LMFDB - Embedded newform 4560.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.17741$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{9} -3.43630 q^{11} +4.91852 q^{13} +1.00000 q^{15} -4.35482 q^{17} +1.00000 q^{19} +2.91852 q^{21} -6.35482 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.27334 q^{29} +2.35482 q^{31} +3.43630 q^{33} +2.91852 q^{35} +4.91852 q^{37} -4.91852 q^{39} +1.43630 q^{41} -6.91852 q^{43} -1.00000 q^{45} +6.35482 q^{47} +1.51777 q^{49} +4.35482 q^{51} -4.35482 q^{53} +3.43630 q^{55} -1.00000 q^{57} -9.83705 q^{59} -6.19186 q^{61} -2.91852 q^{63} -4.91852 q^{65} +8.70964 q^{67} +6.35482 q^{69} -6.87259 q^{71} +10.0000 q^{73} -1.00000 q^{75} +10.0289 q^{77} -8.70964 q^{79} +1.00000 q^{81} +16.9015 q^{83} +4.35482 q^{85} +7.27334 q^{87} -4.40075 q^{89} -14.3548 q^{91} -2.35482 q^{93} -1.00000 q^{95} +12.9185 q^{97} -3.43630 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 4 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 6 q^{39} - 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} - 2 q^{51} + 2 q^{53} + 4 q^{55} - 3 q^{57} - 12 q^{59} + 14 q^{61} - 6 q^{65} - 4 q^{67} + 4 q^{69} - 8 q^{71} + 30 q^{73} - 3 q^{75} - 20 q^{77} + 4 q^{79} + 3 q^{81} - 12 q^{83} - 2 q^{85} - 2 q^{87} - 2 q^{89} - 28 q^{91} + 8 q^{93} - 3 q^{95} + 30 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9 - 4 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 + 3 * q^19 - 4 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 + 6 * q^37 - 6 * q^39 - 2 * q^41 - 12 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 - 2 * q^51 + 2 * q^53 + 4 * q^55 - 3 * q^57 - 12 * q^59 + 14 * q^61 - 6 * q^65 - 4 * q^67 + 4 * q^69 - 8 * q^71 + 30 * q^73 - 3 * q^75 - 20 * q^77 + 4 * q^79 + 3 * q^81 - 12 * q^83 - 2 * q^85 - 2 * q^87 - 2 * q^89 - 28 * q^91 + 8 * q^93 - 3 * q^95 + 30 * q^97 - 4 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.91852	−1.10310	−0.551549	−	0.834143i	$-0.685963\pi$
$7$	−2.91852	−1.10310	−0.551549	+	0.834143i	$0.685963\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.43630	−1.03608	−0.518041	−	0.855356i	$-0.673339\pi$
$11$	−3.43630	−1.03608	−0.518041	+	0.855356i	$0.673339\pi$
$12$	0	0
$13$	4.91852	1.36415	0.682076	−	0.731281i	$-0.261077\pi$
$13$	4.91852	1.36415	0.682076	+	0.731281i	$0.261077\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.35482	−1.05620	−0.528099	−	0.849183i	$-0.677095\pi$
$17$	−4.35482	−1.05620	−0.528099	+	0.849183i	$0.677095\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.91852	0.636874
$22$	0	0
$23$	−6.35482	−1.32507	−0.662536	−	0.749030i	$-0.730520\pi$
$23$	−6.35482	−1.32507	−0.662536	+	0.749030i	$0.730520\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.27334	−1.35063	−0.675313	−	0.737531i	$-0.735991\pi$
$29$	−7.27334	−1.35063	−0.675313	+	0.737531i	$0.735991\pi$
$30$	0	0
$31$	2.35482	0.422938	0.211469	−	0.977385i	$-0.432175\pi$
$31$	2.35482	0.422938	0.211469	+	0.977385i	$0.432175\pi$
$32$	0	0
$33$	3.43630	0.598182
$34$	0	0
$35$	2.91852	0.493320
$36$	0	0
$37$	4.91852	0.808600	0.404300	−	0.914626i	$-0.367515\pi$
$37$	4.91852	0.808600	0.404300	+	0.914626i	$0.367515\pi$
$38$	0	0
$39$	−4.91852	−0.787594
$40$	0	0
$41$	1.43630	0.224312	0.112156	−	0.993691i	$-0.464224\pi$
$41$	1.43630	0.224312	0.112156	+	0.993691i	$0.464224\pi$
$42$	0	0
$43$	−6.91852	−1.05506	−0.527532	−	0.849535i	$-0.676883\pi$
$43$	−6.91852	−1.05506	−0.527532	+	0.849535i	$0.676883\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.35482	0.926946	0.463473	−	0.886111i	$-0.346603\pi$
$47$	6.35482	0.926946	0.463473	+	0.886111i	$0.346603\pi$
$48$	0	0
$49$	1.51777	0.216825
$50$	0	0
$51$	4.35482	0.609797
$52$	0	0
$53$	−4.35482	−0.598180	−0.299090	−	0.954225i	$-0.596683\pi$
$53$	−4.35482	−0.598180	−0.299090	+	0.954225i	$0.596683\pi$
$54$	0	0
$55$	3.43630	0.463350
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−9.83705	−1.28067	−0.640337	−	0.768094i	$-0.721205\pi$
$59$	−9.83705	−1.28067	−0.640337	+	0.768094i	$0.721205\pi$
$60$	0	0
$61$	−6.19186	−0.792787	−0.396394	−	0.918081i	$-0.629738\pi$
$61$	−6.19186	−0.792787	−0.396394	+	0.918081i	$0.629738\pi$
$62$	0	0
$63$	−2.91852	−0.367699
$64$	0	0
$65$	−4.91852	−0.610068
$66$	0	0
$67$	8.70964	1.06405	0.532026	−	0.846728i	$-0.321431\pi$
$67$	8.70964	1.06405	0.532026	+	0.846728i	$0.321431\pi$
$68$	0	0
$69$	6.35482	0.765030
$70$	0	0
$71$	−6.87259	−0.815627	−0.407813	−	0.913065i	$-0.633709\pi$
$71$	−6.87259	−0.815627	−0.407813	+	0.913065i	$0.633709\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	10.0289	1.14290
$78$	0	0
$79$	−8.70964	−0.979911	−0.489955	−	0.871747i	$-0.662987\pi$
$79$	−8.70964	−0.979911	−0.489955	+	0.871747i	$0.662987\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.9015	1.85518	0.927591	−	0.373599i	$-0.121876\pi$
$83$	16.9015	1.85518	0.927591	+	0.373599i	$0.121876\pi$
$84$	0	0
$85$	4.35482	0.472346
$86$	0	0
$87$	7.27334	0.779784
$88$	0	0
$89$	−4.40075	−0.466478	−0.233239	−	0.972419i	$-0.574933\pi$
$89$	−4.40075	−0.466478	−0.233239	+	0.972419i	$0.574933\pi$
$90$	0	0
$91$	−14.3548	−1.50479
$92$	0	0
$93$	−2.35482	−0.244183
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	12.9185	1.31168	0.655839	−	0.754901i	$-0.272315\pi$
$97$	12.9185	1.31168	0.655839	+	0.754901i	$0.272315\pi$
$98$	0	0
$99$	−3.43630	−0.345361
$100$	0	0
$101$	7.83705	0.779815	0.389908	−	0.920854i	$-0.372507\pi$
$101$	7.83705	0.779815	0.389908	+	0.920854i	$0.372507\pi$
$102$	0	0
$103$	−8.70964	−0.858186	−0.429093	−	0.903260i	$-0.641167\pi$
$103$	−8.70964	−0.858186	−0.429093	+	0.903260i	$0.641167\pi$
$104$	0	0
$105$	−2.91852	−0.284819
$106$	0	0
$107$	−8.70964	−0.841993	−0.420996	−	0.907062i	$-0.638319\pi$
$107$	−8.70964	−0.841993	−0.420996	+	0.907062i	$0.638319\pi$
$108$	0	0
$109$	4.16295	0.398739	0.199369	−	0.979924i	$-0.436111\pi$
$109$	4.16295	0.398739	0.199369	+	0.979924i	$0.436111\pi$
$110$	0	0
$111$	−4.91852	−0.466846
$112$	0	0
$113$	−6.19186	−0.582482	−0.291241	−	0.956650i	$-0.594068\pi$
$113$	−6.19186	−0.582482	−0.291241	+	0.956650i	$0.594068\pi$
$114$	0	0
$115$	6.35482	0.592590
$116$	0	0
$117$	4.91852	0.454718
$118$	0	0
$119$	12.7096	1.16509
$120$	0	0
$121$	0.808135	0.0734669
$122$	0	0
$123$	−1.43630	−0.129507
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.709639	−0.0629703	−0.0314851	−	0.999504i	$-0.510024\pi$
$127$	−0.709639	−0.0629703	−0.0314851	+	0.999504i	$0.510024\pi$
$128$	0	0
$129$	6.91852	0.609142
$130$	0	0
$131$	9.27334	0.810216	0.405108	−	0.914269i	$-0.367234\pi$
$131$	9.27334	0.810216	0.405108	+	0.914269i	$0.367234\pi$
$132$	0	0
$133$	−2.91852	−0.253068
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	13.6741	1.16826	0.584128	−	0.811661i	$-0.301437\pi$
$137$	13.6741	1.16826	0.584128	+	0.811661i	$0.301437\pi$
$138$	0	0
$139$	−5.12741	−0.434901	−0.217450	−	0.976071i	$-0.569774\pi$
$139$	−5.12741	−0.434901	−0.217450	+	0.976071i	$0.569774\pi$
$140$	0	0
$141$	−6.35482	−0.535172
$142$	0	0
$143$	−16.9015	−1.41337
$144$	0	0
$145$	7.27334	0.604018
$146$	0	0
$147$	−1.51777	−0.125184
$148$	0	0
$149$	18.3837	1.50605	0.753027	−	0.657990i	$-0.228593\pi$
$149$	18.3837	1.50605	0.753027	+	0.657990i	$0.228593\pi$
$150$	0	0
$151$	22.0289	1.79269	0.896344	−	0.443360i	$-0.146214\pi$
$151$	22.0289	1.79269	0.896344	+	0.443360i	$0.146214\pi$
$152$	0	0
$153$	−4.35482	−0.352066
$154$	0	0
$155$	−2.35482	−0.189144
$156$	0	0
$157$	4.87259	0.388875	0.194438	−	0.980915i	$-0.437712\pi$
$157$	4.87259	0.388875	0.194438	+	0.980915i	$0.437712\pi$
$158$	0	0
$159$	4.35482	0.345360
$160$	0	0
$161$	18.5467	1.46168
$162$	0	0
$163$	−11.6282	−0.910788	−0.455394	−	0.890290i	$-0.650502\pi$
$163$	−11.6282	−0.910788	−0.455394	+	0.890290i	$0.650502\pi$
$164$	0	0
$165$	−3.43630	−0.267515
$166$	0	0
$167$	20.1919	1.56249	0.781247	−	0.624222i	$-0.214584\pi$
$167$	20.1919	1.56249	0.781247	+	0.624222i	$0.214584\pi$
$168$	0	0
$169$	11.1919	0.860913
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	21.1563	1.60848	0.804242	−	0.594301i	$-0.202571\pi$
$173$	21.1563	1.60848	0.804242	+	0.594301i	$0.202571\pi$
$174$	0	0
$175$	−2.91852	−0.220620
$176$	0	0
$177$	9.83705	0.739398
$178$	0	0
$179$	−8.80150	−0.657855	−0.328927	−	0.944355i	$-0.606687\pi$
$179$	−8.80150	−0.657855	−0.328927	+	0.944355i	$0.606687\pi$
$180$	0	0
$181$	−4.87259	−0.362177	−0.181089	−	0.983467i	$-0.557962\pi$
$181$	−4.87259	−0.362177	−0.181089	+	0.983467i	$0.557962\pi$
$182$	0	0
$183$	6.19186	0.457716
$184$	0	0
$185$	−4.91852	−0.361617
$186$	0	0
$187$	14.9645	1.09431
$188$	0	0
$189$	2.91852	0.212291
$190$	0	0
$191$	24.1459	1.74714	0.873569	−	0.486700i	$-0.161799\pi$
$191$	24.1459	1.74714	0.873569	+	0.486700i	$0.161799\pi$
$192$	0	0
$193$	2.37184	0.170729	0.0853643	−	0.996350i	$-0.472795\pi$
$193$	2.37184	0.170729	0.0853643	+	0.996350i	$0.472795\pi$
$194$	0	0
$195$	4.91852	0.352223
$196$	0	0
$197$	−24.0289	−1.71199	−0.855994	−	0.516985i	$-0.827054\pi$
$197$	−24.0289	−1.71199	−0.855994	+	0.516985i	$0.827054\pi$
$198$	0	0
$199$	26.5467	1.88184	0.940922	−	0.338623i	$-0.109961\pi$
$199$	26.5467	1.88184	0.940922	+	0.338623i	$0.109961\pi$
$200$	0	0
$201$	−8.70964	−0.614331
$202$	0	0
$203$	21.2274	1.48987
$204$	0	0
$205$	−1.43630	−0.100315
$206$	0	0
$207$	−6.35482	−0.441690
$208$	0	0
$209$	−3.43630	−0.237694
$210$	0	0
$211$	−21.4193	−1.47456	−0.737282	−	0.675585i	$-0.763891\pi$
$211$	−21.4193	−1.47456	−0.737282	+	0.675585i	$0.763891\pi$
$212$	0	0
$213$	6.87259	0.470902
$214$	0	0
$215$	6.91852	0.471839
$216$	0	0
$217$	−6.87259	−0.466542
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−21.4193	−1.44082
$222$	0	0
$223$	16.7096	1.11896	0.559480	−	0.828844i	$-0.311001\pi$
$223$	16.7096	1.11896	0.559480	+	0.828844i	$0.311001\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.51777	0.299855	0.149928	−	0.988697i	$-0.452096\pi$
$227$	4.51777	0.299855	0.149928	+	0.988697i	$0.452096\pi$
$228$	0	0
$229$	−15.2274	−1.00626	−0.503128	−	0.864212i	$-0.667818\pi$
$229$	−15.2274	−1.00626	−0.503128	+	0.864212i	$0.667818\pi$
$230$	0	0
$231$	−10.0289	−0.659854
$232$	0	0
$233$	15.7452	1.03150	0.515751	−	0.856739i	$-0.327513\pi$
$233$	15.7452	1.03150	0.515751	+	0.856739i	$0.327513\pi$
$234$	0	0
$235$	−6.35482	−0.414543
$236$	0	0
$237$	8.70964	0.565752
$238$	0	0
$239$	3.43630	0.222276	0.111138	−	0.993805i	$-0.464551\pi$
$239$	3.43630	0.222276	0.111138	+	0.993805i	$0.464551\pi$
$240$	0	0
$241$	15.7452	1.01424	0.507118	−	0.861876i	$-0.330711\pi$
$241$	15.7452	1.01424	0.507118	+	0.861876i	$0.330711\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.51777	−0.0969670
$246$	0	0
$247$	4.91852	0.312958
$248$	0	0
$249$	−16.9015	−1.07109
$250$	0	0
$251$	2.40075	0.151534	0.0757669	−	0.997126i	$-0.475859\pi$
$251$	2.40075	0.151534	0.0757669	+	0.997126i	$0.475859\pi$
$252$	0	0
$253$	21.8370	1.37288
$254$	0	0
$255$	−4.35482	−0.272709
$256$	0	0
$257$	−30.1919	−1.88332	−0.941658	−	0.336570i	$-0.890733\pi$
$257$	−30.1919	−1.88332	−0.941658	+	0.336570i	$0.890733\pi$
$258$	0	0
$259$	−14.3548	−0.891965
$260$	0	0
$261$	−7.27334	−0.450209
$262$	0	0
$263$	27.1563	1.67453	0.837265	−	0.546797i	$-0.184153\pi$
$263$	27.1563	1.67453	0.837265	+	0.546797i	$0.184153\pi$
$264$	0	0
$265$	4.35482	0.267514
$266$	0	0
$267$	4.40075	0.269321
$268$	0	0
$269$	−0.400748	−0.0244341	−0.0122170	−	0.999925i	$-0.503889\pi$
$269$	−0.400748	−0.0244341	−0.0122170	+	0.999925i	$0.503889\pi$
$270$	0	0
$271$	7.90814	0.480385	0.240193	−	0.970725i	$-0.422789\pi$
$271$	7.90814	0.480385	0.240193	+	0.970725i	$0.422789\pi$
$272$	0	0
$273$	14.3548	0.868793
$274$	0	0
$275$	−3.43630	−0.207216
$276$	0	0
$277$	11.7452	0.705700	0.352850	−	0.935680i	$-0.385213\pi$
$277$	11.7452	0.705700	0.352850	+	0.935680i	$0.385213\pi$
$278$	0	0
$279$	2.35482	0.140979
$280$	0	0
$281$	17.8200	1.06305	0.531527	−	0.847041i	$-0.321618\pi$
$281$	17.8200	1.06305	0.531527	+	0.847041i	$0.321618\pi$
$282$	0	0
$283$	−10.5926	−0.629665	−0.314833	−	0.949147i	$-0.601948\pi$
$283$	−10.5926	−0.629665	−0.314833	+	0.949147i	$0.601948\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	−4.19186	−0.247438
$288$	0	0
$289$	1.96445	0.115556
$290$	0	0
$291$	−12.9185	−0.757297
$292$	0	0
$293$	2.51777	0.147090	0.0735450	−	0.997292i	$-0.476569\pi$
$293$	2.51777	0.147090	0.0735450	+	0.997292i	$0.476569\pi$
$294$	0	0
$295$	9.83705	0.572735
$296$	0	0
$297$	3.43630	0.199394
$298$	0	0
$299$	−31.2563	−1.80760
$300$	0	0
$301$	20.1919	1.16384
$302$	0	0
$303$	−7.83705	−0.450226
$304$	0	0
$305$	6.19186	0.354545
$306$	0	0
$307$	13.1274	0.749221	0.374610	−	0.927182i	$-0.377776\pi$
$307$	13.1274	0.749221	0.374610	+	0.927182i	$0.377776\pi$
$308$	0	0
$309$	8.70964	0.495474
$310$	0	0
$311$	−24.5297	−1.39095	−0.695475	−	0.718550i	$-0.744806\pi$
$311$	−24.5297	−1.39095	−0.695475	+	0.718550i	$0.744806\pi$
$312$	0	0
$313$	3.12741	0.176771	0.0883857	−	0.996086i	$-0.471829\pi$
$313$	3.12741	0.176771	0.0883857	+	0.996086i	$0.471829\pi$
$314$	0	0
$315$	2.91852	0.164440
$316$	0	0
$317$	−22.9934	−1.29144	−0.645718	−	0.763576i	$-0.723442\pi$
$317$	−22.9934	−1.29144	−0.645718	+	0.763576i	$0.723442\pi$
$318$	0	0
$319$	24.9934	1.39936
$320$	0	0
$321$	8.70964	0.486125
$322$	0	0
$323$	−4.35482	−0.242309
$324$	0	0
$325$	4.91852	0.272831
$326$	0	0
$327$	−4.16295	−0.230212
$328$	0	0
$329$	−18.5467	−1.02251
$330$	0	0
$331$	−4.60963	−0.253368	−0.126684	−	0.991943i	$-0.540433\pi$
$331$	−4.60963	−0.253368	−0.126684	+	0.991943i	$0.540433\pi$
$332$	0	0
$333$	4.91852	0.269533
$334$	0	0
$335$	−8.70964	−0.475858
$336$	0	0
$337$	−25.3023	−1.37830	−0.689151	−	0.724618i	$-0.742016\pi$
$337$	−25.3023	−1.37830	−0.689151	+	0.724618i	$0.742016\pi$
$338$	0	0
$339$	6.19186	0.336296
$340$	0	0
$341$	−8.09186	−0.438199
$342$	0	0
$343$	16.0000	0.863919
$344$	0	0
$345$	−6.35482	−0.342132
$346$	0	0
$347$	12.1919	0.654494	0.327247	−	0.944939i	$-0.393879\pi$
$347$	12.1919	0.654494	0.327247	+	0.944939i	$0.393879\pi$
$348$	0	0
$349$	−2.38373	−0.127598	−0.0637990	−	0.997963i	$-0.520322\pi$
$349$	−2.38373	−0.127598	−0.0637990	+	0.997963i	$0.520322\pi$
$350$	0	0
$351$	−4.91852	−0.262531
$352$	0	0
$353$	−6.38373	−0.339772	−0.169886	−	0.985464i	$-0.554340\pi$
$353$	−6.38373	−0.339772	−0.169886	+	0.985464i	$0.554340\pi$
$354$	0	0
$355$	6.87259	0.364759
$356$	0	0
$357$	−12.7096	−0.672665
$358$	0	0
$359$	−9.27334	−0.489428	−0.244714	−	0.969595i	$-0.578694\pi$
$359$	−9.27334	−0.489428	−0.244714	+	0.969595i	$0.578694\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.808135	−0.0424161
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	35.2104	1.83797	0.918984	−	0.394295i	$-0.129011\pi$
$367$	35.2104	1.83797	0.918984	+	0.394295i	$0.129011\pi$
$368$	0	0
$369$	1.43630	0.0747706
$370$	0	0
$371$	12.7096	0.659852
$372$	0	0
$373$	17.6282	0.912752	0.456376	−	0.889787i	$-0.349147\pi$
$373$	17.6282	0.912752	0.456376	+	0.889787i	$0.349147\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−35.7741	−1.84246
$378$	0	0
$379$	−27.7741	−1.42666	−0.713330	−	0.700829i	$-0.752814\pi$
$379$	−27.7741	−1.42666	−0.713330	+	0.700829i	$0.752814\pi$
$380$	0	0
$381$	0.709639	0.0363559
$382$	0	0
$383$	34.4548	1.76056	0.880280	−	0.474455i	$-0.157355\pi$
$383$	34.4548	1.76056	0.880280	+	0.474455i	$0.157355\pi$
$384$	0	0
$385$	−10.0289	−0.511121
$386$	0	0
$387$	−6.91852	−0.351688
$388$	0	0
$389$	−16.6385	−0.843608	−0.421804	−	0.906687i	$-0.638603\pi$
$389$	−16.6385	−0.843608	−0.421804	+	0.906687i	$0.638603\pi$
$390$	0	0
$391$	27.6741	1.39954
$392$	0	0
$393$	−9.27334	−0.467778
$394$	0	0
$395$	8.70964	0.438229
$396$	0	0
$397$	−5.29036	−0.265516	−0.132758	−	0.991149i	$-0.542383\pi$
$397$	−5.29036	−0.265516	−0.132758	+	0.991149i	$0.542383\pi$
$398$	0	0
$399$	2.91852	0.146109
$400$	0	0
$401$	13.2022	0.659289	0.329644	−	0.944105i	$-0.393071\pi$
$401$	13.2022	0.659289	0.329644	+	0.944105i	$0.393071\pi$
$402$	0	0
$403$	11.5822	0.576952
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−16.9015	−0.837776
$408$	0	0
$409$	−1.19850	−0.0592622	−0.0296311	−	0.999561i	$-0.509433\pi$
$409$	−1.19850	−0.0592622	−0.0296311	+	0.999561i	$0.509433\pi$
$410$	0	0
$411$	−13.6741	−0.674493
$412$	0	0
$413$	28.7096	1.41271
$414$	0	0
$415$	−16.9015	−0.829662
$416$	0	0
$417$	5.12741	0.251090
$418$	0	0
$419$	2.30889	0.112797	0.0563983	−	0.998408i	$-0.482038\pi$
$419$	2.30889	0.112797	0.0563983	+	0.998408i	$0.482038\pi$
$420$	0	0
$421$	36.4548	1.77670	0.888350	−	0.459167i	$-0.151852\pi$
$421$	36.4548	1.77670	0.888350	+	0.459167i	$0.151852\pi$
$422$	0	0
$423$	6.35482	0.308982
$424$	0	0
$425$	−4.35482	−0.211240
$426$	0	0
$427$	18.0711	0.874522
$428$	0	0
$429$	16.9015	0.816012
$430$	0	0
$431$	19.5822	0.943243	0.471621	−	0.881801i	$-0.343669\pi$
$431$	19.5822	0.943243	0.471621	+	0.881801i	$0.343669\pi$
$432$	0	0
$433$	9.72002	0.467114	0.233557	−	0.972343i	$-0.424963\pi$
$433$	9.72002	0.467114	0.233557	+	0.972343i	$0.424963\pi$
$434$	0	0
$435$	−7.27334	−0.348730
$436$	0	0
$437$	−6.35482	−0.303992
$438$	0	0
$439$	−13.0355	−0.622153	−0.311076	−	0.950385i	$-0.600690\pi$
$439$	−13.0355	−0.622153	−0.311076	+	0.950385i	$0.600690\pi$
$440$	0	0
$441$	1.51777	0.0722750
$442$	0	0
$443$	5.31927	0.252726	0.126363	−	0.991984i	$-0.459670\pi$
$443$	5.31927	0.252726	0.126363	+	0.991984i	$0.459670\pi$
$444$	0	0
$445$	4.40075	0.208615
$446$	0	0
$447$	−18.3837	−0.869521
$448$	0	0
$449$	−1.85406	−0.0874987	−0.0437494	−	0.999043i	$-0.513930\pi$
$449$	−1.85406	−0.0874987	−0.0437494	+	0.999043i	$0.513930\pi$
$450$	0	0
$451$	−4.93554	−0.232406
$452$	0	0
$453$	−22.0289	−1.03501
$454$	0	0
$455$	14.3548	0.672964
$456$	0	0
$457$	−25.5822	−1.19669	−0.598343	−	0.801240i	$-0.704174\pi$
$457$	−25.5822	−1.19669	−0.598343	+	0.801240i	$0.704174\pi$
$458$	0	0
$459$	4.35482	0.203266
$460$	0	0
$461$	−11.7452	−0.547028	−0.273514	−	0.961868i	$-0.588186\pi$
$461$	−11.7452	−0.547028	−0.273514	+	0.961868i	$0.588186\pi$
$462$	0	0
$463$	7.72002	0.358780	0.179390	−	0.983778i	$-0.442588\pi$
$463$	7.72002	0.358780	0.179390	+	0.983778i	$0.442588\pi$
$464$	0	0
$465$	2.35482	0.109202
$466$	0	0
$467$	2.68073	0.124049	0.0620247	−	0.998075i	$-0.480244\pi$
$467$	2.68073	0.124049	0.0620247	+	0.998075i	$0.480244\pi$
$468$	0	0
$469$	−25.4193	−1.17375
$470$	0	0
$471$	−4.87259	−0.224517
$472$	0	0
$473$	23.7741	1.09313
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−4.35482	−0.199393
$478$	0	0
$479$	−12.4718	−0.569853	−0.284927	−	0.958549i	$-0.591969\pi$
$479$	−12.4718	−0.569853	−0.284927	+	0.958549i	$0.591969\pi$
$480$	0	0
$481$	24.1919	1.10305
$482$	0	0
$483$	−18.5467	−0.843903
$484$	0	0
$485$	−12.9185	−0.586600
$486$	0	0
$487$	33.8370	1.53330	0.766651	−	0.642064i	$-0.221921\pi$
$487$	33.8370	1.53330	0.766651	+	0.642064i	$0.221921\pi$
$488$	0	0
$489$	11.6282	0.525844
$490$	0	0
$491$	14.7267	0.664605	0.332302	−	0.943173i	$-0.392175\pi$
$491$	14.7267	0.664605	0.332302	+	0.943173i	$0.392175\pi$
$492$	0	0
$493$	31.6741	1.42653
$494$	0	0
$495$	3.43630	0.154450
$496$	0	0
$497$	20.0578	0.899716
$498$	0	0
$499$	−7.19850	−0.322249	−0.161125	−	0.986934i	$-0.551512\pi$
$499$	−7.19850	−0.322249	−0.161125	+	0.986934i	$0.551512\pi$
$500$	0	0
$501$	−20.1919	−0.902106
$502$	0	0
$503$	−21.6111	−0.963593	−0.481797	−	0.876283i	$-0.660016\pi$
$503$	−21.6111	−0.963593	−0.481797	+	0.876283i	$0.660016\pi$
$504$	0	0
$505$	−7.83705	−0.348744
$506$	0	0
$507$	−11.1919	−0.497048
$508$	0	0
$509$	−4.72666	−0.209505	−0.104753	−	0.994498i	$-0.533405\pi$
$509$	−4.72666	−0.209505	−0.104753	+	0.994498i	$0.533405\pi$
$510$	0	0
$511$	−29.1852	−1.29108
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	8.70964	0.383793
$516$	0	0
$517$	−21.8370	−0.960392
$518$	0	0
$519$	−21.1563	−0.928659
$520$	0	0
$521$	1.43630	0.0629253	0.0314627	−	0.999505i	$-0.489983\pi$
$521$	1.43630	0.0629253	0.0314627	+	0.999505i	$0.489983\pi$
$522$	0	0
$523$	14.1630	0.619303	0.309651	−	0.950850i	$-0.399788\pi$
$523$	14.1630	0.619303	0.309651	+	0.950850i	$0.399788\pi$
$524$	0	0
$525$	2.91852	0.127375
$526$	0	0
$527$	−10.2548	−0.446707
$528$	0	0
$529$	17.3837	0.755814
$530$	0	0
$531$	−9.83705	−0.426891
$532$	0	0
$533$	7.06446	0.305996
$534$	0	0
$535$	8.70964	0.376551
$536$	0	0
$537$	8.80150	0.379813
$538$	0	0
$539$	−5.21552	−0.224648
$540$	0	0
$541$	−19.4193	−0.834900	−0.417450	−	0.908700i	$-0.637076\pi$
$541$	−19.4193	−0.834900	−0.417450	+	0.908700i	$0.637076\pi$
$542$	0	0
$543$	4.87259	0.209103
$544$	0	0
$545$	−4.16295	−0.178321
$546$	0	0
$547$	13.1274	0.561287	0.280644	−	0.959812i	$-0.409452\pi$
$547$	13.1274	0.561287	0.280644	+	0.959812i	$0.409452\pi$
$548$	0	0
$549$	−6.19186	−0.264262
$550$	0	0
$551$	−7.27334	−0.309855
$552$	0	0
$553$	25.4193	1.08094
$554$	0	0
$555$	4.91852	0.208780
$556$	0	0
$557$	−0.254813	−0.0107968	−0.00539839	−	0.999985i	$-0.501718\pi$
$557$	−0.254813	−0.0107968	−0.00539839	+	0.999985i	$0.501718\pi$
$558$	0	0
$559$	−34.0289	−1.43927
$560$	0	0
$561$	−14.9645	−0.631800
$562$	0	0
$563$	10.4467	0.440275	0.220137	−	0.975469i	$-0.429349\pi$
$563$	10.4467	0.440275	0.220137	+	0.975469i	$0.429349\pi$
$564$	0	0
$565$	6.19186	0.260494
$566$	0	0
$567$	−2.91852	−0.122566
$568$	0	0
$569$	−12.6926	−0.532102	−0.266051	−	0.963959i	$-0.585719\pi$
$569$	−12.6926	−0.532102	−0.266051	+	0.963959i	$0.585719\pi$
$570$	0	0
$571$	−29.3274	−1.22731	−0.613657	−	0.789573i	$-0.710302\pi$
$571$	−29.3274	−1.22731	−0.613657	+	0.789573i	$0.710302\pi$
$572$	0	0
$573$	−24.1459	−1.00871
$574$	0	0
$575$	−6.35482	−0.265014
$576$	0	0
$577$	41.6401	1.73350	0.866749	−	0.498745i	$-0.166205\pi$
$577$	41.6401	1.73350	0.866749	+	0.498745i	$0.166205\pi$
$578$	0	0
$579$	−2.37184	−0.0985703
$580$	0	0
$581$	−49.3274	−2.04645
$582$	0	0
$583$	14.9645	0.619764
$584$	0	0
$585$	−4.91852	−0.203356
$586$	0	0
$587$	10.0289	0.413937	0.206969	−	0.978348i	$-0.433640\pi$
$587$	10.0289	0.413937	0.206969	+	0.978348i	$0.433640\pi$
$588$	0	0
$589$	2.35482	0.0970286
$590$	0	0
$591$	24.0289	0.988417
$592$	0	0
$593$	−32.4548	−1.33276	−0.666380	−	0.745612i	$-0.732157\pi$
$593$	−32.4548	−1.33276	−0.666380	+	0.745612i	$0.732157\pi$
$594$	0	0
$595$	−12.7096	−0.521044
$596$	0	0
$597$	−26.5467	−1.08648
$598$	0	0
$599$	−2.07110	−0.0846227	−0.0423114	−	0.999104i	$-0.513472\pi$
$599$	−2.07110	−0.0846227	−0.0423114	+	0.999104i	$0.513472\pi$
$600$	0	0
$601$	−9.58223	−0.390867	−0.195434	−	0.980717i	$-0.562611\pi$
$601$	−9.58223	−0.390867	−0.195434	+	0.980717i	$0.562611\pi$
$602$	0	0
$603$	8.70964	0.354684
$604$	0	0
$605$	−0.808135	−0.0328554
$606$	0	0
$607$	24.8015	1.00666	0.503331	−	0.864094i	$-0.332108\pi$
$607$	24.8015	1.00666	0.503331	+	0.864094i	$0.332108\pi$
$608$	0	0
$609$	−21.2274	−0.860178
$610$	0	0
$611$	31.2563	1.26450
$612$	0	0
$613$	25.5822	1.03326	0.516628	−	0.856210i	$-0.327187\pi$
$613$	25.5822	1.03326	0.516628	+	0.856210i	$0.327187\pi$
$614$	0	0
$615$	1.43630	0.0579171
$616$	0	0
$617$	−43.7030	−1.75942	−0.879708	−	0.475514i	$-0.842262\pi$
$617$	−43.7030	−1.75942	−0.879708	+	0.475514i	$0.842262\pi$
$618$	0	0
$619$	1.83705	0.0738371	0.0369185	−	0.999318i	$-0.488246\pi$
$619$	1.83705	0.0738371	0.0369185	+	0.999318i	$0.488246\pi$
$620$	0	0
$621$	6.35482	0.255010
$622$	0	0
$623$	12.8437	0.514571
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.43630	0.137232
$628$	0	0
$629$	−21.4193	−0.854043
$630$	0	0
$631$	43.0223	1.71269	0.856345	−	0.516404i	$-0.172730\pi$
$631$	43.0223	1.71269	0.856345	+	0.516404i	$0.172730\pi$
$632$	0	0
$633$	21.4193	0.851340
$634$	0	0
$635$	0.709639	0.0281612
$636$	0	0
$637$	7.46521	0.295782
$638$	0	0
$639$	−6.87259	−0.271876
$640$	0	0
$641$	17.5282	0.692320	0.346160	−	0.938175i	$-0.387485\pi$
$641$	17.5282	0.692320	0.346160	+	0.938175i	$0.387485\pi$
$642$	0	0
$643$	47.6860	1.88055	0.940276	−	0.340414i	$-0.110567\pi$
$643$	47.6860	1.88055	0.940276	+	0.340414i	$0.110567\pi$
$644$	0	0
$645$	−6.91852	−0.272417
$646$	0	0
$647$	−21.6111	−0.849622	−0.424811	−	0.905282i	$-0.639659\pi$
$647$	−21.6111	−0.849622	−0.424811	+	0.905282i	$0.639659\pi$
$648$	0	0
$649$	33.8030	1.32688
$650$	0	0
$651$	6.87259	0.269358
$652$	0	0
$653$	28.0289	1.09686	0.548428	−	0.836198i	$-0.315226\pi$
$653$	28.0289	1.09686	0.548428	+	0.836198i	$0.315226\pi$
$654$	0	0
$655$	−9.27334	−0.362339
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−9.74519	−0.379619	−0.189809	−	0.981821i	$-0.560787\pi$
$659$	−9.74519	−0.379619	−0.189809	+	0.981821i	$0.560787\pi$
$660$	0	0
$661$	−16.5467	−0.643591	−0.321796	−	0.946809i	$-0.604286\pi$
$661$	−16.5467	−0.643591	−0.321796	+	0.946809i	$0.604286\pi$
$662$	0	0
$663$	21.4193	0.831856
$664$	0	0
$665$	2.91852	0.113175
$666$	0	0
$667$	46.2208	1.78968
$668$	0	0
$669$	−16.7096	−0.646032
$670$	0	0
$671$	21.2771	0.821393
$672$	0	0
$673$	−24.2667	−0.935413	−0.467706	−	0.883884i	$-0.654920\pi$
$673$	−24.2667	−0.935413	−0.467706	+	0.883884i	$0.654920\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−24.6807	−0.948557	−0.474279	−	0.880375i	$-0.657291\pi$
$677$	−24.6807	−0.948557	−0.474279	+	0.880375i	$0.657291\pi$
$678$	0	0
$679$	−37.7030	−1.44691
$680$	0	0
$681$	−4.51777	−0.173121
$682$	0	0
$683$	−2.96445	−0.113432	−0.0567158	−	0.998390i	$-0.518063\pi$
$683$	−2.96445	−0.113432	−0.0567158	+	0.998390i	$0.518063\pi$
$684$	0	0
$685$	−13.6741	−0.522460
$686$	0	0
$687$	15.2274	0.580962
$688$	0	0
$689$	−21.4193	−0.816009
$690$	0	0
$691$	−39.5822	−1.50578	−0.752890	−	0.658147i	$-0.771341\pi$
$691$	−39.5822	−1.50578	−0.752890	+	0.658147i	$0.771341\pi$
$692$	0	0
$693$	10.0289	0.380967
$694$	0	0
$695$	5.12741	0.194494
$696$	0	0
$697$	−6.25481	−0.236918
$698$	0	0
$699$	−15.7452	−0.595538
$700$	0	0
$701$	−44.3126	−1.67367	−0.836833	−	0.547459i	$-0.815595\pi$
$701$	−44.3126	−1.67367	−0.836833	+	0.547459i	$0.815595\pi$
$702$	0	0
$703$	4.91852	0.185506
$704$	0	0
$705$	6.35482	0.239336
$706$	0	0
$707$	−22.8726	−0.860212
$708$	0	0
$709$	48.3208	1.81473	0.907363	−	0.420349i	$-0.138092\pi$
$709$	48.3208	1.81473	0.907363	+	0.420349i	$0.138092\pi$
$710$	0	0
$711$	−8.70964	−0.326637
$712$	0	0
$713$	−14.9645	−0.560423
$714$	0	0
$715$	16.9015	0.632080
$716$	0	0
$717$	−3.43630	−0.128331
$718$	0	0
$719$	12.4718	0.465121	0.232561	−	0.972582i	$-0.425290\pi$
$719$	12.4718	0.465121	0.232561	+	0.972582i	$0.425290\pi$
$720$	0	0
$721$	25.4193	0.946663
$722$	0	0
$723$	−15.7452	−0.585570
$724$	0	0
$725$	−7.27334	−0.270125
$726$	0	0
$727$	−7.72002	−0.286320	−0.143160	−	0.989700i	$-0.545726\pi$
$727$	−7.72002	−0.286320	−0.143160	+	0.989700i	$0.545726\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.1289	1.11436
$732$	0	0
$733$	−0.964452	−0.0356228	−0.0178114	−	0.999841i	$-0.505670\pi$
$733$	−0.964452	−0.0356228	−0.0178114	+	0.999841i	$0.505670\pi$
$734$	0	0
$735$	1.51777	0.0559839
$736$	0	0
$737$	−29.9289	−1.10245
$738$	0	0
$739$	5.41928	0.199351	0.0996757	−	0.995020i	$-0.468219\pi$
$739$	5.41928	0.199351	0.0996757	+	0.995020i	$0.468219\pi$
$740$	0	0
$741$	−4.91852	−0.180686
$742$	0	0
$743$	9.03555	0.331482	0.165741	−	0.986169i	$-0.446998\pi$
$743$	9.03555	0.331482	0.165741	+	0.986169i	$0.446998\pi$
$744$	0	0
$745$	−18.3837	−0.673528
$746$	0	0
$747$	16.9015	0.618394
$748$	0	0
$749$	25.4193	0.928800
$750$	0	0
$751$	−22.0289	−0.803846	−0.401923	−	0.915673i	$-0.631658\pi$
$751$	−22.0289	−0.803846	−0.401923	+	0.915673i	$0.631658\pi$
$752$	0	0
$753$	−2.40075	−0.0874881
$754$	0	0
$755$	−22.0289	−0.801714
$756$	0	0
$757$	−39.0015	−1.41753	−0.708767	−	0.705443i	$-0.750748\pi$
$757$	−39.0015	−1.41753	−0.708767	+	0.705443i	$0.750748\pi$
$758$	0	0
$759$	−21.8370	−0.792635
$760$	0	0
$761$	−46.3837	−1.68141	−0.840704	−	0.541494i	$-0.817859\pi$
$761$	−46.3837	−1.68141	−0.840704	+	0.541494i	$0.817859\pi$
$762$	0	0
$763$	−12.1497	−0.439848
$764$	0	0
$765$	4.35482	0.157449
$766$	0	0
$767$	−48.3837	−1.74704
$768$	0	0
$769$	−43.7452	−1.57749	−0.788746	−	0.614719i	$-0.789269\pi$
$769$	−43.7452	−1.57749	−0.788746	+	0.614719i	$0.789269\pi$
$770$	0	0
$771$	30.1919	1.08733
$772$	0	0
$773$	−3.13555	−0.112778	−0.0563890	−	0.998409i	$-0.517959\pi$
$773$	−3.13555	−0.112778	−0.0563890	+	0.998409i	$0.517959\pi$
$774$	0	0
$775$	2.35482	0.0845876
$776$	0	0
$777$	14.3548	0.514976
$778$	0	0
$779$	1.43630	0.0514607
$780$	0	0
$781$	23.6163	0.845057
$782$	0	0
$783$	7.27334	0.259928
$784$	0	0
$785$	−4.87259	−0.173910
$786$	0	0
$787$	23.5822	0.840616	0.420308	−	0.907382i	$-0.361922\pi$
$787$	23.5822	0.840616	0.420308	+	0.907382i	$0.361922\pi$
$788$	0	0
$789$	−27.1563	−0.966790
$790$	0	0
$791$	18.0711	0.642534
$792$	0	0
$793$	−30.4548	−1.08148
$794$	0	0
$795$	−4.35482	−0.154450
$796$	0	0
$797$	7.22741	0.256008	0.128004	−	0.991774i	$-0.459143\pi$
$797$	7.22741	0.256008	0.128004	+	0.991774i	$0.459143\pi$
$798$	0	0
$799$	−27.6741	−0.979039
$800$	0	0
$801$	−4.40075	−0.155493
$802$	0	0
$803$	−34.3630	−1.21264
$804$	0	0
$805$	−18.5467	−0.653685
$806$	0	0
$807$	0.400748	0.0141070
$808$	0	0
$809$	42.3837	1.49013	0.745066	−	0.666990i	$-0.232418\pi$
$809$	42.3837	1.49013	0.745066	+	0.666990i	$0.232418\pi$
$810$	0	0
$811$	−41.0934	−1.44298	−0.721492	−	0.692423i	$-0.756543\pi$
$811$	−41.0934	−1.44298	−0.721492	+	0.692423i	$0.756543\pi$
$812$	0	0
$813$	−7.90814	−0.277351
$814$	0	0
$815$	11.6282	0.407317
$816$	0	0
$817$	−6.91852	−0.242048
$818$	0	0
$819$	−14.3548	−0.501598
$820$	0	0
$821$	13.9660	0.487415	0.243708	−	0.969849i	$-0.421636\pi$
$821$	13.9660	0.487415	0.243708	+	0.969849i	$0.421636\pi$
$822$	0	0
$823$	−22.4089	−0.781125	−0.390563	−	0.920576i	$-0.627719\pi$
$823$	−22.4089	−0.781125	−0.390563	+	0.920576i	$0.627719\pi$
$824$	0	0
$825$	3.43630	0.119636
$826$	0	0
$827$	−35.8660	−1.24718	−0.623591	−	0.781751i	$-0.714327\pi$
$827$	−35.8660	−1.24718	−0.623591	+	0.781751i	$0.714327\pi$
$828$	0	0
$829$	−38.2919	−1.32993	−0.664966	−	0.746874i	$-0.731554\pi$
$829$	−38.2919	−1.32993	−0.664966	+	0.746874i	$0.731554\pi$
$830$	0	0
$831$	−11.7452	−0.407436
$832$	0	0
$833$	−6.60963	−0.229010
$834$	0	0
$835$	−20.1919	−0.698768
$836$	0	0
$837$	−2.35482	−0.0813945
$838$	0	0
$839$	−1.12741	−0.0389224	−0.0194612	−	0.999811i	$-0.506195\pi$
$839$	−1.12741	−0.0389224	−0.0194612	+	0.999811i	$0.506195\pi$
$840$	0	0
$841$	23.9015	0.824190
$842$	0	0
$843$	−17.8200	−0.613754
$844$	0	0
$845$	−11.1919	−0.385012
$846$	0	0
$847$	−2.35856	−0.0810411
$848$	0	0
$849$	10.5926	0.363538
$850$	0	0
$851$	−31.2563	−1.07145
$852$	0	0
$853$	−42.6756	−1.46118	−0.730592	−	0.682814i	$-0.760756\pi$
$853$	−42.6756	−1.46118	−0.730592	+	0.682814i	$0.760756\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	57.4482	1.96239	0.981196	−	0.193012i	$-0.0618257\pi$
$857$	57.4482	1.96239	0.981196	+	0.193012i	$0.0618257\pi$
$858$	0	0
$859$	6.45483	0.220236	0.110118	−	0.993919i	$-0.464877\pi$
$859$	6.45483	0.220236	0.110118	+	0.993919i	$0.464877\pi$
$860$	0	0
$861$	4.19186	0.142858
$862$	0	0
$863$	17.8030	0.606021	0.303011	−	0.952987i	$-0.402008\pi$
$863$	17.8030	0.606021	0.303011	+	0.952987i	$0.402008\pi$
$864$	0	0
$865$	−21.1563	−0.719336
$866$	0	0
$867$	−1.96445	−0.0667163
$868$	0	0
$869$	29.9289	1.01527
$870$	0	0
$871$	42.8386	1.45153
$872$	0	0
$873$	12.9185	0.437226
$874$	0	0
$875$	2.91852	0.0986641
$876$	0	0
$877$	−39.4312	−1.33150	−0.665748	−	0.746177i	$-0.731887\pi$
$877$	−39.4312	−1.33150	−0.665748	+	0.746177i	$0.731887\pi$
$878$	0	0
$879$	−2.51777	−0.0849224
$880$	0	0
$881$	14.8015	0.498675	0.249338	−	0.968417i	$-0.419787\pi$
$881$	14.8015	0.498675	0.249338	+	0.968417i	$0.419787\pi$
$882$	0	0
$883$	−1.55706	−0.0523994	−0.0261997	−	0.999657i	$-0.508341\pi$
$883$	−1.55706	−0.0523994	−0.0261997	+	0.999657i	$0.508341\pi$
$884$	0	0
$885$	−9.83705	−0.330669
$886$	0	0
$887$	−4.32591	−0.145250	−0.0726249	−	0.997359i	$-0.523138\pi$
$887$	−4.32591	−0.145250	−0.0726249	+	0.997359i	$0.523138\pi$
$888$	0	0
$889$	2.07110	0.0694624
$890$	0	0
$891$	−3.43630	−0.115120
$892$	0	0
$893$	6.35482	0.212656
$894$	0	0
$895$	8.80150	0.294202
$896$	0	0
$897$	31.2563	1.04362
$898$	0	0
$899$	−17.1274	−0.571231
$900$	0	0
$901$	18.9645	0.631797
$902$	0	0
$903$	−20.1919	−0.671943
$904$	0	0
$905$	4.87259	0.161970
$906$	0	0
$907$	−26.2208	−0.870647	−0.435323	−	0.900274i	$-0.643366\pi$
$907$	−26.2208	−0.870647	−0.435323	+	0.900274i	$0.643366\pi$
$908$	0	0
$909$	7.83705	0.259938
$910$	0	0
$911$	32.3837	1.07292	0.536460	−	0.843925i	$-0.319761\pi$
$911$	32.3837	1.07292	0.536460	+	0.843925i	$0.319761\pi$
$912$	0	0
$913$	−58.0786	−1.92212
$914$	0	0
$915$	−6.19186	−0.204697
$916$	0	0
$917$	−27.0645	−0.893747
$918$	0	0
$919$	6.78074	0.223676	0.111838	−	0.993726i	$-0.464326\pi$
$919$	6.78074	0.223676	0.111838	+	0.993726i	$0.464326\pi$
$920$	0	0
$921$	−13.1274	−0.432563
$922$	0	0
$923$	−33.8030	−1.11264
$924$	0	0
$925$	4.91852	0.161720
$926$	0	0
$927$	−8.70964	−0.286062
$928$	0	0
$929$	26.4756	0.868636	0.434318	−	0.900760i	$-0.356989\pi$
$929$	26.4756	0.868636	0.434318	+	0.900760i	$0.356989\pi$
$930$	0	0
$931$	1.51777	0.0497430
$932$	0	0
$933$	24.5297	0.803065
$934$	0	0
$935$	−14.9645	−0.489390
$936$	0	0
$937$	37.9660	1.24029	0.620147	−	0.784486i	$-0.287073\pi$
$937$	37.9660	1.24029	0.620147	+	0.784486i	$0.287073\pi$
$938$	0	0
$939$	−3.12741	−0.102059
$940$	0	0
$941$	34.4378	1.12264	0.561320	−	0.827599i	$-0.310294\pi$
$941$	34.4378	1.12264	0.561320	+	0.827599i	$0.310294\pi$
$942$	0	0
$943$	−9.12741	−0.297229
$944$	0	0
$945$	−2.91852	−0.0949395
$946$	0	0
$947$	0.517774	0.0168254	0.00841270	−	0.999965i	$-0.497322\pi$
$947$	0.517774	0.0168254	0.00841270	+	0.999965i	$0.497322\pi$
$948$	0	0
$949$	49.1852	1.59662
$950$	0	0
$951$	22.9934	0.745611
$952$	0	0
$953$	−29.3563	−0.950945	−0.475472	−	0.879731i	$-0.657723\pi$
$953$	−29.3563	−0.950945	−0.475472	+	0.879731i	$0.657723\pi$
$954$	0	0
$955$	−24.1459	−0.781344
$956$	0	0
$957$	−24.9934	−0.807921
$958$	0	0
$959$	−39.9081	−1.28870
$960$	0	0
$961$	−25.4548	−0.821123
$962$	0	0
$963$	−8.70964	−0.280664
$964$	0	0
$965$	−2.37184	−0.0763522
$966$	0	0
$967$	42.2667	1.35921	0.679603	−	0.733580i	$-0.262152\pi$
$967$	42.2667	1.35921	0.679603	+	0.733580i	$0.262152\pi$
$968$	0	0
$969$	4.35482	0.139897
$970$	0	0
$971$	−43.3482	−1.39111	−0.695555	−	0.718473i	$-0.744841\pi$
$971$	−43.3482	−1.39111	−0.695555	+	0.718473i	$0.744841\pi$
$972$	0	0
$973$	14.9645	0.479738
$974$	0	0
$975$	−4.91852	−0.157519
$976$	0	0
$977$	54.8096	1.75352	0.876758	−	0.480932i	$-0.159702\pi$
$977$	54.8096	1.75352	0.876758	+	0.480932i	$0.159702\pi$
$978$	0	0
$979$	15.1223	0.483310
$980$	0	0
$981$	4.16295	0.132913
$982$	0	0
$983$	6.64669	0.211996	0.105998	−	0.994366i	$-0.466196\pi$
$983$	6.64669	0.211996	0.105998	+	0.994366i	$0.466196\pi$
$984$	0	0
$985$	24.0289	0.765625
$986$	0	0
$987$	18.5467	0.590347
$988$	0	0
$989$	43.9660	1.39804
$990$	0	0
$991$	23.2904	0.739843	0.369921	−	0.929063i	$-0.379385\pi$
$991$	23.2904	0.739843	0.369921	+	0.929063i	$0.379385\pi$
$992$	0	0
$993$	4.60963	0.146282
$994$	0	0
$995$	−26.5467	−0.841586
$996$	0	0
$997$	4.22077	0.133673	0.0668366	−	0.997764i	$-0.478709\pi$
$997$	4.22077	0.133673	0.0668366	+	0.997764i	$0.478709\pi$
$998$	0	0
$999$	−4.91852	−0.155615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bq.1.1		3
4.3	odd	2		2280.2.a.t.1.3	✓	3
12.11	even	2		6840.2.a.bn.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.t.1.3	✓	3	4.3	odd	2
4560.2.a.bq.1.1		3	1.1	even	1	trivial
6840.2.a.bn.1.3		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{9} -3.43630 q^{11} +4.91852 q^{13} +1.00000 q^{15} -4.35482 q^{17} +1.00000 q^{19} +2.91852 q^{21} -6.35482 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.27334 q^{29} +2.35482 q^{31} +3.43630 q^{33} +2.91852 q^{35} +4.91852 q^{37} -4.91852 q^{39} +1.43630 q^{41} -6.91852 q^{43} -1.00000 q^{45} +6.35482 q^{47} +1.51777 q^{49} +4.35482 q^{51} -4.35482 q^{53} +3.43630 q^{55} -1.00000 q^{57} -9.83705 q^{59} -6.19186 q^{61} -2.91852 q^{63} -4.91852 q^{65} +8.70964 q^{67} +6.35482 q^{69} -6.87259 q^{71} +10.0000 q^{73} -1.00000 q^{75} +10.0289 q^{77} -8.70964 q^{79} +1.00000 q^{81} +16.9015 q^{83} +4.35482 q^{85} +7.27334 q^{87} -4.40075 q^{89} -14.3548 q^{91} -2.35482 q^{93} -1.00000 q^{95} +12.9185 q^{97} -3.43630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 4 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 6 q^{39} - 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} - 2 q^{51} + 2 q^{53} + 4 q^{55} - 3 q^{57} - 12 q^{59} + 14 q^{61} - 6 q^{65} - 4 q^{67} + 4 q^{69} - 8 q^{71} + 30 q^{73} - 3 q^{75} - 20 q^{77} + 4 q^{79} + 3 q^{81} - 12 q^{83} - 2 q^{85} - 2 q^{87} - 2 q^{89} - 28 q^{91} + 8 q^{93} - 3 q^{95} + 30 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^9 - 4 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 + 3 * q^19 - 4 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 + 6 * q^37 - 6 * q^39 - 2 * q^41 - 12 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 - 2 * q^51 + 2 * q^53 + 4 * q^55 - 3 * q^57 - 12 * q^59 + 14 * q^61 - 6 * q^65 - 4 * q^67 + 4 * q^69 - 8 * q^71 + 30 * q^73 - 3 * q^75 - 20 * q^77 + 4 * q^79 + 3 * q^81 - 12 * q^83 - 2 * q^85 - 2 * q^87 - 2 * q^89 - 28 * q^91 + 8 * q^93 - 3 * q^95 + 30 * q^97 - 4 * q^99

Introduction

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