# Properties

 Label 4225.2.a.bg.1.3 Level $4225$ Weight $2$ Character 4225.1 Self dual yes Analytic conductor $33.737$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4225 = 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4225.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: $$33.7367948540$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 169) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.24698$$ of defining polynomial Character $$\chi$$ $$=$$ 4225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} +1.24698 q^{6} -2.04892 q^{7} +2.35690 q^{8} -2.69202 q^{9} +O(q^{10})$$ $$q+2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} +1.24698 q^{6} -2.04892 q^{7} +2.35690 q^{8} -2.69202 q^{9} -2.55496 q^{11} +1.69202 q^{12} -4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} -6.04892 q^{18} -5.85086 q^{19} -1.13706 q^{21} -5.74094 q^{22} +1.89008 q^{23} +1.30798 q^{24} -3.15883 q^{27} -6.24698 q^{28} +2.26875 q^{29} -4.26875 q^{31} -6.51573 q^{32} -1.41789 q^{33} +11.8998 q^{34} -8.20775 q^{36} -5.35690 q^{37} -13.1468 q^{38} +1.27413 q^{41} -2.55496 q^{42} -6.13706 q^{43} -7.78986 q^{44} +4.24698 q^{46} +2.95108 q^{47} -0.445042 q^{48} -2.80194 q^{49} +2.93900 q^{51} -5.52111 q^{53} -7.09783 q^{54} -4.82908 q^{56} -3.24698 q^{57} +5.09783 q^{58} -12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} +5.51573 q^{63} -13.0368 q^{64} -3.18598 q^{66} -0.576728 q^{67} +16.1468 q^{68} +1.04892 q^{69} -4.59419 q^{71} -6.34481 q^{72} +10.5526 q^{73} -12.0368 q^{74} -17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} +6.32304 q^{81} +2.86294 q^{82} -7.72348 q^{83} -3.46681 q^{84} -13.7899 q^{86} +1.25906 q^{87} -6.02177 q^{88} +6.61356 q^{89} +5.76271 q^{92} -2.36898 q^{93} +6.63102 q^{94} -3.61596 q^{96} -11.9269 q^{97} -6.29590 q^{98} +6.87800 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 2 q^{3} - q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 + 2 * q^3 - q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 $$3 q + 2 q^{2} + 2 q^{3} - q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} - 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{19} + 2 q^{21} - 3 q^{22} + 5 q^{23} + 9 q^{24} - q^{27} - 14 q^{28} - q^{29} - 5 q^{31} - 7 q^{32} - 10 q^{33} + 13 q^{34} - 7 q^{36} - 12 q^{37} - 12 q^{38} - 7 q^{41} - 8 q^{42} - 13 q^{43} + 8 q^{46} + 18 q^{47} - q^{48} - 4 q^{49} - q^{51} - q^{53} - 3 q^{54} - 4 q^{56} - 5 q^{57} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} + 4 q^{63} - 11 q^{64} + 5 q^{66} + q^{67} + 21 q^{68} - 6 q^{69} - 27 q^{71} + 4 q^{72} - 9 q^{73} - 8 q^{74} - 21 q^{76} - 8 q^{77} - 5 q^{79} - q^{81} + 14 q^{82} + 7 q^{83} - 7 q^{84} - 18 q^{86} + 18 q^{87} - 15 q^{88} - 11 q^{89} - 22 q^{93} + 5 q^{94} - 21 q^{96} - 7 q^{97} - 5 q^{98} + q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 + 2 * q^3 - q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 - 8 * q^11 - 5 * q^14 + 2 * q^16 + 2 * q^17 - 9 * q^18 - 4 * q^19 + 2 * q^21 - 3 * q^22 + 5 * q^23 + 9 * q^24 - q^27 - 14 * q^28 - q^29 - 5 * q^31 - 7 * q^32 - 10 * q^33 + 13 * q^34 - 7 * q^36 - 12 * q^37 - 12 * q^38 - 7 * q^41 - 8 * q^42 - 13 * q^43 + 8 * q^46 + 18 * q^47 - q^48 - 4 * q^49 - q^51 - q^53 - 3 * q^54 - 4 * q^56 - 5 * q^57 - 3 * q^58 - 19 * q^59 + 4 * q^61 - q^62 + 4 * q^63 - 11 * q^64 + 5 * q^66 + q^67 + 21 * q^68 - 6 * q^69 - 27 * q^71 + 4 * q^72 - 9 * q^73 - 8 * q^74 - 21 * q^76 - 8 * q^77 - 5 * q^79 - q^81 + 14 * q^82 + 7 * q^83 - 7 * q^84 - 18 * q^86 + 18 * q^87 - 15 * q^88 - 11 * q^89 - 22 * q^93 + 5 * q^94 - 21 * q^96 - 7 * q^97 - 5 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.24698 1.58885 0.794427 0.607359i $$-0.207771\pi$$
0.794427 + 0.607359i $$0.207771\pi$$
$$3$$ 0.554958 0.320405 0.160203 0.987084i $$-0.448785\pi$$
0.160203 + 0.987084i $$0.448785\pi$$
$$4$$ 3.04892 1.52446
$$5$$ 0 0
$$6$$ 1.24698 0.509077
$$7$$ −2.04892 −0.774418 −0.387209 0.921992i $$-0.626561\pi$$
−0.387209 + 0.921992i $$0.626561\pi$$
$$8$$ 2.35690 0.833289
$$9$$ −2.69202 −0.897340
$$10$$ 0 0
$$11$$ −2.55496 −0.770349 −0.385174 0.922844i $$-0.625859\pi$$
−0.385174 + 0.922844i $$0.625859\pi$$
$$12$$ 1.69202 0.488445
$$13$$ 0 0
$$14$$ −4.60388 −1.23044
$$15$$ 0 0
$$16$$ −0.801938 −0.200484
$$17$$ 5.29590 1.28444 0.642222 0.766519i $$-0.278013\pi$$
0.642222 + 0.766519i $$0.278013\pi$$
$$18$$ −6.04892 −1.42574
$$19$$ −5.85086 −1.34228 −0.671139 0.741331i $$-0.734195\pi$$
−0.671139 + 0.741331i $$0.734195\pi$$
$$20$$ 0 0
$$21$$ −1.13706 −0.248128
$$22$$ −5.74094 −1.22397
$$23$$ 1.89008 0.394110 0.197055 0.980392i $$-0.436862\pi$$
0.197055 + 0.980392i $$0.436862\pi$$
$$24$$ 1.30798 0.266990
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −3.15883 −0.607918
$$28$$ −6.24698 −1.18057
$$29$$ 2.26875 0.421296 0.210648 0.977562i $$-0.432443\pi$$
0.210648 + 0.977562i $$0.432443\pi$$
$$30$$ 0 0
$$31$$ −4.26875 −0.766690 −0.383345 0.923605i $$-0.625228\pi$$
−0.383345 + 0.923605i $$0.625228\pi$$
$$32$$ −6.51573 −1.15183
$$33$$ −1.41789 −0.246824
$$34$$ 11.8998 2.04079
$$35$$ 0 0
$$36$$ −8.20775 −1.36796
$$37$$ −5.35690 −0.880668 −0.440334 0.897834i $$-0.645140\pi$$
−0.440334 + 0.897834i $$0.645140\pi$$
$$38$$ −13.1468 −2.13268
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.27413 0.198985 0.0994926 0.995038i $$-0.468278\pi$$
0.0994926 + 0.995038i $$0.468278\pi$$
$$42$$ −2.55496 −0.394239
$$43$$ −6.13706 −0.935893 −0.467947 0.883757i $$-0.655006\pi$$
−0.467947 + 0.883757i $$0.655006\pi$$
$$44$$ −7.78986 −1.17437
$$45$$ 0 0
$$46$$ 4.24698 0.626183
$$47$$ 2.95108 0.430460 0.215230 0.976563i $$-0.430950\pi$$
0.215230 + 0.976563i $$0.430950\pi$$
$$48$$ −0.445042 −0.0642363
$$49$$ −2.80194 −0.400277
$$50$$ 0 0
$$51$$ 2.93900 0.411542
$$52$$ 0 0
$$53$$ −5.52111 −0.758382 −0.379191 0.925318i $$-0.623798\pi$$
−0.379191 + 0.925318i $$0.623798\pi$$
$$54$$ −7.09783 −0.965893
$$55$$ 0 0
$$56$$ −4.82908 −0.645314
$$57$$ −3.24698 −0.430073
$$58$$ 5.09783 0.669378
$$59$$ −12.2078 −1.58931 −0.794657 0.607059i $$-0.792349\pi$$
−0.794657 + 0.607059i $$0.792349\pi$$
$$60$$ 0 0
$$61$$ 8.56465 1.09659 0.548295 0.836285i $$-0.315277\pi$$
0.548295 + 0.836285i $$0.315277\pi$$
$$62$$ −9.59179 −1.21816
$$63$$ 5.51573 0.694917
$$64$$ −13.0368 −1.62960
$$65$$ 0 0
$$66$$ −3.18598 −0.392167
$$67$$ −0.576728 −0.0704586 −0.0352293 0.999379i $$-0.511216\pi$$
−0.0352293 + 0.999379i $$0.511216\pi$$
$$68$$ 16.1468 1.95808
$$69$$ 1.04892 0.126275
$$70$$ 0 0
$$71$$ −4.59419 −0.545230 −0.272615 0.962123i $$-0.587888\pi$$
−0.272615 + 0.962123i $$0.587888\pi$$
$$72$$ −6.34481 −0.747744
$$73$$ 10.5526 1.23508 0.617542 0.786538i $$-0.288128\pi$$
0.617542 + 0.786538i $$0.288128\pi$$
$$74$$ −12.0368 −1.39925
$$75$$ 0 0
$$76$$ −17.8388 −2.04625
$$77$$ 5.23490 0.596572
$$78$$ 0 0
$$79$$ −15.7778 −1.77514 −0.887569 0.460674i $$-0.847608\pi$$
−0.887569 + 0.460674i $$0.847608\pi$$
$$80$$ 0 0
$$81$$ 6.32304 0.702560
$$82$$ 2.86294 0.316158
$$83$$ −7.72348 −0.847762 −0.423881 0.905718i $$-0.639333\pi$$
−0.423881 + 0.905718i $$0.639333\pi$$
$$84$$ −3.46681 −0.378260
$$85$$ 0 0
$$86$$ −13.7899 −1.48700
$$87$$ 1.25906 0.134986
$$88$$ −6.02177 −0.641923
$$89$$ 6.61356 0.701036 0.350518 0.936556i $$-0.386005\pi$$
0.350518 + 0.936556i $$0.386005\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 5.76271 0.600804
$$93$$ −2.36898 −0.245652
$$94$$ 6.63102 0.683938
$$95$$ 0 0
$$96$$ −3.61596 −0.369052
$$97$$ −11.9269 −1.21100 −0.605498 0.795847i $$-0.707026\pi$$
−0.605498 + 0.795847i $$0.707026\pi$$
$$98$$ −6.29590 −0.635982
$$99$$ 6.87800 0.691265
$$100$$ 0 0
$$101$$ 13.0640 1.29991 0.649957 0.759971i $$-0.274787\pi$$
0.649957 + 0.759971i $$0.274787\pi$$
$$102$$ 6.60388 0.653881
$$103$$ −9.16852 −0.903401 −0.451701 0.892170i $$-0.649182\pi$$
−0.451701 + 0.892170i $$0.649182\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −12.4058 −1.20496
$$107$$ 6.89977 0.667026 0.333513 0.942745i $$-0.391766\pi$$
0.333513 + 0.942745i $$0.391766\pi$$
$$108$$ −9.63102 −0.926746
$$109$$ 0.121998 0.0116853 0.00584264 0.999983i $$-0.498140\pi$$
0.00584264 + 0.999983i $$0.498140\pi$$
$$110$$ 0 0
$$111$$ −2.97285 −0.282171
$$112$$ 1.64310 0.155259
$$113$$ −7.30798 −0.687477 −0.343738 0.939065i $$-0.611693\pi$$
−0.343738 + 0.939065i $$0.611693\pi$$
$$114$$ −7.29590 −0.683323
$$115$$ 0 0
$$116$$ 6.91723 0.642249
$$117$$ 0 0
$$118$$ −27.4306 −2.52519
$$119$$ −10.8509 −0.994696
$$120$$ 0 0
$$121$$ −4.47219 −0.406563
$$122$$ 19.2446 1.74232
$$123$$ 0.707087 0.0637559
$$124$$ −13.0151 −1.16879
$$125$$ 0 0
$$126$$ 12.3937 1.10412
$$127$$ 18.9705 1.68336 0.841678 0.539980i $$-0.181568\pi$$
0.841678 + 0.539980i $$0.181568\pi$$
$$128$$ −16.2620 −1.43738
$$129$$ −3.40581 −0.299865
$$130$$ 0 0
$$131$$ 3.25667 0.284536 0.142268 0.989828i $$-0.454560\pi$$
0.142268 + 0.989828i $$0.454560\pi$$
$$132$$ −4.32304 −0.376273
$$133$$ 11.9879 1.03948
$$134$$ −1.29590 −0.111948
$$135$$ 0 0
$$136$$ 12.4819 1.07031
$$137$$ −0.792249 −0.0676864 −0.0338432 0.999427i $$-0.510775\pi$$
−0.0338432 + 0.999427i $$0.510775\pi$$
$$138$$ 2.35690 0.200632
$$139$$ −11.3394 −0.961799 −0.480899 0.876776i $$-0.659690\pi$$
−0.480899 + 0.876776i $$0.659690\pi$$
$$140$$ 0 0
$$141$$ 1.63773 0.137922
$$142$$ −10.3230 −0.866291
$$143$$ 0 0
$$144$$ 2.15883 0.179903
$$145$$ 0 0
$$146$$ 23.7114 1.96237
$$147$$ −1.55496 −0.128251
$$148$$ −16.3327 −1.34254
$$149$$ 8.40581 0.688631 0.344316 0.938854i $$-0.388111\pi$$
0.344316 + 0.938854i $$0.388111\pi$$
$$150$$ 0 0
$$151$$ 14.1293 1.14983 0.574913 0.818215i $$-0.305036\pi$$
0.574913 + 0.818215i $$0.305036\pi$$
$$152$$ −13.7899 −1.11851
$$153$$ −14.2567 −1.15258
$$154$$ 11.7627 0.947866
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 9.43296 0.752832 0.376416 0.926451i $$-0.377156\pi$$
0.376416 + 0.926451i $$0.377156\pi$$
$$158$$ −35.4523 −2.82044
$$159$$ −3.06398 −0.242990
$$160$$ 0 0
$$161$$ −3.87263 −0.305206
$$162$$ 14.2078 1.11627
$$163$$ −8.70410 −0.681758 −0.340879 0.940107i $$-0.610725\pi$$
−0.340879 + 0.940107i $$0.610725\pi$$
$$164$$ 3.88471 0.303345
$$165$$ 0 0
$$166$$ −17.3545 −1.34697
$$167$$ 23.8538 1.84587 0.922933 0.384961i $$-0.125785\pi$$
0.922933 + 0.384961i $$0.125785\pi$$
$$168$$ −2.67994 −0.206762
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 15.7506 1.20448
$$172$$ −18.7114 −1.42673
$$173$$ 18.8552 1.43353 0.716766 0.697314i $$-0.245622\pi$$
0.716766 + 0.697314i $$0.245622\pi$$
$$174$$ 2.82908 0.214472
$$175$$ 0 0
$$176$$ 2.04892 0.154443
$$177$$ −6.77479 −0.509224
$$178$$ 14.8605 1.11384
$$179$$ 6.02177 0.450088 0.225044 0.974349i $$-0.427747\pi$$
0.225044 + 0.974349i $$0.427747\pi$$
$$180$$ 0 0
$$181$$ −4.77777 −0.355129 −0.177565 0.984109i $$-0.556822\pi$$
−0.177565 + 0.984109i $$0.556822\pi$$
$$182$$ 0 0
$$183$$ 4.75302 0.351353
$$184$$ 4.45473 0.328407
$$185$$ 0 0
$$186$$ −5.32304 −0.390305
$$187$$ −13.5308 −0.989470
$$188$$ 8.99761 0.656218
$$189$$ 6.47219 0.470782
$$190$$ 0 0
$$191$$ 18.4306 1.33359 0.666795 0.745242i $$-0.267666\pi$$
0.666795 + 0.745242i $$0.267666\pi$$
$$192$$ −7.23490 −0.522134
$$193$$ −6.05429 −0.435798 −0.217899 0.975971i $$-0.569920\pi$$
−0.217899 + 0.975971i $$0.569920\pi$$
$$194$$ −26.7995 −1.92410
$$195$$ 0 0
$$196$$ −8.54288 −0.610205
$$197$$ 11.4155 0.813321 0.406660 0.913579i $$-0.366693\pi$$
0.406660 + 0.913579i $$0.366693\pi$$
$$198$$ 15.4547 1.09832
$$199$$ −13.9051 −0.985710 −0.492855 0.870111i $$-0.664047\pi$$
−0.492855 + 0.870111i $$0.664047\pi$$
$$200$$ 0 0
$$201$$ −0.320060 −0.0225753
$$202$$ 29.3545 2.06538
$$203$$ −4.64848 −0.326259
$$204$$ 8.96077 0.627379
$$205$$ 0 0
$$206$$ −20.6015 −1.43537
$$207$$ −5.08815 −0.353651
$$208$$ 0 0
$$209$$ 14.9487 1.03402
$$210$$ 0 0
$$211$$ −13.2446 −0.911795 −0.455897 0.890032i $$-0.650682\pi$$
−0.455897 + 0.890032i $$0.650682\pi$$
$$212$$ −16.8334 −1.15612
$$213$$ −2.54958 −0.174694
$$214$$ 15.5036 1.05981
$$215$$ 0 0
$$216$$ −7.44504 −0.506571
$$217$$ 8.74632 0.593739
$$218$$ 0.274127 0.0185662
$$219$$ 5.85623 0.395727
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −6.67994 −0.448328
$$223$$ 7.33513 0.491196 0.245598 0.969372i $$-0.421016\pi$$
0.245598 + 0.969372i $$0.421016\pi$$
$$224$$ 13.3502 0.891997
$$225$$ 0 0
$$226$$ −16.4209 −1.09230
$$227$$ 8.67456 0.575751 0.287875 0.957668i $$-0.407051\pi$$
0.287875 + 0.957668i $$0.407051\pi$$
$$228$$ −9.89977 −0.655628
$$229$$ −13.6866 −0.904439 −0.452219 0.891907i $$-0.649368\pi$$
−0.452219 + 0.891907i $$0.649368\pi$$
$$230$$ 0 0
$$231$$ 2.90515 0.191145
$$232$$ 5.34721 0.351061
$$233$$ 5.08815 0.333336 0.166668 0.986013i $$-0.446699\pi$$
0.166668 + 0.986013i $$0.446699\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −37.2204 −2.42284
$$237$$ −8.75600 −0.568764
$$238$$ −24.3817 −1.58043
$$239$$ −10.9239 −0.706611 −0.353305 0.935508i $$-0.614942\pi$$
−0.353305 + 0.935508i $$0.614942\pi$$
$$240$$ 0 0
$$241$$ 11.9148 0.767502 0.383751 0.923437i $$-0.374632\pi$$
0.383751 + 0.923437i $$0.374632\pi$$
$$242$$ −10.0489 −0.645969
$$243$$ 12.9855 0.833022
$$244$$ 26.1129 1.67171
$$245$$ 0 0
$$246$$ 1.58881 0.101299
$$247$$ 0 0
$$248$$ −10.0610 −0.638874
$$249$$ −4.28621 −0.271627
$$250$$ 0 0
$$251$$ 22.3478 1.41058 0.705290 0.708919i $$-0.250817\pi$$
0.705290 + 0.708919i $$0.250817\pi$$
$$252$$ 16.8170 1.05937
$$253$$ −4.82908 −0.303602
$$254$$ 42.6262 2.67461
$$255$$ 0 0
$$256$$ −10.4668 −0.654176
$$257$$ 18.6601 1.16398 0.581992 0.813194i $$-0.302273\pi$$
0.581992 + 0.813194i $$0.302273\pi$$
$$258$$ −7.65279 −0.476442
$$259$$ 10.9758 0.682005
$$260$$ 0 0
$$261$$ −6.10752 −0.378046
$$262$$ 7.31767 0.452087
$$263$$ −14.3991 −0.887887 −0.443944 0.896055i $$-0.646421\pi$$
−0.443944 + 0.896055i $$0.646421\pi$$
$$264$$ −3.34183 −0.205675
$$265$$ 0 0
$$266$$ 26.9366 1.65159
$$267$$ 3.67025 0.224616
$$268$$ −1.75840 −0.107411
$$269$$ 0.652793 0.0398015 0.0199007 0.999802i $$-0.493665\pi$$
0.0199007 + 0.999802i $$0.493665\pi$$
$$270$$ 0 0
$$271$$ −1.99569 −0.121229 −0.0606147 0.998161i $$-0.519306\pi$$
−0.0606147 + 0.998161i $$0.519306\pi$$
$$272$$ −4.24698 −0.257511
$$273$$ 0 0
$$274$$ −1.78017 −0.107544
$$275$$ 0 0
$$276$$ 3.19806 0.192501
$$277$$ −11.7845 −0.708061 −0.354030 0.935234i $$-0.615189\pi$$
−0.354030 + 0.935234i $$0.615189\pi$$
$$278$$ −25.4795 −1.52816
$$279$$ 11.4916 0.687982
$$280$$ 0 0
$$281$$ −6.47219 −0.386098 −0.193049 0.981189i $$-0.561838\pi$$
−0.193049 + 0.981189i $$0.561838\pi$$
$$282$$ 3.67994 0.219137
$$283$$ −6.58104 −0.391202 −0.195601 0.980684i $$-0.562666\pi$$
−0.195601 + 0.980684i $$0.562666\pi$$
$$284$$ −14.0073 −0.831180
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.61058 −0.154098
$$288$$ 17.5405 1.03358
$$289$$ 11.0465 0.649796
$$290$$ 0 0
$$291$$ −6.61894 −0.388009
$$292$$ 32.1739 1.88284
$$293$$ 24.3381 1.42185 0.710924 0.703269i $$-0.248277\pi$$
0.710924 + 0.703269i $$0.248277\pi$$
$$294$$ −3.49396 −0.203772
$$295$$ 0 0
$$296$$ −12.6256 −0.733851
$$297$$ 8.07069 0.468309
$$298$$ 18.8877 1.09413
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 12.5743 0.724773
$$302$$ 31.7482 1.82691
$$303$$ 7.24996 0.416500
$$304$$ 4.69202 0.269106
$$305$$ 0 0
$$306$$ −32.0344 −1.83129
$$307$$ 14.0737 0.803227 0.401613 0.915809i $$-0.368450\pi$$
0.401613 + 0.915809i $$0.368450\pi$$
$$308$$ 15.9608 0.909449
$$309$$ −5.08815 −0.289455
$$310$$ 0 0
$$311$$ −29.7700 −1.68810 −0.844051 0.536263i $$-0.819836\pi$$
−0.844051 + 0.536263i $$0.819836\pi$$
$$312$$ 0 0
$$313$$ 7.47889 0.422732 0.211366 0.977407i $$-0.432209\pi$$
0.211366 + 0.977407i $$0.432209\pi$$
$$314$$ 21.1957 1.19614
$$315$$ 0 0
$$316$$ −48.1051 −2.70613
$$317$$ −30.0301 −1.68666 −0.843330 0.537396i $$-0.819408\pi$$
−0.843330 + 0.537396i $$0.819408\pi$$
$$318$$ −6.88471 −0.386075
$$319$$ −5.79656 −0.324545
$$320$$ 0 0
$$321$$ 3.82908 0.213719
$$322$$ −8.70171 −0.484927
$$323$$ −30.9855 −1.72408
$$324$$ 19.2784 1.07102
$$325$$ 0 0
$$326$$ −19.5579 −1.08321
$$327$$ 0.0677037 0.00374402
$$328$$ 3.00298 0.165812
$$329$$ −6.04652 −0.333356
$$330$$ 0 0
$$331$$ −15.7168 −0.863872 −0.431936 0.901904i $$-0.642169\pi$$
−0.431936 + 0.901904i $$0.642169\pi$$
$$332$$ −23.5483 −1.29238
$$333$$ 14.4209 0.790259
$$334$$ 53.5991 2.93281
$$335$$ 0 0
$$336$$ 0.911854 0.0497457
$$337$$ −1.95407 −0.106445 −0.0532224 0.998583i $$-0.516949\pi$$
−0.0532224 + 0.998583i $$0.516949\pi$$
$$338$$ 0 0
$$339$$ −4.05562 −0.220271
$$340$$ 0 0
$$341$$ 10.9065 0.590619
$$342$$ 35.3913 1.91374
$$343$$ 20.0834 1.08440
$$344$$ −14.4644 −0.779869
$$345$$ 0 0
$$346$$ 42.3672 2.27767
$$347$$ 17.1250 0.919317 0.459659 0.888096i $$-0.347972\pi$$
0.459659 + 0.888096i $$0.347972\pi$$
$$348$$ 3.83877 0.205780
$$349$$ −10.4668 −0.560276 −0.280138 0.959960i $$-0.590380\pi$$
−0.280138 + 0.959960i $$0.590380\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 16.6474 0.887310
$$353$$ −15.5308 −0.826621 −0.413310 0.910590i $$-0.635628\pi$$
−0.413310 + 0.910590i $$0.635628\pi$$
$$354$$ −15.2228 −0.809084
$$355$$ 0 0
$$356$$ 20.1642 1.06870
$$357$$ −6.02177 −0.318706
$$358$$ 13.5308 0.715125
$$359$$ −21.4263 −1.13083 −0.565417 0.824805i $$-0.691285\pi$$
−0.565417 + 0.824805i $$0.691285\pi$$
$$360$$ 0 0
$$361$$ 15.2325 0.801711
$$362$$ −10.7356 −0.564249
$$363$$ −2.48188 −0.130265
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 10.6799 0.558249
$$367$$ −34.3032 −1.79061 −0.895306 0.445452i $$-0.853043\pi$$
−0.895306 + 0.445452i $$0.853043\pi$$
$$368$$ −1.51573 −0.0790129
$$369$$ −3.42998 −0.178557
$$370$$ 0 0
$$371$$ 11.3123 0.587305
$$372$$ −7.22282 −0.374486
$$373$$ 12.5961 0.652202 0.326101 0.945335i $$-0.394265\pi$$
0.326101 + 0.945335i $$0.394265\pi$$
$$374$$ −30.4034 −1.57212
$$375$$ 0 0
$$376$$ 6.95539 0.358697
$$377$$ 0 0
$$378$$ 14.5429 0.748005
$$379$$ 16.5386 0.849529 0.424765 0.905304i $$-0.360357\pi$$
0.424765 + 0.905304i $$0.360357\pi$$
$$380$$ 0 0
$$381$$ 10.5278 0.539356
$$382$$ 41.4131 2.11888
$$383$$ −7.53617 −0.385080 −0.192540 0.981289i $$-0.561673\pi$$
−0.192540 + 0.981289i $$0.561673\pi$$
$$384$$ −9.02475 −0.460543
$$385$$ 0 0
$$386$$ −13.6039 −0.692419
$$387$$ 16.5211 0.839815
$$388$$ −36.3642 −1.84611
$$389$$ 35.5555 1.80274 0.901369 0.433052i $$-0.142563\pi$$
0.901369 + 0.433052i $$0.142563\pi$$
$$390$$ 0 0
$$391$$ 10.0097 0.506212
$$392$$ −6.60388 −0.333546
$$393$$ 1.80731 0.0911670
$$394$$ 25.6504 1.29225
$$395$$ 0 0
$$396$$ 20.9705 1.05381
$$397$$ −1.35152 −0.0678308 −0.0339154 0.999425i $$-0.510798\pi$$
−0.0339154 + 0.999425i $$0.510798\pi$$
$$398$$ −31.2446 −1.56615
$$399$$ 6.65279 0.333056
$$400$$ 0 0
$$401$$ 0.579121 0.0289199 0.0144600 0.999895i $$-0.495397\pi$$
0.0144600 + 0.999895i $$0.495397\pi$$
$$402$$ −0.719169 −0.0358689
$$403$$ 0 0
$$404$$ 39.8310 1.98167
$$405$$ 0 0
$$406$$ −10.4450 −0.518379
$$407$$ 13.6866 0.678422
$$408$$ 6.92692 0.342934
$$409$$ −15.1575 −0.749490 −0.374745 0.927128i $$-0.622270\pi$$
−0.374745 + 0.927128i $$0.622270\pi$$
$$410$$ 0 0
$$411$$ −0.439665 −0.0216871
$$412$$ −27.9541 −1.37720
$$413$$ 25.0127 1.23079
$$414$$ −11.4330 −0.561899
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −6.29291 −0.308165
$$418$$ 33.5894 1.64291
$$419$$ −35.7235 −1.74521 −0.872603 0.488430i $$-0.837570\pi$$
−0.872603 + 0.488430i $$0.837570\pi$$
$$420$$ 0 0
$$421$$ −35.0465 −1.70806 −0.854032 0.520221i $$-0.825849\pi$$
−0.854032 + 0.520221i $$0.825849\pi$$
$$422$$ −29.7603 −1.44871
$$423$$ −7.94438 −0.386269
$$424$$ −13.0127 −0.631951
$$425$$ 0 0
$$426$$ −5.72886 −0.277564
$$427$$ −17.5483 −0.849220
$$428$$ 21.0368 1.01685
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.2814 −1.65128 −0.825639 0.564199i $$-0.809185\pi$$
−0.825639 + 0.564199i $$0.809185\pi$$
$$432$$ 2.53319 0.121878
$$433$$ −13.7385 −0.660232 −0.330116 0.943940i $$-0.607088\pi$$
−0.330116 + 0.943940i $$0.607088\pi$$
$$434$$ 19.6528 0.943364
$$435$$ 0 0
$$436$$ 0.371961 0.0178137
$$437$$ −11.0586 −0.529005
$$438$$ 13.1588 0.628753
$$439$$ 10.2403 0.488742 0.244371 0.969682i $$-0.421419\pi$$
0.244371 + 0.969682i $$0.421419\pi$$
$$440$$ 0 0
$$441$$ 7.54288 0.359185
$$442$$ 0 0
$$443$$ −12.1763 −0.578513 −0.289257 0.957252i $$-0.593408\pi$$
−0.289257 + 0.957252i $$0.593408\pi$$
$$444$$ −9.06398 −0.430158
$$445$$ 0 0
$$446$$ 16.4819 0.780440
$$447$$ 4.66487 0.220641
$$448$$ 26.7114 1.26199
$$449$$ −12.9051 −0.609032 −0.304516 0.952507i $$-0.598495\pi$$
−0.304516 + 0.952507i $$0.598495\pi$$
$$450$$ 0 0
$$451$$ −3.25534 −0.153288
$$452$$ −22.2814 −1.04803
$$453$$ 7.84117 0.368410
$$454$$ 19.4916 0.914785
$$455$$ 0 0
$$456$$ −7.65279 −0.358375
$$457$$ −4.65710 −0.217850 −0.108925 0.994050i $$-0.534741\pi$$
−0.108925 + 0.994050i $$0.534741\pi$$
$$458$$ −30.7536 −1.43702
$$459$$ −16.7289 −0.780836
$$460$$ 0 0
$$461$$ −31.5405 −1.46899 −0.734493 0.678616i $$-0.762580\pi$$
−0.734493 + 0.678616i $$0.762580\pi$$
$$462$$ 6.52781 0.303701
$$463$$ −17.6504 −0.820284 −0.410142 0.912022i $$-0.634521\pi$$
−0.410142 + 0.912022i $$0.634521\pi$$
$$464$$ −1.81940 −0.0844633
$$465$$ 0 0
$$466$$ 11.4330 0.529622
$$467$$ 32.1726 1.48877 0.744385 0.667751i $$-0.232743\pi$$
0.744385 + 0.667751i $$0.232743\pi$$
$$468$$ 0 0
$$469$$ 1.18167 0.0545644
$$470$$ 0 0
$$471$$ 5.23490 0.241211
$$472$$ −28.7724 −1.32436
$$473$$ 15.6799 0.720964
$$474$$ −19.6746 −0.903683
$$475$$ 0 0
$$476$$ −33.0834 −1.51637
$$477$$ 14.8629 0.680527
$$478$$ −24.5459 −1.12270
$$479$$ −34.8998 −1.59461 −0.797306 0.603576i $$-0.793742\pi$$
−0.797306 + 0.603576i $$0.793742\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 26.7724 1.21945
$$483$$ −2.14914 −0.0977895
$$484$$ −13.6353 −0.619788
$$485$$ 0 0
$$486$$ 29.1782 1.32355
$$487$$ −41.8351 −1.89573 −0.947864 0.318676i $$-0.896762\pi$$
−0.947864 + 0.318676i $$0.896762\pi$$
$$488$$ 20.1860 0.913776
$$489$$ −4.83041 −0.218439
$$490$$ 0 0
$$491$$ 21.8455 0.985873 0.492936 0.870065i $$-0.335924\pi$$
0.492936 + 0.870065i $$0.335924\pi$$
$$492$$ 2.15585 0.0971932
$$493$$ 12.0151 0.541131
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 3.42327 0.153709
$$497$$ 9.41311 0.422236
$$498$$ −9.63102 −0.431576
$$499$$ 23.5472 1.05412 0.527058 0.849829i $$-0.323295\pi$$
0.527058 + 0.849829i $$0.323295\pi$$
$$500$$ 0 0
$$501$$ 13.2379 0.591425
$$502$$ 50.2150 2.24121
$$503$$ 7.08682 0.315986 0.157993 0.987440i $$-0.449498\pi$$
0.157993 + 0.987440i $$0.449498\pi$$
$$504$$ 13.0000 0.579066
$$505$$ 0 0
$$506$$ −10.8509 −0.482379
$$507$$ 0 0
$$508$$ 57.8394 2.56621
$$509$$ 7.61894 0.337704 0.168852 0.985641i $$-0.445994\pi$$
0.168852 + 0.985641i $$0.445994\pi$$
$$510$$ 0 0
$$511$$ −21.6213 −0.956471
$$512$$ 9.00538 0.397985
$$513$$ 18.4819 0.815995
$$514$$ 41.9288 1.84940
$$515$$ 0 0
$$516$$ −10.3840 −0.457132
$$517$$ −7.53989 −0.331604
$$518$$ 24.6625 1.08361
$$519$$ 10.4638 0.459311
$$520$$ 0 0
$$521$$ −39.5133 −1.73111 −0.865555 0.500813i $$-0.833034\pi$$
−0.865555 + 0.500813i $$0.833034\pi$$
$$522$$ −13.7235 −0.600660
$$523$$ 15.8194 0.691734 0.345867 0.938284i $$-0.387585\pi$$
0.345867 + 0.938284i $$0.387585\pi$$
$$524$$ 9.92931 0.433764
$$525$$ 0 0
$$526$$ −32.3545 −1.41072
$$527$$ −22.6069 −0.984770
$$528$$ 1.13706 0.0494843
$$529$$ −19.4276 −0.844678
$$530$$ 0 0
$$531$$ 32.8635 1.42616
$$532$$ 36.5502 1.58465
$$533$$ 0 0
$$534$$ 8.24698 0.356882
$$535$$ 0 0
$$536$$ −1.35929 −0.0587123
$$537$$ 3.34183 0.144211
$$538$$ 1.46681 0.0632388
$$539$$ 7.15883 0.308353
$$540$$ 0 0
$$541$$ 34.4819 1.48249 0.741246 0.671234i $$-0.234235\pi$$
0.741246 + 0.671234i $$0.234235\pi$$
$$542$$ −4.48427 −0.192616
$$543$$ −2.65146 −0.113785
$$544$$ −34.5066 −1.47946
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −36.8582 −1.57594 −0.787970 0.615713i $$-0.788868\pi$$
−0.787970 + 0.615713i $$0.788868\pi$$
$$548$$ −2.41550 −0.103185
$$549$$ −23.0562 −0.984015
$$550$$ 0 0
$$551$$ −13.2741 −0.565497
$$552$$ 2.47219 0.105223
$$553$$ 32.3274 1.37470
$$554$$ −26.4795 −1.12501
$$555$$ 0 0
$$556$$ −34.5730 −1.46622
$$557$$ −1.27652 −0.0540879 −0.0270439 0.999634i $$-0.508609\pi$$
−0.0270439 + 0.999634i $$0.508609\pi$$
$$558$$ 25.8213 1.09310
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −7.50902 −0.317031
$$562$$ −14.5429 −0.613454
$$563$$ 9.12737 0.384673 0.192336 0.981329i $$-0.438393\pi$$
0.192336 + 0.981329i $$0.438393\pi$$
$$564$$ 4.99330 0.210256
$$565$$ 0 0
$$566$$ −14.7875 −0.621563
$$567$$ −12.9554 −0.544075
$$568$$ −10.8280 −0.454334
$$569$$ −5.72156 −0.239860 −0.119930 0.992782i $$-0.538267\pi$$
−0.119930 + 0.992782i $$0.538267\pi$$
$$570$$ 0 0
$$571$$ 7.60148 0.318112 0.159056 0.987270i $$-0.449155\pi$$
0.159056 + 0.987270i $$0.449155\pi$$
$$572$$ 0 0
$$573$$ 10.2282 0.427289
$$574$$ −5.86592 −0.244839
$$575$$ 0 0
$$576$$ 35.0954 1.46231
$$577$$ −45.1564 −1.87989 −0.939944 0.341330i $$-0.889123\pi$$
−0.939944 + 0.341330i $$0.889123\pi$$
$$578$$ 24.8213 1.03243
$$579$$ −3.35988 −0.139632
$$580$$ 0 0
$$581$$ 15.8248 0.656522
$$582$$ −14.8726 −0.616490
$$583$$ 14.1062 0.584219
$$584$$ 24.8713 1.02918
$$585$$ 0 0
$$586$$ 54.6872 2.25911
$$587$$ 32.4040 1.33746 0.668728 0.743507i $$-0.266839\pi$$
0.668728 + 0.743507i $$0.266839\pi$$
$$588$$ −4.74094 −0.195513
$$589$$ 24.9758 1.02911
$$590$$ 0 0
$$591$$ 6.33513 0.260592
$$592$$ 4.29590 0.176560
$$593$$ −36.6848 −1.50647 −0.753233 0.657754i $$-0.771507\pi$$
−0.753233 + 0.657754i $$0.771507\pi$$
$$594$$ 18.1347 0.744075
$$595$$ 0 0
$$596$$ 25.6286 1.04979
$$597$$ −7.71678 −0.315827
$$598$$ 0 0
$$599$$ −9.99223 −0.408271 −0.204136 0.978943i $$-0.565438\pi$$
−0.204136 + 0.978943i $$0.565438\pi$$
$$600$$ 0 0
$$601$$ −1.81163 −0.0738978 −0.0369489 0.999317i $$-0.511764\pi$$
−0.0369489 + 0.999317i $$0.511764\pi$$
$$602$$ 28.2543 1.15156
$$603$$ 1.55257 0.0632253
$$604$$ 43.0790 1.75286
$$605$$ 0 0
$$606$$ 16.2905 0.661757
$$607$$ −11.2161 −0.455248 −0.227624 0.973749i $$-0.573096\pi$$
−0.227624 + 0.973749i $$0.573096\pi$$
$$608$$ 38.1226 1.54608
$$609$$ −2.57971 −0.104535
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −43.4674 −1.75707
$$613$$ 20.8944 0.843917 0.421958 0.906615i $$-0.361343\pi$$
0.421958 + 0.906615i $$0.361343\pi$$
$$614$$ 31.6233 1.27621
$$615$$ 0 0
$$616$$ 12.3381 0.497117
$$617$$ −12.0992 −0.487094 −0.243547 0.969889i $$-0.578311\pi$$
−0.243547 + 0.969889i $$0.578311\pi$$
$$618$$ −11.4330 −0.459901
$$619$$ −10.5526 −0.424143 −0.212072 0.977254i $$-0.568021\pi$$
−0.212072 + 0.977254i $$0.568021\pi$$
$$620$$ 0 0
$$621$$ −5.97046 −0.239586
$$622$$ −66.8926 −2.68215
$$623$$ −13.5506 −0.542895
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 16.8049 0.671660
$$627$$ 8.29590 0.331306
$$628$$ 28.7603 1.14766
$$629$$ −28.3696 −1.13117
$$630$$ 0 0
$$631$$ −13.8514 −0.551417 −0.275709 0.961241i $$-0.588913\pi$$
−0.275709 + 0.961241i $$0.588913\pi$$
$$632$$ −37.1866 −1.47920
$$633$$ −7.35019 −0.292144
$$634$$ −67.4771 −2.67986
$$635$$ 0 0
$$636$$ −9.34183 −0.370428
$$637$$ 0 0
$$638$$ −13.0248 −0.515655
$$639$$ 12.3676 0.489257
$$640$$ 0 0
$$641$$ 34.9608 1.38087 0.690434 0.723396i $$-0.257420\pi$$
0.690434 + 0.723396i $$0.257420\pi$$
$$642$$ 8.60388 0.339568
$$643$$ −33.3980 −1.31709 −0.658545 0.752541i $$-0.728828\pi$$
−0.658545 + 0.752541i $$0.728828\pi$$
$$644$$ −11.8073 −0.465273
$$645$$ 0 0
$$646$$ −69.6238 −2.73931
$$647$$ −2.32842 −0.0915397 −0.0457698 0.998952i $$-0.514574\pi$$
−0.0457698 + 0.998952i $$0.514574\pi$$
$$648$$ 14.9028 0.585436
$$649$$ 31.1903 1.22433
$$650$$ 0 0
$$651$$ 4.85384 0.190237
$$652$$ −26.5381 −1.03931
$$653$$ −14.5714 −0.570221 −0.285111 0.958495i $$-0.592030\pi$$
−0.285111 + 0.958495i $$0.592030\pi$$
$$654$$ 0.152129 0.00594871
$$655$$ 0 0
$$656$$ −1.02177 −0.0398934
$$657$$ −28.4077 −1.10829
$$658$$ −13.5864 −0.529654
$$659$$ 11.1395 0.433932 0.216966 0.976179i $$-0.430384\pi$$
0.216966 + 0.976179i $$0.430384\pi$$
$$660$$ 0 0
$$661$$ −13.8498 −0.538694 −0.269347 0.963043i $$-0.586808\pi$$
−0.269347 + 0.963043i $$0.586808\pi$$
$$662$$ −35.3153 −1.37257
$$663$$ 0 0
$$664$$ −18.2034 −0.706430
$$665$$ 0 0
$$666$$ 32.4034 1.25561
$$667$$ 4.28813 0.166037
$$668$$ 72.7284 2.81395
$$669$$ 4.07069 0.157382
$$670$$ 0 0
$$671$$ −21.8823 −0.844757
$$672$$ 7.40880 0.285801
$$673$$ 6.52973 0.251703 0.125851 0.992049i $$-0.459834\pi$$
0.125851 + 0.992049i $$0.459834\pi$$
$$674$$ −4.39075 −0.169125
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 11.3104 0.434693 0.217346 0.976095i $$-0.430260\pi$$
0.217346 + 0.976095i $$0.430260\pi$$
$$678$$ −9.11290 −0.349979
$$679$$ 24.4373 0.937816
$$680$$ 0 0
$$681$$ 4.81402 0.184474
$$682$$ 24.5066 0.938407
$$683$$ −14.1793 −0.542555 −0.271277 0.962501i $$-0.587446\pi$$
−0.271277 + 0.962501i $$0.587446\pi$$
$$684$$ 48.0224 1.83618
$$685$$ 0 0
$$686$$ 45.1269 1.72295
$$687$$ −7.59551 −0.289787
$$688$$ 4.92154 0.187632
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 30.7952 1.17151 0.585753 0.810490i $$-0.300799\pi$$
0.585753 + 0.810490i $$0.300799\pi$$
$$692$$ 57.4878 2.18536
$$693$$ −14.0925 −0.535328
$$694$$ 38.4795 1.46066
$$695$$ 0 0
$$696$$ 2.96748 0.112482
$$697$$ 6.74764 0.255585
$$698$$ −23.5187 −0.890196
$$699$$ 2.82371 0.106802
$$700$$ 0 0
$$701$$ 6.73184 0.254258 0.127129 0.991886i $$-0.459424\pi$$
0.127129 + 0.991886i $$0.459424\pi$$
$$702$$ 0 0
$$703$$ 31.3424 1.18210
$$704$$ 33.3086 1.25536
$$705$$ 0 0
$$706$$ −34.8974 −1.31338
$$707$$ −26.7670 −1.00668
$$708$$ −20.6558 −0.776292
$$709$$ −47.6252 −1.78860 −0.894300 0.447467i $$-0.852326\pi$$
−0.894300 + 0.447467i $$0.852326\pi$$
$$710$$ 0 0
$$711$$ 42.4741 1.59290
$$712$$ 15.5875 0.584166
$$713$$ −8.06829 −0.302160
$$714$$ −13.5308 −0.506377
$$715$$ 0 0
$$716$$ 18.3599 0.686141
$$717$$ −6.06233 −0.226402
$$718$$ −48.1444 −1.79673
$$719$$ −5.99330 −0.223512 −0.111756 0.993736i $$-0.535648\pi$$
−0.111756 + 0.993736i $$0.535648\pi$$
$$720$$ 0 0
$$721$$ 18.7855 0.699610
$$722$$ 34.2271 1.27380
$$723$$ 6.61224 0.245912
$$724$$ −14.5670 −0.541380
$$725$$ 0 0
$$726$$ −5.57673 −0.206972
$$727$$ 24.1226 0.894657 0.447329 0.894370i $$-0.352375\pi$$
0.447329 + 0.894370i $$0.352375\pi$$
$$728$$ 0 0
$$729$$ −11.7627 −0.435656
$$730$$ 0 0
$$731$$ −32.5013 −1.20210
$$732$$ 14.4916 0.535624
$$733$$ −36.0646 −1.33208 −0.666038 0.745918i $$-0.732011\pi$$
−0.666038 + 0.745918i $$0.732011\pi$$
$$734$$ −77.0786 −2.84502
$$735$$ 0 0
$$736$$ −12.3153 −0.453947
$$737$$ 1.47352 0.0542777
$$738$$ −7.70709 −0.283702
$$739$$ 27.5254 1.01254 0.506269 0.862375i $$-0.331024\pi$$
0.506269 + 0.862375i $$0.331024\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 25.4185 0.933142
$$743$$ 10.4692 0.384078 0.192039 0.981387i $$-0.438490\pi$$
0.192039 + 0.981387i $$0.438490\pi$$
$$744$$ −5.58343 −0.204699
$$745$$ 0 0
$$746$$ 28.3032 1.03625
$$747$$ 20.7918 0.760731
$$748$$ −41.2543 −1.50841
$$749$$ −14.1371 −0.516557
$$750$$ 0 0
$$751$$ 4.06770 0.148433 0.0742163 0.997242i $$-0.476354\pi$$
0.0742163 + 0.997242i $$0.476354\pi$$
$$752$$ −2.36658 −0.0863005
$$753$$ 12.4021 0.451957
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 19.7332 0.717688
$$757$$ −20.4336 −0.742670 −0.371335 0.928499i $$-0.621100\pi$$
−0.371335 + 0.928499i $$0.621100\pi$$
$$758$$ 37.1618 1.34978
$$759$$ −2.67994 −0.0972757
$$760$$ 0 0
$$761$$ −27.0237 −0.979608 −0.489804 0.871833i $$-0.662932\pi$$
−0.489804 + 0.871833i $$0.662932\pi$$
$$762$$ 23.6558 0.856958
$$763$$ −0.249964 −0.00904929
$$764$$ 56.1933 2.03300
$$765$$ 0 0
$$766$$ −16.9336 −0.611837
$$767$$ 0 0
$$768$$ −5.80864 −0.209601
$$769$$ −37.9407 −1.36818 −0.684088 0.729400i $$-0.739799\pi$$
−0.684088 + 0.729400i $$0.739799\pi$$
$$770$$ 0 0
$$771$$ 10.3556 0.372947
$$772$$ −18.4590 −0.664355
$$773$$ 16.3375 0.587620 0.293810 0.955864i $$-0.405077\pi$$
0.293810 + 0.955864i $$0.405077\pi$$
$$774$$ 37.1226 1.33434
$$775$$ 0 0
$$776$$ −28.1105 −1.00911
$$777$$ 6.09113 0.218518
$$778$$ 79.8926 2.86429
$$779$$ −7.45473 −0.267093
$$780$$ 0 0
$$781$$ 11.7380 0.420017
$$782$$ 22.4916 0.804297
$$783$$ −7.16660 −0.256114
$$784$$ 2.24698 0.0802493
$$785$$ 0 0
$$786$$ 4.06100 0.144851
$$787$$ 18.6907 0.666251 0.333126 0.942882i $$-0.391897\pi$$
0.333126 + 0.942882i $$0.391897\pi$$
$$788$$ 34.8049 1.23987
$$789$$ −7.99090 −0.284484
$$790$$ 0 0
$$791$$ 14.9734 0.532394
$$792$$ 16.2107 0.576023
$$793$$ 0 0
$$794$$ −3.03684 −0.107773
$$795$$ 0 0
$$796$$ −42.3957 −1.50267
$$797$$ −29.2519 −1.03615 −0.518077 0.855334i $$-0.673352\pi$$
−0.518077 + 0.855334i $$0.673352\pi$$
$$798$$ 14.9487 0.529178
$$799$$ 15.6286 0.552901
$$800$$ 0 0
$$801$$ −17.8039 −0.629068
$$802$$ 1.30127 0.0459496
$$803$$ −26.9614 −0.951446
$$804$$ −0.975837 −0.0344151
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0.362273 0.0127526
$$808$$ 30.7904 1.08320
$$809$$ −6.65087 −0.233832 −0.116916 0.993142i $$-0.537301\pi$$
−0.116916 + 0.993142i $$0.537301\pi$$
$$810$$ 0 0
$$811$$ 3.89200 0.136667 0.0683333 0.997663i $$-0.478232\pi$$
0.0683333 + 0.997663i $$0.478232\pi$$
$$812$$ −14.1728 −0.497369
$$813$$ −1.10752 −0.0388425
$$814$$ 30.7536 1.07791
$$815$$ 0 0
$$816$$ −2.35690 −0.0825079
$$817$$ 35.9071 1.25623
$$818$$ −34.0586 −1.19083
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 45.9982 1.60535 0.802674 0.596418i $$-0.203410\pi$$
0.802674 + 0.596418i $$0.203410\pi$$
$$822$$ −0.987918 −0.0344576
$$823$$ −7.95300 −0.277224 −0.138612 0.990347i $$-0.544264\pi$$
−0.138612 + 0.990347i $$0.544264\pi$$
$$824$$ −21.6093 −0.752794
$$825$$ 0 0
$$826$$ 56.2030 1.95555
$$827$$ −27.9648 −0.972432 −0.486216 0.873839i $$-0.661623\pi$$
−0.486216 + 0.873839i $$0.661623\pi$$
$$828$$ −15.5133 −0.539126
$$829$$ 27.6310 0.959665 0.479833 0.877360i $$-0.340697\pi$$
0.479833 + 0.877360i $$0.340697\pi$$
$$830$$ 0 0
$$831$$ −6.53989 −0.226866
$$832$$ 0 0
$$833$$ −14.8388 −0.514133
$$834$$ −14.1400 −0.489630
$$835$$ 0 0
$$836$$ 45.5773 1.57632
$$837$$ 13.4843 0.466085
$$838$$ −80.2699 −2.77288
$$839$$ 28.6848 0.990311 0.495155 0.868804i $$-0.335111\pi$$
0.495155 + 0.868804i $$0.335111\pi$$
$$840$$ 0 0
$$841$$ −23.8528 −0.822509
$$842$$ −78.7488 −2.71386
$$843$$ −3.59179 −0.123708
$$844$$ −40.3817 −1.38999
$$845$$ 0 0
$$846$$ −17.8509 −0.613725
$$847$$ 9.16315 0.314849
$$848$$ 4.42758 0.152044
$$849$$ −3.65220 −0.125343
$$850$$ 0 0
$$851$$ −10.1250 −0.347080
$$852$$ −7.77346 −0.266314
$$853$$ 43.2078 1.47941 0.739703 0.672934i $$-0.234966\pi$$
0.739703 + 0.672934i $$0.234966\pi$$
$$854$$ −39.4306 −1.34929
$$855$$ 0 0
$$856$$ 16.2620 0.555825
$$857$$ −35.1685 −1.20133 −0.600667 0.799499i $$-0.705098\pi$$
−0.600667 + 0.799499i $$0.705098\pi$$
$$858$$ 0 0
$$859$$ 27.3793 0.934168 0.467084 0.884213i $$-0.345305\pi$$
0.467084 + 0.884213i $$0.345305\pi$$
$$860$$ 0 0
$$861$$ −1.44876 −0.0493737
$$862$$ −77.0297 −2.62364
$$863$$ 41.3913 1.40898 0.704489 0.709715i $$-0.251176\pi$$
0.704489 + 0.709715i $$0.251176\pi$$
$$864$$ 20.5821 0.700217
$$865$$ 0 0
$$866$$ −30.8702 −1.04901
$$867$$ 6.13036 0.208198
$$868$$ 26.6668 0.905130
$$869$$ 40.3116 1.36748
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0.287536 0.00973721
$$873$$ 32.1075 1.08668
$$874$$ −24.8485 −0.840512
$$875$$ 0 0
$$876$$ 17.8552 0.603270
$$877$$ 24.7472 0.835653 0.417826 0.908527i $$-0.362792\pi$$
0.417826 + 0.908527i $$0.362792\pi$$
$$878$$ 23.0097 0.776539
$$879$$ 13.5066 0.455567
$$880$$ 0 0
$$881$$ −28.5875 −0.963137 −0.481568 0.876409i $$-0.659933\pi$$
−0.481568 + 0.876409i $$0.659933\pi$$
$$882$$ 16.9487 0.570692
$$883$$ −9.61702 −0.323639 −0.161819 0.986820i $$-0.551736\pi$$
−0.161819 + 0.986820i $$0.551736\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −27.3599 −0.919173
$$887$$ −15.9661 −0.536091 −0.268045 0.963406i $$-0.586378\pi$$
−0.268045 + 0.963406i $$0.586378\pi$$
$$888$$ −7.00670 −0.235130
$$889$$ −38.8689 −1.30362
$$890$$ 0 0
$$891$$ −16.1551 −0.541217
$$892$$ 22.3642 0.748809
$$893$$ −17.2664 −0.577797
$$894$$ 10.4819 0.350566
$$895$$ 0 0
$$896$$ 33.3196 1.11313
$$897$$ 0 0
$$898$$ −28.9976 −0.967663
$$899$$ −9.68473 −0.323004
$$900$$ 0 0
$$901$$ −29.2392 −0.974099
$$902$$ −7.31468 −0.243552
$$903$$ 6.97823 0.232221
$$904$$ −17.2241 −0.572867
$$905$$ 0 0
$$906$$ 17.6189 0.585350
$$907$$ 28.8364 0.957496 0.478748 0.877952i $$-0.341091\pi$$
0.478748 + 0.877952i $$0.341091\pi$$
$$908$$ 26.4480 0.877709
$$909$$ −35.1685 −1.16647
$$910$$ 0 0
$$911$$ 38.5633 1.27766 0.638830 0.769348i $$-0.279419\pi$$
0.638830 + 0.769348i $$0.279419\pi$$
$$912$$ 2.60388 0.0862229
$$913$$ 19.7332 0.653073
$$914$$ −10.4644 −0.346132
$$915$$ 0 0
$$916$$ −41.7294 −1.37878
$$917$$ −6.67264 −0.220350
$$918$$ −37.5894 −1.24064
$$919$$ 8.87502 0.292760 0.146380 0.989228i $$-0.453238\pi$$
0.146380 + 0.989228i $$0.453238\pi$$
$$920$$ 0 0
$$921$$ 7.81030 0.257358
$$922$$ −70.8708 −2.33401
$$923$$ 0 0
$$924$$ 8.85756 0.291392
$$925$$ 0 0
$$926$$ −39.6601 −1.30331
$$927$$ 24.6819 0.810659
$$928$$ −14.7826 −0.485261
$$929$$ 24.2295 0.794945 0.397472 0.917614i $$-0.369887\pi$$
0.397472 + 0.917614i $$0.369887\pi$$
$$930$$ 0 0
$$931$$ 16.3937 0.537283
$$932$$ 15.5133 0.508156
$$933$$ −16.5211 −0.540877
$$934$$ 72.2911 2.36544
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −17.2644 −0.564005 −0.282002 0.959414i $$-0.590999\pi$$
−0.282002 + 0.959414i $$0.590999\pi$$
$$938$$ 2.65519 0.0866949
$$939$$ 4.15047 0.135446
$$940$$ 0 0
$$941$$ 4.34050 0.141496 0.0707482 0.997494i $$-0.477461\pi$$
0.0707482 + 0.997494i $$0.477461\pi$$
$$942$$ 11.7627 0.383250
$$943$$ 2.40821 0.0784220
$$944$$ 9.78986 0.318633
$$945$$ 0 0
$$946$$ 35.2325 1.14551
$$947$$ 45.0146 1.46278 0.731389 0.681961i $$-0.238872\pi$$
0.731389 + 0.681961i $$0.238872\pi$$
$$948$$ −26.6963 −0.867057
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −16.6655 −0.540415
$$952$$ −25.5743 −0.828869
$$953$$ 46.8859 1.51878 0.759391 0.650634i $$-0.225497\pi$$
0.759391 + 0.650634i $$0.225497\pi$$
$$954$$ 33.3967 1.08126
$$955$$ 0 0
$$956$$ −33.3062 −1.07720
$$957$$ −3.21685 −0.103986
$$958$$ −78.4191 −2.53361
$$959$$ 1.62325 0.0524176
$$960$$ 0 0
$$961$$ −12.7778 −0.412186
$$962$$ 0 0
$$963$$ −18.5743 −0.598550
$$964$$ 36.3274 1.17003
$$965$$ 0 0
$$966$$ −4.82908 −0.155373
$$967$$ −6.29457 −0.202420 −0.101210 0.994865i $$-0.532271\pi$$
−0.101210 + 0.994865i $$0.532271\pi$$
$$968$$ −10.5405 −0.338784
$$969$$ −17.1957 −0.552404
$$970$$ 0 0
$$971$$ −41.8068 −1.34165 −0.670823 0.741618i $$-0.734059\pi$$
−0.670823 + 0.741618i $$0.734059\pi$$
$$972$$ 39.5918 1.26991
$$973$$ 23.2336 0.744834
$$974$$ −94.0025 −3.01203
$$975$$ 0 0
$$976$$ −6.86831 −0.219849
$$977$$ −23.7530 −0.759926 −0.379963 0.925002i $$-0.624063\pi$$
−0.379963 + 0.925002i $$0.624063\pi$$
$$978$$ −10.8538 −0.347067
$$979$$ −16.8974 −0.540043
$$980$$ 0 0
$$981$$ −0.328421 −0.0104857
$$982$$ 49.0863 1.56641
$$983$$ 55.7251 1.77736 0.888678 0.458532i $$-0.151625\pi$$
0.888678 + 0.458532i $$0.151625\pi$$
$$984$$ 1.66653 0.0531270
$$985$$ 0 0
$$986$$ 26.9976 0.859779
$$987$$ −3.35557 −0.106809
$$988$$ 0 0
$$989$$ −11.5996 −0.368845
$$990$$ 0 0
$$991$$ −35.5512 −1.12932 −0.564661 0.825323i $$-0.690993\pi$$
−0.564661 + 0.825323i $$0.690993\pi$$
$$992$$ 27.8140 0.883096
$$993$$ −8.72215 −0.276789
$$994$$ 21.1511 0.670871
$$995$$ 0 0
$$996$$ −13.0683 −0.414085
$$997$$ −6.61058 −0.209359 −0.104680 0.994506i $$-0.533382\pi$$
−0.104680 + 0.994506i $$0.533382\pi$$
$$998$$ 52.9101 1.67484
$$999$$ 16.9215 0.535374
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 4225.2.a.bg.1.3 3
5.4 even 2 169.2.a.b.1.1 3
13.12 even 2 4225.2.a.bb.1.1 3
15.14 odd 2 1521.2.a.r.1.3 3
20.19 odd 2 2704.2.a.z.1.2 3
35.34 odd 2 8281.2.a.bf.1.1 3
65.4 even 6 169.2.c.b.146.1 6
65.9 even 6 169.2.c.c.146.3 6
65.19 odd 12 169.2.e.b.23.6 12
65.24 odd 12 169.2.e.b.147.6 12
65.29 even 6 169.2.c.c.22.3 6
65.34 odd 4 169.2.b.b.168.1 6
65.44 odd 4 169.2.b.b.168.6 6
65.49 even 6 169.2.c.b.22.1 6
65.54 odd 12 169.2.e.b.147.1 12
65.59 odd 12 169.2.e.b.23.1 12
65.64 even 2 169.2.a.c.1.3 yes 3
195.44 even 4 1521.2.b.l.1351.1 6
195.164 even 4 1521.2.b.l.1351.6 6
195.194 odd 2 1521.2.a.o.1.1 3
260.99 even 4 2704.2.f.o.337.3 6
260.239 even 4 2704.2.f.o.337.4 6
260.259 odd 2 2704.2.a.ba.1.2 3
455.454 odd 2 8281.2.a.bj.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 5.4 even 2
169.2.a.c.1.3 yes 3 65.64 even 2
169.2.b.b.168.1 6 65.34 odd 4
169.2.b.b.168.6 6 65.44 odd 4
169.2.c.b.22.1 6 65.49 even 6
169.2.c.b.146.1 6 65.4 even 6
169.2.c.c.22.3 6 65.29 even 6
169.2.c.c.146.3 6 65.9 even 6
169.2.e.b.23.1 12 65.59 odd 12
169.2.e.b.23.6 12 65.19 odd 12
169.2.e.b.147.1 12 65.54 odd 12
169.2.e.b.147.6 12 65.24 odd 12
1521.2.a.o.1.1 3 195.194 odd 2
1521.2.a.r.1.3 3 15.14 odd 2
1521.2.b.l.1351.1 6 195.44 even 4
1521.2.b.l.1351.6 6 195.164 even 4
2704.2.a.z.1.2 3 20.19 odd 2
2704.2.a.ba.1.2 3 260.259 odd 2
2704.2.f.o.337.3 6 260.99 even 4
2704.2.f.o.337.4 6 260.239 even 4
4225.2.a.bb.1.1 3 13.12 even 2
4225.2.a.bg.1.3 3 1.1 even 1 trivial
8281.2.a.bf.1.1 3 35.34 odd 2
8281.2.a.bj.1.3 3 455.454 odd 2