LMFDB - Embedded newform 5780.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(1,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5780.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.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:	$46.1535323683$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 206x^{8} - 16x^{7} - 776x^{6} + 152x^{5} + 1226x^{4} - 384x^{3} - 588x^{2} + 200x + 34 $ x^12 - 24x^10 + 206x^8 - 16x^7 - 776x^6 + 152x^5 + 1226x^4 - 384x^3 - 588x^2 + 200*x + 34
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 340)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.68860$ of defining polynomial
Character	$\chi$	$=$	5780.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68860 q^{3} -1.00000 q^{5} +1.08148 q^{7} +4.22856 q^{9} +O(q^{10})$	$q-2.68860 q^{3} -1.00000 q^{5} +1.08148 q^{7} +4.22856 q^{9} -1.14964 q^{11} -2.13266 q^{13} +2.68860 q^{15} +5.51264 q^{19} -2.90767 q^{21} -1.25204 q^{23} +1.00000 q^{25} -3.30310 q^{27} -4.67117 q^{29} -9.69786 q^{31} +3.09092 q^{33} -1.08148 q^{35} +2.24614 q^{37} +5.73386 q^{39} +3.95727 q^{41} +5.72781 q^{43} -4.22856 q^{45} -1.02220 q^{47} -5.83039 q^{49} +5.84557 q^{53} +1.14964 q^{55} -14.8213 q^{57} -3.47291 q^{59} +0.267004 q^{61} +4.57312 q^{63} +2.13266 q^{65} +2.16582 q^{67} +3.36623 q^{69} +12.9196 q^{71} +13.7483 q^{73} -2.68860 q^{75} -1.24332 q^{77} -16.4277 q^{79} -3.80496 q^{81} +17.1273 q^{83} +12.5589 q^{87} -10.4989 q^{89} -2.30643 q^{91} +26.0737 q^{93} -5.51264 q^{95} -12.9935 q^{97} -4.86133 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * 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
$3$	−2.68860	−1.55226	−0.776131	−	0.630571i	$-0.782821\pi$
$3$	−2.68860	−1.55226	−0.776131	+	0.630571i	$0.782821\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.08148	0.408762	0.204381	−	0.978891i	$-0.434482\pi$
$7$	1.08148	0.408762	0.204381	+	0.978891i	$0.434482\pi$
$8$	0	0
$9$	4.22856	1.40952
$10$	0	0
$11$	−1.14964	−0.346630	−0.173315	−	0.984866i	$-0.555448\pi$
$11$	−1.14964	−0.346630	−0.173315	+	0.984866i	$0.555448\pi$
$12$	0	0
$13$	−2.13266	−0.591493	−0.295746	−	0.955267i	$-0.595568\pi$
$13$	−2.13266	−0.591493	−0.295746	+	0.955267i	$0.595568\pi$
$14$	0	0
$15$	2.68860	0.694193
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	5.51264	1.26469	0.632343	−	0.774689i	$-0.282093\pi$
$19$	5.51264	1.26469	0.632343	+	0.774689i	$0.282093\pi$
$20$	0	0
$21$	−2.90767	−0.634506
$22$	0	0
$23$	−1.25204	−0.261068	−0.130534	−	0.991444i	$-0.541669\pi$
$23$	−1.25204	−0.261068	−0.130534	+	0.991444i	$0.541669\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.30310	−0.635682
$28$	0	0
$29$	−4.67117	−0.867415	−0.433707	−	0.901054i	$-0.642795\pi$
$29$	−4.67117	−0.867415	−0.433707	+	0.901054i	$0.642795\pi$
$30$	0	0
$31$	−9.69786	−1.74179	−0.870894	−	0.491471i	$-0.836459\pi$
$31$	−9.69786	−1.74179	−0.870894	+	0.491471i	$0.836459\pi$
$32$	0	0
$33$	3.09092	0.538061
$34$	0	0
$35$	−1.08148	−0.182804
$36$	0	0
$37$	2.24614	0.369263	0.184632	−	0.982808i	$-0.440891\pi$
$37$	2.24614	0.369263	0.184632	+	0.982808i	$0.440891\pi$
$38$	0	0
$39$	5.73386	0.918152
$40$	0	0
$41$	3.95727	0.618022	0.309011	−	0.951059i	$-0.400002\pi$
$41$	3.95727	0.618022	0.309011	+	0.951059i	$0.400002\pi$
$42$	0	0
$43$	5.72781	0.873483	0.436741	−	0.899587i	$-0.356133\pi$
$43$	5.72781	0.873483	0.436741	+	0.899587i	$0.356133\pi$
$44$	0	0
$45$	−4.22856	−0.630356
$46$	0	0
$47$	−1.02220	−0.149104	−0.0745519	−	0.997217i	$-0.523753\pi$
$47$	−1.02220	−0.149104	−0.0745519	+	0.997217i	$0.523753\pi$
$48$	0	0
$49$	−5.83039	−0.832913
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.84557	0.802950	0.401475	−	0.915870i	$-0.368498\pi$
$53$	5.84557	0.802950	0.401475	+	0.915870i	$0.368498\pi$
$54$	0	0
$55$	1.14964	0.155018
$56$	0	0
$57$	−14.8213	−1.96312
$58$	0	0
$59$	−3.47291	−0.452135	−0.226067	−	0.974112i	$-0.572587\pi$
$59$	−3.47291	−0.452135	−0.226067	+	0.974112i	$0.572587\pi$
$60$	0	0
$61$	0.267004	0.0341864	0.0170932	−	0.999854i	$-0.494559\pi$
$61$	0.267004	0.0341864	0.0170932	+	0.999854i	$0.494559\pi$
$62$	0	0
$63$	4.57312	0.576159
$64$	0	0
$65$	2.13266	0.264523
$66$	0	0
$67$	2.16582	0.264597	0.132299	−	0.991210i	$-0.457764\pi$
$67$	2.16582	0.264597	0.132299	+	0.991210i	$0.457764\pi$
$68$	0	0
$69$	3.36623	0.405246
$70$	0	0
$71$	12.9196	1.53328	0.766640	−	0.642077i	$-0.221927\pi$
$71$	12.9196	1.53328	0.766640	+	0.642077i	$0.221927\pi$
$72$	0	0
$73$	13.7483	1.60912	0.804560	−	0.593872i	$-0.202401\pi$
$73$	13.7483	1.60912	0.804560	+	0.593872i	$0.202401\pi$
$74$	0	0
$75$	−2.68860	−0.310453
$76$	0	0
$77$	−1.24332	−0.141689
$78$	0	0
$79$	−16.4277	−1.84826	−0.924132	−	0.382073i	$-0.875210\pi$
$79$	−16.4277	−1.84826	−0.924132	+	0.382073i	$0.875210\pi$
$80$	0	0
$81$	−3.80496	−0.422774
$82$	0	0
$83$	17.1273	1.87997	0.939984	−	0.341220i	$-0.110840\pi$
$83$	17.1273	1.87997	0.939984	+	0.341220i	$0.110840\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.5589	1.34646
$88$	0	0
$89$	−10.4989	−1.11288	−0.556442	−	0.830886i	$-0.687834\pi$
$89$	−10.4989	−1.11288	−0.556442	+	0.830886i	$0.687834\pi$
$90$	0	0
$91$	−2.30643	−0.241780
$92$	0	0
$93$	26.0737	2.70371
$94$	0	0
$95$	−5.51264	−0.565585
$96$	0	0
$97$	−12.9935	−1.31929	−0.659645	−	0.751577i	$-0.729293\pi$
$97$	−12.9935	−1.31929	−0.659645	+	0.751577i	$0.729293\pi$
$98$	0	0
$99$	−4.86133	−0.488582
$100$	0	0
$101$	8.79173	0.874810	0.437405	−	0.899265i	$-0.355898\pi$
$101$	8.79173	0.874810	0.437405	+	0.899265i	$0.355898\pi$
$102$	0	0
$103$	13.2574	1.30629	0.653145	−	0.757233i	$-0.273449\pi$
$103$	13.2574	1.30629	0.653145	+	0.757233i	$0.273449\pi$
$104$	0	0
$105$	2.90767	0.283760
$106$	0	0
$107$	2.07442	0.200542	0.100271	−	0.994960i	$-0.468029\pi$
$107$	2.07442	0.200542	0.100271	+	0.994960i	$0.468029\pi$
$108$	0	0
$109$	−8.00078	−0.766335	−0.383168	−	0.923679i	$-0.625167\pi$
$109$	−8.00078	−0.766335	−0.383168	+	0.923679i	$0.625167\pi$
$110$	0	0
$111$	−6.03897	−0.573193
$112$	0	0
$113$	1.36853	0.128740	0.0643702	−	0.997926i	$-0.479496\pi$
$113$	1.36853	0.128740	0.0643702	+	0.997926i	$0.479496\pi$
$114$	0	0
$115$	1.25204	0.116753
$116$	0	0
$117$	−9.01807	−0.833720
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.67832	−0.879848
$122$	0	0
$123$	−10.6395	−0.959332
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.47192	−0.219347	−0.109674	−	0.993968i	$-0.534981\pi$
$127$	−2.47192	−0.219347	−0.109674	+	0.993968i	$0.534981\pi$
$128$	0	0
$129$	−15.3998	−1.35587
$130$	0	0
$131$	13.0967	1.14426	0.572132	−	0.820161i	$-0.306116\pi$
$131$	13.0967	1.14426	0.572132	+	0.820161i	$0.306116\pi$
$132$	0	0
$133$	5.96183	0.516956
$134$	0	0
$135$	3.30310	0.284286
$136$	0	0
$137$	18.7489	1.60183	0.800914	−	0.598780i	$-0.204347\pi$
$137$	18.7489	1.60183	0.800914	+	0.598780i	$0.204347\pi$
$138$	0	0
$139$	19.3158	1.63834	0.819172	−	0.573548i	$-0.194433\pi$
$139$	19.3158	1.63834	0.819172	+	0.573548i	$0.194433\pi$
$140$	0	0
$141$	2.74830	0.231448
$142$	0	0
$143$	2.45179	0.205029
$144$	0	0
$145$	4.67117	0.387920
$146$	0	0
$147$	15.6756	1.29290
$148$	0	0
$149$	10.9622	0.898057	0.449028	−	0.893517i	$-0.351770\pi$
$149$	10.9622	0.898057	0.449028	+	0.893517i	$0.351770\pi$
$150$	0	0
$151$	−4.49614	−0.365891	−0.182945	−	0.983123i	$-0.558563\pi$
$151$	−4.49614	−0.365891	−0.182945	+	0.983123i	$0.558563\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.69786	0.778951
$156$	0	0
$157$	−13.0360	−1.04039	−0.520194	−	0.854048i	$-0.674140\pi$
$157$	−13.0360	−1.04039	−0.520194	+	0.854048i	$0.674140\pi$
$158$	0	0
$159$	−15.7164	−1.24639
$160$	0	0
$161$	−1.35406	−0.106715
$162$	0	0
$163$	−20.4432	−1.60124	−0.800618	−	0.599175i	$-0.795496\pi$
$163$	−20.4432	−1.60124	−0.800618	+	0.599175i	$0.795496\pi$
$164$	0	0
$165$	−3.09092	−0.240628
$166$	0	0
$167$	−13.1880	−1.02052	−0.510259	−	0.860021i	$-0.670450\pi$
$167$	−13.1880	−1.02052	−0.510259	+	0.860021i	$0.670450\pi$
$168$	0	0
$169$	−8.45178	−0.650137
$170$	0	0
$171$	23.3105	1.78260
$172$	0	0
$173$	−23.3289	−1.77366	−0.886832	−	0.462091i	$-0.847099\pi$
$173$	−23.3289	−1.77366	−0.886832	+	0.462091i	$0.847099\pi$
$174$	0	0
$175$	1.08148	0.0817525
$176$	0	0
$177$	9.33726	0.701832
$178$	0	0
$179$	16.5412	1.23635	0.618173	−	0.786042i	$-0.287873\pi$
$179$	16.5412	1.23635	0.618173	+	0.786042i	$0.287873\pi$
$180$	0	0
$181$	−4.30930	−0.320308	−0.160154	−	0.987092i	$-0.551199\pi$
$181$	−4.30930	−0.320308	−0.160154	+	0.987092i	$0.551199\pi$
$182$	0	0
$183$	−0.717866	−0.0530662
$184$	0	0
$185$	−2.24614	−0.165139
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.57225	−0.259843
$190$	0	0
$191$	−15.6850	−1.13493	−0.567464	−	0.823398i	$-0.692075\pi$
$191$	−15.6850	−1.13493	−0.567464	+	0.823398i	$0.692075\pi$
$192$	0	0
$193$	10.2284	0.736255	0.368127	−	0.929775i	$-0.379999\pi$
$193$	10.2284	0.736255	0.368127	+	0.929775i	$0.379999\pi$
$194$	0	0
$195$	−5.73386	−0.410610
$196$	0	0
$197$	−16.9888	−1.21040	−0.605202	−	0.796072i	$-0.706908\pi$
$197$	−16.9888	−1.21040	−0.605202	+	0.796072i	$0.706908\pi$
$198$	0	0
$199$	−7.71328	−0.546780	−0.273390	−	0.961903i	$-0.588145\pi$
$199$	−7.71328	−0.546780	−0.273390	+	0.961903i	$0.588145\pi$
$200$	0	0
$201$	−5.82302	−0.410724
$202$	0	0
$203$	−5.05180	−0.354567
$204$	0	0
$205$	−3.95727	−0.276388
$206$	0	0
$207$	−5.29432	−0.367980
$208$	0	0
$209$	−6.33756	−0.438378
$210$	0	0
$211$	0.769504	0.0529749	0.0264874	−	0.999649i	$-0.491568\pi$
$211$	0.769504	0.0529749	0.0264874	+	0.999649i	$0.491568\pi$
$212$	0	0
$213$	−34.7357	−2.38005
$214$	0	0
$215$	−5.72781	−0.390633
$216$	0	0
$217$	−10.4881	−0.711977
$218$	0	0
$219$	−36.9637	−2.49778
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.83994	0.525001	0.262501	−	0.964932i	$-0.415453\pi$
$223$	7.83994	0.525001	0.262501	+	0.964932i	$0.415453\pi$
$224$	0	0
$225$	4.22856	0.281904
$226$	0	0
$227$	21.5477	1.43017	0.715087	−	0.699036i	$-0.246387\pi$
$227$	21.5477	1.43017	0.715087	+	0.699036i	$0.246387\pi$
$228$	0	0
$229$	−4.74510	−0.313565	−0.156782	−	0.987633i	$-0.550112\pi$
$229$	−4.74510	−0.313565	−0.156782	+	0.987633i	$0.550112\pi$
$230$	0	0
$231$	3.34278	0.219939
$232$	0	0
$233$	3.63209	0.237946	0.118973	−	0.992897i	$-0.462040\pi$
$233$	3.63209	0.237946	0.118973	+	0.992897i	$0.462040\pi$
$234$	0	0
$235$	1.02220	0.0666813
$236$	0	0
$237$	44.1676	2.86899
$238$	0	0
$239$	−17.1006	−1.10615	−0.553073	−	0.833133i	$-0.686545\pi$
$239$	−17.1006	−1.10615	−0.553073	+	0.833133i	$0.686545\pi$
$240$	0	0
$241$	−11.5947	−0.746883	−0.373441	−	0.927654i	$-0.621822\pi$
$241$	−11.5947	−0.746883	−0.373441	+	0.927654i	$0.621822\pi$
$242$	0	0
$243$	20.1393	1.29194
$244$	0	0
$245$	5.83039	0.372490
$246$	0	0
$247$	−11.7566	−0.748052
$248$	0	0
$249$	−46.0485	−2.91820
$250$	0	0
$251$	−6.89273	−0.435066	−0.217533	−	0.976053i	$-0.569801\pi$
$251$	−6.89273	−0.435066	−0.217533	+	0.976053i	$0.569801\pi$
$252$	0	0
$253$	1.43939	0.0904940
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.65899	0.228241	0.114121	−	0.993467i	$-0.463595\pi$
$257$	3.65899	0.228241	0.114121	+	0.993467i	$0.463595\pi$
$258$	0	0
$259$	2.42916	0.150941
$260$	0	0
$261$	−19.7523	−1.22264
$262$	0	0
$263$	3.41884	0.210815	0.105407	−	0.994429i	$-0.466385\pi$
$263$	3.41884	0.210815	0.105407	+	0.994429i	$0.466385\pi$
$264$	0	0
$265$	−5.84557	−0.359090
$266$	0	0
$267$	28.2274	1.72749
$268$	0	0
$269$	22.3036	1.35987	0.679936	−	0.733271i	$-0.262007\pi$
$269$	22.3036	1.35987	0.679936	+	0.733271i	$0.262007\pi$
$270$	0	0
$271$	−7.40667	−0.449923	−0.224962	−	0.974368i	$-0.572226\pi$
$271$	−7.40667	−0.449923	−0.224962	+	0.974368i	$0.572226\pi$
$272$	0	0
$273$	6.20107	0.375306
$274$	0	0
$275$	−1.14964	−0.0693260
$276$	0	0
$277$	−24.8981	−1.49598	−0.747990	−	0.663710i	$-0.768981\pi$
$277$	−24.8981	−1.49598	−0.747990	+	0.663710i	$0.768981\pi$
$278$	0	0
$279$	−41.0080	−2.45508
$280$	0	0
$281$	−30.3469	−1.81034	−0.905172	−	0.425045i	$-0.860258\pi$
$281$	−30.3469	−1.81034	−0.905172	+	0.425045i	$0.860258\pi$
$282$	0	0
$283$	25.8282	1.53533	0.767664	−	0.640853i	$-0.221419\pi$
$283$	25.8282	1.53533	0.767664	+	0.640853i	$0.221419\pi$
$284$	0	0
$285$	14.8213	0.877936
$286$	0	0
$287$	4.27972	0.252624
$288$	0	0
$289$	0	0
$290$	0	0
$291$	34.9343	2.04789
$292$	0	0
$293$	2.74025	0.160087	0.0800437	−	0.996791i	$-0.474494\pi$
$293$	2.74025	0.160087	0.0800437	+	0.996791i	$0.474494\pi$
$294$	0	0
$295$	3.47291	0.202201
$296$	0	0
$297$	3.79738	0.220347
$298$	0	0
$299$	2.67017	0.154420
$300$	0	0
$301$	6.19453	0.357047
$302$	0	0
$303$	−23.6374	−1.35793
$304$	0	0
$305$	−0.267004	−0.0152886
$306$	0	0
$307$	−6.30443	−0.359813	−0.179906	−	0.983684i	$-0.557579\pi$
$307$	−6.30443	−0.359813	−0.179906	+	0.983684i	$0.557579\pi$
$308$	0	0
$309$	−35.6438	−2.02771
$310$	0	0
$311$	5.57800	0.316300	0.158150	−	0.987415i	$-0.449447\pi$
$311$	5.57800	0.316300	0.158150	+	0.987415i	$0.449447\pi$
$312$	0	0
$313$	2.82456	0.159654	0.0798268	−	0.996809i	$-0.474563\pi$
$313$	2.82456	0.159654	0.0798268	+	0.996809i	$0.474563\pi$
$314$	0	0
$315$	−4.57312	−0.257666
$316$	0	0
$317$	−28.9779	−1.62756	−0.813781	−	0.581172i	$-0.802594\pi$
$317$	−28.9779	−1.62756	−0.813781	+	0.581172i	$0.802594\pi$
$318$	0	0
$319$	5.37018	0.300672
$320$	0	0
$321$	−5.57728	−0.311293
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.13266	−0.118299
$326$	0	0
$327$	21.5109	1.18955
$328$	0	0
$329$	−1.10550	−0.0609480
$330$	0	0
$331$	0.115166	0.00633010	0.00316505	−	0.999995i	$-0.498993\pi$
$331$	0.115166	0.00633010	0.00316505	+	0.999995i	$0.498993\pi$
$332$	0	0
$333$	9.49793	0.520484
$334$	0	0
$335$	−2.16582	−0.118331
$336$	0	0
$337$	−10.8967	−0.593579	−0.296790	−	0.954943i	$-0.595916\pi$
$337$	−10.8967	−0.593579	−0.296790	+	0.954943i	$0.595916\pi$
$338$	0	0
$339$	−3.67943	−0.199839
$340$	0	0
$341$	11.1491	0.603756
$342$	0	0
$343$	−13.8759	−0.749226
$344$	0	0
$345$	−3.36623	−0.181231
$346$	0	0
$347$	3.76662	0.202203	0.101101	−	0.994876i	$-0.467763\pi$
$347$	3.76662	0.202203	0.101101	+	0.994876i	$0.467763\pi$
$348$	0	0
$349$	−12.5148	−0.669901	−0.334950	−	0.942236i	$-0.608720\pi$
$349$	−12.5148	−0.669901	−0.334950	+	0.942236i	$0.608720\pi$
$350$	0	0
$351$	7.04438	0.376001
$352$	0	0
$353$	−24.8592	−1.32312	−0.661561	−	0.749892i	$-0.730106\pi$
$353$	−24.8592	−1.32312	−0.661561	+	0.749892i	$0.730106\pi$
$354$	0	0
$355$	−12.9196	−0.685704
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.0857	1.16564	0.582820	−	0.812601i	$-0.301949\pi$
$359$	22.0857	1.16564	0.582820	+	0.812601i	$0.301949\pi$
$360$	0	0
$361$	11.3892	0.599430
$362$	0	0
$363$	26.0211	1.36575
$364$	0	0
$365$	−13.7483	−0.719620
$366$	0	0
$367$	−18.0451	−0.941944	−0.470972	−	0.882148i	$-0.656097\pi$
$367$	−18.0451	−0.941944	−0.470972	+	0.882148i	$0.656097\pi$
$368$	0	0
$369$	16.7335	0.871114
$370$	0	0
$371$	6.32188	0.328216
$372$	0	0
$373$	−25.5387	−1.32234	−0.661171	−	0.750235i	$-0.729940\pi$
$373$	−25.5387	−1.32234	−0.661171	+	0.750235i	$0.729940\pi$
$374$	0	0
$375$	2.68860	0.138839
$376$	0	0
$377$	9.96201	0.513069
$378$	0	0
$379$	−5.43915	−0.279390	−0.139695	−	0.990195i	$-0.544612\pi$
$379$	−5.43915	−0.279390	−0.139695	+	0.990195i	$0.544612\pi$
$380$	0	0
$381$	6.64600	0.340485
$382$	0	0
$383$	−29.4006	−1.50230	−0.751151	−	0.660130i	$-0.770501\pi$
$383$	−29.4006	−1.50230	−0.751151	+	0.660130i	$0.770501\pi$
$384$	0	0
$385$	1.24332	0.0633654
$386$	0	0
$387$	24.2204	1.23119
$388$	0	0
$389$	−28.2378	−1.43171	−0.715857	−	0.698247i	$-0.753964\pi$
$389$	−28.2378	−1.43171	−0.715857	+	0.698247i	$0.753964\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−35.2118	−1.77620
$394$	0	0
$395$	16.4277	0.826569
$396$	0	0
$397$	−7.24534	−0.363633	−0.181817	−	0.983332i	$-0.558198\pi$
$397$	−7.24534	−0.363633	−0.181817	+	0.983332i	$0.558198\pi$
$398$	0	0
$399$	−16.0290	−0.802451
$400$	0	0
$401$	−5.92391	−0.295826	−0.147913	−	0.989000i	$-0.547256\pi$
$401$	−5.92391	−0.295826	−0.147913	+	0.989000i	$0.547256\pi$
$402$	0	0
$403$	20.6822	1.03025
$404$	0	0
$405$	3.80496	0.189070
$406$	0	0
$407$	−2.58226	−0.127998
$408$	0	0
$409$	−23.3012	−1.15217	−0.576084	−	0.817390i	$-0.695420\pi$
$409$	−23.3012	−1.15217	−0.576084	+	0.817390i	$0.695420\pi$
$410$	0	0
$411$	−50.4083	−2.48646
$412$	0	0
$413$	−3.75590	−0.184816
$414$	0	0
$415$	−17.1273	−0.840747
$416$	0	0
$417$	−51.9324	−2.54314
$418$	0	0
$419$	−25.6536	−1.25326	−0.626631	−	0.779316i	$-0.715567\pi$
$419$	−25.6536	−1.25326	−0.626631	+	0.779316i	$0.715567\pi$
$420$	0	0
$421$	−38.5370	−1.87818	−0.939090	−	0.343670i	$-0.888330\pi$
$421$	−38.5370	−1.87818	−0.939090	+	0.343670i	$0.888330\pi$
$422$	0	0
$423$	−4.32245	−0.210165
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.288760	0.0139741
$428$	0	0
$429$	−6.59188	−0.318259
$430$	0	0
$431$	23.2186	1.11840	0.559199	−	0.829033i	$-0.311109\pi$
$431$	23.2186	1.11840	0.559199	+	0.829033i	$0.311109\pi$
$432$	0	0
$433$	−11.4058	−0.548130	−0.274065	−	0.961711i	$-0.588368\pi$
$433$	−11.4058	−0.548130	−0.274065	+	0.961711i	$0.588368\pi$
$434$	0	0
$435$	−12.5589	−0.602153
$436$	0	0
$437$	−6.90203	−0.330169
$438$	0	0
$439$	−26.1351	−1.24736	−0.623679	−	0.781680i	$-0.714363\pi$
$439$	−26.1351	−1.24736	−0.623679	+	0.781680i	$0.714363\pi$
$440$	0	0
$441$	−24.6542	−1.17401
$442$	0	0
$443$	−10.9481	−0.520161	−0.260080	−	0.965587i	$-0.583749\pi$
$443$	−10.9481	−0.520161	−0.260080	+	0.965587i	$0.583749\pi$
$444$	0	0
$445$	10.4989	0.497697
$446$	0	0
$447$	−29.4729	−1.39402
$448$	0	0
$449$	−37.4447	−1.76713	−0.883563	−	0.468312i	$-0.844862\pi$
$449$	−37.4447	−1.76713	−0.883563	+	0.468312i	$0.844862\pi$
$450$	0	0
$451$	−4.54944	−0.214225
$452$	0	0
$453$	12.0883	0.567959
$454$	0	0
$455$	2.30643	0.108127
$456$	0	0
$457$	4.07286	0.190520	0.0952601	−	0.995452i	$-0.469632\pi$
$457$	4.07286	0.190520	0.0952601	+	0.995452i	$0.469632\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.0889	−0.563037	−0.281519	−	0.959556i	$-0.590838\pi$
$461$	−12.0889	−0.563037	−0.281519	+	0.959556i	$0.590838\pi$
$462$	0	0
$463$	−37.5479	−1.74500	−0.872500	−	0.488615i	$-0.837502\pi$
$463$	−37.5479	−1.74500	−0.872500	+	0.488615i	$0.837502\pi$
$464$	0	0
$465$	−26.0737	−1.20914
$466$	0	0
$467$	7.64758	0.353888	0.176944	−	0.984221i	$-0.443379\pi$
$467$	7.64758	0.353888	0.176944	+	0.984221i	$0.443379\pi$
$468$	0	0
$469$	2.34230	0.108157
$470$	0	0
$471$	35.0487	1.61496
$472$	0	0
$473$	−6.58493	−0.302775
$474$	0	0
$475$	5.51264	0.252937
$476$	0	0
$477$	24.7183	1.13177
$478$	0	0
$479$	3.70829	0.169436	0.0847180	−	0.996405i	$-0.473001\pi$
$479$	3.70829	0.169436	0.0847180	+	0.996405i	$0.473001\pi$
$480$	0	0
$481$	−4.79024	−0.218416
$482$	0	0
$483$	3.64052	0.165649
$484$	0	0
$485$	12.9935	0.590005
$486$	0	0
$487$	36.0696	1.63447	0.817236	−	0.576304i	$-0.195505\pi$
$487$	36.0696	1.63447	0.817236	+	0.576304i	$0.195505\pi$
$488$	0	0
$489$	54.9636	2.48554
$490$	0	0
$491$	−16.9894	−0.766719	−0.383359	−	0.923599i	$-0.625233\pi$
$491$	−16.9894	−0.766719	−0.383359	+	0.923599i	$0.625233\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.86133	0.218500
$496$	0	0
$497$	13.9724	0.626747
$498$	0	0
$499$	24.2752	1.08671	0.543353	−	0.839504i	$-0.317155\pi$
$499$	24.2752	1.08671	0.543353	+	0.839504i	$0.317155\pi$
$500$	0	0
$501$	35.4572	1.58411
$502$	0	0
$503$	15.5400	0.692895	0.346447	−	0.938069i	$-0.387388\pi$
$503$	15.5400	0.692895	0.346447	+	0.938069i	$0.387388\pi$
$504$	0	0
$505$	−8.79173	−0.391227
$506$	0	0
$507$	22.7234	1.00918
$508$	0	0
$509$	−22.2342	−0.985516	−0.492758	−	0.870166i	$-0.664011\pi$
$509$	−22.2342	−0.985516	−0.492758	+	0.870166i	$0.664011\pi$
$510$	0	0
$511$	14.8686	0.657747
$512$	0	0
$513$	−18.2088	−0.803939
$514$	0	0
$515$	−13.2574	−0.584191
$516$	0	0
$517$	1.17517	0.0516839
$518$	0	0
$519$	62.7221	2.75319
$520$	0	0
$521$	24.8179	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$521$	24.8179	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$522$	0	0
$523$	−32.2089	−1.40840	−0.704199	−	0.710003i	$-0.748694\pi$
$523$	−32.2089	−1.40840	−0.704199	+	0.710003i	$0.748694\pi$
$524$	0	0
$525$	−2.90767	−0.126901
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.4324	−0.931844
$530$	0	0
$531$	−14.6854	−0.637293
$532$	0	0
$533$	−8.43950	−0.365555
$534$	0	0
$535$	−2.07442	−0.0896849
$536$	0	0
$537$	−44.4726	−1.91913
$538$	0	0
$539$	6.70286	0.288713
$540$	0	0
$541$	40.7203	1.75070	0.875351	−	0.483488i	$-0.160630\pi$
$541$	40.7203	1.75070	0.875351	+	0.483488i	$0.160630\pi$
$542$	0	0
$543$	11.5860	0.497202
$544$	0	0
$545$	8.00078	0.342716
$546$	0	0
$547$	1.68994	0.0722565	0.0361282	−	0.999347i	$-0.488498\pi$
$547$	1.68994	0.0722565	0.0361282	+	0.999347i	$0.488498\pi$
$548$	0	0
$549$	1.12904	0.0481864
$550$	0	0
$551$	−25.7505	−1.09701
$552$	0	0
$553$	−17.7663	−0.755501
$554$	0	0
$555$	6.03897	0.256340
$556$	0	0
$557$	12.1688	0.515609	0.257805	−	0.966197i	$-0.417001\pi$
$557$	12.1688	0.515609	0.257805	+	0.966197i	$0.417001\pi$
$558$	0	0
$559$	−12.2154	−0.516658
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.4380	1.07208	0.536042	−	0.844192i	$-0.319919\pi$
$563$	25.4380	1.07208	0.536042	+	0.844192i	$0.319919\pi$
$564$	0	0
$565$	−1.36853	−0.0575745
$566$	0	0
$567$	−4.11500	−0.172814
$568$	0	0
$569$	−31.1573	−1.30618	−0.653091	−	0.757280i	$-0.726528\pi$
$569$	−31.1573	−1.30618	−0.653091	+	0.757280i	$0.726528\pi$
$570$	0	0
$571$	−34.4267	−1.44071	−0.720355	−	0.693605i	$-0.756021\pi$
$571$	−34.4267	−1.44071	−0.720355	+	0.693605i	$0.756021\pi$
$572$	0	0
$573$	42.1707	1.76171
$574$	0	0
$575$	−1.25204	−0.0522136
$576$	0	0
$577$	−6.03615	−0.251288	−0.125644	−	0.992075i	$-0.540100\pi$
$577$	−6.03615	−0.251288	−0.125644	+	0.992075i	$0.540100\pi$
$578$	0	0
$579$	−27.5000	−1.14286
$580$	0	0
$581$	18.5229	0.768460
$582$	0	0
$583$	−6.72031	−0.278327
$584$	0	0
$585$	9.01807	0.372851
$586$	0	0
$587$	−14.9804	−0.618307	−0.309154	−	0.951012i	$-0.600046\pi$
$587$	−14.9804	−0.618307	−0.309154	+	0.951012i	$0.600046\pi$
$588$	0	0
$589$	−53.4608	−2.20281
$590$	0	0
$591$	45.6761	1.87887
$592$	0	0
$593$	−0.980533	−0.0402657	−0.0201328	−	0.999797i	$-0.506409\pi$
$593$	−0.980533	−0.0402657	−0.0201328	+	0.999797i	$0.506409\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.7379	0.848746
$598$	0	0
$599$	−18.4545	−0.754030	−0.377015	−	0.926207i	$-0.623050\pi$
$599$	−18.4545	−0.754030	−0.377015	+	0.926207i	$0.623050\pi$
$600$	0	0
$601$	45.0972	1.83955	0.919777	−	0.392442i	$-0.128370\pi$
$601$	45.0972	1.83955	0.919777	+	0.392442i	$0.128370\pi$
$602$	0	0
$603$	9.15830	0.372955
$604$	0	0
$605$	9.67832	0.393480
$606$	0	0
$607$	−4.84727	−0.196745	−0.0983723	−	0.995150i	$-0.531364\pi$
$607$	−4.84727	−0.196745	−0.0983723	+	0.995150i	$0.531364\pi$
$608$	0	0
$609$	13.5822	0.550380
$610$	0	0
$611$	2.18001	0.0881938
$612$	0	0
$613$	45.5944	1.84154	0.920771	−	0.390104i	$-0.127561\pi$
$613$	45.5944	1.84154	0.920771	+	0.390104i	$0.127561\pi$
$614$	0	0
$615$	10.6395	0.429026
$616$	0	0
$617$	32.5277	1.30952	0.654759	−	0.755838i	$-0.272770\pi$
$617$	32.5277	1.30952	0.654759	+	0.755838i	$0.272770\pi$
$618$	0	0
$619$	−8.65520	−0.347882	−0.173941	−	0.984756i	$-0.555650\pi$
$619$	−8.65520	−0.347882	−0.173941	+	0.984756i	$0.555650\pi$
$620$	0	0
$621$	4.13561	0.165956
$622$	0	0
$623$	−11.3544	−0.454905
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	17.0391	0.680478
$628$	0	0
$629$	0	0
$630$	0	0
$631$	11.2572	0.448141	0.224071	−	0.974573i	$-0.428065\pi$
$631$	11.2572	0.448141	0.224071	+	0.974573i	$0.428065\pi$
$632$	0	0
$633$	−2.06889	−0.0822309
$634$	0	0
$635$	2.47192	0.0980952
$636$	0	0
$637$	12.4342	0.492662
$638$	0	0
$639$	54.6315	2.16119
$640$	0	0
$641$	−39.8387	−1.57353	−0.786767	−	0.617250i	$-0.788247\pi$
$641$	−39.8387	−1.57353	−0.786767	+	0.617250i	$0.788247\pi$
$642$	0	0
$643$	−24.6949	−0.973873	−0.486936	−	0.873437i	$-0.661886\pi$
$643$	−24.6949	−0.973873	−0.486936	+	0.873437i	$0.661886\pi$
$644$	0	0
$645$	15.3998	0.606366
$646$	0	0
$647$	−14.9126	−0.586275	−0.293137	−	0.956070i	$-0.594699\pi$
$647$	−14.9126	−0.586275	−0.293137	+	0.956070i	$0.594699\pi$
$648$	0	0
$649$	3.99260	0.156723
$650$	0	0
$651$	28.1982	1.10518
$652$	0	0
$653$	−8.69614	−0.340306	−0.170153	−	0.985418i	$-0.554426\pi$
$653$	−8.69614	−0.340306	−0.170153	+	0.985418i	$0.554426\pi$
$654$	0	0
$655$	−13.0967	−0.511731
$656$	0	0
$657$	58.1356	2.26809
$658$	0	0
$659$	1.11183	0.0433108	0.0216554	−	0.999765i	$-0.493106\pi$
$659$	1.11183	0.0433108	0.0216554	+	0.999765i	$0.493106\pi$
$660$	0	0
$661$	46.5120	1.80911	0.904554	−	0.426358i	$-0.140204\pi$
$661$	46.5120	1.80911	0.904554	+	0.426358i	$0.140204\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.96183	−0.231190
$666$	0	0
$667$	5.84848	0.226454
$668$	0	0
$669$	−21.0785	−0.814940
$670$	0	0
$671$	−0.306959	−0.0118500
$672$	0	0
$673$	2.78629	0.107404	0.0537018	−	0.998557i	$-0.482898\pi$
$673$	2.78629	0.107404	0.0537018	+	0.998557i	$0.482898\pi$
$674$	0	0
$675$	−3.30310	−0.127136
$676$	0	0
$677$	17.2291	0.662169	0.331084	−	0.943601i	$-0.392586\pi$
$677$	17.2291	0.662169	0.331084	+	0.943601i	$0.392586\pi$
$678$	0	0
$679$	−14.0523	−0.539276
$680$	0	0
$681$	−57.9332	−2.22001
$682$	0	0
$683$	36.1732	1.38413	0.692064	−	0.721836i	$-0.256702\pi$
$683$	36.1732	1.38413	0.692064	+	0.721836i	$0.256702\pi$
$684$	0	0
$685$	−18.7489	−0.716359
$686$	0	0
$687$	12.7577	0.486735
$688$	0	0
$689$	−12.4666	−0.474939
$690$	0	0
$691$	7.48339	0.284682	0.142341	−	0.989818i	$-0.454537\pi$
$691$	7.48339	0.284682	0.142341	+	0.989818i	$0.454537\pi$
$692$	0	0
$693$	−5.25745	−0.199714
$694$	0	0
$695$	−19.3158	−0.732690
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−9.76522	−0.369355
$700$	0	0
$701$	19.9157	0.752206	0.376103	−	0.926578i	$-0.377264\pi$
$701$	19.9157	0.752206	0.376103	+	0.926578i	$0.377264\pi$
$702$	0	0
$703$	12.3822	0.467002
$704$	0	0
$705$	−2.74830	−0.103507
$706$	0	0
$707$	9.50811	0.357589
$708$	0	0
$709$	−12.4636	−0.468079	−0.234040	−	0.972227i	$-0.575195\pi$
$709$	−12.4636	−0.468079	−0.234040	+	0.972227i	$0.575195\pi$
$710$	0	0
$711$	−69.4656	−2.60516
$712$	0	0
$713$	12.1421	0.454725
$714$	0	0
$715$	−2.45179	−0.0916918
$716$	0	0
$717$	45.9766	1.71703
$718$	0	0
$719$	38.7328	1.44449	0.722245	−	0.691637i	$-0.243110\pi$
$719$	38.7328	1.44449	0.722245	+	0.691637i	$0.243110\pi$
$720$	0	0
$721$	14.3377	0.533962
$722$	0	0
$723$	31.1736	1.15936
$724$	0	0
$725$	−4.67117	−0.173483
$726$	0	0
$727$	−9.50496	−0.352519	−0.176260	−	0.984344i	$-0.556400\pi$
$727$	−9.50496	−0.352519	−0.176260	+	0.984344i	$0.556400\pi$
$728$	0	0
$729$	−42.7317	−1.58265
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−24.3656	−0.899965	−0.449982	−	0.893037i	$-0.648570\pi$
$733$	−24.3656	−0.899965	−0.449982	+	0.893037i	$0.648570\pi$
$734$	0	0
$735$	−15.6756	−0.578203
$736$	0	0
$737$	−2.48992	−0.0917173
$738$	0	0
$739$	18.5835	0.683605	0.341803	−	0.939772i	$-0.388963\pi$
$739$	18.5835	0.683605	0.341803	+	0.939772i	$0.388963\pi$
$740$	0	0
$741$	31.6087	1.16117
$742$	0	0
$743$	−10.4623	−0.383825	−0.191913	−	0.981412i	$-0.561469\pi$
$743$	−10.4623	−0.383825	−0.191913	+	0.981412i	$0.561469\pi$
$744$	0	0
$745$	−10.9622	−0.401623
$746$	0	0
$747$	72.4239	2.64985
$748$	0	0
$749$	2.24345	0.0819738
$750$	0	0
$751$	−13.5224	−0.493439	−0.246720	−	0.969087i	$-0.579353\pi$
$751$	−13.5224	−0.493439	−0.246720	+	0.969087i	$0.579353\pi$
$752$	0	0
$753$	18.5318	0.675336
$754$	0	0
$755$	4.49614	0.163631
$756$	0	0
$757$	21.4135	0.778289	0.389144	−	0.921177i	$-0.372771\pi$
$757$	21.4135	0.778289	0.389144	+	0.921177i	$0.372771\pi$
$758$	0	0
$759$	−3.86995	−0.140470
$760$	0	0
$761$	2.49472	0.0904335	0.0452167	−	0.998977i	$-0.485602\pi$
$761$	2.49472	0.0904335	0.0452167	+	0.998977i	$0.485602\pi$
$762$	0	0
$763$	−8.65271	−0.313249
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.40653	0.267434
$768$	0	0
$769$	44.4770	1.60388	0.801940	−	0.597405i	$-0.203801\pi$
$769$	44.4770	1.60388	0.801940	+	0.597405i	$0.203801\pi$
$770$	0	0
$771$	−9.83755	−0.354291
$772$	0	0
$773$	34.1173	1.22711	0.613556	−	0.789651i	$-0.289738\pi$
$773$	34.1173	1.22711	0.613556	+	0.789651i	$0.289738\pi$
$774$	0	0
$775$	−9.69786	−0.348358
$776$	0	0
$777$	−6.53104	−0.234300
$778$	0	0
$779$	21.8150	0.781603
$780$	0	0
$781$	−14.8530	−0.531481
$782$	0	0
$783$	15.4294	0.551400
$784$	0	0
$785$	13.0360	0.465276
$786$	0	0
$787$	−31.2593	−1.11427	−0.557136	−	0.830421i	$-0.688100\pi$
$787$	−31.2593	−1.11427	−0.557136	+	0.830421i	$0.688100\pi$
$788$	0	0
$789$	−9.19190	−0.327240
$790$	0	0
$791$	1.48004	0.0526243
$792$	0	0
$793$	−0.569428	−0.0202210
$794$	0	0
$795$	15.7164	0.557402
$796$	0	0
$797$	38.7767	1.37354	0.686771	−	0.726874i	$-0.259028\pi$
$797$	38.7767	1.37354	0.686771	+	0.726874i	$0.259028\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−44.3953	−1.56863
$802$	0	0
$803$	−15.8056	−0.557769
$804$	0	0
$805$	1.35406	0.0477243
$806$	0	0
$807$	−59.9653	−2.11088
$808$	0	0
$809$	−23.8930	−0.840034	−0.420017	−	0.907516i	$-0.637976\pi$
$809$	−23.8930	−0.840034	−0.420017	+	0.907516i	$0.637976\pi$
$810$	0	0
$811$	−17.8765	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$811$	−17.8765	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$812$	0	0
$813$	19.9136	0.698399
$814$	0	0
$815$	20.4432	0.716095
$816$	0	0
$817$	31.5753	1.10468
$818$	0	0
$819$	−9.75289	−0.340793
$820$	0	0
$821$	13.9870	0.488151	0.244075	−	0.969756i	$-0.421516\pi$
$821$	13.9870	0.488151	0.244075	+	0.969756i	$0.421516\pi$
$822$	0	0
$823$	43.4337	1.51400	0.757001	−	0.653413i	$-0.226664\pi$
$823$	43.4337	1.51400	0.757001	+	0.653413i	$0.226664\pi$
$824$	0	0
$825$	3.09092	0.107612
$826$	0	0
$827$	34.1532	1.18762	0.593812	−	0.804604i	$-0.297622\pi$
$827$	34.1532	1.18762	0.593812	+	0.804604i	$0.297622\pi$
$828$	0	0
$829$	−42.1532	−1.46404	−0.732021	−	0.681282i	$-0.761423\pi$
$829$	−42.1532	−1.46404	−0.732021	+	0.681282i	$0.761423\pi$
$830$	0	0
$831$	66.9409	2.32216
$832$	0	0
$833$	0	0
$834$	0	0
$835$	13.1880	0.456389
$836$	0	0
$837$	32.0330	1.10722
$838$	0	0
$839$	−27.6956	−0.956157	−0.478078	−	0.878317i	$-0.658667\pi$
$839$	−27.6956	−0.956157	−0.478078	+	0.878317i	$0.658667\pi$
$840$	0	0
$841$	−7.18015	−0.247591
$842$	0	0
$843$	81.5906	2.81013
$844$	0	0
$845$	8.45178	0.290750
$846$	0	0
$847$	−10.4669	−0.359649
$848$	0	0
$849$	−69.4417	−2.38323
$850$	0	0
$851$	−2.81225	−0.0964027
$852$	0	0
$853$	4.60654	0.157725	0.0788624	−	0.996886i	$-0.474871\pi$
$853$	4.60654	0.157725	0.0788624	+	0.996886i	$0.474871\pi$
$854$	0	0
$855$	−23.3105	−0.797203
$856$	0	0
$857$	54.0952	1.84786	0.923929	−	0.382565i	$-0.124959\pi$
$857$	54.0952	1.84786	0.923929	+	0.382565i	$0.124959\pi$
$858$	0	0
$859$	56.8709	1.94041	0.970206	−	0.242283i	$-0.0778962\pi$
$859$	56.8709	1.94041	0.970206	+	0.242283i	$0.0778962\pi$
$860$	0	0
$861$	−11.5064	−0.392139
$862$	0	0
$863$	−34.9505	−1.18973	−0.594865	−	0.803826i	$-0.702794\pi$
$863$	−34.9505	−1.18973	−0.594865	+	0.803826i	$0.702794\pi$
$864$	0	0
$865$	23.3289	0.793207
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.8860	0.640664
$870$	0	0
$871$	−4.61895	−0.156507
$872$	0	0
$873$	−54.9438	−1.85957
$874$	0	0
$875$	−1.08148	−0.0365608
$876$	0	0
$877$	−32.4795	−1.09676	−0.548378	−	0.836231i	$-0.684754\pi$
$877$	−32.4795	−1.09676	−0.548378	+	0.836231i	$0.684754\pi$
$878$	0	0
$879$	−7.36744	−0.248498
$880$	0	0
$881$	17.3211	0.583561	0.291781	−	0.956485i	$-0.405752\pi$
$881$	17.3211	0.583561	0.291781	+	0.956485i	$0.405752\pi$
$882$	0	0
$883$	−49.1352	−1.65353	−0.826766	−	0.562547i	$-0.809822\pi$
$883$	−49.1352	−1.65353	−0.826766	+	0.562547i	$0.809822\pi$
$884$	0	0
$885$	−9.33726	−0.313869
$886$	0	0
$887$	5.46257	0.183415	0.0917076	−	0.995786i	$-0.470767\pi$
$887$	5.46257	0.183415	0.0917076	+	0.995786i	$0.470767\pi$
$888$	0	0
$889$	−2.67334	−0.0896610
$890$	0	0
$891$	4.37434	0.146546
$892$	0	0
$893$	−5.63504	−0.188570
$894$	0	0
$895$	−16.5412	−0.552911
$896$	0	0
$897$	−7.17900	−0.239700
$898$	0	0
$899$	45.3004	1.51085
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−16.6546	−0.554230
$904$	0	0
$905$	4.30930	0.143246
$906$	0	0
$907$	9.94236	0.330131	0.165065	−	0.986283i	$-0.447217\pi$
$907$	9.94236	0.330131	0.165065	+	0.986283i	$0.447217\pi$
$908$	0	0
$909$	37.1763	1.23306
$910$	0	0
$911$	−12.2621	−0.406260	−0.203130	−	0.979152i	$-0.565111\pi$
$911$	−12.2621	−0.406260	−0.203130	+	0.979152i	$0.565111\pi$
$912$	0	0
$913$	−19.6903	−0.651653
$914$	0	0
$915$	0.717866	0.0237319
$916$	0	0
$917$	14.1639	0.467732
$918$	0	0
$919$	5.90967	0.194942	0.0974710	−	0.995238i	$-0.468925\pi$
$919$	5.90967	0.194942	0.0974710	+	0.995238i	$0.468925\pi$
$920$	0	0
$921$	16.9501	0.558524
$922$	0	0
$923$	−27.5532	−0.906924
$924$	0	0
$925$	2.24614	0.0738526
$926$	0	0
$927$	56.0597	1.84124
$928$	0	0
$929$	−14.8588	−0.487502	−0.243751	−	0.969838i	$-0.578378\pi$
$929$	−14.8588	−0.487502	−0.243751	+	0.969838i	$0.578378\pi$
$930$	0	0
$931$	−32.1408	−1.05337
$932$	0	0
$933$	−14.9970	−0.490980
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.1465	0.952174	0.476087	−	0.879398i	$-0.342055\pi$
$937$	29.1465	0.952174	0.476087	+	0.879398i	$0.342055\pi$
$938$	0	0
$939$	−7.59411	−0.247824
$940$	0	0
$941$	−3.76853	−0.122851	−0.0614253	−	0.998112i	$-0.519565\pi$
$941$	−3.76853	−0.122851	−0.0614253	+	0.998112i	$0.519565\pi$
$942$	0	0
$943$	−4.95465	−0.161346
$944$	0	0
$945$	3.57225	0.116205
$946$	0	0
$947$	−40.0825	−1.30251	−0.651254	−	0.758860i	$-0.725757\pi$
$947$	−40.0825	−1.30251	−0.651254	+	0.758860i	$0.725757\pi$
$948$	0	0
$949$	−29.3205	−0.951782
$950$	0	0
$951$	77.9100	2.52640
$952$	0	0
$953$	−22.3261	−0.723213	−0.361606	−	0.932331i	$-0.617772\pi$
$953$	−22.3261	−0.723213	−0.361606	+	0.932331i	$0.617772\pi$
$954$	0	0
$955$	15.6850	0.507555
$956$	0	0
$957$	−14.4382	−0.466722
$958$	0	0
$959$	20.2766	0.654767
$960$	0	0
$961$	63.0486	2.03382
$962$	0	0
$963$	8.77180	0.282667
$964$	0	0
$965$	−10.2284	−0.329263
$966$	0	0
$967$	29.3866	0.945010	0.472505	−	0.881328i	$-0.343350\pi$
$967$	29.3866	0.945010	0.472505	+	0.881328i	$0.343350\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.86562	−0.252420	−0.126210	−	0.992004i	$-0.540281\pi$
$971$	−7.86562	−0.252420	−0.126210	+	0.992004i	$0.540281\pi$
$972$	0	0
$973$	20.8897	0.669693
$974$	0	0
$975$	5.73386	0.183630
$976$	0	0
$977$	−15.3572	−0.491320	−0.245660	−	0.969356i	$-0.579005\pi$
$977$	−15.3572	−0.491320	−0.245660	+	0.969356i	$0.579005\pi$
$978$	0	0
$979$	12.0700	0.385759
$980$	0	0
$981$	−33.8318	−1.08016
$982$	0	0
$983$	31.0949	0.991774	0.495887	−	0.868387i	$-0.334843\pi$
$983$	31.0949	0.991774	0.495887	+	0.868387i	$0.334843\pi$
$984$	0	0
$985$	16.9888	0.541309
$986$	0	0
$987$	2.97224	0.0946074
$988$	0	0
$989$	−7.17143	−0.228038
$990$	0	0
$991$	7.32995	0.232843	0.116422	−	0.993200i	$-0.462858\pi$
$991$	7.32995	0.232843	0.116422	+	0.993200i	$0.462858\pi$
$992$	0	0
$993$	−0.309635	−0.00982597
$994$	0	0
$995$	7.71328	0.244527
$996$	0	0
$997$	−19.1339	−0.605977	−0.302989	−	0.952994i	$-0.597984\pi$
$997$	−19.1339	−0.605977	−0.302989	+	0.952994i	$0.597984\pi$
$998$	0	0
$999$	−7.41923	−0.234734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.a.q.1.2		12
17.4	even	4		5780.2.c.j.5201.22		24
17.11	odd	16		340.2.u.a.121.6	✓	24
17.13	even	4		5780.2.c.j.5201.3		24
17.14	odd	16		340.2.u.a.281.6	yes	24
17.16	even	2		5780.2.a.r.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.u.a.121.6	✓	24	17.11	odd	16
340.2.u.a.281.6	yes	24	17.14	odd	16
5780.2.a.q.1.2		12	1.1	even	1	trivial
5780.2.a.r.1.11		12	17.16	even	2
5780.2.c.j.5201.3		24	17.13	even	4
5780.2.c.j.5201.22		24	17.4	even	4

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:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68860 q^{3} -1.00000 q^{5} +1.08148 q^{7} +4.22856 q^{9} +O(q^{10})\)	\(q-2.68860 q^{3} -1.00000 q^{5} +1.08148 q^{7} +4.22856 q^{9} -1.14964 q^{11} -2.13266 q^{13} +2.68860 q^{15} +5.51264 q^{19} -2.90767 q^{21} -1.25204 q^{23} +1.00000 q^{25} -3.30310 q^{27} -4.67117 q^{29} -9.69786 q^{31} +3.09092 q^{33} -1.08148 q^{35} +2.24614 q^{37} +5.73386 q^{39} +3.95727 q^{41} +5.72781 q^{43} -4.22856 q^{45} -1.02220 q^{47} -5.83039 q^{49} +5.84557 q^{53} +1.14964 q^{55} -14.8213 q^{57} -3.47291 q^{59} +0.267004 q^{61} +4.57312 q^{63} +2.13266 q^{65} +2.16582 q^{67} +3.36623 q^{69} +12.9196 q^{71} +13.7483 q^{73} -2.68860 q^{75} -1.24332 q^{77} -16.4277 q^{79} -3.80496 q^{81} +17.1273 q^{83} +12.5589 q^{87} -10.4989 q^{89} -2.30643 q^{91} +26.0737 q^{93} -5.51264 q^{95} -12.9935 q^{97} -4.86133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * q^99

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