# Properties

 Label 24.22.a.d.1.2 Level $24$ Weight $22$ Character 24.1 Self dual yes Analytic conductor $67.075$ 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] = [24,22,Mod(1,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("24.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 24.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: $$67.0745626289$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 12529199x - 17012391021$$ x^3 - x^2 - 12529199*x - 17012391021 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{21}\cdot 3^{4}\cdot 7$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$4086.16$$ of defining polynomial Character $$\chi$$ $$=$$ 24.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+59049.0 q^{3} +1.51338e7 q^{5} -8.40758e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$$ $$q+59049.0 q^{3} +1.51338e7 q^{5} -8.40758e8 q^{7} +3.48678e9 q^{9} -3.34735e10 q^{11} -3.01154e11 q^{13} +8.93637e11 q^{15} +5.14303e12 q^{17} +4.56647e13 q^{19} -4.96459e13 q^{21} +2.41579e13 q^{23} -2.47804e14 q^{25} +2.05891e14 q^{27} +7.82910e14 q^{29} +8.00815e15 q^{31} -1.97657e15 q^{33} -1.27239e16 q^{35} +2.06600e16 q^{37} -1.77829e16 q^{39} -7.03094e16 q^{41} +6.95453e15 q^{43} +5.27684e16 q^{45} -2.12675e17 q^{47} +1.48329e17 q^{49} +3.03691e17 q^{51} +8.69454e17 q^{53} -5.06581e17 q^{55} +2.69646e18 q^{57} +9.80538e17 q^{59} +5.62594e18 q^{61} -2.93154e18 q^{63} -4.55762e18 q^{65} +2.36685e18 q^{67} +1.42650e18 q^{69} +4.99695e19 q^{71} +2.63228e19 q^{73} -1.46326e19 q^{75} +2.81431e19 q^{77} +1.05996e20 q^{79} +1.21577e19 q^{81} +8.96505e19 q^{83} +7.78337e19 q^{85} +4.62301e19 q^{87} +4.71064e20 q^{89} +2.53198e20 q^{91} +4.72873e20 q^{93} +6.91082e20 q^{95} -3.93900e20 q^{97} -1.16715e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 177147 q^{3} + 5280498 q^{5} + 852542376 q^{7} + 10460353203 q^{9}+O(q^{10})$$ 3 * q + 177147 * q^3 + 5280498 * q^5 + 852542376 * q^7 + 10460353203 * q^9 $$3 q + 177147 q^{3} + 5280498 q^{5} + 852542376 q^{7} + 10460353203 q^{9} + 62490757668 q^{11} + 203765207802 q^{13} + 311808126402 q^{15} + 695827819926 q^{17} + 4955504123196 q^{19} + 50341774760424 q^{21} + 150867407938152 q^{23} + 678194854969869 q^{25} + 617673396283947 q^{27} + 32\!\cdots\!46 q^{29}+ \cdots + 21\!\cdots\!68 q^{99}+O(q^{100})$$ 3 * q + 177147 * q^3 + 5280498 * q^5 + 852542376 * q^7 + 10460353203 * q^9 + 62490757668 * q^11 + 203765207802 * q^13 + 311808126402 * q^15 + 695827819926 * q^17 + 4955504123196 * q^19 + 50341774760424 * q^21 + 150867407938152 * q^23 + 678194854969869 * q^25 + 617673396283947 * q^27 + 3242224905226746 * q^29 + 652508601550896 * q^31 + 3690016749537732 * q^33 + 15443719285839600 * q^35 - 20693271836916222 * q^37 + 12032131755500298 * q^39 - 47608600312552002 * q^41 - 277110211230415548 * q^43 + 18411958055911698 * q^45 - 535942507326221328 * q^47 + 1192822454236615755 * q^49 + 41087936938810374 * q^51 + 2849297892348987426 * q^53 + 2255425539904757592 * q^55 + 292617562970600604 * q^57 + 881046377434726548 * q^59 + 10620890455615553322 * q^61 + 2972631457828276776 * q^63 + 24923333196516863004 * q^65 + 14840080728867715116 * q^67 + 8908569571339937448 * q^69 + 14608944838311941496 * q^71 + 13575086263949291454 * q^73 + 40046727991115794581 * q^75 + 173887987291330337760 * q^77 + 278096577517755548160 * q^79 + 36472996377170786403 * q^81 + 585716918283456830460 * q^83 + 442542794823708598596 * q^85 + 191450138428734124554 * q^87 + 477818701792662526830 * q^89 + 1318091663263676539056 * q^91 + 38529980412978857904 * q^93 + 698400977547585407208 * q^95 + 722411740013721590502 * q^97 + 217891799043453536868 * 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
$$3$$ 59049.0 0.577350
$$4$$ 0 0
$$5$$ 1.51338e7 0.693049 0.346524 0.938041i $$-0.387362\pi$$
0.346524 + 0.938041i $$0.387362\pi$$
$$6$$ 0 0
$$7$$ −8.40758e8 −1.12497 −0.562486 0.826807i $$-0.690155\pi$$
−0.562486 + 0.826807i $$0.690155\pi$$
$$8$$ 0 0
$$9$$ 3.48678e9 0.333333
$$10$$ 0 0
$$11$$ −3.34735e10 −0.389114 −0.194557 0.980891i $$-0.562327\pi$$
−0.194557 + 0.980891i $$0.562327\pi$$
$$12$$ 0 0
$$13$$ −3.01154e11 −0.605876 −0.302938 0.953010i $$-0.597968\pi$$
−0.302938 + 0.953010i $$0.597968\pi$$
$$14$$ 0 0
$$15$$ 8.93637e11 0.400132
$$16$$ 0 0
$$17$$ 5.14303e12 0.618736 0.309368 0.950942i $$-0.399883\pi$$
0.309368 + 0.950942i $$0.399883\pi$$
$$18$$ 0 0
$$19$$ 4.56647e13 1.70871 0.854355 0.519690i $$-0.173953\pi$$
0.854355 + 0.519690i $$0.173953\pi$$
$$20$$ 0 0
$$21$$ −4.96459e13 −0.649503
$$22$$ 0 0
$$23$$ 2.41579e13 0.121596 0.0607978 0.998150i $$-0.480636\pi$$
0.0607978 + 0.998150i $$0.480636\pi$$
$$24$$ 0 0
$$25$$ −2.47804e14 −0.519684
$$26$$ 0 0
$$27$$ 2.05891e14 0.192450
$$28$$ 0 0
$$29$$ 7.82910e14 0.345568 0.172784 0.984960i $$-0.444724\pi$$
0.172784 + 0.984960i $$0.444724\pi$$
$$30$$ 0 0
$$31$$ 8.00815e15 1.75483 0.877414 0.479734i $$-0.159267\pi$$
0.877414 + 0.479734i $$0.159267\pi$$
$$32$$ 0 0
$$33$$ −1.97657e15 −0.224655
$$34$$ 0 0
$$35$$ −1.27239e16 −0.779660
$$36$$ 0 0
$$37$$ 2.06600e16 0.706339 0.353169 0.935559i $$-0.385104\pi$$
0.353169 + 0.935559i $$0.385104\pi$$
$$38$$ 0 0
$$39$$ −1.77829e16 −0.349803
$$40$$ 0 0
$$41$$ −7.03094e16 −0.818056 −0.409028 0.912522i $$-0.634132\pi$$
−0.409028 + 0.912522i $$0.634132\pi$$
$$42$$ 0 0
$$43$$ 6.95453e15 0.0490737 0.0245369 0.999699i $$-0.492189\pi$$
0.0245369 + 0.999699i $$0.492189\pi$$
$$44$$ 0 0
$$45$$ 5.27684e16 0.231016
$$46$$ 0 0
$$47$$ −2.12675e17 −0.589778 −0.294889 0.955532i $$-0.595283\pi$$
−0.294889 + 0.955532i $$0.595283\pi$$
$$48$$ 0 0
$$49$$ 1.48329e17 0.265562
$$50$$ 0 0
$$51$$ 3.03691e17 0.357227
$$52$$ 0 0
$$53$$ 8.69454e17 0.682889 0.341445 0.939902i $$-0.389084\pi$$
0.341445 + 0.939902i $$0.389084\pi$$
$$54$$ 0 0
$$55$$ −5.06581e17 −0.269675
$$56$$ 0 0
$$57$$ 2.69646e18 0.986524
$$58$$ 0 0
$$59$$ 9.80538e17 0.249757 0.124879 0.992172i $$-0.460146\pi$$
0.124879 + 0.992172i $$0.460146\pi$$
$$60$$ 0 0
$$61$$ 5.62594e18 1.00979 0.504896 0.863180i $$-0.331531\pi$$
0.504896 + 0.863180i $$0.331531\pi$$
$$62$$ 0 0
$$63$$ −2.93154e18 −0.374991
$$64$$ 0 0
$$65$$ −4.55762e18 −0.419902
$$66$$ 0 0
$$67$$ 2.36685e18 0.158630 0.0793152 0.996850i $$-0.474727\pi$$
0.0793152 + 0.996850i $$0.474727\pi$$
$$68$$ 0 0
$$69$$ 1.42650e18 0.0702032
$$70$$ 0 0
$$71$$ 4.99695e19 1.82177 0.910883 0.412666i $$-0.135402\pi$$
0.910883 + 0.412666i $$0.135402\pi$$
$$72$$ 0 0
$$73$$ 2.63228e19 0.716873 0.358437 0.933554i $$-0.383310\pi$$
0.358437 + 0.933554i $$0.383310\pi$$
$$74$$ 0 0
$$75$$ −1.46326e19 −0.300040
$$76$$ 0 0
$$77$$ 2.81431e19 0.437743
$$78$$ 0 0
$$79$$ 1.05996e20 1.25952 0.629762 0.776788i $$-0.283152\pi$$
0.629762 + 0.776788i $$0.283152\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ 0 0
$$83$$ 8.96505e19 0.634209 0.317105 0.948391i $$-0.397289\pi$$
0.317105 + 0.948391i $$0.397289\pi$$
$$84$$ 0 0
$$85$$ 7.78337e19 0.428814
$$86$$ 0 0
$$87$$ 4.62301e19 0.199513
$$88$$ 0 0
$$89$$ 4.71064e20 1.60135 0.800673 0.599102i $$-0.204476\pi$$
0.800673 + 0.599102i $$0.204476\pi$$
$$90$$ 0 0
$$91$$ 2.53198e20 0.681594
$$92$$ 0 0
$$93$$ 4.72873e20 1.01315
$$94$$ 0 0
$$95$$ 6.91082e20 1.18422
$$96$$ 0 0
$$97$$ −3.93900e20 −0.542354 −0.271177 0.962530i $$-0.587413\pi$$
−0.271177 + 0.962530i $$0.587413\pi$$
$$98$$ 0 0
$$99$$ −1.16715e20 −0.129705
$$100$$ 0 0
$$101$$ −1.27087e21 −1.14479 −0.572397 0.819976i $$-0.693986\pi$$
−0.572397 + 0.819976i $$0.693986\pi$$
$$102$$ 0 0
$$103$$ 1.74236e21 1.27746 0.638731 0.769430i $$-0.279460\pi$$
0.638731 + 0.769430i $$0.279460\pi$$
$$104$$ 0 0
$$105$$ −7.51333e20 −0.450137
$$106$$ 0 0
$$107$$ −2.92887e21 −1.43936 −0.719681 0.694305i $$-0.755712\pi$$
−0.719681 + 0.694305i $$0.755712\pi$$
$$108$$ 0 0
$$109$$ 8.72296e19 0.0352928 0.0176464 0.999844i $$-0.494383\pi$$
0.0176464 + 0.999844i $$0.494383\pi$$
$$110$$ 0 0
$$111$$ 1.21995e21 0.407805
$$112$$ 0 0
$$113$$ 5.59051e21 1.54927 0.774636 0.632408i $$-0.217933\pi$$
0.774636 + 0.632408i $$0.217933\pi$$
$$114$$ 0 0
$$115$$ 3.65602e20 0.0842716
$$116$$ 0 0
$$117$$ −1.05006e21 −0.201959
$$118$$ 0 0
$$119$$ −4.32404e21 −0.696060
$$120$$ 0 0
$$121$$ −6.27978e21 −0.848590
$$122$$ 0 0
$$123$$ −4.15170e21 −0.472305
$$124$$ 0 0
$$125$$ −1.09666e22 −1.05321
$$126$$ 0 0
$$127$$ −1.18421e22 −0.962695 −0.481348 0.876530i $$-0.659852\pi$$
−0.481348 + 0.876530i $$0.659852\pi$$
$$128$$ 0 0
$$129$$ 4.10658e20 0.0283327
$$130$$ 0 0
$$131$$ 1.85199e22 1.08715 0.543577 0.839360i $$-0.317070\pi$$
0.543577 + 0.839360i $$0.317070\pi$$
$$132$$ 0 0
$$133$$ −3.83930e22 −1.92225
$$134$$ 0 0
$$135$$ 3.11592e21 0.133377
$$136$$ 0 0
$$137$$ −3.79206e22 −1.39094 −0.695472 0.718553i $$-0.744805\pi$$
−0.695472 + 0.718553i $$0.744805\pi$$
$$138$$ 0 0
$$139$$ 3.59104e21 0.113127 0.0565634 0.998399i $$-0.481986\pi$$
0.0565634 + 0.998399i $$0.481986\pi$$
$$140$$ 0 0
$$141$$ −1.25582e22 −0.340508
$$142$$ 0 0
$$143$$ 1.00807e22 0.235755
$$144$$ 0 0
$$145$$ 1.18484e22 0.239495
$$146$$ 0 0
$$147$$ 8.75865e21 0.153322
$$148$$ 0 0
$$149$$ −6.86199e22 −1.04230 −0.521152 0.853464i $$-0.674498\pi$$
−0.521152 + 0.853464i $$0.674498\pi$$
$$150$$ 0 0
$$151$$ 5.56868e22 0.735351 0.367675 0.929954i $$-0.380154\pi$$
0.367675 + 0.929954i $$0.380154\pi$$
$$152$$ 0 0
$$153$$ 1.79326e22 0.206245
$$154$$ 0 0
$$155$$ 1.21194e23 1.21618
$$156$$ 0 0
$$157$$ 5.36204e22 0.470310 0.235155 0.971958i $$-0.424440\pi$$
0.235155 + 0.971958i $$0.424440\pi$$
$$158$$ 0 0
$$159$$ 5.13404e22 0.394266
$$160$$ 0 0
$$161$$ −2.03110e22 −0.136792
$$162$$ 0 0
$$163$$ 5.37253e22 0.317840 0.158920 0.987291i $$-0.449199\pi$$
0.158920 + 0.987291i $$0.449199\pi$$
$$164$$ 0 0
$$165$$ −2.99131e22 −0.155697
$$166$$ 0 0
$$167$$ 2.18267e23 1.00107 0.500537 0.865715i $$-0.333136\pi$$
0.500537 + 0.865715i $$0.333136\pi$$
$$168$$ 0 0
$$169$$ −1.56371e23 −0.632914
$$170$$ 0 0
$$171$$ 1.59223e23 0.569570
$$172$$ 0 0
$$173$$ 2.08038e23 0.658657 0.329328 0.944215i $$-0.393178\pi$$
0.329328 + 0.944215i $$0.393178\pi$$
$$174$$ 0 0
$$175$$ 2.08344e23 0.584630
$$176$$ 0 0
$$177$$ 5.78998e22 0.144197
$$178$$ 0 0
$$179$$ 2.73346e23 0.605002 0.302501 0.953149i $$-0.402179\pi$$
0.302501 + 0.953149i $$0.402179\pi$$
$$180$$ 0 0
$$181$$ −4.47620e23 −0.881627 −0.440813 0.897599i $$-0.645310\pi$$
−0.440813 + 0.897599i $$0.645310\pi$$
$$182$$ 0 0
$$183$$ 3.32206e23 0.583004
$$184$$ 0 0
$$185$$ 3.12665e23 0.489527
$$186$$ 0 0
$$187$$ −1.72155e23 −0.240759
$$188$$ 0 0
$$189$$ −1.73105e23 −0.216501
$$190$$ 0 0
$$191$$ 7.33413e23 0.821293 0.410647 0.911795i $$-0.365303\pi$$
0.410647 + 0.911795i $$0.365303\pi$$
$$192$$ 0 0
$$193$$ −1.35102e24 −1.35616 −0.678079 0.734989i $$-0.737187\pi$$
−0.678079 + 0.734989i $$0.737187\pi$$
$$194$$ 0 0
$$195$$ −2.69123e23 −0.242430
$$196$$ 0 0
$$197$$ −9.02011e23 −0.729989 −0.364994 0.931010i $$-0.618929\pi$$
−0.364994 + 0.931010i $$0.618929\pi$$
$$198$$ 0 0
$$199$$ 1.91315e24 1.39249 0.696244 0.717806i $$-0.254853\pi$$
0.696244 + 0.717806i $$0.254853\pi$$
$$200$$ 0 0
$$201$$ 1.39760e23 0.0915853
$$202$$ 0 0
$$203$$ −6.58238e23 −0.388754
$$204$$ 0 0
$$205$$ −1.06405e24 −0.566953
$$206$$ 0 0
$$207$$ 8.42336e22 0.0405319
$$208$$ 0 0
$$209$$ −1.52856e24 −0.664884
$$210$$ 0 0
$$211$$ −4.45736e24 −1.75433 −0.877167 0.480185i $$-0.840569\pi$$
−0.877167 + 0.480185i $$0.840569\pi$$
$$212$$ 0 0
$$213$$ 2.95065e24 1.05180
$$214$$ 0 0
$$215$$ 1.05249e23 0.0340105
$$216$$ 0 0
$$217$$ −6.73292e24 −1.97413
$$218$$ 0 0
$$219$$ 1.55434e24 0.413887
$$220$$ 0 0
$$221$$ −1.54884e24 −0.374877
$$222$$ 0 0
$$223$$ −8.25290e24 −1.81721 −0.908606 0.417654i $$-0.862852\pi$$
−0.908606 + 0.417654i $$0.862852\pi$$
$$224$$ 0 0
$$225$$ −8.64041e23 −0.173228
$$226$$ 0 0
$$227$$ −2.17673e24 −0.397680 −0.198840 0.980032i $$-0.563717\pi$$
−0.198840 + 0.980032i $$0.563717\pi$$
$$228$$ 0 0
$$229$$ 2.70002e24 0.449877 0.224939 0.974373i $$-0.427782\pi$$
0.224939 + 0.974373i $$0.427782\pi$$
$$230$$ 0 0
$$231$$ 1.66182e24 0.252731
$$232$$ 0 0
$$233$$ 1.29388e25 1.79745 0.898725 0.438512i $$-0.144494\pi$$
0.898725 + 0.438512i $$0.144494\pi$$
$$234$$ 0 0
$$235$$ −3.21858e24 −0.408745
$$236$$ 0 0
$$237$$ 6.25898e24 0.727187
$$238$$ 0 0
$$239$$ 1.36280e25 1.44962 0.724808 0.688951i $$-0.241929\pi$$
0.724808 + 0.688951i $$0.241929\pi$$
$$240$$ 0 0
$$241$$ 1.58532e24 0.154503 0.0772516 0.997012i $$-0.475386\pi$$
0.0772516 + 0.997012i $$0.475386\pi$$
$$242$$ 0 0
$$243$$ 7.17898e23 0.0641500
$$244$$ 0 0
$$245$$ 2.24478e24 0.184047
$$246$$ 0 0
$$247$$ −1.37521e25 −1.03527
$$248$$ 0 0
$$249$$ 5.29377e24 0.366161
$$250$$ 0 0
$$251$$ 7.10015e23 0.0451537 0.0225768 0.999745i $$-0.492813\pi$$
0.0225768 + 0.999745i $$0.492813\pi$$
$$252$$ 0 0
$$253$$ −8.08650e23 −0.0473146
$$254$$ 0 0
$$255$$ 4.59600e24 0.247576
$$256$$ 0 0
$$257$$ 3.27150e25 1.62349 0.811743 0.584015i $$-0.198519\pi$$
0.811743 + 0.584015i $$0.198519\pi$$
$$258$$ 0 0
$$259$$ −1.73701e25 −0.794611
$$260$$ 0 0
$$261$$ 2.72984e24 0.115189
$$262$$ 0 0
$$263$$ −1.81651e25 −0.707461 −0.353730 0.935347i $$-0.615087\pi$$
−0.353730 + 0.935347i $$0.615087\pi$$
$$264$$ 0 0
$$265$$ 1.31582e25 0.473275
$$266$$ 0 0
$$267$$ 2.78159e25 0.924537
$$268$$ 0 0
$$269$$ 4.75606e25 1.46167 0.730833 0.682556i $$-0.239132\pi$$
0.730833 + 0.682556i $$0.239132\pi$$
$$270$$ 0 0
$$271$$ −1.10127e25 −0.313123 −0.156561 0.987668i $$-0.550041\pi$$
−0.156561 + 0.987668i $$0.550041\pi$$
$$272$$ 0 0
$$273$$ 1.49511e25 0.393518
$$274$$ 0 0
$$275$$ 8.29487e24 0.202216
$$276$$ 0 0
$$277$$ −4.48744e25 −1.01382 −0.506911 0.861999i $$-0.669213\pi$$
−0.506911 + 0.861999i $$0.669213\pi$$
$$278$$ 0 0
$$279$$ 2.79227e25 0.584942
$$280$$ 0 0
$$281$$ −8.48250e24 −0.164857 −0.0824285 0.996597i $$-0.526268\pi$$
−0.0824285 + 0.996597i $$0.526268\pi$$
$$282$$ 0 0
$$283$$ −9.02788e25 −1.62865 −0.814326 0.580407i $$-0.802893\pi$$
−0.814326 + 0.580407i $$0.802893\pi$$
$$284$$ 0 0
$$285$$ 4.08077e25 0.683709
$$286$$ 0 0
$$287$$ 5.91132e25 0.920290
$$288$$ 0 0
$$289$$ −4.26412e25 −0.617166
$$290$$ 0 0
$$291$$ −2.32594e25 −0.313128
$$292$$ 0 0
$$293$$ −1.20579e26 −1.51065 −0.755324 0.655352i $$-0.772520\pi$$
−0.755324 + 0.655352i $$0.772520\pi$$
$$294$$ 0 0
$$295$$ 1.48393e25 0.173094
$$296$$ 0 0
$$297$$ −6.89189e24 −0.0748851
$$298$$ 0 0
$$299$$ −7.27527e24 −0.0736719
$$300$$ 0 0
$$301$$ −5.84708e24 −0.0552065
$$302$$ 0 0
$$303$$ −7.50437e25 −0.660947
$$304$$ 0 0
$$305$$ 8.51420e25 0.699835
$$306$$ 0 0
$$307$$ −1.33188e26 −1.02215 −0.511074 0.859537i $$-0.670752\pi$$
−0.511074 + 0.859537i $$0.670752\pi$$
$$308$$ 0 0
$$309$$ 1.02885e26 0.737543
$$310$$ 0 0
$$311$$ −1.32644e26 −0.888593 −0.444296 0.895880i $$-0.646546\pi$$
−0.444296 + 0.895880i $$0.646546\pi$$
$$312$$ 0 0
$$313$$ −1.86750e26 −1.16962 −0.584811 0.811169i $$-0.698831\pi$$
−0.584811 + 0.811169i $$0.698831\pi$$
$$314$$ 0 0
$$315$$ −4.43655e25 −0.259887
$$316$$ 0 0
$$317$$ −1.99163e26 −1.09166 −0.545829 0.837896i $$-0.683785\pi$$
−0.545829 + 0.837896i $$0.683785\pi$$
$$318$$ 0 0
$$319$$ −2.62067e25 −0.134465
$$320$$ 0 0
$$321$$ −1.72947e26 −0.831016
$$322$$ 0 0
$$323$$ 2.34855e26 1.05724
$$324$$ 0 0
$$325$$ 7.46274e25 0.314864
$$326$$ 0 0
$$327$$ 5.15082e24 0.0203763
$$328$$ 0 0
$$329$$ 1.78808e26 0.663483
$$330$$ 0 0
$$331$$ 3.61887e26 1.26002 0.630012 0.776586i $$-0.283050\pi$$
0.630012 + 0.776586i $$0.283050\pi$$
$$332$$ 0 0
$$333$$ 7.20371e25 0.235446
$$334$$ 0 0
$$335$$ 3.58196e25 0.109939
$$336$$ 0 0
$$337$$ −4.20174e26 −1.21148 −0.605739 0.795664i $$-0.707122\pi$$
−0.605739 + 0.795664i $$0.707122\pi$$
$$338$$ 0 0
$$339$$ 3.30114e26 0.894472
$$340$$ 0 0
$$341$$ −2.68060e26 −0.682829
$$342$$ 0 0
$$343$$ 3.44894e26 0.826222
$$344$$ 0 0
$$345$$ 2.15884e25 0.0486543
$$346$$ 0 0
$$347$$ 7.87467e26 1.67022 0.835108 0.550085i $$-0.185405\pi$$
0.835108 + 0.550085i $$0.185405\pi$$
$$348$$ 0 0
$$349$$ 4.94479e26 0.987372 0.493686 0.869640i $$-0.335649\pi$$
0.493686 + 0.869640i $$0.335649\pi$$
$$350$$ 0 0
$$351$$ −6.20050e25 −0.116601
$$352$$ 0 0
$$353$$ −9.34606e26 −1.65575 −0.827874 0.560914i $$-0.810450\pi$$
−0.827874 + 0.560914i $$0.810450\pi$$
$$354$$ 0 0
$$355$$ 7.56230e26 1.26257
$$356$$ 0 0
$$357$$ −2.55330e26 −0.401871
$$358$$ 0 0
$$359$$ 3.08467e25 0.0457844 0.0228922 0.999738i $$-0.492713\pi$$
0.0228922 + 0.999738i $$0.492713\pi$$
$$360$$ 0 0
$$361$$ 1.37106e27 1.91969
$$362$$ 0 0
$$363$$ −3.70815e26 −0.489934
$$364$$ 0 0
$$365$$ 3.98365e26 0.496828
$$366$$ 0 0
$$367$$ 3.96982e26 0.467495 0.233747 0.972297i $$-0.424901\pi$$
0.233747 + 0.972297i $$0.424901\pi$$
$$368$$ 0 0
$$369$$ −2.45154e26 −0.272685
$$370$$ 0 0
$$371$$ −7.31001e26 −0.768231
$$372$$ 0 0
$$373$$ 2.66510e26 0.264711 0.132355 0.991202i $$-0.457746\pi$$
0.132355 + 0.991202i $$0.457746\pi$$
$$374$$ 0 0
$$375$$ −6.47567e26 −0.608074
$$376$$ 0 0
$$377$$ −2.35777e26 −0.209371
$$378$$ 0 0
$$379$$ −4.17651e26 −0.350834 −0.175417 0.984494i $$-0.556127\pi$$
−0.175417 + 0.984494i $$0.556127\pi$$
$$380$$ 0 0
$$381$$ −6.99263e26 −0.555812
$$382$$ 0 0
$$383$$ 2.09590e26 0.157683 0.0788414 0.996887i $$-0.474878\pi$$
0.0788414 + 0.996887i $$0.474878\pi$$
$$384$$ 0 0
$$385$$ 4.25912e26 0.303377
$$386$$ 0 0
$$387$$ 2.42489e25 0.0163579
$$388$$ 0 0
$$389$$ −1.25020e27 −0.798929 −0.399465 0.916749i $$-0.630804\pi$$
−0.399465 + 0.916749i $$0.630804\pi$$
$$390$$ 0 0
$$391$$ 1.24245e26 0.0752355
$$392$$ 0 0
$$393$$ 1.09358e27 0.627668
$$394$$ 0 0
$$395$$ 1.60413e27 0.872911
$$396$$ 0 0
$$397$$ 2.25158e27 1.16195 0.580974 0.813922i $$-0.302672\pi$$
0.580974 + 0.813922i $$0.302672\pi$$
$$398$$ 0 0
$$399$$ −2.26707e27 −1.10981
$$400$$ 0 0
$$401$$ −2.49719e27 −1.15994 −0.579971 0.814637i $$-0.696936\pi$$
−0.579971 + 0.814637i $$0.696936\pi$$
$$402$$ 0 0
$$403$$ −2.41169e27 −1.06321
$$404$$ 0 0
$$405$$ 1.83992e26 0.0770054
$$406$$ 0 0
$$407$$ −6.91563e26 −0.274847
$$408$$ 0 0
$$409$$ 7.98541e26 0.301441 0.150720 0.988576i $$-0.451841\pi$$
0.150720 + 0.988576i $$0.451841\pi$$
$$410$$ 0 0
$$411$$ −2.23917e27 −0.803062
$$412$$ 0 0
$$413$$ −8.24395e26 −0.280970
$$414$$ 0 0
$$415$$ 1.35675e27 0.439538
$$416$$ 0 0
$$417$$ 2.12048e26 0.0653138
$$418$$ 0 0
$$419$$ −6.23317e27 −1.82584 −0.912918 0.408144i $$-0.866176\pi$$
−0.912918 + 0.408144i $$0.866176\pi$$
$$420$$ 0 0
$$421$$ −1.93368e27 −0.538795 −0.269398 0.963029i $$-0.586825\pi$$
−0.269398 + 0.963029i $$0.586825\pi$$
$$422$$ 0 0
$$423$$ −7.41551e26 −0.196593
$$424$$ 0 0
$$425$$ −1.27447e27 −0.321547
$$426$$ 0 0
$$427$$ −4.73006e27 −1.13599
$$428$$ 0 0
$$429$$ 5.95254e26 0.136113
$$430$$ 0 0
$$431$$ −1.38099e27 −0.300732 −0.150366 0.988630i $$-0.548045\pi$$
−0.150366 + 0.988630i $$0.548045\pi$$
$$432$$ 0 0
$$433$$ 6.39470e27 1.32647 0.663235 0.748411i $$-0.269183\pi$$
0.663235 + 0.748411i $$0.269183\pi$$
$$434$$ 0 0
$$435$$ 6.99638e26 0.138273
$$436$$ 0 0
$$437$$ 1.10317e27 0.207772
$$438$$ 0 0
$$439$$ 6.94609e27 1.24699 0.623495 0.781827i $$-0.285712\pi$$
0.623495 + 0.781827i $$0.285712\pi$$
$$440$$ 0 0
$$441$$ 5.17190e26 0.0885206
$$442$$ 0 0
$$443$$ 8.47333e27 1.38298 0.691488 0.722387i $$-0.256955\pi$$
0.691488 + 0.722387i $$0.256955\pi$$
$$444$$ 0 0
$$445$$ 7.12900e27 1.10981
$$446$$ 0 0
$$447$$ −4.05194e27 −0.601774
$$448$$ 0 0
$$449$$ −5.62902e27 −0.797712 −0.398856 0.917014i $$-0.630593\pi$$
−0.398856 + 0.917014i $$0.630593\pi$$
$$450$$ 0 0
$$451$$ 2.35350e27 0.318317
$$452$$ 0 0
$$453$$ 3.28825e27 0.424555
$$454$$ 0 0
$$455$$ 3.83185e27 0.472378
$$456$$ 0 0
$$457$$ −2.91070e26 −0.0342671 −0.0171335 0.999853i $$-0.505454\pi$$
−0.0171335 + 0.999853i $$0.505454\pi$$
$$458$$ 0 0
$$459$$ 1.05890e27 0.119076
$$460$$ 0 0
$$461$$ 1.31214e28 1.40968 0.704840 0.709367i $$-0.251019\pi$$
0.704840 + 0.709367i $$0.251019\pi$$
$$462$$ 0 0
$$463$$ −1.50940e28 −1.54955 −0.774773 0.632239i $$-0.782136\pi$$
−0.774773 + 0.632239i $$0.782136\pi$$
$$464$$ 0 0
$$465$$ 7.15638e27 0.702162
$$466$$ 0 0
$$467$$ 8.92048e27 0.836683 0.418341 0.908290i $$-0.362612\pi$$
0.418341 + 0.908290i $$0.362612\pi$$
$$468$$ 0 0
$$469$$ −1.98995e27 −0.178455
$$470$$ 0 0
$$471$$ 3.16623e27 0.271533
$$472$$ 0 0
$$473$$ −2.32792e26 −0.0190953
$$474$$ 0 0
$$475$$ −1.13159e28 −0.887989
$$476$$ 0 0
$$477$$ 3.03160e27 0.227630
$$478$$ 0 0
$$479$$ 1.87487e28 1.34725 0.673625 0.739073i $$-0.264736\pi$$
0.673625 + 0.739073i $$0.264736\pi$$
$$480$$ 0 0
$$481$$ −6.22186e27 −0.427954
$$482$$ 0 0
$$483$$ −1.19934e27 −0.0789767
$$484$$ 0 0
$$485$$ −5.96121e27 −0.375878
$$486$$ 0 0
$$487$$ 9.66602e27 0.583705 0.291853 0.956463i $$-0.405728\pi$$
0.291853 + 0.956463i $$0.405728\pi$$
$$488$$ 0 0
$$489$$ 3.17242e27 0.183505
$$490$$ 0 0
$$491$$ 2.43841e28 1.35130 0.675648 0.737224i $$-0.263864\pi$$
0.675648 + 0.737224i $$0.263864\pi$$
$$492$$ 0 0
$$493$$ 4.02653e27 0.213815
$$494$$ 0 0
$$495$$ −1.76634e27 −0.0898917
$$496$$ 0 0
$$497$$ −4.20123e28 −2.04943
$$498$$ 0 0
$$499$$ −3.32191e28 −1.55357 −0.776787 0.629764i $$-0.783152\pi$$
−0.776787 + 0.629764i $$0.783152\pi$$
$$500$$ 0 0
$$501$$ 1.28885e28 0.577970
$$502$$ 0 0
$$503$$ 1.02085e28 0.439036 0.219518 0.975608i $$-0.429552\pi$$
0.219518 + 0.975608i $$0.429552\pi$$
$$504$$ 0 0
$$505$$ −1.92332e28 −0.793398
$$506$$ 0 0
$$507$$ −9.23353e27 −0.365413
$$508$$ 0 0
$$509$$ −2.97174e28 −1.12843 −0.564215 0.825628i $$-0.690821\pi$$
−0.564215 + 0.825628i $$0.690821\pi$$
$$510$$ 0 0
$$511$$ −2.21311e28 −0.806462
$$512$$ 0 0
$$513$$ 9.40197e27 0.328841
$$514$$ 0 0
$$515$$ 2.63686e28 0.885344
$$516$$ 0 0
$$517$$ 7.11896e27 0.229491
$$518$$ 0 0
$$519$$ 1.22845e28 0.380276
$$520$$ 0 0
$$521$$ −6.31442e28 −1.87732 −0.938658 0.344849i $$-0.887930\pi$$
−0.938658 + 0.344849i $$0.887930\pi$$
$$522$$ 0 0
$$523$$ 6.21727e27 0.177554 0.0887772 0.996052i $$-0.471704\pi$$
0.0887772 + 0.996052i $$0.471704\pi$$
$$524$$ 0 0
$$525$$ 1.23025e28 0.337536
$$526$$ 0 0
$$527$$ 4.11861e28 1.08577
$$528$$ 0 0
$$529$$ −3.88880e28 −0.985215
$$530$$ 0 0
$$531$$ 3.41892e27 0.0832524
$$532$$ 0 0
$$533$$ 2.11740e28 0.495641
$$534$$ 0 0
$$535$$ −4.43250e28 −0.997548
$$536$$ 0 0
$$537$$ 1.61408e28 0.349298
$$538$$ 0 0
$$539$$ −4.96507e27 −0.103334
$$540$$ 0 0
$$541$$ −5.85756e28 −1.17259 −0.586294 0.810099i $$-0.699414\pi$$
−0.586294 + 0.810099i $$0.699414\pi$$
$$542$$ 0 0
$$543$$ −2.64315e28 −0.509008
$$544$$ 0 0
$$545$$ 1.32012e27 0.0244596
$$546$$ 0 0
$$547$$ 2.21556e27 0.0395017 0.0197509 0.999805i $$-0.493713\pi$$
0.0197509 + 0.999805i $$0.493713\pi$$
$$548$$ 0 0
$$549$$ 1.96164e28 0.336597
$$550$$ 0 0
$$551$$ 3.57514e28 0.590475
$$552$$ 0 0
$$553$$ −8.91173e28 −1.41693
$$554$$ 0 0
$$555$$ 1.84626e28 0.282629
$$556$$ 0 0
$$557$$ 5.45401e27 0.0803964 0.0401982 0.999192i $$-0.487201\pi$$
0.0401982 + 0.999192i $$0.487201\pi$$
$$558$$ 0 0
$$559$$ −2.09439e27 −0.0297326
$$560$$ 0 0
$$561$$ −1.01656e28 −0.139002
$$562$$ 0 0
$$563$$ 8.27033e27 0.108939 0.0544696 0.998515i $$-0.482653\pi$$
0.0544696 + 0.998515i $$0.482653\pi$$
$$564$$ 0 0
$$565$$ 8.46057e28 1.07372
$$566$$ 0 0
$$567$$ −1.02217e28 −0.124997
$$568$$ 0 0
$$569$$ 8.21967e28 0.968669 0.484335 0.874883i $$-0.339062\pi$$
0.484335 + 0.874883i $$0.339062\pi$$
$$570$$ 0 0
$$571$$ 6.72222e28 0.763544 0.381772 0.924257i $$-0.375314\pi$$
0.381772 + 0.924257i $$0.375314\pi$$
$$572$$ 0 0
$$573$$ 4.33073e28 0.474174
$$574$$ 0 0
$$575$$ −5.98645e27 −0.0631912
$$576$$ 0 0
$$577$$ 1.51324e29 1.54015 0.770075 0.637953i $$-0.220219\pi$$
0.770075 + 0.637953i $$0.220219\pi$$
$$578$$ 0 0
$$579$$ −7.97764e28 −0.782978
$$580$$ 0 0
$$581$$ −7.53744e28 −0.713468
$$582$$ 0 0
$$583$$ −2.91036e28 −0.265722
$$584$$ 0 0
$$585$$ −1.58914e28 −0.139967
$$586$$ 0 0
$$587$$ 7.61361e28 0.646980 0.323490 0.946232i $$-0.395144\pi$$
0.323490 + 0.946232i $$0.395144\pi$$
$$588$$ 0 0
$$589$$ 3.65690e29 2.99849
$$590$$ 0 0
$$591$$ −5.32628e28 −0.421459
$$592$$ 0 0
$$593$$ 6.44095e28 0.491899 0.245950 0.969283i $$-0.420900\pi$$
0.245950 + 0.969283i $$0.420900\pi$$
$$594$$ 0 0
$$595$$ −6.54393e28 −0.482404
$$596$$ 0 0
$$597$$ 1.12969e29 0.803953
$$598$$ 0 0
$$599$$ 5.13552e28 0.352860 0.176430 0.984313i $$-0.443545\pi$$
0.176430 + 0.984313i $$0.443545\pi$$
$$600$$ 0 0
$$601$$ −1.12674e29 −0.747554 −0.373777 0.927519i $$-0.621938\pi$$
−0.373777 + 0.927519i $$0.621938\pi$$
$$602$$ 0 0
$$603$$ 8.25271e27 0.0528768
$$604$$ 0 0
$$605$$ −9.50371e28 −0.588114
$$606$$ 0 0
$$607$$ −1.15513e29 −0.690475 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$608$$ 0 0
$$609$$ −3.88683e28 −0.224447
$$610$$ 0 0
$$611$$ 6.40479e28 0.357332
$$612$$ 0 0
$$613$$ 2.36968e29 1.27748 0.638742 0.769421i $$-0.279455\pi$$
0.638742 + 0.769421i $$0.279455\pi$$
$$614$$ 0 0
$$615$$ −6.28311e28 −0.327330
$$616$$ 0 0
$$617$$ −2.69051e29 −1.35469 −0.677346 0.735664i $$-0.736870\pi$$
−0.677346 + 0.735664i $$0.736870\pi$$
$$618$$ 0 0
$$619$$ 2.88261e29 1.40292 0.701461 0.712708i $$-0.252532\pi$$
0.701461 + 0.712708i $$0.252532\pi$$
$$620$$ 0 0
$$621$$ 4.97391e27 0.0234011
$$622$$ 0 0
$$623$$ −3.96051e29 −1.80147
$$624$$ 0 0
$$625$$ −4.78042e28 −0.210245
$$626$$ 0 0
$$627$$ −9.02598e28 −0.383871
$$628$$ 0 0
$$629$$ 1.06255e29 0.437037
$$630$$ 0 0
$$631$$ −1.60612e29 −0.638953 −0.319477 0.947594i $$-0.603507\pi$$
−0.319477 + 0.947594i $$0.603507\pi$$
$$632$$ 0 0
$$633$$ −2.63203e29 −1.01287
$$634$$ 0 0
$$635$$ −1.79216e29 −0.667194
$$636$$ 0 0
$$637$$ −4.46698e28 −0.160898
$$638$$ 0 0
$$639$$ 1.74233e29 0.607255
$$640$$ 0 0
$$641$$ −5.00919e29 −1.68950 −0.844749 0.535162i $$-0.820250\pi$$
−0.844749 + 0.535162i $$0.820250\pi$$
$$642$$ 0 0
$$643$$ −2.11297e29 −0.689728 −0.344864 0.938653i $$-0.612075\pi$$
−0.344864 + 0.938653i $$0.612075\pi$$
$$644$$ 0 0
$$645$$ 6.21483e27 0.0196359
$$646$$ 0 0
$$647$$ 4.66535e29 1.42689 0.713443 0.700713i $$-0.247135\pi$$
0.713443 + 0.700713i $$0.247135\pi$$
$$648$$ 0 0
$$649$$ −3.28220e28 −0.0971842
$$650$$ 0 0
$$651$$ −3.97572e29 −1.13977
$$652$$ 0 0
$$653$$ 6.87022e29 1.90714 0.953569 0.301175i $$-0.0973787\pi$$
0.953569 + 0.301175i $$0.0973787\pi$$
$$654$$ 0 0
$$655$$ 2.80277e29 0.753450
$$656$$ 0 0
$$657$$ 9.17820e28 0.238958
$$658$$ 0 0
$$659$$ −5.83008e29 −1.47021 −0.735103 0.677956i $$-0.762866\pi$$
−0.735103 + 0.677956i $$0.762866\pi$$
$$660$$ 0 0
$$661$$ 7.08881e29 1.73164 0.865820 0.500355i $$-0.166797\pi$$
0.865820 + 0.500355i $$0.166797\pi$$
$$662$$ 0 0
$$663$$ −9.14577e28 −0.216435
$$664$$ 0 0
$$665$$ −5.81033e29 −1.33221
$$666$$ 0 0
$$667$$ 1.89135e28 0.0420195
$$668$$ 0 0
$$669$$ −4.87326e29 −1.04917
$$670$$ 0 0
$$671$$ −1.88320e29 −0.392925
$$672$$ 0 0
$$673$$ −2.18749e29 −0.442373 −0.221186 0.975232i $$-0.570993\pi$$
−0.221186 + 0.975232i $$0.570993\pi$$
$$674$$ 0 0
$$675$$ −5.10207e28 −0.100013
$$676$$ 0 0
$$677$$ 4.19570e29 0.797302 0.398651 0.917103i $$-0.369478\pi$$
0.398651 + 0.917103i $$0.369478\pi$$
$$678$$ 0 0
$$679$$ 3.31175e29 0.610133
$$680$$ 0 0
$$681$$ −1.28534e29 −0.229601
$$682$$ 0 0
$$683$$ −4.62694e29 −0.801450 −0.400725 0.916198i $$-0.631242\pi$$
−0.400725 + 0.916198i $$0.631242\pi$$
$$684$$ 0 0
$$685$$ −5.73884e29 −0.963991
$$686$$ 0 0
$$687$$ 1.59433e29 0.259737
$$688$$ 0 0
$$689$$ −2.61840e29 −0.413746
$$690$$ 0 0
$$691$$ 7.05176e29 1.08088 0.540441 0.841382i $$-0.318257\pi$$
0.540441 + 0.841382i $$0.318257\pi$$
$$692$$ 0 0
$$693$$ 9.81289e28 0.145914
$$694$$ 0 0
$$695$$ 5.43462e28 0.0784023
$$696$$ 0 0
$$697$$ −3.61603e29 −0.506160
$$698$$ 0 0
$$699$$ 7.64024e29 1.03776
$$700$$ 0 0
$$701$$ 7.25728e29 0.956609 0.478305 0.878194i $$-0.341252\pi$$
0.478305 + 0.878194i $$0.341252\pi$$
$$702$$ 0 0
$$703$$ 9.43435e29 1.20693
$$704$$ 0 0
$$705$$ −1.90054e29 −0.235989
$$706$$ 0 0
$$707$$ 1.06850e30 1.28786
$$708$$ 0 0
$$709$$ −4.71108e29 −0.551233 −0.275616 0.961268i $$-0.588882\pi$$
−0.275616 + 0.961268i $$0.588882\pi$$
$$710$$ 0 0
$$711$$ 3.69586e29 0.419841
$$712$$ 0 0
$$713$$ 1.93460e29 0.213379
$$714$$ 0 0
$$715$$ 1.52559e29 0.163390
$$716$$ 0 0
$$717$$ 8.04717e29 0.836936
$$718$$ 0 0
$$719$$ 5.11299e29 0.516442 0.258221 0.966086i $$-0.416864\pi$$
0.258221 + 0.966086i $$0.416864\pi$$
$$720$$ 0 0
$$721$$ −1.46491e30 −1.43711
$$722$$ 0 0
$$723$$ 9.36115e28 0.0892025
$$724$$ 0 0
$$725$$ −1.94009e29 −0.179586
$$726$$ 0 0
$$727$$ 2.09585e29 0.188473 0.0942366 0.995550i $$-0.469959\pi$$
0.0942366 + 0.995550i $$0.469959\pi$$
$$728$$ 0 0
$$729$$ 4.23912e28 0.0370370
$$730$$ 0 0
$$731$$ 3.57673e28 0.0303637
$$732$$ 0 0
$$733$$ 2.06298e30 1.70178 0.850890 0.525345i $$-0.176064\pi$$
0.850890 + 0.525345i $$0.176064\pi$$
$$734$$ 0 0
$$735$$ 1.32552e29 0.106260
$$736$$ 0 0
$$737$$ −7.92268e28 −0.0617253
$$738$$ 0 0
$$739$$ −8.76008e28 −0.0663348 −0.0331674 0.999450i $$-0.510559\pi$$
−0.0331674 + 0.999450i $$0.510559\pi$$
$$740$$ 0 0
$$741$$ −8.12050e29 −0.597711
$$742$$ 0 0
$$743$$ −5.61196e29 −0.401543 −0.200771 0.979638i $$-0.564345\pi$$
−0.200771 + 0.979638i $$0.564345\pi$$
$$744$$ 0 0
$$745$$ −1.03848e30 −0.722367
$$746$$ 0 0
$$747$$ 3.12592e29 0.211403
$$748$$ 0 0
$$749$$ 2.46247e30 1.61924
$$750$$ 0 0
$$751$$ −1.05895e30 −0.677106 −0.338553 0.940947i $$-0.609937\pi$$
−0.338553 + 0.940947i $$0.609937\pi$$
$$752$$ 0 0
$$753$$ 4.19256e28 0.0260695
$$754$$ 0 0
$$755$$ 8.42755e29 0.509634
$$756$$ 0 0
$$757$$ −1.84921e30 −1.08763 −0.543813 0.839206i $$-0.683020\pi$$
−0.543813 + 0.839206i $$0.683020\pi$$
$$758$$ 0 0
$$759$$ −4.77500e28 −0.0273171
$$760$$ 0 0
$$761$$ −4.53605e29 −0.252429 −0.126214 0.992003i $$-0.540283\pi$$
−0.126214 + 0.992003i $$0.540283\pi$$
$$762$$ 0 0
$$763$$ −7.33390e28 −0.0397034
$$764$$ 0 0
$$765$$ 2.71389e29 0.142938
$$766$$ 0 0
$$767$$ −2.95293e29 −0.151322
$$768$$ 0 0
$$769$$ −1.50798e30 −0.751916 −0.375958 0.926637i $$-0.622686\pi$$
−0.375958 + 0.926637i $$0.622686\pi$$
$$770$$ 0 0
$$771$$ 1.93179e30 0.937320
$$772$$ 0 0
$$773$$ 5.10001e29 0.240817 0.120408 0.992724i $$-0.461580\pi$$
0.120408 + 0.992724i $$0.461580\pi$$
$$774$$ 0 0
$$775$$ −1.98446e30 −0.911955
$$776$$ 0 0
$$777$$ −1.02569e30 −0.458769
$$778$$ 0 0
$$779$$ −3.21066e30 −1.39782
$$780$$ 0 0
$$781$$ −1.67265e30 −0.708875
$$782$$ 0 0
$$783$$ 1.61194e29 0.0665045
$$784$$ 0 0
$$785$$ 8.11482e29 0.325947
$$786$$ 0 0
$$787$$ −3.76408e30 −1.47205 −0.736027 0.676952i $$-0.763300\pi$$
−0.736027 + 0.676952i $$0.763300\pi$$
$$788$$ 0 0
$$789$$ −1.07263e30 −0.408453
$$790$$ 0 0
$$791$$ −4.70026e30 −1.74289
$$792$$ 0 0
$$793$$ −1.69428e30 −0.611809
$$794$$ 0 0
$$795$$ 7.76977e29 0.273246
$$796$$ 0 0
$$797$$ 5.68916e30 1.94866 0.974329 0.225129i $$-0.0722805\pi$$
0.974329 + 0.225129i $$0.0722805\pi$$
$$798$$ 0 0
$$799$$ −1.09379e30 −0.364916
$$800$$ 0 0
$$801$$ 1.64250e30 0.533782
$$802$$ 0 0
$$803$$ −8.81116e29 −0.278946
$$804$$ 0 0
$$805$$ −3.07383e29 −0.0948032
$$806$$ 0 0
$$807$$ 2.80840e30 0.843893
$$808$$ 0 0
$$809$$ −3.33803e29 −0.0977306 −0.0488653 0.998805i $$-0.515561\pi$$
−0.0488653 + 0.998805i $$0.515561\pi$$
$$810$$ 0 0
$$811$$ −2.91268e30 −0.830949 −0.415474 0.909605i $$-0.636384\pi$$
−0.415474 + 0.909605i $$0.636384\pi$$
$$812$$ 0 0
$$813$$ −6.50287e29 −0.180782
$$814$$ 0 0
$$815$$ 8.13069e29 0.220279
$$816$$ 0 0
$$817$$ 3.17577e29 0.0838527
$$818$$ 0 0
$$819$$ 8.82847e29 0.227198
$$820$$ 0 0
$$821$$ 4.82227e30 1.20962 0.604809 0.796370i $$-0.293249\pi$$
0.604809 + 0.796370i $$0.293249\pi$$
$$822$$ 0 0
$$823$$ 2.55917e30 0.625751 0.312875 0.949794i $$-0.398708\pi$$
0.312875 + 0.949794i $$0.398708\pi$$
$$824$$ 0 0
$$825$$ 4.89804e29 0.116750
$$826$$ 0 0
$$827$$ −1.93866e30 −0.450498 −0.225249 0.974301i $$-0.572320\pi$$
−0.225249 + 0.974301i $$0.572320\pi$$
$$828$$ 0 0
$$829$$ 7.77902e30 1.76239 0.881195 0.472753i $$-0.156740\pi$$
0.881195 + 0.472753i $$0.156740\pi$$
$$830$$ 0 0
$$831$$ −2.64979e30 −0.585330
$$832$$ 0 0
$$833$$ 7.62858e29 0.164313
$$834$$ 0 0
$$835$$ 3.30322e30 0.693792
$$836$$ 0 0
$$837$$ 1.64881e30 0.337717
$$838$$ 0 0
$$839$$ −6.16703e30 −1.23190 −0.615950 0.787785i $$-0.711228\pi$$
−0.615950 + 0.787785i $$0.711228\pi$$
$$840$$ 0 0
$$841$$ −4.51989e30 −0.880583
$$842$$ 0 0
$$843$$ −5.00883e29 −0.0951803
$$844$$ 0 0
$$845$$ −2.36649e30 −0.438640
$$846$$ 0 0
$$847$$ 5.27977e30 0.954640
$$848$$ 0 0
$$849$$ −5.33088e30 −0.940303
$$850$$ 0 0
$$851$$ 4.99104e29 0.0858876
$$852$$ 0 0
$$853$$ −1.15935e30 −0.194648 −0.0973239 0.995253i $$-0.531028\pi$$
−0.0973239 + 0.995253i $$0.531028\pi$$
$$854$$ 0 0
$$855$$ 2.40966e30 0.394740
$$856$$ 0 0
$$857$$ 6.59419e30 1.05405 0.527027 0.849848i $$-0.323307\pi$$
0.527027 + 0.849848i $$0.323307\pi$$
$$858$$ 0 0
$$859$$ 6.07395e30 0.947421 0.473710 0.880681i $$-0.342914\pi$$
0.473710 + 0.880681i $$0.342914\pi$$
$$860$$ 0 0
$$861$$ 3.49058e30 0.531330
$$862$$ 0 0
$$863$$ 4.55774e30 0.677075 0.338537 0.940953i $$-0.390068\pi$$
0.338537 + 0.940953i $$0.390068\pi$$
$$864$$ 0 0
$$865$$ 3.14842e30 0.456481
$$866$$ 0 0
$$867$$ −2.51792e30 −0.356321
$$868$$ 0 0
$$869$$ −3.54806e30 −0.490099
$$870$$ 0 0
$$871$$ −7.12788e29 −0.0961103
$$872$$ 0 0
$$873$$ −1.37344e30 −0.180785
$$874$$ 0 0
$$875$$ 9.22026e30 1.18484
$$876$$ 0 0
$$877$$ −6.30080e29 −0.0790496 −0.0395248 0.999219i $$-0.512584\pi$$
−0.0395248 + 0.999219i $$0.512584\pi$$
$$878$$ 0 0
$$879$$ −7.12010e30 −0.872173
$$880$$ 0 0
$$881$$ 4.36026e30 0.521513 0.260757 0.965405i $$-0.416028\pi$$
0.260757 + 0.965405i $$0.416028\pi$$
$$882$$ 0 0
$$883$$ 3.58988e30 0.419269 0.209635 0.977780i $$-0.432773\pi$$
0.209635 + 0.977780i $$0.432773\pi$$
$$884$$ 0 0
$$885$$ 8.76245e29 0.0999358
$$886$$ 0 0
$$887$$ −1.18282e30 −0.131741 −0.0658707 0.997828i $$-0.520982\pi$$
−0.0658707 + 0.997828i $$0.520982\pi$$
$$888$$ 0 0
$$889$$ 9.95632e30 1.08300
$$890$$ 0 0
$$891$$ −4.06959e29 −0.0432349
$$892$$ 0 0
$$893$$ −9.71174e30 −1.00776
$$894$$ 0 0
$$895$$ 4.13678e30 0.419295
$$896$$ 0 0
$$897$$ −4.29597e29 −0.0425345
$$898$$ 0 0
$$899$$ 6.26966e30 0.606411
$$900$$ 0 0
$$901$$ 4.47163e30 0.422528
$$902$$ 0 0
$$903$$ −3.45264e29 −0.0318735
$$904$$ 0 0
$$905$$ −6.77421e30 −0.611010
$$906$$ 0 0
$$907$$ −8.29044e30 −0.730636 −0.365318 0.930883i $$-0.619040\pi$$
−0.365318 + 0.930883i $$0.619040\pi$$
$$908$$ 0 0
$$909$$ −4.43126e30 −0.381598
$$910$$ 0 0
$$911$$ −6.67910e30 −0.562050 −0.281025 0.959700i $$-0.590674\pi$$
−0.281025 + 0.959700i $$0.590674\pi$$
$$912$$ 0 0
$$913$$ −3.00091e30 −0.246780
$$914$$ 0 0
$$915$$ 5.02755e30 0.404050
$$916$$ 0 0
$$917$$ −1.55708e31 −1.22302
$$918$$ 0 0
$$919$$ 1.36155e31 1.04525 0.522626 0.852562i $$-0.324952\pi$$
0.522626 + 0.852562i $$0.324952\pi$$
$$920$$ 0 0
$$921$$ −7.86464e30 −0.590137
$$922$$ 0 0
$$923$$ −1.50485e31 −1.10376
$$924$$ 0 0
$$925$$ −5.11965e30 −0.367073
$$926$$ 0 0
$$927$$ 6.07525e30 0.425821
$$928$$ 0 0
$$929$$ 6.88409e30 0.471717 0.235858 0.971787i $$-0.424210\pi$$
0.235858 + 0.971787i $$0.424210\pi$$
$$930$$ 0 0
$$931$$ 6.77338e30 0.453768
$$932$$ 0 0
$$933$$ −7.83248e30 −0.513029
$$934$$ 0 0
$$935$$ −2.60536e30 −0.166858
$$936$$ 0 0
$$937$$ −5.51478e30 −0.345352 −0.172676 0.984979i $$-0.555241\pi$$
−0.172676 + 0.984979i $$0.555241\pi$$
$$938$$ 0 0
$$939$$ −1.10274e31 −0.675282
$$940$$ 0 0
$$941$$ −2.31900e31 −1.38870 −0.694351 0.719636i $$-0.744309\pi$$
−0.694351 + 0.719636i $$0.744309\pi$$
$$942$$ 0 0
$$943$$ −1.69853e30 −0.0994720
$$944$$ 0 0
$$945$$ −2.61974e30 −0.150046
$$946$$ 0 0
$$947$$ −2.20661e31 −1.23609 −0.618047 0.786141i $$-0.712076\pi$$
−0.618047 + 0.786141i $$0.712076\pi$$
$$948$$ 0 0
$$949$$ −7.92723e30 −0.434336
$$950$$ 0 0
$$951$$ −1.17604e31 −0.630269
$$952$$ 0 0
$$953$$ −2.49350e31 −1.30718 −0.653588 0.756850i $$-0.726737\pi$$
−0.653588 + 0.756850i $$0.726737\pi$$
$$954$$ 0 0
$$955$$ 1.10993e31 0.569196
$$956$$ 0 0
$$957$$ −1.54748e30 −0.0776336
$$958$$ 0 0
$$959$$ 3.18821e31 1.56477
$$960$$ 0 0
$$961$$ 4.33050e31 2.07942
$$962$$ 0 0
$$963$$ −1.02123e31 −0.479787
$$964$$ 0 0
$$965$$ −2.04461e31 −0.939883
$$966$$ 0 0
$$967$$ −4.48307e30 −0.201650 −0.100825 0.994904i $$-0.532148\pi$$
−0.100825 + 0.994904i $$0.532148\pi$$
$$968$$ 0 0
$$969$$ 1.38680e31 0.610398
$$970$$ 0 0
$$971$$ 2.61136e31 1.12477 0.562385 0.826875i $$-0.309884\pi$$
0.562385 + 0.826875i $$0.309884\pi$$
$$972$$ 0 0
$$973$$ −3.01920e30 −0.127264
$$974$$ 0 0
$$975$$ 4.40667e30 0.181787
$$976$$ 0 0
$$977$$ 2.42220e31 0.977949 0.488975 0.872298i $$-0.337371\pi$$
0.488975 + 0.872298i $$0.337371\pi$$
$$978$$ 0 0
$$979$$ −1.57681e31 −0.623107
$$980$$ 0 0
$$981$$ 3.04151e29 0.0117643
$$982$$ 0 0
$$983$$ 2.19828e31 0.832282 0.416141 0.909300i $$-0.363382\pi$$
0.416141 + 0.909300i $$0.363382\pi$$
$$984$$ 0 0
$$985$$ −1.36509e31 −0.505918
$$986$$ 0 0
$$987$$ 1.05584e31 0.383062
$$988$$ 0 0
$$989$$ 1.68007e29 0.00596715
$$990$$ 0 0
$$991$$ −1.58925e31 −0.552609 −0.276304 0.961070i $$-0.589110\pi$$
−0.276304 + 0.961070i $$0.589110\pi$$
$$992$$ 0 0
$$993$$ 2.13690e31 0.727475
$$994$$ 0 0
$$995$$ 2.89532e31 0.965061
$$996$$ 0 0
$$997$$ −3.74264e31 −1.22146 −0.610729 0.791840i $$-0.709123\pi$$
−0.610729 + 0.791840i $$0.709123\pi$$
$$998$$ 0 0
$$999$$ 4.25372e30 0.135935
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 24.22.a.d.1.2 3
3.2 odd 2 72.22.a.c.1.2 3
4.3 odd 2 48.22.a.j.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.d.1.2 3 1.1 even 1 trivial
48.22.a.j.1.2 3 4.3 odd 2
72.22.a.c.1.2 3 3.2 odd 2