# Properties

 Label 48.28.a.g.1.2 Level $48$ Weight $28$ Character 48.1 Self dual yes Analytic conductor $221.691$ 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] = [48,28,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 28, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 48.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: $$221.690675922$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3386644380$$ x^2 - x - 3386644380 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-58194.4$$ of defining polynomial Character $$\chi$$ $$=$$ 48.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.59432e6 q^{3} +4.36927e9 q^{5} -3.11219e11 q^{7} +2.54187e12 q^{9} +O(q^{10})$$ $$q+1.59432e6 q^{3} +4.36927e9 q^{5} -3.11219e11 q^{7} +2.54187e12 q^{9} +6.06174e13 q^{11} +6.46570e14 q^{13} +6.96603e15 q^{15} +2.78493e16 q^{17} +3.23609e17 q^{19} -4.96184e17 q^{21} -2.77958e18 q^{23} +1.16400e19 q^{25} +4.05256e18 q^{27} -4.40804e18 q^{29} -1.35313e20 q^{31} +9.66437e19 q^{33} -1.35980e21 q^{35} +2.42430e21 q^{37} +1.03084e21 q^{39} -1.00752e22 q^{41} -6.87508e21 q^{43} +1.11061e22 q^{45} +7.32039e22 q^{47} +3.11452e22 q^{49} +4.44008e22 q^{51} +2.25305e23 q^{53} +2.64854e23 q^{55} +5.15938e23 q^{57} +6.81895e22 q^{59} +2.06905e24 q^{61} -7.91078e23 q^{63} +2.82504e24 q^{65} -5.18244e24 q^{67} -4.43155e24 q^{69} -9.12560e24 q^{71} -6.99708e24 q^{73} +1.85579e25 q^{75} -1.88653e25 q^{77} -3.46799e25 q^{79} +6.46108e24 q^{81} +5.24073e25 q^{83} +1.21681e26 q^{85} -7.02784e24 q^{87} +2.05205e26 q^{89} -2.01225e26 q^{91} -2.15733e26 q^{93} +1.41394e27 q^{95} +3.43252e26 q^{97} +1.54081e26 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3188646 q^{3} + 291441036 q^{5} - 121646295328 q^{7} + 5083731656658 q^{9}+O(q^{10})$$ 2 * q + 3188646 * q^3 + 291441036 * q^5 - 121646295328 * q^7 + 5083731656658 * q^9 $$2 q + 3188646 q^{3} + 291441036 q^{5} - 121646295328 q^{7} + 5083731656658 q^{9} + 231807361766376 q^{11} - 11\!\cdots\!64 q^{13}+ \cdots + 58\!\cdots\!04 q^{99}+O(q^{100})$$ 2 * q + 3188646 * q^3 + 291441036 * q^5 - 121646295328 * q^7 + 5083731656658 * q^9 + 231807361766376 * q^11 - 1155759271611764 * q^13 + 464651146838628 * q^15 - 46125175777027452 * q^17 + 268519854753239000 * q^19 - 193943486506222944 * q^21 - 1299561926005929936 * q^23 + 20818087556919436046 * q^25 + 8105110306037952534 * q^27 + 60178362995785587900 * q^29 - 203575085639304966544 * q^31 + 369575808433453883448 * q^33 - 2132849735738606710464 * q^35 + 3022337597514982851388 * q^37 - 1842653589193882415772 * q^39 - 10870375798082518905516 * q^41 + 1299568293415606314344 * q^43 + 740804010381201908844 * q^45 + 99372996476917699213632 * q^47 + 1370785920275118717906 * q^49 - 73538428620357738354996 * q^51 - 35770850049954566775924 * q^53 - 433229958906724722129552 * q^55 + 428107380389748262197000 * q^57 - 299672958231389705557080 * q^59 + 2165599091776519863171724 * q^61 - 309208561237060882746912 * q^63 + 10174637395564415468537928 * q^65 - 13568283837486181949525608 * q^67 - 2071921468555552233353328 * q^69 + 4319907418871433623040336 * q^71 - 20032895180752990538485964 * q^73 + 33190755808010466035166858 * q^75 + 13587711578810985760484736 * q^77 - 5633936436869082596599600 * q^79 + 12922163778453346597864482 * q^81 + 46762236922353070723154904 * q^83 + 423336865613342459682869784 * q^85 + 95943748226529865857491700 * q^87 - 191930414076503127699406380 * q^89 - 542898500680491813749530304 * q^91 - 324564441261713612175329712 * q^93 + 1638584179808499075287277840 * q^95 + 544725669490755670759486468 * q^97 + 589223211629049495820465704 * 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
$$3$$ 1.59432e6 0.577350
$$4$$ 0 0
$$5$$ 4.36927e9 1.60072 0.800358 0.599523i $$-0.204643\pi$$
0.800358 + 0.599523i $$0.204643\pi$$
$$6$$ 0 0
$$7$$ −3.11219e11 −1.21407 −0.607034 0.794676i $$-0.707641\pi$$
−0.607034 + 0.794676i $$0.707641\pi$$
$$8$$ 0 0
$$9$$ 2.54187e12 0.333333
$$10$$ 0 0
$$11$$ 6.06174e13 0.529415 0.264707 0.964329i $$-0.414725\pi$$
0.264707 + 0.964329i $$0.414725\pi$$
$$12$$ 0 0
$$13$$ 6.46570e14 0.592080 0.296040 0.955176i $$-0.404334\pi$$
0.296040 + 0.955176i $$0.404334\pi$$
$$14$$ 0 0
$$15$$ 6.96603e15 0.924173
$$16$$ 0 0
$$17$$ 2.78493e16 0.681953 0.340976 0.940072i $$-0.389242\pi$$
0.340976 + 0.940072i $$0.389242\pi$$
$$18$$ 0 0
$$19$$ 3.23609e17 1.76542 0.882709 0.469920i $$-0.155717\pi$$
0.882709 + 0.469920i $$0.155717\pi$$
$$20$$ 0 0
$$21$$ −4.96184e17 −0.700943
$$22$$ 0 0
$$23$$ −2.77958e18 −1.14988 −0.574941 0.818195i $$-0.694975\pi$$
−0.574941 + 0.818195i $$0.694975\pi$$
$$24$$ 0 0
$$25$$ 1.16400e19 1.56229
$$26$$ 0 0
$$27$$ 4.05256e18 0.192450
$$28$$ 0 0
$$29$$ −4.40804e18 −0.0797761 −0.0398880 0.999204i $$-0.512700\pi$$
−0.0398880 + 0.999204i $$0.512700\pi$$
$$30$$ 0 0
$$31$$ −1.35313e20 −0.995309 −0.497654 0.867375i $$-0.665805\pi$$
−0.497654 + 0.867375i $$0.665805\pi$$
$$32$$ 0 0
$$33$$ 9.66437e19 0.305658
$$34$$ 0 0
$$35$$ −1.35980e21 −1.94338
$$36$$ 0 0
$$37$$ 2.42430e21 1.63630 0.818150 0.575005i $$-0.195000\pi$$
0.818150 + 0.575005i $$0.195000\pi$$
$$38$$ 0 0
$$39$$ 1.03084e21 0.341838
$$40$$ 0 0
$$41$$ −1.00752e22 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$42$$ 0 0
$$43$$ −6.87508e21 −0.610174 −0.305087 0.952324i $$-0.598686\pi$$
−0.305087 + 0.952324i $$0.598686\pi$$
$$44$$ 0 0
$$45$$ 1.11061e22 0.533572
$$46$$ 0 0
$$47$$ 7.32039e22 1.95530 0.977650 0.210241i $$-0.0674249\pi$$
0.977650 + 0.210241i $$0.0674249\pi$$
$$48$$ 0 0
$$49$$ 3.11452e22 0.473962
$$50$$ 0 0
$$51$$ 4.44008e22 0.393726
$$52$$ 0 0
$$53$$ 2.25305e23 1.18863 0.594316 0.804232i $$-0.297423\pi$$
0.594316 + 0.804232i $$0.297423\pi$$
$$54$$ 0 0
$$55$$ 2.64854e23 0.847442
$$56$$ 0 0
$$57$$ 5.15938e23 1.01926
$$58$$ 0 0
$$59$$ 6.81895e22 0.0845698 0.0422849 0.999106i $$-0.486536\pi$$
0.0422849 + 0.999106i $$0.486536\pi$$
$$60$$ 0 0
$$61$$ 2.06905e24 1.63613 0.818066 0.575124i $$-0.195046\pi$$
0.818066 + 0.575124i $$0.195046\pi$$
$$62$$ 0 0
$$63$$ −7.91078e23 −0.404689
$$64$$ 0 0
$$65$$ 2.82504e24 0.947752
$$66$$ 0 0
$$67$$ −5.18244e24 −1.15485 −0.577424 0.816444i $$-0.695942\pi$$
−0.577424 + 0.816444i $$0.695942\pi$$
$$68$$ 0 0
$$69$$ −4.43155e24 −0.663885
$$70$$ 0 0
$$71$$ −9.12560e24 −0.929552 −0.464776 0.885428i $$-0.653865\pi$$
−0.464776 + 0.885428i $$0.653865\pi$$
$$72$$ 0 0
$$73$$ −6.99708e24 −0.489845 −0.244922 0.969543i $$-0.578762\pi$$
−0.244922 + 0.969543i $$0.578762\pi$$
$$74$$ 0 0
$$75$$ 1.85579e25 0.901988
$$76$$ 0 0
$$77$$ −1.88653e25 −0.642746
$$78$$ 0 0
$$79$$ −3.46799e25 −0.835818 −0.417909 0.908489i $$-0.637237\pi$$
−0.417909 + 0.908489i $$0.637237\pi$$
$$80$$ 0 0
$$81$$ 6.46108e24 0.111111
$$82$$ 0 0
$$83$$ 5.24073e25 0.648392 0.324196 0.945990i $$-0.394906\pi$$
0.324196 + 0.945990i $$0.394906\pi$$
$$84$$ 0 0
$$85$$ 1.21681e26 1.09161
$$86$$ 0 0
$$87$$ −7.02784e24 −0.0460587
$$88$$ 0 0
$$89$$ 2.05205e26 0.989516 0.494758 0.869031i $$-0.335257\pi$$
0.494758 + 0.869031i $$0.335257\pi$$
$$90$$ 0 0
$$91$$ −2.01225e26 −0.718826
$$92$$ 0 0
$$93$$ −2.15733e26 −0.574642
$$94$$ 0 0
$$95$$ 1.41394e27 2.82593
$$96$$ 0 0
$$97$$ 3.43252e26 0.517839 0.258919 0.965899i $$-0.416634\pi$$
0.258919 + 0.965899i $$0.416634\pi$$
$$98$$ 0 0
$$99$$ 1.54081e26 0.176472
$$100$$ 0 0
$$101$$ −5.43435e26 −0.475126 −0.237563 0.971372i $$-0.576349\pi$$
−0.237563 + 0.971372i $$0.576349\pi$$
$$102$$ 0 0
$$103$$ −5.07992e25 −0.0340843 −0.0170421 0.999855i $$-0.505425\pi$$
−0.0170421 + 0.999855i $$0.505425\pi$$
$$104$$ 0 0
$$105$$ −2.16796e27 −1.12201
$$106$$ 0 0
$$107$$ 1.31347e27 0.526913 0.263456 0.964671i $$-0.415138\pi$$
0.263456 + 0.964671i $$0.415138\pi$$
$$108$$ 0 0
$$109$$ 3.20537e27 1.00143 0.500715 0.865612i $$-0.333071\pi$$
0.500715 + 0.865612i $$0.333071\pi$$
$$110$$ 0 0
$$111$$ 3.86512e27 0.944718
$$112$$ 0 0
$$113$$ 1.64220e27 0.315403 0.157702 0.987487i $$-0.449592\pi$$
0.157702 + 0.987487i $$0.449592\pi$$
$$114$$ 0 0
$$115$$ −1.21447e28 −1.84063
$$116$$ 0 0
$$117$$ 1.64349e27 0.197360
$$118$$ 0 0
$$119$$ −8.66725e27 −0.827937
$$120$$ 0 0
$$121$$ −9.43553e27 −0.719720
$$122$$ 0 0
$$123$$ −1.60632e28 −0.982003
$$124$$ 0 0
$$125$$ 1.83045e28 0.900065
$$126$$ 0 0
$$127$$ −1.82085e28 −0.722644 −0.361322 0.932441i $$-0.617675\pi$$
−0.361322 + 0.932441i $$0.617675\pi$$
$$128$$ 0 0
$$129$$ −1.09611e28 −0.352284
$$130$$ 0 0
$$131$$ 4.91416e28 1.28318 0.641589 0.767049i $$-0.278275\pi$$
0.641589 + 0.767049i $$0.278275\pi$$
$$132$$ 0 0
$$133$$ −1.00714e29 −2.14334
$$134$$ 0 0
$$135$$ 1.77067e28 0.308058
$$136$$ 0 0
$$137$$ 5.37814e28 0.767193 0.383597 0.923501i $$-0.374685\pi$$
0.383597 + 0.923501i $$0.374685\pi$$
$$138$$ 0 0
$$139$$ −4.60592e28 −0.540277 −0.270138 0.962821i $$-0.587069\pi$$
−0.270138 + 0.962821i $$0.587069\pi$$
$$140$$ 0 0
$$141$$ 1.16711e29 1.12889
$$142$$ 0 0
$$143$$ 3.91934e28 0.313456
$$144$$ 0 0
$$145$$ −1.92599e28 −0.127699
$$146$$ 0 0
$$147$$ 4.96555e28 0.273642
$$148$$ 0 0
$$149$$ 6.39934e28 0.293846 0.146923 0.989148i $$-0.453063\pi$$
0.146923 + 0.989148i $$0.453063\pi$$
$$150$$ 0 0
$$151$$ 4.57912e29 1.75628 0.878139 0.478406i $$-0.158785\pi$$
0.878139 + 0.478406i $$0.158785\pi$$
$$152$$ 0 0
$$153$$ 7.07893e28 0.227318
$$154$$ 0 0
$$155$$ −5.91221e29 −1.59321
$$156$$ 0 0
$$157$$ −7.23684e29 −1.64023 −0.820113 0.572202i $$-0.806089\pi$$
−0.820113 + 0.572202i $$0.806089\pi$$
$$158$$ 0 0
$$159$$ 3.59209e29 0.686256
$$160$$ 0 0
$$161$$ 8.65059e29 1.39604
$$162$$ 0 0
$$163$$ −4.01419e29 −0.548360 −0.274180 0.961678i $$-0.588406\pi$$
−0.274180 + 0.961678i $$0.588406\pi$$
$$164$$ 0 0
$$165$$ 4.22263e29 0.489271
$$166$$ 0 0
$$167$$ 1.03017e30 1.01447 0.507233 0.861809i $$-0.330668\pi$$
0.507233 + 0.861809i $$0.330668\pi$$
$$168$$ 0 0
$$169$$ −7.74480e29 −0.649441
$$170$$ 0 0
$$171$$ 8.22572e29 0.588473
$$172$$ 0 0
$$173$$ −2.68290e30 −1.64052 −0.820261 0.571989i $$-0.806172\pi$$
−0.820261 + 0.571989i $$0.806172\pi$$
$$174$$ 0 0
$$175$$ −3.62258e30 −1.89673
$$176$$ 0 0
$$177$$ 1.08716e29 0.0488264
$$178$$ 0 0
$$179$$ −2.21883e30 −0.856264 −0.428132 0.903716i $$-0.640828\pi$$
−0.428132 + 0.903716i $$0.640828\pi$$
$$180$$ 0 0
$$181$$ 4.61780e30 1.53382 0.766911 0.641754i $$-0.221793\pi$$
0.766911 + 0.641754i $$0.221793\pi$$
$$182$$ 0 0
$$183$$ 3.29874e30 0.944622
$$184$$ 0 0
$$185$$ 1.05924e31 2.61925
$$186$$ 0 0
$$187$$ 1.68815e30 0.361036
$$188$$ 0 0
$$189$$ −1.26123e30 −0.233648
$$190$$ 0 0
$$191$$ 4.74656e30 0.762831 0.381415 0.924404i $$-0.375437\pi$$
0.381415 + 0.924404i $$0.375437\pi$$
$$192$$ 0 0
$$193$$ 2.52769e30 0.352940 0.176470 0.984306i $$-0.443532\pi$$
0.176470 + 0.984306i $$0.443532\pi$$
$$194$$ 0 0
$$195$$ 4.50403e30 0.547185
$$196$$ 0 0
$$197$$ −2.31410e30 −0.244956 −0.122478 0.992471i $$-0.539084\pi$$
−0.122478 + 0.992471i $$0.539084\pi$$
$$198$$ 0 0
$$199$$ 3.03480e30 0.280294 0.140147 0.990131i $$-0.455242\pi$$
0.140147 + 0.990131i $$0.455242\pi$$
$$200$$ 0 0
$$201$$ −8.26249e30 −0.666752
$$202$$ 0 0
$$203$$ 1.37187e30 0.0968536
$$204$$ 0 0
$$205$$ −4.40215e31 −2.72262
$$206$$ 0 0
$$207$$ −7.06532e30 −0.383294
$$208$$ 0 0
$$209$$ 1.96164e31 0.934638
$$210$$ 0 0
$$211$$ 1.95873e31 0.820655 0.410328 0.911938i $$-0.365414\pi$$
0.410328 + 0.911938i $$0.365414\pi$$
$$212$$ 0 0
$$213$$ −1.45492e31 −0.536677
$$214$$ 0 0
$$215$$ −3.00391e31 −0.976715
$$216$$ 0 0
$$217$$ 4.21122e31 1.20837
$$218$$ 0 0
$$219$$ −1.11556e31 −0.282812
$$220$$ 0 0
$$221$$ 1.80065e31 0.403771
$$222$$ 0 0
$$223$$ 2.74869e31 0.545771 0.272886 0.962047i $$-0.412022\pi$$
0.272886 + 0.962047i $$0.412022\pi$$
$$224$$ 0 0
$$225$$ 2.95872e31 0.520763
$$226$$ 0 0
$$227$$ 1.10085e30 0.0171940 0.00859701 0.999963i $$-0.497263\pi$$
0.00859701 + 0.999963i $$0.497263\pi$$
$$228$$ 0 0
$$229$$ 7.85209e31 1.08945 0.544724 0.838615i $$-0.316634\pi$$
0.544724 + 0.838615i $$0.316634\pi$$
$$230$$ 0 0
$$231$$ −3.00774e31 −0.371089
$$232$$ 0 0
$$233$$ 1.08140e32 1.18763 0.593817 0.804600i $$-0.297620\pi$$
0.593817 + 0.804600i $$0.297620\pi$$
$$234$$ 0 0
$$235$$ 3.19848e32 3.12988
$$236$$ 0 0
$$237$$ −5.52909e31 −0.482560
$$238$$ 0 0
$$239$$ 3.85813e31 0.300612 0.150306 0.988640i $$-0.451974\pi$$
0.150306 + 0.988640i $$0.451974\pi$$
$$240$$ 0 0
$$241$$ 2.43286e32 1.69389 0.846947 0.531677i $$-0.178438\pi$$
0.846947 + 0.531677i $$0.178438\pi$$
$$242$$ 0 0
$$243$$ 1.03011e31 0.0641500
$$244$$ 0 0
$$245$$ 1.36082e32 0.758678
$$246$$ 0 0
$$247$$ 2.09236e32 1.04527
$$248$$ 0 0
$$249$$ 8.35542e31 0.374349
$$250$$ 0 0
$$251$$ −1.35106e31 −0.0543350 −0.0271675 0.999631i $$-0.508649\pi$$
−0.0271675 + 0.999631i $$0.508649\pi$$
$$252$$ 0 0
$$253$$ −1.68491e32 −0.608765
$$254$$ 0 0
$$255$$ 1.93999e32 0.630243
$$256$$ 0 0
$$257$$ −2.50576e32 −0.732559 −0.366280 0.930505i $$-0.619369\pi$$
−0.366280 + 0.930505i $$0.619369\pi$$
$$258$$ 0 0
$$259$$ −7.54489e32 −1.98658
$$260$$ 0 0
$$261$$ −1.12046e31 −0.0265920
$$262$$ 0 0
$$263$$ 4.15757e32 0.890096 0.445048 0.895507i $$-0.353187\pi$$
0.445048 + 0.895507i $$0.353187\pi$$
$$264$$ 0 0
$$265$$ 9.84420e32 1.90266
$$266$$ 0 0
$$267$$ 3.27163e32 0.571297
$$268$$ 0 0
$$269$$ −1.81575e32 −0.286682 −0.143341 0.989673i $$-0.545785\pi$$
−0.143341 + 0.989673i $$0.545785\pi$$
$$270$$ 0 0
$$271$$ 4.40915e32 0.629897 0.314949 0.949109i $$-0.398013\pi$$
0.314949 + 0.949109i $$0.398013\pi$$
$$272$$ 0 0
$$273$$ −3.20818e32 −0.415014
$$274$$ 0 0
$$275$$ 7.05584e32 0.827099
$$276$$ 0 0
$$277$$ 4.47660e32 0.475852 0.237926 0.971283i $$-0.423532\pi$$
0.237926 + 0.971283i $$0.423532\pi$$
$$278$$ 0 0
$$279$$ −3.43948e32 −0.331770
$$280$$ 0 0
$$281$$ −8.16751e32 −0.715409 −0.357704 0.933835i $$-0.616440\pi$$
−0.357704 + 0.933835i $$0.616440\pi$$
$$282$$ 0 0
$$283$$ 3.19801e32 0.254544 0.127272 0.991868i $$-0.459378\pi$$
0.127272 + 0.991868i $$0.459378\pi$$
$$284$$ 0 0
$$285$$ 2.25427e33 1.63155
$$286$$ 0 0
$$287$$ 3.13561e33 2.06498
$$288$$ 0 0
$$289$$ −8.92126e32 −0.534940
$$290$$ 0 0
$$291$$ 5.47255e32 0.298974
$$292$$ 0 0
$$293$$ 2.58899e33 1.28949 0.644744 0.764399i $$-0.276964\pi$$
0.644744 + 0.764399i $$0.276964\pi$$
$$294$$ 0 0
$$295$$ 2.97939e32 0.135372
$$296$$ 0 0
$$297$$ 2.45655e32 0.101886
$$298$$ 0 0
$$299$$ −1.79719e33 −0.680823
$$300$$ 0 0
$$301$$ 2.13966e33 0.740793
$$302$$ 0 0
$$303$$ −8.66410e32 −0.274314
$$304$$ 0 0
$$305$$ 9.04025e33 2.61898
$$306$$ 0 0
$$307$$ −1.64496e33 −0.436303 −0.218152 0.975915i $$-0.570003\pi$$
−0.218152 + 0.975915i $$0.570003\pi$$
$$308$$ 0 0
$$309$$ −8.09903e31 −0.0196786
$$310$$ 0 0
$$311$$ 6.49387e33 1.44623 0.723117 0.690725i $$-0.242709\pi$$
0.723117 + 0.690725i $$0.242709\pi$$
$$312$$ 0 0
$$313$$ −8.69247e33 −1.77539 −0.887697 0.460428i $$-0.847696\pi$$
−0.887697 + 0.460428i $$0.847696\pi$$
$$314$$ 0 0
$$315$$ −3.45644e33 −0.647793
$$316$$ 0 0
$$317$$ 1.93111e33 0.332282 0.166141 0.986102i $$-0.446869\pi$$
0.166141 + 0.986102i $$0.446869\pi$$
$$318$$ 0 0
$$319$$ −2.67204e32 −0.0422346
$$320$$ 0 0
$$321$$ 2.09409e33 0.304213
$$322$$ 0 0
$$323$$ 9.01231e33 1.20393
$$324$$ 0 0
$$325$$ 7.52605e33 0.925000
$$326$$ 0 0
$$327$$ 5.11039e33 0.578176
$$328$$ 0 0
$$329$$ −2.27825e34 −2.37387
$$330$$ 0 0
$$331$$ 8.64145e33 0.829676 0.414838 0.909895i $$-0.363838\pi$$
0.414838 + 0.909895i $$0.363838\pi$$
$$332$$ 0 0
$$333$$ 6.16224e33 0.545433
$$334$$ 0 0
$$335$$ −2.26435e34 −1.84858
$$336$$ 0 0
$$337$$ −2.03843e33 −0.153565 −0.0767826 0.997048i $$-0.524465\pi$$
−0.0767826 + 0.997048i $$0.524465\pi$$
$$338$$ 0 0
$$339$$ 2.61819e33 0.182098
$$340$$ 0 0
$$341$$ −8.20234e33 −0.526931
$$342$$ 0 0
$$343$$ 1.07580e34 0.638646
$$344$$ 0 0
$$345$$ −1.93626e34 −1.06269
$$346$$ 0 0
$$347$$ −8.12227e33 −0.412315 −0.206158 0.978519i $$-0.566096\pi$$
−0.206158 + 0.978519i $$0.566096\pi$$
$$348$$ 0 0
$$349$$ 4.94258e33 0.232172 0.116086 0.993239i $$-0.462965\pi$$
0.116086 + 0.993239i $$0.462965\pi$$
$$350$$ 0 0
$$351$$ 2.62026e33 0.113946
$$352$$ 0 0
$$353$$ 1.97509e34 0.795478 0.397739 0.917499i $$-0.369795\pi$$
0.397739 + 0.917499i $$0.369795\pi$$
$$354$$ 0 0
$$355$$ −3.98722e34 −1.48795
$$356$$ 0 0
$$357$$ −1.38184e34 −0.478010
$$358$$ 0 0
$$359$$ −1.80645e34 −0.579495 −0.289748 0.957103i $$-0.593571\pi$$
−0.289748 + 0.957103i $$0.593571\pi$$
$$360$$ 0 0
$$361$$ 7.11225e34 2.11670
$$362$$ 0 0
$$363$$ −1.50433e34 −0.415531
$$364$$ 0 0
$$365$$ −3.05722e34 −0.784102
$$366$$ 0 0
$$367$$ 2.27625e34 0.542286 0.271143 0.962539i $$-0.412598\pi$$
0.271143 + 0.962539i $$0.412598\pi$$
$$368$$ 0 0
$$369$$ −2.56099e34 −0.566960
$$370$$ 0 0
$$371$$ −7.01194e34 −1.44308
$$372$$ 0 0
$$373$$ −6.70992e34 −1.28425 −0.642123 0.766601i $$-0.721946\pi$$
−0.642123 + 0.766601i $$0.721946\pi$$
$$374$$ 0 0
$$375$$ 2.91833e34 0.519652
$$376$$ 0 0
$$377$$ −2.85011e33 −0.0472338
$$378$$ 0 0
$$379$$ 3.50560e34 0.540921 0.270460 0.962731i $$-0.412824\pi$$
0.270460 + 0.962731i $$0.412824\pi$$
$$380$$ 0 0
$$381$$ −2.90303e34 −0.417219
$$382$$ 0 0
$$383$$ −3.70361e33 −0.0495955 −0.0247978 0.999692i $$-0.507894\pi$$
−0.0247978 + 0.999692i $$0.507894\pi$$
$$384$$ 0 0
$$385$$ −8.24277e34 −1.02885
$$386$$ 0 0
$$387$$ −1.74755e34 −0.203391
$$388$$ 0 0
$$389$$ −7.02105e34 −0.762224 −0.381112 0.924529i $$-0.624459\pi$$
−0.381112 + 0.924529i $$0.624459\pi$$
$$390$$ 0 0
$$391$$ −7.74095e34 −0.784166
$$392$$ 0 0
$$393$$ 7.83476e34 0.740843
$$394$$ 0 0
$$395$$ −1.51526e35 −1.33791
$$396$$ 0 0
$$397$$ −1.30976e35 −1.08024 −0.540118 0.841589i $$-0.681620\pi$$
−0.540118 + 0.841589i $$0.681620\pi$$
$$398$$ 0 0
$$399$$ −1.60570e35 −1.23746
$$400$$ 0 0
$$401$$ 1.32379e35 0.953612 0.476806 0.879009i $$-0.341794\pi$$
0.476806 + 0.879009i $$0.341794\pi$$
$$402$$ 0 0
$$403$$ −8.74896e34 −0.589303
$$404$$ 0 0
$$405$$ 2.82302e34 0.177857
$$406$$ 0 0
$$407$$ 1.46955e35 0.866281
$$408$$ 0 0
$$409$$ −2.54536e35 −1.40438 −0.702190 0.711989i $$-0.747794\pi$$
−0.702190 + 0.711989i $$0.747794\pi$$
$$410$$ 0 0
$$411$$ 8.57450e34 0.442939
$$412$$ 0 0
$$413$$ −2.12219e34 −0.102674
$$414$$ 0 0
$$415$$ 2.28982e35 1.03789
$$416$$ 0 0
$$417$$ −7.34332e34 −0.311929
$$418$$ 0 0
$$419$$ 2.64873e35 1.05475 0.527373 0.849634i $$-0.323177\pi$$
0.527373 + 0.849634i $$0.323177\pi$$
$$420$$ 0 0
$$421$$ 8.67841e34 0.324065 0.162032 0.986785i $$-0.448195\pi$$
0.162032 + 0.986785i $$0.448195\pi$$
$$422$$ 0 0
$$423$$ 1.86075e35 0.651766
$$424$$ 0 0
$$425$$ 3.24165e35 1.06541
$$426$$ 0 0
$$427$$ −6.43929e35 −1.98638
$$428$$ 0 0
$$429$$ 6.24869e34 0.180974
$$430$$ 0 0
$$431$$ 8.57490e34 0.233231 0.116615 0.993177i $$-0.462796\pi$$
0.116615 + 0.993177i $$0.462796\pi$$
$$432$$ 0 0
$$433$$ −1.22839e35 −0.313870 −0.156935 0.987609i $$-0.550161\pi$$
−0.156935 + 0.987609i $$0.550161\pi$$
$$434$$ 0 0
$$435$$ −3.07065e34 −0.0737269
$$436$$ 0 0
$$437$$ −8.99499e35 −2.03002
$$438$$ 0 0
$$439$$ 2.95742e34 0.0627542 0.0313771 0.999508i $$-0.490011\pi$$
0.0313771 + 0.999508i $$0.490011\pi$$
$$440$$ 0 0
$$441$$ 7.91669e34 0.157987
$$442$$ 0 0
$$443$$ −2.15913e35 −0.405348 −0.202674 0.979246i $$-0.564963\pi$$
−0.202674 + 0.979246i $$0.564963\pi$$
$$444$$ 0 0
$$445$$ 8.96596e35 1.58393
$$446$$ 0 0
$$447$$ 1.02026e35 0.169652
$$448$$ 0 0
$$449$$ 2.22802e35 0.348813 0.174407 0.984674i $$-0.444199\pi$$
0.174407 + 0.984674i $$0.444199\pi$$
$$450$$ 0 0
$$451$$ −6.10735e35 −0.900470
$$452$$ 0 0
$$453$$ 7.30060e35 1.01399
$$454$$ 0 0
$$455$$ −8.79208e35 −1.15064
$$456$$ 0 0
$$457$$ −5.04203e35 −0.621923 −0.310961 0.950423i $$-0.600651\pi$$
−0.310961 + 0.950423i $$0.600651\pi$$
$$458$$ 0 0
$$459$$ 1.12861e35 0.131242
$$460$$ 0 0
$$461$$ −1.02767e36 −1.12691 −0.563456 0.826146i $$-0.690529\pi$$
−0.563456 + 0.826146i $$0.690529\pi$$
$$462$$ 0 0
$$463$$ −1.28910e36 −1.33335 −0.666673 0.745350i $$-0.732282\pi$$
−0.666673 + 0.745350i $$0.732282\pi$$
$$464$$ 0 0
$$465$$ −9.42597e35 −0.919838
$$466$$ 0 0
$$467$$ 1.89841e36 1.74829 0.874143 0.485668i $$-0.161424\pi$$
0.874143 + 0.485668i $$0.161424\pi$$
$$468$$ 0 0
$$469$$ 1.61288e36 1.40206
$$470$$ 0 0
$$471$$ −1.15379e36 −0.946985
$$472$$ 0 0
$$473$$ −4.16750e35 −0.323035
$$474$$ 0 0
$$475$$ 3.76680e36 2.75809
$$476$$ 0 0
$$477$$ 5.72696e35 0.396210
$$478$$ 0 0
$$479$$ 1.33288e36 0.871491 0.435746 0.900070i $$-0.356485\pi$$
0.435746 + 0.900070i $$0.356485\pi$$
$$480$$ 0 0
$$481$$ 1.56748e36 0.968821
$$482$$ 0 0
$$483$$ 1.37918e36 0.806002
$$484$$ 0 0
$$485$$ 1.49976e36 0.828913
$$486$$ 0 0
$$487$$ 3.24682e36 1.69753 0.848764 0.528772i $$-0.177347\pi$$
0.848764 + 0.528772i $$0.177347\pi$$
$$488$$ 0 0
$$489$$ −6.39992e35 −0.316596
$$490$$ 0 0
$$491$$ −5.80339e35 −0.271695 −0.135847 0.990730i $$-0.543376\pi$$
−0.135847 + 0.990730i $$0.543376\pi$$
$$492$$ 0 0
$$493$$ −1.22761e35 −0.0544035
$$494$$ 0 0
$$495$$ 6.73223e35 0.282481
$$496$$ 0 0
$$497$$ 2.84006e36 1.12854
$$498$$ 0 0
$$499$$ 2.77354e36 1.04394 0.521972 0.852963i $$-0.325197\pi$$
0.521972 + 0.852963i $$0.325197\pi$$
$$500$$ 0 0
$$501$$ 1.64243e36 0.585702
$$502$$ 0 0
$$503$$ −1.25022e36 −0.422491 −0.211245 0.977433i $$-0.567752\pi$$
−0.211245 + 0.977433i $$0.567752\pi$$
$$504$$ 0 0
$$505$$ −2.37441e36 −0.760541
$$506$$ 0 0
$$507$$ −1.23477e36 −0.374955
$$508$$ 0 0
$$509$$ 6.18070e36 1.77970 0.889852 0.456250i $$-0.150808\pi$$
0.889852 + 0.456250i $$0.150808\pi$$
$$510$$ 0 0
$$511$$ 2.17763e36 0.594705
$$512$$ 0 0
$$513$$ 1.31145e36 0.339755
$$514$$ 0 0
$$515$$ −2.21955e35 −0.0545592
$$516$$ 0 0
$$517$$ 4.43743e36 1.03516
$$518$$ 0 0
$$519$$ −4.27741e36 −0.947156
$$520$$ 0 0
$$521$$ −4.44502e36 −0.934468 −0.467234 0.884134i $$-0.654749\pi$$
−0.467234 + 0.884134i $$0.654749\pi$$
$$522$$ 0 0
$$523$$ −7.64607e36 −1.52639 −0.763195 0.646169i $$-0.776370\pi$$
−0.763195 + 0.646169i $$0.776370\pi$$
$$524$$ 0 0
$$525$$ −5.77557e36 −1.09508
$$526$$ 0 0
$$527$$ −3.76839e36 −0.678754
$$528$$ 0 0
$$529$$ 1.88286e36 0.322230
$$530$$ 0 0
$$531$$ 1.73329e35 0.0281899
$$532$$ 0 0
$$533$$ −6.51435e36 −1.00706
$$534$$ 0 0
$$535$$ 5.73890e36 0.843437
$$536$$ 0 0
$$537$$ −3.53753e36 −0.494364
$$538$$ 0 0
$$539$$ 1.88794e36 0.250923
$$540$$ 0 0
$$541$$ −1.30472e37 −1.64950 −0.824751 0.565495i $$-0.808685\pi$$
−0.824751 + 0.565495i $$0.808685\pi$$
$$542$$ 0 0
$$543$$ 7.36227e36 0.885552
$$544$$ 0 0
$$545$$ 1.40051e37 1.60300
$$546$$ 0 0
$$547$$ −1.03586e37 −1.12843 −0.564213 0.825629i $$-0.690820\pi$$
−0.564213 + 0.825629i $$0.690820\pi$$
$$548$$ 0 0
$$549$$ 5.25925e36 0.545378
$$550$$ 0 0
$$551$$ −1.42648e36 −0.140838
$$552$$ 0 0
$$553$$ 1.07931e37 1.01474
$$554$$ 0 0
$$555$$ 1.68877e37 1.51222
$$556$$ 0 0
$$557$$ −1.07139e37 −0.913904 −0.456952 0.889491i $$-0.651059\pi$$
−0.456952 + 0.889491i $$0.651059\pi$$
$$558$$ 0 0
$$559$$ −4.44522e36 −0.361272
$$560$$ 0 0
$$561$$ 2.69146e36 0.208444
$$562$$ 0 0
$$563$$ 1.24198e37 0.916749 0.458374 0.888759i $$-0.348432\pi$$
0.458374 + 0.888759i $$0.348432\pi$$
$$564$$ 0 0
$$565$$ 7.17521e36 0.504871
$$566$$ 0 0
$$567$$ −2.01081e36 −0.134896
$$568$$ 0 0
$$569$$ −6.72192e36 −0.430009 −0.215005 0.976613i $$-0.568977\pi$$
−0.215005 + 0.976613i $$0.568977\pi$$
$$570$$ 0 0
$$571$$ −6.75688e36 −0.412248 −0.206124 0.978526i $$-0.566085\pi$$
−0.206124 + 0.978526i $$0.566085\pi$$
$$572$$ 0 0
$$573$$ 7.56754e36 0.440420
$$574$$ 0 0
$$575$$ −3.23542e37 −1.79645
$$576$$ 0 0
$$577$$ −1.83494e37 −0.972184 −0.486092 0.873908i $$-0.661578\pi$$
−0.486092 + 0.873908i $$0.661578\pi$$
$$578$$ 0 0
$$579$$ 4.02996e36 0.203770
$$580$$ 0 0
$$581$$ −1.63102e37 −0.787193
$$582$$ 0 0
$$583$$ 1.36574e37 0.629279
$$584$$ 0 0
$$585$$ 7.18088e36 0.315917
$$586$$ 0 0
$$587$$ −1.38416e37 −0.581528 −0.290764 0.956795i $$-0.593910\pi$$
−0.290764 + 0.956795i $$0.593910\pi$$
$$588$$ 0 0
$$589$$ −4.37887e37 −1.75714
$$590$$ 0 0
$$591$$ −3.68943e36 −0.141426
$$592$$ 0 0
$$593$$ −5.74168e36 −0.210281 −0.105141 0.994457i $$-0.533529\pi$$
−0.105141 + 0.994457i $$0.533529\pi$$
$$594$$ 0 0
$$595$$ −3.78696e37 −1.32529
$$596$$ 0 0
$$597$$ 4.83845e36 0.161828
$$598$$ 0 0
$$599$$ −9.06518e36 −0.289811 −0.144905 0.989446i $$-0.546288\pi$$
−0.144905 + 0.989446i $$0.546288\pi$$
$$600$$ 0 0
$$601$$ 3.75922e36 0.114893 0.0574464 0.998349i $$-0.481704\pi$$
0.0574464 + 0.998349i $$0.481704\pi$$
$$602$$ 0 0
$$603$$ −1.31731e37 −0.384949
$$604$$ 0 0
$$605$$ −4.12264e37 −1.15207
$$606$$ 0 0
$$607$$ 4.26047e37 1.13870 0.569351 0.822094i $$-0.307194\pi$$
0.569351 + 0.822094i $$0.307194\pi$$
$$608$$ 0 0
$$609$$ 2.18720e36 0.0559184
$$610$$ 0 0
$$611$$ 4.73315e37 1.15769
$$612$$ 0 0
$$613$$ 5.38870e37 1.26115 0.630576 0.776127i $$-0.282819\pi$$
0.630576 + 0.776127i $$0.282819\pi$$
$$614$$ 0 0
$$615$$ −7.01844e37 −1.57191
$$616$$ 0 0
$$617$$ −3.49858e37 −0.749967 −0.374983 0.927031i $$-0.622352\pi$$
−0.374983 + 0.927031i $$0.622352\pi$$
$$618$$ 0 0
$$619$$ −3.09916e37 −0.635946 −0.317973 0.948100i $$-0.603002\pi$$
−0.317973 + 0.948100i $$0.603002\pi$$
$$620$$ 0 0
$$621$$ −1.12644e37 −0.221295
$$622$$ 0 0
$$623$$ −6.38638e37 −1.20134
$$624$$ 0 0
$$625$$ −6.74694e36 −0.121542
$$626$$ 0 0
$$627$$ 3.12748e37 0.539614
$$628$$ 0 0
$$629$$ 6.75151e37 1.11588
$$630$$ 0 0
$$631$$ 7.75162e36 0.122743 0.0613714 0.998115i $$-0.480453\pi$$
0.0613714 + 0.998115i $$0.480453\pi$$
$$632$$ 0 0
$$633$$ 3.12284e37 0.473806
$$634$$ 0 0
$$635$$ −7.95580e37 −1.15675
$$636$$ 0 0
$$637$$ 2.01375e37 0.280624
$$638$$ 0 0
$$639$$ −2.31961e37 −0.309851
$$640$$ 0 0
$$641$$ −7.99511e37 −1.02386 −0.511931 0.859027i $$-0.671070\pi$$
−0.511931 + 0.859027i $$0.671070\pi$$
$$642$$ 0 0
$$643$$ −2.47951e37 −0.304451 −0.152226 0.988346i $$-0.548644\pi$$
−0.152226 + 0.988346i $$0.548644\pi$$
$$644$$ 0 0
$$645$$ −4.78920e37 −0.563907
$$646$$ 0 0
$$647$$ 3.52397e37 0.397946 0.198973 0.980005i $$-0.436239\pi$$
0.198973 + 0.980005i $$0.436239\pi$$
$$648$$ 0 0
$$649$$ 4.13347e36 0.0447725
$$650$$ 0 0
$$651$$ 6.71404e37 0.697655
$$652$$ 0 0
$$653$$ 1.58081e37 0.157599 0.0787993 0.996891i $$-0.474891\pi$$
0.0787993 + 0.996891i $$0.474891\pi$$
$$654$$ 0 0
$$655$$ 2.14713e38 2.05400
$$656$$ 0 0
$$657$$ −1.77856e37 −0.163282
$$658$$ 0 0
$$659$$ −2.71928e37 −0.239608 −0.119804 0.992798i $$-0.538227\pi$$
−0.119804 + 0.992798i $$0.538227\pi$$
$$660$$ 0 0
$$661$$ 8.48665e36 0.0717821 0.0358911 0.999356i $$-0.488573\pi$$
0.0358911 + 0.999356i $$0.488573\pi$$
$$662$$ 0 0
$$663$$ 2.87083e37 0.233117
$$664$$ 0 0
$$665$$ −4.40045e38 −3.43087
$$666$$ 0 0
$$667$$ 1.22525e37 0.0917331
$$668$$ 0 0
$$669$$ 4.38230e37 0.315101
$$670$$ 0 0
$$671$$ 1.25421e38 0.866193
$$672$$ 0 0
$$673$$ 2.46452e38 1.63504 0.817522 0.575898i $$-0.195347\pi$$
0.817522 + 0.575898i $$0.195347\pi$$
$$674$$ 0 0
$$675$$ 4.71716e37 0.300663
$$676$$ 0 0
$$677$$ 2.19397e38 1.34364 0.671822 0.740713i $$-0.265512\pi$$
0.671822 + 0.740713i $$0.265512\pi$$
$$678$$ 0 0
$$679$$ −1.06827e38 −0.628692
$$680$$ 0 0
$$681$$ 1.75510e36 0.00992698
$$682$$ 0 0
$$683$$ −1.45078e38 −0.788717 −0.394358 0.918957i $$-0.629033\pi$$
−0.394358 + 0.918957i $$0.629033\pi$$
$$684$$ 0 0
$$685$$ 2.34986e38 1.22806
$$686$$ 0 0
$$687$$ 1.25188e38 0.628993
$$688$$ 0 0
$$689$$ 1.45676e38 0.703765
$$690$$ 0 0
$$691$$ −4.04073e38 −1.87718 −0.938591 0.345031i $$-0.887869\pi$$
−0.938591 + 0.345031i $$0.887869\pi$$
$$692$$ 0 0
$$693$$ −4.79531e37 −0.214249
$$694$$ 0 0
$$695$$ −2.01245e38 −0.864829
$$696$$ 0 0
$$697$$ −2.80589e38 −1.15992
$$698$$ 0 0
$$699$$ 1.72411e38 0.685681
$$700$$ 0 0
$$701$$ 6.40419e37 0.245059 0.122529 0.992465i $$-0.460899\pi$$
0.122529 + 0.992465i $$0.460899\pi$$
$$702$$ 0 0
$$703$$ 7.84526e38 2.88875
$$704$$ 0 0
$$705$$ 5.09941e38 1.80704
$$706$$ 0 0
$$707$$ 1.69127e38 0.576835
$$708$$ 0 0
$$709$$ 7.71389e37 0.253250 0.126625 0.991951i $$-0.459586\pi$$
0.126625 + 0.991951i $$0.459586\pi$$
$$710$$ 0 0
$$711$$ −8.81516e37 −0.278606
$$712$$ 0 0
$$713$$ 3.76114e38 1.14449
$$714$$ 0 0
$$715$$ 1.71247e38 0.501754
$$716$$ 0 0
$$717$$ 6.15111e37 0.173558
$$718$$ 0 0
$$719$$ −6.01226e38 −1.63380 −0.816898 0.576783i $$-0.804308\pi$$
−0.816898 + 0.576783i $$0.804308\pi$$
$$720$$ 0 0
$$721$$ 1.58097e37 0.0413806
$$722$$ 0 0
$$723$$ 3.87876e38 0.977971
$$724$$ 0 0
$$725$$ −5.13094e37 −0.124633
$$726$$ 0 0
$$727$$ 5.96046e38 1.39497 0.697486 0.716599i $$-0.254302\pi$$
0.697486 + 0.716599i $$0.254302\pi$$
$$728$$ 0 0
$$729$$ 1.64232e37 0.0370370
$$730$$ 0 0
$$731$$ −1.91466e38 −0.416110
$$732$$ 0 0
$$733$$ −3.15157e38 −0.660120 −0.330060 0.943960i $$-0.607069\pi$$
−0.330060 + 0.943960i $$0.607069\pi$$
$$734$$ 0 0
$$735$$ 2.16958e38 0.438023
$$736$$ 0 0
$$737$$ −3.14146e38 −0.611393
$$738$$ 0 0
$$739$$ 3.24282e38 0.608447 0.304223 0.952601i $$-0.401603\pi$$
0.304223 + 0.952601i $$0.401603\pi$$
$$740$$ 0 0
$$741$$ 3.33590e38 0.603486
$$742$$ 0 0
$$743$$ −5.80289e38 −1.01227 −0.506134 0.862455i $$-0.668926\pi$$
−0.506134 + 0.862455i $$0.668926\pi$$
$$744$$ 0 0
$$745$$ 2.79604e38 0.470364
$$746$$ 0 0
$$747$$ 1.33212e38 0.216131
$$748$$ 0 0
$$749$$ −4.08777e38 −0.639708
$$750$$ 0 0
$$751$$ −2.21124e38 −0.333808 −0.166904 0.985973i $$-0.553377\pi$$
−0.166904 + 0.985973i $$0.553377\pi$$
$$752$$ 0 0
$$753$$ −2.15403e37 −0.0313703
$$754$$ 0 0
$$755$$ 2.00074e39 2.81130
$$756$$ 0 0
$$757$$ −2.01995e38 −0.273870 −0.136935 0.990580i $$-0.543725\pi$$
−0.136935 + 0.990580i $$0.543725\pi$$
$$758$$ 0 0
$$759$$ −2.68629e38 −0.351470
$$760$$ 0 0
$$761$$ 6.78605e38 0.856888 0.428444 0.903568i $$-0.359062\pi$$
0.428444 + 0.903568i $$0.359062\pi$$
$$762$$ 0 0
$$763$$ −9.97573e38 −1.21580
$$764$$ 0 0
$$765$$ 3.09298e38 0.363871
$$766$$ 0 0
$$767$$ 4.40893e37 0.0500721
$$768$$ 0 0
$$769$$ 9.79619e38 1.07412 0.537059 0.843545i $$-0.319535\pi$$
0.537059 + 0.843545i $$0.319535\pi$$
$$770$$ 0 0
$$771$$ −3.99500e38 −0.422943
$$772$$ 0 0
$$773$$ −1.58678e39 −1.62216 −0.811081 0.584934i $$-0.801120\pi$$
−0.811081 + 0.584934i $$0.801120\pi$$
$$774$$ 0 0
$$775$$ −1.57504e39 −1.55496
$$776$$ 0 0
$$777$$ −1.20290e39 −1.14695
$$778$$ 0 0
$$779$$ −3.26044e39 −3.00276
$$780$$ 0 0
$$781$$ −5.53170e38 −0.492119
$$782$$ 0 0
$$783$$ −1.78638e37 −0.0153529
$$784$$ 0 0
$$785$$ −3.16197e39 −2.62553
$$786$$ 0 0
$$787$$ 1.82547e39 1.46459 0.732296 0.680987i $$-0.238449\pi$$
0.732296 + 0.680987i $$0.238449\pi$$
$$788$$ 0 0
$$789$$ 6.62851e38 0.513897
$$790$$ 0 0
$$791$$ −5.11084e38 −0.382921
$$792$$ 0 0
$$793$$ 1.33779e39 0.968722
$$794$$ 0 0
$$795$$ 1.56948e39 1.09850
$$796$$ 0 0
$$797$$ −6.64299e38 −0.449445 −0.224723 0.974423i $$-0.572148\pi$$
−0.224723 + 0.974423i $$0.572148\pi$$
$$798$$ 0 0
$$799$$ 2.03868e39 1.33342
$$800$$ 0 0
$$801$$ 5.21603e38 0.329839
$$802$$ 0 0
$$803$$ −4.24145e38 −0.259331
$$804$$ 0 0
$$805$$ 3.77968e39 2.23466
$$806$$ 0 0
$$807$$ −2.89489e38 −0.165516
$$808$$ 0 0
$$809$$ −1.75085e39 −0.968150 −0.484075 0.875026i $$-0.660844\pi$$
−0.484075 + 0.875026i $$0.660844\pi$$
$$810$$ 0 0
$$811$$ −1.01209e39 −0.541301 −0.270651 0.962678i $$-0.587239\pi$$
−0.270651 + 0.962678i $$0.587239\pi$$
$$812$$ 0 0
$$813$$ 7.02961e38 0.363671
$$814$$ 0 0
$$815$$ −1.75391e39 −0.877768
$$816$$ 0 0
$$817$$ −2.22484e39 −1.07721
$$818$$ 0 0
$$819$$ −5.11488e38 −0.239609
$$820$$ 0 0
$$821$$ −8.40933e38 −0.381179 −0.190589 0.981670i $$-0.561040\pi$$
−0.190589 + 0.981670i $$0.561040\pi$$
$$822$$ 0 0
$$823$$ −2.05478e39 −0.901297 −0.450649 0.892701i $$-0.648807\pi$$
−0.450649 + 0.892701i $$0.648807\pi$$
$$824$$ 0 0
$$825$$ 1.12493e39 0.477526
$$826$$ 0 0
$$827$$ 9.27685e38 0.381133 0.190566 0.981674i $$-0.438968\pi$$
0.190566 + 0.981674i $$0.438968\pi$$
$$828$$ 0 0
$$829$$ 3.98790e39 1.58584 0.792918 0.609329i $$-0.208561\pi$$
0.792918 + 0.609329i $$0.208561\pi$$
$$830$$ 0 0
$$831$$ 7.13715e38 0.274733
$$832$$ 0 0
$$833$$ 8.67372e38 0.323220
$$834$$ 0 0
$$835$$ 4.50110e39 1.62387
$$836$$ 0 0
$$837$$ −5.48365e38 −0.191547
$$838$$ 0 0
$$839$$ −4.87871e39 −1.65013 −0.825066 0.565036i $$-0.808862\pi$$
−0.825066 + 0.565036i $$0.808862\pi$$
$$840$$ 0 0
$$841$$ −3.03370e39 −0.993636
$$842$$ 0 0
$$843$$ −1.30216e39 −0.413042
$$844$$ 0 0
$$845$$ −3.38391e39 −1.03957
$$846$$ 0 0
$$847$$ 2.93652e39 0.873789
$$848$$ 0 0
$$849$$ 5.09866e38 0.146961
$$850$$ 0 0
$$851$$ −6.73853e39 −1.88155
$$852$$ 0 0
$$853$$ 4.52266e38 0.122344 0.0611720 0.998127i $$-0.480516\pi$$
0.0611720 + 0.998127i $$0.480516\pi$$
$$854$$ 0 0
$$855$$ 3.59404e39 0.941977
$$856$$ 0 0
$$857$$ −7.11106e39 −1.80590 −0.902949 0.429748i $$-0.858602\pi$$
−0.902949 + 0.429748i $$0.858602\pi$$
$$858$$ 0 0
$$859$$ −3.37936e39 −0.831624 −0.415812 0.909451i $$-0.636503\pi$$
−0.415812 + 0.909451i $$0.636503\pi$$
$$860$$ 0 0
$$861$$ 4.99918e39 1.19222
$$862$$ 0 0
$$863$$ 4.33269e39 1.00141 0.500705 0.865618i $$-0.333074\pi$$
0.500705 + 0.865618i $$0.333074\pi$$
$$864$$ 0 0
$$865$$ −1.17223e40 −2.62601
$$866$$ 0 0
$$867$$ −1.42234e39 −0.308848
$$868$$ 0 0
$$869$$ −2.10220e39 −0.442494
$$870$$ 0 0
$$871$$ −3.35081e39 −0.683763
$$872$$ 0 0
$$873$$ 8.72501e38 0.172613
$$874$$ 0 0
$$875$$ −5.69673e39 −1.09274
$$876$$ 0 0
$$877$$ −5.02589e39 −0.934800 −0.467400 0.884046i $$-0.654809\pi$$
−0.467400 + 0.884046i $$0.654809\pi$$
$$878$$ 0 0
$$879$$ 4.12768e39 0.744486
$$880$$ 0 0
$$881$$ −8.13080e39 −1.42219 −0.711097 0.703094i $$-0.751801\pi$$
−0.711097 + 0.703094i $$0.751801\pi$$
$$882$$ 0 0
$$883$$ 8.35207e39 1.41685 0.708427 0.705784i $$-0.249405\pi$$
0.708427 + 0.705784i $$0.249405\pi$$
$$884$$ 0 0
$$885$$ 4.75010e38 0.0781572
$$886$$ 0 0
$$887$$ −3.26135e39 −0.520510 −0.260255 0.965540i $$-0.583807\pi$$
−0.260255 + 0.965540i $$0.583807\pi$$
$$888$$ 0 0
$$889$$ 5.66685e39 0.877339
$$890$$ 0 0
$$891$$ 3.91654e38 0.0588239
$$892$$ 0 0
$$893$$ 2.36895e40 3.45192
$$894$$ 0 0
$$895$$ −9.69467e39 −1.37064
$$896$$ 0 0
$$897$$ −2.86531e39 −0.393073
$$898$$ 0 0
$$899$$ 5.96467e38 0.0794018
$$900$$ 0 0
$$901$$ 6.27460e39 0.810590
$$902$$ 0 0
$$903$$ 3.41131e39 0.427697
$$904$$ 0 0
$$905$$ 2.01764e40 2.45521
$$906$$ 0 0
$$907$$ 1.85496e39 0.219096 0.109548 0.993981i $$-0.465060\pi$$
0.109548 + 0.993981i $$0.465060\pi$$
$$908$$ 0 0
$$909$$ −1.38134e39 −0.158375
$$910$$ 0 0
$$911$$ 1.23539e40 1.37501 0.687503 0.726182i $$-0.258707\pi$$
0.687503 + 0.726182i $$0.258707\pi$$
$$912$$ 0 0
$$913$$ 3.17680e39 0.343268
$$914$$ 0 0
$$915$$ 1.44131e40 1.51207
$$916$$ 0 0
$$917$$ −1.52938e40 −1.55787
$$918$$ 0 0
$$919$$ −9.73771e39 −0.963157 −0.481579 0.876403i $$-0.659936\pi$$
−0.481579 + 0.876403i $$0.659936\pi$$
$$920$$ 0 0
$$921$$ −2.62261e39 −0.251900
$$922$$ 0 0
$$923$$ −5.90034e39 −0.550369
$$924$$ 0 0
$$925$$ 2.82187e40 2.55637
$$926$$ 0 0
$$927$$ −1.29125e38 −0.0113614
$$928$$ 0 0
$$929$$ −6.31554e38 −0.0539757 −0.0269879 0.999636i $$-0.508592\pi$$
−0.0269879 + 0.999636i $$0.508592\pi$$
$$930$$ 0 0
$$931$$ 1.00789e40 0.836741
$$932$$ 0 0
$$933$$ 1.03533e40 0.834984
$$934$$ 0 0
$$935$$ 7.37600e39 0.577916
$$936$$ 0 0
$$937$$ 9.85113e38 0.0749897 0.0374948 0.999297i $$-0.488062\pi$$
0.0374948 + 0.999297i $$0.488062\pi$$
$$938$$ 0 0
$$939$$ −1.38586e40 −1.02502
$$940$$ 0 0
$$941$$ −1.59948e40 −1.14953 −0.574763 0.818320i $$-0.694906\pi$$
−0.574763 + 0.818320i $$0.694906\pi$$
$$942$$ 0 0
$$943$$ 2.80049e40 1.95581
$$944$$ 0 0
$$945$$ −5.51067e39 −0.374003
$$946$$ 0 0
$$947$$ 7.63606e38 0.0503669 0.0251834 0.999683i $$-0.491983\pi$$
0.0251834 + 0.999683i $$0.491983\pi$$
$$948$$ 0 0
$$949$$ −4.52411e39 −0.290027
$$950$$ 0 0
$$951$$ 3.07881e39 0.191843
$$952$$ 0 0
$$953$$ −2.87976e40 −1.74422 −0.872112 0.489307i $$-0.837250\pi$$
−0.872112 + 0.489307i $$0.837250\pi$$
$$954$$ 0 0
$$955$$ 2.07390e40 1.22107
$$956$$ 0 0
$$957$$ −4.26009e38 −0.0243842
$$958$$ 0 0
$$959$$ −1.67378e40 −0.931425
$$960$$ 0 0
$$961$$ −1.73001e38 −0.00936016
$$962$$ 0 0
$$963$$ 3.33866e39 0.175638
$$964$$ 0 0
$$965$$ 1.10442e40 0.564956
$$966$$ 0 0
$$967$$ −2.00353e39 −0.0996639 −0.0498319 0.998758i $$-0.515869\pi$$
−0.0498319 + 0.998758i $$0.515869\pi$$
$$968$$ 0 0
$$969$$ 1.43685e40 0.695090
$$970$$ 0 0
$$971$$ −1.39930e40 −0.658343 −0.329172 0.944270i $$-0.606769\pi$$
−0.329172 + 0.944270i $$0.606769\pi$$
$$972$$ 0 0
$$973$$ 1.43345e40 0.655933
$$974$$ 0 0
$$975$$ 1.19990e40 0.534049
$$976$$ 0 0
$$977$$ −1.69499e39 −0.0733824 −0.0366912 0.999327i $$-0.511682\pi$$
−0.0366912 + 0.999327i $$0.511682\pi$$
$$978$$ 0 0
$$979$$ 1.24390e40 0.523864
$$980$$ 0 0
$$981$$ 8.14762e39 0.333810
$$982$$ 0 0
$$983$$ −1.61199e40 −0.642526 −0.321263 0.946990i $$-0.604107\pi$$
−0.321263 + 0.946990i $$0.604107\pi$$
$$984$$ 0 0
$$985$$ −1.01110e40 −0.392106
$$986$$ 0 0
$$987$$ −3.63226e40 −1.37055
$$988$$ 0 0
$$989$$ 1.91098e40 0.701629
$$990$$ 0 0
$$991$$ −6.12985e39 −0.219006 −0.109503 0.993986i $$-0.534926\pi$$
−0.109503 + 0.993986i $$0.534926\pi$$
$$992$$ 0 0
$$993$$ 1.37773e40 0.479014
$$994$$ 0 0
$$995$$ 1.32599e40 0.448671
$$996$$ 0 0
$$997$$ 1.07857e40 0.355191 0.177596 0.984104i $$-0.443168\pi$$
0.177596 + 0.984104i $$0.443168\pi$$
$$998$$ 0 0
$$999$$ 9.82461e39 0.314906
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 48.28.a.g.1.2 2
4.3 odd 2 6.28.a.d.1.2 2
12.11 even 2 18.28.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
6.28.a.d.1.2 2 4.3 odd 2
18.28.a.g.1.1 2 12.11 even 2
48.28.a.g.1.2 2 1.1 even 1 trivial