LMFDB - Embedded newform 1456.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, 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("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.64268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 6x + 32 $ x^4 - x^3 - 12x^2 + 6x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.18587$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.18587 q^{3} +3.53544 q^{5} +1.00000 q^{7} +7.14978 q^{9} +O(q^{10})$	$q-3.18587 q^{3} +3.53544 q^{5} +1.00000 q^{7} +7.14978 q^{9} +5.11369 q^{11} -1.00000 q^{13} -11.2635 q^{15} +3.03609 q^{17} +7.46326 q^{19} -3.18587 q^{21} -1.65043 q^{23} +7.49935 q^{25} -13.2207 q^{27} -8.87109 q^{29} -0.614337 q^{31} -16.2915 q^{33} +3.53544 q^{35} -9.48543 q^{37} +3.18587 q^{39} +8.84892 q^{41} -1.42847 q^{43} +25.2776 q^{45} +3.91520 q^{47} +1.00000 q^{49} -9.67260 q^{51} -5.80021 q^{53} +18.0791 q^{55} -23.7770 q^{57} -7.37044 q^{59} +5.37845 q^{61} +7.14978 q^{63} -3.53544 q^{65} +6.59371 q^{67} +5.25806 q^{69} -0.806117 q^{71} +2.61434 q^{73} -23.8920 q^{75} +5.11369 q^{77} -0.421756 q^{79} +20.6700 q^{81} -6.23458 q^{83} +10.7339 q^{85} +28.2622 q^{87} +13.7628 q^{89} -1.00000 q^{91} +1.95720 q^{93} +26.3859 q^{95} +11.0583 q^{97} +36.5617 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9}+O(q^{10}) $ 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9	$ 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 10 q^{15} + 16 q^{17} + 10 q^{19} - q^{21} - 7 q^{23} + 14 q^{25} - 13 q^{27} + 4 q^{29} + q^{31} - q^{33} + 2 q^{35} + 5 q^{37} + q^{39} + 19 q^{41} - 14 q^{43} + 10 q^{45} + 13 q^{47} + 4 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 26 q^{59} - q^{61} + 13 q^{63} - 2 q^{65} - 5 q^{67} + 17 q^{69} + 18 q^{71} + 7 q^{73} - q^{75} + q^{77} - 9 q^{79} - 4 q^{81} - 12 q^{83} + 14 q^{85} + 46 q^{87} + 4 q^{89} - 4 q^{91} + 3 q^{93} + 26 q^{95} + 25 q^{97} + 48 q^{99}+O(q^{100}) $ 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9 + q^11 - 4 * q^13 - 10 * q^15 + 16 * q^17 + 10 * q^19 - q^21 - 7 * q^23 + 14 * q^25 - 13 * q^27 + 4 * q^29 + q^31 - q^33 + 2 * q^35 + 5 * q^37 + q^39 + 19 * q^41 - 14 * q^43 + 10 * q^45 + 13 * q^47 + 4 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 26 * q^59 - q^61 + 13 * q^63 - 2 * q^65 - 5 * q^67 + 17 * q^69 + 18 * q^71 + 7 * q^73 - q^75 + q^77 - 9 * q^79 - 4 * q^81 - 12 * q^83 + 14 * q^85 + 46 * q^87 + 4 * q^89 - 4 * q^91 + 3 * q^93 + 26 * q^95 + 25 * q^97 + 48 * 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
$3$	−3.18587	−1.83936	−0.919682	−	0.392664i	$-0.871553\pi$
$3$	−3.18587	−1.83936	−0.919682	+	0.392664i	$0.871553\pi$
$4$	0	0
$5$	3.53544	1.58110	0.790549	−	0.612399i	$-0.209795\pi$
$5$	3.53544	1.58110	0.790549	+	0.612399i	$0.209795\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.14978	2.38326
$10$	0	0
$11$	5.11369	1.54183	0.770917	−	0.636936i	$-0.219798\pi$
$11$	5.11369	1.54183	0.770917	+	0.636936i	$0.219798\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−11.2635	−2.90821
$16$	0	0
$17$	3.03609	0.736361	0.368180	−	0.929754i	$-0.379981\pi$
$17$	3.03609	0.736361	0.368180	+	0.929754i	$0.379981\pi$
$18$	0	0
$19$	7.46326	1.71219	0.856094	−	0.516820i	$-0.172884\pi$
$19$	7.46326	1.71219	0.856094	+	0.516820i	$0.172884\pi$
$20$	0	0
$21$	−3.18587	−0.695214
$22$	0	0
$23$	−1.65043	−0.344138	−0.172069	−	0.985085i	$-0.555045\pi$
$23$	−1.65043	−0.344138	−0.172069	+	0.985085i	$0.555045\pi$
$24$	0	0
$25$	7.49935	1.49987
$26$	0	0
$27$	−13.2207	−2.54432
$28$	0	0
$29$	−8.87109	−1.64732	−0.823660	−	0.567084i	$-0.808072\pi$
$29$	−8.87109	−1.64732	−0.823660	+	0.567084i	$0.808072\pi$
$30$	0	0
$31$	−0.614337	−0.110338	−0.0551691	−	0.998477i	$-0.517570\pi$
$31$	−0.614337	−0.110338	−0.0551691	+	0.998477i	$0.517570\pi$
$32$	0	0
$33$	−16.2915	−2.83599
$34$	0	0
$35$	3.53544	0.597599
$36$	0	0
$37$	−9.48543	−1.55939	−0.779697	−	0.626156i	$-0.784627\pi$
$37$	−9.48543	−1.55939	−0.779697	+	0.626156i	$0.784627\pi$
$38$	0	0
$39$	3.18587	0.510148
$40$	0	0
$41$	8.84892	1.38197	0.690984	−	0.722870i	$-0.257177\pi$
$41$	8.84892	1.38197	0.690984	+	0.722870i	$0.257177\pi$
$42$	0	0
$43$	−1.42847	−0.217839	−0.108919	−	0.994051i	$-0.534739\pi$
$43$	−1.42847	−0.217839	−0.108919	+	0.994051i	$0.534739\pi$
$44$	0	0
$45$	25.2776	3.76817
$46$	0	0
$47$	3.91520	0.571090	0.285545	−	0.958365i	$-0.407825\pi$
$47$	3.91520	0.571090	0.285545	+	0.958365i	$0.407825\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.67260	−1.35444
$52$	0	0
$53$	−5.80021	−0.796720	−0.398360	−	0.917229i	$-0.630421\pi$
$53$	−5.80021	−0.796720	−0.398360	+	0.917229i	$0.630421\pi$
$54$	0	0
$55$	18.0791	2.43779
$56$	0	0
$57$	−23.7770	−3.14934
$58$	0	0
$59$	−7.37044	−0.959550	−0.479775	−	0.877392i	$-0.659282\pi$
$59$	−7.37044	−0.959550	−0.479775	+	0.877392i	$0.659282\pi$
$60$	0	0
$61$	5.37845	0.688640	0.344320	−	0.938852i	$-0.388109\pi$
$61$	5.37845	0.688640	0.344320	+	0.938852i	$0.388109\pi$
$62$	0	0
$63$	7.14978	0.900787
$64$	0	0
$65$	−3.53544	−0.438518
$66$	0	0
$67$	6.59371	0.805550	0.402775	−	0.915299i	$-0.368046\pi$
$67$	6.59371	0.805550	0.402775	+	0.915299i	$0.368046\pi$
$68$	0	0
$69$	5.25806	0.632996
$70$	0	0
$71$	−0.806117	−0.0956685	−0.0478342	−	0.998855i	$-0.515232\pi$
$71$	−0.806117	−0.0956685	−0.0478342	+	0.998855i	$0.515232\pi$
$72$	0	0
$73$	2.61434	0.305985	0.152992	−	0.988227i	$-0.451109\pi$
$73$	2.61434	0.305985	0.152992	+	0.988227i	$0.451109\pi$
$74$	0	0
$75$	−23.8920	−2.75881
$76$	0	0
$77$	5.11369	0.582759
$78$	0	0
$79$	−0.421756	−0.0474513	−0.0237256	−	0.999719i	$-0.507553\pi$
$79$	−0.421756	−0.0474513	−0.0237256	+	0.999719i	$0.507553\pi$
$80$	0	0
$81$	20.6700	2.29667
$82$	0	0
$83$	−6.23458	−0.684334	−0.342167	−	0.939639i	$-0.611161\pi$
$83$	−6.23458	−0.684334	−0.342167	+	0.939639i	$0.611161\pi$
$84$	0	0
$85$	10.7339	1.16426
$86$	0	0
$87$	28.2622	3.03002
$88$	0	0
$89$	13.7628	1.45886	0.729428	−	0.684058i	$-0.239786\pi$
$89$	13.7628	1.45886	0.729428	+	0.684058i	$0.239786\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.95720	0.202952
$94$	0	0
$95$	26.3859	2.70714
$96$	0	0
$97$	11.0583	1.12280	0.561398	−	0.827546i	$-0.310264\pi$
$97$	11.0583	1.12280	0.561398	+	0.827546i	$0.310264\pi$
$98$	0	0
$99$	36.5617	3.67459
$100$	0	0
$101$	−8.51197	−0.846972	−0.423486	−	0.905903i	$-0.639194\pi$
$101$	−8.51197	−0.846972	−0.423486	+	0.905903i	$0.639194\pi$
$102$	0	0
$103$	−8.44393	−0.832005	−0.416003	−	0.909363i	$-0.636569\pi$
$103$	−8.44393	−0.832005	−0.416003	+	0.909363i	$0.636569\pi$
$104$	0	0
$105$	−11.2635	−1.09920
$106$	0	0
$107$	1.14307	0.110505	0.0552524	−	0.998472i	$-0.482404\pi$
$107$	1.14307	0.110505	0.0552524	+	0.998472i	$0.482404\pi$
$108$	0	0
$109$	5.40784	0.517977	0.258988	−	0.965880i	$-0.416611\pi$
$109$	5.40784	0.517977	0.258988	+	0.965880i	$0.416611\pi$
$110$	0	0
$111$	30.2194	2.86829
$112$	0	0
$113$	13.0931	1.23169	0.615846	−	0.787867i	$-0.288815\pi$
$113$	13.0931	1.23169	0.615846	+	0.787867i	$0.288815\pi$
$114$	0	0
$115$	−5.83500	−0.544116
$116$	0	0
$117$	−7.14978	−0.660997
$118$	0	0
$119$	3.03609	0.278318
$120$	0	0
$121$	15.1498	1.37725
$122$	0	0
$123$	−28.1915	−2.54194
$124$	0	0
$125$	8.83630	0.790343
$126$	0	0
$127$	3.99199	0.354232	0.177116	−	0.984190i	$-0.443323\pi$
$127$	3.99199	0.354232	0.177116	+	0.984190i	$0.443323\pi$
$128$	0	0
$129$	4.55091	0.400685
$130$	0	0
$131$	−14.5269	−1.26922	−0.634612	−	0.772831i	$-0.718840\pi$
$131$	−14.5269	−1.26922	−0.634612	+	0.772831i	$0.718840\pi$
$132$	0	0
$133$	7.46326	0.647146
$134$	0	0
$135$	−46.7409	−4.02281
$136$	0	0
$137$	−20.3343	−1.73728	−0.868640	−	0.495443i	$-0.835006\pi$
$137$	−20.3343	−1.73728	−0.868640	+	0.495443i	$0.835006\pi$
$138$	0	0
$139$	7.40784	0.628324	0.314162	−	0.949369i	$-0.398276\pi$
$139$	7.40784	0.628324	0.314162	+	0.949369i	$0.398276\pi$
$140$	0	0
$141$	−12.4733	−1.05044
$142$	0	0
$143$	−5.11369	−0.427628
$144$	0	0
$145$	−31.3632	−2.60457
$146$	0	0
$147$	−3.18587	−0.262766
$148$	0	0
$149$	2.85847	0.234175	0.117088	−	0.993122i	$-0.462644\pi$
$149$	2.85847	0.234175	0.117088	+	0.993122i	$0.462644\pi$
$150$	0	0
$151$	1.48002	0.120443	0.0602213	−	0.998185i	$-0.480819\pi$
$151$	1.48002	0.120443	0.0602213	+	0.998185i	$0.480819\pi$
$152$	0	0
$153$	21.7074	1.75494
$154$	0	0
$155$	−2.17195	−0.174455
$156$	0	0
$157$	18.9281	1.51062	0.755312	−	0.655366i	$-0.227486\pi$
$157$	18.9281	1.51062	0.755312	+	0.655366i	$0.227486\pi$
$158$	0	0
$159$	18.4787	1.46546
$160$	0	0
$161$	−1.65043	−0.130072
$162$	0	0
$163$	−8.52953	−0.668085	−0.334042	−	0.942558i	$-0.608413\pi$
$163$	−8.52953	−0.668085	−0.334042	+	0.942558i	$0.608413\pi$
$164$	0	0
$165$	−57.5978	−4.48398
$166$	0	0
$167$	12.0516	0.932578	0.466289	−	0.884633i	$-0.345591\pi$
$167$	12.0516	0.932578	0.466289	+	0.884633i	$0.345591\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	53.3606	4.08059
$172$	0	0
$173$	−0.664350	−0.0505096	−0.0252548	−	0.999681i	$-0.508040\pi$
$173$	−0.664350	−0.0505096	−0.0252548	+	0.999681i	$0.508040\pi$
$174$	0	0
$175$	7.49935	0.566897
$176$	0	0
$177$	23.4813	1.76496
$178$	0	0
$179$	14.7267	1.10073	0.550363	−	0.834925i	$-0.314489\pi$
$179$	14.7267	1.10073	0.550363	+	0.834925i	$0.314489\pi$
$180$	0	0
$181$	−11.2581	−0.836805	−0.418402	−	0.908262i	$-0.637410\pi$
$181$	−11.2581	−0.836805	−0.418402	+	0.908262i	$0.637410\pi$
$182$	0	0
$183$	−17.1351	−1.26666
$184$	0	0
$185$	−33.5352	−2.46556
$186$	0	0
$187$	15.5256	1.13535
$188$	0	0
$189$	−13.2207	−0.961662
$190$	0	0
$191$	−5.14307	−0.372139	−0.186070	−	0.982537i	$-0.559575\pi$
$191$	−5.14307	−0.372139	−0.186070	+	0.982537i	$0.559575\pi$
$192$	0	0
$193$	−16.2274	−1.16807	−0.584036	−	0.811728i	$-0.698527\pi$
$193$	−16.2274	−1.16807	−0.584036	+	0.811728i	$0.698527\pi$
$194$	0	0
$195$	11.2635	0.806593
$196$	0	0
$197$	−26.5659	−1.89274	−0.946370	−	0.323085i	$-0.895280\pi$
$197$	−26.5659	−1.89274	−0.946370	+	0.323085i	$0.895280\pi$
$198$	0	0
$199$	−5.73523	−0.406560	−0.203280	−	0.979121i	$-0.565160\pi$
$199$	−5.73523	−0.406560	−0.203280	+	0.979121i	$0.565160\pi$
$200$	0	0
$201$	−21.0067	−1.48170
$202$	0	0
$203$	−8.87109	−0.622629
$204$	0	0
$205$	31.2848	2.18503
$206$	0	0
$207$	−11.8002	−0.820171
$208$	0	0
$209$	38.1647	2.63991
$210$	0	0
$211$	−6.34156	−0.436571	−0.218285	−	0.975885i	$-0.570046\pi$
$211$	−6.34156	−0.436571	−0.218285	+	0.975885i	$0.570046\pi$
$212$	0	0
$213$	2.56818	0.175969
$214$	0	0
$215$	−5.05026	−0.344425
$216$	0	0
$217$	−0.614337	−0.0417039
$218$	0	0
$219$	−8.32894	−0.562818
$220$	0	0
$221$	−3.03609	−0.204230
$222$	0	0
$223$	−9.84301	−0.659137	−0.329568	−	0.944132i	$-0.606903\pi$
$223$	−9.84301	−0.659137	−0.329568	+	0.944132i	$0.606903\pi$
$224$	0	0
$225$	53.6187	3.57458
$226$	0	0
$227$	28.9574	1.92197	0.960986	−	0.276596i	$-0.0892062\pi$
$227$	28.9574	1.92197	0.960986	+	0.276596i	$0.0892062\pi$
$228$	0	0
$229$	−15.4426	−1.02048	−0.510239	−	0.860033i	$-0.670443\pi$
$229$	−15.4426	−1.02048	−0.510239	+	0.860033i	$0.670443\pi$
$230$	0	0
$231$	−16.2915	−1.07191
$232$	0	0
$233$	−10.7657	−0.705282	−0.352641	−	0.935759i	$-0.614716\pi$
$233$	−10.7657	−0.705282	−0.352641	+	0.935759i	$0.614716\pi$
$234$	0	0
$235$	13.8419	0.902949
$236$	0	0
$237$	1.34366	0.0872802
$238$	0	0
$239$	−14.3717	−0.929631	−0.464815	−	0.885408i	$-0.653879\pi$
$239$	−14.3717	−0.929631	−0.464815	+	0.885408i	$0.653879\pi$
$240$	0	0
$241$	−4.12374	−0.265634	−0.132817	−	0.991141i	$-0.542402\pi$
$241$	−4.12374	−0.265634	−0.132817	+	0.991141i	$0.542402\pi$
$242$	0	0
$243$	−26.1900	−1.68009
$244$	0	0
$245$	3.53544	0.225871
$246$	0	0
$247$	−7.46326	−0.474876
$248$	0	0
$249$	19.8626	1.25874
$250$	0	0
$251$	20.6285	1.30206	0.651030	−	0.759052i	$-0.274337\pi$
$251$	20.6285	1.30206	0.651030	+	0.759052i	$0.274337\pi$
$252$	0	0
$253$	−8.43978	−0.530604
$254$	0	0
$255$	−34.1969	−2.14149
$256$	0	0
$257$	1.36349	0.0850522	0.0425261	−	0.999095i	$-0.486459\pi$
$257$	1.36349	0.0850522	0.0425261	+	0.999095i	$0.486459\pi$
$258$	0	0
$259$	−9.48543	−0.589396
$260$	0	0
$261$	−63.4263	−3.92599
$262$	0	0
$263$	15.4259	0.951199	0.475600	−	0.879662i	$-0.342231\pi$
$263$	15.4259	0.951199	0.475600	+	0.879662i	$0.342231\pi$
$264$	0	0
$265$	−20.5063	−1.25969
$266$	0	0
$267$	−43.8466	−2.68337
$268$	0	0
$269$	13.1928	0.804381	0.402190	−	0.915556i	$-0.368249\pi$
$269$	13.1928	0.804381	0.402190	+	0.915556i	$0.368249\pi$
$270$	0	0
$271$	−21.8198	−1.32546	−0.662729	−	0.748860i	$-0.730602\pi$
$271$	−21.8198	−1.32546	−0.662729	+	0.748860i	$0.730602\pi$
$272$	0	0
$273$	3.18587	0.192818
$274$	0	0
$275$	38.3493	2.31255
$276$	0	0
$277$	18.5689	1.11570	0.557849	−	0.829942i	$-0.311627\pi$
$277$	18.5689	1.11570	0.557849	+	0.829942i	$0.311627\pi$
$278$	0	0
$279$	−4.39237	−0.262964
$280$	0	0
$281$	7.18047	0.428351	0.214175	−	0.976795i	$-0.431294\pi$
$281$	7.18047	0.428351	0.214175	+	0.976795i	$0.431294\pi$
$282$	0	0
$283$	−6.84892	−0.407126	−0.203563	−	0.979062i	$-0.565252\pi$
$283$	−6.84892	−0.407126	−0.203563	+	0.979062i	$0.565252\pi$
$284$	0	0
$285$	−84.0621	−4.97941
$286$	0	0
$287$	8.84892	0.522335
$288$	0	0
$289$	−7.78214	−0.457773
$290$	0	0
$291$	−35.2302	−2.06523
$292$	0	0
$293$	−30.5063	−1.78220	−0.891099	−	0.453810i	$-0.850065\pi$
$293$	−30.5063	−1.78220	−0.891099	+	0.453810i	$0.850065\pi$
$294$	0	0
$295$	−26.0578	−1.51714
$296$	0	0
$297$	−67.6063	−3.92292
$298$	0	0
$299$	1.65043	0.0954468
$300$	0	0
$301$	−1.42847	−0.0823354
$302$	0	0
$303$	27.1180	1.55789
$304$	0	0
$305$	19.0152	1.08881
$306$	0	0
$307$	−30.8337	−1.75977	−0.879886	−	0.475185i	$-0.842381\pi$
$307$	−30.8337	−1.75977	−0.879886	+	0.475185i	$0.842381\pi$
$308$	0	0
$309$	26.9013	1.53036
$310$	0	0
$311$	21.1057	1.19679	0.598396	−	0.801200i	$-0.295805\pi$
$311$	21.1057	1.19679	0.598396	+	0.801200i	$0.295805\pi$
$312$	0	0
$313$	8.12039	0.458992	0.229496	−	0.973310i	$-0.426292\pi$
$313$	8.12039	0.458992	0.229496	+	0.973310i	$0.426292\pi$
$314$	0	0
$315$	25.2776	1.42423
$316$	0	0
$317$	16.9789	0.953628	0.476814	−	0.879004i	$-0.341791\pi$
$317$	16.9789	0.953628	0.476814	+	0.879004i	$0.341791\pi$
$318$	0	0
$319$	−45.3640	−2.53989
$320$	0	0
$321$	−3.64167	−0.203258
$322$	0	0
$323$	22.6591	1.26079
$324$	0	0
$325$	−7.49935	−0.415989
$326$	0	0
$327$	−17.2287	−0.952748
$328$	0	0
$329$	3.91520	0.215852
$330$	0	0
$331$	−32.3611	−1.77873	−0.889364	−	0.457199i	$-0.848853\pi$
$331$	−32.3611	−1.77873	−0.889364	+	0.457199i	$0.848853\pi$
$332$	0	0
$333$	−67.8187	−3.71644
$334$	0	0
$335$	23.3117	1.27365
$336$	0	0
$337$	13.0931	0.713224	0.356612	−	0.934253i	$-0.383932\pi$
$337$	13.0931	0.713224	0.356612	+	0.934253i	$0.383932\pi$
$338$	0	0
$339$	−41.7128	−2.26553
$340$	0	0
$341$	−3.14153	−0.170123
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	18.5896	1.00083
$346$	0	0
$347$	−1.25391	−0.0673133	−0.0336566	−	0.999433i	$-0.510715\pi$
$347$	−1.25391	−0.0673133	−0.0336566	+	0.999433i	$0.510715\pi$
$348$	0	0
$349$	7.44984	0.398781	0.199390	−	0.979920i	$-0.436104\pi$
$349$	7.44984	0.398781	0.199390	+	0.979920i	$0.436104\pi$
$350$	0	0
$351$	13.2207	0.705667
$352$	0	0
$353$	17.6337	0.938545	0.469273	−	0.883053i	$-0.344516\pi$
$353$	17.6337	0.938545	0.469273	+	0.883053i	$0.344516\pi$
$354$	0	0
$355$	−2.84998	−0.151261
$356$	0	0
$357$	−9.67260	−0.511928
$358$	0	0
$359$	17.4052	0.918613	0.459306	−	0.888278i	$-0.348098\pi$
$359$	17.4052	0.918613	0.459306	+	0.888278i	$0.348098\pi$
$360$	0	0
$361$	36.7002	1.93159
$362$	0	0
$363$	−48.2653	−2.53327
$364$	0	0
$365$	9.24284	0.483792
$366$	0	0
$367$	6.00956	0.313696	0.156848	−	0.987623i	$-0.449867\pi$
$367$	6.00956	0.313696	0.156848	+	0.987623i	$0.449867\pi$
$368$	0	0
$369$	63.2678	3.29359
$370$	0	0
$371$	−5.80021	−0.301132
$372$	0	0
$373$	11.2983	0.585001	0.292501	−	0.956265i	$-0.405513\pi$
$373$	11.2983	0.585001	0.292501	+	0.956265i	$0.405513\pi$
$374$	0	0
$375$	−28.1513	−1.45373
$376$	0	0
$377$	8.87109	0.456884
$378$	0	0
$379$	−6.64993	−0.341584	−0.170792	−	0.985307i	$-0.554633\pi$
$379$	−6.64993	−0.341584	−0.170792	+	0.985307i	$0.554633\pi$
$380$	0	0
$381$	−12.7180	−0.651561
$382$	0	0
$383$	26.7045	1.36454	0.682269	−	0.731101i	$-0.260993\pi$
$383$	26.7045	1.36454	0.682269	+	0.731101i	$0.260993\pi$
$384$	0	0
$385$	18.0791	0.921398
$386$	0	0
$387$	−10.2132	−0.519167
$388$	0	0
$389$	6.93993	0.351868	0.175934	−	0.984402i	$-0.443705\pi$
$389$	6.93993	0.351868	0.175934	+	0.984402i	$0.443705\pi$
$390$	0	0
$391$	−5.01086	−0.253410
$392$	0	0
$393$	46.2809	2.33456
$394$	0	0
$395$	−1.49110	−0.0750251
$396$	0	0
$397$	30.5919	1.53536	0.767682	−	0.640831i	$-0.221410\pi$
$397$	30.5919	1.53536	0.767682	+	0.640831i	$0.221410\pi$
$398$	0	0
$399$	−23.7770	−1.19034
$400$	0	0
$401$	−6.55477	−0.327329	−0.163665	−	0.986516i	$-0.552332\pi$
$401$	−6.55477	−0.327329	−0.163665	+	0.986516i	$0.552332\pi$
$402$	0	0
$403$	0.614337	0.0306023
$404$	0	0
$405$	73.0776	3.63125
$406$	0	0
$407$	−48.5055	−2.40433
$408$	0	0
$409$	−8.97807	−0.443937	−0.221968	−	0.975054i	$-0.571248\pi$
$409$	−8.97807	−0.443937	−0.221968	+	0.975054i	$0.571248\pi$
$410$	0	0
$411$	64.7826	3.19549
$412$	0	0
$413$	−7.37044	−0.362676
$414$	0	0
$415$	−22.0420	−1.08200
$416$	0	0
$417$	−23.6004	−1.15572
$418$	0	0
$419$	14.7007	0.718175	0.359088	−	0.933304i	$-0.383088\pi$
$419$	14.7007	0.718175	0.359088	+	0.933304i	$0.383088\pi$
$420$	0	0
$421$	2.42897	0.118381	0.0591903	−	0.998247i	$-0.481148\pi$
$421$	2.42897	0.118381	0.0591903	+	0.998247i	$0.481148\pi$
$422$	0	0
$423$	27.9928	1.36106
$424$	0	0
$425$	22.7687	1.10445
$426$	0	0
$427$	5.37845	0.260282
$428$	0	0
$429$	16.2915	0.786563
$430$	0	0
$431$	−16.6435	−0.801687	−0.400844	−	0.916146i	$-0.631283\pi$
$431$	−16.6435	−0.801687	−0.400844	+	0.916146i	$0.631283\pi$
$432$	0	0
$433$	−2.85432	−0.137170	−0.0685850	−	0.997645i	$-0.521848\pi$
$433$	−2.85432	−0.137170	−0.0685850	+	0.997645i	$0.521848\pi$
$434$	0	0
$435$	99.9192	4.79076
$436$	0	0
$437$	−12.3176	−0.589230
$438$	0	0
$439$	−2.62695	−0.125378	−0.0626888	−	0.998033i	$-0.519968\pi$
$439$	−2.62695	−0.125378	−0.0626888	+	0.998033i	$0.519968\pi$
$440$	0	0
$441$	7.14978	0.340466
$442$	0	0
$443$	−21.0985	−1.00242	−0.501209	−	0.865326i	$-0.667111\pi$
$443$	−21.0985	−1.00242	−0.501209	+	0.865326i	$0.667111\pi$
$444$	0	0
$445$	48.6576	2.30659
$446$	0	0
$447$	−9.10673	−0.430734
$448$	0	0
$449$	13.3596	0.630480	0.315240	−	0.949012i	$-0.397915\pi$
$449$	13.3596	0.630480	0.315240	+	0.949012i	$0.397915\pi$
$450$	0	0
$451$	45.2506	2.13077
$452$	0	0
$453$	−4.71516	−0.221538
$454$	0	0
$455$	−3.53544	−0.165744
$456$	0	0
$457$	−24.0188	−1.12355	−0.561775	−	0.827290i	$-0.689881\pi$
$457$	−24.0188	−1.12355	−0.561775	+	0.827290i	$0.689881\pi$
$458$	0	0
$459$	−40.1392	−1.87354
$460$	0	0
$461$	−11.8613	−0.552437	−0.276218	−	0.961095i	$-0.589081\pi$
$461$	−11.8613	−0.552437	−0.276218	+	0.961095i	$0.589081\pi$
$462$	0	0
$463$	−23.8213	−1.10707	−0.553535	−	0.832826i	$-0.686722\pi$
$463$	−23.8213	−1.10707	−0.553535	+	0.832826i	$0.686722\pi$
$464$	0	0
$465$	6.91956	0.320887
$466$	0	0
$467$	−13.1217	−0.607200	−0.303600	−	0.952800i	$-0.598189\pi$
$467$	−13.1217	−0.607200	−0.303600	+	0.952800i	$0.598189\pi$
$468$	0	0
$469$	6.59371	0.304469
$470$	0	0
$471$	−60.3024	−2.77859
$472$	0	0
$473$	−7.30472	−0.335871
$474$	0	0
$475$	55.9696	2.56806
$476$	0	0
$477$	−41.4702	−1.89879
$478$	0	0
$479$	16.5207	0.754851	0.377426	−	0.926040i	$-0.376809\pi$
$479$	16.5207	0.754851	0.377426	+	0.926040i	$0.376809\pi$
$480$	0	0
$481$	9.48543	0.432498
$482$	0	0
$483$	5.25806	0.239250
$484$	0	0
$485$	39.0959	1.77525
$486$	0	0
$487$	22.1113	1.00196	0.500980	−	0.865459i	$-0.332973\pi$
$487$	22.1113	1.00196	0.500980	+	0.865459i	$0.332973\pi$
$488$	0	0
$489$	27.1740	1.22885
$490$	0	0
$491$	13.0735	0.589998	0.294999	−	0.955498i	$-0.404681\pi$
$491$	13.0735	0.589998	0.294999	+	0.955498i	$0.404681\pi$
$492$	0	0
$493$	−26.9335	−1.21302
$494$	0	0
$495$	129.262	5.80989
$496$	0	0
$497$	−0.806117	−0.0361593
$498$	0	0
$499$	0.628499	0.0281355	0.0140677	−	0.999901i	$-0.495522\pi$
$499$	0.628499	0.0281355	0.0140677	+	0.999901i	$0.495522\pi$
$500$	0	0
$501$	−38.3947	−1.71535
$502$	0	0
$503$	−5.74218	−0.256031	−0.128016	−	0.991772i	$-0.540861\pi$
$503$	−5.74218	−0.256031	−0.128016	+	0.991772i	$0.540861\pi$
$504$	0	0
$505$	−30.0936	−1.33915
$506$	0	0
$507$	−3.18587	−0.141490
$508$	0	0
$509$	37.1920	1.64851	0.824254	−	0.566221i	$-0.191595\pi$
$509$	37.1920	1.64851	0.824254	+	0.566221i	$0.191595\pi$
$510$	0	0
$511$	2.61434	0.115651
$512$	0	0
$513$	−98.6692	−4.35635
$514$	0	0
$515$	−29.8530	−1.31548
$516$	0	0
$517$	20.0211	0.880526
$518$	0	0
$519$	2.11653	0.0929055
$520$	0	0
$521$	−9.74479	−0.426927	−0.213463	−	0.976951i	$-0.568474\pi$
$521$	−9.74479	−0.426927	−0.213463	+	0.976951i	$0.568474\pi$
$522$	0	0
$523$	−19.8667	−0.868711	−0.434356	−	0.900741i	$-0.643024\pi$
$523$	−19.8667	−0.868711	−0.434356	+	0.900741i	$0.643024\pi$
$524$	0	0
$525$	−23.8920	−1.04273
$526$	0	0
$527$	−1.86518	−0.0812487
$528$	0	0
$529$	−20.2761	−0.881569
$530$	0	0
$531$	−52.6970	−2.28686
$532$	0	0
$533$	−8.84892	−0.383289
$534$	0	0
$535$	4.04126	0.174719
$536$	0	0
$537$	−46.9174	−2.02464
$538$	0	0
$539$	5.11369	0.220262
$540$	0	0
$541$	−12.9891	−0.558447	−0.279223	−	0.960226i	$-0.590077\pi$
$541$	−12.9891	−0.558447	−0.279223	+	0.960226i	$0.590077\pi$
$542$	0	0
$543$	35.8667	1.53919
$544$	0	0
$545$	19.1191	0.818972
$546$	0	0
$547$	14.2828	0.610688	0.305344	−	0.952242i	$-0.401229\pi$
$547$	14.2828	0.610688	0.305344	+	0.952242i	$0.401229\pi$
$548$	0	0
$549$	38.4547	1.64121
$550$	0	0
$551$	−66.2072	−2.82052
$552$	0	0
$553$	−0.421756	−0.0179349
$554$	0	0
$555$	106.839	4.53505
$556$	0	0
$557$	−17.9064	−0.758718	−0.379359	−	0.925250i	$-0.623855\pi$
$557$	−17.9064	−0.758718	−0.379359	+	0.925250i	$0.623855\pi$
$558$	0	0
$559$	1.42847	0.0604176
$560$	0	0
$561$	−49.4627	−2.08831
$562$	0	0
$563$	−21.6646	−0.913054	−0.456527	−	0.889710i	$-0.650907\pi$
$563$	−21.6646	−0.913054	−0.456527	+	0.889710i	$0.650907\pi$
$564$	0	0
$565$	46.2897	1.94742
$566$	0	0
$567$	20.6700	0.868058
$568$	0	0
$569$	24.3496	1.02079	0.510394	−	0.859941i	$-0.329500\pi$
$569$	24.3496	1.02079	0.510394	+	0.859941i	$0.329500\pi$
$570$	0	0
$571$	35.9585	1.50482	0.752408	−	0.658698i	$-0.228892\pi$
$571$	35.9585	1.50482	0.752408	+	0.658698i	$0.228892\pi$
$572$	0	0
$573$	16.3852	0.684500
$574$	0	0
$575$	−12.3771	−0.516163
$576$	0	0
$577$	−22.5991	−0.940813	−0.470407	−	0.882450i	$-0.655893\pi$
$577$	−22.5991	−0.940813	−0.470407	+	0.882450i	$0.655893\pi$
$578$	0	0
$579$	51.6983	2.14851
$580$	0	0
$581$	−6.23458	−0.258654
$582$	0	0
$583$	−29.6604	−1.22841
$584$	0	0
$585$	−25.2776	−1.04510
$586$	0	0
$587$	−3.54726	−0.146411	−0.0732055	−	0.997317i	$-0.523323\pi$
$587$	−3.54726	−0.146411	−0.0732055	+	0.997317i	$0.523323\pi$
$588$	0	0
$589$	−4.58495	−0.188920
$590$	0	0
$591$	84.6354	3.48144
$592$	0	0
$593$	25.4472	1.04499	0.522496	−	0.852642i	$-0.325001\pi$
$593$	25.4472	1.04499	0.522496	+	0.852642i	$0.325001\pi$
$594$	0	0
$595$	10.7339	0.440048
$596$	0	0
$597$	18.2717	0.747812
$598$	0	0
$599$	−16.1330	−0.659177	−0.329588	−	0.944125i	$-0.606910\pi$
$599$	−16.1330	−0.659177	−0.329588	+	0.944125i	$0.606910\pi$
$600$	0	0
$601$	−22.8879	−0.933615	−0.466808	−	0.884359i	$-0.654596\pi$
$601$	−22.8879	−0.933615	−0.466808	+	0.884359i	$0.654596\pi$
$602$	0	0
$603$	47.1436	1.91983
$604$	0	0
$605$	53.5612	2.17757
$606$	0	0
$607$	−23.4666	−0.952480	−0.476240	−	0.879315i	$-0.658001\pi$
$607$	−23.4666	−0.952480	−0.476240	+	0.879315i	$0.658001\pi$
$608$	0	0
$609$	28.2622	1.14524
$610$	0	0
$611$	−3.91520	−0.158392
$612$	0	0
$613$	−15.2650	−0.616548	−0.308274	−	0.951298i	$-0.599751\pi$
$613$	−15.2650	−0.616548	−0.308274	+	0.951298i	$0.599751\pi$
$614$	0	0
$615$	−99.6695	−4.01906
$616$	0	0
$617$	8.74088	0.351895	0.175947	−	0.984400i	$-0.443701\pi$
$617$	8.74088	0.351895	0.175947	+	0.984400i	$0.443701\pi$
$618$	0	0
$619$	16.9544	0.681453	0.340726	−	0.940162i	$-0.389327\pi$
$619$	16.9544	0.681453	0.340726	+	0.940162i	$0.389327\pi$
$620$	0	0
$621$	21.8198	0.875597
$622$	0	0
$623$	13.7628	0.551395
$624$	0	0
$625$	−6.25651	−0.250261
$626$	0	0
$627$	−121.588	−4.85576
$628$	0	0
$629$	−28.7986	−1.14828
$630$	0	0
$631$	41.1539	1.63831	0.819155	−	0.573572i	$-0.194443\pi$
$631$	41.1539	1.63831	0.819155	+	0.573572i	$0.194443\pi$
$632$	0	0
$633$	20.2034	0.803013
$634$	0	0
$635$	14.1134	0.560075
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−5.76356	−0.228003
$640$	0	0
$641$	−46.3609	−1.83114	−0.915572	−	0.402154i	$-0.868262\pi$
$641$	−46.3609	−1.83114	−0.915572	+	0.402154i	$0.868262\pi$
$642$	0	0
$643$	−34.8105	−1.37279	−0.686395	−	0.727229i	$-0.740808\pi$
$643$	−34.8105	−1.37279	−0.686395	+	0.727229i	$0.740808\pi$
$644$	0	0
$645$	16.0895	0.633522
$646$	0	0
$647$	34.2931	1.34820	0.674100	−	0.738640i	$-0.264532\pi$
$647$	34.2931	1.34820	0.674100	+	0.738640i	$0.264532\pi$
$648$	0	0
$649$	−37.6901	−1.47947
$650$	0	0
$651$	1.95720	0.0767086
$652$	0	0
$653$	5.04565	0.197452	0.0987258	−	0.995115i	$-0.468523\pi$
$653$	5.04565	0.197452	0.0987258	+	0.995115i	$0.468523\pi$
$654$	0	0
$655$	−51.3591	−2.00677
$656$	0	0
$657$	18.6919	0.729242
$658$	0	0
$659$	−5.85328	−0.228012	−0.114006	−	0.993480i	$-0.536368\pi$
$659$	−5.85328	−0.228012	−0.114006	+	0.993480i	$0.536368\pi$
$660$	0	0
$661$	5.88195	0.228781	0.114391	−	0.993436i	$-0.463508\pi$
$661$	5.88195	0.228781	0.114391	+	0.993436i	$0.463508\pi$
$662$	0	0
$663$	9.67260	0.375653
$664$	0	0
$665$	26.3859	1.02320
$666$	0	0
$667$	14.6411	0.566906
$668$	0	0
$669$	31.3586	1.21239
$670$	0	0
$671$	27.5037	1.06177
$672$	0	0
$673$	6.79611	0.261971	0.130985	−	0.991384i	$-0.458186\pi$
$673$	6.79611	0.261971	0.130985	+	0.991384i	$0.458186\pi$
$674$	0	0
$675$	−99.1464	−3.81614
$676$	0	0
$677$	−36.1675	−1.39003	−0.695016	−	0.718994i	$-0.744603\pi$
$677$	−36.1675	−1.39003	−0.695016	+	0.718994i	$0.744603\pi$
$678$	0	0
$679$	11.0583	0.424377
$680$	0	0
$681$	−92.2547	−3.53521
$682$	0	0
$683$	47.5924	1.82107	0.910536	−	0.413429i	$-0.135669\pi$
$683$	47.5924	1.82107	0.910536	+	0.413429i	$0.135669\pi$
$684$	0	0
$685$	−71.8909	−2.74681
$686$	0	0
$687$	49.1982	1.87703
$688$	0	0
$689$	5.80021	0.220970
$690$	0	0
$691$	−35.0889	−1.33484	−0.667422	−	0.744679i	$-0.732602\pi$
$691$	−35.0889	−1.33484	−0.667422	+	0.744679i	$0.732602\pi$
$692$	0	0
$693$	36.5617	1.38886
$694$	0	0
$695$	26.1900	0.993442
$696$	0	0
$697$	26.8661	1.01763
$698$	0	0
$699$	34.2980	1.29727
$700$	0	0
$701$	0.726719	0.0274478	0.0137239	−	0.999906i	$-0.495631\pi$
$701$	0.726719	0.0274478	0.0137239	+	0.999906i	$0.495631\pi$
$702$	0	0
$703$	−70.7922	−2.66998
$704$	0	0
$705$	−44.0987	−1.66085
$706$	0	0
$707$	−8.51197	−0.320125
$708$	0	0
$709$	−41.3663	−1.55354	−0.776772	−	0.629782i	$-0.783144\pi$
$709$	−41.3663	−1.55354	−0.776772	+	0.629782i	$0.783144\pi$
$710$	0	0
$711$	−3.01546	−0.113089
$712$	0	0
$713$	1.01392	0.0379716
$714$	0	0
$715$	−18.0791	−0.676121
$716$	0	0
$717$	45.7865	1.70993
$718$	0	0
$719$	−2.14563	−0.0800184	−0.0400092	−	0.999199i	$-0.512739\pi$
$719$	−2.14563	−0.0800184	−0.0400092	+	0.999199i	$0.512739\pi$
$720$	0	0
$721$	−8.44393	−0.314468
$722$	0	0
$723$	13.1377	0.488597
$724$	0	0
$725$	−66.5274	−2.47077
$726$	0	0
$727$	23.2956	0.863988	0.431994	−	0.901877i	$-0.357810\pi$
$727$	23.2956	0.863988	0.431994	+	0.901877i	$0.357810\pi$
$728$	0	0
$729$	21.4279	0.793626
$730$	0	0
$731$	−4.33695	−0.160408
$732$	0	0
$733$	36.0237	1.33057	0.665283	−	0.746591i	$-0.268311\pi$
$733$	36.0237	1.33057	0.665283	+	0.746591i	$0.268311\pi$
$734$	0	0
$735$	−11.2635	−0.415459
$736$	0	0
$737$	33.7181	1.24202
$738$	0	0
$739$	−29.4656	−1.08391	−0.541955	−	0.840408i	$-0.682316\pi$
$739$	−29.4656	−1.08391	−0.541955	+	0.840408i	$0.682316\pi$
$740$	0	0
$741$	23.7770	0.873469
$742$	0	0
$743$	−50.9739	−1.87005	−0.935026	−	0.354578i	$-0.884624\pi$
$743$	−50.9739	−1.87005	−0.935026	+	0.354578i	$0.884624\pi$
$744$	0	0
$745$	10.1060	0.370254
$746$	0	0
$747$	−44.5759	−1.63095
$748$	0	0
$749$	1.14307	0.0417669
$750$	0	0
$751$	−52.0613	−1.89974	−0.949872	−	0.312639i	$-0.898787\pi$
$751$	−52.0613	−1.89974	−0.949872	+	0.312639i	$0.898787\pi$
$752$	0	0
$753$	−65.7198	−2.39496
$754$	0	0
$755$	5.23253	0.190431
$756$	0	0
$757$	−32.9154	−1.19633	−0.598166	−	0.801372i	$-0.704104\pi$
$757$	−32.9154	−1.19633	−0.598166	+	0.801372i	$0.704104\pi$
$758$	0	0
$759$	26.8881	0.975975
$760$	0	0
$761$	−6.19873	−0.224704	−0.112352	−	0.993668i	$-0.535838\pi$
$761$	−6.19873	−0.224704	−0.112352	+	0.993668i	$0.535838\pi$
$762$	0	0
$763$	5.40784	0.195777
$764$	0	0
$765$	76.7452	2.77473
$766$	0	0
$767$	7.37044	0.266131
$768$	0	0
$769$	−50.7283	−1.82931	−0.914654	−	0.404238i	$-0.867537\pi$
$769$	−50.7283	−1.82931	−0.914654	+	0.404238i	$0.867537\pi$
$770$	0	0
$771$	−4.34390	−0.156442
$772$	0	0
$773$	0.726980	0.0261476	0.0130738	−	0.999915i	$-0.495838\pi$
$773$	0.726980	0.0261476	0.0130738	+	0.999915i	$0.495838\pi$
$774$	0	0
$775$	−4.60713	−0.165493
$776$	0	0
$777$	30.2194	1.08411
$778$	0	0
$779$	66.0417	2.36619
$780$	0	0
$781$	−4.12223	−0.147505
$782$	0	0
$783$	117.282	4.19131
$784$	0	0
$785$	66.9190	2.38844
$786$	0	0
$787$	−36.8646	−1.31408	−0.657041	−	0.753855i	$-0.728192\pi$
$787$	−36.8646	−1.31408	−0.657041	+	0.753855i	$0.728192\pi$
$788$	0	0
$789$	−49.1448	−1.74960
$790$	0	0
$791$	13.0931	0.465536
$792$	0	0
$793$	−5.37845	−0.190994
$794$	0	0
$795$	65.3304	2.31703
$796$	0	0
$797$	−11.8358	−0.419246	−0.209623	−	0.977782i	$-0.567224\pi$
$797$	−11.8358	−0.419246	−0.209623	+	0.977782i	$0.567224\pi$
$798$	0	0
$799$	11.8869	0.420528
$800$	0	0
$801$	98.4011	3.47683
$802$	0	0
$803$	13.3689	0.471778
$804$	0	0
$805$	−5.83500	−0.205657
$806$	0	0
$807$	−42.0306	−1.47955
$808$	0	0
$809$	48.5159	1.70573	0.852863	−	0.522134i	$-0.174864\pi$
$809$	48.5159	1.70573	0.852863	+	0.522134i	$0.174864\pi$
$810$	0	0
$811$	5.83140	0.204768	0.102384	−	0.994745i	$-0.467353\pi$
$811$	5.83140	0.204768	0.102384	+	0.994745i	$0.467353\pi$
$812$	0	0
$813$	69.5150	2.43800
$814$	0	0
$815$	−30.1557	−1.05631
$816$	0	0
$817$	−10.6610	−0.372981
$818$	0	0
$819$	−7.14978	−0.249833
$820$	0	0
$821$	29.4148	1.02658	0.513292	−	0.858214i	$-0.328426\pi$
$821$	29.4148	1.02658	0.513292	+	0.858214i	$0.328426\pi$
$822$	0	0
$823$	18.0085	0.627737	0.313868	−	0.949467i	$-0.398375\pi$
$823$	18.0085	0.627737	0.313868	+	0.949467i	$0.398375\pi$
$824$	0	0
$825$	−122.176	−4.25362
$826$	0	0
$827$	−41.7278	−1.45102	−0.725508	−	0.688213i	$-0.758395\pi$
$827$	−41.7278	−1.45102	−0.725508	+	0.688213i	$0.758395\pi$
$828$	0	0
$829$	−11.6604	−0.404984	−0.202492	−	0.979284i	$-0.564904\pi$
$829$	−11.6604	−0.404984	−0.202492	+	0.979284i	$0.564904\pi$
$830$	0	0
$831$	−59.1582	−2.05218
$832$	0	0
$833$	3.03609	0.105194
$834$	0	0
$835$	42.6076	1.47450
$836$	0	0
$837$	8.12194	0.280735
$838$	0	0
$839$	−50.0606	−1.72828	−0.864141	−	0.503249i	$-0.832138\pi$
$839$	−50.0606	−1.72828	−0.864141	+	0.503249i	$0.832138\pi$
$840$	0	0
$841$	49.6963	1.71366
$842$	0	0
$843$	−22.8760	−0.787893
$844$	0	0
$845$	3.53544	0.121623
$846$	0	0
$847$	15.1498	0.520553
$848$	0	0
$849$	21.8198	0.748853
$850$	0	0
$851$	15.6550	0.536648
$852$	0	0
$853$	24.6641	0.844482	0.422241	−	0.906484i	$-0.361244\pi$
$853$	24.6641	0.844482	0.422241	+	0.906484i	$0.361244\pi$
$854$	0	0
$855$	188.653	6.45181
$856$	0	0
$857$	−2.06958	−0.0706955	−0.0353478	−	0.999375i	$-0.511254\pi$
$857$	−2.06958	−0.0706955	−0.0353478	+	0.999375i	$0.511254\pi$
$858$	0	0
$859$	−5.90899	−0.201612	−0.100806	−	0.994906i	$-0.532142\pi$
$859$	−5.90899	−0.201612	−0.100806	+	0.994906i	$0.532142\pi$
$860$	0	0
$861$	−28.1915	−0.960764
$862$	0	0
$863$	5.25391	0.178845	0.0894225	−	0.995994i	$-0.471498\pi$
$863$	5.25391	0.178845	0.0894225	+	0.995994i	$0.471498\pi$
$864$	0	0
$865$	−2.34877	−0.0798606
$866$	0	0
$867$	24.7929	0.842011
$868$	0	0
$869$	−2.15673	−0.0731620
$870$	0	0
$871$	−6.59371	−0.223419
$872$	0	0
$873$	79.0642	2.67592
$874$	0	0
$875$	8.83630	0.298722
$876$	0	0
$877$	41.3411	1.39599	0.697994	−	0.716103i	$-0.254076\pi$
$877$	41.3411	1.39599	0.697994	+	0.716103i	$0.254076\pi$
$878$	0	0
$879$	97.1892	3.27811
$880$	0	0
$881$	−30.0139	−1.01119	−0.505597	−	0.862770i	$-0.668728\pi$
$881$	−30.0139	−1.01119	−0.505597	+	0.862770i	$0.668728\pi$
$882$	0	0
$883$	−16.0252	−0.539292	−0.269646	−	0.962960i	$-0.586907\pi$
$883$	−16.0252	−0.539292	−0.269646	+	0.962960i	$0.586907\pi$
$884$	0	0
$885$	83.0167	2.79058
$886$	0	0
$887$	−37.1126	−1.24612	−0.623060	−	0.782174i	$-0.714111\pi$
$887$	−37.1126	−1.24612	−0.623060	+	0.782174i	$0.714111\pi$
$888$	0	0
$889$	3.99199	0.133887
$890$	0	0
$891$	105.700	3.54108
$892$	0	0
$893$	29.2201	0.977814
$894$	0	0
$895$	52.0655	1.74036
$896$	0	0
$897$	−5.25806	−0.175561
$898$	0	0
$899$	5.44984	0.181762
$900$	0	0
$901$	−17.6100	−0.586673
$902$	0	0
$903$	4.55091	0.151445
$904$	0	0
$905$	−39.8022	−1.32307
$906$	0	0
$907$	−30.8015	−1.02275	−0.511374	−	0.859358i	$-0.670863\pi$
$907$	−30.8015	−1.02275	−0.511374	+	0.859358i	$0.670863\pi$
$908$	0	0
$909$	−60.8587	−2.01855
$910$	0	0
$911$	−13.9672	−0.462754	−0.231377	−	0.972864i	$-0.574323\pi$
$911$	−13.9672	−0.462754	−0.231377	+	0.972864i	$0.574323\pi$
$912$	0	0
$913$	−31.8817	−1.05513
$914$	0	0
$915$	−60.5800	−2.00271
$916$	0	0
$917$	−14.5269	−0.479721
$918$	0	0
$919$	−34.5793	−1.14067	−0.570333	−	0.821414i	$-0.693186\pi$
$919$	−34.5793	−1.14067	−0.570333	+	0.821414i	$0.693186\pi$
$920$	0	0
$921$	98.2322	3.23686
$922$	0	0
$923$	0.806117	0.0265337
$924$	0	0
$925$	−71.1345	−2.33889
$926$	0	0
$927$	−60.3722	−1.98288
$928$	0	0
$929$	−25.3300	−0.831050	−0.415525	−	0.909582i	$-0.636402\pi$
$929$	−25.3300	−0.831050	−0.415525	+	0.909582i	$0.636402\pi$
$930$	0	0
$931$	7.46326	0.244598
$932$	0	0
$933$	−67.2400	−2.20134
$934$	0	0
$935$	54.8899	1.79509
$936$	0	0
$937$	−5.54912	−0.181282	−0.0906409	−	0.995884i	$-0.528892\pi$
$937$	−5.54912	−0.181282	−0.0906409	+	0.995884i	$0.528892\pi$
$938$	0	0
$939$	−25.8705	−0.844253
$940$	0	0
$941$	−9.83760	−0.320697	−0.160348	−	0.987060i	$-0.551262\pi$
$941$	−9.83760	−0.320697	−0.160348	+	0.987060i	$0.551262\pi$
$942$	0	0
$943$	−14.6045	−0.475589
$944$	0	0
$945$	−46.7409	−1.52048
$946$	0	0
$947$	8.92912	0.290157	0.145079	−	0.989420i	$-0.453656\pi$
$947$	8.92912	0.290157	0.145079	+	0.989420i	$0.453656\pi$
$948$	0	0
$949$	−2.61434	−0.0848650
$950$	0	0
$951$	−54.0925	−1.75407
$952$	0	0
$953$	−32.3111	−1.04666	−0.523330	−	0.852130i	$-0.675310\pi$
$953$	−32.3111	−1.04666	−0.523330	+	0.852130i	$0.675310\pi$
$954$	0	0
$955$	−18.1830	−0.588389
$956$	0	0
$957$	144.524	4.67179
$958$	0	0
$959$	−20.3343	−0.656630
$960$	0	0
$961$	−30.6226	−0.987825
$962$	0	0
$963$	8.17270	0.263361
$964$	0	0
$965$	−57.3709	−1.84684
$966$	0	0
$967$	48.9896	1.57540	0.787700	−	0.616059i	$-0.211272\pi$
$967$	48.9896	1.57540	0.787700	+	0.616059i	$0.211272\pi$
$968$	0	0
$969$	−72.1891	−2.31905
$970$	0	0
$971$	49.6450	1.59318	0.796591	−	0.604518i	$-0.206634\pi$
$971$	49.6450	1.59318	0.796591	+	0.604518i	$0.206634\pi$
$972$	0	0
$973$	7.40784	0.237484
$974$	0	0
$975$	23.8920	0.765155
$976$	0	0
$977$	−42.5365	−1.36086	−0.680431	−	0.732812i	$-0.738208\pi$
$977$	−42.5365	−1.36086	−0.680431	+	0.732812i	$0.738208\pi$
$978$	0	0
$979$	70.3787	2.24931
$980$	0	0
$981$	38.6648	1.23447
$982$	0	0
$983$	34.0010	1.08446	0.542232	−	0.840229i	$-0.317580\pi$
$983$	34.0010	1.08446	0.542232	+	0.840229i	$0.317580\pi$
$984$	0	0
$985$	−93.9221	−2.99261
$986$	0	0
$987$	−12.4733	−0.397030
$988$	0	0
$989$	2.35758	0.0749667
$990$	0	0
$991$	−18.5937	−0.590649	−0.295324	−	0.955397i	$-0.595428\pi$
$991$	−18.5937	−0.590649	−0.295324	+	0.955397i	$0.595428\pi$
$992$	0	0
$993$	103.098	3.27173
$994$	0	0
$995$	−20.2766	−0.642811
$996$	0	0
$997$	−47.8290	−1.51476	−0.757380	−	0.652975i	$-0.773521\pi$
$997$	−47.8290	−1.51476	−0.757380	+	0.652975i	$0.773521\pi$
$998$	0	0
$999$	125.404	3.96760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.v.1.1		4
4.3	odd	2		728.2.a.i.1.4	✓	4
8.3	odd	2		5824.2.a.cb.1.1		4
8.5	even	2		5824.2.a.ce.1.4		4
12.11	even	2		6552.2.a.br.1.1		4
28.27	even	2		5096.2.a.s.1.1		4
52.51	odd	2		9464.2.a.z.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.i.1.4	✓	4	4.3	odd	2
1456.2.a.v.1.1		4	1.1	even	1	trivial
5096.2.a.s.1.1		4	28.27	even	2
5824.2.a.cb.1.1		4	8.3	odd	2
5824.2.a.ce.1.4		4	8.5	even	2
6552.2.a.br.1.1		4	12.11	even	2
9464.2.a.z.1.4		4	52.51	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q-3.18587 q^{3} +3.53544 q^{5} +1.00000 q^{7} +7.14978 q^{9} +O(q^{10})\)	\(q-3.18587 q^{3} +3.53544 q^{5} +1.00000 q^{7} +7.14978 q^{9} +5.11369 q^{11} -1.00000 q^{13} -11.2635 q^{15} +3.03609 q^{17} +7.46326 q^{19} -3.18587 q^{21} -1.65043 q^{23} +7.49935 q^{25} -13.2207 q^{27} -8.87109 q^{29} -0.614337 q^{31} -16.2915 q^{33} +3.53544 q^{35} -9.48543 q^{37} +3.18587 q^{39} +8.84892 q^{41} -1.42847 q^{43} +25.2776 q^{45} +3.91520 q^{47} +1.00000 q^{49} -9.67260 q^{51} -5.80021 q^{53} +18.0791 q^{55} -23.7770 q^{57} -7.37044 q^{59} +5.37845 q^{61} +7.14978 q^{63} -3.53544 q^{65} +6.59371 q^{67} +5.25806 q^{69} -0.806117 q^{71} +2.61434 q^{73} -23.8920 q^{75} +5.11369 q^{77} -0.421756 q^{79} +20.6700 q^{81} -6.23458 q^{83} +10.7339 q^{85} +28.2622 q^{87} +13.7628 q^{89} -1.00000 q^{91} +1.95720 q^{93} +26.3859 q^{95} +11.0583 q^{97} +36.5617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9}+O(q^{10}) \) 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9	\( 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 10 q^{15} + 16 q^{17} + 10 q^{19} - q^{21} - 7 q^{23} + 14 q^{25} - 13 q^{27} + 4 q^{29} + q^{31} - q^{33} + 2 q^{35} + 5 q^{37} + q^{39} + 19 q^{41} - 14 q^{43} + 10 q^{45} + 13 q^{47} + 4 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 26 q^{59} - q^{61} + 13 q^{63} - 2 q^{65} - 5 q^{67} + 17 q^{69} + 18 q^{71} + 7 q^{73} - q^{75} + q^{77} - 9 q^{79} - 4 q^{81} - 12 q^{83} + 14 q^{85} + 46 q^{87} + 4 q^{89} - 4 q^{91} + 3 q^{93} + 26 q^{95} + 25 q^{97} + 48 q^{99}+O(q^{100}) \) 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9 + q^11 - 4 * q^13 - 10 * q^15 + 16 * q^17 + 10 * q^19 - q^21 - 7 * q^23 + 14 * q^25 - 13 * q^27 + 4 * q^29 + q^31 - q^33 + 2 * q^35 + 5 * q^37 + q^39 + 19 * q^41 - 14 * q^43 + 10 * q^45 + 13 * q^47 + 4 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 26 * q^59 - q^61 + 13 * q^63 - 2 * q^65 - 5 * q^67 + 17 * q^69 + 18 * q^71 + 7 * q^73 - q^75 + q^77 - 9 * q^79 - 4 * q^81 - 12 * q^83 + 14 * q^85 + 46 * q^87 + 4 * q^89 - 4 * q^91 + 3 * q^93 + 26 * q^95 + 25 * q^97 + 48 * q^99

Introduction

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