Properties

Label 1080.4.a.i.1.3
Level 10801080
Weight 44
Character 1080.1
Self dual yes
Analytic conductor 63.72263.722
Analytic rank 11
Dimension 33
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,4,Mod(1,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 1080=23335 1080 = 2^{3} \cdot 3^{3} \cdot 5
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1080.a (trivial)

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: yes
Analytic conductor: 63.722062806263.7220628062
Analytic rank: 11
Dimension: 33
Coefficient field: 3.3.697.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x37x5 x^{3} - 7x - 5 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 2332 2^{3}\cdot 3^{2}
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.3
Root 0.782816-0.782816 of defining polynomial
Character χ\chi == 1080.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+5.00000q5+11.6263q747.4851q11+84.2727q1326.9488q17152.060q19177.082q23+25.0000q25+60.0913q2992.1613q31+58.1315q35221.798q37+115.099q41+383.827q43+317.275q47207.829q49257.738q53237.426q55642.576q59662.092q61+421.363q65+597.173q67500.152q71+989.949q73552.076q77517.554q79605.867q83134.744q851519.15q89+979.780q91760.301q95+742.835q97+O(q100)q+5.00000 q^{5} +11.6263 q^{7} -47.4851 q^{11} +84.2727 q^{13} -26.9488 q^{17} -152.060 q^{19} -177.082 q^{23} +25.0000 q^{25} +60.0913 q^{29} -92.1613 q^{31} +58.1315 q^{35} -221.798 q^{37} +115.099 q^{41} +383.827 q^{43} +317.275 q^{47} -207.829 q^{49} -257.738 q^{53} -237.426 q^{55} -642.576 q^{59} -662.092 q^{61} +421.363 q^{65} +597.173 q^{67} -500.152 q^{71} +989.949 q^{73} -552.076 q^{77} -517.554 q^{79} -605.867 q^{83} -134.744 q^{85} -1519.15 q^{89} +979.780 q^{91} -760.301 q^{95} +742.835 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+15q524q7+6q11+48q1327q17195q1927q23+75q25+60q29279q31120q35138q37+66q41+222q43264q47237q49507q53+336q97+O(q100) 3 q + 15 q^{5} - 24 q^{7} + 6 q^{11} + 48 q^{13} - 27 q^{17} - 195 q^{19} - 27 q^{23} + 75 q^{25} + 60 q^{29} - 279 q^{31} - 120 q^{35} - 138 q^{37} + 66 q^{41} + 222 q^{43} - 264 q^{47} - 237 q^{49} - 507 q^{53}+ \cdots - 336 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 11.6263 0.627761 0.313881 0.949462i 0.398371π-0.398371\pi
0.313881 + 0.949462i 0.398371π0.398371\pi
88 0 0
99 0 0
1010 0 0
1111 −47.4851 −1.30157 −0.650787 0.759261i 0.725561π-0.725561\pi
−0.650787 + 0.759261i 0.725561π0.725561\pi
1212 0 0
1313 84.2727 1.79793 0.898963 0.438024i 0.144322π-0.144322\pi
0.898963 + 0.438024i 0.144322π0.144322\pi
1414 0 0
1515 0 0
1616 0 0
1717 −26.9488 −0.384474 −0.192237 0.981349i 0.561574π-0.561574\pi
−0.192237 + 0.981349i 0.561574π0.561574\pi
1818 0 0
1919 −152.060 −1.83605 −0.918027 0.396518i 0.870218π-0.870218\pi
−0.918027 + 0.396518i 0.870218π0.870218\pi
2020 0 0
2121 0 0
2222 0 0
2323 −177.082 −1.60539 −0.802697 0.596387i 0.796603π-0.796603\pi
−0.802697 + 0.596387i 0.796603π0.796603\pi
2424 0 0
2525 25.0000 0.200000
2626 0 0
2727 0 0
2828 0 0
2929 60.0913 0.384782 0.192391 0.981318i 0.438376π-0.438376\pi
0.192391 + 0.981318i 0.438376π0.438376\pi
3030 0 0
3131 −92.1613 −0.533957 −0.266978 0.963703i 0.586025π-0.586025\pi
−0.266978 + 0.963703i 0.586025π0.586025\pi
3232 0 0
3333 0 0
3434 0 0
3535 58.1315 0.280743
3636 0 0
3737 −221.798 −0.985496 −0.492748 0.870172i 0.664008π-0.664008\pi
−0.492748 + 0.870172i 0.664008π0.664008\pi
3838 0 0
3939 0 0
4040 0 0
4141 115.099 0.438426 0.219213 0.975677i 0.429651π-0.429651\pi
0.219213 + 0.975677i 0.429651π0.429651\pi
4242 0 0
4343 383.827 1.36123 0.680617 0.732639i 0.261712π-0.261712\pi
0.680617 + 0.732639i 0.261712π0.261712\pi
4444 0 0
4545 0 0
4646 0 0
4747 317.275 0.984666 0.492333 0.870407i 0.336144π-0.336144\pi
0.492333 + 0.870407i 0.336144π0.336144\pi
4848 0 0
4949 −207.829 −0.605916
5050 0 0
5151 0 0
5252 0 0
5353 −257.738 −0.667981 −0.333991 0.942576i 0.608395π-0.608395\pi
−0.333991 + 0.942576i 0.608395π0.608395\pi
5454 0 0
5555 −237.426 −0.582081
5656 0 0
5757 0 0
5858 0 0
5959 −642.576 −1.41790 −0.708951 0.705257i 0.750831π-0.750831\pi
−0.708951 + 0.705257i 0.750831π0.750831\pi
6060 0 0
6161 −662.092 −1.38971 −0.694854 0.719150i 0.744531π-0.744531\pi
−0.694854 + 0.719150i 0.744531π0.744531\pi
6262 0 0
6363 0 0
6464 0 0
6565 421.363 0.804057
6666 0 0
6767 597.173 1.08890 0.544450 0.838794i 0.316739π-0.316739\pi
0.544450 + 0.838794i 0.316739π0.316739\pi
6868 0 0
6969 0 0
7070 0 0
7171 −500.152 −0.836016 −0.418008 0.908443i 0.637272π-0.637272\pi
−0.418008 + 0.908443i 0.637272π0.637272\pi
7272 0 0
7373 989.949 1.58719 0.793594 0.608447i 0.208207π-0.208207\pi
0.793594 + 0.608447i 0.208207π0.208207\pi
7474 0 0
7575 0 0
7676 0 0
7777 −552.076 −0.817077
7878 0 0
7979 −517.554 −0.737081 −0.368540 0.929612i 0.620142π-0.620142\pi
−0.368540 + 0.929612i 0.620142π0.620142\pi
8080 0 0
8181 0 0
8282 0 0
8383 −605.867 −0.801235 −0.400617 0.916245i 0.631204π-0.631204\pi
−0.400617 + 0.916245i 0.631204π0.631204\pi
8484 0 0
8585 −134.744 −0.171942
8686 0 0
8787 0 0
8888 0 0
8989 −1519.15 −1.80932 −0.904658 0.426138i 0.859874π-0.859874\pi
−0.904658 + 0.426138i 0.859874π0.859874\pi
9090 0 0
9191 979.780 1.12867
9292 0 0
9393 0 0
9494 0 0
9595 −760.301 −0.821108
9696 0 0
9797 742.835 0.777561 0.388781 0.921330i 0.372896π-0.372896\pi
0.388781 + 0.921330i 0.372896π0.372896\pi
9898 0 0
9999 0 0
100100 0 0
101101 −1066.26 −1.05047 −0.525233 0.850959i 0.676022π-0.676022\pi
−0.525233 + 0.850959i 0.676022π0.676022\pi
102102 0 0
103103 −471.508 −0.451059 −0.225529 0.974236i 0.572411π-0.572411\pi
−0.225529 + 0.974236i 0.572411π0.572411\pi
104104 0 0
105105 0 0
106106 0 0
107107 1746.59 1.57803 0.789014 0.614375i 0.210592π-0.210592\pi
0.789014 + 0.614375i 0.210592π0.210592\pi
108108 0 0
109109 1596.56 1.40296 0.701480 0.712689i 0.252523π-0.252523\pi
0.701480 + 0.712689i 0.252523π0.252523\pi
110110 0 0
111111 0 0
112112 0 0
113113 1622.95 1.35110 0.675550 0.737315i 0.263906π-0.263906\pi
0.675550 + 0.737315i 0.263906π0.263906\pi
114114 0 0
115115 −885.408 −0.717954
116116 0 0
117117 0 0
118118 0 0
119119 −313.315 −0.241358
120120 0 0
121121 923.837 0.694092
122122 0 0
123123 0 0
124124 0 0
125125 125.000 0.0894427
126126 0 0
127127 −1306.32 −0.912730 −0.456365 0.889793i 0.650849π-0.650849\pi
−0.456365 + 0.889793i 0.650849π0.650849\pi
128128 0 0
129129 0 0
130130 0 0
131131 539.259 0.359659 0.179829 0.983698i 0.442445π-0.442445\pi
0.179829 + 0.983698i 0.442445π0.442445\pi
132132 0 0
133133 −1767.90 −1.15260
134134 0 0
135135 0 0
136136 0 0
137137 −2877.22 −1.79429 −0.897144 0.441738i 0.854362π-0.854362\pi
−0.897144 + 0.441738i 0.854362π0.854362\pi
138138 0 0
139139 −1479.89 −0.903038 −0.451519 0.892261i 0.649118π-0.649118\pi
−0.451519 + 0.892261i 0.649118π0.649118\pi
140140 0 0
141141 0 0
142142 0 0
143143 −4001.70 −2.34013
144144 0 0
145145 300.457 0.172080
146146 0 0
147147 0 0
148148 0 0
149149 −1398.13 −0.768719 −0.384359 0.923184i 0.625578π-0.625578\pi
−0.384359 + 0.923184i 0.625578π0.625578\pi
150150 0 0
151151 −2251.44 −1.21337 −0.606687 0.794941i 0.707502π-0.707502\pi
−0.606687 + 0.794941i 0.707502π0.707502\pi
152152 0 0
153153 0 0
154154 0 0
155155 −460.806 −0.238793
156156 0 0
157157 764.698 0.388723 0.194362 0.980930i 0.437736π-0.437736\pi
0.194362 + 0.980930i 0.437736π0.437736\pi
158158 0 0
159159 0 0
160160 0 0
161161 −2058.81 −1.00780
162162 0 0
163163 −1862.15 −0.894817 −0.447408 0.894330i 0.647653π-0.647653\pi
−0.447408 + 0.894330i 0.647653π0.647653\pi
164164 0 0
165165 0 0
166166 0 0
167167 −3324.33 −1.54038 −0.770192 0.637812i 0.779840π-0.779840\pi
−0.770192 + 0.637812i 0.779840π0.779840\pi
168168 0 0
169169 4904.89 2.23254
170170 0 0
171171 0 0
172172 0 0
173173 −1344.67 −0.590943 −0.295471 0.955352i 0.595477π-0.595477\pi
−0.295471 + 0.955352i 0.595477π0.595477\pi
174174 0 0
175175 290.658 0.125552
176176 0 0
177177 0 0
178178 0 0
179179 −2086.39 −0.871195 −0.435598 0.900142i 0.643463π-0.643463\pi
−0.435598 + 0.900142i 0.643463π0.643463\pi
180180 0 0
181181 698.713 0.286933 0.143467 0.989655i 0.454175π-0.454175\pi
0.143467 + 0.989655i 0.454175π0.454175\pi
182182 0 0
183183 0 0
184184 0 0
185185 −1108.99 −0.440727
186186 0 0
187187 1279.67 0.500421
188188 0 0
189189 0 0
190190 0 0
191191 −2267.66 −0.859067 −0.429534 0.903051i 0.641322π-0.641322\pi
−0.429534 + 0.903051i 0.641322π0.641322\pi
192192 0 0
193193 −3747.64 −1.39772 −0.698862 0.715256i 0.746310π-0.746310\pi
−0.698862 + 0.715256i 0.746310π0.746310\pi
194194 0 0
195195 0 0
196196 0 0
197197 293.859 0.106277 0.0531385 0.998587i 0.483078π-0.483078\pi
0.0531385 + 0.998587i 0.483078π0.483078\pi
198198 0 0
199199 2552.33 0.909197 0.454598 0.890697i 0.349783π-0.349783\pi
0.454598 + 0.890697i 0.349783π0.349783\pi
200200 0 0
201201 0 0
202202 0 0
203203 698.640 0.241551
204204 0 0
205205 575.496 0.196070
206206 0 0
207207 0 0
208208 0 0
209209 7220.60 2.38976
210210 0 0
211211 −5416.57 −1.76726 −0.883630 0.468185i 0.844908π-0.844908\pi
−0.883630 + 0.468185i 0.844908π0.844908\pi
212212 0 0
213213 0 0
214214 0 0
215215 1919.14 0.608763
216216 0 0
217217 −1071.49 −0.335197
218218 0 0
219219 0 0
220220 0 0
221221 −2271.05 −0.691256
222222 0 0
223223 4739.25 1.42316 0.711578 0.702607i 0.247981π-0.247981\pi
0.711578 + 0.702607i 0.247981π0.247981\pi
224224 0 0
225225 0 0
226226 0 0
227227 −2489.12 −0.727790 −0.363895 0.931440i 0.618553π-0.618553\pi
−0.363895 + 0.931440i 0.618553π0.618553\pi
228228 0 0
229229 4239.27 1.22331 0.611657 0.791123i 0.290503π-0.290503\pi
0.611657 + 0.791123i 0.290503π0.290503\pi
230230 0 0
231231 0 0
232232 0 0
233233 −1200.46 −0.337531 −0.168765 0.985656i 0.553978π-0.553978\pi
−0.168765 + 0.985656i 0.553978π0.553978\pi
234234 0 0
235235 1586.37 0.440356
236236 0 0
237237 0 0
238238 0 0
239239 4609.67 1.24759 0.623796 0.781587i 0.285589π-0.285589\pi
0.623796 + 0.781587i 0.285589π0.285589\pi
240240 0 0
241241 −2121.92 −0.567157 −0.283578 0.958949i 0.591522π-0.591522\pi
−0.283578 + 0.958949i 0.591522π0.591522\pi
242242 0 0
243243 0 0
244244 0 0
245245 −1039.15 −0.270974
246246 0 0
247247 −12814.5 −3.30109
248248 0 0
249249 0 0
250250 0 0
251251 5032.56 1.26555 0.632773 0.774337i 0.281916π-0.281916\pi
0.632773 + 0.774337i 0.281916π0.281916\pi
252252 0 0
253253 8408.74 2.08954
254254 0 0
255255 0 0
256256 0 0
257257 −5721.18 −1.38863 −0.694314 0.719672i 0.744292π-0.744292\pi
−0.694314 + 0.719672i 0.744292π0.744292\pi
258258 0 0
259259 −2578.69 −0.618657
260260 0 0
261261 0 0
262262 0 0
263263 3348.24 0.785025 0.392512 0.919747i 0.371606π-0.371606\pi
0.392512 + 0.919747i 0.371606π0.371606\pi
264264 0 0
265265 −1288.69 −0.298730
266266 0 0
267267 0 0
268268 0 0
269269 −1726.71 −0.391373 −0.195686 0.980667i 0.562693π-0.562693\pi
−0.195686 + 0.980667i 0.562693π0.562693\pi
270270 0 0
271271 6347.21 1.42275 0.711375 0.702812i 0.248073π-0.248073\pi
0.711375 + 0.702812i 0.248073π0.248073\pi
272272 0 0
273273 0 0
274274 0 0
275275 −1187.13 −0.260315
276276 0 0
277277 −591.832 −0.128374 −0.0641872 0.997938i 0.520445π-0.520445\pi
−0.0641872 + 0.997938i 0.520445π0.520445\pi
278278 0 0
279279 0 0
280280 0 0
281281 −1405.24 −0.298326 −0.149163 0.988813i 0.547658π-0.547658\pi
−0.149163 + 0.988813i 0.547658π0.547658\pi
282282 0 0
283283 1417.53 0.297750 0.148875 0.988856i 0.452435π-0.452435\pi
0.148875 + 0.988856i 0.452435π0.452435\pi
284284 0 0
285285 0 0
286286 0 0
287287 1338.18 0.275227
288288 0 0
289289 −4186.76 −0.852180
290290 0 0
291291 0 0
292292 0 0
293293 −5127.59 −1.02238 −0.511189 0.859468i 0.670795π-0.670795\pi
−0.511189 + 0.859468i 0.670795π0.670795\pi
294294 0 0
295295 −3212.88 −0.634105
296296 0 0
297297 0 0
298298 0 0
299299 −14923.2 −2.88638
300300 0 0
301301 4462.49 0.854531
302302 0 0
303303 0 0
304304 0 0
305305 −3310.46 −0.621497
306306 0 0
307307 2481.35 0.461296 0.230648 0.973037i 0.425915π-0.425915\pi
0.230648 + 0.973037i 0.425915π0.425915\pi
308308 0 0
309309 0 0
310310 0 0
311311 −201.357 −0.0367135 −0.0183567 0.999832i 0.505843π-0.505843\pi
−0.0183567 + 0.999832i 0.505843π0.505843\pi
312312 0 0
313313 1850.89 0.334244 0.167122 0.985936i 0.446553π-0.446553\pi
0.167122 + 0.985936i 0.446553π0.446553\pi
314314 0 0
315315 0 0
316316 0 0
317317 2257.74 0.400024 0.200012 0.979793i 0.435902π-0.435902\pi
0.200012 + 0.979793i 0.435902π0.435902\pi
318318 0 0
319319 −2853.44 −0.500822
320320 0 0
321321 0 0
322322 0 0
323323 4097.85 0.705915
324324 0 0
325325 2106.82 0.359585
326326 0 0
327327 0 0
328328 0 0
329329 3688.73 0.618135
330330 0 0
331331 −4038.32 −0.670592 −0.335296 0.942113i 0.608836π-0.608836\pi
−0.335296 + 0.942113i 0.608836π0.608836\pi
332332 0 0
333333 0 0
334334 0 0
335335 2985.86 0.486971
336336 0 0
337337 −3552.00 −0.574154 −0.287077 0.957908i 0.592684π-0.592684\pi
−0.287077 + 0.957908i 0.592684π0.592684\pi
338338 0 0
339339 0 0
340340 0 0
341341 4376.29 0.694983
342342 0 0
343343 −6404.11 −1.00813
344344 0 0
345345 0 0
346346 0 0
347347 −781.347 −0.120879 −0.0604394 0.998172i 0.519250π-0.519250\pi
−0.0604394 + 0.998172i 0.519250π0.519250\pi
348348 0 0
349349 2492.19 0.382246 0.191123 0.981566i 0.438787π-0.438787\pi
0.191123 + 0.981566i 0.438787π0.438787\pi
350350 0 0
351351 0 0
352352 0 0
353353 9230.11 1.39170 0.695849 0.718189i 0.255028π-0.255028\pi
0.695849 + 0.718189i 0.255028π0.255028\pi
354354 0 0
355355 −2500.76 −0.373878
356356 0 0
357357 0 0
358358 0 0
359359 925.350 0.136039 0.0680196 0.997684i 0.478332π-0.478332\pi
0.0680196 + 0.997684i 0.478332π0.478332\pi
360360 0 0
361361 16263.3 2.37109
362362 0 0
363363 0 0
364364 0 0
365365 4949.75 0.709812
366366 0 0
367367 −9070.86 −1.29018 −0.645089 0.764108i 0.723180π-0.723180\pi
−0.645089 + 0.764108i 0.723180π0.723180\pi
368368 0 0
369369 0 0
370370 0 0
371371 −2996.54 −0.419333
372372 0 0
373373 9388.07 1.30321 0.651603 0.758560i 0.274097π-0.274097\pi
0.651603 + 0.758560i 0.274097π0.274097\pi
374374 0 0
375375 0 0
376376 0 0
377377 5064.06 0.691810
378378 0 0
379379 10172.2 1.37866 0.689328 0.724449i 0.257906π-0.257906\pi
0.689328 + 0.724449i 0.257906π0.257906\pi
380380 0 0
381381 0 0
382382 0 0
383383 1505.63 0.200873 0.100436 0.994943i 0.467976π-0.467976\pi
0.100436 + 0.994943i 0.467976π0.467976\pi
384384 0 0
385385 −2760.38 −0.365408
386386 0 0
387387 0 0
388388 0 0
389389 −3179.65 −0.414433 −0.207216 0.978295i 0.566440π-0.566440\pi
−0.207216 + 0.978295i 0.566440π0.566440\pi
390390 0 0
391391 4772.15 0.617232
392392 0 0
393393 0 0
394394 0 0
395395 −2587.77 −0.329632
396396 0 0
397397 5360.23 0.677638 0.338819 0.940852i 0.389973π-0.389973\pi
0.338819 + 0.940852i 0.389973π0.389973\pi
398398 0 0
399399 0 0
400400 0 0
401401 12550.4 1.56294 0.781468 0.623945i 0.214471π-0.214471\pi
0.781468 + 0.623945i 0.214471π0.214471\pi
402402 0 0
403403 −7766.68 −0.960015
404404 0 0
405405 0 0
406406 0 0
407407 10532.1 1.28270
408408 0 0
409409 −5395.42 −0.652289 −0.326144 0.945320i 0.605750π-0.605750\pi
−0.326144 + 0.945320i 0.605750π0.605750\pi
410410 0 0
411411 0 0
412412 0 0
413413 −7470.78 −0.890105
414414 0 0
415415 −3029.33 −0.358323
416416 0 0
417417 0 0
418418 0 0
419419 4697.84 0.547743 0.273871 0.961766i 0.411696π-0.411696\pi
0.273871 + 0.961766i 0.411696π0.411696\pi
420420 0 0
421421 8845.86 1.02404 0.512020 0.858974i 0.328897π-0.328897\pi
0.512020 + 0.858974i 0.328897π0.328897\pi
422422 0 0
423423 0 0
424424 0 0
425425 −673.721 −0.0768948
426426 0 0
427427 −7697.69 −0.872406
428428 0 0
429429 0 0
430430 0 0
431431 −5676.98 −0.634456 −0.317228 0.948349i 0.602752π-0.602752\pi
−0.317228 + 0.948349i 0.602752π0.602752\pi
432432 0 0
433433 17273.3 1.91710 0.958549 0.284927i 0.0919695π-0.0919695\pi
0.958549 + 0.284927i 0.0919695π0.0919695\pi
434434 0 0
435435 0 0
436436 0 0
437437 26927.1 2.94759
438438 0 0
439439 −3583.55 −0.389598 −0.194799 0.980843i 0.562405π-0.562405\pi
−0.194799 + 0.980843i 0.562405π0.562405\pi
440440 0 0
441441 0 0
442442 0 0
443443 6937.40 0.744031 0.372016 0.928227i 0.378667π-0.378667\pi
0.372016 + 0.928227i 0.378667π0.378667\pi
444444 0 0
445445 −7595.73 −0.809151
446446 0 0
447447 0 0
448448 0 0
449449 12445.1 1.30806 0.654032 0.756467i 0.273076π-0.273076\pi
0.654032 + 0.756467i 0.273076π0.273076\pi
450450 0 0
451451 −5465.50 −0.570644
452452 0 0
453453 0 0
454454 0 0
455455 4898.90 0.504756
456456 0 0
457457 −12112.2 −1.23979 −0.619896 0.784684i 0.712825π-0.712825\pi
−0.619896 + 0.784684i 0.712825π0.712825\pi
458458 0 0
459459 0 0
460460 0 0
461461 4932.76 0.498355 0.249178 0.968458i 0.419840π-0.419840\pi
0.249178 + 0.968458i 0.419840π0.419840\pi
462462 0 0
463463 −12428.0 −1.24747 −0.623734 0.781637i 0.714385π-0.714385\pi
−0.623734 + 0.781637i 0.714385π0.714385\pi
464464 0 0
465465 0 0
466466 0 0
467467 4549.10 0.450765 0.225383 0.974270i 0.427637π-0.427637\pi
0.225383 + 0.974270i 0.427637π0.427637\pi
468468 0 0
469469 6942.91 0.683569
470470 0 0
471471 0 0
472472 0 0
473473 −18226.1 −1.77175
474474 0 0
475475 −3801.51 −0.367211
476476 0 0
477477 0 0
478478 0 0
479479 5122.02 0.488583 0.244292 0.969702i 0.421445π-0.421445\pi
0.244292 + 0.969702i 0.421445π0.421445\pi
480480 0 0
481481 −18691.5 −1.77185
482482 0 0
483483 0 0
484484 0 0
485485 3714.17 0.347736
486486 0 0
487487 −8650.25 −0.804888 −0.402444 0.915445i 0.631839π-0.631839\pi
−0.402444 + 0.915445i 0.631839π0.631839\pi
488488 0 0
489489 0 0
490490 0 0
491491 985.959 0.0906226 0.0453113 0.998973i 0.485572π-0.485572\pi
0.0453113 + 0.998973i 0.485572π0.485572\pi
492492 0 0
493493 −1619.39 −0.147939
494494 0 0
495495 0 0
496496 0 0
497497 −5814.92 −0.524819
498498 0 0
499499 3131.41 0.280925 0.140462 0.990086i 0.455141π-0.455141\pi
0.140462 + 0.990086i 0.455141π0.455141\pi
500500 0 0
501501 0 0
502502 0 0
503503 13765.8 1.22025 0.610126 0.792304i 0.291119π-0.291119\pi
0.610126 + 0.792304i 0.291119π0.291119\pi
504504 0 0
505505 −5331.31 −0.469783
506506 0 0
507507 0 0
508508 0 0
509509 −8241.51 −0.717679 −0.358839 0.933399i 0.616827π-0.616827\pi
−0.358839 + 0.933399i 0.616827π0.616827\pi
510510 0 0
511511 11509.4 0.996376
512512 0 0
513513 0 0
514514 0 0
515515 −2357.54 −0.201720
516516 0 0
517517 −15065.8 −1.28161
518518 0 0
519519 0 0
520520 0 0
521521 2556.69 0.214991 0.107496 0.994206i 0.465717π-0.465717\pi
0.107496 + 0.994206i 0.465717π0.465717\pi
522522 0 0
523523 8627.71 0.721345 0.360672 0.932693i 0.382547π-0.382547\pi
0.360672 + 0.932693i 0.382547π0.382547\pi
524524 0 0
525525 0 0
526526 0 0
527527 2483.64 0.205292
528528 0 0
529529 19190.9 1.57729
530530 0 0
531531 0 0
532532 0 0
533533 9699.72 0.788258
534534 0 0
535535 8732.94 0.705716
536536 0 0
537537 0 0
538538 0 0
539539 9868.79 0.788643
540540 0 0
541541 1265.74 0.100589 0.0502943 0.998734i 0.483984π-0.483984\pi
0.0502943 + 0.998734i 0.483984π0.483984\pi
542542 0 0
543543 0 0
544544 0 0
545545 7982.79 0.627423
546546 0 0
547547 −6990.29 −0.546405 −0.273202 0.961957i 0.588083π-0.588083\pi
−0.273202 + 0.961957i 0.588083π0.588083\pi
548548 0 0
549549 0 0
550550 0 0
551551 −9137.50 −0.706481
552552 0 0
553553 −6017.24 −0.462711
554554 0 0
555555 0 0
556556 0 0
557557 −19679.0 −1.49699 −0.748496 0.663139i 0.769224π-0.769224\pi
−0.748496 + 0.663139i 0.769224π0.769224\pi
558558 0 0
559559 32346.2 2.44740
560560 0 0
561561 0 0
562562 0 0
563563 12913.2 0.966652 0.483326 0.875440i 0.339429π-0.339429\pi
0.483326 + 0.875440i 0.339429π0.339429\pi
564564 0 0
565565 8114.74 0.604230
566566 0 0
567567 0 0
568568 0 0
569569 19908.8 1.46682 0.733411 0.679786i 0.237927π-0.237927\pi
0.733411 + 0.679786i 0.237927π0.237927\pi
570570 0 0
571571 −13278.9 −0.973212 −0.486606 0.873621i 0.661765π-0.661765\pi
−0.486606 + 0.873621i 0.661765π0.661765\pi
572572 0 0
573573 0 0
574574 0 0
575575 −4427.04 −0.321079
576576 0 0
577577 −2538.04 −0.183120 −0.0915598 0.995800i 0.529185π-0.529185\pi
−0.0915598 + 0.995800i 0.529185π0.529185\pi
578578 0 0
579579 0 0
580580 0 0
581581 −7043.99 −0.502984
582582 0 0
583583 12238.7 0.869426
584584 0 0
585585 0 0
586586 0 0
587587 15312.4 1.07668 0.538341 0.842727i 0.319051π-0.319051\pi
0.538341 + 0.842727i 0.319051π0.319051\pi
588588 0 0
589589 14014.1 0.980373
590590 0 0
591591 0 0
592592 0 0
593593 −196.207 −0.0135873 −0.00679363 0.999977i 0.502162π-0.502162\pi
−0.00679363 + 0.999977i 0.502162π0.502162\pi
594594 0 0
595595 −1566.58 −0.107939
596596 0 0
597597 0 0
598598 0 0
599599 −13738.3 −0.937113 −0.468557 0.883433i 0.655226π-0.655226\pi
−0.468557 + 0.883433i 0.655226π0.655226\pi
600600 0 0
601601 1702.20 0.115531 0.0577655 0.998330i 0.481602π-0.481602\pi
0.0577655 + 0.998330i 0.481602π0.481602\pi
602602 0 0
603603 0 0
604604 0 0
605605 4619.18 0.310408
606606 0 0
607607 −11802.7 −0.789223 −0.394611 0.918848i 0.629121π-0.629121\pi
−0.394611 + 0.918848i 0.629121π0.629121\pi
608608 0 0
609609 0 0
610610 0 0
611611 26737.6 1.77036
612612 0 0
613613 15365.9 1.01244 0.506218 0.862406i 0.331043π-0.331043\pi
0.506218 + 0.862406i 0.331043π0.331043\pi
614614 0 0
615615 0 0
616616 0 0
617617 −17151.5 −1.11911 −0.559555 0.828793i 0.689028π-0.689028\pi
−0.559555 + 0.828793i 0.689028π0.689028\pi
618618 0 0
619619 10081.9 0.654649 0.327324 0.944912i 0.393853π-0.393853\pi
0.327324 + 0.944912i 0.393853π0.393853\pi
620620 0 0
621621 0 0
622622 0 0
623623 −17662.1 −1.13582
624624 0 0
625625 625.000 0.0400000
626626 0 0
627627 0 0
628628 0 0
629629 5977.20 0.378898
630630 0 0
631631 −15510.6 −0.978554 −0.489277 0.872128i 0.662739π-0.662739\pi
−0.489277 + 0.872128i 0.662739π0.662739\pi
632632 0 0
633633 0 0
634634 0 0
635635 −6531.58 −0.408185
636636 0 0
637637 −17514.3 −1.08939
638638 0 0
639639 0 0
640640 0 0
641641 10534.8 0.649143 0.324571 0.945861i 0.394780π-0.394780\pi
0.324571 + 0.945861i 0.394780π0.394780\pi
642642 0 0
643643 −627.807 −0.0385044 −0.0192522 0.999815i 0.506129π-0.506129\pi
−0.0192522 + 0.999815i 0.506129π0.506129\pi
644644 0 0
645645 0 0
646646 0 0
647647 −2880.41 −0.175024 −0.0875121 0.996163i 0.527892π-0.527892\pi
−0.0875121 + 0.996163i 0.527892π0.527892\pi
648648 0 0
649649 30512.8 1.84550
650650 0 0
651651 0 0
652652 0 0
653653 10664.9 0.639127 0.319564 0.947565i 0.396464π-0.396464\pi
0.319564 + 0.947565i 0.396464π0.396464\pi
654654 0 0
655655 2696.30 0.160844
656656 0 0
657657 0 0
658658 0 0
659659 4300.38 0.254202 0.127101 0.991890i 0.459433π-0.459433\pi
0.127101 + 0.991890i 0.459433π0.459433\pi
660660 0 0
661661 −10325.3 −0.607578 −0.303789 0.952739i 0.598252π-0.598252\pi
−0.303789 + 0.952739i 0.598252π0.598252\pi
662662 0 0
663663 0 0
664664 0 0
665665 −8839.49 −0.515460
666666 0 0
667667 −10641.1 −0.617727
668668 0 0
669669 0 0
670670 0 0
671671 31439.5 1.80881
672672 0 0
673673 −13740.9 −0.787030 −0.393515 0.919318i 0.628741π-0.628741\pi
−0.393515 + 0.919318i 0.628741π0.628741\pi
674674 0 0
675675 0 0
676676 0 0
677677 31303.5 1.77709 0.888547 0.458785i 0.151715π-0.151715\pi
0.888547 + 0.458785i 0.151715π0.151715\pi
678678 0 0
679679 8636.42 0.488123
680680 0 0
681681 0 0
682682 0 0
683683 8369.31 0.468877 0.234438 0.972131i 0.424675π-0.424675\pi
0.234438 + 0.972131i 0.424675π0.424675\pi
684684 0 0
685685 −14386.1 −0.802430
686686 0 0
687687 0 0
688688 0 0
689689 −21720.3 −1.20098
690690 0 0
691691 −24106.1 −1.32712 −0.663559 0.748124i 0.730955π-0.730955\pi
−0.663559 + 0.748124i 0.730955π0.730955\pi
692692 0 0
693693 0 0
694694 0 0
695695 −7399.43 −0.403851
696696 0 0
697697 −3101.79 −0.168563
698698 0 0
699699 0 0
700700 0 0
701701 −30713.6 −1.65483 −0.827417 0.561589i 0.810190π-0.810190\pi
−0.827417 + 0.561589i 0.810190π0.810190\pi
702702 0 0
703703 33726.7 1.80942
704704 0 0
705705 0 0
706706 0 0
707707 −12396.7 −0.659442
708708 0 0
709709 −518.034 −0.0274403 −0.0137201 0.999906i 0.504367π-0.504367\pi
−0.0137201 + 0.999906i 0.504367π0.504367\pi
710710 0 0
711711 0 0
712712 0 0
713713 16320.1 0.857211
714714 0 0
715715 −20008.5 −1.04654
716716 0 0
717717 0 0
718718 0 0
719719 −1692.51 −0.0877885 −0.0438943 0.999036i 0.513976π-0.513976\pi
−0.0438943 + 0.999036i 0.513976π0.513976\pi
720720 0 0
721721 −5481.89 −0.283157
722722 0 0
723723 0 0
724724 0 0
725725 1502.28 0.0769564
726726 0 0
727727 −9191.77 −0.468919 −0.234459 0.972126i 0.575332π-0.575332\pi
−0.234459 + 0.972126i 0.575332π0.575332\pi
728728 0 0
729729 0 0
730730 0 0
731731 −10343.7 −0.523359
732732 0 0
733733 −1047.13 −0.0527649 −0.0263824 0.999652i 0.508399π-0.508399\pi
−0.0263824 + 0.999652i 0.508399π0.508399\pi
734734 0 0
735735 0 0
736736 0 0
737737 −28356.8 −1.41728
738738 0 0
739739 21956.9 1.09296 0.546479 0.837473i 0.315968π-0.315968\pi
0.546479 + 0.837473i 0.315968π0.315968\pi
740740 0 0
741741 0 0
742742 0 0
743743 19304.1 0.953160 0.476580 0.879131i 0.341876π-0.341876\pi
0.476580 + 0.879131i 0.341876π0.341876\pi
744744 0 0
745745 −6990.64 −0.343781
746746 0 0
747747 0 0
748748 0 0
749749 20306.4 0.990625
750750 0 0
751751 21883.0 1.06328 0.531640 0.846970i 0.321576π-0.321576\pi
0.531640 + 0.846970i 0.321576π0.321576\pi
752752 0 0
753753 0 0
754754 0 0
755755 −11257.2 −0.542637
756756 0 0
757757 13738.5 0.659624 0.329812 0.944047i 0.393015π-0.393015\pi
0.329812 + 0.944047i 0.393015π0.393015\pi
758758 0 0
759759 0 0
760760 0 0
761761 −20167.9 −0.960689 −0.480345 0.877080i 0.659488π-0.659488\pi
−0.480345 + 0.877080i 0.659488π0.659488\pi
762762 0 0
763763 18562.1 0.880724
764764 0 0
765765 0 0
766766 0 0
767767 −54151.6 −2.54928
768768 0 0
769769 −28655.9 −1.34377 −0.671885 0.740655i 0.734515π-0.734515\pi
−0.671885 + 0.740655i 0.734515π0.734515\pi
770770 0 0
771771 0 0
772772 0 0
773773 −16224.0 −0.754898 −0.377449 0.926030i 0.623199π-0.623199\pi
−0.377449 + 0.926030i 0.623199π0.623199\pi
774774 0 0
775775 −2304.03 −0.106791
776776 0 0
777777 0 0
778778 0 0
779779 −17502.0 −0.804974
780780 0 0
781781 23749.8 1.08814
782782 0 0
783783 0 0
784784 0 0
785785 3823.49 0.173842
786786 0 0
787787 3401.97 0.154088 0.0770440 0.997028i 0.475452π-0.475452\pi
0.0770440 + 0.997028i 0.475452π0.475452\pi
788788 0 0
789789 0 0
790790 0 0
791791 18868.9 0.848168
792792 0 0
793793 −55796.3 −2.49859
794794 0 0
795795 0 0
796796 0 0
797797 9328.18 0.414581 0.207290 0.978279i 0.433535π-0.433535\pi
0.207290 + 0.978279i 0.433535π0.433535\pi
798798 0 0
799799 −8550.19 −0.378578
800800 0 0
801801 0 0
802802 0 0
803803 −47007.9 −2.06584
804804 0 0
805805 −10294.0 −0.450704
806806 0 0
807807 0 0
808808 0 0
809809 21366.3 0.928552 0.464276 0.885690i 0.346314π-0.346314\pi
0.464276 + 0.885690i 0.346314π0.346314\pi
810810 0 0
811811 34119.9 1.47733 0.738664 0.674074i 0.235457π-0.235457\pi
0.738664 + 0.674074i 0.235457π0.235457\pi
812812 0 0
813813 0 0
814814 0 0
815815 −9310.77 −0.400174
816816 0 0
817817 −58364.9 −2.49930
818818 0 0
819819 0 0
820820 0 0
821821 −21760.8 −0.925038 −0.462519 0.886609i 0.653054π-0.653054\pi
−0.462519 + 0.886609i 0.653054π0.653054\pi
822822 0 0
823823 −3225.85 −0.136629 −0.0683147 0.997664i 0.521762π-0.521762\pi
−0.0683147 + 0.997664i 0.521762π0.521762\pi
824824 0 0
825825 0 0
826826 0 0
827827 5208.23 0.218994 0.109497 0.993987i 0.465076π-0.465076\pi
0.109497 + 0.993987i 0.465076π0.465076\pi
828828 0 0
829829 27899.5 1.16886 0.584432 0.811442i 0.301317π-0.301317\pi
0.584432 + 0.811442i 0.301317π0.301317\pi
830830 0 0
831831 0 0
832832 0 0
833833 5600.75 0.232959
834834 0 0
835835 −16621.6 −0.688881
836836 0 0
837837 0 0
838838 0 0
839839 −26244.4 −1.07993 −0.539964 0.841688i 0.681562π-0.681562\pi
−0.539964 + 0.841688i 0.681562π0.681562\pi
840840 0 0
841841 −20778.0 −0.851943
842842 0 0
843843 0 0
844844 0 0
845845 24524.4 0.998422
846846 0 0
847847 10740.8 0.435724
848848 0 0
849849 0 0
850850 0 0
851851 39276.4 1.58211
852852 0 0
853853 37048.2 1.48711 0.743556 0.668674i 0.233138π-0.233138\pi
0.743556 + 0.668674i 0.233138π0.233138\pi
854854 0 0
855855 0 0
856856 0 0
857857 16705.8 0.665879 0.332939 0.942948i 0.391960π-0.391960\pi
0.332939 + 0.942948i 0.391960π0.391960\pi
858858 0 0
859859 13768.5 0.546887 0.273443 0.961888i 0.411837π-0.411837\pi
0.273443 + 0.961888i 0.411837π0.411837\pi
860860 0 0
861861 0 0
862862 0 0
863863 37455.4 1.47740 0.738700 0.674034i 0.235440π-0.235440\pi
0.738700 + 0.674034i 0.235440π0.235440\pi
864864 0 0
865865 −6723.33 −0.264278
866866 0 0
867867 0 0
868868 0 0
869869 24576.1 0.959364
870870 0 0
871871 50325.4 1.95776
872872 0 0
873873 0 0
874874 0 0
875875 1453.29 0.0561487
876876 0 0
877877 −33351.2 −1.28414 −0.642070 0.766646i 0.721924π-0.721924\pi
−0.642070 + 0.766646i 0.721924π0.721924\pi
878878 0 0
879879 0 0
880880 0 0
881881 −45140.2 −1.72623 −0.863116 0.505005i 0.831491π-0.831491\pi
−0.863116 + 0.505005i 0.831491π0.831491\pi
882882 0 0
883883 47316.4 1.80331 0.901656 0.432454i 0.142352π-0.142352\pi
0.901656 + 0.432454i 0.142352π0.142352\pi
884884 0 0
885885 0 0
886886 0 0
887887 −17513.8 −0.662973 −0.331487 0.943460i 0.607550π-0.607550\pi
−0.331487 + 0.943460i 0.607550π0.607550\pi
888888 0 0
889889 −15187.6 −0.572977
890890 0 0
891891 0 0
892892 0 0
893893 −48244.9 −1.80790
894894 0 0
895895 −10431.9 −0.389610
896896 0 0
897897 0 0
898898 0 0
899899 −5538.09 −0.205457
900900 0 0
901901 6945.74 0.256821
902902 0 0
903903 0 0
904904 0 0
905905 3493.57 0.128321
906906 0 0
907907 −10576.5 −0.387196 −0.193598 0.981081i 0.562016π-0.562016\pi
−0.193598 + 0.981081i 0.562016π0.562016\pi
908908 0 0
909909 0 0
910910 0 0
911911 −33191.6 −1.20712 −0.603560 0.797317i 0.706252π-0.706252\pi
−0.603560 + 0.797317i 0.706252π0.706252\pi
912912 0 0
913913 28769.7 1.04287
914914 0 0
915915 0 0
916916 0 0
917917 6269.59 0.225780
918918 0 0
919919 −51842.8 −1.86087 −0.930434 0.366460i 0.880570π-0.880570\pi
−0.930434 + 0.366460i 0.880570π0.880570\pi
920920 0 0
921921 0 0
922922 0 0
923923 −42149.2 −1.50310
924924 0 0
925925 −5544.95 −0.197099
926926 0 0
927927 0 0
928928 0 0
929929 9039.52 0.319243 0.159622 0.987178i 0.448973π-0.448973\pi
0.159622 + 0.987178i 0.448973π0.448973\pi
930930 0 0
931931 31602.5 1.11249
932932 0 0
933933 0 0
934934 0 0
935935 6398.35 0.223795
936936 0 0
937937 −50041.7 −1.74471 −0.872353 0.488876i 0.837407π-0.837407\pi
−0.872353 + 0.488876i 0.837407π0.837407\pi
938938 0 0
939939 0 0
940940 0 0
941941 12305.6 0.426304 0.213152 0.977019i 0.431627π-0.431627\pi
0.213152 + 0.977019i 0.431627π0.431627\pi
942942 0 0
943943 −20382.0 −0.703847
944944 0 0
945945 0 0
946946 0 0
947947 12109.2 0.415519 0.207759 0.978180i 0.433383π-0.433383\pi
0.207759 + 0.978180i 0.433383π0.433383\pi
948948 0 0
949949 83425.7 2.85365
950950 0 0
951951 0 0
952952 0 0
953953 −51371.2 −1.74615 −0.873073 0.487589i 0.837877π-0.837877\pi
−0.873073 + 0.487589i 0.837877π0.837877\pi
954954 0 0
955955 −11338.3 −0.384187
956956 0 0
957957 0 0
958958 0 0
959959 −33451.4 −1.12638
960960 0 0
961961 −21297.3 −0.714890
962962 0 0
963963 0 0
964964 0 0
965965 −18738.2 −0.625082
966966 0 0
967967 −27271.5 −0.906921 −0.453460 0.891276i 0.649811π-0.649811\pi
−0.453460 + 0.891276i 0.649811π0.649811\pi
968968 0 0
969969 0 0
970970 0 0
971971 −41605.1 −1.37505 −0.687524 0.726161i 0.741303π-0.741303\pi
−0.687524 + 0.726161i 0.741303π0.741303\pi
972972 0 0
973973 −17205.6 −0.566893
974974 0 0
975975 0 0
976976 0 0
977977 −41821.2 −1.36948 −0.684738 0.728790i 0.740083π-0.740083\pi
−0.684738 + 0.728790i 0.740083π0.740083\pi
978978 0 0
979979 72136.8 2.35496
980980 0 0
981981 0 0
982982 0 0
983983 37312.9 1.21068 0.605340 0.795967i 0.293037π-0.293037\pi
0.605340 + 0.795967i 0.293037π0.293037\pi
984984 0 0
985985 1469.29 0.0475285
986986 0 0
987987 0 0
988988 0 0
989989 −67968.8 −2.18532
990990 0 0
991991 −32598.1 −1.04492 −0.522459 0.852664i 0.674985π-0.674985\pi
−0.522459 + 0.852664i 0.674985π0.674985\pi
992992 0 0
993993 0 0
994994 0 0
995995 12761.7 0.406605
996996 0 0
997997 41856.8 1.32961 0.664804 0.747018i 0.268515π-0.268515\pi
0.664804 + 0.747018i 0.268515π0.268515\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 1080.4.a.i.1.3 yes 3
3.2 odd 2 1080.4.a.c.1.3 3
4.3 odd 2 2160.4.a.bt.1.1 3
12.11 even 2 2160.4.a.bl.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.c.1.3 3 3.2 odd 2
1080.4.a.i.1.3 yes 3 1.1 even 1 trivial
2160.4.a.bl.1.1 3 12.11 even 2
2160.4.a.bt.1.1 3 4.3 odd 2