LMFDB - Embedded newform 6036.2.a.i.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.34119 q^{5} +2.75130 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.34119 q^{5} +2.75130 q^{7} +1.00000 q^{9} +2.38248 q^{11} +3.79863 q^{13} -3.34119 q^{15} +4.35942 q^{17} -8.55652 q^{19} -2.75130 q^{21} -8.57420 q^{23} +6.16358 q^{25} -1.00000 q^{27} +5.46055 q^{29} -9.44228 q^{31} -2.38248 q^{33} +9.19262 q^{35} +11.3552 q^{37} -3.79863 q^{39} +0.318325 q^{41} +6.22362 q^{43} +3.34119 q^{45} +10.3702 q^{47} +0.569636 q^{49} -4.35942 q^{51} +8.92858 q^{53} +7.96032 q^{55} +8.55652 q^{57} +14.2172 q^{59} +2.18215 q^{61} +2.75130 q^{63} +12.6920 q^{65} +10.2702 q^{67} +8.57420 q^{69} -9.42568 q^{71} -15.9862 q^{73} -6.16358 q^{75} +6.55490 q^{77} -0.894929 q^{79} +1.00000 q^{81} -14.2377 q^{83} +14.5657 q^{85} -5.46055 q^{87} -13.4809 q^{89} +10.4512 q^{91} +9.44228 q^{93} -28.5890 q^{95} +12.1034 q^{97} +2.38248 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.34119	1.49423	0.747114	−	0.664696i	$-0.231439\pi$
$5$	3.34119	1.49423	0.747114	+	0.664696i	$0.231439\pi$
$6$	0	0
$7$	2.75130	1.03989	0.519946	−	0.854199i	$-0.325952\pi$
$7$	2.75130	1.03989	0.519946	+	0.854199i	$0.325952\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.38248	0.718344	0.359172	−	0.933271i	$-0.383059\pi$
$11$	2.38248	0.718344	0.359172	+	0.933271i	$0.383059\pi$
$12$	0	0
$13$	3.79863	1.05355	0.526776	−	0.850004i	$-0.323401\pi$
$13$	3.79863	1.05355	0.526776	+	0.850004i	$0.323401\pi$
$14$	0	0
$15$	−3.34119	−0.862693
$16$	0	0
$17$	4.35942	1.05732	0.528658	−	0.848835i	$-0.322696\pi$
$17$	4.35942	1.05732	0.528658	+	0.848835i	$0.322696\pi$
$18$	0	0
$19$	−8.55652	−1.96300	−0.981500	−	0.191462i	$-0.938677\pi$
$19$	−8.55652	−1.96300	−0.981500	+	0.191462i	$0.938677\pi$
$20$	0	0
$21$	−2.75130	−0.600382
$22$	0	0
$23$	−8.57420	−1.78784	−0.893922	−	0.448223i	$-0.852057\pi$
$23$	−8.57420	−1.78784	−0.893922	+	0.448223i	$0.852057\pi$
$24$	0	0
$25$	6.16358	1.23272
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.46055	1.01400	0.507000	−	0.861946i	$-0.330755\pi$
$29$	5.46055	1.01400	0.507000	+	0.861946i	$0.330755\pi$
$30$	0	0
$31$	−9.44228	−1.69588	−0.847942	−	0.530090i	$-0.822158\pi$
$31$	−9.44228	−1.69588	−0.847942	+	0.530090i	$0.822158\pi$
$32$	0	0
$33$	−2.38248	−0.414736
$34$	0	0
$35$	9.19262	1.55384
$36$	0	0
$37$	11.3552	1.86678	0.933392	−	0.358859i	$-0.116834\pi$
$37$	11.3552	1.86678	0.933392	+	0.358859i	$0.116834\pi$
$38$	0	0
$39$	−3.79863	−0.608268
$40$	0	0
$41$	0.318325	0.0497140	0.0248570	−	0.999691i	$-0.492087\pi$
$41$	0.318325	0.0497140	0.0248570	+	0.999691i	$0.492087\pi$
$42$	0	0
$43$	6.22362	0.949093	0.474547	−	0.880230i	$-0.342612\pi$
$43$	6.22362	0.949093	0.474547	+	0.880230i	$0.342612\pi$
$44$	0	0
$45$	3.34119	0.498076
$46$	0	0
$47$	10.3702	1.51264	0.756322	−	0.654200i	$-0.226994\pi$
$47$	10.3702	1.51264	0.756322	+	0.654200i	$0.226994\pi$
$48$	0	0
$49$	0.569636	0.0813766
$50$	0	0
$51$	−4.35942	−0.610441
$52$	0	0
$53$	8.92858	1.22644	0.613218	−	0.789914i	$-0.289875\pi$
$53$	8.92858	1.22644	0.613218	+	0.789914i	$0.289875\pi$
$54$	0	0
$55$	7.96032	1.07337
$56$	0	0
$57$	8.55652	1.13334
$58$	0	0
$59$	14.2172	1.85092	0.925461	−	0.378843i	$-0.123678\pi$
$59$	14.2172	1.85092	0.925461	+	0.378843i	$0.123678\pi$
$60$	0	0
$61$	2.18215	0.279396	0.139698	−	0.990194i	$-0.455387\pi$
$61$	2.18215	0.279396	0.139698	+	0.990194i	$0.455387\pi$
$62$	0	0
$63$	2.75130	0.346631
$64$	0	0
$65$	12.6920	1.57425
$66$	0	0
$67$	10.2702	1.25470	0.627352	−	0.778736i	$-0.284139\pi$
$67$	10.2702	1.25470	0.627352	+	0.778736i	$0.284139\pi$
$68$	0	0
$69$	8.57420	1.03221
$70$	0	0
$71$	−9.42568	−1.11862	−0.559311	−	0.828958i	$-0.688934\pi$
$71$	−9.42568	−1.11862	−0.559311	+	0.828958i	$0.688934\pi$
$72$	0	0
$73$	−15.9862	−1.87104	−0.935519	−	0.353277i	$-0.885067\pi$
$73$	−15.9862	−1.87104	−0.935519	+	0.353277i	$0.885067\pi$
$74$	0	0
$75$	−6.16358	−0.711709
$76$	0	0
$77$	6.55490	0.747001
$78$	0	0
$79$	−0.894929	−0.100687	−0.0503437	−	0.998732i	$-0.516032\pi$
$79$	−0.894929	−0.100687	−0.0503437	+	0.998732i	$0.516032\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.2377	−1.56279	−0.781396	−	0.624035i	$-0.785492\pi$
$83$	−14.2377	−1.56279	−0.781396	+	0.624035i	$0.785492\pi$
$84$	0	0
$85$	14.5657	1.57987
$86$	0	0
$87$	−5.46055	−0.585433
$88$	0	0
$89$	−13.4809	−1.42898	−0.714488	−	0.699647i	$-0.753340\pi$
$89$	−13.4809	−1.42898	−0.714488	+	0.699647i	$0.753340\pi$
$90$	0	0
$91$	10.4512	1.09558
$92$	0	0
$93$	9.44228	0.979119
$94$	0	0
$95$	−28.5890	−2.93317
$96$	0	0
$97$	12.1034	1.22892	0.614458	−	0.788949i	$-0.289375\pi$
$97$	12.1034	1.22892	0.614458	+	0.788949i	$0.289375\pi$
$98$	0	0
$99$	2.38248	0.239448
$100$	0	0
$101$	0.0702209	0.00698724	0.00349362	−	0.999994i	$-0.498888\pi$
$101$	0.0702209	0.00698724	0.00349362	+	0.999994i	$0.498888\pi$
$102$	0	0
$103$	9.55102	0.941090	0.470545	−	0.882376i	$-0.344057\pi$
$103$	9.55102	0.941090	0.470545	+	0.882376i	$0.344057\pi$
$104$	0	0
$105$	−9.19262	−0.897108
$106$	0	0
$107$	2.92209	0.282489	0.141244	−	0.989975i	$-0.454890\pi$
$107$	2.92209	0.282489	0.141244	+	0.989975i	$0.454890\pi$
$108$	0	0
$109$	−8.81386	−0.844215	−0.422107	−	0.906546i	$-0.638709\pi$
$109$	−8.81386	−0.844215	−0.422107	+	0.906546i	$0.638709\pi$
$110$	0	0
$111$	−11.3552	−1.07779
$112$	0	0
$113$	0.104921	0.00987011	0.00493505	−	0.999988i	$-0.498429\pi$
$113$	0.104921	0.00987011	0.00493505	+	0.999988i	$0.498429\pi$
$114$	0	0
$115$	−28.6481	−2.67144
$116$	0	0
$117$	3.79863	0.351184
$118$	0	0
$119$	11.9941	1.09949
$120$	0	0
$121$	−5.32380	−0.483982
$122$	0	0
$123$	−0.318325	−0.0287024
$124$	0	0
$125$	3.88774	0.347730
$126$	0	0
$127$	−10.1865	−0.903906	−0.451953	−	0.892042i	$-0.649273\pi$
$127$	−10.1865	−0.903906	−0.451953	+	0.892042i	$0.649273\pi$
$128$	0	0
$129$	−6.22362	−0.547959
$130$	0	0
$131$	−3.76921	−0.329317	−0.164658	−	0.986351i	$-0.552652\pi$
$131$	−3.76921	−0.329317	−0.164658	+	0.986351i	$0.552652\pi$
$132$	0	0
$133$	−23.5415	−2.04131
$134$	0	0
$135$	−3.34119	−0.287564
$136$	0	0
$137$	11.5138	0.983691	0.491845	−	0.870683i	$-0.336323\pi$
$137$	11.5138	0.983691	0.491845	+	0.870683i	$0.336323\pi$
$138$	0	0
$139$	19.7901	1.67858	0.839288	−	0.543687i	$-0.182972\pi$
$139$	19.7901	1.67858	0.839288	+	0.543687i	$0.182972\pi$
$140$	0	0
$141$	−10.3702	−0.873325
$142$	0	0
$143$	9.05016	0.756813
$144$	0	0
$145$	18.2448	1.51515
$146$	0	0
$147$	−0.569636	−0.0469828
$148$	0	0
$149$	−7.39253	−0.605620	−0.302810	−	0.953051i	$-0.597925\pi$
$149$	−7.39253	−0.605620	−0.302810	+	0.953051i	$0.597925\pi$
$150$	0	0
$151$	−10.1071	−0.822508	−0.411254	−	0.911521i	$-0.634909\pi$
$151$	−10.1071	−0.822508	−0.411254	+	0.911521i	$0.634909\pi$
$152$	0	0
$153$	4.35942	0.352438
$154$	0	0
$155$	−31.5485	−2.53404
$156$	0	0
$157$	19.4617	1.55321	0.776605	−	0.629988i	$-0.216940\pi$
$157$	19.4617	1.55321	0.776605	+	0.629988i	$0.216940\pi$
$158$	0	0
$159$	−8.92858	−0.708083
$160$	0	0
$161$	−23.5902	−1.85917
$162$	0	0
$163$	8.89534	0.696737	0.348368	−	0.937358i	$-0.386736\pi$
$163$	8.89534	0.696737	0.348368	+	0.937358i	$0.386736\pi$
$164$	0	0
$165$	−7.96032	−0.619710
$166$	0	0
$167$	14.0788	1.08945	0.544725	−	0.838615i	$-0.316634\pi$
$167$	14.0788	1.08945	0.544725	+	0.838615i	$0.316634\pi$
$168$	0	0
$169$	1.42963	0.109971
$170$	0	0
$171$	−8.55652	−0.654333
$172$	0	0
$173$	24.0435	1.82799	0.913997	−	0.405722i	$-0.132980\pi$
$173$	24.0435	1.82799	0.913997	+	0.405722i	$0.132980\pi$
$174$	0	0
$175$	16.9578	1.28189
$176$	0	0
$177$	−14.2172	−1.06863
$178$	0	0
$179$	3.32370	0.248425	0.124213	−	0.992256i	$-0.460360\pi$
$179$	3.32370	0.248425	0.124213	+	0.992256i	$0.460360\pi$
$180$	0	0
$181$	−20.9008	−1.55354	−0.776772	−	0.629782i	$-0.783144\pi$
$181$	−20.9008	−1.55354	−0.776772	+	0.629782i	$0.783144\pi$
$182$	0	0
$183$	−2.18215	−0.161309
$184$	0	0
$185$	37.9399	2.78940
$186$	0	0
$187$	10.3862	0.759516
$188$	0	0
$189$	−2.75130	−0.200127
$190$	0	0
$191$	5.09229	0.368465	0.184232	−	0.982883i	$-0.441020\pi$
$191$	5.09229	0.368465	0.184232	+	0.982883i	$0.441020\pi$
$192$	0	0
$193$	−2.77353	−0.199643	−0.0998214	−	0.995005i	$-0.531827\pi$
$193$	−2.77353	−0.199643	−0.0998214	+	0.995005i	$0.531827\pi$
$194$	0	0
$195$	−12.6920	−0.908891
$196$	0	0
$197$	−24.6932	−1.75932	−0.879659	−	0.475604i	$-0.842230\pi$
$197$	−24.6932	−1.75932	−0.879659	+	0.475604i	$0.842230\pi$
$198$	0	0
$199$	5.61734	0.398203	0.199101	−	0.979979i	$-0.436198\pi$
$199$	5.61734	0.398203	0.199101	+	0.979979i	$0.436198\pi$
$200$	0	0
$201$	−10.2702	−0.724404
$202$	0	0
$203$	15.0236	1.05445
$204$	0	0
$205$	1.06359	0.0742840
$206$	0	0
$207$	−8.57420	−0.595948
$208$	0	0
$209$	−20.3857	−1.41011
$210$	0	0
$211$	−15.7765	−1.08610	−0.543050	−	0.839701i	$-0.682730\pi$
$211$	−15.7765	−1.08610	−0.543050	+	0.839701i	$0.682730\pi$
$212$	0	0
$213$	9.42568	0.645837
$214$	0	0
$215$	20.7943	1.41816
$216$	0	0
$217$	−25.9785	−1.76354
$218$	0	0
$219$	15.9862	1.08024
$220$	0	0
$221$	16.5599	1.11394
$222$	0	0
$223$	−9.99300	−0.669181	−0.334591	−	0.942364i	$-0.608598\pi$
$223$	−9.99300	−0.669181	−0.334591	+	0.942364i	$0.608598\pi$
$224$	0	0
$225$	6.16358	0.410905
$226$	0	0
$227$	−3.27449	−0.217335	−0.108668	−	0.994078i	$-0.534658\pi$
$227$	−3.27449	−0.217335	−0.108668	+	0.994078i	$0.534658\pi$
$228$	0	0
$229$	17.0376	1.12588	0.562939	−	0.826498i	$-0.309670\pi$
$229$	17.0376	1.12588	0.562939	+	0.826498i	$0.309670\pi$
$230$	0	0
$231$	−6.55490	−0.431281
$232$	0	0
$233$	−8.80004	−0.576510	−0.288255	−	0.957554i	$-0.593075\pi$
$233$	−8.80004	−0.576510	−0.288255	+	0.957554i	$0.593075\pi$
$234$	0	0
$235$	34.6487	2.26023
$236$	0	0
$237$	0.894929	0.0581319
$238$	0	0
$239$	21.6734	1.40193	0.700967	−	0.713194i	$-0.252752\pi$
$239$	21.6734	1.40193	0.700967	+	0.713194i	$0.252752\pi$
$240$	0	0
$241$	−23.6003	−1.52023	−0.760113	−	0.649791i	$-0.774857\pi$
$241$	−23.6003	−1.52023	−0.760113	+	0.649791i	$0.774857\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.90327	0.121595
$246$	0	0
$247$	−32.5031	−2.06812
$248$	0	0
$249$	14.2377	0.902279
$250$	0	0
$251$	−18.7082	−1.18085	−0.590425	−	0.807092i	$-0.701040\pi$
$251$	−18.7082	−1.18085	−0.590425	+	0.807092i	$0.701040\pi$
$252$	0	0
$253$	−20.4278	−1.28429
$254$	0	0
$255$	−14.5657	−0.912138
$256$	0	0
$257$	16.1631	1.00823	0.504114	−	0.863637i	$-0.331819\pi$
$257$	16.1631	1.00823	0.504114	+	0.863637i	$0.331819\pi$
$258$	0	0
$259$	31.2415	1.94125
$260$	0	0
$261$	5.46055	0.338000
$262$	0	0
$263$	0.573277	0.0353498	0.0176749	−	0.999844i	$-0.494374\pi$
$263$	0.573277	0.0353498	0.0176749	+	0.999844i	$0.494374\pi$
$264$	0	0
$265$	29.8321	1.83257
$266$	0	0
$267$	13.4809	0.825020
$268$	0	0
$269$	−9.81424	−0.598384	−0.299192	−	0.954193i	$-0.596717\pi$
$269$	−9.81424	−0.598384	−0.299192	+	0.954193i	$0.596717\pi$
$270$	0	0
$271$	23.4592	1.42505	0.712523	−	0.701648i	$-0.247552\pi$
$271$	23.4592	1.42505	0.712523	+	0.701648i	$0.247552\pi$
$272$	0	0
$273$	−10.4512	−0.632534
$274$	0	0
$275$	14.6846	0.885514
$276$	0	0
$277$	5.60572	0.336815	0.168407	−	0.985717i	$-0.446138\pi$
$277$	5.60572	0.336815	0.168407	+	0.985717i	$0.446138\pi$
$278$	0	0
$279$	−9.44228	−0.565294
$280$	0	0
$281$	−5.70978	−0.340617	−0.170309	−	0.985391i	$-0.554476\pi$
$281$	−5.70978	−0.340617	−0.170309	+	0.985391i	$0.554476\pi$
$282$	0	0
$283$	12.3253	0.732661	0.366330	−	0.930485i	$-0.380614\pi$
$283$	12.3253	0.732661	0.366330	+	0.930485i	$0.380614\pi$
$284$	0	0
$285$	28.5890	1.69347
$286$	0	0
$287$	0.875806	0.0516972
$288$	0	0
$289$	2.00457	0.117916
$290$	0	0
$291$	−12.1034	−0.709515
$292$	0	0
$293$	−17.4543	−1.01969	−0.509846	−	0.860266i	$-0.670298\pi$
$293$	−17.4543	−1.01969	−0.509846	+	0.860266i	$0.670298\pi$
$294$	0	0
$295$	47.5024	2.76570
$296$	0	0
$297$	−2.38248	−0.138245
$298$	0	0
$299$	−32.5702	−1.88359
$300$	0	0
$301$	17.1230	0.986955
$302$	0	0
$303$	−0.0702209	−0.00403409
$304$	0	0
$305$	7.29098	0.417480
$306$	0	0
$307$	15.8301	0.903474	0.451737	−	0.892151i	$-0.350805\pi$
$307$	15.8301	0.903474	0.451737	+	0.892151i	$0.350805\pi$
$308$	0	0
$309$	−9.55102	−0.543339
$310$	0	0
$311$	32.6873	1.85353	0.926763	−	0.375646i	$-0.122579\pi$
$311$	32.6873	1.85353	0.926763	+	0.375646i	$0.122579\pi$
$312$	0	0
$313$	−8.86306	−0.500970	−0.250485	−	0.968120i	$-0.580590\pi$
$313$	−8.86306	−0.500970	−0.250485	+	0.968120i	$0.580590\pi$
$314$	0	0
$315$	9.19262	0.517945
$316$	0	0
$317$	−9.99616	−0.561440	−0.280720	−	0.959790i	$-0.590573\pi$
$317$	−9.99616	−0.561440	−0.280720	+	0.959790i	$0.590573\pi$
$318$	0	0
$319$	13.0096	0.728400
$320$	0	0
$321$	−2.92209	−0.163095
$322$	0	0
$323$	−37.3015	−2.07551
$324$	0	0
$325$	23.4132	1.29873
$326$	0	0
$327$	8.81386	0.487407
$328$	0	0
$329$	28.5314	1.57299
$330$	0	0
$331$	9.37191	0.515127	0.257563	−	0.966261i	$-0.417080\pi$
$331$	9.37191	0.515127	0.257563	+	0.966261i	$0.417080\pi$
$332$	0	0
$333$	11.3552	0.622261
$334$	0	0
$335$	34.3147	1.87481
$336$	0	0
$337$	11.2818	0.614561	0.307281	−	0.951619i	$-0.400581\pi$
$337$	11.2818	0.614561	0.307281	+	0.951619i	$0.400581\pi$
$338$	0	0
$339$	−0.104921	−0.00569851
$340$	0	0
$341$	−22.4960	−1.21823
$342$	0	0
$343$	−17.6918	−0.955270
$344$	0	0
$345$	28.6481	1.54236
$346$	0	0
$347$	−23.3047	−1.25106	−0.625532	−	0.780198i	$-0.715118\pi$
$347$	−23.3047	−1.25106	−0.625532	+	0.780198i	$0.715118\pi$
$348$	0	0
$349$	−2.39186	−0.128033	−0.0640166	−	0.997949i	$-0.520391\pi$
$349$	−2.39186	−0.128033	−0.0640166	+	0.997949i	$0.520391\pi$
$350$	0	0
$351$	−3.79863	−0.202756
$352$	0	0
$353$	−1.32631	−0.0705925	−0.0352963	−	0.999377i	$-0.511237\pi$
$353$	−1.32631	−0.0705925	−0.0352963	+	0.999377i	$0.511237\pi$
$354$	0	0
$355$	−31.4930	−1.67148
$356$	0	0
$357$	−11.9941	−0.634793
$358$	0	0
$359$	−19.4270	−1.02532	−0.512658	−	0.858593i	$-0.671339\pi$
$359$	−19.4270	−1.02532	−0.512658	+	0.858593i	$0.671339\pi$
$360$	0	0
$361$	54.2140	2.85337
$362$	0	0
$363$	5.32380	0.279427
$364$	0	0
$365$	−53.4128	−2.79576
$366$	0	0
$367$	−16.7767	−0.875738	−0.437869	−	0.899039i	$-0.644267\pi$
$367$	−16.7767	−0.875738	−0.437869	+	0.899039i	$0.644267\pi$
$368$	0	0
$369$	0.318325	0.0165713
$370$	0	0
$371$	24.5652	1.27536
$372$	0	0
$373$	4.69909	0.243310	0.121655	−	0.992572i	$-0.461180\pi$
$373$	4.69909	0.243310	0.121655	+	0.992572i	$0.461180\pi$
$374$	0	0
$375$	−3.88774	−0.200762
$376$	0	0
$377$	20.7426	1.06830
$378$	0	0
$379$	−4.09942	−0.210573	−0.105286	−	0.994442i	$-0.533576\pi$
$379$	−4.09942	−0.210573	−0.105286	+	0.994442i	$0.533576\pi$
$380$	0	0
$381$	10.1865	0.521870
$382$	0	0
$383$	6.61922	0.338227	0.169113	−	0.985597i	$-0.445910\pi$
$383$	6.61922	0.338227	0.169113	+	0.985597i	$0.445910\pi$
$384$	0	0
$385$	21.9012	1.11619
$386$	0	0
$387$	6.22362	0.316364
$388$	0	0
$389$	−22.5106	−1.14133	−0.570666	−	0.821182i	$-0.693315\pi$
$389$	−22.5106	−1.14133	−0.570666	+	0.821182i	$0.693315\pi$
$390$	0	0
$391$	−37.3786	−1.89031
$392$	0	0
$393$	3.76921	0.190131
$394$	0	0
$395$	−2.99013	−0.150450
$396$	0	0
$397$	−28.0679	−1.40869	−0.704344	−	0.709859i	$-0.748759\pi$
$397$	−28.0679	−1.40869	−0.704344	+	0.709859i	$0.748759\pi$
$398$	0	0
$399$	23.5415	1.17855
$400$	0	0
$401$	−18.6646	−0.932065	−0.466033	−	0.884768i	$-0.654317\pi$
$401$	−18.6646	−0.932065	−0.466033	+	0.884768i	$0.654317\pi$
$402$	0	0
$403$	−35.8678	−1.78670
$404$	0	0
$405$	3.34119	0.166025
$406$	0	0
$407$	27.0535	1.34099
$408$	0	0
$409$	−22.2396	−1.09968	−0.549838	−	0.835271i	$-0.685311\pi$
$409$	−22.2396	−1.09968	−0.549838	+	0.835271i	$0.685311\pi$
$410$	0	0
$411$	−11.5138	−0.567934
$412$	0	0
$413$	39.1157	1.92476
$414$	0	0
$415$	−47.5710	−2.33517
$416$	0	0
$417$	−19.7901	−0.969127
$418$	0	0
$419$	−24.1920	−1.18186	−0.590928	−	0.806724i	$-0.701238\pi$
$419$	−24.1920	−1.18186	−0.590928	+	0.806724i	$0.701238\pi$
$420$	0	0
$421$	15.7615	0.768169	0.384085	−	0.923298i	$-0.374517\pi$
$421$	15.7615	0.768169	0.384085	+	0.923298i	$0.374517\pi$
$422$	0	0
$423$	10.3702	0.504215
$424$	0	0
$425$	26.8696	1.30337
$426$	0	0
$427$	6.00374	0.290541
$428$	0	0
$429$	−9.05016	−0.436946
$430$	0	0
$431$	1.14061	0.0549412	0.0274706	−	0.999623i	$-0.491255\pi$
$431$	1.14061	0.0549412	0.0274706	+	0.999623i	$0.491255\pi$
$432$	0	0
$433$	22.9074	1.10086	0.550429	−	0.834882i	$-0.314464\pi$
$433$	22.9074	1.10086	0.550429	+	0.834882i	$0.314464\pi$
$434$	0	0
$435$	−18.2448	−0.874770
$436$	0	0
$437$	73.3653	3.50954
$438$	0	0
$439$	0.438451	0.0209261	0.0104631	−	0.999945i	$-0.496669\pi$
$439$	0.438451	0.0209261	0.0104631	+	0.999945i	$0.496669\pi$
$440$	0	0
$441$	0.569636	0.0271255
$442$	0	0
$443$	18.3047	0.869684	0.434842	−	0.900507i	$-0.356804\pi$
$443$	18.3047	0.869684	0.434842	+	0.900507i	$0.356804\pi$
$444$	0	0
$445$	−45.0424	−2.13522
$446$	0	0
$447$	7.39253	0.349655
$448$	0	0
$449$	−36.5035	−1.72271	−0.861353	−	0.508007i	$-0.830382\pi$
$449$	−36.5035	−1.72271	−0.861353	+	0.508007i	$0.830382\pi$
$450$	0	0
$451$	0.758402	0.0357118
$452$	0	0
$453$	10.1071	0.474875
$454$	0	0
$455$	34.9194	1.63705
$456$	0	0
$457$	32.6533	1.52746	0.763729	−	0.645537i	$-0.223367\pi$
$457$	32.6533	1.52746	0.763729	+	0.645537i	$0.223367\pi$
$458$	0	0
$459$	−4.35942	−0.203480
$460$	0	0
$461$	2.44606	0.113925	0.0569623	−	0.998376i	$-0.481859\pi$
$461$	2.44606	0.113925	0.0569623	+	0.998376i	$0.481859\pi$
$462$	0	0
$463$	9.73988	0.452651	0.226325	−	0.974052i	$-0.427329\pi$
$463$	9.73988	0.452651	0.226325	+	0.974052i	$0.427329\pi$
$464$	0	0
$465$	31.5485	1.46303
$466$	0	0
$467$	−11.6541	−0.539287	−0.269643	−	0.962960i	$-0.586906\pi$
$467$	−11.6541	−0.539287	−0.269643	+	0.962960i	$0.586906\pi$
$468$	0	0
$469$	28.2564	1.30476
$470$	0	0
$471$	−19.4617	−0.896746
$472$	0	0
$473$	14.8276	0.681776
$474$	0	0
$475$	−52.7388	−2.41982
$476$	0	0
$477$	8.92858	0.408812
$478$	0	0
$479$	−11.6547	−0.532516	−0.266258	−	0.963902i	$-0.585787\pi$
$479$	−11.6547	−0.532516	−0.266258	+	0.963902i	$0.585787\pi$
$480$	0	0
$481$	43.1343	1.96675
$482$	0	0
$483$	23.5902	1.07339
$484$	0	0
$485$	40.4399	1.83628
$486$	0	0
$487$	17.0231	0.771390	0.385695	−	0.922626i	$-0.373962\pi$
$487$	17.0231	0.771390	0.385695	+	0.922626i	$0.373962\pi$
$488$	0	0
$489$	−8.89534	−0.402261
$490$	0	0
$491$	9.21720	0.415966	0.207983	−	0.978132i	$-0.433310\pi$
$491$	9.21720	0.415966	0.207983	+	0.978132i	$0.433310\pi$
$492$	0	0
$493$	23.8049	1.07212
$494$	0	0
$495$	7.96032	0.357790
$496$	0	0
$497$	−25.9329	−1.16325
$498$	0	0
$499$	−22.4665	−1.00574	−0.502870	−	0.864362i	$-0.667723\pi$
$499$	−22.4665	−1.00574	−0.502870	+	0.864362i	$0.667723\pi$
$500$	0	0
$501$	−14.0788	−0.628994
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	0.234622	0.0104405
$506$	0	0
$507$	−1.42963	−0.0634919
$508$	0	0
$509$	1.63111	0.0722978	0.0361489	−	0.999346i	$-0.488491\pi$
$509$	1.63111	0.0722978	0.0361489	+	0.999346i	$0.488491\pi$
$510$	0	0
$511$	−43.9827	−1.94568
$512$	0	0
$513$	8.55652	0.377779
$514$	0	0
$515$	31.9118	1.40620
$516$	0	0
$517$	24.7067	1.08660
$518$	0	0
$519$	−24.0435	−1.05539
$520$	0	0
$521$	0.695383	0.0304653	0.0152326	−	0.999884i	$-0.495151\pi$
$521$	0.695383	0.0304653	0.0152326	+	0.999884i	$0.495151\pi$
$522$	0	0
$523$	34.8502	1.52389	0.761947	−	0.647639i	$-0.224244\pi$
$523$	34.8502	1.52389	0.761947	+	0.647639i	$0.224244\pi$
$524$	0	0
$525$	−16.9578	−0.740100
$526$	0	0
$527$	−41.1629	−1.79308
$528$	0	0
$529$	50.5169	2.19639
$530$	0	0
$531$	14.2172	0.616974
$532$	0	0
$533$	1.20920	0.0523763
$534$	0	0
$535$	9.76326	0.422102
$536$	0	0
$537$	−3.32370	−0.143428
$538$	0	0
$539$	1.35715	0.0584564
$540$	0	0
$541$	−16.4644	−0.707860	−0.353930	−	0.935272i	$-0.615155\pi$
$541$	−16.4644	−0.707860	−0.353930	+	0.935272i	$0.615155\pi$
$542$	0	0
$543$	20.9008	0.896939
$544$	0	0
$545$	−29.4488	−1.26145
$546$	0	0
$547$	−12.1841	−0.520956	−0.260478	−	0.965480i	$-0.583880\pi$
$547$	−12.1841	−0.520956	−0.260478	+	0.965480i	$0.583880\pi$
$548$	0	0
$549$	2.18215	0.0931319
$550$	0	0
$551$	−46.7233	−1.99048
$552$	0	0
$553$	−2.46221	−0.104704
$554$	0	0
$555$	−37.9399	−1.61046
$556$	0	0
$557$	−15.7445	−0.667114	−0.333557	−	0.942730i	$-0.608249\pi$
$557$	−15.7445	−0.667114	−0.333557	+	0.942730i	$0.608249\pi$
$558$	0	0
$559$	23.6413	0.999919
$560$	0	0
$561$	−10.3862	−0.438507
$562$	0	0
$563$	−1.22656	−0.0516934	−0.0258467	−	0.999666i	$-0.508228\pi$
$563$	−1.22656	−0.0516934	−0.0258467	+	0.999666i	$0.508228\pi$
$564$	0	0
$565$	0.350560	0.0147482
$566$	0	0
$567$	2.75130	0.115544
$568$	0	0
$569$	−1.10187	−0.0461926	−0.0230963	−	0.999733i	$-0.507352\pi$
$569$	−1.10187	−0.0461926	−0.0230963	+	0.999733i	$0.507352\pi$
$570$	0	0
$571$	15.1007	0.631946	0.315973	−	0.948768i	$-0.397669\pi$
$571$	15.1007	0.631946	0.315973	+	0.948768i	$0.397669\pi$
$572$	0	0
$573$	−5.09229	−0.212733
$574$	0	0
$575$	−52.8477	−2.20390
$576$	0	0
$577$	1.46883	0.0611481	0.0305740	−	0.999533i	$-0.490266\pi$
$577$	1.46883	0.0611481	0.0305740	+	0.999533i	$0.490266\pi$
$578$	0	0
$579$	2.77353	0.115264
$580$	0	0
$581$	−39.1722	−1.62514
$582$	0	0
$583$	21.2721	0.881002
$584$	0	0
$585$	12.6920	0.524749
$586$	0	0
$587$	36.9278	1.52417	0.762086	−	0.647476i	$-0.224175\pi$
$587$	36.9278	1.52417	0.762086	+	0.647476i	$0.224175\pi$
$588$	0	0
$589$	80.7930	3.32902
$590$	0	0
$591$	24.6932	1.01574
$592$	0	0
$593$	33.0473	1.35709	0.678546	−	0.734558i	$-0.262611\pi$
$593$	33.0473	1.35709	0.678546	+	0.734558i	$0.262611\pi$
$594$	0	0
$595$	40.0745	1.64289
$596$	0	0
$597$	−5.61734	−0.229902
$598$	0	0
$599$	−7.07023	−0.288882	−0.144441	−	0.989513i	$-0.546138\pi$
$599$	−7.07023	−0.288882	−0.144441	+	0.989513i	$0.546138\pi$
$600$	0	0
$601$	23.6586	0.965053	0.482526	−	0.875881i	$-0.339719\pi$
$601$	23.6586	0.965053	0.482526	+	0.875881i	$0.339719\pi$
$602$	0	0
$603$	10.2702	0.418235
$604$	0	0
$605$	−17.7878	−0.723179
$606$	0	0
$607$	−3.82467	−0.155239	−0.0776194	−	0.996983i	$-0.524732\pi$
$607$	−3.82467	−0.155239	−0.0776194	+	0.996983i	$0.524732\pi$
$608$	0	0
$609$	−15.0236	−0.608787
$610$	0	0
$611$	39.3925	1.59365
$612$	0	0
$613$	−19.9315	−0.805027	−0.402514	−	0.915414i	$-0.631863\pi$
$613$	−19.9315	−0.805027	−0.402514	+	0.915414i	$0.631863\pi$
$614$	0	0
$615$	−1.06359	−0.0428879
$616$	0	0
$617$	−1.21450	−0.0488938	−0.0244469	−	0.999701i	$-0.507782\pi$
$617$	−1.21450	−0.0488938	−0.0244469	+	0.999701i	$0.507782\pi$
$618$	0	0
$619$	1.44911	0.0582445	0.0291222	−	0.999576i	$-0.490729\pi$
$619$	1.44911	0.0582445	0.0291222	+	0.999576i	$0.490729\pi$
$620$	0	0
$621$	8.57420	0.344071
$622$	0	0
$623$	−37.0901	−1.48598
$624$	0	0
$625$	−17.8282	−0.713128
$626$	0	0
$627$	20.3857	0.814127
$628$	0	0
$629$	49.5021	1.97378
$630$	0	0
$631$	−25.5160	−1.01578	−0.507888	−	0.861423i	$-0.669574\pi$
$631$	−25.5160	−1.01578	−0.507888	+	0.861423i	$0.669574\pi$
$632$	0	0
$633$	15.7765	0.627060
$634$	0	0
$635$	−34.0351	−1.35064
$636$	0	0
$637$	2.16384	0.0857345
$638$	0	0
$639$	−9.42568	−0.372874
$640$	0	0
$641$	8.83204	0.348844	0.174422	−	0.984671i	$-0.444194\pi$
$641$	8.83204	0.348844	0.174422	+	0.984671i	$0.444194\pi$
$642$	0	0
$643$	38.2953	1.51022	0.755109	−	0.655599i	$-0.227584\pi$
$643$	38.2953	1.51022	0.755109	+	0.655599i	$0.227584\pi$
$644$	0	0
$645$	−20.7943	−0.818776
$646$	0	0
$647$	12.6679	0.498026	0.249013	−	0.968500i	$-0.419894\pi$
$647$	12.6679	0.498026	0.249013	+	0.968500i	$0.419894\pi$
$648$	0	0
$649$	33.8722	1.32960
$650$	0	0
$651$	25.9785	1.01818
$652$	0	0
$653$	−32.9973	−1.29128	−0.645642	−	0.763640i	$-0.723410\pi$
$653$	−32.9973	−1.29128	−0.645642	+	0.763640i	$0.723410\pi$
$654$	0	0
$655$	−12.5936	−0.492074
$656$	0	0
$657$	−15.9862	−0.623679
$658$	0	0
$659$	−36.3071	−1.41432	−0.707162	−	0.707051i	$-0.750025\pi$
$659$	−36.3071	−1.41432	−0.707162	+	0.707051i	$0.750025\pi$
$660$	0	0
$661$	0.312076	0.0121384	0.00606918	−	0.999982i	$-0.498068\pi$
$661$	0.312076	0.0121384	0.00606918	+	0.999982i	$0.498068\pi$
$662$	0	0
$663$	−16.5599	−0.643132
$664$	0	0
$665$	−78.6568	−3.05018
$666$	0	0
$667$	−46.8199	−1.81287
$668$	0	0
$669$	9.99300	0.386352
$670$	0	0
$671$	5.19892	0.200702
$672$	0	0
$673$	26.4960	1.02134	0.510672	−	0.859775i	$-0.329397\pi$
$673$	26.4960	1.02134	0.510672	+	0.859775i	$0.329397\pi$
$674$	0	0
$675$	−6.16358	−0.237236
$676$	0	0
$677$	−16.2196	−0.623370	−0.311685	−	0.950186i	$-0.600893\pi$
$677$	−16.2196	−0.623370	−0.311685	+	0.950186i	$0.600893\pi$
$678$	0	0
$679$	33.3001	1.27794
$680$	0	0
$681$	3.27449	0.125479
$682$	0	0
$683$	14.3426	0.548803	0.274402	−	0.961615i	$-0.411520\pi$
$683$	14.3426	0.548803	0.274402	+	0.961615i	$0.411520\pi$
$684$	0	0
$685$	38.4699	1.46986
$686$	0	0
$687$	−17.0376	−0.650026
$688$	0	0
$689$	33.9164	1.29211
$690$	0	0
$691$	−15.8270	−0.602089	−0.301044	−	0.953610i	$-0.597335\pi$
$691$	−15.8270	−0.602089	−0.301044	+	0.953610i	$0.597335\pi$
$692$	0	0
$693$	6.55490	0.249000
$694$	0	0
$695$	66.1226	2.50818
$696$	0	0
$697$	1.38771	0.0525634
$698$	0	0
$699$	8.80004	0.332848
$700$	0	0
$701$	1.24970	0.0472006	0.0236003	−	0.999721i	$-0.492487\pi$
$701$	1.24970	0.0472006	0.0236003	+	0.999721i	$0.492487\pi$
$702$	0	0
$703$	−97.1610	−3.66450
$704$	0	0
$705$	−34.6487	−1.30495
$706$	0	0
$707$	0.193199	0.00726598
$708$	0	0
$709$	40.7910	1.53194	0.765969	−	0.642878i	$-0.222260\pi$
$709$	40.7910	1.53194	0.765969	+	0.642878i	$0.222260\pi$
$710$	0	0
$711$	−0.894929	−0.0335624
$712$	0	0
$713$	80.9600	3.03197
$714$	0	0
$715$	30.2383	1.13085
$716$	0	0
$717$	−21.6734	−0.809407
$718$	0	0
$719$	−14.1788	−0.528782	−0.264391	−	0.964416i	$-0.585171\pi$
$719$	−14.1788	−0.528782	−0.264391	+	0.964416i	$0.585171\pi$
$720$	0	0
$721$	26.2777	0.978633
$722$	0	0
$723$	23.6003	0.877703
$724$	0	0
$725$	33.6565	1.24997
$726$	0	0
$727$	−22.6774	−0.841057	−0.420528	−	0.907279i	$-0.638155\pi$
$727$	−22.6774	−0.841057	−0.420528	+	0.907279i	$0.638155\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	27.1314	1.00349
$732$	0	0
$733$	−13.5148	−0.499181	−0.249590	−	0.968352i	$-0.580296\pi$
$733$	−13.5148	−0.499181	−0.249590	+	0.968352i	$0.580296\pi$
$734$	0	0
$735$	−1.90327	−0.0702030
$736$	0	0
$737$	24.4685	0.901309
$738$	0	0
$739$	45.0750	1.65811	0.829055	−	0.559166i	$-0.188879\pi$
$739$	45.0750	1.65811	0.829055	+	0.559166i	$0.188879\pi$
$740$	0	0
$741$	32.5031	1.19403
$742$	0	0
$743$	−6.22828	−0.228494	−0.114247	−	0.993452i	$-0.536445\pi$
$743$	−6.22828	−0.228494	−0.114247	+	0.993452i	$0.536445\pi$
$744$	0	0
$745$	−24.6999	−0.904934
$746$	0	0
$747$	−14.2377	−0.520931
$748$	0	0
$749$	8.03953	0.293758
$750$	0	0
$751$	−16.6653	−0.608127	−0.304063	−	0.952652i	$-0.598343\pi$
$751$	−16.6653	−0.608127	−0.304063	+	0.952652i	$0.598343\pi$
$752$	0	0
$753$	18.7082	0.681764
$754$	0	0
$755$	−33.7699	−1.22901
$756$	0	0
$757$	−11.1560	−0.405472	−0.202736	−	0.979233i	$-0.564983\pi$
$757$	−11.1560	−0.405472	−0.202736	+	0.979233i	$0.564983\pi$
$758$	0	0
$759$	20.4278	0.741483
$760$	0	0
$761$	−15.6373	−0.566851	−0.283425	−	0.958994i	$-0.591471\pi$
$761$	−15.6373	−0.566851	−0.283425	+	0.958994i	$0.591471\pi$
$762$	0	0
$763$	−24.2495	−0.877892
$764$	0	0
$765$	14.5657	0.526623
$766$	0	0
$767$	54.0060	1.95004
$768$	0	0
$769$	35.2391	1.27075	0.635377	−	0.772202i	$-0.280845\pi$
$769$	35.2391	1.27075	0.635377	+	0.772202i	$0.280845\pi$
$770$	0	0
$771$	−16.1631	−0.582101
$772$	0	0
$773$	−12.2326	−0.439978	−0.219989	−	0.975502i	$-0.570602\pi$
$773$	−12.2326	−0.439978	−0.219989	+	0.975502i	$0.570602\pi$
$774$	0	0
$775$	−58.1982	−2.09054
$776$	0	0
$777$	−31.2415	−1.12078
$778$	0	0
$779$	−2.72375	−0.0975886
$780$	0	0
$781$	−22.4565	−0.803556
$782$	0	0
$783$	−5.46055	−0.195144
$784$	0	0
$785$	65.0252	2.32085
$786$	0	0
$787$	−18.1639	−0.647471	−0.323736	−	0.946148i	$-0.604939\pi$
$787$	−18.1639	−0.647471	−0.323736	+	0.946148i	$0.604939\pi$
$788$	0	0
$789$	−0.573277	−0.0204092
$790$	0	0
$791$	0.288668	0.0102639
$792$	0	0
$793$	8.28919	0.294358
$794$	0	0
$795$	−29.8321	−1.05804
$796$	0	0
$797$	−32.8453	−1.16344	−0.581719	−	0.813390i	$-0.697620\pi$
$797$	−32.8453	−1.16344	−0.581719	+	0.813390i	$0.697620\pi$
$798$	0	0
$799$	45.2079	1.59934
$800$	0	0
$801$	−13.4809	−0.476326
$802$	0	0
$803$	−38.0866	−1.34405
$804$	0	0
$805$	−78.8193	−2.77802
$806$	0	0
$807$	9.81424	0.345477
$808$	0	0
$809$	7.05654	0.248095	0.124047	−	0.992276i	$-0.460413\pi$
$809$	7.05654	0.248095	0.124047	+	0.992276i	$0.460413\pi$
$810$	0	0
$811$	−10.3362	−0.362953	−0.181476	−	0.983395i	$-0.558088\pi$
$811$	−10.3362	−0.362953	−0.181476	+	0.983395i	$0.558088\pi$
$812$	0	0
$813$	−23.4592	−0.822751
$814$	0	0
$815$	29.7210	1.04108
$816$	0	0
$817$	−53.2525	−1.86307
$818$	0	0
$819$	10.4512	0.365194
$820$	0	0
$821$	−15.1422	−0.528468	−0.264234	−	0.964459i	$-0.585119\pi$
$821$	−15.1422	−0.528468	−0.264234	+	0.964459i	$0.585119\pi$
$822$	0	0
$823$	28.2797	0.985769	0.492884	−	0.870095i	$-0.335943\pi$
$823$	28.2797	0.985769	0.492884	+	0.870095i	$0.335943\pi$
$824$	0	0
$825$	−14.6846	−0.511252
$826$	0	0
$827$	−13.4260	−0.466869	−0.233435	−	0.972372i	$-0.574996\pi$
$827$	−13.4260	−0.466869	−0.233435	+	0.972372i	$0.574996\pi$
$828$	0	0
$829$	−0.795550	−0.0276306	−0.0138153	−	0.999905i	$-0.504398\pi$
$829$	−0.795550	−0.0276306	−0.0138153	+	0.999905i	$0.504398\pi$
$830$	0	0
$831$	−5.60572	−0.194460
$832$	0	0
$833$	2.48329	0.0860408
$834$	0	0
$835$	47.0400	1.62789
$836$	0	0
$837$	9.44228	0.326373
$838$	0	0
$839$	−44.0487	−1.52073	−0.760365	−	0.649496i	$-0.774980\pi$
$839$	−44.0487	−1.52073	−0.760365	+	0.649496i	$0.774980\pi$
$840$	0	0
$841$	0.817640	0.0281945
$842$	0	0
$843$	5.70978	0.196655
$844$	0	0
$845$	4.77666	0.164322
$846$	0	0
$847$	−14.6474	−0.503289
$848$	0	0
$849$	−12.3253	−0.423002
$850$	0	0
$851$	−97.3617	−3.33752
$852$	0	0
$853$	−18.6003	−0.636861	−0.318431	−	0.947946i	$-0.603156\pi$
$853$	−18.6003	−0.636861	−0.318431	+	0.947946i	$0.603156\pi$
$854$	0	0
$855$	−28.5890	−0.977723
$856$	0	0
$857$	−12.9988	−0.444030	−0.222015	−	0.975043i	$-0.571263\pi$
$857$	−12.9988	−0.444030	−0.222015	+	0.975043i	$0.571263\pi$
$858$	0	0
$859$	−14.9617	−0.510487	−0.255244	−	0.966877i	$-0.582156\pi$
$859$	−14.9617	−0.510487	−0.255244	+	0.966877i	$0.582156\pi$
$860$	0	0
$861$	−0.875806	−0.0298474
$862$	0	0
$863$	−8.05689	−0.274260	−0.137130	−	0.990553i	$-0.543788\pi$
$863$	−8.05689	−0.274260	−0.137130	+	0.990553i	$0.543788\pi$
$864$	0	0
$865$	80.3340	2.73144
$866$	0	0
$867$	−2.00457	−0.0680787
$868$	0	0
$869$	−2.13215	−0.0723281
$870$	0	0
$871$	39.0127	1.32190
$872$	0	0
$873$	12.1034	0.409639
$874$	0	0
$875$	10.6963	0.361601
$876$	0	0
$877$	−7.79618	−0.263258	−0.131629	−	0.991299i	$-0.542021\pi$
$877$	−7.79618	−0.263258	−0.131629	+	0.991299i	$0.542021\pi$
$878$	0	0
$879$	17.4543	0.588719
$880$	0	0
$881$	2.57353	0.0867046	0.0433523	−	0.999060i	$-0.486196\pi$
$881$	2.57353	0.0867046	0.0433523	+	0.999060i	$0.486196\pi$
$882$	0	0
$883$	49.0552	1.65084	0.825419	−	0.564520i	$-0.190939\pi$
$883$	49.0552	1.65084	0.825419	+	0.564520i	$0.190939\pi$
$884$	0	0
$885$	−47.5024	−1.59678
$886$	0	0
$887$	−46.3850	−1.55746	−0.778728	−	0.627362i	$-0.784135\pi$
$887$	−46.3850	−1.55746	−0.778728	+	0.627362i	$0.784135\pi$
$888$	0	0
$889$	−28.0261	−0.939965
$890$	0	0
$891$	2.38248	0.0798160
$892$	0	0
$893$	−88.7325	−2.96932
$894$	0	0
$895$	11.1051	0.371204
$896$	0	0
$897$	32.5702	1.08749
$898$	0	0
$899$	−51.5601	−1.71962
$900$	0	0
$901$	38.9235	1.29673
$902$	0	0
$903$	−17.1230	−0.569819
$904$	0	0
$905$	−69.8336	−2.32135
$906$	0	0
$907$	−20.1437	−0.668861	−0.334430	−	0.942420i	$-0.608544\pi$
$907$	−20.1437	−0.668861	−0.334430	+	0.942420i	$0.608544\pi$
$908$	0	0
$909$	0.0702209	0.00232908
$910$	0	0
$911$	−22.7445	−0.753557	−0.376779	−	0.926303i	$-0.622968\pi$
$911$	−22.7445	−0.753557	−0.376779	+	0.926303i	$0.622968\pi$
$912$	0	0
$913$	−33.9210	−1.12262
$914$	0	0
$915$	−7.29098	−0.241032
$916$	0	0
$917$	−10.3702	−0.342454
$918$	0	0
$919$	−10.9491	−0.361177	−0.180589	−	0.983559i	$-0.557800\pi$
$919$	−10.9491	−0.361177	−0.180589	+	0.983559i	$0.557800\pi$
$920$	0	0
$921$	−15.8301	−0.521621
$922$	0	0
$923$	−35.8047	−1.17853
$924$	0	0
$925$	69.9887	2.30121
$926$	0	0
$927$	9.55102	0.313697
$928$	0	0
$929$	13.1855	0.432603	0.216302	−	0.976327i	$-0.430601\pi$
$929$	13.1855	0.432603	0.216302	+	0.976327i	$0.430601\pi$
$930$	0	0
$931$	−4.87410	−0.159742
$932$	0	0
$933$	−32.6873	−1.07013
$934$	0	0
$935$	34.7024	1.13489
$936$	0	0
$937$	3.00650	0.0982181	0.0491090	−	0.998793i	$-0.484362\pi$
$937$	3.00650	0.0982181	0.0491090	+	0.998793i	$0.484362\pi$
$938$	0	0
$939$	8.86306	0.289235
$940$	0	0
$941$	−39.8692	−1.29970	−0.649849	−	0.760064i	$-0.725168\pi$
$941$	−39.8692	−1.29970	−0.649849	+	0.760064i	$0.725168\pi$
$942$	0	0
$943$	−2.72938	−0.0888808
$944$	0	0
$945$	−9.19262	−0.299036
$946$	0	0
$947$	−9.95597	−0.323526	−0.161763	−	0.986830i	$-0.551718\pi$
$947$	−9.95597	−0.323526	−0.161763	+	0.986830i	$0.551718\pi$
$948$	0	0
$949$	−60.7255	−1.97123
$950$	0	0
$951$	9.99616	0.324148
$952$	0	0
$953$	22.3000	0.722369	0.361185	−	0.932494i	$-0.382372\pi$
$953$	22.3000	0.722369	0.361185	+	0.932494i	$0.382372\pi$
$954$	0	0
$955$	17.0143	0.550570
$956$	0	0
$957$	−13.0096	−0.420542
$958$	0	0
$959$	31.6779	1.02293
$960$	0	0
$961$	58.1566	1.87602
$962$	0	0
$963$	2.92209	0.0941629
$964$	0	0
$965$	−9.26689	−0.298312
$966$	0	0
$967$	−31.5963	−1.01607	−0.508034	−	0.861337i	$-0.669627\pi$
$967$	−31.5963	−1.01607	−0.508034	+	0.861337i	$0.669627\pi$
$968$	0	0
$969$	37.3015	1.19830
$970$	0	0
$971$	26.2697	0.843034	0.421517	−	0.906821i	$-0.361498\pi$
$971$	26.2697	0.843034	0.421517	+	0.906821i	$0.361498\pi$
$972$	0	0
$973$	54.4485	1.74554
$974$	0	0
$975$	−23.4132	−0.749822
$976$	0	0
$977$	37.6563	1.20473	0.602365	−	0.798221i	$-0.294225\pi$
$977$	37.6563	1.20473	0.602365	+	0.798221i	$0.294225\pi$
$978$	0	0
$979$	−32.1180	−1.02650
$980$	0	0
$981$	−8.81386	−0.281405
$982$	0	0
$983$	−43.7365	−1.39498	−0.697488	−	0.716596i	$-0.745699\pi$
$983$	−43.7365	−1.39498	−0.697488	+	0.716596i	$0.745699\pi$
$984$	0	0
$985$	−82.5048	−2.62882
$986$	0	0
$987$	−28.5314	−0.908165
$988$	0	0
$989$	−53.3626	−1.69683
$990$	0	0
$991$	0.387635	0.0123136	0.00615681	−	0.999981i	$-0.498040\pi$
$991$	0.387635	0.0123136	0.00615681	+	0.999981i	$0.498040\pi$
$992$	0	0
$993$	−9.37191	−0.297409
$994$	0	0
$995$	18.7686	0.595005
$996$	0	0
$997$	2.70244	0.0855873	0.0427936	−	0.999084i	$-0.486374\pi$
$997$	2.70244	0.0855873	0.0427936	+	0.999084i	$0.486374\pi$
$998$	0	0
$999$	−11.3552	−0.359263

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.23	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.23	✓	26	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.34119 q^{5} +2.75130 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.34119 q^{5} +2.75130 q^{7} +1.00000 q^{9} +2.38248 q^{11} +3.79863 q^{13} -3.34119 q^{15} +4.35942 q^{17} -8.55652 q^{19} -2.75130 q^{21} -8.57420 q^{23} +6.16358 q^{25} -1.00000 q^{27} +5.46055 q^{29} -9.44228 q^{31} -2.38248 q^{33} +9.19262 q^{35} +11.3552 q^{37} -3.79863 q^{39} +0.318325 q^{41} +6.22362 q^{43} +3.34119 q^{45} +10.3702 q^{47} +0.569636 q^{49} -4.35942 q^{51} +8.92858 q^{53} +7.96032 q^{55} +8.55652 q^{57} +14.2172 q^{59} +2.18215 q^{61} +2.75130 q^{63} +12.6920 q^{65} +10.2702 q^{67} +8.57420 q^{69} -9.42568 q^{71} -15.9862 q^{73} -6.16358 q^{75} +6.55490 q^{77} -0.894929 q^{79} +1.00000 q^{81} -14.2377 q^{83} +14.5657 q^{85} -5.46055 q^{87} -13.4809 q^{89} +10.4512 q^{91} +9.44228 q^{93} -28.5890 q^{95} +12.1034 q^{97} +2.38248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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