LMFDB - Embedded newform 4675.2.a.bd.1.3

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			1.3
Root			$-0.452661$ 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)

$f(q)$	$=$	$q+0.452661 q^{2} +2.96565 q^{3} -1.79510 q^{4} +1.34244 q^{6} -1.71789 q^{8} +5.79510 q^{9} +O(q^{10})$	\(q+0.452661 q^{2} +2.96565 q^{3} -1.79510 q^{4} +1.34244 q^{6} -1.71789 q^{8} +5.79510 q^{9} -1.00000 q^{11} -5.32364 q^{12} -5.76075 q^{13} +2.81257 q^{16} -1.00000 q^{17} +2.62322 q^{18} +1.07588 q^{19} -0.452661 q^{22} -5.87098 q^{23} -5.09468 q^{24} -2.60767 q^{26} +8.28929 q^{27} +0.657564 q^{29} -6.69909 q^{31} +4.70893 q^{32} -2.96565 q^{33} -0.452661 q^{34} -10.4028 q^{36} -0.129024 q^{37} +0.487008 q^{38} -17.0844 q^{39} +7.93131 q^{41} -11.5215 q^{43} +1.79510 q^{44} -2.65756 q^{46} -3.51299 q^{47} +8.34111 q^{48} -7.00000 q^{49} -2.96565 q^{51} +10.3411 q^{52} -9.51299 q^{53} +3.75224 q^{54} +3.19068 q^{57} +0.297654 q^{58} +11.8211 q^{59} +0.189355 q^{61} -3.03242 q^{62} -3.49359 q^{64} -1.34244 q^{66} -3.43386 q^{67} +1.79510 q^{68} -17.4113 q^{69} +6.01687 q^{71} -9.95536 q^{72} +7.12199 q^{73} -0.0584043 q^{74} -1.93131 q^{76} -7.73344 q^{78} -17.0844 q^{79} +7.19786 q^{81} +3.59020 q^{82} +3.96240 q^{83} -5.21534 q^{86} +1.95011 q^{87} +1.71789 q^{88} -8.68030 q^{89} +10.5390 q^{92} -19.8672 q^{93} -1.59020 q^{94} +13.9651 q^{96} +3.99164 q^{97} -3.16863 q^{98} -5.79510 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.452661	0.320080	0.160040	−	0.987111i	$-0.448838\pi$
$2$	0.452661	0.320080	0.160040	+	0.987111i	$0.448838\pi$
$3$	2.96565	1.71222	0.856110	−	0.516793i	$-0.172874\pi$
$3$	2.96565	1.71222	0.856110	+	0.516793i	$0.172874\pi$
$4$	−1.79510	−0.897549
$5$	0	0
$5$	0	0
$6$	1.34244	0.548047
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.71789	−0.607367
$9$	5.79510	1.93170
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−5.32364	−1.53680
$13$	−5.76075	−1.59774	−0.798872	−	0.601501i	$-0.794570\pi$
$13$	−5.76075	−1.59774	−0.798872	+	0.601501i	$0.794570\pi$
$14$	0	0
$15$	0	0
$16$	2.81257	0.703143
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.62322	0.618298
$19$	1.07588	0.246823	0.123412	−	0.992356i	$-0.460616\pi$
$19$	1.07588	0.246823	0.123412	+	0.992356i	$0.460616\pi$
$20$	0	0
$21$	0	0
$22$	−0.452661	−0.0965077
$23$	−5.87098	−1.22418	−0.612092	−	0.790787i	$-0.709672\pi$
$23$	−5.87098	−1.22418	−0.612092	+	0.790787i	$0.709672\pi$
$24$	−5.09468	−1.03995
$25$	0	0
$26$	−2.60767	−0.511406
$27$	8.28929	1.59527
$28$	0	0
$29$	0.657564	0.122106	0.0610532	−	0.998135i	$-0.480554\pi$
$29$	0.657564	0.122106	0.0610532	+	0.998135i	$0.480554\pi$
$30$	0	0
$31$	−6.69909	−1.20319	−0.601596	−	0.798800i	$-0.705468\pi$
$31$	−6.69909	−1.20319	−0.601596	+	0.798800i	$0.705468\pi$
$32$	4.70893	0.832429
$33$	−2.96565	−0.516254
$34$	−0.452661	−0.0776308
$35$	0	0
$36$	−10.4028	−1.73379
$37$	−0.129024	−0.0212115	−0.0106057	−	0.999944i	$-0.503376\pi$
$37$	−0.129024	−0.0212115	−0.0106057	+	0.999944i	$0.503376\pi$
$38$	0.487008	0.0790032
$39$	−17.0844	−2.73569
$40$	0	0
$41$	7.93131	1.23866	0.619331	−	0.785130i	$-0.287404\pi$
$41$	7.93131	1.23866	0.619331	+	0.785130i	$0.287404\pi$
$42$	0	0
$43$	−11.5215	−1.75701	−0.878506	−	0.477731i	$-0.841459\pi$
$43$	−11.5215	−1.75701	−0.878506	+	0.477731i	$0.841459\pi$
$44$	1.79510	0.270621
$45$	0	0
$46$	−2.65756	−0.391836
$47$	−3.51299	−0.512423	−0.256211	−	0.966621i	$-0.582474\pi$
$47$	−3.51299	−0.512423	−0.256211	+	0.966621i	$0.582474\pi$
$48$	8.34111	1.20394
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−2.96565	−0.415274
$52$	10.3411	1.43405
$53$	−9.51299	−1.30671	−0.653355	−	0.757052i	$-0.726639\pi$
$53$	−9.51299	−1.30671	−0.653355	+	0.757052i	$0.726639\pi$
$54$	3.75224	0.510615
$55$	0	0
$56$	0	0
$57$	3.19068	0.422616
$58$	0.297654	0.0390838
$59$	11.8211	1.53897	0.769487	−	0.638663i	$-0.220512\pi$
$59$	11.8211	1.53897	0.769487	+	0.638663i	$0.220512\pi$
$60$	0	0
$61$	0.189355	0.0242444	0.0121222	−	0.999927i	$-0.496141\pi$
$61$	0.189355	0.0242444	0.0121222	+	0.999927i	$0.496141\pi$
$62$	−3.03242	−0.385118
$63$	0	0
$64$	−3.49359	−0.436699
$65$	0	0
$66$	−1.34244	−0.165242
$67$	−3.43386	−0.419513	−0.209756	−	0.977754i	$-0.567267\pi$
$67$	−3.43386	−0.419513	−0.209756	+	0.977754i	$0.567267\pi$
$68$	1.79510	0.217688
$69$	−17.4113	−2.09607
$70$	0	0
$71$	6.01687	0.714072	0.357036	−	0.934091i	$-0.383787\pi$
$71$	6.01687	0.714072	0.357036	+	0.934091i	$0.383787\pi$
$72$	−9.95536	−1.17325
$73$	7.12199	0.833565	0.416783	−	0.909006i	$-0.363158\pi$
$73$	7.12199	0.833565	0.416783	+	0.909006i	$0.363158\pi$
$74$	−0.0584043	−0.00678937
$75$	0	0
$76$	−1.93131	−0.221536
$77$	0	0
$78$	−7.73344	−0.875640
$79$	−17.0844	−1.92214	−0.961072	−	0.276298i	$-0.910892\pi$
$79$	−17.0844	−1.92214	−0.961072	+	0.276298i	$0.910892\pi$
$80$	0	0
$81$	7.19786	0.799763
$82$	3.59020	0.396471
$83$	3.96240	0.434930	0.217465	−	0.976068i	$-0.430221\pi$
$83$	3.96240	0.434930	0.217465	+	0.976068i	$0.430221\pi$
$84$	0	0
$85$	0	0
$86$	−5.21534	−0.562384
$87$	1.95011	0.209073
$88$	1.71789	0.183128
$89$	−8.68030	−0.920109	−0.460055	−	0.887891i	$-0.652170\pi$
$89$	−8.68030	−0.920109	−0.460055	+	0.887891i	$0.652170\pi$
$90$	0	0
$91$	0	0
$92$	10.5390	1.09876
$93$	−19.8672	−2.06013
$94$	−1.59020	−0.164016
$95$	0	0
$96$	13.9651	1.42530
$97$	3.99164	0.405289	0.202645	−	0.979252i	$-0.435046\pi$
$97$	3.99164	0.405289	0.202645	+	0.979252i	$0.435046\pi$
$98$	−3.16863	−0.320080
$99$	−5.79510	−0.582429
$100$	0	0
$101$	−3.41699	−0.340003	−0.170002	−	0.985444i	$-0.554377\pi$
$101$	−3.41699	−0.340003	−0.170002	+	0.985444i	$0.554377\pi$
$102$	−1.34244	−0.132921
$103$	−11.0345	−1.08726	−0.543630	−	0.839325i	$-0.682951\pi$
$103$	−11.0345	−1.08726	−0.543630	+	0.839325i	$0.682951\pi$
$104$	9.89636	0.970418
$105$	0	0
$106$	−4.30616	−0.418252
$107$	9.34244	0.903167	0.451584	−	0.892229i	$-0.350859\pi$
$107$	9.34244	0.903167	0.451584	+	0.892229i	$0.350859\pi$
$108$	−14.8801	−1.43184
$109$	5.49419	0.526248	0.263124	−	0.964762i	$-0.415247\pi$
$109$	5.49419	0.526248	0.263124	+	0.964762i	$0.415247\pi$
$110$	0	0
$111$	−0.382641	−0.0363187
$112$	0	0
$113$	−18.2893	−1.72051	−0.860256	−	0.509863i	$-0.829696\pi$
$113$	−18.2893	−1.72051	−0.860256	+	0.509863i	$0.829696\pi$
$114$	1.44430	0.135271
$115$	0	0
$116$	−1.18039	−0.109597
$117$	−33.3841	−3.08636
$118$	5.35095	0.492595
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.0857136	0.00776014
$123$	23.5215	2.12086
$124$	12.0255	1.07992
$125$	0	0
$126$	0	0
$127$	14.4644	1.28351	0.641755	−	0.766910i	$-0.278207\pi$
$127$	14.4644	1.28351	0.641755	+	0.766910i	$0.278207\pi$
$128$	−10.9993	−0.972208
$129$	−34.1688	−3.00839
$130$	0	0
$131$	−7.24643	−0.633124	−0.316562	−	0.948572i	$-0.602529\pi$
$131$	−7.24643	−0.633124	−0.316562	+	0.948572i	$0.602529\pi$
$132$	5.32364	0.463363
$133$	0	0
$134$	−1.55438	−0.134278
$135$	0	0
$136$	1.71789	0.147308
$137$	−13.2134	−1.12890	−0.564449	−	0.825468i	$-0.690911\pi$
$137$	−13.2134	−1.12890	−0.564449	+	0.825468i	$0.690911\pi$
$138$	−7.88141	−0.670910
$139$	−5.56289	−0.471838	−0.235919	−	0.971773i	$-0.575810\pi$
$139$	−5.56289	−0.471838	−0.235919	+	0.971773i	$0.575810\pi$
$140$	0	0
$141$	−10.4183	−0.877380
$142$	2.72361	0.228560
$143$	5.76075	0.481738
$144$	16.2991	1.35826
$145$	0	0
$146$	3.22385	0.266808
$147$	−20.7596	−1.71222
$148$	0.231611	0.0190383
$149$	−8.78673	−0.719837	−0.359919	−	0.932984i	$-0.617196\pi$
$149$	−8.78673	−0.719837	−0.359919	+	0.932984i	$0.617196\pi$
$150$	0	0
$151$	14.1045	1.14781	0.573904	−	0.818922i	$-0.305428\pi$
$151$	14.1045	1.14781	0.573904	+	0.818922i	$0.305428\pi$
$152$	−1.84824	−0.149912
$153$	−5.79510	−0.468506
$154$	0	0
$155$	0	0
$156$	30.6681	2.45542
$157$	8.45985	0.675169	0.337585	−	0.941295i	$-0.390390\pi$
$157$	8.45985	0.675169	0.337585	+	0.941295i	$0.390390\pi$
$158$	−7.73344	−0.615240
$159$	−28.2122	−2.23738
$160$	0	0
$161$	0	0
$162$	3.25819	0.255988
$163$	−7.86261	−0.615847	−0.307924	−	0.951411i	$-0.599634\pi$
$163$	−7.86261	−0.615847	−0.307924	+	0.951411i	$0.599634\pi$
$164$	−14.2375	−1.11176
$165$	0	0
$166$	1.79363	0.139212
$167$	18.7122	1.44799	0.723996	−	0.689804i	$-0.242303\pi$
$167$	18.7122	1.44799	0.723996	+	0.689804i	$0.242303\pi$
$168$	0	0
$169$	20.1862	1.55279
$170$	0	0
$171$	6.23482	0.476788
$172$	20.6822	1.57700
$173$	5.77955	0.439411	0.219706	−	0.975566i	$-0.429490\pi$
$173$	5.77955	0.439411	0.219706	+	0.975566i	$0.429490\pi$
$174$	0.882737	0.0669201
$175$	0	0
$176$	−2.81257	−0.212006
$177$	35.0572	2.63506
$178$	−3.92923	−0.294509
$179$	18.9554	1.41679	0.708395	−	0.705816i	$-0.249419\pi$
$179$	18.9554	1.41679	0.708395	+	0.705816i	$0.249419\pi$
$180$	0	0
$181$	−23.4301	−1.74154	−0.870772	−	0.491687i	$-0.836380\pi$
$181$	−23.4301	−1.74154	−0.870772	+	0.491687i	$0.836380\pi$
$182$	0	0
$183$	0.561560	0.0415117
$184$	10.0857	0.743529
$185$	0	0
$186$	−8.99311	−0.659407
$187$	1.00000	0.0731272
$188$	6.30616	0.459924
$189$	0	0
$190$	0	0
$191$	−2.03567	−0.147296	−0.0736481	−	0.997284i	$-0.523464\pi$
$191$	−2.03567	−0.147296	−0.0736481	+	0.997284i	$0.523464\pi$
$192$	−10.3608	−0.747725
$193$	14.2310	1.02437	0.512186	−	0.858875i	$-0.328836\pi$
$193$	14.2310	1.02437	0.512186	+	0.858875i	$0.328836\pi$
$194$	1.80686	0.129725
$195$	0	0
$196$	12.5657	0.897549
$197$	−15.6835	−1.11741	−0.558703	−	0.829368i	$-0.688701\pi$
$197$	−15.6835	−1.11741	−0.558703	+	0.829368i	$0.688701\pi$
$198$	−2.62322	−0.186424
$199$	−20.8912	−1.48094	−0.740471	−	0.672089i	$-0.765397\pi$
$199$	−20.8912	−1.48094	−0.740471	+	0.672089i	$0.765397\pi$
$200$	0	0
$201$	−10.1836	−0.718299
$202$	−1.54674	−0.108828
$203$	0	0
$204$	5.32364	0.372729
$205$	0	0
$206$	−4.99489	−0.348010
$207$	−34.0229	−2.36475
$208$	−16.2025	−1.12344
$209$	−1.07588	−0.0744200
$210$	0	0
$211$	24.9743	1.71930	0.859651	−	0.510881i	$-0.170681\pi$
$211$	24.9743	1.71930	0.859651	+	0.510881i	$0.170681\pi$
$212$	17.0767	1.17284
$213$	17.8440	1.22265
$214$	4.22896	0.289086
$215$	0	0
$216$	−14.2401	−0.968918
$217$	0	0
$218$	2.48701	0.168441
$219$	21.1213	1.42725
$220$	0	0
$221$	5.76075	0.387510
$222$	−0.173207	−0.0116249
$223$	3.47997	0.233036	0.116518	−	0.993189i	$-0.462827\pi$
$223$	3.47997	0.233036	0.116518	+	0.993189i	$0.462827\pi$
$224$	0	0
$225$	0	0
$226$	−8.27885	−0.550701
$227$	−11.8690	−0.787776	−0.393888	−	0.919158i	$-0.628870\pi$
$227$	−11.8690	−0.787776	−0.393888	+	0.919158i	$0.628870\pi$
$228$	−5.72758	−0.379318
$229$	2.93249	0.193784	0.0968921	−	0.995295i	$-0.469110\pi$
$229$	2.93249	0.193784	0.0968921	+	0.995295i	$0.469110\pi$
$230$	0	0
$231$	0	0
$232$	−1.12962	−0.0741635
$233$	27.5359	1.80393	0.901967	−	0.431804i	$-0.142123\pi$
$233$	27.5359	1.80393	0.901967	+	0.431804i	$0.142123\pi$
$234$	−15.1117	−0.987883
$235$	0	0
$236$	−21.2200	−1.38130
$237$	−50.6664	−3.29113
$238$	0	0
$239$	13.6545	0.883233	0.441617	−	0.897204i	$-0.354405\pi$
$239$	13.6545	0.883233	0.441617	+	0.897204i	$0.354405\pi$
$240$	0	0
$241$	13.2635	0.854374	0.427187	−	0.904163i	$-0.359505\pi$
$241$	13.2635	0.854374	0.427187	+	0.904163i	$0.359505\pi$
$242$	0.452661	0.0290982
$243$	−3.52150	−0.225904
$244$	−0.339910	−0.0217605
$245$	0	0
$246$	10.6473	0.678845
$247$	−6.19786	−0.394361
$248$	11.5083	0.730780
$249$	11.7511	0.744696
$250$	0	0
$251$	−7.74653	−0.488957	−0.244478	−	0.969655i	$-0.578617\pi$
$251$	−7.74653	−0.488957	−0.244478	+	0.969655i	$0.578617\pi$
$252$	0	0
$253$	5.87098	0.369105
$254$	6.54748	0.410826
$255$	0	0
$256$	2.00824	0.125515
$257$	−22.6907	−1.41541	−0.707704	−	0.706509i	$-0.750269\pi$
$257$	−22.6907	−1.41541	−0.707704	+	0.706509i	$0.750269\pi$
$258$	−15.4669	−0.962926
$259$	0	0
$260$	0	0
$261$	3.81065	0.235873
$262$	−3.28018	−0.202650
$263$	1.29633	0.0799350	0.0399675	−	0.999201i	$-0.487275\pi$
$263$	1.29633	0.0799350	0.0399675	+	0.999201i	$0.487275\pi$
$264$	5.09468	0.313556
$265$	0	0
$266$	0	0
$267$	−25.7427	−1.57543
$268$	6.16412	0.376533
$269$	20.0144	1.22030	0.610149	−	0.792287i	$-0.291110\pi$
$269$	20.0144	1.22030	0.610149	+	0.792287i	$0.291110\pi$
$270$	0	0
$271$	−31.1637	−1.89306	−0.946529	−	0.322619i	$-0.895437\pi$
$271$	−31.1637	−1.89306	−0.946529	+	0.322619i	$0.895437\pi$
$272$	−2.81257	−0.170537
$273$	0	0
$274$	−5.98120	−0.361338
$275$	0	0
$276$	31.2549	1.88133
$277$	−17.1428	−1.03001	−0.515005	−	0.857187i	$-0.672210\pi$
$277$	−17.1428	−1.03001	−0.515005	+	0.857187i	$0.672210\pi$
$278$	−2.51810	−0.151026
$279$	−38.8219	−2.32421
$280$	0	0
$281$	−18.0643	−1.07762	−0.538812	−	0.842426i	$-0.681127\pi$
$281$	−18.0643	−1.07762	−0.538812	+	0.842426i	$0.681127\pi$
$282$	−4.71597	−0.280832
$283$	−24.0106	−1.42728	−0.713640	−	0.700512i	$-0.752955\pi$
$283$	−24.0106	−1.42728	−0.713640	+	0.700512i	$0.752955\pi$
$284$	−10.8009	−0.640914
$285$	0	0
$286$	2.60767	0.154195
$287$	0	0
$288$	27.2887	1.60800
$289$	1.00000	0.0588235
$290$	0	0
$291$	11.8378	0.693945
$292$	−12.7847	−0.748166
$293$	7.45281	0.435398	0.217699	−	0.976016i	$-0.430145\pi$
$293$	7.45281	0.435398	0.217699	+	0.976016i	$0.430145\pi$
$294$	−9.39705	−0.548047
$295$	0	0
$296$	0.221650	0.0128832
$297$	−8.28929	−0.480993
$298$	−3.97741	−0.230405
$299$	33.8212	1.95593
$300$	0	0
$301$	0	0
$302$	6.38457	0.367391
$303$	−10.1336	−0.582160
$304$	3.02598	0.173552
$305$	0	0
$306$	−2.62322	−0.149959
$307$	33.0929	1.88871	0.944356	−	0.328926i	$-0.106687\pi$
$307$	33.0929	1.88871	0.944356	+	0.328926i	$0.106687\pi$
$308$	0	0
$309$	−32.7245	−1.86163
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	29.3492	1.66157
$313$	13.0740	0.738983	0.369492	−	0.929234i	$-0.379532\pi$
$313$	13.0740	0.738983	0.369492	+	0.929234i	$0.379532\pi$
$314$	3.82944	0.216108
$315$	0	0
$316$	30.6681	1.72522
$317$	10.9055	0.612512	0.306256	−	0.951949i	$-0.400924\pi$
$317$	10.9055	0.612512	0.306256	+	0.951949i	$0.400924\pi$
$318$	−12.7706	−0.716139
$319$	−0.657564	−0.0368165
$320$	0	0
$321$	27.7064	1.54642
$322$	0	0
$323$	−1.07588	−0.0598634
$324$	−12.9209	−0.717826
$325$	0	0
$326$	−3.55910	−0.197120
$327$	16.2939	0.901053
$328$	−13.6251	−0.752323
$329$	0	0
$330$	0	0
$331$	10.6181	0.583624	0.291812	−	0.956476i	$-0.405742\pi$
$331$	10.6181	0.583624	0.291812	+	0.956476i	$0.405742\pi$
$332$	−7.11290	−0.390371
$333$	−0.747709	−0.0409742
$334$	8.47028	0.463473
$335$	0	0
$336$	0	0
$337$	−9.14664	−0.498249	−0.249125	−	0.968471i	$-0.580143\pi$
$337$	−9.14664	−0.498249	−0.249125	+	0.968471i	$0.580143\pi$
$338$	9.13753	0.497016
$339$	−54.2397	−2.94590
$340$	0	0
$341$	6.69909	0.362776
$342$	2.82226	0.152610
$343$	0	0
$344$	19.7927	1.06715
$345$	0	0
$346$	2.61618	0.140647
$347$	0.957289	0.0513900	0.0256950	−	0.999670i	$-0.491820\pi$
$347$	0.957289	0.0513900	0.0256950	+	0.999670i	$0.491820\pi$
$348$	−3.50063	−0.187653
$349$	0.713316	0.0381829	0.0190915	−	0.999818i	$-0.493923\pi$
$349$	0.713316	0.0381829	0.0190915	+	0.999818i	$0.493923\pi$
$350$	0	0
$351$	−47.7525	−2.54884
$352$	−4.70893	−0.250987
$353$	15.9444	0.848634	0.424317	−	0.905514i	$-0.360514\pi$
$353$	15.9444	0.848634	0.424317	+	0.905514i	$0.360514\pi$
$354$	15.8690	0.843430
$355$	0	0
$356$	15.5820	0.825843
$357$	0	0
$358$	8.58036	0.453486
$359$	−7.72315	−0.407612	−0.203806	−	0.979011i	$-0.565331\pi$
$359$	−7.72315	−0.407612	−0.203806	+	0.979011i	$0.565331\pi$
$360$	0	0
$361$	−17.8425	−0.939078
$362$	−10.6059	−0.557433
$363$	2.96565	0.155656
$364$	0	0
$365$	0	0
$366$	0.254197	0.0132871
$367$	−18.5162	−0.966540	−0.483270	−	0.875471i	$-0.660551\pi$
$367$	−18.5162	−0.966540	−0.483270	+	0.875471i	$0.660551\pi$
$368$	−16.5125	−0.860776
$369$	45.9627	2.39272
$370$	0	0
$371$	0	0
$372$	35.6635	1.84907
$373$	5.54030	0.286866	0.143433	−	0.989660i	$-0.454186\pi$
$373$	5.54030	0.286866	0.143433	+	0.989660i	$0.454186\pi$
$374$	0.452661	0.0234066
$375$	0	0
$376$	6.03495	0.311229
$377$	−3.78806	−0.195095
$378$	0	0
$379$	−19.2492	−0.988767	−0.494383	−	0.869244i	$-0.664606\pi$
$379$	−19.2492	−0.988767	−0.494383	+	0.869244i	$0.664606\pi$
$380$	0	0
$381$	42.8965	2.19765
$382$	−0.921470	−0.0471465
$383$	−4.86965	−0.248827	−0.124414	−	0.992230i	$-0.539705\pi$
$383$	−4.86965	−0.248827	−0.124414	+	0.992230i	$0.539705\pi$
$384$	−32.6200	−1.66463
$385$	0	0
$386$	6.44184	0.327881
$387$	−66.7682	−3.39402
$388$	−7.16538	−0.363767
$389$	−27.5114	−1.39488	−0.697441	−	0.716643i	$-0.745678\pi$
$389$	−27.5114	−1.39488	−0.697441	+	0.716643i	$0.745678\pi$
$390$	0	0
$391$	5.87098	0.296908
$392$	12.0253	0.607367
$393$	−21.4904	−1.08405
$394$	−7.09934	−0.357659
$395$	0	0
$396$	10.4028	0.522759
$397$	13.3547	0.670255	0.335127	−	0.942173i	$-0.391221\pi$
$397$	13.3547	0.670255	0.335127	+	0.942173i	$0.391221\pi$
$398$	−9.45666	−0.474020
$399$	0	0
$400$	0	0
$401$	−5.05122	−0.252246	−0.126123	−	0.992015i	$-0.540253\pi$
$401$	−5.05122	−0.252246	−0.126123	+	0.992015i	$0.540253\pi$
$402$	−4.60974	−0.229913
$403$	38.5918	1.92240
$404$	6.13383	0.305169
$405$	0	0
$406$	0	0
$407$	0.129024	0.00639550
$408$	5.09468	0.252224
$409$	−29.1111	−1.43945	−0.719726	−	0.694258i	$-0.755733\pi$
$409$	−29.1111	−1.43945	−0.719726	+	0.694258i	$0.755733\pi$
$410$	0	0
$411$	−39.1864	−1.93292
$412$	19.8080	0.975870
$413$	0	0
$414$	−15.4008	−0.756910
$415$	0	0
$416$	−27.1270	−1.33001
$417$	−16.4976	−0.807890
$418$	−0.487008	−0.0238204
$419$	26.0314	1.27172	0.635858	−	0.771806i	$-0.280646\pi$
$419$	26.0314	1.27172	0.635858	+	0.771806i	$0.280646\pi$
$420$	0	0
$421$	24.3147	1.18502	0.592512	−	0.805561i	$-0.298136\pi$
$421$	24.3147	1.18502	0.592512	+	0.805561i	$0.298136\pi$
$422$	11.3049	0.550314
$423$	−20.3581	−0.989846
$424$	16.3423	0.793653
$425$	0	0
$426$	8.07727	0.391345
$427$	0	0
$428$	−16.7706	−0.810637
$429$	17.0844	0.824842
$430$	0	0
$431$	23.6319	1.13831	0.569154	−	0.822231i	$-0.307271\pi$
$431$	23.6319	1.13831	0.569154	+	0.822231i	$0.307271\pi$
$432$	23.3142	1.12171
$433$	−0.764003	−0.0367156	−0.0183578	−	0.999831i	$-0.505844\pi$
$433$	−0.764003	−0.0367156	−0.0183578	+	0.999831i	$0.505844\pi$
$434$	0	0
$435$	0	0
$436$	−9.86261	−0.472333
$437$	−6.31645	−0.302157
$438$	9.56081	0.456833
$439$	7.78599	0.371605	0.185802	−	0.982587i	$-0.440512\pi$
$439$	7.78599	0.371605	0.185802	+	0.982587i	$0.440512\pi$
$440$	0	0
$441$	−40.5657	−1.93170
$442$	2.60767	0.124034
$443$	9.71479	0.461564	0.230782	−	0.973006i	$-0.425872\pi$
$443$	9.71479	0.461564	0.230782	+	0.973006i	$0.425872\pi$
$444$	0.686879	0.0325978
$445$	0	0
$446$	1.57525	0.0745902
$447$	−26.0584	−1.23252
$448$	0	0
$449$	8.39494	0.396182	0.198091	−	0.980184i	$-0.436526\pi$
$449$	8.39494	0.396182	0.198091	+	0.980184i	$0.436526\pi$
$450$	0	0
$451$	−7.93131	−0.373471
$452$	32.8311	1.54424
$453$	41.8291	1.96530
$454$	−5.37266	−0.252151
$455$	0	0
$456$	−5.48125	−0.256683
$457$	37.4860	1.75352	0.876760	−	0.480928i	$-0.159700\pi$
$457$	37.4860	1.75352	0.876760	+	0.480928i	$0.159700\pi$
$458$	1.32742	0.0620264
$459$	−8.28929	−0.386911
$460$	0	0
$461$	1.32163	0.0615545	0.0307772	−	0.999526i	$-0.490202\pi$
$461$	1.32163	0.0615545	0.0307772	+	0.999526i	$0.490202\pi$
$462$	0	0
$463$	−41.9523	−1.94969	−0.974843	−	0.222891i	$-0.928451\pi$
$463$	−41.9523	−1.94969	−0.974843	+	0.222891i	$0.928451\pi$
$464$	1.84944	0.0858583
$465$	0	0
$466$	12.4644	0.577403
$467$	32.5370	1.50564	0.752818	−	0.658229i	$-0.228694\pi$
$467$	32.5370	1.50564	0.752818	+	0.658229i	$0.228694\pi$
$468$	59.9277	2.77016
$469$	0	0
$470$	0	0
$471$	25.0890	1.15604
$472$	−20.3074	−0.934722
$473$	11.5215	0.529759
$474$	−22.9347	−1.05343
$475$	0	0
$476$	0	0
$477$	−55.1287	−2.52417
$478$	6.18084	0.282705
$479$	−36.5619	−1.67055	−0.835277	−	0.549830i	$-0.814693\pi$
$479$	−36.5619	−1.67055	−0.835277	+	0.549830i	$0.814693\pi$
$480$	0	0
$481$	0.743277	0.0338905
$482$	6.00385	0.273468
$483$	0	0
$484$	−1.79510	−0.0815954
$485$	0	0
$486$	−1.59405	−0.0723075
$487$	21.8049	0.988076	0.494038	−	0.869440i	$-0.335520\pi$
$487$	21.8049	0.988076	0.494038	+	0.869440i	$0.335520\pi$
$488$	−0.325291	−0.0147252
$489$	−23.3178	−1.05447
$490$	0	0
$491$	−30.6493	−1.38319	−0.691593	−	0.722288i	$-0.743091\pi$
$491$	−30.6493	−1.38319	−0.691593	+	0.722288i	$0.743091\pi$
$492$	−42.2234	−1.90358
$493$	−0.657564	−0.0296152
$494$	−2.80553	−0.126227
$495$	0	0
$496$	−18.8417	−0.846016
$497$	0	0
$498$	5.31927	0.238362
$499$	6.71011	0.300386	0.150193	−	0.988657i	$-0.452011\pi$
$499$	6.71011	0.300386	0.150193	+	0.988657i	$0.452011\pi$
$500$	0	0
$501$	55.4938	2.47928
$502$	−3.50655	−0.156505
$503$	11.6862	0.521062	0.260531	−	0.965466i	$-0.416102\pi$
$503$	11.6862	0.521062	0.260531	+	0.965466i	$0.416102\pi$
$504$	0	0
$505$	0	0
$506$	2.65756	0.118143
$507$	59.8654	2.65872
$508$	−25.9651	−1.15201
$509$	1.27945	0.0567108	0.0283554	−	0.999598i	$-0.490973\pi$
$509$	1.27945	0.0567108	0.0283554	+	0.999598i	$0.490973\pi$
$510$	0	0
$511$	0	0
$512$	22.9076	1.01238
$513$	8.91826	0.393751
$514$	−10.2712	−0.453044
$515$	0	0
$516$	61.3363	2.70018
$517$	3.51299	0.154501
$518$	0	0
$519$	17.1401	0.752369
$520$	0	0
$521$	6.44370	0.282304	0.141152	−	0.989988i	$-0.454919\pi$
$521$	6.44370	0.282304	0.141152	+	0.989988i	$0.454919\pi$
$522$	1.72493	0.0754982
$523$	−14.2440	−0.622845	−0.311423	−	0.950271i	$-0.600806\pi$
$523$	−14.2440	−0.622845	−0.311423	+	0.950271i	$0.600806\pi$
$524$	13.0081	0.568260
$525$	0	0
$526$	0.586797	0.0255856
$527$	6.69909	0.291817
$528$	−8.34111	−0.363000
$529$	11.4684	0.498624
$530$	0	0
$531$	68.5043	2.97283
$532$	0	0
$533$	−45.6903	−1.97907
$534$	−11.6527	−0.504264
$535$	0	0
$536$	5.89901	0.254798
$537$	56.2150	2.42586
$538$	9.05973	0.390593
$539$	7.00000	0.301511
$540$	0	0
$541$	30.7641	1.32265	0.661327	−	0.750098i	$-0.269994\pi$
$541$	30.7641	1.32265	0.661327	+	0.750098i	$0.269994\pi$
$542$	−14.1066	−0.605930
$543$	−69.4855	−2.98191
$544$	−4.70893	−0.201894
$545$	0	0
$546$	0	0
$547$	22.3452	0.955411	0.477706	−	0.878520i	$-0.341469\pi$
$547$	22.3452	0.955411	0.477706	+	0.878520i	$0.341469\pi$
$548$	23.7194	1.01324
$549$	1.09733	0.0468329
$550$	0	0
$551$	0.707458	0.0301387
$552$	29.9107	1.27309
$553$	0	0
$554$	−7.75988	−0.329686
$555$	0	0
$556$	9.98592	0.423498
$557$	−35.9483	−1.52318	−0.761590	−	0.648059i	$-0.775581\pi$
$557$	−35.9483	−1.52318	−0.761590	+	0.648059i	$0.775581\pi$
$558$	−17.5732	−0.743932
$559$	66.3725	2.80726
$560$	0	0
$561$	2.96565	0.125210
$562$	−8.17699	−0.344926
$563$	31.2109	1.31538	0.657691	−	0.753287i	$-0.271533\pi$
$563$	31.2109	1.31538	0.657691	+	0.753287i	$0.271533\pi$
$564$	18.7019	0.787492
$565$	0	0
$566$	−10.8687	−0.456844
$567$	0	0
$568$	−10.3364	−0.433704
$569$	12.1233	0.508236	0.254118	−	0.967173i	$-0.418215\pi$
$569$	12.1233	0.508236	0.254118	+	0.967173i	$0.418215\pi$
$570$	0	0
$571$	19.3906	0.811469	0.405735	−	0.913991i	$-0.367016\pi$
$571$	19.3906	0.811469	0.405735	+	0.913991i	$0.367016\pi$
$572$	−10.3411	−0.432384
$573$	−6.03710	−0.252203
$574$	0	0
$575$	0	0
$576$	−20.2457	−0.843571
$577$	−23.6235	−0.983460	−0.491730	−	0.870748i	$-0.663635\pi$
$577$	−23.6235	−0.983460	−0.491730	+	0.870748i	$0.663635\pi$
$578$	0.452661	0.0188282
$579$	42.2043	1.75395
$580$	0	0
$581$	0	0
$582$	5.35852	0.222118
$583$	9.51299	0.393988
$584$	−12.2348	−0.506280
$585$	0	0
$586$	3.37360	0.139362
$587$	17.2614	0.712456	0.356228	−	0.934399i	$-0.384063\pi$
$587$	17.2614	0.712456	0.356228	+	0.934399i	$0.384063\pi$
$588$	37.2655	1.53680
$589$	−7.20741	−0.296976
$590$	0	0
$591$	−46.5120	−1.91325
$592$	−0.362890	−0.0149147
$593$	36.6285	1.50415	0.752076	−	0.659077i	$-0.229053\pi$
$593$	36.6285	1.50415	0.752076	+	0.659077i	$0.229053\pi$
$594$	−3.75224	−0.153956
$595$	0	0
$596$	15.7730	0.646089
$597$	−61.9562	−2.53570
$598$	15.3096	0.626055
$599$	28.7453	1.17450	0.587252	−	0.809404i	$-0.300210\pi$
$599$	28.7453	1.17450	0.587252	+	0.809404i	$0.300210\pi$
$600$	0	0
$601$	−14.7873	−0.603187	−0.301594	−	0.953437i	$-0.597519\pi$
$601$	−14.7873	−0.603187	−0.301594	+	0.953437i	$0.597519\pi$
$602$	0	0
$603$	−19.8996	−0.810373
$604$	−25.3190	−1.03021
$605$	0	0
$606$	−4.58709	−0.186338
$607$	21.9107	0.889329	0.444664	−	0.895697i	$-0.353323\pi$
$607$	21.9107	0.889329	0.444664	+	0.895697i	$0.353323\pi$
$608$	5.06623	0.205463
$609$	0	0
$610$	0	0
$611$	20.2375	0.818720
$612$	10.4028	0.420507
$613$	18.6616	0.753737	0.376868	−	0.926267i	$-0.377001\pi$
$613$	18.6616	0.753737	0.376868	+	0.926267i	$0.377001\pi$
$614$	14.9799	0.604539
$615$	0	0
$616$	0	0
$617$	9.77039	0.393341	0.196671	−	0.980470i	$-0.436987\pi$
$617$	9.77039	0.393341	0.196671	+	0.980470i	$0.436987\pi$
$618$	−14.8131	−0.595870
$619$	4.23467	0.170206	0.0851029	−	0.996372i	$-0.472878\pi$
$619$	4.23467	0.170206	0.0851029	+	0.996372i	$0.472878\pi$
$620$	0	0
$621$	−48.6662	−1.95291
$622$	0	0
$623$	0	0
$624$	−48.0511	−1.92358
$625$	0	0
$626$	5.91807	0.236534
$627$	−3.19068	−0.127424
$628$	−15.1862	−0.605997
$629$	0.129024	0.00514454
$630$	0	0
$631$	−4.30809	−0.171502	−0.0857512	−	0.996317i	$-0.527329\pi$
$631$	−4.30809	−0.171502	−0.0857512	+	0.996317i	$0.527329\pi$
$632$	29.3492	1.16745
$633$	74.0651	2.94382
$634$	4.93648	0.196053
$635$	0	0
$636$	50.6437	2.00815
$637$	40.3253	1.59774
$638$	−0.297654	−0.0117842
$639$	34.8684	1.37937
$640$	0	0
$641$	17.6649	0.697721	0.348861	−	0.937175i	$-0.386569\pi$
$641$	17.6649	0.697721	0.348861	+	0.937175i	$0.386569\pi$
$642$	12.5416	0.494979
$643$	−17.8049	−0.702158	−0.351079	−	0.936346i	$-0.614185\pi$
$643$	−17.8049	−0.702158	−0.351079	+	0.936346i	$0.614185\pi$
$644$	0	0
$645$	0	0
$646$	−0.487008	−0.0191611
$647$	15.9041	0.625256	0.312628	−	0.949876i	$-0.398791\pi$
$647$	15.9041	0.625256	0.312628	+	0.949876i	$0.398791\pi$
$648$	−12.3652	−0.485750
$649$	−11.8211	−0.464018
$650$	0	0
$651$	0	0
$652$	14.1142	0.552753
$653$	15.8081	0.618620	0.309310	−	0.950961i	$-0.399902\pi$
$653$	15.8081	0.618620	0.309310	+	0.950961i	$0.399902\pi$
$654$	7.37560	0.288409
$655$	0	0
$656$	22.3074	0.870956
$657$	41.2726	1.61020
$658$	0	0
$659$	−26.6000	−1.03619	−0.518095	−	0.855323i	$-0.673359\pi$
$659$	−26.6000	−1.03619	−0.518095	+	0.855323i	$0.673359\pi$
$660$	0	0
$661$	−38.3283	−1.49080	−0.745399	−	0.666618i	$-0.767741\pi$
$661$	−38.3283	−1.49080	−0.745399	+	0.666618i	$0.767741\pi$
$662$	4.80641	0.186806
$663$	17.0844	0.663503
$664$	−6.80698	−0.264162
$665$	0	0
$666$	−0.338459	−0.0131150
$667$	−3.86054	−0.149481
$668$	−33.5902	−1.29964
$669$	10.3204	0.399009
$670$	0	0
$671$	−0.189355	−0.00730996
$672$	0	0
$673$	5.49419	0.211786	0.105893	−	0.994378i	$-0.466230\pi$
$673$	5.49419	0.211786	0.105893	+	0.994378i	$0.466230\pi$
$674$	−4.14033	−0.159480
$675$	0	0
$676$	−36.2363	−1.39370
$677$	−25.8239	−0.992492	−0.496246	−	0.868182i	$-0.665289\pi$
$677$	−25.8239	−0.992492	−0.496246	+	0.868182i	$0.665289\pi$
$678$	−24.5522	−0.942922
$679$	0	0
$680$	0	0
$681$	−35.1995	−1.34885
$682$	3.03242	0.116117
$683$	−4.64462	−0.177722	−0.0888608	−	0.996044i	$-0.528323\pi$
$683$	−4.64462	−0.177722	−0.0888608	+	0.996044i	$0.528323\pi$
$684$	−11.1921	−0.427941
$685$	0	0
$686$	0	0
$687$	8.69674	0.331801
$688$	−32.4050	−1.23543
$689$	54.8020	2.08779
$690$	0	0
$691$	−32.1519	−1.22312	−0.611558	−	0.791200i	$-0.709457\pi$
$691$	−32.1519	−1.22312	−0.611558	+	0.791200i	$0.709457\pi$
$692$	−10.3749	−0.394393
$693$	0	0
$694$	0.433328	0.0164489
$695$	0	0
$696$	−3.35007	−0.126984
$697$	−7.93131	−0.300420
$698$	0.322891	0.0122216
$699$	81.6618	3.08873
$700$	0	0
$701$	−27.2608	−1.02963	−0.514813	−	0.857302i	$-0.672139\pi$
$701$	−27.2608	−1.02963	−0.514813	+	0.857302i	$0.672139\pi$
$702$	−21.6157	−0.815833
$703$	−0.138814	−0.00523549
$704$	3.49359	0.131670
$705$	0	0
$706$	7.21741	0.271631
$707$	0	0
$708$	−62.9311	−2.36510
$709$	−11.6790	−0.438613	−0.219306	−	0.975656i	$-0.570379\pi$
$709$	−11.6790	−0.438613	−0.219306	+	0.975656i	$0.570379\pi$
$710$	0	0
$711$	−99.0057	−3.71300
$712$	14.9118	0.558844
$713$	39.3302	1.47293
$714$	0	0
$715$	0	0
$716$	−34.0267	−1.27164
$717$	40.4944	1.51229
$718$	−3.49597	−0.130469
$719$	−9.10039	−0.339387	−0.169694	−	0.985497i	$-0.554278\pi$
$719$	−9.10039	−0.339387	−0.169694	+	0.985497i	$0.554278\pi$
$720$	0	0
$721$	0	0
$722$	−8.07660	−0.300580
$723$	39.3348	1.46288
$724$	42.0593	1.56312
$725$	0	0
$726$	1.34244	0.0498225
$727$	−25.7439	−0.954788	−0.477394	−	0.878689i	$-0.658418\pi$
$727$	−25.7439	−0.954788	−0.477394	+	0.878689i	$0.658418\pi$
$728$	0	0
$729$	−32.0371	−1.18656
$730$	0	0
$731$	11.5215	0.426138
$732$	−1.00806	−0.0372588
$733$	−11.7602	−0.434372	−0.217186	−	0.976130i	$-0.569688\pi$
$733$	−11.7602	−0.434372	−0.217186	+	0.976130i	$0.569688\pi$
$734$	−8.38159	−0.309370
$735$	0	0
$736$	−27.6460	−1.01905
$737$	3.43386	0.126488
$738$	20.8055	0.765862
$739$	−9.43372	−0.347025	−0.173512	−	0.984832i	$-0.555512\pi$
$739$	−9.43372	−0.347025	−0.173512	+	0.984832i	$0.555512\pi$
$740$	0	0
$741$	−18.3807	−0.675232
$742$	0	0
$743$	−40.6616	−1.49173	−0.745865	−	0.666097i	$-0.767964\pi$
$743$	−40.6616	−1.49173	−0.745865	+	0.666097i	$0.767964\pi$
$744$	34.1297	1.25126
$745$	0	0
$746$	2.50788	0.0918200
$747$	22.9625	0.840154
$748$	−1.79510	−0.0656353
$749$	0	0
$750$	0	0
$751$	13.7251	0.500835	0.250418	−	0.968138i	$-0.419432\pi$
$751$	13.7251	0.500835	0.250418	+	0.968138i	$0.419432\pi$
$752$	−9.88054	−0.360306
$753$	−22.9735	−0.837202
$754$	−1.71471	−0.0624460
$755$	0	0
$756$	0	0
$757$	28.4606	1.03442	0.517209	−	0.855859i	$-0.326971\pi$
$757$	28.4606	1.03442	0.517209	+	0.855859i	$0.326971\pi$
$758$	−8.71338	−0.316484
$759$	17.4113	0.631989
$760$	0	0
$761$	−26.5143	−0.961143	−0.480572	−	0.876955i	$-0.659571\pi$
$761$	−26.5143	−0.961143	−0.480572	+	0.876955i	$0.659571\pi$
$762$	19.4176	0.703424
$763$	0	0
$764$	3.65423	0.132205
$765$	0	0
$766$	−2.20430	−0.0796447
$767$	−68.0983	−2.45889
$768$	5.95574	0.214909
$769$	16.9884	0.612617	0.306308	−	0.951932i	$-0.400906\pi$
$769$	16.9884	0.612617	0.306308	+	0.951932i	$0.400906\pi$
$770$	0	0
$771$	−67.2928	−2.42349
$772$	−25.5461	−0.919424
$773$	17.5793	0.632284	0.316142	−	0.948712i	$-0.397612\pi$
$773$	17.5793	0.632284	0.316142	+	0.948712i	$0.397612\pi$
$774$	−30.2234	−1.08636
$775$	0	0
$776$	−6.85721	−0.246159
$777$	0	0
$778$	−12.4533	−0.446473
$779$	8.53312	0.305731
$780$	0	0
$781$	−6.01687	−0.215301
$782$	2.65756	0.0950343
$783$	5.45074	0.194793
$784$	−19.6880	−0.703143
$785$	0	0
$786$	−9.72788	−0.346982
$787$	−31.4114	−1.11970	−0.559848	−	0.828595i	$-0.689141\pi$
$787$	−31.4114	−1.11970	−0.559848	+	0.828595i	$0.689141\pi$
$788$	28.1535	1.00293
$789$	3.84446	0.136866
$790$	0	0
$791$	0	0
$792$	9.95536	0.353748
$793$	−1.09083	−0.0387363
$794$	6.04517	0.214535
$795$	0	0
$796$	37.5018	1.32922
$797$	−49.6713	−1.75945	−0.879724	−	0.475484i	$-0.842273\pi$
$797$	−49.6713	−1.75945	−0.879724	+	0.475484i	$0.842273\pi$
$798$	0	0
$799$	3.51299	0.124281
$800$	0	0
$801$	−50.3032	−1.77737
$802$	−2.28649	−0.0807388
$803$	−7.12199	−0.251329
$804$	18.2806	0.644708
$805$	0	0
$806$	17.4690	0.615320
$807$	59.3557	2.08942
$808$	5.87002	0.206507
$809$	−19.7044	−0.692770	−0.346385	−	0.938092i	$-0.612591\pi$
$809$	−19.7044	−0.692770	−0.346385	+	0.938092i	$0.612591\pi$
$810$	0	0
$811$	5.71199	0.200575	0.100288	−	0.994958i	$-0.468024\pi$
$811$	5.71199	0.200575	0.100288	+	0.994958i	$0.468024\pi$
$812$	0	0
$813$	−92.4206	−3.24133
$814$	0.0584043	0.00204707
$815$	0	0
$816$	−8.34111	−0.291997
$817$	−12.3957	−0.433672
$818$	−13.1775	−0.460740
$819$	0	0
$820$	0	0
$821$	21.1193	0.737070	0.368535	−	0.929614i	$-0.379859\pi$
$821$	21.1193	0.737070	0.368535	+	0.929614i	$0.379859\pi$
$822$	−17.7382	−0.618690
$823$	−10.4410	−0.363952	−0.181976	−	0.983303i	$-0.558249\pi$
$823$	−10.4410	−0.363952	−0.181976	+	0.983303i	$0.558249\pi$
$824$	18.9561	0.660367
$825$	0	0
$826$	0	0
$827$	−21.4798	−0.746927	−0.373463	−	0.927645i	$-0.621830\pi$
$827$	−21.4798	−0.746927	−0.373463	+	0.927645i	$0.621830\pi$
$828$	61.0744	2.12248
$829$	−0.205649	−0.00714247	−0.00357124	−	0.999994i	$-0.501137\pi$
$829$	−0.205649	−0.00714247	−0.00357124	+	0.999994i	$0.501137\pi$
$830$	0	0
$831$	−50.8396	−1.76361
$832$	20.1257	0.697734
$833$	7.00000	0.242536
$834$	−7.46782	−0.258589
$835$	0	0
$836$	1.93131	0.0667956
$837$	−55.5307	−1.91942
$838$	11.7834	0.407051
$839$	−36.5520	−1.26192	−0.630958	−	0.775817i	$-0.717338\pi$
$839$	−36.5520	−1.26192	−0.630958	+	0.775817i	$0.717338\pi$
$840$	0	0
$841$	−28.5676	−0.985090
$842$	11.0063	0.379303
$843$	−53.5723	−1.84513
$844$	−44.8313	−1.54316
$845$	0	0
$846$	−9.21534	−0.316830
$847$	0	0
$848$	−26.7560	−0.918804
$849$	−71.2071	−2.44382
$850$	0	0
$851$	0.757499	0.0259667
$852$	−32.0316	−1.09739
$853$	1.44873	0.0496035	0.0248018	−	0.999692i	$-0.492105\pi$
$853$	1.44873	0.0496035	0.0248018	+	0.999692i	$0.492105\pi$
$854$	0	0
$855$	0	0
$856$	−16.0493	−0.548554
$857$	−35.4968	−1.21255	−0.606275	−	0.795255i	$-0.707337\pi$
$857$	−35.4968	−1.21255	−0.606275	+	0.795255i	$0.707337\pi$
$858$	7.73344	0.264015
$859$	7.70427	0.262866	0.131433	−	0.991325i	$-0.458042\pi$
$859$	7.70427	0.262866	0.131433	+	0.991325i	$0.458042\pi$
$860$	0	0
$861$	0	0
$862$	10.6972	0.364349
$863$	−30.8971	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$863$	−30.8971	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$864$	39.0337	1.32795
$865$	0	0
$866$	−0.345834	−0.0117519
$867$	2.96565	0.100719
$868$	0	0
$869$	17.0844	0.579548
$870$	0	0
$871$	19.7816	0.670275
$872$	−9.43844	−0.319626
$873$	23.1319	0.782897
$874$	−2.85921	−0.0967144
$875$	0	0
$876$	−37.9149	−1.28102
$877$	27.9043	0.942261	0.471130	−	0.882064i	$-0.343846\pi$
$877$	27.9043	0.942261	0.471130	+	0.882064i	$0.343846\pi$
$878$	3.52442	0.118943
$879$	22.1024	0.745497
$880$	0	0
$881$	−18.8938	−0.636547	−0.318273	−	0.947999i	$-0.603103\pi$
$881$	−18.8938	−0.636547	−0.318273	+	0.947999i	$0.603103\pi$
$882$	−18.3625	−0.618298
$883$	21.9146	0.737484	0.368742	−	0.929532i	$-0.379789\pi$
$883$	21.9146	0.737484	0.368742	+	0.929532i	$0.379789\pi$
$884$	−10.3411	−0.347809
$885$	0	0
$886$	4.39751	0.147737
$887$	−19.8796	−0.667493	−0.333746	−	0.942663i	$-0.608313\pi$
$887$	−19.8796	−0.667493	−0.333746	+	0.942663i	$0.608313\pi$
$888$	0.657337	0.0220588
$889$	0	0
$890$	0	0
$891$	−7.19786	−0.241138
$892$	−6.24689	−0.209161
$893$	−3.77955	−0.126478
$894$	−11.7956	−0.394505
$895$	0	0
$896$	0	0
$897$	100.302	3.34899
$898$	3.80006	0.126810
$899$	−4.40508	−0.146918
$900$	0	0
$901$	9.51299	0.316924
$902$	−3.59020	−0.119540
$903$	0	0
$904$	31.4191	1.04498
$905$	0	0
$906$	18.9344	0.629054
$907$	14.2234	0.472280	0.236140	−	0.971719i	$-0.424118\pi$
$907$	14.2234	0.472280	0.236140	+	0.971719i	$0.424118\pi$
$908$	21.3061	0.707068
$909$	−19.8018	−0.656784
$910$	0	0
$911$	30.4471	1.00876	0.504378	−	0.863483i	$-0.331722\pi$
$911$	30.4471	1.00876	0.504378	+	0.863483i	$0.331722\pi$
$912$	8.97402	0.297159
$913$	−3.96240	−0.131136
$914$	16.9685	0.561266
$915$	0	0
$916$	−5.26410	−0.173931
$917$	0	0
$918$	−3.75224	−0.123842
$919$	35.7874	1.18052	0.590259	−	0.807214i	$-0.299026\pi$
$919$	35.7874	1.18052	0.590259	+	0.807214i	$0.299026\pi$
$920$	0	0
$921$	98.1421	3.23389
$922$	0.598251	0.0197024
$923$	−34.6617	−1.14090
$924$	0	0
$925$	0	0
$926$	−18.9902	−0.624056
$927$	−63.9460	−2.10026
$928$	3.09642	0.101645
$929$	29.7305	0.975427	0.487713	−	0.873004i	$-0.337831\pi$
$929$	29.7305	0.975427	0.487713	+	0.873004i	$0.337831\pi$
$930$	0	0
$931$	−7.53115	−0.246823
$932$	−49.4296	−1.61912
$933$	0	0
$934$	14.7283	0.481924
$935$	0	0
$936$	57.3504	1.87456
$937$	13.7608	0.449544	0.224772	−	0.974411i	$-0.427836\pi$
$937$	13.7608	0.449544	0.224772	+	0.974411i	$0.427836\pi$
$938$	0	0
$939$	38.7728	1.26530
$940$	0	0
$941$	17.3671	0.566151	0.283076	−	0.959098i	$-0.408645\pi$
$941$	17.3671	0.566151	0.283076	+	0.959098i	$0.408645\pi$
$942$	11.3568	0.370025
$943$	−46.5645	−1.51635
$944$	33.2476	1.08212
$945$	0	0
$946$	5.21534	0.169565
$947$	2.09994	0.0682387	0.0341194	−	0.999418i	$-0.489137\pi$
$947$	2.09994	0.0682387	0.0341194	+	0.999418i	$0.489137\pi$
$948$	90.9511	2.95395
$949$	−41.0280	−1.33182
$950$	0	0
$951$	32.3418	1.04876
$952$	0	0
$953$	−32.1195	−1.04045	−0.520226	−	0.854029i	$-0.674152\pi$
$953$	−32.1195	−1.04045	−0.520226	+	0.854029i	$0.674152\pi$
$954$	−24.9546	−0.807936
$955$	0	0
$956$	−24.5111	−0.792745
$957$	−1.95011	−0.0630380
$958$	−16.5501	−0.534711
$959$	0	0
$960$	0	0
$961$	13.8779	0.447673
$962$	0.336453	0.0108477
$963$	54.1403	1.74465
$964$	−23.8092	−0.766843
$965$	0	0
$966$	0	0
$967$	−10.0878	−0.324401	−0.162201	−	0.986758i	$-0.551859\pi$
$967$	−10.0878	−0.324401	−0.162201	+	0.986758i	$0.551859\pi$
$968$	−1.71789	−0.0552152
$969$	−3.19068	−0.102499
$970$	0	0
$971$	−33.9277	−1.08879	−0.544395	−	0.838829i	$-0.683241\pi$
$971$	−33.9277	−1.08879	−0.544395	+	0.838829i	$0.683241\pi$
$972$	6.32144	0.202760
$973$	0	0
$974$	9.87025	0.316263
$975$	0	0
$976$	0.532574	0.0170473
$977$	14.3336	0.458573	0.229287	−	0.973359i	$-0.426361\pi$
$977$	14.3336	0.458573	0.229287	+	0.973359i	$0.426361\pi$
$978$	−10.5551	−0.337513
$979$	8.68030	0.277423
$980$	0	0
$981$	31.8394	1.01655
$982$	−13.8738	−0.442730
$983$	43.4981	1.38737	0.693687	−	0.720276i	$-0.255985\pi$
$983$	43.4981	1.38737	0.693687	+	0.720276i	$0.255985\pi$
$984$	−40.4074	−1.28814
$985$	0	0
$986$	−0.297654	−0.00947922
$987$	0	0
$988$	11.1258	0.353958
$989$	67.6425	2.15090
$990$	0	0
$991$	48.3130	1.53471	0.767356	−	0.641221i	$-0.221572\pi$
$991$	48.3130	1.53471	0.767356	+	0.641221i	$0.221572\pi$
$992$	−31.5456	−1.00157
$993$	31.4896	0.999293
$994$	0	0
$995$	0	0
$996$	−21.0944	−0.668401
$997$	28.9494	0.916835	0.458418	−	0.888737i	$-0.348416\pi$
$997$	28.9494	0.916835	0.458418	+	0.888737i	$0.348416\pi$
$998$	3.03741	0.0961474
$999$	−1.06952	−0.0338381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4675.2.a.bd.1.3		4
5.4	even	2		187.2.a.f.1.2	✓	4
15.14	odd	2		1683.2.a.y.1.3		4
20.19	odd	2		2992.2.a.v.1.4		4
35.34	odd	2		9163.2.a.l.1.2		4
55.54	odd	2		2057.2.a.s.1.3		4
85.84	even	2		3179.2.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.a.f.1.2	✓	4	5.4	even	2
1683.2.a.y.1.3		4	15.14	odd	2
2057.2.a.s.1.3		4	55.54	odd	2
2992.2.a.v.1.4		4	20.19	odd	2
3179.2.a.w.1.2		4	85.84	even	2
4675.2.a.bd.1.3		4	1.1	even	1	trivial
9163.2.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q+0.452661 q^{2} +2.96565 q^{3} -1.79510 q^{4} +1.34244 q^{6} -1.71789 q^{8} +5.79510 q^{9} +O(q^{10})\)	\(q+0.452661 q^{2} +2.96565 q^{3} -1.79510 q^{4} +1.34244 q^{6} -1.71789 q^{8} +5.79510 q^{9} -1.00000 q^{11} -5.32364 q^{12} -5.76075 q^{13} +2.81257 q^{16} -1.00000 q^{17} +2.62322 q^{18} +1.07588 q^{19} -0.452661 q^{22} -5.87098 q^{23} -5.09468 q^{24} -2.60767 q^{26} +8.28929 q^{27} +0.657564 q^{29} -6.69909 q^{31} +4.70893 q^{32} -2.96565 q^{33} -0.452661 q^{34} -10.4028 q^{36} -0.129024 q^{37} +0.487008 q^{38} -17.0844 q^{39} +7.93131 q^{41} -11.5215 q^{43} +1.79510 q^{44} -2.65756 q^{46} -3.51299 q^{47} +8.34111 q^{48} -7.00000 q^{49} -2.96565 q^{51} +10.3411 q^{52} -9.51299 q^{53} +3.75224 q^{54} +3.19068 q^{57} +0.297654 q^{58} +11.8211 q^{59} +0.189355 q^{61} -3.03242 q^{62} -3.49359 q^{64} -1.34244 q^{66} -3.43386 q^{67} +1.79510 q^{68} -17.4113 q^{69} +6.01687 q^{71} -9.95536 q^{72} +7.12199 q^{73} -0.0584043 q^{74} -1.93131 q^{76} -7.73344 q^{78} -17.0844 q^{79} +7.19786 q^{81} +3.59020 q^{82} +3.96240 q^{83} -5.21534 q^{86} +1.95011 q^{87} +1.71789 q^{88} -8.68030 q^{89} +10.5390 q^{92} -19.8672 q^{93} -1.59020 q^{94} +13.9651 q^{96} +3.99164 q^{97} -3.16863 q^{98} -5.79510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 9 q^{8} + 11 q^{9}+O(q^{10}) \) 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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