LMFDB - Embedded newform 4600.2.e.u.4049.7

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 $ x^10 + 28x^8 + 260x^6 + 897x^4 + 1056x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.7
Root			$1.31091i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.u.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31091i q^{3} -4.66212i q^{7} +1.28151 q^{9} +O(q^{10})$	$q+1.31091i q^{3} -4.66212i q^{7} +1.28151 q^{9} +2.23020 q^{11} +2.80072i q^{13} +7.63271i q^{17} +1.36222 q^{19} +6.11163 q^{21} +1.00000i q^{23} +5.61268i q^{27} -8.94362 q^{29} -1.58140 q^{31} +2.92360i q^{33} -1.40251i q^{37} -3.67149 q^{39} +10.7134 q^{41} -7.26391i q^{47} -14.7353 q^{49} -10.0058 q^{51} +8.38212i q^{53} +1.78575i q^{57} +4.88331 q^{59} +4.33282 q^{61} -5.97454i q^{63} +8.54355i q^{67} -1.31091 q^{69} +8.81604 q^{71} +5.26391i q^{73} -10.3974i q^{77} -7.08222 q^{79} -3.51322 q^{81} +4.59749i q^{83} -11.7243i q^{87} +4.70241 q^{89} +13.0573 q^{91} -2.07308i q^{93} +16.5160i q^{97} +2.85802 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 26 q^{9}+O(q^{10}) $ 10 * q - 26 * q^9	$ 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) $ 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * 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
$3$	1.31091i	0.756856i	0.925631	+	0.378428i	$0.123535\pi$
$3$	1.31091i	0.756856i	−0.925631	+	0.378428i	$0.876465\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.66212i	− 1.76211i	−0.473010	−	0.881057i	$-0.656832\pi$
$7$	− 4.66212i	− 1.76211i	0.473010	−	0.881057i	$-0.343168\pi$
$8$	0	0
$9$	1.28151	0.427169
$10$	0	0
$11$	2.23020	0.672430	0.336215	−	0.941785i	$-0.390853\pi$
$11$	2.23020	0.672430	0.336215	+	0.941785i	$0.390853\pi$
$12$	0	0
$13$	2.80072i	0.776779i	0.921495	+	0.388389i	$0.126968\pi$
$13$	2.80072i	0.776779i	−0.921495	+	0.388389i	$0.873032\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.63271i	1.85120i	0.378498	+	0.925602i	$0.376441\pi$
$17$	7.63271i	1.85120i	−0.378498	+	0.925602i	$0.623559\pi$
$18$	0	0
$19$	1.36222	0.312515	0.156258	−	0.987716i	$-0.450057\pi$
$19$	1.36222	0.312515	0.156258	+	0.987716i	$0.450057\pi$
$20$	0	0
$21$	6.11163	1.33367
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.61268i	1.08016i
$28$	0	0
$29$	−8.94362	−1.66079	−0.830395	−	0.557176i	$-0.811885\pi$
$29$	−8.94362	−1.66079	−0.830395	+	0.557176i	$0.811885\pi$
$30$	0	0
$31$	−1.58140	−0.284028	−0.142014	−	0.989865i	$-0.545358\pi$
$31$	−1.58140	−0.284028	−0.142014	+	0.989865i	$0.545358\pi$
$32$	0	0
$33$	2.92360i	0.508933i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.40251i	− 0.230572i	−0.993332	−	0.115286i	$-0.963222\pi$
$37$	− 1.40251i	− 0.230572i	0.993332	−	0.115286i	$-0.0367784\pi$
$38$	0	0
$39$	−3.67149	−0.587909
$40$	0	0
$41$	10.7134	1.67316	0.836578	−	0.547848i	$-0.184553\pi$
$41$	10.7134	1.67316	0.836578	+	0.547848i	$0.184553\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.26391i	− 1.05955i	−0.848138	−	0.529775i	$-0.822276\pi$
$47$	− 7.26391i	− 1.05955i	0.848138	−	0.529775i	$-0.177724\pi$
$48$	0	0
$49$	−14.7353	−2.10505
$50$	0	0
$51$	−10.0058	−1.40109
$52$	0	0
$53$	8.38212i	1.15137i	0.817671	+	0.575686i	$0.195265\pi$
$53$	8.38212i	1.15137i	−0.817671	+	0.575686i	$0.804735\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.78575i	0.236529i
$58$	0	0
$59$	4.88331	0.635752	0.317876	−	0.948132i	$-0.397030\pi$
$59$	4.88331	0.635752	0.317876	+	0.948132i	$0.397030\pi$
$60$	0	0
$61$	4.33282	0.554760	0.277380	−	0.960760i	$-0.410534\pi$
$61$	4.33282	0.554760	0.277380	+	0.960760i	$0.410534\pi$
$62$	0	0
$63$	− 5.97454i	− 0.752721i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.54355i	1.04376i	0.853019	+	0.521880i	$0.174769\pi$
$67$	8.54355i	1.04376i	−0.853019	+	0.521880i	$0.825231\pi$
$68$	0	0
$69$	−1.31091	−0.157815
$70$	0	0
$71$	8.81604	1.04627	0.523136	−	0.852249i	$-0.324762\pi$
$71$	8.81604	1.04627	0.523136	+	0.852249i	$0.324762\pi$
$72$	0	0
$73$	5.26391i	0.616095i	0.951371	+	0.308047i	$0.0996755\pi$
$73$	5.26391i	0.616095i	−0.951371	+	0.308047i	$0.900325\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.3974i	− 1.18490i
$78$	0	0
$79$	−7.08222	−0.796812	−0.398406	−	0.917209i	$-0.630437\pi$
$79$	−7.08222	−0.796812	−0.398406	+	0.917209i	$0.630437\pi$
$80$	0	0
$81$	−3.51322	−0.390357
$82$	0	0
$83$	4.59749i	0.504640i	0.967644	+	0.252320i	$0.0811935\pi$
$83$	4.59749i	0.504640i	−0.967644	+	0.252320i	$0.918806\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 11.7243i	− 1.25698i
$88$	0	0
$89$	4.70241	0.498454	0.249227	−	0.968445i	$-0.419823\pi$
$89$	4.70241	0.498454	0.249227	+	0.968445i	$0.419823\pi$
$90$	0	0
$91$	13.0573	1.36877
$92$	0	0
$93$	− 2.07308i	− 0.214968i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.5160i	1.67695i	0.544942	+	0.838474i	$0.316552\pi$
$97$	16.5160i	1.67695i	−0.544942	+	0.838474i	$0.683448\pi$
$98$	0	0
$99$	2.85802	0.287241
$100$	0	0
$101$	10.7267	1.06735	0.533676	−	0.845689i	$-0.320810\pi$
$101$	10.7267	1.06735	0.533676	+	0.845689i	$0.320810\pi$
$102$	0	0
$103$	− 1.95300i	− 0.192435i	−0.995360	−	0.0962175i	$-0.969326\pi$
$103$	− 1.95300i	− 0.192435i	0.995360	−	0.0962175i	$-0.0306744\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.97566i	− 0.190994i	−0.995430	−	0.0954972i	$-0.969556\pi$
$107$	− 1.97566i	− 0.190994i	0.995430	−	0.0954972i	$-0.0304441\pi$
$108$	0	0
$109$	8.92360	0.854725	0.427363	−	0.904080i	$-0.359443\pi$
$109$	8.92360	0.854725	0.427363	+	0.904080i	$0.359443\pi$
$110$	0	0
$111$	1.83857	0.174510
$112$	0	0
$113$	− 17.3486i	− 1.63202i	−0.578040	−	0.816008i	$-0.696182\pi$
$113$	− 17.3486i	− 1.63202i	0.578040	−	0.816008i	$-0.303818\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.58914i	0.331816i
$118$	0	0
$119$	35.5846	3.26203
$120$	0	0
$121$	−6.02621	−0.547838
$122$	0	0
$123$	14.0444i	1.26634i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.4872i	1.72921i	0.502455	+	0.864603i	$0.332430\pi$
$127$	19.4872i	1.72921i	−0.502455	+	0.864603i	$0.667570\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.68050	0.496308	0.248154	−	0.968721i	$-0.420176\pi$
$131$	5.68050	0.496308	0.248154	+	0.968721i	$0.420176\pi$
$132$	0	0
$133$	− 6.35084i	− 0.550687i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.68645i	− 0.742134i	−0.928606	−	0.371067i	$-0.878992\pi$
$137$	− 8.68645i	− 0.742134i	0.928606	−	0.371067i	$-0.121008\pi$
$138$	0	0
$139$	9.22569	0.782513	0.391256	−	0.920282i	$-0.372041\pi$
$139$	9.22569	0.782513	0.391256	+	0.920282i	$0.372041\pi$
$140$	0	0
$141$	9.52236	0.801927
$142$	0	0
$143$	6.24615i	0.522329i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 19.3167i	− 1.59322i
$148$	0	0
$149$	−18.6056	−1.52423	−0.762115	−	0.647441i	$-0.775839\pi$
$149$	−18.6056	−1.52423	−0.762115	+	0.647441i	$0.775839\pi$
$150$	0	0
$151$	17.4316	1.41856	0.709280	−	0.704927i	$-0.249020\pi$
$151$	17.4316	1.41856	0.709280	+	0.704927i	$0.249020\pi$
$152$	0	0
$153$	9.78138i	0.790778i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.839497i	0.0669992i	0.999439	+	0.0334996i	$0.0106652\pi$
$157$	0.839497i	0.0669992i	−0.999439	+	0.0334996i	$0.989335\pi$
$158$	0	0
$159$	−10.9882	−0.871423
$160$	0	0
$161$	4.66212	0.367426
$162$	0	0
$163$	14.5673i	1.14100i	0.821297	+	0.570500i	$0.193251\pi$
$163$	14.5673i	1.14100i	−0.821297	+	0.570500i	$0.806749\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.03842i	− 0.389884i	−0.980815	−	0.194942i	$-0.937548\pi$
$167$	− 5.03842i	− 0.389884i	0.980815	−	0.194942i	$-0.0624519\pi$
$168$	0	0
$169$	5.15599	0.396615
$170$	0	0
$171$	1.74570	0.133497
$172$	0	0
$173$	11.3124i	0.860067i	0.902813	+	0.430034i	$0.141498\pi$
$173$	11.3124i	0.860067i	−0.902813	+	0.430034i	$0.858502\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.40159i	0.481173i
$178$	0	0
$179$	24.3053	1.81667	0.908333	−	0.418247i	$-0.137355\pi$
$179$	24.3053	1.81667	0.908333	+	0.418247i	$0.137355\pi$
$180$	0	0
$181$	−19.4829	−1.44815	−0.724075	−	0.689721i	$-0.757733\pi$
$181$	−19.4829	−1.44815	−0.724075	+	0.689721i	$0.757733\pi$
$182$	0	0
$183$	5.67994i	0.419874i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17.0225i	1.24481i
$188$	0	0
$189$	26.1670	1.90337
$190$	0	0
$191$	2.62183	0.189709	0.0948543	−	0.995491i	$-0.469761\pi$
$191$	2.62183	0.189709	0.0948543	+	0.995491i	$0.469761\pi$
$192$	0	0
$193$	− 17.4332i	− 1.25487i	−0.778668	−	0.627436i	$-0.784105\pi$
$193$	− 17.4332i	− 1.25487i	0.778668	−	0.627436i	$-0.215895\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.4402i	− 1.67004i	−0.550217	−	0.835022i	$-0.685455\pi$
$197$	− 23.4402i	− 1.67004i	0.550217	−	0.835022i	$-0.314545\pi$
$198$	0	0
$199$	1.29759	0.0919839	0.0459920	−	0.998942i	$-0.485355\pi$
$199$	1.29759	0.0919839	0.0459920	+	0.998942i	$0.485355\pi$
$200$	0	0
$201$	−11.1998	−0.789976
$202$	0	0
$203$	41.6962i	2.92650i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.28151i	0.0890709i
$208$	0	0
$209$	3.03803	0.210145
$210$	0	0
$211$	14.7619	1.01625	0.508127	−	0.861282i	$-0.330338\pi$
$211$	14.7619	1.01625	0.508127	+	0.861282i	$0.330338\pi$
$212$	0	0
$213$	11.5571i	0.791877i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.37268i	0.500490i
$218$	0	0
$219$	−6.90053	−0.466295
$220$	0	0
$221$	−21.3771	−1.43798
$222$	0	0
$223$	− 11.8434i	− 0.793095i	−0.918014	−	0.396548i	$-0.870208\pi$
$223$	− 11.8434i	− 0.793095i	0.918014	−	0.396548i	$-0.129792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.27556i	− 0.350151i	−0.984555	−	0.175075i	$-0.943983\pi$
$227$	− 5.27556i	− 0.350151i	0.984555	−	0.175075i	$-0.0560170\pi$
$228$	0	0
$229$	1.23878	0.0818611	0.0409305	−	0.999162i	$-0.486968\pi$
$229$	1.23878	0.0818611	0.0409305	+	0.999162i	$0.486968\pi$
$230$	0	0
$231$	13.6301	0.896798
$232$	0	0
$233$	− 2.66412i	− 0.174533i	−0.996185	−	0.0872663i	$-0.972187\pi$
$233$	− 2.66412i	− 0.174533i	0.996185	−	0.0872663i	$-0.0278131\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9.28418i	− 0.603072i
$238$	0	0
$239$	−26.2577	−1.69847	−0.849235	−	0.528014i	$-0.822937\pi$
$239$	−26.2577	−1.69847	−0.849235	+	0.528014i	$0.822937\pi$
$240$	0	0
$241$	−2.22326	−0.143213	−0.0716063	−	0.997433i	$-0.522813\pi$
$241$	−2.22326	−0.143213	−0.0716063	+	0.997433i	$0.522813\pi$
$242$	0	0
$243$	12.2325i	0.784717i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.81520i	0.242755i
$248$	0	0
$249$	−6.02690	−0.381940
$250$	0	0
$251$	13.4829	0.851031	0.425515	−	0.904951i	$-0.360093\pi$
$251$	13.4829	0.851031	0.425515	+	0.904951i	$0.360093\pi$
$252$	0	0
$253$	2.23020i	0.140211i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.0281i	1.37408i	0.726620	+	0.687039i	$0.241090\pi$
$257$	22.0281i	1.37408i	−0.726620	+	0.687039i	$0.758910\pi$
$258$	0	0
$259$	−6.53868	−0.406294
$260$	0	0
$261$	−11.4613	−0.709438
$262$	0	0
$263$	− 22.3164i	− 1.37609i	−0.725670	−	0.688043i	$-0.758470\pi$
$263$	− 22.3164i	− 1.37609i	0.725670	−	0.688043i	$-0.241530\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.16445i	0.377258i
$268$	0	0
$269$	−10.5050	−0.640501	−0.320251	−	0.947333i	$-0.603767\pi$
$269$	−10.5050	−0.640501	−0.320251	+	0.947333i	$0.603767\pi$
$270$	0	0
$271$	−3.84696	−0.233686	−0.116843	−	0.993150i	$-0.537277\pi$
$271$	−3.84696	−0.233686	−0.116843	+	0.993150i	$0.537277\pi$
$272$	0	0
$273$	17.1169i	1.03596i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.8653i	0.893172i	0.894741	+	0.446586i	$0.147360\pi$
$277$	14.8653i	0.893172i	−0.894741	+	0.446586i	$0.852640\pi$
$278$	0	0
$279$	−2.02658	−0.121328
$280$	0	0
$281$	4.41659	0.263472	0.131736	−	0.991285i	$-0.457945\pi$
$281$	4.41659	0.263472	0.131736	+	0.991285i	$0.457945\pi$
$282$	0	0
$283$	− 3.88495i	− 0.230936i	−0.993311	−	0.115468i	$-0.963163\pi$
$283$	− 3.88495i	− 0.230936i	0.993311	−	0.115468i	$-0.0368368\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 49.9472i	− 2.94829i
$288$	0	0
$289$	−41.2583	−2.42696
$290$	0	0
$291$	−21.6511	−1.26921
$292$	0	0
$293$	25.0257i	1.46202i	0.682368	+	0.731009i	$0.260950\pi$
$293$	25.0257i	1.46202i	−0.682368	+	0.731009i	$0.739050\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	12.5174i	0.726333i
$298$	0	0
$299$	−2.80072	−0.161970
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.0618i	0.807831i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.0197i	1.25673i	0.777918	+	0.628365i	$0.216276\pi$
$307$	22.0197i	1.25673i	−0.777918	+	0.628365i	$0.783724\pi$
$308$	0	0
$309$	2.56021	0.145645
$310$	0	0
$311$	19.6217	1.11264	0.556322	−	0.830967i	$-0.312212\pi$
$311$	19.6217	1.11264	0.556322	+	0.830967i	$0.312212\pi$
$312$	0	0
$313$	17.9496i	1.01457i	0.861778	+	0.507285i	$0.169351\pi$
$313$	17.9496i	1.01457i	−0.861778	+	0.507285i	$0.830649\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.3185i	1.92752i	0.266768	+	0.963761i	$0.414044\pi$
$317$	34.3185i	1.92752i	−0.266768	+	0.963761i	$0.585956\pi$
$318$	0	0
$319$	−19.9461	−1.11676
$320$	0	0
$321$	2.58992	0.144555
$322$	0	0
$323$	10.3974i	0.578529i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.6981i	0.646904i
$328$	0	0
$329$	−33.8652	−1.86705
$330$	0	0
$331$	−0.299762	−0.0164764	−0.00823821	−	0.999966i	$-0.502622\pi$
$331$	−0.299762	−0.0164764	−0.00823821	+	0.999966i	$0.502622\pi$
$332$	0	0
$333$	− 1.79733i	− 0.0984931i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.2965i	− 1.21457i	−0.794485	−	0.607284i	$-0.792259\pi$
$337$	− 22.2965i	− 1.21457i	0.794485	−	0.607284i	$-0.207741\pi$
$338$	0	0
$339$	22.7425	1.23520
$340$	0	0
$341$	−3.52684	−0.190989
$342$	0	0
$343$	36.0630i	1.94722i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.1240i	1.77819i	0.457725	+	0.889094i	$0.348664\pi$
$347$	33.1240i	1.77819i	−0.457725	+	0.889094i	$0.651336\pi$
$348$	0	0
$349$	22.0041	1.17785	0.588926	−	0.808187i	$-0.299551\pi$
$349$	22.0041	1.17785	0.588926	+	0.808187i	$0.299551\pi$
$350$	0	0
$351$	−15.7195	−0.839046
$352$	0	0
$353$	15.6085i	0.830759i	0.909648	+	0.415379i	$0.136351\pi$
$353$	15.6085i	0.830759i	−0.909648	+	0.415379i	$0.863649\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	46.6483i	2.46889i
$358$	0	0
$359$	−0.263781	−0.0139218	−0.00696092	−	0.999976i	$-0.502216\pi$
$359$	−0.263781	−0.0139218	−0.00696092	+	0.999976i	$0.502216\pi$
$360$	0	0
$361$	−17.1444	−0.902334
$362$	0	0
$363$	− 7.89984i	− 0.414634i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.8615i	0.566967i	0.958977	+	0.283484i	$0.0914902\pi$
$367$	10.8615i	0.566967i	−0.958977	+	0.283484i	$0.908510\pi$
$368$	0	0
$369$	13.7293	0.714721
$370$	0	0
$371$	39.0784	2.02885
$372$	0	0
$373$	− 26.8889i	− 1.39225i	−0.717919	−	0.696127i	$-0.754905\pi$
$373$	− 26.8889i	− 1.39225i	0.717919	−	0.696127i	$-0.245095\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 25.0485i	− 1.29007i
$378$	0	0
$379$	2.15824	0.110861	0.0554306	−	0.998463i	$-0.482347\pi$
$379$	2.15824	0.110861	0.0554306	+	0.998463i	$0.482347\pi$
$380$	0	0
$381$	−25.5460	−1.30876
$382$	0	0
$383$	4.62814i	0.236487i	0.992985	+	0.118244i	$0.0377264\pi$
$383$	4.62814i	0.236487i	−0.992985	+	0.118244i	$0.962274\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.4988	0.937929	0.468964	−	0.883217i	$-0.344627\pi$
$389$	18.4988	0.937929	0.468964	+	0.883217i	$0.344627\pi$
$390$	0	0
$391$	−7.63271	−0.386003
$392$	0	0
$393$	7.44664i	0.375634i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.2685i	− 0.515360i	−0.966230	−	0.257680i	$-0.917042\pi$
$397$	− 10.2685i	− 0.515360i	0.966230	−	0.257680i	$-0.0829581\pi$
$398$	0	0
$399$	8.32539	0.416791
$400$	0	0
$401$	−0.885607	−0.0442251	−0.0221125	−	0.999755i	$-0.507039\pi$
$401$	−0.885607	−0.0442251	−0.0221125	+	0.999755i	$0.507039\pi$
$402$	0	0
$403$	− 4.42906i	− 0.220627i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.12788i	− 0.155043i
$408$	0	0
$409$	−24.3497	−1.20401	−0.602006	−	0.798491i	$-0.705632\pi$
$409$	−24.3497	−1.20401	−0.602006	+	0.798491i	$0.705632\pi$
$410$	0	0
$411$	11.3872	0.561688
$412$	0	0
$413$	− 22.7665i	− 1.12027i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0941i	0.592249i
$418$	0	0
$419$	−25.4535	−1.24348	−0.621742	−	0.783222i	$-0.713575\pi$
$419$	−25.4535	−1.24348	−0.621742	+	0.783222i	$0.713575\pi$
$420$	0	0
$421$	15.2376	0.742634	0.371317	−	0.928506i	$-0.378906\pi$
$421$	15.2376	0.742634	0.371317	+	0.928506i	$0.378906\pi$
$422$	0	0
$423$	− 9.30876i	− 0.452607i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.2001i	− 0.977551i
$428$	0	0
$429$	−8.18816	−0.395328
$430$	0	0
$431$	−34.6166	−1.66742	−0.833711	−	0.552202i	$-0.813788\pi$
$431$	−34.6166	−1.66742	−0.833711	+	0.552202i	$0.813788\pi$
$432$	0	0
$433$	− 28.4774i	− 1.36854i	−0.729230	−	0.684268i	$-0.760122\pi$
$433$	− 28.4774i	− 1.36854i	0.729230	−	0.684268i	$-0.239878\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.36222i	0.0651639i
$438$	0	0
$439$	−21.6009	−1.03095	−0.515477	−	0.856904i	$-0.672385\pi$
$439$	−21.6009	−1.03095	−0.515477	+	0.856904i	$0.672385\pi$
$440$	0	0
$441$	−18.8834	−0.899211
$442$	0	0
$443$	− 40.3700i	− 1.91804i	−0.283346	−	0.959018i	$-0.591445\pi$
$443$	− 40.3700i	− 1.91804i	0.283346	−	0.959018i	$-0.408555\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 24.3903i	− 1.15362i
$448$	0	0
$449$	−5.95484	−0.281026	−0.140513	−	0.990079i	$-0.544875\pi$
$449$	−5.95484	−0.281026	−0.140513	+	0.990079i	$0.544875\pi$
$450$	0	0
$451$	23.8931	1.12508
$452$	0	0
$453$	22.8512i	1.07365i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.66169i	− 0.264843i	−0.991194	−	0.132421i	$-0.957725\pi$
$457$	− 5.66169i	− 0.264843i	0.991194	−	0.132421i	$-0.0422752\pi$
$458$	0	0
$459$	−42.8400	−1.99960
$460$	0	0
$461$	−9.26391	−0.431463	−0.215732	−	0.976453i	$-0.569214\pi$
$461$	−9.26391	−0.431463	−0.215732	+	0.976453i	$0.569214\pi$
$462$	0	0
$463$	− 37.9899i	− 1.76554i	−0.469807	−	0.882769i	$-0.655676\pi$
$463$	− 37.9899i	− 1.76554i	0.469807	−	0.882769i	$-0.344324\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 26.2865i	− 1.21640i	−0.793785	−	0.608198i	$-0.791893\pi$
$467$	− 26.2865i	− 1.21640i	0.793785	−	0.608198i	$-0.208107\pi$
$468$	0	0
$469$	39.8310	1.83922
$470$	0	0
$471$	−1.10051	−0.0507087
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.7417i	0.491831i
$478$	0	0
$479$	31.0770	1.41994	0.709971	−	0.704231i	$-0.248708\pi$
$479$	31.0770	1.41994	0.709971	+	0.704231i	$0.248708\pi$
$480$	0	0
$481$	3.92804	0.179103
$482$	0	0
$483$	6.11163i	0.278089i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.9591i	0.677860i	0.940812	+	0.338930i	$0.110065\pi$
$487$	14.9591i	0.677860i	−0.940812	+	0.338930i	$0.889935\pi$
$488$	0	0
$489$	−19.0965	−0.863573
$490$	0	0
$491$	24.1333	1.08912	0.544561	−	0.838721i	$-0.316696\pi$
$491$	24.1333	1.08912	0.544561	+	0.838721i	$0.316696\pi$
$492$	0	0
$493$	− 68.2641i	− 3.07446i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 41.1014i	− 1.84365i
$498$	0	0
$499$	17.7266	0.793552	0.396776	−	0.917915i	$-0.370129\pi$
$499$	17.7266	0.793552	0.396776	+	0.917915i	$0.370129\pi$
$500$	0	0
$501$	6.60492	0.295086
$502$	0	0
$503$	7.69566i	0.343133i	0.985173	+	0.171566i	$0.0548828\pi$
$503$	7.69566i	0.343133i	−0.985173	+	0.171566i	$0.945117\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.75906i	0.300180i
$508$	0	0
$509$	33.0564	1.46520	0.732600	−	0.680659i	$-0.238307\pi$
$509$	33.0564	1.46520	0.732600	+	0.680659i	$0.238307\pi$
$510$	0	0
$511$	24.5410	1.08563
$512$	0	0
$513$	7.64572i	0.337567i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.2000i	− 0.712474i
$518$	0	0
$519$	−14.8296	−0.650947
$520$	0	0
$521$	10.6047	0.464598	0.232299	−	0.972644i	$-0.425375\pi$
$521$	10.6047	0.464598	0.232299	+	0.972644i	$0.425375\pi$
$522$	0	0
$523$	34.5128i	1.50914i	0.656219	+	0.754570i	$0.272155\pi$
$523$	34.5128i	1.50914i	−0.656219	+	0.754570i	$0.727845\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 12.0704i	− 0.525794i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	6.25799	0.271574
$532$	0	0
$533$	30.0053i	1.29967i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	31.8622i	1.37495i
$538$	0	0
$539$	−32.8627	−1.41550
$540$	0	0
$541$	−27.3344	−1.17520	−0.587598	−	0.809153i	$-0.699926\pi$
$541$	−27.3344	−1.17520	−0.587598	+	0.809153i	$0.699926\pi$
$542$	0	0
$543$	− 25.5403i	− 1.09604i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	34.6190i	1.48020i	0.672496	+	0.740101i	$0.265222\pi$
$547$	34.6190i	1.48020i	−0.672496	+	0.740101i	$0.734778\pi$
$548$	0	0
$549$	5.55254	0.236977
$550$	0	0
$551$	−12.1832	−0.519022
$552$	0	0
$553$	33.0181i	1.40407i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.2490i	1.06983i	0.844905	+	0.534917i	$0.179657\pi$
$557$	25.2490i	1.06983i	−0.844905	+	0.534917i	$0.820343\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−22.3150	−0.942139
$562$	0	0
$563$	− 41.0793i	− 1.73128i	−0.500663	−	0.865642i	$-0.666910\pi$
$563$	− 41.0793i	− 1.73128i	0.500663	−	0.865642i	$-0.333090\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	16.3790i	0.687854i
$568$	0	0
$569$	28.4230	1.19155	0.595776	−	0.803150i	$-0.296844\pi$
$569$	28.4230	1.19155	0.595776	+	0.803150i	$0.296844\pi$
$570$	0	0
$571$	13.4690	0.563659	0.281830	−	0.959464i	$-0.409059\pi$
$571$	13.4690	0.563659	0.281830	+	0.959464i	$0.409059\pi$
$572$	0	0
$573$	3.43698i	0.143582i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.69392i	0.153780i	0.997040	+	0.0768899i	$0.0244990\pi$
$577$	3.69392i	0.153780i	−0.997040	+	0.0768899i	$0.975501\pi$
$578$	0	0
$579$	22.8534	0.949757
$580$	0	0
$581$	21.4340	0.889233
$582$	0	0
$583$	18.6938i	0.774218i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.1042i	0.788513i	0.919000	+	0.394257i	$0.128998\pi$
$587$	19.1042i	0.788513i	−0.919000	+	0.394257i	$0.871002\pi$
$588$	0	0
$589$	−2.15422	−0.0887631
$590$	0	0
$591$	30.7280	1.26398
$592$	0	0
$593$	− 13.9194i	− 0.571602i	−0.958289	−	0.285801i	$-0.907740\pi$
$593$	− 13.9194i	− 0.571602i	0.958289	−	0.285801i	$-0.0922596\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.70103i	0.0696186i
$598$	0	0
$599$	−10.9498	−0.447396	−0.223698	−	0.974659i	$-0.571813\pi$
$599$	−10.9498	−0.447396	−0.223698	+	0.974659i	$0.571813\pi$
$600$	0	0
$601$	28.7034	1.17084	0.585418	−	0.810731i	$-0.300930\pi$
$601$	28.7034	1.17084	0.585418	+	0.810731i	$0.300930\pi$
$602$	0	0
$603$	10.9486i	0.445862i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 43.7745i	− 1.77675i	−0.459117	−	0.888376i	$-0.651834\pi$
$607$	− 43.7745i	− 1.77675i	0.459117	−	0.888376i	$-0.348166\pi$
$608$	0	0
$609$	−54.6601	−2.21494
$610$	0	0
$611$	20.3442	0.823036
$612$	0	0
$613$	− 6.59492i	− 0.266366i	−0.991091	−	0.133183i	$-0.957480\pi$
$613$	− 6.59492i	− 0.266366i	0.991091	−	0.133183i	$-0.0425199\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.3939i	0.821029i	0.911854	+	0.410514i	$0.134651\pi$
$617$	20.3939i	0.821029i	−0.911854	+	0.410514i	$0.865349\pi$
$618$	0	0
$619$	−0.513389	−0.0206348	−0.0103174	−	0.999947i	$-0.503284\pi$
$619$	−0.513389	−0.0206348	−0.0103174	+	0.999947i	$0.503284\pi$
$620$	0	0
$621$	−5.61268	−0.225229
$622$	0	0
$623$	− 21.9232i	− 0.878333i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.98259i	0.159049i
$628$	0	0
$629$	10.7050	0.426835
$630$	0	0
$631$	16.3428	0.650596	0.325298	−	0.945612i	$-0.394535\pi$
$631$	16.3428	0.650596	0.325298	+	0.945612i	$0.394535\pi$
$632$	0	0
$633$	19.3516i	0.769157i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 41.2695i	− 1.63516i
$638$	0	0
$639$	11.2978	0.446935
$640$	0	0
$641$	−33.7798	−1.33422	−0.667110	−	0.744959i	$-0.732469\pi$
$641$	−33.7798	−1.33422	−0.667110	+	0.744959i	$0.732469\pi$
$642$	0	0
$643$	2.18090i	0.0860062i	0.999075	+	0.0430031i	$0.0136925\pi$
$643$	2.18090i	0.0860062i	−0.999075	+	0.0430031i	$0.986307\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 34.7346i	− 1.36556i	−0.730625	−	0.682778i	$-0.760771\pi$
$647$	− 34.7346i	− 1.36556i	0.730625	−	0.682778i	$-0.239229\pi$
$648$	0	0
$649$	10.8907	0.427499
$650$	0	0
$651$	−9.66494	−0.378799
$652$	0	0
$653$	6.68309i	0.261530i	0.991413	+	0.130765i	$0.0417433\pi$
$653$	6.68309i	0.261530i	−0.991413	+	0.130765i	$0.958257\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.74575i	0.263177i
$658$	0	0
$659$	45.8903	1.78763	0.893815	−	0.448435i	$-0.148018\pi$
$659$	45.8903	1.78763	0.893815	+	0.448435i	$0.148018\pi$
$660$	0	0
$661$	13.6251	0.529957	0.264978	−	0.964254i	$-0.414635\pi$
$661$	13.6251	0.529957	0.264978	+	0.964254i	$0.414635\pi$
$662$	0	0
$663$	− 28.0234i	− 1.08834i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.94362i	− 0.346299i
$668$	0	0
$669$	15.5257	0.600259
$670$	0	0
$671$	9.66304	0.373038
$672$	0	0
$673$	25.3475i	0.977075i	0.872543	+	0.488537i	$0.162469\pi$
$673$	25.3475i	0.977075i	−0.872543	+	0.488537i	$0.837531\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.2279i	− 0.662123i	−0.943609	−	0.331062i	$-0.892593\pi$
$677$	− 17.2279i	− 0.662123i	0.943609	−	0.331062i	$-0.107407\pi$
$678$	0	0
$679$	76.9996	2.95497
$680$	0	0
$681$	6.91579	0.265014
$682$	0	0
$683$	− 39.5454i	− 1.51316i	−0.653899	−	0.756582i	$-0.726868\pi$
$683$	− 39.5454i	− 1.51316i	0.653899	−	0.756582i	$-0.273132\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.62394i	0.0619570i
$688$	0	0
$689$	−23.4759	−0.894361
$690$	0	0
$691$	−29.0501	−1.10512	−0.552558	−	0.833474i	$-0.686348\pi$
$691$	−29.0501	−1.10512	−0.552558	+	0.833474i	$0.686348\pi$
$692$	0	0
$693$	− 13.3244i	− 0.506152i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	81.7725i	3.09735i
$698$	0	0
$699$	3.49244	0.132096
$700$	0	0
$701$	−17.7735	−0.671298	−0.335649	−	0.941987i	$-0.608956\pi$
$701$	−17.7735	−0.671298	−0.335649	+	0.941987i	$0.608956\pi$
$702$	0	0
$703$	− 1.91053i	− 0.0720571i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 50.0093i	− 1.88079i
$708$	0	0
$709$	14.8606	0.558103	0.279052	−	0.960276i	$-0.409980\pi$
$709$	14.8606	0.558103	0.279052	+	0.960276i	$0.409980\pi$
$710$	0	0
$711$	−9.07592	−0.340374
$712$	0	0
$713$	− 1.58140i	− 0.0592240i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 34.4216i	− 1.28550i
$718$	0	0
$719$	−27.4365	−1.02321	−0.511604	−	0.859221i	$-0.670949\pi$
$719$	−27.4365	−1.02321	−0.511604	+	0.859221i	$0.670949\pi$
$720$	0	0
$721$	−9.10512	−0.339092
$722$	0	0
$723$	− 2.91449i	− 0.108391i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 23.3674i	− 0.866649i	−0.901238	−	0.433325i	$-0.857340\pi$
$727$	− 23.3674i	− 0.866649i	0.901238	−	0.433325i	$-0.142660\pi$
$728$	0	0
$729$	−26.5754	−0.984275
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2.53381i	0.0935884i	0.998905	+	0.0467942i	$0.0149005\pi$
$733$	2.53381i	0.0935884i	−0.998905	+	0.0467942i	$0.985100\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.0538i	0.701856i
$738$	0	0
$739$	−25.9318	−0.953918	−0.476959	−	0.878925i	$-0.658261\pi$
$739$	−25.9318	−0.953918	−0.476959	+	0.878925i	$0.658261\pi$
$740$	0	0
$741$	−5.00139	−0.183731
$742$	0	0
$743$	2.11370i	0.0775442i	0.999248	+	0.0387721i	$0.0123446\pi$
$743$	2.11370i	0.0775442i	−0.999248	+	0.0387721i	$0.987655\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.89172i	0.215567i
$748$	0	0
$749$	−9.21077	−0.336554
$750$	0	0
$751$	2.54986	0.0930459	0.0465229	−	0.998917i	$-0.485186\pi$
$751$	2.54986	0.0930459	0.0465229	+	0.998917i	$0.485186\pi$
$752$	0	0
$753$	17.6749i	0.644107i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.6897i	1.11544i	0.830030	+	0.557718i	$0.188323\pi$
$757$	30.6897i	1.11544i	−0.830030	+	0.557718i	$0.811677\pi$
$758$	0	0
$759$	−2.92360	−0.106120
$760$	0	0
$761$	19.3112	0.700032	0.350016	−	0.936744i	$-0.386176\pi$
$761$	19.3112	0.700032	0.350016	+	0.936744i	$0.386176\pi$
$762$	0	0
$763$	− 41.6028i	− 1.50612i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.6767i	0.493839i
$768$	0	0
$769$	8.52624	0.307464	0.153732	−	0.988113i	$-0.450871\pi$
$769$	8.52624	0.307464	0.153732	+	0.988113i	$0.450871\pi$
$770$	0	0
$771$	−28.8770	−1.03998
$772$	0	0
$773$	− 29.5107i	− 1.06143i	−0.847551	−	0.530713i	$-0.821924\pi$
$773$	− 29.5107i	− 1.06143i	0.847551	−	0.530713i	$-0.178076\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.57164i	− 0.307506i
$778$	0	0
$779$	14.5941	0.522887
$780$	0	0
$781$	19.6615	0.703545
$782$	0	0
$783$	− 50.1977i	− 1.79392i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 37.1733i	− 1.32508i	−0.749025	−	0.662542i	$-0.769478\pi$
$787$	− 37.1733i	− 1.32508i	0.749025	−	0.662542i	$-0.230522\pi$
$788$	0	0
$789$	29.2548	1.04150
$790$	0	0
$791$	−80.8811	−2.87580
$792$	0	0
$793$	12.1350i	0.430926i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.9954i	0.602008i	0.953623	+	0.301004i	$0.0973217\pi$
$797$	16.9954i	0.602008i	−0.953623	+	0.301004i	$0.902678\pi$
$798$	0	0
$799$	55.4434	1.96144
$800$	0	0
$801$	6.02617	0.212924
$802$	0	0
$803$	11.7396i	0.414281i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.7711i	− 0.484767i
$808$	0	0
$809$	−16.8409	−0.592095	−0.296047	−	0.955173i	$-0.595669\pi$
$809$	−16.8409	−0.592095	−0.296047	+	0.955173i	$0.595669\pi$
$810$	0	0
$811$	4.44804	0.156192	0.0780959	−	0.996946i	$-0.475116\pi$
$811$	4.44804	0.156192	0.0780959	+	0.996946i	$0.475116\pi$
$812$	0	0
$813$	− 5.04303i	− 0.176867i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	16.7330	0.584698
$820$	0	0
$821$	−16.7361	−0.584093	−0.292047	−	0.956404i	$-0.594336\pi$
$821$	−16.7361	−0.584093	−0.292047	+	0.956404i	$0.594336\pi$
$822$	0	0
$823$	43.6605i	1.52191i	0.648805	+	0.760955i	$0.275269\pi$
$823$	43.6605i	1.52191i	−0.648805	+	0.760955i	$0.724731\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 37.9750i	− 1.32052i	−0.751037	−	0.660260i	$-0.770446\pi$
$827$	− 37.9750i	− 1.32052i	0.751037	−	0.660260i	$-0.229554\pi$
$828$	0	0
$829$	−17.9046	−0.621851	−0.310925	−	0.950434i	$-0.600639\pi$
$829$	−17.9046	−0.621851	−0.310925	+	0.950434i	$0.600639\pi$
$830$	0	0
$831$	−19.4872	−0.676002
$832$	0	0
$833$	− 112.471i	− 3.89687i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.87591i	− 0.306796i
$838$	0	0
$839$	42.4503	1.46555	0.732773	−	0.680473i	$-0.238226\pi$
$839$	42.4503	1.46555	0.732773	+	0.680473i	$0.238226\pi$
$840$	0	0
$841$	50.9884	1.75822
$842$	0	0
$843$	5.78976i	0.199410i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	28.0949i	0.965353i
$848$	0	0
$849$	5.09283	0.174785
$850$	0	0
$851$	1.40251	0.0480775
$852$	0	0
$853$	22.9952i	0.787339i	0.919252	+	0.393670i	$0.128795\pi$
$853$	22.9952i	0.787339i	−0.919252	+	0.393670i	$0.871205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.46053i	− 0.323166i	−0.986859	−	0.161583i	$-0.948340\pi$
$857$	− 9.46053i	− 0.323166i	0.986859	−	0.161583i	$-0.0516599\pi$
$858$	0	0
$859$	6.20043	0.211556	0.105778	−	0.994390i	$-0.466267\pi$
$859$	6.20043	0.211556	0.105778	+	0.994390i	$0.466267\pi$
$860$	0	0
$861$	65.4765	2.23143
$862$	0	0
$863$	9.68754i	0.329768i	0.986313	+	0.164884i	$0.0527249\pi$
$863$	9.68754i	0.329768i	−0.986313	+	0.164884i	$0.947275\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 54.0860i	− 1.83686i
$868$	0	0
$869$	−15.7948	−0.535801
$870$	0	0
$871$	−23.9280	−0.810771
$872$	0	0
$873$	21.1654i	0.716340i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.5663i	1.74127i	0.491928	+	0.870636i	$0.336292\pi$
$877$	51.5663i	1.74127i	−0.491928	+	0.870636i	$0.663708\pi$
$878$	0	0
$879$	−32.8065	−1.10654
$880$	0	0
$881$	−33.5969	−1.13191	−0.565954	−	0.824437i	$-0.691492\pi$
$881$	−33.5969	−1.13191	−0.565954	+	0.824437i	$0.691492\pi$
$882$	0	0
$883$	− 11.7914i	− 0.396814i	−0.980120	−	0.198407i	$-0.936423\pi$
$883$	− 11.7914i	− 0.396814i	0.980120	−	0.198407i	$-0.0635767\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 54.9578i	− 1.84530i	−0.385634	−	0.922652i	$-0.626017\pi$
$887$	− 54.9578i	− 1.84530i	0.385634	−	0.922652i	$-0.373983\pi$
$888$	0	0
$889$	90.8515	3.04706
$890$	0	0
$891$	−7.83517	−0.262488
$892$	0	0
$893$	− 9.89506i	− 0.331126i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.67149i	− 0.122588i
$898$	0	0
$899$	14.1435	0.471711
$900$	0	0
$901$	−63.9783	−2.13143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.41429i	0.279392i	0.990194	+	0.139696i	$0.0446125\pi$
$907$	8.41429i	0.279392i	−0.990194	+	0.139696i	$0.955387\pi$
$908$	0	0
$909$	13.7464	0.455940
$910$	0	0
$911$	−26.4143	−0.875146	−0.437573	−	0.899183i	$-0.644162\pi$
$911$	−26.4143	−0.875146	−0.437573	+	0.899183i	$0.644162\pi$
$912$	0	0
$913$	10.2533i	0.339335i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 26.4832i	− 0.874551i
$918$	0	0
$919$	−59.2734	−1.95525	−0.977624	−	0.210359i	$-0.932537\pi$
$919$	−59.2734	−1.95525	−0.977624	+	0.210359i	$0.932537\pi$
$920$	0	0
$921$	−28.8659	−0.951164
$922$	0	0
$923$	24.6912i	0.812722i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.50279i	− 0.0822023i
$928$	0	0
$929$	17.2475	0.565871	0.282935	−	0.959139i	$-0.408692\pi$
$929$	17.2475	0.565871	0.282935	+	0.959139i	$0.408692\pi$
$930$	0	0
$931$	−20.0728	−0.657859
$932$	0	0
$933$	25.7223i	0.842111i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 21.6918i	− 0.708642i	−0.935124	−	0.354321i	$-0.884712\pi$
$937$	− 21.6918i	− 0.708642i	0.935124	−	0.354321i	$-0.115288\pi$
$938$	0	0
$939$	−23.5303	−0.767883
$940$	0	0
$941$	−11.1158	−0.362365	−0.181182	−	0.983449i	$-0.557992\pi$
$941$	−11.1158	−0.362365	−0.181182	+	0.983449i	$0.557992\pi$
$942$	0	0
$943$	10.7134i	0.348877i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 40.8093i	− 1.32613i	−0.748564	−	0.663063i	$-0.769256\pi$
$947$	− 40.8093i	− 1.32613i	0.748564	−	0.663063i	$-0.230744\pi$
$948$	0	0
$949$	−14.7427	−0.478569
$950$	0	0
$951$	−44.9886	−1.45886
$952$	0	0
$953$	− 11.8237i	− 0.383009i	−0.981492	−	0.191504i	$-0.938663\pi$
$953$	− 11.8237i	− 0.383009i	0.981492	−	0.191504i	$-0.0613366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 26.1475i	− 0.845230i
$958$	0	0
$959$	−40.4973	−1.30772
$960$	0	0
$961$	−28.4992	−0.919328
$962$	0	0
$963$	− 2.53183i	− 0.0815869i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	27.8536i	0.895710i	0.894106	+	0.447855i	$0.147812\pi$
$967$	27.8536i	0.895710i	−0.894106	+	0.447855i	$0.852188\pi$
$968$	0	0
$969$	−13.6301	−0.437863
$970$	0	0
$971$	−16.5249	−0.530309	−0.265155	−	0.964206i	$-0.585423\pi$
$971$	−16.5249	−0.530309	−0.265155	+	0.964206i	$0.585423\pi$
$972$	0	0
$973$	− 43.0112i	− 1.37888i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 50.0059i	− 1.59983i	−0.600114	−	0.799915i	$-0.704878\pi$
$977$	− 50.0059i	− 1.59983i	0.600114	−	0.799915i	$-0.295122\pi$
$978$	0	0
$979$	10.4873	0.335176
$980$	0	0
$981$	11.4357	0.365112
$982$	0	0
$983$	− 22.7894i	− 0.726869i	−0.931620	−	0.363434i	$-0.881604\pi$
$983$	− 22.7894i	− 0.726869i	0.931620	−	0.363434i	$-0.118396\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 44.3943i	− 1.41309i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	33.9822	1.07948	0.539740	−	0.841832i	$-0.318522\pi$
$991$	33.9822	1.07948	0.539740	+	0.841832i	$0.318522\pi$
$992$	0	0
$993$	− 0.392962i	− 0.0124703i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 17.7258i	− 0.561382i	−0.959798	−	0.280691i	$-0.909436\pi$
$997$	− 17.7258i	− 0.561382i	0.959798	−	0.280691i	$-0.0905637\pi$
$998$	0	0
$999$	7.87186	0.249055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.u.4049.7		10
5.2	odd	4		4600.2.a.be.1.4		5
5.3	odd	4		920.2.a.j.1.2	✓	5
5.4	even	2	inner	4600.2.e.u.4049.4		10
15.8	even	4		8280.2.a.bs.1.1		5
20.3	even	4		1840.2.a.v.1.4		5
20.7	even	4		9200.2.a.cu.1.2		5
40.3	even	4		7360.2.a.cp.1.2		5
40.13	odd	4		7360.2.a.co.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.2	✓	5	5.3	odd	4
1840.2.a.v.1.4		5	20.3	even	4
4600.2.a.be.1.4		5	5.2	odd	4
4600.2.e.u.4049.4		10	5.4	even	2	inner
4600.2.e.u.4049.7		10	1.1	even	1	trivial
7360.2.a.co.1.4		5	40.13	odd	4
7360.2.a.cp.1.2		5	40.3	even	4
8280.2.a.bs.1.1		5	15.8	even	4
9200.2.a.cu.1.2		5	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)

\(f(q)\)	\(=\)	\(q+1.31091i q^{3} -4.66212i q^{7} +1.28151 q^{9} +O(q^{10})\)	\(q+1.31091i q^{3} -4.66212i q^{7} +1.28151 q^{9} +2.23020 q^{11} +2.80072i q^{13} +7.63271i q^{17} +1.36222 q^{19} +6.11163 q^{21} +1.00000i q^{23} +5.61268i q^{27} -8.94362 q^{29} -1.58140 q^{31} +2.92360i q^{33} -1.40251i q^{37} -3.67149 q^{39} +10.7134 q^{41} -7.26391i q^{47} -14.7353 q^{49} -10.0058 q^{51} +8.38212i q^{53} +1.78575i q^{57} +4.88331 q^{59} +4.33282 q^{61} -5.97454i q^{63} +8.54355i q^{67} -1.31091 q^{69} +8.81604 q^{71} +5.26391i q^{73} -10.3974i q^{77} -7.08222 q^{79} -3.51322 q^{81} +4.59749i q^{83} -11.7243i q^{87} +4.70241 q^{89} +13.0573 q^{91} -2.07308i q^{93} +16.5160i q^{97} +2.85802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 26 q^{9}+O(q^{10}) \) 10 * q - 26 * q^9	\( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100}) \) 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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