LMFDB - Embedded newform 4600.2.e.r.4049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.24681024.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 9 $ x^6 + 12x^4 + 36x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$0.523976i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.r.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} +O(q^{10})$	$q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} -1.67750 q^{11} -2.67750i q^{13} +1.67750i q^{17} -7.92692 q^{19} -0.249425 q^{21} -1.00000i q^{23} +3.00000i q^{27} +2.20147 q^{29} +6.77340 q^{31} -0.878968i q^{33} +4.00000i q^{37} +1.40294 q^{39} +5.97487 q^{41} +0.402945i q^{43} +1.79853i q^{47} +6.77340 q^{49} -0.878968 q^{51} +10.8059i q^{53} -4.15352i q^{57} -9.45090 q^{59} +6.32250 q^{61} +1.29738i q^{63} +14.7100i q^{67} +0.523976 q^{69} +9.97487 q^{71} +6.10557i q^{73} -0.798528i q^{77} +6.60442 q^{81} -4.49885i q^{83} +1.15352i q^{87} +3.59706 q^{89} +1.27455 q^{91} +3.54910i q^{93} -6.08044i q^{97} -4.57193 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0.523976i	0.302518i	0.988494	+	0.151259i	$0.0483327\pi$
$3$	0.523976i	0.302518i	−0.988494	+	0.151259i	$0.951667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.476024i	0.179920i	0.995945	+	0.0899600i	$0.0286739\pi$
$7$	0.476024i	0.179920i	−0.995945	+	0.0899600i	$0.971326\pi$
$8$	0	0
$9$	2.72545	0.908483
$10$	0	0
$11$	−1.67750	−0.505784	−0.252892	−	0.967495i	$-0.581382\pi$
$11$	−1.67750	−0.505784	−0.252892	+	0.967495i	$0.581382\pi$
$12$	0	0
$13$	− 2.67750i	− 0.742604i	−0.928512	−	0.371302i	$-0.878911\pi$
$13$	− 2.67750i	− 0.742604i	0.928512	−	0.371302i	$-0.121089\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.67750i	0.406853i	0.979090	+	0.203426i	$0.0652077\pi$
$17$	1.67750i	0.406853i	−0.979090	+	0.203426i	$0.934792\pi$
$18$	0	0
$19$	−7.92692	−1.81856	−0.909280	−	0.416184i	$-0.863367\pi$
$19$	−7.92692	−1.81856	−0.909280	+	0.416184i	$0.863367\pi$
$20$	0	0
$21$	−0.249425	−0.0544290
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.00000i	0.577350i
$28$	0	0
$29$	2.20147	0.408803	0.204402	−	0.978887i	$-0.434475\pi$
$29$	2.20147	0.408803	0.204402	+	0.978887i	$0.434475\pi$
$30$	0	0
$31$	6.77340	1.21654	0.608269	−	0.793731i	$-0.291864\pi$
$31$	6.77340	1.21654	0.608269	+	0.793731i	$0.291864\pi$
$32$	0	0
$33$	− 0.878968i	− 0.153009i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	0	0
$39$	1.40294	0.224651
$40$	0	0
$41$	5.97487	0.933119	0.466559	−	0.884490i	$-0.345493\pi$
$41$	5.97487	0.933119	0.466559	+	0.884490i	$0.345493\pi$
$42$	0	0
$43$	0.402945i	0.0614485i	0.999528	+	0.0307242i	$0.00978137\pi$
$43$	0.402945i	0.0614485i	−0.999528	+	0.0307242i	$0.990219\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.79853i	0.262342i	0.991360	+	0.131171i	$0.0418737\pi$
$47$	1.79853i	0.262342i	−0.991360	+	0.131171i	$0.958126\pi$
$48$	0	0
$49$	6.77340	0.967629
$50$	0	0
$51$	−0.878968	−0.123080
$52$	0	0
$53$	10.8059i	1.48430i	0.670232	+	0.742152i	$0.266195\pi$
$53$	10.8059i	1.48430i	−0.670232	+	0.742152i	$0.733805\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.15352i	− 0.550147i
$58$	0	0
$59$	−9.45090	−1.23040	−0.615201	−	0.788370i	$-0.710925\pi$
$59$	−9.45090	−1.23040	−0.615201	+	0.788370i	$0.710925\pi$
$60$	0	0
$61$	6.32250	0.809514	0.404757	−	0.914424i	$-0.367356\pi$
$61$	6.32250	0.809514	0.404757	+	0.914424i	$0.367356\pi$
$62$	0	0
$63$	1.29738i	0.163454i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.7100i	1.79711i	0.438860	+	0.898555i	$0.355382\pi$
$67$	14.7100i	1.79711i	−0.438860	+	0.898555i	$0.644618\pi$
$68$	0	0
$69$	0.523976	0.0630793
$70$	0	0
$71$	9.97487	1.18380	0.591900	−	0.806012i	$-0.298378\pi$
$71$	9.97487	1.18380	0.591900	+	0.806012i	$0.298378\pi$
$72$	0	0
$73$	6.10557i	0.714603i	0.933989	+	0.357301i	$0.116303\pi$
$73$	6.10557i	0.714603i	−0.933989	+	0.357301i	$0.883697\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.798528i	− 0.0910007i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	6.60442	0.733824
$82$	0	0
$83$	− 4.49885i	− 0.493813i	−0.969039	−	0.246906i	$-0.920586\pi$
$83$	− 4.49885i	− 0.493813i	0.969039	−	0.246906i	$-0.0794141\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.15352i	0.123670i
$88$	0	0
$89$	3.59706	0.381287	0.190644	−	0.981659i	$-0.438943\pi$
$89$	3.59706	0.381287	0.190644	+	0.981659i	$0.438943\pi$
$90$	0	0
$91$	1.27455	0.133609
$92$	0	0
$93$	3.54910i	0.368025i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.08044i	− 0.617375i	−0.951163	−	0.308688i	$-0.900110\pi$
$97$	− 6.08044i	− 0.617375i	0.951163	−	0.308688i	$-0.0998898\pi$
$98$	0	0
$99$	−4.57193	−0.459496
$100$	0	0
$101$	11.4509	1.13941	0.569703	−	0.821850i	$-0.307058\pi$
$101$	11.4509	1.13941	0.569703	+	0.821850i	$0.307058\pi$
$102$	0	0
$103$	2.32250i	0.228843i	0.993432	+	0.114422i	$0.0365015\pi$
$103$	2.32250i	0.228843i	−0.993432	+	0.114422i	$0.963499\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.85384i	− 0.759260i	−0.925138	−	0.379630i	$-0.876051\pi$
$107$	− 7.85384i	− 0.759260i	0.925138	−	0.379630i	$-0.123949\pi$
$108$	0	0
$109$	−13.9269	−1.33396	−0.666979	−	0.745077i	$-0.732413\pi$
$109$	−13.9269	−1.33396	−0.666979	+	0.745077i	$0.732413\pi$
$110$	0	0
$111$	−2.09591	−0.198935
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 7.29738i	− 0.674643i
$118$	0	0
$119$	−0.798528	−0.0732009
$120$	0	0
$121$	−8.18601	−0.744182
$122$	0	0
$123$	3.13069i	0.282285i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 11.1033i	− 0.985256i	−0.870240	−	0.492628i	$-0.836036\pi$
$127$	− 11.1033i	− 0.985256i	0.870240	−	0.492628i	$-0.163964\pi$
$128$	0	0
$129$	−0.211133	−0.0185893
$130$	0	0
$131$	7.65237	0.668591	0.334295	−	0.942468i	$-0.391502\pi$
$131$	7.65237	0.668591	0.334295	+	0.942468i	$0.391502\pi$
$132$	0	0
$133$	− 3.77340i	− 0.327195i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 21.3778i	− 1.82643i	−0.407478	−	0.913215i	$-0.633592\pi$
$137$	− 21.3778i	− 1.82643i	0.407478	−	0.913215i	$-0.366408\pi$
$138$	0	0
$139$	8.60442	0.729817	0.364909	−	0.931043i	$-0.381100\pi$
$139$	8.60442	0.729817	0.364909	+	0.931043i	$0.381100\pi$
$140$	0	0
$141$	−0.942386	−0.0793632
$142$	0	0
$143$	4.49149i	0.375597i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.54910i	0.292725i
$148$	0	0
$149$	−4.97487	−0.407558	−0.203779	−	0.979017i	$-0.565322\pi$
$149$	−4.97487	−0.407558	−0.203779	+	0.979017i	$0.565322\pi$
$150$	0	0
$151$	−0.926921	−0.0754318	−0.0377159	−	0.999289i	$-0.512008\pi$
$151$	−0.926921	−0.0754318	−0.0377159	+	0.999289i	$0.512008\pi$
$152$	0	0
$153$	4.57193i	0.369619i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.3527i	1.78394i	0.452096	+	0.891970i	$0.350677\pi$
$157$	22.3527i	1.78394i	−0.452096	+	0.891970i	$0.649323\pi$
$158$	0	0
$159$	−5.66203	−0.449028
$160$	0	0
$161$	0.476024	0.0375159
$162$	0	0
$163$	7.93658i	0.621641i	0.950469	+	0.310821i	$0.100604\pi$
$163$	7.93658i	0.621641i	−0.950469	+	0.310821i	$0.899396\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.8059i	0.990949i	0.868622	+	0.495475i	$0.165006\pi$
$167$	12.8059i	0.990949i	−0.868622	+	0.495475i	$0.834994\pi$
$168$	0	0
$169$	5.83102	0.448540
$170$	0	0
$171$	−21.6044	−1.65213
$172$	0	0
$173$	24.6752i	1.87602i	0.346608	+	0.938010i	$0.387334\pi$
$173$	24.6752i	1.87602i	−0.346608	+	0.938010i	$0.612666\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.95205i	− 0.372219i
$178$	0	0
$179$	13.1033	0.979384	0.489692	−	0.871895i	$-0.337109\pi$
$179$	13.1033	0.979384	0.489692	+	0.871895i	$0.337109\pi$
$180$	0	0
$181$	9.52398	0.707912	0.353956	−	0.935262i	$-0.384836\pi$
$181$	9.52398	0.707912	0.353956	+	0.935262i	$0.384836\pi$
$182$	0	0
$183$	3.31284i	0.244892i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.81399i	− 0.205780i
$188$	0	0
$189$	−1.42807	−0.103877
$190$	0	0
$191$	5.85384	0.423569	0.211785	−	0.977316i	$-0.432072\pi$
$191$	5.85384	0.423569	0.211785	+	0.977316i	$0.432072\pi$
$192$	0	0
$193$	− 4.41261i	− 0.317626i	−0.987309	−	0.158813i	$-0.949233\pi$
$193$	− 4.41261i	− 0.317626i	0.987309	−	0.158813i	$-0.0507667\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.4258i	0.814053i	0.913416	+	0.407026i	$0.133434\pi$
$197$	11.4258i	0.814053i	−0.913416	+	0.407026i	$0.866566\pi$
$198$	0	0
$199$	0.307039	0.0217654	0.0108827	−	0.999941i	$-0.496536\pi$
$199$	0.307039	0.0217654	0.0108827	+	0.999941i	$0.496536\pi$
$200$	0	0
$201$	−7.70768	−0.543658
$202$	0	0
$203$	1.04795i	0.0735519i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.72545i	− 0.189432i
$208$	0	0
$209$	13.2974	0.919799
$210$	0	0
$211$	17.3047	1.19131	0.595654	−	0.803241i	$-0.296893\pi$
$211$	17.3047	1.19131	0.595654	+	0.803241i	$0.296893\pi$
$212$	0	0
$213$	5.22660i	0.358121i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.22430i	0.218880i
$218$	0	0
$219$	−3.19917	−0.216180
$220$	0	0
$221$	4.49149	0.302130
$222$	0	0
$223$	1.90409i	0.127508i	0.997966	+	0.0637538i	$0.0203072\pi$
$223$	1.90409i	0.127508i	−0.997966	+	0.0637538i	$0.979693\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.3550i	1.15189i	0.817488	+	0.575946i	$0.195366\pi$
$227$	17.3550i	1.15189i	−0.817488	+	0.575946i	$0.804634\pi$
$228$	0	0
$229$	6.19181	0.409166	0.204583	−	0.978849i	$-0.434416\pi$
$229$	6.19181	0.409166	0.204583	+	0.978849i	$0.434416\pi$
$230$	0	0
$231$	0.418410	0.0275293
$232$	0	0
$233$	15.7985i	1.03500i	0.855684	+	0.517498i	$0.173137\pi$
$233$	15.7985i	1.03500i	−0.855684	+	0.517498i	$0.826863\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.34763	−0.410594	−0.205297	−	0.978700i	$-0.565816\pi$
$239$	−6.34763	−0.410594	−0.205297	+	0.978700i	$0.565816\pi$
$240$	0	0
$241$	5.69296	0.366716	0.183358	−	0.983046i	$-0.441303\pi$
$241$	5.69296	0.366716	0.183358	+	0.983046i	$0.441303\pi$
$242$	0	0
$243$	12.4606i	0.799345i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21.2243i	1.35047i
$248$	0	0
$249$	2.35729	0.149387
$250$	0	0
$251$	−28.9246	−1.82571	−0.912853	−	0.408288i	$-0.866126\pi$
$251$	−28.9246	−1.82571	−0.912853	+	0.408288i	$0.866126\pi$
$252$	0	0
$253$	1.67750i	0.105463i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.5085i	− 0.905016i	−0.891760	−	0.452508i	$-0.850529\pi$
$257$	− 14.5085i	− 0.905016i	0.891760	−	0.452508i	$-0.149471\pi$
$258$	0	0
$259$	−1.90409	−0.118315
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	15.7734i	0.972630i	0.873784	+	0.486315i	$0.161659\pi$
$263$	15.7734i	0.972630i	−0.873784	+	0.486315i	$0.838341\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.88477i	0.115346i
$268$	0	0
$269$	6.20147	0.378110	0.189055	−	0.981966i	$-0.439457\pi$
$269$	6.20147	0.378110	0.189055	+	0.981966i	$0.439457\pi$
$270$	0	0
$271$	−25.5696	−1.55324	−0.776622	−	0.629967i	$-0.783069\pi$
$271$	−25.5696	−1.55324	−0.776622	+	0.629967i	$0.783069\pi$
$272$	0	0
$273$	0.667835i	0.0404192i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.03829i	0.422890i	0.977390	+	0.211445i	$0.0678169\pi$
$277$	7.03829i	0.422890i	−0.977390	+	0.211445i	$0.932183\pi$
$278$	0	0
$279$	18.4606	1.10520
$280$	0	0
$281$	29.0627	1.73373	0.866867	−	0.498540i	$-0.166130\pi$
$281$	29.0627	1.73373	0.866867	+	0.498540i	$0.166130\pi$
$282$	0	0
$283$	18.6141i	1.10649i	0.833018	+	0.553246i	$0.186611\pi$
$283$	18.6141i	1.10649i	−0.833018	+	0.553246i	$0.813389\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.84418i	0.167887i
$288$	0	0
$289$	14.1860	0.834471
$290$	0	0
$291$	3.18601	0.186767
$292$	0	0
$293$	− 23.4006i	− 1.36708i	−0.729913	−	0.683540i	$-0.760439\pi$
$293$	− 23.4006i	− 1.36708i	0.729913	−	0.683540i	$-0.239561\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.03249i	− 0.292015i
$298$	0	0
$299$	−2.67750	−0.154844
$300$	0	0
$301$	−0.191811	−0.0110558
$302$	0	0
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.53134i	− 0.0873981i	−0.999045	−	0.0436990i	$-0.986086\pi$
$307$	− 1.53134i	− 0.0873981i	0.999045	−	0.0436990i	$-0.0139143\pi$
$308$	0	0
$309$	−1.21694	−0.0692291
$310$	0	0
$311$	−7.55646	−0.428488	−0.214244	−	0.976780i	$-0.568729\pi$
$311$	−7.55646	−0.428488	−0.214244	+	0.976780i	$0.568729\pi$
$312$	0	0
$313$	2.32987i	0.131692i	0.997830	+	0.0658459i	$0.0209746\pi$
$313$	2.32987i	0.131692i	−0.997830	+	0.0658459i	$0.979025\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.3778i	− 0.863704i	−0.901944	−	0.431852i	$-0.857860\pi$
$317$	− 15.3778i	− 0.863704i	0.901944	−	0.431852i	$-0.142140\pi$
$318$	0	0
$319$	−3.69296	−0.206766
$320$	0	0
$321$	4.11523	0.229690
$322$	0	0
$323$	− 13.2974i	− 0.739886i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7.29738i	− 0.403546i
$328$	0	0
$329$	−0.856142	−0.0472006
$330$	0	0
$331$	22.9115	1.25933	0.629664	−	0.776868i	$-0.283193\pi$
$331$	22.9115	1.25933	0.629664	+	0.776868i	$0.283193\pi$
$332$	0	0
$333$	10.9018i	0.597415i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.5410i	1.44578i	0.690963	+	0.722890i	$0.257187\pi$
$337$	26.5410i	1.44578i	−0.690963	+	0.722890i	$0.742813\pi$
$338$	0	0
$339$	−5.23976	−0.284585
$340$	0	0
$341$	−11.3624	−0.615306
$342$	0	0
$343$	6.55646i	0.354016i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.27455i	− 0.175787i	−0.996130	−	0.0878936i	$-0.971986\pi$
$347$	− 3.27455i	− 0.175787i	0.996130	−	0.0878936i	$-0.0280135\pi$
$348$	0	0
$349$	−32.0553	−1.71588	−0.857941	−	0.513749i	$-0.828256\pi$
$349$	−32.0553	−1.71588	−0.857941	+	0.513749i	$0.828256\pi$
$350$	0	0
$351$	8.03249	0.428742
$352$	0	0
$353$	− 11.1535i	− 0.593642i	−0.954933	−	0.296821i	$-0.904074\pi$
$353$	− 11.1535i	− 0.593642i	0.954933	−	0.296821i	$-0.0959265\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.418410i	− 0.0221446i
$358$	0	0
$359$	−22.4029	−1.18238	−0.591191	−	0.806532i	$-0.701342\pi$
$359$	−22.4029	−1.18238	−0.591191	+	0.806532i	$0.701342\pi$
$360$	0	0
$361$	43.8361	2.30716
$362$	0	0
$363$	− 4.28927i	− 0.225129i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.2877i	0.745813i	0.927869	+	0.372906i	$0.121639\pi$
$367$	14.2877i	0.745813i	−0.927869	+	0.372906i	$0.878361\pi$
$368$	0	0
$369$	16.2842	0.847722
$370$	0	0
$371$	−5.14386	−0.267056
$372$	0	0
$373$	1.23976i	0.0641925i	0.999485	+	0.0320963i	$0.0102183\pi$
$373$	1.23976i	0.0641925i	−0.999485	+	0.0320963i	$0.989782\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.89443i	− 0.303579i
$378$	0	0
$379$	−4.38748	−0.225370	−0.112685	−	0.993631i	$-0.535945\pi$
$379$	−4.38748	−0.225370	−0.112685	+	0.993631i	$0.535945\pi$
$380$	0	0
$381$	5.81785	0.298057
$382$	0	0
$383$	− 18.4989i	− 0.945247i	−0.881264	−	0.472624i	$-0.843307\pi$
$383$	− 18.4989i	− 0.945247i	0.881264	−	0.472624i	$-0.156693\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.09821i	0.0558249i
$388$	0	0
$389$	−21.6775	−1.09909	−0.549546	−	0.835463i	$-0.685199\pi$
$389$	−21.6775	−1.09909	−0.549546	+	0.835463i	$0.685199\pi$
$390$	0	0
$391$	1.67750	0.0848346
$392$	0	0
$393$	4.00966i	0.202261i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 20.6849i	− 1.03814i	−0.854731	−	0.519072i	$-0.826278\pi$
$397$	− 20.6849i	− 1.03814i	0.854731	−	0.519072i	$-0.173722\pi$
$398$	0	0
$399$	1.97717	0.0989825
$400$	0	0
$401$	−28.5638	−1.42641	−0.713205	−	0.700956i	$-0.752757\pi$
$401$	−28.5638	−1.42641	−0.713205	+	0.700956i	$0.752757\pi$
$402$	0	0
$403$	− 18.1358i	− 0.903406i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.70998i	− 0.332602i
$408$	0	0
$409$	−16.7734	−0.829391	−0.414696	−	0.909960i	$-0.636112\pi$
$409$	−16.7734	−0.829391	−0.414696	+	0.909960i	$0.636112\pi$
$410$	0	0
$411$	11.2015	0.552528
$412$	0	0
$413$	− 4.49885i	− 0.221374i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.50851i	0.220783i
$418$	0	0
$419$	4.59476	0.224469	0.112234	−	0.993682i	$-0.464199\pi$
$419$	4.59476	0.224469	0.112234	+	0.993682i	$0.464199\pi$
$420$	0	0
$421$	31.8310	1.55135	0.775674	−	0.631133i	$-0.217410\pi$
$421$	31.8310	1.55135	0.775674	+	0.631133i	$0.217410\pi$
$422$	0	0
$423$	4.90179i	0.238333i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.00966i	0.145648i
$428$	0	0
$429$	−2.35343	−0.113625
$430$	0	0
$431$	−16.0650	−0.773823	−0.386911	−	0.922117i	$-0.626458\pi$
$431$	−16.0650	−0.773823	−0.386911	+	0.922117i	$0.626458\pi$
$432$	0	0
$433$	7.68486i	0.369311i	0.982803	+	0.184655i	$0.0591169\pi$
$433$	7.68486i	0.369311i	−0.982803	+	0.184655i	$0.940883\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.92692i	0.379196i
$438$	0	0
$439$	−15.9822	−0.762790	−0.381395	−	0.924412i	$-0.624556\pi$
$439$	−15.9822	−0.762790	−0.381395	+	0.924412i	$0.624556\pi$
$440$	0	0
$441$	18.4606	0.879074
$442$	0	0
$443$	3.57929i	0.170057i	0.996379	+	0.0850286i	$0.0270982\pi$
$443$	3.57929i	0.170057i	−0.996379	+	0.0850286i	$0.972902\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2.60672i	− 0.123293i
$448$	0	0
$449$	28.5217	1.34602	0.673011	−	0.739633i	$-0.265001\pi$
$449$	28.5217	1.34602	0.673011	+	0.739633i	$0.265001\pi$
$450$	0	0
$451$	−10.0228	−0.471956
$452$	0	0
$453$	− 0.485685i	− 0.0228195i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4.49885i	− 0.210447i	−0.994449	−	0.105224i	$-0.966444\pi$
$457$	− 4.49885i	− 0.210447i	0.994449	−	0.105224i	$-0.0335559\pi$
$458$	0	0
$459$	−5.03249	−0.234896
$460$	0	0
$461$	2.29738	0.107000	0.0534998	−	0.998568i	$-0.482962\pi$
$461$	2.29738	0.107000	0.0534998	+	0.998568i	$0.482962\pi$
$462$	0	0
$463$	33.4966i	1.55672i	0.627820	+	0.778358i	$0.283947\pi$
$463$	33.4966i	1.55672i	−0.627820	+	0.778358i	$0.716053\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.23976i	− 0.242467i	−0.992624	−	0.121234i	$-0.961315\pi$
$467$	− 5.23976i	− 0.242467i	0.992624	−	0.121234i	$-0.0386850\pi$
$468$	0	0
$469$	−7.00230	−0.323336
$470$	0	0
$471$	−11.7123	−0.539674
$472$	0	0
$473$	− 0.675938i	− 0.0310797i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	29.4509i	1.34846i
$478$	0	0
$479$	2.40294	0.109793	0.0548967	−	0.998492i	$-0.482517\pi$
$479$	2.40294	0.109793	0.0548967	+	0.998492i	$0.482517\pi$
$480$	0	0
$481$	10.7100	0.488333
$482$	0	0
$483$	0.249425i	0.0113492i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.8944i	1.17339i	0.809808	+	0.586694i	$0.199571\pi$
$487$	25.8944i	1.17339i	−0.809808	+	0.586694i	$0.800429\pi$
$488$	0	0
$489$	−4.15858	−0.188058
$490$	0	0
$491$	−0.489189	−0.0220768	−0.0110384	−	0.999939i	$-0.503514\pi$
$491$	−0.489189	−0.0220768	−0.0110384	+	0.999939i	$0.503514\pi$
$492$	0	0
$493$	3.69296i	0.166323i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.74828i	0.212989i
$498$	0	0
$499$	38.4583	1.72163	0.860814	−	0.508920i	$-0.169955\pi$
$499$	38.4583	1.72163	0.860814	+	0.508920i	$0.169955\pi$
$500$	0	0
$501$	−6.70998	−0.299780
$502$	0	0
$503$	− 37.8960i	− 1.68970i	−0.535004	−	0.844849i	$-0.679690\pi$
$503$	− 37.8960i	− 1.68970i	0.535004	−	0.844849i	$-0.320310\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.05531i	0.135691i
$508$	0	0
$509$	−8.70032	−0.385635	−0.192818	−	0.981235i	$-0.561763\pi$
$509$	−8.70032	−0.385635	−0.192818	+	0.981235i	$0.561763\pi$
$510$	0	0
$511$	−2.90639	−0.128571
$512$	0	0
$513$	− 23.7808i	− 1.04995i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.01702i	− 0.132689i
$518$	0	0
$519$	−12.9292	−0.567530
$520$	0	0
$521$	25.1129	1.10022	0.550109	−	0.835093i	$-0.314586\pi$
$521$	25.1129	1.10022	0.550109	+	0.835093i	$0.314586\pi$
$522$	0	0
$523$	− 12.7100i	− 0.555769i	−0.960615	−	0.277884i	$-0.910367\pi$
$523$	− 12.7100i	− 0.555769i	0.960615	−	0.277884i	$-0.0896332\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.3624i	0.494952i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−25.7579	−1.11780
$532$	0	0
$533$	− 15.9977i	− 0.692937i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.86580i	0.296281i
$538$	0	0
$539$	−11.3624	−0.489411
$540$	0	0
$541$	−0.251725	−0.0108225	−0.00541124	−	0.999985i	$-0.501722\pi$
$541$	−0.251725	−0.0108225	−0.00541124	+	0.999985i	$0.501722\pi$
$542$	0	0
$543$	4.99034i	0.214156i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.3395i	1.04068i	0.853958	+	0.520342i	$0.174195\pi$
$547$	24.3395i	1.04068i	−0.853958	+	0.520342i	$0.825805\pi$
$548$	0	0
$549$	17.2317	0.735429
$550$	0	0
$551$	−17.4509	−0.743433
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.4966i	− 0.995581i	−0.867297	−	0.497790i	$-0.834145\pi$
$557$	− 23.4966i	− 0.995581i	0.867297	−	0.497790i	$-0.165855\pi$
$558$	0	0
$559$	1.07888	0.0456319
$560$	0	0
$561$	1.47447	0.0622520
$562$	0	0
$563$	19.5159i	0.822496i	0.911523	+	0.411248i	$0.134907\pi$
$563$	19.5159i	0.822496i	−0.911523	+	0.411248i	$0.865093\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.14386i	0.132030i
$568$	0	0
$569$	8.51817	0.357100	0.178550	−	0.983931i	$-0.442859\pi$
$569$	8.51817	0.357100	0.178550	+	0.983931i	$0.442859\pi$
$570$	0	0
$571$	−27.4811	−1.15005	−0.575024	−	0.818137i	$-0.695007\pi$
$571$	−27.4811	−1.15005	−0.575024	+	0.818137i	$0.695007\pi$
$572$	0	0
$573$	3.06728i	0.128137i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 26.1512i	− 1.08869i	−0.838862	−	0.544345i	$-0.816778\pi$
$577$	− 26.1512i	− 1.08869i	0.838862	−	0.544345i	$-0.183222\pi$
$578$	0	0
$579$	2.31210	0.0960877
$580$	0	0
$581$	2.14156	0.0888468
$582$	0	0
$583$	− 18.1268i	− 0.750737i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 34.3276i	− 1.41685i	−0.705786	−	0.708425i	$-0.749406\pi$
$587$	− 34.3276i	− 1.41685i	0.705786	−	0.708425i	$-0.250594\pi$
$588$	0	0
$589$	−53.6922	−2.21235
$590$	0	0
$591$	−5.98683	−0.246265
$592$	0	0
$593$	− 15.9497i	− 0.654978i	−0.944855	−	0.327489i	$-0.893798\pi$
$593$	− 15.9497i	− 0.654978i	0.944855	−	0.327489i	$-0.106202\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.160881i	0.00658443i
$598$	0	0
$599$	−40.5940	−1.65863	−0.829313	−	0.558784i	$-0.811268\pi$
$599$	−40.5940	−1.65863	−0.829313	+	0.558784i	$0.811268\pi$
$600$	0	0
$601$	−29.0302	−1.18417	−0.592083	−	0.805877i	$-0.701694\pi$
$601$	−29.0302	−1.18417	−0.592083	+	0.805877i	$0.701694\pi$
$602$	0	0
$603$	40.0913i	1.63264i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16.9977i	0.689915i	0.938618	+	0.344958i	$0.112107\pi$
$607$	16.9977i	0.689915i	−0.938618	+	0.344958i	$0.887893\pi$
$608$	0	0
$609$	−0.549103	−0.0222508
$610$	0	0
$611$	4.81555	0.194816
$612$	0	0
$613$	− 30.3024i	− 1.22390i	−0.790895	−	0.611952i	$-0.790385\pi$
$613$	− 30.3024i	− 1.22390i	0.790895	−	0.611952i	$-0.209615\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.13069i	− 0.246812i	−0.992356	−	0.123406i	$-0.960618\pi$
$617$	− 6.13069i	− 0.246812i	0.992356	−	0.123406i	$-0.0393818\pi$
$618$	0	0
$619$	33.7305	1.35574	0.677872	−	0.735180i	$-0.262902\pi$
$619$	33.7305	1.35574	0.677872	+	0.735180i	$0.262902\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	1.71228i	0.0686012i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.96751i	0.278256i
$628$	0	0
$629$	−6.70998	−0.267545
$630$	0	0
$631$	4.85614	0.193320	0.0966600	−	0.995317i	$-0.469184\pi$
$631$	4.85614	0.193320	0.0966600	+	0.995317i	$0.469184\pi$
$632$	0	0
$633$	9.06728i	0.360392i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.1358i	− 0.718565i
$638$	0	0
$639$	27.1860	1.07546
$640$	0	0
$641$	7.78887	0.307642	0.153821	−	0.988099i	$-0.450842\pi$
$641$	7.78887	0.307642	0.153821	+	0.988099i	$0.450842\pi$
$642$	0	0
$643$	− 43.6427i	− 1.72110i	−0.509366	−	0.860550i	$-0.670120\pi$
$643$	− 43.6427i	− 1.72110i	0.509366	−	0.860550i	$-0.329880\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15.4606i	− 0.607817i	−0.952701	−	0.303909i	$-0.901708\pi$
$647$	− 15.4606i	− 0.607817i	0.952701	−	0.303909i	$-0.0982918\pi$
$648$	0	0
$649$	15.8538	0.622318
$650$	0	0
$651$	−1.68946	−0.0662150
$652$	0	0
$653$	9.70613i	0.379830i	0.981801	+	0.189915i	$0.0608213\pi$
$653$	9.70613i	0.379830i	−0.981801	+	0.189915i	$0.939179\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	16.6404i	0.649204i
$658$	0	0
$659$	−50.7100	−1.97538	−0.987690	−	0.156422i	$-0.950004\pi$
$659$	−50.7100	−1.97538	−0.987690	+	0.156422i	$0.950004\pi$
$660$	0	0
$661$	33.0132	1.28406	0.642032	−	0.766678i	$-0.278092\pi$
$661$	33.0132	1.28406	0.642032	+	0.766678i	$0.278092\pi$
$662$	0	0
$663$	2.35343i	0.0913998i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.20147i	− 0.0852413i
$668$	0	0
$669$	−0.997701	−0.0385733
$670$	0	0
$671$	−10.6060	−0.409439
$672$	0	0
$673$	− 19.2494i	− 0.742011i	−0.928631	−	0.371005i	$-0.879013\pi$
$673$	− 19.2494i	− 0.742011i	0.928631	−	0.371005i	$-0.120987\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25.1941i	− 0.968288i	−0.874988	−	0.484144i	$-0.839131\pi$
$677$	− 25.1941i	− 0.968288i	0.874988	−	0.484144i	$-0.160869\pi$
$678$	0	0
$679$	2.89443	0.111078
$680$	0	0
$681$	−9.09361	−0.348468
$682$	0	0
$683$	− 9.48339i	− 0.362872i	−0.983403	−	0.181436i	$-0.941926\pi$
$683$	− 9.48339i	− 0.362872i	0.983403	−	0.181436i	$-0.0580745\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.24436i	0.123780i
$688$	0	0
$689$	28.9327	1.10225
$690$	0	0
$691$	−23.8538	−0.907443	−0.453721	−	0.891144i	$-0.649904\pi$
$691$	−23.8538	−0.907443	−0.453721	+	0.891144i	$0.649904\pi$
$692$	0	0
$693$	− 2.17635i	− 0.0826726i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.0228i	0.379642i
$698$	0	0
$699$	−8.27806	−0.313105
$700$	0	0
$701$	19.8767	0.750731	0.375366	−	0.926877i	$-0.377517\pi$
$701$	19.8767	0.750731	0.375366	+	0.926877i	$0.377517\pi$
$702$	0	0
$703$	− 31.7077i	− 1.19588i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.45090i	0.205002i
$708$	0	0
$709$	6.72545	0.252580	0.126290	−	0.991993i	$-0.459693\pi$
$709$	6.72545	0.252580	0.126290	+	0.991993i	$0.459693\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 6.77340i	− 0.253666i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.32601i	− 0.124212i
$718$	0	0
$719$	38.2294	1.42571	0.712857	−	0.701309i	$-0.247401\pi$
$719$	38.2294	1.42571	0.712857	+	0.701309i	$0.247401\pi$
$720$	0	0
$721$	−1.10557	−0.0411735
$722$	0	0
$723$	2.98298i	0.110938i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3.72315i	− 0.138084i	−0.997614	−	0.0690420i	$-0.978006\pi$
$727$	− 3.72315i	− 0.138084i	0.997614	−	0.0690420i	$-0.0219942\pi$
$728$	0	0
$729$	13.2842	0.492008
$730$	0	0
$731$	−0.675938	−0.0250005
$732$	0	0
$733$	8.49885i	0.313912i	0.987606	+	0.156956i	$0.0501681\pi$
$733$	8.49885i	0.313912i	−0.987606	+	0.156956i	$0.949832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.6759i	− 0.908950i
$738$	0	0
$739$	0.251725	0.00925984	0.00462992	−	0.999989i	$-0.498526\pi$
$739$	0.251725	0.00925984	0.00462992	+	0.999989i	$0.498526\pi$
$740$	0	0
$741$	−11.1210	−0.408541
$742$	0	0
$743$	29.7305i	1.09071i	0.838206	+	0.545353i	$0.183605\pi$
$743$	29.7305i	1.09071i	−0.838206	+	0.545353i	$0.816395\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 12.2614i	− 0.448621i
$748$	0	0
$749$	3.73861	0.136606
$750$	0	0
$751$	10.6597	0.388979	0.194490	−	0.980905i	$-0.437695\pi$
$751$	10.6597	0.388979	0.194490	+	0.980905i	$0.437695\pi$
$752$	0	0
$753$	− 15.1558i	− 0.552309i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 16.1918i	− 0.588501i	−0.955728	−	0.294251i	$-0.904930\pi$
$757$	− 16.1918i	− 0.588501i	0.955728	−	0.294251i	$-0.0950701\pi$
$758$	0	0
$759$	−0.878968	−0.0319045
$760$	0	0
$761$	−46.9343	−1.70137	−0.850683	−	0.525679i	$-0.823811\pi$
$761$	−46.9343	−1.70137	−0.850683	+	0.525679i	$0.823811\pi$
$762$	0	0
$763$	− 6.62954i	− 0.240006i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.3047i	0.913701i
$768$	0	0
$769$	35.9954	1.29803	0.649014	−	0.760777i	$-0.275182\pi$
$769$	35.9954	1.29803	0.649014	+	0.760777i	$0.275182\pi$
$770$	0	0
$771$	7.60212	0.273784
$772$	0	0
$773$	21.8229i	0.784916i	0.919770	+	0.392458i	$0.128375\pi$
$773$	21.8229i	0.784916i	−0.919770	+	0.392458i	$0.871625\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.997701i	− 0.0357923i
$778$	0	0
$779$	−47.3624	−1.69693
$780$	0	0
$781$	−16.7328	−0.598747
$782$	0	0
$783$	6.60442i	0.236023i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.16318i	− 0.255340i	−0.991817	−	0.127670i	$-0.959250\pi$
$787$	− 7.16318i	− 0.255340i	0.991817	−	0.127670i	$-0.0407498\pi$
$788$	0	0
$789$	−8.26489	−0.294238
$790$	0	0
$791$	−4.76024	−0.169255
$792$	0	0
$793$	− 16.9285i	− 0.601148i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 48.0604i	− 1.70239i	−0.524853	−	0.851193i	$-0.675880\pi$
$797$	− 48.0604i	− 1.70239i	0.524853	−	0.851193i	$-0.324120\pi$
$798$	0	0
$799$	−3.01702	−0.106735
$800$	0	0
$801$	9.80359	0.346393
$802$	0	0
$803$	− 10.2421i	− 0.361435i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.24943i	0.114385i
$808$	0	0
$809$	−0.0611183	−0.00214880	−0.00107440	−	0.999999i	$-0.500342\pi$
$809$	−0.0611183	−0.00214880	−0.00107440	+	0.999999i	$0.500342\pi$
$810$	0	0
$811$	25.9092	0.909794	0.454897	−	0.890544i	$-0.349676\pi$
$811$	25.9092	0.909794	0.454897	+	0.890544i	$0.349676\pi$
$812$	0	0
$813$	− 13.3979i	− 0.469884i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.19411i	− 0.111748i
$818$	0	0
$819$	3.47372	0.121382
$820$	0	0
$821$	−21.0936	−0.736172	−0.368086	−	0.929792i	$-0.619987\pi$
$821$	−21.0936	−0.736172	−0.368086	+	0.929792i	$0.619987\pi$
$822$	0	0
$823$	37.2641i	1.29895i	0.760384	+	0.649473i	$0.225011\pi$
$823$	37.2641i	1.29895i	−0.760384	+	0.649473i	$0.774989\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.45320i	− 0.154853i	−0.996998	−	0.0774264i	$-0.975330\pi$
$827$	− 4.45320i	− 0.154853i	0.996998	−	0.0774264i	$-0.0246703\pi$
$828$	0	0
$829$	38.1300	1.32431	0.662154	−	0.749368i	$-0.269642\pi$
$829$	38.1300	1.32431	0.662154	+	0.749368i	$0.269642\pi$
$830$	0	0
$831$	−3.68790	−0.127932
$832$	0	0
$833$	11.3624i	0.393682i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.3202i	0.702369i
$838$	0	0
$839$	7.15858	0.247142	0.123571	−	0.992336i	$-0.460565\pi$
$839$	7.15858	0.247142	0.123571	+	0.992336i	$0.460565\pi$
$840$	0	0
$841$	−24.1535	−0.832880
$842$	0	0
$843$	15.2282i	0.524486i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.89673i	− 0.133893i
$848$	0	0
$849$	−9.75334	−0.334734
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	49.6199i	1.69895i	0.527627	+	0.849476i	$0.323082\pi$
$853$	49.6199i	1.69895i	−0.527627	+	0.849476i	$0.676918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 44.0360i	− 1.50424i	−0.659026	−	0.752120i	$-0.729031\pi$
$857$	− 44.0360i	− 1.50424i	0.659026	−	0.752120i	$-0.270969\pi$
$858$	0	0
$859$	49.5062	1.68913	0.844565	−	0.535453i	$-0.179859\pi$
$859$	49.5062	1.68913	0.844565	+	0.535453i	$0.179859\pi$
$860$	0	0
$861$	−1.49028	−0.0507887
$862$	0	0
$863$	− 20.7962i	− 0.707912i	−0.935262	−	0.353956i	$-0.884836\pi$
$863$	− 20.7962i	− 0.707912i	0.935262	−	0.353956i	$-0.115164\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.43313i	0.252442i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	39.3859	1.33454
$872$	0	0
$873$	− 16.5719i	− 0.560875i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.42071i	− 0.183044i	−0.995803	−	0.0915222i	$-0.970827\pi$
$877$	− 5.42071i	− 0.183044i	0.995803	−	0.0915222i	$-0.0291732\pi$
$878$	0	0
$879$	12.2614	0.413566
$880$	0	0
$881$	−45.8036	−1.54316	−0.771581	−	0.636131i	$-0.780534\pi$
$881$	−45.8036	−1.54316	−0.771581	+	0.636131i	$0.780534\pi$
$882$	0	0
$883$	− 57.1860i	− 1.92446i	−0.272234	−	0.962231i	$-0.587762\pi$
$883$	− 57.1860i	− 1.92446i	0.272234	−	0.962231i	$-0.412238\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 15.5371i	− 0.521686i	−0.965381	−	0.260843i	$-0.915999\pi$
$887$	− 15.5371i	− 0.521686i	0.965381	−	0.260843i	$-0.0840005\pi$
$888$	0	0
$889$	5.28542	0.177267
$890$	0	0
$891$	−11.0789	−0.371157
$892$	0	0
$893$	− 14.2568i	− 0.477085i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.40294i	− 0.0468430i
$898$	0	0
$899$	14.9115	0.497325
$900$	0	0
$901$	−18.1268	−0.603892
$902$	0	0
$903$	− 0.100505i	− 0.00334458i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 32.8515i	− 1.09082i	−0.838170	−	0.545409i	$-0.816374\pi$
$907$	− 32.8515i	− 1.09082i	0.838170	−	0.545409i	$-0.183626\pi$
$908$	0	0
$909$	31.2088	1.03513
$910$	0	0
$911$	8.95205	0.296595	0.148297	−	0.988943i	$-0.452621\pi$
$911$	8.95205	0.296595	0.148297	+	0.988943i	$0.452621\pi$
$912$	0	0
$913$	7.54680i	0.249763i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.64271i	0.120293i
$918$	0	0
$919$	49.8229	1.64351	0.821753	−	0.569844i	$-0.192996\pi$
$919$	49.8229	1.64351	0.821753	+	0.569844i	$0.192996\pi$
$920$	0	0
$921$	0.802385	0.0264395
$922$	0	0
$923$	− 26.7077i	− 0.879094i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.32987i	0.207900i
$928$	0	0
$929$	−25.0723	−0.822597	−0.411298	−	0.911501i	$-0.634925\pi$
$929$	−25.0723	−0.822597	−0.411298	+	0.911501i	$0.634925\pi$
$930$	0	0
$931$	−53.6922	−1.75969
$932$	0	0
$933$	− 3.95941i	− 0.129625i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.3322i	0.696891i	0.937329	+	0.348446i	$0.113290\pi$
$937$	21.3322i	0.696891i	−0.937329	+	0.348446i	$0.886710\pi$
$938$	0	0
$939$	−1.22079	−0.0398391
$940$	0	0
$941$	3.93428	0.128254	0.0641270	−	0.997942i	$-0.479574\pi$
$941$	3.93428	0.128254	0.0641270	+	0.997942i	$0.479574\pi$
$942$	0	0
$943$	− 5.97487i	− 0.194569i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.6420i	1.58065i	0.612687	+	0.790326i	$0.290089\pi$
$947$	48.6420i	1.58065i	−0.612687	+	0.790326i	$0.709911\pi$
$948$	0	0
$949$	16.3476	0.530667
$950$	0	0
$951$	8.05761	0.261286
$952$	0	0
$953$	15.5623i	0.504111i	0.967713	+	0.252056i	$0.0811066\pi$
$953$	15.5623i	0.504111i	−0.967713	+	0.252056i	$0.918893\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.93502i	− 0.0625505i
$958$	0	0
$959$	10.1763	0.328611
$960$	0	0
$961$	14.8790	0.479967
$962$	0	0
$963$	− 21.4052i	− 0.689774i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.80083i	0.218700i	0.994003	+	0.109350i	$0.0348769\pi$
$967$	6.80083i	0.218700i	−0.994003	+	0.109350i	$0.965123\pi$
$968$	0	0
$969$	6.96751	0.223829
$970$	0	0
$971$	6.46406	0.207442	0.103721	−	0.994606i	$-0.466925\pi$
$971$	6.46406	0.207442	0.103721	+	0.994606i	$0.466925\pi$
$972$	0	0
$973$	4.09591i	0.131309i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.18601i	0.293886i	0.989145	+	0.146943i	$0.0469435\pi$
$977$	9.18601i	0.293886i	−0.989145	+	0.146943i	$0.953057\pi$
$978$	0	0
$979$	−6.03405	−0.192849
$980$	0	0
$981$	−37.9571	−1.21188
$982$	0	0
$983$	− 15.3705i	− 0.490241i	−0.969493	−	0.245121i	$-0.921172\pi$
$983$	− 15.3705i	− 0.490241i	0.969493	−	0.245121i	$-0.0788276\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.448598i	− 0.0142790i
$988$	0	0
$989$	0.402945	0.0128129
$990$	0	0
$991$	−52.0109	−1.65218	−0.826090	−	0.563538i	$-0.809440\pi$
$991$	−52.0109	−1.65218	−0.826090	+	0.563538i	$0.809440\pi$
$992$	0	0
$993$	12.0051i	0.380969i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15.7123i	0.497613i	0.968553	+	0.248807i	$0.0800383\pi$
$997$	15.7123i	0.497613i	−0.968553	+	0.248807i	$0.919962\pi$
$998$	0	0
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.r.4049.4		6
5.2	odd	4		4600.2.a.y.1.2		3
5.3	odd	4		920.2.a.g.1.2	✓	3
5.4	even	2	inner	4600.2.e.r.4049.3		6
15.8	even	4		8280.2.a.bo.1.2		3
20.3	even	4		1840.2.a.t.1.2		3
20.7	even	4		9200.2.a.cd.1.2		3
40.3	even	4		7360.2.a.ca.1.2		3
40.13	odd	4		7360.2.a.cb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.g.1.2	✓	3	5.3	odd	4
1840.2.a.t.1.2		3	20.3	even	4
4600.2.a.y.1.2		3	5.2	odd	4
4600.2.e.r.4049.3		6	5.4	even	2	inner
4600.2.e.r.4049.4		6	1.1	even	1	trivial
7360.2.a.ca.1.2		3	40.3	even	4
7360.2.a.cb.1.2		3	40.13	odd	4
8280.2.a.bo.1.2		3	15.8	even	4
9200.2.a.cd.1.2		3	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} +O(q^{10})\)	\(q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} -1.67750 q^{11} -2.67750i q^{13} +1.67750i q^{17} -7.92692 q^{19} -0.249425 q^{21} -1.00000i q^{23} +3.00000i q^{27} +2.20147 q^{29} +6.77340 q^{31} -0.878968i q^{33} +4.00000i q^{37} +1.40294 q^{39} +5.97487 q^{41} +0.402945i q^{43} +1.79853i q^{47} +6.77340 q^{49} -0.878968 q^{51} +10.8059i q^{53} -4.15352i q^{57} -9.45090 q^{59} +6.32250 q^{61} +1.29738i q^{63} +14.7100i q^{67} +0.523976 q^{69} +9.97487 q^{71} +6.10557i q^{73} -0.798528i q^{77} +6.60442 q^{81} -4.49885i q^{83} +1.15352i q^{87} +3.59706 q^{89} +1.27455 q^{91} +3.54910i q^{93} -6.08044i q^{97} -4.57193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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