LMFDB - Embedded newform 4600.2.e.q.4049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$1.86081i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.q.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46260i q^{3} -1.53740i q^{7} +0.860806 q^{9} -0.860806 q^{11} +0.139194i q^{13} -5.50761i q^{17} -5.25901 q^{19} +2.24860 q^{21} -1.00000i q^{23} +5.64681i q^{27} -9.76663 q^{29} +6.78600 q^{31} -1.25901i q^{33} +12.0900i q^{37} -0.203585 q^{39} -9.98062 q^{41} +11.4432i q^{43} +2.32340i q^{47} +4.63640 q^{49} +8.05543 q^{51} +0.149606i q^{53} -7.69182i q^{57} +11.0152 q^{59} +4.43281 q^{61} -1.32340i q^{63} +10.4972i q^{67} +1.46260 q^{69} +7.31299 q^{71} +7.11982i q^{73} +1.32340i q^{77} -6.79641 q^{79} -5.67660 q^{81} +10.6468i q^{83} -14.2847i q^{87} +17.2936 q^{89} +0.213997 q^{91} +9.92520i q^{93} -1.93561i q^{97} -0.740987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89}+ \cdots - 22 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 6 * q^11 - 14 * q^19 - 12 * q^21 + 2 * q^29 + 20 * q^31 - 14 * q^39 - 20 * q^41 - 12 * q^49 + 18 * q^51 - 20 * q^59 - 26 * q^61 + 4 * q^69 + 20 * q^71 - 28 * q^79 - 50 * q^81 + 40 * q^89 + 22 * q^91 - 22 * q^99

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.46260i	0.844432i	0.906495	+	0.422216i	$0.138748\pi$
$3$	1.46260i	0.844432i	−0.906495	+	0.422216i	$0.861252\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.53740i	− 0.581083i	−0.956862	−	0.290542i	$-0.906165\pi$
$7$	− 1.53740i	− 0.581083i	0.956862	−	0.290542i	$-0.0938355\pi$
$8$	0	0
$9$	0.860806	0.286935
$10$	0	0
$11$	−0.860806	−0.259543	−0.129771	−	0.991544i	$-0.541424\pi$
$11$	−0.860806	−0.259543	−0.129771	+	0.991544i	$0.541424\pi$
$12$	0	0
$13$	0.139194i	0.0386055i	0.999814	+	0.0193028i	$0.00614464\pi$
$13$	0.139194i	0.0386055i	−0.999814	+	0.0193028i	$0.993855\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.50761i	− 1.33579i	−0.744254	−	0.667896i	$-0.767195\pi$
$17$	− 5.50761i	− 1.33579i	0.744254	−	0.667896i	$-0.232805\pi$
$18$	0	0
$19$	−5.25901	−1.20650	−0.603250	−	0.797552i	$-0.706128\pi$
$19$	−5.25901	−1.20650	−0.603250	+	0.797552i	$0.706128\pi$
$20$	0	0
$21$	2.24860	0.490685
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.64681i	1.08673i
$28$	0	0
$29$	−9.76663	−1.81362	−0.906809	−	0.421543i	$-0.861489\pi$
$29$	−9.76663	−1.81362	−0.906809	+	0.421543i	$0.861489\pi$
$30$	0	0
$31$	6.78600	1.21880	0.609401	−	0.792862i	$-0.291410\pi$
$31$	6.78600	1.21880	0.609401	+	0.792862i	$0.291410\pi$
$32$	0	0
$33$	− 1.25901i	− 0.219166i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	12.0900i	1.98759i	0.111232	+	0.993795i	$0.464520\pi$
$37$	12.0900i	1.98759i	−0.111232	+	0.993795i	$0.535480\pi$
$38$	0	0
$39$	−0.203585	−0.0325997
$40$	0	0
$41$	−9.98062	−1.55871	−0.779356	−	0.626582i	$-0.784454\pi$
$41$	−9.98062	−1.55871	−0.779356	+	0.626582i	$0.784454\pi$
$42$	0	0
$43$	11.4432i	1.74508i	0.488547	+	0.872538i	$0.337527\pi$
$43$	11.4432i	1.74508i	−0.488547	+	0.872538i	$0.662473\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.32340i	0.338903i	0.985538	+	0.169452i	$0.0541997\pi$
$47$	2.32340i	0.338903i	−0.985538	+	0.169452i	$0.945800\pi$
$48$	0	0
$49$	4.63640	0.662342
$50$	0	0
$51$	8.05543	1.12799
$52$	0	0
$53$	0.149606i	0.0205500i	0.999947	+	0.0102750i	$0.00327069\pi$
$53$	0.149606i	0.0205500i	−0.999947	+	0.0102750i	$0.996729\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 7.69182i	− 1.01881i
$58$	0	0
$59$	11.0152	1.43406	0.717030	−	0.697042i	$-0.245501\pi$
$59$	11.0152	1.43406	0.717030	+	0.697042i	$0.245501\pi$
$60$	0	0
$61$	4.43281	0.567563	0.283782	−	0.958889i	$-0.408411\pi$
$61$	4.43281	0.567563	0.283782	+	0.958889i	$0.408411\pi$
$62$	0	0
$63$	− 1.32340i	− 0.166733i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.4972i	1.28244i	0.767358	+	0.641219i	$0.221571\pi$
$67$	10.4972i	1.28244i	−0.767358	+	0.641219i	$0.778429\pi$
$68$	0	0
$69$	1.46260	0.176076
$70$	0	0
$71$	7.31299	0.867892	0.433946	−	0.900939i	$-0.357121\pi$
$71$	7.31299	0.867892	0.433946	+	0.900939i	$0.357121\pi$
$72$	0	0
$73$	7.11982i	0.833312i	0.909064	+	0.416656i	$0.136798\pi$
$73$	7.11982i	0.833312i	−0.909064	+	0.416656i	$0.863202\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.32340i	0.150816i
$78$	0	0
$79$	−6.79641	−0.764656	−0.382328	−	0.924027i	$-0.624878\pi$
$79$	−6.79641	−0.764656	−0.382328	+	0.924027i	$0.624878\pi$
$80$	0	0
$81$	−5.67660	−0.630733
$82$	0	0
$83$	10.6468i	1.16864i	0.811524	+	0.584320i	$0.198639\pi$
$83$	10.6468i	1.16864i	−0.811524	+	0.584320i	$0.801361\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 14.2847i	− 1.53148i
$88$	0	0
$89$	17.2936	1.83312	0.916560	−	0.399897i	$-0.130954\pi$
$89$	17.2936	1.83312	0.916560	+	0.399897i	$0.130954\pi$
$90$	0	0
$91$	0.213997	0.0224330
$92$	0	0
$93$	9.92520i	1.02919i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.93561i	− 0.196531i	−0.995160	−	0.0982657i	$-0.968671\pi$
$97$	− 1.93561i	− 0.196531i	0.995160	−	0.0982657i	$-0.0313295\pi$
$98$	0	0
$99$	−0.740987	−0.0744720
$100$	0	0
$101$	−15.4224	−1.53459	−0.767293	−	0.641297i	$-0.778397\pi$
$101$	−15.4224	−1.53459	−0.767293	+	0.641297i	$0.778397\pi$
$102$	0	0
$103$	7.91478i	0.779867i	0.920843	+	0.389933i	$0.127502\pi$
$103$	7.91478i	0.779867i	−0.920843	+	0.389933i	$0.872498\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.27839i	0.413607i	0.978382	+	0.206804i	$0.0663061\pi$
$107$	4.27839i	0.413607i	−0.978382	+	0.206804i	$0.933694\pi$
$108$	0	0
$109$	17.7770	1.70273	0.851366	−	0.524572i	$-0.175775\pi$
$109$	17.7770	1.70273	0.851366	+	0.524572i	$0.175775\pi$
$110$	0	0
$111$	−17.6829	−1.67838
$112$	0	0
$113$	− 3.20359i	− 0.301368i	−0.988582	−	0.150684i	$-0.951852\pi$
$113$	− 3.20359i	− 0.301368i	0.988582	−	0.150684i	$-0.0481476\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.119819i	0.0110773i
$118$	0	0
$119$	−8.46742	−0.776207
$120$	0	0
$121$	−10.2590	−0.932638
$122$	0	0
$123$	− 14.5976i	− 1.31623i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.2099i	1.34966i	0.737975	+	0.674828i	$0.235782\pi$
$127$	15.2099i	1.34966i	−0.737975	+	0.674828i	$0.764218\pi$
$128$	0	0
$129$	−16.7368	−1.47360
$130$	0	0
$131$	12.6918	1.10889	0.554445	−	0.832220i	$-0.312931\pi$
$131$	12.6918	1.10889	0.554445	+	0.832220i	$0.312931\pi$
$132$	0	0
$133$	8.08522i	0.701077i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.0346i	1.19906i	0.800353	+	0.599529i	$0.204645\pi$
$137$	14.0346i	1.19906i	−0.800353	+	0.599529i	$0.795355\pi$
$138$	0	0
$139$	−14.2638	−1.20984	−0.604921	−	0.796285i	$-0.706795\pi$
$139$	−14.2638	−1.20984	−0.604921	+	0.796285i	$0.706795\pi$
$140$	0	0
$141$	−3.39821	−0.286181
$142$	0	0
$143$	− 0.119819i	− 0.0100198i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.78119i	0.559303i
$148$	0	0
$149$	4.46260	0.365590	0.182795	−	0.983151i	$-0.441485\pi$
$149$	4.46260	0.365590	0.182795	+	0.983151i	$0.441485\pi$
$150$	0	0
$151$	10.7562	0.875328	0.437664	−	0.899139i	$-0.355806\pi$
$151$	10.7562	0.875328	0.437664	+	0.899139i	$0.355806\pi$
$152$	0	0
$153$	− 4.74099i	− 0.383286i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.5180i	− 1.15866i	−0.815091	−	0.579332i	$-0.803313\pi$
$157$	− 14.5180i	− 1.15866i	0.815091	−	0.579332i	$-0.196687\pi$
$158$	0	0
$159$	−0.218814	−0.0173531
$160$	0	0
$161$	−1.53740	−0.121164
$162$	0	0
$163$	− 1.30403i	− 0.102139i	−0.998695	−	0.0510697i	$-0.983737\pi$
$163$	− 1.30403i	− 0.102139i	0.998695	−	0.0510697i	$-0.0162631\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.44322i	− 0.266445i	−0.991086	−	0.133222i	$-0.957468\pi$
$167$	− 3.44322i	− 0.266445i	0.991086	−	0.133222i	$-0.0425324\pi$
$168$	0	0
$169$	12.9806	0.998510
$170$	0	0
$171$	−4.52699	−0.346188
$172$	0	0
$173$	9.35801i	0.711476i	0.934586	+	0.355738i	$0.115770\pi$
$173$	9.35801i	0.711476i	−0.934586	+	0.355738i	$0.884230\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.1109i	1.21097i
$178$	0	0
$179$	18.4134	1.37628	0.688142	−	0.725576i	$-0.258426\pi$
$179$	18.4134	1.37628	0.688142	+	0.725576i	$0.258426\pi$
$180$	0	0
$181$	−1.94457	−0.144539	−0.0722694	−	0.997385i	$-0.523024\pi$
$181$	−1.94457	−0.144539	−0.0722694	+	0.997385i	$0.523024\pi$
$182$	0	0
$183$	6.48342i	0.479268i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.74099i	0.346695i
$188$	0	0
$189$	8.68141	0.631480
$190$	0	0
$191$	−0.775591	−0.0561198	−0.0280599	−	0.999606i	$-0.508933\pi$
$191$	−0.775591	−0.0561198	−0.0280599	+	0.999606i	$0.508933\pi$
$192$	0	0
$193$	− 11.7666i	− 0.846980i	−0.905901	−	0.423490i	$-0.860805\pi$
$193$	− 11.7666i	− 0.846980i	0.905901	−	0.423490i	$-0.139195\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.68141i	− 0.689772i	−0.938645	−	0.344886i	$-0.887918\pi$
$197$	− 9.68141i	− 0.689772i	0.938645	−	0.344886i	$-0.112082\pi$
$198$	0	0
$199$	3.29362	0.233478	0.116739	−	0.993163i	$-0.462756\pi$
$199$	3.29362	0.233478	0.116739	+	0.993163i	$0.462756\pi$
$200$	0	0
$201$	−15.3532	−1.08293
$202$	0	0
$203$	15.0152i	1.05386i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.860806i	− 0.0598301i
$208$	0	0
$209$	4.52699	0.313138
$210$	0	0
$211$	−16.2396	−1.11798	−0.558991	−	0.829173i	$-0.688812\pi$
$211$	−16.2396	−1.11798	−0.558991	+	0.829173i	$0.688812\pi$
$212$	0	0
$213$	10.6960i	0.732876i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 10.4328i	− 0.708225i
$218$	0	0
$219$	−10.4134	−0.703675
$220$	0	0
$221$	0.766628	0.0515690
$222$	0	0
$223$	21.2936i	1.42593i	0.701202	+	0.712963i	$0.252647\pi$
$223$	21.2936i	1.42593i	−0.701202	+	0.712963i	$0.747353\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29.6620i	1.96874i	0.176117	+	0.984369i	$0.443646\pi$
$227$	29.6620i	1.96874i	−0.176117	+	0.984369i	$0.556354\pi$
$228$	0	0
$229$	−19.6829	−1.30068	−0.650340	−	0.759643i	$-0.725374\pi$
$229$	−19.6829	−1.30068	−0.650340	+	0.759643i	$0.725374\pi$
$230$	0	0
$231$	−1.93561	−0.127354
$232$	0	0
$233$	− 5.52699i	− 0.362085i	−0.983475	−	0.181043i	$-0.942053\pi$
$233$	− 5.52699i	− 0.362085i	0.983475	−	0.181043i	$-0.0579472\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9.94043i	− 0.645700i
$238$	0	0
$239$	17.3982	1.12540	0.562698	−	0.826662i	$-0.309763\pi$
$239$	17.3982	1.12540	0.562698	+	0.826662i	$0.309763\pi$
$240$	0	0
$241$	−16.0900	−1.03645	−0.518225	−	0.855244i	$-0.673407\pi$
$241$	−16.0900	−1.03645	−0.518225	+	0.855244i	$0.673407\pi$
$242$	0	0
$243$	8.63785i	0.554118i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.732024i	− 0.0465776i
$248$	0	0
$249$	−15.5720	−0.986836
$250$	0	0
$251$	−29.0498	−1.83361	−0.916805	−	0.399336i	$-0.869241\pi$
$251$	−29.0498	−1.83361	−0.916805	+	0.399336i	$0.869241\pi$
$252$	0	0
$253$	0.860806i	0.0541184i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.56304i	0.409391i	0.978826	+	0.204696i	$0.0656205\pi$
$257$	6.56304i	0.409391i	−0.978826	+	0.204696i	$0.934380\pi$
$258$	0	0
$259$	18.5872	1.15495
$260$	0	0
$261$	−8.40717	−0.520391
$262$	0	0
$263$	− 0.561593i	− 0.0346293i	−0.999850	−	0.0173147i	$-0.994488\pi$
$263$	− 0.561593i	− 0.0346293i	0.999850	−	0.0173147i	$-0.00551170\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	25.2936i	1.54794i
$268$	0	0
$269$	7.26943	0.443225	0.221612	−	0.975135i	$-0.428868\pi$
$269$	7.26943	0.443225	0.221612	+	0.975135i	$0.428868\pi$
$270$	0	0
$271$	0.184210	0.0111900	0.00559498	−	0.999984i	$-0.498219\pi$
$271$	0.184210	0.0111900	0.00559498	+	0.999984i	$0.498219\pi$
$272$	0	0
$273$	0.312992i	0.0189431i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 25.8954i	− 1.55590i	−0.628323	−	0.777952i	$-0.716259\pi$
$277$	− 25.8954i	− 1.55590i	0.628323	−	0.777952i	$-0.283741\pi$
$278$	0	0
$279$	5.84143	0.349717
$280$	0	0
$281$	−26.7756	−1.59730	−0.798649	−	0.601797i	$-0.794452\pi$
$281$	−26.7756	−1.59730	−0.798649	+	0.601797i	$0.794452\pi$
$282$	0	0
$283$	5.70079i	0.338877i	0.985541	+	0.169438i	$0.0541954\pi$
$283$	5.70079i	0.338877i	−0.985541	+	0.169438i	$0.945805\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.3442i	0.905741i
$288$	0	0
$289$	−13.3338	−0.784342
$290$	0	0
$291$	2.83102	0.165957
$292$	0	0
$293$	1.05398i	0.0615741i	0.999526	+	0.0307871i	$0.00980137\pi$
$293$	1.05398i	0.0615741i	−0.999526	+	0.0307871i	$0.990199\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.86081i	− 0.282053i
$298$	0	0
$299$	0.139194	0.00804981
$300$	0	0
$301$	17.5928	1.01403
$302$	0	0
$303$	− 22.5568i	− 1.29585i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.8760i	− 1.02024i	−0.860103	−	0.510120i	$-0.829601\pi$
$307$	− 17.8760i	− 1.02024i	0.860103	−	0.510120i	$-0.170399\pi$
$308$	0	0
$309$	−11.5762	−0.658544
$310$	0	0
$311$	16.9315	0.960095	0.480048	−	0.877242i	$-0.340619\pi$
$311$	16.9315	0.960095	0.480048	+	0.877242i	$0.340619\pi$
$312$	0	0
$313$	− 19.9959i	− 1.13023i	−0.825011	−	0.565116i	$-0.808831\pi$
$313$	− 19.9959i	− 1.13023i	0.825011	−	0.565116i	$-0.191169\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.29776i	− 0.185221i	−0.995702	−	0.0926104i	$-0.970479\pi$
$317$	− 3.29776i	− 0.185221i	0.995702	−	0.0926104i	$-0.0295211\pi$
$318$	0	0
$319$	8.40717	0.470711
$320$	0	0
$321$	−6.25756	−0.349263
$322$	0	0
$323$	28.9646i	1.61163i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	26.0007i	1.43784i
$328$	0	0
$329$	3.57201	0.196931
$330$	0	0
$331$	−12.8802	−0.707959	−0.353979	−	0.935253i	$-0.615172\pi$
$331$	−12.8802	−0.707959	−0.353979	+	0.935253i	$0.615172\pi$
$332$	0	0
$333$	10.4072i	0.570309i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.22026i	0.502260i	0.967953	+	0.251130i	$0.0808021\pi$
$337$	9.22026i	0.502260i	−0.967953	+	0.251130i	$0.919198\pi$
$338$	0	0
$339$	4.68556	0.254485
$340$	0	0
$341$	−5.84143	−0.316331
$342$	0	0
$343$	− 17.8898i	− 0.965959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 31.2984i	− 1.68019i	−0.542441	−	0.840094i	$-0.682500\pi$
$347$	− 31.2984i	− 1.68019i	0.542441	−	0.840094i	$-0.317500\pi$
$348$	0	0
$349$	−9.89541	−0.529689	−0.264845	−	0.964291i	$-0.585321\pi$
$349$	−9.89541	−0.529689	−0.264845	+	0.964291i	$0.585321\pi$
$350$	0	0
$351$	−0.786003	−0.0419537
$352$	0	0
$353$	19.4287i	1.03408i	0.855960	+	0.517042i	$0.172967\pi$
$353$	19.4287i	1.03408i	−0.855960	+	0.517042i	$0.827033\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.3844i	− 0.655453i
$358$	0	0
$359$	13.1440	0.693714	0.346857	−	0.937918i	$-0.387249\pi$
$359$	13.1440	0.693714	0.346857	+	0.937918i	$0.387249\pi$
$360$	0	0
$361$	8.65722	0.455643
$362$	0	0
$363$	− 15.0048i	− 0.787549i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.7424i	0.821748i	0.911692	+	0.410874i	$0.134776\pi$
$367$	15.7424i	0.821748i	−0.911692	+	0.410874i	$0.865224\pi$
$368$	0	0
$369$	−8.59138	−0.447249
$370$	0	0
$371$	0.230005	0.0119413
$372$	0	0
$373$	27.6620i	1.43229i	0.697954	+	0.716143i	$0.254094\pi$
$373$	27.6620i	1.43229i	−0.697954	+	0.716143i	$0.745906\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.35946i	− 0.0700156i
$378$	0	0
$379$	−28.2140	−1.44926	−0.724628	−	0.689140i	$-0.757988\pi$
$379$	−28.2140	−1.44926	−0.724628	+	0.689140i	$0.757988\pi$
$380$	0	0
$381$	−22.2459	−1.13969
$382$	0	0
$383$	− 21.5928i	− 1.10334i	−0.834062	−	0.551671i	$-0.813990\pi$
$383$	− 21.5928i	− 1.10334i	0.834062	−	0.551671i	$-0.186010\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.85039i	0.500724i
$388$	0	0
$389$	−17.2501	−0.874612	−0.437306	−	0.899313i	$-0.644067\pi$
$389$	−17.2501	−0.874612	−0.437306	+	0.899313i	$0.644067\pi$
$390$	0	0
$391$	−5.50761	−0.278532
$392$	0	0
$393$	18.5630i	0.936382i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.8850i	− 1.75083i	−0.483374	−	0.875414i	$-0.660589\pi$
$397$	− 34.8850i	− 1.75083i	0.483374	−	0.875414i	$-0.339411\pi$
$398$	0	0
$399$	−11.8254	−0.592012
$400$	0	0
$401$	4.21881	0.210678	0.105339	−	0.994436i	$-0.466407\pi$
$401$	4.21881	0.210678	0.105339	+	0.994436i	$0.466407\pi$
$402$	0	0
$403$	0.944572i	0.0470525i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.4072i	− 0.515864i
$408$	0	0
$409$	−38.8165	−1.91935	−0.959675	−	0.281111i	$-0.909297\pi$
$409$	−38.8165	−1.91935	−0.959675	+	0.281111i	$0.909297\pi$
$410$	0	0
$411$	−20.5270	−1.01252
$412$	0	0
$413$	− 16.9348i	− 0.833309i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.8623i	− 1.02163i
$418$	0	0
$419$	−22.0305	−1.07626	−0.538129	−	0.842862i	$-0.680869\pi$
$419$	−22.0305	−1.07626	−0.538129	+	0.842862i	$0.680869\pi$
$420$	0	0
$421$	−17.2203	−0.839264	−0.419632	−	0.907694i	$-0.637841\pi$
$421$	−17.2203	−0.839264	−0.419632	+	0.907694i	$0.637841\pi$
$422$	0	0
$423$	2.00000i	0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.81501i	− 0.329801i
$428$	0	0
$429$	0.175247	0.00846102
$430$	0	0
$431$	−11.8116	−0.568947	−0.284473	−	0.958684i	$-0.591819\pi$
$431$	−11.8116	−0.568947	−0.284473	+	0.958684i	$0.591819\pi$
$432$	0	0
$433$	19.8642i	0.954611i	0.878737	+	0.477306i	$0.158387\pi$
$433$	19.8642i	0.954611i	−0.878737	+	0.477306i	$0.841613\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.25901i	0.251573i
$438$	0	0
$439$	5.04647	0.240855	0.120427	−	0.992722i	$-0.461574\pi$
$439$	5.04647	0.240855	0.120427	+	0.992722i	$0.461574\pi$
$440$	0	0
$441$	3.99104	0.190049
$442$	0	0
$443$	− 14.3193i	− 0.680328i	−0.940366	−	0.340164i	$-0.889517\pi$
$443$	− 14.3193i	− 0.680328i	0.940366	−	0.340164i	$-0.110483\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.52699i	0.308716i
$448$	0	0
$449$	13.3699	0.630963	0.315482	−	0.948932i	$-0.397834\pi$
$449$	13.3699	0.630963	0.315482	+	0.948932i	$0.397834\pi$
$450$	0	0
$451$	8.59138	0.404552
$452$	0	0
$453$	15.7320i	0.739155i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.38924i	− 0.111764i	−0.998437	−	0.0558821i	$-0.982203\pi$
$457$	− 2.38924i	− 0.111764i	0.998437	−	0.0558821i	$-0.0177971\pi$
$458$	0	0
$459$	31.1004	1.45164
$460$	0	0
$461$	21.2278	0.988676	0.494338	−	0.869270i	$-0.335410\pi$
$461$	21.2278	0.988676	0.494338	+	0.869270i	$0.335410\pi$
$462$	0	0
$463$	35.1261i	1.63245i	0.577736	+	0.816224i	$0.303936\pi$
$463$	35.1261i	1.63245i	−0.577736	+	0.816224i	$0.696064\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.7756i	− 0.776282i	−0.921600	−	0.388141i	$-0.873117\pi$
$467$	− 16.7756i	− 0.776282i	0.921600	−	0.388141i	$-0.126883\pi$
$468$	0	0
$469$	16.1384	0.745203
$470$	0	0
$471$	21.2340	0.978413
$472$	0	0
$473$	− 9.85039i	− 0.452922i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.128782i	0.00589652i
$478$	0	0
$479$	5.61076	0.256362	0.128181	−	0.991751i	$-0.459086\pi$
$479$	5.61076	0.256362	0.128181	+	0.991751i	$0.459086\pi$
$480$	0	0
$481$	−1.68286	−0.0767319
$482$	0	0
$483$	− 2.24860i	− 0.102315i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10.2638i	− 0.465099i	−0.972585	−	0.232549i	$-0.925293\pi$
$487$	− 10.2638i	− 0.465099i	0.972585	−	0.232549i	$-0.0747067\pi$
$488$	0	0
$489$	1.90727	0.0862498
$490$	0	0
$491$	9.28735	0.419132	0.209566	−	0.977794i	$-0.432795\pi$
$491$	9.28735	0.419132	0.209566	+	0.977794i	$0.432795\pi$
$492$	0	0
$493$	53.7908i	2.42262i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11.2430i	− 0.504318i
$498$	0	0
$499$	−13.3082	−0.595756	−0.297878	−	0.954604i	$-0.596279\pi$
$499$	−13.3082	−0.595756	−0.297878	+	0.954604i	$0.596279\pi$
$500$	0	0
$501$	5.03605	0.224994
$502$	0	0
$503$	− 37.7383i	− 1.68267i	−0.540516	−	0.841334i	$-0.681771\pi$
$503$	− 37.7383i	− 1.68267i	0.540516	−	0.841334i	$-0.318229\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.9854i	0.843173i
$508$	0	0
$509$	1.11982	0.0496351	0.0248176	−	0.999692i	$-0.492100\pi$
$509$	1.11982	0.0496351	0.0248176	+	0.999692i	$0.492100\pi$
$510$	0	0
$511$	10.9460	0.484223
$512$	0	0
$513$	− 29.6966i	− 1.31114i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.00000i	− 0.0879599i
$518$	0	0
$519$	−13.6870	−0.600793
$520$	0	0
$521$	−13.9100	−0.609407	−0.304703	−	0.952447i	$-0.598557\pi$
$521$	−13.9100	−0.609407	−0.304703	+	0.952447i	$0.598557\pi$
$522$	0	0
$523$	8.19799i	0.358473i	0.983806	+	0.179237i	$0.0573627\pi$
$523$	8.19799i	0.358473i	−0.983806	+	0.179237i	$0.942637\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 37.3747i	− 1.62807i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	9.48197	0.411483
$532$	0	0
$533$	− 1.38924i	− 0.0601749i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	26.9315i	1.16218i
$538$	0	0
$539$	−3.99104	−0.171906
$540$	0	0
$541$	23.5478	1.01240	0.506200	−	0.862416i	$-0.331050\pi$
$541$	23.5478	1.01240	0.506200	+	0.862416i	$0.331050\pi$
$542$	0	0
$543$	− 2.84413i	− 0.122053i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.77077i	− 0.0757128i	−0.999283	−	0.0378564i	$-0.987947\pi$
$547$	− 1.77077i	− 0.0757128i	0.999283	−	0.0378564i	$-0.0120530\pi$
$548$	0	0
$549$	3.81579	0.162854
$550$	0	0
$551$	51.3628	2.18813
$552$	0	0
$553$	10.4488i	0.444329i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.5872i	0.957052i	0.878073	+	0.478526i	$0.158829\pi$
$557$	22.5872i	0.957052i	−0.878073	+	0.478526i	$0.841171\pi$
$558$	0	0
$559$	−1.59283	−0.0673695
$560$	0	0
$561$	−6.93416	−0.292760
$562$	0	0
$563$	21.8504i	0.920884i	0.887690	+	0.460442i	$0.152309\pi$
$563$	21.8504i	0.920884i	−0.887690	+	0.460442i	$0.847691\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	8.72721i	0.366508i
$568$	0	0
$569$	−11.4737	−0.481002	−0.240501	−	0.970649i	$-0.577312\pi$
$569$	−11.4737	−0.481002	−0.240501	+	0.970649i	$0.577312\pi$
$570$	0	0
$571$	27.7085	1.15956	0.579782	−	0.814771i	$-0.303138\pi$
$571$	27.7085	1.15956	0.579782	+	0.814771i	$0.303138\pi$
$572$	0	0
$573$	− 1.13438i	− 0.0473893i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 20.3442i	− 0.846941i	−0.905910	−	0.423471i	$-0.860812\pi$
$577$	− 20.3442i	− 0.846941i	0.905910	−	0.423471i	$-0.139188\pi$
$578$	0	0
$579$	17.2099	0.715217
$580$	0	0
$581$	16.3684	0.679076
$582$	0	0
$583$	− 0.128782i	− 0.00533360i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.1690i	− 0.502268i	−0.967952	−	0.251134i	$-0.919197\pi$
$587$	− 12.1690i	− 0.502268i	0.967952	−	0.251134i	$-0.0808034\pi$
$588$	0	0
$589$	−35.6877	−1.47049
$590$	0	0
$591$	14.1600	0.582465
$592$	0	0
$593$	− 40.5180i	− 1.66388i	−0.554869	−	0.831938i	$-0.687231\pi$
$593$	− 40.5180i	− 1.66388i	0.554869	−	0.831938i	$-0.312769\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.81724i	0.197156i
$598$	0	0
$599$	28.8913	1.18047	0.590233	−	0.807233i	$-0.299036\pi$
$599$	28.8913	1.18047	0.590233	+	0.807233i	$0.299036\pi$
$600$	0	0
$601$	5.34278	0.217937	0.108968	−	0.994045i	$-0.465245\pi$
$601$	5.34278	0.217937	0.108968	+	0.994045i	$0.465245\pi$
$602$	0	0
$603$	9.03605i	0.367977i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.4376i	1.64131i	0.571422	+	0.820656i	$0.306392\pi$
$607$	40.4376i	1.64131i	−0.571422	+	0.820656i	$0.693608\pi$
$608$	0	0
$609$	−21.9612	−0.889915
$610$	0	0
$611$	−0.323404	−0.0130835
$612$	0	0
$613$	− 5.31154i	− 0.214531i	−0.994230	−	0.107266i	$-0.965790\pi$
$613$	− 5.31154i	− 0.214531i	0.994230	−	0.107266i	$-0.0342095\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.71680i	− 0.391183i	−0.980685	−	0.195592i	$-0.937337\pi$
$617$	− 9.71680i	− 0.391183i	0.980685	−	0.195592i	$-0.0626627\pi$
$618$	0	0
$619$	23.7562	0.954843	0.477421	−	0.878674i	$-0.341572\pi$
$619$	23.7562	0.954843	0.477421	+	0.878674i	$0.341572\pi$
$620$	0	0
$621$	5.64681	0.226599
$622$	0	0
$623$	− 26.5872i	− 1.06520i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.62117i	0.264424i
$628$	0	0
$629$	66.5872	2.65501
$630$	0	0
$631$	1.24234	0.0494566	0.0247283	−	0.999694i	$-0.492128\pi$
$631$	1.24234	0.0494566	0.0247283	+	0.999694i	$0.492128\pi$
$632$	0	0
$633$	− 23.7521i	− 0.944060i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.645359i	0.0255701i
$638$	0	0
$639$	6.29507	0.249029
$640$	0	0
$641$	−34.8864	−1.37793	−0.688966	−	0.724794i	$-0.741935\pi$
$641$	−34.8864	−1.37793	−0.688966	+	0.724794i	$0.741935\pi$
$642$	0	0
$643$	32.1592i	1.26824i	0.773236	+	0.634118i	$0.218637\pi$
$643$	32.1592i	1.26824i	−0.773236	+	0.634118i	$0.781363\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 23.9550i	− 0.941768i	−0.882195	−	0.470884i	$-0.843935\pi$
$647$	− 23.9550i	− 0.941768i	0.882195	−	0.470884i	$-0.156065\pi$
$648$	0	0
$649$	−9.48197	−0.372200
$650$	0	0
$651$	15.2590	0.598048
$652$	0	0
$653$	4.31926i	0.169026i	0.996422	+	0.0845128i	$0.0269334\pi$
$653$	4.31926i	0.169026i	−0.996422	+	0.0845128i	$0.973067\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.12878i	0.239107i
$658$	0	0
$659$	6.14961	0.239555	0.119777	−	0.992801i	$-0.461782\pi$
$659$	6.14961	0.239555	0.119777	+	0.992801i	$0.461782\pi$
$660$	0	0
$661$	−15.7473	−0.612497	−0.306249	−	0.951952i	$-0.599074\pi$
$661$	−15.7473	−0.612497	−0.306249	+	0.951952i	$0.599074\pi$
$662$	0	0
$663$	1.12127i	0.0435465i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.76663i	0.378165i
$668$	0	0
$669$	−31.1440	−1.20410
$670$	0	0
$671$	−3.81579	−0.147307
$672$	0	0
$673$	43.5783i	1.67982i	0.542727	+	0.839909i	$0.317392\pi$
$673$	43.5783i	1.67982i	−0.542727	+	0.839909i	$0.682608\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30.1801i	− 1.15991i	−0.814647	−	0.579957i	$-0.803069\pi$
$677$	− 30.1801i	− 1.15991i	0.814647	−	0.579957i	$-0.196931\pi$
$678$	0	0
$679$	−2.97581	−0.114201
$680$	0	0
$681$	−43.3836	−1.66247
$682$	0	0
$683$	29.2832i	1.12049i	0.828327	+	0.560245i	$0.189293\pi$
$683$	29.2832i	1.12049i	−0.828327	+	0.560245i	$0.810707\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 28.7881i	− 1.09834i
$688$	0	0
$689$	−0.0208243	−0.000793344 0
$690$	0	0
$691$	−14.9557	−0.568940	−0.284470	−	0.958685i	$-0.591818\pi$
$691$	−14.9557	−0.568940	−0.284470	+	0.958685i	$0.591818\pi$
$692$	0	0
$693$	1.13919i	0.0432744i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	54.9694i	2.08212i
$698$	0	0
$699$	8.08377	0.305756
$700$	0	0
$701$	24.9627	0.942828	0.471414	−	0.881912i	$-0.343744\pi$
$701$	24.9627	0.942828	0.471414	+	0.881912i	$0.343744\pi$
$702$	0	0
$703$	− 63.5816i	− 2.39803i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23.7104i	0.891722i
$708$	0	0
$709$	38.9717	1.46361	0.731806	−	0.681513i	$-0.238678\pi$
$709$	38.9717	1.46361	0.731806	+	0.681513i	$0.238678\pi$
$710$	0	0
$711$	−5.85039	−0.219407
$712$	0	0
$713$	− 6.78600i	− 0.254138i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.4466i	0.950320i
$718$	0	0
$719$	25.2978	0.943447	0.471724	−	0.881746i	$-0.343632\pi$
$719$	25.2978	0.943447	0.471724	+	0.881746i	$0.343632\pi$
$720$	0	0
$721$	12.1682	0.453168
$722$	0	0
$723$	− 23.5333i	− 0.875211i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 12.7417i	− 0.472562i	−0.971685	−	0.236281i	$-0.924071\pi$
$727$	− 12.7417i	− 0.472562i	0.971685	−	0.236281i	$-0.0759286\pi$
$728$	0	0
$729$	−29.6635	−1.09865
$730$	0	0
$731$	63.0249	2.33106
$732$	0	0
$733$	− 9.83247i	− 0.363170i	−0.983375	−	0.181585i	$-0.941877\pi$
$733$	− 9.83247i	− 0.363170i	0.983375	−	0.181585i	$-0.0581228\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.03605i	− 0.332847i
$738$	0	0
$739$	−22.6918	−0.834732	−0.417366	−	0.908738i	$-0.637047\pi$
$739$	−22.6918	−0.834732	−0.417366	+	0.908738i	$0.637047\pi$
$740$	0	0
$741$	1.07066	0.0393316
$742$	0	0
$743$	3.08858i	0.113309i	0.998394	+	0.0566546i	$0.0180434\pi$
$743$	3.08858i	0.113309i	−0.998394	+	0.0566546i	$0.981957\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.16484i	0.335324i
$748$	0	0
$749$	6.57760	0.240340
$750$	0	0
$751$	−10.7577	−0.392553	−0.196276	−	0.980549i	$-0.562885\pi$
$751$	−10.7577	−0.392553	−0.196276	+	0.980549i	$0.562885\pi$
$752$	0	0
$753$	− 42.4882i	− 1.54836i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19.5333i	0.709948i	0.934876	+	0.354974i	$0.115510\pi$
$757$	19.5333i	0.709948i	−0.934876	+	0.354974i	$0.884490\pi$
$758$	0	0
$759$	−1.25901	−0.0456993
$760$	0	0
$761$	−15.6933	−0.568881	−0.284440	−	0.958694i	$-0.591808\pi$
$761$	−15.6933	−0.568881	−0.284440	+	0.958694i	$0.591808\pi$
$762$	0	0
$763$	− 27.3304i	− 0.989429i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.53326i	0.0553626i
$768$	0	0
$769$	−11.4128	−0.411555	−0.205777	−	0.978599i	$-0.565972\pi$
$769$	−11.4128	−0.411555	−0.205777	+	0.978599i	$0.565972\pi$
$770$	0	0
$771$	−9.59910	−0.345703
$772$	0	0
$773$	− 22.5872i	− 0.812406i	−0.913783	−	0.406203i	$-0.866853\pi$
$773$	− 22.5872i	− 0.812406i	0.913783	−	0.406203i	$-0.133147\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.1857i	0.975280i
$778$	0	0
$779$	52.4882	1.88059
$780$	0	0
$781$	−6.29507	−0.225255
$782$	0	0
$783$	− 55.1503i	− 1.97091i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.42799i	0.157841i	0.996881	+	0.0789205i	$0.0251473\pi$
$787$	4.42799i	0.157841i	−0.996881	+	0.0789205i	$0.974853\pi$
$788$	0	0
$789$	0.821385	0.0292421
$790$	0	0
$791$	−4.92520	−0.175120
$792$	0	0
$793$	0.617021i	0.0219111i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.5124i	− 0.762009i	−0.924573	−	0.381005i	$-0.875578\pi$
$797$	− 21.5124i	− 0.762009i	0.924573	−	0.381005i	$-0.124422\pi$
$798$	0	0
$799$	12.7964	0.452705
$800$	0	0
$801$	14.8864	0.525987
$802$	0	0
$803$	− 6.12878i	− 0.216280i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.6323i	0.374273i
$808$	0	0
$809$	−7.37883	−0.259426	−0.129713	−	0.991552i	$-0.541406\pi$
$809$	−7.37883	−0.259426	−0.129713	+	0.991552i	$0.541406\pi$
$810$	0	0
$811$	29.0423	1.01981	0.509907	−	0.860230i	$-0.329680\pi$
$811$	29.0423	1.01981	0.509907	+	0.860230i	$0.329680\pi$
$812$	0	0
$813$	0.269425i	0.00944916i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 60.1801i	− 2.10543i
$818$	0	0
$819$	0.184210	0.00643682
$820$	0	0
$821$	5.70079	0.198959	0.0994794	−	0.995040i	$-0.468282\pi$
$821$	5.70079	0.198959	0.0994794	+	0.995040i	$0.468282\pi$
$822$	0	0
$823$	− 16.0034i	− 0.557842i	−0.960314	−	0.278921i	$-0.910023\pi$
$823$	− 16.0034i	− 0.557842i	0.960314	−	0.278921i	$-0.0899768\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 39.7937i	− 1.38376i	−0.722011	−	0.691882i	$-0.756782\pi$
$827$	− 39.7937i	− 1.38376i	0.722011	−	0.691882i	$-0.243218\pi$
$828$	0	0
$829$	23.3353	0.810467	0.405234	−	0.914213i	$-0.367190\pi$
$829$	23.3353	0.810467	0.405234	+	0.914213i	$0.367190\pi$
$830$	0	0
$831$	37.8746	1.31385
$832$	0	0
$833$	− 25.5355i	− 0.884752i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	38.3193i	1.32451i
$838$	0	0
$839$	−37.1469	−1.28245	−0.641227	−	0.767351i	$-0.721574\pi$
$839$	−37.1469	−1.28245	−0.641227	+	0.767351i	$0.721574\pi$
$840$	0	0
$841$	66.3870	2.28921
$842$	0	0
$843$	− 39.1619i	− 1.34881i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	15.7722i	0.541940i
$848$	0	0
$849$	−8.33796	−0.286158
$850$	0	0
$851$	12.0900	0.414441
$852$	0	0
$853$	− 53.7658i	− 1.84091i	−0.390851	−	0.920454i	$-0.627819\pi$
$853$	− 53.7658i	− 1.84091i	0.390851	−	0.920454i	$-0.372181\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.9675i	0.818715i	0.912374	+	0.409357i	$0.134247\pi$
$857$	23.9675i	0.818715i	−0.912374	+	0.409357i	$0.865753\pi$
$858$	0	0
$859$	−3.70705	−0.126483	−0.0632415	−	0.997998i	$-0.520144\pi$
$859$	−3.70705	−0.126483	−0.0632415	+	0.997998i	$0.520144\pi$
$860$	0	0
$861$	−22.4424	−0.764836
$862$	0	0
$863$	49.6296i	1.68941i	0.535232	+	0.844705i	$0.320224\pi$
$863$	49.6296i	1.68941i	−0.535232	+	0.844705i	$0.679776\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 19.5020i	− 0.662323i
$868$	0	0
$869$	5.85039	0.198461
$870$	0	0
$871$	−1.46115	−0.0495091
$872$	0	0
$873$	− 1.66618i	− 0.0563918i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.1004i	0.442371i	0.975232	+	0.221185i	$0.0709926\pi$
$877$	13.1004i	0.442371i	−0.975232	+	0.221185i	$0.929007\pi$
$878$	0	0
$879$	−1.54155	−0.0519951
$880$	0	0
$881$	3.88085	0.130749	0.0653746	−	0.997861i	$-0.479176\pi$
$881$	3.88085	0.130749	0.0653746	+	0.997861i	$0.479176\pi$
$882$	0	0
$883$	− 10.8131i	− 0.363890i	−0.983309	−	0.181945i	$-0.941761\pi$
$883$	− 10.8131i	− 0.363890i	0.983309	−	0.181945i	$-0.0582392\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.58387i	0.288218i	0.989562	+	0.144109i	$0.0460316\pi$
$887$	8.58387i	0.288218i	−0.989562	+	0.144109i	$0.953968\pi$
$888$	0	0
$889$	23.3836	0.784262
$890$	0	0
$891$	4.88645	0.163702
$892$	0	0
$893$	− 12.2188i	− 0.408887i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.203585i	0.00679751i
$898$	0	0
$899$	−66.2764	−2.21044
$900$	0	0
$901$	0.823974	0.0274505
$902$	0	0
$903$	25.7312i	0.856282i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	21.2728i	0.706351i	0.935557	+	0.353176i	$0.114898\pi$
$907$	21.2728i	0.706351i	−0.935557	+	0.353176i	$0.885102\pi$
$908$	0	0
$909$	−13.2757	−0.440327
$910$	0	0
$911$	−27.6441	−0.915890	−0.457945	−	0.888980i	$-0.651414\pi$
$911$	−27.6441	−0.915890	−0.457945	+	0.888980i	$0.651414\pi$
$912$	0	0
$913$	− 9.16484i	− 0.303312i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 19.5124i	− 0.644357i
$918$	0	0
$919$	0.994404	0.0328024	0.0164012	−	0.999865i	$-0.494779\pi$
$919$	0.994404	0.0328024	0.0164012	+	0.999865i	$0.494779\pi$
$920$	0	0
$921$	26.1455	0.861522
$922$	0	0
$923$	1.01793i	0.0335054i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.81309i	0.223771i
$928$	0	0
$929$	18.3747	0.602854	0.301427	−	0.953489i	$-0.402537\pi$
$929$	18.3747	0.602854	0.301427	+	0.953489i	$0.402537\pi$
$930$	0	0
$931$	−24.3829	−0.799116
$932$	0	0
$933$	24.7639i	0.810735i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.0942i	0.787122i	0.919298	+	0.393561i	$0.128757\pi$
$937$	24.0942i	0.787122i	−0.919298	+	0.393561i	$0.871243\pi$
$938$	0	0
$939$	29.2459	0.954404
$940$	0	0
$941$	51.9244	1.69269	0.846344	−	0.532637i	$-0.178799\pi$
$941$	51.9244	1.69269	0.846344	+	0.532637i	$0.178799\pi$
$942$	0	0
$943$	9.98062i	0.325014i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.72643i	0.153588i	0.997047	+	0.0767941i	$0.0244684\pi$
$947$	4.72643i	0.153588i	−0.997047	+	0.0767941i	$0.975532\pi$
$948$	0	0
$949$	−0.991037	−0.0321704
$950$	0	0
$951$	4.82330	0.156406
$952$	0	0
$953$	3.57682i	0.115865i	0.998321	+	0.0579323i	$0.0184508\pi$
$953$	3.57682i	0.115865i	−0.998321	+	0.0579323i	$0.981549\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.2963i	0.397483i
$958$	0	0
$959$	21.5768	0.696752
$960$	0	0
$961$	15.0498	0.485478
$962$	0	0
$963$	3.68286i	0.118679i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 33.9646i	− 1.09223i	−0.837711	−	0.546114i	$-0.816106\pi$
$967$	− 33.9646i	− 1.09223i	0.837711	−	0.546114i	$-0.183894\pi$
$968$	0	0
$969$	−42.3636	−1.36092
$970$	0	0
$971$	−56.0561	−1.79893	−0.899463	−	0.436997i	$-0.856042\pi$
$971$	−56.0561	−1.79893	−0.899463	+	0.436997i	$0.856042\pi$
$972$	0	0
$973$	21.9292i	0.703019i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.1219i	1.69952i	0.527169	+	0.849761i	$0.323254\pi$
$977$	53.1219i	1.69952i	−0.527169	+	0.849761i	$0.676746\pi$
$978$	0	0
$979$	−14.8864	−0.475773
$980$	0	0
$981$	15.3026	0.488574
$982$	0	0
$983$	28.7833i	0.918045i	0.888425	+	0.459022i	$0.151800\pi$
$983$	28.7833i	0.918045i	−0.888425	+	0.459022i	$0.848200\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.22441i	0.166295i
$988$	0	0
$989$	11.4432	0.363873
$990$	0	0
$991$	48.1641	1.52998	0.764991	−	0.644041i	$-0.222743\pi$
$991$	48.1641	1.52998	0.764991	+	0.644041i	$0.222743\pi$
$992$	0	0
$993$	− 18.8385i	− 0.597823i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 43.5153i	− 1.37814i	−0.724693	−	0.689072i	$-0.758018\pi$
$997$	− 43.5153i	− 1.37814i	0.724693	−	0.689072i	$-0.241982\pi$
$998$	0	0
$999$	−68.2701	−2.15997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.q.4049.4		6
5.2	odd	4		920.2.a.i.1.2	✓	3
5.3	odd	4		4600.2.a.v.1.2		3
5.4	even	2	inner	4600.2.e.q.4049.3		6
15.2	even	4		8280.2.a.bl.1.2		3
20.3	even	4		9200.2.a.ci.1.2		3
20.7	even	4		1840.2.a.q.1.2		3
40.27	even	4		7360.2.a.cf.1.2		3
40.37	odd	4		7360.2.a.bw.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.i.1.2	✓	3	5.2	odd	4
1840.2.a.q.1.2		3	20.7	even	4
4600.2.a.v.1.2		3	5.3	odd	4
4600.2.e.q.4049.3		6	5.4	even	2	inner
4600.2.e.q.4049.4		6	1.1	even	1	trivial
7360.2.a.bw.1.2		3	40.37	odd	4
7360.2.a.cf.1.2		3	40.27	even	4
8280.2.a.bl.1.2		3	15.2	even	4
9200.2.a.ci.1.2		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)

\(f(q)\)	\(=\)	\(q+1.46260i q^{3} -1.53740i q^{7} +0.860806 q^{9} -0.860806 q^{11} +0.139194i q^{13} -5.50761i q^{17} -5.25901 q^{19} +2.24860 q^{21} -1.00000i q^{23} +5.64681i q^{27} -9.76663 q^{29} +6.78600 q^{31} -1.25901i q^{33} +12.0900i q^{37} -0.203585 q^{39} -9.98062 q^{41} +11.4432i q^{43} +2.32340i q^{47} +4.63640 q^{49} +8.05543 q^{51} +0.149606i q^{53} -7.69182i q^{57} +11.0152 q^{59} +4.43281 q^{61} -1.32340i q^{63} +10.4972i q^{67} +1.46260 q^{69} +7.31299 q^{71} +7.11982i q^{73} +1.32340i q^{77} -6.79641 q^{79} -5.67660 q^{81} +10.6468i q^{83} -14.2847i q^{87} +17.2936 q^{89} +0.213997 q^{91} +9.92520i q^{93} -1.93561i q^{97} -0.740987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89}+ \cdots - 22 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 6 * q^11 - 14 * q^19 - 12 * q^21 + 2 * q^29 + 20 * q^31 - 14 * q^39 - 20 * q^41 - 12 * q^49 + 18 * q^51 - 20 * q^59 - 26 * q^61 + 4 * q^69 + 20 * q^71 - 28 * q^79 - 50 * q^81 + 40 * q^89 + 22 * q^91 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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