LMFDB - Embedded newform 4600.2.e.r.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			4049.2
Root			$-2.14510i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.r.4049.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14510i q^{3} +1.14510i q^{7} -1.60147 q^{9} +O(q^{10})$	$q-2.14510i q^{3} +1.14510i q^{7} -1.60147 q^{9} +5.89167 q^{11} -4.89167i q^{13} +5.89167i q^{17} +2.34803 q^{19} +2.45636 q^{21} +1.00000i q^{23} -3.00000i q^{27} -3.74657 q^{29} +5.68874 q^{31} -12.6382i q^{33} -4.00000i q^{37} -10.4931 q^{39} -1.05783 q^{41} +11.4931i q^{43} -7.74657i q^{47} +5.68874 q^{49} +12.6382 q^{51} +12.9863i q^{53} -5.03677i q^{57} -0.797069 q^{59} +13.8917 q^{61} -1.83384i q^{63} +15.5667i q^{67} +2.14510 q^{69} +2.94217 q^{71} +6.32698i q^{73} +6.74657i q^{77} -11.2397 q^{81} -0.912726i q^{83} +8.03677i q^{87} +15.4931 q^{89} +5.60147 q^{91} -12.2029i q^{93} -13.3848i q^{97} -9.43531 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 2.14510i	− 1.23848i	−0.785204	−	0.619238i	$-0.787442\pi$
$3$	− 2.14510i	− 1.23848i	0.785204	−	0.619238i	$-0.212558\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.14510i	0.432808i	0.976304	+	0.216404i	$0.0694329\pi$
$7$	1.14510i	0.432808i	−0.976304	+	0.216404i	$0.930567\pi$
$8$	0	0
$9$	−1.60147	−0.533822
$10$	0	0
$11$	5.89167	1.77641	0.888203	−	0.459452i	$-0.151954\pi$
$11$	5.89167	1.77641	0.888203	+	0.459452i	$0.151954\pi$
$12$	0	0
$13$	− 4.89167i	− 1.35671i	−0.734736	−	0.678353i	$-0.762694\pi$
$13$	− 4.89167i	− 1.35671i	0.734736	−	0.678353i	$-0.237306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.89167i	1.42894i	0.699666	+	0.714470i	$0.253332\pi$
$17$	5.89167i	1.42894i	−0.699666	+	0.714470i	$0.746668\pi$
$18$	0	0
$19$	2.34803	0.538676	0.269338	−	0.963046i	$-0.413195\pi$
$19$	2.34803	0.538676	0.269338	+	0.963046i	$0.413195\pi$
$20$	0	0
$21$	2.45636	0.536022
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.00000i	− 0.577350i
$28$	0	0
$29$	−3.74657	−0.695720	−0.347860	−	0.937546i	$-0.613092\pi$
$29$	−3.74657	−0.695720	−0.347860	+	0.937546i	$0.613092\pi$
$30$	0	0
$31$	5.68874	1.02173	0.510864	−	0.859662i	$-0.329326\pi$
$31$	5.68874	1.02173	0.510864	+	0.859662i	$0.329326\pi$
$32$	0	0
$33$	− 12.6382i	− 2.20004i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	−10.4931	−1.68025
$40$	0	0
$41$	−1.05783	−0.165205	−0.0826025	−	0.996583i	$-0.526323\pi$
$41$	−1.05783	−0.165205	−0.0826025	+	0.996583i	$0.526323\pi$
$42$	0	0
$43$	11.4931i	1.75269i	0.481687	+	0.876343i	$0.340024\pi$
$43$	11.4931i	1.75269i	−0.481687	+	0.876343i	$0.659976\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.74657i	− 1.12995i	−0.825107	−	0.564977i	$-0.808885\pi$
$47$	− 7.74657i	− 1.12995i	0.825107	−	0.564977i	$-0.191115\pi$
$48$	0	0
$49$	5.68874	0.812677
$50$	0	0
$51$	12.6382	1.76971
$52$	0	0
$53$	12.9863i	1.78380i	0.452231	+	0.891901i	$0.350628\pi$
$53$	12.9863i	1.78380i	−0.452231	+	0.891901i	$0.649372\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.03677i	− 0.667137i
$58$	0	0
$59$	−0.797069	−0.103770	−0.0518848	−	0.998653i	$-0.516523\pi$
$59$	−0.797069	−0.103770	−0.0518848	+	0.998653i	$0.516523\pi$
$60$	0	0
$61$	13.8917	1.77865	0.889323	−	0.457279i	$-0.151176\pi$
$61$	13.8917	1.77865	0.889323	+	0.457279i	$0.151176\pi$
$62$	0	0
$63$	− 1.83384i	− 0.231042i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.5667i	1.90177i	0.309540	+	0.950887i	$0.399825\pi$
$67$	15.5667i	1.90177i	−0.309540	+	0.950887i	$0.600175\pi$
$68$	0	0
$69$	2.14510	0.258240
$70$	0	0
$71$	2.94217	0.349172	0.174586	−	0.984642i	$-0.444141\pi$
$71$	2.94217	0.349172	0.174586	+	0.984642i	$0.444141\pi$
$72$	0	0
$73$	6.32698i	0.740517i	0.928929	+	0.370258i	$0.120731\pi$
$73$	6.32698i	0.740517i	−0.928929	+	0.370258i	$0.879269\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.74657i	0.768843i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−11.2397	−1.24886
$82$	0	0
$83$	− 0.912726i	− 0.100185i	−0.998745	−	0.0500923i	$-0.984048\pi$
$83$	− 0.912726i	− 0.100185i	0.998745	−	0.0500923i	$-0.0159516\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.03677i	0.861633i
$88$	0	0
$89$	15.4931	1.64227	0.821135	−	0.570735i	$-0.193341\pi$
$89$	15.4931	1.64227	0.821135	+	0.570735i	$0.193341\pi$
$90$	0	0
$91$	5.60147	0.587193
$92$	0	0
$93$	− 12.2029i	− 1.26539i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.3848i	− 1.35902i	−0.733666	−	0.679511i	$-0.762192\pi$
$97$	− 13.3848i	− 1.35902i	0.733666	−	0.679511i	$-0.237808\pi$
$98$	0	0
$99$	−9.43531	−0.948284
$100$	0	0
$101$	2.79707	0.278319	0.139159	−	0.990270i	$-0.455560\pi$
$101$	2.79707	0.278319	0.139159	+	0.990270i	$0.455560\pi$
$102$	0	0
$103$	− 9.89167i	− 0.974655i	−0.873219	−	0.487328i	$-0.837972\pi$
$103$	− 9.89167i	− 0.974655i	0.873219	−	0.487328i	$-0.162028\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.6961i	− 1.22738i	−0.789549	−	0.613688i	$-0.789685\pi$
$107$	− 12.6961i	− 1.22738i	0.789549	−	0.613688i	$-0.210315\pi$
$108$	0	0
$109$	−3.65197	−0.349795	−0.174897	−	0.984587i	$-0.555959\pi$
$109$	−3.65197	−0.349795	−0.174897	+	0.984587i	$0.555959\pi$
$110$	0	0
$111$	−8.58041	−0.814417
$112$	0	0
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.83384i	0.724239i
$118$	0	0
$119$	−6.74657	−0.618457
$120$	0	0
$121$	23.7118	2.15562
$122$	0	0
$123$	2.26915i	0.204602i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 12.1524i	− 1.07835i	−0.842193	−	0.539177i	$-0.818735\pi$
$127$	− 12.1524i	− 1.07835i	0.842193	−	0.539177i	$-0.181265\pi$
$128$	0	0
$129$	24.6540	2.17066
$130$	0	0
$131$	−6.94950	−0.607181	−0.303590	−	0.952803i	$-0.598185\pi$
$131$	−6.94950	−0.607181	−0.303590	+	0.952803i	$0.598185\pi$
$132$	0	0
$133$	2.68874i	0.233143i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.44904i	0.209235i	0.994513	+	0.104618i	$0.0333619\pi$
$137$	2.44904i	0.209235i	−0.994513	+	0.104618i	$0.966638\pi$
$138$	0	0
$139$	−9.23970	−0.783702	−0.391851	−	0.920029i	$-0.628165\pi$
$139$	−9.23970	−0.783702	−0.391851	+	0.920029i	$0.628165\pi$
$140$	0	0
$141$	−16.6172	−1.39942
$142$	0	0
$143$	− 28.8201i	− 2.41006i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 12.2029i	− 1.00648i
$148$	0	0
$149$	2.05783	0.168584	0.0842919	−	0.996441i	$-0.473137\pi$
$149$	2.05783	0.168584	0.0842919	+	0.996441i	$0.473137\pi$
$150$	0	0
$151$	9.34803	0.760732	0.380366	−	0.924836i	$-0.375798\pi$
$151$	9.34803	0.760732	0.380366	+	0.924836i	$0.375798\pi$
$152$	0	0
$153$	− 9.43531i	− 0.762799i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.60879i	0.288013i	0.989577	+	0.144007i	$0.0459986\pi$
$157$	3.60879i	0.288013i	−0.989577	+	0.144007i	$0.954001\pi$
$158$	0	0
$159$	27.8569	2.20920
$160$	0	0
$161$	−1.14510	−0.0902467
$162$	0	0
$163$	21.2554i	1.66485i	0.554135	+	0.832427i	$0.313049\pi$
$163$	21.2554i	1.66485i	−0.554135	+	0.832427i	$0.686951\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.9863i	0.850143i	0.905160	+	0.425072i	$0.139751\pi$
$167$	10.9863i	0.850143i	−0.905160	+	0.425072i	$0.860249\pi$
$168$	0	0
$169$	−10.9284	−0.840650
$170$	0	0
$171$	−3.76030	−0.287557
$172$	0	0
$173$	− 6.28288i	− 0.477678i	−0.971059	−	0.238839i	$-0.923233\pi$
$173$	− 6.28288i	− 0.477678i	0.971059	−	0.238839i	$-0.0767669\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.70979i	0.128516i
$178$	0	0
$179$	−10.1524	−0.758828	−0.379414	−	0.925227i	$-0.623874\pi$
$179$	−10.1524	−0.758828	−0.379414	+	0.925227i	$0.623874\pi$
$180$	0	0
$181$	11.1451	0.828409	0.414204	−	0.910184i	$-0.364060\pi$
$181$	11.1451	0.828409	0.414204	+	0.910184i	$0.364060\pi$
$182$	0	0
$183$	− 29.7991i	− 2.20281i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	34.7118i	2.53838i
$188$	0	0
$189$	3.43531	0.249882
$190$	0	0
$191$	−14.6961	−1.06337	−0.531685	−	0.846942i	$-0.678441\pi$
$191$	−14.6961	−1.06337	−0.531685	+	0.846942i	$0.678441\pi$
$192$	0	0
$193$	− 26.4005i	− 1.90035i	−0.311715	−	0.950176i	$-0.600903\pi$
$193$	− 26.4005i	− 1.90035i	0.311715	−	0.950176i	$-0.399097\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.26076i	0.303567i	0.988414	+	0.151783i	$0.0485016\pi$
$197$	4.26076i	0.303567i	−0.988414	+	0.151783i	$0.951498\pi$
$198$	0	0
$199$	−18.0735	−1.28120	−0.640600	−	0.767875i	$-0.721314\pi$
$199$	−18.0735	−1.28120	−0.640600	+	0.767875i	$0.721314\pi$
$200$	0	0
$201$	33.3921	2.35530
$202$	0	0
$203$	− 4.29021i	− 0.301113i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.60147i	− 0.111310i
$208$	0	0
$209$	13.8338	0.956907
$210$	0	0
$211$	−11.8990	−0.819161	−0.409580	−	0.912274i	$-0.634325\pi$
$211$	−11.8990	−0.819161	−0.409580	+	0.912274i	$0.634325\pi$
$212$	0	0
$213$	− 6.31126i	− 0.432440i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.51419i	0.442212i
$218$	0	0
$219$	13.5720	0.917112
$220$	0	0
$221$	28.8201	1.93865
$222$	0	0
$223$	4.58041i	0.306727i	0.988170	+	0.153363i	$0.0490105\pi$
$223$	4.58041i	0.306727i	−0.988170	+	0.153363i	$0.950989\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.21666i	− 0.147125i	−0.997291	−	0.0735624i	$-0.976563\pi$
$227$	− 2.21666i	− 0.147125i	0.997291	−	0.0735624i	$-0.0234368\pi$
$228$	0	0
$229$	19.1608	1.26618	0.633091	−	0.774077i	$-0.281786\pi$
$229$	19.1608	1.26618	0.633091	+	0.774077i	$0.281786\pi$
$230$	0	0
$231$	14.4721	0.952193
$232$	0	0
$233$	− 21.7466i	− 1.42467i	−0.701842	−	0.712333i	$-0.747639\pi$
$233$	− 21.7466i	− 1.42467i	0.701842	−	0.712333i	$-0.252361\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.9495	−1.35511	−0.677555	−	0.735472i	$-0.736961\pi$
$239$	−20.9495	−1.35511	−0.677555	+	0.735472i	$0.736961\pi$
$240$	0	0
$241$	24.0735	1.55071	0.775357	−	0.631523i	$-0.217570\pi$
$241$	24.0735	1.55071	0.775357	+	0.631523i	$0.217570\pi$
$242$	0	0
$243$	15.1103i	0.969328i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 11.4858i	− 0.730825i
$248$	0	0
$249$	−1.95789	−0.124076
$250$	0	0
$251$	−7.82651	−0.494005	−0.247003	−	0.969015i	$-0.579446\pi$
$251$	−7.82651	−0.494005	−0.247003	+	0.969015i	$0.579446\pi$
$252$	0	0
$253$	5.89167i	0.370406i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.82012i	− 0.612562i	−0.951941	−	0.306281i	$-0.900915\pi$
$257$	− 9.82012i	− 0.612562i	0.951941	−	0.306281i	$-0.0990847\pi$
$258$	0	0
$259$	4.58041	0.284613
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	− 14.6887i	− 0.905746i	−0.891575	−	0.452873i	$-0.850399\pi$
$263$	− 14.6887i	− 0.905746i	0.891575	−	0.452873i	$-0.149601\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 33.2344i	− 2.03391i
$268$	0	0
$269$	0.253432	0.0154520	0.00772600	−	0.999970i	$-0.497541\pi$
$269$	0.253432	0.0154520	0.00772600	+	0.999970i	$0.497541\pi$
$270$	0	0
$271$	−19.6099	−1.19121	−0.595607	−	0.803276i	$-0.703088\pi$
$271$	−19.6099	−1.19121	−0.595607	+	0.803276i	$0.703088\pi$
$272$	0	0
$273$	− 12.0157i	− 0.727224i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 29.1976i	− 1.75431i	−0.480204	−	0.877157i	$-0.659437\pi$
$277$	− 29.1976i	− 1.75431i	0.480204	−	0.877157i	$-0.340563\pi$
$278$	0	0
$279$	−9.11032	−0.545421
$280$	0	0
$281$	−27.1755	−1.62115	−0.810577	−	0.585633i	$-0.800846\pi$
$281$	−27.1755	−1.62115	−0.810577	+	0.585633i	$0.800846\pi$
$282$	0	0
$283$	18.1471i	1.07873i	0.842071	+	0.539366i	$0.181336\pi$
$283$	18.1471i	1.07873i	−0.842071	+	0.539366i	$0.818664\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.21132i	− 0.0715021i
$288$	0	0
$289$	−17.7118	−1.04187
$290$	0	0
$291$	−28.7118	−1.68311
$292$	0	0
$293$	0.681412i	0.0398085i	0.999802	+	0.0199043i	$0.00633614\pi$
$293$	0.681412i	0.0398085i	−0.999802	+	0.0199043i	$0.993664\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 17.6750i	− 1.02561i
$298$	0	0
$299$	4.89167	0.282893
$300$	0	0
$301$	−13.1608	−0.758577
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 26.5877i	− 1.51744i	−0.651415	−	0.758721i	$-0.725824\pi$
$307$	− 26.5877i	− 1.51744i	0.651415	−	0.758721i	$-0.274176\pi$
$308$	0	0
$309$	−21.2186	−1.20709
$310$	0	0
$311$	13.5299	0.767211	0.383605	−	0.923497i	$-0.374682\pi$
$311$	13.5299	0.767211	0.383605	+	0.923497i	$0.374682\pi$
$312$	0	0
$313$	19.8412i	1.12149i	0.827989	+	0.560745i	$0.189485\pi$
$313$	19.8412i	1.12149i	−0.827989	+	0.560745i	$0.810515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.55096i	− 0.199442i	−0.995015	−	0.0997210i	$-0.968205\pi$
$317$	− 3.55096i	− 0.199442i	0.995015	−	0.0997210i	$-0.0317950\pi$
$318$	0	0
$319$	−22.0735	−1.23588
$320$	0	0
$321$	−27.2344	−1.52007
$322$	0	0
$323$	13.8338i	0.769736i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.83384i	0.433212i
$328$	0	0
$329$	8.87062	0.489053
$330$	0	0
$331$	−13.3133	−0.731763	−0.365881	−	0.930662i	$-0.619232\pi$
$331$	−13.3133	−0.731763	−0.365881	+	0.930662i	$0.619232\pi$
$332$	0	0
$333$	6.40586i	0.351039i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.4951i	1.11644i	0.829693	+	0.558220i	$0.188516\pi$
$337$	20.4951i	1.11644i	−0.829693	+	0.558220i	$0.811484\pi$
$338$	0	0
$339$	−21.4510	−1.16506
$340$	0	0
$341$	33.5162	1.81500
$342$	0	0
$343$	14.5299i	0.784541i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.60147i	0.408068i	0.978964	+	0.204034i	$0.0654053\pi$
$347$	7.60147i	0.408068i	−0.978964	+	0.204034i	$0.934595\pi$
$348$	0	0
$349$	−5.55736	−0.297479	−0.148739	−	0.988876i	$-0.547522\pi$
$349$	−5.55736	−0.297479	−0.148739	+	0.988876i	$0.547522\pi$
$350$	0	0
$351$	−14.6750	−0.783294
$352$	0	0
$353$	1.96323i	0.104492i	0.998634	+	0.0522460i	$0.0166380\pi$
$353$	1.96323i	0.104492i	−0.998634	+	0.0522460i	$0.983362\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.4721i	0.765944i
$358$	0	0
$359$	−10.5069	−0.554531	−0.277266	−	0.960793i	$-0.589428\pi$
$359$	−10.5069	−0.554531	−0.277266	+	0.960793i	$0.589428\pi$
$360$	0	0
$361$	−13.4867	−0.709828
$362$	0	0
$363$	− 50.8642i	− 2.66968i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 33.7412i	− 1.76128i	−0.473788	−	0.880639i	$-0.657114\pi$
$367$	− 33.7412i	− 1.76128i	0.473788	−	0.880639i	$-0.342886\pi$
$368$	0	0
$369$	1.69408	0.0881901
$370$	0	0
$371$	−14.8706	−0.772044
$372$	0	0
$373$	− 17.4510i	− 0.903580i	−0.892124	−	0.451790i	$-0.850786\pi$
$373$	− 17.4510i	− 0.903580i	0.892124	−	0.451790i	$-0.149214\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.3270i	0.943887i
$378$	0	0
$379$	33.4584	1.71864	0.859320	−	0.511438i	$-0.170887\pi$
$379$	33.4584	1.71864	0.859320	+	0.511438i	$0.170887\pi$
$380$	0	0
$381$	−26.0682	−1.33551
$382$	0	0
$383$	13.0873i	0.668728i	0.942444	+	0.334364i	$0.108522\pi$
$383$	13.0873i	0.668728i	−0.942444	+	0.334364i	$0.891478\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 18.4059i	− 0.935623i
$388$	0	0
$389$	−14.1083	−0.715321	−0.357660	−	0.933852i	$-0.616425\pi$
$389$	−14.1083	−0.715321	−0.357660	+	0.933852i	$0.616425\pi$
$390$	0	0
$391$	−5.89167	−0.297955
$392$	0	0
$393$	14.9074i	0.751978i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 16.6245i	− 0.834360i	−0.908824	−	0.417180i	$-0.863018\pi$
$397$	− 16.6245i	− 0.834360i	0.908824	−	0.417180i	$-0.136982\pi$
$398$	0	0
$399$	5.76762	0.288742
$400$	0	0
$401$	22.2628	1.11175	0.555874	−	0.831266i	$-0.312384\pi$
$401$	22.2628	1.11175	0.555874	+	0.831266i	$0.312384\pi$
$402$	0	0
$403$	− 27.8274i	− 1.38618i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 23.5667i	− 1.16816i
$408$	0	0
$409$	−15.6887	−0.775758	−0.387879	−	0.921710i	$-0.626792\pi$
$409$	−15.6887	−0.775758	−0.387879	+	0.921710i	$0.626792\pi$
$410$	0	0
$411$	5.25343	0.259133
$412$	0	0
$413$	− 0.912726i	− 0.0449123i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.8201i	0.970595i
$418$	0	0
$419$	5.66769	0.276885	0.138442	−	0.990371i	$-0.455790\pi$
$419$	5.66769	0.276885	0.138442	+	0.990371i	$0.455790\pi$
$420$	0	0
$421$	15.0716	0.734543	0.367271	−	0.930114i	$-0.380292\pi$
$421$	15.0716	0.734543	0.367271	+	0.930114i	$0.380292\pi$
$422$	0	0
$423$	12.4059i	0.603194i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.9074i	0.769813i
$428$	0	0
$429$	−61.8221	−2.98480
$430$	0	0
$431$	29.3500	1.41374	0.706870	−	0.707343i	$-0.250106\pi$
$431$	29.3500	1.41374	0.706870	+	0.707343i	$0.250106\pi$
$432$	0	0
$433$	29.6245i	1.42366i	0.702350	+	0.711832i	$0.252134\pi$
$433$	29.6245i	1.42366i	−0.702350	+	0.711832i	$0.747866\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.34803i	0.112322i
$438$	0	0
$439$	20.7907	0.992285	0.496142	−	0.868241i	$-0.334749\pi$
$439$	20.7907	0.992285	0.496142	+	0.868241i	$0.334749\pi$
$440$	0	0
$441$	−9.11032	−0.433825
$442$	0	0
$443$	21.2975i	1.01188i	0.862570	+	0.505938i	$0.168854\pi$
$443$	21.2975i	1.01188i	−0.862570	+	0.505938i	$0.831146\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.41425i	− 0.208787i
$448$	0	0
$449$	19.3197	0.911751	0.455875	−	0.890044i	$-0.349326\pi$
$449$	19.3197	0.911751	0.455875	+	0.890044i	$0.349326\pi$
$450$	0	0
$451$	−6.23238	−0.293471
$452$	0	0
$453$	− 20.0525i	− 0.942148i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.912726i	− 0.0426955i	−0.999772	−	0.0213478i	$-0.993204\pi$
$457$	− 0.912726i	− 0.0426955i	0.999772	−	0.0213478i	$-0.00679572\pi$
$458$	0	0
$459$	17.6750	0.824999
$460$	0	0
$461$	2.83384	0.131985	0.0659926	−	0.997820i	$-0.478979\pi$
$461$	2.83384	0.131985	0.0659926	+	0.997820i	$0.478979\pi$
$462$	0	0
$463$	− 17.2618i	− 0.802225i	−0.916029	−	0.401112i	$-0.868624\pi$
$463$	− 17.2618i	− 0.802225i	0.916029	−	0.401112i	$-0.131376\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.4510i	0.992635i	0.868141	+	0.496318i	$0.165315\pi$
$467$	21.4510i	0.992635i	−0.868141	+	0.496318i	$0.834685\pi$
$468$	0	0
$469$	−17.8255	−0.823103
$470$	0	0
$471$	7.74123	0.356697
$472$	0	0
$473$	67.7138i	3.11348i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 20.7971i	− 0.952232i
$478$	0	0
$479$	−9.49314	−0.433752	−0.216876	−	0.976199i	$-0.569587\pi$
$479$	−9.49314	−0.433752	−0.216876	+	0.976199i	$0.569587\pi$
$480$	0	0
$481$	−19.5667	−0.892164
$482$	0	0
$483$	2.45636i	0.111768i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 38.3270i	− 1.73676i	−0.495898	−	0.868381i	$-0.665161\pi$
$487$	− 38.3270i	− 1.73676i	0.495898	−	0.868381i	$-0.334839\pi$
$488$	0	0
$489$	45.5951	2.06188
$490$	0	0
$491$	−13.9947	−0.631570	−0.315785	−	0.948831i	$-0.602268\pi$
$491$	−13.9947	−0.631570	−0.315785	+	0.948831i	$0.602268\pi$
$492$	0	0
$493$	− 22.0735i	− 0.994143i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.36909i	0.151124i
$498$	0	0
$499$	0.0642277	0.00287522	0.00143761	−	0.999999i	$-0.499542\pi$
$499$	0.0642277	0.00287522	0.00143761	+	0.999999i	$0.499542\pi$
$500$	0	0
$501$	23.5667	1.05288
$502$	0	0
$503$	− 24.2785i	− 1.08252i	−0.840854	−	0.541262i	$-0.817947\pi$
$503$	− 24.2785i	− 1.08252i	0.840854	−	0.541262i	$-0.182053\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.4426i	1.04112i
$508$	0	0
$509$	2.65929	0.117871	0.0589356	−	0.998262i	$-0.481229\pi$
$509$	2.65929	0.117871	0.0589356	+	0.998262i	$0.481229\pi$
$510$	0	0
$511$	−7.24504	−0.320502
$512$	0	0
$513$	− 7.04410i	− 0.311005i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 45.6402i	− 2.00726i
$518$	0	0
$519$	−13.4774	−0.591593
$520$	0	0
$521$	−17.0598	−0.747404	−0.373702	−	0.927549i	$-0.621912\pi$
$521$	−17.0598	−0.747404	−0.373702	+	0.927549i	$0.621912\pi$
$522$	0	0
$523$	− 17.5667i	− 0.768137i	−0.923305	−	0.384069i	$-0.874523\pi$
$523$	− 17.5667i	− 0.768137i	0.923305	−	0.384069i	$-0.125477\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.5162i	1.45999i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	1.27648	0.0553944
$532$	0	0
$533$	5.17455i	0.224135i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.7780i	0.939790i
$538$	0	0
$539$	33.5162	1.44364
$540$	0	0
$541$	−8.36909	−0.359815	−0.179908	−	0.983684i	$-0.557580\pi$
$541$	−8.36909	−0.359815	−0.179908	+	0.983684i	$0.557580\pi$
$542$	0	0
$543$	− 23.9074i	− 1.02596i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.7486i	0.716117i	0.933699	+	0.358058i	$0.116561\pi$
$547$	16.7486i	0.716117i	−0.933699	+	0.358058i	$0.883439\pi$
$548$	0	0
$549$	−22.2470	−0.949480
$550$	0	0
$551$	−8.79707	−0.374768
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.26182i	0.307693i	0.988095	+	0.153847i	$0.0491662\pi$
$557$	7.26182i	0.307693i	−0.988095	+	0.153847i	$0.950834\pi$
$558$	0	0
$559$	56.2206	2.37788
$560$	0	0
$561$	74.4603	3.14372
$562$	0	0
$563$	34.5530i	1.45623i	0.685453	+	0.728117i	$0.259604\pi$
$563$	34.5530i	1.45623i	−0.685453	+	0.728117i	$0.740396\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 12.8706i	− 0.540515i
$568$	0	0
$569$	−34.7275	−1.45585	−0.727926	−	0.685655i	$-0.759516\pi$
$569$	−34.7275	−1.45585	−0.727926	+	0.685655i	$0.759516\pi$
$570$	0	0
$571$	14.7034	0.615318	0.307659	−	0.951497i	$-0.400454\pi$
$571$	14.7034	0.615318	0.307659	+	0.951497i	$0.400454\pi$
$572$	0	0
$573$	31.5246i	1.31696i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.13777i	0.255519i	0.991805	+	0.127759i	$0.0407785\pi$
$577$	6.13777i	0.255519i	−0.991805	+	0.127759i	$0.959221\pi$
$578$	0	0
$579$	−56.6318	−2.35354
$580$	0	0
$581$	1.04516	0.0433607
$582$	0	0
$583$	76.5108i	3.16876i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.33338i	0.0550344i	0.999621	+	0.0275172i	$0.00876010\pi$
$587$	1.33338i	0.0550344i	−0.999621	+	0.0275172i	$0.991240\pi$
$588$	0	0
$589$	13.3574	0.550380
$590$	0	0
$591$	9.13977	0.375960
$592$	0	0
$593$	1.88434i	0.0773807i	0.999251	+	0.0386903i	$0.0123186\pi$
$593$	1.88434i	0.0773807i	−0.999251	+	0.0386903i	$0.987681\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	38.7696i	1.58673i
$598$	0	0
$599$	43.7632	1.78812	0.894058	−	0.447951i	$-0.147846\pi$
$599$	43.7632	1.78812	0.894058	+	0.447951i	$0.147846\pi$
$600$	0	0
$601$	4.50046	0.183578	0.0917889	−	0.995778i	$-0.470742\pi$
$601$	4.50046	0.183578	0.0917889	+	0.995778i	$0.470742\pi$
$602$	0	0
$603$	− 24.9295i	− 1.01521i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 6.17455i	− 0.250617i	−0.992118	−	0.125309i	$-0.960008\pi$
$607$	− 6.17455i	− 0.250617i	0.992118	−	0.125309i	$-0.0399921\pi$
$608$	0	0
$609$	−9.20293	−0.372922
$610$	0	0
$611$	−37.8937	−1.53301
$612$	0	0
$613$	− 9.72445i	− 0.392767i	−0.980527	−	0.196383i	$-0.937080\pi$
$613$	− 9.72445i	− 0.392767i	0.980527	−	0.196383i	$-0.0629197\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.730849i	0.0294229i	0.999892	+	0.0147114i	$0.00468297\pi$
$617$	0.730849i	0.0294229i	−0.999892	+	0.0147114i	$0.995317\pi$
$618$	0	0
$619$	−11.1598	−0.448549	−0.224274	−	0.974526i	$-0.572001\pi$
$619$	−11.1598	−0.448549	−0.224274	+	0.974526i	$0.572001\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	17.7412i	0.710787i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 29.6750i	− 1.18511i
$628$	0	0
$629$	23.5667	0.939665
$630$	0	0
$631$	−4.87062	−0.193896	−0.0969481	−	0.995289i	$-0.530908\pi$
$631$	−4.87062	−0.193896	−0.0969481	+	0.995289i	$0.530908\pi$
$632$	0	0
$633$	25.5246i	1.01451i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 27.8274i	− 1.10256i
$638$	0	0
$639$	−4.71179	−0.186395
$640$	0	0
$641$	32.6540	1.28975	0.644877	−	0.764286i	$-0.276909\pi$
$641$	32.6540	1.28975	0.644877	+	0.764286i	$0.276909\pi$
$642$	0	0
$643$	47.9579i	1.89127i	0.325223	+	0.945637i	$0.394561\pi$
$643$	47.9579i	1.89127i	−0.325223	+	0.945637i	$0.605439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.1103i	− 0.476106i	−0.971252	−	0.238053i	$-0.923491\pi$
$647$	− 12.1103i	− 0.476106i	0.971252	−	0.238053i	$-0.0765092\pi$
$648$	0	0
$649$	−4.69607	−0.184337
$650$	0	0
$651$	13.9736	0.547669
$652$	0	0
$653$	− 43.2133i	− 1.69107i	−0.533922	−	0.845534i	$-0.679282\pi$
$653$	− 43.2133i	− 1.69107i	0.533922	−	0.845534i	$-0.320718\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.1324i	− 0.395304i
$658$	0	0
$659$	−20.4333	−0.795969	−0.397984	−	0.917392i	$-0.630290\pi$
$659$	−20.4333	−0.795969	−0.397984	+	0.917392i	$0.630290\pi$
$660$	0	0
$661$	48.1398	1.87242	0.936210	−	0.351441i	$-0.114308\pi$
$661$	48.1398	1.87242	0.936210	+	0.351441i	$0.114308\pi$
$662$	0	0
$663$	− 61.8221i	− 2.40097i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.74657i	− 0.145068i
$668$	0	0
$669$	9.82545	0.379874
$670$	0	0
$671$	81.8452	3.15960
$672$	0	0
$673$	16.5436i	0.637710i	0.947803	+	0.318855i	$0.103298\pi$
$673$	16.5436i	0.637710i	−0.947803	+	0.318855i	$0.896702\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.9863i	1.88270i	0.337438	+	0.941348i	$0.390440\pi$
$677$	48.9863i	1.88270i	−0.337438	+	0.941348i	$0.609560\pi$
$678$	0	0
$679$	15.3270	0.588195
$680$	0	0
$681$	−4.75496	−0.182210
$682$	0	0
$683$	− 21.8779i	− 0.837136i	−0.908185	−	0.418568i	$-0.862532\pi$
$683$	− 21.8779i	− 0.837136i	0.908185	−	0.418568i	$-0.137468\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 41.1019i	− 1.56814i
$688$	0	0
$689$	63.5246	2.42009
$690$	0	0
$691$	−3.30393	−0.125688	−0.0628438	−	0.998023i	$-0.520017\pi$
$691$	−3.30393	−0.125688	−0.0628438	+	0.998023i	$0.520017\pi$
$692$	0	0
$693$	− 10.8044i	− 0.410425i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.23238i	− 0.236068i
$698$	0	0
$699$	−46.6486	−1.76441
$700$	0	0
$701$	−4.46369	−0.168591	−0.0842956	−	0.996441i	$-0.526864\pi$
$701$	−4.46369	−0.168591	−0.0842956	+	0.996441i	$0.526864\pi$
$702$	0	0
$703$	− 9.39214i	− 0.354231i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.20293i	0.120459i
$708$	0	0
$709$	2.39853	0.0900789	0.0450394	−	0.998985i	$-0.485659\pi$
$709$	2.39853	0.0900789	0.0450394	+	0.998985i	$0.485659\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.68874i	0.213045i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	44.9388i	1.67827i
$718$	0	0
$719$	−12.0725	−0.450228	−0.225114	−	0.974332i	$-0.572275\pi$
$719$	−12.0725	−0.450228	−0.225114	+	0.974332i	$0.572275\pi$
$720$	0	0
$721$	11.3270	0.421839
$722$	0	0
$723$	− 51.6402i	− 1.92052i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 11.4269i	− 0.423801i	−0.977291	−	0.211900i	$-0.932035\pi$
$727$	− 11.4269i	− 0.423801i	0.977291	−	0.211900i	$-0.0679653\pi$
$728$	0	0
$729$	−1.30592	−0.0483676
$730$	0	0
$731$	−67.7138	−2.50448
$732$	0	0
$733$	− 3.08727i	− 0.114031i	−0.998373	−	0.0570155i	$-0.981842\pi$
$733$	− 3.08727i	− 0.114031i	0.998373	−	0.0570155i	$-0.0181585\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	91.7138i	3.37832i
$738$	0	0
$739$	8.36909	0.307862	0.153931	−	0.988082i	$-0.450807\pi$
$739$	8.36909	0.307862	0.153931	+	0.988082i	$0.450807\pi$
$740$	0	0
$741$	−24.6382	−0.905108
$742$	0	0
$743$	15.1598i	0.556158i	0.960558	+	0.278079i	$0.0896976\pi$
$743$	15.1598i	0.556158i	−0.960558	+	0.278079i	$0.910302\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.46170i	0.0534808i
$748$	0	0
$749$	14.5383	0.531218
$750$	0	0
$751$	−33.6823	−1.22909	−0.614543	−	0.788883i	$-0.710660\pi$
$751$	−33.6823	−1.22909	−0.614543	+	0.788883i	$0.710660\pi$
$752$	0	0
$753$	16.7887i	0.611813i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.1608i	1.05987i	0.848039	+	0.529934i	$0.177783\pi$
$757$	29.1608i	1.05987i	−0.848039	+	0.529934i	$0.822217\pi$
$758$	0	0
$759$	12.6382	0.458739
$760$	0	0
$761$	−6.91912	−0.250818	−0.125409	−	0.992105i	$-0.540024\pi$
$761$	−6.91912	−0.250818	−0.125409	+	0.992105i	$0.540024\pi$
$762$	0	0
$763$	− 4.18188i	− 0.151394i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.89900i	0.140785i
$768$	0	0
$769$	14.3491	0.517442	0.258721	−	0.965952i	$-0.416699\pi$
$769$	14.3491	0.517442	0.258721	+	0.965952i	$0.416699\pi$
$770$	0	0
$771$	−21.0652	−0.758643
$772$	0	0
$773$	50.6265i	1.82091i	0.413609	+	0.910454i	$0.364268\pi$
$773$	50.6265i	1.82091i	−0.413609	+	0.910454i	$0.635732\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 9.82545i	− 0.352486i
$778$	0	0
$779$	−2.48382	−0.0889920
$780$	0	0
$781$	17.3343	0.620270
$782$	0	0
$783$	11.2397i	0.401674i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 20.9442i	− 0.746579i	−0.927715	−	0.373289i	$-0.878230\pi$
$787$	− 20.9442i	− 0.746579i	0.927715	−	0.373289i	$-0.121770\pi$
$788$	0	0
$789$	−31.5089	−1.12174
$790$	0	0
$791$	11.4510	0.407152
$792$	0	0
$793$	− 67.9535i	− 2.41310i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.0009i	− 0.673047i	−0.941675	−	0.336524i	$-0.890749\pi$
$797$	− 19.0009i	− 0.673047i	0.941675	−	0.336524i	$-0.109251\pi$
$798$	0	0
$799$	45.6402	1.61464
$800$	0	0
$801$	−24.8117	−0.876679
$802$	0	0
$803$	37.2765i	1.31546i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 0.543637i	− 0.0191369i
$808$	0	0
$809$	−18.4300	−0.647963	−0.323982	−	0.946063i	$-0.605022\pi$
$809$	−18.4300	−0.647963	−0.323982	+	0.946063i	$0.605022\pi$
$810$	0	0
$811$	−21.1387	−0.742280	−0.371140	−	0.928577i	$-0.621033\pi$
$811$	−21.1387	−0.742280	−0.371140	+	0.928577i	$0.621033\pi$
$812$	0	0
$813$	42.0652i	1.47529i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	26.9863i	0.944130i
$818$	0	0
$819$	−8.97055	−0.313457
$820$	0	0
$821$	−16.7550	−0.584752	−0.292376	−	0.956303i	$-0.594446\pi$
$821$	−16.7550	−0.584752	−0.292376	+	0.956303i	$0.594446\pi$
$822$	0	0
$823$	24.9220i	0.868728i	0.900737	+	0.434364i	$0.143027\pi$
$823$	24.9220i	0.868728i	−0.900737	+	0.434364i	$0.856973\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.62252i	0.230288i	0.993349	+	0.115144i	$0.0367329\pi$
$827$	6.62252i	0.230288i	−0.993349	+	0.115144i	$0.963267\pi$
$828$	0	0
$829$	−52.7001	−1.83035	−0.915174	−	0.403058i	$-0.867947\pi$
$829$	−52.7001	−1.83035	−0.915174	+	0.403058i	$0.867947\pi$
$830$	0	0
$831$	−62.6318	−2.17267
$832$	0	0
$833$	33.5162i	1.16127i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 17.0662i	− 0.589895i
$838$	0	0
$839$	−42.5951	−1.47054	−0.735272	−	0.677772i	$-0.762946\pi$
$839$	−42.5951	−1.47054	−0.735272	+	0.677772i	$0.762946\pi$
$840$	0	0
$841$	−14.9632	−0.515973
$842$	0	0
$843$	58.2942i	2.00776i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27.1524i	0.932969i
$848$	0	0
$849$	38.9274	1.33598
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	− 57.7255i	− 1.97648i	−0.152898	−	0.988242i	$-0.548861\pi$
$853$	− 57.7255i	− 1.97648i	0.152898	−	0.988242i	$-0.451139\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	55.3721i	1.89148i	0.324930	+	0.945738i	$0.394659\pi$
$857$	55.3721i	1.89148i	−0.324930	+	0.945738i	$0.605341\pi$
$858$	0	0
$859$	14.3544	0.489767	0.244883	−	0.969553i	$-0.421250\pi$
$859$	14.3544	0.489767	0.244883	+	0.969553i	$0.421250\pi$
$860$	0	0
$861$	−2.59841	−0.0885536
$862$	0	0
$863$	15.9211i	0.541961i	0.962585	+	0.270981i	$0.0873479\pi$
$863$	15.9211i	0.541961i	−0.962585	+	0.270981i	$0.912652\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	37.9936i	1.29033i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	76.1471	2.58015
$872$	0	0
$873$	21.4353i	0.725475i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.2975i	1.02308i	0.859261	+	0.511538i	$0.170924\pi$
$877$	30.2975i	1.02308i	−0.859261	+	0.511538i	$0.829076\pi$
$878$	0	0
$879$	1.46170	0.0493019
$880$	0	0
$881$	−11.1883	−0.376943	−0.188471	−	0.982079i	$-0.560353\pi$
$881$	−11.1883	−0.376943	−0.188471	+	0.982079i	$0.560353\pi$
$882$	0	0
$883$	25.2882i	0.851016i	0.904955	+	0.425508i	$0.139905\pi$
$883$	25.2882i	0.851016i	−0.904955	+	0.425508i	$0.860095\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.2849i	1.08402i	0.840372	+	0.542010i	$0.182336\pi$
$887$	32.2849i	1.08402i	−0.840372	+	0.542010i	$0.817664\pi$
$888$	0	0
$889$	13.9158	0.466720
$890$	0	0
$891$	−66.2206	−2.21847
$892$	0	0
$893$	− 18.1892i	− 0.608679i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 10.4931i	− 0.350356i
$898$	0	0
$899$	−21.3133	−0.710837
$900$	0	0
$901$	−76.5108	−2.54895
$902$	0	0
$903$	28.2313i	0.939479i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.47848i	0.0490922i	0.999699	+	0.0245461i	$0.00781404\pi$
$907$	1.47848i	0.0490922i	−0.999699	+	0.0245461i	$0.992186\pi$
$908$	0	0
$909$	−4.47941	−0.148573
$910$	0	0
$911$	5.70979	0.189174	0.0945870	−	0.995517i	$-0.469847\pi$
$911$	5.70979	0.189174	0.0945870	+	0.995517i	$0.469847\pi$
$912$	0	0
$913$	− 5.37748i	− 0.177969i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.95789i	− 0.262793i
$918$	0	0
$919$	−22.6265	−0.746379	−0.373190	−	0.927755i	$-0.621736\pi$
$919$	−22.6265	−0.746379	−0.373190	+	0.927755i	$0.621736\pi$
$920$	0	0
$921$	−57.0334	−1.87932
$922$	0	0
$923$	− 14.3921i	− 0.473723i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.8412i	0.520292i
$928$	0	0
$929$	50.0829	1.64317	0.821583	−	0.570089i	$-0.193091\pi$
$929$	50.0829	1.64317	0.821583	+	0.570089i	$0.193091\pi$
$930$	0	0
$931$	13.3574	0.437770
$932$	0	0
$933$	− 29.0230i	− 0.950172i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9.98428i	− 0.326172i	−0.986612	−	0.163086i	$-0.947855\pi$
$937$	− 9.98428i	− 0.326172i	0.986612	−	0.163086i	$-0.0521448\pi$
$938$	0	0
$939$	42.5613	1.38894
$940$	0	0
$941$	−36.0809	−1.17620	−0.588101	−	0.808787i	$-0.700124\pi$
$941$	−36.0809	−1.17620	−0.588101	+	0.808787i	$0.700124\pi$
$942$	0	0
$943$	− 1.05783i	− 0.0344476i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.4730i	1.05523i	0.849483	+	0.527616i	$0.176914\pi$
$947$	32.4730i	1.05523i	−0.849483	+	0.527616i	$0.823086\pi$
$948$	0	0
$949$	30.9495	1.00466
$950$	0	0
$951$	−7.61718	−0.247004
$952$	0	0
$953$	− 39.3427i	− 1.27443i	−0.770684	−	0.637217i	$-0.780085\pi$
$953$	− 39.3427i	− 1.27443i	0.770684	−	0.637217i	$-0.219915\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	47.3500i	1.53061i
$958$	0	0
$959$	−2.80440	−0.0905587
$960$	0	0
$961$	1.36176	0.0439278
$962$	0	0
$963$	20.3323i	0.655200i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 23.5720i	− 0.758025i	−0.925392	−	0.379013i	$-0.876264\pi$
$967$	− 23.5720i	− 0.758025i	0.925392	−	0.379013i	$-0.123736\pi$
$968$	0	0
$969$	29.6750	0.953299
$970$	0	0
$971$	12.9368	0.415163	0.207581	−	0.978218i	$-0.433441\pi$
$971$	12.9368	0.415163	0.207581	+	0.978218i	$0.433441\pi$
$972$	0	0
$973$	− 10.5804i	− 0.339192i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.7118i	0.726614i	0.931669	+	0.363307i	$0.118352\pi$
$977$	22.7118i	0.726614i	−0.931669	+	0.363307i	$0.881648\pi$
$978$	0	0
$979$	91.2805	2.91734
$980$	0	0
$981$	5.84850	0.186728
$982$	0	0
$983$	26.1819i	0.835072i	0.908660	+	0.417536i	$0.137106\pi$
$983$	26.1819i	0.835072i	−0.908660	+	0.417536i	$0.862894\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 19.0284i	− 0.605680i
$988$	0	0
$989$	−11.4931	−0.365460
$990$	0	0
$991$	−56.3143	−1.78888	−0.894442	−	0.447185i	$-0.852427\pi$
$991$	−56.3143	−1.78888	−0.894442	+	0.447185i	$0.852427\pi$
$992$	0	0
$993$	28.5583i	0.906270i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.74123i	0.118486i	0.998244	+	0.0592430i	$0.0188687\pi$
$997$	3.74123i	0.118486i	−0.998244	+	0.0592430i	$0.981131\pi$
$998$	0	0
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.r.4049.2		6
5.2	odd	4		920.2.a.g.1.1	✓	3
5.3	odd	4		4600.2.a.y.1.3		3
5.4	even	2	inner	4600.2.e.r.4049.5		6
15.2	even	4		8280.2.a.bo.1.1		3
20.3	even	4		9200.2.a.cd.1.1		3
20.7	even	4		1840.2.a.t.1.3		3
40.27	even	4		7360.2.a.ca.1.1		3
40.37	odd	4		7360.2.a.cb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.g.1.1	✓	3	5.2	odd	4
1840.2.a.t.1.3		3	20.7	even	4
4600.2.a.y.1.3		3	5.3	odd	4
4600.2.e.r.4049.2		6	1.1	even	1	trivial
4600.2.e.r.4049.5		6	5.4	even	2	inner
7360.2.a.ca.1.1		3	40.27	even	4
7360.2.a.cb.1.3		3	40.37	odd	4
8280.2.a.bo.1.1		3	15.2	even	4
9200.2.a.cd.1.1		3	20.3	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)

\(f(q)\)	\(=\)	\(q-2.14510i q^{3} +1.14510i q^{7} -1.60147 q^{9} +O(q^{10})\)	\(q-2.14510i q^{3} +1.14510i q^{7} -1.60147 q^{9} +5.89167 q^{11} -4.89167i q^{13} +5.89167i q^{17} +2.34803 q^{19} +2.45636 q^{21} +1.00000i q^{23} -3.00000i q^{27} -3.74657 q^{29} +5.68874 q^{31} -12.6382i q^{33} -4.00000i q^{37} -10.4931 q^{39} -1.05783 q^{41} +11.4931i q^{43} -7.74657i q^{47} +5.68874 q^{49} +12.6382 q^{51} +12.9863i q^{53} -5.03677i q^{57} -0.797069 q^{59} +13.8917 q^{61} -1.83384i q^{63} +15.5667i q^{67} +2.14510 q^{69} +2.94217 q^{71} +6.32698i q^{73} +6.74657i q^{77} -11.2397 q^{81} -0.912726i q^{83} +8.03677i q^{87} +15.4931 q^{89} +5.60147 q^{91} -12.2029i q^{93} -13.3848i q^{97} -9.43531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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