Properties

Label 1440.4.a.bc.1.2
Level 14401440
Weight 44
Character 1440.1
Self dual yes
Analytic conductor 84.96384.963
Analytic rank 11
Dimension 22
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] = [1440,4,Mod(1,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 1440=25325 1440 = 2^{5} \cdot 3^{2} \cdot 5
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,10,0,0,0,0,0,0,0,-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: yes
Analytic conductor: 84.962750408384.9627504083
Analytic rank: 11
Dimension: 22
Coefficient field: Q(65)\Q(\sqrt{65})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x16 x^{2} - x - 16 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 3.53113-3.53113 of defining polynomial
Character χ\chi == 1440.1

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+5.00000q5+16.1245q7+16.1245q118.00000q1326.0000q1796.7471q19128.996q23+25.0000q2554.0000q29128.996q31+80.6226q35124.000q37152.000q41193.494q4364.4981q4783.0000q4978.0000q53+80.6226q55+596.607q59470.000q6140.0000q65+741.728q67+225.743q71162.000q73+260.000q77+128.996q791193.21q83130.000q85124.000q89128.996q91483.735q95646.000q97+O(q100)q+5.00000 q^{5} +16.1245 q^{7} +16.1245 q^{11} -8.00000 q^{13} -26.0000 q^{17} -96.7471 q^{19} -128.996 q^{23} +25.0000 q^{25} -54.0000 q^{29} -128.996 q^{31} +80.6226 q^{35} -124.000 q^{37} -152.000 q^{41} -193.494 q^{43} -64.4981 q^{47} -83.0000 q^{49} -78.0000 q^{53} +80.6226 q^{55} +596.607 q^{59} -470.000 q^{61} -40.0000 q^{65} +741.728 q^{67} +225.743 q^{71} -162.000 q^{73} +260.000 q^{77} +128.996 q^{79} -1193.21 q^{83} -130.000 q^{85} -124.000 q^{89} -128.996 q^{91} -483.735 q^{95} -646.000 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+10q516q1352q17+50q25108q29248q37304q41166q49156q53940q6180q65324q73+520q77260q85248q891292q97+O(q100) 2 q + 10 q^{5} - 16 q^{13} - 52 q^{17} + 50 q^{25} - 108 q^{29} - 248 q^{37} - 304 q^{41} - 166 q^{49} - 156 q^{53} - 940 q^{61} - 80 q^{65} - 324 q^{73} + 520 q^{77} - 260 q^{85} - 248 q^{89} - 1292 q^{97}+O(q^{100}) Copy content Toggle raw display

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 0
44 0 0
55 5.00000 0.447214
66 0 0
77 16.1245 0.870642 0.435321 0.900275i 0.356635π-0.356635\pi
0.435321 + 0.900275i 0.356635π0.356635\pi
88 0 0
99 0 0
1010 0 0
1111 16.1245 0.441975 0.220987 0.975277i 0.429072π-0.429072\pi
0.220987 + 0.975277i 0.429072π0.429072\pi
1212 0 0
1313 −8.00000 −0.170677 −0.0853385 0.996352i 0.527197π-0.527197\pi
−0.0853385 + 0.996352i 0.527197π0.527197\pi
1414 0 0
1515 0 0
1616 0 0
1717 −26.0000 −0.370937 −0.185468 0.982650i 0.559380π-0.559380\pi
−0.185468 + 0.982650i 0.559380π0.559380\pi
1818 0 0
1919 −96.7471 −1.16817 −0.584087 0.811691i 0.698547π-0.698547\pi
−0.584087 + 0.811691i 0.698547π0.698547\pi
2020 0 0
2121 0 0
2222 0 0
2323 −128.996 −1.16946 −0.584729 0.811228i 0.698799π-0.698799\pi
−0.584729 + 0.811228i 0.698799π0.698799\pi
2424 0 0
2525 25.0000 0.200000
2626 0 0
2727 0 0
2828 0 0
2929 −54.0000 −0.345778 −0.172889 0.984941i 0.555310π-0.555310\pi
−0.172889 + 0.984941i 0.555310π0.555310\pi
3030 0 0
3131 −128.996 −0.747367 −0.373684 0.927556i 0.621905π-0.621905\pi
−0.373684 + 0.927556i 0.621905π0.621905\pi
3232 0 0
3333 0 0
3434 0 0
3535 80.6226 0.389363
3636 0 0
3737 −124.000 −0.550959 −0.275479 0.961307i 0.588837π-0.588837\pi
−0.275479 + 0.961307i 0.588837π0.588837\pi
3838 0 0
3939 0 0
4040 0 0
4141 −152.000 −0.578986 −0.289493 0.957180i 0.593487π-0.593487\pi
−0.289493 + 0.957180i 0.593487π0.593487\pi
4242 0 0
4343 −193.494 −0.686223 −0.343111 0.939295i 0.611481π-0.611481\pi
−0.343111 + 0.939295i 0.611481π0.611481\pi
4444 0 0
4545 0 0
4646 0 0
4747 −64.4981 −0.200170 −0.100085 0.994979i 0.531912π-0.531912\pi
−0.100085 + 0.994979i 0.531912π0.531912\pi
4848 0 0
4949 −83.0000 −0.241983
5050 0 0
5151 0 0
5252 0 0
5353 −78.0000 −0.202153 −0.101077 0.994879i 0.532229π-0.532229\pi
−0.101077 + 0.994879i 0.532229π0.532229\pi
5454 0 0
5555 80.6226 0.197657
5656 0 0
5757 0 0
5858 0 0
5959 596.607 1.31647 0.658234 0.752813i 0.271304π-0.271304\pi
0.658234 + 0.752813i 0.271304π0.271304\pi
6060 0 0
6161 −470.000 −0.986514 −0.493257 0.869884i 0.664194π-0.664194\pi
−0.493257 + 0.869884i 0.664194π0.664194\pi
6262 0 0
6363 0 0
6464 0 0
6565 −40.0000 −0.0763291
6666 0 0
6767 741.728 1.35248 0.676242 0.736679i 0.263607π-0.263607\pi
0.676242 + 0.736679i 0.263607π0.263607\pi
6868 0 0
6969 0 0
7070 0 0
7171 225.743 0.377335 0.188668 0.982041i 0.439583π-0.439583\pi
0.188668 + 0.982041i 0.439583π0.439583\pi
7272 0 0
7373 −162.000 −0.259735 −0.129868 0.991531i 0.541455π-0.541455\pi
−0.129868 + 0.991531i 0.541455π0.541455\pi
7474 0 0
7575 0 0
7676 0 0
7777 260.000 0.384802
7878 0 0
7979 128.996 0.183711 0.0918557 0.995772i 0.470720π-0.470720\pi
0.0918557 + 0.995772i 0.470720π0.470720\pi
8080 0 0
8181 0 0
8282 0 0
8383 −1193.21 −1.57798 −0.788990 0.614407i 0.789395π-0.789395\pi
−0.788990 + 0.614407i 0.789395π0.789395\pi
8484 0 0
8585 −130.000 −0.165888
8686 0 0
8787 0 0
8888 0 0
8989 −124.000 −0.147685 −0.0738425 0.997270i 0.523526π-0.523526\pi
−0.0738425 + 0.997270i 0.523526π0.523526\pi
9090 0 0
9191 −128.996 −0.148599
9292 0 0
9393 0 0
9494 0 0
9595 −483.735 −0.522423
9696 0 0
9797 −646.000 −0.676200 −0.338100 0.941110i 0.609784π-0.609784\pi
−0.338100 + 0.941110i 0.609784π0.609784\pi
9898 0 0
9999 0 0
100100 0 0
101101 530.000 0.522148 0.261074 0.965319i 0.415923π-0.415923\pi
0.261074 + 0.965319i 0.415923π0.415923\pi
102102 0 0
103103 983.595 0.940937 0.470469 0.882417i 0.344085π-0.344085\pi
0.470469 + 0.882417i 0.344085π0.344085\pi
104104 0 0
105105 0 0
106106 0 0
107107 935.222 0.844966 0.422483 0.906371i 0.361159π-0.361159\pi
0.422483 + 0.906371i 0.361159π0.361159\pi
108108 0 0
109109 −1374.00 −1.20739 −0.603694 0.797216i 0.706305π-0.706305\pi
−0.603694 + 0.797216i 0.706305π0.706305\pi
110110 0 0
111111 0 0
112112 0 0
113113 −158.000 −0.131534 −0.0657672 0.997835i 0.520949π-0.520949\pi
−0.0657672 + 0.997835i 0.520949π0.520949\pi
114114 0 0
115115 −644.981 −0.522998
116116 0 0
117117 0 0
118118 0 0
119119 −419.237 −0.322953
120120 0 0
121121 −1071.00 −0.804658
122122 0 0
123123 0 0
124124 0 0
125125 125.000 0.0894427
126126 0 0
127127 −1338.33 −0.935102 −0.467551 0.883966i 0.654864π-0.654864\pi
−0.467551 + 0.883966i 0.654864π0.654864\pi
128128 0 0
129129 0 0
130130 0 0
131131 951.346 0.634500 0.317250 0.948342i 0.397241π-0.397241\pi
0.317250 + 0.948342i 0.397241π0.397241\pi
132132 0 0
133133 −1560.00 −1.01706
134134 0 0
135135 0 0
136136 0 0
137137 474.000 0.295595 0.147798 0.989018i 0.452782π-0.452782\pi
0.147798 + 0.989018i 0.452782π0.452782\pi
138138 0 0
139139 1580.20 0.964252 0.482126 0.876102i 0.339865π-0.339865\pi
0.482126 + 0.876102i 0.339865π0.339865\pi
140140 0 0
141141 0 0
142142 0 0
143143 −128.996 −0.0754349
144144 0 0
145145 −270.000 −0.154636
146146 0 0
147147 0 0
148148 0 0
149149 1270.00 0.698272 0.349136 0.937072i 0.386475π-0.386475\pi
0.349136 + 0.937072i 0.386475π0.386475\pi
150150 0 0
151151 −290.241 −0.156421 −0.0782103 0.996937i 0.524921π-0.524921\pi
−0.0782103 + 0.996937i 0.524921π0.524921\pi
152152 0 0
153153 0 0
154154 0 0
155155 −644.981 −0.334233
156156 0 0
157157 −2496.00 −1.26881 −0.634403 0.773003i 0.718754π-0.718754\pi
−0.634403 + 0.773003i 0.718754π0.718754\pi
158158 0 0
159159 0 0
160160 0 0
161161 −2080.00 −1.01818
162162 0 0
163163 −999.720 −0.480393 −0.240197 0.970724i 0.577212π-0.577212\pi
−0.240197 + 0.970724i 0.577212π0.577212\pi
164164 0 0
165165 0 0
166166 0 0
167167 644.981 0.298863 0.149431 0.988772i 0.452256π-0.452256\pi
0.149431 + 0.988772i 0.452256π0.452256\pi
168168 0 0
169169 −2133.00 −0.970869
170170 0 0
171171 0 0
172172 0 0
173173 2018.00 0.886854 0.443427 0.896311i 0.353763π-0.353763\pi
0.443427 + 0.896311i 0.353763π0.353763\pi
174174 0 0
175175 403.113 0.174128
176176 0 0
177177 0 0
178178 0 0
179179 −2886.29 −1.20520 −0.602601 0.798042i 0.705869π-0.705869\pi
−0.602601 + 0.798042i 0.705869π0.705869\pi
180180 0 0
181181 −1342.00 −0.551105 −0.275553 0.961286i 0.588861π-0.588861\pi
−0.275553 + 0.961286i 0.588861π0.588861\pi
182182 0 0
183183 0 0
184184 0 0
185185 −620.000 −0.246396
186186 0 0
187187 −419.237 −0.163945
188188 0 0
189189 0 0
190190 0 0
191191 4160.12 1.57600 0.788000 0.615675i 0.211117π-0.211117\pi
0.788000 + 0.615675i 0.211117π0.211117\pi
192192 0 0
193193 −4362.00 −1.62686 −0.813429 0.581664i 0.802402π-0.802402\pi
−0.813429 + 0.581664i 0.802402π0.802402\pi
194194 0 0
195195 0 0
196196 0 0
197197 1374.00 0.496921 0.248461 0.968642i 0.420075π-0.420075\pi
0.248461 + 0.968642i 0.420075π0.420075\pi
198198 0 0
199199 −1902.69 −0.677780 −0.338890 0.940826i 0.610052π-0.610052\pi
−0.338890 + 0.940826i 0.610052π0.610052\pi
200200 0 0
201201 0 0
202202 0 0
203203 −870.724 −0.301048
204204 0 0
205205 −760.000 −0.258930
206206 0 0
207207 0 0
208208 0 0
209209 −1560.00 −0.516304
210210 0 0
211211 −1838.19 −0.599747 −0.299873 0.953979i 0.596944π-0.596944\pi
−0.299873 + 0.953979i 0.596944π0.596944\pi
212212 0 0
213213 0 0
214214 0 0
215215 −967.471 −0.306888
216216 0 0
217217 −2080.00 −0.650689
218218 0 0
219219 0 0
220220 0 0
221221 208.000 0.0633104
222222 0 0
223223 −1402.83 −0.421258 −0.210629 0.977566i 0.567551π-0.567551\pi
−0.210629 + 0.977566i 0.567551π0.567551\pi
224224 0 0
225225 0 0
226226 0 0
227227 −322.490 −0.0942927 −0.0471463 0.998888i 0.515013π-0.515013\pi
−0.0471463 + 0.998888i 0.515013π0.515013\pi
228228 0 0
229229 −1450.00 −0.418422 −0.209211 0.977870i 0.567090π-0.567090\pi
−0.209211 + 0.977870i 0.567090π0.567090\pi
230230 0 0
231231 0 0
232232 0 0
233233 −3442.00 −0.967781 −0.483890 0.875129i 0.660777π-0.660777\pi
−0.483890 + 0.875129i 0.660777π0.660777\pi
234234 0 0
235235 −322.490 −0.0895189
236236 0 0
237237 0 0
238238 0 0
239239 3321.65 0.898995 0.449497 0.893282i 0.351603π-0.351603\pi
0.449497 + 0.893282i 0.351603π0.351603\pi
240240 0 0
241241 −4570.00 −1.22149 −0.610746 0.791827i 0.709130π-0.709130\pi
−0.610746 + 0.791827i 0.709130π0.709130\pi
242242 0 0
243243 0 0
244244 0 0
245245 −415.000 −0.108218
246246 0 0
247247 773.977 0.199380
248248 0 0
249249 0 0
250250 0 0
251251 693.354 0.174359 0.0871795 0.996193i 0.472215π-0.472215\pi
0.0871795 + 0.996193i 0.472215π0.472215\pi
252252 0 0
253253 −2080.00 −0.516871
254254 0 0
255255 0 0
256256 0 0
257257 −3214.00 −0.780093 −0.390046 0.920795i 0.627541π-0.627541\pi
−0.390046 + 0.920795i 0.627541π0.627541\pi
258258 0 0
259259 −1999.44 −0.479688
260260 0 0
261261 0 0
262262 0 0
263263 −2966.91 −0.695618 −0.347809 0.937565i 0.613074π-0.613074\pi
−0.347809 + 0.937565i 0.613074π0.613074\pi
264264 0 0
265265 −390.000 −0.0904057
266266 0 0
267267 0 0
268268 0 0
269269 3330.00 0.754772 0.377386 0.926056i 0.376823π-0.376823\pi
0.377386 + 0.926056i 0.376823π0.376823\pi
270270 0 0
271271 8610.49 1.93007 0.965037 0.262113i 0.0844195π-0.0844195\pi
0.965037 + 0.262113i 0.0844195π0.0844195\pi
272272 0 0
273273 0 0
274274 0 0
275275 403.113 0.0883950
276276 0 0
277277 −1924.00 −0.417336 −0.208668 0.977987i 0.566913π-0.566913\pi
−0.208668 + 0.977987i 0.566913π0.566913\pi
278278 0 0
279279 0 0
280280 0 0
281281 −4908.00 −1.04195 −0.520973 0.853573i 0.674431π-0.674431\pi
−0.520973 + 0.853573i 0.674431π0.674431\pi
282282 0 0
283283 4740.61 0.995759 0.497880 0.867246i 0.334112π-0.334112\pi
0.497880 + 0.867246i 0.334112π0.334112\pi
284284 0 0
285285 0 0
286286 0 0
287287 −2450.93 −0.504089
288288 0 0
289289 −4237.00 −0.862406
290290 0 0
291291 0 0
292292 0 0
293293 2542.00 0.506844 0.253422 0.967356i 0.418444π-0.418444\pi
0.253422 + 0.967356i 0.418444π0.418444\pi
294294 0 0
295295 2983.04 0.588742
296296 0 0
297297 0 0
298298 0 0
299299 1031.97 0.199600
300300 0 0
301301 −3120.00 −0.597455
302302 0 0
303303 0 0
304304 0 0
305305 −2350.00 −0.441182
306306 0 0
307307 6449.81 1.19905 0.599527 0.800354i 0.295355π-0.295355\pi
0.599527 + 0.800354i 0.295355π0.295355\pi
308308 0 0
309309 0 0
310310 0 0
311311 −10222.9 −1.86395 −0.931977 0.362517i 0.881918π-0.881918\pi
−0.931977 + 0.362517i 0.881918π0.881918\pi
312312 0 0
313313 −1482.00 −0.267628 −0.133814 0.991006i 0.542722π-0.542722\pi
−0.133814 + 0.991006i 0.542722π0.542722\pi
314314 0 0
315315 0 0
316316 0 0
317317 10014.0 1.77427 0.887133 0.461514i 0.152694π-0.152694\pi
0.887133 + 0.461514i 0.152694π0.152694\pi
318318 0 0
319319 −870.724 −0.152825
320320 0 0
321321 0 0
322322 0 0
323323 2515.42 0.433319
324324 0 0
325325 −200.000 −0.0341354
326326 0 0
327327 0 0
328328 0 0
329329 −1040.00 −0.174277
330330 0 0
331331 −6224.06 −1.03355 −0.516776 0.856121i 0.672868π-0.672868\pi
−0.516776 + 0.856121i 0.672868π0.672868\pi
332332 0 0
333333 0 0
334334 0 0
335335 3708.64 0.604849
336336 0 0
337337 −3546.00 −0.573184 −0.286592 0.958053i 0.592522π-0.592522\pi
−0.286592 + 0.958053i 0.592522π0.592522\pi
338338 0 0
339339 0 0
340340 0 0
341341 −2080.00 −0.330318
342342 0 0
343343 −6869.04 −1.08132
344344 0 0
345345 0 0
346346 0 0
347347 5611.33 0.868104 0.434052 0.900888i 0.357083π-0.357083\pi
0.434052 + 0.900888i 0.357083π0.357083\pi
348348 0 0
349349 −1710.00 −0.262276 −0.131138 0.991364i 0.541863π-0.541863\pi
−0.131138 + 0.991364i 0.541863π0.541863\pi
350350 0 0
351351 0 0
352352 0 0
353353 −2058.00 −0.310301 −0.155151 0.987891i 0.549586π-0.549586\pi
−0.155151 + 0.987891i 0.549586π0.549586\pi
354354 0 0
355355 1128.72 0.168749
356356 0 0
357357 0 0
358358 0 0
359359 3805.39 0.559444 0.279722 0.960081i 0.409758π-0.409758\pi
0.279722 + 0.960081i 0.409758π0.409758\pi
360360 0 0
361361 2501.00 0.364630
362362 0 0
363363 0 0
364364 0 0
365365 −810.000 −0.116157
366366 0 0
367367 −8949.11 −1.27286 −0.636430 0.771335i 0.719590π-0.719590\pi
−0.636430 + 0.771335i 0.719590π0.719590\pi
368368 0 0
369369 0 0
370370 0 0
371371 −1257.71 −0.176003
372372 0 0
373373 2392.00 0.332046 0.166023 0.986122i 0.446907π-0.446907\pi
0.166023 + 0.986122i 0.446907π0.446907\pi
374374 0 0
375375 0 0
376376 0 0
377377 432.000 0.0590163
378378 0 0
379379 3708.64 0.502638 0.251319 0.967904i 0.419136π-0.419136\pi
0.251319 + 0.967904i 0.419136π0.419136\pi
380380 0 0
381381 0 0
382382 0 0
383383 257.992 0.0344198 0.0172099 0.999852i 0.494522π-0.494522\pi
0.0172099 + 0.999852i 0.494522π0.494522\pi
384384 0 0
385385 1300.00 0.172089
386386 0 0
387387 0 0
388388 0 0
389389 9750.00 1.27081 0.635404 0.772180i 0.280833π-0.280833\pi
0.635404 + 0.772180i 0.280833π0.280833\pi
390390 0 0
391391 3353.90 0.433795
392392 0 0
393393 0 0
394394 0 0
395395 644.981 0.0821582
396396 0 0
397397 −2164.00 −0.273572 −0.136786 0.990601i 0.543677π-0.543677\pi
−0.136786 + 0.990601i 0.543677π0.543677\pi
398398 0 0
399399 0 0
400400 0 0
401401 −12500.0 −1.55666 −0.778329 0.627856i 0.783933π-0.783933\pi
−0.778329 + 0.627856i 0.783933π0.783933\pi
402402 0 0
403403 1031.97 0.127558
404404 0 0
405405 0 0
406406 0 0
407407 −1999.44 −0.243510
408408 0 0
409409 −646.000 −0.0780994 −0.0390497 0.999237i 0.512433π-0.512433\pi
−0.0390497 + 0.999237i 0.512433π0.512433\pi
410410 0 0
411411 0 0
412412 0 0
413413 9620.00 1.14617
414414 0 0
415415 −5966.07 −0.705694
416416 0 0
417417 0 0
418418 0 0
419419 3079.78 0.359086 0.179543 0.983750i 0.442538π-0.442538\pi
0.179543 + 0.983750i 0.442538π0.442538\pi
420420 0 0
421421 230.000 0.0266259 0.0133130 0.999911i 0.495762π-0.495762\pi
0.0133130 + 0.999911i 0.495762π0.495762\pi
422422 0 0
423423 0 0
424424 0 0
425425 −650.000 −0.0741874
426426 0 0
427427 −7578.52 −0.858900
428428 0 0
429429 0 0
430430 0 0
431431 2192.93 0.245081 0.122541 0.992464i 0.460896π-0.460896\pi
0.122541 + 0.992464i 0.460896π0.460896\pi
432432 0 0
433433 5902.00 0.655039 0.327520 0.944844i 0.393787π-0.393787\pi
0.327520 + 0.944844i 0.393787π0.393787\pi
434434 0 0
435435 0 0
436436 0 0
437437 12480.0 1.36613
438438 0 0
439439 1902.69 0.206858 0.103429 0.994637i 0.467019π-0.467019\pi
0.103429 + 0.994637i 0.467019π0.467019\pi
440440 0 0
441441 0 0
442442 0 0
443443 6062.82 0.650233 0.325116 0.945674i 0.394597π-0.394597\pi
0.325116 + 0.945674i 0.394597π0.394597\pi
444444 0 0
445445 −620.000 −0.0660468
446446 0 0
447447 0 0
448448 0 0
449449 −7820.00 −0.821935 −0.410967 0.911650i 0.634809π-0.634809\pi
−0.410967 + 0.911650i 0.634809π0.634809\pi
450450 0 0
451451 −2450.93 −0.255897
452452 0 0
453453 0 0
454454 0 0
455455 −644.981 −0.0664553
456456 0 0
457457 12146.0 1.24325 0.621626 0.783314i 0.286472π-0.286472\pi
0.621626 + 0.783314i 0.286472π0.286472\pi
458458 0 0
459459 0 0
460460 0 0
461461 750.000 0.0757722 0.0378861 0.999282i 0.487938π-0.487938\pi
0.0378861 + 0.999282i 0.487938π0.487938\pi
462462 0 0
463463 8755.61 0.878851 0.439425 0.898279i 0.355182π-0.355182\pi
0.439425 + 0.898279i 0.355182π0.355182\pi
464464 0 0
465465 0 0
466466 0 0
467467 −16479.3 −1.63291 −0.816455 0.577409i 0.804064π-0.804064\pi
−0.816455 + 0.577409i 0.804064π0.804064\pi
468468 0 0
469469 11960.0 1.17753
470470 0 0
471471 0 0
472472 0 0
473473 −3120.00 −0.303293
474474 0 0
475475 −2418.68 −0.233635
476476 0 0
477477 0 0
478478 0 0
479479 −17317.7 −1.65191 −0.825957 0.563732i 0.809365π-0.809365\pi
−0.825957 + 0.563732i 0.809365π0.809365\pi
480480 0 0
481481 992.000 0.0940360
482482 0 0
483483 0 0
484484 0 0
485485 −3230.00 −0.302406
486486 0 0
487487 112.872 0.0105025 0.00525123 0.999986i 0.498328π-0.498328\pi
0.00525123 + 0.999986i 0.498328π0.498328\pi
488488 0 0
489489 0 0
490490 0 0
491491 −10271.3 −0.944069 −0.472035 0.881580i 0.656480π-0.656480\pi
−0.472035 + 0.881580i 0.656480π0.656480\pi
492492 0 0
493493 1404.00 0.128262
494494 0 0
495495 0 0
496496 0 0
497497 3640.00 0.328524
498498 0 0
499499 −2612.17 −0.234342 −0.117171 0.993112i 0.537383π-0.537383\pi
−0.117171 + 0.993112i 0.537383π0.537383\pi
500500 0 0
501501 0 0
502502 0 0
503503 21090.9 1.86957 0.934787 0.355210i 0.115591π-0.115591\pi
0.934787 + 0.355210i 0.115591π0.115591\pi
504504 0 0
505505 2650.00 0.233512
506506 0 0
507507 0 0
508508 0 0
509509 11834.0 1.03052 0.515258 0.857035i 0.327696π-0.327696\pi
0.515258 + 0.857035i 0.327696π0.327696\pi
510510 0 0
511511 −2612.17 −0.226136
512512 0 0
513513 0 0
514514 0 0
515515 4917.98 0.420800
516516 0 0
517517 −1040.00 −0.0884703
518518 0 0
519519 0 0
520520 0 0
521521 −17860.0 −1.50184 −0.750922 0.660391i 0.770391π-0.770391\pi
−0.750922 + 0.660391i 0.770391π0.770391\pi
522522 0 0
523523 −5321.09 −0.444885 −0.222443 0.974946i 0.571403π-0.571403\pi
−0.222443 + 0.974946i 0.571403π0.571403\pi
524524 0 0
525525 0 0
526526 0 0
527527 3353.90 0.277226
528528 0 0
529529 4473.00 0.367634
530530 0 0
531531 0 0
532532 0 0
533533 1216.00 0.0988195
534534 0 0
535535 4676.11 0.377880
536536 0 0
537537 0 0
538538 0 0
539539 −1338.33 −0.106950
540540 0 0
541541 11642.0 0.925192 0.462596 0.886569i 0.346918π-0.346918\pi
0.462596 + 0.886569i 0.346918π0.346918\pi
542542 0 0
543543 0 0
544544 0 0
545545 −6870.00 −0.539960
546546 0 0
547547 11416.2 0.892358 0.446179 0.894944i 0.352785π-0.352785\pi
0.446179 + 0.894944i 0.352785π0.352785\pi
548548 0 0
549549 0 0
550550 0 0
551551 5224.34 0.403928
552552 0 0
553553 2080.00 0.159947
554554 0 0
555555 0 0
556556 0 0
557557 20026.0 1.52339 0.761696 0.647935i 0.224367π-0.224367\pi
0.761696 + 0.647935i 0.224367π0.224367\pi
558558 0 0
559559 1547.95 0.117122
560560 0 0
561561 0 0
562562 0 0
563563 7868.76 0.589039 0.294519 0.955646i 0.404840π-0.404840\pi
0.294519 + 0.955646i 0.404840π0.404840\pi
564564 0 0
565565 −790.000 −0.0588240
566566 0 0
567567 0 0
568568 0 0
569569 5060.00 0.372805 0.186403 0.982473i 0.440317π-0.440317\pi
0.186403 + 0.982473i 0.440317π0.440317\pi
570570 0 0
571571 −1709.20 −0.125267 −0.0626337 0.998037i 0.519950π-0.519950\pi
−0.0626337 + 0.998037i 0.519950π0.519950\pi
572572 0 0
573573 0 0
574574 0 0
575575 −3224.90 −0.233892
576576 0 0
577577 23794.0 1.71674 0.858368 0.513034i 0.171478π-0.171478\pi
0.858368 + 0.513034i 0.171478π0.171478\pi
578578 0 0
579579 0 0
580580 0 0
581581 −19240.0 −1.37385
582582 0 0
583583 −1257.71 −0.0893467
584584 0 0
585585 0 0
586586 0 0
587587 1064.22 0.0748296 0.0374148 0.999300i 0.488088π-0.488088\pi
0.0374148 + 0.999300i 0.488088π0.488088\pi
588588 0 0
589589 12480.0 0.873055
590590 0 0
591591 0 0
592592 0 0
593593 −14502.0 −1.00426 −0.502130 0.864792i 0.667450π-0.667450\pi
−0.502130 + 0.864792i 0.667450π0.667450\pi
594594 0 0
595595 −2096.19 −0.144429
596596 0 0
597597 0 0
598598 0 0
599599 −6127.32 −0.417955 −0.208978 0.977920i 0.567014π-0.567014\pi
−0.208978 + 0.977920i 0.567014π0.567014\pi
600600 0 0
601601 13910.0 0.944094 0.472047 0.881573i 0.343515π-0.343515\pi
0.472047 + 0.881573i 0.343515π0.343515\pi
602602 0 0
603603 0 0
604604 0 0
605605 −5355.00 −0.359854
606606 0 0
607607 11529.0 0.770921 0.385460 0.922724i 0.374043π-0.374043\pi
0.385460 + 0.922724i 0.374043π0.374043\pi
608608 0 0
609609 0 0
610610 0 0
611611 515.984 0.0341645
612612 0 0
613613 13108.0 0.863666 0.431833 0.901954i 0.357867π-0.357867\pi
0.431833 + 0.901954i 0.357867π0.357867\pi
614614 0 0
615615 0 0
616616 0 0
617617 13334.0 0.870027 0.435013 0.900424i 0.356744π-0.356744\pi
0.435013 + 0.900424i 0.356744π0.356744\pi
618618 0 0
619619 16027.8 1.04073 0.520364 0.853945i 0.325796π-0.325796\pi
0.520364 + 0.853945i 0.325796π0.325796\pi
620620 0 0
621621 0 0
622622 0 0
623623 −1999.44 −0.128581
624624 0 0
625625 625.000 0.0400000
626626 0 0
627627 0 0
628628 0 0
629629 3224.00 0.204371
630630 0 0
631631 −10255.2 −0.646993 −0.323497 0.946229i 0.604858π-0.604858\pi
−0.323497 + 0.946229i 0.604858π0.604858\pi
632632 0 0
633633 0 0
634634 0 0
635635 −6691.67 −0.418191
636636 0 0
637637 664.000 0.0413008
638638 0 0
639639 0 0
640640 0 0
641641 −28288.0 −1.74307 −0.871536 0.490332i 0.836876π-0.836876\pi
−0.871536 + 0.490332i 0.836876π0.836876\pi
642642 0 0
643643 225.743 0.0138452 0.00692258 0.999976i 0.497796π-0.497796\pi
0.00692258 + 0.999976i 0.497796π0.497796\pi
644644 0 0
645645 0 0
646646 0 0
647647 12448.1 0.756393 0.378197 0.925725i 0.376544π-0.376544\pi
0.378197 + 0.925725i 0.376544π0.376544\pi
648648 0 0
649649 9620.00 0.581846
650650 0 0
651651 0 0
652652 0 0
653653 11542.0 0.691690 0.345845 0.938292i 0.387592π-0.387592\pi
0.345845 + 0.938292i 0.387592π0.387592\pi
654654 0 0
655655 4756.73 0.283757
656656 0 0
657657 0 0
658658 0 0
659659 628.856 0.0371726 0.0185863 0.999827i 0.494083π-0.494083\pi
0.0185863 + 0.999827i 0.494083π0.494083\pi
660660 0 0
661661 13930.0 0.819689 0.409844 0.912155i 0.365583π-0.365583\pi
0.409844 + 0.912155i 0.365583π0.365583\pi
662662 0 0
663663 0 0
664664 0 0
665665 −7800.00 −0.454844
666666 0 0
667667 6965.79 0.404373
668668 0 0
669669 0 0
670670 0 0
671671 −7578.52 −0.436014
672672 0 0
673673 7862.00 0.450309 0.225154 0.974323i 0.427711π-0.427711\pi
0.225154 + 0.974323i 0.427711π0.427711\pi
674674 0 0
675675 0 0
676676 0 0
677677 6526.00 0.370479 0.185240 0.982693i 0.440694π-0.440694\pi
0.185240 + 0.982693i 0.440694π0.440694\pi
678678 0 0
679679 −10416.4 −0.588728
680680 0 0
681681 0 0
682682 0 0
683683 −20284.6 −1.13641 −0.568207 0.822886i 0.692363π-0.692363\pi
−0.568207 + 0.822886i 0.692363π0.692363\pi
684684 0 0
685685 2370.00 0.132194
686686 0 0
687687 0 0
688688 0 0
689689 624.000 0.0345029
690690 0 0
691691 18736.7 1.03152 0.515758 0.856734i 0.327510π-0.327510\pi
0.515758 + 0.856734i 0.327510π0.327510\pi
692692 0 0
693693 0 0
694694 0 0
695695 7901.01 0.431227
696696 0 0
697697 3952.00 0.214767
698698 0 0
699699 0 0
700700 0 0
701701 −8398.00 −0.452479 −0.226240 0.974072i 0.572643π-0.572643\pi
−0.226240 + 0.974072i 0.572643π0.572643\pi
702702 0 0
703703 11996.6 0.643616
704704 0 0
705705 0 0
706706 0 0
707707 8545.99 0.454604
708708 0 0
709709 10550.0 0.558834 0.279417 0.960170i 0.409859π-0.409859\pi
0.279417 + 0.960170i 0.409859π0.409859\pi
710710 0 0
711711 0 0
712712 0 0
713713 16640.0 0.874015
714714 0 0
715715 −644.981 −0.0337355
716716 0 0
717717 0 0
718718 0 0
719719 19671.9 1.02036 0.510180 0.860068i 0.329579π-0.329579\pi
0.510180 + 0.860068i 0.329579π0.329579\pi
720720 0 0
721721 15860.0 0.819220
722722 0 0
723723 0 0
724724 0 0
725725 −1350.00 −0.0691555
726726 0 0
727727 11851.5 0.604606 0.302303 0.953212i 0.402245π-0.402245\pi
0.302303 + 0.953212i 0.402245π0.402245\pi
728728 0 0
729729 0 0
730730 0 0
731731 5030.85 0.254545
732732 0 0
733733 16948.0 0.854009 0.427005 0.904249i 0.359569π-0.359569\pi
0.427005 + 0.904249i 0.359569π0.359569\pi
734734 0 0
735735 0 0
736736 0 0
737737 11960.0 0.597764
738738 0 0
739739 −36989.6 −1.84125 −0.920626 0.390445i 0.872321π-0.872321\pi
−0.920626 + 0.390445i 0.872321π0.872321\pi
740740 0 0
741741 0 0
742742 0 0
743743 −11093.7 −0.547762 −0.273881 0.961764i 0.588307π-0.588307\pi
−0.273881 + 0.961764i 0.588307π0.588307\pi
744744 0 0
745745 6350.00 0.312277
746746 0 0
747747 0 0
748748 0 0
749749 15080.0 0.735662
750750 0 0
751751 −24186.8 −1.17522 −0.587608 0.809145i 0.699930π-0.699930\pi
−0.587608 + 0.809145i 0.699930π0.699930\pi
752752 0 0
753753 0 0
754754 0 0
755755 −1451.21 −0.0699534
756756 0 0
757757 12716.0 0.610530 0.305265 0.952268i 0.401255π-0.401255\pi
0.305265 + 0.952268i 0.401255π0.401255\pi
758758 0 0
759759 0 0
760760 0 0
761761 −20300.0 −0.966983 −0.483492 0.875349i 0.660632π-0.660632\pi
−0.483492 + 0.875349i 0.660632π0.660632\pi
762762 0 0
763763 −22155.1 −1.05120
764764 0 0
765765 0 0
766766 0 0
767767 −4772.86 −0.224691
768768 0 0
769769 −15570.0 −0.730128 −0.365064 0.930982i 0.618953π-0.618953\pi
−0.365064 + 0.930982i 0.618953π0.618953\pi
770770 0 0
771771 0 0
772772 0 0
773773 438.000 0.0203800 0.0101900 0.999948i 0.496756π-0.496756\pi
0.0101900 + 0.999948i 0.496756π0.496756\pi
774774 0 0
775775 −3224.90 −0.149473
776776 0 0
777777 0 0
778778 0 0
779779 14705.6 0.676356
780780 0 0
781781 3640.00 0.166773
782782 0 0
783783 0 0
784784 0 0
785785 −12480.0 −0.567427
786786 0 0
787787 −27572.9 −1.24888 −0.624440 0.781073i 0.714673π-0.714673\pi
−0.624440 + 0.781073i 0.714673π0.714673\pi
788788 0 0
789789 0 0
790790 0 0
791791 −2547.67 −0.114519
792792 0 0
793793 3760.00 0.168375
794794 0 0
795795 0 0
796796 0 0
797797 31746.0 1.41092 0.705459 0.708751i 0.250741π-0.250741\pi
0.705459 + 0.708751i 0.250741π0.250741\pi
798798 0 0
799799 1676.95 0.0742506
800800 0 0
801801 0 0
802802 0 0
803803 −2612.17 −0.114796
804804 0 0
805805 −10400.0 −0.455344
806806 0 0
807807 0 0
808808 0 0
809809 35904.0 1.56034 0.780171 0.625566i 0.215132π-0.215132\pi
0.780171 + 0.625566i 0.215132π0.215132\pi
810810 0 0
811811 18994.7 0.822433 0.411217 0.911538i 0.365104π-0.365104\pi
0.411217 + 0.911538i 0.365104π0.365104\pi
812812 0 0
813813 0 0
814814 0 0
815815 −4998.60 −0.214838
816816 0 0
817817 18720.0 0.801628
818818 0 0
819819 0 0
820820 0 0
821821 −42878.0 −1.82272 −0.911360 0.411611i 0.864966π-0.864966\pi
−0.911360 + 0.411611i 0.864966π0.864966\pi
822822 0 0
823823 −15431.2 −0.653580 −0.326790 0.945097i 0.605967π-0.605967\pi
−0.326790 + 0.945097i 0.605967π0.605967\pi
824824 0 0
825825 0 0
826826 0 0
827827 −20574.9 −0.865125 −0.432563 0.901604i 0.642391π-0.642391\pi
−0.432563 + 0.901604i 0.642391π0.642391\pi
828828 0 0
829829 36634.0 1.53480 0.767401 0.641167i 0.221549π-0.221549\pi
0.767401 + 0.641167i 0.221549π0.221549\pi
830830 0 0
831831 0 0
832832 0 0
833833 2158.00 0.0897602
834834 0 0
835835 3224.90 0.133656
836836 0 0
837837 0 0
838838 0 0
839839 2192.93 0.0902366 0.0451183 0.998982i 0.485634π-0.485634\pi
0.0451183 + 0.998982i 0.485634π0.485634\pi
840840 0 0
841841 −21473.0 −0.880438
842842 0 0
843843 0 0
844844 0 0
845845 −10665.0 −0.434186
846846 0 0
847847 −17269.4 −0.700569
848848 0 0
849849 0 0
850850 0 0
851851 15995.5 0.644324
852852 0 0
853853 9028.00 0.362383 0.181192 0.983448i 0.442005π-0.442005\pi
0.181192 + 0.983448i 0.442005π0.442005\pi
854854 0 0
855855 0 0
856856 0 0
857857 44886.0 1.78912 0.894560 0.446947i 0.147489π-0.147489\pi
0.894560 + 0.446947i 0.147489π0.147489\pi
858858 0 0
859859 24864.0 0.987601 0.493800 0.869575i 0.335607π-0.335607\pi
0.493800 + 0.869575i 0.335607π0.335607\pi
860860 0 0
861861 0 0
862862 0 0
863863 −47599.6 −1.87753 −0.938765 0.344558i 0.888029π-0.888029\pi
−0.938765 + 0.344558i 0.888029π0.888029\pi
864864 0 0
865865 10090.0 0.396613
866866 0 0
867867 0 0
868868 0 0
869869 2080.00 0.0811958
870870 0 0
871871 −5933.82 −0.230838
872872 0 0
873873 0 0
874874 0 0
875875 2015.56 0.0778726
876876 0 0
877877 −25016.0 −0.963204 −0.481602 0.876390i 0.659945π-0.659945\pi
−0.481602 + 0.876390i 0.659945π0.659945\pi
878878 0 0
879879 0 0
880880 0 0
881881 −35532.0 −1.35880 −0.679400 0.733768i 0.737760π-0.737760\pi
−0.679400 + 0.733768i 0.737760π0.737760\pi
882882 0 0
883883 29249.9 1.11476 0.557382 0.830256i 0.311806π-0.311806\pi
0.557382 + 0.830256i 0.311806π0.311806\pi
884884 0 0
885885 0 0
886886 0 0
887887 −13609.1 −0.515162 −0.257581 0.966257i 0.582925π-0.582925\pi
−0.257581 + 0.966257i 0.582925π0.582925\pi
888888 0 0
889889 −21580.0 −0.814139
890890 0 0
891891 0 0
892892 0 0
893893 6240.00 0.233834
894894 0 0
895895 −14431.4 −0.538983
896896 0 0
897897 0 0
898898 0 0
899899 6965.79 0.258423
900900 0 0
901901 2028.00 0.0749861
902902 0 0
903903 0 0
904904 0 0
905905 −6710.00 −0.246462
906906 0 0
907907 −24186.8 −0.885456 −0.442728 0.896656i 0.645989π-0.645989\pi
−0.442728 + 0.896656i 0.645989π0.645989\pi
908908 0 0
909909 0 0
910910 0 0
911911 22574.3 0.820989 0.410494 0.911863i 0.365356π-0.365356\pi
0.410494 + 0.911863i 0.365356π0.365356\pi
912912 0 0
913913 −19240.0 −0.697427
914914 0 0
915915 0 0
916916 0 0
917917 15340.0 0.552422
918918 0 0
919919 48760.5 1.75023 0.875115 0.483915i 0.160786π-0.160786\pi
0.875115 + 0.483915i 0.160786π0.160786\pi
920920 0 0
921921 0 0
922922 0 0
923923 −1805.95 −0.0644024
924924 0 0
925925 −3100.00 −0.110192
926926 0 0
927927 0 0
928928 0 0
929929 2500.00 0.0882910 0.0441455 0.999025i 0.485943π-0.485943\pi
0.0441455 + 0.999025i 0.485943π0.485943\pi
930930 0 0
931931 8030.01 0.282678
932932 0 0
933933 0 0
934934 0 0
935935 −2096.19 −0.0733183
936936 0 0
937937 25114.0 0.875602 0.437801 0.899072i 0.355758π-0.355758\pi
0.437801 + 0.899072i 0.355758π0.355758\pi
938938 0 0
939939 0 0
940940 0 0
941941 −29090.0 −1.00777 −0.503883 0.863772i 0.668096π-0.668096\pi
−0.503883 + 0.863772i 0.668096π0.668096\pi
942942 0 0
943943 19607.4 0.677100
944944 0 0
945945 0 0
946946 0 0
947947 56693.8 1.94541 0.972704 0.232051i 0.0745435π-0.0745435\pi
0.972704 + 0.232051i 0.0745435π0.0745435\pi
948948 0 0
949949 1296.00 0.0443308
950950 0 0
951951 0 0
952952 0 0
953953 17202.0 0.584709 0.292354 0.956310i 0.405561π-0.405561\pi
0.292354 + 0.956310i 0.405561π0.405561\pi
954954 0 0
955955 20800.6 0.704809
956956 0 0
957957 0 0
958958 0 0
959959 7643.02 0.257358
960960 0 0
961961 −13151.0 −0.441442
962962 0 0
963963 0 0
964964 0 0
965965 −21810.0 −0.727553
966966 0 0
967967 3821.51 0.127085 0.0635426 0.997979i 0.479760π-0.479760\pi
0.0635426 + 0.997979i 0.479760π0.479760\pi
968968 0 0
969969 0 0
970970 0 0
971971 38811.7 1.28273 0.641363 0.767238i 0.278369π-0.278369\pi
0.641363 + 0.767238i 0.278369π0.278369\pi
972972 0 0
973973 25480.0 0.839518
974974 0 0
975975 0 0
976976 0 0
977977 29046.0 0.951140 0.475570 0.879678i 0.342242π-0.342242\pi
0.475570 + 0.879678i 0.342242π0.342242\pi
978978 0 0
979979 −1999.44 −0.0652731
980980 0 0
981981 0 0
982982 0 0
983983 −23735.3 −0.770130 −0.385065 0.922889i 0.625821π-0.625821\pi
−0.385065 + 0.922889i 0.625821π0.625821\pi
984984 0 0
985985 6870.00 0.222230
986986 0 0
987987 0 0
988988 0 0
989989 24960.0 0.802509
990990 0 0
991991 33377.7 1.06991 0.534954 0.844881i 0.320329π-0.320329\pi
0.534954 + 0.844881i 0.320329π0.320329\pi
992992 0 0
993993 0 0
994994 0 0
995995 −9513.46 −0.303113
996996 0 0
997997 −52336.0 −1.66249 −0.831243 0.555910i 0.812370π-0.812370\pi
−0.831243 + 0.555910i 0.812370π0.812370\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 1440.4.a.bc.1.2 yes 2
3.2 odd 2 1440.4.a.u.1.2 yes 2
4.3 odd 2 inner 1440.4.a.bc.1.1 yes 2
12.11 even 2 1440.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.4.a.u.1.1 2 12.11 even 2
1440.4.a.u.1.2 yes 2 3.2 odd 2
1440.4.a.bc.1.1 yes 2 4.3 odd 2 inner
1440.4.a.bc.1.2 yes 2 1.1 even 1 trivial