# Properties

 Label 5225.2.a.m.1.2 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11$$ x^7 - x^6 - 10*x^5 + 8*x^4 + 27*x^3 - 16*x^2 - 18*x + 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.97792$$ of defining polynomial Character $$\chi$$ $$=$$ 5225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.97792 q^{2} -2.65412 q^{3} +1.91218 q^{4} +5.24965 q^{6} -1.06653 q^{7} +0.173707 q^{8} +4.04436 q^{9} +O(q^{10})$$ $$q-1.97792 q^{2} -2.65412 q^{3} +1.91218 q^{4} +5.24965 q^{6} -1.06653 q^{7} +0.173707 q^{8} +4.04436 q^{9} +1.00000 q^{11} -5.07515 q^{12} -0.939291 q^{13} +2.10952 q^{14} -4.16793 q^{16} -3.65839 q^{17} -7.99944 q^{18} -1.00000 q^{19} +2.83071 q^{21} -1.97792 q^{22} -5.35035 q^{23} -0.461041 q^{24} +1.85784 q^{26} -2.77187 q^{27} -2.03940 q^{28} -4.46966 q^{29} -2.71979 q^{31} +7.89643 q^{32} -2.65412 q^{33} +7.23600 q^{34} +7.73354 q^{36} -0.932050 q^{37} +1.97792 q^{38} +2.49299 q^{39} -1.21816 q^{41} -5.59893 q^{42} +3.58402 q^{43} +1.91218 q^{44} +10.5826 q^{46} +4.30223 q^{47} +11.0622 q^{48} -5.86250 q^{49} +9.70980 q^{51} -1.79609 q^{52} +12.1575 q^{53} +5.48254 q^{54} -0.185265 q^{56} +2.65412 q^{57} +8.84063 q^{58} -6.15610 q^{59} +2.13701 q^{61} +5.37953 q^{62} -4.31345 q^{63} -7.28267 q^{64} +5.24965 q^{66} -4.93275 q^{67} -6.99548 q^{68} +14.2005 q^{69} +7.32299 q^{71} +0.702536 q^{72} -1.14496 q^{73} +1.84352 q^{74} -1.91218 q^{76} -1.06653 q^{77} -4.93095 q^{78} -6.71067 q^{79} -4.77622 q^{81} +2.40943 q^{82} -2.82737 q^{83} +5.41282 q^{84} -7.08891 q^{86} +11.8630 q^{87} +0.173707 q^{88} -8.77091 q^{89} +1.00179 q^{91} -10.2308 q^{92} +7.21866 q^{93} -8.50948 q^{94} -20.9581 q^{96} -13.3420 q^{97} +11.5956 q^{98} +4.04436 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - q^{2} - 3 q^{3} + 7 q^{4} + 8 q^{6} + q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10})$$ 7 * q - q^2 - 3 * q^3 + 7 * q^4 + 8 * q^6 + q^7 - 3 * q^8 + 2 * q^9 $$7 q - q^{2} - 3 q^{3} + 7 q^{4} + 8 q^{6} + q^{7} - 3 q^{8} + 2 q^{9} + 7 q^{11} - 13 q^{12} - q^{13} + 12 q^{14} + 3 q^{16} - q^{17} - 7 q^{18} - 7 q^{19} + 5 q^{21} - q^{22} + 8 q^{23} + 25 q^{24} - 12 q^{27} - 4 q^{28} + 11 q^{29} + 7 q^{31} - 12 q^{32} - 3 q^{33} - 14 q^{34} + 7 q^{36} + 17 q^{37} + q^{38} + 30 q^{39} + 17 q^{41} - 33 q^{42} + 3 q^{43} + 7 q^{44} + 18 q^{46} - 14 q^{47} + 12 q^{48} + 6 q^{49} + 8 q^{51} + 17 q^{52} - 7 q^{53} - 27 q^{54} + 36 q^{56} + 3 q^{57} + 15 q^{58} + 35 q^{59} + 17 q^{61} - 46 q^{62} + 22 q^{63} + 5 q^{64} + 8 q^{66} - 4 q^{67} + 35 q^{68} - 4 q^{69} + 10 q^{71} - 12 q^{72} - 22 q^{73} - 11 q^{74} - 7 q^{76} + q^{77} + 41 q^{78} + 11 q^{79} - 21 q^{81} + 14 q^{82} - 39 q^{83} + 21 q^{84} - 24 q^{86} + 2 q^{87} - 3 q^{88} + 18 q^{89} - 22 q^{91} + 51 q^{92} - 10 q^{93} + 14 q^{94} - 11 q^{96} + 4 q^{97} + 26 q^{98} + 2 q^{99}+O(q^{100})$$ 7 * q - q^2 - 3 * q^3 + 7 * q^4 + 8 * q^6 + q^7 - 3 * q^8 + 2 * q^9 + 7 * q^11 - 13 * q^12 - q^13 + 12 * q^14 + 3 * q^16 - q^17 - 7 * q^18 - 7 * q^19 + 5 * q^21 - q^22 + 8 * q^23 + 25 * q^24 - 12 * q^27 - 4 * q^28 + 11 * q^29 + 7 * q^31 - 12 * q^32 - 3 * q^33 - 14 * q^34 + 7 * q^36 + 17 * q^37 + q^38 + 30 * q^39 + 17 * q^41 - 33 * q^42 + 3 * q^43 + 7 * q^44 + 18 * q^46 - 14 * q^47 + 12 * q^48 + 6 * q^49 + 8 * q^51 + 17 * q^52 - 7 * q^53 - 27 * q^54 + 36 * q^56 + 3 * q^57 + 15 * q^58 + 35 * q^59 + 17 * q^61 - 46 * q^62 + 22 * q^63 + 5 * q^64 + 8 * q^66 - 4 * q^67 + 35 * q^68 - 4 * q^69 + 10 * q^71 - 12 * q^72 - 22 * q^73 - 11 * q^74 - 7 * q^76 + q^77 + 41 * q^78 + 11 * q^79 - 21 * q^81 + 14 * q^82 - 39 * q^83 + 21 * q^84 - 24 * q^86 + 2 * q^87 - 3 * q^88 + 18 * q^89 - 22 * q^91 + 51 * q^92 - 10 * q^93 + 14 * q^94 - 11 * q^96 + 4 * q^97 + 26 * q^98 + 2 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.97792 −1.39860 −0.699301 0.714827i $$-0.746505\pi$$
−0.699301 + 0.714827i $$0.746505\pi$$
$$3$$ −2.65412 −1.53236 −0.766179 0.642627i $$-0.777844\pi$$
−0.766179 + 0.642627i $$0.777844\pi$$
$$4$$ 1.91218 0.956088
$$5$$ 0 0
$$6$$ 5.24965 2.14316
$$7$$ −1.06653 −0.403112 −0.201556 0.979477i $$-0.564600\pi$$
−0.201556 + 0.979477i $$0.564600\pi$$
$$8$$ 0.173707 0.0614148
$$9$$ 4.04436 1.34812
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −5.07515 −1.46507
$$13$$ −0.939291 −0.260512 −0.130256 0.991480i $$-0.541580\pi$$
−0.130256 + 0.991480i $$0.541580\pi$$
$$14$$ 2.10952 0.563794
$$15$$ 0 0
$$16$$ −4.16793 −1.04198
$$17$$ −3.65839 −0.887289 −0.443645 0.896203i $$-0.646315\pi$$
−0.443645 + 0.896203i $$0.646315\pi$$
$$18$$ −7.99944 −1.88549
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.83071 0.617712
$$22$$ −1.97792 −0.421694
$$23$$ −5.35035 −1.11563 −0.557813 0.829967i $$-0.688359\pi$$
−0.557813 + 0.829967i $$0.688359\pi$$
$$24$$ −0.461041 −0.0941095
$$25$$ 0 0
$$26$$ 1.85784 0.364353
$$27$$ −2.77187 −0.533446
$$28$$ −2.03940 −0.385411
$$29$$ −4.46966 −0.829994 −0.414997 0.909823i $$-0.636217\pi$$
−0.414997 + 0.909823i $$0.636217\pi$$
$$30$$ 0 0
$$31$$ −2.71979 −0.488489 −0.244244 0.969714i $$-0.578540\pi$$
−0.244244 + 0.969714i $$0.578540\pi$$
$$32$$ 7.89643 1.39591
$$33$$ −2.65412 −0.462023
$$34$$ 7.23600 1.24096
$$35$$ 0 0
$$36$$ 7.73354 1.28892
$$37$$ −0.932050 −0.153228 −0.0766141 0.997061i $$-0.524411\pi$$
−0.0766141 + 0.997061i $$0.524411\pi$$
$$38$$ 1.97792 0.320861
$$39$$ 2.49299 0.399198
$$40$$ 0 0
$$41$$ −1.21816 −0.190245 −0.0951227 0.995466i $$-0.530324\pi$$
−0.0951227 + 0.995466i $$0.530324\pi$$
$$42$$ −5.59893 −0.863934
$$43$$ 3.58402 0.546558 0.273279 0.961935i $$-0.411892\pi$$
0.273279 + 0.961935i $$0.411892\pi$$
$$44$$ 1.91218 0.288272
$$45$$ 0 0
$$46$$ 10.5826 1.56032
$$47$$ 4.30223 0.627545 0.313772 0.949498i $$-0.398407\pi$$
0.313772 + 0.949498i $$0.398407\pi$$
$$48$$ 11.0622 1.59669
$$49$$ −5.86250 −0.837501
$$50$$ 0 0
$$51$$ 9.70980 1.35964
$$52$$ −1.79609 −0.249073
$$53$$ 12.1575 1.66996 0.834980 0.550280i $$-0.185479\pi$$
0.834980 + 0.550280i $$0.185479\pi$$
$$54$$ 5.48254 0.746079
$$55$$ 0 0
$$56$$ −0.185265 −0.0247571
$$57$$ 2.65412 0.351547
$$58$$ 8.84063 1.16083
$$59$$ −6.15610 −0.801456 −0.400728 0.916197i $$-0.631243\pi$$
−0.400728 + 0.916197i $$0.631243\pi$$
$$60$$ 0 0
$$61$$ 2.13701 0.273616 0.136808 0.990598i $$-0.456316\pi$$
0.136808 + 0.990598i $$0.456316\pi$$
$$62$$ 5.37953 0.683202
$$63$$ −4.31345 −0.543444
$$64$$ −7.28267 −0.910333
$$65$$ 0 0
$$66$$ 5.24965 0.646187
$$67$$ −4.93275 −0.602631 −0.301315 0.953525i $$-0.597426\pi$$
−0.301315 + 0.953525i $$0.597426\pi$$
$$68$$ −6.99548 −0.848327
$$69$$ 14.2005 1.70954
$$70$$ 0 0
$$71$$ 7.32299 0.869079 0.434539 0.900653i $$-0.356911\pi$$
0.434539 + 0.900653i $$0.356911\pi$$
$$72$$ 0.702536 0.0827946
$$73$$ −1.14496 −0.134008 −0.0670038 0.997753i $$-0.521344\pi$$
−0.0670038 + 0.997753i $$0.521344\pi$$
$$74$$ 1.84352 0.214305
$$75$$ 0 0
$$76$$ −1.91218 −0.219342
$$77$$ −1.06653 −0.121543
$$78$$ −4.93095 −0.558320
$$79$$ −6.71067 −0.755009 −0.377505 0.926008i $$-0.623218\pi$$
−0.377505 + 0.926008i $$0.623218\pi$$
$$80$$ 0 0
$$81$$ −4.77622 −0.530691
$$82$$ 2.40943 0.266078
$$83$$ −2.82737 −0.310344 −0.155172 0.987887i $$-0.549593\pi$$
−0.155172 + 0.987887i $$0.549593\pi$$
$$84$$ 5.41282 0.590588
$$85$$ 0 0
$$86$$ −7.08891 −0.764417
$$87$$ 11.8630 1.27185
$$88$$ 0.173707 0.0185173
$$89$$ −8.77091 −0.929714 −0.464857 0.885386i $$-0.653894\pi$$
−0.464857 + 0.885386i $$0.653894\pi$$
$$90$$ 0 0
$$91$$ 1.00179 0.105016
$$92$$ −10.2308 −1.06664
$$93$$ 7.21866 0.748540
$$94$$ −8.50948 −0.877686
$$95$$ 0 0
$$96$$ −20.9581 −2.13903
$$97$$ −13.3420 −1.35468 −0.677339 0.735671i $$-0.736867\pi$$
−0.677339 + 0.735671i $$0.736867\pi$$
$$98$$ 11.5956 1.17133
$$99$$ 4.04436 0.406474
$$100$$ 0 0
$$101$$ −0.926286 −0.0921689 −0.0460845 0.998938i $$-0.514674\pi$$
−0.0460845 + 0.998938i $$0.514674\pi$$
$$102$$ −19.2052 −1.90160
$$103$$ 10.4344 1.02814 0.514068 0.857749i $$-0.328138\pi$$
0.514068 + 0.857749i $$0.328138\pi$$
$$104$$ −0.163162 −0.0159993
$$105$$ 0 0
$$106$$ −24.0466 −2.33561
$$107$$ −13.5595 −1.31085 −0.655426 0.755260i $$-0.727511\pi$$
−0.655426 + 0.755260i $$0.727511\pi$$
$$108$$ −5.30030 −0.510022
$$109$$ 5.85440 0.560750 0.280375 0.959891i $$-0.409541\pi$$
0.280375 + 0.959891i $$0.409541\pi$$
$$110$$ 0 0
$$111$$ 2.47378 0.234800
$$112$$ 4.44525 0.420036
$$113$$ −8.68686 −0.817191 −0.408595 0.912716i $$-0.633981\pi$$
−0.408595 + 0.912716i $$0.633981\pi$$
$$114$$ −5.24965 −0.491674
$$115$$ 0 0
$$116$$ −8.54677 −0.793548
$$117$$ −3.79883 −0.351202
$$118$$ 12.1763 1.12092
$$119$$ 3.90180 0.357677
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −4.22684 −0.382680
$$123$$ 3.23316 0.291524
$$124$$ −5.20072 −0.467039
$$125$$ 0 0
$$126$$ 8.53168 0.760062
$$127$$ 15.9545 1.41573 0.707866 0.706347i $$-0.249658\pi$$
0.707866 + 0.706347i $$0.249658\pi$$
$$128$$ −1.38832 −0.122711
$$129$$ −9.51242 −0.837522
$$130$$ 0 0
$$131$$ 5.20014 0.454338 0.227169 0.973855i $$-0.427053\pi$$
0.227169 + 0.973855i $$0.427053\pi$$
$$132$$ −5.07515 −0.441735
$$133$$ 1.06653 0.0924803
$$134$$ 9.75659 0.842841
$$135$$ 0 0
$$136$$ −0.635489 −0.0544927
$$137$$ −7.54419 −0.644544 −0.322272 0.946647i $$-0.604447\pi$$
−0.322272 + 0.946647i $$0.604447\pi$$
$$138$$ −28.0875 −2.39096
$$139$$ 0.357301 0.0303059 0.0151529 0.999885i $$-0.495176\pi$$
0.0151529 + 0.999885i $$0.495176\pi$$
$$140$$ 0 0
$$141$$ −11.4186 −0.961624
$$142$$ −14.4843 −1.21550
$$143$$ −0.939291 −0.0785475
$$144$$ −16.8566 −1.40472
$$145$$ 0 0
$$146$$ 2.26465 0.187423
$$147$$ 15.5598 1.28335
$$148$$ −1.78224 −0.146500
$$149$$ −10.7634 −0.881769 −0.440884 0.897564i $$-0.645335\pi$$
−0.440884 + 0.897564i $$0.645335\pi$$
$$150$$ 0 0
$$151$$ 7.13976 0.581025 0.290513 0.956871i $$-0.406174\pi$$
0.290513 + 0.956871i $$0.406174\pi$$
$$152$$ −0.173707 −0.0140895
$$153$$ −14.7958 −1.19617
$$154$$ 2.10952 0.169990
$$155$$ 0 0
$$156$$ 4.76704 0.381669
$$157$$ −12.7164 −1.01488 −0.507440 0.861687i $$-0.669408\pi$$
−0.507440 + 0.861687i $$0.669408\pi$$
$$158$$ 13.2732 1.05596
$$159$$ −32.2675 −2.55898
$$160$$ 0 0
$$161$$ 5.70633 0.449722
$$162$$ 9.44699 0.742225
$$163$$ 4.45308 0.348792 0.174396 0.984676i $$-0.444203\pi$$
0.174396 + 0.984676i $$0.444203\pi$$
$$164$$ −2.32935 −0.181891
$$165$$ 0 0
$$166$$ 5.59231 0.434047
$$167$$ −10.0911 −0.780874 −0.390437 0.920630i $$-0.627676\pi$$
−0.390437 + 0.920630i $$0.627676\pi$$
$$168$$ 0.491716 0.0379367
$$169$$ −12.1177 −0.932133
$$170$$ 0 0
$$171$$ −4.04436 −0.309280
$$172$$ 6.85328 0.522558
$$173$$ −23.8006 −1.80952 −0.904762 0.425917i $$-0.859952\pi$$
−0.904762 + 0.425917i $$0.859952\pi$$
$$174$$ −23.4641 −1.77881
$$175$$ 0 0
$$176$$ −4.16793 −0.314170
$$177$$ 16.3390 1.22812
$$178$$ 17.3482 1.30030
$$179$$ 20.0682 1.49997 0.749985 0.661455i $$-0.230061\pi$$
0.749985 + 0.661455i $$0.230061\pi$$
$$180$$ 0 0
$$181$$ 7.49807 0.557327 0.278663 0.960389i $$-0.410109\pi$$
0.278663 + 0.960389i $$0.410109\pi$$
$$182$$ −1.98146 −0.146875
$$183$$ −5.67188 −0.419278
$$184$$ −0.929396 −0.0685160
$$185$$ 0 0
$$186$$ −14.2779 −1.04691
$$187$$ −3.65839 −0.267528
$$188$$ 8.22663 0.599988
$$189$$ 2.95629 0.215039
$$190$$ 0 0
$$191$$ −4.21863 −0.305250 −0.152625 0.988284i $$-0.548773\pi$$
−0.152625 + 0.988284i $$0.548773\pi$$
$$192$$ 19.3291 1.39496
$$193$$ −7.70714 −0.554772 −0.277386 0.960759i $$-0.589468\pi$$
−0.277386 + 0.960759i $$0.589468\pi$$
$$194$$ 26.3895 1.89465
$$195$$ 0 0
$$196$$ −11.2101 −0.800725
$$197$$ −18.4285 −1.31298 −0.656489 0.754336i $$-0.727959\pi$$
−0.656489 + 0.754336i $$0.727959\pi$$
$$198$$ −7.99944 −0.568495
$$199$$ −21.6489 −1.53465 −0.767324 0.641260i $$-0.778412\pi$$
−0.767324 + 0.641260i $$0.778412\pi$$
$$200$$ 0 0
$$201$$ 13.0921 0.923446
$$202$$ 1.83212 0.128908
$$203$$ 4.76704 0.334581
$$204$$ 18.5669 1.29994
$$205$$ 0 0
$$206$$ −20.6385 −1.43795
$$207$$ −21.6388 −1.50400
$$208$$ 3.91490 0.271450
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 23.9164 1.64647 0.823235 0.567700i $$-0.192167\pi$$
0.823235 + 0.567700i $$0.192167\pi$$
$$212$$ 23.2473 1.59663
$$213$$ −19.4361 −1.33174
$$214$$ 26.8197 1.83336
$$215$$ 0 0
$$216$$ −0.481494 −0.0327615
$$217$$ 2.90075 0.196916
$$218$$ −11.5795 −0.784266
$$219$$ 3.03887 0.205348
$$220$$ 0 0
$$221$$ 3.43629 0.231150
$$222$$ −4.89294 −0.328392
$$223$$ −4.18886 −0.280506 −0.140253 0.990116i $$-0.544792\pi$$
−0.140253 + 0.990116i $$0.544792\pi$$
$$224$$ −8.42182 −0.562707
$$225$$ 0 0
$$226$$ 17.1819 1.14292
$$227$$ 8.74816 0.580636 0.290318 0.956930i $$-0.406239\pi$$
0.290318 + 0.956930i $$0.406239\pi$$
$$228$$ 5.07515 0.336110
$$229$$ −16.8146 −1.11114 −0.555571 0.831469i $$-0.687500\pi$$
−0.555571 + 0.831469i $$0.687500\pi$$
$$230$$ 0 0
$$231$$ 2.83071 0.186247
$$232$$ −0.776412 −0.0509740
$$233$$ 7.41167 0.485554 0.242777 0.970082i $$-0.421942\pi$$
0.242777 + 0.970082i $$0.421942\pi$$
$$234$$ 7.51380 0.491192
$$235$$ 0 0
$$236$$ −11.7716 −0.766263
$$237$$ 17.8109 1.15694
$$238$$ −7.71745 −0.500248
$$239$$ −19.7326 −1.27640 −0.638199 0.769871i $$-0.720320\pi$$
−0.638199 + 0.769871i $$0.720320\pi$$
$$240$$ 0 0
$$241$$ 15.0994 0.972640 0.486320 0.873781i $$-0.338339\pi$$
0.486320 + 0.873781i $$0.338339\pi$$
$$242$$ −1.97792 −0.127146
$$243$$ 20.9923 1.34665
$$244$$ 4.08634 0.261601
$$245$$ 0 0
$$246$$ −6.39493 −0.407726
$$247$$ 0.939291 0.0597657
$$248$$ −0.472448 −0.0300005
$$249$$ 7.50417 0.475558
$$250$$ 0 0
$$251$$ 12.3858 0.781782 0.390891 0.920437i $$-0.372167\pi$$
0.390891 + 0.920437i $$0.372167\pi$$
$$252$$ −8.24809 −0.519581
$$253$$ −5.35035 −0.336374
$$254$$ −31.5567 −1.98005
$$255$$ 0 0
$$256$$ 17.3113 1.08196
$$257$$ −26.5496 −1.65612 −0.828060 0.560639i $$-0.810555\pi$$
−0.828060 + 0.560639i $$0.810555\pi$$
$$258$$ 18.8148 1.17136
$$259$$ 0.994064 0.0617681
$$260$$ 0 0
$$261$$ −18.0769 −1.11893
$$262$$ −10.2855 −0.635438
$$263$$ −15.0108 −0.925604 −0.462802 0.886462i $$-0.653156\pi$$
−0.462802 + 0.886462i $$0.653156\pi$$
$$264$$ −0.461041 −0.0283751
$$265$$ 0 0
$$266$$ −2.10952 −0.129343
$$267$$ 23.2791 1.42466
$$268$$ −9.43228 −0.576168
$$269$$ 11.8307 0.721329 0.360664 0.932696i $$-0.382550\pi$$
0.360664 + 0.932696i $$0.382550\pi$$
$$270$$ 0 0
$$271$$ −13.4154 −0.814927 −0.407464 0.913221i $$-0.633587\pi$$
−0.407464 + 0.913221i $$0.633587\pi$$
$$272$$ 15.2479 0.924540
$$273$$ −2.65886 −0.160922
$$274$$ 14.9218 0.901461
$$275$$ 0 0
$$276$$ 27.1538 1.63447
$$277$$ 0.515992 0.0310030 0.0155015 0.999880i $$-0.495066\pi$$
0.0155015 + 0.999880i $$0.495066\pi$$
$$278$$ −0.706713 −0.0423859
$$279$$ −10.9998 −0.658542
$$280$$ 0 0
$$281$$ 21.9281 1.30812 0.654061 0.756441i $$-0.273064\pi$$
0.654061 + 0.756441i $$0.273064\pi$$
$$282$$ 22.5852 1.34493
$$283$$ −18.6023 −1.10579 −0.552896 0.833250i $$-0.686478\pi$$
−0.552896 + 0.833250i $$0.686478\pi$$
$$284$$ 14.0028 0.830916
$$285$$ 0 0
$$286$$ 1.85784 0.109857
$$287$$ 1.29921 0.0766902
$$288$$ 31.9360 1.88185
$$289$$ −3.61621 −0.212718
$$290$$ 0 0
$$291$$ 35.4114 2.07585
$$292$$ −2.18937 −0.128123
$$293$$ 5.87287 0.343097 0.171548 0.985176i $$-0.445123\pi$$
0.171548 + 0.985176i $$0.445123\pi$$
$$294$$ −30.7761 −1.79490
$$295$$ 0 0
$$296$$ −0.161904 −0.00941048
$$297$$ −2.77187 −0.160840
$$298$$ 21.2891 1.23324
$$299$$ 5.02554 0.290634
$$300$$ 0 0
$$301$$ −3.82248 −0.220324
$$302$$ −14.1219 −0.812623
$$303$$ 2.45848 0.141236
$$304$$ 4.16793 0.239047
$$305$$ 0 0
$$306$$ 29.2650 1.67297
$$307$$ −0.566566 −0.0323356 −0.0161678 0.999869i $$-0.505147\pi$$
−0.0161678 + 0.999869i $$0.505147\pi$$
$$308$$ −2.03940 −0.116206
$$309$$ −27.6943 −1.57547
$$310$$ 0 0
$$311$$ −22.1715 −1.25723 −0.628616 0.777716i $$-0.716378\pi$$
−0.628616 + 0.777716i $$0.716378\pi$$
$$312$$ 0.433051 0.0245167
$$313$$ 30.3230 1.71396 0.856979 0.515351i $$-0.172338\pi$$
0.856979 + 0.515351i $$0.172338\pi$$
$$314$$ 25.1521 1.41941
$$315$$ 0 0
$$316$$ −12.8320 −0.721856
$$317$$ 13.6662 0.767569 0.383785 0.923423i $$-0.374621\pi$$
0.383785 + 0.923423i $$0.374621\pi$$
$$318$$ 63.8225 3.57899
$$319$$ −4.46966 −0.250253
$$320$$ 0 0
$$321$$ 35.9887 2.00869
$$322$$ −11.2867 −0.628982
$$323$$ 3.65839 0.203558
$$324$$ −9.13297 −0.507387
$$325$$ 0 0
$$326$$ −8.80785 −0.487822
$$327$$ −15.5383 −0.859269
$$328$$ −0.211604 −0.0116839
$$329$$ −4.58848 −0.252971
$$330$$ 0 0
$$331$$ 1.04337 0.0573489 0.0286745 0.999589i $$-0.490871\pi$$
0.0286745 + 0.999589i $$0.490871\pi$$
$$332$$ −5.40642 −0.296716
$$333$$ −3.76955 −0.206570
$$334$$ 19.9595 1.09213
$$335$$ 0 0
$$336$$ −11.7982 −0.643646
$$337$$ 32.3936 1.76459 0.882296 0.470696i $$-0.155997\pi$$
0.882296 + 0.470696i $$0.155997\pi$$
$$338$$ 23.9679 1.30368
$$339$$ 23.0560 1.25223
$$340$$ 0 0
$$341$$ −2.71979 −0.147285
$$342$$ 7.99944 0.432560
$$343$$ 13.7183 0.740719
$$344$$ 0.622571 0.0335668
$$345$$ 0 0
$$346$$ 47.0757 2.53081
$$347$$ −0.837115 −0.0449387 −0.0224693 0.999748i $$-0.507153\pi$$
−0.0224693 + 0.999748i $$0.507153\pi$$
$$348$$ 22.6842 1.21600
$$349$$ −1.98491 −0.106250 −0.0531250 0.998588i $$-0.516918\pi$$
−0.0531250 + 0.998588i $$0.516918\pi$$
$$350$$ 0 0
$$351$$ 2.60359 0.138969
$$352$$ 7.89643 0.420881
$$353$$ −0.0707574 −0.00376604 −0.00188302 0.999998i $$-0.500599\pi$$
−0.00188302 + 0.999998i $$0.500599\pi$$
$$354$$ −32.3174 −1.71765
$$355$$ 0 0
$$356$$ −16.7715 −0.888889
$$357$$ −10.3558 −0.548089
$$358$$ −39.6934 −2.09786
$$359$$ −23.4992 −1.24024 −0.620120 0.784507i $$-0.712916\pi$$
−0.620120 + 0.784507i $$0.712916\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −14.8306 −0.779479
$$363$$ −2.65412 −0.139305
$$364$$ 1.91559 0.100404
$$365$$ 0 0
$$366$$ 11.2185 0.586403
$$367$$ 16.4763 0.860057 0.430029 0.902815i $$-0.358503\pi$$
0.430029 + 0.902815i $$0.358503\pi$$
$$368$$ 22.2999 1.16246
$$369$$ −4.92670 −0.256474
$$370$$ 0 0
$$371$$ −12.9664 −0.673181
$$372$$ 13.8033 0.715670
$$373$$ −1.92401 −0.0996217 −0.0498108 0.998759i $$-0.515862\pi$$
−0.0498108 + 0.998759i $$0.515862\pi$$
$$374$$ 7.23600 0.374165
$$375$$ 0 0
$$376$$ 0.747330 0.0385406
$$377$$ 4.19831 0.216224
$$378$$ −5.84732 −0.300753
$$379$$ −29.8162 −1.53156 −0.765779 0.643104i $$-0.777646\pi$$
−0.765779 + 0.643104i $$0.777646\pi$$
$$380$$ 0 0
$$381$$ −42.3452 −2.16941
$$382$$ 8.34413 0.426923
$$383$$ 19.4719 0.994969 0.497484 0.867473i $$-0.334257\pi$$
0.497484 + 0.867473i $$0.334257\pi$$
$$384$$ 3.68477 0.188038
$$385$$ 0 0
$$386$$ 15.2441 0.775906
$$387$$ 14.4951 0.736826
$$388$$ −25.5123 −1.29519
$$389$$ −11.7829 −0.597415 −0.298708 0.954345i $$-0.596556\pi$$
−0.298708 + 0.954345i $$0.596556\pi$$
$$390$$ 0 0
$$391$$ 19.5737 0.989882
$$392$$ −1.01836 −0.0514350
$$393$$ −13.8018 −0.696208
$$394$$ 36.4502 1.83633
$$395$$ 0 0
$$396$$ 7.73354 0.388625
$$397$$ −35.1177 −1.76251 −0.881254 0.472643i $$-0.843300\pi$$
−0.881254 + 0.472643i $$0.843300\pi$$
$$398$$ 42.8198 2.14636
$$399$$ −2.83071 −0.141713
$$400$$ 0 0
$$401$$ 27.0103 1.34883 0.674414 0.738353i $$-0.264396\pi$$
0.674414 + 0.738353i $$0.264396\pi$$
$$402$$ −25.8952 −1.29153
$$403$$ 2.55468 0.127257
$$404$$ −1.77122 −0.0881216
$$405$$ 0 0
$$406$$ −9.42884 −0.467946
$$407$$ −0.932050 −0.0462000
$$408$$ 1.68667 0.0835024
$$409$$ 33.0664 1.63503 0.817515 0.575907i $$-0.195351\pi$$
0.817515 + 0.575907i $$0.195351\pi$$
$$410$$ 0 0
$$411$$ 20.0232 0.987672
$$412$$ 19.9525 0.982989
$$413$$ 6.56570 0.323077
$$414$$ 42.7998 2.10349
$$415$$ 0 0
$$416$$ −7.41705 −0.363651
$$417$$ −0.948320 −0.0464394
$$418$$ 1.97792 0.0967433
$$419$$ 7.98503 0.390094 0.195047 0.980794i $$-0.437514\pi$$
0.195047 + 0.980794i $$0.437514\pi$$
$$420$$ 0 0
$$421$$ −28.8735 −1.40721 −0.703604 0.710592i $$-0.748427\pi$$
−0.703604 + 0.710592i $$0.748427\pi$$
$$422$$ −47.3047 −2.30276
$$423$$ 17.3998 0.846007
$$424$$ 2.11185 0.102560
$$425$$ 0 0
$$426$$ 38.4431 1.86257
$$427$$ −2.27919 −0.110298
$$428$$ −25.9283 −1.25329
$$429$$ 2.49299 0.120363
$$430$$ 0 0
$$431$$ 27.5088 1.32505 0.662527 0.749038i $$-0.269484\pi$$
0.662527 + 0.749038i $$0.269484\pi$$
$$432$$ 11.5530 0.555842
$$433$$ 33.5624 1.61291 0.806453 0.591299i $$-0.201385\pi$$
0.806453 + 0.591299i $$0.201385\pi$$
$$434$$ −5.73746 −0.275407
$$435$$ 0 0
$$436$$ 11.1946 0.536126
$$437$$ 5.35035 0.255942
$$438$$ −6.01065 −0.287200
$$439$$ 25.8608 1.23427 0.617134 0.786858i $$-0.288294\pi$$
0.617134 + 0.786858i $$0.288294\pi$$
$$440$$ 0 0
$$441$$ −23.7101 −1.12905
$$442$$ −6.79672 −0.323287
$$443$$ −2.34708 −0.111513 −0.0557566 0.998444i $$-0.517757\pi$$
−0.0557566 + 0.998444i $$0.517757\pi$$
$$444$$ 4.73030 0.224490
$$445$$ 0 0
$$446$$ 8.28523 0.392317
$$447$$ 28.5673 1.35119
$$448$$ 7.76722 0.366966
$$449$$ 18.4815 0.872195 0.436097 0.899899i $$-0.356360\pi$$
0.436097 + 0.899899i $$0.356360\pi$$
$$450$$ 0 0
$$451$$ −1.21816 −0.0573611
$$452$$ −16.6108 −0.781307
$$453$$ −18.9498 −0.890339
$$454$$ −17.3032 −0.812079
$$455$$ 0 0
$$456$$ 0.461041 0.0215902
$$457$$ −17.1365 −0.801612 −0.400806 0.916163i $$-0.631270\pi$$
−0.400806 + 0.916163i $$0.631270\pi$$
$$458$$ 33.2581 1.55405
$$459$$ 10.1406 0.473321
$$460$$ 0 0
$$461$$ −16.3205 −0.760122 −0.380061 0.924961i $$-0.624097\pi$$
−0.380061 + 0.924961i $$0.624097\pi$$
$$462$$ −5.59893 −0.260486
$$463$$ 18.2993 0.850441 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$464$$ 18.6292 0.864840
$$465$$ 0 0
$$466$$ −14.6597 −0.679098
$$467$$ −3.89977 −0.180460 −0.0902299 0.995921i $$-0.528760\pi$$
−0.0902299 + 0.995921i $$0.528760\pi$$
$$468$$ −7.26404 −0.335780
$$469$$ 5.26094 0.242928
$$470$$ 0 0
$$471$$ 33.7509 1.55516
$$472$$ −1.06936 −0.0492213
$$473$$ 3.58402 0.164793
$$474$$ −35.2286 −1.61811
$$475$$ 0 0
$$476$$ 7.46092 0.341971
$$477$$ 49.1693 2.25131
$$478$$ 39.0296 1.78517
$$479$$ 16.4447 0.751377 0.375688 0.926746i $$-0.377406\pi$$
0.375688 + 0.926746i $$0.377406\pi$$
$$480$$ 0 0
$$481$$ 0.875467 0.0399178
$$482$$ −29.8655 −1.36034
$$483$$ −15.1453 −0.689135
$$484$$ 1.91218 0.0869171
$$485$$ 0 0
$$486$$ −41.5211 −1.88343
$$487$$ 28.4329 1.28842 0.644209 0.764849i $$-0.277187\pi$$
0.644209 + 0.764849i $$0.277187\pi$$
$$488$$ 0.371214 0.0168041
$$489$$ −11.8190 −0.534475
$$490$$ 0 0
$$491$$ −6.25955 −0.282490 −0.141245 0.989975i $$-0.545110\pi$$
−0.141245 + 0.989975i $$0.545110\pi$$
$$492$$ 6.18237 0.278723
$$493$$ 16.3517 0.736445
$$494$$ −1.85784 −0.0835884
$$495$$ 0 0
$$496$$ 11.3359 0.508997
$$497$$ −7.81022 −0.350336
$$498$$ −14.8427 −0.665116
$$499$$ −10.4851 −0.469378 −0.234689 0.972070i $$-0.575407\pi$$
−0.234689 + 0.972070i $$0.575407\pi$$
$$500$$ 0 0
$$501$$ 26.7831 1.19658
$$502$$ −24.4981 −1.09340
$$503$$ −18.4277 −0.821651 −0.410825 0.911714i $$-0.634759\pi$$
−0.410825 + 0.911714i $$0.634759\pi$$
$$504$$ −0.749279 −0.0333755
$$505$$ 0 0
$$506$$ 10.5826 0.470453
$$507$$ 32.1619 1.42836
$$508$$ 30.5078 1.35357
$$509$$ 14.0013 0.620596 0.310298 0.950639i $$-0.399571\pi$$
0.310298 + 0.950639i $$0.399571\pi$$
$$510$$ 0 0
$$511$$ 1.22114 0.0540201
$$512$$ −31.4638 −1.39052
$$513$$ 2.77187 0.122381
$$514$$ 52.5131 2.31625
$$515$$ 0 0
$$516$$ −18.1894 −0.800745
$$517$$ 4.30223 0.189212
$$518$$ −1.96618 −0.0863891
$$519$$ 63.1696 2.77284
$$520$$ 0 0
$$521$$ −36.3013 −1.59039 −0.795195 0.606353i $$-0.792632\pi$$
−0.795195 + 0.606353i $$0.792632\pi$$
$$522$$ 35.7547 1.56494
$$523$$ −23.2657 −1.01734 −0.508669 0.860962i $$-0.669862\pi$$
−0.508669 + 0.860962i $$0.669862\pi$$
$$524$$ 9.94358 0.434387
$$525$$ 0 0
$$526$$ 29.6901 1.29455
$$527$$ 9.95005 0.433431
$$528$$ 11.0622 0.481421
$$529$$ 5.62625 0.244620
$$530$$ 0 0
$$531$$ −24.8975 −1.08046
$$532$$ 2.03940 0.0884193
$$533$$ 1.14421 0.0495613
$$534$$ −46.0442 −1.99253
$$535$$ 0 0
$$536$$ −0.856854 −0.0370105
$$537$$ −53.2635 −2.29849
$$538$$ −23.4001 −1.00885
$$539$$ −5.86250 −0.252516
$$540$$ 0 0
$$541$$ 30.6302 1.31689 0.658447 0.752627i $$-0.271214\pi$$
0.658447 + 0.752627i $$0.271214\pi$$
$$542$$ 26.5346 1.13976
$$543$$ −19.9008 −0.854024
$$544$$ −28.8882 −1.23857
$$545$$ 0 0
$$546$$ 5.25903 0.225066
$$547$$ 14.9225 0.638040 0.319020 0.947748i $$-0.396646\pi$$
0.319020 + 0.947748i $$0.396646\pi$$
$$548$$ −14.4258 −0.616241
$$549$$ 8.64284 0.368867
$$550$$ 0 0
$$551$$ 4.46966 0.190414
$$552$$ 2.46673 0.104991
$$553$$ 7.15716 0.304354
$$554$$ −1.02059 −0.0433608
$$555$$ 0 0
$$556$$ 0.683222 0.0289751
$$557$$ −8.54485 −0.362057 −0.181028 0.983478i $$-0.557943\pi$$
−0.181028 + 0.983478i $$0.557943\pi$$
$$558$$ 21.7568 0.921038
$$559$$ −3.36644 −0.142385
$$560$$ 0 0
$$561$$ 9.70980 0.409948
$$562$$ −43.3721 −1.82954
$$563$$ −17.7855 −0.749571 −0.374786 0.927111i $$-0.622284\pi$$
−0.374786 + 0.927111i $$0.622284\pi$$
$$564$$ −21.8345 −0.919397
$$565$$ 0 0
$$566$$ 36.7939 1.54656
$$567$$ 5.09400 0.213928
$$568$$ 1.27206 0.0533743
$$569$$ −29.7657 −1.24785 −0.623923 0.781486i $$-0.714462\pi$$
−0.623923 + 0.781486i $$0.714462\pi$$
$$570$$ 0 0
$$571$$ 22.2184 0.929810 0.464905 0.885360i $$-0.346088\pi$$
0.464905 + 0.885360i $$0.346088\pi$$
$$572$$ −1.79609 −0.0750983
$$573$$ 11.1968 0.467752
$$574$$ −2.56975 −0.107259
$$575$$ 0 0
$$576$$ −29.4537 −1.22724
$$577$$ 12.0936 0.503464 0.251732 0.967797i $$-0.419000\pi$$
0.251732 + 0.967797i $$0.419000\pi$$
$$578$$ 7.15258 0.297508
$$579$$ 20.4557 0.850110
$$580$$ 0 0
$$581$$ 3.01548 0.125103
$$582$$ −70.0409 −2.90329
$$583$$ 12.1575 0.503512
$$584$$ −0.198888 −0.00823006
$$585$$ 0 0
$$586$$ −11.6161 −0.479856
$$587$$ 14.3554 0.592510 0.296255 0.955109i $$-0.404262\pi$$
0.296255 + 0.955109i $$0.404262\pi$$
$$588$$ 29.7531 1.22700
$$589$$ 2.71979 0.112067
$$590$$ 0 0
$$591$$ 48.9115 2.01195
$$592$$ 3.88472 0.159661
$$593$$ −7.64570 −0.313971 −0.156986 0.987601i $$-0.550178\pi$$
−0.156986 + 0.987601i $$0.550178\pi$$
$$594$$ 5.48254 0.224951
$$595$$ 0 0
$$596$$ −20.5814 −0.843049
$$597$$ 57.4587 2.35163
$$598$$ −9.94012 −0.406482
$$599$$ 22.5450 0.921165 0.460583 0.887617i $$-0.347641\pi$$
0.460583 + 0.887617i $$0.347641\pi$$
$$600$$ 0 0
$$601$$ 8.27574 0.337574 0.168787 0.985653i $$-0.446015\pi$$
0.168787 + 0.985653i $$0.446015\pi$$
$$602$$ 7.56057 0.308146
$$603$$ −19.9498 −0.812419
$$604$$ 13.6525 0.555511
$$605$$ 0 0
$$606$$ −4.86267 −0.197533
$$607$$ 12.8552 0.521778 0.260889 0.965369i $$-0.415984\pi$$
0.260889 + 0.965369i $$0.415984\pi$$
$$608$$ −7.89643 −0.320243
$$609$$ −12.6523 −0.512698
$$610$$ 0 0
$$611$$ −4.04105 −0.163483
$$612$$ −28.2923 −1.14365
$$613$$ 17.0191 0.687394 0.343697 0.939081i $$-0.388321\pi$$
0.343697 + 0.939081i $$0.388321\pi$$
$$614$$ 1.12062 0.0452247
$$615$$ 0 0
$$616$$ −0.185265 −0.00746454
$$617$$ −21.5265 −0.866625 −0.433313 0.901244i $$-0.642655\pi$$
−0.433313 + 0.901244i $$0.642655\pi$$
$$618$$ 54.7771 2.20346
$$619$$ 34.9987 1.40672 0.703359 0.710835i $$-0.251683\pi$$
0.703359 + 0.710835i $$0.251683\pi$$
$$620$$ 0 0
$$621$$ 14.8305 0.595126
$$622$$ 43.8536 1.75837
$$623$$ 9.35448 0.374779
$$624$$ −10.3906 −0.415958
$$625$$ 0 0
$$626$$ −59.9766 −2.39715
$$627$$ 2.65412 0.105995
$$628$$ −24.3160 −0.970315
$$629$$ 3.40980 0.135958
$$630$$ 0 0
$$631$$ 26.6365 1.06038 0.530191 0.847878i $$-0.322120\pi$$
0.530191 + 0.847878i $$0.322120\pi$$
$$632$$ −1.16569 −0.0463688
$$633$$ −63.4769 −2.52298
$$634$$ −27.0306 −1.07352
$$635$$ 0 0
$$636$$ −61.7011 −2.44661
$$637$$ 5.50660 0.218179
$$638$$ 8.84063 0.350004
$$639$$ 29.6168 1.17162
$$640$$ 0 0
$$641$$ −18.3904 −0.726376 −0.363188 0.931716i $$-0.618312\pi$$
−0.363188 + 0.931716i $$0.618312\pi$$
$$642$$ −71.1828 −2.80936
$$643$$ −10.1583 −0.400603 −0.200302 0.979734i $$-0.564192\pi$$
−0.200302 + 0.979734i $$0.564192\pi$$
$$644$$ 10.9115 0.429974
$$645$$ 0 0
$$646$$ −7.23600 −0.284697
$$647$$ −24.9283 −0.980034 −0.490017 0.871713i $$-0.663009\pi$$
−0.490017 + 0.871713i $$0.663009\pi$$
$$648$$ −0.829664 −0.0325923
$$649$$ −6.15610 −0.241648
$$650$$ 0 0
$$651$$ −7.69895 −0.301746
$$652$$ 8.51508 0.333476
$$653$$ −3.18346 −0.124579 −0.0622893 0.998058i $$-0.519840\pi$$
−0.0622893 + 0.998058i $$0.519840\pi$$
$$654$$ 30.7335 1.20178
$$655$$ 0 0
$$656$$ 5.07723 0.198232
$$657$$ −4.63064 −0.180659
$$658$$ 9.07566 0.353806
$$659$$ 44.5078 1.73378 0.866888 0.498503i $$-0.166117\pi$$
0.866888 + 0.498503i $$0.166117\pi$$
$$660$$ 0 0
$$661$$ 5.31054 0.206556 0.103278 0.994653i $$-0.467067\pi$$
0.103278 + 0.994653i $$0.467067\pi$$
$$662$$ −2.06371 −0.0802083
$$663$$ −9.12033 −0.354204
$$664$$ −0.491134 −0.0190597
$$665$$ 0 0
$$666$$ 7.45588 0.288909
$$667$$ 23.9142 0.925963
$$668$$ −19.2960 −0.746585
$$669$$ 11.1177 0.429836
$$670$$ 0 0
$$671$$ 2.13701 0.0824983
$$672$$ 22.3525 0.862268
$$673$$ 27.9535 1.07753 0.538765 0.842456i $$-0.318891\pi$$
0.538765 + 0.842456i $$0.318891\pi$$
$$674$$ −64.0720 −2.46796
$$675$$ 0 0
$$676$$ −23.1712 −0.891202
$$677$$ 41.9276 1.61141 0.805704 0.592319i $$-0.201787\pi$$
0.805704 + 0.592319i $$0.201787\pi$$
$$678$$ −45.6029 −1.75137
$$679$$ 14.2297 0.546087
$$680$$ 0 0
$$681$$ −23.2187 −0.889742
$$682$$ 5.37953 0.205993
$$683$$ −21.6527 −0.828516 −0.414258 0.910159i $$-0.635959\pi$$
−0.414258 + 0.910159i $$0.635959\pi$$
$$684$$ −7.73354 −0.295699
$$685$$ 0 0
$$686$$ −27.1337 −1.03597
$$687$$ 44.6281 1.70267
$$688$$ −14.9380 −0.569504
$$689$$ −11.4194 −0.435045
$$690$$ 0 0
$$691$$ 45.5121 1.73136 0.865681 0.500597i $$-0.166886\pi$$
0.865681 + 0.500597i $$0.166886\pi$$
$$692$$ −45.5109 −1.73007
$$693$$ −4.31345 −0.163855
$$694$$ 1.65575 0.0628514
$$695$$ 0 0
$$696$$ 2.06069 0.0781104
$$697$$ 4.45652 0.168803
$$698$$ 3.92601 0.148602
$$699$$ −19.6715 −0.744043
$$700$$ 0 0
$$701$$ −27.3430 −1.03273 −0.516365 0.856369i $$-0.672715\pi$$
−0.516365 + 0.856369i $$0.672715\pi$$
$$702$$ −5.14970 −0.194363
$$703$$ 0.932050 0.0351529
$$704$$ −7.28267 −0.274476
$$705$$ 0 0
$$706$$ 0.139953 0.00526719
$$707$$ 0.987916 0.0371544
$$708$$ 31.2431 1.17419
$$709$$ 3.38760 0.127224 0.0636120 0.997975i $$-0.479738\pi$$
0.0636120 + 0.997975i $$0.479738\pi$$
$$710$$ 0 0
$$711$$ −27.1404 −1.01784
$$712$$ −1.52357 −0.0570983
$$713$$ 14.5518 0.544970
$$714$$ 20.4831 0.766559
$$715$$ 0 0
$$716$$ 38.3740 1.43410
$$717$$ 52.3728 1.95590
$$718$$ 46.4796 1.73460
$$719$$ 33.5021 1.24942 0.624709 0.780858i $$-0.285218\pi$$
0.624709 + 0.780858i $$0.285218\pi$$
$$720$$ 0 0
$$721$$ −11.1287 −0.414454
$$722$$ −1.97792 −0.0736106
$$723$$ −40.0757 −1.49043
$$724$$ 14.3376 0.532854
$$725$$ 0 0
$$726$$ 5.24965 0.194833
$$727$$ 45.5324 1.68871 0.844353 0.535788i $$-0.179985\pi$$
0.844353 + 0.535788i $$0.179985\pi$$
$$728$$ 0.174018 0.00644953
$$729$$ −41.3874 −1.53287
$$730$$ 0 0
$$731$$ −13.1117 −0.484955
$$732$$ −10.8456 −0.400866
$$733$$ 4.71433 0.174128 0.0870640 0.996203i $$-0.472252\pi$$
0.0870640 + 0.996203i $$0.472252\pi$$
$$734$$ −32.5889 −1.20288
$$735$$ 0 0
$$736$$ −42.2487 −1.55731
$$737$$ −4.93275 −0.181700
$$738$$ 9.74463 0.358705
$$739$$ 35.7265 1.31422 0.657110 0.753795i $$-0.271779\pi$$
0.657110 + 0.753795i $$0.271779\pi$$
$$740$$ 0 0
$$741$$ −2.49299 −0.0915824
$$742$$ 25.6465 0.941513
$$743$$ 26.9341 0.988118 0.494059 0.869428i $$-0.335513\pi$$
0.494059 + 0.869428i $$0.335513\pi$$
$$744$$ 1.25393 0.0459715
$$745$$ 0 0
$$746$$ 3.80555 0.139331
$$747$$ −11.4349 −0.418381
$$748$$ −6.99548 −0.255780
$$749$$ 14.4617 0.528420
$$750$$ 0 0
$$751$$ −27.7767 −1.01358 −0.506792 0.862068i $$-0.669169\pi$$
−0.506792 + 0.862068i $$0.669169\pi$$
$$752$$ −17.9314 −0.653891
$$753$$ −32.8733 −1.19797
$$754$$ −8.30393 −0.302411
$$755$$ 0 0
$$756$$ 5.65295 0.205596
$$757$$ 23.4521 0.852380 0.426190 0.904634i $$-0.359855\pi$$
0.426190 + 0.904634i $$0.359855\pi$$
$$758$$ 58.9742 2.14204
$$759$$ 14.2005 0.515445
$$760$$ 0 0
$$761$$ −15.9940 −0.579783 −0.289892 0.957059i $$-0.593619\pi$$
−0.289892 + 0.957059i $$0.593619\pi$$
$$762$$ 83.7554 3.03414
$$763$$ −6.24392 −0.226045
$$764$$ −8.06678 −0.291846
$$765$$ 0 0
$$766$$ −38.5139 −1.39157
$$767$$ 5.78237 0.208789
$$768$$ −45.9464 −1.65795
$$769$$ 19.9815 0.720552 0.360276 0.932846i $$-0.382683\pi$$
0.360276 + 0.932846i $$0.382683\pi$$
$$770$$ 0 0
$$771$$ 70.4659 2.53777
$$772$$ −14.7374 −0.530411
$$773$$ −53.4366 −1.92198 −0.960991 0.276581i $$-0.910799\pi$$
−0.960991 + 0.276581i $$0.910799\pi$$
$$774$$ −28.6701 −1.03053
$$775$$ 0 0
$$776$$ −2.31761 −0.0831973
$$777$$ −2.63837 −0.0946509
$$778$$ 23.3056 0.835546
$$779$$ 1.21816 0.0436453
$$780$$ 0 0
$$781$$ 7.32299 0.262037
$$782$$ −38.7152 −1.38445
$$783$$ 12.3893 0.442757
$$784$$ 24.4345 0.872662
$$785$$ 0 0
$$786$$ 27.2989 0.973719
$$787$$ −6.90402 −0.246102 −0.123051 0.992400i $$-0.539268\pi$$
−0.123051 + 0.992400i $$0.539268\pi$$
$$788$$ −35.2386 −1.25532
$$789$$ 39.8404 1.41836
$$790$$ 0 0
$$791$$ 9.26484 0.329420
$$792$$ 0.702536 0.0249635
$$793$$ −2.00727 −0.0712804
$$794$$ 69.4601 2.46505
$$795$$ 0 0
$$796$$ −41.3965 −1.46726
$$797$$ −28.5820 −1.01243 −0.506213 0.862409i $$-0.668955\pi$$
−0.506213 + 0.862409i $$0.668955\pi$$
$$798$$ 5.59893 0.198200
$$799$$ −15.7392 −0.556814
$$800$$ 0 0
$$801$$ −35.4727 −1.25337
$$802$$ −53.4242 −1.88647
$$803$$ −1.14496 −0.0404048
$$804$$ 25.0344 0.882896
$$805$$ 0 0
$$806$$ −5.05295 −0.177983
$$807$$ −31.4000 −1.10533
$$808$$ −0.160903 −0.00566054
$$809$$ 45.3303 1.59373 0.796864 0.604158i $$-0.206491\pi$$
0.796864 + 0.604158i $$0.206491\pi$$
$$810$$ 0 0
$$811$$ 11.1668 0.392119 0.196060 0.980592i $$-0.437185\pi$$
0.196060 + 0.980592i $$0.437185\pi$$
$$812$$ 9.11543 0.319889
$$813$$ 35.6061 1.24876
$$814$$ 1.84352 0.0646155
$$815$$ 0 0
$$816$$ −40.4698 −1.41673
$$817$$ −3.58402 −0.125389
$$818$$ −65.4029 −2.28676
$$819$$ 4.05159 0.141574
$$820$$ 0 0
$$821$$ −23.2715 −0.812180 −0.406090 0.913833i $$-0.633108\pi$$
−0.406090 + 0.913833i $$0.633108\pi$$
$$822$$ −39.6044 −1.38136
$$823$$ 36.1529 1.26021 0.630105 0.776510i $$-0.283012\pi$$
0.630105 + 0.776510i $$0.283012\pi$$
$$824$$ 1.81254 0.0631428
$$825$$ 0 0
$$826$$ −12.9864 −0.451856
$$827$$ −10.7797 −0.374848 −0.187424 0.982279i $$-0.560014\pi$$
−0.187424 + 0.982279i $$0.560014\pi$$
$$828$$ −41.3771 −1.43795
$$829$$ −42.2859 −1.46865 −0.734326 0.678797i $$-0.762501\pi$$
−0.734326 + 0.678797i $$0.762501\pi$$
$$830$$ 0 0
$$831$$ −1.36951 −0.0475076
$$832$$ 6.84054 0.237153
$$833$$ 21.4473 0.743105
$$834$$ 1.87570 0.0649503
$$835$$ 0 0
$$836$$ −1.91218 −0.0661340
$$837$$ 7.53890 0.260582
$$838$$ −15.7938 −0.545587
$$839$$ 27.2267 0.939971 0.469986 0.882674i $$-0.344259\pi$$
0.469986 + 0.882674i $$0.344259\pi$$
$$840$$ 0 0
$$841$$ −9.02217 −0.311109
$$842$$ 57.1095 1.96812
$$843$$ −58.1999 −2.00451
$$844$$ 45.7323 1.57417
$$845$$ 0 0
$$846$$ −34.4154 −1.18323
$$847$$ −1.06653 −0.0366466
$$848$$ −50.6716 −1.74007
$$849$$ 49.3728 1.69447
$$850$$ 0 0
$$851$$ 4.98680 0.170945
$$852$$ −37.1653 −1.27326
$$853$$ 11.3022 0.386981 0.193490 0.981102i $$-0.438019\pi$$
0.193490 + 0.981102i $$0.438019\pi$$
$$854$$ 4.50807 0.154263
$$855$$ 0 0
$$856$$ −2.35539 −0.0805057
$$857$$ −1.20558 −0.0411817 −0.0205908 0.999788i $$-0.506555\pi$$
−0.0205908 + 0.999788i $$0.506555\pi$$
$$858$$ −4.93095 −0.168340
$$859$$ 28.1617 0.960865 0.480432 0.877032i $$-0.340480\pi$$
0.480432 + 0.877032i $$0.340480\pi$$
$$860$$ 0 0
$$861$$ −3.44827 −0.117517
$$862$$ −54.4103 −1.85322
$$863$$ −8.48764 −0.288923 −0.144461 0.989510i $$-0.546145\pi$$
−0.144461 + 0.989510i $$0.546145\pi$$
$$864$$ −21.8879 −0.744640
$$865$$ 0 0
$$866$$ −66.3838 −2.25581
$$867$$ 9.59785 0.325960
$$868$$ 5.54675 0.188269
$$869$$ −6.71067 −0.227644
$$870$$ 0 0
$$871$$ 4.63328 0.156993
$$872$$ 1.01695 0.0344383
$$873$$ −53.9600 −1.82627
$$874$$ −10.5826 −0.357961
$$875$$ 0 0
$$876$$ 5.81086 0.196331
$$877$$ −34.1439 −1.15296 −0.576478 0.817112i $$-0.695573\pi$$
−0.576478 + 0.817112i $$0.695573\pi$$
$$878$$ −51.1506 −1.72625
$$879$$ −15.5873 −0.525747
$$880$$ 0 0
$$881$$ −42.0557 −1.41689 −0.708446 0.705765i $$-0.750603\pi$$
−0.708446 + 0.705765i $$0.750603\pi$$
$$882$$ 46.8967 1.57909
$$883$$ 12.9984 0.437430 0.218715 0.975789i $$-0.429813\pi$$
0.218715 + 0.975789i $$0.429813\pi$$
$$884$$ 6.57079 0.221000
$$885$$ 0 0
$$886$$ 4.64235 0.155963
$$887$$ −58.3513 −1.95925 −0.979623 0.200847i $$-0.935631\pi$$
−0.979623 + 0.200847i $$0.935631\pi$$
$$888$$ 0.429713 0.0144202
$$889$$ −17.0160 −0.570699
$$890$$ 0 0
$$891$$ −4.77622 −0.160009
$$892$$ −8.00983 −0.268189
$$893$$ −4.30223 −0.143969
$$894$$ −56.5038 −1.88977
$$895$$ 0 0
$$896$$ 1.48069 0.0494664
$$897$$ −13.3384 −0.445356
$$898$$ −36.5549 −1.21985
$$899$$ 12.1565 0.405443
$$900$$ 0 0
$$901$$ −44.4768 −1.48174
$$902$$ 2.40943 0.0802254
$$903$$ 10.1453 0.337615
$$904$$ −1.50897 −0.0501876
$$905$$ 0 0
$$906$$ 37.4812 1.24523
$$907$$ 58.8388 1.95371 0.976855 0.213904i $$-0.0686181\pi$$
0.976855 + 0.213904i $$0.0686181\pi$$
$$908$$ 16.7280 0.555139
$$909$$ −3.74624 −0.124255
$$910$$ 0 0
$$911$$ −23.1241 −0.766137 −0.383068 0.923720i $$-0.625133\pi$$
−0.383068 + 0.923720i $$0.625133\pi$$
$$912$$ −11.0622 −0.366306
$$913$$ −2.82737 −0.0935721
$$914$$ 33.8947 1.12114
$$915$$ 0 0
$$916$$ −32.1526 −1.06235
$$917$$ −5.54613 −0.183149
$$918$$ −20.0572 −0.661988
$$919$$ −32.2210 −1.06287 −0.531437 0.847098i $$-0.678348\pi$$
−0.531437 + 0.847098i $$0.678348\pi$$
$$920$$ 0 0
$$921$$ 1.50374 0.0495498
$$922$$ 32.2807 1.06311
$$923$$ −6.87842 −0.226406
$$924$$ 5.41282 0.178069
$$925$$ 0 0
$$926$$ −36.1946 −1.18943
$$927$$ 42.2007 1.38605
$$928$$ −35.2943 −1.15859
$$929$$ −0.527392 −0.0173032 −0.00865158 0.999963i $$-0.502754\pi$$
−0.00865158 + 0.999963i $$0.502754\pi$$
$$930$$ 0 0
$$931$$ 5.86250 0.192136
$$932$$ 14.1724 0.464233
$$933$$ 58.8459 1.92653
$$934$$ 7.71344 0.252392
$$935$$ 0 0
$$936$$ −0.659886 −0.0215690
$$937$$ 41.3338 1.35032 0.675158 0.737673i $$-0.264075\pi$$
0.675158 + 0.737673i $$0.264075\pi$$
$$938$$ −10.4057 −0.339759
$$939$$ −80.4810 −2.62640
$$940$$ 0 0
$$941$$ 25.4253 0.828840 0.414420 0.910086i $$-0.363984\pi$$
0.414420 + 0.910086i $$0.363984\pi$$
$$942$$ −66.7567 −2.17505
$$943$$ 6.51761 0.212242
$$944$$ 25.6582 0.835104
$$945$$ 0 0
$$946$$ −7.08891 −0.230480
$$947$$ 13.9372 0.452897 0.226448 0.974023i $$-0.427289\pi$$
0.226448 + 0.974023i $$0.427289\pi$$
$$948$$ 34.0577 1.10614
$$949$$ 1.07545 0.0349107
$$950$$ 0 0
$$951$$ −36.2717 −1.17619
$$952$$ 0.677771 0.0219667
$$953$$ −40.7458 −1.31989 −0.659943 0.751316i $$-0.729420\pi$$
−0.659943 + 0.751316i $$0.729420\pi$$
$$954$$ −97.2531 −3.14868
$$955$$ 0 0
$$956$$ −37.7323 −1.22035
$$957$$ 11.8630 0.383477
$$958$$ −32.5263 −1.05088
$$959$$ 8.04614 0.259824
$$960$$ 0 0
$$961$$ −23.6027 −0.761379
$$962$$ −1.73160 −0.0558292
$$963$$ −54.8397 −1.76719
$$964$$ 28.8728 0.929930
$$965$$ 0 0
$$966$$ 29.9562 0.963826
$$967$$ 21.5806 0.693985 0.346993 0.937868i $$-0.387203\pi$$
0.346993 + 0.937868i $$0.387203\pi$$
$$968$$ 0.173707 0.00558317
$$969$$ −9.70980 −0.311924
$$970$$ 0 0
$$971$$ −40.5213 −1.30039 −0.650194 0.759768i $$-0.725313\pi$$
−0.650194 + 0.759768i $$0.725313\pi$$
$$972$$ 40.1409 1.28752
$$973$$ −0.381074 −0.0122167
$$974$$ −56.2381 −1.80199
$$975$$ 0 0
$$976$$ −8.90691 −0.285103
$$977$$ 51.7112 1.65439 0.827194 0.561917i $$-0.189936\pi$$
0.827194 + 0.561917i $$0.189936\pi$$
$$978$$ 23.3771 0.747518
$$979$$ −8.77091 −0.280319
$$980$$ 0 0
$$981$$ 23.6773 0.755958
$$982$$ 12.3809 0.395090
$$983$$ −39.5850 −1.26256 −0.631282 0.775553i $$-0.717471\pi$$
−0.631282 + 0.775553i $$0.717471\pi$$
$$984$$ 0.561623 0.0179039
$$985$$ 0 0
$$986$$ −32.3425 −1.02999
$$987$$ 12.1784 0.387642
$$988$$ 1.79609 0.0571413
$$989$$ −19.1758 −0.609754
$$990$$ 0 0
$$991$$ 42.2901 1.34339 0.671694 0.740828i $$-0.265567\pi$$
0.671694 + 0.740828i $$0.265567\pi$$
$$992$$ −21.4766 −0.681884
$$993$$ −2.76924 −0.0878791
$$994$$ 15.4480 0.489981
$$995$$ 0 0
$$996$$ 14.3493 0.454675
$$997$$ 40.6774 1.28827 0.644133 0.764914i $$-0.277218\pi$$
0.644133 + 0.764914i $$0.277218\pi$$
$$998$$ 20.7387 0.656473
$$999$$ 2.58352 0.0817389
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 5225.2.a.m.1.2 7
5.4 even 2 1045.2.a.h.1.6 7
15.14 odd 2 9405.2.a.bd.1.2 7

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.h.1.6 7 5.4 even 2
5225.2.a.m.1.2 7 1.1 even 1 trivial
9405.2.a.bd.1.2 7 15.14 odd 2