Properties

 Label 3332.2.a.l.1.1 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ Analytic rank $1$ 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] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.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: $$26.6061539535$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{3} +1.61803 q^{5} -2.61803 q^{9} +O(q^{10})$$ $$q-0.618034 q^{3} +1.61803 q^{5} -2.61803 q^{9} -5.23607 q^{11} +3.23607 q^{13} -1.00000 q^{15} -1.00000 q^{17} -0.472136 q^{19} +5.70820 q^{23} -2.38197 q^{25} +3.47214 q^{27} +7.70820 q^{29} +9.32624 q^{31} +3.23607 q^{33} -8.47214 q^{37} -2.00000 q^{39} -11.0902 q^{41} -0.909830 q^{43} -4.23607 q^{45} -0.472136 q^{47} +0.618034 q^{51} -13.7984 q^{53} -8.47214 q^{55} +0.291796 q^{57} -8.32624 q^{61} +5.23607 q^{65} -1.90983 q^{67} -3.52786 q^{69} -6.94427 q^{71} -13.8541 q^{73} +1.47214 q^{75} -14.9443 q^{79} +5.70820 q^{81} +11.4164 q^{83} -1.61803 q^{85} -4.76393 q^{87} -2.00000 q^{89} -5.76393 q^{93} -0.763932 q^{95} -3.90983 q^{97} +13.7082 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 - 3 * q^9 $$2 q + q^{3} + q^{5} - 3 q^{9} - 6 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{17} + 8 q^{19} - 2 q^{23} - 7 q^{25} - 2 q^{27} + 2 q^{29} + 3 q^{31} + 2 q^{33} - 8 q^{37} - 4 q^{39} - 11 q^{41} - 13 q^{43} - 4 q^{45} + 8 q^{47} - q^{51} - 3 q^{53} - 8 q^{55} + 14 q^{57} - q^{61} + 6 q^{65} - 15 q^{67} - 16 q^{69} + 4 q^{71} - 21 q^{73} - 6 q^{75} - 12 q^{79} - 2 q^{81} - 4 q^{83} - q^{85} - 14 q^{87} - 4 q^{89} - 16 q^{93} - 6 q^{95} - 19 q^{97} + 14 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 - 3 * q^9 - 6 * q^11 + 2 * q^13 - 2 * q^15 - 2 * q^17 + 8 * q^19 - 2 * q^23 - 7 * q^25 - 2 * q^27 + 2 * q^29 + 3 * q^31 + 2 * q^33 - 8 * q^37 - 4 * q^39 - 11 * q^41 - 13 * q^43 - 4 * q^45 + 8 * q^47 - q^51 - 3 * q^53 - 8 * q^55 + 14 * q^57 - q^61 + 6 * q^65 - 15 * q^67 - 16 * q^69 + 4 * q^71 - 21 * q^73 - 6 * q^75 - 12 * q^79 - 2 * q^81 - 4 * q^83 - q^85 - 14 * q^87 - 4 * q^89 - 16 * q^93 - 6 * q^95 - 19 * q^97 + 14 * 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 0
$$3$$ −0.618034 −0.356822 −0.178411 0.983956i $$-0.557096\pi$$
−0.178411 + 0.983956i $$0.557096\pi$$
$$4$$ 0 0
$$5$$ 1.61803 0.723607 0.361803 0.932254i $$-0.382161\pi$$
0.361803 + 0.932254i $$0.382161\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.61803 −0.872678
$$10$$ 0 0
$$11$$ −5.23607 −1.57873 −0.789367 0.613922i $$-0.789591\pi$$
−0.789367 + 0.613922i $$0.789591\pi$$
$$12$$ 0 0
$$13$$ 3.23607 0.897524 0.448762 0.893651i $$-0.351865\pi$$
0.448762 + 0.893651i $$0.351865\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ −0.472136 −0.108315 −0.0541577 0.998532i $$-0.517247\pi$$
−0.0541577 + 0.998532i $$0.517247\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.70820 1.19024 0.595121 0.803636i $$-0.297104\pi$$
0.595121 + 0.803636i $$0.297104\pi$$
$$24$$ 0 0
$$25$$ −2.38197 −0.476393
$$26$$ 0 0
$$27$$ 3.47214 0.668213
$$28$$ 0 0
$$29$$ 7.70820 1.43138 0.715689 0.698419i $$-0.246113\pi$$
0.715689 + 0.698419i $$0.246113\pi$$
$$30$$ 0 0
$$31$$ 9.32624 1.67504 0.837521 0.546405i $$-0.184004\pi$$
0.837521 + 0.546405i $$0.184004\pi$$
$$32$$ 0 0
$$33$$ 3.23607 0.563327
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.47214 −1.39281 −0.696405 0.717649i $$-0.745218\pi$$
−0.696405 + 0.717649i $$0.745218\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −11.0902 −1.73199 −0.865997 0.500050i $$-0.833315\pi$$
−0.865997 + 0.500050i $$0.833315\pi$$
$$42$$ 0 0
$$43$$ −0.909830 −0.138748 −0.0693739 0.997591i $$-0.522100\pi$$
−0.0693739 + 0.997591i $$0.522100\pi$$
$$44$$ 0 0
$$45$$ −4.23607 −0.631476
$$46$$ 0 0
$$47$$ −0.472136 −0.0688681 −0.0344341 0.999407i $$-0.510963\pi$$
−0.0344341 + 0.999407i $$0.510963\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0.618034 0.0865421
$$52$$ 0 0
$$53$$ −13.7984 −1.89535 −0.947676 0.319233i $$-0.896575\pi$$
−0.947676 + 0.319233i $$0.896575\pi$$
$$54$$ 0 0
$$55$$ −8.47214 −1.14238
$$56$$ 0 0
$$57$$ 0.291796 0.0386493
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −8.32624 −1.06607 −0.533033 0.846095i $$-0.678948\pi$$
−0.533033 + 0.846095i $$0.678948\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 5.23607 0.649454
$$66$$ 0 0
$$67$$ −1.90983 −0.233323 −0.116661 0.993172i $$-0.537219\pi$$
−0.116661 + 0.993172i $$0.537219\pi$$
$$68$$ 0 0
$$69$$ −3.52786 −0.424705
$$70$$ 0 0
$$71$$ −6.94427 −0.824133 −0.412067 0.911154i $$-0.635193\pi$$
−0.412067 + 0.911154i $$0.635193\pi$$
$$72$$ 0 0
$$73$$ −13.8541 −1.62150 −0.810750 0.585393i $$-0.800940\pi$$
−0.810750 + 0.585393i $$0.800940\pi$$
$$74$$ 0 0
$$75$$ 1.47214 0.169988
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −14.9443 −1.68136 −0.840681 0.541531i $$-0.817845\pi$$
−0.840681 + 0.541531i $$0.817845\pi$$
$$80$$ 0 0
$$81$$ 5.70820 0.634245
$$82$$ 0 0
$$83$$ 11.4164 1.25311 0.626557 0.779376i $$-0.284464\pi$$
0.626557 + 0.779376i $$0.284464\pi$$
$$84$$ 0 0
$$85$$ −1.61803 −0.175500
$$86$$ 0 0
$$87$$ −4.76393 −0.510747
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −5.76393 −0.597692
$$94$$ 0 0
$$95$$ −0.763932 −0.0783778
$$96$$ 0 0
$$97$$ −3.90983 −0.396983 −0.198492 0.980103i $$-0.563604\pi$$
−0.198492 + 0.980103i $$0.563604\pi$$
$$98$$ 0 0
$$99$$ 13.7082 1.37773
$$100$$ 0 0
$$101$$ −8.47214 −0.843009 −0.421505 0.906826i $$-0.638498\pi$$
−0.421505 + 0.906826i $$0.638498\pi$$
$$102$$ 0 0
$$103$$ 11.4164 1.12489 0.562446 0.826834i $$-0.309860\pi$$
0.562446 + 0.826834i $$0.309860\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.2361 1.47293 0.736463 0.676478i $$-0.236494\pi$$
0.736463 + 0.676478i $$0.236494\pi$$
$$108$$ 0 0
$$109$$ −12.4721 −1.19461 −0.597307 0.802013i $$-0.703763\pi$$
−0.597307 + 0.802013i $$0.703763\pi$$
$$110$$ 0 0
$$111$$ 5.23607 0.496986
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 9.23607 0.861268
$$116$$ 0 0
$$117$$ −8.47214 −0.783249
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ 0 0
$$123$$ 6.85410 0.618014
$$124$$ 0 0
$$125$$ −11.9443 −1.06833
$$126$$ 0 0
$$127$$ −1.14590 −0.101682 −0.0508410 0.998707i $$-0.516190\pi$$
−0.0508410 + 0.998707i $$0.516190\pi$$
$$128$$ 0 0
$$129$$ 0.562306 0.0495083
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 5.61803 0.483523
$$136$$ 0 0
$$137$$ 4.61803 0.394545 0.197273 0.980349i $$-0.436792\pi$$
0.197273 + 0.980349i $$0.436792\pi$$
$$138$$ 0 0
$$139$$ −15.0344 −1.27520 −0.637602 0.770366i $$-0.720074\pi$$
−0.637602 + 0.770366i $$0.720074\pi$$
$$140$$ 0 0
$$141$$ 0.291796 0.0245737
$$142$$ 0 0
$$143$$ −16.9443 −1.41695
$$144$$ 0 0
$$145$$ 12.4721 1.03575
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −11.3820 −0.932447 −0.466223 0.884667i $$-0.654386\pi$$
−0.466223 + 0.884667i $$0.654386\pi$$
$$150$$ 0 0
$$151$$ −21.8541 −1.77846 −0.889231 0.457459i $$-0.848760\pi$$
−0.889231 + 0.457459i $$0.848760\pi$$
$$152$$ 0 0
$$153$$ 2.61803 0.211656
$$154$$ 0 0
$$155$$ 15.0902 1.21207
$$156$$ 0 0
$$157$$ −20.1803 −1.61057 −0.805283 0.592890i $$-0.797987\pi$$
−0.805283 + 0.592890i $$0.797987\pi$$
$$158$$ 0 0
$$159$$ 8.52786 0.676304
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −7.70820 −0.603753 −0.301877 0.953347i $$-0.597613\pi$$
−0.301877 + 0.953347i $$0.597613\pi$$
$$164$$ 0 0
$$165$$ 5.23607 0.407627
$$166$$ 0 0
$$167$$ −16.6180 −1.28594 −0.642971 0.765890i $$-0.722298\pi$$
−0.642971 + 0.765890i $$0.722298\pi$$
$$168$$ 0 0
$$169$$ −2.52786 −0.194451
$$170$$ 0 0
$$171$$ 1.23607 0.0945245
$$172$$ 0 0
$$173$$ 23.0902 1.75551 0.877757 0.479107i $$-0.159039\pi$$
0.877757 + 0.479107i $$0.159039\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −20.6180 −1.54106 −0.770532 0.637401i $$-0.780009\pi$$
−0.770532 + 0.637401i $$0.780009\pi$$
$$180$$ 0 0
$$181$$ 3.52786 0.262224 0.131112 0.991368i $$-0.458145\pi$$
0.131112 + 0.991368i $$0.458145\pi$$
$$182$$ 0 0
$$183$$ 5.14590 0.380396
$$184$$ 0 0
$$185$$ −13.7082 −1.00785
$$186$$ 0 0
$$187$$ 5.23607 0.382899
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.79837 −0.274841 −0.137420 0.990513i $$-0.543881\pi$$
−0.137420 + 0.990513i $$0.543881\pi$$
$$192$$ 0 0
$$193$$ 8.18034 0.588834 0.294417 0.955677i $$-0.404875\pi$$
0.294417 + 0.955677i $$0.404875\pi$$
$$194$$ 0 0
$$195$$ −3.23607 −0.231740
$$196$$ 0 0
$$197$$ −3.70820 −0.264199 −0.132099 0.991236i $$-0.542172\pi$$
−0.132099 + 0.991236i $$0.542172\pi$$
$$198$$ 0 0
$$199$$ 0.909830 0.0644961 0.0322481 0.999480i $$-0.489733\pi$$
0.0322481 + 0.999480i $$0.489733\pi$$
$$200$$ 0 0
$$201$$ 1.18034 0.0832548
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −17.9443 −1.25328
$$206$$ 0 0
$$207$$ −14.9443 −1.03870
$$208$$ 0 0
$$209$$ 2.47214 0.171001
$$210$$ 0 0
$$211$$ −15.7082 −1.08140 −0.540699 0.841216i $$-0.681840\pi$$
−0.540699 + 0.841216i $$0.681840\pi$$
$$212$$ 0 0
$$213$$ 4.29180 0.294069
$$214$$ 0 0
$$215$$ −1.47214 −0.100399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 8.56231 0.578587
$$220$$ 0 0
$$221$$ −3.23607 −0.217681
$$222$$ 0 0
$$223$$ 14.9443 1.00074 0.500371 0.865811i $$-0.333197\pi$$
0.500371 + 0.865811i $$0.333197\pi$$
$$224$$ 0 0
$$225$$ 6.23607 0.415738
$$226$$ 0 0
$$227$$ 16.7426 1.11125 0.555624 0.831434i $$-0.312479\pi$$
0.555624 + 0.831434i $$0.312479\pi$$
$$228$$ 0 0
$$229$$ 14.7639 0.975628 0.487814 0.872948i $$-0.337794\pi$$
0.487814 + 0.872948i $$0.337794\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.47214 −0.161955 −0.0809775 0.996716i $$-0.525804\pi$$
−0.0809775 + 0.996716i $$0.525804\pi$$
$$234$$ 0 0
$$235$$ −0.763932 −0.0498334
$$236$$ 0 0
$$237$$ 9.23607 0.599947
$$238$$ 0 0
$$239$$ 23.0902 1.49358 0.746789 0.665061i $$-0.231594\pi$$
0.746789 + 0.665061i $$0.231594\pi$$
$$240$$ 0 0
$$241$$ 25.0344 1.61261 0.806305 0.591500i $$-0.201464\pi$$
0.806305 + 0.591500i $$0.201464\pi$$
$$242$$ 0 0
$$243$$ −13.9443 −0.894525
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.52786 −0.0972157
$$248$$ 0 0
$$249$$ −7.05573 −0.447139
$$250$$ 0 0
$$251$$ −9.41641 −0.594358 −0.297179 0.954822i $$-0.596046\pi$$
−0.297179 + 0.954822i $$0.596046\pi$$
$$252$$ 0 0
$$253$$ −29.8885 −1.87908
$$254$$ 0 0
$$255$$ 1.00000 0.0626224
$$256$$ 0 0
$$257$$ −12.2918 −0.766741 −0.383371 0.923595i $$-0.625237\pi$$
−0.383371 + 0.923595i $$0.625237\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −20.1803 −1.24913
$$262$$ 0 0
$$263$$ 0.944272 0.0582263 0.0291132 0.999576i $$-0.490732\pi$$
0.0291132 + 0.999576i $$0.490732\pi$$
$$264$$ 0 0
$$265$$ −22.3262 −1.37149
$$266$$ 0 0
$$267$$ 1.23607 0.0756461
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ 9.70820 0.589731 0.294866 0.955539i $$-0.404725\pi$$
0.294866 + 0.955539i $$0.404725\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 12.4721 0.752098
$$276$$ 0 0
$$277$$ 7.23607 0.434773 0.217387 0.976086i $$-0.430247\pi$$
0.217387 + 0.976086i $$0.430247\pi$$
$$278$$ 0 0
$$279$$ −24.4164 −1.46177
$$280$$ 0 0
$$281$$ −20.0902 −1.19848 −0.599240 0.800570i $$-0.704530\pi$$
−0.599240 + 0.800570i $$0.704530\pi$$
$$282$$ 0 0
$$283$$ 24.5623 1.46008 0.730039 0.683406i $$-0.239502\pi$$
0.730039 + 0.683406i $$0.239502\pi$$
$$284$$ 0 0
$$285$$ 0.472136 0.0279669
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 2.41641 0.141652
$$292$$ 0 0
$$293$$ 19.7082 1.15137 0.575683 0.817673i $$-0.304736\pi$$
0.575683 + 0.817673i $$0.304736\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −18.1803 −1.05493
$$298$$ 0 0
$$299$$ 18.4721 1.06827
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 5.23607 0.300804
$$304$$ 0 0
$$305$$ −13.4721 −0.771412
$$306$$ 0 0
$$307$$ −18.9443 −1.08121 −0.540603 0.841278i $$-0.681804\pi$$
−0.540603 + 0.841278i $$0.681804\pi$$
$$308$$ 0 0
$$309$$ −7.05573 −0.401386
$$310$$ 0 0
$$311$$ −24.3820 −1.38257 −0.691287 0.722580i $$-0.742956\pi$$
−0.691287 + 0.722580i $$0.742956\pi$$
$$312$$ 0 0
$$313$$ −32.2705 −1.82404 −0.912019 0.410149i $$-0.865477\pi$$
−0.912019 + 0.410149i $$0.865477\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.81966 −0.326865 −0.163432 0.986555i $$-0.552257\pi$$
−0.163432 + 0.986555i $$0.552257\pi$$
$$318$$ 0 0
$$319$$ −40.3607 −2.25976
$$320$$ 0 0
$$321$$ −9.41641 −0.525573
$$322$$ 0 0
$$323$$ 0.472136 0.0262703
$$324$$ 0 0
$$325$$ −7.70820 −0.427574
$$326$$ 0 0
$$327$$ 7.70820 0.426265
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −11.0344 −0.606508 −0.303254 0.952910i $$-0.598073\pi$$
−0.303254 + 0.952910i $$0.598073\pi$$
$$332$$ 0 0
$$333$$ 22.1803 1.21548
$$334$$ 0 0
$$335$$ −3.09017 −0.168834
$$336$$ 0 0
$$337$$ 18.4721 1.00624 0.503121 0.864216i $$-0.332185\pi$$
0.503121 + 0.864216i $$0.332185\pi$$
$$338$$ 0 0
$$339$$ −3.70820 −0.201402
$$340$$ 0 0
$$341$$ −48.8328 −2.64445
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −5.70820 −0.307319
$$346$$ 0 0
$$347$$ 13.0557 0.700868 0.350434 0.936587i $$-0.386034\pi$$
0.350434 + 0.936587i $$0.386034\pi$$
$$348$$ 0 0
$$349$$ 24.4721 1.30996 0.654982 0.755645i $$-0.272676\pi$$
0.654982 + 0.755645i $$0.272676\pi$$
$$350$$ 0 0
$$351$$ 11.2361 0.599737
$$352$$ 0 0
$$353$$ 17.2361 0.917383 0.458692 0.888595i $$-0.348318\pi$$
0.458692 + 0.888595i $$0.348318\pi$$
$$354$$ 0 0
$$355$$ −11.2361 −0.596349
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −0.562306 −0.0296774 −0.0148387 0.999890i $$-0.504723\pi$$
−0.0148387 + 0.999890i $$0.504723\pi$$
$$360$$ 0 0
$$361$$ −18.7771 −0.988268
$$362$$ 0 0
$$363$$ −10.1459 −0.532522
$$364$$ 0 0
$$365$$ −22.4164 −1.17333
$$366$$ 0 0
$$367$$ −18.5066 −0.966035 −0.483018 0.875611i $$-0.660459\pi$$
−0.483018 + 0.875611i $$0.660459\pi$$
$$368$$ 0 0
$$369$$ 29.0344 1.51147
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.1459 −0.525335 −0.262667 0.964886i $$-0.584602\pi$$
−0.262667 + 0.964886i $$0.584602\pi$$
$$374$$ 0 0
$$375$$ 7.38197 0.381203
$$376$$ 0 0
$$377$$ 24.9443 1.28470
$$378$$ 0 0
$$379$$ −20.6525 −1.06085 −0.530423 0.847733i $$-0.677967\pi$$
−0.530423 + 0.847733i $$0.677967\pi$$
$$380$$ 0 0
$$381$$ 0.708204 0.0362824
$$382$$ 0 0
$$383$$ −0.652476 −0.0333400 −0.0166700 0.999861i $$-0.505306\pi$$
−0.0166700 + 0.999861i $$0.505306\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.38197 0.121082
$$388$$ 0 0
$$389$$ −1.43769 −0.0728940 −0.0364470 0.999336i $$-0.511604\pi$$
−0.0364470 + 0.999336i $$0.511604\pi$$
$$390$$ 0 0
$$391$$ −5.70820 −0.288676
$$392$$ 0 0
$$393$$ −4.94427 −0.249406
$$394$$ 0 0
$$395$$ −24.1803 −1.21664
$$396$$ 0 0
$$397$$ −17.7984 −0.893275 −0.446637 0.894715i $$-0.647379\pi$$
−0.446637 + 0.894715i $$0.647379\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1.52786 −0.0762979 −0.0381489 0.999272i $$-0.512146\pi$$
−0.0381489 + 0.999272i $$0.512146\pi$$
$$402$$ 0 0
$$403$$ 30.1803 1.50339
$$404$$ 0 0
$$405$$ 9.23607 0.458944
$$406$$ 0 0
$$407$$ 44.3607 2.19888
$$408$$ 0 0
$$409$$ −14.1803 −0.701173 −0.350586 0.936530i $$-0.614018\pi$$
−0.350586 + 0.936530i $$0.614018\pi$$
$$410$$ 0 0
$$411$$ −2.85410 −0.140782
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 18.4721 0.906761
$$416$$ 0 0
$$417$$ 9.29180 0.455021
$$418$$ 0 0
$$419$$ 18.4508 0.901383 0.450691 0.892680i $$-0.351177\pi$$
0.450691 + 0.892680i $$0.351177\pi$$
$$420$$ 0 0
$$421$$ 22.2148 1.08268 0.541341 0.840803i $$-0.317917\pi$$
0.541341 + 0.840803i $$0.317917\pi$$
$$422$$ 0 0
$$423$$ 1.23607 0.0600997
$$424$$ 0 0
$$425$$ 2.38197 0.115542
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 10.4721 0.505599
$$430$$ 0 0
$$431$$ 22.9443 1.10519 0.552593 0.833451i $$-0.313638\pi$$
0.552593 + 0.833451i $$0.313638\pi$$
$$432$$ 0 0
$$433$$ 0.944272 0.0453788 0.0226894 0.999743i $$-0.492777\pi$$
0.0226894 + 0.999743i $$0.492777\pi$$
$$434$$ 0 0
$$435$$ −7.70820 −0.369580
$$436$$ 0 0
$$437$$ −2.69505 −0.128922
$$438$$ 0 0
$$439$$ −12.3262 −0.588299 −0.294150 0.955759i $$-0.595036\pi$$
−0.294150 + 0.955759i $$0.595036\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −9.88854 −0.469819 −0.234909 0.972017i $$-0.575479\pi$$
−0.234909 + 0.972017i $$0.575479\pi$$
$$444$$ 0 0
$$445$$ −3.23607 −0.153404
$$446$$ 0 0
$$447$$ 7.03444 0.332718
$$448$$ 0 0
$$449$$ 4.29180 0.202542 0.101271 0.994859i $$-0.467709\pi$$
0.101271 + 0.994859i $$0.467709\pi$$
$$450$$ 0 0
$$451$$ 58.0689 2.73436
$$452$$ 0 0
$$453$$ 13.5066 0.634594
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.61803 −0.0756884 −0.0378442 0.999284i $$-0.512049\pi$$
−0.0378442 + 0.999284i $$0.512049\pi$$
$$458$$ 0 0
$$459$$ −3.47214 −0.162065
$$460$$ 0 0
$$461$$ −4.94427 −0.230278 −0.115139 0.993349i $$-0.536731\pi$$
−0.115139 + 0.993349i $$0.536731\pi$$
$$462$$ 0 0
$$463$$ 16.0344 0.745184 0.372592 0.927995i $$-0.378469\pi$$
0.372592 + 0.927995i $$0.378469\pi$$
$$464$$ 0 0
$$465$$ −9.32624 −0.432494
$$466$$ 0 0
$$467$$ −1.52786 −0.0707011 −0.0353506 0.999375i $$-0.511255\pi$$
−0.0353506 + 0.999375i $$0.511255\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 12.4721 0.574686
$$472$$ 0 0
$$473$$ 4.76393 0.219046
$$474$$ 0 0
$$475$$ 1.12461 0.0516007
$$476$$ 0 0
$$477$$ 36.1246 1.65403
$$478$$ 0 0
$$479$$ −9.27051 −0.423580 −0.211790 0.977315i $$-0.567929\pi$$
−0.211790 + 0.977315i $$0.567929\pi$$
$$480$$ 0 0
$$481$$ −27.4164 −1.25008
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −6.32624 −0.287260
$$486$$ 0 0
$$487$$ 22.3607 1.01326 0.506630 0.862164i $$-0.330891\pi$$
0.506630 + 0.862164i $$0.330891\pi$$
$$488$$ 0 0
$$489$$ 4.76393 0.215432
$$490$$ 0 0
$$491$$ 10.0902 0.455363 0.227681 0.973736i $$-0.426885\pi$$
0.227681 + 0.973736i $$0.426885\pi$$
$$492$$ 0 0
$$493$$ −7.70820 −0.347160
$$494$$ 0 0
$$495$$ 22.1803 0.996932
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 34.9443 1.56432 0.782160 0.623077i $$-0.214118\pi$$
0.782160 + 0.623077i $$0.214118\pi$$
$$500$$ 0 0
$$501$$ 10.2705 0.458853
$$502$$ 0 0
$$503$$ 8.38197 0.373733 0.186867 0.982385i $$-0.440167\pi$$
0.186867 + 0.982385i $$0.440167\pi$$
$$504$$ 0 0
$$505$$ −13.7082 −0.610007
$$506$$ 0 0
$$507$$ 1.56231 0.0693844
$$508$$ 0 0
$$509$$ 5.23607 0.232085 0.116042 0.993244i $$-0.462979\pi$$
0.116042 + 0.993244i $$0.462979\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1.63932 −0.0723778
$$514$$ 0 0
$$515$$ 18.4721 0.813980
$$516$$ 0 0
$$517$$ 2.47214 0.108724
$$518$$ 0 0
$$519$$ −14.2705 −0.626406
$$520$$ 0 0
$$521$$ −15.7426 −0.689698 −0.344849 0.938658i $$-0.612070\pi$$
−0.344849 + 0.938658i $$0.612070\pi$$
$$522$$ 0 0
$$523$$ −11.8197 −0.516838 −0.258419 0.966033i $$-0.583201\pi$$
−0.258419 + 0.966033i $$0.583201\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9.32624 −0.406257
$$528$$ 0 0
$$529$$ 9.58359 0.416678
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −35.8885 −1.55451
$$534$$ 0 0
$$535$$ 24.6525 1.06582
$$536$$ 0 0
$$537$$ 12.7426 0.549886
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −0.763932 −0.0328440 −0.0164220 0.999865i $$-0.505228\pi$$
−0.0164220 + 0.999865i $$0.505228\pi$$
$$542$$ 0 0
$$543$$ −2.18034 −0.0935673
$$544$$ 0 0
$$545$$ −20.1803 −0.864431
$$546$$ 0 0
$$547$$ −9.34752 −0.399671 −0.199836 0.979829i $$-0.564041\pi$$
−0.199836 + 0.979829i $$0.564041\pi$$
$$548$$ 0 0
$$549$$ 21.7984 0.930332
$$550$$ 0 0
$$551$$ −3.63932 −0.155040
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 8.47214 0.359622
$$556$$ 0 0
$$557$$ −10.3607 −0.438996 −0.219498 0.975613i $$-0.570442\pi$$
−0.219498 + 0.975613i $$0.570442\pi$$
$$558$$ 0 0
$$559$$ −2.94427 −0.124529
$$560$$ 0 0
$$561$$ −3.23607 −0.136627
$$562$$ 0 0
$$563$$ −11.1246 −0.468846 −0.234423 0.972135i $$-0.575320\pi$$
−0.234423 + 0.972135i $$0.575320\pi$$
$$564$$ 0 0
$$565$$ 9.70820 0.408427
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5.49342 −0.230296 −0.115148 0.993348i $$-0.536734\pi$$
−0.115148 + 0.993348i $$0.536734\pi$$
$$570$$ 0 0
$$571$$ −0.472136 −0.0197583 −0.00987914 0.999951i $$-0.503145\pi$$
−0.00987914 + 0.999951i $$0.503145\pi$$
$$572$$ 0 0
$$573$$ 2.34752 0.0980692
$$574$$ 0 0
$$575$$ −13.5967 −0.567024
$$576$$ 0 0
$$577$$ 23.1246 0.962690 0.481345 0.876531i $$-0.340148\pi$$
0.481345 + 0.876531i $$0.340148\pi$$
$$578$$ 0 0
$$579$$ −5.05573 −0.210109
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 72.2492 2.99226
$$584$$ 0 0
$$585$$ −13.7082 −0.566764
$$586$$ 0 0
$$587$$ −10.8328 −0.447118 −0.223559 0.974690i $$-0.571768\pi$$
−0.223559 + 0.974690i $$0.571768\pi$$
$$588$$ 0 0
$$589$$ −4.40325 −0.181433
$$590$$ 0 0
$$591$$ 2.29180 0.0942719
$$592$$ 0 0
$$593$$ −22.3607 −0.918243 −0.459122 0.888373i $$-0.651836\pi$$
−0.459122 + 0.888373i $$0.651836\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −0.562306 −0.0230136
$$598$$ 0 0
$$599$$ 24.9098 1.01779 0.508894 0.860829i $$-0.330055\pi$$
0.508894 + 0.860829i $$0.330055\pi$$
$$600$$ 0 0
$$601$$ 32.2492 1.31547 0.657737 0.753248i $$-0.271514\pi$$
0.657737 + 0.753248i $$0.271514\pi$$
$$602$$ 0 0
$$603$$ 5.00000 0.203616
$$604$$ 0 0
$$605$$ 26.5623 1.07991
$$606$$ 0 0
$$607$$ 33.1459 1.34535 0.672675 0.739938i $$-0.265145\pi$$
0.672675 + 0.739938i $$0.265145\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.52786 −0.0618108
$$612$$ 0 0
$$613$$ 2.56231 0.103491 0.0517453 0.998660i $$-0.483522\pi$$
0.0517453 + 0.998660i $$0.483522\pi$$
$$614$$ 0 0
$$615$$ 11.0902 0.447199
$$616$$ 0 0
$$617$$ 41.0132 1.65113 0.825564 0.564309i $$-0.190857\pi$$
0.825564 + 0.564309i $$0.190857\pi$$
$$618$$ 0 0
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 0 0
$$621$$ 19.8197 0.795336
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −7.41641 −0.296656
$$626$$ 0 0
$$627$$ −1.52786 −0.0610170
$$628$$ 0 0
$$629$$ 8.47214 0.337806
$$630$$ 0 0
$$631$$ −25.9098 −1.03145 −0.515727 0.856753i $$-0.672478\pi$$
−0.515727 + 0.856753i $$0.672478\pi$$
$$632$$ 0 0
$$633$$ 9.70820 0.385866
$$634$$ 0 0
$$635$$ −1.85410 −0.0735778
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 18.1803 0.719203
$$640$$ 0 0
$$641$$ 38.9443 1.53821 0.769103 0.639125i $$-0.220703\pi$$
0.769103 + 0.639125i $$0.220703\pi$$
$$642$$ 0 0
$$643$$ −9.85410 −0.388608 −0.194304 0.980941i $$-0.562245\pi$$
−0.194304 + 0.980941i $$0.562245\pi$$
$$644$$ 0 0
$$645$$ 0.909830 0.0358245
$$646$$ 0 0
$$647$$ 25.2361 0.992132 0.496066 0.868285i $$-0.334777\pi$$
0.496066 + 0.868285i $$0.334777\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −23.5279 −0.920716 −0.460358 0.887733i $$-0.652279\pi$$
−0.460358 + 0.887733i $$0.652279\pi$$
$$654$$ 0 0
$$655$$ 12.9443 0.505775
$$656$$ 0 0
$$657$$ 36.2705 1.41505
$$658$$ 0 0
$$659$$ −40.7426 −1.58711 −0.793554 0.608500i $$-0.791772\pi$$
−0.793554 + 0.608500i $$0.791772\pi$$
$$660$$ 0 0
$$661$$ 22.8328 0.888094 0.444047 0.896004i $$-0.353542\pi$$
0.444047 + 0.896004i $$0.353542\pi$$
$$662$$ 0 0
$$663$$ 2.00000 0.0776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 44.0000 1.70369
$$668$$ 0 0
$$669$$ −9.23607 −0.357087
$$670$$ 0 0
$$671$$ 43.5967 1.68303
$$672$$ 0 0
$$673$$ −5.52786 −0.213083 −0.106542 0.994308i $$-0.533978\pi$$
−0.106542 + 0.994308i $$0.533978\pi$$
$$674$$ 0 0
$$675$$ −8.27051 −0.318332
$$676$$ 0 0
$$677$$ −37.7771 −1.45189 −0.725946 0.687752i $$-0.758598\pi$$
−0.725946 + 0.687752i $$0.758598\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −10.3475 −0.396518
$$682$$ 0 0
$$683$$ 5.81966 0.222683 0.111342 0.993782i $$-0.464485\pi$$
0.111342 + 0.993782i $$0.464485\pi$$
$$684$$ 0 0
$$685$$ 7.47214 0.285496
$$686$$ 0 0
$$687$$ −9.12461 −0.348126
$$688$$ 0 0
$$689$$ −44.6525 −1.70112
$$690$$ 0 0
$$691$$ −46.5066 −1.76919 −0.884597 0.466357i $$-0.845566\pi$$
−0.884597 + 0.466357i $$0.845566\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −24.3262 −0.922747
$$696$$ 0 0
$$697$$ 11.0902 0.420070
$$698$$ 0 0
$$699$$ 1.52786 0.0577891
$$700$$ 0 0
$$701$$ −18.5836 −0.701893 −0.350946 0.936396i $$-0.614140\pi$$
−0.350946 + 0.936396i $$0.614140\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0.472136 0.0177817
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 9.34752 0.351054 0.175527 0.984475i $$-0.443837\pi$$
0.175527 + 0.984475i $$0.443837\pi$$
$$710$$ 0 0
$$711$$ 39.1246 1.46729
$$712$$ 0 0
$$713$$ 53.2361 1.99371
$$714$$ 0 0
$$715$$ −27.4164 −1.02532
$$716$$ 0 0
$$717$$ −14.2705 −0.532942
$$718$$ 0 0
$$719$$ 45.8541 1.71007 0.855035 0.518571i $$-0.173536\pi$$
0.855035 + 0.518571i $$0.173536\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −15.4721 −0.575415
$$724$$ 0 0
$$725$$ −18.3607 −0.681899
$$726$$ 0 0
$$727$$ −7.52786 −0.279193 −0.139597 0.990208i $$-0.544581\pi$$
−0.139597 + 0.990208i $$0.544581\pi$$
$$728$$ 0 0
$$729$$ −8.50658 −0.315058
$$730$$ 0 0
$$731$$ 0.909830 0.0336513
$$732$$ 0 0
$$733$$ −36.0000 −1.32969 −0.664845 0.746981i $$-0.731502\pi$$
−0.664845 + 0.746981i $$0.731502\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 10.0000 0.368355
$$738$$ 0 0
$$739$$ −21.5623 −0.793182 −0.396591 0.917995i $$-0.629807\pi$$
−0.396591 + 0.917995i $$0.629807\pi$$
$$740$$ 0 0
$$741$$ 0.944272 0.0346887
$$742$$ 0 0
$$743$$ −26.6525 −0.977785 −0.488892 0.872344i $$-0.662599\pi$$
−0.488892 + 0.872344i $$0.662599\pi$$
$$744$$ 0 0
$$745$$ −18.4164 −0.674725
$$746$$ 0 0
$$747$$ −29.8885 −1.09356
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −6.94427 −0.253400 −0.126700 0.991941i $$-0.540439\pi$$
−0.126700 + 0.991941i $$0.540439\pi$$
$$752$$ 0 0
$$753$$ 5.81966 0.212080
$$754$$ 0 0
$$755$$ −35.3607 −1.28691
$$756$$ 0 0
$$757$$ −29.5623 −1.07446 −0.537230 0.843436i $$-0.680529\pi$$
−0.537230 + 0.843436i $$0.680529\pi$$
$$758$$ 0 0
$$759$$ 18.4721 0.670496
$$760$$ 0 0
$$761$$ −7.81966 −0.283462 −0.141731 0.989905i $$-0.545267\pi$$
−0.141731 + 0.989905i $$0.545267\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 4.23607 0.153155
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −33.1246 −1.19450 −0.597252 0.802054i $$-0.703741\pi$$
−0.597252 + 0.802054i $$0.703741\pi$$
$$770$$ 0 0
$$771$$ 7.59675 0.273590
$$772$$ 0 0
$$773$$ 21.4164 0.770295 0.385147 0.922855i $$-0.374151\pi$$
0.385147 + 0.922855i $$0.374151\pi$$
$$774$$ 0 0
$$775$$ −22.2148 −0.797979
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5.23607 0.187602
$$780$$ 0 0
$$781$$ 36.3607 1.30109
$$782$$ 0 0
$$783$$ 26.7639 0.956465
$$784$$ 0 0
$$785$$ −32.6525 −1.16542
$$786$$ 0 0
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ 0 0
$$789$$ −0.583592 −0.0207764
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −26.9443 −0.956819
$$794$$ 0 0
$$795$$ 13.7984 0.489378
$$796$$ 0 0
$$797$$ 11.4164 0.404390 0.202195 0.979345i $$-0.435193\pi$$
0.202195 + 0.979345i $$0.435193\pi$$
$$798$$ 0 0
$$799$$ 0.472136 0.0167030
$$800$$ 0 0
$$801$$ 5.23607 0.185007
$$802$$ 0 0
$$803$$ 72.5410 2.55992
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.18034 −0.217558
$$808$$ 0 0
$$809$$ 31.4853 1.10696 0.553482 0.832861i $$-0.313299\pi$$
0.553482 + 0.832861i $$0.313299\pi$$
$$810$$ 0 0
$$811$$ 32.0344 1.12488 0.562441 0.826838i $$-0.309862\pi$$
0.562441 + 0.826838i $$0.309862\pi$$
$$812$$ 0 0
$$813$$ −6.00000 −0.210429
$$814$$ 0 0
$$815$$ −12.4721 −0.436880
$$816$$ 0 0
$$817$$ 0.429563 0.0150285
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −30.3607 −1.05960 −0.529798 0.848124i $$-0.677732\pi$$
−0.529798 + 0.848124i $$0.677732\pi$$
$$822$$ 0 0
$$823$$ −37.4853 −1.30666 −0.653328 0.757075i $$-0.726628\pi$$
−0.653328 + 0.757075i $$0.726628\pi$$
$$824$$ 0 0
$$825$$ −7.70820 −0.268365
$$826$$ 0 0
$$827$$ 36.4721 1.26826 0.634130 0.773226i $$-0.281358\pi$$
0.634130 + 0.773226i $$0.281358\pi$$
$$828$$ 0 0
$$829$$ −16.1803 −0.561966 −0.280983 0.959713i $$-0.590661\pi$$
−0.280983 + 0.959713i $$0.590661\pi$$
$$830$$ 0 0
$$831$$ −4.47214 −0.155137
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −26.8885 −0.930516
$$836$$ 0 0
$$837$$ 32.3820 1.11928
$$838$$ 0 0
$$839$$ 22.4721 0.775824 0.387912 0.921696i $$-0.373196\pi$$
0.387912 + 0.921696i $$0.373196\pi$$
$$840$$ 0 0
$$841$$ 30.4164 1.04884
$$842$$ 0 0
$$843$$ 12.4164 0.427644
$$844$$ 0 0
$$845$$ −4.09017 −0.140706
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −15.1803 −0.520988
$$850$$ 0 0
$$851$$ −48.3607 −1.65778
$$852$$ 0 0
$$853$$ −6.36068 −0.217786 −0.108893 0.994054i $$-0.534731\pi$$
−0.108893 + 0.994054i $$0.534731\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ 3.90983 0.133557 0.0667786 0.997768i $$-0.478728\pi$$
0.0667786 + 0.997768i $$0.478728\pi$$
$$858$$ 0 0
$$859$$ 2.94427 0.100457 0.0502286 0.998738i $$-0.484005\pi$$
0.0502286 + 0.998738i $$0.484005\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −30.6738 −1.04415 −0.522074 0.852900i $$-0.674841\pi$$
−0.522074 + 0.852900i $$0.674841\pi$$
$$864$$ 0 0
$$865$$ 37.3607 1.27030
$$866$$ 0 0
$$867$$ −0.618034 −0.0209895
$$868$$ 0 0
$$869$$ 78.2492 2.65442
$$870$$ 0 0
$$871$$ −6.18034 −0.209413
$$872$$ 0 0
$$873$$ 10.2361 0.346438
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 31.7082 1.07071 0.535355 0.844627i $$-0.320178\pi$$
0.535355 + 0.844627i $$0.320178\pi$$
$$878$$ 0 0
$$879$$ −12.1803 −0.410833
$$880$$ 0 0
$$881$$ 35.3262 1.19017 0.595086 0.803662i $$-0.297118\pi$$
0.595086 + 0.803662i $$0.297118\pi$$
$$882$$ 0 0
$$883$$ −31.9787 −1.07617 −0.538085 0.842891i $$-0.680852\pi$$
−0.538085 + 0.842891i $$0.680852\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −21.5066 −0.722120 −0.361060 0.932543i $$-0.617585\pi$$
−0.361060 + 0.932543i $$0.617585\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −29.8885 −1.00130
$$892$$ 0 0
$$893$$ 0.222912 0.00745948
$$894$$ 0 0
$$895$$ −33.3607 −1.11512
$$896$$ 0 0
$$897$$ −11.4164 −0.381183
$$898$$ 0 0
$$899$$ 71.8885 2.39762
$$900$$ 0 0
$$901$$ 13.7984 0.459690
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 5.70820 0.189747
$$906$$ 0 0
$$907$$ 31.3050 1.03946 0.519732 0.854329i $$-0.326032\pi$$
0.519732 + 0.854329i $$0.326032\pi$$
$$908$$ 0 0
$$909$$ 22.1803 0.735675
$$910$$ 0 0
$$911$$ −5.52786 −0.183146 −0.0915732 0.995798i $$-0.529190\pi$$
−0.0915732 + 0.995798i $$0.529190\pi$$
$$912$$ 0 0
$$913$$ −59.7771 −1.97833
$$914$$ 0 0
$$915$$ 8.32624 0.275257
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 39.1591 1.29174 0.645869 0.763448i $$-0.276495\pi$$
0.645869 + 0.763448i $$0.276495\pi$$
$$920$$ 0 0
$$921$$ 11.7082 0.385798
$$922$$ 0 0
$$923$$ −22.4721 −0.739679
$$924$$ 0 0
$$925$$ 20.1803 0.663525
$$926$$ 0 0
$$927$$ −29.8885 −0.981669
$$928$$ 0 0
$$929$$ 12.2705 0.402582 0.201291 0.979531i $$-0.435486\pi$$
0.201291 + 0.979531i $$0.435486\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 15.0689 0.493333
$$934$$ 0 0
$$935$$ 8.47214 0.277068
$$936$$ 0 0
$$937$$ −57.3050 −1.87207 −0.936036 0.351905i $$-0.885534\pi$$
−0.936036 + 0.351905i $$0.885534\pi$$
$$938$$ 0 0
$$939$$ 19.9443 0.650857
$$940$$ 0 0
$$941$$ 0.978714 0.0319052 0.0159526 0.999873i $$-0.494922\pi$$
0.0159526 + 0.999873i $$0.494922\pi$$
$$942$$ 0 0
$$943$$ −63.3050 −2.06149
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 47.1246 1.53134 0.765672 0.643231i $$-0.222407\pi$$
0.765672 + 0.643231i $$0.222407\pi$$
$$948$$ 0 0
$$949$$ −44.8328 −1.45533
$$950$$ 0 0
$$951$$ 3.59675 0.116633
$$952$$ 0 0
$$953$$ 16.6738 0.540116 0.270058 0.962844i $$-0.412957\pi$$
0.270058 + 0.962844i $$0.412957\pi$$
$$954$$ 0 0
$$955$$ −6.14590 −0.198877
$$956$$ 0 0
$$957$$ 24.9443 0.806334
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 55.9787 1.80576
$$962$$ 0 0
$$963$$ −39.8885 −1.28539
$$964$$ 0 0
$$965$$ 13.2361 0.426084
$$966$$ 0 0
$$967$$ 13.2016 0.424536 0.212268 0.977212i $$-0.431915\pi$$
0.212268 + 0.977212i $$0.431915\pi$$
$$968$$ 0 0
$$969$$ −0.291796 −0.00937384
$$970$$ 0 0
$$971$$ 19.1246 0.613738 0.306869 0.951752i $$-0.400719\pi$$
0.306869 + 0.951752i $$0.400719\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 4.76393 0.152568
$$976$$ 0 0
$$977$$ 49.2148 1.57452 0.787260 0.616621i $$-0.211499\pi$$
0.787260 + 0.616621i $$0.211499\pi$$
$$978$$ 0 0
$$979$$ 10.4721 0.334691
$$980$$ 0 0
$$981$$ 32.6525 1.04251
$$982$$ 0 0
$$983$$ −18.0902 −0.576987 −0.288493 0.957482i $$-0.593154\pi$$
−0.288493 + 0.957482i $$0.593154\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −5.19350 −0.165144
$$990$$ 0 0
$$991$$ 52.2492 1.65975 0.829876 0.557948i $$-0.188411\pi$$
0.829876 + 0.557948i $$0.188411\pi$$
$$992$$ 0 0
$$993$$ 6.81966 0.216415
$$994$$ 0 0
$$995$$ 1.47214 0.0466698
$$996$$ 0 0
$$997$$ 0.0344419 0.00109078 0.000545392 1.00000i $$-0.499826\pi$$
0.000545392 1.00000i $$0.499826\pi$$
$$998$$ 0 0
$$999$$ −29.4164 −0.930694
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 3332.2.a.l.1.1 2
7.6 odd 2 476.2.a.b.1.2 2
21.20 even 2 4284.2.a.m.1.2 2
28.27 even 2 1904.2.a.j.1.1 2
56.13 odd 2 7616.2.a.u.1.1 2
56.27 even 2 7616.2.a.p.1.2 2
119.118 odd 2 8092.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.b.1.2 2 7.6 odd 2
1904.2.a.j.1.1 2 28.27 even 2
3332.2.a.l.1.1 2 1.1 even 1 trivial
4284.2.a.m.1.2 2 21.20 even 2
7616.2.a.p.1.2 2 56.27 even 2
7616.2.a.u.1.1 2 56.13 odd 2
8092.2.a.m.1.1 2 119.118 odd 2