LMFDB - Embedded newform 855.2.a.f.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{8} +1.73205 q^{10} -1.26795 q^{11} -2.73205 q^{13} -1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{19} -1.00000 q^{20} +2.19615 q^{22} +3.46410 q^{23} +1.00000 q^{25} +4.73205 q^{26} +0.732051 q^{28} +2.19615 q^{29} -4.92820 q^{31} +5.19615 q^{32} -0.732051 q^{35} +4.19615 q^{37} -1.73205 q^{38} -1.73205 q^{40} -4.73205 q^{41} -6.19615 q^{43} -1.26795 q^{44} -6.00000 q^{46} -3.46410 q^{47} -6.46410 q^{49} -1.73205 q^{50} -2.73205 q^{52} +2.53590 q^{53} +1.26795 q^{55} +1.26795 q^{56} -3.80385 q^{58} -9.46410 q^{59} -13.4641 q^{61} +8.53590 q^{62} +1.00000 q^{64} +2.73205 q^{65} +8.00000 q^{67} +1.26795 q^{70} -16.3923 q^{71} -3.07180 q^{73} -7.26795 q^{74} +1.00000 q^{76} -0.928203 q^{77} +2.92820 q^{79} +5.00000 q^{80} +8.19615 q^{82} -0.928203 q^{83} +10.7321 q^{86} -2.19615 q^{88} -7.26795 q^{89} -2.00000 q^{91} +3.46410 q^{92} +6.00000 q^{94} -1.00000 q^{95} +4.19615 q^{97} +11.1962 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 2 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{19} - 2 q^{20} - 6 q^{22} + 2 q^{25} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{35} - 2 q^{37} - 6 q^{41} - 2 q^{43} - 6 q^{44} - 12 q^{46} - 6 q^{49} - 2 q^{52} + 12 q^{53} + 6 q^{55} + 6 q^{56} - 18 q^{58} - 12 q^{59} - 20 q^{61} + 24 q^{62} + 2 q^{64} + 2 q^{65} + 16 q^{67} + 6 q^{70} - 12 q^{71} - 20 q^{73} - 18 q^{74} + 2 q^{76} + 12 q^{77} - 8 q^{79} + 10 q^{80} + 6 q^{82} + 12 q^{83} + 18 q^{86} + 6 q^{88} - 18 q^{89} - 4 q^{91} + 12 q^{94} - 2 q^{95} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 2 * q^13 - 6 * q^14 - 10 * q^16 + 2 * q^19 - 2 * q^20 - 6 * q^22 + 2 * q^25 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 4 * q^31 + 2 * q^35 - 2 * q^37 - 6 * q^41 - 2 * q^43 - 6 * q^44 - 12 * q^46 - 6 * q^49 - 2 * q^52 + 12 * q^53 + 6 * q^55 + 6 * q^56 - 18 * q^58 - 12 * q^59 - 20 * q^61 + 24 * q^62 + 2 * q^64 + 2 * q^65 + 16 * q^67 + 6 * q^70 - 12 * q^71 - 20 * q^73 - 18 * q^74 + 2 * q^76 + 12 * q^77 - 8 * q^79 + 10 * q^80 + 6 * q^82 + 12 * q^83 + 18 * q^86 + 6 * q^88 - 18 * q^89 - 4 * q^91 + 12 * q^94 - 2 * q^95 - 2 * q^97 + 12 * 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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	1.73205	0.547723
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	−2.73205	−0.757735	−0.378867	−	0.925451i	$-0.623686\pi$
$13$	−2.73205	−0.757735	−0.378867	+	0.925451i	$0.623686\pi$
$14$	−1.26795	−0.338874
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.19615	0.468221
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.73205	0.928032
$27$	0	0
$28$	0.732051	0.138345
$29$	2.19615	0.407815	0.203908	−	0.978990i	$-0.434636\pi$
$29$	2.19615	0.407815	0.203908	+	0.978990i	$0.434636\pi$
$30$	0	0
$31$	−4.92820	−0.885131	−0.442566	−	0.896736i	$-0.645932\pi$
$31$	−4.92820	−0.885131	−0.442566	+	0.896736i	$0.645932\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	4.19615	0.689843	0.344922	−	0.938631i	$-0.387905\pi$
$37$	4.19615	0.689843	0.344922	+	0.938631i	$0.387905\pi$
$38$	−1.73205	−0.280976
$39$	0	0
$40$	−1.73205	−0.273861
$41$	−4.73205	−0.739022	−0.369511	−	0.929226i	$-0.620475\pi$
$41$	−4.73205	−0.739022	−0.369511	+	0.929226i	$0.620475\pi$
$42$	0	0
$43$	−6.19615	−0.944904	−0.472452	−	0.881356i	$-0.656631\pi$
$43$	−6.19615	−0.944904	−0.472452	+	0.881356i	$0.656631\pi$
$44$	−1.26795	−0.191151
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	−1.73205	−0.244949
$51$	0	0
$52$	−2.73205	−0.378867
$53$	2.53590	0.348332	0.174166	−	0.984716i	$-0.444277\pi$
$53$	2.53590	0.348332	0.174166	+	0.984716i	$0.444277\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	1.26795	0.169437
$57$	0	0
$58$	−3.80385	−0.499470
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	0	0
$61$	−13.4641	−1.72390	−0.861951	−	0.506992i	$-0.830757\pi$
$61$	−13.4641	−1.72390	−0.861951	+	0.506992i	$0.830757\pi$
$62$	8.53590	1.08406
$63$	0	0
$64$	1.00000	0.125000
$65$	2.73205	0.338869
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	1.26795	0.151549
$71$	−16.3923	−1.94541	−0.972704	−	0.232048i	$-0.925457\pi$
$71$	−16.3923	−1.94541	−0.972704	+	0.232048i	$0.925457\pi$
$72$	0	0
$73$	−3.07180	−0.359527	−0.179763	−	0.983710i	$-0.557533\pi$
$73$	−3.07180	−0.359527	−0.179763	+	0.983710i	$0.557533\pi$
$74$	−7.26795	−0.844882
$75$	0	0
$76$	1.00000	0.114708
$77$	−0.928203	−0.105779
$78$	0	0
$79$	2.92820	0.329449	0.164724	−	0.986340i	$-0.447327\pi$
$79$	2.92820	0.329449	0.164724	+	0.986340i	$0.447327\pi$
$80$	5.00000	0.559017
$81$	0	0
$82$	8.19615	0.905114
$83$	−0.928203	−0.101884	−0.0509418	−	0.998702i	$-0.516222\pi$
$83$	−0.928203	−0.101884	−0.0509418	+	0.998702i	$0.516222\pi$
$84$	0	0
$85$	0	0
$86$	10.7321	1.15727
$87$	0	0
$88$	−2.19615	−0.234111
$89$	−7.26795	−0.770401	−0.385201	−	0.922833i	$-0.625868\pi$
$89$	−7.26795	−0.770401	−0.385201	+	0.922833i	$0.625868\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	3.46410	0.361158
$93$	0	0
$94$	6.00000	0.618853
$95$	−1.00000	−0.102598
$96$	0	0
$97$	4.19615	0.426055	0.213027	−	0.977046i	$-0.431668\pi$
$97$	4.19615	0.426055	0.213027	+	0.977046i	$0.431668\pi$
$98$	11.1962	1.13098
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	−17.8564	−1.75944	−0.879722	−	0.475488i	$-0.842271\pi$
$103$	−17.8564	−1.75944	−0.879722	+	0.475488i	$0.842271\pi$
$104$	−4.73205	−0.464016
$105$	0	0
$106$	−4.39230	−0.426618
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	6.39230	0.612272	0.306136	−	0.951988i	$-0.400964\pi$
$109$	6.39230	0.612272	0.306136	+	0.951988i	$0.400964\pi$
$110$	−2.19615	−0.209395
$111$	0	0
$112$	−3.66025	−0.345861
$113$	−5.07180	−0.477115	−0.238557	−	0.971128i	$-0.576674\pi$
$113$	−5.07180	−0.477115	−0.238557	+	0.971128i	$0.576674\pi$
$114$	0	0
$115$	−3.46410	−0.323029
$116$	2.19615	0.203908
$117$	0	0
$118$	16.3923	1.50903
$119$	0	0
$120$	0	0
$121$	−9.39230	−0.853846
$122$	23.3205	2.11134
$123$	0	0
$124$	−4.92820	−0.442566
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	−4.73205	−0.415028
$131$	−15.1244	−1.32142	−0.660711	−	0.750641i	$-0.729745\pi$
$131$	−15.1244	−1.32142	−0.660711	+	0.750641i	$0.729745\pi$
$132$	0	0
$133$	0.732051	0.0634769
$134$	−13.8564	−1.19701
$135$	0	0
$136$	0	0
$137$	7.85641	0.671218	0.335609	−	0.942001i	$-0.391058\pi$
$137$	7.85641	0.671218	0.335609	+	0.942001i	$0.391058\pi$
$138$	0	0
$139$	12.3923	1.05110	0.525551	−	0.850762i	$-0.323859\pi$
$139$	12.3923	1.05110	0.525551	+	0.850762i	$0.323859\pi$
$140$	−0.732051	−0.0618696
$141$	0	0
$142$	28.3923	2.38263
$143$	3.46410	0.289683
$144$	0	0
$145$	−2.19615	−0.182381
$146$	5.32051	0.440328
$147$	0	0
$148$	4.19615	0.344922
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	1.60770	0.129552
$155$	4.92820	0.395843
$156$	0	0
$157$	−14.3923	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$157$	−14.3923	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$158$	−5.07180	−0.403490
$159$	0	0
$160$	−5.19615	−0.410792
$161$	2.53590	0.199857
$162$	0	0
$163$	12.7321	0.997251	0.498626	−	0.866817i	$-0.333838\pi$
$163$	12.7321	0.997251	0.498626	+	0.866817i	$0.333838\pi$
$164$	−4.73205	−0.369511
$165$	0	0
$166$	1.60770	0.124781
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	0	0
$172$	−6.19615	−0.472452
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	0.732051	0.0553378
$176$	6.33975	0.477876
$177$	0	0
$178$	12.5885	0.943545
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	0	0
$181$	18.3923	1.36709	0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923	1.36709	0.683545	+	0.729909i	$0.260437\pi$
$182$	3.46410	0.256776
$183$	0	0
$184$	6.00000	0.442326
$185$	−4.19615	−0.308507
$186$	0	0
$187$	0	0
$188$	−3.46410	−0.252646
$189$	0	0
$190$	1.73205	0.125656
$191$	−17.6603	−1.27785	−0.638926	−	0.769269i	$-0.720621\pi$
$191$	−17.6603	−1.27785	−0.638926	+	0.769269i	$0.720621\pi$
$192$	0	0
$193$	−7.12436	−0.512822	−0.256411	−	0.966568i	$-0.582540\pi$
$193$	−7.12436	−0.512822	−0.256411	+	0.966568i	$0.582540\pi$
$194$	−7.26795	−0.521808
$195$	0	0
$196$	−6.46410	−0.461722
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	19.3205	1.36959	0.684797	−	0.728734i	$-0.259891\pi$
$199$	19.3205	1.36959	0.684797	+	0.728734i	$0.259891\pi$
$200$	1.73205	0.122474
$201$	0	0
$202$	18.0000	1.26648
$203$	1.60770	0.112838
$204$	0	0
$205$	4.73205	0.330501
$206$	30.9282	2.15487
$207$	0	0
$208$	13.6603	0.947168
$209$	−1.26795	−0.0877059
$210$	0	0
$211$	14.9282	1.02770	0.513850	−	0.857880i	$-0.328219\pi$
$211$	14.9282	1.02770	0.513850	+	0.857880i	$0.328219\pi$
$212$	2.53590	0.174166
$213$	0	0
$214$	0	0
$215$	6.19615	0.422574
$216$	0	0
$217$	−3.60770	−0.244906
$218$	−11.0718	−0.749877
$219$	0	0
$220$	1.26795	0.0854851
$221$	0	0
$222$	0	0
$223$	9.85641	0.660034	0.330017	−	0.943975i	$-0.392946\pi$
$223$	9.85641	0.660034	0.330017	+	0.943975i	$0.392946\pi$
$224$	3.80385	0.254155
$225$	0	0
$226$	8.78461	0.584344
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	0	0
$229$	−25.4641	−1.68272	−0.841358	−	0.540479i	$-0.818243\pi$
$229$	−25.4641	−1.68272	−0.841358	+	0.540479i	$0.818243\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	3.80385	0.249735
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	−9.46410	−0.616061
$237$	0	0
$238$	0	0
$239$	20.1962	1.30638	0.653190	−	0.757194i	$-0.273430\pi$
$239$	20.1962	1.30638	0.653190	+	0.757194i	$0.273430\pi$
$240$	0	0
$241$	−16.9282	−1.09044	−0.545221	−	0.838293i	$-0.683554\pi$
$241$	−16.9282	−1.09044	−0.545221	+	0.838293i	$0.683554\pi$
$242$	16.2679	1.04574
$243$	0	0
$244$	−13.4641	−0.861951
$245$	6.46410	0.412976
$246$	0	0
$247$	−2.73205	−0.173836
$248$	−8.53590	−0.542030
$249$	0	0
$250$	1.73205	0.109545
$251$	10.0526	0.634512	0.317256	−	0.948340i	$-0.397239\pi$
$251$	10.0526	0.634512	0.317256	+	0.948340i	$0.397239\pi$
$252$	0	0
$253$	−4.39230	−0.276142
$254$	6.92820	0.434714
$255$	0	0
$256$	19.0000	1.18750
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	3.07180	0.190872
$260$	2.73205	0.169435
$261$	0	0
$262$	26.1962	1.61840
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−2.53590	−0.155779
$266$	−1.26795	−0.0777430
$267$	0	0
$268$	8.00000	0.488678
$269$	30.5885	1.86501	0.932506	−	0.361156i	$-0.117618\pi$
$269$	30.5885	1.86501	0.932506	+	0.361156i	$0.117618\pi$
$270$	0	0
$271$	−20.3923	−1.23874	−0.619372	−	0.785098i	$-0.712613\pi$
$271$	−20.3923	−1.23874	−0.619372	+	0.785098i	$0.712613\pi$
$272$	0	0
$273$	0	0
$274$	−13.6077	−0.822071
$275$	−1.26795	−0.0764602
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−21.4641	−1.28733
$279$	0	0
$280$	−1.26795	−0.0757745
$281$	−4.73205	−0.282290	−0.141145	−	0.989989i	$-0.545078\pi$
$281$	−4.73205	−0.282290	−0.141145	+	0.989989i	$0.545078\pi$
$282$	0	0
$283$	−26.9808	−1.60384	−0.801920	−	0.597432i	$-0.796188\pi$
$283$	−26.9808	−1.60384	−0.801920	+	0.597432i	$0.796188\pi$
$284$	−16.3923	−0.972704
$285$	0	0
$286$	−6.00000	−0.354787
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−17.0000	−1.00000
$290$	3.80385	0.223370
$291$	0	0
$292$	−3.07180	−0.179763
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	0	0
$295$	9.46410	0.551021
$296$	7.26795	0.422441
$297$	0	0
$298$	13.6077	0.788273
$299$	−9.46410	−0.547323
$300$	0	0
$301$	−4.53590	−0.261445
$302$	−24.2487	−1.39536
$303$	0	0
$304$	−5.00000	−0.286770
$305$	13.4641	0.770952
$306$	0	0
$307$	−11.6077	−0.662486	−0.331243	−	0.943545i	$-0.607468\pi$
$307$	−11.6077	−0.662486	−0.331243	+	0.943545i	$0.607468\pi$
$308$	−0.928203	−0.0528893
$309$	0	0
$310$	−8.53590	−0.484806
$311$	26.4449	1.49955	0.749775	−	0.661692i	$-0.230162\pi$
$311$	26.4449	1.49955	0.749775	+	0.661692i	$0.230162\pi$
$312$	0	0
$313$	−14.3923	−0.813501	−0.406751	−	0.913539i	$-0.633338\pi$
$313$	−14.3923	−0.813501	−0.406751	+	0.913539i	$0.633338\pi$
$314$	24.9282	1.40678
$315$	0	0
$316$	2.92820	0.164724
$317$	−23.3205	−1.30981	−0.654905	−	0.755711i	$-0.727291\pi$
$317$	−23.3205	−1.30981	−0.654905	+	0.755711i	$0.727291\pi$
$318$	0	0
$319$	−2.78461	−0.155908
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−4.39230	−0.244774
$323$	0	0
$324$	0	0
$325$	−2.73205	−0.151547
$326$	−22.0526	−1.22138
$327$	0	0
$328$	−8.19615	−0.452557
$329$	−2.53590	−0.139809
$330$	0	0
$331$	29.7128	1.63316	0.816582	−	0.577230i	$-0.195866\pi$
$331$	29.7128	1.63316	0.816582	+	0.577230i	$0.195866\pi$
$332$	−0.928203	−0.0509418
$333$	0	0
$334$	−6.00000	−0.328305
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−19.1244	−1.04177	−0.520885	−	0.853627i	$-0.674398\pi$
$337$	−19.1244	−1.04177	−0.520885	+	0.853627i	$0.674398\pi$
$338$	9.58846	0.521543
$339$	0	0
$340$	0	0
$341$	6.24871	0.338387
$342$	0	0
$343$	−9.85641	−0.532196
$344$	−10.7321	−0.578633
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−12.9282	−0.694022	−0.347011	−	0.937861i	$-0.612803\pi$
$347$	−12.9282	−0.694022	−0.347011	+	0.937861i	$0.612803\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−1.26795	−0.0677747
$351$	0	0
$352$	−6.58846	−0.351166
$353$	−26.7846	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$353$	−26.7846	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$354$	0	0
$355$	16.3923	0.870013
$356$	−7.26795	−0.385201
$357$	0	0
$358$	−19.6077	−1.03630
$359$	−17.6603	−0.932073	−0.466036	−	0.884766i	$-0.654318\pi$
$359$	−17.6603	−0.932073	−0.466036	+	0.884766i	$0.654318\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−31.8564	−1.67434
$363$	0	0
$364$	−2.00000	−0.104828
$365$	3.07180	0.160785
$366$	0	0
$367$	5.80385	0.302958	0.151479	−	0.988460i	$-0.451596\pi$
$367$	5.80385	0.302958	0.151479	+	0.988460i	$0.451596\pi$
$368$	−17.3205	−0.902894
$369$	0	0
$370$	7.26795	0.377843
$371$	1.85641	0.0963798
$372$	0	0
$373$	4.19615	0.217269	0.108634	−	0.994082i	$-0.465352\pi$
$373$	4.19615	0.217269	0.108634	+	0.994082i	$0.465352\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−6.00000	−0.309016
$378$	0	0
$379$	7.07180	0.363254	0.181627	−	0.983368i	$-0.441864\pi$
$379$	7.07180	0.363254	0.181627	+	0.983368i	$0.441864\pi$
$380$	−1.00000	−0.0512989
$381$	0	0
$382$	30.5885	1.56504
$383$	30.9282	1.58036	0.790179	−	0.612877i	$-0.209988\pi$
$383$	30.9282	1.58036	0.790179	+	0.612877i	$0.209988\pi$
$384$	0	0
$385$	0.928203	0.0473056
$386$	12.3397	0.628077
$387$	0	0
$388$	4.19615	0.213027
$389$	19.8564	1.00676	0.503380	−	0.864065i	$-0.332090\pi$
$389$	19.8564	1.00676	0.503380	+	0.864065i	$0.332090\pi$
$390$	0	0
$391$	0	0
$392$	−11.1962	−0.565491
$393$	0	0
$394$	−41.5692	−2.09423
$395$	−2.92820	−0.147334
$396$	0	0
$397$	−4.92820	−0.247339	−0.123670	−	0.992323i	$-0.539466\pi$
$397$	−4.92820	−0.247339	−0.123670	+	0.992323i	$0.539466\pi$
$398$	−33.4641	−1.67740
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−4.05256	−0.202375	−0.101188	−	0.994867i	$-0.532264\pi$
$401$	−4.05256	−0.202375	−0.101188	+	0.994867i	$0.532264\pi$
$402$	0	0
$403$	13.4641	0.670695
$404$	−10.3923	−0.517036
$405$	0	0
$406$	−2.78461	−0.138198
$407$	−5.32051	−0.263728
$408$	0	0
$409$	−5.60770	−0.277283	−0.138641	−	0.990343i	$-0.544274\pi$
$409$	−5.60770	−0.277283	−0.138641	+	0.990343i	$0.544274\pi$
$410$	−8.19615	−0.404779
$411$	0	0
$412$	−17.8564	−0.879722
$413$	−6.92820	−0.340915
$414$	0	0
$415$	0.928203	0.0455637
$416$	−14.1962	−0.696024
$417$	0	0
$418$	2.19615	0.107417
$419$	−10.0526	−0.491100	−0.245550	−	0.969384i	$-0.578968\pi$
$419$	−10.0526	−0.491100	−0.245550	+	0.969384i	$0.578968\pi$
$420$	0	0
$421$	22.7846	1.11045	0.555227	−	0.831699i	$-0.312631\pi$
$421$	22.7846	1.11045	0.555227	+	0.831699i	$0.312631\pi$
$422$	−25.8564	−1.25867
$423$	0	0
$424$	4.39230	0.213309
$425$	0	0
$426$	0	0
$427$	−9.85641	−0.476985
$428$	0	0
$429$	0	0
$430$	−10.7321	−0.517545
$431$	−23.3205	−1.12331	−0.561655	−	0.827372i	$-0.689835\pi$
$431$	−23.3205	−1.12331	−0.561655	+	0.827372i	$0.689835\pi$
$432$	0	0
$433$	20.5885	0.989418	0.494709	−	0.869059i	$-0.335275\pi$
$433$	20.5885	0.989418	0.494709	+	0.869059i	$0.335275\pi$
$434$	6.24871	0.299948
$435$	0	0
$436$	6.39230	0.306136
$437$	3.46410	0.165710
$438$	0	0
$439$	13.0718	0.623883	0.311941	−	0.950101i	$-0.399021\pi$
$439$	13.0718	0.623883	0.311941	+	0.950101i	$0.399021\pi$
$440$	2.19615	0.104697
$441$	0	0
$442$	0	0
$443$	−29.3205	−1.39306	−0.696530	−	0.717528i	$-0.745274\pi$
$443$	−29.3205	−1.39306	−0.696530	+	0.717528i	$0.745274\pi$
$444$	0	0
$445$	7.26795	0.344534
$446$	−17.0718	−0.808373
$447$	0	0
$448$	0.732051	0.0345861
$449$	−11.6603	−0.550281	−0.275141	−	0.961404i	$-0.588724\pi$
$449$	−11.6603	−0.550281	−0.275141	+	0.961404i	$0.588724\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−5.07180	−0.238557
$453$	0	0
$454$	−18.0000	−0.844782
$455$	2.00000	0.0937614
$456$	0	0
$457$	11.4641	0.536268	0.268134	−	0.963382i	$-0.413593\pi$
$457$	11.4641	0.536268	0.268134	+	0.963382i	$0.413593\pi$
$458$	44.1051	2.06090
$459$	0	0
$460$	−3.46410	−0.161515
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	9.51666	0.442277	0.221138	−	0.975242i	$-0.429023\pi$
$463$	9.51666	0.442277	0.221138	+	0.975242i	$0.429023\pi$
$464$	−10.9808	−0.509769
$465$	0	0
$466$	34.3923	1.59319
$467$	−27.4641	−1.27089	−0.635444	−	0.772147i	$-0.719183\pi$
$467$	−27.4641	−1.27089	−0.635444	+	0.772147i	$0.719183\pi$
$468$	0	0
$469$	5.85641	0.270424
$470$	−6.00000	−0.276759
$471$	0	0
$472$	−16.3923	−0.754517
$473$	7.85641	0.361238
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	−34.9808	−1.59998
$479$	19.5167	0.891739	0.445869	−	0.895098i	$-0.352895\pi$
$479$	19.5167	0.891739	0.445869	+	0.895098i	$0.352895\pi$
$480$	0	0
$481$	−11.4641	−0.522718
$482$	29.3205	1.33551
$483$	0	0
$484$	−9.39230	−0.426923
$485$	−4.19615	−0.190537
$486$	0	0
$487$	−11.6077	−0.525995	−0.262997	−	0.964797i	$-0.584711\pi$
$487$	−11.6077	−0.525995	−0.262997	+	0.964797i	$0.584711\pi$
$488$	−23.3205	−1.05567
$489$	0	0
$490$	−11.1962	−0.505791
$491$	22.0526	0.995218	0.497609	−	0.867401i	$-0.334211\pi$
$491$	22.0526	0.995218	0.497609	+	0.867401i	$0.334211\pi$
$492$	0	0
$493$	0	0
$494$	4.73205	0.212905
$495$	0	0
$496$	24.6410	1.10641
$497$	−12.0000	−0.538274
$498$	0	0
$499$	17.4641	0.781801	0.390900	−	0.920433i	$-0.372164\pi$
$499$	17.4641	0.781801	0.390900	+	0.920433i	$0.372164\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−17.4115	−0.777115
$503$	36.9282	1.64655	0.823274	−	0.567645i	$-0.192145\pi$
$503$	36.9282	1.64655	0.823274	+	0.567645i	$0.192145\pi$
$504$	0	0
$505$	10.3923	0.462451
$506$	7.60770	0.338203
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−28.0526	−1.24341	−0.621704	−	0.783252i	$-0.713559\pi$
$509$	−28.0526	−1.24341	−0.621704	+	0.783252i	$0.713559\pi$
$510$	0	0
$511$	−2.24871	−0.0994771
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−41.5692	−1.83354
$515$	17.8564	0.786847
$516$	0	0
$517$	4.39230	0.193173
$518$	−5.32051	−0.233770
$519$	0	0
$520$	4.73205	0.207514
$521$	−40.7321	−1.78450	−0.892252	−	0.451538i	$-0.850875\pi$
$521$	−40.7321	−1.78450	−0.892252	+	0.451538i	$0.850875\pi$
$522$	0	0
$523$	43.3205	1.89427	0.947137	−	0.320830i	$-0.103962\pi$
$523$	43.3205	1.89427	0.947137	+	0.320830i	$0.103962\pi$
$524$	−15.1244	−0.660711
$525$	0	0
$526$	10.3923	0.453126
$527$	0	0
$528$	0	0
$529$	−11.0000	−0.478261
$530$	4.39230	0.190790
$531$	0	0
$532$	0.732051	0.0317384
$533$	12.9282	0.559983
$534$	0	0
$535$	0	0
$536$	13.8564	0.598506
$537$	0	0
$538$	−52.9808	−2.28416
$539$	8.19615	0.353033
$540$	0	0
$541$	−13.7128	−0.589560	−0.294780	−	0.955565i	$-0.595246\pi$
$541$	−13.7128	−0.589560	−0.294780	+	0.955565i	$0.595246\pi$
$542$	35.3205	1.51715
$543$	0	0
$544$	0	0
$545$	−6.39230	−0.273816
$546$	0	0
$547$	8.67949	0.371108	0.185554	−	0.982634i	$-0.440592\pi$
$547$	8.67949	0.371108	0.185554	+	0.982634i	$0.440592\pi$
$548$	7.85641	0.335609
$549$	0	0
$550$	2.19615	0.0936443
$551$	2.19615	0.0935592
$552$	0	0
$553$	2.14359	0.0911549
$554$	−3.46410	−0.147176
$555$	0	0
$556$	12.3923	0.525551
$557$	12.9282	0.547786	0.273893	−	0.961760i	$-0.411689\pi$
$557$	12.9282	0.547786	0.273893	+	0.961760i	$0.411689\pi$
$558$	0	0
$559$	16.9282	0.715987
$560$	3.66025	0.154674
$561$	0	0
$562$	8.19615	0.345734
$563$	20.5359	0.865485	0.432742	−	0.901518i	$-0.357546\pi$
$563$	20.5359	0.865485	0.432742	+	0.901518i	$0.357546\pi$
$564$	0	0
$565$	5.07180	0.213372
$566$	46.7321	1.96429
$567$	0	0
$568$	−28.3923	−1.19131
$569$	16.0526	0.672958	0.336479	−	0.941691i	$-0.390764\pi$
$569$	16.0526	0.672958	0.336479	+	0.941691i	$0.390764\pi$
$570$	0	0
$571$	14.2487	0.596290	0.298145	−	0.954521i	$-0.403632\pi$
$571$	14.2487	0.596290	0.298145	+	0.954521i	$0.403632\pi$
$572$	3.46410	0.144841
$573$	0	0
$574$	6.00000	0.250435
$575$	3.46410	0.144463
$576$	0	0
$577$	−47.1769	−1.96400	−0.982000	−	0.188879i	$-0.939515\pi$
$577$	−47.1769	−1.96400	−0.982000	+	0.188879i	$0.939515\pi$
$578$	29.4449	1.22474
$579$	0	0
$580$	−2.19615	−0.0911903
$581$	−0.679492	−0.0281901
$582$	0	0
$583$	−3.21539	−0.133168
$584$	−5.32051	−0.220164
$585$	0	0
$586$	−48.0000	−1.98286
$587$	−3.46410	−0.142979	−0.0714894	−	0.997441i	$-0.522775\pi$
$587$	−3.46410	−0.142979	−0.0714894	+	0.997441i	$0.522775\pi$
$588$	0	0
$589$	−4.92820	−0.203063
$590$	−16.3923	−0.674861
$591$	0	0
$592$	−20.9808	−0.862304
$593$	−2.78461	−0.114350	−0.0571751	−	0.998364i	$-0.518209\pi$
$593$	−2.78461	−0.114350	−0.0571751	+	0.998364i	$0.518209\pi$
$594$	0	0
$595$	0	0
$596$	−7.85641	−0.321811
$597$	0	0
$598$	16.3923	0.670331
$599$	−13.8564	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$599$	−13.8564	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$600$	0	0
$601$	15.1769	0.619079	0.309540	−	0.950887i	$-0.399825\pi$
$601$	15.1769	0.619079	0.309540	+	0.950887i	$0.399825\pi$
$602$	7.85641	0.320203
$603$	0	0
$604$	14.0000	0.569652
$605$	9.39230	0.381851
$606$	0	0
$607$	−32.3923	−1.31476	−0.657382	−	0.753558i	$-0.728336\pi$
$607$	−32.3923	−1.31476	−0.657382	+	0.753558i	$0.728336\pi$
$608$	5.19615	0.210732
$609$	0	0
$610$	−23.3205	−0.944220
$611$	9.46410	0.382877
$612$	0	0
$613$	21.6077	0.872727	0.436363	−	0.899771i	$-0.356266\pi$
$613$	21.6077	0.872727	0.436363	+	0.899771i	$0.356266\pi$
$614$	20.1051	0.811377
$615$	0	0
$616$	−1.60770	−0.0647759
$617$	27.7128	1.11568	0.557838	−	0.829950i	$-0.311631\pi$
$617$	27.7128	1.11568	0.557838	+	0.829950i	$0.311631\pi$
$618$	0	0
$619$	19.3205	0.776557	0.388278	−	0.921542i	$-0.373070\pi$
$619$	19.3205	0.776557	0.388278	+	0.921542i	$0.373070\pi$
$620$	4.92820	0.197921
$621$	0	0
$622$	−45.8038	−1.83657
$623$	−5.32051	−0.213162
$624$	0	0
$625$	1.00000	0.0400000
$626$	24.9282	0.996331
$627$	0	0
$628$	−14.3923	−0.574315
$629$	0	0
$630$	0	0
$631$	−21.0718	−0.838855	−0.419427	−	0.907789i	$-0.637769\pi$
$631$	−21.0718	−0.838855	−0.419427	+	0.907789i	$0.637769\pi$
$632$	5.07180	0.201745
$633$	0	0
$634$	40.3923	1.60418
$635$	4.00000	0.158735
$636$	0	0
$637$	17.6603	0.699725
$638$	4.82309	0.190948
$639$	0	0
$640$	12.1244	0.479257
$641$	−17.4115	−0.687715	−0.343857	−	0.939022i	$-0.611734\pi$
$641$	−17.4115	−0.687715	−0.343857	+	0.939022i	$0.611734\pi$
$642$	0	0
$643$	−1.80385	−0.0711368	−0.0355684	−	0.999367i	$-0.511324\pi$
$643$	−1.80385	−0.0711368	−0.0355684	+	0.999367i	$0.511324\pi$
$644$	2.53590	0.0999284
$645$	0	0
$646$	0	0
$647$	31.8564	1.25240	0.626202	−	0.779661i	$-0.284608\pi$
$647$	31.8564	1.25240	0.626202	+	0.779661i	$0.284608\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	4.73205	0.185606
$651$	0	0
$652$	12.7321	0.498626
$653$	−30.9282	−1.21031	−0.605157	−	0.796106i	$-0.706890\pi$
$653$	−30.9282	−1.21031	−0.605157	+	0.796106i	$0.706890\pi$
$654$	0	0
$655$	15.1244	0.590957
$656$	23.6603	0.923778
$657$	0	0
$658$	4.39230	0.171230
$659$	−18.9282	−0.737338	−0.368669	−	0.929561i	$-0.620186\pi$
$659$	−18.9282	−0.737338	−0.368669	+	0.929561i	$0.620186\pi$
$660$	0	0
$661$	−23.1769	−0.901477	−0.450739	−	0.892656i	$-0.648839\pi$
$661$	−23.1769	−0.901477	−0.450739	+	0.892656i	$0.648839\pi$
$662$	−51.4641	−2.00021
$663$	0	0
$664$	−1.60770	−0.0623907
$665$	−0.732051	−0.0283877
$666$	0	0
$667$	7.60770	0.294571
$668$	3.46410	0.134030
$669$	0	0
$670$	13.8564	0.535320
$671$	17.0718	0.659049
$672$	0	0
$673$	−7.12436	−0.274624	−0.137312	−	0.990528i	$-0.543846\pi$
$673$	−7.12436	−0.274624	−0.137312	+	0.990528i	$0.543846\pi$
$674$	33.1244	1.27590
$675$	0	0
$676$	−5.53590	−0.212919
$677$	−35.3205	−1.35748	−0.678739	−	0.734380i	$-0.737473\pi$
$677$	−35.3205	−1.35748	−0.678739	+	0.734380i	$0.737473\pi$
$678$	0	0
$679$	3.07180	0.117885
$680$	0	0
$681$	0	0
$682$	−10.8231	−0.414437
$683$	18.9282	0.724268	0.362134	−	0.932126i	$-0.382048\pi$
$683$	18.9282	0.724268	0.362134	+	0.932126i	$0.382048\pi$
$684$	0	0
$685$	−7.85641	−0.300178
$686$	17.0718	0.651804
$687$	0	0
$688$	30.9808	1.18113
$689$	−6.92820	−0.263944
$690$	0	0
$691$	−8.39230	−0.319258	−0.159629	−	0.987177i	$-0.551030\pi$
$691$	−8.39230	−0.319258	−0.159629	+	0.987177i	$0.551030\pi$
$692$	6.92820	0.263371
$693$	0	0
$694$	22.3923	0.850000
$695$	−12.3923	−0.470067
$696$	0	0
$697$	0	0
$698$	38.1051	1.44230
$699$	0	0
$700$	0.732051	0.0276689
$701$	−21.7128	−0.820082	−0.410041	−	0.912067i	$-0.634486\pi$
$701$	−21.7128	−0.820082	−0.410041	+	0.912067i	$0.634486\pi$
$702$	0	0
$703$	4.19615	0.158261
$704$	−1.26795	−0.0477876
$705$	0	0
$706$	46.3923	1.74600
$707$	−7.60770	−0.286117
$708$	0	0
$709$	33.1769	1.24599	0.622993	−	0.782228i	$-0.285917\pi$
$709$	33.1769	1.24599	0.622993	+	0.782228i	$0.285917\pi$
$710$	−28.3923	−1.06554
$711$	0	0
$712$	−12.5885	−0.471772
$713$	−17.0718	−0.639344
$714$	0	0
$715$	−3.46410	−0.129550
$716$	11.3205	0.423067
$717$	0	0
$718$	30.5885	1.14155
$719$	−5.66025	−0.211092	−0.105546	−	0.994414i	$-0.533659\pi$
$719$	−5.66025	−0.211092	−0.105546	+	0.994414i	$0.533659\pi$
$720$	0	0
$721$	−13.0718	−0.486819
$722$	−1.73205	−0.0644603
$723$	0	0
$724$	18.3923	0.683545
$725$	2.19615	0.0815631
$726$	0	0
$727$	8.33975	0.309304	0.154652	−	0.987969i	$-0.450574\pi$
$727$	8.33975	0.309304	0.154652	+	0.987969i	$0.450574\pi$
$728$	−3.46410	−0.128388
$729$	0	0
$730$	−5.32051	−0.196921
$731$	0	0
$732$	0	0
$733$	22.7846	0.841569	0.420784	−	0.907161i	$-0.361755\pi$
$733$	22.7846	0.841569	0.420784	+	0.907161i	$0.361755\pi$
$734$	−10.0526	−0.371047
$735$	0	0
$736$	18.0000	0.663489
$737$	−10.1436	−0.373644
$738$	0	0
$739$	33.8564	1.24543	0.622714	−	0.782450i	$-0.286030\pi$
$739$	33.8564	1.24543	0.622714	+	0.782450i	$0.286030\pi$
$740$	−4.19615	−0.154254
$741$	0	0
$742$	−3.21539	−0.118041
$743$	44.7846	1.64299	0.821494	−	0.570217i	$-0.193141\pi$
$743$	44.7846	1.64299	0.821494	+	0.570217i	$0.193141\pi$
$744$	0	0
$745$	7.85641	0.287836
$746$	−7.26795	−0.266099
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	17.3205	0.631614
$753$	0	0
$754$	10.3923	0.378465
$755$	−14.0000	−0.509512
$756$	0	0
$757$	−16.2487	−0.590569	−0.295285	−	0.955409i	$-0.595415\pi$
$757$	−16.2487	−0.590569	−0.295285	+	0.955409i	$0.595415\pi$
$758$	−12.2487	−0.444893
$759$	0	0
$760$	−1.73205	−0.0628281
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	4.67949	0.169409
$764$	−17.6603	−0.638926
$765$	0	0
$766$	−53.5692	−1.93553
$767$	25.8564	0.933621
$768$	0	0
$769$	48.6410	1.75404	0.877020	−	0.480454i	$-0.159528\pi$
$769$	48.6410	1.75404	0.877020	+	0.480454i	$0.159528\pi$
$770$	−1.60770	−0.0579373
$771$	0	0
$772$	−7.12436	−0.256411
$773$	−37.1769	−1.33716	−0.668580	−	0.743640i	$-0.733098\pi$
$773$	−37.1769	−1.33716	−0.668580	+	0.743640i	$0.733098\pi$
$774$	0	0
$775$	−4.92820	−0.177026
$776$	7.26795	0.260904
$777$	0	0
$778$	−34.3923	−1.23302
$779$	−4.73205	−0.169543
$780$	0	0
$781$	20.7846	0.743732
$782$	0	0
$783$	0	0
$784$	32.3205	1.15430
$785$	14.3923	0.513683
$786$	0	0
$787$	43.3205	1.54421	0.772105	−	0.635495i	$-0.219204\pi$
$787$	43.3205	1.54421	0.772105	+	0.635495i	$0.219204\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	5.07180	0.180446
$791$	−3.71281	−0.132012
$792$	0	0
$793$	36.7846	1.30626
$794$	8.53590	0.302928
$795$	0	0
$796$	19.3205	0.684797
$797$	−3.21539	−0.113895	−0.0569475	−	0.998377i	$-0.518137\pi$
$797$	−3.21539	−0.113895	−0.0569475	+	0.998377i	$0.518137\pi$
$798$	0	0
$799$	0	0
$800$	5.19615	0.183712
$801$	0	0
$802$	7.01924	0.247858
$803$	3.89488	0.137447
$804$	0	0
$805$	−2.53590	−0.0893787
$806$	−23.3205	−0.821430
$807$	0	0
$808$	−18.0000	−0.633238
$809$	26.7846	0.941697	0.470848	−	0.882214i	$-0.343948\pi$
$809$	26.7846	0.941697	0.470848	+	0.882214i	$0.343948\pi$
$810$	0	0
$811$	−45.5692	−1.60015	−0.800076	−	0.599899i	$-0.795207\pi$
$811$	−45.5692	−1.60015	−0.800076	+	0.599899i	$0.795207\pi$
$812$	1.60770	0.0564190
$813$	0	0
$814$	9.21539	0.322999
$815$	−12.7321	−0.445984
$816$	0	0
$817$	−6.19615	−0.216776
$818$	9.71281	0.339601
$819$	0	0
$820$	4.73205	0.165250
$821$	39.4641	1.37731	0.688653	−	0.725091i	$-0.258202\pi$
$821$	39.4641	1.37731	0.688653	+	0.725091i	$0.258202\pi$
$822$	0	0
$823$	−38.9808	−1.35878	−0.679392	−	0.733776i	$-0.737756\pi$
$823$	−38.9808	−1.35878	−0.679392	+	0.733776i	$0.737756\pi$
$824$	−30.9282	−1.07744
$825$	0	0
$826$	12.0000	0.417533
$827$	−29.3205	−1.01957	−0.509787	−	0.860301i	$-0.670276\pi$
$827$	−29.3205	−1.01957	−0.509787	+	0.860301i	$0.670276\pi$
$828$	0	0
$829$	−42.1051	−1.46237	−0.731186	−	0.682179i	$-0.761033\pi$
$829$	−42.1051	−1.46237	−0.731186	+	0.682179i	$0.761033\pi$
$830$	−1.60770	−0.0558039
$831$	0	0
$832$	−2.73205	−0.0947168
$833$	0	0
$834$	0	0
$835$	−3.46410	−0.119880
$836$	−1.26795	−0.0438529
$837$	0	0
$838$	17.4115	0.601472
$839$	40.3923	1.39450	0.697249	−	0.716829i	$-0.254407\pi$
$839$	40.3923	1.39450	0.697249	+	0.716829i	$0.254407\pi$
$840$	0	0
$841$	−24.1769	−0.833687
$842$	−39.4641	−1.36002
$843$	0	0
$844$	14.9282	0.513850
$845$	5.53590	0.190441
$846$	0	0
$847$	−6.87564	−0.236250
$848$	−12.6795	−0.435416
$849$	0	0
$850$	0	0
$851$	14.5359	0.498284
$852$	0	0
$853$	−35.1769	−1.20443	−0.602217	−	0.798332i	$-0.705716\pi$
$853$	−35.1769	−1.20443	−0.602217	+	0.798332i	$0.705716\pi$
$854$	17.0718	0.584185
$855$	0	0
$856$	0	0
$857$	−6.24871	−0.213452	−0.106726	−	0.994288i	$-0.534037\pi$
$857$	−6.24871	−0.213452	−0.106726	+	0.994288i	$0.534037\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	6.19615	0.211287
$861$	0	0
$862$	40.3923	1.37577
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	−6.92820	−0.235566
$866$	−35.6603	−1.21178
$867$	0	0
$868$	−3.60770	−0.122453
$869$	−3.71281	−0.125949
$870$	0	0
$871$	−21.8564	−0.740576
$872$	11.0718	0.374938
$873$	0	0
$874$	−6.00000	−0.202953
$875$	−0.732051	−0.0247478
$876$	0	0
$877$	28.8756	0.975061	0.487531	−	0.873106i	$-0.337898\pi$
$877$	28.8756	0.975061	0.487531	+	0.873106i	$0.337898\pi$
$878$	−22.6410	−0.764097
$879$	0	0
$880$	−6.33975	−0.213713
$881$	−15.4641	−0.520999	−0.260499	−	0.965474i	$-0.583887\pi$
$881$	−15.4641	−0.520999	−0.260499	+	0.965474i	$0.583887\pi$
$882$	0	0
$883$	−14.9808	−0.504143	−0.252071	−	0.967709i	$-0.581112\pi$
$883$	−14.9808	−0.504143	−0.252071	+	0.967709i	$0.581112\pi$
$884$	0	0
$885$	0	0
$886$	50.7846	1.70614
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−2.92820	−0.0982088
$890$	−12.5885	−0.421966
$891$	0	0
$892$	9.85641	0.330017
$893$	−3.46410	−0.115922
$894$	0	0
$895$	−11.3205	−0.378403
$896$	−8.87564	−0.296514
$897$	0	0
$898$	20.1962	0.673954
$899$	−10.8231	−0.360970
$900$	0	0
$901$	0	0
$902$	−10.3923	−0.346026
$903$	0	0
$904$	−8.78461	−0.292172
$905$	−18.3923	−0.611381
$906$	0	0
$907$	−11.6077	−0.385427	−0.192714	−	0.981255i	$-0.561729\pi$
$907$	−11.6077	−0.385427	−0.192714	+	0.981255i	$0.561729\pi$
$908$	10.3923	0.344881
$909$	0	0
$910$	−3.46410	−0.114834
$911$	41.0718	1.36077	0.680385	−	0.732855i	$-0.261813\pi$
$911$	41.0718	1.36077	0.680385	+	0.732855i	$0.261813\pi$
$912$	0	0
$913$	1.17691	0.0389502
$914$	−19.8564	−0.656792
$915$	0	0
$916$	−25.4641	−0.841358
$917$	−11.0718	−0.365623
$918$	0	0
$919$	−59.4256	−1.96027	−0.980135	−	0.198330i	$-0.936448\pi$
$919$	−59.4256	−1.96027	−0.980135	+	0.198330i	$0.936448\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	10.3923	0.342252
$923$	44.7846	1.47410
$924$	0	0
$925$	4.19615	0.137969
$926$	−16.4833	−0.541676
$927$	0	0
$928$	11.4115	0.374602
$929$	22.3923	0.734668	0.367334	−	0.930089i	$-0.380271\pi$
$929$	22.3923	0.734668	0.367334	+	0.930089i	$0.380271\pi$
$930$	0	0
$931$	−6.46410	−0.211852
$932$	−19.8564	−0.650418
$933$	0	0
$934$	47.5692	1.55651
$935$	0	0
$936$	0	0
$937$	32.2487	1.05352	0.526760	−	0.850014i	$-0.323407\pi$
$937$	32.2487	1.05352	0.526760	+	0.850014i	$0.323407\pi$
$938$	−10.1436	−0.331200
$939$	0	0
$940$	3.46410	0.112987
$941$	30.5885	0.997155	0.498578	−	0.866845i	$-0.333856\pi$
$941$	30.5885	0.997155	0.498578	+	0.866845i	$0.333856\pi$
$942$	0	0
$943$	−16.3923	−0.533807
$944$	47.3205	1.54015
$945$	0	0
$946$	−13.6077	−0.442424
$947$	−55.8564	−1.81509	−0.907545	−	0.419956i	$-0.862046\pi$
$947$	−55.8564	−1.81509	−0.907545	+	0.419956i	$0.862046\pi$
$948$	0	0
$949$	8.39230	0.272426
$950$	−1.73205	−0.0561951
$951$	0	0
$952$	0	0
$953$	−10.1436	−0.328583	−0.164292	−	0.986412i	$-0.552534\pi$
$953$	−10.1436	−0.328583	−0.164292	+	0.986412i	$0.552534\pi$
$954$	0	0
$955$	17.6603	0.571472
$956$	20.1962	0.653190
$957$	0	0
$958$	−33.8038	−1.09215
$959$	5.75129	0.185719
$960$	0	0
$961$	−6.71281	−0.216542
$962$	19.8564	0.640196
$963$	0	0
$964$	−16.9282	−0.545221
$965$	7.12436	0.229341
$966$	0	0
$967$	29.1244	0.936576	0.468288	−	0.883576i	$-0.344871\pi$
$967$	29.1244	0.936576	0.468288	+	0.883576i	$0.344871\pi$
$968$	−16.2679	−0.522872
$969$	0	0
$970$	7.26795	0.233360
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	0	0
$973$	9.07180	0.290828
$974$	20.1051	0.644210
$975$	0	0
$976$	67.3205	2.15488
$977$	−51.0333	−1.63270	−0.816350	−	0.577557i	$-0.804006\pi$
$977$	−51.0333	−1.63270	−0.816350	+	0.577557i	$0.804006\pi$
$978$	0	0
$979$	9.21539	0.294525
$980$	6.46410	0.206488
$981$	0	0
$982$	−38.1962	−1.21889
$983$	6.67949	0.213043	0.106521	−	0.994310i	$-0.466029\pi$
$983$	6.67949	0.213043	0.106521	+	0.994310i	$0.466029\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	0	0
$988$	−2.73205	−0.0869181
$989$	−21.4641	−0.682519
$990$	0	0
$991$	26.9282	0.855403	0.427701	−	0.903920i	$-0.359324\pi$
$991$	26.9282	0.855403	0.427701	+	0.903920i	$0.359324\pi$
$992$	−25.6077	−0.813045
$993$	0	0
$994$	20.7846	0.659248
$995$	−19.3205	−0.612501
$996$	0	0
$997$	−38.3923	−1.21590	−0.607948	−	0.793977i	$-0.708007\pi$
$997$	−38.3923	−1.21590	−0.607948	+	0.793977i	$0.708007\pi$
$998$	−30.2487	−0.957506
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.f.1.1		2
3.2	odd	2		285.2.a.e.1.2	✓	2
5.4	even	2		4275.2.a.t.1.2		2
12.11	even	2		4560.2.a.bh.1.1		2
15.2	even	4		1425.2.c.k.799.3		4
15.8	even	4		1425.2.c.k.799.2		4
15.14	odd	2		1425.2.a.o.1.1		2
57.56	even	2		5415.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.e.1.2	✓	2	3.2	odd	2
855.2.a.f.1.1		2	1.1	even	1	trivial
1425.2.a.o.1.1		2	15.14	odd	2
1425.2.c.k.799.2		4	15.8	even	4
1425.2.c.k.799.3		4	15.2	even	4
4275.2.a.t.1.2		2	5.4	even	2
4560.2.a.bh.1.1		2	12.11	even	2
5415.2.a.r.1.1		2	57.56	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{8} +1.73205 q^{10} -1.26795 q^{11} -2.73205 q^{13} -1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{19} -1.00000 q^{20} +2.19615 q^{22} +3.46410 q^{23} +1.00000 q^{25} +4.73205 q^{26} +0.732051 q^{28} +2.19615 q^{29} -4.92820 q^{31} +5.19615 q^{32} -0.732051 q^{35} +4.19615 q^{37} -1.73205 q^{38} -1.73205 q^{40} -4.73205 q^{41} -6.19615 q^{43} -1.26795 q^{44} -6.00000 q^{46} -3.46410 q^{47} -6.46410 q^{49} -1.73205 q^{50} -2.73205 q^{52} +2.53590 q^{53} +1.26795 q^{55} +1.26795 q^{56} -3.80385 q^{58} -9.46410 q^{59} -13.4641 q^{61} +8.53590 q^{62} +1.00000 q^{64} +2.73205 q^{65} +8.00000 q^{67} +1.26795 q^{70} -16.3923 q^{71} -3.07180 q^{73} -7.26795 q^{74} +1.00000 q^{76} -0.928203 q^{77} +2.92820 q^{79} +5.00000 q^{80} +8.19615 q^{82} -0.928203 q^{83} +10.7321 q^{86} -2.19615 q^{88} -7.26795 q^{89} -2.00000 q^{91} +3.46410 q^{92} +6.00000 q^{94} -1.00000 q^{95} +4.19615 q^{97} +11.1962 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 2 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{19} - 2 q^{20} - 6 q^{22} + 2 q^{25} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{35} - 2 q^{37} - 6 q^{41} - 2 q^{43} - 6 q^{44} - 12 q^{46} - 6 q^{49} - 2 q^{52} + 12 q^{53} + 6 q^{55} + 6 q^{56} - 18 q^{58} - 12 q^{59} - 20 q^{61} + 24 q^{62} + 2 q^{64} + 2 q^{65} + 16 q^{67} + 6 q^{70} - 12 q^{71} - 20 q^{73} - 18 q^{74} + 2 q^{76} + 12 q^{77} - 8 q^{79} + 10 q^{80} + 6 q^{82} + 12 q^{83} + 18 q^{86} + 6 q^{88} - 18 q^{89} - 4 q^{91} + 12 q^{94} - 2 q^{95} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 2 * q^13 - 6 * q^14 - 10 * q^16 + 2 * q^19 - 2 * q^20 - 6 * q^22 + 2 * q^25 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 4 * q^31 + 2 * q^35 - 2 * q^37 - 6 * q^41 - 2 * q^43 - 6 * q^44 - 12 * q^46 - 6 * q^49 - 2 * q^52 + 12 * q^53 + 6 * q^55 + 6 * q^56 - 18 * q^58 - 12 * q^59 - 20 * q^61 + 24 * q^62 + 2 * q^64 + 2 * q^65 + 16 * q^67 + 6 * q^70 - 12 * q^71 - 20 * q^73 - 18 * q^74 + 2 * q^76 + 12 * q^77 - 8 * q^79 + 10 * q^80 + 6 * q^82 + 12 * q^83 + 18 * q^86 + 6 * q^88 - 18 * q^89 - 4 * q^91 + 12 * q^94 - 2 * q^95 - 2 * q^97 + 12 * q^98

Introduction

L-functions

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