# Properties

 Label 275.4.a.b.1.1 Level $275$ Weight $4$ Character 275.1 Self dual yes Analytic conductor $16.226$ 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] = [275,4,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, 4, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$16.2255252516$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ 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)

 $$f(q)$$ $$=$$ $$q-2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +O(q^{10})$$ $$q-2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} -11.0000 q^{11} -4.24871 q^{12} -5.35898 q^{13} +8.39230 q^{14} -59.4256 q^{16} +41.2154 q^{17} -97.9615 q^{18} +139.923 q^{19} -24.3538 q^{21} +30.0526 q^{22} +111.354 q^{23} +184.890 q^{24} +14.6410 q^{26} +70.2154 q^{27} +1.64617 q^{28} -24.9948 q^{29} +31.4974 q^{31} -24.2102 q^{32} -87.2102 q^{33} -112.603 q^{34} -19.2154 q^{36} -13.1436 q^{37} -382.277 q^{38} -42.4871 q^{39} +261.072 q^{41} +66.5359 q^{42} +57.7128 q^{43} +5.89488 q^{44} -304.224 q^{46} +343.846 q^{47} -471.138 q^{48} -333.564 q^{49} +326.764 q^{51} +2.87187 q^{52} +342.995 q^{53} -191.832 q^{54} -71.6359 q^{56} +1109.34 q^{57} +68.2872 q^{58} +88.3693 q^{59} +738.697 q^{61} -86.0526 q^{62} -110.144 q^{63} +541.549 q^{64} +238.263 q^{66} -342.359 q^{67} -22.0873 q^{68} +882.836 q^{69} -207.364 q^{71} +836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} -74.9845 q^{76} +33.7898 q^{77} +116.077 q^{78} +1294.23 q^{79} -411.441 q^{81} -713.261 q^{82} -441.846 q^{83} +13.0512 q^{84} -157.674 q^{86} -198.164 q^{87} -256.526 q^{88} -1489.11 q^{89} +16.4617 q^{91} -59.6743 q^{92} +249.718 q^{93} -939.405 q^{94} -191.944 q^{96} -1346.42 q^{97} +911.314 q^{98} -394.420 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 $$2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 4 q^{14} - 8 q^{16} + 124 q^{17} - 92 q^{18} + 72 q^{19} + 76 q^{21} + 22 q^{22} + 98 q^{23} + 252 q^{24} - 40 q^{26} + 182 q^{27} + 128 q^{28} + 144 q^{29} - 34 q^{31} + 104 q^{32} - 22 q^{33} - 52 q^{34} - 80 q^{36} - 54 q^{37} - 432 q^{38} + 400 q^{39} + 536 q^{41} + 140 q^{42} + 60 q^{43} + 88 q^{44} - 314 q^{46} + 272 q^{47} - 776 q^{48} - 390 q^{49} - 164 q^{51} + 560 q^{52} + 492 q^{53} - 110 q^{54} + 120 q^{56} + 1512 q^{57} + 192 q^{58} + 634 q^{59} + 840 q^{61} - 134 q^{62} - 248 q^{63} + 224 q^{64} + 286 q^{66} - 754 q^{67} - 640 q^{68} + 962 q^{69} - 678 q^{71} + 744 q^{72} + 400 q^{73} + 6 q^{74} + 432 q^{76} + 220 q^{77} + 440 q^{78} + 316 q^{79} - 1294 q^{81} - 512 q^{82} - 468 q^{83} - 736 q^{84} - 156 q^{86} - 1200 q^{87} - 132 q^{88} - 1842 q^{89} + 1280 q^{91} + 40 q^{92} + 638 q^{93} - 992 q^{94} - 952 q^{96} - 2194 q^{97} + 870 q^{98} - 484 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 - 22 * q^11 + 40 * q^12 - 80 * q^13 - 4 * q^14 - 8 * q^16 + 124 * q^17 - 92 * q^18 + 72 * q^19 + 76 * q^21 + 22 * q^22 + 98 * q^23 + 252 * q^24 - 40 * q^26 + 182 * q^27 + 128 * q^28 + 144 * q^29 - 34 * q^31 + 104 * q^32 - 22 * q^33 - 52 * q^34 - 80 * q^36 - 54 * q^37 - 432 * q^38 + 400 * q^39 + 536 * q^41 + 140 * q^42 + 60 * q^43 + 88 * q^44 - 314 * q^46 + 272 * q^47 - 776 * q^48 - 390 * q^49 - 164 * q^51 + 560 * q^52 + 492 * q^53 - 110 * q^54 + 120 * q^56 + 1512 * q^57 + 192 * q^58 + 634 * q^59 + 840 * q^61 - 134 * q^62 - 248 * q^63 + 224 * q^64 + 286 * q^66 - 754 * q^67 - 640 * q^68 + 962 * q^69 - 678 * q^71 + 744 * q^72 + 400 * q^73 + 6 * q^74 + 432 * q^76 + 220 * q^77 + 440 * q^78 + 316 * q^79 - 1294 * q^81 - 512 * q^82 - 468 * q^83 - 736 * q^84 - 156 * q^86 - 1200 * q^87 - 132 * q^88 - 1842 * q^89 + 1280 * q^91 + 40 * q^92 + 638 * q^93 - 992 * q^94 - 952 * q^96 - 2194 * q^97 + 870 * q^98 - 484 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.73205 −0.965926 −0.482963 0.875641i $$-0.660439\pi$$
−0.482963 + 0.875641i $$0.660439\pi$$
$$3$$ 7.92820 1.52578 0.762892 0.646526i $$-0.223779\pi$$
0.762892 + 0.646526i $$0.223779\pi$$
$$4$$ −0.535898 −0.0669873
$$5$$ 0 0
$$6$$ −21.6603 −1.47379
$$7$$ −3.07180 −0.165861 −0.0829307 0.996555i $$-0.526428\pi$$
−0.0829307 + 0.996555i $$0.526428\pi$$
$$8$$ 23.3205 1.03063
$$9$$ 35.8564 1.32802
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ −4.24871 −0.102208
$$13$$ −5.35898 −0.114332 −0.0571659 0.998365i $$-0.518206\pi$$
−0.0571659 + 0.998365i $$0.518206\pi$$
$$14$$ 8.39230 0.160210
$$15$$ 0 0
$$16$$ −59.4256 −0.928525
$$17$$ 41.2154 0.588012 0.294006 0.955804i $$-0.405011\pi$$
0.294006 + 0.955804i $$0.405011\pi$$
$$18$$ −97.9615 −1.28276
$$19$$ 139.923 1.68950 0.844751 0.535159i $$-0.179748\pi$$
0.844751 + 0.535159i $$0.179748\pi$$
$$20$$ 0 0
$$21$$ −24.3538 −0.253069
$$22$$ 30.0526 0.291238
$$23$$ 111.354 1.00952 0.504758 0.863261i $$-0.331582\pi$$
0.504758 + 0.863261i $$0.331582\pi$$
$$24$$ 184.890 1.57252
$$25$$ 0 0
$$26$$ 14.6410 0.110436
$$27$$ 70.2154 0.500480
$$28$$ 1.64617 0.0111106
$$29$$ −24.9948 −0.160049 −0.0800246 0.996793i $$-0.525500\pi$$
−0.0800246 + 0.996793i $$0.525500\pi$$
$$30$$ 0 0
$$31$$ 31.4974 0.182487 0.0912436 0.995829i $$-0.470916\pi$$
0.0912436 + 0.995829i $$0.470916\pi$$
$$32$$ −24.2102 −0.133744
$$33$$ −87.2102 −0.460041
$$34$$ −112.603 −0.567976
$$35$$ 0 0
$$36$$ −19.2154 −0.0889601
$$37$$ −13.1436 −0.0583998 −0.0291999 0.999574i $$-0.509296\pi$$
−0.0291999 + 0.999574i $$0.509296\pi$$
$$38$$ −382.277 −1.63193
$$39$$ −42.4871 −0.174446
$$40$$ 0 0
$$41$$ 261.072 0.994453 0.497226 0.867621i $$-0.334352\pi$$
0.497226 + 0.867621i $$0.334352\pi$$
$$42$$ 66.5359 0.244446
$$43$$ 57.7128 0.204677 0.102339 0.994750i $$-0.467367\pi$$
0.102339 + 0.994750i $$0.467367\pi$$
$$44$$ 5.89488 0.0201974
$$45$$ 0 0
$$46$$ −304.224 −0.975118
$$47$$ 343.846 1.06713 0.533565 0.845759i $$-0.320852\pi$$
0.533565 + 0.845759i $$0.320852\pi$$
$$48$$ −471.138 −1.41673
$$49$$ −333.564 −0.972490
$$50$$ 0 0
$$51$$ 326.764 0.897179
$$52$$ 2.87187 0.00765879
$$53$$ 342.995 0.888943 0.444471 0.895793i $$-0.353392\pi$$
0.444471 + 0.895793i $$0.353392\pi$$
$$54$$ −191.832 −0.483426
$$55$$ 0 0
$$56$$ −71.6359 −0.170942
$$57$$ 1109.34 2.57782
$$58$$ 68.2872 0.154596
$$59$$ 88.3693 0.194995 0.0974975 0.995236i $$-0.468916\pi$$
0.0974975 + 0.995236i $$0.468916\pi$$
$$60$$ 0 0
$$61$$ 738.697 1.55050 0.775250 0.631654i $$-0.217624\pi$$
0.775250 + 0.631654i $$0.217624\pi$$
$$62$$ −86.0526 −0.176269
$$63$$ −110.144 −0.220266
$$64$$ 541.549 1.05771
$$65$$ 0 0
$$66$$ 238.263 0.444365
$$67$$ −342.359 −0.624266 −0.312133 0.950038i $$-0.601043\pi$$
−0.312133 + 0.950038i $$0.601043\pi$$
$$68$$ −22.0873 −0.0393893
$$69$$ 882.836 1.54030
$$70$$ 0 0
$$71$$ −207.364 −0.346614 −0.173307 0.984868i $$-0.555445\pi$$
−0.173307 + 0.984868i $$0.555445\pi$$
$$72$$ 836.190 1.36869
$$73$$ 1010.60 1.62030 0.810149 0.586224i $$-0.199386\pi$$
0.810149 + 0.586224i $$0.199386\pi$$
$$74$$ 35.9090 0.0564099
$$75$$ 0 0
$$76$$ −74.9845 −0.113175
$$77$$ 33.7898 0.0500091
$$78$$ 116.077 0.168502
$$79$$ 1294.23 1.84319 0.921593 0.388157i $$-0.126888\pi$$
0.921593 + 0.388157i $$0.126888\pi$$
$$80$$ 0 0
$$81$$ −411.441 −0.564391
$$82$$ −713.261 −0.960568
$$83$$ −441.846 −0.584324 −0.292162 0.956369i $$-0.594375\pi$$
−0.292162 + 0.956369i $$0.594375\pi$$
$$84$$ 13.0512 0.0169524
$$85$$ 0 0
$$86$$ −157.674 −0.197703
$$87$$ −198.164 −0.244200
$$88$$ −256.526 −0.310747
$$89$$ −1489.11 −1.77355 −0.886773 0.462205i $$-0.847058\pi$$
−0.886773 + 0.462205i $$0.847058\pi$$
$$90$$ 0 0
$$91$$ 16.4617 0.0189633
$$92$$ −59.6743 −0.0676248
$$93$$ 249.718 0.278436
$$94$$ −939.405 −1.03077
$$95$$ 0 0
$$96$$ −191.944 −0.204064
$$97$$ −1346.42 −1.40936 −0.704679 0.709526i $$-0.748909\pi$$
−0.704679 + 0.709526i $$0.748909\pi$$
$$98$$ 911.314 0.939353
$$99$$ −394.420 −0.400412
$$100$$ 0 0
$$101$$ −161.461 −0.159069 −0.0795347 0.996832i $$-0.525343\pi$$
−0.0795347 + 0.996832i $$0.525343\pi$$
$$102$$ −892.736 −0.866608
$$103$$ 34.7592 0.0332517 0.0166259 0.999862i $$-0.494708\pi$$
0.0166259 + 0.999862i $$0.494708\pi$$
$$104$$ −124.974 −0.117834
$$105$$ 0 0
$$106$$ −937.079 −0.858653
$$107$$ −832.179 −0.751867 −0.375934 0.926647i $$-0.622678\pi$$
−0.375934 + 0.926647i $$0.622678\pi$$
$$108$$ −37.6283 −0.0335258
$$109$$ 1044.26 0.917629 0.458815 0.888532i $$-0.348274\pi$$
0.458815 + 0.888532i $$0.348274\pi$$
$$110$$ 0 0
$$111$$ −104.205 −0.0891055
$$112$$ 182.543 0.154007
$$113$$ −295.082 −0.245654 −0.122827 0.992428i $$-0.539196\pi$$
−0.122827 + 0.992428i $$0.539196\pi$$
$$114$$ −3030.77 −2.48998
$$115$$ 0 0
$$116$$ 13.3947 0.0107213
$$117$$ −192.154 −0.151834
$$118$$ −241.429 −0.188351
$$119$$ −126.605 −0.0975285
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −2018.16 −1.49767
$$123$$ 2069.83 1.51732
$$124$$ −16.8794 −0.0122243
$$125$$ 0 0
$$126$$ 300.918 0.212761
$$127$$ 1317.60 0.920618 0.460309 0.887759i $$-0.347739\pi$$
0.460309 + 0.887759i $$0.347739\pi$$
$$128$$ −1285.86 −0.887928
$$129$$ 457.559 0.312293
$$130$$ 0 0
$$131$$ −1600.71 −1.06759 −0.533797 0.845612i $$-0.679235\pi$$
−0.533797 + 0.845612i $$0.679235\pi$$
$$132$$ 46.7358 0.0308169
$$133$$ −429.815 −0.280223
$$134$$ 935.342 0.602994
$$135$$ 0 0
$$136$$ 961.164 0.606023
$$137$$ −1611.68 −1.00507 −0.502536 0.864556i $$-0.667600\pi$$
−0.502536 + 0.864556i $$0.667600\pi$$
$$138$$ −2411.95 −1.48782
$$139$$ −31.8619 −0.0194424 −0.00972120 0.999953i $$-0.503094\pi$$
−0.00972120 + 0.999953i $$0.503094\pi$$
$$140$$ 0 0
$$141$$ 2726.08 1.62821
$$142$$ 566.529 0.334803
$$143$$ 58.9488 0.0344724
$$144$$ −2130.79 −1.23310
$$145$$ 0 0
$$146$$ −2761.01 −1.56509
$$147$$ −2644.56 −1.48381
$$148$$ 7.04363 0.00391205
$$149$$ −2428.34 −1.33515 −0.667576 0.744542i $$-0.732668\pi$$
−0.667576 + 0.744542i $$0.732668\pi$$
$$150$$ 0 0
$$151$$ −2576.68 −1.38866 −0.694328 0.719659i $$-0.744298\pi$$
−0.694328 + 0.719659i $$0.744298\pi$$
$$152$$ 3263.08 1.74125
$$153$$ 1477.84 0.780889
$$154$$ −92.3154 −0.0483051
$$155$$ 0 0
$$156$$ 22.7688 0.0116856
$$157$$ −2475.94 −1.25861 −0.629305 0.777158i $$-0.716660\pi$$
−0.629305 + 0.777158i $$0.716660\pi$$
$$158$$ −3535.89 −1.78038
$$159$$ 2719.33 1.35633
$$160$$ 0 0
$$161$$ −342.056 −0.167440
$$162$$ 1124.08 0.545160
$$163$$ 2725.11 1.30949 0.654745 0.755850i $$-0.272776\pi$$
0.654745 + 0.755850i $$0.272776\pi$$
$$164$$ −139.908 −0.0666157
$$165$$ 0 0
$$166$$ 1207.15 0.564414
$$167$$ −2737.30 −1.26837 −0.634187 0.773180i $$-0.718665\pi$$
−0.634187 + 0.773180i $$0.718665\pi$$
$$168$$ −567.944 −0.260820
$$169$$ −2168.28 −0.986928
$$170$$ 0 0
$$171$$ 5017.14 2.24368
$$172$$ −30.9282 −0.0137108
$$173$$ −2307.42 −1.01404 −0.507022 0.861933i $$-0.669254\pi$$
−0.507022 + 0.861933i $$0.669254\pi$$
$$174$$ 541.395 0.235879
$$175$$ 0 0
$$176$$ 653.682 0.279961
$$177$$ 700.610 0.297520
$$178$$ 4068.33 1.71311
$$179$$ −1312.15 −0.547905 −0.273953 0.961743i $$-0.588331\pi$$
−0.273953 + 0.961743i $$0.588331\pi$$
$$180$$ 0 0
$$181$$ −803.174 −0.329831 −0.164916 0.986308i $$-0.552735\pi$$
−0.164916 + 0.986308i $$0.552735\pi$$
$$182$$ −44.9742 −0.0183171
$$183$$ 5856.54 2.36573
$$184$$ 2596.83 1.04044
$$185$$ 0 0
$$186$$ −682.242 −0.268949
$$187$$ −453.369 −0.177292
$$188$$ −184.267 −0.0714842
$$189$$ −215.687 −0.0830103
$$190$$ 0 0
$$191$$ 1718.25 0.650932 0.325466 0.945554i $$-0.394479\pi$$
0.325466 + 0.945554i $$0.394479\pi$$
$$192$$ 4293.51 1.61384
$$193$$ −1340.18 −0.499837 −0.249919 0.968267i $$-0.580404\pi$$
−0.249919 + 0.968267i $$0.580404\pi$$
$$194$$ 3678.48 1.36134
$$195$$ 0 0
$$196$$ 178.756 0.0651445
$$197$$ 3518.33 1.27244 0.636220 0.771508i $$-0.280497\pi$$
0.636220 + 0.771508i $$0.280497\pi$$
$$198$$ 1077.58 0.386768
$$199$$ 823.692 0.293417 0.146709 0.989180i $$-0.453132\pi$$
0.146709 + 0.989180i $$0.453132\pi$$
$$200$$ 0 0
$$201$$ −2714.29 −0.952494
$$202$$ 441.121 0.153649
$$203$$ 76.7791 0.0265460
$$204$$ −175.112 −0.0600996
$$205$$ 0 0
$$206$$ −94.9639 −0.0321187
$$207$$ 3992.75 1.34065
$$208$$ 318.461 0.106160
$$209$$ −1539.15 −0.509404
$$210$$ 0 0
$$211$$ −107.343 −0.0350228 −0.0175114 0.999847i $$-0.505574\pi$$
−0.0175114 + 0.999847i $$0.505574\pi$$
$$212$$ −183.810 −0.0595479
$$213$$ −1644.03 −0.528858
$$214$$ 2273.56 0.726248
$$215$$ 0 0
$$216$$ 1637.46 0.515810
$$217$$ −96.7537 −0.0302676
$$218$$ −2852.96 −0.886362
$$219$$ 8012.24 2.47222
$$220$$ 0 0
$$221$$ −220.873 −0.0672285
$$222$$ 284.694 0.0860693
$$223$$ 3933.68 1.18125 0.590625 0.806946i $$-0.298881\pi$$
0.590625 + 0.806946i $$0.298881\pi$$
$$224$$ 74.3689 0.0221830
$$225$$ 0 0
$$226$$ 806.178 0.237284
$$227$$ 1771.90 0.518085 0.259042 0.965866i $$-0.416593\pi$$
0.259042 + 0.965866i $$0.416593\pi$$
$$228$$ −594.493 −0.172681
$$229$$ 1915.37 0.552713 0.276356 0.961055i $$-0.410873\pi$$
0.276356 + 0.961055i $$0.410873\pi$$
$$230$$ 0 0
$$231$$ 267.892 0.0763031
$$232$$ −582.892 −0.164952
$$233$$ −4396.32 −1.23610 −0.618052 0.786137i $$-0.712078\pi$$
−0.618052 + 0.786137i $$0.712078\pi$$
$$234$$ 524.974 0.146661
$$235$$ 0 0
$$236$$ −47.3570 −0.0130622
$$237$$ 10260.9 2.81230
$$238$$ 345.892 0.0942053
$$239$$ −4084.49 −1.10546 −0.552728 0.833362i $$-0.686413\pi$$
−0.552728 + 0.833362i $$0.686413\pi$$
$$240$$ 0 0
$$241$$ 3908.58 1.04471 0.522353 0.852730i $$-0.325054\pi$$
0.522353 + 0.852730i $$0.325054\pi$$
$$242$$ −330.578 −0.0878114
$$243$$ −5157.80 −1.36162
$$244$$ −395.867 −0.103864
$$245$$ 0 0
$$246$$ −5654.88 −1.46562
$$247$$ −749.845 −0.193164
$$248$$ 734.536 0.188077
$$249$$ −3503.05 −0.891552
$$250$$ 0 0
$$251$$ 1094.89 0.275335 0.137667 0.990479i $$-0.456040\pi$$
0.137667 + 0.990479i $$0.456040\pi$$
$$252$$ 59.0258 0.0147551
$$253$$ −1224.89 −0.304381
$$254$$ −3599.76 −0.889249
$$255$$ 0 0
$$256$$ −819.364 −0.200040
$$257$$ −783.179 −0.190091 −0.0950454 0.995473i $$-0.530300\pi$$
−0.0950454 + 0.995473i $$0.530300\pi$$
$$258$$ −1250.07 −0.301652
$$259$$ 40.3744 0.00968628
$$260$$ 0 0
$$261$$ −896.225 −0.212548
$$262$$ 4373.23 1.03122
$$263$$ −6180.06 −1.44897 −0.724484 0.689292i $$-0.757922\pi$$
−0.724484 + 0.689292i $$0.757922\pi$$
$$264$$ −2033.79 −0.474132
$$265$$ 0 0
$$266$$ 1174.28 0.270675
$$267$$ −11806.0 −2.70605
$$268$$ 183.470 0.0418179
$$269$$ 986.965 0.223704 0.111852 0.993725i $$-0.464322\pi$$
0.111852 + 0.993725i $$0.464322\pi$$
$$270$$ 0 0
$$271$$ 4576.99 1.02595 0.512975 0.858404i $$-0.328543\pi$$
0.512975 + 0.858404i $$0.328543\pi$$
$$272$$ −2449.25 −0.545984
$$273$$ 130.512 0.0289338
$$274$$ 4403.18 0.970825
$$275$$ 0 0
$$276$$ −473.110 −0.103181
$$277$$ −567.836 −0.123169 −0.0615847 0.998102i $$-0.519615\pi$$
−0.0615847 + 0.998102i $$0.519615\pi$$
$$278$$ 87.0484 0.0187799
$$279$$ 1129.38 0.242346
$$280$$ 0 0
$$281$$ 5311.01 1.12750 0.563752 0.825944i $$-0.309357\pi$$
0.563752 + 0.825944i $$0.309357\pi$$
$$282$$ −7447.79 −1.57273
$$283$$ 4728.44 0.993204 0.496602 0.867978i $$-0.334581\pi$$
0.496602 + 0.867978i $$0.334581\pi$$
$$284$$ 111.126 0.0232187
$$285$$ 0 0
$$286$$ −161.051 −0.0332977
$$287$$ −801.960 −0.164941
$$288$$ −868.092 −0.177614
$$289$$ −3214.29 −0.654242
$$290$$ 0 0
$$291$$ −10674.7 −2.15038
$$292$$ −541.579 −0.108539
$$293$$ −2328.92 −0.464358 −0.232179 0.972673i $$-0.574585\pi$$
−0.232179 + 0.972673i $$0.574585\pi$$
$$294$$ 7225.08 1.43325
$$295$$ 0 0
$$296$$ −306.515 −0.0601886
$$297$$ −772.369 −0.150900
$$298$$ 6634.36 1.28966
$$299$$ −596.743 −0.115420
$$300$$ 0 0
$$301$$ −177.282 −0.0339481
$$302$$ 7039.61 1.34134
$$303$$ −1280.10 −0.242705
$$304$$ −8315.01 −1.56875
$$305$$ 0 0
$$306$$ −4037.52 −0.754280
$$307$$ 1678.07 0.311962 0.155981 0.987760i $$-0.450146\pi$$
0.155981 + 0.987760i $$0.450146\pi$$
$$308$$ −18.1079 −0.00334997
$$309$$ 275.578 0.0507349
$$310$$ 0 0
$$311$$ 3572.71 0.651413 0.325707 0.945471i $$-0.394398\pi$$
0.325707 + 0.945471i $$0.394398\pi$$
$$312$$ −990.821 −0.179789
$$313$$ −7184.36 −1.29739 −0.648697 0.761047i $$-0.724686\pi$$
−0.648697 + 0.761047i $$0.724686\pi$$
$$314$$ 6764.40 1.21572
$$315$$ 0 0
$$316$$ −693.573 −0.123470
$$317$$ 15.7077 0.00278306 0.00139153 0.999999i $$-0.499557\pi$$
0.00139153 + 0.999999i $$0.499557\pi$$
$$318$$ −7429.36 −1.31012
$$319$$ 274.943 0.0482566
$$320$$ 0 0
$$321$$ −6597.69 −1.14719
$$322$$ 934.515 0.161734
$$323$$ 5766.98 0.993447
$$324$$ 220.491 0.0378070
$$325$$ 0 0
$$326$$ −7445.13 −1.26487
$$327$$ 8279.08 1.40010
$$328$$ 6088.33 1.02491
$$329$$ −1056.23 −0.176996
$$330$$ 0 0
$$331$$ −1318.95 −0.219022 −0.109511 0.993986i $$-0.534928\pi$$
−0.109511 + 0.993986i $$0.534928\pi$$
$$332$$ 236.785 0.0391423
$$333$$ −471.282 −0.0775558
$$334$$ 7478.43 1.22515
$$335$$ 0 0
$$336$$ 1447.24 0.234981
$$337$$ 239.183 0.0386621 0.0193310 0.999813i $$-0.493846\pi$$
0.0193310 + 0.999813i $$0.493846\pi$$
$$338$$ 5923.85 0.953299
$$339$$ −2339.47 −0.374816
$$340$$ 0 0
$$341$$ −346.472 −0.0550220
$$342$$ −13707.1 −2.16723
$$343$$ 2078.27 0.327160
$$344$$ 1345.89 0.210947
$$345$$ 0 0
$$346$$ 6303.98 0.979491
$$347$$ 5862.79 0.907006 0.453503 0.891255i $$-0.350174\pi$$
0.453503 + 0.891255i $$0.350174\pi$$
$$348$$ 106.196 0.0163583
$$349$$ 3491.73 0.535553 0.267776 0.963481i $$-0.413711\pi$$
0.267776 + 0.963481i $$0.413711\pi$$
$$350$$ 0 0
$$351$$ −376.283 −0.0572208
$$352$$ 266.313 0.0403253
$$353$$ 10916.7 1.64600 0.822999 0.568043i $$-0.192299\pi$$
0.822999 + 0.568043i $$0.192299\pi$$
$$354$$ −1914.10 −0.287382
$$355$$ 0 0
$$356$$ 798.013 0.118805
$$357$$ −1003.75 −0.148807
$$358$$ 3584.87 0.529236
$$359$$ −11500.7 −1.69077 −0.845384 0.534160i $$-0.820628\pi$$
−0.845384 + 0.534160i $$0.820628\pi$$
$$360$$ 0 0
$$361$$ 12719.5 1.85442
$$362$$ 2194.31 0.318592
$$363$$ 959.313 0.138708
$$364$$ −8.82180 −0.00127030
$$365$$ 0 0
$$366$$ −16000.4 −2.28512
$$367$$ −6767.01 −0.962493 −0.481246 0.876585i $$-0.659816\pi$$
−0.481246 + 0.876585i $$0.659816\pi$$
$$368$$ −6617.27 −0.937362
$$369$$ 9361.10 1.32065
$$370$$ 0 0
$$371$$ −1053.61 −0.147441
$$372$$ −133.823 −0.0186517
$$373$$ 5310.22 0.737139 0.368569 0.929600i $$-0.379848\pi$$
0.368569 + 0.929600i $$0.379848\pi$$
$$374$$ 1238.63 0.171251
$$375$$ 0 0
$$376$$ 8018.67 1.09982
$$377$$ 133.947 0.0182987
$$378$$ 589.269 0.0801818
$$379$$ −838.267 −0.113612 −0.0568059 0.998385i $$-0.518092\pi$$
−0.0568059 + 0.998385i $$0.518092\pi$$
$$380$$ 0 0
$$381$$ 10446.2 1.40466
$$382$$ −4694.34 −0.628752
$$383$$ 2832.16 0.377851 0.188925 0.981991i $$-0.439500\pi$$
0.188925 + 0.981991i $$0.439500\pi$$
$$384$$ −10194.5 −1.35479
$$385$$ 0 0
$$386$$ 3661.45 0.482806
$$387$$ 2069.37 0.271814
$$388$$ 721.542 0.0944091
$$389$$ 3111.25 0.405519 0.202759 0.979229i $$-0.435009\pi$$
0.202759 + 0.979229i $$0.435009\pi$$
$$390$$ 0 0
$$391$$ 4589.49 0.593608
$$392$$ −7778.88 −1.00228
$$393$$ −12690.8 −1.62892
$$394$$ −9612.25 −1.22908
$$395$$ 0 0
$$396$$ 211.369 0.0268225
$$397$$ −14208.7 −1.79626 −0.898131 0.439728i $$-0.855075\pi$$
−0.898131 + 0.439728i $$0.855075\pi$$
$$398$$ −2250.37 −0.283419
$$399$$ −3407.66 −0.427560
$$400$$ 0 0
$$401$$ −6261.68 −0.779784 −0.389892 0.920861i $$-0.627488\pi$$
−0.389892 + 0.920861i $$0.627488\pi$$
$$402$$ 7415.58 0.920039
$$403$$ −168.794 −0.0208641
$$404$$ 86.5269 0.0106556
$$405$$ 0 0
$$406$$ −209.764 −0.0256415
$$407$$ 144.580 0.0176082
$$408$$ 7620.30 0.924660
$$409$$ −4192.50 −0.506860 −0.253430 0.967354i $$-0.581559\pi$$
−0.253430 + 0.967354i $$0.581559\pi$$
$$410$$ 0 0
$$411$$ −12777.7 −1.53352
$$412$$ −18.6274 −0.00222744
$$413$$ −271.453 −0.0323421
$$414$$ −10908.4 −1.29497
$$415$$ 0 0
$$416$$ 129.742 0.0152912
$$417$$ −252.608 −0.0296649
$$418$$ 4205.05 0.492047
$$419$$ −9287.15 −1.08283 −0.541416 0.840755i $$-0.682112\pi$$
−0.541416 + 0.840755i $$0.682112\pi$$
$$420$$ 0 0
$$421$$ 13146.0 1.52185 0.760923 0.648842i $$-0.224746\pi$$
0.760923 + 0.648842i $$0.224746\pi$$
$$422$$ 293.267 0.0338294
$$423$$ 12329.1 1.41716
$$424$$ 7998.81 0.916172
$$425$$ 0 0
$$426$$ 4491.56 0.510838
$$427$$ −2269.13 −0.257168
$$428$$ 445.964 0.0503656
$$429$$ 467.358 0.0525974
$$430$$ 0 0
$$431$$ 4909.67 0.548701 0.274351 0.961630i $$-0.411537\pi$$
0.274351 + 0.961630i $$0.411537\pi$$
$$432$$ −4172.59 −0.464708
$$433$$ 11743.3 1.30334 0.651671 0.758502i $$-0.274068\pi$$
0.651671 + 0.758502i $$0.274068\pi$$
$$434$$ 264.336 0.0292363
$$435$$ 0 0
$$436$$ −559.615 −0.0614695
$$437$$ 15581.0 1.70558
$$438$$ −21889.8 −2.38798
$$439$$ −11824.2 −1.28551 −0.642754 0.766073i $$-0.722208\pi$$
−0.642754 + 0.766073i $$0.722208\pi$$
$$440$$ 0 0
$$441$$ −11960.4 −1.29148
$$442$$ 603.435 0.0649377
$$443$$ −10102.1 −1.08344 −0.541722 0.840558i $$-0.682228\pi$$
−0.541722 + 0.840558i $$0.682228\pi$$
$$444$$ 55.8433 0.00596894
$$445$$ 0 0
$$446$$ −10747.0 −1.14100
$$447$$ −19252.4 −2.03715
$$448$$ −1663.53 −0.175434
$$449$$ −345.254 −0.0362885 −0.0181443 0.999835i $$-0.505776\pi$$
−0.0181443 + 0.999835i $$0.505776\pi$$
$$450$$ 0 0
$$451$$ −2871.79 −0.299839
$$452$$ 158.134 0.0164557
$$453$$ −20428.4 −2.11879
$$454$$ −4840.93 −0.500431
$$455$$ 0 0
$$456$$ 25870.3 2.65678
$$457$$ 10567.1 1.08164 0.540821 0.841138i $$-0.318114\pi$$
0.540821 + 0.841138i $$0.318114\pi$$
$$458$$ −5232.89 −0.533879
$$459$$ 2893.95 0.294288
$$460$$ 0 0
$$461$$ 4733.96 0.478270 0.239135 0.970986i $$-0.423136\pi$$
0.239135 + 0.970986i $$0.423136\pi$$
$$462$$ −731.895 −0.0737031
$$463$$ −3431.20 −0.344409 −0.172204 0.985061i $$-0.555089\pi$$
−0.172204 + 0.985061i $$0.555089\pi$$
$$464$$ 1485.33 0.148610
$$465$$ 0 0
$$466$$ 12011.0 1.19399
$$467$$ −5116.96 −0.507034 −0.253517 0.967331i $$-0.581587\pi$$
−0.253517 + 0.967331i $$0.581587\pi$$
$$468$$ 102.975 0.0101710
$$469$$ 1051.66 0.103542
$$470$$ 0 0
$$471$$ −19629.8 −1.92037
$$472$$ 2060.82 0.200968
$$473$$ −634.841 −0.0617125
$$474$$ −28033.2 −2.71648
$$475$$ 0 0
$$476$$ 67.8476 0.00653317
$$477$$ 12298.6 1.18053
$$478$$ 11159.0 1.06779
$$479$$ 11566.9 1.10335 0.551675 0.834059i $$-0.313989\pi$$
0.551675 + 0.834059i $$0.313989\pi$$
$$480$$ 0 0
$$481$$ 70.4363 0.00667696
$$482$$ −10678.4 −1.00911
$$483$$ −2711.89 −0.255477
$$484$$ −64.8437 −0.00608975
$$485$$ 0 0
$$486$$ 14091.4 1.31522
$$487$$ 18326.5 1.70525 0.852623 0.522527i $$-0.175010\pi$$
0.852623 + 0.522527i $$0.175010\pi$$
$$488$$ 17226.8 1.59799
$$489$$ 21605.2 1.99800
$$490$$ 0 0
$$491$$ −7617.58 −0.700156 −0.350078 0.936721i $$-0.613845\pi$$
−0.350078 + 0.936721i $$0.613845\pi$$
$$492$$ −1109.22 −0.101641
$$493$$ −1030.17 −0.0941108
$$494$$ 2048.62 0.186582
$$495$$ 0 0
$$496$$ −1871.75 −0.169444
$$497$$ 636.980 0.0574899
$$498$$ 9570.50 0.861173
$$499$$ 12909.1 1.15810 0.579050 0.815292i $$-0.303424\pi$$
0.579050 + 0.815292i $$0.303424\pi$$
$$500$$ 0 0
$$501$$ −21701.8 −1.93526
$$502$$ −2991.30 −0.265953
$$503$$ −10165.7 −0.901121 −0.450561 0.892746i $$-0.648776\pi$$
−0.450561 + 0.892746i $$0.648776\pi$$
$$504$$ −2568.60 −0.227013
$$505$$ 0 0
$$506$$ 3346.47 0.294009
$$507$$ −17190.6 −1.50584
$$508$$ −706.102 −0.0616697
$$509$$ 6449.93 0.561666 0.280833 0.959757i $$-0.409389\pi$$
0.280833 + 0.959757i $$0.409389\pi$$
$$510$$ 0 0
$$511$$ −3104.36 −0.268745
$$512$$ 12525.4 1.08115
$$513$$ 9824.75 0.845562
$$514$$ 2139.68 0.183614
$$515$$ 0 0
$$516$$ −245.205 −0.0209197
$$517$$ −3782.31 −0.321752
$$518$$ −110.305 −0.00935623
$$519$$ −18293.7 −1.54721
$$520$$ 0 0
$$521$$ −19327.4 −1.62524 −0.812620 0.582794i $$-0.801959\pi$$
−0.812620 + 0.582794i $$0.801959\pi$$
$$522$$ 2448.53 0.205305
$$523$$ −6259.09 −0.523310 −0.261655 0.965161i $$-0.584268\pi$$
−0.261655 + 0.965161i $$0.584268\pi$$
$$524$$ 857.819 0.0715153
$$525$$ 0 0
$$526$$ 16884.2 1.39960
$$527$$ 1298.18 0.107305
$$528$$ 5182.52 0.427160
$$529$$ 232.675 0.0191235
$$530$$ 0 0
$$531$$ 3168.61 0.258956
$$532$$ 230.337 0.0187714
$$533$$ −1399.08 −0.113698
$$534$$ 32254.6 2.61384
$$535$$ 0 0
$$536$$ −7983.99 −0.643387
$$537$$ −10403.0 −0.835985
$$538$$ −2696.44 −0.216081
$$539$$ 3669.20 0.293217
$$540$$ 0 0
$$541$$ −14008.2 −1.11323 −0.556616 0.830770i $$-0.687900\pi$$
−0.556616 + 0.830770i $$0.687900\pi$$
$$542$$ −12504.6 −0.990991
$$543$$ −6367.72 −0.503251
$$544$$ −997.834 −0.0786430
$$545$$ 0 0
$$546$$ −356.565 −0.0279479
$$547$$ 4949.45 0.386879 0.193440 0.981112i $$-0.438036\pi$$
0.193440 + 0.981112i $$0.438036\pi$$
$$548$$ 863.695 0.0673270
$$549$$ 26487.0 2.05909
$$550$$ 0 0
$$551$$ −3497.35 −0.270404
$$552$$ 20588.2 1.58748
$$553$$ −3975.60 −0.305714
$$554$$ 1551.36 0.118973
$$555$$ 0 0
$$556$$ 17.0748 0.00130239
$$557$$ 3801.58 0.289188 0.144594 0.989491i $$-0.453812\pi$$
0.144594 + 0.989491i $$0.453812\pi$$
$$558$$ −3085.54 −0.234088
$$559$$ −309.282 −0.0234011
$$560$$ 0 0
$$561$$ −3594.40 −0.270510
$$562$$ −14510.0 −1.08908
$$563$$ 9900.11 0.741101 0.370551 0.928812i $$-0.379169\pi$$
0.370551 + 0.928812i $$0.379169\pi$$
$$564$$ −1460.90 −0.109069
$$565$$ 0 0
$$566$$ −12918.3 −0.959361
$$567$$ 1263.86 0.0936107
$$568$$ −4835.84 −0.357231
$$569$$ 5329.16 0.392636 0.196318 0.980540i $$-0.437102\pi$$
0.196318 + 0.980540i $$0.437102\pi$$
$$570$$ 0 0
$$571$$ −16962.6 −1.24319 −0.621597 0.783337i $$-0.713516\pi$$
−0.621597 + 0.783337i $$0.713516\pi$$
$$572$$ −31.5906 −0.00230921
$$573$$ 13622.6 0.993181
$$574$$ 2190.99 0.159321
$$575$$ 0 0
$$576$$ 19418.0 1.40466
$$577$$ 15487.0 1.11738 0.558692 0.829375i $$-0.311303\pi$$
0.558692 + 0.829375i $$0.311303\pi$$
$$578$$ 8781.61 0.631949
$$579$$ −10625.3 −0.762643
$$580$$ 0 0
$$581$$ 1357.26 0.0969169
$$582$$ 29163.7 2.07710
$$583$$ −3772.94 −0.268026
$$584$$ 23567.7 1.66993
$$585$$ 0 0
$$586$$ 6362.72 0.448535
$$587$$ −11084.2 −0.779373 −0.389686 0.920948i $$-0.627417\pi$$
−0.389686 + 0.920948i $$0.627417\pi$$
$$588$$ 1417.22 0.0993964
$$589$$ 4407.22 0.308313
$$590$$ 0 0
$$591$$ 27894.0 1.94147
$$592$$ 781.066 0.0542257
$$593$$ −4349.68 −0.301214 −0.150607 0.988594i $$-0.548123\pi$$
−0.150607 + 0.988594i $$0.548123\pi$$
$$594$$ 2110.15 0.145759
$$595$$ 0 0
$$596$$ 1301.34 0.0894382
$$597$$ 6530.40 0.447691
$$598$$ 1630.33 0.111487
$$599$$ 13183.9 0.899299 0.449650 0.893205i $$-0.351549\pi$$
0.449650 + 0.893205i $$0.351549\pi$$
$$600$$ 0 0
$$601$$ −18765.0 −1.27361 −0.636806 0.771024i $$-0.719745\pi$$
−0.636806 + 0.771024i $$0.719745\pi$$
$$602$$ 484.344 0.0327913
$$603$$ −12275.8 −0.829034
$$604$$ 1380.84 0.0930223
$$605$$ 0 0
$$606$$ 3497.29 0.234435
$$607$$ −21871.4 −1.46249 −0.731244 0.682116i $$-0.761060\pi$$
−0.731244 + 0.682116i $$0.761060\pi$$
$$608$$ −3387.57 −0.225961
$$609$$ 608.720 0.0405034
$$610$$ 0 0
$$611$$ −1842.67 −0.122007
$$612$$ −791.970 −0.0523096
$$613$$ 3527.85 0.232445 0.116222 0.993223i $$-0.462921\pi$$
0.116222 + 0.993223i $$0.462921\pi$$
$$614$$ −4584.56 −0.301332
$$615$$ 0 0
$$616$$ 787.994 0.0515409
$$617$$ 22728.1 1.48298 0.741490 0.670963i $$-0.234119\pi$$
0.741490 + 0.670963i $$0.234119\pi$$
$$618$$ −752.893 −0.0490062
$$619$$ −21443.3 −1.39237 −0.696187 0.717861i $$-0.745121\pi$$
−0.696187 + 0.717861i $$0.745121\pi$$
$$620$$ 0 0
$$621$$ 7818.75 0.505243
$$622$$ −9760.81 −0.629217
$$623$$ 4574.25 0.294163
$$624$$ 2524.82 0.161977
$$625$$ 0 0
$$626$$ 19628.0 1.25319
$$627$$ −12202.7 −0.777240
$$628$$ 1326.85 0.0843109
$$629$$ −541.718 −0.0343398
$$630$$ 0 0
$$631$$ 21532.0 1.35844 0.679219 0.733936i $$-0.262319\pi$$
0.679219 + 0.733936i $$0.262319\pi$$
$$632$$ 30182.0 1.89964
$$633$$ −851.038 −0.0534372
$$634$$ −42.9141 −0.00268823
$$635$$ 0 0
$$636$$ −1457.29 −0.0908572
$$637$$ 1787.56 0.111187
$$638$$ −751.159 −0.0466123
$$639$$ −7435.33 −0.460309
$$640$$ 0 0
$$641$$ 20148.3 1.24151 0.620756 0.784004i $$-0.286826\pi$$
0.620756 + 0.784004i $$0.286826\pi$$
$$642$$ 18025.2 1.10810
$$643$$ −28869.7 −1.77062 −0.885310 0.465000i $$-0.846054\pi$$
−0.885310 + 0.465000i $$0.846054\pi$$
$$644$$ 183.307 0.0112163
$$645$$ 0 0
$$646$$ −15755.7 −0.959597
$$647$$ 1590.02 0.0966155 0.0483077 0.998833i $$-0.484617\pi$$
0.0483077 + 0.998833i $$0.484617\pi$$
$$648$$ −9595.02 −0.581679
$$649$$ −972.062 −0.0587932
$$650$$ 0 0
$$651$$ −767.083 −0.0461818
$$652$$ −1460.38 −0.0877192
$$653$$ −20028.1 −1.20024 −0.600122 0.799909i $$-0.704881\pi$$
−0.600122 + 0.799909i $$0.704881\pi$$
$$654$$ −22618.9 −1.35240
$$655$$ 0 0
$$656$$ −15514.4 −0.923375
$$657$$ 36236.5 2.15178
$$658$$ 2885.66 0.170965
$$659$$ −10520.7 −0.621897 −0.310948 0.950427i $$-0.600647\pi$$
−0.310948 + 0.950427i $$0.600647\pi$$
$$660$$ 0 0
$$661$$ 3295.83 0.193938 0.0969690 0.995287i $$-0.469085\pi$$
0.0969690 + 0.995287i $$0.469085\pi$$
$$662$$ 3603.45 0.211559
$$663$$ −1751.12 −0.102576
$$664$$ −10304.1 −0.602222
$$665$$ 0 0
$$666$$ 1287.57 0.0749132
$$667$$ −2783.27 −0.161572
$$668$$ 1466.91 0.0849649
$$669$$ 31187.0 1.80233
$$670$$ 0 0
$$671$$ −8125.67 −0.467493
$$672$$ 589.612 0.0338464
$$673$$ 1187.64 0.0680239 0.0340119 0.999421i $$-0.489172\pi$$
0.0340119 + 0.999421i $$0.489172\pi$$
$$674$$ −653.460 −0.0373447
$$675$$ 0 0
$$676$$ 1161.98 0.0661117
$$677$$ −13221.4 −0.750574 −0.375287 0.926909i $$-0.622456\pi$$
−0.375287 + 0.926909i $$0.622456\pi$$
$$678$$ 6391.55 0.362044
$$679$$ 4135.91 0.233758
$$680$$ 0 0
$$681$$ 14048.0 0.790485
$$682$$ 946.578 0.0531471
$$683$$ 13831.4 0.774882 0.387441 0.921894i $$-0.373359\pi$$
0.387441 + 0.921894i $$0.373359\pi$$
$$684$$ −2688.68 −0.150298
$$685$$ 0 0
$$686$$ −5677.93 −0.316012
$$687$$ 15185.4 0.843320
$$688$$ −3429.62 −0.190048
$$689$$ −1838.10 −0.101635
$$690$$ 0 0
$$691$$ −9817.07 −0.540462 −0.270231 0.962796i $$-0.587100\pi$$
−0.270231 + 0.962796i $$0.587100\pi$$
$$692$$ 1236.54 0.0679280
$$693$$ 1211.58 0.0664128
$$694$$ −16017.4 −0.876101
$$695$$ 0 0
$$696$$ −4621.29 −0.251680
$$697$$ 10760.2 0.584750
$$698$$ −9539.58 −0.517304
$$699$$ −34854.9 −1.88603
$$700$$ 0 0
$$701$$ 29949.8 1.61368 0.806838 0.590773i $$-0.201177\pi$$
0.806838 + 0.590773i $$0.201177\pi$$
$$702$$ 1028.02 0.0552711
$$703$$ −1839.09 −0.0986667
$$704$$ −5957.03 −0.318912
$$705$$ 0 0
$$706$$ −29825.0 −1.58991
$$707$$ 495.976 0.0263835
$$708$$ −375.456 −0.0199301
$$709$$ 11307.5 0.598959 0.299479 0.954103i $$-0.403187\pi$$
0.299479 + 0.954103i $$0.403187\pi$$
$$710$$ 0 0
$$711$$ 46406.3 2.44778
$$712$$ −34726.9 −1.82787
$$713$$ 3507.36 0.184224
$$714$$ 2742.30 0.143737
$$715$$ 0 0
$$716$$ 703.181 0.0367027
$$717$$ −32382.7 −1.68669
$$718$$ 31420.6 1.63316
$$719$$ −32623.4 −1.69214 −0.846070 0.533071i $$-0.821038\pi$$
−0.846070 + 0.533071i $$0.821038\pi$$
$$720$$ 0 0
$$721$$ −106.773 −0.00551518
$$722$$ −34750.2 −1.79123
$$723$$ 30988.0 1.59399
$$724$$ 430.420 0.0220945
$$725$$ 0 0
$$726$$ −2620.89 −0.133981
$$727$$ 502.545 0.0256373 0.0128187 0.999918i $$-0.495920\pi$$
0.0128187 + 0.999918i $$0.495920\pi$$
$$728$$ 383.895 0.0195441
$$729$$ −29783.2 −1.51314
$$730$$ 0 0
$$731$$ 2378.66 0.120353
$$732$$ −3138.51 −0.158474
$$733$$ −8631.37 −0.434935 −0.217467 0.976068i $$-0.569780\pi$$
−0.217467 + 0.976068i $$0.569780\pi$$
$$734$$ 18487.8 0.929697
$$735$$ 0 0
$$736$$ −2695.90 −0.135017
$$737$$ 3765.95 0.188223
$$738$$ −25575.0 −1.27565
$$739$$ −18357.5 −0.913792 −0.456896 0.889520i $$-0.651039\pi$$
−0.456896 + 0.889520i $$0.651039\pi$$
$$740$$ 0 0
$$741$$ −5944.93 −0.294726
$$742$$ 2878.52 0.142417
$$743$$ −11182.6 −0.552155 −0.276078 0.961135i $$-0.589035\pi$$
−0.276078 + 0.961135i $$0.589035\pi$$
$$744$$ 5823.55 0.286965
$$745$$ 0 0
$$746$$ −14507.8 −0.712021
$$747$$ −15843.0 −0.775991
$$748$$ 242.960 0.0118763
$$749$$ 2556.29 0.124706
$$750$$ 0 0
$$751$$ 16733.4 0.813063 0.406531 0.913637i $$-0.366738\pi$$
0.406531 + 0.913637i $$0.366738\pi$$
$$752$$ −20433.3 −0.990857
$$753$$ 8680.53 0.420101
$$754$$ −365.950 −0.0176752
$$755$$ 0 0
$$756$$ 115.587 0.00556064
$$757$$ 24402.4 1.17163 0.585813 0.810446i $$-0.300775\pi$$
0.585813 + 0.810446i $$0.300775\pi$$
$$758$$ 2290.19 0.109741
$$759$$ −9711.19 −0.464419
$$760$$ 0 0
$$761$$ 8469.33 0.403434 0.201717 0.979444i $$-0.435348\pi$$
0.201717 + 0.979444i $$0.435348\pi$$
$$762$$ −28539.7 −1.35680
$$763$$ −3207.74 −0.152199
$$764$$ −920.805 −0.0436042
$$765$$ 0 0
$$766$$ −7737.62 −0.364976
$$767$$ −473.570 −0.0222941
$$768$$ −6496.08 −0.305218
$$769$$ 32834.7 1.53973 0.769864 0.638208i $$-0.220324\pi$$
0.769864 + 0.638208i $$0.220324\pi$$
$$770$$ 0 0
$$771$$ −6209.20 −0.290038
$$772$$ 718.202 0.0334827
$$773$$ 35571.4 1.65513 0.827564 0.561371i $$-0.189726\pi$$
0.827564 + 0.561371i $$0.189726\pi$$
$$774$$ −5653.64 −0.262553
$$775$$ 0 0
$$776$$ −31399.1 −1.45253
$$777$$ 320.097 0.0147792
$$778$$ −8500.11 −0.391701
$$779$$ 36530.0 1.68013
$$780$$ 0 0
$$781$$ 2281.01 0.104508
$$782$$ −12538.7 −0.573381
$$783$$ −1755.02 −0.0801014
$$784$$ 19822.3 0.902982
$$785$$ 0 0
$$786$$ 34671.8 1.57341
$$787$$ −15729.6 −0.712452 −0.356226 0.934400i $$-0.615937\pi$$
−0.356226 + 0.934400i $$0.615937\pi$$
$$788$$ −1885.47 −0.0852373
$$789$$ −48996.7 −2.21081
$$790$$ 0 0
$$791$$ 906.431 0.0407446
$$792$$ −9198.09 −0.412676
$$793$$ −3958.67 −0.177272
$$794$$ 38819.0 1.73506
$$795$$ 0 0
$$796$$ −441.415 −0.0196552
$$797$$ −7888.07 −0.350577 −0.175288 0.984517i $$-0.556086\pi$$
−0.175288 + 0.984517i $$0.556086\pi$$
$$798$$ 9309.91 0.412991
$$799$$ 14171.8 0.627485
$$800$$ 0 0
$$801$$ −53394.2 −2.35530
$$802$$ 17107.2 0.753214
$$803$$ −11116.6 −0.488538
$$804$$ 1454.58 0.0638050
$$805$$ 0 0
$$806$$ 461.154 0.0201532
$$807$$ 7824.86 0.341323
$$808$$ −3765.36 −0.163942
$$809$$ 5896.97 0.256275 0.128138 0.991756i $$-0.459100\pi$$
0.128138 + 0.991756i $$0.459100\pi$$
$$810$$ 0 0
$$811$$ 14197.9 0.614744 0.307372 0.951589i $$-0.400550\pi$$
0.307372 + 0.951589i $$0.400550\pi$$
$$812$$ −41.1458 −0.00177824
$$813$$ 36287.3 1.56538
$$814$$ −394.999 −0.0170082
$$815$$ 0 0
$$816$$ −19418.2 −0.833053
$$817$$ 8075.35 0.345803
$$818$$ 11454.1 0.489589
$$819$$ 590.258 0.0251835
$$820$$ 0 0
$$821$$ −19841.7 −0.843459 −0.421729 0.906722i $$-0.638577\pi$$
−0.421729 + 0.906722i $$0.638577\pi$$
$$822$$ 34909.3 1.48127
$$823$$ 28202.2 1.19449 0.597246 0.802058i $$-0.296262\pi$$
0.597246 + 0.802058i $$0.296262\pi$$
$$824$$ 810.602 0.0342702
$$825$$ 0 0
$$826$$ 741.622 0.0312401
$$827$$ −34031.0 −1.43092 −0.715462 0.698651i $$-0.753784\pi$$
−0.715462 + 0.698651i $$0.753784\pi$$
$$828$$ −2139.71 −0.0898067
$$829$$ 4931.55 0.206610 0.103305 0.994650i $$-0.467058\pi$$
0.103305 + 0.994650i $$0.467058\pi$$
$$830$$ 0 0
$$831$$ −4501.92 −0.187930
$$832$$ −2902.15 −0.120930
$$833$$ −13748.0 −0.571836
$$834$$ 690.138 0.0286541
$$835$$ 0 0
$$836$$ 824.830 0.0341236
$$837$$ 2211.60 0.0913312
$$838$$ 25373.0 1.04594
$$839$$ −38189.8 −1.57146 −0.785731 0.618568i $$-0.787713\pi$$
−0.785731 + 0.618568i $$0.787713\pi$$
$$840$$ 0 0
$$841$$ −23764.3 −0.974384
$$842$$ −35915.6 −1.46999
$$843$$ 42106.8 1.72033
$$844$$ 57.5250 0.00234608
$$845$$ 0 0
$$846$$ −33683.7 −1.36888
$$847$$ −371.687 −0.0150783
$$848$$ −20382.7 −0.825406
$$849$$ 37488.0 1.51541
$$850$$ 0 0
$$851$$ −1463.59 −0.0589556
$$852$$ 881.030 0.0354268
$$853$$ −42966.8 −1.72469 −0.862343 0.506325i $$-0.831003\pi$$
−0.862343 + 0.506325i $$0.831003\pi$$
$$854$$ 6199.37 0.248405
$$855$$ 0 0
$$856$$ −19406.8 −0.774898
$$857$$ 17281.5 0.688828 0.344414 0.938818i $$-0.388078\pi$$
0.344414 + 0.938818i $$0.388078\pi$$
$$858$$ −1276.85 −0.0508052
$$859$$ 9316.75 0.370062 0.185031 0.982733i $$-0.440761\pi$$
0.185031 + 0.982733i $$0.440761\pi$$
$$860$$ 0 0
$$861$$ −6358.10 −0.251665
$$862$$ −13413.5 −0.530005
$$863$$ 9647.65 0.380544 0.190272 0.981731i $$-0.439063\pi$$
0.190272 + 0.981731i $$0.439063\pi$$
$$864$$ −1699.93 −0.0669361
$$865$$ 0 0
$$866$$ −32083.3 −1.25893
$$867$$ −25483.6 −0.998232
$$868$$ 51.8501 0.00202754
$$869$$ −14236.5 −0.555742
$$870$$ 0 0
$$871$$ 1834.70 0.0713735
$$872$$ 24352.6 0.945737
$$873$$ −48277.6 −1.87165
$$874$$ −42568.0 −1.64746
$$875$$ 0 0
$$876$$ −4293.75 −0.165608
$$877$$ −19728.7 −0.759624 −0.379812 0.925064i $$-0.624011\pi$$
−0.379812 + 0.925064i $$0.624011\pi$$
$$878$$ 32304.3 1.24171
$$879$$ −18464.1 −0.708509
$$880$$ 0 0
$$881$$ 19473.9 0.744712 0.372356 0.928090i $$-0.378550\pi$$
0.372356 + 0.928090i $$0.378550\pi$$
$$882$$ 32676.4 1.24748
$$883$$ −49092.4 −1.87100 −0.935499 0.353329i $$-0.885050\pi$$
−0.935499 + 0.353329i $$0.885050\pi$$
$$884$$ 118.365 0.00450346
$$885$$ 0 0
$$886$$ 27599.5 1.04653
$$887$$ −9292.86 −0.351774 −0.175887 0.984410i $$-0.556279\pi$$
−0.175887 + 0.984410i $$0.556279\pi$$
$$888$$ −2430.12 −0.0918348
$$889$$ −4047.41 −0.152695
$$890$$ 0 0
$$891$$ 4525.85 0.170170
$$892$$ −2108.05 −0.0791288
$$893$$ 48112.0 1.80292
$$894$$ 52598.5 1.96774
$$895$$ 0 0
$$896$$ 3949.89 0.147273
$$897$$ −4731.10 −0.176106
$$898$$ 943.252 0.0350520
$$899$$ −787.273 −0.0292069
$$900$$ 0 0
$$901$$ 14136.7 0.522709
$$902$$ 7845.88 0.289622
$$903$$ −1405.53 −0.0517974
$$904$$ −6881.46 −0.253179
$$905$$ 0 0
$$906$$ 55811.5 2.04659
$$907$$ −37688.7 −1.37975 −0.689875 0.723928i $$-0.742335\pi$$
−0.689875 + 0.723928i $$0.742335\pi$$
$$908$$ −949.559 −0.0347051
$$909$$ −5789.42 −0.211246
$$910$$ 0 0
$$911$$ 33049.6 1.20196 0.600979 0.799265i $$-0.294778\pi$$
0.600979 + 0.799265i $$0.294778\pi$$
$$912$$ −65923.1 −2.39357
$$913$$ 4860.31 0.176180
$$914$$ −28870.0 −1.04479
$$915$$ 0 0
$$916$$ −1026.44 −0.0370247
$$917$$ 4917.06 0.177073
$$918$$ −7906.43 −0.284260
$$919$$ −23148.0 −0.830883 −0.415442 0.909620i $$-0.636373\pi$$
−0.415442 + 0.909620i $$0.636373\pi$$
$$920$$ 0 0
$$921$$ 13304.1 0.475986
$$922$$ −12933.4 −0.461973
$$923$$ 1111.26 0.0396290
$$924$$ −143.563 −0.00511134
$$925$$ 0 0
$$926$$ 9374.21 0.332673
$$927$$ 1246.34 0.0441588
$$928$$ 605.131 0.0214056
$$929$$ −23177.9 −0.818561 −0.409280 0.912409i $$-0.634220\pi$$
−0.409280 + 0.912409i $$0.634220\pi$$
$$930$$ 0 0
$$931$$ −46673.3 −1.64302
$$932$$ 2355.98 0.0828033
$$933$$ 28325.1 0.993916
$$934$$ 13979.8 0.489757
$$935$$ 0 0
$$936$$ −4481.13 −0.156485
$$937$$ 34574.7 1.20545 0.602724 0.797950i $$-0.294082\pi$$
0.602724 + 0.797950i $$0.294082\pi$$
$$938$$ −2873.18 −0.100014
$$939$$ −56959.1 −1.97954
$$940$$ 0 0
$$941$$ 41831.2 1.44916 0.724578 0.689192i $$-0.242034\pi$$
0.724578 + 0.689192i $$0.242034\pi$$
$$942$$ 53629.6 1.85493
$$943$$ 29071.3 1.00392
$$944$$ −5251.40 −0.181058
$$945$$ 0 0
$$946$$ 1734.42 0.0596097
$$947$$ −27231.2 −0.934419 −0.467209 0.884147i $$-0.654741\pi$$
−0.467209 + 0.884147i $$0.654741\pi$$
$$948$$ −5498.79 −0.188389
$$949$$ −5415.79 −0.185252
$$950$$ 0 0
$$951$$ 124.534 0.00424635
$$952$$ −2952.50 −0.100516
$$953$$ −40939.4 −1.39156 −0.695781 0.718254i $$-0.744942\pi$$
−0.695781 + 0.718254i $$0.744942\pi$$
$$954$$ −33600.3 −1.14030
$$955$$ 0 0
$$956$$ 2188.87 0.0740515
$$957$$ 2179.81 0.0736292
$$958$$ −31601.3 −1.06575
$$959$$ 4950.74 0.166703
$$960$$ 0 0
$$961$$ −28798.9 −0.966698
$$962$$ −192.436 −0.00644945
$$963$$ −29839.0 −0.998491
$$964$$ −2094.60 −0.0699820
$$965$$ 0 0
$$966$$ 7409.03 0.246772
$$967$$ 46173.1 1.53550 0.767750 0.640750i $$-0.221376\pi$$
0.767750 + 0.640750i $$0.221376\pi$$
$$968$$ 2821.78 0.0936937
$$969$$ 45721.8 1.51579
$$970$$ 0 0
$$971$$ −5153.91 −0.170337 −0.0851683 0.996367i $$-0.527143\pi$$
−0.0851683 + 0.996367i $$0.527143\pi$$
$$972$$ 2764.06 0.0912111
$$973$$ 97.8734 0.00322474
$$974$$ −50069.0 −1.64714
$$975$$ 0 0
$$976$$ −43897.6 −1.43968
$$977$$ −9692.13 −0.317378 −0.158689 0.987329i $$-0.550727\pi$$
−0.158689 + 0.987329i $$0.550727\pi$$
$$978$$ −59026.5 −1.92992
$$979$$ 16380.2 0.534744
$$980$$ 0 0
$$981$$ 37443.3 1.21863
$$982$$ 20811.6 0.676299
$$983$$ −32915.7 −1.06800 −0.534002 0.845483i $$-0.679313\pi$$
−0.534002 + 0.845483i $$0.679313\pi$$
$$984$$ 48269.5 1.56380
$$985$$ 0 0
$$986$$ 2814.48 0.0909041
$$987$$ −8373.97 −0.270057
$$988$$ 401.841 0.0129395
$$989$$ 6426.54 0.206625
$$990$$ 0 0
$$991$$ 29477.9 0.944901 0.472451 0.881357i $$-0.343370\pi$$
0.472451 + 0.881357i $$0.343370\pi$$
$$992$$ −762.560 −0.0244066
$$993$$ −10456.9 −0.334180
$$994$$ −1740.26 −0.0555310
$$995$$ 0 0
$$996$$ 1877.28 0.0597227
$$997$$ 31944.4 1.01473 0.507366 0.861731i $$-0.330619\pi$$
0.507366 + 0.861731i $$0.330619\pi$$
$$998$$ −35268.4 −1.11864
$$999$$ −922.883 −0.0292279
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 275.4.a.b.1.1 2
3.2 odd 2 2475.4.a.q.1.2 2
5.2 odd 4 275.4.b.c.199.1 4
5.3 odd 4 275.4.b.c.199.4 4
5.4 even 2 11.4.a.a.1.2 2
15.14 odd 2 99.4.a.c.1.1 2
20.19 odd 2 176.4.a.i.1.2 2
35.34 odd 2 539.4.a.e.1.2 2
40.19 odd 2 704.4.a.n.1.1 2
40.29 even 2 704.4.a.p.1.2 2
55.4 even 10 121.4.c.c.27.2 8
55.9 even 10 121.4.c.c.81.1 8
55.14 even 10 121.4.c.c.9.2 8
55.19 odd 10 121.4.c.f.9.1 8
55.24 odd 10 121.4.c.f.81.2 8
55.29 odd 10 121.4.c.f.27.1 8
55.39 odd 10 121.4.c.f.3.2 8
55.49 even 10 121.4.c.c.3.1 8
55.54 odd 2 121.4.a.c.1.1 2
60.59 even 2 1584.4.a.bc.1.1 2
65.64 even 2 1859.4.a.a.1.1 2
165.164 even 2 1089.4.a.v.1.2 2
220.219 even 2 1936.4.a.w.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 5.4 even 2
99.4.a.c.1.1 2 15.14 odd 2
121.4.a.c.1.1 2 55.54 odd 2
121.4.c.c.3.1 8 55.49 even 10
121.4.c.c.9.2 8 55.14 even 10
121.4.c.c.27.2 8 55.4 even 10
121.4.c.c.81.1 8 55.9 even 10
121.4.c.f.3.2 8 55.39 odd 10
121.4.c.f.9.1 8 55.19 odd 10
121.4.c.f.27.1 8 55.29 odd 10
121.4.c.f.81.2 8 55.24 odd 10
176.4.a.i.1.2 2 20.19 odd 2
275.4.a.b.1.1 2 1.1 even 1 trivial
275.4.b.c.199.1 4 5.2 odd 4
275.4.b.c.199.4 4 5.3 odd 4
539.4.a.e.1.2 2 35.34 odd 2
704.4.a.n.1.1 2 40.19 odd 2
704.4.a.p.1.2 2 40.29 even 2
1089.4.a.v.1.2 2 165.164 even 2
1584.4.a.bc.1.1 2 60.59 even 2
1859.4.a.a.1.1 2 65.64 even 2
1936.4.a.w.1.2 2 220.219 even 2
2475.4.a.q.1.2 2 3.2 odd 2