# Properties

 Label 7616.2.a.p.1.2 Level $7616$ Weight $2$ Character 7616.1 Self dual yes Analytic conductor $60.814$ 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] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7616 = 2^{6} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7616.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: $$60.8140661794$$ Analytic rank: $$0$$ 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 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 7616.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.618034 q^{3} +1.61803 q^{5} +1.00000 q^{7} -2.61803 q^{9} +O(q^{10})$$ $$q+0.618034 q^{3} +1.61803 q^{5} +1.00000 q^{7} -2.61803 q^{9} -5.23607 q^{11} +3.23607 q^{13} +1.00000 q^{15} +1.00000 q^{17} +0.472136 q^{19} +0.618034 q^{21} -5.70820 q^{23} -2.38197 q^{25} -3.47214 q^{27} -7.70820 q^{29} +9.32624 q^{31} -3.23607 q^{33} +1.61803 q^{35} +8.47214 q^{37} +2.00000 q^{39} +11.0902 q^{41} -0.909830 q^{43} -4.23607 q^{45} -0.472136 q^{47} +1.00000 q^{49} +0.618034 q^{51} +13.7984 q^{53} -8.47214 q^{55} +0.291796 q^{57} -8.32624 q^{61} -2.61803 q^{63} +5.23607 q^{65} -1.90983 q^{67} -3.52786 q^{69} +6.94427 q^{71} +13.8541 q^{73} -1.47214 q^{75} -5.23607 q^{77} +14.9443 q^{79} +5.70820 q^{81} -11.4164 q^{83} +1.61803 q^{85} -4.76393 q^{87} +2.00000 q^{89} +3.23607 q^{91} +5.76393 q^{93} +0.763932 q^{95} +3.90983 q^{97} +13.7082 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + q^5 + 2 * q^7 - 3 * q^9 $$2 q - q^{3} + q^{5} + 2 q^{7} - 3 q^{9} - 6 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} - q^{21} + 2 q^{23} - 7 q^{25} + 2 q^{27} - 2 q^{29} + 3 q^{31} - 2 q^{33} + q^{35} + 8 q^{37} + 4 q^{39} + 11 q^{41} - 13 q^{43} - 4 q^{45} + 8 q^{47} + 2 q^{49} - q^{51} + 3 q^{53} - 8 q^{55} + 14 q^{57} - q^{61} - 3 q^{63} + 6 q^{65} - 15 q^{67} - 16 q^{69} - 4 q^{71} + 21 q^{73} + 6 q^{75} - 6 q^{77} + 12 q^{79} - 2 q^{81} + 4 q^{83} + q^{85} - 14 q^{87} + 4 q^{89} + 2 q^{91} + 16 q^{93} + 6 q^{95} + 19 q^{97} + 14 q^{99}+O(q^{100})$$ 2 * q - q^3 + q^5 + 2 * q^7 - 3 * q^9 - 6 * q^11 + 2 * q^13 + 2 * q^15 + 2 * q^17 - 8 * q^19 - q^21 + 2 * q^23 - 7 * q^25 + 2 * q^27 - 2 * q^29 + 3 * q^31 - 2 * q^33 + q^35 + 8 * q^37 + 4 * q^39 + 11 * q^41 - 13 * q^43 - 4 * q^45 + 8 * q^47 + 2 * q^49 - q^51 + 3 * q^53 - 8 * q^55 + 14 * q^57 - q^61 - 3 * q^63 + 6 * q^65 - 15 * q^67 - 16 * q^69 - 4 * q^71 + 21 * q^73 + 6 * q^75 - 6 * q^77 + 12 * q^79 - 2 * q^81 + 4 * q^83 + q^85 - 14 * q^87 + 4 * q^89 + 2 * q^91 + 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 $$-0.442904\pi$$
0.178411 + 0.983956i $$0.442904\pi$$
$$4$$ 0 0
$$5$$ 1.61803 0.723607 0.361803 0.932254i $$-0.382161\pi$$
0.361803 + 0.932254i $$0.382161\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 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 $$-0.482753\pi$$
0.0541577 + 0.998532i $$0.482753\pi$$
$$20$$ 0 0
$$21$$ 0.618034 0.134866
$$22$$ 0 0
$$23$$ −5.70820 −1.19024 −0.595121 0.803636i $$-0.702896\pi$$
−0.595121 + 0.803636i $$0.702896\pi$$
$$24$$ 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 $$-0.753887\pi$$
−0.715689 + 0.698419i $$0.753887\pi$$
$$30$$ 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
$$35$$ 1.61803 0.273498
$$36$$ 0 0
$$37$$ 8.47214 1.39281 0.696405 0.717649i $$-0.254782\pi$$
0.696405 + 0.717649i $$0.254782\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 11.0902 1.73199 0.865997 0.500050i $$-0.166685\pi$$
0.865997 + 0.500050i $$0.166685\pi$$
$$42$$ 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
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0.618034 0.0865421
$$52$$ 0 0
$$53$$ 13.7984 1.89535 0.947676 0.319233i $$-0.103425\pi$$
0.947676 + 0.319233i $$0.103425\pi$$
$$54$$ 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
$$63$$ −2.61803 −0.329841
$$64$$ 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 $$-0.364807\pi$$
0.412067 + 0.911154i $$0.364807\pi$$
$$72$$ 0 0
$$73$$ 13.8541 1.62150 0.810750 0.585393i $$-0.199060\pi$$
0.810750 + 0.585393i $$0.199060\pi$$
$$74$$ 0 0
$$75$$ −1.47214 −0.169988
$$76$$ 0 0
$$77$$ −5.23607 −0.596705
$$78$$ 0 0
$$79$$ 14.9443 1.68136 0.840681 0.541531i $$-0.182155\pi$$
0.840681 + 0.541531i $$0.182155\pi$$
$$80$$ 0 0
$$81$$ 5.70820 0.634245
$$82$$ 0 0
$$83$$ −11.4164 −1.25311 −0.626557 0.779376i $$-0.715536\pi$$
−0.626557 + 0.779376i $$0.715536\pi$$
$$84$$ 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 $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 3.23607 0.339232
$$92$$ 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 $$-0.436396\pi$$
0.198492 + 0.980103i $$0.436396\pi$$
$$98$$ 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
$$105$$ 1.00000 0.0975900
$$106$$ 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 $$-0.296237\pi$$
0.597307 + 0.802013i $$0.296237\pi$$
$$110$$ 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
$$119$$ 1.00000 0.0916698
$$120$$ 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 $$-0.483810\pi$$
0.0508410 + 0.998707i $$0.483810\pi$$
$$128$$ 0 0
$$129$$ −0.562306 −0.0495083
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ 0.472136 0.0409394
$$134$$ 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 $$-0.279926\pi$$
0.637602 + 0.770366i $$0.279926\pi$$
$$140$$ 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
$$147$$ 0.618034 0.0509746
$$148$$ 0 0
$$149$$ 11.3820 0.932447 0.466223 0.884667i $$-0.345614\pi$$
0.466223 + 0.884667i $$0.345614\pi$$
$$150$$ 0 0
$$151$$ 21.8541 1.77846 0.889231 0.457459i $$-0.151240\pi$$
0.889231 + 0.457459i $$0.151240\pi$$
$$152$$ 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
$$161$$ −5.70820 −0.449869
$$162$$ 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
$$175$$ −2.38197 −0.180060
$$176$$ 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
$$189$$ −3.47214 −0.252561
$$190$$ 0 0
$$191$$ 3.79837 0.274841 0.137420 0.990513i $$-0.456119\pi$$
0.137420 + 0.990513i $$0.456119\pi$$
$$192$$ 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 $$-0.457828\pi$$
0.132099 + 0.991236i $$0.457828\pi$$
$$198$$ 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
$$203$$ −7.70820 −0.541010
$$204$$ 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
$$217$$ 9.32624 0.633106
$$218$$ 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 $$-0.687521\pi$$
−0.555624 + 0.831434i $$0.687521\pi$$
$$228$$ 0 0
$$229$$ 14.7639 0.975628 0.487814 0.872948i $$-0.337794\pi$$
0.487814 + 0.872948i $$0.337794\pi$$
$$230$$ 0 0
$$231$$ −3.23607 −0.212918
$$232$$ 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 $$-0.768406\pi$$
−0.746789 + 0.665061i $$0.768406\pi$$
$$240$$ 0 0
$$241$$ −25.0344 −1.61261 −0.806305 0.591500i $$-0.798536\pi$$
−0.806305 + 0.591500i $$0.798536\pi$$
$$242$$ 0 0
$$243$$ 13.9443 0.894525
$$244$$ 0 0
$$245$$ 1.61803 0.103372
$$246$$ 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 $$-0.403954\pi$$
0.297179 + 0.954822i $$0.403954\pi$$
$$252$$ 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 $$-0.374763\pi$$
0.383371 + 0.923595i $$0.374763\pi$$
$$258$$ 0 0
$$259$$ 8.47214 0.526433
$$260$$ 0 0
$$261$$ 20.1803 1.24913
$$262$$ 0 0
$$263$$ −0.944272 −0.0582263 −0.0291132 0.999576i $$-0.509268\pi$$
−0.0291132 + 0.999576i $$0.509268\pi$$
$$264$$ 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
$$273$$ 2.00000 0.121046
$$274$$ 0 0
$$275$$ 12.4721 0.752098
$$276$$ 0 0
$$277$$ −7.23607 −0.434773 −0.217387 0.976086i $$-0.569753\pi$$
−0.217387 + 0.976086i $$0.569753\pi$$
$$278$$ 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 $$-0.760498\pi$$
−0.730039 + 0.683406i $$0.760498\pi$$
$$284$$ 0 0
$$285$$ 0.472136 0.0279669
$$286$$ 0 0
$$287$$ 11.0902 0.654632
$$288$$ 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
$$301$$ −0.909830 −0.0524417
$$302$$ 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 $$-0.318196\pi$$
0.540603 + 0.841278i $$0.318196\pi$$
$$308$$ 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 $$-0.134523\pi$$
0.912019 + 0.410149i $$0.134523\pi$$
$$314$$ 0 0
$$315$$ −4.23607 −0.238675
$$316$$ 0 0
$$317$$ 5.81966 0.326865 0.163432 0.986555i $$-0.447743\pi$$
0.163432 + 0.986555i $$0.447743\pi$$
$$318$$ 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
$$329$$ −0.472136 −0.0260297
$$330$$ 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
$$343$$ 1.00000 0.0539949
$$344$$ 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 $$-0.651682\pi$$
−0.458692 + 0.888595i $$0.651682\pi$$
$$354$$ 0 0
$$355$$ 11.2361 0.596349
$$356$$ 0 0
$$357$$ 0.618034 0.0327098
$$358$$ 0 0
$$359$$ 0.562306 0.0296774 0.0148387 0.999890i $$-0.495277\pi$$
0.0148387 + 0.999890i $$0.495277\pi$$
$$360$$ 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
$$371$$ 13.7984 0.716376
$$372$$ 0 0
$$373$$ 10.1459 0.525335 0.262667 0.964886i $$-0.415398\pi$$
0.262667 + 0.964886i $$0.415398\pi$$
$$374$$ 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
$$385$$ −8.47214 −0.431780
$$386$$ 0 0
$$387$$ 2.38197 0.121082
$$388$$ 0 0
$$389$$ 1.43769 0.0728940 0.0364470 0.999336i $$-0.488396\pi$$
0.0364470 + 0.999336i $$0.488396\pi$$
$$390$$ 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
$$399$$ 0.291796 0.0146081
$$400$$ 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 $$-0.385982\pi$$
0.350586 + 0.936530i $$0.385982\pi$$
$$410$$ 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 $$-0.648823\pi$$
−0.450691 + 0.892680i $$0.648823\pi$$
$$420$$ 0 0
$$421$$ −22.2148 −1.08268 −0.541341 0.840803i $$-0.682083\pi$$
−0.541341 + 0.840803i $$0.682083\pi$$
$$422$$ 0 0
$$423$$ 1.23607 0.0600997
$$424$$ 0 0
$$425$$ −2.38197 −0.115542
$$426$$ 0 0
$$427$$ −8.32624 −0.402935
$$428$$ 0 0
$$429$$ −10.4721 −0.505599
$$430$$ 0 0
$$431$$ −22.9443 −1.10519 −0.552593 0.833451i $$-0.686362\pi$$
−0.552593 + 0.833451i $$0.686362\pi$$
$$432$$ 0 0
$$433$$ −0.944272 −0.0453788 −0.0226894 0.999743i $$-0.507223\pi$$
−0.0226894 + 0.999743i $$0.507223\pi$$
$$434$$ 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
$$441$$ −2.61803 −0.124668
$$442$$ 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
$$455$$ 5.23607 0.245471
$$456$$ 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 $$-0.621531\pi$$
−0.372592 + 0.927995i $$0.621531\pi$$
$$464$$ 0 0
$$465$$ 9.32624 0.432494
$$466$$ 0 0
$$467$$ 1.52786 0.0707011 0.0353506 0.999375i $$-0.488745\pi$$
0.0353506 + 0.999375i $$0.488745\pi$$
$$468$$ 0 0
$$469$$ −1.90983 −0.0881878
$$470$$ 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
$$483$$ −3.52786 −0.160523
$$484$$ 0 0
$$485$$ 6.32624 0.287260
$$486$$ 0 0
$$487$$ −22.3607 −1.01326 −0.506630 0.862164i $$-0.669109\pi$$
−0.506630 + 0.862164i $$0.669109\pi$$
$$488$$ 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
$$497$$ 6.94427 0.311493
$$498$$ 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
$$511$$ 13.8541 0.612869
$$512$$ 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 $$-0.387930\pi$$
0.344849 + 0.938658i $$0.387930\pi$$
$$522$$ 0 0
$$523$$ 11.8197 0.516838 0.258419 0.966033i $$-0.416799\pi$$
0.258419 + 0.966033i $$0.416799\pi$$
$$524$$ 0 0
$$525$$ −1.47214 −0.0642493
$$526$$ 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
$$539$$ −5.23607 −0.225533
$$540$$ 0 0
$$541$$ 0.763932 0.0328440 0.0164220 0.999865i $$-0.494772\pi$$
0.0164220 + 0.999865i $$0.494772\pi$$
$$542$$ 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
$$553$$ 14.9443 0.635495
$$554$$ 0 0
$$555$$ 8.47214 0.359622
$$556$$ 0 0
$$557$$ 10.3607 0.438996 0.219498 0.975613i $$-0.429558\pi$$
0.219498 + 0.975613i $$0.429558\pi$$
$$558$$ 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 $$-0.424680\pi$$
0.234423 + 0.972135i $$0.424680\pi$$
$$564$$ 0 0
$$565$$ 9.70820 0.408427
$$566$$ 0 0
$$567$$ 5.70820 0.239722
$$568$$ 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 $$-0.659852\pi$$
−0.481345 + 0.876531i $$0.659852\pi$$
$$578$$ 0 0
$$579$$ 5.05573 0.210109
$$580$$ 0 0
$$581$$ −11.4164 −0.473632
$$582$$ 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 $$-0.428232\pi$$
0.223559 + 0.974690i $$0.428232\pi$$
$$588$$ 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 $$-0.348164\pi$$
0.459122 + 0.888373i $$0.348164\pi$$
$$594$$ 0 0
$$595$$ 1.61803 0.0663329
$$596$$ 0 0
$$597$$ 0.562306 0.0230136
$$598$$ 0 0
$$599$$ −24.9098 −1.01779 −0.508894 0.860829i $$-0.669945\pi$$
−0.508894 + 0.860829i $$0.669945\pi$$
$$600$$ 0 0
$$601$$ −32.2492 −1.31547 −0.657737 0.753248i $$-0.728486\pi$$
−0.657737 + 0.753248i $$0.728486\pi$$
$$602$$ 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
$$609$$ −4.76393 −0.193044
$$610$$ 0 0
$$611$$ −1.52786 −0.0618108
$$612$$ 0 0
$$613$$ −2.56231 −0.103491 −0.0517453 0.998660i $$-0.516478\pi$$
−0.0517453 + 0.998660i $$0.516478\pi$$
$$614$$ 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 $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 0 0
$$621$$ 19.8197 0.795336
$$622$$ 0 0
$$623$$ 2.00000 0.0801283
$$624$$ 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 $$-0.327522\pi$$
0.515727 + 0.856753i $$0.327522\pi$$
$$632$$ 0 0
$$633$$ −9.70820 −0.385866
$$634$$ 0 0
$$635$$ 1.85410 0.0735778
$$636$$ 0 0
$$637$$ 3.23607 0.128218
$$638$$ 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 $$-0.437755\pi$$
0.194304 + 0.980941i $$0.437755\pi$$
$$644$$ 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
$$651$$ 5.76393 0.225906
$$652$$ 0 0
$$653$$ 23.5279 0.920716 0.460358 0.887733i $$-0.347721\pi$$
0.460358 + 0.887733i $$0.347721\pi$$
$$654$$ 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
$$665$$ 0.763932 0.0296240
$$666$$ 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
$$679$$ 3.90983 0.150046
$$680$$ 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 $$-0.154434\pi$$
0.884597 + 0.466357i $$0.154434\pi$$
$$692$$ 0 0
$$693$$ 13.7082 0.520732
$$694$$ 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 $$-0.385860\pi$$
0.350946 + 0.936396i $$0.385860\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ −0.472136 −0.0177817
$$706$$ 0 0
$$707$$ −8.47214 −0.318627
$$708$$ 0 0
$$709$$ −9.34752 −0.351054 −0.175527 0.984475i $$-0.556163\pi$$
−0.175527 + 0.984475i $$0.556163\pi$$
$$710$$ 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
$$721$$ 11.4164 0.425169
$$722$$ 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
$$735$$ 1.00000 0.0368856
$$736$$ 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 $$-0.337401\pi$$
0.488892 + 0.872344i $$0.337401\pi$$
$$744$$ 0 0
$$745$$ 18.4164 0.674725
$$746$$ 0 0
$$747$$ 29.8885 1.09356
$$748$$ 0 0
$$749$$ 15.2361 0.556714
$$750$$ 0 0
$$751$$ 6.94427 0.253400 0.126700 0.991941i $$-0.459561\pi$$
0.126700 + 0.991941i $$0.459561\pi$$
$$752$$ 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 $$-0.319471\pi$$
0.537230 + 0.843436i $$0.319471\pi$$
$$758$$ 0 0
$$759$$ 18.4721 0.670496
$$760$$ 0 0
$$761$$ 7.81966 0.283462 0.141731 0.989905i $$-0.454733\pi$$
0.141731 + 0.989905i $$0.454733\pi$$
$$762$$ 0 0
$$763$$ 12.4721 0.451522
$$764$$ 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 $$-0.296259\pi$$
0.597252 + 0.802054i $$0.296259\pi$$
$$770$$ 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
$$777$$ 5.23607 0.187843
$$778$$ 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 $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ 0 0
$$789$$ −0.583592 −0.0207764
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 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
$$805$$ −9.23607 −0.325529
$$806$$ 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 $$-0.690138\pi$$
−0.562441 + 0.826838i $$0.690138\pi$$
$$812$$ 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
$$819$$ −8.47214 −0.296040
$$820$$ 0 0
$$821$$ 30.3607 1.05960 0.529798 0.848124i $$-0.322268\pi$$
0.529798 + 0.848124i $$0.322268\pi$$
$$822$$ 0 0
$$823$$ 37.4853 1.30666 0.653328 0.757075i $$-0.273372\pi$$
0.653328 + 0.757075i $$0.273372\pi$$
$$824$$ 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
$$833$$ 1.00000 0.0346479
$$834$$ 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
$$847$$ 16.4164 0.564074
$$848$$ 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 $$-0.521272\pi$$
−0.0667786 + 0.997768i $$0.521272\pi$$
$$858$$ 0 0
$$859$$ −2.94427 −0.100457 −0.0502286 0.998738i $$-0.515995\pi$$
−0.0502286 + 0.998738i $$0.515995\pi$$
$$860$$ 0 0
$$861$$ 6.85410 0.233587
$$862$$ 0 0
$$863$$ 30.6738 1.04415 0.522074 0.852900i $$-0.325159\pi$$
0.522074 + 0.852900i $$0.325159\pi$$
$$864$$ 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
$$875$$ −11.9443 −0.403790
$$876$$ 0 0
$$877$$ −31.7082 −1.07071 −0.535355 0.844627i $$-0.679822\pi$$
−0.535355 + 0.844627i $$0.679822\pi$$
$$878$$ 0 0
$$879$$ 12.1803 0.410833
$$880$$ 0 0
$$881$$ −35.3262 −1.19017 −0.595086 0.803662i $$-0.702882\pi$$
−0.595086 + 0.803662i $$0.702882\pi$$
$$882$$ 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
$$889$$ 1.14590 0.0384322
$$890$$ 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
$$903$$ −0.562306 −0.0187124
$$904$$ 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 $$-0.470810\pi$$
0.0915732 + 0.995798i $$0.470810\pi$$
$$912$$ 0 0
$$913$$ 59.7771 1.97833
$$914$$ 0 0
$$915$$ −8.32624 −0.275257
$$916$$ 0 0
$$917$$ −8.00000 −0.264183
$$918$$ 0 0
$$919$$ −39.1591 −1.29174 −0.645869 0.763448i $$-0.723505\pi$$
−0.645869 + 0.763448i $$0.723505\pi$$
$$920$$ 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 $$-0.564514\pi$$
−0.201291 + 0.979531i $$0.564514\pi$$
$$930$$ 0 0
$$931$$ 0.472136 0.0154736
$$932$$ 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 $$-0.114466\pi$$
0.936036 + 0.351905i $$0.114466\pi$$
$$938$$ 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
$$945$$ −5.61803 −0.182755
$$946$$ 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
$$959$$ 4.61803 0.149124
$$960$$ 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 $$-0.568085\pi$$
−0.212268 + 0.977212i $$0.568085\pi$$
$$968$$ 0 0
$$969$$ 0.291796 0.00937384
$$970$$ 0 0
$$971$$ −19.1246 −0.613738 −0.306869 0.951752i $$-0.599281\pi$$
−0.306869 + 0.951752i $$0.599281\pi$$
$$972$$ 0 0
$$973$$ 15.0344 0.481982
$$974$$ 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
$$987$$ −0.291796 −0.00928797
$$988$$ 0 0
$$989$$ 5.19350 0.165144
$$990$$ 0 0
$$991$$ −52.2492 −1.65975 −0.829876 0.557948i $$-0.811589\pi$$
−0.829876 + 0.557948i $$0.811589\pi$$
$$992$$ 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 7616.2.a.p.1.2 2
4.3 odd 2 7616.2.a.u.1.1 2
8.3 odd 2 476.2.a.b.1.2 2
8.5 even 2 1904.2.a.j.1.1 2
24.11 even 2 4284.2.a.m.1.2 2
56.27 even 2 3332.2.a.l.1.1 2
136.67 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 8.3 odd 2
1904.2.a.j.1.1 2 8.5 even 2
3332.2.a.l.1.1 2 56.27 even 2
4284.2.a.m.1.2 2 24.11 even 2
7616.2.a.p.1.2 2 1.1 even 1 trivial
7616.2.a.u.1.1 2 4.3 odd 2
8092.2.a.m.1.1 2 136.67 odd 2