# Properties

 Label 275.2.a.f.1.1 Level $275$ Weight $2$ Character 275.1 Self dual yes Analytic conductor $2.196$ 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] = [275,2,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, 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("275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.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: $$2.19588605559$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.30278 q^{2} +2.30278 q^{3} -0.302776 q^{4} -3.00000 q^{6} +0.697224 q^{7} +3.00000 q^{8} +2.30278 q^{9} +O(q^{10})$$ $$q-1.30278 q^{2} +2.30278 q^{3} -0.302776 q^{4} -3.00000 q^{6} +0.697224 q^{7} +3.00000 q^{8} +2.30278 q^{9} -1.00000 q^{11} -0.697224 q^{12} +5.00000 q^{13} -0.908327 q^{14} -3.30278 q^{16} +6.90833 q^{17} -3.00000 q^{18} -1.00000 q^{19} +1.60555 q^{21} +1.30278 q^{22} -7.30278 q^{23} +6.90833 q^{24} -6.51388 q^{26} -1.60555 q^{27} -0.211103 q^{28} +0.908327 q^{29} +10.2111 q^{31} -1.69722 q^{32} -2.30278 q^{33} -9.00000 q^{34} -0.697224 q^{36} +2.39445 q^{37} +1.30278 q^{38} +11.5139 q^{39} -5.60555 q^{41} -2.09167 q^{42} +7.21110 q^{43} +0.302776 q^{44} +9.51388 q^{46} -3.00000 q^{47} -7.60555 q^{48} -6.51388 q^{49} +15.9083 q^{51} -1.51388 q^{52} -1.30278 q^{53} +2.09167 q^{54} +2.09167 q^{56} -2.30278 q^{57} -1.18335 q^{58} -14.2111 q^{59} -7.90833 q^{61} -13.3028 q^{62} +1.60555 q^{63} +8.81665 q^{64} +3.00000 q^{66} -4.00000 q^{67} -2.09167 q^{68} -16.8167 q^{69} -2.60555 q^{71} +6.90833 q^{72} -7.90833 q^{73} -3.11943 q^{74} +0.302776 q^{76} -0.697224 q^{77} -15.0000 q^{78} -10.9083 q^{79} -10.6056 q^{81} +7.30278 q^{82} -3.51388 q^{83} -0.486122 q^{84} -9.39445 q^{86} +2.09167 q^{87} -3.00000 q^{88} +1.69722 q^{89} +3.48612 q^{91} +2.21110 q^{92} +23.5139 q^{93} +3.90833 q^{94} -3.90833 q^{96} +15.3028 q^{97} +8.48612 q^{98} -2.30278 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} + 3 q^{4} - 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 + 3 * q^4 - 6 * q^6 + 5 * q^7 + 6 * q^8 + q^9 $$2 q + q^{2} + q^{3} + 3 q^{4} - 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9} - 2 q^{11} - 5 q^{12} + 10 q^{13} + 9 q^{14} - 3 q^{16} + 3 q^{17} - 6 q^{18} - 2 q^{19} - 4 q^{21} - q^{22} - 11 q^{23} + 3 q^{24} + 5 q^{26} + 4 q^{27} + 14 q^{28} - 9 q^{29} + 6 q^{31} - 7 q^{32} - q^{33} - 18 q^{34} - 5 q^{36} + 12 q^{37} - q^{38} + 5 q^{39} - 4 q^{41} - 15 q^{42} - 3 q^{44} + q^{46} - 6 q^{47} - 8 q^{48} + 5 q^{49} + 21 q^{51} + 15 q^{52} + q^{53} + 15 q^{54} + 15 q^{56} - q^{57} - 24 q^{58} - 14 q^{59} - 5 q^{61} - 23 q^{62} - 4 q^{63} - 4 q^{64} + 6 q^{66} - 8 q^{67} - 15 q^{68} - 12 q^{69} + 2 q^{71} + 3 q^{72} - 5 q^{73} + 19 q^{74} - 3 q^{76} - 5 q^{77} - 30 q^{78} - 11 q^{79} - 14 q^{81} + 11 q^{82} + 11 q^{83} - 19 q^{84} - 26 q^{86} + 15 q^{87} - 6 q^{88} + 7 q^{89} + 25 q^{91} - 10 q^{92} + 29 q^{93} - 3 q^{94} + 3 q^{96} + 27 q^{97} + 35 q^{98} - q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 + 3 * q^4 - 6 * q^6 + 5 * q^7 + 6 * q^8 + q^9 - 2 * q^11 - 5 * q^12 + 10 * q^13 + 9 * q^14 - 3 * q^16 + 3 * q^17 - 6 * q^18 - 2 * q^19 - 4 * q^21 - q^22 - 11 * q^23 + 3 * q^24 + 5 * q^26 + 4 * q^27 + 14 * q^28 - 9 * q^29 + 6 * q^31 - 7 * q^32 - q^33 - 18 * q^34 - 5 * q^36 + 12 * q^37 - q^38 + 5 * q^39 - 4 * q^41 - 15 * q^42 - 3 * q^44 + q^46 - 6 * q^47 - 8 * q^48 + 5 * q^49 + 21 * q^51 + 15 * q^52 + q^53 + 15 * q^54 + 15 * q^56 - q^57 - 24 * q^58 - 14 * q^59 - 5 * q^61 - 23 * q^62 - 4 * q^63 - 4 * q^64 + 6 * q^66 - 8 * q^67 - 15 * q^68 - 12 * q^69 + 2 * q^71 + 3 * q^72 - 5 * q^73 + 19 * q^74 - 3 * q^76 - 5 * q^77 - 30 * q^78 - 11 * q^79 - 14 * q^81 + 11 * q^82 + 11 * q^83 - 19 * q^84 - 26 * q^86 + 15 * q^87 - 6 * q^88 + 7 * q^89 + 25 * q^91 - 10 * q^92 + 29 * q^93 - 3 * q^94 + 3 * q^96 + 27 * q^97 + 35 * q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.30278 −0.921201 −0.460601 0.887607i $$-0.652366\pi$$
−0.460601 + 0.887607i $$0.652366\pi$$
$$3$$ 2.30278 1.32951 0.664754 0.747062i $$-0.268536\pi$$
0.664754 + 0.747062i $$0.268536\pi$$
$$4$$ −0.302776 −0.151388
$$5$$ 0 0
$$6$$ −3.00000 −1.22474
$$7$$ 0.697224 0.263526 0.131763 0.991281i $$-0.457936\pi$$
0.131763 + 0.991281i $$0.457936\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 2.30278 0.767592
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −0.697224 −0.201271
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ −0.908327 −0.242761
$$15$$ 0 0
$$16$$ −3.30278 −0.825694
$$17$$ 6.90833 1.67552 0.837758 0.546042i $$-0.183866\pi$$
0.837758 + 0.546042i $$0.183866\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 1.60555 0.350360
$$22$$ 1.30278 0.277753
$$23$$ −7.30278 −1.52273 −0.761367 0.648321i $$-0.775471\pi$$
−0.761367 + 0.648321i $$0.775471\pi$$
$$24$$ 6.90833 1.41016
$$25$$ 0 0
$$26$$ −6.51388 −1.27748
$$27$$ −1.60555 −0.308988
$$28$$ −0.211103 −0.0398946
$$29$$ 0.908327 0.168672 0.0843360 0.996437i $$-0.473123\pi$$
0.0843360 + 0.996437i $$0.473123\pi$$
$$30$$ 0 0
$$31$$ 10.2111 1.83397 0.916984 0.398924i $$-0.130616\pi$$
0.916984 + 0.398924i $$0.130616\pi$$
$$32$$ −1.69722 −0.300030
$$33$$ −2.30278 −0.400862
$$34$$ −9.00000 −1.54349
$$35$$ 0 0
$$36$$ −0.697224 −0.116204
$$37$$ 2.39445 0.393645 0.196822 0.980439i $$-0.436938\pi$$
0.196822 + 0.980439i $$0.436938\pi$$
$$38$$ 1.30278 0.211338
$$39$$ 11.5139 1.84370
$$40$$ 0 0
$$41$$ −5.60555 −0.875440 −0.437720 0.899111i $$-0.644214\pi$$
−0.437720 + 0.899111i $$0.644214\pi$$
$$42$$ −2.09167 −0.322752
$$43$$ 7.21110 1.09968 0.549841 0.835269i $$-0.314688\pi$$
0.549841 + 0.835269i $$0.314688\pi$$
$$44$$ 0.302776 0.0456451
$$45$$ 0 0
$$46$$ 9.51388 1.40274
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ −7.60555 −1.09777
$$49$$ −6.51388 −0.930554
$$50$$ 0 0
$$51$$ 15.9083 2.22761
$$52$$ −1.51388 −0.209937
$$53$$ −1.30278 −0.178950 −0.0894750 0.995989i $$-0.528519\pi$$
−0.0894750 + 0.995989i $$0.528519\pi$$
$$54$$ 2.09167 0.284641
$$55$$ 0 0
$$56$$ 2.09167 0.279512
$$57$$ −2.30278 −0.305010
$$58$$ −1.18335 −0.155381
$$59$$ −14.2111 −1.85013 −0.925064 0.379811i $$-0.875989\pi$$
−0.925064 + 0.379811i $$0.875989\pi$$
$$60$$ 0 0
$$61$$ −7.90833 −1.01256 −0.506279 0.862370i $$-0.668979\pi$$
−0.506279 + 0.862370i $$0.668979\pi$$
$$62$$ −13.3028 −1.68945
$$63$$ 1.60555 0.202280
$$64$$ 8.81665 1.10208
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −2.09167 −0.253653
$$69$$ −16.8167 −2.02449
$$70$$ 0 0
$$71$$ −2.60555 −0.309222 −0.154611 0.987975i $$-0.549412\pi$$
−0.154611 + 0.987975i $$0.549412\pi$$
$$72$$ 6.90833 0.814154
$$73$$ −7.90833 −0.925600 −0.462800 0.886463i $$-0.653155\pi$$
−0.462800 + 0.886463i $$0.653155\pi$$
$$74$$ −3.11943 −0.362626
$$75$$ 0 0
$$76$$ 0.302776 0.0347307
$$77$$ −0.697224 −0.0794561
$$78$$ −15.0000 −1.69842
$$79$$ −10.9083 −1.22728 −0.613641 0.789585i $$-0.710296\pi$$
−0.613641 + 0.789585i $$0.710296\pi$$
$$80$$ 0 0
$$81$$ −10.6056 −1.17839
$$82$$ 7.30278 0.806457
$$83$$ −3.51388 −0.385698 −0.192849 0.981228i $$-0.561773\pi$$
−0.192849 + 0.981228i $$0.561773\pi$$
$$84$$ −0.486122 −0.0530402
$$85$$ 0 0
$$86$$ −9.39445 −1.01303
$$87$$ 2.09167 0.224251
$$88$$ −3.00000 −0.319801
$$89$$ 1.69722 0.179905 0.0899527 0.995946i $$-0.471328\pi$$
0.0899527 + 0.995946i $$0.471328\pi$$
$$90$$ 0 0
$$91$$ 3.48612 0.365445
$$92$$ 2.21110 0.230523
$$93$$ 23.5139 2.43828
$$94$$ 3.90833 0.403113
$$95$$ 0 0
$$96$$ −3.90833 −0.398892
$$97$$ 15.3028 1.55376 0.776881 0.629648i $$-0.216801\pi$$
0.776881 + 0.629648i $$0.216801\pi$$
$$98$$ 8.48612 0.857228
$$99$$ −2.30278 −0.231438
$$100$$ 0 0
$$101$$ −0.513878 −0.0511328 −0.0255664 0.999673i $$-0.508139\pi$$
−0.0255664 + 0.999673i $$0.508139\pi$$
$$102$$ −20.7250 −2.05208
$$103$$ 2.90833 0.286566 0.143283 0.989682i $$-0.454234\pi$$
0.143283 + 0.989682i $$0.454234\pi$$
$$104$$ 15.0000 1.47087
$$105$$ 0 0
$$106$$ 1.69722 0.164849
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0.486122 0.0467771
$$109$$ 11.5139 1.10283 0.551415 0.834231i $$-0.314088\pi$$
0.551415 + 0.834231i $$0.314088\pi$$
$$110$$ 0 0
$$111$$ 5.51388 0.523354
$$112$$ −2.30278 −0.217592
$$113$$ −10.8167 −1.01755 −0.508773 0.860901i $$-0.669901\pi$$
−0.508773 + 0.860901i $$0.669901\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 0 0
$$116$$ −0.275019 −0.0255349
$$117$$ 11.5139 1.06446
$$118$$ 18.5139 1.70434
$$119$$ 4.81665 0.441542
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 10.3028 0.932769
$$123$$ −12.9083 −1.16390
$$124$$ −3.09167 −0.277640
$$125$$ 0 0
$$126$$ −2.09167 −0.186341
$$127$$ 8.11943 0.720483 0.360241 0.932859i $$-0.382694\pi$$
0.360241 + 0.932859i $$0.382694\pi$$
$$128$$ −8.09167 −0.715210
$$129$$ 16.6056 1.46204
$$130$$ 0 0
$$131$$ −9.90833 −0.865695 −0.432847 0.901467i $$-0.642491\pi$$
−0.432847 + 0.901467i $$0.642491\pi$$
$$132$$ 0.697224 0.0606856
$$133$$ −0.697224 −0.0604570
$$134$$ 5.21110 0.450171
$$135$$ 0 0
$$136$$ 20.7250 1.77715
$$137$$ −12.9083 −1.10283 −0.551416 0.834230i $$-0.685912\pi$$
−0.551416 + 0.834230i $$0.685912\pi$$
$$138$$ 21.9083 1.86496
$$139$$ −6.21110 −0.526819 −0.263409 0.964684i $$-0.584847\pi$$
−0.263409 + 0.964684i $$0.584847\pi$$
$$140$$ 0 0
$$141$$ −6.90833 −0.581786
$$142$$ 3.39445 0.284856
$$143$$ −5.00000 −0.418121
$$144$$ −7.60555 −0.633796
$$145$$ 0 0
$$146$$ 10.3028 0.852664
$$147$$ −15.0000 −1.23718
$$148$$ −0.724981 −0.0595930
$$149$$ 17.2111 1.40999 0.704994 0.709213i $$-0.250950\pi$$
0.704994 + 0.709213i $$0.250950\pi$$
$$150$$ 0 0
$$151$$ 0.816654 0.0664583 0.0332292 0.999448i $$-0.489421\pi$$
0.0332292 + 0.999448i $$0.489421\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ 15.9083 1.28611
$$154$$ 0.908327 0.0731951
$$155$$ 0 0
$$156$$ −3.48612 −0.279113
$$157$$ 19.2111 1.53321 0.766606 0.642117i $$-0.221944\pi$$
0.766606 + 0.642117i $$0.221944\pi$$
$$158$$ 14.2111 1.13057
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ −5.09167 −0.401280
$$162$$ 13.8167 1.08554
$$163$$ 9.30278 0.728650 0.364325 0.931272i $$-0.381300\pi$$
0.364325 + 0.931272i $$0.381300\pi$$
$$164$$ 1.69722 0.132531
$$165$$ 0 0
$$166$$ 4.57779 0.355306
$$167$$ 13.4222 1.03864 0.519321 0.854579i $$-0.326185\pi$$
0.519321 + 0.854579i $$0.326185\pi$$
$$168$$ 4.81665 0.371613
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −2.30278 −0.176098
$$172$$ −2.18335 −0.166479
$$173$$ −4.81665 −0.366203 −0.183102 0.983094i $$-0.558614\pi$$
−0.183102 + 0.983094i $$0.558614\pi$$
$$174$$ −2.72498 −0.206580
$$175$$ 0 0
$$176$$ 3.30278 0.248956
$$177$$ −32.7250 −2.45976
$$178$$ −2.21110 −0.165729
$$179$$ 12.5139 0.935331 0.467666 0.883905i $$-0.345095\pi$$
0.467666 + 0.883905i $$0.345095\pi$$
$$180$$ 0 0
$$181$$ −19.9083 −1.47977 −0.739887 0.672731i $$-0.765121\pi$$
−0.739887 + 0.672731i $$0.765121\pi$$
$$182$$ −4.54163 −0.336648
$$183$$ −18.2111 −1.34620
$$184$$ −21.9083 −1.61510
$$185$$ 0 0
$$186$$ −30.6333 −2.24614
$$187$$ −6.90833 −0.505187
$$188$$ 0.908327 0.0662465
$$189$$ −1.11943 −0.0814265
$$190$$ 0 0
$$191$$ 10.3028 0.745483 0.372741 0.927935i $$-0.378418\pi$$
0.372741 + 0.927935i $$0.378418\pi$$
$$192$$ 20.3028 1.46523
$$193$$ 13.2111 0.950956 0.475478 0.879728i $$-0.342275\pi$$
0.475478 + 0.879728i $$0.342275\pi$$
$$194$$ −19.9361 −1.43133
$$195$$ 0 0
$$196$$ 1.97224 0.140875
$$197$$ 13.3028 0.947784 0.473892 0.880583i $$-0.342849\pi$$
0.473892 + 0.880583i $$0.342849\pi$$
$$198$$ 3.00000 0.213201
$$199$$ −6.48612 −0.459789 −0.229894 0.973216i $$-0.573838\pi$$
−0.229894 + 0.973216i $$0.573838\pi$$
$$200$$ 0 0
$$201$$ −9.21110 −0.649701
$$202$$ 0.669468 0.0471036
$$203$$ 0.633308 0.0444495
$$204$$ −4.81665 −0.337233
$$205$$ 0 0
$$206$$ −3.78890 −0.263985
$$207$$ −16.8167 −1.16884
$$208$$ −16.5139 −1.14503
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −25.2389 −1.73751 −0.868757 0.495238i $$-0.835081\pi$$
−0.868757 + 0.495238i $$0.835081\pi$$
$$212$$ 0.394449 0.0270908
$$213$$ −6.00000 −0.411113
$$214$$ 3.90833 0.267168
$$215$$ 0 0
$$216$$ −4.81665 −0.327732
$$217$$ 7.11943 0.483298
$$218$$ −15.0000 −1.01593
$$219$$ −18.2111 −1.23059
$$220$$ 0 0
$$221$$ 34.5416 2.32352
$$222$$ −7.18335 −0.482115
$$223$$ −22.6333 −1.51564 −0.757819 0.652465i $$-0.773735\pi$$
−0.757819 + 0.652465i $$0.773735\pi$$
$$224$$ −1.18335 −0.0790656
$$225$$ 0 0
$$226$$ 14.0917 0.937364
$$227$$ 1.69722 0.112649 0.0563244 0.998413i $$-0.482062\pi$$
0.0563244 + 0.998413i $$0.482062\pi$$
$$228$$ 0.697224 0.0461748
$$229$$ −18.7250 −1.23738 −0.618691 0.785635i $$-0.712337\pi$$
−0.618691 + 0.785635i $$0.712337\pi$$
$$230$$ 0 0
$$231$$ −1.60555 −0.105638
$$232$$ 2.72498 0.178904
$$233$$ 15.9083 1.04219 0.521095 0.853499i $$-0.325524\pi$$
0.521095 + 0.853499i $$0.325524\pi$$
$$234$$ −15.0000 −0.980581
$$235$$ 0 0
$$236$$ 4.30278 0.280087
$$237$$ −25.1194 −1.63168
$$238$$ −6.27502 −0.406749
$$239$$ 21.1194 1.36610 0.683051 0.730371i $$-0.260653\pi$$
0.683051 + 0.730371i $$0.260653\pi$$
$$240$$ 0 0
$$241$$ 21.9361 1.41303 0.706514 0.707699i $$-0.250267\pi$$
0.706514 + 0.707699i $$0.250267\pi$$
$$242$$ −1.30278 −0.0837456
$$243$$ −19.6056 −1.25770
$$244$$ 2.39445 0.153289
$$245$$ 0 0
$$246$$ 16.8167 1.07219
$$247$$ −5.00000 −0.318142
$$248$$ 30.6333 1.94522
$$249$$ −8.09167 −0.512789
$$250$$ 0 0
$$251$$ 6.90833 0.436050 0.218025 0.975943i $$-0.430039\pi$$
0.218025 + 0.975943i $$0.430039\pi$$
$$252$$ −0.486122 −0.0306228
$$253$$ 7.30278 0.459122
$$254$$ −10.5778 −0.663710
$$255$$ 0 0
$$256$$ −7.09167 −0.443230
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −21.6333 −1.34683
$$259$$ 1.66947 0.103736
$$260$$ 0 0
$$261$$ 2.09167 0.129471
$$262$$ 12.9083 0.797479
$$263$$ −22.8167 −1.40694 −0.703468 0.710727i $$-0.748366\pi$$
−0.703468 + 0.710727i $$0.748366\pi$$
$$264$$ −6.90833 −0.425178
$$265$$ 0 0
$$266$$ 0.908327 0.0556931
$$267$$ 3.90833 0.239186
$$268$$ 1.21110 0.0739799
$$269$$ 8.72498 0.531971 0.265986 0.963977i $$-0.414303\pi$$
0.265986 + 0.963977i $$0.414303\pi$$
$$270$$ 0 0
$$271$$ −0.211103 −0.0128236 −0.00641178 0.999979i $$-0.502041\pi$$
−0.00641178 + 0.999979i $$0.502041\pi$$
$$272$$ −22.8167 −1.38346
$$273$$ 8.02776 0.485862
$$274$$ 16.8167 1.01593
$$275$$ 0 0
$$276$$ 5.09167 0.306483
$$277$$ 14.3944 0.864879 0.432439 0.901663i $$-0.357653\pi$$
0.432439 + 0.901663i $$0.357653\pi$$
$$278$$ 8.09167 0.485306
$$279$$ 23.5139 1.40774
$$280$$ 0 0
$$281$$ −1.18335 −0.0705925 −0.0352963 0.999377i $$-0.511237\pi$$
−0.0352963 + 0.999377i $$0.511237\pi$$
$$282$$ 9.00000 0.535942
$$283$$ 6.30278 0.374661 0.187331 0.982297i $$-0.440016\pi$$
0.187331 + 0.982297i $$0.440016\pi$$
$$284$$ 0.788897 0.0468125
$$285$$ 0 0
$$286$$ 6.51388 0.385174
$$287$$ −3.90833 −0.230701
$$288$$ −3.90833 −0.230300
$$289$$ 30.7250 1.80735
$$290$$ 0 0
$$291$$ 35.2389 2.06574
$$292$$ 2.39445 0.140125
$$293$$ 0.788897 0.0460879 0.0230439 0.999734i $$-0.492664\pi$$
0.0230439 + 0.999734i $$0.492664\pi$$
$$294$$ 19.5416 1.13969
$$295$$ 0 0
$$296$$ 7.18335 0.417524
$$297$$ 1.60555 0.0931635
$$298$$ −22.4222 −1.29888
$$299$$ −36.5139 −2.11165
$$300$$ 0 0
$$301$$ 5.02776 0.289795
$$302$$ −1.06392 −0.0612215
$$303$$ −1.18335 −0.0679815
$$304$$ 3.30278 0.189427
$$305$$ 0 0
$$306$$ −20.7250 −1.18477
$$307$$ −16.9083 −0.965009 −0.482505 0.875893i $$-0.660273\pi$$
−0.482505 + 0.875893i $$0.660273\pi$$
$$308$$ 0.211103 0.0120287
$$309$$ 6.69722 0.380992
$$310$$ 0 0
$$311$$ 4.81665 0.273127 0.136564 0.990631i $$-0.456394\pi$$
0.136564 + 0.990631i $$0.456394\pi$$
$$312$$ 34.5416 1.95553
$$313$$ 0.183346 0.0103633 0.00518167 0.999987i $$-0.498351\pi$$
0.00518167 + 0.999987i $$0.498351\pi$$
$$314$$ −25.0278 −1.41240
$$315$$ 0 0
$$316$$ 3.30278 0.185796
$$317$$ −0.908327 −0.0510167 −0.0255084 0.999675i $$-0.508120\pi$$
−0.0255084 + 0.999675i $$0.508120\pi$$
$$318$$ 3.90833 0.219168
$$319$$ −0.908327 −0.0508565
$$320$$ 0 0
$$321$$ −6.90833 −0.385585
$$322$$ 6.63331 0.369660
$$323$$ −6.90833 −0.384390
$$324$$ 3.21110 0.178395
$$325$$ 0 0
$$326$$ −12.1194 −0.671233
$$327$$ 26.5139 1.46622
$$328$$ −16.8167 −0.928544
$$329$$ −2.09167 −0.115318
$$330$$ 0 0
$$331$$ −21.6056 −1.18755 −0.593774 0.804632i $$-0.702363\pi$$
−0.593774 + 0.804632i $$0.702363\pi$$
$$332$$ 1.06392 0.0583900
$$333$$ 5.51388 0.302159
$$334$$ −17.4861 −0.956798
$$335$$ 0 0
$$336$$ −5.30278 −0.289290
$$337$$ −30.8444 −1.68020 −0.840101 0.542430i $$-0.817504\pi$$
−0.840101 + 0.542430i $$0.817504\pi$$
$$338$$ −15.6333 −0.850340
$$339$$ −24.9083 −1.35283
$$340$$ 0 0
$$341$$ −10.2111 −0.552962
$$342$$ 3.00000 0.162221
$$343$$ −9.42221 −0.508751
$$344$$ 21.6333 1.16639
$$345$$ 0 0
$$346$$ 6.27502 0.337347
$$347$$ −12.5139 −0.671780 −0.335890 0.941901i $$-0.609037\pi$$
−0.335890 + 0.941901i $$0.609037\pi$$
$$348$$ −0.633308 −0.0339489
$$349$$ −5.18335 −0.277458 −0.138729 0.990330i $$-0.544302\pi$$
−0.138729 + 0.990330i $$0.544302\pi$$
$$350$$ 0 0
$$351$$ −8.02776 −0.428490
$$352$$ 1.69722 0.0904624
$$353$$ 18.6333 0.991751 0.495875 0.868394i $$-0.334847\pi$$
0.495875 + 0.868394i $$0.334847\pi$$
$$354$$ 42.6333 2.26593
$$355$$ 0 0
$$356$$ −0.513878 −0.0272355
$$357$$ 11.0917 0.587034
$$358$$ −16.3028 −0.861628
$$359$$ 0.788897 0.0416364 0.0208182 0.999783i $$-0.493373\pi$$
0.0208182 + 0.999783i $$0.493373\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 25.9361 1.36317
$$363$$ 2.30278 0.120864
$$364$$ −1.05551 −0.0553239
$$365$$ 0 0
$$366$$ 23.7250 1.24012
$$367$$ −20.6972 −1.08039 −0.540193 0.841541i $$-0.681649\pi$$
−0.540193 + 0.841541i $$0.681649\pi$$
$$368$$ 24.1194 1.25731
$$369$$ −12.9083 −0.671981
$$370$$ 0 0
$$371$$ −0.908327 −0.0471580
$$372$$ −7.11943 −0.369125
$$373$$ 27.4222 1.41987 0.709934 0.704268i $$-0.248725\pi$$
0.709934 + 0.704268i $$0.248725\pi$$
$$374$$ 9.00000 0.465379
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ 4.54163 0.233906
$$378$$ 1.45837 0.0750102
$$379$$ 3.18335 0.163518 0.0817588 0.996652i $$-0.473946\pi$$
0.0817588 + 0.996652i $$0.473946\pi$$
$$380$$ 0 0
$$381$$ 18.6972 0.957888
$$382$$ −13.4222 −0.686740
$$383$$ 21.6333 1.10541 0.552705 0.833377i $$-0.313596\pi$$
0.552705 + 0.833377i $$0.313596\pi$$
$$384$$ −18.6333 −0.950877
$$385$$ 0 0
$$386$$ −17.2111 −0.876022
$$387$$ 16.6056 0.844108
$$388$$ −4.63331 −0.235221
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ −50.4500 −2.55136
$$392$$ −19.5416 −0.987002
$$393$$ −22.8167 −1.15095
$$394$$ −17.3305 −0.873100
$$395$$ 0 0
$$396$$ 0.697224 0.0350368
$$397$$ 21.6972 1.08895 0.544476 0.838776i $$-0.316728\pi$$
0.544476 + 0.838776i $$0.316728\pi$$
$$398$$ 8.44996 0.423558
$$399$$ −1.60555 −0.0803781
$$400$$ 0 0
$$401$$ −12.7889 −0.638647 −0.319324 0.947646i $$-0.603456\pi$$
−0.319324 + 0.947646i $$0.603456\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 51.0555 2.54326
$$404$$ 0.155590 0.00774088
$$405$$ 0 0
$$406$$ −0.825058 −0.0409469
$$407$$ −2.39445 −0.118688
$$408$$ 47.7250 2.36274
$$409$$ −6.21110 −0.307119 −0.153560 0.988139i $$-0.549074\pi$$
−0.153560 + 0.988139i $$0.549074\pi$$
$$410$$ 0 0
$$411$$ −29.7250 −1.46623
$$412$$ −0.880571 −0.0433826
$$413$$ −9.90833 −0.487557
$$414$$ 21.9083 1.07674
$$415$$ 0 0
$$416$$ −8.48612 −0.416066
$$417$$ −14.3028 −0.700410
$$418$$ −1.30278 −0.0637208
$$419$$ 6.39445 0.312389 0.156195 0.987726i $$-0.450077\pi$$
0.156195 + 0.987726i $$0.450077\pi$$
$$420$$ 0 0
$$421$$ 0.697224 0.0339806 0.0169903 0.999856i $$-0.494592\pi$$
0.0169903 + 0.999856i $$0.494592\pi$$
$$422$$ 32.8806 1.60060
$$423$$ −6.90833 −0.335894
$$424$$ −3.90833 −0.189805
$$425$$ 0 0
$$426$$ 7.81665 0.378718
$$427$$ −5.51388 −0.266835
$$428$$ 0.908327 0.0439056
$$429$$ −11.5139 −0.555895
$$430$$ 0 0
$$431$$ 33.0000 1.58955 0.794777 0.606902i $$-0.207588\pi$$
0.794777 + 0.606902i $$0.207588\pi$$
$$432$$ 5.30278 0.255130
$$433$$ 5.00000 0.240285 0.120142 0.992757i $$-0.461665\pi$$
0.120142 + 0.992757i $$0.461665\pi$$
$$434$$ −9.27502 −0.445215
$$435$$ 0 0
$$436$$ −3.48612 −0.166955
$$437$$ 7.30278 0.349339
$$438$$ 23.7250 1.13362
$$439$$ 24.3028 1.15991 0.579954 0.814649i $$-0.303070\pi$$
0.579954 + 0.814649i $$0.303070\pi$$
$$440$$ 0 0
$$441$$ −15.0000 −0.714286
$$442$$ −45.0000 −2.14043
$$443$$ −8.60555 −0.408862 −0.204431 0.978881i $$-0.565534\pi$$
−0.204431 + 0.978881i $$0.565534\pi$$
$$444$$ −1.66947 −0.0792294
$$445$$ 0 0
$$446$$ 29.4861 1.39621
$$447$$ 39.6333 1.87459
$$448$$ 6.14719 0.290427
$$449$$ 23.4861 1.10838 0.554189 0.832391i $$-0.313028\pi$$
0.554189 + 0.832391i $$0.313028\pi$$
$$450$$ 0 0
$$451$$ 5.60555 0.263955
$$452$$ 3.27502 0.154044
$$453$$ 1.88057 0.0883569
$$454$$ −2.21110 −0.103772
$$455$$ 0 0
$$456$$ −6.90833 −0.323512
$$457$$ −20.6972 −0.968175 −0.484088 0.875020i $$-0.660848\pi$$
−0.484088 + 0.875020i $$0.660848\pi$$
$$458$$ 24.3944 1.13988
$$459$$ −11.0917 −0.517715
$$460$$ 0 0
$$461$$ 32.2111 1.50022 0.750110 0.661313i $$-0.230000\pi$$
0.750110 + 0.661313i $$0.230000\pi$$
$$462$$ 2.09167 0.0973134
$$463$$ 11.7889 0.547877 0.273938 0.961747i $$-0.411674\pi$$
0.273938 + 0.961747i $$0.411674\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 0 0
$$466$$ −20.7250 −0.960066
$$467$$ −18.6333 −0.862247 −0.431123 0.902293i $$-0.641883\pi$$
−0.431123 + 0.902293i $$0.641883\pi$$
$$468$$ −3.48612 −0.161146
$$469$$ −2.78890 −0.128779
$$470$$ 0 0
$$471$$ 44.2389 2.03842
$$472$$ −42.6333 −1.96236
$$473$$ −7.21110 −0.331567
$$474$$ 32.7250 1.50311
$$475$$ 0 0
$$476$$ −1.45837 −0.0668441
$$477$$ −3.00000 −0.137361
$$478$$ −27.5139 −1.25846
$$479$$ −34.8167 −1.59081 −0.795407 0.606076i $$-0.792743\pi$$
−0.795407 + 0.606076i $$0.792743\pi$$
$$480$$ 0 0
$$481$$ 11.9722 0.545887
$$482$$ −28.5778 −1.30168
$$483$$ −11.7250 −0.533505
$$484$$ −0.302776 −0.0137625
$$485$$ 0 0
$$486$$ 25.5416 1.15859
$$487$$ 4.21110 0.190823 0.0954116 0.995438i $$-0.469583\pi$$
0.0954116 + 0.995438i $$0.469583\pi$$
$$488$$ −23.7250 −1.07398
$$489$$ 21.4222 0.968746
$$490$$ 0 0
$$491$$ −9.78890 −0.441767 −0.220883 0.975300i $$-0.570894\pi$$
−0.220883 + 0.975300i $$0.570894\pi$$
$$492$$ 3.90833 0.176201
$$493$$ 6.27502 0.282613
$$494$$ 6.51388 0.293073
$$495$$ 0 0
$$496$$ −33.7250 −1.51430
$$497$$ −1.81665 −0.0814881
$$498$$ 10.5416 0.472382
$$499$$ −3.48612 −0.156060 −0.0780301 0.996951i $$-0.524863\pi$$
−0.0780301 + 0.996951i $$0.524863\pi$$
$$500$$ 0 0
$$501$$ 30.9083 1.38088
$$502$$ −9.00000 −0.401690
$$503$$ 9.39445 0.418878 0.209439 0.977822i $$-0.432836\pi$$
0.209439 + 0.977822i $$0.432836\pi$$
$$504$$ 4.81665 0.214551
$$505$$ 0 0
$$506$$ −9.51388 −0.422943
$$507$$ 27.6333 1.22724
$$508$$ −2.45837 −0.109072
$$509$$ −22.6972 −1.00604 −0.503018 0.864276i $$-0.667777\pi$$
−0.503018 + 0.864276i $$0.667777\pi$$
$$510$$ 0 0
$$511$$ −5.51388 −0.243920
$$512$$ 25.4222 1.12351
$$513$$ 1.60555 0.0708868
$$514$$ 23.4500 1.03433
$$515$$ 0 0
$$516$$ −5.02776 −0.221335
$$517$$ 3.00000 0.131940
$$518$$ −2.17494 −0.0955615
$$519$$ −11.0917 −0.486870
$$520$$ 0 0
$$521$$ −41.4500 −1.81596 −0.907978 0.419018i $$-0.862374\pi$$
−0.907978 + 0.419018i $$0.862374\pi$$
$$522$$ −2.72498 −0.119269
$$523$$ −32.4222 −1.41772 −0.708862 0.705347i $$-0.750791\pi$$
−0.708862 + 0.705347i $$0.750791\pi$$
$$524$$ 3.00000 0.131056
$$525$$ 0 0
$$526$$ 29.7250 1.29607
$$527$$ 70.5416 3.07284
$$528$$ 7.60555 0.330989
$$529$$ 30.3305 1.31872
$$530$$ 0 0
$$531$$ −32.7250 −1.42014
$$532$$ 0.211103 0.00915246
$$533$$ −28.0278 −1.21402
$$534$$ −5.09167 −0.220338
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 28.8167 1.24353
$$538$$ −11.3667 −0.490053
$$539$$ 6.51388 0.280573
$$540$$ 0 0
$$541$$ 25.7250 1.10600 0.553002 0.833180i $$-0.313482\pi$$
0.553002 + 0.833180i $$0.313482\pi$$
$$542$$ 0.275019 0.0118131
$$543$$ −45.8444 −1.96737
$$544$$ −11.7250 −0.502704
$$545$$ 0 0
$$546$$ −10.4584 −0.447577
$$547$$ −7.11943 −0.304405 −0.152202 0.988349i $$-0.548637\pi$$
−0.152202 + 0.988349i $$0.548637\pi$$
$$548$$ 3.90833 0.166955
$$549$$ −18.2111 −0.777231
$$550$$ 0 0
$$551$$ −0.908327 −0.0386960
$$552$$ −50.4500 −2.14729
$$553$$ −7.60555 −0.323421
$$554$$ −18.7527 −0.796727
$$555$$ 0 0
$$556$$ 1.88057 0.0797540
$$557$$ 19.4222 0.822945 0.411473 0.911422i $$-0.365015\pi$$
0.411473 + 0.911422i $$0.365015\pi$$
$$558$$ −30.6333 −1.29681
$$559$$ 36.0555 1.52499
$$560$$ 0 0
$$561$$ −15.9083 −0.671650
$$562$$ 1.54163 0.0650299
$$563$$ 8.09167 0.341023 0.170512 0.985356i $$-0.445458\pi$$
0.170512 + 0.985356i $$0.445458\pi$$
$$564$$ 2.09167 0.0880753
$$565$$ 0 0
$$566$$ −8.21110 −0.345138
$$567$$ −7.39445 −0.310538
$$568$$ −7.81665 −0.327980
$$569$$ −46.1472 −1.93459 −0.967295 0.253653i $$-0.918368\pi$$
−0.967295 + 0.253653i $$0.918368\pi$$
$$570$$ 0 0
$$571$$ 22.3305 0.934504 0.467252 0.884124i $$-0.345244\pi$$
0.467252 + 0.884124i $$0.345244\pi$$
$$572$$ 1.51388 0.0632984
$$573$$ 23.7250 0.991125
$$574$$ 5.09167 0.212522
$$575$$ 0 0
$$576$$ 20.3028 0.845949
$$577$$ 44.3583 1.84666 0.923330 0.384008i $$-0.125456\pi$$
0.923330 + 0.384008i $$0.125456\pi$$
$$578$$ −40.0278 −1.66494
$$579$$ 30.4222 1.26430
$$580$$ 0 0
$$581$$ −2.44996 −0.101642
$$582$$ −45.9083 −1.90296
$$583$$ 1.30278 0.0539555
$$584$$ −23.7250 −0.981747
$$585$$ 0 0
$$586$$ −1.02776 −0.0424562
$$587$$ 16.5416 0.682746 0.341373 0.939928i $$-0.389108\pi$$
0.341373 + 0.939928i $$0.389108\pi$$
$$588$$ 4.54163 0.187294
$$589$$ −10.2111 −0.420741
$$590$$ 0 0
$$591$$ 30.6333 1.26009
$$592$$ −7.90833 −0.325030
$$593$$ −6.39445 −0.262589 −0.131294 0.991343i $$-0.541913\pi$$
−0.131294 + 0.991343i $$0.541913\pi$$
$$594$$ −2.09167 −0.0858224
$$595$$ 0 0
$$596$$ −5.21110 −0.213455
$$597$$ −14.9361 −0.611293
$$598$$ 47.5694 1.94526
$$599$$ −24.9083 −1.01773 −0.508863 0.860847i $$-0.669934\pi$$
−0.508863 + 0.860847i $$0.669934\pi$$
$$600$$ 0 0
$$601$$ −1.90833 −0.0778423 −0.0389211 0.999242i $$-0.512392\pi$$
−0.0389211 + 0.999242i $$0.512392\pi$$
$$602$$ −6.55004 −0.266960
$$603$$ −9.21110 −0.375105
$$604$$ −0.247263 −0.0100610
$$605$$ 0 0
$$606$$ 1.54163 0.0626246
$$607$$ 7.21110 0.292690 0.146345 0.989234i $$-0.453249\pi$$
0.146345 + 0.989234i $$0.453249\pi$$
$$608$$ 1.69722 0.0688315
$$609$$ 1.45837 0.0590959
$$610$$ 0 0
$$611$$ −15.0000 −0.606835
$$612$$ −4.81665 −0.194702
$$613$$ −15.8806 −0.641410 −0.320705 0.947179i $$-0.603920\pi$$
−0.320705 + 0.947179i $$0.603920\pi$$
$$614$$ 22.0278 0.888968
$$615$$ 0 0
$$616$$ −2.09167 −0.0842759
$$617$$ 3.39445 0.136655 0.0683277 0.997663i $$-0.478234\pi$$
0.0683277 + 0.997663i $$0.478234\pi$$
$$618$$ −8.72498 −0.350970
$$619$$ −11.4222 −0.459097 −0.229549 0.973297i $$-0.573725\pi$$
−0.229549 + 0.973297i $$0.573725\pi$$
$$620$$ 0 0
$$621$$ 11.7250 0.470507
$$622$$ −6.27502 −0.251605
$$623$$ 1.18335 0.0474098
$$624$$ −38.0278 −1.52233
$$625$$ 0 0
$$626$$ −0.238859 −0.00954672
$$627$$ 2.30278 0.0919640
$$628$$ −5.81665 −0.232110
$$629$$ 16.5416 0.659558
$$630$$ 0 0
$$631$$ 6.93608 0.276121 0.138061 0.990424i $$-0.455913\pi$$
0.138061 + 0.990424i $$0.455913\pi$$
$$632$$ −32.7250 −1.30173
$$633$$ −58.1194 −2.31004
$$634$$ 1.18335 0.0469967
$$635$$ 0 0
$$636$$ 0.908327 0.0360175
$$637$$ −32.5694 −1.29045
$$638$$ 1.18335 0.0468491
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 27.7889 1.09760 0.548798 0.835955i $$-0.315086\pi$$
0.548798 + 0.835955i $$0.315086\pi$$
$$642$$ 9.00000 0.355202
$$643$$ −22.0000 −0.867595 −0.433798 0.901010i $$-0.642827\pi$$
−0.433798 + 0.901010i $$0.642827\pi$$
$$644$$ 1.54163 0.0607489
$$645$$ 0 0
$$646$$ 9.00000 0.354100
$$647$$ −33.2389 −1.30675 −0.653377 0.757033i $$-0.726648\pi$$
−0.653377 + 0.757033i $$0.726648\pi$$
$$648$$ −31.8167 −1.24988
$$649$$ 14.2111 0.557835
$$650$$ 0 0
$$651$$ 16.3944 0.642549
$$652$$ −2.81665 −0.110309
$$653$$ 6.11943 0.239472 0.119736 0.992806i $$-0.461795\pi$$
0.119736 + 0.992806i $$0.461795\pi$$
$$654$$ −34.5416 −1.35068
$$655$$ 0 0
$$656$$ 18.5139 0.722846
$$657$$ −18.2111 −0.710483
$$658$$ 2.72498 0.106231
$$659$$ −30.9083 −1.20402 −0.602009 0.798489i $$-0.705633\pi$$
−0.602009 + 0.798489i $$0.705633\pi$$
$$660$$ 0 0
$$661$$ −8.81665 −0.342928 −0.171464 0.985190i $$-0.554850\pi$$
−0.171464 + 0.985190i $$0.554850\pi$$
$$662$$ 28.1472 1.09397
$$663$$ 79.5416 3.08914
$$664$$ −10.5416 −0.409095
$$665$$ 0 0
$$666$$ −7.18335 −0.278349
$$667$$ −6.63331 −0.256843
$$668$$ −4.06392 −0.157238
$$669$$ −52.1194 −2.01505
$$670$$ 0 0
$$671$$ 7.90833 0.305298
$$672$$ −2.72498 −0.105118
$$673$$ 30.0278 1.15748 0.578742 0.815510i $$-0.303544\pi$$
0.578742 + 0.815510i $$0.303544\pi$$
$$674$$ 40.1833 1.54780
$$675$$ 0 0
$$676$$ −3.63331 −0.139743
$$677$$ 24.2389 0.931575 0.465788 0.884897i $$-0.345771\pi$$
0.465788 + 0.884897i $$0.345771\pi$$
$$678$$ 32.4500 1.24623
$$679$$ 10.6695 0.409457
$$680$$ 0 0
$$681$$ 3.90833 0.149767
$$682$$ 13.3028 0.509390
$$683$$ 47.8444 1.83072 0.915358 0.402642i $$-0.131908\pi$$
0.915358 + 0.402642i $$0.131908\pi$$
$$684$$ 0.697224 0.0266590
$$685$$ 0 0
$$686$$ 12.2750 0.468662
$$687$$ −43.1194 −1.64511
$$688$$ −23.8167 −0.908001
$$689$$ −6.51388 −0.248159
$$690$$ 0 0
$$691$$ 27.5416 1.04773 0.523867 0.851800i $$-0.324489\pi$$
0.523867 + 0.851800i $$0.324489\pi$$
$$692$$ 1.45837 0.0554387
$$693$$ −1.60555 −0.0609898
$$694$$ 16.3028 0.618845
$$695$$ 0 0
$$696$$ 6.27502 0.237854
$$697$$ −38.7250 −1.46681
$$698$$ 6.75274 0.255595
$$699$$ 36.6333 1.38560
$$700$$ 0 0
$$701$$ −41.2111 −1.55652 −0.778261 0.627941i $$-0.783898\pi$$
−0.778261 + 0.627941i $$0.783898\pi$$
$$702$$ 10.4584 0.394726
$$703$$ −2.39445 −0.0903083
$$704$$ −8.81665 −0.332290
$$705$$ 0 0
$$706$$ −24.2750 −0.913602
$$707$$ −0.358288 −0.0134748
$$708$$ 9.90833 0.372378
$$709$$ −31.6333 −1.18801 −0.594007 0.804460i $$-0.702455\pi$$
−0.594007 + 0.804460i $$0.702455\pi$$
$$710$$ 0 0
$$711$$ −25.1194 −0.942052
$$712$$ 5.09167 0.190819
$$713$$ −74.5694 −2.79265
$$714$$ −14.4500 −0.540776
$$715$$ 0 0
$$716$$ −3.78890 −0.141598
$$717$$ 48.6333 1.81624
$$718$$ −1.02776 −0.0383555
$$719$$ 7.18335 0.267894 0.133947 0.990989i $$-0.457235\pi$$
0.133947 + 0.990989i $$0.457235\pi$$
$$720$$ 0 0
$$721$$ 2.02776 0.0755176
$$722$$ 23.4500 0.872717
$$723$$ 50.5139 1.87863
$$724$$ 6.02776 0.224020
$$725$$ 0 0
$$726$$ −3.00000 −0.111340
$$727$$ −39.3305 −1.45869 −0.729344 0.684147i $$-0.760175\pi$$
−0.729344 + 0.684147i $$0.760175\pi$$
$$728$$ 10.4584 0.387613
$$729$$ −13.3305 −0.493723
$$730$$ 0 0
$$731$$ 49.8167 1.84254
$$732$$ 5.51388 0.203799
$$733$$ 19.6056 0.724148 0.362074 0.932149i $$-0.382069\pi$$
0.362074 + 0.932149i $$0.382069\pi$$
$$734$$ 26.9638 0.995253
$$735$$ 0 0
$$736$$ 12.3944 0.456865
$$737$$ 4.00000 0.147342
$$738$$ 16.8167 0.619030
$$739$$ 35.1194 1.29189 0.645945 0.763384i $$-0.276464\pi$$
0.645945 + 0.763384i $$0.276464\pi$$
$$740$$ 0 0
$$741$$ −11.5139 −0.422973
$$742$$ 1.18335 0.0434420
$$743$$ 40.6972 1.49304 0.746518 0.665365i $$-0.231724\pi$$
0.746518 + 0.665365i $$0.231724\pi$$
$$744$$ 70.5416 2.58618
$$745$$ 0 0
$$746$$ −35.7250 −1.30798
$$747$$ −8.09167 −0.296059
$$748$$ 2.09167 0.0764791
$$749$$ −2.09167 −0.0764281
$$750$$ 0 0
$$751$$ −45.3305 −1.65413 −0.827067 0.562103i $$-0.809992\pi$$
−0.827067 + 0.562103i $$0.809992\pi$$
$$752$$ 9.90833 0.361320
$$753$$ 15.9083 0.579732
$$754$$ −5.91673 −0.215475
$$755$$ 0 0
$$756$$ 0.338936 0.0123270
$$757$$ 49.0555 1.78295 0.891476 0.453067i $$-0.149670\pi$$
0.891476 + 0.453067i $$0.149670\pi$$
$$758$$ −4.14719 −0.150633
$$759$$ 16.8167 0.610406
$$760$$ 0 0
$$761$$ −13.5778 −0.492195 −0.246097 0.969245i $$-0.579148\pi$$
−0.246097 + 0.969245i $$0.579148\pi$$
$$762$$ −24.3583 −0.882408
$$763$$ 8.02776 0.290624
$$764$$ −3.11943 −0.112857
$$765$$ 0 0
$$766$$ −28.1833 −1.01831
$$767$$ −71.0555 −2.56567
$$768$$ −16.3305 −0.589277
$$769$$ −5.18335 −0.186916 −0.0934581 0.995623i $$-0.529792\pi$$
−0.0934581 + 0.995623i $$0.529792\pi$$
$$770$$ 0 0
$$771$$ −41.4500 −1.49278
$$772$$ −4.00000 −0.143963
$$773$$ 3.11943 0.112198 0.0560990 0.998425i $$-0.482134\pi$$
0.0560990 + 0.998425i $$0.482134\pi$$
$$774$$ −21.6333 −0.777593
$$775$$ 0 0
$$776$$ 45.9083 1.64801
$$777$$ 3.84441 0.137917
$$778$$ −15.6333 −0.560481
$$779$$ 5.60555 0.200840
$$780$$ 0 0
$$781$$ 2.60555 0.0932340
$$782$$ 65.7250 2.35032
$$783$$ −1.45837 −0.0521177
$$784$$ 21.5139 0.768353
$$785$$ 0 0
$$786$$ 29.7250 1.06025
$$787$$ 10.2111 0.363986 0.181993 0.983300i $$-0.441745\pi$$
0.181993 + 0.983300i $$0.441745\pi$$
$$788$$ −4.02776 −0.143483
$$789$$ −52.5416 −1.87053
$$790$$ 0 0
$$791$$ −7.54163 −0.268150
$$792$$ −6.90833 −0.245477
$$793$$ −39.5416 −1.40416
$$794$$ −28.2666 −1.00314
$$795$$ 0 0
$$796$$ 1.96384 0.0696065
$$797$$ 3.51388 0.124468 0.0622340 0.998062i $$-0.480178\pi$$
0.0622340 + 0.998062i $$0.480178\pi$$
$$798$$ 2.09167 0.0740444
$$799$$ −20.7250 −0.733197
$$800$$ 0 0
$$801$$ 3.90833 0.138094
$$802$$ 16.6611 0.588323
$$803$$ 7.90833 0.279079
$$804$$ 2.78890 0.0983568
$$805$$ 0 0
$$806$$ −66.5139 −2.34285
$$807$$ 20.0917 0.707260
$$808$$ −1.54163 −0.0542345
$$809$$ 39.6333 1.39343 0.696716 0.717347i $$-0.254644\pi$$
0.696716 + 0.717347i $$0.254644\pi$$
$$810$$ 0 0
$$811$$ 38.8722 1.36499 0.682493 0.730892i $$-0.260896\pi$$
0.682493 + 0.730892i $$0.260896\pi$$
$$812$$ −0.191750 −0.00672911
$$813$$ −0.486122 −0.0170490
$$814$$ 3.11943 0.109336
$$815$$ 0 0
$$816$$ −52.5416 −1.83933
$$817$$ −7.21110 −0.252285
$$818$$ 8.09167 0.282919
$$819$$ 8.02776 0.280513
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ 38.7250 1.35069
$$823$$ 18.4222 0.642158 0.321079 0.947052i $$-0.395955\pi$$
0.321079 + 0.947052i $$0.395955\pi$$
$$824$$ 8.72498 0.303949
$$825$$ 0 0
$$826$$ 12.9083 0.449138
$$827$$ 13.8167 0.480452 0.240226 0.970717i $$-0.422778\pi$$
0.240226 + 0.970717i $$0.422778\pi$$
$$828$$ 5.09167 0.176948
$$829$$ 29.7527 1.03336 0.516678 0.856180i $$-0.327169\pi$$
0.516678 + 0.856180i $$0.327169\pi$$
$$830$$ 0 0
$$831$$ 33.1472 1.14986
$$832$$ 44.0833 1.52831
$$833$$ −45.0000 −1.55916
$$834$$ 18.6333 0.645219
$$835$$ 0 0
$$836$$ −0.302776 −0.0104717
$$837$$ −16.3944 −0.566675
$$838$$ −8.33053 −0.287773
$$839$$ −9.11943 −0.314838 −0.157419 0.987532i $$-0.550317\pi$$
−0.157419 + 0.987532i $$0.550317\pi$$
$$840$$ 0 0
$$841$$ −28.1749 −0.971550
$$842$$ −0.908327 −0.0313030
$$843$$ −2.72498 −0.0938533
$$844$$ 7.64171 0.263039
$$845$$ 0 0
$$846$$ 9.00000 0.309426
$$847$$ 0.697224 0.0239569
$$848$$ 4.30278 0.147758
$$849$$ 14.5139 0.498115
$$850$$ 0 0
$$851$$ −17.4861 −0.599417
$$852$$ 1.81665 0.0622375
$$853$$ −12.7250 −0.435695 −0.217848 0.975983i $$-0.569904\pi$$
−0.217848 + 0.975983i $$0.569904\pi$$
$$854$$ 7.18335 0.245809
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ −3.00000 −0.102478 −0.0512390 0.998686i $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 15.0000 0.512092
$$859$$ 41.3944 1.41236 0.706180 0.708032i $$-0.250417\pi$$
0.706180 + 0.708032i $$0.250417\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ −42.9916 −1.46430
$$863$$ −12.3944 −0.421912 −0.210956 0.977496i $$-0.567658\pi$$
−0.210956 + 0.977496i $$0.567658\pi$$
$$864$$ 2.72498 0.0927057
$$865$$ 0 0
$$866$$ −6.51388 −0.221351
$$867$$ 70.7527 2.40289
$$868$$ −2.15559 −0.0731655
$$869$$ 10.9083 0.370040
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 34.5416 1.16973
$$873$$ 35.2389 1.19265
$$874$$ −9.51388 −0.321812
$$875$$ 0 0
$$876$$ 5.51388 0.186297
$$877$$ 20.0000 0.675352 0.337676 0.941262i $$-0.390359\pi$$
0.337676 + 0.941262i $$0.390359\pi$$
$$878$$ −31.6611 −1.06851
$$879$$ 1.81665 0.0612742
$$880$$ 0 0
$$881$$ 19.5416 0.658374 0.329187 0.944265i $$-0.393225\pi$$
0.329187 + 0.944265i $$0.393225\pi$$
$$882$$ 19.5416 0.658001
$$883$$ 52.4500 1.76508 0.882541 0.470236i $$-0.155831\pi$$
0.882541 + 0.470236i $$0.155831\pi$$
$$884$$ −10.4584 −0.351753
$$885$$ 0 0
$$886$$ 11.2111 0.376644
$$887$$ −3.23886 −0.108750 −0.0543751 0.998521i $$-0.517317\pi$$
−0.0543751 + 0.998521i $$0.517317\pi$$
$$888$$ 16.5416 0.555101
$$889$$ 5.66106 0.189866
$$890$$ 0 0
$$891$$ 10.6056 0.355299
$$892$$ 6.85281 0.229449
$$893$$ 3.00000 0.100391
$$894$$ −51.6333 −1.72688
$$895$$ 0 0
$$896$$ −5.64171 −0.188476
$$897$$ −84.0833 −2.80746
$$898$$ −30.5971 −1.02104
$$899$$ 9.27502 0.309339
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ −7.30278 −0.243156
$$903$$ 11.5778 0.385285
$$904$$ −32.4500 −1.07927
$$905$$ 0 0
$$906$$ −2.44996 −0.0813945
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ −0.513878 −0.0170536
$$909$$ −1.18335 −0.0392491
$$910$$ 0 0
$$911$$ −24.7889 −0.821293 −0.410646 0.911795i $$-0.634697\pi$$
−0.410646 + 0.911795i $$0.634697\pi$$
$$912$$ 7.60555 0.251845
$$913$$ 3.51388 0.116292
$$914$$ 26.9638 0.891885
$$915$$ 0 0
$$916$$ 5.66947 0.187324
$$917$$ −6.90833 −0.228133
$$918$$ 14.4500 0.476920
$$919$$ 26.7889 0.883684 0.441842 0.897093i $$-0.354325\pi$$
0.441842 + 0.897093i $$0.354325\pi$$
$$920$$ 0 0
$$921$$ −38.9361 −1.28299
$$922$$ −41.9638 −1.38201
$$923$$ −13.0278 −0.428814
$$924$$ 0.486122 0.0159922
$$925$$ 0 0
$$926$$ −15.3583 −0.504705
$$927$$ 6.69722 0.219966
$$928$$ −1.54163 −0.0506066
$$929$$ 53.6056 1.75874 0.879371 0.476138i $$-0.157964\pi$$
0.879371 + 0.476138i $$0.157964\pi$$
$$930$$ 0 0
$$931$$ 6.51388 0.213484
$$932$$ −4.81665 −0.157775
$$933$$ 11.0917 0.363125
$$934$$ 24.2750 0.794303
$$935$$ 0 0
$$936$$ 34.5416 1.12903
$$937$$ −9.21110 −0.300914 −0.150457 0.988617i $$-0.548074\pi$$
−0.150457 + 0.988617i $$0.548074\pi$$
$$938$$ 3.63331 0.118632
$$939$$ 0.422205 0.0137781
$$940$$ 0 0
$$941$$ 59.6056 1.94309 0.971543 0.236864i $$-0.0761197\pi$$
0.971543 + 0.236864i $$0.0761197\pi$$
$$942$$ −57.6333 −1.87779
$$943$$ 40.9361 1.33306
$$944$$ 46.9361 1.52764
$$945$$ 0 0
$$946$$ 9.39445 0.305440
$$947$$ 6.63331 0.215554 0.107777 0.994175i $$-0.465627\pi$$
0.107777 + 0.994175i $$0.465627\pi$$
$$948$$ 7.60555 0.247017
$$949$$ −39.5416 −1.28358
$$950$$ 0 0
$$951$$ −2.09167 −0.0678271
$$952$$ 14.4500 0.468326
$$953$$ 37.2666 1.20718 0.603592 0.797293i $$-0.293736\pi$$
0.603592 + 0.797293i $$0.293736\pi$$
$$954$$ 3.90833 0.126537
$$955$$ 0 0
$$956$$ −6.39445 −0.206811
$$957$$ −2.09167 −0.0676142
$$958$$ 45.3583 1.46546
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ 73.2666 2.36344
$$962$$ −15.5971 −0.502872
$$963$$ −6.90833 −0.222618
$$964$$ −6.64171 −0.213915
$$965$$ 0 0
$$966$$ 15.2750 0.491466
$$967$$ 14.9083 0.479419 0.239710 0.970845i $$-0.422948\pi$$
0.239710 + 0.970845i $$0.422948\pi$$
$$968$$ 3.00000 0.0964237
$$969$$ −15.9083 −0.511049
$$970$$ 0 0
$$971$$ 45.3583 1.45562 0.727808 0.685781i $$-0.240539\pi$$
0.727808 + 0.685781i $$0.240539\pi$$
$$972$$ 5.93608 0.190400
$$973$$ −4.33053 −0.138830
$$974$$ −5.48612 −0.175787
$$975$$ 0 0
$$976$$ 26.1194 0.836063
$$977$$ −52.0278 −1.66452 −0.832258 0.554389i $$-0.812952\pi$$
−0.832258 + 0.554389i $$0.812952\pi$$
$$978$$ −27.9083 −0.892410
$$979$$ −1.69722 −0.0542435
$$980$$ 0 0
$$981$$ 26.5139 0.846523
$$982$$ 12.7527 0.406956
$$983$$ 8.84441 0.282093 0.141046 0.990003i $$-0.454953\pi$$
0.141046 + 0.990003i $$0.454953\pi$$
$$984$$ −38.7250 −1.23451
$$985$$ 0 0
$$986$$ −8.17494 −0.260343
$$987$$ −4.81665 −0.153316
$$988$$ 1.51388 0.0481629
$$989$$ −52.6611 −1.67452
$$990$$ 0 0
$$991$$ −16.9083 −0.537111 −0.268555 0.963264i $$-0.586546\pi$$
−0.268555 + 0.963264i $$0.586546\pi$$
$$992$$ −17.3305 −0.550245
$$993$$ −49.7527 −1.57886
$$994$$ 2.36669 0.0750669
$$995$$ 0 0
$$996$$ 2.44996 0.0776300
$$997$$ 46.7250 1.47979 0.739897 0.672720i $$-0.234874\pi$$
0.739897 + 0.672720i $$0.234874\pi$$
$$998$$ 4.54163 0.143763
$$999$$ −3.84441 −0.121632
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 275.2.a.f.1.1 yes 2
3.2 odd 2 2475.2.a.o.1.2 2
4.3 odd 2 4400.2.a.bh.1.1 2
5.2 odd 4 275.2.b.c.199.2 4
5.3 odd 4 275.2.b.c.199.3 4
5.4 even 2 275.2.a.e.1.2 2
11.10 odd 2 3025.2.a.h.1.2 2
15.2 even 4 2475.2.c.k.199.3 4
15.8 even 4 2475.2.c.k.199.2 4
15.14 odd 2 2475.2.a.t.1.1 2
20.3 even 4 4400.2.b.y.4049.1 4
20.7 even 4 4400.2.b.y.4049.4 4
20.19 odd 2 4400.2.a.bs.1.2 2
55.54 odd 2 3025.2.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 5.4 even 2
275.2.a.f.1.1 yes 2 1.1 even 1 trivial
275.2.b.c.199.2 4 5.2 odd 4
275.2.b.c.199.3 4 5.3 odd 4
2475.2.a.o.1.2 2 3.2 odd 2
2475.2.a.t.1.1 2 15.14 odd 2
2475.2.c.k.199.2 4 15.8 even 4
2475.2.c.k.199.3 4 15.2 even 4
3025.2.a.h.1.2 2 11.10 odd 2
3025.2.a.n.1.1 2 55.54 odd 2
4400.2.a.bh.1.1 2 4.3 odd 2
4400.2.a.bs.1.2 2 20.19 odd 2
4400.2.b.y.4049.1 4 20.3 even 4
4400.2.b.y.4049.4 4 20.7 even 4