LMFDB - Embedded newform 3380.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.21969$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33225 q^{3} -1.00000 q^{5} -0.399804 q^{7} +2.43937 q^{9} +O(q^{10})$	$q-2.33225 q^{3} -1.00000 q^{5} -0.399804 q^{7} +2.43937 q^{9} +1.73205 q^{11} +2.33225 q^{15} +0.692481 q^{17} -5.37182 q^{19} +0.932442 q^{21} -0.107127 q^{23} +1.00000 q^{25} +1.30752 q^{27} -4.90348 q^{29} +7.86488 q^{31} -4.03957 q^{33} +0.399804 q^{35} +2.26469 q^{37} -7.73205 q^{41} +6.01060 q^{43} -2.43937 q^{45} +3.46410 q^{47} -6.84016 q^{49} -1.61504 q^{51} +11.7189 q^{53} -1.73205 q^{55} +12.5284 q^{57} +7.27529 q^{59} +8.68922 q^{61} -0.975272 q^{63} +1.32801 q^{67} +0.249847 q^{69} -3.87803 q^{71} -10.2251 q^{73} -2.33225 q^{75} -0.692481 q^{77} -13.1533 q^{79} -10.3676 q^{81} +14.0791 q^{83} -0.692481 q^{85} +11.4361 q^{87} +0.347088 q^{89} -18.3428 q^{93} +5.37182 q^{95} -8.85004 q^{97} +4.22512 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9	$ 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{15} + 6 q^{17} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 2 q^{27} - 6 q^{33} + 6 q^{35} - 18 q^{37} - 24 q^{41} + 10 q^{43} - 4 q^{45} - 4 q^{49} + 12 q^{53} + 18 q^{57} - 12 q^{59} + 4 q^{61} - 12 q^{63} - 18 q^{67} - 24 q^{69} - 12 q^{71} - 24 q^{73} + 2 q^{75} - 6 q^{77} - 8 q^{79} - 8 q^{81} + 36 q^{83} - 6 q^{85} + 6 q^{87} - 12 q^{89} - 48 q^{93} - 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^15 + 6 * q^17 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 2 * q^27 - 6 * q^33 + 6 * q^35 - 18 * q^37 - 24 * q^41 + 10 * q^43 - 4 * q^45 - 4 * q^49 + 12 * q^53 + 18 * q^57 - 12 * q^59 + 4 * q^61 - 12 * q^63 - 18 * q^67 - 24 * q^69 - 12 * q^71 - 24 * q^73 + 2 * q^75 - 6 * q^77 - 8 * q^79 - 8 * q^81 + 36 * q^83 - 6 * q^85 + 6 * q^87 - 12 * q^89 - 48 * q^93 - 6 * 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
$3$	−2.33225	−1.34652	−0.673262	−	0.739404i	$-0.735107\pi$
$3$	−2.33225	−1.34652	−0.673262	+	0.739404i	$0.735107\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.399804	−0.151112	−0.0755559	−	0.997142i	$-0.524073\pi$
$7$	−0.399804	−0.151112	−0.0755559	+	0.997142i	$0.524073\pi$
$8$	0	0
$9$	2.43937	0.813125
$10$	0	0
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.33225	0.602183
$16$	0	0
$17$	0.692481	0.167951	0.0839757	−	0.996468i	$-0.473238\pi$
$17$	0.692481	0.167951	0.0839757	+	0.996468i	$0.473238\pi$
$18$	0	0
$19$	−5.37182	−1.23238	−0.616190	−	0.787598i	$-0.711324\pi$
$19$	−5.37182	−1.23238	−0.616190	+	0.787598i	$0.711324\pi$
$20$	0	0
$21$	0.932442	0.203476
$22$	0	0
$23$	−0.107127	−0.0223376	−0.0111688	−	0.999938i	$-0.503555\pi$
$23$	−0.107127	−0.0223376	−0.0111688	+	0.999938i	$0.503555\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.30752	0.251632
$28$	0	0
$29$	−4.90348	−0.910553	−0.455276	−	0.890350i	$-0.650460\pi$
$29$	−4.90348	−0.910553	−0.455276	+	0.890350i	$0.650460\pi$
$30$	0	0
$31$	7.86488	1.41257	0.706287	−	0.707925i	$-0.250369\pi$
$31$	7.86488	1.41257	0.706287	+	0.707925i	$0.250369\pi$
$32$	0	0
$33$	−4.03957	−0.703199
$34$	0	0
$35$	0.399804	0.0675793
$36$	0	0
$37$	2.26469	0.372313	0.186156	−	0.982520i	$-0.440397\pi$
$37$	2.26469	0.372313	0.186156	+	0.982520i	$0.440397\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.73205	−1.20754	−0.603772	−	0.797157i	$-0.706336\pi$
$41$	−7.73205	−1.20754	−0.603772	+	0.797157i	$0.706336\pi$
$42$	0	0
$43$	6.01060	0.916608	0.458304	−	0.888795i	$-0.348457\pi$
$43$	6.01060	0.916608	0.458304	+	0.888795i	$0.348457\pi$
$44$	0	0
$45$	−2.43937	−0.363640
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−6.84016	−0.977165
$50$	0	0
$51$	−1.61504	−0.226150
$52$	0	0
$53$	11.7189	1.60972	0.804858	−	0.593468i	$-0.202242\pi$
$53$	11.7189	1.60972	0.804858	+	0.593468i	$0.202242\pi$
$54$	0	0
$55$	−1.73205	−0.233550
$56$	0	0
$57$	12.5284	1.65943
$58$	0	0
$59$	7.27529	0.947162	0.473581	−	0.880750i	$-0.342961\pi$
$59$	7.27529	0.947162	0.473581	+	0.880750i	$0.342961\pi$
$60$	0	0
$61$	8.68922	1.11254	0.556270	−	0.831001i	$-0.312232\pi$
$61$	8.68922	1.11254	0.556270	+	0.831001i	$0.312232\pi$
$62$	0	0
$63$	−0.975272	−0.122873
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.32801	0.162242	0.0811210	−	0.996704i	$-0.474150\pi$
$67$	1.32801	0.162242	0.0811210	+	0.996704i	$0.474150\pi$
$68$	0	0
$69$	0.249847	0.0300781
$70$	0	0
$71$	−3.87803	−0.460238	−0.230119	−	0.973163i	$-0.573911\pi$
$71$	−3.87803	−0.460238	−0.230119	+	0.973163i	$0.573911\pi$
$72$	0	0
$73$	−10.2251	−1.19676	−0.598380	−	0.801213i	$-0.704189\pi$
$73$	−10.2251	−1.19676	−0.598380	+	0.801213i	$0.704189\pi$
$74$	0	0
$75$	−2.33225	−0.269305
$76$	0	0
$77$	−0.692481	−0.0789156
$78$	0	0
$79$	−13.1533	−1.47986	−0.739932	−	0.672681i	$-0.765142\pi$
$79$	−13.1533	−1.47986	−0.739932	+	0.672681i	$0.765142\pi$
$80$	0	0
$81$	−10.3676	−1.15195
$82$	0	0
$83$	14.0791	1.54539	0.772693	−	0.634780i	$-0.218909\pi$
$83$	14.0791	1.54539	0.772693	+	0.634780i	$0.218909\pi$
$84$	0	0
$85$	−0.692481	−0.0751102
$86$	0	0
$87$	11.4361	1.22608
$88$	0	0
$89$	0.347088	0.0367913	0.0183956	−	0.999831i	$-0.494144\pi$
$89$	0.347088	0.0367913	0.0183956	+	0.999831i	$0.494144\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18.3428	−1.90206
$94$	0	0
$95$	5.37182	0.551137
$96$	0	0
$97$	−8.85004	−0.898586	−0.449293	−	0.893385i	$-0.648324\pi$
$97$	−8.85004	−0.898586	−0.449293	+	0.893385i	$0.648324\pi$
$98$	0	0
$99$	4.22512	0.424640
$100$	0	0
$101$	4.10387	0.408350	0.204175	−	0.978934i	$-0.434549\pi$
$101$	4.10387	0.408350	0.204175	+	0.978934i	$0.434549\pi$
$102$	0	0
$103$	11.2325	1.10677	0.553384	−	0.832926i	$-0.313336\pi$
$103$	11.2325	1.10677	0.553384	+	0.832926i	$0.313336\pi$
$104$	0	0
$105$	−0.932442	−0.0909970
$106$	0	0
$107$	−17.6033	−1.70178	−0.850888	−	0.525348i	$-0.823935\pi$
$107$	−17.6033	−1.70178	−0.850888	+	0.525348i	$0.823935\pi$
$108$	0	0
$109$	−15.1830	−1.45427	−0.727134	−	0.686495i	$-0.759148\pi$
$109$	−15.1830	−1.45427	−0.727134	+	0.686495i	$0.759148\pi$
$110$	0	0
$111$	−5.28181	−0.501327
$112$	0	0
$113$	8.90021	0.837262	0.418631	−	0.908156i	$-0.362510\pi$
$113$	8.90021	0.837262	0.418631	+	0.908156i	$0.362510\pi$
$114$	0	0
$115$	0.107127	0.00998967
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.276857	−0.0253794
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	18.0330	1.62599
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.1566	1.07872	0.539361	−	0.842075i	$-0.318666\pi$
$127$	12.1566	1.07872	0.539361	+	0.842075i	$0.318666\pi$
$128$	0	0
$129$	−14.0182	−1.23423
$130$	0	0
$131$	−2.11773	−0.185027	−0.0925135	−	0.995711i	$-0.529490\pi$
$131$	−2.11773	−0.185027	−0.0925135	+	0.995711i	$0.529490\pi$
$132$	0	0
$133$	2.14768	0.186227
$134$	0	0
$135$	−1.30752	−0.112533
$136$	0	0
$137$	−18.0330	−1.54067	−0.770334	−	0.637641i	$-0.779910\pi$
$137$	−18.0330	−1.54067	−0.770334	+	0.637641i	$0.779910\pi$
$138$	0	0
$139$	−9.84016	−0.834631	−0.417316	−	0.908762i	$-0.637029\pi$
$139$	−9.84016	−0.834631	−0.417316	+	0.908762i	$0.637029\pi$
$140$	0	0
$141$	−8.07914	−0.680386
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.90348	0.407211
$146$	0	0
$147$	15.9529	1.31578
$148$	0	0
$149$	−8.88299	−0.727723	−0.363861	−	0.931453i	$-0.618542\pi$
$149$	−8.88299	−0.727723	−0.363861	+	0.931453i	$0.618542\pi$
$150$	0	0
$151$	−4.43937	−0.361271	−0.180636	−	0.983550i	$-0.557815\pi$
$151$	−4.43937	−0.361271	−0.180636	+	0.983550i	$0.557815\pi$
$152$	0	0
$153$	1.68922	0.136565
$154$	0	0
$155$	−7.86488	−0.631723
$156$	0	0
$157$	−4.16719	−0.332578	−0.166289	−	0.986077i	$-0.553178\pi$
$157$	−4.16719	−0.332578	−0.166289	+	0.986077i	$0.553178\pi$
$158$	0	0
$159$	−27.3314	−2.16752
$160$	0	0
$161$	0.0428299	0.00337547
$162$	0	0
$163$	−3.69672	−0.289549	−0.144775	−	0.989465i	$-0.546246\pi$
$163$	−3.69672	−0.289549	−0.144775	+	0.989465i	$0.546246\pi$
$164$	0	0
$165$	4.03957	0.314480
$166$	0	0
$167$	−21.4972	−1.66350	−0.831750	−	0.555151i	$-0.812661\pi$
$167$	−21.4972	−1.66350	−0.831750	+	0.555151i	$0.812661\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−13.1039	−1.00208
$172$	0	0
$173$	−11.6118	−0.882827	−0.441414	−	0.897304i	$-0.645523\pi$
$173$	−11.6118	−0.882827	−0.441414	+	0.897304i	$0.645523\pi$
$174$	0	0
$175$	−0.399804	−0.0302224
$176$	0	0
$177$	−16.9678	−1.27538
$178$	0	0
$179$	−4.97032	−0.371499	−0.185749	−	0.982597i	$-0.559471\pi$
$179$	−4.97032	−0.371499	−0.185749	+	0.982597i	$0.559471\pi$
$180$	0	0
$181$	17.3695	1.29107	0.645534	−	0.763732i	$-0.276635\pi$
$181$	17.3695	1.29107	0.645534	+	0.763732i	$0.276635\pi$
$182$	0	0
$183$	−20.2654	−1.49806
$184$	0	0
$185$	−2.26469	−0.166503
$186$	0	0
$187$	1.19941	0.0877098
$188$	0	0
$189$	−0.522752	−0.0380246
$190$	0	0
$191$	−20.5046	−1.48366	−0.741832	−	0.670586i	$-0.766043\pi$
$191$	−20.5046	−1.48366	−0.741832	+	0.670586i	$0.766043\pi$
$192$	0	0
$193$	−26.6075	−1.91525	−0.957626	−	0.288014i	$-0.907005\pi$
$193$	−26.6075	−1.91525	−0.957626	+	0.288014i	$0.907005\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.408282	−0.0290889	−0.0145445	−	0.999894i	$-0.504630\pi$
$197$	−0.408282	−0.0290889	−0.0145445	+	0.999894i	$0.504630\pi$
$198$	0	0
$199$	24.3996	1.72965	0.864823	−	0.502078i	$-0.167431\pi$
$199$	24.3996	1.72965	0.864823	+	0.502078i	$0.167431\pi$
$200$	0	0
$201$	−3.09724	−0.218463
$202$	0	0
$203$	1.96043	0.137595
$204$	0	0
$205$	7.73205	0.540030
$206$	0	0
$207$	−0.261323	−0.0181632
$208$	0	0
$209$	−9.30426	−0.643589
$210$	0	0
$211$	1.76102	0.121233	0.0606167	−	0.998161i	$-0.480693\pi$
$211$	1.76102	0.121233	0.0606167	+	0.998161i	$0.480693\pi$
$212$	0	0
$213$	9.04452	0.619721
$214$	0	0
$215$	−6.01060	−0.409920
$216$	0	0
$217$	−3.14441	−0.213457
$218$	0	0
$219$	23.8475	1.61146
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−11.3191	−0.757983	−0.378991	−	0.925400i	$-0.623729\pi$
$223$	−11.3191	−0.757983	−0.378991	+	0.925400i	$0.623729\pi$
$224$	0	0
$225$	2.43937	0.162625
$226$	0	0
$227$	−8.06077	−0.535012	−0.267506	−	0.963556i	$-0.586200\pi$
$227$	−8.06077	−0.535012	−0.267506	+	0.963556i	$0.586200\pi$
$228$	0	0
$229$	−11.5715	−0.764666	−0.382333	−	0.924025i	$-0.624879\pi$
$229$	−11.5715	−0.764666	−0.382333	+	0.924025i	$0.624879\pi$
$230$	0	0
$231$	1.61504	0.106262
$232$	0	0
$233$	24.0900	1.57819	0.789094	−	0.614272i	$-0.210550\pi$
$233$	24.0900	1.57819	0.789094	+	0.614272i	$0.210550\pi$
$234$	0	0
$235$	−3.46410	−0.225973
$236$	0	0
$237$	30.6768	1.99267
$238$	0	0
$239$	−30.7089	−1.98639	−0.993196	−	0.116459i	$-0.962846\pi$
$239$	−30.7089	−1.98639	−0.993196	+	0.116459i	$0.962846\pi$
$240$	0	0
$241$	−7.92749	−0.510654	−0.255327	−	0.966855i	$-0.582183\pi$
$241$	−7.92749	−0.510654	−0.255327	+	0.966855i	$0.582183\pi$
$242$	0	0
$243$	20.2572	1.29950
$244$	0	0
$245$	6.84016	0.437002
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−32.8360	−2.08090
$250$	0	0
$251$	22.6224	1.42791	0.713956	−	0.700191i	$-0.246902\pi$
$251$	22.6224	1.42791	0.713956	+	0.700191i	$0.246902\pi$
$252$	0	0
$253$	−0.185550	−0.0116654
$254$	0	0
$255$	1.61504	0.101138
$256$	0	0
$257$	25.7963	1.60913	0.804566	−	0.593863i	$-0.202398\pi$
$257$	25.7963	1.60913	0.804566	+	0.593863i	$0.202398\pi$
$258$	0	0
$259$	−0.905432	−0.0562608
$260$	0	0
$261$	−11.9614	−0.740393
$262$	0	0
$263$	1.59057	0.0980788	0.0490394	−	0.998797i	$-0.484384\pi$
$263$	1.59057	0.0980788	0.0490394	+	0.998797i	$0.484384\pi$
$264$	0	0
$265$	−11.7189	−0.719887
$266$	0	0
$267$	−0.809495	−0.0495403
$268$	0	0
$269$	−27.8228	−1.69638	−0.848192	−	0.529689i	$-0.822309\pi$
$269$	−27.8228	−1.69638	−0.848192	+	0.529689i	$0.822309\pi$
$270$	0	0
$271$	−23.5004	−1.42755	−0.713774	−	0.700376i	$-0.753016\pi$
$271$	−23.5004	−1.42755	−0.713774	+	0.700376i	$0.753016\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.73205	0.104447
$276$	0	0
$277$	−2.99674	−0.180057	−0.0900283	−	0.995939i	$-0.528696\pi$
$277$	−2.99674	−0.180057	−0.0900283	+	0.995939i	$0.528696\pi$
$278$	0	0
$279$	19.1854	1.14860
$280$	0	0
$281$	−24.6085	−1.46802	−0.734011	−	0.679138i	$-0.762354\pi$
$281$	−24.6085	−1.46802	−0.734011	+	0.679138i	$0.762354\pi$
$282$	0	0
$283$	8.16888	0.485590	0.242795	−	0.970078i	$-0.421936\pi$
$283$	8.16888	0.485590	0.242795	+	0.970078i	$0.421936\pi$
$284$	0	0
$285$	−12.5284	−0.742118
$286$	0	0
$287$	3.09131	0.182474
$288$	0	0
$289$	−16.5205	−0.971792
$290$	0	0
$291$	20.6405	1.20997
$292$	0	0
$293$	7.50367	0.438369	0.219185	−	0.975683i	$-0.429660\pi$
$293$	7.50367	0.438369	0.219185	+	0.975683i	$0.429660\pi$
$294$	0	0
$295$	−7.27529	−0.423584
$296$	0	0
$297$	2.26469	0.131411
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−2.40306	−0.138510
$302$	0	0
$303$	−9.57123	−0.549853
$304$	0	0
$305$	−8.68922	−0.497543
$306$	0	0
$307$	−5.95293	−0.339752	−0.169876	−	0.985465i	$-0.554337\pi$
$307$	−5.95293	−0.339752	−0.169876	+	0.985465i	$0.554337\pi$
$308$	0	0
$309$	−26.1969	−1.49029
$310$	0	0
$311$	17.9247	1.01642	0.508208	−	0.861235i	$-0.330308\pi$
$311$	17.9247	1.01642	0.508208	+	0.861235i	$0.330308\pi$
$312$	0	0
$313$	−17.0073	−0.961312	−0.480656	−	0.876909i	$-0.659601\pi$
$313$	−17.0073	−0.961312	−0.480656	+	0.876909i	$0.659601\pi$
$314$	0	0
$315$	0.975272	0.0549504
$316$	0	0
$317$	3.09300	0.173720	0.0868601	−	0.996221i	$-0.472317\pi$
$317$	3.09300	0.173720	0.0868601	+	0.996221i	$0.472317\pi$
$318$	0	0
$319$	−8.49307	−0.475521
$320$	0	0
$321$	41.0552	2.29148
$322$	0	0
$323$	−3.71988	−0.206980
$324$	0	0
$325$	0	0
$326$	0	0
$327$	35.4105	1.95821
$328$	0	0
$329$	−1.38496	−0.0763555
$330$	0	0
$331$	22.5722	1.24068	0.620340	−	0.784333i	$-0.286994\pi$
$331$	22.5722	1.24068	0.620340	+	0.784333i	$0.286994\pi$
$332$	0	0
$333$	5.52442	0.302736
$334$	0	0
$335$	−1.32801	−0.0725568
$336$	0	0
$337$	−18.0603	−0.983808	−0.491904	−	0.870649i	$-0.663699\pi$
$337$	−18.0603	−0.983808	−0.491904	+	0.870649i	$0.663699\pi$
$338$	0	0
$339$	−20.7575	−1.12739
$340$	0	0
$341$	13.6224	0.737693
$342$	0	0
$343$	5.53335	0.298773
$344$	0	0
$345$	−0.249847	−0.0134513
$346$	0	0
$347$	26.4717	1.42108	0.710538	−	0.703659i	$-0.248452\pi$
$347$	26.4717	1.42108	0.710538	+	0.703659i	$0.248452\pi$
$348$	0	0
$349$	12.3772	0.662536	0.331268	−	0.943537i	$-0.392524\pi$
$349$	12.3772	0.662536	0.331268	+	0.943537i	$0.392524\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.2809	−1.02622	−0.513110	−	0.858323i	$-0.671507\pi$
$353$	−19.2809	−1.02622	−0.513110	+	0.858323i	$0.671507\pi$
$354$	0	0
$355$	3.87803	0.205825
$356$	0	0
$357$	0.645699	0.0341740
$358$	0	0
$359$	26.5506	1.40129	0.700643	−	0.713512i	$-0.252897\pi$
$359$	26.5506	1.40129	0.700643	+	0.713512i	$0.252897\pi$
$360$	0	0
$361$	9.85641	0.518758
$362$	0	0
$363$	18.6580	0.979290
$364$	0	0
$365$	10.2251	0.535207
$366$	0	0
$367$	7.37436	0.384938	0.192469	−	0.981303i	$-0.438350\pi$
$367$	7.37436	0.384938	0.192469	+	0.981303i	$0.438350\pi$
$368$	0	0
$369$	−18.8614	−0.981883
$370$	0	0
$371$	−4.68527	−0.243247
$372$	0	0
$373$	−28.3149	−1.46609	−0.733044	−	0.680181i	$-0.761901\pi$
$373$	−28.3149	−1.46609	−0.733044	+	0.680181i	$0.761901\pi$
$374$	0	0
$375$	2.33225	0.120437
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−10.4267	−0.535581	−0.267791	−	0.963477i	$-0.586294\pi$
$379$	−10.4267	−0.535581	−0.267791	+	0.963477i	$0.586294\pi$
$380$	0	0
$381$	−28.3521	−1.45252
$382$	0	0
$383$	−10.8520	−0.554511	−0.277256	−	0.960796i	$-0.589425\pi$
$383$	−10.8520	−0.554511	−0.277256	+	0.960796i	$0.589425\pi$
$384$	0	0
$385$	0.692481	0.0352921
$386$	0	0
$387$	14.6621	0.745317
$388$	0	0
$389$	−20.2893	−1.02871	−0.514353	−	0.857578i	$-0.671968\pi$
$389$	−20.2893	−1.02871	−0.514353	+	0.857578i	$0.671968\pi$
$390$	0	0
$391$	−0.0741836	−0.00375163
$392$	0	0
$393$	4.93907	0.249143
$394$	0	0
$395$	13.1533	0.661815
$396$	0	0
$397$	−25.2527	−1.26740	−0.633698	−	0.773581i	$-0.718464\pi$
$397$	−25.2527	−1.26740	−0.633698	+	0.773581i	$0.718464\pi$
$398$	0	0
$399$	−5.00891	−0.250759
$400$	0	0
$401$	12.5143	0.624933	0.312467	−	0.949929i	$-0.398845\pi$
$401$	12.5143	0.624933	0.312467	+	0.949929i	$0.398845\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	10.3676	0.515169
$406$	0	0
$407$	3.92256	0.194434
$408$	0	0
$409$	5.99576	0.296471	0.148236	−	0.988952i	$-0.452641\pi$
$409$	5.99576	0.296471	0.148236	+	0.988952i	$0.452641\pi$
$410$	0	0
$411$	42.0575	2.07454
$412$	0	0
$413$	−2.90869	−0.143127
$414$	0	0
$415$	−14.0791	−0.691118
$416$	0	0
$417$	22.9497	1.12385
$418$	0	0
$419$	−10.9703	−0.535935	−0.267968	−	0.963428i	$-0.586352\pi$
$419$	−10.9703	−0.535935	−0.267968	+	0.963428i	$0.586352\pi$
$420$	0	0
$421$	−36.6085	−1.78419	−0.892095	−	0.451848i	$-0.850765\pi$
$421$	−36.6085	−1.78419	−0.892095	+	0.451848i	$0.850765\pi$
$422$	0	0
$423$	8.45024	0.410865
$424$	0	0
$425$	0.692481	0.0335903
$426$	0	0
$427$	−3.47399	−0.168118
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.1097	0.583302	0.291651	−	0.956525i	$-0.405795\pi$
$431$	12.1097	0.583302	0.291651	+	0.956525i	$0.405795\pi$
$432$	0	0
$433$	−9.01794	−0.433375	−0.216687	−	0.976241i	$-0.569525\pi$
$433$	−9.01794	−0.433375	−0.216687	+	0.976241i	$0.569525\pi$
$434$	0	0
$435$	−11.4361	−0.548320
$436$	0	0
$437$	0.575468	0.0275284
$438$	0	0
$439$	−12.1509	−0.579933	−0.289966	−	0.957037i	$-0.593644\pi$
$439$	−12.1509	−0.579933	−0.289966	+	0.957037i	$0.593644\pi$
$440$	0	0
$441$	−16.6857	−0.794557
$442$	0	0
$443$	15.3116	0.727476	0.363738	−	0.931501i	$-0.381500\pi$
$443$	15.3116	0.727476	0.363738	+	0.931501i	$0.381500\pi$
$444$	0	0
$445$	−0.347088	−0.0164536
$446$	0	0
$447$	20.7173	0.979895
$448$	0	0
$449$	22.7394	1.07314	0.536569	−	0.843856i	$-0.319720\pi$
$449$	22.7394	1.07314	0.536569	+	0.843856i	$0.319720\pi$
$450$	0	0
$451$	−13.3923	−0.630619
$452$	0	0
$453$	10.3537	0.486460
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.7450	0.502632	0.251316	−	0.967905i	$-0.419137\pi$
$457$	10.7450	0.502632	0.251316	+	0.967905i	$0.419137\pi$
$458$	0	0
$459$	0.905432	0.0422620
$460$	0	0
$461$	11.8903	0.553788	0.276894	−	0.960900i	$-0.410695\pi$
$461$	11.8903	0.553788	0.276894	+	0.960900i	$0.410695\pi$
$462$	0	0
$463$	−3.39726	−0.157884	−0.0789420	−	0.996879i	$-0.525154\pi$
$463$	−3.39726	−0.157884	−0.0789420	+	0.996879i	$0.525154\pi$
$464$	0	0
$465$	18.3428	0.850629
$466$	0	0
$467$	6.39426	0.295891	0.147946	−	0.988996i	$-0.452734\pi$
$467$	6.39426	0.295891	0.147946	+	0.988996i	$0.452734\pi$
$468$	0	0
$469$	−0.530943	−0.0245167
$470$	0	0
$471$	9.71890	0.447823
$472$	0	0
$473$	10.4107	0.478683
$474$	0	0
$475$	−5.37182	−0.246476
$476$	0	0
$477$	28.5868	1.30890
$478$	0	0
$479$	16.3194	0.745651	0.372825	−	0.927902i	$-0.378389\pi$
$479$	16.3194	0.745651	0.372825	+	0.927902i	$0.378389\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.0998900	−0.00454515
$484$	0	0
$485$	8.85004	0.401860
$486$	0	0
$487$	−11.9346	−0.540807	−0.270403	−	0.962747i	$-0.587157\pi$
$487$	−11.9346	−0.540807	−0.270403	+	0.962747i	$0.587157\pi$
$488$	0	0
$489$	8.62166	0.389885
$490$	0	0
$491$	−34.0517	−1.53673	−0.768366	−	0.640011i	$-0.778930\pi$
$491$	−34.0517	−1.53673	−0.768366	+	0.640011i	$0.778930\pi$
$492$	0	0
$493$	−3.39557	−0.152929
$494$	0	0
$495$	−4.22512	−0.189905
$496$	0	0
$497$	1.55045	0.0695473
$498$	0	0
$499$	−12.5854	−0.563398	−0.281699	−	0.959503i	$-0.590898\pi$
$499$	−12.5854	−0.563398	−0.281699	+	0.959503i	$0.590898\pi$
$500$	0	0
$501$	50.1367	2.23994
$502$	0	0
$503$	−18.1887	−0.810992	−0.405496	−	0.914097i	$-0.632901\pi$
$503$	−18.1887	−0.810992	−0.405496	+	0.914097i	$0.632901\pi$
$504$	0	0
$505$	−4.10387	−0.182620
$506$	0	0
$507$	0	0
$508$	0	0
$509$	29.0135	1.28600	0.643001	−	0.765865i	$-0.277689\pi$
$509$	29.0135	1.28600	0.643001	+	0.765865i	$0.277689\pi$
$510$	0	0
$511$	4.08805	0.180845
$512$	0	0
$513$	−7.02375	−0.310106
$514$	0	0
$515$	−11.2325	−0.494961
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	27.0815	1.18875
$520$	0	0
$521$	35.0240	1.53443	0.767214	−	0.641391i	$-0.221642\pi$
$521$	35.0240	1.53443	0.767214	+	0.641391i	$0.221642\pi$
$522$	0	0
$523$	−9.27741	−0.405673	−0.202836	−	0.979213i	$-0.565016\pi$
$523$	−9.27741	−0.405673	−0.202836	+	0.979213i	$0.565016\pi$
$524$	0	0
$525$	0.932442	0.0406951
$526$	0	0
$527$	5.44629	0.237244
$528$	0	0
$529$	−22.9885	−0.999501
$530$	0	0
$531$	17.7472	0.770161
$532$	0	0
$533$	0	0
$534$	0	0
$535$	17.6033	0.761057
$536$	0	0
$537$	11.5920	0.500232
$538$	0	0
$539$	−11.8475	−0.510308
$540$	0	0
$541$	−24.3814	−1.04824	−0.524120	−	0.851644i	$-0.675606\pi$
$541$	−24.3814	−1.04824	−0.524120	+	0.851644i	$0.675606\pi$
$542$	0	0
$543$	−40.5100	−1.73845
$544$	0	0
$545$	15.1830	0.650369
$546$	0	0
$547$	26.5270	1.13421	0.567106	−	0.823645i	$-0.308063\pi$
$547$	26.5270	1.13421	0.567106	+	0.823645i	$0.308063\pi$
$548$	0	0
$549$	21.1963	0.904634
$550$	0	0
$551$	26.3406	1.12215
$552$	0	0
$553$	5.25875	0.223625
$554$	0	0
$555$	5.28181	0.224200
$556$	0	0
$557$	−3.21171	−0.136085	−0.0680423	−	0.997682i	$-0.521675\pi$
$557$	−3.21171	−0.136085	−0.0680423	+	0.997682i	$0.521675\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.79733	−0.118103
$562$	0	0
$563$	−34.2852	−1.44495	−0.722474	−	0.691398i	$-0.756995\pi$
$563$	−34.2852	−1.44495	−0.722474	+	0.691398i	$0.756995\pi$
$564$	0	0
$565$	−8.90021	−0.374435
$566$	0	0
$567$	4.14500	0.174074
$568$	0	0
$569$	−35.6457	−1.49434	−0.747172	−	0.664630i	$-0.768589\pi$
$569$	−35.6457	−1.49434	−0.747172	+	0.664630i	$0.768589\pi$
$570$	0	0
$571$	4.53590	0.189821	0.0949107	−	0.995486i	$-0.469743\pi$
$571$	4.53590	0.189821	0.0949107	+	0.995486i	$0.469743\pi$
$572$	0	0
$573$	47.8219	1.99779
$574$	0	0
$575$	−0.107127	−0.00446751
$576$	0	0
$577$	18.4475	0.767981	0.383991	−	0.923337i	$-0.374550\pi$
$577$	18.4475	0.767981	0.383991	+	0.923337i	$0.374550\pi$
$578$	0	0
$579$	62.0553	2.57893
$580$	0	0
$581$	−5.62890	−0.233526
$582$	0	0
$583$	20.2977	0.840646
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.9914	1.77444	0.887222	−	0.461343i	$-0.152632\pi$
$587$	42.9914	1.77444	0.887222	+	0.461343i	$0.152632\pi$
$588$	0	0
$589$	−42.2487	−1.74083
$590$	0	0
$591$	0.952215	0.0391689
$592$	0	0
$593$	12.8614	0.528153	0.264076	−	0.964502i	$-0.414933\pi$
$593$	12.8614	0.528153	0.264076	+	0.964502i	$0.414933\pi$
$594$	0	0
$595$	0.276857	0.0113500
$596$	0	0
$597$	−56.9060	−2.32901
$598$	0	0
$599$	28.3170	1.15700	0.578500	−	0.815682i	$-0.303638\pi$
$599$	28.3170	1.15700	0.578500	+	0.815682i	$0.303638\pi$
$600$	0	0
$601$	−7.13469	−0.291030	−0.145515	−	0.989356i	$-0.546484\pi$
$601$	−7.13469	−0.291030	−0.145515	+	0.989356i	$0.546484\pi$
$602$	0	0
$603$	3.23951	0.131923
$604$	0	0
$605$	8.00000	0.325246
$606$	0	0
$607$	−22.1802	−0.900266	−0.450133	−	0.892962i	$-0.648623\pi$
$607$	−22.1802	−0.900266	−0.450133	+	0.892962i	$0.648623\pi$
$608$	0	0
$609$	−4.57221	−0.185275
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0.322622	0.0130306	0.00651530	−	0.999979i	$-0.497926\pi$
$613$	0.322622	0.0130306	0.00651530	+	0.999979i	$0.497926\pi$
$614$	0	0
$615$	−18.0330	−0.727163
$616$	0	0
$617$	−37.1987	−1.49756	−0.748781	−	0.662817i	$-0.769361\pi$
$617$	−37.1987	−1.49756	−0.748781	+	0.662817i	$0.769361\pi$
$618$	0	0
$619$	3.94911	0.158728	0.0793641	−	0.996846i	$-0.474711\pi$
$619$	3.94911	0.158728	0.0793641	+	0.996846i	$0.474711\pi$
$620$	0	0
$621$	−0.140071	−0.00562085
$622$	0	0
$623$	−0.138767	−0.00555959
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.6998	0.866607
$628$	0	0
$629$	1.56825	0.0625304
$630$	0	0
$631$	−33.5815	−1.33686	−0.668429	−	0.743776i	$-0.733033\pi$
$631$	−33.5815	−1.33686	−0.668429	+	0.743776i	$0.733033\pi$
$632$	0	0
$633$	−4.10713	−0.163244
$634$	0	0
$635$	−12.1566	−0.482419
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.45997	−0.374231
$640$	0	0
$641$	33.1800	1.31053	0.655266	−	0.755398i	$-0.272557\pi$
$641$	33.1800	1.31053	0.655266	+	0.755398i	$0.272557\pi$
$642$	0	0
$643$	31.3666	1.23698	0.618489	−	0.785793i	$-0.287745\pi$
$643$	31.3666	1.23698	0.618489	+	0.785793i	$0.287745\pi$
$644$	0	0
$645$	14.0182	0.551966
$646$	0	0
$647$	19.5906	0.770185	0.385092	−	0.922878i	$-0.374170\pi$
$647$	19.5906	0.770185	0.385092	+	0.922878i	$0.374170\pi$
$648$	0	0
$649$	12.6012	0.494639
$650$	0	0
$651$	7.33355	0.287424
$652$	0	0
$653$	7.11251	0.278334	0.139167	−	0.990269i	$-0.455557\pi$
$653$	7.11251	0.278334	0.139167	+	0.990269i	$0.455557\pi$
$654$	0	0
$655$	2.11773	0.0827466
$656$	0	0
$657$	−24.9429	−0.973115
$658$	0	0
$659$	18.5842	0.723938	0.361969	−	0.932190i	$-0.382105\pi$
$659$	18.5842	0.723938	0.361969	+	0.932190i	$0.382105\pi$
$660$	0	0
$661$	16.7908	0.653087	0.326543	−	0.945182i	$-0.394116\pi$
$661$	16.7908	0.653087	0.326543	+	0.945182i	$0.394116\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.14768	−0.0832833
$666$	0	0
$667$	0.525296	0.0203395
$668$	0	0
$669$	26.3989	1.02064
$670$	0	0
$671$	15.0502	0.581005
$672$	0	0
$673$	22.5914	0.870834	0.435417	−	0.900229i	$-0.356601\pi$
$673$	22.5914	0.870834	0.435417	+	0.900229i	$0.356601\pi$
$674$	0	0
$675$	1.30752	0.0503264
$676$	0	0
$677$	5.31616	0.204317	0.102158	−	0.994768i	$-0.467425\pi$
$677$	5.31616	0.204317	0.102158	+	0.994768i	$0.467425\pi$
$678$	0	0
$679$	3.53829	0.135787
$680$	0	0
$681$	18.7997	0.720407
$682$	0	0
$683$	−3.95391	−0.151292	−0.0756461	−	0.997135i	$-0.524102\pi$
$683$	−3.95391	−0.151292	−0.0756461	+	0.997135i	$0.524102\pi$
$684$	0	0
$685$	18.0330	0.689007
$686$	0	0
$687$	26.9876	1.02964
$688$	0	0
$689$	0	0
$690$	0	0
$691$	12.8741	0.489753	0.244877	−	0.969554i	$-0.421253\pi$
$691$	12.8741	0.489753	0.244877	+	0.969554i	$0.421253\pi$
$692$	0	0
$693$	−1.68922	−0.0641682
$694$	0	0
$695$	9.84016	0.373258
$696$	0	0
$697$	−5.35430	−0.202809
$698$	0	0
$699$	−56.1838	−2.12507
$700$	0	0
$701$	−34.9777	−1.32109	−0.660544	−	0.750787i	$-0.729674\pi$
$701$	−34.9777	−1.32109	−0.660544	+	0.750787i	$0.729674\pi$
$702$	0	0
$703$	−12.1655	−0.458830
$704$	0	0
$705$	8.07914	0.304278
$706$	0	0
$707$	−1.64074	−0.0617065
$708$	0	0
$709$	19.7665	0.742347	0.371173	−	0.928564i	$-0.378956\pi$
$709$	19.7665	0.742347	0.371173	+	0.928564i	$0.378956\pi$
$710$	0	0
$711$	−32.0859	−1.20331
$712$	0	0
$713$	−0.842543	−0.0315535
$714$	0	0
$715$	0	0
$716$	0	0
$717$	71.6206	2.67472
$718$	0	0
$719$	−44.2467	−1.65012	−0.825062	−	0.565042i	$-0.808860\pi$
$719$	−44.2467	−1.65012	−0.825062	+	0.565042i	$0.808860\pi$
$720$	0	0
$721$	−4.49079	−0.167246
$722$	0	0
$723$	18.4889	0.687608
$724$	0	0
$725$	−4.90348	−0.182111
$726$	0	0
$727$	31.4877	1.16781	0.583907	−	0.811821i	$-0.301523\pi$
$727$	31.4877	1.16781	0.583907	+	0.811821i	$0.301523\pi$
$728$	0	0
$729$	−16.1420	−0.597853
$730$	0	0
$731$	4.16223	0.153946
$732$	0	0
$733$	−42.4714	−1.56872	−0.784359	−	0.620307i	$-0.787008\pi$
$733$	−42.4714	−1.56872	−0.784359	+	0.620307i	$0.787008\pi$
$734$	0	0
$735$	−15.9529	−0.588433
$736$	0	0
$737$	2.30018	0.0847281
$738$	0	0
$739$	−13.7835	−0.507033	−0.253516	−	0.967331i	$-0.581587\pi$
$739$	−13.7835	−0.507033	−0.253516	+	0.967331i	$0.581587\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.9392	0.584753	0.292377	−	0.956303i	$-0.405554\pi$
$743$	15.9392	0.584753	0.292377	+	0.956303i	$0.405554\pi$
$744$	0	0
$745$	8.88299	0.325447
$746$	0	0
$747$	34.3443	1.25659
$748$	0	0
$749$	7.03787	0.257158
$750$	0	0
$751$	−1.51708	−0.0553590	−0.0276795	−	0.999617i	$-0.508812\pi$
$751$	−1.51708	−0.0553590	−0.0276795	+	0.999617i	$0.508812\pi$
$752$	0	0
$753$	−52.7610	−1.92272
$754$	0	0
$755$	4.43937	0.161565
$756$	0	0
$757$	47.4829	1.72579	0.862897	−	0.505380i	$-0.168648\pi$
$757$	47.4829	1.72579	0.862897	+	0.505380i	$0.168648\pi$
$758$	0	0
$759$	0.432748	0.0157078
$760$	0	0
$761$	38.5445	1.39724	0.698618	−	0.715495i	$-0.253799\pi$
$761$	38.5445	1.39724	0.698618	+	0.715495i	$0.253799\pi$
$762$	0	0
$763$	6.07023	0.219757
$764$	0	0
$765$	−1.68922	−0.0610739
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−39.8645	−1.43755	−0.718775	−	0.695243i	$-0.755297\pi$
$769$	−39.8645	−1.43755	−0.718775	+	0.695243i	$0.755297\pi$
$770$	0	0
$771$	−60.1634	−2.16673
$772$	0	0
$773$	32.9547	1.18530	0.592650	−	0.805460i	$-0.298082\pi$
$773$	32.9547	1.18530	0.592650	+	0.805460i	$0.298082\pi$
$774$	0	0
$775$	7.86488	0.282515
$776$	0	0
$777$	2.11169	0.0757565
$778$	0	0
$779$	41.5352	1.48815
$780$	0	0
$781$	−6.71695	−0.240351
$782$	0	0
$783$	−6.41139	−0.229124
$784$	0	0
$785$	4.16719	0.148733
$786$	0	0
$787$	1.07930	0.0384728	0.0192364	−	0.999815i	$-0.493876\pi$
$787$	1.07930	0.0384728	0.0192364	+	0.999815i	$0.493876\pi$
$788$	0	0
$789$	−3.70960	−0.132065
$790$	0	0
$791$	−3.55834	−0.126520
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27.3314	0.969344
$796$	0	0
$797$	−19.1014	−0.676607	−0.338304	−	0.941037i	$-0.609853\pi$
$797$	−19.1014	−0.676607	−0.338304	+	0.941037i	$0.609853\pi$
$798$	0	0
$799$	2.39883	0.0848644
$800$	0	0
$801$	0.846678	0.0299159
$802$	0	0
$803$	−17.7104	−0.624987
$804$	0	0
$805$	−0.0428299	−0.00150956
$806$	0	0
$807$	64.8896	2.28422
$808$	0	0
$809$	1.76340	0.0619980	0.0309990	−	0.999519i	$-0.490131\pi$
$809$	1.76340	0.0619980	0.0309990	+	0.999519i	$0.490131\pi$
$810$	0	0
$811$	−52.3298	−1.83755	−0.918774	−	0.394784i	$-0.870819\pi$
$811$	−52.3298	−1.83755	−0.918774	+	0.394784i	$0.870819\pi$
$812$	0	0
$813$	54.8088	1.92223
$814$	0	0
$815$	3.69672	0.129490
$816$	0	0
$817$	−32.2879	−1.12961
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.2703	0.498038	0.249019	−	0.968499i	$-0.419892\pi$
$821$	14.2703	0.498038	0.249019	+	0.968499i	$0.419892\pi$
$822$	0	0
$823$	43.7145	1.52379	0.761896	−	0.647699i	$-0.224269\pi$
$823$	43.7145	1.52379	0.761896	+	0.647699i	$0.224269\pi$
$824$	0	0
$825$	−4.03957	−0.140640
$826$	0	0
$827$	22.5962	0.785748	0.392874	−	0.919592i	$-0.371481\pi$
$827$	22.5962	0.785748	0.392874	+	0.919592i	$0.371481\pi$
$828$	0	0
$829$	54.0946	1.87878	0.939392	−	0.342844i	$-0.111390\pi$
$829$	54.0946	1.87878	0.939392	+	0.342844i	$0.111390\pi$
$830$	0	0
$831$	6.98914	0.242450
$832$	0	0
$833$	−4.73668	−0.164116
$834$	0	0
$835$	21.4972	0.743940
$836$	0	0
$837$	10.2835	0.355449
$838$	0	0
$839$	−22.2337	−0.767593	−0.383796	−	0.923418i	$-0.625384\pi$
$839$	−22.2337	−0.767593	−0.383796	+	0.923418i	$0.625384\pi$
$840$	0	0
$841$	−4.95593	−0.170894
$842$	0	0
$843$	57.3931	1.97672
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.19843	0.109900
$848$	0	0
$849$	−19.0518	−0.653858
$850$	0	0
$851$	−0.242610	−0.00831656
$852$	0	0
$853$	16.5312	0.566019	0.283009	−	0.959117i	$-0.408667\pi$
$853$	16.5312	0.566019	0.283009	+	0.959117i	$0.408667\pi$
$854$	0	0
$855$	13.1039	0.448143
$856$	0	0
$857$	−27.5842	−0.942259	−0.471129	−	0.882064i	$-0.656154\pi$
$857$	−27.5842	−0.942259	−0.471129	+	0.882064i	$0.656154\pi$
$858$	0	0
$859$	−32.5016	−1.10894	−0.554469	−	0.832204i	$-0.687079\pi$
$859$	−32.5016	−1.10894	−0.554469	+	0.832204i	$0.687079\pi$
$860$	0	0
$861$	−7.20969	−0.245706
$862$	0	0
$863$	−29.7986	−1.01436	−0.507178	−	0.861842i	$-0.669311\pi$
$863$	−29.7986	−1.01436	−0.507178	+	0.861842i	$0.669311\pi$
$864$	0	0
$865$	11.6118	0.394812
$866$	0	0
$867$	38.5298	1.30854
$868$	0	0
$869$	−22.7822	−0.772834
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−21.5886	−0.730662
$874$	0	0
$875$	0.399804	0.0135159
$876$	0	0
$877$	−14.4036	−0.486376	−0.243188	−	0.969979i	$-0.578193\pi$
$877$	−14.4036	−0.486376	−0.243188	+	0.969979i	$0.578193\pi$
$878$	0	0
$879$	−17.5004	−0.590274
$880$	0	0
$881$	−7.36914	−0.248273	−0.124136	−	0.992265i	$-0.539616\pi$
$881$	−7.36914	−0.248273	−0.124136	+	0.992265i	$0.539616\pi$
$882$	0	0
$883$	−24.3646	−0.819933	−0.409967	−	0.912101i	$-0.634460\pi$
$883$	−24.3646	−0.819933	−0.409967	+	0.912101i	$0.634460\pi$
$884$	0	0
$885$	16.9678	0.570365
$886$	0	0
$887$	−24.0644	−0.808004	−0.404002	−	0.914758i	$-0.632381\pi$
$887$	−24.0644	−0.808004	−0.404002	+	0.914758i	$0.632381\pi$
$888$	0	0
$889$	−4.86025	−0.163008
$890$	0	0
$891$	−17.9572	−0.601588
$892$	0	0
$893$	−18.6085	−0.622710
$894$	0	0
$895$	4.97032	0.166139
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38.5653	−1.28622
$900$	0	0
$901$	8.11512	0.270354
$902$	0	0
$903$	5.60454	0.186507
$904$	0	0
$905$	−17.3695	−0.577383
$906$	0	0
$907$	37.5073	1.24541	0.622705	−	0.782457i	$-0.286034\pi$
$907$	37.5073	1.24541	0.622705	+	0.782457i	$0.286034\pi$
$908$	0	0
$909$	10.0109	0.332039
$910$	0	0
$911$	12.6000	0.417458	0.208729	−	0.977974i	$-0.433067\pi$
$911$	12.6000	0.417458	0.208729	+	0.977974i	$0.433067\pi$
$912$	0	0
$913$	24.3858	0.807052
$914$	0	0
$915$	20.2654	0.669954
$916$	0	0
$917$	0.846678	0.0279598
$918$	0	0
$919$	−31.6664	−1.04458	−0.522288	−	0.852769i	$-0.674922\pi$
$919$	−31.6664	−1.04458	−0.522288	+	0.852769i	$0.674922\pi$
$920$	0	0
$921$	13.8837	0.457484
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.26469	0.0744625
$926$	0	0
$927$	27.4002	0.899940
$928$	0	0
$929$	29.7378	0.975666	0.487833	−	0.872937i	$-0.337787\pi$
$929$	29.7378	0.975666	0.487833	+	0.872937i	$0.337787\pi$
$930$	0	0
$931$	36.7441	1.20424
$932$	0	0
$933$	−41.8048	−1.36863
$934$	0	0
$935$	−1.19941	−0.0392250
$936$	0	0
$937$	−52.3124	−1.70897	−0.854486	−	0.519474i	$-0.826128\pi$
$937$	−52.3124	−1.70897	−0.854486	+	0.519474i	$0.826128\pi$
$938$	0	0
$939$	39.6653	1.29443
$940$	0	0
$941$	−29.5767	−0.964174	−0.482087	−	0.876123i	$-0.660121\pi$
$941$	−29.5767	−0.964174	−0.482087	+	0.876123i	$0.660121\pi$
$942$	0	0
$943$	0.828313	0.0269736
$944$	0	0
$945$	0.522752	0.0170051
$946$	0	0
$947$	48.0330	1.56086	0.780432	−	0.625240i	$-0.214999\pi$
$947$	48.0330	1.56086	0.780432	+	0.625240i	$0.214999\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−7.21364	−0.233918
$952$	0	0
$953$	29.7824	0.964745	0.482373	−	0.875966i	$-0.339775\pi$
$953$	29.7824	0.964745	0.482373	+	0.875966i	$0.339775\pi$
$954$	0	0
$955$	20.5046	0.663515
$956$	0	0
$957$	19.8079	0.640299
$958$	0	0
$959$	7.20969	0.232813
$960$	0	0
$961$	30.8564	0.995368
$962$	0	0
$963$	−42.9410	−1.38376
$964$	0	0
$965$	26.6075	0.856527
$966$	0	0
$967$	−20.6730	−0.664798	−0.332399	−	0.943139i	$-0.607858\pi$
$967$	−20.6730	−0.664798	−0.332399	+	0.943139i	$0.607858\pi$
$968$	0	0
$969$	8.67568	0.278703
$970$	0	0
$971$	−19.9861	−0.641386	−0.320693	−	0.947183i	$-0.603916\pi$
$971$	−19.9861	−0.641386	−0.320693	+	0.947183i	$0.603916\pi$
$972$	0	0
$973$	3.93414	0.126123
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.6276	1.45976	0.729878	−	0.683577i	$-0.239577\pi$
$977$	45.6276	1.45976	0.729878	+	0.683577i	$0.239577\pi$
$978$	0	0
$979$	0.601174	0.0192136
$980$	0	0
$981$	−37.0370	−1.18250
$982$	0	0
$983$	−54.1966	−1.72860	−0.864301	−	0.502975i	$-0.832239\pi$
$983$	−54.1966	−1.72860	−0.864301	+	0.502975i	$0.832239\pi$
$984$	0	0
$985$	0.408282	0.0130090
$986$	0	0
$987$	3.23007	0.102814
$988$	0	0
$989$	−0.643899	−0.0204748
$990$	0	0
$991$	46.5278	1.47800	0.739002	−	0.673703i	$-0.235297\pi$
$991$	46.5278	1.47800	0.739002	+	0.673703i	$0.235297\pi$
$992$	0	0
$993$	−52.6440	−1.67061
$994$	0	0
$995$	−24.3996	−0.773521
$996$	0	0
$997$	−39.1029	−1.23840	−0.619200	−	0.785233i	$-0.712543\pi$
$997$	−39.1029	−1.23840	−0.619200	+	0.785233i	$0.712543\pi$
$998$	0	0
$999$	2.96112	0.0936858

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.p.1.1		4
13.5	odd	4		3380.2.f.i.3041.2		8
13.6	odd	12		260.2.x.a.101.4	✓	8
13.8	odd	4		3380.2.f.i.3041.1		8
13.11	odd	12		260.2.x.a.121.4	yes	8
13.12	even	2		3380.2.a.q.1.1		4
39.11	even	12		2340.2.dj.d.901.3		8
39.32	even	12		2340.2.dj.d.361.1		8
52.11	even	12		1040.2.da.c.641.1		8
52.19	even	12		1040.2.da.c.881.1		8
65.19	odd	12		1300.2.y.b.101.1		8
65.24	odd	12		1300.2.y.b.901.1		8
65.32	even	12		1300.2.ba.c.49.3		8
65.37	even	12		1300.2.ba.b.849.2		8
65.58	even	12		1300.2.ba.b.49.2		8
65.63	even	12		1300.2.ba.c.849.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.x.a.101.4	✓	8	13.6	odd	12
260.2.x.a.121.4	yes	8	13.11	odd	12
1040.2.da.c.641.1		8	52.11	even	12
1040.2.da.c.881.1		8	52.19	even	12
1300.2.y.b.101.1		8	65.19	odd	12
1300.2.y.b.901.1		8	65.24	odd	12
1300.2.ba.b.49.2		8	65.58	even	12
1300.2.ba.b.849.2		8	65.37	even	12
1300.2.ba.c.49.3		8	65.32	even	12
1300.2.ba.c.849.3		8	65.63	even	12
2340.2.dj.d.361.1		8	39.32	even	12
2340.2.dj.d.901.3		8	39.11	even	12
3380.2.a.p.1.1		4	1.1	even	1	trivial
3380.2.a.q.1.1		4	13.12	even	2
3380.2.f.i.3041.1		8	13.8	odd	4
3380.2.f.i.3041.2		8	13.5	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33225 q^{3} -1.00000 q^{5} -0.399804 q^{7} +2.43937 q^{9} +O(q^{10})\)	\(q-2.33225 q^{3} -1.00000 q^{5} -0.399804 q^{7} +2.43937 q^{9} +1.73205 q^{11} +2.33225 q^{15} +0.692481 q^{17} -5.37182 q^{19} +0.932442 q^{21} -0.107127 q^{23} +1.00000 q^{25} +1.30752 q^{27} -4.90348 q^{29} +7.86488 q^{31} -4.03957 q^{33} +0.399804 q^{35} +2.26469 q^{37} -7.73205 q^{41} +6.01060 q^{43} -2.43937 q^{45} +3.46410 q^{47} -6.84016 q^{49} -1.61504 q^{51} +11.7189 q^{53} -1.73205 q^{55} +12.5284 q^{57} +7.27529 q^{59} +8.68922 q^{61} -0.975272 q^{63} +1.32801 q^{67} +0.249847 q^{69} -3.87803 q^{71} -10.2251 q^{73} -2.33225 q^{75} -0.692481 q^{77} -13.1533 q^{79} -10.3676 q^{81} +14.0791 q^{83} -0.692481 q^{85} +11.4361 q^{87} +0.347088 q^{89} -18.3428 q^{93} +5.37182 q^{95} -8.85004 q^{97} +4.22512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9	\( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{15} + 6 q^{17} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 2 q^{27} - 6 q^{33} + 6 q^{35} - 18 q^{37} - 24 q^{41} + 10 q^{43} - 4 q^{45} - 4 q^{49} + 12 q^{53} + 18 q^{57} - 12 q^{59} + 4 q^{61} - 12 q^{63} - 18 q^{67} - 24 q^{69} - 12 q^{71} - 24 q^{73} + 2 q^{75} - 6 q^{77} - 8 q^{79} - 8 q^{81} + 36 q^{83} - 6 q^{85} + 6 q^{87} - 12 q^{89} - 48 q^{93} - 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^15 + 6 * q^17 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 2 * q^27 - 6 * q^33 + 6 * q^35 - 18 * q^37 - 24 * q^41 + 10 * q^43 - 4 * q^45 - 4 * q^49 + 12 * q^53 + 18 * q^57 - 12 * q^59 + 4 * q^61 - 12 * q^63 - 18 * q^67 - 24 * q^69 - 12 * q^71 - 24 * q^73 + 2 * q^75 - 6 * q^77 - 8 * q^79 - 8 * q^81 + 36 * q^83 - 6 * q^85 + 6 * q^87 - 12 * q^89 - 48 * q^93 - 6 * q^97

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