# Properties

 Label 175.2.a.e.1.2 Level $175$ Weight $2$ Character 175.1 Self dual yes Analytic conductor $1.397$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,2,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^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 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} +1.23607 q^{3} +0.618034 q^{4} +2.00000 q^{6} +1.00000 q^{7} -2.23607 q^{8} -1.47214 q^{9} +O(q^{10})$$ $$q+1.61803 q^{2} +1.23607 q^{3} +0.618034 q^{4} +2.00000 q^{6} +1.00000 q^{7} -2.23607 q^{8} -1.47214 q^{9} +4.23607 q^{11} +0.763932 q^{12} -3.23607 q^{13} +1.61803 q^{14} -4.85410 q^{16} -6.47214 q^{17} -2.38197 q^{18} +4.47214 q^{19} +1.23607 q^{21} +6.85410 q^{22} +1.76393 q^{23} -2.76393 q^{24} -5.23607 q^{26} -5.52786 q^{27} +0.618034 q^{28} +5.00000 q^{29} -9.70820 q^{31} -3.38197 q^{32} +5.23607 q^{33} -10.4721 q^{34} -0.909830 q^{36} +3.00000 q^{37} +7.23607 q^{38} -4.00000 q^{39} +9.23607 q^{41} +2.00000 q^{42} +6.23607 q^{43} +2.61803 q^{44} +2.85410 q^{46} -2.00000 q^{47} -6.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} -2.00000 q^{52} -0.472136 q^{53} -8.94427 q^{54} -2.23607 q^{56} +5.52786 q^{57} +8.09017 q^{58} -1.70820 q^{59} +3.70820 q^{61} -15.7082 q^{62} -1.47214 q^{63} +4.23607 q^{64} +8.47214 q^{66} +0.236068 q^{67} -4.00000 q^{68} +2.18034 q^{69} -4.70820 q^{71} +3.29180 q^{72} -13.2361 q^{73} +4.85410 q^{74} +2.76393 q^{76} +4.23607 q^{77} -6.47214 q^{78} +11.1803 q^{79} -2.41641 q^{81} +14.9443 q^{82} +5.70820 q^{83} +0.763932 q^{84} +10.0902 q^{86} +6.18034 q^{87} -9.47214 q^{88} +12.7639 q^{89} -3.23607 q^{91} +1.09017 q^{92} -12.0000 q^{93} -3.23607 q^{94} -4.18034 q^{96} +0.763932 q^{97} +1.61803 q^{98} -6.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + q^2 - 2 * q^3 - q^4 + 4 * q^6 + 2 * q^7 + 6 * q^9 $$2 q + q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 2 q^{7} + 6 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{18} - 2 q^{21} + 7 q^{22} + 8 q^{23} - 10 q^{24} - 6 q^{26} - 20 q^{27} - q^{28} + 10 q^{29} - 6 q^{31} - 9 q^{32} + 6 q^{33} - 12 q^{34} - 13 q^{36} + 6 q^{37} + 10 q^{38} - 8 q^{39} + 14 q^{41} + 4 q^{42} + 8 q^{43} + 3 q^{44} - q^{46} - 4 q^{47} - 12 q^{48} + 2 q^{49} - 16 q^{51} - 4 q^{52} + 8 q^{53} + 20 q^{57} + 5 q^{58} + 10 q^{59} - 6 q^{61} - 18 q^{62} + 6 q^{63} + 4 q^{64} + 8 q^{66} - 4 q^{67} - 8 q^{68} - 18 q^{69} + 4 q^{71} + 20 q^{72} - 22 q^{73} + 3 q^{74} + 10 q^{76} + 4 q^{77} - 4 q^{78} + 22 q^{81} + 12 q^{82} - 2 q^{83} + 6 q^{84} + 9 q^{86} - 10 q^{87} - 10 q^{88} + 30 q^{89} - 2 q^{91} - 9 q^{92} - 24 q^{93} - 2 q^{94} + 14 q^{96} + 6 q^{97} + q^{98} - 8 q^{99}+O(q^{100})$$ 2 * q + q^2 - 2 * q^3 - q^4 + 4 * q^6 + 2 * q^7 + 6 * q^9 + 4 * q^11 + 6 * q^12 - 2 * q^13 + q^14 - 3 * q^16 - 4 * q^17 - 7 * q^18 - 2 * q^21 + 7 * q^22 + 8 * q^23 - 10 * q^24 - 6 * q^26 - 20 * q^27 - q^28 + 10 * q^29 - 6 * q^31 - 9 * q^32 + 6 * q^33 - 12 * q^34 - 13 * q^36 + 6 * q^37 + 10 * q^38 - 8 * q^39 + 14 * q^41 + 4 * q^42 + 8 * q^43 + 3 * q^44 - q^46 - 4 * q^47 - 12 * q^48 + 2 * q^49 - 16 * q^51 - 4 * q^52 + 8 * q^53 + 20 * q^57 + 5 * q^58 + 10 * q^59 - 6 * q^61 - 18 * q^62 + 6 * q^63 + 4 * q^64 + 8 * q^66 - 4 * q^67 - 8 * q^68 - 18 * q^69 + 4 * q^71 + 20 * q^72 - 22 * q^73 + 3 * q^74 + 10 * q^76 + 4 * q^77 - 4 * q^78 + 22 * q^81 + 12 * q^82 - 2 * q^83 + 6 * q^84 + 9 * q^86 - 10 * q^87 - 10 * q^88 + 30 * q^89 - 2 * q^91 - 9 * q^92 - 24 * q^93 - 2 * q^94 + 14 * q^96 + 6 * q^97 + q^98 - 8 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803 1.14412 0.572061 0.820211i $$-0.306144\pi$$
0.572061 + 0.820211i $$0.306144\pi$$
$$3$$ 1.23607 0.713644 0.356822 0.934172i $$-0.383860\pi$$
0.356822 + 0.934172i $$0.383860\pi$$
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 1.00000 0.377964
$$8$$ −2.23607 −0.790569
$$9$$ −1.47214 −0.490712
$$10$$ 0 0
$$11$$ 4.23607 1.27722 0.638611 0.769529i $$-0.279509\pi$$
0.638611 + 0.769529i $$0.279509\pi$$
$$12$$ 0.763932 0.220528
$$13$$ −3.23607 −0.897524 −0.448762 0.893651i $$-0.648135\pi$$
−0.448762 + 0.893651i $$0.648135\pi$$
$$14$$ 1.61803 0.432438
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ −6.47214 −1.56972 −0.784862 0.619671i $$-0.787266\pi$$
−0.784862 + 0.619671i $$0.787266\pi$$
$$18$$ −2.38197 −0.561435
$$19$$ 4.47214 1.02598 0.512989 0.858395i $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ 0 0
$$21$$ 1.23607 0.269732
$$22$$ 6.85410 1.46130
$$23$$ 1.76393 0.367805 0.183903 0.982944i $$-0.441127\pi$$
0.183903 + 0.982944i $$0.441127\pi$$
$$24$$ −2.76393 −0.564185
$$25$$ 0 0
$$26$$ −5.23607 −1.02688
$$27$$ −5.52786 −1.06384
$$28$$ 0.618034 0.116797
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −9.70820 −1.74364 −0.871822 0.489822i $$-0.837062\pi$$
−0.871822 + 0.489822i $$0.837062\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ 5.23607 0.911482
$$34$$ −10.4721 −1.79596
$$35$$ 0 0
$$36$$ −0.909830 −0.151638
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 7.23607 1.17385
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 9.23607 1.44243 0.721216 0.692711i $$-0.243584\pi$$
0.721216 + 0.692711i $$0.243584\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 6.23607 0.950991 0.475496 0.879718i $$-0.342269\pi$$
0.475496 + 0.879718i $$0.342269\pi$$
$$44$$ 2.61803 0.394683
$$45$$ 0 0
$$46$$ 2.85410 0.420814
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ −6.00000 −0.866025
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ −2.00000 −0.277350
$$53$$ −0.472136 −0.0648529 −0.0324264 0.999474i $$-0.510323\pi$$
−0.0324264 + 0.999474i $$0.510323\pi$$
$$54$$ −8.94427 −1.21716
$$55$$ 0 0
$$56$$ −2.23607 −0.298807
$$57$$ 5.52786 0.732183
$$58$$ 8.09017 1.06229
$$59$$ −1.70820 −0.222389 −0.111195 0.993799i $$-0.535468\pi$$
−0.111195 + 0.993799i $$0.535468\pi$$
$$60$$ 0 0
$$61$$ 3.70820 0.474787 0.237393 0.971414i $$-0.423707\pi$$
0.237393 + 0.971414i $$0.423707\pi$$
$$62$$ −15.7082 −1.99494
$$63$$ −1.47214 −0.185472
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 8.47214 1.04285
$$67$$ 0.236068 0.0288403 0.0144201 0.999896i $$-0.495410\pi$$
0.0144201 + 0.999896i $$0.495410\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 2.18034 0.262482
$$70$$ 0 0
$$71$$ −4.70820 −0.558761 −0.279381 0.960180i $$-0.590129\pi$$
−0.279381 + 0.960180i $$0.590129\pi$$
$$72$$ 3.29180 0.387942
$$73$$ −13.2361 −1.54916 −0.774582 0.632473i $$-0.782040\pi$$
−0.774582 + 0.632473i $$0.782040\pi$$
$$74$$ 4.85410 0.564278
$$75$$ 0 0
$$76$$ 2.76393 0.317045
$$77$$ 4.23607 0.482745
$$78$$ −6.47214 −0.732825
$$79$$ 11.1803 1.25789 0.628943 0.777451i $$-0.283488\pi$$
0.628943 + 0.777451i $$0.283488\pi$$
$$80$$ 0 0
$$81$$ −2.41641 −0.268490
$$82$$ 14.9443 1.65032
$$83$$ 5.70820 0.626557 0.313278 0.949661i $$-0.398573\pi$$
0.313278 + 0.949661i $$0.398573\pi$$
$$84$$ 0.763932 0.0833518
$$85$$ 0 0
$$86$$ 10.0902 1.08805
$$87$$ 6.18034 0.662602
$$88$$ −9.47214 −1.00973
$$89$$ 12.7639 1.35297 0.676487 0.736455i $$-0.263501\pi$$
0.676487 + 0.736455i $$0.263501\pi$$
$$90$$ 0 0
$$91$$ −3.23607 −0.339232
$$92$$ 1.09017 0.113658
$$93$$ −12.0000 −1.24434
$$94$$ −3.23607 −0.333775
$$95$$ 0 0
$$96$$ −4.18034 −0.426654
$$97$$ 0.763932 0.0775655 0.0387828 0.999248i $$-0.487652\pi$$
0.0387828 + 0.999248i $$0.487652\pi$$
$$98$$ 1.61803 0.163446
$$99$$ −6.23607 −0.626748
$$100$$ 0 0
$$101$$ 9.23607 0.919023 0.459512 0.888172i $$-0.348024\pi$$
0.459512 + 0.888172i $$0.348024\pi$$
$$102$$ −12.9443 −1.28167
$$103$$ −0.472136 −0.0465209 −0.0232605 0.999729i $$-0.507405\pi$$
−0.0232605 + 0.999729i $$0.507405\pi$$
$$104$$ 7.23607 0.709555
$$105$$ 0 0
$$106$$ −0.763932 −0.0741996
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ −3.41641 −0.328744
$$109$$ −18.4164 −1.76397 −0.881986 0.471276i $$-0.843794\pi$$
−0.881986 + 0.471276i $$0.843794\pi$$
$$110$$ 0 0
$$111$$ 3.70820 0.351967
$$112$$ −4.85410 −0.458670
$$113$$ 12.4164 1.16804 0.584019 0.811740i $$-0.301479\pi$$
0.584019 + 0.811740i $$0.301479\pi$$
$$114$$ 8.94427 0.837708
$$115$$ 0 0
$$116$$ 3.09017 0.286915
$$117$$ 4.76393 0.440426
$$118$$ −2.76393 −0.254441
$$119$$ −6.47214 −0.593300
$$120$$ 0 0
$$121$$ 6.94427 0.631297
$$122$$ 6.00000 0.543214
$$123$$ 11.4164 1.02938
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ −2.38197 −0.212202
$$127$$ −17.6525 −1.56640 −0.783202 0.621767i $$-0.786415\pi$$
−0.783202 + 0.621767i $$0.786415\pi$$
$$128$$ 13.6180 1.20368
$$129$$ 7.70820 0.678670
$$130$$ 0 0
$$131$$ 0.944272 0.0825014 0.0412507 0.999149i $$-0.486866\pi$$
0.0412507 + 0.999149i $$0.486866\pi$$
$$132$$ 3.23607 0.281664
$$133$$ 4.47214 0.387783
$$134$$ 0.381966 0.0329968
$$135$$ 0 0
$$136$$ 14.4721 1.24098
$$137$$ 6.94427 0.593289 0.296645 0.954988i $$-0.404132\pi$$
0.296645 + 0.954988i $$0.404132\pi$$
$$138$$ 3.52786 0.300312
$$139$$ −20.6525 −1.75172 −0.875860 0.482565i $$-0.839705\pi$$
−0.875860 + 0.482565i $$0.839705\pi$$
$$140$$ 0 0
$$141$$ −2.47214 −0.208191
$$142$$ −7.61803 −0.639291
$$143$$ −13.7082 −1.14634
$$144$$ 7.14590 0.595492
$$145$$ 0 0
$$146$$ −21.4164 −1.77243
$$147$$ 1.23607 0.101949
$$148$$ 1.85410 0.152406
$$149$$ −13.9443 −1.14236 −0.571180 0.820825i $$-0.693514\pi$$
−0.571180 + 0.820825i $$0.693514\pi$$
$$150$$ 0 0
$$151$$ −15.7639 −1.28285 −0.641425 0.767185i $$-0.721657\pi$$
−0.641425 + 0.767185i $$0.721657\pi$$
$$152$$ −10.0000 −0.811107
$$153$$ 9.52786 0.770282
$$154$$ 6.85410 0.552319
$$155$$ 0 0
$$156$$ −2.47214 −0.197929
$$157$$ 5.23607 0.417884 0.208942 0.977928i $$-0.432998\pi$$
0.208942 + 0.977928i $$0.432998\pi$$
$$158$$ 18.0902 1.43918
$$159$$ −0.583592 −0.0462819
$$160$$ 0 0
$$161$$ 1.76393 0.139017
$$162$$ −3.90983 −0.307185
$$163$$ −10.4721 −0.820241 −0.410120 0.912031i $$-0.634513\pi$$
−0.410120 + 0.912031i $$0.634513\pi$$
$$164$$ 5.70820 0.445736
$$165$$ 0 0
$$166$$ 9.23607 0.716858
$$167$$ 0.763932 0.0591148 0.0295574 0.999563i $$-0.490590\pi$$
0.0295574 + 0.999563i $$0.490590\pi$$
$$168$$ −2.76393 −0.213242
$$169$$ −2.52786 −0.194451
$$170$$ 0 0
$$171$$ −6.58359 −0.503460
$$172$$ 3.85410 0.293873
$$173$$ −20.4721 −1.55647 −0.778234 0.627975i $$-0.783884\pi$$
−0.778234 + 0.627975i $$0.783884\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ −20.5623 −1.54994
$$177$$ −2.11146 −0.158707
$$178$$ 20.6525 1.54797
$$179$$ 3.41641 0.255354 0.127677 0.991816i $$-0.459248\pi$$
0.127677 + 0.991816i $$0.459248\pi$$
$$180$$ 0 0
$$181$$ −14.1803 −1.05402 −0.527008 0.849860i $$-0.676686\pi$$
−0.527008 + 0.849860i $$0.676686\pi$$
$$182$$ −5.23607 −0.388123
$$183$$ 4.58359 0.338829
$$184$$ −3.94427 −0.290776
$$185$$ 0 0
$$186$$ −19.4164 −1.42368
$$187$$ −27.4164 −2.00489
$$188$$ −1.23607 −0.0901495
$$189$$ −5.52786 −0.402093
$$190$$ 0 0
$$191$$ −2.47214 −0.178877 −0.0894387 0.995992i $$-0.528507\pi$$
−0.0894387 + 0.995992i $$0.528507\pi$$
$$192$$ 5.23607 0.377881
$$193$$ −14.4164 −1.03772 −0.518858 0.854861i $$-0.673643\pi$$
−0.518858 + 0.854861i $$0.673643\pi$$
$$194$$ 1.23607 0.0887445
$$195$$ 0 0
$$196$$ 0.618034 0.0441453
$$197$$ 7.47214 0.532368 0.266184 0.963922i $$-0.414237\pi$$
0.266184 + 0.963922i $$0.414237\pi$$
$$198$$ −10.0902 −0.717077
$$199$$ 2.76393 0.195930 0.0979650 0.995190i $$-0.468767\pi$$
0.0979650 + 0.995190i $$0.468767\pi$$
$$200$$ 0 0
$$201$$ 0.291796 0.0205817
$$202$$ 14.9443 1.05148
$$203$$ 5.00000 0.350931
$$204$$ −4.94427 −0.346168
$$205$$ 0 0
$$206$$ −0.763932 −0.0532257
$$207$$ −2.59675 −0.180486
$$208$$ 15.7082 1.08917
$$209$$ 18.9443 1.31040
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −0.291796 −0.0200406
$$213$$ −5.81966 −0.398757
$$214$$ 12.9443 0.884852
$$215$$ 0 0
$$216$$ 12.3607 0.841038
$$217$$ −9.70820 −0.659036
$$218$$ −29.7984 −2.01820
$$219$$ −16.3607 −1.10555
$$220$$ 0 0
$$221$$ 20.9443 1.40886
$$222$$ 6.00000 0.402694
$$223$$ −2.18034 −0.146006 −0.0730032 0.997332i $$-0.523258\pi$$
−0.0730032 + 0.997332i $$0.523258\pi$$
$$224$$ −3.38197 −0.225967
$$225$$ 0 0
$$226$$ 20.0902 1.33638
$$227$$ −5.41641 −0.359500 −0.179750 0.983712i $$-0.557529\pi$$
−0.179750 + 0.983712i $$0.557529\pi$$
$$228$$ 3.41641 0.226257
$$229$$ 4.47214 0.295527 0.147764 0.989023i $$-0.452793\pi$$
0.147764 + 0.989023i $$0.452793\pi$$
$$230$$ 0 0
$$231$$ 5.23607 0.344508
$$232$$ −11.1803 −0.734025
$$233$$ −9.94427 −0.651471 −0.325735 0.945461i $$-0.605612\pi$$
−0.325735 + 0.945461i $$0.605612\pi$$
$$234$$ 7.70820 0.503901
$$235$$ 0 0
$$236$$ −1.05573 −0.0687220
$$237$$ 13.8197 0.897683
$$238$$ −10.4721 −0.678808
$$239$$ −14.4721 −0.936125 −0.468062 0.883695i $$-0.655048\pi$$
−0.468062 + 0.883695i $$0.655048\pi$$
$$240$$ 0 0
$$241$$ −12.4721 −0.803401 −0.401700 0.915771i $$-0.631581\pi$$
−0.401700 + 0.915771i $$0.631581\pi$$
$$242$$ 11.2361 0.722282
$$243$$ 13.5967 0.872232
$$244$$ 2.29180 0.146717
$$245$$ 0 0
$$246$$ 18.4721 1.17774
$$247$$ −14.4721 −0.920840
$$248$$ 21.7082 1.37847
$$249$$ 7.05573 0.447139
$$250$$ 0 0
$$251$$ −2.47214 −0.156040 −0.0780199 0.996952i $$-0.524860\pi$$
−0.0780199 + 0.996952i $$0.524860\pi$$
$$252$$ −0.909830 −0.0573139
$$253$$ 7.47214 0.469769
$$254$$ −28.5623 −1.79216
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ 18.6525 1.16351 0.581755 0.813364i $$-0.302366\pi$$
0.581755 + 0.813364i $$0.302366\pi$$
$$258$$ 12.4721 0.776481
$$259$$ 3.00000 0.186411
$$260$$ 0 0
$$261$$ −7.36068 −0.455615
$$262$$ 1.52786 0.0943918
$$263$$ 11.7639 0.725395 0.362698 0.931907i $$-0.381856\pi$$
0.362698 + 0.931907i $$0.381856\pi$$
$$264$$ −11.7082 −0.720590
$$265$$ 0 0
$$266$$ 7.23607 0.443672
$$267$$ 15.7771 0.965542
$$268$$ 0.145898 0.00891214
$$269$$ 1.70820 0.104151 0.0520755 0.998643i $$-0.483416\pi$$
0.0520755 + 0.998643i $$0.483416\pi$$
$$270$$ 0 0
$$271$$ 10.2918 0.625182 0.312591 0.949888i $$-0.398803\pi$$
0.312591 + 0.949888i $$0.398803\pi$$
$$272$$ 31.4164 1.90490
$$273$$ −4.00000 −0.242091
$$274$$ 11.2361 0.678796
$$275$$ 0 0
$$276$$ 1.34752 0.0811114
$$277$$ 15.8885 0.954650 0.477325 0.878727i $$-0.341606\pi$$
0.477325 + 0.878727i $$0.341606\pi$$
$$278$$ −33.4164 −2.00418
$$279$$ 14.2918 0.855627
$$280$$ 0 0
$$281$$ 29.3607 1.75151 0.875756 0.482755i $$-0.160364\pi$$
0.875756 + 0.482755i $$0.160364\pi$$
$$282$$ −4.00000 −0.238197
$$283$$ −9.41641 −0.559747 −0.279874 0.960037i $$-0.590293\pi$$
−0.279874 + 0.960037i $$0.590293\pi$$
$$284$$ −2.90983 −0.172667
$$285$$ 0 0
$$286$$ −22.1803 −1.31155
$$287$$ 9.23607 0.545188
$$288$$ 4.97871 0.293374
$$289$$ 24.8885 1.46403
$$290$$ 0 0
$$291$$ 0.944272 0.0553542
$$292$$ −8.18034 −0.478718
$$293$$ 9.12461 0.533066 0.266533 0.963826i $$-0.414122\pi$$
0.266533 + 0.963826i $$0.414122\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ −6.70820 −0.389906
$$297$$ −23.4164 −1.35876
$$298$$ −22.5623 −1.30700
$$299$$ −5.70820 −0.330114
$$300$$ 0 0
$$301$$ 6.23607 0.359441
$$302$$ −25.5066 −1.46774
$$303$$ 11.4164 0.655855
$$304$$ −21.7082 −1.24505
$$305$$ 0 0
$$306$$ 15.4164 0.881297
$$307$$ 31.4164 1.79303 0.896515 0.443014i $$-0.146091\pi$$
0.896515 + 0.443014i $$0.146091\pi$$
$$308$$ 2.61803 0.149176
$$309$$ −0.583592 −0.0331994
$$310$$ 0 0
$$311$$ −20.3607 −1.15455 −0.577274 0.816550i $$-0.695884\pi$$
−0.577274 + 0.816550i $$0.695884\pi$$
$$312$$ 8.94427 0.506370
$$313$$ 28.4721 1.60934 0.804670 0.593722i $$-0.202342\pi$$
0.804670 + 0.593722i $$0.202342\pi$$
$$314$$ 8.47214 0.478110
$$315$$ 0 0
$$316$$ 6.90983 0.388708
$$317$$ −19.3607 −1.08740 −0.543702 0.839278i $$-0.682978\pi$$
−0.543702 + 0.839278i $$0.682978\pi$$
$$318$$ −0.944272 −0.0529521
$$319$$ 21.1803 1.18587
$$320$$ 0 0
$$321$$ 9.88854 0.551925
$$322$$ 2.85410 0.159053
$$323$$ −28.9443 −1.61050
$$324$$ −1.49342 −0.0829679
$$325$$ 0 0
$$326$$ −16.9443 −0.938456
$$327$$ −22.7639 −1.25885
$$328$$ −20.6525 −1.14034
$$329$$ −2.00000 −0.110264
$$330$$ 0 0
$$331$$ −11.2918 −0.620653 −0.310327 0.950630i $$-0.600438\pi$$
−0.310327 + 0.950630i $$0.600438\pi$$
$$332$$ 3.52786 0.193617
$$333$$ −4.41641 −0.242018
$$334$$ 1.23607 0.0676346
$$335$$ 0 0
$$336$$ −6.00000 −0.327327
$$337$$ −7.52786 −0.410069 −0.205034 0.978755i $$-0.565731\pi$$
−0.205034 + 0.978755i $$0.565731\pi$$
$$338$$ −4.09017 −0.222476
$$339$$ 15.3475 0.833563
$$340$$ 0 0
$$341$$ −41.1246 −2.22702
$$342$$ −10.6525 −0.576020
$$343$$ 1.00000 0.0539949
$$344$$ −13.9443 −0.751825
$$345$$ 0 0
$$346$$ −33.1246 −1.78079
$$347$$ 15.7639 0.846252 0.423126 0.906071i $$-0.360933\pi$$
0.423126 + 0.906071i $$0.360933\pi$$
$$348$$ 3.81966 0.204755
$$349$$ −4.47214 −0.239388 −0.119694 0.992811i $$-0.538191\pi$$
−0.119694 + 0.992811i $$0.538191\pi$$
$$350$$ 0 0
$$351$$ 17.8885 0.954820
$$352$$ −14.3262 −0.763591
$$353$$ 20.1803 1.07409 0.537046 0.843553i $$-0.319540\pi$$
0.537046 + 0.843553i $$0.319540\pi$$
$$354$$ −3.41641 −0.181580
$$355$$ 0 0
$$356$$ 7.88854 0.418092
$$357$$ −8.00000 −0.423405
$$358$$ 5.52786 0.292157
$$359$$ 10.1246 0.534357 0.267178 0.963647i $$-0.413909\pi$$
0.267178 + 0.963647i $$0.413909\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −22.9443 −1.20592
$$363$$ 8.58359 0.450522
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 7.41641 0.387662
$$367$$ 3.12461 0.163103 0.0815517 0.996669i $$-0.474012\pi$$
0.0815517 + 0.996669i $$0.474012\pi$$
$$368$$ −8.56231 −0.446341
$$369$$ −13.5967 −0.707818
$$370$$ 0 0
$$371$$ −0.472136 −0.0245121
$$372$$ −7.41641 −0.384523
$$373$$ 15.8328 0.819792 0.409896 0.912132i $$-0.365565\pi$$
0.409896 + 0.912132i $$0.365565\pi$$
$$374$$ −44.3607 −2.29384
$$375$$ 0 0
$$376$$ 4.47214 0.230633
$$377$$ −16.1803 −0.833330
$$378$$ −8.94427 −0.460044
$$379$$ 11.1803 0.574295 0.287148 0.957886i $$-0.407293\pi$$
0.287148 + 0.957886i $$0.407293\pi$$
$$380$$ 0 0
$$381$$ −21.8197 −1.11786
$$382$$ −4.00000 −0.204658
$$383$$ −28.7639 −1.46977 −0.734884 0.678193i $$-0.762763\pi$$
−0.734884 + 0.678193i $$0.762763\pi$$
$$384$$ 16.8328 0.858996
$$385$$ 0 0
$$386$$ −23.3262 −1.18727
$$387$$ −9.18034 −0.466663
$$388$$ 0.472136 0.0239691
$$389$$ −32.8885 −1.66752 −0.833758 0.552131i $$-0.813815\pi$$
−0.833758 + 0.552131i $$0.813815\pi$$
$$390$$ 0 0
$$391$$ −11.4164 −0.577353
$$392$$ −2.23607 −0.112938
$$393$$ 1.16718 0.0588767
$$394$$ 12.0902 0.609094
$$395$$ 0 0
$$396$$ −3.85410 −0.193676
$$397$$ 26.9443 1.35229 0.676147 0.736767i $$-0.263648\pi$$
0.676147 + 0.736767i $$0.263648\pi$$
$$398$$ 4.47214 0.224168
$$399$$ 5.52786 0.276739
$$400$$ 0 0
$$401$$ 11.4721 0.572891 0.286446 0.958097i $$-0.407526\pi$$
0.286446 + 0.958097i $$0.407526\pi$$
$$402$$ 0.472136 0.0235480
$$403$$ 31.4164 1.56496
$$404$$ 5.70820 0.283994
$$405$$ 0 0
$$406$$ 8.09017 0.401508
$$407$$ 12.7082 0.629922
$$408$$ 17.8885 0.885615
$$409$$ 15.5279 0.767803 0.383902 0.923374i $$-0.374580\pi$$
0.383902 + 0.923374i $$0.374580\pi$$
$$410$$ 0 0
$$411$$ 8.58359 0.423397
$$412$$ −0.291796 −0.0143758
$$413$$ −1.70820 −0.0840552
$$414$$ −4.20163 −0.206499
$$415$$ 0 0
$$416$$ 10.9443 0.536587
$$417$$ −25.5279 −1.25010
$$418$$ 30.6525 1.49926
$$419$$ −3.81966 −0.186603 −0.0933013 0.995638i $$-0.529742\pi$$
−0.0933013 + 0.995638i $$0.529742\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ 19.4164 0.945176
$$423$$ 2.94427 0.143155
$$424$$ 1.05573 0.0512707
$$425$$ 0 0
$$426$$ −9.41641 −0.456226
$$427$$ 3.70820 0.179453
$$428$$ 4.94427 0.238990
$$429$$ −16.9443 −0.818077
$$430$$ 0 0
$$431$$ 26.4721 1.27512 0.637559 0.770402i $$-0.279944\pi$$
0.637559 + 0.770402i $$0.279944\pi$$
$$432$$ 26.8328 1.29099
$$433$$ 16.3607 0.786244 0.393122 0.919486i $$-0.371395\pi$$
0.393122 + 0.919486i $$0.371395\pi$$
$$434$$ −15.7082 −0.754018
$$435$$ 0 0
$$436$$ −11.3820 −0.545097
$$437$$ 7.88854 0.377360
$$438$$ −26.4721 −1.26489
$$439$$ −21.7082 −1.03608 −0.518038 0.855358i $$-0.673337\pi$$
−0.518038 + 0.855358i $$0.673337\pi$$
$$440$$ 0 0
$$441$$ −1.47214 −0.0701017
$$442$$ 33.8885 1.61191
$$443$$ 7.41641 0.352364 0.176182 0.984358i $$-0.443625\pi$$
0.176182 + 0.984358i $$0.443625\pi$$
$$444$$ 2.29180 0.108764
$$445$$ 0 0
$$446$$ −3.52786 −0.167049
$$447$$ −17.2361 −0.815238
$$448$$ 4.23607 0.200135
$$449$$ 29.4721 1.39088 0.695438 0.718586i $$-0.255210\pi$$
0.695438 + 0.718586i $$0.255210\pi$$
$$450$$ 0 0
$$451$$ 39.1246 1.84231
$$452$$ 7.67376 0.360943
$$453$$ −19.4853 −0.915499
$$454$$ −8.76393 −0.411312
$$455$$ 0 0
$$456$$ −12.3607 −0.578842
$$457$$ −21.4721 −1.00442 −0.502212 0.864744i $$-0.667480\pi$$
−0.502212 + 0.864744i $$0.667480\pi$$
$$458$$ 7.23607 0.338119
$$459$$ 35.7771 1.66993
$$460$$ 0 0
$$461$$ 8.18034 0.380996 0.190498 0.981688i $$-0.438990\pi$$
0.190498 + 0.981688i $$0.438990\pi$$
$$462$$ 8.47214 0.394159
$$463$$ 21.8885 1.01725 0.508623 0.860989i $$-0.330155\pi$$
0.508623 + 0.860989i $$0.330155\pi$$
$$464$$ −24.2705 −1.12673
$$465$$ 0 0
$$466$$ −16.0902 −0.745363
$$467$$ −10.9443 −0.506441 −0.253220 0.967409i $$-0.581490\pi$$
−0.253220 + 0.967409i $$0.581490\pi$$
$$468$$ 2.94427 0.136099
$$469$$ 0.236068 0.0109006
$$470$$ 0 0
$$471$$ 6.47214 0.298220
$$472$$ 3.81966 0.175814
$$473$$ 26.4164 1.21463
$$474$$ 22.3607 1.02706
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ 0.695048 0.0318241
$$478$$ −23.4164 −1.07104
$$479$$ −3.81966 −0.174525 −0.0872624 0.996185i $$-0.527812\pi$$
−0.0872624 + 0.996185i $$0.527812\pi$$
$$480$$ 0 0
$$481$$ −9.70820 −0.442656
$$482$$ −20.1803 −0.919189
$$483$$ 2.18034 0.0992089
$$484$$ 4.29180 0.195082
$$485$$ 0 0
$$486$$ 22.0000 0.997940
$$487$$ 10.2361 0.463841 0.231920 0.972735i $$-0.425499\pi$$
0.231920 + 0.972735i $$0.425499\pi$$
$$488$$ −8.29180 −0.375352
$$489$$ −12.9443 −0.585360
$$490$$ 0 0
$$491$$ −10.2361 −0.461947 −0.230974 0.972960i $$-0.574191\pi$$
−0.230974 + 0.972960i $$0.574191\pi$$
$$492$$ 7.05573 0.318097
$$493$$ −32.3607 −1.45745
$$494$$ −23.4164 −1.05355
$$495$$ 0 0
$$496$$ 47.1246 2.11596
$$497$$ −4.70820 −0.211192
$$498$$ 11.4164 0.511581
$$499$$ 28.9443 1.29572 0.647862 0.761758i $$-0.275663\pi$$
0.647862 + 0.761758i $$0.275663\pi$$
$$500$$ 0 0
$$501$$ 0.944272 0.0421870
$$502$$ −4.00000 −0.178529
$$503$$ −43.8885 −1.95689 −0.978447 0.206499i $$-0.933793\pi$$
−0.978447 + 0.206499i $$0.933793\pi$$
$$504$$ 3.29180 0.146628
$$505$$ 0 0
$$506$$ 12.0902 0.537474
$$507$$ −3.12461 −0.138769
$$508$$ −10.9098 −0.484045
$$509$$ 9.34752 0.414322 0.207161 0.978307i $$-0.433578\pi$$
0.207161 + 0.978307i $$0.433578\pi$$
$$510$$ 0 0
$$511$$ −13.2361 −0.585529
$$512$$ −5.29180 −0.233867
$$513$$ −24.7214 −1.09147
$$514$$ 30.1803 1.33120
$$515$$ 0 0
$$516$$ 4.76393 0.209720
$$517$$ −8.47214 −0.372604
$$518$$ 4.85410 0.213277
$$519$$ −25.3050 −1.11076
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ −11.9098 −0.521279
$$523$$ −28.3607 −1.24013 −0.620063 0.784552i $$-0.712893\pi$$
−0.620063 + 0.784552i $$0.712893\pi$$
$$524$$ 0.583592 0.0254943
$$525$$ 0 0
$$526$$ 19.0344 0.829941
$$527$$ 62.8328 2.73704
$$528$$ −25.4164 −1.10611
$$529$$ −19.8885 −0.864719
$$530$$ 0 0
$$531$$ 2.51471 0.109129
$$532$$ 2.76393 0.119832
$$533$$ −29.8885 −1.29462
$$534$$ 25.5279 1.10470
$$535$$ 0 0
$$536$$ −0.527864 −0.0228003
$$537$$ 4.22291 0.182232
$$538$$ 2.76393 0.119162
$$539$$ 4.23607 0.182460
$$540$$ 0 0
$$541$$ −1.94427 −0.0835908 −0.0417954 0.999126i $$-0.513308\pi$$
−0.0417954 + 0.999126i $$0.513308\pi$$
$$542$$ 16.6525 0.715285
$$543$$ −17.5279 −0.752193
$$544$$ 21.8885 0.938464
$$545$$ 0 0
$$546$$ −6.47214 −0.276982
$$547$$ −14.2361 −0.608690 −0.304345 0.952562i $$-0.598438\pi$$
−0.304345 + 0.952562i $$0.598438\pi$$
$$548$$ 4.29180 0.183336
$$549$$ −5.45898 −0.232984
$$550$$ 0 0
$$551$$ 22.3607 0.952597
$$552$$ −4.87539 −0.207510
$$553$$ 11.1803 0.475436
$$554$$ 25.7082 1.09224
$$555$$ 0 0
$$556$$ −12.7639 −0.541311
$$557$$ −44.8885 −1.90199 −0.950994 0.309208i $$-0.899936\pi$$
−0.950994 + 0.309208i $$0.899936\pi$$
$$558$$ 23.1246 0.978943
$$559$$ −20.1803 −0.853537
$$560$$ 0 0
$$561$$ −33.8885 −1.43078
$$562$$ 47.5066 2.00394
$$563$$ −9.41641 −0.396854 −0.198427 0.980116i $$-0.563583\pi$$
−0.198427 + 0.980116i $$0.563583\pi$$
$$564$$ −1.52786 −0.0643347
$$565$$ 0 0
$$566$$ −15.2361 −0.640420
$$567$$ −2.41641 −0.101480
$$568$$ 10.5279 0.441739
$$569$$ 13.9443 0.584574 0.292287 0.956331i $$-0.405584\pi$$
0.292287 + 0.956331i $$0.405584\pi$$
$$570$$ 0 0
$$571$$ −12.5967 −0.527157 −0.263579 0.964638i $$-0.584903\pi$$
−0.263579 + 0.964638i $$0.584903\pi$$
$$572$$ −8.47214 −0.354238
$$573$$ −3.05573 −0.127655
$$574$$ 14.9443 0.623762
$$575$$ 0 0
$$576$$ −6.23607 −0.259836
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 40.2705 1.67503
$$579$$ −17.8197 −0.740560
$$580$$ 0 0
$$581$$ 5.70820 0.236816
$$582$$ 1.52786 0.0633320
$$583$$ −2.00000 −0.0828315
$$584$$ 29.5967 1.22472
$$585$$ 0 0
$$586$$ 14.7639 0.609892
$$587$$ −29.2361 −1.20670 −0.603351 0.797476i $$-0.706168\pi$$
−0.603351 + 0.797476i $$0.706168\pi$$
$$588$$ 0.763932 0.0315040
$$589$$ −43.4164 −1.78894
$$590$$ 0 0
$$591$$ 9.23607 0.379921
$$592$$ −14.5623 −0.598507
$$593$$ 25.3050 1.03915 0.519575 0.854425i $$-0.326090\pi$$
0.519575 + 0.854425i $$0.326090\pi$$
$$594$$ −37.8885 −1.55459
$$595$$ 0 0
$$596$$ −8.61803 −0.353008
$$597$$ 3.41641 0.139824
$$598$$ −9.23607 −0.377691
$$599$$ 11.1803 0.456816 0.228408 0.973565i $$-0.426648\pi$$
0.228408 + 0.973565i $$0.426648\pi$$
$$600$$ 0 0
$$601$$ −19.0557 −0.777299 −0.388650 0.921386i $$-0.627058\pi$$
−0.388650 + 0.921386i $$0.627058\pi$$
$$602$$ 10.0902 0.411245
$$603$$ −0.347524 −0.0141523
$$604$$ −9.74265 −0.396423
$$605$$ 0 0
$$606$$ 18.4721 0.750379
$$607$$ 33.1246 1.34449 0.672243 0.740330i $$-0.265331\pi$$
0.672243 + 0.740330i $$0.265331\pi$$
$$608$$ −15.1246 −0.613384
$$609$$ 6.18034 0.250440
$$610$$ 0 0
$$611$$ 6.47214 0.261835
$$612$$ 5.88854 0.238030
$$613$$ −17.5836 −0.710195 −0.355097 0.934829i $$-0.615552\pi$$
−0.355097 + 0.934829i $$0.615552\pi$$
$$614$$ 50.8328 2.05145
$$615$$ 0 0
$$616$$ −9.47214 −0.381643
$$617$$ 11.9443 0.480858 0.240429 0.970667i $$-0.422712\pi$$
0.240429 + 0.970667i $$0.422712\pi$$
$$618$$ −0.944272 −0.0379842
$$619$$ 1.70820 0.0686585 0.0343293 0.999411i $$-0.489071\pi$$
0.0343293 + 0.999411i $$0.489071\pi$$
$$620$$ 0 0
$$621$$ −9.75078 −0.391285
$$622$$ −32.9443 −1.32094
$$623$$ 12.7639 0.511376
$$624$$ 19.4164 0.777278
$$625$$ 0 0
$$626$$ 46.0689 1.84128
$$627$$ 23.4164 0.935161
$$628$$ 3.23607 0.129133
$$629$$ −19.4164 −0.774183
$$630$$ 0 0
$$631$$ −3.65248 −0.145403 −0.0727014 0.997354i $$-0.523162\pi$$
−0.0727014 + 0.997354i $$0.523162\pi$$
$$632$$ −25.0000 −0.994447
$$633$$ 14.8328 0.589551
$$634$$ −31.3262 −1.24412
$$635$$ 0 0
$$636$$ −0.360680 −0.0143019
$$637$$ −3.23607 −0.128218
$$638$$ 34.2705 1.35678
$$639$$ 6.93112 0.274191
$$640$$ 0 0
$$641$$ −9.83282 −0.388373 −0.194186 0.980965i $$-0.562207\pi$$
−0.194186 + 0.980965i $$0.562207\pi$$
$$642$$ 16.0000 0.631470
$$643$$ 9.52786 0.375742 0.187871 0.982194i $$-0.439841\pi$$
0.187871 + 0.982194i $$0.439841\pi$$
$$644$$ 1.09017 0.0429587
$$645$$ 0 0
$$646$$ −46.8328 −1.84261
$$647$$ 15.8885 0.624643 0.312322 0.949976i $$-0.398893\pi$$
0.312322 + 0.949976i $$0.398893\pi$$
$$648$$ 5.40325 0.212260
$$649$$ −7.23607 −0.284041
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ −6.47214 −0.253468
$$653$$ 42.9443 1.68054 0.840270 0.542169i $$-0.182397\pi$$
0.840270 + 0.542169i $$0.182397\pi$$
$$654$$ −36.8328 −1.44028
$$655$$ 0 0
$$656$$ −44.8328 −1.75043
$$657$$ 19.4853 0.760194
$$658$$ −3.23607 −0.126155
$$659$$ −17.8885 −0.696839 −0.348419 0.937339i $$-0.613281\pi$$
−0.348419 + 0.937339i $$0.613281\pi$$
$$660$$ 0 0
$$661$$ 46.7214 1.81725 0.908625 0.417613i $$-0.137133\pi$$
0.908625 + 0.417613i $$0.137133\pi$$
$$662$$ −18.2705 −0.710104
$$663$$ 25.8885 1.00543
$$664$$ −12.7639 −0.495337
$$665$$ 0 0
$$666$$ −7.14590 −0.276898
$$667$$ 8.81966 0.341499
$$668$$ 0.472136 0.0182675
$$669$$ −2.69505 −0.104197
$$670$$ 0 0
$$671$$ 15.7082 0.606408
$$672$$ −4.18034 −0.161260
$$673$$ 28.4721 1.09752 0.548760 0.835980i $$-0.315100\pi$$
0.548760 + 0.835980i $$0.315100\pi$$
$$674$$ −12.1803 −0.469169
$$675$$ 0 0
$$676$$ −1.56231 −0.0600887
$$677$$ 30.3607 1.16686 0.583428 0.812165i $$-0.301711\pi$$
0.583428 + 0.812165i $$0.301711\pi$$
$$678$$ 24.8328 0.953699
$$679$$ 0.763932 0.0293170
$$680$$ 0 0
$$681$$ −6.69505 −0.256555
$$682$$ −66.5410 −2.54799
$$683$$ −26.1246 −0.999630 −0.499815 0.866132i $$-0.666599\pi$$
−0.499815 + 0.866132i $$0.666599\pi$$
$$684$$ −4.06888 −0.155578
$$685$$ 0 0
$$686$$ 1.61803 0.0617768
$$687$$ 5.52786 0.210901
$$688$$ −30.2705 −1.15405
$$689$$ 1.52786 0.0582070
$$690$$ 0 0
$$691$$ 18.1803 0.691613 0.345806 0.938306i $$-0.387605\pi$$
0.345806 + 0.938306i $$0.387605\pi$$
$$692$$ −12.6525 −0.480975
$$693$$ −6.23607 −0.236889
$$694$$ 25.5066 0.968216
$$695$$ 0 0
$$696$$ −13.8197 −0.523833
$$697$$ −59.7771 −2.26422
$$698$$ −7.23607 −0.273889
$$699$$ −12.2918 −0.464918
$$700$$ 0 0
$$701$$ −46.9443 −1.77306 −0.886530 0.462670i $$-0.846891\pi$$
−0.886530 + 0.462670i $$0.846891\pi$$
$$702$$ 28.9443 1.09243
$$703$$ 13.4164 0.506009
$$704$$ 17.9443 0.676300
$$705$$ 0 0
$$706$$ 32.6525 1.22889
$$707$$ 9.23607 0.347358
$$708$$ −1.30495 −0.0490431
$$709$$ −47.8885 −1.79849 −0.899246 0.437443i $$-0.855884\pi$$
−0.899246 + 0.437443i $$0.855884\pi$$
$$710$$ 0 0
$$711$$ −16.4590 −0.617260
$$712$$ −28.5410 −1.06962
$$713$$ −17.1246 −0.641322
$$714$$ −12.9443 −0.484427
$$715$$ 0 0
$$716$$ 2.11146 0.0789088
$$717$$ −17.8885 −0.668060
$$718$$ 16.3820 0.611370
$$719$$ −6.18034 −0.230488 −0.115244 0.993337i $$-0.536765\pi$$
−0.115244 + 0.993337i $$0.536765\pi$$
$$720$$ 0 0
$$721$$ −0.472136 −0.0175833
$$722$$ 1.61803 0.0602170
$$723$$ −15.4164 −0.573342
$$724$$ −8.76393 −0.325709
$$725$$ 0 0
$$726$$ 13.8885 0.515452
$$727$$ −20.9443 −0.776780 −0.388390 0.921495i $$-0.626969\pi$$
−0.388390 + 0.921495i $$0.626969\pi$$
$$728$$ 7.23607 0.268187
$$729$$ 24.0557 0.890953
$$730$$ 0 0
$$731$$ −40.3607 −1.49279
$$732$$ 2.83282 0.104704
$$733$$ 4.00000 0.147743 0.0738717 0.997268i $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ 5.05573 0.186610
$$735$$ 0 0
$$736$$ −5.96556 −0.219893
$$737$$ 1.00000 0.0368355
$$738$$ −22.0000 −0.809831
$$739$$ −5.65248 −0.207930 −0.103965 0.994581i $$-0.533153\pi$$
−0.103965 + 0.994581i $$0.533153\pi$$
$$740$$ 0 0
$$741$$ −17.8885 −0.657152
$$742$$ −0.763932 −0.0280448
$$743$$ −1.52786 −0.0560519 −0.0280259 0.999607i $$-0.508922\pi$$
−0.0280259 + 0.999607i $$0.508922\pi$$
$$744$$ 26.8328 0.983739
$$745$$ 0 0
$$746$$ 25.6180 0.937943
$$747$$ −8.40325 −0.307459
$$748$$ −16.9443 −0.619544
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ 20.9443 0.764267 0.382134 0.924107i $$-0.375189\pi$$
0.382134 + 0.924107i $$0.375189\pi$$
$$752$$ 9.70820 0.354022
$$753$$ −3.05573 −0.111357
$$754$$ −26.1803 −0.953432
$$755$$ 0 0
$$756$$ −3.41641 −0.124254
$$757$$ 46.4164 1.68703 0.843517 0.537103i $$-0.180481\pi$$
0.843517 + 0.537103i $$0.180481\pi$$
$$758$$ 18.0902 0.657065
$$759$$ 9.23607 0.335248
$$760$$ 0 0
$$761$$ −43.7771 −1.58692 −0.793459 0.608624i $$-0.791722\pi$$
−0.793459 + 0.608624i $$0.791722\pi$$
$$762$$ −35.3050 −1.27896
$$763$$ −18.4164 −0.666719
$$764$$ −1.52786 −0.0552762
$$765$$ 0 0
$$766$$ −46.5410 −1.68160
$$767$$ 5.52786 0.199600
$$768$$ 16.7639 0.604916
$$769$$ 33.0132 1.19048 0.595242 0.803546i $$-0.297056\pi$$
0.595242 + 0.803546i $$0.297056\pi$$
$$770$$ 0 0
$$771$$ 23.0557 0.830332
$$772$$ −8.90983 −0.320672
$$773$$ 27.8197 1.00060 0.500302 0.865851i $$-0.333222\pi$$
0.500302 + 0.865851i $$0.333222\pi$$
$$774$$ −14.8541 −0.533920
$$775$$ 0 0
$$776$$ −1.70820 −0.0613209
$$777$$ 3.70820 0.133031
$$778$$ −53.2148 −1.90784
$$779$$ 41.3050 1.47990
$$780$$ 0 0
$$781$$ −19.9443 −0.713662
$$782$$ −18.4721 −0.660562
$$783$$ −27.6393 −0.987749
$$784$$ −4.85410 −0.173361
$$785$$ 0 0
$$786$$ 1.88854 0.0673621
$$787$$ 45.2361 1.61249 0.806246 0.591581i $$-0.201496\pi$$
0.806246 + 0.591581i $$0.201496\pi$$
$$788$$ 4.61803 0.164511
$$789$$ 14.5410 0.517674
$$790$$ 0 0
$$791$$ 12.4164 0.441477
$$792$$ 13.9443 0.495488
$$793$$ −12.0000 −0.426132
$$794$$ 43.5967 1.54719
$$795$$ 0 0
$$796$$ 1.70820 0.0605457
$$797$$ −8.58359 −0.304046 −0.152023 0.988377i $$-0.548579\pi$$
−0.152023 + 0.988377i $$0.548579\pi$$
$$798$$ 8.94427 0.316624
$$799$$ 12.9443 0.457935
$$800$$ 0 0
$$801$$ −18.7902 −0.663921
$$802$$ 18.5623 0.655458
$$803$$ −56.0689 −1.97863
$$804$$ 0.180340 0.00636010
$$805$$ 0 0
$$806$$ 50.8328 1.79051
$$807$$ 2.11146 0.0743268
$$808$$ −20.6525 −0.726552
$$809$$ 20.5279 0.721721 0.360861 0.932620i $$-0.382483\pi$$
0.360861 + 0.932620i $$0.382483\pi$$
$$810$$ 0 0
$$811$$ 46.7214 1.64061 0.820304 0.571927i $$-0.193804\pi$$
0.820304 + 0.571927i $$0.193804\pi$$
$$812$$ 3.09017 0.108444
$$813$$ 12.7214 0.446158
$$814$$ 20.5623 0.720708
$$815$$ 0 0
$$816$$ 38.8328 1.35942
$$817$$ 27.8885 0.975697
$$818$$ 25.1246 0.878461
$$819$$ 4.76393 0.166465
$$820$$ 0 0
$$821$$ −24.8328 −0.866671 −0.433336 0.901233i $$-0.642664\pi$$
−0.433336 + 0.901233i $$0.642664\pi$$
$$822$$ 13.8885 0.484419
$$823$$ −0.347524 −0.0121139 −0.00605697 0.999982i $$-0.501928\pi$$
−0.00605697 + 0.999982i $$0.501928\pi$$
$$824$$ 1.05573 0.0367780
$$825$$ 0 0
$$826$$ −2.76393 −0.0961695
$$827$$ −25.5410 −0.888148 −0.444074 0.895990i $$-0.646467\pi$$
−0.444074 + 0.895990i $$0.646467\pi$$
$$828$$ −1.60488 −0.0557734
$$829$$ −52.3607 −1.81856 −0.909281 0.416183i $$-0.863368\pi$$
−0.909281 + 0.416183i $$0.863368\pi$$
$$830$$ 0 0
$$831$$ 19.6393 0.681280
$$832$$ −13.7082 −0.475246
$$833$$ −6.47214 −0.224246
$$834$$ −41.3050 −1.43027
$$835$$ 0 0
$$836$$ 11.7082 0.404937
$$837$$ 53.6656 1.85496
$$838$$ −6.18034 −0.213496
$$839$$ 0.652476 0.0225260 0.0112630 0.999937i $$-0.496415\pi$$
0.0112630 + 0.999937i $$0.496415\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −21.0344 −0.724895
$$843$$ 36.2918 1.24996
$$844$$ 7.41641 0.255283
$$845$$ 0 0
$$846$$ 4.76393 0.163787
$$847$$ 6.94427 0.238608
$$848$$ 2.29180 0.0787006
$$849$$ −11.6393 −0.399460
$$850$$ 0 0
$$851$$ 5.29180 0.181400
$$852$$ −3.59675 −0.123223
$$853$$ 0.583592 0.0199818 0.00999091 0.999950i $$-0.496820\pi$$
0.00999091 + 0.999950i $$0.496820\pi$$
$$854$$ 6.00000 0.205316
$$855$$ 0 0
$$856$$ −17.8885 −0.611418
$$857$$ −38.1803 −1.30422 −0.652108 0.758126i $$-0.726115\pi$$
−0.652108 + 0.758126i $$0.726115\pi$$
$$858$$ −27.4164 −0.935981
$$859$$ −22.3607 −0.762937 −0.381468 0.924382i $$-0.624581\pi$$
−0.381468 + 0.924382i $$0.624581\pi$$
$$860$$ 0 0
$$861$$ 11.4164 0.389070
$$862$$ 42.8328 1.45889
$$863$$ 49.6525 1.69019 0.845095 0.534616i $$-0.179544\pi$$
0.845095 + 0.534616i $$0.179544\pi$$
$$864$$ 18.6950 0.636018
$$865$$ 0 0
$$866$$ 26.4721 0.899560
$$867$$ 30.7639 1.04480
$$868$$ −6.00000 −0.203653
$$869$$ 47.3607 1.60660
$$870$$ 0 0
$$871$$ −0.763932 −0.0258848
$$872$$ 41.1803 1.39454
$$873$$ −1.12461 −0.0380623
$$874$$ 12.7639 0.431746
$$875$$ 0 0
$$876$$ −10.1115 −0.341634
$$877$$ −14.3607 −0.484926 −0.242463 0.970161i $$-0.577955\pi$$
−0.242463 + 0.970161i $$0.577955\pi$$
$$878$$ −35.1246 −1.18540
$$879$$ 11.2786 0.380419
$$880$$ 0 0
$$881$$ 28.1803 0.949420 0.474710 0.880142i $$-0.342553\pi$$
0.474710 + 0.880142i $$0.342553\pi$$
$$882$$ −2.38197 −0.0802050
$$883$$ −50.5967 −1.70272 −0.851358 0.524585i $$-0.824220\pi$$
−0.851358 + 0.524585i $$0.824220\pi$$
$$884$$ 12.9443 0.435363
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ −52.6525 −1.76790 −0.883949 0.467584i $$-0.845124\pi$$
−0.883949 + 0.467584i $$0.845124\pi$$
$$888$$ −8.29180 −0.278254
$$889$$ −17.6525 −0.592045
$$890$$ 0 0
$$891$$ −10.2361 −0.342921
$$892$$ −1.34752 −0.0451184
$$893$$ −8.94427 −0.299309
$$894$$ −27.8885 −0.932732
$$895$$ 0 0
$$896$$ 13.6180 0.454947
$$897$$ −7.05573 −0.235584
$$898$$ 47.6869 1.59133
$$899$$ −48.5410 −1.61893
$$900$$ 0 0
$$901$$ 3.05573 0.101801
$$902$$ 63.3050 2.10782
$$903$$ 7.70820 0.256513
$$904$$ −27.7639 −0.923415
$$905$$ 0 0
$$906$$ −31.5279 −1.04744
$$907$$ −18.8328 −0.625333 −0.312667 0.949863i $$-0.601222\pi$$
−0.312667 + 0.949863i $$0.601222\pi$$
$$908$$ −3.34752 −0.111091
$$909$$ −13.5967 −0.450976
$$910$$ 0 0
$$911$$ 23.1803 0.767999 0.383999 0.923333i $$-0.374546\pi$$
0.383999 + 0.923333i $$0.374546\pi$$
$$912$$ −26.8328 −0.888523
$$913$$ 24.1803 0.800252
$$914$$ −34.7426 −1.14918
$$915$$ 0 0
$$916$$ 2.76393 0.0913229
$$917$$ 0.944272 0.0311826
$$918$$ 57.8885 1.91061
$$919$$ −32.2361 −1.06337 −0.531685 0.846942i $$-0.678441\pi$$
−0.531685 + 0.846942i $$0.678441\pi$$
$$920$$ 0 0
$$921$$ 38.8328 1.27958
$$922$$ 13.2361 0.435907
$$923$$ 15.2361 0.501501
$$924$$ 3.23607 0.106459
$$925$$ 0 0
$$926$$ 35.4164 1.16386
$$927$$ 0.695048 0.0228284
$$928$$ −16.9098 −0.555092
$$929$$ 51.7082 1.69649 0.848246 0.529603i $$-0.177659\pi$$
0.848246 + 0.529603i $$0.177659\pi$$
$$930$$ 0 0
$$931$$ 4.47214 0.146568
$$932$$ −6.14590 −0.201316
$$933$$ −25.1672 −0.823937
$$934$$ −17.7082 −0.579430
$$935$$ 0 0
$$936$$ −10.6525 −0.348187
$$937$$ 30.7639 1.00501 0.502507 0.864573i $$-0.332411\pi$$
0.502507 + 0.864573i $$0.332411\pi$$
$$938$$ 0.381966 0.0124716
$$939$$ 35.1935 1.14850
$$940$$ 0 0
$$941$$ −0.763932 −0.0249035 −0.0124517 0.999922i $$-0.503964\pi$$
−0.0124517 + 0.999922i $$0.503964\pi$$
$$942$$ 10.4721 0.341201
$$943$$ 16.2918 0.530534
$$944$$ 8.29180 0.269875
$$945$$ 0 0
$$946$$ 42.7426 1.38968
$$947$$ −18.8328 −0.611984 −0.305992 0.952034i $$-0.598988\pi$$
−0.305992 + 0.952034i $$0.598988\pi$$
$$948$$ 8.54102 0.277399
$$949$$ 42.8328 1.39041
$$950$$ 0 0
$$951$$ −23.9311 −0.776020
$$952$$ 14.4721 0.469045
$$953$$ −5.47214 −0.177260 −0.0886299 0.996065i $$-0.528249\pi$$
−0.0886299 + 0.996065i $$0.528249\pi$$
$$954$$ 1.12461 0.0364107
$$955$$ 0 0
$$956$$ −8.94427 −0.289278
$$957$$ 26.1803 0.846290
$$958$$ −6.18034 −0.199678
$$959$$ 6.94427 0.224242
$$960$$ 0 0
$$961$$ 63.2492 2.04030
$$962$$ −15.7082 −0.506453
$$963$$ −11.7771 −0.379511
$$964$$ −7.70820 −0.248265
$$965$$ 0 0
$$966$$ 3.52786 0.113507
$$967$$ −49.8885 −1.60431 −0.802154 0.597118i $$-0.796313\pi$$
−0.802154 + 0.597118i $$0.796313\pi$$
$$968$$ −15.5279 −0.499084
$$969$$ −35.7771 −1.14933
$$970$$ 0 0
$$971$$ −18.0000 −0.577647 −0.288824 0.957382i $$-0.593264\pi$$
−0.288824 + 0.957382i $$0.593264\pi$$
$$972$$ 8.40325 0.269534
$$973$$ −20.6525 −0.662088
$$974$$ 16.5623 0.530691
$$975$$ 0 0
$$976$$ −18.0000 −0.576166
$$977$$ −2.52786 −0.0808735 −0.0404368 0.999182i $$-0.512875\pi$$
−0.0404368 + 0.999182i $$0.512875\pi$$
$$978$$ −20.9443 −0.669724
$$979$$ 54.0689 1.72805
$$980$$ 0 0
$$981$$ 27.1115 0.865602
$$982$$ −16.5623 −0.528524
$$983$$ 32.5410 1.03790 0.518949 0.854805i $$-0.326324\pi$$
0.518949 + 0.854805i $$0.326324\pi$$
$$984$$ −25.5279 −0.813799
$$985$$ 0 0
$$986$$ −52.3607 −1.66750
$$987$$ −2.47214 −0.0786890
$$988$$ −8.94427 −0.284555
$$989$$ 11.0000 0.349780
$$990$$ 0 0
$$991$$ −9.18034 −0.291623 −0.145812 0.989312i $$-0.546579\pi$$
−0.145812 + 0.989312i $$0.546579\pi$$
$$992$$ 32.8328 1.04244
$$993$$ −13.9574 −0.442926
$$994$$ −7.61803 −0.241629
$$995$$ 0 0
$$996$$ 4.36068 0.138173
$$997$$ −18.5836 −0.588548 −0.294274 0.955721i $$-0.595078\pi$$
−0.294274 + 0.955721i $$0.595078\pi$$
$$998$$ 46.8328 1.48247
$$999$$ −16.5836 −0.524682
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 175.2.a.e.1.2 yes 2
3.2 odd 2 1575.2.a.n.1.1 2
4.3 odd 2 2800.2.a.bp.1.1 2
5.2 odd 4 175.2.b.c.99.4 4
5.3 odd 4 175.2.b.c.99.1 4
5.4 even 2 175.2.a.d.1.1 2
7.6 odd 2 1225.2.a.u.1.2 2
15.2 even 4 1575.2.d.k.1324.1 4
15.8 even 4 1575.2.d.k.1324.4 4
15.14 odd 2 1575.2.a.s.1.2 2
20.3 even 4 2800.2.g.s.449.2 4
20.7 even 4 2800.2.g.s.449.3 4
20.19 odd 2 2800.2.a.bh.1.2 2
35.13 even 4 1225.2.b.k.99.1 4
35.27 even 4 1225.2.b.k.99.4 4
35.34 odd 2 1225.2.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.1 2 5.4 even 2
175.2.a.e.1.2 yes 2 1.1 even 1 trivial
175.2.b.c.99.1 4 5.3 odd 4
175.2.b.c.99.4 4 5.2 odd 4
1225.2.a.n.1.1 2 35.34 odd 2
1225.2.a.u.1.2 2 7.6 odd 2
1225.2.b.k.99.1 4 35.13 even 4
1225.2.b.k.99.4 4 35.27 even 4
1575.2.a.n.1.1 2 3.2 odd 2
1575.2.a.s.1.2 2 15.14 odd 2
1575.2.d.k.1324.1 4 15.2 even 4
1575.2.d.k.1324.4 4 15.8 even 4
2800.2.a.bh.1.2 2 20.19 odd 2
2800.2.a.bp.1.1 2 4.3 odd 2
2800.2.g.s.449.2 4 20.3 even 4
2800.2.g.s.449.3 4 20.7 even 4