Properties

Label 115.7.c.a.114.1
Level 115115
Weight 77
Character 115.114
Self dual yes
Analytic conductor 26.45626.456
Analytic rank 00
Dimension 11
CM discriminant -115
Inner twists 22

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,7,Mod(114,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: N N == 115=523 115 = 5 \cdot 23
Weight: k k == 7 7
Character orbit: [χ][\chi] == 115.c (of order 22, degree 11, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 26.456219616326.4562196163
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: U(1)[D2]\mathrm{U}(1)[D_{2}]

Embedding invariants

Embedding label 114.1
Character χ\chi == 115.114

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+64.0000q4125.000q5594.000q7+729.000q9+4096.00q16+8206.00q178000.00q20+12167.0q23+15625.0q2538016.0q2841382.0q29+3922.00q31+74250.0q35+46656.0q36+76806.0q37+130482.q4133066.0q4391125.0q45+235187.q49+179174.q5331302.0q59433026.q63+262144.q64+127206.q67+525184.q68388638.q71512000.q80+531441.q81778426.q831.02575e6q85+778688.q92+356526.q97+O(q100)q+64.0000 q^{4} -125.000 q^{5} -594.000 q^{7} +729.000 q^{9} +4096.00 q^{16} +8206.00 q^{17} -8000.00 q^{20} +12167.0 q^{23} +15625.0 q^{25} -38016.0 q^{28} -41382.0 q^{29} +3922.00 q^{31} +74250.0 q^{35} +46656.0 q^{36} +76806.0 q^{37} +130482. q^{41} -33066.0 q^{43} -91125.0 q^{45} +235187. q^{49} +179174. q^{53} -31302.0 q^{59} -433026. q^{63} +262144. q^{64} +127206. q^{67} +525184. q^{68} -388638. q^{71} -512000. q^{80} +531441. q^{81} -778426. q^{83} -1.02575e6 q^{85} +778688. q^{92} +356526. q^{97} +O(q^{100})

Character values

We give the values of χ\chi on generators for (Z/115Z)×\left(\mathbb{Z}/115\mathbb{Z}\right)^\times.

nn 4747 5151
χ(n)\chi(n) 1-1 1-1

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0 0 1.00000 00
−1.00000 π\pi
33 0 0 1.00000 00
−1.00000 π\pi
44 64.0000 1.00000
55 −125.000 −1.00000
66 0 0
77 −594.000 −1.73178 −0.865889 0.500236i 0.833247π-0.833247\pi
−0.865889 + 0.500236i 0.833247π0.833247\pi
88 0 0
99 729.000 1.00000
1010 0 0
1111 0 0 1.00000 00
−1.00000 π\pi
1212 0 0
1313 0 0 1.00000 00
−1.00000 π\pi
1414 0 0
1515 0 0
1616 4096.00 1.00000
1717 8206.00 1.67026 0.835131 0.550051i 0.185392π-0.185392\pi
0.835131 + 0.550051i 0.185392π0.185392\pi
1818 0 0
1919 0 0 1.00000 00
−1.00000 π\pi
2020 −8000.00 −1.00000
2121 0 0
2222 0 0
2323 12167.0 1.00000
2424 0 0
2525 15625.0 1.00000
2626 0 0
2727 0 0
2828 −38016.0 −1.73178
2929 −41382.0 −1.69675 −0.848374 0.529397i 0.822418π-0.822418\pi
−0.848374 + 0.529397i 0.822418π0.822418\pi
3030 0 0
3131 3922.00 0.131650 0.0658252 0.997831i 0.479032π-0.479032\pi
0.0658252 + 0.997831i 0.479032π0.479032\pi
3232 0 0
3333 0 0
3434 0 0
3535 74250.0 1.73178
3636 46656.0 1.00000
3737 76806.0 1.51632 0.758158 0.652070i 0.226099π-0.226099\pi
0.758158 + 0.652070i 0.226099π0.226099\pi
3838 0 0
3939 0 0
4040 0 0
4141 130482. 1.89321 0.946606 0.322394i 0.104488π-0.104488\pi
0.946606 + 0.322394i 0.104488π0.104488\pi
4242 0 0
4343 −33066.0 −0.415888 −0.207944 0.978141i 0.566677π-0.566677\pi
−0.207944 + 0.978141i 0.566677π0.566677\pi
4444 0 0
4545 −91125.0 −1.00000
4646 0 0
4747 0 0 1.00000 00
−1.00000 π\pi
4848 0 0
4949 235187. 1.99906
5050 0 0
5151 0 0
5252 0 0
5353 179174. 1.20350 0.601752 0.798683i 0.294470π-0.294470\pi
0.601752 + 0.798683i 0.294470π0.294470\pi
5454 0 0
5555 0 0
5656 0 0
5757 0 0
5858 0 0
5959 −31302.0 −0.152411 −0.0762055 0.997092i 0.524280π-0.524280\pi
−0.0762055 + 0.997092i 0.524280π0.524280\pi
6060 0 0
6161 0 0 1.00000 00
−1.00000 π\pi
6262 0 0
6363 −433026. −1.73178
6464 262144. 1.00000
6565 0 0
6666 0 0
6767 127206. 0.422944 0.211472 0.977384i 0.432174π-0.432174\pi
0.211472 + 0.977384i 0.432174π0.432174\pi
6868 525184. 1.67026
6969 0 0
7070 0 0
7171 −388638. −1.08585 −0.542925 0.839781i 0.682683π-0.682683\pi
−0.542925 + 0.839781i 0.682683π0.682683\pi
7272 0 0
7373 0 0 1.00000 00
−1.00000 π\pi
7474 0 0
7575 0 0
7676 0 0
7777 0 0
7878 0 0
7979 0 0 1.00000 00
−1.00000 π\pi
8080 −512000. −1.00000
8181 531441. 1.00000
8282 0 0
8383 −778426. −1.36139 −0.680696 0.732566i 0.738322π-0.738322\pi
−0.680696 + 0.732566i 0.738322π0.738322\pi
8484 0 0
8585 −1.02575e6 −1.67026
8686 0 0
8787 0 0
8888 0 0
8989 0 0 1.00000 00
−1.00000 π\pi
9090 0 0
9191 0 0
9292 778688. 1.00000
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 356526. 0.390639 0.195320 0.980740i 0.437426π-0.437426\pi
0.195320 + 0.980740i 0.437426π0.437426\pi
9898 0 0
9999 0 0
100100 1.00000e6 1.00000
101101 −2.00396e6 −1.94502 −0.972511 0.232857i 0.925193π-0.925193\pi
−0.972511 + 0.232857i 0.925193π0.925193\pi
102102 0 0
103103 2.14673e6 1.96457 0.982283 0.187404i 0.0600075π-0.0600075\pi
0.982283 + 0.187404i 0.0600075π0.0600075\pi
104104 0 0
105105 0 0
106106 0 0
107107 −407594. −0.332718 −0.166359 0.986065i 0.553201π-0.553201\pi
−0.166359 + 0.986065i 0.553201π0.553201\pi
108108 0 0
109109 0 0 1.00000 00
−1.00000 π\pi
110110 0 0
111111 0 0
112112 −2.43302e6 −1.73178
113113 720974. 0.499671 0.249836 0.968288i 0.419623π-0.419623\pi
0.249836 + 0.968288i 0.419623π0.419623\pi
114114 0 0
115115 −1.52088e6 −1.00000
116116 −2.64845e6 −1.69675
117117 0 0
118118 0 0
119119 −4.87436e6 −2.89252
120120 0 0
121121 1.77156e6 1.00000
122122 0 0
123123 0 0
124124 251008. 0.131650
125125 −1.95312e6 −1.00000
126126 0 0
127127 0 0 1.00000 00
−1.00000 π\pi
128128 0 0
129129 0 0
130130 0 0
131131 2.43124e6 1.08147 0.540735 0.841193i 0.318146π-0.318146\pi
0.540735 + 0.841193i 0.318146π0.318146\pi
132132 0 0
133133 0 0
134134 0 0
135135 0 0
136136 0 0
137137 −4.46227e6 −1.73538 −0.867690 0.497106i 0.834396π-0.834396\pi
−0.867690 + 0.497106i 0.834396π0.834396\pi
138138 0 0
139139 −5.11722e6 −1.90542 −0.952708 0.303887i 0.901715π-0.901715\pi
−0.952708 + 0.303887i 0.901715π0.901715\pi
140140 4.75200e6 1.73178
141141 0 0
142142 0 0
143143 0 0
144144 2.98598e6 1.00000
145145 5.17275e6 1.69675
146146 0 0
147147 0 0
148148 4.91558e6 1.51632
149149 0 0 1.00000 00
−1.00000 π\pi
150150 0 0
151151 6.86336e6 1.99345 0.996727 0.0808455i 0.0257620π-0.0257620\pi
0.996727 + 0.0808455i 0.0257620π0.0257620\pi
152152 0 0
153153 5.98217e6 1.67026
154154 0 0
155155 −490250. −0.131650
156156 0 0
157157 −3.12359e6 −0.807153 −0.403576 0.914946i 0.632233π-0.632233\pi
−0.403576 + 0.914946i 0.632233π0.632233\pi
158158 0 0
159159 0 0
160160 0 0
161161 −7.22720e6 −1.73178
162162 0 0
163163 0 0 1.00000 00
−1.00000 π\pi
164164 8.35085e6 1.89321
165165 0 0
166166 0 0
167167 0 0 1.00000 00
−1.00000 π\pi
168168 0 0
169169 4.82681e6 1.00000
170170 0 0
171171 0 0
172172 −2.11622e6 −0.415888
173173 0 0 1.00000 00
−1.00000 π\pi
174174 0 0
175175 −9.28125e6 −1.73178
176176 0 0
177177 0 0
178178 0 0
179179 8.74334e6 1.52447 0.762234 0.647302i 0.224103π-0.224103\pi
0.762234 + 0.647302i 0.224103π0.224103\pi
180180 −5.83200e6 −1.00000
181181 0 0 1.00000 00
−1.00000 π\pi
182182 0 0
183183 0 0
184184 0 0
185185 −9.60075e6 −1.51632
186186 0 0
187187 0 0
188188 0 0
189189 0 0
190190 0 0
191191 0 0 1.00000 00
−1.00000 π\pi
192192 0 0
193193 0 0 1.00000 00
−1.00000 π\pi
194194 0 0
195195 0 0
196196 1.50520e7 1.99906
197197 0 0 1.00000 00
−1.00000 π\pi
198198 0 0
199199 0 0 1.00000 00
−1.00000 π\pi
200200 0 0
201201 0 0
202202 0 0
203203 2.45809e7 2.93839
204204 0 0
205205 −1.63102e7 −1.89321
206206 0 0
207207 8.86974e6 1.00000
208208 0 0
209209 0 0
210210 0 0
211211 −1.20694e7 −1.28481 −0.642404 0.766366i 0.722063π-0.722063\pi
−0.642404 + 0.766366i 0.722063π0.722063\pi
212212 1.14671e7 1.20350
213213 0 0
214214 0 0
215215 4.13325e6 0.415888
216216 0 0
217217 −2.32967e6 −0.227989
218218 0 0
219219 0 0
220220 0 0
221221 0 0
222222 0 0
223223 0 0 1.00000 00
−1.00000 π\pi
224224 0 0
225225 1.13906e7 1.00000
226226 0 0
227227 −5.50191e6 −0.470366 −0.235183 0.971951i 0.575569π-0.575569\pi
−0.235183 + 0.971951i 0.575569π0.575569\pi
228228 0 0
229229 0 0 1.00000 00
−1.00000 π\pi
230230 0 0
231231 0 0
232232 0 0
233233 0 0 1.00000 00
−1.00000 π\pi
234234 0 0
235235 0 0
236236 −2.00333e6 −0.152411
237237 0 0
238238 0 0
239239 −1.43726e7 −1.05279 −0.526396 0.850240i 0.676457π-0.676457\pi
−0.526396 + 0.850240i 0.676457π0.676457\pi
240240 0 0
241241 0 0 1.00000 00
−1.00000 π\pi
242242 0 0
243243 0 0
244244 0 0
245245 −2.93984e7 −1.99906
246246 0 0
247247 0 0
248248 0 0
249249 0 0
250250 0 0
251251 0 0 1.00000 00
−1.00000 π\pi
252252 −2.77137e7 −1.73178
253253 0 0
254254 0 0
255255 0 0
256256 1.67772e7 1.00000
257257 0 0 1.00000 00
−1.00000 π\pi
258258 0 0
259259 −4.56228e7 −2.62592
260260 0 0
261261 −3.01675e7 −1.69675
262262 0 0
263263 3.37330e7 1.85433 0.927166 0.374651i 0.122238π-0.122238\pi
0.927166 + 0.374651i 0.122238π0.122238\pi
264264 0 0
265265 −2.23968e7 −1.20350
266266 0 0
267267 0 0
268268 8.14118e6 0.422944
269269 −1.61391e7 −0.829132 −0.414566 0.910019i 0.636067π-0.636067\pi
−0.414566 + 0.910019i 0.636067π0.636067\pi
270270 0 0
271271 −1.12039e7 −0.562940 −0.281470 0.959570i 0.590822π-0.590822\pi
−0.281470 + 0.959570i 0.590822π0.590822\pi
272272 3.36118e7 1.67026
273273 0 0
274274 0 0
275275 0 0
276276 0 0
277277 0 0 1.00000 00
−1.00000 π\pi
278278 0 0
279279 2.85914e6 0.131650
280280 0 0
281281 0 0 1.00000 00
−1.00000 π\pi
282282 0 0
283283 4.17687e7 1.84286 0.921428 0.388548i 0.127023π-0.127023\pi
0.921428 + 0.388548i 0.127023π0.127023\pi
284284 −2.48728e7 −1.08585
285285 0 0
286286 0 0
287287 −7.75063e7 −3.27862
288288 0 0
289289 4.32009e7 1.78978
290290 0 0
291291 0 0
292292 0 0
293293 1.90075e7 0.755652 0.377826 0.925877i 0.376672π-0.376672\pi
0.377826 + 0.925877i 0.376672π0.376672\pi
294294 0 0
295295 3.91275e6 0.152411
296296 0 0
297297 0 0
298298 0 0
299299 0 0
300300 0 0
301301 1.96412e7 0.720226
302302 0 0
303303 0 0
304304 0 0
305305 0 0
306306 0 0
307307 0 0 1.00000 00
−1.00000 π\pi
308308 0 0
309309 0 0
310310 0 0
311311 −4.27549e7 −1.42136 −0.710681 0.703515i 0.751613π-0.751613\pi
−0.710681 + 0.703515i 0.751613π0.751613\pi
312312 0 0
313313 −2.14959e7 −0.701008 −0.350504 0.936561i 0.613990π-0.613990\pi
−0.350504 + 0.936561i 0.613990π0.613990\pi
314314 0 0
315315 5.41282e7 1.73178
316316 0 0
317317 0 0 1.00000 00
−1.00000 π\pi
318318 0 0
319319 0 0
320320 −3.27680e7 −1.00000
321321 0 0
322322 0 0
323323 0 0
324324 3.40122e7 1.00000
325325 0 0
326326 0 0
327327 0 0
328328 0 0
329329 0 0
330330 0 0
331331 7.07036e7 1.94966 0.974828 0.222960i 0.0715720π-0.0715720\pi
0.974828 + 0.222960i 0.0715720π0.0715720\pi
332332 −4.98193e7 −1.36139
333333 5.59916e7 1.51632
334334 0 0
335335 −1.59008e7 −0.422944
336336 0 0
337337 7.56037e7 1.97539 0.987696 0.156383i 0.0499835π-0.0499835\pi
0.987696 + 0.156383i 0.0499835π0.0499835\pi
338338 0 0
339339 0 0
340340 −6.56480e7 −1.67026
341341 0 0
342342 0 0
343343 −6.98176e7 −1.73014
344344 0 0
345345 0 0
346346 0 0
347347 0 0 1.00000 00
−1.00000 π\pi
348348 0 0
349349 8.48681e7 1.99649 0.998247 0.0591865i 0.0188507π-0.0188507\pi
0.998247 + 0.0591865i 0.0188507π0.0188507\pi
350350 0 0
351351 0 0
352352 0 0
353353 0 0 1.00000 00
−1.00000 π\pi
354354 0 0
355355 4.85798e7 1.08585
356356 0 0
357357 0 0
358358 0 0
359359 0 0 1.00000 00
−1.00000 π\pi
360360 0 0
361361 4.70459e7 1.00000
362362 0 0
363363 0 0
364364 0 0
365365 0 0
366366 0 0
367367 −7.21338e7 −1.45929 −0.729643 0.683828i 0.760314π-0.760314\pi
−0.729643 + 0.683828i 0.760314π0.760314\pi
368368 4.98360e7 1.00000
369369 9.51214e7 1.89321
370370 0 0
371371 −1.06429e8 −2.08420
372372 0 0
373373 7.96342e7 1.53452 0.767261 0.641335i 0.221619π-0.221619\pi
0.767261 + 0.641335i 0.221619π0.221619\pi
374374 0 0
375375 0 0
376376 0 0
377377 0 0
378378 0 0
379379 0 0 1.00000 00
−1.00000 π\pi
380380 0 0
381381 0 0
382382 0 0
383383 −1.11818e8 −1.99029 −0.995146 0.0984144i 0.968623π-0.968623\pi
−0.995146 + 0.0984144i 0.968623π0.968623\pi
384384 0 0
385385 0 0
386386 0 0
387387 −2.41051e7 −0.415888
388388 2.28177e7 0.390639
389389 0 0 1.00000 00
−1.00000 π\pi
390390 0 0
391391 9.98424e7 1.67026
392392 0 0
393393 0 0
394394 0 0
395395 0 0
396396 0 0
397397 0 0 1.00000 00
−1.00000 π\pi
398398 0 0
399399 0 0
400400 6.40000e7 1.00000
401401 0 0 1.00000 00
−1.00000 π\pi
402402 0 0
403403 0 0
404404 −1.28253e8 −1.94502
405405 −6.64301e7 −1.00000
406406 0 0
407407 0 0
408408 0 0
409409 −5.36670e6 −0.0784400 −0.0392200 0.999231i 0.512487π-0.512487\pi
−0.0392200 + 0.999231i 0.512487π0.512487\pi
410410 0 0
411411 0 0
412412 1.37391e8 1.96457
413413 1.85934e7 0.263942
414414 0 0
415415 9.73032e7 1.36139
416416 0 0
417417 0 0
418418 0 0
419419 0 0 1.00000 00
−1.00000 π\pi
420420 0 0
421421 0 0 1.00000 00
−1.00000 π\pi
422422 0 0
423423 0 0
424424 0 0
425425 1.28219e8 1.67026
426426 0 0
427427 0 0
428428 −2.60860e7 −0.332718
429429 0 0
430430 0 0
431431 0 0 1.00000 00
−1.00000 π\pi
432432 0 0
433433 −1.29024e8 −1.58931 −0.794654 0.607063i 0.792347π-0.792347\pi
−0.794654 + 0.607063i 0.792347π0.792347\pi
434434 0 0
435435 0 0
436436 0 0
437437 0 0
438438 0 0
439439 −1.68638e8 −1.99324 −0.996622 0.0821217i 0.973830π-0.973830\pi
−0.996622 + 0.0821217i 0.973830π0.973830\pi
440440 0 0
441441 1.71451e8 1.99906
442442 0 0
443443 0 0 1.00000 00
−1.00000 π\pi
444444 0 0
445445 0 0
446446 0 0
447447 0 0
448448 −1.55714e8 −1.73178
449449 6.48794e6 0.0716750 0.0358375 0.999358i 0.488590π-0.488590\pi
0.0358375 + 0.999358i 0.488590π0.488590\pi
450450 0 0
451451 0 0
452452 4.61423e7 0.499671
453453 0 0
454454 0 0
455455 0 0
456456 0 0
457457 −1.49095e8 −1.56212 −0.781058 0.624458i 0.785320π-0.785320\pi
−0.781058 + 0.624458i 0.785320π0.785320\pi
458458 0 0
459459 0 0
460460 −9.73360e7 −1.00000
461461 −1.88338e8 −1.92236 −0.961180 0.275922i 0.911017π-0.911017\pi
−0.961180 + 0.275922i 0.911017π0.911017\pi
462462 0 0
463463 0 0 1.00000 00
−1.00000 π\pi
464464 −1.69501e8 −1.69675
465465 0 0
466466 0 0
467467 1.93951e8 1.90433 0.952163 0.305589i 0.0988535π-0.0988535\pi
0.952163 + 0.305589i 0.0988535π0.0988535\pi
468468 0 0
469469 −7.55604e7 −0.732446
470470 0 0
471471 0 0
472472 0 0
473473 0 0
474474 0 0
475475 0 0
476476 −3.11959e8 −2.89252
477477 1.30618e8 1.20350
478478 0 0
479479 0 0 1.00000 00
−1.00000 π\pi
480480 0 0
481481 0 0
482482 0 0
483483 0 0
484484 1.13380e8 1.00000
485485 −4.45658e7 −0.390639
486486 0 0
487487 0 0 1.00000 00
−1.00000 π\pi
488488 0 0
489489 0 0
490490 0 0
491491 2.46627e7 0.208351 0.104176 0.994559i 0.466780π-0.466780\pi
0.104176 + 0.994559i 0.466780π0.466780\pi
492492 0 0
493493 −3.39581e8 −2.83402
494494 0 0
495495 0 0
496496 1.60645e7 0.131650
497497 2.30851e8 1.88045
498498 0 0
499499 2.75885e7 0.222037 0.111019 0.993818i 0.464589π-0.464589\pi
0.111019 + 0.993818i 0.464589π0.464589\pi
500500 −1.25000e8 −1.00000
501501 0 0
502502 0 0
503503 9.63802e7 0.757328 0.378664 0.925534i 0.376384π-0.376384\pi
0.378664 + 0.925534i 0.376384π0.376384\pi
504504 0 0
505505 2.50495e8 1.94502
506506 0 0
507507 0 0
508508 0 0
509509 8.34815e7 0.633048 0.316524 0.948584i 0.397484π-0.397484\pi
0.316524 + 0.948584i 0.397484π0.397484\pi
510510 0 0
511511 0 0
512512 0 0
513513 0 0
514514 0 0
515515 −2.68342e8 −1.96457
516516 0 0
517517 0 0
518518 0 0
519519 0 0
520520 0 0
521521 0 0 1.00000 00
−1.00000 π\pi
522522 0 0
523523 −8.54107e7 −0.597045 −0.298522 0.954403i 0.596494π-0.596494\pi
−0.298522 + 0.954403i 0.596494π0.596494\pi
524524 1.55599e8 1.08147
525525 0 0
526526 0 0
527527 3.21839e7 0.219891
528528 0 0
529529 1.48036e8 1.00000
530530 0 0
531531 −2.28192e7 −0.152411
532532 0 0
533533 0 0
534534 0 0
535535 5.09492e7 0.332718
536536 0 0
537537 0 0
538538 0 0
539539 0 0
540540 0 0
541541 2.30037e8 1.45280 0.726400 0.687272i 0.241192π-0.241192\pi
0.726400 + 0.687272i 0.241192π0.241192\pi
542542 0 0
543543 0 0
544544 0 0
545545 0 0
546546 0 0
547547 0 0 1.00000 00
−1.00000 π\pi
548548 −2.85586e8 −1.73538
549549 0 0
550550 0 0
551551 0 0
552552 0 0
553553 0 0
554554 0 0
555555 0 0
556556 −3.27502e8 −1.90542
557557 −2.83152e8 −1.63853 −0.819263 0.573418i 0.805617π-0.805617\pi
−0.819263 + 0.573418i 0.805617π0.805617\pi
558558 0 0
559559 0 0
560560 3.04128e8 1.73178
561561 0 0
562562 0 0
563563 −950906. −0.00532859 −0.00266430 0.999996i 0.500848π-0.500848\pi
−0.00266430 + 0.999996i 0.500848π0.500848\pi
564564 0 0
565565 −9.01218e7 −0.499671
566566 0 0
567567 −3.15676e8 −1.73178
568568 0 0
569569 0 0 1.00000 00
−1.00000 π\pi
570570 0 0
571571 0 0 1.00000 00
−1.00000 π\pi
572572 0 0
573573 0 0
574574 0 0
575575 1.90109e8 1.00000
576576 1.91103e8 1.00000
577577 0 0 1.00000 00
−1.00000 π\pi
578578 0 0
579579 0 0
580580 3.31056e8 1.69675
581581 4.62385e8 2.35763
582582 0 0
583583 0 0
584584 0 0
585585 0 0
586586 0 0
587587 0 0 1.00000 00
−1.00000 π\pi
588588 0 0
589589 0 0
590590 0 0
591591 0 0
592592 3.14597e8 1.51632
593593 0 0 1.00000 00
−1.00000 π\pi
594594 0 0
595595 6.09296e8 2.89252
596596 0 0
597597 0 0
598598 0 0
599599 −3.92438e8 −1.82596 −0.912979 0.408007i 0.866224π-0.866224\pi
−0.912979 + 0.408007i 0.866224π0.866224\pi
600600 0 0
601601 −1.76304e8 −0.812156 −0.406078 0.913838i 0.633104π-0.633104\pi
−0.406078 + 0.913838i 0.633104π0.633104\pi
602602 0 0
603603 9.27332e7 0.422944
604604 4.39255e8 1.99345
605605 −2.21445e8 −1.00000
606606 0 0
607607 0 0 1.00000 00
−1.00000 π\pi
608608 0 0
609609 0 0
610610 0 0
611611 0 0
612612 3.82859e8 1.67026
613613 −2.62431e8 −1.13929 −0.569643 0.821892i 0.692919π-0.692919\pi
−0.569643 + 0.821892i 0.692919π0.692919\pi
614614 0 0
615615 0 0
616616 0 0
617617 −4.69502e8 −1.99886 −0.999429 0.0337908i 0.989242π-0.989242\pi
−0.999429 + 0.0337908i 0.989242π0.989242\pi
618618 0 0
619619 0 0 1.00000 00
−1.00000 π\pi
620620 −3.13760e7 −0.131650
621621 0 0
622622 0 0
623623 0 0
624624 0 0
625625 2.44141e8 1.00000
626626 0 0
627627 0 0
628628 −1.99910e8 −0.807153
629629 6.30270e8 2.53265
630630 0 0
631631 0 0 1.00000 00
−1.00000 π\pi
632632 0 0
633633 0 0
634634 0 0
635635 0 0
636636 0 0
637637 0 0
638638 0 0
639639 −2.83317e8 −1.08585
640640 0 0
641641 0 0 1.00000 00
−1.00000 π\pi
642642 0 0
643643 1.93195e8 0.726715 0.363357 0.931650i 0.381630π-0.381630\pi
0.363357 + 0.931650i 0.381630π0.381630\pi
644644 −4.62541e8 −1.73178
645645 0 0
646646 0 0
647647 0 0 1.00000 00
−1.00000 π\pi
648648 0 0
649649 0 0
650650 0 0
651651 0 0
652652 0 0
653653 0 0 1.00000 00
−1.00000 π\pi
654654 0 0
655655 −3.03905e8 −1.08147
656656 5.34454e8 1.89321
657657 0 0
658658 0 0
659659 0 0 1.00000 00
−1.00000 π\pi
660660 0 0
661661 0 0 1.00000 00
−1.00000 π\pi
662662 0 0
663663 0 0
664664 0 0
665665 0 0
666666 0 0
667667 −5.03495e8 −1.69675
668668 0 0
669669 0 0
670670 0 0
671671 0 0
672672 0 0
673673 0 0 1.00000 00
−1.00000 π\pi
674674 0 0
675675 0 0
676676 3.08916e8 1.00000
677677 −5.67996e8 −1.83054 −0.915270 0.402840i 0.868023π-0.868023\pi
−0.915270 + 0.402840i 0.868023π0.868023\pi
678678 0 0
679679 −2.11776e8 −0.676501
680680 0 0
681681 0 0
682682 0 0
683683 0 0 1.00000 00
−1.00000 π\pi
684684 0 0
685685 5.57784e8 1.73538
686686 0 0
687687 0 0
688688 −1.35438e8 −0.415888
689689 0 0
690690 0 0
691691 −5.36935e8 −1.62737 −0.813687 0.581303i 0.802543π-0.802543\pi
−0.813687 + 0.581303i 0.802543π0.802543\pi
692692 0 0
693693 0 0
694694 0 0
695695 6.39653e8 1.90542
696696 0 0
697697 1.07074e9 3.16216
698698 0 0
699699 0 0
700700 −5.94000e8 −1.73178
701701 0 0 1.00000 00
−1.00000 π\pi
702702 0 0
703703 0 0
704704 0 0
705705 0 0
706706 0 0
707707 1.19035e9 3.36835
708708 0 0
709709 0 0 1.00000 00
−1.00000 π\pi
710710 0 0
711711 0 0
712712 0 0
713713 4.77190e7 0.131650
714714 0 0
715715 0 0
716716 5.59574e8 1.52447
717717 0 0
718718 0 0
719719 −7.38742e8 −1.98749 −0.993747 0.111654i 0.964385π-0.964385\pi
−0.993747 + 0.111654i 0.964385π0.964385\pi
720720 −3.73248e8 −1.00000
721721 −1.27516e9 −3.40219
722722 0 0
723723 0 0
724724 0 0
725725 −6.46594e8 −1.69675
726726 0 0
727727 2.40089e8 0.624841 0.312420 0.949944i 0.398860π-0.398860\pi
0.312420 + 0.949944i 0.398860π0.398860\pi
728728 0 0
729729 3.87420e8 1.00000
730730 0 0
731731 −2.71340e8 −0.694642
732732 0 0
733733 5.07465e8 1.28853 0.644264 0.764803i 0.277164π-0.277164\pi
0.644264 + 0.764803i 0.277164π0.277164\pi
734734 0 0
735735 0 0
736736 0 0
737737 0 0
738738 0 0
739739 −4.37605e8 −1.08430 −0.542149 0.840282i 0.682389π-0.682389\pi
−0.542149 + 0.840282i 0.682389π0.682389\pi
740740 −6.14448e8 −1.51632
741741 0 0
742742 0 0
743743 −4.00202e8 −0.975692 −0.487846 0.872930i 0.662217π-0.662217\pi
−0.487846 + 0.872930i 0.662217π0.662217\pi
744744 0 0
745745 0 0
746746 0 0
747747 −5.67473e8 −1.36139
748748 0 0
749749 2.42111e8 0.576194
750750 0 0
751751 0 0 1.00000 00
−1.00000 π\pi
752752 0 0
753753 0 0
754754 0 0
755755 −8.57920e8 −1.99345
756756 0 0
757757 −1.38049e8 −0.318234 −0.159117 0.987260i 0.550865π-0.550865\pi
−0.159117 + 0.987260i 0.550865π0.550865\pi
758758 0 0
759759 0 0
760760 0 0
761761 −1.26162e8 −0.286269 −0.143134 0.989703i 0.545718π-0.545718\pi
−0.143134 + 0.989703i 0.545718π0.545718\pi
762762 0 0
763763 0 0
764764 0 0
765765 −7.47772e8 −1.67026
766766 0 0
767767 0 0
768768 0 0
769769 0 0 1.00000 00
−1.00000 π\pi
770770 0 0
771771 0 0
772772 0 0
773773 1.32246e8 0.286315 0.143158 0.989700i 0.454274π-0.454274\pi
0.143158 + 0.989700i 0.454274π0.454274\pi
774774 0 0
775775 6.12812e7 0.131650
776776 0 0
777777 0 0
778778 0 0
779779 0 0
780780 0 0
781781 0 0
782782 0 0
783783 0 0
784784 9.63326e8 1.99906
785785 3.90449e8 0.807153
786786 0 0
787787 −8.49163e8 −1.74208 −0.871038 0.491216i 0.836553π-0.836553\pi
−0.871038 + 0.491216i 0.836553π0.836553\pi
788788 0 0
789789 0 0
790790 0 0
791791 −4.28259e8 −0.865320
792792 0 0
793793 0 0
794794 0 0
795795 0 0
796796 0 0
797797 −5.84876e8 −1.15528 −0.577642 0.816290i 0.696027π-0.696027\pi
−0.577642 + 0.816290i 0.696027π0.696027\pi
798798 0 0
799799 0 0
800800 0 0
801801 0 0
802802 0 0
803803 0 0
804804 0 0
805805 9.03400e8 1.73178
806806 0 0
807807 0 0
808808 0 0
809809 4.81926e8 0.910196 0.455098 0.890441i 0.349604π-0.349604\pi
0.455098 + 0.890441i 0.349604π0.349604\pi
810810 0 0
811811 −9.55985e8 −1.79221 −0.896104 0.443844i 0.853614π-0.853614\pi
−0.896104 + 0.443844i 0.853614π0.853614\pi
812812 1.57318e9 2.93839
813813 0 0
814814 0 0
815815 0 0
816816 0 0
817817 0 0
818818 0 0
819819 0 0
820820 −1.04386e9 −1.89321
821821 3.92618e8 0.709481 0.354740 0.934965i 0.384569π-0.384569\pi
0.354740 + 0.934965i 0.384569π0.384569\pi
822822 0 0
823823 0 0 1.00000 00
−1.00000 π\pi
824824 0 0
825825 0 0
826826 0 0
827827 −4.19694e8 −0.742021 −0.371011 0.928629i 0.620989π-0.620989\pi
−0.371011 + 0.928629i 0.620989π0.620989\pi
828828 5.67664e8 1.00000
829829 7.06632e8 1.24031 0.620154 0.784480i 0.287070π-0.287070\pi
0.620154 + 0.784480i 0.287070π0.287070\pi
830830 0 0
831831 0 0
832832 0 0
833833 1.92994e9 3.33895
834834 0 0
835835 0 0
836836 0 0
837837 0 0
838838 0 0
839839 0 0 1.00000 00
−1.00000 π\pi
840840 0 0
841841 1.11765e9 1.87896
842842 0 0
843843 0 0
844844 −7.72441e8 −1.28481
845845 −6.03351e8 −1.00000
846846 0 0
847847 −1.05231e9 −1.73178
848848 7.33897e8 1.20350
849849 0 0
850850 0 0
851851 9.34499e8 1.51632
852852 0 0
853853 0 0 1.00000 00
−1.00000 π\pi
854854 0 0
855855 0 0
856856 0 0
857857 0 0 1.00000 00
−1.00000 π\pi
858858 0 0
859859 −1.19331e9 −1.88266 −0.941331 0.337484i 0.890424π-0.890424\pi
−0.941331 + 0.337484i 0.890424π0.890424\pi
860860 2.64528e8 0.415888
861861 0 0
862862 0 0
863863 0 0 1.00000 00
−1.00000 π\pi
864864 0 0
865865 0 0
866866 0 0
867867 0 0
868868 −1.49099e8 −0.227989
869869 0 0
870870 0 0
871871 0 0
872872 0 0
873873 2.59907e8 0.390639
874874 0 0
875875 1.16016e9 1.73178
876876 0 0
877877 0 0 1.00000 00
−1.00000 π\pi
878878 0 0
879879 0 0
880880 0 0
881881 0 0 1.00000 00
−1.00000 π\pi
882882 0 0
883883 0 0 1.00000 00
−1.00000 π\pi
884884 0 0
885885 0 0
886886 0 0
887887 0 0 1.00000 00
−1.00000 π\pi
888888 0 0
889889 0 0
890890 0 0
891891 0 0
892892 0 0
893893 0 0
894894 0 0
895895 −1.09292e9 −1.52447
896896 0 0
897897 0 0
898898 0 0
899899 −1.62300e8 −0.223378
900900 7.29000e8 1.00000
901901 1.47030e9 2.01017
902902 0 0
903903 0 0
904904 0 0
905905 0 0
906906 0 0
907907 −1.48159e9 −1.98566 −0.992831 0.119525i 0.961863π-0.961863\pi
−0.992831 + 0.119525i 0.961863π0.961863\pi
908908 −3.52122e8 −0.470366
909909 −1.46089e9 −1.94502
910910 0 0
911911 0 0 1.00000 00
−1.00000 π\pi
912912 0 0
913913 0 0
914914 0 0
915915 0 0
916916 0 0
917917 −1.44416e9 −1.87287
918918 0 0
919919 0 0 1.00000 00
−1.00000 π\pi
920920 0 0
921921 0 0
922922 0 0
923923 0 0
924924 0 0
925925 1.20009e9 1.51632
926926 0 0
927927 1.56497e9 1.96457
928928 0 0
929929 1.58127e9 1.97223 0.986116 0.166060i 0.0531046π-0.0531046\pi
0.986116 + 0.166060i 0.0531046π0.0531046\pi
930930 0 0
931931 0 0
932932 0 0
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 5.51002e8 0.669783 0.334892 0.942257i 0.391300π-0.391300\pi
0.334892 + 0.942257i 0.391300π0.391300\pi
938938 0 0
939939 0 0
940940 0 0
941941 0 0 1.00000 00
−1.00000 π\pi
942942 0 0
943943 1.58757e9 1.89321
944944 −1.28213e8 −0.152411
945945 0 0
946946 0 0
947947 0 0 1.00000 00
−1.00000 π\pi
948948 0 0
949949 0 0
950950 0 0
951951 0 0
952952 0 0
953953 −1.05139e9 −1.21475 −0.607375 0.794415i 0.707777π-0.707777\pi
−0.607375 + 0.794415i 0.707777π0.707777\pi
954954 0 0
955955 0 0
956956 −9.19848e8 −1.05279
957957 0 0
958958 0 0
959959 2.65059e9 3.00529
960960 0 0
961961 −8.72122e8 −0.982668
962962 0 0
963963 −2.97136e8 −0.332718
964964 0 0
965965 0 0
966966 0 0
967967 0 0 1.00000 00
−1.00000 π\pi
968968 0 0
969969 0 0
970970 0 0
971971 0 0 1.00000 00
−1.00000 π\pi
972972 0 0
973973 3.03963e9 3.29976
974974 0 0
975975 0 0
976976 0 0
977977 −1.80419e9 −1.93463 −0.967317 0.253570i 0.918395π-0.918395\pi
−0.967317 + 0.253570i 0.918395π0.918395\pi
978978 0 0
979979 0 0
980980 −1.88150e9 −1.99906
981981 0 0
982982 0 0
983983 1.52023e9 1.60047 0.800237 0.599684i 0.204707π-0.204707\pi
0.800237 + 0.599684i 0.204707π0.204707\pi
984984 0 0
985985 0 0
986986 0 0
987987 0 0
988988 0 0
989989 −4.02314e8 −0.415888
990990 0 0
991991 1.91887e9 1.97163 0.985815 0.167837i 0.0536784π-0.0536784\pi
0.985815 + 0.167837i 0.0536784π0.0536784\pi
992992 0 0
993993 0 0
994994 0 0
995995 0 0
996996 0 0
997997 0 0 1.00000 00
−1.00000 π\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 115.7.c.a.114.1 1
5.4 even 2 115.7.c.b.114.1 yes 1
23.22 odd 2 115.7.c.b.114.1 yes 1
115.114 odd 2 CM 115.7.c.a.114.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.7.c.a.114.1 1 1.1 even 1 trivial
115.7.c.a.114.1 1 115.114 odd 2 CM
115.7.c.b.114.1 yes 1 5.4 even 2
115.7.c.b.114.1 yes 1 23.22 odd 2