LMFDB - Embedded newform 6021.2.a.k.1.1

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			1.1
Root			$0.874032$ 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)

$f(q)$	$=$	$q-1.41421 q^{2} -2.28825 q^{5} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} -2.28825 q^{5} +1.00000 q^{7} +2.82843 q^{8} +3.23607 q^{10} -4.57649 q^{11} +1.00000 q^{13} -1.41421 q^{14} -4.00000 q^{16} +5.99070 q^{17} -6.70820 q^{19} +6.47214 q^{22} +8.27895 q^{23} +0.236068 q^{25} -1.41421 q^{26} +6.86474 q^{29} -7.23607 q^{31} -8.47214 q^{34} -2.28825 q^{35} -9.00000 q^{37} +9.48683 q^{38} -6.47214 q^{40} -1.95440 q^{41} +8.47214 q^{43} -11.7082 q^{46} -13.0618 q^{47} -6.00000 q^{49} -0.333851 q^{50} +7.61125 q^{53} +10.4721 q^{55} +2.82843 q^{56} -9.70820 q^{58} +13.7295 q^{59} +2.70820 q^{61} +10.2333 q^{62} +8.00000 q^{64} -2.28825 q^{65} -9.00000 q^{67} +3.23607 q^{70} +12.3153 q^{71} +13.1803 q^{73} +12.7279 q^{74} -4.57649 q^{77} +5.76393 q^{79} +9.15298 q^{80} +2.76393 q^{82} -11.4412 q^{83} -13.7082 q^{85} -11.9814 q^{86} -12.9443 q^{88} +4.57649 q^{89} +1.00000 q^{91} +18.4721 q^{94} +15.3500 q^{95} +2.70820 q^{97} +8.48528 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.28825	−1.02333	−0.511667	−	0.859184i	$-0.670972\pi$
$5$	−2.28825	−1.02333	−0.511667	+	0.859184i	$0.670972\pi$
$6$	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
$10$	3.23607	1.02333
$11$	−4.57649	−1.37986	−0.689932	−	0.723874i	$-0.742360\pi$
$11$	−4.57649	−1.37986	−0.689932	+	0.723874i	$0.742360\pi$
$12$	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
$17$	5.99070	1.45296	0.726480	−	0.687188i	$-0.241155\pi$
$17$	5.99070	1.45296	0.726480	+	0.687188i	$0.241155\pi$
$18$	0	0
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	0	0
$22$	6.47214	1.37986
$23$	8.27895	1.72628	0.863140	−	0.504964i	$-0.168494\pi$
$23$	8.27895	1.72628	0.863140	+	0.504964i	$0.168494\pi$
$24$	0	0
$25$	0.236068	0.0472136
$26$	−1.41421	−0.277350
$27$	0	0
$28$	0	0
$29$	6.86474	1.27475	0.637375	−	0.770554i	$-0.280020\pi$
$29$	6.86474	1.27475	0.637375	+	0.770554i	$0.280020\pi$
$30$	0	0
$31$	−7.23607	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$31$	−7.23607	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$32$	0	0
$33$	0	0
$34$	−8.47214	−1.45296
$35$	−2.28825	−0.386784
$36$	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
$40$	−6.47214	−1.02333
$41$	−1.95440	−0.305225	−0.152613	−	0.988286i	$-0.548769\pi$
$41$	−1.95440	−0.305225	−0.152613	+	0.988286i	$0.548769\pi$
$42$	0	0
$43$	8.47214	1.29199	0.645994	−	0.763342i	$-0.276443\pi$
$43$	8.47214	1.29199	0.645994	+	0.763342i	$0.276443\pi$
$44$	0	0
$45$	0	0
$46$	−11.7082	−1.72628
$47$	−13.0618	−1.90526	−0.952628	−	0.304139i	$-0.901631\pi$
$47$	−13.0618	−1.90526	−0.952628	+	0.304139i	$0.901631\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−0.333851	−0.0472136
$51$	0	0
$52$	0	0
$53$	7.61125	1.04549	0.522743	−	0.852490i	$-0.324909\pi$
$53$	7.61125	1.04549	0.522743	+	0.852490i	$0.324909\pi$
$54$	0	0
$55$	10.4721	1.41206
$56$	2.82843	0.377964
$57$	0	0
$58$	−9.70820	−1.27475
$59$	13.7295	1.78743	0.893713	−	0.448640i	$-0.148091\pi$
$59$	13.7295	1.78743	0.893713	+	0.448640i	$0.148091\pi$
$60$	0	0
$61$	2.70820	0.346750	0.173375	−	0.984856i	$-0.444533\pi$
$61$	2.70820	0.346750	0.173375	+	0.984856i	$0.444533\pi$
$62$	10.2333	1.29964
$63$	0	0
$64$	8.00000	1.00000
$65$	−2.28825	−0.283822
$66$	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
$70$	3.23607	0.386784
$71$	12.3153	1.46155	0.730776	−	0.682617i	$-0.239158\pi$
$71$	12.3153	1.46155	0.730776	+	0.682617i	$0.239158\pi$
$72$	0	0
$73$	13.1803	1.54264	0.771321	−	0.636446i	$-0.219596\pi$
$73$	13.1803	1.54264	0.771321	+	0.636446i	$0.219596\pi$
$74$	12.7279	1.47959
$75$	0	0
$76$	0	0
$77$	−4.57649	−0.521540
$78$	0	0
$79$	5.76393	0.648493	0.324247	−	0.945973i	$-0.394889\pi$
$79$	5.76393	0.648493	0.324247	+	0.945973i	$0.394889\pi$
$80$	9.15298	1.02333
$81$	0	0
$82$	2.76393	0.305225
$83$	−11.4412	−1.25584	−0.627919	−	0.778279i	$-0.716093\pi$
$83$	−11.4412	−1.25584	−0.627919	+	0.778279i	$0.716093\pi$
$84$	0	0
$85$	−13.7082	−1.48686
$86$	−11.9814	−1.29199
$87$	0	0
$88$	−12.9443	−1.37986
$89$	4.57649	0.485107	0.242554	−	0.970138i	$-0.422015\pi$
$89$	4.57649	0.485107	0.242554	+	0.970138i	$0.422015\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	18.4721	1.90526
$95$	15.3500	1.57488
$96$	0	0
$97$	2.70820	0.274976	0.137488	−	0.990503i	$-0.456097\pi$
$97$	2.70820	0.274976	0.137488	+	0.990503i	$0.456097\pi$
$98$	8.48528	0.857143
$99$	0	0
$100$	0	0
$101$	−13.5231	−1.34560	−0.672801	−	0.739823i	$-0.734909\pi$
$101$	−13.5231	−1.34560	−0.672801	+	0.739823i	$0.734909\pi$
$102$	0	0
$103$	5.47214	0.539186	0.269593	−	0.962974i	$-0.413111\pi$
$103$	5.47214	0.539186	0.269593	+	0.962974i	$0.413111\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−10.7639	−1.04549
$107$	−2.28825	−0.221213	−0.110607	−	0.993864i	$-0.535279\pi$
$107$	−2.28825	−0.221213	−0.110607	+	0.993864i	$0.535279\pi$
$108$	0	0
$109$	13.4164	1.28506	0.642529	−	0.766261i	$-0.277885\pi$
$109$	13.4164	1.28506	0.642529	+	0.766261i	$0.277885\pi$
$110$	−14.8098	−1.41206
$111$	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
$115$	−18.9443	−1.76656
$116$	0	0
$117$	0	0
$118$	−19.4164	−1.78743
$119$	5.99070	0.549167
$120$	0	0
$121$	9.94427	0.904025
$122$	−3.82998	−0.346750
$123$	0	0
$124$	0	0
$125$	10.9010	0.975019
$126$	0	0
$127$	−4.76393	−0.422731	−0.211365	−	0.977407i	$-0.567791\pi$
$127$	−4.76393	−0.422731	−0.211365	+	0.977407i	$0.567791\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	3.23607	0.283822
$131$	−15.6839	−1.37031	−0.685153	−	0.728399i	$-0.740265\pi$
$131$	−15.6839	−1.37031	−0.685153	+	0.728399i	$0.740265\pi$
$132$	0	0
$133$	−6.70820	−0.581675
$134$	12.7279	1.09952
$135$	0	0
$136$	16.9443	1.45296
$137$	−5.99070	−0.511820	−0.255910	−	0.966701i	$-0.582375\pi$
$137$	−5.99070	−0.511820	−0.255910	+	0.966701i	$0.582375\pi$
$138$	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
$142$	−17.4164	−1.46155
$143$	−4.57649	−0.382705
$144$	0	0
$145$	−15.7082	−1.30450
$146$	−18.6398	−1.54264
$147$	0	0
$148$	0	0
$149$	0.874032	0.0716035	0.0358017	−	0.999359i	$-0.488602\pi$
$149$	0.874032	0.0716035	0.0358017	+	0.999359i	$0.488602\pi$
$150$	0	0
$151$	1.94427	0.158223	0.0791113	−	0.996866i	$-0.474792\pi$
$151$	1.94427	0.158223	0.0791113	+	0.996866i	$0.474792\pi$
$152$	−18.9737	−1.53897
$153$	0	0
$154$	6.47214	0.521540
$155$	16.5579	1.32996
$156$	0	0
$157$	−13.4164	−1.07075	−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164	−1.07075	−0.535373	+	0.844616i	$0.679829\pi$
$158$	−8.15143	−0.648493
$159$	0	0
$160$	0	0
$161$	8.27895	0.652473
$162$	0	0
$163$	−0.708204	−0.0554708	−0.0277354	−	0.999615i	$-0.508830\pi$
$163$	−0.708204	−0.0554708	−0.0277354	+	0.999615i	$0.508830\pi$
$164$	0	0
$165$	0	0
$166$	16.1803	1.25584
$167$	−6.86474	−0.531209	−0.265605	−	0.964082i	$-0.585572\pi$
$167$	−6.86474	−0.531209	−0.265605	+	0.964082i	$0.585572\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	19.3863	1.48686
$171$	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
$175$	0.236068	0.0178451
$176$	18.3060	1.37986
$177$	0	0
$178$	−6.47214	−0.485107
$179$	0.952843	0.0712189	0.0356094	−	0.999366i	$-0.488663\pi$
$179$	0.952843	0.0712189	0.0356094	+	0.999366i	$0.488663\pi$
$180$	0	0
$181$	11.6525	0.866122	0.433061	−	0.901365i	$-0.357434\pi$
$181$	11.6525	0.866122	0.433061	+	0.901365i	$0.357434\pi$
$182$	−1.41421	−0.104828
$183$	0	0
$184$	23.4164	1.72628
$185$	20.5942	1.51412
$186$	0	0
$187$	−27.4164	−2.00489
$188$	0	0
$189$	0	0
$190$	−21.7082	−1.57488
$191$	2.49458	0.180501	0.0902506	−	0.995919i	$-0.471233\pi$
$191$	2.49458	0.180501	0.0902506	+	0.995919i	$0.471233\pi$
$192$	0	0
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	−3.82998	−0.274976
$195$	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
$199$	−1.76393	−0.125042	−0.0625209	−	0.998044i	$-0.519914\pi$
$199$	−1.76393	−0.125042	−0.0625209	+	0.998044i	$0.519914\pi$
$200$	0.667701	0.0472136
$201$	0	0
$202$	19.1246	1.34560
$203$	6.86474	0.481810
$204$	0	0
$205$	4.47214	0.312348
$206$	−7.73877	−0.539186
$207$	0	0
$208$	−4.00000	−0.277350
$209$	30.7000	2.12357
$210$	0	0
$211$	−6.23607	−0.429309	−0.214654	−	0.976690i	$-0.568862\pi$
$211$	−6.23607	−0.429309	−0.214654	+	0.976690i	$0.568862\pi$
$212$	0	0
$213$	0	0
$214$	3.23607	0.221213
$215$	−19.3863	−1.32214
$216$	0	0
$217$	−7.23607	−0.491216
$218$	−18.9737	−1.28506
$219$	0	0
$220$	0	0
$221$	5.99070	0.402978
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	−8.00000	−0.532152
$227$	28.1266	1.86683	0.933416	−	0.358797i	$-0.116813\pi$
$227$	28.1266	1.86683	0.933416	+	0.358797i	$0.116813\pi$
$228$	0	0
$229$	−6.47214	−0.427691	−0.213845	−	0.976868i	$-0.568599\pi$
$229$	−6.47214	−0.427691	−0.213845	+	0.976868i	$0.568599\pi$
$230$	26.7912	1.76656
$231$	0	0
$232$	19.4164	1.27475
$233$	29.4922	1.93210	0.966048	−	0.258364i	$-0.0831834\pi$
$233$	29.4922	1.93210	0.966048	+	0.258364i	$0.0831834\pi$
$234$	0	0
$235$	29.8885	1.94971
$236$	0	0
$237$	0	0
$238$	−8.47214	−0.549167
$239$	−18.7186	−1.21081	−0.605404	−	0.795919i	$-0.706988\pi$
$239$	−18.7186	−1.21081	−0.605404	+	0.795919i	$0.706988\pi$
$240$	0	0
$241$	28.4164	1.83046	0.915231	−	0.402930i	$-0.132008\pi$
$241$	28.4164	1.83046	0.915231	+	0.402930i	$0.132008\pi$
$242$	−14.0633	−0.904025
$243$	0	0
$244$	0	0
$245$	13.7295	0.877144
$246$	0	0
$247$	−6.70820	−0.426833
$248$	−20.4667	−1.29964
$249$	0	0
$250$	−15.4164	−0.975019
$251$	−27.0463	−1.70715	−0.853573	−	0.520973i	$-0.825569\pi$
$251$	−27.0463	−1.70715	−0.853573	+	0.520973i	$0.825569\pi$
$252$	0	0
$253$	−37.8885	−2.38203
$254$	6.73722	0.422731
$255$	0	0
$256$	0	0
$257$	−10.1058	−0.630384	−0.315192	−	0.949028i	$-0.602069\pi$
$257$	−10.1058	−0.630384	−0.315192	+	0.949028i	$0.602069\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	0	0
$262$	22.1803	1.37031
$263$	−7.65996	−0.472333	−0.236167	−	0.971713i	$-0.575891\pi$
$263$	−7.65996	−0.472333	−0.236167	+	0.971713i	$0.575891\pi$
$264$	0	0
$265$	−17.4164	−1.06988
$266$	9.48683	0.581675
$267$	0	0
$268$	0	0
$269$	4.44897	0.271259	0.135629	−	0.990760i	$-0.456694\pi$
$269$	4.44897	0.271259	0.135629	+	0.990760i	$0.456694\pi$
$270$	0	0
$271$	−12.8885	−0.782923	−0.391462	−	0.920194i	$-0.628030\pi$
$271$	−12.8885	−0.782923	−0.391462	+	0.920194i	$0.628030\pi$
$272$	−23.9628	−1.45296
$273$	0	0
$274$	8.47214	0.511820
$275$	−1.08036	−0.0651483
$276$	0	0
$277$	−14.3607	−0.862850	−0.431425	−	0.902149i	$-0.641989\pi$
$277$	−14.3607	−0.862850	−0.431425	+	0.902149i	$0.641989\pi$
$278$	−4.24264	−0.254457
$279$	0	0
$280$	−6.47214	−0.386784
$281$	12.5216	0.746975	0.373488	−	0.927635i	$-0.378162\pi$
$281$	12.5216	0.746975	0.373488	+	0.927635i	$0.378162\pi$
$282$	0	0
$283$	−11.7082	−0.695980	−0.347990	−	0.937498i	$-0.613136\pi$
$283$	−11.7082	−0.695980	−0.347990	+	0.937498i	$0.613136\pi$
$284$	0	0
$285$	0	0
$286$	6.47214	0.382705
$287$	−1.95440	−0.115364
$288$	0	0
$289$	18.8885	1.11109
$290$	22.2148	1.30450
$291$	0	0
$292$	0	0
$293$	1.66925	0.0975188	0.0487594	−	0.998811i	$-0.484473\pi$
$293$	1.66925	0.0975188	0.0487594	+	0.998811i	$0.484473\pi$
$294$	0	0
$295$	−31.4164	−1.82913
$296$	−25.4558	−1.47959
$297$	0	0
$298$	−1.23607	−0.0716035
$299$	8.27895	0.478784
$300$	0	0
$301$	8.47214	0.488326
$302$	−2.74962	−0.158223
$303$	0	0
$304$	26.8328	1.53897
$305$	−6.19704	−0.354841
$306$	0	0
$307$	−19.4164	−1.10815	−0.554076	−	0.832466i	$-0.686928\pi$
$307$	−19.4164	−1.10815	−0.554076	+	0.832466i	$0.686928\pi$
$308$	0	0
$309$	0	0
$310$	−23.4164	−1.32996
$311$	13.3168	0.755127	0.377564	−	0.925984i	$-0.376762\pi$
$311$	13.3168	0.755127	0.377564	+	0.925984i	$0.376762\pi$
$312$	0	0
$313$	−26.7082	−1.50964	−0.754818	−	0.655934i	$-0.772275\pi$
$313$	−26.7082	−1.50964	−0.754818	+	0.655934i	$0.772275\pi$
$314$	18.9737	1.07075
$315$	0	0
$316$	0	0
$317$	−22.9613	−1.28963	−0.644817	−	0.764337i	$-0.723066\pi$
$317$	−22.9613	−1.28963	−0.644817	+	0.764337i	$0.723066\pi$
$318$	0	0
$319$	−31.4164	−1.75898
$320$	−18.3060	−1.02333
$321$	0	0
$322$	−11.7082	−0.652473
$323$	−40.1869	−2.23606
$324$	0	0
$325$	0.236068	0.0130947
$326$	1.00155	0.0554708
$327$	0	0
$328$	−5.52786	−0.305225
$329$	−13.0618	−0.720119
$330$	0	0
$331$	3.94427	0.216797	0.108398	−	0.994108i	$-0.465428\pi$
$331$	3.94427	0.216797	0.108398	+	0.994108i	$0.465428\pi$
$332$	0	0
$333$	0	0
$334$	9.70820	0.531209
$335$	20.5942	1.12518
$336$	0	0
$337$	−7.76393	−0.422928	−0.211464	−	0.977386i	$-0.567823\pi$
$337$	−7.76393	−0.422928	−0.211464	+	0.977386i	$0.567823\pi$
$338$	16.9706	0.923077
$339$	0	0
$340$	0	0
$341$	33.1158	1.79332
$342$	0	0
$343$	−13.0000	−0.701934
$344$	23.9628	1.29199
$345$	0	0
$346$	22.0000	1.18273
$347$	21.4195	1.14986	0.574930	−	0.818202i	$-0.305029\pi$
$347$	21.4195	1.14986	0.574930	+	0.818202i	$0.305029\pi$
$348$	0	0
$349$	−10.4164	−0.557578	−0.278789	−	0.960352i	$-0.589933\pi$
$349$	−10.4164	−0.557578	−0.278789	+	0.960352i	$0.589933\pi$
$350$	−0.333851	−0.0178451
$351$	0	0
$352$	0	0
$353$	−24.2480	−1.29059	−0.645294	−	0.763934i	$-0.723265\pi$
$353$	−24.2480	−1.29059	−0.645294	+	0.763934i	$0.723265\pi$
$354$	0	0
$355$	−28.1803	−1.49566
$356$	0	0
$357$	0	0
$358$	−1.34752	−0.0712189
$359$	−10.7735	−0.568605	−0.284303	−	0.958735i	$-0.591762\pi$
$359$	−10.7735	−0.568605	−0.284303	+	0.958735i	$0.591762\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	−16.4791	−0.866122
$363$	0	0
$364$	0	0
$365$	−30.1599	−1.57864
$366$	0	0
$367$	−32.7082	−1.70735	−0.853677	−	0.520803i	$-0.825633\pi$
$367$	−32.7082	−1.70735	−0.853677	+	0.520803i	$0.825633\pi$
$368$	−33.1158	−1.72628
$369$	0	0
$370$	−29.1246	−1.51412
$371$	7.61125	0.395156
$372$	0	0
$373$	4.70820	0.243782	0.121891	−	0.992544i	$-0.461104\pi$
$373$	4.70820	0.243782	0.121891	+	0.992544i	$0.461104\pi$
$374$	38.7727	2.00489
$375$	0	0
$376$	−36.9443	−1.90526
$377$	6.86474	0.353552
$378$	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
$382$	−3.52786	−0.180501
$383$	9.61435	0.491270	0.245635	−	0.969362i	$-0.421003\pi$
$383$	9.61435	0.491270	0.245635	+	0.969362i	$0.421003\pi$
$384$	0	0
$385$	10.4721	0.533709
$386$	−12.7279	−0.647834
$387$	0	0
$388$	0	0
$389$	−23.0100	−1.16665	−0.583326	−	0.812238i	$-0.698249\pi$
$389$	−23.0100	−1.16665	−0.583326	+	0.812238i	$0.698249\pi$
$390$	0	0
$391$	49.5967	2.50822
$392$	−16.9706	−0.857143
$393$	0	0
$394$	−28.0000	−1.41062
$395$	−13.1893	−0.663625
$396$	0	0
$397$	−29.4164	−1.47637	−0.738184	−	0.674600i	$-0.764316\pi$
$397$	−29.4164	−1.47637	−0.738184	+	0.674600i	$0.764316\pi$
$398$	2.49458	0.125042
$399$	0	0
$400$	−0.944272	−0.0472136
$401$	−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
$403$	−7.23607	−0.360454
$404$	0	0
$405$	0	0
$406$	−9.70820	−0.481810
$407$	41.1884	2.04163
$408$	0	0
$409$	7.58359	0.374984	0.187492	−	0.982266i	$-0.439964\pi$
$409$	7.58359	0.374984	0.187492	+	0.982266i	$0.439964\pi$
$410$	−6.32456	−0.312348
$411$	0	0
$412$	0	0
$413$	13.7295	0.675583
$414$	0	0
$415$	26.1803	1.28514
$416$	0	0
$417$	0	0
$418$	−43.4164	−2.12357
$419$	5.91189	0.288815	0.144407	−	0.989518i	$-0.453872\pi$
$419$	5.91189	0.288815	0.144407	+	0.989518i	$0.453872\pi$
$420$	0	0
$421$	6.12461	0.298495	0.149248	−	0.988800i	$-0.452315\pi$
$421$	6.12461	0.298495	0.149248	+	0.988800i	$0.452315\pi$
$422$	8.81913	0.429309
$423$	0	0
$424$	21.5279	1.04549
$425$	1.41421	0.0685994
$426$	0	0
$427$	2.70820	0.131059
$428$	0	0
$429$	0	0
$430$	27.4164	1.32214
$431$	−2.28825	−0.110221	−0.0551105	−	0.998480i	$-0.517551\pi$
$431$	−2.28825	−0.110221	−0.0551105	+	0.998480i	$0.517551\pi$
$432$	0	0
$433$	−1.41641	−0.0680682	−0.0340341	−	0.999421i	$-0.510835\pi$
$433$	−1.41641	−0.0680682	−0.0340341	+	0.999421i	$0.510835\pi$
$434$	10.2333	0.491216
$435$	0	0
$436$	0	0
$437$	−55.5369	−2.65669
$438$	0	0
$439$	−0.180340	−0.00860715	−0.00430358	−	0.999991i	$-0.501370\pi$
$439$	−0.180340	−0.00860715	−0.00430358	+	0.999991i	$0.501370\pi$
$440$	29.6197	1.41206
$441$	0	0
$442$	−8.47214	−0.402978
$443$	−31.9079	−1.51599	−0.757995	−	0.652260i	$-0.773821\pi$
$443$	−31.9079	−1.51599	−0.757995	+	0.652260i	$0.773821\pi$
$444$	0	0
$445$	−10.4721	−0.496427
$446$	−1.41421	−0.0669650
$447$	0	0
$448$	8.00000	0.377964
$449$	−1.82688	−0.0862156	−0.0431078	−	0.999070i	$-0.513726\pi$
$449$	−1.82688	−0.0862156	−0.0431078	+	0.999070i	$0.513726\pi$
$450$	0	0
$451$	8.94427	0.421169
$452$	0	0
$453$	0	0
$454$	−39.7771	−1.86683
$455$	−2.28825	−0.107275
$456$	0	0
$457$	7.12461	0.333275	0.166638	−	0.986018i	$-0.446709\pi$
$457$	7.12461	0.333275	0.166638	+	0.986018i	$0.446709\pi$
$458$	9.15298	0.427691
$459$	0	0
$460$	0	0
$461$	−20.1815	−0.939948	−0.469974	−	0.882680i	$-0.655737\pi$
$461$	−20.1815	−0.939948	−0.469974	+	0.882680i	$0.655737\pi$
$462$	0	0
$463$	−28.2361	−1.31224	−0.656121	−	0.754656i	$-0.727804\pi$
$463$	−28.2361	−1.31224	−0.656121	+	0.754656i	$0.727804\pi$
$464$	−27.4589	−1.27475
$465$	0	0
$466$	−41.7082	−1.93210
$467$	−24.6305	−1.13976	−0.569882	−	0.821726i	$-0.693011\pi$
$467$	−24.6305	−1.13976	−0.569882	+	0.821726i	$0.693011\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	−42.2688	−1.94971
$471$	0	0
$472$	38.8328	1.78743
$473$	−38.7727	−1.78277
$474$	0	0
$475$	−1.58359	−0.0726602
$476$	0	0
$477$	0	0
$478$	26.4721	1.21081
$479$	−5.86319	−0.267896	−0.133948	−	0.990988i	$-0.542765\pi$
$479$	−5.86319	−0.267896	−0.133948	+	0.990988i	$0.542765\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	−40.1869	−1.83046
$483$	0	0
$484$	0	0
$485$	−6.19704	−0.281393
$486$	0	0
$487$	5.29180	0.239794	0.119897	−	0.992786i	$-0.461744\pi$
$487$	5.29180	0.239794	0.119897	+	0.992786i	$0.461744\pi$
$488$	7.65996	0.346750
$489$	0	0
$490$	−19.4164	−0.877144
$491$	−3.28980	−0.148466	−0.0742332	−	0.997241i	$-0.523651\pi$
$491$	−3.28980	−0.148466	−0.0742332	+	0.997241i	$0.523651\pi$
$492$	0	0
$493$	41.1246	1.85216
$494$	9.48683	0.426833
$495$	0	0
$496$	28.9443	1.29964
$497$	12.3153	0.552415
$498$	0	0
$499$	−33.7082	−1.50899	−0.754493	−	0.656308i	$-0.772117\pi$
$499$	−33.7082	−1.50899	−0.754493	+	0.656308i	$0.772117\pi$
$500$	0	0
$501$	0	0
$502$	38.2492	1.70715
$503$	−30.4149	−1.35613	−0.678067	−	0.735001i	$-0.737182\pi$
$503$	−30.4149	−1.35613	−0.678067	+	0.735001i	$0.737182\pi$
$504$	0	0
$505$	30.9443	1.37700
$506$	53.5825	2.38203
$507$	0	0
$508$	0	0
$509$	−17.6383	−0.781802	−0.390901	−	0.920433i	$-0.627837\pi$
$509$	−17.6383	−0.781802	−0.390901	+	0.920433i	$0.627837\pi$
$510$	0	0
$511$	13.1803	0.583064
$512$	22.6274	1.00000
$513$	0	0
$514$	14.2918	0.630384
$515$	−12.5216	−0.551767
$516$	0	0
$517$	59.7771	2.62899
$518$	12.7279	0.559233
$519$	0	0
$520$	−6.47214	−0.283822
$521$	31.3677	1.37425	0.687123	−	0.726541i	$-0.258873\pi$
$521$	31.3677	1.37425	0.687123	+	0.726541i	$0.258873\pi$
$522$	0	0
$523$	−16.8885	−0.738484	−0.369242	−	0.929333i	$-0.620383\pi$
$523$	−16.8885	−0.738484	−0.369242	+	0.929333i	$0.620383\pi$
$524$	0	0
$525$	0	0
$526$	10.8328	0.472333
$527$	−43.3491	−1.88832
$528$	0	0
$529$	45.5410	1.98004
$530$	24.6305	1.06988
$531$	0	0
$532$	0	0
$533$	−1.95440	−0.0846542
$534$	0	0
$535$	5.23607	0.226375
$536$	−25.4558	−1.09952
$537$	0	0
$538$	−6.29180	−0.271259
$539$	27.4589	1.18274
$540$	0	0
$541$	17.4721	0.751186	0.375593	−	0.926785i	$-0.377439\pi$
$541$	17.4721	0.751186	0.375593	+	0.926785i	$0.377439\pi$
$542$	18.2272	0.782923
$543$	0	0
$544$	0	0
$545$	−30.7000	−1.31505
$546$	0	0
$547$	−0.236068	−0.0100935	−0.00504677	−	0.999987i	$-0.501606\pi$
$547$	−0.236068	−0.0100935	−0.00504677	+	0.999987i	$0.501606\pi$
$548$	0	0
$549$	0	0
$550$	1.52786	0.0651483
$551$	−46.0501	−1.96180
$552$	0	0
$553$	5.76393	0.245107
$554$	20.3091	0.862850
$555$	0	0
$556$	0	0
$557$	−21.3894	−0.906299	−0.453150	−	0.891434i	$-0.649700\pi$
$557$	−21.3894	−0.906299	−0.453150	+	0.891434i	$0.649700\pi$
$558$	0	0
$559$	8.47214	0.358333
$560$	9.15298	0.386784
$561$	0	0
$562$	−17.7082	−0.746975
$563$	18.1784	0.766130	0.383065	−	0.923721i	$-0.374869\pi$
$563$	18.1784	0.766130	0.383065	+	0.923721i	$0.374869\pi$
$564$	0	0
$565$	−12.9443	−0.544570
$566$	16.5579	0.695980
$567$	0	0
$568$	34.8328	1.46155
$569$	−7.19859	−0.301780	−0.150890	−	0.988551i	$-0.548214\pi$
$569$	−7.19859	−0.301780	−0.150890	+	0.988551i	$0.548214\pi$
$570$	0	0
$571$	24.7082	1.03401	0.517003	−	0.855984i	$-0.327048\pi$
$571$	24.7082	1.03401	0.517003	+	0.855984i	$0.327048\pi$
$572$	0	0
$573$	0	0
$574$	2.76393	0.115364
$575$	1.95440	0.0815039
$576$	0	0
$577$	−34.2361	−1.42527	−0.712633	−	0.701537i	$-0.752498\pi$
$577$	−34.2361	−1.42527	−0.712633	+	0.701537i	$0.752498\pi$
$578$	−26.7124	−1.11109
$579$	0	0
$580$	0	0
$581$	−11.4412	−0.474662
$582$	0	0
$583$	−34.8328	−1.44263
$584$	37.2796	1.54264
$585$	0	0
$586$	−2.36068	−0.0975188
$587$	11.4899	0.474240	0.237120	−	0.971480i	$-0.423796\pi$
$587$	11.4899	0.474240	0.237120	+	0.971480i	$0.423796\pi$
$588$	0	0
$589$	48.5410	2.00010
$590$	44.4295	1.82913
$591$	0	0
$592$	36.0000	1.47959
$593$	18.9737	0.779155	0.389578	−	0.920994i	$-0.372621\pi$
$593$	18.9737	0.779155	0.389578	+	0.920994i	$0.372621\pi$
$594$	0	0
$595$	−13.7082	−0.561982
$596$	0	0
$597$	0	0
$598$	−11.7082	−0.478784
$599$	−33.5285	−1.36994	−0.684968	−	0.728573i	$-0.740184\pi$
$599$	−33.5285	−1.36994	−0.684968	+	0.728573i	$0.740184\pi$
$600$	0	0
$601$	−35.4164	−1.44467	−0.722333	−	0.691546i	$-0.756930\pi$
$601$	−35.4164	−1.44467	−0.722333	+	0.691546i	$0.756930\pi$
$602$	−11.9814	−0.488326
$603$	0	0
$604$	0	0
$605$	−22.7549	−0.925120
$606$	0	0
$607$	32.1246	1.30390	0.651949	−	0.758263i	$-0.273952\pi$
$607$	32.1246	1.30390	0.651949	+	0.758263i	$0.273952\pi$
$608$	0	0
$609$	0	0
$610$	8.76393	0.354841
$611$	−13.0618	−0.528423
$612$	0	0
$613$	5.29180	0.213734	0.106867	−	0.994273i	$-0.465918\pi$
$613$	5.29180	0.213734	0.106867	+	0.994273i	$0.465918\pi$
$614$	27.4589	1.10815
$615$	0	0
$616$	−12.9443	−0.521540
$617$	22.5486	0.907773	0.453886	−	0.891060i	$-0.350037\pi$
$617$	22.5486	0.907773	0.453886	+	0.891060i	$0.350037\pi$
$618$	0	0
$619$	−21.5836	−0.867518	−0.433759	−	0.901029i	$-0.642813\pi$
$619$	−21.5836	−0.867518	−0.433759	+	0.901029i	$0.642813\pi$
$620$	0	0
$621$	0	0
$622$	−18.8328	−0.755127
$623$	4.57649	0.183353
$624$	0	0
$625$	−26.1246	−1.04498
$626$	37.7711	1.50964
$627$	0	0
$628$	0	0
$629$	−53.9163	−2.14979
$630$	0	0
$631$	−37.6525	−1.49892	−0.749461	−	0.662049i	$-0.769687\pi$
$631$	−37.6525	−1.49892	−0.749461	+	0.662049i	$0.769687\pi$
$632$	16.3029	0.648493
$633$	0	0
$634$	32.4721	1.28963
$635$	10.9010	0.432595
$636$	0	0
$637$	−6.00000	−0.237729
$638$	44.4295	1.75898
$639$	0	0
$640$	25.8885	1.02333
$641$	5.45052	0.215283	0.107641	−	0.994190i	$-0.465670\pi$
$641$	5.45052	0.215283	0.107641	+	0.994190i	$0.465670\pi$
$642$	0	0
$643$	−33.8885	−1.33643	−0.668217	−	0.743967i	$-0.732942\pi$
$643$	−33.8885	−1.33643	−0.668217	+	0.743967i	$0.732942\pi$
$644$	0	0
$645$	0	0
$646$	56.8328	2.23606
$647$	−27.0764	−1.06448	−0.532241	−	0.846593i	$-0.678650\pi$
$647$	−27.0764	−1.06448	−0.532241	+	0.846593i	$0.678650\pi$
$648$	0	0
$649$	−62.8328	−2.46640
$650$	−0.333851	−0.0130947
$651$	0	0
$652$	0	0
$653$	30.4937	1.19331	0.596655	−	0.802498i	$-0.296496\pi$
$653$	30.4937	1.19331	0.596655	+	0.802498i	$0.296496\pi$
$654$	0	0
$655$	35.8885	1.40228
$656$	7.81758	0.305225
$657$	0	0
$658$	18.4721	0.720119
$659$	20.4667	0.797269	0.398635	−	0.917110i	$-0.369484\pi$
$659$	20.4667	0.797269	0.398635	+	0.917110i	$0.369484\pi$
$660$	0	0
$661$	−45.8328	−1.78269	−0.891345	−	0.453326i	$-0.850237\pi$
$661$	−45.8328	−1.78269	−0.891345	+	0.453326i	$0.850237\pi$
$662$	−5.57804	−0.216797
$663$	0	0
$664$	−32.3607	−1.25584
$665$	15.3500	0.595248
$666$	0	0
$667$	56.8328	2.20058
$668$	0	0
$669$	0	0
$670$	−29.1246	−1.12518
$671$	−12.3941	−0.478468
$672$	0	0
$673$	6.52786	0.251631	0.125815	−	0.992054i	$-0.459845\pi$
$673$	6.52786	0.251631	0.125815	+	0.992054i	$0.459845\pi$
$674$	10.9799	0.422928
$675$	0	0
$676$	0	0
$677$	−17.3531	−0.666935	−0.333467	−	0.942762i	$-0.608219\pi$
$677$	−17.3531	−0.666935	−0.333467	+	0.942762i	$0.608219\pi$
$678$	0	0
$679$	2.70820	0.103931
$680$	−38.7727	−1.48686
$681$	0	0
$682$	−46.8328	−1.79332
$683$	15.0649	0.576441	0.288221	−	0.957564i	$-0.406936\pi$
$683$	15.0649	0.576441	0.288221	+	0.957564i	$0.406936\pi$
$684$	0	0
$685$	13.7082	0.523764
$686$	18.3848	0.701934
$687$	0	0
$688$	−33.8885	−1.29199
$689$	7.61125	0.289966
$690$	0	0
$691$	−14.3607	−0.546306	−0.273153	−	0.961971i	$-0.588067\pi$
$691$	−14.3607	−0.546306	−0.273153	+	0.961971i	$0.588067\pi$
$692$	0	0
$693$	0	0
$694$	−30.2918	−1.14986
$695$	−6.86474	−0.260394
$696$	0	0
$697$	−11.7082	−0.443480
$698$	14.7310	0.557578
$699$	0	0
$700$	0	0
$701$	−30.9064	−1.16732	−0.583659	−	0.811999i	$-0.698379\pi$
$701$	−30.9064	−1.16732	−0.583659	+	0.811999i	$0.698379\pi$
$702$	0	0
$703$	60.3738	2.27704
$704$	−36.6119	−1.37986
$705$	0	0
$706$	34.2918	1.29059
$707$	−13.5231	−0.508590
$708$	0	0
$709$	−46.1246	−1.73225	−0.866123	−	0.499831i	$-0.833396\pi$
$709$	−46.1246	−1.73225	−0.866123	+	0.499831i	$0.833396\pi$
$710$	39.8530	1.49566
$711$	0	0
$712$	12.9443	0.485107
$713$	−59.9070	−2.24354
$714$	0	0
$715$	10.4721	0.391636
$716$	0	0
$717$	0	0
$718$	15.2361	0.568605
$719$	30.1111	1.12296	0.561478	−	0.827492i	$-0.310233\pi$
$719$	30.1111	1.12296	0.561478	+	0.827492i	$0.310233\pi$
$720$	0	0
$721$	5.47214	0.203793
$722$	−36.7696	−1.36842
$723$	0	0
$724$	0	0
$725$	1.62054	0.0601855
$726$	0	0
$727$	−15.4164	−0.571763	−0.285881	−	0.958265i	$-0.592286\pi$
$727$	−15.4164	−0.571763	−0.285881	+	0.958265i	$0.592286\pi$
$728$	2.82843	0.104828
$729$	0	0
$730$	42.6525	1.57864
$731$	50.7541	1.87721
$732$	0	0
$733$	47.1246	1.74059	0.870294	−	0.492533i	$-0.163929\pi$
$733$	47.1246	1.74059	0.870294	+	0.492533i	$0.163929\pi$
$734$	46.2564	1.70735
$735$	0	0
$736$	0	0
$737$	41.1884	1.51719
$738$	0	0
$739$	−43.4164	−1.59710	−0.798549	−	0.601930i	$-0.794399\pi$
$739$	−43.4164	−1.59710	−0.798549	+	0.601930i	$0.794399\pi$
$740$	0	0
$741$	0	0
$742$	−10.7639	−0.395156
$743$	39.1552	1.43647	0.718233	−	0.695803i	$-0.244951\pi$
$743$	39.1552	1.43647	0.718233	+	0.695803i	$0.244951\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	−6.65841	−0.243782
$747$	0	0
$748$	0	0
$749$	−2.28825	−0.0836107
$750$	0	0
$751$	−9.94427	−0.362872	−0.181436	−	0.983403i	$-0.558074\pi$
$751$	−9.94427	−0.362872	−0.181436	+	0.983403i	$0.558074\pi$
$752$	52.2471	1.90526
$753$	0	0
$754$	−9.70820	−0.353552
$755$	−4.44897	−0.161915
$756$	0	0
$757$	6.52786	0.237259	0.118630	−	0.992939i	$-0.462150\pi$
$757$	6.52786	0.237259	0.118630	+	0.992939i	$0.462150\pi$
$758$	7.07107	0.256833
$759$	0	0
$760$	43.4164	1.57488
$761$	−46.8453	−1.69814	−0.849070	−	0.528280i	$-0.822837\pi$
$761$	−46.8453	−1.69814	−0.849070	+	0.528280i	$0.822837\pi$
$762$	0	0
$763$	13.4164	0.485707
$764$	0	0
$765$	0	0
$766$	−13.5967	−0.491270
$767$	13.7295	0.495743
$768$	0	0
$769$	5.76393	0.207853	0.103926	−	0.994585i	$-0.466859\pi$
$769$	5.76393	0.207853	0.103926	+	0.994585i	$0.466859\pi$
$770$	−14.8098	−0.533709
$771$	0	0
$772$	0	0
$773$	−10.8222	−0.389249	−0.194624	−	0.980878i	$-0.562349\pi$
$773$	−10.8222	−0.389249	−0.194624	+	0.980878i	$0.562349\pi$
$774$	0	0
$775$	−1.70820	−0.0613605
$776$	7.65996	0.274976
$777$	0	0
$778$	32.5410	1.16665
$779$	13.1105	0.469732
$780$	0	0
$781$	−56.3607	−2.01674
$782$	−70.1404	−2.50822
$783$	0	0
$784$	24.0000	0.857143
$785$	30.7000	1.09573
$786$	0	0
$787$	−25.1803	−0.897582	−0.448791	−	0.893637i	$-0.648145\pi$
$787$	−25.1803	−0.897582	−0.448791	+	0.893637i	$0.648145\pi$
$788$	0	0
$789$	0	0
$790$	18.6525	0.663625
$791$	5.65685	0.201135
$792$	0	0
$793$	2.70820	0.0961711
$794$	41.6011	1.47637
$795$	0	0
$796$	0	0
$797$	−14.5548	−0.515557	−0.257779	−	0.966204i	$-0.582991\pi$
$797$	−14.5548	−0.515557	−0.257779	+	0.966204i	$0.582991\pi$
$798$	0	0
$799$	−78.2492	−2.76826
$800$	0	0
$801$	0	0
$802$	20.0000	0.706225
$803$	−60.3197	−2.12864
$804$	0	0
$805$	−18.9443	−0.667698
$806$	10.2333	0.360454
$807$	0	0
$808$	−38.2492	−1.34560
$809$	−26.2511	−0.922938	−0.461469	−	0.887156i	$-0.652677\pi$
$809$	−26.2511	−0.922938	−0.461469	+	0.887156i	$0.652677\pi$
$810$	0	0
$811$	−15.4164	−0.541343	−0.270672	−	0.962672i	$-0.587246\pi$
$811$	−15.4164	−0.541343	−0.270672	+	0.962672i	$0.587246\pi$
$812$	0	0
$813$	0	0
$814$	−58.2492	−2.04163
$815$	1.62054	0.0567652
$816$	0	0
$817$	−56.8328	−1.98833
$818$	−10.7248	−0.374984
$819$	0	0
$820$	0	0
$821$	43.0640	1.50294	0.751472	−	0.659765i	$-0.229344\pi$
$821$	43.0640	1.50294	0.751472	+	0.659765i	$0.229344\pi$
$822$	0	0
$823$	44.9574	1.56712	0.783559	−	0.621318i	$-0.213402\pi$
$823$	44.9574	1.56712	0.783559	+	0.621318i	$0.213402\pi$
$824$	15.4775	0.539186
$825$	0	0
$826$	−19.4164	−0.675583
$827$	−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	$0.472327\pi$
$830$	−37.0246	−1.28514
$831$	0	0
$832$	8.00000	0.277350
$833$	−35.9442	−1.24539
$834$	0	0
$835$	15.7082	0.543605
$836$	0	0
$837$	0	0
$838$	−8.36068	−0.288815
$839$	40.1382	1.38572	0.692862	−	0.721071i	$-0.256350\pi$
$839$	40.1382	1.38572	0.692862	+	0.721071i	$0.256350\pi$
$840$	0	0
$841$	18.1246	0.624987
$842$	−8.66151	−0.298495
$843$	0	0
$844$	0	0
$845$	27.4589	0.944617
$846$	0	0
$847$	9.94427	0.341689
$848$	−30.4450	−1.04549
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−74.5106	−2.55419
$852$	0	0
$853$	−2.70820	−0.0927271	−0.0463636	−	0.998925i	$-0.514763\pi$
$853$	−2.70820	−0.0927271	−0.0463636	+	0.998925i	$0.514763\pi$
$854$	−3.82998	−0.131059
$855$	0	0
$856$	−6.47214	−0.221213
$857$	12.1390	0.414661	0.207331	−	0.978271i	$-0.433522\pi$
$857$	12.1390	0.414661	0.207331	+	0.978271i	$0.433522\pi$
$858$	0	0
$859$	−4.05573	−0.138380	−0.0691898	−	0.997604i	$-0.522041\pi$
$859$	−4.05573	−0.138380	−0.0691898	+	0.997604i	$0.522041\pi$
$860$	0	0
$861$	0	0
$862$	3.23607	0.110221
$863$	13.8570	0.471698	0.235849	−	0.971790i	$-0.424213\pi$
$863$	13.8570	0.471698	0.235849	+	0.971790i	$0.424213\pi$
$864$	0	0
$865$	35.5967	1.21033
$866$	2.00310	0.0680682
$867$	0	0
$868$	0	0
$869$	−26.3786	−0.894832
$870$	0	0
$871$	−9.00000	−0.304953
$872$	37.9473	1.28506
$873$	0	0
$874$	78.5410	2.65669
$875$	10.9010	0.368523
$876$	0	0
$877$	−40.0132	−1.35115	−0.675574	−	0.737292i	$-0.736104\pi$
$877$	−40.0132	−1.35115	−0.675574	+	0.737292i	$0.736104\pi$
$878$	0.255039	0.00860715
$879$	0	0
$880$	−41.8885	−1.41206
$881$	−50.5778	−1.70401	−0.852005	−	0.523533i	$-0.824614\pi$
$881$	−50.5778	−1.70401	−0.852005	+	0.523533i	$0.824614\pi$
$882$	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
$886$	45.1246	1.51599
$887$	−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
$889$	−4.76393	−0.159777
$890$	14.8098	0.496427
$891$	0	0
$892$	0	0
$893$	87.6210	2.93213
$894$	0	0
$895$	−2.18034	−0.0728807
$896$	−11.3137	−0.377964
$897$	0	0
$898$	2.58359	0.0862156
$899$	−49.6737	−1.65671
$900$	0	0
$901$	45.5967	1.51905
$902$	−12.6491	−0.421169
$903$	0	0
$904$	16.0000	0.532152
$905$	−26.6637	−0.886332
$906$	0	0
$907$	−15.5836	−0.517445	−0.258722	−	0.965952i	$-0.583301\pi$
$907$	−15.5836	−0.517445	−0.258722	+	0.965952i	$0.583301\pi$
$908$	0	0
$909$	0	0
$910$	3.23607	0.107275
$911$	49.0547	1.62526	0.812628	−	0.582783i	$-0.198036\pi$
$911$	49.0547	1.62526	0.812628	+	0.582783i	$0.198036\pi$
$912$	0	0
$913$	52.3607	1.73289
$914$	−10.0757	−0.333275
$915$	0	0
$916$	0	0
$917$	−15.6839	−0.517927
$918$	0	0
$919$	16.9443	0.558940	0.279470	−	0.960154i	$-0.409841\pi$
$919$	16.9443	0.558940	0.279470	+	0.960154i	$0.409841\pi$
$920$	−53.5825	−1.76656
$921$	0	0
$922$	28.5410	0.939948
$923$	12.3153	0.405362
$924$	0	0
$925$	−2.12461	−0.0698568
$926$	39.9318	1.31224
$927$	0	0
$928$	0	0
$929$	3.00465	0.0985795	0.0492898	−	0.998785i	$-0.484304\pi$
$929$	3.00465	0.0985795	0.0492898	+	0.998785i	$0.484304\pi$
$930$	0	0
$931$	40.2492	1.31912
$932$	0	0
$933$	0	0
$934$	34.8328	1.13976
$935$	62.7355	2.05167
$936$	0	0
$937$	−25.5410	−0.834389	−0.417194	−	0.908817i	$-0.636987\pi$
$937$	−25.5410	−0.834389	−0.417194	+	0.908817i	$0.636987\pi$
$938$	12.7279	0.415581
$939$	0	0
$940$	0	0
$941$	26.4574	0.862486	0.431243	−	0.902236i	$-0.358075\pi$
$941$	26.4574	0.862486	0.431243	+	0.902236i	$0.358075\pi$
$942$	0	0
$943$	−16.1803	−0.526904
$944$	−54.9179	−1.78743
$945$	0	0
$946$	54.8328	1.78277
$947$	20.2117	0.656790	0.328395	−	0.944540i	$-0.393492\pi$
$947$	20.2117	0.656790	0.328395	+	0.944540i	$0.393492\pi$
$948$	0	0
$949$	13.1803	0.427852
$950$	2.23954	0.0726602
$951$	0	0
$952$	16.9443	0.549167
$953$	4.65530	0.150800	0.0754000	−	0.997153i	$-0.475977\pi$
$953$	4.65530	0.150800	0.0754000	+	0.997153i	$0.475977\pi$
$954$	0	0
$955$	−5.70820	−0.184713
$956$	0	0
$957$	0	0
$958$	8.29180	0.267896
$959$	−5.99070	−0.193450
$960$	0	0
$961$	21.3607	0.689054
$962$	12.7279	0.410365
$963$	0	0
$964$	0	0
$965$	−20.5942	−0.662951
$966$	0	0
$967$	−16.4164	−0.527916	−0.263958	−	0.964534i	$-0.585028\pi$
$967$	−16.4164	−0.527916	−0.263958	+	0.964534i	$0.585028\pi$
$968$	28.1266	0.904025
$969$	0	0
$970$	8.76393	0.281393
$971$	43.3979	1.39270	0.696352	−	0.717701i	$-0.254805\pi$
$971$	43.3979	1.39270	0.696352	+	0.717701i	$0.254805\pi$
$972$	0	0
$973$	3.00000	0.0961756
$974$	−7.48373	−0.239794
$975$	0	0
$976$	−10.8328	−0.346750
$977$	7.19859	0.230303	0.115152	−	0.993348i	$-0.463265\pi$
$977$	7.19859	0.230303	0.115152	+	0.993348i	$0.463265\pi$
$978$	0	0
$979$	−20.9443	−0.669382
$980$	0	0
$981$	0	0
$982$	4.65248	0.148466
$983$	18.1784	0.579802	0.289901	−	0.957057i	$-0.406378\pi$
$983$	18.1784	0.579802	0.289901	+	0.957057i	$0.406378\pi$
$984$	0	0
$985$	−45.3050	−1.44354
$986$	−58.1590	−1.85216
$987$	0	0
$988$	0	0
$989$	70.1404	2.23033
$990$	0	0
$991$	10.7082	0.340157	0.170079	−	0.985430i	$-0.445598\pi$
$991$	10.7082	0.340157	0.170079	+	0.985430i	$0.445598\pi$
$992$	0	0
$993$	0	0
$994$	−17.4164	−0.552415
$995$	4.03631	0.127960
$996$	0	0
$997$	34.9443	1.10670	0.553348	−	0.832950i	$-0.313350\pi$
$997$	34.9443	1.10670	0.553348	+	0.832950i	$0.313350\pi$
$998$	47.6706	1.50899
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.k.1.1	✓	4
3.2	odd	2	inner	6021.2.a.k.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.k.1.1	✓	4	1.1	even	1	trivial
6021.2.a.k.1.4	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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -2.28825 q^{5} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} -2.28825 q^{5} +1.00000 q^{7} +2.82843 q^{8} +3.23607 q^{10} -4.57649 q^{11} +1.00000 q^{13} -1.41421 q^{14} -4.00000 q^{16} +5.99070 q^{17} -6.70820 q^{19} +6.47214 q^{22} +8.27895 q^{23} +0.236068 q^{25} -1.41421 q^{26} +6.86474 q^{29} -7.23607 q^{31} -8.47214 q^{34} -2.28825 q^{35} -9.00000 q^{37} +9.48683 q^{38} -6.47214 q^{40} -1.95440 q^{41} +8.47214 q^{43} -11.7082 q^{46} -13.0618 q^{47} -6.00000 q^{49} -0.333851 q^{50} +7.61125 q^{53} +10.4721 q^{55} +2.82843 q^{56} -9.70820 q^{58} +13.7295 q^{59} +2.70820 q^{61} +10.2333 q^{62} +8.00000 q^{64} -2.28825 q^{65} -9.00000 q^{67} +3.23607 q^{70} +12.3153 q^{71} +13.1803 q^{73} +12.7279 q^{74} -4.57649 q^{77} +5.76393 q^{79} +9.15298 q^{80} +2.76393 q^{82} -11.4412 q^{83} -13.7082 q^{85} -11.9814 q^{86} -12.9443 q^{88} +4.57649 q^{89} +1.00000 q^{91} +18.4721 q^{94} +15.3500 q^{95} +2.70820 q^{97} +8.48528 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 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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