Properties

Label 252.8.e.a.71.8
Level 252252
Weight 88
Character 252.71
Analytic conductor 78.72178.721
Analytic rank 00
Dimension 8484
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(71,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.71");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 252=22327 252 = 2^{2} \cdot 3^{2} \cdot 7
Weight: k k == 8 8
Character orbit: [χ][\chi] == 252.e (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: no
Analytic conductor: 78.721026422078.7210264220
Analytic rank: 00
Dimension: 8484
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 71.8
Character χ\chi == 252.71
Dual form 252.8.e.a.71.7

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+(10.8566+3.18337i)q2+(107.73269.1213i)q4512.680iq5343.000iq7+(949.569+1093.38i)q8+(1632.05+5565.97i)q10+6950.61q11968.015q13+(1091.90+3723.82i)q14+(6828.4914893.2i)q16+6796.54iq17+37771.6iq19+(35437.155232.2i)q20+(75460.1+22126.4i)q22+2978.61q23184716.q25+(10509.43081.55i)q26+(23708.636952.2i)q28+179257.iq29+200289.iq31+(26723.7+183427.i)q32+(21635.973787.5i)q34175849.q35425727.q37+(120241.410072.i)q38+(560552.+486826.i)q40+244050.iq4188312.9iq43+(748805.480435.i)q44+(32337.6+9482.02i)q46+453536.q47117649.q49+(2.00539e6588020.i)q50+(104287.+66910.5i)q52+1.57314e6iq533.56344e6iq55+(375028.+325702.i)q56+(570643.1.94613e6i)q58+77228.5q591.19278e6q61+(637593.2.17446e6i)q62+(293788.2.07647e6i)q64+496282.iq651.71428e6iq67+(469786.+732207.i)q68+(1.90913e6559794.i)q70+1.52612e6q715.69727e6q73+(4.62196e61.35525e6i)q74+(2.61082e6+4.06922e6i)q762.38406e6iq77+8.63828e6iq79+(7.63545e63.50083e6i)q80+(776900.2.64955e6i)q82+866901.q83+3.48445e6q85+(281133.+958779.i)q86+(6.60009e6+7.59963e6i)q88+6.05763e6iq89+332029.iq91+(320892.205885.i)q92+(4.92387e6+1.44377e6i)q94+1.93648e7q95+1.19866e7q97+(1.27727e6374521.i)q98+O(q100)q+(-10.8566 + 3.18337i) q^{2} +(107.732 - 69.1213i) q^{4} -512.680i q^{5} -343.000i q^{7} +(-949.569 + 1093.38i) q^{8} +(1632.05 + 5565.97i) q^{10} +6950.61 q^{11} -968.015 q^{13} +(1091.90 + 3723.82i) q^{14} +(6828.49 - 14893.2i) q^{16} +6796.54i q^{17} +37771.6i q^{19} +(-35437.1 - 55232.2i) q^{20} +(-75460.1 + 22126.4i) q^{22} +2978.61 q^{23} -184716. q^{25} +(10509.4 - 3081.55i) q^{26} +(-23708.6 - 36952.2i) q^{28} +179257. i q^{29} +200289. i q^{31} +(-26723.7 + 183427. i) q^{32} +(-21635.9 - 73787.5i) q^{34} -175849. q^{35} -425727. q^{37} +(-120241. - 410072. i) q^{38} +(560552. + 486826. i) q^{40} +244050. i q^{41} -88312.9i q^{43} +(748805. - 480435. i) q^{44} +(-32337.6 + 9482.02i) q^{46} +453536. q^{47} -117649. q^{49} +(2.00539e6 - 588020. i) q^{50} +(-104287. + 66910.5i) q^{52} +1.57314e6i q^{53} -3.56344e6i q^{55} +(375028. + 325702. i) q^{56} +(-570643. - 1.94613e6i) q^{58} +77228.5 q^{59} -1.19278e6 q^{61} +(-637593. - 2.17446e6i) q^{62} +(-293788. - 2.07647e6i) q^{64} +496282. i q^{65} -1.71428e6i q^{67} +(469786. + 732207. i) q^{68} +(1.90913e6 - 559794. i) q^{70} +1.52612e6 q^{71} -5.69727e6 q^{73} +(4.62196e6 - 1.35525e6i) q^{74} +(2.61082e6 + 4.06922e6i) q^{76} -2.38406e6i q^{77} +8.63828e6i q^{79} +(-7.63545e6 - 3.50083e6i) q^{80} +(-776900. - 2.64955e6i) q^{82} +866901. q^{83} +3.48445e6 q^{85} +(281133. + 958779. i) q^{86} +(-6.60009e6 + 7.59963e6i) q^{88} +6.05763e6i q^{89} +332029. i q^{91} +(320892. - 205885. i) q^{92} +(-4.92387e6 + 1.44377e6i) q^{94} +1.93648e7 q^{95} +1.19866e7 q^{97} +(1.27727e6 - 374521. i) q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 84q+104q45816q10+16684q16201700q221312500q25+39788q28400048q34165544q371477624q40+1616724q469882516q493744592q52+12053212q58++73638456q94+O(q100) 84 q + 104 q^{4} - 5816 q^{10} + 16684 q^{16} - 201700 q^{22} - 1312500 q^{25} + 39788 q^{28} - 400048 q^{34} - 165544 q^{37} - 1477624 q^{40} + 1616724 q^{46} - 9882516 q^{49} - 3744592 q^{52} + 12053212 q^{58}+ \cdots + 73638456 q^{94}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/252Z)×\left(\mathbb{Z}/252\mathbb{Z}\right)^\times.

nn 2929 7373 127127
χ(n)\chi(n) 1-1 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 −10.8566 + 3.18337i −0.959598 + 0.281373i
33 0 0
44 107.732 69.1213i 0.841658 0.540010i
55 512.680i 1.83422i −0.398633 0.917110i 0.630515π-0.630515\pi
0.398633 0.917110i 0.369485π-0.369485\pi
66 0 0
77 343.000i 0.377964i
88 −949.569 + 1093.38i −0.655710 + 0.755013i
99 0 0
1010 1632.05 + 5565.97i 0.516100 + 1.76012i
1111 6950.61 1.57452 0.787260 0.616621i 0.211499π-0.211499\pi
0.787260 + 0.616621i 0.211499π0.211499\pi
1212 0 0
1313 −968.015 −0.122203 −0.0611013 0.998132i 0.519461π-0.519461\pi
−0.0611013 + 0.998132i 0.519461π0.519461\pi
1414 1091.90 + 3723.82i 0.106349 + 0.362694i
1515 0 0
1616 6828.49 14893.2i 0.416778 0.909008i
1717 6796.54i 0.335519i 0.985828 + 0.167759i 0.0536532π0.0536532\pi
−0.985828 + 0.167759i 0.946347π0.946347\pi
1818 0 0
1919 37771.6i 1.26336i 0.775228 + 0.631681i 0.217635π0.217635\pi
−0.775228 + 0.631681i 0.782365π0.782365\pi
2020 −35437.1 55232.2i −0.990498 1.54379i
2121 0 0
2222 −75460.1 + 22126.4i −1.51091 + 0.443028i
2323 2978.61 0.0510465 0.0255233 0.999674i 0.491875π-0.491875\pi
0.0255233 + 0.999674i 0.491875π0.491875\pi
2424 0 0
2525 −184716. −2.36437
2626 10509.4 3081.55i 0.117265 0.0343845i
2727 0 0
2828 −23708.6 36952.2i −0.204105 0.318117i
2929 179257.i 1.36485i 0.730957 + 0.682423i 0.239074π0.239074\pi
−0.730957 + 0.682423i 0.760926π0.760926\pi
3030 0 0
3131 200289.i 1.20751i 0.797170 + 0.603755i 0.206329π0.206329\pi
−0.797170 + 0.603755i 0.793671π0.793671\pi
3232 −26723.7 + 183427.i −0.144169 + 0.989553i
3333 0 0
3434 −21635.9 73787.5i −0.0944060 0.321963i
3535 −175849. −0.693270
3636 0 0
3737 −425727. −1.38174 −0.690869 0.722980i 0.742772π-0.742772\pi
−0.690869 + 0.722980i 0.742772π0.742772\pi
3838 −120241. 410072.i −0.355476 1.21232i
3939 0 0
4040 560552. + 486826.i 1.38486 + 1.20272i
4141 244050.i 0.553012i 0.961012 + 0.276506i 0.0891766π0.0891766\pi
−0.961012 + 0.276506i 0.910823π0.910823\pi
4242 0 0
4343 88312.9i 0.169389i −0.996407 0.0846943i 0.973009π-0.973009\pi
0.996407 0.0846943i 0.0269914π-0.0269914\pi
4444 748805. 480435.i 1.32521 0.850257i
4545 0 0
4646 −32337.6 + 9482.02i −0.0489842 + 0.0143631i
4747 453536. 0.637191 0.318595 0.947891i 0.396789π-0.396789\pi
0.318595 + 0.947891i 0.396789π0.396789\pi
4848 0 0
4949 −117649. −0.142857
5050 2.00539e6 588020.i 2.26884 0.665269i
5151 0 0
5252 −104287. + 66910.5i −0.102853 + 0.0659907i
5353 1.57314e6i 1.45145i 0.687984 + 0.725726i 0.258496π0.258496\pi
−0.687984 + 0.725726i 0.741504π0.741504\pi
5454 0 0
5555 3.56344e6i 2.88802i
5656 375028. + 325702.i 0.285368 + 0.247835i
5757 0 0
5858 −570643. 1.94613e6i −0.384031 1.30970i
5959 77228.5 0.0489548 0.0244774 0.999700i 0.492208π-0.492208\pi
0.0244774 + 0.999700i 0.492208π0.492208\pi
6060 0 0
6161 −1.19278e6 −0.672828 −0.336414 0.941714i 0.609214π-0.609214\pi
−0.336414 + 0.941714i 0.609214π0.609214\pi
6262 −637593. 2.17446e6i −0.339760 1.15872i
6363 0 0
6464 −293788. 2.07647e6i −0.140089 0.990139i
6565 496282.i 0.224147i
6666 0 0
6767 1.71428e6i 0.696338i −0.937432 0.348169i 0.886804π-0.886804\pi
0.937432 0.348169i 0.113196π-0.113196\pi
6868 469786. + 732207.i 0.181184 + 0.282392i
6969 0 0
7070 1.90913e6 559794.i 0.665261 0.195068i
7171 1.52612e6 0.506039 0.253020 0.967461i 0.418576π-0.418576\pi
0.253020 + 0.967461i 0.418576π0.418576\pi
7272 0 0
7373 −5.69727e6 −1.71410 −0.857051 0.515232i 0.827706π-0.827706\pi
−0.857051 + 0.515232i 0.827706π0.827706\pi
7474 4.62196e6 1.35525e6i 1.32591 0.388784i
7575 0 0
7676 2.61082e6 + 4.06922e6i 0.682228 + 1.06332i
7777 2.38406e6i 0.595113i
7878 0 0
7979 8.63828e6i 1.97121i 0.169068 + 0.985604i 0.445924π0.445924\pi
−0.169068 + 0.985604i 0.554076π0.554076\pi
8080 −7.63545e6 3.50083e6i −1.66732 0.764463i
8181 0 0
8282 −776900. 2.64955e6i −0.155603 0.530669i
8383 866901. 0.166416 0.0832082 0.996532i 0.473483π-0.473483\pi
0.0832082 + 0.996532i 0.473483π0.473483\pi
8484 0 0
8585 3.48445e6 0.615416
8686 281133. + 958779.i 0.0476614 + 0.162545i
8787 0 0
8888 −6.60009e6 + 7.59963e6i −1.03243 + 1.18878i
8989 6.05763e6i 0.910831i 0.890279 + 0.455416i 0.150509π0.150509\pi
−0.890279 + 0.455416i 0.849491π0.849491\pi
9090 0 0
9191 332029.i 0.0461883i
9292 320892. 205885.i 0.0429637 0.0275656i
9393 0 0
9494 −4.92387e6 + 1.44377e6i −0.611447 + 0.179288i
9595 1.93648e7 2.31729
9696 0 0
9797 1.19866e7 1.33351 0.666756 0.745276i 0.267683π-0.267683\pi
0.666756 + 0.745276i 0.267683π0.267683\pi
9898 1.27727e6 374521.i 0.137085 0.0401961i
9999 0 0
100100 −1.98999e7 + 1.27678e7i −1.98999 + 1.27678i
101101 9.58096e6i 0.925303i −0.886540 0.462652i 0.846898π-0.846898\pi
0.886540 0.462652i 0.153102π-0.153102\pi
102102 0 0
103103 1.28909e7i 1.16240i 0.813762 + 0.581198i 0.197416π0.197416\pi
−0.813762 + 0.581198i 0.802584π0.802584\pi
104104 919198. 1.05840e6i 0.0801295 0.0922646i
105105 0 0
106106 −5.00790e6 1.70790e7i −0.408399 1.39281i
107107 −6.50775e6 −0.513556 −0.256778 0.966470i 0.582661π-0.582661\pi
−0.256778 + 0.966470i 0.582661π0.582661\pi
108108 0 0
109109 −2.28730e7 −1.69173 −0.845865 0.533397i 0.820915π-0.820915\pi
−0.845865 + 0.533397i 0.820915π0.820915\pi
110110 1.13438e7 + 3.86869e7i 0.812611 + 2.77134i
111111 0 0
112112 −5.10836e6 2.34217e6i −0.343573 0.157527i
113113 1.68905e7i 1.10121i −0.834767 0.550603i 0.814398π-0.814398\pi
0.834767 0.550603i 0.185602π-0.185602\pi
114114 0 0
115115 1.52707e6i 0.0936306i
116116 1.23905e7 + 1.93118e7i 0.737031 + 1.14873i
117117 0 0
118118 −838440. + 245847.i −0.0469770 + 0.0137746i
119119 2.33121e6 0.126814
120120 0 0
121121 2.88238e7 1.47912
122122 1.29495e7 3.79705e6i 0.645645 0.189316i
123123 0 0
124124 1.38442e7 + 2.15776e7i 0.652067 + 1.01631i
125125 5.46472e7i 2.50255i
126126 0 0
127127 341188.i 0.0147802i −0.999973 0.00739010i 0.997648π-0.997648\pi
0.999973 0.00739010i 0.00235236π-0.00235236\pi
128128 9.79973e6 + 2.16082e7i 0.413028 + 0.910719i
129129 0 0
130130 −1.57985e6 5.38795e6i −0.0630688 0.215091i
131131 9.47174e6 0.368112 0.184056 0.982916i 0.441077π-0.441077\pi
0.184056 + 0.982916i 0.441077π0.441077\pi
132132 0 0
133133 1.29557e7 0.477506
134134 5.45719e6 + 1.86113e7i 0.195931 + 0.668205i
135135 0 0
136136 −7.43118e6 6.45379e6i −0.253321 0.220003i
137137 2.45922e7i 0.817099i 0.912736 + 0.408550i 0.133965π0.133965\pi
−0.912736 + 0.408550i 0.866035π0.866035\pi
138138 0 0
139139 5.44958e7i 1.72112i 0.509348 + 0.860561i 0.329887π0.329887\pi
−0.509348 + 0.860561i 0.670113π0.670113\pi
140140 −1.89447e7 + 1.21549e7i −0.583497 + 0.374373i
141141 0 0
142142 −1.65685e7 + 4.85821e6i −0.485595 + 0.142386i
143143 −6.72830e6 −0.192411
144144 0 0
145145 9.19017e7 2.50343
146146 6.18530e7 1.81365e7i 1.64485 0.482302i
147147 0 0
148148 −4.58646e7 + 2.94268e7i −1.16295 + 0.746152i
149149 4.71462e6i 0.116760i 0.998294 + 0.0583801i 0.0185935π0.0185935\pi
−0.998294 + 0.0583801i 0.981406π0.981406\pi
150150 0 0
151151 6.10409e7i 1.44279i 0.692526 + 0.721393i 0.256498π0.256498\pi
−0.692526 + 0.721393i 0.743502π0.743502\pi
152152 −4.12986e7 3.58668e7i −0.953855 0.828399i
153153 0 0
154154 7.58935e6 + 2.58828e7i 0.167449 + 0.571070i
155155 1.02684e8 2.21484
156156 0 0
157157 −4.67384e7 −0.963884 −0.481942 0.876203i 0.660068π-0.660068\pi
−0.481942 + 0.876203i 0.660068π0.660068\pi
158158 −2.74989e7 9.37825e7i −0.554645 1.89157i
159159 0 0
160160 9.40396e7 + 1.37007e7i 1.81506 + 0.264438i
161161 1.02166e6i 0.0192938i
162162 0 0
163163 4.74541e7i 0.858257i −0.903243 0.429128i 0.858821π-0.858821\pi
0.903243 0.429128i 0.141179π-0.141179\pi
164164 1.68690e7 + 2.62920e7i 0.298632 + 0.465447i
165165 0 0
166166 −9.41161e6 + 2.75967e6i −0.159693 + 0.0468251i
167167 −7.81920e7 −1.29914 −0.649568 0.760304i 0.725050π-0.725050\pi
−0.649568 + 0.760304i 0.725050π0.725050\pi
168168 0 0
169169 −6.18115e7 −0.985067
170170 −3.78294e7 + 1.10923e7i −0.590552 + 0.173161i
171171 0 0
172172 −6.10430e6 9.51415e6i −0.0914716 0.142567i
173173 9.24523e7i 1.35755i 0.734346 + 0.678776i 0.237489π0.237489\pi
−0.734346 + 0.678776i 0.762511π0.762511\pi
174174 0 0
175175 6.33576e7i 0.893646i
176176 4.74622e7 1.03517e8i 0.656226 1.43125i
177177 0 0
178178 −1.92837e7 6.57654e7i −0.256283 0.874032i
179179 −3.31204e7 −0.431628 −0.215814 0.976434i 0.569241π-0.569241\pi
−0.215814 + 0.976434i 0.569241π0.569241\pi
180180 0 0
181181 −3.97750e7 −0.498581 −0.249290 0.968429i 0.580197π-0.580197\pi
−0.249290 + 0.968429i 0.580197π0.580197\pi
182182 −1.05697e6 3.60472e6i −0.0129961 0.0443222i
183183 0 0
184184 −2.82840e6 + 3.25674e6i −0.0334717 + 0.0385408i
185185 2.18262e8i 2.53441i
186186 0 0
187187 4.72401e7i 0.528281i
188188 4.88605e7 3.13490e7i 0.536297 0.344090i
189189 0 0
190190 −2.10236e8 + 6.16452e7i −2.22366 + 0.652022i
191191 −9.10602e7 −0.945610 −0.472805 0.881167i 0.656758π-0.656758\pi
−0.472805 + 0.881167i 0.656758π0.656758\pi
192192 0 0
193193 7.26301e7 0.727221 0.363610 0.931551i 0.381544π-0.381544\pi
0.363610 + 0.931551i 0.381544π0.381544\pi
194194 −1.30134e8 + 3.81580e7i −1.27964 + 0.375214i
195195 0 0
196196 −1.26746e7 + 8.13205e6i −0.120237 + 0.0771443i
197197 1.49839e8i 1.39635i −0.715927 0.698175i 0.753996π-0.753996\pi
0.715927 0.698175i 0.246004π-0.246004\pi
198198 0 0
199199 7.77032e7i 0.698961i −0.936944 0.349481i 0.886358π-0.886358\pi
0.936944 0.349481i 0.113642π-0.113642\pi
200200 1.75401e8 2.01964e8i 1.55034 1.78513i
201201 0 0
202202 3.04998e7 + 1.04017e8i 0.260355 + 0.887920i
203203 6.14853e7 0.515864
204204 0 0
205205 1.25119e8 1.01435
206206 −4.10366e7 1.39952e8i −0.327067 1.11543i
207207 0 0
208208 −6.61008e6 + 1.44168e7i −0.0509314 + 0.111083i
209209 2.62536e8i 1.98919i
210210 0 0
211211 1.31549e7i 0.0964050i −0.998838 0.0482025i 0.984651π-0.984651\pi
0.998838 0.0482025i 0.0153493π-0.0153493\pi
212212 1.08738e8 + 1.69478e8i 0.783799 + 1.22163i
213213 0 0
214214 7.06522e7 2.07166e7i 0.492808 0.144501i
215215 −4.52763e7 −0.310696
216216 0 0
217217 6.86990e7 0.456396
218218 2.48324e8 7.28134e7i 1.62338 0.476007i
219219 0 0
220220 −2.46310e8 3.83898e8i −1.55956 2.43073i
221221 6.57916e6i 0.0410013i
222222 0 0
223223 1.23865e8i 0.747968i −0.927435 0.373984i 0.877991π-0.877991\pi
0.927435 0.373984i 0.122009π-0.122009\pi
224224 6.29156e7 + 9.16624e6i 0.374016 + 0.0544908i
225225 0 0
226226 5.37688e7 + 1.83374e8i 0.309849 + 1.05671i
227227 1.65072e8 0.936661 0.468330 0.883553i 0.344856π-0.344856\pi
0.468330 + 0.883553i 0.344856π0.344856\pi
228228 0 0
229229 4.70535e7 0.258922 0.129461 0.991585i 0.458675π-0.458675\pi
0.129461 + 0.991585i 0.458675π0.458675\pi
230230 4.86125e6 + 1.65789e7i 0.0263451 + 0.0898478i
231231 0 0
232232 −1.95996e8 1.70217e8i −1.03048 0.894944i
233233 1.57640e8i 0.816436i −0.912885 0.408218i 0.866150π-0.866150\pi
0.912885 0.408218i 0.133850π-0.133850\pi
234234 0 0
235235 2.32519e8i 1.16875i
236236 8.32000e6 5.33813e6i 0.0412032 0.0264361i
237237 0 0
238238 −2.53091e7 + 7.42112e6i −0.121691 + 0.0356821i
239239 −2.25713e8 −1.06946 −0.534730 0.845023i 0.679587π-0.679587\pi
−0.534730 + 0.845023i 0.679587π0.679587\pi
240240 0 0
241241 1.45737e8 0.670673 0.335336 0.942098i 0.391150π-0.391150\pi
0.335336 + 0.942098i 0.391150π0.391150\pi
242242 −3.12929e8 + 9.17569e7i −1.41936 + 0.416183i
243243 0 0
244244 −1.28500e8 + 8.24462e7i −0.566292 + 0.363334i
245245 6.03163e7i 0.262032i
246246 0 0
247247 3.65635e7i 0.154386i
248248 −2.18991e8 1.90188e8i −0.911685 0.791776i
249249 0 0
250250 −1.73962e8 5.93283e8i −0.704150 2.40144i
251251 1.30634e8 0.521432 0.260716 0.965416i 0.416041π-0.416041\pi
0.260716 + 0.965416i 0.416041π0.416041\pi
252252 0 0
253253 2.07031e7 0.0803738
254254 1.08613e6 + 3.70415e6i 0.00415875 + 0.0141831i
255255 0 0
256256 −1.75179e8 2.03396e8i −0.652592 0.757709i
257257 1.00167e8i 0.368095i −0.982917 0.184048i 0.941080π-0.941080\pi
0.982917 0.184048i 0.0589200π-0.0589200\pi
258258 0 0
259259 1.46024e8i 0.522248i
260260 3.43037e7 + 5.34656e7i 0.121041 + 0.188655i
261261 0 0
262262 −1.02831e8 + 3.01521e7i −0.353240 + 0.103577i
263263 4.92409e8 1.66909 0.834547 0.550937i 0.185730π-0.185730\pi
0.834547 + 0.550937i 0.185730π0.185730\pi
264264 0 0
265265 8.06519e8 2.66228
266266 −1.40655e8 + 4.12427e7i −0.458214 + 0.134357i
267267 0 0
268268 −1.18493e8 1.84683e8i −0.376030 0.586079i
269269 1.03348e8i 0.323719i −0.986814 0.161859i 0.948251π-0.948251\pi
0.986814 0.161859i 0.0517491π-0.0517491\pi
270270 0 0
271271 6.11019e8i 1.86493i 0.361264 + 0.932464i 0.382345π0.382345\pi
−0.361264 + 0.932464i 0.617655π0.617655\pi
272272 1.01222e8 + 4.64101e7i 0.304989 + 0.139837i
273273 0 0
274274 −7.82860e7 2.66988e8i −0.229910 0.784087i
275275 −1.28389e9 −3.72274
276276 0 0
277277 6.69007e8 1.89126 0.945630 0.325245i 0.105447π-0.105447\pi
0.945630 + 0.325245i 0.105447π0.105447\pi
278278 −1.73481e8 5.91640e8i −0.484277 1.65159i
279279 0 0
280280 1.66981e8 1.92269e8i 0.454584 0.523428i
281281 3.40848e8i 0.916408i −0.888847 0.458204i 0.848493π-0.848493\pi
0.888847 0.458204i 0.151507π-0.151507\pi
282282 0 0
283283 1.30260e8i 0.341633i 0.985303 + 0.170816i 0.0546405π0.0546405\pi
−0.985303 + 0.170816i 0.945359π0.945359\pi
284284 1.64412e8 1.05487e8i 0.425912 0.273266i
285285 0 0
286286 7.30465e7 2.14187e7i 0.184637 0.0541391i
287287 8.37090e7 0.209019
288288 0 0
289289 3.64146e8 0.887427
290290 −9.97742e8 + 2.92557e8i −2.40229 + 0.704398i
291291 0 0
292292 −6.13779e8 + 3.93802e8i −1.44269 + 0.925632i
293293 4.55641e8i 1.05825i −0.848545 0.529123i 0.822521π-0.822521\pi
0.848545 0.529123i 0.177479π-0.177479\pi
294294 0 0
295295 3.95935e7i 0.0897939i
296296 4.04258e8 4.65480e8i 0.906019 1.04323i
297297 0 0
298298 −1.50084e7 5.11848e7i −0.0328532 0.112043i
299299 −2.88334e6 −0.00623802
300300 0 0
301301 −3.02913e7 −0.0640229
302302 −1.94316e8 6.62698e8i −0.405961 1.38450i
303303 0 0
304304 5.62540e8 + 2.57923e8i 1.14841 + 0.526541i
305305 6.11513e8i 1.23412i
306306 0 0
307307 9.01188e8i 1.77759i −0.458307 0.888794i 0.651544π-0.651544\pi
0.458307 0.888794i 0.348456π-0.348456\pi
308308 −1.64789e8 2.56840e8i −0.321367 0.500882i
309309 0 0
310310 −1.11480e9 + 3.26882e8i −2.12536 + 0.623196i
311311 −8.22596e8 −1.55069 −0.775345 0.631538i 0.782424π-0.782424\pi
−0.775345 + 0.631538i 0.782424π0.782424\pi
312312 0 0
313313 −3.31209e8 −0.610515 −0.305258 0.952270i 0.598743π-0.598743\pi
−0.305258 + 0.952270i 0.598743π0.598743\pi
314314 5.07420e8 1.48786e8i 0.924941 0.271211i
315315 0 0
316316 5.97089e8 + 9.30622e8i 1.06447 + 1.65908i
317317 3.47756e8i 0.613150i 0.951847 + 0.306575i 0.0991830π0.0991830\pi
−0.951847 + 0.306575i 0.900817π0.900817\pi
318318 0 0
319319 1.24595e9i 2.14898i
320320 −1.06457e9 + 1.50619e8i −1.81613 + 0.256954i
321321 0 0
322322 3.25233e6 + 1.10918e7i 0.00542875 + 0.0185143i
323323 −2.56716e8 −0.423882
324324 0 0
325325 1.78808e8 0.288932
326326 1.51064e8 + 5.15191e8i 0.241490 + 0.823582i
327327 0 0
328328 −2.66838e8 2.31742e8i −0.417531 0.362615i
329329 1.55563e8i 0.240835i
330330 0 0
331331 1.14708e9i 1.73858i −0.494298 0.869292i 0.664575π-0.664575\pi
0.494298 0.869292i 0.335425π-0.335425\pi
332332 9.33932e7 5.99213e7i 0.140066 0.0898665i
333333 0 0
334334 8.48900e8 2.48914e8i 1.24665 0.365542i
335335 −8.78878e8 −1.27724
336336 0 0
337337 8.31129e8 1.18294 0.591471 0.806326i 0.298547π-0.298547\pi
0.591471 + 0.806326i 0.298547π0.298547\pi
338338 6.71063e8 1.96769e8i 0.945268 0.277171i
339339 0 0
340340 3.75388e8 2.40850e8i 0.517970 0.332331i
341341 1.39213e9i 1.90125i
342342 0 0
343343 4.03536e7i 0.0539949i
344344 9.65591e7 + 8.38592e7i 0.127891 + 0.111070i
345345 0 0
346346 −2.94310e8 1.00372e9i −0.381978 1.30270i
347347 9.66197e8 1.24140 0.620701 0.784047i 0.286848π-0.286848\pi
0.620701 + 0.784047i 0.286848π0.286848\pi
348348 0 0
349349 1.56923e9 1.97605 0.988027 0.154280i 0.0493058π-0.0493058\pi
0.988027 + 0.154280i 0.0493058π0.0493058\pi
350350 −2.01691e8 6.87850e8i −0.251448 0.857542i
351351 0 0
352352 −1.85746e8 + 1.27493e9i −0.226997 + 1.55807i
353353 7.01860e8i 0.849257i 0.905368 + 0.424629i 0.139595π0.139595\pi
−0.905368 + 0.424629i 0.860405π0.860405\pi
354354 0 0
355355 7.82411e8i 0.928188i
356356 4.18712e8 + 6.52603e8i 0.491858 + 0.766609i
357357 0 0
358358 3.59576e8 1.05435e8i 0.414190 0.121449i
359359 −1.49448e9 −1.70474 −0.852370 0.522939i 0.824836π-0.824836\pi
−0.852370 + 0.522939i 0.824836π0.824836\pi
360360 0 0
361361 −5.32823e8 −0.596084
362362 4.31822e8 1.26619e8i 0.478437 0.140287i
363363 0 0
364364 2.29503e7 + 3.57703e7i 0.0249421 + 0.0388747i
365365 2.92088e9i 3.14404i
366366 0 0
367367 1.38437e9i 1.46191i 0.682425 + 0.730955i 0.260925π0.260925\pi
−0.682425 + 0.730955i 0.739075π0.739075\pi
368368 2.03394e7 4.43610e7i 0.0212751 0.0464017i
369369 0 0
370370 −6.94809e8 2.36959e9i −0.713115 2.43202i
371371 5.39588e8 0.548597
372372 0 0
373373 −7.73503e8 −0.771758 −0.385879 0.922549i 0.626102π-0.626102\pi
−0.385879 + 0.922549i 0.626102π0.626102\pi
374374 −1.50383e8 5.12868e8i −0.148644 0.506938i
375375 0 0
376376 −4.30664e8 + 4.95885e8i −0.417812 + 0.481087i
377377 1.73524e8i 0.166788i
378378 0 0
379379 6.35484e8i 0.599608i 0.954001 + 0.299804i 0.0969213π0.0969213\pi
−0.954001 + 0.299804i 0.903079π0.903079\pi
380380 2.08621e9 1.33852e9i 1.95036 1.25136i
381381 0 0
382382 9.88606e8 2.89879e8i 0.907406 0.266069i
383383 1.60965e9 1.46399 0.731993 0.681312i 0.238590π-0.238590\pi
0.731993 + 0.681312i 0.238590π0.238590\pi
384384 0 0
385385 −1.22226e9 −1.09157
386386 −7.88518e8 + 2.31209e8i −0.697840 + 0.204620i
387387 0 0
388388 1.29135e9 8.28533e8i 1.12236 0.720110i
389389 6.09066e8i 0.524615i −0.964984 0.262308i 0.915516π-0.915516\pi
0.964984 0.262308i 0.0844835π-0.0844835\pi
390390 0 0
391391 2.02442e7i 0.0171271i
392392 1.11716e8 1.28635e8i 0.0936728 0.107859i
393393 0 0
394394 4.76995e8 + 1.62675e9i 0.392895 + 1.33994i
395395 4.42868e9 3.61563
396396 0 0
397397 −2.25970e9 −1.81252 −0.906262 0.422717i 0.861076π-0.861076\pi
−0.906262 + 0.422717i 0.861076π0.861076\pi
398398 2.47358e8 + 8.43594e8i 0.196669 + 0.670722i
399399 0 0
400400 −1.26133e9 + 2.75101e9i −0.985416 + 2.14923i
401401 1.34647e9i 1.04278i −0.853319 0.521389i 0.825414π-0.825414\pi
0.853319 0.521389i 0.174586π-0.174586\pi
402402 0 0
403403 1.93883e8i 0.147561i
404404 −6.62248e8 1.03218e9i −0.499673 0.778789i
405405 0 0
406406 −6.67522e8 + 1.95730e8i −0.495022 + 0.145150i
407407 −2.95906e9 −2.17557
408408 0 0
409409 1.51324e9 1.09364 0.546821 0.837250i 0.315838π-0.315838\pi
0.546821 + 0.837250i 0.315838π0.315838\pi
410410 −1.35837e9 + 3.98302e8i −0.973365 + 0.285410i
411411 0 0
412412 8.91038e8 + 1.38877e9i 0.627706 + 0.978340i
413413 2.64894e7i 0.0185032i
414414 0 0
415415 4.44443e8i 0.305244i
416416 2.58690e7 1.77560e8i 0.0176178 0.120926i
417417 0 0
418418 −8.35749e8 2.85025e9i −0.559704 1.90882i
419419 7.19570e7 0.0477886 0.0238943 0.999714i 0.492393π-0.492393\pi
0.0238943 + 0.999714i 0.492393π0.492393\pi
420420 0 0
421421 −1.76118e8 −0.115031 −0.0575156 0.998345i 0.518318π-0.518318\pi
−0.0575156 + 0.998345i 0.518318π0.518318\pi
422422 4.18770e7 + 1.42818e8i 0.0271258 + 0.0925101i
423423 0 0
424424 −1.72004e9 1.49381e9i −1.09586 0.951731i
425425 1.25543e9i 0.793290i
426426 0 0
427427 4.09122e8i 0.254305i
428428 −7.01095e8 + 4.49824e8i −0.432239 + 0.277326i
429429 0 0
430430 4.91547e8 1.44131e8i 0.298144 0.0874215i
431431 −1.07945e9 −0.649430 −0.324715 0.945812i 0.605268π-0.605268\pi
−0.324715 + 0.945812i 0.605268π0.605268\pi
432432 0 0
433433 −2.09096e9 −1.23777 −0.618884 0.785483i 0.712415π-0.712415\pi
−0.618884 + 0.785483i 0.712415π0.712415\pi
434434 −7.45839e8 + 2.18695e8i −0.437957 + 0.128417i
435435 0 0
436436 −2.46416e9 + 1.58101e9i −1.42386 + 0.913551i
437437 1.12507e8i 0.0644902i
438438 0 0
439439 2.09571e9i 1.18224i 0.806585 + 0.591118i 0.201313π0.201313\pi
−0.806585 + 0.591118i 0.798687π0.798687\pi
440440 3.89618e9 + 3.38373e9i 2.18049 + 1.89370i
441441 0 0
442442 2.09439e7 + 7.14274e7i 0.0115367 + 0.0393448i
443443 −4.95891e8 −0.271002 −0.135501 0.990777i 0.543264π-0.543264\pi
−0.135501 + 0.990777i 0.543264π0.543264\pi
444444 0 0
445445 3.10563e9 1.67067
446446 3.94310e8 + 1.34476e9i 0.210458 + 0.717749i
447447 0 0
448448 −7.12230e8 + 1.00769e8i −0.374237 + 0.0529487i
449449 2.38831e9i 1.24517i −0.782551 0.622586i 0.786082π-0.786082\pi
0.782551 0.622586i 0.213918π-0.213918\pi
450450 0 0
451451 1.69629e9i 0.870729i
452452 −1.16749e9 1.81965e9i −0.594662 0.926839i
453453 0 0
454454 −1.79212e9 + 5.25485e8i −0.898818 + 0.263551i
455455 1.70225e8 0.0847195
456456 0 0
457457 −1.91773e9 −0.939899 −0.469949 0.882693i 0.655728π-0.655728\pi
−0.469949 + 0.882693i 0.655728π0.655728\pi
458458 −5.10842e8 + 1.49789e8i −0.248461 + 0.0728535i
459459 0 0
460460 −1.05553e8 1.64515e8i −0.0505615 0.0788050i
461461 6.71193e8i 0.319076i −0.987192 0.159538i 0.949000π-0.949000\pi
0.987192 0.159538i 0.0510005π-0.0510005\pi
462462 0 0
463463 1.48992e9i 0.697635i 0.937191 + 0.348817i 0.113417π0.113417\pi
−0.937191 + 0.348817i 0.886583π0.886583\pi
464464 2.66971e9 + 1.22406e9i 1.24066 + 0.568838i
465465 0 0
466466 5.01828e8 + 1.71144e9i 0.229723 + 0.783451i
467467 1.72985e9 0.785956 0.392978 0.919548i 0.371445π-0.371445\pi
0.392978 + 0.919548i 0.371445π0.371445\pi
468468 0 0
469469 −5.87998e8 −0.263191
470470 7.40195e8 + 2.52437e9i 0.328854 + 1.12153i
471471 0 0
472472 −7.33338e7 + 8.44397e7i −0.0321002 + 0.0369615i
473473 6.13828e8i 0.266706i
474474 0 0
475475 6.97702e9i 2.98705i
476476 2.51147e8 1.61137e8i 0.106734 0.0684810i
477477 0 0
478478 2.45048e9 7.18529e8i 1.02625 0.300917i
479479 3.39296e9 1.41060 0.705301 0.708908i 0.250812π-0.250812\pi
0.705301 + 0.708908i 0.250812π0.250812\pi
480480 0 0
481481 4.12111e8 0.168852
482482 −1.58221e9 + 4.63936e8i −0.643577 + 0.188709i
483483 0 0
484484 3.10525e9 1.99234e9i 1.24491 0.798738i
485485 6.14532e9i 2.44595i
486486 0 0
487487 2.58539e9i 1.01432i −0.861852 0.507159i 0.830696π-0.830696\pi
0.861852 0.507159i 0.169304π-0.169304\pi
488488 1.13262e9 1.30415e9i 0.441180 0.507994i
489489 0 0
490490 −1.92009e8 6.54831e8i −0.0737286 0.251445i
491491 1.16509e7 0.00444194 0.00222097 0.999998i 0.499293π-0.499293\pi
0.00222097 + 0.999998i 0.499293π0.499293\pi
492492 0 0
493493 −1.21833e9 −0.457932
494494 1.16395e8 + 3.96956e8i 0.0434401 + 0.148149i
495495 0 0
496496 2.98294e9 + 1.36767e9i 1.09764 + 0.503263i
497497 5.23459e8i 0.191265i
498498 0 0
499499 6.10284e8i 0.219877i −0.993938 0.109939i 0.964935π-0.964935\pi
0.993938 0.109939i 0.0350654π-0.0350654\pi
500500 3.77728e9 + 5.88726e9i 1.35140 + 2.10629i
501501 0 0
502502 −1.41824e9 + 4.15856e8i −0.500365 + 0.146717i
503503 5.36664e9 1.88025 0.940123 0.340835i 0.110710π-0.110710\pi
0.940123 + 0.340835i 0.110710π0.110710\pi
504504 0 0
505505 −4.91197e9 −1.69721
506506 −2.24766e8 + 6.59058e7i −0.0771266 + 0.0226150i
507507 0 0
508508 −2.35834e7 3.67570e7i −0.00798146 0.0124399i
509509 1.27945e9i 0.430043i 0.976609 + 0.215022i 0.0689822π0.0689822\pi
−0.976609 + 0.215022i 0.931018π0.931018\pi
510510 0 0
511511 1.95416e9i 0.647869i
512512 2.54934e9 + 1.65053e9i 0.839425 + 0.543475i
513513 0 0
514514 3.18870e8 + 1.08748e9i 0.103572 + 0.353224i
515515 6.60893e9 2.13209
516516 0 0
517517 3.15235e9 1.00327
518518 −4.64850e8 1.58533e9i −0.146946 0.501148i
519519 0 0
520520 −5.42623e8 4.71255e8i −0.169234 0.146975i
521521 1.78970e9i 0.554431i 0.960808 + 0.277216i 0.0894116π0.0894116\pi
−0.960808 + 0.277216i 0.910588π0.910588\pi
522522 0 0
523523 1.61533e9i 0.493748i 0.969048 + 0.246874i 0.0794035π0.0794035\pi
−0.969048 + 0.246874i 0.920597π0.920597\pi
524524 1.02041e9 6.54699e8i 0.309825 0.198784i
525525 0 0
526526 −5.34589e9 + 1.56752e9i −1.60166 + 0.469638i
527527 −1.36127e9 −0.405142
528528 0 0
529529 −3.39595e9 −0.997394
530530 −8.75607e9 + 2.56745e9i −2.55472 + 0.749094i
531531 0 0
532532 1.39574e9 8.95512e8i 0.401897 0.257858i
533533 2.36244e8i 0.0675795i
534534 0 0
535535 3.33640e9i 0.941975i
536536 1.87435e9 + 1.62783e9i 0.525744 + 0.456596i
537537 0 0
538538 3.28994e8 + 1.12201e9i 0.0910857 + 0.310640i
539539 −8.17732e8 −0.224932
540540 0 0
541541 −2.11635e9 −0.574642 −0.287321 0.957834i 0.592765π-0.592765\pi
−0.287321 + 0.957834i 0.592765π0.592765\pi
542542 −1.94510e9 6.63360e9i −0.524740 1.78958i
543543 0 0
544544 −1.24667e9 1.81629e8i −0.332014 0.0483715i
545545 1.17266e10i 3.10301i
546546 0 0
547547 3.87336e9i 1.01189i −0.862566 0.505944i 0.831144π-0.831144\pi
0.862566 0.505944i 0.168856π-0.168856\pi
548548 1.69984e9 + 2.64937e9i 0.441242 + 0.687719i
549549 0 0
550550 1.39387e10 4.08710e9i 3.57234 1.04748i
551551 −6.77084e9 −1.72430
552552 0 0
553553 2.96293e9 0.745047
554554 −7.26315e9 + 2.12970e9i −1.81485 + 0.532149i
555555 0 0
556556 3.76682e9 + 5.87096e9i 0.929423 + 1.44860i
557557 3.18813e9i 0.781705i 0.920453 + 0.390853i 0.127820π0.127820\pi
−0.920453 + 0.390853i 0.872180π0.872180\pi
558558 0 0
559559 8.54882e7i 0.0206997i
560560 −1.20079e9 + 2.61896e9i −0.288940 + 0.630189i
561561 0 0
562562 1.08505e9 + 3.70046e9i 0.257853 + 0.879384i
563563 −1.86008e9 −0.439292 −0.219646 0.975580i 0.570490π-0.570490\pi
−0.219646 + 0.975580i 0.570490π0.570490\pi
564564 0 0
565565 −8.65943e9 −2.01985
566566 −4.14667e8 1.41419e9i −0.0961263 0.327830i
567567 0 0
568568 −1.44916e9 + 1.66862e9i −0.331815 + 0.382066i
569569 4.37966e9i 0.996661i −0.866987 0.498331i 0.833947π-0.833947\pi
0.866987 0.498331i 0.166053π-0.166053\pi
570570 0 0
571571 1.50649e8i 0.0338642i −0.999857 0.0169321i 0.994610π-0.994610\pi
0.999857 0.0169321i 0.00538991π-0.00538991\pi
572572 −7.24855e8 + 4.65069e8i −0.161944 + 0.103904i
573573 0 0
574574 −9.08796e8 + 2.66477e8i −0.200574 + 0.0588123i
575575 −5.50197e8 −0.120693
576576 0 0
577577 −2.10867e9 −0.456975 −0.228488 0.973547i 0.573378π-0.573378\pi
−0.228488 + 0.973547i 0.573378π0.573378\pi
578578 −3.95339e9 + 1.15921e9i −0.851574 + 0.249698i
579579 0 0
580580 9.90078e9 6.35237e9i 2.10703 1.35188i
581581 2.97347e8i 0.0628995i
582582 0 0
583583 1.09343e10i 2.28534i
584584 5.40995e9 6.22925e9i 1.12395 1.29417i
585585 0 0
586586 1.45048e9 + 4.94672e9i 0.297762 + 1.01549i
587587 −8.38246e9 −1.71056 −0.855280 0.518167i 0.826615π-0.826615\pi
−0.855280 + 0.518167i 0.826615π0.826615\pi
588588 0 0
589589 −7.56522e9 −1.52552
590590 1.26041e8 + 4.29852e8i 0.0252656 + 0.0861661i
591591 0 0
592592 −2.90708e9 + 6.34044e9i −0.575878 + 1.25601i
593593 6.67307e8i 0.131412i 0.997839 + 0.0657059i 0.0209299π0.0209299\pi
−0.997839 + 0.0657059i 0.979070π0.979070\pi
594594 0 0
595595 1.19517e9i 0.232605i
596596 3.25881e8 + 5.07917e8i 0.0630517 + 0.0982722i
597597 0 0
598598 3.13033e7 9.17874e6i 0.00598599 0.00175521i
599599 −4.39677e9 −0.835872 −0.417936 0.908476i 0.637246π-0.637246\pi
−0.417936 + 0.908476i 0.637246π0.637246\pi
600600 0 0
601601 −7.73210e9 −1.45290 −0.726452 0.687217i 0.758832π-0.758832\pi
−0.726452 + 0.687217i 0.758832π0.758832\pi
602602 3.28861e8 9.64285e7i 0.0614363 0.0180143i
603603 0 0
604604 4.21923e9 + 6.57608e9i 0.779119 + 1.21433i
605605 1.47774e10i 2.71303i
606606 0 0
607607 4.36451e9i 0.792091i −0.918231 0.396045i 0.870382π-0.870382\pi
0.918231 0.396045i 0.129618π-0.129618\pi
608608 −6.92834e9 1.00940e9i −1.25016 0.182138i
609609 0 0
610610 −1.94667e9 6.63896e9i −0.347247 1.18426i
611611 −4.39030e8 −0.0778664
612612 0 0
613613 1.20038e9 0.210478 0.105239 0.994447i 0.466439π-0.466439\pi
0.105239 + 0.994447i 0.466439π0.466439\pi
614614 2.86882e9 + 9.78385e9i 0.500165 + 1.70577i
615615 0 0
616616 2.60667e9 + 2.26383e9i 0.449318 + 0.390221i
617617 8.52396e9i 1.46098i −0.682925 0.730488i 0.739293π-0.739293\pi
0.682925 0.730488i 0.260707π-0.260707\pi
618618 0 0
619619 4.84688e9i 0.821381i −0.911775 0.410691i 0.865288π-0.865288\pi
0.911775 0.410691i 0.134712π-0.134712\pi
620620 1.10624e10 7.09766e9i 1.86414 1.19604i
621621 0 0
622622 8.93061e9 2.61863e9i 1.48804 0.436322i
623623 2.07777e9 0.344262
624624 0 0
625625 1.35856e10 2.22586
626626 3.59581e9 1.05436e9i 0.585850 0.171783i
627627 0 0
628628 −5.03523e9 + 3.23062e9i −0.811261 + 0.520507i
629629 2.89348e9i 0.463599i
630630 0 0
631631 9.50221e9i 1.50564i −0.658225 0.752821i 0.728692π-0.728692\pi
0.658225 0.752821i 0.271308π-0.271308\pi
632632 −9.44489e9 8.20265e9i −1.48829 1.29254i
633633 0 0
634634 −1.10704e9 3.77545e9i −0.172524 0.588378i
635635 −1.74920e8 −0.0271102
636636 0 0
637637 1.13886e8 0.0174575
638638 −3.96631e9 1.35268e10i −0.604665 2.06216i
639639 0 0
640640 1.10781e10 5.02413e9i 1.67046 0.757584i
641641 8.25267e8i 0.123763i 0.998083 + 0.0618816i 0.0197101π0.0197101\pi
−0.998083 + 0.0618816i 0.980290π0.980290\pi
642642 0 0
643643 1.46353e9i 0.217102i −0.994091 0.108551i 0.965379π-0.965379\pi
0.994091 0.108551i 0.0346211π-0.0346211\pi
644644 −7.06187e7 1.10066e8i −0.0104188 0.0162388i
645645 0 0
646646 2.78707e9 8.17224e8i 0.406756 0.119269i
647647 −8.14995e9 −1.18301 −0.591507 0.806300i 0.701467π-0.701467\pi
−0.591507 + 0.806300i 0.701467π0.701467\pi
648648 0 0
649649 5.36785e8 0.0770804
650650 −1.94125e9 + 5.69213e8i −0.277259 + 0.0812976i
651651 0 0
652652 −3.28009e9 5.11234e9i −0.463468 0.722359i
653653 7.90957e9i 1.11162i 0.831309 + 0.555811i 0.187592π0.187592\pi
−0.831309 + 0.555811i 0.812408π0.812408\pi
654654 0 0
655655 4.85598e9i 0.675199i
656656 3.63468e9 + 1.66649e9i 0.502692 + 0.230483i
657657 0 0
658658 4.95215e8 + 1.68889e9i 0.0677646 + 0.231105i
659659 1.79941e9 0.244924 0.122462 0.992473i 0.460921π-0.460921\pi
0.122462 + 0.992473i 0.460921π0.460921\pi
660660 0 0
661661 −2.56731e9 −0.345759 −0.172879 0.984943i 0.555307π-0.555307\pi
−0.172879 + 0.984943i 0.555307π0.555307\pi
662662 3.65158e9 + 1.24534e10i 0.489191 + 1.66834i
663663 0 0
664664 −8.23182e8 + 9.47848e8i −0.109121 + 0.125646i
665665 6.64211e9i 0.875852i
666666 0 0
667667 5.33937e8i 0.0696707i
668668 −8.42380e9 + 5.40473e9i −1.09343 + 0.701546i
669669 0 0
670670 9.54164e9 2.79780e9i 1.22564 0.359380i
671671 −8.29052e9 −1.05938
672672 0 0
673673 1.58520e9 0.200462 0.100231 0.994964i 0.468042π-0.468042\pi
0.100231 + 0.994964i 0.468042π0.468042\pi
674674 −9.02325e9 + 2.64579e9i −1.13515 + 0.332848i
675675 0 0
676676 −6.65909e9 + 4.27249e9i −0.829090 + 0.531946i
677677 1.07207e9i 0.132789i 0.997793 + 0.0663947i 0.0211497π0.0211497\pi
−0.997793 + 0.0663947i 0.978850π0.978850\pi
678678 0 0
679679 4.11142e9i 0.504020i
680680 −3.30873e9 + 3.80982e9i −0.403534 + 0.464647i
681681 0 0
682682 −4.43166e9 1.51138e10i −0.534960 1.82444i
683683 4.50598e9 0.541149 0.270575 0.962699i 0.412786π-0.412786\pi
0.270575 + 0.962699i 0.412786π0.412786\pi
684684 0 0
685685 1.26079e10 1.49874
686686 −1.28461e8 4.38104e8i −0.0151927 0.0518134i
687687 0 0
688688 −1.31526e9 6.03043e8i −0.153976 0.0705975i
689689 1.52283e9i 0.177371i
690690 0 0
691691 2.55258e9i 0.294311i 0.989113 + 0.147155i 0.0470117π0.0470117\pi
−0.989113 + 0.147155i 0.952988π0.952988\pi
692692 6.39042e9 + 9.96009e9i 0.733092 + 1.14259i
693693 0 0
694694 −1.04896e10 + 3.07577e9i −1.19125 + 0.349297i
695695 2.79389e10 3.15692
696696 0 0
697697 −1.65869e9 −0.185546
698698 −1.70366e10 + 4.99546e9i −1.89622 + 0.556008i
699699 0 0
700700 4.37936e9 + 6.82566e9i 0.482578 + 0.752145i
701701 3.27800e9i 0.359414i −0.983720 0.179707i 0.942485π-0.942485\pi
0.983720 0.179707i 0.0575150π-0.0575150\pi
702702 0 0
703703 1.60804e10i 1.74563i
704704 −2.04201e9 1.44327e10i −0.220573 1.55899i
705705 0 0
706706 −2.23428e9 7.61983e9i −0.238958 0.814946i
707707 −3.28627e9 −0.349732
708708 0 0
709709 −1.00606e10 −1.06014 −0.530069 0.847954i 0.677834π-0.677834\pi
−0.530069 + 0.847954i 0.677834π0.677834\pi
710710 2.49071e9 + 8.49434e9i 0.261167 + 0.890688i
711711 0 0
712712 −6.62327e9 5.75214e9i −0.687689 0.597241i
713713 5.96582e8i 0.0616391i
714714 0 0
715715 3.44947e9i 0.352924i
716716 −3.56814e9 + 2.28933e9i −0.363284 + 0.233084i
717717 0 0
718718 1.62249e10 4.75747e9i 1.63587 0.479668i
719719 −1.81069e10 −1.81675 −0.908373 0.418161i 0.862675π-0.862675\pi
−0.908373 + 0.418161i 0.862675π0.862675\pi
720720 0 0
721721 4.42159e9 0.439344
722722 5.78465e9 1.69617e9i 0.572001 0.167722i
723723 0 0
724724 −4.28505e9 + 2.74930e9i −0.419635 + 0.269239i
725725 3.31117e10i 3.22700i
726726 0 0
727727 1.39140e10i 1.34301i 0.740998 + 0.671507i 0.234353π0.234353\pi
−0.740998 + 0.671507i 0.765647π0.765647\pi
728728 −3.63033e8 3.15285e8i −0.0348727 0.0302861i
729729 0 0
730730 −9.29823e9 3.17108e10i −0.884648 3.01702i
731731 6.00222e8 0.0568331
732732 0 0
733733 −5.67481e8 −0.0532215 −0.0266107 0.999646i 0.508471π-0.508471\pi
−0.0266107 + 0.999646i 0.508471π0.508471\pi
734734 −4.40696e9 1.50296e10i −0.411342 1.40285i
735735 0 0
736736 −7.95995e7 + 5.46358e8i −0.00735933 + 0.0505132i
737737 1.19153e10i 1.09640i
738738 0 0
739739 2.66240e9i 0.242671i 0.992612 + 0.121335i 0.0387177π0.0387177\pi
−0.992612 + 0.121335i 0.961282π0.961282\pi
740740 1.50866e10 + 2.35139e10i 1.36861 + 2.13311i
741741 0 0
742742 −5.85810e9 + 1.71771e9i −0.526433 + 0.154360i
743743 1.17550e10 1.05139 0.525693 0.850674i 0.323806π-0.323806\pi
0.525693 + 0.850674i 0.323806π0.323806\pi
744744 0 0
745745 2.41709e9 0.214164
746746 8.39763e9 2.46235e9i 0.740578 0.217152i
747747 0 0
748748 3.26530e9 + 5.08929e9i 0.285277 + 0.444633i
749749 2.23216e9i 0.194106i
750750 0 0
751751 3.68001e9i 0.317037i 0.987356 + 0.158518i 0.0506717π0.0506717\pi
−0.987356 + 0.158518i 0.949328π0.949328\pi
752752 3.09697e9 6.75460e9i 0.265567 0.579212i
753753 0 0
754754 5.52391e8 + 1.88388e9i 0.0469296 + 0.160049i
755755 3.12945e10 2.64639
756756 0 0
757757 −6.00364e9 −0.503013 −0.251507 0.967856i 0.580926π-0.580926\pi
−0.251507 + 0.967856i 0.580926π0.580926\pi
758758 −2.02298e9 6.89921e9i −0.168714 0.575383i
759759 0 0
760760 −1.83882e10 + 2.11730e10i −1.51947 + 1.74958i
761761 2.05678e10i 1.69177i 0.533366 + 0.845885i 0.320927π0.320927\pi
−0.533366 + 0.845885i 0.679073π0.679073\pi
762762 0 0
763763 7.84545e9i 0.639414i
764764 −9.81013e9 + 6.29420e9i −0.795881 + 0.510639i
765765 0 0
766766 −1.74754e10 + 5.12413e9i −1.40484 + 0.411926i
767767 −7.47584e7 −0.00598241
768768 0 0
769769 −3.33896e9 −0.264770 −0.132385 0.991198i 0.542264π-0.542264\pi
−0.132385 + 0.991198i 0.542264π0.542264\pi
770770 1.32696e10 3.89091e9i 1.04747 0.307138i
771771 0 0
772772 7.82461e9 5.02029e9i 0.612072 0.392707i
773773 4.49786e9i 0.350250i 0.984546 + 0.175125i 0.0560329π0.0560329\pi
−0.984546 + 0.175125i 0.943967π0.943967\pi
774774 0 0
775775 3.69965e10i 2.85499i
776776 −1.13822e10 + 1.31059e10i −0.874396 + 1.00682i
777777 0 0
778778 1.93888e9 + 6.61240e9i 0.147613 + 0.503420i
779779 −9.21814e9 −0.698654
780780 0 0
781781 1.06075e10 0.796770
782782 −6.44450e7 2.19784e8i −0.00481909 0.0164351i
783783 0 0
784784 −8.03365e8 + 1.75217e9i −0.0595397 + 0.129858i
785785 2.39618e10i 1.76798i
786786 0 0
787787 1.52635e10i 1.11620i 0.829773 + 0.558101i 0.188470π0.188470\pi
−0.829773 + 0.558101i 0.811530π0.811530\pi
788788 −1.03571e10 1.61425e10i −0.754043 1.17525i
789789 0 0
790790 −4.80805e10 + 1.40981e10i −3.46956 + 1.01734i
791791 −5.79345e9 −0.416216
792792 0 0
793793 1.15463e9 0.0822214
794794 2.45327e10 7.19346e9i 1.73929 0.509995i
795795 0 0
796796 −5.37095e9 8.37115e9i −0.377446 0.588287i
797797 2.04580e8i 0.0143139i −0.999974 0.00715696i 0.997722π-0.997722\pi
0.999974 0.00715696i 0.00227815π-0.00227815\pi
798798 0 0
799799 3.08248e9i 0.213790i
800800 4.93630e9 3.38820e10i 0.340869 2.33967i
801801 0 0
802802 4.28632e9 + 1.46181e10i 0.293410 + 1.00065i
803803 −3.95995e10 −2.69889
804804 0 0
805805 −5.23786e8 −0.0353890
806806 6.17200e8 + 2.10491e9i 0.0415196 + 0.141599i
807807 0 0
808808 1.04756e10 + 9.09779e9i 0.698616 + 0.606731i
809809 8.93941e9i 0.593594i −0.954941 0.296797i 0.904082π-0.904082\pi
0.954941 0.296797i 0.0959184π-0.0959184\pi
810810 0 0
811811 5.70590e9i 0.375622i −0.982205 0.187811i 0.939861π-0.939861\pi
0.982205 0.187811i 0.0601393π-0.0601393\pi
812812 6.62395e9 4.24994e9i 0.434181 0.278572i
813813 0 0
814814 3.21254e10 9.41980e9i 2.08768 0.612148i
815815 −2.43288e10 −1.57423
816816 0 0
817817 3.33572e9 0.213999
818818 −1.64286e10 + 4.81719e9i −1.04946 + 0.307721i
819819 0 0
820820 1.34794e10 8.64842e9i 0.853733 0.547757i
821821 1.17803e10i 0.742942i 0.928445 + 0.371471i 0.121146π0.121146\pi
−0.928445 + 0.371471i 0.878854π0.878854\pi
822822 0 0
823823 7.47614e9i 0.467496i 0.972297 + 0.233748i 0.0750991π0.0750991\pi
−0.972297 + 0.233748i 0.924901π0.924901\pi
824824 −1.40946e10 1.22408e10i −0.877624 0.762194i
825825 0 0
826826 8.43255e7 + 2.87585e8i 0.00520630 + 0.0177556i
827827 −5.68386e9 −0.349441 −0.174721 0.984618i 0.555902π-0.555902\pi
−0.174721 + 0.984618i 0.555902π0.555902\pi
828828 0 0
829829 1.59689e10 0.973498 0.486749 0.873542i 0.338183π-0.338183\pi
0.486749 + 0.873542i 0.338183π0.338183\pi
830830 1.41483e9 + 4.82515e9i 0.0858875 + 0.292912i
831831 0 0
832832 2.84391e8 + 2.01006e9i 0.0171193 + 0.120998i
833833 7.99607e8i 0.0479313i
834834 0 0
835835 4.00875e10i 2.38290i
836836 1.81468e10 + 2.82836e10i 1.07418 + 1.67422i
837837 0 0
838838 −7.81210e8 + 2.29066e8i −0.0458578 + 0.0134464i
839839 1.98205e10 1.15864 0.579318 0.815101i 0.303319π-0.303319\pi
0.579318 + 0.815101i 0.303319π0.303319\pi
840840 0 0
841841 −1.48833e10 −0.862807
842842 1.91204e9 5.60648e8i 0.110384 0.0323666i
843843 0 0
844844 −9.09285e8 1.41721e9i −0.0520597 0.0811401i
845845 3.16895e10i 1.80683i
846846 0 0
847847 9.88656e9i 0.559053i
848848 2.34291e10 + 1.07422e10i 1.31938 + 0.604933i
849849 0 0
850850 3.99651e9 + 1.36297e10i 0.223210 + 0.761239i
851851 −1.26808e9 −0.0705329
852852 0 0
853853 −1.55847e10 −0.859762 −0.429881 0.902886i 0.641444π-0.641444\pi
−0.429881 + 0.902886i 0.641444π0.641444\pi
854854 −1.30239e9 4.44168e9i −0.0715546 0.244031i
855855 0 0
856856 6.17956e9 7.11542e9i 0.336744 0.387742i
857857 1.07475e10i 0.583277i −0.956529 0.291638i 0.905800π-0.905800\pi
0.956529 0.291638i 0.0942004π-0.0942004\pi
858858 0 0
859859 1.22437e9i 0.0659077i −0.999457 0.0329538i 0.989509π-0.989509\pi
0.999457 0.0329538i 0.0104914π-0.0104914\pi
860860 −4.87772e9 + 3.12955e9i −0.261500 + 0.167779i
861861 0 0
862862 1.17192e10 3.43629e9i 0.623192 0.182732i
863863 2.77391e10 1.46911 0.734555 0.678549i 0.237391π-0.237391\pi
0.734555 + 0.678549i 0.237391π0.237391\pi
864864 0 0
865865 4.73985e10 2.49005
866866 2.27008e10 6.65632e9i 1.18776 0.348274i
867867 0 0
868868 7.40110e9 4.74857e9i 0.384129 0.246458i
869869 6.00413e10i 3.10371i
870870 0 0
871871 1.65945e9i 0.0850943i
872872 2.17195e10 2.50088e10i 1.10928 1.27728i
873873 0 0
874874 −3.58151e8 1.22144e9i −0.0181458 0.0618847i
875875 1.87440e10 0.945875
876876 0 0
877877 −4.17436e9 −0.208974 −0.104487 0.994526i 0.533320π-0.533320\pi
−0.104487 + 0.994526i 0.533320π0.533320\pi
878878 −6.67141e9 2.27523e10i −0.332649 1.13447i
879879 0 0
880880 −5.30710e10 2.43329e10i −2.62523 1.20366i
881881 4.37768e9i 0.215689i 0.994168 + 0.107845i 0.0343949π0.0343949\pi
−0.994168 + 0.107845i 0.965605π0.965605\pi
882882 0 0
883883 2.27246e10i 1.11080i 0.831585 + 0.555398i 0.187434π0.187434\pi
−0.831585 + 0.555398i 0.812566π0.812566\pi
884884 −4.54760e8 7.08788e8i −0.0221411 0.0345091i
885885 0 0
886886 5.38370e9 1.57861e9i 0.260054 0.0762528i
887887 2.78475e9 0.133984 0.0669921 0.997754i 0.478660π-0.478660\pi
0.0669921 + 0.997754i 0.478660π0.478660\pi
888888 0 0
889889 −1.17027e8 −0.00558639
890890 −3.37166e10 + 9.88637e9i −1.60317 + 0.470080i
891891 0 0
892892 −8.56174e9 1.33443e10i −0.403910 0.629534i
893893 1.71308e10i 0.805003i
894894 0 0
895895 1.69802e10i 0.791702i
896896 7.41162e9 3.36131e9i 0.344219 0.156110i
897897 0 0
898898 7.60289e9 + 2.59290e10i 0.350358 + 1.19487i
899899 −3.59032e10 −1.64807
900900 0 0
901901 −1.06919e10 −0.486989
902902 −5.39993e9 1.84160e10i −0.245000 0.835550i
903903 0 0
904904 1.84677e10 + 1.60387e10i 0.831424 + 0.722071i
905905 2.03919e10i 0.914507i
906906 0 0
907907 2.37580e9i 0.105726i −0.998602 0.0528632i 0.983165π-0.983165\pi
0.998602 0.0528632i 0.0168347π-0.0168347\pi
908908 1.77836e10 1.14100e10i 0.788348 0.505806i
909909 0 0
910910 −1.84807e9 + 5.41889e8i −0.0812967 + 0.0238378i
911911 2.73821e10 1.19992 0.599961 0.800030i 0.295183π-0.295183\pi
0.599961 + 0.800030i 0.295183π0.295183\pi
912912 0 0
913913 6.02549e9 0.262026
914914 2.08201e10 6.10485e9i 0.901926 0.264462i
915915 0 0
916916 5.06919e9 3.25240e9i 0.217923 0.139820i
917917 3.24881e9i 0.139133i
918918 0 0
919919 1.23130e9i 0.0523311i −0.999658 0.0261656i 0.991670π-0.991670\pi
0.999658 0.0261656i 0.00832970π-0.00832970\pi
920920 1.66967e9 + 1.45006e9i 0.0706923 + 0.0613945i
921921 0 0
922922 2.13666e9 + 7.28689e9i 0.0897794 + 0.306185i
923923 −1.47731e9 −0.0618393
924924 0 0
925925 7.86387e10 3.26693
926926 −4.74295e9 1.61754e10i −0.196296 0.669449i
927927 0 0
928928 −3.28807e10 4.79042e9i −1.35059 0.196769i
929929 2.93114e10i 1.19945i 0.800207 + 0.599723i 0.204723π0.204723\pi
−0.800207 + 0.599723i 0.795277π0.795277\pi
930930 0 0
931931 4.44379e9i 0.180480i
932932 −1.08963e10 1.69830e10i −0.440884 0.687160i
933933 0 0
934934 −1.87803e10 + 5.50674e9i −0.754202 + 0.221147i
935935 2.42191e10 0.968985
936936 0 0
937937 −2.39661e9 −0.0951720 −0.0475860 0.998867i 0.515153π-0.515153\pi
−0.0475860 + 0.998867i 0.515153π0.515153\pi
938938 6.38367e9 1.87182e9i 0.252558 0.0740549i
939939 0 0
940940 −1.60720e10 2.50498e10i −0.631136 0.983687i
941941 1.91604e10i 0.749620i 0.927102 + 0.374810i 0.122292π0.122292\pi
−0.927102 + 0.374810i 0.877708π0.877708\pi
942942 0 0
943943 7.26928e8i 0.0282293i
944944 5.27354e8 1.15018e9i 0.0204033 0.0445003i
945945 0 0
946946 1.95404e9 + 6.66410e9i 0.0750439 + 0.255931i
947947 3.09806e10 1.18540 0.592700 0.805423i 0.298062π-0.298062\pi
0.592700 + 0.805423i 0.298062π0.298062\pi
948948 0 0
949949 5.51504e9 0.209468
950950 2.22105e10 + 7.57469e10i 0.840475 + 2.86637i
951951 0 0
952952 −2.21365e9 + 2.54889e9i −0.0831533 + 0.0957464i
953953 2.24750e10i 0.841152i 0.907257 + 0.420576i 0.138172π0.138172\pi
−0.907257 + 0.420576i 0.861828π0.861828\pi
954954 0 0
955955 4.66848e10i 1.73446i
956956 −2.43166e10 + 1.56016e10i −0.900120 + 0.577519i
957957 0 0
958958 −3.68361e10 + 1.08011e10i −1.35361 + 0.396906i
959959 8.43511e9 0.308835
960960 0 0
961961 −1.26029e10 −0.458079
962962 −4.47413e9 + 1.31190e9i −0.162030 + 0.0475104i
963963 0 0
964964 1.57006e10 1.00735e10i 0.564477 0.362170i
965965 3.72360e10i 1.33388i
966966 0 0
967967 4.14573e10i 1.47437i −0.675688 0.737187i 0.736153π-0.736153\pi
0.675688 0.737187i 0.263847π-0.263847\pi
968968 −2.73702e10 + 3.15152e10i −0.969871 + 1.11675i
969969 0 0
970970 1.95628e10 + 6.67174e10i 0.688225 + 2.34713i
971971 3.03308e9 0.106320 0.0531602 0.998586i 0.483071π-0.483071\pi
0.0531602 + 0.998586i 0.483071π0.483071\pi
972972 0 0
973973 1.86921e10 0.650523
974974 8.23025e9 + 2.80686e10i 0.285402 + 0.973339i
975975 0 0
976976 −8.14486e9 + 1.77642e10i −0.280420 + 0.611607i
977977 2.89511e10i 0.993194i −0.867981 0.496597i 0.834583π-0.834583\pi
0.867981 0.496597i 0.165417π-0.165417\pi
978978 0 0
979979 4.21042e10i 1.43412i
980980 4.16914e9 + 6.49802e9i 0.141500 + 0.220541i
981981 0 0
982982 −1.26489e8 + 3.70890e7i −0.00426248 + 0.00124984i
983983 −3.71869e10 −1.24868 −0.624341 0.781152i 0.714633π-0.714633\pi
−0.624341 + 0.781152i 0.714633π0.714633\pi
984984 0 0
985985 −7.68197e10 −2.56121
986986 1.32269e10 3.87840e9i 0.439431 0.128850i
987987 0 0
988988 −2.52732e9 3.93907e9i −0.0833701 0.129940i
989989 2.63049e8i 0.00864670i
990990 0 0
991991 2.03370e10i 0.663788i 0.943317 + 0.331894i 0.107688π0.107688\pi
−0.943317 + 0.331894i 0.892312π0.892312\pi
992992 −3.67384e10 5.35246e9i −1.19489 0.174086i
993993 0 0
994994 1.66636e9 + 5.68299e9i 0.0538168 + 0.183538i
995995 −3.98369e10 −1.28205
996996 0 0
997997 2.69713e10 0.861923 0.430961 0.902370i 0.358175π-0.358175\pi
0.430961 + 0.902370i 0.358175π0.358175\pi
998998 1.94276e9 + 6.62562e9i 0.0618675 + 0.210994i
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 252.8.e.a.71.8 yes 84
3.2 odd 2 inner 252.8.e.a.71.77 yes 84
4.3 odd 2 inner 252.8.e.a.71.78 yes 84
12.11 even 2 inner 252.8.e.a.71.7 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.8.e.a.71.7 84 12.11 even 2 inner
252.8.e.a.71.8 yes 84 1.1 even 1 trivial
252.8.e.a.71.77 yes 84 3.2 odd 2 inner
252.8.e.a.71.78 yes 84 4.3 odd 2 inner