LMFDB - Embedded newform 4304.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4913.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + x + 1 $ x^4 - x^3 - 6*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.344151$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.04948 q^{3} +1.48793 q^{5} -0.344151 q^{7} +6.29934 q^{9} +O(q^{10})$	$q-3.04948 q^{3} +1.48793 q^{5} -0.344151 q^{7} +6.29934 q^{9} +0.905704 q^{11} -2.24985 q^{13} -4.53741 q^{15} +2.58570 q^{17} +0.688301 q^{19} +1.04948 q^{21} -1.46259 q^{23} -2.78607 q^{25} -10.0613 q^{27} -4.24274 q^{29} -1.92985 q^{31} -2.76193 q^{33} -0.512072 q^{35} -1.84911 q^{37} +6.86089 q^{39} +1.21740 q^{41} +6.59036 q^{43} +9.37296 q^{45} +6.42244 q^{47} -6.88156 q^{49} -7.88503 q^{51} +2.87689 q^{53} +1.34762 q^{55} -2.09896 q^{57} -11.0118 q^{59} -1.75015 q^{61} -2.16792 q^{63} -3.34762 q^{65} -4.34415 q^{67} +4.46014 q^{69} -5.29586 q^{71} +5.90206 q^{73} +8.49607 q^{75} -0.311699 q^{77} +15.1107 q^{79} +11.7836 q^{81} -15.7701 q^{83} +3.84733 q^{85} +12.9382 q^{87} -1.61934 q^{89} +0.774289 q^{91} +5.88503 q^{93} +1.02414 q^{95} +14.3436 q^{97} +5.70533 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9	$ 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} - 7 q^{11} + 4 q^{13} - 8 q^{15} + 4 q^{17} - 2 q^{19} - 5 q^{21} - 16 q^{23} - q^{25} - 6 q^{27} + q^{31} + q^{33} - 3 q^{35} - 2 q^{37} - 3 q^{39} - q^{41} - 9 q^{43} + 8 q^{45} - 13 q^{47} - 15 q^{49} - 3 q^{51} + 28 q^{53} - 13 q^{55} + 10 q^{57} - 19 q^{59} - 20 q^{61} - 12 q^{63} + 5 q^{65} - 15 q^{67} - 5 q^{69} - 15 q^{71} - 7 q^{73} - 12 q^{75} - 6 q^{77} + 17 q^{79} + 4 q^{81} - 6 q^{83} - 29 q^{85} + 34 q^{87} + 20 q^{89} + 18 q^{91} - 5 q^{93} + 6 q^{95} + 3 q^{97} + 16 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9 - 7 * q^11 + 4 * q^13 - 8 * q^15 + 4 * q^17 - 2 * q^19 - 5 * q^21 - 16 * q^23 - q^25 - 6 * q^27 + q^31 + q^33 - 3 * q^35 - 2 * q^37 - 3 * q^39 - q^41 - 9 * q^43 + 8 * q^45 - 13 * q^47 - 15 * q^49 - 3 * q^51 + 28 * q^53 - 13 * q^55 + 10 * q^57 - 19 * q^59 - 20 * q^61 - 12 * q^63 + 5 * q^65 - 15 * q^67 - 5 * q^69 - 15 * q^71 - 7 * q^73 - 12 * q^75 - 6 * q^77 + 17 * q^79 + 4 * q^81 - 6 * q^83 - 29 * q^85 + 34 * q^87 + 20 * q^89 + 18 * q^91 - 5 * q^93 + 6 * q^95 + 3 * q^97 + 16 * 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	0
$2$	0	0
$3$	−3.04948	−1.76062	−0.880309	−	0.474400i	$-0.842665\pi$
$3$	−3.04948	−1.76062	−0.880309	+	0.474400i	$0.842665\pi$
$4$	0	0
$5$	1.48793	0.665422	0.332711	−	0.943029i	$-0.392037\pi$
$5$	1.48793	0.665422	0.332711	+	0.943029i	$0.392037\pi$
$6$	0	0
$7$	−0.344151	−0.130077	−0.0650384	−	0.997883i	$-0.520717\pi$
$7$	−0.344151	−0.130077	−0.0650384	+	0.997883i	$0.520717\pi$
$8$	0	0
$9$	6.29934	2.09978
$10$	0	0
$11$	0.905704	0.273080	0.136540	−	0.990635i	$-0.456402\pi$
$11$	0.905704	0.273080	0.136540	+	0.990635i	$0.456402\pi$
$12$	0	0
$13$	−2.24985	−0.623997	−0.311999	−	0.950083i	$-0.600998\pi$
$13$	−2.24985	−0.623997	−0.311999	+	0.950083i	$0.600998\pi$
$14$	0	0
$15$	−4.53741	−1.17155
$16$	0	0
$17$	2.58570	0.627123	0.313562	−	0.949568i	$-0.398478\pi$
$17$	2.58570	0.627123	0.313562	+	0.949568i	$0.398478\pi$
$18$	0	0
$19$	0.688301	0.157907	0.0789536	−	0.996878i	$-0.474842\pi$
$19$	0.688301	0.157907	0.0789536	+	0.996878i	$0.474842\pi$
$20$	0	0
$21$	1.04948	0.229016
$22$	0	0
$23$	−1.46259	−0.304971	−0.152486	−	0.988306i	$-0.548728\pi$
$23$	−1.46259	−0.304971	−0.152486	+	0.988306i	$0.548728\pi$
$24$	0	0
$25$	−2.78607	−0.557214
$26$	0	0
$27$	−10.0613	−1.93629
$28$	0	0
$29$	−4.24274	−0.787857	−0.393929	−	0.919141i	$-0.628884\pi$
$29$	−4.24274	−0.787857	−0.393929	+	0.919141i	$0.628884\pi$
$30$	0	0
$31$	−1.92985	−0.346611	−0.173305	−	0.984868i	$-0.555445\pi$
$31$	−1.92985	−0.346611	−0.173305	+	0.984868i	$0.555445\pi$
$32$	0	0
$33$	−2.76193	−0.480790
$34$	0	0
$35$	−0.512072	−0.0865559
$36$	0	0
$37$	−1.84911	−0.303991	−0.151996	−	0.988381i	$-0.548570\pi$
$37$	−1.84911	−0.303991	−0.151996	+	0.988381i	$0.548570\pi$
$38$	0	0
$39$	6.86089	1.09862
$40$	0	0
$41$	1.21740	0.190126	0.0950631	−	0.995471i	$-0.469695\pi$
$41$	1.21740	0.190126	0.0950631	+	0.995471i	$0.469695\pi$
$42$	0	0
$43$	6.59036	1.00502	0.502510	−	0.864571i	$-0.332410\pi$
$43$	6.59036	1.00502	0.502510	+	0.864571i	$0.332410\pi$
$44$	0	0
$45$	9.37296	1.39724
$46$	0	0
$47$	6.42244	0.936809	0.468405	−	0.883514i	$-0.344829\pi$
$47$	6.42244	0.936809	0.468405	+	0.883514i	$0.344829\pi$
$48$	0	0
$49$	−6.88156	−0.983080
$50$	0	0
$51$	−7.88503	−1.10413
$52$	0	0
$53$	2.87689	0.395172	0.197586	−	0.980286i	$-0.436690\pi$
$53$	2.87689	0.395172	0.197586	+	0.980286i	$0.436690\pi$
$54$	0	0
$55$	1.34762	0.181713
$56$	0	0
$57$	−2.09896	−0.278014
$58$	0	0
$59$	−11.0118	−1.43361	−0.716806	−	0.697273i	$-0.754397\pi$
$59$	−11.0118	−1.43361	−0.716806	+	0.697273i	$0.754397\pi$
$60$	0	0
$61$	−1.75015	−0.224083	−0.112042	−	0.993704i	$-0.535739\pi$
$61$	−1.75015	−0.224083	−0.112042	+	0.993704i	$0.535739\pi$
$62$	0	0
$63$	−2.16792	−0.273132
$64$	0	0
$65$	−3.34762	−0.415221
$66$	0	0
$67$	−4.34415	−0.530722	−0.265361	−	0.964149i	$-0.585491\pi$
$67$	−4.34415	−0.530722	−0.265361	+	0.964149i	$0.585491\pi$
$68$	0	0
$69$	4.46014	0.536938
$70$	0	0
$71$	−5.29586	−0.628503	−0.314252	−	0.949340i	$-0.601754\pi$
$71$	−5.29586	−0.628503	−0.314252	+	0.949340i	$0.601754\pi$
$72$	0	0
$73$	5.90206	0.690784	0.345392	−	0.938459i	$-0.387746\pi$
$73$	5.90206	0.690784	0.345392	+	0.938459i	$0.387746\pi$
$74$	0	0
$75$	8.49607	0.981041
$76$	0	0
$77$	−0.311699	−0.0355213
$78$	0	0
$79$	15.1107	1.70009	0.850046	−	0.526709i	$-0.176574\pi$
$79$	15.1107	1.70009	0.850046	+	0.526709i	$0.176574\pi$
$80$	0	0
$81$	11.7836	1.30929
$82$	0	0
$83$	−15.7701	−1.73099	−0.865495	−	0.500918i	$-0.832996\pi$
$83$	−15.7701	−1.73099	−0.865495	+	0.500918i	$0.832996\pi$
$84$	0	0
$85$	3.84733	0.417302
$86$	0	0
$87$	12.9382	1.38712
$88$	0	0
$89$	−1.61934	−0.171650	−0.0858250	−	0.996310i	$-0.527353\pi$
$89$	−1.61934	−0.171650	−0.0858250	+	0.996310i	$0.527353\pi$
$90$	0	0
$91$	0.774289	0.0811675
$92$	0	0
$93$	5.88503	0.610249
$94$	0	0
$95$	1.02414	0.105075
$96$	0	0
$97$	14.3436	1.45637	0.728184	−	0.685381i	$-0.240364\pi$
$97$	14.3436	1.45637	0.728184	+	0.685381i	$0.240364\pi$
$98$	0	0
$99$	5.70533	0.573407
$100$	0	0
$101$	2.90690	0.289247	0.144624	−	0.989487i	$-0.453803\pi$
$101$	2.90690	0.289247	0.144624	+	0.989487i	$0.453803\pi$
$102$	0	0
$103$	12.0866	1.19093	0.595464	−	0.803382i	$-0.296968\pi$
$103$	12.0866	1.19093	0.595464	+	0.803382i	$0.296968\pi$
$104$	0	0
$105$	1.56155	0.152392
$106$	0	0
$107$	7.03712	0.680304	0.340152	−	0.940370i	$-0.389521\pi$
$107$	7.03712	0.680304	0.340152	+	0.940370i	$0.389521\pi$
$108$	0	0
$109$	−4.91926	−0.471180	−0.235590	−	0.971853i	$-0.575702\pi$
$109$	−4.91926	−0.471180	−0.235590	+	0.971853i	$0.575702\pi$
$110$	0	0
$111$	5.63882	0.535213
$112$	0	0
$113$	10.5108	0.988775	0.494387	−	0.869242i	$-0.335392\pi$
$113$	10.5108	0.988775	0.494387	+	0.869242i	$0.335392\pi$
$114$	0	0
$115$	−2.17623	−0.202934
$116$	0	0
$117$	−14.1726	−1.31026
$118$	0	0
$119$	−0.889869	−0.0815742
$120$	0	0
$121$	−10.1797	−0.925427
$122$	0	0
$123$	−3.71244	−0.334740
$124$	0	0
$125$	−11.5851	−1.03620
$126$	0	0
$127$	20.0830	1.78207	0.891037	−	0.453930i	$-0.149979\pi$
$127$	20.0830	1.78207	0.891037	+	0.453930i	$0.149979\pi$
$128$	0	0
$129$	−20.0972	−1.76946
$130$	0	0
$131$	8.54572	0.746643	0.373321	−	0.927702i	$-0.378219\pi$
$131$	8.54572	0.746643	0.373321	+	0.927702i	$0.378219\pi$
$132$	0	0
$133$	−0.236879	−0.0205401
$134$	0	0
$135$	−14.9704	−1.28845
$136$	0	0
$137$	−15.0625	−1.28687	−0.643436	−	0.765500i	$-0.722492\pi$
$137$	−15.0625	−1.28687	−0.643436	+	0.765500i	$0.722492\pi$
$138$	0	0
$139$	2.85030	0.241760	0.120880	−	0.992667i	$-0.461428\pi$
$139$	2.85030	0.241760	0.120880	+	0.992667i	$0.461428\pi$
$140$	0	0
$141$	−19.5851	−1.64936
$142$	0	0
$143$	−2.03770	−0.170401
$144$	0	0
$145$	−6.31289	−0.524257
$146$	0	0
$147$	20.9852	1.73083
$148$	0	0
$149$	−5.24502	−0.429689	−0.214844	−	0.976648i	$-0.568924\pi$
$149$	−5.24502	−0.429689	−0.214844	+	0.976648i	$0.568924\pi$
$150$	0	0
$151$	−12.0613	−0.981532	−0.490766	−	0.871292i	$-0.663283\pi$
$151$	−12.0613	−0.981532	−0.490766	+	0.871292i	$0.663283\pi$
$152$	0	0
$153$	16.2882	1.31682
$154$	0	0
$155$	−2.87147	−0.230642
$156$	0	0
$157$	−14.4190	−1.15076	−0.575380	−	0.817887i	$-0.695146\pi$
$157$	−14.4190	−1.15076	−0.575380	+	0.817887i	$0.695146\pi$
$158$	0	0
$159$	−8.77304	−0.695747
$160$	0	0
$161$	0.503352	0.0396697
$162$	0	0
$163$	−20.9562	−1.64142	−0.820708	−	0.571347i	$-0.806421\pi$
$163$	−20.9562	−1.64142	−0.820708	+	0.571347i	$0.806421\pi$
$164$	0	0
$165$	−4.10955	−0.319928
$166$	0	0
$167$	2.59520	0.200823	0.100411	−	0.994946i	$-0.467984\pi$
$167$	2.59520	0.200823	0.100411	+	0.994946i	$0.467984\pi$
$168$	0	0
$169$	−7.93816	−0.610627
$170$	0	0
$171$	4.33584	0.331570
$172$	0	0
$173$	−15.6629	−1.19083	−0.595414	−	0.803419i	$-0.703012\pi$
$173$	−15.6629	−1.19083	−0.595414	+	0.803419i	$0.703012\pi$
$174$	0	0
$175$	0.958828	0.0724806
$176$	0	0
$177$	33.5802	2.52404
$178$	0	0
$179$	−25.1868	−1.88255	−0.941273	−	0.337646i	$-0.890369\pi$
$179$	−25.1868	−1.88255	−0.941273	+	0.337646i	$0.890369\pi$
$180$	0	0
$181$	6.06957	0.451148	0.225574	−	0.974226i	$-0.427574\pi$
$181$	6.06957	0.451148	0.225574	+	0.974226i	$0.427574\pi$
$182$	0	0
$183$	5.33704	0.394525
$184$	0	0
$185$	−2.75134	−0.202283
$186$	0	0
$187$	2.34187	0.171255
$188$	0	0
$189$	3.46259	0.251866
$190$	0	0
$191$	−8.11319	−0.587050	−0.293525	−	0.955951i	$-0.594828\pi$
$191$	−8.11319	−0.587050	−0.293525	+	0.955951i	$0.594828\pi$
$192$	0	0
$193$	−12.8968	−0.928333	−0.464166	−	0.885748i	$-0.653646\pi$
$193$	−12.8968	−0.928333	−0.464166	+	0.885748i	$0.653646\pi$
$194$	0	0
$195$	10.2085	0.731047
$196$	0	0
$197$	14.4920	1.03251	0.516257	−	0.856434i	$-0.327325\pi$
$197$	14.4920	1.03251	0.516257	+	0.856434i	$0.327325\pi$
$198$	0	0
$199$	−20.5344	−1.45564	−0.727822	−	0.685766i	$-0.759467\pi$
$199$	−20.5344	−1.45564	−0.727822	+	0.685766i	$0.759467\pi$
$200$	0	0
$201$	13.2474	0.934400
$202$	0	0
$203$	1.46014	0.102482
$204$	0	0
$205$	1.81141	0.126514
$206$	0	0
$207$	−9.21335	−0.640372
$208$	0	0
$209$	0.623397	0.0431213
$210$	0	0
$211$	6.99356	0.481456	0.240728	−	0.970593i	$-0.422614\pi$
$211$	6.99356	0.481456	0.240728	+	0.970593i	$0.422614\pi$
$212$	0	0
$213$	16.1496	1.10655
$214$	0	0
$215$	9.80599	0.668763
$216$	0	0
$217$	0.664158	0.0450860
$218$	0	0
$219$	−17.9982	−1.21621
$220$	0	0
$221$	−5.81744	−0.391323
$222$	0	0
$223$	8.26630	0.553552	0.276776	−	0.960934i	$-0.410734\pi$
$223$	8.26630	0.553552	0.276776	+	0.960934i	$0.410734\pi$
$224$	0	0
$225$	−17.5504	−1.17003
$226$	0	0
$227$	3.27586	0.217427	0.108713	−	0.994073i	$-0.465327\pi$
$227$	3.27586	0.217427	0.108713	+	0.994073i	$0.465327\pi$
$228$	0	0
$229$	−3.04601	−0.201286	−0.100643	−	0.994923i	$-0.532090\pi$
$229$	−3.04601	−0.201286	−0.100643	+	0.994923i	$0.532090\pi$
$230$	0	0
$231$	0.950519	0.0625395
$232$	0	0
$233$	12.9540	0.848644	0.424322	−	0.905511i	$-0.360512\pi$
$233$	12.9540	0.848644	0.424322	+	0.905511i	$0.360512\pi$
$234$	0	0
$235$	9.55613	0.623373
$236$	0	0
$237$	−46.0799	−2.99321
$238$	0	0
$239$	−8.68652	−0.561885	−0.280942	−	0.959725i	$-0.590647\pi$
$239$	−8.68652	−0.561885	−0.280942	+	0.959725i	$0.590647\pi$
$240$	0	0
$241$	−15.8491	−1.02093	−0.510465	−	0.859898i	$-0.670527\pi$
$241$	−15.8491	−1.02093	−0.510465	+	0.859898i	$0.670527\pi$
$242$	0	0
$243$	−5.75015	−0.368872
$244$	0	0
$245$	−10.2393	−0.654163
$246$	0	0
$247$	−1.54858	−0.0985337
$248$	0	0
$249$	48.0905	3.04761
$250$	0	0
$251$	−4.61103	−0.291046	−0.145523	−	0.989355i	$-0.546486\pi$
$251$	−4.61103	−0.291046	−0.145523	+	0.989355i	$0.546486\pi$
$252$	0	0
$253$	−1.32467	−0.0832815
$254$	0	0
$255$	−11.7324	−0.734709
$256$	0	0
$257$	20.2182	1.26118	0.630588	−	0.776118i	$-0.282814\pi$
$257$	20.2182	1.26118	0.630588	+	0.776118i	$0.282814\pi$
$258$	0	0
$259$	0.636372	0.0395422
$260$	0	0
$261$	−26.7264	−1.65433
$262$	0	0
$263$	−15.7471	−0.971009	−0.485504	−	0.874234i	$-0.661364\pi$
$263$	−15.7471	−0.971009	−0.485504	+	0.874234i	$0.661364\pi$
$264$	0	0
$265$	4.28061	0.262956
$266$	0	0
$267$	4.93816	0.302210
$268$	0	0
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−17.3316	−1.05282	−0.526410	−	0.850231i	$-0.676462\pi$
$271$	−17.3316	−1.05282	−0.526410	+	0.850231i	$0.676462\pi$
$272$	0	0
$273$	−2.36118	−0.142905
$274$	0	0
$275$	−2.52335	−0.152164
$276$	0	0
$277$	−23.1578	−1.39142	−0.695708	−	0.718325i	$-0.744909\pi$
$277$	−23.1578	−1.39142	−0.695708	+	0.718325i	$0.744909\pi$
$278$	0	0
$279$	−12.1568	−0.727806
$280$	0	0
$281$	−14.0106	−0.835801	−0.417901	−	0.908493i	$-0.637234\pi$
$281$	−14.0106	−0.835801	−0.417901	+	0.908493i	$0.637234\pi$
$282$	0	0
$283$	−26.1135	−1.55229	−0.776145	−	0.630555i	$-0.782827\pi$
$283$	−26.1135	−1.55229	−0.776145	+	0.630555i	$0.782827\pi$
$284$	0	0
$285$	−3.12311	−0.184997
$286$	0	0
$287$	−0.418970	−0.0247310
$288$	0	0
$289$	−10.3142	−0.606716
$290$	0	0
$291$	−43.7404	−2.56411
$292$	0	0
$293$	−8.14964	−0.476107	−0.238053	−	0.971252i	$-0.576509\pi$
$293$	−8.14964	−0.476107	−0.238053	+	0.971252i	$0.576509\pi$
$294$	0	0
$295$	−16.3847	−0.953956
$296$	0	0
$297$	−9.11252	−0.528762
$298$	0	0
$299$	3.29062	0.190301
$300$	0	0
$301$	−2.26808	−0.130730
$302$	0	0
$303$	−8.86453	−0.509254
$304$	0	0
$305$	−2.60409	−0.149110
$306$	0	0
$307$	−0.809130	−0.0461795	−0.0230898	−	0.999733i	$-0.507350\pi$
$307$	−0.809130	−0.0461795	−0.0230898	+	0.999733i	$0.507350\pi$
$308$	0	0
$309$	−36.8579	−2.09677
$310$	0	0
$311$	0.444367	0.0251977	0.0125989	−	0.999921i	$-0.495990\pi$
$311$	0.444367	0.0251977	0.0125989	+	0.999921i	$0.495990\pi$
$312$	0	0
$313$	1.05085	0.0593974	0.0296987	−	0.999559i	$-0.490545\pi$
$313$	1.05085	0.0593974	0.0296987	+	0.999559i	$0.490545\pi$
$314$	0	0
$315$	−3.22571	−0.181748
$316$	0	0
$317$	6.46784	0.363270	0.181635	−	0.983366i	$-0.441861\pi$
$317$	6.46784	0.363270	0.181635	+	0.983366i	$0.441861\pi$
$318$	0	0
$319$	−3.84266	−0.215148
$320$	0	0
$321$	−21.4596	−1.19776
$322$	0	0
$323$	1.77974	0.0990273
$324$	0	0
$325$	6.26825	0.347700
$326$	0	0
$327$	15.0012	0.829568
$328$	0	0
$329$	−2.21029	−0.121857
$330$	0	0
$331$	−22.0153	−1.21007	−0.605034	−	0.796200i	$-0.706840\pi$
$331$	−22.0153	−1.21007	−0.605034	+	0.796200i	$0.706840\pi$
$332$	0	0
$333$	−11.6482	−0.638315
$334$	0	0
$335$	−6.46379	−0.353154
$336$	0	0
$337$	14.2647	0.777047	0.388524	−	0.921439i	$-0.372985\pi$
$337$	14.2647	0.777047	0.388524	+	0.921439i	$0.372985\pi$
$338$	0	0
$339$	−32.0525	−1.74086
$340$	0	0
$341$	−1.74787	−0.0946524
$342$	0	0
$343$	4.77735	0.257953
$344$	0	0
$345$	6.63637	0.357290
$346$	0	0
$347$	−26.0825	−1.40018	−0.700092	−	0.714052i	$-0.746858\pi$
$347$	−26.0825	−1.40018	−0.700092	+	0.714052i	$0.746858\pi$
$348$	0	0
$349$	24.5684	1.31512	0.657559	−	0.753403i	$-0.271589\pi$
$349$	24.5684	1.31512	0.657559	+	0.753403i	$0.271589\pi$
$350$	0	0
$351$	22.6364	1.20824
$352$	0	0
$353$	8.36823	0.445396	0.222698	−	0.974887i	$-0.428514\pi$
$353$	8.36823	0.445396	0.222698	+	0.974887i	$0.428514\pi$
$354$	0	0
$355$	−7.87987	−0.418220
$356$	0	0
$357$	2.71364	0.143621
$358$	0	0
$359$	2.12835	0.112330	0.0561651	−	0.998421i	$-0.482113\pi$
$359$	2.12835	0.112330	0.0561651	+	0.998421i	$0.482113\pi$
$360$	0	0
$361$	−18.5262	−0.975065
$362$	0	0
$363$	31.0428	1.62932
$364$	0	0
$365$	8.78184	0.459663
$366$	0	0
$367$	−9.53496	−0.497721	−0.248860	−	0.968539i	$-0.580056\pi$
$367$	−9.53496	−0.497721	−0.248860	+	0.968539i	$0.580056\pi$
$368$	0	0
$369$	7.66882	0.399223
$370$	0	0
$371$	−0.990085	−0.0514027
$372$	0	0
$373$	16.7808	0.868875	0.434437	−	0.900702i	$-0.356947\pi$
$373$	16.7808	0.868875	0.434437	+	0.900702i	$0.356947\pi$
$374$	0	0
$375$	35.3286	1.82436
$376$	0	0
$377$	9.54555	0.491621
$378$	0	0
$379$	29.0341	1.49138	0.745690	−	0.666293i	$-0.232120\pi$
$379$	29.0341	1.49138	0.745690	+	0.666293i	$0.232120\pi$
$380$	0	0
$381$	−61.2426	−3.13755
$382$	0	0
$383$	−35.5798	−1.81804	−0.909022	−	0.416749i	$-0.863169\pi$
$383$	−35.5798	−1.81804	−0.909022	+	0.416749i	$0.863169\pi$
$384$	0	0
$385$	−0.463785	−0.0236367
$386$	0	0
$387$	41.5149	2.11032
$388$	0	0
$389$	−37.5839	−1.90558	−0.952789	−	0.303634i	$-0.901800\pi$
$389$	−37.5839	−1.90558	−0.952789	+	0.303634i	$0.901800\pi$
$390$	0	0
$391$	−3.78181	−0.191255
$392$	0	0
$393$	−26.0600	−1.31455
$394$	0	0
$395$	22.4837	1.13128
$396$	0	0
$397$	−16.8913	−0.847750	−0.423875	−	0.905721i	$-0.639330\pi$
$397$	−16.8913	−0.847750	−0.423875	+	0.905721i	$0.639330\pi$
$398$	0	0
$399$	0.722359	0.0361632
$400$	0	0
$401$	3.68585	0.184063	0.0920314	−	0.995756i	$-0.470664\pi$
$401$	3.68585	0.184063	0.0920314	+	0.995756i	$0.470664\pi$
$402$	0	0
$403$	4.34187	0.216284
$404$	0	0
$405$	17.5332	0.871231
$406$	0	0
$407$	−1.67474	−0.0830140
$408$	0	0
$409$	18.5784	0.918643	0.459322	−	0.888270i	$-0.348093\pi$
$409$	18.5784	0.918643	0.459322	+	0.888270i	$0.348093\pi$
$410$	0	0
$411$	45.9327	2.26569
$412$	0	0
$413$	3.78971	0.186480
$414$	0	0
$415$	−23.4647	−1.15184
$416$	0	0
$417$	−8.69194	−0.425646
$418$	0	0
$419$	−3.30890	−0.161650	−0.0808251	−	0.996728i	$-0.525756\pi$
$419$	−3.30890	−0.161650	−0.0808251	+	0.996728i	$0.525756\pi$
$420$	0	0
$421$	7.23994	0.352853	0.176427	−	0.984314i	$-0.443546\pi$
$421$	7.23994	0.352853	0.176427	+	0.984314i	$0.443546\pi$
$422$	0	0
$423$	40.4571	1.96709
$424$	0	0
$425$	−7.20393	−0.349442
$426$	0	0
$427$	0.602314	0.0291480
$428$	0	0
$429$	6.21393	0.300011
$430$	0	0
$431$	11.1645	0.537777	0.268888	−	0.963171i	$-0.413344\pi$
$431$	11.1645	0.537777	0.268888	+	0.963171i	$0.413344\pi$
$432$	0	0
$433$	28.8143	1.38473	0.692363	−	0.721549i	$-0.256570\pi$
$433$	28.8143	1.38473	0.692363	+	0.721549i	$0.256570\pi$
$434$	0	0
$435$	19.2510	0.923017
$436$	0	0
$437$	−1.00670	−0.0481571
$438$	0	0
$439$	−20.4897	−0.977922	−0.488961	−	0.872306i	$-0.662624\pi$
$439$	−20.4897	−0.977922	−0.488961	+	0.872306i	$0.662624\pi$
$440$	0	0
$441$	−43.3493	−2.06425
$442$	0	0
$443$	−9.58503	−0.455398	−0.227699	−	0.973732i	$-0.573120\pi$
$443$	−9.58503	−0.455398	−0.227699	+	0.973732i	$0.573120\pi$
$444$	0	0
$445$	−2.40947	−0.114220
$446$	0	0
$447$	15.9946	0.756518
$448$	0	0
$449$	−21.4476	−1.01218	−0.506088	−	0.862482i	$-0.668909\pi$
$449$	−21.4476	−1.01218	−0.506088	+	0.862482i	$0.668909\pi$
$450$	0	0
$451$	1.10261	0.0519197
$452$	0	0
$453$	36.7806	1.72810
$454$	0	0
$455$	1.15209	0.0540107
$456$	0	0
$457$	10.7176	0.501350	0.250675	−	0.968071i	$-0.419347\pi$
$457$	10.7176	0.501350	0.250675	+	0.968071i	$0.419347\pi$
$458$	0	0
$459$	−26.0154	−1.21429
$460$	0	0
$461$	−19.6658	−0.915926	−0.457963	−	0.888971i	$-0.651421\pi$
$461$	−19.6658	−0.915926	−0.457963	+	0.888971i	$0.651421\pi$
$462$	0	0
$463$	−22.6226	−1.05136	−0.525682	−	0.850681i	$-0.676190\pi$
$463$	−22.6226	−1.05136	−0.525682	+	0.850681i	$0.676190\pi$
$464$	0	0
$465$	8.75651	0.406073
$466$	0	0
$467$	−30.0503	−1.39056	−0.695282	−	0.718737i	$-0.744720\pi$
$467$	−30.0503	−1.39056	−0.695282	+	0.718737i	$0.744720\pi$
$468$	0	0
$469$	1.49504	0.0690347
$470$	0	0
$471$	43.9704	2.02605
$472$	0	0
$473$	5.96891	0.274451
$474$	0	0
$475$	−1.91766	−0.0879881
$476$	0	0
$477$	18.1225	0.829773
$478$	0	0
$479$	32.6251	1.49068	0.745339	−	0.666685i	$-0.232288\pi$
$479$	32.6251	1.49068	0.745339	+	0.666685i	$0.232288\pi$
$480$	0	0
$481$	4.16022	0.189690
$482$	0	0
$483$	−1.53496	−0.0698432
$484$	0	0
$485$	21.3422	0.969099
$486$	0	0
$487$	4.12981	0.187139	0.0935697	−	0.995613i	$-0.470172\pi$
$487$	4.12981	0.187139	0.0935697	+	0.995613i	$0.470172\pi$
$488$	0	0
$489$	63.9056	2.88991
$490$	0	0
$491$	35.2225	1.58957	0.794785	−	0.606891i	$-0.207584\pi$
$491$	35.2225	1.58957	0.794785	+	0.606891i	$0.207584\pi$
$492$	0	0
$493$	−10.9704	−0.494084
$494$	0	0
$495$	8.48912	0.381558
$496$	0	0
$497$	1.82258	0.0817537
$498$	0	0
$499$	−16.3253	−0.730819	−0.365409	−	0.930847i	$-0.619071\pi$
$499$	−16.3253	−0.730819	−0.365409	+	0.930847i	$0.619071\pi$
$500$	0	0
$501$	−7.91401	−0.353572
$502$	0	0
$503$	−39.0571	−1.74147	−0.870736	−	0.491751i	$-0.836357\pi$
$503$	−39.0571	−1.74147	−0.870736	+	0.491751i	$0.836357\pi$
$504$	0	0
$505$	4.32526	0.192471
$506$	0	0
$507$	24.2073	1.07508
$508$	0	0
$509$	8.19139	0.363077	0.181539	−	0.983384i	$-0.441892\pi$
$509$	8.19139	0.363077	0.181539	+	0.983384i	$0.441892\pi$
$510$	0	0
$511$	−2.03120	−0.0898549
$512$	0	0
$513$	−6.92518	−0.305754
$514$	0	0
$515$	17.9840	0.792469
$516$	0	0
$517$	5.81683	0.255824
$518$	0	0
$519$	47.7637	2.09660
$520$	0	0
$521$	−9.81960	−0.430205	−0.215102	−	0.976592i	$-0.569009\pi$
$521$	−9.81960	−0.430205	−0.215102	+	0.976592i	$0.569009\pi$
$522$	0	0
$523$	−4.42973	−0.193698	−0.0968492	−	0.995299i	$-0.530876\pi$
$523$	−4.42973	−0.193698	−0.0968492	+	0.995299i	$0.530876\pi$
$524$	0	0
$525$	−2.92393	−0.127611
$526$	0	0
$527$	−4.99000	−0.217368
$528$	0	0
$529$	−20.8608	−0.906993
$530$	0	0
$531$	−69.3669	−3.01027
$532$	0	0
$533$	−2.73898	−0.118638
$534$	0	0
$535$	10.4707	0.452689
$536$	0	0
$537$	76.8065	3.31445
$538$	0	0
$539$	−6.23265	−0.268459
$540$	0	0
$541$	3.01458	0.129607	0.0648035	−	0.997898i	$-0.479358\pi$
$541$	3.01458	0.129607	0.0648035	+	0.997898i	$0.479358\pi$
$542$	0	0
$543$	−18.5090	−0.794299
$544$	0	0
$545$	−7.31951	−0.313533
$546$	0	0
$547$	−35.5411	−1.51963	−0.759813	−	0.650141i	$-0.774710\pi$
$547$	−35.5411	−1.51963	−0.759813	+	0.650141i	$0.774710\pi$
$548$	0	0
$549$	−11.0248	−0.470525
$550$	0	0
$551$	−2.92028	−0.124408
$552$	0	0
$553$	−5.20037	−0.221142
$554$	0	0
$555$	8.39016	0.356142
$556$	0	0
$557$	21.5510	0.913144	0.456572	−	0.889687i	$-0.349077\pi$
$557$	21.5510	0.913144	0.456572	+	0.889687i	$0.349077\pi$
$558$	0	0
$559$	−14.8274	−0.627130
$560$	0	0
$561$	−7.14150	−0.301514
$562$	0	0
$563$	−30.3310	−1.27830	−0.639150	−	0.769082i	$-0.720714\pi$
$563$	−30.3310	−1.27830	−0.639150	+	0.769082i	$0.720714\pi$
$564$	0	0
$565$	15.6393	0.657952
$566$	0	0
$567$	−4.05534	−0.170308
$568$	0	0
$569$	3.78038	0.158482	0.0792409	−	0.996855i	$-0.474750\pi$
$569$	3.78038	0.158482	0.0792409	+	0.996855i	$0.474750\pi$
$570$	0	0
$571$	−18.4336	−0.771424	−0.385712	−	0.922619i	$-0.626044\pi$
$571$	−18.4336	−0.771424	−0.385712	+	0.922619i	$0.626044\pi$
$572$	0	0
$573$	24.7410	1.03357
$574$	0	0
$575$	4.07488	0.169934
$576$	0	0
$577$	30.3922	1.26524	0.632622	−	0.774461i	$-0.281979\pi$
$577$	30.3922	1.26524	0.632622	+	0.774461i	$0.281979\pi$
$578$	0	0
$579$	39.3286	1.63444
$580$	0	0
$581$	5.42728	0.225161
$582$	0	0
$583$	2.60561	0.107913
$584$	0	0
$585$	−21.0878	−0.871873
$586$	0	0
$587$	38.4248	1.58596	0.792981	−	0.609246i	$-0.208528\pi$
$587$	38.4248	1.58596	0.792981	+	0.609246i	$0.208528\pi$
$588$	0	0
$589$	−1.32832	−0.0547323
$590$	0	0
$591$	−44.1931	−1.81786
$592$	0	0
$593$	15.0010	0.616018	0.308009	−	0.951383i	$-0.400337\pi$
$593$	15.0010	0.616018	0.308009	+	0.951383i	$0.400337\pi$
$594$	0	0
$595$	−1.32406	−0.0542812
$596$	0	0
$597$	62.6192	2.56283
$598$	0	0
$599$	8.67407	0.354413	0.177207	−	0.984174i	$-0.443294\pi$
$599$	8.67407	0.354413	0.177207	+	0.984174i	$0.443294\pi$
$600$	0	0
$601$	−20.1244	−0.820891	−0.410445	−	0.911885i	$-0.634627\pi$
$601$	−20.1244	−0.820891	−0.410445	+	0.911885i	$0.634627\pi$
$602$	0	0
$603$	−27.3653	−1.11440
$604$	0	0
$605$	−15.1467	−0.615800
$606$	0	0
$607$	−5.35730	−0.217446	−0.108723	−	0.994072i	$-0.534676\pi$
$607$	−5.35730	−0.217446	−0.108723	+	0.994072i	$0.534676\pi$
$608$	0	0
$609$	−4.45268	−0.180432
$610$	0	0
$611$	−14.4496	−0.584566
$612$	0	0
$613$	21.6009	0.872454	0.436227	−	0.899837i	$-0.356314\pi$
$613$	21.6009	0.872454	0.436227	+	0.899837i	$0.356314\pi$
$614$	0	0
$615$	−5.52385	−0.222743
$616$	0	0
$617$	−36.2231	−1.45829	−0.729144	−	0.684360i	$-0.760082\pi$
$617$	−36.2231	−1.45829	−0.729144	+	0.684360i	$0.760082\pi$
$618$	0	0
$619$	−31.9927	−1.28590	−0.642948	−	0.765910i	$-0.722289\pi$
$619$	−31.9927	−1.28590	−0.642948	+	0.765910i	$0.722289\pi$
$620$	0	0
$621$	14.7155	0.590513
$622$	0	0
$623$	0.557298	0.0223277
$624$	0	0
$625$	−3.30747	−0.132299
$626$	0	0
$627$	−1.90104	−0.0759201
$628$	0	0
$629$	−4.78123	−0.190640
$630$	0	0
$631$	22.3539	0.889894	0.444947	−	0.895557i	$-0.353222\pi$
$631$	22.3539	0.889894	0.444947	+	0.895557i	$0.353222\pi$
$632$	0	0
$633$	−21.3267	−0.847661
$634$	0	0
$635$	29.8820	1.18583
$636$	0	0
$637$	15.4825	0.613439
$638$	0	0
$639$	−33.3604	−1.31972
$640$	0	0
$641$	−8.18573	−0.323317	−0.161659	−	0.986847i	$-0.551684\pi$
$641$	−8.18573	−0.323317	−0.161659	+	0.986847i	$0.551684\pi$
$642$	0	0
$643$	−17.4703	−0.688961	−0.344480	−	0.938793i	$-0.611945\pi$
$643$	−17.4703	−0.688961	−0.344480	+	0.938793i	$0.611945\pi$
$644$	0	0
$645$	−29.9032	−1.17744
$646$	0	0
$647$	44.5260	1.75050	0.875249	−	0.483673i	$-0.160697\pi$
$647$	44.5260	1.75050	0.875249	+	0.483673i	$0.160697\pi$
$648$	0	0
$649$	−9.97341	−0.391491
$650$	0	0
$651$	−2.02534	−0.0793793
$652$	0	0
$653$	37.0875	1.45134	0.725672	−	0.688041i	$-0.241529\pi$
$653$	37.0875	1.45134	0.725672	+	0.688041i	$0.241529\pi$
$654$	0	0
$655$	12.7154	0.496832
$656$	0	0
$657$	37.1791	1.45049
$658$	0	0
$659$	−15.6928	−0.611305	−0.305652	−	0.952143i	$-0.598874\pi$
$659$	−15.6928	−0.611305	−0.305652	+	0.952143i	$0.598874\pi$
$660$	0	0
$661$	4.10083	0.159504	0.0797519	−	0.996815i	$-0.474587\pi$
$661$	4.10083	0.159504	0.0797519	+	0.996815i	$0.474587\pi$
$662$	0	0
$663$	17.7402	0.688971
$664$	0	0
$665$	−0.352460	−0.0136678
$666$	0	0
$667$	6.20539	0.240274
$668$	0	0
$669$	−25.2079	−0.974595
$670$	0	0
$671$	−1.58511	−0.0611926
$672$	0	0
$673$	8.56587	0.330190	0.165095	−	0.986278i	$-0.447207\pi$
$673$	8.56587	0.330190	0.165095	+	0.986278i	$0.447207\pi$
$674$	0	0
$675$	28.0314	1.07893
$676$	0	0
$677$	15.2255	0.585165	0.292583	−	0.956240i	$-0.405485\pi$
$677$	15.2255	0.585165	0.292583	+	0.956240i	$0.405485\pi$
$678$	0	0
$679$	−4.93635	−0.189440
$680$	0	0
$681$	−9.98968	−0.382805
$682$	0	0
$683$	34.5265	1.32112	0.660560	−	0.750774i	$-0.270319\pi$
$683$	34.5265	1.32112	0.660560	+	0.750774i	$0.270319\pi$
$684$	0	0
$685$	−22.4119	−0.856313
$686$	0	0
$687$	9.28875	0.354388
$688$	0	0
$689$	−6.47259	−0.246586
$690$	0	0
$691$	−32.7649	−1.24644	−0.623218	−	0.782048i	$-0.714175\pi$
$691$	−32.7649	−1.24644	−0.623218	+	0.782048i	$0.714175\pi$
$692$	0	0
$693$	−1.96349	−0.0745870
$694$	0	0
$695$	4.24105	0.160872
$696$	0	0
$697$	3.14783	0.119233
$698$	0	0
$699$	−39.5029	−1.49414
$700$	0	0
$701$	−4.80166	−0.181356	−0.0906782	−	0.995880i	$-0.528903\pi$
$701$	−4.80166	−0.181356	−0.0906782	+	0.995880i	$0.528903\pi$
$702$	0	0
$703$	−1.27274	−0.0480024
$704$	0	0
$705$	−29.1412	−1.09752
$706$	0	0
$707$	−1.00041	−0.0376243
$708$	0	0
$709$	−34.4618	−1.29424	−0.647120	−	0.762389i	$-0.724027\pi$
$709$	−34.4618	−1.29424	−0.647120	+	0.762389i	$0.724027\pi$
$710$	0	0
$711$	95.1876	3.56982
$712$	0	0
$713$	2.82258	0.105706
$714$	0	0
$715$	−3.03195	−0.113389
$716$	0	0
$717$	26.4894	0.989264
$718$	0	0
$719$	41.1653	1.53521	0.767604	−	0.640925i	$-0.221449\pi$
$719$	41.1653	1.53521	0.767604	+	0.640925i	$0.221449\pi$
$720$	0	0
$721$	−4.15961	−0.154912
$722$	0	0
$723$	48.3316	1.79747
$724$	0	0
$725$	11.8206	0.439005
$726$	0	0
$727$	50.5549	1.87498	0.937489	−	0.348015i	$-0.113144\pi$
$727$	50.5549	1.87498	0.937489	+	0.348015i	$0.113144\pi$
$728$	0	0
$729$	−17.8159	−0.659848
$730$	0	0
$731$	17.0407	0.630272
$732$	0	0
$733$	−29.3797	−1.08516	−0.542581	−	0.840003i	$-0.682553\pi$
$733$	−29.3797	−1.08516	−0.542581	+	0.840003i	$0.682553\pi$
$734$	0	0
$735$	31.2245	1.15173
$736$	0	0
$737$	−3.93451	−0.144930
$738$	0	0
$739$	22.4975	0.827584	0.413792	−	0.910371i	$-0.364204\pi$
$739$	22.4975	0.827584	0.413792	+	0.910371i	$0.364204\pi$
$740$	0	0
$741$	4.72236	0.173480
$742$	0	0
$743$	7.49123	0.274826	0.137413	−	0.990514i	$-0.456121\pi$
$743$	7.49123	0.274826	0.137413	+	0.990514i	$0.456121\pi$
$744$	0	0
$745$	−7.80421	−0.285924
$746$	0	0
$747$	−99.3409	−3.63469
$748$	0	0
$749$	−2.42183	−0.0884917
$750$	0	0
$751$	−31.3441	−1.14376	−0.571881	−	0.820337i	$-0.693786\pi$
$751$	−31.3441	−1.14376	−0.571881	+	0.820337i	$0.693786\pi$
$752$	0	0
$753$	14.0613	0.512421
$754$	0	0
$755$	−17.9463	−0.653132
$756$	0	0
$757$	−44.8626	−1.63056	−0.815280	−	0.579067i	$-0.803417\pi$
$757$	−44.8626	−1.63056	−0.815280	+	0.579067i	$0.803417\pi$
$758$	0	0
$759$	4.03957	0.146627
$760$	0	0
$761$	−18.5983	−0.674189	−0.337094	−	0.941471i	$-0.609444\pi$
$761$	−18.5983	−0.674189	−0.337094	+	0.941471i	$0.609444\pi$
$762$	0	0
$763$	1.69297	0.0612895
$764$	0	0
$765$	24.2356	0.876241
$766$	0	0
$767$	24.7749	0.894570
$768$	0	0
$769$	52.1929	1.88213	0.941063	−	0.338232i	$-0.109829\pi$
$769$	52.1929	1.88213	0.941063	+	0.338232i	$0.109829\pi$
$770$	0	0
$771$	−61.6550	−2.22045
$772$	0	0
$773$	41.7206	1.50059	0.750293	−	0.661105i	$-0.229912\pi$
$773$	41.7206	1.50059	0.750293	+	0.661105i	$0.229912\pi$
$774$	0	0
$775$	5.37669	0.193136
$776$	0	0
$777$	−1.94060	−0.0696188
$778$	0	0
$779$	0.837940	0.0300223
$780$	0	0
$781$	−4.79648	−0.171632
$782$	0	0
$783$	42.6873	1.52552
$784$	0	0
$785$	−21.4544	−0.765740
$786$	0	0
$787$	−20.8879	−0.744574	−0.372287	−	0.928118i	$-0.621426\pi$
$787$	−20.8879	−0.744574	−0.372287	+	0.928118i	$0.621426\pi$
$788$	0	0
$789$	48.0205	1.70958
$790$	0	0
$791$	−3.61731	−0.128617
$792$	0	0
$793$	3.93757	0.139827
$794$	0	0
$795$	−13.0536	−0.462965
$796$	0	0
$797$	20.6075	0.729955	0.364978	−	0.931016i	$-0.381077\pi$
$797$	20.6075	0.729955	0.364978	+	0.931016i	$0.381077\pi$
$798$	0	0
$799$	16.6065	0.587495
$800$	0	0
$801$	−10.2008	−0.360427
$802$	0	0
$803$	5.34552	0.188639
$804$	0	0
$805$	0.748951	0.0263971
$806$	0	0
$807$	−3.04948	−0.107347
$808$	0	0
$809$	8.27012	0.290762	0.145381	−	0.989376i	$-0.453559\pi$
$809$	8.27012	0.290762	0.145381	+	0.989376i	$0.453559\pi$
$810$	0	0
$811$	−26.6200	−0.934753	−0.467377	−	0.884058i	$-0.654801\pi$
$811$	−26.6200	−0.934753	−0.467377	+	0.884058i	$0.654801\pi$
$812$	0	0
$813$	52.8524	1.85362
$814$	0	0
$815$	−31.1813	−1.09223
$816$	0	0
$817$	4.53616	0.158700
$818$	0	0
$819$	4.87751	0.170434
$820$	0	0
$821$	29.4026	1.02616	0.513078	−	0.858342i	$-0.328505\pi$
$821$	29.4026	1.02616	0.513078	+	0.858342i	$0.328505\pi$
$822$	0	0
$823$	−15.4716	−0.539305	−0.269653	−	0.962958i	$-0.586909\pi$
$823$	−15.4716	−0.539305	−0.269653	+	0.962958i	$0.586909\pi$
$824$	0	0
$825$	7.69492	0.267903
$826$	0	0
$827$	24.5579	0.853963	0.426982	−	0.904260i	$-0.359577\pi$
$827$	24.5579	0.853963	0.426982	+	0.904260i	$0.359577\pi$
$828$	0	0
$829$	20.1760	0.700741	0.350371	−	0.936611i	$-0.386056\pi$
$829$	20.1760	0.700741	0.350371	+	0.936611i	$0.386056\pi$
$830$	0	0
$831$	70.6192	2.44975
$832$	0	0
$833$	−17.7936	−0.616513
$834$	0	0
$835$	3.86147	0.133632
$836$	0	0
$837$	19.4167	0.671139
$838$	0	0
$839$	39.9874	1.38052	0.690259	−	0.723562i	$-0.257496\pi$
$839$	39.9874	1.38052	0.690259	+	0.723562i	$0.257496\pi$
$840$	0	0
$841$	−10.9992	−0.379281
$842$	0	0
$843$	42.7250	1.47153
$844$	0	0
$845$	−11.8114	−0.406325
$846$	0	0
$847$	3.50335	0.120377
$848$	0	0
$849$	79.6328	2.73299
$850$	0	0
$851$	2.70449	0.0927086
$852$	0	0
$853$	−38.2866	−1.31091	−0.655453	−	0.755236i	$-0.727522\pi$
$853$	−38.2866	−1.31091	−0.655453	+	0.755236i	$0.727522\pi$
$854$	0	0
$855$	6.45142	0.220634
$856$	0	0
$857$	10.4287	0.356239	0.178120	−	0.984009i	$-0.442999\pi$
$857$	10.4287	0.356239	0.178120	+	0.984009i	$0.442999\pi$
$858$	0	0
$859$	31.3752	1.07051	0.535254	−	0.844691i	$-0.320216\pi$
$859$	31.3752	1.07051	0.535254	+	0.844691i	$0.320216\pi$
$860$	0	0
$861$	1.27764	0.0435419
$862$	0	0
$863$	−22.9420	−0.780954	−0.390477	−	0.920613i	$-0.627690\pi$
$863$	−22.9420	−0.780954	−0.390477	+	0.920613i	$0.627690\pi$
$864$	0	0
$865$	−23.3053	−0.792403
$866$	0	0
$867$	31.4529	1.06820
$868$	0	0
$869$	13.6859	0.464261
$870$	0	0
$871$	9.77371	0.331169
$872$	0	0
$873$	90.3549	3.05805
$874$	0	0
$875$	3.98703	0.134786
$876$	0	0
$877$	13.8521	0.467754	0.233877	−	0.972266i	$-0.424859\pi$
$877$	13.8521	0.467754	0.233877	+	0.972266i	$0.424859\pi$
$878$	0	0
$879$	24.8522	0.838243
$880$	0	0
$881$	20.0782	0.676451	0.338225	−	0.941065i	$-0.390173\pi$
$881$	20.0782	0.676451	0.338225	+	0.941065i	$0.390173\pi$
$882$	0	0
$883$	−55.3481	−1.86261	−0.931306	−	0.364237i	$-0.881330\pi$
$883$	−55.3481	−1.86261	−0.931306	+	0.364237i	$0.881330\pi$
$884$	0	0
$885$	49.9650	1.67955
$886$	0	0
$887$	−1.28823	−0.0432544	−0.0216272	−	0.999766i	$-0.506885\pi$
$887$	−1.28823	−0.0432544	−0.0216272	+	0.999766i	$0.506885\pi$
$888$	0	0
$889$	−6.91156	−0.231806
$890$	0	0
$891$	10.6725	0.357541
$892$	0	0
$893$	4.42058	0.147929
$894$	0	0
$895$	−37.4761	−1.25269
$896$	0	0
$897$	−10.0347	−0.335048
$898$	0	0
$899$	8.18784	0.273080
$900$	0	0
$901$	7.43877	0.247822
$902$	0	0
$903$	6.91646	0.230165
$904$	0	0
$905$	9.03109	0.300203
$906$	0	0
$907$	−11.6760	−0.387695	−0.193848	−	0.981032i	$-0.562097\pi$
$907$	−11.6760	−0.387695	−0.193848	+	0.981032i	$0.562097\pi$
$908$	0	0
$909$	18.3115	0.607355
$910$	0	0
$911$	−31.6479	−1.04854	−0.524271	−	0.851551i	$-0.675662\pi$
$911$	−31.6479	−1.04854	−0.524271	+	0.851551i	$0.675662\pi$
$912$	0	0
$913$	−14.2830	−0.472698
$914$	0	0
$915$	7.94113	0.262526
$916$	0	0
$917$	−2.94102	−0.0971209
$918$	0	0
$919$	49.3145	1.62674	0.813368	−	0.581749i	$-0.197632\pi$
$919$	49.3145	1.62674	0.813368	+	0.581749i	$0.197632\pi$
$920$	0	0
$921$	2.46743	0.0813045
$922$	0	0
$923$	11.9149	0.392184
$924$	0	0
$925$	5.15174	0.169388
$926$	0	0
$927$	76.1375	2.50069
$928$	0	0
$929$	−25.7689	−0.845450	−0.422725	−	0.906258i	$-0.638926\pi$
$929$	−25.7689	−0.845450	−0.422725	+	0.906258i	$0.638926\pi$
$930$	0	0
$931$	−4.73659	−0.155235
$932$	0	0
$933$	−1.35509	−0.0443636
$934$	0	0
$935$	3.48454	0.113957
$936$	0	0
$937$	−6.22816	−0.203465	−0.101733	−	0.994812i	$-0.532439\pi$
$937$	−6.22816	−0.203465	−0.101733	+	0.994812i	$0.532439\pi$
$938$	0	0
$939$	−3.20454	−0.104576
$940$	0	0
$941$	15.3930	0.501799	0.250900	−	0.968013i	$-0.419274\pi$
$941$	15.3930	0.501799	0.250900	+	0.968013i	$0.419274\pi$
$942$	0	0
$943$	−1.78056	−0.0579830
$944$	0	0
$945$	5.15209	0.167597
$946$	0	0
$947$	19.0683	0.619635	0.309818	−	0.950796i	$-0.399732\pi$
$947$	19.0683	0.619635	0.309818	+	0.950796i	$0.399732\pi$
$948$	0	0
$949$	−13.2788	−0.431047
$950$	0	0
$951$	−19.7236	−0.639580
$952$	0	0
$953$	−39.7141	−1.28647	−0.643234	−	0.765670i	$-0.722408\pi$
$953$	−39.7141	−1.28647	−0.643234	+	0.765670i	$0.722408\pi$
$954$	0	0
$955$	−12.0718	−0.390636
$956$	0	0
$957$	11.7181	0.378793
$958$	0	0
$959$	5.18376	0.167392
$960$	0	0
$961$	−27.2757	−0.879861
$962$	0	0
$963$	44.3292	1.42849
$964$	0	0
$965$	−19.1895	−0.617733
$966$	0	0
$967$	54.5429	1.75398	0.876991	−	0.480507i	$-0.159547\pi$
$967$	54.5429	1.75398	0.876991	+	0.480507i	$0.159547\pi$
$968$	0	0
$969$	−5.42728	−0.174349
$970$	0	0
$971$	20.6894	0.663955	0.331977	−	0.943287i	$-0.392284\pi$
$971$	20.6894	0.663955	0.331977	+	0.943287i	$0.392284\pi$
$972$	0	0
$973$	−0.980934	−0.0314473
$974$	0	0
$975$	−19.1149	−0.612167
$976$	0	0
$977$	3.49749	0.111895	0.0559473	−	0.998434i	$-0.482182\pi$
$977$	3.49749	0.111895	0.0559473	+	0.998434i	$0.482182\pi$
$978$	0	0
$979$	−1.46664	−0.0468742
$980$	0	0
$981$	−30.9881	−0.989373
$982$	0	0
$983$	−16.5884	−0.529086	−0.264543	−	0.964374i	$-0.585221\pi$
$983$	−16.5884	−0.529086	−0.264543	+	0.964374i	$0.585221\pi$
$984$	0	0
$985$	21.5631	0.687057
$986$	0	0
$987$	6.74023	0.214544
$988$	0	0
$989$	−9.63900	−0.306502
$990$	0	0
$991$	27.1297	0.861805	0.430902	−	0.902399i	$-0.358195\pi$
$991$	27.1297	0.861805	0.430902	+	0.902399i	$0.358195\pi$
$992$	0	0
$993$	67.1351	2.13047
$994$	0	0
$995$	−30.5537	−0.968617
$996$	0	0
$997$	56.5029	1.78946	0.894732	−	0.446603i	$-0.147366\pi$
$997$	56.5029	1.78946	0.894732	+	0.446603i	$0.147366\pi$
$998$	0	0
$999$	18.6044	0.588616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.e.1.1		4
4.3	odd	2		538.2.a.c.1.4	✓	4
12.11	even	2		4842.2.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.c.1.4	✓	4	4.3	odd	2
4304.2.a.e.1.1		4	1.1	even	1	trivial
4842.2.a.j.1.2		4	12.11	even	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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q-3.04948 q^{3} +1.48793 q^{5} -0.344151 q^{7} +6.29934 q^{9} +O(q^{10})\)	\(q-3.04948 q^{3} +1.48793 q^{5} -0.344151 q^{7} +6.29934 q^{9} +0.905704 q^{11} -2.24985 q^{13} -4.53741 q^{15} +2.58570 q^{17} +0.688301 q^{19} +1.04948 q^{21} -1.46259 q^{23} -2.78607 q^{25} -10.0613 q^{27} -4.24274 q^{29} -1.92985 q^{31} -2.76193 q^{33} -0.512072 q^{35} -1.84911 q^{37} +6.86089 q^{39} +1.21740 q^{41} +6.59036 q^{43} +9.37296 q^{45} +6.42244 q^{47} -6.88156 q^{49} -7.88503 q^{51} +2.87689 q^{53} +1.34762 q^{55} -2.09896 q^{57} -11.0118 q^{59} -1.75015 q^{61} -2.16792 q^{63} -3.34762 q^{65} -4.34415 q^{67} +4.46014 q^{69} -5.29586 q^{71} +5.90206 q^{73} +8.49607 q^{75} -0.311699 q^{77} +15.1107 q^{79} +11.7836 q^{81} -15.7701 q^{83} +3.84733 q^{85} +12.9382 q^{87} -1.61934 q^{89} +0.774289 q^{91} +5.88503 q^{93} +1.02414 q^{95} +14.3436 q^{97} +5.70533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9	\( 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} - 7 q^{11} + 4 q^{13} - 8 q^{15} + 4 q^{17} - 2 q^{19} - 5 q^{21} - 16 q^{23} - q^{25} - 6 q^{27} + q^{31} + q^{33} - 3 q^{35} - 2 q^{37} - 3 q^{39} - q^{41} - 9 q^{43} + 8 q^{45} - 13 q^{47} - 15 q^{49} - 3 q^{51} + 28 q^{53} - 13 q^{55} + 10 q^{57} - 19 q^{59} - 20 q^{61} - 12 q^{63} + 5 q^{65} - 15 q^{67} - 5 q^{69} - 15 q^{71} - 7 q^{73} - 12 q^{75} - 6 q^{77} + 17 q^{79} + 4 q^{81} - 6 q^{83} - 29 q^{85} + 34 q^{87} + 20 q^{89} + 18 q^{91} - 5 q^{93} + 6 q^{95} + 3 q^{97} + 16 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9 - 7 * q^11 + 4 * q^13 - 8 * q^15 + 4 * q^17 - 2 * q^19 - 5 * q^21 - 16 * q^23 - q^25 - 6 * q^27 + q^31 + q^33 - 3 * q^35 - 2 * q^37 - 3 * q^39 - q^41 - 9 * q^43 + 8 * q^45 - 13 * q^47 - 15 * q^49 - 3 * q^51 + 28 * q^53 - 13 * q^55 + 10 * q^57 - 19 * q^59 - 20 * q^61 - 12 * q^63 + 5 * q^65 - 15 * q^67 - 5 * q^69 - 15 * q^71 - 7 * q^73 - 12 * q^75 - 6 * q^77 + 17 * q^79 + 4 * q^81 - 6 * q^83 - 29 * q^85 + 34 * q^87 + 20 * q^89 + 18 * q^91 - 5 * q^93 + 6 * q^95 + 3 * q^97 + 16 * q^99

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