LMFDB - Embedded newform 3364.2.a.q.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3364,2,Mod(1,3364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3364, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3364.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3364 = 2^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3364.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:	$26.8616752400$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.3266578125.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1 $ x^8 - x^7 - 10x^6 + 6x^5 + 29x^4 - 9x^3 - 25*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.944572$ of defining polynomial
Character	$\chi$	$=$	3364.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54007 q^{3} +2.86661 q^{5} -4.47762 q^{7} +3.45197 q^{9} +O(q^{10})$	$q+2.54007 q^{3} +2.86661 q^{5} -4.47762 q^{7} +3.45197 q^{9} -4.32400 q^{11} -6.10684 q^{13} +7.28140 q^{15} -3.82253 q^{17} +1.09874 q^{19} -11.3735 q^{21} -2.39010 q^{23} +3.21745 q^{25} +1.14803 q^{27} +3.71243 q^{31} -10.9833 q^{33} -12.8356 q^{35} -1.69068 q^{37} -15.5118 q^{39} -5.13396 q^{41} +2.00170 q^{43} +9.89545 q^{45} +9.28904 q^{47} +13.0491 q^{49} -9.70950 q^{51} +1.66810 q^{53} -12.3952 q^{55} +2.79087 q^{57} +0.0901690 q^{59} -8.52727 q^{61} -15.4566 q^{63} -17.5059 q^{65} -8.77154 q^{67} -6.07102 q^{69} +9.23764 q^{71} -2.10072 q^{73} +8.17257 q^{75} +19.3612 q^{77} -5.42047 q^{79} -7.43982 q^{81} -6.23899 q^{83} -10.9577 q^{85} +3.62161 q^{89} +27.3441 q^{91} +9.42984 q^{93} +3.14965 q^{95} -6.64300 q^{97} -14.9263 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9	$ 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} + 17 q^{15} - 15 q^{17} - 11 q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 16 q^{27} + 9 q^{31} - 16 q^{33} + 9 q^{35} + 2 q^{37} - 9 q^{39} - 38 q^{41} + 3 q^{43} + 3 q^{45} + 7 q^{47} + 12 q^{49} + 3 q^{51} - 5 q^{53} - 34 q^{55} + 34 q^{57} - 7 q^{59} - 39 q^{61} - 6 q^{63} - 21 q^{65} - 26 q^{67} - 34 q^{69} + 52 q^{71} + 34 q^{73} - 10 q^{77} - 31 q^{79} + 24 q^{81} - 17 q^{83} - 18 q^{85} - 24 q^{89} + 54 q^{91} + 6 q^{93} - 9 q^{95} - 22 q^{97} + 11 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 + 17 * q^15 - 15 * q^17 - 11 * q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 16 * q^27 + 9 * q^31 - 16 * q^33 + 9 * q^35 + 2 * q^37 - 9 * q^39 - 38 * q^41 + 3 * q^43 + 3 * q^45 + 7 * q^47 + 12 * q^49 + 3 * q^51 - 5 * q^53 - 34 * q^55 + 34 * q^57 - 7 * q^59 - 39 * q^61 - 6 * q^63 - 21 * q^65 - 26 * q^67 - 34 * q^69 + 52 * q^71 + 34 * q^73 - 10 * q^77 - 31 * q^79 + 24 * q^81 - 17 * q^83 - 18 * q^85 - 24 * q^89 + 54 * q^91 + 6 * q^93 - 9 * q^95 - 22 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.54007	1.46651	0.733256	−	0.679953i	$-0.238000\pi$
$3$	2.54007	1.46651	0.733256	+	0.679953i	$0.238000\pi$
$4$	0	0
$5$	2.86661	1.28199	0.640994	−	0.767546i	$-0.278523\pi$
$5$	2.86661	1.28199	0.640994	+	0.767546i	$0.278523\pi$
$6$	0	0
$7$	−4.47762	−1.69238	−0.846190	−	0.532881i	$-0.821109\pi$
$7$	−4.47762	−1.69238	−0.846190	+	0.532881i	$0.821109\pi$
$8$	0	0
$9$	3.45197	1.15066
$10$	0	0
$11$	−4.32400	−1.30374	−0.651868	−	0.758332i	$-0.726014\pi$
$11$	−4.32400	−1.30374	−0.651868	+	0.758332i	$0.726014\pi$
$12$	0	0
$13$	−6.10684	−1.69373	−0.846866	−	0.531806i	$-0.821514\pi$
$13$	−6.10684	−1.69373	−0.846866	+	0.531806i	$0.821514\pi$
$14$	0	0
$15$	7.28140	1.88005
$16$	0	0
$17$	−3.82253	−0.927099	−0.463550	−	0.886071i	$-0.653424\pi$
$17$	−3.82253	−0.927099	−0.463550	+	0.886071i	$0.653424\pi$
$18$	0	0
$19$	1.09874	0.252068	0.126034	−	0.992026i	$-0.459775\pi$
$19$	1.09874	0.252068	0.126034	+	0.992026i	$0.459775\pi$
$20$	0	0
$21$	−11.3735	−2.48190
$22$	0	0
$23$	−2.39010	−0.498370	−0.249185	−	0.968456i	$-0.580163\pi$
$23$	−2.39010	−0.498370	−0.249185	+	0.968456i	$0.580163\pi$
$24$	0	0
$25$	3.21745	0.643491
$26$	0	0
$27$	1.14803	0.220939
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	3.71243	0.666772	0.333386	−	0.942790i	$-0.391809\pi$
$31$	3.71243	0.666772	0.333386	+	0.942790i	$0.391809\pi$
$32$	0	0
$33$	−10.9833	−1.91194
$34$	0	0
$35$	−12.8356	−2.16961
$36$	0	0
$37$	−1.69068	−0.277946	−0.138973	−	0.990296i	$-0.544380\pi$
$37$	−1.69068	−0.277946	−0.138973	+	0.990296i	$0.544380\pi$
$38$	0	0
$39$	−15.5118	−2.48388
$40$	0	0
$41$	−5.13396	−0.801790	−0.400895	−	0.916124i	$-0.631301\pi$
$41$	−5.13396	−0.801790	−0.400895	+	0.916124i	$0.631301\pi$
$42$	0	0
$43$	2.00170	0.305256	0.152628	−	0.988284i	$-0.451226\pi$
$43$	2.00170	0.305256	0.152628	+	0.988284i	$0.451226\pi$
$44$	0	0
$45$	9.89545	1.47513
$46$	0	0
$47$	9.28904	1.35495	0.677473	−	0.735547i	$-0.263075\pi$
$47$	9.28904	1.35495	0.677473	+	0.735547i	$0.263075\pi$
$48$	0	0
$49$	13.0491	1.86415
$50$	0	0
$51$	−9.70950	−1.35960
$52$	0	0
$53$	1.66810	0.229130	0.114565	−	0.993416i	$-0.463453\pi$
$53$	1.66810	0.229130	0.114565	+	0.993416i	$0.463453\pi$
$54$	0	0
$55$	−12.3952	−1.67137
$56$	0	0
$57$	2.79087	0.369660
$58$	0	0
$59$	0.0901690	0.0117390	0.00586950	−	0.999983i	$-0.498132\pi$
$59$	0.0901690	0.0117390	0.00586950	+	0.999983i	$0.498132\pi$
$60$	0	0
$61$	−8.52727	−1.09181	−0.545903	−	0.837849i	$-0.683813\pi$
$61$	−8.52727	−1.09181	−0.545903	+	0.837849i	$0.683813\pi$
$62$	0	0
$63$	−15.4566	−1.94735
$64$	0	0
$65$	−17.5059	−2.17134
$66$	0	0
$67$	−8.77154	−1.07161	−0.535807	−	0.844340i	$-0.679993\pi$
$67$	−8.77154	−1.07161	−0.535807	+	0.844340i	$0.679993\pi$
$68$	0	0
$69$	−6.07102	−0.730865
$70$	0	0
$71$	9.23764	1.09631	0.548153	−	0.836378i	$-0.315331\pi$
$71$	9.23764	1.09631	0.548153	+	0.836378i	$0.315331\pi$
$72$	0	0
$73$	−2.10072	−0.245871	−0.122936	−	0.992415i	$-0.539231\pi$
$73$	−2.10072	−0.245871	−0.122936	+	0.992415i	$0.539231\pi$
$74$	0	0
$75$	8.17257	0.943687
$76$	0	0
$77$	19.3612	2.20642
$78$	0	0
$79$	−5.42047	−0.609851	−0.304925	−	0.952376i	$-0.598632\pi$
$79$	−5.42047	−0.609851	−0.304925	+	0.952376i	$0.598632\pi$
$80$	0	0
$81$	−7.43982	−0.826646
$82$	0	0
$83$	−6.23899	−0.684818	−0.342409	−	0.939551i	$-0.611243\pi$
$83$	−6.23899	−0.684818	−0.342409	+	0.939551i	$0.611243\pi$
$84$	0	0
$85$	−10.9577	−1.18853
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.62161	0.383890	0.191945	−	0.981406i	$-0.438520\pi$
$89$	3.62161	0.383890	0.191945	+	0.981406i	$0.438520\pi$
$90$	0	0
$91$	27.3441	2.86644
$92$	0	0
$93$	9.42984	0.977829
$94$	0	0
$95$	3.14965	0.323148
$96$	0	0
$97$	−6.64300	−0.674495	−0.337247	−	0.941416i	$-0.609496\pi$
$97$	−6.64300	−0.674495	−0.337247	+	0.941416i	$0.609496\pi$
$98$	0	0
$99$	−14.9263	−1.50015
$100$	0	0
$101$	−13.3066	−1.32406	−0.662030	−	0.749477i	$-0.730305\pi$
$101$	−13.3066	−1.32406	−0.662030	+	0.749477i	$0.730305\pi$
$102$	0	0
$103$	13.1162	1.29237	0.646187	−	0.763179i	$-0.276363\pi$
$103$	13.1162	1.29237	0.646187	+	0.763179i	$0.276363\pi$
$104$	0	0
$105$	−32.6033	−3.18176
$106$	0	0
$107$	−0.431361	−0.0417012	−0.0208506	−	0.999783i	$-0.506637\pi$
$107$	−0.431361	−0.0417012	−0.0208506	+	0.999783i	$0.506637\pi$
$108$	0	0
$109$	−11.3350	−1.08569	−0.542846	−	0.839832i	$-0.682653\pi$
$109$	−11.3350	−1.08569	−0.542846	+	0.839832i	$0.682653\pi$
$110$	0	0
$111$	−4.29445	−0.407611
$112$	0	0
$113$	−13.5162	−1.27150	−0.635749	−	0.771896i	$-0.719309\pi$
$113$	−13.5162	−1.27150	−0.635749	+	0.771896i	$0.719309\pi$
$114$	0	0
$115$	−6.85148	−0.638904
$116$	0	0
$117$	−21.0806	−1.94890
$118$	0	0
$119$	17.1158	1.56901
$120$	0	0
$121$	7.69700	0.699727
$122$	0	0
$123$	−13.0406	−1.17583
$124$	0	0
$125$	−5.10986	−0.457040
$126$	0	0
$127$	17.0701	1.51473	0.757363	−	0.652994i	$-0.226487\pi$
$127$	17.0701	1.51473	0.757363	+	0.652994i	$0.226487\pi$
$128$	0	0
$129$	5.08446	0.447662
$130$	0	0
$131$	−3.17246	−0.277179	−0.138589	−	0.990350i	$-0.544257\pi$
$131$	−3.17246	−0.277179	−0.138589	+	0.990350i	$0.544257\pi$
$132$	0	0
$133$	−4.91973	−0.426595
$134$	0	0
$135$	3.29097	0.283241
$136$	0	0
$137$	15.9993	1.36691	0.683455	−	0.729993i	$-0.260477\pi$
$137$	15.9993	1.36691	0.683455	+	0.729993i	$0.260477\pi$
$138$	0	0
$139$	2.91997	0.247669	0.123835	−	0.992303i	$-0.460481\pi$
$139$	2.91997	0.247669	0.123835	+	0.992303i	$0.460481\pi$
$140$	0	0
$141$	23.5948	1.98704
$142$	0	0
$143$	26.4060	2.20818
$144$	0	0
$145$	0	0
$146$	0	0
$147$	33.1456	2.73380
$148$	0	0
$149$	−20.1018	−1.64680	−0.823402	−	0.567459i	$-0.807927\pi$
$149$	−20.1018	−1.64680	−0.823402	+	0.567459i	$0.807927\pi$
$150$	0	0
$151$	21.2129	1.72629	0.863143	−	0.504960i	$-0.168493\pi$
$151$	21.2129	1.72629	0.863143	+	0.504960i	$0.168493\pi$
$152$	0	0
$153$	−13.1953	−1.06677
$154$	0	0
$155$	10.6421	0.854793
$156$	0	0
$157$	−9.25613	−0.738720	−0.369360	−	0.929286i	$-0.620423\pi$
$157$	−9.25613	−0.738720	−0.369360	+	0.929286i	$0.620423\pi$
$158$	0	0
$159$	4.23708	0.336023
$160$	0	0
$161$	10.7019	0.843432
$162$	0	0
$163$	−21.4376	−1.67912	−0.839562	−	0.543264i	$-0.817188\pi$
$163$	−21.4376	−1.67912	−0.839562	+	0.543264i	$0.817188\pi$
$164$	0	0
$165$	−31.4848	−2.45109
$166$	0	0
$167$	23.8164	1.84297	0.921486	−	0.388411i	$-0.126976\pi$
$167$	23.8164	1.84297	0.921486	+	0.388411i	$0.126976\pi$
$168$	0	0
$169$	24.2935	1.86873
$170$	0	0
$171$	3.79281	0.290043
$172$	0	0
$173$	22.9900	1.74789	0.873947	−	0.486021i	$-0.161552\pi$
$173$	22.9900	1.74789	0.873947	+	0.486021i	$0.161552\pi$
$174$	0	0
$175$	−14.4065	−1.08903
$176$	0	0
$177$	0.229036	0.0172154
$178$	0	0
$179$	−1.39397	−0.104190	−0.0520950	−	0.998642i	$-0.516590\pi$
$179$	−1.39397	−0.104190	−0.0520950	+	0.998642i	$0.516590\pi$
$180$	0	0
$181$	5.50670	0.409310	0.204655	−	0.978834i	$-0.434393\pi$
$181$	5.50670	0.409310	0.204655	+	0.978834i	$0.434393\pi$
$182$	0	0
$183$	−21.6599	−1.60115
$184$	0	0
$185$	−4.84652	−0.356324
$186$	0	0
$187$	16.5286	1.20869
$188$	0	0
$189$	−5.14046	−0.373913
$190$	0	0
$191$	4.50876	0.326242	0.163121	−	0.986606i	$-0.447844\pi$
$191$	4.50876	0.326242	0.163121	+	0.986606i	$0.447844\pi$
$192$	0	0
$193$	−18.6121	−1.33972	−0.669862	−	0.742485i	$-0.733647\pi$
$193$	−18.6121	−1.33972	−0.669862	+	0.742485i	$0.733647\pi$
$194$	0	0
$195$	−44.4663	−3.18430
$196$	0	0
$197$	−1.93897	−0.138146	−0.0690730	−	0.997612i	$-0.522004\pi$
$197$	−1.93897	−0.138146	−0.0690730	+	0.997612i	$0.522004\pi$
$198$	0	0
$199$	−8.60268	−0.609828	−0.304914	−	0.952380i	$-0.598628\pi$
$199$	−8.60268	−0.609828	−0.304914	+	0.952380i	$0.598628\pi$
$200$	0	0
$201$	−22.2804	−1.57154
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−14.7171	−1.02788
$206$	0	0
$207$	−8.25054	−0.573452
$208$	0	0
$209$	−4.75095	−0.328630
$210$	0	0
$211$	−12.4158	−0.854740	−0.427370	−	0.904077i	$-0.640560\pi$
$211$	−12.4158	−0.854740	−0.427370	+	0.904077i	$0.640560\pi$
$212$	0	0
$213$	23.4643	1.60775
$214$	0	0
$215$	5.73809	0.391335
$216$	0	0
$217$	−16.6228	−1.12843
$218$	0	0
$219$	−5.33599	−0.360573
$220$	0	0
$221$	23.3436	1.57026
$222$	0	0
$223$	4.53663	0.303795	0.151897	−	0.988396i	$-0.451462\pi$
$223$	4.53663	0.303795	0.151897	+	0.988396i	$0.451462\pi$
$224$	0	0
$225$	11.1066	0.740437
$226$	0	0
$227$	22.0483	1.46340	0.731698	−	0.681629i	$-0.238728\pi$
$227$	22.0483	1.46340	0.731698	+	0.681629i	$0.238728\pi$
$228$	0	0
$229$	−15.2327	−1.00661	−0.503304	−	0.864110i	$-0.667882\pi$
$229$	−15.2327	−1.00661	−0.503304	+	0.864110i	$0.667882\pi$
$230$	0	0
$231$	49.1789	3.23574
$232$	0	0
$233$	23.8550	1.56279	0.781396	−	0.624035i	$-0.214508\pi$
$233$	23.8550	1.56279	0.781396	+	0.624035i	$0.214508\pi$
$234$	0	0
$235$	26.6281	1.73702
$236$	0	0
$237$	−13.7684	−0.894353
$238$	0	0
$239$	8.43908	0.545879	0.272939	−	0.962031i	$-0.412004\pi$
$239$	8.43908	0.545879	0.272939	+	0.962031i	$0.412004\pi$
$240$	0	0
$241$	−4.58894	−0.295599	−0.147800	−	0.989017i	$-0.547219\pi$
$241$	−4.58894	−0.295599	−0.147800	+	0.989017i	$0.547219\pi$
$242$	0	0
$243$	−22.3418	−1.43323
$244$	0	0
$245$	37.4066	2.38982
$246$	0	0
$247$	−6.70981	−0.426935
$248$	0	0
$249$	−15.8475	−1.00429
$250$	0	0
$251$	−24.6139	−1.55362	−0.776808	−	0.629737i	$-0.783163\pi$
$251$	−24.6139	−1.55362	−0.776808	+	0.629737i	$0.783163\pi$
$252$	0	0
$253$	10.3348	0.649743
$254$	0	0
$255$	−27.8334	−1.74299
$256$	0	0
$257$	27.7988	1.73404	0.867021	−	0.498272i	$-0.166032\pi$
$257$	27.7988	1.73404	0.867021	+	0.498272i	$0.166032\pi$
$258$	0	0
$259$	7.57023	0.470391
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.94396	0.489846	0.244923	−	0.969543i	$-0.421237\pi$
$263$	7.94396	0.489846	0.244923	+	0.969543i	$0.421237\pi$
$264$	0	0
$265$	4.78178	0.293742
$266$	0	0
$267$	9.19916	0.562979
$268$	0	0
$269$	−16.8863	−1.02958	−0.514789	−	0.857317i	$-0.672130\pi$
$269$	−16.8863	−1.02958	−0.514789	+	0.857317i	$0.672130\pi$
$270$	0	0
$271$	−8.46327	−0.514107	−0.257053	−	0.966397i	$-0.582752\pi$
$271$	−8.46327	−0.514107	−0.257053	+	0.966397i	$0.582752\pi$
$272$	0	0
$273$	69.4560	4.20367
$274$	0	0
$275$	−13.9123	−0.838942
$276$	0	0
$277$	5.99603	0.360266	0.180133	−	0.983642i	$-0.442347\pi$
$277$	5.99603	0.360266	0.180133	+	0.983642i	$0.442347\pi$
$278$	0	0
$279$	12.8152	0.767225
$280$	0	0
$281$	7.92833	0.472965	0.236482	−	0.971636i	$-0.424005\pi$
$281$	7.92833	0.472965	0.236482	+	0.971636i	$0.424005\pi$
$282$	0	0
$283$	−0.461103	−0.0274097	−0.0137049	−	0.999906i	$-0.504363\pi$
$283$	−0.461103	−0.0274097	−0.0137049	+	0.999906i	$0.504363\pi$
$284$	0	0
$285$	8.00035	0.473900
$286$	0	0
$287$	22.9879	1.35693
$288$	0	0
$289$	−2.38827	−0.140487
$290$	0	0
$291$	−16.8737	−0.989155
$292$	0	0
$293$	17.1593	1.00246	0.501229	−	0.865314i	$-0.332881\pi$
$293$	17.1593	1.00246	0.501229	+	0.865314i	$0.332881\pi$
$294$	0	0
$295$	0.258479	0.0150492
$296$	0	0
$297$	−4.96410	−0.288046
$298$	0	0
$299$	14.5959	0.844105
$300$	0	0
$301$	−8.96284	−0.516610
$302$	0	0
$303$	−33.7998	−1.94175
$304$	0	0
$305$	−24.4444	−1.39968
$306$	0	0
$307$	18.4233	1.05148	0.525738	−	0.850647i	$-0.323789\pi$
$307$	18.4233	1.05148	0.525738	+	0.850647i	$0.323789\pi$
$308$	0	0
$309$	33.3160	1.89528
$310$	0	0
$311$	6.38060	0.361811	0.180905	−	0.983501i	$-0.442097\pi$
$311$	6.38060	0.361811	0.180905	+	0.983501i	$0.442097\pi$
$312$	0	0
$313$	25.4903	1.44080	0.720400	−	0.693559i	$-0.243958\pi$
$313$	25.4903	1.44080	0.720400	+	0.693559i	$0.243958\pi$
$314$	0	0
$315$	−44.3080	−2.49648
$316$	0	0
$317$	31.4695	1.76750	0.883751	−	0.467958i	$-0.155010\pi$
$317$	31.4695	1.76750	0.883751	+	0.467958i	$0.155010\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.09569	−0.0611554
$322$	0	0
$323$	−4.19996	−0.233692
$324$	0	0
$325$	−19.6485	−1.08990
$326$	0	0
$327$	−28.7916	−1.59218
$328$	0	0
$329$	−41.5928	−2.29309
$330$	0	0
$331$	5.51681	0.303231	0.151616	−	0.988440i	$-0.451552\pi$
$331$	5.51681	0.303231	0.151616	+	0.988440i	$0.451552\pi$
$332$	0	0
$333$	−5.83618	−0.319821
$334$	0	0
$335$	−25.1446	−1.37380
$336$	0	0
$337$	−31.4322	−1.71222	−0.856111	−	0.516792i	$-0.827126\pi$
$337$	−31.4322	−1.71222	−0.856111	+	0.516792i	$0.827126\pi$
$338$	0	0
$339$	−34.3321	−1.86467
$340$	0	0
$341$	−16.0526	−0.869295
$342$	0	0
$343$	−27.0854	−1.46248
$344$	0	0
$345$	−17.4033	−0.936960
$346$	0	0
$347$	−11.3510	−0.609352	−0.304676	−	0.952456i	$-0.598548\pi$
$347$	−11.3510	−0.609352	−0.304676	+	0.952456i	$0.598548\pi$
$348$	0	0
$349$	−30.2358	−1.61848	−0.809242	−	0.587476i	$-0.800122\pi$
$349$	−30.2358	−1.61848	−0.809242	+	0.587476i	$0.800122\pi$
$350$	0	0
$351$	−7.01086	−0.374212
$352$	0	0
$353$	−19.6670	−1.04677	−0.523385	−	0.852096i	$-0.675331\pi$
$353$	−19.6670	−1.04677	−0.523385	+	0.852096i	$0.675331\pi$
$354$	0	0
$355$	26.4807	1.40545
$356$	0	0
$357$	43.4754	2.30096
$358$	0	0
$359$	2.82863	0.149289	0.0746446	−	0.997210i	$-0.476218\pi$
$359$	2.82863	0.149289	0.0746446	+	0.997210i	$0.476218\pi$
$360$	0	0
$361$	−17.7928	−0.936462
$362$	0	0
$363$	19.5509	1.02616
$364$	0	0
$365$	−6.02195	−0.315203
$366$	0	0
$367$	−20.9002	−1.09098	−0.545490	−	0.838117i	$-0.683656\pi$
$367$	−20.9002	−1.09098	−0.545490	+	0.838117i	$0.683656\pi$
$368$	0	0
$369$	−17.7223	−0.922584
$370$	0	0
$371$	−7.46909	−0.387776
$372$	0	0
$373$	25.9106	1.34160	0.670801	−	0.741638i	$-0.265951\pi$
$373$	25.9106	1.34160	0.670801	+	0.741638i	$0.265951\pi$
$374$	0	0
$375$	−12.9794	−0.670255
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7.58517	−0.389624	−0.194812	−	0.980841i	$-0.562410\pi$
$379$	−7.58517	−0.389624	−0.194812	+	0.980841i	$0.562410\pi$
$380$	0	0
$381$	43.3593	2.22136
$382$	0	0
$383$	−19.8197	−1.01274	−0.506370	−	0.862316i	$-0.669013\pi$
$383$	−19.8197	−1.01274	−0.506370	+	0.862316i	$0.669013\pi$
$384$	0	0
$385$	55.5011	2.82860
$386$	0	0
$387$	6.90980	0.351245
$388$	0	0
$389$	−4.72989	−0.239815	−0.119907	−	0.992785i	$-0.538260\pi$
$389$	−4.72989	−0.239815	−0.119907	+	0.992785i	$0.538260\pi$
$390$	0	0
$391$	9.13622	0.462038
$392$	0	0
$393$	−8.05827	−0.406486
$394$	0	0
$395$	−15.5384	−0.781821
$396$	0	0
$397$	−2.98587	−0.149856	−0.0749281	−	0.997189i	$-0.523873\pi$
$397$	−2.98587	−0.149856	−0.0749281	+	0.997189i	$0.523873\pi$
$398$	0	0
$399$	−12.4965	−0.625606
$400$	0	0
$401$	−0.398244	−0.0198874	−0.00994369	−	0.999951i	$-0.503165\pi$
$401$	−0.398244	−0.0198874	−0.00994369	+	0.999951i	$0.503165\pi$
$402$	0	0
$403$	−22.6712	−1.12933
$404$	0	0
$405$	−21.3271	−1.05975
$406$	0	0
$407$	7.31051	0.362369
$408$	0	0
$409$	−23.3127	−1.15274	−0.576368	−	0.817190i	$-0.695531\pi$
$409$	−23.3127	−1.15274	−0.576368	+	0.817190i	$0.695531\pi$
$410$	0	0
$411$	40.6393	2.00459
$412$	0	0
$413$	−0.403742	−0.0198669
$414$	0	0
$415$	−17.8847	−0.877928
$416$	0	0
$417$	7.41695	0.363209
$418$	0	0
$419$	−16.7523	−0.818404	−0.409202	−	0.912444i	$-0.634193\pi$
$419$	−16.7523	−0.818404	−0.409202	+	0.912444i	$0.634193\pi$
$420$	0	0
$421$	16.7982	0.818696	0.409348	−	0.912378i	$-0.365756\pi$
$421$	16.7982	0.818696	0.409348	+	0.912378i	$0.365756\pi$
$422$	0	0
$423$	32.0655	1.55908
$424$	0	0
$425$	−12.2988	−0.596580
$426$	0	0
$427$	38.1819	1.84775
$428$	0	0
$429$	67.0731	3.23832
$430$	0	0
$431$	−3.46854	−0.167074	−0.0835368	−	0.996505i	$-0.526622\pi$
$431$	−3.46854	−0.167074	−0.0835368	+	0.996505i	$0.526622\pi$
$432$	0	0
$433$	8.04976	0.386847	0.193423	−	0.981115i	$-0.438041\pi$
$433$	8.04976	0.386847	0.193423	+	0.981115i	$0.438041\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.62609	−0.125623
$438$	0	0
$439$	7.46048	0.356069	0.178035	−	0.984024i	$-0.443026\pi$
$439$	7.46048	0.356069	0.178035	+	0.984024i	$0.443026\pi$
$440$	0	0
$441$	45.0450	2.14500
$442$	0	0
$443$	−18.9218	−0.899002	−0.449501	−	0.893280i	$-0.648398\pi$
$443$	−18.9218	−0.899002	−0.449501	+	0.893280i	$0.648398\pi$
$444$	0	0
$445$	10.3818	0.492142
$446$	0	0
$447$	−51.0600	−2.41506
$448$	0	0
$449$	12.5079	0.590285	0.295143	−	0.955453i	$-0.404633\pi$
$449$	12.5079	0.590285	0.295143	+	0.955453i	$0.404633\pi$
$450$	0	0
$451$	22.1993	1.04532
$452$	0	0
$453$	53.8824	2.53162
$454$	0	0
$455$	78.3849	3.67474
$456$	0	0
$457$	−24.7065	−1.15572	−0.577860	−	0.816136i	$-0.696112\pi$
$457$	−24.7065	−1.15572	−0.577860	+	0.816136i	$0.696112\pi$
$458$	0	0
$459$	−4.38839	−0.204833
$460$	0	0
$461$	−24.0499	−1.12012	−0.560059	−	0.828453i	$-0.689221\pi$
$461$	−24.0499	−1.12012	−0.560059	+	0.828453i	$0.689221\pi$
$462$	0	0
$463$	−0.384490	−0.0178687	−0.00893437	−	0.999960i	$-0.502844\pi$
$463$	−0.384490	−0.0178687	−0.00893437	+	0.999960i	$0.502844\pi$
$464$	0	0
$465$	27.0317	1.25356
$466$	0	0
$467$	−29.5370	−1.36681	−0.683404	−	0.730041i	$-0.739501\pi$
$467$	−29.5370	−1.36681	−0.683404	+	0.730041i	$0.739501\pi$
$468$	0	0
$469$	39.2756	1.81358
$470$	0	0
$471$	−23.5112	−1.08334
$472$	0	0
$473$	−8.65535	−0.397973
$474$	0	0
$475$	3.53514	0.162203
$476$	0	0
$477$	5.75821	0.263650
$478$	0	0
$479$	−22.3313	−1.02034	−0.510171	−	0.860073i	$-0.670418\pi$
$479$	−22.3313	−1.02034	−0.510171	+	0.860073i	$0.670418\pi$
$480$	0	0
$481$	10.3247	0.470767
$482$	0	0
$483$	27.1837	1.23690
$484$	0	0
$485$	−19.0429	−0.864694
$486$	0	0
$487$	3.11160	0.141000	0.0705000	−	0.997512i	$-0.477541\pi$
$487$	3.11160	0.141000	0.0705000	+	0.997512i	$0.477541\pi$
$488$	0	0
$489$	−54.4531	−2.46245
$490$	0	0
$491$	−26.0152	−1.17405	−0.587025	−	0.809569i	$-0.699701\pi$
$491$	−26.0152	−1.17405	−0.587025	+	0.809569i	$0.699701\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−42.7880	−1.92318
$496$	0	0
$497$	−41.3626	−1.85537
$498$	0	0
$499$	−32.3847	−1.44974	−0.724868	−	0.688888i	$-0.758099\pi$
$499$	−32.3847	−1.44974	−0.724868	+	0.688888i	$0.758099\pi$
$500$	0	0
$501$	60.4955	2.70274
$502$	0	0
$503$	3.68250	0.164195	0.0820974	−	0.996624i	$-0.473838\pi$
$503$	3.68250	0.164195	0.0820974	+	0.996624i	$0.473838\pi$
$504$	0	0
$505$	−38.1449	−1.69743
$506$	0	0
$507$	61.7072	2.74051
$508$	0	0
$509$	−21.8431	−0.968177	−0.484089	−	0.875019i	$-0.660849\pi$
$509$	−21.8431	−0.968177	−0.484089	+	0.875019i	$0.660849\pi$
$510$	0	0
$511$	9.40624	0.416107
$512$	0	0
$513$	1.26139	0.0556916
$514$	0	0
$515$	37.5989	1.65681
$516$	0	0
$517$	−40.1659	−1.76649
$518$	0	0
$519$	58.3962	2.56331
$520$	0	0
$521$	−13.6886	−0.599710	−0.299855	−	0.953985i	$-0.596938\pi$
$521$	−13.6886	−0.599710	−0.299855	+	0.953985i	$0.596938\pi$
$522$	0	0
$523$	10.0308	0.438614	0.219307	−	0.975656i	$-0.429620\pi$
$523$	10.0308	0.438614	0.219307	+	0.975656i	$0.429620\pi$
$524$	0	0
$525$	−36.5936	−1.59708
$526$	0	0
$527$	−14.1909	−0.618164
$528$	0	0
$529$	−17.2874	−0.751627
$530$	0	0
$531$	0.311260	0.0135076
$532$	0	0
$533$	31.3523	1.35802
$534$	0	0
$535$	−1.23654	−0.0534605
$536$	0	0
$537$	−3.54078	−0.152796
$538$	0	0
$539$	−56.4242	−2.43036
$540$	0	0
$541$	4.62008	0.198633	0.0993163	−	0.995056i	$-0.468334\pi$
$541$	4.62008	0.198633	0.0993163	+	0.995056i	$0.468334\pi$
$542$	0	0
$543$	13.9874	0.600258
$544$	0	0
$545$	−32.4929	−1.39184
$546$	0	0
$547$	2.68607	0.114848	0.0574241	−	0.998350i	$-0.481711\pi$
$547$	2.68607	0.114848	0.0574241	+	0.998350i	$0.481711\pi$
$548$	0	0
$549$	−29.4359	−1.25629
$550$	0	0
$551$	0	0
$552$	0	0
$553$	24.2708	1.03210
$554$	0	0
$555$	−12.3105	−0.522553
$556$	0	0
$557$	12.3114	0.521651	0.260826	−	0.965386i	$-0.416005\pi$
$557$	12.3114	0.521651	0.260826	+	0.965386i	$0.416005\pi$
$558$	0	0
$559$	−12.2241	−0.517022
$560$	0	0
$561$	41.9839	1.77256
$562$	0	0
$563$	−28.1906	−1.18809	−0.594046	−	0.804431i	$-0.702470\pi$
$563$	−28.1906	−1.18809	−0.594046	+	0.804431i	$0.702470\pi$
$564$	0	0
$565$	−38.7457	−1.63004
$566$	0	0
$567$	33.3127	1.39900
$568$	0	0
$569$	4.10036	0.171896	0.0859480	−	0.996300i	$-0.472608\pi$
$569$	4.10036	0.171896	0.0859480	+	0.996300i	$0.472608\pi$
$570$	0	0
$571$	12.3376	0.516313	0.258157	−	0.966103i	$-0.416885\pi$
$571$	12.3376	0.516313	0.258157	+	0.966103i	$0.416885\pi$
$572$	0	0
$573$	11.4526	0.478438
$574$	0	0
$575$	−7.69003	−0.320697
$576$	0	0
$577$	37.1453	1.54638	0.773190	−	0.634174i	$-0.218660\pi$
$577$	37.1453	1.54638	0.773190	+	0.634174i	$0.218660\pi$
$578$	0	0
$579$	−47.2760	−1.96472
$580$	0	0
$581$	27.9358	1.15897
$582$	0	0
$583$	−7.21285	−0.298726
$584$	0	0
$585$	−60.4299	−2.49847
$586$	0	0
$587$	33.3927	1.37827	0.689133	−	0.724635i	$-0.257992\pi$
$587$	33.3927	1.37827	0.689133	+	0.724635i	$0.257992\pi$
$588$	0	0
$589$	4.07899	0.168072
$590$	0	0
$591$	−4.92513	−0.202593
$592$	0	0
$593$	−10.6651	−0.437965	−0.218983	−	0.975729i	$-0.570274\pi$
$593$	−10.6651	−0.437965	−0.218983	+	0.975729i	$0.570274\pi$
$594$	0	0
$595$	49.0644	2.01144
$596$	0	0
$597$	−21.8514	−0.894319
$598$	0	0
$599$	−3.74463	−0.153002	−0.0765008	−	0.997070i	$-0.524375\pi$
$599$	−3.74463	−0.153002	−0.0765008	+	0.997070i	$0.524375\pi$
$600$	0	0
$601$	43.8423	1.78836	0.894181	−	0.447705i	$-0.147758\pi$
$601$	43.8423	1.78836	0.894181	+	0.447705i	$0.147758\pi$
$602$	0	0
$603$	−30.2791	−1.23306
$604$	0	0
$605$	22.0643	0.897041
$606$	0	0
$607$	15.8709	0.644180	0.322090	−	0.946709i	$-0.395615\pi$
$607$	15.8709	0.644180	0.322090	+	0.946709i	$0.395615\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−56.7267	−2.29492
$612$	0	0
$613$	−24.6412	−0.995247	−0.497624	−	0.867393i	$-0.665794\pi$
$613$	−24.6412	−0.995247	−0.497624	+	0.867393i	$0.665794\pi$
$614$	0	0
$615$	−37.3824	−1.50740
$616$	0	0
$617$	26.8503	1.08095	0.540476	−	0.841359i	$-0.318244\pi$
$617$	26.8503	1.08095	0.540476	+	0.841359i	$0.318244\pi$
$618$	0	0
$619$	−40.9215	−1.64477	−0.822386	−	0.568929i	$-0.807358\pi$
$619$	−40.9215	−1.64477	−0.822386	+	0.568929i	$0.807358\pi$
$620$	0	0
$621$	−2.74391	−0.110109
$622$	0	0
$623$	−16.2162	−0.649688
$624$	0	0
$625$	−30.7353	−1.22941
$626$	0	0
$627$	−12.0677	−0.481939
$628$	0	0
$629$	6.46268	0.257684
$630$	0	0
$631$	−2.47464	−0.0985137	−0.0492569	−	0.998786i	$-0.515685\pi$
$631$	−2.47464	−0.0985137	−0.0492569	+	0.998786i	$0.515685\pi$
$632$	0	0
$633$	−31.5371	−1.25349
$634$	0	0
$635$	48.9333	1.94186
$636$	0	0
$637$	−79.6886	−3.15738
$638$	0	0
$639$	31.8881	1.26147
$640$	0	0
$641$	−20.4284	−0.806872	−0.403436	−	0.915008i	$-0.632184\pi$
$641$	−20.4284	−0.806872	−0.403436	+	0.915008i	$0.632184\pi$
$642$	0	0
$643$	35.6119	1.40440	0.702198	−	0.711981i	$-0.252202\pi$
$643$	35.6119	1.40440	0.702198	+	0.711981i	$0.252202\pi$
$644$	0	0
$645$	14.5752	0.573897
$646$	0	0
$647$	−45.6943	−1.79643	−0.898214	−	0.439558i	$-0.855135\pi$
$647$	−45.6943	−1.79643	−0.898214	+	0.439558i	$0.855135\pi$
$648$	0	0
$649$	−0.389891	−0.0153046
$650$	0	0
$651$	−42.2232	−1.65486
$652$	0	0
$653$	50.1369	1.96201	0.981005	−	0.193983i	$-0.0621408\pi$
$653$	50.1369	1.96201	0.981005	+	0.193983i	$0.0621408\pi$
$654$	0	0
$655$	−9.09420	−0.355340
$656$	0	0
$657$	−7.25163	−0.282913
$658$	0	0
$659$	10.8024	0.420801	0.210401	−	0.977615i	$-0.432523\pi$
$659$	10.8024	0.420801	0.210401	+	0.977615i	$0.432523\pi$
$660$	0	0
$661$	−39.8040	−1.54820	−0.774098	−	0.633065i	$-0.781796\pi$
$661$	−39.8040	−1.54820	−0.774098	+	0.633065i	$0.781796\pi$
$662$	0	0
$663$	59.2944	2.30280
$664$	0	0
$665$	−14.1029	−0.546889
$666$	0	0
$667$	0	0
$668$	0	0
$669$	11.5234	0.445519
$670$	0	0
$671$	36.8720	1.42343
$672$	0	0
$673$	3.98519	0.153618	0.0768088	−	0.997046i	$-0.475527\pi$
$673$	3.98519	0.153618	0.0768088	+	0.997046i	$0.475527\pi$
$674$	0	0
$675$	3.69375	0.142172
$676$	0	0
$677$	−36.5729	−1.40561	−0.702806	−	0.711382i	$-0.748070\pi$
$677$	−36.5729	−1.40561	−0.702806	+	0.711382i	$0.748070\pi$
$678$	0	0
$679$	29.7448	1.14150
$680$	0	0
$681$	56.0042	2.14609
$682$	0	0
$683$	−17.6178	−0.674126	−0.337063	−	0.941482i	$-0.609434\pi$
$683$	−17.6178	−0.674126	−0.337063	+	0.941482i	$0.609434\pi$
$684$	0	0
$685$	45.8636	1.75236
$686$	0	0
$687$	−38.6923	−1.47620
$688$	0	0
$689$	−10.1868	−0.388086
$690$	0	0
$691$	25.5735	0.972862	0.486431	−	0.873719i	$-0.338298\pi$
$691$	25.5735	0.972862	0.486431	+	0.873719i	$0.338298\pi$
$692$	0	0
$693$	66.8344	2.53883
$694$	0	0
$695$	8.37043	0.317508
$696$	0	0
$697$	19.6247	0.743339
$698$	0	0
$699$	60.5934	2.29185
$700$	0	0
$701$	−10.3556	−0.391125	−0.195563	−	0.980691i	$-0.562653\pi$
$701$	−10.3556	−0.391125	−0.195563	+	0.980691i	$0.562653\pi$
$702$	0	0
$703$	−1.85762	−0.0700613
$704$	0	0
$705$	67.6372	2.54737
$706$	0	0
$707$	59.5820	2.24081
$708$	0	0
$709$	25.2348	0.947713	0.473856	−	0.880602i	$-0.342862\pi$
$709$	25.2348	0.947713	0.473856	+	0.880602i	$0.342862\pi$
$710$	0	0
$711$	−18.7113	−0.701729
$712$	0	0
$713$	−8.87307	−0.332299
$714$	0	0
$715$	75.6957	2.83086
$716$	0	0
$717$	21.4359	0.800538
$718$	0	0
$719$	−31.7480	−1.18400	−0.592000	−	0.805938i	$-0.701661\pi$
$719$	−31.7480	−1.18400	−0.592000	+	0.805938i	$0.701661\pi$
$720$	0	0
$721$	−58.7291	−2.18719
$722$	0	0
$723$	−11.6562	−0.433500
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.1191	1.48794	0.743968	−	0.668215i	$-0.232941\pi$
$727$	40.1191	1.48794	0.743968	+	0.668215i	$0.232941\pi$
$728$	0	0
$729$	−34.4303	−1.27520
$730$	0	0
$731$	−7.65155	−0.283003
$732$	0	0
$733$	−15.2548	−0.563447	−0.281724	−	0.959496i	$-0.590906\pi$
$733$	−15.2548	−0.563447	−0.281724	+	0.959496i	$0.590906\pi$
$734$	0	0
$735$	95.0155	3.50470
$736$	0	0
$737$	37.9282	1.39710
$738$	0	0
$739$	48.3414	1.77827	0.889133	−	0.457649i	$-0.151308\pi$
$739$	48.3414	1.77827	0.889133	+	0.457649i	$0.151308\pi$
$740$	0	0
$741$	−17.0434	−0.626105
$742$	0	0
$743$	37.9370	1.39177	0.695886	−	0.718152i	$-0.255012\pi$
$743$	37.9370	1.39177	0.695886	+	0.718152i	$0.255012\pi$
$744$	0	0
$745$	−57.6240	−2.11118
$746$	0	0
$747$	−21.5368	−0.787990
$748$	0	0
$749$	1.93147	0.0705744
$750$	0	0
$751$	−34.5571	−1.26100	−0.630502	−	0.776187i	$-0.717151\pi$
$751$	−34.5571	−1.26100	−0.630502	+	0.776187i	$0.717151\pi$
$752$	0	0
$753$	−62.5211	−2.27840
$754$	0	0
$755$	60.8092	2.21308
$756$	0	0
$757$	27.4426	0.997418	0.498709	−	0.866770i	$-0.333808\pi$
$757$	27.4426	0.997418	0.498709	+	0.866770i	$0.333808\pi$
$758$	0	0
$759$	26.2511	0.952855
$760$	0	0
$761$	−36.3063	−1.31610	−0.658052	−	0.752973i	$-0.728619\pi$
$761$	−36.3063	−1.31610	−0.658052	+	0.752973i	$0.728619\pi$
$762$	0	0
$763$	50.7536	1.83740
$764$	0	0
$765$	−37.8256	−1.36759
$766$	0	0
$767$	−0.550647	−0.0198827
$768$	0	0
$769$	−4.95322	−0.178618	−0.0893089	−	0.996004i	$-0.528466\pi$
$769$	−4.95322	−0.178618	−0.0893089	+	0.996004i	$0.528466\pi$
$770$	0	0
$771$	70.6110	2.54299
$772$	0	0
$773$	−53.5778	−1.92706	−0.963530	−	0.267600i	$-0.913769\pi$
$773$	−53.5778	−1.92706	−0.963530	+	0.267600i	$0.913769\pi$
$774$	0	0
$775$	11.9446	0.429062
$776$	0	0
$777$	19.2289	0.689834
$778$	0	0
$779$	−5.64087	−0.202105
$780$	0	0
$781$	−39.9436	−1.42929
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−26.5337	−0.947029
$786$	0	0
$787$	−0.705956	−0.0251646	−0.0125823	−	0.999921i	$-0.504005\pi$
$787$	−0.705956	−0.0251646	−0.0125823	+	0.999921i	$0.504005\pi$
$788$	0	0
$789$	20.1782	0.718364
$790$	0	0
$791$	60.5204	2.15186
$792$	0	0
$793$	52.0747	1.84923
$794$	0	0
$795$	12.1461	0.430777
$796$	0	0
$797$	−8.43839	−0.298903	−0.149452	−	0.988769i	$-0.547751\pi$
$797$	−8.43839	−0.298903	−0.149452	+	0.988769i	$0.547751\pi$
$798$	0	0
$799$	−35.5076	−1.25617
$800$	0	0
$801$	12.5017	0.441726
$802$	0	0
$803$	9.08353	0.320551
$804$	0	0
$805$	30.6783	1.08127
$806$	0	0
$807$	−42.8925	−1.50989
$808$	0	0
$809$	−40.1191	−1.41051	−0.705256	−	0.708953i	$-0.749168\pi$
$809$	−40.1191	−1.41051	−0.705256	+	0.708953i	$0.749168\pi$
$810$	0	0
$811$	47.7449	1.67655	0.838274	−	0.545249i	$-0.183565\pi$
$811$	47.7449	1.67655	0.838274	+	0.545249i	$0.183565\pi$
$812$	0	0
$813$	−21.4973	−0.753944
$814$	0	0
$815$	−61.4533	−2.15261
$816$	0	0
$817$	2.19934	0.0769452
$818$	0	0
$819$	94.3910	3.29829
$820$	0	0
$821$	−40.1280	−1.40048	−0.700238	−	0.713909i	$-0.746923\pi$
$821$	−40.1280	−1.40048	−0.700238	+	0.713909i	$0.746923\pi$
$822$	0	0
$823$	−10.5866	−0.369025	−0.184513	−	0.982830i	$-0.559071\pi$
$823$	−10.5866	−0.369025	−0.184513	+	0.982830i	$0.559071\pi$
$824$	0	0
$825$	−35.3382	−1.23032
$826$	0	0
$827$	−9.19893	−0.319878	−0.159939	−	0.987127i	$-0.551130\pi$
$827$	−9.19893	−0.319878	−0.159939	+	0.987127i	$0.551130\pi$
$828$	0	0
$829$	3.02848	0.105183	0.0525917	−	0.998616i	$-0.483252\pi$
$829$	3.02848	0.105183	0.0525917	+	0.998616i	$0.483252\pi$
$830$	0	0
$831$	15.2303	0.528335
$832$	0	0
$833$	−49.8804	−1.72825
$834$	0	0
$835$	68.2725	2.36267
$836$	0	0
$837$	4.26199	0.147316
$838$	0	0
$839$	30.0647	1.03795	0.518975	−	0.854789i	$-0.326314\pi$
$839$	30.0647	1.03795	0.518975	+	0.854789i	$0.326314\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	20.1385	0.693608
$844$	0	0
$845$	69.6399	2.39569
$846$	0	0
$847$	−34.4642	−1.18421
$848$	0	0
$849$	−1.17123	−0.0401967
$850$	0	0
$851$	4.04089	0.138520
$852$	0	0
$853$	23.0425	0.788959	0.394479	−	0.918905i	$-0.370925\pi$
$853$	23.0425	0.788959	0.394479	+	0.918905i	$0.370925\pi$
$854$	0	0
$855$	10.8725	0.371832
$856$	0	0
$857$	2.50916	0.0857112	0.0428556	−	0.999081i	$-0.486354\pi$
$857$	2.50916	0.0857112	0.0428556	+	0.999081i	$0.486354\pi$
$858$	0	0
$859$	−30.1508	−1.02873	−0.514367	−	0.857570i	$-0.671973\pi$
$859$	−30.1508	−1.02873	−0.514367	+	0.857570i	$0.671973\pi$
$860$	0	0
$861$	58.3910	1.98996
$862$	0	0
$863$	−6.17741	−0.210281	−0.105141	−	0.994457i	$-0.533529\pi$
$863$	−6.17741	−0.210281	−0.105141	+	0.994457i	$0.533529\pi$
$864$	0	0
$865$	65.9032	2.24078
$866$	0	0
$867$	−6.06638	−0.206025
$868$	0	0
$869$	23.4381	0.795084
$870$	0	0
$871$	53.5664	1.81503
$872$	0	0
$873$	−22.9314	−0.776112
$874$	0	0
$875$	22.8800	0.773486
$876$	0	0
$877$	36.3707	1.22815	0.614075	−	0.789247i	$-0.289529\pi$
$877$	36.3707	1.22815	0.614075	+	0.789247i	$0.289529\pi$
$878$	0	0
$879$	43.5860	1.47012
$880$	0	0
$881$	−27.3364	−0.920986	−0.460493	−	0.887663i	$-0.652327\pi$
$881$	−27.3364	−0.920986	−0.460493	+	0.887663i	$0.652327\pi$
$882$	0	0
$883$	22.8431	0.768733	0.384367	−	0.923181i	$-0.374420\pi$
$883$	22.8431	0.768733	0.384367	+	0.923181i	$0.374420\pi$
$884$	0	0
$885$	0.656556	0.0220699
$886$	0	0
$887$	−21.4514	−0.720269	−0.360134	−	0.932900i	$-0.617269\pi$
$887$	−21.4514	−0.720269	−0.360134	+	0.932900i	$0.617269\pi$
$888$	0	0
$889$	−76.4333	−2.56349
$890$	0	0
$891$	32.1698	1.07773
$892$	0	0
$893$	10.2062	0.341538
$894$	0	0
$895$	−3.99596	−0.133570
$896$	0	0
$897$	37.0748	1.23789
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−6.37634	−0.212427
$902$	0	0
$903$	−22.7663	−0.757614
$904$	0	0
$905$	15.7856	0.524730
$906$	0	0
$907$	15.0698	0.500385	0.250192	−	0.968196i	$-0.419506\pi$
$907$	15.0698	0.500385	0.250192	+	0.968196i	$0.419506\pi$
$908$	0	0
$909$	−45.9341	−1.52354
$910$	0	0
$911$	23.9314	0.792881	0.396441	−	0.918060i	$-0.370245\pi$
$911$	23.9314	0.792881	0.396441	+	0.918060i	$0.370245\pi$
$912$	0	0
$913$	26.9774	0.892822
$914$	0	0
$915$	−62.0905	−2.05265
$916$	0	0
$917$	14.2051	0.469092
$918$	0	0
$919$	6.74894	0.222627	0.111313	−	0.993785i	$-0.464494\pi$
$919$	6.74894	0.222627	0.111313	+	0.993785i	$0.464494\pi$
$920$	0	0
$921$	46.7966	1.54200
$922$	0	0
$923$	−56.4128	−1.85685
$924$	0	0
$925$	−5.43969	−0.178856
$926$	0	0
$927$	45.2766	1.48708
$928$	0	0
$929$	−52.3748	−1.71836	−0.859181	−	0.511671i	$-0.829027\pi$
$929$	−52.3748	−1.71836	−0.859181	+	0.511671i	$0.829027\pi$
$930$	0	0
$931$	14.3375	0.469893
$932$	0	0
$933$	16.2072	0.530600
$934$	0	0
$935$	47.3811	1.54953
$936$	0	0
$937$	−26.1242	−0.853440	−0.426720	−	0.904384i	$-0.640331\pi$
$937$	−26.1242	−0.853440	−0.426720	+	0.904384i	$0.640331\pi$
$938$	0	0
$939$	64.7473	2.11295
$940$	0	0
$941$	38.2936	1.24834	0.624168	−	0.781291i	$-0.285438\pi$
$941$	38.2936	1.24834	0.624168	+	0.781291i	$0.285438\pi$
$942$	0	0
$943$	12.2707	0.399588
$944$	0	0
$945$	−14.7357	−0.479352
$946$	0	0
$947$	0.0508504	0.00165242	0.000826208	−	1.00000i	$-0.499737\pi$
$947$	0.0508504	0.00165242	0.000826208		1.00000i	$0.499737\pi$
$948$	0	0
$949$	12.8288	0.416440
$950$	0	0
$951$	79.9347	2.59206
$952$	0	0
$953$	17.9962	0.582953	0.291476	−	0.956578i	$-0.405853\pi$
$953$	17.9962	0.582953	0.291476	+	0.956578i	$0.405853\pi$
$954$	0	0
$955$	12.9249	0.418239
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−71.6386	−2.31333
$960$	0	0
$961$	−17.2179	−0.555415
$962$	0	0
$963$	−1.48904	−0.0479838
$964$	0	0
$965$	−53.3535	−1.71751
$966$	0	0
$967$	−12.5848	−0.404701	−0.202350	−	0.979313i	$-0.564858\pi$
$967$	−12.5848	−0.404701	−0.202350	+	0.979313i	$0.564858\pi$
$968$	0	0
$969$	−10.6682	−0.342712
$970$	0	0
$971$	−35.0649	−1.12529	−0.562644	−	0.826700i	$-0.690216\pi$
$971$	−35.0649	−1.12529	−0.562644	+	0.826700i	$0.690216\pi$
$972$	0	0
$973$	−13.0745	−0.419150
$974$	0	0
$975$	−49.9086	−1.59835
$976$	0	0
$977$	−23.3649	−0.747509	−0.373754	−	0.927528i	$-0.621930\pi$
$977$	−23.3649	−0.747509	−0.373754	+	0.927528i	$0.621930\pi$
$978$	0	0
$979$	−15.6599	−0.500491
$980$	0	0
$981$	−39.1279	−1.24926
$982$	0	0
$983$	58.5673	1.86801	0.934003	−	0.357265i	$-0.116291\pi$
$983$	58.5673	1.86801	0.934003	+	0.357265i	$0.116291\pi$
$984$	0	0
$985$	−5.55828	−0.177101
$986$	0	0
$987$	−105.649	−3.36284
$988$	0	0
$989$	−4.78426	−0.152131
$990$	0	0
$991$	−5.95075	−0.189032	−0.0945158	−	0.995523i	$-0.530130\pi$
$991$	−5.95075	−0.189032	−0.0945158	+	0.995523i	$0.530130\pi$
$992$	0	0
$993$	14.0131	0.444692
$994$	0	0
$995$	−24.6605	−0.781791
$996$	0	0
$997$	8.69974	0.275523	0.137762	−	0.990465i	$-0.456009\pi$
$997$	8.69974	0.275523	0.137762	+	0.990465i	$0.456009\pi$
$998$	0	0
$999$	−1.94096	−0.0614092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3364.2.a.q.1.8	✓	8
29.12	odd	4		3364.2.c.k.1681.15		16
29.17	odd	4		3364.2.c.k.1681.2		16
29.28	even	2		3364.2.a.r.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3364.2.a.q.1.8	✓	8	1.1	even	1	trivial
3364.2.a.r.1.1	yes	8	29.28	even	2
3364.2.c.k.1681.2		16	29.17	odd	4
3364.2.c.k.1681.15		16	29.12	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:	\( N \)	\(=\)	\( 3364 = 2^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3364.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54007 q^{3} +2.86661 q^{5} -4.47762 q^{7} +3.45197 q^{9} +O(q^{10})\)	\(q+2.54007 q^{3} +2.86661 q^{5} -4.47762 q^{7} +3.45197 q^{9} -4.32400 q^{11} -6.10684 q^{13} +7.28140 q^{15} -3.82253 q^{17} +1.09874 q^{19} -11.3735 q^{21} -2.39010 q^{23} +3.21745 q^{25} +1.14803 q^{27} +3.71243 q^{31} -10.9833 q^{33} -12.8356 q^{35} -1.69068 q^{37} -15.5118 q^{39} -5.13396 q^{41} +2.00170 q^{43} +9.89545 q^{45} +9.28904 q^{47} +13.0491 q^{49} -9.70950 q^{51} +1.66810 q^{53} -12.3952 q^{55} +2.79087 q^{57} +0.0901690 q^{59} -8.52727 q^{61} -15.4566 q^{63} -17.5059 q^{65} -8.77154 q^{67} -6.07102 q^{69} +9.23764 q^{71} -2.10072 q^{73} +8.17257 q^{75} +19.3612 q^{77} -5.42047 q^{79} -7.43982 q^{81} -6.23899 q^{83} -10.9577 q^{85} +3.62161 q^{89} +27.3441 q^{91} +9.42984 q^{93} +3.14965 q^{95} -6.64300 q^{97} -14.9263 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9	\( 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} + 17 q^{15} - 15 q^{17} - 11 q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 16 q^{27} + 9 q^{31} - 16 q^{33} + 9 q^{35} + 2 q^{37} - 9 q^{39} - 38 q^{41} + 3 q^{43} + 3 q^{45} + 7 q^{47} + 12 q^{49} + 3 q^{51} - 5 q^{53} - 34 q^{55} + 34 q^{57} - 7 q^{59} - 39 q^{61} - 6 q^{63} - 21 q^{65} - 26 q^{67} - 34 q^{69} + 52 q^{71} + 34 q^{73} - 10 q^{77} - 31 q^{79} + 24 q^{81} - 17 q^{83} - 18 q^{85} - 24 q^{89} + 54 q^{91} + 6 q^{93} - 9 q^{95} - 22 q^{97} + 11 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 + 17 * q^15 - 15 * q^17 - 11 * q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 16 * q^27 + 9 * q^31 - 16 * q^33 + 9 * q^35 + 2 * q^37 - 9 * q^39 - 38 * q^41 + 3 * q^43 + 3 * q^45 + 7 * q^47 + 12 * q^49 + 3 * q^51 - 5 * q^53 - 34 * q^55 + 34 * q^57 - 7 * q^59 - 39 * q^61 - 6 * q^63 - 21 * q^65 - 26 * q^67 - 34 * q^69 + 52 * q^71 + 34 * q^73 - 10 * q^77 - 31 * q^79 + 24 * q^81 - 17 * q^83 - 18 * q^85 - 24 * q^89 + 54 * q^91 + 6 * q^93 - 9 * q^95 - 22 * q^97 + 11 * q^99

Introduction

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