LMFDB - Embedded newform 8004.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.76979$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.76979 q^{5} -3.16982 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.76979 q^{5} -3.16982 q^{7} +1.00000 q^{9} +5.49016 q^{11} -2.24096 q^{13} -2.76979 q^{15} +5.82218 q^{17} +6.74968 q^{19} -3.16982 q^{21} -1.00000 q^{23} +2.67175 q^{25} +1.00000 q^{27} +1.00000 q^{29} -4.12077 q^{31} +5.49016 q^{33} +8.77975 q^{35} -11.9056 q^{37} -2.24096 q^{39} -5.91919 q^{41} -2.75817 q^{43} -2.76979 q^{45} +6.49640 q^{47} +3.04776 q^{49} +5.82218 q^{51} -6.54972 q^{53} -15.2066 q^{55} +6.74968 q^{57} +3.93478 q^{59} +2.42327 q^{61} -3.16982 q^{63} +6.20701 q^{65} +5.15868 q^{67} -1.00000 q^{69} +9.58968 q^{71} -1.17317 q^{73} +2.67175 q^{75} -17.4028 q^{77} +2.46474 q^{79} +1.00000 q^{81} -12.6753 q^{83} -16.1262 q^{85} +1.00000 q^{87} +6.96986 q^{89} +7.10346 q^{91} -4.12077 q^{93} -18.6952 q^{95} -11.8574 q^{97} +5.49016 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * 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
$5$	−2.76979	−1.23869	−0.619345	−	0.785119i	$-0.712602\pi$
$5$	−2.76979	−1.23869	−0.619345	+	0.785119i	$0.712602\pi$
$6$	0	0
$7$	−3.16982	−1.19808	−0.599040	−	0.800719i	$-0.704451\pi$
$7$	−3.16982	−1.19808	−0.599040	+	0.800719i	$0.704451\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.49016	1.65535	0.827673	−	0.561211i	$-0.189664\pi$
$11$	5.49016	1.65535	0.827673	+	0.561211i	$0.189664\pi$
$12$	0	0
$13$	−2.24096	−0.621532	−0.310766	−	0.950486i	$-0.600586\pi$
$13$	−2.24096	−0.621532	−0.310766	+	0.950486i	$0.600586\pi$
$14$	0	0
$15$	−2.76979	−0.715157
$16$	0	0
$17$	5.82218	1.41209	0.706043	−	0.708169i	$-0.250478\pi$
$17$	5.82218	1.41209	0.706043	+	0.708169i	$0.250478\pi$
$18$	0	0
$19$	6.74968	1.54848	0.774242	−	0.632890i	$-0.218132\pi$
$19$	6.74968	1.54848	0.774242	+	0.632890i	$0.218132\pi$
$20$	0	0
$21$	−3.16982	−0.691712
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.67175	0.534351
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.12077	−0.740112	−0.370056	−	0.929009i	$-0.620662\pi$
$31$	−4.12077	−0.740112	−0.370056	+	0.929009i	$0.620662\pi$
$32$	0	0
$33$	5.49016	0.955714
$34$	0	0
$35$	8.77975	1.48405
$36$	0	0
$37$	−11.9056	−1.95727	−0.978636	−	0.205600i	$-0.934085\pi$
$37$	−11.9056	−1.95727	−0.978636	+	0.205600i	$0.934085\pi$
$38$	0	0
$39$	−2.24096	−0.358842
$40$	0	0
$41$	−5.91919	−0.924423	−0.462211	−	0.886770i	$-0.652944\pi$
$41$	−5.91919	−0.924423	−0.462211	+	0.886770i	$0.652944\pi$
$42$	0	0
$43$	−2.75817	−0.420617	−0.210309	−	0.977635i	$-0.567447\pi$
$43$	−2.75817	−0.420617	−0.210309	+	0.977635i	$0.567447\pi$
$44$	0	0
$45$	−2.76979	−0.412896
$46$	0	0
$47$	6.49640	0.947597	0.473799	−	0.880633i	$-0.342882\pi$
$47$	6.49640	0.947597	0.473799	+	0.880633i	$0.342882\pi$
$48$	0	0
$49$	3.04776	0.435395
$50$	0	0
$51$	5.82218	0.815269
$52$	0	0
$53$	−6.54972	−0.899674	−0.449837	−	0.893111i	$-0.648518\pi$
$53$	−6.54972	−0.899674	−0.449837	+	0.893111i	$0.648518\pi$
$54$	0	0
$55$	−15.2066	−2.05046
$56$	0	0
$57$	6.74968	0.894017
$58$	0	0
$59$	3.93478	0.512265	0.256133	−	0.966642i	$-0.417552\pi$
$59$	3.93478	0.512265	0.256133	+	0.966642i	$0.417552\pi$
$60$	0	0
$61$	2.42327	0.310267	0.155134	−	0.987893i	$-0.450419\pi$
$61$	2.42327	0.310267	0.155134	+	0.987893i	$0.450419\pi$
$62$	0	0
$63$	−3.16982	−0.399360
$64$	0	0
$65$	6.20701	0.769885
$66$	0	0
$67$	5.15868	0.630233	0.315116	−	0.949053i	$-0.397956\pi$
$67$	5.15868	0.630233	0.315116	+	0.949053i	$0.397956\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	9.58968	1.13809	0.569043	−	0.822308i	$-0.307314\pi$
$71$	9.58968	1.13809	0.569043	+	0.822308i	$0.307314\pi$
$72$	0	0
$73$	−1.17317	−0.137309	−0.0686545	−	0.997640i	$-0.521871\pi$
$73$	−1.17317	−0.137309	−0.0686545	+	0.997640i	$0.521871\pi$
$74$	0	0
$75$	2.67175	0.308508
$76$	0	0
$77$	−17.4028	−1.98324
$78$	0	0
$79$	2.46474	0.277305	0.138652	−	0.990341i	$-0.455723\pi$
$79$	2.46474	0.277305	0.138652	+	0.990341i	$0.455723\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.6753	−1.39129	−0.695646	−	0.718385i	$-0.744882\pi$
$83$	−12.6753	−1.39129	−0.695646	+	0.718385i	$0.744882\pi$
$84$	0	0
$85$	−16.1262	−1.74914
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	6.96986	0.738804	0.369402	−	0.929270i	$-0.379563\pi$
$89$	6.96986	0.738804	0.369402	+	0.929270i	$0.379563\pi$
$90$	0	0
$91$	7.10346	0.744645
$92$	0	0
$93$	−4.12077	−0.427304
$94$	0	0
$95$	−18.6952	−1.91809
$96$	0	0
$97$	−11.8574	−1.20394	−0.601968	−	0.798520i	$-0.705617\pi$
$97$	−11.8574	−1.20394	−0.601968	+	0.798520i	$0.705617\pi$
$98$	0	0
$99$	5.49016	0.551782
$100$	0	0
$101$	18.3401	1.82490	0.912452	−	0.409185i	$-0.134187\pi$
$101$	18.3401	1.82490	0.912452	+	0.409185i	$0.134187\pi$
$102$	0	0
$103$	12.8280	1.26398	0.631992	−	0.774975i	$-0.282238\pi$
$103$	12.8280	1.26398	0.631992	+	0.774975i	$0.282238\pi$
$104$	0	0
$105$	8.77975	0.856816
$106$	0	0
$107$	−12.3449	−1.19343	−0.596715	−	0.802453i	$-0.703528\pi$
$107$	−12.3449	−1.19343	−0.596715	+	0.802453i	$0.703528\pi$
$108$	0	0
$109$	10.0065	0.958445	0.479223	−	0.877693i	$-0.340919\pi$
$109$	10.0065	0.958445	0.479223	+	0.877693i	$0.340919\pi$
$110$	0	0
$111$	−11.9056	−1.13003
$112$	0	0
$113$	−4.54898	−0.427932	−0.213966	−	0.976841i	$-0.568638\pi$
$113$	−4.54898	−0.427932	−0.213966	+	0.976841i	$0.568638\pi$
$114$	0	0
$115$	2.76979	0.258285
$116$	0	0
$117$	−2.24096	−0.207177
$118$	0	0
$119$	−18.4553	−1.69179
$120$	0	0
$121$	19.1419	1.74017
$122$	0	0
$123$	−5.91919	−0.533716
$124$	0	0
$125$	6.44876	0.576795
$126$	0	0
$127$	5.16851	0.458631	0.229315	−	0.973352i	$-0.426351\pi$
$127$	5.16851	0.458631	0.229315	+	0.973352i	$0.426351\pi$
$128$	0	0
$129$	−2.75817	−0.242843
$130$	0	0
$131$	20.3889	1.78139	0.890694	−	0.454604i	$-0.150219\pi$
$131$	20.3889	1.78139	0.890694	+	0.454604i	$0.150219\pi$
$132$	0	0
$133$	−21.3953	−1.85521
$134$	0	0
$135$	−2.76979	−0.238386
$136$	0	0
$137$	−5.98386	−0.511236	−0.255618	−	0.966778i	$-0.582279\pi$
$137$	−5.98386	−0.511236	−0.255618	+	0.966778i	$0.582279\pi$
$138$	0	0
$139$	6.94816	0.589336	0.294668	−	0.955600i	$-0.404791\pi$
$139$	6.94816	0.589336	0.294668	+	0.955600i	$0.404791\pi$
$140$	0	0
$141$	6.49640	0.547096
$142$	0	0
$143$	−12.3033	−1.02885
$144$	0	0
$145$	−2.76979	−0.230019
$146$	0	0
$147$	3.04776	0.251375
$148$	0	0
$149$	−0.196403	−0.0160899	−0.00804496	−	0.999968i	$-0.502561\pi$
$149$	−0.196403	−0.0160899	−0.00804496	+	0.999968i	$0.502561\pi$
$150$	0	0
$151$	13.3391	1.08552	0.542759	−	0.839888i	$-0.317380\pi$
$151$	13.3391	1.08552	0.542759	+	0.839888i	$0.317380\pi$
$152$	0	0
$153$	5.82218	0.470696
$154$	0	0
$155$	11.4137	0.916769
$156$	0	0
$157$	0.686698	0.0548045	0.0274022	−	0.999624i	$-0.491277\pi$
$157$	0.686698	0.0548045	0.0274022	+	0.999624i	$0.491277\pi$
$158$	0	0
$159$	−6.54972	−0.519427
$160$	0	0
$161$	3.16982	0.249817
$162$	0	0
$163$	5.69021	0.445691	0.222846	−	0.974854i	$-0.428465\pi$
$163$	5.69021	0.445691	0.222846	+	0.974854i	$0.428465\pi$
$164$	0	0
$165$	−15.2066	−1.18383
$166$	0	0
$167$	−8.04965	−0.622901	−0.311450	−	0.950262i	$-0.600815\pi$
$167$	−8.04965	−0.622901	−0.311450	+	0.950262i	$0.600815\pi$
$168$	0	0
$169$	−7.97808	−0.613698
$170$	0	0
$171$	6.74968	0.516161
$172$	0	0
$173$	17.8663	1.35835	0.679174	−	0.733978i	$-0.262338\pi$
$173$	17.8663	1.35835	0.679174	+	0.733978i	$0.262338\pi$
$174$	0	0
$175$	−8.46898	−0.640195
$176$	0	0
$177$	3.93478	0.295756
$178$	0	0
$179$	−5.00570	−0.374144	−0.187072	−	0.982346i	$-0.559900\pi$
$179$	−5.00570	−0.374144	−0.187072	+	0.982346i	$0.559900\pi$
$180$	0	0
$181$	6.11998	0.454895	0.227447	−	0.973790i	$-0.426962\pi$
$181$	6.11998	0.454895	0.227447	+	0.973790i	$0.426962\pi$
$182$	0	0
$183$	2.42327	0.179133
$184$	0	0
$185$	32.9761	2.42445
$186$	0	0
$187$	31.9647	2.33749
$188$	0	0
$189$	−3.16982	−0.230571
$190$	0	0
$191$	1.52978	0.110691	0.0553456	−	0.998467i	$-0.482374\pi$
$191$	1.52978	0.110691	0.0553456	+	0.998467i	$0.482374\pi$
$192$	0	0
$193$	−7.47365	−0.537965	−0.268983	−	0.963145i	$-0.586687\pi$
$193$	−7.47365	−0.537965	−0.268983	+	0.963145i	$0.586687\pi$
$194$	0	0
$195$	6.20701	0.444493
$196$	0	0
$197$	−15.0874	−1.07494	−0.537468	−	0.843284i	$-0.680619\pi$
$197$	−15.0874	−1.07494	−0.537468	+	0.843284i	$0.680619\pi$
$198$	0	0
$199$	16.8893	1.19725	0.598626	−	0.801029i	$-0.295714\pi$
$199$	16.8893	1.19725	0.598626	+	0.801029i	$0.295714\pi$
$200$	0	0
$201$	5.15868	0.363865
$202$	0	0
$203$	−3.16982	−0.222478
$204$	0	0
$205$	16.3949	1.14507
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	37.0568	2.56327
$210$	0	0
$211$	9.64944	0.664295	0.332148	−	0.943227i	$-0.392227\pi$
$211$	9.64944	0.664295	0.332148	+	0.943227i	$0.392227\pi$
$212$	0	0
$213$	9.58968	0.657074
$214$	0	0
$215$	7.63956	0.521014
$216$	0	0
$217$	13.0621	0.886714
$218$	0	0
$219$	−1.17317	−0.0792754
$220$	0	0
$221$	−13.0473	−0.877657
$222$	0	0
$223$	−2.04044	−0.136638	−0.0683191	−	0.997664i	$-0.521764\pi$
$223$	−2.04044	−0.136638	−0.0683191	+	0.997664i	$0.521764\pi$
$224$	0	0
$225$	2.67175	0.178117
$226$	0	0
$227$	23.4455	1.55613	0.778065	−	0.628184i	$-0.216201\pi$
$227$	23.4455	1.55613	0.778065	+	0.628184i	$0.216201\pi$
$228$	0	0
$229$	−1.43125	−0.0945796	−0.0472898	−	0.998881i	$-0.515058\pi$
$229$	−1.43125	−0.0945796	−0.0472898	+	0.998881i	$0.515058\pi$
$230$	0	0
$231$	−17.4028	−1.14502
$232$	0	0
$233$	18.1113	1.18651	0.593255	−	0.805015i	$-0.297843\pi$
$233$	18.1113	1.18651	0.593255	+	0.805015i	$0.297843\pi$
$234$	0	0
$235$	−17.9937	−1.17378
$236$	0	0
$237$	2.46474	0.160102
$238$	0	0
$239$	11.1101	0.718650	0.359325	−	0.933212i	$-0.383007\pi$
$239$	11.1101	0.718650	0.359325	+	0.933212i	$0.383007\pi$
$240$	0	0
$241$	2.18864	0.140982	0.0704912	−	0.997512i	$-0.477543\pi$
$241$	2.18864	0.140982	0.0704912	+	0.997512i	$0.477543\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−8.44167	−0.539319
$246$	0	0
$247$	−15.1258	−0.962431
$248$	0	0
$249$	−12.6753	−0.803263
$250$	0	0
$251$	−21.1501	−1.33498	−0.667490	−	0.744619i	$-0.732631\pi$
$251$	−21.1501	−1.33498	−0.667490	+	0.744619i	$0.732631\pi$
$252$	0	0
$253$	−5.49016	−0.345163
$254$	0	0
$255$	−16.1262	−1.00986
$256$	0	0
$257$	−23.8657	−1.48870	−0.744350	−	0.667789i	$-0.767241\pi$
$257$	−23.8657	−1.48870	−0.744350	+	0.667789i	$0.767241\pi$
$258$	0	0
$259$	37.7387	2.34497
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	16.6916	1.02925	0.514625	−	0.857416i	$-0.327931\pi$
$263$	16.6916	1.02925	0.514625	+	0.857416i	$0.327931\pi$
$264$	0	0
$265$	18.1414	1.11442
$266$	0	0
$267$	6.96986	0.426549
$268$	0	0
$269$	10.1935	0.621509	0.310755	−	0.950490i	$-0.399418\pi$
$269$	10.1935	0.621509	0.310755	+	0.950490i	$0.399418\pi$
$270$	0	0
$271$	−7.96044	−0.483562	−0.241781	−	0.970331i	$-0.577732\pi$
$271$	−7.96044	−0.483562	−0.241781	+	0.970331i	$0.577732\pi$
$272$	0	0
$273$	7.10346	0.429921
$274$	0	0
$275$	14.6684	0.884535
$276$	0	0
$277$	−21.2425	−1.27634	−0.638168	−	0.769897i	$-0.720307\pi$
$277$	−21.2425	−1.27634	−0.638168	+	0.769897i	$0.720307\pi$
$278$	0	0
$279$	−4.12077	−0.246704
$280$	0	0
$281$	18.2543	1.08896	0.544479	−	0.838774i	$-0.316727\pi$
$281$	18.2543	1.08896	0.544479	+	0.838774i	$0.316727\pi$
$282$	0	0
$283$	31.1156	1.84963	0.924814	−	0.380420i	$-0.124221\pi$
$283$	31.1156	1.84963	0.924814	+	0.380420i	$0.124221\pi$
$284$	0	0
$285$	−18.6952	−1.10741
$286$	0	0
$287$	18.7628	1.10753
$288$	0	0
$289$	16.8978	0.993989
$290$	0	0
$291$	−11.8574	−0.695093
$292$	0	0
$293$	−33.0397	−1.93020	−0.965100	−	0.261883i	$-0.915657\pi$
$293$	−33.0397	−1.93020	−0.965100	+	0.261883i	$0.915657\pi$
$294$	0	0
$295$	−10.8985	−0.634537
$296$	0	0
$297$	5.49016	0.318571
$298$	0	0
$299$	2.24096	0.129598
$300$	0	0
$301$	8.74291	0.503933
$302$	0	0
$303$	18.3401	1.05361
$304$	0	0
$305$	−6.71195	−0.384325
$306$	0	0
$307$	25.5928	1.46066	0.730330	−	0.683094i	$-0.239366\pi$
$307$	25.5928	1.46066	0.730330	+	0.683094i	$0.239366\pi$
$308$	0	0
$309$	12.8280	0.729761
$310$	0	0
$311$	−26.1235	−1.48133	−0.740663	−	0.671876i	$-0.765489\pi$
$311$	−26.1235	−1.48133	−0.740663	+	0.671876i	$0.765489\pi$
$312$	0	0
$313$	9.79501	0.553647	0.276823	−	0.960921i	$-0.410718\pi$
$313$	9.79501	0.553647	0.276823	+	0.960921i	$0.410718\pi$
$314$	0	0
$315$	8.77975	0.494683
$316$	0	0
$317$	26.6180	1.49502	0.747508	−	0.664252i	$-0.231250\pi$
$317$	26.6180	1.49502	0.747508	+	0.664252i	$0.231250\pi$
$318$	0	0
$319$	5.49016	0.307390
$320$	0	0
$321$	−12.3449	−0.689028
$322$	0	0
$323$	39.2979	2.18659
$324$	0	0
$325$	−5.98731	−0.332116
$326$	0	0
$327$	10.0065	0.553359
$328$	0	0
$329$	−20.5924	−1.13530
$330$	0	0
$331$	−22.1818	−1.21922	−0.609610	−	0.792701i	$-0.708674\pi$
$331$	−22.1818	−1.21922	−0.609610	+	0.792701i	$0.708674\pi$
$332$	0	0
$333$	−11.9056	−0.652424
$334$	0	0
$335$	−14.2885	−0.780662
$336$	0	0
$337$	18.8517	1.02692	0.513459	−	0.858114i	$-0.328364\pi$
$337$	18.8517	1.02692	0.513459	+	0.858114i	$0.328364\pi$
$338$	0	0
$339$	−4.54898	−0.247067
$340$	0	0
$341$	−22.6237	−1.22514
$342$	0	0
$343$	12.5279	0.676442
$344$	0	0
$345$	2.76979	0.149121
$346$	0	0
$347$	−22.4332	−1.20428	−0.602139	−	0.798391i	$-0.705685\pi$
$347$	−22.4332	−1.20428	−0.602139	+	0.798391i	$0.705685\pi$
$348$	0	0
$349$	1.03861	0.0555958	0.0277979	−	0.999614i	$-0.491151\pi$
$349$	1.03861	0.0555958	0.0277979	+	0.999614i	$0.491151\pi$
$350$	0	0
$351$	−2.24096	−0.119614
$352$	0	0
$353$	7.85330	0.417989	0.208995	−	0.977917i	$-0.432981\pi$
$353$	7.85330	0.417989	0.208995	+	0.977917i	$0.432981\pi$
$354$	0	0
$355$	−26.5614	−1.40973
$356$	0	0
$357$	−18.4553	−0.976757
$358$	0	0
$359$	25.2943	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$359$	25.2943	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$360$	0	0
$361$	26.5582	1.39780
$362$	0	0
$363$	19.1419	1.00469
$364$	0	0
$365$	3.24944	0.170083
$366$	0	0
$367$	6.71377	0.350456	0.175228	−	0.984528i	$-0.443934\pi$
$367$	6.71377	0.350456	0.175228	+	0.984528i	$0.443934\pi$
$368$	0	0
$369$	−5.91919	−0.308141
$370$	0	0
$371$	20.7615	1.07788
$372$	0	0
$373$	32.3840	1.67678	0.838389	−	0.545072i	$-0.183498\pi$
$373$	32.3840	1.67678	0.838389	+	0.545072i	$0.183498\pi$
$374$	0	0
$375$	6.44876	0.333013
$376$	0	0
$377$	−2.24096	−0.115416
$378$	0	0
$379$	31.9951	1.64348	0.821740	−	0.569863i	$-0.193004\pi$
$379$	31.9951	1.64348	0.821740	+	0.569863i	$0.193004\pi$
$380$	0	0
$381$	5.16851	0.264791
$382$	0	0
$383$	32.5892	1.66523	0.832614	−	0.553853i	$-0.186843\pi$
$383$	32.5892	1.66523	0.832614	+	0.553853i	$0.186843\pi$
$384$	0	0
$385$	48.2022	2.45661
$386$	0	0
$387$	−2.75817	−0.140206
$388$	0	0
$389$	23.1790	1.17522	0.587611	−	0.809144i	$-0.300069\pi$
$389$	23.1790	1.17522	0.587611	+	0.809144i	$0.300069\pi$
$390$	0	0
$391$	−5.82218	−0.294440
$392$	0	0
$393$	20.3889	1.02848
$394$	0	0
$395$	−6.82681	−0.343494
$396$	0	0
$397$	−5.28780	−0.265387	−0.132694	−	0.991157i	$-0.542363\pi$
$397$	−5.28780	−0.265387	−0.132694	+	0.991157i	$0.542363\pi$
$398$	0	0
$399$	−21.3953	−1.07110
$400$	0	0
$401$	23.3569	1.16639	0.583194	−	0.812333i	$-0.301803\pi$
$401$	23.3569	1.16639	0.583194	+	0.812333i	$0.301803\pi$
$402$	0	0
$403$	9.23450	0.460003
$404$	0	0
$405$	−2.76979	−0.137632
$406$	0	0
$407$	−65.3638	−3.23996
$408$	0	0
$409$	−12.0709	−0.596866	−0.298433	−	0.954431i	$-0.596464\pi$
$409$	−12.0709	−0.596866	−0.298433	+	0.954431i	$0.596464\pi$
$410$	0	0
$411$	−5.98386	−0.295162
$412$	0	0
$413$	−12.4726	−0.613734
$414$	0	0
$415$	35.1079	1.72338
$416$	0	0
$417$	6.94816	0.340253
$418$	0	0
$419$	−39.5649	−1.93287	−0.966436	−	0.256907i	$-0.917297\pi$
$419$	−39.5649	−1.93287	−0.966436	+	0.256907i	$0.917297\pi$
$420$	0	0
$421$	−0.268848	−0.0131028	−0.00655142	−	0.999979i	$-0.502085\pi$
$421$	−0.268848	−0.0131028	−0.00655142	+	0.999979i	$0.502085\pi$
$422$	0	0
$423$	6.49640	0.315866
$424$	0	0
$425$	15.5554	0.754549
$426$	0	0
$427$	−7.68132	−0.371725
$428$	0	0
$429$	−12.3033	−0.594007
$430$	0	0
$431$	6.17055	0.297225	0.148612	−	0.988896i	$-0.452519\pi$
$431$	6.17055	0.297225	0.148612	+	0.988896i	$0.452519\pi$
$432$	0	0
$433$	9.85578	0.473639	0.236819	−	0.971554i	$-0.423895\pi$
$433$	9.85578	0.473639	0.236819	+	0.971554i	$0.423895\pi$
$434$	0	0
$435$	−2.76979	−0.132801
$436$	0	0
$437$	−6.74968	−0.322881
$438$	0	0
$439$	21.9081	1.04562	0.522809	−	0.852450i	$-0.324884\pi$
$439$	21.9081	1.04562	0.522809	+	0.852450i	$0.324884\pi$
$440$	0	0
$441$	3.04776	0.145132
$442$	0	0
$443$	22.6035	1.07393	0.536963	−	0.843606i	$-0.319572\pi$
$443$	22.6035	1.07393	0.536963	+	0.843606i	$0.319572\pi$
$444$	0	0
$445$	−19.3051	−0.915149
$446$	0	0
$447$	−0.196403	−0.00928952
$448$	0	0
$449$	−25.2214	−1.19027	−0.595137	−	0.803625i	$-0.702902\pi$
$449$	−25.2214	−1.19027	−0.595137	+	0.803625i	$0.702902\pi$
$450$	0	0
$451$	−32.4973	−1.53024
$452$	0	0
$453$	13.3391	0.626724
$454$	0	0
$455$	−19.6751	−0.922383
$456$	0	0
$457$	39.6305	1.85384	0.926918	−	0.375265i	$-0.122448\pi$
$457$	39.6305	1.85384	0.926918	+	0.375265i	$0.122448\pi$
$458$	0	0
$459$	5.82218	0.271756
$460$	0	0
$461$	−18.3139	−0.852965	−0.426482	−	0.904496i	$-0.640247\pi$
$461$	−18.3139	−0.852965	−0.426482	+	0.904496i	$0.640247\pi$
$462$	0	0
$463$	−31.6609	−1.47140	−0.735702	−	0.677305i	$-0.763148\pi$
$463$	−31.6609	−1.47140	−0.735702	+	0.677305i	$0.763148\pi$
$464$	0	0
$465$	11.4137	0.529297
$466$	0	0
$467$	2.36223	0.109311	0.0546555	−	0.998505i	$-0.482594\pi$
$467$	2.36223	0.109311	0.0546555	+	0.998505i	$0.482594\pi$
$468$	0	0
$469$	−16.3521	−0.755069
$470$	0	0
$471$	0.686698	0.0316414
$472$	0	0
$473$	−15.1428	−0.696266
$474$	0	0
$475$	18.0335	0.827433
$476$	0	0
$477$	−6.54972	−0.299891
$478$	0	0
$479$	23.2788	1.06363	0.531817	−	0.846859i	$-0.321509\pi$
$479$	23.2788	1.06363	0.531817	+	0.846859i	$0.321509\pi$
$480$	0	0
$481$	26.6801	1.21651
$482$	0	0
$483$	3.16982	0.144232
$484$	0	0
$485$	32.8425	1.49130
$486$	0	0
$487$	−11.5194	−0.521994	−0.260997	−	0.965340i	$-0.584051\pi$
$487$	−11.5194	−0.521994	−0.260997	+	0.965340i	$0.584051\pi$
$488$	0	0
$489$	5.69021	0.257320
$490$	0	0
$491$	−2.90881	−0.131273	−0.0656363	−	0.997844i	$-0.520908\pi$
$491$	−2.90881	−0.131273	−0.0656363	+	0.997844i	$0.520908\pi$
$492$	0	0
$493$	5.82218	0.262218
$494$	0	0
$495$	−15.2066	−0.683486
$496$	0	0
$497$	−30.3976	−1.36352
$498$	0	0
$499$	1.42174	0.0636460	0.0318230	−	0.999494i	$-0.489869\pi$
$499$	1.42174	0.0636460	0.0318230	+	0.999494i	$0.489869\pi$
$500$	0	0
$501$	−8.04965	−0.359632
$502$	0	0
$503$	−13.3035	−0.593172	−0.296586	−	0.955006i	$-0.595848\pi$
$503$	−13.3035	−0.593172	−0.296586	+	0.955006i	$0.595848\pi$
$504$	0	0
$505$	−50.7981	−2.26049
$506$	0	0
$507$	−7.97808	−0.354319
$508$	0	0
$509$	−18.0156	−0.798526	−0.399263	−	0.916836i	$-0.630734\pi$
$509$	−18.0156	−0.798526	−0.399263	+	0.916836i	$0.630734\pi$
$510$	0	0
$511$	3.71874	0.164507
$512$	0	0
$513$	6.74968	0.298006
$514$	0	0
$515$	−35.5310	−1.56568
$516$	0	0
$517$	35.6663	1.56860
$518$	0	0
$519$	17.8663	0.784242
$520$	0	0
$521$	0.956011	0.0418836	0.0209418	−	0.999781i	$-0.493334\pi$
$521$	0.956011	0.0418836	0.0209418	+	0.999781i	$0.493334\pi$
$522$	0	0
$523$	−2.90772	−0.127146	−0.0635729	−	0.997977i	$-0.520250\pi$
$523$	−2.90772	−0.127146	−0.0635729	+	0.997977i	$0.520250\pi$
$524$	0	0
$525$	−8.46898	−0.369617
$526$	0	0
$527$	−23.9919	−1.04510
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.93478	0.170755
$532$	0	0
$533$	13.2647	0.574558
$534$	0	0
$535$	34.1929	1.47829
$536$	0	0
$537$	−5.00570	−0.216012
$538$	0	0
$539$	16.7327	0.720728
$540$	0	0
$541$	−32.2821	−1.38792	−0.693958	−	0.720015i	$-0.744135\pi$
$541$	−32.2821	−1.38792	−0.693958	+	0.720015i	$0.744135\pi$
$542$	0	0
$543$	6.11998	0.262634
$544$	0	0
$545$	−27.7158	−1.18722
$546$	0	0
$547$	29.9950	1.28249	0.641246	−	0.767335i	$-0.278418\pi$
$547$	29.9950	1.28249	0.641246	+	0.767335i	$0.278418\pi$
$548$	0	0
$549$	2.42327	0.103422
$550$	0	0
$551$	6.74968	0.287546
$552$	0	0
$553$	−7.81278	−0.332233
$554$	0	0
$555$	32.9761	1.39976
$556$	0	0
$557$	6.30193	0.267021	0.133511	−	0.991047i	$-0.457375\pi$
$557$	6.30193	0.267021	0.133511	+	0.991047i	$0.457375\pi$
$558$	0	0
$559$	6.18096	0.261427
$560$	0	0
$561$	31.9647	1.34955
$562$	0	0
$563$	−37.4658	−1.57900	−0.789498	−	0.613754i	$-0.789659\pi$
$563$	−37.4658	−1.57900	−0.789498	+	0.613754i	$0.789659\pi$
$564$	0	0
$565$	12.5997	0.530075
$566$	0	0
$567$	−3.16982	−0.133120
$568$	0	0
$569$	−18.7324	−0.785303	−0.392652	−	0.919687i	$-0.628442\pi$
$569$	−18.7324	−0.785303	−0.392652	+	0.919687i	$0.628442\pi$
$570$	0	0
$571$	16.4723	0.689345	0.344673	−	0.938723i	$-0.387990\pi$
$571$	16.4723	0.689345	0.344673	+	0.938723i	$0.387990\pi$
$572$	0	0
$573$	1.52978	0.0639076
$574$	0	0
$575$	−2.67175	−0.111420
$576$	0	0
$577$	15.7973	0.657650	0.328825	−	0.944391i	$-0.393347\pi$
$577$	15.7973	0.657650	0.328825	+	0.944391i	$0.393347\pi$
$578$	0	0
$579$	−7.47365	−0.310594
$580$	0	0
$581$	40.1783	1.66688
$582$	0	0
$583$	−35.9590	−1.48927
$584$	0	0
$585$	6.20701	0.256628
$586$	0	0
$587$	−22.7942	−0.940818	−0.470409	−	0.882449i	$-0.655894\pi$
$587$	−22.7942	−0.940818	−0.470409	+	0.882449i	$0.655894\pi$
$588$	0	0
$589$	−27.8139	−1.14605
$590$	0	0
$591$	−15.0874	−0.620615
$592$	0	0
$593$	−23.5701	−0.967909	−0.483954	−	0.875093i	$-0.660800\pi$
$593$	−23.5701	−0.967909	−0.483954	+	0.875093i	$0.660800\pi$
$594$	0	0
$595$	51.1173	2.09560
$596$	0	0
$597$	16.8893	0.691233
$598$	0	0
$599$	25.2908	1.03335	0.516677	−	0.856181i	$-0.327169\pi$
$599$	25.2908	1.03335	0.516677	+	0.856181i	$0.327169\pi$
$600$	0	0
$601$	26.7649	1.09176	0.545882	−	0.837862i	$-0.316195\pi$
$601$	26.7649	1.09176	0.545882	+	0.837862i	$0.316195\pi$
$602$	0	0
$603$	5.15868	0.210078
$604$	0	0
$605$	−53.0190	−2.15553
$606$	0	0
$607$	0.756781	0.0307168	0.0153584	−	0.999882i	$-0.495111\pi$
$607$	0.756781	0.0307168	0.0153584	+	0.999882i	$0.495111\pi$
$608$	0	0
$609$	−3.16982	−0.128448
$610$	0	0
$611$	−14.5582	−0.588962
$612$	0	0
$613$	7.95980	0.321493	0.160747	−	0.986996i	$-0.448610\pi$
$613$	7.95980	0.321493	0.160747	+	0.986996i	$0.448610\pi$
$614$	0	0
$615$	16.3949	0.661108
$616$	0	0
$617$	25.1269	1.01157	0.505785	−	0.862659i	$-0.331203\pi$
$617$	25.1269	1.01157	0.505785	+	0.862659i	$0.331203\pi$
$618$	0	0
$619$	−29.8969	−1.20166	−0.600829	−	0.799377i	$-0.705163\pi$
$619$	−29.8969	−1.20166	−0.600829	+	0.799377i	$0.705163\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−22.0932	−0.885146
$624$	0	0
$625$	−31.2205	−1.24882
$626$	0	0
$627$	37.0568	1.47991
$628$	0	0
$629$	−69.3167	−2.76384
$630$	0	0
$631$	43.5364	1.73316	0.866579	−	0.499040i	$-0.166314\pi$
$631$	43.5364	1.73316	0.866579	+	0.499040i	$0.166314\pi$
$632$	0	0
$633$	9.64944	0.383531
$634$	0	0
$635$	−14.3157	−0.568101
$636$	0	0
$637$	−6.82993	−0.270612
$638$	0	0
$639$	9.58968	0.379362
$640$	0	0
$641$	−34.4738	−1.36163	−0.680817	−	0.732453i	$-0.738375\pi$
$641$	−34.4738	−1.36163	−0.680817	+	0.732453i	$0.738375\pi$
$642$	0	0
$643$	−16.8635	−0.665032	−0.332516	−	0.943098i	$-0.607898\pi$
$643$	−16.8635	−0.665032	−0.332516	+	0.943098i	$0.607898\pi$
$644$	0	0
$645$	7.63956	0.300807
$646$	0	0
$647$	−44.2121	−1.73816	−0.869078	−	0.494675i	$-0.835287\pi$
$647$	−44.2121	−1.73816	−0.869078	+	0.494675i	$0.835287\pi$
$648$	0	0
$649$	21.6026	0.847976
$650$	0	0
$651$	13.0621	0.511944
$652$	0	0
$653$	−40.9585	−1.60283	−0.801415	−	0.598109i	$-0.795919\pi$
$653$	−40.9585	−1.60283	−0.801415	+	0.598109i	$0.795919\pi$
$654$	0	0
$655$	−56.4731	−2.20659
$656$	0	0
$657$	−1.17317	−0.0457697
$658$	0	0
$659$	−12.6332	−0.492120	−0.246060	−	0.969255i	$-0.579136\pi$
$659$	−12.6332	−0.492120	−0.246060	+	0.969255i	$0.579136\pi$
$660$	0	0
$661$	−22.1947	−0.863275	−0.431638	−	0.902047i	$-0.642064\pi$
$661$	−22.1947	−0.863275	−0.431638	+	0.902047i	$0.642064\pi$
$662$	0	0
$663$	−13.0473	−0.506715
$664$	0	0
$665$	59.2605	2.29802
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−2.04044	−0.0788881
$670$	0	0
$671$	13.3041	0.513600
$672$	0	0
$673$	23.3518	0.900145	0.450073	−	0.892992i	$-0.351398\pi$
$673$	23.3518	0.900145	0.450073	+	0.892992i	$0.351398\pi$
$674$	0	0
$675$	2.67175	0.102836
$676$	0	0
$677$	22.2656	0.855737	0.427869	−	0.903841i	$-0.359265\pi$
$677$	22.2656	0.855737	0.427869	+	0.903841i	$0.359265\pi$
$678$	0	0
$679$	37.5858	1.44241
$680$	0	0
$681$	23.4455	0.898432
$682$	0	0
$683$	−1.11977	−0.0428467	−0.0214233	−	0.999770i	$-0.506820\pi$
$683$	−1.11977	−0.0428467	−0.0214233	+	0.999770i	$0.506820\pi$
$684$	0	0
$685$	16.5741	0.633262
$686$	0	0
$687$	−1.43125	−0.0546056
$688$	0	0
$689$	14.6777	0.559176
$690$	0	0
$691$	21.2447	0.808187	0.404093	−	0.914718i	$-0.367587\pi$
$691$	21.2447	0.808187	0.404093	+	0.914718i	$0.367587\pi$
$692$	0	0
$693$	−17.4028	−0.661078
$694$	0	0
$695$	−19.2450	−0.730004
$696$	0	0
$697$	−34.4626	−1.30537
$698$	0	0
$699$	18.1113	0.685032
$700$	0	0
$701$	−30.5989	−1.15571	−0.577853	−	0.816141i	$-0.696109\pi$
$701$	−30.5989	−1.15571	−0.577853	+	0.816141i	$0.696109\pi$
$702$	0	0
$703$	−80.3592	−3.03080
$704$	0	0
$705$	−17.9937	−0.677681
$706$	0	0
$707$	−58.1347	−2.18638
$708$	0	0
$709$	21.4942	0.807233	0.403617	−	0.914928i	$-0.367753\pi$
$709$	21.4942	0.807233	0.403617	+	0.914928i	$0.367753\pi$
$710$	0	0
$711$	2.46474	0.0924349
$712$	0	0
$713$	4.12077	0.154324
$714$	0	0
$715$	34.0775	1.27442
$716$	0	0
$717$	11.1101	0.414913
$718$	0	0
$719$	−2.16440	−0.0807184	−0.0403592	−	0.999185i	$-0.512850\pi$
$719$	−2.16440	−0.0807184	−0.0403592	+	0.999185i	$0.512850\pi$
$720$	0	0
$721$	−40.6625	−1.51435
$722$	0	0
$723$	2.18864	0.0813962
$724$	0	0
$725$	2.67175	0.0992264
$726$	0	0
$727$	43.7730	1.62345	0.811725	−	0.584040i	$-0.198529\pi$
$727$	43.7730	1.62345	0.811725	+	0.584040i	$0.198529\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0586	−0.593948
$732$	0	0
$733$	−52.5826	−1.94218	−0.971091	−	0.238708i	$-0.923276\pi$
$733$	−52.5826	−1.94218	−0.971091	+	0.238708i	$0.923276\pi$
$734$	0	0
$735$	−8.44167	−0.311376
$736$	0	0
$737$	28.3220	1.04325
$738$	0	0
$739$	−31.0022	−1.14044	−0.570218	−	0.821493i	$-0.693141\pi$
$739$	−31.0022	−1.14044	−0.570218	+	0.821493i	$0.693141\pi$
$740$	0	0
$741$	−15.1258	−0.555660
$742$	0	0
$743$	17.3787	0.637564	0.318782	−	0.947828i	$-0.396726\pi$
$743$	17.3787	0.637564	0.318782	+	0.947828i	$0.396726\pi$
$744$	0	0
$745$	0.543994	0.0199304
$746$	0	0
$747$	−12.6753	−0.463764
$748$	0	0
$749$	39.1313	1.42982
$750$	0	0
$751$	−16.8025	−0.613132	−0.306566	−	0.951849i	$-0.599180\pi$
$751$	−16.8025	−0.613132	−0.306566	+	0.951849i	$0.599180\pi$
$752$	0	0
$753$	−21.1501	−0.770751
$754$	0	0
$755$	−36.9465	−1.34462
$756$	0	0
$757$	7.32776	0.266332	0.133166	−	0.991094i	$-0.457486\pi$
$757$	7.32776	0.266332	0.133166	+	0.991094i	$0.457486\pi$
$758$	0	0
$759$	−5.49016	−0.199280
$760$	0	0
$761$	−22.0238	−0.798364	−0.399182	−	0.916872i	$-0.630706\pi$
$761$	−22.0238	−0.798364	−0.399182	+	0.916872i	$0.630706\pi$
$762$	0	0
$763$	−31.7187	−1.14829
$764$	0	0
$765$	−16.1262	−0.583045
$766$	0	0
$767$	−8.81771	−0.318389
$768$	0	0
$769$	−32.8703	−1.18533	−0.592666	−	0.805448i	$-0.701925\pi$
$769$	−32.8703	−1.18533	−0.592666	+	0.805448i	$0.701925\pi$
$770$	0	0
$771$	−23.8657	−0.859502
$772$	0	0
$773$	40.8858	1.47056	0.735279	−	0.677764i	$-0.237051\pi$
$773$	40.8858	1.47056	0.735279	+	0.677764i	$0.237051\pi$
$774$	0	0
$775$	−11.0097	−0.395480
$776$	0	0
$777$	37.7387	1.35387
$778$	0	0
$779$	−39.9527	−1.43145
$780$	0	0
$781$	52.6489	1.88393
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−1.90201	−0.0678857
$786$	0	0
$787$	17.2915	0.616374	0.308187	−	0.951326i	$-0.400278\pi$
$787$	17.2915	0.616374	0.308187	+	0.951326i	$0.400278\pi$
$788$	0	0
$789$	16.6916	0.594237
$790$	0	0
$791$	14.4194	0.512696
$792$	0	0
$793$	−5.43045	−0.192841
$794$	0	0
$795$	18.1414	0.643408
$796$	0	0
$797$	35.7004	1.26457	0.632286	−	0.774735i	$-0.282117\pi$
$797$	35.7004	1.26457	0.632286	+	0.774735i	$0.282117\pi$
$798$	0	0
$799$	37.8232	1.33809
$800$	0	0
$801$	6.96986	0.246268
$802$	0	0
$803$	−6.44089	−0.227294
$804$	0	0
$805$	−8.77975	−0.309445
$806$	0	0
$807$	10.1935	0.358828
$808$	0	0
$809$	25.9639	0.912842	0.456421	−	0.889764i	$-0.349131\pi$
$809$	25.9639	0.912842	0.456421	+	0.889764i	$0.349131\pi$
$810$	0	0
$811$	1.85227	0.0650419	0.0325209	−	0.999471i	$-0.489646\pi$
$811$	1.85227	0.0650419	0.0325209	+	0.999471i	$0.489646\pi$
$812$	0	0
$813$	−7.96044	−0.279185
$814$	0	0
$815$	−15.7607	−0.552073
$816$	0	0
$817$	−18.6168	−0.651318
$818$	0	0
$819$	7.10346	0.248215
$820$	0	0
$821$	−44.2411	−1.54403	−0.772013	−	0.635607i	$-0.780750\pi$
$821$	−44.2411	−1.54403	−0.772013	+	0.635607i	$0.780750\pi$
$822$	0	0
$823$	22.3856	0.780314	0.390157	−	0.920748i	$-0.372421\pi$
$823$	22.3856	0.780314	0.390157	+	0.920748i	$0.372421\pi$
$824$	0	0
$825$	14.6684	0.510686
$826$	0	0
$827$	10.1460	0.352810	0.176405	−	0.984318i	$-0.443553\pi$
$827$	10.1460	0.352810	0.176405	+	0.984318i	$0.443553\pi$
$828$	0	0
$829$	22.8148	0.792390	0.396195	−	0.918166i	$-0.370330\pi$
$829$	22.8148	0.792390	0.396195	+	0.918166i	$0.370330\pi$
$830$	0	0
$831$	−21.2425	−0.736893
$832$	0	0
$833$	17.7446	0.614815
$834$	0	0
$835$	22.2959	0.771581
$836$	0	0
$837$	−4.12077	−0.142435
$838$	0	0
$839$	−25.7668	−0.889569	−0.444784	−	0.895638i	$-0.646720\pi$
$839$	−25.7668	−0.889569	−0.444784	+	0.895638i	$0.646720\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	18.2543	0.628711
$844$	0	0
$845$	22.0976	0.760181
$846$	0	0
$847$	−60.6762	−2.08486
$848$	0	0
$849$	31.1156	1.06788
$850$	0	0
$851$	11.9056	0.408119
$852$	0	0
$853$	23.8945	0.818132	0.409066	−	0.912505i	$-0.365855\pi$
$853$	23.8945	0.818132	0.409066	+	0.912505i	$0.365855\pi$
$854$	0	0
$855$	−18.6952	−0.639363
$856$	0	0
$857$	21.6568	0.739780	0.369890	−	0.929075i	$-0.379395\pi$
$857$	21.6568	0.739780	0.369890	+	0.929075i	$0.379395\pi$
$858$	0	0
$859$	10.5305	0.359295	0.179647	−	0.983731i	$-0.442504\pi$
$859$	10.5305	0.359295	0.179647	+	0.983731i	$0.442504\pi$
$860$	0	0
$861$	18.7628	0.639434
$862$	0	0
$863$	−57.7783	−1.96680	−0.983399	−	0.181458i	$-0.941918\pi$
$863$	−57.7783	−1.96680	−0.983399	+	0.181458i	$0.941918\pi$
$864$	0	0
$865$	−49.4859	−1.68257
$866$	0	0
$867$	16.8978	0.573880
$868$	0	0
$869$	13.5318	0.459035
$870$	0	0
$871$	−11.5604	−0.391710
$872$	0	0
$873$	−11.8574	−0.401312
$874$	0	0
$875$	−20.4414	−0.691046
$876$	0	0
$877$	22.6742	0.765654	0.382827	−	0.923820i	$-0.374951\pi$
$877$	22.6742	0.765654	0.382827	+	0.923820i	$0.374951\pi$
$878$	0	0
$879$	−33.0397	−1.11440
$880$	0	0
$881$	30.7312	1.03536	0.517681	−	0.855574i	$-0.326796\pi$
$881$	30.7312	1.03536	0.517681	+	0.855574i	$0.326796\pi$
$882$	0	0
$883$	−46.6694	−1.57055	−0.785274	−	0.619148i	$-0.787478\pi$
$883$	−46.6694	−1.57055	−0.785274	+	0.619148i	$0.787478\pi$
$884$	0	0
$885$	−10.8985	−0.366350
$886$	0	0
$887$	−32.5708	−1.09362	−0.546809	−	0.837257i	$-0.684158\pi$
$887$	−32.5708	−1.09362	−0.546809	+	0.837257i	$0.684158\pi$
$888$	0	0
$889$	−16.3832	−0.549476
$890$	0	0
$891$	5.49016	0.183927
$892$	0	0
$893$	43.8486	1.46734
$894$	0	0
$895$	13.8648	0.463448
$896$	0	0
$897$	2.24096	0.0748236
$898$	0	0
$899$	−4.12077	−0.137435
$900$	0	0
$901$	−38.1337	−1.27042
$902$	0	0
$903$	8.74291	0.290946
$904$	0	0
$905$	−16.9511	−0.563473
$906$	0	0
$907$	39.0545	1.29678	0.648391	−	0.761307i	$-0.275442\pi$
$907$	39.0545	1.29678	0.648391	+	0.761307i	$0.275442\pi$
$908$	0	0
$909$	18.3401	0.608301
$910$	0	0
$911$	34.5687	1.14531	0.572657	−	0.819795i	$-0.305913\pi$
$911$	34.5687	1.14531	0.572657	+	0.819795i	$0.305913\pi$
$912$	0	0
$913$	−69.5893	−2.30307
$914$	0	0
$915$	−6.71195	−0.221890
$916$	0	0
$917$	−64.6292	−2.13424
$918$	0	0
$919$	−57.2573	−1.88874	−0.944372	−	0.328879i	$-0.893329\pi$
$919$	−57.2573	−1.88874	−0.944372	+	0.328879i	$0.893329\pi$
$920$	0	0
$921$	25.5928	0.843313
$922$	0	0
$923$	−21.4901	−0.707357
$924$	0	0
$925$	−31.8089	−1.04587
$926$	0	0
$927$	12.8280	0.421328
$928$	0	0
$929$	9.13416	0.299682	0.149841	−	0.988710i	$-0.452124\pi$
$929$	9.13416	0.299682	0.149841	+	0.988710i	$0.452124\pi$
$930$	0	0
$931$	20.5714	0.674201
$932$	0	0
$933$	−26.1235	−0.855244
$934$	0	0
$935$	−88.5356	−2.89542
$936$	0	0
$937$	−10.1072	−0.330189	−0.165095	−	0.986278i	$-0.552793\pi$
$937$	−10.1072	−0.330189	−0.165095	+	0.986278i	$0.552793\pi$
$938$	0	0
$939$	9.79501	0.319648
$940$	0	0
$941$	45.6362	1.48770	0.743849	−	0.668347i	$-0.232998\pi$
$941$	45.6362	1.48770	0.743849	+	0.668347i	$0.232998\pi$
$942$	0	0
$943$	5.91919	0.192755
$944$	0	0
$945$	8.77975	0.285605
$946$	0	0
$947$	23.4085	0.760673	0.380336	−	0.924848i	$-0.375808\pi$
$947$	23.4085	0.760673	0.380336	+	0.924848i	$0.375808\pi$
$948$	0	0
$949$	2.62903	0.0853419
$950$	0	0
$951$	26.6180	0.863148
$952$	0	0
$953$	18.6949	0.605587	0.302793	−	0.953056i	$-0.402081\pi$
$953$	18.6949	0.605587	0.302793	+	0.953056i	$0.402081\pi$
$954$	0	0
$955$	−4.23718	−0.137112
$956$	0	0
$957$	5.49016	0.177472
$958$	0	0
$959$	18.9678	0.612501
$960$	0	0
$961$	−14.0192	−0.452234
$962$	0	0
$963$	−12.3449	−0.397810
$964$	0	0
$965$	20.7005	0.666372
$966$	0	0
$967$	17.7664	0.571330	0.285665	−	0.958330i	$-0.407786\pi$
$967$	17.7664	0.571330	0.285665	+	0.958330i	$0.407786\pi$
$968$	0	0
$969$	39.2979	1.26243
$970$	0	0
$971$	23.9272	0.767861	0.383931	−	0.923362i	$-0.374570\pi$
$971$	23.9272	0.767861	0.383931	+	0.923362i	$0.374570\pi$
$972$	0	0
$973$	−22.0244	−0.706071
$974$	0	0
$975$	−5.98731	−0.191747
$976$	0	0
$977$	−23.9111	−0.764984	−0.382492	−	0.923959i	$-0.624934\pi$
$977$	−23.9111	−0.764984	−0.382492	+	0.923959i	$0.624934\pi$
$978$	0	0
$979$	38.2657	1.22298
$980$	0	0
$981$	10.0065	0.319482
$982$	0	0
$983$	−32.6222	−1.04049	−0.520243	−	0.854018i	$-0.674159\pi$
$983$	−32.6222	−1.04049	−0.520243	+	0.854018i	$0.674159\pi$
$984$	0	0
$985$	41.7891	1.33151
$986$	0	0
$987$	−20.5924	−0.655464
$988$	0	0
$989$	2.75817	0.0877047
$990$	0	0
$991$	−52.6244	−1.67167	−0.835835	−	0.548980i	$-0.815016\pi$
$991$	−52.6244	−1.67167	−0.835835	+	0.548980i	$0.815016\pi$
$992$	0	0
$993$	−22.1818	−0.703917
$994$	0	0
$995$	−46.7799	−1.48302
$996$	0	0
$997$	50.0395	1.58477	0.792383	−	0.610024i	$-0.208840\pi$
$997$	50.0395	1.58477	0.792383	+	0.610024i	$0.208840\pi$
$998$	0	0
$999$	−11.9056	−0.376677

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.4	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.4	✓	18	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.76979 q^{5} -3.16982 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.76979 q^{5} -3.16982 q^{7} +1.00000 q^{9} +5.49016 q^{11} -2.24096 q^{13} -2.76979 q^{15} +5.82218 q^{17} +6.74968 q^{19} -3.16982 q^{21} -1.00000 q^{23} +2.67175 q^{25} +1.00000 q^{27} +1.00000 q^{29} -4.12077 q^{31} +5.49016 q^{33} +8.77975 q^{35} -11.9056 q^{37} -2.24096 q^{39} -5.91919 q^{41} -2.75817 q^{43} -2.76979 q^{45} +6.49640 q^{47} +3.04776 q^{49} +5.82218 q^{51} -6.54972 q^{53} -15.2066 q^{55} +6.74968 q^{57} +3.93478 q^{59} +2.42327 q^{61} -3.16982 q^{63} +6.20701 q^{65} +5.15868 q^{67} -1.00000 q^{69} +9.58968 q^{71} -1.17317 q^{73} +2.67175 q^{75} -17.4028 q^{77} +2.46474 q^{79} +1.00000 q^{81} -12.6753 q^{83} -16.1262 q^{85} +1.00000 q^{87} +6.96986 q^{89} +7.10346 q^{91} -4.12077 q^{93} -18.6952 q^{95} -11.8574 q^{97} +5.49016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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