Properties

Label 75.12.b.d.49.3
Level 7575
Weight 1212
Character 75.49
Analytic conductor 57.62657.626
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,12,Mod(49,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.49"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Level: N N == 75=352 75 = 3 \cdot 5^{2}
Weight: k k == 12 12
Character orbit: [χ][\chi] == 75.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,6222] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 57.625738542057.6257385420
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,1801)\Q(i, \sqrt{1801})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4+901x2+202500 x^{4} + 901x^{2} + 202500 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 49.3
Root 20.7191i20.7191i of defining polynomial
Character χ\chi == 75.49
Dual form 75.12.b.d.49.2

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q+14.7191iq2+243.000iq3+1831.35q43576.74q6+79941.2iq7+57100.5iq859049.0q9+805067.q11+445018.iq12+1.19767e6iq131.17666e6q14+2.91013e6q16+2.63319e6iq17869148.iq181.16061e7q191.94257e7q21+1.18499e7iq221.84216e7iq231.38754e7q241.76286e7q261.43489e7iq27+1.46400e8iq28+1.90527e8q29+1.01127e8q31+1.59776e8iq32+1.95631e8iq333.87581e7q341.08139e8q36+8.06675e7iq371.70831e8iq382.91034e8q39+2.26316e8q412.85929e8iq421.67149e9iq43+1.47436e9q44+2.71150e8q46+8.58507e8iq47+7.07162e8iq484.41327e9q496.39864e8q51+2.19335e9iq52+3.52750e9iq53+2.11203e8q544.56468e9q562.82028e9iq57+2.80438e9iq584.35760e9q591.65393e9q61+1.48849e9iq624.72045e9iq63+3.60819e9q642.87951e9q66+7.58610e9iq67+4.82228e9iq68+4.47645e9q692.75809e10q713.37173e9iq723.22368e10iq731.18735e9q742.12548e10q76+6.43580e10iq774.28376e9iq78+2.43149e9q79+3.48678e9q81+3.33116e9iq821.20729e10iq833.55753e10q84+2.46028e10q86+4.62981e10iq87+4.59697e10iq884.44073e9q899.57433e10q913.37364e10iq92+2.45737e10iq931.26364e10q943.88257e10q962.04453e10iq976.49594e10iq984.75384e10q99+O(q100)q+14.7191i q^{2} +243.000i q^{3} +1831.35 q^{4} -3576.74 q^{6} +79941.2i q^{7} +57100.5i q^{8} -59049.0 q^{9} +805067. q^{11} +445018. i q^{12} +1.19767e6i q^{13} -1.17666e6 q^{14} +2.91013e6 q^{16} +2.63319e6i q^{17} -869148. i q^{18} -1.16061e7 q^{19} -1.94257e7 q^{21} +1.18499e7i q^{22} -1.84216e7i q^{23} -1.38754e7 q^{24} -1.76286e7 q^{26} -1.43489e7i q^{27} +1.46400e8i q^{28} +1.90527e8 q^{29} +1.01127e8 q^{31} +1.59776e8i q^{32} +1.95631e8i q^{33} -3.87581e7 q^{34} -1.08139e8 q^{36} +8.06675e7i q^{37} -1.70831e8i q^{38} -2.91034e8 q^{39} +2.26316e8 q^{41} -2.85929e8i q^{42} -1.67149e9i q^{43} +1.47436e9 q^{44} +2.71150e8 q^{46} +8.58507e8i q^{47} +7.07162e8i q^{48} -4.41327e9 q^{49} -6.39864e8 q^{51} +2.19335e9i q^{52} +3.52750e9i q^{53} +2.11203e8 q^{54} -4.56468e9 q^{56} -2.82028e9i q^{57} +2.80438e9i q^{58} -4.35760e9 q^{59} -1.65393e9 q^{61} +1.48849e9i q^{62} -4.72045e9i q^{63} +3.60819e9 q^{64} -2.87951e9 q^{66} +7.58610e9i q^{67} +4.82228e9i q^{68} +4.47645e9 q^{69} -2.75809e10 q^{71} -3.37173e9i q^{72} -3.22368e10i q^{73} -1.18735e9 q^{74} -2.12548e10 q^{76} +6.43580e10i q^{77} -4.28376e9i q^{78} +2.43149e9 q^{79} +3.48678e9 q^{81} +3.33116e9i q^{82} -1.20729e10i q^{83} -3.55753e10 q^{84} +2.46028e10 q^{86} +4.62981e10i q^{87} +4.59697e10i q^{88} -4.44073e9 q^{89} -9.57433e10 q^{91} -3.37364e10i q^{92} +2.45737e10i q^{93} -1.26364e10 q^{94} -3.88257e10 q^{96} -2.04453e10i q^{97} -6.49594e10i q^{98} -4.75384e10 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+6222q4+6318q6236196q9+591136q116353592q14+5948130q1635255952q193783024q21+17077554q24138104132q26+403763896q29142114016q31+34905989664q99+O(q100) 4 q + 6222 q^{4} + 6318 q^{6} - 236196 q^{9} + 591136 q^{11} - 6353592 q^{14} + 5948130 q^{16} - 35255952 q^{19} - 3783024 q^{21} + 17077554 q^{24} - 138104132 q^{26} + 403763896 q^{29} - 142114016 q^{31}+ \cdots - 34905989664 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/75Z)×\left(\mathbb{Z}/75\mathbb{Z}\right)^\times.

nn 2626 5252
χ(n)\chi(n) 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 14.7191i 0.325249i 0.986688 + 0.162625i 0.0519959π0.0519959\pi
−0.986688 + 0.162625i 0.948004π0.948004\pi
33 243.000i 0.577350i
44 1831.35 0.894213
55 0 0
66 −3576.74 −0.187783
77 79941.2i 1.79776i 0.438195 + 0.898880i 0.355618π0.355618\pi
−0.438195 + 0.898880i 0.644382π0.644382\pi
88 57100.5i 0.616091i
99 −59049.0 −0.333333
1010 0 0
1111 805067. 1.50720 0.753602 0.657331i 0.228315π-0.228315\pi
0.753602 + 0.657331i 0.228315π0.228315\pi
1212 445018.i 0.516274i
1313 1.19767e6i 0.894641i 0.894374 + 0.447321i 0.147622π0.147622\pi
−0.894374 + 0.447321i 0.852378π0.852378\pi
1414 −1.17666e6 −0.584720
1515 0 0
1616 2.91013e6 0.693830
1717 2.63319e6i 0.449793i 0.974383 + 0.224896i 0.0722044π0.0722044\pi
−0.974383 + 0.224896i 0.927796π0.927796\pi
1818 − 869148.i − 0.108416i
1919 −1.16061e7 −1.07533 −0.537664 0.843159i 0.680693π-0.680693\pi
−0.537664 + 0.843159i 0.680693π0.680693\pi
2020 0 0
2121 −1.94257e7 −1.03794
2222 1.18499e7i 0.490217i
2323 − 1.84216e7i − 0.596794i −0.954442 0.298397i 0.903548π-0.903548\pi
0.954442 0.298397i 0.0964520π-0.0964520\pi
2424 −1.38754e7 −0.355700
2525 0 0
2626 −1.76286e7 −0.290981
2727 − 1.43489e7i − 0.192450i
2828 1.46400e8i 1.60758i
2929 1.90527e8 1.72491 0.862457 0.506130i 0.168924π-0.168924\pi
0.862457 + 0.506130i 0.168924π0.168924\pi
3030 0 0
3131 1.01127e8 0.634419 0.317209 0.948356i 0.397254π-0.397254\pi
0.317209 + 0.948356i 0.397254π0.397254\pi
3232 1.59776e8i 0.841759i
3333 1.95631e8i 0.870185i
3434 −3.87581e7 −0.146295
3535 0 0
3636 −1.08139e8 −0.298071
3737 8.06675e7i 0.191245i 0.995418 + 0.0956223i 0.0304841π0.0304841\pi
−0.995418 + 0.0956223i 0.969516π0.969516\pi
3838 − 1.70831e8i − 0.349749i
3939 −2.91034e8 −0.516522
4040 0 0
4141 2.26316e8 0.305073 0.152537 0.988298i 0.451256π-0.451256\pi
0.152537 + 0.988298i 0.451256π0.451256\pi
4242 − 2.85929e8i − 0.337588i
4343 − 1.67149e9i − 1.73391i −0.498385 0.866956i 0.666073π-0.666073\pi
0.498385 0.866956i 0.333927π-0.333927\pi
4444 1.47436e9 1.34776
4545 0 0
4646 2.71150e8 0.194107
4747 8.58507e8i 0.546016i 0.962012 + 0.273008i 0.0880186π0.0880186\pi
−0.962012 + 0.273008i 0.911981π0.911981\pi
4848 7.07162e8i 0.400583i
4949 −4.41327e9 −2.23194
5050 0 0
5151 −6.39864e8 −0.259688
5252 2.19335e9i 0.800000i
5353 3.52750e9i 1.15864i 0.815099 + 0.579322i 0.196683π0.196683\pi
−0.815099 + 0.579322i 0.803317π0.803317\pi
5454 2.11203e8 0.0625942
5555 0 0
5656 −4.56468e9 −1.10758
5757 − 2.82028e9i − 0.620841i
5858 2.80438e9i 0.561027i
5959 −4.35760e9 −0.793527 −0.396763 0.917921i 0.629867π-0.629867\pi
−0.396763 + 0.917921i 0.629867π0.629867\pi
6060 0 0
6161 −1.65393e9 −0.250727 −0.125364 0.992111i 0.540010π-0.540010\pi
−0.125364 + 0.992111i 0.540010π0.540010\pi
6262 1.48849e9i 0.206344i
6363 − 4.72045e9i − 0.599253i
6464 3.60819e9 0.420049
6565 0 0
6666 −2.87951e9 −0.283027
6767 7.58610e9i 0.686447i 0.939254 + 0.343224i 0.111519π0.111519\pi
−0.939254 + 0.343224i 0.888481π0.888481\pi
6868 4.82228e9i 0.402211i
6969 4.47645e9 0.344559
7070 0 0
7171 −2.75809e10 −1.81421 −0.907104 0.420905i 0.861712π-0.861712\pi
−0.907104 + 0.420905i 0.861712π0.861712\pi
7272 − 3.37173e9i − 0.205364i
7373 − 3.22368e10i − 1.82002i −0.414584 0.910011i 0.636073π-0.636073\pi
0.414584 0.910011i 0.363927π-0.363927\pi
7474 −1.18735e9 −0.0622022
7575 0 0
7676 −2.12548e10 −0.961572
7777 6.43580e10i 2.70959i
7878 − 4.28376e9i − 0.167998i
7979 2.43149e9 0.0889046 0.0444523 0.999012i 0.485846π-0.485846\pi
0.0444523 + 0.999012i 0.485846π0.485846\pi
8080 0 0
8181 3.48678e9 0.111111
8282 3.33116e9i 0.0992247i
8383 − 1.20729e10i − 0.336421i −0.985751 0.168210i 0.946201π-0.946201\pi
0.985751 0.168210i 0.0537988π-0.0537988\pi
8484 −3.55753e10 −0.928137
8585 0 0
8686 2.46028e10 0.563953
8787 4.62981e10i 0.995880i
8888 4.59697e10i 0.928575i
8989 −4.44073e9 −0.0842964 −0.0421482 0.999111i 0.513420π-0.513420\pi
−0.0421482 + 0.999111i 0.513420π0.513420\pi
9090 0 0
9191 −9.57433e10 −1.60835
9292 − 3.37364e10i − 0.533661i
9393 2.45737e10i 0.366282i
9494 −1.26364e10 −0.177591
9595 0 0
9696 −3.88257e10 −0.485990
9797 − 2.04453e10i − 0.241740i −0.992668 0.120870i 0.961432π-0.961432\pi
0.992668 0.120870i 0.0385684π-0.0385684\pi
9898 − 6.49594e10i − 0.725937i
9999 −4.75384e10 −0.502401
100100 0 0
101101 −1.55947e11 −1.47642 −0.738209 0.674572i 0.764328π-0.764328\pi
−0.738209 + 0.674572i 0.764328π0.764328\pi
102102 − 9.41823e9i − 0.0844633i
103103 5.10325e10i 0.433752i 0.976199 + 0.216876i 0.0695868π0.0695868\pi
−0.976199 + 0.216876i 0.930413π0.930413\pi
104104 −6.83876e10 −0.551181
105105 0 0
106106 −5.19217e10 −0.376848
107107 1.06665e10i 0.0735207i 0.999324 + 0.0367603i 0.0117038π0.0117038\pi
−0.999324 + 0.0367603i 0.988296π0.988296\pi
108108 − 2.62778e10i − 0.172091i
109109 3.12513e10 0.194546 0.0972729 0.995258i 0.468988π-0.468988\pi
0.0972729 + 0.995258i 0.468988π0.468988\pi
110110 0 0
111111 −1.96022e10 −0.110415
112112 2.32640e11i 1.24734i
113113 3.96346e10i 0.202369i 0.994868 + 0.101184i 0.0322632π0.0322632\pi
−0.994868 + 0.101184i 0.967737π0.967737\pi
114114 4.15119e10 0.201928
115115 0 0
116116 3.48921e11 1.54244
117117 − 7.07213e10i − 0.298214i
118118 − 6.41400e10i − 0.258094i
119119 −2.10500e11 −0.808620
120120 0 0
121121 3.62821e11 1.27166
122122 − 2.43443e10i − 0.0815489i
123123 5.49948e10i 0.176134i
124124 1.85198e11 0.567305
125125 0 0
126126 6.94808e10 0.194907
127127 2.63460e10i 0.0707611i 0.999374 + 0.0353806i 0.0112643π0.0112643\pi
−0.999374 + 0.0353806i 0.988736π0.988736\pi
128128 3.80331e11i 0.978379i
129129 4.06172e11 1.00107
130130 0 0
131131 −2.19917e11 −0.498044 −0.249022 0.968498i 0.580109π-0.580109\pi
−0.249022 + 0.968498i 0.580109π0.580109\pi
132132 3.58269e11i 0.778131i
133133 − 9.27805e11i − 1.93318i
134134 −1.11661e11 −0.223266
135135 0 0
136136 −1.50356e11 −0.277113
137137 − 5.54041e11i − 0.980795i −0.871499 0.490398i 0.836852π-0.836852\pi
0.871499 0.490398i 0.163148π-0.163148\pi
138138 6.58894e10i 0.112068i
139139 6.17540e11 1.00945 0.504723 0.863281i 0.331594π-0.331594\pi
0.504723 + 0.863281i 0.331594π0.331594\pi
140140 0 0
141141 −2.08617e11 −0.315243
142142 − 4.05966e11i − 0.590070i
143143 9.64205e11i 1.34841i
144144 −1.71840e11 −0.231277
145145 0 0
146146 4.74497e11 0.591961
147147 − 1.07243e12i − 1.28861i
148148 1.47730e11i 0.171013i
149149 6.68742e11 0.745992 0.372996 0.927833i 0.378330π-0.378330\pi
0.372996 + 0.927833i 0.378330π0.378330\pi
150150 0 0
151151 1.38243e12 1.43308 0.716541 0.697545i 0.245724π-0.245724\pi
0.716541 + 0.697545i 0.245724π0.245724\pi
152152 − 6.62713e11i − 0.662500i
153153 − 1.55487e11i − 0.149931i
154154 −9.47292e11 −0.881292
155155 0 0
156156 −5.32985e11 −0.461880
157157 − 7.93661e11i − 0.664029i −0.943274 0.332014i 0.892272π-0.892272\pi
0.943274 0.332014i 0.107728π-0.107728\pi
158158 3.57894e10i 0.0289161i
159159 −8.57184e11 −0.668944
160160 0 0
161161 1.47265e12 1.07289
162162 5.13223e10i 0.0361388i
163163 4.67399e11i 0.318168i 0.987265 + 0.159084i 0.0508540π0.0508540\pi
−0.987265 + 0.159084i 0.949146π0.949146\pi
164164 4.14463e11 0.272800
165165 0 0
166166 1.77702e11 0.109421
167167 2.87482e12i 1.71266i 0.516432 + 0.856328i 0.327260π0.327260\pi
−0.516432 + 0.856328i 0.672740π0.672740\pi
168168 − 1.10922e12i − 0.639464i
169169 3.57745e11 0.199617
170170 0 0
171171 6.85328e11 0.358443
172172 − 3.06108e12i − 1.55049i
173173 − 1.53085e12i − 0.751068i −0.926809 0.375534i 0.877459π-0.877459\pi
0.926809 0.375534i 0.122541π-0.122541\pi
174174 −6.81465e11 −0.323909
175175 0 0
176176 2.34285e12 1.04574
177177 − 1.05890e12i − 0.458143i
178178 − 6.53635e10i − 0.0274173i
179179 1.91337e12 0.778228 0.389114 0.921190i 0.372781π-0.372781\pi
0.389114 + 0.921190i 0.372781π0.372781\pi
180180 0 0
181181 −1.70819e12 −0.653587 −0.326794 0.945096i 0.605968π-0.605968\pi
−0.326794 + 0.945096i 0.605968π0.605968\pi
182182 − 1.40925e12i − 0.523115i
183183 − 4.01904e11i − 0.144758i
184184 1.05188e12 0.367680
185185 0 0
186186 −3.61703e11 −0.119133
187187 2.11989e12i 0.677930i
188188 1.57223e12i 0.488255i
189189 1.14707e12 0.345979
190190 0 0
191191 3.16020e12 0.899562 0.449781 0.893139i 0.351502π-0.351502\pi
0.449781 + 0.893139i 0.351502π0.351502\pi
192192 8.76790e11i 0.242515i
193193 − 2.77296e12i − 0.745382i −0.927956 0.372691i 0.878435π-0.878435\pi
0.927956 0.372691i 0.121565π-0.121565\pi
194194 3.00936e11 0.0786256
195195 0 0
196196 −8.08224e12 −1.99583
197197 3.86504e12i 0.928089i 0.885812 + 0.464044i 0.153602π0.153602\pi
−0.885812 + 0.464044i 0.846398π0.846398\pi
198198 − 6.99722e11i − 0.163406i
199199 1.01839e12 0.231325 0.115663 0.993289i 0.463101π-0.463101\pi
0.115663 + 0.993289i 0.463101π0.463101\pi
200200 0 0
201201 −1.84342e12 −0.396321
202202 − 2.29540e12i − 0.480204i
203203 1.52310e13i 3.10098i
204204 −1.17181e12 −0.232216
205205 0 0
206206 −7.51152e11 −0.141078
207207 1.08778e12i 0.198931i
208208 3.48538e12i 0.620729i
209209 −9.34367e12 −1.62074
210210 0 0
211211 2.05668e12 0.338543 0.169271 0.985569i 0.445859π-0.445859\pi
0.169271 + 0.985569i 0.445859π0.445859\pi
212212 6.46009e12i 1.03608i
213213 − 6.70216e12i − 1.04743i
214214 −1.57001e11 −0.0239125
215215 0 0
216216 8.19330e11 0.118567
217217 8.08418e12i 1.14053i
218218 4.59990e11i 0.0632759i
219219 7.83355e12 1.05079
220220 0 0
221221 −3.15369e12 −0.402403
222222 − 2.88527e11i − 0.0359124i
223223 − 9.99986e12i − 1.21428i −0.794597 0.607138i 0.792318π-0.792318\pi
0.794597 0.607138i 0.207682π-0.207682\pi
224224 −1.27727e13 −1.51328
225225 0 0
226226 −5.83386e11 −0.0658202
227227 − 6.77123e12i − 0.745634i −0.927905 0.372817i 0.878392π-0.878392\pi
0.927905 0.372817i 0.121608π-0.121608\pi
228228 − 5.16491e12i − 0.555164i
229229 −1.01933e13 −1.06959 −0.534796 0.844981i 0.679611π-0.679611\pi
−0.534796 + 0.844981i 0.679611π0.679611\pi
230230 0 0
231231 −1.56390e13 −1.56438
232232 1.08792e13i 1.06270i
233233 − 6.74446e12i − 0.643412i −0.946840 0.321706i 0.895744π-0.895744\pi
0.946840 0.321706i 0.104256π-0.104256\pi
234234 1.04095e12 0.0969938
235235 0 0
236236 −7.98029e12 −0.709582
237237 5.90853e11i 0.0513291i
238238 − 3.09837e12i − 0.263003i
239239 1.76501e13 1.46406 0.732030 0.681272i 0.238573π-0.238573\pi
0.732030 + 0.681272i 0.238573π0.238573\pi
240240 0 0
241241 −1.37662e13 −1.09074 −0.545369 0.838196i 0.683611π-0.683611\pi
−0.545369 + 0.838196i 0.683611π0.683611\pi
242242 5.34039e12i 0.413608i
243243 8.47289e11i 0.0641500i
244244 −3.02891e12 −0.224204
245245 0 0
246246 −8.09473e11 −0.0572874
247247 − 1.39003e13i − 0.962033i
248248 5.77438e12i 0.390860i
249249 2.93372e12 0.194233
250250 0 0
251251 3.02252e13 1.91498 0.957490 0.288467i 0.0931454π-0.0931454\pi
0.957490 + 0.288467i 0.0931454π0.0931454\pi
252252 − 8.64479e12i − 0.535860i
253253 − 1.48306e13i − 0.899491i
254254 −3.87790e11 −0.0230150
255255 0 0
256256 1.79144e12 0.101832
257257 6.88263e12i 0.382933i 0.981499 + 0.191466i 0.0613243π0.0613243\pi
−0.981499 + 0.191466i 0.938676π0.938676\pi
258258 5.97848e12i 0.325599i
259259 −6.44866e12 −0.343812
260260 0 0
261261 −1.12504e13 −0.574971
262262 − 3.23698e12i − 0.161988i
263263 3.29948e13i 1.61692i 0.588549 + 0.808462i 0.299699π0.299699\pi
−0.588549 + 0.808462i 0.700301π0.700301\pi
264264 −1.11706e13 −0.536113
265265 0 0
266266 1.36564e13 0.628765
267267 − 1.07910e12i − 0.0486686i
268268 1.38928e13i 0.613830i
269269 2.97478e13 1.28771 0.643855 0.765148i 0.277334π-0.277334\pi
0.643855 + 0.765148i 0.277334π0.277334\pi
270270 0 0
271271 −4.30410e13 −1.78876 −0.894378 0.447312i 0.852381π-0.852381\pi
−0.894378 + 0.447312i 0.852381π0.852381\pi
272272 7.66293e12i 0.312080i
273273 − 2.32656e13i − 0.928582i
274274 8.15498e12 0.319003
275275 0 0
276276 8.19795e12 0.308110
277277 1.46706e13i 0.540518i 0.962788 + 0.270259i 0.0871094π0.0871094\pi
−0.962788 + 0.270259i 0.912891π0.912891\pi
278278 9.08963e12i 0.328322i
279279 −5.97142e12 −0.211473
280280 0 0
281281 4.18378e13 1.42457 0.712286 0.701890i 0.247660π-0.247660\pi
0.712286 + 0.701890i 0.247660π0.247660\pi
282282 − 3.07066e12i − 0.102532i
283283 − 2.34242e13i − 0.767078i −0.923525 0.383539i 0.874705π-0.874705\pi
0.923525 0.383539i 0.125295π-0.125295\pi
284284 −5.05102e13 −1.62229
285285 0 0
286286 −1.41922e13 −0.438568
287287 1.80920e13i 0.548448i
288288 − 9.43463e12i − 0.280586i
289289 2.73382e13 0.797686
290290 0 0
291291 4.96820e12 0.139568
292292 − 5.90369e13i − 1.62749i
293293 7.31258e12i 0.197833i 0.995096 + 0.0989166i 0.0315377π0.0315377\pi
−0.995096 + 0.0989166i 0.968462π0.968462\pi
294294 1.57851e13 0.419120
295295 0 0
296296 −4.60616e12 −0.117824
297297 − 1.15518e13i − 0.290062i
298298 9.84328e12i 0.242633i
299299 2.20630e13 0.533917
300300 0 0
301301 1.33621e14 3.11716
302302 2.03482e13i 0.466109i
303303 − 3.78951e13i − 0.852410i
304304 −3.37752e13 −0.746094
305305 0 0
306306 2.28863e12 0.0487649
307307 − 5.37239e13i − 1.12436i −0.827014 0.562181i 0.809962π-0.809962\pi
0.827014 0.562181i 0.190038π-0.190038\pi
308308 1.17862e14i 2.42295i
309309 −1.24009e13 −0.250427
310310 0 0
311311 5.41406e13 1.05521 0.527607 0.849489i 0.323089π-0.323089\pi
0.527607 + 0.849489i 0.323089π0.323089\pi
312312 − 1.66182e13i − 0.318224i
313313 4.29721e12i 0.0808524i 0.999183 + 0.0404262i 0.0128716π0.0128716\pi
−0.999183 + 0.0404262i 0.987128π0.987128\pi
314314 1.16820e13 0.215975
315315 0 0
316316 4.45291e12 0.0794996
317317 − 2.81928e13i − 0.494666i −0.968931 0.247333i 0.920446π-0.920446\pi
0.968931 0.247333i 0.0795541π-0.0795541\pi
318318 − 1.26170e13i − 0.217573i
319319 1.53387e14 2.59980
320320 0 0
321321 −2.59195e12 −0.0424472
322322 2.16760e13i 0.348958i
323323 − 3.05610e13i − 0.483675i
324324 6.38552e12 0.0993570
325325 0 0
326326 −6.87969e12 −0.103484
327327 7.59406e12i 0.112321i
328328 1.29227e13i 0.187953i
329329 −6.86301e13 −0.981606
330330 0 0
331331 −9.95469e13 −1.37713 −0.688563 0.725176i 0.741758π-0.741758\pi
−0.688563 + 0.725176i 0.741758π0.741758\pi
332332 − 2.21097e13i − 0.300832i
333333 − 4.76334e12i − 0.0637482i
334334 −4.23148e13 −0.557040
335335 0 0
336336 −5.65314e13 −0.720152
337337 − 9.05175e13i − 1.13440i −0.823579 0.567202i 0.808026π-0.808026\pi
0.823579 0.567202i 0.191974π-0.191974\pi
338338 5.26568e12i 0.0649251i
339339 −9.63121e12 −0.116838
340340 0 0
341341 8.14136e13 0.956198
342342 1.00874e13i 0.116583i
343343 − 1.94733e14i − 2.21473i
344344 9.54428e13 1.06825
345345 0 0
346346 2.25327e13 0.244284
347347 − 6.36892e13i − 0.679601i −0.940498 0.339801i 0.889640π-0.889640\pi
0.940498 0.339801i 0.110360π-0.110360\pi
348348 8.47879e13i 0.890529i
349349 −7.08598e13 −0.732588 −0.366294 0.930499i 0.619374π-0.619374\pi
−0.366294 + 0.930499i 0.619374π0.619374\pi
350350 0 0
351351 1.71853e13 0.172174
352352 1.28631e14i 1.26870i
353353 − 6.62893e13i − 0.643699i −0.946791 0.321849i 0.895696π-0.895696\pi
0.946791 0.321849i 0.104304π-0.104304\pi
354354 1.55860e13 0.149011
355355 0 0
356356 −8.13252e12 −0.0753790
357357 − 5.11516e13i − 0.466857i
358358 2.81630e13i 0.253118i
359359 −6.80217e13 −0.602044 −0.301022 0.953617i 0.597328π-0.597328\pi
−0.301022 + 0.953617i 0.597328π0.597328\pi
360360 0 0
361361 1.82109e13 0.156330
362362 − 2.51430e13i − 0.212579i
363363 8.81654e13i 0.734196i
364364 −1.75339e14 −1.43821
365365 0 0
366366 5.91566e12 0.0470823
367367 2.19771e14i 1.72309i 0.507685 + 0.861543i 0.330501π0.330501\pi
−0.507685 + 0.861543i 0.669499π0.669499\pi
368368 − 5.36094e13i − 0.414074i
369369 −1.33637e13 −0.101691
370370 0 0
371371 −2.81993e14 −2.08296
372372 4.50031e13i 0.327534i
373373 1.67232e14i 1.19928i 0.800271 + 0.599639i 0.204689π0.204689\pi
−0.800271 + 0.599639i 0.795311π0.795311\pi
374374 −3.12029e13 −0.220496
375375 0 0
376376 −4.90212e13 −0.336396
377377 2.28189e14i 1.54318i
378378 1.68838e13i 0.112529i
379379 1.57778e14 1.03641 0.518205 0.855256i 0.326600π-0.326600\pi
0.518205 + 0.855256i 0.326600π0.326600\pi
380380 0 0
381381 −6.40209e12 −0.0408540
382382 4.65153e13i 0.292582i
383383 − 5.49255e13i − 0.340550i −0.985397 0.170275i 0.945534π-0.945534\pi
0.985397 0.170275i 0.0544655π-0.0544655\pi
384384 −9.24205e13 −0.564867
385385 0 0
386386 4.08155e13 0.242435
387387 9.86997e13i 0.577971i
388388 − 3.74424e13i − 0.216167i
389389 2.32940e14 1.32593 0.662965 0.748650i 0.269298π-0.269298\pi
0.662965 + 0.748650i 0.269298π0.269298\pi
390390 0 0
391391 4.85076e13 0.268434
392392 − 2.52000e14i − 1.37508i
393393 − 5.34399e13i − 0.287546i
394394 −5.68899e13 −0.301860
395395 0 0
396396 −8.70593e13 −0.449254
397397 − 4.76134e13i − 0.242316i −0.992633 0.121158i 0.961339π-0.961339\pi
0.992633 0.121158i 0.0386607π-0.0386607\pi
398398 1.49898e13i 0.0752383i
399399 2.25456e14 1.11612
400400 0 0
401401 −2.99074e14 −1.44040 −0.720202 0.693764i 0.755951π-0.755951\pi
−0.720202 + 0.693764i 0.755951π0.755951\pi
402402 − 2.71335e13i − 0.128903i
403403 1.21116e14i 0.567577i
404404 −2.85593e14 −1.32023
405405 0 0
406406 −2.24186e14 −1.00859
407407 6.49428e13i 0.288245i
408408 − 3.65366e13i − 0.159992i
409409 2.55879e14 1.10550 0.552748 0.833349i 0.313579π-0.313579\pi
0.552748 + 0.833349i 0.313579π0.313579\pi
410410 0 0
411411 1.34632e14 0.566262
412412 9.34582e13i 0.387867i
413413 − 3.48352e14i − 1.42657i
414414 −1.60111e13 −0.0647023
415415 0 0
416416 −1.91359e14 −0.753072
417417 1.50062e14i 0.582804i
418418 − 1.37530e14i − 0.527144i
419419 −2.21825e14 −0.839139 −0.419570 0.907723i 0.637819π-0.637819\pi
−0.419570 + 0.907723i 0.637819π0.637819\pi
420420 0 0
421421 9.78949e13 0.360752 0.180376 0.983598i 0.442269π-0.442269\pi
0.180376 + 0.983598i 0.442269π0.442269\pi
422422 3.02725e13i 0.110111i
423423 − 5.06940e13i − 0.182005i
424424 −2.01422e14 −0.713831
425425 0 0
426426 9.86497e13 0.340677
427427 − 1.32217e14i − 0.450748i
428428 1.95340e13i 0.0657432i
429429 −2.34302e14 −0.778503
430430 0 0
431431 1.11415e14 0.360842 0.180421 0.983589i 0.442254π-0.442254\pi
0.180421 + 0.983589i 0.442254π0.442254\pi
432432 − 4.17572e13i − 0.133528i
433433 3.42045e14i 1.07994i 0.841684 + 0.539970i 0.181564π0.181564\pi
−0.841684 + 0.539970i 0.818436π0.818436\pi
434434 −1.18992e14 −0.370957
435435 0 0
436436 5.72320e13 0.173965
437437 2.13803e14i 0.641750i
438438 1.15303e14i 0.341769i
439439 −1.20947e14 −0.354029 −0.177015 0.984208i 0.556644π-0.556644\pi
−0.177015 + 0.984208i 0.556644π0.556644\pi
440440 0 0
441441 2.60599e14 0.743980
442442 − 4.64195e13i − 0.130881i
443443 1.27316e14i 0.354539i 0.984162 + 0.177270i 0.0567264π0.0567264\pi
−0.984162 + 0.177270i 0.943274π0.943274\pi
444444 −3.58985e13 −0.0987347
445445 0 0
446446 1.47189e14 0.394942
447447 1.62504e14i 0.430699i
448448 2.88443e14i 0.755147i
449449 1.61837e14 0.418526 0.209263 0.977859i 0.432894π-0.432894\pi
0.209263 + 0.977859i 0.432894π0.432894\pi
450450 0 0
451451 1.82199e14 0.459807
452452 7.25848e13i 0.180961i
453453 3.35931e14i 0.827390i
454454 9.96664e13 0.242517
455455 0 0
456456 1.61039e14 0.382494
457457 − 3.49165e14i − 0.819391i −0.912222 0.409696i 0.865635π-0.865635\pi
0.912222 0.409696i 0.134365π-0.134365\pi
458458 − 1.50036e14i − 0.347884i
459459 3.77834e13 0.0865627
460460 0 0
461461 −4.49619e14 −1.00575 −0.502875 0.864359i 0.667724π-0.667724\pi
−0.502875 + 0.864359i 0.667724π0.667724\pi
462462 − 2.30192e14i − 0.508814i
463463 − 4.21098e13i − 0.0919787i −0.998942 0.0459894i 0.985356π-0.985356\pi
0.998942 0.0459894i 0.0146440π-0.0146440\pi
464464 5.54459e14 1.19680
465465 0 0
466466 9.92723e13 0.209269
467467 − 7.91358e13i − 0.164866i −0.996597 0.0824328i 0.973731π-0.973731\pi
0.996597 0.0824328i 0.0262690π-0.0262690\pi
468468 − 1.29515e14i − 0.266667i
469469 −6.06442e14 −1.23407
470470 0 0
471471 1.92860e14 0.383377
472472 − 2.48821e14i − 0.488885i
473473 − 1.34566e15i − 2.61336i
474474 −8.69682e12 −0.0166947
475475 0 0
476476 −3.85499e14 −0.723078
477477 − 2.08296e14i − 0.386215i
478478 2.59794e14i 0.476184i
479479 9.62318e14 1.74371 0.871853 0.489768i 0.162919π-0.162919\pi
0.871853 + 0.489768i 0.162919π0.162919\pi
480480 0 0
481481 −9.66132e13 −0.171095
482482 − 2.02626e14i − 0.354762i
483483 3.57853e14i 0.619435i
484484 6.64451e14 1.13714
485485 0 0
486486 −1.24713e13 −0.0208647
487487 4.57117e14i 0.756168i 0.925771 + 0.378084i 0.123417π0.123417\pi
−0.925771 + 0.378084i 0.876583π0.876583\pi
488488 − 9.44399e13i − 0.154471i
489489 −1.13578e14 −0.183694
490490 0 0
491491 4.40122e14 0.696025 0.348013 0.937490i 0.386857π-0.386857\pi
0.348013 + 0.937490i 0.386857π0.386857\pi
492492 1.00715e14i 0.157501i
493493 5.01693e14i 0.775854i
494494 2.04599e14 0.312900
495495 0 0
496496 2.94292e14 0.440179
497497 − 2.20485e15i − 3.26151i
498498 4.31817e13i 0.0631740i
499499 −6.36085e14 −0.920369 −0.460185 0.887823i 0.652217π-0.652217\pi
−0.460185 + 0.887823i 0.652217π0.652217\pi
500500 0 0
501501 −6.98581e14 −0.988802
502502 4.44888e14i 0.622846i
503503 4.81639e12i 0.00666957i 0.999994 + 0.00333478i 0.00106150π0.00106150\pi
−0.999994 + 0.00333478i 0.998939π0.998939\pi
504504 2.69540e14 0.369195
505505 0 0
506506 2.18294e14 0.292559
507507 8.69320e13i 0.115249i
508508 4.82488e13i 0.0632755i
509509 5.73829e14 0.744449 0.372224 0.928143i 0.378595π-0.378595\pi
0.372224 + 0.928143i 0.378595π0.378595\pi
510510 0 0
511511 2.57705e15 3.27196
512512 8.05287e14i 1.01150i
513513 1.66535e14i 0.206947i
514514 −1.01306e14 −0.124548
515515 0 0
516516 7.43842e14 0.895174
517517 6.91155e14i 0.822958i
518518 − 9.49185e13i − 0.111825i
519519 3.71997e14 0.433630
520520 0 0
521521 5.14705e14 0.587422 0.293711 0.955894i 0.405110π-0.405110\pi
0.293711 + 0.955894i 0.405110π0.405110\pi
522522 − 1.65596e14i − 0.187009i
523523 − 1.20146e15i − 1.34261i −0.741182 0.671304i 0.765735π-0.765735\pi
0.741182 0.671304i 0.234265π-0.234265\pi
524524 −4.02745e14 −0.445357
525525 0 0
526526 −4.85654e14 −0.525903
527527 2.66285e14i 0.285357i
528528 5.69313e14i 0.603760i
529529 6.13454e14 0.643836
530530 0 0
531531 2.57312e14 0.264509
532532 − 1.69913e15i − 1.72868i
533533 2.71052e14i 0.272931i
534534 1.58833e13 0.0158294
535535 0 0
536536 −4.33170e14 −0.422914
537537 4.64948e14i 0.449310i
538538 4.37861e14i 0.418826i
539539 −3.55298e15 −3.36399
540540 0 0
541541 −1.89107e15 −1.75438 −0.877190 0.480143i 0.840585π-0.840585\pi
−0.877190 + 0.480143i 0.840585π0.840585\pi
542542 − 6.33524e14i − 0.581791i
543543 − 4.15090e14i − 0.377349i
544544 −4.20721e14 −0.378617
545545 0 0
546546 3.42449e14 0.302020
547547 1.37400e14i 0.119966i 0.998199 + 0.0599830i 0.0191046π0.0191046\pi
−0.998199 + 0.0599830i 0.980895π0.980895\pi
548548 − 1.01464e15i − 0.877040i
549549 9.76626e13 0.0835758
550550 0 0
551551 −2.21127e15 −1.85485
552552 2.55608e14i 0.212280i
553553 1.94377e14i 0.159829i
554554 −2.15939e14 −0.175803
555555 0 0
556556 1.13093e15 0.902661
557557 − 1.18622e15i − 0.937483i −0.883335 0.468742i 0.844708π-0.844708\pi
0.883335 0.468742i 0.155292π-0.155292\pi
558558 − 8.78939e13i − 0.0687814i
559559 2.00189e15 1.55123
560560 0 0
561561 −5.15134e14 −0.391403
562562 6.15815e14i 0.463340i
563563 2.58208e15i 1.92386i 0.273298 + 0.961929i 0.411885π0.411885\pi
−0.273298 + 0.961929i 0.588115π0.588115\pi
564564 −3.82051e14 −0.281894
565565 0 0
566566 3.44783e14 0.249491
567567 2.78738e14i 0.199751i
568568 − 1.57488e15i − 1.11772i
569569 −6.86455e13 −0.0482497 −0.0241249 0.999709i 0.507680π-0.507680\pi
−0.0241249 + 0.999709i 0.507680π0.507680\pi
570570 0 0
571571 −1.48521e15 −1.02397 −0.511986 0.858994i 0.671090π-0.671090\pi
−0.511986 + 0.858994i 0.671090π0.671090\pi
572572 1.76579e15i 1.20576i
573573 7.67929e14i 0.519363i
574574 −2.66297e14 −0.178382
575575 0 0
576576 −2.13060e14 −0.140016
577577 − 1.11682e14i − 0.0726966i −0.999339 0.0363483i 0.988427π-0.988427\pi
0.999339 0.0363483i 0.0115726π-0.0115726\pi
578578 4.02394e14i 0.259447i
579579 6.73830e14 0.430346
580580 0 0
581581 9.65124e14 0.604803
582582 7.31274e13i 0.0453945i
583583 2.83988e15i 1.74631i
584584 1.84074e15 1.12130
585585 0 0
586586 −1.07635e14 −0.0643450
587587 1.03228e15i 0.611347i 0.952136 + 0.305673i 0.0988815π0.0988815\pi
−0.952136 + 0.305673i 0.901118π0.901118\pi
588588 − 1.96398e15i − 1.15229i
589589 −1.17368e15 −0.682208
590590 0 0
591591 −9.39204e14 −0.535832
592592 2.34753e14i 0.132691i
593593 − 5.05946e14i − 0.283337i −0.989914 0.141668i 0.954753π-0.954753\pi
0.989914 0.141668i 0.0452467π-0.0452467\pi
594594 1.70032e14 0.0943423
595595 0 0
596596 1.22470e15 0.667076
597597 2.47469e14i 0.133556i
598598 3.24748e14i 0.173656i
599599 1.86439e14 0.0987846 0.0493923 0.998779i 0.484272π-0.484272\pi
0.0493923 + 0.998779i 0.484272π0.484272\pi
600600 0 0
601601 −6.39018e14 −0.332433 −0.166216 0.986089i 0.553155π-0.553155\pi
−0.166216 + 0.986089i 0.553155π0.553155\pi
602602 1.96678e15i 1.01385i
603603 − 4.47952e14i − 0.228816i
604604 2.53172e15 1.28148
605605 0 0
606606 5.57782e14 0.277246
607607 3.07440e15i 1.51434i 0.653219 + 0.757169i 0.273418π0.273418\pi
−0.653219 + 0.757169i 0.726582π0.726582\pi
608608 − 1.85438e15i − 0.905166i
609609 −3.70112e15 −1.79035
610610 0 0
611611 −1.02821e15 −0.488489
612612 − 2.84751e14i − 0.134070i
613613 − 1.45823e15i − 0.680447i −0.940345 0.340223i 0.889497π-0.889497\pi
0.940345 0.340223i 0.110503π-0.110503\pi
614614 7.90767e14 0.365698
615615 0 0
616616 −3.67488e15 −1.66935
617617 2.52038e15i 1.13474i 0.823462 + 0.567372i 0.192040π0.192040\pi
−0.823462 + 0.567372i 0.807960π0.807960\pi
618618 − 1.82530e14i − 0.0814512i
619619 −2.76117e15 −1.22122 −0.610612 0.791930i 0.709076π-0.709076\pi
−0.610612 + 0.791930i 0.709076π0.709076\pi
620620 0 0
621621 −2.64330e14 −0.114853
622622 7.96900e14i 0.343207i
623623 − 3.54997e14i − 0.151545i
624624 −8.46948e14 −0.358378
625625 0 0
626626 −6.32511e13 −0.0262972
627627 − 2.27051e15i − 0.935734i
628628 − 1.45347e15i − 0.593783i
629629 −2.12413e14 −0.0860205
630630 0 0
631631 −5.89126e14 −0.234448 −0.117224 0.993105i 0.537400π-0.537400\pi
−0.117224 + 0.993105i 0.537400π0.537400\pi
632632 1.38839e14i 0.0547733i
633633 4.99774e14i 0.195458i
634634 4.14972e14 0.160890
635635 0 0
636636 −1.56980e15 −0.598178
637637 − 5.28565e15i − 1.99679i
638638 2.25772e15i 0.845582i
639639 1.62862e15 0.604736
640640 0 0
641641 1.59787e15 0.583208 0.291604 0.956539i 0.405811π-0.405811\pi
0.291604 + 0.956539i 0.405811π0.405811\pi
642642 − 3.81512e13i − 0.0138059i
643643 − 4.44445e15i − 1.59462i −0.603569 0.797310i 0.706255π-0.706255\pi
0.603569 0.797310i 0.293745π-0.293745\pi
644644 2.69693e15 0.959395
645645 0 0
646646 4.49830e14 0.157315
647647 − 8.54247e14i − 0.296217i −0.988971 0.148108i 0.952682π-0.952682\pi
0.988971 0.148108i 0.0473185π-0.0473185\pi
648648 1.99097e14i 0.0684546i
649649 −3.50816e15 −1.19601
650650 0 0
651651 −1.96446e15 −0.658486
652652 8.55971e14i 0.284510i
653653 2.86428e15i 0.944045i 0.881587 + 0.472022i 0.156476π0.156476\pi
−0.881587 + 0.472022i 0.843524π0.843524\pi
654654 −1.11778e14 −0.0365323
655655 0 0
656656 6.58609e14 0.211669
657657 1.90355e15i 0.606674i
658658 − 1.01017e15i − 0.319267i
659659 −4.78135e15 −1.49858 −0.749291 0.662241i 0.769606π-0.769606\pi
−0.749291 + 0.662241i 0.769606π0.769606\pi
660660 0 0
661661 −5.52275e14 −0.170234 −0.0851172 0.996371i 0.527126π-0.527126\pi
−0.0851172 + 0.996371i 0.527126π0.527126\pi
662662 − 1.46524e15i − 0.447909i
663663 − 7.66347e14i − 0.232328i
664664 6.89369e14 0.207266
665665 0 0
666666 7.01120e13 0.0207341
667667 − 3.50982e15i − 1.02942i
668668 5.26480e15i 1.53148i
669669 2.42997e15 0.701062
670670 0 0
671671 −1.33152e15 −0.377897
672672 − 3.10377e15i − 0.873693i
673673 3.31175e15i 0.924645i 0.886712 + 0.462323i 0.152984π0.152984\pi
−0.886712 + 0.462323i 0.847016π0.847016\pi
674674 1.33234e15 0.368964
675675 0 0
676676 6.55156e14 0.178500
677677 6.69181e15i 1.80845i 0.427060 + 0.904223i 0.359549π0.359549\pi
−0.427060 + 0.904223i 0.640451π0.640451\pi
678678 − 1.41763e14i − 0.0380013i
679679 1.63442e15 0.434590
680680 0 0
681681 1.64541e15 0.430492
682682 1.19833e15i 0.311003i
683683 7.20810e15i 1.85570i 0.372959 + 0.927848i 0.378343π0.378343\pi
−0.372959 + 0.927848i 0.621657π0.621657\pi
684684 1.25507e15 0.320524
685685 0 0
686686 2.86629e15 0.720340
687687 − 2.47696e15i − 0.617529i
688688 − 4.86426e15i − 1.20304i
689689 −4.22479e15 −1.03657
690690 0 0
691691 7.79467e15 1.88221 0.941106 0.338112i 0.109788π-0.109788\pi
0.941106 + 0.338112i 0.109788π0.109788\pi
692692 − 2.80352e15i − 0.671615i
693693 − 3.80028e15i − 0.903197i
694694 9.37448e14 0.221040
695695 0 0
696696 −2.64364e15 −0.613553
697697 5.95932e14i 0.137220i
698698 − 1.04299e15i − 0.238274i
699699 1.63890e15 0.371474
700700 0 0
701701 6.71290e15 1.49782 0.748912 0.662669i 0.230577π-0.230577\pi
0.748912 + 0.662669i 0.230577π0.230577\pi
702702 2.52952e14i 0.0559994i
703703 − 9.36234e14i − 0.205651i
704704 2.90483e15 0.633099
705705 0 0
706706 9.75719e14 0.209362
707707 − 1.24666e16i − 2.65424i
708708 − 1.93921e15i − 0.409677i
709709 4.10214e15 0.859917 0.429959 0.902849i 0.358528π-0.358528\pi
0.429959 + 0.902849i 0.358528π0.358528\pi
710710 0 0
711711 −1.43577e14 −0.0296349
712712 − 2.53568e14i − 0.0519343i
713713 − 1.86291e15i − 0.378617i
714714 7.52905e14 0.151845
715715 0 0
716716 3.50404e15 0.695902
717717 4.28898e15i 0.845276i
718718 − 1.00122e15i − 0.195814i
719719 3.31818e15 0.644008 0.322004 0.946738i 0.395644π-0.395644\pi
0.322004 + 0.946738i 0.395644π0.395644\pi
720720 0 0
721721 −4.07960e15 −0.779783
722722 2.68047e14i 0.0508460i
723723 − 3.34519e15i − 0.629738i
724724 −3.12829e15 −0.584446
725725 0 0
726726 −1.29772e15 −0.238797
727727 3.34273e15i 0.610467i 0.952278 + 0.305233i 0.0987344π0.0987344\pi
−0.952278 + 0.305233i 0.901266π0.901266\pi
728728 − 5.46699e15i − 0.990890i
729729 −2.05891e14 −0.0370370
730730 0 0
731731 4.40134e15 0.779901
732732 − 7.36026e14i − 0.129444i
733733 6.14966e15i 1.07344i 0.843759 + 0.536722i 0.180338π0.180338\pi
−0.843759 + 0.536722i 0.819662π0.819662\pi
734734 −3.23483e15 −0.560432
735735 0 0
736736 2.94334e15 0.502357
737737 6.10732e15i 1.03462i
738738 − 1.96702e14i − 0.0330749i
739739 4.15701e15 0.693804 0.346902 0.937901i 0.387234π-0.387234\pi
0.346902 + 0.937901i 0.387234π0.387234\pi
740740 0 0
741741 3.37776e15 0.555430
742742 − 4.15068e15i − 0.677482i
743743 4.90364e15i 0.794474i 0.917716 + 0.397237i 0.130031π0.130031\pi
−0.917716 + 0.397237i 0.869969π0.869969\pi
744744 −1.40317e15 −0.225663
745745 0 0
746746 −2.46150e15 −0.390064
747747 7.12893e14i 0.112140i
748748 3.88226e15i 0.606214i
749749 −8.52690e14 −0.132173
750750 0 0
751751 −2.00580e15 −0.306385 −0.153192 0.988196i 0.548955π-0.548955\pi
−0.153192 + 0.988196i 0.548955π0.548955\pi
752752 2.49837e15i 0.378842i
753753 7.34473e15i 1.10561i
754754 −3.35873e15 −0.501918
755755 0 0
756756 2.10068e15 0.309379
757757 2.85016e15i 0.416718i 0.978052 + 0.208359i 0.0668122π0.0668122\pi
−0.978052 + 0.208359i 0.933188π0.933188\pi
758758 2.32235e15i 0.337091i
759759 3.60384e15 0.519321
760760 0 0
761761 8.65962e15 1.22994 0.614969 0.788551i 0.289168π-0.289168\pi
0.614969 + 0.788551i 0.289168π0.289168\pi
762762 − 9.42329e13i − 0.0132877i
763763 2.49827e15i 0.349747i
764764 5.78743e15 0.804400
765765 0 0
766766 8.08453e14 0.110763
767767 − 5.21897e15i − 0.709922i
768768 4.35320e14i 0.0587925i
769769 7.49727e15 1.00533 0.502665 0.864481i 0.332353π-0.332353\pi
0.502665 + 0.864481i 0.332353π0.332353\pi
770770 0 0
771771 −1.67248e15 −0.221086
772772 − 5.07826e15i − 0.666530i
773773 − 2.07867e15i − 0.270894i −0.990785 0.135447i 0.956753π-0.956753\pi
0.990785 0.135447i 0.0432470π-0.0432470\pi
774774 −1.45277e15 −0.187984
775775 0 0
776776 1.16743e15 0.148934
777777 − 1.56703e15i − 0.198500i
778778 3.42866e15i 0.431258i
779779 −2.62664e15 −0.328053
780780 0 0
781781 −2.22045e16 −2.73438
782782 7.13988e14i 0.0873079i
783783 − 2.73385e15i − 0.331960i
784784 −1.28432e16 −1.54859
785785 0 0
786786 7.86587e14 0.0935239
787787 − 1.58421e16i − 1.87047i −0.354026 0.935235i 0.615188π-0.615188\pi
0.354026 0.935235i 0.384812π-0.384812\pi
788788 7.07823e15i 0.829909i
789789 −8.01774e15 −0.933531
790790 0 0
791791 −3.16844e15 −0.363810
792792 − 2.71447e15i − 0.309525i
793793 − 1.98086e15i − 0.224311i
794794 7.00826e14 0.0788129
795795 0 0
796796 1.86503e15 0.206854
797797 − 4.58469e15i − 0.504997i −0.967597 0.252499i 0.918748π-0.918748\pi
0.967597 0.252499i 0.0812523π-0.0812523\pi
798798 3.31852e15i 0.363018i
799799 −2.26061e15 −0.245594
800800 0 0
801801 2.62221e14 0.0280988
802802 − 4.40210e15i − 0.468490i
803803 − 2.59528e16i − 2.74315i
804804 −3.37595e15 −0.354395
805805 0 0
806806 −1.78272e15 −0.184604
807807 7.22872e15i 0.743459i
808808 − 8.90465e15i − 0.909608i
809809 1.72813e15 0.175331 0.0876656 0.996150i 0.472059π-0.472059\pi
0.0876656 + 0.996150i 0.472059π0.472059\pi
810810 0 0
811811 4.24638e14 0.0425015 0.0212507 0.999774i 0.493235π-0.493235\pi
0.0212507 + 0.999774i 0.493235π0.493235\pi
812812 2.78932e16i 2.77294i
813813 − 1.04590e16i − 1.03274i
814814 −9.55899e14 −0.0937514
815815 0 0
816816 −1.86209e15 −0.180179
817817 1.93994e16i 1.86452i
818818 3.76631e15i 0.359561i
819819 5.65355e15 0.536117
820820 0 0
821821 −1.96517e16 −1.83871 −0.919354 0.393431i 0.871288π-0.871288\pi
−0.919354 + 0.393431i 0.871288π0.871288\pi
822822 1.98166e15i 0.184176i
823823 − 5.95701e14i − 0.0549957i −0.999622 0.0274979i 0.991246π-0.991246\pi
0.999622 0.0274979i 0.00875394π-0.00875394\pi
824824 −2.91398e15 −0.267231
825825 0 0
826826 5.12743e15 0.463991
827827 − 1.40975e16i − 1.26725i −0.773639 0.633626i 0.781566π-0.781566\pi
0.773639 0.633626i 0.218434π-0.218434\pi
828828 1.99210e15i 0.177887i
829829 7.87187e15 0.698277 0.349139 0.937071i 0.386474π-0.386474\pi
0.349139 + 0.937071i 0.386474π0.386474\pi
830830 0 0
831831 −3.56497e15 −0.312068
832832 4.32142e15i 0.375793i
833833 − 1.16210e16i − 1.00391i
834834 −2.20878e15 −0.189557
835835 0 0
836836 −1.71115e16 −1.44929
837837 − 1.45106e15i − 0.122094i
838838 − 3.26507e15i − 0.272929i
839839 −6.12117e15 −0.508327 −0.254164 0.967161i 0.581800π-0.581800\pi
−0.254164 + 0.967161i 0.581800π0.581800\pi
840840 0 0
841841 2.41000e16 1.97533
842842 1.44092e15i 0.117334i
843843 1.01666e16i 0.822476i
844844 3.76650e15 0.302729
845845 0 0
846846 7.46170e14 0.0591971
847847 2.90043e16i 2.28615i
848848 1.02655e16i 0.803902i
849849 5.69208e15 0.442873
850850 0 0
851851 1.48603e15 0.114134
852852 − 1.22740e16i − 0.936629i
853853 − 2.83032e15i − 0.214594i −0.994227 0.107297i 0.965781π-0.965781\pi
0.994227 0.107297i 0.0342195π-0.0342195\pi
854854 1.94611e15 0.146605
855855 0 0
856856 −6.09060e14 −0.0452954
857857 − 4.83193e14i − 0.0357048i −0.999841 0.0178524i 0.994317π-0.994317\pi
0.999841 0.0178524i 0.00568289π-0.00568289\pi
858858 − 3.44871e15i − 0.253208i
859859 −1.23886e16 −0.903775 −0.451888 0.892075i 0.649249π-0.649249\pi
−0.451888 + 0.892075i 0.649249π0.649249\pi
860860 0 0
861861 −4.39635e15 −0.316647
862862 1.63992e15i 0.117364i
863863 2.26458e16i 1.61038i 0.593016 + 0.805191i 0.297937π0.297937\pi
−0.593016 + 0.805191i 0.702063π0.702063\pi
864864 2.29262e15 0.161997
865865 0 0
866866 −5.03459e15 −0.351249
867867 6.64319e15i 0.460544i
868868 1.48050e16i 1.01988i
869869 1.95751e15 0.133997
870870 0 0
871871 −9.08565e15 −0.614124
872872 1.78446e15i 0.119858i
873873 1.20727e15i 0.0805799i
874874 −3.14698e15 −0.208728
875875 0 0
876876 1.43460e16 0.939630
877877 − 8.89622e15i − 0.579039i −0.957172 0.289519i 0.906505π-0.906505\pi
0.957172 0.289519i 0.0934955π-0.0934955\pi
878878 − 1.78023e15i − 0.115148i
879879 −1.77696e15 −0.114219
880880 0 0
881881 −1.82720e16 −1.15989 −0.579946 0.814655i 0.696927π-0.696927\pi
−0.579946 + 0.814655i 0.696927π0.696927\pi
882882 3.83579e15i 0.241979i
883883 − 8.75758e15i − 0.549035i −0.961582 0.274518i 0.911482π-0.911482\pi
0.961582 0.274518i 0.0885182π-0.0885182\pi
884884 −5.77551e15 −0.359834
885885 0 0
886886 −1.87398e15 −0.115314
887887 8.52159e15i 0.521124i 0.965457 + 0.260562i 0.0839078π0.0839078\pi
−0.965457 + 0.260562i 0.916092π0.916092\pi
888888 − 1.11930e15i − 0.0680258i
889889 −2.10613e15 −0.127212
890890 0 0
891891 2.80709e15 0.167467
892892 − 1.83132e16i − 1.08582i
893893 − 9.96390e15i − 0.587146i
894894 −2.39192e15 −0.140084
895895 0 0
896896 −3.04042e16 −1.75889
897897 5.36132e15i 0.308257i
898898 2.38209e15i 0.136125i
899899 1.92673e16 1.09432
900900 0 0
901901 −9.28858e15 −0.521150
902902 2.68181e15i 0.149552i
903903 3.24699e16i 1.79969i
904904 −2.26316e15 −0.124677
905905 0 0
906906 −4.94461e15 −0.269108
907907 2.10550e16i 1.13898i 0.821998 + 0.569490i 0.192859π0.192859\pi
−0.821998 + 0.569490i 0.807141π0.807141\pi
908908 − 1.24005e16i − 0.666756i
909909 9.20851e15 0.492139
910910 0 0
911911 3.04479e16 1.60771 0.803854 0.594827i 0.202780π-0.202780\pi
0.803854 + 0.594827i 0.202780π0.202780\pi
912912 − 8.20739e15i − 0.430758i
913913 − 9.71950e15i − 0.507055i
914914 5.13939e15 0.266506
915915 0 0
916916 −1.86674e16 −0.956443
917917 − 1.75805e16i − 0.895363i
918918 5.56137e14i 0.0281544i
919919 4.47695e15 0.225293 0.112646 0.993635i 0.464067π-0.464067\pi
0.112646 + 0.993635i 0.464067π0.464067\pi
920920 0 0
921921 1.30549e16 0.649151
922922 − 6.61799e15i − 0.327119i
923923 − 3.30328e16i − 1.62307i
924924 −2.86405e16 −1.39889
925925 0 0
926926 6.19818e14 0.0299160
927927 − 3.01342e15i − 0.144584i
928928 3.04417e16i 1.45196i
929929 −2.63310e16 −1.24848 −0.624238 0.781234i 0.714591π-0.714591\pi
−0.624238 + 0.781234i 0.714591π0.714591\pi
930930 0 0
931931 5.12208e16 2.40007
932932 − 1.23515e16i − 0.575348i
933933 1.31562e16i 0.609228i
934934 1.16481e15 0.0536224
935935 0 0
936936 4.03822e15 0.183727
937937 1.18259e16i 0.534894i 0.963573 + 0.267447i 0.0861800π0.0861800\pi
−0.963573 + 0.267447i 0.913820π0.913820\pi
938938 − 8.92628e15i − 0.401379i
939939 −1.04422e15 −0.0466802
940940 0 0
941941 6.96003e15 0.307517 0.153758 0.988108i 0.450862π-0.450862\pi
0.153758 + 0.988108i 0.450862π0.450862\pi
942942 2.83872e15i 0.124693i
943943 − 4.16911e15i − 0.182066i
944944 −1.26812e16 −0.550573
945945 0 0
946946 1.98069e16 0.849993
947947 − 3.38501e16i − 1.44423i −0.691775 0.722113i 0.743171π-0.743171\pi
0.691775 0.722113i 0.256829π-0.256829\pi
948948 1.08206e15i 0.0458991i
949949 3.86091e16 1.62827
950950 0 0
951951 6.85084e15 0.285595
952952 − 1.20197e16i − 0.498183i
953953 − 4.50062e16i − 1.85465i −0.374262 0.927323i 0.622104π-0.622104\pi
0.374262 0.927323i 0.377896π-0.377896\pi
954954 3.06592e15 0.125616
955955 0 0
956956 3.23235e16 1.30918
957957 3.72730e16i 1.50099i
958958 1.41644e16i 0.567139i
959959 4.42907e16 1.76323
960960 0 0
961961 −1.51819e16 −0.597513
962962 − 1.42206e15i − 0.0556486i
963963 − 6.29844e14i − 0.0245069i
964964 −2.52107e16 −0.975353
965965 0 0
966966 −5.26728e15 −0.201471
967967 − 1.48359e16i − 0.564246i −0.959378 0.282123i 0.908961π-0.908961\pi
0.959378 0.282123i 0.0910386π-0.0910386\pi
968968 2.07172e16i 0.783461i
969969 7.42632e15 0.279250
970970 0 0
971971 1.14414e16 0.425378 0.212689 0.977120i 0.431778π-0.431778\pi
0.212689 + 0.977120i 0.431778π0.431778\pi
972972 1.55168e15i 0.0573638i
973973 4.93669e16i 1.81474i
974974 −6.72835e15 −0.245943
975975 0 0
976976 −4.81314e15 −0.173962
977977 − 2.64437e16i − 0.950390i −0.879881 0.475195i 0.842378π-0.842378\pi
0.879881 0.475195i 0.157622π-0.157622\pi
978978 − 1.67177e15i − 0.0597464i
979979 −3.57508e15 −0.127052
980980 0 0
981981 −1.84536e15 −0.0648486
982982 6.47820e15i 0.226382i
983983 − 3.63656e16i − 1.26371i −0.775088 0.631854i 0.782294π-0.782294\pi
0.775088 0.631854i 0.217706π-0.217706\pi
984984 −3.14023e15 −0.108515
985985 0 0
986986 −7.38447e15 −0.252346
987987 − 1.66771e16i − 0.566731i
988988 − 2.54562e16i − 0.860262i
989989 −3.07915e16 −1.03479
990990 0 0
991991 5.09067e16 1.69188 0.845941 0.533276i 0.179039π-0.179039\pi
0.845941 + 0.533276i 0.179039π0.179039\pi
992992 1.61576e16i 0.534027i
993993 − 2.41899e16i − 0.795084i
994994 3.24534e16 1.06080
995995 0 0
996996 5.37266e15 0.173685
997997 − 4.28291e16i − 1.37694i −0.725264 0.688471i 0.758282π-0.758282\pi
0.725264 0.688471i 0.241718π-0.241718\pi
998998 − 9.36259e15i − 0.299349i
999999 1.15749e15 0.0368051
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 75.12.b.d.49.3 4
3.2 odd 2 225.12.b.i.199.2 4
5.2 odd 4 75.12.a.c.1.1 2
5.3 odd 4 15.12.a.c.1.2 2
5.4 even 2 inner 75.12.b.d.49.2 4
15.2 even 4 225.12.a.i.1.2 2
15.8 even 4 45.12.a.c.1.1 2
15.14 odd 2 225.12.b.i.199.3 4
20.3 even 4 240.12.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.12.a.c.1.2 2 5.3 odd 4
45.12.a.c.1.1 2 15.8 even 4
75.12.a.c.1.1 2 5.2 odd 4
75.12.b.d.49.2 4 5.4 even 2 inner
75.12.b.d.49.3 4 1.1 even 1 trivial
225.12.a.i.1.2 2 15.2 even 4
225.12.b.i.199.2 4 3.2 odd 2
225.12.b.i.199.3 4 15.14 odd 2
240.12.a.m.1.1 2 20.3 even 4