# Properties

 Label 7600.2.a.cb.1.3 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 950) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.713538$$ of defining polynomial Character $$\chi$$ $$=$$ 7600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.77733 q^{3} -4.69527 q^{7} +4.71354 q^{9} +O(q^{10})$$ $$q+2.77733 q^{3} -4.69527 q^{7} +4.71354 q^{9} -6.40880 q^{11} -1.06379 q^{13} -1.91794 q^{17} +1.00000 q^{19} -13.0403 q^{21} +1.79560 q^{23} +4.75905 q^{27} +2.93621 q^{29} +5.55465 q^{31} -17.7993 q^{33} +11.4088 q^{37} -2.95449 q^{39} -1.14585 q^{41} -3.55465 q^{43} +10.8359 q^{47} +15.0455 q^{49} -5.32674 q^{51} +8.69527 q^{53} +2.77733 q^{57} +5.63148 q^{59} -3.39053 q^{61} -22.1313 q^{63} -8.82284 q^{67} +4.98696 q^{69} +1.42708 q^{71} +12.6132 q^{73} +30.0910 q^{77} +1.96345 q^{79} -0.923174 q^{81} +16.2447 q^{83} +8.15482 q^{87} -10.0000 q^{89} +4.99477 q^{91} +15.4271 q^{93} +14.9452 q^{97} -30.2081 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 2 q^{7} + 13 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9 $$3 q + 2 q^{3} + 2 q^{7} + 13 q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{19} - 11 q^{21} + 14 q^{23} - 7 q^{27} + 14 q^{29} + 4 q^{31} - 4 q^{33} + 17 q^{37} - 29 q^{39} - 8 q^{41} + 2 q^{43} + 13 q^{47} + 25 q^{49} + 11 q^{51} + 10 q^{53} + 2 q^{57} + 6 q^{59} + 22 q^{61} + 2 q^{63} + 8 q^{69} + 2 q^{71} + 12 q^{73} + 50 q^{77} - 24 q^{79} - q^{81} + 12 q^{83} - 21 q^{87} - 30 q^{89} + 7 q^{91} + 44 q^{93} - 24 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9 - 2 * q^11 + 2 * q^13 + 4 * q^17 + 3 * q^19 - 11 * q^21 + 14 * q^23 - 7 * q^27 + 14 * q^29 + 4 * q^31 - 4 * q^33 + 17 * q^37 - 29 * q^39 - 8 * q^41 + 2 * q^43 + 13 * q^47 + 25 * q^49 + 11 * q^51 + 10 * q^53 + 2 * q^57 + 6 * q^59 + 22 * q^61 + 2 * q^63 + 8 * q^69 + 2 * q^71 + 12 * q^73 + 50 * q^77 - 24 * q^79 - q^81 + 12 * q^83 - 21 * q^87 - 30 * q^89 + 7 * q^91 + 44 * q^93 - 24 * 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
$$3$$ 2.77733 1.60349 0.801745 0.597666i $$-0.203905\pi$$
0.801745 + 0.597666i $$0.203905\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.69527 −1.77464 −0.887322 0.461151i $$-0.847437\pi$$
−0.887322 + 0.461151i $$0.847437\pi$$
$$8$$ 0 0
$$9$$ 4.71354 1.57118
$$10$$ 0 0
$$11$$ −6.40880 −1.93233 −0.966163 0.257931i $$-0.916959\pi$$
−0.966163 + 0.257931i $$0.916959\pi$$
$$12$$ 0 0
$$13$$ −1.06379 −0.295042 −0.147521 0.989059i $$-0.547129\pi$$
−0.147521 + 0.989059i $$0.547129\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.91794 −0.465169 −0.232584 0.972576i $$-0.574718\pi$$
−0.232584 + 0.972576i $$0.574718\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −13.0403 −2.84562
$$22$$ 0 0
$$23$$ 1.79560 0.374408 0.187204 0.982321i $$-0.440057\pi$$
0.187204 + 0.982321i $$0.440057\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.75905 0.915880
$$28$$ 0 0
$$29$$ 2.93621 0.545241 0.272620 0.962122i $$-0.412110\pi$$
0.272620 + 0.962122i $$0.412110\pi$$
$$30$$ 0 0
$$31$$ 5.55465 0.997645 0.498822 0.866704i $$-0.333766\pi$$
0.498822 + 0.866704i $$0.333766\pi$$
$$32$$ 0 0
$$33$$ −17.7993 −3.09847
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 11.4088 1.87560 0.937798 0.347182i $$-0.112861\pi$$
0.937798 + 0.347182i $$0.112861\pi$$
$$38$$ 0 0
$$39$$ −2.95449 −0.473096
$$40$$ 0 0
$$41$$ −1.14585 −0.178951 −0.0894757 0.995989i $$-0.528519\pi$$
−0.0894757 + 0.995989i $$0.528519\pi$$
$$42$$ 0 0
$$43$$ −3.55465 −0.542079 −0.271040 0.962568i $$-0.587367\pi$$
−0.271040 + 0.962568i $$0.587367\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.8359 1.58058 0.790288 0.612736i $$-0.209931\pi$$
0.790288 + 0.612736i $$0.209931\pi$$
$$48$$ 0 0
$$49$$ 15.0455 2.14936
$$50$$ 0 0
$$51$$ −5.32674 −0.745893
$$52$$ 0 0
$$53$$ 8.69527 1.19439 0.597193 0.802097i $$-0.296283\pi$$
0.597193 + 0.802097i $$0.296283\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.77733 0.367866
$$58$$ 0 0
$$59$$ 5.63148 0.733156 0.366578 0.930387i $$-0.380529\pi$$
0.366578 + 0.930387i $$0.380529\pi$$
$$60$$ 0 0
$$61$$ −3.39053 −0.434113 −0.217056 0.976159i $$-0.569646\pi$$
−0.217056 + 0.976159i $$0.569646\pi$$
$$62$$ 0 0
$$63$$ −22.1313 −2.78828
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.82284 −1.07788 −0.538941 0.842344i $$-0.681175\pi$$
−0.538941 + 0.842344i $$0.681175\pi$$
$$68$$ 0 0
$$69$$ 4.98696 0.600360
$$70$$ 0 0
$$71$$ 1.42708 0.169363 0.0846814 0.996408i $$-0.473013\pi$$
0.0846814 + 0.996408i $$0.473013\pi$$
$$72$$ 0 0
$$73$$ 12.6132 1.47626 0.738132 0.674656i $$-0.235708\pi$$
0.738132 + 0.674656i $$0.235708\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 30.0910 3.42919
$$78$$ 0 0
$$79$$ 1.96345 0.220906 0.110453 0.993881i $$-0.464770\pi$$
0.110453 + 0.993881i $$0.464770\pi$$
$$80$$ 0 0
$$81$$ −0.923174 −0.102575
$$82$$ 0 0
$$83$$ 16.2447 1.78309 0.891543 0.452937i $$-0.149624\pi$$
0.891543 + 0.452937i $$0.149624\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 8.15482 0.874288
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 4.99477 0.523594
$$92$$ 0 0
$$93$$ 15.4271 1.59971
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.9452 1.51745 0.758727 0.651409i $$-0.225822\pi$$
0.758727 + 0.651409i $$0.225822\pi$$
$$98$$ 0 0
$$99$$ −30.2081 −3.03603
$$100$$ 0 0
$$101$$ 10.5364 1.04841 0.524204 0.851592i $$-0.324363\pi$$
0.524204 + 0.851592i $$0.324363\pi$$
$$102$$ 0 0
$$103$$ −16.9817 −1.67326 −0.836630 0.547769i $$-0.815477\pi$$
−0.836630 + 0.547769i $$0.815477\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1.79036 −0.173081 −0.0865405 0.996248i $$-0.527581\pi$$
−0.0865405 + 0.996248i $$0.527581\pi$$
$$108$$ 0 0
$$109$$ 2.41404 0.231223 0.115611 0.993295i $$-0.463117\pi$$
0.115611 + 0.993295i $$0.463117\pi$$
$$110$$ 0 0
$$111$$ 31.6860 3.00750
$$112$$ 0 0
$$113$$ −7.14585 −0.672225 −0.336112 0.941822i $$-0.609112\pi$$
−0.336112 + 0.941822i $$0.609112\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −5.01420 −0.463563
$$118$$ 0 0
$$119$$ 9.00523 0.825508
$$120$$ 0 0
$$121$$ 30.0728 2.73389
$$122$$ 0 0
$$123$$ −3.18239 −0.286947
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 9.26295 0.821954 0.410977 0.911646i $$-0.365188\pi$$
0.410977 + 0.911646i $$0.365188\pi$$
$$128$$ 0 0
$$129$$ −9.87242 −0.869219
$$130$$ 0 0
$$131$$ −11.5181 −1.00634 −0.503171 0.864187i $$-0.667833\pi$$
−0.503171 + 0.864187i $$0.667833\pi$$
$$132$$ 0 0
$$133$$ −4.69527 −0.407131
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −15.1041 −1.29043 −0.645214 0.764002i $$-0.723232\pi$$
−0.645214 + 0.764002i $$0.723232\pi$$
$$138$$ 0 0
$$139$$ 0.700500 0.0594156 0.0297078 0.999559i $$-0.490542\pi$$
0.0297078 + 0.999559i $$0.490542\pi$$
$$140$$ 0 0
$$141$$ 30.0948 2.53444
$$142$$ 0 0
$$143$$ 6.81761 0.570117
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 41.7863 3.44648
$$148$$ 0 0
$$149$$ 12.9452 1.06051 0.530255 0.847838i $$-0.322096\pi$$
0.530255 + 0.847838i $$0.322096\pi$$
$$150$$ 0 0
$$151$$ −5.70830 −0.464535 −0.232268 0.972652i $$-0.574615\pi$$
−0.232268 + 0.972652i $$0.574615\pi$$
$$152$$ 0 0
$$153$$ −9.04028 −0.730863
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −9.26295 −0.739264 −0.369632 0.929178i $$-0.620516\pi$$
−0.369632 + 0.929178i $$0.620516\pi$$
$$158$$ 0 0
$$159$$ 24.1496 1.91519
$$160$$ 0 0
$$161$$ −8.43081 −0.664441
$$162$$ 0 0
$$163$$ 4.28123 0.335332 0.167666 0.985844i $$-0.446377\pi$$
0.167666 + 0.985844i $$0.446377\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −15.2264 −1.17825 −0.589127 0.808040i $$-0.700528\pi$$
−0.589127 + 0.808040i $$0.700528\pi$$
$$168$$ 0 0
$$169$$ −11.8684 −0.912950
$$170$$ 0 0
$$171$$ 4.71354 0.360453
$$172$$ 0 0
$$173$$ 12.2630 0.932335 0.466168 0.884696i $$-0.345634\pi$$
0.466168 + 0.884696i $$0.345634\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 15.6404 1.17561
$$178$$ 0 0
$$179$$ 1.57292 0.117566 0.0587829 0.998271i $$-0.481278\pi$$
0.0587829 + 0.998271i $$0.481278\pi$$
$$180$$ 0 0
$$181$$ 11.4088 0.848010 0.424005 0.905660i $$-0.360624\pi$$
0.424005 + 0.905660i $$0.360624\pi$$
$$182$$ 0 0
$$183$$ −9.41661 −0.696096
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.2917 0.898858
$$188$$ 0 0
$$189$$ −22.3450 −1.62536
$$190$$ 0 0
$$191$$ 6.35805 0.460053 0.230026 0.973184i $$-0.426119\pi$$
0.230026 + 0.973184i $$0.426119\pi$$
$$192$$ 0 0
$$193$$ −14.4998 −1.04372 −0.521860 0.853031i $$-0.674762\pi$$
−0.521860 + 0.853031i $$0.674762\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.14585 0.366627 0.183313 0.983055i $$-0.441318\pi$$
0.183313 + 0.983055i $$0.441318\pi$$
$$198$$ 0 0
$$199$$ −3.87766 −0.274880 −0.137440 0.990510i $$-0.543887\pi$$
−0.137440 + 0.990510i $$0.543887\pi$$
$$200$$ 0 0
$$201$$ −24.5039 −1.72837
$$202$$ 0 0
$$203$$ −13.7863 −0.967608
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.46362 0.588262
$$208$$ 0 0
$$209$$ −6.40880 −0.443306
$$210$$ 0 0
$$211$$ −17.1496 −1.18063 −0.590313 0.807174i $$-0.700996\pi$$
−0.590313 + 0.807174i $$0.700996\pi$$
$$212$$ 0 0
$$213$$ 3.96345 0.271571
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −26.0806 −1.77046
$$218$$ 0 0
$$219$$ 35.0310 2.36717
$$220$$ 0 0
$$221$$ 2.04028 0.137244
$$222$$ 0 0
$$223$$ 7.84635 0.525430 0.262715 0.964873i $$-0.415382\pi$$
0.262715 + 0.964873i $$0.415382\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 6.28646 0.417247 0.208624 0.977996i $$-0.433102\pi$$
0.208624 + 0.977996i $$0.433102\pi$$
$$228$$ 0 0
$$229$$ −3.14585 −0.207884 −0.103942 0.994583i $$-0.533146\pi$$
−0.103942 + 0.994583i $$0.533146\pi$$
$$230$$ 0 0
$$231$$ 83.5726 5.49867
$$232$$ 0 0
$$233$$ −0.182394 −0.0119490 −0.00597451 0.999982i $$-0.501902\pi$$
−0.00597451 + 0.999982i $$0.501902\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 5.45315 0.354220
$$238$$ 0 0
$$239$$ 11.5039 0.744126 0.372063 0.928208i $$-0.378651\pi$$
0.372063 + 0.928208i $$0.378651\pi$$
$$240$$ 0 0
$$241$$ −0.445349 −0.0286874 −0.0143437 0.999897i $$-0.504566\pi$$
−0.0143437 + 0.999897i $$0.504566\pi$$
$$242$$ 0 0
$$243$$ −16.8411 −1.08036
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.06379 −0.0676872
$$248$$ 0 0
$$249$$ 45.1168 2.85916
$$250$$ 0 0
$$251$$ −13.2630 −0.837150 −0.418575 0.908182i $$-0.637470\pi$$
−0.418575 + 0.908182i $$0.637470\pi$$
$$252$$ 0 0
$$253$$ −11.5076 −0.723479
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 23.4816 1.46474 0.732370 0.680907i $$-0.238414\pi$$
0.732370 + 0.680907i $$0.238414\pi$$
$$258$$ 0 0
$$259$$ −53.5674 −3.32851
$$260$$ 0 0
$$261$$ 13.8399 0.856671
$$262$$ 0 0
$$263$$ 27.9269 1.72205 0.861023 0.508565i $$-0.169824\pi$$
0.861023 + 0.508565i $$0.169824\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −27.7733 −1.69970
$$268$$ 0 0
$$269$$ 16.4816 1.00490 0.502449 0.864607i $$-0.332432\pi$$
0.502449 + 0.864607i $$0.332432\pi$$
$$270$$ 0 0
$$271$$ 1.46736 0.0891356 0.0445678 0.999006i $$-0.485809\pi$$
0.0445678 + 0.999006i $$0.485809\pi$$
$$272$$ 0 0
$$273$$ 13.8721 0.839577
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −15.8359 −0.951486 −0.475743 0.879584i $$-0.657821\pi$$
−0.475743 + 0.879584i $$0.657821\pi$$
$$278$$ 0 0
$$279$$ 26.1821 1.56748
$$280$$ 0 0
$$281$$ −6.25515 −0.373151 −0.186576 0.982441i $$-0.559739\pi$$
−0.186576 + 0.982441i $$0.559739\pi$$
$$282$$ 0 0
$$283$$ −16.7005 −0.992742 −0.496371 0.868111i $$-0.665334\pi$$
−0.496371 + 0.868111i $$0.665334\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 5.38006 0.317575
$$288$$ 0 0
$$289$$ −13.3215 −0.783618
$$290$$ 0 0
$$291$$ 41.5076 2.43322
$$292$$ 0 0
$$293$$ −0.644516 −0.0376530 −0.0188265 0.999823i $$-0.505993\pi$$
−0.0188265 + 0.999823i $$0.505993\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −30.4998 −1.76978
$$298$$ 0 0
$$299$$ −1.91014 −0.110466
$$300$$ 0 0
$$301$$ 16.6900 0.961997
$$302$$ 0 0
$$303$$ 29.2630 1.68111
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 26.6457 1.52075 0.760375 0.649485i $$-0.225015\pi$$
0.760375 + 0.649485i $$0.225015\pi$$
$$308$$ 0 0
$$309$$ −47.1638 −2.68305
$$310$$ 0 0
$$311$$ −11.5494 −0.654907 −0.327454 0.944867i $$-0.606191\pi$$
−0.327454 + 0.944867i $$0.606191\pi$$
$$312$$ 0 0
$$313$$ 23.0638 1.30364 0.651821 0.758373i $$-0.274005\pi$$
0.651821 + 0.758373i $$0.274005\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 13.9232 0.782003 0.391002 0.920390i $$-0.372129\pi$$
0.391002 + 0.920390i $$0.372129\pi$$
$$318$$ 0 0
$$319$$ −18.8176 −1.05358
$$320$$ 0 0
$$321$$ −4.97242 −0.277534
$$322$$ 0 0
$$323$$ −1.91794 −0.106717
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 6.70457 0.370763
$$328$$ 0 0
$$329$$ −50.8773 −2.80496
$$330$$ 0 0
$$331$$ 0.735546 0.0404292 0.0202146 0.999796i $$-0.493565\pi$$
0.0202146 + 0.999796i $$0.493565\pi$$
$$332$$ 0 0
$$333$$ 53.7758 2.94690
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.28123 0.124266 0.0621332 0.998068i $$-0.480210\pi$$
0.0621332 + 0.998068i $$0.480210\pi$$
$$338$$ 0 0
$$339$$ −19.8463 −1.07791
$$340$$ 0 0
$$341$$ −35.5987 −1.92778
$$342$$ 0 0
$$343$$ −37.7758 −2.03970
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2.56246 −0.137560 −0.0687799 0.997632i $$-0.521911\pi$$
−0.0687799 + 0.997632i $$0.521911\pi$$
$$348$$ 0 0
$$349$$ −5.67176 −0.303602 −0.151801 0.988411i $$-0.548507\pi$$
−0.151801 + 0.988411i $$0.548507\pi$$
$$350$$ 0 0
$$351$$ −5.06262 −0.270223
$$352$$ 0 0
$$353$$ 23.6665 1.25964 0.629821 0.776740i $$-0.283128\pi$$
0.629821 + 0.776740i $$0.283128\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 25.0105 1.32369
$$358$$ 0 0
$$359$$ −31.9724 −1.68744 −0.843720 0.536784i $$-0.819639\pi$$
−0.843720 + 0.536784i $$0.819639\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 83.5218 4.38376
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.14585 0.0598128 0.0299064 0.999553i $$-0.490479\pi$$
0.0299064 + 0.999553i $$0.490479\pi$$
$$368$$ 0 0
$$369$$ −5.40100 −0.281165
$$370$$ 0 0
$$371$$ −40.8266 −2.11961
$$372$$ 0 0
$$373$$ −32.8579 −1.70132 −0.850658 0.525719i $$-0.823796\pi$$
−0.850658 + 0.525719i $$0.823796\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3.12351 −0.160869
$$378$$ 0 0
$$379$$ 18.2865 0.939312 0.469656 0.882849i $$-0.344378\pi$$
0.469656 + 0.882849i $$0.344378\pi$$
$$380$$ 0 0
$$381$$ 25.7262 1.31800
$$382$$ 0 0
$$383$$ 20.8542 1.06560 0.532799 0.846242i $$-0.321140\pi$$
0.532799 + 0.846242i $$0.321140\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −16.7550 −0.851704
$$388$$ 0 0
$$389$$ −24.9086 −1.26292 −0.631459 0.775409i $$-0.717544\pi$$
−0.631459 + 0.775409i $$0.717544\pi$$
$$390$$ 0 0
$$391$$ −3.44385 −0.174163
$$392$$ 0 0
$$393$$ −31.9895 −1.61366
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 24.7445 1.24189 0.620946 0.783853i $$-0.286749\pi$$
0.620946 + 0.783853i $$0.286749\pi$$
$$398$$ 0 0
$$399$$ −13.0403 −0.652831
$$400$$ 0 0
$$401$$ 15.6457 0.781308 0.390654 0.920538i $$-0.372249\pi$$
0.390654 + 0.920538i $$0.372249\pi$$
$$402$$ 0 0
$$403$$ −5.90897 −0.294347
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −73.1168 −3.62426
$$408$$ 0 0
$$409$$ 30.5804 1.51210 0.756052 0.654512i $$-0.227126\pi$$
0.756052 + 0.654512i $$0.227126\pi$$
$$410$$ 0 0
$$411$$ −41.9489 −2.06919
$$412$$ 0 0
$$413$$ −26.4413 −1.30109
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 1.94552 0.0952723
$$418$$ 0 0
$$419$$ −1.96345 −0.0959210 −0.0479605 0.998849i $$-0.515272\pi$$
−0.0479605 + 0.998849i $$0.515272\pi$$
$$420$$ 0 0
$$421$$ −25.5949 −1.24742 −0.623710 0.781656i $$-0.714376\pi$$
−0.623710 + 0.781656i $$0.714376\pi$$
$$422$$ 0 0
$$423$$ 51.0753 2.48337
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 15.9194 0.770396
$$428$$ 0 0
$$429$$ 18.9347 0.914177
$$430$$ 0 0
$$431$$ 15.3540 0.739575 0.369788 0.929116i $$-0.379430\pi$$
0.369788 + 0.929116i $$0.379430\pi$$
$$432$$ 0 0
$$433$$ −16.6640 −0.800819 −0.400409 0.916336i $$-0.631132\pi$$
−0.400409 + 0.916336i $$0.631132\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.79560 0.0858951
$$438$$ 0 0
$$439$$ 19.5987 0.935393 0.467697 0.883889i $$-0.345084\pi$$
0.467697 + 0.883889i $$0.345084\pi$$
$$440$$ 0 0
$$441$$ 70.9176 3.37703
$$442$$ 0 0
$$443$$ 13.0183 0.618517 0.309258 0.950978i $$-0.399919\pi$$
0.309258 + 0.950978i $$0.399919\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 35.9530 1.70052
$$448$$ 0 0
$$449$$ 6.77359 0.319666 0.159833 0.987144i $$-0.448905\pi$$
0.159833 + 0.987144i $$0.448905\pi$$
$$450$$ 0 0
$$451$$ 7.34352 0.345793
$$452$$ 0 0
$$453$$ −15.8538 −0.744877
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 27.3189 1.27793 0.638963 0.769237i $$-0.279364\pi$$
0.638963 + 0.769237i $$0.279364\pi$$
$$458$$ 0 0
$$459$$ −9.12758 −0.426039
$$460$$ 0 0
$$461$$ −27.7993 −1.29474 −0.647372 0.762174i $$-0.724132\pi$$
−0.647372 + 0.762174i $$0.724132\pi$$
$$462$$ 0 0
$$463$$ 38.2369 1.77702 0.888509 0.458859i $$-0.151742\pi$$
0.888509 + 0.458859i $$0.151742\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 23.4711 1.08611 0.543056 0.839696i $$-0.317267\pi$$
0.543056 + 0.839696i $$0.317267\pi$$
$$468$$ 0 0
$$469$$ 41.4256 1.91286
$$470$$ 0 0
$$471$$ −25.7262 −1.18540
$$472$$ 0 0
$$473$$ 22.7811 1.04747
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 40.9855 1.87660
$$478$$ 0 0
$$479$$ 19.6900 0.899660 0.449830 0.893114i $$-0.351484\pi$$
0.449830 + 0.893114i $$0.351484\pi$$
$$480$$ 0 0
$$481$$ −12.1365 −0.553379
$$482$$ 0 0
$$483$$ −23.4151 −1.06542
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.3357 0.740242 0.370121 0.928984i $$-0.379316\pi$$
0.370121 + 0.928984i $$0.379316\pi$$
$$488$$ 0 0
$$489$$ 11.8904 0.537701
$$490$$ 0 0
$$491$$ −27.1093 −1.22343 −0.611713 0.791080i $$-0.709519\pi$$
−0.611713 + 0.791080i $$0.709519\pi$$
$$492$$ 0 0
$$493$$ −5.63148 −0.253629
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −6.70050 −0.300558
$$498$$ 0 0
$$499$$ 4.69003 0.209955 0.104977 0.994475i $$-0.466523\pi$$
0.104977 + 0.994475i $$0.466523\pi$$
$$500$$ 0 0
$$501$$ −42.2887 −1.88932
$$502$$ 0 0
$$503$$ 5.19136 0.231471 0.115736 0.993280i $$-0.463077\pi$$
0.115736 + 0.993280i $$0.463077\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −32.9623 −1.46391
$$508$$ 0 0
$$509$$ −4.51811 −0.200262 −0.100131 0.994974i $$-0.531926\pi$$
−0.100131 + 0.994974i $$0.531926\pi$$
$$510$$ 0 0
$$511$$ −59.2223 −2.61984
$$512$$ 0 0
$$513$$ 4.75905 0.210117
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −69.4450 −3.05419
$$518$$ 0 0
$$519$$ 34.0582 1.49499
$$520$$ 0 0
$$521$$ 14.4453 0.632862 0.316431 0.948615i $$-0.397515\pi$$
0.316431 + 0.948615i $$0.397515\pi$$
$$522$$ 0 0
$$523$$ −18.2134 −0.796415 −0.398208 0.917295i $$-0.630368\pi$$
−0.398208 + 0.917295i $$0.630368\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −10.6535 −0.464073
$$528$$ 0 0
$$529$$ −19.7758 −0.859819
$$530$$ 0 0
$$531$$ 26.5442 1.15192
$$532$$ 0 0
$$533$$ 1.21894 0.0527981
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4.36852 0.188516
$$538$$ 0 0
$$539$$ −96.4237 −4.15326
$$540$$ 0 0
$$541$$ 37.3905 1.60754 0.803772 0.594937i $$-0.202823\pi$$
0.803772 + 0.594937i $$0.202823\pi$$
$$542$$ 0 0
$$543$$ 31.6860 1.35977
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 29.5621 1.26399 0.631993 0.774974i $$-0.282237\pi$$
0.631993 + 0.774974i $$0.282237\pi$$
$$548$$ 0 0
$$549$$ −15.9814 −0.682069
$$550$$ 0 0
$$551$$ 2.93621 0.125087
$$552$$ 0 0
$$553$$ −9.21894 −0.392029
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −14.3723 −0.608972 −0.304486 0.952517i $$-0.598485\pi$$
−0.304486 + 0.952517i $$0.598485\pi$$
$$558$$ 0 0
$$559$$ 3.78139 0.159936
$$560$$ 0 0
$$561$$ 34.1380 1.44131
$$562$$ 0 0
$$563$$ −6.39053 −0.269329 −0.134664 0.990891i $$-0.542996\pi$$
−0.134664 + 0.990891i $$0.542996\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 4.33455 0.182034
$$568$$ 0 0
$$569$$ −44.9086 −1.88267 −0.941334 0.337476i $$-0.890427\pi$$
−0.941334 + 0.337476i $$0.890427\pi$$
$$570$$ 0 0
$$571$$ −12.4193 −0.519730 −0.259865 0.965645i $$-0.583678\pi$$
−0.259865 + 0.965645i $$0.583678\pi$$
$$572$$ 0 0
$$573$$ 17.6584 0.737690
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 4.20440 0.175032 0.0875158 0.996163i $$-0.472107\pi$$
0.0875158 + 0.996163i $$0.472107\pi$$
$$578$$ 0 0
$$579$$ −40.2708 −1.67360
$$580$$ 0 0
$$581$$ −76.2731 −3.16434
$$582$$ 0 0
$$583$$ −55.7262 −2.30795
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.7915 1.02326 0.511628 0.859207i $$-0.329043\pi$$
0.511628 + 0.859207i $$0.329043\pi$$
$$588$$ 0 0
$$589$$ 5.55465 0.228875
$$590$$ 0 0
$$591$$ 14.2917 0.587882
$$592$$ 0 0
$$593$$ −4.67176 −0.191846 −0.0959231 0.995389i $$-0.530580\pi$$
−0.0959231 + 0.995389i $$0.530580\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −10.7695 −0.440767
$$598$$ 0 0
$$599$$ −39.2369 −1.60318 −0.801588 0.597877i $$-0.796011\pi$$
−0.801588 + 0.597877i $$0.796011\pi$$
$$600$$ 0 0
$$601$$ −42.4267 −1.73062 −0.865311 0.501235i $$-0.832879\pi$$
−0.865311 + 0.501235i $$0.832879\pi$$
$$602$$ 0 0
$$603$$ −41.5868 −1.69355
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −2.98173 −0.121025 −0.0605123 0.998167i $$-0.519273\pi$$
−0.0605123 + 0.998167i $$0.519273\pi$$
$$608$$ 0 0
$$609$$ −38.2890 −1.55155
$$610$$ 0 0
$$611$$ −11.5271 −0.466336
$$612$$ 0 0
$$613$$ 28.9452 1.16908 0.584542 0.811363i $$-0.301274\pi$$
0.584542 + 0.811363i $$0.301274\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 15.0365 0.605349 0.302674 0.953094i $$-0.402121\pi$$
0.302674 + 0.953094i $$0.402121\pi$$
$$618$$ 0 0
$$619$$ −8.25515 −0.331803 −0.165901 0.986142i $$-0.553053\pi$$
−0.165901 + 0.986142i $$0.553053\pi$$
$$620$$ 0 0
$$621$$ 8.54535 0.342913
$$622$$ 0 0
$$623$$ 46.9527 1.88112
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −17.7993 −0.710837
$$628$$ 0 0
$$629$$ −21.8814 −0.872468
$$630$$ 0 0
$$631$$ 9.09103 0.361908 0.180954 0.983492i $$-0.442081\pi$$
0.180954 + 0.983492i $$0.442081\pi$$
$$632$$ 0 0
$$633$$ −47.6300 −1.89312
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −16.0052 −0.634150
$$638$$ 0 0
$$639$$ 6.72658 0.266099
$$640$$ 0 0
$$641$$ 30.1171 1.18955 0.594777 0.803891i $$-0.297240\pi$$
0.594777 + 0.803891i $$0.297240\pi$$
$$642$$ 0 0
$$643$$ −35.3174 −1.39278 −0.696392 0.717662i $$-0.745212\pi$$
−0.696392 + 0.717662i $$0.745212\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 4.12234 0.162066 0.0810330 0.996711i $$-0.474178\pi$$
0.0810330 + 0.996711i $$0.474178\pi$$
$$648$$ 0 0
$$649$$ −36.0910 −1.41670
$$650$$ 0 0
$$651$$ −72.4342 −2.83892
$$652$$ 0 0
$$653$$ −8.62741 −0.337617 −0.168808 0.985649i $$-0.553992\pi$$
−0.168808 + 0.985649i $$0.553992\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 59.4528 2.31948
$$658$$ 0 0
$$659$$ 37.3853 1.45632 0.728162 0.685405i $$-0.240375\pi$$
0.728162 + 0.685405i $$0.240375\pi$$
$$660$$ 0 0
$$661$$ −12.8997 −0.501739 −0.250869 0.968021i $$-0.580716\pi$$
−0.250869 + 0.968021i $$0.580716\pi$$
$$662$$ 0 0
$$663$$ 5.66652 0.220070
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.27226 0.204143
$$668$$ 0 0
$$669$$ 21.7919 0.842522
$$670$$ 0 0
$$671$$ 21.7292 0.838848
$$672$$ 0 0
$$673$$ −10.4349 −0.402235 −0.201118 0.979567i $$-0.564457\pi$$
−0.201118 + 0.979567i $$0.564457\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −12.4401 −0.478112 −0.239056 0.971006i $$-0.576838\pi$$
−0.239056 + 0.971006i $$0.576838\pi$$
$$678$$ 0 0
$$679$$ −70.1716 −2.69294
$$680$$ 0 0
$$681$$ 17.4596 0.669052
$$682$$ 0 0
$$683$$ 17.5364 0.671011 0.335505 0.942038i $$-0.391093\pi$$
0.335505 + 0.942038i $$0.391093\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −8.73705 −0.333339
$$688$$ 0 0
$$689$$ −9.24992 −0.352394
$$690$$ 0 0
$$691$$ 25.2734 0.961446 0.480723 0.876872i $$-0.340374\pi$$
0.480723 + 0.876872i $$0.340374\pi$$
$$692$$ 0 0
$$693$$ 141.835 5.38787
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.19767 0.0832426
$$698$$ 0 0
$$699$$ −0.506567 −0.0191601
$$700$$ 0 0
$$701$$ −16.0545 −0.606370 −0.303185 0.952932i $$-0.598050\pi$$
−0.303185 + 0.952932i $$0.598050\pi$$
$$702$$ 0 0
$$703$$ 11.4088 0.430291
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −49.4711 −1.86055
$$708$$ 0 0
$$709$$ 13.5076 0.507290 0.253645 0.967297i $$-0.418371\pi$$
0.253645 + 0.967297i $$0.418371\pi$$
$$710$$ 0 0
$$711$$ 9.25482 0.347083
$$712$$ 0 0
$$713$$ 9.97392 0.373526
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 31.9501 1.19320
$$718$$ 0 0
$$719$$ −0.803402 −0.0299619 −0.0149809 0.999888i $$-0.504769\pi$$
−0.0149809 + 0.999888i $$0.504769\pi$$
$$720$$ 0 0
$$721$$ 79.7337 2.96944
$$722$$ 0 0
$$723$$ −1.23688 −0.0460000
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 51.6860 1.91693 0.958463 0.285217i $$-0.0920655\pi$$
0.958463 + 0.285217i $$0.0920655\pi$$
$$728$$ 0 0
$$729$$ −44.0037 −1.62977
$$730$$ 0 0
$$731$$ 6.81761 0.252158
$$732$$ 0 0
$$733$$ 22.3723 0.826338 0.413169 0.910654i $$-0.364422\pi$$
0.413169 + 0.910654i $$0.364422\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 56.5438 2.08282
$$738$$ 0 0
$$739$$ −33.4271 −1.22963 −0.614817 0.788669i $$-0.710770\pi$$
−0.614817 + 0.788669i $$0.710770\pi$$
$$740$$ 0 0
$$741$$ −2.95449 −0.108536
$$742$$ 0 0
$$743$$ 14.4998 0.531947 0.265974 0.963980i $$-0.414307\pi$$
0.265974 + 0.963980i $$0.414307\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 76.5699 2.80155
$$748$$ 0 0
$$749$$ 8.40623 0.307157
$$750$$ 0 0
$$751$$ 26.6169 0.971266 0.485633 0.874163i $$-0.338589\pi$$
0.485633 + 0.874163i $$0.338589\pi$$
$$752$$ 0 0
$$753$$ −36.8355 −1.34236
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 31.0362 1.12803 0.564015 0.825764i $$-0.309256\pi$$
0.564015 + 0.825764i $$0.309256\pi$$
$$758$$ 0 0
$$759$$ −31.9605 −1.16009
$$760$$ 0 0
$$761$$ −27.8083 −1.00805 −0.504025 0.863689i $$-0.668148\pi$$
−0.504025 + 0.863689i $$0.668148\pi$$
$$762$$ 0 0
$$763$$ −11.3345 −0.410338
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.99070 −0.216312
$$768$$ 0 0
$$769$$ −19.2279 −0.693376 −0.346688 0.937980i $$-0.612694\pi$$
−0.346688 + 0.937980i $$0.612694\pi$$
$$770$$ 0 0
$$771$$ 65.2159 2.34869
$$772$$ 0 0
$$773$$ −6.00373 −0.215939 −0.107970 0.994154i $$-0.534435\pi$$
−0.107970 + 0.994154i $$0.534435\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −148.774 −5.33724
$$778$$ 0 0
$$779$$ −1.14585 −0.0410543
$$780$$ 0 0
$$781$$ −9.14585 −0.327264
$$782$$ 0 0
$$783$$ 13.9736 0.499375
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22.2797 0.794187 0.397093 0.917778i $$-0.370019\pi$$
0.397093 + 0.917778i $$0.370019\pi$$
$$788$$ 0 0
$$789$$ 77.5621 2.76128
$$790$$ 0 0
$$791$$ 33.5517 1.19296
$$792$$ 0 0
$$793$$ 3.60680 0.128081
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −26.6259 −0.943138 −0.471569 0.881829i $$-0.656312\pi$$
−0.471569 + 0.881829i $$0.656312\pi$$
$$798$$ 0 0
$$799$$ −20.7826 −0.735234
$$800$$ 0 0
$$801$$ −47.1354 −1.66545
$$802$$ 0 0
$$803$$ −80.8355 −2.85262
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 45.7747 1.61134
$$808$$ 0 0
$$809$$ 0.651250 0.0228967 0.0114484 0.999934i $$-0.496356\pi$$
0.0114484 + 0.999934i $$0.496356\pi$$
$$810$$ 0 0
$$811$$ 13.0780 0.459230 0.229615 0.973281i $$-0.426253\pi$$
0.229615 + 0.973281i $$0.426253\pi$$
$$812$$ 0 0
$$813$$ 4.07533 0.142928
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −3.55465 −0.124362
$$818$$ 0 0
$$819$$ 23.5430 0.822660
$$820$$ 0 0
$$821$$ 14.6640 0.511776 0.255888 0.966706i $$-0.417632\pi$$
0.255888 + 0.966706i $$0.417632\pi$$
$$822$$ 0 0
$$823$$ −18.2850 −0.637374 −0.318687 0.947860i $$-0.603242\pi$$
−0.318687 + 0.947860i $$0.603242\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 40.1910 1.39758 0.698790 0.715327i $$-0.253722\pi$$
0.698790 + 0.715327i $$0.253722\pi$$
$$828$$ 0 0
$$829$$ 28.3760 0.985539 0.492769 0.870160i $$-0.335985\pi$$
0.492769 + 0.870160i $$0.335985\pi$$
$$830$$ 0 0
$$831$$ −43.9814 −1.52570
$$832$$ 0 0
$$833$$ −28.8564 −0.999815
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 26.4349 0.913723
$$838$$ 0 0
$$839$$ −6.93471 −0.239413 −0.119706 0.992809i $$-0.538195\pi$$
−0.119706 + 0.992809i $$0.538195\pi$$
$$840$$ 0 0
$$841$$ −20.3787 −0.702712
$$842$$ 0 0
$$843$$ −17.3726 −0.598344
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −141.200 −4.85167
$$848$$ 0 0
$$849$$ −46.3827 −1.59185
$$850$$ 0 0
$$851$$ 20.4856 0.702238
$$852$$ 0 0
$$853$$ −21.9739 −0.752373 −0.376186 0.926544i $$-0.622765\pi$$
−0.376186 + 0.926544i $$0.622765\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −13.6091 −0.464879 −0.232440 0.972611i $$-0.574671\pi$$
−0.232440 + 0.972611i $$0.574671\pi$$
$$858$$ 0 0
$$859$$ −5.17192 −0.176464 −0.0882319 0.996100i $$-0.528122\pi$$
−0.0882319 + 0.996100i $$0.528122\pi$$
$$860$$ 0 0
$$861$$ 14.9422 0.509228
$$862$$ 0 0
$$863$$ 15.9635 0.543402 0.271701 0.962382i $$-0.412414\pi$$
0.271701 + 0.962382i $$0.412414\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −36.9982 −1.25652
$$868$$ 0 0
$$869$$ −12.5834 −0.426862
$$870$$ 0 0
$$871$$ 9.38563 0.318020
$$872$$ 0 0
$$873$$ 70.4447 2.38419
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 15.8866 0.536453 0.268227 0.963356i $$-0.413562\pi$$
0.268227 + 0.963356i $$0.413562\pi$$
$$878$$ 0 0
$$879$$ −1.79003 −0.0603762
$$880$$ 0 0
$$881$$ −16.1458 −0.543967 −0.271984 0.962302i $$-0.587680\pi$$
−0.271984 + 0.962302i $$0.587680\pi$$
$$882$$ 0 0
$$883$$ 38.2887 1.28852 0.644259 0.764808i $$-0.277166\pi$$
0.644259 + 0.764808i $$0.277166\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 19.2809 0.647389 0.323695 0.946162i $$-0.395075\pi$$
0.323695 + 0.946162i $$0.395075\pi$$
$$888$$ 0 0
$$889$$ −43.4920 −1.45868
$$890$$ 0 0
$$891$$ 5.91644 0.198208
$$892$$ 0 0
$$893$$ 10.8359 0.362609
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −5.30507 −0.177131
$$898$$ 0 0
$$899$$ 16.3096 0.543957
$$900$$ 0 0
$$901$$ −16.6770 −0.555591
$$902$$ 0 0
$$903$$ 46.3537 1.54255
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 43.0205 1.42847 0.714236 0.699905i $$-0.246774\pi$$
0.714236 + 0.699905i $$0.246774\pi$$
$$908$$ 0 0
$$909$$ 49.6636 1.64724
$$910$$ 0 0
$$911$$ 25.7733 0.853906 0.426953 0.904274i $$-0.359587\pi$$
0.426953 + 0.904274i $$0.359587\pi$$
$$912$$ 0 0
$$913$$ −104.109 −3.44550
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 54.0806 1.78590
$$918$$ 0 0
$$919$$ 29.5897 0.976074 0.488037 0.872823i $$-0.337713\pi$$
0.488037 + 0.872823i $$0.337713\pi$$
$$920$$ 0 0
$$921$$ 74.0037 2.43851
$$922$$ 0 0
$$923$$ −1.51811 −0.0499691
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −80.0440 −2.62899
$$928$$ 0 0
$$929$$ −1.42334 −0.0466983 −0.0233491 0.999727i $$-0.507433\pi$$
−0.0233491 + 0.999727i $$0.507433\pi$$
$$930$$ 0 0
$$931$$ 15.0455 0.493097
$$932$$ 0 0
$$933$$ −32.0765 −1.05014
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −28.2276 −0.922155 −0.461077 0.887360i $$-0.652537\pi$$
−0.461077 + 0.887360i $$0.652537\pi$$
$$938$$ 0 0
$$939$$ 64.0556 2.09038
$$940$$ 0 0
$$941$$ −15.8672 −0.517256 −0.258628 0.965977i $$-0.583270\pi$$
−0.258628 + 0.965977i $$0.583270\pi$$
$$942$$ 0 0
$$943$$ −2.05748 −0.0670009
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 43.6718 1.41914 0.709571 0.704634i $$-0.248889\pi$$
0.709571 + 0.704634i $$0.248889\pi$$
$$948$$ 0 0
$$949$$ −13.4178 −0.435559
$$950$$ 0 0
$$951$$ 38.6692 1.25393
$$952$$ 0 0
$$953$$ −16.0261 −0.519136 −0.259568 0.965725i $$-0.583580\pi$$
−0.259568 + 0.965725i $$0.583580\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −52.2626 −1.68941
$$958$$ 0 0
$$959$$ 70.9176 2.29005
$$960$$ 0 0
$$961$$ −0.145848 −0.00470478
$$962$$ 0 0
$$963$$ −8.43895 −0.271941
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −11.5987 −0.372988 −0.186494 0.982456i $$-0.559712\pi$$
−0.186494 + 0.982456i $$0.559712\pi$$
$$968$$ 0 0
$$969$$ −5.32674 −0.171120
$$970$$ 0 0
$$971$$ 27.0362 0.867633 0.433817 0.901001i $$-0.357167\pi$$
0.433817 + 0.901001i $$0.357167\pi$$
$$972$$ 0 0
$$973$$ −3.28903 −0.105442
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −14.1537 −0.452815 −0.226408 0.974033i $$-0.572698\pi$$
−0.226408 + 0.974033i $$0.572698\pi$$
$$978$$ 0 0
$$979$$ 64.0880 2.04826
$$980$$ 0 0
$$981$$ 11.3787 0.363293
$$982$$ 0 0
$$983$$ −32.8542 −1.04788 −0.523942 0.851754i $$-0.675539\pi$$
−0.523942 + 0.851754i $$0.675539\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −141.303 −4.49772
$$988$$ 0 0
$$989$$ −6.38273 −0.202959
$$990$$ 0 0
$$991$$ −47.9709 −1.52385 −0.761923 0.647667i $$-0.775745\pi$$
−0.761923 + 0.647667i $$0.775745\pi$$
$$992$$ 0 0
$$993$$ 2.04285 0.0648279
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −44.1716 −1.39893 −0.699464 0.714668i $$-0.746578\pi$$
−0.699464 + 0.714668i $$0.746578\pi$$
$$998$$ 0 0
$$999$$ 54.2951 1.71782
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 7600.2.a.cb.1.3 3
4.3 odd 2 950.2.a.k.1.1 3
5.4 even 2 7600.2.a.bm.1.1 3
12.11 even 2 8550.2.a.co.1.3 3
20.3 even 4 950.2.b.g.799.4 6
20.7 even 4 950.2.b.g.799.3 6
20.19 odd 2 950.2.a.m.1.3 yes 3
60.59 even 2 8550.2.a.cj.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.1 3 4.3 odd 2
950.2.a.m.1.3 yes 3 20.19 odd 2
950.2.b.g.799.3 6 20.7 even 4
950.2.b.g.799.4 6 20.3 even 4
7600.2.a.bm.1.1 3 5.4 even 2
7600.2.a.cb.1.3 3 1.1 even 1 trivial
8550.2.a.cj.1.1 3 60.59 even 2
8550.2.a.co.1.3 3 12.11 even 2