LMFDB - Embedded newform 3332.2.a.g.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-2.61803 q^{3} -3.61803 q^{5} +3.85410 q^{9} +O(q^{10})$	$q-2.61803 q^{3} -3.61803 q^{5} +3.85410 q^{9} -1.23607 q^{13} +9.47214 q^{15} -1.00000 q^{17} -6.47214 q^{19} +0.763932 q^{23} +8.09017 q^{25} -2.23607 q^{27} +7.23607 q^{29} +10.5623 q^{31} -9.70820 q^{37} +3.23607 q^{39} +7.85410 q^{41} -1.85410 q^{43} -13.9443 q^{45} +6.94427 q^{47} +2.61803 q^{51} -3.32624 q^{53} +16.9443 q^{57} -0.763932 q^{59} +7.56231 q^{61} +4.47214 q^{65} +6.09017 q^{67} -2.00000 q^{69} -9.23607 q^{71} -3.85410 q^{73} -21.1803 q^{75} +11.7082 q^{79} -5.70820 q^{81} +10.4721 q^{83} +3.61803 q^{85} -18.9443 q^{87} -8.76393 q^{89} -27.6525 q^{93} +23.4164 q^{95} +2.85410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 5 q^{5} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 5 * q^5 + q^9	$ 2 q - 3 q^{3} - 5 q^{5} + q^{9} + 2 q^{13} + 10 q^{15} - 2 q^{17} - 4 q^{19} + 6 q^{23} + 5 q^{25} + 10 q^{29} + q^{31} - 6 q^{37} + 2 q^{39} + 9 q^{41} + 3 q^{43} - 10 q^{45} - 4 q^{47} + 3 q^{51} + 9 q^{53} + 16 q^{57} - 6 q^{59} - 5 q^{61} + q^{67} - 4 q^{69} - 14 q^{71} - q^{73} - 20 q^{75} + 10 q^{79} + 2 q^{81} + 12 q^{83} + 5 q^{85} - 20 q^{87} - 22 q^{89} - 24 q^{93} + 20 q^{95} - q^{97}+O(q^{100}) $ 2 * q - 3 * q^3 - 5 * q^5 + q^9 + 2 * q^13 + 10 * q^15 - 2 * q^17 - 4 * q^19 + 6 * q^23 + 5 * q^25 + 10 * q^29 + q^31 - 6 * q^37 + 2 * q^39 + 9 * q^41 + 3 * q^43 - 10 * q^45 - 4 * q^47 + 3 * q^51 + 9 * q^53 + 16 * q^57 - 6 * q^59 - 5 * q^61 + q^67 - 4 * q^69 - 14 * q^71 - q^73 - 20 * q^75 + 10 * q^79 + 2 * q^81 + 12 * q^83 + 5 * q^85 - 20 * q^87 - 22 * q^89 - 24 * q^93 + 20 * q^95 - 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
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	0	0
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	0	0
$15$	9.47214	2.44569
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.763932	0.159291	0.0796454	−	0.996823i	$-0.474621\pi$
$23$	0.763932	0.159291	0.0796454	+	0.996823i	$0.474621\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	7.23607	1.34370	0.671852	−	0.740685i	$-0.265499\pi$
$29$	7.23607	1.34370	0.671852	+	0.740685i	$0.265499\pi$
$30$	0	0
$31$	10.5623	1.89705	0.948523	−	0.316708i	$-0.102578\pi$
$31$	10.5623	1.89705	0.948523	+	0.316708i	$0.102578\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.70820	−1.59602	−0.798009	−	0.602645i	$-0.794114\pi$
$37$	−9.70820	−1.59602	−0.798009	+	0.602645i	$0.794114\pi$
$38$	0	0
$39$	3.23607	0.518186
$40$	0	0
$41$	7.85410	1.22660	0.613302	−	0.789848i	$-0.289841\pi$
$41$	7.85410	1.22660	0.613302	+	0.789848i	$0.289841\pi$
$42$	0	0
$43$	−1.85410	−0.282748	−0.141374	−	0.989956i	$-0.545152\pi$
$43$	−1.85410	−0.282748	−0.141374	+	0.989956i	$0.545152\pi$
$44$	0	0
$45$	−13.9443	−2.07869
$46$	0	0
$47$	6.94427	1.01293	0.506463	−	0.862262i	$-0.330953\pi$
$47$	6.94427	1.01293	0.506463	+	0.862262i	$0.330953\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.61803	0.366598
$52$	0	0
$53$	−3.32624	−0.456894	−0.228447	−	0.973556i	$-0.573365\pi$
$53$	−3.32624	−0.456894	−0.228447	+	0.973556i	$0.573365\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.9443	2.24432
$58$	0	0
$59$	−0.763932	−0.0994555	−0.0497277	−	0.998763i	$-0.515835\pi$
$59$	−0.763932	−0.0994555	−0.0497277	+	0.998763i	$0.515835\pi$
$60$	0	0
$61$	7.56231	0.968254	0.484127	−	0.874998i	$-0.339137\pi$
$61$	7.56231	0.968254	0.484127	+	0.874998i	$0.339137\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	6.09017	0.744033	0.372016	−	0.928226i	$-0.378667\pi$
$67$	6.09017	0.744033	0.372016	+	0.928226i	$0.378667\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−9.23607	−1.09612	−0.548060	−	0.836439i	$-0.684633\pi$
$71$	−9.23607	−1.09612	−0.548060	+	0.836439i	$0.684633\pi$
$72$	0	0
$73$	−3.85410	−0.451089	−0.225544	−	0.974233i	$-0.572416\pi$
$73$	−3.85410	−0.451089	−0.225544	+	0.974233i	$0.572416\pi$
$74$	0	0
$75$	−21.1803	−2.44569
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.7082	1.31728	0.658638	−	0.752460i	$-0.271133\pi$
$79$	11.7082	1.31728	0.658638	+	0.752460i	$0.271133\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	10.4721	1.14947	0.574733	−	0.818341i	$-0.305106\pi$
$83$	10.4721	1.14947	0.574733	+	0.818341i	$0.305106\pi$
$84$	0	0
$85$	3.61803	0.392431
$86$	0	0
$87$	−18.9443	−2.03104
$88$	0	0
$89$	−8.76393	−0.928975	−0.464487	−	0.885580i	$-0.653761\pi$
$89$	−8.76393	−0.928975	−0.464487	+	0.885580i	$0.653761\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−27.6525	−2.86743
$94$	0	0
$95$	23.4164	2.40247
$96$	0	0
$97$	2.85410	0.289790	0.144895	−	0.989447i	$-0.453716\pi$
$97$	2.85410	0.289790	0.144895	+	0.989447i	$0.453716\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.52786	−0.749050	−0.374525	−	0.927217i	$-0.622194\pi$
$101$	−7.52786	−0.749050	−0.374525	+	0.927217i	$0.622194\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	12.1803	1.16666	0.583332	−	0.812233i	$-0.301748\pi$
$109$	12.1803	1.16666	0.583332	+	0.812233i	$0.301748\pi$
$110$	0	0
$111$	25.4164	2.41242
$112$	0	0
$113$	12.9443	1.21769	0.608847	−	0.793287i	$-0.291632\pi$
$113$	12.9443	1.21769	0.608847	+	0.793287i	$0.291632\pi$
$114$	0	0
$115$	−2.76393	−0.257738
$116$	0	0
$117$	−4.76393	−0.440426
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−20.5623	−1.85404
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−15.6180	−1.38588	−0.692938	−	0.720997i	$-0.743684\pi$
$127$	−15.6180	−1.38588	−0.692938	+	0.720997i	$0.743684\pi$
$128$	0	0
$129$	4.85410	0.427380
$130$	0	0
$131$	−6.47214	−0.565473	−0.282737	−	0.959198i	$-0.591242\pi$
$131$	−6.47214	−0.565473	−0.282737	+	0.959198i	$0.591242\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.09017	0.696291
$136$	0	0
$137$	11.0902	0.947497	0.473749	−	0.880660i	$-0.342901\pi$
$137$	11.0902	0.947497	0.473749	+	0.880660i	$0.342901\pi$
$138$	0	0
$139$	13.6180	1.15507	0.577533	−	0.816367i	$-0.304015\pi$
$139$	13.6180	1.15507	0.577533	+	0.816367i	$0.304015\pi$
$140$	0	0
$141$	−18.1803	−1.53106
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−26.1803	−2.17416
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.38197	−0.277061	−0.138531	−	0.990358i	$-0.544238\pi$
$149$	−3.38197	−0.277061	−0.138531	+	0.990358i	$0.544238\pi$
$150$	0	0
$151$	−21.2705	−1.73097	−0.865485	−	0.500935i	$-0.832989\pi$
$151$	−21.2705	−1.73097	−0.865485	+	0.500935i	$0.832989\pi$
$152$	0	0
$153$	−3.85410	−0.311586
$154$	0	0
$155$	−38.2148	−3.06949
$156$	0	0
$157$	−23.7082	−1.89212	−0.946060	−	0.323991i	$-0.894975\pi$
$157$	−23.7082	−1.89212	−0.946060	+	0.323991i	$0.894975\pi$
$158$	0	0
$159$	8.70820	0.690605
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.437694	0.0338698	0.0169349	−	0.999857i	$-0.494609\pi$
$167$	0.437694	0.0338698	0.0169349	+	0.999857i	$0.494609\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−24.9443	−1.90754
$172$	0	0
$173$	−22.3262	−1.69743	−0.848716	−	0.528849i	$-0.822624\pi$
$173$	−22.3262	−1.69743	−0.848716	+	0.528849i	$0.822624\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	−3.09017	−0.230970	−0.115485	−	0.993309i	$-0.536842\pi$
$179$	−3.09017	−0.230970	−0.115485	+	0.993309i	$0.536842\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−19.7984	−1.46354
$184$	0	0
$185$	35.1246	2.58241
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.0902	1.59839	0.799194	−	0.601073i	$-0.205260\pi$
$191$	22.0902	1.59839	0.799194	+	0.601073i	$0.205260\pi$
$192$	0	0
$193$	−5.52786	−0.397904	−0.198952	−	0.980009i	$-0.563754\pi$
$193$	−5.52786	−0.397904	−0.198952	+	0.980009i	$0.563754\pi$
$194$	0	0
$195$	−11.7082	−0.838442
$196$	0	0
$197$	−15.7082	−1.11916	−0.559582	−	0.828775i	$-0.689038\pi$
$197$	−15.7082	−1.11916	−0.559582	+	0.828775i	$0.689038\pi$
$198$	0	0
$199$	−12.1459	−0.861000	−0.430500	−	0.902591i	$-0.641663\pi$
$199$	−12.1459	−0.861000	−0.430500	+	0.902591i	$0.641663\pi$
$200$	0	0
$201$	−15.9443	−1.12462
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−28.4164	−1.98469
$206$	0	0
$207$	2.94427	0.204641
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.5967	1.62447	0.812234	−	0.583332i	$-0.198251\pi$
$211$	23.5967	1.62447	0.812234	+	0.583332i	$0.198251\pi$
$212$	0	0
$213$	24.1803	1.65681
$214$	0	0
$215$	6.70820	0.457496
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0902	0.681830
$220$	0	0
$221$	1.23607	0.0831469
$222$	0	0
$223$	−15.8885	−1.06398	−0.531988	−	0.846752i	$-0.678555\pi$
$223$	−15.8885	−1.06398	−0.531988	+	0.846752i	$0.678555\pi$
$224$	0	0
$225$	31.1803	2.07869
$226$	0	0
$227$	7.14590	0.474290	0.237145	−	0.971474i	$-0.423788\pi$
$227$	7.14590	0.474290	0.237145	+	0.971474i	$0.423788\pi$
$228$	0	0
$229$	17.4164	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$229$	17.4164	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.18034	−0.142839	−0.0714194	−	0.997446i	$-0.522753\pi$
$233$	−2.18034	−0.142839	−0.0714194	+	0.997446i	$0.522753\pi$
$234$	0	0
$235$	−25.1246	−1.63895
$236$	0	0
$237$	−30.6525	−1.99109
$238$	0	0
$239$	−19.3820	−1.25372	−0.626858	−	0.779134i	$-0.715659\pi$
$239$	−19.3820	−1.25372	−0.626858	+	0.779134i	$0.715659\pi$
$240$	0	0
$241$	−3.61803	−0.233058	−0.116529	−	0.993187i	$-0.537177\pi$
$241$	−3.61803	−0.233058	−0.116529	+	0.993187i	$0.537177\pi$
$242$	0	0
$243$	21.6525	1.38901
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	−27.4164	−1.73744
$250$	0	0
$251$	20.1803	1.27377	0.636886	−	0.770958i	$-0.280222\pi$
$251$	20.1803	1.27377	0.636886	+	0.770958i	$0.280222\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.47214	−0.593168
$256$	0	0
$257$	−1.52786	−0.0953055	−0.0476528	−	0.998864i	$-0.515174\pi$
$257$	−1.52786	−0.0953055	−0.0476528	+	0.998864i	$0.515174\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	27.8885	1.72626
$262$	0	0
$263$	19.4164	1.19727	0.598633	−	0.801023i	$-0.295711\pi$
$263$	19.4164	1.19727	0.598633	+	0.801023i	$0.295711\pi$
$264$	0	0
$265$	12.0344	0.739270
$266$	0	0
$267$	22.9443	1.40417
$268$	0	0
$269$	−13.4164	−0.818013	−0.409006	−	0.912532i	$-0.634125\pi$
$269$	−13.4164	−0.818013	−0.409006	+	0.912532i	$0.634125\pi$
$270$	0	0
$271$	20.4721	1.24359	0.621797	−	0.783179i	$-0.286403\pi$
$271$	20.4721	1.24359	0.621797	+	0.783179i	$0.286403\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−31.1246	−1.87010	−0.935048	−	0.354520i	$-0.884644\pi$
$277$	−31.1246	−1.87010	−0.935048	+	0.354520i	$0.884644\pi$
$278$	0	0
$279$	40.7082	2.43714
$280$	0	0
$281$	−23.5066	−1.40228	−0.701142	−	0.713021i	$-0.747326\pi$
$281$	−23.5066	−1.40228	−0.701142	+	0.713021i	$0.747326\pi$
$282$	0	0
$283$	−19.1459	−1.13811	−0.569053	−	0.822301i	$-0.692690\pi$
$283$	−19.1459	−1.13811	−0.569053	+	0.822301i	$0.692690\pi$
$284$	0	0
$285$	−61.3050	−3.63139
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.47214	−0.438024
$292$	0	0
$293$	27.5967	1.61222	0.806110	−	0.591766i	$-0.201569\pi$
$293$	27.5967	1.61222	0.806110	+	0.591766i	$0.201569\pi$
$294$	0	0
$295$	2.76393	0.160922
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.944272	−0.0546087
$300$	0	0
$301$	0	0
$302$	0	0
$303$	19.7082	1.13221
$304$	0	0
$305$	−27.3607	−1.56667
$306$	0	0
$307$	9.41641	0.537423	0.268711	−	0.963221i	$-0.413402\pi$
$307$	9.41641	0.537423	0.268711	+	0.963221i	$0.413402\pi$
$308$	0	0
$309$	15.7082	0.893609
$310$	0	0
$311$	30.7426	1.74326	0.871628	−	0.490168i	$-0.163065\pi$
$311$	30.7426	1.74326	0.871628	+	0.490168i	$0.163065\pi$
$312$	0	0
$313$	−17.1459	−0.969143	−0.484572	−	0.874752i	$-0.661025\pi$
$313$	−17.1459	−0.969143	−0.484572	+	0.874752i	$0.661025\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.8885	1.34171	0.670857	−	0.741587i	$-0.265926\pi$
$317$	23.8885	1.34171	0.670857	+	0.741587i	$0.265926\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.47214	0.360119
$324$	0	0
$325$	−10.0000	−0.554700
$326$	0	0
$327$	−31.8885	−1.76344
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.3262	0.732476	0.366238	−	0.930521i	$-0.380646\pi$
$331$	13.3262	0.732476	0.366238	+	0.930521i	$0.380646\pi$
$332$	0	0
$333$	−37.4164	−2.05041
$334$	0	0
$335$	−22.0344	−1.20387
$336$	0	0
$337$	−13.4164	−0.730838	−0.365419	−	0.930843i	$-0.619074\pi$
$337$	−13.4164	−0.730838	−0.365419	+	0.930843i	$0.619074\pi$
$338$	0	0
$339$	−33.8885	−1.84057
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7.23607	0.389577
$346$	0	0
$347$	25.8885	1.38977	0.694885	−	0.719121i	$-0.255455\pi$
$347$	25.8885	1.38977	0.694885	+	0.719121i	$0.255455\pi$
$348$	0	0
$349$	−8.47214	−0.453503	−0.226752	−	0.973953i	$-0.572811\pi$
$349$	−8.47214	−0.453503	−0.226752	+	0.973953i	$0.572811\pi$
$350$	0	0
$351$	2.76393	0.147528
$352$	0	0
$353$	−34.0689	−1.81330	−0.906652	−	0.421880i	$-0.861370\pi$
$353$	−34.0689	−1.81330	−0.906652	+	0.421880i	$0.861370\pi$
$354$	0	0
$355$	33.4164	1.77356
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.8541	0.572858	0.286429	−	0.958102i	$-0.407532\pi$
$359$	10.8541	0.572858	0.286429	+	0.958102i	$0.407532\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	28.7984	1.51152
$364$	0	0
$365$	13.9443	0.729877
$366$	0	0
$367$	13.3820	0.698533	0.349266	−	0.937023i	$-0.386431\pi$
$367$	13.3820	0.698533	0.349266	+	0.937023i	$0.386431\pi$
$368$	0	0
$369$	30.2705	1.57582
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.5066	−1.57957	−0.789785	−	0.613383i	$-0.789808\pi$
$373$	−30.5066	−1.57957	−0.789785	+	0.613383i	$0.789808\pi$
$374$	0	0
$375$	29.2705	1.51152
$376$	0	0
$377$	−8.94427	−0.460653
$378$	0	0
$379$	−9.12461	−0.468700	−0.234350	−	0.972152i	$-0.575296\pi$
$379$	−9.12461	−0.468700	−0.234350	+	0.972152i	$0.575296\pi$
$380$	0	0
$381$	40.8885	2.09478
$382$	0	0
$383$	9.88854	0.505281	0.252640	−	0.967560i	$-0.418701\pi$
$383$	9.88854	0.505281	0.252640	+	0.967560i	$0.418701\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.14590	−0.363246
$388$	0	0
$389$	27.5066	1.39464	0.697319	−	0.716761i	$-0.254376\pi$
$389$	27.5066	1.39464	0.697319	+	0.716761i	$0.254376\pi$
$390$	0	0
$391$	−0.763932	−0.0386337
$392$	0	0
$393$	16.9443	0.854725
$394$	0	0
$395$	−42.3607	−2.13140
$396$	0	0
$397$	17.3262	0.869579	0.434789	−	0.900532i	$-0.356823\pi$
$397$	17.3262	0.869579	0.434789	+	0.900532i	$0.356823\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.0557	0.552097	0.276048	−	0.961144i	$-0.410975\pi$
$401$	11.0557	0.552097	0.276048	+	0.961144i	$0.410975\pi$
$402$	0	0
$403$	−13.0557	−0.650352
$404$	0	0
$405$	20.6525	1.02623
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	0	0
$411$	−29.0344	−1.43216
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−37.8885	−1.85988
$416$	0	0
$417$	−35.6525	−1.74591
$418$	0	0
$419$	−35.5066	−1.73461	−0.867305	−	0.497777i	$-0.834150\pi$
$419$	−35.5066	−1.73461	−0.867305	+	0.497777i	$0.834150\pi$
$420$	0	0
$421$	−10.1459	−0.494481	−0.247240	−	0.968954i	$-0.579524\pi$
$421$	−10.1459	−0.494481	−0.247240	+	0.968954i	$0.579524\pi$
$422$	0	0
$423$	26.7639	1.30131
$424$	0	0
$425$	−8.09017	−0.392431
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.94427	−0.141821	−0.0709103	−	0.997483i	$-0.522590\pi$
$431$	−2.94427	−0.141821	−0.0709103	+	0.997483i	$0.522590\pi$
$432$	0	0
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	68.5410	3.28629
$436$	0	0
$437$	−4.94427	−0.236517
$438$	0	0
$439$	16.1459	0.770602	0.385301	−	0.922791i	$-0.374098\pi$
$439$	16.1459	0.770602	0.385301	+	0.922791i	$0.374098\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.05573	−0.335228	−0.167614	−	0.985853i	$-0.553606\pi$
$443$	−7.05573	−0.335228	−0.167614	+	0.985853i	$0.553606\pi$
$444$	0	0
$445$	31.7082	1.50311
$446$	0	0
$447$	8.85410	0.418785
$448$	0	0
$449$	0.944272	0.0445629	0.0222815	−	0.999752i	$-0.492907\pi$
$449$	0.944272	0.0445629	0.0222815	+	0.999752i	$0.492907\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	55.6869	2.61640
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.2705	0.573990	0.286995	−	0.957932i	$-0.407344\pi$
$457$	12.2705	0.573990	0.286995	+	0.957932i	$0.407344\pi$
$458$	0	0
$459$	2.23607	0.104371
$460$	0	0
$461$	26.8328	1.24973	0.624864	−	0.780733i	$-0.285154\pi$
$461$	26.8328	1.24973	0.624864	+	0.780733i	$0.285154\pi$
$462$	0	0
$463$	−2.79837	−0.130051	−0.0650257	−	0.997884i	$-0.520713\pi$
$463$	−2.79837	−0.130051	−0.0650257	+	0.997884i	$0.520713\pi$
$464$	0	0
$465$	100.048	4.63960
$466$	0	0
$467$	28.7639	1.33104	0.665518	−	0.746382i	$-0.268211\pi$
$467$	28.7639	1.33104	0.665518	+	0.746382i	$0.268211\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	62.0689	2.85998
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−52.3607	−2.40247
$476$	0	0
$477$	−12.8197	−0.586972
$478$	0	0
$479$	10.6180	0.485150	0.242575	−	0.970133i	$-0.422008\pi$
$479$	10.6180	0.485150	0.242575	+	0.970133i	$0.422008\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.3262	−0.468890
$486$	0	0
$487$	19.0557	0.863497	0.431749	−	0.901994i	$-0.357897\pi$
$487$	19.0557	0.863497	0.431749	+	0.901994i	$0.357897\pi$
$488$	0	0
$489$	15.7082	0.710350
$490$	0	0
$491$	−39.7984	−1.79608	−0.898038	−	0.439918i	$-0.855007\pi$
$491$	−39.7984	−1.79608	−0.898038	+	0.439918i	$0.855007\pi$
$492$	0	0
$493$	−7.23607	−0.325896
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−35.3050	−1.58047	−0.790233	−	0.612806i	$-0.790041\pi$
$499$	−35.3050	−1.58047	−0.790233	+	0.612806i	$0.790041\pi$
$500$	0	0
$501$	−1.14590	−0.0511949
$502$	0	0
$503$	−12.2705	−0.547115	−0.273557	−	0.961856i	$-0.588200\pi$
$503$	−12.2705	−0.547115	−0.273557	+	0.961856i	$0.588200\pi$
$504$	0	0
$505$	27.2361	1.21199
$506$	0	0
$507$	30.0344	1.33388
$508$	0	0
$509$	23.2361	1.02992	0.514960	−	0.857214i	$-0.327807\pi$
$509$	23.2361	1.02992	0.514960	+	0.857214i	$0.327807\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	14.4721	0.638960
$514$	0	0
$515$	21.7082	0.956578
$516$	0	0
$517$	0	0
$518$	0	0
$519$	58.4508	2.56571
$520$	0	0
$521$	13.0902	0.573491	0.286745	−	0.958007i	$-0.407427\pi$
$521$	13.0902	0.573491	0.286745	+	0.958007i	$0.407427\pi$
$522$	0	0
$523$	7.34752	0.321285	0.160642	−	0.987013i	$-0.448643\pi$
$523$	7.34752	0.321285	0.160642	+	0.987013i	$0.448643\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.5623	−0.460101
$528$	0	0
$529$	−22.4164	−0.974626
$530$	0	0
$531$	−2.94427	−0.127771
$532$	0	0
$533$	−9.70820	−0.420509
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.09017	0.349117
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	36.6525	1.57291
$544$	0	0
$545$	−44.0689	−1.88770
$546$	0	0
$547$	−13.1246	−0.561168	−0.280584	−	0.959829i	$-0.590528\pi$
$547$	−13.1246	−0.561168	−0.280584	+	0.959829i	$0.590528\pi$
$548$	0	0
$549$	29.1459	1.24392
$550$	0	0
$551$	−46.8328	−1.99515
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−91.9574	−3.90338
$556$	0	0
$557$	24.4721	1.03692	0.518459	−	0.855103i	$-0.326506\pi$
$557$	24.4721	1.03692	0.518459	+	0.855103i	$0.326506\pi$
$558$	0	0
$559$	2.29180	0.0969326
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.1246	0.721716	0.360858	−	0.932621i	$-0.382484\pi$
$563$	17.1246	0.721716	0.360858	+	0.932621i	$0.382484\pi$
$564$	0	0
$565$	−46.8328	−1.97027
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.09017	0.297235	0.148618	−	0.988895i	$-0.452518\pi$
$569$	7.09017	0.297235	0.148618	+	0.988895i	$0.452518\pi$
$570$	0	0
$571$	28.6525	1.19907	0.599534	−	0.800349i	$-0.295352\pi$
$571$	28.6525	1.19907	0.599534	+	0.800349i	$0.295352\pi$
$572$	0	0
$573$	−57.8328	−2.41600
$574$	0	0
$575$	6.18034	0.257738
$576$	0	0
$577$	−45.4164	−1.89071	−0.945355	−	0.326043i	$-0.894285\pi$
$577$	−45.4164	−1.89071	−0.945355	+	0.326043i	$0.894285\pi$
$578$	0	0
$579$	14.4721	0.601441
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	17.2361	0.712624
$586$	0	0
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	0	0
$589$	−68.3607	−2.81675
$590$	0	0
$591$	41.1246	1.69164
$592$	0	0
$593$	−29.5967	−1.21539	−0.607696	−	0.794169i	$-0.707906\pi$
$593$	−29.5967	−1.21539	−0.607696	+	0.794169i	$0.707906\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.7984	1.30142
$598$	0	0
$599$	−12.0344	−0.491714	−0.245857	−	0.969306i	$-0.579069\pi$
$599$	−12.0344	−0.491714	−0.245857	+	0.969306i	$0.579069\pi$
$600$	0	0
$601$	−23.5279	−0.959722	−0.479861	−	0.877345i	$-0.659313\pi$
$601$	−23.5279	−0.959722	−0.479861	+	0.877345i	$0.659313\pi$
$602$	0	0
$603$	23.4721	0.955859
$604$	0	0
$605$	39.7984	1.61803
$606$	0	0
$607$	24.8541	1.00880	0.504398	−	0.863471i	$-0.331714\pi$
$607$	24.8541	1.00880	0.504398	+	0.863471i	$0.331714\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.58359	−0.347255
$612$	0	0
$613$	−0.854102	−0.0344969	−0.0172484	−	0.999851i	$-0.505491\pi$
$613$	−0.854102	−0.0344969	−0.0172484	+	0.999851i	$0.505491\pi$
$614$	0	0
$615$	74.3951	2.99990
$616$	0	0
$617$	45.5967	1.83566	0.917828	−	0.396978i	$-0.129941\pi$
$617$	45.5967	1.83566	0.917828	+	0.396978i	$0.129941\pi$
$618$	0	0
$619$	−46.8328	−1.88237	−0.941185	−	0.337892i	$-0.890286\pi$
$619$	−46.8328	−1.88237	−0.941185	+	0.337892i	$0.890286\pi$
$620$	0	0
$621$	−1.70820	−0.0685479
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.70820	0.387091
$630$	0	0
$631$	15.9787	0.636103	0.318051	−	0.948074i	$-0.396972\pi$
$631$	15.9787	0.636103	0.318051	+	0.948074i	$0.396972\pi$
$632$	0	0
$633$	−61.7771	−2.45542
$634$	0	0
$635$	56.5066	2.24240
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−35.5967	−1.40819
$640$	0	0
$641$	41.7771	1.65010	0.825048	−	0.565063i	$-0.191148\pi$
$641$	41.7771	1.65010	0.825048	+	0.565063i	$0.191148\pi$
$642$	0	0
$643$	−29.3820	−1.15871	−0.579356	−	0.815075i	$-0.696696\pi$
$643$	−29.3820	−1.15871	−0.579356	+	0.815075i	$0.696696\pi$
$644$	0	0
$645$	−17.5623	−0.691515
$646$	0	0
$647$	−27.5967	−1.08494	−0.542470	−	0.840075i	$-0.682511\pi$
$647$	−27.5967	−1.08494	−0.542470	+	0.840075i	$0.682511\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.2361	−0.987564	−0.493782	−	0.869586i	$-0.664386\pi$
$653$	−25.2361	−0.987564	−0.493782	+	0.869586i	$0.664386\pi$
$654$	0	0
$655$	23.4164	0.914955
$656$	0	0
$657$	−14.8541	−0.579514
$658$	0	0
$659$	−9.90983	−0.386032	−0.193016	−	0.981196i	$-0.561827\pi$
$659$	−9.90983	−0.386032	−0.193016	+	0.981196i	$0.561827\pi$
$660$	0	0
$661$	4.58359	0.178281	0.0891405	−	0.996019i	$-0.471588\pi$
$661$	4.58359	0.178281	0.0891405	+	0.996019i	$0.471588\pi$
$662$	0	0
$663$	−3.23607	−0.125678
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.52786	0.214040
$668$	0	0
$669$	41.5967	1.60822
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0.944272	0.0363990	0.0181995	−	0.999834i	$-0.494207\pi$
$673$	0.944272	0.0363990	0.0181995	+	0.999834i	$0.494207\pi$
$674$	0	0
$675$	−18.0902	−0.696291
$676$	0	0
$677$	20.8328	0.800670	0.400335	−	0.916369i	$-0.368894\pi$
$677$	20.8328	0.800670	0.400335	+	0.916369i	$0.368894\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−18.7082	−0.716900
$682$	0	0
$683$	−45.8885	−1.75588	−0.877938	−	0.478774i	$-0.841081\pi$
$683$	−45.8885	−1.75588	−0.877938	+	0.478774i	$0.841081\pi$
$684$	0	0
$685$	−40.1246	−1.53308
$686$	0	0
$687$	−45.5967	−1.73962
$688$	0	0
$689$	4.11146	0.156634
$690$	0	0
$691$	−10.9787	−0.417650	−0.208825	−	0.977953i	$-0.566964\pi$
$691$	−10.9787	−0.417650	−0.208825	+	0.977953i	$0.566964\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−49.2705	−1.86894
$696$	0	0
$697$	−7.85410	−0.297495
$698$	0	0
$699$	5.70820	0.215904
$700$	0	0
$701$	−23.3050	−0.880216	−0.440108	−	0.897945i	$-0.645060\pi$
$701$	−23.3050	−0.880216	−0.440108	+	0.897945i	$0.645060\pi$
$702$	0	0
$703$	62.8328	2.36978
$704$	0	0
$705$	65.7771	2.47731
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.65248	−0.0996158	−0.0498079	−	0.998759i	$-0.515861\pi$
$709$	−2.65248	−0.0996158	−0.0498079	+	0.998759i	$0.515861\pi$
$710$	0	0
$711$	45.1246	1.69231
$712$	0	0
$713$	8.06888	0.302182
$714$	0	0
$715$	0	0
$716$	0	0
$717$	50.7426	1.89502
$718$	0	0
$719$	10.9098	0.406868	0.203434	−	0.979089i	$-0.434790\pi$
$719$	10.9098	0.406868	0.203434	+	0.979089i	$0.434790\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	9.47214	0.352273
$724$	0	0
$725$	58.5410	2.17416
$726$	0	0
$727$	13.7082	0.508409	0.254205	−	0.967150i	$-0.418186\pi$
$727$	13.7082	0.508409	0.254205	+	0.967150i	$0.418186\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	1.85410	0.0685764
$732$	0	0
$733$	−46.2492	−1.70825	−0.854127	−	0.520064i	$-0.825908\pi$
$733$	−46.2492	−1.70825	−0.854127	+	0.520064i	$0.825908\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	7.38197	0.271550	0.135775	−	0.990740i	$-0.456648\pi$
$739$	7.38197	0.271550	0.135775	+	0.990740i	$0.456648\pi$
$740$	0	0
$741$	−20.9443	−0.769407
$742$	0	0
$743$	0.111456	0.00408893	0.00204447	−	0.999998i	$-0.499349\pi$
$743$	0.111456	0.00408893	0.00204447	+	0.999998i	$0.499349\pi$
$744$	0	0
$745$	12.2361	0.448295
$746$	0	0
$747$	40.3607	1.47672
$748$	0	0
$749$	0	0
$750$	0	0
$751$	50.8328	1.85492	0.927458	−	0.373928i	$-0.121989\pi$
$751$	50.8328	1.85492	0.927458	+	0.373928i	$0.121989\pi$
$752$	0	0
$753$	−52.8328	−1.92533
$754$	0	0
$755$	76.9574	2.80077
$756$	0	0
$757$	−5.56231	−0.202165	−0.101083	−	0.994878i	$-0.532231\pi$
$757$	−5.56231	−0.202165	−0.101083	+	0.994878i	$0.532231\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.1115	−0.656540	−0.328270	−	0.944584i	$-0.606466\pi$
$761$	−18.1115	−0.656540	−0.328270	+	0.944584i	$0.606466\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	13.9443	0.504156
$766$	0	0
$767$	0.944272	0.0340957
$768$	0	0
$769$	−35.5967	−1.28365	−0.641826	−	0.766850i	$-0.721823\pi$
$769$	−35.5967	−1.28365	−0.641826	+	0.766850i	$0.721823\pi$
$770$	0	0
$771$	4.00000	0.144056
$772$	0	0
$773$	11.5279	0.414628	0.207314	−	0.978274i	$-0.433528\pi$
$773$	11.5279	0.414628	0.207314	+	0.978274i	$0.433528\pi$
$774$	0	0
$775$	85.4508	3.06949
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−50.8328	−1.82127
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−16.1803	−0.578238
$784$	0	0
$785$	85.7771	3.06152
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	0	0
$789$	−50.8328	−1.80970
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.34752	−0.331940
$794$	0	0
$795$	−31.5066	−1.11742
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−6.94427	−0.245671
$800$	0	0
$801$	−33.7771	−1.19345
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	35.1246	1.23644
$808$	0	0
$809$	−41.1246	−1.44586	−0.722932	−	0.690919i	$-0.757206\pi$
$809$	−41.1246	−1.44586	−0.722932	+	0.690919i	$0.757206\pi$
$810$	0	0
$811$	14.9787	0.525974	0.262987	−	0.964799i	$-0.415292\pi$
$811$	14.9787	0.525974	0.262987	+	0.964799i	$0.415292\pi$
$812$	0	0
$813$	−53.5967	−1.87972
$814$	0	0
$815$	21.7082	0.760405
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.5279	1.30973	0.654866	−	0.755745i	$-0.272725\pi$
$821$	37.5279	1.30973	0.654866	+	0.755745i	$0.272725\pi$
$822$	0	0
$823$	−42.5410	−1.48289	−0.741443	−	0.671015i	$-0.765858\pi$
$823$	−42.5410	−1.48289	−0.741443	+	0.671015i	$0.765858\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.7639	−1.27841	−0.639204	−	0.769038i	$-0.720736\pi$
$827$	−36.7639	−1.27841	−0.639204	+	0.769038i	$0.720736\pi$
$828$	0	0
$829$	11.5967	0.402772	0.201386	−	0.979512i	$-0.435455\pi$
$829$	11.5967	0.402772	0.201386	+	0.979512i	$0.435455\pi$
$830$	0	0
$831$	81.4853	2.82669
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.58359	−0.0548025
$836$	0	0
$837$	−23.6180	−0.816359
$838$	0	0
$839$	32.9443	1.13736	0.568681	−	0.822558i	$-0.307454\pi$
$839$	32.9443	1.13736	0.568681	+	0.822558i	$0.307454\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	0	0
$843$	61.5410	2.11959
$844$	0	0
$845$	41.5066	1.42787
$846$	0	0
$847$	0	0
$848$	0	0
$849$	50.1246	1.72027
$850$	0	0
$851$	−7.41641	−0.254231
$852$	0	0
$853$	25.7771	0.882591	0.441295	−	0.897362i	$-0.354519\pi$
$853$	25.7771	0.882591	0.441295	+	0.897362i	$0.354519\pi$
$854$	0	0
$855$	90.2492	3.08646
$856$	0	0
$857$	20.2016	0.690074	0.345037	−	0.938589i	$-0.387866\pi$
$857$	20.2016	0.690074	0.345037	+	0.938589i	$0.387866\pi$
$858$	0	0
$859$	−16.8328	−0.574328	−0.287164	−	0.957881i	$-0.592713\pi$
$859$	−16.8328	−0.574328	−0.287164	+	0.957881i	$0.592713\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.6738	−0.363339	−0.181670	−	0.983360i	$-0.558150\pi$
$863$	−10.6738	−0.363339	−0.181670	+	0.983360i	$0.558150\pi$
$864$	0	0
$865$	80.7771	2.74650
$866$	0	0
$867$	−2.61803	−0.0889131
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−7.52786	−0.255072
$872$	0	0
$873$	11.0000	0.372294
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.3050	1.52984	0.764920	−	0.644126i	$-0.222779\pi$
$877$	45.3050	1.52984	0.764920	+	0.644126i	$0.222779\pi$
$878$	0	0
$879$	−72.2492	−2.43691
$880$	0	0
$881$	−48.1591	−1.62252	−0.811260	−	0.584686i	$-0.801218\pi$
$881$	−48.1591	−1.62252	−0.811260	+	0.584686i	$0.801218\pi$
$882$	0	0
$883$	45.6869	1.53749	0.768744	−	0.639557i	$-0.220882\pi$
$883$	45.6869	1.53749	0.768744	+	0.639557i	$0.220882\pi$
$884$	0	0
$885$	−7.23607	−0.243238
$886$	0	0
$887$	−30.7426	−1.03224	−0.516119	−	0.856517i	$-0.672624\pi$
$887$	−30.7426	−1.03224	−0.516119	+	0.856517i	$0.672624\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−44.9443	−1.50400
$894$	0	0
$895$	11.1803	0.373718
$896$	0	0
$897$	2.47214	0.0825422
$898$	0	0
$899$	76.4296	2.54907
$900$	0	0
$901$	3.32624	0.110813
$902$	0	0
$903$	0	0
$904$	0	0
$905$	50.6525	1.68375
$906$	0	0
$907$	−13.8197	−0.458874	−0.229437	−	0.973323i	$-0.573689\pi$
$907$	−13.8197	−0.458874	−0.229437	+	0.973323i	$0.573689\pi$
$908$	0	0
$909$	−29.0132	−0.962306
$910$	0	0
$911$	−38.7214	−1.28290	−0.641448	−	0.767167i	$-0.721666\pi$
$911$	−38.7214	−1.28290	−0.641448	+	0.767167i	$0.721666\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	71.6312	2.36805
$916$	0	0
$917$	0	0
$918$	0	0
$919$	36.6869	1.21019	0.605095	−	0.796153i	$-0.293135\pi$
$919$	36.6869	1.21019	0.605095	+	0.796153i	$0.293135\pi$
$920$	0	0
$921$	−24.6525	−0.812327
$922$	0	0
$923$	11.4164	0.375776
$924$	0	0
$925$	−78.5410	−2.58241
$926$	0	0
$927$	−23.1246	−0.759512
$928$	0	0
$929$	20.2016	0.662794	0.331397	−	0.943491i	$-0.392480\pi$
$929$	20.2016	0.662794	0.331397	+	0.943491i	$0.392480\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−80.4853	−2.63497
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.58359	−0.149739	−0.0748697	−	0.997193i	$-0.523854\pi$
$937$	−4.58359	−0.149739	−0.0748697	+	0.997193i	$0.523854\pi$
$938$	0	0
$939$	44.8885	1.46488
$940$	0	0
$941$	19.2016	0.625955	0.312978	−	0.949761i	$-0.398674\pi$
$941$	19.2016	0.625955	0.312978	+	0.949761i	$0.398674\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.1935	−1.33861	−0.669304	−	0.742988i	$-0.733408\pi$
$947$	−41.1935	−1.33861	−0.669304	+	0.742988i	$0.733408\pi$
$948$	0	0
$949$	4.76393	0.154644
$950$	0	0
$951$	−62.5410	−2.02803
$952$	0	0
$953$	−19.9098	−0.644943	−0.322471	−	0.946579i	$-0.604514\pi$
$953$	−19.9098	−0.644943	−0.322471	+	0.946579i	$0.604514\pi$
$954$	0	0
$955$	−79.9230	−2.58625
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	80.5623	2.59878
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.0000	0.643823
$966$	0	0
$967$	−24.9098	−0.801046	−0.400523	−	0.916287i	$-0.631172\pi$
$967$	−24.9098	−0.801046	−0.400523	+	0.916287i	$0.631172\pi$
$968$	0	0
$969$	−16.9443	−0.544328
$970$	0	0
$971$	−22.7639	−0.730529	−0.365265	−	0.930904i	$-0.619022\pi$
$971$	−22.7639	−0.730529	−0.365265	+	0.930904i	$0.619022\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	26.1803	0.838442
$976$	0	0
$977$	−36.4508	−1.16617	−0.583083	−	0.812413i	$-0.698154\pi$
$977$	−36.4508	−1.16617	−0.583083	+	0.812413i	$0.698154\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	46.9443	1.49882
$982$	0	0
$983$	−0.270510	−0.00862792	−0.00431396	−	0.999991i	$-0.501373\pi$
$983$	−0.270510	−0.00862792	−0.00431396	+	0.999991i	$0.501373\pi$
$984$	0	0
$985$	56.8328	1.81084
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.41641	−0.0450391
$990$	0	0
$991$	8.54102	0.271314	0.135657	−	0.990756i	$-0.456685\pi$
$991$	8.54102	0.271314	0.135657	+	0.990756i	$0.456685\pi$
$992$	0	0
$993$	−34.8885	−1.10715
$994$	0	0
$995$	43.9443	1.39313
$996$	0	0
$997$	−29.7426	−0.941959	−0.470980	−	0.882144i	$-0.656099\pi$
$997$	−29.7426	−0.941959	−0.470980	+	0.882144i	$0.656099\pi$
$998$	0	0
$999$	21.7082	0.686817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.g.1.1	✓	2
7.6	odd	2		3332.2.a.o.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.g.1.1	✓	2	1.1	even	1	trivial
3332.2.a.o.1.2	yes	2	7.6	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)

\(f(q)\)	\(=\)	\(q-2.61803 q^{3} -3.61803 q^{5} +3.85410 q^{9} +O(q^{10})\)	\(q-2.61803 q^{3} -3.61803 q^{5} +3.85410 q^{9} -1.23607 q^{13} +9.47214 q^{15} -1.00000 q^{17} -6.47214 q^{19} +0.763932 q^{23} +8.09017 q^{25} -2.23607 q^{27} +7.23607 q^{29} +10.5623 q^{31} -9.70820 q^{37} +3.23607 q^{39} +7.85410 q^{41} -1.85410 q^{43} -13.9443 q^{45} +6.94427 q^{47} +2.61803 q^{51} -3.32624 q^{53} +16.9443 q^{57} -0.763932 q^{59} +7.56231 q^{61} +4.47214 q^{65} +6.09017 q^{67} -2.00000 q^{69} -9.23607 q^{71} -3.85410 q^{73} -21.1803 q^{75} +11.7082 q^{79} -5.70820 q^{81} +10.4721 q^{83} +3.61803 q^{85} -18.9443 q^{87} -8.76393 q^{89} -27.6525 q^{93} +23.4164 q^{95} +2.85410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 5 q^{5} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 5 * q^5 + q^9	\( 2 q - 3 q^{3} - 5 q^{5} + q^{9} + 2 q^{13} + 10 q^{15} - 2 q^{17} - 4 q^{19} + 6 q^{23} + 5 q^{25} + 10 q^{29} + q^{31} - 6 q^{37} + 2 q^{39} + 9 q^{41} + 3 q^{43} - 10 q^{45} - 4 q^{47} + 3 q^{51} + 9 q^{53} + 16 q^{57} - 6 q^{59} - 5 q^{61} + q^{67} - 4 q^{69} - 14 q^{71} - q^{73} - 20 q^{75} + 10 q^{79} + 2 q^{81} + 12 q^{83} + 5 q^{85} - 20 q^{87} - 22 q^{89} - 24 q^{93} + 20 q^{95} - q^{97}+O(q^{100}) \) 2 * q - 3 * q^3 - 5 * q^5 + q^9 + 2 * q^13 + 10 * q^15 - 2 * q^17 - 4 * q^19 + 6 * q^23 + 5 * q^25 + 10 * q^29 + q^31 - 6 * q^37 + 2 * q^39 + 9 * q^41 + 3 * q^43 - 10 * q^45 - 4 * q^47 + 3 * q^51 + 9 * q^53 + 16 * q^57 - 6 * q^59 - 5 * q^61 + q^67 - 4 * q^69 - 14 * q^71 - q^73 - 20 * q^75 + 10 * q^79 + 2 * q^81 + 12 * q^83 + 5 * q^85 - 20 * q^87 - 22 * q^89 - 24 * q^93 + 20 * q^95 - q^97

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