LMFDB - Embedded newform 2112.2.a.bh.1.3

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			1.3
Root			$2.11491$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.22982 q^{5} -4.94567 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.22982 q^{5} -4.94567 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.22982 q^{13} +4.22982 q^{15} +3.28415 q^{17} +1.28415 q^{19} -4.94567 q^{21} -2.22982 q^{23} +12.8913 q^{25} +1.00000 q^{27} -3.28415 q^{29} -2.56829 q^{31} +1.00000 q^{33} -20.9193 q^{35} -0.568295 q^{37} +4.22982 q^{39} -5.17548 q^{41} +11.1755 q^{43} +4.22982 q^{45} +10.2298 q^{47} +17.4596 q^{49} +3.28415 q^{51} -10.1212 q^{53} +4.22982 q^{55} +1.28415 q^{57} +8.45963 q^{59} -5.66152 q^{61} -4.94567 q^{63} +17.8913 q^{65} +14.3510 q^{67} -2.22982 q^{69} +5.77018 q^{71} -12.3510 q^{73} +12.8913 q^{75} -4.94567 q^{77} -0.486038 q^{79} +1.00000 q^{81} -8.00000 q^{83} +13.8913 q^{85} -3.28415 q^{87} +2.00000 q^{89} -20.9193 q^{91} -2.56829 q^{93} +5.43171 q^{95} -6.45963 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.22982	1.89163	0.945815	−	0.324705i	$-0.105265\pi$
$5$	4.22982	1.89163	0.945815	+	0.324705i	$0.105265\pi$
$6$	0	0
$7$	−4.94567	−1.86929	−0.934643	−	0.355587i	$-0.884281\pi$
$7$	−4.94567	−1.86929	−0.934643	+	0.355587i	$0.884281\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.22982	1.17314	0.586570	−	0.809899i	$-0.300478\pi$
$13$	4.22982	1.17314	0.586570	+	0.809899i	$0.300478\pi$
$14$	0	0
$15$	4.22982	1.09213
$16$	0	0
$17$	3.28415	0.796523	0.398261	−	0.917272i	$-0.369614\pi$
$17$	3.28415	0.796523	0.398261	+	0.917272i	$0.369614\pi$
$18$	0	0
$19$	1.28415	0.294604	0.147302	−	0.989092i	$-0.452941\pi$
$19$	1.28415	0.294604	0.147302	+	0.989092i	$0.452941\pi$
$20$	0	0
$21$	−4.94567	−1.07923
$22$	0	0
$23$	−2.22982	−0.464949	−0.232474	−	0.972603i	$-0.574682\pi$
$23$	−2.22982	−0.464949	−0.232474	+	0.972603i	$0.574682\pi$
$24$	0	0
$25$	12.8913	2.57827
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.28415	−0.609851	−0.304925	−	0.952376i	$-0.598632\pi$
$29$	−3.28415	−0.609851	−0.304925	+	0.952376i	$0.598632\pi$
$30$	0	0
$31$	−2.56829	−0.461279	−0.230640	−	0.973039i	$-0.574082\pi$
$31$	−2.56829	−0.461279	−0.230640	+	0.973039i	$0.574082\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−20.9193	−3.53600
$36$	0	0
$37$	−0.568295	−0.0934270	−0.0467135	−	0.998908i	$-0.514875\pi$
$37$	−0.568295	−0.0934270	−0.0467135	+	0.998908i	$0.514875\pi$
$38$	0	0
$39$	4.22982	0.677312
$40$	0	0
$41$	−5.17548	−0.808275	−0.404137	−	0.914698i	$-0.632428\pi$
$41$	−5.17548	−0.808275	−0.404137	+	0.914698i	$0.632428\pi$
$42$	0	0
$43$	11.1755	1.70425	0.852123	−	0.523342i	$-0.175315\pi$
$43$	11.1755	1.70425	0.852123	+	0.523342i	$0.175315\pi$
$44$	0	0
$45$	4.22982	0.630544
$46$	0	0
$47$	10.2298	1.49217	0.746086	−	0.665850i	$-0.231931\pi$
$47$	10.2298	1.49217	0.746086	+	0.665850i	$0.231931\pi$
$48$	0	0
$49$	17.4596	2.49423
$50$	0	0
$51$	3.28415	0.459873
$52$	0	0
$53$	−10.1212	−1.39025	−0.695123	−	0.718890i	$-0.744650\pi$
$53$	−10.1212	−1.39025	−0.695123	+	0.718890i	$0.744650\pi$
$54$	0	0
$55$	4.22982	0.570348
$56$	0	0
$57$	1.28415	0.170089
$58$	0	0
$59$	8.45963	1.10135	0.550675	−	0.834720i	$-0.314370\pi$
$59$	8.45963	1.10135	0.550675	+	0.834720i	$0.314370\pi$
$60$	0	0
$61$	−5.66152	−0.724883	−0.362442	−	0.932006i	$-0.618057\pi$
$61$	−5.66152	−0.724883	−0.362442	+	0.932006i	$0.618057\pi$
$62$	0	0
$63$	−4.94567	−0.623096
$64$	0	0
$65$	17.8913	2.21915
$66$	0	0
$67$	14.3510	1.75325	0.876625	−	0.481175i	$-0.159790\pi$
$67$	14.3510	1.75325	0.876625	+	0.481175i	$0.159790\pi$
$68$	0	0
$69$	−2.22982	−0.268438
$70$	0	0
$71$	5.77018	0.684795	0.342397	−	0.939555i	$-0.388761\pi$
$71$	5.77018	0.684795	0.342397	+	0.939555i	$0.388761\pi$
$72$	0	0
$73$	−12.3510	−1.44557	−0.722786	−	0.691072i	$-0.757139\pi$
$73$	−12.3510	−1.44557	−0.722786	+	0.691072i	$0.757139\pi$
$74$	0	0
$75$	12.8913	1.48856
$76$	0	0
$77$	−4.94567	−0.563611
$78$	0	0
$79$	−0.486038	−0.0546835	−0.0273418	−	0.999626i	$-0.508704\pi$
$79$	−0.486038	−0.0546835	−0.0273418	+	0.999626i	$0.508704\pi$
$80$	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
$85$	13.8913	1.50673
$86$	0	0
$87$	−3.28415	−0.352098
$88$	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
$91$	−20.9193	−2.19293
$92$	0	0
$93$	−2.56829	−0.266320
$94$	0	0
$95$	5.43171	0.557281
$96$	0	0
$97$	−6.45963	−0.655876	−0.327938	−	0.944699i	$-0.606354\pi$
$97$	−6.45963	−0.655876	−0.327938	+	0.944699i	$0.606354\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	4.71585	0.469245	0.234622	−	0.972087i	$-0.424615\pi$
$101$	4.71585	0.469245	0.234622	+	0.972087i	$0.424615\pi$
$102$	0	0
$103$	−3.54037	−0.348843	−0.174422	−	0.984671i	$-0.555806\pi$
$103$	−3.54037	−0.348843	−0.174422	+	0.984671i	$0.555806\pi$
$104$	0	0
$105$	−20.9193	−2.04151
$106$	0	0
$107$	1.43171	0.138408	0.0692041	−	0.997603i	$-0.477954\pi$
$107$	1.43171	0.138408	0.0692041	+	0.997603i	$0.477954\pi$
$108$	0	0
$109$	4.22982	0.405143	0.202571	−	0.979267i	$-0.435070\pi$
$109$	4.22982	0.405143	0.202571	+	0.979267i	$0.435070\pi$
$110$	0	0
$111$	−0.568295	−0.0539401
$112$	0	0
$113$	7.43171	0.699116	0.349558	−	0.936915i	$-0.386332\pi$
$113$	7.43171	0.699116	0.349558	+	0.936915i	$0.386332\pi$
$114$	0	0
$115$	−9.43171	−0.879511
$116$	0	0
$117$	4.22982	0.391047
$118$	0	0
$119$	−16.2423	−1.48893
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.17548	−0.466658
$124$	0	0
$125$	33.3789	2.98550
$126$	0	0
$127$	−0.486038	−0.0431289	−0.0215644	−	0.999767i	$-0.506865\pi$
$127$	−0.486038	−0.0431289	−0.0215644	+	0.999767i	$0.506865\pi$
$128$	0	0
$129$	11.1755	0.983946
$130$	0	0
$131$	2.56829	0.224393	0.112196	−	0.993686i	$-0.464211\pi$
$131$	2.56829	0.224393	0.112196	+	0.993686i	$0.464211\pi$
$132$	0	0
$133$	−6.35097	−0.550699
$134$	0	0
$135$	4.22982	0.364045
$136$	0	0
$137$	−13.0279	−1.11305	−0.556525	−	0.830831i	$-0.687866\pi$
$137$	−13.0279	−1.11305	−0.556525	+	0.830831i	$0.687866\pi$
$138$	0	0
$139$	−15.6351	−1.32615	−0.663076	−	0.748552i	$-0.730750\pi$
$139$	−15.6351	−1.32615	−0.663076	+	0.748552i	$0.730750\pi$
$140$	0	0
$141$	10.2298	0.861506
$142$	0	0
$143$	4.22982	0.353715
$144$	0	0
$145$	−13.8913	−1.15361
$146$	0	0
$147$	17.4596	1.44005
$148$	0	0
$149$	−22.0947	−1.81007	−0.905036	−	0.425335i	$-0.860156\pi$
$149$	−22.0947	−1.81007	−0.905036	+	0.425335i	$0.860156\pi$
$150$	0	0
$151$	19.2966	1.57034	0.785169	−	0.619282i	$-0.212576\pi$
$151$	19.2966	1.57034	0.785169	+	0.619282i	$0.212576\pi$
$152$	0	0
$153$	3.28415	0.265508
$154$	0	0
$155$	−10.8634	−0.872570
$156$	0	0
$157$	−2.45963	−0.196300	−0.0981499	−	0.995172i	$-0.531292\pi$
$157$	−2.45963	−0.196300	−0.0981499	+	0.995172i	$0.531292\pi$
$158$	0	0
$159$	−10.1212	−0.802659
$160$	0	0
$161$	11.0279	0.869122
$162$	0	0
$163$	−18.3510	−1.43736	−0.718679	−	0.695342i	$-0.755253\pi$
$163$	−18.3510	−1.43736	−0.718679	+	0.695342i	$0.755253\pi$
$164$	0	0
$165$	4.22982	0.329291
$166$	0	0
$167$	−7.02792	−0.543837	−0.271919	−	0.962320i	$-0.587658\pi$
$167$	−7.02792	−0.543837	−0.271919	+	0.962320i	$0.587658\pi$
$168$	0	0
$169$	4.89134	0.376257
$170$	0	0
$171$	1.28415	0.0982012
$172$	0	0
$173$	−3.28415	−0.249689	−0.124845	−	0.992176i	$-0.539843\pi$
$173$	−3.28415	−0.249689	−0.124845	+	0.992176i	$0.539843\pi$
$174$	0	0
$175$	−63.7563	−4.81952
$176$	0	0
$177$	8.45963	0.635865
$178$	0	0
$179$	−16.4596	−1.23025	−0.615125	−	0.788429i	$-0.710895\pi$
$179$	−16.4596	−1.23025	−0.615125	+	0.788429i	$0.710895\pi$
$180$	0	0
$181$	14.4596	1.07478	0.537388	−	0.843335i	$-0.319411\pi$
$181$	14.4596	1.07478	0.537388	+	0.843335i	$0.319411\pi$
$182$	0	0
$183$	−5.66152	−0.418512
$184$	0	0
$185$	−2.40378	−0.176729
$186$	0	0
$187$	3.28415	0.240161
$188$	0	0
$189$	−4.94567	−0.359744
$190$	0	0
$191$	19.1491	1.38558	0.692789	−	0.721140i	$-0.256382\pi$
$191$	19.1491	1.38558	0.692789	+	0.721140i	$0.256382\pi$
$192$	0	0
$193$	−5.32304	−0.383161	−0.191580	−	0.981477i	$-0.561361\pi$
$193$	−5.32304	−0.383161	−0.191580	+	0.981477i	$0.561361\pi$
$194$	0	0
$195$	17.8913	1.28123
$196$	0	0
$197$	12.7159	0.905967	0.452983	−	0.891519i	$-0.350360\pi$
$197$	12.7159	0.905967	0.452983	+	0.891519i	$0.350360\pi$
$198$	0	0
$199$	−9.89134	−0.701178	−0.350589	−	0.936529i	$-0.614019\pi$
$199$	−9.89134	−0.701178	−0.350589	+	0.936529i	$0.614019\pi$
$200$	0	0
$201$	14.3510	1.01224
$202$	0	0
$203$	16.2423	1.13999
$204$	0	0
$205$	−21.8913	−1.52896
$206$	0	0
$207$	−2.22982	−0.154983
$208$	0	0
$209$	1.28415	0.0888263
$210$	0	0
$211$	−16.3121	−1.12297	−0.561485	−	0.827487i	$-0.689770\pi$
$211$	−16.3121	−1.12297	−0.561485	+	0.827487i	$0.689770\pi$
$212$	0	0
$213$	5.77018	0.395367
$214$	0	0
$215$	47.2702	3.22380
$216$	0	0
$217$	12.7019	0.862263
$218$	0	0
$219$	−12.3510	−0.834601
$220$	0	0
$221$	13.8913	0.934432
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	12.8913	0.859422
$226$	0	0
$227$	21.2144	1.40805	0.704024	−	0.710176i	$-0.251385\pi$
$227$	21.2144	1.40805	0.704024	+	0.710176i	$0.251385\pi$
$228$	0	0
$229$	17.0279	1.12524	0.562618	−	0.826717i	$-0.309794\pi$
$229$	17.0279	1.12524	0.562618	+	0.826717i	$0.309794\pi$
$230$	0	0
$231$	−4.94567	−0.325401
$232$	0	0
$233$	−1.39281	−0.0912461	−0.0456231	−	0.998959i	$-0.514527\pi$
$233$	−1.39281	−0.0912461	−0.0456231	+	0.998959i	$0.514527\pi$
$234$	0	0
$235$	43.2702	2.82264
$236$	0	0
$237$	−0.486038	−0.0315715
$238$	0	0
$239$	−1.89134	−0.122340	−0.0611702	−	0.998127i	$-0.519483\pi$
$239$	−1.89134	−0.122340	−0.0611702	+	0.998127i	$0.519483\pi$
$240$	0	0
$241$	11.8913	0.765988	0.382994	−	0.923751i	$-0.374893\pi$
$241$	11.8913	0.765988	0.382994	+	0.923751i	$0.374893\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	73.8510	4.71817
$246$	0	0
$247$	5.43171	0.345611
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	4.67696	0.295207	0.147604	−	0.989047i	$-0.452844\pi$
$251$	4.67696	0.295207	0.147604	+	0.989047i	$0.452844\pi$
$252$	0	0
$253$	−2.22982	−0.140187
$254$	0	0
$255$	13.8913	0.869909
$256$	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
$259$	2.81060	0.174642
$260$	0	0
$261$	−3.28415	−0.203284
$262$	0	0
$263$	−18.5683	−1.14497	−0.572485	−	0.819915i	$-0.694021\pi$
$263$	−18.5683	−1.14497	−0.572485	+	0.819915i	$0.694021\pi$
$264$	0	0
$265$	−42.8106	−2.62983
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	−14.3385	−0.874233	−0.437116	−	0.899405i	$-0.644000\pi$
$269$	−14.3385	−0.874233	−0.437116	+	0.899405i	$0.644000\pi$
$270$	0	0
$271$	−32.4332	−1.97018	−0.985089	−	0.172046i	$-0.944962\pi$
$271$	−32.4332	−1.97018	−0.985089	+	0.172046i	$0.944962\pi$
$272$	0	0
$273$	−20.9193	−1.26609
$274$	0	0
$275$	12.8913	0.777377
$276$	0	0
$277$	7.77018	0.466865	0.233433	−	0.972373i	$-0.425004\pi$
$277$	7.77018	0.466865	0.233433	+	0.972373i	$0.425004\pi$
$278$	0	0
$279$	−2.56829	−0.153760
$280$	0	0
$281$	9.63511	0.574783	0.287391	−	0.957813i	$-0.407212\pi$
$281$	9.63511	0.574783	0.287391	+	0.957813i	$0.407212\pi$
$282$	0	0
$283$	−9.52645	−0.566289	−0.283144	−	0.959077i	$-0.591378\pi$
$283$	−9.52645	−0.566289	−0.283144	+	0.959077i	$0.591378\pi$
$284$	0	0
$285$	5.43171	0.321746
$286$	0	0
$287$	25.5962	1.51090
$288$	0	0
$289$	−6.21438	−0.365552
$290$	0	0
$291$	−6.45963	−0.378670
$292$	0	0
$293$	−8.03889	−0.469637	−0.234819	−	0.972039i	$-0.575450\pi$
$293$	−8.03889	−0.469637	−0.234819	+	0.972039i	$0.575450\pi$
$294$	0	0
$295$	35.7827	2.08335
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−9.43171	−0.545450
$300$	0	0
$301$	−55.2702	−3.18572
$302$	0	0
$303$	4.71585	0.270919
$304$	0	0
$305$	−23.9472	−1.37121
$306$	0	0
$307$	−20.7717	−1.18550	−0.592752	−	0.805385i	$-0.701959\pi$
$307$	−20.7717	−1.18550	−0.592752	+	0.805385i	$0.701959\pi$
$308$	0	0
$309$	−3.54037	−0.201405
$310$	0	0
$311$	5.77018	0.327197	0.163599	−	0.986527i	$-0.447690\pi$
$311$	5.77018	0.327197	0.163599	+	0.986527i	$0.447690\pi$
$312$	0	0
$313$	13.4876	0.762362	0.381181	−	0.924500i	$-0.375518\pi$
$313$	13.4876	0.762362	0.381181	+	0.924500i	$0.375518\pi$
$314$	0	0
$315$	−20.9193	−1.17867
$316$	0	0
$317$	−17.1491	−0.963188	−0.481594	−	0.876394i	$-0.659942\pi$
$317$	−17.1491	−0.963188	−0.481594	+	0.876394i	$0.659942\pi$
$318$	0	0
$319$	−3.28415	−0.183877
$320$	0	0
$321$	1.43171	0.0799100
$322$	0	0
$323$	4.21733	0.234658
$324$	0	0
$325$	54.5280	3.02467
$326$	0	0
$327$	4.22982	0.233909
$328$	0	0
$329$	−50.5933	−2.78930
$330$	0	0
$331$	−22.5683	−1.24047	−0.620233	−	0.784418i	$-0.712962\pi$
$331$	−22.5683	−1.24047	−0.620233	+	0.784418i	$0.712962\pi$
$332$	0	0
$333$	−0.568295	−0.0311423
$334$	0	0
$335$	60.7019	3.31650
$336$	0	0
$337$	−5.70488	−0.310765	−0.155382	−	0.987854i	$-0.549661\pi$
$337$	−5.70488	−0.310765	−0.155382	+	0.987854i	$0.549661\pi$
$338$	0	0
$339$	7.43171	0.403635
$340$	0	0
$341$	−2.56829	−0.139081
$342$	0	0
$343$	−51.7299	−2.79315
$344$	0	0
$345$	−9.43171	−0.507786
$346$	0	0
$347$	0.919260	0.0493485	0.0246742	−	0.999696i	$-0.492145\pi$
$347$	0.919260	0.0493485	0.0246742	+	0.999696i	$0.492145\pi$
$348$	0	0
$349$	−1.20189	−0.0643357	−0.0321679	−	0.999482i	$-0.510241\pi$
$349$	−1.20189	−0.0643357	−0.0321679	+	0.999482i	$0.510241\pi$
$350$	0	0
$351$	4.22982	0.225771
$352$	0	0
$353$	21.7827	1.15937	0.579687	−	0.814839i	$-0.303175\pi$
$353$	21.7827	1.15937	0.579687	+	0.814839i	$0.303175\pi$
$354$	0	0
$355$	24.4068	1.29538
$356$	0	0
$357$	−16.2423	−0.859634
$358$	0	0
$359$	24.2423	1.27946	0.639730	−	0.768600i	$-0.279046\pi$
$359$	24.2423	1.27946	0.639730	+	0.768600i	$0.279046\pi$
$360$	0	0
$361$	−17.3510	−0.913209
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−52.2423	−2.73449
$366$	0	0
$367$	0.919260	0.0479850	0.0239925	−	0.999712i	$-0.492362\pi$
$367$	0.919260	0.0479850	0.0239925	+	0.999712i	$0.492362\pi$
$368$	0	0
$369$	−5.17548	−0.269425
$370$	0	0
$371$	50.0558	2.59877
$372$	0	0
$373$	1.66152	0.0860303	0.0430151	−	0.999074i	$-0.486304\pi$
$373$	1.66152	0.0860303	0.0430151	+	0.999074i	$0.486304\pi$
$374$	0	0
$375$	33.3789	1.72368
$376$	0	0
$377$	−13.8913	−0.715440
$378$	0	0
$379$	−34.3510	−1.76449	−0.882245	−	0.470790i	$-0.843969\pi$
$379$	−34.3510	−1.76449	−0.882245	+	0.470790i	$0.843969\pi$
$380$	0	0
$381$	−0.486038	−0.0249005
$382$	0	0
$383$	2.90677	0.148529	0.0742646	−	0.997239i	$-0.476339\pi$
$383$	2.90677	0.148529	0.0742646	+	0.997239i	$0.476339\pi$
$384$	0	0
$385$	−20.9193	−1.06614
$386$	0	0
$387$	11.1755	0.568082
$388$	0	0
$389$	−20.6894	−1.04900	−0.524498	−	0.851412i	$-0.675747\pi$
$389$	−20.6894	−1.04900	−0.524498	+	0.851412i	$0.675747\pi$
$390$	0	0
$391$	−7.32304	−0.370342
$392$	0	0
$393$	2.56829	0.129553
$394$	0	0
$395$	−2.05585	−0.103441
$396$	0	0
$397$	14.7547	0.740520	0.370260	−	0.928928i	$-0.379269\pi$
$397$	14.7547	0.740520	0.370260	+	0.928928i	$0.379269\pi$
$398$	0	0
$399$	−6.35097	−0.317946
$400$	0	0
$401$	−24.8106	−1.23898	−0.619491	−	0.785004i	$-0.712661\pi$
$401$	−24.8106	−1.23898	−0.619491	+	0.785004i	$0.712661\pi$
$402$	0	0
$403$	−10.8634	−0.541145
$404$	0	0
$405$	4.22982	0.210181
$406$	0	0
$407$	−0.568295	−0.0281693
$408$	0	0
$409$	−1.24525	−0.0615738	−0.0307869	−	0.999526i	$-0.509801\pi$
$409$	−1.24525	−0.0615738	−0.0307869	+	0.999526i	$0.509801\pi$
$410$	0	0
$411$	−13.0279	−0.642620
$412$	0	0
$413$	−41.8385	−2.05874
$414$	0	0
$415$	−33.8385	−1.66107
$416$	0	0
$417$	−15.6351	−0.765655
$418$	0	0
$419$	−24.4596	−1.19493	−0.597466	−	0.801895i	$-0.703826\pi$
$419$	−24.4596	−1.19493	−0.597466	+	0.801895i	$0.703826\pi$
$420$	0	0
$421$	−28.0558	−1.36736	−0.683679	−	0.729783i	$-0.739621\pi$
$421$	−28.0558	−1.36736	−0.683679	+	0.729783i	$0.739621\pi$
$422$	0	0
$423$	10.2298	0.497391
$424$	0	0
$425$	42.3370	2.05365
$426$	0	0
$427$	28.0000	1.35501
$428$	0	0
$429$	4.22982	0.204217
$430$	0	0
$431$	−17.2144	−0.829187	−0.414594	−	0.910007i	$-0.636076\pi$
$431$	−17.2144	−0.829187	−0.414594	+	0.910007i	$0.636076\pi$
$432$	0	0
$433$	23.4317	1.12606	0.563028	−	0.826438i	$-0.309636\pi$
$433$	23.4317	1.12606	0.563028	+	0.826438i	$0.309636\pi$
$434$	0	0
$435$	−13.8913	−0.666039
$436$	0	0
$437$	−2.86341	−0.136976
$438$	0	0
$439$	−13.9177	−0.664258	−0.332129	−	0.943234i	$-0.607767\pi$
$439$	−13.9177	−0.664258	−0.332129	+	0.943234i	$0.607767\pi$
$440$	0	0
$441$	17.4596	0.831411
$442$	0	0
$443$	28.2423	1.34183	0.670916	−	0.741533i	$-0.265901\pi$
$443$	28.2423	1.34183	0.670916	+	0.741533i	$0.265901\pi$
$444$	0	0
$445$	8.45963	0.401025
$446$	0	0
$447$	−22.0947	−1.04505
$448$	0	0
$449$	34.2423	1.61599	0.807997	−	0.589187i	$-0.200552\pi$
$449$	34.2423	1.61599	0.807997	+	0.589187i	$0.200552\pi$
$450$	0	0
$451$	−5.17548	−0.243704
$452$	0	0
$453$	19.2966	0.906635
$454$	0	0
$455$	−88.4846	−4.14822
$456$	0	0
$457$	20.8106	0.973479	0.486739	−	0.873547i	$-0.338186\pi$
$457$	20.8106	0.973479	0.486739	+	0.873547i	$0.338186\pi$
$458$	0	0
$459$	3.28415	0.153291
$460$	0	0
$461$	−12.2562	−0.570829	−0.285415	−	0.958404i	$-0.592131\pi$
$461$	−12.2562	−0.570829	−0.285415	+	0.958404i	$0.592131\pi$
$462$	0	0
$463$	8.24230	0.383052	0.191526	−	0.981488i	$-0.438656\pi$
$463$	8.24230	0.383052	0.191526	+	0.981488i	$0.438656\pi$
$464$	0	0
$465$	−10.8634	−0.503779
$466$	0	0
$467$	−12.9193	−0.597832	−0.298916	−	0.954279i	$-0.596625\pi$
$467$	−12.9193	−0.597832	−0.298916	+	0.954279i	$0.596625\pi$
$468$	0	0
$469$	−70.9751	−3.27733
$470$	0	0
$471$	−2.45963	−0.113334
$472$	0	0
$473$	11.1755	0.513849
$474$	0	0
$475$	16.5544	0.759567
$476$	0	0
$477$	−10.1212	−0.463416
$478$	0	0
$479$	−36.4068	−1.66347	−0.831735	−	0.555173i	$-0.812652\pi$
$479$	−36.4068	−1.66347	−0.831735	+	0.555173i	$0.812652\pi$
$480$	0	0
$481$	−2.40378	−0.109603
$482$	0	0
$483$	11.0279	0.501788
$484$	0	0
$485$	−27.3230	−1.24068
$486$	0	0
$487$	33.8913	1.53576	0.767882	−	0.640592i	$-0.221311\pi$
$487$	33.8913	1.53576	0.767882	+	0.640592i	$0.221311\pi$
$488$	0	0
$489$	−18.3510	−0.829859
$490$	0	0
$491$	−10.3510	−0.467133	−0.233566	−	0.972341i	$-0.575040\pi$
$491$	−10.3510	−0.467133	−0.233566	+	0.972341i	$0.575040\pi$
$492$	0	0
$493$	−10.7856	−0.485760
$494$	0	0
$495$	4.22982	0.190116
$496$	0	0
$497$	−28.5374	−1.28008
$498$	0	0
$499$	−4.91926	−0.220216	−0.110108	−	0.993920i	$-0.535120\pi$
$499$	−4.91926	−0.220216	−0.110108	+	0.993920i	$0.535120\pi$
$500$	0	0
$501$	−7.02792	−0.313985
$502$	0	0
$503$	−9.89134	−0.441033	−0.220516	−	0.975383i	$-0.570774\pi$
$503$	−9.89134	−0.441033	−0.220516	+	0.975383i	$0.570774\pi$
$504$	0	0
$505$	19.9472	0.887638
$506$	0	0
$507$	4.89134	0.217232
$508$	0	0
$509$	−12.9846	−0.575531	−0.287765	−	0.957701i	$-0.592912\pi$
$509$	−12.9846	−0.575531	−0.287765	+	0.957701i	$0.592912\pi$
$510$	0	0
$511$	61.0838	2.70219
$512$	0	0
$513$	1.28415	0.0566965
$514$	0	0
$515$	−14.9751	−0.659882
$516$	0	0
$517$	10.2298	0.449907
$518$	0	0
$519$	−3.28415	−0.144158
$520$	0	0
$521$	−21.0279	−0.921250	−0.460625	−	0.887595i	$-0.652375\pi$
$521$	−21.0279	−0.921250	−0.460625	+	0.887595i	$0.652375\pi$
$522$	0	0
$523$	−29.7438	−1.30060	−0.650302	−	0.759676i	$-0.725358\pi$
$523$	−29.7438	−1.30060	−0.650302	+	0.759676i	$0.725358\pi$
$524$	0	0
$525$	−63.7563	−2.78255
$526$	0	0
$527$	−8.43466	−0.367419
$528$	0	0
$529$	−18.0279	−0.783823
$530$	0	0
$531$	8.45963	0.367117
$532$	0	0
$533$	−21.8913	−0.948219
$534$	0	0
$535$	6.05585	0.261817
$536$	0	0
$537$	−16.4596	−0.710285
$538$	0	0
$539$	17.4596	0.752040
$540$	0	0
$541$	−15.5529	−0.668670	−0.334335	−	0.942454i	$-0.608512\pi$
$541$	−15.5529	−0.668670	−0.334335	+	0.942454i	$0.608512\pi$
$542$	0	0
$543$	14.4596	0.620522
$544$	0	0
$545$	17.8913	0.766381
$546$	0	0
$547$	18.4985	0.790940	0.395470	−	0.918479i	$-0.370582\pi$
$547$	18.4985	0.790940	0.395470	+	0.918479i	$0.370582\pi$
$548$	0	0
$549$	−5.66152	−0.241628
$550$	0	0
$551$	−4.21733	−0.179664
$552$	0	0
$553$	2.40378	0.102219
$554$	0	0
$555$	−2.40378	−0.102035
$556$	0	0
$557$	39.5264	1.67479	0.837395	−	0.546599i	$-0.184078\pi$
$557$	39.5264	1.67479	0.837395	+	0.546599i	$0.184078\pi$
$558$	0	0
$559$	47.2702	1.99932
$560$	0	0
$561$	3.28415	0.138657
$562$	0	0
$563$	−18.6461	−0.785839	−0.392919	−	0.919573i	$-0.628535\pi$
$563$	−18.6461	−0.785839	−0.392919	+	0.919573i	$0.628535\pi$
$564$	0	0
$565$	31.4347	1.32247
$566$	0	0
$567$	−4.94567	−0.207699
$568$	0	0
$569$	22.1476	0.928474	0.464237	−	0.885711i	$-0.346329\pi$
$569$	22.1476	0.928474	0.464237	+	0.885711i	$0.346329\pi$
$570$	0	0
$571$	19.8524	0.830799	0.415399	−	0.909639i	$-0.363642\pi$
$571$	19.8524	0.830799	0.415399	+	0.909639i	$0.363642\pi$
$572$	0	0
$573$	19.1491	0.799964
$574$	0	0
$575$	−28.7453	−1.19876
$576$	0	0
$577$	2.75475	0.114682	0.0573408	−	0.998355i	$-0.481738\pi$
$577$	2.75475	0.114682	0.0573408	+	0.998355i	$0.481738\pi$
$578$	0	0
$579$	−5.32304	−0.221218
$580$	0	0
$581$	39.5653	1.64145
$582$	0	0
$583$	−10.1212	−0.419175
$584$	0	0
$585$	17.8913	0.739716
$586$	0	0
$587$	−3.32304	−0.137157	−0.0685783	−	0.997646i	$-0.521846\pi$
$587$	−3.32304	−0.137157	−0.0685783	+	0.997646i	$0.521846\pi$
$588$	0	0
$589$	−3.29807	−0.135895
$590$	0	0
$591$	12.7159	0.523060
$592$	0	0
$593$	33.6351	1.38123	0.690614	−	0.723223i	$-0.257340\pi$
$593$	33.6351	1.38123	0.690614	+	0.723223i	$0.257340\pi$
$594$	0	0
$595$	−68.7019	−2.81650
$596$	0	0
$597$	−9.89134	−0.404825
$598$	0	0
$599$	5.77018	0.235763	0.117882	−	0.993028i	$-0.462390\pi$
$599$	5.77018	0.235763	0.117882	+	0.993028i	$0.462390\pi$
$600$	0	0
$601$	−26.7019	−1.08919	−0.544597	−	0.838698i	$-0.683317\pi$
$601$	−26.7019	−1.08919	−0.544597	+	0.838698i	$0.683317\pi$
$602$	0	0
$603$	14.3510	0.584417
$604$	0	0
$605$	4.22982	0.171966
$606$	0	0
$607$	6.16004	0.250028	0.125014	−	0.992155i	$-0.460102\pi$
$607$	6.16004	0.250028	0.125014	+	0.992155i	$0.460102\pi$
$608$	0	0
$609$	16.2423	0.658171
$610$	0	0
$611$	43.2702	1.75053
$612$	0	0
$613$	18.3385	0.740684	0.370342	−	0.928895i	$-0.379240\pi$
$613$	18.3385	0.740684	0.370342	+	0.928895i	$0.379240\pi$
$614$	0	0
$615$	−21.8913	−0.882744
$616$	0	0
$617$	−10.0778	−0.405716	−0.202858	−	0.979208i	$-0.565023\pi$
$617$	−10.0778	−0.405716	−0.202858	+	0.979208i	$0.565023\pi$
$618$	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
$621$	−2.22982	−0.0894794
$622$	0	0
$623$	−9.89134	−0.396288
$624$	0	0
$625$	76.7299	3.06919
$626$	0	0
$627$	1.28415	0.0512839
$628$	0	0
$629$	−1.86636	−0.0744168
$630$	0	0
$631$	−7.70488	−0.306727	−0.153363	−	0.988170i	$-0.549010\pi$
$631$	−7.70488	−0.306727	−0.153363	+	0.988170i	$0.549010\pi$
$632$	0	0
$633$	−16.3121	−0.648347
$634$	0	0
$635$	−2.05585	−0.0815839
$636$	0	0
$637$	73.8510	2.92608
$638$	0	0
$639$	5.77018	0.228265
$640$	0	0
$641$	−39.1616	−1.54679	−0.773394	−	0.633925i	$-0.781443\pi$
$641$	−39.1616	−1.54679	−0.773394	+	0.633925i	$0.781443\pi$
$642$	0	0
$643$	−22.0558	−0.869798	−0.434899	−	0.900479i	$-0.643216\pi$
$643$	−22.0558	−0.869798	−0.434899	+	0.900479i	$0.643216\pi$
$644$	0	0
$645$	47.2702	1.86126
$646$	0	0
$647$	25.5529	1.00459	0.502293	−	0.864697i	$-0.332490\pi$
$647$	25.5529	1.00459	0.502293	+	0.864697i	$0.332490\pi$
$648$	0	0
$649$	8.45963	0.332070
$650$	0	0
$651$	12.7019	0.497828
$652$	0	0
$653$	−9.87885	−0.386589	−0.193295	−	0.981141i	$-0.561917\pi$
$653$	−9.87885	−0.386589	−0.193295	+	0.981141i	$0.561917\pi$
$654$	0	0
$655$	10.8634	0.424469
$656$	0	0
$657$	−12.3510	−0.481857
$658$	0	0
$659$	23.7827	0.926441	0.463221	−	0.886243i	$-0.346694\pi$
$659$	23.7827	0.926441	0.463221	+	0.886243i	$0.346694\pi$
$660$	0	0
$661$	−34.0250	−1.32342	−0.661709	−	0.749761i	$-0.730169\pi$
$661$	−34.0250	−1.32342	−0.661709	+	0.749761i	$0.730169\pi$
$662$	0	0
$663$	13.8913	0.539495
$664$	0	0
$665$	−26.8634	−1.04172
$666$	0	0
$667$	7.32304	0.283549
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.66152	−0.218561
$672$	0	0
$673$	36.5155	1.40757	0.703784	−	0.710414i	$-0.251492\pi$
$673$	36.5155	1.40757	0.703784	+	0.710414i	$0.251492\pi$
$674$	0	0
$675$	12.8913	0.496188
$676$	0	0
$677$	48.4985	1.86395	0.931975	−	0.362523i	$-0.118085\pi$
$677$	48.4985	1.86395	0.931975	+	0.362523i	$0.118085\pi$
$678$	0	0
$679$	31.9472	1.22602
$680$	0	0
$681$	21.2144	0.812937
$682$	0	0
$683$	−38.5155	−1.47375	−0.736877	−	0.676027i	$-0.763700\pi$
$683$	−38.5155	−1.47375	−0.736877	+	0.676027i	$0.763700\pi$
$684$	0	0
$685$	−55.1057	−2.10548
$686$	0	0
$687$	17.0279	0.649656
$688$	0	0
$689$	−42.8106	−1.63095
$690$	0	0
$691$	28.7019	1.09187	0.545936	−	0.837827i	$-0.316174\pi$
$691$	28.7019	1.09187	0.545936	+	0.837827i	$0.316174\pi$
$692$	0	0
$693$	−4.94567	−0.187870
$694$	0	0
$695$	−66.1336	−2.50859
$696$	0	0
$697$	−16.9970	−0.643809
$698$	0	0
$699$	−1.39281	−0.0526810
$700$	0	0
$701$	14.6072	0.551706	0.275853	−	0.961200i	$-0.411040\pi$
$701$	14.6072	0.551706	0.275853	+	0.961200i	$0.411040\pi$
$702$	0	0
$703$	−0.729774	−0.0275239
$704$	0	0
$705$	43.2702	1.62965
$706$	0	0
$707$	−23.3230	−0.877153
$708$	0	0
$709$	−32.8106	−1.23223	−0.616114	−	0.787657i	$-0.711294\pi$
$709$	−32.8106	−1.23223	−0.616114	+	0.787657i	$0.711294\pi$
$710$	0	0
$711$	−0.486038	−0.0182278
$712$	0	0
$713$	5.72682	0.214471
$714$	0	0
$715$	17.8913	0.669098
$716$	0	0
$717$	−1.89134	−0.0706332
$718$	0	0
$719$	−7.36640	−0.274721	−0.137360	−	0.990521i	$-0.543862\pi$
$719$	−7.36640	−0.274721	−0.137360	+	0.990521i	$0.543862\pi$
$720$	0	0
$721$	17.5095	0.652088
$722$	0	0
$723$	11.8913	0.442244
$724$	0	0
$725$	−42.3370	−1.57236
$726$	0	0
$727$	33.8913	1.25696	0.628480	−	0.777826i	$-0.283677\pi$
$727$	33.8913	1.25696	0.628480	+	0.777826i	$0.283677\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	36.7019	1.35747
$732$	0	0
$733$	20.2298	0.747205	0.373603	−	0.927589i	$-0.378122\pi$
$733$	20.2298	0.747205	0.373603	+	0.927589i	$0.378122\pi$
$734$	0	0
$735$	73.8510	2.72404
$736$	0	0
$737$	14.3510	0.528625
$738$	0	0
$739$	−24.6072	−0.905190	−0.452595	−	0.891716i	$-0.649502\pi$
$739$	−24.6072	−0.905190	−0.452595	+	0.891716i	$0.649502\pi$
$740$	0	0
$741$	5.43171	0.199539
$742$	0	0
$743$	−31.3230	−1.14913	−0.574565	−	0.818459i	$-0.694829\pi$
$743$	−31.3230	−1.14913	−0.574565	+	0.818459i	$0.694829\pi$
$744$	0	0
$745$	−93.4567	−3.42399
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	−7.08074	−0.258725
$750$	0	0
$751$	−50.1336	−1.82940	−0.914701	−	0.404131i	$-0.867574\pi$
$751$	−50.1336	−1.82940	−0.914701	+	0.404131i	$0.867574\pi$
$752$	0	0
$753$	4.67696	0.170438
$754$	0	0
$755$	81.6212	2.97050
$756$	0	0
$757$	−45.5653	−1.65610	−0.828050	−	0.560654i	$-0.810550\pi$
$757$	−45.5653	−1.65610	−0.828050	+	0.560654i	$0.810550\pi$
$758$	0	0
$759$	−2.22982	−0.0809372
$760$	0	0
$761$	34.8495	1.26329	0.631647	−	0.775257i	$-0.282379\pi$
$761$	34.8495	1.26329	0.631647	+	0.775257i	$0.282379\pi$
$762$	0	0
$763$	−20.9193	−0.757328
$764$	0	0
$765$	13.8913	0.502242
$766$	0	0
$767$	35.7827	1.29204
$768$	0	0
$769$	23.3789	0.843064	0.421532	−	0.906813i	$-0.361492\pi$
$769$	23.3789	0.843064	0.421532	+	0.906813i	$0.361492\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−3.04041	−0.109356	−0.0546780	−	0.998504i	$-0.517413\pi$
$773$	−3.04041	−0.109356	−0.0546780	+	0.998504i	$0.517413\pi$
$774$	0	0
$775$	−33.1087	−1.18930
$776$	0	0
$777$	2.81060	0.100830
$778$	0	0
$779$	−6.64608	−0.238121
$780$	0	0
$781$	5.77018	0.206473
$782$	0	0
$783$	−3.28415	−0.117366
$784$	0	0
$785$	−10.4038	−0.371327
$786$	0	0
$787$	−28.3899	−1.01199	−0.505995	−	0.862537i	$-0.668874\pi$
$787$	−28.3899	−1.01199	−0.505995	+	0.862537i	$0.668874\pi$
$788$	0	0
$789$	−18.5683	−0.661049
$790$	0	0
$791$	−36.7547	−1.30685
$792$	0	0
$793$	−23.9472	−0.850389
$794$	0	0
$795$	−42.8106	−1.51834
$796$	0	0
$797$	12.2298	0.433202	0.216601	−	0.976260i	$-0.430503\pi$
$797$	12.2298	0.433202	0.216601	+	0.976260i	$0.430503\pi$
$798$	0	0
$799$	33.5962	1.18855
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−12.3510	−0.435856
$804$	0	0
$805$	46.6461	1.64406
$806$	0	0
$807$	−14.3385	−0.504738
$808$	0	0
$809$	0.256223	0.00900831	0.00450415	−	0.999990i	$-0.498566\pi$
$809$	0.256223	0.00900831	0.00450415	+	0.999990i	$0.498566\pi$
$810$	0	0
$811$	29.7438	1.04445	0.522223	−	0.852809i	$-0.325103\pi$
$811$	29.7438	1.04445	0.522223	+	0.852809i	$0.325103\pi$
$812$	0	0
$813$	−32.4332	−1.13748
$814$	0	0
$815$	−77.6212	−2.71895
$816$	0	0
$817$	14.3510	0.502077
$818$	0	0
$819$	−20.9193	−0.730978
$820$	0	0
$821$	32.4985	1.13421	0.567103	−	0.823647i	$-0.308064\pi$
$821$	32.4985	1.13421	0.567103	+	0.823647i	$0.308064\pi$
$822$	0	0
$823$	−23.5653	−0.821436	−0.410718	−	0.911762i	$-0.634722\pi$
$823$	−23.5653	−0.821436	−0.410718	+	0.911762i	$0.634722\pi$
$824$	0	0
$825$	12.8913	0.448819
$826$	0	0
$827$	37.6212	1.30822	0.654109	−	0.756401i	$-0.273044\pi$
$827$	37.6212	1.30822	0.654109	+	0.756401i	$0.273044\pi$
$828$	0	0
$829$	−22.6241	−0.785769	−0.392884	−	0.919588i	$-0.628523\pi$
$829$	−22.6241	−0.785769	−0.392884	+	0.919588i	$0.628523\pi$
$830$	0	0
$831$	7.77018	0.269545
$832$	0	0
$833$	57.3400	1.98671
$834$	0	0
$835$	−29.7268	−1.02874
$836$	0	0
$837$	−2.56829	−0.0887732
$838$	0	0
$839$	37.4223	1.29196	0.645980	−	0.763354i	$-0.276449\pi$
$839$	37.4223	1.29196	0.645980	+	0.763354i	$0.276449\pi$
$840$	0	0
$841$	−18.2144	−0.628082
$842$	0	0
$843$	9.63511	0.331851
$844$	0	0
$845$	20.6894	0.711739
$846$	0	0
$847$	−4.94567	−0.169935
$848$	0	0
$849$	−9.52645	−0.326947
$850$	0	0
$851$	1.26719	0.0434388
$852$	0	0
$853$	16.9846	0.581540	0.290770	−	0.956793i	$-0.406089\pi$
$853$	16.9846	0.581540	0.290770	+	0.956793i	$0.406089\pi$
$854$	0	0
$855$	5.43171	0.185760
$856$	0	0
$857$	−36.8804	−1.25981	−0.629905	−	0.776672i	$-0.716906\pi$
$857$	−36.8804	−1.25981	−0.629905	+	0.776672i	$0.716906\pi$
$858$	0	0
$859$	26.3510	0.899083	0.449542	−	0.893259i	$-0.351587\pi$
$859$	26.3510	0.899083	0.449542	+	0.893259i	$0.351587\pi$
$860$	0	0
$861$	25.5962	0.872317
$862$	0	0
$863$	40.8759	1.39143	0.695716	−	0.718317i	$-0.255087\pi$
$863$	40.8759	1.39143	0.695716	+	0.718317i	$0.255087\pi$
$864$	0	0
$865$	−13.8913	−0.472320
$866$	0	0
$867$	−6.21438	−0.211051
$868$	0	0
$869$	−0.486038	−0.0164877
$870$	0	0
$871$	60.7019	2.05681
$872$	0	0
$873$	−6.45963	−0.218625
$874$	0	0
$875$	−165.081	−5.58075
$876$	0	0
$877$	5.87885	0.198515	0.0992573	−	0.995062i	$-0.468353\pi$
$877$	5.87885	0.198515	0.0992573	+	0.995062i	$0.468353\pi$
$878$	0	0
$879$	−8.03889	−0.271145
$880$	0	0
$881$	49.2702	1.65996	0.829978	−	0.557796i	$-0.188353\pi$
$881$	49.2702	1.65996	0.829978	+	0.557796i	$0.188353\pi$
$882$	0	0
$883$	−31.7049	−1.06695	−0.533477	−	0.845814i	$-0.679115\pi$
$883$	−31.7049	−1.06695	−0.533477	+	0.845814i	$0.679115\pi$
$884$	0	0
$885$	35.7827	1.20282
$886$	0	0
$887$	28.0250	0.940986	0.470493	−	0.882404i	$-0.344076\pi$
$887$	28.0250	0.940986	0.470493	+	0.882404i	$0.344076\pi$
$888$	0	0
$889$	2.40378	0.0806202
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	13.1366	0.439599
$894$	0	0
$895$	−69.6212	−2.32718
$896$	0	0
$897$	−9.43171	−0.314915
$898$	0	0
$899$	8.43466	0.281312
$900$	0	0
$901$	−33.2393	−1.10736
$902$	0	0
$903$	−55.2702	−1.83928
$904$	0	0
$905$	61.1616	2.03308
$906$	0	0
$907$	35.1924	1.16855	0.584273	−	0.811557i	$-0.301380\pi$
$907$	35.1924	1.16855	0.584273	+	0.811557i	$0.301380\pi$
$908$	0	0
$909$	4.71585	0.156415
$910$	0	0
$911$	−38.9317	−1.28987	−0.644933	−	0.764239i	$-0.723115\pi$
$911$	−38.9317	−1.28987	−0.644933	+	0.764239i	$0.723115\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	−23.9472	−0.791670
$916$	0	0
$917$	−12.7019	−0.419455
$918$	0	0
$919$	−4.65055	−0.153408	−0.0767038	−	0.997054i	$-0.524440\pi$
$919$	−4.65055	−0.153408	−0.0767038	+	0.997054i	$0.524440\pi$
$920$	0	0
$921$	−20.7717	−0.684451
$922$	0	0
$923$	24.4068	0.803360
$924$	0	0
$925$	−7.32608	−0.240880
$926$	0	0
$927$	−3.54037	−0.116281
$928$	0	0
$929$	−12.1087	−0.397272	−0.198636	−	0.980073i	$-0.563651\pi$
$929$	−12.1087	−0.397272	−0.198636	+	0.980073i	$0.563651\pi$
$930$	0	0
$931$	22.4207	0.734810
$932$	0	0
$933$	5.77018	0.188907
$934$	0	0
$935$	13.8913	0.454295
$936$	0	0
$937$	−42.7019	−1.39501	−0.697506	−	0.716579i	$-0.745707\pi$
$937$	−42.7019	−1.39501	−0.697506	+	0.716579i	$0.745707\pi$
$938$	0	0
$939$	13.4876	0.440150
$940$	0	0
$941$	17.8524	0.581973	0.290986	−	0.956727i	$-0.406017\pi$
$941$	17.8524	0.581973	0.290986	+	0.956727i	$0.406017\pi$
$942$	0	0
$943$	11.5404	0.375806
$944$	0	0
$945$	−20.9193	−0.680504
$946$	0	0
$947$	−39.7827	−1.29276	−0.646382	−	0.763014i	$-0.723719\pi$
$947$	−39.7827	−1.29276	−0.646382	+	0.763014i	$0.723719\pi$
$948$	0	0
$949$	−52.2423	−1.69586
$950$	0	0
$951$	−17.1491	−0.556097
$952$	0	0
$953$	−28.4457	−0.921447	−0.460723	−	0.887544i	$-0.652410\pi$
$953$	−28.4457	−0.921447	−0.460723	+	0.887544i	$0.652410\pi$
$954$	0	0
$955$	80.9970	2.62100
$956$	0	0
$957$	−3.28415	−0.106161
$958$	0	0
$959$	64.4318	2.08061
$960$	0	0
$961$	−24.4039	−0.787221
$962$	0	0
$963$	1.43171	0.0461361
$964$	0	0
$965$	−22.5155	−0.724799
$966$	0	0
$967$	22.1600	0.712619	0.356309	−	0.934368i	$-0.384035\pi$
$967$	22.1600	0.712619	0.356309	+	0.934368i	$0.384035\pi$
$968$	0	0
$969$	4.21733	0.135480
$970$	0	0
$971$	3.75770	0.120590	0.0602951	−	0.998181i	$-0.480796\pi$
$971$	3.75770	0.120590	0.0602951	+	0.998181i	$0.480796\pi$
$972$	0	0
$973$	77.3261	2.47896
$974$	0	0
$975$	54.5280	1.74629
$976$	0	0
$977$	−0.325993	−0.0104294	−0.00521472	−	0.999986i	$-0.501660\pi$
$977$	−0.325993	−0.0104294	−0.00521472	+	0.999986i	$0.501660\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	4.22982	0.135048
$982$	0	0
$983$	18.4721	0.589169	0.294585	−	0.955625i	$-0.404819\pi$
$983$	18.4721	0.589169	0.294585	+	0.955625i	$0.404819\pi$
$984$	0	0
$985$	53.7857	1.71376
$986$	0	0
$987$	−50.5933	−1.61040
$988$	0	0
$989$	−24.9193	−0.792386
$990$	0	0
$991$	−26.1336	−0.830162	−0.415081	−	0.909784i	$-0.636247\pi$
$991$	−26.1336	−0.830162	−0.415081	+	0.909784i	$0.636247\pi$
$992$	0	0
$993$	−22.5683	−0.716183
$994$	0	0
$995$	−41.8385	−1.32637
$996$	0	0
$997$	17.6615	0.559346	0.279673	−	0.960095i	$-0.409774\pi$
$997$	17.6615	0.559346	0.279673	+	0.960095i	$0.409774\pi$
$998$	0	0
$999$	−0.568295	−0.0179800

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.22982 q^{5} -4.94567 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.22982 q^{5} -4.94567 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.22982 q^{13} +4.22982 q^{15} +3.28415 q^{17} +1.28415 q^{19} -4.94567 q^{21} -2.22982 q^{23} +12.8913 q^{25} +1.00000 q^{27} -3.28415 q^{29} -2.56829 q^{31} +1.00000 q^{33} -20.9193 q^{35} -0.568295 q^{37} +4.22982 q^{39} -5.17548 q^{41} +11.1755 q^{43} +4.22982 q^{45} +10.2298 q^{47} +17.4596 q^{49} +3.28415 q^{51} -10.1212 q^{53} +4.22982 q^{55} +1.28415 q^{57} +8.45963 q^{59} -5.66152 q^{61} -4.94567 q^{63} +17.8913 q^{65} +14.3510 q^{67} -2.22982 q^{69} +5.77018 q^{71} -12.3510 q^{73} +12.8913 q^{75} -4.94567 q^{77} -0.486038 q^{79} +1.00000 q^{81} -8.00000 q^{83} +13.8913 q^{85} -3.28415 q^{87} +2.00000 q^{89} -20.9193 q^{91} -2.56829 q^{93} +5.43171 q^{95} -6.45963 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 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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