LMFDB - Embedded newform 6004.2.a.e.1.12

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:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.33751 q^{3} -1.75116 q^{5} -1.54762 q^{7} -1.21107 q^{9} +O(q^{10})$	$q-1.33751 q^{3} -1.75116 q^{5} -1.54762 q^{7} -1.21107 q^{9} -0.541862 q^{11} -2.33172 q^{13} +2.34220 q^{15} -4.76711 q^{17} +1.00000 q^{19} +2.06996 q^{21} -7.12294 q^{23} -1.93343 q^{25} +5.63234 q^{27} -8.75476 q^{29} -5.40070 q^{31} +0.724745 q^{33} +2.71014 q^{35} -2.01663 q^{37} +3.11870 q^{39} -0.778494 q^{41} -2.07804 q^{43} +2.12078 q^{45} +2.55721 q^{47} -4.60487 q^{49} +6.37605 q^{51} +5.31127 q^{53} +0.948888 q^{55} -1.33751 q^{57} +9.33467 q^{59} +3.53264 q^{61} +1.87428 q^{63} +4.08323 q^{65} -14.2713 q^{67} +9.52699 q^{69} -14.5531 q^{71} -9.88909 q^{73} +2.58598 q^{75} +0.838597 q^{77} +1.00000 q^{79} -3.90010 q^{81} +3.42095 q^{83} +8.34799 q^{85} +11.7096 q^{87} +0.144112 q^{89} +3.60862 q^{91} +7.22349 q^{93} -1.75116 q^{95} -6.99434 q^{97} +0.656233 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * 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
$3$	−1.33751	−0.772211	−0.386105	−	0.922455i	$-0.626180\pi$
$3$	−1.33751	−0.772211	−0.386105	+	0.922455i	$0.626180\pi$
$4$	0	0
$5$	−1.75116	−0.783144	−0.391572	−	0.920147i	$-0.628069\pi$
$5$	−1.75116	−0.783144	−0.391572	+	0.920147i	$0.628069\pi$
$6$	0	0
$7$	−1.54762	−0.584946	−0.292473	−	0.956274i	$-0.594478\pi$
$7$	−1.54762	−0.584946	−0.292473	+	0.956274i	$0.594478\pi$
$8$	0	0
$9$	−1.21107	−0.403690
$10$	0	0
$11$	−0.541862	−0.163377	−0.0816887	−	0.996658i	$-0.526031\pi$
$11$	−0.541862	−0.163377	−0.0816887	+	0.996658i	$0.526031\pi$
$12$	0	0
$13$	−2.33172	−0.646703	−0.323352	−	0.946279i	$-0.604810\pi$
$13$	−2.33172	−0.646703	−0.323352	+	0.946279i	$0.604810\pi$
$14$	0	0
$15$	2.34220	0.604752
$16$	0	0
$17$	−4.76711	−1.15619	−0.578097	−	0.815968i	$-0.696204\pi$
$17$	−4.76711	−1.15619	−0.578097	+	0.815968i	$0.696204\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.06996	0.451702
$22$	0	0
$23$	−7.12294	−1.48523	−0.742617	−	0.669716i	$-0.766416\pi$
$23$	−7.12294	−1.48523	−0.742617	+	0.669716i	$0.766416\pi$
$24$	0	0
$25$	−1.93343	−0.386686
$26$	0	0
$27$	5.63234	1.08394
$28$	0	0
$29$	−8.75476	−1.62572	−0.812859	−	0.582460i	$-0.802090\pi$
$29$	−8.75476	−1.62572	−0.812859	+	0.582460i	$0.802090\pi$
$30$	0	0
$31$	−5.40070	−0.969995	−0.484998	−	0.874516i	$-0.661180\pi$
$31$	−5.40070	−0.969995	−0.484998	+	0.874516i	$0.661180\pi$
$32$	0	0
$33$	0.724745	0.126162
$34$	0	0
$35$	2.71014	0.458097
$36$	0	0
$37$	−2.01663	−0.331531	−0.165766	−	0.986165i	$-0.553010\pi$
$37$	−2.01663	−0.331531	−0.165766	+	0.986165i	$0.553010\pi$
$38$	0	0
$39$	3.11870	0.499392
$40$	0	0
$41$	−0.778494	−0.121580	−0.0607901	−	0.998151i	$-0.519362\pi$
$41$	−0.778494	−0.121580	−0.0607901	+	0.998151i	$0.519362\pi$
$42$	0	0
$43$	−2.07804	−0.316897	−0.158449	−	0.987367i	$-0.550649\pi$
$43$	−2.07804	−0.316897	−0.158449	+	0.987367i	$0.550649\pi$
$44$	0	0
$45$	2.12078	0.316148
$46$	0	0
$47$	2.55721	0.373008	0.186504	−	0.982454i	$-0.440284\pi$
$47$	2.55721	0.373008	0.186504	+	0.982454i	$0.440284\pi$
$48$	0	0
$49$	−4.60487	−0.657838
$50$	0	0
$51$	6.37605	0.892826
$52$	0	0
$53$	5.31127	0.729559	0.364780	−	0.931094i	$-0.381144\pi$
$53$	5.31127	0.729559	0.364780	+	0.931094i	$0.381144\pi$
$54$	0	0
$55$	0.948888	0.127948
$56$	0	0
$57$	−1.33751	−0.177157
$58$	0	0
$59$	9.33467	1.21527	0.607636	−	0.794216i	$-0.292118\pi$
$59$	9.33467	1.21527	0.607636	+	0.794216i	$0.292118\pi$
$60$	0	0
$61$	3.53264	0.452308	0.226154	−	0.974092i	$-0.427385\pi$
$61$	3.53264	0.452308	0.226154	+	0.974092i	$0.427385\pi$
$62$	0	0
$63$	1.87428	0.236137
$64$	0	0
$65$	4.08323	0.506462
$66$	0	0
$67$	−14.2713	−1.74352	−0.871760	−	0.489933i	$-0.837021\pi$
$67$	−14.2713	−1.74352	−0.871760	+	0.489933i	$0.837021\pi$
$68$	0	0
$69$	9.52699	1.14691
$70$	0	0
$71$	−14.5531	−1.72713	−0.863565	−	0.504237i	$-0.831774\pi$
$71$	−14.5531	−1.72713	−0.863565	+	0.504237i	$0.831774\pi$
$72$	0	0
$73$	−9.88909	−1.15743	−0.578715	−	0.815530i	$-0.696446\pi$
$73$	−9.88909	−1.15743	−0.578715	+	0.815530i	$0.696446\pi$
$74$	0	0
$75$	2.58598	0.298603
$76$	0	0
$77$	0.838597	0.0955670
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−3.90010	−0.433344
$82$	0	0
$83$	3.42095	0.375498	0.187749	−	0.982217i	$-0.439881\pi$
$83$	3.42095	0.375498	0.187749	+	0.982217i	$0.439881\pi$
$84$	0	0
$85$	8.34799	0.905466
$86$	0	0
$87$	11.7096	1.25540
$88$	0	0
$89$	0.144112	0.0152759	0.00763795	−	0.999971i	$-0.497569\pi$
$89$	0.144112	0.0152759	0.00763795	+	0.999971i	$0.497569\pi$
$90$	0	0
$91$	3.60862	0.378287
$92$	0	0
$93$	7.22349	0.749041
$94$	0	0
$95$	−1.75116	−0.179666
$96$	0	0
$97$	−6.99434	−0.710168	−0.355084	−	0.934834i	$-0.615548\pi$
$97$	−6.99434	−0.710168	−0.355084	+	0.934834i	$0.615548\pi$
$98$	0	0
$99$	0.656233	0.0659539
$100$	0	0
$101$	−19.5493	−1.94522	−0.972612	−	0.232434i	$-0.925331\pi$
$101$	−19.5493	−1.94522	−0.972612	+	0.232434i	$0.925331\pi$
$102$	0	0
$103$	−1.81648	−0.178983	−0.0894916	−	0.995988i	$-0.528524\pi$
$103$	−1.81648	−0.178983	−0.0894916	+	0.995988i	$0.528524\pi$
$104$	0	0
$105$	−3.62483	−0.353748
$106$	0	0
$107$	−20.1453	−1.94752	−0.973760	−	0.227578i	$-0.926919\pi$
$107$	−20.1453	−1.94752	−0.973760	+	0.227578i	$0.926919\pi$
$108$	0	0
$109$	−3.20813	−0.307283	−0.153642	−	0.988127i	$-0.549100\pi$
$109$	−3.20813	−0.307283	−0.153642	+	0.988127i	$0.549100\pi$
$110$	0	0
$111$	2.69725	0.256012
$112$	0	0
$113$	12.0940	1.13770	0.568852	−	0.822440i	$-0.307388\pi$
$113$	12.0940	1.13770	0.568852	+	0.822440i	$0.307388\pi$
$114$	0	0
$115$	12.4734	1.16315
$116$	0	0
$117$	2.82388	0.261068
$118$	0	0
$119$	7.37768	0.676311
$120$	0	0
$121$	−10.7064	−0.973308
$122$	0	0
$123$	1.04124	0.0938856
$124$	0	0
$125$	12.1416	1.08597
$126$	0	0
$127$	−3.55627	−0.315568	−0.157784	−	0.987474i	$-0.550435\pi$
$127$	−3.55627	−0.315568	−0.157784	+	0.987474i	$0.550435\pi$
$128$	0	0
$129$	2.77939	0.244712
$130$	0	0
$131$	6.86407	0.599717	0.299858	−	0.953984i	$-0.403061\pi$
$131$	6.86407	0.599717	0.299858	+	0.953984i	$0.403061\pi$
$132$	0	0
$133$	−1.54762	−0.134196
$134$	0	0
$135$	−9.86315	−0.848885
$136$	0	0
$137$	−5.94830	−0.508197	−0.254099	−	0.967178i	$-0.581779\pi$
$137$	−5.94830	−0.508197	−0.254099	+	0.967178i	$0.581779\pi$
$138$	0	0
$139$	−10.1485	−0.860788	−0.430394	−	0.902641i	$-0.641625\pi$
$139$	−10.1485	−0.860788	−0.430394	+	0.902641i	$0.641625\pi$
$140$	0	0
$141$	−3.42029	−0.288041
$142$	0	0
$143$	1.26347	0.105657
$144$	0	0
$145$	15.3310	1.27317
$146$	0	0
$147$	6.15905	0.507990
$148$	0	0
$149$	18.6372	1.52682	0.763410	−	0.645914i	$-0.223524\pi$
$149$	18.6372	1.52682	0.763410	+	0.645914i	$0.223524\pi$
$150$	0	0
$151$	8.22784	0.669572	0.334786	−	0.942294i	$-0.391336\pi$
$151$	8.22784	0.669572	0.334786	+	0.942294i	$0.391336\pi$
$152$	0	0
$153$	5.77331	0.466744
$154$	0	0
$155$	9.45751	0.759646
$156$	0	0
$157$	15.9832	1.27560	0.637799	−	0.770203i	$-0.279845\pi$
$157$	15.9832	1.27560	0.637799	+	0.770203i	$0.279845\pi$
$158$	0	0
$159$	−7.10387	−0.563374
$160$	0	0
$161$	11.0236	0.868782
$162$	0	0
$163$	−12.0669	−0.945150	−0.472575	−	0.881290i	$-0.656675\pi$
$163$	−12.0669	−0.945150	−0.472575	+	0.881290i	$0.656675\pi$
$164$	0	0
$165$	−1.26915	−0.0988029
$166$	0	0
$167$	−0.919392	−0.0711447	−0.0355723	−	0.999367i	$-0.511325\pi$
$167$	−0.919392	−0.0711447	−0.0355723	+	0.999367i	$0.511325\pi$
$168$	0	0
$169$	−7.56307	−0.581775
$170$	0	0
$171$	−1.21107	−0.0926129
$172$	0	0
$173$	4.52652	0.344145	0.172073	−	0.985084i	$-0.444954\pi$
$173$	4.52652	0.344145	0.172073	+	0.985084i	$0.444954\pi$
$174$	0	0
$175$	2.99222	0.226190
$176$	0	0
$177$	−12.4852	−0.938446
$178$	0	0
$179$	−13.2695	−0.991806	−0.495903	−	0.868378i	$-0.665163\pi$
$179$	−13.2695	−0.991806	−0.495903	+	0.868378i	$0.665163\pi$
$180$	0	0
$181$	−0.386105	−0.0286989	−0.0143495	−	0.999897i	$-0.504568\pi$
$181$	−0.386105	−0.0286989	−0.0143495	+	0.999897i	$0.504568\pi$
$182$	0	0
$183$	−4.72494	−0.349278
$184$	0	0
$185$	3.53144	0.259637
$186$	0	0
$187$	2.58311	0.188896
$188$	0	0
$189$	−8.71674	−0.634049
$190$	0	0
$191$	24.4591	1.76980	0.884898	−	0.465785i	$-0.154228\pi$
$191$	24.4591	1.76980	0.884898	+	0.465785i	$0.154228\pi$
$192$	0	0
$193$	10.6271	0.764958	0.382479	−	0.923964i	$-0.375070\pi$
$193$	10.6271	0.764958	0.382479	+	0.923964i	$0.375070\pi$
$194$	0	0
$195$	−5.46135	−0.391095
$196$	0	0
$197$	17.8355	1.27073	0.635365	−	0.772212i	$-0.280850\pi$
$197$	17.8355	1.27073	0.635365	+	0.772212i	$0.280850\pi$
$198$	0	0
$199$	21.3368	1.51252	0.756261	−	0.654270i	$-0.227024\pi$
$199$	21.3368	1.51252	0.756261	+	0.654270i	$0.227024\pi$
$200$	0	0
$201$	19.0880	1.34637
$202$	0	0
$203$	13.5491	0.950958
$204$	0	0
$205$	1.36327	0.0952149
$206$	0	0
$207$	8.62638	0.599575
$208$	0	0
$209$	−0.541862	−0.0374813
$210$	0	0
$211$	−1.49217	−0.102725	−0.0513626	−	0.998680i	$-0.516356\pi$
$211$	−1.49217	−0.102725	−0.0513626	+	0.998680i	$0.516356\pi$
$212$	0	0
$213$	19.4648	1.33371
$214$	0	0
$215$	3.63898	0.248176
$216$	0	0
$217$	8.35825	0.567395
$218$	0	0
$219$	13.2267	0.893780
$220$	0	0
$221$	11.1156	0.747715
$222$	0	0
$223$	−25.8269	−1.72950	−0.864750	−	0.502203i	$-0.832523\pi$
$223$	−25.8269	−1.72950	−0.864750	+	0.502203i	$0.832523\pi$
$224$	0	0
$225$	2.34152	0.156101
$226$	0	0
$227$	−1.66556	−0.110547	−0.0552735	−	0.998471i	$-0.517603\pi$
$227$	−1.66556	−0.110547	−0.0552735	+	0.998471i	$0.517603\pi$
$228$	0	0
$229$	−18.5114	−1.22327	−0.611633	−	0.791142i	$-0.709487\pi$
$229$	−18.5114	−1.22327	−0.611633	+	0.791142i	$0.709487\pi$
$230$	0	0
$231$	−1.12163	−0.0737979
$232$	0	0
$233$	11.5429	0.756203	0.378101	−	0.925764i	$-0.376577\pi$
$233$	11.5429	0.756203	0.378101	+	0.925764i	$0.376577\pi$
$234$	0	0
$235$	−4.47810	−0.292119
$236$	0	0
$237$	−1.33751	−0.0868805
$238$	0	0
$239$	6.90939	0.446932	0.223466	−	0.974712i	$-0.428263\pi$
$239$	6.90939	0.446932	0.223466	+	0.974712i	$0.428263\pi$
$240$	0	0
$241$	3.79854	0.244686	0.122343	−	0.992488i	$-0.460959\pi$
$241$	3.79854	0.244686	0.122343	+	0.992488i	$0.460959\pi$
$242$	0	0
$243$	−11.6806	−0.749312
$244$	0	0
$245$	8.06387	0.515182
$246$	0	0
$247$	−2.33172	−0.148364
$248$	0	0
$249$	−4.57555	−0.289964
$250$	0	0
$251$	−11.9985	−0.757341	−0.378670	−	0.925532i	$-0.623619\pi$
$251$	−11.9985	−0.757341	−0.378670	+	0.925532i	$0.623619\pi$
$252$	0	0
$253$	3.85965	0.242654
$254$	0	0
$255$	−11.1655	−0.699211
$256$	0	0
$257$	−5.82860	−0.363578	−0.181789	−	0.983338i	$-0.558189\pi$
$257$	−5.82860	−0.363578	−0.181789	+	0.983338i	$0.558189\pi$
$258$	0	0
$259$	3.12097	0.193928
$260$	0	0
$261$	10.6026	0.656287
$262$	0	0
$263$	−30.3155	−1.86933	−0.934667	−	0.355526i	$-0.884302\pi$
$263$	−30.3155	−1.86933	−0.934667	+	0.355526i	$0.884302\pi$
$264$	0	0
$265$	−9.30090	−0.571350
$266$	0	0
$267$	−0.192752	−0.0117962
$268$	0	0
$269$	12.2663	0.747891	0.373946	−	0.927451i	$-0.378005\pi$
$269$	12.2663	0.747891	0.373946	+	0.927451i	$0.378005\pi$
$270$	0	0
$271$	22.9105	1.39172	0.695858	−	0.718179i	$-0.255024\pi$
$271$	22.9105	1.39172	0.695858	+	0.718179i	$0.255024\pi$
$272$	0	0
$273$	−4.82657	−0.292117
$274$	0	0
$275$	1.04765	0.0631757
$276$	0	0
$277$	5.61342	0.337278	0.168639	−	0.985678i	$-0.446063\pi$
$277$	5.61342	0.337278	0.168639	+	0.985678i	$0.446063\pi$
$278$	0	0
$279$	6.54063	0.391577
$280$	0	0
$281$	20.7330	1.23683	0.618414	−	0.785853i	$-0.287776\pi$
$281$	20.7330	1.23683	0.618414	+	0.785853i	$0.287776\pi$
$282$	0	0
$283$	−16.1173	−0.958071	−0.479036	−	0.877795i	$-0.659014\pi$
$283$	−16.1173	−0.958071	−0.479036	+	0.877795i	$0.659014\pi$
$284$	0	0
$285$	2.34220	0.138740
$286$	0	0
$287$	1.20481	0.0711179
$288$	0	0
$289$	5.72533	0.336784
$290$	0	0
$291$	9.35499	0.548399
$292$	0	0
$293$	−16.9488	−0.990159	−0.495080	−	0.868848i	$-0.664861\pi$
$293$	−16.9488	−0.990159	−0.495080	+	0.868848i	$0.664861\pi$
$294$	0	0
$295$	−16.3465	−0.951732
$296$	0	0
$297$	−3.05195	−0.177092
$298$	0	0
$299$	16.6087	0.960507
$300$	0	0
$301$	3.21601	0.185368
$302$	0	0
$303$	26.1473	1.50212
$304$	0	0
$305$	−6.18623	−0.354223
$306$	0	0
$307$	−5.51119	−0.314540	−0.157270	−	0.987556i	$-0.550269\pi$
$307$	−5.51119	−0.314540	−0.157270	+	0.987556i	$0.550269\pi$
$308$	0	0
$309$	2.42956	0.138213
$310$	0	0
$311$	31.9646	1.81254	0.906272	−	0.422695i	$-0.138916\pi$
$311$	31.9646	1.81254	0.906272	+	0.422695i	$0.138916\pi$
$312$	0	0
$313$	−0.0907824	−0.00513132	−0.00256566	−	0.999997i	$-0.500817\pi$
$313$	−0.0907824	−0.00513132	−0.00256566	+	0.999997i	$0.500817\pi$
$314$	0	0
$315$	−3.28217	−0.184929
$316$	0	0
$317$	15.3693	0.863228	0.431614	−	0.902058i	$-0.357944\pi$
$317$	15.3693	0.863228	0.431614	+	0.902058i	$0.357944\pi$
$318$	0	0
$319$	4.74387	0.265606
$320$	0	0
$321$	26.9445	1.50390
$322$	0	0
$323$	−4.76711	−0.265249
$324$	0	0
$325$	4.50822	0.250071
$326$	0	0
$327$	4.29090	0.237288
$328$	0	0
$329$	−3.95760	−0.218189
$330$	0	0
$331$	−16.7629	−0.921370	−0.460685	−	0.887564i	$-0.652396\pi$
$331$	−16.7629	−0.921370	−0.460685	+	0.887564i	$0.652396\pi$
$332$	0	0
$333$	2.44228	0.133836
$334$	0	0
$335$	24.9914	1.36543
$336$	0	0
$337$	2.84714	0.155093	0.0775467	−	0.996989i	$-0.475291\pi$
$337$	2.84714	0.155093	0.0775467	+	0.996989i	$0.475291\pi$
$338$	0	0
$339$	−16.1758	−0.878547
$340$	0	0
$341$	2.92643	0.158475
$342$	0	0
$343$	17.9599	0.969746
$344$	0	0
$345$	−16.6833	−0.898199
$346$	0	0
$347$	−7.35280	−0.394719	−0.197360	−	0.980331i	$-0.563237\pi$
$347$	−7.35280	−0.394719	−0.197360	+	0.980331i	$0.563237\pi$
$348$	0	0
$349$	−21.6480	−1.15879	−0.579395	−	0.815047i	$-0.696711\pi$
$349$	−21.6480	−1.15879	−0.579395	+	0.815047i	$0.696711\pi$
$350$	0	0
$351$	−13.1331	−0.700991
$352$	0	0
$353$	6.45352	0.343486	0.171743	−	0.985142i	$-0.445060\pi$
$353$	6.45352	0.343486	0.171743	+	0.985142i	$0.445060\pi$
$354$	0	0
$355$	25.4848	1.35259
$356$	0	0
$357$	−9.86771	−0.522255
$358$	0	0
$359$	−11.8908	−0.627574	−0.313787	−	0.949493i	$-0.601598\pi$
$359$	−11.8908	−0.627574	−0.313787	+	0.949493i	$0.601598\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	14.3199	0.751599
$364$	0	0
$365$	17.3174	0.906434
$366$	0	0
$367$	19.9773	1.04281	0.521403	−	0.853311i	$-0.325409\pi$
$367$	19.9773	1.04281	0.521403	+	0.853311i	$0.325409\pi$
$368$	0	0
$369$	0.942811	0.0490808
$370$	0	0
$371$	−8.21984	−0.426753
$372$	0	0
$373$	1.40732	0.0728681	0.0364341	−	0.999336i	$-0.488400\pi$
$373$	1.40732	0.0728681	0.0364341	+	0.999336i	$0.488400\pi$
$374$	0	0
$375$	−16.2394	−0.838601
$376$	0	0
$377$	20.4137	1.05136
$378$	0	0
$379$	−8.07028	−0.414542	−0.207271	−	0.978284i	$-0.566458\pi$
$379$	−8.07028	−0.414542	−0.207271	+	0.978284i	$0.566458\pi$
$380$	0	0
$381$	4.75654	0.243685
$382$	0	0
$383$	−18.7724	−0.959224	−0.479612	−	0.877481i	$-0.659223\pi$
$383$	−18.7724	−0.959224	−0.479612	+	0.877481i	$0.659223\pi$
$384$	0	0
$385$	−1.46852	−0.0748427
$386$	0	0
$387$	2.51665	0.127928
$388$	0	0
$389$	−16.4146	−0.832254	−0.416127	−	0.909306i	$-0.636613\pi$
$389$	−16.4146	−0.832254	−0.416127	+	0.909306i	$0.636613\pi$
$390$	0	0
$391$	33.9558	1.71722
$392$	0	0
$393$	−9.18075	−0.463108
$394$	0	0
$395$	−1.75116	−0.0881106
$396$	0	0
$397$	6.87808	0.345201	0.172600	−	0.984992i	$-0.444783\pi$
$397$	6.87808	0.345201	0.172600	+	0.984992i	$0.444783\pi$
$398$	0	0
$399$	2.06996	0.103628
$400$	0	0
$401$	18.3222	0.914969	0.457484	−	0.889218i	$-0.348751\pi$
$401$	18.3222	0.914969	0.457484	+	0.889218i	$0.348751\pi$
$402$	0	0
$403$	12.5929	0.627299
$404$	0	0
$405$	6.82970	0.339371
$406$	0	0
$407$	1.09273	0.0541647
$408$	0	0
$409$	−15.9052	−0.786463	−0.393231	−	0.919440i	$-0.628643\pi$
$409$	−15.9052	−0.786463	−0.393231	+	0.919440i	$0.628643\pi$
$410$	0	0
$411$	7.95590	0.392436
$412$	0	0
$413$	−14.4465	−0.710868
$414$	0	0
$415$	−5.99064	−0.294069
$416$	0	0
$417$	13.5738	0.664710
$418$	0	0
$419$	−10.9881	−0.536802	−0.268401	−	0.963307i	$-0.586495\pi$
$419$	−10.9881	−0.536802	−0.268401	+	0.963307i	$0.586495\pi$
$420$	0	0
$421$	25.4816	1.24190	0.620948	−	0.783852i	$-0.286748\pi$
$421$	25.4816	1.24190	0.620948	+	0.783852i	$0.286748\pi$
$422$	0	0
$423$	−3.09696	−0.150580
$424$	0	0
$425$	9.21686	0.447083
$426$	0	0
$427$	−5.46719	−0.264576
$428$	0	0
$429$	−1.68990	−0.0815893
$430$	0	0
$431$	−17.1912	−0.828071	−0.414035	−	0.910261i	$-0.635881\pi$
$431$	−17.1912	−0.828071	−0.414035	+	0.910261i	$0.635881\pi$
$432$	0	0
$433$	−4.92756	−0.236803	−0.118402	−	0.992966i	$-0.537777\pi$
$433$	−4.92756	−0.236803	−0.118402	+	0.992966i	$0.537777\pi$
$434$	0	0
$435$	−20.5054	−0.983157
$436$	0	0
$437$	−7.12294	−0.340736
$438$	0	0
$439$	0.312691	0.0149239	0.00746196	−	0.999972i	$-0.497625\pi$
$439$	0.312691	0.0149239	0.00746196	+	0.999972i	$0.497625\pi$
$440$	0	0
$441$	5.57682	0.265563
$442$	0	0
$443$	4.48066	0.212883	0.106441	−	0.994319i	$-0.466054\pi$
$443$	4.48066	0.212883	0.106441	+	0.994319i	$0.466054\pi$
$444$	0	0
$445$	−0.252364	−0.0119632
$446$	0	0
$447$	−24.9274	−1.17903
$448$	0	0
$449$	28.5886	1.34918	0.674590	−	0.738193i	$-0.264320\pi$
$449$	28.5886	1.34918	0.674590	+	0.738193i	$0.264320\pi$
$450$	0	0
$451$	0.421836	0.0198635
$452$	0	0
$453$	−11.0048	−0.517051
$454$	0	0
$455$	−6.31929	−0.296253
$456$	0	0
$457$	−4.57977	−0.214233	−0.107116	−	0.994246i	$-0.534162\pi$
$457$	−4.57977	−0.214233	−0.107116	+	0.994246i	$0.534162\pi$
$458$	0	0
$459$	−26.8500	−1.25325
$460$	0	0
$461$	−26.9803	−1.25660	−0.628300	−	0.777971i	$-0.716249\pi$
$461$	−26.9803	−1.25660	−0.628300	+	0.777971i	$0.716249\pi$
$462$	0	0
$463$	−8.69640	−0.404156	−0.202078	−	0.979369i	$-0.564769\pi$
$463$	−8.69640	−0.404156	−0.202078	+	0.979369i	$0.564769\pi$
$464$	0	0
$465$	−12.6495	−0.586607
$466$	0	0
$467$	12.7670	0.590786	0.295393	−	0.955376i	$-0.404549\pi$
$467$	12.7670	0.590786	0.295393	+	0.955376i	$0.404549\pi$
$468$	0	0
$469$	22.0866	1.01987
$470$	0	0
$471$	−21.3777	−0.985031
$472$	0	0
$473$	1.12601	0.0517739
$474$	0	0
$475$	−1.93343	−0.0887118
$476$	0	0
$477$	−6.43232	−0.294516
$478$	0	0
$479$	24.1912	1.10532	0.552661	−	0.833406i	$-0.313612\pi$
$479$	24.1912	1.10532	0.552661	+	0.833406i	$0.313612\pi$
$480$	0	0
$481$	4.70221	0.214402
$482$	0	0
$483$	−14.7442	−0.670883
$484$	0	0
$485$	12.2482	0.556163
$486$	0	0
$487$	−6.56372	−0.297430	−0.148715	−	0.988880i	$-0.547514\pi$
$487$	−6.56372	−0.297430	−0.148715	+	0.988880i	$0.547514\pi$
$488$	0	0
$489$	16.1395	0.729855
$490$	0	0
$491$	6.24050	0.281630	0.140815	−	0.990036i	$-0.455028\pi$
$491$	6.24050	0.281630	0.140815	+	0.990036i	$0.455028\pi$
$492$	0	0
$493$	41.7349	1.87965
$494$	0	0
$495$	−1.14917	−0.0516514
$496$	0	0
$497$	22.5226	1.01028
$498$	0	0
$499$	2.99507	0.134078	0.0670388	−	0.997750i	$-0.478645\pi$
$499$	2.99507	0.134078	0.0670388	+	0.997750i	$0.478645\pi$
$500$	0	0
$501$	1.22969	0.0549387
$502$	0	0
$503$	−37.7770	−1.68439	−0.842197	−	0.539171i	$-0.818738\pi$
$503$	−37.7770	−1.68439	−0.842197	+	0.539171i	$0.818738\pi$
$504$	0	0
$505$	34.2340	1.52339
$506$	0	0
$507$	10.1157	0.449253
$508$	0	0
$509$	9.75466	0.432368	0.216184	−	0.976353i	$-0.430639\pi$
$509$	9.75466	0.432368	0.216184	+	0.976353i	$0.430639\pi$
$510$	0	0
$511$	15.3046	0.677034
$512$	0	0
$513$	5.63234	0.248674
$514$	0	0
$515$	3.18095	0.140170
$516$	0	0
$517$	−1.38565	−0.0609410
$518$	0	0
$519$	−6.05427	−0.265753
$520$	0	0
$521$	25.6210	1.12248	0.561239	−	0.827654i	$-0.310325\pi$
$521$	25.6210	1.12248	0.561239	+	0.827654i	$0.310325\pi$
$522$	0	0
$523$	−0.0949741	−0.00415293	−0.00207646	−	0.999998i	$-0.500661\pi$
$523$	−0.0949741	−0.00415293	−0.00207646	+	0.999998i	$0.500661\pi$
$524$	0	0
$525$	−4.00211	−0.174667
$526$	0	0
$527$	25.7457	1.12150
$528$	0	0
$529$	27.7362	1.20592
$530$	0	0
$531$	−11.3049	−0.490593
$532$	0	0
$533$	1.81523	0.0786264
$534$	0	0
$535$	35.2777	1.52519
$536$	0	0
$537$	17.7480	0.765883
$538$	0	0
$539$	2.49520	0.107476
$540$	0	0
$541$	36.8045	1.58235	0.791173	−	0.611592i	$-0.209471\pi$
$541$	36.8045	1.58235	0.791173	+	0.611592i	$0.209471\pi$
$542$	0	0
$543$	0.516418	0.0221616
$544$	0	0
$545$	5.61796	0.240647
$546$	0	0
$547$	11.9362	0.510355	0.255178	−	0.966894i	$-0.417866\pi$
$547$	11.9362	0.510355	0.255178	+	0.966894i	$0.417866\pi$
$548$	0	0
$549$	−4.27828	−0.182592
$550$	0	0
$551$	−8.75476	−0.372965
$552$	0	0
$553$	−1.54762	−0.0658116
$554$	0	0
$555$	−4.72333	−0.200494
$556$	0	0
$557$	10.7833	0.456904	0.228452	−	0.973555i	$-0.426634\pi$
$557$	10.7833	0.456904	0.228452	+	0.973555i	$0.426634\pi$
$558$	0	0
$559$	4.84540	0.204939
$560$	0	0
$561$	−3.45494	−0.145868
$562$	0	0
$563$	−30.1439	−1.27041	−0.635206	−	0.772343i	$-0.719085\pi$
$563$	−30.1439	−1.27041	−0.635206	+	0.772343i	$0.719085\pi$
$564$	0	0
$565$	−21.1785	−0.890985
$566$	0	0
$567$	6.03587	0.253483
$568$	0	0
$569$	−25.4108	−1.06528	−0.532638	−	0.846343i	$-0.678799\pi$
$569$	−25.4108	−1.06528	−0.532638	+	0.846343i	$0.678799\pi$
$570$	0	0
$571$	22.8275	0.955301	0.477651	−	0.878550i	$-0.341488\pi$
$571$	22.8275	0.955301	0.477651	+	0.878550i	$0.341488\pi$
$572$	0	0
$573$	−32.7142	−1.36666
$574$	0	0
$575$	13.7717	0.574319
$576$	0	0
$577$	−15.2566	−0.635139	−0.317570	−	0.948235i	$-0.602867\pi$
$577$	−15.2566	−0.635139	−0.317570	+	0.948235i	$0.602867\pi$
$578$	0	0
$579$	−14.2139	−0.590709
$580$	0	0
$581$	−5.29433	−0.219646
$582$	0	0
$583$	−2.87797	−0.119193
$584$	0	0
$585$	−4.94508	−0.204454
$586$	0	0
$587$	−32.2113	−1.32950	−0.664750	−	0.747065i	$-0.731462\pi$
$587$	−32.2113	−1.32950	−0.664750	+	0.747065i	$0.731462\pi$
$588$	0	0
$589$	−5.40070	−0.222532
$590$	0	0
$591$	−23.8552	−0.981272
$592$	0	0
$593$	18.0292	0.740372	0.370186	−	0.928958i	$-0.379294\pi$
$593$	18.0292	0.740372	0.370186	+	0.928958i	$0.379294\pi$
$594$	0	0
$595$	−12.9195	−0.529649
$596$	0	0
$597$	−28.5381	−1.16799
$598$	0	0
$599$	−35.6799	−1.45784	−0.728922	−	0.684597i	$-0.759978\pi$
$599$	−35.6799	−1.45784	−0.728922	+	0.684597i	$0.759978\pi$
$600$	0	0
$601$	4.76977	0.194563	0.0972815	−	0.995257i	$-0.468985\pi$
$601$	4.76977	0.194563	0.0972815	+	0.995257i	$0.468985\pi$
$602$	0	0
$603$	17.2836	0.703842
$604$	0	0
$605$	18.7486	0.762240
$606$	0	0
$607$	−13.4602	−0.546331	−0.273166	−	0.961967i	$-0.588071\pi$
$607$	−13.4602	−0.546331	−0.273166	+	0.961967i	$0.588071\pi$
$608$	0	0
$609$	−18.1220	−0.734340
$610$	0	0
$611$	−5.96271	−0.241225
$612$	0	0
$613$	−7.05172	−0.284816	−0.142408	−	0.989808i	$-0.545485\pi$
$613$	−7.05172	−0.284816	−0.142408	+	0.989808i	$0.545485\pi$
$614$	0	0
$615$	−1.82338	−0.0735260
$616$	0	0
$617$	−48.9145	−1.96922	−0.984611	−	0.174759i	$-0.944085\pi$
$617$	−48.9145	−1.96922	−0.984611	+	0.174759i	$0.944085\pi$
$618$	0	0
$619$	−38.4951	−1.54725	−0.773624	−	0.633645i	$-0.781558\pi$
$619$	−38.4951	−1.54725	−0.773624	+	0.633645i	$0.781558\pi$
$620$	0	0
$621$	−40.1188	−1.60991
$622$	0	0
$623$	−0.223032	−0.00893557
$624$	0	0
$625$	−11.5947	−0.463789
$626$	0	0
$627$	0.724745	0.0289435
$628$	0	0
$629$	9.61348	0.383314
$630$	0	0
$631$	−11.4107	−0.454252	−0.227126	−	0.973865i	$-0.572933\pi$
$631$	−11.4107	−0.454252	−0.227126	+	0.973865i	$0.572933\pi$
$632$	0	0
$633$	1.99579	0.0793255
$634$	0	0
$635$	6.22761	0.247135
$636$	0	0
$637$	10.7373	0.425426
$638$	0	0
$639$	17.6248	0.697226
$640$	0	0
$641$	−5.45672	−0.215528	−0.107764	−	0.994177i	$-0.534369\pi$
$641$	−5.45672	−0.215528	−0.107764	+	0.994177i	$0.534369\pi$
$642$	0	0
$643$	−2.99229	−0.118005	−0.0590023	−	0.998258i	$-0.518792\pi$
$643$	−2.99229	−0.118005	−0.0590023	+	0.998258i	$0.518792\pi$
$644$	0	0
$645$	−4.86717	−0.191644
$646$	0	0
$647$	−15.8390	−0.622694	−0.311347	−	0.950296i	$-0.600780\pi$
$647$	−15.8390	−0.622694	−0.311347	+	0.950296i	$0.600780\pi$
$648$	0	0
$649$	−5.05810	−0.198548
$650$	0	0
$651$	−11.1792	−0.438149
$652$	0	0
$653$	15.9934	0.625870	0.312935	−	0.949775i	$-0.398688\pi$
$653$	15.9934	0.625870	0.312935	+	0.949775i	$0.398688\pi$
$654$	0	0
$655$	−12.0201	−0.469664
$656$	0	0
$657$	11.9764	0.467243
$658$	0	0
$659$	−3.02073	−0.117671	−0.0588355	−	0.998268i	$-0.518739\pi$
$659$	−3.02073	−0.117671	−0.0588355	+	0.998268i	$0.518739\pi$
$660$	0	0
$661$	−31.6117	−1.22955	−0.614776	−	0.788701i	$-0.710754\pi$
$661$	−31.6117	−1.22955	−0.614776	+	0.788701i	$0.710754\pi$
$662$	0	0
$663$	−14.8672	−0.577393
$664$	0	0
$665$	2.71014	0.105095
$666$	0	0
$667$	62.3596	2.41457
$668$	0	0
$669$	34.5438	1.33554
$670$	0	0
$671$	−1.91420	−0.0738970
$672$	0	0
$673$	22.0675	0.850639	0.425319	−	0.905043i	$-0.360162\pi$
$673$	22.0675	0.850639	0.425319	+	0.905043i	$0.360162\pi$
$674$	0	0
$675$	−10.8897	−0.419146
$676$	0	0
$677$	−18.5010	−0.711050	−0.355525	−	0.934667i	$-0.615698\pi$
$677$	−18.5010	−0.711050	−0.355525	+	0.934667i	$0.615698\pi$
$678$	0	0
$679$	10.8246	0.415410
$680$	0	0
$681$	2.22770	0.0853656
$682$	0	0
$683$	12.1578	0.465205	0.232602	−	0.972572i	$-0.425276\pi$
$683$	12.1578	0.465205	0.232602	+	0.972572i	$0.425276\pi$
$684$	0	0
$685$	10.4164	0.397992
$686$	0	0
$687$	24.7591	0.944619
$688$	0	0
$689$	−12.3844	−0.471808
$690$	0	0
$691$	−16.7329	−0.636548	−0.318274	−	0.947999i	$-0.603103\pi$
$691$	−16.7329	−0.636548	−0.318274	+	0.947999i	$0.603103\pi$
$692$	0	0
$693$	−1.01560	−0.0385794
$694$	0	0
$695$	17.7718	0.674121
$696$	0	0
$697$	3.71116	0.140570
$698$	0	0
$699$	−15.4388	−0.583948
$700$	0	0
$701$	42.4866	1.60470	0.802348	−	0.596856i	$-0.203584\pi$
$701$	42.4866	1.60470	0.802348	+	0.596856i	$0.203584\pi$
$702$	0	0
$703$	−2.01663	−0.0760585
$704$	0	0
$705$	5.98949	0.225577
$706$	0	0
$707$	30.2549	1.13785
$708$	0	0
$709$	−18.6095	−0.698896	−0.349448	−	0.936956i	$-0.613631\pi$
$709$	−18.6095	−0.698896	−0.349448	+	0.936956i	$0.613631\pi$
$710$	0	0
$711$	−1.21107	−0.0454187
$712$	0	0
$713$	38.4689	1.44067
$714$	0	0
$715$	−2.21254	−0.0827444
$716$	0	0
$717$	−9.24137	−0.345125
$718$	0	0
$719$	25.0314	0.933515	0.466757	−	0.884385i	$-0.345422\pi$
$719$	25.0314	0.933515	0.466757	+	0.884385i	$0.345422\pi$
$720$	0	0
$721$	2.81122	0.104695
$722$	0	0
$723$	−5.08058	−0.188949
$724$	0	0
$725$	16.9267	0.628642
$726$	0	0
$727$	−3.82106	−0.141715	−0.0708576	−	0.997486i	$-0.522574\pi$
$727$	−3.82106	−0.141715	−0.0708576	+	0.997486i	$0.522574\pi$
$728$	0	0
$729$	27.3232	1.01197
$730$	0	0
$731$	9.90622	0.366395
$732$	0	0
$733$	−32.5215	−1.20121	−0.600604	−	0.799547i	$-0.705073\pi$
$733$	−32.5215	−1.20121	−0.600604	+	0.799547i	$0.705073\pi$
$734$	0	0
$735$	−10.7855	−0.397829
$736$	0	0
$737$	7.73308	0.284852
$738$	0	0
$739$	15.1845	0.558571	0.279286	−	0.960208i	$-0.409902\pi$
$739$	15.1845	0.558571	0.279286	+	0.960208i	$0.409902\pi$
$740$	0	0
$741$	3.11870	0.114568
$742$	0	0
$743$	−5.74020	−0.210587	−0.105294	−	0.994441i	$-0.533578\pi$
$743$	−5.74020	−0.210587	−0.105294	+	0.994441i	$0.533578\pi$
$744$	0	0
$745$	−32.6368	−1.19572
$746$	0	0
$747$	−4.14301	−0.151585
$748$	0	0
$749$	31.1773	1.13919
$750$	0	0
$751$	9.48868	0.346247	0.173123	−	0.984900i	$-0.444614\pi$
$751$	9.48868	0.346247	0.173123	+	0.984900i	$0.444614\pi$
$752$	0	0
$753$	16.0481	0.584827
$754$	0	0
$755$	−14.4083	−0.524371
$756$	0	0
$757$	3.02065	0.109787	0.0548937	−	0.998492i	$-0.482518\pi$
$757$	3.02065	0.109787	0.0548937	+	0.998492i	$0.482518\pi$
$758$	0	0
$759$	−5.16231	−0.187380
$760$	0	0
$761$	0.352893	0.0127924	0.00639618	−	0.999980i	$-0.497964\pi$
$761$	0.352893	0.0127924	0.00639618	+	0.999980i	$0.497964\pi$
$762$	0	0
$763$	4.96498	0.179744
$764$	0	0
$765$	−10.1100	−0.365528
$766$	0	0
$767$	−21.7659	−0.785920
$768$	0	0
$769$	−13.4791	−0.486069	−0.243035	−	0.970018i	$-0.578143\pi$
$769$	−13.4791	−0.486069	−0.243035	+	0.970018i	$0.578143\pi$
$770$	0	0
$771$	7.79581	0.280759
$772$	0	0
$773$	−49.7284	−1.78861	−0.894303	−	0.447461i	$-0.852328\pi$
$773$	−49.7284	−1.78861	−0.894303	+	0.447461i	$0.852328\pi$
$774$	0	0
$775$	10.4419	0.375083
$776$	0	0
$777$	−4.17433	−0.149753
$778$	0	0
$779$	−0.778494	−0.0278924
$780$	0	0
$781$	7.88574	0.282174
$782$	0	0
$783$	−49.3098	−1.76219
$784$	0	0
$785$	−27.9892	−0.998977
$786$	0	0
$787$	−4.34006	−0.154706	−0.0773532	−	0.997004i	$-0.524647\pi$
$787$	−4.34006	−0.154706	−0.0773532	+	0.997004i	$0.524647\pi$
$788$	0	0
$789$	40.5472	1.44352
$790$	0	0
$791$	−18.7169	−0.665495
$792$	0	0
$793$	−8.23714	−0.292509
$794$	0	0
$795$	12.4400	0.441203
$796$	0	0
$797$	−40.1706	−1.42291	−0.711457	−	0.702729i	$-0.751965\pi$
$797$	−40.1706	−1.42291	−0.711457	+	0.702729i	$0.751965\pi$
$798$	0	0
$799$	−12.1905	−0.431269
$800$	0	0
$801$	−0.174530	−0.00616673
$802$	0	0
$803$	5.35852	0.189098
$804$	0	0
$805$	−19.3041	−0.680382
$806$	0	0
$807$	−16.4063	−0.577530
$808$	0	0
$809$	42.1952	1.48351	0.741753	−	0.670673i	$-0.233995\pi$
$809$	42.1952	1.48351	0.741753	+	0.670673i	$0.233995\pi$
$810$	0	0
$811$	−27.7919	−0.975905	−0.487953	−	0.872870i	$-0.662256\pi$
$811$	−27.7919	−0.975905	−0.487953	+	0.872870i	$0.662256\pi$
$812$	0	0
$813$	−30.6430	−1.07470
$814$	0	0
$815$	21.1311	0.740189
$816$	0	0
$817$	−2.07804	−0.0727012
$818$	0	0
$819$	−4.37030	−0.152711
$820$	0	0
$821$	−49.8829	−1.74092	−0.870462	−	0.492235i	$-0.836180\pi$
$821$	−49.8829	−1.74092	−0.870462	+	0.492235i	$0.836180\pi$
$822$	0	0
$823$	1.25715	0.0438215	0.0219107	−	0.999760i	$-0.493025\pi$
$823$	1.25715	0.0438215	0.0219107	+	0.999760i	$0.493025\pi$
$824$	0	0
$825$	−1.40124	−0.0487850
$826$	0	0
$827$	6.25344	0.217454	0.108727	−	0.994072i	$-0.465323\pi$
$827$	6.25344	0.217454	0.108727	+	0.994072i	$0.465323\pi$
$828$	0	0
$829$	−23.7760	−0.825773	−0.412887	−	0.910782i	$-0.635479\pi$
$829$	−23.7760	−0.825773	−0.412887	+	0.910782i	$0.635479\pi$
$830$	0	0
$831$	−7.50799	−0.260449
$832$	0	0
$833$	21.9519	0.760588
$834$	0	0
$835$	1.61001	0.0557165
$836$	0	0
$837$	−30.4186	−1.05142
$838$	0	0
$839$	46.5067	1.60559	0.802795	−	0.596255i	$-0.203345\pi$
$839$	46.5067	1.60559	0.802795	+	0.596255i	$0.203345\pi$
$840$	0	0
$841$	47.6458	1.64296
$842$	0	0
$843$	−27.7306	−0.955092
$844$	0	0
$845$	13.2442	0.455613
$846$	0	0
$847$	16.5694	0.569333
$848$	0	0
$849$	21.5570	0.739833
$850$	0	0
$851$	14.3643	0.492402
$852$	0	0
$853$	29.8194	1.02100	0.510498	−	0.859879i	$-0.329461\pi$
$853$	29.8194	1.02100	0.510498	+	0.859879i	$0.329461\pi$
$854$	0	0
$855$	2.12078	0.0725292
$856$	0	0
$857$	11.3571	0.387952	0.193976	−	0.981006i	$-0.437862\pi$
$857$	11.3571	0.387952	0.193976	+	0.981006i	$0.437862\pi$
$858$	0	0
$859$	−41.3283	−1.41010	−0.705052	−	0.709155i	$-0.749076\pi$
$859$	−41.3283	−1.41010	−0.705052	+	0.709155i	$0.749076\pi$
$860$	0	0
$861$	−1.61145	−0.0549180
$862$	0	0
$863$	−20.6582	−0.703215	−0.351607	−	0.936148i	$-0.614365\pi$
$863$	−20.6582	−0.703215	−0.351607	+	0.936148i	$0.614365\pi$
$864$	0	0
$865$	−7.92668	−0.269515
$866$	0	0
$867$	−7.65768	−0.260068
$868$	0	0
$869$	−0.541862	−0.0183814
$870$	0	0
$871$	33.2768	1.12754
$872$	0	0
$873$	8.47064	0.286688
$874$	0	0
$875$	−18.7905	−0.635236
$876$	0	0
$877$	−39.8806	−1.34667	−0.673337	−	0.739336i	$-0.735140\pi$
$877$	−39.8806	−1.34667	−0.673337	+	0.739336i	$0.735140\pi$
$878$	0	0
$879$	22.6692	0.764612
$880$	0	0
$881$	−34.6277	−1.16664	−0.583318	−	0.812244i	$-0.698246\pi$
$881$	−34.6277	−1.16664	−0.583318	+	0.812244i	$0.698246\pi$
$882$	0	0
$883$	−38.4543	−1.29409	−0.647045	−	0.762452i	$-0.723996\pi$
$883$	−38.4543	−1.29409	−0.647045	+	0.762452i	$0.723996\pi$
$884$	0	0
$885$	21.8636	0.734938
$886$	0	0
$887$	−38.1596	−1.28127	−0.640637	−	0.767844i	$-0.721329\pi$
$887$	−38.1596	−1.28127	−0.640637	+	0.767844i	$0.721329\pi$
$888$	0	0
$889$	5.50376	0.184590
$890$	0	0
$891$	2.11331	0.0707986
$892$	0	0
$893$	2.55721	0.0855738
$894$	0	0
$895$	23.2370	0.776727
$896$	0	0
$897$	−22.2143	−0.741714
$898$	0	0
$899$	47.2819	1.57694
$900$	0	0
$901$	−25.3194	−0.843512
$902$	0	0
$903$	−4.30144	−0.143143
$904$	0	0
$905$	0.676132	0.0224754
$906$	0	0
$907$	−37.6971	−1.25171	−0.625855	−	0.779939i	$-0.715250\pi$
$907$	−37.6971	−1.25171	−0.625855	+	0.779939i	$0.715250\pi$
$908$	0	0
$909$	23.6755	0.785268
$910$	0	0
$911$	37.0588	1.22781	0.613906	−	0.789379i	$-0.289598\pi$
$911$	37.0588	1.22781	0.613906	+	0.789379i	$0.289598\pi$
$912$	0	0
$913$	−1.85368	−0.0613479
$914$	0	0
$915$	8.27414	0.273535
$916$	0	0
$917$	−10.6230	−0.350802
$918$	0	0
$919$	−55.0014	−1.81433	−0.907164	−	0.420776i	$-0.861758\pi$
$919$	−55.0014	−1.81433	−0.907164	+	0.420776i	$0.861758\pi$
$920$	0	0
$921$	7.37126	0.242891
$922$	0	0
$923$	33.9337	1.11694
$924$	0	0
$925$	3.89900	0.128198
$926$	0	0
$927$	2.19989	0.0722537
$928$	0	0
$929$	4.96719	0.162968	0.0814841	−	0.996675i	$-0.474034\pi$
$929$	4.96719	0.162968	0.0814841	+	0.996675i	$0.474034\pi$
$930$	0	0
$931$	−4.60487	−0.150918
$932$	0	0
$933$	−42.7529	−1.39967
$934$	0	0
$935$	−4.52345	−0.147933
$936$	0	0
$937$	−51.6942	−1.68877	−0.844387	−	0.535733i	$-0.820035\pi$
$937$	−51.6942	−1.68877	−0.844387	+	0.535733i	$0.820035\pi$
$938$	0	0
$939$	0.121422	0.00396246
$940$	0	0
$941$	19.3263	0.630021	0.315010	−	0.949088i	$-0.397992\pi$
$941$	19.3263	0.630021	0.315010	+	0.949088i	$0.397992\pi$
$942$	0	0
$943$	5.54516	0.180575
$944$	0	0
$945$	15.2644	0.496552
$946$	0	0
$947$	−8.27481	−0.268895	−0.134448	−	0.990921i	$-0.542926\pi$
$947$	−8.27481	−0.268895	−0.134448	+	0.990921i	$0.542926\pi$
$948$	0	0
$949$	23.0586	0.748514
$950$	0	0
$951$	−20.5566	−0.666594
$952$	0	0
$953$	35.7457	1.15792	0.578958	−	0.815357i	$-0.303459\pi$
$953$	35.7457	1.15792	0.578958	+	0.815357i	$0.303459\pi$
$954$	0	0
$955$	−42.8318	−1.38600
$956$	0	0
$957$	−6.34496	−0.205104
$958$	0	0
$959$	9.20572	0.297268
$960$	0	0
$961$	−1.83239	−0.0591095
$962$	0	0
$963$	24.3974	0.786195
$964$	0	0
$965$	−18.6099	−0.599072
$966$	0	0
$967$	−49.3716	−1.58768	−0.793842	−	0.608123i	$-0.791923\pi$
$967$	−49.3716	−1.58768	−0.793842	+	0.608123i	$0.791923\pi$
$968$	0	0
$969$	6.37605	0.204828
$970$	0	0
$971$	36.9475	1.18570	0.592851	−	0.805312i	$-0.298002\pi$
$971$	36.9475	1.18570	0.592851	+	0.805312i	$0.298002\pi$
$972$	0	0
$973$	15.7061	0.503515
$974$	0	0
$975$	−6.02978	−0.193108
$976$	0	0
$977$	0.419491	0.0134207	0.00671035	−	0.999977i	$-0.497864\pi$
$977$	0.419491	0.0134207	0.00671035	+	0.999977i	$0.497864\pi$
$978$	0	0
$979$	−0.0780890	−0.00249574
$980$	0	0
$981$	3.88527	0.124047
$982$	0	0
$983$	23.2996	0.743141	0.371570	−	0.928405i	$-0.378819\pi$
$983$	23.2996	0.743141	0.371570	+	0.928405i	$0.378819\pi$
$984$	0	0
$985$	−31.2329	−0.995164
$986$	0	0
$987$	5.29332	0.168488
$988$	0	0
$989$	14.8017	0.470667
$990$	0	0
$991$	40.0982	1.27376	0.636881	−	0.770962i	$-0.280224\pi$
$991$	40.0982	1.27376	0.636881	+	0.770962i	$0.280224\pi$
$992$	0	0
$993$	22.4205	0.711492
$994$	0	0
$995$	−37.3641	−1.18452
$996$	0	0
$997$	15.1781	0.480695	0.240348	−	0.970687i	$-0.422739\pi$
$997$	15.1781	0.480695	0.240348	+	0.970687i	$0.422739\pi$
$998$	0	0
$999$	−11.3583	−0.359362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.12	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.12	✓	24	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)

\(f(q)\)	\(=\)	\(q-1.33751 q^{3} -1.75116 q^{5} -1.54762 q^{7} -1.21107 q^{9} +O(q^{10})\)	\(q-1.33751 q^{3} -1.75116 q^{5} -1.54762 q^{7} -1.21107 q^{9} -0.541862 q^{11} -2.33172 q^{13} +2.34220 q^{15} -4.76711 q^{17} +1.00000 q^{19} +2.06996 q^{21} -7.12294 q^{23} -1.93343 q^{25} +5.63234 q^{27} -8.75476 q^{29} -5.40070 q^{31} +0.724745 q^{33} +2.71014 q^{35} -2.01663 q^{37} +3.11870 q^{39} -0.778494 q^{41} -2.07804 q^{43} +2.12078 q^{45} +2.55721 q^{47} -4.60487 q^{49} +6.37605 q^{51} +5.31127 q^{53} +0.948888 q^{55} -1.33751 q^{57} +9.33467 q^{59} +3.53264 q^{61} +1.87428 q^{63} +4.08323 q^{65} -14.2713 q^{67} +9.52699 q^{69} -14.5531 q^{71} -9.88909 q^{73} +2.58598 q^{75} +0.838597 q^{77} +1.00000 q^{79} -3.90010 q^{81} +3.42095 q^{83} +8.34799 q^{85} +11.7096 q^{87} +0.144112 q^{89} +3.60862 q^{91} +7.22349 q^{93} -1.75116 q^{95} -6.99434 q^{97} +0.656233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more