# Properties

 Label 2575.2.a.g.1.2 Level $2575$ Weight $2$ Character 2575.1 Self dual yes Analytic conductor $20.561$ 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] = [2575,2,Mod(1,2575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2575, 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("2575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2575 = 5^{2} \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2575.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: $$20.5614785205$$ 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: no (minimal twist has level 103) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 2575.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} +2.61803 q^{6} +1.00000 q^{7} +7.47214 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} +2.61803 q^{6} +1.00000 q^{7} +7.47214 q^{8} -2.00000 q^{9} -2.61803 q^{11} +4.85410 q^{12} +4.85410 q^{13} +2.61803 q^{14} +9.85410 q^{16} +5.61803 q^{17} -5.23607 q^{18} +5.85410 q^{19} +1.00000 q^{21} -6.85410 q^{22} -4.47214 q^{23} +7.47214 q^{24} +12.7082 q^{26} -5.00000 q^{27} +4.85410 q^{28} -5.23607 q^{29} -6.70820 q^{31} +10.8541 q^{32} -2.61803 q^{33} +14.7082 q^{34} -9.70820 q^{36} -6.70820 q^{37} +15.3262 q^{38} +4.85410 q^{39} +8.94427 q^{41} +2.61803 q^{42} +8.70820 q^{43} -12.7082 q^{44} -11.7082 q^{46} -4.09017 q^{47} +9.85410 q^{48} -6.00000 q^{49} +5.61803 q^{51} +23.5623 q^{52} -1.09017 q^{53} -13.0902 q^{54} +7.47214 q^{56} +5.85410 q^{57} -13.7082 q^{58} +6.38197 q^{59} +4.14590 q^{61} -17.5623 q^{62} -2.00000 q^{63} +8.70820 q^{64} -6.85410 q^{66} -14.4164 q^{67} +27.2705 q^{68} -4.47214 q^{69} -4.09017 q^{71} -14.9443 q^{72} +10.8541 q^{73} -17.5623 q^{74} +28.4164 q^{76} -2.61803 q^{77} +12.7082 q^{78} -6.56231 q^{79} +1.00000 q^{81} +23.4164 q^{82} +6.32624 q^{83} +4.85410 q^{84} +22.7984 q^{86} -5.23607 q^{87} -19.5623 q^{88} -2.29180 q^{89} +4.85410 q^{91} -21.7082 q^{92} -6.70820 q^{93} -10.7082 q^{94} +10.8541 q^{96} +1.70820 q^{97} -15.7082 q^{98} +5.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{6} + 2 q^{7} + 6 q^{8} - 4 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^6 + 2 * q^7 + 6 * q^8 - 4 * q^9 $$2 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{6} + 2 q^{7} + 6 q^{8} - 4 q^{9} - 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{14} + 13 q^{16} + 9 q^{17} - 6 q^{18} + 5 q^{19} + 2 q^{21} - 7 q^{22} + 6 q^{24} + 12 q^{26} - 10 q^{27} + 3 q^{28} - 6 q^{29} + 15 q^{32} - 3 q^{33} + 16 q^{34} - 6 q^{36} + 15 q^{38} + 3 q^{39} + 3 q^{42} + 4 q^{43} - 12 q^{44} - 10 q^{46} + 3 q^{47} + 13 q^{48} - 12 q^{49} + 9 q^{51} + 27 q^{52} + 9 q^{53} - 15 q^{54} + 6 q^{56} + 5 q^{57} - 14 q^{58} + 15 q^{59} + 15 q^{61} - 15 q^{62} - 4 q^{63} + 4 q^{64} - 7 q^{66} - 2 q^{67} + 21 q^{68} + 3 q^{71} - 12 q^{72} + 15 q^{73} - 15 q^{74} + 30 q^{76} - 3 q^{77} + 12 q^{78} + 7 q^{79} + 2 q^{81} + 20 q^{82} - 3 q^{83} + 3 q^{84} + 21 q^{86} - 6 q^{87} - 19 q^{88} - 18 q^{89} + 3 q^{91} - 30 q^{92} - 8 q^{94} + 15 q^{96} - 10 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^6 + 2 * q^7 + 6 * q^8 - 4 * q^9 - 3 * q^11 + 3 * q^12 + 3 * q^13 + 3 * q^14 + 13 * q^16 + 9 * q^17 - 6 * q^18 + 5 * q^19 + 2 * q^21 - 7 * q^22 + 6 * q^24 + 12 * q^26 - 10 * q^27 + 3 * q^28 - 6 * q^29 + 15 * q^32 - 3 * q^33 + 16 * q^34 - 6 * q^36 + 15 * q^38 + 3 * q^39 + 3 * q^42 + 4 * q^43 - 12 * q^44 - 10 * q^46 + 3 * q^47 + 13 * q^48 - 12 * q^49 + 9 * q^51 + 27 * q^52 + 9 * q^53 - 15 * q^54 + 6 * q^56 + 5 * q^57 - 14 * q^58 + 15 * q^59 + 15 * q^61 - 15 * q^62 - 4 * q^63 + 4 * q^64 - 7 * q^66 - 2 * q^67 + 21 * q^68 + 3 * q^71 - 12 * q^72 + 15 * q^73 - 15 * q^74 + 30 * q^76 - 3 * q^77 + 12 * q^78 + 7 * q^79 + 2 * q^81 + 20 * q^82 - 3 * q^83 + 3 * q^84 + 21 * q^86 - 6 * q^87 - 19 * q^88 - 18 * q^89 + 3 * q^91 - 30 * q^92 - 8 * q^94 + 15 * q^96 - 10 * q^97 - 18 * q^98 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.61803 1.85123 0.925615 0.378467i $$-0.123549\pi$$
0.925615 + 0.378467i $$0.123549\pi$$
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 4.85410 2.42705
$$5$$ 0 0
$$6$$ 2.61803 1.06881
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 7.47214 2.64180
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −2.61803 −0.789367 −0.394683 0.918817i $$-0.629146\pi$$
−0.394683 + 0.918817i $$0.629146\pi$$
$$12$$ 4.85410 1.40126
$$13$$ 4.85410 1.34629 0.673143 0.739512i $$-0.264944\pi$$
0.673143 + 0.739512i $$0.264944\pi$$
$$14$$ 2.61803 0.699699
$$15$$ 0 0
$$16$$ 9.85410 2.46353
$$17$$ 5.61803 1.36257 0.681287 0.732017i $$-0.261421\pi$$
0.681287 + 0.732017i $$0.261421\pi$$
$$18$$ −5.23607 −1.23415
$$19$$ 5.85410 1.34302 0.671512 0.740994i $$-0.265645\pi$$
0.671512 + 0.740994i $$0.265645\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ −6.85410 −1.46130
$$23$$ −4.47214 −0.932505 −0.466252 0.884652i $$-0.654396\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ 7.47214 1.52524
$$25$$ 0 0
$$26$$ 12.7082 2.49228
$$27$$ −5.00000 −0.962250
$$28$$ 4.85410 0.917339
$$29$$ −5.23607 −0.972313 −0.486157 0.873872i $$-0.661602\pi$$
−0.486157 + 0.873872i $$0.661602\pi$$
$$30$$ 0 0
$$31$$ −6.70820 −1.20483 −0.602414 0.798183i $$-0.705795\pi$$
−0.602414 + 0.798183i $$0.705795\pi$$
$$32$$ 10.8541 1.91875
$$33$$ −2.61803 −0.455741
$$34$$ 14.7082 2.52244
$$35$$ 0 0
$$36$$ −9.70820 −1.61803
$$37$$ −6.70820 −1.10282 −0.551411 0.834234i $$-0.685910\pi$$
−0.551411 + 0.834234i $$0.685910\pi$$
$$38$$ 15.3262 2.48624
$$39$$ 4.85410 0.777278
$$40$$ 0 0
$$41$$ 8.94427 1.39686 0.698430 0.715678i $$-0.253882\pi$$
0.698430 + 0.715678i $$0.253882\pi$$
$$42$$ 2.61803 0.403971
$$43$$ 8.70820 1.32799 0.663994 0.747738i $$-0.268860\pi$$
0.663994 + 0.747738i $$0.268860\pi$$
$$44$$ −12.7082 −1.91583
$$45$$ 0 0
$$46$$ −11.7082 −1.72628
$$47$$ −4.09017 −0.596613 −0.298306 0.954470i $$-0.596422\pi$$
−0.298306 + 0.954470i $$0.596422\pi$$
$$48$$ 9.85410 1.42232
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 5.61803 0.786682
$$52$$ 23.5623 3.26750
$$53$$ −1.09017 −0.149746 −0.0748732 0.997193i $$-0.523855\pi$$
−0.0748732 + 0.997193i $$0.523855\pi$$
$$54$$ −13.0902 −1.78135
$$55$$ 0 0
$$56$$ 7.47214 0.998506
$$57$$ 5.85410 0.775395
$$58$$ −13.7082 −1.79998
$$59$$ 6.38197 0.830861 0.415431 0.909625i $$-0.363631\pi$$
0.415431 + 0.909625i $$0.363631\pi$$
$$60$$ 0 0
$$61$$ 4.14590 0.530828 0.265414 0.964135i $$-0.414491\pi$$
0.265414 + 0.964135i $$0.414491\pi$$
$$62$$ −17.5623 −2.23042
$$63$$ −2.00000 −0.251976
$$64$$ 8.70820 1.08853
$$65$$ 0 0
$$66$$ −6.85410 −0.843682
$$67$$ −14.4164 −1.76124 −0.880622 0.473819i $$-0.842875\pi$$
−0.880622 + 0.473819i $$0.842875\pi$$
$$68$$ 27.2705 3.30704
$$69$$ −4.47214 −0.538382
$$70$$ 0 0
$$71$$ −4.09017 −0.485414 −0.242707 0.970100i $$-0.578035\pi$$
−0.242707 + 0.970100i $$0.578035\pi$$
$$72$$ −14.9443 −1.76120
$$73$$ 10.8541 1.27038 0.635188 0.772357i $$-0.280923\pi$$
0.635188 + 0.772357i $$0.280923\pi$$
$$74$$ −17.5623 −2.04158
$$75$$ 0 0
$$76$$ 28.4164 3.25959
$$77$$ −2.61803 −0.298353
$$78$$ 12.7082 1.43892
$$79$$ −6.56231 −0.738317 −0.369159 0.929366i $$-0.620354\pi$$
−0.369159 + 0.929366i $$0.620354\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 23.4164 2.58591
$$83$$ 6.32624 0.694395 0.347197 0.937792i $$-0.387133\pi$$
0.347197 + 0.937792i $$0.387133\pi$$
$$84$$ 4.85410 0.529626
$$85$$ 0 0
$$86$$ 22.7984 2.45841
$$87$$ −5.23607 −0.561365
$$88$$ −19.5623 −2.08535
$$89$$ −2.29180 −0.242930 −0.121465 0.992596i $$-0.538759\pi$$
−0.121465 + 0.992596i $$0.538759\pi$$
$$90$$ 0 0
$$91$$ 4.85410 0.508848
$$92$$ −21.7082 −2.26324
$$93$$ −6.70820 −0.695608
$$94$$ −10.7082 −1.10447
$$95$$ 0 0
$$96$$ 10.8541 1.10779
$$97$$ 1.70820 0.173442 0.0867209 0.996233i $$-0.472361\pi$$
0.0867209 + 0.996233i $$0.472361\pi$$
$$98$$ −15.7082 −1.58677
$$99$$ 5.23607 0.526245
$$100$$ 0 0
$$101$$ 1.90983 0.190035 0.0950176 0.995476i $$-0.469709\pi$$
0.0950176 + 0.995476i $$0.469709\pi$$
$$102$$ 14.7082 1.45633
$$103$$ 1.00000 0.0985329
$$104$$ 36.2705 3.55662
$$105$$ 0 0
$$106$$ −2.85410 −0.277215
$$107$$ 7.09017 0.685433 0.342716 0.939439i $$-0.388653\pi$$
0.342716 + 0.939439i $$0.388653\pi$$
$$108$$ −24.2705 −2.33543
$$109$$ −9.56231 −0.915903 −0.457951 0.888977i $$-0.651417\pi$$
−0.457951 + 0.888977i $$0.651417\pi$$
$$110$$ 0 0
$$111$$ −6.70820 −0.636715
$$112$$ 9.85410 0.931125
$$113$$ 15.0000 1.41108 0.705541 0.708669i $$-0.250704\pi$$
0.705541 + 0.708669i $$0.250704\pi$$
$$114$$ 15.3262 1.43543
$$115$$ 0 0
$$116$$ −25.4164 −2.35985
$$117$$ −9.70820 −0.897524
$$118$$ 16.7082 1.53811
$$119$$ 5.61803 0.515004
$$120$$ 0 0
$$121$$ −4.14590 −0.376900
$$122$$ 10.8541 0.982684
$$123$$ 8.94427 0.806478
$$124$$ −32.5623 −2.92418
$$125$$ 0 0
$$126$$ −5.23607 −0.466466
$$127$$ −15.2705 −1.35504 −0.677519 0.735505i $$-0.736945\pi$$
−0.677519 + 0.735505i $$0.736945\pi$$
$$128$$ 1.09017 0.0963583
$$129$$ 8.70820 0.766715
$$130$$ 0 0
$$131$$ −2.23607 −0.195366 −0.0976831 0.995218i $$-0.531143\pi$$
−0.0976831 + 0.995218i $$0.531143\pi$$
$$132$$ −12.7082 −1.10611
$$133$$ 5.85410 0.507615
$$134$$ −37.7426 −3.26047
$$135$$ 0 0
$$136$$ 41.9787 3.59965
$$137$$ 12.7082 1.08574 0.542868 0.839818i $$-0.317339\pi$$
0.542868 + 0.839818i $$0.317339\pi$$
$$138$$ −11.7082 −0.996669
$$139$$ −11.1459 −0.945383 −0.472691 0.881228i $$-0.656717\pi$$
−0.472691 + 0.881228i $$0.656717\pi$$
$$140$$ 0 0
$$141$$ −4.09017 −0.344454
$$142$$ −10.7082 −0.898613
$$143$$ −12.7082 −1.06271
$$144$$ −19.7082 −1.64235
$$145$$ 0 0
$$146$$ 28.4164 2.35176
$$147$$ −6.00000 −0.494872
$$148$$ −32.5623 −2.67661
$$149$$ −7.47214 −0.612141 −0.306071 0.952009i $$-0.599014\pi$$
−0.306071 + 0.952009i $$0.599014\pi$$
$$150$$ 0 0
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 43.7426 3.54800
$$153$$ −11.2361 −0.908382
$$154$$ −6.85410 −0.552319
$$155$$ 0 0
$$156$$ 23.5623 1.88649
$$157$$ −16.7082 −1.33346 −0.666730 0.745299i $$-0.732307\pi$$
−0.666730 + 0.745299i $$0.732307\pi$$
$$158$$ −17.1803 −1.36679
$$159$$ −1.09017 −0.0864561
$$160$$ 0 0
$$161$$ −4.47214 −0.352454
$$162$$ 2.61803 0.205692
$$163$$ −2.70820 −0.212123 −0.106061 0.994360i $$-0.533824\pi$$
−0.106061 + 0.994360i $$0.533824\pi$$
$$164$$ 43.4164 3.39025
$$165$$ 0 0
$$166$$ 16.5623 1.28548
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 7.47214 0.576488
$$169$$ 10.5623 0.812485
$$170$$ 0 0
$$171$$ −11.7082 −0.895349
$$172$$ 42.2705 3.22310
$$173$$ −16.0344 −1.21908 −0.609538 0.792757i $$-0.708645\pi$$
−0.609538 + 0.792757i $$0.708645\pi$$
$$174$$ −13.7082 −1.03922
$$175$$ 0 0
$$176$$ −25.7984 −1.94463
$$177$$ 6.38197 0.479698
$$178$$ −6.00000 −0.449719
$$179$$ −7.85410 −0.587043 −0.293522 0.955952i $$-0.594827\pi$$
−0.293522 + 0.955952i $$0.594827\pi$$
$$180$$ 0 0
$$181$$ 3.85410 0.286473 0.143237 0.989688i $$-0.454249\pi$$
0.143237 + 0.989688i $$0.454249\pi$$
$$182$$ 12.7082 0.941995
$$183$$ 4.14590 0.306474
$$184$$ −33.4164 −2.46349
$$185$$ 0 0
$$186$$ −17.5623 −1.28773
$$187$$ −14.7082 −1.07557
$$188$$ −19.8541 −1.44801
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 5.61803 0.406507 0.203253 0.979126i $$-0.434848\pi$$
0.203253 + 0.979126i $$0.434848\pi$$
$$192$$ 8.70820 0.628460
$$193$$ −20.1246 −1.44860 −0.724301 0.689484i $$-0.757837\pi$$
−0.724301 + 0.689484i $$0.757837\pi$$
$$194$$ 4.47214 0.321081
$$195$$ 0 0
$$196$$ −29.1246 −2.08033
$$197$$ 16.4164 1.16962 0.584810 0.811170i $$-0.301169\pi$$
0.584810 + 0.811170i $$0.301169\pi$$
$$198$$ 13.7082 0.974200
$$199$$ −23.4164 −1.65995 −0.829973 0.557804i $$-0.811644\pi$$
−0.829973 + 0.557804i $$0.811644\pi$$
$$200$$ 0 0
$$201$$ −14.4164 −1.01686
$$202$$ 5.00000 0.351799
$$203$$ −5.23607 −0.367500
$$204$$ 27.2705 1.90932
$$205$$ 0 0
$$206$$ 2.61803 0.182407
$$207$$ 8.94427 0.621670
$$208$$ 47.8328 3.31661
$$209$$ −15.3262 −1.06014
$$210$$ 0 0
$$211$$ 14.8541 1.02260 0.511299 0.859403i $$-0.329164\pi$$
0.511299 + 0.859403i $$0.329164\pi$$
$$212$$ −5.29180 −0.363442
$$213$$ −4.09017 −0.280254
$$214$$ 18.5623 1.26889
$$215$$ 0 0
$$216$$ −37.3607 −2.54207
$$217$$ −6.70820 −0.455383
$$218$$ −25.0344 −1.69555
$$219$$ 10.8541 0.733452
$$220$$ 0 0
$$221$$ 27.2705 1.83441
$$222$$ −17.5623 −1.17870
$$223$$ −7.70820 −0.516180 −0.258090 0.966121i $$-0.583093\pi$$
−0.258090 + 0.966121i $$0.583093\pi$$
$$224$$ 10.8541 0.725220
$$225$$ 0 0
$$226$$ 39.2705 2.61224
$$227$$ −2.94427 −0.195418 −0.0977091 0.995215i $$-0.531151\pi$$
−0.0977091 + 0.995215i $$0.531151\pi$$
$$228$$ 28.4164 1.88192
$$229$$ 6.70820 0.443291 0.221645 0.975127i $$-0.428857\pi$$
0.221645 + 0.975127i $$0.428857\pi$$
$$230$$ 0 0
$$231$$ −2.61803 −0.172254
$$232$$ −39.1246 −2.56866
$$233$$ 26.8885 1.76153 0.880764 0.473556i $$-0.157030\pi$$
0.880764 + 0.473556i $$0.157030\pi$$
$$234$$ −25.4164 −1.66152
$$235$$ 0 0
$$236$$ 30.9787 2.01654
$$237$$ −6.56231 −0.426268
$$238$$ 14.7082 0.953391
$$239$$ 8.67376 0.561059 0.280530 0.959845i $$-0.409490\pi$$
0.280530 + 0.959845i $$0.409490\pi$$
$$240$$ 0 0
$$241$$ −8.27051 −0.532750 −0.266375 0.963869i $$-0.585826\pi$$
−0.266375 + 0.963869i $$0.585826\pi$$
$$242$$ −10.8541 −0.697728
$$243$$ 16.0000 1.02640
$$244$$ 20.1246 1.28835
$$245$$ 0 0
$$246$$ 23.4164 1.49298
$$247$$ 28.4164 1.80809
$$248$$ −50.1246 −3.18292
$$249$$ 6.32624 0.400909
$$250$$ 0 0
$$251$$ 11.2361 0.709214 0.354607 0.935015i $$-0.384615\pi$$
0.354607 + 0.935015i $$0.384615\pi$$
$$252$$ −9.70820 −0.611559
$$253$$ 11.7082 0.736088
$$254$$ −39.9787 −2.50849
$$255$$ 0 0
$$256$$ −14.5623 −0.910144
$$257$$ 13.4721 0.840369 0.420184 0.907439i $$-0.361965\pi$$
0.420184 + 0.907439i $$0.361965\pi$$
$$258$$ 22.7984 1.41936
$$259$$ −6.70820 −0.416828
$$260$$ 0 0
$$261$$ 10.4721 0.648209
$$262$$ −5.85410 −0.361668
$$263$$ 20.6180 1.27136 0.635681 0.771952i $$-0.280719\pi$$
0.635681 + 0.771952i $$0.280719\pi$$
$$264$$ −19.5623 −1.20398
$$265$$ 0 0
$$266$$ 15.3262 0.939712
$$267$$ −2.29180 −0.140256
$$268$$ −69.9787 −4.27463
$$269$$ 12.3262 0.751544 0.375772 0.926712i $$-0.377378\pi$$
0.375772 + 0.926712i $$0.377378\pi$$
$$270$$ 0 0
$$271$$ 1.00000 0.0607457 0.0303728 0.999539i $$-0.490331\pi$$
0.0303728 + 0.999539i $$0.490331\pi$$
$$272$$ 55.3607 3.35673
$$273$$ 4.85410 0.293784
$$274$$ 33.2705 2.00995
$$275$$ 0 0
$$276$$ −21.7082 −1.30668
$$277$$ −4.70820 −0.282889 −0.141444 0.989946i $$-0.545175\pi$$
−0.141444 + 0.989946i $$0.545175\pi$$
$$278$$ −29.1803 −1.75012
$$279$$ 13.4164 0.803219
$$280$$ 0 0
$$281$$ −31.4721 −1.87747 −0.938735 0.344640i $$-0.888001\pi$$
−0.938735 + 0.344640i $$0.888001\pi$$
$$282$$ −10.7082 −0.637664
$$283$$ 19.7082 1.17153 0.585766 0.810481i $$-0.300794\pi$$
0.585766 + 0.810481i $$0.300794\pi$$
$$284$$ −19.8541 −1.17812
$$285$$ 0 0
$$286$$ −33.2705 −1.96733
$$287$$ 8.94427 0.527964
$$288$$ −21.7082 −1.27917
$$289$$ 14.5623 0.856606
$$290$$ 0 0
$$291$$ 1.70820 0.100137
$$292$$ 52.6869 3.08327
$$293$$ 21.6525 1.26495 0.632476 0.774580i $$-0.282039\pi$$
0.632476 + 0.774580i $$0.282039\pi$$
$$294$$ −15.7082 −0.916121
$$295$$ 0 0
$$296$$ −50.1246 −2.91343
$$297$$ 13.0902 0.759569
$$298$$ −19.5623 −1.13321
$$299$$ −21.7082 −1.25542
$$300$$ 0 0
$$301$$ 8.70820 0.501933
$$302$$ −49.7426 −2.86237
$$303$$ 1.90983 0.109717
$$304$$ 57.6869 3.30857
$$305$$ 0 0
$$306$$ −29.4164 −1.68162
$$307$$ 2.85410 0.162892 0.0814461 0.996678i $$-0.474046\pi$$
0.0814461 + 0.996678i $$0.474046\pi$$
$$308$$ −12.7082 −0.724117
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 2.88854 0.163794 0.0818971 0.996641i $$-0.473902\pi$$
0.0818971 + 0.996641i $$0.473902\pi$$
$$312$$ 36.2705 2.05341
$$313$$ −16.2918 −0.920867 −0.460433 0.887694i $$-0.652306\pi$$
−0.460433 + 0.887694i $$0.652306\pi$$
$$314$$ −43.7426 −2.46854
$$315$$ 0 0
$$316$$ −31.8541 −1.79193
$$317$$ −28.4164 −1.59602 −0.798012 0.602641i $$-0.794115\pi$$
−0.798012 + 0.602641i $$0.794115\pi$$
$$318$$ −2.85410 −0.160050
$$319$$ 13.7082 0.767512
$$320$$ 0 0
$$321$$ 7.09017 0.395735
$$322$$ −11.7082 −0.652473
$$323$$ 32.8885 1.82997
$$324$$ 4.85410 0.269672
$$325$$ 0 0
$$326$$ −7.09017 −0.392688
$$327$$ −9.56231 −0.528797
$$328$$ 66.8328 3.69022
$$329$$ −4.09017 −0.225498
$$330$$ 0 0
$$331$$ −16.1459 −0.887459 −0.443729 0.896161i $$-0.646345\pi$$
−0.443729 + 0.896161i $$0.646345\pi$$
$$332$$ 30.7082 1.68533
$$333$$ 13.4164 0.735215
$$334$$ −23.5623 −1.28927
$$335$$ 0 0
$$336$$ 9.85410 0.537585
$$337$$ 2.43769 0.132790 0.0663948 0.997793i $$-0.478850\pi$$
0.0663948 + 0.997793i $$0.478850\pi$$
$$338$$ 27.6525 1.50410
$$339$$ 15.0000 0.814688
$$340$$ 0 0
$$341$$ 17.5623 0.951052
$$342$$ −30.6525 −1.65750
$$343$$ −13.0000 −0.701934
$$344$$ 65.0689 3.50828
$$345$$ 0 0
$$346$$ −41.9787 −2.25679
$$347$$ 1.47214 0.0790284 0.0395142 0.999219i $$-0.487419\pi$$
0.0395142 + 0.999219i $$0.487419\pi$$
$$348$$ −25.4164 −1.36246
$$349$$ 15.4164 0.825221 0.412611 0.910907i $$-0.364617\pi$$
0.412611 + 0.910907i $$0.364617\pi$$
$$350$$ 0 0
$$351$$ −24.2705 −1.29546
$$352$$ −28.4164 −1.51460
$$353$$ 25.0344 1.33245 0.666224 0.745751i $$-0.267909\pi$$
0.666224 + 0.745751i $$0.267909\pi$$
$$354$$ 16.7082 0.888031
$$355$$ 0 0
$$356$$ −11.1246 −0.589603
$$357$$ 5.61803 0.297338
$$358$$ −20.5623 −1.08675
$$359$$ 14.6738 0.774452 0.387226 0.921985i $$-0.373433\pi$$
0.387226 + 0.921985i $$0.373433\pi$$
$$360$$ 0 0
$$361$$ 15.2705 0.803711
$$362$$ 10.0902 0.530328
$$363$$ −4.14590 −0.217603
$$364$$ 23.5623 1.23500
$$365$$ 0 0
$$366$$ 10.8541 0.567353
$$367$$ −16.4377 −0.858041 −0.429020 0.903295i $$-0.641141\pi$$
−0.429020 + 0.903295i $$0.641141\pi$$
$$368$$ −44.0689 −2.29725
$$369$$ −17.8885 −0.931240
$$370$$ 0 0
$$371$$ −1.09017 −0.0565988
$$372$$ −32.5623 −1.68828
$$373$$ 22.6869 1.17468 0.587342 0.809339i $$-0.300174\pi$$
0.587342 + 0.809339i $$0.300174\pi$$
$$374$$ −38.5066 −1.99113
$$375$$ 0 0
$$376$$ −30.5623 −1.57613
$$377$$ −25.4164 −1.30901
$$378$$ −13.0902 −0.673286
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ −15.2705 −0.782332
$$382$$ 14.7082 0.752537
$$383$$ −0.819660 −0.0418827 −0.0209413 0.999781i $$-0.506666\pi$$
−0.0209413 + 0.999781i $$0.506666\pi$$
$$384$$ 1.09017 0.0556325
$$385$$ 0 0
$$386$$ −52.6869 −2.68169
$$387$$ −17.4164 −0.885326
$$388$$ 8.29180 0.420952
$$389$$ 7.41641 0.376027 0.188013 0.982166i $$-0.439795\pi$$
0.188013 + 0.982166i $$0.439795\pi$$
$$390$$ 0 0
$$391$$ −25.1246 −1.27061
$$392$$ −44.8328 −2.26440
$$393$$ −2.23607 −0.112795
$$394$$ 42.9787 2.16524
$$395$$ 0 0
$$396$$ 25.4164 1.27722
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ −61.3050 −3.07294
$$399$$ 5.85410 0.293072
$$400$$ 0 0
$$401$$ −23.8885 −1.19294 −0.596468 0.802637i $$-0.703430\pi$$
−0.596468 + 0.802637i $$0.703430\pi$$
$$402$$ −37.7426 −1.88243
$$403$$ −32.5623 −1.62204
$$404$$ 9.27051 0.461225
$$405$$ 0 0
$$406$$ −13.7082 −0.680327
$$407$$ 17.5623 0.870531
$$408$$ 41.9787 2.07826
$$409$$ 36.7082 1.81510 0.907552 0.419940i $$-0.137949\pi$$
0.907552 + 0.419940i $$0.137949\pi$$
$$410$$ 0 0
$$411$$ 12.7082 0.626849
$$412$$ 4.85410 0.239144
$$413$$ 6.38197 0.314036
$$414$$ 23.4164 1.15085
$$415$$ 0 0
$$416$$ 52.6869 2.58319
$$417$$ −11.1459 −0.545817
$$418$$ −40.1246 −1.96256
$$419$$ −4.09017 −0.199818 −0.0999089 0.994997i $$-0.531855\pi$$
−0.0999089 + 0.994997i $$0.531855\pi$$
$$420$$ 0 0
$$421$$ 3.00000 0.146211 0.0731055 0.997324i $$-0.476709\pi$$
0.0731055 + 0.997324i $$0.476709\pi$$
$$422$$ 38.8885 1.89306
$$423$$ 8.18034 0.397742
$$424$$ −8.14590 −0.395600
$$425$$ 0 0
$$426$$ −10.7082 −0.518814
$$427$$ 4.14590 0.200634
$$428$$ 34.4164 1.66358
$$429$$ −12.7082 −0.613558
$$430$$ 0 0
$$431$$ 34.3607 1.65510 0.827548 0.561395i $$-0.189735\pi$$
0.827548 + 0.561395i $$0.189735\pi$$
$$432$$ −49.2705 −2.37053
$$433$$ −14.4164 −0.692808 −0.346404 0.938085i $$-0.612597\pi$$
−0.346404 + 0.938085i $$0.612597\pi$$
$$434$$ −17.5623 −0.843018
$$435$$ 0 0
$$436$$ −46.4164 −2.22294
$$437$$ −26.1803 −1.25238
$$438$$ 28.4164 1.35779
$$439$$ 29.5623 1.41093 0.705466 0.708744i $$-0.250738\pi$$
0.705466 + 0.708744i $$0.250738\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 71.3951 3.39592
$$443$$ 15.4377 0.733467 0.366733 0.930326i $$-0.380476\pi$$
0.366733 + 0.930326i $$0.380476\pi$$
$$444$$ −32.5623 −1.54534
$$445$$ 0 0
$$446$$ −20.1803 −0.955567
$$447$$ −7.47214 −0.353420
$$448$$ 8.70820 0.411424
$$449$$ −13.3607 −0.630529 −0.315265 0.949004i $$-0.602093\pi$$
−0.315265 + 0.949004i $$0.602093\pi$$
$$450$$ 0 0
$$451$$ −23.4164 −1.10264
$$452$$ 72.8115 3.42477
$$453$$ −19.0000 −0.892698
$$454$$ −7.70820 −0.361764
$$455$$ 0 0
$$456$$ 43.7426 2.04844
$$457$$ 9.85410 0.460955 0.230478 0.973078i $$-0.425971\pi$$
0.230478 + 0.973078i $$0.425971\pi$$
$$458$$ 17.5623 0.820633
$$459$$ −28.0902 −1.31114
$$460$$ 0 0
$$461$$ 12.2148 0.568899 0.284450 0.958691i $$-0.408189\pi$$
0.284450 + 0.958691i $$0.408189\pi$$
$$462$$ −6.85410 −0.318882
$$463$$ 21.4164 0.995305 0.497652 0.867377i $$-0.334196\pi$$
0.497652 + 0.867377i $$0.334196\pi$$
$$464$$ −51.5967 −2.39532
$$465$$ 0 0
$$466$$ 70.3951 3.26099
$$467$$ 3.65248 0.169016 0.0845082 0.996423i $$-0.473068\pi$$
0.0845082 + 0.996423i $$0.473068\pi$$
$$468$$ −47.1246 −2.17834
$$469$$ −14.4164 −0.665688
$$470$$ 0 0
$$471$$ −16.7082 −0.769873
$$472$$ 47.6869 2.19497
$$473$$ −22.7984 −1.04827
$$474$$ −17.1803 −0.789119
$$475$$ 0 0
$$476$$ 27.2705 1.24994
$$477$$ 2.18034 0.0998309
$$478$$ 22.7082 1.03865
$$479$$ 8.18034 0.373769 0.186885 0.982382i $$-0.440161\pi$$
0.186885 + 0.982382i $$0.440161\pi$$
$$480$$ 0 0
$$481$$ −32.5623 −1.48471
$$482$$ −21.6525 −0.986243
$$483$$ −4.47214 −0.203489
$$484$$ −20.1246 −0.914755
$$485$$ 0 0
$$486$$ 41.8885 1.90010
$$487$$ 23.0000 1.04223 0.521115 0.853487i $$-0.325516\pi$$
0.521115 + 0.853487i $$0.325516\pi$$
$$488$$ 30.9787 1.40234
$$489$$ −2.70820 −0.122469
$$490$$ 0 0
$$491$$ 35.7771 1.61460 0.807299 0.590143i $$-0.200929\pi$$
0.807299 + 0.590143i $$0.200929\pi$$
$$492$$ 43.4164 1.95736
$$493$$ −29.4164 −1.32485
$$494$$ 74.3951 3.34719
$$495$$ 0 0
$$496$$ −66.1033 −2.96813
$$497$$ −4.09017 −0.183469
$$498$$ 16.5623 0.742175
$$499$$ 13.2705 0.594070 0.297035 0.954867i $$-0.404002\pi$$
0.297035 + 0.954867i $$0.404002\pi$$
$$500$$ 0 0
$$501$$ −9.00000 −0.402090
$$502$$ 29.4164 1.31292
$$503$$ −13.3607 −0.595723 −0.297862 0.954609i $$-0.596273\pi$$
−0.297862 + 0.954609i $$0.596273\pi$$
$$504$$ −14.9443 −0.665671
$$505$$ 0 0
$$506$$ 30.6525 1.36267
$$507$$ 10.5623 0.469088
$$508$$ −74.1246 −3.28875
$$509$$ 26.6180 1.17982 0.589912 0.807468i $$-0.299163\pi$$
0.589912 + 0.807468i $$0.299163\pi$$
$$510$$ 0 0
$$511$$ 10.8541 0.480157
$$512$$ −40.3050 −1.78124
$$513$$ −29.2705 −1.29232
$$514$$ 35.2705 1.55572
$$515$$ 0 0
$$516$$ 42.2705 1.86086
$$517$$ 10.7082 0.470946
$$518$$ −17.5623 −0.771643
$$519$$ −16.0344 −0.703834
$$520$$ 0 0
$$521$$ 17.1803 0.752684 0.376342 0.926481i $$-0.377182\pi$$
0.376342 + 0.926481i $$0.377182\pi$$
$$522$$ 27.4164 1.19998
$$523$$ 10.5836 0.462788 0.231394 0.972860i $$-0.425671\pi$$
0.231394 + 0.972860i $$0.425671\pi$$
$$524$$ −10.8541 −0.474164
$$525$$ 0 0
$$526$$ 53.9787 2.35358
$$527$$ −37.6869 −1.64167
$$528$$ −25.7984 −1.12273
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ −12.7639 −0.553907
$$532$$ 28.4164 1.23201
$$533$$ 43.4164 1.88057
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ −107.721 −4.65285
$$537$$ −7.85410 −0.338930
$$538$$ 32.2705 1.39128
$$539$$ 15.7082 0.676600
$$540$$ 0 0
$$541$$ 9.14590 0.393213 0.196606 0.980482i $$-0.437008\pi$$
0.196606 + 0.980482i $$0.437008\pi$$
$$542$$ 2.61803 0.112454
$$543$$ 3.85410 0.165395
$$544$$ 60.9787 2.61444
$$545$$ 0 0
$$546$$ 12.7082 0.543861
$$547$$ 2.27051 0.0970800 0.0485400 0.998821i $$-0.484543\pi$$
0.0485400 + 0.998821i $$0.484543\pi$$
$$548$$ 61.6869 2.63513
$$549$$ −8.29180 −0.353885
$$550$$ 0 0
$$551$$ −30.6525 −1.30584
$$552$$ −33.4164 −1.42230
$$553$$ −6.56231 −0.279058
$$554$$ −12.3262 −0.523692
$$555$$ 0 0
$$556$$ −54.1033 −2.29449
$$557$$ 31.6869 1.34262 0.671309 0.741178i $$-0.265732\pi$$
0.671309 + 0.741178i $$0.265732\pi$$
$$558$$ 35.1246 1.48694
$$559$$ 42.2705 1.78785
$$560$$ 0 0
$$561$$ −14.7082 −0.620981
$$562$$ −82.3951 −3.47563
$$563$$ 1.20163 0.0506425 0.0253213 0.999679i $$-0.491939\pi$$
0.0253213 + 0.999679i $$0.491939\pi$$
$$564$$ −19.8541 −0.836009
$$565$$ 0 0
$$566$$ 51.5967 2.16877
$$567$$ 1.00000 0.0419961
$$568$$ −30.5623 −1.28237
$$569$$ −27.1591 −1.13857 −0.569283 0.822141i $$-0.692779\pi$$
−0.569283 + 0.822141i $$0.692779\pi$$
$$570$$ 0 0
$$571$$ −22.5623 −0.944203 −0.472102 0.881544i $$-0.656504\pi$$
−0.472102 + 0.881544i $$0.656504\pi$$
$$572$$ −61.6869 −2.57926
$$573$$ 5.61803 0.234697
$$574$$ 23.4164 0.977382
$$575$$ 0 0
$$576$$ −17.4164 −0.725684
$$577$$ 36.8328 1.53337 0.766685 0.642023i $$-0.221905\pi$$
0.766685 + 0.642023i $$0.221905\pi$$
$$578$$ 38.1246 1.58577
$$579$$ −20.1246 −0.836350
$$580$$ 0 0
$$581$$ 6.32624 0.262457
$$582$$ 4.47214 0.185376
$$583$$ 2.85410 0.118205
$$584$$ 81.1033 3.35608
$$585$$ 0 0
$$586$$ 56.6869 2.34171
$$587$$ 47.0689 1.94274 0.971370 0.237570i $$-0.0763510\pi$$
0.971370 + 0.237570i $$0.0763510\pi$$
$$588$$ −29.1246 −1.20108
$$589$$ −39.2705 −1.61811
$$590$$ 0 0
$$591$$ 16.4164 0.675281
$$592$$ −66.1033 −2.71683
$$593$$ −2.18034 −0.0895358 −0.0447679 0.998997i $$-0.514255\pi$$
−0.0447679 + 0.998997i $$0.514255\pi$$
$$594$$ 34.2705 1.40614
$$595$$ 0 0
$$596$$ −36.2705 −1.48570
$$597$$ −23.4164 −0.958370
$$598$$ −56.8328 −2.32407
$$599$$ −35.4508 −1.44848 −0.724241 0.689547i $$-0.757810\pi$$
−0.724241 + 0.689547i $$0.757810\pi$$
$$600$$ 0 0
$$601$$ 3.56231 0.145309 0.0726547 0.997357i $$-0.476853\pi$$
0.0726547 + 0.997357i $$0.476853\pi$$
$$602$$ 22.7984 0.929192
$$603$$ 28.8328 1.17416
$$604$$ −92.2279 −3.75270
$$605$$ 0 0
$$606$$ 5.00000 0.203111
$$607$$ −5.70820 −0.231689 −0.115844 0.993267i $$-0.536957\pi$$
−0.115844 + 0.993267i $$0.536957\pi$$
$$608$$ 63.5410 2.57693
$$609$$ −5.23607 −0.212176
$$610$$ 0 0
$$611$$ −19.8541 −0.803211
$$612$$ −54.5410 −2.20469
$$613$$ −24.4164 −0.986169 −0.493085 0.869981i $$-0.664131\pi$$
−0.493085 + 0.869981i $$0.664131\pi$$
$$614$$ 7.47214 0.301551
$$615$$ 0 0
$$616$$ −19.5623 −0.788188
$$617$$ −3.27051 −0.131666 −0.0658329 0.997831i $$-0.520970\pi$$
−0.0658329 + 0.997831i $$0.520970\pi$$
$$618$$ 2.61803 0.105313
$$619$$ −28.6869 −1.15302 −0.576512 0.817088i $$-0.695587\pi$$
−0.576512 + 0.817088i $$0.695587\pi$$
$$620$$ 0 0
$$621$$ 22.3607 0.897303
$$622$$ 7.56231 0.303221
$$623$$ −2.29180 −0.0918189
$$624$$ 47.8328 1.91485
$$625$$ 0 0
$$626$$ −42.6525 −1.70474
$$627$$ −15.3262 −0.612071
$$628$$ −81.1033 −3.23638
$$629$$ −37.6869 −1.50268
$$630$$ 0 0
$$631$$ 42.2705 1.68276 0.841381 0.540442i $$-0.181743\pi$$
0.841381 + 0.540442i $$0.181743\pi$$
$$632$$ −49.0344 −1.95049
$$633$$ 14.8541 0.590398
$$634$$ −74.3951 −2.95461
$$635$$ 0 0
$$636$$ −5.29180 −0.209833
$$637$$ −29.1246 −1.15396
$$638$$ 35.8885 1.42084
$$639$$ 8.18034 0.323609
$$640$$ 0 0
$$641$$ −15.0000 −0.592464 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$642$$ 18.5623 0.732596
$$643$$ −5.00000 −0.197181 −0.0985904 0.995128i $$-0.531433\pi$$
−0.0985904 + 0.995128i $$0.531433\pi$$
$$644$$ −21.7082 −0.855423
$$645$$ 0 0
$$646$$ 86.1033 3.38769
$$647$$ −1.25735 −0.0494317 −0.0247158 0.999695i $$-0.507868\pi$$
−0.0247158 + 0.999695i $$0.507868\pi$$
$$648$$ 7.47214 0.293533
$$649$$ −16.7082 −0.655854
$$650$$ 0 0
$$651$$ −6.70820 −0.262915
$$652$$ −13.1459 −0.514833
$$653$$ 5.23607 0.204903 0.102452 0.994738i $$-0.467331\pi$$
0.102452 + 0.994738i $$0.467331\pi$$
$$654$$ −25.0344 −0.978924
$$655$$ 0 0
$$656$$ 88.1378 3.44120
$$657$$ −21.7082 −0.846918
$$658$$ −10.7082 −0.417449
$$659$$ 36.5967 1.42561 0.712803 0.701364i $$-0.247425\pi$$
0.712803 + 0.701364i $$0.247425\pi$$
$$660$$ 0 0
$$661$$ −31.5623 −1.22763 −0.613816 0.789449i $$-0.710366\pi$$
−0.613816 + 0.789449i $$0.710366\pi$$
$$662$$ −42.2705 −1.64289
$$663$$ 27.2705 1.05910
$$664$$ 47.2705 1.83445
$$665$$ 0 0
$$666$$ 35.1246 1.36105
$$667$$ 23.4164 0.906687
$$668$$ −43.6869 −1.69030
$$669$$ −7.70820 −0.298016
$$670$$ 0 0
$$671$$ −10.8541 −0.419018
$$672$$ 10.8541 0.418706
$$673$$ 24.2918 0.936380 0.468190 0.883628i $$-0.344906\pi$$
0.468190 + 0.883628i $$0.344906\pi$$
$$674$$ 6.38197 0.245824
$$675$$ 0 0
$$676$$ 51.2705 1.97194
$$677$$ −40.0344 −1.53865 −0.769324 0.638858i $$-0.779407\pi$$
−0.769324 + 0.638858i $$0.779407\pi$$
$$678$$ 39.2705 1.50817
$$679$$ 1.70820 0.0655549
$$680$$ 0 0
$$681$$ −2.94427 −0.112825
$$682$$ 45.9787 1.76062
$$683$$ −47.2361 −1.80744 −0.903719 0.428126i $$-0.859174\pi$$
−0.903719 + 0.428126i $$0.859174\pi$$
$$684$$ −56.8328 −2.17306
$$685$$ 0 0
$$686$$ −34.0344 −1.29944
$$687$$ 6.70820 0.255934
$$688$$ 85.8115 3.27153
$$689$$ −5.29180 −0.201601
$$690$$ 0 0
$$691$$ 7.85410 0.298784 0.149392 0.988778i $$-0.452268\pi$$
0.149392 + 0.988778i $$0.452268\pi$$
$$692$$ −77.8328 −2.95876
$$693$$ 5.23607 0.198902
$$694$$ 3.85410 0.146300
$$695$$ 0 0
$$696$$ −39.1246 −1.48301
$$697$$ 50.2492 1.90333
$$698$$ 40.3607 1.52767
$$699$$ 26.8885 1.01702
$$700$$ 0 0
$$701$$ −25.7984 −0.974391 −0.487196 0.873293i $$-0.661980\pi$$
−0.487196 + 0.873293i $$0.661980\pi$$
$$702$$ −63.5410 −2.39820
$$703$$ −39.2705 −1.48112
$$704$$ −22.7984 −0.859246
$$705$$ 0 0
$$706$$ 65.5410 2.46667
$$707$$ 1.90983 0.0718266
$$708$$ 30.9787 1.16425
$$709$$ −1.02129 −0.0383552 −0.0191776 0.999816i $$-0.506105\pi$$
−0.0191776 + 0.999816i $$0.506105\pi$$
$$710$$ 0 0
$$711$$ 13.1246 0.492211
$$712$$ −17.1246 −0.641772
$$713$$ 30.0000 1.12351
$$714$$ 14.7082 0.550441
$$715$$ 0 0
$$716$$ −38.1246 −1.42478
$$717$$ 8.67376 0.323928
$$718$$ 38.4164 1.43369
$$719$$ 39.3262 1.46662 0.733311 0.679894i $$-0.237974\pi$$
0.733311 + 0.679894i $$0.237974\pi$$
$$720$$ 0 0
$$721$$ 1.00000 0.0372419
$$722$$ 39.9787 1.48785
$$723$$ −8.27051 −0.307584
$$724$$ 18.7082 0.695285
$$725$$ 0 0
$$726$$ −10.8541 −0.402834
$$727$$ 4.72949 0.175407 0.0877035 0.996147i $$-0.472047\pi$$
0.0877035 + 0.996147i $$0.472047\pi$$
$$728$$ 36.2705 1.34427
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 48.9230 1.80948
$$732$$ 20.1246 0.743827
$$733$$ −14.7082 −0.543260 −0.271630 0.962402i $$-0.587563\pi$$
−0.271630 + 0.962402i $$0.587563\pi$$
$$734$$ −43.0344 −1.58843
$$735$$ 0 0
$$736$$ −48.5410 −1.78925
$$737$$ 37.7426 1.39027
$$738$$ −46.8328 −1.72394
$$739$$ −16.8328 −0.619205 −0.309603 0.950866i $$-0.600196\pi$$
−0.309603 + 0.950866i $$0.600196\pi$$
$$740$$ 0 0
$$741$$ 28.4164 1.04390
$$742$$ −2.85410 −0.104777
$$743$$ −30.7082 −1.12657 −0.563287 0.826261i $$-0.690464\pi$$
−0.563287 + 0.826261i $$0.690464\pi$$
$$744$$ −50.1246 −1.83766
$$745$$ 0 0
$$746$$ 59.3951 2.17461
$$747$$ −12.6525 −0.462930
$$748$$ −71.3951 −2.61046
$$749$$ 7.09017 0.259069
$$750$$ 0 0
$$751$$ −44.1246 −1.61013 −0.805065 0.593187i $$-0.797870\pi$$
−0.805065 + 0.593187i $$0.797870\pi$$
$$752$$ −40.3050 −1.46977
$$753$$ 11.2361 0.409465
$$754$$ −66.5410 −2.42328
$$755$$ 0 0
$$756$$ −24.2705 −0.882710
$$757$$ 21.2918 0.773863 0.386932 0.922108i $$-0.373535\pi$$
0.386932 + 0.922108i $$0.373535\pi$$
$$758$$ 13.0902 0.475456
$$759$$ 11.7082 0.424981
$$760$$ 0 0
$$761$$ −7.47214 −0.270865 −0.135432 0.990787i $$-0.543242\pi$$
−0.135432 + 0.990787i $$0.543242\pi$$
$$762$$ −39.9787 −1.44828
$$763$$ −9.56231 −0.346179
$$764$$ 27.2705 0.986612
$$765$$ 0 0
$$766$$ −2.14590 −0.0775344
$$767$$ 30.9787 1.11858
$$768$$ −14.5623 −0.525472
$$769$$ 31.3951 1.13214 0.566069 0.824358i $$-0.308464\pi$$
0.566069 + 0.824358i $$0.308464\pi$$
$$770$$ 0 0
$$771$$ 13.4721 0.485187
$$772$$ −97.6869 −3.51583
$$773$$ 16.5279 0.594466 0.297233 0.954805i $$-0.403936\pi$$
0.297233 + 0.954805i $$0.403936\pi$$
$$774$$ −45.5967 −1.63894
$$775$$ 0 0
$$776$$ 12.7639 0.458198
$$777$$ −6.70820 −0.240655
$$778$$ 19.4164 0.696112
$$779$$ 52.3607 1.87602
$$780$$ 0 0
$$781$$ 10.7082 0.383170
$$782$$ −65.7771 −2.35218
$$783$$ 26.1803 0.935609
$$784$$ −59.1246 −2.11159
$$785$$ 0 0
$$786$$ −5.85410 −0.208809
$$787$$ −43.4164 −1.54763 −0.773814 0.633413i $$-0.781653\pi$$
−0.773814 + 0.633413i $$0.781653\pi$$
$$788$$ 79.6869 2.83873
$$789$$ 20.6180 0.734021
$$790$$ 0 0
$$791$$ 15.0000 0.533339
$$792$$ 39.1246 1.39023
$$793$$ 20.1246 0.714646
$$794$$ −52.3607 −1.85821
$$795$$ 0 0
$$796$$ −113.666 −4.02877
$$797$$ 50.1246 1.77550 0.887752 0.460321i $$-0.152266\pi$$
0.887752 + 0.460321i $$0.152266\pi$$
$$798$$ 15.3262 0.542543
$$799$$ −22.9787 −0.812928
$$800$$ 0 0
$$801$$ 4.58359 0.161953
$$802$$ −62.5410 −2.20840
$$803$$ −28.4164 −1.00279
$$804$$ −69.9787 −2.46796
$$805$$ 0 0
$$806$$ −85.2492 −3.00278
$$807$$ 12.3262 0.433904
$$808$$ 14.2705 0.502035
$$809$$ 2.94427 0.103515 0.0517575 0.998660i $$-0.483518\pi$$
0.0517575 + 0.998660i $$0.483518\pi$$
$$810$$ 0 0
$$811$$ 21.5410 0.756408 0.378204 0.925722i $$-0.376542\pi$$
0.378204 + 0.925722i $$0.376542\pi$$
$$812$$ −25.4164 −0.891941
$$813$$ 1.00000 0.0350715
$$814$$ 45.9787 1.61155
$$815$$ 0 0
$$816$$ 55.3607 1.93801
$$817$$ 50.9787 1.78352
$$818$$ 96.1033 3.36017
$$819$$ −9.70820 −0.339232
$$820$$ 0 0
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ 33.2705 1.16044
$$823$$ −3.43769 −0.119830 −0.0599152 0.998203i $$-0.519083\pi$$
−0.0599152 + 0.998203i $$0.519083\pi$$
$$824$$ 7.47214 0.260304
$$825$$ 0 0
$$826$$ 16.7082 0.581353
$$827$$ −6.70820 −0.233267 −0.116634 0.993175i $$-0.537210\pi$$
−0.116634 + 0.993175i $$0.537210\pi$$
$$828$$ 43.4164 1.50882
$$829$$ 14.2705 0.495635 0.247818 0.968807i $$-0.420287\pi$$
0.247818 + 0.968807i $$0.420287\pi$$
$$830$$ 0 0
$$831$$ −4.70820 −0.163326
$$832$$ 42.2705 1.46547
$$833$$ −33.7082 −1.16792
$$834$$ −29.1803 −1.01043
$$835$$ 0 0
$$836$$ −74.3951 −2.57301
$$837$$ 33.5410 1.15935
$$838$$ −10.7082 −0.369909
$$839$$ 23.6180 0.815385 0.407693 0.913119i $$-0.366334\pi$$
0.407693 + 0.913119i $$0.366334\pi$$
$$840$$ 0 0
$$841$$ −1.58359 −0.0546066
$$842$$ 7.85410 0.270670
$$843$$ −31.4721 −1.08396
$$844$$ 72.1033 2.48190
$$845$$ 0 0
$$846$$ 21.4164 0.736311
$$847$$ −4.14590 −0.142455
$$848$$ −10.7426 −0.368904
$$849$$ 19.7082 0.676384
$$850$$ 0 0
$$851$$ 30.0000 1.02839
$$852$$ −19.8541 −0.680190
$$853$$ 44.2705 1.51579 0.757897 0.652375i $$-0.226227\pi$$
0.757897 + 0.652375i $$0.226227\pi$$
$$854$$ 10.8541 0.371420
$$855$$ 0 0
$$856$$ 52.9787 1.81078
$$857$$ −8.23607 −0.281339 −0.140669 0.990057i $$-0.544925\pi$$
−0.140669 + 0.990057i $$0.544925\pi$$
$$858$$ −33.2705 −1.13584
$$859$$ 10.5623 0.360381 0.180191 0.983632i $$-0.442329\pi$$
0.180191 + 0.983632i $$0.442329\pi$$
$$860$$ 0 0
$$861$$ 8.94427 0.304820
$$862$$ 89.9574 3.06396
$$863$$ 21.7082 0.738956 0.369478 0.929240i $$-0.379537\pi$$
0.369478 + 0.929240i $$0.379537\pi$$
$$864$$ −54.2705 −1.84632
$$865$$ 0 0
$$866$$ −37.7426 −1.28255
$$867$$ 14.5623 0.494562
$$868$$ −32.5623 −1.10524
$$869$$ 17.1803 0.582803
$$870$$ 0 0
$$871$$ −69.9787 −2.37114
$$872$$ −71.4508 −2.41963
$$873$$ −3.41641 −0.115628
$$874$$ −68.5410 −2.31843
$$875$$ 0 0
$$876$$ 52.6869 1.78013
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ 77.3951 2.61196
$$879$$ 21.6525 0.730320
$$880$$ 0 0
$$881$$ 29.8885 1.00697 0.503485 0.864004i $$-0.332051\pi$$
0.503485 + 0.864004i $$0.332051\pi$$
$$882$$ 31.4164 1.05785
$$883$$ −5.87539 −0.197723 −0.0988613 0.995101i $$-0.531520\pi$$
−0.0988613 + 0.995101i $$0.531520\pi$$
$$884$$ 132.374 4.45221
$$885$$ 0 0
$$886$$ 40.4164 1.35782
$$887$$ −40.1935 −1.34957 −0.674783 0.738016i $$-0.735763\pi$$
−0.674783 + 0.738016i $$0.735763\pi$$
$$888$$ −50.1246 −1.68207
$$889$$ −15.2705 −0.512156
$$890$$ 0 0
$$891$$ −2.61803 −0.0877074
$$892$$ −37.4164 −1.25279
$$893$$ −23.9443 −0.801265
$$894$$ −19.5623 −0.654261
$$895$$ 0 0
$$896$$ 1.09017 0.0364200
$$897$$ −21.7082 −0.724816
$$898$$ −34.9787 −1.16725
$$899$$ 35.1246 1.17147
$$900$$ 0 0
$$901$$ −6.12461 −0.204040
$$902$$ −61.3050 −2.04123
$$903$$ 8.70820 0.289791
$$904$$ 112.082 3.72779
$$905$$ 0 0
$$906$$ −49.7426 −1.65259
$$907$$ 7.12461 0.236569 0.118284 0.992980i $$-0.462261\pi$$
0.118284 + 0.992980i $$0.462261\pi$$
$$908$$ −14.2918 −0.474290
$$909$$ −3.81966 −0.126690
$$910$$ 0 0
$$911$$ −10.0344 −0.332456 −0.166228 0.986087i $$-0.553159\pi$$
−0.166228 + 0.986087i $$0.553159\pi$$
$$912$$ 57.6869 1.91020
$$913$$ −16.5623 −0.548132
$$914$$ 25.7984 0.853334
$$915$$ 0 0
$$916$$ 32.5623 1.07589
$$917$$ −2.23607 −0.0738415
$$918$$ −73.5410 −2.42722
$$919$$ −27.9787 −0.922933 −0.461466 0.887158i $$-0.652676\pi$$
−0.461466 + 0.887158i $$0.652676\pi$$
$$920$$ 0 0
$$921$$ 2.85410 0.0940459
$$922$$ 31.9787 1.05316
$$923$$ −19.8541 −0.653506
$$924$$ −12.7082 −0.418069
$$925$$ 0 0
$$926$$ 56.0689 1.84254
$$927$$ −2.00000 −0.0656886
$$928$$ −56.8328 −1.86563
$$929$$ 14.9443 0.490306 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$930$$ 0 0
$$931$$ −35.1246 −1.15116
$$932$$ 130.520 4.27532
$$933$$ 2.88854 0.0945667
$$934$$ 9.56231 0.312888
$$935$$ 0 0
$$936$$ −72.5410 −2.37108
$$937$$ 11.0000 0.359354 0.179677 0.983726i $$-0.442495\pi$$
0.179677 + 0.983726i $$0.442495\pi$$
$$938$$ −37.7426 −1.23234
$$939$$ −16.2918 −0.531663
$$940$$ 0 0
$$941$$ −23.3951 −0.762659 −0.381330 0.924439i $$-0.624534\pi$$
−0.381330 + 0.924439i $$0.624534\pi$$
$$942$$ −43.7426 −1.42521
$$943$$ −40.0000 −1.30258
$$944$$ 62.8885 2.04685
$$945$$ 0 0
$$946$$ −59.6869 −1.94059
$$947$$ 41.0132 1.33275 0.666374 0.745617i $$-0.267845\pi$$
0.666374 + 0.745617i $$0.267845\pi$$
$$948$$ −31.8541 −1.03457
$$949$$ 52.6869 1.71029
$$950$$ 0 0
$$951$$ −28.4164 −0.921465
$$952$$ 41.9787 1.36054
$$953$$ −13.3607 −0.432795 −0.216397 0.976305i $$-0.569431\pi$$
−0.216397 + 0.976305i $$0.569431\pi$$
$$954$$ 5.70820 0.184810
$$955$$ 0 0
$$956$$ 42.1033 1.36172
$$957$$ 13.7082 0.443123
$$958$$ 21.4164 0.691933
$$959$$ 12.7082 0.410369
$$960$$ 0 0
$$961$$ 14.0000 0.451613
$$962$$ −85.2492 −2.74855
$$963$$ −14.1803 −0.456955
$$964$$ −40.1459 −1.29301
$$965$$ 0 0
$$966$$ −11.7082 −0.376705
$$967$$ 14.4164 0.463600 0.231800 0.972763i $$-0.425538\pi$$
0.231800 + 0.972763i $$0.425538\pi$$
$$968$$ −30.9787 −0.995694
$$969$$ 32.8885 1.05653
$$970$$ 0 0
$$971$$ −53.0132 −1.70127 −0.850637 0.525754i $$-0.823783\pi$$
−0.850637 + 0.525754i $$0.823783\pi$$
$$972$$ 77.6656 2.49113
$$973$$ −11.1459 −0.357321
$$974$$ 60.2148 1.92941
$$975$$ 0 0
$$976$$ 40.8541 1.30771
$$977$$ −46.7426 −1.49543 −0.747715 0.664020i $$-0.768849\pi$$
−0.747715 + 0.664020i $$0.768849\pi$$
$$978$$ −7.09017 −0.226719
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ 19.1246 0.610602
$$982$$ 93.6656 2.98899
$$983$$ −18.6525 −0.594922 −0.297461 0.954734i $$-0.596140\pi$$
−0.297461 + 0.954734i $$0.596140\pi$$
$$984$$ 66.8328 2.13055
$$985$$ 0 0
$$986$$ −77.0132 −2.45260
$$987$$ −4.09017 −0.130192
$$988$$ 137.936 4.38833
$$989$$ −38.9443 −1.23836
$$990$$ 0 0
$$991$$ 24.2705 0.770978 0.385489 0.922712i $$-0.374033\pi$$
0.385489 + 0.922712i $$0.374033\pi$$
$$992$$ −72.8115 −2.31177
$$993$$ −16.1459 −0.512375
$$994$$ −10.7082 −0.339644
$$995$$ 0 0
$$996$$ 30.7082 0.973027
$$997$$ 39.2918 1.24438 0.622192 0.782865i $$-0.286242\pi$$
0.622192 + 0.782865i $$0.286242\pi$$
$$998$$ 34.7426 1.09976
$$999$$ 33.5410 1.06119
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 2575.2.a.g.1.2 2
5.4 even 2 103.2.a.a.1.1 2
15.14 odd 2 927.2.a.b.1.2 2
20.19 odd 2 1648.2.a.f.1.2 2
35.34 odd 2 5047.2.a.a.1.1 2
40.19 odd 2 6592.2.a.h.1.1 2
40.29 even 2 6592.2.a.t.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.1 2 5.4 even 2
927.2.a.b.1.2 2 15.14 odd 2
1648.2.a.f.1.2 2 20.19 odd 2
2575.2.a.g.1.2 2 1.1 even 1 trivial
5047.2.a.a.1.1 2 35.34 odd 2
6592.2.a.h.1.1 2 40.19 odd 2
6592.2.a.t.1.1 2 40.29 even 2