LMFDB - Embedded newform 1856.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.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:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.230224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.66240$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37051 q^{3} +2.20308 q^{5} -3.32479 q^{7} +2.61930 q^{9} +O(q^{10})$	$q-2.37051 q^{3} +2.20308 q^{5} -3.32479 q^{7} +2.61930 q^{9} -5.40004 q^{11} -5.97363 q^{13} -5.22242 q^{15} +3.82239 q^{17} -0.472877 q^{19} +7.88144 q^{21} +2.58378 q^{23} -0.146427 q^{25} +0.902442 q^{27} +1.00000 q^{29} +8.74955 q^{31} +12.8008 q^{33} -7.32479 q^{35} -3.29767 q^{37} +14.1605 q^{39} -6.06581 q^{41} -5.40004 q^{43} +5.77054 q^{45} +5.92476 q^{47} +4.05425 q^{49} -9.06100 q^{51} +9.97363 q^{53} -11.8967 q^{55} +1.12096 q^{57} +4.06581 q^{59} -0.989943 q^{61} -8.70864 q^{63} -13.1604 q^{65} +9.70383 q^{67} -6.12487 q^{69} +9.55334 q^{71} -10.9424 q^{73} +0.347107 q^{75} +17.9540 q^{77} +3.55727 q^{79} -9.99716 q^{81} -15.1201 q^{83} +8.42103 q^{85} -2.37051 q^{87} +9.47528 q^{89} +19.8611 q^{91} -20.7409 q^{93} -1.04179 q^{95} -15.7696 q^{97} -14.1443 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9	$ 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9} - 8 q^{11} + 6 q^{13} + 6 q^{15} + 6 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{23} + 7 q^{25} - 22 q^{27} + 5 q^{29} + 8 q^{31} - 16 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} + 28 q^{47} + 13 q^{49} - 24 q^{51} + 14 q^{53} + 10 q^{55} + 20 q^{57} - 16 q^{59} + 18 q^{61} + 20 q^{63} - 8 q^{65} + 28 q^{69} - 4 q^{71} + 2 q^{73} + 6 q^{75} + 4 q^{77} - 12 q^{79} + 9 q^{81} - 32 q^{83} + 32 q^{85} - 4 q^{87} + 30 q^{89} + 40 q^{91} - 4 q^{93} + 4 q^{95} + 6 q^{97} + 22 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9 - 8 * q^11 + 6 * q^13 + 6 * q^15 + 6 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^23 + 7 * q^25 - 22 * q^27 + 5 * q^29 + 8 * q^31 - 16 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 + 28 * q^47 + 13 * q^49 - 24 * q^51 + 14 * q^53 + 10 * q^55 + 20 * q^57 - 16 * q^59 + 18 * q^61 + 20 * q^63 - 8 * q^65 + 28 * q^69 - 4 * q^71 + 2 * q^73 + 6 * q^75 + 4 * q^77 - 12 * q^79 + 9 * q^81 - 32 * q^83 + 32 * q^85 - 4 * q^87 + 30 * q^89 + 40 * q^91 - 4 * q^93 + 4 * q^95 + 6 * q^97 + 22 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−2.37051	−1.36861	−0.684306	−	0.729195i	$-0.739895\pi$
$3$	−2.37051	−1.36861	−0.684306	+	0.729195i	$0.739895\pi$
$4$	0	0
$5$	2.20308	0.985248	0.492624	−	0.870242i	$-0.336038\pi$
$5$	2.20308	0.985248	0.492624	+	0.870242i	$0.336038\pi$
$6$	0	0
$7$	−3.32479	−1.25665	−0.628327	−	0.777950i	$-0.716260\pi$
$7$	−3.32479	−1.25665	−0.628327	+	0.777950i	$0.716260\pi$
$8$	0	0
$9$	2.61930	0.873101
$10$	0	0
$11$	−5.40004	−1.62817	−0.814086	−	0.580744i	$-0.802762\pi$
$11$	−5.40004	−1.62817	−0.814086	+	0.580744i	$0.802762\pi$
$12$	0	0
$13$	−5.97363	−1.65679	−0.828393	−	0.560148i	$-0.810744\pi$
$13$	−5.97363	−1.65679	−0.828393	+	0.560148i	$0.810744\pi$
$14$	0	0
$15$	−5.22242	−1.34842
$16$	0	0
$17$	3.82239	0.927065	0.463532	−	0.886080i	$-0.346582\pi$
$17$	3.82239	0.927065	0.463532	+	0.886080i	$0.346582\pi$
$18$	0	0
$19$	−0.472877	−0.108485	−0.0542427	−	0.998528i	$-0.517274\pi$
$19$	−0.472877	−0.108485	−0.0542427	+	0.998528i	$0.517274\pi$
$20$	0	0
$21$	7.88144	1.71987
$22$	0	0
$23$	2.58378	0.538755	0.269378	−	0.963035i	$-0.413182\pi$
$23$	2.58378	0.538755	0.269378	+	0.963035i	$0.413182\pi$
$24$	0	0
$25$	−0.146427	−0.0292855
$26$	0	0
$27$	0.902442	0.173675
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	8.74955	1.57146	0.785732	−	0.618567i	$-0.212286\pi$
$31$	8.74955	1.57146	0.785732	+	0.618567i	$0.212286\pi$
$32$	0	0
$33$	12.8008	2.22834
$34$	0	0
$35$	−7.32479	−1.23812
$36$	0	0
$37$	−3.29767	−0.542133	−0.271067	−	0.962561i	$-0.587376\pi$
$37$	−3.29767	−0.542133	−0.271067	+	0.962561i	$0.587376\pi$
$38$	0	0
$39$	14.1605	2.26750
$40$	0	0
$41$	−6.06581	−0.947320	−0.473660	−	0.880708i	$-0.657067\pi$
$41$	−6.06581	−0.947320	−0.473660	+	0.880708i	$0.657067\pi$
$42$	0	0
$43$	−5.40004	−0.823498	−0.411749	−	0.911297i	$-0.635082\pi$
$43$	−5.40004	−0.823498	−0.411749	+	0.911297i	$0.635082\pi$
$44$	0	0
$45$	5.77054	0.860222
$46$	0	0
$47$	5.92476	0.864215	0.432107	−	0.901822i	$-0.357770\pi$
$47$	5.92476	0.864215	0.432107	+	0.901822i	$0.357770\pi$
$48$	0	0
$49$	4.05425	0.579178
$50$	0	0
$51$	−9.06100	−1.26879
$52$	0	0
$53$	9.97363	1.36998	0.684991	−	0.728551i	$-0.259806\pi$
$53$	9.97363	1.36998	0.684991	+	0.728551i	$0.259806\pi$
$54$	0	0
$55$	−11.8967	−1.60415
$56$	0	0
$57$	1.12096	0.148475
$58$	0	0
$59$	4.06581	0.529323	0.264661	−	0.964341i	$-0.414740\pi$
$59$	4.06581	0.529323	0.264661	+	0.964341i	$0.414740\pi$
$60$	0	0
$61$	−0.989943	−0.126749	−0.0633746	−	0.997990i	$-0.520186\pi$
$61$	−0.989943	−0.126749	−0.0633746	+	0.997990i	$0.520186\pi$
$62$	0	0
$63$	−8.70864	−1.09719
$64$	0	0
$65$	−13.1604	−1.63235
$66$	0	0
$67$	9.70383	1.18551	0.592756	−	0.805382i	$-0.298040\pi$
$67$	9.70383	1.18551	0.592756	+	0.805382i	$0.298040\pi$
$68$	0	0
$69$	−6.12487	−0.737347
$70$	0	0
$71$	9.55334	1.13377	0.566887	−	0.823796i	$-0.308148\pi$
$71$	9.55334	1.13377	0.566887	+	0.823796i	$0.308148\pi$
$72$	0	0
$73$	−10.9424	−1.28072	−0.640358	−	0.768077i	$-0.721214\pi$
$73$	−10.9424	−1.28072	−0.640358	+	0.768077i	$0.721214\pi$
$74$	0	0
$75$	0.347107	0.0400805
$76$	0	0
$77$	17.9540	2.04605
$78$	0	0
$79$	3.55727	0.400224	0.200112	−	0.979773i	$-0.435869\pi$
$79$	3.55727	0.400224	0.200112	+	0.979773i	$0.435869\pi$
$80$	0	0
$81$	−9.99716	−1.11080
$82$	0	0
$83$	−15.1201	−1.65964	−0.829821	−	0.558030i	$-0.811557\pi$
$83$	−15.1201	−1.65964	−0.829821	+	0.558030i	$0.811557\pi$
$84$	0	0
$85$	8.42103	0.913389
$86$	0	0
$87$	−2.37051	−0.254145
$88$	0	0
$89$	9.47528	1.00438	0.502189	−	0.864758i	$-0.332528\pi$
$89$	9.47528	1.00438	0.502189	+	0.864758i	$0.332528\pi$
$90$	0	0
$91$	19.8611	2.08201
$92$	0	0
$93$	−20.7409	−2.15073
$94$	0	0
$95$	−1.04179	−0.106885
$96$	0	0
$97$	−15.7696	−1.60116	−0.800582	−	0.599223i	$-0.795476\pi$
$97$	−15.7696	−1.60116	−0.800582	+	0.599223i	$0.795476\pi$
$98$	0	0
$99$	−14.1443	−1.42156
$100$	0	0
$101$	18.8930	1.87992	0.939962	−	0.341279i	$-0.110860\pi$
$101$	18.8930	1.87992	0.939962	+	0.341279i	$0.110860\pi$
$102$	0	0
$103$	−6.90376	−0.680248	−0.340124	−	0.940381i	$-0.610469\pi$
$103$	−6.90376	−0.680248	−0.340124	+	0.940381i	$0.610469\pi$
$104$	0	0
$105$	17.3635	1.69450
$106$	0	0
$107$	10.8659	1.05044	0.525222	−	0.850965i	$-0.323982\pi$
$107$	10.8659	1.05044	0.525222	+	0.850965i	$0.323982\pi$
$108$	0	0
$109$	12.7984	1.22587	0.612933	−	0.790135i	$-0.289990\pi$
$109$	12.7984	1.22587	0.612933	+	0.790135i	$0.289990\pi$
$110$	0	0
$111$	7.81714	0.741970
$112$	0	0
$113$	18.3534	1.72654	0.863272	−	0.504739i	$-0.168411\pi$
$113$	18.3534	1.72654	0.863272	+	0.504739i	$0.168411\pi$
$114$	0	0
$115$	5.69228	0.530808
$116$	0	0
$117$	−15.6467	−1.44654
$118$	0	0
$119$	−12.7086	−1.16500
$120$	0	0
$121$	18.1604	1.65094
$122$	0	0
$123$	14.3790	1.29651
$124$	0	0
$125$	−11.3380	−1.01410
$126$	0	0
$127$	5.36438	0.476012	0.238006	−	0.971264i	$-0.423506\pi$
$127$	5.36438	0.476012	0.238006	+	0.971264i	$0.423506\pi$
$128$	0	0
$129$	12.8008	1.12705
$130$	0	0
$131$	13.8215	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$131$	13.8215	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$132$	0	0
$133$	1.57222	0.136329
$134$	0	0
$135$	1.98815	0.171113
$136$	0	0
$137$	9.05575	0.773685	0.386842	−	0.922146i	$-0.373566\pi$
$137$	9.05575	0.773685	0.386842	+	0.922146i	$0.373566\pi$
$138$	0	0
$139$	−3.39611	−0.288054	−0.144027	−	0.989574i	$-0.546005\pi$
$139$	−3.39611	−0.288054	−0.144027	+	0.989574i	$0.546005\pi$
$140$	0	0
$141$	−14.0447	−1.18278
$142$	0	0
$143$	32.2578	2.69753
$144$	0	0
$145$	2.20308	0.182956
$146$	0	0
$147$	−9.61062	−0.792671
$148$	0	0
$149$	−9.39660	−0.769799	−0.384900	−	0.922958i	$-0.625764\pi$
$149$	−9.39660	−0.769799	−0.384900	+	0.922958i	$0.625764\pi$
$150$	0	0
$151$	−16.9792	−1.38175	−0.690873	−	0.722976i	$-0.742774\pi$
$151$	−16.9792	−1.38175	−0.690873	+	0.722976i	$0.742774\pi$
$152$	0	0
$153$	10.0120	0.809422
$154$	0	0
$155$	19.2760	1.54828
$156$	0	0
$157$	15.5478	1.24085	0.620426	−	0.784265i	$-0.286960\pi$
$157$	15.5478	1.24085	0.620426	+	0.784265i	$0.286960\pi$
$158$	0	0
$159$	−23.6426	−1.87498
$160$	0	0
$161$	−8.59053	−0.677028
$162$	0	0
$163$	−4.32851	−0.339035	−0.169518	−	0.985527i	$-0.554221\pi$
$163$	−4.32851	−0.339035	−0.169518	+	0.985527i	$0.554221\pi$
$164$	0	0
$165$	28.2013	2.19547
$166$	0	0
$167$	7.25899	0.561717	0.280859	−	0.959749i	$-0.409381\pi$
$167$	7.25899	0.561717	0.280859	+	0.959749i	$0.409381\pi$
$168$	0	0
$169$	22.6842	1.74494
$170$	0	0
$171$	−1.23861	−0.0947188
$172$	0	0
$173$	−4.83244	−0.367404	−0.183702	−	0.982982i	$-0.558808\pi$
$173$	−4.83244	−0.367404	−0.183702	+	0.982982i	$0.558808\pi$
$174$	0	0
$175$	0.486840	0.0368017
$176$	0	0
$177$	−9.63802	−0.724438
$178$	0	0
$179$	−2.26380	−0.169204	−0.0846021	−	0.996415i	$-0.526962\pi$
$179$	−2.26380	−0.169204	−0.0846021	+	0.996415i	$0.526962\pi$
$180$	0	0
$181$	20.9828	1.55964	0.779819	−	0.626005i	$-0.215311\pi$
$181$	20.9828	1.55964	0.779819	+	0.626005i	$0.215311\pi$
$182$	0	0
$183$	2.34667	0.173471
$184$	0	0
$185$	−7.26503	−0.534136
$186$	0	0
$187$	−20.6410	−1.50942
$188$	0	0
$189$	−3.00043	−0.218249
$190$	0	0
$191$	5.95491	0.430882	0.215441	−	0.976517i	$-0.430881\pi$
$191$	5.95491	0.430882	0.215441	+	0.976517i	$0.430881\pi$
$192$	0	0
$193$	−11.9031	−0.856801	−0.428401	−	0.903589i	$-0.640923\pi$
$193$	−11.9031	−0.856801	−0.428401	+	0.903589i	$0.640923\pi$
$194$	0	0
$195$	31.1968	2.23405
$196$	0	0
$197$	−10.4868	−0.747156	−0.373578	−	0.927599i	$-0.621869\pi$
$197$	−10.4868	−0.747156	−0.373578	+	0.927599i	$0.621869\pi$
$198$	0	0
$199$	−3.62577	−0.257024	−0.128512	−	0.991708i	$-0.541020\pi$
$199$	−3.62577	−0.257024	−0.128512	+	0.991708i	$0.541020\pi$
$200$	0	0
$201$	−23.0030	−1.62251
$202$	0	0
$203$	−3.32479	−0.233355
$204$	0	0
$205$	−13.3635	−0.933345
$206$	0	0
$207$	6.76770	0.470388
$208$	0	0
$209$	2.55355	0.176633
$210$	0	0
$211$	8.88848	0.611908	0.305954	−	0.952046i	$-0.401025\pi$
$211$	8.88848	0.611908	0.305954	+	0.952046i	$0.401025\pi$
$212$	0	0
$213$	−22.6463	−1.55170
$214$	0	0
$215$	−11.8967	−0.811350
$216$	0	0
$217$	−29.0904	−1.97479
$218$	0	0
$219$	25.9391	1.75280
$220$	0	0
$221$	−22.8335	−1.53595
$222$	0	0
$223$	4.66184	0.312180	0.156090	−	0.987743i	$-0.450111\pi$
$223$	4.66184	0.312180	0.156090	+	0.987743i	$0.450111\pi$
$224$	0	0
$225$	−0.383538	−0.0255692
$226$	0	0
$227$	−27.6124	−1.83270	−0.916350	−	0.400379i	$-0.868879\pi$
$227$	−27.6124	−1.83270	−0.916350	+	0.400379i	$0.868879\pi$
$228$	0	0
$229$	22.8781	1.51183	0.755915	−	0.654670i	$-0.227192\pi$
$229$	22.8781	1.51183	0.755915	+	0.654670i	$0.227192\pi$
$230$	0	0
$231$	−42.5601	−2.80025
$232$	0	0
$233$	−5.14643	−0.337154	−0.168577	−	0.985689i	$-0.553917\pi$
$233$	−5.14643	−0.337154	−0.168577	+	0.985689i	$0.553917\pi$
$234$	0	0
$235$	13.0527	0.851466
$236$	0	0
$237$	−8.43254	−0.547752
$238$	0	0
$239$	4.95731	0.320662	0.160331	−	0.987063i	$-0.448744\pi$
$239$	4.95731	0.320662	0.160331	+	0.987063i	$0.448744\pi$
$240$	0	0
$241$	10.8008	0.695742	0.347871	−	0.937542i	$-0.386905\pi$
$241$	10.8008	0.695742	0.347871	+	0.937542i	$0.386905\pi$
$242$	0	0
$243$	20.9910	1.34657
$244$	0	0
$245$	8.93184	0.570634
$246$	0	0
$247$	2.82479	0.179737
$248$	0	0
$249$	35.8422	2.27141
$250$	0	0
$251$	−7.12858	−0.449952	−0.224976	−	0.974364i	$-0.572230\pi$
$251$	−7.12858	−0.449952	−0.224976	+	0.974364i	$0.572230\pi$
$252$	0	0
$253$	−13.9525	−0.877186
$254$	0	0
$255$	−19.9621	−1.25008
$256$	0	0
$257$	−8.56506	−0.534274	−0.267137	−	0.963659i	$-0.586078\pi$
$257$	−8.56506	−0.534274	−0.267137	+	0.963659i	$0.586078\pi$
$258$	0	0
$259$	10.9641	0.681273
$260$	0	0
$261$	2.61930	0.162131
$262$	0	0
$263$	−20.7307	−1.27831	−0.639154	−	0.769079i	$-0.720715\pi$
$263$	−20.7307	−1.27831	−0.639154	+	0.769079i	$0.720715\pi$
$264$	0	0
$265$	21.9727	1.34977
$266$	0	0
$267$	−22.4612	−1.37460
$268$	0	0
$269$	12.5805	0.767044	0.383522	−	0.923532i	$-0.374711\pi$
$269$	12.5805	0.767044	0.383522	+	0.923532i	$0.374711\pi$
$270$	0	0
$271$	−1.46000	−0.0886886	−0.0443443	−	0.999016i	$-0.514120\pi$
$271$	−1.46000	−0.0886886	−0.0443443	+	0.999016i	$0.514120\pi$
$272$	0	0
$273$	−47.0808	−2.84946
$274$	0	0
$275$	0.790713	0.0476818
$276$	0	0
$277$	8.15173	0.489790	0.244895	−	0.969550i	$-0.421247\pi$
$277$	8.15173	0.489790	0.244895	+	0.969550i	$0.421247\pi$
$278$	0	0
$279$	22.9177	1.37205
$280$	0	0
$281$	11.6659	0.695929	0.347965	−	0.937508i	$-0.386873\pi$
$281$	11.6659	0.695929	0.347965	+	0.937508i	$0.386873\pi$
$282$	0	0
$283$	−25.7202	−1.52891	−0.764454	−	0.644679i	$-0.776991\pi$
$283$	−25.7202	−1.52891	−0.764454	+	0.644679i	$0.776991\pi$
$284$	0	0
$285$	2.46956	0.146284
$286$	0	0
$287$	20.1676	1.19045
$288$	0	0
$289$	−2.38936	−0.140551
$290$	0	0
$291$	37.3820	2.19137
$292$	0	0
$293$	−4.92564	−0.287759	−0.143879	−	0.989595i	$-0.545958\pi$
$293$	−4.92564	−0.287759	−0.143879	+	0.989595i	$0.545958\pi$
$294$	0	0
$295$	8.95731	0.521515
$296$	0	0
$297$	−4.87322	−0.282773
$298$	0	0
$299$	−15.4345	−0.892602
$300$	0	0
$301$	17.9540	1.03485
$302$	0	0
$303$	−44.7860	−2.57289
$304$	0	0
$305$	−2.18093	−0.124880
$306$	0	0
$307$	23.0307	1.31443	0.657217	−	0.753701i	$-0.271734\pi$
$307$	23.0307	1.31443	0.657217	+	0.753701i	$0.271734\pi$
$308$	0	0
$309$	16.3654	0.930996
$310$	0	0
$311$	−2.04509	−0.115967	−0.0579833	−	0.998318i	$-0.518467\pi$
$311$	−2.04509	−0.115967	−0.0579833	+	0.998318i	$0.518467\pi$
$312$	0	0
$313$	28.0534	1.58567	0.792836	−	0.609435i	$-0.208604\pi$
$313$	28.0534	1.58567	0.792836	+	0.609435i	$0.208604\pi$
$314$	0	0
$315$	−19.1859	−1.08100
$316$	0	0
$317$	8.25497	0.463646	0.231823	−	0.972758i	$-0.425531\pi$
$317$	8.25497	0.463646	0.231823	+	0.972758i	$0.425531\pi$
$318$	0	0
$319$	−5.40004	−0.302344
$320$	0	0
$321$	−25.7576	−1.43765
$322$	0	0
$323$	−1.80752	−0.100573
$324$	0	0
$325$	0.874702	0.0485197
$326$	0	0
$327$	−30.3387	−1.67774
$328$	0	0
$329$	−19.6986	−1.08602
$330$	0	0
$331$	1.32107	0.0726124	0.0363062	−	0.999341i	$-0.488441\pi$
$331$	1.32107	0.0726124	0.0363062	+	0.999341i	$0.488441\pi$
$332$	0	0
$333$	−8.63759	−0.473337
$334$	0	0
$335$	21.3783	1.16802
$336$	0	0
$337$	−31.1264	−1.69556	−0.847781	−	0.530347i	$-0.822062\pi$
$337$	−31.1264	−1.69556	−0.847781	+	0.530347i	$0.822062\pi$
$338$	0	0
$339$	−43.5069	−2.36297
$340$	0	0
$341$	−47.2479	−2.55862
$342$	0	0
$343$	9.79402	0.528827
$344$	0	0
$345$	−13.4936	−0.726470
$346$	0	0
$347$	11.8885	0.638206	0.319103	−	0.947720i	$-0.396618\pi$
$347$	11.8885	0.638206	0.319103	+	0.947720i	$0.396618\pi$
$348$	0	0
$349$	−4.59679	−0.246060	−0.123030	−	0.992403i	$-0.539261\pi$
$349$	−4.59679	−0.246060	−0.123030	+	0.992403i	$0.539261\pi$
$350$	0	0
$351$	−5.39085	−0.287742
$352$	0	0
$353$	−11.7446	−0.625101	−0.312551	−	0.949901i	$-0.601183\pi$
$353$	−11.7446	−0.625101	−0.312551	+	0.949901i	$0.601183\pi$
$354$	0	0
$355$	21.0468	1.11705
$356$	0	0
$357$	30.1259	1.59443
$358$	0	0
$359$	30.4829	1.60883	0.804413	−	0.594070i	$-0.202480\pi$
$359$	30.4829	1.60883	0.804413	+	0.594070i	$0.202480\pi$
$360$	0	0
$361$	−18.7764	−0.988231
$362$	0	0
$363$	−43.0493	−2.25950
$364$	0	0
$365$	−24.1071	−1.26182
$366$	0	0
$367$	−11.9597	−0.624292	−0.312146	−	0.950034i	$-0.601048\pi$
$367$	−11.9597	−0.624292	−0.312146	+	0.950034i	$0.601048\pi$
$368$	0	0
$369$	−15.8882	−0.827106
$370$	0	0
$371$	−33.1602	−1.72159
$372$	0	0
$373$	−32.7082	−1.69357	−0.846784	−	0.531937i	$-0.821464\pi$
$373$	−32.7082	−1.69357	−0.846784	+	0.531937i	$0.821464\pi$
$374$	0	0
$375$	26.8768	1.38791
$376$	0	0
$377$	−5.97363	−0.307657
$378$	0	0
$379$	13.1273	0.674302	0.337151	−	0.941451i	$-0.390537\pi$
$379$	13.1273	0.674302	0.337151	+	0.941451i	$0.390537\pi$
$380$	0	0
$381$	−12.7163	−0.651476
$382$	0	0
$383$	−16.9879	−0.868040	−0.434020	−	0.900903i	$-0.642905\pi$
$383$	−16.9879	−0.868040	−0.434020	+	0.900903i	$0.642905\pi$
$384$	0	0
$385$	39.5541	2.01587
$386$	0	0
$387$	−14.1443	−0.718997
$388$	0	0
$389$	13.8978	0.704647	0.352324	−	0.935878i	$-0.385392\pi$
$389$	13.8978	0.704647	0.352324	+	0.935878i	$0.385392\pi$
$390$	0	0
$391$	9.87620	0.499461
$392$	0	0
$393$	−32.7639	−1.65272
$394$	0	0
$395$	7.83696	0.394320
$396$	0	0
$397$	20.0418	1.00587	0.502936	−	0.864324i	$-0.332253\pi$
$397$	20.0418	1.00587	0.502936	+	0.864324i	$0.332253\pi$
$398$	0	0
$399$	−3.72695	−0.186581
$400$	0	0
$401$	3.70432	0.184985	0.0924924	−	0.995713i	$-0.470517\pi$
$401$	3.70432	0.184985	0.0924924	+	0.995713i	$0.470517\pi$
$402$	0	0
$403$	−52.2665	−2.60358
$404$	0	0
$405$	−22.0246	−1.09441
$406$	0	0
$407$	17.8075	0.882686
$408$	0	0
$409$	11.7830	0.582632	0.291316	−	0.956627i	$-0.405907\pi$
$409$	11.7830	0.582632	0.291316	+	0.956627i	$0.405907\pi$
$410$	0	0
$411$	−21.4667	−1.05888
$412$	0	0
$413$	−13.5180	−0.665176
$414$	0	0
$415$	−33.3107	−1.63516
$416$	0	0
$417$	8.05050	0.394235
$418$	0	0
$419$	11.7051	0.571830	0.285915	−	0.958255i	$-0.407702\pi$
$419$	11.7051	0.571830	0.285915	+	0.958255i	$0.407702\pi$
$420$	0	0
$421$	−32.0227	−1.56069	−0.780345	−	0.625349i	$-0.784957\pi$
$421$	−32.0227	−1.56069	−0.780345	+	0.625349i	$0.784957\pi$
$422$	0	0
$423$	15.5187	0.754547
$424$	0	0
$425$	−0.559702	−0.0271495
$426$	0	0
$427$	3.29136	0.159280
$428$	0	0
$429$	−76.4673	−3.69188
$430$	0	0
$431$	−14.6117	−0.703821	−0.351911	−	0.936034i	$-0.614468\pi$
$431$	−14.6117	−0.703821	−0.351911	+	0.936034i	$0.614468\pi$
$432$	0	0
$433$	30.7351	1.47703	0.738517	−	0.674235i	$-0.235527\pi$
$433$	30.7351	1.47703	0.738517	+	0.674235i	$0.235527\pi$
$434$	0	0
$435$	−5.22242	−0.250396
$436$	0	0
$437$	−1.22181	−0.0584471
$438$	0	0
$439$	1.78495	0.0851908	0.0425954	−	0.999092i	$-0.486437\pi$
$439$	1.78495	0.0851908	0.0425954	+	0.999092i	$0.486437\pi$
$440$	0	0
$441$	10.6193	0.505681
$442$	0	0
$443$	20.5205	0.974958	0.487479	−	0.873135i	$-0.337916\pi$
$443$	20.5205	0.974958	0.487479	+	0.873135i	$0.337916\pi$
$444$	0	0
$445$	20.8748	0.989562
$446$	0	0
$447$	22.2747	1.05356
$448$	0	0
$449$	−28.4212	−1.34128	−0.670639	−	0.741784i	$-0.733980\pi$
$449$	−28.4212	−1.34128	−0.670639	+	0.741784i	$0.733980\pi$
$450$	0	0
$451$	32.7556	1.54240
$452$	0	0
$453$	40.2493	1.89108
$454$	0	0
$455$	43.7556	2.05129
$456$	0	0
$457$	−27.0341	−1.26460	−0.632302	−	0.774722i	$-0.717890\pi$
$457$	−27.0341	−1.26460	−0.632302	+	0.774722i	$0.717890\pi$
$458$	0	0
$459$	3.44948	0.161008
$460$	0	0
$461$	13.0932	0.609811	0.304905	−	0.952383i	$-0.401375\pi$
$461$	13.0932	0.609811	0.304905	+	0.952383i	$0.401375\pi$
$462$	0	0
$463$	−7.71239	−0.358425	−0.179213	−	0.983810i	$-0.557355\pi$
$463$	−7.71239	−0.358425	−0.179213	+	0.983810i	$0.557355\pi$
$464$	0	0
$465$	−45.6938	−2.11900
$466$	0	0
$467$	−14.5811	−0.674732	−0.337366	−	0.941373i	$-0.609536\pi$
$467$	−14.5811	−0.674732	−0.337366	+	0.941373i	$0.609536\pi$
$468$	0	0
$469$	−32.2632	−1.48978
$470$	0	0
$471$	−36.8563	−1.69825
$472$	0	0
$473$	29.1604	1.34080
$474$	0	0
$475$	0.0692421	0.00317705
$476$	0	0
$477$	26.1240	1.19613
$478$	0	0
$479$	9.49237	0.433717	0.216859	−	0.976203i	$-0.430419\pi$
$479$	9.49237	0.433717	0.216859	+	0.976203i	$0.430419\pi$
$480$	0	0
$481$	19.6990	0.898198
$482$	0	0
$483$	20.3639	0.926590
$484$	0	0
$485$	−34.7418	−1.57754
$486$	0	0
$487$	27.0382	1.22522	0.612610	−	0.790385i	$-0.290120\pi$
$487$	27.0382	1.22522	0.612610	+	0.790385i	$0.290120\pi$
$488$	0	0
$489$	10.2608	0.464008
$490$	0	0
$491$	20.1812	0.910766	0.455383	−	0.890296i	$-0.349502\pi$
$491$	20.1812	0.910766	0.455383	+	0.890296i	$0.349502\pi$
$492$	0	0
$493$	3.82239	0.172152
$494$	0	0
$495$	−31.1611	−1.40059
$496$	0	0
$497$	−31.7629	−1.42476
$498$	0	0
$499$	−13.9525	−0.624600	−0.312300	−	0.949984i	$-0.601099\pi$
$499$	−13.9525	−0.624600	−0.312300	+	0.949984i	$0.601099\pi$
$500$	0	0
$501$	−17.2075	−0.768773
$502$	0	0
$503$	33.7556	1.50509	0.752544	−	0.658542i	$-0.228827\pi$
$503$	33.7556	1.50509	0.752544	+	0.658542i	$0.228827\pi$
$504$	0	0
$505$	41.6229	1.85219
$506$	0	0
$507$	−53.7731	−2.38815
$508$	0	0
$509$	−21.0788	−0.934302	−0.467151	−	0.884178i	$-0.654720\pi$
$509$	−21.0788	−0.934302	−0.467151	+	0.884178i	$0.654720\pi$
$510$	0	0
$511$	36.3813	1.60942
$512$	0	0
$513$	−0.426744	−0.0188412
$514$	0	0
$515$	−15.2096	−0.670213
$516$	0	0
$517$	−31.9939	−1.40709
$518$	0	0
$519$	11.4553	0.502834
$520$	0	0
$521$	2.58336	0.113179	0.0565896	−	0.998398i	$-0.481977\pi$
$521$	2.58336	0.113179	0.0565896	+	0.998398i	$0.481977\pi$
$522$	0	0
$523$	−42.3861	−1.85342	−0.926708	−	0.375781i	$-0.877374\pi$
$523$	−42.3861	−1.85342	−0.926708	+	0.375781i	$0.877374\pi$
$524$	0	0
$525$	−1.15406	−0.0503673
$526$	0	0
$527$	33.4441	1.45685
$528$	0	0
$529$	−16.3241	−0.709743
$530$	0	0
$531$	10.6496	0.462153
$532$	0	0
$533$	36.2349	1.56951
$534$	0	0
$535$	23.9384	1.03495
$536$	0	0
$537$	5.36635	0.231575
$538$	0	0
$539$	−21.8931	−0.943002
$540$	0	0
$541$	23.6511	1.01684	0.508420	−	0.861109i	$-0.330230\pi$
$541$	23.6511	1.01684	0.508420	+	0.861109i	$0.330230\pi$
$542$	0	0
$543$	−49.7398	−2.13454
$544$	0	0
$545$	28.1960	1.20778
$546$	0	0
$547$	37.9318	1.62185	0.810924	−	0.585152i	$-0.198965\pi$
$547$	37.9318	1.62185	0.810924	+	0.585152i	$0.198965\pi$
$548$	0	0
$549$	−2.59296	−0.110665
$550$	0	0
$551$	−0.472877	−0.0201452
$552$	0	0
$553$	−11.8272	−0.502943
$554$	0	0
$555$	17.2218	0.731025
$556$	0	0
$557$	−19.7581	−0.837177	−0.418588	−	0.908176i	$-0.637475\pi$
$557$	−19.7581	−0.837177	−0.418588	+	0.908176i	$0.637475\pi$
$558$	0	0
$559$	32.2578	1.36436
$560$	0	0
$561$	48.9297	2.06581
$562$	0	0
$563$	33.7269	1.42142	0.710710	−	0.703485i	$-0.248374\pi$
$563$	33.7269	1.42142	0.710710	+	0.703485i	$0.248374\pi$
$564$	0	0
$565$	40.4341	1.70108
$566$	0	0
$567$	33.2385	1.39588
$568$	0	0
$569$	18.8915	0.791973	0.395986	−	0.918256i	$-0.370403\pi$
$569$	18.8915	0.791973	0.395986	+	0.918256i	$0.370403\pi$
$570$	0	0
$571$	29.6544	1.24100	0.620499	−	0.784207i	$-0.286930\pi$
$571$	29.6544	1.24100	0.620499	+	0.784207i	$0.286930\pi$
$572$	0	0
$573$	−14.1161	−0.589710
$574$	0	0
$575$	−0.378336	−0.0157777
$576$	0	0
$577$	37.4965	1.56100	0.780499	−	0.625156i	$-0.214965\pi$
$577$	37.4965	1.56100	0.780499	+	0.625156i	$0.214965\pi$
$578$	0	0
$579$	28.2163	1.17263
$580$	0	0
$581$	50.2710	2.08559
$582$	0	0
$583$	−53.8579	−2.23057
$584$	0	0
$585$	−34.4711	−1.42520
$586$	0	0
$587$	−13.7202	−0.566293	−0.283147	−	0.959077i	$-0.591378\pi$
$587$	−13.7202	−0.566293	−0.283147	+	0.959077i	$0.591378\pi$
$588$	0	0
$589$	−4.13746	−0.170481
$590$	0	0
$591$	24.8591	1.02257
$592$	0	0
$593$	−16.7341	−0.687188	−0.343594	−	0.939118i	$-0.611644\pi$
$593$	−16.7341	−0.687188	−0.343594	+	0.939118i	$0.611644\pi$
$594$	0	0
$595$	−27.9982	−1.14781
$596$	0	0
$597$	8.59491	0.351766
$598$	0	0
$599$	−29.6579	−1.21179	−0.605895	−	0.795545i	$-0.707185\pi$
$599$	−29.6579	−1.21179	−0.605895	+	0.795545i	$0.707185\pi$
$600$	0	0
$601$	25.6069	1.04453	0.522263	−	0.852784i	$-0.325088\pi$
$601$	25.6069	1.04453	0.522263	+	0.852784i	$0.325088\pi$
$602$	0	0
$603$	25.4173	1.03507
$604$	0	0
$605$	40.0088	1.62659
$606$	0	0
$607$	20.0907	0.815457	0.407728	−	0.913103i	$-0.366321\pi$
$607$	20.0907	0.815457	0.407728	+	0.913103i	$0.366321\pi$
$608$	0	0
$609$	7.88144	0.319372
$610$	0	0
$611$	−35.3923	−1.43182
$612$	0	0
$613$	16.5517	0.668517	0.334258	−	0.942482i	$-0.391514\pi$
$613$	16.5517	0.668517	0.334258	+	0.942482i	$0.391514\pi$
$614$	0	0
$615$	31.6782	1.27739
$616$	0	0
$617$	−8.36154	−0.336623	−0.168311	−	0.985734i	$-0.553831\pi$
$617$	−8.36154	−0.336623	−0.168311	+	0.985734i	$0.553831\pi$
$618$	0	0
$619$	35.5252	1.42788	0.713940	−	0.700207i	$-0.246909\pi$
$619$	35.5252	1.42788	0.713940	+	0.700207i	$0.246909\pi$
$620$	0	0
$621$	2.33171	0.0935683
$622$	0	0
$623$	−31.5033	−1.26215
$624$	0	0
$625$	−24.2464	−0.969857
$626$	0	0
$627$	−6.05321	−0.241742
$628$	0	0
$629$	−12.6050	−0.502593
$630$	0	0
$631$	−22.9317	−0.912896	−0.456448	−	0.889750i	$-0.650879\pi$
$631$	−22.9317	−0.912896	−0.456448	+	0.889750i	$0.650879\pi$
$632$	0	0
$633$	−21.0702	−0.837465
$634$	0	0
$635$	11.8182	0.468990
$636$	0	0
$637$	−24.2186	−0.959574
$638$	0	0
$639$	25.0231	0.989899
$640$	0	0
$641$	−17.8455	−0.704855	−0.352428	−	0.935839i	$-0.614644\pi$
$641$	−17.8455	−0.704855	−0.352428	+	0.935839i	$0.614644\pi$
$642$	0	0
$643$	40.7805	1.60823	0.804113	−	0.594477i	$-0.202641\pi$
$643$	40.7805	1.60823	0.804113	+	0.594477i	$0.202641\pi$
$644$	0	0
$645$	28.2013	1.11042
$646$	0	0
$647$	−37.0841	−1.45793	−0.728963	−	0.684553i	$-0.759998\pi$
$647$	−37.0841	−1.45793	−0.728963	+	0.684553i	$0.759998\pi$
$648$	0	0
$649$	−21.9555	−0.861829
$650$	0	0
$651$	68.9591	2.70272
$652$	0	0
$653$	8.88294	0.347616	0.173808	−	0.984780i	$-0.444393\pi$
$653$	8.88294	0.347616	0.173808	+	0.984780i	$0.444393\pi$
$654$	0	0
$655$	30.4499	1.18977
$656$	0	0
$657$	−28.6616	−1.11819
$658$	0	0
$659$	41.8792	1.63138	0.815691	−	0.578487i	$-0.196357\pi$
$659$	41.8792	1.63138	0.815691	+	0.578487i	$0.196357\pi$
$660$	0	0
$661$	33.7611	1.31315	0.656577	−	0.754259i	$-0.272004\pi$
$661$	33.7611	1.31315	0.656577	+	0.754259i	$0.272004\pi$
$662$	0	0
$663$	54.1270	2.10212
$664$	0	0
$665$	3.46373	0.134317
$666$	0	0
$667$	2.58378	0.100044
$668$	0	0
$669$	−11.0509	−0.427254
$670$	0	0
$671$	5.34573	0.206370
$672$	0	0
$673$	3.03493	0.116988	0.0584939	−	0.998288i	$-0.481370\pi$
$673$	3.03493	0.116988	0.0584939	+	0.998288i	$0.481370\pi$
$674$	0	0
$675$	−0.132142	−0.00508615
$676$	0	0
$677$	−32.7020	−1.25684	−0.628421	−	0.777874i	$-0.716298\pi$
$677$	−32.7020	−1.25684	−0.628421	+	0.777874i	$0.716298\pi$
$678$	0	0
$679$	52.4308	2.01211
$680$	0	0
$681$	65.4554	2.50826
$682$	0	0
$683$	−34.1609	−1.30713	−0.653566	−	0.756870i	$-0.726728\pi$
$683$	−34.1609	−1.30713	−0.653566	+	0.756870i	$0.726728\pi$
$684$	0	0
$685$	19.9506	0.762272
$686$	0	0
$687$	−54.2328	−2.06911
$688$	0	0
$689$	−59.5787	−2.26977
$690$	0	0
$691$	−41.3376	−1.57256	−0.786278	−	0.617873i	$-0.787995\pi$
$691$	−41.3376	−1.57256	−0.786278	+	0.617873i	$0.787995\pi$
$692$	0	0
$693$	47.0270	1.78641
$694$	0	0
$695$	−7.48191	−0.283805
$696$	0	0
$697$	−23.1859	−0.878227
$698$	0	0
$699$	12.1996	0.461433
$700$	0	0
$701$	−30.0755	−1.13594	−0.567968	−	0.823051i	$-0.692270\pi$
$701$	−30.0755	−1.13594	−0.567968	+	0.823051i	$0.692270\pi$
$702$	0	0
$703$	1.55939	0.0588135
$704$	0	0
$705$	−30.9416	−1.16533
$706$	0	0
$707$	−62.8153	−2.36241
$708$	0	0
$709$	−7.13937	−0.268125	−0.134062	−	0.990973i	$-0.542802\pi$
$709$	−7.13937	−0.268125	−0.134062	+	0.990973i	$0.542802\pi$
$710$	0	0
$711$	9.31758	0.349436
$712$	0	0
$713$	22.6069	0.846635
$714$	0	0
$715$	71.0666	2.65774
$716$	0	0
$717$	−11.7513	−0.438862
$718$	0	0
$719$	14.0354	0.523431	0.261715	−	0.965145i	$-0.415712\pi$
$719$	14.0354	0.523431	0.261715	+	0.965145i	$0.415712\pi$
$720$	0	0
$721$	22.9536	0.854836
$722$	0	0
$723$	−25.6034	−0.952202
$724$	0	0
$725$	−0.146427	−0.00543817
$726$	0	0
$727$	−15.9732	−0.592414	−0.296207	−	0.955124i	$-0.595722\pi$
$727$	−15.9732	−0.592414	−0.296207	+	0.955124i	$0.595722\pi$
$728$	0	0
$729$	−19.7679	−0.732143
$730$	0	0
$731$	−20.6410	−0.763436
$732$	0	0
$733$	18.7614	0.692968	0.346484	−	0.938056i	$-0.387376\pi$
$733$	18.7614	0.692968	0.346484	+	0.938056i	$0.387376\pi$
$734$	0	0
$735$	−21.1730	−0.780977
$736$	0	0
$737$	−52.4010	−1.93022
$738$	0	0
$739$	−21.0336	−0.773733	−0.386867	−	0.922136i	$-0.626443\pi$
$739$	−21.0336	−0.773733	−0.386867	+	0.922136i	$0.626443\pi$
$740$	0	0
$741$	−6.69618	−0.245990
$742$	0	0
$743$	−6.93960	−0.254589	−0.127295	−	0.991865i	$-0.540629\pi$
$743$	−6.93960	−0.254589	−0.127295	+	0.991865i	$0.540629\pi$
$744$	0	0
$745$	−20.7015	−0.758443
$746$	0	0
$747$	−39.6040	−1.44904
$748$	0	0
$749$	−36.1268	−1.32004
$750$	0	0
$751$	−16.7873	−0.612579	−0.306290	−	0.951938i	$-0.599088\pi$
$751$	−16.7873	−0.612579	−0.306290	+	0.951938i	$0.599088\pi$
$752$	0	0
$753$	16.8984	0.615811
$754$	0	0
$755$	−37.4066	−1.36136
$756$	0	0
$757$	28.8350	1.04803	0.524013	−	0.851710i	$-0.324434\pi$
$757$	28.8350	1.04803	0.524013	+	0.851710i	$0.324434\pi$
$758$	0	0
$759$	33.0745	1.20053
$760$	0	0
$761$	−9.06774	−0.328705	−0.164353	−	0.986402i	$-0.552554\pi$
$761$	−9.06774	−0.328705	−0.164353	+	0.986402i	$0.552554\pi$
$762$	0	0
$763$	−42.5521	−1.54049
$764$	0	0
$765$	22.0572	0.797482
$766$	0	0
$767$	−24.2876	−0.876975
$768$	0	0
$769$	40.1101	1.44641	0.723204	−	0.690634i	$-0.242669\pi$
$769$	40.1101	1.44641	0.723204	+	0.690634i	$0.242669\pi$
$770$	0	0
$771$	20.3035	0.731214
$772$	0	0
$773$	−18.9276	−0.680777	−0.340389	−	0.940285i	$-0.610559\pi$
$773$	−18.9276	−0.680777	−0.340389	+	0.940285i	$0.610559\pi$
$774$	0	0
$775$	−1.28117	−0.0460211
$776$	0	0
$777$	−25.9904	−0.932400
$778$	0	0
$779$	2.86838	0.102770
$780$	0	0
$781$	−51.5884	−1.84598
$782$	0	0
$783$	0.902442	0.0322507
$784$	0	0
$785$	34.2532	1.22255
$786$	0	0
$787$	39.0743	1.39285	0.696425	−	0.717630i	$-0.254773\pi$
$787$	39.0743	1.39285	0.696425	+	0.717630i	$0.254773\pi$
$788$	0	0
$789$	49.1422	1.74951
$790$	0	0
$791$	−61.0213	−2.16967
$792$	0	0
$793$	5.91355	0.209996
$794$	0	0
$795$	−52.0865	−1.84732
$796$	0	0
$797$	−0.279490	−0.00990006	−0.00495003	−	0.999988i	$-0.501576\pi$
$797$	−0.279490	−0.00990006	−0.00495003	+	0.999988i	$0.501576\pi$
$798$	0	0
$799$	22.6467	0.801183
$800$	0	0
$801$	24.8186	0.876924
$802$	0	0
$803$	59.0896	2.08523
$804$	0	0
$805$	−18.9256	−0.667041
$806$	0	0
$807$	−29.8221	−1.04979
$808$	0	0
$809$	31.5474	1.10915	0.554574	−	0.832135i	$-0.312881\pi$
$809$	31.5474	1.10915	0.554574	+	0.832135i	$0.312881\pi$
$810$	0	0
$811$	−8.26874	−0.290355	−0.145177	−	0.989406i	$-0.546375\pi$
$811$	−8.26874	−0.290355	−0.145177	+	0.989406i	$0.546375\pi$
$812$	0	0
$813$	3.46094	0.121380
$814$	0	0
$815$	−9.53607	−0.334034
$816$	0	0
$817$	2.55355	0.0893375
$818$	0	0
$819$	52.0222	1.81780
$820$	0	0
$821$	−52.7285	−1.84024	−0.920119	−	0.391640i	$-0.871908\pi$
$821$	−52.7285	−1.84024	−0.920119	+	0.391640i	$0.871908\pi$
$822$	0	0
$823$	−40.0895	−1.39743	−0.698716	−	0.715399i	$-0.746245\pi$
$823$	−40.0895	−1.39743	−0.698716	+	0.715399i	$0.746245\pi$
$824$	0	0
$825$	−1.87439	−0.0652579
$826$	0	0
$827$	−30.6526	−1.06590	−0.532948	−	0.846148i	$-0.678916\pi$
$827$	−30.6526	−1.06590	−0.532948	+	0.846148i	$0.678916\pi$
$828$	0	0
$829$	20.7384	0.720275	0.360137	−	0.932899i	$-0.382730\pi$
$829$	20.7384	0.720275	0.360137	+	0.932899i	$0.382730\pi$
$830$	0	0
$831$	−19.3237	−0.670333
$832$	0	0
$833$	15.4969	0.536936
$834$	0	0
$835$	15.9921	0.553431
$836$	0	0
$837$	7.89596	0.272924
$838$	0	0
$839$	1.46765	0.0506690	0.0253345	−	0.999679i	$-0.491935\pi$
$839$	1.46765	0.0506690	0.0253345	+	0.999679i	$0.491935\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−27.6541	−0.952458
$844$	0	0
$845$	49.9752	1.71920
$846$	0	0
$847$	−60.3795	−2.07467
$848$	0	0
$849$	60.9699	2.09248
$850$	0	0
$851$	−8.52044	−0.292077
$852$	0	0
$853$	6.49304	0.222317	0.111159	−	0.993803i	$-0.464544\pi$
$853$	6.49304	0.222317	0.111159	+	0.993803i	$0.464544\pi$
$854$	0	0
$855$	−2.72876	−0.0933215
$856$	0	0
$857$	29.1173	0.994627	0.497313	−	0.867571i	$-0.334320\pi$
$857$	29.1173	0.994627	0.497313	+	0.867571i	$0.334320\pi$
$858$	0	0
$859$	48.0478	1.63937	0.819685	−	0.572814i	$-0.194148\pi$
$859$	48.0478	1.63937	0.819685	+	0.572814i	$0.194148\pi$
$860$	0	0
$861$	−47.8073	−1.62927
$862$	0	0
$863$	1.54466	0.0525810	0.0262905	−	0.999654i	$-0.491631\pi$
$863$	1.54466	0.0525810	0.0262905	+	0.999654i	$0.491631\pi$
$864$	0	0
$865$	−10.6463	−0.361984
$866$	0	0
$867$	5.66399	0.192359
$868$	0	0
$869$	−19.2094	−0.651634
$870$	0	0
$871$	−57.9671	−1.96414
$872$	0	0
$873$	−41.3055	−1.39798
$874$	0	0
$875$	37.6965	1.27437
$876$	0	0
$877$	−30.6343	−1.03445	−0.517224	−	0.855850i	$-0.673035\pi$
$877$	−30.6343	−1.03445	−0.517224	+	0.855850i	$0.673035\pi$
$878$	0	0
$879$	11.6763	0.393831
$880$	0	0
$881$	−15.5459	−0.523754	−0.261877	−	0.965101i	$-0.584342\pi$
$881$	−15.5459	−0.523754	−0.261877	+	0.965101i	$0.584342\pi$
$882$	0	0
$883$	24.5960	0.827723	0.413861	−	0.910340i	$-0.364180\pi$
$883$	24.5960	0.827723	0.413861	+	0.910340i	$0.364180\pi$
$884$	0	0
$885$	−21.2334	−0.713752
$886$	0	0
$887$	30.4582	1.02269	0.511343	−	0.859377i	$-0.329148\pi$
$887$	30.4582	1.02269	0.511343	+	0.859377i	$0.329148\pi$
$888$	0	0
$889$	−17.8354	−0.598182
$890$	0	0
$891$	53.9850	1.80857
$892$	0	0
$893$	−2.80168	−0.0937547
$894$	0	0
$895$	−4.98733	−0.166708
$896$	0	0
$897$	36.5877	1.22163
$898$	0	0
$899$	8.74955	0.291814
$900$	0	0
$901$	38.1231	1.27006
$902$	0	0
$903$	−42.5601	−1.41631
$904$	0	0
$905$	46.2268	1.53663
$906$	0	0
$907$	−45.3578	−1.50608	−0.753040	−	0.657975i	$-0.771413\pi$
$907$	−45.3578	−1.50608	−0.753040	+	0.657975i	$0.771413\pi$
$908$	0	0
$909$	49.4865	1.64136
$910$	0	0
$911$	−2.11098	−0.0699398	−0.0349699	−	0.999388i	$-0.511134\pi$
$911$	−2.11098	−0.0699398	−0.0349699	+	0.999388i	$0.511134\pi$
$912$	0	0
$913$	81.6488	2.70218
$914$	0	0
$915$	5.16990	0.170912
$916$	0	0
$917$	−45.9536	−1.51752
$918$	0	0
$919$	−23.4151	−0.772393	−0.386197	−	0.922416i	$-0.626211\pi$
$919$	−23.4151	−0.772393	−0.386197	+	0.922416i	$0.626211\pi$
$920$	0	0
$921$	−54.5945	−1.79895
$922$	0	0
$923$	−57.0681	−1.87842
$924$	0	0
$925$	0.482868	0.0158766
$926$	0	0
$927$	−18.0830	−0.593925
$928$	0	0
$929$	11.0888	0.363811	0.181906	−	0.983316i	$-0.441773\pi$
$929$	11.0888	0.363811	0.181906	+	0.983316i	$0.441773\pi$
$930$	0	0
$931$	−1.91716	−0.0628324
$932$	0	0
$933$	4.84791	0.158713
$934$	0	0
$935$	−45.4739	−1.48716
$936$	0	0
$937$	61.0939	1.99585	0.997925	−	0.0643912i	$-0.0205106\pi$
$937$	61.0939	1.99585	0.997925	+	0.0643912i	$0.0205106\pi$
$938$	0	0
$939$	−66.5008	−2.17017
$940$	0	0
$941$	33.0250	1.07659	0.538293	−	0.842758i	$-0.319069\pi$
$941$	33.0250	1.07659	0.538293	+	0.842758i	$0.319069\pi$
$942$	0	0
$943$	−15.6727	−0.510373
$944$	0	0
$945$	−6.61020	−0.215030
$946$	0	0
$947$	−18.4213	−0.598613	−0.299307	−	0.954157i	$-0.596755\pi$
$947$	−18.4213	−0.598613	−0.299307	+	0.954157i	$0.596755\pi$
$948$	0	0
$949$	65.3660	2.12187
$950$	0	0
$951$	−19.5685	−0.634551
$952$	0	0
$953$	32.8421	1.06386	0.531931	−	0.846788i	$-0.321467\pi$
$953$	32.8421	1.06386	0.531931	+	0.846788i	$0.321467\pi$
$954$	0	0
$955$	13.1191	0.424526
$956$	0	0
$957$	12.8008	0.413792
$958$	0	0
$959$	−30.1085	−0.972254
$960$	0	0
$961$	45.5546	1.46950
$962$	0	0
$963$	28.4610	0.917144
$964$	0	0
$965$	−26.2234	−0.844162
$966$	0	0
$967$	−8.73585	−0.280926	−0.140463	−	0.990086i	$-0.544859\pi$
$967$	−8.73585	−0.280926	−0.140463	+	0.990086i	$0.544859\pi$
$968$	0	0
$969$	4.28474	0.137646
$970$	0	0
$971$	−22.2046	−0.712581	−0.356290	−	0.934375i	$-0.615959\pi$
$971$	−22.2046	−0.712581	−0.356290	+	0.934375i	$0.615959\pi$
$972$	0	0
$973$	11.2914	0.361984
$974$	0	0
$975$	−2.07349	−0.0664047
$976$	0	0
$977$	16.8522	0.539151	0.269575	−	0.962979i	$-0.413117\pi$
$977$	16.8522	0.539151	0.269575	+	0.962979i	$0.413117\pi$
$978$	0	0
$979$	−51.1669	−1.63530
$980$	0	0
$981$	33.5229	1.07031
$982$	0	0
$983$	1.01976	0.0325252	0.0162626	−	0.999868i	$-0.494823\pi$
$983$	1.01976	0.0325252	0.0162626	+	0.999868i	$0.494823\pi$
$984$	0	0
$985$	−23.1034	−0.736135
$986$	0	0
$987$	46.6956	1.48634
$988$	0	0
$989$	−13.9525	−0.443664
$990$	0	0
$991$	10.8543	0.344799	0.172399	−	0.985027i	$-0.444848\pi$
$991$	10.8543	0.344799	0.172399	+	0.985027i	$0.444848\pi$
$992$	0	0
$993$	−3.13160	−0.0993783
$994$	0	0
$995$	−7.98786	−0.253232
$996$	0	0
$997$	−11.8060	−0.373900	−0.186950	−	0.982369i	$-0.559860\pi$
$997$	−11.8060	−0.373900	−0.186950	+	0.982369i	$0.559860\pi$
$998$	0	0
$999$	−2.97595	−0.0941550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.ba.1.2		5
4.3	odd	2		1856.2.a.bb.1.4		5
8.3	odd	2		928.2.a.g.1.2	✓	5
8.5	even	2		928.2.a.h.1.4	yes	5
24.5	odd	2		8352.2.a.bh.1.4		5
24.11	even	2		8352.2.a.bg.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.g.1.2	✓	5	8.3	odd	2
928.2.a.h.1.4	yes	5	8.5	even	2
1856.2.a.ba.1.2		5	1.1	even	1	trivial
1856.2.a.bb.1.4		5	4.3	odd	2
8352.2.a.bg.1.4		5	24.11	even	2
8352.2.a.bh.1.4		5	24.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37051 q^{3} +2.20308 q^{5} -3.32479 q^{7} +2.61930 q^{9} +O(q^{10})\)	\(q-2.37051 q^{3} +2.20308 q^{5} -3.32479 q^{7} +2.61930 q^{9} -5.40004 q^{11} -5.97363 q^{13} -5.22242 q^{15} +3.82239 q^{17} -0.472877 q^{19} +7.88144 q^{21} +2.58378 q^{23} -0.146427 q^{25} +0.902442 q^{27} +1.00000 q^{29} +8.74955 q^{31} +12.8008 q^{33} -7.32479 q^{35} -3.29767 q^{37} +14.1605 q^{39} -6.06581 q^{41} -5.40004 q^{43} +5.77054 q^{45} +5.92476 q^{47} +4.05425 q^{49} -9.06100 q^{51} +9.97363 q^{53} -11.8967 q^{55} +1.12096 q^{57} +4.06581 q^{59} -0.989943 q^{61} -8.70864 q^{63} -13.1604 q^{65} +9.70383 q^{67} -6.12487 q^{69} +9.55334 q^{71} -10.9424 q^{73} +0.347107 q^{75} +17.9540 q^{77} +3.55727 q^{79} -9.99716 q^{81} -15.1201 q^{83} +8.42103 q^{85} -2.37051 q^{87} +9.47528 q^{89} +19.8611 q^{91} -20.7409 q^{93} -1.04179 q^{95} -15.7696 q^{97} -14.1443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9	\( 5 q - 4 q^{3} + 2 q^{5} + 4 q^{7} + 9 q^{9} - 8 q^{11} + 6 q^{13} + 6 q^{15} + 6 q^{17} - 6 q^{19} + 4 q^{21} + 8 q^{23} + 7 q^{25} - 22 q^{27} + 5 q^{29} + 8 q^{31} - 16 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} + 28 q^{47} + 13 q^{49} - 24 q^{51} + 14 q^{53} + 10 q^{55} + 20 q^{57} - 16 q^{59} + 18 q^{61} + 20 q^{63} - 8 q^{65} + 28 q^{69} - 4 q^{71} + 2 q^{73} + 6 q^{75} + 4 q^{77} - 12 q^{79} + 9 q^{81} - 32 q^{83} + 32 q^{85} - 4 q^{87} + 30 q^{89} + 40 q^{91} - 4 q^{93} + 4 q^{95} + 6 q^{97} + 22 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 + 2 * q^5 + 4 * q^7 + 9 * q^9 - 8 * q^11 + 6 * q^13 + 6 * q^15 + 6 * q^17 - 6 * q^19 + 4 * q^21 + 8 * q^23 + 7 * q^25 - 22 * q^27 + 5 * q^29 + 8 * q^31 - 16 * q^35 + 14 * q^37 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 + 28 * q^47 + 13 * q^49 - 24 * q^51 + 14 * q^53 + 10 * q^55 + 20 * q^57 - 16 * q^59 + 18 * q^61 + 20 * q^63 - 8 * q^65 + 28 * q^69 - 4 * q^71 + 2 * q^73 + 6 * q^75 + 4 * q^77 - 12 * q^79 + 9 * q^81 - 32 * q^83 + 32 * q^85 - 4 * q^87 + 30 * q^89 + 40 * q^91 - 4 * q^93 + 4 * q^95 + 6 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more