# Properties

 Label 475.2.a.h.1.1 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.24698$$ 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)

 $$f(q)$$ $$=$$ $$q-0.246980 q^{2} -0.801938 q^{3} -1.93900 q^{4} +0.198062 q^{6} +1.69202 q^{7} +0.972853 q^{8} -2.35690 q^{9} +O(q^{10})$$ $$q-0.246980 q^{2} -0.801938 q^{3} -1.93900 q^{4} +0.198062 q^{6} +1.69202 q^{7} +0.972853 q^{8} -2.35690 q^{9} -0.911854 q^{11} +1.55496 q^{12} +1.55496 q^{13} -0.417895 q^{14} +3.63773 q^{16} +5.29590 q^{17} +0.582105 q^{18} -1.00000 q^{19} -1.35690 q^{21} +0.225209 q^{22} +4.24698 q^{23} -0.780167 q^{24} -0.384043 q^{26} +4.29590 q^{27} -3.28083 q^{28} +5.00969 q^{29} +1.82908 q^{31} -2.84415 q^{32} +0.731250 q^{33} -1.30798 q^{34} +4.57002 q^{36} +6.29590 q^{37} +0.246980 q^{38} -1.24698 q^{39} +4.18060 q^{41} +0.335126 q^{42} +7.31767 q^{43} +1.76809 q^{44} -1.04892 q^{46} -2.04892 q^{47} -2.91723 q^{48} -4.13706 q^{49} -4.24698 q^{51} -3.01507 q^{52} -2.70171 q^{53} -1.06100 q^{54} +1.64609 q^{56} +0.801938 q^{57} -1.23729 q^{58} +9.87800 q^{59} +0.542877 q^{61} -0.451747 q^{62} -3.98792 q^{63} -6.57301 q^{64} -0.180604 q^{66} -13.9976 q^{67} -10.2687 q^{68} -3.40581 q^{69} -12.8780 q^{71} -2.29291 q^{72} -2.80731 q^{73} -1.55496 q^{74} +1.93900 q^{76} -1.54288 q^{77} +0.307979 q^{78} +1.59419 q^{79} +3.62565 q^{81} -1.03252 q^{82} +12.2349 q^{83} +2.63102 q^{84} -1.80731 q^{86} -4.01746 q^{87} -0.887100 q^{88} +2.91723 q^{89} +2.63102 q^{91} -8.23490 q^{92} -1.46681 q^{93} +0.506041 q^{94} +2.28083 q^{96} +1.55496 q^{97} +1.02177 q^{98} +2.14914 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{6} + 9 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 + 5 * q^6 + 9 * q^8 - 3 * q^9 $$3 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{6} + 9 q^{8} - 3 q^{9} + q^{11} + 5 q^{12} + 5 q^{13} - 7 q^{14} + 18 q^{16} + 2 q^{17} - 4 q^{18} - 3 q^{19} - q^{22} + 8 q^{23} - q^{24} + 9 q^{26} - q^{27} - 21 q^{28} - 7 q^{29} - 5 q^{31} + 27 q^{32} + 10 q^{33} - 9 q^{34} - 11 q^{36} + 5 q^{37} - 4 q^{38} + q^{39} + q^{41} + 5 q^{43} - 15 q^{44} + 6 q^{46} + 3 q^{47} - 2 q^{48} - 7 q^{49} - 8 q^{51} + 16 q^{52} + 19 q^{53} - 13 q^{54} - 35 q^{56} - 2 q^{57} - 21 q^{58} + 10 q^{59} - 17 q^{61} - 23 q^{62} + 7 q^{63} + 49 q^{64} + 11 q^{66} - q^{67} - 23 q^{68} + 3 q^{69} - 19 q^{71} - 37 q^{72} - q^{73} - 5 q^{74} - 4 q^{76} + 14 q^{77} + 6 q^{78} + 18 q^{79} - q^{81} + 6 q^{82} + 13 q^{83} - 7 q^{84} + 2 q^{86} - 28 q^{87} - 46 q^{88} + 2 q^{89} - 7 q^{91} - q^{92} - q^{93} + 11 q^{94} + 18 q^{96} + 5 q^{97} + 20 q^{99}+O(q^{100})$$ 3 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 + 5 * q^6 + 9 * q^8 - 3 * q^9 + q^11 + 5 * q^12 + 5 * q^13 - 7 * q^14 + 18 * q^16 + 2 * q^17 - 4 * q^18 - 3 * q^19 - q^22 + 8 * q^23 - q^24 + 9 * q^26 - q^27 - 21 * q^28 - 7 * q^29 - 5 * q^31 + 27 * q^32 + 10 * q^33 - 9 * q^34 - 11 * q^36 + 5 * q^37 - 4 * q^38 + q^39 + q^41 + 5 * q^43 - 15 * q^44 + 6 * q^46 + 3 * q^47 - 2 * q^48 - 7 * q^49 - 8 * q^51 + 16 * q^52 + 19 * q^53 - 13 * q^54 - 35 * q^56 - 2 * q^57 - 21 * q^58 + 10 * q^59 - 17 * q^61 - 23 * q^62 + 7 * q^63 + 49 * q^64 + 11 * q^66 - q^67 - 23 * q^68 + 3 * q^69 - 19 * q^71 - 37 * q^72 - q^73 - 5 * q^74 - 4 * q^76 + 14 * q^77 + 6 * q^78 + 18 * q^79 - q^81 + 6 * q^82 + 13 * q^83 - 7 * q^84 + 2 * q^86 - 28 * q^87 - 46 * q^88 + 2 * q^89 - 7 * q^91 - q^92 - q^93 + 11 * q^94 + 18 * q^96 + 5 * q^97 + 20 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.246980 −0.174641 −0.0873205 0.996180i $$-0.527830\pi$$
−0.0873205 + 0.996180i $$0.527830\pi$$
$$3$$ −0.801938 −0.462999 −0.231499 0.972835i $$-0.574363\pi$$
−0.231499 + 0.972835i $$0.574363\pi$$
$$4$$ −1.93900 −0.969501
$$5$$ 0 0
$$6$$ 0.198062 0.0808586
$$7$$ 1.69202 0.639524 0.319762 0.947498i $$-0.396397\pi$$
0.319762 + 0.947498i $$0.396397\pi$$
$$8$$ 0.972853 0.343955
$$9$$ −2.35690 −0.785632
$$10$$ 0 0
$$11$$ −0.911854 −0.274934 −0.137467 0.990506i $$-0.543896\pi$$
−0.137467 + 0.990506i $$0.543896\pi$$
$$12$$ 1.55496 0.448878
$$13$$ 1.55496 0.431268 0.215634 0.976474i $$-0.430818\pi$$
0.215634 + 0.976474i $$0.430818\pi$$
$$14$$ −0.417895 −0.111687
$$15$$ 0 0
$$16$$ 3.63773 0.909432
$$17$$ 5.29590 1.28444 0.642222 0.766519i $$-0.278013\pi$$
0.642222 + 0.766519i $$0.278013\pi$$
$$18$$ 0.582105 0.137204
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.35690 −0.296099
$$22$$ 0.225209 0.0480148
$$23$$ 4.24698 0.885556 0.442778 0.896631i $$-0.353993\pi$$
0.442778 + 0.896631i $$0.353993\pi$$
$$24$$ −0.780167 −0.159251
$$25$$ 0 0
$$26$$ −0.384043 −0.0753170
$$27$$ 4.29590 0.826746
$$28$$ −3.28083 −0.620019
$$29$$ 5.00969 0.930276 0.465138 0.885238i $$-0.346005\pi$$
0.465138 + 0.885238i $$0.346005\pi$$
$$30$$ 0 0
$$31$$ 1.82908 0.328513 0.164257 0.986418i $$-0.447477\pi$$
0.164257 + 0.986418i $$0.447477\pi$$
$$32$$ −2.84415 −0.502779
$$33$$ 0.731250 0.127294
$$34$$ −1.30798 −0.224316
$$35$$ 0 0
$$36$$ 4.57002 0.761671
$$37$$ 6.29590 1.03504 0.517520 0.855671i $$-0.326855\pi$$
0.517520 + 0.855671i $$0.326855\pi$$
$$38$$ 0.246980 0.0400654
$$39$$ −1.24698 −0.199677
$$40$$ 0 0
$$41$$ 4.18060 0.652901 0.326450 0.945214i $$-0.394147\pi$$
0.326450 + 0.945214i $$0.394147\pi$$
$$42$$ 0.335126 0.0517110
$$43$$ 7.31767 1.11593 0.557967 0.829863i $$-0.311582\pi$$
0.557967 + 0.829863i $$0.311582\pi$$
$$44$$ 1.76809 0.266549
$$45$$ 0 0
$$46$$ −1.04892 −0.154654
$$47$$ −2.04892 −0.298865 −0.149433 0.988772i $$-0.547745\pi$$
−0.149433 + 0.988772i $$0.547745\pi$$
$$48$$ −2.91723 −0.421066
$$49$$ −4.13706 −0.591009
$$50$$ 0 0
$$51$$ −4.24698 −0.594696
$$52$$ −3.01507 −0.418114
$$53$$ −2.70171 −0.371108 −0.185554 0.982634i $$-0.559408\pi$$
−0.185554 + 0.982634i $$0.559408\pi$$
$$54$$ −1.06100 −0.144384
$$55$$ 0 0
$$56$$ 1.64609 0.219968
$$57$$ 0.801938 0.106219
$$58$$ −1.23729 −0.162464
$$59$$ 9.87800 1.28601 0.643003 0.765864i $$-0.277688\pi$$
0.643003 + 0.765864i $$0.277688\pi$$
$$60$$ 0 0
$$61$$ 0.542877 0.0695082 0.0347541 0.999396i $$-0.488935\pi$$
0.0347541 + 0.999396i $$0.488935\pi$$
$$62$$ −0.451747 −0.0573719
$$63$$ −3.98792 −0.502430
$$64$$ −6.57301 −0.821626
$$65$$ 0 0
$$66$$ −0.180604 −0.0222308
$$67$$ −13.9976 −1.71008 −0.855040 0.518562i $$-0.826467\pi$$
−0.855040 + 0.518562i $$0.826467\pi$$
$$68$$ −10.2687 −1.24527
$$69$$ −3.40581 −0.410012
$$70$$ 0 0
$$71$$ −12.8780 −1.52834 −0.764169 0.645016i $$-0.776851\pi$$
−0.764169 + 0.645016i $$0.776851\pi$$
$$72$$ −2.29291 −0.270222
$$73$$ −2.80731 −0.328571 −0.164286 0.986413i $$-0.552532\pi$$
−0.164286 + 0.986413i $$0.552532\pi$$
$$74$$ −1.55496 −0.180760
$$75$$ 0 0
$$76$$ 1.93900 0.222419
$$77$$ −1.54288 −0.175827
$$78$$ 0.307979 0.0348717
$$79$$ 1.59419 0.179360 0.0896800 0.995971i $$-0.471416\pi$$
0.0896800 + 0.995971i $$0.471416\pi$$
$$80$$ 0 0
$$81$$ 3.62565 0.402850
$$82$$ −1.03252 −0.114023
$$83$$ 12.2349 1.34295 0.671477 0.741025i $$-0.265660\pi$$
0.671477 + 0.741025i $$0.265660\pi$$
$$84$$ 2.63102 0.287068
$$85$$ 0 0
$$86$$ −1.80731 −0.194888
$$87$$ −4.01746 −0.430717
$$88$$ −0.887100 −0.0945652
$$89$$ 2.91723 0.309226 0.154613 0.987975i $$-0.450587\pi$$
0.154613 + 0.987975i $$0.450587\pi$$
$$90$$ 0 0
$$91$$ 2.63102 0.275806
$$92$$ −8.23490 −0.858547
$$93$$ −1.46681 −0.152101
$$94$$ 0.506041 0.0521941
$$95$$ 0 0
$$96$$ 2.28083 0.232786
$$97$$ 1.55496 0.157882 0.0789410 0.996879i $$-0.474846\pi$$
0.0789410 + 0.996879i $$0.474846\pi$$
$$98$$ 1.02177 0.103214
$$99$$ 2.14914 0.215997
$$100$$ 0 0
$$101$$ −16.6015 −1.65191 −0.825955 0.563737i $$-0.809363\pi$$
−0.825955 + 0.563737i $$0.809363\pi$$
$$102$$ 1.04892 0.103858
$$103$$ −4.84548 −0.477439 −0.238720 0.971089i $$-0.576728\pi$$
−0.238720 + 0.971089i $$0.576728\pi$$
$$104$$ 1.51275 0.148337
$$105$$ 0 0
$$106$$ 0.667267 0.0648107
$$107$$ −4.46681 −0.431823 −0.215912 0.976413i $$-0.569272\pi$$
−0.215912 + 0.976413i $$0.569272\pi$$
$$108$$ −8.32975 −0.801530
$$109$$ 18.8267 1.80327 0.901635 0.432498i $$-0.142368\pi$$
0.901635 + 0.432498i $$0.142368\pi$$
$$110$$ 0 0
$$111$$ −5.04892 −0.479222
$$112$$ 6.15511 0.581603
$$113$$ 20.0368 1.88491 0.942453 0.334337i $$-0.108512\pi$$
0.942453 + 0.334337i $$0.108512\pi$$
$$114$$ −0.198062 −0.0185502
$$115$$ 0 0
$$116$$ −9.71379 −0.901903
$$117$$ −3.66487 −0.338818
$$118$$ −2.43967 −0.224589
$$119$$ 8.96077 0.821433
$$120$$ 0 0
$$121$$ −10.1685 −0.924411
$$122$$ −0.134079 −0.0121390
$$123$$ −3.35258 −0.302292
$$124$$ −3.54660 −0.318494
$$125$$ 0 0
$$126$$ 0.984935 0.0877449
$$127$$ −17.8702 −1.58573 −0.792863 0.609399i $$-0.791411\pi$$
−0.792863 + 0.609399i $$0.791411\pi$$
$$128$$ 7.31170 0.646269
$$129$$ −5.86831 −0.516676
$$130$$ 0 0
$$131$$ 7.44265 0.650267 0.325134 0.945668i $$-0.394591\pi$$
0.325134 + 0.945668i $$0.394591\pi$$
$$132$$ −1.41789 −0.123412
$$133$$ −1.69202 −0.146717
$$134$$ 3.45712 0.298650
$$135$$ 0 0
$$136$$ 5.15213 0.441791
$$137$$ 5.68664 0.485843 0.242921 0.970046i $$-0.421894\pi$$
0.242921 + 0.970046i $$0.421894\pi$$
$$138$$ 0.841166 0.0716048
$$139$$ 3.61596 0.306701 0.153351 0.988172i $$-0.450994\pi$$
0.153351 + 0.988172i $$0.450994\pi$$
$$140$$ 0 0
$$141$$ 1.64310 0.138374
$$142$$ 3.18060 0.266910
$$143$$ −1.41789 −0.118570
$$144$$ −8.57374 −0.714479
$$145$$ 0 0
$$146$$ 0.693349 0.0573820
$$147$$ 3.31767 0.273637
$$148$$ −12.2078 −1.00347
$$149$$ 3.29052 0.269570 0.134785 0.990875i $$-0.456966\pi$$
0.134785 + 0.990875i $$0.456966\pi$$
$$150$$ 0 0
$$151$$ −10.2131 −0.831133 −0.415566 0.909563i $$-0.636417\pi$$
−0.415566 + 0.909563i $$0.636417\pi$$
$$152$$ −0.972853 −0.0789088
$$153$$ −12.4819 −1.00910
$$154$$ 0.381059 0.0307066
$$155$$ 0 0
$$156$$ 2.41789 0.193587
$$157$$ 14.3448 1.14484 0.572420 0.819960i $$-0.306005\pi$$
0.572420 + 0.819960i $$0.306005\pi$$
$$158$$ −0.393732 −0.0313236
$$159$$ 2.16660 0.171823
$$160$$ 0 0
$$161$$ 7.18598 0.566335
$$162$$ −0.895461 −0.0703540
$$163$$ −19.5308 −1.52977 −0.764885 0.644167i $$-0.777204\pi$$
−0.764885 + 0.644167i $$0.777204\pi$$
$$164$$ −8.10620 −0.632988
$$165$$ 0 0
$$166$$ −3.02177 −0.234535
$$167$$ 11.8823 0.919481 0.459741 0.888053i $$-0.347942\pi$$
0.459741 + 0.888053i $$0.347942\pi$$
$$168$$ −1.32006 −0.101845
$$169$$ −10.5821 −0.814008
$$170$$ 0 0
$$171$$ 2.35690 0.180236
$$172$$ −14.1890 −1.08190
$$173$$ 15.4722 1.17633 0.588164 0.808741i $$-0.299851\pi$$
0.588164 + 0.808741i $$0.299851\pi$$
$$174$$ 0.992230 0.0752208
$$175$$ 0 0
$$176$$ −3.31708 −0.250034
$$177$$ −7.92154 −0.595420
$$178$$ −0.720497 −0.0540035
$$179$$ 2.16421 0.161761 0.0808803 0.996724i $$-0.474227\pi$$
0.0808803 + 0.996724i $$0.474227\pi$$
$$180$$ 0 0
$$181$$ 16.8974 1.25597 0.627986 0.778225i $$-0.283879\pi$$
0.627986 + 0.778225i $$0.283879\pi$$
$$182$$ −0.649809 −0.0481670
$$183$$ −0.435353 −0.0321822
$$184$$ 4.13169 0.304592
$$185$$ 0 0
$$186$$ 0.362273 0.0265631
$$187$$ −4.82908 −0.353138
$$188$$ 3.97285 0.289750
$$189$$ 7.26875 0.528724
$$190$$ 0 0
$$191$$ 5.92394 0.428641 0.214320 0.976763i $$-0.431246\pi$$
0.214320 + 0.976763i $$0.431246\pi$$
$$192$$ 5.27114 0.380412
$$193$$ −4.43535 −0.319264 −0.159632 0.987177i $$-0.551031\pi$$
−0.159632 + 0.987177i $$0.551031\pi$$
$$194$$ −0.384043 −0.0275727
$$195$$ 0 0
$$196$$ 8.02177 0.572984
$$197$$ 16.4722 1.17359 0.586797 0.809734i $$-0.300388\pi$$
0.586797 + 0.809734i $$0.300388\pi$$
$$198$$ −0.530795 −0.0377220
$$199$$ −24.4131 −1.73060 −0.865300 0.501255i $$-0.832872\pi$$
−0.865300 + 0.501255i $$0.832872\pi$$
$$200$$ 0 0
$$201$$ 11.2252 0.791765
$$202$$ 4.10023 0.288491
$$203$$ 8.47650 0.594934
$$204$$ 8.23490 0.576558
$$205$$ 0 0
$$206$$ 1.19673 0.0833804
$$207$$ −10.0097 −0.695721
$$208$$ 5.65651 0.392209
$$209$$ 0.911854 0.0630743
$$210$$ 0 0
$$211$$ −4.34050 −0.298813 −0.149406 0.988776i $$-0.547736\pi$$
−0.149406 + 0.988776i $$0.547736\pi$$
$$212$$ 5.23862 0.359790
$$213$$ 10.3274 0.707619
$$214$$ 1.10321 0.0754140
$$215$$ 0 0
$$216$$ 4.17928 0.284364
$$217$$ 3.09485 0.210092
$$218$$ −4.64981 −0.314925
$$219$$ 2.25129 0.152128
$$220$$ 0 0
$$221$$ 8.23490 0.553939
$$222$$ 1.24698 0.0836918
$$223$$ 26.2379 1.75702 0.878509 0.477725i $$-0.158539\pi$$
0.878509 + 0.477725i $$0.158539\pi$$
$$224$$ −4.81236 −0.321540
$$225$$ 0 0
$$226$$ −4.94869 −0.329182
$$227$$ 14.6853 0.974699 0.487349 0.873207i $$-0.337964\pi$$
0.487349 + 0.873207i $$0.337964\pi$$
$$228$$ −1.55496 −0.102980
$$229$$ −21.6407 −1.43006 −0.715029 0.699095i $$-0.753587\pi$$
−0.715029 + 0.699095i $$0.753587\pi$$
$$230$$ 0 0
$$231$$ 1.23729 0.0814078
$$232$$ 4.87369 0.319973
$$233$$ 27.1183 1.77658 0.888289 0.459286i $$-0.151895\pi$$
0.888289 + 0.459286i $$0.151895\pi$$
$$234$$ 0.905149 0.0591715
$$235$$ 0 0
$$236$$ −19.1535 −1.24678
$$237$$ −1.27844 −0.0830435
$$238$$ −2.21313 −0.143456
$$239$$ −11.5308 −0.745865 −0.372933 0.927858i $$-0.621648\pi$$
−0.372933 + 0.927858i $$0.621648\pi$$
$$240$$ 0 0
$$241$$ 11.8194 0.761354 0.380677 0.924708i $$-0.375691\pi$$
0.380677 + 0.924708i $$0.375691\pi$$
$$242$$ 2.51142 0.161440
$$243$$ −15.7952 −1.01326
$$244$$ −1.05264 −0.0673883
$$245$$ 0 0
$$246$$ 0.828020 0.0527926
$$247$$ −1.55496 −0.0989396
$$248$$ 1.77943 0.112994
$$249$$ −9.81163 −0.621787
$$250$$ 0 0
$$251$$ −9.66487 −0.610041 −0.305021 0.952346i $$-0.598663\pi$$
−0.305021 + 0.952346i $$0.598663\pi$$
$$252$$ 7.73258 0.487107
$$253$$ −3.87263 −0.243470
$$254$$ 4.41358 0.276933
$$255$$ 0 0
$$256$$ 11.3402 0.708761
$$257$$ 14.1860 0.884897 0.442449 0.896794i $$-0.354110\pi$$
0.442449 + 0.896794i $$0.354110\pi$$
$$258$$ 1.44935 0.0902328
$$259$$ 10.6528 0.661932
$$260$$ 0 0
$$261$$ −11.8073 −0.730854
$$262$$ −1.83818 −0.113563
$$263$$ −21.7942 −1.34389 −0.671943 0.740603i $$-0.734540\pi$$
−0.671943 + 0.740603i $$0.734540\pi$$
$$264$$ 0.711399 0.0437836
$$265$$ 0 0
$$266$$ 0.417895 0.0256228
$$267$$ −2.33944 −0.143171
$$268$$ 27.1414 1.65792
$$269$$ −24.7265 −1.50760 −0.753800 0.657104i $$-0.771781\pi$$
−0.753800 + 0.657104i $$0.771781\pi$$
$$270$$ 0 0
$$271$$ −13.2295 −0.803636 −0.401818 0.915720i $$-0.631622\pi$$
−0.401818 + 0.915720i $$0.631622\pi$$
$$272$$ 19.2650 1.16811
$$273$$ −2.10992 −0.127698
$$274$$ −1.40449 −0.0848481
$$275$$ 0 0
$$276$$ 6.60388 0.397507
$$277$$ −0.560335 −0.0336673 −0.0168336 0.999858i $$-0.505359\pi$$
−0.0168336 + 0.999858i $$0.505359\pi$$
$$278$$ −0.893068 −0.0535626
$$279$$ −4.31096 −0.258091
$$280$$ 0 0
$$281$$ 27.6039 1.64671 0.823355 0.567527i $$-0.192100\pi$$
0.823355 + 0.567527i $$0.192100\pi$$
$$282$$ −0.405813 −0.0241658
$$283$$ −15.9608 −0.948769 −0.474385 0.880318i $$-0.657329\pi$$
−0.474385 + 0.880318i $$0.657329\pi$$
$$284$$ 24.9705 1.48172
$$285$$ 0 0
$$286$$ 0.350191 0.0207072
$$287$$ 7.07367 0.417546
$$288$$ 6.70337 0.395000
$$289$$ 11.0465 0.649796
$$290$$ 0 0
$$291$$ −1.24698 −0.0730992
$$292$$ 5.44339 0.318550
$$293$$ 25.3327 1.47995 0.739977 0.672632i $$-0.234836\pi$$
0.739977 + 0.672632i $$0.234836\pi$$
$$294$$ −0.819396 −0.0477882
$$295$$ 0 0
$$296$$ 6.12498 0.356007
$$297$$ −3.91723 −0.227301
$$298$$ −0.812691 −0.0470779
$$299$$ 6.60388 0.381912
$$300$$ 0 0
$$301$$ 12.3817 0.713666
$$302$$ 2.52243 0.145150
$$303$$ 13.3134 0.764832
$$304$$ −3.63773 −0.208638
$$305$$ 0 0
$$306$$ 3.08277 0.176230
$$307$$ −11.5574 −0.659613 −0.329806 0.944049i $$-0.606983\pi$$
−0.329806 + 0.944049i $$0.606983\pi$$
$$308$$ 2.99164 0.170464
$$309$$ 3.88577 0.221054
$$310$$ 0 0
$$311$$ −18.3424 −1.04010 −0.520052 0.854135i $$-0.674087\pi$$
−0.520052 + 0.854135i $$0.674087\pi$$
$$312$$ −1.21313 −0.0686798
$$313$$ −18.9119 −1.06896 −0.534481 0.845181i $$-0.679493\pi$$
−0.534481 + 0.845181i $$0.679493\pi$$
$$314$$ −3.54288 −0.199936
$$315$$ 0 0
$$316$$ −3.09113 −0.173890
$$317$$ 20.2784 1.13895 0.569475 0.822008i $$-0.307146\pi$$
0.569475 + 0.822008i $$0.307146\pi$$
$$318$$ −0.535107 −0.0300073
$$319$$ −4.56810 −0.255765
$$320$$ 0 0
$$321$$ 3.58211 0.199934
$$322$$ −1.77479 −0.0989052
$$323$$ −5.29590 −0.294672
$$324$$ −7.03013 −0.390563
$$325$$ 0 0
$$326$$ 4.82371 0.267160
$$327$$ −15.0978 −0.834912
$$328$$ 4.06711 0.224569
$$329$$ −3.46681 −0.191132
$$330$$ 0 0
$$331$$ −4.77479 −0.262446 −0.131223 0.991353i $$-0.541890\pi$$
−0.131223 + 0.991353i $$0.541890\pi$$
$$332$$ −23.7235 −1.30200
$$333$$ −14.8388 −0.813160
$$334$$ −2.93469 −0.160579
$$335$$ 0 0
$$336$$ −4.93602 −0.269282
$$337$$ 24.3967 1.32897 0.664487 0.747300i $$-0.268650\pi$$
0.664487 + 0.747300i $$0.268650\pi$$
$$338$$ 2.61356 0.142159
$$339$$ −16.0683 −0.872710
$$340$$ 0 0
$$341$$ −1.66786 −0.0903196
$$342$$ −0.582105 −0.0314766
$$343$$ −18.8442 −1.01749
$$344$$ 7.11901 0.383832
$$345$$ 0 0
$$346$$ −3.82132 −0.205435
$$347$$ −9.99761 −0.536700 −0.268350 0.963322i $$-0.586478\pi$$
−0.268350 + 0.963322i $$0.586478\pi$$
$$348$$ 7.78986 0.417580
$$349$$ 21.9584 1.17541 0.587703 0.809077i $$-0.300033\pi$$
0.587703 + 0.809077i $$0.300033\pi$$
$$350$$ 0 0
$$351$$ 6.67994 0.356549
$$352$$ 2.59345 0.138231
$$353$$ −36.8786 −1.96285 −0.981425 0.191848i $$-0.938552\pi$$
−0.981425 + 0.191848i $$0.938552\pi$$
$$354$$ 1.95646 0.103985
$$355$$ 0 0
$$356$$ −5.65651 −0.299795
$$357$$ −7.18598 −0.380322
$$358$$ −0.534516 −0.0282500
$$359$$ 16.5187 0.871824 0.435912 0.899989i $$-0.356426\pi$$
0.435912 + 0.899989i $$0.356426\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −4.17331 −0.219344
$$363$$ 8.15452 0.428001
$$364$$ −5.10156 −0.267394
$$365$$ 0 0
$$366$$ 0.107523 0.00562034
$$367$$ −6.83148 −0.356600 −0.178300 0.983976i $$-0.557060\pi$$
−0.178300 + 0.983976i $$0.557060\pi$$
$$368$$ 15.4494 0.805353
$$369$$ −9.85325 −0.512940
$$370$$ 0 0
$$371$$ −4.57135 −0.237333
$$372$$ 2.84415 0.147462
$$373$$ −4.76271 −0.246604 −0.123302 0.992369i $$-0.539348\pi$$
−0.123302 + 0.992369i $$0.539348\pi$$
$$374$$ 1.19269 0.0616723
$$375$$ 0 0
$$376$$ −1.99330 −0.102796
$$377$$ 7.78986 0.401198
$$378$$ −1.79523 −0.0923368
$$379$$ 14.3773 0.738514 0.369257 0.929327i $$-0.379612\pi$$
0.369257 + 0.929327i $$0.379612\pi$$
$$380$$ 0 0
$$381$$ 14.3308 0.734190
$$382$$ −1.46309 −0.0748583
$$383$$ −30.6708 −1.56721 −0.783603 0.621261i $$-0.786621\pi$$
−0.783603 + 0.621261i $$0.786621\pi$$
$$384$$ −5.86353 −0.299222
$$385$$ 0 0
$$386$$ 1.09544 0.0557565
$$387$$ −17.2470 −0.876713
$$388$$ −3.01507 −0.153067
$$389$$ −12.9215 −0.655148 −0.327574 0.944825i $$-0.606231\pi$$
−0.327574 + 0.944825i $$0.606231\pi$$
$$390$$ 0 0
$$391$$ 22.4916 1.13745
$$392$$ −4.02475 −0.203281
$$393$$ −5.96854 −0.301073
$$394$$ −4.06829 −0.204958
$$395$$ 0 0
$$396$$ −4.16719 −0.209409
$$397$$ −29.8471 −1.49798 −0.748992 0.662579i $$-0.769462\pi$$
−0.748992 + 0.662579i $$0.769462\pi$$
$$398$$ 6.02954 0.302234
$$399$$ 1.35690 0.0679298
$$400$$ 0 0
$$401$$ −28.7101 −1.43371 −0.716856 0.697221i $$-0.754420\pi$$
−0.716856 + 0.697221i $$0.754420\pi$$
$$402$$ −2.77240 −0.138275
$$403$$ 2.84415 0.141677
$$404$$ 32.1903 1.60153
$$405$$ 0 0
$$406$$ −2.09352 −0.103900
$$407$$ −5.74094 −0.284568
$$408$$ −4.13169 −0.204549
$$409$$ −13.6203 −0.673479 −0.336739 0.941598i $$-0.609324\pi$$
−0.336739 + 0.941598i $$0.609324\pi$$
$$410$$ 0 0
$$411$$ −4.56033 −0.224945
$$412$$ 9.39539 0.462878
$$413$$ 16.7138 0.822432
$$414$$ 2.47219 0.121501
$$415$$ 0 0
$$416$$ −4.42253 −0.216833
$$417$$ −2.89977 −0.142002
$$418$$ −0.225209 −0.0110153
$$419$$ −2.13946 −0.104519 −0.0522596 0.998634i $$-0.516642\pi$$
−0.0522596 + 0.998634i $$0.516642\pi$$
$$420$$ 0 0
$$421$$ −15.0562 −0.733795 −0.366897 0.930261i $$-0.619580\pi$$
−0.366897 + 0.930261i $$0.619580\pi$$
$$422$$ 1.07202 0.0521849
$$423$$ 4.82908 0.234798
$$424$$ −2.62837 −0.127645
$$425$$ 0 0
$$426$$ −2.55065 −0.123579
$$427$$ 0.918559 0.0444522
$$428$$ 8.66115 0.418653
$$429$$ 1.13706 0.0548979
$$430$$ 0 0
$$431$$ 4.37435 0.210705 0.105353 0.994435i $$-0.466403\pi$$
0.105353 + 0.994435i $$0.466403\pi$$
$$432$$ 15.6273 0.751869
$$433$$ −33.1564 −1.59340 −0.796698 0.604377i $$-0.793422\pi$$
−0.796698 + 0.604377i $$0.793422\pi$$
$$434$$ −0.764365 −0.0366907
$$435$$ 0 0
$$436$$ −36.5050 −1.74827
$$437$$ −4.24698 −0.203161
$$438$$ −0.556023 −0.0265678
$$439$$ 32.2368 1.53858 0.769290 0.638900i $$-0.220610\pi$$
0.769290 + 0.638900i $$0.220610\pi$$
$$440$$ 0 0
$$441$$ 9.75063 0.464316
$$442$$ −2.03385 −0.0967405
$$443$$ 21.7362 1.03272 0.516358 0.856373i $$-0.327287\pi$$
0.516358 + 0.856373i $$0.327287\pi$$
$$444$$ 9.78986 0.464606
$$445$$ 0 0
$$446$$ −6.48022 −0.306847
$$447$$ −2.63879 −0.124811
$$448$$ −11.1217 −0.525450
$$449$$ −24.9584 −1.17786 −0.588929 0.808185i $$-0.700450\pi$$
−0.588929 + 0.808185i $$0.700450\pi$$
$$450$$ 0 0
$$451$$ −3.81210 −0.179505
$$452$$ −38.8514 −1.82742
$$453$$ 8.19029 0.384814
$$454$$ −3.62697 −0.170222
$$455$$ 0 0
$$456$$ 0.780167 0.0365347
$$457$$ 13.1347 0.614414 0.307207 0.951643i $$-0.400606\pi$$
0.307207 + 0.951643i $$0.400606\pi$$
$$458$$ 5.34481 0.249747
$$459$$ 22.7506 1.06191
$$460$$ 0 0
$$461$$ −35.3726 −1.64746 −0.823732 0.566979i $$-0.808112\pi$$
−0.823732 + 0.566979i $$0.808112\pi$$
$$462$$ −0.305586 −0.0142171
$$463$$ 14.6963 0.682997 0.341498 0.939882i $$-0.389066\pi$$
0.341498 + 0.939882i $$0.389066\pi$$
$$464$$ 18.2239 0.846022
$$465$$ 0 0
$$466$$ −6.69766 −0.310263
$$467$$ −33.2121 −1.53687 −0.768435 0.639927i $$-0.778964\pi$$
−0.768435 + 0.639927i $$0.778964\pi$$
$$468$$ 7.10620 0.328484
$$469$$ −23.6843 −1.09364
$$470$$ 0 0
$$471$$ −11.5036 −0.530060
$$472$$ 9.60984 0.442329
$$473$$ −6.67264 −0.306809
$$474$$ 0.315748 0.0145028
$$475$$ 0 0
$$476$$ −17.3749 −0.796379
$$477$$ 6.36765 0.291555
$$478$$ 2.84787 0.130259
$$479$$ 24.6219 1.12500 0.562502 0.826796i $$-0.309839\pi$$
0.562502 + 0.826796i $$0.309839\pi$$
$$480$$ 0 0
$$481$$ 9.78986 0.446379
$$482$$ −2.91915 −0.132964
$$483$$ −5.76271 −0.262212
$$484$$ 19.7168 0.896217
$$485$$ 0 0
$$486$$ 3.90110 0.176958
$$487$$ −29.5646 −1.33970 −0.669851 0.742496i $$-0.733642\pi$$
−0.669851 + 0.742496i $$0.733642\pi$$
$$488$$ 0.528139 0.0239077
$$489$$ 15.6625 0.708282
$$490$$ 0 0
$$491$$ 36.2978 1.63810 0.819049 0.573724i $$-0.194502\pi$$
0.819049 + 0.573724i $$0.194502\pi$$
$$492$$ 6.50066 0.293073
$$493$$ 26.5308 1.19489
$$494$$ 0.384043 0.0172789
$$495$$ 0 0
$$496$$ 6.65371 0.298760
$$497$$ −21.7899 −0.977409
$$498$$ 2.42327 0.108589
$$499$$ 7.84415 0.351152 0.175576 0.984466i $$-0.443821\pi$$
0.175576 + 0.984466i $$0.443821\pi$$
$$500$$ 0 0
$$501$$ −9.52888 −0.425719
$$502$$ 2.38703 0.106538
$$503$$ −20.4166 −0.910330 −0.455165 0.890407i $$-0.650420\pi$$
−0.455165 + 0.890407i $$0.650420\pi$$
$$504$$ −3.87966 −0.172814
$$505$$ 0 0
$$506$$ 0.956459 0.0425198
$$507$$ 8.48619 0.376885
$$508$$ 34.6504 1.53736
$$509$$ −14.7530 −0.653916 −0.326958 0.945039i $$-0.606024\pi$$
−0.326958 + 0.945039i $$0.606024\pi$$
$$510$$ 0 0
$$511$$ −4.75004 −0.210129
$$512$$ −17.4242 −0.770048
$$513$$ −4.29590 −0.189668
$$514$$ −3.50365 −0.154539
$$515$$ 0 0
$$516$$ 11.3787 0.500918
$$517$$ 1.86831 0.0821683
$$518$$ −2.63102 −0.115600
$$519$$ −12.4077 −0.544639
$$520$$ 0 0
$$521$$ 37.1487 1.62751 0.813756 0.581206i $$-0.197419\pi$$
0.813756 + 0.581206i $$0.197419\pi$$
$$522$$ 2.91617 0.127637
$$523$$ −16.3623 −0.715472 −0.357736 0.933823i $$-0.616451\pi$$
−0.357736 + 0.933823i $$0.616451\pi$$
$$524$$ −14.4313 −0.630434
$$525$$ 0 0
$$526$$ 5.38271 0.234698
$$527$$ 9.68664 0.421957
$$528$$ 2.66009 0.115765
$$529$$ −4.96316 −0.215790
$$530$$ 0 0
$$531$$ −23.2814 −1.01033
$$532$$ 3.28083 0.142242
$$533$$ 6.50066 0.281575
$$534$$ 0.577793 0.0250036
$$535$$ 0 0
$$536$$ −13.6176 −0.588191
$$537$$ −1.73556 −0.0748950
$$538$$ 6.10693 0.263289
$$539$$ 3.77240 0.162489
$$540$$ 0 0
$$541$$ −2.08947 −0.0898335 −0.0449168 0.998991i $$-0.514302\pi$$
−0.0449168 + 0.998991i $$0.514302\pi$$
$$542$$ 3.26742 0.140348
$$543$$ −13.5506 −0.581514
$$544$$ −15.0623 −0.645792
$$545$$ 0 0
$$546$$ 0.521106 0.0223013
$$547$$ −1.86054 −0.0795511 −0.0397756 0.999209i $$-0.512664\pi$$
−0.0397756 + 0.999209i $$0.512664\pi$$
$$548$$ −11.0264 −0.471025
$$549$$ −1.27950 −0.0546079
$$550$$ 0 0
$$551$$ −5.00969 −0.213420
$$552$$ −3.31336 −0.141026
$$553$$ 2.69740 0.114705
$$554$$ 0.138391 0.00587968
$$555$$ 0 0
$$556$$ −7.01134 −0.297347
$$557$$ 9.80061 0.415265 0.207633 0.978207i $$-0.433424\pi$$
0.207633 + 0.978207i $$0.433424\pi$$
$$558$$ 1.06472 0.0450732
$$559$$ 11.3787 0.481266
$$560$$ 0 0
$$561$$ 3.87263 0.163502
$$562$$ −6.81759 −0.287583
$$563$$ −38.0170 −1.60222 −0.801112 0.598514i $$-0.795758\pi$$
−0.801112 + 0.598514i $$0.795758\pi$$
$$564$$ −3.18598 −0.134154
$$565$$ 0 0
$$566$$ 3.94198 0.165694
$$567$$ 6.13467 0.257632
$$568$$ −12.5284 −0.525680
$$569$$ −7.32975 −0.307279 −0.153640 0.988127i $$-0.549099\pi$$
−0.153640 + 0.988127i $$0.549099\pi$$
$$570$$ 0 0
$$571$$ 37.8775 1.58513 0.792563 0.609791i $$-0.208746\pi$$
0.792563 + 0.609791i $$0.208746\pi$$
$$572$$ 2.74930 0.114954
$$573$$ −4.75063 −0.198460
$$574$$ −1.74705 −0.0729206
$$575$$ 0 0
$$576$$ 15.4919 0.645496
$$577$$ −28.6993 −1.19477 −0.597384 0.801955i $$-0.703793\pi$$
−0.597384 + 0.801955i $$0.703793\pi$$
$$578$$ −2.72827 −0.113481
$$579$$ 3.55688 0.147819
$$580$$ 0 0
$$581$$ 20.7017 0.858852
$$582$$ 0.307979 0.0127661
$$583$$ 2.46357 0.102030
$$584$$ −2.73110 −0.113014
$$585$$ 0 0
$$586$$ −6.25667 −0.258461
$$587$$ −3.72348 −0.153684 −0.0768422 0.997043i $$-0.524484\pi$$
−0.0768422 + 0.997043i $$0.524484\pi$$
$$588$$ −6.43296 −0.265291
$$589$$ −1.82908 −0.0753661
$$590$$ 0 0
$$591$$ −13.2097 −0.543373
$$592$$ 22.9028 0.941297
$$593$$ −27.7399 −1.13914 −0.569570 0.821943i $$-0.692890\pi$$
−0.569570 + 0.821943i $$0.692890\pi$$
$$594$$ 0.967476 0.0396960
$$595$$ 0 0
$$596$$ −6.38032 −0.261348
$$597$$ 19.5778 0.801266
$$598$$ −1.63102 −0.0666975
$$599$$ −23.5579 −0.962551 −0.481276 0.876569i $$-0.659826\pi$$
−0.481276 + 0.876569i $$0.659826\pi$$
$$600$$ 0 0
$$601$$ −7.36898 −0.300587 −0.150293 0.988641i $$-0.548022\pi$$
−0.150293 + 0.988641i $$0.548022\pi$$
$$602$$ −3.05802 −0.124635
$$603$$ 32.9909 1.34349
$$604$$ 19.8033 0.805783
$$605$$ 0 0
$$606$$ −3.28813 −0.133571
$$607$$ 28.4198 1.15352 0.576762 0.816912i $$-0.304316\pi$$
0.576762 + 0.816912i $$0.304316\pi$$
$$608$$ 2.84415 0.115346
$$609$$ −6.79763 −0.275454
$$610$$ 0 0
$$611$$ −3.18598 −0.128891
$$612$$ 24.2024 0.978323
$$613$$ 6.48129 0.261777 0.130888 0.991397i $$-0.458217\pi$$
0.130888 + 0.991397i $$0.458217\pi$$
$$614$$ 2.85443 0.115195
$$615$$ 0 0
$$616$$ −1.50099 −0.0604767
$$617$$ −24.4650 −0.984924 −0.492462 0.870334i $$-0.663903\pi$$
−0.492462 + 0.870334i $$0.663903\pi$$
$$618$$ −0.959706 −0.0386051
$$619$$ −22.2457 −0.894128 −0.447064 0.894502i $$-0.647530\pi$$
−0.447064 + 0.894502i $$0.647530\pi$$
$$620$$ 0 0
$$621$$ 18.2446 0.732130
$$622$$ 4.53020 0.181645
$$623$$ 4.93602 0.197757
$$624$$ −4.53617 −0.181592
$$625$$ 0 0
$$626$$ 4.67084 0.186684
$$627$$ −0.731250 −0.0292033
$$628$$ −27.8146 −1.10992
$$629$$ 33.3424 1.32945
$$630$$ 0 0
$$631$$ −15.5888 −0.620581 −0.310290 0.950642i $$-0.600426\pi$$
−0.310290 + 0.950642i $$0.600426\pi$$
$$632$$ 1.55091 0.0616919
$$633$$ 3.48081 0.138350
$$634$$ −5.00836 −0.198907
$$635$$ 0 0
$$636$$ −4.20105 −0.166582
$$637$$ −6.43296 −0.254883
$$638$$ 1.12823 0.0446670
$$639$$ 30.3521 1.20071
$$640$$ 0 0
$$641$$ 8.17496 0.322892 0.161446 0.986882i $$-0.448384\pi$$
0.161446 + 0.986882i $$0.448384\pi$$
$$642$$ −0.884707 −0.0349166
$$643$$ 11.1836 0.441038 0.220519 0.975383i $$-0.429225\pi$$
0.220519 + 0.975383i $$0.429225\pi$$
$$644$$ −13.9336 −0.549062
$$645$$ 0 0
$$646$$ 1.30798 0.0514617
$$647$$ 16.2121 0.637362 0.318681 0.947862i $$-0.396760\pi$$
0.318681 + 0.947862i $$0.396760\pi$$
$$648$$ 3.52722 0.138562
$$649$$ −9.00730 −0.353567
$$650$$ 0 0
$$651$$ −2.48188 −0.0972725
$$652$$ 37.8702 1.48311
$$653$$ 24.7952 0.970312 0.485156 0.874427i $$-0.338763\pi$$
0.485156 + 0.874427i $$0.338763\pi$$
$$654$$ 3.72886 0.145810
$$655$$ 0 0
$$656$$ 15.2079 0.593769
$$657$$ 6.61655 0.258136
$$658$$ 0.856232 0.0333794
$$659$$ −20.2868 −0.790262 −0.395131 0.918625i $$-0.629301\pi$$
−0.395131 + 0.918625i $$0.629301\pi$$
$$660$$ 0 0
$$661$$ 20.4222 0.794332 0.397166 0.917747i $$-0.369994\pi$$
0.397166 + 0.917747i $$0.369994\pi$$
$$662$$ 1.17928 0.0458339
$$663$$ −6.60388 −0.256473
$$664$$ 11.9028 0.461917
$$665$$ 0 0
$$666$$ 3.66487 0.142011
$$667$$ 21.2760 0.823812
$$668$$ −23.0398 −0.891437
$$669$$ −21.0411 −0.813498
$$670$$ 0 0
$$671$$ −0.495024 −0.0191102
$$672$$ 3.85922 0.148872
$$673$$ 28.7254 1.10728 0.553641 0.832755i $$-0.313238\pi$$
0.553641 + 0.832755i $$0.313238\pi$$
$$674$$ −6.02549 −0.232093
$$675$$ 0 0
$$676$$ 20.5187 0.789181
$$677$$ 44.0062 1.69130 0.845648 0.533740i $$-0.179214\pi$$
0.845648 + 0.533740i $$0.179214\pi$$
$$678$$ 3.96854 0.152411
$$679$$ 2.63102 0.100969
$$680$$ 0 0
$$681$$ −11.7767 −0.451284
$$682$$ 0.411927 0.0157735
$$683$$ 8.48427 0.324642 0.162321 0.986738i $$-0.448102\pi$$
0.162321 + 0.986738i $$0.448102\pi$$
$$684$$ −4.57002 −0.174739
$$685$$ 0 0
$$686$$ 4.65412 0.177695
$$687$$ 17.3545 0.662116
$$688$$ 26.6197 1.01487
$$689$$ −4.20105 −0.160047
$$690$$ 0 0
$$691$$ 20.2586 0.770673 0.385336 0.922776i $$-0.374085\pi$$
0.385336 + 0.922776i $$0.374085\pi$$
$$692$$ −30.0006 −1.14045
$$693$$ 3.63640 0.138135
$$694$$ 2.46921 0.0937297
$$695$$ 0 0
$$696$$ −3.90840 −0.148147
$$697$$ 22.1400 0.838614
$$698$$ −5.42327 −0.205274
$$699$$ −21.7472 −0.822553
$$700$$ 0 0
$$701$$ −23.4101 −0.884188 −0.442094 0.896969i $$-0.645764\pi$$
−0.442094 + 0.896969i $$0.645764\pi$$
$$702$$ −1.64981 −0.0622680
$$703$$ −6.29590 −0.237454
$$704$$ 5.99362 0.225893
$$705$$ 0 0
$$706$$ 9.10826 0.342794
$$707$$ −28.0901 −1.05644
$$708$$ 15.3599 0.577260
$$709$$ 30.3472 1.13971 0.569857 0.821744i $$-0.306999\pi$$
0.569857 + 0.821744i $$0.306999\pi$$
$$710$$ 0 0
$$711$$ −3.75733 −0.140911
$$712$$ 2.83804 0.106360
$$713$$ 7.76809 0.290917
$$714$$ 1.77479 0.0664199
$$715$$ 0 0
$$716$$ −4.19641 −0.156827
$$717$$ 9.24698 0.345335
$$718$$ −4.07979 −0.152256
$$719$$ −38.3230 −1.42921 −0.714604 0.699529i $$-0.753393\pi$$
−0.714604 + 0.699529i $$0.753393\pi$$
$$720$$ 0 0
$$721$$ −8.19865 −0.305334
$$722$$ −0.246980 −0.00919163
$$723$$ −9.47842 −0.352506
$$724$$ −32.7640 −1.21767
$$725$$ 0 0
$$726$$ −2.01400 −0.0747466
$$727$$ 2.69069 0.0997923 0.0498961 0.998754i $$-0.484111\pi$$
0.0498961 + 0.998754i $$0.484111\pi$$
$$728$$ 2.55960 0.0948650
$$729$$ 1.78986 0.0662910
$$730$$ 0 0
$$731$$ 38.7536 1.43335
$$732$$ 0.844150 0.0312007
$$733$$ 18.9651 0.700491 0.350246 0.936658i $$-0.386098\pi$$
0.350246 + 0.936658i $$0.386098\pi$$
$$734$$ 1.68724 0.0622770
$$735$$ 0 0
$$736$$ −12.0790 −0.445240
$$737$$ 12.7638 0.470160
$$738$$ 2.43355 0.0895803
$$739$$ 29.8278 1.09723 0.548616 0.836075i $$-0.315155\pi$$
0.548616 + 0.836075i $$0.315155\pi$$
$$740$$ 0 0
$$741$$ 1.24698 0.0458089
$$742$$ 1.12903 0.0414480
$$743$$ 8.78448 0.322271 0.161136 0.986932i $$-0.448484\pi$$
0.161136 + 0.986932i $$0.448484\pi$$
$$744$$ −1.42699 −0.0523161
$$745$$ 0 0
$$746$$ 1.17629 0.0430671
$$747$$ −28.8364 −1.05507
$$748$$ 9.36360 0.342367
$$749$$ −7.55794 −0.276161
$$750$$ 0 0
$$751$$ −18.1142 −0.660998 −0.330499 0.943806i $$-0.607217\pi$$
−0.330499 + 0.943806i $$0.607217\pi$$
$$752$$ −7.45340 −0.271798
$$753$$ 7.75063 0.282449
$$754$$ −1.92394 −0.0700656
$$755$$ 0 0
$$756$$ −14.0941 −0.512598
$$757$$ −0.222816 −0.00809840 −0.00404920 0.999992i $$-0.501289\pi$$
−0.00404920 + 0.999992i $$0.501289\pi$$
$$758$$ −3.55091 −0.128975
$$759$$ 3.10560 0.112726
$$760$$ 0 0
$$761$$ −6.07798 −0.220327 −0.110163 0.993913i $$-0.535137\pi$$
−0.110163 + 0.993913i $$0.535137\pi$$
$$762$$ −3.53942 −0.128220
$$763$$ 31.8552 1.15323
$$764$$ −11.4865 −0.415568
$$765$$ 0 0
$$766$$ 7.57507 0.273698
$$767$$ 15.3599 0.554613
$$768$$ −9.09411 −0.328156
$$769$$ −14.6267 −0.527453 −0.263726 0.964598i $$-0.584952\pi$$
−0.263726 + 0.964598i $$0.584952\pi$$
$$770$$ 0 0
$$771$$ −11.3763 −0.409706
$$772$$ 8.60015 0.309526
$$773$$ −4.05728 −0.145930 −0.0729651 0.997334i $$-0.523246\pi$$
−0.0729651 + 0.997334i $$0.523246\pi$$
$$774$$ 4.25965 0.153110
$$775$$ 0 0
$$776$$ 1.51275 0.0543044
$$777$$ −8.54288 −0.306474
$$778$$ 3.19136 0.114416
$$779$$ −4.18060 −0.149786
$$780$$ 0 0
$$781$$ 11.7429 0.420192
$$782$$ −5.55496 −0.198645
$$783$$ 21.5211 0.769102
$$784$$ −15.0495 −0.537482
$$785$$ 0 0
$$786$$ 1.47411 0.0525797
$$787$$ 19.5657 0.697442 0.348721 0.937227i $$-0.386616\pi$$
0.348721 + 0.937227i $$0.386616\pi$$
$$788$$ −31.9396 −1.13780
$$789$$ 17.4776 0.622218
$$790$$ 0 0
$$791$$ 33.9028 1.20544
$$792$$ 2.09080 0.0742934
$$793$$ 0.844150 0.0299767
$$794$$ 7.37163 0.261609
$$795$$ 0 0
$$796$$ 47.3370 1.67782
$$797$$ 38.9051 1.37809 0.689046 0.724718i $$-0.258030\pi$$
0.689046 + 0.724718i $$0.258030\pi$$
$$798$$ −0.335126 −0.0118633
$$799$$ −10.8509 −0.383876
$$800$$ 0 0
$$801$$ −6.87561 −0.242938
$$802$$ 7.09080 0.250385
$$803$$ 2.55986 0.0903355
$$804$$ −21.7657 −0.767617
$$805$$ 0 0
$$806$$ −0.702447 −0.0247426
$$807$$ 19.8291 0.698017
$$808$$ −16.1508 −0.568183
$$809$$ −22.5730 −0.793625 −0.396812 0.917900i $$-0.629884\pi$$
−0.396812 + 0.917900i $$0.629884\pi$$
$$810$$ 0 0
$$811$$ −46.3605 −1.62794 −0.813968 0.580909i $$-0.802697\pi$$
−0.813968 + 0.580909i $$0.802697\pi$$
$$812$$ −16.4359 −0.576789
$$813$$ 10.6093 0.372083
$$814$$ 1.41789 0.0496972
$$815$$ 0 0
$$816$$ −15.4494 −0.540836
$$817$$ −7.31767 −0.256013
$$818$$ 3.36393 0.117617
$$819$$ −6.20105 −0.216682
$$820$$ 0 0
$$821$$ −17.8194 −0.621901 −0.310951 0.950426i $$-0.600647\pi$$
−0.310951 + 0.950426i $$0.600647\pi$$
$$822$$ 1.12631 0.0392846
$$823$$ −32.5394 −1.13425 −0.567126 0.823631i $$-0.691945\pi$$
−0.567126 + 0.823631i $$0.691945\pi$$
$$824$$ −4.71394 −0.164218
$$825$$ 0 0
$$826$$ −4.12797 −0.143630
$$827$$ 39.8256 1.38487 0.692436 0.721479i $$-0.256537\pi$$
0.692436 + 0.721479i $$0.256537\pi$$
$$828$$ 19.4088 0.674502
$$829$$ 38.7525 1.34593 0.672966 0.739674i $$-0.265020\pi$$
0.672966 + 0.739674i $$0.265020\pi$$
$$830$$ 0 0
$$831$$ 0.449354 0.0155879
$$832$$ −10.2208 −0.354341
$$833$$ −21.9095 −0.759118
$$834$$ 0.716185 0.0247994
$$835$$ 0 0
$$836$$ −1.76809 −0.0611505
$$837$$ 7.85756 0.271597
$$838$$ 0.528402 0.0182533
$$839$$ 2.76749 0.0955445 0.0477723 0.998858i $$-0.484788\pi$$
0.0477723 + 0.998858i $$0.484788\pi$$
$$840$$ 0 0
$$841$$ −3.90302 −0.134587
$$842$$ 3.71858 0.128151
$$843$$ −22.1366 −0.762425
$$844$$ 8.41624 0.289699
$$845$$ 0 0
$$846$$ −1.19269 −0.0410054
$$847$$ −17.2054 −0.591183
$$848$$ −9.82808 −0.337498
$$849$$ 12.7995 0.439279
$$850$$ 0 0
$$851$$ 26.7385 0.916586
$$852$$ −20.0248 −0.686037
$$853$$ 24.1390 0.826503 0.413252 0.910617i $$-0.364393\pi$$
0.413252 + 0.910617i $$0.364393\pi$$
$$854$$ −0.226865 −0.00776317
$$855$$ 0 0
$$856$$ −4.34555 −0.148528
$$857$$ −3.16229 −0.108022 −0.0540109 0.998540i $$-0.517201\pi$$
−0.0540109 + 0.998540i $$0.517201\pi$$
$$858$$ −0.280831 −0.00958743
$$859$$ 4.86426 0.165967 0.0829833 0.996551i $$-0.473555\pi$$
0.0829833 + 0.996551i $$0.473555\pi$$
$$860$$ 0 0
$$861$$ −5.67264 −0.193323
$$862$$ −1.08038 −0.0367978
$$863$$ 4.54958 0.154870 0.0774348 0.996997i $$-0.475327\pi$$
0.0774348 + 0.996997i $$0.475327\pi$$
$$864$$ −12.2182 −0.415671
$$865$$ 0 0
$$866$$ 8.18896 0.278272
$$867$$ −8.85862 −0.300855
$$868$$ −6.00092 −0.203684
$$869$$ −1.45367 −0.0493122
$$870$$ 0 0
$$871$$ −21.7657 −0.737502
$$872$$ 18.3156 0.620245
$$873$$ −3.66487 −0.124037
$$874$$ 1.04892 0.0354802
$$875$$ 0 0
$$876$$ −4.36526 −0.147488
$$877$$ −18.6595 −0.630086 −0.315043 0.949077i $$-0.602019\pi$$
−0.315043 + 0.949077i $$0.602019\pi$$
$$878$$ −7.96184 −0.268699
$$879$$ −20.3153 −0.685217
$$880$$ 0 0
$$881$$ 19.4306 0.654632 0.327316 0.944915i $$-0.393856\pi$$
0.327316 + 0.944915i $$0.393856\pi$$
$$882$$ −2.40821 −0.0810885
$$883$$ 25.6437 0.862979 0.431490 0.902118i $$-0.357988\pi$$
0.431490 + 0.902118i $$0.357988\pi$$
$$884$$ −15.9675 −0.537044
$$885$$ 0 0
$$886$$ −5.36839 −0.180354
$$887$$ −31.0062 −1.04109 −0.520544 0.853835i $$-0.674271\pi$$
−0.520544 + 0.853835i $$0.674271\pi$$
$$888$$ −4.91185 −0.164831
$$889$$ −30.2368 −1.01411
$$890$$ 0 0
$$891$$ −3.30606 −0.110757
$$892$$ −50.8753 −1.70343
$$893$$ 2.04892 0.0685644
$$894$$ 0.651728 0.0217970
$$895$$ 0 0
$$896$$ 12.3716 0.413305
$$897$$ −5.29590 −0.176825
$$898$$ 6.16421 0.205702
$$899$$ 9.16315 0.305608
$$900$$ 0 0
$$901$$ −14.3080 −0.476668
$$902$$ 0.941511 0.0313489
$$903$$ −9.92931 −0.330427
$$904$$ 19.4929 0.648324
$$905$$ 0 0
$$906$$ −2.02284 −0.0672042
$$907$$ −17.7676 −0.589964 −0.294982 0.955503i $$-0.595314\pi$$
−0.294982 + 0.955503i $$0.595314\pi$$
$$908$$ −28.4748 −0.944971
$$909$$ 39.1280 1.29779
$$910$$ 0 0
$$911$$ 41.7313 1.38262 0.691309 0.722559i $$-0.257034\pi$$
0.691309 + 0.722559i $$0.257034\pi$$
$$912$$ 2.91723 0.0965992
$$913$$ −11.1564 −0.369224
$$914$$ −3.24400 −0.107302
$$915$$ 0 0
$$916$$ 41.9614 1.38644
$$917$$ 12.5931 0.415862
$$918$$ −5.61894 −0.185453
$$919$$ 30.4088 1.00309 0.501547 0.865130i $$-0.332764\pi$$
0.501547 + 0.865130i $$0.332764\pi$$
$$920$$ 0 0
$$921$$ 9.26828 0.305400
$$922$$ 8.73630 0.287715
$$923$$ −20.0248 −0.659123
$$924$$ −2.39911 −0.0789249
$$925$$ 0 0
$$926$$ −3.62969 −0.119279
$$927$$ 11.4203 0.375091
$$928$$ −14.2483 −0.467724
$$929$$ −28.8219 −0.945616 −0.472808 0.881165i $$-0.656760\pi$$
−0.472808 + 0.881165i $$0.656760\pi$$
$$930$$ 0 0
$$931$$ 4.13706 0.135587
$$932$$ −52.5824 −1.72239
$$933$$ 14.7095 0.481567
$$934$$ 8.20270 0.268401
$$935$$ 0 0
$$936$$ −3.56538 −0.116538
$$937$$ −50.4601 −1.64846 −0.824230 0.566255i $$-0.808392\pi$$
−0.824230 + 0.566255i $$0.808392\pi$$
$$938$$ 5.84953 0.190994
$$939$$ 15.1661 0.494928
$$940$$ 0 0
$$941$$ 1.10082 0.0358857 0.0179428 0.999839i $$-0.494288\pi$$
0.0179428 + 0.999839i $$0.494288\pi$$
$$942$$ 2.84117 0.0925702
$$943$$ 17.7549 0.578180
$$944$$ 35.9335 1.16954
$$945$$ 0 0
$$946$$ 1.64801 0.0535813
$$947$$ −13.9353 −0.452836 −0.226418 0.974030i $$-0.572701\pi$$
−0.226418 + 0.974030i $$0.572701\pi$$
$$948$$ 2.47889 0.0805107
$$949$$ −4.36526 −0.141702
$$950$$ 0 0
$$951$$ −16.2620 −0.527333
$$952$$ 8.71751 0.282536
$$953$$ −41.3400 −1.33913 −0.669567 0.742751i $$-0.733520\pi$$
−0.669567 + 0.742751i $$0.733520\pi$$
$$954$$ −1.57268 −0.0509174
$$955$$ 0 0
$$956$$ 22.3582 0.723117
$$957$$ 3.66334 0.118419
$$958$$ −6.08111 −0.196472
$$959$$ 9.62192 0.310708
$$960$$ 0 0
$$961$$ −27.6544 −0.892079
$$962$$ −2.41789 −0.0779561
$$963$$ 10.5278 0.339254
$$964$$ −22.9178 −0.738133
$$965$$ 0 0
$$966$$ 1.42327 0.0457930
$$967$$ −5.26875 −0.169432 −0.0847158 0.996405i $$-0.526998\pi$$
−0.0847158 + 0.996405i $$0.526998\pi$$
$$968$$ −9.89248 −0.317956
$$969$$ 4.24698 0.136433
$$970$$ 0 0
$$971$$ −5.15346 −0.165382 −0.0826911 0.996575i $$-0.526351\pi$$
−0.0826911 + 0.996575i $$0.526351\pi$$
$$972$$ 30.6270 0.982361
$$973$$ 6.11828 0.196143
$$974$$ 7.30186 0.233967
$$975$$ 0 0
$$976$$ 1.97484 0.0632130
$$977$$ 4.77612 0.152802 0.0764008 0.997077i $$-0.475657\pi$$
0.0764008 + 0.997077i $$0.475657\pi$$
$$978$$ −3.86831 −0.123695
$$979$$ −2.66009 −0.0850168
$$980$$ 0 0
$$981$$ −44.3726 −1.41671
$$982$$ −8.96482 −0.286079
$$983$$ −28.9758 −0.924186 −0.462093 0.886832i $$-0.652901\pi$$
−0.462093 + 0.886832i $$0.652901\pi$$
$$984$$ −3.26157 −0.103975
$$985$$ 0 0
$$986$$ −6.55257 −0.208676
$$987$$ 2.78017 0.0884937
$$988$$ 3.01507 0.0959220
$$989$$ 31.0780 0.988222
$$990$$ 0 0
$$991$$ −38.5042 −1.22313 −0.611564 0.791195i $$-0.709459\pi$$
−0.611564 + 0.791195i $$0.709459\pi$$
$$992$$ −5.20219 −0.165170
$$993$$ 3.82908 0.121512
$$994$$ 5.38165 0.170696
$$995$$ 0 0
$$996$$ 19.0248 0.602822
$$997$$ 10.1491 0.321427 0.160713 0.987001i $$-0.448621\pi$$
0.160713 + 0.987001i $$0.448621\pi$$
$$998$$ −1.93735 −0.0613256
$$999$$ 27.0465 0.855714
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.h.1.1 yes 3
3.2 odd 2 4275.2.a.z.1.3 3
4.3 odd 2 7600.2.a.bn.1.3 3
5.2 odd 4 475.2.b.c.324.3 6
5.3 odd 4 475.2.b.c.324.4 6
5.4 even 2 475.2.a.d.1.3 3
15.14 odd 2 4275.2.a.bn.1.1 3
19.18 odd 2 9025.2.a.w.1.3 3
20.19 odd 2 7600.2.a.bw.1.1 3
95.94 odd 2 9025.2.a.be.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.3 3 5.4 even 2
475.2.a.h.1.1 yes 3 1.1 even 1 trivial
475.2.b.c.324.3 6 5.2 odd 4
475.2.b.c.324.4 6 5.3 odd 4
4275.2.a.z.1.3 3 3.2 odd 2
4275.2.a.bn.1.1 3 15.14 odd 2
7600.2.a.bn.1.3 3 4.3 odd 2
7600.2.a.bw.1.1 3 20.19 odd 2
9025.2.a.w.1.3 3 19.18 odd 2
9025.2.a.be.1.1 3 95.94 odd 2