Properties

Label 1225.2.a.q.1.1
Level 12251225
Weight 22
Character 1225.1
Self dual yes
Analytic conductor 9.7829.782
Analytic rank 11
Dimension 22
CM discriminant -35
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,2,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1225=5272 1225 = 5^{2} \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1225.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: 9.781674247619.78167424761
Analytic rank: 11
Dimension: 22
Coefficient field: Q(ζ10)+\Q(\zeta_{10})^+
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x1 x^{2} - x - 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 245)
Fricke sign: +1+1
Sato-Tate group: N(U(1))N(\mathrm{U}(1))

Embedding invariants

Embedding label 1.1
Root 1.618031.61803 of defining polynomial
Character χ\chi == 1225.1

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) == q2.23607q32.00000q4+2.00000q93.00000q11+4.47214q12+6.70820q13+4.00000q16+2.23607q17+2.23607q279.00000q29+6.70820q334.00000q3615.0000q39+6.00000q4411.1803q478.94427q485.00000q5113.4164q528.00000q644.47214q6812.0000q71+13.4164q731.00000q7911.0000q818.94427q83+20.1246q876.70820q976.00000q99+O(q100)q-2.23607 q^{3} -2.00000 q^{4} +2.00000 q^{9} -3.00000 q^{11} +4.47214 q^{12} +6.70820 q^{13} +4.00000 q^{16} +2.23607 q^{17} +2.23607 q^{27} -9.00000 q^{29} +6.70820 q^{33} -4.00000 q^{36} -15.0000 q^{39} +6.00000 q^{44} -11.1803 q^{47} -8.94427 q^{48} -5.00000 q^{51} -13.4164 q^{52} -8.00000 q^{64} -4.47214 q^{68} -12.0000 q^{71} +13.4164 q^{73} -1.00000 q^{79} -11.0000 q^{81} -8.94427 q^{83} +20.1246 q^{87} -6.70820 q^{97} -6.00000 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q4q4+4q96q11+8q1618q298q3630q39+12q4410q5116q6424q712q7922q8112q99+O(q100) 2 q - 4 q^{4} + 4 q^{9} - 6 q^{11} + 8 q^{16} - 18 q^{29} - 8 q^{36} - 30 q^{39} + 12 q^{44} - 10 q^{51} - 16 q^{64} - 24 q^{71} - 2 q^{79} - 22 q^{81} - 12 q^{99}+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 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
33 −2.23607 −1.29099 −0.645497 0.763763i 0.723350π-0.723350\pi
−0.645497 + 0.763763i 0.723350π0.723350\pi
44 −2.00000 −1.00000
55 0 0
66 0 0
77 0 0
88 0 0
99 2.00000 0.666667
1010 0 0
1111 −3.00000 −0.904534 −0.452267 0.891883i 0.649385π-0.649385\pi
−0.452267 + 0.891883i 0.649385π0.649385\pi
1212 4.47214 1.29099
1313 6.70820 1.86052 0.930261 0.366900i 0.119581π-0.119581\pi
0.930261 + 0.366900i 0.119581π0.119581\pi
1414 0 0
1515 0 0
1616 4.00000 1.00000
1717 2.23607 0.542326 0.271163 0.962533i 0.412592π-0.412592\pi
0.271163 + 0.962533i 0.412592π0.412592\pi
1818 0 0
1919 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
2020 0 0
2121 0 0
2222 0 0
2323 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
2424 0 0
2525 0 0
2626 0 0
2727 2.23607 0.430331
2828 0 0
2929 −9.00000 −1.67126 −0.835629 0.549294i 0.814897π-0.814897\pi
−0.835629 + 0.549294i 0.814897π0.814897\pi
3030 0 0
3131 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
3232 0 0
3333 6.70820 1.16775
3434 0 0
3535 0 0
3636 −4.00000 −0.666667
3737 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
3838 0 0
3939 −15.0000 −2.40192
4040 0 0
4141 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
4242 0 0
4343 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
4444 6.00000 0.904534
4545 0 0
4646 0 0
4747 −11.1803 −1.63082 −0.815410 0.578884i 0.803489π-0.803489\pi
−0.815410 + 0.578884i 0.803489π0.803489\pi
4848 −8.94427 −1.29099
4949 0 0
5050 0 0
5151 −5.00000 −0.700140
5252 −13.4164 −1.86052
5353 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
5454 0 0
5555 0 0
5656 0 0
5757 0 0
5858 0 0
5959 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
6060 0 0
6161 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
6262 0 0
6363 0 0
6464 −8.00000 −1.00000
6565 0 0
6666 0 0
6767 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
6868 −4.47214 −0.542326
6969 0 0
7070 0 0
7171 −12.0000 −1.42414 −0.712069 0.702109i 0.752242π-0.752242\pi
−0.712069 + 0.702109i 0.752242π0.752242\pi
7272 0 0
7373 13.4164 1.57027 0.785136 0.619324i 0.212593π-0.212593\pi
0.785136 + 0.619324i 0.212593π0.212593\pi
7474 0 0
7575 0 0
7676 0 0
7777 0 0
7878 0 0
7979 −1.00000 −0.112509 −0.0562544 0.998416i 0.517916π-0.517916\pi
−0.0562544 + 0.998416i 0.517916π0.517916\pi
8080 0 0
8181 −11.0000 −1.22222
8282 0 0
8383 −8.94427 −0.981761 −0.490881 0.871227i 0.663325π-0.663325\pi
−0.490881 + 0.871227i 0.663325π0.663325\pi
8484 0 0
8585 0 0
8686 0 0
8787 20.1246 2.15758
8888 0 0
8989 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
9090 0 0
9191 0 0
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 −6.70820 −0.681115 −0.340557 0.940224i 0.610616π-0.610616\pi
−0.340557 + 0.940224i 0.610616π0.610616\pi
9898 0 0
9999 −6.00000 −0.603023
100100 0 0
101101 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
102102 0 0
103103 −20.1246 −1.98294 −0.991468 0.130347i 0.958391π-0.958391\pi
−0.991468 + 0.130347i 0.958391π0.958391\pi
104104 0 0
105105 0 0
106106 0 0
107107 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
108108 −4.47214 −0.430331
109109 −11.0000 −1.05361 −0.526804 0.849987i 0.676610π-0.676610\pi
−0.526804 + 0.849987i 0.676610π0.676610\pi
110110 0 0
111111 0 0
112112 0 0
113113 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
114114 0 0
115115 0 0
116116 18.0000 1.67126
117117 13.4164 1.24035
118118 0 0
119119 0 0
120120 0 0
121121 −2.00000 −0.181818
122122 0 0
123123 0 0
124124 0 0
125125 0 0
126126 0 0
127127 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
128128 0 0
129129 0 0
130130 0 0
131131 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
132132 −13.4164 −1.16775
133133 0 0
134134 0 0
135135 0 0
136136 0 0
137137 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
138138 0 0
139139 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
140140 0 0
141141 25.0000 2.10538
142142 0 0
143143 −20.1246 −1.68290
144144 8.00000 0.666667
145145 0 0
146146 0 0
147147 0 0
148148 0 0
149149 6.00000 0.491539 0.245770 0.969328i 0.420959π-0.420959\pi
0.245770 + 0.969328i 0.420959π0.420959\pi
150150 0 0
151151 −17.0000 −1.38344 −0.691720 0.722166i 0.743147π-0.743147\pi
−0.691720 + 0.722166i 0.743147π0.743147\pi
152152 0 0
153153 4.47214 0.361551
154154 0 0
155155 0 0
156156 30.0000 2.40192
157157 −13.4164 −1.07075 −0.535373 0.844616i 0.679829π-0.679829\pi
−0.535373 + 0.844616i 0.679829π0.679829\pi
158158 0 0
159159 0 0
160160 0 0
161161 0 0
162162 0 0
163163 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
164164 0 0
165165 0 0
166166 0 0
167167 24.5967 1.90335 0.951677 0.307102i 0.0993591π-0.0993591\pi
0.951677 + 0.307102i 0.0993591π0.0993591\pi
168168 0 0
169169 32.0000 2.46154
170170 0 0
171171 0 0
172172 0 0
173173 11.1803 0.850026 0.425013 0.905187i 0.360270π-0.360270\pi
0.425013 + 0.905187i 0.360270π0.360270\pi
174174 0 0
175175 0 0
176176 −12.0000 −0.904534
177177 0 0
178178 0 0
179179 24.0000 1.79384 0.896922 0.442189i 0.145798π-0.145798\pi
0.896922 + 0.442189i 0.145798π0.145798\pi
180180 0 0
181181 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
182182 0 0
183183 0 0
184184 0 0
185185 0 0
186186 0 0
187187 −6.70820 −0.490552
188188 22.3607 1.63082
189189 0 0
190190 0 0
191191 −27.0000 −1.95365 −0.976826 0.214036i 0.931339π-0.931339\pi
−0.976826 + 0.214036i 0.931339π0.931339\pi
192192 17.8885 1.29099
193193 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
194194 0 0
195195 0 0
196196 0 0
197197 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
198198 0 0
199199 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
200200 0 0
201201 0 0
202202 0 0
203203 0 0
204204 10.0000 0.700140
205205 0 0
206206 0 0
207207 0 0
208208 26.8328 1.86052
209209 0 0
210210 0 0
211211 23.0000 1.58339 0.791693 0.610920i 0.209200π-0.209200\pi
0.791693 + 0.610920i 0.209200π0.209200\pi
212212 0 0
213213 26.8328 1.83855
214214 0 0
215215 0 0
216216 0 0
217217 0 0
218218 0 0
219219 −30.0000 −2.02721
220220 0 0
221221 15.0000 1.00901
222222 0 0
223223 6.70820 0.449215 0.224607 0.974449i 0.427890π-0.427890\pi
0.224607 + 0.974449i 0.427890π0.427890\pi
224224 0 0
225225 0 0
226226 0 0
227227 −29.0689 −1.92937 −0.964685 0.263407i 0.915154π-0.915154\pi
−0.964685 + 0.263407i 0.915154π0.915154\pi
228228 0 0
229229 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
230230 0 0
231231 0 0
232232 0 0
233233 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
234234 0 0
235235 0 0
236236 0 0
237237 2.23607 0.145248
238238 0 0
239239 −9.00000 −0.582162 −0.291081 0.956698i 0.594015π-0.594015\pi
−0.291081 + 0.956698i 0.594015π0.594015\pi
240240 0 0
241241 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
242242 0 0
243243 17.8885 1.14755
244244 0 0
245245 0 0
246246 0 0
247247 0 0
248248 0 0
249249 20.0000 1.26745
250250 0 0
251251 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
252252 0 0
253253 0 0
254254 0 0
255255 0 0
256256 16.0000 1.00000
257257 4.47214 0.278964 0.139482 0.990225i 0.455456π-0.455456\pi
0.139482 + 0.990225i 0.455456π0.455456\pi
258258 0 0
259259 0 0
260260 0 0
261261 −18.0000 −1.11417
262262 0 0
263263 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
264264 0 0
265265 0 0
266266 0 0
267267 0 0
268268 0 0
269269 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
270270 0 0
271271 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
272272 8.94427 0.542326
273273 0 0
274274 0 0
275275 0 0
276276 0 0
277277 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
278278 0 0
279279 0 0
280280 0 0
281281 −33.0000 −1.96861 −0.984307 0.176462i 0.943535π-0.943535\pi
−0.984307 + 0.176462i 0.943535π0.943535\pi
282282 0 0
283283 −33.5410 −1.99381 −0.996903 0.0786368i 0.974943π-0.974943\pi
−0.996903 + 0.0786368i 0.974943π0.974943\pi
284284 24.0000 1.42414
285285 0 0
286286 0 0
287287 0 0
288288 0 0
289289 −12.0000 −0.705882
290290 0 0
291291 15.0000 0.879316
292292 −26.8328 −1.57027
293293 −24.5967 −1.43696 −0.718479 0.695549i 0.755161π-0.755161\pi
−0.718479 + 0.695549i 0.755161π0.755161\pi
294294 0 0
295295 0 0
296296 0 0
297297 −6.70820 −0.389249
298298 0 0
299299 0 0
300300 0 0
301301 0 0
302302 0 0
303303 0 0
304304 0 0
305305 0 0
306306 0 0
307307 −6.70820 −0.382857 −0.191429 0.981507i 0.561312π-0.561312\pi
−0.191429 + 0.981507i 0.561312π0.561312\pi
308308 0 0
309309 45.0000 2.55996
310310 0 0
311311 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
312312 0 0
313313 −20.1246 −1.13751 −0.568755 0.822507i 0.692575π-0.692575\pi
−0.568755 + 0.822507i 0.692575π0.692575\pi
314314 0 0
315315 0 0
316316 2.00000 0.112509
317317 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
318318 0 0
319319 27.0000 1.51171
320320 0 0
321321 0 0
322322 0 0
323323 0 0
324324 22.0000 1.22222
325325 0 0
326326 0 0
327327 24.5967 1.36020
328328 0 0
329329 0 0
330330 0 0
331331 8.00000 0.439720 0.219860 0.975531i 0.429440π-0.429440\pi
0.219860 + 0.975531i 0.429440π0.429440\pi
332332 17.8885 0.981761
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
338338 0 0
339339 0 0
340340 0 0
341341 0 0
342342 0 0
343343 0 0
344344 0 0
345345 0 0
346346 0 0
347347 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
348348 −40.2492 −2.15758
349349 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
350350 0 0
351351 15.0000 0.800641
352352 0 0
353353 29.0689 1.54718 0.773590 0.633686i 0.218459π-0.218459\pi
0.773590 + 0.633686i 0.218459π0.218459\pi
354354 0 0
355355 0 0
356356 0 0
357357 0 0
358358 0 0
359359 −36.0000 −1.90001 −0.950004 0.312239i 0.898921π-0.898921\pi
−0.950004 + 0.312239i 0.898921π0.898921\pi
360360 0 0
361361 −19.0000 −1.00000
362362 0 0
363363 4.47214 0.234726
364364 0 0
365365 0 0
366366 0 0
367367 33.5410 1.75083 0.875413 0.483375i 0.160589π-0.160589\pi
0.875413 + 0.483375i 0.160589π0.160589\pi
368368 0 0
369369 0 0
370370 0 0
371371 0 0
372372 0 0
373373 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
374374 0 0
375375 0 0
376376 0 0
377377 −60.3738 −3.10941
378378 0 0
379379 −16.0000 −0.821865 −0.410932 0.911666i 0.634797π-0.634797\pi
−0.410932 + 0.911666i 0.634797π0.634797\pi
380380 0 0
381381 0 0
382382 0 0
383383 −35.7771 −1.82812 −0.914062 0.405575i 0.867071π-0.867071\pi
−0.914062 + 0.405575i 0.867071π0.867071\pi
384384 0 0
385385 0 0
386386 0 0
387387 0 0
388388 13.4164 0.681115
389389 −39.0000 −1.97738 −0.988689 0.149979i 0.952080π-0.952080\pi
−0.988689 + 0.149979i 0.952080π0.952080\pi
390390 0 0
391391 0 0
392392 0 0
393393 0 0
394394 0 0
395395 0 0
396396 12.0000 0.603023
397397 20.1246 1.01003 0.505013 0.863112i 0.331488π-0.331488\pi
0.505013 + 0.863112i 0.331488π0.331488\pi
398398 0 0
399399 0 0
400400 0 0
401401 −27.0000 −1.34832 −0.674158 0.738587i 0.735493π-0.735493\pi
−0.674158 + 0.738587i 0.735493π0.735493\pi
402402 0 0
403403 0 0
404404 0 0
405405 0 0
406406 0 0
407407 0 0
408408 0 0
409409 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
410410 0 0
411411 0 0
412412 40.2492 1.98294
413413 0 0
414414 0 0
415415 0 0
416416 0 0
417417 0 0
418418 0 0
419419 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
420420 0 0
421421 37.0000 1.80327 0.901635 0.432498i 0.142368π-0.142368\pi
0.901635 + 0.432498i 0.142368π0.142368\pi
422422 0 0
423423 −22.3607 −1.08721
424424 0 0
425425 0 0
426426 0 0
427427 0 0
428428 0 0
429429 45.0000 2.17262
430430 0 0
431431 −3.00000 −0.144505 −0.0722525 0.997386i 0.523019π-0.523019\pi
−0.0722525 + 0.997386i 0.523019π0.523019\pi
432432 8.94427 0.430331
433433 −40.2492 −1.93425 −0.967127 0.254293i 0.918157π-0.918157\pi
−0.967127 + 0.254293i 0.918157π0.918157\pi
434434 0 0
435435 0 0
436436 22.0000 1.05361
437437 0 0
438438 0 0
439439 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
440440 0 0
441441 0 0
442442 0 0
443443 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
444444 0 0
445445 0 0
446446 0 0
447447 −13.4164 −0.634574
448448 0 0
449449 −9.00000 −0.424736 −0.212368 0.977190i 0.568118π-0.568118\pi
−0.212368 + 0.977190i 0.568118π0.568118\pi
450450 0 0
451451 0 0
452452 0 0
453453 38.0132 1.78601
454454 0 0
455455 0 0
456456 0 0
457457 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
458458 0 0
459459 5.00000 0.233380
460460 0 0
461461 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
462462 0 0
463463 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
464464 −36.0000 −1.67126
465465 0 0
466466 0 0
467467 −42.4853 −1.96598 −0.982992 0.183646i 0.941210π-0.941210\pi
−0.982992 + 0.183646i 0.941210π0.941210\pi
468468 −26.8328 −1.24035
469469 0 0
470470 0 0
471471 30.0000 1.38233
472472 0 0
473473 0 0
474474 0 0
475475 0 0
476476 0 0
477477 0 0
478478 0 0
479479 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
480480 0 0
481481 0 0
482482 0 0
483483 0 0
484484 4.00000 0.181818
485485 0 0
486486 0 0
487487 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
488488 0 0
489489 0 0
490490 0 0
491491 −33.0000 −1.48927 −0.744635 0.667472i 0.767376π-0.767376\pi
−0.744635 + 0.667472i 0.767376π0.767376\pi
492492 0 0
493493 −20.1246 −0.906367
494494 0 0
495495 0 0
496496 0 0
497497 0 0
498498 0 0
499499 41.0000 1.83541 0.917706 0.397260i 0.130039π-0.130039\pi
0.917706 + 0.397260i 0.130039π0.130039\pi
500500 0 0
501501 −55.0000 −2.45722
502502 0 0
503503 38.0132 1.69492 0.847461 0.530857i 0.178130π-0.178130\pi
0.847461 + 0.530857i 0.178130π0.178130\pi
504504 0 0
505505 0 0
506506 0 0
507507 −71.5542 −3.17783
508508 0 0
509509 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
510510 0 0
511511 0 0
512512 0 0
513513 0 0
514514 0 0
515515 0 0
516516 0 0
517517 33.5410 1.47513
518518 0 0
519519 −25.0000 −1.09738
520520 0 0
521521 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
522522 0 0
523523 26.8328 1.17332 0.586659 0.809834i 0.300443π-0.300443\pi
0.586659 + 0.809834i 0.300443π0.300443\pi
524524 0 0
525525 0 0
526526 0 0
527527 0 0
528528 26.8328 1.16775
529529 −23.0000 −1.00000
530530 0 0
531531 0 0
532532 0 0
533533 0 0
534534 0 0
535535 0 0
536536 0 0
537537 −53.6656 −2.31584
538538 0 0
539539 0 0
540540 0 0
541541 43.0000 1.84871 0.924357 0.381528i 0.124602π-0.124602\pi
0.924357 + 0.381528i 0.124602π0.124602\pi
542542 0 0
543543 0 0
544544 0 0
545545 0 0
546546 0 0
547547 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
548548 0 0
549549 0 0
550550 0 0
551551 0 0
552552 0 0
553553 0 0
554554 0 0
555555 0 0
556556 0 0
557557 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
558558 0 0
559559 0 0
560560 0 0
561561 15.0000 0.633300
562562 0 0
563563 44.7214 1.88478 0.942390 0.334515i 0.108573π-0.108573\pi
0.942390 + 0.334515i 0.108573π0.108573\pi
564564 −50.0000 −2.10538
565565 0 0
566566 0 0
567567 0 0
568568 0 0
569569 6.00000 0.251533 0.125767 0.992060i 0.459861π-0.459861\pi
0.125767 + 0.992060i 0.459861π0.459861\pi
570570 0 0
571571 32.0000 1.33916 0.669579 0.742741i 0.266474π-0.266474\pi
0.669579 + 0.742741i 0.266474π0.266474\pi
572572 40.2492 1.68290
573573 60.3738 2.52215
574574 0 0
575575 0 0
576576 −16.0000 −0.666667
577577 33.5410 1.39633 0.698165 0.715936i 0.254000π-0.254000\pi
0.698165 + 0.715936i 0.254000π0.254000\pi
578578 0 0
579579 0 0
580580 0 0
581581 0 0
582582 0 0
583583 0 0
584584 0 0
585585 0 0
586586 0 0
587587 8.94427 0.369170 0.184585 0.982817i 0.440906π-0.440906\pi
0.184585 + 0.982817i 0.440906π0.440906\pi
588588 0 0
589589 0 0
590590 0 0
591591 0 0
592592 0 0
593593 42.4853 1.74466 0.872331 0.488916i 0.162608π-0.162608\pi
0.872331 + 0.488916i 0.162608π0.162608\pi
594594 0 0
595595 0 0
596596 −12.0000 −0.491539
597597 0 0
598598 0 0
599599 −39.0000 −1.59350 −0.796748 0.604311i 0.793448π-0.793448\pi
−0.796748 + 0.604311i 0.793448π0.793448\pi
600600 0 0
601601 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
602602 0 0
603603 0 0
604604 34.0000 1.38344
605605 0 0
606606 0 0
607607 20.1246 0.816833 0.408416 0.912796i 0.366081π-0.366081\pi
0.408416 + 0.912796i 0.366081π0.366081\pi
608608 0 0
609609 0 0
610610 0 0
611611 −75.0000 −3.03418
612612 −8.94427 −0.361551
613613 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
614614 0 0
615615 0 0
616616 0 0
617617 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
618618 0 0
619619 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
620620 0 0
621621 0 0
622622 0 0
623623 0 0
624624 −60.0000 −2.40192
625625 0 0
626626 0 0
627627 0 0
628628 26.8328 1.07075
629629 0 0
630630 0 0
631631 −47.0000 −1.87104 −0.935520 0.353273i 0.885069π-0.885069\pi
−0.935520 + 0.353273i 0.885069π0.885069\pi
632632 0 0
633633 −51.4296 −2.04414
634634 0 0
635635 0 0
636636 0 0
637637 0 0
638638 0 0
639639 −24.0000 −0.949425
640640 0 0
641641 18.0000 0.710957 0.355479 0.934684i 0.384318π-0.384318\pi
0.355479 + 0.934684i 0.384318π0.384318\pi
642642 0 0
643643 6.70820 0.264546 0.132273 0.991213i 0.457772π-0.457772\pi
0.132273 + 0.991213i 0.457772π0.457772\pi
644644 0 0
645645 0 0
646646 0 0
647647 17.8885 0.703271 0.351636 0.936137i 0.385626π-0.385626\pi
0.351636 + 0.936137i 0.385626π0.385626\pi
648648 0 0
649649 0 0
650650 0 0
651651 0 0
652652 0 0
653653 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
654654 0 0
655655 0 0
656656 0 0
657657 26.8328 1.04685
658658 0 0
659659 −51.0000 −1.98668 −0.993339 0.115229i 0.963240π-0.963240\pi
−0.993339 + 0.115229i 0.963240π0.963240\pi
660660 0 0
661661 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
662662 0 0
663663 −33.5410 −1.30263
664664 0 0
665665 0 0
666666 0 0
667667 0 0
668668 −49.1935 −1.90335
669669 −15.0000 −0.579934
670670 0 0
671671 0 0
672672 0 0
673673 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
674674 0 0
675675 0 0
676676 −64.0000 −2.46154
677677 51.4296 1.97660 0.988299 0.152527i 0.0487410π-0.0487410\pi
0.988299 + 0.152527i 0.0487410π0.0487410\pi
678678 0 0
679679 0 0
680680 0 0
681681 65.0000 2.49081
682682 0 0
683683 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
684684 0 0
685685 0 0
686686 0 0
687687 0 0
688688 0 0
689689 0 0
690690 0 0
691691 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
692692 −22.3607 −0.850026
693693 0 0
694694 0 0
695695 0 0
696696 0 0
697697 0 0
698698 0 0
699699 0 0
700700 0 0
701701 −33.0000 −1.24639 −0.623196 0.782065i 0.714166π-0.714166\pi
−0.623196 + 0.782065i 0.714166π0.714166\pi
702702 0 0
703703 0 0
704704 24.0000 0.904534
705705 0 0
706706 0 0
707707 0 0
708708 0 0
709709 −1.00000 −0.0375558 −0.0187779 0.999824i 0.505978π-0.505978\pi
−0.0187779 + 0.999824i 0.505978π0.505978\pi
710710 0 0
711711 −2.00000 −0.0750059
712712 0 0
713713 0 0
714714 0 0
715715 0 0
716716 −48.0000 −1.79384
717717 20.1246 0.751567
718718 0 0
719719 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
720720 0 0
721721 0 0
722722 0 0
723723 0 0
724724 0 0
725725 0 0
726726 0 0
727727 −53.6656 −1.99035 −0.995174 0.0981255i 0.968715π-0.968715\pi
−0.995174 + 0.0981255i 0.968715π0.968715\pi
728728 0 0
729729 −7.00000 −0.259259
730730 0 0
731731 0 0
732732 0 0
733733 −20.1246 −0.743319 −0.371660 0.928369i 0.621211π-0.621211\pi
−0.371660 + 0.928369i 0.621211π0.621211\pi
734734 0 0
735735 0 0
736736 0 0
737737 0 0
738738 0 0
739739 −11.0000 −0.404642 −0.202321 0.979319i 0.564848π-0.564848\pi
−0.202321 + 0.979319i 0.564848π0.564848\pi
740740 0 0
741741 0 0
742742 0 0
743743 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
744744 0 0
745745 0 0
746746 0 0
747747 −17.8885 −0.654508
748748 13.4164 0.490552
749749 0 0
750750 0 0
751751 −13.0000 −0.474377 −0.237188 0.971464i 0.576226π-0.576226\pi
−0.237188 + 0.971464i 0.576226π0.576226\pi
752752 −44.7214 −1.63082
753753 0 0
754754 0 0
755755 0 0
756756 0 0
757757 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
758758 0 0
759759 0 0
760760 0 0
761761 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
762762 0 0
763763 0 0
764764 54.0000 1.95365
765765 0 0
766766 0 0
767767 0 0
768768 −35.7771 −1.29099
769769 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
770770 0 0
771771 −10.0000 −0.360141
772772 0 0
773773 −2.23607 −0.0804258 −0.0402129 0.999191i 0.512804π-0.512804\pi
−0.0402129 + 0.999191i 0.512804π0.512804\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 0 0
779779 0 0
780780 0 0
781781 36.0000 1.28818
782782 0 0
783783 −20.1246 −0.719195
784784 0 0
785785 0 0
786786 0 0
787787 33.5410 1.19561 0.597804 0.801642i 0.296040π-0.296040\pi
0.597804 + 0.801642i 0.296040π0.296040\pi
788788 0 0
789789 0 0
790790 0 0
791791 0 0
792792 0 0
793793 0 0
794794 0 0
795795 0 0
796796 0 0
797797 55.9017 1.98014 0.990070 0.140576i 0.0448954π-0.0448954\pi
0.990070 + 0.140576i 0.0448954π0.0448954\pi
798798 0 0
799799 −25.0000 −0.884436
800800 0 0
801801 0 0
802802 0 0
803803 −40.2492 −1.42036
804804 0 0
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 −39.0000 −1.37117 −0.685583 0.727994i 0.740453π-0.740453\pi
−0.685583 + 0.727994i 0.740453π0.740453\pi
810810 0 0
811811 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
812812 0 0
813813 0 0
814814 0 0
815815 0 0
816816 −20.0000 −0.700140
817817 0 0
818818 0 0
819819 0 0
820820 0 0
821821 57.0000 1.98931 0.994657 0.103236i 0.0329198π-0.0329198\pi
0.994657 + 0.103236i 0.0329198π0.0329198\pi
822822 0 0
823823 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
824824 0 0
825825 0 0
826826 0 0
827827 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
828828 0 0
829829 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
830830 0 0
831831 0 0
832832 −53.6656 −1.86052
833833 0 0
834834 0 0
835835 0 0
836836 0 0
837837 0 0
838838 0 0
839839 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
840840 0 0
841841 52.0000 1.79310
842842 0 0
843843 73.7902 2.54147
844844 −46.0000 −1.58339
845845 0 0
846846 0 0
847847 0 0
848848 0 0
849849 75.0000 2.57399
850850 0 0
851851 0 0
852852 −53.6656 −1.83855
853853 −40.2492 −1.37811 −0.689054 0.724710i 0.741974π-0.741974\pi
−0.689054 + 0.724710i 0.741974π0.741974\pi
854854 0 0
855855 0 0
856856 0 0
857857 49.1935 1.68042 0.840209 0.542263i 0.182432π-0.182432\pi
0.840209 + 0.542263i 0.182432π0.182432\pi
858858 0 0
859859 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
860860 0 0
861861 0 0
862862 0 0
863863 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
864864 0 0
865865 0 0
866866 0 0
867867 26.8328 0.911290
868868 0 0
869869 3.00000 0.101768
870870 0 0
871871 0 0
872872 0 0
873873 −13.4164 −0.454077
874874 0 0
875875 0 0
876876 60.0000 2.02721
877877 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
878878 0 0
879879 55.0000 1.85510
880880 0 0
881881 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
882882 0 0
883883 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
884884 −30.0000 −1.00901
885885 0 0
886886 0 0
887887 35.7771 1.20128 0.600639 0.799521i 0.294913π-0.294913\pi
0.600639 + 0.799521i 0.294913π0.294913\pi
888888 0 0
889889 0 0
890890 0 0
891891 33.0000 1.10554
892892 −13.4164 −0.449215
893893 0 0
894894 0 0
895895 0 0
896896 0 0
897897 0 0
898898 0 0
899899 0 0
900900 0 0
901901 0 0
902902 0 0
903903 0 0
904904 0 0
905905 0 0
906906 0 0
907907 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
908908 58.1378 1.92937
909909 0 0
910910 0 0
911911 −12.0000 −0.397578 −0.198789 0.980042i 0.563701π-0.563701\pi
−0.198789 + 0.980042i 0.563701π0.563701\pi
912912 0 0
913913 26.8328 0.888037
914914 0 0
915915 0 0
916916 0 0
917917 0 0
918918 0 0
919919 −29.0000 −0.956622 −0.478311 0.878191i 0.658751π-0.658751\pi
−0.478311 + 0.878191i 0.658751π0.658751\pi
920920 0 0
921921 15.0000 0.494267
922922 0 0
923923 −80.4984 −2.64964
924924 0 0
925925 0 0
926926 0 0
927927 −40.2492 −1.32196
928928 0 0
929929 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
930930 0 0
931931 0 0
932932 0 0
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 −6.70820 −0.219147 −0.109574 0.993979i 0.534949π-0.534949\pi
−0.109574 + 0.993979i 0.534949π0.534949\pi
938938 0 0
939939 45.0000 1.46852
940940 0 0
941941 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
942942 0 0
943943 0 0
944944 0 0
945945 0 0
946946 0 0
947947 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
948948 −4.47214 −0.145248
949949 90.0000 2.92152
950950 0 0
951951 0 0
952952 0 0
953953 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
954954 0 0
955955 0 0
956956 18.0000 0.582162
957957 −60.3738 −1.95161
958958 0 0
959959 0 0
960960 0 0
961961 −31.0000 −1.00000
962962 0 0
963963 0 0
964964 0 0
965965 0 0
966966 0 0
967967 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
968968 0 0
969969 0 0
970970 0 0
971971 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
972972 −35.7771 −1.14755
973973 0 0
974974 0 0
975975 0 0
976976 0 0
977977 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
978978 0 0
979979 0 0
980980 0 0
981981 −22.0000 −0.702406
982982 0 0
983983 29.0689 0.927153 0.463577 0.886057i 0.346566π-0.346566\pi
0.463577 + 0.886057i 0.346566π0.346566\pi
984984 0 0
985985 0 0
986986 0 0
987987 0 0
988988 0 0
989989 0 0
990990 0 0
991991 −52.0000 −1.65183 −0.825917 0.563791i 0.809342π-0.809342\pi
−0.825917 + 0.563791i 0.809342π0.809342\pi
992992 0 0
993993 −17.8885 −0.567676
994994 0 0
995995 0 0
996996 −40.0000 −1.26745
997997 −60.3738 −1.91206 −0.956029 0.293271i 0.905256π-0.905256\pi
−0.956029 + 0.293271i 0.905256π0.905256\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 1225.2.a.q.1.1 2
5.2 odd 4 245.2.b.d.99.2 yes 2
5.3 odd 4 245.2.b.d.99.1 2
5.4 even 2 inner 1225.2.a.q.1.2 2
7.6 odd 2 inner 1225.2.a.q.1.2 2
15.2 even 4 2205.2.d.h.1324.1 2
15.8 even 4 2205.2.d.h.1324.2 2
35.2 odd 12 245.2.j.b.214.1 4
35.3 even 12 245.2.j.b.79.2 4
35.12 even 12 245.2.j.b.214.2 4
35.13 even 4 245.2.b.d.99.2 yes 2
35.17 even 12 245.2.j.b.79.1 4
35.18 odd 12 245.2.j.b.79.1 4
35.23 odd 12 245.2.j.b.214.2 4
35.27 even 4 245.2.b.d.99.1 2
35.32 odd 12 245.2.j.b.79.2 4
35.33 even 12 245.2.j.b.214.1 4
35.34 odd 2 CM 1225.2.a.q.1.1 2
105.62 odd 4 2205.2.d.h.1324.2 2
105.83 odd 4 2205.2.d.h.1324.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.b.d.99.1 2 5.3 odd 4
245.2.b.d.99.1 2 35.27 even 4
245.2.b.d.99.2 yes 2 5.2 odd 4
245.2.b.d.99.2 yes 2 35.13 even 4
245.2.j.b.79.1 4 35.17 even 12
245.2.j.b.79.1 4 35.18 odd 12
245.2.j.b.79.2 4 35.3 even 12
245.2.j.b.79.2 4 35.32 odd 12
245.2.j.b.214.1 4 35.2 odd 12
245.2.j.b.214.1 4 35.33 even 12
245.2.j.b.214.2 4 35.12 even 12
245.2.j.b.214.2 4 35.23 odd 12
1225.2.a.q.1.1 2 1.1 even 1 trivial
1225.2.a.q.1.1 2 35.34 odd 2 CM
1225.2.a.q.1.2 2 5.4 even 2 inner
1225.2.a.q.1.2 2 7.6 odd 2 inner
2205.2.d.h.1324.1 2 15.2 even 4
2205.2.d.h.1324.1 2 105.83 odd 4
2205.2.d.h.1324.2 2 15.8 even 4
2205.2.d.h.1324.2 2 105.62 odd 4