522233i 0.174901π0.174901\pi
−0.852803 + 0.522233i 0.825099π0.825099\pi
1212 1.73205i 0.500000i
1313 0 0 1.00000 00
−1.00000 π\pi
1414 3.00000 3.46410i 0.801784 0.925820i
1515 1.73205i 0.447214i
1616 −5.00000 −1.25000
1717 6.00000 1.45521 0.727607 0.685994i 0.240633π-0.240633\pi
0.727607 + 0.685994i 0.240633π0.240633\pi
1818 5.19615i 1.22474i
1919 3.46410i 0.794719i −0.917663 0.397360i 0.869927π-0.869927\pi
0.917663 0.397360i 0.130073π-0.130073\pi
2020 1.00000 0.223607
2121 3.00000 3.46410i 0.654654 0.755929i
2222 6.00000 1.27920
2323 3.46410i 0.722315i −0.932505 0.361158i 0.882382π-0.882382\pi
0.932505 0.361158i 0.117618π-0.117618\pi
2424 −3.00000 −0.612372
2525 1.00000 0.200000
2626 0 0
2727 5.19615i 1.00000i
2828 −2.00000 1.73205i −0.377964 0.327327i
2929 6.92820i 1.28654i 0.765641 + 0.643268i 0.222422π0.222422\pi
−0.765641 + 0.643268i 0.777578π0.777578\pi
3030 3.00000 0.547723
3131 3.46410i 0.622171i 0.950382 + 0.311086i 0.100693π0.100693\pi
−0.950382 + 0.311086i 0.899307π0.899307\pi
3232 5.19615i 0.918559i
3333 6.00000 1.04447
3434 10.3923i 1.78227i
3535 −2.00000 1.73205i −0.338062 0.292770i
3636 3.00000 0.500000
3737 −2.00000 −0.328798 −0.164399 0.986394i 0.552568π-0.552568\pi
−0.164399 + 0.986394i 0.552568π0.552568\pi
3838 −6.00000 −0.973329
3939 0 0
4040 1.73205i 0.273861i
4141 6.00000 0.937043 0.468521 0.883452i 0.344787π-0.344787\pi
0.468521 + 0.883452i 0.344787π0.344787\pi
4242 −6.00000 5.19615i −0.925820 0.801784i
4343 −8.00000 −1.21999 −0.609994 0.792406i 0.708828π-0.708828\pi
−0.609994 + 0.792406i 0.708828π0.708828\pi
4444 3.46410i 0.522233i
4545 3.00000 0.447214
4646 −6.00000 −0.884652
4747 −12.0000 −1.75038 −0.875190 0.483779i 0.839264π-0.839264\pi
−0.875190 + 0.483779i 0.839264π0.839264\pi
4848 8.66025i 1.25000i
4949 1.00000 + 6.92820i 0.142857 + 0.989743i
5050 1.73205i 0.244949i
5151 10.3923i 1.45521i
5252 0 0
5353 0 0 1.00000 00
−1.00000 π\pi
5454 9.00000 1.22474
5555 3.46410i 0.467099i
5656 3.00000 3.46410i 0.400892 0.462910i
5757 −6.00000 −0.794719
5858 12.0000 1.57568
5959 −12.0000 −1.56227 −0.781133 0.624364i 0.785358π-0.785358\pi
−0.781133 + 0.624364i 0.785358π0.785358\pi
6060 1.73205i 0.223607i
6161 6.92820i 0.887066i 0.896258 + 0.443533i 0.146275π0.146275\pi
−0.896258 + 0.443533i 0.853725π0.853725\pi
6262 6.00000 0.762001
6363 −6.00000 5.19615i −0.755929 0.654654i
6464 −1.00000 −0.125000
6565 0 0
6666 10.3923i 1.27920i
6767 8.00000 0.977356 0.488678 0.872464i 0.337479π-0.337479\pi
0.488678 + 0.872464i 0.337479π0.337479\pi
6868 −6.00000 −0.727607
6969 −6.00000 −0.722315
7070 −3.00000 + 3.46410i −0.358569 + 0.414039i
7171 3.46410i 0.411113i −0.978645 0.205557i 0.934100π-0.934100\pi
0.978645 0.205557i 0.0659005π-0.0659005\pi
7272 5.19615i 0.612372i
7373 6.92820i 0.810885i −0.914121 0.405442i 0.867117π-0.867117\pi
0.914121 0.405442i 0.132883π-0.132883\pi
7474 3.46410i 0.402694i
7575 1.73205i 0.200000i
7676 3.46410i 0.397360i
7777 −6.00000 + 6.92820i −0.683763 + 0.789542i
7878 0 0
7979 8.00000 0.900070 0.450035 0.893011i 0.351411π-0.351411\pi
0.450035 + 0.893011i 0.351411π0.351411\pi
8080 5.00000 0.559017
8181 9.00000 1.00000
8282 10.3923i 1.14764i
8383 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
8484 −3.00000 + 3.46410i −0.327327 + 0.377964i
8585 −6.00000 −0.650791
8686 13.8564i 1.49417i
8787 12.0000 1.28654
8888 6.00000 0.639602
8989 6.00000 0.635999 0.317999 0.948091i 0.396989π-0.396989\pi
0.317999 + 0.948091i 0.396989π0.396989\pi
9090 5.19615i 0.547723i
9191 0 0
9292 3.46410i 0.361158i
9393 6.00000 0.622171
9494 20.7846i 2.14377i
9595 3.46410i 0.355409i
9696 9.00000 0.918559
9797 6.92820i 0.703452i −0.936103 0.351726i 0.885595π-0.885595\pi
0.936103 0.351726i 0.114405π-0.114405\pi
9898 12.0000 1.73205i 1.21218 0.174964i
9999 10.3923i 1.04447i
100100 −1.00000 −0.100000
101101 −6.00000 −0.597022 −0.298511 0.954406i 0.596490π-0.596490\pi
−0.298511 + 0.954406i 0.596490π0.596490\pi
102102 −18.0000 −1.78227
103103 3.46410i 0.341328i 0.985329 + 0.170664i 0.0545913π0.0545913\pi
−0.985329 + 0.170664i 0.945409π0.945409\pi
104104 0 0
105105 −3.00000 + 3.46410i −0.292770 + 0.338062i
106106 0 0
107107 10.3923i 1.00466i −0.864675 0.502331i 0.832476π-0.832476\pi
0.864675 0.502331i 0.167524π-0.167524\pi
108108 5.19615i 0.500000i
109109 −2.00000 −0.191565 −0.0957826 0.995402i 0.530535π-0.530535\pi
−0.0957826 + 0.995402i 0.530535π0.530535\pi
110110 −6.00000 −0.572078
111111 3.46410i 0.328798i
112112 −10.0000 8.66025i −0.944911 0.818317i
113113 6.92820i 0.651751i 0.945413 + 0.325875i 0.105659π0.105659\pi
−0.945413 + 0.325875i 0.894341π0.894341\pi
114114 10.3923i 0.973329i
115115 3.46410i 0.323029i
116116 6.92820i 0.643268i
117117 0 0
118118 20.7846i 1.91338i
119119 12.0000 + 10.3923i 1.10004 + 0.952661i
120120 3.00000 0.273861
121121 −1.00000 −0.0909091
122122 12.0000 1.08643
123123 10.3923i 0.937043i
124124 3.46410i 0.311086i
125125 −1.00000 −0.0894427
126126 −9.00000 + 10.3923i −0.801784 + 0.925820i
127127 −4.00000 −0.354943 −0.177471 0.984126i 0.556792π-0.556792\pi
−0.177471 + 0.984126i 0.556792π0.556792\pi
128128 12.1244i 1.07165i
129129 13.8564i 1.21999i
130130 0 0
131131 −12.0000 −1.04844 −0.524222 0.851581i 0.675644π-0.675644\pi
−0.524222 + 0.851581i 0.675644π0.675644\pi
132132 −6.00000 −0.522233
133133 6.00000 6.92820i 0.520266 0.600751i
134134 13.8564i 1.19701i
135135 5.19615i 0.447214i
136136 10.3923i 0.891133i
137137 20.7846i 1.77575i −0.460086 0.887875i 0.652181π-0.652181\pi
0.460086 0.887875i 0.347819π-0.347819\pi
138138 10.3923i 0.884652i
139139 17.3205i 1.46911i −0.678551 0.734553i 0.737392π-0.737392\pi
0.678551 0.734553i 0.262608π-0.262608\pi
140140 2.00000 + 1.73205i 0.169031 + 0.146385i
141141 20.7846i 1.75038i
142142 −6.00000 −0.503509
143143 0 0
144144 15.0000 1.25000
145145 6.92820i 0.575356i
146146 −12.0000 −0.993127
147147 12.0000 1.73205i 0.989743 0.142857i
148148 2.00000 0.164399
149149 6.92820i 0.567581i −0.958886 0.283790i 0.908408π-0.908408\pi
0.958886 0.283790i 0.0915919π-0.0915919\pi
150150 −3.00000 −0.244949
151151 −8.00000 −0.651031 −0.325515 0.945537i 0.605538π-0.605538\pi
−0.325515 + 0.945537i 0.605538π0.605538\pi
152152 −6.00000 −0.486664
153153 −18.0000 −1.45521
154154 12.0000 + 10.3923i 0.966988 + 0.837436i
155155 3.46410i 0.278243i
156156 0 0
157157 13.8564i 1.10586i −0.833227 0.552931i 0.813509π-0.813509\pi
0.833227 0.552931i 0.186491π-0.186491\pi
158158 13.8564i 1.10236i
159159 0 0
160160 5.19615i 0.410792i
161161 6.00000 6.92820i 0.472866 0.546019i
162162 15.5885i 1.22474i
163163 −16.0000 −1.25322 −0.626608 0.779334i 0.715557π-0.715557\pi
−0.626608 + 0.779334i 0.715557π0.715557\pi
164164 −6.00000 −0.468521
165165 −6.00000 −0.467099
166166 0 0
167167 12.0000 0.928588 0.464294 0.885681i 0.346308π-0.346308\pi
0.464294 + 0.885681i 0.346308π0.346308\pi
168168 −6.00000 5.19615i −0.462910 0.400892i
169169 13.0000 1.00000
170170 10.3923i 0.797053i
171171 10.3923i 0.794719i
172172 8.00000 0.609994
173173 18.0000 1.36851 0.684257 0.729241i 0.260127π-0.260127\pi
0.684257 + 0.729241i 0.260127π0.260127\pi
174174 20.7846i 1.57568i
175175 2.00000 + 1.73205i 0.151186 + 0.130931i
176176 17.3205i 1.30558i
177177 20.7846i 1.56227i
178178 10.3923i 0.778936i
179179 10.3923i 0.776757i −0.921500 0.388379i 0.873035π-0.873035\pi
0.921500 0.388379i 0.126965π-0.126965\pi
180180 −3.00000 −0.223607
181181 20.7846i 1.54491i 0.635071 + 0.772454i 0.280971π0.280971\pi
−0.635071 + 0.772454i 0.719029π0.719029\pi
182182 0 0
183183 12.0000 0.887066
184184 −6.00000 −0.442326
185185 2.00000 0.147043
186186 10.3923i 0.762001i
187187 20.7846i 1.51992i
188188 12.0000 0.875190
189189 −9.00000 + 10.3923i −0.654654 + 0.755929i
190190 6.00000 0.435286
191191 10.3923i 0.751961i 0.926628 + 0.375980i 0.122694π0.122694\pi
−0.926628 + 0.375980i 0.877306π0.877306\pi
192192 1.73205i 0.125000i
193193 −14.0000 −1.00774 −0.503871 0.863779i 0.668091π-0.668091\pi
−0.503871 + 0.863779i 0.668091π0.668091\pi
194194 −12.0000 −0.861550
195195 0 0
196196 −1.00000 6.92820i −0.0714286 0.494872i
197197 13.8564i 0.987228i 0.869681 + 0.493614i 0.164324π0.164324\pi
−0.869681 + 0.493614i 0.835676π0.835676\pi
198198 −18.0000 −1.27920
199199 10.3923i 0.736691i −0.929689 0.368345i 0.879924π-0.879924\pi
0.929689 0.368345i 0.120076π-0.120076\pi
200200 1.73205i 0.122474i
201201 13.8564i 0.977356i
202202 10.3923i 0.731200i
203203 −12.0000 + 13.8564i −0.842235 + 0.972529i
204204 10.3923i 0.727607i
205205 −6.00000 −0.419058
206206 6.00000 0.418040
207207 10.3923i 0.722315i
208208 0 0
209209 12.0000 0.830057
210210 6.00000 + 5.19615i 0.414039 + 0.358569i
211211 4.00000 0.275371 0.137686 0.990476i 0.456034π-0.456034\pi
0.137686 + 0.990476i 0.456034π0.456034\pi
212212 0 0
213213 −6.00000 −0.411113
214214 −18.0000 −1.23045
215215 8.00000 0.545595
216216 9.00000 0.612372
217217 −6.00000 + 6.92820i −0.407307 + 0.470317i
218218 3.46410i 0.234619i
219219 −12.0000 −0.810885
220220 3.46410i 0.233550i
221221 0 0
222222 6.00000 0.402694
223223 17.3205i 1.15987i 0.814664 + 0.579934i 0.196921π0.196921\pi
−0.814664 + 0.579934i 0.803079π0.803079\pi
224224 −9.00000 + 10.3923i −0.601338 + 0.694365i
225225 −3.00000 −0.200000
226226 12.0000 0.798228
227227 −24.0000 −1.59294 −0.796468 0.604681i 0.793301π-0.793301\pi
−0.796468 + 0.604681i 0.793301π0.793301\pi
228228 6.00000 0.397360
229229 6.92820i 0.457829i 0.973447 + 0.228914i 0.0735176π0.0735176\pi
−0.973447 + 0.228914i 0.926482π0.926482\pi
230230 6.00000 0.395628
231231 12.0000 + 10.3923i 0.789542 + 0.683763i
232232 12.0000 0.787839
233233 6.92820i 0.453882i −0.973909 0.226941i 0.927128π-0.927128\pi
0.973909 0.226941i 0.0728724π-0.0728724\pi
234234 0 0
235235 12.0000 0.782794
236236 12.0000 0.781133
237237 13.8564i 0.900070i
238238 18.0000 20.7846i 1.16677 1.34727i
239239 10.3923i 0.672222i 0.941822 + 0.336111i 0.109112π0.109112\pi
−0.941822 + 0.336111i 0.890888π0.890888\pi
240240 8.66025i 0.559017i
241241 0 0 1.00000 00
−1.00000 π\pi
242242 1.73205i 0.111340i
243243 15.5885i 1.00000i
244244 6.92820i 0.443533i
245245 −1.00000 6.92820i −0.0638877 0.442627i
246246 −18.0000 −1.14764
247247 0 0
248248 6.00000 0.381000
249249 0 0
250250 1.73205i 0.109545i
251251 12.0000 0.757433 0.378717 0.925513i 0.376365π-0.376365\pi
0.378717 + 0.925513i 0.376365π0.376365\pi
252252 6.00000 + 5.19615i 0.377964 + 0.327327i
253253 12.0000 0.754434
254254 6.92820i 0.434714i
255255 10.3923i 0.650791i
256256 19.0000 1.18750
257257 6.00000 0.374270 0.187135 0.982334i 0.440080π-0.440080\pi
0.187135 + 0.982334i 0.440080π0.440080\pi
258258 24.0000 1.49417
259259 −4.00000 3.46410i −0.248548 0.215249i
260260 0 0
261261 20.7846i 1.28654i
262262 20.7846i 1.28408i
263263 24.2487i 1.49524i 0.664127 + 0.747620i 0.268803π0.268803\pi
−0.664127 + 0.747620i 0.731197π0.731197\pi
264264 10.3923i 0.639602i
265265 0 0
266266 −12.0000 10.3923i −0.735767 0.637193i
267267 10.3923i 0.635999i
268268 −8.00000 −0.488678
269269 18.0000 1.09748 0.548740 0.835993i 0.315108π-0.315108\pi
0.548740 + 0.835993i 0.315108π0.315108\pi
270270 −9.00000 −0.547723
271271 24.2487i 1.47300i −0.676435 0.736502i 0.736476π-0.736476\pi
0.676435 0.736502i 0.263524π-0.263524\pi
272272 −30.0000 −1.81902
273273 0 0
274274 −36.0000 −2.17484
275275 3.46410i 0.208893i
276276 6.00000 0.361158
277277 14.0000 0.841178 0.420589 0.907251i 0.361823π-0.361823\pi
0.420589 + 0.907251i 0.361823π0.361823\pi
278278 −30.0000 −1.79928
279279 10.3923i 0.622171i
280280 −3.00000 + 3.46410i −0.179284 + 0.207020i
281281 13.8564i 0.826604i −0.910594 0.413302i 0.864375π-0.864375\pi
0.910594 0.413302i 0.135625π-0.135625\pi
282282 36.0000 2.14377
283283 17.3205i 1.02960i −0.857311 0.514799i 0.827867π-0.827867\pi
0.857311 0.514799i 0.172133π-0.172133\pi
284284 3.46410i 0.205557i
285285 6.00000 0.355409
286286 0 0
287287 12.0000 + 10.3923i 0.708338 + 0.613438i
288288 15.5885i 0.918559i
289289 19.0000 1.11765
290290 −12.0000 −0.704664
291291 −12.0000 −0.703452
292292 6.92820i 0.405442i
293293 −6.00000 −0.350524 −0.175262 0.984522i 0.556077π-0.556077\pi
−0.175262 + 0.984522i 0.556077π0.556077\pi
294294 −3.00000 20.7846i −0.174964 1.21218i
295295 12.0000 0.698667
296296 3.46410i 0.201347i
297297 −18.0000 −1.04447
298298 −12.0000 −0.695141
299299 0 0
300300 1.73205i 0.100000i
301301 −16.0000 13.8564i −0.922225 0.798670i
302302 13.8564i 0.797347i
303303 10.3923i 0.597022i
304304 17.3205i 0.993399i
305305 6.92820i 0.396708i
306306 31.1769i 1.78227i
307307 24.2487i 1.38395i 0.721923 + 0.691974i 0.243259π0.243259\pi
−0.721923 + 0.691974i 0.756741π0.756741\pi
308308 6.00000 6.92820i 0.341882 0.394771i
309309 6.00000 0.341328
310310 −6.00000 −0.340777
311311 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
312312 0 0
313313 20.7846i 1.17482i 0.809291 + 0.587408i 0.199852π0.199852\pi
−0.809291 + 0.587408i 0.800148π0.800148\pi
314314 −24.0000 −1.35440
315315 6.00000 + 5.19615i 0.338062 + 0.292770i
316316 −8.00000 −0.450035
317317 27.7128i 1.55651i −0.627950 0.778253i 0.716106π-0.716106\pi
0.627950 0.778253i 0.283894π-0.283894\pi
318318 0 0
319319 −24.0000 −1.34374
320320 1.00000 0.0559017
321321 −18.0000 −1.00466
322322 −12.0000 10.3923i −0.668734 0.579141i
323323 20.7846i 1.15649i
324324 −9.00000 −0.500000
325325 0 0
326326 27.7128i 1.53487i
327327 3.46410i 0.191565i
328328 10.3923i 0.573819i
329329 −24.0000 20.7846i −1.32316 1.14589i
330330 10.3923i 0.572078i
331331 28.0000 1.53902 0.769510 0.638635i 0.220501π-0.220501\pi
0.769510 + 0.638635i 0.220501π0.220501\pi
332332 0 0
333333 6.00000 0.328798
334334 20.7846i 1.13728i
335335 −8.00000 −0.437087
336336 −15.0000 + 17.3205i −0.818317 + 0.944911i
337337 −14.0000 −0.762629 −0.381314 0.924445i 0.624528π-0.624528\pi
−0.381314 + 0.924445i 0.624528π0.624528\pi
338338 22.5167i 1.22474i
339339 12.0000 0.651751
340340 6.00000 0.325396
341341 −12.0000 −0.649836
342342 18.0000 0.973329
343343 −10.0000 + 15.5885i −0.539949 + 0.841698i
344344 13.8564i 0.747087i
345345 6.00000 0.323029
346346 31.1769i 1.67608i
347347 17.3205i 0.929814i 0.885360 + 0.464907i 0.153912π0.153912\pi
−0.885360 + 0.464907i 0.846088π0.846088\pi
348348 −12.0000 −0.643268
349349 6.92820i 0.370858i −0.982658 0.185429i 0.940632π-0.940632\pi
0.982658 0.185429i 0.0593675π-0.0593675\pi
350350 3.00000 3.46410i 0.160357 0.185164i
351351 0 0
352352 −18.0000 −0.959403
353353 6.00000 0.319348 0.159674 0.987170i 0.448956π-0.448956\pi
0.159674 + 0.987170i 0.448956π0.448956\pi
354354 36.0000 1.91338
355355 3.46410i 0.183855i
356356 −6.00000 −0.317999
357357 18.0000 20.7846i 0.952661 1.10004i
358358 −18.0000 −0.951330
359359 3.46410i 0.182828i −0.995813 0.0914141i 0.970861π-0.970861\pi
0.995813 0.0914141i 0.0291387π-0.0291387\pi
360360 5.19615i 0.273861i
361361 7.00000 0.368421
362362 36.0000 1.89212
363363 1.73205i 0.0909091i
364364 0 0
365365 6.92820i 0.362639i
366366 20.7846i 1.08643i
367367 10.3923i 0.542474i −0.962513 0.271237i 0.912567π-0.912567\pi
0.962513 0.271237i 0.0874327π-0.0874327\pi
368368 17.3205i 0.902894i
369369 −18.0000 −0.937043
370370 3.46410i 0.180090i
371371 0 0
372372 −6.00000 −0.311086
373373 14.0000 0.724893 0.362446 0.932005i 0.381942π-0.381942\pi
0.362446 + 0.932005i 0.381942π0.381942\pi
374374 36.0000 1.86152
375375 1.73205i 0.0894427i
376376 20.7846i 1.07188i
377377 0 0
378378 18.0000 + 15.5885i 0.925820 + 0.801784i
379379 4.00000 0.205466 0.102733 0.994709i 0.467241π-0.467241\pi
0.102733 + 0.994709i 0.467241π0.467241\pi
380380 3.46410i 0.177705i
381381 6.92820i 0.354943i
382382 18.0000 0.920960
383383 12.0000 0.613171 0.306586 0.951843i 0.400813π-0.400813\pi
0.306586 + 0.951843i 0.400813π0.400813\pi
384384 21.0000 1.07165
385385 6.00000 6.92820i 0.305788 0.353094i
386386 24.2487i 1.23423i
387387 24.0000 1.21999
388388 6.92820i 0.351726i
389389 6.92820i 0.351274i −0.984455 0.175637i 0.943802π-0.943802\pi
0.984455 0.175637i 0.0561985π-0.0561985\pi
390390 0 0
391391 20.7846i 1.05112i
392392 12.0000 1.73205i 0.606092 0.0874818i
393393 20.7846i 1.04844i
394394 24.0000 1.20910
395395 −8.00000 −0.402524
396396 10.3923i 0.522233i
397397 13.8564i 0.695433i −0.937600 0.347717i 0.886957π-0.886957\pi
0.937600 0.347717i 0.113043π-0.113043\pi
398398 −18.0000 −0.902258
399399 −12.0000 10.3923i −0.600751 0.520266i
400400 −5.00000 −0.250000
401401 27.7128i 1.38391i 0.721940 + 0.691956i 0.243251π0.243251\pi
−0.721940 + 0.691956i 0.756749π0.756749\pi
402402 −24.0000 −1.19701
403403 0 0
404404 6.00000 0.298511
405405 −9.00000 −0.447214
406406 24.0000 + 20.7846i 1.19110 + 1.03152i
407407 6.92820i 0.343418i
408408 −18.0000 −0.891133
409409 0 0 1.00000 00
−1.00000 π\pi
410410 10.3923i 0.513239i
411411 −36.0000 −1.77575
412412 3.46410i 0.170664i
413413 −24.0000 20.7846i −1.18096 1.02274i
414414 18.0000 0.884652
415415 0 0
416416 0 0
417417 −30.0000 −1.46911
418418 20.7846i 1.01661i
419419 12.0000 0.586238 0.293119 0.956076i 0.405307π-0.405307\pi
0.293119 + 0.956076i 0.405307π0.405307\pi
420420 3.00000 3.46410i 0.146385 0.169031i
421421 −10.0000 −0.487370 −0.243685 0.969854i 0.578356π-0.578356\pi
−0.243685 + 0.969854i 0.578356π0.578356\pi
422422 6.92820i 0.337260i
423423 36.0000 1.75038
424424 0 0
425425 6.00000 0.291043
426426 10.3923i 0.503509i
427427 −12.0000 + 13.8564i −0.580721 + 0.670559i
428428 10.3923i 0.502331i
429429 0 0
430430 13.8564i 0.668215i
431431 10.3923i 0.500580i 0.968171 + 0.250290i 0.0805259π0.0805259\pi
−0.968171 + 0.250290i 0.919474π0.919474\pi
432432 25.9808i 1.25000i
433433 6.92820i 0.332948i 0.986046 + 0.166474i 0.0532382π0.0532382\pi
−0.986046 + 0.166474i 0.946762π0.946762\pi
434434 12.0000 + 10.3923i 0.576018 + 0.498847i
435435 −12.0000 −0.575356
436436 2.00000 0.0957826
437437 −12.0000 −0.574038
438438 20.7846i 0.993127i
439439 17.3205i 0.826663i 0.910581 + 0.413331i 0.135635π0.135635\pi
−0.910581 + 0.413331i 0.864365π0.864365\pi
440440 −6.00000 −0.286039
441441 −3.00000 20.7846i −0.142857 0.989743i
442442 0 0
443443 10.3923i 0.493753i −0.969047 0.246877i 0.920596π-0.920596\pi
0.969047 0.246877i 0.0794043π-0.0794043\pi
444444 3.46410i 0.164399i
445445 −6.00000 −0.284427
446446 30.0000 1.42054
447447 −12.0000 −0.567581
448448 −2.00000 1.73205i −0.0944911 0.0818317i
449449 13.8564i 0.653924i 0.945037 + 0.326962i 0.106025π0.106025\pi
−0.945037 + 0.326962i 0.893975π0.893975\pi
450450 5.19615i 0.244949i
451451 20.7846i 0.978709i
452452 6.92820i 0.325875i
453453 13.8564i 0.651031i
454454 41.5692i 1.95094i
455455 0 0
456456 10.3923i 0.486664i
457457 −38.0000 −1.77757 −0.888783 0.458329i 0.848448π-0.848448\pi
−0.888783 + 0.458329i 0.848448π0.848448\pi
458458 12.0000 0.560723
459459 31.1769i 1.45521i
460460 3.46410i 0.161515i
461461 18.0000 0.838344 0.419172 0.907907i 0.362320π-0.362320\pi
0.419172 + 0.907907i 0.362320π0.362320\pi
462462 18.0000 20.7846i 0.837436 0.966988i
463463 20.0000 0.929479 0.464739 0.885448i 0.346148π-0.346148\pi
0.464739 + 0.885448i 0.346148π0.346148\pi
464464 34.6410i 1.60817i
465465 −6.00000 −0.278243
466466 −12.0000 −0.555889
467467 24.0000 1.11059 0.555294 0.831654i 0.312606π-0.312606\pi
0.555294 + 0.831654i 0.312606π0.312606\pi
468468 0 0
469469 16.0000 + 13.8564i 0.738811 + 0.639829i
470470 20.7846i 0.958723i
471471 −24.0000 −1.10586
472472 20.7846i 0.956689i
473473 27.7128i 1.27424i
474474 −24.0000 −1.10236
475475 3.46410i 0.158944i
476476 −12.0000 10.3923i −0.550019 0.476331i
477477 0 0
478478 18.0000 0.823301
479479 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
480480 −9.00000 −0.410792
481481 0 0
482482 0 0
483483 −12.0000 10.3923i −0.546019 0.472866i
484484 1.00000 0.0454545
485485 6.92820i 0.314594i
486486 −27.0000 −1.22474
487487 4.00000 0.181257 0.0906287 0.995885i 0.471112π-0.471112\pi
0.0906287 + 0.995885i 0.471112π0.471112\pi
488488 12.0000 0.543214
489489 27.7128i 1.25322i
490490 −12.0000 + 1.73205i −0.542105 + 0.0782461i
491491 24.2487i 1.09433i −0.837025 0.547165i 0.815707π-0.815707\pi
0.837025 0.547165i 0.184293π-0.184293\pi
492492 10.3923i 0.468521i
493493 41.5692i 1.87218i
494494 0 0
495495 10.3923i 0.467099i
496496 17.3205i 0.777714i
497497 6.00000 6.92820i 0.269137 0.310772i
498498 0 0
499499 −20.0000 −0.895323 −0.447661 0.894203i 0.647743π-0.647743\pi
−0.447661 + 0.894203i 0.647743π0.647743\pi
500500 1.00000 0.0447214
501501 20.7846i 0.928588i
502502 20.7846i 0.927663i
503503 −12.0000 −0.535054 −0.267527 0.963550i 0.586206π-0.586206\pi
−0.267527 + 0.963550i 0.586206π0.586206\pi
504504 −9.00000 + 10.3923i −0.400892 + 0.462910i
505505 6.00000 0.266996
506506 20.7846i 0.923989i
507507 22.5167i 1.00000i
508508 4.00000 0.177471
509509 −30.0000 −1.32973 −0.664863 0.746965i 0.731510π-0.731510\pi
−0.664863 + 0.746965i 0.731510π0.731510\pi
510510 18.0000 0.797053
511511 12.0000 13.8564i 0.530849 0.612971i
512512 8.66025i 0.382733i
513513 18.0000 0.794719
514514 10.3923i 0.458385i
515515 3.46410i 0.152647i
516516 13.8564i 0.609994i
517517 41.5692i 1.82821i
518518 −6.00000 + 6.92820i −0.263625 + 0.304408i
519519 31.1769i 1.36851i
520520 0 0
521521 6.00000 0.262865 0.131432 0.991325i 0.458042π-0.458042\pi
0.131432 + 0.991325i 0.458042π0.458042\pi
522522 −36.0000 −1.57568
523523 17.3205i 0.757373i −0.925525 0.378686i 0.876376π-0.876376\pi
0.925525 0.378686i 0.123624π-0.123624\pi
524524 12.0000 0.524222
525525 3.00000 3.46410i 0.130931 0.151186i
526526 42.0000 1.83129
527527 20.7846i 0.905392i
528528 −30.0000 −1.30558
529529 11.0000 0.478261
530530 0 0
531531 36.0000 1.56227
532532 −6.00000 + 6.92820i −0.260133 + 0.300376i
533533 0 0
534534 −18.0000 −0.778936
535535 10.3923i 0.449299i
536536 13.8564i 0.598506i
537537 −18.0000 −0.776757
538538 31.1769i 1.34413i
539539 −24.0000 + 3.46410i −1.03375 + 0.149209i
540540 5.19615i 0.223607i
541541 −34.0000 −1.46177 −0.730887 0.682498i 0.760893π-0.760893\pi
−0.730887 + 0.682498i 0.760893π0.760893\pi
542542 −42.0000 −1.80405
543543 36.0000 1.54491
544544 31.1769i 1.33670i
545545 2.00000 0.0856706
546546 0 0
547547 8.00000 0.342055 0.171028 0.985266i 0.445291π-0.445291\pi
0.171028 + 0.985266i 0.445291π0.445291\pi
548548 20.7846i 0.887875i
549549 20.7846i 0.887066i
550550 6.00000 0.255841
551551 24.0000 1.02243
552552 10.3923i 0.442326i
553553 16.0000 + 13.8564i 0.680389 + 0.589234i
554554 24.2487i 1.03023i
555555 3.46410i 0.147043i
556556 17.3205i 0.734553i
557557 27.7128i 1.17423i 0.809504 + 0.587115i 0.199736π0.199736\pi
−0.809504 + 0.587115i 0.800264π0.800264\pi
558558 −18.0000 −0.762001
559559 0 0
560560 10.0000 + 8.66025i 0.422577 + 0.365963i
561561 36.0000 1.51992
562562 −24.0000 −1.01238
563563 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
564564 20.7846i 0.875190i
565565 6.92820i 0.291472i
566566 −30.0000 −1.26099
567567 18.0000 + 15.5885i 0.755929 + 0.654654i
568568 −6.00000 −0.251754
569569 27.7128i 1.16178i 0.813982 + 0.580891i 0.197296π0.197296\pi
−0.813982 + 0.580891i 0.802704π0.802704\pi
570570 10.3923i 0.435286i
571571 −4.00000 −0.167395 −0.0836974 0.996491i 0.526673π-0.526673\pi
−0.0836974 + 0.996491i 0.526673π0.526673\pi
572572 0 0
573573 18.0000 0.751961
574574 18.0000 20.7846i 0.751305 0.867533i
575575 3.46410i 0.144463i
576576 3.00000 0.125000
577577 34.6410i 1.44212i −0.692870 0.721062i 0.743654π-0.743654\pi
0.692870 0.721062i 0.256346π-0.256346\pi
578578 32.9090i 1.36883i
579579 24.2487i 1.00774i
580580 6.92820i 0.287678i
581581 0 0
582582 20.7846i 0.861550i
583583 0 0
584584 −12.0000 −0.496564
585585 0 0
586586 10.3923i 0.429302i
587587 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
588588 −12.0000 + 1.73205i −0.494872 + 0.0714286i
589589 12.0000 0.494451
590590 20.7846i 0.855689i
591591 24.0000 0.987228
592592 10.0000 0.410997
593593 6.00000 0.246390 0.123195 0.992382i 0.460686π-0.460686\pi
0.123195 + 0.992382i 0.460686π0.460686\pi
594594 31.1769i 1.27920i
595595 −12.0000 10.3923i −0.491952 0.426043i
596596 6.92820i 0.283790i
597597 −18.0000 −0.736691
598598 0 0
599599 24.2487i 0.990775i 0.868672 + 0.495388i 0.164974π0.164974\pi
−0.868672 + 0.495388i 0.835026π0.835026\pi
600600 −3.00000 −0.122474
601601 13.8564i 0.565215i −0.959236 0.282607i 0.908801π-0.908801\pi
0.959236 0.282607i 0.0911993π-0.0911993\pi
602602 −24.0000 + 27.7128i −0.978167 + 1.12949i
603603 −24.0000 −0.977356
604604 8.00000 0.325515
605605 1.00000 0.0406558
606606 18.0000 0.731200
607607 17.3205i 0.703018i 0.936185 + 0.351509i 0.114331π0.114331\pi
−0.936185 + 0.351509i 0.885669π0.885669\pi
608608 18.0000 0.729996
609609 24.0000 + 20.7846i 0.972529 + 0.842235i
610610 −12.0000 −0.485866
611611 0 0
612612 18.0000 0.727607
613613 −34.0000 −1.37325 −0.686624 0.727013i 0.740908π-0.740908\pi
−0.686624 + 0.727013i 0.740908π0.740908\pi
614614 42.0000 1.69498
615615 10.3923i 0.419058i
616616 12.0000 + 10.3923i 0.483494 + 0.418718i
617617 6.92820i 0.278919i 0.990228 + 0.139459i 0.0445365π0.0445365\pi
−0.990228 + 0.139459i 0.955464π0.955464\pi
618618 10.3923i 0.418040i
619619 17.3205i 0.696170i −0.937463 0.348085i 0.886832π-0.886832\pi
0.937463 0.348085i 0.113168π-0.113168\pi
620620 3.46410i 0.139122i
621621 18.0000 0.722315
622622 0 0
623623 12.0000 + 10.3923i 0.480770 + 0.416359i
624624 0 0
625625 1.00000 0.0400000
626626 36.0000 1.43885
627627 20.7846i 0.830057i
628628 13.8564i 0.552931i
629629 −12.0000 −0.478471
630630 9.00000 10.3923i 0.358569 0.414039i
631631 −40.0000 −1.59237 −0.796187 0.605050i 0.793153π-0.793153\pi
−0.796187 + 0.605050i 0.793153π0.793153\pi
632632 13.8564i 0.551178i
633633 6.92820i 0.275371i
634634 −48.0000 −1.90632
635635 4.00000 0.158735
636636 0 0
637637 0 0
638638 41.5692i 1.64574i
639639 10.3923i 0.411113i
640640 12.1244i 0.479257i
641641 27.7128i 1.09459i −0.836940 0.547295i 0.815658π-0.815658\pi
0.836940 0.547295i 0.184342π-0.184342\pi
642642 31.1769i 1.23045i
643643 31.1769i 1.22950i −0.788723 0.614749i 0.789257π-0.789257\pi
0.788723 0.614749i 0.210743π-0.210743\pi
644644 −6.00000 + 6.92820i −0.236433 + 0.273009i
645645 13.8564i 0.545595i
646646 −36.0000 −1.41640
647647 12.0000 0.471769 0.235884 0.971781i 0.424201π-0.424201\pi
0.235884 + 0.971781i 0.424201π0.424201\pi
648648 15.5885i 0.612372i
649649 41.5692i 1.63173i
650650 0 0
651651 12.0000 + 10.3923i 0.470317 + 0.407307i
652652 16.0000 0.626608
653653 41.5692i 1.62673i 0.581754 + 0.813365i 0.302367π0.302367\pi
−0.581754 + 0.813365i 0.697633π0.697633\pi
654654 6.00000 0.234619
655655 12.0000 0.468879
656656 −30.0000 −1.17130
657657 20.7846i 0.810885i
658658 −36.0000 + 41.5692i −1.40343 + 1.62054i
659659 10.3923i 0.404827i −0.979300 0.202413i 0.935122π-0.935122\pi
0.979300 0.202413i 0.0648785π-0.0648785\pi
660660 6.00000 0.233550
661661 48.4974i 1.88633i 0.332323 + 0.943166i 0.392168π0.392168\pi
−0.332323 + 0.943166i 0.607832π0.607832\pi
662662 48.4974i 1.88491i
663663 0 0
664664 0 0
665665 −6.00000 + 6.92820i −0.232670 + 0.268664i
666666 10.3923i 0.402694i
667667 24.0000 0.929284
668668 −12.0000 −0.464294
669669 30.0000 1.15987
670670 13.8564i 0.535320i
671671 −24.0000 −0.926510
672672 18.0000 + 15.5885i 0.694365 + 0.601338i
673673 −14.0000 −0.539660 −0.269830 0.962908i 0.586968π-0.586968\pi
−0.269830 + 0.962908i 0.586968π0.586968\pi
674674 24.2487i 0.934025i
675675 5.19615i 0.200000i
676676 −13.0000 −0.500000
677677 −6.00000 −0.230599 −0.115299 0.993331i 0.536783π-0.536783\pi
−0.115299 + 0.993331i 0.536783π0.536783\pi
678678 20.7846i 0.798228i
679679 12.0000 13.8564i 0.460518 0.531760i
680680 10.3923i 0.398527i
681681 41.5692i 1.59294i
682682 20.7846i 0.795884i
683683 17.3205i 0.662751i 0.943499 + 0.331375i 0.107513π0.107513\pi
−0.943499 + 0.331375i 0.892487π0.892487\pi
684684 10.3923i 0.397360i
685685 20.7846i 0.794139i
686686 27.0000 + 17.3205i 1.03086 + 0.661300i
687687 12.0000 0.457829
688688 40.0000 1.52499
689689 0 0
690690 10.3923i 0.395628i
691691 31.1769i 1.18603i −0.805193 0.593013i 0.797938π-0.797938\pi
0.805193 0.593013i 0.202062π-0.202062\pi
692692 −18.0000 −0.684257
693693 18.0000 20.7846i 0.683763 0.789542i
694694 30.0000 1.13878
695695 17.3205i 0.657004i
696696 20.7846i 0.787839i
697697 36.0000 1.36360
698698 −12.0000 −0.454207
699699 −12.0000 −0.453882
700700 −2.00000 1.73205i −0.0755929 0.0654654i
701701 20.7846i 0.785024i 0.919747 + 0.392512i 0.128394π0.128394\pi
−0.919747 + 0.392512i 0.871606π0.871606\pi
702702 0 0
703703 6.92820i 0.261302i
704704 3.46410i 0.130558i
705705 20.7846i 0.782794i
706706 10.3923i 0.391120i
707707 −12.0000 10.3923i −0.451306 0.390843i
708708 20.7846i 0.781133i
709709 22.0000 0.826227 0.413114 0.910679i 0.364441π-0.364441\pi
0.413114 + 0.910679i 0.364441π0.364441\pi
710710 6.00000 0.225176
711711 −24.0000 −0.900070
712712 10.3923i 0.389468i
713713 12.0000 0.449404
714714 −36.0000 31.1769i −1.34727 1.16677i
715715 0 0
716716 10.3923i 0.388379i
717717 18.0000 0.672222
718718 −6.00000 −0.223918
719719 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
720720 −15.0000 −0.559017
721721 −6.00000 + 6.92820i −0.223452 + 0.258020i
722722 12.1244i 0.451222i
723723 0 0
724724 20.7846i 0.772454i
725725 6.92820i 0.257307i
726726 3.00000 0.111340
727727 3.46410i 0.128476i 0.997935 + 0.0642382i 0.0204617π0.0204617\pi
−0.997935 + 0.0642382i 0.979538π0.979538\pi
728728 0 0
729729 −27.0000 −1.00000
730730 12.0000 0.444140
731731 −48.0000 −1.77534
732732 −12.0000 −0.443533
733733 0 0 1.00000 00
−1.00000 π\pi
734734 −18.0000 −0.664392
735735 −12.0000 + 1.73205i −0.442627 + 0.0638877i
736736 18.0000 0.663489
737737 27.7128i 1.02081i
738738 31.1769i 1.14764i
739739 −4.00000 −0.147142 −0.0735712 0.997290i 0.523440π-0.523440\pi
−0.0735712 + 0.997290i 0.523440π0.523440\pi
740740 −2.00000 −0.0735215
741741 0 0
742742 0 0
743743 31.1769i 1.14377i −0.820334 0.571885i 0.806212π-0.806212\pi
0.820334 0.571885i 0.193788π-0.193788\pi
744744 10.3923i 0.381000i
745745 6.92820i 0.253830i
746746 24.2487i 0.887808i
747747 0 0
748748 20.7846i 0.759961i
749749 18.0000 20.7846i 0.657706 0.759453i
750750 3.00000 0.109545
751751 16.0000 0.583848 0.291924 0.956441i 0.405705π-0.405705\pi
0.291924 + 0.956441i 0.405705π0.405705\pi
752752 60.0000 2.18797
753753 20.7846i 0.757433i
754754 0 0
755755 8.00000 0.291150
756756 9.00000 10.3923i 0.327327 0.377964i
757757 −2.00000 −0.0726912 −0.0363456 0.999339i 0.511572π-0.511572\pi
−0.0363456 + 0.999339i 0.511572π0.511572\pi
758758 6.92820i 0.251644i
759759 20.7846i 0.754434i
760760 6.00000 0.217643
761761 −42.0000 −1.52250 −0.761249 0.648459i 0.775414π-0.775414\pi
−0.761249 + 0.648459i 0.775414π0.775414\pi
762762 12.0000 0.434714
763763 −4.00000 3.46410i −0.144810 0.125409i
764764 10.3923i 0.375980i
765765 18.0000 0.650791
766766 20.7846i 0.750978i
767767 0 0
768768 32.9090i 1.18750i
769769 41.5692i 1.49902i 0.661991 + 0.749512i 0.269712π0.269712\pi
−0.661991 + 0.749512i 0.730288π0.730288\pi
770770 −12.0000 10.3923i −0.432450 0.374513i
771771 10.3923i 0.374270i
772772 14.0000 0.503871
773773 −6.00000 −0.215805 −0.107903 0.994161i 0.534413π-0.534413\pi
−0.107903 + 0.994161i 0.534413π0.534413\pi
774774 41.5692i 1.49417i
775775 3.46410i 0.124434i
776776 −12.0000 −0.430775
777777 −6.00000 + 6.92820i −0.215249 + 0.248548i
778778 −12.0000 −0.430221
779779 20.7846i 0.744686i
780780 0 0
781781 12.0000 0.429394
782782 −36.0000 −1.28736
783783 −36.0000 −1.28654
784784 −5.00000 34.6410i −0.178571 1.23718i
785785 13.8564i 0.494556i
786786 36.0000 1.28408
787787 24.2487i 0.864373i 0.901784 + 0.432187i 0.142258π0.142258\pi
−0.901784 + 0.432187i 0.857742π0.857742\pi
788788 13.8564i 0.493614i
789789 42.0000 1.49524
790790 13.8564i 0.492989i
791791 −12.0000 + 13.8564i −0.426671 + 0.492677i
792792 −18.0000 −0.639602
793793 0 0
794794 −24.0000 −0.851728
795795 0 0
796796 10.3923i 0.368345i
797797 18.0000 0.637593 0.318796 0.947823i 0.396721π-0.396721\pi
0.318796 + 0.947823i 0.396721π0.396721\pi
798798 −18.0000 + 20.7846i −0.637193 + 0.735767i
799799 −72.0000 −2.54718
800800 5.19615i 0.183712i
801801 −18.0000 −0.635999
802802 48.0000 1.69494
803803 24.0000 0.846942
804804 13.8564i 0.488678i
805805 −6.00000 + 6.92820i −0.211472 + 0.244187i
806806 0 0
807807 31.1769i 1.09748i
808808 10.3923i 0.365600i
809809 55.4256i 1.94866i −0.225122 0.974331i 0.572278π-0.572278\pi
0.225122 0.974331i 0.427722π-0.427722\pi
810810 15.5885i 0.547723i
811811 38.1051i 1.33805i 0.743239 + 0.669026i 0.233288π0.233288\pi
−0.743239 + 0.669026i 0.766712π0.766712\pi
812812 12.0000 13.8564i 0.421117 0.486265i
813813 −42.0000 −1.47300
814814 −12.0000 −0.420600
815815 16.0000 0.560456
816816 51.9615i 1.81902i
817817 27.7128i 0.969549i
818818 0 0
819819 0 0
820820 6.00000 0.209529
821821 20.7846i 0.725388i −0.931908 0.362694i 0.881857π-0.881857\pi
0.931908 0.362694i 0.118143π-0.118143\pi
822822 62.3538i 2.17484i
823823 44.0000 1.53374 0.766872 0.641800i 0.221812π-0.221812\pi
0.766872 + 0.641800i 0.221812π0.221812\pi
824824 6.00000 0.209020
825825 6.00000 0.208893
826826 −36.0000 + 41.5692i −1.25260 + 1.44638i
827827 38.1051i 1.32504i −0.749042 0.662522i 0.769486π-0.769486\pi
0.749042 0.662522i 0.230514π-0.230514\pi
828828 10.3923i 0.361158i
829829 34.6410i 1.20313i −0.798823 0.601566i 0.794544π-0.794544\pi
0.798823 0.601566i 0.205456π-0.205456\pi
830830 0 0
831831 24.2487i 0.841178i
832832 0 0
833833 6.00000 + 41.5692i 0.207888 + 1.44029i
834834 51.9615i 1.79928i
835835 −12.0000 −0.415277
836836 −12.0000 −0.415029
837837 −18.0000 −0.622171
838838 20.7846i 0.717992i
839839 −24.0000 −0.828572 −0.414286 0.910147i 0.635969π-0.635969\pi
−0.414286 + 0.910147i 0.635969π0.635969\pi
840840 6.00000 + 5.19615i 0.207020 + 0.179284i
841841 −19.0000 −0.655172
842842 17.3205i 0.596904i
843843 −24.0000 −0.826604
844844 −4.00000 −0.137686
845845 −13.0000 −0.447214
846846 62.3538i 2.14377i
847847 −2.00000 1.73205i −0.0687208 0.0595140i
848848 0 0
849849 −30.0000 −1.02960
850850 10.3923i 0.356453i
851851 6.92820i 0.237496i
852852 6.00000 0.205557
853853 41.5692i 1.42330i −0.702533 0.711651i 0.747948π-0.747948\pi
0.702533 0.711651i 0.252052π-0.252052\pi
854854 24.0000 + 20.7846i 0.821263 + 0.711235i
855855 10.3923i 0.355409i
856856 −18.0000 −0.615227
857857 30.0000 1.02478 0.512390 0.858753i 0.328760π-0.328760\pi
0.512390 + 0.858753i 0.328760π0.328760\pi
858858 0 0
859859 38.1051i 1.30013i 0.759879 + 0.650065i 0.225258π0.225258\pi
−0.759879 + 0.650065i 0.774742π0.774742\pi
860860 −8.00000 −0.272798
861861 18.0000 20.7846i 0.613438 0.708338i
862862 18.0000 0.613082
863863 17.3205i 0.589597i −0.955559 0.294798i 0.904747π-0.904747\pi
0.955559 0.294798i 0.0952525π-0.0952525\pi
864864 −27.0000 −0.918559
865865 −18.0000 −0.612018
866866 12.0000 0.407777
867867 32.9090i 1.11765i
868868 6.00000 6.92820i 0.203653 0.235159i
869869 27.7128i 0.940093i
870870 20.7846i 0.704664i
871871 0 0
872872 3.46410i 0.117309i
873873 20.7846i 0.703452i
874874 20.7846i 0.703050i
875875 −2.00000 1.73205i −0.0676123 0.0585540i
876876 12.0000 0.405442
877877 38.0000 1.28317 0.641584 0.767052i 0.278277π-0.278277\pi
0.641584 + 0.767052i 0.278277π0.278277\pi
878878 30.0000 1.01245
879879 10.3923i 0.350524i
880880 17.3205i 0.583874i
881881 30.0000 1.01073 0.505363 0.862907i 0.331359π-0.331359\pi
0.505363 + 0.862907i 0.331359π0.331359\pi
882882 −36.0000 + 5.19615i −1.21218 + 0.174964i
883883 16.0000 0.538443 0.269221 0.963078i 0.413234π-0.413234\pi
0.269221 + 0.963078i 0.413234π0.413234\pi
884884 0 0
885885 20.7846i 0.698667i
886886 −18.0000 −0.604722
887887 −36.0000 −1.20876 −0.604381 0.796696i 0.706579π-0.706579\pi
−0.604381 + 0.796696i 0.706579π0.706579\pi
888888 6.00000 0.201347
889889 −8.00000 6.92820i −0.268311 0.232364i
890890 10.3923i 0.348351i
891891 31.1769i 1.04447i
892892 17.3205i 0.579934i
893893 41.5692i 1.39106i
894894 20.7846i 0.695141i
895895 10.3923i 0.347376i
896896 −21.0000 + 24.2487i −0.701561 + 0.810093i
897897 0 0
898898 24.0000 0.800890
899899 −24.0000 −0.800445
900900 3.00000 0.100000
901901 0 0
902902 36.0000 1.19867
903903 −24.0000 + 27.7128i −0.798670 + 0.922225i
904904 12.0000 0.399114
905905 20.7846i 0.690904i
906906 24.0000 0.797347
907907 −32.0000 −1.06254 −0.531271 0.847202i 0.678286π-0.678286\pi
−0.531271 + 0.847202i 0.678286π0.678286\pi
908908 24.0000 0.796468
909909 18.0000 0.597022
910910 0 0
911911 17.3205i 0.573854i −0.957952 0.286927i 0.907366π-0.907366\pi
0.957952 0.286927i 0.0926337π-0.0926337\pi
912912 30.0000 0.993399
913913 0 0
914914 65.8179i 2.17706i
915915 −12.0000 −0.396708
916916 6.92820i 0.228914i
917917 −24.0000 20.7846i −0.792550 0.686368i
918918 54.0000 1.78227
919919 16.0000 0.527791 0.263896 0.964551i 0.414993π-0.414993\pi
0.263896 + 0.964551i 0.414993π0.414993\pi
920920 6.00000 0.197814
921921 42.0000 1.38395
922922 31.1769i 1.02676i
923923 0 0
924924 −12.0000 10.3923i −0.394771 0.341882i
925925 −2.00000 −0.0657596
926926 34.6410i 1.13837i
927927 10.3923i 0.341328i
928928 −36.0000 −1.18176
929929 30.0000 0.984268 0.492134 0.870519i 0.336217π-0.336217\pi
0.492134 + 0.870519i 0.336217π0.336217\pi
930930 10.3923i 0.340777i
931931 24.0000 3.46410i 0.786568 0.113531i
932932 6.92820i 0.226941i
933933 0 0
934934 41.5692i 1.36019i
935935 20.7846i 0.679729i
936936 0 0
937937 6.92820i 0.226335i 0.993576 + 0.113167i 0.0360996π0.0360996\pi
−0.993576 + 0.113167i 0.963900π0.963900\pi
938938 24.0000 27.7128i 0.783628 0.904855i
939939 36.0000 1.17482
940940 −12.0000 −0.391397
941941 18.0000 0.586783 0.293392 0.955992i 0.405216π-0.405216\pi
0.293392 + 0.955992i 0.405216π0.405216\pi
942942 41.5692i 1.35440i
943943 20.7846i 0.676840i
944944 60.0000 1.95283
945945 9.00000 10.3923i 0.292770 0.338062i
946946 −48.0000 −1.56061
947947 31.1769i 1.01311i 0.862207 + 0.506557i 0.169082π0.169082\pi
−0.862207 + 0.506557i 0.830918π0.830918\pi
948948 13.8564i 0.450035i
949949 0 0
950950 −6.00000 −0.194666
951951 −48.0000 −1.55651
952952 18.0000 20.7846i 0.583383 0.673633i
953953 34.6410i 1.12213i −0.827771 0.561066i 0.810391π-0.810391\pi
0.827771 0.561066i 0.189609π-0.189609\pi
954954 0 0
955955 10.3923i 0.336287i
956956 10.3923i 0.336111i
957957 41.5692i 1.34374i
958958 0 0
959959 36.0000 41.5692i 1.16250 1.34234i
960960 1.73205i 0.0559017i
961961 19.0000 0.612903
962962 0 0
963963 31.1769i 1.00466i
964964 0 0
965965 14.0000 0.450676
966966 −18.0000 + 20.7846i −0.579141 + 0.668734i
967967 20.0000 0.643157 0.321578 0.946883i 0.395787π-0.395787\pi
0.321578 + 0.946883i 0.395787π0.395787\pi
968968 1.73205i 0.0556702i
969969 −36.0000 −1.15649
970970 12.0000 0.385297
971971 60.0000 1.92549 0.962746 0.270408i 0.0871586π-0.0871586\pi
0.962746 + 0.270408i 0.0871586π0.0871586\pi
972972 15.5885i 0.500000i
973973 30.0000 34.6410i 0.961756 1.11054i
974974 6.92820i 0.221994i
975975 0 0
976976 34.6410i 1.10883i
977977 6.92820i 0.221653i −0.993840 0.110826i 0.964650π-0.964650\pi
0.993840 0.110826i 0.0353498π-0.0353498\pi
978978 48.0000 1.53487
979979 20.7846i 0.664279i
980980 1.00000 + 6.92820i 0.0319438 + 0.221313i
981981 6.00000 0.191565
982982 −42.0000 −1.34027
983983 −60.0000 −1.91370 −0.956851 0.290578i 0.906153π-0.906153\pi
−0.956851 + 0.290578i 0.906153π0.906153\pi
984984 −18.0000 −0.573819
985985 13.8564i 0.441502i
986986 72.0000 2.29295
987987 −36.0000 + 41.5692i −1.14589 + 1.32316i
988988 0 0
989989 27.7128i 0.881216i
990990 18.0000 0.572078
991991 16.0000 0.508257 0.254128 0.967170i 0.418211π-0.418211\pi
0.254128 + 0.967170i 0.418211π0.418211\pi
992992 −18.0000 −0.571501
993993 48.4974i 1.53902i
994994 −12.0000 10.3923i −0.380617 0.329624i
995995 10.3923i 0.329458i
996996 0 0
997997 0 0 1.00000 00
−1.00000 π\pi
998998 34.6410i 1.09654i
999999 10.3923i 0.328798i
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 105.2.b.a.41.1 2
3.2 odd 2 105.2.b.b.41.2 yes 2
4.3 odd 2 1680.2.f.b.881.2 2
5.2 odd 4 525.2.g.b.524.4 4
5.3 odd 4 525.2.g.b.524.1 4
5.4 even 2 525.2.b.a.251.2 2
7.2 even 3 735.2.s.b.521.1 2
7.3 odd 6 735.2.s.f.656.1 2
7.4 even 3 735.2.s.d.656.1 2
7.5 odd 6 735.2.s.a.521.1 2
7.6 odd 2 105.2.b.b.41.1 yes 2
12.11 even 2 1680.2.f.c.881.2 2
15.2 even 4 525.2.g.c.524.2 4
15.8 even 4 525.2.g.c.524.3 4
15.14 odd 2 525.2.b.b.251.1 2
21.2 odd 6 735.2.s.f.521.1 2
21.5 even 6 735.2.s.d.521.1 2
21.11 odd 6 735.2.s.a.656.1 2
21.17 even 6 735.2.s.b.656.1 2
21.20 even 2 inner 105.2.b.a.41.2 yes 2
28.27 even 2 1680.2.f.c.881.1 2
35.13 even 4 525.2.g.c.524.1 4
35.27 even 4 525.2.g.c.524.4 4
35.34 odd 2 525.2.b.b.251.2 2
84.83 odd 2 1680.2.f.b.881.1 2
105.62 odd 4 525.2.g.b.524.2 4
105.83 odd 4 525.2.g.b.524.3 4
105.104 even 2 525.2.b.a.251.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.b.a.41.1 2 1.1 even 1 trivial
105.2.b.a.41.2 yes 2 21.20 even 2 inner
105.2.b.b.41.1 yes 2 7.6 odd 2
105.2.b.b.41.2 yes 2 3.2 odd 2
525.2.b.a.251.1 2 105.104 even 2
525.2.b.a.251.2 2 5.4 even 2
525.2.b.b.251.1 2 15.14 odd 2
525.2.b.b.251.2 2 35.34 odd 2
525.2.g.b.524.1 4 5.3 odd 4
525.2.g.b.524.2 4 105.62 odd 4
525.2.g.b.524.3 4 105.83 odd 4
525.2.g.b.524.4 4 5.2 odd 4
525.2.g.c.524.1 4 35.13 even 4
525.2.g.c.524.2 4 15.2 even 4
525.2.g.c.524.3 4 15.8 even 4
525.2.g.c.524.4 4 35.27 even 4
735.2.s.a.521.1 2 7.5 odd 6
735.2.s.a.656.1 2 21.11 odd 6
735.2.s.b.521.1 2 7.2 even 3
735.2.s.b.656.1 2 21.17 even 6
735.2.s.d.521.1 2 21.5 even 6
735.2.s.d.656.1 2 7.4 even 3
735.2.s.f.521.1 2 21.2 odd 6
735.2.s.f.656.1 2 7.3 odd 6
1680.2.f.b.881.1 2 84.83 odd 2
1680.2.f.b.881.2 2 4.3 odd 2
1680.2.f.c.881.1 2 28.27 even 2
1680.2.f.c.881.2 2 12.11 even 2