# Properties

 Label 435.2.a.j.1.1 Level $435$ Weight $2$ Character 435.1 Self dual yes Analytic conductor $3.473$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("435.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 435.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: $$3.47349248793$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.75660$$ of defining polynomial Character $$\chi$$ $$=$$ 435.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.75660 q^{2} -1.00000 q^{3} +5.59883 q^{4} -1.00000 q^{5} +2.75660 q^{6} +0.393832 q^{7} -9.92054 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.75660 q^{2} -1.00000 q^{3} +5.59883 q^{4} -1.00000 q^{5} +2.75660 q^{6} +0.393832 q^{7} -9.92054 q^{8} +1.00000 q^{9} +2.75660 q^{10} -0.393832 q^{11} -5.59883 q^{12} -2.56511 q^{13} -1.08564 q^{14} +1.00000 q^{15} +16.1493 q^{16} +2.07830 q^{17} -2.75660 q^{18} -0.958939 q^{19} -5.59883 q^{20} -0.393832 q^{21} +1.08564 q^{22} +6.15661 q^{23} +9.92054 q^{24} +1.00000 q^{25} +7.07097 q^{26} -1.00000 q^{27} +2.20500 q^{28} -1.00000 q^{29} -2.75660 q^{30} -10.1566 q^{31} -24.6760 q^{32} +0.393832 q^{33} -5.72905 q^{34} -0.393832 q^{35} +5.59883 q^{36} -7.34192 q^{37} +2.64341 q^{38} +2.56511 q^{39} +9.92054 q^{40} -1.65745 q^{41} +1.08564 q^{42} +10.3279 q^{43} -2.20500 q^{44} -1.00000 q^{45} -16.9713 q^{46} -11.5915 q^{47} -16.1493 q^{48} -6.84490 q^{49} -2.75660 q^{50} -2.07830 q^{51} -14.3616 q^{52} -12.3279 q^{53} +2.75660 q^{54} +0.393832 q^{55} -3.90703 q^{56} +0.958939 q^{57} +2.75660 q^{58} -9.54022 q^{59} +5.59883 q^{60} -6.25340 q^{61} +27.9977 q^{62} +0.393832 q^{63} +35.7232 q^{64} +2.56511 q^{65} -1.08564 q^{66} -7.42023 q^{67} +11.6361 q^{68} -6.15661 q^{69} +1.08564 q^{70} +5.98533 q^{71} -9.92054 q^{72} +3.34192 q^{73} +20.2387 q^{74} -1.00000 q^{75} -5.36894 q^{76} -0.155104 q^{77} -7.07097 q^{78} -2.06745 q^{79} -16.1493 q^{80} +1.00000 q^{81} +4.56892 q^{82} +6.41000 q^{83} -2.20500 q^{84} -2.07830 q^{85} -28.4698 q^{86} +1.00000 q^{87} +3.90703 q^{88} +15.8302 q^{89} +2.75660 q^{90} -1.01022 q^{91} +34.4698 q^{92} +10.1566 q^{93} +31.9531 q^{94} +0.958939 q^{95} +24.6760 q^{96} -18.4575 q^{97} +18.8686 q^{98} -0.393832 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 3 * q^2 - 4 * q^3 + 5 * q^4 - 4 * q^5 + 3 * q^6 + 2 * q^7 - 12 * q^8 + 4 * q^9 $$4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9} + 3 q^{10} - 2 q^{11} - 5 q^{12} - 8 q^{13} - 3 q^{14} + 4 q^{15} + 11 q^{16} - 10 q^{17} - 3 q^{18} - 2 q^{19} - 5 q^{20} - 2 q^{21} + 3 q^{22} - 12 q^{23} + 12 q^{24} + 4 q^{25} - 7 q^{26} - 4 q^{27} - 9 q^{28} - 4 q^{29} - 3 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} - q^{34} - 2 q^{35} + 5 q^{36} - 16 q^{37} - 10 q^{38} + 8 q^{39} + 12 q^{40} - 12 q^{41} + 3 q^{42} + 2 q^{43} + 9 q^{44} - 4 q^{45} - 8 q^{46} - 12 q^{47} - 11 q^{48} + 6 q^{49} - 3 q^{50} + 10 q^{51} - 3 q^{52} - 10 q^{53} + 3 q^{54} + 2 q^{55} + 2 q^{57} + 3 q^{58} + 2 q^{59} + 5 q^{60} - 26 q^{61} + 20 q^{62} + 2 q^{63} + 34 q^{64} + 8 q^{65} - 3 q^{66} + 2 q^{67} + 9 q^{68} + 12 q^{69} + 3 q^{70} - 10 q^{71} - 12 q^{72} + 48 q^{74} - 4 q^{75} + 16 q^{76} - 34 q^{77} + 7 q^{78} + 22 q^{79} - 11 q^{80} + 4 q^{81} + 38 q^{82} - 10 q^{83} + 9 q^{84} + 10 q^{85} - 4 q^{86} + 4 q^{87} - 4 q^{89} + 3 q^{90} - 8 q^{91} + 28 q^{92} + 4 q^{93} + 39 q^{94} + 2 q^{95} + 17 q^{96} - 22 q^{97} + 34 q^{98} - 2 q^{99}+O(q^{100})$$ 4 * q - 3 * q^2 - 4 * q^3 + 5 * q^4 - 4 * q^5 + 3 * q^6 + 2 * q^7 - 12 * q^8 + 4 * q^9 + 3 * q^10 - 2 * q^11 - 5 * q^12 - 8 * q^13 - 3 * q^14 + 4 * q^15 + 11 * q^16 - 10 * q^17 - 3 * q^18 - 2 * q^19 - 5 * q^20 - 2 * q^21 + 3 * q^22 - 12 * q^23 + 12 * q^24 + 4 * q^25 - 7 * q^26 - 4 * q^27 - 9 * q^28 - 4 * q^29 - 3 * q^30 - 4 * q^31 - 17 * q^32 + 2 * q^33 - q^34 - 2 * q^35 + 5 * q^36 - 16 * q^37 - 10 * q^38 + 8 * q^39 + 12 * q^40 - 12 * q^41 + 3 * q^42 + 2 * q^43 + 9 * q^44 - 4 * q^45 - 8 * q^46 - 12 * q^47 - 11 * q^48 + 6 * q^49 - 3 * q^50 + 10 * q^51 - 3 * q^52 - 10 * q^53 + 3 * q^54 + 2 * q^55 + 2 * q^57 + 3 * q^58 + 2 * q^59 + 5 * q^60 - 26 * q^61 + 20 * q^62 + 2 * q^63 + 34 * q^64 + 8 * q^65 - 3 * q^66 + 2 * q^67 + 9 * q^68 + 12 * q^69 + 3 * q^70 - 10 * q^71 - 12 * q^72 + 48 * q^74 - 4 * q^75 + 16 * q^76 - 34 * q^77 + 7 * q^78 + 22 * q^79 - 11 * q^80 + 4 * q^81 + 38 * q^82 - 10 * q^83 + 9 * q^84 + 10 * q^85 - 4 * q^86 + 4 * q^87 - 4 * q^89 + 3 * q^90 - 8 * q^91 + 28 * q^92 + 4 * q^93 + 39 * q^94 + 2 * q^95 + 17 * q^96 - 22 * q^97 + 34 * q^98 - 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.75660 −1.94921 −0.974605 0.223932i $$-0.928110\pi$$
−0.974605 + 0.223932i $$0.928110\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.59883 2.79942
$$5$$ −1.00000 −0.447214
$$6$$ 2.75660 1.12538
$$7$$ 0.393832 0.148855 0.0744273 0.997226i $$-0.476287\pi$$
0.0744273 + 0.997226i $$0.476287\pi$$
$$8$$ −9.92054 −3.50744
$$9$$ 1.00000 0.333333
$$10$$ 2.75660 0.871713
$$11$$ −0.393832 −0.118745 −0.0593725 0.998236i $$-0.518910\pi$$
−0.0593725 + 0.998236i $$0.518910\pi$$
$$12$$ −5.59883 −1.61624
$$13$$ −2.56511 −0.711433 −0.355716 0.934594i $$-0.615763\pi$$
−0.355716 + 0.934594i $$0.615763\pi$$
$$14$$ −1.08564 −0.290149
$$15$$ 1.00000 0.258199
$$16$$ 16.1493 4.03732
$$17$$ 2.07830 0.504063 0.252031 0.967719i $$-0.418901\pi$$
0.252031 + 0.967719i $$0.418901\pi$$
$$18$$ −2.75660 −0.649736
$$19$$ −0.958939 −0.219996 −0.109998 0.993932i $$-0.535084\pi$$
−0.109998 + 0.993932i $$0.535084\pi$$
$$20$$ −5.59883 −1.25194
$$21$$ −0.393832 −0.0859413
$$22$$ 1.08564 0.231459
$$23$$ 6.15661 1.28374 0.641871 0.766813i $$-0.278159\pi$$
0.641871 + 0.766813i $$0.278159\pi$$
$$24$$ 9.92054 2.02502
$$25$$ 1.00000 0.200000
$$26$$ 7.07097 1.38673
$$27$$ −1.00000 −0.192450
$$28$$ 2.20500 0.416706
$$29$$ −1.00000 −0.185695
$$30$$ −2.75660 −0.503284
$$31$$ −10.1566 −1.82418 −0.912090 0.409989i $$-0.865532\pi$$
−0.912090 + 0.409989i $$0.865532\pi$$
$$32$$ −24.6760 −4.36214
$$33$$ 0.393832 0.0685574
$$34$$ −5.72905 −0.982524
$$35$$ −0.393832 −0.0665698
$$36$$ 5.59883 0.933139
$$37$$ −7.34192 −1.20700 −0.603502 0.797361i $$-0.706229\pi$$
−0.603502 + 0.797361i $$0.706229\pi$$
$$38$$ 2.64341 0.428818
$$39$$ 2.56511 0.410746
$$40$$ 9.92054 1.56857
$$41$$ −1.65745 −0.258850 −0.129425 0.991589i $$-0.541313\pi$$
−0.129425 + 0.991589i $$0.541313\pi$$
$$42$$ 1.08564 0.167517
$$43$$ 10.3279 1.57499 0.787494 0.616323i $$-0.211378\pi$$
0.787494 + 0.616323i $$0.211378\pi$$
$$44$$ −2.20500 −0.332417
$$45$$ −1.00000 −0.149071
$$46$$ −16.9713 −2.50228
$$47$$ −11.5915 −1.69079 −0.845397 0.534138i $$-0.820636\pi$$
−0.845397 + 0.534138i $$0.820636\pi$$
$$48$$ −16.1493 −2.33095
$$49$$ −6.84490 −0.977842
$$50$$ −2.75660 −0.389842
$$51$$ −2.07830 −0.291021
$$52$$ −14.3616 −1.99160
$$53$$ −12.3279 −1.69336 −0.846682 0.532099i $$-0.821404\pi$$
−0.846682 + 0.532099i $$0.821404\pi$$
$$54$$ 2.75660 0.375126
$$55$$ 0.393832 0.0531043
$$56$$ −3.90703 −0.522099
$$57$$ 0.958939 0.127015
$$58$$ 2.75660 0.361959
$$59$$ −9.54022 −1.24203 −0.621015 0.783798i $$-0.713280\pi$$
−0.621015 + 0.783798i $$0.713280\pi$$
$$60$$ 5.59883 0.722806
$$61$$ −6.25340 −0.800665 −0.400333 0.916370i $$-0.631105\pi$$
−0.400333 + 0.916370i $$0.631105\pi$$
$$62$$ 27.9977 3.55571
$$63$$ 0.393832 0.0496182
$$64$$ 35.7232 4.46540
$$65$$ 2.56511 0.318162
$$66$$ −1.08564 −0.133633
$$67$$ −7.42023 −0.906525 −0.453262 0.891377i $$-0.649740\pi$$
−0.453262 + 0.891377i $$0.649740\pi$$
$$68$$ 11.6361 1.41108
$$69$$ −6.15661 −0.741168
$$70$$ 1.08564 0.129758
$$71$$ 5.98533 0.710328 0.355164 0.934804i $$-0.384425\pi$$
0.355164 + 0.934804i $$0.384425\pi$$
$$72$$ −9.92054 −1.16915
$$73$$ 3.34192 0.391142 0.195571 0.980690i $$-0.437344\pi$$
0.195571 + 0.980690i $$0.437344\pi$$
$$74$$ 20.2387 2.35270
$$75$$ −1.00000 −0.115470
$$76$$ −5.36894 −0.615860
$$77$$ −0.155104 −0.0176757
$$78$$ −7.07097 −0.800630
$$79$$ −2.06745 −0.232607 −0.116303 0.993214i $$-0.537104\pi$$
−0.116303 + 0.993214i $$0.537104\pi$$
$$80$$ −16.1493 −1.80554
$$81$$ 1.00000 0.111111
$$82$$ 4.56892 0.504553
$$83$$ 6.41000 0.703589 0.351795 0.936077i $$-0.385572\pi$$
0.351795 + 0.936077i $$0.385572\pi$$
$$84$$ −2.20500 −0.240585
$$85$$ −2.07830 −0.225424
$$86$$ −28.4698 −3.06998
$$87$$ 1.00000 0.107211
$$88$$ 3.90703 0.416491
$$89$$ 15.8302 1.67800 0.839000 0.544131i $$-0.183140\pi$$
0.839000 + 0.544131i $$0.183140\pi$$
$$90$$ 2.75660 0.290571
$$91$$ −1.01022 −0.105900
$$92$$ 34.4698 3.59373
$$93$$ 10.1566 1.05319
$$94$$ 31.9531 3.29571
$$95$$ 0.958939 0.0983851
$$96$$ 24.6760 2.51848
$$97$$ −18.4575 −1.87407 −0.937036 0.349233i $$-0.886442\pi$$
−0.937036 + 0.349233i $$0.886442\pi$$
$$98$$ 18.8686 1.90602
$$99$$ −0.393832 −0.0395816
$$100$$ 5.59883 0.559883
$$101$$ −12.8038 −1.27403 −0.637015 0.770852i $$-0.719831\pi$$
−0.637015 + 0.770852i $$0.719831\pi$$
$$102$$ 5.72905 0.567260
$$103$$ 4.86979 0.479834 0.239917 0.970793i $$-0.422880\pi$$
0.239917 + 0.970793i $$0.422880\pi$$
$$104$$ 25.4472 2.49531
$$105$$ 0.393832 0.0384341
$$106$$ 33.9830 3.30072
$$107$$ −2.34255 −0.226463 −0.113231 0.993569i $$-0.536120\pi$$
−0.113231 + 0.993569i $$0.536120\pi$$
$$108$$ −5.59883 −0.538748
$$109$$ 8.55044 0.818984 0.409492 0.912314i $$-0.365706\pi$$
0.409492 + 0.912314i $$0.365706\pi$$
$$110$$ −1.08564 −0.103511
$$111$$ 7.34192 0.696864
$$112$$ 6.36011 0.600973
$$113$$ −11.2085 −1.05441 −0.527204 0.849739i $$-0.676760\pi$$
−0.527204 + 0.849739i $$0.676760\pi$$
$$114$$ −2.64341 −0.247578
$$115$$ −6.15661 −0.574107
$$116$$ −5.59883 −0.519839
$$117$$ −2.56511 −0.237144
$$118$$ 26.2985 2.42098
$$119$$ 0.818503 0.0750321
$$120$$ −9.92054 −0.905617
$$121$$ −10.8449 −0.985900
$$122$$ 17.2381 1.56066
$$123$$ 1.65745 0.149447
$$124$$ −56.8652 −5.10664
$$125$$ −1.00000 −0.0894427
$$126$$ −1.08564 −0.0967163
$$127$$ 20.1566 1.78861 0.894305 0.447458i $$-0.147671\pi$$
0.894305 + 0.447458i $$0.147671\pi$$
$$128$$ −49.1226 −4.34187
$$129$$ −10.3279 −0.909319
$$130$$ −7.07097 −0.620165
$$131$$ −5.50235 −0.480742 −0.240371 0.970681i $$-0.577269\pi$$
−0.240371 + 0.970681i $$0.577269\pi$$
$$132$$ 2.20500 0.191921
$$133$$ −0.377661 −0.0327474
$$134$$ 20.4546 1.76701
$$135$$ 1.00000 0.0860663
$$136$$ −20.6179 −1.76797
$$137$$ 7.49853 0.640643 0.320321 0.947309i $$-0.396209\pi$$
0.320321 + 0.947309i $$0.396209\pi$$
$$138$$ 16.9713 1.44469
$$139$$ −9.35277 −0.793292 −0.396646 0.917972i $$-0.629826\pi$$
−0.396646 + 0.917972i $$0.629826\pi$$
$$140$$ −2.20500 −0.186357
$$141$$ 11.5915 0.976180
$$142$$ −16.4992 −1.38458
$$143$$ 1.01022 0.0844790
$$144$$ 16.1493 1.34577
$$145$$ 1.00000 0.0830455
$$146$$ −9.21234 −0.762418
$$147$$ 6.84490 0.564558
$$148$$ −41.1062 −3.37891
$$149$$ −11.5402 −0.945411 −0.472706 0.881220i $$-0.656723\pi$$
−0.472706 + 0.881220i $$0.656723\pi$$
$$150$$ 2.75660 0.225075
$$151$$ 6.08212 0.494956 0.247478 0.968894i $$-0.420398\pi$$
0.247478 + 0.968894i $$0.420398\pi$$
$$152$$ 9.51320 0.771622
$$153$$ 2.07830 0.168021
$$154$$ 0.427559 0.0344537
$$155$$ 10.1566 0.815798
$$156$$ 14.3616 1.14985
$$157$$ −11.5953 −0.925407 −0.462704 0.886513i $$-0.653121\pi$$
−0.462704 + 0.886513i $$0.653121\pi$$
$$158$$ 5.69914 0.453399
$$159$$ 12.3279 0.977665
$$160$$ 24.6760 1.95081
$$161$$ 2.42467 0.191091
$$162$$ −2.75660 −0.216579
$$163$$ −0.855118 −0.0669780 −0.0334890 0.999439i $$-0.510662\pi$$
−0.0334890 + 0.999439i $$0.510662\pi$$
$$164$$ −9.27979 −0.724630
$$165$$ −0.393832 −0.0306598
$$166$$ −17.6698 −1.37144
$$167$$ 7.73194 0.598315 0.299158 0.954204i $$-0.403294\pi$$
0.299158 + 0.954204i $$0.403294\pi$$
$$168$$ 3.90703 0.301434
$$169$$ −6.42023 −0.493863
$$170$$ 5.72905 0.439398
$$171$$ −0.958939 −0.0733319
$$172$$ 57.8241 4.40905
$$173$$ 8.07449 0.613892 0.306946 0.951727i $$-0.400693\pi$$
0.306946 + 0.951727i $$0.400693\pi$$
$$174$$ −2.75660 −0.208977
$$175$$ 0.393832 0.0297709
$$176$$ −6.36011 −0.479411
$$177$$ 9.54022 0.717087
$$178$$ −43.6376 −3.27077
$$179$$ 12.4100 0.927567 0.463784 0.885949i $$-0.346492\pi$$
0.463784 + 0.885949i $$0.346492\pi$$
$$180$$ −5.59883 −0.417312
$$181$$ −2.49765 −0.185649 −0.0928246 0.995682i $$-0.529590\pi$$
−0.0928246 + 0.995682i $$0.529590\pi$$
$$182$$ 2.78478 0.206421
$$183$$ 6.25340 0.462264
$$184$$ −61.0769 −4.50265
$$185$$ 7.34192 0.539789
$$186$$ −27.9977 −2.05289
$$187$$ −0.818503 −0.0598549
$$188$$ −64.8989 −4.73324
$$189$$ −0.393832 −0.0286471
$$190$$ −2.64341 −0.191773
$$191$$ −6.32085 −0.457361 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$192$$ −35.7232 −2.57810
$$193$$ 25.2921 1.82057 0.910284 0.413984i $$-0.135863\pi$$
0.910284 + 0.413984i $$0.135863\pi$$
$$194$$ 50.8798 3.65296
$$195$$ −2.56511 −0.183691
$$196$$ −38.3234 −2.73739
$$197$$ −21.2047 −1.51077 −0.755386 0.655280i $$-0.772551\pi$$
−0.755386 + 0.655280i $$0.772551\pi$$
$$198$$ 1.08564 0.0771529
$$199$$ −2.64723 −0.187657 −0.0938285 0.995588i $$-0.529911\pi$$
−0.0938285 + 0.995588i $$0.529911\pi$$
$$200$$ −9.92054 −0.701488
$$201$$ 7.42023 0.523382
$$202$$ 35.2950 2.48335
$$203$$ −0.393832 −0.0276416
$$204$$ −11.6361 −0.814688
$$205$$ 1.65745 0.115761
$$206$$ −13.4240 −0.935297
$$207$$ 6.15661 0.427914
$$208$$ −41.4246 −2.87228
$$209$$ 0.377661 0.0261234
$$210$$ −1.08564 −0.0749161
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ −69.0218 −4.74043
$$213$$ −5.98533 −0.410108
$$214$$ 6.45747 0.441423
$$215$$ −10.3279 −0.704356
$$216$$ 9.92054 0.675007
$$217$$ −4.00000 −0.271538
$$218$$ −23.5701 −1.59637
$$219$$ −3.34192 −0.225826
$$220$$ 2.20500 0.148661
$$221$$ −5.33107 −0.358607
$$222$$ −20.2387 −1.35833
$$223$$ 12.5504 0.840440 0.420220 0.907422i $$-0.361953\pi$$
0.420220 + 0.907422i $$0.361953\pi$$
$$224$$ −9.71820 −0.649324
$$225$$ 1.00000 0.0666667
$$226$$ 30.8974 2.05526
$$227$$ 1.30149 0.0863829 0.0431914 0.999067i $$-0.486247\pi$$
0.0431914 + 0.999067i $$0.486247\pi$$
$$228$$ 5.36894 0.355567
$$229$$ 3.28682 0.217199 0.108600 0.994086i $$-0.465363\pi$$
0.108600 + 0.994086i $$0.465363\pi$$
$$230$$ 16.9713 1.11905
$$231$$ 0.155104 0.0102051
$$232$$ 9.92054 0.651315
$$233$$ 17.7115 1.16032 0.580159 0.814503i $$-0.302990\pi$$
0.580159 + 0.814503i $$0.302990\pi$$
$$234$$ 7.07097 0.462244
$$235$$ 11.5915 0.756146
$$236$$ −53.4141 −3.47696
$$237$$ 2.06745 0.134296
$$238$$ −2.25628 −0.146253
$$239$$ 21.2651 1.37553 0.687763 0.725935i $$-0.258593\pi$$
0.687763 + 0.725935i $$0.258593\pi$$
$$240$$ 16.1493 1.04243
$$241$$ 15.3177 0.986697 0.493349 0.869832i $$-0.335773\pi$$
0.493349 + 0.869832i $$0.335773\pi$$
$$242$$ 29.8950 1.92172
$$243$$ −1.00000 −0.0641500
$$244$$ −35.0117 −2.24140
$$245$$ 6.84490 0.437304
$$246$$ −4.56892 −0.291304
$$247$$ 2.45978 0.156512
$$248$$ 100.759 6.39820
$$249$$ −6.41000 −0.406217
$$250$$ 2.75660 0.174343
$$251$$ 26.8917 1.69739 0.848696 0.528882i $$-0.177388\pi$$
0.848696 + 0.528882i $$0.177388\pi$$
$$252$$ 2.20500 0.138902
$$253$$ −2.42467 −0.152438
$$254$$ −55.5637 −3.48637
$$255$$ 2.07830 0.130148
$$256$$ 63.9648 3.99780
$$257$$ −6.85512 −0.427611 −0.213805 0.976876i $$-0.568586\pi$$
−0.213805 + 0.976876i $$0.568586\pi$$
$$258$$ 28.4698 1.77245
$$259$$ −2.89149 −0.179668
$$260$$ 14.3616 0.890669
$$261$$ −1.00000 −0.0618984
$$262$$ 15.1678 0.937067
$$263$$ −13.6294 −0.840423 −0.420212 0.907426i $$-0.638044\pi$$
−0.420212 + 0.907426i $$0.638044\pi$$
$$264$$ −3.90703 −0.240461
$$265$$ 12.3279 0.757296
$$266$$ 1.04106 0.0638315
$$267$$ −15.8302 −0.968794
$$268$$ −41.5446 −2.53774
$$269$$ −8.46129 −0.515894 −0.257947 0.966159i $$-0.583046\pi$$
−0.257947 + 0.966159i $$0.583046\pi$$
$$270$$ −2.75660 −0.167761
$$271$$ 6.34255 0.385282 0.192641 0.981269i $$-0.438295\pi$$
0.192641 + 0.981269i $$0.438295\pi$$
$$272$$ 33.5631 2.03506
$$273$$ 1.01022 0.0611414
$$274$$ −20.6704 −1.24875
$$275$$ −0.393832 −0.0237490
$$276$$ −34.4698 −2.07484
$$277$$ −17.1464 −1.03023 −0.515113 0.857122i $$-0.672250\pi$$
−0.515113 + 0.857122i $$0.672250\pi$$
$$278$$ 25.7818 1.54629
$$279$$ −10.1566 −0.608060
$$280$$ 3.90703 0.233490
$$281$$ −12.2985 −0.733670 −0.366835 0.930286i $$-0.619559\pi$$
−0.366835 + 0.930286i $$0.619559\pi$$
$$282$$ −31.9531 −1.90278
$$283$$ 25.1830 1.49697 0.748487 0.663149i $$-0.230781\pi$$
0.748487 + 0.663149i $$0.230781\pi$$
$$284$$ 33.5109 1.98851
$$285$$ −0.958939 −0.0568027
$$286$$ −2.78478 −0.164667
$$287$$ −0.652757 −0.0385311
$$288$$ −24.6760 −1.45405
$$289$$ −12.6807 −0.745921
$$290$$ −2.75660 −0.161873
$$291$$ 18.4575 1.08200
$$292$$ 18.7109 1.09497
$$293$$ 12.3170 0.719569 0.359784 0.933035i $$-0.382850\pi$$
0.359784 + 0.933035i $$0.382850\pi$$
$$294$$ −18.8686 −1.10044
$$295$$ 9.54022 0.555453
$$296$$ 72.8358 4.23350
$$297$$ 0.393832 0.0228525
$$298$$ 31.8117 1.84280
$$299$$ −15.7924 −0.913296
$$300$$ −5.59883 −0.323249
$$301$$ 4.06745 0.234444
$$302$$ −16.7660 −0.964773
$$303$$ 12.8038 0.735561
$$304$$ −15.4862 −0.888193
$$305$$ 6.25340 0.358068
$$306$$ −5.72905 −0.327508
$$307$$ −22.7379 −1.29772 −0.648860 0.760908i $$-0.724754\pi$$
−0.648860 + 0.760908i $$0.724754\pi$$
$$308$$ −0.868401 −0.0494817
$$309$$ −4.86979 −0.277032
$$310$$ −27.9977 −1.59016
$$311$$ −5.33810 −0.302696 −0.151348 0.988481i $$-0.548361\pi$$
−0.151348 + 0.988481i $$0.548361\pi$$
$$312$$ −25.4472 −1.44067
$$313$$ −1.96338 −0.110977 −0.0554885 0.998459i $$-0.517672\pi$$
−0.0554885 + 0.998459i $$0.517672\pi$$
$$314$$ 31.9636 1.80381
$$315$$ −0.393832 −0.0221899
$$316$$ −11.5753 −0.651163
$$317$$ 1.29064 0.0724895 0.0362448 0.999343i $$-0.488460\pi$$
0.0362448 + 0.999343i $$0.488460\pi$$
$$318$$ −33.9830 −1.90567
$$319$$ 0.393832 0.0220504
$$320$$ −35.7232 −1.99699
$$321$$ 2.34255 0.130748
$$322$$ −6.68384 −0.372476
$$323$$ −1.99297 −0.110892
$$324$$ 5.59883 0.311046
$$325$$ −2.56511 −0.142287
$$326$$ 2.35722 0.130554
$$327$$ −8.55044 −0.472840
$$328$$ 16.4428 0.907902
$$329$$ −4.56511 −0.251683
$$330$$ 1.08564 0.0597624
$$331$$ 28.9971 1.59382 0.796911 0.604096i $$-0.206466\pi$$
0.796911 + 0.604096i $$0.206466\pi$$
$$332$$ 35.8885 1.96964
$$333$$ −7.34192 −0.402335
$$334$$ −21.3138 −1.16624
$$335$$ 7.42023 0.405410
$$336$$ −6.36011 −0.346972
$$337$$ −16.7854 −0.914356 −0.457178 0.889375i $$-0.651140\pi$$
−0.457178 + 0.889375i $$0.651140\pi$$
$$338$$ 17.6980 0.962643
$$339$$ 11.2085 0.608763
$$340$$ −11.6361 −0.631055
$$341$$ 4.00000 0.216612
$$342$$ 2.64341 0.142939
$$343$$ −5.45257 −0.294411
$$344$$ −102.458 −5.52417
$$345$$ 6.15661 0.331461
$$346$$ −22.2581 −1.19660
$$347$$ −17.8681 −0.959210 −0.479605 0.877485i $$-0.659220\pi$$
−0.479605 + 0.877485i $$0.659220\pi$$
$$348$$ 5.59883 0.300129
$$349$$ 8.73789 0.467728 0.233864 0.972269i $$-0.424863\pi$$
0.233864 + 0.972269i $$0.424863\pi$$
$$350$$ −1.08564 −0.0580298
$$351$$ 2.56511 0.136915
$$352$$ 9.71820 0.517982
$$353$$ −22.6017 −1.20297 −0.601484 0.798885i $$-0.705424\pi$$
−0.601484 + 0.798885i $$0.705424\pi$$
$$354$$ −26.2985 −1.39775
$$355$$ −5.98533 −0.317668
$$356$$ 88.6308 4.69742
$$357$$ −0.818503 −0.0433198
$$358$$ −34.2094 −1.80802
$$359$$ −13.1830 −0.695772 −0.347886 0.937537i $$-0.613100\pi$$
−0.347886 + 0.937537i $$0.613100\pi$$
$$360$$ 9.92054 0.522858
$$361$$ −18.0804 −0.951602
$$362$$ 6.88503 0.361869
$$363$$ 10.8449 0.569209
$$364$$ −5.65607 −0.296458
$$365$$ −3.34192 −0.174924
$$366$$ −17.2381 −0.901050
$$367$$ 17.4511 0.910938 0.455469 0.890252i $$-0.349472\pi$$
0.455469 + 0.890252i $$0.349472\pi$$
$$368$$ 99.4247 5.18287
$$369$$ −1.65745 −0.0862834
$$370$$ −20.2387 −1.05216
$$371$$ −4.85512 −0.252065
$$372$$ 56.8652 2.94832
$$373$$ −32.2018 −1.66734 −0.833672 0.552260i $$-0.813766\pi$$
−0.833672 + 0.552260i $$0.813766\pi$$
$$374$$ 2.25628 0.116670
$$375$$ 1.00000 0.0516398
$$376$$ 114.994 5.93036
$$377$$ 2.56511 0.132110
$$378$$ 1.08564 0.0558392
$$379$$ 32.3660 1.66253 0.831265 0.555876i $$-0.187617\pi$$
0.831265 + 0.555876i $$0.187617\pi$$
$$380$$ 5.36894 0.275421
$$381$$ −20.1566 −1.03265
$$382$$ 17.4240 0.891492
$$383$$ −25.8739 −1.32209 −0.661047 0.750345i $$-0.729887\pi$$
−0.661047 + 0.750345i $$0.729887\pi$$
$$384$$ 49.1226 2.50678
$$385$$ 0.155104 0.00790483
$$386$$ −69.7203 −3.54867
$$387$$ 10.3279 0.524996
$$388$$ −103.340 −5.24631
$$389$$ 32.6103 1.65341 0.826703 0.562639i $$-0.190214\pi$$
0.826703 + 0.562639i $$0.190214\pi$$
$$390$$ 7.07097 0.358052
$$391$$ 12.7953 0.647086
$$392$$ 67.9051 3.42972
$$393$$ 5.50235 0.277557
$$394$$ 58.4528 2.94481
$$395$$ 2.06745 0.104025
$$396$$ −2.20500 −0.110806
$$397$$ 4.39534 0.220596 0.110298 0.993899i $$-0.464820\pi$$
0.110298 + 0.993899i $$0.464820\pi$$
$$398$$ 7.29734 0.365783
$$399$$ 0.377661 0.0189067
$$400$$ 16.1493 0.807464
$$401$$ −19.0658 −0.952099 −0.476049 0.879418i $$-0.657932\pi$$
−0.476049 + 0.879418i $$0.657932\pi$$
$$402$$ −20.4546 −1.02018
$$403$$ 26.0528 1.29778
$$404$$ −71.6865 −3.56654
$$405$$ −1.00000 −0.0496904
$$406$$ 1.08564 0.0538793
$$407$$ 2.89149 0.143326
$$408$$ 20.6179 1.02074
$$409$$ −1.38235 −0.0683530 −0.0341765 0.999416i $$-0.510881\pi$$
−0.0341765 + 0.999416i $$0.510881\pi$$
$$410$$ −4.56892 −0.225643
$$411$$ −7.49853 −0.369875
$$412$$ 27.2651 1.34326
$$413$$ −3.75725 −0.184882
$$414$$ −16.9713 −0.834094
$$415$$ −6.41000 −0.314655
$$416$$ 63.2965 3.10337
$$417$$ 9.35277 0.458007
$$418$$ −1.04106 −0.0509199
$$419$$ 2.86979 0.140198 0.0700991 0.997540i $$-0.477668\pi$$
0.0700991 + 0.997540i $$0.477668\pi$$
$$420$$ 2.20500 0.107593
$$421$$ 13.6440 0.664970 0.332485 0.943109i $$-0.392113\pi$$
0.332485 + 0.943109i $$0.392113\pi$$
$$422$$ −5.51320 −0.268378
$$423$$ −11.5915 −0.563598
$$424$$ 122.299 5.93938
$$425$$ 2.07830 0.100813
$$426$$ 16.4992 0.799387
$$427$$ −2.46279 −0.119183
$$428$$ −13.1155 −0.633964
$$429$$ −1.01022 −0.0487740
$$430$$ 28.4698 1.37294
$$431$$ −11.9707 −0.576607 −0.288303 0.957539i $$-0.593091\pi$$
−0.288303 + 0.957539i $$0.593091\pi$$
$$432$$ −16.1493 −0.776982
$$433$$ 17.9907 0.864576 0.432288 0.901736i $$-0.357706\pi$$
0.432288 + 0.901736i $$0.357706\pi$$
$$434$$ 11.0264 0.529284
$$435$$ −1.00000 −0.0479463
$$436$$ 47.8725 2.29268
$$437$$ −5.90381 −0.282418
$$438$$ 9.21234 0.440182
$$439$$ −16.2226 −0.774260 −0.387130 0.922025i $$-0.626534\pi$$
−0.387130 + 0.922025i $$0.626534\pi$$
$$440$$ −3.90703 −0.186260
$$441$$ −6.84490 −0.325947
$$442$$ 14.6956 0.698999
$$443$$ −35.3762 −1.68078 −0.840388 0.541986i $$-0.817673\pi$$
−0.840388 + 0.541986i $$0.817673\pi$$
$$444$$ 41.1062 1.95081
$$445$$ −15.8302 −0.750425
$$446$$ −34.5965 −1.63819
$$447$$ 11.5402 0.545834
$$448$$ 14.0690 0.664696
$$449$$ 41.6971 1.96781 0.983903 0.178702i $$-0.0571898\pi$$
0.983903 + 0.178702i $$0.0571898\pi$$
$$450$$ −2.75660 −0.129947
$$451$$ 0.652757 0.0307371
$$452$$ −62.7546 −2.95173
$$453$$ −6.08212 −0.285763
$$454$$ −3.58768 −0.168378
$$455$$ 1.01022 0.0473599
$$456$$ −9.51320 −0.445496
$$457$$ −16.1111 −0.753646 −0.376823 0.926285i $$-0.622983\pi$$
−0.376823 + 0.926285i $$0.622983\pi$$
$$458$$ −9.06045 −0.423367
$$459$$ −2.07830 −0.0970069
$$460$$ −34.4698 −1.60716
$$461$$ 17.3396 0.807586 0.403793 0.914850i $$-0.367692\pi$$
0.403793 + 0.914850i $$0.367692\pi$$
$$462$$ −0.427559 −0.0198918
$$463$$ 19.3177 0.897768 0.448884 0.893590i $$-0.351822\pi$$
0.448884 + 0.893590i $$0.351822\pi$$
$$464$$ −16.1493 −0.749711
$$465$$ −10.1566 −0.471001
$$466$$ −48.8235 −2.26170
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ −14.3616 −0.663866
$$469$$ −2.92232 −0.134940
$$470$$ −31.9531 −1.47389
$$471$$ 11.5953 0.534284
$$472$$ 94.6441 4.35635
$$473$$ −4.06745 −0.187022
$$474$$ −5.69914 −0.261770
$$475$$ −0.958939 −0.0439992
$$476$$ 4.58266 0.210046
$$477$$ −12.3279 −0.564455
$$478$$ −58.6194 −2.68119
$$479$$ −40.3877 −1.84536 −0.922681 0.385565i $$-0.874006\pi$$
−0.922681 + 0.385565i $$0.874006\pi$$
$$480$$ −24.6760 −1.12630
$$481$$ 18.8328 0.858702
$$482$$ −42.2246 −1.92328
$$483$$ −2.42467 −0.110326
$$484$$ −60.7188 −2.75994
$$485$$ 18.4575 0.838110
$$486$$ 2.75660 0.125042
$$487$$ 2.84171 0.128770 0.0643850 0.997925i $$-0.479491\pi$$
0.0643850 + 0.997925i $$0.479491\pi$$
$$488$$ 62.0371 2.80829
$$489$$ 0.855118 0.0386698
$$490$$ −18.8686 −0.852398
$$491$$ 0.157863 0.00712428 0.00356214 0.999994i $$-0.498866\pi$$
0.00356214 + 0.999994i $$0.498866\pi$$
$$492$$ 9.27979 0.418365
$$493$$ −2.07830 −0.0936021
$$494$$ −6.78063 −0.305075
$$495$$ 0.393832 0.0177014
$$496$$ −164.022 −7.36480
$$497$$ 2.35722 0.105736
$$498$$ 17.6698 0.791803
$$499$$ 2.64723 0.118506 0.0592531 0.998243i $$-0.481128\pi$$
0.0592531 + 0.998243i $$0.481128\pi$$
$$500$$ −5.59883 −0.250387
$$501$$ −7.73194 −0.345437
$$502$$ −74.1297 −3.30857
$$503$$ 13.9545 0.622200 0.311100 0.950377i $$-0.399303\pi$$
0.311100 + 0.950377i $$0.399303\pi$$
$$504$$ −3.90703 −0.174033
$$505$$ 12.8038 0.569763
$$506$$ 6.68384 0.297133
$$507$$ 6.42023 0.285132
$$508$$ 112.853 5.00706
$$509$$ −16.4921 −0.731001 −0.365500 0.930811i $$-0.619102\pi$$
−0.365500 + 0.930811i $$0.619102\pi$$
$$510$$ −5.72905 −0.253687
$$511$$ 1.31616 0.0582233
$$512$$ −78.0802 −3.45069
$$513$$ 0.958939 0.0423382
$$514$$ 18.8968 0.833502
$$515$$ −4.86979 −0.214588
$$516$$ −57.8241 −2.54556
$$517$$ 4.56511 0.200773
$$518$$ 7.97066 0.350211
$$519$$ −8.07449 −0.354431
$$520$$ −25.4472 −1.11594
$$521$$ −12.2253 −0.535601 −0.267800 0.963474i $$-0.586297\pi$$
−0.267800 + 0.963474i $$0.586297\pi$$
$$522$$ 2.75660 0.120653
$$523$$ 1.66659 0.0728748 0.0364374 0.999336i $$-0.488399\pi$$
0.0364374 + 0.999336i $$0.488399\pi$$
$$524$$ −30.8067 −1.34580
$$525$$ −0.393832 −0.0171883
$$526$$ 37.5707 1.63816
$$527$$ −21.1085 −0.919501
$$528$$ 6.36011 0.276788
$$529$$ 14.9038 0.647992
$$530$$ −33.9830 −1.47613
$$531$$ −9.54022 −0.414010
$$532$$ −2.11446 −0.0916736
$$533$$ 4.25154 0.184155
$$534$$ 43.6376 1.88838
$$535$$ 2.34255 0.101277
$$536$$ 73.6126 3.17958
$$537$$ −12.4100 −0.535531
$$538$$ 23.3244 1.00558
$$539$$ 2.69574 0.116114
$$540$$ 5.59883 0.240935
$$541$$ −34.6558 −1.48997 −0.744984 0.667083i $$-0.767543\pi$$
−0.744984 + 0.667083i $$0.767543\pi$$
$$542$$ −17.4839 −0.750996
$$543$$ 2.49765 0.107185
$$544$$ −51.2842 −2.19879
$$545$$ −8.55044 −0.366261
$$546$$ −2.78478 −0.119177
$$547$$ 38.2530 1.63558 0.817791 0.575515i $$-0.195199\pi$$
0.817791 + 0.575515i $$0.195199\pi$$
$$548$$ 41.9830 1.79343
$$549$$ −6.25340 −0.266888
$$550$$ 1.08564 0.0462917
$$551$$ 0.958939 0.0408522
$$552$$ 61.0769 2.59960
$$553$$ −0.814230 −0.0346246
$$554$$ 47.2657 2.00813
$$555$$ −7.34192 −0.311647
$$556$$ −52.3646 −2.22075
$$557$$ −28.7760 −1.21928 −0.609639 0.792679i $$-0.708686\pi$$
−0.609639 + 0.792679i $$0.708686\pi$$
$$558$$ 27.9977 1.18524
$$559$$ −26.4921 −1.12050
$$560$$ −6.36011 −0.268764
$$561$$ 0.818503 0.0345572
$$562$$ 33.9022 1.43008
$$563$$ −32.9809 −1.38998 −0.694989 0.719020i $$-0.744591\pi$$
−0.694989 + 0.719020i $$0.744591\pi$$
$$564$$ 64.8989 2.73274
$$565$$ 11.2085 0.471546
$$566$$ −69.4194 −2.91792
$$567$$ 0.393832 0.0165394
$$568$$ −59.3777 −2.49143
$$569$$ 16.8038 0.704453 0.352227 0.935915i $$-0.385425\pi$$
0.352227 + 0.935915i $$0.385425\pi$$
$$570$$ 2.64341 0.110720
$$571$$ −6.21703 −0.260175 −0.130087 0.991503i $$-0.541526\pi$$
−0.130087 + 0.991503i $$0.541526\pi$$
$$572$$ 5.65607 0.236492
$$573$$ 6.32085 0.264057
$$574$$ 1.79939 0.0751051
$$575$$ 6.15661 0.256748
$$576$$ 35.7232 1.48847
$$577$$ 11.9977 0.499470 0.249735 0.968314i $$-0.419656\pi$$
0.249735 + 0.968314i $$0.419656\pi$$
$$578$$ 34.9555 1.45396
$$579$$ −25.2921 −1.05111
$$580$$ 5.59883 0.232479
$$581$$ 2.52447 0.104733
$$582$$ −50.8798 −2.10904
$$583$$ 4.85512 0.201078
$$584$$ −33.1537 −1.37191
$$585$$ 2.56511 0.106054
$$586$$ −33.9531 −1.40259
$$587$$ −11.2194 −0.463073 −0.231536 0.972826i $$-0.574375\pi$$
−0.231536 + 0.972826i $$0.574375\pi$$
$$588$$ 38.3234 1.58043
$$589$$ 9.73957 0.401312
$$590$$ −26.2985 −1.08269
$$591$$ 21.2047 0.872245
$$592$$ −118.567 −4.87306
$$593$$ −7.69682 −0.316071 −0.158035 0.987433i $$-0.550516\pi$$
−0.158035 + 0.987433i $$0.550516\pi$$
$$594$$ −1.08564 −0.0445442
$$595$$ −0.818503 −0.0335554
$$596$$ −64.6118 −2.64660
$$597$$ 2.64723 0.108344
$$598$$ 43.5332 1.78020
$$599$$ 20.0543 0.819396 0.409698 0.912221i $$-0.365634\pi$$
0.409698 + 0.912221i $$0.365634\pi$$
$$600$$ 9.92054 0.405004
$$601$$ 3.10851 0.126799 0.0633995 0.997988i $$-0.479806\pi$$
0.0633995 + 0.997988i $$0.479806\pi$$
$$602$$ −11.2123 −0.456981
$$603$$ −7.42023 −0.302175
$$604$$ 34.0528 1.38559
$$605$$ 10.8449 0.440908
$$606$$ −35.2950 −1.43376
$$607$$ 37.0441 1.50357 0.751786 0.659407i $$-0.229193\pi$$
0.751786 + 0.659407i $$0.229193\pi$$
$$608$$ 23.6628 0.959652
$$609$$ 0.393832 0.0159589
$$610$$ −17.2381 −0.697950
$$611$$ 29.7334 1.20289
$$612$$ 11.6361 0.470361
$$613$$ −13.3839 −0.540569 −0.270284 0.962781i $$-0.587118\pi$$
−0.270284 + 0.962781i $$0.587118\pi$$
$$614$$ 62.6792 2.52953
$$615$$ −1.65745 −0.0668348
$$616$$ 1.53871 0.0619966
$$617$$ −12.6364 −0.508721 −0.254361 0.967109i $$-0.581865\pi$$
−0.254361 + 0.967109i $$0.581865\pi$$
$$618$$ 13.4240 0.539994
$$619$$ −41.2447 −1.65776 −0.828882 0.559424i $$-0.811022\pi$$
−0.828882 + 0.559424i $$0.811022\pi$$
$$620$$ 56.8652 2.28376
$$621$$ −6.15661 −0.247056
$$622$$ 14.7150 0.590018
$$623$$ 6.23446 0.249778
$$624$$ 41.4246 1.65831
$$625$$ 1.00000 0.0400000
$$626$$ 5.41226 0.216318
$$627$$ −0.377661 −0.0150823
$$628$$ −64.9203 −2.59060
$$629$$ −15.2587 −0.608406
$$630$$ 1.08564 0.0432528
$$631$$ 29.8009 1.18635 0.593177 0.805072i $$-0.297873\pi$$
0.593177 + 0.805072i $$0.297873\pi$$
$$632$$ 20.5103 0.815854
$$633$$ −2.00000 −0.0794929
$$634$$ −3.55777 −0.141297
$$635$$ −20.1566 −0.799891
$$636$$ 69.0218 2.73689
$$637$$ 17.5579 0.695669
$$638$$ −1.08564 −0.0429808
$$639$$ 5.98533 0.236776
$$640$$ 49.1226 1.94174
$$641$$ 17.7774 0.702167 0.351083 0.936344i $$-0.385813\pi$$
0.351083 + 0.936344i $$0.385813\pi$$
$$642$$ −6.45747 −0.254856
$$643$$ −44.5534 −1.75702 −0.878508 0.477727i $$-0.841461\pi$$
−0.878508 + 0.477727i $$0.841461\pi$$
$$644$$ 13.5753 0.534943
$$645$$ 10.3279 0.406660
$$646$$ 5.49381 0.216151
$$647$$ 42.9339 1.68790 0.843952 0.536418i $$-0.180223\pi$$
0.843952 + 0.536418i $$0.180223\pi$$
$$648$$ −9.92054 −0.389716
$$649$$ 3.75725 0.147485
$$650$$ 7.07097 0.277346
$$651$$ 4.00000 0.156772
$$652$$ −4.78766 −0.187499
$$653$$ −19.8426 −0.776500 −0.388250 0.921554i $$-0.626920\pi$$
−0.388250 + 0.921554i $$0.626920\pi$$
$$654$$ 23.5701 0.921665
$$655$$ 5.50235 0.214994
$$656$$ −26.7666 −1.04506
$$657$$ 3.34192 0.130381
$$658$$ 12.5842 0.490582
$$659$$ 23.0483 0.897836 0.448918 0.893573i $$-0.351810\pi$$
0.448918 + 0.893573i $$0.351810\pi$$
$$660$$ −2.20500 −0.0858296
$$661$$ −36.2079 −1.40832 −0.704162 0.710040i $$-0.748677\pi$$
−0.704162 + 0.710040i $$0.748677\pi$$
$$662$$ −79.9332 −3.10669
$$663$$ 5.33107 0.207042
$$664$$ −63.5907 −2.46780
$$665$$ 0.377661 0.0146451
$$666$$ 20.2387 0.784235
$$667$$ −6.15661 −0.238385
$$668$$ 43.2898 1.67493
$$669$$ −12.5504 −0.485228
$$670$$ −20.4546 −0.790229
$$671$$ 2.46279 0.0950749
$$672$$ 9.71820 0.374888
$$673$$ 22.9000 0.882731 0.441365 0.897327i $$-0.354494\pi$$
0.441365 + 0.897327i $$0.354494\pi$$
$$674$$ 46.2705 1.78227
$$675$$ −1.00000 −0.0384900
$$676$$ −35.9458 −1.38253
$$677$$ 36.0068 1.38385 0.691927 0.721967i $$-0.256762\pi$$
0.691927 + 0.721967i $$0.256762\pi$$
$$678$$ −30.8974 −1.18661
$$679$$ −7.26915 −0.278964
$$680$$ 20.6179 0.790660
$$681$$ −1.30149 −0.0498732
$$682$$ −11.0264 −0.422222
$$683$$ 9.12896 0.349310 0.174655 0.984630i $$-0.444119\pi$$
0.174655 + 0.984630i $$0.444119\pi$$
$$684$$ −5.36894 −0.205287
$$685$$ −7.49853 −0.286504
$$686$$ 15.0305 0.573869
$$687$$ −3.28682 −0.125400
$$688$$ 166.788 6.35873
$$689$$ 31.6223 1.20472
$$690$$ −16.9713 −0.646086
$$691$$ 5.85956 0.222908 0.111454 0.993770i $$-0.464449\pi$$
0.111454 + 0.993770i $$0.464449\pi$$
$$692$$ 45.2077 1.71854
$$693$$ −0.155104 −0.00589191
$$694$$ 49.2552 1.86970
$$695$$ 9.35277 0.354771
$$696$$ −9.92054 −0.376037
$$697$$ −3.44469 −0.130477
$$698$$ −24.0868 −0.911700
$$699$$ −17.7115 −0.669910
$$700$$ 2.20500 0.0833412
$$701$$ −7.56955 −0.285898 −0.142949 0.989730i $$-0.545658\pi$$
−0.142949 + 0.989730i $$0.545658\pi$$
$$702$$ −7.07097 −0.266877
$$703$$ 7.04046 0.265536
$$704$$ −14.0690 −0.530244
$$705$$ −11.5915 −0.436561
$$706$$ 62.3039 2.34484
$$707$$ −5.04256 −0.189645
$$708$$ 53.4141 2.00742
$$709$$ 14.5209 0.545342 0.272671 0.962107i $$-0.412093\pi$$
0.272671 + 0.962107i $$0.412093\pi$$
$$710$$ 16.4992 0.619202
$$711$$ −2.06745 −0.0775356
$$712$$ −157.044 −5.88549
$$713$$ −62.5302 −2.34178
$$714$$ 2.25628 0.0844393
$$715$$ −1.01022 −0.0377802
$$716$$ 69.4815 2.59665
$$717$$ −21.2651 −0.794161
$$718$$ 36.3402 1.35621
$$719$$ 12.2164 0.455596 0.227798 0.973708i $$-0.426847\pi$$
0.227798 + 0.973708i $$0.426847\pi$$
$$720$$ −16.1493 −0.601848
$$721$$ 1.91788 0.0714255
$$722$$ 49.8405 1.85487
$$723$$ −15.3177 −0.569670
$$724$$ −13.9839 −0.519709
$$725$$ −1.00000 −0.0371391
$$726$$ −29.8950 −1.10951
$$727$$ −40.8856 −1.51636 −0.758182 0.652044i $$-0.773912\pi$$
−0.758182 + 0.652044i $$0.773912\pi$$
$$728$$ 10.0219 0.371438
$$729$$ 1.00000 0.0370370
$$730$$ 9.21234 0.340964
$$731$$ 21.4645 0.793892
$$732$$ 35.0117 1.29407
$$733$$ 25.4915 0.941550 0.470775 0.882253i $$-0.343974\pi$$
0.470775 + 0.882253i $$0.343974\pi$$
$$734$$ −48.1056 −1.77561
$$735$$ −6.84490 −0.252478
$$736$$ −151.920 −5.59986
$$737$$ 2.92232 0.107645
$$738$$ 4.56892 0.168184
$$739$$ 9.56830 0.351975 0.175988 0.984392i $$-0.443688\pi$$
0.175988 + 0.984392i $$0.443688\pi$$
$$740$$ 41.1062 1.51109
$$741$$ −2.45978 −0.0903624
$$742$$ 13.3836 0.491328
$$743$$ −18.0455 −0.662025 −0.331013 0.943626i $$-0.607390\pi$$
−0.331013 + 0.943626i $$0.607390\pi$$
$$744$$ −100.759 −3.69400
$$745$$ 11.5402 0.422801
$$746$$ 88.7673 3.25000
$$747$$ 6.41000 0.234530
$$748$$ −4.58266 −0.167559
$$749$$ −0.922572 −0.0337100
$$750$$ −2.75660 −0.100657
$$751$$ 11.9255 0.435168 0.217584 0.976042i $$-0.430182\pi$$
0.217584 + 0.976042i $$0.430182\pi$$
$$752$$ −187.194 −6.82627
$$753$$ −26.8917 −0.979989
$$754$$ −7.07097 −0.257510
$$755$$ −6.08212 −0.221351
$$756$$ −2.20500 −0.0801951
$$757$$ 5.86152 0.213041 0.106520 0.994311i $$-0.466029\pi$$
0.106520 + 0.994311i $$0.466029\pi$$
$$758$$ −89.2201 −3.24062
$$759$$ 2.42467 0.0880100
$$760$$ −9.51320 −0.345080
$$761$$ 49.8739 1.80793 0.903963 0.427610i $$-0.140644\pi$$
0.903963 + 0.427610i $$0.140644\pi$$
$$762$$ 55.5637 2.01286
$$763$$ 3.36744 0.121909
$$764$$ −35.3894 −1.28034
$$765$$ −2.07830 −0.0751412
$$766$$ 71.3239 2.57704
$$767$$ 24.4717 0.883621
$$768$$ −63.9648 −2.30813
$$769$$ 29.5121 1.06423 0.532117 0.846671i $$-0.321396\pi$$
0.532117 + 0.846671i $$0.321396\pi$$
$$770$$ −0.427559 −0.0154082
$$771$$ 6.85512 0.246881
$$772$$ 141.607 5.09653
$$773$$ −2.41935 −0.0870180 −0.0435090 0.999053i $$-0.513854\pi$$
−0.0435090 + 0.999053i $$0.513854\pi$$
$$774$$ −28.4698 −1.02333
$$775$$ −10.1566 −0.364836
$$776$$ 183.108 6.57320
$$777$$ 2.89149 0.103731
$$778$$ −89.8934 −3.22283
$$779$$ 1.58939 0.0569460
$$780$$ −14.3616 −0.514228
$$781$$ −2.35722 −0.0843479
$$782$$ −35.2715 −1.26131
$$783$$ 1.00000 0.0357371
$$784$$ −110.540 −3.94786
$$785$$ 11.5953 0.413855
$$786$$ −15.1678 −0.541016
$$787$$ −51.1549 −1.82348 −0.911738 0.410772i $$-0.865259\pi$$
−0.911738 + 0.410772i $$0.865259\pi$$
$$788$$ −118.722 −4.22928
$$789$$ 13.6294 0.485218
$$790$$ −5.69914 −0.202766
$$791$$ −4.41428 −0.156954
$$792$$ 3.90703 0.138830
$$793$$ 16.0406 0.569619
$$794$$ −12.1162 −0.429987
$$795$$ −12.3279 −0.437225
$$796$$ −14.8214 −0.525330
$$797$$ 28.8662 1.02249 0.511247 0.859434i $$-0.329184\pi$$
0.511247 + 0.859434i $$0.329184\pi$$
$$798$$ −1.04106 −0.0368531
$$799$$ −24.0907 −0.852266
$$800$$ −24.6760 −0.872428
$$801$$ 15.8302 0.559334
$$802$$ 52.5567 1.85584
$$803$$ −1.31616 −0.0464462
$$804$$ 41.5446 1.46517
$$805$$ −2.42467 −0.0854584
$$806$$ −71.8171 −2.52965
$$807$$ 8.46129 0.297851
$$808$$ 127.021 4.46858
$$809$$ 19.0426 0.669501 0.334750 0.942307i $$-0.391348\pi$$
0.334750 + 0.942307i $$0.391348\pi$$
$$810$$ 2.75660 0.0968570
$$811$$ −20.8783 −0.733137 −0.366569 0.930391i $$-0.619467\pi$$
−0.366569 + 0.930391i $$0.619467\pi$$
$$812$$ −2.20500 −0.0773804
$$813$$ −6.34255 −0.222443
$$814$$ −7.97066 −0.279372
$$815$$ 0.855118 0.0299535
$$816$$ −33.5631 −1.17494
$$817$$ −9.90381 −0.346491
$$818$$ 3.81060 0.133234
$$819$$ −1.01022 −0.0353000
$$820$$ 9.27979 0.324064
$$821$$ 3.60466 0.125804 0.0629018 0.998020i $$-0.479965\pi$$
0.0629018 + 0.998020i $$0.479965\pi$$
$$822$$ 20.6704 0.720964
$$823$$ 27.2288 0.949135 0.474567 0.880219i $$-0.342605\pi$$
0.474567 + 0.880219i $$0.342605\pi$$
$$824$$ −48.3109 −1.68299
$$825$$ 0.393832 0.0137115
$$826$$ 10.3572 0.360374
$$827$$ 47.3936 1.64804 0.824019 0.566562i $$-0.191727\pi$$
0.824019 + 0.566562i $$0.191727\pi$$
$$828$$ 34.4698 1.19791
$$829$$ −41.8388 −1.45312 −0.726560 0.687103i $$-0.758882\pi$$
−0.726560 + 0.687103i $$0.758882\pi$$
$$830$$ 17.6698 0.613328
$$831$$ 17.1464 0.594802
$$832$$ −91.6339 −3.17683
$$833$$ −14.2258 −0.492894
$$834$$ −25.7818 −0.892752
$$835$$ −7.73194 −0.267575
$$836$$ 2.11446 0.0731302
$$837$$ 10.1566 0.351064
$$838$$ −7.91085 −0.273276
$$839$$ −31.6385 −1.09228 −0.546141 0.837693i $$-0.683904\pi$$
−0.546141 + 0.837693i $$0.683904\pi$$
$$840$$ −3.90703 −0.134805
$$841$$ 1.00000 0.0344828
$$842$$ −37.6111 −1.29617
$$843$$ 12.2985 0.423584
$$844$$ 11.1977 0.385440
$$845$$ 6.42023 0.220862
$$846$$ 31.9531 1.09857
$$847$$ −4.27107 −0.146756
$$848$$ −199.086 −6.83665
$$849$$ −25.1830 −0.864278
$$850$$ −5.72905 −0.196505
$$851$$ −45.2013 −1.54948
$$852$$ −33.5109 −1.14806
$$853$$ 44.2000 1.51338 0.756690 0.653773i $$-0.226815\pi$$
0.756690 + 0.653773i $$0.226815\pi$$
$$854$$ 6.78892 0.232312
$$855$$ 0.958939 0.0327950
$$856$$ 23.2394 0.794305
$$857$$ 14.4775 0.494541 0.247270 0.968947i $$-0.420466\pi$$
0.247270 + 0.968947i $$0.420466\pi$$
$$858$$ 2.78478 0.0950707
$$859$$ 22.5883 0.770703 0.385352 0.922770i $$-0.374080\pi$$
0.385352 + 0.922770i $$0.374080\pi$$
$$860$$ −57.8241 −1.97179
$$861$$ 0.652757 0.0222459
$$862$$ 32.9983 1.12393
$$863$$ −22.9900 −0.782590 −0.391295 0.920265i $$-0.627973\pi$$
−0.391295 + 0.920265i $$0.627973\pi$$
$$864$$ 24.6760 0.839494
$$865$$ −8.07449 −0.274541
$$866$$ −49.5930 −1.68524
$$867$$ 12.6807 0.430658
$$868$$ −22.3953 −0.760147
$$869$$ 0.814230 0.0276209
$$870$$ 2.75660 0.0934574
$$871$$ 19.0337 0.644931
$$872$$ −84.8250 −2.87254
$$873$$ −18.4575 −0.624691
$$874$$ 16.2744 0.550491
$$875$$ −0.393832 −0.0133140
$$876$$ −18.7109 −0.632182
$$877$$ 13.5741 0.458364 0.229182 0.973384i $$-0.426395\pi$$
0.229182 + 0.973384i $$0.426395\pi$$
$$878$$ 44.7191 1.50920
$$879$$ −12.3170 −0.415443
$$880$$ 6.36011 0.214399
$$881$$ 9.91235 0.333956 0.166978 0.985961i $$-0.446599\pi$$
0.166978 + 0.985961i $$0.446599\pi$$
$$882$$ 18.8686 0.635340
$$883$$ 39.3192 1.32320 0.661598 0.749859i $$-0.269879\pi$$
0.661598 + 0.749859i $$0.269879\pi$$
$$884$$ −29.8478 −1.00389
$$885$$ −9.54022 −0.320691
$$886$$ 97.5180 3.27618
$$887$$ −59.2520 −1.98949 −0.994743 0.102403i $$-0.967347\pi$$
−0.994743 + 0.102403i $$0.967347\pi$$
$$888$$ −72.8358 −2.44421
$$889$$ 7.93832 0.266243
$$890$$ 43.6376 1.46274
$$891$$ −0.393832 −0.0131939
$$892$$ 70.2678 2.35274
$$893$$ 11.1155 0.371968
$$894$$ −31.8117 −1.06394
$$895$$ −12.4100 −0.414821
$$896$$ −19.3461 −0.646307
$$897$$ 15.7924 0.527291
$$898$$ −114.942 −3.83567
$$899$$ 10.1566 0.338742
$$900$$ 5.59883 0.186628
$$901$$ −25.6211 −0.853562
$$902$$ −1.79939 −0.0599131
$$903$$ −4.06745 −0.135356
$$904$$ 111.195 3.69828
$$905$$ 2.49765 0.0830248
$$906$$ 16.7660 0.557012
$$907$$ −5.91210 −0.196308 −0.0981541 0.995171i $$-0.531294\pi$$
−0.0981541 + 0.995171i $$0.531294\pi$$
$$908$$ 7.28682 0.241822
$$909$$ −12.8038 −0.424676
$$910$$ −2.78478 −0.0923144
$$911$$ −12.4185 −0.411445 −0.205722 0.978610i $$-0.565954\pi$$
−0.205722 + 0.978610i $$0.565954\pi$$
$$912$$ 15.4862 0.512799
$$913$$ −2.52447 −0.0835476
$$914$$ 44.4118 1.46901
$$915$$ −6.25340 −0.206731
$$916$$ 18.4024 0.608031
$$917$$ −2.16700 −0.0715607
$$918$$ 5.72905 0.189087
$$919$$ 16.4332 0.542081 0.271041 0.962568i $$-0.412632\pi$$
0.271041 + 0.962568i $$0.412632\pi$$
$$920$$ 61.0769 2.01364
$$921$$ 22.7379 0.749239
$$922$$ −47.7983 −1.57415
$$923$$ −15.3530 −0.505351
$$924$$ 0.868401 0.0285683
$$925$$ −7.34192 −0.241401
$$926$$ −53.2510 −1.74994
$$927$$ 4.86979 0.159945
$$928$$ 24.6760 0.810029
$$929$$ 13.8358 0.453936 0.226968 0.973902i $$-0.427119\pi$$
0.226968 + 0.973902i $$0.427119\pi$$
$$930$$ 27.9977 0.918080
$$931$$ 6.56384 0.215121
$$932$$ 99.1637 3.24822
$$933$$ 5.33810 0.174762
$$934$$ 22.0528 0.721589
$$935$$ 0.818503 0.0267679
$$936$$ 25.4472 0.831769
$$937$$ 20.7528 0.677964 0.338982 0.940793i $$-0.389917\pi$$
0.338982 + 0.940793i $$0.389917\pi$$
$$938$$ 8.05567 0.263027
$$939$$ 1.96338 0.0640726
$$940$$ 64.8989 2.11677
$$941$$ 40.1666 1.30940 0.654698 0.755891i $$-0.272796\pi$$
0.654698 + 0.755891i $$0.272796\pi$$
$$942$$ −31.9636 −1.04143
$$943$$ −10.2043 −0.332297
$$944$$ −154.068 −5.01447
$$945$$ 0.393832 0.0128114
$$946$$ 11.2123 0.364544
$$947$$ −17.4144 −0.565894 −0.282947 0.959136i $$-0.591312\pi$$
−0.282947 + 0.959136i $$0.591312\pi$$
$$948$$ 11.5753 0.375949
$$949$$ −8.57239 −0.278271
$$950$$ 2.64341 0.0857636
$$951$$ −1.29064 −0.0418518
$$952$$ −8.11999 −0.263170
$$953$$ 13.5121 0.437701 0.218851 0.975758i $$-0.429769\pi$$
0.218851 + 0.975758i $$0.429769\pi$$
$$954$$ 33.9830 1.10024
$$955$$ 6.32085 0.204538
$$956$$ 119.060 3.85067
$$957$$ −0.393832 −0.0127308
$$958$$ 111.333 3.59700
$$959$$ 2.95316 0.0953626
$$960$$ 35.7232 1.15296
$$961$$ 72.1567 2.32763
$$962$$ −51.9145 −1.67379
$$963$$ −2.34255 −0.0754876
$$964$$ 85.7610 2.76218
$$965$$ −25.2921 −0.814183
$$966$$ 6.68384 0.215049
$$967$$ −24.0598 −0.773712 −0.386856 0.922140i $$-0.626439\pi$$
−0.386856 + 0.922140i $$0.626439\pi$$
$$968$$ 107.587 3.45798
$$969$$ 1.99297 0.0640233
$$970$$ −50.8798 −1.63365
$$971$$ 34.7877 1.11639 0.558195 0.829710i $$-0.311494\pi$$
0.558195 + 0.829710i $$0.311494\pi$$
$$972$$ −5.59883 −0.179583
$$973$$ −3.68342 −0.118085
$$974$$ −7.83344 −0.251000
$$975$$ 2.56511 0.0821492
$$976$$ −100.988 −3.23254
$$977$$ 35.7566 1.14396 0.571978 0.820269i $$-0.306176\pi$$
0.571978 + 0.820269i $$0.306176\pi$$
$$978$$ −2.35722 −0.0753755
$$979$$ −6.23446 −0.199254
$$980$$ 38.3234 1.22420
$$981$$ 8.55044 0.272995
$$982$$ −0.435166 −0.0138867
$$983$$ −7.19064 −0.229346 −0.114673 0.993403i $$-0.536582\pi$$
−0.114673 + 0.993403i $$0.536582\pi$$
$$984$$ −16.4428 −0.524177
$$985$$ 21.2047 0.675638
$$986$$ 5.72905 0.182450
$$987$$ 4.56511 0.145309
$$988$$ 13.7719 0.438143
$$989$$ 63.5847 2.02188
$$990$$ −1.08564 −0.0345038
$$991$$ −19.9123 −0.632537 −0.316268 0.948670i $$-0.602430\pi$$
−0.316268 + 0.948670i $$0.602430\pi$$
$$992$$ 250.624 7.95733
$$993$$ −28.9971 −0.920194
$$994$$ −6.49790 −0.206101
$$995$$ 2.64723 0.0839228
$$996$$ −35.8885 −1.13717
$$997$$ −1.57302 −0.0498179 −0.0249089 0.999690i $$-0.507930\pi$$
−0.0249089 + 0.999690i $$0.507930\pi$$
$$998$$ −7.29734 −0.230993
$$999$$ 7.34192 0.232288
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 435.2.a.j.1.1 4
3.2 odd 2 1305.2.a.r.1.4 4
4.3 odd 2 6960.2.a.co.1.2 4
5.2 odd 4 2175.2.c.n.349.1 8
5.3 odd 4 2175.2.c.n.349.8 8
5.4 even 2 2175.2.a.v.1.4 4
15.14 odd 2 6525.2.a.bi.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 1.1 even 1 trivial
1305.2.a.r.1.4 4 3.2 odd 2
2175.2.a.v.1.4 4 5.4 even 2
2175.2.c.n.349.1 8 5.2 odd 4
2175.2.c.n.349.8 8 5.3 odd 4
6525.2.a.bi.1.1 4 15.14 odd 2
6960.2.a.co.1.2 4 4.3 odd 2