Properties

 Label 95.7.d.a.94.2 Level $95$ Weight $7$ Character 95.94 Analytic conductor $21.855$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,7,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 95.d (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: $$21.8551379439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

 Embedding label 94.2 Root $$0.500000 + 2.17945i$$ of defining polynomial Character $$\chi$$ $$=$$ 95.94 Dual form 95.7.d.a.94.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-64.0000 q^{4} +(-27.0000 + 122.049i) q^{5} +313.841i q^{7} -729.000 q^{9} +O(q^{10})$$ $$q-64.0000 q^{4} +(-27.0000 + 122.049i) q^{5} +313.841i q^{7} -729.000 q^{9} +1062.00 q^{11} +4096.00 q^{16} -1952.79i q^{17} -6859.00 q^{19} +(1728.00 - 7811.15i) q^{20} -12937.2i q^{23} +(-14167.0 - 6590.66i) q^{25} -20085.8i q^{28} +(-38304.0 - 8473.70i) q^{35} +46656.0 q^{36} -70300.3i q^{43} -67968.0 q^{44} +(19683.0 - 88973.8i) q^{45} -193570. i q^{47} +19153.0 q^{49} +(-28674.0 + 129616. i) q^{55} +57062.0 q^{61} -228790. i q^{63} -262144. q^{64} +124978. i q^{68} +676641. i q^{73} +438976. q^{76} +333299. i q^{77} +(-110592. + 499913. i) q^{80} +531441. q^{81} +168916. i q^{83} +(238336. + 52725.2i) q^{85} +827982. i q^{92} +(185193. - 837135. i) q^{95} -774198. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9}+O(q^{10})$$ 2 * q - 128 * q^4 - 54 * q^5 - 1458 * q^9 $$2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9} + 2124 q^{11} + 8192 q^{16} - 13718 q^{19} + 3456 q^{20} - 28334 q^{25} - 76608 q^{35} + 93312 q^{36} - 135936 q^{44} + 39366 q^{45} + 38306 q^{49} - 57348 q^{55} + 114124 q^{61} - 524288 q^{64} + 877952 q^{76} - 221184 q^{80} + 1062882 q^{81} + 476672 q^{85} + 370386 q^{95} - 1548396 q^{99}+O(q^{100})$$ 2 * q - 128 * q^4 - 54 * q^5 - 1458 * q^9 + 2124 * q^11 + 8192 * q^16 - 13718 * q^19 + 3456 * q^20 - 28334 * q^25 - 76608 * q^35 + 93312 * q^36 - 135936 * q^44 + 39366 * q^45 + 38306 * q^49 - 57348 * q^55 + 114124 * q^61 - 524288 * q^64 + 877952 * q^76 - 221184 * q^80 + 1062882 * q^81 + 476672 * q^85 + 370386 * q^95 - 1548396 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −64.0000 −1.00000
$$5$$ −27.0000 + 122.049i −0.216000 + 0.976393i
$$6$$ 0 0
$$7$$ 313.841i 0.914988i 0.889213 + 0.457494i $$0.151253\pi$$
−0.889213 + 0.457494i $$0.848747\pi$$
$$8$$ 0 0
$$9$$ −729.000 −1.00000
$$10$$ 0 0
$$11$$ 1062.00 0.797896 0.398948 0.916973i $$-0.369375\pi$$
0.398948 + 0.916973i $$0.369375\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4096.00 1.00000
$$17$$ 1952.79i 0.397473i −0.980053 0.198737i $$-0.936316\pi$$
0.980053 0.198737i $$-0.0636839\pi$$
$$18$$ 0 0
$$19$$ −6859.00 −1.00000
$$20$$ 1728.00 7811.15i 0.216000 0.976393i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 12937.2i 1.06330i −0.846963 0.531652i $$-0.821572\pi$$
0.846963 0.531652i $$-0.178428\pi$$
$$24$$ 0 0
$$25$$ −14167.0 6590.66i −0.906688 0.421802i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 20085.8i 0.914988i
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −38304.0 8473.70i −0.893388 0.197637i
$$36$$ 46656.0 1.00000
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 70300.3i 0.884203i −0.896965 0.442101i $$-0.854233\pi$$
0.896965 0.442101i $$-0.145767\pi$$
$$44$$ −67968.0 −0.797896
$$45$$ 19683.0 88973.8i 0.216000 0.976393i
$$46$$ 0 0
$$47$$ 193570.i 1.86442i −0.361914 0.932211i $$-0.617877\pi$$
0.361914 0.932211i $$-0.382123\pi$$
$$48$$ 0 0
$$49$$ 19153.0 0.162798
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −28674.0 + 129616.i −0.172346 + 0.779061i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 57062.0 0.251395 0.125698 0.992069i $$-0.459883\pi$$
0.125698 + 0.992069i $$0.459883\pi$$
$$62$$ 0 0
$$63$$ 228790.i 0.914988i
$$64$$ −262144. −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 124978.i 0.397473i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 676641.i 1.73936i 0.493616 + 0.869680i $$0.335675\pi$$
−0.493616 + 0.869680i $$0.664325\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 438976. 1.00000
$$77$$ 333299.i 0.730065i
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ −110592. + 499913.i −0.216000 + 0.976393i
$$81$$ 531441. 1.00000
$$82$$ 0 0
$$83$$ 168916.i 0.295418i 0.989031 + 0.147709i $$0.0471899\pi$$
−0.989031 + 0.147709i $$0.952810\pi$$
$$84$$ 0 0
$$85$$ 238336. + 52725.2i 0.388090 + 0.0858543i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 827982.i 1.06330i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 185193. 837135.i 0.216000 0.976393i
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ −774198. −0.797896
$$100$$ 906688. + 421802.i 0.906688 + 0.421802i
$$101$$ −2.06030e6 −1.99970 −0.999852 0.0171767i $$-0.994532\pi$$
−0.999852 + 0.0171767i $$0.994532\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.28549e6i 0.914988i
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 1.57898e6 + 349305.i 1.03820 + 0.229674i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 612864. 0.363683
$$120$$ 0 0
$$121$$ −643717. −0.363361
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.18689e6 1.55112e6i 0.607689 0.794175i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.30228e6 0.579283 0.289642 0.957135i $$-0.406464\pi$$
0.289642 + 0.957135i $$0.406464\pi$$
$$132$$ 0 0
$$133$$ 2.15263e6i 0.914988i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.63738e6i 1.80348i −0.432280 0.901739i $$-0.642291\pi$$
0.432280 0.901739i $$-0.357709\pi$$
$$138$$ 0 0
$$139$$ −3.77334e6 −1.40502 −0.702508 0.711676i $$-0.747937\pi$$
−0.702508 + 0.711676i $$0.747937\pi$$
$$140$$ 2.45146e6 + 542317.i 0.893388 + 0.197637i
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −2.98598e6 −1.00000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.24350e6 −1.88742 −0.943711 0.330770i $$-0.892692\pi$$
−0.943711 + 0.330770i $$0.892692\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 1.42358e6i 0.397473i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.70448e6i 1.99088i −0.0954122 0.995438i $$-0.530417\pi$$
0.0954122 0.995438i $$-0.469583\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4.06022e6 0.972909
$$162$$ 0 0
$$163$$ 7.51774e6i 1.73590i −0.496652 0.867950i $$-0.665438\pi$$
0.496652 0.867950i $$-0.334562\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −4.82681e6 −1.00000
$$170$$ 0 0
$$171$$ 5.00021e6 1.00000
$$172$$ 4.49922e6i 0.884203i
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 2.06842e6 4.44618e6i 0.385944 0.829608i
$$176$$ 4.34995e6 0.797896
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ −1.25971e6 + 5.69433e6i −0.216000 + 0.976393i
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.07386e6i 0.317143i
$$188$$ 1.23885e7i 1.86442i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 9.37384e6 1.34529 0.672647 0.739963i $$-0.265157\pi$$
0.672647 + 0.739963i $$0.265157\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −1.22579e6 −0.162798
$$197$$ 1.17795e7i 1.54073i 0.637603 + 0.770365i $$0.279926\pi$$
−0.637603 + 0.770365i $$0.720074\pi$$
$$198$$ 0 0
$$199$$ −1.52712e7 −1.93782 −0.968911 0.247410i $$-0.920421\pi$$
−0.968911 + 0.247410i $$0.920421\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 9.43123e6i 1.06330i
$$208$$ 0 0
$$209$$ −7.28426e6 −0.797896
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.58010e6 + 1.89811e6i 0.863330 + 0.190988i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 1.83514e6 8.29544e6i 0.172346 0.779061i
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 1.03277e7 + 4.80459e6i 0.906688 + 0.421802i
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −2.66958e6 −0.222298 −0.111149 0.993804i $$-0.535453\pi$$
−0.111149 + 0.993804i $$0.535453\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 4.94104e6i 0.390616i 0.980742 + 0.195308i $$0.0625707\pi$$
−0.980742 + 0.195308i $$0.937429\pi$$
$$234$$ 0 0
$$235$$ 2.36251e7 + 5.22639e6i 1.82041 + 0.402715i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1.53322e7 −1.12308 −0.561541 0.827449i $$-0.689791\pi$$
−0.561541 + 0.827449i $$0.689791\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −3.65197e6 −0.251395
$$245$$ −517131. + 2.33761e6i −0.0351643 + 0.158955i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5.08340e6 −0.321464 −0.160732 0.986998i $$-0.551386\pi$$
−0.160732 + 0.986998i $$0.551386\pi$$
$$252$$ 1.46426e7i 0.914988i
$$253$$ 1.37393e7i 0.848406i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.67772e7 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 3.18370e7i 1.75011i 0.484024 + 0.875054i $$0.339175\pi$$
−0.484024 + 0.875054i $$0.660825\pi$$
$$264$$ 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
$$271$$ −2.84226e7 −1.42809 −0.714045 0.700100i $$-0.753139\pi$$
−0.714045 + 0.700100i $$0.753139\pi$$
$$272$$ 7.99861e6i 0.397473i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.50454e7 6.99928e6i −0.723443 0.336554i
$$276$$ 0 0
$$277$$ 3.01347e7i 1.41784i −0.705289 0.708920i $$-0.749183\pi$$
0.705289 0.708920i $$-0.250817\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 3.07828e7i 1.35815i −0.734068 0.679076i $$-0.762381\pi$$
0.734068 0.679076i $$-0.237619\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.03242e7 0.842015
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 4.33050e7i 1.73936i
$$293$$ 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
$$301$$ 2.20631e7 0.809035
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −2.80945e7 −1.00000
$$305$$ −1.54067e6 + 6.96437e6i −0.0543014 + 0.245461i
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 2.13311e7i 0.730065i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.42879e7 −1.47233 −0.736164 0.676804i $$-0.763365\pi$$
−0.736164 + 0.676804i $$0.763365\pi$$
$$312$$ 0 0
$$313$$ 5.23342e7i 1.70668i −0.521353 0.853341i $$-0.674573\pi$$
0.521353 0.853341i $$-0.325427\pi$$
$$314$$ 0 0
$$315$$ 2.79236e7 + 6.17733e6i 0.893388 + 0.197637i
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 7.07789e6 3.19945e7i 0.216000 0.976393i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1.33942e7i 0.397473i
$$324$$ −3.40122e7 −1.00000
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 6.07501e7 1.70592
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 1.08106e7i 0.295418i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −1.52535e7 3.37442e6i −0.388090 0.0858543i
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 4.29340e7i 1.06395i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5.40634e7i 1.29394i 0.762515 + 0.646971i $$0.223965\pi$$
−0.762515 + 0.646971i $$0.776035\pi$$
$$348$$ 0 0
$$349$$ −4.62042e7 −1.08694 −0.543469 0.839429i $$-0.682890\pi$$
−0.543469 + 0.839429i $$0.682890\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.61463e7i 1.50377i 0.659295 + 0.751885i $$0.270855\pi$$
−0.659295 + 0.751885i $$0.729145\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 7.96053e7 1.72052 0.860258 0.509858i $$-0.170302\pi$$
0.860258 + 0.509858i $$0.170302\pi$$
$$360$$ 0 0
$$361$$ 4.70459e7 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.25834e7 1.82693e7i −1.69830 0.375702i
$$366$$ 0 0
$$367$$ 9.68013e7i 1.95832i 0.203095 + 0.979159i $$0.434900\pi$$
−0.203095 + 0.979159i $$0.565100\pi$$
$$368$$ 5.29908e7i 1.06330i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ −1.18524e7 + 5.35767e7i −0.216000 + 0.976393i
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −4.06788e7 8.99907e6i −0.712831 0.157694i
$$386$$ 0 0
$$387$$ 5.12489e7i 0.884203i
$$388$$ 0 0
$$389$$ 6.58748e7 1.11910 0.559552 0.828795i $$-0.310973\pi$$
0.559552 + 0.828795i $$0.310973\pi$$
$$390$$ 0 0
$$391$$ −2.52636e7 −0.422635
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 4.95487e7 0.797896
$$397$$ 1.09135e8i 1.74418i −0.489346 0.872090i $$-0.662764\pi$$
0.489346 0.872090i $$-0.337236\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −5.80280e7 2.69953e7i −0.906688 0.421802i
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 1.31859e8 1.99970
$$405$$ −1.43489e7 + 6.48619e7i −0.216000 + 0.976393i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −2.06161e7 4.56073e6i −0.288444 0.0638102i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 4.11183e7 0.558976 0.279488 0.960149i $$-0.409835\pi$$
0.279488 + 0.960149i $$0.409835\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 1.41113e8i 1.86442i
$$424$$ 0 0
$$425$$ −1.28701e7 + 2.76651e7i −0.167655 + 0.360384i
$$426$$ 0 0
$$427$$ 1.79084e7i 0.230024i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 8.87363e7i 1.06330i
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ −1.39625e7 −0.162798
$$442$$ 0 0
$$443$$ 1.71860e8i 1.97681i 0.151846 + 0.988404i $$0.451478\pi$$
−0.151846 + 0.988404i $$0.548522\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 8.22715e7i 0.914988i
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.21229e8i 1.27015i −0.772449 0.635077i $$-0.780968\pi$$
0.772449 0.635077i $$-0.219032\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ −1.01054e8 2.23555e7i −1.03820 0.229674i
$$461$$ −1.95676e8 −1.99726 −0.998631 0.0523154i $$-0.983340\pi$$
−0.998631 + 0.0523154i $$0.983340\pi$$
$$462$$ 0 0
$$463$$ 1.90683e8i 1.92118i −0.277961 0.960592i $$-0.589659\pi$$
0.277961 0.960592i $$-0.410341\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1.16046e8i 1.13941i −0.821849 0.569706i $$-0.807057\pi$$
0.821849 0.569706i $$-0.192943\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 7.46589e7i 0.705502i
$$474$$ 0 0
$$475$$ 9.71715e7 + 4.52053e7i 0.906688 + 0.421802i
$$476$$ −3.92233e7 −0.363683
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 1.87497e8 1.70603 0.853015 0.521886i $$-0.174771\pi$$
0.853015 + 0.521886i $$0.174771\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 4.11979e7 0.363361
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.09684e8 −0.926610 −0.463305 0.886199i $$-0.653337\pi$$
−0.463305 + 0.886199i $$0.653337\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 2.09033e7 9.44902e7i 0.172346 0.779061i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2.47627e8 1.99295 0.996475 0.0838961i $$-0.0267364\pi$$
0.996475 + 0.0838961i $$0.0267364\pi$$
$$500$$ −7.59612e7 + 9.92719e7i −0.607689 + 0.794175i
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 2.34711e8i 1.84429i −0.386843 0.922146i $$-0.626434\pi$$
0.386843 0.922146i $$-0.373566\pi$$
$$504$$ 0 0
$$505$$ 5.56280e7 2.51458e8i 0.431936 1.95250i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ −2.12357e8 −1.59149
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 2.05571e8i 1.48762i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ −8.33460e7 −0.579283
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −1.93356e7 −0.130614
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 1.37769e8i 0.914988i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 2.03405e7 0.129896
$$540$$ 0 0
$$541$$ −3.05822e8 −1.93142 −0.965709 0.259626i $$-0.916401\pi$$
−0.965709 + 0.259626i $$0.916401\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 2.96792e8i 1.80348i
$$549$$ −4.15982e7 −0.251395
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 2.41494e8 1.40502
$$557$$ 1.82082e8i 1.05366i 0.849970 + 0.526831i $$0.176620\pi$$
−0.849970 + 0.526831i $$0.823380\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −1.56893e8 3.47083e7i −0.893388 0.197637i
$$561$$ 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
$$567$$ 1.66788e8i 0.914988i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 3.51908e8 1.89026 0.945130 0.326696i $$-0.105935\pi$$
0.945130 + 0.326696i $$0.105935\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −8.52647e7 + 1.83281e8i −0.448503 + 0.964084i
$$576$$ 1.91103e8 1.00000
$$577$$ 1.40693e8i 0.732394i −0.930537 0.366197i $$-0.880660\pi$$
0.930537 0.366197i $$-0.119340\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −5.30127e7 −0.270304
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 3.09139e8i 1.52841i 0.644975 + 0.764204i $$0.276868\pi$$
−0.644975 + 0.764204i $$0.723132\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 4.15855e8i 1.99424i −0.0758206 0.997121i $$-0.524158\pi$$
0.0758206 0.997121i $$-0.475842\pi$$
$$594$$ 0 0
$$595$$ −1.65473e7 + 7.47995e7i −0.0785556 + 0.355098i
$$596$$ 3.99584e8 1.88742
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1.73804e7 7.85651e7i 0.0784861 0.354784i
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 9.11092e7i 0.397473i
$$613$$ 3.43599e8i 1.49166i 0.666134 + 0.745832i $$0.267948\pi$$
−0.666134 + 0.745832i $$0.732052\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.69696e8i 1.99968i −0.0177783 0.999842i $$-0.505659\pi$$
0.0177783 0.999842i $$-0.494341\pi$$
$$618$$ 0 0
$$619$$ −4.70840e8 −1.98519 −0.992594 0.121480i $$-0.961236\pi$$
−0.992594 + 0.121480i $$0.961236\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.57267e8 + 1.86740e8i 0.644166 + 0.764886i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 4.93086e8i 1.99088i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.23521e8 −0.491647 −0.245824 0.969315i $$-0.579058\pi$$
−0.245824 + 0.969315i $$0.579058\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 5.31686e8i 1.99996i 0.00603535 + 0.999982i $$0.498079\pi$$
−0.00603535 + 0.999982i $$0.501921\pi$$
$$644$$ −2.59854e8 −0.972909
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 4.82072e8i 1.77991i 0.456046 + 0.889956i $$0.349265\pi$$
−0.456046 + 0.889956i $$0.650735\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.81135e8i 1.73590i
$$653$$ 3.57517e8i 1.28398i −0.766714 0.641988i $$-0.778110\pi$$
0.766714 0.641988i $$-0.221890\pi$$
$$654$$ 0 0
$$655$$ −3.51616e7 + 1.58942e8i −0.125125 + 0.565609i
$$656$$ 0 0
$$657$$ 4.93271e8i 1.73936i
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.62727e8 + 5.81211e7i 0.893388 + 0.197637i
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6.05998e7 0.200588
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 3.08916e8 1.00000
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ −3.20014e8 −1.00000
$$685$$ 5.65988e8 + 1.25209e8i 1.76090 + 0.389551i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.87950e8i 0.884203i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −2.21024e8 −0.669892 −0.334946 0.942237i $$-0.608718\pi$$
−0.334946 + 0.942237i $$0.608718\pi$$
$$692$$ 0 0
$$693$$ 2.42975e8i 0.730065i
$$694$$ 0 0
$$695$$ 1.01880e8 4.60533e8i 0.303483 1.37185i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −1.32379e8 + 2.84556e8i −0.385944 + 0.829608i
$$701$$ 2.94922e8 0.856156 0.428078 0.903742i $$-0.359191\pi$$
0.428078 + 0.903742i $$0.359191\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −2.78397e8 −0.797896
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6.46605e8i 1.82971i
$$708$$ 0 0
$$709$$ −3.01929e8 −0.847161 −0.423580 0.905859i $$-0.639227\pi$$
−0.423580 + 0.905859i $$0.639227\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 6.00444e8 1.61542 0.807711 0.589579i $$-0.200706\pi$$
0.807711 + 0.589579i $$0.200706\pi$$
$$720$$ 8.06216e7 3.64437e8i 0.216000 0.976393i
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 7.56680e8i 1.96929i 0.174579 + 0.984643i $$0.444144\pi$$
−0.174579 + 0.984643i $$0.555856\pi$$
$$728$$ 0 0
$$729$$ −3.87420e8 −1.00000
$$730$$ 0 0
$$731$$ −1.37282e8 −0.351447
$$732$$ 0 0
$$733$$ 7.87665e8i 2.00000i −0.00166557 0.999999i $$-0.500530\pi$$
0.00166557 0.999999i $$-0.499470\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −7.32517e8 −1.81503 −0.907517 0.420016i $$-0.862024\pi$$
−0.907517 + 0.420016i $$0.862024\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 1.68574e8 7.62014e8i 0.407683 1.84287i
$$746$$ 0 0
$$747$$ 1.23140e8i 0.295418i
$$748$$ 1.32727e8i 0.317143i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 7.92863e8i 1.86442i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 5.58953e7i 0.128851i 0.997923 + 0.0644255i $$0.0205215\pi$$
−0.997923 + 0.0644255i $$0.979479\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −7.84034e8 −1.77902 −0.889510 0.456915i $$-0.848954\pi$$
−0.889510 + 0.456915i $$0.848954\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −5.99926e8 −1.34529
$$765$$ −1.73747e8 3.84367e7i −0.388090 0.0858543i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 6.84693e8 1.50563 0.752813 0.658235i $$-0.228697\pi$$
0.752813 + 0.658235i $$0.228697\pi$$
$$770$$ 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
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 7.84507e7 0.162798
$$785$$ 9.40325e8 + 2.08021e8i 1.94388 + 0.430029i
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 7.53885e8i 1.54073i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 9.77357e8 1.93782
$$797$$ 0 0