LMFDB - Embedded newform 4675.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4675,2,Mod(1,4675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4675, 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("4675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4675 = 5^{2} \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4675.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:	$37.3300629449$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.33844.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2x + 2 $ x^4 - x^3 - 6x^2 + 2x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 187)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79774$ of defining polynomial
Character	$\chi$	$=$	4675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79774 q^{2} +1.08288 q^{3} +5.82737 q^{4} -3.02962 q^{6} -10.7080 q^{8} -1.82737 q^{9} +O(q^{10})$	\(q-2.79774 q^{2} +1.08288 q^{3} +5.82737 q^{4} -3.02962 q^{6} -10.7080 q^{8} -1.82737 q^{9} -1.00000 q^{11} +6.31035 q^{12} +3.74449 q^{13} +18.3035 q^{16} -1.00000 q^{17} +5.11251 q^{18} +0.314763 q^{19} +2.79774 q^{22} +2.51261 q^{23} -11.5955 q^{24} -10.4761 q^{26} -5.22747 q^{27} +5.02962 q^{29} -8.42727 q^{31} -29.7925 q^{32} -1.08288 q^{33} +2.79774 q^{34} -10.6487 q^{36} -8.51261 q^{37} -0.880625 q^{38} +4.05483 q^{39} +4.16576 q^{41} +7.48897 q^{43} -5.82737 q^{44} -7.02962 q^{46} -4.88062 q^{47} +19.8205 q^{48} -7.00000 q^{49} -1.08288 q^{51} +21.8205 q^{52} -10.8806 q^{53} +14.6251 q^{54} +0.340851 q^{57} -14.0716 q^{58} +6.93388 q^{59} +13.1910 q^{61} +23.5773 q^{62} +46.7447 q^{64} +3.02962 q^{66} +7.07847 q^{67} -5.82737 q^{68} +2.72085 q^{69} -15.2137 q^{71} +19.5674 q^{72} +0.506613 q^{73} +23.8161 q^{74} +1.83424 q^{76} -11.3444 q^{78} +4.05483 q^{79} -0.178624 q^{81} -11.6547 q^{82} -10.5614 q^{83} -20.9522 q^{86} +5.44649 q^{87} +10.7080 q^{88} -3.14654 q^{89} +14.6419 q^{92} -9.12573 q^{93} +13.6547 q^{94} -32.2617 q^{96} +4.84413 q^{97} +19.5842 q^{98} +1.82737 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 9 q^{8} + 11 q^{9}+O(q^{10}) $ 4 * q - q^2 - q^3 + 5 * q^4 - 4 * q^6 - 9 * q^8 + 11 * q^9	$ 4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 9 q^{8} + 11 q^{9} - 4 q^{11} + 2 q^{13} + 19 q^{16} - 4 q^{17} + 7 q^{18} - 2 q^{19} + q^{22} - 5 q^{23} - 26 q^{24} - 6 q^{26} - q^{27} + 12 q^{29} - 17 q^{31} - 39 q^{32} + q^{33} + q^{34} - 25 q^{36} - 19 q^{37} + 12 q^{38} - 22 q^{39} + 6 q^{41} + 4 q^{43} - 5 q^{44} - 20 q^{46} - 4 q^{47} + 32 q^{48} - 28 q^{49} + q^{51} + 40 q^{52} - 28 q^{53} + 30 q^{54} + 16 q^{57} + 15 q^{59} + 12 q^{61} - 6 q^{62} + 35 q^{64} + 4 q^{66} + q^{67} - 5 q^{68} - 13 q^{69} + 17 q^{71} + 25 q^{72} + 6 q^{73} + 26 q^{74} + 18 q^{76} - 34 q^{78} - 22 q^{79} - 10 q^{82} - 8 q^{83} - 12 q^{86} - 6 q^{87} + 9 q^{88} - 13 q^{89} + 12 q^{92} - 31 q^{93} + 18 q^{94} + 16 q^{96} - 17 q^{97} + 7 q^{98} - 11 q^{99}+O(q^{100}) $ 4 * q - q^2 - q^3 + 5 * q^4 - 4 * q^6 - 9 * q^8 + 11 * q^9 - 4 * q^11 + 2 * q^13 + 19 * q^16 - 4 * q^17 + 7 * q^18 - 2 * q^19 + q^22 - 5 * q^23 - 26 * q^24 - 6 * q^26 - q^27 + 12 * q^29 - 17 * q^31 - 39 * q^32 + q^33 + q^34 - 25 * q^36 - 19 * q^37 + 12 * q^38 - 22 * q^39 + 6 * q^41 + 4 * q^43 - 5 * q^44 - 20 * q^46 - 4 * q^47 + 32 * q^48 - 28 * q^49 + q^51 + 40 * q^52 - 28 * q^53 + 30 * q^54 + 16 * q^57 + 15 * q^59 + 12 * q^61 - 6 * q^62 + 35 * q^64 + 4 * q^66 + q^67 - 5 * q^68 - 13 * q^69 + 17 * q^71 + 25 * q^72 + 6 * q^73 + 26 * q^74 + 18 * q^76 - 34 * q^78 - 22 * q^79 - 10 * q^82 - 8 * q^83 - 12 * q^86 - 6 * q^87 + 9 * q^88 - 13 * q^89 + 12 * q^92 - 31 * q^93 + 18 * q^94 + 16 * q^96 - 17 * q^97 + 7 * q^98 - 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 $

$2$	−2.79774	−1.97830	−0.989152	−	0.146898i	$-0.953071\pi$
$2$	−2.79774	−1.97830	−0.989152	+	0.146898i	$0.953071\pi$
$3$	1.08288	0.625202	0.312601	−	0.949885i	$-0.398800\pi$
$3$	1.08288	0.625202	0.312601	+	0.949885i	$0.398800\pi$
$4$	5.82737	2.91368
$5$	0	0
$5$	0	0
$6$	−3.02962	−1.23684
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−10.7080	−3.78585
$9$	−1.82737	−0.609123
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	6.31035	1.82164
$13$	3.74449	1.03853	0.519267	−	0.854612i	$-0.326205\pi$
$13$	3.74449	1.03853	0.519267	+	0.854612i	$0.326205\pi$
$14$	0	0
$15$	0	0
$16$	18.3035	4.57587
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	5.11251	1.20503
$19$	0.314763	0.0722115	0.0361057	−	0.999348i	$-0.488505\pi$
$19$	0.314763	0.0722115	0.0361057	+	0.999348i	$0.488505\pi$
$20$	0	0
$21$	0	0
$22$	2.79774	0.596481
$23$	2.51261	0.523914	0.261957	−	0.965079i	$-0.415632\pi$
$23$	2.51261	0.523914	0.261957	+	0.965079i	$0.415632\pi$
$24$	−11.5955	−2.36692
$25$	0	0
$26$	−10.4761	−2.05453
$27$	−5.22747	−1.00603
$28$	0	0
$29$	5.02962	0.933978	0.466989	−	0.884263i	$-0.345339\pi$
$29$	5.02962	0.933978	0.466989	+	0.884263i	$0.345339\pi$
$30$	0	0
$31$	−8.42727	−1.51358	−0.756791	−	0.653657i	$-0.773234\pi$
$31$	−8.42727	−1.51358	−0.756791	+	0.653657i	$0.773234\pi$
$32$	−29.7925	−5.26661
$33$	−1.08288	−0.188505
$34$	2.79774	0.479809
$35$	0	0
$36$	−10.6487	−1.77479
$37$	−8.51261	−1.39946	−0.699732	−	0.714406i	$-0.746697\pi$
$37$	−8.51261	−1.39946	−0.699732	+	0.714406i	$0.746697\pi$
$38$	−0.880625	−0.142856
$39$	4.05483	0.649293
$40$	0	0
$41$	4.16576	0.650583	0.325291	−	0.945614i	$-0.394538\pi$
$41$	4.16576	0.650583	0.325291	+	0.945614i	$0.394538\pi$
$42$	0	0
$43$	7.48897	1.14206	0.571029	−	0.820930i	$-0.306544\pi$
$43$	7.48897	1.14206	0.571029	+	0.820930i	$0.306544\pi$
$44$	−5.82737	−0.878509
$45$	0	0
$46$	−7.02962	−1.03646
$47$	−4.88062	−0.711912	−0.355956	−	0.934503i	$-0.615845\pi$
$47$	−4.88062	−0.711912	−0.355956	+	0.934503i	$0.615845\pi$
$48$	19.8205	2.86084
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−1.08288	−0.151634
$52$	21.8205	3.02596
$53$	−10.8806	−1.49457	−0.747284	−	0.664504i	$-0.768643\pi$
$53$	−10.8806	−1.49457	−0.747284	+	0.664504i	$0.768643\pi$
$54$	14.6251	1.99023
$55$	0	0
$56$	0	0
$57$	0.340851	0.0451468
$58$	−14.0716	−1.84769
$59$	6.93388	0.902715	0.451357	−	0.892343i	$-0.350940\pi$
$59$	6.93388	0.902715	0.451357	+	0.892343i	$0.350940\pi$
$60$	0	0
$61$	13.1910	1.68893	0.844466	−	0.535610i	$-0.179918\pi$
$61$	13.1910	1.68893	0.844466	+	0.535610i	$0.179918\pi$
$62$	23.5773	2.99432
$63$	0	0
$64$	46.7447	5.84308
$65$	0	0
$66$	3.02962	0.372921
$67$	7.07847	0.864772	0.432386	−	0.901689i	$-0.357672\pi$
$67$	7.07847	0.864772	0.432386	+	0.901689i	$0.357672\pi$
$68$	−5.82737	−0.706672
$69$	2.72085	0.327552
$70$	0	0
$71$	−15.2137	−1.80554	−0.902769	−	0.430126i	$-0.858469\pi$
$71$	−15.2137	−1.80554	−0.902769	+	0.430126i	$0.858469\pi$
$72$	19.5674	2.30604
$73$	0.506613	0.0592946	0.0296473	−	0.999560i	$-0.490562\pi$
$73$	0.506613	0.0592946	0.0296473	+	0.999560i	$0.490562\pi$
$74$	23.8161	2.76856
$75$	0	0
$76$	1.83424	0.210401
$77$	0	0
$78$	−11.3444	−1.28450
$79$	4.05483	0.456205	0.228102	−	0.973637i	$-0.426748\pi$
$79$	4.05483	0.456205	0.228102	+	0.973637i	$0.426748\pi$
$80$	0	0
$81$	−0.178624	−0.0198471
$82$	−11.6547	−1.28705
$83$	−10.5614	−1.15927	−0.579635	−	0.814876i	$-0.696805\pi$
$83$	−10.5614	−1.15927	−0.579635	+	0.814876i	$0.696805\pi$
$84$	0	0
$85$	0	0
$86$	−20.9522	−2.25934
$87$	5.44649	0.583925
$88$	10.7080	1.14148
$89$	−3.14654	−0.333533	−0.166767	−	0.985996i	$-0.553333\pi$
$89$	−3.14654	−0.333533	−0.166767	+	0.985996i	$0.553333\pi$
$90$	0	0
$91$	0	0
$92$	14.6419	1.52652
$93$	−9.12573	−0.946294
$94$	13.6547	1.40838
$95$	0	0
$96$	−32.2617	−3.29270
$97$	4.84413	0.491847	0.245923	−	0.969289i	$-0.420909\pi$
$97$	4.84413	0.491847	0.245923	+	0.969289i	$0.420909\pi$
$98$	19.5842	1.97830
$99$	1.82737	0.183657
$100$	0	0
$101$	−14.1353	−1.40651	−0.703256	−	0.710937i	$-0.748271\pi$
$101$	−14.1353	−1.40651	−0.703256	+	0.710937i	$0.748271\pi$
$102$	3.02962	0.299978
$103$	6.60835	0.651140	0.325570	−	0.945518i	$-0.394444\pi$
$103$	6.60835	0.651140	0.325570	+	0.945518i	$0.394444\pi$
$104$	−40.0959	−3.93173
$105$	0	0
$106$	30.4412	2.95671
$107$	4.97038	0.480504	0.240252	−	0.970711i	$-0.422770\pi$
$107$	4.97038	0.480504	0.240252	+	0.970711i	$0.422770\pi$
$108$	−30.4624	−2.93124
$109$	−0.400099	−0.0383226	−0.0191613	−	0.999816i	$-0.506100\pi$
$109$	−0.400099	−0.0383226	−0.0191613	+	0.999816i	$0.506100\pi$
$110$	0	0
$111$	−9.21814	−0.874947
$112$	0	0
$113$	−4.77253	−0.448962	−0.224481	−	0.974478i	$-0.572069\pi$
$113$	−4.77253	−0.448962	−0.224481	+	0.974478i	$0.572069\pi$
$114$	−0.953612	−0.0893140
$115$	0	0
$116$	29.3095	2.72132
$117$	−6.84255	−0.632594
$118$	−19.3992	−1.78584
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−36.9050	−3.34122
$123$	4.51103	0.406746
$124$	−49.1088	−4.41010
$125$	0	0
$126$	0	0
$127$	3.47699	0.308533	0.154266	−	0.988029i	$-0.450699\pi$
$127$	3.47699	0.308533	0.154266	+	0.988029i	$0.450699\pi$
$128$	−71.1947	−6.29278
$129$	8.10967	0.714017
$130$	0	0
$131$	−12.2250	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$131$	−12.2250	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$132$	−6.31035	−0.549245
$133$	0	0
$134$	−19.8037	−1.71078
$135$	0	0
$136$	10.7080	0.918203
$137$	−0.457770	−0.0391100	−0.0195550	−	0.999809i	$-0.506225\pi$
$137$	−0.457770	−0.0391100	−0.0195550	+	0.999809i	$0.506225\pi$
$138$	−7.61225	−0.647998
$139$	−3.43414	−0.291280	−0.145640	−	0.989338i	$-0.546524\pi$
$139$	−3.43414	−0.291280	−0.145640	+	0.989338i	$0.546524\pi$
$140$	0	0
$141$	−5.28514	−0.445089
$142$	42.5641	3.57190
$143$	−3.74449	−0.313130
$144$	−33.4472	−2.78727
$145$	0	0
$146$	−1.41737	−0.117303
$147$	−7.58017	−0.625202
$148$	−49.6061	−4.07759
$149$	−2.01676	−0.165220	−0.0826098	−	0.996582i	$-0.526326\pi$
$149$	−2.01676	−0.165220	−0.0826098	+	0.996582i	$0.526326\pi$
$150$	0	0
$151$	−15.6242	−1.27148	−0.635741	−	0.771902i	$-0.719305\pi$
$151$	−15.6242	−1.27148	−0.635741	+	0.771902i	$0.719305\pi$
$152$	−3.37047	−0.273382
$153$	1.82737	0.147734
$154$	0	0
$155$	0	0
$156$	23.6290	1.89184
$157$	0.682782	0.0544919	0.0272460	−	0.999629i	$-0.491326\pi$
$157$	0.682782	0.0544919	0.0272460	+	0.999629i	$0.491326\pi$
$158$	−11.3444	−0.902511
$159$	−11.7824	−0.934407
$160$	0	0
$161$	0	0
$162$	0.499744	0.0392636
$163$	−0.331526	−0.0259671	−0.0129836	−	0.999916i	$-0.504133\pi$
$163$	−0.331526	−0.0259671	−0.0129836	+	0.999916i	$0.504133\pi$
$164$	24.2754	1.89559
$165$	0	0
$166$	29.5482	2.29339
$167$	−3.14812	−0.243609	−0.121804	−	0.992554i	$-0.538868\pi$
$167$	−3.14812	−0.243609	−0.121804	+	0.992554i	$0.538868\pi$
$168$	0	0
$169$	1.02118	0.0785521
$170$	0	0
$171$	−0.575187	−0.0439856
$172$	43.6410	3.32759
$173$	3.53624	0.268855	0.134428	−	0.990923i	$-0.457080\pi$
$173$	3.53624	0.268855	0.134428	+	0.990923i	$0.457080\pi$
$174$	−15.2379	−1.15518
$175$	0	0
$176$	−18.3035	−1.37968
$177$	7.50857	0.564379
$178$	8.80322	0.659829
$179$	−10.5674	−0.789848	−0.394924	−	0.918714i	$-0.629229\pi$
$179$	−10.5674	−0.789848	−0.394924	+	0.918714i	$0.629229\pi$
$180$	0	0
$181$	−10.5599	−0.784909	−0.392454	−	0.919771i	$-0.628374\pi$
$181$	−10.5599	−0.784909	−0.392454	+	0.919771i	$0.628374\pi$
$182$	0	0
$183$	14.2843	1.05592
$184$	−26.9050	−1.98346
$185$	0	0
$186$	25.5315	1.87206
$187$	1.00000	0.0731272
$188$	−28.4412	−2.07429
$189$	0	0
$190$	0	0
$191$	11.9330	0.863442	0.431721	−	0.902007i	$-0.357907\pi$
$191$	11.9330	0.863442	0.431721	+	0.902007i	$0.357907\pi$
$192$	50.6189	3.65311
$193$	5.06315	0.364454	0.182227	−	0.983257i	$-0.441669\pi$
$193$	5.06315	0.364454	0.182227	+	0.983257i	$0.441669\pi$
$194$	−13.5526	−0.973023
$195$	0	0
$196$	−40.7916	−2.91368
$197$	−22.7909	−1.62378	−0.811891	−	0.583809i	$-0.801562\pi$
$197$	−22.7909	−1.62378	−0.811891	+	0.583809i	$0.801562\pi$
$198$	−5.11251	−0.363330
$199$	15.6075	1.10638	0.553192	−	0.833054i	$-0.313410\pi$
$199$	15.6075	1.10638	0.553192	+	0.833054i	$0.313410\pi$
$200$	0	0
$201$	7.66514	0.540657
$202$	39.5468	2.78251
$203$	0	0
$204$	−6.31035	−0.441813
$205$	0	0
$206$	−18.4885	−1.28815
$207$	−4.59145	−0.319128
$208$	68.5371	4.75219
$209$	−0.314763	−0.0217726
$210$	0	0
$211$	−16.8122	−1.15740	−0.578699	−	0.815541i	$-0.696439\pi$
$211$	−16.8122	−1.15740	−0.578699	+	0.815541i	$0.696439\pi$
$212$	−63.4054	−4.35470
$213$	−16.4747	−1.12883
$214$	−13.9058	−0.950583
$215$	0	0
$216$	55.9757	3.80866
$217$	0	0
$218$	1.11938	0.0758137
$219$	0.548602	0.0370711
$220$	0	0
$221$	−3.74449	−0.251881
$222$	25.7900	1.73091
$223$	−12.8866	−0.862952	−0.431476	−	0.902124i	$-0.642007\pi$
$223$	−12.8866	−0.862952	−0.431476	+	0.902124i	$0.642007\pi$
$224$	0	0
$225$	0	0
$226$	13.3523	0.888183
$227$	25.0071	1.65978	0.829888	−	0.557929i	$-0.188404\pi$
$227$	25.0071	1.65978	0.829888	+	0.557929i	$0.188404\pi$
$228$	1.98626	0.131543
$229$	2.84111	0.187746	0.0938728	−	0.995584i	$-0.470075\pi$
$229$	2.84111	0.187746	0.0938728	+	0.995584i	$0.470075\pi$
$230$	0	0
$231$	0	0
$232$	−53.8572	−3.53590
$233$	−0.527922	−0.0345853	−0.0172926	−	0.999850i	$-0.505505\pi$
$233$	−0.527922	−0.0345853	−0.0172926	+	0.999850i	$0.505505\pi$
$234$	19.1437	1.25146
$235$	0	0
$236$	40.4063	2.63022
$237$	4.39091	0.285220
$238$	0	0
$239$	−14.1402	−0.914652	−0.457326	−	0.889299i	$-0.651193\pi$
$239$	−14.1402	−0.914652	−0.457326	+	0.889299i	$0.651193\pi$
$240$	0	0
$241$	−22.5142	−1.45027	−0.725133	−	0.688609i	$-0.758222\pi$
$241$	−22.5142	−1.45027	−0.725133	+	0.688609i	$0.758222\pi$
$242$	−2.79774	−0.179846
$243$	15.4890	0.993618
$244$	76.8687	4.92101
$245$	0	0
$246$	−12.6207	−0.804666
$247$	1.17862	0.0749940
$248$	90.2391	5.73019
$249$	−11.4368	−0.724778
$250$	0	0
$251$	12.2310	0.772014	0.386007	−	0.922496i	$-0.373854\pi$
$251$	12.2310	0.772014	0.386007	+	0.922496i	$0.373854\pi$
$252$	0	0
$253$	−2.51261	−0.157966
$254$	−9.72772	−0.610372
$255$	0	0
$256$	105.695	6.60595
$257$	−25.2714	−1.57639	−0.788193	−	0.615428i	$-0.788983\pi$
$257$	−25.2714	−1.57639	−0.788193	+	0.615428i	$0.788983\pi$
$258$	−22.6888	−1.41254
$259$	0	0
$260$	0	0
$261$	−9.19097	−0.568907
$262$	34.2024	2.11303
$263$	2.77852	0.171331	0.0856656	−	0.996324i	$-0.472698\pi$
$263$	2.77852	0.171331	0.0856656	+	0.996324i	$0.472698\pi$
$264$	11.5955	0.713653
$265$	0	0
$266$	0	0
$267$	−3.40733	−0.208525
$268$	41.2488	2.51967
$269$	10.9611	0.668307	0.334154	−	0.942519i	$-0.391550\pi$
$269$	10.9611	0.668307	0.334154	+	0.942519i	$0.391550\pi$
$270$	0	0
$271$	−2.37879	−0.144501	−0.0722506	−	0.997387i	$-0.523018\pi$
$271$	−2.37879	−0.144501	−0.0722506	+	0.997387i	$0.523018\pi$
$272$	−18.3035	−1.10981
$273$	0	0
$274$	1.28072	0.0773714
$275$	0	0
$276$	15.8554	0.954384
$277$	27.8709	1.67460	0.837301	−	0.546743i	$-0.184132\pi$
$277$	27.8709	1.67460	0.837301	+	0.546743i	$0.184132\pi$
$278$	9.60784	0.576240
$279$	15.3997	0.921957
$280$	0	0
$281$	−5.51456	−0.328971	−0.164486	−	0.986379i	$-0.552596\pi$
$281$	−5.51456	−0.328971	−0.164486	+	0.986379i	$0.552596\pi$
$282$	14.7865	0.880521
$283$	−12.5994	−0.748956	−0.374478	−	0.927236i	$-0.622178\pi$
$283$	−12.5994	−0.748956	−0.374478	+	0.927236i	$0.622178\pi$
$284$	−88.6560	−5.26077
$285$	0	0
$286$	10.4761	0.619465
$287$	0	0
$288$	54.4418	3.20801
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.24562	0.307504
$292$	2.95222	0.172766
$293$	−15.3232	−0.895191	−0.447596	−	0.894236i	$-0.647720\pi$
$293$	−15.3232	−0.895191	−0.447596	+	0.894236i	$0.647720\pi$
$294$	21.2074	1.23684
$295$	0	0
$296$	91.1529	5.29816
$297$	5.22747	0.303328
$298$	5.64239	0.326855
$299$	9.40842	0.544103
$300$	0	0
$301$	0	0
$302$	43.7126	2.51538
$303$	−15.3068	−0.879353
$304$	5.76125	0.330430
$305$	0	0
$306$	−5.11251	−0.292263
$307$	−8.42443	−0.480808	−0.240404	−	0.970673i	$-0.577280\pi$
$307$	−8.42443	−0.480808	−0.240404	+	0.970673i	$0.577280\pi$
$308$	0	0
$309$	7.15606	0.407094
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−43.4191	−2.45812
$313$	−16.1797	−0.914530	−0.457265	−	0.889330i	$-0.651171\pi$
$313$	−16.1797	−0.914530	−0.457265	+	0.889330i	$0.651171\pi$
$314$	−1.91025	−0.107802
$315$	0	0
$316$	23.6290	1.32924
$317$	−15.1210	−0.849277	−0.424639	−	0.905363i	$-0.639599\pi$
$317$	−15.1210	−0.849277	−0.424639	+	0.905363i	$0.639599\pi$
$318$	32.9642	1.84854
$319$	−5.02962	−0.281605
$320$	0	0
$321$	5.38233	0.300412
$322$	0	0
$323$	−0.314763	−0.0175139
$324$	−1.04091	−0.0578282
$325$	0	0
$326$	0.927524	0.0513708
$327$	−0.433260	−0.0239593
$328$	−44.6070	−2.46301
$329$	0	0
$330$	0	0
$331$	26.6010	1.46212	0.731061	−	0.682312i	$-0.239026\pi$
$331$	26.6010	1.46212	0.731061	+	0.682312i	$0.239026\pi$
$332$	−61.5454	−3.37774
$333$	15.5557	0.852445
$334$	8.80764	0.481932
$335$	0	0
$336$	0	0
$337$	−21.1180	−1.15037	−0.575185	−	0.818023i	$-0.695070\pi$
$337$	−21.1180	−1.15037	−0.575185	+	0.818023i	$0.695070\pi$
$338$	−2.85699	−0.155400
$339$	−5.16809	−0.280692
$340$	0	0
$341$	8.42727	0.456362
$342$	1.60923	0.0870169
$343$	0	0
$344$	−80.1919	−4.32366
$345$	0	0
$346$	−9.89349	−0.531877
$347$	−0.0729875	−0.00391817	−0.00195909	−	0.999998i	$-0.500624\pi$
$347$	−0.0729875	−0.00391817	−0.00195909	+	0.999998i	$0.500624\pi$
$348$	31.7387	1.70137
$349$	12.9138	0.691259	0.345630	−	0.938371i	$-0.387665\pi$
$349$	12.9138	0.691259	0.345630	+	0.938371i	$0.387665\pi$
$350$	0	0
$351$	−19.5742	−1.04479
$352$	29.7925	1.58794
$353$	−11.4096	−0.607273	−0.303637	−	0.952788i	$-0.598201\pi$
$353$	−11.4096	−0.607273	−0.303637	+	0.952788i	$0.598201\pi$
$354$	−21.0071	−1.11651
$355$	0	0
$356$	−18.3361	−0.971810
$357$	0	0
$358$	29.5650	1.56256
$359$	16.3059	0.860594	0.430297	−	0.902687i	$-0.358409\pi$
$359$	16.3059	0.860594	0.430297	+	0.902687i	$0.358409\pi$
$360$	0	0
$361$	−18.9009	−0.994786
$362$	29.5438	1.55279
$363$	1.08288	0.0568365
$364$	0	0
$365$	0	0
$366$	−39.9637	−2.08894
$367$	−32.5250	−1.69779	−0.848894	−	0.528562i	$-0.822731\pi$
$367$	−32.5250	−1.69779	−0.848894	+	0.528562i	$0.822731\pi$
$368$	45.9894	2.39736
$369$	−7.61238	−0.396285
$370$	0	0
$371$	0	0
$372$	−53.1790	−2.75720
$373$	−6.20825	−0.321451	−0.160725	−	0.986999i	$-0.551383\pi$
$373$	−6.20825	−0.321451	−0.160725	+	0.986999i	$0.551383\pi$
$374$	−2.79774	−0.144668
$375$	0	0
$376$	52.2617	2.69519
$377$	18.8334	0.969967
$378$	0	0
$379$	27.0007	1.38693	0.693466	−	0.720489i	$-0.256083\pi$
$379$	27.0007	1.38693	0.693466	+	0.720489i	$0.256083\pi$
$380$	0	0
$381$	3.76517	0.192895
$382$	−33.3855	−1.70815
$383$	−12.3375	−0.630418	−0.315209	−	0.949022i	$-0.602075\pi$
$383$	−12.3375	−0.630418	−0.315209	+	0.949022i	$0.602075\pi$
$384$	−77.0954	−3.93426
$385$	0	0
$386$	−14.1654	−0.721000
$387$	−13.6851	−0.695653
$388$	28.2285	1.43309
$389$	38.6648	1.96038	0.980191	−	0.198057i	$-0.0634630\pi$
$389$	38.6648	1.96038	0.980191	+	0.198057i	$0.0634630\pi$
$390$	0	0
$391$	−2.51261	−0.127068
$392$	74.9559	3.78585
$393$	−13.2382	−0.667781
$394$	63.7630	3.21233
$395$	0	0
$396$	10.6487	0.535120
$397$	−9.03756	−0.453582	−0.226791	−	0.973943i	$-0.572823\pi$
$397$	−9.03756	−0.453582	−0.226791	+	0.973943i	$0.572823\pi$
$398$	−43.6657	−2.18876
$399$	0	0
$400$	0	0
$401$	14.2966	0.713939	0.356969	−	0.934116i	$-0.383810\pi$
$401$	14.2966	0.713939	0.356969	+	0.934116i	$0.383810\pi$
$402$	−21.4451	−1.06958
$403$	−31.5558	−1.57191
$404$	−82.3714	−4.09813
$405$	0	0
$406$	0	0
$407$	8.51261	0.421954
$408$	11.5955	0.574062
$409$	−35.5255	−1.75662	−0.878312	−	0.478088i	$-0.841330\pi$
$409$	−35.5255	−1.75662	−0.878312	+	0.478088i	$0.841330\pi$
$410$	0	0
$411$	−0.495711	−0.0244516
$412$	38.5093	1.89722
$413$	0	0
$414$	12.8457	0.631332
$415$	0	0
$416$	−111.557	−5.46955
$417$	−3.71876	−0.182109
$418$	0.880625	0.0430728
$419$	−23.7781	−1.16164	−0.580819	−	0.814033i	$-0.697268\pi$
$419$	−23.7781	−1.16164	−0.580819	+	0.814033i	$0.697268\pi$
$420$	0	0
$421$	−30.8108	−1.50163	−0.750813	−	0.660515i	$-0.770338\pi$
$421$	−30.8108	−1.50163	−0.750813	+	0.660515i	$0.770338\pi$
$422$	47.0362	2.28968
$423$	8.91870	0.433642
$424$	116.510	5.65821
$425$	0	0
$426$	46.0919	2.23316
$427$	0	0
$428$	28.9642	1.40004
$429$	−4.05483	−0.195769
$430$	0	0
$431$	−13.7826	−0.663882	−0.331941	−	0.943300i	$-0.607704\pi$
$431$	−13.7826	−0.663882	−0.331941	+	0.943300i	$0.607704\pi$
$432$	−95.6808	−4.60345
$433$	−3.89984	−0.187415	−0.0937073	−	0.995600i	$-0.529872\pi$
$433$	−3.89984	−0.187415	−0.0937073	+	0.995600i	$0.529872\pi$
$434$	0	0
$435$	0	0
$436$	−2.33153	−0.111660
$437$	0.790874	0.0378326
$438$	−1.53485	−0.0733379
$439$	−23.8023	−1.13602	−0.568012	−	0.823020i	$-0.692287\pi$
$439$	−23.8023	−1.13602	−0.568012	+	0.823020i	$0.692287\pi$
$440$	0	0
$441$	12.7916	0.609123
$442$	10.4761	0.498298
$443$	−13.4618	−0.639590	−0.319795	−	0.947487i	$-0.603614\pi$
$443$	−13.4618	−0.639590	−0.319795	+	0.947487i	$0.603614\pi$
$444$	−53.7175	−2.54932
$445$	0	0
$446$	36.0534	1.70718
$447$	−2.18392	−0.103296
$448$	0	0
$449$	−0.789793	−0.0372726	−0.0186363	−	0.999826i	$-0.505932\pi$
$449$	−0.789793	−0.0372726	−0.0186363	+	0.999826i	$0.505932\pi$
$450$	0	0
$451$	−4.16576	−0.196158
$452$	−27.8113	−1.30813
$453$	−16.9192	−0.794933
$454$	−69.9633	−3.28354
$455$	0	0
$456$	−3.64982	−0.170919
$457$	12.9186	0.604305	0.302153	−	0.953260i	$-0.402295\pi$
$457$	12.9186	0.604305	0.302153	+	0.953260i	$0.402295\pi$
$458$	−7.94869	−0.371418
$459$	5.22747	0.243997
$460$	0	0
$461$	35.3479	1.64632	0.823158	−	0.567812i	$-0.192210\pi$
$461$	35.3479	1.64632	0.823158	+	0.567812i	$0.192210\pi$
$462$	0	0
$463$	19.7372	0.917267	0.458634	−	0.888625i	$-0.348339\pi$
$463$	19.7372	0.917267	0.458634	+	0.888625i	$0.348339\pi$
$464$	92.0596	4.27376
$465$	0	0
$466$	1.47699	0.0684202
$467$	8.14742	0.377018	0.188509	−	0.982071i	$-0.439635\pi$
$467$	8.14742	0.377018	0.188509	+	0.982071i	$0.439635\pi$
$468$	−39.8741	−1.84318
$469$	0	0
$470$	0	0
$471$	0.739372	0.0340685
$472$	−74.2480	−3.41754
$473$	−7.48897	−0.344343
$474$	−12.2846	−0.564252
$475$	0	0
$476$	0	0
$477$	19.8829	0.910376
$478$	39.5606	1.80946
$479$	−11.2333	−0.513264	−0.256632	−	0.966509i	$-0.582613\pi$
$479$	−11.2333	−0.513264	−0.256632	+	0.966509i	$0.582613\pi$
$480$	0	0
$481$	−31.8753	−1.45339
$482$	62.9889	2.86907
$483$	0	0
$484$	5.82737	0.264880
$485$	0	0
$486$	−43.3342	−1.96568
$487$	−22.0471	−0.999049	−0.499524	−	0.866300i	$-0.666492\pi$
$487$	−22.0471	−0.999049	−0.499524	+	0.866300i	$0.666492\pi$
$488$	−141.249	−6.39403
$489$	−0.359003	−0.0162347
$490$	0	0
$491$	−16.3483	−0.737788	−0.368894	−	0.929471i	$-0.620263\pi$
$491$	−16.3483	−0.737788	−0.368894	+	0.929471i	$0.620263\pi$
$492$	26.2874	1.18513
$493$	−5.02962	−0.226523
$494$	−3.29749	−0.148361
$495$	0	0
$496$	−154.248	−6.92595
$497$	0	0
$498$	31.9972	1.43383
$499$	−24.1171	−1.07963	−0.539815	−	0.841784i	$-0.681506\pi$
$499$	−24.1171	−1.07963	−0.539815	+	0.841784i	$0.681506\pi$
$500$	0	0
$501$	−3.40904	−0.152305
$502$	−34.2192	−1.52728
$503$	−12.9094	−0.575600	−0.287800	−	0.957690i	$-0.592924\pi$
$503$	−12.9094	−0.575600	−0.287800	+	0.957690i	$0.592924\pi$
$504$	0	0
$505$	0	0
$506$	7.02962	0.312505
$507$	1.10581	0.0491109
$508$	20.2617	0.898967
$509$	23.9923	1.06344	0.531719	−	0.846921i	$-0.321546\pi$
$509$	23.9923	1.06344	0.531719	+	0.846921i	$0.321546\pi$
$510$	0	0
$511$	0	0
$512$	−153.318	−6.77578
$513$	−1.64541	−0.0726467
$514$	70.7029	3.11857
$515$	0	0
$516$	47.2580	2.08042
$517$	4.88062	0.214650
$518$	0	0
$519$	3.82933	0.168089
$520$	0	0
$521$	−40.2982	−1.76550	−0.882748	−	0.469847i	$-0.844309\pi$
$521$	−40.2982	−1.76550	−0.882748	+	0.469847i	$0.844309\pi$
$522$	25.7140	1.12547
$523$	−1.01323	−0.0443053	−0.0221527	−	0.999755i	$-0.507052\pi$
$523$	−1.01323	−0.0443053	−0.0221527	+	0.999755i	$0.507052\pi$
$524$	−71.2396	−3.11212
$525$	0	0
$526$	−7.77360	−0.338945
$527$	8.42727	0.367098
$528$	−19.8205	−0.862576
$529$	−16.6868	−0.725514
$530$	0	0
$531$	−12.6708	−0.549864
$532$	0	0
$533$	15.5986	0.675652
$534$	9.53285	0.412527
$535$	0	0
$536$	−75.7962	−3.27390
$537$	−11.4433	−0.493814
$538$	−30.6662	−1.32211
$539$	7.00000	0.301511
$540$	0	0
$541$	14.3744	0.618003	0.309001	−	0.951062i	$-0.400005\pi$
$541$	14.3744	0.618003	0.309001	+	0.951062i	$0.400005\pi$
$542$	6.65525	0.285867
$543$	−11.4351	−0.490727
$544$	29.7925	1.27734
$545$	0	0
$546$	0	0
$547$	−2.86877	−0.122660	−0.0613299	−	0.998118i	$-0.519534\pi$
$547$	−2.86877	−0.122660	−0.0613299	+	0.998118i	$0.519534\pi$
$548$	−2.66760	−0.113954
$549$	−24.1048	−1.02877
$550$	0	0
$551$	1.58314	0.0674439
$552$	−29.1349	−1.24006
$553$	0	0
$554$	−77.9757	−3.31287
$555$	0	0
$556$	−20.0120	−0.848697
$557$	8.57343	0.363268	0.181634	−	0.983366i	$-0.441861\pi$
$557$	8.57343	0.363268	0.181634	+	0.983366i	$0.441861\pi$
$558$	−43.0845	−1.82391
$559$	28.0424	1.18607
$560$	0	0
$561$	1.08288	0.0457193
$562$	15.4283	0.650805
$563$	30.6325	1.29101	0.645504	−	0.763757i	$-0.276647\pi$
$563$	30.6325	1.29101	0.645504	+	0.763757i	$0.276647\pi$
$564$	−30.7984	−1.29685
$565$	0	0
$566$	35.2499	1.48166
$567$	0	0
$568$	162.908	6.83549
$569$	−10.3435	−0.433622	−0.216811	−	0.976214i	$-0.569566\pi$
$569$	−10.3435	−0.433622	−0.216811	+	0.976214i	$0.569566\pi$
$570$	0	0
$571$	−36.4960	−1.52731	−0.763656	−	0.645624i	$-0.776597\pi$
$571$	−36.4960	−1.52731	−0.763656	+	0.645624i	$0.776597\pi$
$572$	−21.8205	−0.912361
$573$	12.9220	0.539826
$574$	0	0
$575$	0	0
$576$	−85.4197	−3.55915
$577$	12.9384	0.538634	0.269317	−	0.963052i	$-0.413202\pi$
$577$	12.9384	0.538634	0.269317	+	0.963052i	$0.413202\pi$
$578$	−2.79774	−0.116371
$579$	5.48279	0.227857
$580$	0	0
$581$	0	0
$582$	−14.6759	−0.608336
$583$	10.8806	0.450629
$584$	−5.42481	−0.224480
$585$	0	0
$586$	42.8704	1.77096
$587$	27.1441	1.12036	0.560178	−	0.828372i	$-0.310733\pi$
$587$	27.1441	1.12036	0.560178	+	0.828372i	$0.310733\pi$
$588$	−44.1724	−1.82164
$589$	−2.65259	−0.109298
$590$	0	0
$591$	−24.6798	−1.01519
$592$	−155.810	−6.40376
$593$	6.09858	0.250439	0.125219	−	0.992129i	$-0.460037\pi$
$593$	6.09858	0.250439	0.125219	+	0.992129i	$0.460037\pi$
$594$	−14.6251	−0.600076
$595$	0	0
$596$	−11.7524	−0.481398
$597$	16.9010	0.691714
$598$	−26.3223	−1.07640
$599$	5.09365	0.208121	0.104061	−	0.994571i	$-0.466816\pi$
$599$	5.09365	0.208121	0.104061	+	0.994571i	$0.466816\pi$
$600$	0	0
$601$	32.6525	1.33192	0.665961	−	0.745987i	$-0.268022\pi$
$601$	32.6525	1.33192	0.665961	+	0.745987i	$0.268022\pi$
$602$	0	0
$603$	−12.9350	−0.526752
$604$	−91.0482	−3.70470
$605$	0	0
$606$	42.8245	1.73963
$607$	−37.1349	−1.50726	−0.753629	−	0.657300i	$-0.771699\pi$
$607$	−37.1349	−1.50726	−0.753629	+	0.657300i	$0.771699\pi$
$608$	−9.37755	−0.380310
$609$	0	0
$610$	0	0
$611$	−18.2754	−0.739345
$612$	10.6487	0.430450
$613$	−13.6596	−0.551708	−0.275854	−	0.961200i	$-0.588961\pi$
$613$	−13.6596	−0.551708	−0.275854	+	0.961200i	$0.588961\pi$
$614$	23.5694	0.951184
$615$	0	0
$616$	0	0
$617$	13.9478	0.561518	0.280759	−	0.959778i	$-0.409414\pi$
$617$	13.9478	0.561518	0.280759	+	0.959778i	$0.409414\pi$
$618$	−20.0208	−0.805355
$619$	16.9503	0.681289	0.340645	−	0.940192i	$-0.389355\pi$
$619$	16.9503	0.681289	0.340645	+	0.940192i	$0.389355\pi$
$620$	0	0
$621$	−13.1346	−0.527072
$622$	0	0
$623$	0	0
$624$	74.2176	2.97108
$625$	0	0
$626$	45.2666	1.80922
$627$	−0.340851	−0.0136123
$628$	3.97882	0.158772
$629$	8.51261	0.339420
$630$	0	0
$631$	1.94674	0.0774986	0.0387493	−	0.999249i	$-0.487663\pi$
$631$	1.94674	0.0774986	0.0387493	+	0.999249i	$0.487663\pi$
$632$	−43.4191	−1.72712
$633$	−18.2056	−0.723608
$634$	42.3045	1.68013
$635$	0	0
$636$	−68.6605	−2.72257
$637$	−26.2114	−1.03853
$638$	14.0716	0.557100
$639$	27.8011	1.09979
$640$	0	0
$641$	−2.01532	−0.0796002	−0.0398001	−	0.999208i	$-0.512672\pi$
$641$	−2.01532	−0.0796002	−0.0398001	+	0.999208i	$0.512672\pi$
$642$	−15.0584	−0.594307
$643$	26.0471	1.02720	0.513598	−	0.858031i	$-0.328312\pi$
$643$	26.0471	1.02720	0.513598	+	0.858031i	$0.328312\pi$
$644$	0	0
$645$	0	0
$646$	0.880625	0.0346477
$647$	5.72917	0.225237	0.112618	−	0.993638i	$-0.464076\pi$
$647$	5.72917	0.225237	0.112618	+	0.993638i	$0.464076\pi$
$648$	1.91270	0.0751381
$649$	−6.93388	−0.272179
$650$	0	0
$651$	0	0
$652$	−1.93192	−0.0756599
$653$	14.9838	0.586362	0.293181	−	0.956057i	$-0.405286\pi$
$653$	14.9838	0.586362	0.293181	+	0.956057i	$0.405286\pi$
$654$	1.21215	0.0473988
$655$	0	0
$656$	76.2480	2.97698
$657$	−0.925769	−0.0361177
$658$	0	0
$659$	24.8745	0.968971	0.484486	−	0.874799i	$-0.339007\pi$
$659$	24.8745	0.968971	0.484486	+	0.874799i	$0.339007\pi$
$660$	0	0
$661$	−34.3450	−1.33586	−0.667932	−	0.744222i	$-0.732820\pi$
$661$	−34.3450	−1.33586	−0.667932	+	0.744222i	$0.732820\pi$
$662$	−74.4227	−2.89252
$663$	−4.05483	−0.157477
$664$	113.092	4.38882
$665$	0	0
$666$	−43.5207	−1.68639
$667$	12.6375	0.489324
$668$	−18.3453	−0.709800
$669$	−13.9547	−0.539519
$670$	0	0
$671$	−13.1910	−0.509232
$672$	0	0
$673$	−0.400099	−0.0154227	−0.00771135	−	0.999970i	$-0.502455\pi$
$673$	−0.400099	−0.0154227	−0.00771135	+	0.999970i	$0.502455\pi$
$674$	59.0827	2.27578
$675$	0	0
$676$	5.95078	0.228876
$677$	30.2918	1.16421	0.582105	−	0.813114i	$-0.302229\pi$
$677$	30.2918	1.16421	0.582105	+	0.813114i	$0.302229\pi$
$678$	14.4590	0.555294
$679$	0	0
$680$	0	0
$681$	27.0797	1.03770
$682$	−23.5773	−0.902823
$683$	−13.0795	−0.500475	−0.250238	−	0.968184i	$-0.580509\pi$
$683$	−13.0795	−0.500475	−0.250238	+	0.968184i	$0.580509\pi$
$684$	−3.35183	−0.128160
$685$	0	0
$686$	0	0
$687$	3.07658	0.117379
$688$	137.074	5.22591
$689$	−40.7424	−1.55216
$690$	0	0
$691$	−11.1041	−0.422418	−0.211209	−	0.977441i	$-0.567740\pi$
$691$	−11.1041	−0.422418	−0.211209	+	0.977441i	$0.567740\pi$
$692$	20.6070	0.783359
$693$	0	0
$694$	0.204200	0.00775133
$695$	0	0
$696$	−58.3209	−2.21065
$697$	−4.16576	−0.157790
$698$	−36.1295	−1.36752
$699$	−0.571676	−0.0216228
$700$	0	0
$701$	−23.1861	−0.875726	−0.437863	−	0.899042i	$-0.644264\pi$
$701$	−23.1861	−0.875726	−0.437863	+	0.899042i	$0.644264\pi$
$702$	54.7635	2.06692
$703$	−2.67945	−0.101057
$704$	−46.7447	−1.76176
$705$	0	0
$706$	31.9212	1.20137
$707$	0	0
$708$	43.7552	1.64442
$709$	−21.9967	−0.826102	−0.413051	−	0.910708i	$-0.635537\pi$
$709$	−21.9967	−0.826102	−0.413051	+	0.910708i	$0.635537\pi$
$710$	0	0
$711$	−7.40967	−0.277884
$712$	33.6932	1.26271
$713$	−21.1744	−0.792987
$714$	0	0
$715$	0	0
$716$	−61.5804	−2.30137
$717$	−15.3121	−0.571842
$718$	−45.6198	−1.70252
$719$	−46.4516	−1.73235	−0.866176	−	0.499739i	$-0.833429\pi$
$719$	−46.4516	−1.73235	−0.866176	+	0.499739i	$0.833429\pi$
$720$	0	0
$721$	0	0
$722$	52.8799	1.96799
$723$	−24.3802	−0.906709
$724$	−61.5362	−2.28698
$725$	0	0
$726$	−3.02962	−0.112440
$727$	−37.4692	−1.38966	−0.694829	−	0.719175i	$-0.744520\pi$
$727$	−37.4692	−1.38966	−0.694829	+	0.719175i	$0.744520\pi$
$728$	0	0
$729$	17.3086	0.641059
$730$	0	0
$731$	−7.48897	−0.276990
$732$	83.2396	3.07663
$733$	−42.9247	−1.58546	−0.792731	−	0.609572i	$-0.791341\pi$
$733$	−42.9247	−1.58546	−0.792731	+	0.609572i	$0.791341\pi$
$734$	90.9965	3.35874
$735$	0	0
$736$	−74.8567	−2.75925
$737$	−7.07847	−0.260739
$738$	21.2975	0.783971
$739$	−18.4470	−0.678584	−0.339292	−	0.940681i	$-0.610187\pi$
$739$	−18.4470	−0.678584	−0.339292	+	0.940681i	$0.610187\pi$
$740$	0	0
$741$	1.27631	0.0468864
$742$	0	0
$743$	−8.34035	−0.305978	−0.152989	−	0.988228i	$-0.548890\pi$
$743$	−8.34035	−0.305978	−0.152989	+	0.988228i	$0.548890\pi$
$744$	97.7183	3.58253
$745$	0	0
$746$	17.3691	0.635927
$747$	19.2996	0.706137
$748$	5.82737	0.213070
$749$	0	0
$750$	0	0
$751$	18.1885	0.663708	0.331854	−	0.943331i	$-0.392326\pi$
$751$	18.1885	0.663708	0.331854	+	0.943331i	$0.392326\pi$
$752$	−89.3324	−3.25762
$753$	13.2447	0.482665
$754$	−52.6909	−1.91889
$755$	0	0
$756$	0	0
$757$	−39.5119	−1.43609	−0.718043	−	0.695999i	$-0.754962\pi$
$757$	−39.5119	−1.43609	−0.718043	+	0.695999i	$0.754962\pi$
$758$	−75.5410	−2.74377
$759$	−2.72085	−0.0987607
$760$	0	0
$761$	−12.0305	−0.436105	−0.218053	−	0.975937i	$-0.569970\pi$
$761$	−12.0305	−0.436105	−0.218053	+	0.975937i	$0.569970\pi$
$762$	−10.5340	−0.381606
$763$	0	0
$764$	69.5380	2.51580
$765$	0	0
$766$	34.5172	1.24716
$767$	25.9638	0.937499
$768$	114.455	4.13005
$769$	5.19980	0.187510	0.0937548	−	0.995595i	$-0.470113\pi$
$769$	5.19980	0.187510	0.0937548	+	0.995595i	$0.470113\pi$
$770$	0	0
$771$	−27.3659	−0.985560
$772$	29.5048	1.06190
$773$	15.3642	0.552611	0.276305	−	0.961070i	$-0.410890\pi$
$773$	15.3642	0.552611	0.276305	+	0.961070i	$0.410890\pi$
$774$	38.2874	1.37621
$775$	0	0
$776$	−51.8709	−1.86206
$777$	0	0
$778$	−108.174	−3.87823
$779$	1.31123	0.0469796
$780$	0	0
$781$	15.2137	0.544390
$782$	7.02962	0.251379
$783$	−26.2922	−0.939606
$784$	−128.124	−4.57587
$785$	0	0
$786$	37.0372	1.32107
$787$	8.24632	0.293950	0.146975	−	0.989140i	$-0.453046\pi$
$787$	8.24632	0.293950	0.146975	+	0.989140i	$0.453046\pi$
$788$	−132.811	−4.73119
$789$	3.00881	0.107117
$790$	0	0
$791$	0	0
$792$	−19.5674	−0.695299
$793$	49.3934	1.75401
$794$	25.2848	0.897323
$795$	0	0
$796$	90.9505	3.22365
$797$	−0.646100	−0.0228860	−0.0114430	−	0.999935i	$-0.503643\pi$
$797$	−0.646100	−0.0228860	−0.0114430	+	0.999935i	$0.503643\pi$
$798$	0	0
$799$	4.88062	0.172664
$800$	0	0
$801$	5.74989	0.203162
$802$	−39.9982	−1.41239
$803$	−0.506613	−0.0178780
$804$	44.6676	1.57530
$805$	0	0
$806$	88.2850	3.10971
$807$	11.8695	0.417827
$808$	151.360	5.32484
$809$	−43.0406	−1.51323	−0.756613	−	0.653863i	$-0.773147\pi$
$809$	−43.0406	−1.51323	−0.756613	+	0.653863i	$0.773147\pi$
$810$	0	0
$811$	33.7639	1.18561	0.592806	−	0.805346i	$-0.298020\pi$
$811$	33.7639	1.18561	0.592806	+	0.805346i	$0.298020\pi$
$812$	0	0
$813$	−2.57595	−0.0903425
$814$	−23.8161	−0.834753
$815$	0	0
$816$	−19.8205	−0.693856
$817$	2.35725	0.0824697
$818$	99.3913	3.47514
$819$	0	0
$820$	0	0
$821$	46.2069	1.61263	0.806315	−	0.591486i	$-0.201459\pi$
$821$	46.2069	1.61263	0.806315	+	0.591486i	$0.201459\pi$
$822$	1.38687	0.0483727
$823$	4.59794	0.160274	0.0801371	−	0.996784i	$-0.474464\pi$
$823$	4.59794	0.160274	0.0801371	+	0.996784i	$0.474464\pi$
$824$	−70.7621	−2.46512
$825$	0	0
$826$	0	0
$827$	−24.6388	−0.856777	−0.428388	−	0.903595i	$-0.640918\pi$
$827$	−24.6388	−0.856777	−0.428388	+	0.903595i	$0.640918\pi$
$828$	−26.7561	−0.929838
$829$	−32.6465	−1.13386	−0.566930	−	0.823766i	$-0.691869\pi$
$829$	−32.6465	−1.13386	−0.566930	+	0.823766i	$0.691869\pi$
$830$	0	0
$831$	30.1809	1.04696
$832$	175.035	6.06824
$833$	7.00000	0.242536
$834$	10.4041	0.360266
$835$	0	0
$836$	−1.83424	−0.0634384
$837$	44.0533	1.52270
$838$	66.5251	2.29807
$839$	37.5608	1.29674	0.648371	−	0.761325i	$-0.275451\pi$
$839$	37.5608	1.29674	0.648371	+	0.761325i	$0.275451\pi$
$840$	0	0
$841$	−3.70288	−0.127685
$842$	86.2007	2.97067
$843$	−5.97162	−0.205673
$844$	−97.9708	−3.37229
$845$	0	0
$846$	−24.9522	−0.857875
$847$	0	0
$848$	−199.153	−6.83895
$849$	−13.6436	−0.468249
$850$	0	0
$851$	−21.3888	−0.733199
$852$	−96.0039	−3.28904
$853$	15.3661	0.526124	0.263062	−	0.964779i	$-0.415268\pi$
$853$	15.3661	0.526124	0.263062	+	0.964779i	$0.415268\pi$
$854$	0	0
$855$	0	0
$856$	−53.2227	−1.81912
$857$	2.10034	0.0717464	0.0358732	−	0.999356i	$-0.488579\pi$
$857$	2.10034	0.0717464	0.0358732	+	0.999356i	$0.488579\pi$
$858$	11.3444	0.387291
$859$	50.5661	1.72529	0.862646	−	0.505809i	$-0.168806\pi$
$859$	50.5661	1.72529	0.862646	+	0.505809i	$0.168806\pi$
$860$	0	0
$861$	0	0
$862$	38.5601	1.31336
$863$	−5.72318	−0.194819	−0.0974096	−	0.995244i	$-0.531056\pi$
$863$	−5.72318	−0.194819	−0.0974096	+	0.995244i	$0.531056\pi$
$864$	155.739	5.29835
$865$	0	0
$866$	10.9108	0.370763
$867$	1.08288	0.0367766
$868$	0	0
$869$	−4.05483	−0.137551
$870$	0	0
$871$	26.5052	0.898095
$872$	4.28426	0.145083
$873$	−8.85201	−0.299595
$874$	−2.21266	−0.0748444
$875$	0	0
$876$	3.19691	0.108013
$877$	−1.79630	−0.0606566	−0.0303283	−	0.999540i	$-0.509655\pi$
$877$	−1.79630	−0.0606566	−0.0303283	+	0.999540i	$0.509655\pi$
$878$	66.5929	2.24740
$879$	−16.5932	−0.559675
$880$	0	0
$881$	29.7822	1.00339	0.501695	−	0.865045i	$-0.332710\pi$
$881$	29.7822	1.00339	0.501695	+	0.865045i	$0.332710\pi$
$882$	−35.7875	−1.20503
$883$	19.8540	0.668141	0.334071	−	0.942548i	$-0.391578\pi$
$883$	19.8540	0.668141	0.334071	+	0.942548i	$0.391578\pi$
$884$	−21.8205	−0.733903
$885$	0	0
$886$	37.6627	1.26530
$887$	28.4077	0.953836	0.476918	−	0.878948i	$-0.341754\pi$
$887$	28.4077	0.953836	0.476918	+	0.878948i	$0.341754\pi$
$888$	98.7078	3.31242
$889$	0	0
$890$	0	0
$891$	0.178624	0.00598413
$892$	−75.0951	−2.51437
$893$	−1.53624	−0.0514082
$894$	6.11004	0.204350
$895$	0	0
$896$	0	0
$897$	10.1882	0.340174
$898$	2.20964	0.0737366
$899$	−42.3860	−1.41365
$900$	0	0
$901$	10.8806	0.362486
$902$	11.6547	0.388060
$903$	0	0
$904$	51.1042	1.69970
$905$	0	0
$906$	47.3356	1.57262
$907$	−54.2874	−1.80258	−0.901292	−	0.433212i	$-0.857380\pi$
$907$	−54.2874	−1.80258	−0.901292	+	0.433212i	$0.857380\pi$
$908$	145.725	4.83606
$909$	25.8303	0.856738
$910$	0	0
$911$	7.20724	0.238787	0.119393	−	0.992847i	$-0.461905\pi$
$911$	7.20724	0.238787	0.119393	+	0.992847i	$0.461905\pi$
$912$	6.23875	0.206586
$913$	10.5614	0.349533
$914$	−36.1428	−1.19550
$915$	0	0
$916$	16.5562	0.547031
$917$	0	0
$918$	−14.6251	−0.482701
$919$	−0.791371	−0.0261049	−0.0130525	−	0.999915i	$-0.504155\pi$
$919$	−0.791371	−0.0261049	−0.0130525	+	0.999915i	$0.504155\pi$
$920$	0	0
$921$	−9.12266	−0.300602
$922$	−98.8944	−3.25691
$923$	−56.9676	−1.87511
$924$	0	0
$925$	0	0
$926$	−55.2197	−1.81463
$927$	−12.0759	−0.396624
$928$	−149.845	−4.91890
$929$	−48.7374	−1.59902	−0.799512	−	0.600650i	$-0.794908\pi$
$929$	−48.7374	−1.59902	−0.799512	+	0.600650i	$0.794908\pi$
$930$	0	0
$931$	−2.20334	−0.0722115
$932$	−3.07639	−0.100771
$933$	0	0
$934$	−22.7944	−0.745855
$935$	0	0
$936$	73.2700	2.39491
$937$	4.25551	0.139022	0.0695108	−	0.997581i	$-0.477856\pi$
$937$	4.25551	0.139022	0.0695108	+	0.997581i	$0.477856\pi$
$938$	0	0
$939$	−17.5207	−0.571766
$940$	0	0
$941$	31.5817	1.02954	0.514768	−	0.857330i	$-0.327878\pi$
$941$	31.5817	1.02954	0.514768	+	0.857330i	$0.327878\pi$
$942$	−2.06857	−0.0673978
$943$	10.4669	0.340850
$944$	126.914	4.13070
$945$	0	0
$946$	20.9522	0.681216
$947$	−24.4184	−0.793493	−0.396746	−	0.917928i	$-0.629861\pi$
$947$	−24.4184	−0.793493	−0.396746	+	0.917928i	$0.629861\pi$
$948$	25.5874	0.831041
$949$	1.89701	0.0615795
$950$	0	0
$951$	−16.3742	−0.530970
$952$	0	0
$953$	47.3324	1.53325	0.766624	−	0.642097i	$-0.221935\pi$
$953$	47.3324	1.53325	0.766624	+	0.642097i	$0.221935\pi$
$954$	−55.6273	−1.80100
$955$	0	0
$956$	−82.4000	−2.66501
$957$	−5.44649	−0.176060
$958$	31.4280	1.01539
$959$	0	0
$960$	0	0
$961$	40.0189	1.29093
$962$	89.1790	2.87525
$963$	−9.08270	−0.292686
$964$	−131.198	−4.22562
$965$	0	0
$966$	0	0
$967$	17.9360	0.576782	0.288391	−	0.957513i	$-0.406880\pi$
$967$	17.9360	0.576782	0.288391	+	0.957513i	$0.406880\pi$
$968$	−10.7080	−0.344168
$969$	−0.340851	−0.0109497
$970$	0	0
$971$	−8.27863	−0.265674	−0.132837	−	0.991138i	$-0.542409\pi$
$971$	−8.27863	−0.265674	−0.132837	+	0.991138i	$0.542409\pi$
$972$	90.2599	2.89509
$973$	0	0
$974$	61.6821	1.97642
$975$	0	0
$976$	241.441	7.72833
$977$	−53.0555	−1.69740	−0.848698	−	0.528877i	$-0.822613\pi$
$977$	−53.0555	−1.69740	−0.848698	+	0.528877i	$0.822613\pi$
$978$	1.00440	0.0321171
$979$	3.14654	0.100564
$980$	0	0
$981$	0.731129	0.0233431
$982$	45.7383	1.45957
$983$	20.4361	0.651810	0.325905	−	0.945402i	$-0.394331\pi$
$983$	20.4361	0.651810	0.325905	+	0.945402i	$0.394331\pi$
$984$	−48.3040	−1.53988
$985$	0	0
$986$	14.0716	0.448131
$987$	0	0
$988$	6.86828	0.218509
$989$	18.8168	0.598340
$990$	0	0
$991$	−0.203469	−0.00646339	−0.00323170	−	0.999995i	$-0.501029\pi$
$991$	−0.203469	−0.00646339	−0.00323170	+	0.999995i	$0.501029\pi$
$992$	251.069	7.97145
$993$	28.8057	0.914121
$994$	0	0
$995$	0	0
$996$	−66.6464	−2.11177
$997$	7.62738	0.241561	0.120781	−	0.992679i	$-0.461460\pi$
$997$	7.62738	0.241561	0.120781	+	0.992679i	$0.461460\pi$
$998$	67.4735	2.13584
$999$	44.4994	1.40790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4675.2.a.bd.1.1		4
5.4	even	2		187.2.a.f.1.4	✓	4
15.14	odd	2		1683.2.a.y.1.1		4
20.19	odd	2		2992.2.a.v.1.3		4
35.34	odd	2		9163.2.a.l.1.4		4
55.54	odd	2		2057.2.a.s.1.1		4
85.84	even	2		3179.2.a.w.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.a.f.1.4	✓	4	5.4	even	2
1683.2.a.y.1.1		4	15.14	odd	2
2057.2.a.s.1.1		4	55.54	odd	2
2992.2.a.v.1.3		4	20.19	odd	2
3179.2.a.w.1.4		4	85.84	even	2
4675.2.a.bd.1.1		4	1.1	even	1	trivial
9163.2.a.l.1.4		4	35.34	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:	\( N \)	\(=\)	\( 4675 = 5^{2} \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4675.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79774 q^{2} +1.08288 q^{3} +5.82737 q^{4} -3.02962 q^{6} -10.7080 q^{8} -1.82737 q^{9} +O(q^{10})\)	\(q-2.79774 q^{2} +1.08288 q^{3} +5.82737 q^{4} -3.02962 q^{6} -10.7080 q^{8} -1.82737 q^{9} -1.00000 q^{11} +6.31035 q^{12} +3.74449 q^{13} +18.3035 q^{16} -1.00000 q^{17} +5.11251 q^{18} +0.314763 q^{19} +2.79774 q^{22} +2.51261 q^{23} -11.5955 q^{24} -10.4761 q^{26} -5.22747 q^{27} +5.02962 q^{29} -8.42727 q^{31} -29.7925 q^{32} -1.08288 q^{33} +2.79774 q^{34} -10.6487 q^{36} -8.51261 q^{37} -0.880625 q^{38} +4.05483 q^{39} +4.16576 q^{41} +7.48897 q^{43} -5.82737 q^{44} -7.02962 q^{46} -4.88062 q^{47} +19.8205 q^{48} -7.00000 q^{49} -1.08288 q^{51} +21.8205 q^{52} -10.8806 q^{53} +14.6251 q^{54} +0.340851 q^{57} -14.0716 q^{58} +6.93388 q^{59} +13.1910 q^{61} +23.5773 q^{62} +46.7447 q^{64} +3.02962 q^{66} +7.07847 q^{67} -5.82737 q^{68} +2.72085 q^{69} -15.2137 q^{71} +19.5674 q^{72} +0.506613 q^{73} +23.8161 q^{74} +1.83424 q^{76} -11.3444 q^{78} +4.05483 q^{79} -0.178624 q^{81} -11.6547 q^{82} -10.5614 q^{83} -20.9522 q^{86} +5.44649 q^{87} +10.7080 q^{88} -3.14654 q^{89} +14.6419 q^{92} -9.12573 q^{93} +13.6547 q^{94} -32.2617 q^{96} +4.84413 q^{97} +19.5842 q^{98} +1.82737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 9 q^{8} + 11 q^{9}+O(q^{10}) \) 4 * q - q^2 - q^3 + 5 * q^4 - 4 * q^6 - 9 * q^8 + 11 * q^9	\( 4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 9 q^{8} + 11 q^{9} - 4 q^{11} + 2 q^{13} + 19 q^{16} - 4 q^{17} + 7 q^{18} - 2 q^{19} + q^{22} - 5 q^{23} - 26 q^{24} - 6 q^{26} - q^{27} + 12 q^{29} - 17 q^{31} - 39 q^{32} + q^{33} + q^{34} - 25 q^{36} - 19 q^{37} + 12 q^{38} - 22 q^{39} + 6 q^{41} + 4 q^{43} - 5 q^{44} - 20 q^{46} - 4 q^{47} + 32 q^{48} - 28 q^{49} + q^{51} + 40 q^{52} - 28 q^{53} + 30 q^{54} + 16 q^{57} + 15 q^{59} + 12 q^{61} - 6 q^{62} + 35 q^{64} + 4 q^{66} + q^{67} - 5 q^{68} - 13 q^{69} + 17 q^{71} + 25 q^{72} + 6 q^{73} + 26 q^{74} + 18 q^{76} - 34 q^{78} - 22 q^{79} - 10 q^{82} - 8 q^{83} - 12 q^{86} - 6 q^{87} + 9 q^{88} - 13 q^{89} + 12 q^{92} - 31 q^{93} + 18 q^{94} + 16 q^{96} - 17 q^{97} + 7 q^{98} - 11 q^{99}+O(q^{100}) \) 4 * q - q^2 - q^3 + 5 * q^4 - 4 * q^6 - 9 * q^8 + 11 * q^9 - 4 * q^11 + 2 * q^13 + 19 * q^16 - 4 * q^17 + 7 * q^18 - 2 * q^19 + q^22 - 5 * q^23 - 26 * q^24 - 6 * q^26 - q^27 + 12 * q^29 - 17 * q^31 - 39 * q^32 + q^33 + q^34 - 25 * q^36 - 19 * q^37 + 12 * q^38 - 22 * q^39 + 6 * q^41 + 4 * q^43 - 5 * q^44 - 20 * q^46 - 4 * q^47 + 32 * q^48 - 28 * q^49 + q^51 + 40 * q^52 - 28 * q^53 + 30 * q^54 + 16 * q^57 + 15 * q^59 + 12 * q^61 - 6 * q^62 + 35 * q^64 + 4 * q^66 + q^67 - 5 * q^68 - 13 * q^69 + 17 * q^71 + 25 * q^72 + 6 * q^73 + 26 * q^74 + 18 * q^76 - 34 * q^78 - 22 * q^79 - 10 * q^82 - 8 * q^83 - 12 * q^86 - 6 * q^87 + 9 * q^88 - 13 * q^89 + 12 * q^92 - 31 * q^93 + 18 * q^94 + 16 * q^96 - 17 * q^97 + 7 * q^98 - 11 * q^99

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