# Properties

 Label 475.2.a.j.1.4 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.66064384.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 9x^{4} + 13x^{2} - 1$$ x^6 - 9*x^4 + 13*x^2 - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-0.285442$$ of defining polynomial Character $$\chi$$ $$=$$ 475.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.906968 q^{2} -3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} -2.59637 q^{7} -2.88181 q^{8} +7.35482 q^{9} +O(q^{10})$$ $$q+0.906968 q^{2} -3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} -2.59637 q^{7} -2.88181 q^{8} +7.35482 q^{9} +0.741113 q^{11} +3.78878 q^{12} +3.78878 q^{13} -2.35482 q^{14} -0.258887 q^{16} +3.16725 q^{17} +6.67058 q^{18} -1.00000 q^{19} +8.35482 q^{21} +0.672165 q^{22} -0.570885 q^{23} +9.27334 q^{24} +3.43630 q^{26} -14.0133 q^{27} +3.05699 q^{28} +6.00000 q^{29} +5.83705 q^{31} +5.52881 q^{32} -2.38482 q^{33} +2.87259 q^{34} -8.65964 q^{36} +1.40396 q^{37} -0.906968 q^{38} -12.1919 q^{39} -3.83705 q^{41} +7.57755 q^{42} +2.59637 q^{43} -0.872594 q^{44} -0.517774 q^{46} -5.08247 q^{47} +0.833070 q^{48} -0.258887 q^{49} -10.1919 q^{51} -4.46094 q^{52} +0.160905 q^{53} -12.7096 q^{54} +7.48223 q^{56} +3.21789 q^{57} +5.44181 q^{58} +8.35482 q^{59} -8.57816 q^{61} +5.29401 q^{62} -19.0958 q^{63} +5.53223 q^{64} -2.16295 q^{66} +14.8464 q^{67} -3.72915 q^{68} +1.83705 q^{69} +3.64518 q^{71} -21.1952 q^{72} +10.8461 q^{73} +1.27334 q^{74} +1.17741 q^{76} -1.92420 q^{77} -11.0576 q^{78} +1.83705 q^{79} +23.0289 q^{81} -3.48008 q^{82} +4.19876 q^{83} -9.83705 q^{84} +2.35482 q^{86} -19.3073 q^{87} -2.13574 q^{88} -16.9015 q^{89} -9.83705 q^{91} +0.672165 q^{92} -18.7830 q^{93} -4.60963 q^{94} -17.7911 q^{96} +3.78878 q^{97} -0.234802 q^{98} +5.45075 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{4} + 14 q^{9}+O(q^{10})$$ 6 * q + 8 * q^4 + 14 * q^9 $$6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99}+O(q^{100})$$ 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.906968 0.641323 0.320661 0.947194i $$-0.396095\pi$$
0.320661 + 0.947194i $$0.396095\pi$$
$$3$$ −3.21789 −1.85785 −0.928925 0.370268i $$-0.879266\pi$$
−0.928925 + 0.370268i $$0.879266\pi$$
$$4$$ −1.17741 −0.588705
$$5$$ 0 0
$$6$$ −2.91852 −1.19148
$$7$$ −2.59637 −0.981334 −0.490667 0.871347i $$-0.663247\pi$$
−0.490667 + 0.871347i $$0.663247\pi$$
$$8$$ −2.88181 −1.01887
$$9$$ 7.35482 2.45161
$$10$$ 0 0
$$11$$ 0.741113 0.223454 0.111727 0.993739i $$-0.464362\pi$$
0.111727 + 0.993739i $$0.464362\pi$$
$$12$$ 3.78878 1.09373
$$13$$ 3.78878 1.05082 0.525409 0.850850i $$-0.323912\pi$$
0.525409 + 0.850850i $$0.323912\pi$$
$$14$$ −2.35482 −0.629352
$$15$$ 0 0
$$16$$ −0.258887 −0.0647218
$$17$$ 3.16725 0.768171 0.384086 0.923298i $$-0.374517\pi$$
0.384086 + 0.923298i $$0.374517\pi$$
$$18$$ 6.67058 1.57227
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 8.35482 1.82317
$$22$$ 0.672165 0.143306
$$23$$ −0.570885 −0.119038 −0.0595189 0.998227i $$-0.518957\pi$$
−0.0595189 + 0.998227i $$0.518957\pi$$
$$24$$ 9.27334 1.89291
$$25$$ 0 0
$$26$$ 3.43630 0.673913
$$27$$ −14.0133 −2.69687
$$28$$ 3.05699 0.577716
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 5.83705 1.04836 0.524182 0.851606i $$-0.324371\pi$$
0.524182 + 0.851606i $$0.324371\pi$$
$$32$$ 5.52881 0.977365
$$33$$ −2.38482 −0.415144
$$34$$ 2.87259 0.492646
$$35$$ 0 0
$$36$$ −8.65964 −1.44327
$$37$$ 1.40396 0.230809 0.115404 0.993319i $$-0.463184\pi$$
0.115404 + 0.993319i $$0.463184\pi$$
$$38$$ −0.906968 −0.147130
$$39$$ −12.1919 −1.95226
$$40$$ 0 0
$$41$$ −3.83705 −0.599246 −0.299623 0.954058i $$-0.596861\pi$$
−0.299623 + 0.954058i $$0.596861\pi$$
$$42$$ 7.57755 1.16924
$$43$$ 2.59637 0.395942 0.197971 0.980208i $$-0.436565\pi$$
0.197971 + 0.980208i $$0.436565\pi$$
$$44$$ −0.872594 −0.131548
$$45$$ 0 0
$$46$$ −0.517774 −0.0763416
$$47$$ −5.08247 −0.741354 −0.370677 0.928762i $$-0.620874\pi$$
−0.370677 + 0.928762i $$0.620874\pi$$
$$48$$ 0.833070 0.120243
$$49$$ −0.258887 −0.0369839
$$50$$ 0 0
$$51$$ −10.1919 −1.42715
$$52$$ −4.46094 −0.618621
$$53$$ 0.160905 0.0221020 0.0110510 0.999939i $$-0.496482\pi$$
0.0110510 + 0.999939i $$0.496482\pi$$
$$54$$ −12.7096 −1.72956
$$55$$ 0 0
$$56$$ 7.48223 0.999854
$$57$$ 3.21789 0.426220
$$58$$ 5.44181 0.714544
$$59$$ 8.35482 1.08770 0.543852 0.839181i $$-0.316965\pi$$
0.543852 + 0.839181i $$0.316965\pi$$
$$60$$ 0 0
$$61$$ −8.57816 −1.09832 −0.549160 0.835717i $$-0.685052\pi$$
−0.549160 + 0.835717i $$0.685052\pi$$
$$62$$ 5.29401 0.672340
$$63$$ −19.0958 −2.40584
$$64$$ 5.53223 0.691529
$$65$$ 0 0
$$66$$ −2.16295 −0.266241
$$67$$ 14.8464 1.81378 0.906888 0.421371i $$-0.138451\pi$$
0.906888 + 0.421371i $$0.138451\pi$$
$$68$$ −3.72915 −0.452226
$$69$$ 1.83705 0.221154
$$70$$ 0 0
$$71$$ 3.64518 0.432603 0.216302 0.976327i $$-0.430601\pi$$
0.216302 + 0.976327i $$0.430601\pi$$
$$72$$ −21.1952 −2.49788
$$73$$ 10.8461 1.26944 0.634719 0.772743i $$-0.281116\pi$$
0.634719 + 0.772743i $$0.281116\pi$$
$$74$$ 1.27334 0.148023
$$75$$ 0 0
$$76$$ 1.17741 0.135058
$$77$$ −1.92420 −0.219283
$$78$$ −11.0576 −1.25203
$$79$$ 1.83705 0.206684 0.103342 0.994646i $$-0.467046\pi$$
0.103342 + 0.994646i $$0.467046\pi$$
$$80$$ 0 0
$$81$$ 23.0289 2.55877
$$82$$ −3.48008 −0.384310
$$83$$ 4.19876 0.460873 0.230437 0.973087i $$-0.425985\pi$$
0.230437 + 0.973087i $$0.425985\pi$$
$$84$$ −9.83705 −1.07331
$$85$$ 0 0
$$86$$ 2.35482 0.253927
$$87$$ −19.3073 −2.06996
$$88$$ −2.13574 −0.227671
$$89$$ −16.9015 −1.79156 −0.895778 0.444502i $$-0.853381\pi$$
−0.895778 + 0.444502i $$0.853381\pi$$
$$90$$ 0 0
$$91$$ −9.83705 −1.03120
$$92$$ 0.672165 0.0700781
$$93$$ −18.7830 −1.94770
$$94$$ −4.60963 −0.475447
$$95$$ 0 0
$$96$$ −17.7911 −1.81580
$$97$$ 3.78878 0.384692 0.192346 0.981327i $$-0.438390\pi$$
0.192346 + 0.981327i $$0.438390\pi$$
$$98$$ −0.234802 −0.0237186
$$99$$ 5.45075 0.547821
$$100$$ 0 0
$$101$$ 8.35482 0.831336 0.415668 0.909517i $$-0.363548\pi$$
0.415668 + 0.909517i $$0.363548\pi$$
$$102$$ −9.24369 −0.915262
$$103$$ −2.07612 −0.204566 −0.102283 0.994755i $$-0.532615\pi$$
−0.102283 + 0.994755i $$0.532615\pi$$
$$104$$ −10.9185 −1.07065
$$105$$ 0 0
$$106$$ 0.145935 0.0141745
$$107$$ 5.70399 0.551426 0.275713 0.961240i $$-0.411086\pi$$
0.275713 + 0.961240i $$0.411086\pi$$
$$108$$ 16.4994 1.58766
$$109$$ 1.64518 0.157580 0.0787899 0.996891i $$-0.474894\pi$$
0.0787899 + 0.996891i $$0.474894\pi$$
$$110$$ 0 0
$$111$$ −4.51777 −0.428808
$$112$$ 0.672165 0.0635137
$$113$$ 3.89006 0.365946 0.182973 0.983118i $$-0.441428\pi$$
0.182973 + 0.983118i $$0.441428\pi$$
$$114$$ 2.91852 0.273345
$$115$$ 0 0
$$116$$ −7.06446 −0.655918
$$117$$ 27.8658 2.57619
$$118$$ 7.57755 0.697570
$$119$$ −8.22334 −0.753832
$$120$$ 0 0
$$121$$ −10.4508 −0.950068
$$122$$ −7.78011 −0.704378
$$123$$ 12.3472 1.11331
$$124$$ −6.87259 −0.617177
$$125$$ 0 0
$$126$$ −17.3193 −1.54292
$$127$$ 14.4233 1.27986 0.639931 0.768432i $$-0.278963\pi$$
0.639931 + 0.768432i $$0.278963\pi$$
$$128$$ −6.04007 −0.533872
$$129$$ −8.35482 −0.735601
$$130$$ 0 0
$$131$$ 9.96853 0.870954 0.435477 0.900200i $$-0.356580\pi$$
0.435477 + 0.900200i $$0.356580\pi$$
$$132$$ 2.80791 0.244397
$$133$$ 2.59637 0.225133
$$134$$ 13.4652 1.16322
$$135$$ 0 0
$$136$$ −9.12741 −0.782669
$$137$$ 9.70431 0.829095 0.414548 0.910028i $$-0.363940\pi$$
0.414548 + 0.910028i $$0.363940\pi$$
$$138$$ 1.66614 0.141831
$$139$$ −13.4508 −1.14088 −0.570439 0.821340i $$-0.693227\pi$$
−0.570439 + 0.821340i $$0.693227\pi$$
$$140$$ 0 0
$$141$$ 16.3548 1.37732
$$142$$ 3.30606 0.277438
$$143$$ 2.80791 0.234809
$$144$$ −1.90407 −0.158672
$$145$$ 0 0
$$146$$ 9.83705 0.814120
$$147$$ 0.833070 0.0687105
$$148$$ −1.65303 −0.135878
$$149$$ 15.0959 1.23671 0.618353 0.785900i $$-0.287800\pi$$
0.618353 + 0.785900i $$0.287800\pi$$
$$150$$ 0 0
$$151$$ 14.1919 1.15492 0.577459 0.816420i $$-0.304044\pi$$
0.577459 + 0.816420i $$0.304044\pi$$
$$152$$ 2.88181 0.233745
$$153$$ 23.2946 1.88325
$$154$$ −1.74519 −0.140631
$$155$$ 0 0
$$156$$ 14.3548 1.14931
$$157$$ 7.57755 0.604754 0.302377 0.953188i $$-0.402220\pi$$
0.302377 + 0.953188i $$0.402220\pi$$
$$158$$ 1.66614 0.132551
$$159$$ −0.517774 −0.0410622
$$160$$ 0 0
$$161$$ 1.48223 0.116816
$$162$$ 20.8865 1.64100
$$163$$ −19.6757 −1.54112 −0.770559 0.637369i $$-0.780023\pi$$
−0.770559 + 0.637369i $$0.780023\pi$$
$$164$$ 4.51777 0.352779
$$165$$ 0 0
$$166$$ 3.80814 0.295569
$$167$$ −10.7954 −0.835376 −0.417688 0.908590i $$-0.637160\pi$$
−0.417688 + 0.908590i $$0.637160\pi$$
$$168$$ −24.0770 −1.85758
$$169$$ 1.35482 0.104217
$$170$$ 0 0
$$171$$ −7.35482 −0.562437
$$172$$ −3.05699 −0.233093
$$173$$ 20.3895 1.55018 0.775092 0.631848i $$-0.217703\pi$$
0.775092 + 0.631848i $$0.217703\pi$$
$$174$$ −17.5111 −1.32752
$$175$$ 0 0
$$176$$ −0.191865 −0.0144623
$$177$$ −26.8849 −2.02079
$$178$$ −15.3291 −1.14897
$$179$$ 25.0645 1.87341 0.936703 0.350126i $$-0.113861\pi$$
0.936703 + 0.350126i $$0.113861\pi$$
$$180$$ 0 0
$$181$$ −19.4193 −1.44342 −0.721712 0.692194i $$-0.756644\pi$$
−0.721712 + 0.692194i $$0.756644\pi$$
$$182$$ −8.92188 −0.661334
$$183$$ 27.6036 2.04051
$$184$$ 1.64518 0.121284
$$185$$ 0 0
$$186$$ −17.0355 −1.24911
$$187$$ 2.34729 0.171651
$$188$$ 5.98414 0.436439
$$189$$ 36.3837 2.64653
$$190$$ 0 0
$$191$$ 11.4508 0.828547 0.414274 0.910152i $$-0.364036\pi$$
0.414274 + 0.910152i $$0.364036\pi$$
$$192$$ −17.8021 −1.28476
$$193$$ 3.78878 0.272722 0.136361 0.990659i $$-0.456459\pi$$
0.136361 + 0.990659i $$0.456459\pi$$
$$194$$ 3.43630 0.246712
$$195$$ 0 0
$$196$$ 0.304816 0.0217726
$$197$$ 2.28354 0.162695 0.0813477 0.996686i $$-0.474078\pi$$
0.0813477 + 0.996686i $$0.474078\pi$$
$$198$$ 4.94366 0.351330
$$199$$ −19.4508 −1.37883 −0.689414 0.724368i $$-0.742132\pi$$
−0.689414 + 0.724368i $$0.742132\pi$$
$$200$$ 0 0
$$201$$ −47.7741 −3.36972
$$202$$ 7.57755 0.533155
$$203$$ −15.5782 −1.09337
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ −1.88297 −0.131193
$$207$$ −4.19876 −0.291834
$$208$$ −0.980865 −0.0680108
$$209$$ −0.741113 −0.0512639
$$210$$ 0 0
$$211$$ 11.2274 0.772927 0.386463 0.922305i $$-0.373697\pi$$
0.386463 + 0.922305i $$0.373697\pi$$
$$212$$ −0.189451 −0.0130115
$$213$$ −11.7298 −0.803712
$$214$$ 5.17334 0.353642
$$215$$ 0 0
$$216$$ 40.3837 2.74776
$$217$$ −15.1551 −1.02880
$$218$$ 1.49213 0.101059
$$219$$ −34.9015 −2.35843
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ −4.09748 −0.275005
$$223$$ −4.03785 −0.270394 −0.135197 0.990819i $$-0.543167\pi$$
−0.135197 + 0.990819i $$0.543167\pi$$
$$224$$ −14.3548 −0.959122
$$225$$ 0 0
$$226$$ 3.52815 0.234689
$$227$$ −11.2185 −0.744600 −0.372300 0.928112i $$-0.621431\pi$$
−0.372300 + 0.928112i $$0.621431\pi$$
$$228$$ −3.78878 −0.250918
$$229$$ 16.1315 1.06600 0.532999 0.846116i $$-0.321065\pi$$
0.532999 + 0.846116i $$0.321065\pi$$
$$230$$ 0 0
$$231$$ 6.19186 0.407395
$$232$$ −17.2908 −1.13520
$$233$$ −2.12676 −0.139329 −0.0696644 0.997570i $$-0.522193\pi$$
−0.0696644 + 0.997570i $$0.522193\pi$$
$$234$$ 25.2733 1.65217
$$235$$ 0 0
$$236$$ −9.83705 −0.640337
$$237$$ −5.91141 −0.383987
$$238$$ −7.45830 −0.483450
$$239$$ −14.4152 −0.932442 −0.466221 0.884668i $$-0.654385\pi$$
−0.466221 + 0.884668i $$0.654385\pi$$
$$240$$ 0 0
$$241$$ −0.162955 −0.0104968 −0.00524842 0.999986i $$-0.501671\pi$$
−0.00524842 + 0.999986i $$0.501671\pi$$
$$242$$ −9.47849 −0.609301
$$243$$ −32.0645 −2.05694
$$244$$ 10.1000 0.646587
$$245$$ 0 0
$$246$$ 11.1985 0.713990
$$247$$ −3.78878 −0.241074
$$248$$ −16.8212 −1.06815
$$249$$ −13.5111 −0.856233
$$250$$ 0 0
$$251$$ 12.9330 0.816322 0.408161 0.912910i $$-0.366170\pi$$
0.408161 + 0.912910i $$0.366170\pi$$
$$252$$ 22.4836 1.41633
$$253$$ −0.423090 −0.0265995
$$254$$ 13.0815 0.820805
$$255$$ 0 0
$$256$$ −16.5426 −1.03391
$$257$$ −11.0445 −0.688938 −0.344469 0.938798i $$-0.611941\pi$$
−0.344469 + 0.938798i $$0.611941\pi$$
$$258$$ −7.57755 −0.471758
$$259$$ −3.64518 −0.226501
$$260$$ 0 0
$$261$$ 44.1289 2.73151
$$262$$ 9.04113 0.558563
$$263$$ 17.8527 1.10085 0.550424 0.834885i $$-0.314466\pi$$
0.550424 + 0.834885i $$0.314466\pi$$
$$264$$ 6.87259 0.422979
$$265$$ 0 0
$$266$$ 2.35482 0.144383
$$267$$ 54.3872 3.32844
$$268$$ −17.4803 −1.06778
$$269$$ 24.9934 1.52387 0.761936 0.647652i $$-0.224249\pi$$
0.761936 + 0.647652i $$0.224249\pi$$
$$270$$ 0 0
$$271$$ −23.8660 −1.44975 −0.724877 0.688879i $$-0.758103\pi$$
−0.724877 + 0.688879i $$0.758103\pi$$
$$272$$ −0.819960 −0.0497174
$$273$$ 31.6545 1.91582
$$274$$ 8.80150 0.531718
$$275$$ 0 0
$$276$$ −2.16295 −0.130195
$$277$$ −21.2315 −1.27568 −0.637840 0.770169i $$-0.720172\pi$$
−0.637840 + 0.770169i $$0.720172\pi$$
$$278$$ −12.1994 −0.731671
$$279$$ 42.9304 2.57018
$$280$$ 0 0
$$281$$ 3.83705 0.228899 0.114449 0.993429i $$-0.463490\pi$$
0.114449 + 0.993429i $$0.463490\pi$$
$$282$$ 14.8333 0.883310
$$283$$ 0.211545 0.0125751 0.00628753 0.999980i $$-0.497999\pi$$
0.00628753 + 0.999980i $$0.497999\pi$$
$$284$$ −4.29187 −0.254676
$$285$$ 0 0
$$286$$ 2.54668 0.150589
$$287$$ 9.96237 0.588060
$$288$$ 40.6634 2.39612
$$289$$ −6.96853 −0.409913
$$290$$ 0 0
$$291$$ −12.1919 −0.714700
$$292$$ −12.7703 −0.747324
$$293$$ −14.9942 −0.875970 −0.437985 0.898982i $$-0.644308\pi$$
−0.437985 + 0.898982i $$0.644308\pi$$
$$294$$ 0.755568 0.0440656
$$295$$ 0 0
$$296$$ −4.04593 −0.235165
$$297$$ −10.3855 −0.602626
$$298$$ 13.6915 0.793129
$$299$$ −2.16295 −0.125087
$$300$$ 0 0
$$301$$ −6.74111 −0.388551
$$302$$ 12.8716 0.740675
$$303$$ −26.8849 −1.54450
$$304$$ 0.258887 0.0148482
$$305$$ 0 0
$$306$$ 21.1274 1.20777
$$307$$ 1.65303 0.0943434 0.0471717 0.998887i $$-0.484979\pi$$
0.0471717 + 0.998887i $$0.484979\pi$$
$$308$$ 2.26557 0.129093
$$309$$ 6.68073 0.380053
$$310$$ 0 0
$$311$$ −0.741113 −0.0420247 −0.0210123 0.999779i $$-0.506689\pi$$
−0.0210123 + 0.999779i $$0.506689\pi$$
$$312$$ 35.1346 1.98911
$$313$$ −26.8849 −1.51962 −0.759812 0.650143i $$-0.774709\pi$$
−0.759812 + 0.650143i $$0.774709\pi$$
$$314$$ 6.87259 0.387843
$$315$$ 0 0
$$316$$ −2.16295 −0.121676
$$317$$ −8.16155 −0.458398 −0.229199 0.973380i $$-0.573611\pi$$
−0.229199 + 0.973380i $$0.573611\pi$$
$$318$$ −0.469604 −0.0263341
$$319$$ 4.44668 0.248966
$$320$$ 0 0
$$321$$ −18.3548 −1.02447
$$322$$ 1.34433 0.0749166
$$323$$ −3.16725 −0.176231
$$324$$ −27.1145 −1.50636
$$325$$ 0 0
$$326$$ −17.8452 −0.988354
$$327$$ −5.29401 −0.292759
$$328$$ 11.0576 0.610555
$$329$$ 13.1959 0.727516
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ −4.94366 −0.271318
$$333$$ 10.3258 0.565852
$$334$$ −9.79112 −0.535746
$$335$$ 0 0
$$336$$ −2.16295 −0.117999
$$337$$ −9.90275 −0.539437 −0.269718 0.962939i $$-0.586931\pi$$
−0.269718 + 0.962939i $$0.586931\pi$$
$$338$$ 1.22878 0.0668367
$$339$$ −12.5178 −0.679872
$$340$$ 0 0
$$341$$ 4.32591 0.234261
$$342$$ −6.67058 −0.360704
$$343$$ 18.8467 1.01763
$$344$$ −7.48223 −0.403415
$$345$$ 0 0
$$346$$ 18.4926 0.994169
$$347$$ −21.2781 −1.14227 −0.571133 0.820858i $$-0.693496\pi$$
−0.571133 + 0.820858i $$0.693496\pi$$
$$348$$ 22.7327 1.21860
$$349$$ −16.4152 −0.878686 −0.439343 0.898319i $$-0.644789\pi$$
−0.439343 + 0.898319i $$0.644789\pi$$
$$350$$ 0 0
$$351$$ −53.0934 −2.83391
$$352$$ 4.09748 0.218396
$$353$$ 23.8744 1.27071 0.635354 0.772221i $$-0.280854\pi$$
0.635354 + 0.772221i $$0.280854\pi$$
$$354$$ −24.3837 −1.29598
$$355$$ 0 0
$$356$$ 19.9000 1.05470
$$357$$ 26.4618 1.40051
$$358$$ 22.7327 1.20146
$$359$$ 2.22334 0.117343 0.0586717 0.998277i $$-0.481313\pi$$
0.0586717 + 0.998277i $$0.481313\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −17.6127 −0.925701
$$363$$ 33.6294 1.76508
$$364$$ 11.5822 0.607074
$$365$$ 0 0
$$366$$ 25.0355 1.30863
$$367$$ −4.52057 −0.235972 −0.117986 0.993015i $$-0.537644\pi$$
−0.117986 + 0.993015i $$0.537644\pi$$
$$368$$ 0.147795 0.00770433
$$369$$ −28.2208 −1.46911
$$370$$ 0 0
$$371$$ −0.417768 −0.0216894
$$372$$ 22.1153 1.14662
$$373$$ 15.5186 0.803521 0.401760 0.915745i $$-0.368398\pi$$
0.401760 + 0.915745i $$0.368398\pi$$
$$374$$ 2.12892 0.110084
$$375$$ 0 0
$$376$$ 14.6467 0.755345
$$377$$ 22.7327 1.17079
$$378$$ 32.9989 1.69728
$$379$$ −18.9015 −0.970905 −0.485453 0.874263i $$-0.661345\pi$$
−0.485453 + 0.874263i $$0.661345\pi$$
$$380$$ 0 0
$$381$$ −46.4126 −2.37779
$$382$$ 10.3855 0.531366
$$383$$ −13.7046 −0.700274 −0.350137 0.936699i $$-0.613865\pi$$
−0.350137 + 0.936699i $$0.613865\pi$$
$$384$$ 19.4363 0.991854
$$385$$ 0 0
$$386$$ 3.43630 0.174903
$$387$$ 19.0958 0.970694
$$388$$ −4.46094 −0.226470
$$389$$ 12.7411 0.646000 0.323000 0.946399i $$-0.395309\pi$$
0.323000 + 0.946399i $$0.395309\pi$$
$$390$$ 0 0
$$391$$ −1.80814 −0.0914413
$$392$$ 0.746063 0.0376819
$$393$$ −32.0776 −1.61810
$$394$$ 2.07110 0.104340
$$395$$ 0 0
$$396$$ −6.41777 −0.322505
$$397$$ −38.6522 −1.93990 −0.969950 0.243306i $$-0.921768\pi$$
−0.969950 + 0.243306i $$0.921768\pi$$
$$398$$ −17.6412 −0.884274
$$399$$ −8.35482 −0.418264
$$400$$ 0 0
$$401$$ 31.8660 1.59131 0.795655 0.605750i $$-0.207127\pi$$
0.795655 + 0.605750i $$0.207127\pi$$
$$402$$ −43.3296 −2.16108
$$403$$ 22.1153 1.10164
$$404$$ −9.83705 −0.489411
$$405$$ 0 0
$$406$$ −14.1289 −0.701206
$$407$$ 1.04049 0.0515751
$$408$$ 29.3710 1.45408
$$409$$ 11.0645 0.547102 0.273551 0.961857i $$-0.411802\pi$$
0.273551 + 0.961857i $$0.411802\pi$$
$$410$$ 0 0
$$411$$ −31.2274 −1.54033
$$412$$ 2.44444 0.120429
$$413$$ −21.6922 −1.06740
$$414$$ −3.80814 −0.187160
$$415$$ 0 0
$$416$$ 20.9474 1.02703
$$417$$ 43.2830 2.11958
$$418$$ −0.672165 −0.0328767
$$419$$ −25.7452 −1.25773 −0.628867 0.777513i $$-0.716481\pi$$
−0.628867 + 0.777513i $$0.716481\pi$$
$$420$$ 0 0
$$421$$ 27.4482 1.33774 0.668871 0.743378i $$-0.266778\pi$$
0.668871 + 0.743378i $$0.266778\pi$$
$$422$$ 10.1829 0.495696
$$423$$ −37.3806 −1.81751
$$424$$ −0.463697 −0.0225191
$$425$$ 0 0
$$426$$ −10.6385 −0.515439
$$427$$ 22.2720 1.07782
$$428$$ −6.71593 −0.324627
$$429$$ −9.03555 −0.436240
$$430$$ 0 0
$$431$$ 1.74519 0.0840627 0.0420314 0.999116i $$-0.486617\pi$$
0.0420314 + 0.999116i $$0.486617\pi$$
$$432$$ 3.62787 0.174546
$$433$$ −18.5208 −0.890052 −0.445026 0.895518i $$-0.646806\pi$$
−0.445026 + 0.895518i $$0.646806\pi$$
$$434$$ −13.7452 −0.659790
$$435$$ 0 0
$$436$$ −1.93705 −0.0927679
$$437$$ 0.570885 0.0273091
$$438$$ −31.6545 −1.51251
$$439$$ 29.4482 1.40549 0.702743 0.711444i $$-0.251959\pi$$
0.702743 + 0.711444i $$0.251959\pi$$
$$440$$ 0 0
$$441$$ −1.90407 −0.0906699
$$442$$ 10.8836 0.517681
$$443$$ 11.7388 0.557726 0.278863 0.960331i $$-0.410042\pi$$
0.278863 + 0.960331i $$0.410042\pi$$
$$444$$ 5.31927 0.252441
$$445$$ 0 0
$$446$$ −3.66220 −0.173410
$$447$$ −48.5771 −2.29762
$$448$$ −14.3637 −0.678620
$$449$$ −7.06446 −0.333392 −0.166696 0.986008i $$-0.553310\pi$$
−0.166696 + 0.986008i $$0.553310\pi$$
$$450$$ 0 0
$$451$$ −2.84368 −0.133904
$$452$$ −4.58019 −0.215434
$$453$$ −45.6679 −2.14566
$$454$$ −10.1748 −0.477529
$$455$$ 0 0
$$456$$ −9.27334 −0.434264
$$457$$ −34.5000 −1.61384 −0.806920 0.590660i $$-0.798867\pi$$
−0.806920 + 0.590660i $$0.798867\pi$$
$$458$$ 14.6307 0.683649
$$459$$ −44.3837 −2.07166
$$460$$ 0 0
$$461$$ 8.03147 0.374063 0.187032 0.982354i $$-0.440113\pi$$
0.187032 + 0.982354i $$0.440113\pi$$
$$462$$ 5.61582 0.261272
$$463$$ −25.3290 −1.17714 −0.588570 0.808447i $$-0.700309\pi$$
−0.588570 + 0.808447i $$0.700309\pi$$
$$464$$ −1.55332 −0.0721112
$$465$$ 0 0
$$466$$ −1.92890 −0.0893547
$$467$$ 26.8759 1.24367 0.621834 0.783149i $$-0.286388\pi$$
0.621834 + 0.783149i $$0.286388\pi$$
$$468$$ −32.8094 −1.51662
$$469$$ −38.5467 −1.77992
$$470$$ 0 0
$$471$$ −24.3837 −1.12354
$$472$$ −24.0770 −1.10823
$$473$$ 1.92420 0.0884748
$$474$$ −5.36146 −0.246260
$$475$$ 0 0
$$476$$ 9.68224 0.443785
$$477$$ 1.18343 0.0541854
$$478$$ −13.0741 −0.597996
$$479$$ −28.9015 −1.32054 −0.660272 0.751027i $$-0.729559\pi$$
−0.660272 + 0.751027i $$0.729559\pi$$
$$480$$ 0 0
$$481$$ 5.31927 0.242538
$$482$$ −0.147795 −0.00673187
$$483$$ −4.76964 −0.217026
$$484$$ 12.3048 0.559310
$$485$$ 0 0
$$486$$ −29.0815 −1.31916
$$487$$ −17.7294 −0.803395 −0.401697 0.915773i $$-0.631580\pi$$
−0.401697 + 0.915773i $$0.631580\pi$$
$$488$$ 24.7206 1.11905
$$489$$ 63.3141 2.86316
$$490$$ 0 0
$$491$$ −35.1645 −1.58695 −0.793475 0.608603i $$-0.791730\pi$$
−0.793475 + 0.608603i $$0.791730\pi$$
$$492$$ −14.5377 −0.655410
$$493$$ 19.0035 0.855875
$$494$$ −3.43630 −0.154606
$$495$$ 0 0
$$496$$ −1.51114 −0.0678520
$$497$$ −9.46422 −0.424528
$$498$$ −12.2542 −0.549122
$$499$$ 21.4508 0.960268 0.480134 0.877195i $$-0.340588\pi$$
0.480134 + 0.877195i $$0.340588\pi$$
$$500$$ 0 0
$$501$$ 34.7385 1.55200
$$502$$ 11.7298 0.523526
$$503$$ −5.34053 −0.238122 −0.119061 0.992887i $$-0.537988\pi$$
−0.119061 + 0.992887i $$0.537988\pi$$
$$504$$ 55.0304 2.45125
$$505$$ 0 0
$$506$$ −0.383729 −0.0170588
$$507$$ −4.35966 −0.193619
$$508$$ −16.9821 −0.753461
$$509$$ 36.1919 1.60418 0.802088 0.597206i $$-0.203722\pi$$
0.802088 + 0.597206i $$0.203722\pi$$
$$510$$ 0 0
$$511$$ −28.1604 −1.24574
$$512$$ −2.92346 −0.129200
$$513$$ 14.0133 0.618704
$$514$$ −10.0170 −0.441832
$$515$$ 0 0
$$516$$ 9.83705 0.433052
$$517$$ −3.76668 −0.165658
$$518$$ −3.30606 −0.145260
$$519$$ −65.6111 −2.88001
$$520$$ 0 0
$$521$$ −2.77259 −0.121469 −0.0607346 0.998154i $$-0.519344\pi$$
−0.0607346 + 0.998154i $$0.519344\pi$$
$$522$$ 40.0235 1.75178
$$523$$ −20.5373 −0.898033 −0.449016 0.893524i $$-0.648226\pi$$
−0.449016 + 0.893524i $$0.648226\pi$$
$$524$$ −11.7370 −0.512735
$$525$$ 0 0
$$526$$ 16.1919 0.705999
$$527$$ 18.4874 0.805323
$$528$$ 0.617399 0.0268688
$$529$$ −22.6741 −0.985830
$$530$$ 0 0
$$531$$ 61.4482 2.66662
$$532$$ −3.05699 −0.132537
$$533$$ −14.5377 −0.629698
$$534$$ 49.3274 2.13461
$$535$$ 0 0
$$536$$ −42.7845 −1.84801
$$537$$ −80.6547 −3.48051
$$538$$ 22.6682 0.977294
$$539$$ −0.191865 −0.00826419
$$540$$ 0 0
$$541$$ 35.4797 1.52539 0.762695 0.646758i $$-0.223876\pi$$
0.762695 + 0.646758i $$0.223876\pi$$
$$542$$ −21.6456 −0.929760
$$543$$ 62.4891 2.68166
$$544$$ 17.5111 0.750784
$$545$$ 0 0
$$546$$ 28.7096 1.22866
$$547$$ 43.0756 1.84178 0.920890 0.389822i $$-0.127463\pi$$
0.920890 + 0.389822i $$0.127463\pi$$
$$548$$ −11.4260 −0.488092
$$549$$ −63.0908 −2.69265
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ −5.29401 −0.225328
$$553$$ −4.76964 −0.202826
$$554$$ −19.2563 −0.818123
$$555$$ 0 0
$$556$$ 15.8370 0.671640
$$557$$ 40.4376 1.71340 0.856698 0.515818i $$-0.172512\pi$$
0.856698 + 0.515818i $$0.172512\pi$$
$$558$$ 38.9365 1.64831
$$559$$ 9.83705 0.416063
$$560$$ 0 0
$$561$$ −7.55332 −0.318902
$$562$$ 3.48008 0.146798
$$563$$ 19.7173 0.830986 0.415493 0.909596i $$-0.363609\pi$$
0.415493 + 0.909596i $$0.363609\pi$$
$$564$$ −19.2563 −0.810837
$$565$$ 0 0
$$566$$ 0.191865 0.00806467
$$567$$ −59.7915 −2.51101
$$568$$ −10.5047 −0.440768
$$569$$ 18.6807 0.783137 0.391568 0.920149i $$-0.371933\pi$$
0.391568 + 0.920149i $$0.371933\pi$$
$$570$$ 0 0
$$571$$ −29.9371 −1.25283 −0.626413 0.779491i $$-0.715478\pi$$
−0.626413 + 0.779491i $$0.715478\pi$$
$$572$$ −3.30606 −0.138233
$$573$$ −36.8473 −1.53932
$$574$$ 9.03555 0.377137
$$575$$ 0 0
$$576$$ 40.6885 1.69536
$$577$$ −0.156779 −0.00652679 −0.00326339 0.999995i $$-0.501039\pi$$
−0.00326339 + 0.999995i $$0.501039\pi$$
$$578$$ −6.32023 −0.262887
$$579$$ −12.1919 −0.506677
$$580$$ 0 0
$$581$$ −10.9015 −0.452271
$$582$$ −11.0576 −0.458353
$$583$$ 0.119249 0.00493877
$$584$$ −31.2563 −1.29340
$$585$$ 0 0
$$586$$ −13.5993 −0.561780
$$587$$ 31.1474 1.28559 0.642795 0.766038i $$-0.277775\pi$$
0.642795 + 0.766038i $$0.277775\pi$$
$$588$$ −0.980865 −0.0404502
$$589$$ −5.83705 −0.240511
$$590$$ 0 0
$$591$$ −7.34818 −0.302264
$$592$$ −0.363466 −0.0149384
$$593$$ 28.8728 1.18567 0.592833 0.805326i $$-0.298009\pi$$
0.592833 + 0.805326i $$0.298009\pi$$
$$594$$ −9.41928 −0.386478
$$595$$ 0 0
$$596$$ −17.7741 −0.728055
$$597$$ 62.5904 2.56165
$$598$$ −1.96173 −0.0802211
$$599$$ 25.3274 1.03485 0.517425 0.855728i $$-0.326891\pi$$
0.517425 + 0.855728i $$0.326891\pi$$
$$600$$ 0 0
$$601$$ −19.8370 −0.809170 −0.404585 0.914500i $$-0.632584\pi$$
−0.404585 + 0.914500i $$0.632584\pi$$
$$602$$ −6.11397 −0.249187
$$603$$ 109.193 4.44667
$$604$$ −16.7096 −0.679906
$$605$$ 0 0
$$606$$ −24.3837 −0.990521
$$607$$ −2.49921 −0.101440 −0.0507199 0.998713i $$-0.516152\pi$$
−0.0507199 + 0.998713i $$0.516152\pi$$
$$608$$ −5.52881 −0.224223
$$609$$ 50.1289 2.03133
$$610$$ 0 0
$$611$$ −19.2563 −0.779027
$$612$$ −27.4272 −1.10868
$$613$$ −0.883711 −0.0356927 −0.0178464 0.999841i $$-0.505681\pi$$
−0.0178464 + 0.999841i $$0.505681\pi$$
$$614$$ 1.49925 0.0605046
$$615$$ 0 0
$$616$$ 5.54517 0.223421
$$617$$ −29.4085 −1.18394 −0.591971 0.805959i $$-0.701650\pi$$
−0.591971 + 0.805959i $$0.701650\pi$$
$$618$$ 6.05921 0.243737
$$619$$ 30.3208 1.21870 0.609348 0.792903i $$-0.291431\pi$$
0.609348 + 0.792903i $$0.291431\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ −0.672165 −0.0269514
$$623$$ 43.8825 1.75811
$$624$$ 3.15632 0.126354
$$625$$ 0 0
$$626$$ −24.3837 −0.974570
$$627$$ 2.38482 0.0952405
$$628$$ −8.92188 −0.356022
$$629$$ 4.44668 0.177301
$$630$$ 0 0
$$631$$ 17.7767 0.707678 0.353839 0.935306i $$-0.384876\pi$$
0.353839 + 0.935306i $$0.384876\pi$$
$$632$$ −5.29401 −0.210584
$$633$$ −36.1286 −1.43598
$$634$$ −7.40226 −0.293981
$$635$$ 0 0
$$636$$ 0.609632 0.0241735
$$637$$ −0.980865 −0.0388633
$$638$$ 4.03299 0.159668
$$639$$ 26.8096 1.06057
$$640$$ 0 0
$$641$$ −32.6675 −1.29029 −0.645143 0.764062i $$-0.723202\pi$$
−0.645143 + 0.764062i $$0.723202\pi$$
$$642$$ −16.6472 −0.657014
$$643$$ 31.8661 1.25668 0.628338 0.777941i $$-0.283736\pi$$
0.628338 + 0.777941i $$0.283736\pi$$
$$644$$ −1.74519 −0.0687700
$$645$$ 0 0
$$646$$ −2.87259 −0.113021
$$647$$ 21.2601 0.835820 0.417910 0.908488i $$-0.362763\pi$$
0.417910 + 0.908488i $$0.362763\pi$$
$$648$$ −66.3649 −2.60706
$$649$$ 6.19186 0.243052
$$650$$ 0 0
$$651$$ 48.7675 1.91135
$$652$$ 23.1663 0.907263
$$653$$ 12.8340 0.502234 0.251117 0.967957i $$-0.419202\pi$$
0.251117 + 0.967957i $$0.419202\pi$$
$$654$$ −4.80150 −0.187753
$$655$$ 0 0
$$656$$ 0.993361 0.0387842
$$657$$ 79.7710 3.11216
$$658$$ 11.9683 0.466573
$$659$$ 20.3548 0.792911 0.396456 0.918054i $$-0.370240\pi$$
0.396456 + 0.918054i $$0.370240\pi$$
$$660$$ 0 0
$$661$$ −30.7385 −1.19559 −0.597795 0.801649i $$-0.703957\pi$$
−0.597795 + 0.801649i $$0.703957\pi$$
$$662$$ 7.25574 0.282002
$$663$$ −38.6147 −1.49967
$$664$$ −12.1000 −0.469571
$$665$$ 0 0
$$666$$ 9.36520 0.362894
$$667$$ −3.42531 −0.132629
$$668$$ 12.7107 0.491790
$$669$$ 12.9934 0.502352
$$670$$ 0 0
$$671$$ −6.35738 −0.245424
$$672$$ 46.1922 1.78190
$$673$$ 21.2094 0.817564 0.408782 0.912632i $$-0.365954\pi$$
0.408782 + 0.912632i $$0.365954\pi$$
$$674$$ −8.98147 −0.345953
$$675$$ 0 0
$$676$$ −1.59518 −0.0613530
$$677$$ 11.2650 0.432951 0.216475 0.976288i $$-0.430544\pi$$
0.216475 + 0.976288i $$0.430544\pi$$
$$678$$ −11.3532 −0.436018
$$679$$ −9.83705 −0.377511
$$680$$ 0 0
$$681$$ 36.1000 1.38336
$$682$$ 3.92346 0.150237
$$683$$ −12.3603 −0.472954 −0.236477 0.971637i $$-0.575993\pi$$
−0.236477 + 0.971637i $$0.575993\pi$$
$$684$$ 8.65964 0.331109
$$685$$ 0 0
$$686$$ 17.0934 0.652628
$$687$$ −51.9093 −1.98046
$$688$$ −0.672165 −0.0256261
$$689$$ 0.609632 0.0232251
$$690$$ 0 0
$$691$$ 22.7493 0.865423 0.432711 0.901533i $$-0.357557\pi$$
0.432711 + 0.901533i $$0.357557\pi$$
$$692$$ −24.0068 −0.912601
$$693$$ −14.1521 −0.537595
$$694$$ −19.2985 −0.732561
$$695$$ 0 0
$$696$$ 55.6401 2.10903
$$697$$ −12.1529 −0.460323
$$698$$ −14.8881 −0.563521
$$699$$ 6.84368 0.258852
$$700$$ 0 0
$$701$$ −16.0289 −0.605404 −0.302702 0.953085i $$-0.597889\pi$$
−0.302702 + 0.953085i $$0.597889\pi$$
$$702$$ −48.1540 −1.81745
$$703$$ −1.40396 −0.0529512
$$704$$ 4.10001 0.154525
$$705$$ 0 0
$$706$$ 21.6533 0.814934
$$707$$ −21.6922 −0.815818
$$708$$ 31.6545 1.18965
$$709$$ 31.4193 1.17998 0.589988 0.807412i $$-0.299133\pi$$
0.589988 + 0.807412i $$0.299133\pi$$
$$710$$ 0 0
$$711$$ 13.5111 0.506707
$$712$$ 48.7069 1.82537
$$713$$ −3.33228 −0.124795
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ −29.5111 −1.10288
$$717$$ 46.3865 1.73234
$$718$$ 2.01650 0.0752550
$$719$$ 11.2589 0.419886 0.209943 0.977714i $$-0.432672\pi$$
0.209943 + 0.977714i $$0.432672\pi$$
$$720$$ 0 0
$$721$$ 5.39037 0.200748
$$722$$ 0.906968 0.0337538
$$723$$ 0.524371 0.0195016
$$724$$ 22.8644 0.849750
$$725$$ 0 0
$$726$$ 30.5008 1.13199
$$727$$ 48.9829 1.81668 0.908338 0.418237i $$-0.137352\pi$$
0.908338 + 0.418237i $$0.137352\pi$$
$$728$$ 28.3485 1.05066
$$729$$ 34.0934 1.26272
$$730$$ 0 0
$$731$$ 8.22334 0.304151
$$732$$ −32.5007 −1.20126
$$733$$ −35.9260 −1.32696 −0.663479 0.748195i $$-0.730921\pi$$
−0.663479 + 0.748195i $$0.730921\pi$$
$$734$$ −4.10001 −0.151334
$$735$$ 0 0
$$736$$ −3.15632 −0.116343
$$737$$ 11.0029 0.405296
$$738$$ −25.5953 −0.942177
$$739$$ 14.3523 0.527956 0.263978 0.964529i $$-0.414965\pi$$
0.263978 + 0.964529i $$0.414965\pi$$
$$740$$ 0 0
$$741$$ 12.1919 0.447879
$$742$$ −0.378902 −0.0139099
$$743$$ 12.5629 0.460887 0.230443 0.973086i $$-0.425982\pi$$
0.230443 + 0.973086i $$0.425982\pi$$
$$744$$ 54.1289 1.98446
$$745$$ 0 0
$$746$$ 14.0748 0.515316
$$747$$ 30.8811 1.12988
$$748$$ −2.76372 −0.101052
$$749$$ −14.8096 −0.541133
$$750$$ 0 0
$$751$$ 26.4548 0.965350 0.482675 0.875799i $$-0.339665\pi$$
0.482675 + 0.875799i $$0.339665\pi$$
$$752$$ 1.31578 0.0479817
$$753$$ −41.6169 −1.51660
$$754$$ 20.6178 0.750855
$$755$$ 0 0
$$756$$ −42.8386 −1.55802
$$757$$ −15.7350 −0.571897 −0.285949 0.958245i $$-0.592309\pi$$
−0.285949 + 0.958245i $$0.592309\pi$$
$$758$$ −17.1431 −0.622664
$$759$$ 1.36146 0.0494178
$$760$$ 0 0
$$761$$ 16.9619 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$762$$ −42.0948 −1.52493
$$763$$ −4.27149 −0.154638
$$764$$ −13.4822 −0.487770
$$765$$ 0 0
$$766$$ −12.4297 −0.449102
$$767$$ 31.6545 1.14298
$$768$$ 53.2323 1.92086
$$769$$ 41.9974 1.51447 0.757233 0.653145i $$-0.226551\pi$$
0.757233 + 0.653145i $$0.226551\pi$$
$$770$$ 0 0
$$771$$ 35.5400 1.27994
$$772$$ −4.46094 −0.160553
$$773$$ −40.9579 −1.47315 −0.736576 0.676355i $$-0.763559\pi$$
−0.736576 + 0.676355i $$0.763559\pi$$
$$774$$ 17.3193 0.622528
$$775$$ 0 0
$$776$$ −10.9185 −0.391952
$$777$$ 11.7298 0.420804
$$778$$ 11.5558 0.414295
$$779$$ 3.83705 0.137476
$$780$$ 0 0
$$781$$ 2.70149 0.0966669
$$782$$ −1.63992 −0.0586434
$$783$$ −84.0800 −3.00477
$$784$$ 0.0670225 0.00239366
$$785$$ 0 0
$$786$$ −29.0934 −1.03773
$$787$$ −28.5379 −1.01727 −0.508634 0.860983i $$-0.669849\pi$$
−0.508634 + 0.860983i $$0.669849\pi$$
$$788$$ −2.68866 −0.0957796
$$789$$ −57.4482 −2.04521
$$790$$ 0 0
$$791$$ −10.1000 −0.359115
$$792$$ −15.7080 −0.558160
$$793$$ −32.5007 −1.15413
$$794$$ −35.0563 −1.24410
$$795$$ 0 0
$$796$$ 22.9015 0.811722
$$797$$ −33.2790 −1.17880 −0.589402 0.807840i $$-0.700636\pi$$
−0.589402 + 0.807840i $$0.700636\pi$$
$$798$$ −7.57755 −0.268242
$$799$$ −16.0974 −0.569487
$$800$$ 0 0
$$801$$ −124.308 −4.39219
$$802$$ 28.9014 1.02054
$$803$$ 8.03817 0.283661
$$804$$ 56.2497 1.98377
$$805$$ 0 0
$$806$$ 20.0578 0.706507
$$807$$ −80.4259 −2.83113
$$808$$ −24.0770 −0.847025
$$809$$ −34.4234 −1.21026 −0.605130 0.796126i $$-0.706879\pi$$
−0.605130 + 0.796126i $$0.706879\pi$$
$$810$$ 0 0
$$811$$ −11.2563 −0.395263 −0.197631 0.980276i $$-0.563325\pi$$
−0.197631 + 0.980276i $$0.563325\pi$$
$$812$$ 18.3419 0.643675
$$813$$ 76.7980 2.69342
$$814$$ 0.943690 0.0330763
$$815$$ 0 0
$$816$$ 2.63854 0.0923674
$$817$$ −2.59637 −0.0908353
$$818$$ 10.0351 0.350869
$$819$$ −72.3497 −2.52810
$$820$$ 0 0
$$821$$ −31.3167 −1.09296 −0.546480 0.837472i $$-0.684033\pi$$
−0.546480 + 0.837472i $$0.684033\pi$$
$$822$$ −28.3223 −0.987852
$$823$$ 13.4800 0.469882 0.234941 0.972010i $$-0.424510\pi$$
0.234941 + 0.972010i $$0.424510\pi$$
$$824$$ 5.98298 0.208427
$$825$$ 0 0
$$826$$ −19.6741 −0.684549
$$827$$ −15.9882 −0.555963 −0.277982 0.960586i $$-0.589665\pi$$
−0.277982 + 0.960586i $$0.589665\pi$$
$$828$$ 4.94366 0.171804
$$829$$ −48.4837 −1.68391 −0.841955 0.539548i $$-0.818595\pi$$
−0.841955 + 0.539548i $$0.818595\pi$$
$$830$$ 0 0
$$831$$ 68.3208 2.37002
$$832$$ 20.9604 0.726670
$$833$$ −0.819960 −0.0284099
$$834$$ 39.2563 1.35934
$$835$$ 0 0
$$836$$ 0.872594 0.0301793
$$837$$ −81.7965 −2.82730
$$838$$ −23.3501 −0.806614
$$839$$ −18.9934 −0.655724 −0.327862 0.944726i $$-0.606328\pi$$
−0.327862 + 0.944726i $$0.606328\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 24.8946 0.857925
$$843$$ −12.3472 −0.425260
$$844$$ −13.2193 −0.455026
$$845$$ 0 0
$$846$$ −33.9030 −1.16561
$$847$$ 27.1340 0.932334
$$848$$ −0.0416562 −0.00143048
$$849$$ −0.680729 −0.0233626
$$850$$ 0 0
$$851$$ −0.801497 −0.0274750
$$852$$ 13.8108 0.473149
$$853$$ 44.1210 1.51067 0.755337 0.655337i $$-0.227473\pi$$
0.755337 + 0.655337i $$0.227473\pi$$
$$854$$ 20.2000 0.691230
$$855$$ 0 0
$$856$$ −16.4378 −0.561833
$$857$$ 32.4149 1.10727 0.553635 0.832759i $$-0.313240\pi$$
0.553635 + 0.832759i $$0.313240\pi$$
$$858$$ −8.19495 −0.279771
$$859$$ 6.29444 0.214763 0.107382 0.994218i $$-0.465753\pi$$
0.107382 + 0.994218i $$0.465753\pi$$
$$860$$ 0 0
$$861$$ −32.0578 −1.09253
$$862$$ 1.58283 0.0539113
$$863$$ −17.4338 −0.593453 −0.296726 0.954963i $$-0.595895\pi$$
−0.296726 + 0.954963i $$0.595895\pi$$
$$864$$ −77.4771 −2.63582
$$865$$ 0 0
$$866$$ −16.7978 −0.570811
$$867$$ 22.4240 0.761557
$$868$$ 17.8438 0.605657
$$869$$ 1.36146 0.0461843
$$870$$ 0 0
$$871$$ 56.2497 1.90595
$$872$$ −4.74109 −0.160554
$$873$$ 27.8658 0.943113
$$874$$ 0.517774 0.0175140
$$875$$ 0 0
$$876$$ 41.0934 1.38842
$$877$$ 23.2904 0.786462 0.393231 0.919440i $$-0.371357\pi$$
0.393231 + 0.919440i $$0.371357\pi$$
$$878$$ 26.7086 0.901370
$$879$$ 48.2497 1.62742
$$880$$ 0 0
$$881$$ 28.9619 0.975751 0.487875 0.872913i $$-0.337772\pi$$
0.487875 + 0.872913i $$0.337772\pi$$
$$882$$ −1.72693 −0.0581487
$$883$$ −37.3627 −1.25735 −0.628677 0.777667i $$-0.716403\pi$$
−0.628677 + 0.777667i $$0.716403\pi$$
$$884$$ −14.1289 −0.475207
$$885$$ 0 0
$$886$$ 10.6467 0.357683
$$887$$ −23.0234 −0.773050 −0.386525 0.922279i $$-0.626325\pi$$
−0.386525 + 0.922279i $$0.626325\pi$$
$$888$$ 13.0194 0.436901
$$889$$ −37.4482 −1.25597
$$890$$ 0 0
$$891$$ 17.0670 0.571767
$$892$$ 4.75420 0.159183
$$893$$ 5.08247 0.170078
$$894$$ −44.0578 −1.47351
$$895$$ 0 0
$$896$$ 15.6822 0.523907
$$897$$ 6.96015 0.232393
$$898$$ −6.40723 −0.213812
$$899$$ 35.0223 1.16806
$$900$$ 0 0
$$901$$ 0.509626 0.0169781
$$902$$ −2.57913 −0.0858756
$$903$$ 21.6922 0.721870
$$904$$ −11.2104 −0.372852
$$905$$ 0 0
$$906$$ −41.4193 −1.37606
$$907$$ −35.2693 −1.17110 −0.585549 0.810637i $$-0.699121\pi$$
−0.585549 + 0.810637i $$0.699121\pi$$
$$908$$ 13.2088 0.438350
$$909$$ 61.4482 2.03811
$$910$$ 0 0
$$911$$ −4.97260 −0.164750 −0.0823748 0.996601i $$-0.526250\pi$$
−0.0823748 + 0.996601i $$0.526250\pi$$
$$912$$ −0.833070 −0.0275857
$$913$$ 3.11175 0.102984
$$914$$ −31.2904 −1.03499
$$915$$ 0 0
$$916$$ −18.9934 −0.627558
$$917$$ −25.8819 −0.854697
$$918$$ −40.2546 −1.32860
$$919$$ −48.7096 −1.60678 −0.803391 0.595451i $$-0.796973\pi$$
−0.803391 + 0.595451i $$0.796973\pi$$
$$920$$ 0 0
$$921$$ −5.31927 −0.175276
$$922$$ 7.28429 0.239895
$$923$$ 13.8108 0.454587
$$924$$ −7.29036 −0.239835
$$925$$ 0 0
$$926$$ −22.9726 −0.754926
$$927$$ −15.2695 −0.501516
$$928$$ 33.1729 1.08895
$$929$$ 32.5126 1.06671 0.533353 0.845893i $$-0.320932\pi$$
0.533353 + 0.845893i $$0.320932\pi$$
$$930$$ 0 0
$$931$$ 0.258887 0.00848468
$$932$$ 2.50407 0.0820235
$$933$$ 2.38482 0.0780755
$$934$$ 24.3756 0.797593
$$935$$ 0 0
$$936$$ −80.3038 −2.62481
$$937$$ −0.385560 −0.0125957 −0.00629785 0.999980i $$-0.502005\pi$$
−0.00629785 + 0.999980i $$0.502005\pi$$
$$938$$ −34.9606 −1.14150
$$939$$ 86.5126 2.82323
$$940$$ 0 0
$$941$$ 45.3482 1.47831 0.739154 0.673536i $$-0.235225\pi$$
0.739154 + 0.673536i $$0.235225\pi$$
$$942$$ −22.1153 −0.720554
$$943$$ 2.19051 0.0713329
$$944$$ −2.16295 −0.0703982
$$945$$ 0 0
$$946$$ 1.74519 0.0567409
$$947$$ 9.61202 0.312349 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$948$$ 6.96015 0.226055
$$949$$ 41.0934 1.33395
$$950$$ 0 0
$$951$$ 26.2630 0.851635
$$952$$ 23.6981 0.768059
$$953$$ −59.8421 −1.93848 −0.969238 0.246126i $$-0.920842\pi$$
−0.969238 + 0.246126i $$0.920842\pi$$
$$954$$ 1.07333 0.0347503
$$955$$ 0 0
$$956$$ 16.9726 0.548933
$$957$$ −14.3089 −0.462542
$$958$$ −26.2127 −0.846895
$$959$$ −25.1959 −0.813619
$$960$$ 0 0
$$961$$ 3.07110 0.0990676
$$962$$ 4.82441 0.155545
$$963$$ 41.9518 1.35188
$$964$$ 0.191865 0.00617954
$$965$$ 0 0
$$966$$ −4.32591 −0.139184
$$967$$ −0.368324 −0.0118445 −0.00592225 0.999982i $$-0.501885\pi$$
−0.00592225 + 0.999982i $$0.501885\pi$$
$$968$$ 30.1171 0.967999
$$969$$ 10.1919 0.327410
$$970$$ 0 0
$$971$$ −45.8370 −1.47098 −0.735490 0.677535i $$-0.763048\pi$$
−0.735490 + 0.677535i $$0.763048\pi$$
$$972$$ 37.7531 1.21093
$$973$$ 34.9231 1.11958
$$974$$ −16.0800 −0.515235
$$975$$ 0 0
$$976$$ 2.22077 0.0710853
$$977$$ 29.0337 0.928872 0.464436 0.885607i $$-0.346257\pi$$
0.464436 + 0.885607i $$0.346257\pi$$
$$978$$ 57.4239 1.83621
$$979$$ −12.5259 −0.400330
$$980$$ 0 0
$$981$$ 12.1000 0.386323
$$982$$ −31.8930 −1.01775
$$983$$ −11.0160 −0.351355 −0.175677 0.984448i $$-0.556212\pi$$
−0.175677 + 0.984448i $$0.556212\pi$$
$$984$$ −35.5822 −1.13432
$$985$$ 0 0
$$986$$ 17.2356 0.548892
$$987$$ −42.4631 −1.35161
$$988$$ 4.46094 0.141921
$$989$$ −1.48223 −0.0471320
$$990$$ 0 0
$$991$$ −25.6822 −0.815823 −0.407912 0.913021i $$-0.633743\pi$$
−0.407912 + 0.913021i $$0.633743\pi$$
$$992$$ 32.2719 1.02463
$$993$$ −25.7431 −0.816933
$$994$$ −8.58374 −0.272260
$$995$$ 0 0
$$996$$ 15.9081 0.504069
$$997$$ 20.3854 0.645611 0.322805 0.946465i $$-0.395374\pi$$
0.322805 + 0.946465i $$0.395374\pi$$
$$998$$ 19.4551 0.615842
$$999$$ −19.6741 −0.622461
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 475.2.a.j.1.4 6
3.2 odd 2 4275.2.a.br.1.3 6
4.3 odd 2 7600.2.a.ck.1.6 6
5.2 odd 4 95.2.b.b.39.4 yes 6
5.3 odd 4 95.2.b.b.39.3 6
5.4 even 2 inner 475.2.a.j.1.3 6
15.2 even 4 855.2.c.d.514.3 6
15.8 even 4 855.2.c.d.514.4 6
15.14 odd 2 4275.2.a.br.1.4 6
19.18 odd 2 9025.2.a.bx.1.3 6
20.3 even 4 1520.2.d.h.609.6 6
20.7 even 4 1520.2.d.h.609.1 6
20.19 odd 2 7600.2.a.ck.1.1 6
95.18 even 4 1805.2.b.e.1084.4 6
95.37 even 4 1805.2.b.e.1084.3 6
95.94 odd 2 9025.2.a.bx.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.3 6 5.3 odd 4
95.2.b.b.39.4 yes 6 5.2 odd 4
475.2.a.j.1.3 6 5.4 even 2 inner
475.2.a.j.1.4 6 1.1 even 1 trivial
855.2.c.d.514.3 6 15.2 even 4
855.2.c.d.514.4 6 15.8 even 4
1520.2.d.h.609.1 6 20.7 even 4
1520.2.d.h.609.6 6 20.3 even 4
1805.2.b.e.1084.3 6 95.37 even 4
1805.2.b.e.1084.4 6 95.18 even 4
4275.2.a.br.1.3 6 3.2 odd 2
4275.2.a.br.1.4 6 15.14 odd 2
7600.2.a.ck.1.1 6 20.19 odd 2
7600.2.a.ck.1.6 6 4.3 odd 2
9025.2.a.bx.1.3 6 19.18 odd 2
9025.2.a.bx.1.4 6 95.94 odd 2