LMFDB - Embedded newform 6004.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68125 q^{3} -3.32036 q^{5} +3.49635 q^{7} -0.173410 q^{9} +O(q^{10})$	$q-1.68125 q^{3} -3.32036 q^{5} +3.49635 q^{7} -0.173410 q^{9} +1.26661 q^{11} -1.98312 q^{13} +5.58235 q^{15} -1.03307 q^{17} +1.00000 q^{19} -5.87823 q^{21} -6.45830 q^{23} +6.02481 q^{25} +5.33528 q^{27} +5.24328 q^{29} -0.427070 q^{31} -2.12948 q^{33} -11.6092 q^{35} +8.20179 q^{37} +3.33411 q^{39} -12.4681 q^{41} -4.70973 q^{43} +0.575785 q^{45} +9.04823 q^{47} +5.22450 q^{49} +1.73684 q^{51} +0.909060 q^{53} -4.20559 q^{55} -1.68125 q^{57} -10.3555 q^{59} +15.1296 q^{61} -0.606304 q^{63} +6.58467 q^{65} -3.76031 q^{67} +10.8580 q^{69} +5.33012 q^{71} +5.75154 q^{73} -10.1292 q^{75} +4.42851 q^{77} -1.00000 q^{79} -8.44970 q^{81} -8.11305 q^{83} +3.43016 q^{85} -8.81525 q^{87} -2.12521 q^{89} -6.93369 q^{91} +0.718010 q^{93} -3.32036 q^{95} +14.4726 q^{97} -0.219643 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	−1.68125	−0.970668	−0.485334	−	0.874329i	$-0.661302\pi$
$3$	−1.68125	−0.970668	−0.485334	+	0.874329i	$0.661302\pi$
$4$	0	0
$5$	−3.32036	−1.48491	−0.742456	−	0.669895i	$-0.766339\pi$
$5$	−3.32036	−1.48491	−0.742456	+	0.669895i	$0.766339\pi$
$6$	0	0
$7$	3.49635	1.32150	0.660749	−	0.750607i	$-0.270239\pi$
$7$	3.49635	1.32150	0.660749	+	0.750607i	$0.270239\pi$
$8$	0	0
$9$	−0.173410	−0.0578034
$10$	0	0
$11$	1.26661	0.381896	0.190948	−	0.981600i	$-0.438844\pi$
$11$	1.26661	0.381896	0.190948	+	0.981600i	$0.438844\pi$
$12$	0	0
$13$	−1.98312	−0.550018	−0.275009	−	0.961442i	$-0.588681\pi$
$13$	−1.98312	−0.550018	−0.275009	+	0.961442i	$0.588681\pi$
$14$	0	0
$15$	5.58235	1.44136
$16$	0	0
$17$	−1.03307	−0.250556	−0.125278	−	0.992122i	$-0.539982\pi$
$17$	−1.03307	−0.250556	−0.125278	+	0.992122i	$0.539982\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−5.87823	−1.28274
$22$	0	0
$23$	−6.45830	−1.34665	−0.673325	−	0.739347i	$-0.735134\pi$
$23$	−6.45830	−1.34665	−0.673325	+	0.739347i	$0.735134\pi$
$24$	0	0
$25$	6.02481	1.20496
$26$	0	0
$27$	5.33528	1.02678
$28$	0	0
$29$	5.24328	0.973653	0.486826	−	0.873499i	$-0.338154\pi$
$29$	5.24328	0.973653	0.486826	+	0.873499i	$0.338154\pi$
$30$	0	0
$31$	−0.427070	−0.0767041	−0.0383520	−	0.999264i	$-0.512211\pi$
$31$	−0.427070	−0.0767041	−0.0383520	+	0.999264i	$0.512211\pi$
$32$	0	0
$33$	−2.12948	−0.370694
$34$	0	0
$35$	−11.6092	−1.96231
$36$	0	0
$37$	8.20179	1.34837	0.674183	−	0.738564i	$-0.264496\pi$
$37$	8.20179	1.34837	0.674183	+	0.738564i	$0.264496\pi$
$38$	0	0
$39$	3.33411	0.533885
$40$	0	0
$41$	−12.4681	−1.94719	−0.973593	−	0.228289i	$-0.926687\pi$
$41$	−12.4681	−1.94719	−0.973593	+	0.228289i	$0.926687\pi$
$42$	0	0
$43$	−4.70973	−0.718227	−0.359114	−	0.933294i	$-0.616921\pi$
$43$	−4.70973	−0.718227	−0.359114	+	0.933294i	$0.616921\pi$
$44$	0	0
$45$	0.575785	0.0858330
$46$	0	0
$47$	9.04823	1.31982	0.659910	−	0.751345i	$-0.270594\pi$
$47$	9.04823	1.31982	0.659910	+	0.751345i	$0.270594\pi$
$48$	0	0
$49$	5.22450	0.746357
$50$	0	0
$51$	1.73684	0.243206
$52$	0	0
$53$	0.909060	0.124869	0.0624345	−	0.998049i	$-0.480114\pi$
$53$	0.909060	0.124869	0.0624345	+	0.998049i	$0.480114\pi$
$54$	0	0
$55$	−4.20559	−0.567082
$56$	0	0
$57$	−1.68125	−0.222687
$58$	0	0
$59$	−10.3555	−1.34818	−0.674088	−	0.738651i	$-0.735463\pi$
$59$	−10.3555	−1.34818	−0.674088	+	0.738651i	$0.735463\pi$
$60$	0	0
$61$	15.1296	1.93714	0.968572	−	0.248732i	$-0.0800139\pi$
$61$	15.1296	1.93714	0.968572	+	0.248732i	$0.0800139\pi$
$62$	0	0
$63$	−0.606304	−0.0763871
$64$	0	0
$65$	6.58467	0.816728
$66$	0	0
$67$	−3.76031	−0.459395	−0.229698	−	0.973262i	$-0.573774\pi$
$67$	−3.76031	−0.459395	−0.229698	+	0.973262i	$0.573774\pi$
$68$	0	0
$69$	10.8580	1.30715
$70$	0	0
$71$	5.33012	0.632569	0.316284	−	0.948664i	$-0.397565\pi$
$71$	5.33012	0.632569	0.316284	+	0.948664i	$0.397565\pi$
$72$	0	0
$73$	5.75154	0.673167	0.336584	−	0.941654i	$-0.390729\pi$
$73$	5.75154	0.673167	0.336584	+	0.941654i	$0.390729\pi$
$74$	0	0
$75$	−10.1292	−1.16962
$76$	0	0
$77$	4.42851	0.504675
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−8.44970	−0.938855
$82$	0	0
$83$	−8.11305	−0.890523	−0.445261	−	0.895401i	$-0.646889\pi$
$83$	−8.11305	−0.890523	−0.445261	+	0.895401i	$0.646889\pi$
$84$	0	0
$85$	3.43016	0.372053
$86$	0	0
$87$	−8.81525	−0.945094
$88$	0	0
$89$	−2.12521	−0.225271	−0.112636	−	0.993636i	$-0.535929\pi$
$89$	−2.12521	−0.225271	−0.112636	+	0.993636i	$0.535929\pi$
$90$	0	0
$91$	−6.93369	−0.726848
$92$	0	0
$93$	0.718010	0.0744542
$94$	0	0
$95$	−3.32036	−0.340662
$96$	0	0
$97$	14.4726	1.46947	0.734734	−	0.678355i	$-0.237307\pi$
$97$	14.4726	1.46947	0.734734	+	0.678355i	$0.237307\pi$
$98$	0	0
$99$	−0.219643	−0.0220749
$100$	0	0
$101$	18.5493	1.84572	0.922861	−	0.385134i	$-0.125845\pi$
$101$	18.5493	1.84572	0.922861	+	0.385134i	$0.125845\pi$
$102$	0	0
$103$	−0.132652	−0.0130706	−0.00653528	−	0.999979i	$-0.502080\pi$
$103$	−0.132652	−0.0130706	−0.00653528	+	0.999979i	$0.502080\pi$
$104$	0	0
$105$	19.5179	1.90475
$106$	0	0
$107$	−0.807610	−0.0780746	−0.0390373	−	0.999238i	$-0.512429\pi$
$107$	−0.807610	−0.0780746	−0.0390373	+	0.999238i	$0.512429\pi$
$108$	0	0
$109$	4.29642	0.411522	0.205761	−	0.978602i	$-0.434033\pi$
$109$	4.29642	0.411522	0.205761	+	0.978602i	$0.434033\pi$
$110$	0	0
$111$	−13.7892	−1.30882
$112$	0	0
$113$	−15.2214	−1.43191	−0.715956	−	0.698145i	$-0.754009\pi$
$113$	−15.2214	−1.43191	−0.715956	+	0.698145i	$0.754009\pi$
$114$	0	0
$115$	21.4439	1.99965
$116$	0	0
$117$	0.343893	0.0317930
$118$	0	0
$119$	−3.61197	−0.331109
$120$	0	0
$121$	−9.39571	−0.854155
$122$	0	0
$123$	20.9619	1.89007
$124$	0	0
$125$	−3.40273	−0.304350
$126$	0	0
$127$	0.371561	0.0329707	0.0164853	−	0.999864i	$-0.494752\pi$
$127$	0.371561	0.0329707	0.0164853	+	0.999864i	$0.494752\pi$
$128$	0	0
$129$	7.91822	0.697160
$130$	0	0
$131$	4.99094	0.436060	0.218030	−	0.975942i	$-0.430037\pi$
$131$	4.99094	0.436060	0.218030	+	0.975942i	$0.430037\pi$
$132$	0	0
$133$	3.49635	0.303172
$134$	0	0
$135$	−17.7151	−1.52467
$136$	0	0
$137$	−12.2861	−1.04967	−0.524837	−	0.851203i	$-0.675874\pi$
$137$	−12.2861	−1.04967	−0.524837	+	0.851203i	$0.675874\pi$
$138$	0	0
$139$	10.3930	0.881520	0.440760	−	0.897625i	$-0.354709\pi$
$139$	10.3930	0.881520	0.440760	+	0.897625i	$0.354709\pi$
$140$	0	0
$141$	−15.2123	−1.28111
$142$	0	0
$143$	−2.51183	−0.210050
$144$	0	0
$145$	−17.4096	−1.44579
$146$	0	0
$147$	−8.78367	−0.724465
$148$	0	0
$149$	5.52981	0.453020	0.226510	−	0.974009i	$-0.427268\pi$
$149$	5.52981	0.453020	0.226510	+	0.974009i	$0.427268\pi$
$150$	0	0
$151$	−2.05849	−0.167517	−0.0837586	−	0.996486i	$-0.526692\pi$
$151$	−2.05849	−0.167517	−0.0837586	+	0.996486i	$0.526692\pi$
$152$	0	0
$153$	0.179145	0.0144830
$154$	0	0
$155$	1.41803	0.113899
$156$	0	0
$157$	−4.42056	−0.352799	−0.176400	−	0.984319i	$-0.556445\pi$
$157$	−4.42056	−0.352799	−0.176400	+	0.984319i	$0.556445\pi$
$158$	0	0
$159$	−1.52835	−0.121206
$160$	0	0
$161$	−22.5805	−1.77959
$162$	0	0
$163$	−5.01264	−0.392620	−0.196310	−	0.980542i	$-0.562896\pi$
$163$	−5.01264	−0.392620	−0.196310	+	0.980542i	$0.562896\pi$
$164$	0	0
$165$	7.07064	0.550448
$166$	0	0
$167$	23.8475	1.84538	0.922689	−	0.385544i	$-0.125986\pi$
$167$	23.8475	1.84538	0.922689	+	0.385544i	$0.125986\pi$
$168$	0	0
$169$	−9.06724	−0.697480
$170$	0	0
$171$	−0.173410	−0.0132610
$172$	0	0
$173$	−8.06203	−0.612945	−0.306472	−	0.951880i	$-0.599149\pi$
$173$	−8.06203	−0.612945	−0.306472	+	0.951880i	$0.599149\pi$
$174$	0	0
$175$	21.0649	1.59235
$176$	0	0
$177$	17.4102	1.30863
$178$	0	0
$179$	−3.39075	−0.253436	−0.126718	−	0.991939i	$-0.540444\pi$
$179$	−3.39075	−0.253436	−0.126718	+	0.991939i	$0.540444\pi$
$180$	0	0
$181$	5.58804	0.415356	0.207678	−	0.978197i	$-0.433409\pi$
$181$	5.58804	0.415356	0.207678	+	0.978197i	$0.433409\pi$
$182$	0	0
$183$	−25.4366	−1.88032
$184$	0	0
$185$	−27.2329	−2.00220
$186$	0	0
$187$	−1.30849	−0.0956863
$188$	0	0
$189$	18.6540	1.35688
$190$	0	0
$191$	−14.7560	−1.06770	−0.533852	−	0.845578i	$-0.679256\pi$
$191$	−14.7560	−1.06770	−0.533852	+	0.845578i	$0.679256\pi$
$192$	0	0
$193$	−7.77834	−0.559897	−0.279949	−	0.960015i	$-0.590317\pi$
$193$	−7.77834	−0.559897	−0.279949	+	0.960015i	$0.590317\pi$
$194$	0	0
$195$	−11.0705	−0.792772
$196$	0	0
$197$	−14.6711	−1.04527	−0.522636	−	0.852556i	$-0.675051\pi$
$197$	−14.6711	−1.04527	−0.522636	+	0.852556i	$0.675051\pi$
$198$	0	0
$199$	2.33881	0.165794	0.0828969	−	0.996558i	$-0.473583\pi$
$199$	2.33881	0.165794	0.0828969	+	0.996558i	$0.473583\pi$
$200$	0	0
$201$	6.32201	0.445920
$202$	0	0
$203$	18.3324	1.28668
$204$	0	0
$205$	41.3985	2.89140
$206$	0	0
$207$	1.11994	0.0778410
$208$	0	0
$209$	1.26661	0.0876130
$210$	0	0
$211$	19.5048	1.34277	0.671384	−	0.741110i	$-0.265700\pi$
$211$	19.5048	1.34277	0.671384	+	0.741110i	$0.265700\pi$
$212$	0	0
$213$	−8.96125	−0.614014
$214$	0	0
$215$	15.6380	1.06650
$216$	0	0
$217$	−1.49319	−0.101364
$218$	0	0
$219$	−9.66976	−0.653422
$220$	0	0
$221$	2.04870	0.137810
$222$	0	0
$223$	−23.8667	−1.59823	−0.799116	−	0.601177i	$-0.794699\pi$
$223$	−23.8667	−1.59823	−0.799116	+	0.601177i	$0.794699\pi$
$224$	0	0
$225$	−1.04476	−0.0696509
$226$	0	0
$227$	19.8376	1.31666	0.658332	−	0.752727i	$-0.271262\pi$
$227$	19.8376	1.31666	0.658332	+	0.752727i	$0.271262\pi$
$228$	0	0
$229$	−13.8457	−0.914949	−0.457474	−	0.889223i	$-0.651246\pi$
$229$	−13.8457	−0.914949	−0.457474	+	0.889223i	$0.651246\pi$
$230$	0	0
$231$	−7.44541	−0.489872
$232$	0	0
$233$	20.6041	1.34982	0.674911	−	0.737899i	$-0.264182\pi$
$233$	20.6041	1.34982	0.674911	+	0.737899i	$0.264182\pi$
$234$	0	0
$235$	−30.0434	−1.95982
$236$	0	0
$237$	1.68125	0.109209
$238$	0	0
$239$	−23.1289	−1.49608	−0.748042	−	0.663651i	$-0.769006\pi$
$239$	−23.1289	−1.49608	−0.748042	+	0.663651i	$0.769006\pi$
$240$	0	0
$241$	−19.9271	−1.28362	−0.641810	−	0.766863i	$-0.721816\pi$
$241$	−19.9271	−1.28362	−0.641810	+	0.766863i	$0.721816\pi$
$242$	0	0
$243$	−1.79983	−0.115459
$244$	0	0
$245$	−17.3472	−1.10827
$246$	0	0
$247$	−1.98312	−0.126183
$248$	0	0
$249$	13.6400	0.864402
$250$	0	0
$251$	−26.8450	−1.69444	−0.847220	−	0.531242i	$-0.821725\pi$
$251$	−26.8450	−1.69444	−0.847220	+	0.531242i	$0.821725\pi$
$252$	0	0
$253$	−8.18013	−0.514280
$254$	0	0
$255$	−5.76694	−0.361140
$256$	0	0
$257$	18.7548	1.16989	0.584946	−	0.811072i	$-0.301116\pi$
$257$	18.7548	1.16989	0.584946	+	0.811072i	$0.301116\pi$
$258$	0	0
$259$	28.6764	1.78186
$260$	0	0
$261$	−0.909239	−0.0562805
$262$	0	0
$263$	−21.4425	−1.32220	−0.661101	−	0.750297i	$-0.729910\pi$
$263$	−21.4425	−1.32220	−0.661101	+	0.750297i	$0.729910\pi$
$264$	0	0
$265$	−3.01841	−0.185419
$266$	0	0
$267$	3.57299	0.218664
$268$	0	0
$269$	−26.0706	−1.58955	−0.794776	−	0.606903i	$-0.792412\pi$
$269$	−26.0706	−1.58955	−0.794776	+	0.606903i	$0.792412\pi$
$270$	0	0
$271$	−13.8688	−0.842470	−0.421235	−	0.906952i	$-0.638403\pi$
$271$	−13.8688	−0.842470	−0.421235	+	0.906952i	$0.638403\pi$
$272$	0	0
$273$	11.6572	0.705528
$274$	0	0
$275$	7.63106	0.460170
$276$	0	0
$277$	−7.24715	−0.435439	−0.217719	−	0.976011i	$-0.569862\pi$
$277$	−7.24715	−0.435439	−0.217719	+	0.976011i	$0.569862\pi$
$278$	0	0
$279$	0.0740584	0.00443376
$280$	0	0
$281$	−15.8410	−0.944993	−0.472496	−	0.881333i	$-0.656647\pi$
$281$	−15.8410	−0.944993	−0.472496	+	0.881333i	$0.656647\pi$
$282$	0	0
$283$	−18.3513	−1.09087	−0.545436	−	0.838153i	$-0.683636\pi$
$283$	−18.3513	−1.09087	−0.545436	+	0.838153i	$0.683636\pi$
$284$	0	0
$285$	5.58235	0.330670
$286$	0	0
$287$	−43.5928	−2.57320
$288$	0	0
$289$	−15.9328	−0.937222
$290$	0	0
$291$	−24.3320	−1.42637
$292$	0	0
$293$	−11.0088	−0.643143	−0.321572	−	0.946885i	$-0.604211\pi$
$293$	−11.0088	−0.643143	−0.321572	+	0.946885i	$0.604211\pi$
$294$	0	0
$295$	34.3841	2.00192
$296$	0	0
$297$	6.75771	0.392122
$298$	0	0
$299$	12.8076	0.740682
$300$	0	0
$301$	−16.4669	−0.949136
$302$	0	0
$303$	−31.1859	−1.79158
$304$	0	0
$305$	−50.2357	−2.87649
$306$	0	0
$307$	20.0621	1.14500	0.572502	−	0.819903i	$-0.305973\pi$
$307$	20.0621	1.14500	0.572502	+	0.819903i	$0.305973\pi$
$308$	0	0
$309$	0.223020	0.0126872
$310$	0	0
$311$	32.1654	1.82393	0.911965	−	0.410268i	$-0.134565\pi$
$311$	32.1654	1.82393	0.911965	+	0.410268i	$0.134565\pi$
$312$	0	0
$313$	−25.4898	−1.44077	−0.720385	−	0.693574i	$-0.756035\pi$
$313$	−25.4898	−1.44077	−0.720385	+	0.693574i	$0.756035\pi$
$314$	0	0
$315$	2.01315	0.113428
$316$	0	0
$317$	−18.3911	−1.03295	−0.516475	−	0.856302i	$-0.672756\pi$
$317$	−18.3911	−1.03295	−0.516475	+	0.856302i	$0.672756\pi$
$318$	0	0
$319$	6.64117	0.371834
$320$	0	0
$321$	1.35779	0.0757846
$322$	0	0
$323$	−1.03307	−0.0574814
$324$	0	0
$325$	−11.9479	−0.662751
$326$	0	0
$327$	−7.22334	−0.399452
$328$	0	0
$329$	31.6358	1.74414
$330$	0	0
$331$	−28.1213	−1.54569	−0.772844	−	0.634596i	$-0.781166\pi$
$331$	−28.1213	−1.54569	−0.772844	+	0.634596i	$0.781166\pi$
$332$	0	0
$333$	−1.42228	−0.0779402
$334$	0	0
$335$	12.4856	0.682161
$336$	0	0
$337$	0.794119	0.0432584	0.0216292	−	0.999766i	$-0.493115\pi$
$337$	0.794119	0.0432584	0.0216292	+	0.999766i	$0.493115\pi$
$338$	0	0
$339$	25.5910	1.38991
$340$	0	0
$341$	−0.540930	−0.0292930
$342$	0	0
$343$	−6.20779	−0.335189
$344$	0	0
$345$	−36.0525	−1.94100
$346$	0	0
$347$	14.7652	0.792635	0.396317	−	0.918114i	$-0.370288\pi$
$347$	14.7652	0.792635	0.396317	+	0.918114i	$0.370288\pi$
$348$	0	0
$349$	7.21238	0.386070	0.193035	−	0.981192i	$-0.438167\pi$
$349$	7.21238	0.386070	0.193035	+	0.981192i	$0.438167\pi$
$350$	0	0
$351$	−10.5805	−0.564746
$352$	0	0
$353$	24.3047	1.29361	0.646804	−	0.762656i	$-0.276105\pi$
$353$	24.3047	1.29361	0.646804	+	0.762656i	$0.276105\pi$
$354$	0	0
$355$	−17.6979	−0.939309
$356$	0	0
$357$	6.07261	0.321397
$358$	0	0
$359$	1.46627	0.0773870	0.0386935	−	0.999251i	$-0.487680\pi$
$359$	1.46627	0.0773870	0.0386935	+	0.999251i	$0.487680\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	15.7965	0.829101
$364$	0	0
$365$	−19.0972	−0.999593
$366$	0	0
$367$	0.999961	0.0521975	0.0260988	−	0.999659i	$-0.491692\pi$
$367$	0.999961	0.0521975	0.0260988	+	0.999659i	$0.491692\pi$
$368$	0	0
$369$	2.16209	0.112554
$370$	0	0
$371$	3.17840	0.165014
$372$	0	0
$373$	−9.61737	−0.497968	−0.248984	−	0.968508i	$-0.580097\pi$
$373$	−9.61737	−0.497968	−0.248984	+	0.968508i	$0.580097\pi$
$374$	0	0
$375$	5.72083	0.295422
$376$	0	0
$377$	−10.3980	−0.535527
$378$	0	0
$379$	−1.63781	−0.0841289	−0.0420645	−	0.999115i	$-0.513393\pi$
$379$	−1.63781	−0.0841289	−0.0420645	+	0.999115i	$0.513393\pi$
$380$	0	0
$381$	−0.624685	−0.0320036
$382$	0	0
$383$	−31.0318	−1.58565	−0.792825	−	0.609449i	$-0.791391\pi$
$383$	−31.0318	−1.58565	−0.792825	+	0.609449i	$0.791391\pi$
$384$	0	0
$385$	−14.7042	−0.749398
$386$	0	0
$387$	0.816716	0.0415160
$388$	0	0
$389$	−24.5350	−1.24398	−0.621988	−	0.783027i	$-0.713675\pi$
$389$	−24.5350	−1.24398	−0.621988	+	0.783027i	$0.713675\pi$
$390$	0	0
$391$	6.67186	0.337411
$392$	0	0
$393$	−8.39100	−0.423270
$394$	0	0
$395$	3.32036	0.167066
$396$	0	0
$397$	−17.9648	−0.901627	−0.450813	−	0.892618i	$-0.648866\pi$
$397$	−17.9648	−0.901627	−0.450813	+	0.892618i	$0.648866\pi$
$398$	0	0
$399$	−5.87823	−0.294280
$400$	0	0
$401$	−33.2471	−1.66028	−0.830140	−	0.557555i	$-0.811740\pi$
$401$	−33.2471	−1.66028	−0.830140	+	0.557555i	$0.811740\pi$
$402$	0	0
$403$	0.846931	0.0421886
$404$	0	0
$405$	28.0561	1.39412
$406$	0	0
$407$	10.3884	0.514936
$408$	0	0
$409$	−8.24436	−0.407657	−0.203829	−	0.979007i	$-0.565339\pi$
$409$	−8.24436	−0.407657	−0.203829	+	0.979007i	$0.565339\pi$
$410$	0	0
$411$	20.6560	1.01888
$412$	0	0
$413$	−36.2066	−1.78161
$414$	0	0
$415$	26.9383	1.32235
$416$	0	0
$417$	−17.4731	−0.855663
$418$	0	0
$419$	−12.3445	−0.603068	−0.301534	−	0.953455i	$-0.597499\pi$
$419$	−12.3445	−0.603068	−0.301534	+	0.953455i	$0.597499\pi$
$420$	0	0
$421$	−21.8527	−1.06504	−0.532518	−	0.846419i	$-0.678754\pi$
$421$	−21.8527	−1.06504	−0.532518	+	0.846419i	$0.678754\pi$
$422$	0	0
$423$	−1.56906	−0.0762901
$424$	0	0
$425$	−6.22403	−0.301910
$426$	0	0
$427$	52.8984	2.55993
$428$	0	0
$429$	4.22301	0.203889
$430$	0	0
$431$	−29.0252	−1.39809	−0.699047	−	0.715076i	$-0.746392\pi$
$431$	−29.0252	−1.39809	−0.699047	+	0.715076i	$0.746392\pi$
$432$	0	0
$433$	27.2078	1.30753	0.653763	−	0.756700i	$-0.273190\pi$
$433$	27.2078	1.30753	0.653763	+	0.756700i	$0.273190\pi$
$434$	0	0
$435$	29.2698	1.40338
$436$	0	0
$437$	−6.45830	−0.308943
$438$	0	0
$439$	−26.5741	−1.26831	−0.634156	−	0.773206i	$-0.718652\pi$
$439$	−26.5741	−1.26831	−0.634156	+	0.773206i	$0.718652\pi$
$440$	0	0
$441$	−0.905982	−0.0431420
$442$	0	0
$443$	30.8090	1.46378	0.731890	−	0.681423i	$-0.238638\pi$
$443$	30.8090	1.46378	0.731890	+	0.681423i	$0.238638\pi$
$444$	0	0
$445$	7.05645	0.334508
$446$	0	0
$447$	−9.29698	−0.439732
$448$	0	0
$449$	10.9251	0.515587	0.257793	−	0.966200i	$-0.417005\pi$
$449$	10.9251	0.515587	0.257793	+	0.966200i	$0.417005\pi$
$450$	0	0
$451$	−15.7922	−0.743623
$452$	0	0
$453$	3.46082	0.162604
$454$	0	0
$455$	23.0224	1.07930
$456$	0	0
$457$	14.3313	0.670388	0.335194	−	0.942149i	$-0.391198\pi$
$457$	14.3313	0.670388	0.335194	+	0.942149i	$0.391198\pi$
$458$	0	0
$459$	−5.51171	−0.257265
$460$	0	0
$461$	24.6189	1.14661	0.573307	−	0.819340i	$-0.305660\pi$
$461$	24.6189	1.14661	0.573307	+	0.819340i	$0.305660\pi$
$462$	0	0
$463$	−16.6334	−0.773020	−0.386510	−	0.922285i	$-0.626320\pi$
$463$	−16.6334	−0.773020	−0.386510	+	0.922285i	$0.626320\pi$
$464$	0	0
$465$	−2.38405	−0.110558
$466$	0	0
$467$	−31.8559	−1.47412	−0.737059	−	0.675829i	$-0.763786\pi$
$467$	−31.8559	−1.47412	−0.737059	+	0.675829i	$0.763786\pi$
$468$	0	0
$469$	−13.1474	−0.607090
$470$	0	0
$471$	7.43206	0.342451
$472$	0	0
$473$	−5.96538	−0.274288
$474$	0	0
$475$	6.02481	0.276437
$476$	0	0
$477$	−0.157640	−0.00721786
$478$	0	0
$479$	10.7382	0.490643	0.245321	−	0.969442i	$-0.421106\pi$
$479$	10.7382	0.490643	0.245321	+	0.969442i	$0.421106\pi$
$480$	0	0
$481$	−16.2651	−0.741626
$482$	0	0
$483$	37.9634	1.72740
$484$	0	0
$485$	−48.0542	−2.18203
$486$	0	0
$487$	11.9175	0.540035	0.270018	−	0.962855i	$-0.412970\pi$
$487$	11.9175	0.540035	0.270018	+	0.962855i	$0.412970\pi$
$488$	0	0
$489$	8.42748	0.381104
$490$	0	0
$491$	31.7813	1.43427	0.717134	−	0.696935i	$-0.245454\pi$
$491$	31.7813	1.43427	0.717134	+	0.696935i	$0.245454\pi$
$492$	0	0
$493$	−5.41666	−0.243954
$494$	0	0
$495$	0.729293	0.0327793
$496$	0	0
$497$	18.6360	0.835938
$498$	0	0
$499$	13.8203	0.618681	0.309341	−	0.950951i	$-0.399892\pi$
$499$	13.8203	0.618681	0.309341	+	0.950951i	$0.399892\pi$
$500$	0	0
$501$	−40.0936	−1.79125
$502$	0	0
$503$	−22.7226	−1.01315	−0.506575	−	0.862196i	$-0.669089\pi$
$503$	−22.7226	−1.01315	−0.506575	+	0.862196i	$0.669089\pi$
$504$	0	0
$505$	−61.5903	−2.74073
$506$	0	0
$507$	15.2443	0.677021
$508$	0	0
$509$	18.6081	0.824789	0.412395	−	0.911005i	$-0.364692\pi$
$509$	18.6081	0.824789	0.412395	+	0.911005i	$0.364692\pi$
$510$	0	0
$511$	20.1094	0.889589
$512$	0	0
$513$	5.33528	0.235559
$514$	0	0
$515$	0.440452	0.0194086
$516$	0	0
$517$	11.4605	0.504034
$518$	0	0
$519$	13.5543	0.594966
$520$	0	0
$521$	5.63400	0.246830	0.123415	−	0.992355i	$-0.460615\pi$
$521$	5.63400	0.246830	0.123415	+	0.992355i	$0.460615\pi$
$522$	0	0
$523$	−13.0347	−0.569969	−0.284985	−	0.958532i	$-0.591989\pi$
$523$	−13.0347	−0.569969	−0.284985	+	0.958532i	$0.591989\pi$
$524$	0	0
$525$	−35.4152	−1.54565
$526$	0	0
$527$	0.441192	0.0192186
$528$	0	0
$529$	18.7097	0.813464
$530$	0	0
$531$	1.79576	0.0779292
$532$	0	0
$533$	24.7257	1.07099
$534$	0	0
$535$	2.68156	0.115934
$536$	0	0
$537$	5.70068	0.246003
$538$	0	0
$539$	6.61738	0.285031
$540$	0	0
$541$	14.0611	0.604535	0.302267	−	0.953223i	$-0.402256\pi$
$541$	14.0611	0.604535	0.302267	+	0.953223i	$0.402256\pi$
$542$	0	0
$543$	−9.39487	−0.403173
$544$	0	0
$545$	−14.2657	−0.611074
$546$	0	0
$547$	−7.59072	−0.324556	−0.162278	−	0.986745i	$-0.551884\pi$
$547$	−7.59072	−0.324556	−0.162278	+	0.986745i	$0.551884\pi$
$548$	0	0
$549$	−2.62363	−0.111974
$550$	0	0
$551$	5.24328	0.223371
$552$	0	0
$553$	−3.49635	−0.148680
$554$	0	0
$555$	45.7853	1.94348
$556$	0	0
$557$	−19.3102	−0.818201	−0.409100	−	0.912489i	$-0.634157\pi$
$557$	−19.3102	−0.818201	−0.409100	+	0.912489i	$0.634157\pi$
$558$	0	0
$559$	9.33996	0.395038
$560$	0	0
$561$	2.19989	0.0928796
$562$	0	0
$563$	−11.2777	−0.475299	−0.237649	−	0.971351i	$-0.576377\pi$
$563$	−11.2777	−0.475299	−0.237649	+	0.971351i	$0.576377\pi$
$564$	0	0
$565$	50.5407	2.12626
$566$	0	0
$567$	−29.5431	−1.24070
$568$	0	0
$569$	14.6778	0.615327	0.307663	−	0.951495i	$-0.400453\pi$
$569$	14.6778	0.615327	0.307663	+	0.951495i	$0.400453\pi$
$570$	0	0
$571$	23.2476	0.972883	0.486441	−	0.873713i	$-0.338295\pi$
$571$	23.2476	0.972883	0.486441	+	0.873713i	$0.338295\pi$
$572$	0	0
$573$	24.8084	1.03639
$574$	0	0
$575$	−38.9100	−1.62266
$576$	0	0
$577$	−2.79889	−0.116519	−0.0582596	−	0.998301i	$-0.518555\pi$
$577$	−2.79889	−0.116519	−0.0582596	+	0.998301i	$0.518555\pi$
$578$	0	0
$579$	13.0773	0.543475
$580$	0	0
$581$	−28.3661	−1.17682
$582$	0	0
$583$	1.15142	0.0476870
$584$	0	0
$585$	−1.14185	−0.0472097
$586$	0	0
$587$	44.4760	1.83572	0.917860	−	0.396905i	$-0.129916\pi$
$587$	44.4760	1.83572	0.917860	+	0.396905i	$0.129916\pi$
$588$	0	0
$589$	−0.427070	−0.0175971
$590$	0	0
$591$	24.6657	1.01461
$592$	0	0
$593$	−34.2530	−1.40660	−0.703300	−	0.710893i	$-0.748291\pi$
$593$	−34.2530	−1.40660	−0.703300	+	0.710893i	$0.748291\pi$
$594$	0	0
$595$	11.9931	0.491667
$596$	0	0
$597$	−3.93211	−0.160931
$598$	0	0
$599$	−40.6935	−1.66269	−0.831346	−	0.555755i	$-0.812429\pi$
$599$	−40.6935	−1.66269	−0.831346	+	0.555755i	$0.812429\pi$
$600$	0	0
$601$	28.4266	1.15954	0.579772	−	0.814778i	$-0.303141\pi$
$601$	28.4266	1.15954	0.579772	+	0.814778i	$0.303141\pi$
$602$	0	0
$603$	0.652077	0.0265546
$604$	0	0
$605$	31.1972	1.26834
$606$	0	0
$607$	−13.1611	−0.534191	−0.267095	−	0.963670i	$-0.586064\pi$
$607$	−13.1611	−0.534191	−0.267095	+	0.963670i	$0.586064\pi$
$608$	0	0
$609$	−30.8212	−1.24894
$610$	0	0
$611$	−17.9437	−0.725925
$612$	0	0
$613$	−16.6195	−0.671257	−0.335628	−	0.941995i	$-0.608949\pi$
$613$	−16.6195	−0.671257	−0.335628	+	0.941995i	$0.608949\pi$
$614$	0	0
$615$	−69.6012	−2.80659
$616$	0	0
$617$	−42.4142	−1.70753	−0.853765	−	0.520658i	$-0.825687\pi$
$617$	−42.4142	−1.70753	−0.853765	+	0.520658i	$0.825687\pi$
$618$	0	0
$619$	−25.6570	−1.03124	−0.515622	−	0.856816i	$-0.672439\pi$
$619$	−25.6570	−1.03124	−0.515622	+	0.856816i	$0.672439\pi$
$620$	0	0
$621$	−34.4569	−1.38271
$622$	0	0
$623$	−7.43047	−0.297696
$624$	0	0
$625$	−18.8257	−0.753029
$626$	0	0
$627$	−2.12948	−0.0850431
$628$	0	0
$629$	−8.47301	−0.337841
$630$	0	0
$631$	11.9840	0.477077	0.238538	−	0.971133i	$-0.423332\pi$
$631$	11.9840	0.477077	0.238538	+	0.971133i	$0.423332\pi$
$632$	0	0
$633$	−32.7924	−1.30338
$634$	0	0
$635$	−1.23372	−0.0489586
$636$	0	0
$637$	−10.3608	−0.410510
$638$	0	0
$639$	−0.924298	−0.0365647
$640$	0	0
$641$	−37.7830	−1.49234	−0.746170	−	0.665756i	$-0.768109\pi$
$641$	−37.7830	−1.49234	−0.746170	+	0.665756i	$0.768109\pi$
$642$	0	0
$643$	18.0679	0.712528	0.356264	−	0.934385i	$-0.384050\pi$
$643$	18.0679	0.712528	0.356264	+	0.934385i	$0.384050\pi$
$644$	0	0
$645$	−26.2914	−1.03522
$646$	0	0
$647$	14.8721	0.584681	0.292341	−	0.956314i	$-0.405566\pi$
$647$	14.8721	0.584681	0.292341	+	0.956314i	$0.405566\pi$
$648$	0	0
$649$	−13.1164	−0.514863
$650$	0	0
$651$	2.51042	0.0983911
$652$	0	0
$653$	21.0504	0.823766	0.411883	−	0.911237i	$-0.364871\pi$
$653$	21.0504	0.823766	0.411883	+	0.911237i	$0.364871\pi$
$654$	0	0
$655$	−16.5717	−0.647511
$656$	0	0
$657$	−0.997377	−0.0389114
$658$	0	0
$659$	−24.7380	−0.963657	−0.481829	−	0.876265i	$-0.660027\pi$
$659$	−24.7380	−0.963657	−0.481829	+	0.876265i	$0.660027\pi$
$660$	0	0
$661$	28.5182	1.10923	0.554615	−	0.832107i	$-0.312866\pi$
$661$	28.5182	1.10923	0.554615	+	0.832107i	$0.312866\pi$
$662$	0	0
$663$	−3.44436	−0.133768
$664$	0	0
$665$	−11.6092	−0.450184
$666$	0	0
$667$	−33.8627	−1.31117
$668$	0	0
$669$	40.1258	1.55135
$670$	0	0
$671$	19.1632	0.739788
$672$	0	0
$673$	9.05670	0.349110	0.174555	−	0.984647i	$-0.444151\pi$
$673$	9.05670	0.349110	0.174555	+	0.984647i	$0.444151\pi$
$674$	0	0
$675$	32.1441	1.23723
$676$	0	0
$677$	−4.32881	−0.166370	−0.0831848	−	0.996534i	$-0.526509\pi$
$677$	−4.32881	−0.166370	−0.0831848	+	0.996534i	$0.526509\pi$
$678$	0	0
$679$	50.6013	1.94190
$680$	0	0
$681$	−33.3518	−1.27804
$682$	0	0
$683$	−2.08891	−0.0799299	−0.0399650	−	0.999201i	$-0.512725\pi$
$683$	−2.08891	−0.0799299	−0.0399650	+	0.999201i	$0.512725\pi$
$684$	0	0
$685$	40.7944	1.55867
$686$	0	0
$687$	23.2780	0.888111
$688$	0	0
$689$	−1.80277	−0.0686802
$690$	0	0
$691$	17.9220	0.681785	0.340892	−	0.940102i	$-0.389271\pi$
$691$	17.9220	0.681785	0.340892	+	0.940102i	$0.389271\pi$
$692$	0	0
$693$	−0.767949	−0.0291720
$694$	0	0
$695$	−34.5084	−1.30898
$696$	0	0
$697$	12.8804	0.487879
$698$	0	0
$699$	−34.6406	−1.31023
$700$	0	0
$701$	−27.2178	−1.02800	−0.514002	−	0.857789i	$-0.671838\pi$
$701$	−27.2178	−1.02800	−0.514002	+	0.857789i	$0.671838\pi$
$702$	0	0
$703$	8.20179	0.309337
$704$	0	0
$705$	50.5104	1.90233
$706$	0	0
$707$	64.8548	2.43912
$708$	0	0
$709$	−16.0913	−0.604321	−0.302160	−	0.953257i	$-0.597708\pi$
$709$	−16.0913	−0.604321	−0.302160	+	0.953257i	$0.597708\pi$
$710$	0	0
$711$	0.173410	0.00650340
$712$	0	0
$713$	2.75815	0.103294
$714$	0	0
$715$	8.34019	0.311905
$716$	0	0
$717$	38.8854	1.45220
$718$	0	0
$719$	9.17704	0.342246	0.171123	−	0.985250i	$-0.445260\pi$
$719$	9.17704	0.342246	0.171123	+	0.985250i	$0.445260\pi$
$720$	0	0
$721$	−0.463797	−0.0172727
$722$	0	0
$723$	33.5024	1.24597
$724$	0	0
$725$	31.5898	1.17321
$726$	0	0
$727$	39.6531	1.47065	0.735326	−	0.677713i	$-0.237029\pi$
$727$	39.6531	1.47065	0.735326	+	0.677713i	$0.237029\pi$
$728$	0	0
$729$	28.3751	1.05093
$730$	0	0
$731$	4.86547	0.179956
$732$	0	0
$733$	22.2952	0.823493	0.411746	−	0.911298i	$-0.364919\pi$
$733$	22.2952	0.823493	0.411746	+	0.911298i	$0.364919\pi$
$734$	0	0
$735$	29.1650	1.07577
$736$	0	0
$737$	−4.76284	−0.175441
$738$	0	0
$739$	13.5829	0.499656	0.249828	−	0.968290i	$-0.419626\pi$
$739$	13.5829	0.499656	0.249828	+	0.968290i	$0.419626\pi$
$740$	0	0
$741$	3.33411	0.122482
$742$	0	0
$743$	8.05890	0.295653	0.147826	−	0.989013i	$-0.452772\pi$
$743$	8.05890	0.295653	0.147826	+	0.989013i	$0.452772\pi$
$744$	0	0
$745$	−18.3610	−0.672694
$746$	0	0
$747$	1.40689	0.0514753
$748$	0	0
$749$	−2.82369	−0.103175
$750$	0	0
$751$	24.5015	0.894072	0.447036	−	0.894516i	$-0.352480\pi$
$751$	24.5015	0.894072	0.447036	+	0.894516i	$0.352480\pi$
$752$	0	0
$753$	45.1330	1.64474
$754$	0	0
$755$	6.83492	0.248748
$756$	0	0
$757$	−9.28504	−0.337471	−0.168735	−	0.985661i	$-0.553968\pi$
$757$	−9.28504	−0.337471	−0.168735	+	0.985661i	$0.553968\pi$
$758$	0	0
$759$	13.7528	0.499195
$760$	0	0
$761$	−1.08497	−0.0393303	−0.0196651	−	0.999807i	$-0.506260\pi$
$761$	−1.08497	−0.0393303	−0.0196651	+	0.999807i	$0.506260\pi$
$762$	0	0
$763$	15.0218	0.543826
$764$	0	0
$765$	−0.594825	−0.0215059
$766$	0	0
$767$	20.5363	0.741521
$768$	0	0
$769$	−38.3327	−1.38231	−0.691156	−	0.722705i	$-0.742898\pi$
$769$	−38.3327	−1.38231	−0.691156	+	0.722705i	$0.742898\pi$
$770$	0	0
$771$	−31.5314	−1.13558
$772$	0	0
$773$	20.9782	0.754532	0.377266	−	0.926105i	$-0.376864\pi$
$773$	20.9782	0.754532	0.377266	+	0.926105i	$0.376864\pi$
$774$	0	0
$775$	−2.57302	−0.0924255
$776$	0	0
$777$	−48.2121	−1.72960
$778$	0	0
$779$	−12.4681	−0.446715
$780$	0	0
$781$	6.75117	0.241576
$782$	0	0
$783$	27.9744	0.999723
$784$	0	0
$785$	14.6779	0.523876
$786$	0	0
$787$	30.0647	1.07169	0.535846	−	0.844316i	$-0.319993\pi$
$787$	30.0647	1.07169	0.535846	+	0.844316i	$0.319993\pi$
$788$	0	0
$789$	36.0501	1.28342
$790$	0	0
$791$	−53.2195	−1.89227
$792$	0	0
$793$	−30.0038	−1.06546
$794$	0	0
$795$	5.07469	0.179981
$796$	0	0
$797$	18.5740	0.657925	0.328963	−	0.944343i	$-0.393301\pi$
$797$	18.5740	0.657925	0.328963	+	0.944343i	$0.393301\pi$
$798$	0	0
$799$	−9.34743	−0.330688
$800$	0	0
$801$	0.368533	0.0130215
$802$	0	0
$803$	7.28494	0.257080
$804$	0	0
$805$	74.9755	2.64254
$806$	0	0
$807$	43.8311	1.54293
$808$	0	0
$809$	8.15637	0.286763	0.143381	−	0.989668i	$-0.454202\pi$
$809$	8.15637	0.286763	0.143381	+	0.989668i	$0.454202\pi$
$810$	0	0
$811$	20.4763	0.719022	0.359511	−	0.933141i	$-0.382944\pi$
$811$	20.4763	0.719022	0.359511	+	0.933141i	$0.382944\pi$
$812$	0	0
$813$	23.3169	0.817759
$814$	0	0
$815$	16.6438	0.583006
$816$	0	0
$817$	−4.70973	−0.164773
$818$	0	0
$819$	1.20237	0.0420143
$820$	0	0
$821$	30.2547	1.05590	0.527949	−	0.849276i	$-0.322961\pi$
$821$	30.2547	1.05590	0.527949	+	0.849276i	$0.322961\pi$
$822$	0	0
$823$	44.4332	1.54884	0.774422	−	0.632669i	$-0.218041\pi$
$823$	44.4332	1.54884	0.774422	+	0.632669i	$0.218041\pi$
$824$	0	0
$825$	−12.8297	−0.446673
$826$	0	0
$827$	−15.4952	−0.538820	−0.269410	−	0.963026i	$-0.586829\pi$
$827$	−15.4952	−0.538820	−0.269410	+	0.963026i	$0.586829\pi$
$828$	0	0
$829$	21.5222	0.747499	0.373749	−	0.927530i	$-0.378072\pi$
$829$	21.5222	0.747499	0.373749	+	0.927530i	$0.378072\pi$
$830$	0	0
$831$	12.1842	0.422667
$832$	0	0
$833$	−5.39726	−0.187004
$834$	0	0
$835$	−79.1825	−2.74022
$836$	0	0
$837$	−2.27854	−0.0787579
$838$	0	0
$839$	−15.6891	−0.541649	−0.270825	−	0.962629i	$-0.587296\pi$
$839$	−15.6891	−0.541649	−0.270825	+	0.962629i	$0.587296\pi$
$840$	0	0
$841$	−1.50801	−0.0520004
$842$	0	0
$843$	26.6326	0.917274
$844$	0	0
$845$	30.1065	1.03570
$846$	0	0
$847$	−32.8507	−1.12876
$848$	0	0
$849$	30.8531	1.05887
$850$	0	0
$851$	−52.9697	−1.81578
$852$	0	0
$853$	34.9636	1.19713	0.598565	−	0.801074i	$-0.295738\pi$
$853$	34.9636	1.19713	0.598565	+	0.801074i	$0.295738\pi$
$854$	0	0
$855$	0.575785	0.0196914
$856$	0	0
$857$	−18.8079	−0.642466	−0.321233	−	0.947000i	$-0.604097\pi$
$857$	−18.8079	−0.642466	−0.321233	+	0.947000i	$0.604097\pi$
$858$	0	0
$859$	−10.2523	−0.349803	−0.174901	−	0.984586i	$-0.555961\pi$
$859$	−10.2523	−0.349803	−0.174901	+	0.984586i	$0.555961\pi$
$860$	0	0
$861$	73.2903	2.49773
$862$	0	0
$863$	−33.0872	−1.12630	−0.563151	−	0.826354i	$-0.690411\pi$
$863$	−33.0872	−1.12630	−0.563151	+	0.826354i	$0.690411\pi$
$864$	0	0
$865$	26.7689	0.910169
$866$	0	0
$867$	26.7869	0.909731
$868$	0	0
$869$	−1.26661	−0.0429667
$870$	0	0
$871$	7.45715	0.252676
$872$	0	0
$873$	−2.50970	−0.0849403
$874$	0	0
$875$	−11.8972	−0.402197
$876$	0	0
$877$	12.9270	0.436514	0.218257	−	0.975891i	$-0.429963\pi$
$877$	12.9270	0.436514	0.218257	+	0.975891i	$0.429963\pi$
$878$	0	0
$879$	18.5086	0.624279
$880$	0	0
$881$	52.1846	1.75814	0.879072	−	0.476689i	$-0.158163\pi$
$881$	52.1846	1.75814	0.879072	+	0.476689i	$0.158163\pi$
$882$	0	0
$883$	39.5787	1.33193	0.665965	−	0.745983i	$-0.268020\pi$
$883$	39.5787	1.33193	0.665965	+	0.745983i	$0.268020\pi$
$884$	0	0
$885$	−57.8082	−1.94320
$886$	0	0
$887$	−32.1055	−1.07800	−0.538998	−	0.842307i	$-0.681197\pi$
$887$	−32.1055	−1.07800	−0.538998	+	0.842307i	$0.681197\pi$
$888$	0	0
$889$	1.29911	0.0435707
$890$	0	0
$891$	−10.7024	−0.358545
$892$	0	0
$893$	9.04823	0.302787
$894$	0	0
$895$	11.2585	0.376331
$896$	0	0
$897$	−21.5327	−0.718956
$898$	0	0
$899$	−2.23925	−0.0746831
$900$	0	0
$901$	−0.939120	−0.0312866
$902$	0	0
$903$	27.6849	0.921296
$904$	0	0
$905$	−18.5543	−0.616766
$906$	0	0
$907$	−40.4698	−1.34378	−0.671889	−	0.740651i	$-0.734517\pi$
$907$	−40.4698	−1.34378	−0.671889	+	0.740651i	$0.734517\pi$
$908$	0	0
$909$	−3.21664	−0.106689
$910$	0	0
$911$	13.5309	0.448297	0.224149	−	0.974555i	$-0.428040\pi$
$911$	13.5309	0.448297	0.224149	+	0.974555i	$0.428040\pi$
$912$	0	0
$913$	−10.2760	−0.340087
$914$	0	0
$915$	84.4586	2.79211
$916$	0	0
$917$	17.4501	0.576253
$918$	0	0
$919$	40.0076	1.31973	0.659864	−	0.751385i	$-0.270614\pi$
$919$	40.0076	1.31973	0.659864	+	0.751385i	$0.270614\pi$
$920$	0	0
$921$	−33.7293	−1.11142
$922$	0	0
$923$	−10.5703	−0.347924
$924$	0	0
$925$	49.4142	1.62473
$926$	0	0
$927$	0.0230032	0.000755523 0
$928$	0	0
$929$	−14.3525	−0.470889	−0.235444	−	0.971888i	$-0.575655\pi$
$929$	−14.3525	−0.470889	−0.235444	+	0.971888i	$0.575655\pi$
$930$	0	0
$931$	5.22450	0.171226
$932$	0	0
$933$	−54.0779	−1.77043
$934$	0	0
$935$	4.34466	0.142086
$936$	0	0
$937$	−3.23488	−0.105679	−0.0528394	−	0.998603i	$-0.516827\pi$
$937$	−3.23488	−0.105679	−0.0528394	+	0.998603i	$0.516827\pi$
$938$	0	0
$939$	42.8547	1.39851
$940$	0	0
$941$	−47.5835	−1.55118	−0.775589	−	0.631238i	$-0.782547\pi$
$941$	−47.5835	−1.55118	−0.775589	+	0.631238i	$0.782547\pi$
$942$	0	0
$943$	80.5226	2.62218
$944$	0	0
$945$	−61.9382	−2.01485
$946$	0	0
$947$	−12.9219	−0.419905	−0.209953	−	0.977712i	$-0.567331\pi$
$947$	−12.9219	−0.419905	−0.209953	+	0.977712i	$0.567331\pi$
$948$	0	0
$949$	−11.4060	−0.370254
$950$	0	0
$951$	30.9200	1.00265
$952$	0	0
$953$	−53.1361	−1.72125	−0.860624	−	0.509241i	$-0.829926\pi$
$953$	−53.1361	−1.72125	−0.860624	+	0.509241i	$0.829926\pi$
$954$	0	0
$955$	48.9951	1.58545
$956$	0	0
$957$	−11.1654	−0.360928
$958$	0	0
$959$	−42.9566	−1.38714
$960$	0	0
$961$	−30.8176	−0.994116
$962$	0	0
$963$	0.140048	0.00451298
$964$	0	0
$965$	25.8269	0.831398
$966$	0	0
$967$	−22.1088	−0.710972	−0.355486	−	0.934682i	$-0.615685\pi$
$967$	−22.1088	−0.710972	−0.355486	+	0.934682i	$0.615685\pi$
$968$	0	0
$969$	1.73684	0.0557954
$970$	0	0
$971$	−0.955264	−0.0306559	−0.0153279	−	0.999883i	$-0.504879\pi$
$971$	−0.955264	−0.0306559	−0.0153279	+	0.999883i	$0.504879\pi$
$972$	0	0
$973$	36.3375	1.16493
$974$	0	0
$975$	20.0874	0.643311
$976$	0	0
$977$	20.5005	0.655868	0.327934	−	0.944701i	$-0.393648\pi$
$977$	20.5005	0.655868	0.327934	+	0.944701i	$0.393648\pi$
$978$	0	0
$979$	−2.69180	−0.0860303
$980$	0	0
$981$	−0.745044	−0.0237874
$982$	0	0
$983$	−38.5967	−1.23104	−0.615522	−	0.788120i	$-0.711055\pi$
$983$	−38.5967	−1.23104	−0.615522	+	0.788120i	$0.711055\pi$
$984$	0	0
$985$	48.7133	1.55213
$986$	0	0
$987$	−53.1876	−1.69298
$988$	0	0
$989$	30.4169	0.967200
$990$	0	0
$991$	−30.8674	−0.980535	−0.490267	−	0.871572i	$-0.663101\pi$
$991$	−30.8674	−0.980535	−0.490267	+	0.871572i	$0.663101\pi$
$992$	0	0
$993$	47.2789	1.50035
$994$	0	0
$995$	−7.76569	−0.246189
$996$	0	0
$997$	5.75663	0.182314	0.0911571	−	0.995837i	$-0.470943\pi$
$997$	5.75663	0.182314	0.0911571	+	0.995837i	$0.470943\pi$
$998$	0	0
$999$	43.7589	1.38447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.8	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.8	✓	27	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.68125 q^{3} -3.32036 q^{5} +3.49635 q^{7} -0.173410 q^{9} +O(q^{10})\)	\(q-1.68125 q^{3} -3.32036 q^{5} +3.49635 q^{7} -0.173410 q^{9} +1.26661 q^{11} -1.98312 q^{13} +5.58235 q^{15} -1.03307 q^{17} +1.00000 q^{19} -5.87823 q^{21} -6.45830 q^{23} +6.02481 q^{25} +5.33528 q^{27} +5.24328 q^{29} -0.427070 q^{31} -2.12948 q^{33} -11.6092 q^{35} +8.20179 q^{37} +3.33411 q^{39} -12.4681 q^{41} -4.70973 q^{43} +0.575785 q^{45} +9.04823 q^{47} +5.22450 q^{49} +1.73684 q^{51} +0.909060 q^{53} -4.20559 q^{55} -1.68125 q^{57} -10.3555 q^{59} +15.1296 q^{61} -0.606304 q^{63} +6.58467 q^{65} -3.76031 q^{67} +10.8580 q^{69} +5.33012 q^{71} +5.75154 q^{73} -10.1292 q^{75} +4.42851 q^{77} -1.00000 q^{79} -8.44970 q^{81} -8.11305 q^{83} +3.43016 q^{85} -8.81525 q^{87} -2.12521 q^{89} -6.93369 q^{91} +0.718010 q^{93} -3.32036 q^{95} +14.4726 q^{97} -0.219643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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