LMFDB - Embedded newform 2112.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.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:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1056)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	2112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.72161 q^{5} -2.92520 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.72161 q^{5} -2.92520 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.72161 q^{13} -3.72161 q^{15} -2.64681 q^{17} -4.64681 q^{19} -2.92520 q^{21} +5.72161 q^{23} +8.85039 q^{25} +1.00000 q^{27} +2.64681 q^{29} +9.29362 q^{31} +1.00000 q^{33} +10.8864 q^{35} +11.2936 q^{37} -3.72161 q^{39} +4.79641 q^{41} +1.20359 q^{43} -3.72161 q^{45} +2.27839 q^{47} +1.55678 q^{49} -2.64681 q^{51} +1.87122 q^{53} -3.72161 q^{55} -4.64681 q^{57} -7.44322 q^{59} -9.57201 q^{61} -2.92520 q^{63} +13.8504 q^{65} -5.59283 q^{67} +5.72161 q^{69} +13.7216 q^{71} +7.59283 q^{73} +8.85039 q^{75} -2.92520 q^{77} -14.3684 q^{79} +1.00000 q^{81} -8.00000 q^{83} +9.85039 q^{85} +2.64681 q^{87} +2.00000 q^{89} +10.8864 q^{91} +9.29362 q^{93} +17.2936 q^{95} +9.44322 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 4 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 4 q^{7} + 3 q^{9} + 3 q^{11} + 8 q^{17} + 2 q^{19} - 4 q^{21} + 6 q^{23} + 17 q^{25} + 3 q^{27} - 8 q^{29} - 4 q^{31} + 3 q^{33} - 12 q^{35} + 2 q^{37} + 8 q^{41} + 10 q^{43} + 18 q^{47} + 27 q^{49} + 8 q^{51} + 4 q^{53} + 2 q^{57} - 8 q^{61} - 4 q^{63} + 32 q^{65} - 4 q^{67} + 6 q^{69} + 30 q^{71} + 10 q^{73} + 17 q^{75} - 4 q^{77} - 16 q^{79} + 3 q^{81} - 24 q^{83} + 20 q^{85} - 8 q^{87} + 6 q^{89} - 12 q^{91} - 4 q^{93} + 20 q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 4 * q^7 + 3 * q^9 + 3 * q^11 + 8 * q^17 + 2 * q^19 - 4 * q^21 + 6 * q^23 + 17 * q^25 + 3 * q^27 - 8 * q^29 - 4 * q^31 + 3 * q^33 - 12 * q^35 + 2 * q^37 + 8 * q^41 + 10 * q^43 + 18 * q^47 + 27 * q^49 + 8 * q^51 + 4 * q^53 + 2 * q^57 - 8 * q^61 - 4 * q^63 + 32 * q^65 - 4 * q^67 + 6 * q^69 + 30 * q^71 + 10 * q^73 + 17 * q^75 - 4 * q^77 - 16 * q^79 + 3 * q^81 - 24 * q^83 + 20 * q^85 - 8 * q^87 + 6 * q^89 - 12 * q^91 - 4 * q^93 + 20 * q^95 + 6 * q^97 + 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.72161	−1.66436	−0.832178	−	0.554509i	$-0.812906\pi$
$5$	−3.72161	−1.66436	−0.832178	+	0.554509i	$0.812906\pi$
$6$	0	0
$7$	−2.92520	−1.10562	−0.552810	−	0.833307i	$-0.686445\pi$
$7$	−2.92520	−1.10562	−0.552810	+	0.833307i	$0.686445\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.72161	−1.03219	−0.516095	−	0.856532i	$-0.672615\pi$
$13$	−3.72161	−1.03219	−0.516095	+	0.856532i	$0.672615\pi$
$14$	0	0
$15$	−3.72161	−0.960916
$16$	0	0
$17$	−2.64681	−0.641945	−0.320973	−	0.947088i	$-0.604010\pi$
$17$	−2.64681	−0.641945	−0.320973	+	0.947088i	$0.604010\pi$
$18$	0	0
$19$	−4.64681	−1.06605	−0.533025	−	0.846099i	$-0.678945\pi$
$19$	−4.64681	−1.06605	−0.533025	+	0.846099i	$0.678945\pi$
$20$	0	0
$21$	−2.92520	−0.638330
$22$	0	0
$23$	5.72161	1.19304	0.596519	−	0.802599i	$-0.296550\pi$
$23$	5.72161	1.19304	0.596519	+	0.802599i	$0.296550\pi$
$24$	0	0
$25$	8.85039	1.77008
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.64681	0.491500	0.245750	−	0.969333i	$-0.420966\pi$
$29$	2.64681	0.491500	0.245750	+	0.969333i	$0.420966\pi$
$30$	0	0
$31$	9.29362	1.66918	0.834591	−	0.550869i	$-0.185704\pi$
$31$	9.29362	1.66918	0.834591	+	0.550869i	$0.185704\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	10.8864	1.84015
$36$	0	0
$37$	11.2936	1.85666	0.928330	−	0.371758i	$-0.121245\pi$
$37$	11.2936	1.85666	0.928330	+	0.371758i	$0.121245\pi$
$38$	0	0
$39$	−3.72161	−0.595935
$40$	0	0
$41$	4.79641	0.749074	0.374537	−	0.927212i	$-0.377802\pi$
$41$	4.79641	0.749074	0.374537	+	0.927212i	$0.377802\pi$
$42$	0	0
$43$	1.20359	0.183545	0.0917725	−	0.995780i	$-0.470747\pi$
$43$	1.20359	0.183545	0.0917725	+	0.995780i	$0.470747\pi$
$44$	0	0
$45$	−3.72161	−0.554785
$46$	0	0
$47$	2.27839	0.332337	0.166169	−	0.986097i	$-0.446860\pi$
$47$	2.27839	0.332337	0.166169	+	0.986097i	$0.446860\pi$
$48$	0	0
$49$	1.55678	0.222397
$50$	0	0
$51$	−2.64681	−0.370627
$52$	0	0
$53$	1.87122	0.257032	0.128516	−	0.991707i	$-0.458979\pi$
$53$	1.87122	0.257032	0.128516	+	0.991707i	$0.458979\pi$
$54$	0	0
$55$	−3.72161	−0.501822
$56$	0	0
$57$	−4.64681	−0.615485
$58$	0	0
$59$	−7.44322	−0.969025	−0.484513	−	0.874784i	$-0.661003\pi$
$59$	−7.44322	−0.969025	−0.484513	+	0.874784i	$0.661003\pi$
$60$	0	0
$61$	−9.57201	−1.22557	−0.612785	−	0.790250i	$-0.709951\pi$
$61$	−9.57201	−1.22557	−0.612785	+	0.790250i	$0.709951\pi$
$62$	0	0
$63$	−2.92520	−0.368540
$64$	0	0
$65$	13.8504	1.71793
$66$	0	0
$67$	−5.59283	−0.683273	−0.341636	−	0.939832i	$-0.610981\pi$
$67$	−5.59283	−0.683273	−0.341636	+	0.939832i	$0.610981\pi$
$68$	0	0
$69$	5.72161	0.688801
$70$	0	0
$71$	13.7216	1.62846	0.814228	−	0.580545i	$-0.197161\pi$
$71$	13.7216	1.62846	0.814228	+	0.580545i	$0.197161\pi$
$72$	0	0
$73$	7.59283	0.888674	0.444337	−	0.895860i	$-0.353439\pi$
$73$	7.59283	0.888674	0.444337	+	0.895860i	$0.353439\pi$
$74$	0	0
$75$	8.85039	1.02196
$76$	0	0
$77$	−2.92520	−0.333357
$78$	0	0
$79$	−14.3684	−1.61657	−0.808287	−	0.588789i	$-0.799605\pi$
$79$	−14.3684	−1.61657	−0.808287	+	0.588789i	$0.799605\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	9.85039	1.06843
$86$	0	0
$87$	2.64681	0.283768
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	10.8864	1.14121
$92$	0	0
$93$	9.29362	0.963703
$94$	0	0
$95$	17.2936	1.77429
$96$	0	0
$97$	9.44322	0.958814	0.479407	−	0.877593i	$-0.340852\pi$
$97$	9.44322	0.958814	0.479407	+	0.877593i	$0.340852\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	10.6468	1.05940	0.529699	−	0.848186i	$-0.322305\pi$
$101$	10.6468	1.05940	0.529699	+	0.848186i	$0.322305\pi$
$102$	0	0
$103$	−19.4432	−1.91580	−0.957899	−	0.287106i	$-0.907307\pi$
$103$	−19.4432	−1.91580	−0.957899	+	0.287106i	$0.907307\pi$
$104$	0	0
$105$	10.8864	1.06241
$106$	0	0
$107$	13.2936	1.28514	0.642571	−	0.766226i	$-0.277868\pi$
$107$	13.2936	1.28514	0.642571	+	0.766226i	$0.277868\pi$
$108$	0	0
$109$	−3.72161	−0.356466	−0.178233	−	0.983988i	$-0.557038\pi$
$109$	−3.72161	−0.356466	−0.178233	+	0.983988i	$0.557038\pi$
$110$	0	0
$111$	11.2936	1.07194
$112$	0	0
$113$	19.2936	1.81499	0.907495	−	0.420062i	$-0.137992\pi$
$113$	19.2936	1.81499	0.907495	+	0.420062i	$0.137992\pi$
$114$	0	0
$115$	−21.2936	−1.98564
$116$	0	0
$117$	−3.72161	−0.344063
$118$	0	0
$119$	7.74244	0.709748
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.79641	0.432478
$124$	0	0
$125$	−14.3297	−1.28168
$126$	0	0
$127$	−14.3684	−1.27499	−0.637496	−	0.770454i	$-0.720030\pi$
$127$	−14.3684	−1.27499	−0.637496	+	0.770454i	$0.720030\pi$
$128$	0	0
$129$	1.20359	0.105970
$130$	0	0
$131$	−9.29362	−0.811987	−0.405994	−	0.913876i	$-0.633074\pi$
$131$	−9.29362	−0.811987	−0.405994	+	0.913876i	$0.633074\pi$
$132$	0	0
$133$	13.5928	1.17865
$134$	0	0
$135$	−3.72161	−0.320305
$136$	0	0
$137$	14.7368	1.25905	0.629527	−	0.776979i	$-0.283249\pi$
$137$	14.7368	1.25905	0.629527	+	0.776979i	$0.283249\pi$
$138$	0	0
$139$	10.2396	0.868515	0.434257	−	0.900789i	$-0.357011\pi$
$139$	10.2396	0.868515	0.434257	+	0.900789i	$0.357011\pi$
$140$	0	0
$141$	2.27839	0.191875
$142$	0	0
$143$	−3.72161	−0.311217
$144$	0	0
$145$	−9.85039	−0.818031
$146$	0	0
$147$	1.55678	0.128401
$148$	0	0
$149$	19.6829	1.61248	0.806241	−	0.591587i	$-0.201498\pi$
$149$	19.6829	1.61248	0.806241	+	0.591587i	$0.201498\pi$
$150$	0	0
$151$	−2.66763	−0.217089	−0.108544	−	0.994092i	$-0.534619\pi$
$151$	−2.66763	−0.217089	−0.108544	+	0.994092i	$0.534619\pi$
$152$	0	0
$153$	−2.64681	−0.213982
$154$	0	0
$155$	−34.5872	−2.77811
$156$	0	0
$157$	13.4432	1.07289	0.536443	−	0.843937i	$-0.319768\pi$
$157$	13.4432	1.07289	0.536443	+	0.843937i	$0.319768\pi$
$158$	0	0
$159$	1.87122	0.148397
$160$	0	0
$161$	−16.7368	−1.31905
$162$	0	0
$163$	1.59283	0.124760	0.0623800	−	0.998052i	$-0.480131\pi$
$163$	1.59283	0.124760	0.0623800	+	0.998052i	$0.480131\pi$
$164$	0	0
$165$	−3.72161	−0.289727
$166$	0	0
$167$	20.7368	1.60466	0.802332	−	0.596877i	$-0.203592\pi$
$167$	20.7368	1.60466	0.802332	+	0.596877i	$0.203592\pi$
$168$	0	0
$169$	0.850394	0.0654149
$170$	0	0
$171$	−4.64681	−0.355350
$172$	0	0
$173$	2.64681	0.201233	0.100617	−	0.994925i	$-0.467918\pi$
$173$	2.64681	0.201233	0.100617	+	0.994925i	$0.467918\pi$
$174$	0	0
$175$	−25.8891	−1.95704
$176$	0	0
$177$	−7.44322	−0.559467
$178$	0	0
$179$	−0.556777	−0.0416154	−0.0208077	−	0.999783i	$-0.506624\pi$
$179$	−0.556777	−0.0416154	−0.0208077	+	0.999783i	$0.506624\pi$
$180$	0	0
$181$	−1.44322	−0.107274	−0.0536370	−	0.998561i	$-0.517081\pi$
$181$	−1.44322	−0.107274	−0.0536370	+	0.998561i	$0.517081\pi$
$182$	0	0
$183$	−9.57201	−0.707583
$184$	0	0
$185$	−42.0305	−3.09014
$186$	0	0
$187$	−2.64681	−0.193554
$188$	0	0
$189$	−2.92520	−0.212777
$190$	0	0
$191$	−20.6081	−1.49115	−0.745573	−	0.666424i	$-0.767824\pi$
$191$	−20.6081	−1.49115	−0.745573	+	0.666424i	$0.767824\pi$
$192$	0	0
$193$	−13.1440	−0.946127	−0.473063	−	0.881028i	$-0.656852\pi$
$193$	−13.1440	−0.946127	−0.473063	+	0.881028i	$0.656852\pi$
$194$	0	0
$195$	13.8504	0.991847
$196$	0	0
$197$	18.6468	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$197$	18.6468	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$198$	0	0
$199$	−5.85039	−0.414723	−0.207362	−	0.978264i	$-0.566488\pi$
$199$	−5.85039	−0.414723	−0.207362	+	0.978264i	$0.566488\pi$
$200$	0	0
$201$	−5.59283	−0.394488
$202$	0	0
$203$	−7.74244	−0.543412
$204$	0	0
$205$	−17.8504	−1.24673
$206$	0	0
$207$	5.72161	0.397680
$208$	0	0
$209$	−4.64681	−0.321426
$210$	0	0
$211$	17.3836	1.19674	0.598370	−	0.801220i	$-0.295815\pi$
$211$	17.3836	1.19674	0.598370	+	0.801220i	$0.295815\pi$
$212$	0	0
$213$	13.7216	0.940189
$214$	0	0
$215$	−4.47928	−0.305484
$216$	0	0
$217$	−27.1857	−1.84548
$218$	0	0
$219$	7.59283	0.513076
$220$	0	0
$221$	9.85039	0.662609
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	8.85039	0.590026
$226$	0	0
$227$	24.9944	1.65894	0.829468	−	0.558554i	$-0.188643\pi$
$227$	24.9944	1.65894	0.829468	+	0.558554i	$0.188643\pi$
$228$	0	0
$229$	−10.7368	−0.709510	−0.354755	−	0.934959i	$-0.615436\pi$
$229$	−10.7368	−0.709510	−0.354755	+	0.934959i	$0.615436\pi$
$230$	0	0
$231$	−2.92520	−0.192464
$232$	0	0
$233$	0.497202	0.0325728	0.0162864	−	0.999867i	$-0.494816\pi$
$233$	0.497202	0.0325728	0.0162864	+	0.999867i	$0.494816\pi$
$234$	0	0
$235$	−8.47928	−0.553127
$236$	0	0
$237$	−14.3684	−0.933329
$238$	0	0
$239$	2.14961	0.139046	0.0695232	−	0.997580i	$-0.477852\pi$
$239$	2.14961	0.139046	0.0695232	+	0.997580i	$0.477852\pi$
$240$	0	0
$241$	7.85039	0.505688	0.252844	−	0.967507i	$-0.418634\pi$
$241$	7.85039	0.505688	0.252844	+	0.967507i	$0.418634\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.79372	−0.370147
$246$	0	0
$247$	17.2936	1.10037
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−3.14401	−0.198448	−0.0992241	−	0.995065i	$-0.531636\pi$
$251$	−3.14401	−0.198448	−0.0992241	+	0.995065i	$0.531636\pi$
$252$	0	0
$253$	5.72161	0.359715
$254$	0	0
$255$	9.85039	0.616856
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−33.0361	−2.05276
$260$	0	0
$261$	2.64681	0.163833
$262$	0	0
$263$	−6.70638	−0.413533	−0.206767	−	0.978390i	$-0.566294\pi$
$263$	−6.70638	−0.413533	−0.206767	+	0.978390i	$0.566294\pi$
$264$	0	0
$265$	−6.96395	−0.427792
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	−10.4280	−0.635806	−0.317903	−	0.948123i	$-0.602979\pi$
$269$	−10.4280	−0.635806	−0.317903	+	0.948123i	$0.602979\pi$
$270$	0	0
$271$	13.2549	0.805176	0.402588	−	0.915381i	$-0.368111\pi$
$271$	13.2549	0.805176	0.402588	+	0.915381i	$0.368111\pi$
$272$	0	0
$273$	10.8864	0.658878
$274$	0	0
$275$	8.85039	0.533699
$276$	0	0
$277$	15.7216	0.944620	0.472310	−	0.881432i	$-0.343420\pi$
$277$	15.7216	0.944620	0.472310	+	0.881432i	$0.343420\pi$
$278$	0	0
$279$	9.29362	0.556394
$280$	0	0
$281$	−16.2396	−0.968776	−0.484388	−	0.874853i	$-0.660958\pi$
$281$	−16.2396	−0.968776	−0.484388	+	0.874853i	$0.660958\pi$
$282$	0	0
$283$	20.3892	1.21201	0.606007	−	0.795459i	$-0.292770\pi$
$283$	20.3892	1.21201	0.606007	+	0.795459i	$0.292770\pi$
$284$	0	0
$285$	17.2936	1.02439
$286$	0	0
$287$	−14.0305	−0.828192
$288$	0	0
$289$	−9.99440	−0.587906
$290$	0	0
$291$	9.44322	0.553572
$292$	0	0
$293$	−21.7908	−1.27303	−0.636517	−	0.771263i	$-0.719625\pi$
$293$	−21.7908	−1.27303	−0.636517	+	0.771263i	$0.719625\pi$
$294$	0	0
$295$	27.7008	1.61280
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−21.2936	−1.23144
$300$	0	0
$301$	−3.52072	−0.202931
$302$	0	0
$303$	10.6468	0.611643
$304$	0	0
$305$	35.6233	2.03978
$306$	0	0
$307$	28.8269	1.64524	0.822618	−	0.568594i	$-0.192512\pi$
$307$	28.8269	1.64524	0.822618	+	0.568594i	$0.192512\pi$
$308$	0	0
$309$	−19.4432	−1.10609
$310$	0	0
$311$	13.7216	0.778081	0.389041	−	0.921221i	$-0.372807\pi$
$311$	13.7216	0.778081	0.389041	+	0.921221i	$0.372807\pi$
$312$	0	0
$313$	−30.1801	−1.70588	−0.852939	−	0.522011i	$-0.825182\pi$
$313$	−30.1801	−1.70588	−0.852939	+	0.522011i	$0.825182\pi$
$314$	0	0
$315$	10.8864	0.613382
$316$	0	0
$317$	22.6081	1.26980	0.634898	−	0.772596i	$-0.281042\pi$
$317$	22.6081	1.26980	0.634898	+	0.772596i	$0.281042\pi$
$318$	0	0
$319$	2.64681	0.148193
$320$	0	0
$321$	13.2936	0.741977
$322$	0	0
$323$	12.2992	0.684346
$324$	0	0
$325$	−32.9377	−1.82706
$326$	0	0
$327$	−3.72161	−0.205806
$328$	0	0
$329$	−6.66473	−0.367439
$330$	0	0
$331$	−10.7064	−0.588476	−0.294238	−	0.955732i	$-0.595066\pi$
$331$	−10.7064	−0.588476	−0.294238	+	0.955732i	$0.595066\pi$
$332$	0	0
$333$	11.2936	0.618886
$334$	0	0
$335$	20.8143	1.13721
$336$	0	0
$337$	29.8809	1.62771	0.813857	−	0.581065i	$-0.197364\pi$
$337$	29.8809	1.62771	0.813857	+	0.581065i	$0.197364\pi$
$338$	0	0
$339$	19.2936	1.04789
$340$	0	0
$341$	9.29362	0.503278
$342$	0	0
$343$	15.9225	0.859734
$344$	0	0
$345$	−21.2936	−1.14641
$346$	0	0
$347$	−30.8864	−1.65807	−0.829036	−	0.559196i	$-0.811110\pi$
$347$	−30.8864	−1.65807	−0.829036	+	0.559196i	$0.811110\pi$
$348$	0	0
$349$	−21.0152	−1.12492	−0.562460	−	0.826825i	$-0.690145\pi$
$349$	−21.0152	−1.12492	−0.562460	+	0.826825i	$0.690145\pi$
$350$	0	0
$351$	−3.72161	−0.198645
$352$	0	0
$353$	13.7008	0.729219	0.364610	−	0.931160i	$-0.381202\pi$
$353$	13.7008	0.729219	0.364610	+	0.931160i	$0.381202\pi$
$354$	0	0
$355$	−51.0665	−2.71033
$356$	0	0
$357$	7.74244	0.409773
$358$	0	0
$359$	0.257564	0.0135937	0.00679685	−	0.999977i	$-0.497836\pi$
$359$	0.257564	0.0135937	0.00679685	+	0.999977i	$0.497836\pi$
$360$	0	0
$361$	2.59283	0.136465
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−28.2576	−1.47907
$366$	0	0
$367$	−30.8864	−1.61226	−0.806130	−	0.591739i	$-0.798442\pi$
$367$	−30.8864	−1.61226	−0.806130	+	0.591739i	$0.798442\pi$
$368$	0	0
$369$	4.79641	0.249691
$370$	0	0
$371$	−5.47368	−0.284179
$372$	0	0
$373$	5.57201	0.288508	0.144254	−	0.989541i	$-0.453922\pi$
$373$	5.57201	0.288508	0.144254	+	0.989541i	$0.453922\pi$
$374$	0	0
$375$	−14.3297	−0.739981
$376$	0	0
$377$	−9.85039	−0.507321
$378$	0	0
$379$	−14.4072	−0.740047	−0.370023	−	0.929022i	$-0.620650\pi$
$379$	−14.4072	−0.740047	−0.370023	+	0.929022i	$0.620650\pi$
$380$	0	0
$381$	−14.3684	−0.736116
$382$	0	0
$383$	−12.8656	−0.657403	−0.328701	−	0.944434i	$-0.606611\pi$
$383$	−12.8656	−0.657403	−0.328701	+	0.944434i	$0.606611\pi$
$384$	0	0
$385$	10.8864	0.554825
$386$	0	0
$387$	1.20359	0.0611817
$388$	0	0
$389$	3.16484	0.160464	0.0802318	−	0.996776i	$-0.474434\pi$
$389$	3.16484	0.160464	0.0802318	+	0.996776i	$0.474434\pi$
$390$	0	0
$391$	−15.1440	−0.765866
$392$	0	0
$393$	−9.29362	−0.468801
$394$	0	0
$395$	53.4737	2.69055
$396$	0	0
$397$	34.4376	1.72837	0.864187	−	0.503170i	$-0.167833\pi$
$397$	34.4376	1.72837	0.864187	+	0.503170i	$0.167833\pi$
$398$	0	0
$399$	13.5928	0.680493
$400$	0	0
$401$	11.0361	0.551114	0.275557	−	0.961285i	$-0.411138\pi$
$401$	11.0361	0.551114	0.275557	+	0.961285i	$0.411138\pi$
$402$	0	0
$403$	−34.5872	−1.72291
$404$	0	0
$405$	−3.72161	−0.184928
$406$	0	0
$407$	11.2936	0.559804
$408$	0	0
$409$	18.4376	0.911682	0.455841	−	0.890061i	$-0.349339\pi$
$409$	18.4376	0.911682	0.455841	+	0.890061i	$0.349339\pi$
$410$	0	0
$411$	14.7368	0.726915
$412$	0	0
$413$	21.7729	1.07137
$414$	0	0
$415$	29.7729	1.46149
$416$	0	0
$417$	10.2396	0.501437
$418$	0	0
$419$	−8.55678	−0.418026	−0.209013	−	0.977913i	$-0.567025\pi$
$419$	−8.55678	−0.418026	−0.209013	+	0.977913i	$0.567025\pi$
$420$	0	0
$421$	27.4737	1.33899	0.669493	−	0.742819i	$-0.266512\pi$
$421$	27.4737	1.33899	0.669493	+	0.742819i	$0.266512\pi$
$422$	0	0
$423$	2.27839	0.110779
$424$	0	0
$425$	−23.4253	−1.13629
$426$	0	0
$427$	28.0000	1.35501
$428$	0	0
$429$	−3.72161	−0.179681
$430$	0	0
$431$	−20.9944	−1.01126	−0.505632	−	0.862749i	$-0.668741\pi$
$431$	−20.9944	−1.01126	−0.505632	+	0.862749i	$0.668741\pi$
$432$	0	0
$433$	35.2936	1.69610	0.848051	−	0.529914i	$-0.177776\pi$
$433$	35.2936	1.69610	0.848051	+	0.529914i	$0.177776\pi$
$434$	0	0
$435$	−9.85039	−0.472290
$436$	0	0
$437$	−26.5872	−1.27184
$438$	0	0
$439$	−39.6620	−1.89297	−0.946483	−	0.322754i	$-0.895391\pi$
$439$	−39.6620	−1.89297	−0.946483	+	0.322754i	$0.895391\pi$
$440$	0	0
$441$	1.55678	0.0741322
$442$	0	0
$443$	4.25756	0.202283	0.101141	−	0.994872i	$-0.467751\pi$
$443$	4.25756	0.202283	0.101141	+	0.994872i	$0.467751\pi$
$444$	0	0
$445$	−7.44322	−0.352843
$446$	0	0
$447$	19.6829	0.930967
$448$	0	0
$449$	10.2576	0.484084	0.242042	−	0.970266i	$-0.422183\pi$
$449$	10.2576	0.484084	0.242042	+	0.970266i	$0.422183\pi$
$450$	0	0
$451$	4.79641	0.225854
$452$	0	0
$453$	−2.66763	−0.125336
$454$	0	0
$455$	−40.5151	−1.89938
$456$	0	0
$457$	−15.0361	−0.703357	−0.351678	−	0.936121i	$-0.614389\pi$
$457$	−15.0361	−0.703357	−0.351678	+	0.936121i	$0.614389\pi$
$458$	0	0
$459$	−2.64681	−0.123542
$460$	0	0
$461$	−34.0900	−1.58773	−0.793866	−	0.608093i	$-0.791935\pi$
$461$	−34.0900	−1.58773	−0.793866	+	0.608093i	$0.791935\pi$
$462$	0	0
$463$	−15.7424	−0.731613	−0.365807	−	0.930691i	$-0.619207\pi$
$463$	−15.7424	−0.731613	−0.365807	+	0.930691i	$0.619207\pi$
$464$	0	0
$465$	−34.5872	−1.60394
$466$	0	0
$467$	18.8864	0.873961	0.436980	−	0.899471i	$-0.356048\pi$
$467$	18.8864	0.873961	0.436980	+	0.899471i	$0.356048\pi$
$468$	0	0
$469$	16.3601	0.755441
$470$	0	0
$471$	13.4432	0.619431
$472$	0	0
$473$	1.20359	0.0553409
$474$	0	0
$475$	−41.1261	−1.88699
$476$	0	0
$477$	1.87122	0.0856772
$478$	0	0
$479$	39.0665	1.78499	0.892497	−	0.451053i	$-0.148951\pi$
$479$	39.0665	1.78499	0.892497	+	0.451053i	$0.148951\pi$
$480$	0	0
$481$	−42.0305	−1.91642
$482$	0	0
$483$	−16.7368	−0.761553
$484$	0	0
$485$	−35.1440	−1.59581
$486$	0	0
$487$	29.8504	1.35265	0.676325	−	0.736603i	$-0.263571\pi$
$487$	29.8504	1.35265	0.676325	+	0.736603i	$0.263571\pi$
$488$	0	0
$489$	1.59283	0.0720302
$490$	0	0
$491$	9.59283	0.432918	0.216459	−	0.976292i	$-0.430549\pi$
$491$	9.59283	0.432918	0.216459	+	0.976292i	$0.430549\pi$
$492$	0	0
$493$	−7.00560	−0.315516
$494$	0	0
$495$	−3.72161	−0.167274
$496$	0	0
$497$	−40.1384	−1.80045
$498$	0	0
$499$	26.8864	1.20360	0.601801	−	0.798646i	$-0.294450\pi$
$499$	26.8864	1.20360	0.601801	+	0.798646i	$0.294450\pi$
$500$	0	0
$501$	20.7368	0.926454
$502$	0	0
$503$	−5.85039	−0.260856	−0.130428	−	0.991458i	$-0.541635\pi$
$503$	−5.85039	−0.260856	−0.130428	+	0.991458i	$0.541635\pi$
$504$	0	0
$505$	−39.6233	−1.76321
$506$	0	0
$507$	0.850394	0.0377673
$508$	0	0
$509$	−24.7160	−1.09552	−0.547759	−	0.836636i	$-0.684519\pi$
$509$	−24.7160	−1.09552	−0.547759	+	0.836636i	$0.684519\pi$
$510$	0	0
$511$	−22.2105	−0.982536
$512$	0	0
$513$	−4.64681	−0.205162
$514$	0	0
$515$	72.3601	3.18857
$516$	0	0
$517$	2.27839	0.100203
$518$	0	0
$519$	2.64681	0.116182
$520$	0	0
$521$	6.73684	0.295146	0.147573	−	0.989051i	$-0.452854\pi$
$521$	6.73684	0.295146	0.147573	+	0.989051i	$0.452854\pi$
$522$	0	0
$523$	−7.90997	−0.345879	−0.172939	−	0.984932i	$-0.555326\pi$
$523$	−7.90997	−0.345879	−0.172939	+	0.984932i	$0.555326\pi$
$524$	0	0
$525$	−25.8891	−1.12989
$526$	0	0
$527$	−24.5984	−1.07152
$528$	0	0
$529$	9.73684	0.423341
$530$	0	0
$531$	−7.44322	−0.323008
$532$	0	0
$533$	−17.8504	−0.773186
$534$	0	0
$535$	−49.4737	−2.13893
$536$	0	0
$537$	−0.556777	−0.0240267
$538$	0	0
$539$	1.55678	0.0670551
$540$	0	0
$541$	−15.4224	−0.663061	−0.331530	−	0.943445i	$-0.607565\pi$
$541$	−15.4224	−0.663061	−0.331530	+	0.943445i	$0.607565\pi$
$542$	0	0
$543$	−1.44322	−0.0619346
$544$	0	0
$545$	13.8504	0.593286
$546$	0	0
$547$	16.3476	0.698973	0.349486	−	0.936941i	$-0.386356\pi$
$547$	16.3476	0.698973	0.349486	+	0.936941i	$0.386356\pi$
$548$	0	0
$549$	−9.57201	−0.408523
$550$	0	0
$551$	−12.2992	−0.523964
$552$	0	0
$553$	42.0305	1.78732
$554$	0	0
$555$	−42.0305	−1.78409
$556$	0	0
$557$	9.61076	0.407221	0.203610	−	0.979052i	$-0.434732\pi$
$557$	9.61076	0.407221	0.203610	+	0.979052i	$0.434732\pi$
$558$	0	0
$559$	−4.47928	−0.189453
$560$	0	0
$561$	−2.64681	−0.111748
$562$	0	0
$563$	−34.2880	−1.44507	−0.722534	−	0.691335i	$-0.757023\pi$
$563$	−34.2880	−1.44507	−0.722534	+	0.691335i	$0.757023\pi$
$564$	0	0
$565$	−71.8034	−3.02079
$566$	0	0
$567$	−2.92520	−0.122847
$568$	0	0
$569$	39.9404	1.67439	0.837195	−	0.546905i	$-0.184194\pi$
$569$	39.9404	1.67439	0.837195	+	0.546905i	$0.184194\pi$
$570$	0	0
$571$	2.05957	0.0861905	0.0430953	−	0.999071i	$-0.486278\pi$
$571$	2.05957	0.0861905	0.0430953	+	0.999071i	$0.486278\pi$
$572$	0	0
$573$	−20.6081	−0.860914
$574$	0	0
$575$	50.6385	2.11177
$576$	0	0
$577$	22.4376	0.934091	0.467045	−	0.884233i	$-0.345318\pi$
$577$	22.4376	0.934091	0.467045	+	0.884233i	$0.345318\pi$
$578$	0	0
$579$	−13.1440	−0.546246
$580$	0	0
$581$	23.4016	0.970861
$582$	0	0
$583$	1.87122	0.0774979
$584$	0	0
$585$	13.8504	0.572643
$586$	0	0
$587$	−11.1440	−0.459963	−0.229981	−	0.973195i	$-0.573866\pi$
$587$	−11.1440	−0.459963	−0.229981	+	0.973195i	$0.573866\pi$
$588$	0	0
$589$	−43.1857	−1.77943
$590$	0	0
$591$	18.6468	0.767027
$592$	0	0
$593$	7.76036	0.318680	0.159340	−	0.987224i	$-0.449063\pi$
$593$	7.76036	0.318680	0.159340	+	0.987224i	$0.449063\pi$
$594$	0	0
$595$	−28.8143	−1.18127
$596$	0	0
$597$	−5.85039	−0.239441
$598$	0	0
$599$	13.7216	0.560650	0.280325	−	0.959905i	$-0.409558\pi$
$599$	13.7216	0.560650	0.280325	+	0.959905i	$0.409558\pi$
$600$	0	0
$601$	13.1857	0.537854	0.268927	−	0.963161i	$-0.413331\pi$
$601$	13.1857	0.537854	0.268927	+	0.963161i	$0.413331\pi$
$602$	0	0
$603$	−5.59283	−0.227758
$604$	0	0
$605$	−3.72161	−0.151305
$606$	0	0
$607$	7.91960	0.321447	0.160723	−	0.986999i	$-0.448617\pi$
$607$	7.91960	0.321447	0.160723	+	0.986999i	$0.448617\pi$
$608$	0	0
$609$	−7.74244	−0.313739
$610$	0	0
$611$	−8.47928	−0.343035
$612$	0	0
$613$	14.4280	0.582741	0.291371	−	0.956610i	$-0.405889\pi$
$613$	14.4280	0.582741	0.291371	+	0.956610i	$0.405889\pi$
$614$	0	0
$615$	−17.8504	−0.719797
$616$	0	0
$617$	−37.5816	−1.51298	−0.756490	−	0.654005i	$-0.773087\pi$
$617$	−37.5816	−1.51298	−0.756490	+	0.654005i	$0.773087\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	5.72161	0.229600
$622$	0	0
$623$	−5.85039	−0.234391
$624$	0	0
$625$	9.07750	0.363100
$626$	0	0
$627$	−4.64681	−0.185576
$628$	0	0
$629$	−29.8920	−1.19187
$630$	0	0
$631$	27.8809	1.10992	0.554960	−	0.831877i	$-0.312734\pi$
$631$	27.8809	1.10992	0.554960	+	0.831877i	$0.312734\pi$
$632$	0	0
$633$	17.3836	0.690938
$634$	0	0
$635$	53.4737	2.12204
$636$	0	0
$637$	−5.79372	−0.229555
$638$	0	0
$639$	13.7216	0.542819
$640$	0	0
$641$	16.6289	0.656801	0.328401	−	0.944539i	$-0.393490\pi$
$641$	16.6289	0.656801	0.328401	+	0.944539i	$0.393490\pi$
$642$	0	0
$643$	33.4737	1.32007	0.660037	−	0.751233i	$-0.270541\pi$
$643$	33.4737	1.32007	0.660037	+	0.751233i	$0.270541\pi$
$644$	0	0
$645$	−4.47928	−0.176371
$646$	0	0
$647$	25.4224	0.999458	0.499729	−	0.866182i	$-0.333433\pi$
$647$	25.4224	0.999458	0.499729	+	0.866182i	$0.333433\pi$
$648$	0	0
$649$	−7.44322	−0.292172
$650$	0	0
$651$	−27.1857	−1.06549
$652$	0	0
$653$	−21.8712	−0.855887	−0.427943	−	0.903806i	$-0.640762\pi$
$653$	−21.8712	−0.855887	−0.427943	+	0.903806i	$0.640762\pi$
$654$	0	0
$655$	34.5872	1.35144
$656$	0	0
$657$	7.59283	0.296225
$658$	0	0
$659$	15.7008	0.611616	0.305808	−	0.952093i	$-0.401073\pi$
$659$	15.7008	0.611616	0.305808	+	0.952093i	$0.401073\pi$
$660$	0	0
$661$	−1.95835	−0.0761710	−0.0380855	−	0.999274i	$-0.512126\pi$
$661$	−1.95835	−0.0761710	−0.0380855	+	0.999274i	$0.512126\pi$
$662$	0	0
$663$	9.85039	0.382558
$664$	0	0
$665$	−50.5872	−1.96169
$666$	0	0
$667$	15.1440	0.586378
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.57201	−0.369523
$672$	0	0
$673$	−34.9169	−1.34595	−0.672974	−	0.739666i	$-0.734983\pi$
$673$	−34.9169	−1.34595	−0.672974	+	0.739666i	$0.734983\pi$
$674$	0	0
$675$	8.85039	0.340652
$676$	0	0
$677$	46.3476	1.78128	0.890641	−	0.454706i	$-0.150256\pi$
$677$	46.3476	1.78128	0.890641	+	0.454706i	$0.150256\pi$
$678$	0	0
$679$	−27.6233	−1.06008
$680$	0	0
$681$	24.9944	0.957788
$682$	0	0
$683$	32.9169	1.25953	0.629765	−	0.776786i	$-0.283151\pi$
$683$	32.9169	1.25953	0.629765	+	0.776786i	$0.283151\pi$
$684$	0	0
$685$	−54.8448	−2.09551
$686$	0	0
$687$	−10.7368	−0.409636
$688$	0	0
$689$	−6.96395	−0.265305
$690$	0	0
$691$	−11.1857	−0.425523	−0.212761	−	0.977104i	$-0.568246\pi$
$691$	−11.1857	−0.425523	−0.212761	+	0.977104i	$0.568246\pi$
$692$	0	0
$693$	−2.92520	−0.111119
$694$	0	0
$695$	−38.1080	−1.44552
$696$	0	0
$697$	−12.6952	−0.480865
$698$	0	0
$699$	0.497202	0.0188059
$700$	0	0
$701$	16.4972	0.623091	0.311545	−	0.950231i	$-0.399153\pi$
$701$	16.4972	0.623091	0.311545	+	0.950231i	$0.399153\pi$
$702$	0	0
$703$	−52.4793	−1.97929
$704$	0	0
$705$	−8.47928	−0.319348
$706$	0	0
$707$	−31.1440	−1.17129
$708$	0	0
$709$	3.03605	0.114021	0.0570107	−	0.998374i	$-0.481843\pi$
$709$	3.03605	0.114021	0.0570107	+	0.998374i	$0.481843\pi$
$710$	0	0
$711$	−14.3684	−0.538858
$712$	0	0
$713$	53.1745	1.99140
$714$	0	0
$715$	13.8504	0.517975
$716$	0	0
$717$	2.14961	0.0802785
$718$	0	0
$719$	24.3088	0.906567	0.453283	−	0.891366i	$-0.350253\pi$
$719$	24.3088	0.906567	0.453283	+	0.891366i	$0.350253\pi$
$720$	0	0
$721$	56.8753	2.11815
$722$	0	0
$723$	7.85039	0.291959
$724$	0	0
$725$	23.4253	0.869994
$726$	0	0
$727$	29.8504	1.10709	0.553545	−	0.832819i	$-0.313275\pi$
$727$	29.8504	1.10709	0.553545	+	0.832819i	$0.313275\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.18566	−0.117826
$732$	0	0
$733$	12.2784	0.453513	0.226756	−	0.973952i	$-0.427188\pi$
$733$	12.2784	0.453513	0.226756	+	0.973952i	$0.427188\pi$
$734$	0	0
$735$	−5.79372	−0.213705
$736$	0	0
$737$	−5.59283	−0.206015
$738$	0	0
$739$	−26.4972	−0.974715	−0.487358	−	0.873202i	$-0.662039\pi$
$739$	−26.4972	−0.974715	−0.487358	+	0.873202i	$0.662039\pi$
$740$	0	0
$741$	17.2936	0.635297
$742$	0	0
$743$	−39.1440	−1.43605	−0.718027	−	0.696015i	$-0.754955\pi$
$743$	−39.1440	−1.43605	−0.718027	+	0.696015i	$0.754955\pi$
$744$	0	0
$745$	−73.2520	−2.68374
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	−38.8864	−1.42088
$750$	0	0
$751$	−22.1080	−0.806731	−0.403365	−	0.915039i	$-0.632160\pi$
$751$	−22.1080	−0.806731	−0.403365	+	0.915039i	$0.632160\pi$
$752$	0	0
$753$	−3.14401	−0.114574
$754$	0	0
$755$	9.92789	0.361313
$756$	0	0
$757$	−29.4016	−1.06862	−0.534309	−	0.845289i	$-0.679428\pi$
$757$	−29.4016	−1.06862	−0.534309	+	0.845289i	$0.679428\pi$
$758$	0	0
$759$	5.72161	0.207681
$760$	0	0
$761$	12.7548	0.462360	0.231180	−	0.972911i	$-0.425741\pi$
$761$	12.7548	0.462360	0.231180	+	0.972911i	$0.425741\pi$
$762$	0	0
$763$	10.8864	0.394116
$764$	0	0
$765$	9.85039	0.356142
$766$	0	0
$767$	27.7008	1.00022
$768$	0	0
$769$	−24.3297	−0.877350	−0.438675	−	0.898646i	$-0.644552\pi$
$769$	−24.3297	−0.877350	−0.438675	+	0.898646i	$0.644552\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	40.7577	1.46595	0.732976	−	0.680255i	$-0.238131\pi$
$773$	40.7577	1.46595	0.732976	+	0.680255i	$0.238131\pi$
$774$	0	0
$775$	82.2522	2.95459
$776$	0	0
$777$	−33.0361	−1.18516
$778$	0	0
$779$	−22.2880	−0.798551
$780$	0	0
$781$	13.7216	0.490998
$782$	0	0
$783$	2.64681	0.0945892
$784$	0	0
$785$	−50.0305	−1.78566
$786$	0	0
$787$	−22.1980	−0.791273	−0.395636	−	0.918407i	$-0.629476\pi$
$787$	−22.1980	−0.791273	−0.395636	+	0.918407i	$0.629476\pi$
$788$	0	0
$789$	−6.70638	−0.238754
$790$	0	0
$791$	−56.4376	−2.00669
$792$	0	0
$793$	35.6233	1.26502
$794$	0	0
$795$	−6.96395	−0.246986
$796$	0	0
$797$	4.27839	0.151548	0.0757741	−	0.997125i	$-0.475857\pi$
$797$	4.27839	0.151548	0.0757741	+	0.997125i	$0.475857\pi$
$798$	0	0
$799$	−6.03046	−0.213342
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	7.59283	0.267945
$804$	0	0
$805$	62.2880	2.19536
$806$	0	0
$807$	−10.4280	−0.367083
$808$	0	0
$809$	22.0900	0.776644	0.388322	−	0.921524i	$-0.373055\pi$
$809$	22.0900	0.776644	0.388322	+	0.921524i	$0.373055\pi$
$810$	0	0
$811$	7.90997	0.277757	0.138878	−	0.990309i	$-0.455650\pi$
$811$	7.90997	0.277757	0.138878	+	0.990309i	$0.455650\pi$
$812$	0	0
$813$	13.2549	0.464868
$814$	0	0
$815$	−5.92789	−0.207645
$816$	0	0
$817$	−5.59283	−0.195668
$818$	0	0
$819$	10.8864	0.380403
$820$	0	0
$821$	30.3476	1.05914	0.529569	−	0.848267i	$-0.322354\pi$
$821$	30.3476	1.05914	0.529569	+	0.848267i	$0.322354\pi$
$822$	0	0
$823$	−7.40157	−0.258003	−0.129001	−	0.991644i	$-0.541177\pi$
$823$	−7.40157	−0.258003	−0.129001	+	0.991644i	$0.541177\pi$
$824$	0	0
$825$	8.85039	0.308131
$826$	0	0
$827$	−34.0721	−1.18480	−0.592402	−	0.805643i	$-0.701820\pi$
$827$	−34.0721	−1.18480	−0.592402	+	0.805643i	$0.701820\pi$
$828$	0	0
$829$	44.7673	1.55483	0.777416	−	0.628986i	$-0.216530\pi$
$829$	44.7673	1.55483	0.777416	+	0.628986i	$0.216530\pi$
$830$	0	0
$831$	15.7216	0.545377
$832$	0	0
$833$	−4.12049	−0.142767
$834$	0	0
$835$	−77.1745	−2.67073
$836$	0	0
$837$	9.29362	0.321234
$838$	0	0
$839$	−49.7825	−1.71868	−0.859342	−	0.511402i	$-0.829126\pi$
$839$	−49.7825	−1.71868	−0.859342	+	0.511402i	$0.829126\pi$
$840$	0	0
$841$	−21.9944	−0.758428
$842$	0	0
$843$	−16.2396	−0.559323
$844$	0	0
$845$	−3.16484	−0.108874
$846$	0	0
$847$	−2.92520	−0.100511
$848$	0	0
$849$	20.3892	0.699757
$850$	0	0
$851$	64.6177	2.21507
$852$	0	0
$853$	28.7160	0.983218	0.491609	−	0.870816i	$-0.336409\pi$
$853$	28.7160	0.983218	0.491609	+	0.870816i	$0.336409\pi$
$854$	0	0
$855$	17.2936	0.591429
$856$	0	0
$857$	8.67727	0.296410	0.148205	−	0.988957i	$-0.452650\pi$
$857$	8.67727	0.296410	0.148205	+	0.988957i	$0.452650\pi$
$858$	0	0
$859$	6.40717	0.218610	0.109305	−	0.994008i	$-0.465138\pi$
$859$	6.40717	0.218610	0.109305	+	0.994008i	$0.465138\pi$
$860$	0	0
$861$	−14.0305	−0.478157
$862$	0	0
$863$	48.5664	1.65322	0.826610	−	0.562775i	$-0.190266\pi$
$863$	48.5664	1.65322	0.826610	+	0.562775i	$0.190266\pi$
$864$	0	0
$865$	−9.85039	−0.334923
$866$	0	0
$867$	−9.99440	−0.339428
$868$	0	0
$869$	−14.3684	−0.487415
$870$	0	0
$871$	20.8143	0.705267
$872$	0	0
$873$	9.44322	0.319605
$874$	0	0
$875$	41.9171	1.41706
$876$	0	0
$877$	17.8712	0.603468	0.301734	−	0.953392i	$-0.402435\pi$
$877$	17.8712	0.603468	0.301734	+	0.953392i	$0.402435\pi$
$878$	0	0
$879$	−21.7908	−0.734986
$880$	0	0
$881$	−2.47928	−0.0835289	−0.0417645	−	0.999127i	$-0.513298\pi$
$881$	−2.47928	−0.0835289	−0.0417645	+	0.999127i	$0.513298\pi$
$882$	0	0
$883$	3.88085	0.130601	0.0653005	−	0.997866i	$-0.479199\pi$
$883$	3.88085	0.130601	0.0653005	+	0.997866i	$0.479199\pi$
$884$	0	0
$885$	27.7008	0.931152
$886$	0	0
$887$	−4.04165	−0.135705	−0.0678526	−	0.997695i	$-0.521615\pi$
$887$	−4.04165	−0.135705	−0.0678526	+	0.997695i	$0.521615\pi$
$888$	0	0
$889$	42.0305	1.40966
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−10.5872	−0.354288
$894$	0	0
$895$	2.07211	0.0692629
$896$	0	0
$897$	−21.2936	−0.710973
$898$	0	0
$899$	24.5984	0.820403
$900$	0	0
$901$	−4.95276	−0.165000
$902$	0	0
$903$	−3.52072	−0.117162
$904$	0	0
$905$	5.37112	0.178542
$906$	0	0
$907$	−44.0609	−1.46302	−0.731509	−	0.681831i	$-0.761184\pi$
$907$	−44.0609	−1.46302	−0.731509	+	0.681831i	$0.761184\pi$
$908$	0	0
$909$	10.6468	0.353132
$910$	0	0
$911$	8.90727	0.295111	0.147556	−	0.989054i	$-0.452859\pi$
$911$	8.90727	0.295111	0.147556	+	0.989054i	$0.452859\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	35.6233	1.17767
$916$	0	0
$917$	27.1857	0.897750
$918$	0	0
$919$	32.9557	1.08711	0.543553	−	0.839375i	$-0.317079\pi$
$919$	32.9557	1.08711	0.543553	+	0.839375i	$0.317079\pi$
$920$	0	0
$921$	28.8269	0.949878
$922$	0	0
$923$	−51.0665	−1.68087
$924$	0	0
$925$	99.9530	3.28643
$926$	0	0
$927$	−19.4432	−0.638599
$928$	0	0
$929$	−16.1496	−0.529851	−0.264926	−	0.964269i	$-0.585347\pi$
$929$	−16.1496	−0.529851	−0.264926	+	0.964269i	$0.585347\pi$
$930$	0	0
$931$	−7.23404	−0.237086
$932$	0	0
$933$	13.7216	0.449225
$934$	0	0
$935$	9.85039	0.322142
$936$	0	0
$937$	−2.81434	−0.0919405	−0.0459702	−	0.998943i	$-0.514638\pi$
$937$	−2.81434	−0.0919405	−0.0459702	+	0.998943i	$0.514638\pi$
$938$	0	0
$939$	−30.1801	−0.984889
$940$	0	0
$941$	0.0595743	0.00194207	0.000971034	−	1.00000i	$-0.499691\pi$
$941$	0.0595743	0.00194207	0.000971034		1.00000i	$0.499691\pi$
$942$	0	0
$943$	27.4432	0.893674
$944$	0	0
$945$	10.8864	0.354136
$946$	0	0
$947$	−31.7008	−1.03014	−0.515069	−	0.857149i	$-0.672234\pi$
$947$	−31.7008	−1.03014	−0.515069	+	0.857149i	$0.672234\pi$
$948$	0	0
$949$	−28.2576	−0.917279
$950$	0	0
$951$	22.6081	0.733117
$952$	0	0
$953$	33.2757	1.07791	0.538953	−	0.842336i	$-0.318820\pi$
$953$	33.2757	1.07791	0.538953	+	0.842336i	$0.318820\pi$
$954$	0	0
$955$	76.6952	2.48180
$956$	0	0
$957$	2.64681	0.0855592
$958$	0	0
$959$	−43.1082	−1.39204
$960$	0	0
$961$	55.3713	1.78617
$962$	0	0
$963$	13.2936	0.428381
$964$	0	0
$965$	48.9169	1.57469
$966$	0	0
$967$	23.9196	0.769203	0.384601	−	0.923083i	$-0.374339\pi$
$967$	23.9196	0.769203	0.384601	+	0.923083i	$0.374339\pi$
$968$	0	0
$969$	12.2992	0.395108
$970$	0	0
$971$	27.7424	0.890297	0.445149	−	0.895457i	$-0.353151\pi$
$971$	27.7424	0.890297	0.445149	+	0.895457i	$0.353151\pi$
$972$	0	0
$973$	−29.9530	−0.960248
$974$	0	0
$975$	−32.9377	−1.05485
$976$	0	0
$977$	−12.4488	−0.398273	−0.199137	−	0.979972i	$-0.563814\pi$
$977$	−12.4488	−0.398273	−0.199137	+	0.979972i	$0.563814\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	−3.72161	−0.118822
$982$	0	0
$983$	−13.4640	−0.429436	−0.214718	−	0.976676i	$-0.568883\pi$
$983$	−13.4640	−0.429436	−0.214718	+	0.976676i	$0.568883\pi$
$984$	0	0
$985$	−69.3962	−2.21115
$986$	0	0
$987$	−6.66473	−0.212141
$988$	0	0
$989$	6.88645	0.218976
$990$	0	0
$991$	1.89204	0.0601027	0.0300514	−	0.999548i	$-0.490433\pi$
$991$	1.89204	0.0601027	0.0300514	+	0.999548i	$0.490433\pi$
$992$	0	0
$993$	−10.7064	−0.339757
$994$	0	0
$995$	21.7729	0.690247
$996$	0	0
$997$	21.5720	0.683192	0.341596	−	0.939847i	$-0.389033\pi$
$997$	21.5720	0.683192	0.341596	+	0.939847i	$0.389033\pi$
$998$	0	0
$999$	11.2936	0.357314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.a.bh.1.1		3
3.2	odd	2		6336.2.a.cy.1.3		3
4.3	odd	2		2112.2.a.bg.1.1		3
8.3	odd	2		1056.2.a.n.1.3	yes	3
8.5	even	2		1056.2.a.m.1.3	✓	3
12.11	even	2		6336.2.a.cz.1.3		3
24.5	odd	2		3168.2.a.bg.1.1		3
24.11	even	2		3168.2.a.bh.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1056.2.a.m.1.3	✓	3	8.5	even	2
1056.2.a.n.1.3	yes	3	8.3	odd	2
2112.2.a.bg.1.1		3	4.3	odd	2
2112.2.a.bh.1.1		3	1.1	even	1	trivial
3168.2.a.bg.1.1		3	24.5	odd	2
3168.2.a.bh.1.1		3	24.11	even	2
6336.2.a.cy.1.3		3	3.2	odd	2
6336.2.a.cz.1.3		3	12.11	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.72161 q^{5} -2.92520 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.72161 q^{5} -2.92520 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.72161 q^{13} -3.72161 q^{15} -2.64681 q^{17} -4.64681 q^{19} -2.92520 q^{21} +5.72161 q^{23} +8.85039 q^{25} +1.00000 q^{27} +2.64681 q^{29} +9.29362 q^{31} +1.00000 q^{33} +10.8864 q^{35} +11.2936 q^{37} -3.72161 q^{39} +4.79641 q^{41} +1.20359 q^{43} -3.72161 q^{45} +2.27839 q^{47} +1.55678 q^{49} -2.64681 q^{51} +1.87122 q^{53} -3.72161 q^{55} -4.64681 q^{57} -7.44322 q^{59} -9.57201 q^{61} -2.92520 q^{63} +13.8504 q^{65} -5.59283 q^{67} +5.72161 q^{69} +13.7216 q^{71} +7.59283 q^{73} +8.85039 q^{75} -2.92520 q^{77} -14.3684 q^{79} +1.00000 q^{81} -8.00000 q^{83} +9.85039 q^{85} +2.64681 q^{87} +2.00000 q^{89} +10.8864 q^{91} +9.29362 q^{93} +17.2936 q^{95} +9.44322 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 4 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 4 q^{7} + 3 q^{9} + 3 q^{11} + 8 q^{17} + 2 q^{19} - 4 q^{21} + 6 q^{23} + 17 q^{25} + 3 q^{27} - 8 q^{29} - 4 q^{31} + 3 q^{33} - 12 q^{35} + 2 q^{37} + 8 q^{41} + 10 q^{43} + 18 q^{47} + 27 q^{49} + 8 q^{51} + 4 q^{53} + 2 q^{57} - 8 q^{61} - 4 q^{63} + 32 q^{65} - 4 q^{67} + 6 q^{69} + 30 q^{71} + 10 q^{73} + 17 q^{75} - 4 q^{77} - 16 q^{79} + 3 q^{81} - 24 q^{83} + 20 q^{85} - 8 q^{87} + 6 q^{89} - 12 q^{91} - 4 q^{93} + 20 q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 4 * q^7 + 3 * q^9 + 3 * q^11 + 8 * q^17 + 2 * q^19 - 4 * q^21 + 6 * q^23 + 17 * q^25 + 3 * q^27 - 8 * q^29 - 4 * q^31 + 3 * q^33 - 12 * q^35 + 2 * q^37 + 8 * q^41 + 10 * q^43 + 18 * q^47 + 27 * q^49 + 8 * q^51 + 4 * q^53 + 2 * q^57 - 8 * q^61 - 4 * q^63 + 32 * q^65 - 4 * q^67 + 6 * q^69 + 30 * q^71 + 10 * q^73 + 17 * q^75 - 4 * q^77 - 16 * q^79 + 3 * q^81 - 24 * q^83 + 20 * q^85 - 8 * q^87 + 6 * q^89 - 12 * q^91 - 4 * q^93 + 20 * q^95 + 6 * q^97 + 3 * q^99

Introduction

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