Properties

 Label 1008.6.a.be.1.1 Level $1008$ Weight $6$ Character 1008.1 Self dual yes Analytic conductor $161.667$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,6,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1008.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: $$161.666890371$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{429})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 107$$ x^2 - x - 107 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 504) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$10.8562$$ of defining polynomial Character $$\chi$$ $$=$$ 1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-81.4246 q^{5} -49.0000 q^{7} +O(q^{10})$$ $$q-81.4246 q^{5} -49.0000 q^{7} +340.274 q^{11} -1100.79 q^{13} -197.369 q^{17} +2338.79 q^{19} -2606.58 q^{23} +3504.97 q^{25} +7915.83 q^{29} +9044.32 q^{31} +3989.81 q^{35} -5472.73 q^{37} -15249.0 q^{41} +3828.88 q^{43} -1940.82 q^{47} +2401.00 q^{49} +26291.9 q^{53} -27706.7 q^{55} +45841.3 q^{59} -43407.8 q^{61} +89631.8 q^{65} -15855.3 q^{67} +24120.6 q^{71} +69057.5 q^{73} -16673.4 q^{77} -60934.9 q^{79} -52245.0 q^{83} +16070.7 q^{85} +100507. q^{89} +53938.9 q^{91} -190435. q^{95} +64941.4 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 80 q^{5} - 98 q^{7}+O(q^{10})$$ 2 * q - 80 * q^5 - 98 * q^7 $$2 q - 80 q^{5} - 98 q^{7} + 432 q^{11} - 876 q^{13} + 848 q^{17} + 3352 q^{19} - 5296 q^{23} + 382 q^{25} + 256 q^{29} + 856 q^{31} + 3920 q^{35} + 3636 q^{37} + 3056 q^{41} + 26216 q^{43} + 2912 q^{47} + 4802 q^{49} + 27232 q^{53} - 27576 q^{55} + 59040 q^{59} - 40420 q^{61} + 89952 q^{65} - 52920 q^{67} - 52752 q^{71} + 18812 q^{73} - 21168 q^{77} - 50288 q^{79} - 66048 q^{83} + 17560 q^{85} + 131504 q^{89} + 42924 q^{91} - 188992 q^{95} + 164348 q^{97}+O(q^{100})$$ 2 * q - 80 * q^5 - 98 * q^7 + 432 * q^11 - 876 * q^13 + 848 * q^17 + 3352 * q^19 - 5296 * q^23 + 382 * q^25 + 256 * q^29 + 856 * q^31 + 3920 * q^35 + 3636 * q^37 + 3056 * q^41 + 26216 * q^43 + 2912 * q^47 + 4802 * q^49 + 27232 * q^53 - 27576 * q^55 + 59040 * q^59 - 40420 * q^61 + 89952 * q^65 - 52920 * q^67 - 52752 * q^71 + 18812 * q^73 - 21168 * q^77 - 50288 * q^79 - 66048 * q^83 + 17560 * q^85 + 131504 * q^89 + 42924 * q^91 - 188992 * q^95 + 164348 * q^97

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −81.4246 −1.45657 −0.728284 0.685275i $$-0.759682\pi$$
−0.728284 + 0.685275i $$0.759682\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 340.274 0.847904 0.423952 0.905685i $$-0.360642\pi$$
0.423952 + 0.905685i $$0.360642\pi$$
$$12$$ 0 0
$$13$$ −1100.79 −1.80654 −0.903270 0.429072i $$-0.858841\pi$$
−0.903270 + 0.429072i $$0.858841\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −197.369 −0.165637 −0.0828186 0.996565i $$-0.526392\pi$$
−0.0828186 + 0.996565i $$0.526392\pi$$
$$18$$ 0 0
$$19$$ 2338.79 1.48631 0.743153 0.669122i $$-0.233330\pi$$
0.743153 + 0.669122i $$0.233330\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2606.58 −1.02743 −0.513713 0.857962i $$-0.671730\pi$$
−0.513713 + 0.857962i $$0.671730\pi$$
$$24$$ 0 0
$$25$$ 3504.97 1.12159
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7915.83 1.74784 0.873920 0.486070i $$-0.161570\pi$$
0.873920 + 0.486070i $$0.161570\pi$$
$$30$$ 0 0
$$31$$ 9044.32 1.69033 0.845166 0.534504i $$-0.179502\pi$$
0.845166 + 0.534504i $$0.179502\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3989.81 0.550531
$$36$$ 0 0
$$37$$ −5472.73 −0.657204 −0.328602 0.944469i $$-0.606577\pi$$
−0.328602 + 0.944469i $$0.606577\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −15249.0 −1.41671 −0.708355 0.705856i $$-0.750562\pi$$
−0.708355 + 0.705856i $$0.750562\pi$$
$$42$$ 0 0
$$43$$ 3828.88 0.315792 0.157896 0.987456i $$-0.449529\pi$$
0.157896 + 0.987456i $$0.449529\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1940.82 −0.128156 −0.0640782 0.997945i $$-0.520411\pi$$
−0.0640782 + 0.997945i $$0.520411\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 26291.9 1.28568 0.642840 0.766000i $$-0.277756\pi$$
0.642840 + 0.766000i $$0.277756\pi$$
$$54$$ 0 0
$$55$$ −27706.7 −1.23503
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 45841.3 1.71446 0.857229 0.514935i $$-0.172184\pi$$
0.857229 + 0.514935i $$0.172184\pi$$
$$60$$ 0 0
$$61$$ −43407.8 −1.49363 −0.746815 0.665032i $$-0.768418\pi$$
−0.746815 + 0.665032i $$0.768418\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 89631.8 2.63135
$$66$$ 0 0
$$67$$ −15855.3 −0.431506 −0.215753 0.976448i $$-0.569221\pi$$
−0.215753 + 0.976448i $$0.569221\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 24120.6 0.567862 0.283931 0.958845i $$-0.408361\pi$$
0.283931 + 0.958845i $$0.408361\pi$$
$$72$$ 0 0
$$73$$ 69057.5 1.51671 0.758357 0.651840i $$-0.226003\pi$$
0.758357 + 0.651840i $$0.226003\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −16673.4 −0.320478
$$78$$ 0 0
$$79$$ −60934.9 −1.09850 −0.549248 0.835660i $$-0.685086\pi$$
−0.549248 + 0.835660i $$0.685086\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −52245.0 −0.832434 −0.416217 0.909265i $$-0.636644\pi$$
−0.416217 + 0.909265i $$0.636644\pi$$
$$84$$ 0 0
$$85$$ 16070.7 0.241262
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 100507. 1.34500 0.672500 0.740097i $$-0.265220\pi$$
0.672500 + 0.740097i $$0.265220\pi$$
$$90$$ 0 0
$$91$$ 53938.9 0.682808
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −190435. −2.16490
$$96$$ 0 0
$$97$$ 64941.4 0.700797 0.350398 0.936601i $$-0.386046\pi$$
0.350398 + 0.936601i $$0.386046\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 35889.9 0.350081 0.175041 0.984561i $$-0.443994\pi$$
0.175041 + 0.984561i $$0.443994\pi$$
$$102$$ 0 0
$$103$$ −121284. −1.12645 −0.563223 0.826305i $$-0.690439\pi$$
−0.563223 + 0.826305i $$0.690439\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 54535.5 0.460490 0.230245 0.973133i $$-0.426047\pi$$
0.230245 + 0.973133i $$0.426047\pi$$
$$108$$ 0 0
$$109$$ −105311. −0.848999 −0.424499 0.905428i $$-0.639550\pi$$
−0.424499 + 0.905428i $$0.639550\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −68204.7 −0.502479 −0.251240 0.967925i $$-0.580838\pi$$
−0.251240 + 0.967925i $$0.580838\pi$$
$$114$$ 0 0
$$115$$ 212239. 1.49652
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 9671.10 0.0626049
$$120$$ 0 0
$$121$$ −45264.7 −0.281058
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −30939.0 −0.177105
$$126$$ 0 0
$$127$$ 213149. 1.17266 0.586332 0.810071i $$-0.300572\pi$$
0.586332 + 0.810071i $$0.300572\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −285781. −1.45497 −0.727486 0.686123i $$-0.759311\pi$$
−0.727486 + 0.686123i $$0.759311\pi$$
$$132$$ 0 0
$$133$$ −114601. −0.561771
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 340712. 1.55091 0.775453 0.631405i $$-0.217521\pi$$
0.775453 + 0.631405i $$0.217521\pi$$
$$138$$ 0 0
$$139$$ −335029. −1.47077 −0.735387 0.677648i $$-0.763000\pi$$
−0.735387 + 0.677648i $$0.763000\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −374571. −1.53177
$$144$$ 0 0
$$145$$ −644544. −2.54585
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −66133.2 −0.244036 −0.122018 0.992528i $$-0.538937\pi$$
−0.122018 + 0.992528i $$0.538937\pi$$
$$150$$ 0 0
$$151$$ −29502.2 −0.105296 −0.0526480 0.998613i $$-0.516766\pi$$
−0.0526480 + 0.998613i $$0.516766\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −736431. −2.46208
$$156$$ 0 0
$$157$$ −426148. −1.37978 −0.689892 0.723912i $$-0.742342\pi$$
−0.689892 + 0.723912i $$0.742342\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 127722. 0.388331
$$162$$ 0 0
$$163$$ −640985. −1.88964 −0.944819 0.327593i $$-0.893763\pi$$
−0.944819 + 0.327593i $$0.893763\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −211785. −0.587629 −0.293815 0.955862i $$-0.594925\pi$$
−0.293815 + 0.955862i $$0.594925\pi$$
$$168$$ 0 0
$$169$$ 840455. 2.26359
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −311274. −0.790730 −0.395365 0.918524i $$-0.629382\pi$$
−0.395365 + 0.918524i $$0.629382\pi$$
$$174$$ 0 0
$$175$$ −171744. −0.423921
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 803487. 1.87433 0.937165 0.348886i $$-0.113440\pi$$
0.937165 + 0.348886i $$0.113440\pi$$
$$180$$ 0 0
$$181$$ 36730.7 0.0833359 0.0416680 0.999132i $$-0.486733\pi$$
0.0416680 + 0.999132i $$0.486733\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 445615. 0.957262
$$186$$ 0 0
$$187$$ −67159.7 −0.140444
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −352538. −0.699234 −0.349617 0.936893i $$-0.613688\pi$$
−0.349617 + 0.936893i $$0.613688\pi$$
$$192$$ 0 0
$$193$$ −467781. −0.903961 −0.451981 0.892028i $$-0.649282\pi$$
−0.451981 + 0.892028i $$0.649282\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 200981. 0.368969 0.184485 0.982835i $$-0.440938\pi$$
0.184485 + 0.982835i $$0.440938\pi$$
$$198$$ 0 0
$$199$$ 140758. 0.251966 0.125983 0.992032i $$-0.459792\pi$$
0.125983 + 0.992032i $$0.459792\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −387876. −0.660621
$$204$$ 0 0
$$205$$ 1.24164e6 2.06353
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 795831. 1.26024
$$210$$ 0 0
$$211$$ 178072. 0.275352 0.137676 0.990477i $$-0.456037\pi$$
0.137676 + 0.990477i $$0.456037\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −311765. −0.459972
$$216$$ 0 0
$$217$$ −443172. −0.638885
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 217263. 0.299230
$$222$$ 0 0
$$223$$ −577918. −0.778223 −0.389111 0.921191i $$-0.627218\pi$$
−0.389111 + 0.921191i $$0.627218\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 46602.0 0.0600261 0.0300130 0.999550i $$-0.490445\pi$$
0.0300130 + 0.999550i $$0.490445\pi$$
$$228$$ 0 0
$$229$$ −447131. −0.563438 −0.281719 0.959497i $$-0.590905\pi$$
−0.281719 + 0.959497i $$0.590905\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1.07326e6 1.29513 0.647567 0.762009i $$-0.275787\pi$$
0.647567 + 0.762009i $$0.275787\pi$$
$$234$$ 0 0
$$235$$ 158031. 0.186669
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 375795. 0.425555 0.212778 0.977101i $$-0.431749\pi$$
0.212778 + 0.977101i $$0.431749\pi$$
$$240$$ 0 0
$$241$$ −573941. −0.636539 −0.318269 0.948000i $$-0.603102\pi$$
−0.318269 + 0.948000i $$0.603102\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −195501. −0.208081
$$246$$ 0 0
$$247$$ −2.57453e6 −2.68507
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.56414e6 −1.56708 −0.783542 0.621338i $$-0.786589\pi$$
−0.783542 + 0.621338i $$0.786589\pi$$
$$252$$ 0 0
$$253$$ −886950. −0.871159
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −254654. −0.240502 −0.120251 0.992744i $$-0.538370\pi$$
−0.120251 + 0.992744i $$0.538370\pi$$
$$258$$ 0 0
$$259$$ 268164. 0.248400
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1.93115e6 1.72157 0.860787 0.508965i $$-0.169972\pi$$
0.860787 + 0.508965i $$0.169972\pi$$
$$264$$ 0 0
$$265$$ −2.14081e6 −1.87268
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −1.90099e6 −1.60177 −0.800883 0.598821i $$-0.795636\pi$$
−0.800883 + 0.598821i $$0.795636\pi$$
$$270$$ 0 0
$$271$$ 1.09964e6 0.909555 0.454778 0.890605i $$-0.349719\pi$$
0.454778 + 0.890605i $$0.349719\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.19265e6 0.951002
$$276$$ 0 0
$$277$$ −1.12622e6 −0.881908 −0.440954 0.897530i $$-0.645360\pi$$
−0.440954 + 0.897530i $$0.645360\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 619287. 0.467871 0.233936 0.972252i $$-0.424840\pi$$
0.233936 + 0.972252i $$0.424840\pi$$
$$282$$ 0 0
$$283$$ 354811. 0.263348 0.131674 0.991293i $$-0.457965\pi$$
0.131674 + 0.991293i $$0.457965\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 747200. 0.535466
$$288$$ 0 0
$$289$$ −1.38090e6 −0.972564
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1.74574e6 −1.18798 −0.593992 0.804471i $$-0.702449\pi$$
−0.593992 + 0.804471i $$0.702449\pi$$
$$294$$ 0 0
$$295$$ −3.73261e6 −2.49723
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2.86930e6 1.85609
$$300$$ 0 0
$$301$$ −187615. −0.119358
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 3.53446e6 2.17557
$$306$$ 0 0
$$307$$ −2.40315e6 −1.45524 −0.727621 0.685980i $$-0.759374\pi$$
−0.727621 + 0.685980i $$0.759374\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −738921. −0.433208 −0.216604 0.976260i $$-0.569498\pi$$
−0.216604 + 0.976260i $$0.569498\pi$$
$$312$$ 0 0
$$313$$ 410140. 0.236631 0.118316 0.992976i $$-0.462251\pi$$
0.118316 + 0.992976i $$0.462251\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.48927e6 −0.832385 −0.416193 0.909276i $$-0.636636\pi$$
−0.416193 + 0.909276i $$0.636636\pi$$
$$318$$ 0 0
$$319$$ 2.69355e6 1.48200
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −461607. −0.246187
$$324$$ 0 0
$$325$$ −3.85825e6 −2.02620
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 95100.2 0.0484386
$$330$$ 0 0
$$331$$ 397114. 0.199226 0.0996129 0.995026i $$-0.468240\pi$$
0.0996129 + 0.995026i $$0.468240\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 1.29101e6 0.628519
$$336$$ 0 0
$$337$$ −1.31077e6 −0.628712 −0.314356 0.949305i $$-0.601789\pi$$
−0.314356 + 0.949305i $$0.601789\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.07755e6 1.43324
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2.01372e6 −0.897793 −0.448896 0.893584i $$-0.648183\pi$$
−0.448896 + 0.893584i $$0.648183\pi$$
$$348$$ 0 0
$$349$$ 2.13139e6 0.936697 0.468349 0.883544i $$-0.344849\pi$$
0.468349 + 0.883544i $$0.344849\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2.59455e6 −1.10822 −0.554109 0.832444i $$-0.686941\pi$$
−0.554109 + 0.832444i $$0.686941\pi$$
$$354$$ 0 0
$$355$$ −1.96401e6 −0.827129
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 2.92478e6 1.19772 0.598862 0.800852i $$-0.295620\pi$$
0.598862 + 0.800852i $$0.295620\pi$$
$$360$$ 0 0
$$361$$ 2.99386e6 1.20910
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −5.62298e6 −2.20920
$$366$$ 0 0
$$367$$ −2.86220e6 −1.10926 −0.554631 0.832096i $$-0.687141\pi$$
−0.554631 + 0.832096i $$0.687141\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.28830e6 −0.485941
$$372$$ 0 0
$$373$$ 1.82182e6 0.678005 0.339002 0.940786i $$-0.389911\pi$$
0.339002 + 0.940786i $$0.389911\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −8.71370e6 −3.15754
$$378$$ 0 0
$$379$$ −111694. −0.0399421 −0.0199711 0.999801i $$-0.506357\pi$$
−0.0199711 + 0.999801i $$0.506357\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −502665. −0.175098 −0.0875490 0.996160i $$-0.527903\pi$$
−0.0875490 + 0.996160i $$0.527903\pi$$
$$384$$ 0 0
$$385$$ 1.35763e6 0.466798
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 1.12977e6 0.378544 0.189272 0.981925i $$-0.439387\pi$$
0.189272 + 0.981925i $$0.439387\pi$$
$$390$$ 0 0
$$391$$ 514458. 0.170180
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4.96160e6 1.60003
$$396$$ 0 0
$$397$$ −3.17615e6 −1.01140 −0.505701 0.862709i $$-0.668766\pi$$
−0.505701 + 0.862709i $$0.668766\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 344088. 0.106858 0.0534291 0.998572i $$-0.482985\pi$$
0.0534291 + 0.998572i $$0.482985\pi$$
$$402$$ 0 0
$$403$$ −9.95594e6 −3.05365
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.86223e6 −0.557246
$$408$$ 0 0
$$409$$ 4.09290e6 1.20983 0.604913 0.796292i $$-0.293208\pi$$
0.604913 + 0.796292i $$0.293208\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2.24622e6 −0.648004
$$414$$ 0 0
$$415$$ 4.25403e6 1.21250
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −2.40179e6 −0.668345 −0.334172 0.942512i $$-0.608457\pi$$
−0.334172 + 0.942512i $$0.608457\pi$$
$$420$$ 0 0
$$421$$ −6.67831e6 −1.83638 −0.918188 0.396146i $$-0.870347\pi$$
−0.918188 + 0.396146i $$0.870347\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −691774. −0.185777
$$426$$ 0 0
$$427$$ 2.12698e6 0.564539
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4.47258e6 1.15975 0.579876 0.814705i $$-0.303101\pi$$
0.579876 + 0.814705i $$0.303101\pi$$
$$432$$ 0 0
$$433$$ 365188. 0.0936045 0.0468023 0.998904i $$-0.485097\pi$$
0.0468023 + 0.998904i $$0.485097\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.09624e6 −1.52707
$$438$$ 0 0
$$439$$ −89090.3 −0.0220632 −0.0110316 0.999939i $$-0.503512\pi$$
−0.0110316 + 0.999939i $$0.503512\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 489817. 0.118583 0.0592917 0.998241i $$-0.481116\pi$$
0.0592917 + 0.998241i $$0.481116\pi$$
$$444$$ 0 0
$$445$$ −8.18377e6 −1.95908
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −744805. −0.174352 −0.0871760 0.996193i $$-0.527784\pi$$
−0.0871760 + 0.996193i $$0.527784\pi$$
$$450$$ 0 0
$$451$$ −5.18883e6 −1.20123
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.39196e6 −0.994557
$$456$$ 0 0
$$457$$ −5.14670e6 −1.15276 −0.576379 0.817182i $$-0.695535\pi$$
−0.576379 + 0.817182i $$0.695535\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 7.01082e6 1.53644 0.768222 0.640184i $$-0.221142\pi$$
0.768222 + 0.640184i $$0.221142\pi$$
$$462$$ 0 0
$$463$$ 222601. 0.0482585 0.0241293 0.999709i $$-0.492319\pi$$
0.0241293 + 0.999709i $$0.492319\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 5.47129e6 1.16091 0.580454 0.814293i $$-0.302875\pi$$
0.580454 + 0.814293i $$0.302875\pi$$
$$468$$ 0 0
$$469$$ 776909. 0.163094
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1.30287e6 0.267761
$$474$$ 0 0
$$475$$ 8.19740e6 1.66703
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −4.46045e6 −0.888260 −0.444130 0.895962i $$-0.646487\pi$$
−0.444130 + 0.895962i $$0.646487\pi$$
$$480$$ 0 0
$$481$$ 6.02435e6 1.18727
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −5.28783e6 −1.02076
$$486$$ 0 0
$$487$$ 966691. 0.184699 0.0923497 0.995727i $$-0.470562\pi$$
0.0923497 + 0.995727i $$0.470562\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.43131e6 −0.267936 −0.133968 0.990986i $$-0.542772\pi$$
−0.133968 + 0.990986i $$0.542772\pi$$
$$492$$ 0 0
$$493$$ −1.56234e6 −0.289507
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.18191e6 −0.214632
$$498$$ 0 0
$$499$$ 6.88711e6 1.23819 0.619093 0.785318i $$-0.287500\pi$$
0.619093 + 0.785318i $$0.287500\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −5.00139e6 −0.881395 −0.440697 0.897656i $$-0.645269\pi$$
−0.440697 + 0.897656i $$0.645269\pi$$
$$504$$ 0 0
$$505$$ −2.92232e6 −0.509917
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 381400. 0.0652508 0.0326254 0.999468i $$-0.489613\pi$$
0.0326254 + 0.999468i $$0.489613\pi$$
$$510$$ 0 0
$$511$$ −3.38382e6 −0.573264
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 9.87550e6 1.64075
$$516$$ 0 0
$$517$$ −660410. −0.108664
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.45886e6 −1.04246 −0.521232 0.853415i $$-0.674527\pi$$
−0.521232 + 0.853415i $$0.674527\pi$$
$$522$$ 0 0
$$523$$ −1.57762e6 −0.252202 −0.126101 0.992017i $$-0.540246\pi$$
−0.126101 + 0.992017i $$0.540246\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.78507e6 −0.279982
$$528$$ 0 0
$$529$$ 357892. 0.0556049
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1.67860e7 2.55934
$$534$$ 0 0
$$535$$ −4.44053e6 −0.670734
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 816998. 0.121129
$$540$$ 0 0
$$541$$ −5.98175e6 −0.878690 −0.439345 0.898319i $$-0.644789\pi$$
−0.439345 + 0.898319i $$0.644789\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 8.57490e6 1.23662
$$546$$ 0 0
$$547$$ 3.52416e6 0.503602 0.251801 0.967779i $$-0.418977\pi$$
0.251801 + 0.967779i $$0.418977\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.85135e7 2.59782
$$552$$ 0 0
$$553$$ 2.98581e6 0.415192
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.23769e7 1.69034 0.845172 0.534495i $$-0.179498\pi$$
0.845172 + 0.534495i $$0.179498\pi$$
$$558$$ 0 0
$$559$$ −4.21481e6 −0.570491
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −630148. −0.0837860 −0.0418930 0.999122i $$-0.513339\pi$$
−0.0418930 + 0.999122i $$0.513339\pi$$
$$564$$ 0 0
$$565$$ 5.55354e6 0.731895
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 751980. 0.0973701 0.0486851 0.998814i $$-0.484497\pi$$
0.0486851 + 0.998814i $$0.484497\pi$$
$$570$$ 0 0
$$571$$ −1.16905e7 −1.50052 −0.750259 0.661144i $$-0.770071\pi$$
−0.750259 + 0.661144i $$0.770071\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −9.13597e6 −1.15235
$$576$$ 0 0
$$577$$ 8.52927e6 1.06653 0.533264 0.845949i $$-0.320965\pi$$
0.533264 + 0.845949i $$0.320965\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.56001e6 0.314630
$$582$$ 0 0
$$583$$ 8.94646e6 1.09013
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −5.62232e6 −0.673473 −0.336737 0.941599i $$-0.609323\pi$$
−0.336737 + 0.941599i $$0.609323\pi$$
$$588$$ 0 0
$$589$$ 2.11528e7 2.51235
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 2.24396e6 0.262047 0.131023 0.991379i $$-0.458174\pi$$
0.131023 + 0.991379i $$0.458174\pi$$
$$594$$ 0 0
$$595$$ −787466. −0.0911884
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −2.13424e6 −0.243039 −0.121520 0.992589i $$-0.538777\pi$$
−0.121520 + 0.992589i $$0.538777\pi$$
$$600$$ 0 0
$$601$$ −6.01631e6 −0.679429 −0.339715 0.940529i $$-0.610330\pi$$
−0.339715 + 0.940529i $$0.610330\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 3.68566e6 0.409380
$$606$$ 0 0
$$607$$ −4.33356e6 −0.477390 −0.238695 0.971095i $$-0.576720\pi$$
−0.238695 + 0.971095i $$0.576720\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2.13644e6 0.231520
$$612$$ 0 0
$$613$$ 1.25876e7 1.35299 0.676493 0.736449i $$-0.263499\pi$$
0.676493 + 0.736449i $$0.263499\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.91124e6 −0.202116 −0.101058 0.994881i $$-0.532223\pi$$
−0.101058 + 0.994881i $$0.532223\pi$$
$$618$$ 0 0
$$619$$ 1.84569e7 1.93612 0.968058 0.250727i $$-0.0806696\pi$$
0.968058 + 0.250727i $$0.0806696\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −4.92486e6 −0.508362
$$624$$ 0 0
$$625$$ −8.43384e6 −0.863625
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.08015e6 0.108857
$$630$$ 0 0
$$631$$ 2.56432e6 0.256389 0.128194 0.991749i $$-0.459082\pi$$
0.128194 + 0.991749i $$0.459082\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −1.73556e7 −1.70806
$$636$$ 0 0
$$637$$ −2.64301e6 −0.258077
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.60595e6 0.154378 0.0771891 0.997016i $$-0.475405\pi$$
0.0771891 + 0.997016i $$0.475405\pi$$
$$642$$ 0 0
$$643$$ 7.63307e6 0.728068 0.364034 0.931386i $$-0.381399\pi$$
0.364034 + 0.931386i $$0.381399\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 5.79911e6 0.544629 0.272314 0.962208i $$-0.412211\pi$$
0.272314 + 0.962208i $$0.412211\pi$$
$$648$$ 0 0
$$649$$ 1.55986e7 1.45370
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 8.55249e6 0.784892 0.392446 0.919775i $$-0.371629\pi$$
0.392446 + 0.919775i $$0.371629\pi$$
$$654$$ 0 0
$$655$$ 2.32696e7 2.11926
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −1.06831e7 −0.958260 −0.479130 0.877744i $$-0.659048\pi$$
−0.479130 + 0.877744i $$0.659048\pi$$
$$660$$ 0 0
$$661$$ 8.79509e6 0.782955 0.391477 0.920188i $$-0.371964\pi$$
0.391477 + 0.920188i $$0.371964\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 9.33134e6 0.818257
$$666$$ 0 0
$$667$$ −2.06332e7 −1.79578
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −1.47705e7 −1.26646
$$672$$ 0 0
$$673$$ 1.77607e6 0.151155 0.0755775 0.997140i $$-0.475920\pi$$
0.0755775 + 0.997140i $$0.475920\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1.19599e7 1.00290 0.501448 0.865188i $$-0.332801\pi$$
0.501448 + 0.865188i $$0.332801\pi$$
$$678$$ 0 0
$$679$$ −3.18213e6 −0.264876
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.52201e7 −1.24844 −0.624218 0.781250i $$-0.714582\pi$$
−0.624218 + 0.781250i $$0.714582\pi$$
$$684$$ 0 0
$$685$$ −2.77423e7 −2.25900
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −2.89420e7 −2.32263
$$690$$ 0 0
$$691$$ −24706.9 −0.00196844 −0.000984221 1.00000i $$-0.500313\pi$$
−0.000984221 1.00000i $$0.500313\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 2.72796e7 2.14228
$$696$$ 0 0
$$697$$ 3.00968e6 0.234660
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −466834. −0.0358813 −0.0179406 0.999839i $$-0.505711\pi$$
−0.0179406 + 0.999839i $$0.505711\pi$$
$$702$$ 0 0
$$703$$ −1.27996e7 −0.976805
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.75861e6 −0.132318
$$708$$ 0 0
$$709$$ 9.10361e6 0.680140 0.340070 0.940400i $$-0.389549\pi$$
0.340070 + 0.940400i $$0.389549\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −2.35747e7 −1.73669
$$714$$ 0 0
$$715$$ 3.04993e7 2.23113
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −1.88266e7 −1.35815 −0.679077 0.734067i $$-0.737620\pi$$
−0.679077 + 0.734067i $$0.737620\pi$$
$$720$$ 0 0
$$721$$ 5.94291e6 0.425757
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 2.77448e7 1.96036
$$726$$ 0 0
$$727$$ −1.60515e7 −1.12636 −0.563182 0.826333i $$-0.690423\pi$$
−0.563182 + 0.826333i $$0.690423\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −755705. −0.0523069
$$732$$ 0 0
$$733$$ −2.72193e7 −1.87119 −0.935593 0.353080i $$-0.885135\pi$$
−0.935593 + 0.353080i $$0.885135\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −5.39514e6 −0.365876
$$738$$ 0 0
$$739$$ 1.30938e7 0.881974 0.440987 0.897514i $$-0.354629\pi$$
0.440987 + 0.897514i $$0.354629\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −3.36637e6 −0.223712 −0.111856 0.993724i $$-0.535680\pi$$
−0.111856 + 0.993724i $$0.535680\pi$$
$$744$$ 0 0
$$745$$ 5.38487e6 0.355455
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −2.67224e6 −0.174049
$$750$$ 0 0
$$751$$ 2.06113e7 1.33354 0.666769 0.745264i $$-0.267677\pi$$
0.666769 + 0.745264i $$0.267677\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 2.40220e6 0.153371
$$756$$ 0 0
$$757$$ 157073. 0.00996238 0.00498119 0.999988i $$-0.498414\pi$$
0.00498119 + 0.999988i $$0.498414\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3.11164e6 0.194772 0.0973862 0.995247i $$-0.468952\pi$$
0.0973862 + 0.995247i $$0.468952\pi$$
$$762$$ 0 0
$$763$$ 5.16023e6 0.320891
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.04618e7 −3.09724
$$768$$ 0 0
$$769$$ 2.28503e7 1.39340 0.696702 0.717361i $$-0.254650\pi$$
0.696702 + 0.717361i $$0.254650\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −1.44437e7 −0.869420 −0.434710 0.900570i $$-0.643149\pi$$
−0.434710 + 0.900570i $$0.643149\pi$$
$$774$$ 0 0
$$775$$ 3.17001e7 1.89586
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −3.56642e7 −2.10566
$$780$$ 0 0
$$781$$ 8.20762e6 0.481493
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 3.46989e7 2.00975
$$786$$ 0 0
$$787$$ 1.13089e7 0.650854 0.325427 0.945567i $$-0.394492\pi$$
0.325427 + 0.945567i $$0.394492\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3.34203e6 0.189919
$$792$$ 0 0
$$793$$ 4.77830e7 2.69830
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.96145e7 1.65143 0.825713 0.564091i $$-0.190773\pi$$
0.825713 + 0.564091i $$0.190773\pi$$
$$798$$ 0 0
$$799$$ 383059. 0.0212275
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 2.34985e7 1.28603
$$804$$ 0 0
$$805$$ −1.03997e7 −0.565630
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 6.90828e6 0.371106 0.185553 0.982634i $$-0.440592\pi$$
0.185553 + 0.982634i $$0.440592\pi$$
$$810$$ 0 0
$$811$$ 463040. 0.0247210 0.0123605 0.999924i $$-0.496065\pi$$
0.0123605 + 0.999924i $$0.496065\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 5.21919e7 2.75239
$$816$$ 0 0
$$817$$ 8.95497e6 0.469363
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.79933e7 −1.44942 −0.724712 0.689052i $$-0.758027\pi$$
−0.724712 + 0.689052i $$0.758027\pi$$
$$822$$ 0 0
$$823$$ 1.71925e7 0.884786 0.442393 0.896821i $$-0.354130\pi$$
0.442393 + 0.896821i $$0.354130\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −3.87619e6 −0.197079 −0.0985397 0.995133i $$-0.531417\pi$$
−0.0985397 + 0.995133i $$0.531417\pi$$
$$828$$ 0 0
$$829$$ −2.36921e7 −1.19734 −0.598670 0.800996i $$-0.704304\pi$$
−0.598670 + 0.800996i $$0.704304\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −473884. −0.0236624
$$834$$ 0 0
$$835$$ 1.72445e7 0.855922
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 7.48831e6 0.367265 0.183632 0.982995i $$-0.441214\pi$$
0.183632 + 0.982995i $$0.441214\pi$$
$$840$$ 0 0
$$841$$ 4.21492e7 2.05494
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −6.84337e7 −3.29707
$$846$$ 0 0
$$847$$ 2.21797e6 0.106230
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.42651e7 0.675229
$$852$$ 0 0
$$853$$ 2.19918e6 0.103488 0.0517438 0.998660i $$-0.483522\pi$$
0.0517438 + 0.998660i $$0.483522\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1.10099e7 0.512071 0.256035 0.966667i $$-0.417584\pi$$
0.256035 + 0.966667i $$0.417584\pi$$
$$858$$ 0 0
$$859$$ −3.31391e7 −1.53235 −0.766175 0.642632i $$-0.777842\pi$$
−0.766175 + 0.642632i $$0.777842\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 3.12254e7 1.42719 0.713594 0.700559i $$-0.247066\pi$$
0.713594 + 0.700559i $$0.247066\pi$$
$$864$$ 0 0
$$865$$ 2.53454e7 1.15175
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −2.07345e7 −0.931419
$$870$$ 0 0
$$871$$ 1.74534e7 0.779534
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 1.51601e6 0.0669394
$$876$$ 0 0
$$877$$ 5.05675e6 0.222010 0.111005 0.993820i $$-0.464593\pi$$
0.111005 + 0.993820i $$0.464593\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1.22651e7 0.532392 0.266196 0.963919i $$-0.414233\pi$$
0.266196 + 0.963919i $$0.414233\pi$$
$$882$$ 0 0
$$883$$ −1.39549e7 −0.602318 −0.301159 0.953574i $$-0.597374\pi$$
−0.301159 + 0.953574i $$0.597374\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −2.78677e7 −1.18930 −0.594651 0.803984i $$-0.702710\pi$$
−0.594651 + 0.803984i $$0.702710\pi$$
$$888$$ 0 0
$$889$$ −1.04443e7 −0.443225
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −4.53918e6 −0.190480
$$894$$ 0 0
$$895$$ −6.54236e7 −2.73009
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 7.15933e7 2.95443
$$900$$ 0 0
$$901$$ −5.18923e6 −0.212956
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −2.99078e6 −0.121384
$$906$$ 0 0
$$907$$ −4.03006e7 −1.62665 −0.813323 0.581812i $$-0.802344\pi$$
−0.813323 + 0.581812i $$0.802344\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −3.94303e7 −1.57411 −0.787053 0.616886i $$-0.788394\pi$$
−0.787053 + 0.616886i $$0.788394\pi$$
$$912$$ 0 0
$$913$$ −1.77776e7 −0.705824
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.40032e7 0.549927
$$918$$ 0 0
$$919$$ −2.51340e7 −0.981687 −0.490843 0.871248i $$-0.663311\pi$$
−0.490843 + 0.871248i $$0.663311\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −2.65518e7 −1.02587
$$924$$ 0 0
$$925$$ −1.91818e7 −0.737114
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −4.21100e6 −0.160083 −0.0800416 0.996792i $$-0.525505\pi$$
−0.0800416 + 0.996792i $$0.525505\pi$$
$$930$$ 0 0
$$931$$ 5.61544e6 0.212329
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 5.46845e6 0.204567
$$936$$ 0 0
$$937$$ 2.46140e7 0.915870 0.457935 0.888986i $$-0.348589\pi$$
0.457935 + 0.888986i $$0.348589\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1.00570e7 −0.370250 −0.185125 0.982715i $$-0.559269\pi$$
−0.185125 + 0.982715i $$0.559269\pi$$
$$942$$ 0 0
$$943$$ 3.97476e7 1.45557
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.34193e7 0.486246 0.243123 0.969996i $$-0.421828\pi$$
0.243123 + 0.969996i $$0.421828\pi$$
$$948$$ 0 0
$$949$$ −7.60181e7 −2.74000
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.38524e7 0.850747 0.425373 0.905018i $$-0.360143\pi$$
0.425373 + 0.905018i $$0.360143\pi$$
$$954$$ 0 0
$$955$$ 2.87053e7 1.01848
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1.66949e7 −0.586187
$$960$$ 0 0
$$961$$ 5.31706e7 1.85722
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 3.80889e7 1.31668
$$966$$ 0 0
$$967$$ 1.07883e7 0.371012 0.185506 0.982643i $$-0.440608\pi$$
0.185506 + 0.982643i $$0.440608\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 4.97812e7 1.69440 0.847202 0.531270i $$-0.178285\pi$$
0.847202 + 0.531270i $$0.178285\pi$$
$$972$$ 0 0
$$973$$ 1.64164e7 0.555900
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −4.09847e7 −1.37368 −0.686840 0.726809i $$-0.741003\pi$$
−0.686840 + 0.726809i $$0.741003\pi$$
$$978$$ 0 0
$$979$$ 3.42000e7 1.14043
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 4.69345e7 1.54920 0.774602 0.632449i $$-0.217950\pi$$
0.774602 + 0.632449i $$0.217950\pi$$
$$984$$ 0 0
$$985$$ −1.63648e7 −0.537429
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9.98027e6 −0.324453
$$990$$ 0 0
$$991$$ −3.80764e6 −0.123161 −0.0615804 0.998102i $$-0.519614\pi$$
−0.0615804 + 0.998102i $$0.519614\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1.14612e7 −0.367006
$$996$$ 0 0
$$997$$ 1.32091e7 0.420859 0.210430 0.977609i $$-0.432514\pi$$
0.210430 + 0.977609i $$0.432514\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 1008.6.a.be.1.1 2
3.2 odd 2 1008.6.a.bx.1.2 2
4.3 odd 2 504.6.a.j.1.1 2
12.11 even 2 504.6.a.w.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
504.6.a.j.1.1 2 4.3 odd 2
504.6.a.w.1.2 yes 2 12.11 even 2
1008.6.a.be.1.1 2 1.1 even 1 trivial
1008.6.a.bx.1.2 2 3.2 odd 2