# Properties

 Label 300.9.b.b.149.1 Level $300$ Weight $9$ Character 300.149 Analytic conductor $122.214$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(149,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 300.b (of order $$2$$, degree $$1$$, not 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: $$122.213583018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 149.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 300.149 Dual form 300.9.b.b.149.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-81.0000i q^{3} +4034.00i q^{7} -6561.00 q^{9} +O(q^{10})$$ $$q-81.0000i q^{3} +4034.00i q^{7} -6561.00 q^{9} +35806.0i q^{13} +258526. q^{19} +326754. q^{21} +531441. i q^{27} -1.80941e6 q^{31} +503522. i q^{37} +2.90029e6 q^{39} -3.49219e6i q^{43} -1.05084e7 q^{49} -2.09406e7i q^{57} -2.38265e7 q^{61} -2.64671e7i q^{63} -5.42141e6i q^{67} -1.61693e7i q^{73} +1.88870e7 q^{79} +4.30467e7 q^{81} -1.44441e8 q^{91} +1.46562e8i q^{93} +1.76908e8i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 13122 q^{9}+O(q^{10})$$ 2 * q - 13122 * q^9 $$2 q - 13122 q^{9} + 517052 q^{19} + 653508 q^{21} - 3618812 q^{31} + 5800572 q^{39} - 21016710 q^{49} - 47653052 q^{61} + 37774076 q^{79} + 86093442 q^{81} - 288882808 q^{91}+O(q^{100})$$ 2 * q - 13122 * q^9 + 517052 * q^19 + 653508 * q^21 - 3618812 * q^31 + 5800572 * q^39 - 21016710 * q^49 - 47653052 * q^61 + 37774076 * q^79 + 86093442 * q^81 - 288882808 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 81.0000i − 1.00000i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4034.00i 1.68013i 0.542483 + 0.840067i $$0.317484\pi$$
−0.542483 + 0.840067i $$0.682516\pi$$
$$8$$ 0 0
$$9$$ −6561.00 −1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 35806.0i 1.25367i 0.779153 + 0.626834i $$0.215650\pi$$
−0.779153 + 0.626834i $$0.784350\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 258526. 1.98376 0.991882 0.127165i $$-0.0405878\pi$$
0.991882 + 0.127165i $$0.0405878\pi$$
$$20$$ 0 0
$$21$$ 326754. 1.68013
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 531441.i 1.00000i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −1.80941e6 −1.95925 −0.979624 0.200842i $$-0.935632\pi$$
−0.979624 + 0.200842i $$0.935632\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 503522.i 0.268665i 0.990936 + 0.134333i $$0.0428891\pi$$
−0.990936 + 0.134333i $$0.957111\pi$$
$$38$$ 0 0
$$39$$ 2.90029e6 1.25367
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ − 3.49219e6i − 1.02147i −0.859739 0.510734i $$-0.829374\pi$$
0.859739 0.510734i $$-0.170626\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −1.05084e7 −1.82285
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.09406e7i − 1.98376i
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ −2.38265e7 −1.72084 −0.860422 0.509583i $$-0.829800\pi$$
−0.860422 + 0.509583i $$0.829800\pi$$
$$62$$ 0 0
$$63$$ − 2.64671e7i − 1.68013i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 5.42141e6i − 0.269037i −0.990911 0.134519i $$-0.957051\pi$$
0.990911 0.134519i $$-0.0429488\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ − 1.61693e7i − 0.569376i −0.958620 0.284688i $$-0.908110\pi$$
0.958620 0.284688i $$-0.0918900\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.88870e7 0.484904 0.242452 0.970163i $$-0.422048\pi$$
0.242452 + 0.970163i $$0.422048\pi$$
$$80$$ 0 0
$$81$$ 4.30467e7 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ −1.44441e8 −2.10633
$$92$$ 0 0
$$93$$ 1.46562e8i 1.95925i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.76908e8i 1.99830i 0.0412262 + 0.999150i $$0.486874\pi$$
−0.0412262 + 0.999150i $$0.513126\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ − 4.44490e7i − 0.394923i −0.980311 0.197462i $$-0.936730\pi$$
0.980311 0.197462i $$-0.0632698\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −2.03181e8 −1.43938 −0.719692 0.694293i $$-0.755717\pi$$
−0.719692 + 0.694293i $$0.755717\pi$$
$$110$$ 0 0
$$111$$ 4.07853e7 0.268665
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.34923e8i − 1.25367i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 2.14359e8 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 4.00562e8i − 1.53977i −0.638185 0.769883i $$-0.720314\pi$$
0.638185 0.769883i $$-0.279686\pi$$
$$128$$ 0 0
$$129$$ −2.82868e8 −1.02147
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 1.04289e9i 3.33299i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ −7.09431e8 −1.90043 −0.950213 0.311602i $$-0.899135\pi$$
−0.950213 + 0.311602i $$0.899135\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 8.51177e8i 1.82285i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 2.70234e8 0.519796 0.259898 0.965636i $$-0.416311\pi$$
0.259898 + 0.965636i $$0.416311\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3.61735e7i 0.0595376i 0.999557 + 0.0297688i $$0.00947711\pi$$
−0.999557 + 0.0297688i $$0.990523\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1.17139e9i 1.65940i 0.558212 + 0.829698i $$0.311488\pi$$
−0.558212 + 0.829698i $$0.688512\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −4.66339e8 −0.571682
$$170$$ 0 0
$$171$$ −1.69619e9 −1.98376
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ −1.05268e9 −0.980801 −0.490400 0.871497i $$-0.663149\pi$$
−0.490400 + 0.871497i $$0.663149\pi$$
$$182$$ 0 0
$$183$$ 1.92995e9i 1.72084i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −2.14383e9 −1.68013
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ − 2.32670e9i − 1.67691i −0.544968 0.838457i $$-0.683458\pi$$
0.544968 0.838457i $$-0.316542\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −1.73472e9 −1.10616 −0.553080 0.833128i $$-0.686548\pi$$
−0.553080 + 0.833128i $$0.686548\pi$$
$$200$$ 0 0
$$201$$ −4.39134e8 −0.269037
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1.83711e8 −0.0926840 −0.0463420 0.998926i $$-0.514756\pi$$
−0.0463420 + 0.998926i $$0.514756\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 7.29914e9i − 3.29180i
$$218$$ 0 0
$$219$$ −1.30971e9 −0.569376
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 4.72732e9i 1.91159i 0.294027 + 0.955797i $$0.405004\pi$$
−0.294027 + 0.955797i $$0.594996\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −5.35877e9 −1.94860 −0.974302 0.225247i $$-0.927681\pi$$
−0.974302 + 0.225247i $$0.927681\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 1.52985e9i − 0.484904i
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −5.56578e9 −1.64990 −0.824951 0.565204i $$-0.808797\pi$$
−0.824951 + 0.565204i $$0.808797\pi$$
$$242$$ 0 0
$$243$$ − 3.48678e9i − 1.00000i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9.25678e9i 2.48698i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ −2.03121e9 −0.451393
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ −2.98709e9 −0.553824 −0.276912 0.960895i $$-0.589311\pi$$
−0.276912 + 0.960895i $$0.589311\pi$$
$$272$$ 0 0
$$273$$ 1.16998e10i 2.10633i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.42807e9i 1.26170i 0.775904 + 0.630852i $$0.217294\pi$$
−0.775904 + 0.630852i $$0.782706\pi$$
$$278$$ 0 0
$$279$$ 1.18715e10 1.95925
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 1.03697e10i 1.61667i 0.588723 + 0.808335i $$0.299631\pi$$
−0.588723 + 0.808335i $$0.700369\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −6.97576e9 −1.00000
$$290$$ 0 0
$$291$$ 1.43296e10 1.99830
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.40875e10 1.71620
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1.41256e10i − 1.59020i −0.606478 0.795101i $$-0.707418\pi$$
0.606478 0.795101i $$-0.292582\pi$$
$$308$$ 0 0
$$309$$ −3.60037e9 −0.394923
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ − 1.17006e10i − 1.21908i −0.792755 0.609541i $$-0.791354\pi$$
0.792755 0.609541i $$-0.208646\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1.64576e10i 1.43938i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.62495e10 −1.35372 −0.676858 0.736113i $$-0.736659\pi$$
−0.676858 + 0.736113i $$0.736659\pi$$
$$332$$ 0 0
$$333$$ − 3.30361e9i − 0.268665i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 2.57689e10i − 1.99791i −0.0456520 0.998957i $$-0.514537\pi$$
0.0456520 0.998957i $$-0.485463\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 1.91355e10i − 1.38249i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 2.96004e10 1.99524 0.997621 0.0689403i $$-0.0219618\pi$$
0.997621 + 0.0689403i $$0.0219618\pi$$
$$350$$ 0 0
$$351$$ −1.90288e10 −1.25367
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 4.98521e10 2.93532
$$362$$ 0 0
$$363$$ − 1.73631e10i − 1.00000i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 2.43056e10i 1.33981i 0.742448 + 0.669903i $$0.233664\pi$$
−0.742448 + 0.669903i $$0.766336\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 4.94467e9i − 0.255448i −0.991810 0.127724i $$-0.959233\pi$$
0.991810 0.127724i $$-0.0407672\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 3.49200e9 0.169245 0.0846227 0.996413i $$-0.473032\pi$$
0.0846227 + 0.996413i $$0.473032\pi$$
$$380$$ 0 0
$$381$$ −3.24455e10 −1.53977
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.29123e10i 1.02147i
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 1.57611e10i − 0.634491i −0.948343 0.317245i $$-0.897242\pi$$
0.948343 0.317245i $$-0.102758\pi$$
$$398$$ 0 0
$$399$$ 8.44744e10 3.33299
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ − 6.47876e10i − 2.45624i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 5.26810e10 1.88261 0.941305 0.337556i $$-0.109600\pi$$
0.941305 + 0.337556i $$0.109600\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 5.74639e10i 1.90043i
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ −1.16089e10 −0.369541 −0.184771 0.982782i $$-0.559154\pi$$
−0.184771 + 0.982782i $$0.559154\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 9.61162e10i − 2.89125i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ − 6.51683e10i − 1.85389i −0.375192 0.926947i $$-0.622423\pi$$
0.375192 0.926947i $$-0.377577\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −6.75493e10 −1.81871 −0.909354 0.416024i $$-0.863423\pi$$
−0.909354 + 0.416024i $$0.863423\pi$$
$$440$$ 0 0
$$441$$ 6.89453e10 1.82285
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 2.18890e10i − 0.519796i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 2.72608e10i − 0.624991i −0.949919 0.312495i $$-0.898835\pi$$
0.949919 0.312495i $$-0.101165\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 6.87853e10i 1.49683i 0.663232 + 0.748413i $$0.269184\pi$$
−0.663232 + 0.748413i $$0.730816\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 2.18700e10 0.452019
$$470$$ 0 0
$$471$$ 2.93005e9 0.0595376
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ −1.80291e10 −0.336817
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 9.09984e10i 1.61777i 0.587965 + 0.808887i $$0.299929\pi$$
−0.587965 + 0.808887i $$0.700071\pi$$
$$488$$ 0 0
$$489$$ 9.48824e10 1.65940
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −1.23330e11 −1.98915 −0.994574 0.104032i $$-0.966826\pi$$
−0.994574 + 0.104032i $$0.966826\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3.77735e10i 0.571682i
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 6.52269e10 0.956628
$$512$$ 0 0
$$513$$ 1.37391e11i 1.98376i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ − 2.44747e10i − 0.327122i −0.986533 0.163561i $$-0.947702\pi$$
0.986533 0.163561i $$-0.0522981\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.83110e10 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 6.27323e10 0.732322 0.366161 0.930551i $$-0.380672\pi$$
0.366161 + 0.930551i $$0.380672\pi$$
$$542$$ 0 0
$$543$$ 8.52668e10i 0.980801i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.18266e10i − 0.690599i −0.938492 0.345300i $$-0.887777\pi$$
0.938492 0.345300i $$-0.112223\pi$$
$$548$$ 0 0
$$549$$ 1.56326e11 1.72084
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 7.61903e10i 0.814703i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 1.25041e11 1.28058
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.73650e11i 1.68013i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ −1.94939e11 −1.83381 −0.916907 0.399102i $$-0.869322\pi$$
−0.916907 + 0.399102i $$0.869322\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 1.54299e11i − 1.39206i −0.718012 0.696031i $$-0.754948\pi$$
0.718012 0.696031i $$-0.245052\pi$$
$$578$$ 0 0
$$579$$ −1.88462e11 −1.67691
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ −4.67778e11 −3.88668
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.40513e11i 1.10616i
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −6.22607e10 −0.477217 −0.238608 0.971116i $$-0.576691\pi$$
−0.238608 + 0.971116i $$0.576691\pi$$
$$602$$ 0 0
$$603$$ 3.55698e10i 0.269037i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 2.65989e11i − 1.95933i −0.200634 0.979666i $$-0.564300\pi$$
0.200634 0.979666i $$-0.435700\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 1.57998e10i 0.111894i 0.998434 + 0.0559472i $$0.0178179\pi$$
−0.998434 + 0.0559472i $$0.982182\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ −2.25533e11 −1.53620 −0.768100 0.640330i $$-0.778797\pi$$
−0.768100 + 0.640330i $$0.778797\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −2.00069e11 −1.26201 −0.631004 0.775780i $$-0.717357\pi$$
−0.631004 + 0.775780i $$0.717357\pi$$
$$632$$ 0 0
$$633$$ 1.48806e10i 0.0926840i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 3.76262e11i − 2.28525i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ − 1.88544e11i − 1.10298i −0.834181 0.551490i $$-0.814059\pi$$
0.834181 0.551490i $$-0.185941\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −5.91231e11 −3.29180
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1.06087e11i 0.569376i
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 3.56009e11 1.86490 0.932449 0.361302i $$-0.117668\pi$$
0.932449 + 0.361302i $$0.117668\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 3.82913e11 1.91159
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 2.28934e11i 1.11597i 0.829853 + 0.557983i $$0.188424\pi$$
−0.829853 + 0.557983i $$0.811576\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ −7.13647e11 −3.35741
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 4.34061e11i 1.94860i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1.55801e11 0.683374 0.341687 0.939814i $$-0.389002\pi$$
0.341687 + 0.939814i $$0.389002\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 1.30174e11i 0.532968i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 4.87686e11 1.92999 0.964995 0.262268i $$-0.0844703\pi$$
0.964995 + 0.262268i $$0.0844703\pi$$
$$710$$ 0 0
$$711$$ −1.23918e11 −0.484904
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 1.79307e11 0.663524
$$722$$ 0 0
$$723$$ 4.50828e11i 1.64990i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.21500e11i 0.434951i 0.976066 + 0.217475i $$0.0697822\pi$$
−0.976066 + 0.217475i $$0.930218\pi$$
$$728$$ 0 0
$$729$$ −2.82430e11 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 5.77330e11i 1.99990i 0.0100913 + 0.999949i $$0.496788\pi$$
−0.0100913 + 0.999949i $$0.503212\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 4.35662e11 1.46074 0.730368 0.683054i $$-0.239349\pi$$
0.730368 + 0.683054i $$0.239349\pi$$
$$740$$ 0 0
$$741$$ 7.49799e11 2.48698
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −5.35831e11 −1.68449 −0.842244 0.539097i $$-0.818766\pi$$
−0.842244 + 0.539097i $$0.818766\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 4.59764e11i − 1.40008i −0.714106 0.700038i $$-0.753167\pi$$
0.714106 0.700038i $$-0.246833\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ − 8.19631e11i − 2.41836i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −6.70500e11 −1.91732 −0.958658 0.284562i $$-0.908152\pi$$
−0.958658 + 0.284562i $$0.908152\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1.64528e11i 0.451393i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 6.75420e11i 1.76066i 0.474364 + 0.880329i $$0.342678\pi$$
−0.474364 + 0.880329i $$0.657322\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 8.53133e11i − 2.15737i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 8.88545e10 0.205398 0.102699 0.994712i $$-0.467252\pi$$
0.102699 + 0.994712i $$0.467252\pi$$
$$812$$ 0 0
$$813$$ 2.41954e11i 0.553824i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 9.02823e11i − 2.02635i
$$818$$ 0 0
$$819$$ 9.47680e11 2.10633
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 8.62186e11i 1.87932i 0.342105 + 0.939662i $$0.388860\pi$$
−0.342105 + 0.939662i $$0.611140\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ −4.11968e11 −0.872258 −0.436129 0.899884i $$-0.643651\pi$$
−0.436129 + 0.899884i $$0.643651\pi$$
$$830$$ 0 0
$$831$$ 6.01674e11 1.26170
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 9.61593e11i − 1.95925i
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 5.00246e11 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8.64724e11i 1.68013i
$$848$$ 0 0
$$849$$ 8.39948e11 1.61667
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 6.14949e11i − 1.16156i −0.814059 0.580782i $$-0.802747\pi$$
0.814059 0.580782i $$-0.197253\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 8.02752e11 1.47438 0.737189 0.675686i $$-0.236153\pi$$
0.737189 + 0.675686i $$0.236153\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 5.65036e11i 1.00000i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 1.94119e11 0.337284
$$872$$ 0 0
$$873$$ − 1.16069e12i − 1.99830i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.10825e11i 0.356389i 0.983995 + 0.178194i $$0.0570256\pi$$
−0.983995 + 0.178194i $$0.942974\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ − 5.72878e11i − 0.942366i −0.882035 0.471183i $$-0.843827\pi$$
0.882035 0.471183i $$-0.156173\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 1.61587e12 2.58701
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ − 1.14109e12i − 1.71620i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.20490e12i 1.78042i 0.455547 + 0.890212i $$0.349444\pi$$
−0.455547 + 0.890212i $$0.650556\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 5.44534e11 0.763419 0.381709 0.924282i $$-0.375336\pi$$
0.381709 + 0.924282i $$0.375336\pi$$
$$920$$ 0 0
$$921$$ −1.14417e12 −1.59020
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.91630e11i 0.394923i
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ −2.71668e12 −3.61610
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 1.43879e12i − 1.86655i −0.359168 0.933273i $$-0.616939\pi$$
0.359168 0.933273i $$-0.383061\pi$$
$$938$$ 0 0
$$939$$ −9.47753e11 −1.21908
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 5.78957e11 0.713808
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 2.42106e12 2.83865
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.51552e12i − 1.73322i −0.498984 0.866611i $$-0.666293\pi$$
0.498984 0.866611i $$-0.333707\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 0 0
$$973$$ − 2.86184e12i − 3.19297i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 1.33307e12 1.43938
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 1.92066e12 1.99139 0.995693 0.0927105i $$-0.0295531\pi$$
0.995693 + 0.0927105i $$0.0295531\pi$$
$$992$$ 0 0
$$993$$ 1.31621e12i 1.35372i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 5.81390e11i 0.588420i 0.955741 + 0.294210i $$0.0950565\pi$$
−0.955741 + 0.294210i $$0.904944\pi$$
$$998$$ 0 0
$$999$$ −2.67592e11 −0.268665
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 300.9.b.b.149.1 2
3.2 odd 2 CM 300.9.b.b.149.1 2
5.2 odd 4 300.9.g.a.101.1 1
5.3 odd 4 12.9.c.a.5.1 1
5.4 even 2 inner 300.9.b.b.149.2 2
15.2 even 4 300.9.g.a.101.1 1
15.8 even 4 12.9.c.a.5.1 1
15.14 odd 2 inner 300.9.b.b.149.2 2
20.3 even 4 48.9.e.a.17.1 1
40.3 even 4 192.9.e.b.65.1 1
40.13 odd 4 192.9.e.a.65.1 1
45.13 odd 12 324.9.g.a.269.1 2
45.23 even 12 324.9.g.a.269.1 2
45.38 even 12 324.9.g.a.53.1 2
45.43 odd 12 324.9.g.a.53.1 2
60.23 odd 4 48.9.e.a.17.1 1
120.53 even 4 192.9.e.a.65.1 1
120.83 odd 4 192.9.e.b.65.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
12.9.c.a.5.1 1 5.3 odd 4
12.9.c.a.5.1 1 15.8 even 4
48.9.e.a.17.1 1 20.3 even 4
48.9.e.a.17.1 1 60.23 odd 4
192.9.e.a.65.1 1 40.13 odd 4
192.9.e.a.65.1 1 120.53 even 4
192.9.e.b.65.1 1 40.3 even 4
192.9.e.b.65.1 1 120.83 odd 4
300.9.b.b.149.1 2 1.1 even 1 trivial
300.9.b.b.149.1 2 3.2 odd 2 CM
300.9.b.b.149.2 2 5.4 even 2 inner
300.9.b.b.149.2 2 15.14 odd 2 inner
300.9.g.a.101.1 1 5.2 odd 4
300.9.g.a.101.1 1 15.2 even 4
324.9.g.a.53.1 2 45.38 even 12
324.9.g.a.53.1 2 45.43 odd 12
324.9.g.a.269.1 2 45.13 odd 12
324.9.g.a.269.1 2 45.23 even 12