LMFDB - Embedded newform 6900.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.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:	$55.0967773947$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -1.67513 q^{11} -0.869067 q^{13} +1.86907 q^{17} +0.869067 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{27} +2.44358 q^{29} -4.19394 q^{31} +1.67513 q^{33} -2.76845 q^{37} +0.869067 q^{39} -10.5999 q^{41} +7.11871 q^{43} +2.71274 q^{47} -6.00000 q^{49} -1.86907 q^{51} +3.63752 q^{53} -0.869067 q^{57} +3.48119 q^{59} -1.78067 q^{61} +1.00000 q^{63} -11.0811 q^{67} +1.00000 q^{69} -0.100615 q^{71} -2.16854 q^{73} -1.67513 q^{77} +5.08840 q^{79} +1.00000 q^{81} +9.09332 q^{83} -2.44358 q^{87} +7.92478 q^{89} -0.869067 q^{91} +4.19394 q^{93} +4.99271 q^{97} -1.67513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{13} + q^{17} - 2 q^{19} - 3 q^{21} - 3 q^{23} - 3 q^{27} - 9 q^{29} - 13 q^{31} + 3 q^{37} - 2 q^{39} - 5 q^{41} + 14 q^{47} - 18 q^{49} - q^{51} - 5 q^{53} + 2 q^{57} + 5 q^{59} - 20 q^{61} + 3 q^{63} - q^{67} + 3 q^{69} - 7 q^{71} - 22 q^{73} - 4 q^{79} + 3 q^{81} + 21 q^{83} + 9 q^{87} + 2 q^{89} + 2 q^{91} + 13 q^{93} + 2 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^13 + q^17 - 2 * q^19 - 3 * q^21 - 3 * q^23 - 3 * q^27 - 9 * q^29 - 13 * q^31 + 3 * q^37 - 2 * q^39 - 5 * q^41 + 14 * q^47 - 18 * q^49 - q^51 - 5 * q^53 + 2 * q^57 + 5 * q^59 - 20 * q^61 + 3 * q^63 - q^67 + 3 * q^69 - 7 * q^71 - 22 * q^73 - 4 * q^79 + 3 * q^81 + 21 * q^83 + 9 * q^87 + 2 * q^89 + 2 * q^91 + 13 * q^93 + 2 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.67513	−0.505071	−0.252535	−	0.967588i	$-0.581264\pi$
$11$	−1.67513	−0.505071	−0.252535	+	0.967588i	$0.581264\pi$
$12$	0	0
$13$	−0.869067	−0.241036	−0.120518	−	0.992711i	$-0.538456\pi$
$13$	−0.869067	−0.241036	−0.120518	+	0.992711i	$0.538456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.86907	0.453315	0.226658	−	0.973974i	$-0.427220\pi$
$17$	1.86907	0.453315	0.226658	+	0.973974i	$0.427220\pi$
$18$	0	0
$19$	0.869067	0.199378	0.0996889	−	0.995019i	$-0.468215\pi$
$19$	0.869067	0.199378	0.0996889	+	0.995019i	$0.468215\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.44358	0.453762	0.226881	−	0.973922i	$-0.427147\pi$
$29$	2.44358	0.453762	0.226881	+	0.973922i	$0.427147\pi$
$30$	0	0
$31$	−4.19394	−0.753253	−0.376627	−	0.926365i	$-0.622916\pi$
$31$	−4.19394	−0.753253	−0.376627	+	0.926365i	$0.622916\pi$
$32$	0	0
$33$	1.67513	0.291603
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.76845	−0.455131	−0.227565	−	0.973763i	$-0.573077\pi$
$37$	−2.76845	−0.455131	−0.227565	+	0.973763i	$0.573077\pi$
$38$	0	0
$39$	0.869067	0.139162
$40$	0	0
$41$	−10.5999	−1.65543	−0.827714	−	0.561151i	$-0.810359\pi$
$41$	−10.5999	−1.65543	−0.827714	+	0.561151i	$0.810359\pi$
$42$	0	0
$43$	7.11871	1.08559	0.542797	−	0.839864i	$-0.317365\pi$
$43$	7.11871	1.08559	0.542797	+	0.839864i	$0.317365\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.71274	0.395694	0.197847	−	0.980233i	$-0.436605\pi$
$47$	2.71274	0.395694	0.197847	+	0.980233i	$0.436605\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.86907	−0.261722
$52$	0	0
$53$	3.63752	0.499652	0.249826	−	0.968291i	$-0.419627\pi$
$53$	3.63752	0.499652	0.249826	+	0.968291i	$0.419627\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.869067	−0.115111
$58$	0	0
$59$	3.48119	0.453213	0.226606	−	0.973986i	$-0.427237\pi$
$59$	3.48119	0.453213	0.226606	+	0.973986i	$0.427237\pi$
$60$	0	0
$61$	−1.78067	−0.227992	−0.113996	−	0.993481i	$-0.536365\pi$
$61$	−1.78067	−0.227992	−0.113996	+	0.993481i	$0.536365\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.0811	−1.35377	−0.676886	−	0.736088i	$-0.736671\pi$
$67$	−11.0811	−1.35377	−0.676886	+	0.736088i	$0.736671\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−0.100615	−0.0119409	−0.00597043	−	0.999982i	$-0.501900\pi$
$71$	−0.100615	−0.0119409	−0.00597043	+	0.999982i	$0.501900\pi$
$72$	0	0
$73$	−2.16854	−0.253809	−0.126904	−	0.991915i	$-0.540504\pi$
$73$	−2.16854	−0.253809	−0.126904	+	0.991915i	$0.540504\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.67513	−0.190899
$78$	0	0
$79$	5.08840	0.572489	0.286245	−	0.958157i	$-0.407593\pi$
$79$	5.08840	0.572489	0.286245	+	0.958157i	$0.407593\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.09332	0.998122	0.499061	−	0.866567i	$-0.333678\pi$
$83$	9.09332	0.998122	0.499061	+	0.866567i	$0.333678\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.44358	−0.261980
$88$	0	0
$89$	7.92478	0.840025	0.420012	−	0.907518i	$-0.362026\pi$
$89$	7.92478	0.840025	0.420012	+	0.907518i	$0.362026\pi$
$90$	0	0
$91$	−0.869067	−0.0911030
$92$	0	0
$93$	4.19394	0.434891
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.99271	0.506932	0.253466	−	0.967344i	$-0.418429\pi$
$97$	4.99271	0.506932	0.253466	+	0.967344i	$0.418429\pi$
$98$	0	0
$99$	−1.67513	−0.168357
$100$	0	0
$101$	−8.28726	−0.824613	−0.412306	−	0.911045i	$-0.635277\pi$
$101$	−8.28726	−0.824613	−0.412306	+	0.911045i	$0.635277\pi$
$102$	0	0
$103$	−2.80606	−0.276490	−0.138245	−	0.990398i	$-0.544146\pi$
$103$	−2.80606	−0.276490	−0.138245	+	0.990398i	$0.544146\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.45088	0.333609	0.166804	−	0.985990i	$-0.446655\pi$
$107$	3.45088	0.333609	0.166804	+	0.985990i	$0.446655\pi$
$108$	0	0
$109$	−13.3684	−1.28046	−0.640228	−	0.768185i	$-0.721160\pi$
$109$	−13.3684	−1.28046	−0.640228	+	0.768185i	$0.721160\pi$
$110$	0	0
$111$	2.76845	0.262770
$112$	0	0
$113$	−8.98778	−0.845499	−0.422750	−	0.906246i	$-0.638935\pi$
$113$	−8.98778	−0.845499	−0.422750	+	0.906246i	$0.638935\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.869067	−0.0803453
$118$	0	0
$119$	1.86907	0.171337
$120$	0	0
$121$	−8.19394	−0.744903
$122$	0	0
$123$	10.5999	0.955762
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.99508	−0.265770	−0.132885	−	0.991131i	$-0.542424\pi$
$127$	−2.99508	−0.265770	−0.132885	+	0.991131i	$0.542424\pi$
$128$	0	0
$129$	−7.11871	−0.626768
$130$	0	0
$131$	−20.9380	−1.82936	−0.914679	−	0.404182i	$-0.867556\pi$
$131$	−20.9380	−1.82936	−0.914679	+	0.404182i	$0.867556\pi$
$132$	0	0
$133$	0.869067	0.0753577
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.8945	1.01621	0.508106	−	0.861294i	$-0.330346\pi$
$137$	11.8945	1.01621	0.508106	+	0.861294i	$0.330346\pi$
$138$	0	0
$139$	6.35026	0.538622	0.269311	−	0.963053i	$-0.413204\pi$
$139$	6.35026	0.538622	0.269311	+	0.963053i	$0.413204\pi$
$140$	0	0
$141$	−2.71274	−0.228454
$142$	0	0
$143$	1.45580	0.121740
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−9.67513	−0.792618	−0.396309	−	0.918117i	$-0.629709\pi$
$149$	−9.67513	−0.792618	−0.396309	+	0.918117i	$0.629709\pi$
$150$	0	0
$151$	2.15633	0.175479	0.0877396	−	0.996143i	$-0.472036\pi$
$151$	2.15633	0.175479	0.0877396	+	0.996143i	$0.472036\pi$
$152$	0	0
$153$	1.86907	0.151105
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.42548	0.353192	0.176596	−	0.984283i	$-0.443491\pi$
$157$	4.42548	0.353192	0.176596	+	0.984283i	$0.443491\pi$
$158$	0	0
$159$	−3.63752	−0.288474
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	4.31265	0.337793	0.168896	−	0.985634i	$-0.445980\pi$
$163$	4.31265	0.337793	0.168896	+	0.985634i	$0.445980\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.08603	0.703098	0.351549	−	0.936170i	$-0.385655\pi$
$167$	9.08603	0.703098	0.351549	+	0.936170i	$0.385655\pi$
$168$	0	0
$169$	−12.2447	−0.941902
$170$	0	0
$171$	0.869067	0.0664592
$172$	0	0
$173$	−6.49929	−0.494132	−0.247066	−	0.968999i	$-0.579466\pi$
$173$	−6.49929	−0.494132	−0.247066	+	0.968999i	$0.579466\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.48119	−0.261663
$178$	0	0
$179$	2.54420	0.190162	0.0950812	−	0.995470i	$-0.469689\pi$
$179$	2.54420	0.190162	0.0950812	+	0.995470i	$0.469689\pi$
$180$	0	0
$181$	−8.12601	−0.604001	−0.302001	−	0.953308i	$-0.597655\pi$
$181$	−8.12601	−0.604001	−0.302001	+	0.953308i	$0.597655\pi$
$182$	0	0
$183$	1.78067	0.131631
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.13093	−0.228956
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−7.28726	−0.527287	−0.263644	−	0.964620i	$-0.584924\pi$
$191$	−7.28726	−0.527287	−0.263644	+	0.964620i	$0.584924\pi$
$192$	0	0
$193$	−13.5877	−0.978063	−0.489032	−	0.872266i	$-0.662650\pi$
$193$	−13.5877	−0.978063	−0.489032	+	0.872266i	$0.662650\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.9380	1.20678	0.603390	−	0.797447i	$-0.293816\pi$
$197$	16.9380	1.20678	0.603390	+	0.797447i	$0.293816\pi$
$198$	0	0
$199$	16.4241	1.16427	0.582136	−	0.813092i	$-0.302217\pi$
$199$	16.4241	1.16427	0.582136	+	0.813092i	$0.302217\pi$
$200$	0	0
$201$	11.0811	0.781601
$202$	0	0
$203$	2.44358	0.171506
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−1.45580	−0.100700
$210$	0	0
$211$	−20.1490	−1.38712	−0.693558	−	0.720401i	$-0.743958\pi$
$211$	−20.1490	−1.38712	−0.693558	+	0.720401i	$0.743958\pi$
$212$	0	0
$213$	0.100615	0.00689405
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.19394	−0.284703
$218$	0	0
$219$	2.16854	0.146537
$220$	0	0
$221$	−1.62435	−0.109265
$222$	0	0
$223$	−17.9248	−1.20033	−0.600166	−	0.799876i	$-0.704899\pi$
$223$	−17.9248	−1.20033	−0.600166	+	0.799876i	$0.704899\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.14903	−0.341753	−0.170877	−	0.985292i	$-0.554660\pi$
$227$	−5.14903	−0.341753	−0.170877	+	0.985292i	$0.554660\pi$
$228$	0	0
$229$	−20.7513	−1.37129	−0.685643	−	0.727938i	$-0.740479\pi$
$229$	−20.7513	−1.37129	−0.685643	+	0.727938i	$0.740479\pi$
$230$	0	0
$231$	1.67513	0.110216
$232$	0	0
$233$	−14.1768	−0.928753	−0.464376	−	0.885638i	$-0.653721\pi$
$233$	−14.1768	−0.928753	−0.464376	+	0.885638i	$0.653721\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.08840	−0.330527
$238$	0	0
$239$	−1.26423	−0.0817766	−0.0408883	−	0.999164i	$-0.513019\pi$
$239$	−1.26423	−0.0817766	−0.0408883	+	0.999164i	$0.513019\pi$
$240$	0	0
$241$	6.45676	0.415916	0.207958	−	0.978138i	$-0.433318\pi$
$241$	6.45676	0.415916	0.207958	+	0.978138i	$0.433318\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.755278	−0.0480572
$248$	0	0
$249$	−9.09332	−0.576266
$250$	0	0
$251$	7.66291	0.483679	0.241839	−	0.970316i	$-0.422249\pi$
$251$	7.66291	0.483679	0.241839	+	0.970316i	$0.422249\pi$
$252$	0	0
$253$	1.67513	0.105315
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.65069	0.601994	0.300997	−	0.953625i	$-0.402681\pi$
$257$	9.65069	0.601994	0.300997	+	0.953625i	$0.402681\pi$
$258$	0	0
$259$	−2.76845	−0.172023
$260$	0	0
$261$	2.44358	0.151254
$262$	0	0
$263$	15.8994	0.980398	0.490199	−	0.871611i	$-0.336924\pi$
$263$	15.8994	0.980398	0.490199	+	0.871611i	$0.336924\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.92478	−0.484988
$268$	0	0
$269$	7.85448	0.478896	0.239448	−	0.970909i	$-0.423034\pi$
$269$	7.85448	0.478896	0.239448	+	0.970909i	$0.423034\pi$
$270$	0	0
$271$	−9.96239	−0.605172	−0.302586	−	0.953122i	$-0.597850\pi$
$271$	−9.96239	−0.605172	−0.302586	+	0.953122i	$0.597850\pi$
$272$	0	0
$273$	0.869067	0.0525984
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.61213	0.337200	0.168600	−	0.985685i	$-0.446075\pi$
$277$	5.61213	0.337200	0.168600	+	0.985685i	$0.446075\pi$
$278$	0	0
$279$	−4.19394	−0.251084
$280$	0	0
$281$	−6.96002	−0.415200	−0.207600	−	0.978214i	$-0.566565\pi$
$281$	−6.96002	−0.415200	−0.207600	+	0.978214i	$0.566565\pi$
$282$	0	0
$283$	9.05079	0.538013	0.269007	−	0.963138i	$-0.413305\pi$
$283$	9.05079	0.538013	0.269007	+	0.963138i	$0.413305\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.5999	−0.625693
$288$	0	0
$289$	−13.5066	−0.794505
$290$	0	0
$291$	−4.99271	−0.292678
$292$	0	0
$293$	−2.18172	−0.127457	−0.0637287	−	0.997967i	$-0.520299\pi$
$293$	−2.18172	−0.127457	−0.0637287	+	0.997967i	$0.520299\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.67513	0.0972010
$298$	0	0
$299$	0.869067	0.0502595
$300$	0	0
$301$	7.11871	0.410316
$302$	0	0
$303$	8.28726	0.476091
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.91985	0.280791	0.140395	−	0.990096i	$-0.455163\pi$
$307$	4.91985	0.280791	0.140395	+	0.990096i	$0.455163\pi$
$308$	0	0
$309$	2.80606	0.159631
$310$	0	0
$311$	−14.2619	−0.808716	−0.404358	−	0.914601i	$-0.632505\pi$
$311$	−14.2619	−0.808716	−0.404358	+	0.914601i	$0.632505\pi$
$312$	0	0
$313$	−4.32979	−0.244734	−0.122367	−	0.992485i	$-0.539049\pi$
$313$	−4.32979	−0.244734	−0.122367	+	0.992485i	$0.539049\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.0616	1.74459	0.872296	−	0.488978i	$-0.162630\pi$
$317$	31.0616	1.74459	0.872296	+	0.488978i	$0.162630\pi$
$318$	0	0
$319$	−4.09332	−0.229182
$320$	0	0
$321$	−3.45088	−0.192609
$322$	0	0
$323$	1.62435	0.0903810
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.3684	0.739272
$328$	0	0
$329$	2.71274	0.149558
$330$	0	0
$331$	3.80606	0.209200	0.104600	−	0.994514i	$-0.466644\pi$
$331$	3.80606	0.209200	0.104600	+	0.994514i	$0.466644\pi$
$332$	0	0
$333$	−2.76845	−0.151710
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.62530	0.360903	0.180452	−	0.983584i	$-0.442244\pi$
$337$	6.62530	0.360903	0.180452	+	0.983584i	$0.442244\pi$
$338$	0	0
$339$	8.98778	0.488149
$340$	0	0
$341$	7.02539	0.380446
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.9756	−0.857613	−0.428807	−	0.903396i	$-0.641066\pi$
$347$	−15.9756	−0.857613	−0.428807	+	0.903396i	$0.641066\pi$
$348$	0	0
$349$	−22.2750	−1.19236	−0.596178	−	0.802852i	$-0.703315\pi$
$349$	−22.2750	−1.19236	−0.596178	+	0.802852i	$0.703315\pi$
$350$	0	0
$351$	0.869067	0.0463874
$352$	0	0
$353$	−9.47390	−0.504245	−0.252122	−	0.967695i	$-0.581129\pi$
$353$	−9.47390	−0.504245	−0.252122	+	0.967695i	$0.581129\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.86907	−0.0989215
$358$	0	0
$359$	−27.5853	−1.45590	−0.727949	−	0.685632i	$-0.759526\pi$
$359$	−27.5853	−1.45590	−0.727949	+	0.685632i	$0.759526\pi$
$360$	0	0
$361$	−18.2447	−0.960249
$362$	0	0
$363$	8.19394	0.430070
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.3199	−0.538697	−0.269348	−	0.963043i	$-0.586808\pi$
$367$	−10.3199	−0.538697	−0.269348	+	0.963043i	$0.586808\pi$
$368$	0	0
$369$	−10.5999	−0.551809
$370$	0	0
$371$	3.63752	0.188851
$372$	0	0
$373$	−25.9551	−1.34390	−0.671952	−	0.740595i	$-0.734544\pi$
$373$	−25.9551	−1.34390	−0.671952	+	0.740595i	$0.734544\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.12364	−0.109373
$378$	0	0
$379$	−7.84955	−0.403205	−0.201602	−	0.979467i	$-0.564615\pi$
$379$	−7.84955	−0.403205	−0.201602	+	0.979467i	$0.564615\pi$
$380$	0	0
$381$	2.99508	0.153442
$382$	0	0
$383$	−27.8300	−1.42205	−0.711024	−	0.703167i	$-0.751768\pi$
$383$	−27.8300	−1.42205	−0.711024	+	0.703167i	$0.751768\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.11871	0.361865
$388$	0	0
$389$	−16.3733	−0.830158	−0.415079	−	0.909785i	$-0.636246\pi$
$389$	−16.3733	−0.830158	−0.415079	+	0.909785i	$0.636246\pi$
$390$	0	0
$391$	−1.86907	−0.0945228
$392$	0	0
$393$	20.9380	1.05618
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−19.0884	−0.958019	−0.479010	−	0.877810i	$-0.659004\pi$
$397$	−19.0884	−0.958019	−0.479010	+	0.877810i	$0.659004\pi$
$398$	0	0
$399$	−0.869067	−0.0435078
$400$	0	0
$401$	−12.2619	−0.612328	−0.306164	−	0.951979i	$-0.599046\pi$
$401$	−12.2619	−0.612328	−0.306164	+	0.951979i	$0.599046\pi$
$402$	0	0
$403$	3.64481	0.181561
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.63752	0.229873
$408$	0	0
$409$	20.8251	1.02974	0.514868	−	0.857270i	$-0.327841\pi$
$409$	20.8251	1.02974	0.514868	+	0.857270i	$0.327841\pi$
$410$	0	0
$411$	−11.8945	−0.586710
$412$	0	0
$413$	3.48119	0.171298
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.35026	−0.310974
$418$	0	0
$419$	28.4363	1.38920	0.694602	−	0.719394i	$-0.255581\pi$
$419$	28.4363	1.38920	0.694602	+	0.719394i	$0.255581\pi$
$420$	0	0
$421$	18.2555	0.889720	0.444860	−	0.895600i	$-0.353253\pi$
$421$	18.2555	0.889720	0.444860	+	0.895600i	$0.353253\pi$
$422$	0	0
$423$	2.71274	0.131898
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.78067	−0.0861727
$428$	0	0
$429$	−1.45580	−0.0702868
$430$	0	0
$431$	6.62530	0.319130	0.159565	−	0.987187i	$-0.448991\pi$
$431$	6.62530	0.319130	0.159565	+	0.987187i	$0.448991\pi$
$432$	0	0
$433$	−2.93937	−0.141257	−0.0706284	−	0.997503i	$-0.522500\pi$
$433$	−2.93937	−0.141257	−0.0706284	+	0.997503i	$0.522500\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.869067	−0.0415731
$438$	0	0
$439$	−26.7572	−1.27705	−0.638525	−	0.769601i	$-0.720455\pi$
$439$	−26.7572	−1.27705	−0.638525	+	0.769601i	$0.720455\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	12.2398	0.581530	0.290765	−	0.956794i	$-0.406090\pi$
$443$	12.2398	0.581530	0.290765	+	0.956794i	$0.406090\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.67513	0.457618
$448$	0	0
$449$	−26.8315	−1.26625	−0.633127	−	0.774048i	$-0.718229\pi$
$449$	−26.8315	−1.26625	−0.633127	+	0.774048i	$0.718229\pi$
$450$	0	0
$451$	17.7562	0.836108
$452$	0	0
$453$	−2.15633	−0.101313
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.8568	−0.835308	−0.417654	−	0.908606i	$-0.637148\pi$
$457$	−17.8568	−0.835308	−0.417654	+	0.908606i	$0.637148\pi$
$458$	0	0
$459$	−1.86907	−0.0872406
$460$	0	0
$461$	−18.3127	−0.852905	−0.426453	−	0.904510i	$-0.640237\pi$
$461$	−18.3127	−0.852905	−0.426453	+	0.904510i	$0.640237\pi$
$462$	0	0
$463$	−28.7840	−1.33771	−0.668853	−	0.743395i	$-0.733214\pi$
$463$	−28.7840	−1.33771	−0.668853	+	0.743395i	$0.733214\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.7767	−0.776333	−0.388167	−	0.921589i	$-0.626891\pi$
$467$	−16.7767	−0.776333	−0.388167	+	0.921589i	$0.626891\pi$
$468$	0	0
$469$	−11.0811	−0.511678
$470$	0	0
$471$	−4.42548	−0.203916
$472$	0	0
$473$	−11.9248	−0.548302
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.63752	0.166551
$478$	0	0
$479$	11.6751	0.533450	0.266725	−	0.963773i	$-0.414058\pi$
$479$	11.6751	0.533450	0.266725	+	0.963773i	$0.414058\pi$
$480$	0	0
$481$	2.40597	0.109703
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.16854	0.0982661	0.0491331	−	0.998792i	$-0.484354\pi$
$487$	2.16854	0.0982661	0.0491331	+	0.998792i	$0.484354\pi$
$488$	0	0
$489$	−4.31265	−0.195025
$490$	0	0
$491$	−12.0155	−0.542254	−0.271127	−	0.962544i	$-0.587396\pi$
$491$	−12.0155	−0.542254	−0.271127	+	0.962544i	$0.587396\pi$
$492$	0	0
$493$	4.56722	0.205697
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.100615	−0.00451322
$498$	0	0
$499$	34.5936	1.54862	0.774310	−	0.632806i	$-0.218097\pi$
$499$	34.5936	1.54862	0.774310	+	0.632806i	$0.218097\pi$
$500$	0	0
$501$	−9.08603	−0.405934
$502$	0	0
$503$	5.35519	0.238776	0.119388	−	0.992848i	$-0.461907\pi$
$503$	5.35519	0.238776	0.119388	+	0.992848i	$0.461907\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.2447	0.543807
$508$	0	0
$509$	−6.74543	−0.298986	−0.149493	−	0.988763i	$-0.547764\pi$
$509$	−6.74543	−0.298986	−0.149493	+	0.988763i	$0.547764\pi$
$510$	0	0
$511$	−2.16854	−0.0959307
$512$	0	0
$513$	−0.869067	−0.0383703
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.54420	−0.199854
$518$	0	0
$519$	6.49929	0.285287
$520$	0	0
$521$	−12.2252	−0.535596	−0.267798	−	0.963475i	$-0.586296\pi$
$521$	−12.2252	−0.535596	−0.267798	+	0.963475i	$0.586296\pi$
$522$	0	0
$523$	3.51247	0.153589	0.0767947	−	0.997047i	$-0.475531\pi$
$523$	3.51247	0.153589	0.0767947	+	0.997047i	$0.475531\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.83875	−0.341461
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.48119	0.151071
$532$	0	0
$533$	9.21203	0.399018
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.54420	−0.109790
$538$	0	0
$539$	10.0508	0.432918
$540$	0	0
$541$	−6.56864	−0.282408	−0.141204	−	0.989981i	$-0.545097\pi$
$541$	−6.56864	−0.282408	−0.141204	+	0.989981i	$0.545097\pi$
$542$	0	0
$543$	8.12601	0.348720
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−21.9102	−0.936812	−0.468406	−	0.883513i	$-0.655172\pi$
$547$	−21.9102	−0.936812	−0.468406	+	0.883513i	$0.655172\pi$
$548$	0	0
$549$	−1.78067	−0.0759972
$550$	0	0
$551$	2.12364	0.0904700
$552$	0	0
$553$	5.08840	0.216381
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.6883	−1.51216	−0.756081	−	0.654478i	$-0.772888\pi$
$557$	−35.6883	−1.51216	−0.756081	+	0.654478i	$0.772888\pi$
$558$	0	0
$559$	−6.18664	−0.261667
$560$	0	0
$561$	3.13093	0.132188
$562$	0	0
$563$	38.5510	1.62473	0.812366	−	0.583148i	$-0.198179\pi$
$563$	38.5510	1.62473	0.812366	+	0.583148i	$0.198179\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−6.27645	−0.263123	−0.131561	−	0.991308i	$-0.541999\pi$
$569$	−6.27645	−0.263123	−0.131561	+	0.991308i	$0.541999\pi$
$570$	0	0
$571$	−22.4323	−0.938763	−0.469382	−	0.882995i	$-0.655523\pi$
$571$	−22.4323	−0.938763	−0.469382	+	0.882995i	$0.655523\pi$
$572$	0	0
$573$	7.28726	0.304430
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.9525	−1.03879	−0.519394	−	0.854535i	$-0.673842\pi$
$577$	−24.9525	−1.03879	−0.519394	+	0.854535i	$0.673842\pi$
$578$	0	0
$579$	13.5877	0.564685
$580$	0	0
$581$	9.09332	0.377255
$582$	0	0
$583$	−6.09332	−0.252360
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.68735	0.0696444	0.0348222	−	0.999394i	$-0.488914\pi$
$587$	1.68735	0.0696444	0.0348222	+	0.999394i	$0.488914\pi$
$588$	0	0
$589$	−3.64481	−0.150182
$590$	0	0
$591$	−16.9380	−0.696734
$592$	0	0
$593$	−37.7523	−1.55030	−0.775150	−	0.631777i	$-0.782326\pi$
$593$	−37.7523	−1.55030	−0.775150	+	0.631777i	$0.782326\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.4241	−0.672192
$598$	0	0
$599$	−13.5271	−0.552700	−0.276350	−	0.961057i	$-0.589125\pi$
$599$	−13.5271	−0.552700	−0.276350	+	0.961057i	$0.589125\pi$
$600$	0	0
$601$	−29.4544	−1.20147	−0.600735	−	0.799448i	$-0.705125\pi$
$601$	−29.4544	−1.20147	−0.600735	+	0.799448i	$0.705125\pi$
$602$	0	0
$603$	−11.0811	−0.451257
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.6942	1.16466	0.582331	−	0.812952i	$-0.302141\pi$
$607$	28.6942	1.16466	0.582331	+	0.812952i	$0.302141\pi$
$608$	0	0
$609$	−2.44358	−0.0990190
$610$	0	0
$611$	−2.35756	−0.0953765
$612$	0	0
$613$	−22.9018	−0.924993	−0.462497	−	0.886621i	$-0.653046\pi$
$613$	−22.9018	−0.924993	−0.462497	+	0.886621i	$0.653046\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.0411	−0.967859	−0.483930	−	0.875107i	$-0.660791\pi$
$617$	−24.0411	−0.967859	−0.483930	+	0.875107i	$0.660791\pi$
$618$	0	0
$619$	34.5647	1.38927	0.694636	−	0.719362i	$-0.255566\pi$
$619$	34.5647	1.38927	0.694636	+	0.719362i	$0.255566\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	7.92478	0.317499
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.45580	0.0581391
$628$	0	0
$629$	−5.17442	−0.206318
$630$	0	0
$631$	−29.8677	−1.18901	−0.594506	−	0.804091i	$-0.702652\pi$
$631$	−29.8677	−1.18901	−0.594506	+	0.804091i	$0.702652\pi$
$632$	0	0
$633$	20.1490	0.800852
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.21440	0.206602
$638$	0	0
$639$	−0.100615	−0.00398028
$640$	0	0
$641$	−30.9257	−1.22149	−0.610746	−	0.791826i	$-0.709131\pi$
$641$	−30.9257	−1.22149	−0.610746	+	0.791826i	$0.709131\pi$
$642$	0	0
$643$	−36.2955	−1.43136	−0.715678	−	0.698431i	$-0.753882\pi$
$643$	−36.2955	−1.43136	−0.715678	+	0.698431i	$0.753882\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.67513	0.223112	0.111556	−	0.993758i	$-0.464416\pi$
$647$	5.67513	0.223112	0.111556	+	0.993758i	$0.464416\pi$
$648$	0	0
$649$	−5.83146	−0.228905
$650$	0	0
$651$	4.19394	0.164373
$652$	0	0
$653$	50.7245	1.98500	0.992502	−	0.122232i	$-0.0390052\pi$
$653$	50.7245	1.98500	0.992502	+	0.122232i	$0.0390052\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.16854	−0.0846030
$658$	0	0
$659$	23.4641	0.914030	0.457015	−	0.889459i	$-0.348919\pi$
$659$	23.4641	0.914030	0.457015	+	0.889459i	$0.348919\pi$
$660$	0	0
$661$	−18.9135	−0.735650	−0.367825	−	0.929895i	$-0.619898\pi$
$661$	−18.9135	−0.735650	−0.367825	+	0.929895i	$0.619898\pi$
$662$	0	0
$663$	1.62435	0.0630844
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.44358	−0.0946159
$668$	0	0
$669$	17.9248	0.693012
$670$	0	0
$671$	2.98286	0.115152
$672$	0	0
$673$	−15.8559	−0.611200	−0.305600	−	0.952160i	$-0.598857\pi$
$673$	−15.8559	−0.611200	−0.305600	+	0.952160i	$0.598857\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.52469	−0.0970315	−0.0485158	−	0.998822i	$-0.515449\pi$
$677$	−2.52469	−0.0970315	−0.0485158	+	0.998822i	$0.515449\pi$
$678$	0	0
$679$	4.99271	0.191602
$680$	0	0
$681$	5.14903	0.197311
$682$	0	0
$683$	24.5477	0.939292	0.469646	−	0.882855i	$-0.344382\pi$
$683$	24.5477	0.939292	0.469646	+	0.882855i	$0.344382\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.7513	0.791712
$688$	0	0
$689$	−3.16125	−0.120434
$690$	0	0
$691$	23.7685	0.904195	0.452097	−	0.891969i	$-0.350676\pi$
$691$	23.7685	0.904195	0.452097	+	0.891969i	$0.350676\pi$
$692$	0	0
$693$	−1.67513	−0.0636330
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.8119	−0.750431
$698$	0	0
$699$	14.1768	0.536216
$700$	0	0
$701$	11.9633	0.451849	0.225925	−	0.974145i	$-0.427460\pi$
$701$	11.9633	0.451849	0.225925	+	0.974145i	$0.427460\pi$
$702$	0	0
$703$	−2.40597	−0.0907429
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.28726	−0.311674
$708$	0	0
$709$	29.4093	1.10449	0.552245	−	0.833682i	$-0.313771\pi$
$709$	29.4093	1.10449	0.552245	+	0.833682i	$0.313771\pi$
$710$	0	0
$711$	5.08840	0.190830
$712$	0	0
$713$	4.19394	0.157064
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.26423	0.0472137
$718$	0	0
$719$	−25.6190	−0.955426	−0.477713	−	0.878516i	$-0.658534\pi$
$719$	−25.6190	−0.955426	−0.477713	+	0.878516i	$0.658534\pi$
$720$	0	0
$721$	−2.80606	−0.104503
$722$	0	0
$723$	−6.45676	−0.240129
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−31.4894	−1.16788	−0.583939	−	0.811797i	$-0.698489\pi$
$727$	−31.4894	−1.16788	−0.583939	+	0.811797i	$0.698489\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.3054	0.492116
$732$	0	0
$733$	−37.0059	−1.36684	−0.683422	−	0.730024i	$-0.739509\pi$
$733$	−37.0059	−1.36684	−0.683422	+	0.730024i	$0.739509\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.5623	0.683751
$738$	0	0
$739$	15.7816	0.580536	0.290268	−	0.956945i	$-0.406255\pi$
$739$	15.7816	0.580536	0.290268	+	0.956945i	$0.406255\pi$
$740$	0	0
$741$	0.755278	0.0277458
$742$	0	0
$743$	45.9657	1.68632	0.843159	−	0.537664i	$-0.180693\pi$
$743$	45.9657	1.68632	0.843159	+	0.537664i	$0.180693\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.09332	0.332707
$748$	0	0
$749$	3.45088	0.126092
$750$	0	0
$751$	−15.3538	−0.560267	−0.280134	−	0.959961i	$-0.590379\pi$
$751$	−15.3538	−0.560267	−0.280134	+	0.959961i	$0.590379\pi$
$752$	0	0
$753$	−7.66291	−0.279252
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−24.6385	−0.895501	−0.447750	−	0.894159i	$-0.647775\pi$
$757$	−24.6385	−0.895501	−0.447750	+	0.894159i	$0.647775\pi$
$758$	0	0
$759$	−1.67513	−0.0608034
$760$	0	0
$761$	11.0787	0.401604	0.200802	−	0.979632i	$-0.435645\pi$
$761$	11.0787	0.401604	0.200802	+	0.979632i	$0.435645\pi$
$762$	0	0
$763$	−13.3684	−0.483967
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.02539	−0.109241
$768$	0	0
$769$	22.3961	0.807625	0.403812	−	0.914842i	$-0.367685\pi$
$769$	22.3961	0.807625	0.403812	+	0.914842i	$0.367685\pi$
$770$	0	0
$771$	−9.65069	−0.347561
$772$	0	0
$773$	33.7645	1.21442	0.607212	−	0.794540i	$-0.292288\pi$
$773$	33.7645	1.21442	0.607212	+	0.794540i	$0.292288\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.76845	0.0993177
$778$	0	0
$779$	−9.21203	−0.330055
$780$	0	0
$781$	0.168544	0.00603098
$782$	0	0
$783$	−2.44358	−0.0873265
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.7718	0.918665	0.459332	−	0.888265i	$-0.348089\pi$
$787$	25.7718	0.918665	0.459332	+	0.888265i	$0.348089\pi$
$788$	0	0
$789$	−15.8994	−0.566033
$790$	0	0
$791$	−8.98778	−0.319569
$792$	0	0
$793$	1.54752	0.0549542
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.9633	1.16762	0.583811	−	0.811890i	$-0.301561\pi$
$797$	32.9633	1.16762	0.583811	+	0.811890i	$0.301561\pi$
$798$	0	0
$799$	5.07030	0.179374
$800$	0	0
$801$	7.92478	0.280008
$802$	0	0
$803$	3.63259	0.128191
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.85448	−0.276491
$808$	0	0
$809$	27.6639	0.972610	0.486305	−	0.873789i	$-0.338344\pi$
$809$	27.6639	0.972610	0.486305	+	0.873789i	$0.338344\pi$
$810$	0	0
$811$	13.2228	0.464317	0.232158	−	0.972678i	$-0.425421\pi$
$811$	13.2228	0.464317	0.232158	+	0.972678i	$0.425421\pi$
$812$	0	0
$813$	9.96239	0.349396
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.18664	0.216443
$818$	0	0
$819$	−0.869067	−0.0303677
$820$	0	0
$821$	−10.1319	−0.353605	−0.176803	−	0.984246i	$-0.556575\pi$
$821$	−10.1319	−0.353605	−0.176803	+	0.984246i	$0.556575\pi$
$822$	0	0
$823$	−13.9394	−0.485896	−0.242948	−	0.970039i	$-0.578114\pi$
$823$	−13.9394	−0.485896	−0.242948	+	0.970039i	$0.578114\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.18901	−0.180440	−0.0902198	−	0.995922i	$-0.528757\pi$
$827$	−5.18901	−0.180440	−0.0902198	+	0.995922i	$0.528757\pi$
$828$	0	0
$829$	29.1173	1.01129	0.505643	−	0.862743i	$-0.331255\pi$
$829$	29.1173	1.01129	0.505643	+	0.862743i	$0.331255\pi$
$830$	0	0
$831$	−5.61213	−0.194683
$832$	0	0
$833$	−11.2144	−0.388556
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.19394	0.144964
$838$	0	0
$839$	−24.2882	−0.838522	−0.419261	−	0.907866i	$-0.637711\pi$
$839$	−24.2882	−0.838522	−0.419261	+	0.907866i	$0.637711\pi$
$840$	0	0
$841$	−23.0289	−0.794100
$842$	0	0
$843$	6.96002	0.239716
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.19394	−0.281547
$848$	0	0
$849$	−9.05079	−0.310622
$850$	0	0
$851$	2.76845	0.0949013
$852$	0	0
$853$	15.8397	0.542341	0.271171	−	0.962531i	$-0.412589\pi$
$853$	15.8397	0.542341	0.271171	+	0.962531i	$0.412589\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.2605	−0.931199	−0.465600	−	0.884995i	$-0.654161\pi$
$857$	−27.2605	−0.931199	−0.465600	+	0.884995i	$0.654161\pi$
$858$	0	0
$859$	22.5223	0.768451	0.384226	−	0.923239i	$-0.374468\pi$
$859$	22.5223	0.768451	0.384226	+	0.923239i	$0.374468\pi$
$860$	0	0
$861$	10.5999	0.361244
$862$	0	0
$863$	45.5026	1.54893	0.774464	−	0.632619i	$-0.218020\pi$
$863$	45.5026	1.54893	0.774464	+	0.632619i	$0.218020\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.5066	0.458708
$868$	0	0
$869$	−8.52373	−0.289148
$870$	0	0
$871$	9.63023	0.326308
$872$	0	0
$873$	4.99271	0.168977
$874$	0	0
$875$	0	0
$876$	0	0
$877$	58.0019	1.95859	0.979293	−	0.202450i	$-0.0648903\pi$
$877$	58.0019	1.95859	0.979293	+	0.202450i	$0.0648903\pi$
$878$	0	0
$879$	2.18172	0.0735875
$880$	0	0
$881$	26.2351	0.883882	0.441941	−	0.897044i	$-0.354290\pi$
$881$	26.2351	0.883882	0.441941	+	0.897044i	$0.354290\pi$
$882$	0	0
$883$	−27.8578	−0.937490	−0.468745	−	0.883334i	$-0.655294\pi$
$883$	−27.8578	−0.937490	−0.468745	+	0.883334i	$0.655294\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.1866	0.744955	0.372477	−	0.928041i	$-0.378508\pi$
$887$	22.1866	0.744955	0.372477	+	0.928041i	$0.378508\pi$
$888$	0	0
$889$	−2.99508	−0.100452
$890$	0	0
$891$	−1.67513	−0.0561190
$892$	0	0
$893$	2.35756	0.0788926
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.869067	−0.0290173
$898$	0	0
$899$	−10.2482	−0.341798
$900$	0	0
$901$	6.79877	0.226500
$902$	0	0
$903$	−7.11871	−0.236896
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22.0943	−0.733628	−0.366814	−	0.930294i	$-0.619552\pi$
$907$	−22.0943	−0.733628	−0.366814	+	0.930294i	$0.619552\pi$
$908$	0	0
$909$	−8.28726	−0.274871
$910$	0	0
$911$	54.0381	1.79036	0.895181	−	0.445702i	$-0.147046\pi$
$911$	54.0381	1.79036	0.895181	+	0.445702i	$0.147046\pi$
$912$	0	0
$913$	−15.2325	−0.504122
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.9380	−0.691432
$918$	0	0
$919$	23.9854	0.791206	0.395603	−	0.918422i	$-0.370536\pi$
$919$	23.9854	0.791206	0.395603	+	0.918422i	$0.370536\pi$
$920$	0	0
$921$	−4.91985	−0.162115
$922$	0	0
$923$	0.0874416	0.00287817
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.80606	−0.0921632
$928$	0	0
$929$	−11.8183	−0.387745	−0.193873	−	0.981027i	$-0.562105\pi$
$929$	−11.8183	−0.387745	−0.193873	+	0.981027i	$0.562105\pi$
$930$	0	0
$931$	−5.21440	−0.170895
$932$	0	0
$933$	14.2619	0.466913
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.70782	0.186466	0.0932331	−	0.995644i	$-0.470280\pi$
$937$	5.70782	0.186466	0.0932331	+	0.995644i	$0.470280\pi$
$938$	0	0
$939$	4.32979	0.141297
$940$	0	0
$941$	25.0616	0.816984	0.408492	−	0.912762i	$-0.366055\pi$
$941$	25.0616	0.816984	0.408492	+	0.912762i	$0.366055\pi$
$942$	0	0
$943$	10.5999	0.345180
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.7626	0.512215	0.256107	−	0.966648i	$-0.417560\pi$
$947$	15.7626	0.512215	0.256107	+	0.966648i	$0.417560\pi$
$948$	0	0
$949$	1.88461	0.0611771
$950$	0	0
$951$	−31.0616	−1.00724
$952$	0	0
$953$	10.2520	0.332095	0.166048	−	0.986118i	$-0.446899\pi$
$953$	10.2520	0.332095	0.166048	+	0.986118i	$0.446899\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.09332	0.132318
$958$	0	0
$959$	11.8945	0.384092
$960$	0	0
$961$	−13.4109	−0.432610
$962$	0	0
$963$	3.45088	0.111203
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.5075	0.466531	0.233266	−	0.972413i	$-0.425059\pi$
$967$	14.5075	0.466531	0.233266	+	0.972413i	$0.425059\pi$
$968$	0	0
$969$	−1.62435	−0.0521815
$970$	0	0
$971$	53.2144	1.70773	0.853866	−	0.520493i	$-0.174252\pi$
$971$	53.2144	1.70773	0.853866	+	0.520493i	$0.174252\pi$
$972$	0	0
$973$	6.35026	0.203580
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.0590	−0.737724	−0.368862	−	0.929484i	$-0.620252\pi$
$977$	−23.0590	−0.737724	−0.368862	+	0.929484i	$0.620252\pi$
$978$	0	0
$979$	−13.2750	−0.424272
$980$	0	0
$981$	−13.3684	−0.426819
$982$	0	0
$983$	14.7708	0.471116	0.235558	−	0.971860i	$-0.424308\pi$
$983$	14.7708	0.471116	0.235558	+	0.971860i	$0.424308\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.71274	−0.0863475
$988$	0	0
$989$	−7.11871	−0.226362
$990$	0	0
$991$	52.8905	1.68012	0.840061	−	0.542492i	$-0.182519\pi$
$991$	52.8905	1.68012	0.840061	+	0.542492i	$0.182519\pi$
$992$	0	0
$993$	−3.80606	−0.120782
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.87399	0.186031	0.0930156	−	0.995665i	$-0.470349\pi$
$997$	5.87399	0.186031	0.0930156	+	0.995665i	$0.470349\pi$
$998$	0	0
$999$	2.76845	0.0875899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.y.1.1		3
5.2	odd	4		1380.2.f.a.829.4	yes	6
5.3	odd	4		1380.2.f.a.829.1	✓	6
5.4	even	2		6900.2.a.z.1.1		3
15.2	even	4		4140.2.f.a.829.5		6
15.8	even	4		4140.2.f.a.829.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.a.829.1	✓	6	5.3	odd	4
1380.2.f.a.829.4	yes	6	5.2	odd	4
4140.2.f.a.829.5		6	15.2	even	4
4140.2.f.a.829.6		6	15.8	even	4
6900.2.a.y.1.1		3	1.1	even	1	trivial
6900.2.a.z.1.1		3	5.4	even	2

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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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -1.67513 q^{11} -0.869067 q^{13} +1.86907 q^{17} +0.869067 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{27} +2.44358 q^{29} -4.19394 q^{31} +1.67513 q^{33} -2.76845 q^{37} +0.869067 q^{39} -10.5999 q^{41} +7.11871 q^{43} +2.71274 q^{47} -6.00000 q^{49} -1.86907 q^{51} +3.63752 q^{53} -0.869067 q^{57} +3.48119 q^{59} -1.78067 q^{61} +1.00000 q^{63} -11.0811 q^{67} +1.00000 q^{69} -0.100615 q^{71} -2.16854 q^{73} -1.67513 q^{77} +5.08840 q^{79} +1.00000 q^{81} +9.09332 q^{83} -2.44358 q^{87} +7.92478 q^{89} -0.869067 q^{91} +4.19394 q^{93} +4.99271 q^{97} -1.67513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{13} + q^{17} - 2 q^{19} - 3 q^{21} - 3 q^{23} - 3 q^{27} - 9 q^{29} - 13 q^{31} + 3 q^{37} - 2 q^{39} - 5 q^{41} + 14 q^{47} - 18 q^{49} - q^{51} - 5 q^{53} + 2 q^{57} + 5 q^{59} - 20 q^{61} + 3 q^{63} - q^{67} + 3 q^{69} - 7 q^{71} - 22 q^{73} - 4 q^{79} + 3 q^{81} + 21 q^{83} + 9 q^{87} + 2 q^{89} + 2 q^{91} + 13 q^{93} + 2 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^13 + q^17 - 2 * q^19 - 3 * q^21 - 3 * q^23 - 3 * q^27 - 9 * q^29 - 13 * q^31 + 3 * q^37 - 2 * q^39 - 5 * q^41 + 14 * q^47 - 18 * q^49 - q^51 - 5 * q^53 + 2 * q^57 + 5 * q^59 - 20 * q^61 + 3 * q^63 - q^67 + 3 * q^69 - 7 * q^71 - 22 * q^73 - 4 * q^79 + 3 * q^81 + 21 * q^83 + 9 * q^87 + 2 * q^89 + 2 * q^91 + 13 * q^93 + 2 * q^97

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