LMFDB - Embedded newform 855.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.53919 q^{8} +O(q^{10})$	\(q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.53919 q^{8} +2.17009 q^{10} +6.34017 q^{11} +1.36910 q^{13} -2.34017 q^{14} -2.07838 q^{16} -3.26180 q^{17} -1.00000 q^{19} -2.70928 q^{20} -13.7587 q^{22} -2.34017 q^{23} +1.00000 q^{25} -2.97107 q^{26} +2.92162 q^{28} -1.41855 q^{29} +8.68035 q^{31} +7.58864 q^{32} +7.07838 q^{34} -1.07838 q^{35} +5.36910 q^{37} +2.17009 q^{38} +1.53919 q^{40} +3.26180 q^{41} -11.9155 q^{43} +17.1773 q^{44} +5.07838 q^{46} -1.07838 q^{47} -5.83710 q^{49} -2.17009 q^{50} +3.70928 q^{52} -6.63090 q^{53} -6.34017 q^{55} -1.65983 q^{56} +3.07838 q^{58} +11.4186 q^{59} +5.60197 q^{61} -18.8371 q^{62} -12.3112 q^{64} -1.36910 q^{65} +10.3896 q^{67} -8.83710 q^{68} +2.34017 q^{70} +10.8371 q^{71} +5.41855 q^{73} -11.6514 q^{74} -2.70928 q^{76} +6.83710 q^{77} +14.2557 q^{79} +2.07838 q^{80} -7.07838 q^{82} +14.3402 q^{83} +3.26180 q^{85} +25.8576 q^{86} -9.75872 q^{88} -7.57531 q^{89} +1.47641 q^{91} -6.34017 q^{92} +2.34017 q^{94} +1.00000 q^{95} -8.88655 q^{97} +12.6670 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8	$ 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} + 8 q^{11} + 8 q^{13} + 4 q^{14} - 3 q^{16} - 2 q^{17} - 3 q^{19} - q^{20} - 16 q^{22} + 4 q^{23} + 3 q^{25} + 6 q^{26} + 12 q^{28} + 10 q^{29} + 4 q^{31} + 3 q^{32} + 18 q^{34} + 20 q^{37} + q^{38} + 3 q^{40} + 2 q^{41} - 4 q^{43} + 12 q^{44} + 12 q^{46} + 11 q^{49} - q^{50} + 4 q^{52} - 16 q^{53} - 8 q^{55} - 16 q^{56} + 6 q^{58} + 20 q^{59} - 2 q^{61} - 28 q^{62} - 11 q^{64} - 8 q^{65} + 2 q^{67} + 2 q^{68} - 4 q^{70} + 4 q^{71} + 2 q^{73} + 2 q^{74} - q^{76} - 8 q^{77} + 3 q^{80} - 18 q^{82} + 32 q^{83} + 2 q^{85} + 16 q^{86} - 4 q^{88} - 2 q^{89} + 20 q^{91} - 8 q^{92} - 4 q^{94} + 3 q^{95} + 20 q^{97} + 15 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 + 8 * q^11 + 8 * q^13 + 4 * q^14 - 3 * q^16 - 2 * q^17 - 3 * q^19 - q^20 - 16 * q^22 + 4 * q^23 + 3 * q^25 + 6 * q^26 + 12 * q^28 + 10 * q^29 + 4 * q^31 + 3 * q^32 + 18 * q^34 + 20 * q^37 + q^38 + 3 * q^40 + 2 * q^41 - 4 * q^43 + 12 * q^44 + 12 * q^46 + 11 * q^49 - q^50 + 4 * q^52 - 16 * q^53 - 8 * q^55 - 16 * q^56 + 6 * q^58 + 20 * q^59 - 2 * q^61 - 28 * q^62 - 11 * q^64 - 8 * q^65 + 2 * q^67 + 2 * q^68 - 4 * q^70 + 4 * q^71 + 2 * q^73 + 2 * q^74 - q^76 - 8 * q^77 + 3 * q^80 - 18 * q^82 + 32 * q^83 + 2 * q^85 + 16 * q^86 - 4 * q^88 - 2 * q^89 + 20 * q^91 - 8 * q^92 - 4 * q^94 + 3 * q^95 + 20 * q^97 + 15 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.17009	−1.53448	−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009	−1.53448	−0.767241	+	0.641358i	$0.778371\pi$
$3$	0	0
$3$	0	0
$4$	2.70928	1.35464
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.07838	0.407588	0.203794	−	0.979014i	$-0.434673\pi$
$7$	1.07838	0.407588	0.203794	+	0.979014i	$0.434673\pi$
$8$	−1.53919	−0.544185
$9$	0	0
$10$	2.17009	0.686242
$11$	6.34017	1.91163	0.955817	−	0.293962i	$-0.0949740\pi$
$11$	6.34017	1.91163	0.955817	+	0.293962i	$0.0949740\pi$
$12$	0	0
$13$	1.36910	0.379721	0.189860	−	0.981811i	$-0.439196\pi$
$13$	1.36910	0.379721	0.189860	+	0.981811i	$0.439196\pi$
$14$	−2.34017	−0.625438
$15$	0	0
$16$	−2.07838	−0.519594
$17$	−3.26180	−0.791102	−0.395551	−	0.918444i	$-0.629446\pi$
$17$	−3.26180	−0.791102	−0.395551	+	0.918444i	$0.629446\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.70928	−0.605812
$21$	0	0
$22$	−13.7587	−2.93337
$23$	−2.34017	−0.487960	−0.243980	−	0.969780i	$-0.578453\pi$
$23$	−2.34017	−0.487960	−0.243980	+	0.969780i	$0.578453\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.97107	−0.582675
$27$	0	0
$28$	2.92162	0.552135
$29$	−1.41855	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$29$	−1.41855	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$30$	0	0
$31$	8.68035	1.55904	0.779518	−	0.626380i	$-0.215464\pi$
$31$	8.68035	1.55904	0.779518	+	0.626380i	$0.215464\pi$
$32$	7.58864	1.34149
$33$	0	0
$34$	7.07838	1.21393
$35$	−1.07838	−0.182279
$36$	0	0
$37$	5.36910	0.882675	0.441337	−	0.897341i	$-0.354504\pi$
$37$	5.36910	0.882675	0.441337	+	0.897341i	$0.354504\pi$
$38$	2.17009	0.352035
$39$	0	0
$40$	1.53919	0.243367
$41$	3.26180	0.509407	0.254703	−	0.967019i	$-0.418022\pi$
$41$	3.26180	0.509407	0.254703	+	0.967019i	$0.418022\pi$
$42$	0	0
$43$	−11.9155	−1.81709	−0.908547	−	0.417783i	$-0.862807\pi$
$43$	−11.9155	−1.81709	−0.908547	+	0.417783i	$0.862807\pi$
$44$	17.1773	2.58957
$45$	0	0
$46$	5.07838	0.748766
$47$	−1.07838	−0.157298	−0.0786488	−	0.996902i	$-0.525061\pi$
$47$	−1.07838	−0.157298	−0.0786488	+	0.996902i	$0.525061\pi$
$48$	0	0
$49$	−5.83710	−0.833872
$50$	−2.17009	−0.306897
$51$	0	0
$52$	3.70928	0.514384
$53$	−6.63090	−0.910824	−0.455412	−	0.890281i	$-0.650508\pi$
$53$	−6.63090	−0.910824	−0.455412	+	0.890281i	$0.650508\pi$
$54$	0	0
$55$	−6.34017	−0.854909
$56$	−1.65983	−0.221804
$57$	0	0
$58$	3.07838	0.404211
$59$	11.4186	1.48657	0.743284	−	0.668976i	$-0.233267\pi$
$59$	11.4186	1.48657	0.743284	+	0.668976i	$0.233267\pi$
$60$	0	0
$61$	5.60197	0.717259	0.358629	−	0.933480i	$-0.383244\pi$
$61$	5.60197	0.717259	0.358629	+	0.933480i	$0.383244\pi$
$62$	−18.8371	−2.39231
$63$	0	0
$64$	−12.3112	−1.53891
$65$	−1.36910	−0.169816
$66$	0	0
$67$	10.3896	1.26929	0.634647	−	0.772802i	$-0.281145\pi$
$67$	10.3896	1.26929	0.634647	+	0.772802i	$0.281145\pi$
$68$	−8.83710	−1.07166
$69$	0	0
$70$	2.34017	0.279704
$71$	10.8371	1.28613	0.643064	−	0.765813i	$-0.277663\pi$
$71$	10.8371	1.28613	0.643064	+	0.765813i	$0.277663\pi$
$72$	0	0
$73$	5.41855	0.634193	0.317097	−	0.948393i	$-0.397292\pi$
$73$	5.41855	0.634193	0.317097	+	0.948393i	$0.397292\pi$
$74$	−11.6514	−1.35445
$75$	0	0
$76$	−2.70928	−0.310775
$77$	6.83710	0.779160
$78$	0	0
$79$	14.2557	1.60389	0.801943	−	0.597400i	$-0.203800\pi$
$79$	14.2557	1.60389	0.801943	+	0.597400i	$0.203800\pi$
$80$	2.07838	0.232370
$81$	0	0
$82$	−7.07838	−0.781676
$83$	14.3402	1.57404	0.787019	−	0.616928i	$-0.211623\pi$
$83$	14.3402	1.57404	0.787019	+	0.616928i	$0.211623\pi$
$84$	0	0
$85$	3.26180	0.353791
$86$	25.8576	2.78830
$87$	0	0
$88$	−9.75872	−1.04028
$89$	−7.57531	−0.802981	−0.401490	−	0.915863i	$-0.631508\pi$
$89$	−7.57531	−0.802981	−0.401490	+	0.915863i	$0.631508\pi$
$90$	0	0
$91$	1.47641	0.154770
$92$	−6.34017	−0.661009
$93$	0	0
$94$	2.34017	0.241370
$95$	1.00000	0.102598
$96$	0	0
$97$	−8.88655	−0.902292	−0.451146	−	0.892450i	$-0.648985\pi$
$97$	−8.88655	−0.902292	−0.451146	+	0.892450i	$0.648985\pi$
$98$	12.6670	1.27956
$99$	0	0
$100$	2.70928	0.270928
$101$	4.92162	0.489720	0.244860	−	0.969558i	$-0.421258\pi$
$101$	4.92162	0.489720	0.244860	+	0.969558i	$0.421258\pi$
$102$	0	0
$103$	6.38962	0.629588	0.314794	−	0.949160i	$-0.398065\pi$
$103$	6.38962	0.629588	0.314794	+	0.949160i	$0.398065\pi$
$104$	−2.10731	−0.206638
$105$	0	0
$106$	14.3896	1.39764
$107$	−2.29072	−0.221453	−0.110726	−	0.993851i	$-0.535318\pi$
$107$	−2.29072	−0.221453	−0.110726	+	0.993851i	$0.535318\pi$
$108$	0	0
$109$	−12.8371	−1.22957	−0.614786	−	0.788694i	$-0.710757\pi$
$109$	−12.8371	−1.22957	−0.614786	+	0.788694i	$0.710757\pi$
$110$	13.7587	1.31184
$111$	0	0
$112$	−2.24128	−0.211781
$113$	12.8865	1.21226	0.606132	−	0.795364i	$-0.292720\pi$
$113$	12.8865	1.21226	0.606132	+	0.795364i	$0.292720\pi$
$114$	0	0
$115$	2.34017	0.218222
$116$	−3.84324	−0.356836
$117$	0	0
$118$	−24.7792	−2.28111
$119$	−3.51745	−0.322444
$120$	0	0
$121$	29.1978	2.65434
$122$	−12.1568	−1.10062
$123$	0	0
$124$	23.5174	2.11193
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.23287	−0.730549	−0.365274	−	0.930900i	$-0.619025\pi$
$127$	−8.23287	−0.730549	−0.365274	+	0.930900i	$0.619025\pi$
$128$	11.5392	1.01993
$129$	0	0
$130$	2.97107	0.260580
$131$	−1.47641	−0.128995	−0.0644973	−	0.997918i	$-0.520544\pi$
$131$	−1.47641	−0.128995	−0.0644973	+	0.997918i	$0.520544\pi$
$132$	0	0
$133$	−1.07838	−0.0935072
$134$	−22.5464	−1.94771
$135$	0	0
$136$	5.02052	0.430506
$137$	3.94214	0.336800	0.168400	−	0.985719i	$-0.446140\pi$
$137$	3.94214	0.336800	0.168400	+	0.985719i	$0.446140\pi$
$138$	0	0
$139$	8.86376	0.751815	0.375907	−	0.926657i	$-0.377331\pi$
$139$	8.86376	0.751815	0.375907	+	0.926657i	$0.377331\pi$
$140$	−2.92162	−0.246922
$141$	0	0
$142$	−23.5174	−1.97354
$143$	8.68035	0.725887
$144$	0	0
$145$	1.41855	0.117804
$146$	−11.7587	−0.973159
$147$	0	0
$148$	14.5464	1.19570
$149$	19.7587	1.61870	0.809349	−	0.587328i	$-0.199820\pi$
$149$	19.7587	1.61870	0.809349	+	0.587328i	$0.199820\pi$
$150$	0	0
$151$	3.41855	0.278198	0.139099	−	0.990279i	$-0.455579\pi$
$151$	3.41855	0.278198	0.139099	+	0.990279i	$0.455579\pi$
$152$	1.53919	0.124845
$153$	0	0
$154$	−14.8371	−1.19561
$155$	−8.68035	−0.697222
$156$	0	0
$157$	9.41855	0.751682	0.375841	−	0.926684i	$-0.377354\pi$
$157$	9.41855	0.751682	0.375841	+	0.926684i	$0.377354\pi$
$158$	−30.9360	−2.46114
$159$	0	0
$160$	−7.58864	−0.599934
$161$	−2.52359	−0.198887
$162$	0	0
$163$	−2.92162	−0.228839	−0.114420	−	0.993433i	$-0.536501\pi$
$163$	−2.92162	−0.228839	−0.114420	+	0.993433i	$0.536501\pi$
$164$	8.83710	0.690062
$165$	0	0
$166$	−31.1194	−2.41534
$167$	−20.9132	−1.61831	−0.809156	−	0.587593i	$-0.800076\pi$
$167$	−20.9132	−1.61831	−0.809156	+	0.587593i	$0.800076\pi$
$168$	0	0
$169$	−11.1256	−0.855812
$170$	−7.07838	−0.542887
$171$	0	0
$172$	−32.2823	−2.46150
$173$	1.05559	0.0802551	0.0401276	−	0.999195i	$-0.487224\pi$
$173$	1.05559	0.0802551	0.0401276	+	0.999195i	$0.487224\pi$
$174$	0	0
$175$	1.07838	0.0815177
$176$	−13.1773	−0.993274
$177$	0	0
$178$	16.4391	1.23216
$179$	−0.894960	−0.0668925	−0.0334462	−	0.999441i	$-0.510648\pi$
$179$	−0.894960	−0.0668925	−0.0334462	+	0.999441i	$0.510648\pi$
$180$	0	0
$181$	−0.837101	−0.0622213	−0.0311106	−	0.999516i	$-0.509904\pi$
$181$	−0.837101	−0.0622213	−0.0311106	+	0.999516i	$0.509904\pi$
$182$	−3.20394	−0.237492
$183$	0	0
$184$	3.60197	0.265541
$185$	−5.36910	−0.394744
$186$	0	0
$187$	−20.6803	−1.51230
$188$	−2.92162	−0.213081
$189$	0	0
$190$	−2.17009	−0.157435
$191$	−22.0410	−1.59483	−0.797417	−	0.603429i	$-0.793801\pi$
$191$	−22.0410	−1.59483	−0.797417	+	0.603429i	$0.793801\pi$
$192$	0	0
$193$	12.7877	0.920475	0.460238	−	0.887796i	$-0.347764\pi$
$193$	12.7877	0.920475	0.460238	+	0.887796i	$0.347764\pi$
$194$	19.2846	1.38455
$195$	0	0
$196$	−15.8143	−1.12959
$197$	9.20394	0.655753	0.327877	−	0.944721i	$-0.393667\pi$
$197$	9.20394	0.655753	0.327877	+	0.944721i	$0.393667\pi$
$198$	0	0
$199$	−16.1978	−1.14823	−0.574116	−	0.818774i	$-0.694654\pi$
$199$	−16.1978	−1.14823	−0.574116	+	0.818774i	$0.694654\pi$
$200$	−1.53919	−0.108837
$201$	0	0
$202$	−10.6803	−0.751467
$203$	−1.52973	−0.107366
$204$	0	0
$205$	−3.26180	−0.227814
$206$	−13.8660	−0.966092
$207$	0	0
$208$	−2.84551	−0.197301
$209$	−6.34017	−0.438559
$210$	0	0
$211$	7.78539	0.535968	0.267984	−	0.963423i	$-0.413643\pi$
$211$	7.78539	0.535968	0.267984	+	0.963423i	$0.413643\pi$
$212$	−17.9649	−1.23384
$213$	0	0
$214$	4.97107	0.339815
$215$	11.9155	0.812629
$216$	0	0
$217$	9.36069	0.635445
$218$	27.8576	1.88676
$219$	0	0
$220$	−17.1773	−1.15809
$221$	−4.46573	−0.300398
$222$	0	0
$223$	12.5464	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$223$	12.5464	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$224$	8.18342	0.546778
$225$	0	0
$226$	−27.9649	−1.86020
$227$	−2.29072	−0.152041	−0.0760204	−	0.997106i	$-0.524221\pi$
$227$	−2.29072	−0.152041	−0.0760204	+	0.997106i	$0.524221\pi$
$228$	0	0
$229$	−5.91548	−0.390906	−0.195453	−	0.980713i	$-0.562618\pi$
$229$	−5.91548	−0.390906	−0.195453	+	0.980713i	$0.562618\pi$
$230$	−5.07838	−0.334858
$231$	0	0
$232$	2.18342	0.143348
$233$	−13.5174	−0.885557	−0.442779	−	0.896631i	$-0.646007\pi$
$233$	−13.5174	−0.885557	−0.442779	+	0.896631i	$0.646007\pi$
$234$	0	0
$235$	1.07838	0.0703456
$236$	30.9360	2.01376
$237$	0	0
$238$	7.63317	0.494785
$239$	−13.8432	−0.895445	−0.447723	−	0.894173i	$-0.647765\pi$
$239$	−13.8432	−0.895445	−0.447723	+	0.894173i	$0.647765\pi$
$240$	0	0
$241$	7.26180	0.467773	0.233887	−	0.972264i	$-0.424856\pi$
$241$	7.26180	0.467773	0.233887	+	0.972264i	$0.424856\pi$
$242$	−63.3617	−4.07305
$243$	0	0
$244$	15.1773	0.971625
$245$	5.83710	0.372919
$246$	0	0
$247$	−1.36910	−0.0871139
$248$	−13.3607	−0.848405
$249$	0	0
$250$	2.17009	0.137248
$251$	−10.4703	−0.660877	−0.330439	−	0.943827i	$-0.607197\pi$
$251$	−10.4703	−0.660877	−0.330439	+	0.943827i	$0.607197\pi$
$252$	0	0
$253$	−14.8371	−0.932801
$254$	17.8660	1.12101
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	23.6248	1.47367	0.736836	−	0.676072i	$-0.236319\pi$
$257$	23.6248	1.47367	0.736836	+	0.676072i	$0.236319\pi$
$258$	0	0
$259$	5.78992	0.359768
$260$	−3.70928	−0.230039
$261$	0	0
$262$	3.20394	0.197940
$263$	−5.65983	−0.349000	−0.174500	−	0.984657i	$-0.555831\pi$
$263$	−5.65983	−0.349000	−0.174500	+	0.984657i	$0.555831\pi$
$264$	0	0
$265$	6.63090	0.407333
$266$	2.34017	0.143485
$267$	0	0
$268$	28.1483	1.71943
$269$	2.31351	0.141057	0.0705286	−	0.997510i	$-0.477531\pi$
$269$	2.31351	0.141057	0.0705286	+	0.997510i	$0.477531\pi$
$270$	0	0
$271$	−19.7009	−1.19674	−0.598371	−	0.801219i	$-0.704185\pi$
$271$	−19.7009	−1.19674	−0.598371	+	0.801219i	$0.704185\pi$
$272$	6.77924	0.411052
$273$	0	0
$274$	−8.55479	−0.516814
$275$	6.34017	0.382327
$276$	0	0
$277$	−25.7321	−1.54609	−0.773045	−	0.634351i	$-0.781267\pi$
$277$	−25.7321	−1.54609	−0.773045	+	0.634351i	$0.781267\pi$
$278$	−19.2351	−1.15365
$279$	0	0
$280$	1.65983	0.0991936
$281$	6.58145	0.392616	0.196308	−	0.980542i	$-0.437105\pi$
$281$	6.58145	0.392616	0.196308	+	0.980542i	$0.437105\pi$
$282$	0	0
$283$	−0.496928	−0.0295393	−0.0147697	−	0.999891i	$-0.504702\pi$
$283$	−0.496928	−0.0295393	−0.0147697	+	0.999891i	$0.504702\pi$
$284$	29.3607	1.74224
$285$	0	0
$286$	−18.8371	−1.11386
$287$	3.51745	0.207628
$288$	0	0
$289$	−6.36069	−0.374158
$290$	−3.07838	−0.180769
$291$	0	0
$292$	14.6803	0.859102
$293$	−6.63090	−0.387381	−0.193691	−	0.981063i	$-0.562046\pi$
$293$	−6.63090	−0.387381	−0.193691	+	0.981063i	$0.562046\pi$
$294$	0	0
$295$	−11.4186	−0.664814
$296$	−8.26406	−0.480339
$297$	0	0
$298$	−42.8781	−2.48386
$299$	−3.20394	−0.185288
$300$	0	0
$301$	−12.8494	−0.740626
$302$	−7.41855	−0.426890
$303$	0	0
$304$	2.07838	0.119203
$305$	−5.60197	−0.320768
$306$	0	0
$307$	14.6042	0.833508	0.416754	−	0.909019i	$-0.363168\pi$
$307$	14.6042	0.833508	0.416754	+	0.909019i	$0.363168\pi$
$308$	18.5236	1.05548
$309$	0	0
$310$	18.8371	1.06988
$311$	−19.3340	−1.09633	−0.548166	−	0.836369i	$-0.684674\pi$
$311$	−19.3340	−1.09633	−0.548166	+	0.836369i	$0.684674\pi$
$312$	0	0
$313$	30.6803	1.73416	0.867078	−	0.498173i	$-0.165995\pi$
$313$	30.6803	1.73416	0.867078	+	0.498173i	$0.165995\pi$
$314$	−20.4391	−1.15344
$315$	0	0
$316$	38.6225	2.17268
$317$	−18.7298	−1.05197	−0.525985	−	0.850494i	$-0.676303\pi$
$317$	−18.7298	−1.05197	−0.525985	+	0.850494i	$0.676303\pi$
$318$	0	0
$319$	−8.99386	−0.503559
$320$	12.3112	0.688219
$321$	0	0
$322$	5.47641	0.305188
$323$	3.26180	0.181491
$324$	0	0
$325$	1.36910	0.0759441
$326$	6.34017	0.351150
$327$	0	0
$328$	−5.02052	−0.277212
$329$	−1.16290	−0.0641127
$330$	0	0
$331$	−2.73820	−0.150505	−0.0752527	−	0.997164i	$-0.523976\pi$
$331$	−2.73820	−0.150505	−0.0752527	+	0.997164i	$0.523976\pi$
$332$	38.8515	2.13225
$333$	0	0
$334$	45.3835	2.48327
$335$	−10.3896	−0.567646
$336$	0	0
$337$	−6.04945	−0.329534	−0.164767	−	0.986332i	$-0.552687\pi$
$337$	−6.04945	−0.329534	−0.164767	+	0.986332i	$0.552687\pi$
$338$	24.1434	1.31323
$339$	0	0
$340$	8.83710	0.479259
$341$	55.0349	2.98031
$342$	0	0
$343$	−13.8432	−0.747465
$344$	18.3402	0.988836
$345$	0	0
$346$	−2.29072	−0.123150
$347$	5.97334	0.320666	0.160333	−	0.987063i	$-0.448743\pi$
$347$	5.97334	0.320666	0.160333	+	0.987063i	$0.448743\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−2.34017	−0.125088
$351$	0	0
$352$	48.1133	2.56445
$353$	14.0989	0.750409	0.375204	−	0.926942i	$-0.377573\pi$
$353$	14.0989	0.750409	0.375204	+	0.926942i	$0.377573\pi$
$354$	0	0
$355$	−10.8371	−0.575174
$356$	−20.5236	−1.08775
$357$	0	0
$358$	1.94214	0.102645
$359$	−6.02666	−0.318075	−0.159038	−	0.987273i	$-0.550839\pi$
$359$	−6.02666	−0.318075	−0.159038	+	0.987273i	$0.550839\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.81658	0.0954775
$363$	0	0
$364$	4.00000	0.209657
$365$	−5.41855	−0.283620
$366$	0	0
$367$	1.07838	0.0562909	0.0281454	−	0.999604i	$-0.491040\pi$
$367$	1.07838	0.0562909	0.0281454	+	0.999604i	$0.491040\pi$
$368$	4.86376	0.253541
$369$	0	0
$370$	11.6514	0.605728
$371$	−7.15061	−0.371241
$372$	0	0
$373$	12.3051	0.637134	0.318567	−	0.947900i	$-0.396798\pi$
$373$	12.3051	0.637134	0.318567	+	0.947900i	$0.396798\pi$
$374$	44.8781	2.32059
$375$	0	0
$376$	1.65983	0.0855990
$377$	−1.94214	−0.100025
$378$	0	0
$379$	1.04718	0.0537901	0.0268950	−	0.999638i	$-0.491438\pi$
$379$	1.04718	0.0537901	0.0268950	+	0.999638i	$0.491438\pi$
$380$	2.70928	0.138983
$381$	0	0
$382$	47.8310	2.44724
$383$	−0.0806452	−0.00412078	−0.00206039	−	0.999998i	$-0.500656\pi$
$383$	−0.0806452	−0.00412078	−0.00206039	+	0.999998i	$0.500656\pi$
$384$	0	0
$385$	−6.83710	−0.348451
$386$	−27.7503	−1.41245
$387$	0	0
$388$	−24.0761	−1.22228
$389$	20.5236	1.04059	0.520294	−	0.853987i	$-0.325822\pi$
$389$	20.5236	1.04059	0.520294	+	0.853987i	$0.325822\pi$
$390$	0	0
$391$	7.63317	0.386026
$392$	8.98440	0.453781
$393$	0	0
$394$	−19.9733	−1.00624
$395$	−14.2557	−0.717280
$396$	0	0
$397$	39.4596	1.98042	0.990210	−	0.139586i	$-0.0445771\pi$
$397$	39.4596	1.98042	0.990210	+	0.139586i	$0.0445771\pi$
$398$	35.1506	1.76194
$399$	0	0
$400$	−2.07838	−0.103919
$401$	−0.470266	−0.0234840	−0.0117420	−	0.999931i	$-0.503738\pi$
$401$	−0.470266	−0.0234840	−0.0117420	+	0.999931i	$0.503738\pi$
$402$	0	0
$403$	11.8843	0.591998
$404$	13.3340	0.663393
$405$	0	0
$406$	3.31965	0.164752
$407$	34.0410	1.68735
$408$	0	0
$409$	10.4826	0.518329	0.259164	−	0.965833i	$-0.416553\pi$
$409$	10.4826	0.518329	0.259164	+	0.965833i	$0.416553\pi$
$410$	7.07838	0.349576
$411$	0	0
$412$	17.3112	0.852864
$413$	12.3135	0.605908
$414$	0	0
$415$	−14.3402	−0.703931
$416$	10.3896	0.509393
$417$	0	0
$418$	13.7587	0.672961
$419$	−34.6681	−1.69365	−0.846823	−	0.531875i	$-0.821488\pi$
$419$	−34.6681	−1.69365	−0.846823	+	0.531875i	$0.821488\pi$
$420$	0	0
$421$	−34.6102	−1.68680	−0.843399	−	0.537288i	$-0.819449\pi$
$421$	−34.6102	−1.68680	−0.843399	+	0.537288i	$0.819449\pi$
$422$	−16.8950	−0.822434
$423$	0	0
$424$	10.2062	0.495657
$425$	−3.26180	−0.158220
$426$	0	0
$427$	6.04104	0.292346
$428$	−6.20620	−0.299988
$429$	0	0
$430$	−25.8576	−1.24697
$431$	−6.73820	−0.324568	−0.162284	−	0.986744i	$-0.551886\pi$
$431$	−6.73820	−0.324568	−0.162284	+	0.986744i	$0.551886\pi$
$432$	0	0
$433$	20.4741	0.983924	0.491962	−	0.870617i	$-0.336280\pi$
$433$	20.4741	0.983924	0.491962	+	0.870617i	$0.336280\pi$
$434$	−20.3135	−0.975080
$435$	0	0
$436$	−34.7792	−1.66562
$437$	2.34017	0.111946
$438$	0	0
$439$	−21.4596	−1.02421	−0.512105	−	0.858923i	$-0.671134\pi$
$439$	−21.4596	−1.02421	−0.512105	+	0.858923i	$0.671134\pi$
$440$	9.75872	0.465229
$441$	0	0
$442$	9.69102	0.460955
$443$	21.5441	1.02359	0.511796	−	0.859107i	$-0.328980\pi$
$443$	21.5441	1.02359	0.511796	+	0.859107i	$0.328980\pi$
$444$	0	0
$445$	7.57531	0.359104
$446$	−27.2267	−1.28922
$447$	0	0
$448$	−13.2762	−0.627240
$449$	8.47027	0.399737	0.199868	−	0.979823i	$-0.435949\pi$
$449$	8.47027	0.399737	0.199868	+	0.979823i	$0.435949\pi$
$450$	0	0
$451$	20.6803	0.973799
$452$	34.9132	1.64218
$453$	0	0
$454$	4.97107	0.233304
$455$	−1.47641	−0.0692151
$456$	0	0
$457$	11.3607	0.531431	0.265715	−	0.964052i	$-0.414392\pi$
$457$	11.3607	0.531431	0.265715	+	0.964052i	$0.414392\pi$
$458$	12.8371	0.599838
$459$	0	0
$460$	6.34017	0.295612
$461$	−3.04718	−0.141921	−0.0709607	−	0.997479i	$-0.522606\pi$
$461$	−3.04718	−0.141921	−0.0709607	+	0.997479i	$0.522606\pi$
$462$	0	0
$463$	−9.97334	−0.463500	−0.231750	−	0.972775i	$-0.574445\pi$
$463$	−9.97334	−0.463500	−0.231750	+	0.972775i	$0.574445\pi$
$464$	2.94828	0.136871
$465$	0	0
$466$	29.3340	1.35887
$467$	−1.49079	−0.0689853	−0.0344927	−	0.999405i	$-0.510982\pi$
$467$	−1.49079	−0.0689853	−0.0344927	+	0.999405i	$0.510982\pi$
$468$	0	0
$469$	11.2039	0.517350
$470$	−2.34017	−0.107944
$471$	0	0
$472$	−17.5753	−0.808969
$473$	−75.5462	−3.47362
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−9.52973	−0.436795
$477$	0	0
$478$	30.0410	1.37405
$479$	10.1711	0.464731	0.232365	−	0.972629i	$-0.425353\pi$
$479$	10.1711	0.464731	0.232365	+	0.972629i	$0.425353\pi$
$480$	0	0
$481$	7.35085	0.335170
$482$	−15.7587	−0.717790
$483$	0	0
$484$	79.1049	3.59568
$485$	8.88655	0.403517
$486$	0	0
$487$	−24.2784	−1.10016	−0.550081	−	0.835112i	$-0.685403\pi$
$487$	−24.2784	−1.10016	−0.550081	+	0.835112i	$0.685403\pi$
$488$	−8.62249	−0.390322
$489$	0	0
$490$	−12.6670	−0.572237
$491$	−19.2039	−0.866662	−0.433331	−	0.901235i	$-0.642662\pi$
$491$	−19.2039	−0.866662	−0.433331	+	0.901235i	$0.642662\pi$
$492$	0	0
$493$	4.62702	0.208391
$494$	2.97107	0.133675
$495$	0	0
$496$	−18.0410	−0.810067
$497$	11.6865	0.524211
$498$	0	0
$499$	7.33403	0.328316	0.164158	−	0.986434i	$-0.447509\pi$
$499$	7.33403	0.328316	0.164158	+	0.986434i	$0.447509\pi$
$500$	−2.70928	−0.121162
$501$	0	0
$502$	22.7214	1.01410
$503$	−29.0616	−1.29579	−0.647895	−	0.761729i	$-0.724351\pi$
$503$	−29.0616	−1.29579	−0.647895	+	0.761729i	$0.724351\pi$
$504$	0	0
$505$	−4.92162	−0.219009
$506$	32.1978	1.43137
$507$	0	0
$508$	−22.3051	−0.989629
$509$	−26.0456	−1.15445	−0.577225	−	0.816585i	$-0.695864\pi$
$509$	−26.0456	−1.15445	−0.577225	+	0.816585i	$0.695864\pi$
$510$	0	0
$511$	5.84324	0.258490
$512$	−22.1701	−0.979789
$513$	0	0
$514$	−51.2678	−2.26132
$515$	−6.38962	−0.281560
$516$	0	0
$517$	−6.83710	−0.300695
$518$	−12.5646	−0.552058
$519$	0	0
$520$	2.10731	0.0924115
$521$	38.8248	1.70095	0.850473	−	0.526019i	$-0.176316\pi$
$521$	38.8248	1.70095	0.850473	+	0.526019i	$0.176316\pi$
$522$	0	0
$523$	4.59970	0.201131	0.100565	−	0.994930i	$-0.467935\pi$
$523$	4.59970	0.201131	0.100565	+	0.994930i	$0.467935\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	12.2823	0.535534
$527$	−28.3135	−1.23336
$528$	0	0
$529$	−17.5236	−0.761895
$530$	−14.3896	−0.625045
$531$	0	0
$532$	−2.92162	−0.126668
$533$	4.46573	0.193432
$534$	0	0
$535$	2.29072	0.0990367
$536$	−15.9916	−0.690731
$537$	0	0
$538$	−5.02052	−0.216450
$539$	−37.0082	−1.59406
$540$	0	0
$541$	−12.1256	−0.521318	−0.260659	−	0.965431i	$-0.583940\pi$
$541$	−12.1256	−0.521318	−0.260659	+	0.965431i	$0.583940\pi$
$542$	42.7526	1.83638
$543$	0	0
$544$	−24.7526	−1.06126
$545$	12.8371	0.549881
$546$	0	0
$547$	−9.54023	−0.407911	−0.203955	−	0.978980i	$-0.565380\pi$
$547$	−9.54023	−0.407911	−0.203955	+	0.978980i	$0.565380\pi$
$548$	10.6803	0.456242
$549$	0	0
$550$	−13.7587	−0.586674
$551$	1.41855	0.0604323
$552$	0	0
$553$	15.3730	0.653726
$554$	55.8408	2.37245
$555$	0	0
$556$	24.0144	1.01844
$557$	−19.9421	−0.844976	−0.422488	−	0.906369i	$-0.638843\pi$
$557$	−19.9421	−0.844976	−0.422488	+	0.906369i	$0.638843\pi$
$558$	0	0
$559$	−16.3135	−0.689988
$560$	2.24128	0.0947112
$561$	0	0
$562$	−14.2823	−0.602463
$563$	−23.5525	−0.992620	−0.496310	−	0.868145i	$-0.665312\pi$
$563$	−23.5525	−0.992620	−0.496310	+	0.868145i	$0.665312\pi$
$564$	0	0
$565$	−12.8865	−0.542141
$566$	1.07838	0.0453276
$567$	0	0
$568$	−16.6803	−0.699892
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−23.5031	−0.983573	−0.491786	−	0.870716i	$-0.663656\pi$
$571$	−23.5031	−0.983573	−0.491786	+	0.870716i	$0.663656\pi$
$572$	23.5174	0.983314
$573$	0	0
$574$	−7.63317	−0.318602
$575$	−2.34017	−0.0975920
$576$	0	0
$577$	16.4703	0.685666	0.342833	−	0.939396i	$-0.388613\pi$
$577$	16.4703	0.685666	0.342833	+	0.939396i	$0.388613\pi$
$578$	13.8033	0.574140
$579$	0	0
$580$	3.84324	0.159582
$581$	15.4641	0.641560
$582$	0	0
$583$	−42.0410	−1.74116
$584$	−8.34017	−0.345119
$585$	0	0
$586$	14.3896	0.594430
$587$	28.8104	1.18913	0.594567	−	0.804046i	$-0.297323\pi$
$587$	28.8104	1.18913	0.594567	+	0.804046i	$0.297323\pi$
$588$	0	0
$589$	−8.68035	−0.357667
$590$	24.7792	1.02015
$591$	0	0
$592$	−11.1590	−0.458633
$593$	43.2450	1.77586	0.887929	−	0.459980i	$-0.152144\pi$
$593$	43.2450	1.77586	0.887929	+	0.459980i	$0.152144\pi$
$594$	0	0
$595$	3.51745	0.144201
$596$	53.5318	2.19275
$597$	0	0
$598$	6.95282	0.284322
$599$	44.2967	1.80991	0.904957	−	0.425503i	$-0.139903\pi$
$599$	44.2967	1.80991	0.904957	+	0.425503i	$0.139903\pi$
$600$	0	0
$601$	−24.3090	−0.991584	−0.495792	−	0.868441i	$-0.665122\pi$
$601$	−24.3090	−0.991584	−0.495792	+	0.868441i	$0.665122\pi$
$602$	27.8843	1.13648
$603$	0	0
$604$	9.26180	0.376857
$605$	−29.1978	−1.18706
$606$	0	0
$607$	−6.29072	−0.255333	−0.127666	−	0.991817i	$-0.540749\pi$
$607$	−6.29072	−0.255333	−0.127666	+	0.991817i	$0.540749\pi$
$608$	−7.58864	−0.307760
$609$	0	0
$610$	12.1568	0.492213
$611$	−1.47641	−0.0597291
$612$	0	0
$613$	−12.7915	−0.516645	−0.258322	−	0.966059i	$-0.583170\pi$
$613$	−12.7915	−0.516645	−0.258322	+	0.966059i	$0.583170\pi$
$614$	−31.6925	−1.27900
$615$	0	0
$616$	−10.5236	−0.424008
$617$	−17.9299	−0.721829	−0.360914	−	0.932599i	$-0.617535\pi$
$617$	−17.9299	−0.721829	−0.360914	+	0.932599i	$0.617535\pi$
$618$	0	0
$619$	26.8515	1.07925	0.539626	−	0.841905i	$-0.318566\pi$
$619$	26.8515	1.07925	0.539626	+	0.841905i	$0.318566\pi$
$620$	−23.5174	−0.944483
$621$	0	0
$622$	41.9565	1.68230
$623$	−8.16904	−0.327286
$624$	0	0
$625$	1.00000	0.0400000
$626$	−66.5790	−2.66103
$627$	0	0
$628$	25.5174	1.01826
$629$	−17.5129	−0.698286
$630$	0	0
$631$	35.5318	1.41450	0.707250	−	0.706964i	$-0.249936\pi$
$631$	35.5318	1.41450	0.707250	+	0.706964i	$0.249936\pi$
$632$	−21.9421	−0.872812
$633$	0	0
$634$	40.6453	1.61423
$635$	8.23287	0.326711
$636$	0	0
$637$	−7.99159	−0.316638
$638$	19.5174	0.772703
$639$	0	0
$640$	−11.5392	−0.456126
$641$	−12.9360	−0.510941	−0.255471	−	0.966817i	$-0.582230\pi$
$641$	−12.9360	−0.510941	−0.255471	+	0.966817i	$0.582230\pi$
$642$	0	0
$643$	8.49693	0.335086	0.167543	−	0.985865i	$-0.446417\pi$
$643$	8.49693	0.335086	0.167543	+	0.985865i	$0.446417\pi$
$644$	−6.83710	−0.269420
$645$	0	0
$646$	−7.07838	−0.278495
$647$	45.4908	1.78843	0.894214	−	0.447640i	$-0.147736\pi$
$647$	45.4908	1.78843	0.894214	+	0.447640i	$0.147736\pi$
$648$	0	0
$649$	72.3956	2.84178
$650$	−2.97107	−0.116535
$651$	0	0
$652$	−7.91548	−0.309994
$653$	−35.9421	−1.40652	−0.703262	−	0.710930i	$-0.748274\pi$
$653$	−35.9421	−1.40652	−0.703262	+	0.710930i	$0.748274\pi$
$654$	0	0
$655$	1.47641	0.0576881
$656$	−6.77924	−0.264685
$657$	0	0
$658$	2.52359	0.0983798
$659$	−30.9360	−1.20510	−0.602548	−	0.798083i	$-0.705848\pi$
$659$	−30.9360	−1.20510	−0.602548	+	0.798083i	$0.705848\pi$
$660$	0	0
$661$	10.0989	0.392802	0.196401	−	0.980524i	$-0.437075\pi$
$661$	10.0989	0.392802	0.196401	+	0.980524i	$0.437075\pi$
$662$	5.94214	0.230948
$663$	0	0
$664$	−22.0722	−0.856569
$665$	1.07838	0.0418177
$666$	0	0
$667$	3.31965	0.128538
$668$	−56.6596	−2.19223
$669$	0	0
$670$	22.5464	0.871043
$671$	35.5174	1.37114
$672$	0	0
$673$	−8.67194	−0.334279	−0.167139	−	0.985933i	$-0.553453\pi$
$673$	−8.67194	−0.334279	−0.167139	+	0.985933i	$0.553453\pi$
$674$	13.1278	0.505665
$675$	0	0
$676$	−30.1422	−1.15932
$677$	−41.9793	−1.61340	−0.806698	−	0.590964i	$-0.798747\pi$
$677$	−41.9793	−1.61340	−0.806698	+	0.590964i	$0.798747\pi$
$678$	0	0
$679$	−9.58306	−0.367764
$680$	−5.02052	−0.192528
$681$	0	0
$682$	−119.430	−4.57323
$683$	−46.3896	−1.77505	−0.887525	−	0.460760i	$-0.847577\pi$
$683$	−46.3896	−1.77505	−0.887525	+	0.460760i	$0.847577\pi$
$684$	0	0
$685$	−3.94214	−0.150621
$686$	30.0410	1.14697
$687$	0	0
$688$	24.7649	0.944152
$689$	−9.07838	−0.345859
$690$	0	0
$691$	−34.8515	−1.32581	−0.662906	−	0.748702i	$-0.730677\pi$
$691$	−34.8515	−1.32581	−0.662906	+	0.748702i	$0.730677\pi$
$692$	2.85989	0.108717
$693$	0	0
$694$	−12.9627	−0.492056
$695$	−8.86376	−0.336222
$696$	0	0
$697$	−10.6393	−0.402993
$698$	21.7009	0.821390
$699$	0	0
$700$	2.92162	0.110427
$701$	−35.6430	−1.34622	−0.673109	−	0.739543i	$-0.735041\pi$
$701$	−35.6430	−1.34622	−0.673109	+	0.739543i	$0.735041\pi$
$702$	0	0
$703$	−5.36910	−0.202500
$704$	−78.0554	−2.94182
$705$	0	0
$706$	−30.5958	−1.15149
$707$	5.30737	0.199604
$708$	0	0
$709$	16.7214	0.627985	0.313992	−	0.949426i	$-0.398333\pi$
$709$	16.7214	0.627985	0.313992	+	0.949426i	$0.398333\pi$
$710$	23.5174	0.882594
$711$	0	0
$712$	11.6598	0.436970
$713$	−20.3135	−0.760747
$714$	0	0
$715$	−8.68035	−0.324627
$716$	−2.42469	−0.0906151
$717$	0	0
$718$	13.0784	0.488081
$719$	6.85148	0.255517	0.127758	−	0.991805i	$-0.459222\pi$
$719$	6.85148	0.255517	0.127758	+	0.991805i	$0.459222\pi$
$720$	0	0
$721$	6.89043	0.256613
$722$	−2.17009	−0.0807623
$723$	0	0
$724$	−2.26794	−0.0842873
$725$	−1.41855	−0.0526837
$726$	0	0
$727$	34.4391	1.27727	0.638637	−	0.769508i	$-0.279498\pi$
$727$	34.4391	1.27727	0.638637	+	0.769508i	$0.279498\pi$
$728$	−2.27247	−0.0842235
$729$	0	0
$730$	11.7587	0.435210
$731$	38.8659	1.43751
$732$	0	0
$733$	−19.3607	−0.715103	−0.357552	−	0.933893i	$-0.616388\pi$
$733$	−19.3607	−0.715103	−0.357552	+	0.933893i	$0.616388\pi$
$734$	−2.34017	−0.0863774
$735$	0	0
$736$	−17.7587	−0.654595
$737$	65.8720	2.42643
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−14.5464	−0.534735
$741$	0	0
$742$	15.5174	0.569663
$743$	12.7154	0.466483	0.233242	−	0.972419i	$-0.425067\pi$
$743$	12.7154	0.466483	0.233242	+	0.972419i	$0.425067\pi$
$744$	0	0
$745$	−19.7587	−0.723904
$746$	−26.7031	−0.977671
$747$	0	0
$748$	−56.0288	−2.04861
$749$	−2.47027	−0.0902616
$750$	0	0
$751$	3.15836	0.115250	0.0576252	−	0.998338i	$-0.481647\pi$
$751$	3.15836	0.115250	0.0576252	+	0.998338i	$0.481647\pi$
$752$	2.24128	0.0817309
$753$	0	0
$754$	4.21461	0.153487
$755$	−3.41855	−0.124414
$756$	0	0
$757$	28.0410	1.01917	0.509584	−	0.860421i	$-0.329799\pi$
$757$	28.0410	1.01917	0.509584	+	0.860421i	$0.329799\pi$
$758$	−2.27247	−0.0825399
$759$	0	0
$760$	−1.53919	−0.0558322
$761$	16.4924	0.597849	0.298924	−	0.954277i	$-0.403372\pi$
$761$	16.4924	0.597849	0.298924	+	0.954277i	$0.403372\pi$
$762$	0	0
$763$	−13.8432	−0.501159
$764$	−59.7152	−2.16042
$765$	0	0
$766$	0.175007	0.00632326
$767$	15.6332	0.564481
$768$	0	0
$769$	26.9627	0.972298	0.486149	−	0.873876i	$-0.338401\pi$
$769$	26.9627	0.972298	0.486149	+	0.873876i	$0.338401\pi$
$770$	14.8371	0.534692
$771$	0	0
$772$	34.6453	1.24691
$773$	−42.1939	−1.51761	−0.758805	−	0.651318i	$-0.774216\pi$
$773$	−42.1939	−1.51761	−0.758805	+	0.651318i	$0.774216\pi$
$774$	0	0
$775$	8.68035	0.311807
$776$	13.6781	0.491014
$777$	0	0
$778$	−44.5380	−1.59676
$779$	−3.26180	−0.116866
$780$	0	0
$781$	68.7091	2.45860
$782$	−16.5646	−0.592350
$783$	0	0
$784$	12.1317	0.433275
$785$	−9.41855	−0.336162
$786$	0	0
$787$	−35.4368	−1.26319	−0.631593	−	0.775300i	$-0.717599\pi$
$787$	−35.4368	−1.26319	−0.631593	+	0.775300i	$0.717599\pi$
$788$	24.9360	0.888308
$789$	0	0
$790$	30.9360	1.10065
$791$	13.8966	0.494105
$792$	0	0
$793$	7.66967	0.272358
$794$	−85.6307	−3.03892
$795$	0	0
$796$	−43.8843	−1.55544
$797$	−15.2579	−0.540463	−0.270232	−	0.962795i	$-0.587100\pi$
$797$	−15.2579	−0.540463	−0.270232	+	0.962795i	$0.587100\pi$
$798$	0	0
$799$	3.51745	0.124438
$800$	7.58864	0.268299
$801$	0	0
$802$	1.02052	0.0360358
$803$	34.3545	1.21235
$804$	0	0
$805$	2.52359	0.0889449
$806$	−25.7899	−0.908411
$807$	0	0
$808$	−7.57531	−0.266498
$809$	7.16290	0.251834	0.125917	−	0.992041i	$-0.459813\pi$
$809$	7.16290	0.251834	0.125917	+	0.992041i	$0.459813\pi$
$810$	0	0
$811$	−50.3545	−1.76819	−0.884094	−	0.467310i	$-0.845223\pi$
$811$	−50.3545	−1.76819	−0.884094	+	0.467310i	$0.845223\pi$
$812$	−4.14447	−0.145442
$813$	0	0
$814$	−73.8720	−2.58921
$815$	2.92162	0.102340
$816$	0	0
$817$	11.9155	0.416870
$818$	−22.7480	−0.795367
$819$	0	0
$820$	−8.83710	−0.308605
$821$	−0.952819	−0.0332536	−0.0166268	−	0.999862i	$-0.505293\pi$
$821$	−0.952819	−0.0332536	−0.0166268	+	0.999862i	$0.505293\pi$
$822$	0	0
$823$	−44.2290	−1.54173	−0.770863	−	0.637001i	$-0.780175\pi$
$823$	−44.2290	−1.54173	−0.770863	+	0.637001i	$0.780175\pi$
$824$	−9.83483	−0.342613
$825$	0	0
$826$	−26.7214	−0.929756
$827$	37.8615	1.31657	0.658287	−	0.752767i	$-0.271281\pi$
$827$	37.8615	1.31657	0.658287	+	0.752767i	$0.271281\pi$
$828$	0	0
$829$	−56.7214	−1.97002	−0.985008	−	0.172511i	$-0.944812\pi$
$829$	−56.7214	−1.97002	−0.985008	+	0.172511i	$0.944812\pi$
$830$	31.1194	1.08017
$831$	0	0
$832$	−16.8554	−0.584354
$833$	19.0394	0.659677
$834$	0	0
$835$	20.9132	0.723732
$836$	−17.1773	−0.594088
$837$	0	0
$838$	75.2327	2.59887
$839$	28.3591	0.979064	0.489532	−	0.871985i	$-0.337168\pi$
$839$	28.3591	0.979064	0.489532	+	0.871985i	$0.337168\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	75.1071	2.58836
$843$	0	0
$844$	21.0928	0.726043
$845$	11.1256	0.382731
$846$	0	0
$847$	31.4863	1.08188
$848$	13.7815	0.473259
$849$	0	0
$850$	7.07838	0.242786
$851$	−12.5646	−0.430710
$852$	0	0
$853$	−17.0061	−0.582279	−0.291140	−	0.956681i	$-0.594034\pi$
$853$	−17.0061	−0.582279	−0.291140	+	0.956681i	$0.594034\pi$
$854$	−13.1096	−0.448600
$855$	0	0
$856$	3.52586	0.120511
$857$	−15.4101	−0.526400	−0.263200	−	0.964741i	$-0.584778\pi$
$857$	−15.4101	−0.526400	−0.263200	+	0.964741i	$0.584778\pi$
$858$	0	0
$859$	−37.7275	−1.28725	−0.643623	−	0.765342i	$-0.722570\pi$
$859$	−37.7275	−1.28725	−0.643623	+	0.765342i	$0.722570\pi$
$860$	32.2823	1.10082
$861$	0	0
$862$	14.6225	0.498044
$863$	−32.1340	−1.09385	−0.546927	−	0.837181i	$-0.684202\pi$
$863$	−32.1340	−1.09385	−0.546927	+	0.837181i	$0.684202\pi$
$864$	0	0
$865$	−1.05559	−0.0358912
$866$	−44.4307	−1.50982
$867$	0	0
$868$	25.3607	0.860798
$869$	90.3833	3.06604
$870$	0	0
$871$	14.2245	0.481977
$872$	19.7587	0.669115
$873$	0	0
$874$	−5.07838	−0.171779
$875$	−1.07838	−0.0364558
$876$	0	0
$877$	−19.8927	−0.671729	−0.335864	−	0.941910i	$-0.609028\pi$
$877$	−19.8927	−0.671729	−0.335864	+	0.941910i	$0.609028\pi$
$878$	46.5692	1.57163
$879$	0	0
$880$	13.1773	0.444206
$881$	−24.0722	−0.811014	−0.405507	−	0.914092i	$-0.632905\pi$
$881$	−24.0722	−0.811014	−0.405507	+	0.914092i	$0.632905\pi$
$882$	0	0
$883$	−31.1727	−1.04905	−0.524523	−	0.851396i	$-0.675756\pi$
$883$	−31.1727	−1.04905	−0.524523	+	0.851396i	$0.675756\pi$
$884$	−12.0989	−0.406930
$885$	0	0
$886$	−46.7526	−1.57068
$887$	−4.86764	−0.163439	−0.0817197	−	0.996655i	$-0.526041\pi$
$887$	−4.86764	−0.163439	−0.0817197	+	0.996655i	$0.526041\pi$
$888$	0	0
$889$	−8.87814	−0.297763
$890$	−16.4391	−0.551039
$891$	0	0
$892$	33.9916	1.13812
$893$	1.07838	0.0360865
$894$	0	0
$895$	0.894960	0.0299152
$896$	12.4436	0.415712
$897$	0	0
$898$	−18.3812	−0.613389
$899$	−12.3135	−0.410679
$900$	0	0
$901$	21.6286	0.720554
$902$	−44.8781	−1.49428
$903$	0	0
$904$	−19.8348	−0.659697
$905$	0.837101	0.0278262
$906$	0	0
$907$	−45.4778	−1.51007	−0.755033	−	0.655686i	$-0.772379\pi$
$907$	−45.4778	−1.51007	−0.755033	+	0.655686i	$0.772379\pi$
$908$	−6.20620	−0.205960
$909$	0	0
$910$	3.20394	0.106209
$911$	20.9483	0.694048	0.347024	−	0.937856i	$-0.387192\pi$
$911$	20.9483	0.694048	0.347024	+	0.937856i	$0.387192\pi$
$912$	0	0
$913$	90.9192	3.00899
$914$	−24.6537	−0.815471
$915$	0	0
$916$	−16.0267	−0.529536
$917$	−1.59213	−0.0525767
$918$	0	0
$919$	−59.5174	−1.96330	−0.981650	−	0.190693i	$-0.938926\pi$
$919$	−59.5174	−1.96330	−0.981650	+	0.190693i	$0.938926\pi$
$920$	−3.60197	−0.118753
$921$	0	0
$922$	6.61265	0.217776
$923$	14.8371	0.488369
$924$	0	0
$925$	5.36910	0.176535
$926$	21.6430	0.711233
$927$	0	0
$928$	−10.7649	−0.353374
$929$	16.7214	0.548611	0.274305	−	0.961643i	$-0.411552\pi$
$929$	16.7214	0.548611	0.274305	+	0.961643i	$0.411552\pi$
$930$	0	0
$931$	5.83710	0.191303
$932$	−36.6225	−1.19961
$933$	0	0
$934$	3.23513	0.105857
$935$	20.6803	0.676320
$936$	0	0
$937$	−32.4534	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$937$	−32.4534	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$938$	−24.3135	−0.793864
$939$	0	0
$940$	2.92162	0.0952928
$941$	23.6742	0.771757	0.385878	−	0.922550i	$-0.373898\pi$
$941$	23.6742	0.771757	0.385878	+	0.922550i	$0.373898\pi$
$942$	0	0
$943$	−7.63317	−0.248570
$944$	−23.7321	−0.772413
$945$	0	0
$946$	163.942	5.33021
$947$	−21.9733	−0.714038	−0.357019	−	0.934097i	$-0.616207\pi$
$947$	−21.9733	−0.714038	−0.357019	+	0.934097i	$0.616207\pi$
$948$	0	0
$949$	7.41855	0.240816
$950$	2.17009	0.0704069
$951$	0	0
$952$	5.41402	0.175469
$953$	53.2990	1.72652	0.863261	−	0.504757i	$-0.168418\pi$
$953$	53.2990	1.72652	0.863261	+	0.504757i	$0.168418\pi$
$954$	0	0
$955$	22.0410	0.713231
$956$	−37.5052	−1.21300
$957$	0	0
$958$	−22.0722	−0.713122
$959$	4.25112	0.137276
$960$	0	0
$961$	44.3484	1.43059
$962$	−15.9520	−0.514313
$963$	0	0
$964$	19.6742	0.633663
$965$	−12.7877	−0.411649
$966$	0	0
$967$	15.8166	0.508627	0.254314	−	0.967122i	$-0.418150\pi$
$967$	15.8166	0.508627	0.254314	+	0.967122i	$0.418150\pi$
$968$	−44.9409	−1.44446
$969$	0	0
$970$	−19.2846	−0.619191
$971$	43.1506	1.38477	0.692385	−	0.721529i	$-0.256560\pi$
$971$	43.1506	1.38477	0.692385	+	0.721529i	$0.256560\pi$
$972$	0	0
$973$	9.55849	0.306431
$974$	52.6863	1.68818
$975$	0	0
$976$	−11.6430	−0.372684
$977$	8.47414	0.271112	0.135556	−	0.990770i	$-0.456718\pi$
$977$	8.47414	0.271112	0.135556	+	0.990770i	$0.456718\pi$
$978$	0	0
$979$	−48.0288	−1.53501
$980$	15.8143	0.505170
$981$	0	0
$982$	41.6742	1.32988
$983$	5.59356	0.178407	0.0892034	−	0.996013i	$-0.471568\pi$
$983$	5.59356	0.178407	0.0892034	+	0.996013i	$0.471568\pi$
$984$	0	0
$985$	−9.20394	−0.293262
$986$	−10.0410	−0.319772
$987$	0	0
$988$	−3.70928	−0.118008
$989$	27.8843	0.886669
$990$	0	0
$991$	32.8950	1.04494	0.522471	−	0.852657i	$-0.325010\pi$
$991$	32.8950	1.04494	0.522471	+	0.852657i	$0.325010\pi$
$992$	65.8720	2.09144
$993$	0	0
$994$	−25.3607	−0.804392
$995$	16.1978	0.513505
$996$	0	0
$997$	23.2618	0.736708	0.368354	−	0.929686i	$-0.379921\pi$
$997$	23.2618	0.736708	0.368354	+	0.929686i	$0.379921\pi$
$998$	−15.9155	−0.503796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.i.1.1		3
3.2	odd	2		95.2.a.a.1.3	✓	3
5.4	even	2		4275.2.a.bk.1.3		3
12.11	even	2		1520.2.a.p.1.3		3
15.2	even	4		475.2.b.d.324.6		6
15.8	even	4		475.2.b.d.324.1		6
15.14	odd	2		475.2.a.f.1.1		3
21.20	even	2		4655.2.a.u.1.3		3
24.5	odd	2		6080.2.a.bo.1.3		3
24.11	even	2		6080.2.a.by.1.1		3
57.56	even	2		1805.2.a.f.1.1		3
60.59	even	2		7600.2.a.bx.1.1		3
285.284	even	2		9025.2.a.bb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.3	✓	3	3.2	odd	2
475.2.a.f.1.1		3	15.14	odd	2
475.2.b.d.324.1		6	15.8	even	4
475.2.b.d.324.6		6	15.2	even	4
855.2.a.i.1.1		3	1.1	even	1	trivial
1520.2.a.p.1.3		3	12.11	even	2
1805.2.a.f.1.1		3	57.56	even	2
4275.2.a.bk.1.3		3	5.4	even	2
4655.2.a.u.1.3		3	21.20	even	2
6080.2.a.bo.1.3		3	24.5	odd	2
6080.2.a.by.1.1		3	24.11	even	2
7600.2.a.bx.1.1		3	60.59	even	2
9025.2.a.bb.1.3		3	285.284	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.53919 q^{8} +O(q^{10})\)	\(q-2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.53919 q^{8} +2.17009 q^{10} +6.34017 q^{11} +1.36910 q^{13} -2.34017 q^{14} -2.07838 q^{16} -3.26180 q^{17} -1.00000 q^{19} -2.70928 q^{20} -13.7587 q^{22} -2.34017 q^{23} +1.00000 q^{25} -2.97107 q^{26} +2.92162 q^{28} -1.41855 q^{29} +8.68035 q^{31} +7.58864 q^{32} +7.07838 q^{34} -1.07838 q^{35} +5.36910 q^{37} +2.17009 q^{38} +1.53919 q^{40} +3.26180 q^{41} -11.9155 q^{43} +17.1773 q^{44} +5.07838 q^{46} -1.07838 q^{47} -5.83710 q^{49} -2.17009 q^{50} +3.70928 q^{52} -6.63090 q^{53} -6.34017 q^{55} -1.65983 q^{56} +3.07838 q^{58} +11.4186 q^{59} +5.60197 q^{61} -18.8371 q^{62} -12.3112 q^{64} -1.36910 q^{65} +10.3896 q^{67} -8.83710 q^{68} +2.34017 q^{70} +10.8371 q^{71} +5.41855 q^{73} -11.6514 q^{74} -2.70928 q^{76} +6.83710 q^{77} +14.2557 q^{79} +2.07838 q^{80} -7.07838 q^{82} +14.3402 q^{83} +3.26180 q^{85} +25.8576 q^{86} -9.75872 q^{88} -7.57531 q^{89} +1.47641 q^{91} -6.34017 q^{92} +2.34017 q^{94} +1.00000 q^{95} -8.88655 q^{97} +12.6670 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8	\( 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} + 8 q^{11} + 8 q^{13} + 4 q^{14} - 3 q^{16} - 2 q^{17} - 3 q^{19} - q^{20} - 16 q^{22} + 4 q^{23} + 3 q^{25} + 6 q^{26} + 12 q^{28} + 10 q^{29} + 4 q^{31} + 3 q^{32} + 18 q^{34} + 20 q^{37} + q^{38} + 3 q^{40} + 2 q^{41} - 4 q^{43} + 12 q^{44} + 12 q^{46} + 11 q^{49} - q^{50} + 4 q^{52} - 16 q^{53} - 8 q^{55} - 16 q^{56} + 6 q^{58} + 20 q^{59} - 2 q^{61} - 28 q^{62} - 11 q^{64} - 8 q^{65} + 2 q^{67} + 2 q^{68} - 4 q^{70} + 4 q^{71} + 2 q^{73} + 2 q^{74} - q^{76} - 8 q^{77} + 3 q^{80} - 18 q^{82} + 32 q^{83} + 2 q^{85} + 16 q^{86} - 4 q^{88} - 2 q^{89} + 20 q^{91} - 8 q^{92} - 4 q^{94} + 3 q^{95} + 20 q^{97} + 15 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 + 8 * q^11 + 8 * q^13 + 4 * q^14 - 3 * q^16 - 2 * q^17 - 3 * q^19 - q^20 - 16 * q^22 + 4 * q^23 + 3 * q^25 + 6 * q^26 + 12 * q^28 + 10 * q^29 + 4 * q^31 + 3 * q^32 + 18 * q^34 + 20 * q^37 + q^38 + 3 * q^40 + 2 * q^41 - 4 * q^43 + 12 * q^44 + 12 * q^46 + 11 * q^49 - q^50 + 4 * q^52 - 16 * q^53 - 8 * q^55 - 16 * q^56 + 6 * q^58 + 20 * q^59 - 2 * q^61 - 28 * q^62 - 11 * q^64 - 8 * q^65 + 2 * q^67 + 2 * q^68 - 4 * q^70 + 4 * q^71 + 2 * q^73 + 2 * q^74 - q^76 - 8 * q^77 + 3 * q^80 - 18 * q^82 + 32 * q^83 + 2 * q^85 + 16 * q^86 - 4 * q^88 - 2 * q^89 + 20 * q^91 - 8 * q^92 - 4 * q^94 + 3 * q^95 + 20 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more