LMFDB - Embedded newform 6004.2.a.g.1.26

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			1.26
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)

$f(q)$	$=$	$q+2.71988 q^{3} -0.592160 q^{5} -2.94575 q^{7} +4.39773 q^{9} +O(q^{10})$	$q+2.71988 q^{3} -0.592160 q^{5} -2.94575 q^{7} +4.39773 q^{9} +2.40684 q^{11} +3.66568 q^{13} -1.61060 q^{15} -7.66581 q^{17} +1.00000 q^{19} -8.01207 q^{21} -7.91853 q^{23} -4.64935 q^{25} +3.80164 q^{27} -8.53016 q^{29} -0.177489 q^{31} +6.54631 q^{33} +1.74435 q^{35} +4.75848 q^{37} +9.97019 q^{39} +4.20422 q^{41} -5.39793 q^{43} -2.60416 q^{45} +0.161608 q^{47} +1.67743 q^{49} -20.8501 q^{51} -11.3149 q^{53} -1.42523 q^{55} +2.71988 q^{57} -3.61194 q^{59} +10.3097 q^{61} -12.9546 q^{63} -2.17067 q^{65} +10.7657 q^{67} -21.5374 q^{69} -9.67502 q^{71} +6.62788 q^{73} -12.6456 q^{75} -7.08994 q^{77} -1.00000 q^{79} -2.85318 q^{81} -4.26142 q^{83} +4.53939 q^{85} -23.2010 q^{87} -17.0062 q^{89} -10.7982 q^{91} -0.482747 q^{93} -0.592160 q^{95} +14.0216 q^{97} +10.5846 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.71988	1.57032	0.785161	−	0.619292i	$-0.212580\pi$
$3$	2.71988	1.57032	0.785161	+	0.619292i	$0.212580\pi$
$4$	0	0
$5$	−0.592160	−0.264822	−0.132411	−	0.991195i	$-0.542272\pi$
$5$	−0.592160	−0.264822	−0.132411	+	0.991195i	$0.542272\pi$
$6$	0	0
$7$	−2.94575	−1.11339	−0.556694	−	0.830718i	$-0.687930\pi$
$7$	−2.94575	−1.11339	−0.556694	+	0.830718i	$0.687930\pi$
$8$	0	0
$9$	4.39773	1.46591
$10$	0	0
$11$	2.40684	0.725690	0.362845	−	0.931850i	$-0.381806\pi$
$11$	2.40684	0.725690	0.362845	+	0.931850i	$0.381806\pi$
$12$	0	0
$13$	3.66568	1.01668	0.508338	−	0.861158i	$-0.330260\pi$
$13$	3.66568	1.01668	0.508338	+	0.861158i	$0.330260\pi$
$14$	0	0
$15$	−1.61060	−0.415855
$16$	0	0
$17$	−7.66581	−1.85923	−0.929616	−	0.368529i	$-0.879862\pi$
$17$	−7.66581	−1.85923	−0.929616	+	0.368529i	$0.879862\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−8.01207	−1.74838
$22$	0	0
$23$	−7.91853	−1.65113	−0.825564	−	0.564309i	$-0.809143\pi$
$23$	−7.91853	−1.65113	−0.825564	+	0.564309i	$0.809143\pi$
$24$	0	0
$25$	−4.64935	−0.929869
$26$	0	0
$27$	3.80164	0.731627
$28$	0	0
$29$	−8.53016	−1.58401	−0.792006	−	0.610514i	$-0.790963\pi$
$29$	−8.53016	−1.58401	−0.792006	+	0.610514i	$0.790963\pi$
$30$	0	0
$31$	−0.177489	−0.0318779	−0.0159390	−	0.999873i	$-0.505074\pi$
$31$	−0.177489	−0.0318779	−0.0159390	+	0.999873i	$0.505074\pi$
$32$	0	0
$33$	6.54631	1.13957
$34$	0	0
$35$	1.74435	0.294849
$36$	0	0
$37$	4.75848	0.782289	0.391145	−	0.920329i	$-0.372079\pi$
$37$	4.75848	0.782289	0.391145	+	0.920329i	$0.372079\pi$
$38$	0	0
$39$	9.97019	1.59651
$40$	0	0
$41$	4.20422	0.656589	0.328294	−	0.944575i	$-0.393526\pi$
$41$	4.20422	0.656589	0.328294	+	0.944575i	$0.393526\pi$
$42$	0	0
$43$	−5.39793	−0.823177	−0.411588	−	0.911370i	$-0.635026\pi$
$43$	−5.39793	−0.823177	−0.411588	+	0.911370i	$0.635026\pi$
$44$	0	0
$45$	−2.60416	−0.388205
$46$	0	0
$47$	0.161608	0.0235730	0.0117865	−	0.999931i	$-0.496248\pi$
$47$	0.161608	0.0235730	0.0117865	+	0.999931i	$0.496248\pi$
$48$	0	0
$49$	1.67743	0.239632
$50$	0	0
$51$	−20.8501	−2.91959
$52$	0	0
$53$	−11.3149	−1.55422	−0.777109	−	0.629366i	$-0.783315\pi$
$53$	−11.3149	−1.55422	−0.777109	+	0.629366i	$0.783315\pi$
$54$	0	0
$55$	−1.42523	−0.192178
$56$	0	0
$57$	2.71988	0.360256
$58$	0	0
$59$	−3.61194	−0.470235	−0.235117	−	0.971967i	$-0.575547\pi$
$59$	−3.61194	−0.470235	−0.235117	+	0.971967i	$0.575547\pi$
$60$	0	0
$61$	10.3097	1.32002	0.660010	−	0.751257i	$-0.270552\pi$
$61$	10.3097	1.32002	0.660010	+	0.751257i	$0.270552\pi$
$62$	0	0
$63$	−12.9546	−1.63213
$64$	0	0
$65$	−2.17067	−0.269238
$66$	0	0
$67$	10.7657	1.31525	0.657623	−	0.753347i	$-0.271562\pi$
$67$	10.7657	1.31525	0.657623	+	0.753347i	$0.271562\pi$
$68$	0	0
$69$	−21.5374	−2.59280
$70$	0	0
$71$	−9.67502	−1.14821	−0.574107	−	0.818780i	$-0.694651\pi$
$71$	−9.67502	−1.14821	−0.574107	+	0.818780i	$0.694651\pi$
$72$	0	0
$73$	6.62788	0.775735	0.387868	−	0.921715i	$-0.373212\pi$
$73$	6.62788	0.775735	0.387868	+	0.921715i	$0.373212\pi$
$74$	0	0
$75$	−12.6456	−1.46019
$76$	0	0
$77$	−7.08994	−0.807974
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−2.85318	−0.317020
$82$	0	0
$83$	−4.26142	−0.467752	−0.233876	−	0.972266i	$-0.575141\pi$
$83$	−4.26142	−0.467752	−0.233876	+	0.972266i	$0.575141\pi$
$84$	0	0
$85$	4.53939	0.492365
$86$	0	0
$87$	−23.2010	−2.48741
$88$	0	0
$89$	−17.0062	−1.80266	−0.901329	−	0.433136i	$-0.857407\pi$
$89$	−17.0062	−1.80266	−0.901329	+	0.433136i	$0.857407\pi$
$90$	0	0
$91$	−10.7982	−1.13195
$92$	0	0
$93$	−0.482747	−0.0500586
$94$	0	0
$95$	−0.592160	−0.0607543
$96$	0	0
$97$	14.0216	1.42368	0.711838	−	0.702344i	$-0.247863\pi$
$97$	14.0216	1.42368	0.711838	+	0.702344i	$0.247863\pi$
$98$	0	0
$99$	10.5846	1.06379
$100$	0	0
$101$	−0.260123	−0.0258832	−0.0129416	−	0.999916i	$-0.504120\pi$
$101$	−0.260123	−0.0258832	−0.0129416	+	0.999916i	$0.504120\pi$
$102$	0	0
$103$	−14.3067	−1.40968	−0.704842	−	0.709365i	$-0.748982\pi$
$103$	−14.3067	−1.40968	−0.704842	+	0.709365i	$0.748982\pi$
$104$	0	0
$105$	4.74442	0.463008
$106$	0	0
$107$	3.17406	0.306848	0.153424	−	0.988160i	$-0.450970\pi$
$107$	3.17406	0.306848	0.153424	+	0.988160i	$0.450970\pi$
$108$	0	0
$109$	8.92663	0.855017	0.427508	−	0.904011i	$-0.359391\pi$
$109$	8.92663	0.855017	0.427508	+	0.904011i	$0.359391\pi$
$110$	0	0
$111$	12.9425	1.22845
$112$	0	0
$113$	−3.65897	−0.344207	−0.172103	−	0.985079i	$-0.555056\pi$
$113$	−3.65897	−0.344207	−0.172103	+	0.985079i	$0.555056\pi$
$114$	0	0
$115$	4.68903	0.437255
$116$	0	0
$117$	16.1206	1.49035
$118$	0	0
$119$	22.5815	2.07005
$120$	0	0
$121$	−5.20712	−0.473375
$122$	0	0
$123$	11.4350	1.03106
$124$	0	0
$125$	5.71395	0.511072
$126$	0	0
$127$	−0.414767	−0.0368047	−0.0184023	−	0.999831i	$-0.505858\pi$
$127$	−0.414767	−0.0368047	−0.0184023	+	0.999831i	$0.505858\pi$
$128$	0	0
$129$	−14.6817	−1.29265
$130$	0	0
$131$	9.27389	0.810264	0.405132	−	0.914258i	$-0.367226\pi$
$131$	9.27389	0.810264	0.405132	+	0.914258i	$0.367226\pi$
$132$	0	0
$133$	−2.94575	−0.255429
$134$	0	0
$135$	−2.25118	−0.193751
$136$	0	0
$137$	2.07279	0.177090	0.0885451	−	0.996072i	$-0.471778\pi$
$137$	2.07279	0.177090	0.0885451	+	0.996072i	$0.471778\pi$
$138$	0	0
$139$	16.2159	1.37541	0.687707	−	0.725988i	$-0.258617\pi$
$139$	16.2159	1.37541	0.687707	+	0.725988i	$0.258617\pi$
$140$	0	0
$141$	0.439554	0.0370172
$142$	0	0
$143$	8.82270	0.737791
$144$	0	0
$145$	5.05122	0.419481
$146$	0	0
$147$	4.56239	0.376300
$148$	0	0
$149$	−3.12243	−0.255800	−0.127900	−	0.991787i	$-0.540824\pi$
$149$	−3.12243	−0.255800	−0.127900	+	0.991787i	$0.540824\pi$
$150$	0	0
$151$	−19.9817	−1.62608	−0.813042	−	0.582205i	$-0.802190\pi$
$151$	−19.9817	−1.62608	−0.813042	+	0.582205i	$0.802190\pi$
$152$	0	0
$153$	−33.7122	−2.72547
$154$	0	0
$155$	0.105102	0.00844197
$156$	0	0
$157$	−21.6630	−1.72890	−0.864448	−	0.502721i	$-0.832332\pi$
$157$	−21.6630	−1.72890	−0.864448	+	0.502721i	$0.832332\pi$
$158$	0	0
$159$	−30.7751	−2.44062
$160$	0	0
$161$	23.3260	1.83835
$162$	0	0
$163$	6.13425	0.480471	0.240236	−	0.970715i	$-0.422775\pi$
$163$	6.13425	0.480471	0.240236	+	0.970715i	$0.422775\pi$
$164$	0	0
$165$	−3.87646	−0.301782
$166$	0	0
$167$	22.8505	1.76823	0.884113	−	0.467273i	$-0.154764\pi$
$167$	22.8505	1.76823	0.884113	+	0.467273i	$0.154764\pi$
$168$	0	0
$169$	0.437185	0.0336296
$170$	0	0
$171$	4.39773	0.336303
$172$	0	0
$173$	−22.3280	−1.69756	−0.848782	−	0.528743i	$-0.822663\pi$
$173$	−22.3280	−1.69756	−0.848782	+	0.528743i	$0.822663\pi$
$174$	0	0
$175$	13.6958	1.03531
$176$	0	0
$177$	−9.82403	−0.738419
$178$	0	0
$179$	20.4749	1.53037	0.765183	−	0.643812i	$-0.222648\pi$
$179$	20.4749	1.53037	0.765183	+	0.643812i	$0.222648\pi$
$180$	0	0
$181$	25.3923	1.88740	0.943699	−	0.330805i	$-0.107320\pi$
$181$	25.3923	1.88740	0.943699	+	0.330805i	$0.107320\pi$
$182$	0	0
$183$	28.0411	2.07285
$184$	0	0
$185$	−2.81778	−0.207167
$186$	0	0
$187$	−18.4504	−1.34923
$188$	0	0
$189$	−11.1987	−0.814584
$190$	0	0
$191$	3.56106	0.257670	0.128835	−	0.991666i	$-0.458876\pi$
$191$	3.56106	0.257670	0.128835	+	0.991666i	$0.458876\pi$
$192$	0	0
$193$	10.2826	0.740158	0.370079	−	0.929000i	$-0.379331\pi$
$193$	10.2826	0.740158	0.370079	+	0.929000i	$0.379331\pi$
$194$	0	0
$195$	−5.90394	−0.422790
$196$	0	0
$197$	1.29975	0.0926034	0.0463017	−	0.998928i	$-0.485256\pi$
$197$	1.29975	0.0926034	0.0463017	+	0.998928i	$0.485256\pi$
$198$	0	0
$199$	−22.1902	−1.57302	−0.786511	−	0.617577i	$-0.788114\pi$
$199$	−22.1902	−1.57302	−0.786511	+	0.617577i	$0.788114\pi$
$200$	0	0
$201$	29.2815	2.06536
$202$	0	0
$203$	25.1277	1.76362
$204$	0	0
$205$	−2.48957	−0.173879
$206$	0	0
$207$	−34.8235	−2.42040
$208$	0	0
$209$	2.40684	0.166485
$210$	0	0
$211$	−21.3451	−1.46946	−0.734728	−	0.678362i	$-0.762690\pi$
$211$	−21.3451	−1.46946	−0.734728	+	0.678362i	$0.762690\pi$
$212$	0	0
$213$	−26.3149	−1.80306
$214$	0	0
$215$	3.19644	0.217995
$216$	0	0
$217$	0.522837	0.0354925
$218$	0	0
$219$	18.0270	1.21815
$220$	0	0
$221$	−28.1004	−1.89024
$222$	0	0
$223$	9.26500	0.620430	0.310215	−	0.950666i	$-0.399599\pi$
$223$	9.26500	0.620430	0.310215	+	0.950666i	$0.399599\pi$
$224$	0	0
$225$	−20.4466	−1.36310
$226$	0	0
$227$	−0.840113	−0.0557602	−0.0278801	−	0.999611i	$-0.508876\pi$
$227$	−0.840113	−0.0557602	−0.0278801	+	0.999611i	$0.508876\pi$
$228$	0	0
$229$	−14.6086	−0.965361	−0.482680	−	0.875797i	$-0.660337\pi$
$229$	−14.6086	−0.965361	−0.482680	+	0.875797i	$0.660337\pi$
$230$	0	0
$231$	−19.2838	−1.26878
$232$	0	0
$233$	−0.183054	−0.0119923	−0.00599614	−	0.999982i	$-0.501909\pi$
$233$	−0.183054	−0.0119923	−0.00599614	+	0.999982i	$0.501909\pi$
$234$	0	0
$235$	−0.0956979	−0.00624264
$236$	0	0
$237$	−2.71988	−0.176675
$238$	0	0
$239$	−14.2417	−0.921217	−0.460608	−	0.887604i	$-0.652369\pi$
$239$	−14.2417	−0.921217	−0.460608	+	0.887604i	$0.652369\pi$
$240$	0	0
$241$	−22.3242	−1.43803	−0.719014	−	0.694995i	$-0.755406\pi$
$241$	−22.3242	−1.43803	−0.719014	+	0.694995i	$0.755406\pi$
$242$	0	0
$243$	−19.1652	−1.22945
$244$	0	0
$245$	−0.993305	−0.0634599
$246$	0	0
$247$	3.66568	0.233241
$248$	0	0
$249$	−11.5905	−0.734521
$250$	0	0
$251$	−24.3574	−1.53743	−0.768713	−	0.639594i	$-0.779102\pi$
$251$	−24.3574	−1.53743	−0.768713	+	0.639594i	$0.779102\pi$
$252$	0	0
$253$	−19.0586	−1.19821
$254$	0	0
$255$	12.3466	0.773172
$256$	0	0
$257$	3.09664	0.193163	0.0965817	−	0.995325i	$-0.469209\pi$
$257$	3.09664	0.193163	0.0965817	+	0.995325i	$0.469209\pi$
$258$	0	0
$259$	−14.0173	−0.870991
$260$	0	0
$261$	−37.5133	−2.32202
$262$	0	0
$263$	−11.3064	−0.697181	−0.348590	−	0.937275i	$-0.613340\pi$
$263$	−11.3064	−0.697181	−0.348590	+	0.937275i	$0.613340\pi$
$264$	0	0
$265$	6.70022	0.411591
$266$	0	0
$267$	−46.2549	−2.83075
$268$	0	0
$269$	13.0366	0.794855	0.397428	−	0.917634i	$-0.369903\pi$
$269$	13.0366	0.794855	0.397428	+	0.917634i	$0.369903\pi$
$270$	0	0
$271$	−9.63428	−0.585241	−0.292620	−	0.956229i	$-0.594527\pi$
$271$	−9.63428	−0.585241	−0.292620	+	0.956229i	$0.594527\pi$
$272$	0	0
$273$	−29.3697	−1.77753
$274$	0	0
$275$	−11.1902	−0.674797
$276$	0	0
$277$	−0.298797	−0.0179530	−0.00897648	−	0.999960i	$-0.502857\pi$
$277$	−0.298797	−0.0179530	−0.00897648	+	0.999960i	$0.502857\pi$
$278$	0	0
$279$	−0.780547	−0.0467301
$280$	0	0
$281$	5.74039	0.342443	0.171221	−	0.985233i	$-0.445229\pi$
$281$	5.74039	0.342443	0.171221	+	0.985233i	$0.445229\pi$
$282$	0	0
$283$	25.1521	1.49514	0.747570	−	0.664183i	$-0.231221\pi$
$283$	25.1521	1.49514	0.747570	+	0.664183i	$0.231221\pi$
$284$	0	0
$285$	−1.61060	−0.0954038
$286$	0	0
$287$	−12.3846	−0.731038
$288$	0	0
$289$	41.7647	2.45675
$290$	0	0
$291$	38.1369	2.23563
$292$	0	0
$293$	−22.6981	−1.32604	−0.663020	−	0.748602i	$-0.730725\pi$
$293$	−22.6981	−1.32604	−0.663020	+	0.748602i	$0.730725\pi$
$294$	0	0
$295$	2.13885	0.124528
$296$	0	0
$297$	9.14995	0.530934
$298$	0	0
$299$	−29.0268	−1.67866
$300$	0	0
$301$	15.9009	0.916515
$302$	0	0
$303$	−0.707502	−0.0406449
$304$	0	0
$305$	−6.10498	−0.349570
$306$	0	0
$307$	25.1284	1.43415	0.717077	−	0.696994i	$-0.245480\pi$
$307$	25.1284	1.43415	0.717077	+	0.696994i	$0.245480\pi$
$308$	0	0
$309$	−38.9125	−2.21366
$310$	0	0
$311$	−26.1490	−1.48277	−0.741387	−	0.671078i	$-0.765832\pi$
$311$	−26.1490	−1.48277	−0.741387	+	0.671078i	$0.765832\pi$
$312$	0	0
$313$	0.369964	0.0209116	0.0104558	−	0.999945i	$-0.496672\pi$
$313$	0.369964	0.0209116	0.0104558	+	0.999945i	$0.496672\pi$
$314$	0	0
$315$	7.67119	0.432222
$316$	0	0
$317$	29.1437	1.63687	0.818437	−	0.574596i	$-0.194841\pi$
$317$	29.1437	1.63687	0.818437	+	0.574596i	$0.194841\pi$
$318$	0	0
$319$	−20.5307	−1.14950
$320$	0	0
$321$	8.63304	0.481850
$322$	0	0
$323$	−7.66581	−0.426537
$324$	0	0
$325$	−17.0430	−0.945376
$326$	0	0
$327$	24.2793	1.34265
$328$	0	0
$329$	−0.476057	−0.0262459
$330$	0	0
$331$	31.7588	1.74562	0.872811	−	0.488058i	$-0.162295\pi$
$331$	31.7588	1.74562	0.872811	+	0.488058i	$0.162295\pi$
$332$	0	0
$333$	20.9265	1.14676
$334$	0	0
$335$	−6.37504	−0.348306
$336$	0	0
$337$	14.7789	0.805058	0.402529	−	0.915407i	$-0.368131\pi$
$337$	14.7789	0.805058	0.402529	+	0.915407i	$0.368131\pi$
$338$	0	0
$339$	−9.95194	−0.540515
$340$	0	0
$341$	−0.427187	−0.0231335
$342$	0	0
$343$	15.6790	0.846584
$344$	0	0
$345$	12.7536	0.686630
$346$	0	0
$347$	−20.6479	−1.10844	−0.554220	−	0.832370i	$-0.686983\pi$
$347$	−20.6479	−1.10844	−0.554220	+	0.832370i	$0.686983\pi$
$348$	0	0
$349$	−34.8066	−1.86315	−0.931576	−	0.363546i	$-0.881566\pi$
$349$	−34.8066	−1.86315	−0.931576	+	0.363546i	$0.881566\pi$
$350$	0	0
$351$	13.9356	0.743827
$352$	0	0
$353$	11.3935	0.606415	0.303207	−	0.952925i	$-0.401942\pi$
$353$	11.3935	0.606415	0.303207	+	0.952925i	$0.401942\pi$
$354$	0	0
$355$	5.72916	0.304072
$356$	0	0
$357$	61.4190	3.25064
$358$	0	0
$359$	22.8684	1.20695	0.603474	−	0.797383i	$-0.293783\pi$
$359$	22.8684	1.20695	0.603474	+	0.797383i	$0.293783\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−14.1627	−0.743350
$364$	0	0
$365$	−3.92477	−0.205432
$366$	0	0
$367$	−6.76747	−0.353259	−0.176629	−	0.984277i	$-0.556519\pi$
$367$	−6.76747	−0.353259	−0.176629	+	0.984277i	$0.556519\pi$
$368$	0	0
$369$	18.4890	0.962500
$370$	0	0
$371$	33.3308	1.73045
$372$	0	0
$373$	−26.8854	−1.39207	−0.696036	−	0.718007i	$-0.745055\pi$
$373$	−26.8854	−1.39207	−0.696036	+	0.718007i	$0.745055\pi$
$374$	0	0
$375$	15.5412	0.802547
$376$	0	0
$377$	−31.2688	−1.61043
$378$	0	0
$379$	−31.9811	−1.64276	−0.821379	−	0.570383i	$-0.806795\pi$
$379$	−31.9811	−1.64276	−0.821379	+	0.570383i	$0.806795\pi$
$380$	0	0
$381$	−1.12812	−0.0577951
$382$	0	0
$383$	31.4537	1.60721	0.803604	−	0.595165i	$-0.202913\pi$
$383$	31.4537	1.60721	0.803604	+	0.595165i	$0.202913\pi$
$384$	0	0
$385$	4.19838	0.213969
$386$	0	0
$387$	−23.7386	−1.20670
$388$	0	0
$389$	−15.9989	−0.811175	−0.405588	−	0.914056i	$-0.632933\pi$
$389$	−15.9989	−0.811175	−0.405588	+	0.914056i	$0.632933\pi$
$390$	0	0
$391$	60.7020	3.06983
$392$	0	0
$393$	25.2238	1.27237
$394$	0	0
$395$	0.592160	0.0297948
$396$	0	0
$397$	−10.3054	−0.517214	−0.258607	−	0.965983i	$-0.583263\pi$
$397$	−10.3054	−0.517214	−0.258607	+	0.965983i	$0.583263\pi$
$398$	0	0
$399$	−8.01207	−0.401105
$400$	0	0
$401$	33.0946	1.65266	0.826332	−	0.563183i	$-0.190424\pi$
$401$	33.0946	1.65266	0.826332	+	0.563183i	$0.190424\pi$
$402$	0	0
$403$	−0.650616	−0.0324095
$404$	0	0
$405$	1.68954	0.0839538
$406$	0	0
$407$	11.4529	0.567699
$408$	0	0
$409$	−19.7646	−0.977295	−0.488647	−	0.872481i	$-0.662510\pi$
$409$	−19.7646	−0.977295	−0.488647	+	0.872481i	$0.662510\pi$
$410$	0	0
$411$	5.63772	0.278088
$412$	0	0
$413$	10.6399	0.523554
$414$	0	0
$415$	2.52344	0.123871
$416$	0	0
$417$	44.1052	2.15984
$418$	0	0
$419$	−31.1608	−1.52231	−0.761153	−	0.648573i	$-0.775366\pi$
$419$	−31.1608	−1.52231	−0.761153	+	0.648573i	$0.775366\pi$
$420$	0	0
$421$	30.0185	1.46301	0.731506	−	0.681835i	$-0.238818\pi$
$421$	30.0185	1.46301	0.731506	+	0.681835i	$0.238818\pi$
$422$	0	0
$423$	0.710709	0.0345559
$424$	0	0
$425$	35.6410	1.72884
$426$	0	0
$427$	−30.3697	−1.46969
$428$	0	0
$429$	23.9966	1.15857
$430$	0	0
$431$	2.56995	0.123790	0.0618952	−	0.998083i	$-0.480286\pi$
$431$	2.56995	0.123790	0.0618952	+	0.998083i	$0.480286\pi$
$432$	0	0
$433$	−11.3974	−0.547723	−0.273861	−	0.961769i	$-0.588301\pi$
$433$	−11.3974	−0.547723	−0.273861	+	0.961769i	$0.588301\pi$
$434$	0	0
$435$	13.7387	0.658720
$436$	0	0
$437$	−7.91853	−0.378795
$438$	0	0
$439$	−28.6807	−1.36886	−0.684428	−	0.729080i	$-0.739948\pi$
$439$	−28.6807	−1.36886	−0.684428	+	0.729080i	$0.739948\pi$
$440$	0	0
$441$	7.37687	0.351279
$442$	0	0
$443$	19.3206	0.917950	0.458975	−	0.888449i	$-0.348217\pi$
$443$	19.3206	0.917950	0.458975	+	0.888449i	$0.348217\pi$
$444$	0	0
$445$	10.0704	0.477383
$446$	0	0
$447$	−8.49263	−0.401688
$448$	0	0
$449$	22.0959	1.04277	0.521386	−	0.853321i	$-0.325415\pi$
$449$	22.0959	1.04277	0.521386	+	0.853321i	$0.325415\pi$
$450$	0	0
$451$	10.1189	0.476480
$452$	0	0
$453$	−54.3476	−2.55347
$454$	0	0
$455$	6.39423	0.299766
$456$	0	0
$457$	17.4439	0.815991	0.407996	−	0.912984i	$-0.366228\pi$
$457$	17.4439	0.815991	0.407996	+	0.912984i	$0.366228\pi$
$458$	0	0
$459$	−29.1427	−1.36026
$460$	0	0
$461$	−28.5439	−1.32942	−0.664712	−	0.747100i	$-0.731446\pi$
$461$	−28.5439	−1.32942	−0.664712	+	0.747100i	$0.731446\pi$
$462$	0	0
$463$	12.3512	0.574011	0.287005	−	0.957929i	$-0.407340\pi$
$463$	12.3512	0.574011	0.287005	+	0.957929i	$0.407340\pi$
$464$	0	0
$465$	0.285864	0.0132566
$466$	0	0
$467$	−29.7802	−1.37807	−0.689033	−	0.724730i	$-0.741964\pi$
$467$	−29.7802	−1.37807	−0.689033	+	0.724730i	$0.741964\pi$
$468$	0	0
$469$	−31.7132	−1.46438
$470$	0	0
$471$	−58.9207	−2.71492
$472$	0	0
$473$	−12.9920	−0.597371
$474$	0	0
$475$	−4.64935	−0.213327
$476$	0	0
$477$	−49.7597	−2.27834
$478$	0	0
$479$	−2.69120	−0.122964	−0.0614821	−	0.998108i	$-0.519583\pi$
$479$	−2.69120	−0.122964	−0.0614821	+	0.998108i	$0.519583\pi$
$480$	0	0
$481$	17.4430	0.795334
$482$	0	0
$483$	63.4438	2.88679
$484$	0	0
$485$	−8.30301	−0.377020
$486$	0	0
$487$	40.6509	1.84207	0.921034	−	0.389482i	$-0.127346\pi$
$487$	40.6509	1.84207	0.921034	+	0.389482i	$0.127346\pi$
$488$	0	0
$489$	16.6844	0.754494
$490$	0	0
$491$	7.70487	0.347716	0.173858	−	0.984771i	$-0.444377\pi$
$491$	7.70487	0.347716	0.173858	+	0.984771i	$0.444377\pi$
$492$	0	0
$493$	65.3906	2.94505
$494$	0	0
$495$	−6.26779	−0.281716
$496$	0	0
$497$	28.5002	1.27841
$498$	0	0
$499$	−18.8438	−0.843566	−0.421783	−	0.906697i	$-0.638596\pi$
$499$	−18.8438	−0.843566	−0.421783	+	0.906697i	$0.638596\pi$
$500$	0	0
$501$	62.1506	2.77668
$502$	0	0
$503$	12.2068	0.544273	0.272136	−	0.962259i	$-0.412270\pi$
$503$	12.2068	0.544273	0.272136	+	0.962259i	$0.412270\pi$
$504$	0	0
$505$	0.154034	0.00685444
$506$	0	0
$507$	1.18909	0.0528093
$508$	0	0
$509$	−18.5816	−0.823613	−0.411807	−	0.911271i	$-0.635102\pi$
$509$	−18.5816	−0.823613	−0.411807	+	0.911271i	$0.635102\pi$
$510$	0	0
$511$	−19.5241	−0.863694
$512$	0	0
$513$	3.80164	0.167847
$514$	0	0
$515$	8.47186	0.373315
$516$	0	0
$517$	0.388965	0.0171067
$518$	0	0
$519$	−60.7293	−2.66572
$520$	0	0
$521$	16.0856	0.704725	0.352362	−	0.935864i	$-0.385378\pi$
$521$	16.0856	0.704725	0.352362	+	0.935864i	$0.385378\pi$
$522$	0	0
$523$	−5.11684	−0.223744	−0.111872	−	0.993723i	$-0.535685\pi$
$523$	−5.11684	−0.223744	−0.111872	+	0.993723i	$0.535685\pi$
$524$	0	0
$525$	37.2509	1.62576
$526$	0	0
$527$	1.36060	0.0592684
$528$	0	0
$529$	39.7031	1.72622
$530$	0	0
$531$	−15.8843	−0.689321
$532$	0	0
$533$	15.4113	0.667538
$534$	0	0
$535$	−1.87955	−0.0812600
$536$	0	0
$537$	55.6892	2.40317
$538$	0	0
$539$	4.03730	0.173899
$540$	0	0
$541$	−17.3642	−0.746547	−0.373274	−	0.927721i	$-0.621765\pi$
$541$	−17.3642	−0.746547	−0.373274	+	0.927721i	$0.621765\pi$
$542$	0	0
$543$	69.0640	2.96382
$544$	0	0
$545$	−5.28599	−0.226427
$546$	0	0
$547$	−19.6927	−0.841998	−0.420999	−	0.907061i	$-0.638320\pi$
$547$	−19.6927	−0.841998	−0.420999	+	0.907061i	$0.638320\pi$
$548$	0	0
$549$	45.3392	1.93503
$550$	0	0
$551$	−8.53016	−0.363397
$552$	0	0
$553$	2.94575	0.125266
$554$	0	0
$555$	−7.66401	−0.325319
$556$	0	0
$557$	−41.4631	−1.75685	−0.878423	−	0.477883i	$-0.841404\pi$
$557$	−41.4631	−1.75685	−0.878423	+	0.477883i	$0.841404\pi$
$558$	0	0
$559$	−19.7871	−0.836904
$560$	0	0
$561$	−50.1828	−2.11872
$562$	0	0
$563$	−17.7121	−0.746477	−0.373238	−	0.927735i	$-0.621753\pi$
$563$	−17.7121	−0.746477	−0.373238	+	0.927735i	$0.621753\pi$
$564$	0	0
$565$	2.16669	0.0911535
$566$	0	0
$567$	8.40474	0.352966
$568$	0	0
$569$	−5.44969	−0.228463	−0.114232	−	0.993454i	$-0.536441\pi$
$569$	−5.44969	−0.228463	−0.114232	+	0.993454i	$0.536441\pi$
$570$	0	0
$571$	−8.56064	−0.358252	−0.179126	−	0.983826i	$-0.557327\pi$
$571$	−8.56064	−0.358252	−0.179126	+	0.983826i	$0.557327\pi$
$572$	0	0
$573$	9.68566	0.404624
$574$	0	0
$575$	36.8160	1.53533
$576$	0	0
$577$	19.4759	0.810794	0.405397	−	0.914141i	$-0.367133\pi$
$577$	19.4759	0.810794	0.405397	+	0.914141i	$0.367133\pi$
$578$	0	0
$579$	27.9674	1.16229
$580$	0	0
$581$	12.5531	0.520789
$582$	0	0
$583$	−27.2331	−1.12788
$584$	0	0
$585$	−9.54600	−0.394678
$586$	0	0
$587$	−2.10064	−0.0867027	−0.0433513	−	0.999060i	$-0.513803\pi$
$587$	−2.10064	−0.0867027	−0.0433513	+	0.999060i	$0.513803\pi$
$588$	0	0
$589$	−0.177489	−0.00731329
$590$	0	0
$591$	3.53516	0.145417
$592$	0	0
$593$	38.5015	1.58107	0.790534	−	0.612418i	$-0.209803\pi$
$593$	38.5015	1.58107	0.790534	+	0.612418i	$0.209803\pi$
$594$	0	0
$595$	−13.3719	−0.548194
$596$	0	0
$597$	−60.3546	−2.47015
$598$	0	0
$599$	25.3020	1.03381	0.516905	−	0.856043i	$-0.327084\pi$
$599$	25.3020	1.03381	0.516905	+	0.856043i	$0.327084\pi$
$600$	0	0
$601$	−13.8150	−0.563527	−0.281763	−	0.959484i	$-0.590919\pi$
$601$	−13.8150	−0.563527	−0.281763	+	0.959484i	$0.590919\pi$
$602$	0	0
$603$	47.3448	1.92803
$604$	0	0
$605$	3.08345	0.125360
$606$	0	0
$607$	−25.6662	−1.04176	−0.520879	−	0.853631i	$-0.674396\pi$
$607$	−25.6662	−1.04176	−0.520879	+	0.853631i	$0.674396\pi$
$608$	0	0
$609$	68.3442	2.76945
$610$	0	0
$611$	0.592404	0.0239661
$612$	0	0
$613$	30.6385	1.23748	0.618738	−	0.785597i	$-0.287644\pi$
$613$	30.6385	1.23748	0.618738	+	0.785597i	$0.287644\pi$
$614$	0	0
$615$	−6.77132	−0.273046
$616$	0	0
$617$	−6.40157	−0.257718	−0.128859	−	0.991663i	$-0.541131\pi$
$617$	−6.40157	−0.257718	−0.128859	+	0.991663i	$0.541131\pi$
$618$	0	0
$619$	−11.3704	−0.457014	−0.228507	−	0.973542i	$-0.573384\pi$
$619$	−11.3704	−0.457014	−0.228507	+	0.973542i	$0.573384\pi$
$620$	0	0
$621$	−30.1034	−1.20801
$622$	0	0
$623$	50.0961	2.00706
$624$	0	0
$625$	19.8632	0.794526
$626$	0	0
$627$	6.54631	0.261434
$628$	0	0
$629$	−36.4776	−1.45446
$630$	0	0
$631$	31.5680	1.25670	0.628352	−	0.777929i	$-0.283730\pi$
$631$	31.5680	1.25670	0.628352	+	0.777929i	$0.283730\pi$
$632$	0	0
$633$	−58.0560	−2.30752
$634$	0	0
$635$	0.245609	0.00974668
$636$	0	0
$637$	6.14890	0.243628
$638$	0	0
$639$	−42.5481	−1.68318
$640$	0	0
$641$	30.0046	1.18511	0.592556	−	0.805529i	$-0.298119\pi$
$641$	30.0046	1.18511	0.592556	+	0.805529i	$0.298119\pi$
$642$	0	0
$643$	7.71725	0.304339	0.152169	−	0.988354i	$-0.451374\pi$
$643$	7.71725	0.304339	0.152169	+	0.988354i	$0.451374\pi$
$644$	0	0
$645$	8.69391	0.342322
$646$	0	0
$647$	−9.12449	−0.358721	−0.179360	−	0.983783i	$-0.557403\pi$
$647$	−9.12449	−0.358721	−0.179360	+	0.983783i	$0.557403\pi$
$648$	0	0
$649$	−8.69336	−0.341244
$650$	0	0
$651$	1.42205	0.0557346
$652$	0	0
$653$	26.6714	1.04373	0.521866	−	0.853028i	$-0.325236\pi$
$653$	26.6714	1.04373	0.521866	+	0.853028i	$0.325236\pi$
$654$	0	0
$655$	−5.49162	−0.214576
$656$	0	0
$657$	29.1476	1.13716
$658$	0	0
$659$	30.8681	1.20245	0.601225	−	0.799080i	$-0.294680\pi$
$659$	30.8681	1.20245	0.601225	+	0.799080i	$0.294680\pi$
$660$	0	0
$661$	−35.0682	−1.36399	−0.681997	−	0.731355i	$-0.738888\pi$
$661$	−35.0682	−1.36399	−0.681997	+	0.731355i	$0.738888\pi$
$662$	0	0
$663$	−76.4296	−2.96828
$664$	0	0
$665$	1.74435	0.0676431
$666$	0	0
$667$	67.5463	2.61540
$668$	0	0
$669$	25.1997	0.974275
$670$	0	0
$671$	24.8138	0.957925
$672$	0	0
$673$	27.1644	1.04711	0.523556	−	0.851991i	$-0.324605\pi$
$673$	27.1644	1.04711	0.523556	+	0.851991i	$0.324605\pi$
$674$	0	0
$675$	−17.6752	−0.680317
$676$	0	0
$677$	36.2990	1.39508	0.697541	−	0.716544i	$-0.254277\pi$
$677$	36.2990	1.39508	0.697541	+	0.716544i	$0.254277\pi$
$678$	0	0
$679$	−41.3040	−1.58510
$680$	0	0
$681$	−2.28500	−0.0875615
$682$	0	0
$683$	37.9871	1.45354	0.726768	−	0.686883i	$-0.241022\pi$
$683$	37.9871	1.45354	0.726768	+	0.686883i	$0.241022\pi$
$684$	0	0
$685$	−1.22742	−0.0468973
$686$	0	0
$687$	−39.7335	−1.51593
$688$	0	0
$689$	−41.4767	−1.58014
$690$	0	0
$691$	6.16824	0.234651	0.117325	−	0.993094i	$-0.462568\pi$
$691$	6.16824	0.234651	0.117325	+	0.993094i	$0.462568\pi$
$692$	0	0
$693$	−31.1796	−1.18442
$694$	0	0
$695$	−9.60240	−0.364240
$696$	0	0
$697$	−32.2288	−1.22075
$698$	0	0
$699$	−0.497885	−0.0188317
$700$	0	0
$701$	−4.13482	−0.156170	−0.0780851	−	0.996947i	$-0.524881\pi$
$701$	−4.13482	−0.156170	−0.0780851	+	0.996947i	$0.524881\pi$
$702$	0	0
$703$	4.75848	0.179469
$704$	0	0
$705$	−0.260286	−0.00980295
$706$	0	0
$707$	0.766256	0.0288180
$708$	0	0
$709$	−2.21199	−0.0830730	−0.0415365	−	0.999137i	$-0.513225\pi$
$709$	−2.21199	−0.0830730	−0.0415365	+	0.999137i	$0.513225\pi$
$710$	0	0
$711$	−4.39773	−0.164928
$712$	0	0
$713$	1.40545	0.0526345
$714$	0	0
$715$	−5.22445	−0.195383
$716$	0	0
$717$	−38.7356	−1.44661
$718$	0	0
$719$	18.4892	0.689532	0.344766	−	0.938689i	$-0.387958\pi$
$719$	18.4892	0.689532	0.344766	+	0.938689i	$0.387958\pi$
$720$	0	0
$721$	42.1440	1.56952
$722$	0	0
$723$	−60.7191	−2.25817
$724$	0	0
$725$	39.6597	1.47292
$726$	0	0
$727$	−18.0772	−0.670447	−0.335224	−	0.942139i	$-0.608812\pi$
$727$	−18.0772	−0.670447	−0.335224	+	0.942139i	$0.608812\pi$
$728$	0	0
$729$	−43.5675	−1.61361
$730$	0	0
$731$	41.3795	1.53048
$732$	0	0
$733$	−10.5169	−0.388452	−0.194226	−	0.980957i	$-0.562220\pi$
$733$	−10.5169	−0.388452	−0.194226	+	0.980957i	$0.562220\pi$
$734$	0	0
$735$	−2.70167	−0.0996524
$736$	0	0
$737$	25.9114	0.954460
$738$	0	0
$739$	−21.8505	−0.803784	−0.401892	−	0.915687i	$-0.631647\pi$
$739$	−21.8505	−0.803784	−0.401892	+	0.915687i	$0.631647\pi$
$740$	0	0
$741$	9.97019	0.366264
$742$	0	0
$743$	44.1742	1.62059	0.810297	−	0.586019i	$-0.199306\pi$
$743$	44.1742	1.62059	0.810297	+	0.586019i	$0.199306\pi$
$744$	0	0
$745$	1.84898	0.0677414
$746$	0	0
$747$	−18.7406	−0.685682
$748$	0	0
$749$	−9.34997	−0.341641
$750$	0	0
$751$	−22.2521	−0.811991	−0.405995	−	0.913875i	$-0.633075\pi$
$751$	−22.2521	−0.811991	−0.405995	+	0.913875i	$0.633075\pi$
$752$	0	0
$753$	−66.2491	−2.41425
$754$	0	0
$755$	11.8323	0.430623
$756$	0	0
$757$	−15.9029	−0.578002	−0.289001	−	0.957329i	$-0.593323\pi$
$757$	−15.9029	−0.578002	−0.289001	+	0.957329i	$0.593323\pi$
$758$	0	0
$759$	−51.8371	−1.88157
$760$	0	0
$761$	−6.41785	−0.232647	−0.116323	−	0.993211i	$-0.537111\pi$
$761$	−6.41785	−0.232647	−0.116323	+	0.993211i	$0.537111\pi$
$762$	0	0
$763$	−26.2956	−0.951965
$764$	0	0
$765$	19.9630	0.721763
$766$	0	0
$767$	−13.2402	−0.478076
$768$	0	0
$769$	−21.1031	−0.760999	−0.380499	−	0.924781i	$-0.624248\pi$
$769$	−21.1031	−0.760999	−0.380499	+	0.924781i	$0.624248\pi$
$770$	0	0
$771$	8.42249	0.303329
$772$	0	0
$773$	−18.1093	−0.651348	−0.325674	−	0.945482i	$-0.605591\pi$
$773$	−18.1093	−0.651348	−0.325674	+	0.945482i	$0.605591\pi$
$774$	0	0
$775$	0.825206	0.0296423
$776$	0	0
$777$	−38.1253	−1.36774
$778$	0	0
$779$	4.20422	0.150632
$780$	0	0
$781$	−23.2862	−0.833247
$782$	0	0
$783$	−32.4286	−1.15891
$784$	0	0
$785$	12.8280	0.457850
$786$	0	0
$787$	41.2633	1.47088	0.735440	−	0.677590i	$-0.236976\pi$
$787$	41.2633	1.47088	0.735440	+	0.677590i	$0.236976\pi$
$788$	0	0
$789$	−30.7519	−1.09480
$790$	0	0
$791$	10.7784	0.383236
$792$	0	0
$793$	37.7920	1.34203
$794$	0	0
$795$	18.2238	0.646330
$796$	0	0
$797$	37.4139	1.32527	0.662635	−	0.748943i	$-0.269438\pi$
$797$	37.4139	1.32527	0.662635	+	0.748943i	$0.269438\pi$
$798$	0	0
$799$	−1.23886	−0.0438277
$800$	0	0
$801$	−74.7888	−2.64253
$802$	0	0
$803$	15.9523	0.562943
$804$	0	0
$805$	−13.8127	−0.486834
$806$	0	0
$807$	35.4579	1.24818
$808$	0	0
$809$	−56.7517	−1.99528	−0.997642	−	0.0686296i	$-0.978137\pi$
$809$	−56.7517	−1.99528	−0.997642	+	0.0686296i	$0.978137\pi$
$810$	0	0
$811$	11.8384	0.415702	0.207851	−	0.978161i	$-0.433353\pi$
$811$	11.8384	0.415702	0.207851	+	0.978161i	$0.433353\pi$
$812$	0	0
$813$	−26.2040	−0.919016
$814$	0	0
$815$	−3.63245	−0.127239
$816$	0	0
$817$	−5.39793	−0.188850
$818$	0	0
$819$	−47.4873	−1.65934
$820$	0	0
$821$	−21.1755	−0.739031	−0.369515	−	0.929225i	$-0.620476\pi$
$821$	−21.1755	−0.739031	−0.369515	+	0.929225i	$0.620476\pi$
$822$	0	0
$823$	9.02961	0.314752	0.157376	−	0.987539i	$-0.449696\pi$
$823$	9.02961	0.314752	0.157376	+	0.987539i	$0.449696\pi$
$824$	0	0
$825$	−30.4361	−1.05965
$826$	0	0
$827$	8.40374	0.292227	0.146113	−	0.989268i	$-0.453324\pi$
$827$	8.40374	0.292227	0.146113	+	0.989268i	$0.453324\pi$
$828$	0	0
$829$	37.3540	1.29736	0.648679	−	0.761062i	$-0.275322\pi$
$829$	37.3540	1.29736	0.648679	+	0.761062i	$0.275322\pi$
$830$	0	0
$831$	−0.812691	−0.0281919
$832$	0	0
$833$	−12.8588	−0.445532
$834$	0	0
$835$	−13.5312	−0.468265
$836$	0	0
$837$	−0.674749	−0.0233227
$838$	0	0
$839$	−9.96301	−0.343961	−0.171981	−	0.985100i	$-0.555017\pi$
$839$	−9.96301	−0.343961	−0.171981	+	0.985100i	$0.555017\pi$
$840$	0	0
$841$	43.7637	1.50909
$842$	0	0
$843$	15.6131	0.537745
$844$	0	0
$845$	−0.258883	−0.00890586
$846$	0	0
$847$	15.3389	0.527050
$848$	0	0
$849$	68.4107	2.34785
$850$	0	0
$851$	−37.6802	−1.29166
$852$	0	0
$853$	−45.2551	−1.54951	−0.774753	−	0.632264i	$-0.782126\pi$
$853$	−45.2551	−1.54951	−0.774753	+	0.632264i	$0.782126\pi$
$854$	0	0
$855$	−2.60416	−0.0890603
$856$	0	0
$857$	−31.0656	−1.06118	−0.530590	−	0.847629i	$-0.678030\pi$
$857$	−31.0656	−1.06118	−0.530590	+	0.847629i	$0.678030\pi$
$858$	0	0
$859$	35.6714	1.21709	0.608546	−	0.793519i	$-0.291753\pi$
$859$	35.6714	1.21709	0.608546	+	0.793519i	$0.291753\pi$
$860$	0	0
$861$	−33.6845	−1.14796
$862$	0	0
$863$	−48.4201	−1.64824	−0.824120	−	0.566415i	$-0.808330\pi$
$863$	−48.4201	−1.64824	−0.824120	+	0.566415i	$0.808330\pi$
$864$	0	0
$865$	13.2217	0.449552
$866$	0	0
$867$	113.595	3.85788
$868$	0	0
$869$	−2.40684	−0.0816465
$870$	0	0
$871$	39.4637	1.33718
$872$	0	0
$873$	61.6631	2.08698
$874$	0	0
$875$	−16.8319	−0.569021
$876$	0	0
$877$	36.4341	1.23029	0.615146	−	0.788413i	$-0.289097\pi$
$877$	36.4341	1.23029	0.615146	+	0.788413i	$0.289097\pi$
$878$	0	0
$879$	−61.7361	−2.08231
$880$	0	0
$881$	31.9437	1.07621	0.538106	−	0.842877i	$-0.319140\pi$
$881$	31.9437	1.07621	0.538106	+	0.842877i	$0.319140\pi$
$882$	0	0
$883$	36.3025	1.22168	0.610838	−	0.791756i	$-0.290833\pi$
$883$	36.3025	1.22168	0.610838	+	0.791756i	$0.290833\pi$
$884$	0	0
$885$	5.81740	0.195550
$886$	0	0
$887$	−19.6571	−0.660022	−0.330011	−	0.943977i	$-0.607052\pi$
$887$	−19.6571	−0.660022	−0.330011	+	0.943977i	$0.607052\pi$
$888$	0	0
$889$	1.22180	0.0409779
$890$	0	0
$891$	−6.86714	−0.230058
$892$	0	0
$893$	0.161608	0.00540801
$894$	0	0
$895$	−12.1244	−0.405275
$896$	0	0
$897$	−78.9492	−2.63604
$898$	0	0
$899$	1.51401	0.0504950
$900$	0	0
$901$	86.7377	2.88965
$902$	0	0
$903$	43.2486	1.43922
$904$	0	0
$905$	−15.0363	−0.499824
$906$	0	0
$907$	12.0743	0.400921	0.200461	−	0.979702i	$-0.435756\pi$
$907$	12.0743	0.400921	0.200461	+	0.979702i	$0.435756\pi$
$908$	0	0
$909$	−1.14395	−0.0379424
$910$	0	0
$911$	42.3744	1.40393	0.701963	−	0.712213i	$-0.252307\pi$
$911$	42.3744	1.40393	0.701963	+	0.712213i	$0.252307\pi$
$912$	0	0
$913$	−10.2566	−0.339443
$914$	0	0
$915$	−16.6048	−0.548937
$916$	0	0
$917$	−27.3185	−0.902138
$918$	0	0
$919$	−38.5524	−1.27172	−0.635862	−	0.771802i	$-0.719355\pi$
$919$	−38.5524	−1.27172	−0.635862	+	0.771802i	$0.719355\pi$
$920$	0	0
$921$	68.3462	2.25208
$922$	0	0
$923$	−35.4655	−1.16736
$924$	0	0
$925$	−22.1238	−0.727427
$926$	0	0
$927$	−62.9171	−2.06647
$928$	0	0
$929$	−0.245863	−0.00806651	−0.00403326	−	0.999992i	$-0.501284\pi$
$929$	−0.245863	−0.00806651	−0.00403326	+	0.999992i	$0.501284\pi$
$930$	0	0
$931$	1.67743	0.0549754
$932$	0	0
$933$	−71.1220	−2.32843
$934$	0	0
$935$	10.9256	0.357305
$936$	0	0
$937$	−29.6860	−0.969798	−0.484899	−	0.874570i	$-0.661144\pi$
$937$	−29.6860	−0.969798	−0.484899	+	0.874570i	$0.661144\pi$
$938$	0	0
$939$	1.00626	0.0328380
$940$	0	0
$941$	−22.7390	−0.741270	−0.370635	−	0.928779i	$-0.620860\pi$
$941$	−22.7390	−0.741270	−0.370635	+	0.928779i	$0.620860\pi$
$942$	0	0
$943$	−33.2912	−1.08411
$944$	0	0
$945$	6.63141	0.215720
$946$	0	0
$947$	−31.4595	−1.02230	−0.511149	−	0.859492i	$-0.670780\pi$
$947$	−31.4595	−1.02230	−0.511149	+	0.859492i	$0.670780\pi$
$948$	0	0
$949$	24.2957	0.788671
$950$	0	0
$951$	79.2673	2.57042
$952$	0	0
$953$	−11.8602	−0.384191	−0.192095	−	0.981376i	$-0.561528\pi$
$953$	−11.8602	−0.384191	−0.192095	+	0.981376i	$0.561528\pi$
$954$	0	0
$955$	−2.10872	−0.0682365
$956$	0	0
$957$	−55.8411	−1.80509
$958$	0	0
$959$	−6.10591	−0.197170
$960$	0	0
$961$	−30.9685	−0.998984
$962$	0	0
$963$	13.9586	0.449811
$964$	0	0
$965$	−6.08894	−0.196010
$966$	0	0
$967$	55.9859	1.80038	0.900192	−	0.435494i	$-0.143426\pi$
$967$	55.9859	1.80038	0.900192	+	0.435494i	$0.143426\pi$
$968$	0	0
$969$	−20.8501	−0.669800
$970$	0	0
$971$	−22.5950	−0.725108	−0.362554	−	0.931963i	$-0.618095\pi$
$971$	−22.5950	−0.725108	−0.362554	+	0.931963i	$0.618095\pi$
$972$	0	0
$973$	−47.7679	−1.53137
$974$	0	0
$975$	−46.3549	−1.48454
$976$	0	0
$977$	20.4344	0.653753	0.326876	−	0.945067i	$-0.394004\pi$
$977$	20.4344	0.653753	0.326876	+	0.945067i	$0.394004\pi$
$978$	0	0
$979$	−40.9313	−1.30817
$980$	0	0
$981$	39.2569	1.25338
$982$	0	0
$983$	45.8372	1.46198	0.730990	−	0.682388i	$-0.239058\pi$
$983$	45.8372	1.46198	0.730990	+	0.682388i	$0.239058\pi$
$984$	0	0
$985$	−0.769660	−0.0245234
$986$	0	0
$987$	−1.29482	−0.0412145
$988$	0	0
$989$	42.7437	1.35917
$990$	0	0
$991$	16.4265	0.521804	0.260902	−	0.965365i	$-0.415980\pi$
$991$	16.4265	0.521804	0.260902	+	0.965365i	$0.415980\pi$
$992$	0	0
$993$	86.3800	2.74119
$994$	0	0
$995$	13.1401	0.416570
$996$	0	0
$997$	23.1395	0.732836	0.366418	−	0.930450i	$-0.380584\pi$
$997$	23.1395	0.732836	0.366418	+	0.930450i	$0.380584\pi$
$998$	0	0
$999$	18.0900	0.572344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.26	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.26	✓	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)

\(f(q)\)	\(=\)	\(q+2.71988 q^{3} -0.592160 q^{5} -2.94575 q^{7} +4.39773 q^{9} +O(q^{10})\)	\(q+2.71988 q^{3} -0.592160 q^{5} -2.94575 q^{7} +4.39773 q^{9} +2.40684 q^{11} +3.66568 q^{13} -1.61060 q^{15} -7.66581 q^{17} +1.00000 q^{19} -8.01207 q^{21} -7.91853 q^{23} -4.64935 q^{25} +3.80164 q^{27} -8.53016 q^{29} -0.177489 q^{31} +6.54631 q^{33} +1.74435 q^{35} +4.75848 q^{37} +9.97019 q^{39} +4.20422 q^{41} -5.39793 q^{43} -2.60416 q^{45} +0.161608 q^{47} +1.67743 q^{49} -20.8501 q^{51} -11.3149 q^{53} -1.42523 q^{55} +2.71988 q^{57} -3.61194 q^{59} +10.3097 q^{61} -12.9546 q^{63} -2.17067 q^{65} +10.7657 q^{67} -21.5374 q^{69} -9.67502 q^{71} +6.62788 q^{73} -12.6456 q^{75} -7.08994 q^{77} -1.00000 q^{79} -2.85318 q^{81} -4.26142 q^{83} +4.53939 q^{85} -23.2010 q^{87} -17.0062 q^{89} -10.7982 q^{91} -0.482747 q^{93} -0.592160 q^{95} +14.0216 q^{97} +10.5846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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