LMFDB - Embedded newform 276.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [276,2,Mod(1,276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(276, 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("276.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 276 = 2^{2} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	276.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:	$2.20387109579$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	276.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.16228 q^{5} +5.16228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.16228 q^{5} +5.16228 q^{7} +1.00000 q^{9} +4.00000 q^{13} +3.16228 q^{15} +7.16228 q^{17} -1.16228 q^{19} -5.16228 q^{21} -1.00000 q^{23} +5.00000 q^{25} -1.00000 q^{27} +8.32456 q^{29} -6.32456 q^{31} -16.3246 q^{35} -8.32456 q^{37} -4.00000 q^{39} -2.00000 q^{41} +1.16228 q^{43} -3.16228 q^{45} +0.324555 q^{47} +19.6491 q^{49} -7.16228 q^{51} +5.48683 q^{53} +1.16228 q^{57} -8.32456 q^{59} +0.324555 q^{61} +5.16228 q^{63} -12.6491 q^{65} +1.16228 q^{67} +1.00000 q^{69} -14.6491 q^{73} -5.00000 q^{75} +13.1623 q^{79} +1.00000 q^{81} -4.00000 q^{83} -22.6491 q^{85} -8.32456 q^{87} -9.48683 q^{89} +20.6491 q^{91} +6.32456 q^{93} +3.67544 q^{95} -3.67544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9} + 8 q^{13} + 8 q^{17} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 10 q^{25} - 2 q^{27} + 4 q^{29} - 20 q^{35} - 4 q^{37} - 8 q^{39} - 4 q^{41} - 4 q^{43} - 12 q^{47} + 14 q^{49} - 8 q^{51} - 8 q^{53} - 4 q^{57} - 4 q^{59} - 12 q^{61} + 4 q^{63} - 4 q^{67} + 2 q^{69} - 4 q^{73} - 10 q^{75} + 20 q^{79} + 2 q^{81} - 8 q^{83} - 20 q^{85} - 4 q^{87} + 16 q^{91} + 20 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9 + 8 * q^13 + 8 * q^17 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 10 * q^25 - 2 * q^27 + 4 * q^29 - 20 * q^35 - 4 * q^37 - 8 * q^39 - 4 * q^41 - 4 * q^43 - 12 * q^47 + 14 * q^49 - 8 * q^51 - 8 * q^53 - 4 * q^57 - 4 * q^59 - 12 * q^61 + 4 * q^63 - 4 * q^67 + 2 * q^69 - 4 * q^73 - 10 * q^75 + 20 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 - 4 * q^87 + 16 * q^91 + 20 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.16228	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$5$	−3.16228	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$6$	0	0
$7$	5.16228	1.95116	0.975579	−	0.219650i	$-0.0704915\pi$
$7$	5.16228	1.95116	0.975579	+	0.219650i	$0.0704915\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	3.16228	0.816497
$16$	0	0
$17$	7.16228	1.73711	0.868554	−	0.495595i	$-0.165050\pi$
$17$	7.16228	1.73711	0.868554	+	0.495595i	$0.165050\pi$
$18$	0	0
$19$	−1.16228	−0.266645	−0.133322	−	0.991073i	$-0.542565\pi$
$19$	−1.16228	−0.266645	−0.133322	+	0.991073i	$0.542565\pi$
$20$	0	0
$21$	−5.16228	−1.12650
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.32456	1.54583	0.772916	−	0.634509i	$-0.218798\pi$
$29$	8.32456	1.54583	0.772916	+	0.634509i	$0.218798\pi$
$30$	0	0
$31$	−6.32456	−1.13592	−0.567962	−	0.823055i	$-0.692268\pi$
$31$	−6.32456	−1.13592	−0.567962	+	0.823055i	$0.692268\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−16.3246	−2.75935
$36$	0	0
$37$	−8.32456	−1.36855	−0.684274	−	0.729225i	$-0.739881\pi$
$37$	−8.32456	−1.36855	−0.684274	+	0.729225i	$0.739881\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.16228	0.177246	0.0886228	−	0.996065i	$-0.471753\pi$
$43$	1.16228	0.177246	0.0886228	+	0.996065i	$0.471753\pi$
$44$	0	0
$45$	−3.16228	−0.471405
$46$	0	0
$47$	0.324555	0.0473413	0.0236706	−	0.999720i	$-0.492465\pi$
$47$	0.324555	0.0473413	0.0236706	+	0.999720i	$0.492465\pi$
$48$	0	0
$49$	19.6491	2.80702
$50$	0	0
$51$	−7.16228	−1.00292
$52$	0	0
$53$	5.48683	0.753674	0.376837	−	0.926279i	$-0.377012\pi$
$53$	5.48683	0.753674	0.376837	+	0.926279i	$0.377012\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.16228	0.153947
$58$	0	0
$59$	−8.32456	−1.08376	−0.541882	−	0.840454i	$-0.682288\pi$
$59$	−8.32456	−1.08376	−0.541882	+	0.840454i	$0.682288\pi$
$60$	0	0
$61$	0.324555	0.0415551	0.0207775	−	0.999784i	$-0.493386\pi$
$61$	0.324555	0.0415551	0.0207775	+	0.999784i	$0.493386\pi$
$62$	0	0
$63$	5.16228	0.650386
$64$	0	0
$65$	−12.6491	−1.56893
$66$	0	0
$67$	1.16228	0.141995	0.0709974	−	0.997477i	$-0.477382\pi$
$67$	1.16228	0.141995	0.0709974	+	0.997477i	$0.477382\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−14.6491	−1.71455	−0.857274	−	0.514860i	$-0.827844\pi$
$73$	−14.6491	−1.71455	−0.857274	+	0.514860i	$0.827844\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.1623	1.48087	0.740436	−	0.672127i	$-0.234619\pi$
$79$	13.1623	1.48087	0.740436	+	0.672127i	$0.234619\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−22.6491	−2.45664
$86$	0	0
$87$	−8.32456	−0.892486
$88$	0	0
$89$	−9.48683	−1.00560	−0.502801	−	0.864402i	$-0.667697\pi$
$89$	−9.48683	−1.00560	−0.502801	+	0.864402i	$0.667697\pi$
$90$	0	0
$91$	20.6491	2.16461
$92$	0	0
$93$	6.32456	0.655826
$94$	0	0
$95$	3.67544	0.377093
$96$	0	0
$97$	−3.67544	−0.373185	−0.186592	−	0.982437i	$-0.559744\pi$
$97$	−3.67544	−0.373185	−0.186592	+	0.982437i	$0.559744\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−7.48683	−0.737700	−0.368850	−	0.929489i	$-0.620248\pi$
$103$	−7.48683	−0.737700	−0.368850	+	0.929489i	$0.620248\pi$
$104$	0	0
$105$	16.3246	1.59311
$106$	0	0
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	8.32456	0.790132
$112$	0	0
$113$	11.1623	1.05006	0.525029	−	0.851084i	$-0.324054\pi$
$113$	11.1623	1.05006	0.525029	+	0.851084i	$0.324054\pi$
$114$	0	0
$115$	3.16228	0.294884
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	36.9737	3.38937
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.32456	0.206271	0.103135	−	0.994667i	$-0.467112\pi$
$127$	2.32456	0.206271	0.103135	+	0.994667i	$0.467112\pi$
$128$	0	0
$129$	−1.16228	−0.102333
$130$	0	0
$131$	8.64911	0.755676	0.377838	−	0.925872i	$-0.376668\pi$
$131$	8.64911	0.755676	0.377838	+	0.925872i	$0.376668\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	3.16228	0.272166
$136$	0	0
$137$	4.83772	0.413315	0.206657	−	0.978413i	$-0.433741\pi$
$137$	4.83772	0.413315	0.206657	+	0.978413i	$0.433741\pi$
$138$	0	0
$139$	−16.6491	−1.41216	−0.706080	−	0.708133i	$-0.749538\pi$
$139$	−16.6491	−1.41216	−0.706080	+	0.708133i	$0.749538\pi$
$140$	0	0
$141$	−0.324555	−0.0273325
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−26.3246	−2.18614
$146$	0	0
$147$	−19.6491	−1.62063
$148$	0	0
$149$	19.1623	1.56983	0.784917	−	0.619600i	$-0.212705\pi$
$149$	19.1623	1.56983	0.784917	+	0.619600i	$0.212705\pi$
$150$	0	0
$151$	−4.64911	−0.378339	−0.189170	−	0.981944i	$-0.560580\pi$
$151$	−4.64911	−0.378339	−0.189170	+	0.981944i	$0.560580\pi$
$152$	0	0
$153$	7.16228	0.579036
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	12.3246	0.983607	0.491803	−	0.870706i	$-0.336338\pi$
$157$	12.3246	0.983607	0.491803	+	0.870706i	$0.336338\pi$
$158$	0	0
$159$	−5.48683	−0.435134
$160$	0	0
$161$	−5.16228	−0.406844
$162$	0	0
$163$	−2.32456	−0.182073	−0.0910366	−	0.995848i	$-0.529018\pi$
$163$	−2.32456	−0.182073	−0.0910366	+	0.995848i	$0.529018\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.6491	−1.59788	−0.798938	−	0.601413i	$-0.794605\pi$
$167$	−20.6491	−1.59788	−0.798938	+	0.601413i	$0.794605\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.16228	−0.0888816
$172$	0	0
$173$	−20.3246	−1.54525	−0.772624	−	0.634864i	$-0.781056\pi$
$173$	−20.3246	−1.54525	−0.772624	+	0.634864i	$0.781056\pi$
$174$	0	0
$175$	25.8114	1.95116
$176$	0	0
$177$	8.32456	0.625712
$178$	0	0
$179$	−7.67544	−0.573690	−0.286845	−	0.957977i	$-0.592606\pi$
$179$	−7.67544	−0.573690	−0.286845	+	0.957977i	$0.592606\pi$
$180$	0	0
$181$	6.64911	0.494225	0.247112	−	0.968987i	$-0.420518\pi$
$181$	6.64911	0.494225	0.247112	+	0.968987i	$0.420518\pi$
$182$	0	0
$183$	−0.324555	−0.0239918
$184$	0	0
$185$	26.3246	1.93542
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.16228	−0.375500
$190$	0	0
$191$	−18.9737	−1.37289	−0.686443	−	0.727183i	$-0.740829\pi$
$191$	−18.9737	−1.37289	−0.686443	+	0.727183i	$0.740829\pi$
$192$	0	0
$193$	0.649111	0.0467240	0.0233620	−	0.999727i	$-0.492563\pi$
$193$	0.649111	0.0467240	0.0233620	+	0.999727i	$0.492563\pi$
$194$	0	0
$195$	12.6491	0.905822
$196$	0	0
$197$	−3.67544	−0.261865	−0.130932	−	0.991391i	$-0.541797\pi$
$197$	−3.67544	−0.261865	−0.130932	+	0.991391i	$0.541797\pi$
$198$	0	0
$199$	−9.81139	−0.695511	−0.347755	−	0.937585i	$-0.613056\pi$
$199$	−9.81139	−0.695511	−0.347755	+	0.937585i	$0.613056\pi$
$200$	0	0
$201$	−1.16228	−0.0819807
$202$	0	0
$203$	42.9737	3.01616
$204$	0	0
$205$	6.32456	0.441726
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.3246	0.710772	0.355386	−	0.934720i	$-0.384349\pi$
$211$	10.3246	0.710772	0.355386	+	0.934720i	$0.384349\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.67544	−0.250663
$216$	0	0
$217$	−32.6491	−2.21637
$218$	0	0
$219$	14.6491	0.989895
$220$	0	0
$221$	28.6491	1.92715
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	−18.3246	−1.21624	−0.608122	−	0.793844i	$-0.708077\pi$
$227$	−18.3246	−1.21624	−0.608122	+	0.793844i	$0.708077\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.64911	−0.435598	−0.217799	−	0.975994i	$-0.569888\pi$
$233$	−6.64911	−0.435598	−0.217799	+	0.975994i	$0.569888\pi$
$234$	0	0
$235$	−1.02633	−0.0669507
$236$	0	0
$237$	−13.1623	−0.854982
$238$	0	0
$239$	3.35089	0.216751	0.108376	−	0.994110i	$-0.465435\pi$
$239$	3.35089	0.216751	0.108376	+	0.994110i	$0.465435\pi$
$240$	0	0
$241$	−20.3246	−1.30922	−0.654610	−	0.755967i	$-0.727167\pi$
$241$	−20.3246	−1.30922	−0.654610	+	0.755967i	$0.727167\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−62.1359	−3.96972
$246$	0	0
$247$	−4.64911	−0.295816
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−14.3246	−0.904158	−0.452079	−	0.891978i	$-0.649317\pi$
$251$	−14.3246	−0.904158	−0.452079	+	0.891978i	$0.649317\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	22.6491	1.41834
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−42.9737	−2.67025
$260$	0	0
$261$	8.32456	0.515277
$262$	0	0
$263$	30.3246	1.86989	0.934946	−	0.354790i	$-0.115448\pi$
$263$	30.3246	1.86989	0.934946	+	0.354790i	$0.115448\pi$
$264$	0	0
$265$	−17.3509	−1.06586
$266$	0	0
$267$	9.48683	0.580585
$268$	0	0
$269$	−3.67544	−0.224096	−0.112048	−	0.993703i	$-0.535741\pi$
$269$	−3.67544	−0.224096	−0.112048	+	0.993703i	$0.535741\pi$
$270$	0	0
$271$	−5.67544	−0.344759	−0.172379	−	0.985031i	$-0.555145\pi$
$271$	−5.67544	−0.344759	−0.172379	+	0.985031i	$0.555145\pi$
$272$	0	0
$273$	−20.6491	−1.24974
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.2982	−1.15952	−0.579759	−	0.814788i	$-0.696853\pi$
$277$	−19.2982	−1.15952	−0.579759	+	0.814788i	$0.696853\pi$
$278$	0	0
$279$	−6.32456	−0.378641
$280$	0	0
$281$	−3.16228	−0.188646	−0.0943228	−	0.995542i	$-0.530069\pi$
$281$	−3.16228	−0.188646	−0.0943228	+	0.995542i	$0.530069\pi$
$282$	0	0
$283$	−3.48683	−0.207271	−0.103635	−	0.994615i	$-0.533047\pi$
$283$	−3.48683	−0.207271	−0.103635	+	0.994615i	$0.533047\pi$
$284$	0	0
$285$	−3.67544	−0.217715
$286$	0	0
$287$	−10.3246	−0.609439
$288$	0	0
$289$	34.2982	2.01754
$290$	0	0
$291$	3.67544	0.215458
$292$	0	0
$293$	2.51317	0.146821	0.0734104	−	0.997302i	$-0.476612\pi$
$293$	2.51317	0.146821	0.0734104	+	0.997302i	$0.476612\pi$
$294$	0	0
$295$	26.3246	1.53267
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−1.02633	−0.0587677
$306$	0	0
$307$	6.32456	0.360961	0.180481	−	0.983579i	$-0.442235\pi$
$307$	6.32456	0.360961	0.180481	+	0.983579i	$0.442235\pi$
$308$	0	0
$309$	7.48683	0.425911
$310$	0	0
$311$	28.9737	1.64295	0.821473	−	0.570248i	$-0.193153\pi$
$311$	28.9737	1.64295	0.821473	+	0.570248i	$0.193153\pi$
$312$	0	0
$313$	−6.64911	−0.375830	−0.187915	−	0.982185i	$-0.560173\pi$
$313$	−6.64911	−0.375830	−0.187915	+	0.982185i	$0.560173\pi$
$314$	0	0
$315$	−16.3246	−0.919784
$316$	0	0
$317$	−11.6754	−0.655758	−0.327879	−	0.944720i	$-0.606334\pi$
$317$	−11.6754	−0.655758	−0.327879	+	0.944720i	$0.606334\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	−8.32456	−0.463191
$324$	0	0
$325$	20.0000	1.10940
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	1.67544	0.0923703
$330$	0	0
$331$	6.32456	0.347629	0.173814	−	0.984778i	$-0.444391\pi$
$331$	6.32456	0.347629	0.173814	+	0.984778i	$0.444391\pi$
$332$	0	0
$333$	−8.32456	−0.456183
$334$	0	0
$335$	−3.67544	−0.200811
$336$	0	0
$337$	4.32456	0.235574	0.117787	−	0.993039i	$-0.462420\pi$
$337$	4.32456	0.235574	0.117787	+	0.993039i	$0.462420\pi$
$338$	0	0
$339$	−11.1623	−0.606252
$340$	0	0
$341$	0	0
$342$	0	0
$343$	65.2982	3.52577
$344$	0	0
$345$	−3.16228	−0.170251
$346$	0	0
$347$	−28.3246	−1.52054	−0.760271	−	0.649606i	$-0.774934\pi$
$347$	−28.3246	−1.52054	−0.760271	+	0.649606i	$0.774934\pi$
$348$	0	0
$349$	−8.64911	−0.462976	−0.231488	−	0.972838i	$-0.574359\pi$
$349$	−8.64911	−0.462976	−0.231488	+	0.972838i	$0.574359\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	16.9737	0.903417	0.451709	−	0.892166i	$-0.350815\pi$
$353$	16.9737	0.903417	0.451709	+	0.892166i	$0.350815\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−36.9737	−1.95685
$358$	0	0
$359$	−18.9737	−1.00139	−0.500696	−	0.865623i	$-0.666923\pi$
$359$	−18.9737	−1.00139	−0.500696	+	0.865623i	$0.666923\pi$
$360$	0	0
$361$	−17.6491	−0.928901
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	46.3246	2.42474
$366$	0	0
$367$	29.1623	1.52226	0.761129	−	0.648600i	$-0.224645\pi$
$367$	29.1623	1.52226	0.761129	+	0.648600i	$0.224645\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	28.3246	1.47054
$372$	0	0
$373$	−16.3246	−0.845253	−0.422627	−	0.906304i	$-0.638892\pi$
$373$	−16.3246	−0.845253	−0.422627	+	0.906304i	$0.638892\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.2982	1.71495
$378$	0	0
$379$	25.1623	1.29250	0.646250	−	0.763126i	$-0.276336\pi$
$379$	25.1623	1.29250	0.646250	+	0.763126i	$0.276336\pi$
$380$	0	0
$381$	−2.32456	−0.119091
$382$	0	0
$383$	22.9737	1.17390	0.586950	−	0.809623i	$-0.300329\pi$
$383$	22.9737	1.17390	0.586950	+	0.809623i	$0.300329\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.16228	0.0590819
$388$	0	0
$389$	7.16228	0.363142	0.181571	−	0.983378i	$-0.441882\pi$
$389$	7.16228	0.363142	0.181571	+	0.983378i	$0.441882\pi$
$390$	0	0
$391$	−7.16228	−0.362212
$392$	0	0
$393$	−8.64911	−0.436290
$394$	0	0
$395$	−41.6228	−2.09427
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	−5.48683	−0.273999	−0.137000	−	0.990571i	$-0.543746\pi$
$401$	−5.48683	−0.273999	−0.137000	+	0.990571i	$0.543746\pi$
$402$	0	0
$403$	−25.2982	−1.26019
$404$	0	0
$405$	−3.16228	−0.157135
$406$	0	0
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	−4.83772	−0.238627
$412$	0	0
$413$	−42.9737	−2.11460
$414$	0	0
$415$	12.6491	0.620920
$416$	0	0
$417$	16.6491	0.815310
$418$	0	0
$419$	8.64911	0.422537	0.211268	−	0.977428i	$-0.432241\pi$
$419$	8.64911	0.422537	0.211268	+	0.977428i	$0.432241\pi$
$420$	0	0
$421$	14.6491	0.713954	0.356977	−	0.934113i	$-0.383807\pi$
$421$	14.6491	0.713954	0.356977	+	0.934113i	$0.383807\pi$
$422$	0	0
$423$	0.324555	0.0157804
$424$	0	0
$425$	35.8114	1.73711
$426$	0	0
$427$	1.67544	0.0810805
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−18.6491	−0.896219	−0.448110	−	0.893979i	$-0.647903\pi$
$433$	−18.6491	−0.896219	−0.448110	+	0.893979i	$0.647903\pi$
$434$	0	0
$435$	26.3246	1.26217
$436$	0	0
$437$	1.16228	0.0555993
$438$	0	0
$439$	12.6491	0.603709	0.301855	−	0.953354i	$-0.402394\pi$
$439$	12.6491	0.603709	0.301855	+	0.953354i	$0.402394\pi$
$440$	0	0
$441$	19.6491	0.935672
$442$	0	0
$443$	−16.6491	−0.791023	−0.395512	−	0.918461i	$-0.629433\pi$
$443$	−16.6491	−0.791023	−0.395512	+	0.918461i	$0.629433\pi$
$444$	0	0
$445$	30.0000	1.42214
$446$	0	0
$447$	−19.1623	−0.906345
$448$	0	0
$449$	−26.6491	−1.25765	−0.628825	−	0.777547i	$-0.716464\pi$
$449$	−26.6491	−1.25765	−0.628825	+	0.777547i	$0.716464\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	4.64911	0.218434
$454$	0	0
$455$	−65.2982	−3.06123
$456$	0	0
$457$	16.3246	0.763630	0.381815	−	0.924239i	$-0.375299\pi$
$457$	16.3246	0.763630	0.381815	+	0.924239i	$0.375299\pi$
$458$	0	0
$459$	−7.16228	−0.334306
$460$	0	0
$461$	−0.973666	−0.0453481	−0.0226741	−	0.999743i	$-0.507218\pi$
$461$	−0.973666	−0.0453481	−0.0226741	+	0.999743i	$0.507218\pi$
$462$	0	0
$463$	−8.64911	−0.401958	−0.200979	−	0.979596i	$-0.564412\pi$
$463$	−8.64911	−0.401958	−0.200979	+	0.979596i	$0.564412\pi$
$464$	0	0
$465$	−20.0000	−0.927478
$466$	0	0
$467$	18.9737	0.877997	0.438998	−	0.898488i	$-0.355333\pi$
$467$	18.9737	0.877997	0.438998	+	0.898488i	$0.355333\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	−12.3246	−0.567886
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.81139	−0.266645
$476$	0	0
$477$	5.48683	0.251225
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−33.2982	−1.51827
$482$	0	0
$483$	5.16228	0.234892
$484$	0	0
$485$	11.6228	0.527763
$486$	0	0
$487$	0.649111	0.0294140	0.0147070	−	0.999892i	$-0.495318\pi$
$487$	0.649111	0.0294140	0.0147070	+	0.999892i	$0.495318\pi$
$488$	0	0
$489$	2.32456	0.105120
$490$	0	0
$491$	−20.9737	−0.946528	−0.473264	−	0.880921i	$-0.656924\pi$
$491$	−20.9737	−0.946528	−0.473264	+	0.880921i	$0.656924\pi$
$492$	0	0
$493$	59.6228	2.68527
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.02633	−0.0459450	−0.0229725	−	0.999736i	$-0.507313\pi$
$499$	−1.02633	−0.0459450	−0.0229725	+	0.999736i	$0.507313\pi$
$500$	0	0
$501$	20.6491	0.922534
$502$	0	0
$503$	21.2982	0.949641	0.474820	−	0.880083i	$-0.342513\pi$
$503$	21.2982	0.949641	0.474820	+	0.880083i	$0.342513\pi$
$504$	0	0
$505$	−18.9737	−0.844317
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	28.3246	1.25546	0.627732	−	0.778430i	$-0.283983\pi$
$509$	28.3246	1.25546	0.627732	+	0.778430i	$0.283983\pi$
$510$	0	0
$511$	−75.6228	−3.34535
$512$	0	0
$513$	1.16228	0.0513158
$514$	0	0
$515$	23.6754	1.04326
$516$	0	0
$517$	0	0
$518$	0	0
$519$	20.3246	0.892149
$520$	0	0
$521$	−8.18861	−0.358750	−0.179375	−	0.983781i	$-0.557407\pi$
$521$	−8.18861	−0.358750	−0.179375	+	0.983781i	$0.557407\pi$
$522$	0	0
$523$	13.8114	0.603930	0.301965	−	0.953319i	$-0.402357\pi$
$523$	13.8114	0.603930	0.301965	+	0.953319i	$0.402357\pi$
$524$	0	0
$525$	−25.8114	−1.12650
$526$	0	0
$527$	−45.2982	−1.97322
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.32456	−0.361255
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	50.5964	2.18747
$536$	0	0
$537$	7.67544	0.331220
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	−6.64911	−0.285341
$544$	0	0
$545$	−44.2719	−1.89640
$546$	0	0
$547$	22.9737	0.982283	0.491142	−	0.871080i	$-0.336580\pi$
$547$	22.9737	0.982283	0.491142	+	0.871080i	$0.336580\pi$
$548$	0	0
$549$	0.324555	0.0138517
$550$	0	0
$551$	−9.67544	−0.412188
$552$	0	0
$553$	67.9473	2.88941
$554$	0	0
$555$	−26.3246	−1.11742
$556$	0	0
$557$	−30.7851	−1.30440	−0.652202	−	0.758045i	$-0.726155\pi$
$557$	−30.7851	−1.30440	−0.652202	+	0.758045i	$0.726155\pi$
$558$	0	0
$559$	4.64911	0.196636
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.3246	1.44661	0.723304	−	0.690530i	$-0.242623\pi$
$563$	34.3246	1.44661	0.723304	+	0.690530i	$0.242623\pi$
$564$	0	0
$565$	−35.2982	−1.48501
$566$	0	0
$567$	5.16228	0.216795
$568$	0	0
$569$	13.4868	0.565397	0.282699	−	0.959209i	$-0.408770\pi$
$569$	13.4868	0.565397	0.282699	+	0.959209i	$0.408770\pi$
$570$	0	0
$571$	−42.4605	−1.77692	−0.888458	−	0.458957i	$-0.848223\pi$
$571$	−42.4605	−1.77692	−0.888458	+	0.458957i	$0.848223\pi$
$572$	0	0
$573$	18.9737	0.792636
$574$	0	0
$575$	−5.00000	−0.208514
$576$	0	0
$577$	15.3509	0.639066	0.319533	−	0.947575i	$-0.396474\pi$
$577$	15.3509	0.639066	0.319533	+	0.947575i	$0.396474\pi$
$578$	0	0
$579$	−0.649111	−0.0269761
$580$	0	0
$581$	−20.6491	−0.856669
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−12.6491	−0.522976
$586$	0	0
$587$	8.64911	0.356987	0.178494	−	0.983941i	$-0.442878\pi$
$587$	8.64911	0.356987	0.178494	+	0.983941i	$0.442878\pi$
$588$	0	0
$589$	7.35089	0.302888
$590$	0	0
$591$	3.67544	0.151188
$592$	0	0
$593$	−34.6491	−1.42287	−0.711434	−	0.702753i	$-0.751954\pi$
$593$	−34.6491	−1.42287	−0.711434	+	0.702753i	$0.751954\pi$
$594$	0	0
$595$	−116.921	−4.79329
$596$	0	0
$597$	9.81139	0.401553
$598$	0	0
$599$	−3.35089	−0.136914	−0.0684568	−	0.997654i	$-0.521808\pi$
$599$	−3.35089	−0.136914	−0.0684568	+	0.997654i	$0.521808\pi$
$600$	0	0
$601$	8.64911	0.352805	0.176402	−	0.984318i	$-0.443554\pi$
$601$	8.64911	0.352805	0.176402	+	0.984318i	$0.443554\pi$
$602$	0	0
$603$	1.16228	0.0473316
$604$	0	0
$605$	34.7851	1.41421
$606$	0	0
$607$	−44.6491	−1.81225	−0.906126	−	0.423008i	$-0.860974\pi$
$607$	−44.6491	−1.81225	−0.906126	+	0.423008i	$0.860974\pi$
$608$	0	0
$609$	−42.9737	−1.74138
$610$	0	0
$611$	1.29822	0.0525204
$612$	0	0
$613$	37.6228	1.51957	0.759785	−	0.650175i	$-0.225304\pi$
$613$	37.6228	1.51957	0.759785	+	0.650175i	$0.225304\pi$
$614$	0	0
$615$	−6.32456	−0.255031
$616$	0	0
$617$	35.1623	1.41558	0.707790	−	0.706423i	$-0.249692\pi$
$617$	35.1623	1.41558	0.707790	+	0.706423i	$0.249692\pi$
$618$	0	0
$619$	−5.81139	−0.233579	−0.116790	−	0.993157i	$-0.537260\pi$
$619$	−5.81139	−0.233579	−0.116790	+	0.993157i	$0.537260\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−48.9737	−1.96209
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−59.6228	−2.37732
$630$	0	0
$631$	−25.8114	−1.02754	−0.513768	−	0.857929i	$-0.671751\pi$
$631$	−25.8114	−1.02754	−0.513768	+	0.857929i	$0.671751\pi$
$632$	0	0
$633$	−10.3246	−0.410364
$634$	0	0
$635$	−7.35089	−0.291711
$636$	0	0
$637$	78.5964	3.11410
$638$	0	0
$639$	0	0
$640$	0	0
$641$	47.1623	1.86280	0.931399	−	0.364000i	$-0.118589\pi$
$641$	47.1623	1.86280	0.931399	+	0.364000i	$0.118589\pi$
$642$	0	0
$643$	9.16228	0.361325	0.180662	−	0.983545i	$-0.442176\pi$
$643$	9.16228	0.361325	0.180662	+	0.983545i	$0.442176\pi$
$644$	0	0
$645$	3.67544	0.144720
$646$	0	0
$647$	42.2719	1.66188	0.830940	−	0.556363i	$-0.187803\pi$
$647$	42.2719	1.66188	0.830940	+	0.556363i	$0.187803\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	32.6491	1.27962
$652$	0	0
$653$	−35.9473	−1.40673	−0.703364	−	0.710830i	$-0.748320\pi$
$653$	−35.9473	−1.40673	−0.703364	+	0.710830i	$0.748320\pi$
$654$	0	0
$655$	−27.3509	−1.06869
$656$	0	0
$657$	−14.6491	−0.571516
$658$	0	0
$659$	47.6228	1.85512	0.927560	−	0.373674i	$-0.121902\pi$
$659$	47.6228	1.85512	0.927560	+	0.373674i	$0.121902\pi$
$660$	0	0
$661$	−19.6754	−0.765286	−0.382643	−	0.923896i	$-0.624986\pi$
$661$	−19.6754	−0.765286	−0.382643	+	0.923896i	$0.624986\pi$
$662$	0	0
$663$	−28.6491	−1.11264
$664$	0	0
$665$	18.9737	0.735767
$666$	0	0
$667$	−8.32456	−0.322328
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−20.6491	−0.795965	−0.397982	−	0.917393i	$-0.630289\pi$
$673$	−20.6491	−0.795965	−0.397982	+	0.917393i	$0.630289\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−28.8377	−1.10832	−0.554162	−	0.832409i	$-0.686961\pi$
$677$	−28.8377	−1.10832	−0.554162	+	0.832409i	$0.686961\pi$
$678$	0	0
$679$	−18.9737	−0.728142
$680$	0	0
$681$	18.3246	0.702198
$682$	0	0
$683$	−21.2982	−0.814954	−0.407477	−	0.913216i	$-0.633591\pi$
$683$	−21.2982	−0.814954	−0.407477	+	0.913216i	$0.633591\pi$
$684$	0	0
$685$	−15.2982	−0.584515
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	21.9473	0.836127
$690$	0	0
$691$	−9.02633	−0.343378	−0.171689	−	0.985151i	$-0.554922\pi$
$691$	−9.02633	−0.343378	−0.171689	+	0.985151i	$0.554922\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	52.6491	1.99709
$696$	0	0
$697$	−14.3246	−0.542581
$698$	0	0
$699$	6.64911	0.251492
$700$	0	0
$701$	36.8377	1.39134	0.695671	−	0.718361i	$-0.255107\pi$
$701$	36.8377	1.39134	0.695671	+	0.718361i	$0.255107\pi$
$702$	0	0
$703$	9.67544	0.364916
$704$	0	0
$705$	1.02633	0.0386540
$706$	0	0
$707$	30.9737	1.16488
$708$	0	0
$709$	35.9473	1.35003	0.675015	−	0.737804i	$-0.264137\pi$
$709$	35.9473	1.35003	0.675015	+	0.737804i	$0.264137\pi$
$710$	0	0
$711$	13.1623	0.493624
$712$	0	0
$713$	6.32456	0.236856
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.35089	−0.125141
$718$	0	0
$719$	48.9737	1.82641	0.913205	−	0.407501i	$-0.133600\pi$
$719$	48.9737	1.82641	0.913205	+	0.407501i	$0.133600\pi$
$720$	0	0
$721$	−38.6491	−1.43937
$722$	0	0
$723$	20.3246	0.755878
$724$	0	0
$725$	41.6228	1.54583
$726$	0	0
$727$	−8.51317	−0.315736	−0.157868	−	0.987460i	$-0.550462\pi$
$727$	−8.51317	−0.315736	−0.157868	+	0.987460i	$0.550462\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.32456	0.307895
$732$	0	0
$733$	18.0000	0.664845	0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000	0.664845	0.332423	+	0.943131i	$0.392134\pi$
$734$	0	0
$735$	62.1359	2.29192
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	4.64911	0.170789
$742$	0	0
$743$	−48.2719	−1.77092	−0.885462	−	0.464712i	$-0.846158\pi$
$743$	−48.2719	−1.77092	−0.885462	+	0.464712i	$0.846158\pi$
$744$	0	0
$745$	−60.5964	−2.22008
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−82.5964	−3.01801
$750$	0	0
$751$	−23.4868	−0.857047	−0.428523	−	0.903531i	$-0.640966\pi$
$751$	−23.4868	−0.857047	−0.428523	+	0.903531i	$0.640966\pi$
$752$	0	0
$753$	14.3246	0.522016
$754$	0	0
$755$	14.7018	0.535053
$756$	0	0
$757$	−27.9473	−1.01576	−0.507882	−	0.861427i	$-0.669571\pi$
$757$	−27.9473	−1.01576	−0.507882	+	0.861427i	$0.669571\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.0263	0.399704	0.199852	−	0.979826i	$-0.435954\pi$
$761$	11.0263	0.399704	0.199852	+	0.979826i	$0.435954\pi$
$762$	0	0
$763$	72.2719	2.61642
$764$	0	0
$765$	−22.6491	−0.818880
$766$	0	0
$767$	−33.2982	−1.20233
$768$	0	0
$769$	−30.6491	−1.10524	−0.552618	−	0.833435i	$-0.686371\pi$
$769$	−30.6491	−1.10524	−0.552618	+	0.833435i	$0.686371\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	−38.1359	−1.37165	−0.685827	−	0.727764i	$-0.740559\pi$
$773$	−38.1359	−1.37165	−0.685827	+	0.727764i	$0.740559\pi$
$774$	0	0
$775$	−31.6228	−1.13592
$776$	0	0
$777$	42.9737	1.54167
$778$	0	0
$779$	2.32456	0.0832858
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−8.32456	−0.297495
$784$	0	0
$785$	−38.9737	−1.39103
$786$	0	0
$787$	33.1623	1.18211	0.591054	−	0.806632i	$-0.298712\pi$
$787$	33.1623	1.18211	0.591054	+	0.806632i	$0.298712\pi$
$788$	0	0
$789$	−30.3246	−1.07958
$790$	0	0
$791$	57.6228	2.04883
$792$	0	0
$793$	1.29822	0.0461012
$794$	0	0
$795$	17.3509	0.615373
$796$	0	0
$797$	−35.8114	−1.26850	−0.634252	−	0.773126i	$-0.718692\pi$
$797$	−35.8114	−1.26850	−0.634252	+	0.773126i	$0.718692\pi$
$798$	0	0
$799$	2.32456	0.0822369
$800$	0	0
$801$	−9.48683	−0.335201
$802$	0	0
$803$	0	0
$804$	0	0
$805$	16.3246	0.575365
$806$	0	0
$807$	3.67544	0.129382
$808$	0	0
$809$	−8.97367	−0.315497	−0.157749	−	0.987479i	$-0.550424\pi$
$809$	−8.97367	−0.315497	−0.157749	+	0.987479i	$0.550424\pi$
$810$	0	0
$811$	−5.67544	−0.199292	−0.0996459	−	0.995023i	$-0.531771\pi$
$811$	−5.67544	−0.199292	−0.0996459	+	0.995023i	$0.531771\pi$
$812$	0	0
$813$	5.67544	0.199047
$814$	0	0
$815$	7.35089	0.257490
$816$	0	0
$817$	−1.35089	−0.0472616
$818$	0	0
$819$	20.6491	0.721538
$820$	0	0
$821$	39.2982	1.37152	0.685759	−	0.727829i	$-0.259471\pi$
$821$	39.2982	1.37152	0.685759	+	0.727829i	$0.259471\pi$
$822$	0	0
$823$	−35.6228	−1.24173	−0.620866	−	0.783917i	$-0.713219\pi$
$823$	−35.6228	−1.24173	−0.620866	+	0.783917i	$0.713219\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.02633	0.174783	0.0873914	−	0.996174i	$-0.472147\pi$
$827$	5.02633	0.174783	0.0873914	+	0.996174i	$0.472147\pi$
$828$	0	0
$829$	33.2982	1.15650	0.578248	−	0.815861i	$-0.303737\pi$
$829$	33.2982	1.15650	0.578248	+	0.815861i	$0.303737\pi$
$830$	0	0
$831$	19.2982	0.669448
$832$	0	0
$833$	140.732	4.87609
$834$	0	0
$835$	65.2982	2.25974
$836$	0	0
$837$	6.32456	0.218609
$838$	0	0
$839$	−23.6228	−0.815549	−0.407774	−	0.913083i	$-0.633695\pi$
$839$	−23.6228	−0.815549	−0.407774	+	0.913083i	$0.633695\pi$
$840$	0	0
$841$	40.2982	1.38959
$842$	0	0
$843$	3.16228	0.108915
$844$	0	0
$845$	−9.48683	−0.326357
$846$	0	0
$847$	−56.7851	−1.95116
$848$	0	0
$849$	3.48683	0.119668
$850$	0	0
$851$	8.32456	0.285362
$852$	0	0
$853$	27.2982	0.934673	0.467337	−	0.884079i	$-0.345214\pi$
$853$	27.2982	0.934673	0.467337	+	0.884079i	$0.345214\pi$
$854$	0	0
$855$	3.67544	0.125698
$856$	0	0
$857$	9.35089	0.319420	0.159710	−	0.987164i	$-0.448944\pi$
$857$	9.35089	0.319420	0.159710	+	0.987164i	$0.448944\pi$
$858$	0	0
$859$	41.2982	1.40908	0.704539	−	0.709666i	$-0.251154\pi$
$859$	41.2982	1.40908	0.704539	+	0.709666i	$0.251154\pi$
$860$	0	0
$861$	10.3246	0.351860
$862$	0	0
$863$	7.02633	0.239179	0.119590	−	0.992823i	$-0.461842\pi$
$863$	7.02633	0.239179	0.119590	+	0.992823i	$0.461842\pi$
$864$	0	0
$865$	64.2719	2.18531
$866$	0	0
$867$	−34.2982	−1.16483
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.64911	0.157529
$872$	0	0
$873$	−3.67544	−0.124395
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.9473	−0.943714	−0.471857	−	0.881675i	$-0.656416\pi$
$877$	−27.9473	−0.943714	−0.471857	+	0.881675i	$0.656416\pi$
$878$	0	0
$879$	−2.51317	−0.0847670
$880$	0	0
$881$	−3.16228	−0.106540	−0.0532699	−	0.998580i	$-0.516964\pi$
$881$	−3.16228	−0.106540	−0.0532699	+	0.998580i	$0.516964\pi$
$882$	0	0
$883$	14.3246	0.482060	0.241030	−	0.970518i	$-0.422515\pi$
$883$	14.3246	0.482060	0.241030	+	0.970518i	$0.422515\pi$
$884$	0	0
$885$	−26.3246	−0.884890
$886$	0	0
$887$	−19.6754	−0.660637	−0.330318	−	0.943870i	$-0.607156\pi$
$887$	−19.6754	−0.660637	−0.330318	+	0.943870i	$0.607156\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.377223	−0.0126233
$894$	0	0
$895$	24.2719	0.811320
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	−52.6491	−1.75595
$900$	0	0
$901$	39.2982	1.30921
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	−21.0263	−0.698939
$906$	0	0
$907$	−6.83772	−0.227043	−0.113521	−	0.993536i	$-0.536213\pi$
$907$	−6.83772	−0.227043	−0.113521	+	0.993536i	$0.536213\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1.02633	0.0339296
$916$	0	0
$917$	44.6491	1.47444
$918$	0	0
$919$	−30.4605	−1.00480	−0.502400	−	0.864636i	$-0.667549\pi$
$919$	−30.4605	−1.00480	−0.502400	+	0.864636i	$0.667549\pi$
$920$	0	0
$921$	−6.32456	−0.208401
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−41.6228	−1.36855
$926$	0	0
$927$	−7.48683	−0.245900
$928$	0	0
$929$	−22.6491	−0.743093	−0.371547	−	0.928414i	$-0.621172\pi$
$929$	−22.6491	−0.743093	−0.371547	+	0.928414i	$0.621172\pi$
$930$	0	0
$931$	−22.8377	−0.748476
$932$	0	0
$933$	−28.9737	−0.948555
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	6.64911	0.216986
$940$	0	0
$941$	−43.1623	−1.40705	−0.703525	−	0.710670i	$-0.748392\pi$
$941$	−43.1623	−1.40705	−0.703525	+	0.710670i	$0.748392\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	0	0
$945$	16.3246	0.531038
$946$	0	0
$947$	−16.9737	−0.551570	−0.275785	−	0.961219i	$-0.588938\pi$
$947$	−16.9737	−0.551570	−0.275785	+	0.961219i	$0.588938\pi$
$948$	0	0
$949$	−58.5964	−1.90212
$950$	0	0
$951$	11.6754	0.378602
$952$	0	0
$953$	26.1359	0.846626	0.423313	−	0.905983i	$-0.360867\pi$
$953$	26.1359	0.846626	0.423313	+	0.905983i	$0.360867\pi$
$954$	0	0
$955$	60.0000	1.94155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.9737	0.806442
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	−16.0000	−0.515593
$964$	0	0
$965$	−2.05267	−0.0660777
$966$	0	0
$967$	10.9737	0.352889	0.176445	−	0.984311i	$-0.443540\pi$
$967$	10.9737	0.352889	0.176445	+	0.984311i	$0.443540\pi$
$968$	0	0
$969$	8.32456	0.267423
$970$	0	0
$971$	−3.62278	−0.116260	−0.0581302	−	0.998309i	$-0.518514\pi$
$971$	−3.62278	−0.116260	−0.0581302	+	0.998309i	$0.518514\pi$
$972$	0	0
$973$	−85.9473	−2.75534
$974$	0	0
$975$	−20.0000	−0.640513
$976$	0	0
$977$	−14.1359	−0.452249	−0.226124	−	0.974098i	$-0.572606\pi$
$977$	−14.1359	−0.452249	−0.226124	+	0.974098i	$0.572606\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	−53.2982	−1.69995	−0.849975	−	0.526824i	$-0.823383\pi$
$983$	−53.2982	−1.69995	−0.849975	+	0.526824i	$0.823383\pi$
$984$	0	0
$985$	11.6228	0.370332
$986$	0	0
$987$	−1.67544	−0.0533300
$988$	0	0
$989$	−1.16228	−0.0369583
$990$	0	0
$991$	−42.3246	−1.34448	−0.672242	−	0.740332i	$-0.734669\pi$
$991$	−42.3246	−1.34448	−0.672242	+	0.740332i	$0.734669\pi$
$992$	0	0
$993$	−6.32456	−0.200704
$994$	0	0
$995$	31.0263	0.983601
$996$	0	0
$997$	−0.649111	−0.0205575	−0.0102788	−	0.999947i	$-0.503272\pi$
$997$	−0.649111	−0.0205575	−0.0102788	+	0.999947i	$0.503272\pi$
$998$	0	0
$999$	8.32456	0.263377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	276.2.a.a.1.1	✓	2
3.2	odd	2		828.2.a.f.1.2		2
4.3	odd	2		1104.2.a.n.1.1		2
5.2	odd	4		6900.2.f.k.6349.4		4
5.3	odd	4		6900.2.f.k.6349.1		4
5.4	even	2		6900.2.a.p.1.1		2
8.3	odd	2		4416.2.a.be.1.2		2
8.5	even	2		4416.2.a.bk.1.2		2
12.11	even	2		3312.2.a.y.1.2		2
23.22	odd	2		6348.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.a.a.1.1	✓	2	1.1	even	1	trivial
828.2.a.f.1.2		2	3.2	odd	2
1104.2.a.n.1.1		2	4.3	odd	2
3312.2.a.y.1.2		2	12.11	even	2
4416.2.a.be.1.2		2	8.3	odd	2
4416.2.a.bk.1.2		2	8.5	even	2
6348.2.a.e.1.2		2	23.22	odd	2
6900.2.a.p.1.1		2	5.4	even	2
6900.2.f.k.6349.1		4	5.3	odd	4
6900.2.f.k.6349.4		4	5.2	odd	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:	\( N \)	\(=\)	\( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	276.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.16228 q^{5} +5.16228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.16228 q^{5} +5.16228 q^{7} +1.00000 q^{9} +4.00000 q^{13} +3.16228 q^{15} +7.16228 q^{17} -1.16228 q^{19} -5.16228 q^{21} -1.00000 q^{23} +5.00000 q^{25} -1.00000 q^{27} +8.32456 q^{29} -6.32456 q^{31} -16.3246 q^{35} -8.32456 q^{37} -4.00000 q^{39} -2.00000 q^{41} +1.16228 q^{43} -3.16228 q^{45} +0.324555 q^{47} +19.6491 q^{49} -7.16228 q^{51} +5.48683 q^{53} +1.16228 q^{57} -8.32456 q^{59} +0.324555 q^{61} +5.16228 q^{63} -12.6491 q^{65} +1.16228 q^{67} +1.00000 q^{69} -14.6491 q^{73} -5.00000 q^{75} +13.1623 q^{79} +1.00000 q^{81} -4.00000 q^{83} -22.6491 q^{85} -8.32456 q^{87} -9.48683 q^{89} +20.6491 q^{91} +6.32456 q^{93} +3.67544 q^{95} -3.67544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 4 q^{7} + 2 q^{9} + 8 q^{13} + 8 q^{17} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 10 q^{25} - 2 q^{27} + 4 q^{29} - 20 q^{35} - 4 q^{37} - 8 q^{39} - 4 q^{41} - 4 q^{43} - 12 q^{47} + 14 q^{49} - 8 q^{51} - 8 q^{53} - 4 q^{57} - 4 q^{59} - 12 q^{61} + 4 q^{63} - 4 q^{67} + 2 q^{69} - 4 q^{73} - 10 q^{75} + 20 q^{79} + 2 q^{81} - 8 q^{83} - 20 q^{85} - 4 q^{87} + 16 q^{91} + 20 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^9 + 8 * q^13 + 8 * q^17 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 10 * q^25 - 2 * q^27 + 4 * q^29 - 20 * q^35 - 4 * q^37 - 8 * q^39 - 4 * q^41 - 4 * q^43 - 12 * q^47 + 14 * q^49 - 8 * q^51 - 8 * q^53 - 4 * q^57 - 4 * q^59 - 12 * q^61 + 4 * q^63 - 4 * q^67 + 2 * q^69 - 4 * q^73 - 10 * q^75 + 20 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 - 4 * q^87 + 16 * q^91 + 20 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more