LMFDB - Embedded newform 2112.2.a.be.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 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			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.23607 q^{5} -1.23607 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.23607 q^{5} -1.23607 q^{7} +1.00000 q^{9} +1.00000 q^{11} -0.763932 q^{13} +3.23607 q^{15} +4.47214 q^{17} +2.47214 q^{19} +1.23607 q^{21} +7.70820 q^{23} +5.47214 q^{25} -1.00000 q^{27} +0.472136 q^{29} -1.00000 q^{33} +4.00000 q^{35} -6.00000 q^{37} +0.763932 q^{39} -6.94427 q^{41} -4.94427 q^{43} -3.23607 q^{45} -7.70820 q^{47} -5.47214 q^{49} -4.47214 q^{51} -11.2361 q^{53} -3.23607 q^{55} -2.47214 q^{57} +14.4721 q^{59} +1.70820 q^{61} -1.23607 q^{63} +2.47214 q^{65} -8.00000 q^{67} -7.70820 q^{69} +7.70820 q^{71} -6.00000 q^{73} -5.47214 q^{75} -1.23607 q^{77} -11.7082 q^{79} +1.00000 q^{81} +12.9443 q^{83} -14.4721 q^{85} -0.472136 q^{87} +2.00000 q^{89} +0.944272 q^{91} -8.00000 q^{95} +0.472136 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 6 q^{13} + 2 q^{15} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 2 q^{33} + 8 q^{35} - 12 q^{37} + 6 q^{39} + 4 q^{41} + 8 q^{43} - 2 q^{45} - 2 q^{47} - 2 q^{49} - 18 q^{53} - 2 q^{55} + 4 q^{57} + 20 q^{59} - 10 q^{61} + 2 q^{63} - 4 q^{65} - 16 q^{67} - 2 q^{69} + 2 q^{71} - 12 q^{73} - 2 q^{75} + 2 q^{77} - 10 q^{79} + 2 q^{81} + 8 q^{83} - 20 q^{85} + 8 q^{87} + 4 q^{89} - 16 q^{91} - 16 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 6 * q^13 + 2 * q^15 - 4 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 - 2 * q^33 + 8 * q^35 - 12 * q^37 + 6 * q^39 + 4 * q^41 + 8 * q^43 - 2 * q^45 - 2 * q^47 - 2 * q^49 - 18 * q^53 - 2 * q^55 + 4 * q^57 + 20 * q^59 - 10 * q^61 + 2 * q^63 - 4 * q^65 - 16 * q^67 - 2 * q^69 + 2 * q^71 - 12 * q^73 - 2 * q^75 + 2 * q^77 - 10 * q^79 + 2 * q^81 + 8 * q^83 - 20 * q^85 + 8 * q^87 + 4 * q^89 - 16 * q^91 - 16 * q^95 - 8 * q^97 + 2 * 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
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	0	0
$15$	3.23607	0.835549
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	0	0
$23$	7.70820	1.60727	0.803636	−	0.595121i	$-0.202896\pi$
$23$	7.70820	1.60727	0.803636	+	0.595121i	$0.202896\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0.763932	0.122327
$40$	0	0
$41$	−6.94427	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$41$	−6.94427	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$42$	0	0
$43$	−4.94427	−0.753994	−0.376997	−	0.926214i	$-0.623043\pi$
$43$	−4.94427	−0.753994	−0.376997	+	0.926214i	$0.623043\pi$
$44$	0	0
$45$	−3.23607	−0.482405
$46$	0	0
$47$	−7.70820	−1.12436	−0.562179	−	0.827016i	$-0.690037\pi$
$47$	−7.70820	−1.12436	−0.562179	+	0.827016i	$0.690037\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	−11.2361	−1.54339	−0.771696	−	0.635991i	$-0.780591\pi$
$53$	−11.2361	−1.54339	−0.771696	+	0.635991i	$0.780591\pi$
$54$	0	0
$55$	−3.23607	−0.436351
$56$	0	0
$57$	−2.47214	−0.327442
$58$	0	0
$59$	14.4721	1.88411	0.942056	−	0.335456i	$-0.108890\pi$
$59$	14.4721	1.88411	0.942056	+	0.335456i	$0.108890\pi$
$60$	0	0
$61$	1.70820	0.218713	0.109357	−	0.994003i	$-0.465121\pi$
$61$	1.70820	0.218713	0.109357	+	0.994003i	$0.465121\pi$
$62$	0	0
$63$	−1.23607	−0.155730
$64$	0	0
$65$	2.47214	0.306631
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−7.70820	−0.927959
$70$	0	0
$71$	7.70820	0.914796	0.457398	−	0.889262i	$-0.348782\pi$
$71$	7.70820	0.914796	0.457398	+	0.889262i	$0.348782\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	−5.47214	−0.631868
$76$	0	0
$77$	−1.23607	−0.140863
$78$	0	0
$79$	−11.7082	−1.31728	−0.658638	−	0.752460i	$-0.728867\pi$
$79$	−11.7082	−1.31728	−0.658638	+	0.752460i	$0.728867\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9443	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$83$	12.9443	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$84$	0	0
$85$	−14.4721	−1.56972
$86$	0	0
$87$	−0.472136	−0.0506183
$88$	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
$91$	0.944272	0.0989866
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−12.4721	−1.24102	−0.620512	−	0.784197i	$-0.713075\pi$
$101$	−12.4721	−1.24102	−0.620512	+	0.784197i	$0.713075\pi$
$102$	0	0
$103$	2.47214	0.243587	0.121793	−	0.992555i	$-0.461135\pi$
$103$	2.47214	0.243587	0.121793	+	0.992555i	$0.461135\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−0.944272	−0.0912862	−0.0456431	−	0.998958i	$-0.514534\pi$
$107$	−0.944272	−0.0912862	−0.0456431	+	0.998958i	$0.514534\pi$
$108$	0	0
$109$	−10.6525	−1.02032	−0.510161	−	0.860079i	$-0.670414\pi$
$109$	−10.6525	−1.02032	−0.510161	+	0.860079i	$0.670414\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−10.9443	−1.02955	−0.514775	−	0.857325i	$-0.672125\pi$
$113$	−10.9443	−1.02955	−0.514775	+	0.857325i	$0.672125\pi$
$114$	0	0
$115$	−24.9443	−2.32607
$116$	0	0
$117$	−0.763932	−0.0706255
$118$	0	0
$119$	−5.52786	−0.506738
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.94427	0.626144
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	14.1803	1.25830	0.629151	−	0.777283i	$-0.283403\pi$
$127$	14.1803	1.25830	0.629151	+	0.777283i	$0.283403\pi$
$128$	0	0
$129$	4.94427	0.435319
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	−3.05573	−0.264965
$134$	0	0
$135$	3.23607	0.278516
$136$	0	0
$137$	−16.4721	−1.40731	−0.703655	−	0.710542i	$-0.748450\pi$
$137$	−16.4721	−1.40731	−0.703655	+	0.710542i	$0.748450\pi$
$138$	0	0
$139$	−10.4721	−0.888235	−0.444117	−	0.895969i	$-0.646483\pi$
$139$	−10.4721	−0.888235	−0.444117	+	0.895969i	$0.646483\pi$
$140$	0	0
$141$	7.70820	0.649148
$142$	0	0
$143$	−0.763932	−0.0638832
$144$	0	0
$145$	−1.52786	−0.126882
$146$	0	0
$147$	5.47214	0.451334
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−6.76393	−0.550441	−0.275220	−	0.961381i	$-0.588751\pi$
$151$	−6.76393	−0.550441	−0.275220	+	0.961381i	$0.588751\pi$
$152$	0	0
$153$	4.47214	0.361551
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.52786	0.600789	0.300394	−	0.953815i	$-0.402882\pi$
$157$	7.52786	0.600789	0.300394	+	0.953815i	$0.402882\pi$
$158$	0	0
$159$	11.2361	0.891078
$160$	0	0
$161$	−9.52786	−0.750901
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	3.23607	0.251928
$166$	0	0
$167$	−20.3607	−1.57556	−0.787778	−	0.615959i	$-0.788769\pi$
$167$	−20.3607	−1.57556	−0.787778	+	0.615959i	$0.788769\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	2.47214	0.189049
$172$	0	0
$173$	16.4721	1.25235	0.626177	−	0.779681i	$-0.284619\pi$
$173$	16.4721	1.25235	0.626177	+	0.779681i	$0.284619\pi$
$174$	0	0
$175$	−6.76393	−0.511305
$176$	0	0
$177$	−14.4721	−1.08779
$178$	0	0
$179$	1.52786	0.114198	0.0570990	−	0.998369i	$-0.481815\pi$
$179$	1.52786	0.114198	0.0570990	+	0.998369i	$0.481815\pi$
$180$	0	0
$181$	−0.472136	−0.0350936	−0.0175468	−	0.999846i	$-0.505586\pi$
$181$	−0.472136	−0.0350936	−0.0175468	+	0.999846i	$0.505586\pi$
$182$	0	0
$183$	−1.70820	−0.126274
$184$	0	0
$185$	19.4164	1.42752
$186$	0	0
$187$	4.47214	0.327035
$188$	0	0
$189$	1.23607	0.0899107
$190$	0	0
$191$	−2.76393	−0.199991	−0.0999956	−	0.994988i	$-0.531883\pi$
$191$	−2.76393	−0.199991	−0.0999956	+	0.994988i	$0.531883\pi$
$192$	0	0
$193$	−16.4721	−1.18569	−0.592845	−	0.805316i	$-0.701995\pi$
$193$	−16.4721	−1.18569	−0.592845	+	0.805316i	$0.701995\pi$
$194$	0	0
$195$	−2.47214	−0.177033
$196$	0	0
$197$	21.4164	1.52586	0.762928	−	0.646484i	$-0.223761\pi$
$197$	21.4164	1.52586	0.762928	+	0.646484i	$0.223761\pi$
$198$	0	0
$199$	23.4164	1.65995	0.829973	−	0.557804i	$-0.188356\pi$
$199$	23.4164	1.65995	0.829973	+	0.557804i	$0.188356\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	−0.583592	−0.0409601
$204$	0	0
$205$	22.4721	1.56952
$206$	0	0
$207$	7.70820	0.535757
$208$	0	0
$209$	2.47214	0.171001
$210$	0	0
$211$	−20.9443	−1.44186	−0.720932	−	0.693006i	$-0.756286\pi$
$211$	−20.9443	−1.44186	−0.720932	+	0.693006i	$0.756286\pi$
$212$	0	0
$213$	−7.70820	−0.528157
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−3.41641	−0.229812
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	17.4164	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$229$	17.4164	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$230$	0	0
$231$	1.23607	0.0813273
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	24.9443	1.62718
$236$	0	0
$237$	11.7082	0.760530
$238$	0	0
$239$	−26.4721	−1.71234	−0.856170	−	0.516694i	$-0.827162\pi$
$239$	−26.4721	−1.71234	−0.856170	+	0.516694i	$0.827162\pi$
$240$	0	0
$241$	−26.3607	−1.69804	−0.849020	−	0.528360i	$-0.822807\pi$
$241$	−26.3607	−1.69804	−0.849020	+	0.528360i	$0.822807\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	17.7082	1.13134
$246$	0	0
$247$	−1.88854	−0.120165
$248$	0	0
$249$	−12.9443	−0.820310
$250$	0	0
$251$	−24.3607	−1.53763	−0.768816	−	0.639470i	$-0.779154\pi$
$251$	−24.3607	−1.53763	−0.768816	+	0.639470i	$0.779154\pi$
$252$	0	0
$253$	7.70820	0.484611
$254$	0	0
$255$	14.4721	0.906280
$256$	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
$259$	7.41641	0.460833
$260$	0	0
$261$	0.472136	0.0292245
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	36.3607	2.23362
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	17.7082	1.07969	0.539844	−	0.841765i	$-0.318483\pi$
$269$	17.7082	1.07969	0.539844	+	0.841765i	$0.318483\pi$
$270$	0	0
$271$	−17.2361	−1.04702	−0.523508	−	0.852021i	$-0.675377\pi$
$271$	−17.2361	−1.04702	−0.523508	+	0.852021i	$0.675377\pi$
$272$	0	0
$273$	−0.944272	−0.0571499
$274$	0	0
$275$	5.47214	0.329982
$276$	0	0
$277$	−3.23607	−0.194436	−0.0972182	−	0.995263i	$-0.530994\pi$
$277$	−3.23607	−0.194436	−0.0972182	+	0.995263i	$0.530994\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.3607	1.81117	0.905583	−	0.424169i	$-0.139434\pi$
$281$	30.3607	1.81117	0.905583	+	0.424169i	$0.139434\pi$
$282$	0	0
$283$	−3.05573	−0.181644	−0.0908221	−	0.995867i	$-0.528949\pi$
$283$	−3.05573	−0.181644	−0.0908221	+	0.995867i	$0.528949\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	8.58359	0.506673
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−0.472136	−0.0276771
$292$	0	0
$293$	−27.8885	−1.62927	−0.814633	−	0.579977i	$-0.803062\pi$
$293$	−27.8885	−1.62927	−0.814633	+	0.579977i	$0.803062\pi$
$294$	0	0
$295$	−46.8328	−2.72671
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−5.88854	−0.340543
$300$	0	0
$301$	6.11146	0.352258
$302$	0	0
$303$	12.4721	0.716505
$304$	0	0
$305$	−5.52786	−0.316525
$306$	0	0
$307$	21.5279	1.22866	0.614330	−	0.789049i	$-0.289426\pi$
$307$	21.5279	1.22866	0.614330	+	0.789049i	$0.289426\pi$
$308$	0	0
$309$	−2.47214	−0.140635
$310$	0	0
$311$	15.7082	0.890731	0.445365	−	0.895349i	$-0.353074\pi$
$311$	15.7082	0.890731	0.445365	+	0.895349i	$0.353074\pi$
$312$	0	0
$313$	19.8885	1.12417	0.562083	−	0.827081i	$-0.310000\pi$
$313$	19.8885	1.12417	0.562083	+	0.827081i	$0.310000\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	−5.70820	−0.320605	−0.160302	−	0.987068i	$-0.551247\pi$
$317$	−5.70820	−0.320605	−0.160302	+	0.987068i	$0.551247\pi$
$318$	0	0
$319$	0.472136	0.0264345
$320$	0	0
$321$	0.944272	0.0527041
$322$	0	0
$323$	11.0557	0.615157
$324$	0	0
$325$	−4.18034	−0.231884
$326$	0	0
$327$	10.6525	0.589083
$328$	0	0
$329$	9.52786	0.525288
$330$	0	0
$331$	−0.944272	−0.0519019	−0.0259509	−	0.999663i	$-0.508261\pi$
$331$	−0.944272	−0.0519019	−0.0259509	+	0.999663i	$0.508261\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	25.8885	1.41444
$336$	0	0
$337$	−9.05573	−0.493297	−0.246648	−	0.969105i	$-0.579329\pi$
$337$	−9.05573	−0.493297	−0.246648	+	0.969105i	$0.579329\pi$
$338$	0	0
$339$	10.9443	0.594411
$340$	0	0
$341$	0	0
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	24.9443	1.34295
$346$	0	0
$347$	−17.8885	−0.960307	−0.480154	−	0.877184i	$-0.659419\pi$
$347$	−17.8885	−0.960307	−0.480154	+	0.877184i	$0.659419\pi$
$348$	0	0
$349$	−29.7082	−1.59024	−0.795122	−	0.606450i	$-0.792593\pi$
$349$	−29.7082	−1.59024	−0.795122	+	0.606450i	$0.792593\pi$
$350$	0	0
$351$	0.763932	0.0407757
$352$	0	0
$353$	−28.8328	−1.53462	−0.767308	−	0.641279i	$-0.778404\pi$
$353$	−28.8328	−1.53462	−0.767308	+	0.641279i	$0.778404\pi$
$354$	0	0
$355$	−24.9443	−1.32390
$356$	0	0
$357$	5.52786	0.292566
$358$	0	0
$359$	15.4164	0.813647	0.406823	−	0.913507i	$-0.366636\pi$
$359$	15.4164	0.813647	0.406823	+	0.913507i	$0.366636\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	19.4164	1.01630
$366$	0	0
$367$	−19.0557	−0.994701	−0.497350	−	0.867550i	$-0.665694\pi$
$367$	−19.0557	−0.994701	−0.497350	+	0.867550i	$0.665694\pi$
$368$	0	0
$369$	−6.94427	−0.361504
$370$	0	0
$371$	13.8885	0.721057
$372$	0	0
$373$	4.18034	0.216450	0.108225	−	0.994126i	$-0.465483\pi$
$373$	4.18034	0.216450	0.108225	+	0.994126i	$0.465483\pi$
$374$	0	0
$375$	1.52786	0.0788986
$376$	0	0
$377$	−0.360680	−0.0185760
$378$	0	0
$379$	5.88854	0.302474	0.151237	−	0.988498i	$-0.451674\pi$
$379$	5.88854	0.302474	0.151237	+	0.988498i	$0.451674\pi$
$380$	0	0
$381$	−14.1803	−0.726481
$382$	0	0
$383$	−28.0689	−1.43425	−0.717126	−	0.696943i	$-0.754543\pi$
$383$	−28.0689	−1.43425	−0.717126	+	0.696943i	$0.754543\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	−4.94427	−0.251331
$388$	0	0
$389$	−14.2918	−0.724623	−0.362311	−	0.932057i	$-0.618012\pi$
$389$	−14.2918	−0.724623	−0.362311	+	0.932057i	$0.618012\pi$
$390$	0	0
$391$	34.4721	1.74333
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	37.8885	1.90638
$396$	0	0
$397$	30.3607	1.52376	0.761879	−	0.647719i	$-0.224277\pi$
$397$	30.3607	1.52376	0.761879	+	0.647719i	$0.224277\pi$
$398$	0	0
$399$	3.05573	0.152978
$400$	0	0
$401$	−13.4164	−0.669983	−0.334992	−	0.942221i	$-0.608734\pi$
$401$	−13.4164	−0.669983	−0.334992	+	0.942221i	$0.608734\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.23607	−0.160802
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	23.5279	1.16338	0.581689	−	0.813411i	$-0.302392\pi$
$409$	23.5279	1.16338	0.581689	+	0.813411i	$0.302392\pi$
$410$	0	0
$411$	16.4721	0.812511
$412$	0	0
$413$	−17.8885	−0.880238
$414$	0	0
$415$	−41.8885	−2.05623
$416$	0	0
$417$	10.4721	0.512823
$418$	0	0
$419$	19.4164	0.948554	0.474277	−	0.880376i	$-0.342710\pi$
$419$	19.4164	0.948554	0.474277	+	0.880376i	$0.342710\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−7.70820	−0.374786
$424$	0	0
$425$	24.4721	1.18707
$426$	0	0
$427$	−2.11146	−0.102181
$428$	0	0
$429$	0.763932	0.0368830
$430$	0	0
$431$	36.9443	1.77954	0.889771	−	0.456406i	$-0.150864\pi$
$431$	36.9443	1.77954	0.889771	+	0.456406i	$0.150864\pi$
$432$	0	0
$433$	16.8328	0.808933	0.404467	−	0.914553i	$-0.367457\pi$
$433$	16.8328	0.808933	0.404467	+	0.914553i	$0.367457\pi$
$434$	0	0
$435$	1.52786	0.0732555
$436$	0	0
$437$	19.0557	0.911559
$438$	0	0
$439$	−22.7639	−1.08646	−0.543232	−	0.839583i	$-0.682799\pi$
$439$	−22.7639	−1.08646	−0.543232	+	0.839583i	$0.682799\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	−38.4721	−1.82787	−0.913933	−	0.405865i	$-0.866970\pi$
$443$	−38.4721	−1.82787	−0.913933	+	0.405865i	$0.866970\pi$
$444$	0	0
$445$	−6.47214	−0.306809
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	20.4721	0.966140	0.483070	−	0.875582i	$-0.339522\pi$
$449$	20.4721	0.966140	0.483070	+	0.875582i	$0.339522\pi$
$450$	0	0
$451$	−6.94427	−0.326993
$452$	0	0
$453$	6.76393	0.317797
$454$	0	0
$455$	−3.05573	−0.143255
$456$	0	0
$457$	−31.3050	−1.46438	−0.732192	−	0.681098i	$-0.761503\pi$
$457$	−31.3050	−1.46438	−0.732192	+	0.681098i	$0.761503\pi$
$458$	0	0
$459$	−4.47214	−0.208741
$460$	0	0
$461$	1.05573	0.0491702	0.0245851	−	0.999698i	$-0.492174\pi$
$461$	1.05573	0.0491702	0.0245851	+	0.999698i	$0.492174\pi$
$462$	0	0
$463$	5.52786	0.256902	0.128451	−	0.991716i	$-0.459000\pi$
$463$	5.52786	0.256902	0.128451	+	0.991716i	$0.459000\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.9443	−1.15428	−0.577142	−	0.816644i	$-0.695832\pi$
$467$	−24.9443	−1.15428	−0.577142	+	0.816644i	$0.695832\pi$
$468$	0	0
$469$	9.88854	0.456611
$470$	0	0
$471$	−7.52786	−0.346866
$472$	0	0
$473$	−4.94427	−0.227338
$474$	0	0
$475$	13.5279	0.620701
$476$	0	0
$477$	−11.2361	−0.514464
$478$	0	0
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	0	0
$481$	4.58359	0.208994
$482$	0	0
$483$	9.52786	0.433533
$484$	0	0
$485$	−1.52786	−0.0693767
$486$	0	0
$487$	10.4721	0.474538	0.237269	−	0.971444i	$-0.423748\pi$
$487$	10.4721	0.474538	0.237269	+	0.971444i	$0.423748\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	5.88854	0.265746	0.132873	−	0.991133i	$-0.457580\pi$
$491$	5.88854	0.265746	0.132873	+	0.991133i	$0.457580\pi$
$492$	0	0
$493$	2.11146	0.0950952
$494$	0	0
$495$	−3.23607	−0.145450
$496$	0	0
$497$	−9.52786	−0.427383
$498$	0	0
$499$	0.944272	0.0422714	0.0211357	−	0.999777i	$-0.493272\pi$
$499$	0.944272	0.0422714	0.0211357	+	0.999777i	$0.493272\pi$
$500$	0	0
$501$	20.3607	0.909648
$502$	0	0
$503$	7.41641	0.330681	0.165341	−	0.986237i	$-0.447128\pi$
$503$	7.41641	0.330681	0.165341	+	0.986237i	$0.447128\pi$
$504$	0	0
$505$	40.3607	1.79603
$506$	0	0
$507$	12.4164	0.551432
$508$	0	0
$509$	22.6525	1.00405	0.502027	−	0.864852i	$-0.332588\pi$
$509$	22.6525	1.00405	0.502027	+	0.864852i	$0.332588\pi$
$510$	0	0
$511$	7.41641	0.328083
$512$	0	0
$513$	−2.47214	−0.109147
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−7.70820	−0.339006
$518$	0	0
$519$	−16.4721	−0.723047
$520$	0	0
$521$	7.52786	0.329802	0.164901	−	0.986310i	$-0.447270\pi$
$521$	7.52786	0.329802	0.164901	+	0.986310i	$0.447270\pi$
$522$	0	0
$523$	−25.8885	−1.13203	−0.566013	−	0.824396i	$-0.691515\pi$
$523$	−25.8885	−1.13203	−0.566013	+	0.824396i	$0.691515\pi$
$524$	0	0
$525$	6.76393	0.295202
$526$	0	0
$527$	0	0
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	14.4721	0.628037
$532$	0	0
$533$	5.30495	0.229783
$534$	0	0
$535$	3.05573	0.132111
$536$	0	0
$537$	−1.52786	−0.0659322
$538$	0	0
$539$	−5.47214	−0.235702
$540$	0	0
$541$	−21.7082	−0.933309	−0.466654	−	0.884440i	$-0.654541\pi$
$541$	−21.7082	−0.933309	−0.466654	+	0.884440i	$0.654541\pi$
$542$	0	0
$543$	0.472136	0.0202613
$544$	0	0
$545$	34.4721	1.47662
$546$	0	0
$547$	−23.4164	−1.00121	−0.500607	−	0.865675i	$-0.666890\pi$
$547$	−23.4164	−1.00121	−0.500607	+	0.865675i	$0.666890\pi$
$548$	0	0
$549$	1.70820	0.0729044
$550$	0	0
$551$	1.16718	0.0497237
$552$	0	0
$553$	14.4721	0.615418
$554$	0	0
$555$	−19.4164	−0.824181
$556$	0	0
$557$	−3.88854	−0.164763	−0.0823814	−	0.996601i	$-0.526253\pi$
$557$	−3.88854	−0.164763	−0.0823814	+	0.996601i	$0.526253\pi$
$558$	0	0
$559$	3.77709	0.159754
$560$	0	0
$561$	−4.47214	−0.188814
$562$	0	0
$563$	−24.9443	−1.05128	−0.525638	−	0.850708i	$-0.676173\pi$
$563$	−24.9443	−1.05128	−0.525638	+	0.850708i	$0.676173\pi$
$564$	0	0
$565$	35.4164	1.48998
$566$	0	0
$567$	−1.23607	−0.0519100
$568$	0	0
$569$	28.4721	1.19361	0.596807	−	0.802385i	$-0.296436\pi$
$569$	28.4721	1.19361	0.596807	+	0.802385i	$0.296436\pi$
$570$	0	0
$571$	39.4164	1.64953	0.824763	−	0.565479i	$-0.191309\pi$
$571$	39.4164	1.64953	0.824763	+	0.565479i	$0.191309\pi$
$572$	0	0
$573$	2.76393	0.115465
$574$	0	0
$575$	42.1803	1.75904
$576$	0	0
$577$	−9.41641	−0.392010	−0.196005	−	0.980603i	$-0.562797\pi$
$577$	−9.41641	−0.392010	−0.196005	+	0.980603i	$0.562797\pi$
$578$	0	0
$579$	16.4721	0.684559
$580$	0	0
$581$	−16.0000	−0.663792
$582$	0	0
$583$	−11.2361	−0.465350
$584$	0	0
$585$	2.47214	0.102210
$586$	0	0
$587$	27.4164	1.13160	0.565798	−	0.824544i	$-0.308568\pi$
$587$	27.4164	1.13160	0.565798	+	0.824544i	$0.308568\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−21.4164	−0.880953
$592$	0	0
$593$	12.4721	0.512169	0.256085	−	0.966654i	$-0.417567\pi$
$593$	12.4721	0.512169	0.256085	+	0.966654i	$0.417567\pi$
$594$	0	0
$595$	17.8885	0.733359
$596$	0	0
$597$	−23.4164	−0.958370
$598$	0	0
$599$	−20.0689	−0.819992	−0.409996	−	0.912087i	$-0.634470\pi$
$599$	−20.0689	−0.819992	−0.409996	+	0.912087i	$0.634470\pi$
$600$	0	0
$601$	27.8885	1.13760	0.568799	−	0.822477i	$-0.307408\pi$
$601$	27.8885	1.13760	0.568799	+	0.822477i	$0.307408\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	−3.23607	−0.131565
$606$	0	0
$607$	43.1246	1.75037	0.875187	−	0.483785i	$-0.160738\pi$
$607$	43.1246	1.75037	0.875187	+	0.483785i	$0.160738\pi$
$608$	0	0
$609$	0.583592	0.0236483
$610$	0	0
$611$	5.88854	0.238225
$612$	0	0
$613$	−48.1803	−1.94599	−0.972993	−	0.230835i	$-0.925854\pi$
$613$	−48.1803	−1.94599	−0.972993	+	0.230835i	$0.925854\pi$
$614$	0	0
$615$	−22.4721	−0.906164
$616$	0	0
$617$	11.8885	0.478615	0.239307	−	0.970944i	$-0.423080\pi$
$617$	11.8885	0.478615	0.239307	+	0.970944i	$0.423080\pi$
$618$	0	0
$619$	−43.7771	−1.75955	−0.879775	−	0.475391i	$-0.842307\pi$
$619$	−43.7771	−1.75955	−0.879775	+	0.475391i	$0.842307\pi$
$620$	0	0
$621$	−7.70820	−0.309320
$622$	0	0
$623$	−2.47214	−0.0990440
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	−2.47214	−0.0987276
$628$	0	0
$629$	−26.8328	−1.06989
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	20.9443	0.832460
$634$	0	0
$635$	−45.8885	−1.82103
$636$	0	0
$637$	4.18034	0.165631
$638$	0	0
$639$	7.70820	0.304932
$640$	0	0
$641$	40.2492	1.58975	0.794874	−	0.606774i	$-0.207537\pi$
$641$	40.2492	1.58975	0.794874	+	0.606774i	$0.207537\pi$
$642$	0	0
$643$	−44.9443	−1.77243	−0.886215	−	0.463275i	$-0.846674\pi$
$643$	−44.9443	−1.77243	−0.886215	+	0.463275i	$0.846674\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−11.3475	−0.446117	−0.223059	−	0.974805i	$-0.571604\pi$
$647$	−11.3475	−0.446117	−0.223059	+	0.974805i	$0.571604\pi$
$648$	0	0
$649$	14.4721	0.568081
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.12461	0.0440095	0.0220047	−	0.999758i	$-0.492995\pi$
$653$	1.12461	0.0440095	0.0220047	+	0.999758i	$0.492995\pi$
$654$	0	0
$655$	−51.7771	−2.02310
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	31.0557	1.20976	0.604880	−	0.796317i	$-0.293221\pi$
$659$	31.0557	1.20976	0.604880	+	0.796317i	$0.293221\pi$
$660$	0	0
$661$	−16.4721	−0.640692	−0.320346	−	0.947301i	$-0.603799\pi$
$661$	−16.4721	−0.640692	−0.320346	+	0.947301i	$0.603799\pi$
$662$	0	0
$663$	3.41641	0.132682
$664$	0	0
$665$	9.88854	0.383461
$666$	0	0
$667$	3.63932	0.140915
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.70820	0.0659445
$672$	0	0
$673$	7.52786	0.290178	0.145089	−	0.989419i	$-0.453653\pi$
$673$	7.52786	0.290178	0.145089	+	0.989419i	$0.453653\pi$
$674$	0	0
$675$	−5.47214	−0.210623
$676$	0	0
$677$	16.4721	0.633076	0.316538	−	0.948580i	$-0.397480\pi$
$677$	16.4721	0.633076	0.316538	+	0.948580i	$0.397480\pi$
$678$	0	0
$679$	−0.583592	−0.0223962
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−11.4164	−0.436837	−0.218418	−	0.975855i	$-0.570090\pi$
$683$	−11.4164	−0.436837	−0.218418	+	0.975855i	$0.570090\pi$
$684$	0	0
$685$	53.3050	2.03668
$686$	0	0
$687$	−17.4164	−0.664477
$688$	0	0
$689$	8.58359	0.327009
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−1.23607	−0.0469543
$694$	0	0
$695$	33.8885	1.28547
$696$	0	0
$697$	−31.0557	−1.17632
$698$	0	0
$699$	2.00000	0.0756469
$700$	0	0
$701$	36.8328	1.39116	0.695578	−	0.718450i	$-0.255148\pi$
$701$	36.8328	1.39116	0.695578	+	0.718450i	$0.255148\pi$
$702$	0	0
$703$	−14.8328	−0.559430
$704$	0	0
$705$	−24.9443	−0.939456
$706$	0	0
$707$	15.4164	0.579794
$708$	0	0
$709$	−34.3607	−1.29044	−0.645221	−	0.763996i	$-0.723235\pi$
$709$	−34.3607	−1.29044	−0.645221	+	0.763996i	$0.723235\pi$
$710$	0	0
$711$	−11.7082	−0.439092
$712$	0	0
$713$	0	0
$714$	0	0
$715$	2.47214	0.0924526
$716$	0	0
$717$	26.4721	0.988620
$718$	0	0
$719$	−20.0689	−0.748443	−0.374222	−	0.927339i	$-0.622090\pi$
$719$	−20.0689	−0.748443	−0.374222	+	0.927339i	$0.622090\pi$
$720$	0	0
$721$	−3.05573	−0.113801
$722$	0	0
$723$	26.3607	0.980364
$724$	0	0
$725$	2.58359	0.0959522
$726$	0	0
$727$	0.583592	0.0216442	0.0108221	−	0.999941i	$-0.496555\pi$
$727$	0.583592	0.0216442	0.0108221	+	0.999941i	$0.496555\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.1115	−0.817822
$732$	0	0
$733$	9.12461	0.337025	0.168513	−	0.985699i	$-0.446104\pi$
$733$	9.12461	0.337025	0.168513	+	0.985699i	$0.446104\pi$
$734$	0	0
$735$	−17.7082	−0.653177
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	25.8885	0.952325	0.476163	−	0.879357i	$-0.342027\pi$
$739$	25.8885	0.952325	0.476163	+	0.879357i	$0.342027\pi$
$740$	0	0
$741$	1.88854	0.0693774
$742$	0	0
$743$	25.3050	0.928349	0.464174	−	0.885744i	$-0.346351\pi$
$743$	25.3050	0.928349	0.464174	+	0.885744i	$0.346351\pi$
$744$	0	0
$745$	6.47214	0.237121
$746$	0	0
$747$	12.9443	0.473606
$748$	0	0
$749$	1.16718	0.0426480
$750$	0	0
$751$	−1.88854	−0.0689139	−0.0344570	−	0.999406i	$-0.510970\pi$
$751$	−1.88854	−0.0689139	−0.0344570	+	0.999406i	$0.510970\pi$
$752$	0	0
$753$	24.3607	0.887753
$754$	0	0
$755$	21.8885	0.796606
$756$	0	0
$757$	21.0557	0.765283	0.382642	−	0.923897i	$-0.375014\pi$
$757$	21.0557	0.765283	0.382642	+	0.923897i	$0.375014\pi$
$758$	0	0
$759$	−7.70820	−0.279790
$760$	0	0
$761$	−45.4164	−1.64634	−0.823172	−	0.567792i	$-0.807798\pi$
$761$	−45.4164	−1.64634	−0.823172	+	0.567792i	$0.807798\pi$
$762$	0	0
$763$	13.1672	0.476684
$764$	0	0
$765$	−14.4721	−0.523241
$766$	0	0
$767$	−11.0557	−0.399199
$768$	0	0
$769$	15.5279	0.559949	0.279975	−	0.960007i	$-0.409674\pi$
$769$	15.5279	0.559949	0.279975	+	0.960007i	$0.409674\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	−32.1803	−1.15745	−0.578723	−	0.815524i	$-0.696449\pi$
$773$	−32.1803	−1.15745	−0.578723	+	0.815524i	$0.696449\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.41641	−0.266062
$778$	0	0
$779$	−17.1672	−0.615078
$780$	0	0
$781$	7.70820	0.275821
$782$	0	0
$783$	−0.472136	−0.0168728
$784$	0	0
$785$	−24.3607	−0.869470
$786$	0	0
$787$	−30.8328	−1.09907	−0.549536	−	0.835470i	$-0.685195\pi$
$787$	−30.8328	−1.09907	−0.549536	+	0.835470i	$0.685195\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	13.5279	0.480995
$792$	0	0
$793$	−1.30495	−0.0463402
$794$	0	0
$795$	−36.3607	−1.28958
$796$	0	0
$797$	−37.1246	−1.31502	−0.657511	−	0.753445i	$-0.728391\pi$
$797$	−37.1246	−1.31502	−0.657511	+	0.753445i	$0.728391\pi$
$798$	0	0
$799$	−34.4721	−1.21954
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	30.8328	1.08671
$806$	0	0
$807$	−17.7082	−0.623358
$808$	0	0
$809$	−19.8885	−0.699244	−0.349622	−	0.936891i	$-0.613690\pi$
$809$	−19.8885	−0.699244	−0.349622	+	0.936891i	$0.613690\pi$
$810$	0	0
$811$	6.11146	0.214602	0.107301	−	0.994227i	$-0.465779\pi$
$811$	6.11146	0.214602	0.107301	+	0.994227i	$0.465779\pi$
$812$	0	0
$813$	17.2361	0.604495
$814$	0	0
$815$	64.7214	2.26709
$816$	0	0
$817$	−12.2229	−0.427626
$818$	0	0
$819$	0.944272	0.0329955
$820$	0	0
$821$	−5.63932	−0.196814	−0.0984068	−	0.995146i	$-0.531375\pi$
$821$	−5.63932	−0.196814	−0.0984068	+	0.995146i	$0.531375\pi$
$822$	0	0
$823$	9.88854	0.344693	0.172346	−	0.985036i	$-0.444865\pi$
$823$	9.88854	0.344693	0.172346	+	0.985036i	$0.444865\pi$
$824$	0	0
$825$	−5.47214	−0.190515
$826$	0	0
$827$	51.7771	1.80047	0.900233	−	0.435409i	$-0.143396\pi$
$827$	51.7771	1.80047	0.900233	+	0.435409i	$0.143396\pi$
$828$	0	0
$829$	43.8885	1.52431	0.762156	−	0.647393i	$-0.224141\pi$
$829$	43.8885	1.52431	0.762156	+	0.647393i	$0.224141\pi$
$830$	0	0
$831$	3.23607	0.112258
$832$	0	0
$833$	−24.4721	−0.847909
$834$	0	0
$835$	65.8885	2.28017
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.7639	−0.647803	−0.323901	−	0.946091i	$-0.604995\pi$
$839$	−18.7639	−0.647803	−0.323901	+	0.946091i	$0.604995\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	−30.3607	−1.04568
$844$	0	0
$845$	40.1803	1.38225
$846$	0	0
$847$	−1.23607	−0.0424718
$848$	0	0
$849$	3.05573	0.104872
$850$	0	0
$851$	−46.2492	−1.58540
$852$	0	0
$853$	−56.9017	−1.94828	−0.974139	−	0.225952i	$-0.927451\pi$
$853$	−56.9017	−1.94828	−0.974139	+	0.225952i	$0.927451\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−51.8885	−1.77248	−0.886239	−	0.463227i	$-0.846691\pi$
$857$	−51.8885	−1.77248	−0.886239	+	0.463227i	$0.846691\pi$
$858$	0	0
$859$	5.88854	0.200915	0.100457	−	0.994941i	$-0.467969\pi$
$859$	5.88854	0.200915	0.100457	+	0.994941i	$0.467969\pi$
$860$	0	0
$861$	−8.58359	−0.292528
$862$	0	0
$863$	21.2361	0.722884	0.361442	−	0.932395i	$-0.382285\pi$
$863$	21.2361	0.722884	0.361442	+	0.932395i	$0.382285\pi$
$864$	0	0
$865$	−53.3050	−1.81242
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	−11.7082	−0.397174
$870$	0	0
$871$	6.11146	0.207079
$872$	0	0
$873$	0.472136	0.0159794
$874$	0	0
$875$	1.88854	0.0638444
$876$	0	0
$877$	−24.7639	−0.836219	−0.418109	−	0.908397i	$-0.637307\pi$
$877$	−24.7639	−0.836219	−0.418109	+	0.908397i	$0.637307\pi$
$878$	0	0
$879$	27.8885	0.940657
$880$	0	0
$881$	−25.0557	−0.844149	−0.422074	−	0.906561i	$-0.638698\pi$
$881$	−25.0557	−0.844149	−0.422074	+	0.906561i	$0.638698\pi$
$882$	0	0
$883$	48.7214	1.63960	0.819802	−	0.572647i	$-0.194083\pi$
$883$	48.7214	1.63960	0.819802	+	0.572647i	$0.194083\pi$
$884$	0	0
$885$	46.8328	1.57427
$886$	0	0
$887$	20.3607	0.683645	0.341822	−	0.939765i	$-0.388956\pi$
$887$	20.3607	0.683645	0.341822	+	0.939765i	$0.388956\pi$
$888$	0	0
$889$	−17.5279	−0.587866
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−19.0557	−0.637676
$894$	0	0
$895$	−4.94427	−0.165269
$896$	0	0
$897$	5.88854	0.196613
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−50.2492	−1.67404
$902$	0	0
$903$	−6.11146	−0.203377
$904$	0	0
$905$	1.52786	0.0507879
$906$	0	0
$907$	20.9443	0.695443	0.347722	−	0.937598i	$-0.386955\pi$
$907$	20.9443	0.695443	0.347722	+	0.937598i	$0.386955\pi$
$908$	0	0
$909$	−12.4721	−0.413675
$910$	0	0
$911$	−2.18034	−0.0722379	−0.0361189	−	0.999347i	$-0.511500\pi$
$911$	−2.18034	−0.0722379	−0.0361189	+	0.999347i	$0.511500\pi$
$912$	0	0
$913$	12.9443	0.428393
$914$	0	0
$915$	5.52786	0.182746
$916$	0	0
$917$	−19.7771	−0.653097
$918$	0	0
$919$	27.7082	0.914009	0.457005	−	0.889464i	$-0.348922\pi$
$919$	27.7082	0.914009	0.457005	+	0.889464i	$0.348922\pi$
$920$	0	0
$921$	−21.5279	−0.709367
$922$	0	0
$923$	−5.88854	−0.193824
$924$	0	0
$925$	−32.8328	−1.07954
$926$	0	0
$927$	2.47214	0.0811956
$928$	0	0
$929$	6.36068	0.208687	0.104344	−	0.994541i	$-0.466726\pi$
$929$	6.36068	0.208687	0.104344	+	0.994541i	$0.466726\pi$
$930$	0	0
$931$	−13.5279	−0.443358
$932$	0	0
$933$	−15.7082	−0.514264
$934$	0	0
$935$	−14.4721	−0.473289
$936$	0	0
$937$	−7.88854	−0.257707	−0.128854	−	0.991664i	$-0.541130\pi$
$937$	−7.88854	−0.257707	−0.128854	+	0.991664i	$0.541130\pi$
$938$	0	0
$939$	−19.8885	−0.649038
$940$	0	0
$941$	21.4164	0.698155	0.349077	−	0.937094i	$-0.386495\pi$
$941$	21.4164	0.698155	0.349077	+	0.937094i	$0.386495\pi$
$942$	0	0
$943$	−53.5279	−1.74311
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−52.7214	−1.71321	−0.856607	−	0.515969i	$-0.827432\pi$
$947$	−52.7214	−1.71321	−0.856607	+	0.515969i	$0.827432\pi$
$948$	0	0
$949$	4.58359	0.148790
$950$	0	0
$951$	5.70820	0.185101
$952$	0	0
$953$	−3.88854	−0.125962	−0.0629811	−	0.998015i	$-0.520061\pi$
$953$	−3.88854	−0.125962	−0.0629811	+	0.998015i	$0.520061\pi$
$954$	0	0
$955$	8.94427	0.289430
$956$	0	0
$957$	−0.472136	−0.0152620
$958$	0	0
$959$	20.3607	0.657481
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−0.944272	−0.0304287
$964$	0	0
$965$	53.3050	1.71595
$966$	0	0
$967$	59.1246	1.90132	0.950660	−	0.310236i	$-0.100408\pi$
$967$	59.1246	1.90132	0.950660	+	0.310236i	$0.100408\pi$
$968$	0	0
$969$	−11.0557	−0.355161
$970$	0	0
$971$	0.360680	0.0115748	0.00578738	−	0.999983i	$-0.498158\pi$
$971$	0.360680	0.0115748	0.00578738	+	0.999983i	$0.498158\pi$
$972$	0	0
$973$	12.9443	0.414974
$974$	0	0
$975$	4.18034	0.133878
$976$	0	0
$977$	−32.4721	−1.03888	−0.519438	−	0.854508i	$-0.673859\pi$
$977$	−32.4721	−1.03888	−0.519438	+	0.854508i	$0.673859\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	−10.6525	−0.340107
$982$	0	0
$983$	−45.9574	−1.46581	−0.732907	−	0.680329i	$-0.761837\pi$
$983$	−45.9574	−1.46581	−0.732907	+	0.680329i	$0.761837\pi$
$984$	0	0
$985$	−69.3050	−2.20824
$986$	0	0
$987$	−9.52786	−0.303275
$988$	0	0
$989$	−38.1115	−1.21187
$990$	0	0
$991$	−9.88854	−0.314120	−0.157060	−	0.987589i	$-0.550202\pi$
$991$	−9.88854	−0.314120	−0.157060	+	0.987589i	$0.550202\pi$
$992$	0	0
$993$	0.944272	0.0299656
$994$	0	0
$995$	−75.7771	−2.40230
$996$	0	0
$997$	26.2918	0.832670	0.416335	−	0.909211i	$-0.363314\pi$
$997$	26.2918	0.832670	0.416335	+	0.909211i	$0.363314\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.a.be.1.1		2
3.2	odd	2		6336.2.a.ct.1.2		2
4.3	odd	2		2112.2.a.bf.1.1		2
8.3	odd	2		1056.2.a.k.1.2	✓	2
8.5	even	2		1056.2.a.l.1.2	yes	2
12.11	even	2		6336.2.a.cs.1.2		2
24.5	odd	2		3168.2.a.bf.1.1		2
24.11	even	2		3168.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1056.2.a.k.1.2	✓	2	8.3	odd	2
1056.2.a.l.1.2	yes	2	8.5	even	2
2112.2.a.be.1.1		2	1.1	even	1	trivial
2112.2.a.bf.1.1		2	4.3	odd	2
3168.2.a.be.1.1		2	24.11	even	2
3168.2.a.bf.1.1		2	24.5	odd	2
6336.2.a.cs.1.2		2	12.11	even	2
6336.2.a.ct.1.2		2	3.2	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.23607 q^{5} -1.23607 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.23607 q^{5} -1.23607 q^{7} +1.00000 q^{9} +1.00000 q^{11} -0.763932 q^{13} +3.23607 q^{15} +4.47214 q^{17} +2.47214 q^{19} +1.23607 q^{21} +7.70820 q^{23} +5.47214 q^{25} -1.00000 q^{27} +0.472136 q^{29} -1.00000 q^{33} +4.00000 q^{35} -6.00000 q^{37} +0.763932 q^{39} -6.94427 q^{41} -4.94427 q^{43} -3.23607 q^{45} -7.70820 q^{47} -5.47214 q^{49} -4.47214 q^{51} -11.2361 q^{53} -3.23607 q^{55} -2.47214 q^{57} +14.4721 q^{59} +1.70820 q^{61} -1.23607 q^{63} +2.47214 q^{65} -8.00000 q^{67} -7.70820 q^{69} +7.70820 q^{71} -6.00000 q^{73} -5.47214 q^{75} -1.23607 q^{77} -11.7082 q^{79} +1.00000 q^{81} +12.9443 q^{83} -14.4721 q^{85} -0.472136 q^{87} +2.00000 q^{89} +0.944272 q^{91} -8.00000 q^{95} +0.472136 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 6 q^{13} + 2 q^{15} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 2 q^{33} + 8 q^{35} - 12 q^{37} + 6 q^{39} + 4 q^{41} + 8 q^{43} - 2 q^{45} - 2 q^{47} - 2 q^{49} - 18 q^{53} - 2 q^{55} + 4 q^{57} + 20 q^{59} - 10 q^{61} + 2 q^{63} - 4 q^{65} - 16 q^{67} - 2 q^{69} + 2 q^{71} - 12 q^{73} - 2 q^{75} + 2 q^{77} - 10 q^{79} + 2 q^{81} + 8 q^{83} - 20 q^{85} + 8 q^{87} + 4 q^{89} - 16 q^{91} - 16 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 6 * q^13 + 2 * q^15 - 4 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 - 2 * q^33 + 8 * q^35 - 12 * q^37 + 6 * q^39 + 4 * q^41 + 8 * q^43 - 2 * q^45 - 2 * q^47 - 2 * q^49 - 18 * q^53 - 2 * q^55 + 4 * q^57 + 20 * q^59 - 10 * q^61 + 2 * q^63 - 4 * q^65 - 16 * q^67 - 2 * q^69 + 2 * q^71 - 12 * q^73 - 2 * q^75 + 2 * q^77 - 10 * q^79 + 2 * q^81 + 8 * q^83 - 20 * q^85 + 8 * q^87 + 4 * q^89 - 16 * q^91 - 16 * q^95 - 8 * q^97 + 2 * q^99

Introduction

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