Properties

Label 825.2.c.f.199.1
Level 825825
Weight 22
Character 825.199
Analytic conductor 6.5886.588
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 825=35211 825 = 3 \cdot 5^{2} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 825.c (of order 22, degree 11, not 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: no
Analytic conductor: 6.587658166766.58765816676
Analytic rank: 00
Dimension: 66
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x64x3+25x220x+8 x^{6} - 4x^{3} + 25x^{2} - 20x + 8 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: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 199.1
Root 0.432320+0.432320i0.432320 + 0.432320i of defining polynomial
Character χ\chi == 825.199
Dual form 825.2.c.f.199.6

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) == q2.76156iq2+1.00000iq35.62620q4+2.76156q6+1.86464iq7+10.0140iq81.00000q9+1.00000q115.62620iq124.62620iq13+5.14931q14+16.4017q16+2.49084iq17+2.76156iq18+5.38776q191.86464q212.76156iq227.14931iq2310.0140q2412.7755q261.00000iq2710.4908iq28+3.52311q29+8.62620q3125.2663iq32+1.00000iq33+6.87859q34+5.62620q368.87859iq3714.8786iq38+4.62620q390.761557q41+5.14931iq42+7.40171iq435.62620q4419.7432q460.373802iq47+16.4017iq48+3.52311q492.49084q51+26.0279iq52+5.45856iq532.76156q5418.6724q56+5.38776iq579.72928iq585.14931q59+4.42003q6123.8217iq621.86464iq6336.9711q64+2.76156q66+11.9431iq6714.0140iq68+7.14931q69+11.6262q7110.0140iq726.77551iq7324.5187q7430.3126q76+1.86464iq7712.7755iq78+6.01395q79+1.00000q81+2.10308iq8214.5693iq83+10.4908q84+20.4402q86+3.52311iq87+10.0140iq889.04623q89+8.62620q91+40.2234iq92+8.62620iq931.03228q94+25.2663q96+16.3169iq979.72928iq981.00000q99+O(q100)q-2.76156i q^{2} +1.00000i q^{3} -5.62620 q^{4} +2.76156 q^{6} +1.86464i q^{7} +10.0140i q^{8} -1.00000 q^{9} +1.00000 q^{11} -5.62620i q^{12} -4.62620i q^{13} +5.14931 q^{14} +16.4017 q^{16} +2.49084i q^{17} +2.76156i q^{18} +5.38776 q^{19} -1.86464 q^{21} -2.76156i q^{22} -7.14931i q^{23} -10.0140 q^{24} -12.7755 q^{26} -1.00000i q^{27} -10.4908i q^{28} +3.52311 q^{29} +8.62620 q^{31} -25.2663i q^{32} +1.00000i q^{33} +6.87859 q^{34} +5.62620 q^{36} -8.87859i q^{37} -14.8786i q^{38} +4.62620 q^{39} -0.761557 q^{41} +5.14931i q^{42} +7.40171i q^{43} -5.62620 q^{44} -19.7432 q^{46} -0.373802i q^{47} +16.4017i q^{48} +3.52311 q^{49} -2.49084 q^{51} +26.0279i q^{52} +5.45856i q^{53} -2.76156 q^{54} -18.6724 q^{56} +5.38776i q^{57} -9.72928i q^{58} -5.14931 q^{59} +4.42003 q^{61} -23.8217i q^{62} -1.86464i q^{63} -36.9711 q^{64} +2.76156 q^{66} +11.9431i q^{67} -14.0140i q^{68} +7.14931 q^{69} +11.6262 q^{71} -10.0140i q^{72} -6.77551i q^{73} -24.5187 q^{74} -30.3126 q^{76} +1.86464i q^{77} -12.7755i q^{78} +6.01395 q^{79} +1.00000 q^{81} +2.10308i q^{82} -14.5693i q^{83} +10.4908 q^{84} +20.4402 q^{86} +3.52311i q^{87} +10.0140i q^{88} -9.04623 q^{89} +8.62620 q^{91} +40.2234i q^{92} +8.62620i q^{93} -1.03228 q^{94} +25.2663 q^{96} +16.3169i q^{97} -9.72928i q^{98} -1.00000 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q16q4+4q66q9+6q1112q14+20q16+2q196q2112q2416q264q29+34q3112q34+16q36+10q39+8q4116q4460q46+6q99+O(q100) 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46}+ \cdots - 6 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/825Z)×\left(\mathbb{Z}/825\mathbb{Z}\right)^\times.

nn 376376 551551 727727
χ(n)\chi(n) 11 11 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 − 2.76156i − 1.95272i −0.216160 0.976358i 0.569353π-0.569353\pi
0.216160 0.976358i 0.430647π-0.430647\pi
33 1.00000i 0.577350i
44 −5.62620 −2.81310
55 0 0
66 2.76156 1.12740
77 1.86464i 0.704768i 0.935855 + 0.352384i 0.114629π0.114629\pi
−0.935855 + 0.352384i 0.885371π0.885371\pi
88 10.0140i 3.54047i
99 −1.00000 −0.333333
1010 0 0
1111 1.00000 0.301511
1212 − 5.62620i − 1.62414i
1313 − 4.62620i − 1.28308i −0.767091 0.641538i 0.778297π-0.778297\pi
0.767091 0.641538i 0.221703π-0.221703\pi
1414 5.14931 1.37621
1515 0 0
1616 16.4017 4.10043
1717 2.49084i 0.604117i 0.953289 + 0.302059i 0.0976738π0.0976738\pi
−0.953289 + 0.302059i 0.902326π0.902326\pi
1818 2.76156i 0.650905i
1919 5.38776 1.23604 0.618018 0.786164i 0.287936π-0.287936\pi
0.618018 + 0.786164i 0.287936π0.287936\pi
2020 0 0
2121 −1.86464 −0.406898
2222 − 2.76156i − 0.588766i
2323 − 7.14931i − 1.49073i −0.666654 0.745367i 0.732274π-0.732274\pi
0.666654 0.745367i 0.267726π-0.267726\pi
2424 −10.0140 −2.04409
2525 0 0
2626 −12.7755 −2.50548
2727 − 1.00000i − 0.192450i
2828 − 10.4908i − 1.98258i
2929 3.52311 0.654226 0.327113 0.944985i 0.393924π-0.393924\pi
0.327113 + 0.944985i 0.393924π0.393924\pi
3030 0 0
3131 8.62620 1.54931 0.774655 0.632384i 0.217923π-0.217923\pi
0.774655 + 0.632384i 0.217923π0.217923\pi
3232 − 25.2663i − 4.46650i
3333 1.00000i 0.174078i
3434 6.87859 1.17967
3535 0 0
3636 5.62620 0.937700
3737 − 8.87859i − 1.45963i −0.683644 0.729816i 0.739606π-0.739606\pi
0.683644 0.729816i 0.260394π-0.260394\pi
3838 − 14.8786i − 2.41363i
3939 4.62620 0.740785
4040 0 0
4141 −0.761557 −0.118935 −0.0594676 0.998230i 0.518940π-0.518940\pi
−0.0594676 + 0.998230i 0.518940π0.518940\pi
4242 5.14931i 0.794556i
4343 7.40171i 1.12875i 0.825519 + 0.564375i 0.190883π0.190883\pi
−0.825519 + 0.564375i 0.809117π0.809117\pi
4444 −5.62620 −0.848181
4545 0 0
4646 −19.7432 −2.91098
4747 − 0.373802i − 0.0545246i −0.999628 0.0272623i 0.991321π-0.991321\pi
0.999628 0.0272623i 0.00867894π-0.00867894\pi
4848 16.4017i 2.36738i
4949 3.52311 0.503302
5050 0 0
5151 −2.49084 −0.348787
5252 26.0279i 3.60942i
5353 5.45856i 0.749791i 0.927067 + 0.374896i 0.122321π0.122321\pi
−0.927067 + 0.374896i 0.877679π0.877679\pi
5454 −2.76156 −0.375800
5555 0 0
5656 −18.6724 −2.49521
5757 5.38776i 0.713626i
5858 − 9.72928i − 1.27752i
5959 −5.14931 −0.670383 −0.335192 0.942150i 0.608801π-0.608801\pi
−0.335192 + 0.942150i 0.608801π0.608801\pi
6060 0 0
6161 4.42003 0.565927 0.282963 0.959131i 0.408682π-0.408682\pi
0.282963 + 0.959131i 0.408682π0.408682\pi
6262 − 23.8217i − 3.02536i
6363 − 1.86464i − 0.234923i
6464 −36.9711 −4.62138
6565 0 0
6666 2.76156 0.339924
6767 11.9431i 1.45909i 0.683934 + 0.729544i 0.260268π0.260268\pi
−0.683934 + 0.729544i 0.739732π0.739732\pi
6868 − 14.0140i − 1.69944i
6969 7.14931 0.860676
7070 0 0
7171 11.6262 1.37978 0.689888 0.723916i 0.257660π-0.257660\pi
0.689888 + 0.723916i 0.257660π0.257660\pi
7272 − 10.0140i − 1.18016i
7373 − 6.77551i − 0.793014i −0.918032 0.396507i 0.870222π-0.870222\pi
0.918032 0.396507i 0.129778π-0.129778\pi
7474 −24.5187 −2.85025
7575 0 0
7676 −30.3126 −3.47709
7777 1.86464i 0.212496i
7878 − 12.7755i − 1.44654i
7979 6.01395 0.676623 0.338311 0.941034i 0.390144π-0.390144\pi
0.338311 + 0.941034i 0.390144π0.390144\pi
8080 0 0
8181 1.00000 0.111111
8282 2.10308i 0.232247i
8383 − 14.5693i − 1.59919i −0.600538 0.799597i 0.705047π-0.705047\pi
0.600538 0.799597i 0.294953π-0.294953\pi
8484 10.4908 1.14464
8585 0 0
8686 20.4402 2.20413
8787 3.52311i 0.377718i
8888 10.0140i 1.06749i
8989 −9.04623 −0.958898 −0.479449 0.877570i 0.659164π-0.659164\pi
−0.479449 + 0.877570i 0.659164π0.659164\pi
9090 0 0
9191 8.62620 0.904271
9292 40.2234i 4.19358i
9393 8.62620i 0.894495i
9494 −1.03228 −0.106471
9595 0 0
9696 25.2663 2.57874
9797 16.3169i 1.65673i 0.560185 + 0.828367i 0.310730π0.310730\pi
−0.560185 + 0.828367i 0.689270π0.689270\pi
9898 − 9.72928i − 0.982806i
9999 −1.00000 −0.100504
100100 0 0
101101 −1.03228 −0.102715 −0.0513576 0.998680i 0.516355π-0.516355\pi
−0.0513576 + 0.998680i 0.516355π0.516355\pi
102102 6.87859i 0.681082i
103103 − 3.04623i − 0.300154i −0.988674 0.150077i 0.952048π-0.952048\pi
0.988674 0.150077i 0.0479521π-0.0479521\pi
104104 46.3265 4.54269
105105 0 0
106106 15.0741 1.46413
107107 − 8.50479i − 0.822189i −0.911593 0.411095i 0.865147π-0.865147\pi
0.911593 0.411095i 0.134853π-0.134853\pi
108108 5.62620i 0.541381i
109109 6.14931 0.588997 0.294499 0.955652i 0.404847π-0.404847\pi
0.294499 + 0.955652i 0.404847π0.404847\pi
110110 0 0
111111 8.87859 0.842719
112112 30.5833i 2.88985i
113113 6.00000i 0.564433i 0.959351 + 0.282216i 0.0910696π0.0910696\pi
−0.959351 + 0.282216i 0.908930π0.908930\pi
114114 14.8786 1.39351
115115 0 0
116116 −19.8217 −1.84040
117117 4.62620i 0.427692i
118118 14.2201i 1.30907i
119119 −4.64452 −0.425762
120120 0 0
121121 1.00000 0.0909091
122122 − 12.2062i − 1.10509i
123123 − 0.761557i − 0.0686673i
124124 −48.5327 −4.35837
125125 0 0
126126 −5.14931 −0.458737
127127 − 9.26635i − 0.822256i −0.911578 0.411128i 0.865135π-0.865135\pi
0.911578 0.411128i 0.134865π-0.134865\pi
128128 51.5650i 4.55774i
129129 −7.40171 −0.651684
130130 0 0
131131 4.06455 0.355121 0.177561 0.984110i 0.443179π-0.443179\pi
0.177561 + 0.984110i 0.443179π0.443179\pi
132132 − 5.62620i − 0.489698i
133133 10.0462i 0.871119i
134134 32.9817 2.84918
135135 0 0
136136 −24.9431 −2.13886
137137 5.93545i 0.507100i 0.967322 + 0.253550i 0.0815982π0.0815982\pi
−0.967322 + 0.253550i 0.918402π0.918402\pi
138138 − 19.7432i − 1.68066i
139139 −6.71096 −0.569216 −0.284608 0.958644i 0.591863π-0.591863\pi
−0.284608 + 0.958644i 0.591863π0.591863\pi
140140 0 0
141141 0.373802 0.0314798
142142 − 32.1064i − 2.69431i
143143 − 4.62620i − 0.386862i
144144 −16.4017 −1.36681
145145 0 0
146146 −18.7110 −1.54853
147147 3.52311i 0.290582i
148148 49.9527i 4.10609i
149149 −5.74324 −0.470504 −0.235252 0.971934i 0.575592π-0.575592\pi
−0.235252 + 0.971934i 0.575592π0.575592\pi
150150 0 0
151151 10.5616 0.859495 0.429747 0.902949i 0.358603π-0.358603\pi
0.429747 + 0.902949i 0.358603π0.358603\pi
152152 53.9527i 4.37614i
153153 − 2.49084i − 0.201372i
154154 5.14931 0.414943
155155 0 0
156156 −26.0279 −2.08390
157157 3.10308i 0.247653i 0.992304 + 0.123827i 0.0395166π0.0395166\pi
−0.992304 + 0.123827i 0.960483π0.960483\pi
158158 − 16.6079i − 1.32125i
159159 −5.45856 −0.432892
160160 0 0
161161 13.3309 1.05062
162162 − 2.76156i − 0.216968i
163163 − 15.6724i − 1.22756i −0.789477 0.613780i 0.789648π-0.789648\pi
0.789477 0.613780i 0.210352π-0.210352\pi
164164 4.28467 0.334577
165165 0 0
166166 −40.2341 −3.12277
167167 − 8.98168i − 0.695023i −0.937676 0.347512i 0.887027π-0.887027\pi
0.937676 0.347512i 0.112973π-0.112973\pi
168168 − 18.6724i − 1.44061i
169169 −8.40171 −0.646285
170170 0 0
171171 −5.38776 −0.412012
172172 − 41.6435i − 3.17529i
173173 11.5092i 0.875025i 0.899212 + 0.437513i 0.144140π0.144140\pi
−0.899212 + 0.437513i 0.855860π0.855860\pi
174174 9.72928 0.737575
175175 0 0
176176 16.4017 1.23633
177177 − 5.14931i − 0.387046i
178178 24.9817i 1.87246i
179179 −10.3738 −0.775374 −0.387687 0.921791i 0.626726π-0.626726\pi
−0.387687 + 0.921791i 0.626726π0.626726\pi
180180 0 0
181181 −2.66473 −0.198068 −0.0990339 0.995084i 0.531575π-0.531575\pi
−0.0990339 + 0.995084i 0.531575π0.531575\pi
182182 − 23.8217i − 1.76578i
183183 4.42003i 0.326738i
184184 71.5929 5.27790
185185 0 0
186186 23.8217 1.74669
187187 2.49084i 0.182148i
188188 2.10308i 0.153383i
189189 1.86464 0.135633
190190 0 0
191191 14.6724 1.06166 0.530830 0.847478i 0.321880π-0.321880\pi
0.530830 + 0.847478i 0.321880π0.321880\pi
192192 − 36.9711i − 2.66816i
193193 − 5.10308i − 0.367328i −0.982989 0.183664i 0.941204π-0.941204\pi
0.982989 0.183664i 0.0587958π-0.0587958\pi
194194 45.0602 3.23513
195195 0 0
196196 −19.8217 −1.41584
197197 3.74324i 0.266694i 0.991069 + 0.133347i 0.0425725π0.0425725\pi
−0.991069 + 0.133347i 0.957427π0.957427\pi
198198 2.76156i 0.196255i
199199 8.08476 0.573114 0.286557 0.958063i 0.407489π-0.407489\pi
0.286557 + 0.958063i 0.407489π0.407489\pi
200200 0 0
201201 −11.9431 −0.842404
202202 2.85069i 0.200574i
203203 6.56934i 0.461077i
204204 14.0140 0.981173
205205 0 0
206206 −8.41233 −0.586115
207207 7.14931i 0.496912i
208208 − 75.8776i − 5.26116i
209209 5.38776 0.372679
210210 0 0
211211 −15.9431 −1.09757 −0.548786 0.835963i 0.684910π-0.684910\pi
−0.548786 + 0.835963i 0.684910π0.684910\pi
212212 − 30.7110i − 2.10924i
213213 11.6262i 0.796614i
214214 −23.4865 −1.60550
215215 0 0
216216 10.0140 0.681363
217217 16.0848i 1.09190i
218218 − 16.9817i − 1.15014i
219219 6.77551 0.457847
220220 0 0
221221 11.5231 0.775129
222222 − 24.5187i − 1.64559i
223223 22.1772i 1.48510i 0.669793 + 0.742548i 0.266383π0.266383\pi
−0.669793 + 0.742548i 0.733617π0.733617\pi
224224 47.1127 3.14785
225225 0 0
226226 16.5693 1.10218
227227 5.93545i 0.393950i 0.980409 + 0.196975i 0.0631117π0.0631117\pi
−0.980409 + 0.196975i 0.936888π0.936888\pi
228228 − 30.3126i − 2.00750i
229229 8.31695 0.549599 0.274800 0.961502i 0.411388π-0.411388\pi
0.274800 + 0.961502i 0.411388π0.411388\pi
230230 0 0
231231 −1.86464 −0.122684
232232 35.2803i 2.31627i
233233 − 28.5187i − 1.86833i −0.356848 0.934163i 0.616148π-0.616148\pi
0.356848 0.934163i 0.383852π-0.383852\pi
234234 12.7755 0.835161
235235 0 0
236236 28.9711 1.88585
237237 6.01395i 0.390648i
238238 12.8261i 0.831393i
239239 −24.0925 −1.55841 −0.779206 0.626768i 0.784377π-0.784377\pi
−0.779206 + 0.626768i 0.784377π0.784377\pi
240240 0 0
241241 −5.16763 −0.332877 −0.166438 0.986052i 0.553227π-0.553227\pi
−0.166438 + 0.986052i 0.553227π0.553227\pi
242242 − 2.76156i − 0.177520i
243243 1.00000i 0.0641500i
244244 −24.8680 −1.59201
245245 0 0
246246 −2.10308 −0.134088
247247 − 24.9248i − 1.58593i
248248 86.3823i 5.48528i
249249 14.5693 0.923295
250250 0 0
251251 −9.25240 −0.584006 −0.292003 0.956417i 0.594322π-0.594322\pi
−0.292003 + 0.956417i 0.594322π0.594322\pi
252252 10.4908i 0.660861i
253253 − 7.14931i − 0.449473i
254254 −25.5896 −1.60563
255255 0 0
256256 68.4575 4.27860
257257 − 25.2158i − 1.57292i −0.617644 0.786458i 0.711913π-0.711913\pi
0.617644 0.786458i 0.288087π-0.288087\pi
258258 20.4402i 1.27255i
259259 16.5554 1.02870
260260 0 0
261261 −3.52311 −0.218075
262262 − 11.2245i − 0.693451i
263263 − 5.45856i − 0.336589i −0.985737 0.168295i 0.946174π-0.946174\pi
0.985737 0.168295i 0.0538260π-0.0538260\pi
264264 −10.0140 −0.616316
265265 0 0
266266 27.7432 1.70105
267267 − 9.04623i − 0.553620i
268268 − 67.1945i − 4.10456i
269269 −17.0741 −1.04103 −0.520514 0.853853i 0.674260π-0.674260\pi
−0.520514 + 0.853853i 0.674260π0.674260\pi
270270 0 0
271271 9.77988 0.594085 0.297043 0.954864i 0.404000π-0.404000\pi
0.297043 + 0.954864i 0.404000π0.404000\pi
272272 40.8540i 2.47714i
273273 8.62620i 0.522081i
274274 16.3911 0.990221
275275 0 0
276276 −40.2234 −2.42117
277277 5.91524i 0.355412i 0.984084 + 0.177706i 0.0568676π0.0568676\pi
−0.984084 + 0.177706i 0.943132π0.943132\pi
278278 18.5327i 1.11152i
279279 −8.62620 −0.516437
280280 0 0
281281 −6.76156 −0.403361 −0.201680 0.979451i 0.564640π-0.564640\pi
−0.201680 + 0.979451i 0.564640π0.564640\pi
282282 − 1.03228i − 0.0614711i
283283 − 8.81841i − 0.524200i −0.965041 0.262100i 0.915585π-0.915585\pi
0.965041 0.262100i 0.0844151π-0.0844151\pi
284284 −65.4113 −3.88145
285285 0 0
286286 −12.7755 −0.755432
287287 − 1.42003i − 0.0838218i
288288 25.2663i 1.48883i
289289 10.7957 0.635042
290290 0 0
291291 −16.3169 −0.956516
292292 38.1204i 2.23083i
293293 30.4908i 1.78129i 0.454696 + 0.890647i 0.349748π0.349748\pi
−0.454696 + 0.890647i 0.650252π0.650252\pi
294294 9.72928 0.567423
295295 0 0
296296 88.9098 5.16778
297297 − 1.00000i − 0.0580259i
298298 15.8603i 0.918761i
299299 −33.0741 −1.91273
300300 0 0
301301 −13.8015 −0.795507
302302 − 29.1666i − 1.67835i
303303 − 1.03228i − 0.0593027i
304304 88.3684 5.06827
305305 0 0
306306 −6.87859 −0.393223
307307 0.767815i 0.0438215i 0.999760 + 0.0219107i 0.00697497π0.00697497\pi
−0.999760 + 0.0219107i 0.993025π0.993025\pi
308308 − 10.4908i − 0.597771i
309309 3.04623 0.173294
310310 0 0
311311 −33.8496 −1.91944 −0.959719 0.280963i 0.909346π-0.909346\pi
−0.959719 + 0.280963i 0.909346π0.909346\pi
312312 46.3265i 2.62272i
313313 8.11078i 0.458448i 0.973374 + 0.229224i 0.0736189π0.0736189\pi
−0.973374 + 0.229224i 0.926381π0.926381\pi
314314 8.56934 0.483596
315315 0 0
316316 −33.8357 −1.90341
317317 8.06455i 0.452950i 0.974017 + 0.226475i 0.0727202π0.0727202\pi
−0.974017 + 0.226475i 0.927280π0.927280\pi
318318 15.0741i 0.845316i
319319 3.52311 0.197257
320320 0 0
321321 8.50479 0.474691
322322 − 36.8140i − 2.05157i
323323 13.4200i 0.746710i
324324 −5.62620 −0.312567
325325 0 0
326326 −43.2803 −2.39707
327327 6.14931i 0.340058i
328328 − 7.62620i − 0.421086i
329329 0.697006 0.0384272
330330 0 0
331331 22.4769 1.23544 0.617721 0.786398i 0.288056π-0.288056\pi
0.617721 + 0.786398i 0.288056π0.288056\pi
332332 81.9700i 4.49869i
333333 8.87859i 0.486544i
334334 −24.8034 −1.35718
335335 0 0
336336 −30.5833 −1.66846
337337 6.32757i 0.344685i 0.985037 + 0.172342i 0.0551336π0.0551336\pi
−0.985037 + 0.172342i 0.944866π0.944866\pi
338338 23.2018i 1.26201i
339339 −6.00000 −0.325875
340340 0 0
341341 8.62620 0.467135
342342 14.8786i 0.804542i
343343 19.6218i 1.05948i
344344 −74.1204 −3.99630
345345 0 0
346346 31.7832 1.70868
347347 − 0.178261i − 0.00956954i −0.999989 0.00478477i 0.998477π-0.998477\pi
0.999989 0.00478477i 0.00152305π-0.00152305\pi
348348 − 19.8217i − 1.06256i
349349 19.7293 1.05608 0.528042 0.849218i 0.322926π-0.322926\pi
0.528042 + 0.849218i 0.322926π0.322926\pi
350350 0 0
351351 −4.62620 −0.246928
352352 − 25.2663i − 1.34670i
353353 19.5231i 1.03911i 0.854437 + 0.519555i 0.173902π0.173902\pi
−0.854437 + 0.519555i 0.826098π0.826098\pi
354354 −14.2201 −0.755791
355355 0 0
356356 50.8959 2.69748
357357 − 4.64452i − 0.245814i
358358 28.6478i 1.51409i
359359 −35.6926 −1.88379 −0.941893 0.335914i 0.890955π-0.890955\pi
−0.941893 + 0.335914i 0.890955π0.890955\pi
360360 0 0
361361 10.0279 0.527785
362362 7.35881i 0.386770i
363363 1.00000i 0.0524864i
364364 −48.5327 −2.54380
365365 0 0
366366 12.2062 0.638027
367367 20.1127i 1.04987i 0.851141 + 0.524936i 0.175911π0.175911\pi
−0.851141 + 0.524936i 0.824089π0.824089\pi
368368 − 117.261i − 6.11265i
369369 0.761557 0.0396451
370370 0 0
371371 −10.1783 −0.528429
372372 − 48.5327i − 2.51630i
373373 2.56165i 0.132637i 0.997799 + 0.0663185i 0.0211253π0.0211253\pi
−0.997799 + 0.0663185i 0.978875π0.978875\pi
374374 6.87859 0.355684
375375 0 0
376376 3.74324 0.193043
377377 − 16.2986i − 0.839422i
378378 − 5.14931i − 0.264852i
379379 −23.2601 −1.19479 −0.597395 0.801947i 0.703798π-0.703798\pi
−0.597395 + 0.801947i 0.703798π0.703798\pi
380380 0 0
381381 9.26635 0.474729
382382 − 40.5187i − 2.07312i
383383 − 25.7938i − 1.31800i −0.752142 0.659002i 0.770979π-0.770979\pi
0.752142 0.659002i 0.229021π-0.229021\pi
384384 −51.5650 −2.63141
385385 0 0
386386 −14.0925 −0.717287
387387 − 7.40171i − 0.376250i
388388 − 91.8024i − 4.66056i
389389 −2.33527 −0.118403 −0.0592014 0.998246i 0.518855π-0.518855\pi
−0.0592014 + 0.998246i 0.518855π0.518855\pi
390390 0 0
391391 17.8078 0.900578
392392 35.2803i 1.78192i
393393 4.06455i 0.205029i
394394 10.3372 0.520778
395395 0 0
396396 5.62620 0.282727
397397 17.1955i 0.863019i 0.902108 + 0.431510i 0.142019π0.142019\pi
−0.902108 + 0.431510i 0.857981π0.857981\pi
398398 − 22.3265i − 1.11913i
399399 −10.0462 −0.502941
400400 0 0
401401 −26.5693 −1.32681 −0.663405 0.748261i 0.730889π-0.730889\pi
−0.663405 + 0.748261i 0.730889π0.730889\pi
402402 32.9817i 1.64498i
403403 − 39.9065i − 1.98788i
404404 5.80779 0.288948
405405 0 0
406406 18.1416 0.900353
407407 − 8.87859i − 0.440096i
408408 − 24.9431i − 1.23487i
409409 20.4200 1.00971 0.504853 0.863205i 0.331547π-0.331547\pi
0.504853 + 0.863205i 0.331547π0.331547\pi
410410 0 0
411411 −5.93545 −0.292774
412412 17.1387i 0.844362i
413413 − 9.60162i − 0.472465i
414414 19.7432 0.970327
415415 0 0
416416 −116.887 −5.73086
417417 − 6.71096i − 0.328637i
418418 − 14.8786i − 0.727736i
419419 13.5896 0.663893 0.331947 0.943298i 0.392295π-0.392295\pi
0.331947 + 0.943298i 0.392295π0.392295\pi
420420 0 0
421421 −39.6175 −1.93084 −0.965418 0.260705i 0.916045π-0.916045\pi
−0.965418 + 0.260705i 0.916045π0.916045\pi
422422 44.0279i 2.14324i
423423 0.373802i 0.0181749i
424424 −54.6618 −2.65461
425425 0 0
426426 32.1064 1.55556
427427 8.24177i 0.398847i
428428 47.8496i 2.31290i
429429 4.62620 0.223355
430430 0 0
431431 −6.56934 −0.316434 −0.158217 0.987404i 0.550575π-0.550575\pi
−0.158217 + 0.987404i 0.550575π0.550575\pi
432432 − 16.4017i − 0.789128i
433433 36.4113i 1.74982i 0.484290 + 0.874908i 0.339078π0.339078\pi
−0.484290 + 0.874908i 0.660922π0.660922\pi
434434 44.4190 2.13218
435435 0 0
436436 −34.5972 −1.65691
437437 − 38.5187i − 1.84260i
438438 − 18.7110i − 0.894044i
439439 −25.3878 −1.21169 −0.605846 0.795582i 0.707165π-0.707165\pi
−0.605846 + 0.795582i 0.707165π0.707165\pi
440440 0 0
441441 −3.52311 −0.167767
442442 − 31.8217i − 1.51361i
443443 20.9065i 0.993298i 0.867952 + 0.496649i 0.165436π0.165436\pi
−0.867952 + 0.496649i 0.834564π0.834564\pi
444444 −49.9527 −2.37065
445445 0 0
446446 61.2437 2.89997
447447 − 5.74324i − 0.271646i
448448 − 68.9377i − 3.25700i
449449 34.9450 1.64916 0.824579 0.565747i 0.191412π-0.191412\pi
0.824579 + 0.565747i 0.191412π0.191412\pi
450450 0 0
451451 −0.761557 −0.0358603
452452 − 33.7572i − 1.58780i
453453 10.5616i 0.496229i
454454 16.3911 0.769272
455455 0 0
456456 −53.9527 −2.52657
457457 7.31695i 0.342272i 0.985247 + 0.171136i 0.0547438π0.0547438\pi
−0.985247 + 0.171136i 0.945256π0.945256\pi
458458 − 22.9677i − 1.07321i
459459 2.49084 0.116262
460460 0 0
461461 −14.0279 −0.653345 −0.326672 0.945138i 0.605927π-0.605927\pi
−0.326672 + 0.945138i 0.605927π0.605927\pi
462462 5.14931i 0.239568i
463463 4.33527i 0.201477i 0.994913 + 0.100739i 0.0321206π0.0321206\pi
−0.994913 + 0.100739i 0.967879π0.967879\pi
464464 57.7851 2.68261
465465 0 0
466466 −78.7561 −3.64831
467467 − 3.70138i − 0.171279i −0.996326 0.0856396i 0.972707π-0.972707\pi
0.996326 0.0856396i 0.0272934π-0.0272934\pi
468468 − 26.0279i − 1.20314i
469469 −22.2697 −1.02832
470470 0 0
471471 −3.10308 −0.142983
472472 − 51.5650i − 2.37347i
473473 7.40171i 0.340331i
474474 16.6079 0.762825
475475 0 0
476476 26.1310 1.19771
477477 − 5.45856i − 0.249930i
478478 66.5327i 3.04313i
479479 22.8680 1.04486 0.522432 0.852681i 0.325025π-0.325025\pi
0.522432 + 0.852681i 0.325025π0.325025\pi
480480 0 0
481481 −41.0741 −1.87282
482482 14.2707i 0.650013i
483483 13.3309i 0.606577i
484484 −5.62620 −0.255736
485485 0 0
486486 2.76156 0.125267
487487 5.42962i 0.246039i 0.992404 + 0.123020i 0.0392578π0.0392578\pi
−0.992404 + 0.123020i 0.960742π0.960742\pi
488488 44.2620i 2.00365i
489489 15.6724 0.708732
490490 0 0
491491 21.2803 0.960367 0.480183 0.877168i 0.340570π-0.340570\pi
0.480183 + 0.877168i 0.340570π0.340570\pi
492492 4.28467i 0.193168i
493493 8.77551i 0.395229i
494494 −68.8313 −3.09687
495495 0 0
496496 141.484 6.35284
497497 21.6787i 0.972422i
498498 − 40.2341i − 1.80293i
499499 16.0077 0.716603 0.358301 0.933606i 0.383356π-0.383356\pi
0.358301 + 0.933606i 0.383356π0.383356\pi
500500 0 0
501501 8.98168 0.401272
502502 25.5510i 1.14040i
503503 12.0925i 0.539176i 0.962976 + 0.269588i 0.0868875π0.0868875\pi
−0.962976 + 0.269588i 0.913112π0.913112\pi
504504 18.6724 0.831736
505505 0 0
506506 −19.7432 −0.877694
507507 − 8.40171i − 0.373133i
508508 52.1343i 2.31309i
509509 −14.7110 −0.652052 −0.326026 0.945361i 0.605710π-0.605710\pi
−0.326026 + 0.945361i 0.605710π0.605710\pi
510510 0 0
511511 12.6339 0.558891
512512 − 85.9194i − 3.79714i
513513 − 5.38776i − 0.237875i
514514 −69.6347 −3.07146
515515 0 0
516516 41.6435 1.83325
517517 − 0.373802i − 0.0164398i
518518 − 45.7187i − 2.00876i
519519 −11.5092 −0.505196
520520 0 0
521521 −17.2803 −0.757064 −0.378532 0.925588i 0.623571π-0.623571\pi
−0.378532 + 0.925588i 0.623571π0.623571\pi
522522 9.72928i 0.425839i
523523 − 42.7326i − 1.86857i −0.356532 0.934283i 0.616041π-0.616041\pi
0.356532 0.934283i 0.383959π-0.383959\pi
524524 −22.8680 −0.998992
525525 0 0
526526 −15.0741 −0.657264
527527 21.4865i 0.935965i
528528 16.4017i 0.713793i
529529 −28.1127 −1.22229
530530 0 0
531531 5.14931 0.223461
532532 − 56.5221i − 2.45054i
533533 3.52311i 0.152603i
534534 −24.9817 −1.08106
535535 0 0
536536 −119.598 −5.16585
537537 − 10.3738i − 0.447663i
538538 47.1512i 2.03283i
539539 3.52311 0.151751
540540 0 0
541541 8.69075 0.373644 0.186822 0.982394i 0.440181π-0.440181\pi
0.186822 + 0.982394i 0.440181π0.440181\pi
542542 − 27.0077i − 1.16008i
543543 − 2.66473i − 0.114355i
544544 62.9344 2.69829
545545 0 0
546546 23.8217 1.01948
547547 − 44.1064i − 1.88585i −0.333000 0.942927i 0.608061π-0.608061\pi
0.333000 0.942927i 0.391939π-0.391939\pi
548548 − 33.3940i − 1.42652i
549549 −4.42003 −0.188642
550550 0 0
551551 18.9817 0.808647
552552 71.5929i 3.04720i
553553 11.2139i 0.476862i
554554 16.3353 0.694019
555555 0 0
556556 37.7572 1.60126
557557 − 7.69264i − 0.325948i −0.986630 0.162974i 0.947891π-0.947891\pi
0.986630 0.162974i 0.0521086π-0.0521086\pi
558558 23.8217i 1.00845i
559559 34.2418 1.44827
560560 0 0
561561 −2.49084 −0.105163
562562 18.6724i 0.787649i
563563 − 9.25240i − 0.389942i −0.980809 0.194971i 0.937539π-0.937539\pi
0.980809 0.194971i 0.0624613π-0.0624613\pi
564564 −2.10308 −0.0885558
565565 0 0
566566 −24.3525 −1.02361
567567 1.86464i 0.0783076i
568568 116.424i 4.88505i
569569 −4.96772 −0.208258 −0.104129 0.994564i 0.533205π-0.533205\pi
−0.104129 + 0.994564i 0.533205π0.533205\pi
570570 0 0
571571 6.02021 0.251938 0.125969 0.992034i 0.459796π-0.459796\pi
0.125969 + 0.992034i 0.459796π0.459796\pi
572572 26.0279i 1.08828i
573573 14.6724i 0.612949i
574574 −3.92150 −0.163680
575575 0 0
576576 36.9711 1.54046
577577 8.25240i 0.343552i 0.985136 + 0.171776i 0.0549505π0.0549505\pi
−0.985136 + 0.171776i 0.945050π0.945050\pi
578578 − 29.8130i − 1.24006i
579579 5.10308 0.212077
580580 0 0
581581 27.1666 1.12706
582582 45.0602i 1.86780i
583583 5.45856i 0.226071i
584584 67.8496 2.80764
585585 0 0
586586 84.2022 3.47836
587587 − 11.4846i − 0.474019i −0.971507 0.237010i 0.923833π-0.923833\pi
0.971507 0.237010i 0.0761673π-0.0761673\pi
588588 − 19.8217i − 0.817435i
589589 46.4758 1.91500
590590 0 0
591591 −3.74324 −0.153976
592592 − 145.624i − 5.98511i
593593 39.0375i 1.60308i 0.597943 + 0.801539i 0.295985π0.295985\pi
−0.597943 + 0.801539i 0.704015π0.704015\pi
594594 −2.76156 −0.113308
595595 0 0
596596 32.3126 1.32357
597597 8.08476i 0.330887i
598598 91.3361i 3.73501i
599599 30.1589 1.23226 0.616130 0.787645i 0.288700π-0.288700\pi
0.616130 + 0.787645i 0.288700π0.288700\pi
600600 0 0
601601 19.1310 0.780369 0.390185 0.920737i 0.372411π-0.372411\pi
0.390185 + 0.920737i 0.372411π0.372411\pi
602602 38.1137i 1.55340i
603603 − 11.9431i − 0.486362i
604604 −59.4219 −2.41784
605605 0 0
606606 −2.85069 −0.115801
607607 4.20617i 0.170723i 0.996350 + 0.0853615i 0.0272045π0.0272045\pi
−0.996350 + 0.0853615i 0.972795π0.972795\pi
608608 − 136.129i − 5.52076i
609609 −6.56934 −0.266203
610610 0 0
611611 −1.72928 −0.0699593
612612 14.0140i 0.566480i
613613 − 5.45856i − 0.220469i −0.993906 0.110235i 0.964840π-0.964840\pi
0.993906 0.110235i 0.0351602π-0.0351602\pi
614614 2.12036 0.0855709
615615 0 0
616616 −18.6724 −0.752334
617617 30.1974i 1.21570i 0.794051 + 0.607851i 0.207968π0.207968\pi
−0.794051 + 0.607851i 0.792032π0.792032\pi
618618 − 8.41233i − 0.338394i
619619 −23.1955 −0.932308 −0.466154 0.884704i 0.654361π-0.654361\pi
−0.466154 + 0.884704i 0.654361π0.654361\pi
620620 0 0
621621 −7.14931 −0.286892
622622 93.4777i 3.74812i
623623 − 16.8680i − 0.675801i
624624 75.8776 3.03753
625625 0 0
626626 22.3984 0.895219
627627 5.38776i 0.215166i
628628 − 17.4586i − 0.696673i
629629 22.1151 0.881789
630630 0 0
631631 −3.19554 −0.127212 −0.0636062 0.997975i 0.520260π-0.520260\pi
−0.0636062 + 0.997975i 0.520260π0.520260\pi
632632 60.2234i 2.39556i
633633 − 15.9431i − 0.633683i
634634 22.2707 0.884483
635635 0 0
636636 30.7110 1.21777
637637 − 16.2986i − 0.645775i
638638 − 9.72928i − 0.385186i
639639 −11.6262 −0.459925
640640 0 0
641641 28.1974 1.11373 0.556866 0.830603i 0.312004π-0.312004\pi
0.556866 + 0.830603i 0.312004π0.312004\pi
642642 − 23.4865i − 0.926937i
643643 − 17.9634i − 0.708406i −0.935169 0.354203i 0.884752π-0.884752\pi
0.935169 0.354203i 0.115248π-0.115248\pi
644644 −75.0023 −2.95550
645645 0 0
646646 37.0602 1.45811
647647 − 40.9990i − 1.61184i −0.592028 0.805918i 0.701672π-0.701672\pi
0.592028 0.805918i 0.298328π-0.298328\pi
648648 10.0140i 0.393385i
649649 −5.14931 −0.202128
650650 0 0
651651 −16.0848 −0.630412
652652 88.1762i 3.45325i
653653 − 11.6926i − 0.457568i −0.973477 0.228784i 0.926525π-0.926525\pi
0.973477 0.228784i 0.0734750π-0.0734750\pi
654654 16.9817 0.664036
655655 0 0
656656 −12.4908 −0.487685
657657 6.77551i 0.264338i
658658 − 1.92482i − 0.0750374i
659659 1.72928 0.0673633 0.0336816 0.999433i 0.489277π-0.489277\pi
0.0336816 + 0.999433i 0.489277π0.489277\pi
660660 0 0
661661 0.232185 0.00903096 0.00451548 0.999990i 0.498563π-0.498563\pi
0.00451548 + 0.999990i 0.498563π0.498563\pi
662662 − 62.0712i − 2.41247i
663663 11.5231i 0.447521i
664664 145.897 5.66189
665665 0 0
666666 24.5187 0.950082
667667 − 25.1878i − 0.975277i
668668 50.5327i 1.95517i
669669 −22.1772 −0.857421
670670 0 0
671671 4.42003 0.170633
672672 47.1127i 1.81741i
673673 − 33.5144i − 1.29188i −0.763386 0.645942i 0.776465π-0.776465\pi
0.763386 0.645942i 0.223535π-0.223535\pi
674674 17.4740 0.673072
675675 0 0
676676 47.2697 1.81806
677677 − 13.0183i − 0.500335i −0.968203 0.250167i 0.919514π-0.919514\pi
0.968203 0.250167i 0.0804857π-0.0804857\pi
678678 16.5693i 0.636342i
679679 −30.4252 −1.16761
680680 0 0
681681 −5.93545 −0.227447
682682 − 23.8217i − 0.912182i
683683 4.54333i 0.173846i 0.996215 + 0.0869228i 0.0277033π0.0277033\pi
−0.996215 + 0.0869228i 0.972297π0.972297\pi
684684 30.3126 1.15903
685685 0 0
686686 54.1868 2.06886
687687 8.31695i 0.317311i
688688 121.401i 4.62836i
689689 25.2524 0.962040
690690 0 0
691691 −19.8988 −0.756986 −0.378493 0.925604i 0.623558π-0.623558\pi
−0.378493 + 0.925604i 0.623558π0.623558\pi
692692 − 64.7528i − 2.46153i
693693 − 1.86464i − 0.0708319i
694694 −0.492277 −0.0186866
695695 0 0
696696 −35.2803 −1.33730
697697 − 1.89692i − 0.0718508i
698698 − 54.4835i − 2.06223i
699699 28.5187 1.07868
700700 0 0
701701 −15.8444 −0.598436 −0.299218 0.954185i 0.596726π-0.596726\pi
−0.299218 + 0.954185i 0.596726π0.596726\pi
702702 12.7755i 0.482181i
703703 − 47.8357i − 1.80416i
704704 −36.9711 −1.39340
705705 0 0
706706 53.9142 2.02909
707707 − 1.92482i − 0.0723904i
708708 28.9711i 1.08880i
709709 −8.82174 −0.331307 −0.165654 0.986184i 0.552973π-0.552973\pi
−0.165654 + 0.986184i 0.552973π0.552973\pi
710710 0 0
711711 −6.01395 −0.225541
712712 − 90.5885i − 3.39495i
713713 − 61.6714i − 2.30961i
714714 −12.8261 −0.480005
715715 0 0
716716 58.3651 2.18120
717717 − 24.0925i − 0.899749i
718718 98.5673i 3.67850i
719719 −36.7668 −1.37117 −0.685585 0.727993i 0.740453π-0.740453\pi
−0.685585 + 0.727993i 0.740453π0.740453\pi
720720 0 0
721721 5.68012 0.211539
722722 − 27.6926i − 1.03061i
723723 − 5.16763i − 0.192186i
724724 14.9923 0.557185
725725 0 0
726726 2.76156 0.102491
727727 − 5.27261i − 0.195550i −0.995209 0.0977751i 0.968827π-0.968827\pi
0.995209 0.0977751i 0.0311726π-0.0311726\pi
728728 86.3823i 3.20154i
729729 −1.00000 −0.0370370
730730 0 0
731731 −18.4365 −0.681897
732732 − 24.8680i − 0.919147i
733733 − 37.9508i − 1.40175i −0.713286 0.700873i 0.752794π-0.752794\pi
0.713286 0.700873i 0.247206π-0.247206\pi
734734 55.5423 2.05010
735735 0 0
736736 −180.637 −6.65837
737737 11.9431i 0.439931i
738738 − 2.10308i − 0.0774156i
739739 −37.2943 −1.37189 −0.685946 0.727653i 0.740611π-0.740611\pi
−0.685946 + 0.727653i 0.740611π0.740611\pi
740740 0 0
741741 24.9248 0.915636
742742 28.1078i 1.03187i
743743 − 22.5693i − 0.827989i −0.910279 0.413994i 0.864133π-0.864133\pi
0.910279 0.413994i 0.135867π-0.135867\pi
744744 −86.3823 −3.16693
745745 0 0
746746 7.07414 0.259002
747747 14.5693i 0.533064i
748748 − 14.0140i − 0.512401i
749749 15.8584 0.579453
750750 0 0
751751 −9.55102 −0.348522 −0.174261 0.984700i 0.555754π-0.555754\pi
−0.174261 + 0.984700i 0.555754π0.555754\pi
752752 − 6.13099i − 0.223574i
753753 − 9.25240i − 0.337176i
754754 −45.0096 −1.63915
755755 0 0
756756 −10.4908 −0.381548
757757 30.3188i 1.10196i 0.834519 + 0.550978i 0.185745π0.185745\pi
−0.834519 + 0.550978i 0.814255π0.814255\pi
758758 64.2341i 2.33309i
759759 7.14931 0.259504
760760 0 0
761761 −1.43066 −0.0518613 −0.0259306 0.999664i 0.508255π-0.508255\pi
−0.0259306 + 0.999664i 0.508255π0.508255\pi
762762 − 25.5896i − 0.927012i
763763 11.4663i 0.415106i
764764 −82.5500 −2.98655
765765 0 0
766766 −71.2311 −2.57369
767767 23.8217i 0.860153i
768768 68.4575i 2.47025i
769769 35.4017 1.27662 0.638309 0.769780i 0.279634π-0.279634\pi
0.638309 + 0.769780i 0.279634π0.279634\pi
770770 0 0
771771 25.2158 0.908123
772772 28.7110i 1.03333i
773773 13.4094i 0.482303i 0.970488 + 0.241151i 0.0775250π0.0775250\pi
−0.970488 + 0.241151i 0.922475π0.922475\pi
774774 −20.4402 −0.734709
775775 0 0
776776 −163.397 −5.86562
777777 16.5554i 0.593921i
778778 6.44898i 0.231207i
779779 −4.10308 −0.147008
780780 0 0
781781 11.6262 0.416018
782782 − 49.1772i − 1.75857i
783783 − 3.52311i − 0.125906i
784784 57.7851 2.06375
785785 0 0
786786 11.2245 0.400364
787787 − 37.3232i − 1.33043i −0.746653 0.665214i 0.768340π-0.768340\pi
0.746653 0.665214i 0.231660π-0.231660\pi
788788 − 21.0602i − 0.750238i
789789 5.45856 0.194330
790790 0 0
791791 −11.1878 −0.397794
792792 − 10.0140i − 0.355830i
793793 − 20.4479i − 0.726128i
794794 47.4865 1.68523
795795 0 0
796796 −45.4865 −1.61223
797797 − 8.05581i − 0.285352i −0.989769 0.142676i 0.954429π-0.954429\pi
0.989769 0.142676i 0.0455706π-0.0455706\pi
798798 27.7432i 0.982100i
799799 0.931080 0.0329393
800800 0 0
801801 9.04623 0.319633
802802 73.3728i 2.59088i
803803 − 6.77551i − 0.239103i
804804 67.1945 2.36977
805805 0 0
806806 −110.204 −3.88177
807807 − 17.0741i − 0.601038i
808808 − 10.3372i − 0.363660i
809809 42.3771 1.48990 0.744950 0.667120i 0.232473π-0.232473\pi
0.744950 + 0.667120i 0.232473π0.232473\pi
810810 0 0
811811 −21.3878 −0.751026 −0.375513 0.926817i 0.622533π-0.622533\pi
−0.375513 + 0.926817i 0.622533π0.622533\pi
812812 − 36.9604i − 1.29706i
813813 9.77988i 0.342995i
814814 −24.5187 −0.859382
815815 0 0
816816 −40.8540 −1.43018
817817 39.8786i 1.39518i
818818 − 56.3911i − 1.97167i
819819 −8.62620 −0.301424
820820 0 0
821821 −47.8776 −1.67094 −0.835469 0.549537i 0.814804π-0.814804\pi
−0.835469 + 0.549537i 0.814804π0.814804\pi
822822 16.3911i 0.571705i
823823 − 16.3555i − 0.570116i −0.958510 0.285058i 0.907987π-0.907987\pi
0.958510 0.285058i 0.0920129π-0.0920129\pi
824824 30.5048 1.06268
825825 0 0
826826 −26.5154 −0.922589
827827 − 44.7110i − 1.55475i −0.629036 0.777376i 0.716550π-0.716550\pi
0.629036 0.777376i 0.283450π-0.283450\pi
828828 − 40.2234i − 1.39786i
829829 38.5972 1.34054 0.670269 0.742118i 0.266179π-0.266179\pi
0.670269 + 0.742118i 0.266179π0.266179\pi
830830 0 0
831831 −5.91524 −0.205197
832832 171.035i 5.92959i
833833 8.77551i 0.304053i
834834 −18.5327 −0.641735
835835 0 0
836836 −30.3126 −1.04838
837837 − 8.62620i − 0.298165i
838838 − 37.5283i − 1.29639i
839839 0.710960 0.0245451 0.0122725 0.999925i 0.496093π-0.496093\pi
0.0122725 + 0.999925i 0.496093π0.496093\pi
840840 0 0
841841 −16.5877 −0.571988
842842 109.406i 3.77038i
843843 − 6.76156i − 0.232880i
844844 89.6993 3.08758
845845 0 0
846846 1.03228 0.0354904
847847 1.86464i 0.0640698i
848848 89.5298i 3.07446i
849849 8.81841 0.302647
850850 0 0
851851 −63.4758 −2.17592
852852 − 65.4113i − 2.24095i
853853 − 52.7187i − 1.80505i −0.430635 0.902526i 0.641710π-0.641710\pi
0.430635 0.902526i 0.358290π-0.358290\pi
854854 22.7601 0.778835
855855 0 0
856856 85.1666 2.91093
857857 16.7124i 0.570885i 0.958396 + 0.285442i 0.0921405π0.0921405\pi
−0.958396 + 0.285442i 0.907859π0.907859\pi
858858 − 12.7755i − 0.436149i
859859 41.9142 1.43009 0.715047 0.699076i 0.246405π-0.246405\pi
0.715047 + 0.699076i 0.246405π0.246405\pi
860860 0 0
861861 1.42003 0.0483945
862862 18.1416i 0.617906i
863863 41.4219i 1.41002i 0.709198 + 0.705009i 0.249057π0.249057\pi
−0.709198 + 0.705009i 0.750943π0.750943\pi
864864 −25.2663 −0.859579
865865 0 0
866866 100.552 3.41689
867867 10.7957i 0.366642i
868868 − 90.4961i − 3.07164i
869869 6.01395 0.204009
870870 0 0
871871 55.2514 1.87212
872872 61.5789i 2.08533i
873873 − 16.3169i − 0.552245i
874874 −106.372 −3.59808
875875 0 0
876876 −38.1204 −1.28797
877877 − 12.0848i − 0.408073i −0.978963 0.204037i 0.934594π-0.934594\pi
0.978963 0.204037i 0.0654062π-0.0654062\pi
878878 70.1097i 2.36609i
879879 −30.4908 −1.02843
880880 0 0
881881 −31.8130 −1.07181 −0.535904 0.844279i 0.680029π-0.680029\pi
−0.535904 + 0.844279i 0.680029π0.680029\pi
882882 9.72928i 0.327602i
883883 20.9894i 0.706349i 0.935558 + 0.353174i 0.114898π0.114898\pi
−0.935558 + 0.353174i 0.885102π0.885102\pi
884884 −64.8313 −2.18051
885885 0 0
886886 57.7345 1.93963
887887 49.2032i 1.65208i 0.563609 + 0.826042i 0.309412π0.309412\pi
−0.563609 + 0.826042i 0.690588π0.690588\pi
888888 88.9098i 2.98362i
889889 17.2784 0.579499
890890 0 0
891891 1.00000 0.0335013
892892 − 124.773i − 4.17772i
893893 − 2.01395i − 0.0673944i
894894 −15.8603 −0.530447
895895 0 0
896896 −96.1502 −3.21215
897897 − 33.0741i − 1.10431i
898898 − 96.5027i − 3.22034i
899899 30.3911 1.01360
900900 0 0
901901 −13.5964 −0.452962
902902 2.10308i 0.0700250i
903903 − 13.8015i − 0.459286i
904904 −60.0837 −1.99835
905905 0 0
906906 29.1666 0.968995
907907 − 24.8526i − 0.825216i −0.910909 0.412608i 0.864618π-0.864618\pi
0.910909 0.412608i 0.135382π-0.135382\pi
908908 − 33.3940i − 1.10822i
909909 1.03228 0.0342384
910910 0 0
911911 45.4758 1.50668 0.753341 0.657630i 0.228441π-0.228441\pi
0.753341 + 0.657630i 0.228441π0.228441\pi
912912 88.3684i 2.92617i
913913 − 14.5693i − 0.482175i
914914 20.2062 0.668361
915915 0 0
916916 −46.7928 −1.54608
917917 7.57893i 0.250278i
918918 − 6.87859i − 0.227027i
919919 15.7355 0.519068 0.259534 0.965734i 0.416431π-0.416431\pi
0.259534 + 0.965734i 0.416431π0.416431\pi
920920 0 0
921921 −0.767815 −0.0253004
922922 38.7389i 1.27580i
923923 − 53.7851i − 1.77036i
924924 10.4908 0.345123
925925 0 0
926926 11.9721 0.393427
927927 3.04623i 0.100051i
928928 − 89.0162i − 2.92210i
929929 2.06455 0.0677357 0.0338679 0.999426i 0.489217π-0.489217\pi
0.0338679 + 0.999426i 0.489217π0.489217\pi
930930 0 0
931931 18.9817 0.622099
932932 160.452i 5.25578i
933933 − 33.8496i − 1.10819i
934934 −10.2216 −0.334460
935935 0 0
936936 −46.3265 −1.51423
937937 40.1493i 1.31162i 0.754926 + 0.655810i 0.227673π0.227673\pi
−0.754926 + 0.655810i 0.772327π0.772327\pi
938938 61.4990i 2.00801i
939939 −8.11078 −0.264685
940940 0 0
941941 26.4050 0.860780 0.430390 0.902643i 0.358376π-0.358376\pi
0.430390 + 0.902643i 0.358376π0.358376\pi
942942 8.56934i 0.279204i
943943 5.44461i 0.177301i
944944 −84.4575 −2.74886
945945 0 0
946946 20.4402 0.664570
947947 8.67243i 0.281816i 0.990023 + 0.140908i 0.0450022π0.0450022\pi
−0.990023 + 0.140908i 0.954998π0.954998\pi
948948 − 33.8357i − 1.09893i
949949 −31.3449 −1.01750
950950 0 0
951951 −8.06455 −0.261511
952952 − 46.5100i − 1.50740i
953953 26.8540i 0.869887i 0.900458 + 0.434943i 0.143232π0.143232\pi
−0.900458 + 0.434943i 0.856768π0.856768\pi
954954 −15.0741 −0.488043
955955 0 0
956956 135.549 4.38397
957957 3.52311i 0.113886i
958958 − 63.1512i − 2.04032i
959959 −11.0675 −0.357388
960960 0 0
961961 43.4113 1.40036
962962 113.429i 3.65708i
963963 8.50479i 0.274063i
964964 29.0741 0.936415
965965 0 0
966966 36.8140 1.18447
967967 − 55.8496i − 1.79600i −0.439992 0.898002i 0.645019π-0.645019\pi
0.439992 0.898002i 0.354981π-0.354981\pi
968968 10.0140i 0.321861i
969969 −13.4200 −0.431113
970970 0 0
971971 30.0664 0.964878 0.482439 0.875930i 0.339751π-0.339751\pi
0.482439 + 0.875930i 0.339751π0.339751\pi
972972 − 5.62620i − 0.180460i
973973 − 12.5135i − 0.401165i
974974 14.9942 0.480445
975975 0 0
976976 72.4961 2.32054
977977 15.7014i 0.502331i 0.967944 + 0.251166i 0.0808139π0.0808139\pi
−0.967944 + 0.251166i 0.919186π0.919186\pi
978978 − 43.2803i − 1.38395i
979979 −9.04623 −0.289119
980980 0 0
981981 −6.14931 −0.196332
982982 − 58.7668i − 1.87532i
983983 51.9894i 1.65820i 0.559098 + 0.829102i 0.311148π0.311148\pi
−0.559098 + 0.829102i 0.688852π0.688852\pi
984984 7.62620 0.243114
985985 0 0
986986 24.2341 0.771770
987987 0.697006i 0.0221860i
988988 140.232i 4.46137i
989989 52.9171 1.68267
990990 0 0
991991 27.6445 0.878157 0.439079 0.898449i 0.355305π-0.355305\pi
0.439079 + 0.898449i 0.355305π0.355305\pi
992992 − 217.953i − 6.92000i
993993 22.4769i 0.713282i
994994 59.8669 1.89886
995995 0 0
996996 −81.9700 −2.59732
997997 40.5048i 1.28280i 0.767207 + 0.641400i 0.221646π0.221646\pi
−0.767207 + 0.641400i 0.778354π0.778354\pi
998998 − 44.2062i − 1.39932i
999999 −8.87859 −0.280906
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 825.2.c.f.199.1 6
3.2 odd 2 2475.2.c.q.199.6 6
5.2 odd 4 825.2.a.m.1.3 yes 3
5.3 odd 4 825.2.a.i.1.1 3
5.4 even 2 inner 825.2.c.f.199.6 6
15.2 even 4 2475.2.a.z.1.1 3
15.8 even 4 2475.2.a.bd.1.3 3
15.14 odd 2 2475.2.c.q.199.1 6
55.32 even 4 9075.2.a.cd.1.1 3
55.43 even 4 9075.2.a.cj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.1 3 5.3 odd 4
825.2.a.m.1.3 yes 3 5.2 odd 4
825.2.c.f.199.1 6 1.1 even 1 trivial
825.2.c.f.199.6 6 5.4 even 2 inner
2475.2.a.z.1.1 3 15.2 even 4
2475.2.a.bd.1.3 3 15.8 even 4
2475.2.c.q.199.1 6 15.14 odd 2
2475.2.c.q.199.6 6 3.2 odd 2
9075.2.a.cd.1.1 3 55.32 even 4
9075.2.a.cj.1.3 3 55.43 even 4