# Properties

 Label 18.7.b.a.17.2 Level $18$ Weight $7$ Character 18.17 Analytic conductor $4.141$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,7,Mod(17,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("18.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 18.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$4.14097350516$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 17.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 18.17 Dual form 18.7.b.a.17.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.65685i q^{2} -32.0000 q^{4} +173.948i q^{5} -484.000 q^{7} -181.019i q^{8} +O(q^{10})$$ $$q+5.65685i q^{2} -32.0000 q^{4} +173.948i q^{5} -484.000 q^{7} -181.019i q^{8} -984.000 q^{10} +1340.67i q^{11} +3368.00 q^{13} -2737.92i q^{14} +1024.00 q^{16} +12.7279i q^{17} +5744.00 q^{19} -5566.34i q^{20} -7584.00 q^{22} -3377.14i q^{23} -14633.0 q^{25} +19052.3i q^{26} +15488.0 q^{28} +29354.8i q^{29} -39796.0 q^{31} +5792.62i q^{32} -72.0000 q^{34} -84191.0i q^{35} +52526.0 q^{37} +32493.0i q^{38} +31488.0 q^{40} -37042.5i q^{41} +3800.00 q^{43} -42901.6i q^{44} +19104.0 q^{46} +76791.8i q^{47} +116607. q^{49} -82776.7i q^{50} -107776. q^{52} +238738. i q^{53} -233208. q^{55} +87613.4i q^{56} -166056. q^{58} -249841. i q^{59} +13250.0 q^{61} -225120. i q^{62} -32768.0 q^{64} +585858. i q^{65} +168968. q^{67} -407.294i q^{68} +476256. q^{70} -531467. i q^{71} +236144. q^{73} +297132. i q^{74} -183808. q^{76} -648886. i q^{77} -35116.0 q^{79} +178123. i q^{80} +209544. q^{82} -10980.0i q^{83} -2214.00 q^{85} +21496.0i q^{86} +242688. q^{88} -129328. i q^{89} -1.63011e6 q^{91} +108069. i q^{92} -434400. q^{94} +999159. i q^{95} -321424. q^{97} +659629. i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 64 q^{4} - 968 q^{7}+O(q^{10})$$ 2 * q - 64 * q^4 - 968 * q^7 $$2 q - 64 q^{4} - 968 q^{7} - 1968 q^{10} + 6736 q^{13} + 2048 q^{16} + 11488 q^{19} - 15168 q^{22} - 29266 q^{25} + 30976 q^{28} - 79592 q^{31} - 144 q^{34} + 105052 q^{37} + 62976 q^{40} + 7600 q^{43} + 38208 q^{46} + 233214 q^{49} - 215552 q^{52} - 466416 q^{55} - 332112 q^{58} + 26500 q^{61} - 65536 q^{64} + 337936 q^{67} + 952512 q^{70} + 472288 q^{73} - 367616 q^{76} - 70232 q^{79} + 419088 q^{82} - 4428 q^{85} + 485376 q^{88} - 3260224 q^{91} - 868800 q^{94} - 642848 q^{97}+O(q^{100})$$ 2 * q - 64 * q^4 - 968 * q^7 - 1968 * q^10 + 6736 * q^13 + 2048 * q^16 + 11488 * q^19 - 15168 * q^22 - 29266 * q^25 + 30976 * q^28 - 79592 * q^31 - 144 * q^34 + 105052 * q^37 + 62976 * q^40 + 7600 * q^43 + 38208 * q^46 + 233214 * q^49 - 215552 * q^52 - 466416 * q^55 - 332112 * q^58 + 26500 * q^61 - 65536 * q^64 + 337936 * q^67 + 952512 * q^70 + 472288 * q^73 - 367616 * q^76 - 70232 * q^79 + 419088 * q^82 - 4428 * q^85 + 485376 * q^88 - 3260224 * q^91 - 868800 * q^94 - 642848 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.65685i 0.707107i
$$3$$ 0 0
$$4$$ −32.0000 −0.500000
$$5$$ 173.948i 1.39159i 0.718242 + 0.695793i $$0.244947\pi$$
−0.718242 + 0.695793i $$0.755053\pi$$
$$6$$ 0 0
$$7$$ −484.000 −1.41108 −0.705539 0.708671i $$-0.749295\pi$$
−0.705539 + 0.708671i $$0.749295\pi$$
$$8$$ − 181.019i − 0.353553i
$$9$$ 0 0
$$10$$ −984.000 −0.984000
$$11$$ 1340.67i 1.00727i 0.863917 + 0.503634i $$0.168004\pi$$
−0.863917 + 0.503634i $$0.831996\pi$$
$$12$$ 0 0
$$13$$ 3368.00 1.53300 0.766500 0.642245i $$-0.221997\pi$$
0.766500 + 0.642245i $$0.221997\pi$$
$$14$$ − 2737.92i − 0.997783i
$$15$$ 0 0
$$16$$ 1024.00 0.250000
$$17$$ 12.7279i 0.00259066i 0.999999 + 0.00129533i $$0.000412317\pi$$
−0.999999 + 0.00129533i $$0.999588\pi$$
$$18$$ 0 0
$$19$$ 5744.00 0.837440 0.418720 0.908115i $$-0.362479\pi$$
0.418720 + 0.908115i $$0.362479\pi$$
$$20$$ − 5566.34i − 0.695793i
$$21$$ 0 0
$$22$$ −7584.00 −0.712246
$$23$$ − 3377.14i − 0.277566i −0.990323 0.138783i $$-0.955681\pi$$
0.990323 0.138783i $$-0.0443190\pi$$
$$24$$ 0 0
$$25$$ −14633.0 −0.936512
$$26$$ 19052.3i 1.08399i
$$27$$ 0 0
$$28$$ 15488.0 0.705539
$$29$$ 29354.8i 1.20361i 0.798643 + 0.601805i $$0.205551\pi$$
−0.798643 + 0.601805i $$0.794449\pi$$
$$30$$ 0 0
$$31$$ −39796.0 −1.33584 −0.667920 0.744233i $$-0.732815\pi$$
−0.667920 + 0.744233i $$0.732815\pi$$
$$32$$ 5792.62i 0.176777i
$$33$$ 0 0
$$34$$ −72.0000 −0.00183187
$$35$$ − 84191.0i − 1.96364i
$$36$$ 0 0
$$37$$ 52526.0 1.03698 0.518489 0.855085i $$-0.326495\pi$$
0.518489 + 0.855085i $$0.326495\pi$$
$$38$$ 32493.0i 0.592159i
$$39$$ 0 0
$$40$$ 31488.0 0.492000
$$41$$ − 37042.5i − 0.537463i −0.963215 0.268732i $$-0.913396\pi$$
0.963215 0.268732i $$-0.0866045\pi$$
$$42$$ 0 0
$$43$$ 3800.00 0.0477945 0.0238973 0.999714i $$-0.492393\pi$$
0.0238973 + 0.999714i $$0.492393\pi$$
$$44$$ − 42901.6i − 0.503634i
$$45$$ 0 0
$$46$$ 19104.0 0.196269
$$47$$ 76791.8i 0.739641i 0.929103 + 0.369821i $$0.120581\pi$$
−0.929103 + 0.369821i $$0.879419\pi$$
$$48$$ 0 0
$$49$$ 116607. 0.991143
$$50$$ − 82776.7i − 0.662214i
$$51$$ 0 0
$$52$$ −107776. −0.766500
$$53$$ 238738.i 1.60359i 0.597599 + 0.801795i $$0.296121\pi$$
−0.597599 + 0.801795i $$0.703879\pi$$
$$54$$ 0 0
$$55$$ −233208. −1.40170
$$56$$ 87613.4i 0.498892i
$$57$$ 0 0
$$58$$ −166056. −0.851080
$$59$$ − 249841.i − 1.21649i −0.793751 0.608243i $$-0.791875\pi$$
0.793751 0.608243i $$-0.208125\pi$$
$$60$$ 0 0
$$61$$ 13250.0 0.0583749 0.0291875 0.999574i $$-0.490708\pi$$
0.0291875 + 0.999574i $$0.490708\pi$$
$$62$$ − 225120.i − 0.944581i
$$63$$ 0 0
$$64$$ −32768.0 −0.125000
$$65$$ 585858.i 2.13330i
$$66$$ 0 0
$$67$$ 168968. 0.561798 0.280899 0.959737i $$-0.409367\pi$$
0.280899 + 0.959737i $$0.409367\pi$$
$$68$$ − 407.294i − 0.00129533i
$$69$$ 0 0
$$70$$ 476256. 1.38850
$$71$$ − 531467.i − 1.48491i −0.669894 0.742457i $$-0.733660\pi$$
0.669894 0.742457i $$-0.266340\pi$$
$$72$$ 0 0
$$73$$ 236144. 0.607027 0.303514 0.952827i $$-0.401840\pi$$
0.303514 + 0.952827i $$0.401840\pi$$
$$74$$ 297132.i 0.733254i
$$75$$ 0 0
$$76$$ −183808. −0.418720
$$77$$ − 648886.i − 1.42134i
$$78$$ 0 0
$$79$$ −35116.0 −0.0712236 −0.0356118 0.999366i $$-0.511338\pi$$
−0.0356118 + 0.999366i $$0.511338\pi$$
$$80$$ 178123.i 0.347897i
$$81$$ 0 0
$$82$$ 209544. 0.380044
$$83$$ − 10980.0i − 0.0192029i −0.999954 0.00960144i $$-0.996944\pi$$
0.999954 0.00960144i $$-0.00305628\pi$$
$$84$$ 0 0
$$85$$ −2214.00 −0.00360513
$$86$$ 21496.0i 0.0337958i
$$87$$ 0 0
$$88$$ 242688. 0.356123
$$89$$ − 129328.i − 0.183453i −0.995784 0.0917263i $$-0.970762\pi$$
0.995784 0.0917263i $$-0.0292385\pi$$
$$90$$ 0 0
$$91$$ −1.63011e6 −2.16318
$$92$$ 108069.i 0.138783i
$$93$$ 0 0
$$94$$ −434400. −0.523005
$$95$$ 999159.i 1.16537i
$$96$$ 0 0
$$97$$ −321424. −0.352179 −0.176089 0.984374i $$-0.556345\pi$$
−0.176089 + 0.984374i $$0.556345\pi$$
$$98$$ 659629.i 0.700844i
$$99$$ 0 0
$$100$$ 468256. 0.468256
$$101$$ 668780.i 0.649111i 0.945867 + 0.324556i $$0.105215\pi$$
−0.945867 + 0.324556i $$0.894785\pi$$
$$102$$ 0 0
$$103$$ 1.99341e6 1.82425 0.912127 0.409907i $$-0.134439\pi$$
0.912127 + 0.409907i $$0.134439\pi$$
$$104$$ − 609673.i − 0.541997i
$$105$$ 0 0
$$106$$ −1.35050e6 −1.13391
$$107$$ − 260668.i − 0.212783i −0.994324 0.106391i $$-0.966070\pi$$
0.994324 0.106391i $$-0.0339296\pi$$
$$108$$ 0 0
$$109$$ 194456. 0.150156 0.0750779 0.997178i $$-0.476079\pi$$
0.0750779 + 0.997178i $$0.476079\pi$$
$$110$$ − 1.31922e6i − 0.991152i
$$111$$ 0 0
$$112$$ −495616. −0.352770
$$113$$ − 821897.i − 0.569616i −0.958585 0.284808i $$-0.908070\pi$$
0.958585 0.284808i $$-0.0919298\pi$$
$$114$$ 0 0
$$115$$ 587448. 0.386257
$$116$$ − 939355.i − 0.601805i
$$117$$ 0 0
$$118$$ 1.41331e6 0.860185
$$119$$ − 6160.31i − 0.00365563i
$$120$$ 0 0
$$121$$ −25847.0 −0.0145900
$$122$$ 74953.3i 0.0412773i
$$123$$ 0 0
$$124$$ 1.27347e6 0.667920
$$125$$ 172557.i 0.0883490i
$$126$$ 0 0
$$127$$ 3.05721e6 1.49250 0.746250 0.665666i $$-0.231852\pi$$
0.746250 + 0.665666i $$0.231852\pi$$
$$128$$ − 185364.i − 0.0883883i
$$129$$ 0 0
$$130$$ −3.31411e6 −1.50847
$$131$$ 3.07388e6i 1.36733i 0.729797 + 0.683664i $$0.239615\pi$$
−0.729797 + 0.683664i $$0.760385\pi$$
$$132$$ 0 0
$$133$$ −2.78010e6 −1.18169
$$134$$ 955827.i 0.397251i
$$135$$ 0 0
$$136$$ 2304.00 0.000915937 0
$$137$$ − 4.48412e6i − 1.74388i −0.489617 0.871938i $$-0.662863\pi$$
0.489617 0.871938i $$-0.337137\pi$$
$$138$$ 0 0
$$139$$ −1.09233e6 −0.406732 −0.203366 0.979103i $$-0.565188\pi$$
−0.203366 + 0.979103i $$0.565188\pi$$
$$140$$ 2.69411e6i 0.981819i
$$141$$ 0 0
$$142$$ 3.00643e6 1.04999
$$143$$ 4.51539e6i 1.54414i
$$144$$ 0 0
$$145$$ −5.10622e6 −1.67493
$$146$$ 1.33583e6i 0.429233i
$$147$$ 0 0
$$148$$ −1.68083e6 −0.518489
$$149$$ − 2.22087e6i − 0.671375i −0.941973 0.335687i $$-0.891031\pi$$
0.941973 0.335687i $$-0.108969\pi$$
$$150$$ 0 0
$$151$$ −4.07871e6 −1.18465 −0.592327 0.805697i $$-0.701791\pi$$
−0.592327 + 0.805697i $$0.701791\pi$$
$$152$$ − 1.03978e6i − 0.296080i
$$153$$ 0 0
$$154$$ 3.67066e6 1.00504
$$155$$ − 6.92245e6i − 1.85894i
$$156$$ 0 0
$$157$$ 6.15568e6 1.59066 0.795329 0.606178i $$-0.207298\pi$$
0.795329 + 0.606178i $$0.207298\pi$$
$$158$$ − 198646.i − 0.0503627i
$$159$$ 0 0
$$160$$ −1.00762e6 −0.246000
$$161$$ 1.63454e6i 0.391667i
$$162$$ 0 0
$$163$$ 800696. 0.184886 0.0924432 0.995718i $$-0.470532\pi$$
0.0924432 + 0.995718i $$0.470532\pi$$
$$164$$ 1.18536e6i 0.268732i
$$165$$ 0 0
$$166$$ 62112.0 0.0135785
$$167$$ 4.80467e6i 1.03161i 0.856707 + 0.515804i $$0.172507\pi$$
−0.856707 + 0.515804i $$0.827493\pi$$
$$168$$ 0 0
$$169$$ 6.51661e6 1.35009
$$170$$ − 12524.3i − 0.00254921i
$$171$$ 0 0
$$172$$ −121600. −0.0238973
$$173$$ − 3.56992e6i − 0.689478i −0.938699 0.344739i $$-0.887967\pi$$
0.938699 0.344739i $$-0.112033\pi$$
$$174$$ 0 0
$$175$$ 7.08237e6 1.32149
$$176$$ 1.37285e6i 0.251817i
$$177$$ 0 0
$$178$$ 731592. 0.129721
$$179$$ − 7.43698e6i − 1.29669i −0.761345 0.648347i $$-0.775461\pi$$
0.761345 0.648347i $$-0.224539\pi$$
$$180$$ 0 0
$$181$$ −1.03812e7 −1.75070 −0.875350 0.483491i $$-0.839369\pi$$
−0.875350 + 0.483491i $$0.839369\pi$$
$$182$$ − 9.22131e6i − 1.52960i
$$183$$ 0 0
$$184$$ −611328. −0.0981343
$$185$$ 9.13681e6i 1.44304i
$$186$$ 0 0
$$187$$ −17064.0 −0.00260949
$$188$$ − 2.45734e6i − 0.369821i
$$189$$ 0 0
$$190$$ −5.65210e6 −0.824041
$$191$$ 1.29941e7i 1.86485i 0.361360 + 0.932426i $$0.382313\pi$$
−0.361360 + 0.932426i $$0.617687\pi$$
$$192$$ 0 0
$$193$$ −3.93195e6 −0.546936 −0.273468 0.961881i $$-0.588171\pi$$
−0.273468 + 0.961881i $$0.588171\pi$$
$$194$$ − 1.81825e6i − 0.249028i
$$195$$ 0 0
$$196$$ −3.73142e6 −0.495572
$$197$$ 5.37967e6i 0.703651i 0.936066 + 0.351825i $$0.114439\pi$$
−0.936066 + 0.351825i $$0.885561\pi$$
$$198$$ 0 0
$$199$$ −565900. −0.0718093 −0.0359046 0.999355i $$-0.511431\pi$$
−0.0359046 + 0.999355i $$0.511431\pi$$
$$200$$ 2.64886e6i 0.331107i
$$201$$ 0 0
$$202$$ −3.78319e6 −0.458991
$$203$$ − 1.42077e7i − 1.69839i
$$204$$ 0 0
$$205$$ 6.44348e6 0.747926
$$206$$ 1.12764e7i 1.28994i
$$207$$ 0 0
$$208$$ 3.44883e6 0.383250
$$209$$ 7.70083e6i 0.843527i
$$210$$ 0 0
$$211$$ −1.35165e7 −1.43885 −0.719427 0.694568i $$-0.755596\pi$$
−0.719427 + 0.694568i $$0.755596\pi$$
$$212$$ − 7.63960e6i − 0.801795i
$$213$$ 0 0
$$214$$ 1.47456e6 0.150460
$$215$$ 661003.i 0.0665102i
$$216$$ 0 0
$$217$$ 1.92613e7 1.88497
$$218$$ 1.10001e6i 0.106176i
$$219$$ 0 0
$$220$$ 7.46266e6 0.700850
$$221$$ 42867.6i 0.00397148i
$$222$$ 0 0
$$223$$ −5.35484e6 −0.482872 −0.241436 0.970417i $$-0.577618\pi$$
−0.241436 + 0.970417i $$0.577618\pi$$
$$224$$ − 2.80363e6i − 0.249446i
$$225$$ 0 0
$$226$$ 4.64935e6 0.402779
$$227$$ − 1.36063e7i − 1.16322i −0.813466 0.581612i $$-0.802422\pi$$
0.813466 0.581612i $$-0.197578\pi$$
$$228$$ 0 0
$$229$$ 4.34641e6 0.361930 0.180965 0.983490i $$-0.442078\pi$$
0.180965 + 0.983490i $$0.442078\pi$$
$$230$$ 3.32311e6i 0.273125i
$$231$$ 0 0
$$232$$ 5.31379e6 0.425540
$$233$$ − 2.02333e7i − 1.59956i −0.600297 0.799778i $$-0.704951\pi$$
0.600297 0.799778i $$-0.295049\pi$$
$$234$$ 0 0
$$235$$ −1.33578e7 −1.02927
$$236$$ 7.99490e6i 0.608243i
$$237$$ 0 0
$$238$$ 34848.0 0.00258492
$$239$$ − 2.03947e7i − 1.49391i −0.664877 0.746953i $$-0.731516\pi$$
0.664877 0.746953i $$-0.268484\pi$$
$$240$$ 0 0
$$241$$ −3.12093e6 −0.222963 −0.111481 0.993767i $$-0.535560\pi$$
−0.111481 + 0.993767i $$0.535560\pi$$
$$242$$ − 146213.i − 0.0103167i
$$243$$ 0 0
$$244$$ −424000. −0.0291875
$$245$$ 2.02836e7i 1.37926i
$$246$$ 0 0
$$247$$ 1.93458e7 1.28379
$$248$$ 7.20385e6i 0.472291i
$$249$$ 0 0
$$250$$ −976128. −0.0624722
$$251$$ 5.09519e6i 0.322210i 0.986937 + 0.161105i $$0.0515058\pi$$
−0.986937 + 0.161105i $$0.948494\pi$$
$$252$$ 0 0
$$253$$ 4.52765e6 0.279583
$$254$$ 1.72942e7i 1.05536i
$$255$$ 0 0
$$256$$ 1.04858e6 0.0625000
$$257$$ 1.44374e7i 0.850529i 0.905069 + 0.425264i $$0.139819\pi$$
−0.905069 + 0.425264i $$0.860181\pi$$
$$258$$ 0 0
$$259$$ −2.54226e7 −1.46326
$$260$$ − 1.87474e7i − 1.06665i
$$261$$ 0 0
$$262$$ −1.73885e7 −0.966847
$$263$$ 3.12567e7i 1.71821i 0.511801 + 0.859104i $$0.328978\pi$$
−0.511801 + 0.859104i $$0.671022\pi$$
$$264$$ 0 0
$$265$$ −4.15280e7 −2.23153
$$266$$ − 1.57266e7i − 0.835584i
$$267$$ 0 0
$$268$$ −5.40698e6 −0.280899
$$269$$ 251338.i 0.0129122i 0.999979 + 0.00645612i $$0.00205506\pi$$
−0.999979 + 0.00645612i $$0.997945\pi$$
$$270$$ 0 0
$$271$$ 2.96399e7 1.48925 0.744627 0.667481i $$-0.232627\pi$$
0.744627 + 0.667481i $$0.232627\pi$$
$$272$$ 13033.4i 0 0.000647665i
$$273$$ 0 0
$$274$$ 2.53660e7 1.23311
$$275$$ − 1.96181e7i − 0.943319i
$$276$$ 0 0
$$277$$ 1.32213e7 0.622062 0.311031 0.950400i $$-0.399326\pi$$
0.311031 + 0.950400i $$0.399326\pi$$
$$278$$ − 6.17914e6i − 0.287603i
$$279$$ 0 0
$$280$$ −1.52402e7 −0.694251
$$281$$ − 6.12360e6i − 0.275987i −0.990433 0.137993i $$-0.955935\pi$$
0.990433 0.137993i $$-0.0440652\pi$$
$$282$$ 0 0
$$283$$ −6.74325e6 −0.297516 −0.148758 0.988874i $$-0.547527\pi$$
−0.148758 + 0.988874i $$0.547527\pi$$
$$284$$ 1.70069e7i 0.742457i
$$285$$ 0 0
$$286$$ −2.55429e7 −1.09187
$$287$$ 1.79286e7i 0.758403i
$$288$$ 0 0
$$289$$ 2.41374e7 0.999993
$$290$$ − 2.88852e7i − 1.18435i
$$291$$ 0 0
$$292$$ −7.55661e6 −0.303514
$$293$$ − 1.00239e7i − 0.398505i −0.979948 0.199253i $$-0.936149\pi$$
0.979948 0.199253i $$-0.0638514\pi$$
$$294$$ 0 0
$$295$$ 4.34593e7 1.69284
$$296$$ − 9.50822e6i − 0.366627i
$$297$$ 0 0
$$298$$ 1.25632e7 0.474734
$$299$$ − 1.13742e7i − 0.425508i
$$300$$ 0 0
$$301$$ −1.83920e6 −0.0674418
$$302$$ − 2.30727e7i − 0.837677i
$$303$$ 0 0
$$304$$ 5.88186e6 0.209360
$$305$$ 2.30481e6i 0.0812337i
$$306$$ 0 0
$$307$$ −5.23060e6 −0.180774 −0.0903871 0.995907i $$-0.528810\pi$$
−0.0903871 + 0.995907i $$0.528810\pi$$
$$308$$ 2.07644e7i 0.710668i
$$309$$ 0 0
$$310$$ 3.91593e7 1.31447
$$311$$ 3.12221e7i 1.03796i 0.854786 + 0.518981i $$0.173688\pi$$
−0.854786 + 0.518981i $$0.826312\pi$$
$$312$$ 0 0
$$313$$ 2.24778e7 0.733029 0.366515 0.930412i $$-0.380551\pi$$
0.366515 + 0.930412i $$0.380551\pi$$
$$314$$ 3.48218e7i 1.12477i
$$315$$ 0 0
$$316$$ 1.12371e6 0.0356118
$$317$$ − 2.76211e7i − 0.867088i −0.901132 0.433544i $$-0.857263\pi$$
0.901132 0.433544i $$-0.142737\pi$$
$$318$$ 0 0
$$319$$ −3.93553e7 −1.21236
$$320$$ − 5.69994e6i − 0.173948i
$$321$$ 0 0
$$322$$ −9.24634e6 −0.276950
$$323$$ 73109.2i 0.00216952i
$$324$$ 0 0
$$325$$ −4.92839e7 −1.43567
$$326$$ 4.52942e6i 0.130734i
$$327$$ 0 0
$$328$$ −6.70541e6 −0.190022
$$329$$ − 3.71672e7i − 1.04369i
$$330$$ 0 0
$$331$$ −5.76138e6 −0.158870 −0.0794352 0.996840i $$-0.525312\pi$$
−0.0794352 + 0.996840i $$0.525312\pi$$
$$332$$ 351359.i 0.00960144i
$$333$$ 0 0
$$334$$ −2.71793e7 −0.729456
$$335$$ 2.93917e7i 0.781790i
$$336$$ 0 0
$$337$$ −4.01052e7 −1.04788 −0.523939 0.851756i $$-0.675538\pi$$
−0.523939 + 0.851756i $$0.675538\pi$$
$$338$$ 3.68635e7i 0.954656i
$$339$$ 0 0
$$340$$ 70848.0 0.00180256
$$341$$ − 5.33535e7i − 1.34555i
$$342$$ 0 0
$$343$$ 504328. 0.0124977
$$344$$ − 687873.i − 0.0168979i
$$345$$ 0 0
$$346$$ 2.01945e7 0.487535
$$347$$ 6.78127e7i 1.62302i 0.584341 + 0.811508i $$0.301353\pi$$
−0.584341 + 0.811508i $$0.698647\pi$$
$$348$$ 0 0
$$349$$ −4.20638e7 −0.989538 −0.494769 0.869024i $$-0.664747\pi$$
−0.494769 + 0.869024i $$0.664747\pi$$
$$350$$ 4.00639e7i 0.934436i
$$351$$ 0 0
$$352$$ −7.76602e6 −0.178062
$$353$$ − 1.75976e7i − 0.400063i −0.979789 0.200032i $$-0.935896\pi$$
0.979789 0.200032i $$-0.0641045\pi$$
$$354$$ 0 0
$$355$$ 9.24478e7 2.06639
$$356$$ 4.13851e6i 0.0917263i
$$357$$ 0 0
$$358$$ 4.20699e7 0.916901
$$359$$ − 1.39920e7i − 0.302410i −0.988502 0.151205i $$-0.951685\pi$$
0.988502 0.151205i $$-0.0483154\pi$$
$$360$$ 0 0
$$361$$ −1.40523e7 −0.298694
$$362$$ − 5.87249e7i − 1.23793i
$$363$$ 0 0
$$364$$ 5.21636e7 1.08159
$$365$$ 4.10768e7i 0.844731i
$$366$$ 0 0
$$367$$ −2.65855e7 −0.537832 −0.268916 0.963164i $$-0.586665\pi$$
−0.268916 + 0.963164i $$0.586665\pi$$
$$368$$ − 3.45819e6i − 0.0693914i
$$369$$ 0 0
$$370$$ −5.16856e7 −1.02039
$$371$$ − 1.15549e8i − 2.26279i
$$372$$ 0 0
$$373$$ 1.78829e7 0.344598 0.172299 0.985045i $$-0.444881\pi$$
0.172299 + 0.985045i $$0.444881\pi$$
$$374$$ − 96528.6i − 0.00184519i
$$375$$ 0 0
$$376$$ 1.39008e7 0.261503
$$377$$ 9.88671e7i 1.84513i
$$378$$ 0 0
$$379$$ 7.20978e7 1.32435 0.662177 0.749347i $$-0.269633\pi$$
0.662177 + 0.749347i $$0.269633\pi$$
$$380$$ − 3.19731e7i − 0.582685i
$$381$$ 0 0
$$382$$ −7.35055e7 −1.31865
$$383$$ 8.68648e6i 0.154614i 0.997007 + 0.0773068i $$0.0246321\pi$$
−0.997007 + 0.0773068i $$0.975368\pi$$
$$384$$ 0 0
$$385$$ 1.12873e8 1.97791
$$386$$ − 2.22425e7i − 0.386742i
$$387$$ 0 0
$$388$$ 1.02856e7 0.176089
$$389$$ − 4.94411e7i − 0.839923i −0.907542 0.419962i $$-0.862044\pi$$
0.907542 0.419962i $$-0.137956\pi$$
$$390$$ 0 0
$$391$$ 42984.0 0.000719079 0
$$392$$ − 2.11081e7i − 0.350422i
$$393$$ 0 0
$$394$$ −3.04320e7 −0.497556
$$395$$ − 6.10837e6i − 0.0991137i
$$396$$ 0 0
$$397$$ 1.56911e7 0.250774 0.125387 0.992108i $$-0.459983\pi$$
0.125387 + 0.992108i $$0.459983\pi$$
$$398$$ − 3.20121e6i − 0.0507768i
$$399$$ 0 0
$$400$$ −1.49842e7 −0.234128
$$401$$ − 4.74514e7i − 0.735895i −0.929847 0.367947i $$-0.880061\pi$$
0.929847 0.367947i $$-0.119939\pi$$
$$402$$ 0 0
$$403$$ −1.34033e8 −2.04784
$$404$$ − 2.14010e7i − 0.324556i
$$405$$ 0 0
$$406$$ 8.03711e7 1.20094
$$407$$ 7.04203e7i 1.04451i
$$408$$ 0 0
$$409$$ −1.15512e8 −1.68832 −0.844162 0.536088i $$-0.819901\pi$$
−0.844162 + 0.536088i $$0.819901\pi$$
$$410$$ 3.64498e7i 0.528864i
$$411$$ 0 0
$$412$$ −6.37892e7 −0.912127
$$413$$ 1.20923e8i 1.71656i
$$414$$ 0 0
$$415$$ 1.90994e6 0.0267225
$$416$$ 1.95095e7i 0.270999i
$$417$$ 0 0
$$418$$ −4.35625e7 −0.596464
$$419$$ 1.46693e8i 1.99420i 0.0761306 + 0.997098i $$0.475743\pi$$
−0.0761306 + 0.997098i $$0.524257\pi$$
$$420$$ 0 0
$$421$$ 1.39239e8 1.86601 0.933005 0.359863i $$-0.117176\pi$$
0.933005 + 0.359863i $$0.117176\pi$$
$$422$$ − 7.64609e7i − 1.01742i
$$423$$ 0 0
$$424$$ 4.32161e7 0.566955
$$425$$ − 186248.i − 0.00242619i
$$426$$ 0 0
$$427$$ −6.41300e6 −0.0823716
$$428$$ 8.34137e6i 0.106391i
$$429$$ 0 0
$$430$$ −3.73920e6 −0.0470298
$$431$$ − 1.00392e8i − 1.25391i −0.779056 0.626954i $$-0.784301\pi$$
0.779056 0.626954i $$-0.215699\pi$$
$$432$$ 0 0
$$433$$ −4.00631e7 −0.493493 −0.246747 0.969080i $$-0.579362\pi$$
−0.246747 + 0.969080i $$0.579362\pi$$
$$434$$ 1.08958e8i 1.33288i
$$435$$ 0 0
$$436$$ −6.22259e6 −0.0750779
$$437$$ − 1.93983e7i − 0.232445i
$$438$$ 0 0
$$439$$ −1.38592e8 −1.63811 −0.819057 0.573712i $$-0.805503\pi$$
−0.819057 + 0.573712i $$0.805503\pi$$
$$440$$ 4.22152e7i 0.495576i
$$441$$ 0 0
$$442$$ −242496. −0.00280826
$$443$$ − 1.11443e8i − 1.28186i −0.767600 0.640929i $$-0.778549\pi$$
0.767600 0.640929i $$-0.221451\pi$$
$$444$$ 0 0
$$445$$ 2.24965e7 0.255290
$$446$$ − 3.02915e7i − 0.341442i
$$447$$ 0 0
$$448$$ 1.58597e7 0.176385
$$449$$ − 6.11166e7i − 0.675181i −0.941293 0.337591i $$-0.890388\pi$$
0.941293 0.337591i $$-0.109612\pi$$
$$450$$ 0 0
$$451$$ 4.96619e7 0.541370
$$452$$ 2.63007e7i 0.284808i
$$453$$ 0 0
$$454$$ 7.69691e7 0.822524
$$455$$ − 2.83555e8i − 3.01026i
$$456$$ 0 0
$$457$$ 3.56665e7 0.373690 0.186845 0.982389i $$-0.440174\pi$$
0.186845 + 0.982389i $$0.440174\pi$$
$$458$$ 2.45870e7i 0.255923i
$$459$$ 0 0
$$460$$ −1.87983e7 −0.193128
$$461$$ 1.51983e8i 1.55128i 0.631173 + 0.775642i $$0.282574\pi$$
−0.631173 + 0.775642i $$0.717426\pi$$
$$462$$ 0 0
$$463$$ 1.14978e8 1.15844 0.579218 0.815173i $$-0.303358\pi$$
0.579218 + 0.815173i $$0.303358\pi$$
$$464$$ 3.00593e7i 0.300902i
$$465$$ 0 0
$$466$$ 1.14457e8 1.13106
$$467$$ 8.81705e7i 0.865711i 0.901463 + 0.432855i $$0.142494\pi$$
−0.901463 + 0.432855i $$0.857506\pi$$
$$468$$ 0 0
$$469$$ −8.17805e7 −0.792741
$$470$$ − 7.55631e7i − 0.727807i
$$471$$ 0 0
$$472$$ −4.52260e7 −0.430093
$$473$$ 5.09456e6i 0.0481419i
$$474$$ 0 0
$$475$$ −8.40520e7 −0.784272
$$476$$ 197130.i 0.00182781i
$$477$$ 0 0
$$478$$ 1.15370e8 1.05635
$$479$$ − 8.94388e7i − 0.813803i −0.913472 0.406902i $$-0.866609\pi$$
0.913472 0.406902i $$-0.133391\pi$$
$$480$$ 0 0
$$481$$ 1.76908e8 1.58969
$$482$$ − 1.76546e7i − 0.157659i
$$483$$ 0 0
$$484$$ 827104. 0.00729498
$$485$$ − 5.59111e7i − 0.490087i
$$486$$ 0 0
$$487$$ −7.51688e7 −0.650805 −0.325403 0.945576i $$-0.605500\pi$$
−0.325403 + 0.945576i $$0.605500\pi$$
$$488$$ − 2.39851e6i − 0.0206387i
$$489$$ 0 0
$$490$$ −1.14741e8 −0.975285
$$491$$ 4.50822e7i 0.380856i 0.981701 + 0.190428i $$0.0609876\pi$$
−0.981701 + 0.190428i $$0.939012\pi$$
$$492$$ 0 0
$$493$$ −373626. −0.00311815
$$494$$ 1.09436e8i 0.907780i
$$495$$ 0 0
$$496$$ −4.07511e7 −0.333960
$$497$$ 2.57230e8i 2.09533i
$$498$$ 0 0
$$499$$ 9.15458e7 0.736778 0.368389 0.929672i $$-0.379909\pi$$
0.368389 + 0.929672i $$0.379909\pi$$
$$500$$ − 5.52181e6i − 0.0441745i
$$501$$ 0 0
$$502$$ −2.88228e7 −0.227837
$$503$$ − 1.61043e8i − 1.26543i −0.774386 0.632713i $$-0.781941\pi$$
0.774386 0.632713i $$-0.218059\pi$$
$$504$$ 0 0
$$505$$ −1.16333e8 −0.903295
$$506$$ 2.56122e7i 0.197695i
$$507$$ 0 0
$$508$$ −9.78308e7 −0.746250
$$509$$ 2.39995e7i 0.181990i 0.995851 + 0.0909951i $$0.0290048\pi$$
−0.995851 + 0.0909951i $$0.970995\pi$$
$$510$$ 0 0
$$511$$ −1.14294e8 −0.856564
$$512$$ 5.93164e6i 0.0441942i
$$513$$ 0 0
$$514$$ −8.16702e7 −0.601415
$$515$$ 3.46751e8i 2.53861i
$$516$$ 0 0
$$517$$ −1.02953e8 −0.745018
$$518$$ − 1.43812e8i − 1.03468i
$$519$$ 0 0
$$520$$ 1.06052e8 0.754236
$$521$$ − 9.00897e7i − 0.637033i −0.947917 0.318517i $$-0.896815\pi$$
0.947917 0.318517i $$-0.103185\pi$$
$$522$$ 0 0
$$523$$ −3.77691e7 −0.264016 −0.132008 0.991249i $$-0.542143\pi$$
−0.132008 + 0.991249i $$0.542143\pi$$
$$524$$ − 9.83641e7i − 0.683664i
$$525$$ 0 0
$$526$$ −1.76815e8 −1.21496
$$527$$ − 506520.i − 0.00346071i
$$528$$ 0 0
$$529$$ 1.36631e8 0.922957
$$530$$ − 2.34918e8i − 1.57793i
$$531$$ 0 0
$$532$$ 8.89631e7 0.590847
$$533$$ − 1.24759e8i − 0.823931i
$$534$$ 0 0
$$535$$ 4.53427e7 0.296105
$$536$$ − 3.05865e7i − 0.198626i
$$537$$ 0 0
$$538$$ −1.42178e6 −0.00913034
$$539$$ 1.56332e8i 0.998347i
$$540$$ 0 0
$$541$$ 2.54800e7 0.160919 0.0804595 0.996758i $$-0.474361\pi$$
0.0804595 + 0.996758i $$0.474361\pi$$
$$542$$ 1.67669e8i 1.05306i
$$543$$ 0 0
$$544$$ −73728.0 −0.000457969 0
$$545$$ 3.38253e7i 0.208955i
$$546$$ 0 0
$$547$$ 2.05216e8 1.25386 0.626930 0.779076i $$-0.284311\pi$$
0.626930 + 0.779076i $$0.284311\pi$$
$$548$$ 1.43492e8i 0.871938i
$$549$$ 0 0
$$550$$ 1.10977e8 0.667027
$$551$$ 1.68614e8i 1.00795i
$$552$$ 0 0
$$553$$ 1.69961e7 0.100502
$$554$$ 7.47908e7i 0.439865i
$$555$$ 0 0
$$556$$ 3.49545e7 0.203366
$$557$$ − 2.41143e8i − 1.39543i −0.716375 0.697715i $$-0.754200\pi$$
0.716375 0.697715i $$-0.245800\pi$$
$$558$$ 0 0
$$559$$ 1.27984e7 0.0732690
$$560$$ − 8.62115e7i − 0.490909i
$$561$$ 0 0
$$562$$ 3.46403e7 0.195152
$$563$$ − 1.68877e8i − 0.946337i −0.880972 0.473168i $$-0.843110\pi$$
0.880972 0.473168i $$-0.156890\pi$$
$$564$$ 0 0
$$565$$ 1.42968e8 0.792670
$$566$$ − 3.81456e7i − 0.210375i
$$567$$ 0 0
$$568$$ −9.62058e7 −0.524996
$$569$$ − 2.43995e8i − 1.32448i −0.749293 0.662238i $$-0.769607\pi$$
0.749293 0.662238i $$-0.230393\pi$$
$$570$$ 0 0
$$571$$ 2.41502e8 1.29722 0.648608 0.761123i $$-0.275352\pi$$
0.648608 + 0.761123i $$0.275352\pi$$
$$572$$ − 1.44493e8i − 0.772071i
$$573$$ 0 0
$$574$$ −1.01419e8 −0.536272
$$575$$ 4.94177e7i 0.259944i
$$576$$ 0 0
$$577$$ −4.93979e7 −0.257147 −0.128573 0.991700i $$-0.541040\pi$$
−0.128573 + 0.991700i $$0.541040\pi$$
$$578$$ 1.36542e8i 0.707102i
$$579$$ 0 0
$$580$$ 1.63399e8 0.837463
$$581$$ 5.31430e6i 0.0270968i
$$582$$ 0 0
$$583$$ −3.20069e8 −1.61525
$$584$$ − 4.27466e7i − 0.214617i
$$585$$ 0 0
$$586$$ 5.67038e7 0.281786
$$587$$ − 1.72052e8i − 0.850639i −0.905043 0.425320i $$-0.860162\pi$$
0.905043 0.425320i $$-0.139838\pi$$
$$588$$ 0 0
$$589$$ −2.28588e8 −1.11869
$$590$$ 2.45843e8i 1.19702i
$$591$$ 0 0
$$592$$ 5.37866e7 0.259244
$$593$$ 2.70643e8i 1.29788i 0.760841 + 0.648938i $$0.224787\pi$$
−0.760841 + 0.648938i $$0.775213\pi$$
$$594$$ 0 0
$$595$$ 1.07158e6 0.00508712
$$596$$ 7.10680e7i 0.335687i
$$597$$ 0 0
$$598$$ 6.43423e7 0.300880
$$599$$ − 1.73299e8i − 0.806337i −0.915126 0.403169i $$-0.867909\pi$$
0.915126 0.403169i $$-0.132091\pi$$
$$600$$ 0 0
$$601$$ −4.31090e8 −1.98584 −0.992921 0.118775i $$-0.962103\pi$$
−0.992921 + 0.118775i $$0.962103\pi$$
$$602$$ − 1.04041e7i − 0.0476886i
$$603$$ 0 0
$$604$$ 1.30519e8 0.592327
$$605$$ − 4.49604e6i − 0.0203032i
$$606$$ 0 0
$$607$$ 1.66991e7 0.0746665 0.0373332 0.999303i $$-0.488114\pi$$
0.0373332 + 0.999303i $$0.488114\pi$$
$$608$$ 3.32728e7i 0.148040i
$$609$$ 0 0
$$610$$ −1.30380e7 −0.0574409
$$611$$ 2.58635e8i 1.13387i
$$612$$ 0 0
$$613$$ −1.92321e8 −0.834920 −0.417460 0.908695i $$-0.637080\pi$$
−0.417460 + 0.908695i $$0.637080\pi$$
$$614$$ − 2.95887e7i − 0.127827i
$$615$$ 0 0
$$616$$ −1.17461e8 −0.502518
$$617$$ − 1.87023e8i − 0.796233i −0.917335 0.398117i $$-0.869664\pi$$
0.917335 0.398117i $$-0.130336\pi$$
$$618$$ 0 0
$$619$$ 2.54873e8 1.07461 0.537307 0.843387i $$-0.319442\pi$$
0.537307 + 0.843387i $$0.319442\pi$$
$$620$$ 2.21518e8i 0.929468i
$$621$$ 0 0
$$622$$ −1.76619e8 −0.733950
$$623$$ 6.25950e7i 0.258866i
$$624$$ 0 0
$$625$$ −2.58657e8 −1.05946
$$626$$ 1.27154e8i 0.518330i
$$627$$ 0 0
$$628$$ −1.96982e8 −0.795329
$$629$$ 668547.i 0.00268646i
$$630$$ 0 0
$$631$$ 9.23602e7 0.367618 0.183809 0.982962i $$-0.441157\pi$$
0.183809 + 0.982962i $$0.441157\pi$$
$$632$$ 6.35668e6i 0.0251813i
$$633$$ 0 0
$$634$$ 1.56249e8 0.613124
$$635$$ 5.31797e8i 2.07694i
$$636$$ 0 0
$$637$$ 3.92732e8 1.51942
$$638$$ − 2.22627e8i − 0.857267i
$$639$$ 0 0
$$640$$ 3.22437e7 0.123000
$$641$$ − 4.24666e8i − 1.61240i −0.591643 0.806200i $$-0.701520\pi$$
0.591643 0.806200i $$-0.298480\pi$$
$$642$$ 0 0
$$643$$ 3.75946e8 1.41414 0.707071 0.707143i $$-0.250016\pi$$
0.707071 + 0.707143i $$0.250016\pi$$
$$644$$ − 5.23052e7i − 0.195834i
$$645$$ 0 0
$$646$$ −413568. −0.00153408
$$647$$ 2.63747e7i 0.0973813i 0.998814 + 0.0486906i $$0.0155048\pi$$
−0.998814 + 0.0486906i $$0.984495\pi$$
$$648$$ 0 0
$$649$$ 3.34955e8 1.22533
$$650$$ − 2.78792e8i − 1.01517i
$$651$$ 0 0
$$652$$ −2.56223e7 −0.0924432
$$653$$ 2.58756e8i 0.929291i 0.885497 + 0.464645i $$0.153818\pi$$
−0.885497 + 0.464645i $$0.846182\pi$$
$$654$$ 0 0
$$655$$ −5.34696e8 −1.90275
$$656$$ − 3.79315e7i − 0.134366i
$$657$$ 0 0
$$658$$ 2.10250e8 0.738002
$$659$$ − 1.39345e8i − 0.486895i −0.969914 0.243447i $$-0.921722\pi$$
0.969914 0.243447i $$-0.0782783\pi$$
$$660$$ 0 0
$$661$$ −4.72545e8 −1.63621 −0.818104 0.575070i $$-0.804975\pi$$
−0.818104 + 0.575070i $$0.804975\pi$$
$$662$$ − 3.25913e7i − 0.112338i
$$663$$ 0 0
$$664$$ −1.98758e6 −0.00678924
$$665$$ − 4.83593e8i − 1.64443i
$$666$$ 0 0
$$667$$ 9.91354e7 0.334081
$$668$$ − 1.53749e8i − 0.515804i
$$669$$ 0 0
$$670$$ −1.66265e8 −0.552809
$$671$$ 1.77639e7i 0.0587992i
$$672$$ 0 0
$$673$$ 5.48833e8 1.80051 0.900254 0.435364i $$-0.143380\pi$$
0.900254 + 0.435364i $$0.143380\pi$$
$$674$$ − 2.26869e8i − 0.740961i
$$675$$ 0 0
$$676$$ −2.08532e8 −0.675044
$$677$$ − 1.00760e8i − 0.324731i −0.986731 0.162365i $$-0.948088\pi$$
0.986731 0.162365i $$-0.0519123\pi$$
$$678$$ 0 0
$$679$$ 1.55569e8 0.496952
$$680$$ 400777.i 0.00127461i
$$681$$ 0 0
$$682$$ 3.01813e8 0.951447
$$683$$ − 313056.i 0 0.000982562i −1.00000 0.000491281i $$-0.999844\pi$$
1.00000 0.000491281i $$-0.000156380\pi$$
$$684$$ 0 0
$$685$$ 7.80005e8 2.42675
$$686$$ 2.85291e6i 0.00883722i
$$687$$ 0 0
$$688$$ 3.89120e6 0.0119486
$$689$$ 8.04068e8i 2.45830i
$$690$$ 0 0
$$691$$ −3.72812e8 −1.12994 −0.564971 0.825111i $$-0.691113\pi$$
−0.564971 + 0.825111i $$0.691113\pi$$
$$692$$ 1.14238e8i 0.344739i
$$693$$ 0 0
$$694$$ −3.83607e8 −1.14765
$$695$$ − 1.90009e8i − 0.566003i
$$696$$ 0 0
$$697$$ 471474. 0.00139239
$$698$$ − 2.37949e8i − 0.699709i
$$699$$ 0 0
$$700$$ −2.26636e8 −0.660746
$$701$$ 6.21170e8i 1.80325i 0.432517 + 0.901626i $$0.357626\pi$$
−0.432517 + 0.901626i $$0.642374\pi$$
$$702$$ 0 0
$$703$$ 3.01709e8 0.868406
$$704$$ − 4.39312e7i − 0.125909i
$$705$$ 0 0
$$706$$ 9.95469e7 0.282887
$$707$$ − 3.23690e8i − 0.915947i
$$708$$ 0 0
$$709$$ −2.46510e8 −0.691666 −0.345833 0.938296i $$-0.612404\pi$$
−0.345833 + 0.938296i $$0.612404\pi$$
$$710$$ 5.22964e8i 1.46116i
$$711$$ 0 0
$$712$$ −2.34109e7 −0.0648603
$$713$$ 1.34397e8i 0.370783i
$$714$$ 0 0
$$715$$ −7.85445e8 −2.14881
$$716$$ 2.37983e8i 0.648347i
$$717$$ 0 0
$$718$$ 7.91508e7 0.213836
$$719$$ − 9.60389e7i − 0.258381i −0.991620 0.129191i $$-0.958762\pi$$
0.991620 0.129191i $$-0.0412379\pi$$
$$720$$ 0 0
$$721$$ −9.64811e8 −2.57417
$$722$$ − 7.94921e7i − 0.211209i
$$723$$ 0 0
$$724$$ 3.32198e8 0.875350
$$725$$ − 4.29549e8i − 1.12719i
$$726$$ 0 0
$$727$$ 3.91371e8 1.01856 0.509278 0.860602i $$-0.329912\pi$$
0.509278 + 0.860602i $$0.329912\pi$$
$$728$$ 2.95082e8i 0.764801i
$$729$$ 0 0
$$730$$ −2.32366e8 −0.597315
$$731$$ 48366.1i 0 0.000123819i
$$732$$ 0 0
$$733$$ 3.49078e7 0.0886361 0.0443181 0.999017i $$-0.485889\pi$$
0.0443181 + 0.999017i $$0.485889\pi$$
$$734$$ − 1.50390e8i − 0.380304i
$$735$$ 0 0
$$736$$ 1.95625e7 0.0490671
$$737$$ 2.26531e8i 0.565881i
$$738$$ 0 0
$$739$$ −3.02999e8 −0.750773 −0.375386 0.926868i $$-0.622490\pi$$
−0.375386 + 0.926868i $$0.622490\pi$$
$$740$$ − 2.92378e8i − 0.721521i
$$741$$ 0 0
$$742$$ 6.53644e8 1.60004
$$743$$ 2.45628e8i 0.598842i 0.954121 + 0.299421i $$0.0967935\pi$$
−0.954121 + 0.299421i $$0.903207\pi$$
$$744$$ 0 0
$$745$$ 3.86317e8 0.934276
$$746$$ 1.01161e8i 0.243667i
$$747$$ 0 0
$$748$$ 546048. 0.00130475
$$749$$ 1.26163e8i 0.300253i
$$750$$ 0 0
$$751$$ −8.23270e7 −0.194367 −0.0971835 0.995266i $$-0.530983\pi$$
−0.0971835 + 0.995266i $$0.530983\pi$$
$$752$$ 7.86348e7i 0.184910i
$$753$$ 0 0
$$754$$ −5.59277e8 −1.30471
$$755$$ − 7.09484e8i − 1.64855i
$$756$$ 0 0
$$757$$ −6.03579e8 −1.39138 −0.695691 0.718341i $$-0.744902\pi$$
−0.695691 + 0.718341i $$0.744902\pi$$
$$758$$ 4.07847e8i 0.936460i
$$759$$ 0 0
$$760$$ 1.80867e8 0.412020
$$761$$ − 2.32982e8i − 0.528651i −0.964434 0.264325i $$-0.914851\pi$$
0.964434 0.264325i $$-0.0851493\pi$$
$$762$$ 0 0
$$763$$ −9.41167e7 −0.211882
$$764$$ − 4.15810e8i − 0.932426i
$$765$$ 0 0
$$766$$ −4.91382e7 −0.109328
$$767$$ − 8.41463e8i − 1.86487i
$$768$$ 0 0
$$769$$ 8.15796e8 1.79392 0.896958 0.442115i $$-0.145772\pi$$
0.896958 + 0.442115i $$0.145772\pi$$
$$770$$ 6.38504e8i 1.39859i
$$771$$ 0 0
$$772$$ 1.25823e8 0.273468
$$773$$ − 3.66587e8i − 0.793667i −0.917891 0.396833i $$-0.870109\pi$$
0.917891 0.396833i $$-0.129891\pi$$
$$774$$ 0 0
$$775$$ 5.82335e8 1.25103
$$776$$ 5.81840e7i 0.124514i
$$777$$ 0 0
$$778$$ 2.79681e8 0.593915
$$779$$ − 2.12772e8i − 0.450093i
$$780$$ 0 0
$$781$$ 7.12524e8 1.49571
$$782$$ 243154.i 0 0.000508466i
$$783$$ 0 0
$$784$$ 1.19406e8 0.247786
$$785$$ 1.07077e9i 2.21354i
$$786$$ 0 0
$$787$$ −4.02462e8 −0.825659 −0.412830 0.910808i $$-0.635460\pi$$
−0.412830 + 0.910808i $$0.635460\pi$$
$$788$$ − 1.72150e8i − 0.351825i
$$789$$ 0 0
$$790$$ 3.45541e7 0.0700840
$$791$$ 3.97798e8i 0.803773i
$$792$$ 0 0
$$793$$ 4.46260e7 0.0894887
$$794$$ 8.87623e7i 0.177324i
$$795$$ 0 0
$$796$$ 1.81088e7 0.0359046
$$797$$ − 5.18940e8i − 1.02504i −0.858675 0.512521i $$-0.828712\pi$$
0.858675 0.512521i $$-0.171288\pi$$
$$798$$ 0 0
$$799$$ −977400. −0.00191616
$$800$$ − 8.47634e7i − 0.165553i
$$801$$ 0 0
$$802$$ 2.68426e8 0.520356
$$803$$ 3.16592e8i 0.611440i
$$804$$ 0 0
$$805$$ −2.84325e8 −0.545038
$$806$$ − 7.58205e8i − 1.44804i
$$807$$ 0 0
$$808$$ 1.21062e8 0.229496
$$809$$ − 3.04036e6i − 0.00574221i −0.999996 0.00287110i $$-0.999086\pi$$
0.999996 0.00287110i $$-0.000913902\pi$$
$$810$$ 0 0
$$811$$ 2.25521e8 0.422790 0.211395 0.977401i $$-0.432199\pi$$
0.211395 + 0.977401i $$0.432199\pi$$
$$812$$ 4.54648e8i 0.849194i
$$813$$ 0 0
$$814$$ −3.98357e8 −0.738583
$$815$$ 1.39280e8i 0.257285i
$$816$$ 0 0
$$817$$ 2.18272e7 0.0400250
$$818$$ − 6.53432e8i − 1.19383i
$$819$$ 0 0
$$820$$ −2.06191e8 −0.373963
$$821$$ 2.77035e8i 0.500617i 0.968166 + 0.250309i $$0.0805321\pi$$
−0.968166 + 0.250309i $$0.919468\pi$$
$$822$$ 0 0
$$823$$ −7.07336e8 −1.26890 −0.634448 0.772965i $$-0.718773\pi$$
−0.634448 + 0.772965i $$0.718773\pi$$
$$824$$ − 3.60846e8i − 0.644971i
$$825$$ 0 0
$$826$$ −6.84043e8 −1.21379
$$827$$ − 2.66346e8i − 0.470900i −0.971886 0.235450i $$-0.924344\pi$$
0.971886 0.235450i $$-0.0756564\pi$$
$$828$$ 0 0
$$829$$ 5.03826e8 0.884336 0.442168 0.896932i $$-0.354209\pi$$
0.442168 + 0.896932i $$0.354209\pi$$
$$830$$ 1.08043e7i 0.0188956i
$$831$$ 0 0
$$832$$ −1.10363e8 −0.191625
$$833$$ 1.48416e6i 0.00256772i
$$834$$ 0 0
$$835$$ −8.35764e8 −1.43557
$$836$$ − 2.46427e8i − 0.421763i
$$837$$ 0 0
$$838$$ −8.29822e8 −1.41011
$$839$$ 7.63364e8i 1.29255i 0.763106 + 0.646273i $$0.223673\pi$$
−0.763106 + 0.646273i $$0.776327\pi$$
$$840$$ 0 0
$$841$$ −2.66883e8 −0.448676
$$842$$ 7.87654e8i 1.31947i
$$843$$ 0 0
$$844$$ 4.32528e8 0.719427
$$845$$ 1.13355e9i 1.87876i
$$846$$ 0 0
$$847$$ 1.25099e7 0.0205876
$$848$$ 2.44467e8i 0.400897i
$$849$$ 0 0
$$850$$ 1.05358e6 0.00171557
$$851$$ − 1.77388e8i − 0.287829i
$$852$$ 0 0
$$853$$ −1.87985e7 −0.0302884 −0.0151442 0.999885i $$-0.504821\pi$$
−0.0151442 + 0.999885i $$0.504821\pi$$
$$854$$ − 3.62774e7i − 0.0582455i
$$855$$ 0 0
$$856$$ −4.71859e7 −0.0752300
$$857$$ − 6.86427e8i − 1.09057i −0.838252 0.545283i $$-0.816422\pi$$
0.838252 0.545283i $$-0.183578\pi$$
$$858$$ 0 0
$$859$$ 5.51932e8 0.870775 0.435387 0.900243i $$-0.356611\pi$$
0.435387 + 0.900243i $$0.356611\pi$$
$$860$$ − 2.11521e7i − 0.0332551i
$$861$$ 0 0
$$862$$ 5.67901e8 0.886647
$$863$$ 3.65665e8i 0.568920i 0.958688 + 0.284460i $$0.0918142\pi$$
−0.958688 + 0.284460i $$0.908186\pi$$
$$864$$ 0 0
$$865$$ 6.20982e8 0.959468
$$866$$ − 2.26631e8i − 0.348952i
$$867$$ 0 0
$$868$$ −6.16360e8 −0.942487
$$869$$ − 4.70791e7i − 0.0717413i
$$870$$ 0 0
$$871$$ 5.69084e8 0.861236
$$872$$ − 3.52003e7i − 0.0530881i
$$873$$ 0 0
$$874$$ 1.09733e8 0.164363
$$875$$ − 8.35174e7i − 0.124667i
$$876$$ 0 0
$$877$$ −5.85387e8 −0.867849 −0.433925 0.900949i $$-0.642872\pi$$
−0.433925 + 0.900949i $$0.642872\pi$$
$$878$$ − 7.83994e8i − 1.15832i
$$879$$ 0 0
$$880$$ −2.38805e8 −0.350425
$$881$$ − 4.29761e8i − 0.628491i −0.949342 0.314246i $$-0.898248\pi$$
0.949342 0.314246i $$-0.101752\pi$$
$$882$$ 0 0
$$883$$ 2.20085e8 0.319675 0.159837 0.987143i $$-0.448903\pi$$
0.159837 + 0.987143i $$0.448903\pi$$
$$884$$ − 1.37176e6i − 0.00198574i
$$885$$ 0 0
$$886$$ 6.30415e8 0.906411
$$887$$ 1.17196e9i 1.67936i 0.543084 + 0.839678i $$0.317256\pi$$
−0.543084 + 0.839678i $$0.682744\pi$$
$$888$$ 0 0
$$889$$ −1.47969e9 −2.10604
$$890$$ 1.27259e8i 0.180517i
$$891$$ 0 0
$$892$$ 1.71355e8 0.241436
$$893$$ 4.41092e8i 0.619405i
$$894$$ 0 0
$$895$$ 1.29365e9 1.80446
$$896$$ 8.97161e7i 0.124723i
$$897$$ 0 0
$$898$$ 3.45728e8 0.477425
$$899$$ − 1.16820e9i − 1.60783i
$$900$$ 0 0
$$901$$ −3.03863e6 −0.00415436
$$902$$ 2.80930e8i 0.382806i
$$903$$ 0 0
$$904$$ −1.48779e8 −0.201390
$$905$$ − 1.80579e9i − 2.43625i
$$906$$ 0 0
$$907$$ 7.31614e8 0.980529 0.490264 0.871574i $$-0.336900\pi$$
0.490264 + 0.871574i $$0.336900\pi$$
$$908$$ 4.35403e8i 0.581612i
$$909$$ 0 0
$$910$$ 1.60403e9 2.12857
$$911$$ 9.18595e8i 1.21498i 0.794327 + 0.607490i $$0.207823\pi$$
−0.794327 + 0.607490i $$0.792177\pi$$
$$912$$ 0 0
$$913$$ 1.47205e7 0.0193425
$$914$$ 2.01760e8i 0.264239i
$$915$$ 0 0
$$916$$ −1.39085e8 −0.180965
$$917$$ − 1.48776e9i − 1.92941i
$$918$$ 0 0
$$919$$ −2.15987e8 −0.278279 −0.139139 0.990273i $$-0.544434\pi$$
−0.139139 + 0.990273i $$0.544434\pi$$
$$920$$ − 1.06339e8i − 0.136562i
$$921$$ 0 0
$$922$$ −8.59744e8 −1.09692
$$923$$ − 1.78998e9i − 2.27637i
$$924$$ 0 0
$$925$$ −7.68613e8 −0.971141
$$926$$ 6.50414e8i 0.819138i
$$927$$ 0 0
$$928$$ −1.70041e8 −0.212770
$$929$$ 3.10124e8i 0.386802i 0.981120 + 0.193401i $$0.0619518\pi$$
−0.981120 + 0.193401i $$0.938048\pi$$
$$930$$ 0 0
$$931$$ 6.69791e8 0.830023
$$932$$ 6.47466e8i 0.799778i
$$933$$ 0 0
$$934$$ −4.98768e8 −0.612150
$$935$$ − 2.96825e6i − 0.00363133i
$$936$$ 0 0
$$937$$ −7.42448e8 −0.902501 −0.451250 0.892397i $$-0.649022\pi$$
−0.451250 + 0.892397i $$0.649022\pi$$
$$938$$ − 4.62620e8i − 0.560553i
$$939$$ 0 0
$$940$$ 4.27450e8 0.514637
$$941$$ 1.81766e8i 0.218144i 0.994034 + 0.109072i $$0.0347879\pi$$
−0.994034 + 0.109072i $$0.965212\pi$$
$$942$$ 0 0
$$943$$ −1.25098e8 −0.149181
$$944$$ − 2.55837e8i − 0.304121i
$$945$$ 0 0
$$946$$ −2.88192e7 −0.0340415
$$947$$ 8.59189e8i 1.01167i 0.862630 + 0.505835i $$0.168816\pi$$
−0.862630 + 0.505835i $$0.831184\pi$$
$$948$$ 0 0
$$949$$ 7.95333e8 0.930573
$$950$$ − 4.75470e8i − 0.554564i
$$951$$ 0 0
$$952$$ −1.11514e6 −0.00129246
$$953$$ 6.86819e8i 0.793530i 0.917920 + 0.396765i $$0.129867\pi$$
−0.917920 + 0.396765i $$0.870133\pi$$
$$954$$ 0 0
$$955$$ −2.26029e9 −2.59510
$$956$$ 6.52630e8i 0.746953i
$$957$$ 0 0
$$958$$ 5.05942e8 0.575446
$$959$$ 2.17031e9i 2.46075i
$$960$$ 0 0
$$961$$ 6.96218e8 0.784468
$$962$$ 1.00074e9i 1.12408i
$$963$$ 0 0
$$964$$ 9.98697e7 0.111481
$$965$$ − 6.83957e8i − 0.761109i
$$966$$ 0 0
$$967$$ −1.09411e9 −1.20999 −0.604995 0.796230i $$-0.706825\pi$$
−0.604995 + 0.796230i $$0.706825\pi$$
$$968$$ 4.67881e6i 0.00515833i
$$969$$ 0 0
$$970$$ 3.16281e8 0.346544
$$971$$ 4.43115e8i 0.484014i 0.970274 + 0.242007i $$0.0778058\pi$$
−0.970274 + 0.242007i $$0.922194\pi$$
$$972$$ 0 0
$$973$$ 5.28687e8 0.573931
$$974$$ − 4.25219e8i − 0.460189i
$$975$$ 0 0
$$976$$ 1.35680e7 0.0145937
$$977$$ 1.19004e9i 1.27608i 0.770001 + 0.638042i $$0.220256\pi$$
−0.770001 + 0.638042i $$0.779744\pi$$
$$978$$ 0 0
$$979$$ 1.73387e8 0.184786
$$980$$ − 6.49075e8i − 0.689631i
$$981$$ 0 0
$$982$$ −2.55024e8 −0.269306
$$983$$ − 1.18187e9i − 1.24425i −0.782918 0.622125i $$-0.786270\pi$$
0.782918 0.622125i $$-0.213730\pi$$
$$984$$ 0 0
$$985$$ −9.35785e8 −0.979191
$$986$$ − 2.11355e6i − 0.00220486i
$$987$$ 0 0
$$988$$ −6.19065e8 −0.641897
$$989$$ − 1.28331e7i − 0.0132661i
$$990$$ 0 0
$$991$$ 5.09602e8 0.523613 0.261806 0.965120i $$-0.415682\pi$$
0.261806 + 0.965120i $$0.415682\pi$$
$$992$$ − 2.30523e8i − 0.236145i
$$993$$ 0 0
$$994$$ −1.45511e9 −1.48162
$$995$$ − 9.84373e7i − 0.0999288i
$$996$$ 0 0
$$997$$ −9.90780e8 −0.999751 −0.499875 0.866097i $$-0.666621\pi$$
−0.499875 + 0.866097i $$0.666621\pi$$
$$998$$ 5.17861e8i 0.520981i
$$999$$ 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 18.7.b.a.17.2 yes 2
3.2 odd 2 inner 18.7.b.a.17.1 2
4.3 odd 2 144.7.e.d.17.2 2
5.2 odd 4 450.7.b.a.449.1 4
5.3 odd 4 450.7.b.a.449.4 4
5.4 even 2 450.7.d.a.251.1 2
8.3 odd 2 576.7.e.k.449.1 2
8.5 even 2 576.7.e.b.449.1 2
9.2 odd 6 162.7.d.d.53.1 4
9.4 even 3 162.7.d.d.107.1 4
9.5 odd 6 162.7.d.d.107.2 4
9.7 even 3 162.7.d.d.53.2 4
12.11 even 2 144.7.e.d.17.1 2
15.2 even 4 450.7.b.a.449.3 4
15.8 even 4 450.7.b.a.449.2 4
15.14 odd 2 450.7.d.a.251.2 2
24.5 odd 2 576.7.e.b.449.2 2
24.11 even 2 576.7.e.k.449.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.b.a.17.1 2 3.2 odd 2 inner
18.7.b.a.17.2 yes 2 1.1 even 1 trivial
144.7.e.d.17.1 2 12.11 even 2
144.7.e.d.17.2 2 4.3 odd 2
162.7.d.d.53.1 4 9.2 odd 6
162.7.d.d.53.2 4 9.7 even 3
162.7.d.d.107.1 4 9.4 even 3
162.7.d.d.107.2 4 9.5 odd 6
450.7.b.a.449.1 4 5.2 odd 4
450.7.b.a.449.2 4 15.8 even 4
450.7.b.a.449.3 4 15.2 even 4
450.7.b.a.449.4 4 5.3 odd 4
450.7.d.a.251.1 2 5.4 even 2
450.7.d.a.251.2 2 15.14 odd 2
576.7.e.b.449.1 2 8.5 even 2
576.7.e.b.449.2 2 24.5 odd 2
576.7.e.k.449.1 2 8.3 odd 2
576.7.e.k.449.2 2 24.11 even 2