LMFDB - Embedded newform 3332.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.30278 q^{3} -0.302776 q^{5} +7.90833 q^{9} +O(q^{10})$	$q+3.30278 q^{3} -0.302776 q^{5} +7.90833 q^{9} -2.60555 q^{11} -0.605551 q^{13} -1.00000 q^{15} +1.00000 q^{17} +6.00000 q^{19} +4.60555 q^{23} -4.90833 q^{25} +16.2111 q^{27} -1.39445 q^{29} +10.3028 q^{31} -8.60555 q^{33} -7.21110 q^{37} -2.00000 q^{39} +8.51388 q^{41} -0.697224 q^{43} -2.39445 q^{45} -10.0000 q^{47} +3.30278 q^{51} +4.30278 q^{53} +0.788897 q^{55} +19.8167 q^{57} +9.21110 q^{59} +0.697224 q^{61} +0.183346 q^{65} +6.30278 q^{67} +15.2111 q^{69} -2.00000 q^{71} -4.51388 q^{73} -16.2111 q^{75} -6.00000 q^{79} +29.8167 q^{81} +5.21110 q^{83} -0.302776 q^{85} -4.60555 q^{87} +2.00000 q^{89} +34.0278 q^{93} -1.81665 q^{95} +12.9083 q^{97} -20.6056 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9	$ 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 12 q^{19} + 2 q^{23} + q^{25} + 18 q^{27} - 10 q^{29} + 17 q^{31} - 10 q^{33} - 4 q^{39} - q^{41} - 5 q^{43} - 12 q^{45} - 20 q^{47} + 3 q^{51} + 5 q^{53} + 16 q^{55} + 18 q^{57} + 4 q^{59} + 5 q^{61} + 22 q^{65} + 9 q^{67} + 16 q^{69} - 4 q^{71} + 9 q^{73} - 18 q^{75} - 12 q^{79} + 38 q^{81} - 4 q^{83} + 3 q^{85} - 2 q^{87} + 4 q^{89} + 32 q^{93} + 18 q^{95} + 15 q^{97} - 34 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 + 2 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 12 * q^19 + 2 * q^23 + q^25 + 18 * q^27 - 10 * q^29 + 17 * q^31 - 10 * q^33 - 4 * q^39 - q^41 - 5 * q^43 - 12 * q^45 - 20 * q^47 + 3 * q^51 + 5 * q^53 + 16 * q^55 + 18 * q^57 + 4 * q^59 + 5 * q^61 + 22 * q^65 + 9 * q^67 + 16 * q^69 - 4 * q^71 + 9 * q^73 - 18 * q^75 - 12 * q^79 + 38 * q^81 - 4 * q^83 + 3 * q^85 - 2 * q^87 + 4 * q^89 + 32 * q^93 + 18 * q^95 + 15 * q^97 - 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.30278	1.90686	0.953429	−	0.301617i	$-0.0975264\pi$
$3$	3.30278	1.90686	0.953429	+	0.301617i	$0.0975264\pi$
$4$	0	0
$5$	−0.302776	−0.135405	−0.0677027	−	0.997706i	$-0.521567\pi$
$5$	−0.302776	−0.135405	−0.0677027	+	0.997706i	$0.521567\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	−2.60555	−0.785603	−0.392802	−	0.919623i	$-0.628494\pi$
$11$	−2.60555	−0.785603	−0.392802	+	0.919623i	$0.628494\pi$
$12$	0	0
$13$	−0.605551	−0.167950	−0.0839749	−	0.996468i	$-0.526762\pi$
$13$	−0.605551	−0.167950	−0.0839749	+	0.996468i	$0.526762\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.60555	0.960324	0.480162	−	0.877180i	$-0.340578\pi$
$23$	4.60555	0.960324	0.480162	+	0.877180i	$0.340578\pi$
$24$	0	0
$25$	−4.90833	−0.981665
$26$	0	0
$27$	16.2111	3.11983
$28$	0	0
$29$	−1.39445	−0.258943	−0.129471	−	0.991583i	$-0.541328\pi$
$29$	−1.39445	−0.258943	−0.129471	+	0.991583i	$0.541328\pi$
$30$	0	0
$31$	10.3028	1.85043	0.925217	−	0.379439i	$-0.123883\pi$
$31$	10.3028	1.85043	0.925217	+	0.379439i	$0.123883\pi$
$32$	0	0
$33$	−8.60555	−1.49803
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.21110	−1.18550	−0.592749	−	0.805387i	$-0.701957\pi$
$37$	−7.21110	−1.18550	−0.592749	+	0.805387i	$0.701957\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	8.51388	1.32964	0.664822	−	0.747002i	$-0.268507\pi$
$41$	8.51388	1.32964	0.664822	+	0.747002i	$0.268507\pi$
$42$	0	0
$43$	−0.697224	−0.106326	−0.0531629	−	0.998586i	$-0.516930\pi$
$43$	−0.697224	−0.106326	−0.0531629	+	0.998586i	$0.516930\pi$
$44$	0	0
$45$	−2.39445	−0.356943
$46$	0	0
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.30278	0.462481
$52$	0	0
$53$	4.30278	0.591032	0.295516	−	0.955338i	$-0.404508\pi$
$53$	4.30278	0.591032	0.295516	+	0.955338i	$0.404508\pi$
$54$	0	0
$55$	0.788897	0.106375
$56$	0	0
$57$	19.8167	2.62478
$58$	0	0
$59$	9.21110	1.19918	0.599592	−	0.800306i	$-0.295330\pi$
$59$	9.21110	1.19918	0.599592	+	0.800306i	$0.295330\pi$
$60$	0	0
$61$	0.697224	0.0892704	0.0446352	−	0.999003i	$-0.485787\pi$
$61$	0.697224	0.0892704	0.0446352	+	0.999003i	$0.485787\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.183346	0.0227413
$66$	0	0
$67$	6.30278	0.770007	0.385003	−	0.922915i	$-0.374200\pi$
$67$	6.30278	0.770007	0.385003	+	0.922915i	$0.374200\pi$
$68$	0	0
$69$	15.2111	1.83120
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−4.51388	−0.528309	−0.264155	−	0.964480i	$-0.585093\pi$
$73$	−4.51388	−0.528309	−0.264155	+	0.964480i	$0.585093\pi$
$74$	0	0
$75$	−16.2111	−1.87190
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	5.21110	0.571993	0.285996	−	0.958231i	$-0.407675\pi$
$83$	5.21110	0.571993	0.285996	+	0.958231i	$0.407675\pi$
$84$	0	0
$85$	−0.302776	−0.0328406
$86$	0	0
$87$	−4.60555	−0.493767
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	34.0278	3.52851
$94$	0	0
$95$	−1.81665	−0.186385
$96$	0	0
$97$	12.9083	1.31064	0.655321	−	0.755350i	$-0.272533\pi$
$97$	12.9083	1.31064	0.655321	+	0.755350i	$0.272533\pi$
$98$	0	0
$99$	−20.6056	−2.07094
$100$	0	0
$101$	−16.4222	−1.63407	−0.817035	−	0.576588i	$-0.804384\pi$
$101$	−16.4222	−1.63407	−0.817035	+	0.576588i	$0.804384\pi$
$102$	0	0
$103$	1.21110	0.119333	0.0596667	−	0.998218i	$-0.480996\pi$
$103$	1.21110	0.119333	0.0596667	+	0.998218i	$0.480996\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.6056	−1.60532	−0.802660	−	0.596437i	$-0.796582\pi$
$107$	−16.6056	−1.60532	−0.802660	+	0.596437i	$0.796582\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−23.8167	−2.26058
$112$	0	0
$113$	4.78890	0.450502	0.225251	−	0.974301i	$-0.427680\pi$
$113$	4.78890	0.450502	0.225251	+	0.974301i	$0.427680\pi$
$114$	0	0
$115$	−1.39445	−0.130033
$116$	0	0
$117$	−4.78890	−0.442734
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.21110	−0.382828
$122$	0	0
$123$	28.1194	2.53544
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	20.1194	1.78531	0.892655	−	0.450740i	$-0.148840\pi$
$127$	20.1194	1.78531	0.892655	+	0.450740i	$0.148840\pi$
$128$	0	0
$129$	−2.30278	−0.202748
$130$	0	0
$131$	−18.4222	−1.60956	−0.804778	−	0.593576i	$-0.797716\pi$
$131$	−18.4222	−1.60956	−0.804778	+	0.593576i	$0.797716\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.90833	−0.422442
$136$	0	0
$137$	−13.9083	−1.18827	−0.594134	−	0.804366i	$-0.702505\pi$
$137$	−13.9083	−1.18827	−0.594134	+	0.804366i	$0.702505\pi$
$138$	0	0
$139$	19.5139	1.65515	0.827573	−	0.561358i	$-0.189721\pi$
$139$	19.5139	1.65515	0.827573	+	0.561358i	$0.189721\pi$
$140$	0	0
$141$	−33.0278	−2.78144
$142$	0	0
$143$	1.57779	0.131942
$144$	0	0
$145$	0.422205	0.0350622
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.90833	−0.484029	−0.242015	−	0.970273i	$-0.577808\pi$
$149$	−5.90833	−0.484029	−0.242015	+	0.970273i	$0.577808\pi$
$150$	0	0
$151$	−11.1194	−0.904886	−0.452443	−	0.891793i	$-0.649447\pi$
$151$	−11.1194	−0.904886	−0.452443	+	0.891793i	$0.649447\pi$
$152$	0	0
$153$	7.90833	0.639350
$154$	0	0
$155$	−3.11943	−0.250559
$156$	0	0
$157$	21.8167	1.74116	0.870579	−	0.492028i	$-0.163744\pi$
$157$	21.8167	1.74116	0.870579	+	0.492028i	$0.163744\pi$
$158$	0	0
$159$	14.2111	1.12701
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.183346	−0.0143608	−0.00718039	−	0.999974i	$-0.502286\pi$
$163$	−0.183346	−0.0143608	−0.00718039	+	0.999974i	$0.502286\pi$
$164$	0	0
$165$	2.60555	0.202842
$166$	0	0
$167$	−23.1194	−1.78904	−0.894518	−	0.447033i	$-0.852481\pi$
$167$	−23.1194	−1.78904	−0.894518	+	0.447033i	$0.852481\pi$
$168$	0	0
$169$	−12.6333	−0.971793
$170$	0	0
$171$	47.4500	3.62859
$172$	0	0
$173$	5.90833	0.449202	0.224601	−	0.974451i	$-0.427892\pi$
$173$	5.90833	0.449202	0.224601	+	0.974451i	$0.427892\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	30.4222	2.28667
$178$	0	0
$179$	21.9083	1.63751	0.818753	−	0.574146i	$-0.194666\pi$
$179$	21.9083	1.63751	0.818753	+	0.574146i	$0.194666\pi$
$180$	0	0
$181$	−21.6333	−1.60799	−0.803996	−	0.594635i	$-0.797296\pi$
$181$	−21.6333	−1.60799	−0.803996	+	0.594635i	$0.797296\pi$
$182$	0	0
$183$	2.30278	0.170226
$184$	0	0
$185$	2.18335	0.160523
$186$	0	0
$187$	−2.60555	−0.190537
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.72498	0.631317	0.315659	−	0.948873i	$-0.397775\pi$
$191$	8.72498	0.631317	0.315659	+	0.948873i	$0.397775\pi$
$192$	0	0
$193$	−12.6056	−0.907367	−0.453684	−	0.891163i	$-0.649890\pi$
$193$	−12.6056	−0.907367	−0.453684	+	0.891163i	$0.649890\pi$
$194$	0	0
$195$	0.605551	0.0433644
$196$	0	0
$197$	13.0278	0.928189	0.464095	−	0.885786i	$-0.346380\pi$
$197$	13.0278	0.928189	0.464095	+	0.885786i	$0.346380\pi$
$198$	0	0
$199$	4.51388	0.319980	0.159990	−	0.987119i	$-0.448854\pi$
$199$	4.51388	0.319980	0.159990	+	0.987119i	$0.448854\pi$
$200$	0	0
$201$	20.8167	1.46829
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.57779	−0.180041
$206$	0	0
$207$	36.4222	2.53152
$208$	0	0
$209$	−15.6333	−1.08138
$210$	0	0
$211$	9.02776	0.621496	0.310748	−	0.950492i	$-0.399420\pi$
$211$	9.02776	0.621496	0.310748	+	0.950492i	$0.399420\pi$
$212$	0	0
$213$	−6.60555	−0.452605
$214$	0	0
$215$	0.211103	0.0143971
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−14.9083	−1.00741
$220$	0	0
$221$	−0.605551	−0.0407338
$222$	0	0
$223$	−21.6333	−1.44867	−0.724337	−	0.689446i	$-0.757854\pi$
$223$	−21.6333	−1.44867	−0.724337	+	0.689446i	$0.757854\pi$
$224$	0	0
$225$	−38.8167	−2.58778
$226$	0	0
$227$	−10.9083	−0.724011	−0.362006	−	0.932176i	$-0.617908\pi$
$227$	−10.9083	−0.724011	−0.362006	+	0.932176i	$0.617908\pi$
$228$	0	0
$229$	−15.8167	−1.04519	−0.522597	−	0.852580i	$-0.675037\pi$
$229$	−15.8167	−1.04519	−0.522597	+	0.852580i	$0.675037\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.21110	−0.0793420	−0.0396710	−	0.999213i	$-0.512631\pi$
$233$	−1.21110	−0.0793420	−0.0396710	+	0.999213i	$0.512631\pi$
$234$	0	0
$235$	3.02776	0.197509
$236$	0	0
$237$	−19.8167	−1.28723
$238$	0	0
$239$	−8.69722	−0.562577	−0.281288	−	0.959623i	$-0.590762\pi$
$239$	−8.69722	−0.562577	−0.281288	+	0.959623i	$0.590762\pi$
$240$	0	0
$241$	−3.09167	−0.199152	−0.0995761	−	0.995030i	$-0.531749\pi$
$241$	−3.09167	−0.199152	−0.0995761	+	0.995030i	$0.531749\pi$
$242$	0	0
$243$	49.8444	3.19752
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.63331	−0.231182
$248$	0	0
$249$	17.2111	1.09071
$250$	0	0
$251$	−23.2111	−1.46507	−0.732536	−	0.680728i	$-0.761663\pi$
$251$	−23.2111	−1.46507	−0.732536	+	0.680728i	$0.761663\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	−7.81665	−0.487589	−0.243795	−	0.969827i	$-0.578392\pi$
$257$	−7.81665	−0.487589	−0.243795	+	0.969827i	$0.578392\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11.0278	−0.682601
$262$	0	0
$263$	−6.42221	−0.396010	−0.198005	−	0.980201i	$-0.563446\pi$
$263$	−6.42221	−0.396010	−0.198005	+	0.980201i	$0.563446\pi$
$264$	0	0
$265$	−1.30278	−0.0800289
$266$	0	0
$267$	6.60555	0.404253
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	10.1833	0.618594	0.309297	−	0.950965i	$-0.399906\pi$
$271$	10.1833	0.618594	0.309297	+	0.950965i	$0.399906\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.7889	0.771200
$276$	0	0
$277$	−11.0278	−0.662594	−0.331297	−	0.943527i	$-0.607486\pi$
$277$	−11.0278	−0.662594	−0.331297	+	0.943527i	$0.607486\pi$
$278$	0	0
$279$	81.4777	4.87794
$280$	0	0
$281$	−20.3028	−1.21116	−0.605581	−	0.795784i	$-0.707059\pi$
$281$	−20.3028	−1.21116	−0.605581	+	0.795784i	$0.707059\pi$
$282$	0	0
$283$	−10.3028	−0.612436	−0.306218	−	0.951961i	$-0.599064\pi$
$283$	−10.3028	−0.612436	−0.306218	+	0.951961i	$0.599064\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	42.6333	2.49921
$292$	0	0
$293$	19.8167	1.15770	0.578851	−	0.815434i	$-0.303501\pi$
$293$	19.8167	1.15770	0.578851	+	0.815434i	$0.303501\pi$
$294$	0	0
$295$	−2.78890	−0.162376
$296$	0	0
$297$	−42.2389	−2.45095
$298$	0	0
$299$	−2.78890	−0.161286
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−54.2389	−3.11594
$304$	0	0
$305$	−0.211103	−0.0120877
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−16.7250	−0.948387	−0.474193	−	0.880421i	$-0.657260\pi$
$311$	−16.7250	−0.948387	−0.474193	+	0.880421i	$0.657260\pi$
$312$	0	0
$313$	−26.7250	−1.51059	−0.755293	−	0.655388i	$-0.772505\pi$
$313$	−26.7250	−1.51059	−0.755293	+	0.655388i	$0.772505\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.39445	−0.302982	−0.151491	−	0.988459i	$-0.548408\pi$
$317$	−5.39445	−0.302982	−0.151491	+	0.988459i	$0.548408\pi$
$318$	0	0
$319$	3.63331	0.203426
$320$	0	0
$321$	−54.8444	−3.06112
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	2.97224	0.164870
$326$	0	0
$327$	−6.60555	−0.365288
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−32.7250	−1.79873	−0.899364	−	0.437201i	$-0.855970\pi$
$331$	−32.7250	−1.79873	−0.899364	+	0.437201i	$0.855970\pi$
$332$	0	0
$333$	−57.0278	−3.12510
$334$	0	0
$335$	−1.90833	−0.104263
$336$	0	0
$337$	25.2111	1.37334	0.686668	−	0.726971i	$-0.259073\pi$
$337$	25.2111	1.37334	0.686668	+	0.726971i	$0.259073\pi$
$338$	0	0
$339$	15.8167	0.859043
$340$	0	0
$341$	−26.8444	−1.45371
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.60555	−0.247955
$346$	0	0
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	−4.78890	−0.256344	−0.128172	−	0.991752i	$-0.540911\pi$
$349$	−4.78890	−0.256344	−0.128172	+	0.991752i	$0.540911\pi$
$350$	0	0
$351$	−9.81665	−0.523974
$352$	0	0
$353$	2.60555	0.138680	0.0693398	−	0.997593i	$-0.477911\pi$
$353$	2.60555	0.138680	0.0693398	+	0.997593i	$0.477911\pi$
$354$	0	0
$355$	0.605551	0.0321393
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−35.5139	−1.87435	−0.937175	−	0.348859i	$-0.886569\pi$
$359$	−35.5139	−1.87435	−0.937175	+	0.348859i	$0.886569\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−13.9083	−0.729998
$364$	0	0
$365$	1.36669	0.0715359
$366$	0	0
$367$	5.72498	0.298842	0.149421	−	0.988774i	$-0.452259\pi$
$367$	5.72498	0.298842	0.149421	+	0.988774i	$0.452259\pi$
$368$	0	0
$369$	67.3305	3.50509
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.69722	0.450325	0.225163	−	0.974321i	$-0.427709\pi$
$373$	8.69722	0.450325	0.225163	+	0.974321i	$0.427709\pi$
$374$	0	0
$375$	9.90833	0.511664
$376$	0	0
$377$	0.844410	0.0434893
$378$	0	0
$379$	22.2389	1.14233	0.571167	−	0.820834i	$-0.306491\pi$
$379$	22.2389	1.14233	0.571167	+	0.820834i	$0.306491\pi$
$380$	0	0
$381$	66.4500	3.40433
$382$	0	0
$383$	−33.0278	−1.68764	−0.843820	−	0.536627i	$-0.819698\pi$
$383$	−33.0278	−1.68764	−0.843820	+	0.536627i	$0.819698\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.51388	−0.280286
$388$	0	0
$389$	−24.9083	−1.26290	−0.631451	−	0.775416i	$-0.717540\pi$
$389$	−24.9083	−1.26290	−0.631451	+	0.775416i	$0.717540\pi$
$390$	0	0
$391$	4.60555	0.232913
$392$	0	0
$393$	−60.8444	−3.06919
$394$	0	0
$395$	1.81665	0.0914058
$396$	0	0
$397$	16.9083	0.848605	0.424302	−	0.905521i	$-0.360519\pi$
$397$	16.9083	0.848605	0.424302	+	0.905521i	$0.360519\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−6.23886	−0.310780
$404$	0	0
$405$	−9.02776	−0.448593
$406$	0	0
$407$	18.7889	0.931331
$408$	0	0
$409$	−33.0278	−1.63312	−0.816559	−	0.577262i	$-0.804121\pi$
$409$	−33.0278	−1.63312	−0.816559	+	0.577262i	$0.804121\pi$
$410$	0	0
$411$	−45.9361	−2.26586
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.57779	−0.0774509
$416$	0	0
$417$	64.4500	3.15613
$418$	0	0
$419$	−2.30278	−0.112498	−0.0562490	−	0.998417i	$-0.517914\pi$
$419$	−2.30278	−0.112498	−0.0562490	+	0.998417i	$0.517914\pi$
$420$	0	0
$421$	9.90833	0.482902	0.241451	−	0.970413i	$-0.422377\pi$
$421$	9.90833	0.482902	0.241451	+	0.970413i	$0.422377\pi$
$422$	0	0
$423$	−79.0833	−3.84516
$424$	0	0
$425$	−4.90833	−0.238089
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.21110	0.251594
$430$	0	0
$431$	−38.8444	−1.87107	−0.935535	−	0.353235i	$-0.885082\pi$
$431$	−38.8444	−1.87107	−0.935535	+	0.353235i	$0.885082\pi$
$432$	0	0
$433$	31.6333	1.52020	0.760100	−	0.649806i	$-0.225150\pi$
$433$	31.6333	1.52020	0.760100	+	0.649806i	$0.225150\pi$
$434$	0	0
$435$	1.39445	0.0668587
$436$	0	0
$437$	27.6333	1.32188
$438$	0	0
$439$	−3.30278	−0.157633	−0.0788164	−	0.996889i	$-0.525114\pi$
$439$	−3.30278	−0.157633	−0.0788164	+	0.996889i	$0.525114\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.4222	0.875265	0.437633	−	0.899154i	$-0.355817\pi$
$443$	18.4222	0.875265	0.437633	+	0.899154i	$0.355817\pi$
$444$	0	0
$445$	−0.605551	−0.0287059
$446$	0	0
$447$	−19.5139	−0.922975
$448$	0	0
$449$	18.2389	0.860745	0.430372	−	0.902651i	$-0.358382\pi$
$449$	18.2389	0.860745	0.430372	+	0.902651i	$0.358382\pi$
$450$	0	0
$451$	−22.1833	−1.04457
$452$	0	0
$453$	−36.7250	−1.72549
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.9083	0.790938	0.395469	−	0.918479i	$-0.370582\pi$
$457$	16.9083	0.790938	0.395469	+	0.918479i	$0.370582\pi$
$458$	0	0
$459$	16.2111	0.756669
$460$	0	0
$461$	5.21110	0.242705	0.121353	−	0.992609i	$-0.461277\pi$
$461$	5.21110	0.242705	0.121353	+	0.992609i	$0.461277\pi$
$462$	0	0
$463$	−17.5416	−0.815229	−0.407614	−	0.913154i	$-0.633639\pi$
$463$	−17.5416	−0.815229	−0.407614	+	0.913154i	$0.633639\pi$
$464$	0	0
$465$	−10.3028	−0.477780
$466$	0	0
$467$	−2.78890	−0.129055	−0.0645274	−	0.997916i	$-0.520554\pi$
$467$	−2.78890	−0.129055	−0.0645274	+	0.997916i	$0.520554\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	72.0555	3.32014
$472$	0	0
$473$	1.81665	0.0835298
$474$	0	0
$475$	−29.4500	−1.35126
$476$	0	0
$477$	34.0278	1.55802
$478$	0	0
$479$	20.6972	0.945680	0.472840	−	0.881148i	$-0.343229\pi$
$479$	20.6972	0.945680	0.472840	+	0.881148i	$0.343229\pi$
$480$	0	0
$481$	4.36669	0.199104
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.90833	−0.177468
$486$	0	0
$487$	−29.6333	−1.34281	−0.671407	−	0.741089i	$-0.734310\pi$
$487$	−29.6333	−1.34281	−0.671407	+	0.741089i	$0.734310\pi$
$488$	0	0
$489$	−0.605551	−0.0273840
$490$	0	0
$491$	34.3028	1.54806	0.774031	−	0.633147i	$-0.218237\pi$
$491$	34.3028	1.54806	0.774031	+	0.633147i	$0.218237\pi$
$492$	0	0
$493$	−1.39445	−0.0628028
$494$	0	0
$495$	6.23886	0.280416
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	−76.3583	−3.41144
$502$	0	0
$503$	−36.1194	−1.61049	−0.805243	−	0.592945i	$-0.797965\pi$
$503$	−36.1194	−1.61049	−0.805243	+	0.592945i	$0.797965\pi$
$504$	0	0
$505$	4.97224	0.221262
$506$	0	0
$507$	−41.7250	−1.85307
$508$	0	0
$509$	−10.6056	−0.470083	−0.235041	−	0.971985i	$-0.575523\pi$
$509$	−10.6056	−0.470083	−0.235041	+	0.971985i	$0.575523\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	97.2666	4.29443
$514$	0	0
$515$	−0.366692	−0.0161584
$516$	0	0
$517$	26.0555	1.14592
$518$	0	0
$519$	19.5139	0.856564
$520$	0	0
$521$	32.3305	1.41643	0.708213	−	0.705999i	$-0.249502\pi$
$521$	32.3305	1.41643	0.708213	+	0.705999i	$0.249502\pi$
$522$	0	0
$523$	19.0278	0.832026	0.416013	−	0.909359i	$-0.363427\pi$
$523$	19.0278	0.832026	0.416013	+	0.909359i	$0.363427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.3028	0.448796
$528$	0	0
$529$	−1.78890	−0.0777781
$530$	0	0
$531$	72.8444	3.16118
$532$	0	0
$533$	−5.15559	−0.223313
$534$	0	0
$535$	5.02776	0.217369
$536$	0	0
$537$	72.3583	3.12249
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.8167	−0.765998	−0.382999	−	0.923749i	$-0.625109\pi$
$541$	−17.8167	−0.765998	−0.382999	+	0.923749i	$0.625109\pi$
$542$	0	0
$543$	−71.4500	−3.06621
$544$	0	0
$545$	0.605551	0.0259390
$546$	0	0
$547$	25.4500	1.08816	0.544081	−	0.839033i	$-0.316878\pi$
$547$	25.4500	1.08816	0.544081	+	0.839033i	$0.316878\pi$
$548$	0	0
$549$	5.51388	0.235327
$550$	0	0
$551$	−8.36669	−0.356433
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.21110	0.306094
$556$	0	0
$557$	−3.21110	−0.136059	−0.0680294	−	0.997683i	$-0.521671\pi$
$557$	−3.21110	−0.136059	−0.0680294	+	0.997683i	$0.521671\pi$
$558$	0	0
$559$	0.422205	0.0178574
$560$	0	0
$561$	−8.60555	−0.363327
$562$	0	0
$563$	18.2389	0.768676	0.384338	−	0.923192i	$-0.374430\pi$
$563$	18.2389	0.768676	0.384338	+	0.923192i	$0.374430\pi$
$564$	0	0
$565$	−1.44996	−0.0610003
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.4861	−0.984589	−0.492295	−	0.870429i	$-0.663842\pi$
$569$	−23.4861	−0.984589	−0.492295	+	0.870429i	$0.663842\pi$
$570$	0	0
$571$	9.63331	0.403141	0.201571	−	0.979474i	$-0.435395\pi$
$571$	9.63331	0.403141	0.201571	+	0.979474i	$0.435395\pi$
$572$	0	0
$573$	28.8167	1.20383
$574$	0	0
$575$	−22.6056	−0.942717
$576$	0	0
$577$	−2.97224	−0.123736	−0.0618681	−	0.998084i	$-0.519706\pi$
$577$	−2.97224	−0.123736	−0.0618681	+	0.998084i	$0.519706\pi$
$578$	0	0
$579$	−41.6333	−1.73022
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.2111	−0.464316
$584$	0	0
$585$	1.44996	0.0599485
$586$	0	0
$587$	2.78890	0.115110	0.0575551	−	0.998342i	$-0.481670\pi$
$587$	2.78890	0.115110	0.0575551	+	0.998342i	$0.481670\pi$
$588$	0	0
$589$	61.8167	2.54711
$590$	0	0
$591$	43.0278	1.76993
$592$	0	0
$593$	33.6333	1.38115	0.690577	−	0.723259i	$-0.257357\pi$
$593$	33.6333	1.38115	0.690577	+	0.723259i	$0.257357\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.9083	0.610157
$598$	0	0
$599$	21.5416	0.880167	0.440084	−	0.897957i	$-0.354949\pi$
$599$	21.5416	0.880167	0.440084	+	0.897957i	$0.354949\pi$
$600$	0	0
$601$	−4.78890	−0.195343	−0.0976716	−	0.995219i	$-0.531139\pi$
$601$	−4.78890	−0.195343	−0.0976716	+	0.995219i	$0.531139\pi$
$602$	0	0
$603$	49.8444	2.02982
$604$	0	0
$605$	1.27502	0.0518369
$606$	0	0
$607$	26.9083	1.09218	0.546088	−	0.837728i	$-0.316117\pi$
$607$	26.9083	1.09218	0.546088	+	0.837728i	$0.316117\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.05551	0.244980
$612$	0	0
$613$	15.9361	0.643652	0.321826	−	0.946799i	$-0.395703\pi$
$613$	15.9361	0.643652	0.321826	+	0.946799i	$0.395703\pi$
$614$	0	0
$615$	−8.51388	−0.343313
$616$	0	0
$617$	−2.60555	−0.104896	−0.0524478	−	0.998624i	$-0.516702\pi$
$617$	−2.60555	−0.104896	−0.0524478	+	0.998624i	$0.516702\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	74.6611	2.99605
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.6333	0.945332
$626$	0	0
$627$	−51.6333	−2.06204
$628$	0	0
$629$	−7.21110	−0.287525
$630$	0	0
$631$	−12.1194	−0.482467	−0.241233	−	0.970467i	$-0.577552\pi$
$631$	−12.1194	−0.482467	−0.241233	+	0.970467i	$0.577552\pi$
$632$	0	0
$633$	29.8167	1.18511
$634$	0	0
$635$	−6.09167	−0.241741
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−15.8167	−0.625697
$640$	0	0
$641$	−36.4222	−1.43859	−0.719295	−	0.694704i	$-0.755535\pi$
$641$	−36.4222	−1.43859	−0.719295	+	0.694704i	$0.755535\pi$
$642$	0	0
$643$	12.3305	0.486269	0.243134	−	0.969993i	$-0.421824\pi$
$643$	12.3305	0.486269	0.243134	+	0.969993i	$0.421824\pi$
$644$	0	0
$645$	0.697224	0.0274532
$646$	0	0
$647$	34.6056	1.36048	0.680242	−	0.732987i	$-0.261875\pi$
$647$	34.6056	1.36048	0.680242	+	0.732987i	$0.261875\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	5.57779	0.217942
$656$	0	0
$657$	−35.6972	−1.39268
$658$	0	0
$659$	−46.5416	−1.81300	−0.906502	−	0.422201i	$-0.861258\pi$
$659$	−46.5416	−1.81300	−0.906502	+	0.422201i	$0.861258\pi$
$660$	0	0
$661$	27.6333	1.07481	0.537406	−	0.843324i	$-0.319404\pi$
$661$	27.6333	1.07481	0.537406	+	0.843324i	$0.319404\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.42221	−0.248669
$668$	0	0
$669$	−71.4500	−2.76242
$670$	0	0
$671$	−1.81665	−0.0701311
$672$	0	0
$673$	36.8444	1.42025	0.710124	−	0.704077i	$-0.248639\pi$
$673$	36.8444	1.42025	0.710124	+	0.704077i	$0.248639\pi$
$674$	0	0
$675$	−79.5694	−3.06263
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−36.0278	−1.38059
$682$	0	0
$683$	25.0278	0.957660	0.478830	−	0.877908i	$-0.341061\pi$
$683$	25.0278	0.957660	0.478830	+	0.877908i	$0.341061\pi$
$684$	0	0
$685$	4.21110	0.160898
$686$	0	0
$687$	−52.2389	−1.99304
$688$	0	0
$689$	−2.60555	−0.0992636
$690$	0	0
$691$	−0.697224	−0.0265237	−0.0132618	−	0.999912i	$-0.504221\pi$
$691$	−0.697224	−0.0265237	−0.0132618	+	0.999912i	$0.504221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.90833	−0.224116
$696$	0	0
$697$	8.51388	0.322486
$698$	0	0
$699$	−4.00000	−0.151294
$700$	0	0
$701$	3.21110	0.121282	0.0606408	−	0.998160i	$-0.480686\pi$
$701$	3.21110	0.121282	0.0606408	+	0.998160i	$0.480686\pi$
$702$	0	0
$703$	−43.2666	−1.63183
$704$	0	0
$705$	10.0000	0.376622
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−45.8167	−1.72068	−0.860340	−	0.509720i	$-0.829749\pi$
$709$	−45.8167	−1.72068	−0.860340	+	0.509720i	$0.829749\pi$
$710$	0	0
$711$	−47.4500	−1.77951
$712$	0	0
$713$	47.4500	1.77702
$714$	0	0
$715$	−0.477718	−0.0178656
$716$	0	0
$717$	−28.7250	−1.07275
$718$	0	0
$719$	20.5139	0.765039	0.382519	−	0.923948i	$-0.375057\pi$
$719$	20.5139	0.765039	0.382519	+	0.923948i	$0.375057\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−10.2111	−0.379755
$724$	0	0
$725$	6.84441	0.254195
$726$	0	0
$727$	37.2666	1.38214	0.691071	−	0.722787i	$-0.257139\pi$
$727$	37.2666	1.38214	0.691071	+	0.722787i	$0.257139\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	−0.697224	−0.0257878
$732$	0	0
$733$	−27.6333	−1.02066	−0.510330	−	0.859979i	$-0.670477\pi$
$733$	−27.6333	−1.02066	−0.510330	+	0.859979i	$0.670477\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.4222	−0.604920
$738$	0	0
$739$	−14.0917	−0.518371	−0.259185	−	0.965828i	$-0.583454\pi$
$739$	−14.0917	−0.518371	−0.259185	+	0.965828i	$0.583454\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−40.6056	−1.48967	−0.744837	−	0.667247i	$-0.767473\pi$
$743$	−40.6056	−1.48967	−0.744837	+	0.667247i	$0.767473\pi$
$744$	0	0
$745$	1.78890	0.0655401
$746$	0	0
$747$	41.2111	1.50784
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−36.7889	−1.34245	−0.671223	−	0.741256i	$-0.734231\pi$
$751$	−36.7889	−1.34245	−0.671223	+	0.741256i	$0.734231\pi$
$752$	0	0
$753$	−76.6611	−2.79368
$754$	0	0
$755$	3.36669	0.122526
$756$	0	0
$757$	33.9083	1.23242	0.616210	−	0.787582i	$-0.288667\pi$
$757$	33.9083	1.23242	0.616210	+	0.787582i	$0.288667\pi$
$758$	0	0
$759$	−39.6333	−1.43860
$760$	0	0
$761$	−39.3944	−1.42805	−0.714024	−	0.700121i	$-0.753129\pi$
$761$	−39.3944	−1.42805	−0.714024	+	0.700121i	$0.753129\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.39445	−0.0865715
$766$	0	0
$767$	−5.57779	−0.201403
$768$	0	0
$769$	9.81665	0.353998	0.176999	−	0.984211i	$-0.443361\pi$
$769$	9.81665	0.353998	0.176999	+	0.984211i	$0.443361\pi$
$770$	0	0
$771$	−25.8167	−0.929764
$772$	0	0
$773$	−50.8444	−1.82875	−0.914373	−	0.404872i	$-0.867316\pi$
$773$	−50.8444	−1.82875	−0.914373	+	0.404872i	$0.867316\pi$
$774$	0	0
$775$	−50.5694	−1.81651
$776$	0	0
$777$	0	0
$778$	0	0
$779$	51.0833	1.83025
$780$	0	0
$781$	5.21110	0.186468
$782$	0	0
$783$	−22.6056	−0.807856
$784$	0	0
$785$	−6.60555	−0.235762
$786$	0	0
$787$	−14.4222	−0.514096	−0.257048	−	0.966399i	$-0.582750\pi$
$787$	−14.4222	−0.514096	−0.257048	+	0.966399i	$0.582750\pi$
$788$	0	0
$789$	−21.2111	−0.755135
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.422205	−0.0149929
$794$	0	0
$795$	−4.30278	−0.152604
$796$	0	0
$797$	−29.2111	−1.03471	−0.517355	−	0.855771i	$-0.673083\pi$
$797$	−29.2111	−1.03471	−0.517355	+	0.855771i	$0.673083\pi$
$798$	0	0
$799$	−10.0000	−0.353775
$800$	0	0
$801$	15.8167	0.558854
$802$	0	0
$803$	11.7611	0.415042
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.8167	0.697579
$808$	0	0
$809$	−19.4500	−0.683824	−0.341912	−	0.939732i	$-0.611075\pi$
$809$	−19.4500	−0.683824	−0.341912	+	0.939732i	$0.611075\pi$
$810$	0	0
$811$	−37.3583	−1.31183	−0.655913	−	0.754836i	$-0.727716\pi$
$811$	−37.3583	−1.31183	−0.655913	+	0.754836i	$0.727716\pi$
$812$	0	0
$813$	33.6333	1.17957
$814$	0	0
$815$	0.0555128	0.00194453
$816$	0	0
$817$	−4.18335	−0.146357
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.5778	0.404068	0.202034	−	0.979379i	$-0.435245\pi$
$821$	11.5778	0.404068	0.202034	+	0.979379i	$0.435245\pi$
$822$	0	0
$823$	31.0278	1.08156	0.540780	−	0.841164i	$-0.318129\pi$
$823$	31.0278	1.08156	0.540780	+	0.841164i	$0.318129\pi$
$824$	0	0
$825$	42.2389	1.47057
$826$	0	0
$827$	−38.8444	−1.35075	−0.675376	−	0.737473i	$-0.736019\pi$
$827$	−38.8444	−1.35075	−0.675376	+	0.737473i	$0.736019\pi$
$828$	0	0
$829$	36.6056	1.27136	0.635682	−	0.771951i	$-0.280719\pi$
$829$	36.6056	1.27136	0.635682	+	0.771951i	$0.280719\pi$
$830$	0	0
$831$	−36.4222	−1.26347
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.00000	0.242245
$836$	0	0
$837$	167.019	5.77303
$838$	0	0
$839$	−6.78890	−0.234379	−0.117189	−	0.993110i	$-0.537388\pi$
$839$	−6.78890	−0.234379	−0.117189	+	0.993110i	$0.537388\pi$
$840$	0	0
$841$	−27.0555	−0.932949
$842$	0	0
$843$	−67.0555	−2.30951
$844$	0	0
$845$	3.82506	0.131586
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−34.0278	−1.16783
$850$	0	0
$851$	−33.2111	−1.13846
$852$	0	0
$853$	4.78890	0.163969	0.0819844	−	0.996634i	$-0.473874\pi$
$853$	4.78890	0.163969	0.0819844	+	0.996634i	$0.473874\pi$
$854$	0	0
$855$	−14.3667	−0.491331
$856$	0	0
$857$	5.51388	0.188350	0.0941752	−	0.995556i	$-0.469979\pi$
$857$	5.51388	0.188350	0.0941752	+	0.995556i	$0.469979\pi$
$858$	0	0
$859$	2.36669	0.0807505	0.0403753	−	0.999185i	$-0.487145\pi$
$859$	2.36669	0.0807505	0.0403753	+	0.999185i	$0.487145\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.9361	−1.29136	−0.645680	−	0.763608i	$-0.723426\pi$
$863$	−37.9361	−1.29136	−0.645680	+	0.763608i	$0.723426\pi$
$864$	0	0
$865$	−1.78890	−0.0608243
$866$	0	0
$867$	3.30278	0.112168
$868$	0	0
$869$	15.6333	0.530324
$870$	0	0
$871$	−3.81665	−0.129322
$872$	0	0
$873$	102.083	3.45500
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.8167	−0.939302	−0.469651	−	0.882852i	$-0.655620\pi$
$877$	−27.8167	−0.939302	−0.469651	+	0.882852i	$0.655620\pi$
$878$	0	0
$879$	65.4500	2.20757
$880$	0	0
$881$	−18.1194	−0.610459	−0.305230	−	0.952279i	$-0.598733\pi$
$881$	−18.1194	−0.610459	−0.305230	+	0.952279i	$0.598733\pi$
$882$	0	0
$883$	23.2750	0.783267	0.391633	−	0.920121i	$-0.371910\pi$
$883$	23.2750	0.783267	0.391633	+	0.920121i	$0.371910\pi$
$884$	0	0
$885$	−9.21110	−0.309628
$886$	0	0
$887$	44.7250	1.50172	0.750859	−	0.660463i	$-0.229640\pi$
$887$	44.7250	1.50172	0.750859	+	0.660463i	$0.229640\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−77.6888	−2.60267
$892$	0	0
$893$	−60.0000	−2.00782
$894$	0	0
$895$	−6.63331	−0.221727
$896$	0	0
$897$	−9.21110	−0.307550
$898$	0	0
$899$	−14.3667	−0.479156
$900$	0	0
$901$	4.30278	0.143346
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.55004	0.217731
$906$	0	0
$907$	−32.4222	−1.07656	−0.538281	−	0.842766i	$-0.680926\pi$
$907$	−32.4222	−1.07656	−0.538281	+	0.842766i	$0.680926\pi$
$908$	0	0
$909$	−129.872	−4.30759
$910$	0	0
$911$	18.4222	0.610355	0.305177	−	0.952296i	$-0.401284\pi$
$911$	18.4222	0.610355	0.305177	+	0.952296i	$0.401284\pi$
$912$	0	0
$913$	−13.5778	−0.449359
$914$	0	0
$915$	−0.697224	−0.0230495
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−28.5139	−0.940586	−0.470293	−	0.882510i	$-0.655852\pi$
$919$	−28.5139	−0.940586	−0.470293	+	0.882510i	$0.655852\pi$
$920$	0	0
$921$	85.8722	2.82958
$922$	0	0
$923$	1.21110	0.0398639
$924$	0	0
$925$	35.3944	1.16376
$926$	0	0
$927$	9.57779	0.314576
$928$	0	0
$929$	28.3028	0.928584	0.464292	−	0.885682i	$-0.346309\pi$
$929$	28.3028	0.928584	0.464292	+	0.885682i	$0.346309\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−55.2389	−1.80844
$934$	0	0
$935$	0.788897	0.0257997
$936$	0	0
$937$	10.7889	0.352458	0.176229	−	0.984349i	$-0.443610\pi$
$937$	10.7889	0.352458	0.176229	+	0.984349i	$0.443610\pi$
$938$	0	0
$939$	−88.2666	−2.88047
$940$	0	0
$941$	40.3305	1.31474	0.657369	−	0.753569i	$-0.271669\pi$
$941$	40.3305	1.31474	0.657369	+	0.753569i	$0.271669\pi$
$942$	0	0
$943$	39.2111	1.27689
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.2389	−0.982631	−0.491315	−	0.870982i	$-0.663484\pi$
$947$	−30.2389	−0.982631	−0.491315	+	0.870982i	$0.663484\pi$
$948$	0	0
$949$	2.73338	0.0887294
$950$	0	0
$951$	−17.8167	−0.577745
$952$	0	0
$953$	−41.7527	−1.35250	−0.676252	−	0.736670i	$-0.736397\pi$
$953$	−41.7527	−1.35250	−0.676252	+	0.736670i	$0.736397\pi$
$954$	0	0
$955$	−2.64171	−0.0854838
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	0	0
$960$	0	0
$961$	75.1472	2.42410
$962$	0	0
$963$	−131.322	−4.23180
$964$	0	0
$965$	3.81665	0.122862
$966$	0	0
$967$	−16.6972	−0.536947	−0.268473	−	0.963287i	$-0.586519\pi$
$967$	−16.6972	−0.536947	−0.268473	+	0.963287i	$0.586519\pi$
$968$	0	0
$969$	19.8167	0.636603
$970$	0	0
$971$	−30.2389	−0.970411	−0.485206	−	0.874400i	$-0.661255\pi$
$971$	−30.2389	−0.970411	−0.485206	+	0.874400i	$0.661255\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	9.81665	0.314385
$976$	0	0
$977$	−5.51388	−0.176405	−0.0882023	−	0.996103i	$-0.528112\pi$
$977$	−5.51388	−0.176405	−0.0882023	+	0.996103i	$0.528112\pi$
$978$	0	0
$979$	−5.21110	−0.166548
$980$	0	0
$981$	−15.8167	−0.504987
$982$	0	0
$983$	3.51388	0.112075	0.0560377	−	0.998429i	$-0.482153\pi$
$983$	3.51388	0.112075	0.0560377	+	0.998429i	$0.482153\pi$
$984$	0	0
$985$	−3.94449	−0.125682
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.21110	−0.102107
$990$	0	0
$991$	6.00000	0.190596	0.0952981	−	0.995449i	$-0.469620\pi$
$991$	6.00000	0.190596	0.0952981	+	0.995449i	$0.469620\pi$
$992$	0	0
$993$	−108.083	−3.42992
$994$	0	0
$995$	−1.36669	−0.0433271
$996$	0	0
$997$	−22.0917	−0.699650	−0.349825	−	0.936815i	$-0.613759\pi$
$997$	−22.0917	−0.699650	−0.349825	+	0.936815i	$0.613759\pi$
$998$	0	0
$999$	−116.900	−3.69855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.n.1.2		2
7.6	odd	2		476.2.a.a.1.1	✓	2
21.20	even	2		4284.2.a.p.1.1		2
28.27	even	2		1904.2.a.l.1.2		2
56.13	odd	2		7616.2.a.z.1.2		2
56.27	even	2		7616.2.a.m.1.1		2
119.118	odd	2		8092.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.a.1.1	✓	2	7.6	odd	2
1904.2.a.l.1.2		2	28.27	even	2
3332.2.a.n.1.2		2	1.1	even	1	trivial
4284.2.a.p.1.1		2	21.20	even	2
7616.2.a.m.1.1		2	56.27	even	2
7616.2.a.z.1.2		2	56.13	odd	2
8092.2.a.n.1.2		2	119.118	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q+3.30278 q^{3} -0.302776 q^{5} +7.90833 q^{9} +O(q^{10})\)	\(q+3.30278 q^{3} -0.302776 q^{5} +7.90833 q^{9} -2.60555 q^{11} -0.605551 q^{13} -1.00000 q^{15} +1.00000 q^{17} +6.00000 q^{19} +4.60555 q^{23} -4.90833 q^{25} +16.2111 q^{27} -1.39445 q^{29} +10.3028 q^{31} -8.60555 q^{33} -7.21110 q^{37} -2.00000 q^{39} +8.51388 q^{41} -0.697224 q^{43} -2.39445 q^{45} -10.0000 q^{47} +3.30278 q^{51} +4.30278 q^{53} +0.788897 q^{55} +19.8167 q^{57} +9.21110 q^{59} +0.697224 q^{61} +0.183346 q^{65} +6.30278 q^{67} +15.2111 q^{69} -2.00000 q^{71} -4.51388 q^{73} -16.2111 q^{75} -6.00000 q^{79} +29.8167 q^{81} +5.21110 q^{83} -0.302776 q^{85} -4.60555 q^{87} +2.00000 q^{89} +34.0278 q^{93} -1.81665 q^{95} +12.9083 q^{97} -20.6056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9	\( 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 12 q^{19} + 2 q^{23} + q^{25} + 18 q^{27} - 10 q^{29} + 17 q^{31} - 10 q^{33} - 4 q^{39} - q^{41} - 5 q^{43} - 12 q^{45} - 20 q^{47} + 3 q^{51} + 5 q^{53} + 16 q^{55} + 18 q^{57} + 4 q^{59} + 5 q^{61} + 22 q^{65} + 9 q^{67} + 16 q^{69} - 4 q^{71} + 9 q^{73} - 18 q^{75} - 12 q^{79} + 38 q^{81} - 4 q^{83} + 3 q^{85} - 2 q^{87} + 4 q^{89} + 32 q^{93} + 18 q^{95} + 15 q^{97} - 34 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 + 2 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 12 * q^19 + 2 * q^23 + q^25 + 18 * q^27 - 10 * q^29 + 17 * q^31 - 10 * q^33 - 4 * q^39 - q^41 - 5 * q^43 - 12 * q^45 - 20 * q^47 + 3 * q^51 + 5 * q^53 + 16 * q^55 + 18 * q^57 + 4 * q^59 + 5 * q^61 + 22 * q^65 + 9 * q^67 + 16 * q^69 - 4 * q^71 + 9 * q^73 - 18 * q^75 - 12 * q^79 + 38 * q^81 - 4 * q^83 + 3 * q^85 - 2 * q^87 + 4 * q^89 + 32 * q^93 + 18 * q^95 + 15 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more