Properties

 Label 5225.2.a.bb.1.12 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $22$ Inner twists $2$

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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: $$22$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.12 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)

 $$f(q)$$ $$=$$ $$q+0.104144 q^{2} +0.696995 q^{3} -1.98915 q^{4} +0.0725876 q^{6} -3.21657 q^{7} -0.415445 q^{8} -2.51420 q^{9} +O(q^{10})$$ $$q+0.104144 q^{2} +0.696995 q^{3} -1.98915 q^{4} +0.0725876 q^{6} -3.21657 q^{7} -0.415445 q^{8} -2.51420 q^{9} -1.00000 q^{11} -1.38643 q^{12} -4.65405 q^{13} -0.334986 q^{14} +3.93504 q^{16} +0.00602394 q^{17} -0.261838 q^{18} -1.00000 q^{19} -2.24194 q^{21} -0.104144 q^{22} +1.65858 q^{23} -0.289563 q^{24} -0.484690 q^{26} -3.84337 q^{27} +6.39826 q^{28} +3.49988 q^{29} -3.34225 q^{31} +1.24070 q^{32} -0.696995 q^{33} +0.000627356 q^{34} +5.00113 q^{36} -6.13742 q^{37} -0.104144 q^{38} -3.24385 q^{39} -9.38032 q^{41} -0.233483 q^{42} -8.61740 q^{43} +1.98915 q^{44} +0.172731 q^{46} -0.893231 q^{47} +2.74270 q^{48} +3.34635 q^{49} +0.00419866 q^{51} +9.25763 q^{52} -1.78431 q^{53} -0.400263 q^{54} +1.33631 q^{56} -0.696995 q^{57} +0.364491 q^{58} +2.57022 q^{59} -3.68861 q^{61} -0.348074 q^{62} +8.08710 q^{63} -7.74087 q^{64} -0.0725876 q^{66} -5.98644 q^{67} -0.0119826 q^{68} +1.15602 q^{69} +0.852858 q^{71} +1.04451 q^{72} -4.72786 q^{73} -0.639174 q^{74} +1.98915 q^{76} +3.21657 q^{77} -0.337827 q^{78} -4.94098 q^{79} +4.86379 q^{81} -0.976902 q^{82} +15.0339 q^{83} +4.45956 q^{84} -0.897448 q^{86} +2.43940 q^{87} +0.415445 q^{88} +5.11449 q^{89} +14.9701 q^{91} -3.29917 q^{92} -2.32953 q^{93} -0.0930244 q^{94} +0.864762 q^{96} +16.5462 q^{97} +0.348501 q^{98} +2.51420 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$22 q + 32 q^{4} - 12 q^{6} + 34 q^{9}+O(q^{10})$$ 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 $$22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44} + 88 q^{46} + 28 q^{49} + 8 q^{51} + 38 q^{54} + 44 q^{56} + 40 q^{59} - 6 q^{61} + 140 q^{64} + 12 q^{66} + 74 q^{69} + 62 q^{71} - 26 q^{74} - 32 q^{76} + 102 q^{79} + 94 q^{81} - 38 q^{84} + 28 q^{86} + 54 q^{89} + 88 q^{91} + 36 q^{94} + 2 q^{96} - 34 q^{99}+O(q^{100})$$ 22 * q + 32 * q^4 - 12 * q^6 + 34 * q^9 - 22 * q^11 - 8 * q^14 + 40 * q^16 - 22 * q^19 - 22 * q^21 - 22 * q^24 + 16 * q^26 - 10 * q^29 + 76 * q^31 + 56 * q^34 + 104 * q^36 - 8 * q^39 + 6 * q^41 - 32 * q^44 + 88 * q^46 + 28 * q^49 + 8 * q^51 + 38 * q^54 + 44 * q^56 + 40 * q^59 - 6 * q^61 + 140 * q^64 + 12 * q^66 + 74 * q^69 + 62 * q^71 - 26 * q^74 - 32 * q^76 + 102 * q^79 + 94 * q^81 - 38 * q^84 + 28 * q^86 + 54 * q^89 + 88 * q^91 + 36 * q^94 + 2 * q^96 - 34 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.104144 0.0736407 0.0368204 0.999322i $$-0.488277\pi$$
0.0368204 + 0.999322i $$0.488277\pi$$
$$3$$ 0.696995 0.402410 0.201205 0.979549i $$-0.435514\pi$$
0.201205 + 0.979549i $$0.435514\pi$$
$$4$$ −1.98915 −0.994577
$$5$$ 0 0
$$6$$ 0.0725876 0.0296338
$$7$$ −3.21657 −1.21575 −0.607875 0.794033i $$-0.707978\pi$$
−0.607875 + 0.794033i $$0.707978\pi$$
$$8$$ −0.415445 −0.146882
$$9$$ −2.51420 −0.838066
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −1.38643 −0.400228
$$13$$ −4.65405 −1.29080 −0.645401 0.763844i $$-0.723310\pi$$
−0.645401 + 0.763844i $$0.723310\pi$$
$$14$$ −0.334986 −0.0895287
$$15$$ 0 0
$$16$$ 3.93504 0.983761
$$17$$ 0.00602394 0.00146102 0.000730510 1.00000i $$-0.499767\pi$$
0.000730510 1.00000i $$0.499767\pi$$
$$18$$ −0.261838 −0.0617158
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −2.24194 −0.489230
$$22$$ −0.104144 −0.0222035
$$23$$ 1.65858 0.345838 0.172919 0.984936i $$-0.444680\pi$$
0.172919 + 0.984936i $$0.444680\pi$$
$$24$$ −0.289563 −0.0591069
$$25$$ 0 0
$$26$$ −0.484690 −0.0950556
$$27$$ −3.84337 −0.739657
$$28$$ 6.39826 1.20916
$$29$$ 3.49988 0.649912 0.324956 0.945729i $$-0.394650\pi$$
0.324956 + 0.945729i $$0.394650\pi$$
$$30$$ 0 0
$$31$$ −3.34225 −0.600285 −0.300143 0.953894i $$-0.597034\pi$$
−0.300143 + 0.953894i $$0.597034\pi$$
$$32$$ 1.24070 0.219327
$$33$$ −0.696995 −0.121331
$$34$$ 0.000627356 0 0.000107591 0
$$35$$ 0 0
$$36$$ 5.00113 0.833521
$$37$$ −6.13742 −1.00899 −0.504493 0.863416i $$-0.668320\pi$$
−0.504493 + 0.863416i $$0.668320\pi$$
$$38$$ −0.104144 −0.0168943
$$39$$ −3.24385 −0.519432
$$40$$ 0 0
$$41$$ −9.38032 −1.46496 −0.732480 0.680788i $$-0.761637\pi$$
−0.732480 + 0.680788i $$0.761637\pi$$
$$42$$ −0.233483 −0.0360273
$$43$$ −8.61740 −1.31414 −0.657071 0.753829i $$-0.728205\pi$$
−0.657071 + 0.753829i $$0.728205\pi$$
$$44$$ 1.98915 0.299876
$$45$$ 0 0
$$46$$ 0.172731 0.0254677
$$47$$ −0.893231 −0.130291 −0.0651456 0.997876i $$-0.520751\pi$$
−0.0651456 + 0.997876i $$0.520751\pi$$
$$48$$ 2.74270 0.395875
$$49$$ 3.34635 0.478049
$$50$$ 0 0
$$51$$ 0.00419866 0.000587930 0
$$52$$ 9.25763 1.28380
$$53$$ −1.78431 −0.245093 −0.122547 0.992463i $$-0.539106\pi$$
−0.122547 + 0.992463i $$0.539106\pi$$
$$54$$ −0.400263 −0.0544688
$$55$$ 0 0
$$56$$ 1.33631 0.178572
$$57$$ −0.696995 −0.0923192
$$58$$ 0.364491 0.0478600
$$59$$ 2.57022 0.334614 0.167307 0.985905i $$-0.446493\pi$$
0.167307 + 0.985905i $$0.446493\pi$$
$$60$$ 0 0
$$61$$ −3.68861 −0.472278 −0.236139 0.971719i $$-0.575882\pi$$
−0.236139 + 0.971719i $$0.575882\pi$$
$$62$$ −0.348074 −0.0442054
$$63$$ 8.08710 1.01888
$$64$$ −7.74087 −0.967609
$$65$$ 0 0
$$66$$ −0.0725876 −0.00893492
$$67$$ −5.98644 −0.731361 −0.365680 0.930741i $$-0.619164\pi$$
−0.365680 + 0.930741i $$0.619164\pi$$
$$68$$ −0.0119826 −0.00145310
$$69$$ 1.15602 0.139169
$$70$$ 0 0
$$71$$ 0.852858 0.101216 0.0506078 0.998719i $$-0.483884\pi$$
0.0506078 + 0.998719i $$0.483884\pi$$
$$72$$ 1.04451 0.123097
$$73$$ −4.72786 −0.553354 −0.276677 0.960963i $$-0.589233\pi$$
−0.276677 + 0.960963i $$0.589233\pi$$
$$74$$ −0.639174 −0.0743024
$$75$$ 0 0
$$76$$ 1.98915 0.228172
$$77$$ 3.21657 0.366563
$$78$$ −0.337827 −0.0382513
$$79$$ −4.94098 −0.555903 −0.277952 0.960595i $$-0.589656\pi$$
−0.277952 + 0.960595i $$0.589656\pi$$
$$80$$ 0 0
$$81$$ 4.86379 0.540421
$$82$$ −0.976902 −0.107881
$$83$$ 15.0339 1.65019 0.825093 0.564997i $$-0.191123\pi$$
0.825093 + 0.564997i $$0.191123\pi$$
$$84$$ 4.45956 0.486577
$$85$$ 0 0
$$86$$ −0.897448 −0.0967743
$$87$$ 2.43940 0.261531
$$88$$ 0.415445 0.0442866
$$89$$ 5.11449 0.542135 0.271067 0.962560i $$-0.412623\pi$$
0.271067 + 0.962560i $$0.412623\pi$$
$$90$$ 0 0
$$91$$ 14.9701 1.56929
$$92$$ −3.29917 −0.343962
$$93$$ −2.32953 −0.241561
$$94$$ −0.0930244 −0.00959474
$$95$$ 0 0
$$96$$ 0.864762 0.0882594
$$97$$ 16.5462 1.68001 0.840005 0.542578i $$-0.182552\pi$$
0.840005 + 0.542578i $$0.182552\pi$$
$$98$$ 0.348501 0.0352039
$$99$$ 2.51420 0.252686
$$100$$ 0 0
$$101$$ 5.62840 0.560047 0.280023 0.959993i $$-0.409658\pi$$
0.280023 + 0.959993i $$0.409658\pi$$
$$102$$ 0.000437264 0 4.32956e−5 0
$$103$$ 7.56366 0.745270 0.372635 0.927978i $$-0.378454\pi$$
0.372635 + 0.927978i $$0.378454\pi$$
$$104$$ 1.93350 0.189596
$$105$$ 0 0
$$106$$ −0.185824 −0.0180488
$$107$$ −5.01113 −0.484444 −0.242222 0.970221i $$-0.577876\pi$$
−0.242222 + 0.970221i $$0.577876\pi$$
$$108$$ 7.64505 0.735645
$$109$$ 3.73155 0.357418 0.178709 0.983902i $$-0.442808\pi$$
0.178709 + 0.983902i $$0.442808\pi$$
$$110$$ 0 0
$$111$$ −4.27775 −0.406026
$$112$$ −12.6574 −1.19601
$$113$$ 16.9702 1.59642 0.798209 0.602380i $$-0.205781\pi$$
0.798209 + 0.602380i $$0.205781\pi$$
$$114$$ −0.0725876 −0.00679846
$$115$$ 0 0
$$116$$ −6.96181 −0.646388
$$117$$ 11.7012 1.08178
$$118$$ 0.267672 0.0246412
$$119$$ −0.0193765 −0.00177624
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −0.384146 −0.0347789
$$123$$ −6.53804 −0.589515
$$124$$ 6.64824 0.597030
$$125$$ 0 0
$$126$$ 0.842221 0.0750310
$$127$$ −0.870581 −0.0772516 −0.0386258 0.999254i $$-0.512298\pi$$
−0.0386258 + 0.999254i $$0.512298\pi$$
$$128$$ −3.28756 −0.290582
$$129$$ −6.00628 −0.528824
$$130$$ 0 0
$$131$$ −0.332706 −0.0290687 −0.0145343 0.999894i $$-0.504627\pi$$
−0.0145343 + 0.999894i $$0.504627\pi$$
$$132$$ 1.38643 0.120673
$$133$$ 3.21657 0.278912
$$134$$ −0.623450 −0.0538579
$$135$$ 0 0
$$136$$ −0.00250262 −0.000214598 0
$$137$$ 13.0314 1.11335 0.556673 0.830732i $$-0.312078\pi$$
0.556673 + 0.830732i $$0.312078\pi$$
$$138$$ 0.120392 0.0102485
$$139$$ 12.6827 1.07573 0.537867 0.843030i $$-0.319230\pi$$
0.537867 + 0.843030i $$0.319230\pi$$
$$140$$ 0 0
$$141$$ −0.622578 −0.0524305
$$142$$ 0.0888197 0.00745359
$$143$$ 4.65405 0.389192
$$144$$ −9.89348 −0.824456
$$145$$ 0 0
$$146$$ −0.492376 −0.0407494
$$147$$ 2.33239 0.192372
$$148$$ 12.2083 1.00351
$$149$$ 17.2444 1.41272 0.706358 0.707855i $$-0.250337\pi$$
0.706358 + 0.707855i $$0.250337\pi$$
$$150$$ 0 0
$$151$$ 22.5478 1.83491 0.917456 0.397836i $$-0.130239\pi$$
0.917456 + 0.397836i $$0.130239\pi$$
$$152$$ 0.415445 0.0336971
$$153$$ −0.0151454 −0.00122443
$$154$$ 0.334986 0.0269939
$$155$$ 0 0
$$156$$ 6.45252 0.516615
$$157$$ −12.1662 −0.970972 −0.485486 0.874244i $$-0.661357\pi$$
−0.485486 + 0.874244i $$0.661357\pi$$
$$158$$ −0.514572 −0.0409371
$$159$$ −1.24365 −0.0986280
$$160$$ 0 0
$$161$$ −5.33494 −0.420452
$$162$$ 0.506533 0.0397970
$$163$$ −21.8086 −1.70818 −0.854090 0.520125i $$-0.825885\pi$$
−0.854090 + 0.520125i $$0.825885\pi$$
$$164$$ 18.6589 1.45702
$$165$$ 0 0
$$166$$ 1.56569 0.121521
$$167$$ 15.5829 1.20584 0.602919 0.797802i $$-0.294004\pi$$
0.602919 + 0.797802i $$0.294004\pi$$
$$168$$ 0.931402 0.0718592
$$169$$ 8.66022 0.666170
$$170$$ 0 0
$$171$$ 2.51420 0.192266
$$172$$ 17.1413 1.30701
$$173$$ −10.4213 −0.792315 −0.396157 0.918183i $$-0.629656\pi$$
−0.396157 + 0.918183i $$0.629656\pi$$
$$174$$ 0.254048 0.0192594
$$175$$ 0 0
$$176$$ −3.93504 −0.296615
$$177$$ 1.79143 0.134652
$$178$$ 0.532642 0.0399232
$$179$$ 20.5356 1.53490 0.767452 0.641107i $$-0.221524\pi$$
0.767452 + 0.641107i $$0.221524\pi$$
$$180$$ 0 0
$$181$$ 0.0667973 0.00496500 0.00248250 0.999997i $$-0.499210\pi$$
0.00248250 + 0.999997i $$0.499210\pi$$
$$182$$ 1.55904 0.115564
$$183$$ −2.57094 −0.190050
$$184$$ −0.689049 −0.0507974
$$185$$ 0 0
$$186$$ −0.242606 −0.0177887
$$187$$ −0.00602394 −0.000440514 0
$$188$$ 1.77677 0.129585
$$189$$ 12.3625 0.899238
$$190$$ 0 0
$$191$$ −2.09641 −0.151691 −0.0758456 0.997120i $$-0.524166\pi$$
−0.0758456 + 0.997120i $$0.524166\pi$$
$$192$$ −5.39535 −0.389376
$$193$$ −25.1960 −1.81365 −0.906825 0.421507i $$-0.861501\pi$$
−0.906825 + 0.421507i $$0.861501\pi$$
$$194$$ 1.72318 0.123717
$$195$$ 0 0
$$196$$ −6.65640 −0.475457
$$197$$ −5.70885 −0.406739 −0.203369 0.979102i $$-0.565189\pi$$
−0.203369 + 0.979102i $$0.565189\pi$$
$$198$$ 0.261838 0.0186080
$$199$$ −10.2003 −0.723081 −0.361541 0.932356i $$-0.617749\pi$$
−0.361541 + 0.932356i $$0.617749\pi$$
$$200$$ 0 0
$$201$$ −4.17252 −0.294307
$$202$$ 0.586162 0.0412422
$$203$$ −11.2576 −0.790131
$$204$$ −0.00835178 −0.000584741 0
$$205$$ 0 0
$$206$$ 0.787708 0.0548822
$$207$$ −4.17000 −0.289835
$$208$$ −18.3139 −1.26984
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ 1.03962 0.0715706 0.0357853 0.999360i $$-0.488607\pi$$
0.0357853 + 0.999360i $$0.488607\pi$$
$$212$$ 3.54926 0.243764
$$213$$ 0.594437 0.0407302
$$214$$ −0.521877 −0.0356748
$$215$$ 0 0
$$216$$ 1.59671 0.108642
$$217$$ 10.7506 0.729797
$$218$$ 0.388618 0.0263205
$$219$$ −3.29529 −0.222675
$$220$$ 0 0
$$221$$ −0.0280358 −0.00188589
$$222$$ −0.445501 −0.0299001
$$223$$ −6.42634 −0.430339 −0.215170 0.976577i $$-0.569030\pi$$
−0.215170 + 0.976577i $$0.569030\pi$$
$$224$$ −3.99080 −0.266647
$$225$$ 0 0
$$226$$ 1.76734 0.117561
$$227$$ −6.25815 −0.415368 −0.207684 0.978196i $$-0.566593\pi$$
−0.207684 + 0.978196i $$0.566593\pi$$
$$228$$ 1.38643 0.0918186
$$229$$ 26.2671 1.73578 0.867890 0.496757i $$-0.165476\pi$$
0.867890 + 0.496757i $$0.165476\pi$$
$$230$$ 0 0
$$231$$ 2.24194 0.147509
$$232$$ −1.45401 −0.0954605
$$233$$ −2.94990 −0.193255 −0.0966273 0.995321i $$-0.530805\pi$$
−0.0966273 + 0.995321i $$0.530805\pi$$
$$234$$ 1.21861 0.0796629
$$235$$ 0 0
$$236$$ −5.11257 −0.332800
$$237$$ −3.44384 −0.223701
$$238$$ −0.00201794 −0.000130803 0
$$239$$ 18.5942 1.20276 0.601378 0.798965i $$-0.294619\pi$$
0.601378 + 0.798965i $$0.294619\pi$$
$$240$$ 0 0
$$241$$ −8.68183 −0.559246 −0.279623 0.960110i $$-0.590209\pi$$
−0.279623 + 0.960110i $$0.590209\pi$$
$$242$$ 0.104144 0.00669461
$$243$$ 14.9201 0.957127
$$244$$ 7.33722 0.469717
$$245$$ 0 0
$$246$$ −0.680896 −0.0434123
$$247$$ 4.65405 0.296130
$$248$$ 1.38852 0.0881711
$$249$$ 10.4786 0.664052
$$250$$ 0 0
$$251$$ −27.7431 −1.75113 −0.875565 0.483100i $$-0.839511\pi$$
−0.875565 + 0.483100i $$0.839511\pi$$
$$252$$ −16.0865 −1.01335
$$253$$ −1.65858 −0.104274
$$254$$ −0.0906655 −0.00568886
$$255$$ 0 0
$$256$$ 15.1394 0.946210
$$257$$ 17.7003 1.10412 0.552059 0.833805i $$-0.313842\pi$$
0.552059 + 0.833805i $$0.313842\pi$$
$$258$$ −0.625517 −0.0389430
$$259$$ 19.7415 1.22667
$$260$$ 0 0
$$261$$ −8.79940 −0.544669
$$262$$ −0.0346492 −0.00214064
$$263$$ −22.4026 −1.38140 −0.690702 0.723139i $$-0.742698\pi$$
−0.690702 + 0.723139i $$0.742698\pi$$
$$264$$ 0.289563 0.0178214
$$265$$ 0 0
$$266$$ 0.334986 0.0205393
$$267$$ 3.56477 0.218161
$$268$$ 11.9080 0.727394
$$269$$ 16.1724 0.986046 0.493023 0.870016i $$-0.335892\pi$$
0.493023 + 0.870016i $$0.335892\pi$$
$$270$$ 0 0
$$271$$ 2.09973 0.127550 0.0637748 0.997964i $$-0.479686\pi$$
0.0637748 + 0.997964i $$0.479686\pi$$
$$272$$ 0.0237045 0.00143729
$$273$$ 10.4341 0.631500
$$274$$ 1.35714 0.0819876
$$275$$ 0 0
$$276$$ −2.29950 −0.138414
$$277$$ −29.9790 −1.80127 −0.900633 0.434580i $$-0.856897\pi$$
−0.900633 + 0.434580i $$0.856897\pi$$
$$278$$ 1.32082 0.0792178
$$279$$ 8.40307 0.503078
$$280$$ 0 0
$$281$$ −0.469805 −0.0280262 −0.0140131 0.999902i $$-0.504461\pi$$
−0.0140131 + 0.999902i $$0.504461\pi$$
$$282$$ −0.0648376 −0.00386102
$$283$$ 13.2122 0.785386 0.392693 0.919670i $$-0.371543\pi$$
0.392693 + 0.919670i $$0.371543\pi$$
$$284$$ −1.69647 −0.100667
$$285$$ 0 0
$$286$$ 0.484690 0.0286603
$$287$$ 30.1725 1.78103
$$288$$ −3.11937 −0.183810
$$289$$ −17.0000 −0.999998
$$290$$ 0 0
$$291$$ 11.5326 0.676054
$$292$$ 9.40443 0.550353
$$293$$ −5.47146 −0.319646 −0.159823 0.987146i $$-0.551092\pi$$
−0.159823 + 0.987146i $$0.551092\pi$$
$$294$$ 0.242903 0.0141664
$$295$$ 0 0
$$296$$ 2.54976 0.148202
$$297$$ 3.84337 0.223015
$$298$$ 1.79590 0.104033
$$299$$ −7.71912 −0.446408
$$300$$ 0 0
$$301$$ 27.7185 1.59767
$$302$$ 2.34821 0.135124
$$303$$ 3.92297 0.225369
$$304$$ −3.93504 −0.225690
$$305$$ 0 0
$$306$$ −0.00157730 −9.01681e−5 0
$$307$$ −30.8787 −1.76234 −0.881169 0.472801i $$-0.843243\pi$$
−0.881169 + 0.472801i $$0.843243\pi$$
$$308$$ −6.39826 −0.364575
$$309$$ 5.27183 0.299904
$$310$$ 0 0
$$311$$ 24.2528 1.37525 0.687625 0.726066i $$-0.258653\pi$$
0.687625 + 0.726066i $$0.258653\pi$$
$$312$$ 1.34764 0.0762953
$$313$$ 12.2922 0.694796 0.347398 0.937718i $$-0.387065\pi$$
0.347398 + 0.937718i $$0.387065\pi$$
$$314$$ −1.26704 −0.0715031
$$315$$ 0 0
$$316$$ 9.82837 0.552889
$$317$$ 20.0634 1.12687 0.563436 0.826160i $$-0.309479\pi$$
0.563436 + 0.826160i $$0.309479\pi$$
$$318$$ −0.129519 −0.00726304
$$319$$ −3.49988 −0.195956
$$320$$ 0 0
$$321$$ −3.49273 −0.194945
$$322$$ −0.555601 −0.0309624
$$323$$ −0.00602394 −0.000335181 0
$$324$$ −9.67482 −0.537490
$$325$$ 0 0
$$326$$ −2.27123 −0.125792
$$327$$ 2.60087 0.143829
$$328$$ 3.89701 0.215176
$$329$$ 2.87314 0.158402
$$330$$ 0 0
$$331$$ −28.3147 −1.55632 −0.778159 0.628067i $$-0.783846\pi$$
−0.778159 + 0.628067i $$0.783846\pi$$
$$332$$ −29.9048 −1.64124
$$333$$ 15.4307 0.845596
$$334$$ 1.62286 0.0887988
$$335$$ 0 0
$$336$$ −8.82211 −0.481286
$$337$$ −35.6067 −1.93962 −0.969810 0.243863i $$-0.921585\pi$$
−0.969810 + 0.243863i $$0.921585\pi$$
$$338$$ 0.901907 0.0490573
$$339$$ 11.8281 0.642415
$$340$$ 0 0
$$341$$ 3.34225 0.180993
$$342$$ 0.261838 0.0141586
$$343$$ 11.7522 0.634562
$$344$$ 3.58006 0.193024
$$345$$ 0 0
$$346$$ −1.08531 −0.0583466
$$347$$ −8.36165 −0.448877 −0.224438 0.974488i $$-0.572055\pi$$
−0.224438 + 0.974488i $$0.572055\pi$$
$$348$$ −4.85235 −0.260113
$$349$$ −26.3753 −1.41184 −0.705919 0.708293i $$-0.749466\pi$$
−0.705919 + 0.708293i $$0.749466\pi$$
$$350$$ 0 0
$$351$$ 17.8872 0.954750
$$352$$ −1.24070 −0.0661296
$$353$$ −13.3537 −0.710745 −0.355372 0.934725i $$-0.615646\pi$$
−0.355372 + 0.934725i $$0.615646\pi$$
$$354$$ 0.186566 0.00991589
$$355$$ 0 0
$$356$$ −10.1735 −0.539195
$$357$$ −0.0135053 −0.000714776 0
$$358$$ 2.13865 0.113031
$$359$$ 14.7872 0.780439 0.390220 0.920722i $$-0.372399\pi$$
0.390220 + 0.920722i $$0.372399\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0.00695651 0.000365626 0
$$363$$ 0.696995 0.0365827
$$364$$ −29.7778 −1.56078
$$365$$ 0 0
$$366$$ −0.267748 −0.0139954
$$367$$ −26.2183 −1.36858 −0.684291 0.729209i $$-0.739888\pi$$
−0.684291 + 0.729209i $$0.739888\pi$$
$$368$$ 6.52658 0.340221
$$369$$ 23.5840 1.22773
$$370$$ 0 0
$$371$$ 5.73935 0.297972
$$372$$ 4.63379 0.240251
$$373$$ 1.30489 0.0675648 0.0337824 0.999429i $$-0.489245\pi$$
0.0337824 + 0.999429i $$0.489245\pi$$
$$374$$ −0.000627356 0 −3.24398e−5 0
$$375$$ 0 0
$$376$$ 0.371089 0.0191374
$$377$$ −16.2887 −0.838908
$$378$$ 1.28747 0.0662205
$$379$$ −10.0189 −0.514634 −0.257317 0.966327i $$-0.582838\pi$$
−0.257317 + 0.966327i $$0.582838\pi$$
$$380$$ 0 0
$$381$$ −0.606791 −0.0310868
$$382$$ −0.218328 −0.0111706
$$383$$ −5.44686 −0.278322 −0.139161 0.990270i $$-0.544440\pi$$
−0.139161 + 0.990270i $$0.544440\pi$$
$$384$$ −2.29142 −0.116933
$$385$$ 0 0
$$386$$ −2.62401 −0.133559
$$387$$ 21.6659 1.10134
$$388$$ −32.9129 −1.67090
$$389$$ −7.59394 −0.385028 −0.192514 0.981294i $$-0.561664\pi$$
−0.192514 + 0.981294i $$0.561664\pi$$
$$390$$ 0 0
$$391$$ 0.00999119 0.000505276 0
$$392$$ −1.39022 −0.0702169
$$393$$ −0.231894 −0.0116975
$$394$$ −0.594541 −0.0299525
$$395$$ 0 0
$$396$$ −5.00113 −0.251316
$$397$$ 6.27453 0.314910 0.157455 0.987526i $$-0.449671\pi$$
0.157455 + 0.987526i $$0.449671\pi$$
$$398$$ −1.06230 −0.0532482
$$399$$ 2.24194 0.112237
$$400$$ 0 0
$$401$$ 17.8048 0.889131 0.444565 0.895746i $$-0.353358\pi$$
0.444565 + 0.895746i $$0.353358\pi$$
$$402$$ −0.434542 −0.0216730
$$403$$ 15.5550 0.774849
$$404$$ −11.1958 −0.557010
$$405$$ 0 0
$$406$$ −1.17241 −0.0581858
$$407$$ 6.13742 0.304221
$$408$$ −0.00174431 −8.63564e−5 0
$$409$$ 8.47192 0.418909 0.209455 0.977818i $$-0.432831\pi$$
0.209455 + 0.977818i $$0.432831\pi$$
$$410$$ 0 0
$$411$$ 9.08281 0.448022
$$412$$ −15.0453 −0.741228
$$413$$ −8.26731 −0.406807
$$414$$ −0.434279 −0.0213436
$$415$$ 0 0
$$416$$ −5.77429 −0.283108
$$417$$ 8.83978 0.432886
$$418$$ 0.104144 0.00509384
$$419$$ 10.4558 0.510799 0.255399 0.966836i $$-0.417793\pi$$
0.255399 + 0.966836i $$0.417793\pi$$
$$420$$ 0 0
$$421$$ 32.4814 1.58305 0.791523 0.611139i $$-0.209288\pi$$
0.791523 + 0.611139i $$0.209288\pi$$
$$422$$ 0.108270 0.00527051
$$423$$ 2.24576 0.109193
$$424$$ 0.741281 0.0359998
$$425$$ 0 0
$$426$$ 0.0619069 0.00299940
$$427$$ 11.8647 0.574173
$$428$$ 9.96790 0.481817
$$429$$ 3.24385 0.156615
$$430$$ 0 0
$$431$$ −2.28726 −0.110174 −0.0550868 0.998482i $$-0.517544\pi$$
−0.0550868 + 0.998482i $$0.517544\pi$$
$$432$$ −15.1238 −0.727645
$$433$$ −36.5224 −1.75515 −0.877576 0.479437i $$-0.840841\pi$$
−0.877576 + 0.479437i $$0.840841\pi$$
$$434$$ 1.11961 0.0537428
$$435$$ 0 0
$$436$$ −7.42264 −0.355480
$$437$$ −1.65858 −0.0793406
$$438$$ −0.343184 −0.0163980
$$439$$ −23.0323 −1.09927 −0.549636 0.835404i $$-0.685233\pi$$
−0.549636 + 0.835404i $$0.685233\pi$$
$$440$$ 0 0
$$441$$ −8.41338 −0.400637
$$442$$ −0.00291975 −0.000138878 0
$$443$$ −3.63563 −0.172734 −0.0863671 0.996263i $$-0.527526\pi$$
−0.0863671 + 0.996263i $$0.527526\pi$$
$$444$$ 8.50910 0.403824
$$445$$ 0 0
$$446$$ −0.669263 −0.0316905
$$447$$ 12.0193 0.568492
$$448$$ 24.8991 1.17637
$$449$$ 25.5726 1.20685 0.603423 0.797422i $$-0.293803\pi$$
0.603423 + 0.797422i $$0.293803\pi$$
$$450$$ 0 0
$$451$$ 9.38032 0.441702
$$452$$ −33.7563 −1.58776
$$453$$ 15.7157 0.738388
$$454$$ −0.651747 −0.0305880
$$455$$ 0 0
$$456$$ 0.289563 0.0135600
$$457$$ −6.45150 −0.301789 −0.150894 0.988550i $$-0.548215\pi$$
−0.150894 + 0.988550i $$0.548215\pi$$
$$458$$ 2.73555 0.127824
$$459$$ −0.0231522 −0.00108065
$$460$$ 0 0
$$461$$ −23.1293 −1.07724 −0.538619 0.842550i $$-0.681054\pi$$
−0.538619 + 0.842550i $$0.681054\pi$$
$$462$$ 0.233483 0.0108626
$$463$$ −36.4434 −1.69367 −0.846834 0.531858i $$-0.821494\pi$$
−0.846834 + 0.531858i $$0.821494\pi$$
$$464$$ 13.7722 0.639358
$$465$$ 0 0
$$466$$ −0.307214 −0.0142314
$$467$$ 36.0540 1.66838 0.834191 0.551476i $$-0.185935\pi$$
0.834191 + 0.551476i $$0.185935\pi$$
$$468$$ −23.2755 −1.07591
$$469$$ 19.2558 0.889152
$$470$$ 0 0
$$471$$ −8.47981 −0.390729
$$472$$ −1.06779 −0.0491488
$$473$$ 8.61740 0.396228
$$474$$ −0.358654 −0.0164735
$$475$$ 0 0
$$476$$ 0.0385428 0.00176660
$$477$$ 4.48610 0.205404
$$478$$ 1.93646 0.0885718
$$479$$ 17.2250 0.787032 0.393516 0.919318i $$-0.371259\pi$$
0.393516 + 0.919318i $$0.371259\pi$$
$$480$$ 0 0
$$481$$ 28.5639 1.30240
$$482$$ −0.904158 −0.0411832
$$483$$ −3.71843 −0.169194
$$484$$ −1.98915 −0.0904161
$$485$$ 0 0
$$486$$ 1.55384 0.0704835
$$487$$ 31.6437 1.43391 0.716957 0.697117i $$-0.245534\pi$$
0.716957 + 0.697117i $$0.245534\pi$$
$$488$$ 1.53242 0.0693692
$$489$$ −15.2005 −0.687389
$$490$$ 0 0
$$491$$ 38.9066 1.75583 0.877916 0.478814i $$-0.158933\pi$$
0.877916 + 0.478814i $$0.158933\pi$$
$$492$$ 13.0052 0.586318
$$493$$ 0.0210831 0.000949536 0
$$494$$ 0.484690 0.0218073
$$495$$ 0 0
$$496$$ −13.1519 −0.590537
$$497$$ −2.74328 −0.123053
$$498$$ 1.09128 0.0489013
$$499$$ 16.4194 0.735033 0.367516 0.930017i $$-0.380208\pi$$
0.367516 + 0.930017i $$0.380208\pi$$
$$500$$ 0 0
$$501$$ 10.8612 0.485242
$$502$$ −2.88927 −0.128954
$$503$$ −4.73308 −0.211038 −0.105519 0.994417i $$-0.533650\pi$$
−0.105519 + 0.994417i $$0.533650\pi$$
$$504$$ −3.35975 −0.149655
$$505$$ 0 0
$$506$$ −0.172731 −0.00767881
$$507$$ 6.03613 0.268074
$$508$$ 1.73172 0.0768326
$$509$$ −32.8750 −1.45716 −0.728580 0.684961i $$-0.759819\pi$$
−0.728580 + 0.684961i $$0.759819\pi$$
$$510$$ 0 0
$$511$$ 15.2075 0.672740
$$512$$ 8.15180 0.360262
$$513$$ 3.84337 0.169689
$$514$$ 1.84338 0.0813080
$$515$$ 0 0
$$516$$ 11.9474 0.525956
$$517$$ 0.893231 0.0392843
$$518$$ 2.05595 0.0903332
$$519$$ −7.26357 −0.318835
$$520$$ 0 0
$$521$$ 7.50334 0.328727 0.164364 0.986400i $$-0.447443\pi$$
0.164364 + 0.986400i $$0.447443\pi$$
$$522$$ −0.916402 −0.0401098
$$523$$ −11.7845 −0.515300 −0.257650 0.966238i $$-0.582948\pi$$
−0.257650 + 0.966238i $$0.582948\pi$$
$$524$$ 0.661803 0.0289110
$$525$$ 0 0
$$526$$ −2.33309 −0.101728
$$527$$ −0.0201335 −0.000877029 0
$$528$$ −2.74270 −0.119361
$$529$$ −20.2491 −0.880396
$$530$$ 0 0
$$531$$ −6.46204 −0.280429
$$532$$ −6.39826 −0.277400
$$533$$ 43.6565 1.89097
$$534$$ 0.371249 0.0160655
$$535$$ 0 0
$$536$$ 2.48704 0.107424
$$537$$ 14.3132 0.617661
$$538$$ 1.68425 0.0726131
$$539$$ −3.34635 −0.144137
$$540$$ 0 0
$$541$$ 18.8061 0.808535 0.404268 0.914641i $$-0.367526\pi$$
0.404268 + 0.914641i $$0.367526\pi$$
$$542$$ 0.218674 0.00939285
$$543$$ 0.0465574 0.00199797
$$544$$ 0.00747391 0.000320441 0
$$545$$ 0 0
$$546$$ 1.08664 0.0465041
$$547$$ −11.1690 −0.477550 −0.238775 0.971075i $$-0.576746\pi$$
−0.238775 + 0.971075i $$0.576746\pi$$
$$548$$ −25.9214 −1.10731
$$549$$ 9.27390 0.395800
$$550$$ 0 0
$$551$$ −3.49988 −0.149100
$$552$$ −0.480264 −0.0204414
$$553$$ 15.8930 0.675840
$$554$$ −3.12213 −0.132647
$$555$$ 0 0
$$556$$ −25.2279 −1.06990
$$557$$ −26.3044 −1.11455 −0.557276 0.830327i $$-0.688154\pi$$
−0.557276 + 0.830327i $$0.688154\pi$$
$$558$$ 0.875127 0.0370471
$$559$$ 40.1058 1.69630
$$560$$ 0 0
$$561$$ −0.00419866 −0.000177267 0
$$562$$ −0.0489273 −0.00206387
$$563$$ −27.3566 −1.15294 −0.576472 0.817117i $$-0.695571\pi$$
−0.576472 + 0.817117i $$0.695571\pi$$
$$564$$ 1.23840 0.0521462
$$565$$ 0 0
$$566$$ 1.37597 0.0578364
$$567$$ −15.6447 −0.657017
$$568$$ −0.354316 −0.0148668
$$569$$ −17.7921 −0.745882 −0.372941 0.927855i $$-0.621651\pi$$
−0.372941 + 0.927855i $$0.621651\pi$$
$$570$$ 0 0
$$571$$ 25.9440 1.08572 0.542862 0.839822i $$-0.317341\pi$$
0.542862 + 0.839822i $$0.317341\pi$$
$$572$$ −9.25763 −0.387081
$$573$$ −1.46119 −0.0610421
$$574$$ 3.14228 0.131156
$$575$$ 0 0
$$576$$ 19.4621 0.810920
$$577$$ 13.7994 0.574476 0.287238 0.957859i $$-0.407263\pi$$
0.287238 + 0.957859i $$0.407263\pi$$
$$578$$ −1.77044 −0.0736406
$$579$$ −17.5615 −0.729831
$$580$$ 0 0
$$581$$ −48.3577 −2.00621
$$582$$ 1.20105 0.0497851
$$583$$ 1.78431 0.0738984
$$584$$ 1.96417 0.0812777
$$585$$ 0 0
$$586$$ −0.569818 −0.0235390
$$587$$ −31.3230 −1.29284 −0.646420 0.762982i $$-0.723734\pi$$
−0.646420 + 0.762982i $$0.723734\pi$$
$$588$$ −4.63947 −0.191329
$$589$$ 3.34225 0.137715
$$590$$ 0 0
$$591$$ −3.97904 −0.163676
$$592$$ −24.1510 −0.992600
$$593$$ 33.8912 1.39174 0.695872 0.718166i $$-0.255018\pi$$
0.695872 + 0.718166i $$0.255018\pi$$
$$594$$ 0.400263 0.0164230
$$595$$ 0 0
$$596$$ −34.3018 −1.40506
$$597$$ −7.10957 −0.290975
$$598$$ −0.803897 −0.0328738
$$599$$ −20.5165 −0.838280 −0.419140 0.907922i $$-0.637668\pi$$
−0.419140 + 0.907922i $$0.637668\pi$$
$$600$$ 0 0
$$601$$ 23.2037 0.946501 0.473250 0.880928i $$-0.343081\pi$$
0.473250 + 0.880928i $$0.343081\pi$$
$$602$$ 2.88671 0.117653
$$603$$ 15.0511 0.612928
$$604$$ −44.8510 −1.82496
$$605$$ 0 0
$$606$$ 0.408552 0.0165963
$$607$$ −2.28113 −0.0925881 −0.0462941 0.998928i $$-0.514741\pi$$
−0.0462941 + 0.998928i $$0.514741\pi$$
$$608$$ −1.24070 −0.0503170
$$609$$ −7.84652 −0.317957
$$610$$ 0 0
$$611$$ 4.15715 0.168180
$$612$$ 0.0301265 0.00121779
$$613$$ −2.06670 −0.0834732 −0.0417366 0.999129i $$-0.513289\pi$$
−0.0417366 + 0.999129i $$0.513289\pi$$
$$614$$ −3.21582 −0.129780
$$615$$ 0 0
$$616$$ −1.33631 −0.0538415
$$617$$ −39.7201 −1.59907 −0.799536 0.600618i $$-0.794921\pi$$
−0.799536 + 0.600618i $$0.794921\pi$$
$$618$$ 0.549028 0.0220852
$$619$$ 3.94724 0.158653 0.0793266 0.996849i $$-0.474723\pi$$
0.0793266 + 0.996849i $$0.474723\pi$$
$$620$$ 0 0
$$621$$ −6.37453 −0.255801
$$622$$ 2.52578 0.101274
$$623$$ −16.4511 −0.659101
$$624$$ −12.7647 −0.510997
$$625$$ 0 0
$$626$$ 1.28016 0.0511653
$$627$$ 0.696995 0.0278353
$$628$$ 24.2005 0.965707
$$629$$ −0.0369715 −0.00147415
$$630$$ 0 0
$$631$$ −12.3504 −0.491663 −0.245832 0.969313i $$-0.579061\pi$$
−0.245832 + 0.969313i $$0.579061\pi$$
$$632$$ 2.05271 0.0816523
$$633$$ 0.724612 0.0288007
$$634$$ 2.08947 0.0829836
$$635$$ 0 0
$$636$$ 2.47382 0.0980932
$$637$$ −15.5741 −0.617067
$$638$$ −0.364491 −0.0144303
$$639$$ −2.14425 −0.0848253
$$640$$ 0 0
$$641$$ 44.1646 1.74440 0.872198 0.489153i $$-0.162694\pi$$
0.872198 + 0.489153i $$0.162694\pi$$
$$642$$ −0.363746 −0.0143559
$$643$$ 7.27326 0.286829 0.143415 0.989663i $$-0.454192\pi$$
0.143415 + 0.989663i $$0.454192\pi$$
$$644$$ 10.6120 0.418172
$$645$$ 0 0
$$646$$ −0.000627356 0 −2.46830e−5 0
$$647$$ 11.7738 0.462878 0.231439 0.972849i $$-0.425657\pi$$
0.231439 + 0.972849i $$0.425657\pi$$
$$648$$ −2.02064 −0.0793781
$$649$$ −2.57022 −0.100890
$$650$$ 0 0
$$651$$ 7.49310 0.293678
$$652$$ 43.3806 1.69892
$$653$$ 31.7958 1.24427 0.622134 0.782911i $$-0.286266\pi$$
0.622134 + 0.782911i $$0.286266\pi$$
$$654$$ 0.270865 0.0105916
$$655$$ 0 0
$$656$$ −36.9120 −1.44117
$$657$$ 11.8868 0.463747
$$658$$ 0.299220 0.0116648
$$659$$ −23.9090 −0.931361 −0.465680 0.884953i $$-0.654190\pi$$
−0.465680 + 0.884953i $$0.654190\pi$$
$$660$$ 0 0
$$661$$ −39.9343 −1.55327 −0.776633 0.629953i $$-0.783074\pi$$
−0.776633 + 0.629953i $$0.783074\pi$$
$$662$$ −2.94880 −0.114608
$$663$$ −0.0195408 −0.000758901 0
$$664$$ −6.24577 −0.242383
$$665$$ 0 0
$$666$$ 1.60701 0.0622703
$$667$$ 5.80484 0.224764
$$668$$ −30.9967 −1.19930
$$669$$ −4.47913 −0.173173
$$670$$ 0 0
$$671$$ 3.68861 0.142397
$$672$$ −2.78157 −0.107301
$$673$$ 15.3378 0.591228 0.295614 0.955308i $$-0.404476\pi$$
0.295614 + 0.955308i $$0.404476\pi$$
$$674$$ −3.70821 −0.142835
$$675$$ 0 0
$$676$$ −17.2265 −0.662558
$$677$$ 20.2183 0.777054 0.388527 0.921437i $$-0.372984\pi$$
0.388527 + 0.921437i $$0.372984\pi$$
$$678$$ 1.23182 0.0473079
$$679$$ −53.2220 −2.04247
$$680$$ 0 0
$$681$$ −4.36190 −0.167148
$$682$$ 0.348074 0.0133284
$$683$$ 9.75882 0.373411 0.186705 0.982416i $$-0.440219\pi$$
0.186705 + 0.982416i $$0.440219\pi$$
$$684$$ −5.00113 −0.191223
$$685$$ 0 0
$$686$$ 1.22392 0.0467296
$$687$$ 18.3080 0.698495
$$688$$ −33.9098 −1.29280
$$689$$ 8.30425 0.316367
$$690$$ 0 0
$$691$$ −3.10119 −0.117975 −0.0589873 0.998259i $$-0.518787\pi$$
−0.0589873 + 0.998259i $$0.518787\pi$$
$$692$$ 20.7295 0.788018
$$693$$ −8.08710 −0.307204
$$694$$ −0.870813 −0.0330556
$$695$$ 0 0
$$696$$ −1.01344 −0.0384143
$$697$$ −0.0565066 −0.00214034
$$698$$ −2.74682 −0.103969
$$699$$ −2.05607 −0.0777676
$$700$$ 0 0
$$701$$ 1.49983 0.0566478 0.0283239 0.999599i $$-0.490983\pi$$
0.0283239 + 0.999599i $$0.490983\pi$$
$$702$$ 1.86284 0.0703085
$$703$$ 6.13742 0.231477
$$704$$ 7.74087 0.291745
$$705$$ 0 0
$$706$$ −1.39070 −0.0523397
$$707$$ −18.1042 −0.680877
$$708$$ −3.56343 −0.133922
$$709$$ −33.3647 −1.25304 −0.626518 0.779407i $$-0.715520\pi$$
−0.626518 + 0.779407i $$0.715520\pi$$
$$710$$ 0 0
$$711$$ 12.4226 0.465884
$$712$$ −2.12479 −0.0796299
$$713$$ −5.54338 −0.207601
$$714$$ −0.00140649 −5.26366e−5 0
$$715$$ 0 0
$$716$$ −40.8485 −1.52658
$$717$$ 12.9600 0.484001
$$718$$ 1.53999 0.0574721
$$719$$ −11.9782 −0.446711 −0.223355 0.974737i $$-0.571701\pi$$
−0.223355 + 0.974737i $$0.571701\pi$$
$$720$$ 0 0
$$721$$ −24.3291 −0.906062
$$722$$ 0.104144 0.00387583
$$723$$ −6.05119 −0.225046
$$724$$ −0.132870 −0.00493808
$$725$$ 0 0
$$726$$ 0.0725876 0.00269398
$$727$$ 4.56220 0.169203 0.0846014 0.996415i $$-0.473038\pi$$
0.0846014 + 0.996415i $$0.473038\pi$$
$$728$$ −6.21926 −0.230501
$$729$$ −4.19210 −0.155263
$$730$$ 0 0
$$731$$ −0.0519107 −0.00191999
$$732$$ 5.11400 0.189019
$$733$$ −22.2267 −0.820963 −0.410482 0.911869i $$-0.634639\pi$$
−0.410482 + 0.911869i $$0.634639\pi$$
$$734$$ −2.73047 −0.100783
$$735$$ 0 0
$$736$$ 2.05780 0.0758515
$$737$$ 5.98644 0.220513
$$738$$ 2.45612 0.0904112
$$739$$ 14.4390 0.531149 0.265574 0.964090i $$-0.414438\pi$$
0.265574 + 0.964090i $$0.414438\pi$$
$$740$$ 0 0
$$741$$ 3.24385 0.119166
$$742$$ 0.597717 0.0219429
$$743$$ 43.0239 1.57839 0.789197 0.614141i $$-0.210497\pi$$
0.789197 + 0.614141i $$0.210497\pi$$
$$744$$ 0.967792 0.0354810
$$745$$ 0 0
$$746$$ 0.135896 0.00497552
$$747$$ −37.7982 −1.38297
$$748$$ 0.0119826 0.000438126 0
$$749$$ 16.1187 0.588963
$$750$$ 0 0
$$751$$ 8.50342 0.310294 0.155147 0.987891i $$-0.450415\pi$$
0.155147 + 0.987891i $$0.450415\pi$$
$$752$$ −3.51490 −0.128175
$$753$$ −19.3368 −0.704673
$$754$$ −1.69636 −0.0617778
$$755$$ 0 0
$$756$$ −24.5909 −0.894361
$$757$$ 18.2012 0.661533 0.330766 0.943713i $$-0.392693\pi$$
0.330766 + 0.943713i $$0.392693\pi$$
$$758$$ −1.04340 −0.0378980
$$759$$ −1.15602 −0.0419609
$$760$$ 0 0
$$761$$ 45.3295 1.64319 0.821597 0.570069i $$-0.193083\pi$$
0.821597 + 0.570069i $$0.193083\pi$$
$$762$$ −0.0631934 −0.00228926
$$763$$ −12.0028 −0.434531
$$764$$ 4.17009 0.150869
$$765$$ 0 0
$$766$$ −0.567256 −0.0204958
$$767$$ −11.9619 −0.431921
$$768$$ 10.5521 0.380765
$$769$$ −27.6341 −0.996510 −0.498255 0.867031i $$-0.666026\pi$$
−0.498255 + 0.867031i $$0.666026\pi$$
$$770$$ 0 0
$$771$$ 12.3371 0.444308
$$772$$ 50.1188 1.80382
$$773$$ 27.7758 0.999027 0.499513 0.866306i $$-0.333512\pi$$
0.499513 + 0.866306i $$0.333512\pi$$
$$774$$ 2.25636 0.0811033
$$775$$ 0 0
$$776$$ −6.87404 −0.246764
$$777$$ 13.7597 0.493626
$$778$$ −0.790861 −0.0283538
$$779$$ 9.38032 0.336085
$$780$$ 0 0
$$781$$ −0.852858 −0.0305176
$$782$$ 0.00104052 3.72089e−5 0
$$783$$ −13.4513 −0.480712
$$784$$ 13.1680 0.470286
$$785$$ 0 0
$$786$$ −0.0241503 −0.000861414 0
$$787$$ −1.98342 −0.0707012 −0.0353506 0.999375i $$-0.511255\pi$$
−0.0353506 + 0.999375i $$0.511255\pi$$
$$788$$ 11.3558 0.404533
$$789$$ −15.6145 −0.555891
$$790$$ 0 0
$$791$$ −54.5858 −1.94085
$$792$$ −1.04451 −0.0371151
$$793$$ 17.1670 0.609618
$$794$$ 0.653453 0.0231902
$$795$$ 0 0
$$796$$ 20.2900 0.719160
$$797$$ −30.9790 −1.09733 −0.548666 0.836041i $$-0.684864\pi$$
−0.548666 + 0.836041i $$0.684864\pi$$
$$798$$ 0.233483 0.00826523
$$799$$ −0.00538078 −0.000190358 0
$$800$$ 0 0
$$801$$ −12.8588 −0.454345
$$802$$ 1.85426 0.0654762
$$803$$ 4.72786 0.166842
$$804$$ 8.29979 0.292711
$$805$$ 0 0
$$806$$ 1.61995 0.0570605
$$807$$ 11.2721 0.396795
$$808$$ −2.33829 −0.0822608
$$809$$ 33.4595 1.17637 0.588186 0.808725i $$-0.299842\pi$$
0.588186 + 0.808725i $$0.299842\pi$$
$$810$$ 0 0
$$811$$ −33.7987 −1.18683 −0.593416 0.804896i $$-0.702221\pi$$
−0.593416 + 0.804896i $$0.702221\pi$$
$$812$$ 22.3932 0.785846
$$813$$ 1.46350 0.0513273
$$814$$ 0.639174 0.0224030
$$815$$ 0 0
$$816$$ 0.0165219 0.000578382 0
$$817$$ 8.61740 0.301485
$$818$$ 0.882297 0.0308488
$$819$$ −37.6378 −1.31517
$$820$$ 0 0
$$821$$ −17.6687 −0.616641 −0.308320 0.951283i $$-0.599767\pi$$
−0.308320 + 0.951283i $$0.599767\pi$$
$$822$$ 0.945917 0.0329927
$$823$$ 46.0491 1.60517 0.802585 0.596538i $$-0.203457\pi$$
0.802585 + 0.596538i $$0.203457\pi$$
$$824$$ −3.14229 −0.109467
$$825$$ 0 0
$$826$$ −0.860988 −0.0299576
$$827$$ 23.9121 0.831503 0.415752 0.909478i $$-0.363519\pi$$
0.415752 + 0.909478i $$0.363519\pi$$
$$828$$ 8.29476 0.288263
$$829$$ −10.2688 −0.356649 −0.178324 0.983972i $$-0.557068\pi$$
−0.178324 + 0.983972i $$0.557068\pi$$
$$830$$ 0 0
$$831$$ −20.8952 −0.724848
$$832$$ 36.0264 1.24899
$$833$$ 0.0201582 0.000698440 0
$$834$$ 0.920608 0.0318780
$$835$$ 0 0
$$836$$ −1.98915 −0.0687963
$$837$$ 12.8455 0.444005
$$838$$ 1.08890 0.0376156
$$839$$ −40.5454 −1.39978 −0.699890 0.714250i $$-0.746768\pi$$
−0.699890 + 0.714250i $$0.746768\pi$$
$$840$$ 0 0
$$841$$ −16.7508 −0.577614
$$842$$ 3.38273 0.116577
$$843$$ −0.327452 −0.0112780
$$844$$ −2.06797 −0.0711825
$$845$$ 0 0
$$846$$ 0.233882 0.00804102
$$847$$ −3.21657 −0.110523
$$848$$ −7.02132 −0.241113
$$849$$ 9.20886 0.316047
$$850$$ 0 0
$$851$$ −10.1794 −0.348945
$$852$$ −1.18243 −0.0405093
$$853$$ 1.99694 0.0683740 0.0341870 0.999415i $$-0.489116\pi$$
0.0341870 + 0.999415i $$0.489116\pi$$
$$854$$ 1.23563 0.0422825
$$855$$ 0 0
$$856$$ 2.08185 0.0711561
$$857$$ −25.1830 −0.860235 −0.430118 0.902773i $$-0.641528\pi$$
−0.430118 + 0.902773i $$0.641528\pi$$
$$858$$ 0.337827 0.0115332
$$859$$ −0.915472 −0.0312355 −0.0156178 0.999878i $$-0.504971\pi$$
−0.0156178 + 0.999878i $$0.504971\pi$$
$$860$$ 0 0
$$861$$ 21.0301 0.716703
$$862$$ −0.238204 −0.00811326
$$863$$ 49.8812 1.69797 0.848987 0.528413i $$-0.177213\pi$$
0.848987 + 0.528413i $$0.177213\pi$$
$$864$$ −4.76847 −0.162227
$$865$$ 0 0
$$866$$ −3.80357 −0.129251
$$867$$ −11.8489 −0.402409
$$868$$ −21.3846 −0.725839
$$869$$ 4.94098 0.167611
$$870$$ 0 0
$$871$$ 27.8612 0.944042
$$872$$ −1.55026 −0.0524983
$$873$$ −41.6004 −1.40796
$$874$$ −0.172731 −0.00584270
$$875$$ 0 0
$$876$$ 6.55484 0.221468
$$877$$ −18.6769 −0.630673 −0.315337 0.948980i $$-0.602117\pi$$
−0.315337 + 0.948980i $$0.602117\pi$$
$$878$$ −2.39867 −0.0809511
$$879$$ −3.81358 −0.128629
$$880$$ 0 0
$$881$$ −46.9216 −1.58083 −0.790414 0.612573i $$-0.790134\pi$$
−0.790414 + 0.612573i $$0.790134\pi$$
$$882$$ −0.876200 −0.0295032
$$883$$ 44.8499 1.50932 0.754659 0.656117i $$-0.227802\pi$$
0.754659 + 0.656117i $$0.227802\pi$$
$$884$$ 0.0557674 0.00187566
$$885$$ 0 0
$$886$$ −0.378628 −0.0127203
$$887$$ −14.5451 −0.488376 −0.244188 0.969728i $$-0.578521\pi$$
−0.244188 + 0.969728i $$0.578521\pi$$
$$888$$ 1.77717 0.0596380
$$889$$ 2.80029 0.0939186
$$890$$ 0 0
$$891$$ −4.86379 −0.162943
$$892$$ 12.7830 0.428006
$$893$$ 0.893231 0.0298908
$$894$$ 1.25173 0.0418641
$$895$$ 0 0
$$896$$ 10.5747 0.353276
$$897$$ −5.38018 −0.179639
$$898$$ 2.66322 0.0888730
$$899$$ −11.6975 −0.390133
$$900$$ 0 0
$$901$$ −0.0107486 −0.000358086 0
$$902$$ 0.976902 0.0325273
$$903$$ 19.3197 0.642918
$$904$$ −7.05017 −0.234485
$$905$$ 0 0
$$906$$ 1.63669 0.0543754
$$907$$ −56.2751 −1.86858 −0.934292 0.356510i $$-0.883967\pi$$
−0.934292 + 0.356510i $$0.883967\pi$$
$$908$$ 12.4484 0.413115
$$909$$ −14.1509 −0.469356
$$910$$ 0 0
$$911$$ 11.8656 0.393123 0.196562 0.980491i $$-0.437022\pi$$
0.196562 + 0.980491i $$0.437022\pi$$
$$912$$ −2.74270 −0.0908200
$$913$$ −15.0339 −0.497550
$$914$$ −0.671883 −0.0222239
$$915$$ 0 0
$$916$$ −52.2493 −1.72637
$$917$$ 1.07017 0.0353402
$$918$$ −0.00241116 −7.95801e−5 0
$$919$$ 17.4567 0.575844 0.287922 0.957654i $$-0.407036\pi$$
0.287922 + 0.957654i $$0.407036\pi$$
$$920$$ 0 0
$$921$$ −21.5223 −0.709183
$$922$$ −2.40877 −0.0793286
$$923$$ −3.96924 −0.130649
$$924$$ −4.45956 −0.146709
$$925$$ 0 0
$$926$$ −3.79535 −0.124723
$$927$$ −19.0165 −0.624585
$$928$$ 4.34231 0.142543
$$929$$ 6.75452 0.221609 0.110804 0.993842i $$-0.464657\pi$$
0.110804 + 0.993842i $$0.464657\pi$$
$$930$$ 0 0
$$931$$ −3.34635 −0.109672
$$932$$ 5.86781 0.192206
$$933$$ 16.9041 0.553415
$$934$$ 3.75480 0.122861
$$935$$ 0 0
$$936$$ −4.86121 −0.158894
$$937$$ 45.1286 1.47429 0.737143 0.675736i $$-0.236174\pi$$
0.737143 + 0.675736i $$0.236174\pi$$
$$938$$ 2.00537 0.0654778
$$939$$ 8.56760 0.279593
$$940$$ 0 0
$$941$$ 26.7484 0.871972 0.435986 0.899954i $$-0.356400\pi$$
0.435986 + 0.899954i $$0.356400\pi$$
$$942$$ −0.883119 −0.0287736
$$943$$ −15.5580 −0.506639
$$944$$ 10.1139 0.329180
$$945$$ 0 0
$$946$$ 0.897448 0.0291786
$$947$$ 16.2900 0.529355 0.264677 0.964337i $$-0.414735\pi$$
0.264677 + 0.964337i $$0.414735\pi$$
$$948$$ 6.85032 0.222488
$$949$$ 22.0037 0.714270
$$950$$ 0 0
$$951$$ 13.9841 0.453465
$$952$$ 0.00804986 0.000260897 0
$$953$$ −4.10009 −0.132815 −0.0664075 0.997793i $$-0.521154\pi$$
−0.0664075 + 0.997793i $$0.521154\pi$$
$$954$$ 0.467199 0.0151261
$$955$$ 0 0
$$956$$ −36.9866 −1.19623
$$957$$ −2.43940 −0.0788547
$$958$$ 1.79388 0.0579576
$$959$$ −41.9164 −1.35355
$$960$$ 0 0
$$961$$ −19.8294 −0.639658
$$962$$ 2.97475 0.0959097
$$963$$ 12.5990 0.405996
$$964$$ 17.2695 0.556213
$$965$$ 0 0
$$966$$ −0.387251 −0.0124596
$$967$$ 34.0890 1.09623 0.548115 0.836403i $$-0.315346\pi$$
0.548115 + 0.836403i $$0.315346\pi$$
$$968$$ −0.415445 −0.0133529
$$969$$ −0.00419866 −0.000134880 0
$$970$$ 0 0
$$971$$ 40.8253 1.31015 0.655073 0.755565i $$-0.272638\pi$$
0.655073 + 0.755565i $$0.272638\pi$$
$$972$$ −29.6785 −0.951937
$$973$$ −40.7949 −1.30782
$$974$$ 3.29550 0.105594
$$975$$ 0 0
$$976$$ −14.5148 −0.464609
$$977$$ 14.7093 0.470591 0.235296 0.971924i $$-0.424394\pi$$
0.235296 + 0.971924i $$0.424394\pi$$
$$978$$ −1.58303 −0.0506198
$$979$$ −5.11449 −0.163460
$$980$$ 0 0
$$981$$ −9.38187 −0.299540
$$982$$ 4.05188 0.129301
$$983$$ −31.0823 −0.991372 −0.495686 0.868502i $$-0.665083\pi$$
−0.495686 + 0.868502i $$0.665083\pi$$
$$984$$ 2.71620 0.0865892
$$985$$ 0 0
$$986$$ 0.00219567 6.99245e−5 0
$$987$$ 2.00257 0.0637424
$$988$$ −9.25763 −0.294524
$$989$$ −14.2926 −0.454479
$$990$$ 0 0
$$991$$ 45.1569 1.43446 0.717228 0.696838i $$-0.245411\pi$$
0.717228 + 0.696838i $$0.245411\pi$$
$$992$$ −4.14673 −0.131659
$$993$$ −19.7352 −0.626279
$$994$$ −0.285695 −0.00906170
$$995$$ 0 0
$$996$$ −20.8435 −0.660451
$$997$$ 46.8604 1.48409 0.742043 0.670353i $$-0.233857\pi$$
0.742043 + 0.670353i $$0.233857\pi$$
$$998$$ 1.70998 0.0541283
$$999$$ 23.5884 0.746303
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.bb.1.12 22
5.2 odd 4 1045.2.b.d.419.12 yes 22
5.3 odd 4 1045.2.b.d.419.11 22
5.4 even 2 inner 5225.2.a.bb.1.11 22

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.d.419.11 22 5.3 odd 4
1045.2.b.d.419.12 yes 22 5.2 odd 4
5225.2.a.bb.1.11 22 5.4 even 2 inner
5225.2.a.bb.1.12 22 1.1 even 1 trivial