LMFDB - Embedded newform 4600.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 $ x^7 - 3x^6 - 7x^5 + 24x^4 + x^3 - 35x^2 + 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.49931$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} +O(q^{10})$	$q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} -4.10909 q^{11} -0.0122971 q^{13} +0.155378 q^{17} +4.32471 q^{19} -7.31743 q^{21} -1.00000 q^{23} -0.616262 q^{27} -6.79978 q^{29} +2.20713 q^{31} +10.2699 q^{33} -4.60741 q^{37} +0.0307342 q^{39} -7.67826 q^{41} +8.38997 q^{43} +6.38116 q^{47} +1.57186 q^{49} -0.388339 q^{51} -7.80358 q^{53} -10.8088 q^{57} -14.1275 q^{59} +7.05297 q^{61} +9.50523 q^{63} +7.31363 q^{67} +2.49931 q^{69} -5.84061 q^{71} +0.727674 q^{73} -12.0305 q^{77} +4.81003 q^{79} -8.19948 q^{81} +7.75797 q^{83} +16.9948 q^{87} +6.77560 q^{89} -0.0360030 q^{91} -5.51631 q^{93} +14.8705 q^{97} -13.3405 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} - 7 q^{11} - 7 q^{13} - 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} - 10 q^{31} - 19 q^{33} - 19 q^{37} - 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} - 7 q^{51} - 15 q^{53} - 8 q^{57} - 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} - 14 q^{71} - 10 q^{73} - 6 q^{77} - 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} - 7 q^{91} - 26 q^{93} + 7 q^{97} - 61 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 - 7 * q^11 - 7 * q^13 - 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 - 10 * q^31 - 19 * q^33 - 19 * q^37 - 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 - 7 * q^51 - 15 * q^53 - 8 * q^57 - 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 - 14 * q^71 - 10 * q^73 - 6 * q^77 - 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 - 7 * q^91 - 26 * q^93 + 7 * q^97 - 61 * 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
$3$	−2.49931	−1.44298	−0.721490	−	0.692425i	$-0.756542\pi$
$3$	−2.49931	−1.44298	−0.721490	+	0.692425i	$0.756542\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.92777	1.10659	0.553297	−	0.832984i	$-0.313369\pi$
$7$	2.92777	1.10659	0.553297	+	0.832984i	$0.313369\pi$
$8$	0	0
$9$	3.24657	1.08219
$10$	0	0
$11$	−4.10909	−1.23894	−0.619468	−	0.785022i	$-0.712652\pi$
$11$	−4.10909	−1.23894	−0.619468	+	0.785022i	$0.712652\pi$
$12$	0	0
$13$	−0.0122971	−0.00341059	−0.00170530	−	0.999999i	$-0.500543\pi$
$13$	−0.0122971	−0.00341059	−0.00170530	+	0.999999i	$0.500543\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.155378	0.0376847	0.0188424	−	0.999822i	$-0.494002\pi$
$17$	0.155378	0.0376847	0.0188424	+	0.999822i	$0.494002\pi$
$18$	0	0
$19$	4.32471	0.992158	0.496079	−	0.868277i	$-0.334773\pi$
$19$	4.32471	0.992158	0.496079	+	0.868277i	$0.334773\pi$
$20$	0	0
$21$	−7.31743	−1.59679
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.616262	−0.118600
$28$	0	0
$29$	−6.79978	−1.26269	−0.631344	−	0.775503i	$-0.717496\pi$
$29$	−6.79978	−1.26269	−0.631344	+	0.775503i	$0.717496\pi$
$30$	0	0
$31$	2.20713	0.396412	0.198206	−	0.980160i	$-0.436489\pi$
$31$	2.20713	0.396412	0.198206	+	0.980160i	$0.436489\pi$
$32$	0	0
$33$	10.2699	1.78776
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.60741	−0.757453	−0.378726	−	0.925509i	$-0.623638\pi$
$37$	−4.60741	−0.757453	−0.378726	+	0.925509i	$0.623638\pi$
$38$	0	0
$39$	0.0307342	0.00492142
$40$	0	0
$41$	−7.67826	−1.19914	−0.599571	−	0.800321i	$-0.704662\pi$
$41$	−7.67826	−1.19914	−0.599571	+	0.800321i	$0.704662\pi$
$42$	0	0
$43$	8.38997	1.27946	0.639729	−	0.768600i	$-0.279046\pi$
$43$	8.38997	1.27946	0.639729	+	0.768600i	$0.279046\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.38116	0.930787	0.465394	−	0.885104i	$-0.345913\pi$
$47$	6.38116	0.930787	0.465394	+	0.885104i	$0.345913\pi$
$48$	0	0
$49$	1.57186	0.224551
$50$	0	0
$51$	−0.388339	−0.0543783
$52$	0	0
$53$	−7.80358	−1.07190	−0.535952	−	0.844248i	$-0.680047\pi$
$53$	−7.80358	−1.07190	−0.535952	+	0.844248i	$0.680047\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−10.8088	−1.43166
$58$	0	0
$59$	−14.1275	−1.83924	−0.919622	−	0.392805i	$-0.871505\pi$
$59$	−14.1275	−1.83924	−0.919622	+	0.392805i	$0.871505\pi$
$60$	0	0
$61$	7.05297	0.903040	0.451520	−	0.892261i	$-0.350882\pi$
$61$	7.05297	0.903040	0.451520	+	0.892261i	$0.350882\pi$
$62$	0	0
$63$	9.50523	1.19755
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.31363	0.893502	0.446751	−	0.894658i	$-0.352581\pi$
$67$	7.31363	0.893502	0.446751	+	0.894658i	$0.352581\pi$
$68$	0	0
$69$	2.49931	0.300882
$70$	0	0
$71$	−5.84061	−0.693153	−0.346576	−	0.938022i	$-0.612656\pi$
$71$	−5.84061	−0.693153	−0.346576	+	0.938022i	$0.612656\pi$
$72$	0	0
$73$	0.727674	0.0851678	0.0425839	−	0.999093i	$-0.486441\pi$
$73$	0.727674	0.0851678	0.0425839	+	0.999093i	$0.486441\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.0305	−1.37100
$78$	0	0
$79$	4.81003	0.541171	0.270586	−	0.962696i	$-0.412783\pi$
$79$	4.81003	0.541171	0.270586	+	0.962696i	$0.412783\pi$
$80$	0	0
$81$	−8.19948	−0.911054
$82$	0	0
$83$	7.75797	0.851548	0.425774	−	0.904830i	$-0.360002\pi$
$83$	7.75797	0.851548	0.425774	+	0.904830i	$0.360002\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.9948	1.82203
$88$	0	0
$89$	6.77560	0.718212	0.359106	−	0.933297i	$-0.383082\pi$
$89$	6.77560	0.718212	0.359106	+	0.933297i	$0.383082\pi$
$90$	0	0
$91$	−0.0360030	−0.00377414
$92$	0	0
$93$	−5.51631	−0.572014
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.8705	1.50987	0.754935	−	0.655799i	$-0.227668\pi$
$97$	14.8705	1.50987	0.754935	+	0.655799i	$0.227668\pi$
$98$	0	0
$99$	−13.3405	−1.34077
$100$	0	0
$101$	−13.0023	−1.29378	−0.646889	−	0.762584i	$-0.723930\pi$
$101$	−13.0023	−1.29378	−0.646889	+	0.762584i	$0.723930\pi$
$102$	0	0
$103$	−1.85814	−0.183088	−0.0915442	−	0.995801i	$-0.529180\pi$
$103$	−1.85814	−0.183088	−0.0915442	+	0.995801i	$0.529180\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.471267	0.0455591	0.0227795	−	0.999741i	$-0.492748\pi$
$107$	0.471267	0.0455591	0.0227795	+	0.999741i	$0.492748\pi$
$108$	0	0
$109$	−1.83022	−0.175303	−0.0876515	−	0.996151i	$-0.527936\pi$
$109$	−1.83022	−0.175303	−0.0876515	+	0.996151i	$0.527936\pi$
$110$	0	0
$111$	11.5154	1.09299
$112$	0	0
$113$	7.81151	0.734845	0.367422	−	0.930054i	$-0.380240\pi$
$113$	7.81151	0.734845	0.367422	+	0.930054i	$0.380240\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.0399233	−0.00369091
$118$	0	0
$119$	0.454912	0.0417017
$120$	0	0
$121$	5.88461	0.534964
$122$	0	0
$123$	19.1904	1.73034
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.62237	−0.676376	−0.338188	−	0.941079i	$-0.609814\pi$
$127$	−7.62237	−0.676376	−0.338188	+	0.941079i	$0.609814\pi$
$128$	0	0
$129$	−20.9692	−1.84623
$130$	0	0
$131$	19.4575	1.70001	0.850007	−	0.526772i	$-0.176598\pi$
$131$	19.4575	1.70001	0.850007	+	0.526772i	$0.176598\pi$
$132$	0	0
$133$	12.6618	1.09792
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.3163	1.39399	0.696997	−	0.717074i	$-0.254519\pi$
$137$	16.3163	1.39399	0.696997	+	0.717074i	$0.254519\pi$
$138$	0	0
$139$	−19.9289	−1.69035	−0.845175	−	0.534490i	$-0.820504\pi$
$139$	−19.9289	−1.69035	−0.845175	+	0.534490i	$0.820504\pi$
$140$	0	0
$141$	−15.9485	−1.34311
$142$	0	0
$143$	0.0505297	0.00422551
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.92857	−0.324023
$148$	0	0
$149$	−8.04540	−0.659105	−0.329552	−	0.944137i	$-0.606898\pi$
$149$	−8.04540	−0.659105	−0.329552	+	0.944137i	$0.606898\pi$
$150$	0	0
$151$	3.40963	0.277472	0.138736	−	0.990329i	$-0.455696\pi$
$151$	3.40963	0.277472	0.138736	+	0.990329i	$0.455696\pi$
$152$	0	0
$153$	0.504446	0.0407821
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−21.1868	−1.69089	−0.845446	−	0.534061i	$-0.820665\pi$
$157$	−21.1868	−1.69089	−0.845446	+	0.534061i	$0.820665\pi$
$158$	0	0
$159$	19.5036	1.54674
$160$	0	0
$161$	−2.92777	−0.230741
$162$	0	0
$163$	−20.9487	−1.64083	−0.820415	−	0.571769i	$-0.806257\pi$
$163$	−20.9487	−1.64083	−0.820415	+	0.571769i	$0.806257\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.9861	0.850131	0.425065	−	0.905163i	$-0.360251\pi$
$167$	10.9861	0.850131	0.425065	+	0.905163i	$0.360251\pi$
$168$	0	0
$169$	−12.9998	−0.999988
$170$	0	0
$171$	14.0405	1.07370
$172$	0	0
$173$	−4.31110	−0.327767	−0.163884	−	0.986480i	$-0.552402\pi$
$173$	−4.31110	−0.327767	−0.163884	+	0.986480i	$0.552402\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	35.3091	2.65399
$178$	0	0
$179$	−19.0791	−1.42604	−0.713018	−	0.701146i	$-0.752672\pi$
$179$	−19.0791	−1.42604	−0.713018	+	0.701146i	$0.752672\pi$
$180$	0	0
$181$	4.09312	0.304239	0.152120	−	0.988362i	$-0.451390\pi$
$181$	4.09312	0.304239	0.152120	+	0.988362i	$0.451390\pi$
$182$	0	0
$183$	−17.6276	−1.30307
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.638462	−0.0466890
$188$	0	0
$189$	−1.80428	−0.131242
$190$	0	0
$191$	6.21032	0.449363	0.224682	−	0.974432i	$-0.427866\pi$
$191$	6.21032	0.449363	0.224682	+	0.974432i	$0.427866\pi$
$192$	0	0
$193$	−13.2674	−0.955005	−0.477503	−	0.878630i	$-0.658458\pi$
$193$	−13.2674	−0.955005	−0.477503	+	0.878630i	$0.658458\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.8836	−0.989163	−0.494582	−	0.869131i	$-0.664679\pi$
$197$	−13.8836	−0.989163	−0.494582	+	0.869131i	$0.664679\pi$
$198$	0	0
$199$	−22.8004	−1.61628	−0.808139	−	0.588992i	$-0.799525\pi$
$199$	−22.8004	−1.61628	−0.808139	+	0.588992i	$0.799525\pi$
$200$	0	0
$201$	−18.2791	−1.28931
$202$	0	0
$203$	−19.9082	−1.39728
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.24657	−0.225652
$208$	0	0
$209$	−17.7706	−1.22922
$210$	0	0
$211$	−0.698944	−0.0481173	−0.0240587	−	0.999711i	$-0.507659\pi$
$211$	−0.698944	−0.0481173	−0.0240587	+	0.999711i	$0.507659\pi$
$212$	0	0
$213$	14.5975	1.00021
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.46197	0.438667
$218$	0	0
$219$	−1.81869	−0.122895
$220$	0	0
$221$	−0.00191069	−0.000128527 0
$222$	0	0
$223$	27.3719	1.83296	0.916480	−	0.400082i	$-0.131018\pi$
$223$	27.3719	1.83296	0.916480	+	0.400082i	$0.131018\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.6070	−0.704011	−0.352005	−	0.935998i	$-0.614500\pi$
$227$	−10.6070	−0.704011	−0.352005	+	0.935998i	$0.614500\pi$
$228$	0	0
$229$	3.11433	0.205801	0.102900	−	0.994692i	$-0.467188\pi$
$229$	3.11433	0.205801	0.102900	+	0.994692i	$0.467188\pi$
$230$	0	0
$231$	30.0680	1.97833
$232$	0	0
$233$	−23.5295	−1.54147	−0.770735	−	0.637156i	$-0.780111\pi$
$233$	−23.5295	−1.54147	−0.770735	+	0.637156i	$0.780111\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.0218	−0.780899
$238$	0	0
$239$	24.4214	1.57969	0.789844	−	0.613308i	$-0.210162\pi$
$239$	24.4214	1.57969	0.789844	+	0.613308i	$0.210162\pi$
$240$	0	0
$241$	27.5525	1.77481	0.887407	−	0.460986i	$-0.152504\pi$
$241$	27.5525	1.77481	0.887407	+	0.460986i	$0.152504\pi$
$242$	0	0
$243$	22.3419	1.43323
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.0531813	−0.00338385
$248$	0	0
$249$	−19.3896	−1.22877
$250$	0	0
$251$	−25.3692	−1.60129	−0.800645	−	0.599139i	$-0.795510\pi$
$251$	−25.3692	−1.60129	−0.800645	+	0.599139i	$0.795510\pi$
$252$	0	0
$253$	4.10909	0.258336
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.8501	−1.23822	−0.619108	−	0.785306i	$-0.712506\pi$
$257$	−19.8501	−1.23822	−0.619108	+	0.785306i	$0.712506\pi$
$258$	0	0
$259$	−13.4894	−0.838193
$260$	0	0
$261$	−22.0760	−1.36647
$262$	0	0
$263$	−22.9087	−1.41261	−0.706306	−	0.707906i	$-0.749640\pi$
$263$	−22.9087	−1.41261	−0.706306	+	0.707906i	$0.749640\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−16.9344	−1.03637
$268$	0	0
$269$	1.03048	0.0628295	0.0314148	−	0.999506i	$-0.489999\pi$
$269$	1.03048	0.0628295	0.0314148	+	0.999506i	$0.489999\pi$
$270$	0	0
$271$	−3.79576	−0.230576	−0.115288	−	0.993332i	$-0.536779\pi$
$271$	−3.79576	−0.230576	−0.115288	+	0.993332i	$0.536779\pi$
$272$	0	0
$273$	0.0899829	0.00544601
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.48782	0.0893946	0.0446973	−	0.999001i	$-0.485768\pi$
$277$	1.48782	0.0893946	0.0446973	+	0.999001i	$0.485768\pi$
$278$	0	0
$279$	7.16560	0.428993
$280$	0	0
$281$	−10.2796	−0.613228	−0.306614	−	0.951834i	$-0.599196\pi$
$281$	−10.2796	−0.613228	−0.306614	+	0.951834i	$0.599196\pi$
$282$	0	0
$283$	4.62427	0.274884	0.137442	−	0.990510i	$-0.456112\pi$
$283$	4.62427	0.274884	0.137442	+	0.990510i	$0.456112\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−22.4802	−1.32697
$288$	0	0
$289$	−16.9759	−0.998580
$290$	0	0
$291$	−37.1661	−2.17871
$292$	0	0
$293$	−16.3067	−0.952650	−0.476325	−	0.879269i	$-0.658031\pi$
$293$	−16.3067	−0.952650	−0.476325	+	0.879269i	$0.658031\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.53228	0.146938
$298$	0	0
$299$	0.0122971	0.000711158 0
$300$	0	0
$301$	24.5639	1.41584
$302$	0	0
$303$	32.4969	1.86690
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.28659	−0.530014	−0.265007	−	0.964246i	$-0.585374\pi$
$307$	−9.28659	−0.530014	−0.265007	+	0.964246i	$0.585374\pi$
$308$	0	0
$309$	4.64409	0.264193
$310$	0	0
$311$	−24.3713	−1.38197	−0.690984	−	0.722870i	$-0.742822\pi$
$311$	−24.3713	−1.38197	−0.690984	+	0.722870i	$0.742822\pi$
$312$	0	0
$313$	−15.0048	−0.848119	−0.424060	−	0.905634i	$-0.639395\pi$
$313$	−15.0048	−0.848119	−0.424060	+	0.905634i	$0.639395\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.84747	0.440758	0.220379	−	0.975414i	$-0.429271\pi$
$317$	7.84747	0.440758	0.220379	+	0.975414i	$0.429271\pi$
$318$	0	0
$319$	27.9409	1.56439
$320$	0	0
$321$	−1.17784	−0.0657408
$322$	0	0
$323$	0.671966	0.0373892
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.57429	0.252959
$328$	0	0
$329$	18.6826	1.03000
$330$	0	0
$331$	9.28156	0.510160	0.255080	−	0.966920i	$-0.417898\pi$
$331$	9.28156	0.510160	0.255080	+	0.966920i	$0.417898\pi$
$332$	0	0
$333$	−14.9583	−0.819709
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.3760	−1.43679	−0.718397	−	0.695634i	$-0.755124\pi$
$337$	−26.3760	−1.43679	−0.718397	+	0.695634i	$0.755124\pi$
$338$	0	0
$339$	−19.5234	−1.06037
$340$	0	0
$341$	−9.06928	−0.491129
$342$	0	0
$343$	−15.8924	−0.858107
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.3316	−0.984093	−0.492047	−	0.870569i	$-0.663751\pi$
$347$	−18.3316	−0.984093	−0.492047	+	0.870569i	$0.663751\pi$
$348$	0	0
$349$	−32.2759	−1.72769	−0.863845	−	0.503758i	$-0.831950\pi$
$349$	−32.2759	−1.72769	−0.863845	+	0.503758i	$0.831950\pi$
$350$	0	0
$351$	0.00757822	0.000404495 0
$352$	0	0
$353$	−29.5653	−1.57360	−0.786800	−	0.617208i	$-0.788264\pi$
$353$	−29.5653	−1.57360	−0.786800	+	0.617208i	$0.788264\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.13697	−0.0601747
$358$	0	0
$359$	33.4169	1.76368	0.881838	−	0.471552i	$-0.156306\pi$
$359$	33.4169	1.76368	0.881838	+	0.471552i	$0.156306\pi$
$360$	0	0
$361$	−0.296842	−0.0156232
$362$	0	0
$363$	−14.7075	−0.771943
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.4764	−1.32986	−0.664928	−	0.746908i	$-0.731538\pi$
$367$	−25.4764	−1.32986	−0.664928	+	0.746908i	$0.731538\pi$
$368$	0	0
$369$	−24.9280	−1.29770
$370$	0	0
$371$	−22.8471	−1.18616
$372$	0	0
$373$	−3.28172	−0.169921	−0.0849605	−	0.996384i	$-0.527076\pi$
$373$	−3.28172	−0.169921	−0.0849605	+	0.996384i	$0.527076\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.0836174	0.00430651
$378$	0	0
$379$	5.66762	0.291126	0.145563	−	0.989349i	$-0.453501\pi$
$379$	5.66762	0.291126	0.145563	+	0.989349i	$0.453501\pi$
$380$	0	0
$381$	19.0507	0.975997
$382$	0	0
$383$	34.4343	1.75951	0.879755	−	0.475428i	$-0.157707\pi$
$383$	34.4343	1.75951	0.879755	+	0.475428i	$0.157707\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	27.2387	1.38462
$388$	0	0
$389$	−14.3737	−0.728773	−0.364387	−	0.931248i	$-0.618721\pi$
$389$	−14.3737	−0.728773	−0.364387	+	0.931248i	$0.618721\pi$
$390$	0	0
$391$	−0.155378	−0.00785781
$392$	0	0
$393$	−48.6305	−2.45309
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−14.6149	−0.733500	−0.366750	−	0.930319i	$-0.619530\pi$
$397$	−14.6149	−0.733500	−0.366750	+	0.930319i	$0.619530\pi$
$398$	0	0
$399$	−31.6458	−1.58427
$400$	0	0
$401$	−3.65553	−0.182548	−0.0912742	−	0.995826i	$-0.529094\pi$
$401$	−3.65553	−0.182548	−0.0912742	+	0.995826i	$0.529094\pi$
$402$	0	0
$403$	−0.0271412	−0.00135200
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.9322	0.938436
$408$	0	0
$409$	4.40926	0.218024	0.109012	−	0.994040i	$-0.465231\pi$
$409$	4.40926	0.218024	0.109012	+	0.994040i	$0.465231\pi$
$410$	0	0
$411$	−40.7795	−2.01150
$412$	0	0
$413$	−41.3621	−2.03530
$414$	0	0
$415$	0	0
$416$	0	0
$417$	49.8087	2.43914
$418$	0	0
$419$	32.5906	1.59216	0.796078	−	0.605195i	$-0.206905\pi$
$419$	32.5906	1.59216	0.796078	+	0.605195i	$0.206905\pi$
$420$	0	0
$421$	4.23954	0.206622	0.103311	−	0.994649i	$-0.467056\pi$
$421$	4.23954	0.206622	0.103311	+	0.994649i	$0.467056\pi$
$422$	0	0
$423$	20.7169	1.00729
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.6495	0.999299
$428$	0	0
$429$	−0.126290	−0.00609732
$430$	0	0
$431$	1.81123	0.0872436	0.0436218	−	0.999048i	$-0.486110\pi$
$431$	1.81123	0.0872436	0.0436218	+	0.999048i	$0.486110\pi$
$432$	0	0
$433$	−1.76263	−0.0847068	−0.0423534	−	0.999103i	$-0.513486\pi$
$433$	−1.76263	−0.0847068	−0.0423534	+	0.999103i	$0.513486\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.32471	−0.206879
$438$	0	0
$439$	−26.7848	−1.27837	−0.639184	−	0.769054i	$-0.720728\pi$
$439$	−26.7848	−1.27837	−0.639184	+	0.769054i	$0.720728\pi$
$440$	0	0
$441$	5.10316	0.243008
$442$	0	0
$443$	16.6631	0.791686	0.395843	−	0.918318i	$-0.370452\pi$
$443$	16.6631	0.791686	0.395843	+	0.918318i	$0.370452\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.1080	0.951075
$448$	0	0
$449$	−34.1236	−1.61039	−0.805196	−	0.593008i	$-0.797940\pi$
$449$	−34.1236	−1.61039	−0.805196	+	0.593008i	$0.797940\pi$
$450$	0	0
$451$	31.5507	1.48566
$452$	0	0
$453$	−8.52174	−0.400386
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.5980	−1.15065	−0.575324	−	0.817926i	$-0.695124\pi$
$457$	−24.5980	−1.15065	−0.575324	+	0.817926i	$0.695124\pi$
$458$	0	0
$459$	−0.0957537	−0.00446940
$460$	0	0
$461$	−10.7297	−0.499733	−0.249866	−	0.968280i	$-0.580387\pi$
$461$	−10.7297	−0.499733	−0.249866	+	0.968280i	$0.580387\pi$
$462$	0	0
$463$	−22.3124	−1.03695	−0.518473	−	0.855094i	$-0.673499\pi$
$463$	−22.3124	−1.03695	−0.518473	+	0.855094i	$0.673499\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.0218	−0.926497	−0.463248	−	0.886229i	$-0.653316\pi$
$467$	−20.0218	−0.926497	−0.463248	+	0.886229i	$0.653316\pi$
$468$	0	0
$469$	21.4127	0.988744
$470$	0	0
$471$	52.9525	2.43992
$472$	0	0
$473$	−34.4751	−1.58517
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.3349	−1.16001
$478$	0	0
$479$	−7.61905	−0.348123	−0.174062	−	0.984735i	$-0.555689\pi$
$479$	−7.61905	−0.348123	−0.174062	+	0.984735i	$0.555689\pi$
$480$	0	0
$481$	0.0566576	0.00258336
$482$	0	0
$483$	7.31743	0.332954
$484$	0	0
$485$	0	0
$486$	0	0
$487$	21.5163	0.974997	0.487499	−	0.873124i	$-0.337909\pi$
$487$	21.5163	0.974997	0.487499	+	0.873124i	$0.337909\pi$
$488$	0	0
$489$	52.3574	2.36768
$490$	0	0
$491$	−10.3705	−0.468015	−0.234007	−	0.972235i	$-0.575184\pi$
$491$	−10.3705	−0.468015	−0.234007	+	0.972235i	$0.575184\pi$
$492$	0	0
$493$	−1.05654	−0.0475841
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.1000	−0.767039
$498$	0	0
$499$	−16.2850	−0.729018	−0.364509	−	0.931200i	$-0.618763\pi$
$499$	−16.2850	−0.729018	−0.364509	+	0.931200i	$0.618763\pi$
$500$	0	0
$501$	−27.4577	−1.22672
$502$	0	0
$503$	16.9326	0.754986	0.377493	−	0.926012i	$-0.376786\pi$
$503$	16.9326	0.754986	0.377493	+	0.926012i	$0.376786\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.4907	1.44296
$508$	0	0
$509$	39.0075	1.72898	0.864488	−	0.502654i	$-0.167643\pi$
$509$	39.0075	1.72898	0.864488	+	0.502654i	$0.167643\pi$
$510$	0	0
$511$	2.13046	0.0942462
$512$	0	0
$513$	−2.66516	−0.117670
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−26.2207	−1.15319
$518$	0	0
$519$	10.7748	0.472961
$520$	0	0
$521$	33.1380	1.45180	0.725902	−	0.687798i	$-0.241423\pi$
$521$	33.1380	1.45180	0.725902	+	0.687798i	$0.241423\pi$
$522$	0	0
$523$	12.4056	0.542460	0.271230	−	0.962515i	$-0.412570\pi$
$523$	12.4056	0.542460	0.271230	+	0.962515i	$0.412570\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.342939	0.0149387
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−45.8660	−1.99041
$532$	0	0
$533$	0.0944201	0.00408979
$534$	0	0
$535$	0	0
$536$	0	0
$537$	47.6846	2.05774
$538$	0	0
$539$	−6.45891	−0.278205
$540$	0	0
$541$	−8.53472	−0.366936	−0.183468	−	0.983026i	$-0.558732\pi$
$541$	−8.53472	−0.366936	−0.183468	+	0.983026i	$0.558732\pi$
$542$	0	0
$543$	−10.2300	−0.439011
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.45828	0.276136	0.138068	−	0.990423i	$-0.455911\pi$
$547$	6.45828	0.276136	0.138068	+	0.990423i	$0.455911\pi$
$548$	0	0
$549$	22.8980	0.977262
$550$	0	0
$551$	−29.4071	−1.25279
$552$	0	0
$553$	14.0827	0.598857
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.08413	−0.173050	−0.0865251	−	0.996250i	$-0.527576\pi$
$557$	−4.08413	−0.173050	−0.0865251	+	0.996250i	$0.527576\pi$
$558$	0	0
$559$	−0.103172	−0.00436371
$560$	0	0
$561$	1.59572	0.0673713
$562$	0	0
$563$	−15.4494	−0.651114	−0.325557	−	0.945522i	$-0.605552\pi$
$563$	−15.4494	−0.651114	−0.325557	+	0.945522i	$0.605552\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−24.0062	−1.00817
$568$	0	0
$569$	42.2633	1.77177	0.885886	−	0.463903i	$-0.153552\pi$
$569$	42.2633	1.77177	0.885886	+	0.463903i	$0.153552\pi$
$570$	0	0
$571$	−2.74497	−0.114873	−0.0574367	−	0.998349i	$-0.518293\pi$
$571$	−2.74497	−0.114873	−0.0574367	+	0.998349i	$0.518293\pi$
$572$	0	0
$573$	−15.5215	−0.648422
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.93871	0.372123	0.186062	−	0.982538i	$-0.440428\pi$
$577$	8.93871	0.372123	0.186062	+	0.982538i	$0.440428\pi$
$578$	0	0
$579$	33.1593	1.37805
$580$	0	0
$581$	22.7136	0.942318
$582$	0	0
$583$	32.0656	1.32802
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.90821	0.243858	0.121929	−	0.992539i	$-0.461092\pi$
$587$	5.90821	0.243858	0.121929	+	0.992539i	$0.461092\pi$
$588$	0	0
$589$	9.54520	0.393303
$590$	0	0
$591$	34.6994	1.42734
$592$	0	0
$593$	27.8823	1.14499	0.572494	−	0.819909i	$-0.305976\pi$
$593$	27.8823	1.14499	0.572494	+	0.819909i	$0.305976\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	56.9854	2.33226
$598$	0	0
$599$	10.5644	0.431648	0.215824	−	0.976432i	$-0.430756\pi$
$599$	10.5644	0.431648	0.215824	+	0.976432i	$0.430756\pi$
$600$	0	0
$601$	35.8910	1.46403	0.732013	−	0.681291i	$-0.238581\pi$
$601$	35.8910	1.46403	0.732013	+	0.681291i	$0.238581\pi$
$602$	0	0
$603$	23.7442	0.966940
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.0640	−1.17967	−0.589835	−	0.807524i	$-0.700807\pi$
$607$	−29.0640	−1.17967	−0.589835	+	0.807524i	$0.700807\pi$
$608$	0	0
$609$	49.7569	2.01625
$610$	0	0
$611$	−0.0784695	−0.00317454
$612$	0	0
$613$	24.4282	0.986644	0.493322	−	0.869847i	$-0.335782\pi$
$613$	24.4282	0.986644	0.493322	+	0.869847i	$0.335782\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.5614	1.19010	0.595049	−	0.803689i	$-0.297133\pi$
$617$	29.5614	1.19010	0.595049	+	0.803689i	$0.297133\pi$
$618$	0	0
$619$	−30.5880	−1.22944	−0.614718	−	0.788747i	$-0.710730\pi$
$619$	−30.5880	−1.22944	−0.614718	+	0.788747i	$0.710730\pi$
$620$	0	0
$621$	0.616262	0.0247298
$622$	0	0
$623$	19.8374	0.794769
$624$	0	0
$625$	0	0
$626$	0	0
$627$	44.4144	1.77374
$628$	0	0
$629$	−0.715890	−0.0285444
$630$	0	0
$631$	10.5713	0.420835	0.210418	−	0.977612i	$-0.432518\pi$
$631$	10.5713	0.420835	0.210418	+	0.977612i	$0.432518\pi$
$632$	0	0
$633$	1.74688	0.0694323
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.0193293	−0.000765853 0
$638$	0	0
$639$	−18.9620	−0.750123
$640$	0	0
$641$	−20.4738	−0.808666	−0.404333	−	0.914612i	$-0.632496\pi$
$641$	−20.4738	−0.808666	−0.404333	+	0.914612i	$0.632496\pi$
$642$	0	0
$643$	18.4989	0.729525	0.364763	−	0.931101i	$-0.381150\pi$
$643$	18.4989	0.729525	0.364763	+	0.931101i	$0.381150\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.6328	0.732531	0.366266	−	0.930510i	$-0.380636\pi$
$647$	18.6328	0.732531	0.366266	+	0.930510i	$0.380636\pi$
$648$	0	0
$649$	58.0511	2.27871
$650$	0	0
$651$	−16.1505	−0.632988
$652$	0	0
$653$	30.5290	1.19469	0.597347	−	0.801983i	$-0.296222\pi$
$653$	30.5290	1.19469	0.597347	+	0.801983i	$0.296222\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.36245	0.0921678
$658$	0	0
$659$	−26.6231	−1.03709	−0.518545	−	0.855051i	$-0.673526\pi$
$659$	−26.6231	−1.03709	−0.518545	+	0.855051i	$0.673526\pi$
$660$	0	0
$661$	−13.0529	−0.507700	−0.253850	−	0.967244i	$-0.581697\pi$
$661$	−13.0529	−0.507700	−0.253850	+	0.967244i	$0.581697\pi$
$662$	0	0
$663$	0.00477543	0.000185462 0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.79978	0.263289
$668$	0	0
$669$	−68.4110	−2.64492
$670$	0	0
$671$	−28.9813	−1.11881
$672$	0	0
$673$	28.9120	1.11448	0.557238	−	0.830353i	$-0.311861\pi$
$673$	28.9120	1.11448	0.557238	+	0.830353i	$0.311861\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.4824	−0.479738	−0.239869	−	0.970805i	$-0.577104\pi$
$677$	−12.4824	−0.479738	−0.239869	+	0.970805i	$0.577104\pi$
$678$	0	0
$679$	43.5375	1.67081
$680$	0	0
$681$	26.5102	1.01587
$682$	0	0
$683$	40.0241	1.53148	0.765740	−	0.643150i	$-0.222373\pi$
$683$	40.0241	1.53148	0.765740	+	0.643150i	$0.222373\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7.78369	−0.296966
$688$	0	0
$689$	0.0959612	0.00365583
$690$	0	0
$691$	−38.9938	−1.48340	−0.741698	−	0.670734i	$-0.765979\pi$
$691$	−38.9938	−1.48340	−0.741698	+	0.670734i	$0.765979\pi$
$692$	0	0
$693$	−39.0578	−1.48368
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.19303	−0.0451894
$698$	0	0
$699$	58.8077	2.22431
$700$	0	0
$701$	21.9042	0.827309	0.413655	−	0.910434i	$-0.364252\pi$
$701$	21.9042	0.827309	0.413655	+	0.910434i	$0.364252\pi$
$702$	0	0
$703$	−19.9257	−0.751513
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−38.0678	−1.43169
$708$	0	0
$709$	32.1360	1.20689	0.603447	−	0.797403i	$-0.293793\pi$
$709$	32.1360	1.20689	0.603447	+	0.797403i	$0.293793\pi$
$710$	0	0
$711$	15.6161	0.585650
$712$	0	0
$713$	−2.20713	−0.0826576
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−61.0367	−2.27946
$718$	0	0
$719$	−24.4018	−0.910033	−0.455017	−	0.890483i	$-0.650367\pi$
$719$	−24.4018	−0.910033	−0.455017	+	0.890483i	$0.650367\pi$
$720$	0	0
$721$	−5.44023	−0.202605
$722$	0	0
$723$	−68.8625	−2.56102
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.8573	−1.29279	−0.646393	−	0.763005i	$-0.723723\pi$
$727$	−34.8573	−1.29279	−0.646393	+	0.763005i	$0.723723\pi$
$728$	0	0
$729$	−31.2409	−1.15707
$730$	0	0
$731$	1.30362	0.0482161
$732$	0	0
$733$	30.7343	1.13520	0.567598	−	0.823306i	$-0.307873\pi$
$733$	30.7343	1.13520	0.567598	+	0.823306i	$0.307873\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30.0523	−1.10699
$738$	0	0
$739$	−38.7573	−1.42571	−0.712855	−	0.701312i	$-0.752598\pi$
$739$	−38.7573	−1.42571	−0.712855	+	0.701312i	$0.752598\pi$
$740$	0	0
$741$	0.132917	0.00488282
$742$	0	0
$743$	43.0371	1.57888	0.789438	−	0.613830i	$-0.210372\pi$
$743$	43.0371	1.57888	0.789438	+	0.613830i	$0.210372\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	25.1868	0.921538
$748$	0	0
$749$	1.37976	0.0504154
$750$	0	0
$751$	−35.3761	−1.29089	−0.645447	−	0.763805i	$-0.723329\pi$
$751$	−35.3761	−1.29089	−0.645447	+	0.763805i	$0.723329\pi$
$752$	0	0
$753$	63.4056	2.31063
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−51.2605	−1.86309	−0.931547	−	0.363621i	$-0.881540\pi$
$757$	−51.2605	−1.86309	−0.931547	+	0.363621i	$0.881540\pi$
$758$	0	0
$759$	−10.2699	−0.372774
$760$	0	0
$761$	−13.4331	−0.486950	−0.243475	−	0.969907i	$-0.578287\pi$
$761$	−13.4331	−0.486950	−0.243475	+	0.969907i	$0.578287\pi$
$762$	0	0
$763$	−5.35846	−0.193989
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.173727	0.00627291
$768$	0	0
$769$	−24.7779	−0.893512	−0.446756	−	0.894656i	$-0.647421\pi$
$769$	−24.7779	−0.893512	−0.446756	+	0.894656i	$0.647421\pi$
$770$	0	0
$771$	49.6117	1.78672
$772$	0	0
$773$	11.9946	0.431417	0.215708	−	0.976458i	$-0.430794\pi$
$773$	11.9946	0.431417	0.215708	+	0.976458i	$0.430794\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	33.7144	1.20950
$778$	0	0
$779$	−33.2063	−1.18974
$780$	0	0
$781$	23.9996	0.858772
$782$	0	0
$783$	4.19045	0.149755
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10.4110	0.371114	0.185557	−	0.982634i	$-0.440591\pi$
$787$	10.4110	0.371114	0.185557	+	0.982634i	$0.440591\pi$
$788$	0	0
$789$	57.2561	2.03837
$790$	0	0
$791$	22.8703	0.813175
$792$	0	0
$793$	−0.0867308	−0.00307990
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.0848	0.994814	0.497407	−	0.867517i	$-0.334286\pi$
$797$	28.0848	0.994814	0.497407	+	0.867517i	$0.334286\pi$
$798$	0	0
$799$	0.991492	0.0350765
$800$	0	0
$801$	21.9975	0.777243
$802$	0	0
$803$	−2.99008	−0.105517
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.57550	−0.0906618
$808$	0	0
$809$	−11.1238	−0.391092	−0.195546	−	0.980695i	$-0.562648\pi$
$809$	−11.1238	−0.391092	−0.195546	+	0.980695i	$0.562648\pi$
$810$	0	0
$811$	−28.7572	−1.00980	−0.504900	−	0.863178i	$-0.668471\pi$
$811$	−28.7572	−1.00980	−0.504900	+	0.863178i	$0.668471\pi$
$812$	0	0
$813$	9.48679	0.332716
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.2842	1.26942
$818$	0	0
$819$	−0.116886	−0.00408434
$820$	0	0
$821$	37.0808	1.29413	0.647064	−	0.762435i	$-0.275996\pi$
$821$	37.0808	1.29413	0.647064	+	0.762435i	$0.275996\pi$
$822$	0	0
$823$	−10.9501	−0.381698	−0.190849	−	0.981619i	$-0.561124\pi$
$823$	−10.9501	−0.381698	−0.190849	+	0.981619i	$0.561124\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.9973	0.486732	0.243366	−	0.969934i	$-0.421748\pi$
$827$	13.9973	0.486732	0.243366	+	0.969934i	$0.421748\pi$
$828$	0	0
$829$	16.0622	0.557864	0.278932	−	0.960311i	$-0.410020\pi$
$829$	16.0622	0.557864	0.278932	+	0.960311i	$0.410020\pi$
$830$	0	0
$831$	−3.71853	−0.128995
$832$	0	0
$833$	0.244233	0.00846216
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.36017	−0.0470144
$838$	0	0
$839$	−9.49187	−0.327696	−0.163848	−	0.986486i	$-0.552391\pi$
$839$	−9.49187	−0.327696	−0.163848	+	0.986486i	$0.552391\pi$
$840$	0	0
$841$	17.2371	0.594381
$842$	0	0
$843$	25.6919	0.884876
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.2288	0.591988
$848$	0	0
$849$	−11.5575	−0.396653
$850$	0	0
$851$	4.60741	0.157940
$852$	0	0
$853$	−3.45965	−0.118456	−0.0592280	−	0.998244i	$-0.518864\pi$
$853$	−3.45965	−0.118456	−0.0592280	+	0.998244i	$0.518864\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.729973	−0.0249354	−0.0124677	−	0.999922i	$-0.503969\pi$
$857$	−0.729973	−0.0249354	−0.0124677	+	0.999922i	$0.503969\pi$
$858$	0	0
$859$	0.357241	0.0121889	0.00609446	−	0.999981i	$-0.498060\pi$
$859$	0.357241	0.0121889	0.00609446	+	0.999981i	$0.498060\pi$
$860$	0	0
$861$	56.1851	1.91478
$862$	0	0
$863$	−16.8358	−0.573098	−0.286549	−	0.958066i	$-0.592508\pi$
$863$	−16.8358	−0.573098	−0.286549	+	0.958066i	$0.592508\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	42.4280	1.44093
$868$	0	0
$869$	−19.7649	−0.670477
$870$	0	0
$871$	−0.0899362	−0.00304737
$872$	0	0
$873$	48.2782	1.63397
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.06147	0.272217	0.136108	−	0.990694i	$-0.456540\pi$
$877$	8.06147	0.272217	0.136108	+	0.990694i	$0.456540\pi$
$878$	0	0
$879$	40.7557	1.37466
$880$	0	0
$881$	−37.7548	−1.27199	−0.635996	−	0.771693i	$-0.719410\pi$
$881$	−37.7548	−1.27199	−0.635996	+	0.771693i	$0.719410\pi$
$882$	0	0
$883$	35.2103	1.18492	0.592460	−	0.805600i	$-0.298157\pi$
$883$	35.2103	1.18492	0.592460	+	0.805600i	$0.298157\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.4225	−0.719297	−0.359649	−	0.933088i	$-0.617103\pi$
$887$	−21.4225	−0.719297	−0.359649	+	0.933088i	$0.617103\pi$
$888$	0	0
$889$	−22.3166	−0.748474
$890$	0	0
$891$	33.6924	1.12874
$892$	0	0
$893$	27.5967	0.923488
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.0307342	−0.00102619
$898$	0	0
$899$	−15.0080	−0.500545
$900$	0	0
$901$	−1.21251	−0.0403944
$902$	0	0
$903$	−61.3930	−2.04303
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.7517	1.28673	0.643364	−	0.765560i	$-0.277538\pi$
$907$	38.7517	1.28673	0.643364	+	0.765560i	$0.277538\pi$
$908$	0	0
$909$	−42.2129	−1.40012
$910$	0	0
$911$	−47.0069	−1.55741	−0.778704	−	0.627391i	$-0.784123\pi$
$911$	−47.0069	−1.55741	−0.778704	+	0.627391i	$0.784123\pi$
$912$	0	0
$913$	−31.8782	−1.05501
$914$	0	0
$915$	0	0
$916$	0	0
$917$	56.9673	1.88123
$918$	0	0
$919$	12.2360	0.403629	0.201814	−	0.979424i	$-0.435316\pi$
$919$	12.2360	0.403629	0.201814	+	0.979424i	$0.435316\pi$
$920$	0	0
$921$	23.2101	0.764799
$922$	0	0
$923$	0.0718223	0.00236406
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6.03260	−0.198137
$928$	0	0
$929$	−13.0143	−0.426984	−0.213492	−	0.976945i	$-0.568484\pi$
$929$	−13.0143	−0.426984	−0.213492	+	0.976945i	$0.568484\pi$
$930$	0	0
$931$	6.79785	0.222790
$932$	0	0
$933$	60.9114	1.99415
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.6856	−1.26380	−0.631902	−	0.775048i	$-0.717726\pi$
$937$	−38.6856	−1.26380	−0.631902	+	0.775048i	$0.717726\pi$
$938$	0	0
$939$	37.5016	1.22382
$940$	0	0
$941$	−43.8241	−1.42862	−0.714312	−	0.699828i	$-0.753260\pi$
$941$	−43.8241	−1.42862	−0.714312	+	0.699828i	$0.753260\pi$
$942$	0	0
$943$	7.67826	0.250039
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.3854	−1.40984	−0.704918	−	0.709289i	$-0.749016\pi$
$947$	−43.3854	−1.40984	−0.704918	+	0.709289i	$0.749016\pi$
$948$	0	0
$949$	−0.00894825	−0.000290473 0
$950$	0	0
$951$	−19.6133	−0.636005
$952$	0	0
$953$	2.80245	0.0907802	0.0453901	−	0.998969i	$-0.485547\pi$
$953$	2.80245	0.0907802	0.0453901	+	0.998969i	$0.485547\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−69.8331	−2.25738
$958$	0	0
$959$	47.7704	1.54259
$960$	0	0
$961$	−26.1286	−0.842858
$962$	0	0
$963$	1.53000	0.0493036
$964$	0	0
$965$	0	0
$966$	0	0
$967$	30.0015	0.964783	0.482391	−	0.875956i	$-0.339768\pi$
$967$	30.0015	0.964783	0.482391	+	0.875956i	$0.339768\pi$
$968$	0	0
$969$	−1.67945	−0.0539518
$970$	0	0
$971$	53.6632	1.72213	0.861067	−	0.508492i	$-0.169797\pi$
$971$	53.6632	1.72213	0.861067	+	0.508492i	$0.169797\pi$
$972$	0	0
$973$	−58.3474	−1.87053
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.4590	1.48636	0.743178	−	0.669094i	$-0.233318\pi$
$977$	46.4590	1.48636	0.743178	+	0.669094i	$0.233318\pi$
$978$	0	0
$979$	−27.8415	−0.889819
$980$	0	0
$981$	−5.94193	−0.189711
$982$	0	0
$983$	−50.8346	−1.62137	−0.810686	−	0.585481i	$-0.800906\pi$
$983$	−50.8346	−1.62137	−0.810686	+	0.585481i	$0.800906\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−46.6936	−1.48627
$988$	0	0
$989$	−8.38997	−0.266786
$990$	0	0
$991$	4.30720	0.136823	0.0684114	−	0.997657i	$-0.478207\pi$
$991$	4.30720	0.136823	0.0684114	+	0.997657i	$0.478207\pi$
$992$	0	0
$993$	−23.1975	−0.736151
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−13.2358	−0.419181	−0.209590	−	0.977789i	$-0.567213\pi$
$997$	−13.2358	−0.419181	−0.209590	+	0.977789i	$0.567213\pi$
$998$	0	0
$999$	2.83937	0.0898337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bi.1.1		7
4.3	odd	2		9200.2.a.cz.1.7		7
5.2	odd	4		920.2.e.b.369.13	yes	14
5.3	odd	4		920.2.e.b.369.2	✓	14
5.4	even	2		4600.2.a.bh.1.7		7
20.3	even	4		1840.2.e.g.369.13		14
20.7	even	4		1840.2.e.g.369.2		14
20.19	odd	2		9200.2.a.dc.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.2	✓	14	5.3	odd	4
920.2.e.b.369.13	yes	14	5.2	odd	4
1840.2.e.g.369.2		14	20.7	even	4
1840.2.e.g.369.13		14	20.3	even	4
4600.2.a.bh.1.7		7	5.4	even	2
4600.2.a.bi.1.1		7	1.1	even	1	trivial
9200.2.a.cz.1.7		7	4.3	odd	2
9200.2.a.dc.1.1		7	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} +O(q^{10})\)	\(q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} -4.10909 q^{11} -0.0122971 q^{13} +0.155378 q^{17} +4.32471 q^{19} -7.31743 q^{21} -1.00000 q^{23} -0.616262 q^{27} -6.79978 q^{29} +2.20713 q^{31} +10.2699 q^{33} -4.60741 q^{37} +0.0307342 q^{39} -7.67826 q^{41} +8.38997 q^{43} +6.38116 q^{47} +1.57186 q^{49} -0.388339 q^{51} -7.80358 q^{53} -10.8088 q^{57} -14.1275 q^{59} +7.05297 q^{61} +9.50523 q^{63} +7.31363 q^{67} +2.49931 q^{69} -5.84061 q^{71} +0.727674 q^{73} -12.0305 q^{77} +4.81003 q^{79} -8.19948 q^{81} +7.75797 q^{83} +16.9948 q^{87} +6.77560 q^{89} -0.0360030 q^{91} -5.51631 q^{93} +14.8705 q^{97} -13.3405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} - 7 q^{11} - 7 q^{13} - 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} - 10 q^{31} - 19 q^{33} - 19 q^{37} - 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} - 7 q^{51} - 15 q^{53} - 8 q^{57} - 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} - 14 q^{71} - 10 q^{73} - 6 q^{77} - 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} - 7 q^{91} - 26 q^{93} + 7 q^{97} - 61 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 - 7 * q^11 - 7 * q^13 - 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 - 10 * q^31 - 19 * q^33 - 19 * q^37 - 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 - 7 * q^51 - 15 * q^53 - 8 * q^57 - 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 - 14 * q^71 - 10 * q^73 - 6 * q^77 - 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 - 7 * q^91 - 26 * q^93 + 7 * q^97 - 61 * q^99

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