LMFDB - Embedded newform 6021.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.0779270570$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.28825$ of defining polynomial
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +0.874032 q^{5} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +0.874032 q^{5} +1.00000 q^{7} +2.82843 q^{8} -1.23607 q^{10} +1.74806 q^{11} +1.00000 q^{13} -1.41421 q^{14} -4.00000 q^{16} -0.333851 q^{17} +6.70820 q^{19} -2.47214 q^{22} -1.20788 q^{23} -4.23607 q^{25} -1.41421 q^{26} -2.62210 q^{29} -2.76393 q^{31} +0.472136 q^{34} +0.874032 q^{35} -9.00000 q^{37} -9.48683 q^{38} +2.47214 q^{40} -5.11667 q^{41} -0.472136 q^{43} +1.70820 q^{46} -6.73722 q^{47} -6.00000 q^{49} +5.99070 q^{50} +10.7735 q^{53} +1.52786 q^{55} +2.82843 q^{56} +3.70820 q^{58} -5.24419 q^{59} -10.7082 q^{61} +3.90879 q^{62} +8.00000 q^{64} +0.874032 q^{65} -9.00000 q^{67} -1.23607 q^{70} -6.65841 q^{71} -9.18034 q^{73} +12.7279 q^{74} +1.74806 q^{77} +10.2361 q^{79} -3.49613 q^{80} +7.23607 q^{82} +4.37016 q^{83} -0.291796 q^{85} +0.667701 q^{86} +4.94427 q^{88} -1.74806 q^{89} +1.00000 q^{91} +9.52786 q^{94} +5.86319 q^{95} -10.7082 q^{97} +8.48528 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 4 q^{10} + 4 q^{13} - 16 q^{16} + 8 q^{22} - 8 q^{25} - 20 q^{31} - 16 q^{34} - 36 q^{37} - 8 q^{40} + 16 q^{43} - 20 q^{46} - 24 q^{49} + 24 q^{55} - 12 q^{58} - 16 q^{61} + 32 q^{64} - 36 q^{67} + 4 q^{70} + 8 q^{73} + 32 q^{79} + 20 q^{82} - 28 q^{85} - 16 q^{88} + 4 q^{91} + 56 q^{94} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 4 * q^10 + 4 * q^13 - 16 * q^16 + 8 * q^22 - 8 * q^25 - 20 * q^31 - 16 * q^34 - 36 * q^37 - 8 * q^40 + 16 * q^43 - 20 * q^46 - 24 * q^49 + 24 * q^55 - 12 * q^58 - 16 * q^61 + 32 * q^64 - 36 * q^67 + 4 * q^70 + 8 * q^73 + 32 * q^79 + 20 * q^82 - 28 * q^85 - 16 * q^88 + 4 * q^91 + 56 * q^94 - 16 * q^97

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.874032	0.390879	0.195440	−	0.980716i	$-0.437387\pi$
$5$	0.874032	0.390879	0.195440	+	0.980716i	$0.437387\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	2.82843	1.00000
$9$	0	0
$10$	−1.23607	−0.390879
$11$	1.74806	0.527061	0.263531	−	0.964651i	$-0.415113\pi$
$11$	1.74806	0.527061	0.263531	+	0.964651i	$0.415113\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.41421	−0.377964
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−0.333851	−0.0809706	−0.0404853	−	0.999180i	$-0.512890\pi$
$17$	−0.333851	−0.0809706	−0.0404853	+	0.999180i	$0.512890\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	0	0
$22$	−2.47214	−0.527061
$23$	−1.20788	−0.251861	−0.125930	−	0.992039i	$-0.540192\pi$
$23$	−1.20788	−0.251861	−0.125930	+	0.992039i	$0.540192\pi$
$24$	0	0
$25$	−4.23607	−0.847214
$26$	−1.41421	−0.277350
$27$	0	0
$28$	0	0
$29$	−2.62210	−0.486911	−0.243456	−	0.969912i	$-0.578281\pi$
$29$	−2.62210	−0.486911	−0.243456	+	0.969912i	$0.578281\pi$
$30$	0	0
$31$	−2.76393	−0.496417	−0.248208	−	0.968707i	$-0.579842\pi$
$31$	−2.76393	−0.496417	−0.248208	+	0.968707i	$0.579842\pi$
$32$	0	0
$33$	0	0
$34$	0.472136	0.0809706
$35$	0.874032	0.147738
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	−9.48683	−1.53897
$39$	0	0
$40$	2.47214	0.390879
$41$	−5.11667	−0.799090	−0.399545	−	0.916714i	$-0.630832\pi$
$41$	−5.11667	−0.799090	−0.399545	+	0.916714i	$0.630832\pi$
$42$	0	0
$43$	−0.472136	−0.0720001	−0.0360000	−	0.999352i	$-0.511462\pi$
$43$	−0.472136	−0.0720001	−0.0360000	+	0.999352i	$0.511462\pi$
$44$	0	0
$45$	0	0
$46$	1.70820	0.251861
$47$	−6.73722	−0.982724	−0.491362	−	0.870955i	$-0.663501\pi$
$47$	−6.73722	−0.982724	−0.491362	+	0.870955i	$0.663501\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	5.99070	0.847214
$51$	0	0
$52$	0	0
$53$	10.7735	1.47986	0.739929	−	0.672685i	$-0.234859\pi$
$53$	10.7735	1.47986	0.739929	+	0.672685i	$0.234859\pi$
$54$	0	0
$55$	1.52786	0.206017
$56$	2.82843	0.377964
$57$	0	0
$58$	3.70820	0.486911
$59$	−5.24419	−0.682736	−0.341368	−	0.939930i	$-0.610890\pi$
$59$	−5.24419	−0.682736	−0.341368	+	0.939930i	$0.610890\pi$
$60$	0	0
$61$	−10.7082	−1.37105	−0.685523	−	0.728051i	$-0.740426\pi$
$61$	−10.7082	−1.37105	−0.685523	+	0.728051i	$0.740426\pi$
$62$	3.90879	0.496417
$63$	0	0
$64$	8.00000	1.00000
$65$	0.874032	0.108410
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	0	0
$70$	−1.23607	−0.147738
$71$	−6.65841	−0.790207	−0.395104	−	0.918637i	$-0.629291\pi$
$71$	−6.65841	−0.790207	−0.395104	+	0.918637i	$0.629291\pi$
$72$	0	0
$73$	−9.18034	−1.07448	−0.537239	−	0.843430i	$-0.680533\pi$
$73$	−9.18034	−1.07448	−0.537239	+	0.843430i	$0.680533\pi$
$74$	12.7279	1.47959
$75$	0	0
$76$	0	0
$77$	1.74806	0.199210
$78$	0	0
$79$	10.2361	1.15165	0.575824	−	0.817574i	$-0.304681\pi$
$79$	10.2361	1.15165	0.575824	+	0.817574i	$0.304681\pi$
$80$	−3.49613	−0.390879
$81$	0	0
$82$	7.23607	0.799090
$83$	4.37016	0.479687	0.239844	−	0.970812i	$-0.422904\pi$
$83$	4.37016	0.479687	0.239844	+	0.970812i	$0.422904\pi$
$84$	0	0
$85$	−0.291796	−0.0316497
$86$	0.667701	0.0720001
$87$	0	0
$88$	4.94427	0.527061
$89$	−1.74806	−0.185294	−0.0926472	−	0.995699i	$-0.529533\pi$
$89$	−1.74806	−0.185294	−0.0926472	+	0.995699i	$0.529533\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	9.52786	0.982724
$95$	5.86319	0.601550
$96$	0	0
$97$	−10.7082	−1.08725	−0.543627	−	0.839327i	$-0.682949\pi$
$97$	−10.7082	−1.08725	−0.543627	+	0.839327i	$0.682949\pi$
$98$	8.48528	0.857143
$99$	0	0
$100$	0	0
$101$	14.9374	1.48632	0.743161	−	0.669112i	$-0.233326\pi$
$101$	14.9374	1.48632	0.743161	+	0.669112i	$0.233326\pi$
$102$	0	0
$103$	−3.47214	−0.342120	−0.171060	−	0.985261i	$-0.554719\pi$
$103$	−3.47214	−0.342120	−0.171060	+	0.985261i	$0.554719\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−15.2361	−1.47986
$107$	0.874032	0.0844959	0.0422479	−	0.999107i	$-0.486548\pi$
$107$	0.874032	0.0844959	0.0422479	+	0.999107i	$0.486548\pi$
$108$	0	0
$109$	−13.4164	−1.28506	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164	−1.28506	−0.642529	+	0.766261i	$0.722115\pi$
$110$	−2.16073	−0.206017
$111$	0	0
$112$	−4.00000	−0.377964
$113$	5.65685	0.532152	0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685	0.532152	0.266076	+	0.963952i	$0.414273\pi$
$114$	0	0
$115$	−1.05573	−0.0984472
$116$	0	0
$117$	0	0
$118$	7.41641	0.682736
$119$	−0.333851	−0.0306040
$120$	0	0
$121$	−7.94427	−0.722207
$122$	15.1437	1.37105
$123$	0	0
$124$	0	0
$125$	−8.07262	−0.722037
$126$	0	0
$127$	−9.23607	−0.819569	−0.409784	−	0.912182i	$-0.634396\pi$
$127$	−9.23607	−0.819569	−0.409784	+	0.912182i	$0.634396\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	−1.23607	−0.108410
$131$	0.127520	0.0111414	0.00557072	−	0.999984i	$-0.498227\pi$
$131$	0.127520	0.0111414	0.00557072	+	0.999984i	$0.498227\pi$
$132$	0	0
$133$	6.70820	0.581675
$134$	12.7279	1.09952
$135$	0	0
$136$	−0.944272	−0.0809706
$137$	0.333851	0.0285228	0.0142614	−	0.999898i	$-0.495460\pi$
$137$	0.333851	0.0285228	0.0142614	+	0.999898i	$0.495460\pi$
$138$	0	0
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	0	0
$141$	0	0
$142$	9.41641	0.790207
$143$	1.74806	0.146180
$144$	0	0
$145$	−2.29180	−0.190323
$146$	12.9830	1.07448
$147$	0	0
$148$	0	0
$149$	−2.28825	−0.187460	−0.0937302	−	0.995598i	$-0.529879\pi$
$149$	−2.28825	−0.187460	−0.0937302	+	0.995598i	$0.529879\pi$
$150$	0	0
$151$	−15.9443	−1.29753	−0.648763	−	0.760990i	$-0.724713\pi$
$151$	−15.9443	−1.29753	−0.648763	+	0.760990i	$0.724713\pi$
$152$	18.9737	1.53897
$153$	0	0
$154$	−2.47214	−0.199210
$155$	−2.41577	−0.194039
$156$	0	0
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	−14.4760	−1.15165
$159$	0	0
$160$	0	0
$161$	−1.20788	−0.0951945
$162$	0	0
$163$	12.7082	0.995383	0.497692	−	0.867354i	$-0.334181\pi$
$163$	12.7082	0.995383	0.497692	+	0.867354i	$0.334181\pi$
$164$	0	0
$165$	0	0
$166$	−6.18034	−0.479687
$167$	2.62210	0.202904	0.101452	−	0.994840i	$-0.467651\pi$
$167$	2.62210	0.202904	0.101452	+	0.994840i	$0.467651\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0.412662	0.0316497
$171$	0	0
$172$	0	0
$173$	−15.5563	−1.18273	−0.591364	−	0.806405i	$-0.701410\pi$
$173$	−15.5563	−1.18273	−0.591364	+	0.806405i	$0.701410\pi$
$174$	0	0
$175$	−4.23607	−0.320217
$176$	−6.99226	−0.527061
$177$	0	0
$178$	2.47214	0.185294
$179$	23.0888	1.72574	0.862868	−	0.505429i	$-0.168666\pi$
$179$	23.0888	1.72574	0.862868	+	0.505429i	$0.168666\pi$
$180$	0	0
$181$	−19.6525	−1.46076	−0.730379	−	0.683043i	$-0.760656\pi$
$181$	−19.6525	−1.46076	−0.730379	+	0.683043i	$0.760656\pi$
$182$	−1.41421	−0.104828
$183$	0	0
$184$	−3.41641	−0.251861
$185$	−7.86629	−0.578341
$186$	0	0
$187$	−0.583592	−0.0426765
$188$	0	0
$189$	0	0
$190$	−8.29180	−0.601550
$191$	8.81913	0.638130	0.319065	−	0.947733i	$-0.396631\pi$
$191$	8.81913	0.638130	0.319065	+	0.947733i	$0.396631\pi$
$192$	0	0
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	15.1437	1.08725
$195$	0	0
$196$	0	0
$197$	19.7990	1.41062	0.705310	−	0.708899i	$-0.250808\pi$
$197$	19.7990	1.41062	0.705310	+	0.708899i	$0.250808\pi$
$198$	0	0
$199$	−6.23607	−0.442063	−0.221032	−	0.975267i	$-0.570942\pi$
$199$	−6.23607	−0.442063	−0.221032	+	0.975267i	$0.570942\pi$
$200$	−11.9814	−0.847214
$201$	0	0
$202$	−21.1246	−1.48632
$203$	−2.62210	−0.184035
$204$	0	0
$205$	−4.47214	−0.312348
$206$	4.91034	0.342120
$207$	0	0
$208$	−4.00000	−0.277350
$209$	11.7264	0.811130
$210$	0	0
$211$	−1.76393	−0.121434	−0.0607170	−	0.998155i	$-0.519339\pi$
$211$	−1.76393	−0.121434	−0.0607170	+	0.998155i	$0.519339\pi$
$212$	0	0
$213$	0	0
$214$	−1.23607	−0.0844959
$215$	−0.412662	−0.0281433
$216$	0	0
$217$	−2.76393	−0.187628
$218$	18.9737	1.28506
$219$	0	0
$220$	0	0
$221$	−0.333851	−0.0224572
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	−8.00000	−0.532152
$227$	−22.4698	−1.49137	−0.745686	−	0.666297i	$-0.767878\pi$
$227$	−22.4698	−1.49137	−0.745686	+	0.666297i	$0.767878\pi$
$228$	0	0
$229$	2.47214	0.163363	0.0816817	−	0.996658i	$-0.473971\pi$
$229$	2.47214	0.163363	0.0816817	+	0.996658i	$0.473971\pi$
$230$	1.49302	0.0984472
$231$	0	0
$232$	−7.41641	−0.486911
$233$	20.0053	1.31059	0.655296	−	0.755372i	$-0.272544\pi$
$233$	20.0053	1.31059	0.655296	+	0.755372i	$0.272544\pi$
$234$	0	0
$235$	−5.88854	−0.384126
$236$	0	0
$237$	0	0
$238$	0.472136	0.0306040
$239$	−12.3941	−0.801706	−0.400853	−	0.916142i	$-0.631286\pi$
$239$	−12.3941	−0.801706	−0.400853	+	0.916142i	$0.631286\pi$
$240$	0	0
$241$	1.58359	0.102008	0.0510041	−	0.998698i	$-0.483758\pi$
$241$	1.58359	0.102008	0.0510041	+	0.998698i	$0.483758\pi$
$242$	11.2349	0.722207
$243$	0	0
$244$	0	0
$245$	−5.24419	−0.335039
$246$	0	0
$247$	6.70820	0.426833
$248$	−7.81758	−0.496417
$249$	0	0
$250$	11.4164	0.722037
$251$	29.8747	1.88568	0.942838	−	0.333253i	$-0.108146\pi$
$251$	29.8747	1.88568	0.942838	+	0.333253i	$0.108146\pi$
$252$	0	0
$253$	−2.11146	−0.132746
$254$	13.0618	0.819569
$255$	0	0
$256$	0	0
$257$	−19.5927	−1.22216	−0.611078	−	0.791570i	$-0.709264\pi$
$257$	−19.5927	−1.22216	−0.611078	+	0.791570i	$0.709264\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	0	0
$262$	−0.180340	−0.0111414
$263$	30.2874	1.86760	0.933800	−	0.357796i	$-0.116472\pi$
$263$	30.2874	1.86760	0.933800	+	0.357796i	$0.116472\pi$
$264$	0	0
$265$	9.41641	0.578445
$266$	−9.48683	−0.581675
$267$	0	0
$268$	0	0
$269$	13.9358	0.849681	0.424841	−	0.905268i	$-0.360330\pi$
$269$	13.9358	0.849681	0.424841	+	0.905268i	$0.360330\pi$
$270$	0	0
$271$	22.8885	1.39038	0.695190	−	0.718826i	$-0.255320\pi$
$271$	22.8885	1.39038	0.695190	+	0.718826i	$0.255320\pi$
$272$	1.33540	0.0809706
$273$	0	0
$274$	−0.472136	−0.0285228
$275$	−7.40492	−0.446533
$276$	0	0
$277$	30.3607	1.82420	0.912098	−	0.409972i	$-0.134461\pi$
$277$	30.3607	1.82420	0.912098	+	0.409972i	$0.134461\pi$
$278$	−4.24264	−0.254457
$279$	0	0
$280$	2.47214	0.147738
$281$	3.03476	0.181038	0.0905192	−	0.995895i	$-0.471147\pi$
$281$	3.03476	0.181038	0.0905192	+	0.995895i	$0.471147\pi$
$282$	0	0
$283$	1.70820	0.101542	0.0507711	−	0.998710i	$-0.483832\pi$
$283$	1.70820	0.101542	0.0507711	+	0.998710i	$0.483832\pi$
$284$	0	0
$285$	0	0
$286$	−2.47214	−0.146180
$287$	−5.11667	−0.302028
$288$	0	0
$289$	−16.8885	−0.993444
$290$	3.24109	0.190323
$291$	0	0
$292$	0	0
$293$	−29.9535	−1.74990	−0.874952	−	0.484210i	$-0.839107\pi$
$293$	−29.9535	−1.74990	−0.874952	+	0.484210i	$0.839107\pi$
$294$	0	0
$295$	−4.58359	−0.266867
$296$	−25.4558	−1.47959
$297$	0	0
$298$	3.23607	0.187460
$299$	−1.20788	−0.0698537
$300$	0	0
$301$	−0.472136	−0.0272135
$302$	22.5486	1.29753
$303$	0	0
$304$	−26.8328	−1.53897
$305$	−9.35931	−0.535913
$306$	0	0
$307$	7.41641	0.423277	0.211638	−	0.977348i	$-0.432120\pi$
$307$	7.41641	0.423277	0.211638	+	0.977348i	$0.432120\pi$
$308$	0	0
$309$	0	0
$310$	3.41641	0.194039
$311$	−24.6305	−1.39667	−0.698334	−	0.715772i	$-0.746075\pi$
$311$	−24.6305	−1.39667	−0.698334	+	0.715772i	$0.746075\pi$
$312$	0	0
$313$	−13.2918	−0.751297	−0.375648	−	0.926762i	$-0.622580\pi$
$313$	−13.2918	−0.751297	−0.375648	+	0.926762i	$0.622580\pi$
$314$	−18.9737	−1.07075
$315$	0	0
$316$	0	0
$317$	−16.6367	−0.934411	−0.467205	−	0.884149i	$-0.654739\pi$
$317$	−16.6367	−0.934411	−0.467205	+	0.884149i	$0.654739\pi$
$318$	0	0
$319$	−4.58359	−0.256632
$320$	6.99226	0.390879
$321$	0	0
$322$	1.70820	0.0951945
$323$	−2.23954	−0.124611
$324$	0	0
$325$	−4.23607	−0.234975
$326$	−17.9721	−0.995383
$327$	0	0
$328$	−14.4721	−0.799090
$329$	−6.73722	−0.371435
$330$	0	0
$331$	−13.9443	−0.766447	−0.383223	−	0.923656i	$-0.625186\pi$
$331$	−13.9443	−0.766447	−0.383223	+	0.923656i	$0.625186\pi$
$332$	0	0
$333$	0	0
$334$	−3.70820	−0.202904
$335$	−7.86629	−0.429781
$336$	0	0
$337$	−12.2361	−0.666541	−0.333271	−	0.942831i	$-0.608152\pi$
$337$	−12.2361	−0.666541	−0.333271	+	0.942831i	$0.608152\pi$
$338$	16.9706	0.923077
$339$	0	0
$340$	0	0
$341$	−4.83153	−0.261642
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−1.33540	−0.0720001
$345$	0	0
$346$	22.0000	1.18273
$347$	30.9064	1.65914	0.829570	−	0.558402i	$-0.188585\pi$
$347$	30.9064	1.65914	0.829570	+	0.558402i	$0.188585\pi$
$348$	0	0
$349$	16.4164	0.878750	0.439375	−	0.898304i	$-0.355200\pi$
$349$	16.4164	0.878750	0.439375	+	0.898304i	$0.355200\pi$
$350$	5.99070	0.320217
$351$	0	0
$352$	0	0
$353$	−33.7348	−1.79552	−0.897761	−	0.440483i	$-0.854807\pi$
$353$	−33.7348	−1.79552	−0.897761	+	0.440483i	$0.854807\pi$
$354$	0	0
$355$	−5.81966	−0.308875
$356$	0	0
$357$	0	0
$358$	−32.6525	−1.72574
$359$	−7.61125	−0.401706	−0.200853	−	0.979621i	$-0.564371\pi$
$359$	−7.61125	−0.401706	−0.200853	+	0.979621i	$0.564371\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	27.7928	1.46076
$363$	0	0
$364$	0	0
$365$	−8.02391	−0.419991
$366$	0	0
$367$	−19.2918	−1.00702	−0.503512	−	0.863988i	$-0.667959\pi$
$367$	−19.2918	−1.00702	−0.503512	+	0.863988i	$0.667959\pi$
$368$	4.83153	0.251861
$369$	0	0
$370$	11.1246	0.578341
$371$	10.7735	0.559334
$372$	0	0
$373$	−8.70820	−0.450894	−0.225447	−	0.974255i	$-0.572384\pi$
$373$	−8.70820	−0.450894	−0.225447	+	0.974255i	$0.572384\pi$
$374$	0.825324	0.0426765
$375$	0	0
$376$	−19.0557	−0.982724
$377$	−2.62210	−0.135045
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	0	0
$382$	−12.4721	−0.638130
$383$	−25.1707	−1.28616	−0.643081	−	0.765798i	$-0.722344\pi$
$383$	−25.1707	−1.28616	−0.643081	+	0.765798i	$0.722344\pi$
$384$	0	0
$385$	1.52786	0.0778672
$386$	−12.7279	−0.647834
$387$	0	0
$388$	0	0
$389$	24.4242	1.23836	0.619178	−	0.785251i	$-0.287466\pi$
$389$	24.4242	1.23836	0.619178	+	0.785251i	$0.287466\pi$
$390$	0	0
$391$	0.403252	0.0203933
$392$	−16.9706	−0.857143
$393$	0	0
$394$	−28.0000	−1.41062
$395$	8.94665	0.450155
$396$	0	0
$397$	−2.58359	−0.129667	−0.0648334	−	0.997896i	$-0.520652\pi$
$397$	−2.58359	−0.129667	−0.0648334	+	0.997896i	$0.520652\pi$
$398$	8.81913	0.442063
$399$	0	0
$400$	16.9443	0.847214
$401$	−14.1421	−0.706225	−0.353112	−	0.935581i	$-0.614877\pi$
$401$	−14.1421	−0.706225	−0.353112	+	0.935581i	$0.614877\pi$
$402$	0	0
$403$	−2.76393	−0.137681
$404$	0	0
$405$	0	0
$406$	3.70820	0.184035
$407$	−15.7326	−0.779835
$408$	0	0
$409$	34.4164	1.70178	0.850891	−	0.525342i	$-0.176063\pi$
$409$	34.4164	1.70178	0.850891	+	0.525342i	$0.176063\pi$
$410$	6.32456	0.312348
$411$	0	0
$412$	0	0
$413$	−5.24419	−0.258050
$414$	0	0
$415$	3.81966	0.187500
$416$	0	0
$417$	0	0
$418$	−16.5836	−0.811130
$419$	−25.7109	−1.25606	−0.628029	−	0.778190i	$-0.716138\pi$
$419$	−25.7109	−1.25606	−0.628029	+	0.778190i	$0.716138\pi$
$420$	0	0
$421$	−34.1246	−1.66313	−0.831566	−	0.555426i	$-0.812555\pi$
$421$	−34.1246	−1.66313	−0.831566	+	0.555426i	$0.812555\pi$
$422$	2.49458	0.121434
$423$	0	0
$424$	30.4721	1.47986
$425$	1.41421	0.0685994
$426$	0	0
$427$	−10.7082	−0.518206
$428$	0	0
$429$	0	0
$430$	0.583592	0.0281433
$431$	0.874032	0.0421006	0.0210503	−	0.999778i	$-0.493299\pi$
$431$	0.874032	0.0421006	0.0210503	+	0.999778i	$0.493299\pi$
$432$	0	0
$433$	25.4164	1.22143	0.610717	−	0.791849i	$-0.290881\pi$
$433$	25.4164	1.22143	0.610717	+	0.791849i	$0.290881\pi$
$434$	3.90879	0.187628
$435$	0	0
$436$	0	0
$437$	−8.10272	−0.387606
$438$	0	0
$439$	22.1803	1.05861	0.529305	−	0.848432i	$-0.322453\pi$
$439$	22.1803	1.05861	0.529305	+	0.848432i	$0.322453\pi$
$440$	4.32145	0.206017
$441$	0	0
$442$	0.472136	0.0224572
$443$	−3.44742	−0.163792	−0.0818959	−	0.996641i	$-0.526098\pi$
$443$	−3.44742	−0.163792	−0.0818959	+	0.996641i	$0.526098\pi$
$444$	0	0
$445$	−1.52786	−0.0724277
$446$	−1.41421	−0.0669650
$447$	0	0
$448$	8.00000	0.377964
$449$	−20.8005	−0.981638	−0.490819	−	0.871261i	$-0.663302\pi$
$449$	−20.8005	−0.981638	−0.490819	+	0.871261i	$0.663302\pi$
$450$	0	0
$451$	−8.94427	−0.421169
$452$	0	0
$453$	0	0
$454$	31.7771	1.49137
$455$	0.874032	0.0409753
$456$	0	0
$457$	−33.1246	−1.54950	−0.774752	−	0.632265i	$-0.782125\pi$
$457$	−33.1246	−1.54950	−0.774752	+	0.632265i	$0.782125\pi$
$458$	−3.49613	−0.163363
$459$	0	0
$460$	0	0
$461$	27.2526	1.26928	0.634640	−	0.772808i	$-0.281148\pi$
$461$	27.2526	1.26928	0.634640	+	0.772808i	$0.281148\pi$
$462$	0	0
$463$	−23.7639	−1.10440	−0.552202	−	0.833710i	$-0.686212\pi$
$463$	−23.7639	−1.10440	−0.552202	+	0.833710i	$0.686212\pi$
$464$	10.4884	0.486911
$465$	0	0
$466$	−28.2918	−1.31059
$467$	13.3168	0.616229	0.308114	−	0.951349i	$-0.400302\pi$
$467$	13.3168	0.616229	0.308114	+	0.951349i	$0.400302\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	8.32766	0.384126
$471$	0	0
$472$	−14.8328	−0.682736
$473$	−0.825324	−0.0379484
$474$	0	0
$475$	−28.4164	−1.30383
$476$	0	0
$477$	0	0
$478$	17.5279	0.801706
$479$	−15.3500	−0.701360	−0.350680	−	0.936495i	$-0.614050\pi$
$479$	−15.3500	−0.701360	−0.350680	+	0.936495i	$0.614050\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	−2.23954	−0.102008
$483$	0	0
$484$	0	0
$485$	−9.35931	−0.424985
$486$	0	0
$487$	18.7082	0.847750	0.423875	−	0.905721i	$-0.360670\pi$
$487$	18.7082	0.847750	0.423875	+	0.905721i	$0.360670\pi$
$488$	−30.2874	−1.37105
$489$	0	0
$490$	7.41641	0.335039
$491$	18.8461	0.850515	0.425257	−	0.905072i	$-0.360184\pi$
$491$	18.8461	0.850515	0.425257	+	0.905072i	$0.360184\pi$
$492$	0	0
$493$	0.875388	0.0394255
$494$	−9.48683	−0.426833
$495$	0	0
$496$	11.0557	0.496417
$497$	−6.65841	−0.298670
$498$	0	0
$499$	−20.2918	−0.908386	−0.454193	−	0.890903i	$-0.650072\pi$
$499$	−20.2918	−0.908386	−0.454193	+	0.890903i	$0.650072\pi$
$500$	0	0
$501$	0	0
$502$	−42.2492	−1.88568
$503$	23.3438	1.04085	0.520425	−	0.853907i	$-0.325774\pi$
$503$	23.3438	1.04085	0.520425	+	0.853907i	$0.325774\pi$
$504$	0	0
$505$	13.0557	0.580972
$506$	2.98605	0.132746
$507$	0	0
$508$	0	0
$509$	−4.98915	−0.221140	−0.110570	−	0.993868i	$-0.535268\pi$
$509$	−4.98915	−0.221140	−0.110570	+	0.993868i	$0.535268\pi$
$510$	0	0
$511$	−9.18034	−0.406114
$512$	22.6274	1.00000
$513$	0	0
$514$	27.7082	1.22216
$515$	−3.03476	−0.133727
$516$	0	0
$517$	−11.7771	−0.517956
$518$	12.7279	0.559233
$519$	0	0
$520$	2.47214	0.108410
$521$	−0.255039	−0.0111735	−0.00558673	−	0.999984i	$-0.501778\pi$
$521$	−0.255039	−0.0111735	−0.00558673	+	0.999984i	$0.501778\pi$
$522$	0	0
$523$	18.8885	0.825938	0.412969	−	0.910745i	$-0.364492\pi$
$523$	18.8885	0.825938	0.412969	+	0.910745i	$0.364492\pi$
$524$	0	0
$525$	0	0
$526$	−42.8328	−1.86760
$527$	0.922740	0.0401952
$528$	0	0
$529$	−21.5410	−0.936566
$530$	−13.3168	−0.578445
$531$	0	0
$532$	0	0
$533$	−5.11667	−0.221628
$534$	0	0
$535$	0.763932	0.0330277
$536$	−25.4558	−1.09952
$537$	0	0
$538$	−19.7082	−0.849681
$539$	−10.4884	−0.451767
$540$	0	0
$541$	8.52786	0.366642	0.183321	−	0.983053i	$-0.441315\pi$
$541$	8.52786	0.366642	0.183321	+	0.983053i	$0.441315\pi$
$542$	−32.3693	−1.39038
$543$	0	0
$544$	0	0
$545$	−11.7264	−0.502303
$546$	0	0
$547$	4.23607	0.181121	0.0905606	−	0.995891i	$-0.471134\pi$
$547$	4.23607	0.181121	0.0905606	+	0.995891i	$0.471134\pi$
$548$	0	0
$549$	0	0
$550$	10.4721	0.446533
$551$	−17.5896	−0.749340
$552$	0	0
$553$	10.2361	0.435282
$554$	−42.9365	−1.82420
$555$	0	0
$556$	0	0
$557$	35.5316	1.50552	0.752760	−	0.658295i	$-0.228722\pi$
$557$	35.5316	1.50552	0.752760	+	0.658295i	$0.228722\pi$
$558$	0	0
$559$	−0.472136	−0.0199692
$560$	−3.49613	−0.147738
$561$	0	0
$562$	−4.29180	−0.181038
$563$	8.69161	0.366308	0.183154	−	0.983084i	$-0.441369\pi$
$563$	8.69161	0.366308	0.183154	+	0.983084i	$0.441369\pi$
$564$	0	0
$565$	4.94427	0.208007
$566$	−2.41577	−0.101542
$567$	0	0
$568$	−18.8328	−0.790207
$569$	8.61280	0.361067	0.180534	−	0.983569i	$-0.442218\pi$
$569$	8.61280	0.361067	0.180534	+	0.983569i	$0.442218\pi$
$570$	0	0
$571$	11.2918	0.472547	0.236273	−	0.971687i	$-0.424074\pi$
$571$	11.2918	0.472547	0.236273	+	0.971687i	$0.424074\pi$
$572$	0	0
$573$	0	0
$574$	7.23607	0.302028
$575$	5.11667	0.213380
$576$	0	0
$577$	−29.7639	−1.23909	−0.619544	−	0.784962i	$-0.712683\pi$
$577$	−29.7639	−1.23909	−0.619544	+	0.784962i	$0.712683\pi$
$578$	23.8840	0.993444
$579$	0	0
$580$	0	0
$581$	4.37016	0.181305
$582$	0	0
$583$	18.8328	0.779976
$584$	−25.9659	−1.07448
$585$	0	0
$586$	42.3607	1.74990
$587$	−45.4311	−1.87514	−0.937570	−	0.347796i	$-0.886930\pi$
$587$	−45.4311	−1.87514	−0.937570	+	0.347796i	$0.886930\pi$
$588$	0	0
$589$	−18.5410	−0.763969
$590$	6.48218	0.266867
$591$	0	0
$592$	36.0000	1.47959
$593$	−18.9737	−0.779155	−0.389578	−	0.920994i	$-0.627379\pi$
$593$	−18.9737	−0.779155	−0.389578	+	0.920994i	$0.627379\pi$
$594$	0	0
$595$	−0.291796	−0.0119625
$596$	0	0
$597$	0	0
$598$	1.70820	0.0698537
$599$	−14.5548	−0.594693	−0.297346	−	0.954770i	$-0.596102\pi$
$599$	−14.5548	−0.594693	−0.297346	+	0.954770i	$0.596102\pi$
$600$	0	0
$601$	−8.58359	−0.350132	−0.175066	−	0.984557i	$-0.556014\pi$
$601$	−8.58359	−0.350132	−0.175066	+	0.984557i	$0.556014\pi$
$602$	0.667701	0.0272135
$603$	0	0
$604$	0	0
$605$	−6.94355	−0.282295
$606$	0	0
$607$	−8.12461	−0.329768	−0.164884	−	0.986313i	$-0.552725\pi$
$607$	−8.12461	−0.329768	−0.164884	+	0.986313i	$0.552725\pi$
$608$	0	0
$609$	0	0
$610$	13.2361	0.535913
$611$	−6.73722	−0.272559
$612$	0	0
$613$	18.7082	0.755617	0.377809	−	0.925884i	$-0.376678\pi$
$613$	18.7082	0.755617	0.377809	+	0.925884i	$0.376678\pi$
$614$	−10.4884	−0.423277
$615$	0	0
$616$	4.94427	0.199210
$617$	−2.74962	−0.110695	−0.0553477	−	0.998467i	$-0.517627\pi$
$617$	−2.74962	−0.110695	−0.0553477	+	0.998467i	$0.517627\pi$
$618$	0	0
$619$	−48.4164	−1.94602	−0.973010	−	0.230764i	$-0.925878\pi$
$619$	−48.4164	−1.94602	−0.973010	+	0.230764i	$0.925878\pi$
$620$	0	0
$621$	0	0
$622$	34.8328	1.39667
$623$	−1.74806	−0.0700347
$624$	0	0
$625$	14.1246	0.564984
$626$	18.7974	0.751297
$627$	0	0
$628$	0	0
$629$	3.00465	0.119803
$630$	0	0
$631$	−6.34752	−0.252691	−0.126345	−	0.991986i	$-0.540325\pi$
$631$	−6.34752	−0.252691	−0.126345	+	0.991986i	$0.540325\pi$
$632$	28.9520	1.15165
$633$	0	0
$634$	23.5279	0.934411
$635$	−8.07262	−0.320352
$636$	0	0
$637$	−6.00000	−0.237729
$638$	6.48218	0.256632
$639$	0	0
$640$	−9.88854	−0.390879
$641$	−4.03631	−0.159425	−0.0797123	−	0.996818i	$-0.525400\pi$
$641$	−4.03631	−0.159425	−0.0797123	+	0.996818i	$0.525400\pi$
$642$	0	0
$643$	1.88854	0.0744769	0.0372384	−	0.999306i	$-0.488144\pi$
$643$	1.88854	0.0744769	0.0372384	+	0.999306i	$0.488144\pi$
$644$	0	0
$645$	0	0
$646$	3.16718	0.124611
$647$	−36.5632	−1.43745	−0.718724	−	0.695295i	$-0.755274\pi$
$647$	−36.5632	−1.43745	−0.718724	+	0.695295i	$0.755274\pi$
$648$	0	0
$649$	−9.16718	−0.359843
$650$	5.99070	0.234975
$651$	0	0
$652$	0	0
$653$	2.03321	0.0795655	0.0397828	−	0.999208i	$-0.487333\pi$
$653$	2.03321	0.0795655	0.0397828	+	0.999208i	$0.487333\pi$
$654$	0	0
$655$	0.111456	0.00435495
$656$	20.4667	0.799090
$657$	0	0
$658$	9.52786	0.371435
$659$	7.81758	0.304530	0.152265	−	0.988340i	$-0.451343\pi$
$659$	7.81758	0.304530	0.152265	+	0.988340i	$0.451343\pi$
$660$	0	0
$661$	7.83282	0.304661	0.152331	−	0.988330i	$-0.451322\pi$
$661$	7.83282	0.304661	0.152331	+	0.988330i	$0.451322\pi$
$662$	19.7202	0.766447
$663$	0	0
$664$	12.3607	0.479687
$665$	5.86319	0.227365
$666$	0	0
$667$	3.16718	0.122634
$668$	0	0
$669$	0	0
$670$	11.1246	0.429781
$671$	−18.7186	−0.722625
$672$	0	0
$673$	15.4721	0.596407	0.298204	−	0.954502i	$-0.403613\pi$
$673$	15.4721	0.596407	0.298204	+	0.954502i	$0.403613\pi$
$674$	17.3044	0.666541
$675$	0	0
$676$	0	0
$677$	30.0810	1.15611	0.578054	−	0.815998i	$-0.303812\pi$
$677$	30.0810	1.15611	0.578054	+	0.815998i	$0.303812\pi$
$678$	0	0
$679$	−10.7082	−0.410943
$680$	−0.825324	−0.0316497
$681$	0	0
$682$	6.83282	0.261642
$683$	−29.2070	−1.11758	−0.558788	−	0.829311i	$-0.688733\pi$
$683$	−29.2070	−1.11758	−0.558788	+	0.829311i	$0.688733\pi$
$684$	0	0
$685$	0.291796	0.0111490
$686$	18.3848	0.701934
$687$	0	0
$688$	1.88854	0.0720001
$689$	10.7735	0.410439
$690$	0	0
$691$	30.3607	1.15497	0.577487	−	0.816400i	$-0.304033\pi$
$691$	30.3607	1.15497	0.577487	+	0.816400i	$0.304033\pi$
$692$	0	0
$693$	0	0
$694$	−43.7082	−1.65914
$695$	2.62210	0.0994618
$696$	0	0
$697$	1.70820	0.0647028
$698$	−23.2163	−0.878750
$699$	0	0
$700$	0	0
$701$	−21.4195	−0.809005	−0.404502	−	0.914537i	$-0.632555\pi$
$701$	−21.4195	−0.809005	−0.404502	+	0.914537i	$0.632555\pi$
$702$	0	0
$703$	−60.3738	−2.27704
$704$	13.9845	0.527061
$705$	0	0
$706$	47.7082	1.79552
$707$	14.9374	0.561777
$708$	0	0
$709$	−5.87539	−0.220655	−0.110327	−	0.993895i	$-0.535190\pi$
$709$	−5.87539	−0.220655	−0.110327	+	0.993895i	$0.535190\pi$
$710$	8.23024	0.308875
$711$	0	0
$712$	−4.94427	−0.185294
$713$	3.33851	0.125028
$714$	0	0
$715$	1.52786	0.0571389
$716$	0	0
$717$	0	0
$718$	10.7639	0.401706
$719$	49.0848	1.83055	0.915277	−	0.402824i	$-0.131971\pi$
$719$	49.0848	1.83055	0.915277	+	0.402824i	$0.131971\pi$
$720$	0	0
$721$	−3.47214	−0.129309
$722$	−36.7696	−1.36842
$723$	0	0
$724$	0	0
$725$	11.1074	0.412518
$726$	0	0
$727$	11.4164	0.423411	0.211706	−	0.977333i	$-0.432098\pi$
$727$	11.4164	0.423411	0.211706	+	0.977333i	$0.432098\pi$
$728$	2.82843	0.104828
$729$	0	0
$730$	11.3475	0.419991
$731$	0.157623	0.00582989
$732$	0	0
$733$	6.87539	0.253948	0.126974	−	0.991906i	$-0.459473\pi$
$733$	6.87539	0.253948	0.126974	+	0.991906i	$0.459473\pi$
$734$	27.2827	1.00702
$735$	0	0
$736$	0	0
$737$	−15.7326	−0.579517
$738$	0	0
$739$	−16.5836	−0.610037	−0.305019	−	0.952346i	$-0.598663\pi$
$739$	−16.5836	−0.610037	−0.305019	+	0.952346i	$0.598663\pi$
$740$	0	0
$741$	0	0
$742$	−15.2361	−0.559334
$743$	−46.2263	−1.69588	−0.847939	−	0.530094i	$-0.822157\pi$
$743$	−46.2263	−1.69588	−0.847939	+	0.530094i	$0.822157\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	12.3153	0.450894
$747$	0	0
$748$	0	0
$749$	0.874032	0.0319364
$750$	0	0
$751$	7.94427	0.289891	0.144945	−	0.989440i	$-0.453699\pi$
$751$	7.94427	0.289891	0.144945	+	0.989440i	$0.453699\pi$
$752$	26.9489	0.982724
$753$	0	0
$754$	3.70820	0.135045
$755$	−13.9358	−0.507176
$756$	0	0
$757$	15.4721	0.562344	0.281172	−	0.959657i	$-0.409277\pi$
$757$	15.4721	0.562344	0.281172	+	0.959657i	$0.409277\pi$
$758$	7.07107	0.256833
$759$	0	0
$760$	16.5836	0.601550
$761$	10.0757	0.365245	0.182622	−	0.983183i	$-0.441541\pi$
$761$	10.0757	0.365245	0.182622	+	0.983183i	$0.441541\pi$
$762$	0	0
$763$	−13.4164	−0.485707
$764$	0	0
$765$	0	0
$766$	35.5967	1.28616
$767$	−5.24419	−0.189357
$768$	0	0
$769$	10.2361	0.369122	0.184561	−	0.982821i	$-0.440914\pi$
$769$	10.2361	0.369122	0.184561	+	0.982821i	$0.440914\pi$
$770$	−2.16073	−0.0778672
$771$	0	0
$772$	0	0
$773$	33.4497	1.20310	0.601550	−	0.798835i	$-0.294550\pi$
$773$	33.4497	1.20310	0.601550	+	0.798835i	$0.294550\pi$
$774$	0	0
$775$	11.7082	0.420571
$776$	−30.2874	−1.08725
$777$	0	0
$778$	−34.5410	−1.23836
$779$	−34.3237	−1.22977
$780$	0	0
$781$	−11.6393	−0.416488
$782$	−0.570285	−0.0203933
$783$	0	0
$784$	24.0000	0.857143
$785$	11.7264	0.418532
$786$	0	0
$787$	−2.81966	−0.100510	−0.0502550	−	0.998736i	$-0.516003\pi$
$787$	−2.81966	−0.100510	−0.0502550	+	0.998736i	$0.516003\pi$
$788$	0	0
$789$	0	0
$790$	−12.6525	−0.450155
$791$	5.65685	0.201135
$792$	0	0
$793$	−10.7082	−0.380259
$794$	3.65375	0.129667
$795$	0	0
$796$	0	0
$797$	−33.5285	−1.18764	−0.593820	−	0.804598i	$-0.702381\pi$
$797$	−33.5285	−1.18764	−0.593820	+	0.804598i	$0.702381\pi$
$798$	0	0
$799$	2.24922	0.0795718
$800$	0	0
$801$	0	0
$802$	20.0000	0.706225
$803$	−16.0478	−0.566315
$804$	0	0
$805$	−1.05573	−0.0372095
$806$	3.90879	0.137681
$807$	0	0
$808$	42.2492	1.48632
$809$	2.20943	0.0776796	0.0388398	−	0.999245i	$-0.487634\pi$
$809$	2.20943	0.0776796	0.0388398	+	0.999245i	$0.487634\pi$
$810$	0	0
$811$	11.4164	0.400884	0.200442	−	0.979706i	$-0.435762\pi$
$811$	11.4164	0.400884	0.200442	+	0.979706i	$0.435762\pi$
$812$	0	0
$813$	0	0
$814$	22.2492	0.779835
$815$	11.1074	0.389074
$816$	0	0
$817$	−3.16718	−0.110806
$818$	−48.6722	−1.70178
$819$	0	0
$820$	0	0
$821$	−35.9929	−1.25616	−0.628081	−	0.778148i	$-0.716159\pi$
$821$	−35.9929	−1.25616	−0.628081	+	0.778148i	$0.716159\pi$
$822$	0	0
$823$	−48.9574	−1.70655	−0.853274	−	0.521462i	$-0.825387\pi$
$823$	−48.9574	−1.70655	−0.853274	+	0.521462i	$0.825387\pi$
$824$	−9.82068	−0.342120
$825$	0	0
$826$	7.41641	0.258050
$827$	−12.7279	−0.442593	−0.221297	−	0.975207i	$-0.571029\pi$
$827$	−12.7279	−0.442593	−0.221297	+	0.975207i	$0.571029\pi$
$828$	0	0
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	−5.40182	−0.187500
$831$	0	0
$832$	8.00000	0.277350
$833$	2.00310	0.0694034
$834$	0	0
$835$	2.29180	0.0793109
$836$	0	0
$837$	0	0
$838$	36.3607	1.25606
$839$	43.3004	1.49490	0.747449	−	0.664320i	$-0.231279\pi$
$839$	43.3004	1.49490	0.747449	+	0.664320i	$0.231279\pi$
$840$	0	0
$841$	−22.1246	−0.762918
$842$	48.2595	1.66313
$843$	0	0
$844$	0	0
$845$	−10.4884	−0.360811
$846$	0	0
$847$	−7.94427	−0.272968
$848$	−43.0941	−1.47986
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	10.8709	0.372651
$852$	0	0
$853$	10.7082	0.366642	0.183321	−	0.983053i	$-0.441315\pi$
$853$	10.7082	0.366642	0.183321	+	0.983053i	$0.441315\pi$
$854$	15.1437	0.518206
$855$	0	0
$856$	2.47214	0.0844959
$857$	50.0864	1.71092	0.855459	−	0.517871i	$-0.173275\pi$
$857$	50.0864	1.71092	0.855459	+	0.517871i	$0.173275\pi$
$858$	0	0
$859$	−21.9443	−0.748729	−0.374364	−	0.927282i	$-0.622139\pi$
$859$	−21.9443	−0.748729	−0.374364	+	0.927282i	$0.622139\pi$
$860$	0	0
$861$	0	0
$862$	−1.23607	−0.0421006
$863$	−20.9281	−0.712399	−0.356200	−	0.934410i	$-0.615928\pi$
$863$	−20.9281	−0.712399	−0.356200	+	0.934410i	$0.615928\pi$
$864$	0	0
$865$	−13.5967	−0.462303
$866$	−35.9442	−1.22143
$867$	0	0
$868$	0	0
$869$	17.8933	0.606989
$870$	0	0
$871$	−9.00000	−0.304953
$872$	−37.9473	−1.28506
$873$	0	0
$874$	11.4590	0.387606
$875$	−8.07262	−0.272904
$876$	0	0
$877$	36.0132	1.21608	0.608039	−	0.793907i	$-0.291956\pi$
$877$	36.0132	1.21608	0.608039	+	0.793907i	$0.291956\pi$
$878$	−31.3677	−1.05861
$879$	0	0
$880$	−6.11146	−0.206017
$881$	−56.9024	−1.91709	−0.958545	−	0.284941i	$-0.908026\pi$
$881$	−56.9024	−1.91709	−0.958545	+	0.284941i	$0.908026\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	0	0
$885$	0	0
$886$	4.87539	0.163792
$887$	−2.82843	−0.0949693	−0.0474846	−	0.998872i	$-0.515121\pi$
$887$	−2.82843	−0.0949693	−0.0474846	+	0.998872i	$0.515121\pi$
$888$	0	0
$889$	−9.23607	−0.309768
$890$	2.16073	0.0724277
$891$	0	0
$892$	0	0
$893$	−45.1946	−1.51238
$894$	0	0
$895$	20.1803	0.674554
$896$	−11.3137	−0.377964
$897$	0	0
$898$	29.4164	0.981638
$899$	7.24730	0.241711
$900$	0	0
$901$	−3.59675	−0.119825
$902$	12.6491	0.421169
$903$	0	0
$904$	16.0000	0.532152
$905$	−17.1769	−0.570979
$906$	0	0
$907$	−42.4164	−1.40841	−0.704207	−	0.709995i	$-0.748697\pi$
$907$	−42.4164	−1.40841	−0.704207	+	0.709995i	$0.748697\pi$
$908$	0	0
$909$	0	0
$910$	−1.23607	−0.0409753
$911$	−36.3268	−1.20356	−0.601780	−	0.798662i	$-0.705542\pi$
$911$	−36.3268	−1.20356	−0.601780	+	0.798662i	$0.705542\pi$
$912$	0	0
$913$	7.63932	0.252825
$914$	46.8453	1.54950
$915$	0	0
$916$	0	0
$917$	0.127520	0.00421107
$918$	0	0
$919$	−0.944272	−0.0311487	−0.0155743	−	0.999879i	$-0.504958\pi$
$919$	−0.944272	−0.0311487	−0.0155743	+	0.999879i	$0.504958\pi$
$920$	−2.98605	−0.0984472
$921$	0	0
$922$	−38.5410	−1.26928
$923$	−6.65841	−0.219164
$924$	0	0
$925$	38.1246	1.25353
$926$	33.6073	1.10440
$927$	0	0
$928$	0	0
$929$	−53.9163	−1.76894	−0.884469	−	0.466599i	$-0.845479\pi$
$929$	−53.9163	−1.76894	−0.884469	+	0.466599i	$0.845479\pi$
$930$	0	0
$931$	−40.2492	−1.31912
$932$	0	0
$933$	0	0
$934$	−18.8328	−0.616229
$935$	−0.510078	−0.0166813
$936$	0	0
$937$	41.5410	1.35709	0.678543	−	0.734561i	$-0.262612\pi$
$937$	41.5410	1.35709	0.678543	+	0.734561i	$0.262612\pi$
$938$	12.7279	0.415581
$939$	0	0
$940$	0	0
$941$	7.48373	0.243963	0.121981	−	0.992532i	$-0.461075\pi$
$941$	7.48373	0.243963	0.121981	+	0.992532i	$0.461075\pi$
$942$	0	0
$943$	6.18034	0.201260
$944$	20.9768	0.682736
$945$	0	0
$946$	1.16718	0.0379484
$947$	39.1853	1.27335	0.636676	−	0.771132i	$-0.280309\pi$
$947$	39.1853	1.27335	0.636676	+	0.771132i	$0.280309\pi$
$948$	0	0
$949$	−9.18034	−0.298006
$950$	40.1869	1.30383
$951$	0	0
$952$	−0.944272	−0.0306040
$953$	23.6290	0.765417	0.382709	−	0.923869i	$-0.374991\pi$
$953$	23.6290	0.765417	0.382709	+	0.923869i	$0.374991\pi$
$954$	0	0
$955$	7.70820	0.249432
$956$	0	0
$957$	0	0
$958$	21.7082	0.701360
$959$	0.333851	0.0107806
$960$	0	0
$961$	−23.3607	−0.753570
$962$	12.7279	0.410365
$963$	0	0
$964$	0	0
$965$	7.86629	0.253225
$966$	0	0
$967$	10.4164	0.334969	0.167485	−	0.985875i	$-0.446436\pi$
$967$	10.4164	0.334969	0.167485	+	0.985875i	$0.446436\pi$
$968$	−22.4698	−0.722207
$969$	0	0
$970$	13.2361	0.424985
$971$	−41.9836	−1.34732	−0.673660	−	0.739042i	$-0.735279\pi$
$971$	−41.9836	−1.34732	−0.673660	+	0.739042i	$0.735279\pi$
$972$	0	0
$973$	3.00000	0.0961756
$974$	−26.4574	−0.847750
$975$	0	0
$976$	42.8328	1.37105
$977$	−8.61280	−0.275548	−0.137774	−	0.990464i	$-0.543995\pi$
$977$	−8.61280	−0.275548	−0.137774	+	0.990464i	$0.543995\pi$
$978$	0	0
$979$	−3.05573	−0.0976615
$980$	0	0
$981$	0	0
$982$	−26.6525	−0.850515
$983$	8.69161	0.277219	0.138610	−	0.990347i	$-0.455737\pi$
$983$	8.69161	0.277219	0.138610	+	0.990347i	$0.455737\pi$
$984$	0	0
$985$	17.3050	0.551382
$986$	−1.23799	−0.0394255
$987$	0	0
$988$	0	0
$989$	0.570285	0.0181340
$990$	0	0
$991$	−2.70820	−0.0860289	−0.0430145	−	0.999074i	$-0.513696\pi$
$991$	−2.70820	−0.0860289	−0.0430145	+	0.999074i	$0.513696\pi$
$992$	0	0
$993$	0	0
$994$	9.41641	0.298670
$995$	−5.45052	−0.172793
$996$	0	0
$997$	17.0557	0.540160	0.270080	−	0.962838i	$-0.412950\pi$
$997$	17.0557	0.540160	0.270080	+	0.962838i	$0.412950\pi$
$998$	28.6969	0.908386
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.k.1.2	✓	4
3.2	odd	2	inner	6021.2.a.k.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.k.1.2	✓	4	1.1	even	1	trivial
6021.2.a.k.1.3	yes	4	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +0.874032 q^{5} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +0.874032 q^{5} +1.00000 q^{7} +2.82843 q^{8} -1.23607 q^{10} +1.74806 q^{11} +1.00000 q^{13} -1.41421 q^{14} -4.00000 q^{16} -0.333851 q^{17} +6.70820 q^{19} -2.47214 q^{22} -1.20788 q^{23} -4.23607 q^{25} -1.41421 q^{26} -2.62210 q^{29} -2.76393 q^{31} +0.472136 q^{34} +0.874032 q^{35} -9.00000 q^{37} -9.48683 q^{38} +2.47214 q^{40} -5.11667 q^{41} -0.472136 q^{43} +1.70820 q^{46} -6.73722 q^{47} -6.00000 q^{49} +5.99070 q^{50} +10.7735 q^{53} +1.52786 q^{55} +2.82843 q^{56} +3.70820 q^{58} -5.24419 q^{59} -10.7082 q^{61} +3.90879 q^{62} +8.00000 q^{64} +0.874032 q^{65} -9.00000 q^{67} -1.23607 q^{70} -6.65841 q^{71} -9.18034 q^{73} +12.7279 q^{74} +1.74806 q^{77} +10.2361 q^{79} -3.49613 q^{80} +7.23607 q^{82} +4.37016 q^{83} -0.291796 q^{85} +0.667701 q^{86} +4.94427 q^{88} -1.74806 q^{89} +1.00000 q^{91} +9.52786 q^{94} +5.86319 q^{95} -10.7082 q^{97} +8.48528 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 4 q^{10} + 4 q^{13} - 16 q^{16} + 8 q^{22} - 8 q^{25} - 20 q^{31} - 16 q^{34} - 36 q^{37} - 8 q^{40} + 16 q^{43} - 20 q^{46} - 24 q^{49} + 24 q^{55} - 12 q^{58} - 16 q^{61} + 32 q^{64} - 36 q^{67} + 4 q^{70} + 8 q^{73} + 32 q^{79} + 20 q^{82} - 28 q^{85} - 16 q^{88} + 4 q^{91} + 56 q^{94} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 4 * q^10 + 4 * q^13 - 16 * q^16 + 8 * q^22 - 8 * q^25 - 20 * q^31 - 16 * q^34 - 36 * q^37 - 8 * q^40 + 16 * q^43 - 20 * q^46 - 24 * q^49 + 24 * q^55 - 12 * q^58 - 16 * q^61 + 32 * q^64 - 36 * q^67 + 4 * q^70 + 8 * q^73 + 32 * q^79 + 20 * q^82 - 28 * q^85 - 16 * q^88 + 4 * q^91 + 56 * q^94 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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