Properties

Label 2352.2.q.bd.961.1
Level 23522352
Weight 22
Character 2352.961
Analytic conductor 18.78118.781
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2352,2,Mod(961,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) 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("2352.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 2352=24372 2352 = 2^{4} \cdot 3 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2352.q (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,-4,0,0,0,-2,0,-4,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 18.780814555418.7808145554
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(2,3)\Q(\sqrt{2}, \sqrt{-3})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4+2x2+4 x^{4} + 2x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 961.1
Root 0.7071071.22474i-0.707107 - 1.22474i of defining polynomial
Character χ\chi == 2352.961
Dual form 2352.2.q.bd.1537.1

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+(0.500000+0.866025i)q3+(1.70711+2.95680i)q5+(0.500000+0.866025i)q9+(1.000001.73205i)q11+2.58579q133.41421q15+(1.12132+1.94218i)q17+(1.41421+2.44949i)q19+(3.82843+6.63103i)q23+(3.328435.76500i)q251.00000q276.82843q29+(0.5857861.01461i)q31+(1.000001.73205i)q33+(2.000003.46410i)q37+(1.29289+2.23936i)q39+6.24264q415.65685q43+(1.707112.95680i)q45+(1.41421+2.44949i)q47+(1.12132+1.94218i)q51+(1.00000+1.73205i)q53+6.82843q552.82843q57+(0.5857861.01461i)q59+(6.12132+10.6024i)q61+(4.41421+7.64564i)q65+(2.828434.89898i)q677.65685q699.31371q71+(6.9497512.0373i)q73+(3.328435.76500i)q75+(6.8284311.8272i)q79+(0.5000000.866025i)q817.31371q837.65685q85+(3.414215.91359i)q87+(7.1213212.3345i)q89+(0.5857861.01461i)q93+(4.828438.36308i)q95+2.58579q97+2.00000q99+O(q100)q+(0.500000 + 0.866025i) q^{3} +(-1.70711 + 2.95680i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-1.00000 - 1.73205i) q^{11} +2.58579 q^{13} -3.41421 q^{15} +(1.12132 + 1.94218i) q^{17} +(-1.41421 + 2.44949i) q^{19} +(-3.82843 + 6.63103i) q^{23} +(-3.32843 - 5.76500i) q^{25} -1.00000 q^{27} -6.82843 q^{29} +(-0.585786 - 1.01461i) q^{31} +(1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{37} +(1.29289 + 2.23936i) q^{39} +6.24264 q^{41} -5.65685 q^{43} +(-1.70711 - 2.95680i) q^{45} +(-1.41421 + 2.44949i) q^{47} +(-1.12132 + 1.94218i) q^{51} +(1.00000 + 1.73205i) q^{53} +6.82843 q^{55} -2.82843 q^{57} +(-0.585786 - 1.01461i) q^{59} +(-6.12132 + 10.6024i) q^{61} +(-4.41421 + 7.64564i) q^{65} +(-2.82843 - 4.89898i) q^{67} -7.65685 q^{69} -9.31371 q^{71} +(-6.94975 - 12.0373i) q^{73} +(3.32843 - 5.76500i) q^{75} +(6.82843 - 11.8272i) q^{79} +(-0.500000 - 0.866025i) q^{81} -7.31371 q^{83} -7.65685 q^{85} +(-3.41421 - 5.91359i) q^{87} +(7.12132 - 12.3345i) q^{89} +(0.585786 - 1.01461i) q^{93} +(-4.82843 - 8.36308i) q^{95} +2.58579 q^{97} +2.00000 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+2q34q52q94q11+16q138q154q174q232q254q2716q298q31+4q33+8q37+8q39+8q414q45+4q51+4q53++8q99+O(q100) 4 q + 2 q^{3} - 4 q^{5} - 2 q^{9} - 4 q^{11} + 16 q^{13} - 8 q^{15} - 4 q^{17} - 4 q^{23} - 2 q^{25} - 4 q^{27} - 16 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{39} + 8 q^{41} - 4 q^{45} + 4 q^{51} + 4 q^{53}+ \cdots + 8 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2352Z)×\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times.

nn 785785 14711471 17651765 22572257
χ(n)\chi(n) 11 11 11 e(13)e\left(\frac{1}{3}\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 0.500000 + 0.866025i 0.288675 + 0.500000i
44 0 0
55 −1.70711 + 2.95680i −0.763441 + 1.32232i 0.177625 + 0.984098i 0.443158π0.443158\pi
−0.941067 + 0.338221i 0.890175π0.890175\pi
66 0 0
77 0 0
88 0 0
99 −0.500000 + 0.866025i −0.166667 + 0.288675i
1010 0 0
1111 −1.00000 1.73205i −0.301511 0.522233i 0.674967 0.737848i 0.264158π-0.264158\pi
−0.976478 + 0.215615i 0.930824π0.930824\pi
1212 0 0
1313 2.58579 0.717168 0.358584 0.933497i 0.383260π-0.383260\pi
0.358584 + 0.933497i 0.383260π0.383260\pi
1414 0 0
1515 −3.41421 −0.881546
1616 0 0
1717 1.12132 + 1.94218i 0.271960 + 0.471049i 0.969364 0.245630i 0.0789948π-0.0789948\pi
−0.697404 + 0.716679i 0.745661π0.745661\pi
1818 0 0
1919 −1.41421 + 2.44949i −0.324443 + 0.561951i −0.981399 0.191977i 0.938510π-0.938510\pi
0.656957 + 0.753928i 0.271843π0.271843\pi
2020 0 0
2121 0 0
2222 0 0
2323 −3.82843 + 6.63103i −0.798282 + 1.38267i 0.122452 + 0.992474i 0.460924π0.460924\pi
−0.920734 + 0.390191i 0.872409π0.872409\pi
2424 0 0
2525 −3.32843 5.76500i −0.665685 1.15300i
2626 0 0
2727 −1.00000 −0.192450
2828 0 0
2929 −6.82843 −1.26801 −0.634004 0.773330i 0.718590π-0.718590\pi
−0.634004 + 0.773330i 0.718590π0.718590\pi
3030 0 0
3131 −0.585786 1.01461i −0.105210 0.182230i 0.808614 0.588340i 0.200218π-0.200218\pi
−0.913824 + 0.406110i 0.866885π0.866885\pi
3232 0 0
3333 1.00000 1.73205i 0.174078 0.301511i
3434 0 0
3535 0 0
3636 0 0
3737 2.00000 3.46410i 0.328798 0.569495i −0.653476 0.756948i 0.726690π-0.726690\pi
0.982274 + 0.187453i 0.0600231π0.0600231\pi
3838 0 0
3939 1.29289 + 2.23936i 0.207029 + 0.358584i
4040 0 0
4141 6.24264 0.974937 0.487468 0.873141i 0.337920π-0.337920\pi
0.487468 + 0.873141i 0.337920π0.337920\pi
4242 0 0
4343 −5.65685 −0.862662 −0.431331 0.902194i 0.641956π-0.641956\pi
−0.431331 + 0.902194i 0.641956π0.641956\pi
4444 0 0
4545 −1.70711 2.95680i −0.254480 0.440773i
4646 0 0
4747 −1.41421 + 2.44949i −0.206284 + 0.357295i −0.950541 0.310599i 0.899470π-0.899470\pi
0.744257 + 0.667893i 0.232804π0.232804\pi
4848 0 0
4949 0 0
5050 0 0
5151 −1.12132 + 1.94218i −0.157016 + 0.271960i
5252 0 0
5353 1.00000 + 1.73205i 0.137361 + 0.237915i 0.926497 0.376303i 0.122805π-0.122805\pi
−0.789136 + 0.614218i 0.789471π0.789471\pi
5454 0 0
5555 6.82843 0.920745
5656 0 0
5757 −2.82843 −0.374634
5858 0 0
5959 −0.585786 1.01461i −0.0762629 0.132091i 0.825372 0.564589i 0.190965π-0.190965\pi
−0.901635 + 0.432498i 0.857632π0.857632\pi
6060 0 0
6161 −6.12132 + 10.6024i −0.783755 + 1.35750i 0.145985 + 0.989287i 0.453365π0.453365\pi
−0.929740 + 0.368216i 0.879969π0.879969\pi
6262 0 0
6363 0 0
6464 0 0
6565 −4.41421 + 7.64564i −0.547516 + 0.948325i
6666 0 0
6767 −2.82843 4.89898i −0.345547 0.598506i 0.639906 0.768453i 0.278973π-0.278973\pi
−0.985453 + 0.169948i 0.945640π0.945640\pi
6868 0 0
6969 −7.65685 −0.921777
7070 0 0
7171 −9.31371 −1.10533 −0.552667 0.833402i 0.686390π-0.686390\pi
−0.552667 + 0.833402i 0.686390π0.686390\pi
7272 0 0
7373 −6.94975 12.0373i −0.813406 1.40886i −0.910467 0.413583i 0.864277π-0.864277\pi
0.0970601 0.995279i 0.469056π-0.469056\pi
7474 0 0
7575 3.32843 5.76500i 0.384334 0.665685i
7676 0 0
7777 0 0
7878 0 0
7979 6.82843 11.8272i 0.768258 1.33066i −0.170249 0.985401i 0.554457π-0.554457\pi
0.938507 0.345261i 0.112210π-0.112210\pi
8080 0 0
8181 −0.500000 0.866025i −0.0555556 0.0962250i
8282 0 0
8383 −7.31371 −0.802784 −0.401392 0.915906i 0.631473π-0.631473\pi
−0.401392 + 0.915906i 0.631473π0.631473\pi
8484 0 0
8585 −7.65685 −0.830502
8686 0 0
8787 −3.41421 5.91359i −0.366042 0.634004i
8888 0 0
8989 7.12132 12.3345i 0.754858 1.30745i −0.190586 0.981670i 0.561039π-0.561039\pi
0.945445 0.325783i 0.105628π-0.105628\pi
9090 0 0
9191 0 0
9292 0 0
9393 0.585786 1.01461i 0.0607432 0.105210i
9494 0 0
9595 −4.82843 8.36308i −0.495386 0.858034i
9696 0 0
9797 2.58579 0.262547 0.131273 0.991346i 0.458093π-0.458093\pi
0.131273 + 0.991346i 0.458093π0.458093\pi
9898 0 0
9999 2.00000 0.201008
100100 0 0
101101 −1.46447 2.53653i −0.145720 0.252394i 0.783921 0.620860i 0.213216π-0.213216\pi
−0.929641 + 0.368466i 0.879883π0.879883\pi
102102 0 0
103103 2.24264 3.88437i 0.220974 0.382738i −0.734130 0.679009i 0.762410π-0.762410\pi
0.955104 + 0.296271i 0.0957431π0.0957431\pi
104104 0 0
105105 0 0
106106 0 0
107107 −0.171573 + 0.297173i −0.0165866 + 0.0287288i −0.874200 0.485567i 0.838613π-0.838613\pi
0.857613 + 0.514296i 0.171947π0.171947\pi
108108 0 0
109109 2.82843 + 4.89898i 0.270914 + 0.469237i 0.969096 0.246683i 0.0793407π-0.0793407\pi
−0.698182 + 0.715920i 0.746007π0.746007\pi
110110 0 0
111111 4.00000 0.379663
112112 0 0
113113 −5.31371 −0.499872 −0.249936 0.968262i 0.580410π-0.580410\pi
−0.249936 + 0.968262i 0.580410π0.580410\pi
114114 0 0
115115 −13.0711 22.6398i −1.21888 2.11117i
116116 0 0
117117 −1.29289 + 2.23936i −0.119528 + 0.207029i
118118 0 0
119119 0 0
120120 0 0
121121 3.50000 6.06218i 0.318182 0.551107i
122122 0 0
123123 3.12132 + 5.40629i 0.281440 + 0.487468i
124124 0 0
125125 5.65685 0.505964
126126 0 0
127127 1.65685 0.147022 0.0735110 0.997294i 0.476580π-0.476580\pi
0.0735110 + 0.997294i 0.476580π0.476580\pi
128128 0 0
129129 −2.82843 4.89898i −0.249029 0.431331i
130130 0 0
131131 −7.65685 + 13.2621i −0.668982 + 1.15871i 0.309207 + 0.950995i 0.399937π0.399937\pi
−0.978189 + 0.207717i 0.933397π0.933397\pi
132132 0 0
133133 0 0
134134 0 0
135135 1.70711 2.95680i 0.146924 0.254480i
136136 0 0
137137 −7.07107 12.2474i −0.604122 1.04637i −0.992190 0.124739i 0.960191π-0.960191\pi
0.388067 0.921631i 0.373143π-0.373143\pi
138138 0 0
139139 17.6569 1.49763 0.748817 0.662776i 0.230622π-0.230622\pi
0.748817 + 0.662776i 0.230622π0.230622\pi
140140 0 0
141141 −2.82843 −0.238197
142142 0 0
143143 −2.58579 4.47871i −0.216234 0.374529i
144144 0 0
145145 11.6569 20.1903i 0.968049 1.67671i
146146 0 0
147147 0 0
148148 0 0
149149 −8.65685 + 14.9941i −0.709197 + 1.22837i 0.255958 + 0.966688i 0.417609π0.417609\pi
−0.965155 + 0.261678i 0.915724π0.915724\pi
150150 0 0
151151 6.00000 + 10.3923i 0.488273 + 0.845714i 0.999909 0.0134886i 0.00429367π-0.00429367\pi
−0.511636 + 0.859202i 0.670960π0.670960\pi
152152 0 0
153153 −2.24264 −0.181307
154154 0 0
155155 4.00000 0.321288
156156 0 0
157157 −5.87868 10.1822i −0.469170 0.812626i 0.530209 0.847867i 0.322113π-0.322113\pi
−0.999379 + 0.0352411i 0.988780π0.988780\pi
158158 0 0
159159 −1.00000 + 1.73205i −0.0793052 + 0.137361i
160160 0 0
161161 0 0
162162 0 0
163163 −5.65685 + 9.79796i −0.443079 + 0.767435i −0.997916 0.0645236i 0.979447π-0.979447\pi
0.554837 + 0.831959i 0.312781π0.312781\pi
164164 0 0
165165 3.41421 + 5.91359i 0.265796 + 0.460372i
166166 0 0
167167 19.7990 1.53209 0.766046 0.642786i 0.222221π-0.222221\pi
0.766046 + 0.642786i 0.222221π0.222221\pi
168168 0 0
169169 −6.31371 −0.485670
170170 0 0
171171 −1.41421 2.44949i −0.108148 0.187317i
172172 0 0
173173 −10.5355 + 18.2481i −0.801002 + 1.38738i 0.117956 + 0.993019i 0.462366π0.462366\pi
−0.918957 + 0.394357i 0.870967π0.870967\pi
174174 0 0
175175 0 0
176176 0 0
177177 0.585786 1.01461i 0.0440304 0.0762629i
178178 0 0
179179 −9.82843 17.0233i −0.734611 1.27238i −0.954894 0.296948i 0.904031π-0.904031\pi
0.220283 0.975436i 0.429302π-0.429302\pi
180180 0 0
181181 −2.58579 −0.192200 −0.0961000 0.995372i 0.530637π-0.530637\pi
−0.0961000 + 0.995372i 0.530637π0.530637\pi
182182 0 0
183183 −12.2426 −0.905002
184184 0 0
185185 6.82843 + 11.8272i 0.502036 + 0.869552i
186186 0 0
187187 2.24264 3.88437i 0.163998 0.284053i
188188 0 0
189189 0 0
190190 0 0
191191 −9.00000 + 15.5885i −0.651217 + 1.12794i 0.331611 + 0.943416i 0.392408π0.392408\pi
−0.982828 + 0.184525i 0.940925π0.940925\pi
192192 0 0
193193 −2.65685 4.60181i −0.191245 0.331245i 0.754418 0.656394i 0.227919π-0.227919\pi
−0.945663 + 0.325149i 0.894586π0.894586\pi
194194 0 0
195195 −8.82843 −0.632217
196196 0 0
197197 2.00000 0.142494 0.0712470 0.997459i 0.477302π-0.477302\pi
0.0712470 + 0.997459i 0.477302π0.477302\pi
198198 0 0
199199 10.8284 + 18.7554i 0.767607 + 1.32953i 0.938857 + 0.344307i 0.111886π0.111886\pi
−0.171250 + 0.985228i 0.554781π0.554781\pi
200200 0 0
201201 2.82843 4.89898i 0.199502 0.345547i
202202 0 0
203203 0 0
204204 0 0
205205 −10.6569 + 18.4582i −0.744307 + 1.28918i
206206 0 0
207207 −3.82843 6.63103i −0.266094 0.460888i
208208 0 0
209209 5.65685 0.391293
210210 0 0
211211 −12.9706 −0.892930 −0.446465 0.894801i 0.647317π-0.647317\pi
−0.446465 + 0.894801i 0.647317π0.647317\pi
212212 0 0
213213 −4.65685 8.06591i −0.319082 0.552667i
214214 0 0
215215 9.65685 16.7262i 0.658592 1.14071i
216216 0 0
217217 0 0
218218 0 0
219219 6.94975 12.0373i 0.469620 0.813406i
220220 0 0
221221 2.89949 + 5.02207i 0.195041 + 0.337821i
222222 0 0
223223 −24.9706 −1.67215 −0.836076 0.548613i 0.815156π-0.815156\pi
−0.836076 + 0.548613i 0.815156π0.815156\pi
224224 0 0
225225 6.65685 0.443790
226226 0 0
227227 11.8995 + 20.6105i 0.789797 + 1.36797i 0.926091 + 0.377300i 0.123148π0.123148\pi
−0.136294 + 0.990668i 0.543519π0.543519\pi
228228 0 0
229229 0.121320 0.210133i 0.00801707 0.0138860i −0.861989 0.506927i 0.830781π-0.830781\pi
0.870006 + 0.493041i 0.164115π0.164115\pi
230230 0 0
231231 0 0
232232 0 0
233233 3.07107 5.31925i 0.201192 0.348475i −0.747721 0.664014i 0.768852π-0.768852\pi
0.948913 + 0.315538i 0.102185π0.102185\pi
234234 0 0
235235 −4.82843 8.36308i −0.314972 0.545547i
236236 0 0
237237 13.6569 0.887108
238238 0 0
239239 15.6569 1.01276 0.506379 0.862311i 0.330984π-0.330984\pi
0.506379 + 0.862311i 0.330984π0.330984\pi
240240 0 0
241241 8.12132 + 14.0665i 0.523140 + 0.906105i 0.999637 + 0.0269294i 0.00857294π0.00857294\pi
−0.476497 + 0.879176i 0.658094π0.658094\pi
242242 0 0
243243 0.500000 0.866025i 0.0320750 0.0555556i
244244 0 0
245245 0 0
246246 0 0
247247 −3.65685 + 6.33386i −0.232680 + 0.403014i
248248 0 0
249249 −3.65685 6.33386i −0.231744 0.401392i
250250 0 0
251251 12.4853 0.788064 0.394032 0.919097i 0.371080π-0.371080\pi
0.394032 + 0.919097i 0.371080π0.371080\pi
252252 0 0
253253 15.3137 0.962765
254254 0 0
255255 −3.82843 6.63103i −0.239745 0.415251i
256256 0 0
257257 −11.6066 + 20.1032i −0.724000 + 1.25400i 0.235384 + 0.971902i 0.424365π0.424365\pi
−0.959384 + 0.282102i 0.908968π0.908968\pi
258258 0 0
259259 0 0
260260 0 0
261261 3.41421 5.91359i 0.211335 0.366042i
262262 0 0
263263 2.65685 + 4.60181i 0.163829 + 0.283760i 0.936239 0.351365i 0.114282π-0.114282\pi
−0.772410 + 0.635124i 0.780949π0.780949\pi
264264 0 0
265265 −6.82843 −0.419467
266266 0 0
267267 14.2426 0.871635
268268 0 0
269269 7.36396 + 12.7548i 0.448989 + 0.777671i 0.998320 0.0579332i 0.0184510π-0.0184510\pi
−0.549332 + 0.835604i 0.685118π0.685118\pi
270270 0 0
271271 −5.07107 + 8.78335i −0.308045 + 0.533550i −0.977935 0.208911i 0.933008π-0.933008\pi
0.669889 + 0.742461i 0.266342π0.266342\pi
272272 0 0
273273 0 0
274274 0 0
275275 −6.65685 + 11.5300i −0.401423 + 0.695286i
276276 0 0
277277 4.65685 + 8.06591i 0.279803 + 0.484633i 0.971336 0.237712i 0.0763974π-0.0763974\pi
−0.691532 + 0.722345i 0.743064π0.743064\pi
278278 0 0
279279 1.17157 0.0701402
280280 0 0
281281 0.485281 0.0289495 0.0144747 0.999895i 0.495392π-0.495392\pi
0.0144747 + 0.999895i 0.495392π0.495392\pi
282282 0 0
283283 −4.24264 7.34847i −0.252199 0.436821i 0.711932 0.702248i 0.247820π-0.247820\pi
−0.964131 + 0.265427i 0.914487π0.914487\pi
284284 0 0
285285 4.82843 8.36308i 0.286011 0.495386i
286286 0 0
287287 0 0
288288 0 0
289289 5.98528 10.3668i 0.352075 0.609812i
290290 0 0
291291 1.29289 + 2.23936i 0.0757907 + 0.131273i
292292 0 0
293293 16.5858 0.968952 0.484476 0.874805i 0.339010π-0.339010\pi
0.484476 + 0.874805i 0.339010π0.339010\pi
294294 0 0
295295 4.00000 0.232889
296296 0 0
297297 1.00000 + 1.73205i 0.0580259 + 0.100504i
298298 0 0
299299 −9.89949 + 17.1464i −0.572503 + 0.991604i
300300 0 0
301301 0 0
302302 0 0
303303 1.46447 2.53653i 0.0841314 0.145720i
304304 0 0
305305 −20.8995 36.1990i −1.19670 2.07275i
306306 0 0
307307 −30.1421 −1.72030 −0.860151 0.510039i 0.829631π-0.829631\pi
−0.860151 + 0.510039i 0.829631π0.829631\pi
308308 0 0
309309 4.48528 0.255159
310310 0 0
311311 3.07107 + 5.31925i 0.174144 + 0.301627i 0.939865 0.341547i 0.110951π-0.110951\pi
−0.765721 + 0.643173i 0.777617π0.777617\pi
312312 0 0
313313 −0.949747 + 1.64501i −0.0536829 + 0.0929815i −0.891618 0.452788i 0.850429π-0.850429\pi
0.837935 + 0.545770i 0.183763π0.183763\pi
314314 0 0
315315 0 0
316316 0 0
317317 −5.00000 + 8.66025i −0.280828 + 0.486408i −0.971589 0.236675i 0.923942π-0.923942\pi
0.690761 + 0.723083i 0.257276π0.257276\pi
318318 0 0
319319 6.82843 + 11.8272i 0.382319 + 0.662195i
320320 0 0
321321 −0.343146 −0.0191525
322322 0 0
323323 −6.34315 −0.352942
324324 0 0
325325 −8.60660 14.9071i −0.477408 0.826896i
326326 0 0
327327 −2.82843 + 4.89898i −0.156412 + 0.270914i
328328 0 0
329329 0 0
330330 0 0
331331 −2.00000 + 3.46410i −0.109930 + 0.190404i −0.915742 0.401768i 0.868396π-0.868396\pi
0.805812 + 0.592172i 0.201729π0.201729\pi
332332 0 0
333333 2.00000 + 3.46410i 0.109599 + 0.189832i
334334 0 0
335335 19.3137 1.05522
336336 0 0
337337 −29.6569 −1.61551 −0.807756 0.589517i 0.799318π-0.799318\pi
−0.807756 + 0.589517i 0.799318π0.799318\pi
338338 0 0
339339 −2.65685 4.60181i −0.144301 0.249936i
340340 0 0
341341 −1.17157 + 2.02922i −0.0634442 + 0.109889i
342342 0 0
343343 0 0
344344 0 0
345345 13.0711 22.6398i 0.703723 1.21888i
346346 0 0
347347 16.6569 + 28.8505i 0.894187 + 1.54878i 0.834808 + 0.550541i 0.185579π0.185579\pi
0.0593789 + 0.998236i 0.481088π0.481088\pi
348348 0 0
349349 −9.89949 −0.529908 −0.264954 0.964261i 0.585357π-0.585357\pi
−0.264954 + 0.964261i 0.585357π0.585357\pi
350350 0 0
351351 −2.58579 −0.138019
352352 0 0
353353 7.36396 + 12.7548i 0.391944 + 0.678867i 0.992706 0.120561i 0.0384693π-0.0384693\pi
−0.600762 + 0.799428i 0.705136π0.705136\pi
354354 0 0
355355 15.8995 27.5387i 0.843858 1.46160i
356356 0 0
357357 0 0
358358 0 0
359359 −0.171573 + 0.297173i −0.00905527 + 0.0156842i −0.870518 0.492137i 0.836216π-0.836216\pi
0.861462 + 0.507822i 0.169549π0.169549\pi
360360 0 0
361361 5.50000 + 9.52628i 0.289474 + 0.501383i
362362 0 0
363363 7.00000 0.367405
364364 0 0
365365 47.4558 2.48395
366366 0 0
367367 −1.65685 2.86976i −0.0864871 0.149800i 0.819537 0.573027i 0.194231π-0.194231\pi
−0.906024 + 0.423226i 0.860897π0.860897\pi
368368 0 0
369369 −3.12132 + 5.40629i −0.162489 + 0.281440i
370370 0 0
371371 0 0
372372 0 0
373373 5.34315 9.25460i 0.276658 0.479185i −0.693894 0.720077i 0.744107π-0.744107\pi
0.970552 + 0.240892i 0.0774399π0.0774399\pi
374374 0 0
375375 2.82843 + 4.89898i 0.146059 + 0.252982i
376376 0 0
377377 −17.6569 −0.909374
378378 0 0
379379 −8.68629 −0.446185 −0.223092 0.974797i 0.571615π-0.571615\pi
−0.223092 + 0.974797i 0.571615π0.571615\pi
380380 0 0
381381 0.828427 + 1.43488i 0.0424416 + 0.0735110i
382382 0 0
383383 −9.17157 + 15.8856i −0.468645 + 0.811718i −0.999358 0.0358343i 0.988591π-0.988591\pi
0.530712 + 0.847552i 0.321924π0.321924\pi
384384 0 0
385385 0 0
386386 0 0
387387 2.82843 4.89898i 0.143777 0.249029i
388388 0 0
389389 9.07107 + 15.7116i 0.459921 + 0.796607i 0.998956 0.0456762i 0.0145442π-0.0145442\pi
−0.539035 + 0.842283i 0.681211π0.681211\pi
390390 0 0
391391 −17.1716 −0.868404
392392 0 0
393393 −15.3137 −0.772474
394394 0 0
395395 23.3137 + 40.3805i 1.17304 + 2.03176i
396396 0 0
397397 1.19239 2.06528i 0.0598442 0.103653i −0.834551 0.550931i 0.814273π-0.814273\pi
0.894395 + 0.447277i 0.147606π0.147606\pi
398398 0 0
399399 0 0
400400 0 0
401401 3.07107 5.31925i 0.153362 0.265630i −0.779100 0.626900i 0.784323π-0.784323\pi
0.932461 + 0.361270i 0.117657π0.117657\pi
402402 0 0
403403 −1.51472 2.62357i −0.0754535 0.130689i
404404 0 0
405405 3.41421 0.169654
406406 0 0
407407 −8.00000 −0.396545
408408 0 0
409409 10.7071 + 18.5453i 0.529432 + 0.917004i 0.999411 + 0.0343258i 0.0109284π0.0109284\pi
−0.469978 + 0.882678i 0.655738π0.655738\pi
410410 0 0
411411 7.07107 12.2474i 0.348790 0.604122i
412412 0 0
413413 0 0
414414 0 0
415415 12.4853 21.6251i 0.612878 1.06154i
416416 0 0
417417 8.82843 + 15.2913i 0.432330 + 0.748817i
418418 0 0
419419 −33.1716 −1.62054 −0.810269 0.586059i 0.800679π-0.800679\pi
−0.810269 + 0.586059i 0.800679π0.800679\pi
420420 0 0
421421 16.6274 0.810371 0.405185 0.914235i 0.367207π-0.367207\pi
0.405185 + 0.914235i 0.367207π0.367207\pi
422422 0 0
423423 −1.41421 2.44949i −0.0687614 0.119098i
424424 0 0
425425 7.46447 12.9288i 0.362080 0.627141i
426426 0 0
427427 0 0
428428 0 0
429429 2.58579 4.47871i 0.124843 0.216234i
430430 0 0
431431 −13.4853 23.3572i −0.649563 1.12508i −0.983227 0.182384i 0.941618π-0.941618\pi
0.333664 0.942692i 0.391715π-0.391715\pi
432432 0 0
433433 −20.2426 −0.972799 −0.486400 0.873736i 0.661690π-0.661690\pi
−0.486400 + 0.873736i 0.661690π0.661690\pi
434434 0 0
435435 23.3137 1.11781
436436 0 0
437437 −10.8284 18.7554i −0.517994 0.897192i
438438 0 0
439439 6.34315 10.9867i 0.302742 0.524364i −0.674014 0.738718i 0.735431π-0.735431\pi
0.976756 + 0.214354i 0.0687647π0.0687647\pi
440440 0 0
441441 0 0
442442 0 0
443443 17.4853 30.2854i 0.830751 1.43890i −0.0666929 0.997774i 0.521245π-0.521245\pi
0.897444 0.441129i 0.145422π-0.145422\pi
444444 0 0
445445 24.3137 + 42.1126i 1.15258 + 1.99633i
446446 0 0
447447 −17.3137 −0.818910
448448 0 0
449449 −5.31371 −0.250769 −0.125385 0.992108i 0.540017π-0.540017\pi
−0.125385 + 0.992108i 0.540017π0.540017\pi
450450 0 0
451451 −6.24264 10.8126i −0.293954 0.509144i
452452 0 0
453453 −6.00000 + 10.3923i −0.281905 + 0.488273i
454454 0 0
455455 0 0
456456 0 0
457457 9.00000 15.5885i 0.421002 0.729197i −0.575036 0.818128i 0.695012π-0.695012\pi
0.996038 + 0.0889312i 0.0283451π0.0283451\pi
458458 0 0
459459 −1.12132 1.94218i −0.0523388 0.0906534i
460460 0 0
461461 −16.5858 −0.772477 −0.386239 0.922399i 0.626226π-0.626226\pi
−0.386239 + 0.922399i 0.626226π0.626226\pi
462462 0 0
463463 26.6274 1.23748 0.618741 0.785595i 0.287643π-0.287643\pi
0.618741 + 0.785595i 0.287643π0.287643\pi
464464 0 0
465465 2.00000 + 3.46410i 0.0927478 + 0.160644i
466466 0 0
467467 0.100505 0.174080i 0.00465082 0.00805546i −0.863691 0.504022i 0.831853π-0.831853\pi
0.868341 + 0.495967i 0.165186π0.165186\pi
468468 0 0
469469 0 0
470470 0 0
471471 5.87868 10.1822i 0.270875 0.469170i
472472 0 0
473473 5.65685 + 9.79796i 0.260102 + 0.450511i
474474 0 0
475475 18.8284 0.863907
476476 0 0
477477 −2.00000 −0.0915737
478478 0 0
479479 0.928932 + 1.60896i 0.0424440 + 0.0735152i 0.886467 0.462792i 0.153152π-0.153152\pi
−0.844023 + 0.536307i 0.819819π0.819819\pi
480480 0 0
481481 5.17157 8.95743i 0.235803 0.408424i
482482 0 0
483483 0 0
484484 0 0
485485 −4.41421 + 7.64564i −0.200439 + 0.347171i
486486 0 0
487487 13.3137 + 23.0600i 0.603302 + 1.04495i 0.992317 + 0.123718i 0.0394819π0.0394819\pi
−0.389016 + 0.921231i 0.627185π0.627185\pi
488488 0 0
489489 −11.3137 −0.511624
490490 0 0
491491 −5.02944 −0.226975 −0.113488 0.993539i 0.536202π-0.536202\pi
−0.113488 + 0.993539i 0.536202π0.536202\pi
492492 0 0
493493 −7.65685 13.2621i −0.344847 0.597293i
494494 0 0
495495 −3.41421 + 5.91359i −0.153457 + 0.265796i
496496 0 0
497497 0 0
498498 0 0
499499 1.65685 2.86976i 0.0741710 0.128468i −0.826554 0.562857i 0.809702π-0.809702\pi
0.900725 + 0.434389i 0.143036π0.143036\pi
500500 0 0
501501 9.89949 + 17.1464i 0.442277 + 0.766046i
502502 0 0
503503 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
504504 0 0
505505 10.0000 0.444994
506506 0 0
507507 −3.15685 5.46783i −0.140201 0.242835i
508508 0 0
509509 −2.77817 + 4.81194i −0.123140 + 0.213285i −0.921005 0.389552i 0.872630π-0.872630\pi
0.797864 + 0.602837i 0.205963π0.205963\pi
510510 0 0
511511 0 0
512512 0 0
513513 1.41421 2.44949i 0.0624391 0.108148i
514514 0 0
515515 7.65685 + 13.2621i 0.337401 + 0.584396i
516516 0 0
517517 5.65685 0.248788
518518 0 0
519519 −21.0711 −0.924917
520520 0 0
521521 17.7071 + 30.6696i 0.775762 + 1.34366i 0.934365 + 0.356317i 0.115968π0.115968\pi
−0.158603 + 0.987343i 0.550699π0.550699\pi
522522 0 0
523523 −12.8284 + 22.2195i −0.560948 + 0.971590i 0.436466 + 0.899721i 0.356230π0.356230\pi
−0.997414 + 0.0718696i 0.977103π0.977103\pi
524524 0 0
525525 0 0
526526 0 0
527527 1.31371 2.27541i 0.0572260 0.0991184i
528528 0 0
529529 −17.8137 30.8542i −0.774509 1.34149i
530530 0 0
531531 1.17157 0.0508419
532532 0 0
533533 16.1421 0.699194
534534 0 0
535535 −0.585786 1.01461i −0.0253258 0.0438655i
536536 0 0
537537 9.82843 17.0233i 0.424128 0.734611i
538538 0 0
539539 0 0
540540 0 0
541541 −8.65685 + 14.9941i −0.372187 + 0.644647i −0.989902 0.141755i 0.954725π-0.954725\pi
0.617715 + 0.786402i 0.288059π0.288059\pi
542542 0 0
543543 −1.29289 2.23936i −0.0554834 0.0961000i
544544 0 0
545545 −19.3137 −0.827308
546546 0 0
547547 36.9706 1.58075 0.790374 0.612625i 0.209886π-0.209886\pi
0.790374 + 0.612625i 0.209886π0.209886\pi
548548 0 0
549549 −6.12132 10.6024i −0.261252 0.452501i
550550 0 0
551551 9.65685 16.7262i 0.411396 0.712558i
552552 0 0
553553 0 0
554554 0 0
555555 −6.82843 + 11.8272i −0.289851 + 0.502036i
556556 0 0
557557 −13.0000 22.5167i −0.550828 0.954062i −0.998215 0.0597213i 0.980979π-0.980979\pi
0.447387 0.894340i 0.352355π-0.352355\pi
558558 0 0
559559 −14.6274 −0.618674
560560 0 0
561561 4.48528 0.189369
562562 0 0
563563 −0.585786 1.01461i −0.0246880 0.0427608i 0.853417 0.521228i 0.174526π-0.174526\pi
−0.878105 + 0.478467i 0.841193π0.841193\pi
564564 0 0
565565 9.07107 15.7116i 0.381623 0.660990i
566566 0 0
567567 0 0
568568 0 0
569569 8.24264 14.2767i 0.345549 0.598509i −0.639904 0.768455i 0.721026π-0.721026\pi
0.985453 + 0.169946i 0.0543592π0.0543592\pi
570570 0 0
571571 11.1716 + 19.3497i 0.467516 + 0.809761i 0.999311 0.0371118i 0.0118158π-0.0118158\pi
−0.531795 + 0.846873i 0.678482π0.678482\pi
572572 0 0
573573 −18.0000 −0.751961
574574 0 0
575575 50.9706 2.12562
576576 0 0
577577 16.9497 + 29.3578i 0.705627 + 1.22218i 0.966465 + 0.256799i 0.0826680π0.0826680\pi
−0.260837 + 0.965383i 0.583999π0.583999\pi
578578 0 0
579579 2.65685 4.60181i 0.110415 0.191245i
580580 0 0
581581 0 0
582582 0 0
583583 2.00000 3.46410i 0.0828315 0.143468i
584584 0 0
585585 −4.41421 7.64564i −0.182505 0.316108i
586586 0 0
587587 22.8284 0.942230 0.471115 0.882072i 0.343852π-0.343852\pi
0.471115 + 0.882072i 0.343852π0.343852\pi
588588 0 0
589589 3.31371 0.136539
590590 0 0
591591 1.00000 + 1.73205i 0.0411345 + 0.0712470i
592592 0 0
593593 3.46447 6.00063i 0.142269 0.246416i −0.786082 0.618122i 0.787894π-0.787894\pi
0.928351 + 0.371706i 0.121227π0.121227\pi
594594 0 0
595595 0 0
596596 0 0
597597 −10.8284 + 18.7554i −0.443178 + 0.767607i
598598 0 0
599599 −1.00000 1.73205i −0.0408589 0.0707697i 0.844873 0.534967i 0.179676π-0.179676\pi
−0.885732 + 0.464198i 0.846343π0.846343\pi
600600 0 0
601601 −15.0711 −0.614762 −0.307381 0.951587i 0.599453π-0.599453\pi
−0.307381 + 0.951587i 0.599453π0.599453\pi
602602 0 0
603603 5.65685 0.230365
604604 0 0
605605 11.9497 + 20.6976i 0.485826 + 0.841476i
606606 0 0
607607 −9.17157 + 15.8856i −0.372263 + 0.644778i −0.989913 0.141675i 0.954751π-0.954751\pi
0.617651 + 0.786453i 0.288085π0.288085\pi
608608 0 0
609609 0 0
610610 0 0
611611 −3.65685 + 6.33386i −0.147940 + 0.256240i
612612 0 0
613613 −2.34315 4.05845i −0.0946388 0.163919i 0.814819 0.579715i 0.196836π-0.196836\pi
−0.909458 + 0.415796i 0.863503π0.863503\pi
614614 0 0
615615 −21.3137 −0.859452
616616 0 0
617617 −24.4853 −0.985740 −0.492870 0.870103i 0.664052π-0.664052\pi
−0.492870 + 0.870103i 0.664052π0.664052\pi
618618 0 0
619619 14.4853 + 25.0892i 0.582213 + 1.00842i 0.995217 + 0.0976926i 0.0311462π0.0311462\pi
−0.413004 + 0.910729i 0.635520π0.635520\pi
620620 0 0
621621 3.82843 6.63103i 0.153629 0.266094i
622622 0 0
623623 0 0
624624 0 0
625625 6.98528 12.0989i 0.279411 0.483954i
626626 0 0
627627 2.82843 + 4.89898i 0.112956 + 0.195646i
628628 0 0
629629 8.97056 0.357680
630630 0 0
631631 −23.3137 −0.928104 −0.464052 0.885808i 0.653605π-0.653605\pi
−0.464052 + 0.885808i 0.653605π0.653605\pi
632632 0 0
633633 −6.48528 11.2328i −0.257767 0.446465i
634634 0 0
635635 −2.82843 + 4.89898i −0.112243 + 0.194410i
636636 0 0
637637 0 0
638638 0 0
639639 4.65685 8.06591i 0.184222 0.319082i
640640 0 0
641641 5.41421 + 9.37769i 0.213849 + 0.370397i 0.952916 0.303235i 0.0980668π-0.0980668\pi
−0.739067 + 0.673632i 0.764733π0.764733\pi
642642 0 0
643643 34.4264 1.35764 0.678822 0.734302i 0.262491π-0.262491\pi
0.678822 + 0.734302i 0.262491π0.262491\pi
644644 0 0
645645 19.3137 0.760477
646646 0 0
647647 13.4142 + 23.2341i 0.527367 + 0.913427i 0.999491 + 0.0318946i 0.0101541π0.0101541\pi
−0.472124 + 0.881532i 0.656513π0.656513\pi
648648 0 0
649649 −1.17157 + 2.02922i −0.0459883 + 0.0796540i
650650 0 0
651651 0 0
652652 0 0
653653 −18.2426 + 31.5972i −0.713890 + 1.23649i 0.249497 + 0.968376i 0.419735π0.419735\pi
−0.963386 + 0.268118i 0.913598π0.913598\pi
654654 0 0
655655 −26.1421 45.2795i −1.02146 1.76922i
656656 0 0
657657 13.8995 0.542271
658658 0 0
659659 −9.31371 −0.362811 −0.181405 0.983408i 0.558065π-0.558065\pi
−0.181405 + 0.983408i 0.558065π0.558065\pi
660660 0 0
661661 11.7782 + 20.4004i 0.458118 + 0.793483i 0.998862 0.0477040i 0.0151904π-0.0151904\pi
−0.540744 + 0.841187i 0.681857π0.681857\pi
662662 0 0
663663 −2.89949 + 5.02207i −0.112607 + 0.195041i
664664 0 0
665665 0 0
666666 0 0
667667 26.1421 45.2795i 1.01223 1.75323i
668668 0 0
669669 −12.4853 21.6251i −0.482709 0.836076i
670670 0 0
671671 24.4853 0.945244
672672 0 0
673673 23.3137 0.898677 0.449339 0.893361i 0.351660π-0.351660\pi
0.449339 + 0.893361i 0.351660π0.351660\pi
674674 0 0
675675 3.32843 + 5.76500i 0.128111 + 0.221895i
676676 0 0
677677 −15.7071 + 27.2055i −0.603673 + 1.04559i 0.388587 + 0.921412i 0.372963π0.372963\pi
−0.992260 + 0.124180i 0.960370π0.960370\pi
678678 0 0
679679 0 0
680680 0 0
681681 −11.8995 + 20.6105i −0.455990 + 0.789797i
682682 0 0
683683 −9.82843 17.0233i −0.376074 0.651380i 0.614413 0.788985i 0.289393π-0.289393\pi
−0.990487 + 0.137605i 0.956060π0.956060\pi
684684 0 0
685685 48.2843 1.84485
686686 0 0
687687 0.242641 0.00925732
688688 0 0
689689 2.58579 + 4.47871i 0.0985106 + 0.170625i
690690 0 0
691691 −0.343146 + 0.594346i −0.0130539 + 0.0226100i −0.872479 0.488652i 0.837489π-0.837489\pi
0.859425 + 0.511262i 0.170822π0.170822\pi
692692 0 0
693693 0 0
694694 0 0
695695 −30.1421 + 52.2077i −1.14336 + 1.98035i
696696 0 0
697697 7.00000 + 12.1244i 0.265144 + 0.459243i
698698 0 0
699699 6.14214 0.232317
700700 0 0
701701 −17.1716 −0.648561 −0.324281 0.945961i 0.605122π-0.605122\pi
−0.324281 + 0.945961i 0.605122π0.605122\pi
702702 0 0
703703 5.65685 + 9.79796i 0.213352 + 0.369537i
704704 0 0
705705 4.82843 8.36308i 0.181849 0.314972i
706706 0 0
707707 0 0
708708 0 0
709709 18.1421 31.4231i 0.681342 1.18012i −0.293229 0.956042i 0.594730π-0.594730\pi
0.974571 0.224077i 0.0719368π-0.0719368\pi
710710 0 0
711711 6.82843 + 11.8272i 0.256086 + 0.443554i
712712 0 0
713713 8.97056 0.335950
714714 0 0
715715 17.6569 0.660329
716716 0 0
717717 7.82843 + 13.5592i 0.292358 + 0.506379i
718718 0 0
719719 20.9706 36.3221i 0.782070 1.35459i −0.148664 0.988888i 0.547497π-0.547497\pi
0.930734 0.365697i 0.119169π-0.119169\pi
720720 0 0
721721 0 0
722722 0 0
723723 −8.12132 + 14.0665i −0.302035 + 0.523140i
724724 0 0
725725 22.7279 + 39.3659i 0.844094 + 1.46201i
726726 0 0
727727 −12.4853 −0.463053 −0.231527 0.972829i 0.574372π-0.574372\pi
−0.231527 + 0.972829i 0.574372π0.574372\pi
728728 0 0
729729 1.00000 0.0370370
730730 0 0
731731 −6.34315 10.9867i −0.234610 0.406356i
732732 0 0
733733 −24.8492 + 43.0402i −0.917828 + 1.58972i −0.115120 + 0.993352i 0.536725π0.536725\pi
−0.802708 + 0.596373i 0.796608π0.796608\pi
734734 0 0
735735 0 0
736736 0 0
737737 −5.65685 + 9.79796i −0.208373 + 0.360912i
738738 0 0
739739 2.34315 + 4.05845i 0.0861940 + 0.149292i 0.905899 0.423493i 0.139196π-0.139196\pi
−0.819705 + 0.572785i 0.805863π0.805863\pi
740740 0 0
741741 −7.31371 −0.268676
742742 0 0
743743 50.9706 1.86993 0.934964 0.354742i 0.115431π-0.115431\pi
0.934964 + 0.354742i 0.115431π0.115431\pi
744744 0 0
745745 −29.5563 51.1931i −1.08286 1.87557i
746746 0 0
747747 3.65685 6.33386i 0.133797 0.231744i
748748 0 0
749749 0 0
750750 0 0
751751 6.82843 11.8272i 0.249173 0.431580i −0.714124 0.700020i 0.753175π-0.753175\pi
0.963297 + 0.268440i 0.0865079π0.0865079\pi
752752 0 0
753753 6.24264 + 10.8126i 0.227494 + 0.394032i
754754 0 0
755755 −40.9706 −1.49107
756756 0 0
757757 26.3431 0.957458 0.478729 0.877963i 0.341098π-0.341098\pi
0.478729 + 0.877963i 0.341098π0.341098\pi
758758 0 0
759759 7.65685 + 13.2621i 0.277926 + 0.481382i
760760 0 0
761761 9.26346 16.0448i 0.335800 0.581623i −0.647838 0.761778i 0.724327π-0.724327\pi
0.983638 + 0.180155i 0.0576600π0.0576600\pi
762762 0 0
763763 0 0
764764 0 0
765765 3.82843 6.63103i 0.138417 0.239745i
766766 0 0
767767 −1.51472 2.62357i −0.0546933 0.0947316i
768768 0 0
769769 29.6985 1.07095 0.535477 0.844550i 0.320132π-0.320132\pi
0.535477 + 0.844550i 0.320132π0.320132\pi
770770 0 0
771771 −23.2132 −0.836003
772772 0 0
773773 4.77817 + 8.27604i 0.171859 + 0.297669i 0.939070 0.343727i 0.111689π-0.111689\pi
−0.767211 + 0.641395i 0.778356π0.778356\pi
774774 0 0
775775 −3.89949 + 6.75412i −0.140074 + 0.242615i
776776 0 0
777777 0 0
778778 0 0
779779 −8.82843 + 15.2913i −0.316311 + 0.547867i
780780 0 0
781781 9.31371 + 16.1318i 0.333271 + 0.577242i
782782 0 0
783783 6.82843 0.244028
784784 0 0
785785 40.1421 1.43273
786786 0 0
787787 12.3431 + 21.3790i 0.439986 + 0.762077i 0.997688 0.0679637i 0.0216502π-0.0216502\pi
−0.557702 + 0.830041i 0.688317π0.688317\pi
788788 0 0
789789 −2.65685 + 4.60181i −0.0945865 + 0.163829i
790790 0 0
791791 0 0
792792 0 0
793793 −15.8284 + 27.4156i −0.562084 + 0.973558i
794794 0 0
795795 −3.41421 5.91359i −0.121090 0.209733i
796796 0 0
797797 −8.38478 −0.297004 −0.148502 0.988912i 0.547445π-0.547445\pi
−0.148502 + 0.988912i 0.547445π0.547445\pi
798798 0 0
799799 −6.34315 −0.224404
800800 0 0
801801 7.12132 + 12.3345i 0.251619 + 0.435818i
802802 0 0
803803 −13.8995 + 24.0746i −0.490503 + 0.849575i
804804 0 0
805805 0 0
806806 0 0
807807 −7.36396 + 12.7548i −0.259224 + 0.448989i
808808 0 0
809809 −9.97056 17.2695i −0.350546 0.607164i 0.635799 0.771855i 0.280671π-0.280671\pi
−0.986345 + 0.164691i 0.947337π0.947337\pi
810810 0 0
811811 17.6569 0.620016 0.310008 0.950734i 0.399668π-0.399668\pi
0.310008 + 0.950734i 0.399668π0.399668\pi
812812 0 0
813813 −10.1421 −0.355700
814814 0 0
815815 −19.3137 33.4523i −0.676530 1.17178i
816816 0 0
817817 8.00000 13.8564i 0.279885 0.484774i
818818 0 0
819819 0 0
820820 0 0
821821 5.34315 9.25460i 0.186477 0.322988i −0.757596 0.652724i 0.773626π-0.773626\pi
0.944073 + 0.329736i 0.106960π0.106960\pi
822822 0 0
823823 4.48528 + 7.76874i 0.156347 + 0.270801i 0.933549 0.358451i 0.116695π-0.116695\pi
−0.777202 + 0.629252i 0.783361π0.783361\pi
824824 0 0
825825 −13.3137 −0.463524
826826 0 0
827827 −47.6569 −1.65719 −0.828596 0.559848i 0.810860π-0.810860\pi
−0.828596 + 0.559848i 0.810860π0.810860\pi
828828 0 0
829829 0.363961 + 0.630399i 0.0126409 + 0.0218947i 0.872277 0.489013i 0.162643π-0.162643\pi
−0.859636 + 0.510907i 0.829310π0.829310\pi
830830 0 0
831831 −4.65685 + 8.06591i −0.161544 + 0.279803i
832832 0 0
833833 0 0
834834 0 0
835835 −33.7990 + 58.5416i −1.16966 + 2.02591i
836836 0 0
837837 0.585786 + 1.01461i 0.0202477 + 0.0350701i
838838 0 0
839839 −50.8284 −1.75479 −0.877396 0.479767i 0.840721π-0.840721\pi
−0.877396 + 0.479767i 0.840721π0.840721\pi
840840 0 0
841841 17.6274 0.607842
842842 0 0
843843 0.242641 + 0.420266i 0.00835699 + 0.0144747i
844844 0 0
845845 10.7782 18.6683i 0.370780 0.642211i
846846 0 0
847847 0 0
848848 0 0
849849 4.24264 7.34847i 0.145607 0.252199i
850850 0 0
851851 15.3137 + 26.5241i 0.524947 + 0.909235i
852852 0 0
853853 −49.4975 −1.69476 −0.847381 0.530986i 0.821822π-0.821822\pi
−0.847381 + 0.530986i 0.821822π0.821822\pi
854854 0 0
855855 9.65685 0.330257
856856 0 0
857857 −7.70711 13.3491i −0.263270 0.455997i 0.703839 0.710359i 0.251468π-0.251468\pi
−0.967109 + 0.254363i 0.918134π0.918134\pi
858858 0 0
859859 28.7279 49.7582i 0.980184 1.69773i 0.318542 0.947909i 0.396807π-0.396807\pi
0.661642 0.749820i 0.269860π-0.269860\pi
860860 0 0
861861 0 0
862862 0 0
863863 8.65685 14.9941i 0.294683 0.510405i −0.680228 0.733000i 0.738119π-0.738119\pi
0.974911 + 0.222595i 0.0714527π0.0714527\pi
864864 0 0
865865 −35.9706 62.3028i −1.22304 2.11836i
866866 0 0
867867 11.9706 0.406542
868868 0 0
869869 −27.3137 −0.926554
870870 0 0
871871 −7.31371 12.6677i −0.247816 0.429229i
872872 0 0
873873 −1.29289 + 2.23936i −0.0437578 + 0.0757907i
874874 0 0
875875 0 0
876876 0 0
877877 5.65685 9.79796i 0.191018 0.330854i −0.754570 0.656220i 0.772154π-0.772154\pi
0.945588 + 0.325366i 0.105488π0.105488\pi
878878 0 0
879879 8.29289 + 14.3637i 0.279712 + 0.484476i
880880 0 0
881881 21.7574 0.733024 0.366512 0.930413i 0.380552π-0.380552\pi
0.366512 + 0.930413i 0.380552π0.380552\pi
882882 0 0
883883 4.68629 0.157706 0.0788531 0.996886i 0.474874π-0.474874\pi
0.0788531 + 0.996886i 0.474874π0.474874\pi
884884 0 0
885885 2.00000 + 3.46410i 0.0672293 + 0.116445i
886886 0 0
887887 −1.41421 + 2.44949i −0.0474846 + 0.0822458i −0.888791 0.458313i 0.848454π-0.848454\pi
0.841306 + 0.540559i 0.181787π0.181787\pi
888888 0 0
889889 0 0
890890 0 0
891891 −1.00000 + 1.73205i −0.0335013 + 0.0580259i
892892 0 0
893893 −4.00000 6.92820i −0.133855 0.231843i
894894 0 0
895895 67.1127 2.24333
896896 0 0
897897 −19.7990 −0.661069
898898 0 0
899899 4.00000 + 6.92820i 0.133407 + 0.231069i
900900 0 0
901901 −2.24264 + 3.88437i −0.0747132 + 0.129407i
902902 0 0
903903 0 0
904904 0 0
905905 4.41421 7.64564i 0.146733 0.254150i
906906 0 0
907907 −8.00000 13.8564i −0.265636 0.460094i 0.702094 0.712084i 0.252248π-0.252248\pi
−0.967730 + 0.251990i 0.918915π0.918915\pi
908908 0 0
909909 2.92893 0.0971465
910910 0 0
911911 1.02944 0.0341068 0.0170534 0.999855i 0.494571π-0.494571\pi
0.0170534 + 0.999855i 0.494571π0.494571\pi
912912 0 0
913913 7.31371 + 12.6677i 0.242048 + 0.419240i
914914 0 0
915915 20.8995 36.1990i 0.690916 1.19670i
916916 0 0
917917 0 0
918918 0 0
919919 −4.14214 + 7.17439i −0.136636 + 0.236661i −0.926221 0.376980i 0.876963π-0.876963\pi
0.789585 + 0.613641i 0.210296π0.210296\pi
920920 0 0
921921 −15.0711 26.1039i −0.496609 0.860151i
922922 0 0
923923 −24.0833 −0.792710
924924 0 0
925925 −26.6274 −0.875504
926926 0 0
927927 2.24264 + 3.88437i 0.0736580 + 0.127579i
928928 0 0
929929 19.6066 33.9596i 0.643272 1.11418i −0.341426 0.939909i 0.610910π-0.610910\pi
0.984698 0.174271i 0.0557568π-0.0557568\pi
930930 0 0
931931 0 0
932932 0 0
933933 −3.07107 + 5.31925i −0.100542 + 0.174144i
934934 0 0
935935 7.65685 + 13.2621i 0.250406 + 0.433716i
936936 0 0
937937 30.5858 0.999194 0.499597 0.866258i 0.333481π-0.333481\pi
0.499597 + 0.866258i 0.333481π0.333481\pi
938938 0 0
939939 −1.89949 −0.0619877
940940 0 0
941941 −17.6066 30.4955i −0.573959 0.994126i −0.996154 0.0876208i 0.972074π-0.972074\pi
0.422195 0.906505i 0.361260π-0.361260\pi
942942 0 0
943943 −23.8995 + 41.3951i −0.778275 + 1.34801i
944944 0 0
945945 0 0
946946 0 0
947947 15.3431 26.5751i 0.498585 0.863575i −0.501413 0.865208i 0.667186π-0.667186\pi
0.999999 + 0.00163285i 0.000519752π0.000519752\pi
948948 0 0
949949 −17.9706 31.1259i −0.583349 1.01039i
950950 0 0
951951 −10.0000 −0.324272
952952 0 0
953953 2.00000 0.0647864 0.0323932 0.999475i 0.489687π-0.489687\pi
0.0323932 + 0.999475i 0.489687π0.489687\pi
954954 0 0
955955 −30.7279 53.2223i −0.994332 1.72223i
956956 0 0
957957 −6.82843 + 11.8272i −0.220732 + 0.382319i
958958 0 0
959959 0 0
960960 0 0
961961 14.8137 25.6581i 0.477862 0.827681i
962962 0 0
963963 −0.171573 0.297173i −0.00552886 0.00957626i
964964 0 0
965965 18.1421 0.584016
966966 0 0
967967 −33.6569 −1.08233 −0.541166 0.840916i 0.682017π-0.682017\pi
−0.541166 + 0.840916i 0.682017π0.682017\pi
968968 0 0
969969 −3.17157 5.49333i −0.101886 0.176471i
970970 0 0
971971 −25.3137 + 43.8446i −0.812356 + 1.40704i 0.0988557 + 0.995102i 0.468482π0.468482\pi
−0.911211 + 0.411939i 0.864852π0.864852\pi
972972 0 0
973973 0 0
974974 0 0
975975 8.60660 14.9071i 0.275632 0.477408i
976976 0 0
977977 10.5858 + 18.3351i 0.338669 + 0.586592i 0.984183 0.177157i 0.0566899π-0.0566899\pi
−0.645513 + 0.763749i 0.723357π0.723357\pi
978978 0 0
979979 −28.4853 −0.910394
980980 0 0
981981 −5.65685 −0.180609
982982 0 0
983983 −26.6274 46.1200i −0.849283 1.47100i −0.881849 0.471531i 0.843701π-0.843701\pi
0.0325667 0.999470i 0.489632π-0.489632\pi
984984 0 0
985985 −3.41421 + 5.91359i −0.108786 + 0.188423i
986986 0 0
987987 0 0
988988 0 0
989989 21.6569 37.5108i 0.688648 1.19277i
990990 0 0
991991 −6.48528 11.2328i −0.206012 0.356823i 0.744443 0.667686i 0.232715π-0.232715\pi
−0.950455 + 0.310863i 0.899382π0.899382\pi
992992 0 0
993993 −4.00000 −0.126936
994994 0 0
995995 −73.9411 −2.34409
996996 0 0
997997 −13.1924 22.8499i −0.417807 0.723663i 0.577911 0.816099i 0.303868π-0.303868\pi
−0.995719 + 0.0924363i 0.970535π0.970535\pi
998998 0 0
999999 −2.00000 + 3.46410i −0.0632772 + 0.109599i
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 2352.2.q.bd.961.1 4
4.3 odd 2 147.2.e.d.79.1 4
7.2 even 3 2352.2.a.bc.1.2 2
7.3 odd 6 2352.2.q.bb.1537.2 4
7.4 even 3 inner 2352.2.q.bd.1537.1 4
7.5 odd 6 2352.2.a.be.1.1 2
7.6 odd 2 2352.2.q.bb.961.2 4
12.11 even 2 441.2.e.g.226.2 4
21.2 odd 6 7056.2.a.cf.1.1 2
21.5 even 6 7056.2.a.cv.1.2 2
28.3 even 6 147.2.e.e.67.1 4
28.11 odd 6 147.2.e.d.67.1 4
28.19 even 6 147.2.a.d.1.2 2
28.23 odd 6 147.2.a.e.1.2 yes 2
28.27 even 2 147.2.e.e.79.1 4
56.5 odd 6 9408.2.a.dq.1.2 2
56.19 even 6 9408.2.a.ef.1.2 2
56.37 even 6 9408.2.a.dt.1.1 2
56.51 odd 6 9408.2.a.di.1.1 2
84.11 even 6 441.2.e.g.361.2 4
84.23 even 6 441.2.a.i.1.1 2
84.47 odd 6 441.2.a.j.1.1 2
84.59 odd 6 441.2.e.f.361.2 4
84.83 odd 2 441.2.e.f.226.2 4
140.19 even 6 3675.2.a.bf.1.1 2
140.79 odd 6 3675.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.2 2 28.19 even 6
147.2.a.e.1.2 yes 2 28.23 odd 6
147.2.e.d.67.1 4 28.11 odd 6
147.2.e.d.79.1 4 4.3 odd 2
147.2.e.e.67.1 4 28.3 even 6
147.2.e.e.79.1 4 28.27 even 2
441.2.a.i.1.1 2 84.23 even 6
441.2.a.j.1.1 2 84.47 odd 6
441.2.e.f.226.2 4 84.83 odd 2
441.2.e.f.361.2 4 84.59 odd 6
441.2.e.g.226.2 4 12.11 even 2
441.2.e.g.361.2 4 84.11 even 6
2352.2.a.bc.1.2 2 7.2 even 3
2352.2.a.be.1.1 2 7.5 odd 6
2352.2.q.bb.961.2 4 7.6 odd 2
2352.2.q.bb.1537.2 4 7.3 odd 6
2352.2.q.bd.961.1 4 1.1 even 1 trivial
2352.2.q.bd.1537.1 4 7.4 even 3 inner
3675.2.a.bd.1.1 2 140.79 odd 6
3675.2.a.bf.1.1 2 140.19 even 6
7056.2.a.cf.1.1 2 21.2 odd 6
7056.2.a.cv.1.2 2 21.5 even 6
9408.2.a.di.1.1 2 56.51 odd 6
9408.2.a.dq.1.2 2 56.5 odd 6
9408.2.a.dt.1.1 2 56.37 even 6
9408.2.a.ef.1.2 2 56.19 even 6