LMFDB - Embedded newform 170.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [170,2,Mod(1,170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(170, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$170 = 2 \cdot 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	170.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.35745683436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 4$ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	170.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.56155 q^{6} -3.12311 q^{7} +1.00000 q^{8} -0.561553 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.56155 q^{12} +0.438447 q^{13} -3.12311 q^{14} +1.56155 q^{15} +1.00000 q^{16} +1.00000 q^{17} -0.561553 q^{18} +1.56155 q^{19} +1.00000 q^{20} -4.87689 q^{21} -4.00000 q^{22} +3.12311 q^{23} +1.56155 q^{24} +1.00000 q^{25} +0.438447 q^{26} -5.56155 q^{27} -3.12311 q^{28} +6.68466 q^{29} +1.56155 q^{30} -2.43845 q^{31} +1.00000 q^{32} -6.24621 q^{33} +1.00000 q^{34} -3.12311 q^{35} -0.561553 q^{36} +1.12311 q^{37} +1.56155 q^{38} +0.684658 q^{39} +1.00000 q^{40} -12.2462 q^{41} -4.87689 q^{42} +7.12311 q^{43} -4.00000 q^{44} -0.561553 q^{45} +3.12311 q^{46} +2.43845 q^{47} +1.56155 q^{48} +2.75379 q^{49} +1.00000 q^{50} +1.56155 q^{51} +0.438447 q^{52} +3.56155 q^{53} -5.56155 q^{54} -4.00000 q^{55} -3.12311 q^{56} +2.43845 q^{57} +6.68466 q^{58} -12.6847 q^{59} +1.56155 q^{60} +11.5616 q^{61} -2.43845 q^{62} +1.75379 q^{63} +1.00000 q^{64} +0.438447 q^{65} -6.24621 q^{66} -0.876894 q^{67} +1.00000 q^{68} +4.87689 q^{69} -3.12311 q^{70} -16.6847 q^{71} -0.561553 q^{72} +13.8078 q^{73} +1.12311 q^{74} +1.56155 q^{75} +1.56155 q^{76} +12.4924 q^{77} +0.684658 q^{78} +1.00000 q^{80} -7.00000 q^{81} -12.2462 q^{82} +10.2462 q^{83} -4.87689 q^{84} +1.00000 q^{85} +7.12311 q^{86} +10.4384 q^{87} -4.00000 q^{88} +2.68466 q^{89} -0.561553 q^{90} -1.36932 q^{91} +3.12311 q^{92} -3.80776 q^{93} +2.43845 q^{94} +1.56155 q^{95} +1.56155 q^{96} +10.6847 q^{97} +2.75379 q^{98} +2.24621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} + 3 q^{9} + 2 q^{10} - 8 q^{11} - q^{12} + 5 q^{13} + 2 q^{14} - q^{15} + 2 q^{16} + 2 q^{17} + 3 q^{18} - q^{19} + 2 q^{20} - 18 q^{21}+ \cdots - 12 q^{99}+O(q^{100})$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 2 * q^5 - q^6 + 2 * q^7 + 2 * q^8 + 3 * q^9 + 2 * q^10 - 8 * q^11 - q^12 + 5 * q^13 + 2 * q^14 - q^15 + 2 * q^16 + 2 * q^17 + 3 * q^18 - q^19 + 2 * q^20 - 18 * q^21 - 8 * q^22 - 2 * q^23 - q^24 + 2 * q^25 + 5 * q^26 - 7 * q^27 + 2 * q^28 + q^29 - q^30 - 9 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^34 + 2 * q^35 + 3 * q^36 - 6 * q^37 - q^38 - 11 * q^39 + 2 * q^40 - 8 * q^41 - 18 * q^42 + 6 * q^43 - 8 * q^44 + 3 * q^45 - 2 * q^46 + 9 * q^47 - q^48 + 22 * q^49 + 2 * q^50 - q^51 + 5 * q^52 + 3 * q^53 - 7 * q^54 - 8 * q^55 + 2 * q^56 + 9 * q^57 + q^58 - 13 * q^59 - q^60 + 19 * q^61 - 9 * q^62 + 20 * q^63 + 2 * q^64 + 5 * q^65 + 4 * q^66 - 10 * q^67 + 2 * q^68 + 18 * q^69 + 2 * q^70 - 21 * q^71 + 3 * q^72 + 7 * q^73 - 6 * q^74 - q^75 - q^76 - 8 * q^77 - 11 * q^78 + 2 * q^80 - 14 * q^81 - 8 * q^82 + 4 * q^83 - 18 * q^84 + 2 * q^85 + 6 * q^86 + 25 * q^87 - 8 * q^88 - 7 * q^89 + 3 * q^90 + 22 * q^91 - 2 * q^92 + 13 * q^93 + 9 * q^94 - q^95 - q^96 + 9 * q^97 + 22 * q^98 - 12 * q^99

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.56155	0.637501
$7$	−3.12311	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$7$	−3.12311	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$8$	1.00000	0.353553
$9$	−0.561553	−0.187184
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.56155	0.450781
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	−3.12311	−0.834685
$15$	1.56155	0.403191
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−0.561553	−0.132359
$19$	1.56155	0.358245	0.179122	−	0.983827i	$-0.442674\pi$
$19$	1.56155	0.358245	0.179122	+	0.983827i	$0.442674\pi$
$20$	1.00000	0.223607
$21$	−4.87689	−1.06423
$22$	−4.00000	−0.852803
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	1.56155	0.318751
$25$	1.00000	0.200000
$26$	0.438447	0.0859866
$27$	−5.56155	−1.07032
$28$	−3.12311	−0.590211
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	1.56155	0.285099
$31$	−2.43845	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$31$	−2.43845	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$32$	1.00000	0.176777
$33$	−6.24621	−1.08733
$34$	1.00000	0.171499
$35$	−3.12311	−0.527901
$36$	−0.561553	−0.0935921
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	1.56155	0.253317
$39$	0.684658	0.109633
$40$	1.00000	0.158114
$41$	−12.2462	−1.91254	−0.956268	−	0.292490i	$-0.905516\pi$
$41$	−12.2462	−1.91254	−0.956268	+	0.292490i	$0.905516\pi$
$42$	−4.87689	−0.752521
$43$	7.12311	1.08626	0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311	1.08626	0.543132	+	0.839648i	$0.317238\pi$
$44$	−4.00000	−0.603023
$45$	−0.561553	−0.0837114
$46$	3.12311	0.460477
$47$	2.43845	0.355684	0.177842	−	0.984059i	$-0.443088\pi$
$47$	2.43845	0.355684	0.177842	+	0.984059i	$0.443088\pi$
$48$	1.56155	0.225391
$49$	2.75379	0.393398
$50$	1.00000	0.141421
$51$	1.56155	0.218661
$52$	0.438447	0.0608017
$53$	3.56155	0.489217	0.244608	−	0.969622i	$-0.421341\pi$
$53$	3.56155	0.489217	0.244608	+	0.969622i	$0.421341\pi$
$54$	−5.56155	−0.756831
$55$	−4.00000	−0.539360
$56$	−3.12311	−0.417343
$57$	2.43845	0.322980
$58$	6.68466	0.877739
$59$	−12.6847	−1.65140	−0.825701	−	0.564108i	$-0.809220\pi$
$59$	−12.6847	−1.65140	−0.825701	+	0.564108i	$0.809220\pi$
$60$	1.56155	0.201596
$61$	11.5616	1.48031	0.740153	−	0.672439i	$-0.234753\pi$
$61$	11.5616	1.48031	0.740153	+	0.672439i	$0.234753\pi$
$62$	−2.43845	−0.309683
$63$	1.75379	0.220957
$64$	1.00000	0.125000
$65$	0.438447	0.0543827
$66$	−6.24621	−0.768855
$67$	−0.876894	−0.107130	−0.0535648	−	0.998564i	$-0.517058\pi$
$67$	−0.876894	−0.107130	−0.0535648	+	0.998564i	$0.517058\pi$
$68$	1.00000	0.121268
$69$	4.87689	0.587109
$70$	−3.12311	−0.373283
$71$	−16.6847	−1.98010	−0.990052	−	0.140700i	$-0.955065\pi$
$71$	−16.6847	−1.98010	−0.990052	+	0.140700i	$0.955065\pi$
$72$	−0.561553	−0.0661796
$73$	13.8078	1.61608	0.808038	−	0.589130i	$-0.200529\pi$
$73$	13.8078	1.61608	0.808038	+	0.589130i	$0.200529\pi$
$74$	1.12311	0.130558
$75$	1.56155	0.180313
$76$	1.56155	0.179122
$77$	12.4924	1.42364
$78$	0.684658	0.0775223
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.00000	0.111803
$81$	−7.00000	−0.777778
$82$	−12.2462	−1.35237
$83$	10.2462	1.12467	0.562334	−	0.826910i	$-0.309904\pi$
$83$	10.2462	1.12467	0.562334	+	0.826910i	$0.309904\pi$
$84$	−4.87689	−0.532113
$85$	1.00000	0.108465
$86$	7.12311	0.768104
$87$	10.4384	1.11912
$88$	−4.00000	−0.426401
$89$	2.68466	0.284573	0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466	0.284573	0.142287	+	0.989825i	$0.454555\pi$
$90$	−0.561553	−0.0591929
$91$	−1.36932	−0.143543
$92$	3.12311	0.325606
$93$	−3.80776	−0.394847
$94$	2.43845	0.251507
$95$	1.56155	0.160212
$96$	1.56155	0.159375
$97$	10.6847	1.08486	0.542431	−	0.840100i	$-0.317504\pi$
$97$	10.6847	1.08486	0.542431	+	0.840100i	$0.317504\pi$
$98$	2.75379	0.278175
$99$	2.24621	0.225753
$100$	1.00000	0.100000
$101$	18.4924	1.84006	0.920032	−	0.391842i	$-0.128162\pi$
$101$	18.4924	1.84006	0.920032	+	0.391842i	$0.128162\pi$
$102$	1.56155	0.154617
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0.438447	0.0429933
$105$	−4.87689	−0.475936
$106$	3.56155	0.345929
$107$	−16.4924	−1.59438	−0.797191	−	0.603727i	$-0.793682\pi$
$107$	−16.4924	−1.59438	−0.797191	+	0.603727i	$0.793682\pi$
$108$	−5.56155	−0.535161
$109$	−4.43845	−0.425126	−0.212563	−	0.977147i	$-0.568181\pi$
$109$	−4.43845	−0.425126	−0.212563	+	0.977147i	$0.568181\pi$
$110$	−4.00000	−0.381385
$111$	1.75379	0.166462
$112$	−3.12311	−0.295106
$113$	−17.8078	−1.67521	−0.837607	−	0.546274i	$-0.816046\pi$
$113$	−17.8078	−1.67521	−0.837607	+	0.546274i	$0.816046\pi$
$114$	2.43845	0.228382
$115$	3.12311	0.291231
$116$	6.68466	0.620655
$117$	−0.246211	−0.0227622
$118$	−12.6847	−1.16772
$119$	−3.12311	−0.286295
$120$	1.56155	0.142550
$121$	5.00000	0.454545
$122$	11.5616	1.04673
$123$	−19.1231	−1.72427
$124$	−2.43845	−0.218979
$125$	1.00000	0.0894427
$126$	1.75379	0.156240
$127$	13.5616	1.20339	0.601697	−	0.798725i	$-0.294492\pi$
$127$	13.5616	1.20339	0.601697	+	0.798725i	$0.294492\pi$
$128$	1.00000	0.0883883
$129$	11.1231	0.979335
$130$	0.438447	0.0384544
$131$	−7.12311	−0.622349	−0.311174	−	0.950353i	$-0.600722\pi$
$131$	−7.12311	−0.622349	−0.311174	+	0.950353i	$0.600722\pi$
$132$	−6.24621	−0.543663
$133$	−4.87689	−0.422880
$134$	−0.876894	−0.0757521
$135$	−5.56155	−0.478662
$136$	1.00000	0.0857493
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	4.87689	0.415149
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	−3.12311	−0.263951
$141$	3.80776	0.320672
$142$	−16.6847	−1.40015
$143$	−1.75379	−0.146659
$144$	−0.561553	−0.0467961
$145$	6.68466	0.555131
$146$	13.8078	1.14274
$147$	4.30019	0.354673
$148$	1.12311	0.0923187
$149$	−5.12311	−0.419701	−0.209851	−	0.977733i	$-0.567298\pi$
$149$	−5.12311	−0.419701	−0.209851	+	0.977733i	$0.567298\pi$
$150$	1.56155	0.127500
$151$	9.36932	0.762464	0.381232	−	0.924479i	$-0.375500\pi$
$151$	9.36932	0.762464	0.381232	+	0.924479i	$0.375500\pi$
$152$	1.56155	0.126659
$153$	−0.561553	−0.0453989
$154$	12.4924	1.00667
$155$	−2.43845	−0.195861
$156$	0.684658	0.0548165
$157$	−14.4924	−1.15662	−0.578311	−	0.815817i	$-0.696288\pi$
$157$	−14.4924	−1.15662	−0.578311	+	0.815817i	$0.696288\pi$
$158$	0	0
$159$	5.56155	0.441060
$160$	1.00000	0.0790569
$161$	−9.75379	−0.768706
$162$	−7.00000	−0.549972
$163$	−2.24621	−0.175937	−0.0879684	−	0.996123i	$-0.528037\pi$
$163$	−2.24621	−0.175937	−0.0879684	+	0.996123i	$0.528037\pi$
$164$	−12.2462	−0.956268
$165$	−6.24621	−0.486267
$166$	10.2462	0.795260
$167$	−3.12311	−0.241673	−0.120837	−	0.992672i	$-0.538558\pi$
$167$	−3.12311	−0.241673	−0.120837	+	0.992672i	$0.538558\pi$
$168$	−4.87689	−0.376261
$169$	−12.8078	−0.985213
$170$	1.00000	0.0766965
$171$	−0.876894	−0.0670578
$172$	7.12311	0.543132
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	10.4384	0.791337
$175$	−3.12311	−0.236085
$176$	−4.00000	−0.301511
$177$	−19.8078	−1.48884
$178$	2.68466	0.201224
$179$	−24.4924	−1.83065	−0.915325	−	0.402716i	$-0.868066\pi$
$179$	−24.4924	−1.83065	−0.915325	+	0.402716i	$0.868066\pi$
$180$	−0.561553	−0.0418557
$181$	−0.246211	−0.0183007	−0.00915037	−	0.999958i	$-0.502913\pi$
$181$	−0.246211	−0.0183007	−0.00915037	+	0.999958i	$0.502913\pi$
$182$	−1.36932	−0.101501
$183$	18.0540	1.33459
$184$	3.12311	0.230238
$185$	1.12311	0.0825724
$186$	−3.80776	−0.279199
$187$	−4.00000	−0.292509
$188$	2.43845	0.177842
$189$	17.3693	1.26343
$190$	1.56155	0.113287
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.56155	0.112695
$193$	−7.75379	−0.558130	−0.279065	−	0.960272i	$-0.590024\pi$
$193$	−7.75379	−0.558130	−0.279065	+	0.960272i	$0.590024\pi$
$194$	10.6847	0.767114
$195$	0.684658	0.0490294
$196$	2.75379	0.196699
$197$	−11.3693	−0.810030	−0.405015	−	0.914310i	$-0.632734\pi$
$197$	−11.3693	−0.810030	−0.405015	+	0.914310i	$0.632734\pi$
$198$	2.24621	0.159631
$199$	10.4384	0.739962	0.369981	−	0.929039i	$-0.379364\pi$
$199$	10.4384	0.739962	0.369981	+	0.929039i	$0.379364\pi$
$200$	1.00000	0.0707107
$201$	−1.36932	−0.0965842
$202$	18.4924	1.30112
$203$	−20.8769	−1.46527
$204$	1.56155	0.109331
$205$	−12.2462	−0.855312
$206$	8.00000	0.557386
$207$	−1.75379	−0.121897
$208$	0.438447	0.0304008
$209$	−6.24621	−0.432059
$210$	−4.87689	−0.336538
$211$	21.3693	1.47112	0.735562	−	0.677457i	$-0.236918\pi$
$211$	21.3693	1.47112	0.735562	+	0.677457i	$0.236918\pi$
$212$	3.56155	0.244608
$213$	−26.0540	−1.78519
$214$	−16.4924	−1.12740
$215$	7.12311	0.485792
$216$	−5.56155	−0.378416
$217$	7.61553	0.516976
$218$	−4.43845	−0.300610
$219$	21.5616	1.45699
$220$	−4.00000	−0.269680
$221$	0.438447	0.0294931
$222$	1.75379	0.117707
$223$	−8.68466	−0.581568	−0.290784	−	0.956789i	$-0.593916\pi$
$223$	−8.68466	−0.581568	−0.290784	+	0.956789i	$0.593916\pi$
$224$	−3.12311	−0.208671
$225$	−0.561553	−0.0374369
$226$	−17.8078	−1.18455
$227$	28.6847	1.90387	0.951934	−	0.306304i	$-0.0990923\pi$
$227$	28.6847	1.90387	0.951934	+	0.306304i	$0.0990923\pi$
$228$	2.43845	0.161490
$229$	18.4924	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$229$	18.4924	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$230$	3.12311	0.205931
$231$	19.5076	1.28350
$232$	6.68466	0.438869
$233$	18.6847	1.22407	0.612036	−	0.790830i	$-0.290351\pi$
$233$	18.6847	1.22407	0.612036	+	0.790830i	$0.290351\pi$
$234$	−0.246211	−0.0160953
$235$	2.43845	0.159067
$236$	−12.6847	−0.825701
$237$	0	0
$238$	−3.12311	−0.202441
$239$	1.36932	0.0885737	0.0442869	−	0.999019i	$-0.485898\pi$
$239$	1.36932	0.0885737	0.0442869	+	0.999019i	$0.485898\pi$
$240$	1.56155	0.100798
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	5.00000	0.321412
$243$	5.75379	0.369106
$244$	11.5616	0.740153
$245$	2.75379	0.175933
$246$	−19.1231	−1.21924
$247$	0.684658	0.0435638
$248$	−2.43845	−0.154842
$249$	16.0000	1.01396
$250$	1.00000	0.0632456
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	1.75379	0.110478
$253$	−12.4924	−0.785392
$254$	13.5616	0.850928
$255$	1.56155	0.0977882
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	11.1231	0.692494
$259$	−3.50758	−0.217950
$260$	0.438447	0.0271913
$261$	−3.75379	−0.232354
$262$	−7.12311	−0.440067
$263$	4.19224	0.258504	0.129252	−	0.991612i	$-0.458742\pi$
$263$	4.19224	0.258504	0.129252	+	0.991612i	$0.458742\pi$
$264$	−6.24621	−0.384428
$265$	3.56155	0.218784
$266$	−4.87689	−0.299022
$267$	4.19224	0.256561
$268$	−0.876894	−0.0535648
$269$	17.8078	1.08576	0.542879	−	0.839811i	$-0.317334\pi$
$269$	17.8078	1.08576	0.542879	+	0.839811i	$0.317334\pi$
$270$	−5.56155	−0.338465
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	1.00000	0.0606339
$273$	−2.13826	−0.129413
$274$	10.0000	0.604122
$275$	−4.00000	−0.241209
$276$	4.87689	0.293555
$277$	−32.2462	−1.93749	−0.968744	−	0.248064i	$-0.920206\pi$
$277$	−32.2462	−1.93749	−0.968744	+	0.248064i	$0.920206\pi$
$278$	−16.4924	−0.989150
$279$	1.36932	0.0819789
$280$	−3.12311	−0.186641
$281$	12.4384	0.742016	0.371008	−	0.928630i	$-0.379012\pi$
$281$	12.4384	0.742016	0.371008	+	0.928630i	$0.379012\pi$
$282$	3.80776	0.226749
$283$	20.6847	1.22958	0.614788	−	0.788693i	$-0.289242\pi$
$283$	20.6847	1.22958	0.614788	+	0.788693i	$0.289242\pi$
$284$	−16.6847	−0.990052
$285$	2.43845	0.144441
$286$	−1.75379	−0.103704
$287$	38.2462	2.25760
$288$	−0.561553	−0.0330898
$289$	1.00000	0.0588235
$290$	6.68466	0.392537
$291$	16.6847	0.978072
$292$	13.8078	0.808038
$293$	−12.4384	−0.726662	−0.363331	−	0.931660i	$-0.618361\pi$
$293$	−12.4384	−0.726662	−0.363331	+	0.931660i	$0.618361\pi$
$294$	4.30019	0.250792
$295$	−12.6847	−0.738529
$296$	1.12311	0.0652792
$297$	22.2462	1.29086
$298$	−5.12311	−0.296774
$299$	1.36932	0.0791896
$300$	1.56155	0.0901563
$301$	−22.2462	−1.28225
$302$	9.36932	0.539144
$303$	28.8769	1.65893
$304$	1.56155	0.0895612
$305$	11.5616	0.662013
$306$	−0.561553	−0.0321018
$307$	−34.2462	−1.95453	−0.977267	−	0.212011i	$-0.931999\pi$
$307$	−34.2462	−1.95453	−0.977267	+	0.212011i	$0.931999\pi$
$308$	12.4924	0.711822
$309$	12.4924	0.710669
$310$	−2.43845	−0.138494
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0.684658	0.0387612
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−14.4924	−0.817855
$315$	1.75379	0.0988148
$316$	0	0
$317$	−9.61553	−0.540062	−0.270031	−	0.962852i	$-0.587034\pi$
$317$	−9.61553	−0.540062	−0.270031	+	0.962852i	$0.587034\pi$
$318$	5.56155	0.311876
$319$	−26.7386	−1.49708
$320$	1.00000	0.0559017
$321$	−25.7538	−1.43744
$322$	−9.75379	−0.543557
$323$	1.56155	0.0868871
$324$	−7.00000	−0.388889
$325$	0.438447	0.0243207
$326$	−2.24621	−0.124406
$327$	−6.93087	−0.383278
$328$	−12.2462	−0.676184
$329$	−7.61553	−0.419858
$330$	−6.24621	−0.343843
$331$	−14.0540	−0.772476	−0.386238	−	0.922399i	$-0.626226\pi$
$331$	−14.0540	−0.772476	−0.386238	+	0.922399i	$0.626226\pi$
$332$	10.2462	0.562334
$333$	−0.630683	−0.0345612
$334$	−3.12311	−0.170889
$335$	−0.876894	−0.0479099
$336$	−4.87689	−0.266056
$337$	−22.6847	−1.23571	−0.617856	−	0.786291i	$-0.711999\pi$
$337$	−22.6847	−1.23571	−0.617856	+	0.786291i	$0.711999\pi$
$338$	−12.8078	−0.696651
$339$	−27.8078	−1.51031
$340$	1.00000	0.0542326
$341$	9.75379	0.528197
$342$	−0.876894	−0.0474170
$343$	13.2614	0.716046
$344$	7.12311	0.384052
$345$	4.87689	0.262563
$346$	−18.0000	−0.967686
$347$	−6.43845	−0.345634	−0.172817	−	0.984954i	$-0.555287\pi$
$347$	−6.43845	−0.345634	−0.172817	+	0.984954i	$0.555287\pi$
$348$	10.4384	0.559560
$349$	−6.87689	−0.368112	−0.184056	−	0.982916i	$-0.558923\pi$
$349$	−6.87689	−0.368112	−0.184056	+	0.982916i	$0.558923\pi$
$350$	−3.12311	−0.166937
$351$	−2.43845	−0.130155
$352$	−4.00000	−0.213201
$353$	−10.4924	−0.558455	−0.279228	−	0.960225i	$-0.590078\pi$
$353$	−10.4924	−0.558455	−0.279228	+	0.960225i	$0.590078\pi$
$354$	−19.8078	−1.05277
$355$	−16.6847	−0.885530
$356$	2.68466	0.142287
$357$	−4.87689	−0.258113
$358$	−24.4924	−1.29446
$359$	−19.1231	−1.00928	−0.504639	−	0.863330i	$-0.668375\pi$
$359$	−19.1231	−1.00928	−0.504639	+	0.863330i	$0.668375\pi$
$360$	−0.561553	−0.0295964
$361$	−16.5616	−0.871661
$362$	−0.246211	−0.0129406
$363$	7.80776	0.409801
$364$	−1.36932	−0.0717717
$365$	13.8078	0.722731
$366$	18.0540	0.943696
$367$	7.61553	0.397527	0.198764	−	0.980047i	$-0.436307\pi$
$367$	7.61553	0.397527	0.198764	+	0.980047i	$0.436307\pi$
$368$	3.12311	0.162803
$369$	6.87689	0.357997
$370$	1.12311	0.0583875
$371$	−11.1231	−0.577483
$372$	−3.80776	−0.197423
$373$	24.7386	1.28092	0.640459	−	0.767992i	$-0.278744\pi$
$373$	24.7386	1.28092	0.640459	+	0.767992i	$0.278744\pi$
$374$	−4.00000	−0.206835
$375$	1.56155	0.0806382
$376$	2.43845	0.125753
$377$	2.93087	0.150947
$378$	17.3693	0.893381
$379$	2.24621	0.115380	0.0576901	−	0.998335i	$-0.481626\pi$
$379$	2.24621	0.115380	0.0576901	+	0.998335i	$0.481626\pi$
$380$	1.56155	0.0801060
$381$	21.1771	1.08493
$382$	0	0
$383$	3.80776	0.194568	0.0972838	−	0.995257i	$-0.468985\pi$
$383$	3.80776	0.194568	0.0972838	+	0.995257i	$0.468985\pi$
$384$	1.56155	0.0796877
$385$	12.4924	0.636673
$386$	−7.75379	−0.394657
$387$	−4.00000	−0.203331
$388$	10.6847	0.542431
$389$	−11.3693	−0.576447	−0.288224	−	0.957563i	$-0.593065\pi$
$389$	−11.3693	−0.576447	−0.288224	+	0.957563i	$0.593065\pi$
$390$	0.684658	0.0346690
$391$	3.12311	0.157942
$392$	2.75379	0.139087
$393$	−11.1231	−0.561086
$394$	−11.3693	−0.572778
$395$	0	0
$396$	2.24621	0.112876
$397$	−3.36932	−0.169101	−0.0845506	−	0.996419i	$-0.526945\pi$
$397$	−3.36932	−0.169101	−0.0845506	+	0.996419i	$0.526945\pi$
$398$	10.4384	0.523232
$399$	−7.61553	−0.381253
$400$	1.00000	0.0500000
$401$	35.3693	1.76626	0.883130	−	0.469129i	$-0.155432\pi$
$401$	35.3693	1.76626	0.883130	+	0.469129i	$0.155432\pi$
$402$	−1.36932	−0.0682953
$403$	−1.06913	−0.0532572
$404$	18.4924	0.920032
$405$	−7.00000	−0.347833
$406$	−20.8769	−1.03610
$407$	−4.49242	−0.222681
$408$	1.56155	0.0773084
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	−12.2462	−0.604797
$411$	15.6155	0.770257
$412$	8.00000	0.394132
$413$	39.6155	1.94935
$414$	−1.75379	−0.0861940
$415$	10.2462	0.502967
$416$	0.438447	0.0214966
$417$	−25.7538	−1.26117
$418$	−6.24621	−0.305512
$419$	−16.8769	−0.824490	−0.412245	−	0.911073i	$-0.635255\pi$
$419$	−16.8769	−0.824490	−0.412245	+	0.911073i	$0.635255\pi$
$420$	−4.87689	−0.237968
$421$	−33.6155	−1.63832	−0.819160	−	0.573565i	$-0.805560\pi$
$421$	−33.6155	−1.63832	−0.819160	+	0.573565i	$0.805560\pi$
$422$	21.3693	1.04024
$423$	−1.36932	−0.0665785
$424$	3.56155	0.172964
$425$	1.00000	0.0485071
$426$	−26.0540	−1.26232
$427$	−36.1080	−1.74739
$428$	−16.4924	−0.797191
$429$	−2.73863	−0.132222
$430$	7.12311	0.343507
$431$	12.4924	0.601739	0.300869	−	0.953665i	$-0.402723\pi$
$431$	12.4924	0.601739	0.300869	+	0.953665i	$0.402723\pi$
$432$	−5.56155	−0.267580
$433$	30.4924	1.46537	0.732686	−	0.680567i	$-0.238266\pi$
$433$	30.4924	1.46537	0.732686	+	0.680567i	$0.238266\pi$
$434$	7.61553	0.365557
$435$	10.4384	0.500485
$436$	−4.43845	−0.212563
$437$	4.87689	0.233293
$438$	21.5616	1.03025
$439$	32.9848	1.57428	0.787140	−	0.616774i	$-0.211561\pi$
$439$	32.9848	1.57428	0.787140	+	0.616774i	$0.211561\pi$
$440$	−4.00000	−0.190693
$441$	−1.54640	−0.0736380
$442$	0.438447	0.0208548
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	1.75379	0.0832311
$445$	2.68466	0.127265
$446$	−8.68466	−0.411230
$447$	−8.00000	−0.378387
$448$	−3.12311	−0.147553
$449$	−25.1231	−1.18563	−0.592816	−	0.805338i	$-0.701984\pi$
$449$	−25.1231	−1.18563	−0.592816	+	0.805338i	$0.701984\pi$
$450$	−0.561553	−0.0264719
$451$	48.9848	2.30661
$452$	−17.8078	−0.837607
$453$	14.6307	0.687409
$454$	28.6847	1.34624
$455$	−1.36932	−0.0641946
$456$	2.43845	0.114191
$457$	0.246211	0.0115173	0.00575864	−	0.999983i	$-0.498167\pi$
$457$	0.246211	0.0115173	0.00575864	+	0.999983i	$0.498167\pi$
$458$	18.4924	0.864094
$459$	−5.56155	−0.259591
$460$	3.12311	0.145616
$461$	−30.4924	−1.42017	−0.710087	−	0.704114i	$-0.751344\pi$
$461$	−30.4924	−1.42017	−0.710087	+	0.704114i	$0.751344\pi$
$462$	19.5076	0.907575
$463$	14.9309	0.693896	0.346948	−	0.937884i	$-0.387218\pi$
$463$	14.9309	0.693896	0.346948	+	0.937884i	$0.387218\pi$
$464$	6.68466	0.310327
$465$	−3.80776	−0.176581
$466$	18.6847	0.865550
$467$	0.492423	0.0227866	0.0113933	−	0.999935i	$-0.496373\pi$
$467$	0.492423	0.0227866	0.0113933	+	0.999935i	$0.496373\pi$
$468$	−0.246211	−0.0113811
$469$	2.73863	0.126458
$470$	2.43845	0.112477
$471$	−22.6307	−1.04277
$472$	−12.6847	−0.583859
$473$	−28.4924	−1.31008
$474$	0	0
$475$	1.56155	0.0716490
$476$	−3.12311	−0.143147
$477$	−2.00000	−0.0915737
$478$	1.36932	0.0626311
$479$	34.4384	1.57353	0.786766	−	0.617251i	$-0.211754\pi$
$479$	34.4384	1.57353	0.786766	+	0.617251i	$0.211754\pi$
$480$	1.56155	0.0712748
$481$	0.492423	0.0224525
$482$	−4.24621	−0.193410
$483$	−15.2311	−0.693037
$484$	5.00000	0.227273
$485$	10.6847	0.485165
$486$	5.75379	0.260997
$487$	−15.6155	−0.707607	−0.353804	−	0.935320i	$-0.615112\pi$
$487$	−15.6155	−0.707607	−0.353804	+	0.935320i	$0.615112\pi$
$488$	11.5616	0.523367
$489$	−3.50758	−0.158618
$490$	2.75379	0.124403
$491$	−1.56155	−0.0704719	−0.0352359	−	0.999379i	$-0.511218\pi$
$491$	−1.56155	−0.0704719	−0.0352359	+	0.999379i	$0.511218\pi$
$492$	−19.1231	−0.862136
$493$	6.68466	0.301062
$494$	0.684658	0.0308042
$495$	2.24621	0.100960
$496$	−2.43845	−0.109490
$497$	52.1080	2.33736
$498$	16.0000	0.716977
$499$	−0.876894	−0.0392552	−0.0196276	−	0.999807i	$-0.506248\pi$
$499$	−0.876894	−0.0392552	−0.0196276	+	0.999807i	$0.506248\pi$
$500$	1.00000	0.0447214
$501$	−4.87689	−0.217884
$502$	−4.00000	−0.178529
$503$	−26.7386	−1.19222	−0.596108	−	0.802904i	$-0.703287\pi$
$503$	−26.7386	−1.19222	−0.596108	+	0.802904i	$0.703287\pi$
$504$	1.75379	0.0781200
$505$	18.4924	0.822902
$506$	−12.4924	−0.555356
$507$	−20.0000	−0.888231
$508$	13.5616	0.601697
$509$	25.1231	1.11356	0.556781	−	0.830659i	$-0.312036\pi$
$509$	25.1231	1.11356	0.556781	+	0.830659i	$0.312036\pi$
$510$	1.56155	0.0691467
$511$	−43.1231	−1.90765
$512$	1.00000	0.0441942
$513$	−8.68466	−0.383437
$514$	2.00000	0.0882162
$515$	8.00000	0.352522
$516$	11.1231	0.489667
$517$	−9.75379	−0.428971
$518$	−3.50758	−0.154114
$519$	−28.1080	−1.23380
$520$	0.438447	0.0192272
$521$	2.38447	0.104466	0.0522328	−	0.998635i	$-0.483366\pi$
$521$	2.38447	0.104466	0.0522328	+	0.998635i	$0.483366\pi$
$522$	−3.75379	−0.164299
$523$	−17.8617	−0.781039	−0.390520	−	0.920595i	$-0.627705\pi$
$523$	−17.8617	−0.781039	−0.390520	+	0.920595i	$0.627705\pi$
$524$	−7.12311	−0.311174
$525$	−4.87689	−0.212845
$526$	4.19224	0.182790
$527$	−2.43845	−0.106220
$528$	−6.24621	−0.271831
$529$	−13.2462	−0.575922
$530$	3.56155	0.154704
$531$	7.12311	0.309116
$532$	−4.87689	−0.211440
$533$	−5.36932	−0.232571
$534$	4.19224	0.181416
$535$	−16.4924	−0.713030
$536$	−0.876894	−0.0378761
$537$	−38.2462	−1.65045
$538$	17.8078	0.767747
$539$	−11.0152	−0.474456
$540$	−5.56155	−0.239331
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	16.0000	0.687259
$543$	−0.384472	−0.0164993
$544$	1.00000	0.0428746
$545$	−4.43845	−0.190122
$546$	−2.13826	−0.0915091
$547$	33.5616	1.43499	0.717494	−	0.696564i	$-0.245289\pi$
$547$	33.5616	1.43499	0.717494	+	0.696564i	$0.245289\pi$
$548$	10.0000	0.427179
$549$	−6.49242	−0.277090
$550$	−4.00000	−0.170561
$551$	10.4384	0.444693
$552$	4.87689	0.207574
$553$	0	0
$554$	−32.2462	−1.37001
$555$	1.75379	0.0744442
$556$	−16.4924	−0.699435
$557$	32.4384	1.37446	0.687231	−	0.726439i	$-0.258826\pi$
$557$	32.4384	1.37446	0.687231	+	0.726439i	$0.258826\pi$
$558$	1.36932	0.0579678
$559$	3.12311	0.132093
$560$	−3.12311	−0.131975
$561$	−6.24621	−0.263715
$562$	12.4384	0.524684
$563$	−29.3693	−1.23777	−0.618885	−	0.785482i	$-0.712415\pi$
$563$	−29.3693	−1.23777	−0.618885	+	0.785482i	$0.712415\pi$
$564$	3.80776	0.160336
$565$	−17.8078	−0.749178
$566$	20.6847	0.869441
$567$	21.8617	0.918107
$568$	−16.6847	−0.700073
$569$	24.9309	1.04516	0.522578	−	0.852591i	$-0.324970\pi$
$569$	24.9309	1.04516	0.522578	+	0.852591i	$0.324970\pi$
$570$	2.43845	0.102135
$571$	16.8769	0.706276	0.353138	−	0.935571i	$-0.385115\pi$
$571$	16.8769	0.706276	0.353138	+	0.935571i	$0.385115\pi$
$572$	−1.75379	−0.0733296
$573$	0	0
$574$	38.2462	1.59637
$575$	3.12311	0.130243
$576$	−0.561553	−0.0233980
$577$	19.3693	0.806355	0.403178	−	0.915122i	$-0.367906\pi$
$577$	19.3693	0.806355	0.403178	+	0.915122i	$0.367906\pi$
$578$	1.00000	0.0415945
$579$	−12.1080	−0.503189
$580$	6.68466	0.277565
$581$	−32.0000	−1.32758
$582$	16.6847	0.691601
$583$	−14.2462	−0.590018
$584$	13.8078	0.571369
$585$	−0.246211	−0.0101796
$586$	−12.4384	−0.513828
$587$	−15.1231	−0.624197	−0.312099	−	0.950050i	$-0.601032\pi$
$587$	−15.1231	−0.624197	−0.312099	+	0.950050i	$0.601032\pi$
$588$	4.30019	0.177337
$589$	−3.80776	−0.156896
$590$	−12.6847	−0.522219
$591$	−17.7538	−0.730293
$592$	1.12311	0.0461594
$593$	8.24621	0.338631	0.169316	−	0.985562i	$-0.445844\pi$
$593$	8.24621	0.338631	0.169316	+	0.985562i	$0.445844\pi$
$594$	22.2462	0.912773
$595$	−3.12311	−0.128035
$596$	−5.12311	−0.209851
$597$	16.3002	0.667122
$598$	1.36932	0.0559955
$599$	31.6155	1.29178	0.645888	−	0.763432i	$-0.276487\pi$
$599$	31.6155	1.29178	0.645888	+	0.763432i	$0.276487\pi$
$600$	1.56155	0.0637501
$601$	33.6155	1.37121	0.685603	−	0.727976i	$-0.259539\pi$
$601$	33.6155	1.37121	0.685603	+	0.727976i	$0.259539\pi$
$602$	−22.2462	−0.906688
$603$	0.492423	0.0200530
$604$	9.36932	0.381232
$605$	5.00000	0.203279
$606$	28.8769	1.17304
$607$	13.8617	0.562631	0.281315	−	0.959615i	$-0.409229\pi$
$607$	13.8617	0.562631	0.281315	+	0.959615i	$0.409229\pi$
$608$	1.56155	0.0633293
$609$	−32.6004	−1.32103
$610$	11.5616	0.468114
$611$	1.06913	0.0432524
$612$	−0.561553	−0.0226994
$613$	4.93087	0.199156	0.0995780	−	0.995030i	$-0.468251\pi$
$613$	4.93087	0.199156	0.0995780	+	0.995030i	$0.468251\pi$
$614$	−34.2462	−1.38206
$615$	−19.1231	−0.771118
$616$	12.4924	0.503334
$617$	18.6847	0.752216	0.376108	−	0.926576i	$-0.377262\pi$
$617$	18.6847	0.752216	0.376108	+	0.926576i	$0.377262\pi$
$618$	12.4924	0.502519
$619$	0.876894	0.0352454	0.0176227	−	0.999845i	$-0.494390\pi$
$619$	0.876894	0.0352454	0.0176227	+	0.999845i	$0.494390\pi$
$620$	−2.43845	−0.0979304
$621$	−17.3693	−0.697007
$622$	8.00000	0.320771
$623$	−8.38447	−0.335917
$624$	0.684658	0.0274083
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	−9.75379	−0.389529
$628$	−14.4924	−0.578311
$629$	1.12311	0.0447812
$630$	1.75379	0.0698726
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	33.3693	1.32631
$634$	−9.61553	−0.381881
$635$	13.5616	0.538174
$636$	5.56155	0.220530
$637$	1.20739	0.0478386
$638$	−26.7386	−1.05859
$639$	9.36932	0.370644
$640$	1.00000	0.0395285
$641$	−7.75379	−0.306256	−0.153128	−	0.988206i	$-0.548935\pi$
$641$	−7.75379	−0.306256	−0.153128	+	0.988206i	$0.548935\pi$
$642$	−25.7538	−1.01642
$643$	7.50758	0.296070	0.148035	−	0.988982i	$-0.452705\pi$
$643$	7.50758	0.296070	0.148035	+	0.988982i	$0.452705\pi$
$644$	−9.75379	−0.384353
$645$	11.1231	0.437972
$646$	1.56155	0.0614385
$647$	−0.684658	−0.0269167	−0.0134584	−	0.999909i	$-0.504284\pi$
$647$	−0.684658	−0.0269167	−0.0134584	+	0.999909i	$0.504284\pi$
$648$	−7.00000	−0.274986
$649$	50.7386	1.99167
$650$	0.438447	0.0171973
$651$	11.8920	0.466086
$652$	−2.24621	−0.0879684
$653$	−5.50758	−0.215528	−0.107764	−	0.994176i	$-0.534369\pi$
$653$	−5.50758	−0.215528	−0.107764	+	0.994176i	$0.534369\pi$
$654$	−6.93087	−0.271018
$655$	−7.12311	−0.278323
$656$	−12.2462	−0.478134
$657$	−7.75379	−0.302504
$658$	−7.61553	−0.296884
$659$	−13.0691	−0.509101	−0.254551	−	0.967059i	$-0.581928\pi$
$659$	−13.0691	−0.509101	−0.254551	+	0.967059i	$0.581928\pi$
$660$	−6.24621	−0.243133
$661$	−39.8617	−1.55044	−0.775221	−	0.631690i	$-0.782362\pi$
$661$	−39.8617	−1.55044	−0.775221	+	0.631690i	$0.782362\pi$
$662$	−14.0540	−0.546223
$663$	0.684658	0.0265899
$664$	10.2462	0.397630
$665$	−4.87689	−0.189118
$666$	−0.630683	−0.0244385
$667$	20.8769	0.808357
$668$	−3.12311	−0.120837
$669$	−13.5616	−0.524320
$670$	−0.876894	−0.0338774
$671$	−46.2462	−1.78532
$672$	−4.87689	−0.188130
$673$	−41.4233	−1.59675	−0.798375	−	0.602160i	$-0.794307\pi$
$673$	−41.4233	−1.59675	−0.798375	+	0.602160i	$0.794307\pi$
$674$	−22.6847	−0.873780
$675$	−5.56155	−0.214064
$676$	−12.8078	−0.492606
$677$	8.73863	0.335853	0.167926	−	0.985800i	$-0.446293\pi$
$677$	8.73863	0.335853	0.167926	+	0.985800i	$0.446293\pi$
$678$	−27.8078	−1.06795
$679$	−33.3693	−1.28060
$680$	1.00000	0.0383482
$681$	44.7926	1.71646
$682$	9.75379	0.373492
$683$	12.3002	0.470654	0.235327	−	0.971916i	$-0.424384\pi$
$683$	12.3002	0.470654	0.235327	+	0.971916i	$0.424384\pi$
$684$	−0.876894	−0.0335289
$685$	10.0000	0.382080
$686$	13.2614	0.506321
$687$	28.8769	1.10172
$688$	7.12311	0.271566
$689$	1.56155	0.0594904
$690$	4.87689	0.185660
$691$	26.2462	0.998453	0.499226	−	0.866472i	$-0.333618\pi$
$691$	26.2462	0.998453	0.499226	+	0.866472i	$0.333618\pi$
$692$	−18.0000	−0.684257
$693$	−7.01515	−0.266484
$694$	−6.43845	−0.244400
$695$	−16.4924	−0.625593
$696$	10.4384	0.395668
$697$	−12.2462	−0.463858
$698$	−6.87689	−0.260294
$699$	29.1771	1.10358
$700$	−3.12311	−0.118042
$701$	−44.3542	−1.67523	−0.837617	−	0.546258i	$-0.816052\pi$
$701$	−44.3542	−1.67523	−0.837617	+	0.546258i	$0.816052\pi$
$702$	−2.43845	−0.0920333
$703$	1.75379	0.0661454
$704$	−4.00000	−0.150756
$705$	3.80776	0.143409
$706$	−10.4924	−0.394888
$707$	−57.7538	−2.17205
$708$	−19.8078	−0.744421
$709$	−32.5464	−1.22231	−0.611153	−	0.791513i	$-0.709294\pi$
$709$	−32.5464	−1.22231	−0.611153	+	0.791513i	$0.709294\pi$
$710$	−16.6847	−0.626164
$711$	0	0
$712$	2.68466	0.100612
$713$	−7.61553	−0.285204
$714$	−4.87689	−0.182513
$715$	−1.75379	−0.0655880
$716$	−24.4924	−0.915325
$717$	2.13826	0.0798548
$718$	−19.1231	−0.713668
$719$	35.8078	1.33540	0.667702	−	0.744429i	$-0.267278\pi$
$719$	35.8078	1.33540	0.667702	+	0.744429i	$0.267278\pi$
$720$	−0.561553	−0.0209278
$721$	−24.9848	−0.930484
$722$	−16.5616	−0.616357
$723$	−6.63068	−0.246598
$724$	−0.246211	−0.00915037
$725$	6.68466	0.248262
$726$	7.80776	0.289773
$727$	42.4384	1.57395	0.786977	−	0.616982i	$-0.211645\pi$
$727$	42.4384	1.57395	0.786977	+	0.616982i	$0.211645\pi$
$728$	−1.36932	−0.0507503
$729$	29.9848	1.11055
$730$	13.8078	0.511048
$731$	7.12311	0.263458
$732$	18.0540	0.667294
$733$	10.4924	0.387546	0.193773	−	0.981046i	$-0.437927\pi$
$733$	10.4924	0.387546	0.193773	+	0.981046i	$0.437927\pi$
$734$	7.61553	0.281094
$735$	4.30019	0.158615
$736$	3.12311	0.115119
$737$	3.50758	0.129203
$738$	6.87689	0.253142
$739$	25.1771	0.926154	0.463077	−	0.886318i	$-0.346745\pi$
$739$	25.1771	0.926154	0.463077	+	0.886318i	$0.346745\pi$
$740$	1.12311	0.0412862
$741$	1.06913	0.0392755
$742$	−11.1231	−0.408342
$743$	−12.8769	−0.472407	−0.236204	−	0.971704i	$-0.575903\pi$
$743$	−12.8769	−0.472407	−0.236204	+	0.971704i	$0.575903\pi$
$744$	−3.80776	−0.139599
$745$	−5.12311	−0.187696
$746$	24.7386	0.905746
$747$	−5.75379	−0.210520
$748$	−4.00000	−0.146254
$749$	51.5076	1.88205
$750$	1.56155	0.0570198
$751$	21.1771	0.772763	0.386381	−	0.922339i	$-0.373725\pi$
$751$	21.1771	0.772763	0.386381	+	0.922339i	$0.373725\pi$
$752$	2.43845	0.0889210
$753$	−6.24621	−0.227625
$754$	2.93087	0.106736
$755$	9.36932	0.340984
$756$	17.3693	0.631716
$757$	−38.7926	−1.40994	−0.704971	−	0.709236i	$-0.749040\pi$
$757$	−38.7926	−1.40994	−0.704971	+	0.709236i	$0.749040\pi$
$758$	2.24621	0.0815861
$759$	−19.5076	−0.708080
$760$	1.56155	0.0566435
$761$	28.7386	1.04177	0.520887	−	0.853625i	$-0.325601\pi$
$761$	28.7386	1.04177	0.520887	+	0.853625i	$0.325601\pi$
$762$	21.1771	0.767165
$763$	13.8617	0.501829
$764$	0	0
$765$	−0.561553	−0.0203030
$766$	3.80776	0.137580
$767$	−5.56155	−0.200816
$768$	1.56155	0.0563477
$769$	15.5616	0.561164	0.280582	−	0.959830i	$-0.409473\pi$
$769$	15.5616	0.561164	0.280582	+	0.959830i	$0.409473\pi$
$770$	12.4924	0.450196
$771$	3.12311	0.112476
$772$	−7.75379	−0.279065
$773$	−28.7386	−1.03366	−0.516828	−	0.856089i	$-0.672887\pi$
$773$	−28.7386	−1.03366	−0.516828	+	0.856089i	$0.672887\pi$
$774$	−4.00000	−0.143777
$775$	−2.43845	−0.0875916
$776$	10.6847	0.383557
$777$	−5.47727	−0.196496
$778$	−11.3693	−0.407610
$779$	−19.1231	−0.685156
$780$	0.684658	0.0245147
$781$	66.7386	2.38810
$782$	3.12311	0.111682
$783$	−37.1771	−1.32860
$784$	2.75379	0.0983496
$785$	−14.4924	−0.517257
$786$	−11.1231	−0.396748
$787$	42.5464	1.51662	0.758308	−	0.651897i	$-0.226027\pi$
$787$	42.5464	1.51662	0.758308	+	0.651897i	$0.226027\pi$
$788$	−11.3693	−0.405015
$789$	6.54640	0.233058
$790$	0	0
$791$	55.6155	1.97746
$792$	2.24621	0.0798156
$793$	5.06913	0.180010
$794$	−3.36932	−0.119573
$795$	5.56155	0.197248
$796$	10.4384	0.369981
$797$	−14.4924	−0.513348	−0.256674	−	0.966498i	$-0.582627\pi$
$797$	−14.4924	−0.513348	−0.256674	+	0.966498i	$0.582627\pi$
$798$	−7.61553	−0.269587
$799$	2.43845	0.0862661
$800$	1.00000	0.0353553
$801$	−1.50758	−0.0532676
$802$	35.3693	1.24893
$803$	−55.2311	−1.94906
$804$	−1.36932	−0.0482921
$805$	−9.75379	−0.343776
$806$	−1.06913	−0.0376585
$807$	27.8078	0.978880
$808$	18.4924	0.650561
$809$	−46.9848	−1.65190	−0.825950	−	0.563744i	$-0.809360\pi$
$809$	−46.9848	−1.65190	−0.825950	+	0.563744i	$0.809360\pi$
$810$	−7.00000	−0.245955
$811$	−21.3693	−0.750378	−0.375189	−	0.926948i	$-0.622422\pi$
$811$	−21.3693	−0.750378	−0.375189	+	0.926948i	$0.622422\pi$
$812$	−20.8769	−0.732635
$813$	24.9848	0.876257
$814$	−4.49242	−0.157459
$815$	−2.24621	−0.0786813
$816$	1.56155	0.0546653
$817$	11.1231	0.389148
$818$	−14.6847	−0.513437
$819$	0.768944	0.0268691
$820$	−12.2462	−0.427656
$821$	38.3002	1.33669	0.668343	−	0.743853i	$-0.267004\pi$
$821$	38.3002	1.33669	0.668343	+	0.743853i	$0.267004\pi$
$822$	15.6155	0.544654
$823$	25.3693	0.884319	0.442159	−	0.896936i	$-0.354213\pi$
$823$	25.3693	0.884319	0.442159	+	0.896936i	$0.354213\pi$
$824$	8.00000	0.278693
$825$	−6.24621	−0.217465
$826$	39.6155	1.37840
$827$	−32.4924	−1.12987	−0.564936	−	0.825135i	$-0.691099\pi$
$827$	−32.4924	−1.12987	−0.564936	+	0.825135i	$0.691099\pi$
$828$	−1.75379	−0.0609484
$829$	15.3693	0.533798	0.266899	−	0.963724i	$-0.414001\pi$
$829$	15.3693	0.533798	0.266899	+	0.963724i	$0.414001\pi$
$830$	10.2462	0.355651
$831$	−50.3542	−1.74677
$832$	0.438447	0.0152004
$833$	2.75379	0.0954131
$834$	−25.7538	−0.891781
$835$	−3.12311	−0.108080
$836$	−6.24621	−0.216030
$837$	13.5616	0.468756
$838$	−16.8769	−0.583003
$839$	−2.05398	−0.0709111	−0.0354556	−	0.999371i	$-0.511288\pi$
$839$	−2.05398	−0.0709111	−0.0354556	+	0.999371i	$0.511288\pi$
$840$	−4.87689	−0.168269
$841$	15.6847	0.540850
$842$	−33.6155	−1.15847
$843$	19.4233	0.668974
$844$	21.3693	0.735562
$845$	−12.8078	−0.440600
$846$	−1.36932	−0.0470781
$847$	−15.6155	−0.536556
$848$	3.56155	0.122304
$849$	32.3002	1.10854
$850$	1.00000	0.0342997
$851$	3.50758	0.120238
$852$	−26.0540	−0.892594
$853$	31.7538	1.08723	0.543615	−	0.839335i	$-0.317055\pi$
$853$	31.7538	1.08723	0.543615	+	0.839335i	$0.317055\pi$
$854$	−36.1080	−1.23559
$855$	−0.876894	−0.0299892
$856$	−16.4924	−0.563699
$857$	47.1771	1.61154	0.805769	−	0.592230i	$-0.201752\pi$
$857$	47.1771	1.61154	0.805769	+	0.592230i	$0.201752\pi$
$858$	−2.73863	−0.0934954
$859$	34.5464	1.17871	0.589354	−	0.807875i	$-0.299382\pi$
$859$	34.5464	1.17871	0.589354	+	0.807875i	$0.299382\pi$
$860$	7.12311	0.242896
$861$	59.7235	2.03537
$862$	12.4924	0.425494
$863$	44.4924	1.51454	0.757270	−	0.653102i	$-0.226533\pi$
$863$	44.4924	1.51454	0.757270	+	0.653102i	$0.226533\pi$
$864$	−5.56155	−0.189208
$865$	−18.0000	−0.612018
$866$	30.4924	1.03617
$867$	1.56155	0.0530331
$868$	7.61553	0.258488
$869$	0	0
$870$	10.4384	0.353897
$871$	−0.384472	−0.0130273
$872$	−4.43845	−0.150305
$873$	−6.00000	−0.203069
$874$	4.87689	0.164963
$875$	−3.12311	−0.105580
$876$	21.5616	0.728497
$877$	−10.3845	−0.350659	−0.175329	−	0.984510i	$-0.556099\pi$
$877$	−10.3845	−0.350659	−0.175329	+	0.984510i	$0.556099\pi$
$878$	32.9848	1.11318
$879$	−19.4233	−0.655131
$880$	−4.00000	−0.134840
$881$	20.7386	0.698702	0.349351	−	0.936992i	$-0.386402\pi$
$881$	20.7386	0.698702	0.349351	+	0.936992i	$0.386402\pi$
$882$	−1.54640	−0.0520699
$883$	2.63068	0.0885295	0.0442648	−	0.999020i	$-0.485905\pi$
$883$	2.63068	0.0885295	0.0442648	+	0.999020i	$0.485905\pi$
$884$	0.438447	0.0147466
$885$	−19.8078	−0.665831
$886$	28.0000	0.940678
$887$	−56.6004	−1.90045	−0.950227	−	0.311558i	$-0.899149\pi$
$887$	−56.6004	−1.90045	−0.950227	+	0.311558i	$0.899149\pi$
$888$	1.75379	0.0588533
$889$	−42.3542	−1.42051
$890$	2.68466	0.0899900
$891$	28.0000	0.938035
$892$	−8.68466	−0.290784
$893$	3.80776	0.127422
$894$	−8.00000	−0.267560
$895$	−24.4924	−0.818691
$896$	−3.12311	−0.104336
$897$	2.13826	0.0713944
$898$	−25.1231	−0.838369
$899$	−16.3002	−0.543642
$900$	−0.561553	−0.0187184
$901$	3.56155	0.118653
$902$	48.9848	1.63102
$903$	−34.7386	−1.15603
$904$	−17.8078	−0.592277
$905$	−0.246211	−0.00818434
$906$	14.6307	0.486072
$907$	14.4384	0.479421	0.239710	−	0.970844i	$-0.422948\pi$
$907$	14.4384	0.479421	0.239710	+	0.970844i	$0.422948\pi$
$908$	28.6847	0.951934
$909$	−10.3845	−0.344431
$910$	−1.36932	−0.0453924
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	2.43845	0.0807451
$913$	−40.9848	−1.35640
$914$	0.246211	0.00814394
$915$	18.0540	0.596846
$916$	18.4924	0.611007
$917$	22.2462	0.734635
$918$	−5.56155	−0.183559
$919$	−12.8769	−0.424770	−0.212385	−	0.977186i	$-0.568123\pi$
$919$	−12.8769	−0.424770	−0.212385	+	0.977186i	$0.568123\pi$
$920$	3.12311	0.102966
$921$	−53.4773	−1.76214
$922$	−30.4924	−1.00421
$923$	−7.31534	−0.240787
$924$	19.5076	0.641752
$925$	1.12311	0.0369275
$926$	14.9309	0.490659
$927$	−4.49242	−0.147551
$928$	6.68466	0.219435
$929$	39.4773	1.29521	0.647604	−	0.761977i	$-0.275771\pi$
$929$	39.4773	1.29521	0.647604	+	0.761977i	$0.275771\pi$
$930$	−3.80776	−0.124862
$931$	4.30019	0.140933
$932$	18.6847	0.612036
$933$	12.4924	0.408984
$934$	0.492423	0.0161126
$935$	−4.00000	−0.130814
$936$	−0.246211	−0.00804767
$937$	30.1080	0.983584	0.491792	−	0.870713i	$-0.336342\pi$
$937$	30.1080	0.983584	0.491792	+	0.870713i	$0.336342\pi$
$938$	2.73863	0.0894196
$939$	−9.36932	−0.305756
$940$	2.43845	0.0795334
$941$	−31.5616	−1.02888	−0.514439	−	0.857527i	$-0.672000\pi$
$941$	−31.5616	−1.02888	−0.514439	+	0.857527i	$0.672000\pi$
$942$	−22.6307	−0.737348
$943$	−38.2462	−1.24547
$944$	−12.6847	−0.412850
$945$	17.3693	0.565024
$946$	−28.4924	−0.926369
$947$	32.1922	1.04611	0.523054	−	0.852300i	$-0.324793\pi$
$947$	32.1922	1.04611	0.523054	+	0.852300i	$0.324793\pi$
$948$	0	0
$949$	6.05398	0.196520
$950$	1.56155	0.0506635
$951$	−15.0152	−0.486900
$952$	−3.12311	−0.101220
$953$	30.8769	1.00020	0.500100	−	0.865967i	$-0.333296\pi$
$953$	30.8769	1.00020	0.500100	+	0.865967i	$0.333296\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	1.36932	0.0442869
$957$	−41.7538	−1.34971
$958$	34.4384	1.11266
$959$	−31.2311	−1.00850
$960$	1.56155	0.0503989
$961$	−25.0540	−0.808193
$962$	0.492423	0.0158763
$963$	9.26137	0.298443
$964$	−4.24621	−0.136761
$965$	−7.75379	−0.249603
$966$	−15.2311	−0.490051
$967$	52.4924	1.68804	0.844021	−	0.536310i	$-0.180182\pi$
$967$	52.4924	1.68804	0.844021	+	0.536310i	$0.180182\pi$
$968$	5.00000	0.160706
$969$	2.43845	0.0783342
$970$	10.6847	0.343064
$971$	3.31534	0.106394	0.0531972	−	0.998584i	$-0.483059\pi$
$971$	3.31534	0.106394	0.0531972	+	0.998584i	$0.483059\pi$
$972$	5.75379	0.184553
$973$	51.5076	1.65126
$974$	−15.6155	−0.500354
$975$	0.684658	0.0219266
$976$	11.5616	0.370076
$977$	−18.8769	−0.603925	−0.301963	−	0.953320i	$-0.597642\pi$
$977$	−18.8769	−0.603925	−0.301963	+	0.953320i	$0.597642\pi$
$978$	−3.50758	−0.112160
$979$	−10.7386	−0.343208
$980$	2.75379	0.0879666
$981$	2.49242	0.0795769
$982$	−1.56155	−0.0498312
$983$	−28.8769	−0.921030	−0.460515	−	0.887652i	$-0.652335\pi$
$983$	−28.8769	−0.921030	−0.460515	+	0.887652i	$0.652335\pi$
$984$	−19.1231	−0.609622
$985$	−11.3693	−0.362257
$986$	6.68466	0.212883
$987$	−11.8920	−0.378528
$988$	0.684658	0.0217819
$989$	22.2462	0.707388
$990$	2.24621	0.0713893
$991$	−27.4233	−0.871130	−0.435565	−	0.900157i	$-0.643451\pi$
$991$	−27.4233	−0.871130	−0.435565	+	0.900157i	$0.643451\pi$
$992$	−2.43845	−0.0774208
$993$	−21.9460	−0.696436
$994$	52.1080	1.65276
$995$	10.4384	0.330921
$996$	16.0000	0.506979
$997$	28.2462	0.894566	0.447283	−	0.894392i	$-0.352392\pi$
$997$	28.2462	0.894566	0.447283	+	0.894392i	$0.352392\pi$
$998$	−0.876894	−0.0277576
$999$	−6.24621	−0.197621

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.2.a.f.1.2	✓	2
3.2	odd	2		1530.2.a.r.1.1		2
4.3	odd	2		1360.2.a.m.1.1		2
5.2	odd	4		850.2.c.i.749.3		4
5.3	odd	4		850.2.c.i.749.2		4
5.4	even	2		850.2.a.n.1.1		2
7.6	odd	2		8330.2.a.bq.1.1		2
8.3	odd	2		5440.2.a.bd.1.2		2
8.5	even	2		5440.2.a.bj.1.1		2
15.14	odd	2		7650.2.a.de.1.2		2
17.4	even	4		2890.2.b.i.2311.2		4
17.13	even	4		2890.2.b.i.2311.3		4
17.16	even	2		2890.2.a.u.1.1		2
20.19	odd	2		6800.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.f.1.2	✓	2	1.1	even	1	trivial
850.2.a.n.1.1		2	5.4	even	2
850.2.c.i.749.2		4	5.3	odd	4
850.2.c.i.749.3		4	5.2	odd	4
1360.2.a.m.1.1		2	4.3	odd	2
1530.2.a.r.1.1		2	3.2	odd	2
2890.2.a.u.1.1		2	17.16	even	2
2890.2.b.i.2311.2		4	17.4	even	4
2890.2.b.i.2311.3		4	17.13	even	4
5440.2.a.bd.1.2		2	8.3	odd	2
5440.2.a.bj.1.1		2	8.5	even	2
6800.2.a.be.1.2		2	20.19	odd	2
7650.2.a.de.1.2		2	15.14	odd	2
8330.2.a.bq.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	170.2.a.f.1.2

Level	$170$
Weight	$2$
Character	170.1
Self dual	yes
Analytic conductor	$1.357$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$