LMFDB - Embedded newform 6012.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.495342$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.03761 q^{5} -3.68648 q^{7} +O(q^{10})$	$q-4.03761 q^{5} -3.68648 q^{7} +3.69397 q^{11} -5.90947 q^{13} +1.76032 q^{17} +3.61578 q^{19} +8.06467 q^{23} +11.3023 q^{25} -3.96378 q^{29} +0.214905 q^{31} +14.8846 q^{35} -7.87923 q^{37} +1.82086 q^{41} -0.119717 q^{43} +10.0699 q^{47} +6.59017 q^{49} -10.8096 q^{53} -14.9148 q^{55} +6.28572 q^{59} +10.8163 q^{61} +23.8601 q^{65} -0.658388 q^{67} +8.18447 q^{71} -6.26279 q^{73} -13.6178 q^{77} +9.58289 q^{79} -6.61862 q^{83} -7.10747 q^{85} +13.3218 q^{89} +21.7852 q^{91} -14.5991 q^{95} -10.7262 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) $ 7 * q + 2 * q^5 - 12 * q^7	$ 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) $ 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.03761	−1.80567	−0.902837	−	0.429982i	$-0.858520\pi$
$5$	−4.03761	−1.80567	−0.902837	+	0.429982i	$0.858520\pi$
$6$	0	0
$7$	−3.68648	−1.39336	−0.696680	−	0.717382i	$-0.745340\pi$
$7$	−3.68648	−1.39336	−0.696680	+	0.717382i	$0.745340\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.69397	1.11378	0.556888	−	0.830588i	$-0.311995\pi$
$11$	3.69397	1.11378	0.556888	+	0.830588i	$0.311995\pi$
$12$	0	0
$13$	−5.90947	−1.63899	−0.819496	−	0.573085i	$-0.805746\pi$
$13$	−5.90947	−1.63899	−0.819496	+	0.573085i	$0.805746\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.76032	0.426939	0.213470	−	0.976950i	$-0.431524\pi$
$17$	1.76032	0.426939	0.213470	+	0.976950i	$0.431524\pi$
$18$	0	0
$19$	3.61578	0.829517	0.414759	−	0.909931i	$-0.363866\pi$
$19$	3.61578	0.829517	0.414759	+	0.909931i	$0.363866\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.06467	1.68160	0.840800	−	0.541346i	$-0.182085\pi$
$23$	8.06467	1.68160	0.840800	+	0.541346i	$0.182085\pi$
$24$	0	0
$25$	11.3023	2.26046
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.96378	−0.736056	−0.368028	−	0.929815i	$-0.619967\pi$
$29$	−3.96378	−0.736056	−0.368028	+	0.929815i	$0.619967\pi$
$30$	0	0
$31$	0.214905	0.0385980	0.0192990	−	0.999814i	$-0.493857\pi$
$31$	0.214905	0.0385980	0.0192990	+	0.999814i	$0.493857\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.8846	2.51596
$36$	0	0
$37$	−7.87923	−1.29534	−0.647669	−	0.761922i	$-0.724256\pi$
$37$	−7.87923	−1.29534	−0.647669	+	0.761922i	$0.724256\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.82086	0.284370	0.142185	−	0.989840i	$-0.454587\pi$
$41$	1.82086	0.284370	0.142185	+	0.989840i	$0.454587\pi$
$42$	0	0
$43$	−0.119717	−0.0182567	−0.00912835	−	0.999958i	$-0.502906\pi$
$43$	−0.119717	−0.0182567	−0.00912835	+	0.999958i	$0.502906\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0699	1.46885	0.734425	−	0.678690i	$-0.237452\pi$
$47$	10.0699	1.46885	0.734425	+	0.678690i	$0.237452\pi$
$48$	0	0
$49$	6.59017	0.941453
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.8096	−1.48482	−0.742408	−	0.669948i	$-0.766316\pi$
$53$	−10.8096	−1.48482	−0.742408	+	0.669948i	$0.766316\pi$
$54$	0	0
$55$	−14.9148	−2.01112
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.28572	0.818331	0.409166	−	0.912460i	$-0.365820\pi$
$59$	6.28572	0.818331	0.409166	+	0.912460i	$0.365820\pi$
$60$	0	0
$61$	10.8163	1.38489	0.692443	−	0.721473i	$-0.256535\pi$
$61$	10.8163	1.38489	0.692443	+	0.721473i	$0.256535\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23.8601	2.95949
$66$	0	0
$67$	−0.658388	−0.0804348	−0.0402174	−	0.999191i	$-0.512805\pi$
$67$	−0.658388	−0.0804348	−0.0402174	+	0.999191i	$0.512805\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.18447	0.971318	0.485659	−	0.874148i	$-0.338580\pi$
$71$	8.18447	0.971318	0.485659	+	0.874148i	$0.338580\pi$
$72$	0	0
$73$	−6.26279	−0.733004	−0.366502	−	0.930417i	$-0.619445\pi$
$73$	−6.26279	−0.733004	−0.366502	+	0.930417i	$0.619445\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.6178	−1.55189
$78$	0	0
$79$	9.58289	1.07816	0.539080	−	0.842255i	$-0.318772\pi$
$79$	9.58289	1.07816	0.539080	+	0.842255i	$0.318772\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.61862	−0.726488	−0.363244	−	0.931694i	$-0.618331\pi$
$83$	−6.61862	−0.726488	−0.363244	+	0.931694i	$0.618331\pi$
$84$	0	0
$85$	−7.10747	−0.770913
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.3218	1.41211	0.706054	−	0.708158i	$-0.250474\pi$
$89$	13.3218	1.41211	0.706054	+	0.708158i	$0.250474\pi$
$90$	0	0
$91$	21.7852	2.28371
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−14.5991	−1.49784
$96$	0	0
$97$	−10.7262	−1.08908	−0.544539	−	0.838736i	$-0.683295\pi$
$97$	−10.7262	−1.08908	−0.544539	+	0.838736i	$0.683295\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.888544	0.0884134	0.0442067	−	0.999022i	$-0.485924\pi$
$101$	0.888544	0.0884134	0.0442067	+	0.999022i	$0.485924\pi$
$102$	0	0
$103$	4.38251	0.431821	0.215911	−	0.976413i	$-0.430728\pi$
$103$	4.38251	0.431821	0.215911	+	0.976413i	$0.430728\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.55846	−0.827378	−0.413689	−	0.910418i	$-0.635760\pi$
$107$	−8.55846	−0.827378	−0.413689	+	0.910418i	$0.635760\pi$
$108$	0	0
$109$	−18.4638	−1.76851	−0.884254	−	0.467007i	$-0.845332\pi$
$109$	−18.4638	−1.76851	−0.884254	+	0.467007i	$0.845332\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.1961	−1.14731	−0.573655	−	0.819097i	$-0.694475\pi$
$113$	−12.1961	−1.14731	−0.573655	+	0.819097i	$0.694475\pi$
$114$	0	0
$115$	−32.5620	−3.03642
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.48938	−0.594880
$120$	0	0
$121$	2.64545	0.240495
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−25.4463	−2.27598
$126$	0	0
$127$	10.8001	0.958353	0.479177	−	0.877719i	$-0.340935\pi$
$127$	10.8001	0.958353	0.479177	+	0.877719i	$0.340935\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.317204	0.0277142	0.0138571	−	0.999904i	$-0.495589\pi$
$131$	0.317204	0.0277142	0.0138571	+	0.999904i	$0.495589\pi$
$132$	0	0
$133$	−13.3295	−1.15582
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.68972	−0.486105	−0.243053	−	0.970013i	$-0.578149\pi$
$137$	−5.68972	−0.486105	−0.243053	+	0.970013i	$0.578149\pi$
$138$	0	0
$139$	21.6380	1.83531	0.917654	−	0.397379i	$-0.130080\pi$
$139$	21.6380	1.83531	0.917654	+	0.397379i	$0.130080\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−21.8294	−1.82547
$144$	0	0
$145$	16.0042	1.32908
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.31055	0.189288	0.0946439	−	0.995511i	$-0.469829\pi$
$149$	2.31055	0.189288	0.0946439	+	0.995511i	$0.469829\pi$
$150$	0	0
$151$	−4.68639	−0.381373	−0.190686	−	0.981651i	$-0.561071\pi$
$151$	−4.68639	−0.381373	−0.190686	+	0.981651i	$0.561071\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.867702	−0.0696955
$156$	0	0
$157$	−6.50267	−0.518970	−0.259485	−	0.965747i	$-0.583553\pi$
$157$	−6.50267	−0.518970	−0.259485	+	0.965747i	$0.583553\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−29.7303	−2.34307
$162$	0	0
$163$	−24.4612	−1.91595	−0.957976	−	0.286848i	$-0.907393\pi$
$163$	−24.4612	−1.91595	−0.957976	+	0.286848i	$0.907393\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	21.9218	1.68629
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.4338	1.47752	0.738762	−	0.673966i	$-0.235411\pi$
$173$	19.4338	1.47752	0.738762	+	0.673966i	$0.235411\pi$
$174$	0	0
$175$	−41.6658	−3.14964
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.26629	0.692595	0.346298	−	0.938125i	$-0.387439\pi$
$179$	9.26629	0.692595	0.346298	+	0.938125i	$0.387439\pi$
$180$	0	0
$181$	−12.8107	−0.952213	−0.476107	−	0.879388i	$-0.657952\pi$
$181$	−12.8107	−0.952213	−0.476107	+	0.879388i	$0.657952\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	31.8133	2.33896
$186$	0	0
$187$	6.50256	0.475514
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−23.3718	−1.69113	−0.845564	−	0.533875i	$-0.820735\pi$
$191$	−23.3718	−1.69113	−0.845564	+	0.533875i	$0.820735\pi$
$192$	0	0
$193$	−10.5001	−0.755811	−0.377906	−	0.925844i	$-0.623356\pi$
$193$	−10.5001	−0.755811	−0.377906	+	0.925844i	$0.623356\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.5171	−0.963055	−0.481527	−	0.876431i	$-0.659918\pi$
$197$	−13.5171	−0.963055	−0.481527	+	0.876431i	$0.659918\pi$
$198$	0	0
$199$	−8.05525	−0.571022	−0.285511	−	0.958376i	$-0.592163\pi$
$199$	−8.05525	−0.571022	−0.285511	+	0.958376i	$0.592163\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.6124	1.02559
$204$	0	0
$205$	−7.35192	−0.513480
$206$	0	0
$207$	0	0
$208$	0	0
$209$	13.3566	0.923896
$210$	0	0
$211$	−8.37523	−0.576575	−0.288287	−	0.957544i	$-0.593086\pi$
$211$	−8.37523	−0.576575	−0.288287	+	0.957544i	$0.593086\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.483371	0.0329657
$216$	0	0
$217$	−0.792243	−0.0537810
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.4025	−0.699750
$222$	0	0
$223$	2.99656	0.200664	0.100332	−	0.994954i	$-0.468009\pi$
$223$	2.99656	0.200664	0.100332	+	0.994954i	$0.468009\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.31007	−0.0869527	−0.0434764	−	0.999054i	$-0.513843\pi$
$227$	−1.31007	−0.0869527	−0.0434764	+	0.999054i	$0.513843\pi$
$228$	0	0
$229$	−8.89808	−0.588002	−0.294001	−	0.955805i	$-0.594987\pi$
$229$	−8.89808	−0.588002	−0.294001	+	0.955805i	$0.594987\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.7038	−1.61840	−0.809200	−	0.587534i	$-0.800099\pi$
$233$	−24.7038	−1.61840	−0.809200	+	0.587534i	$0.800099\pi$
$234$	0	0
$235$	−40.6584	−2.65226
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.4984	1.00251	0.501253	−	0.865301i	$-0.332873\pi$
$239$	15.4984	1.00251	0.501253	+	0.865301i	$0.332873\pi$
$240$	0	0
$241$	6.29706	0.405630	0.202815	−	0.979217i	$-0.434991\pi$
$241$	6.29706	0.405630	0.202815	+	0.979217i	$0.434991\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−26.6085	−1.69996
$246$	0	0
$247$	−21.3673	−1.35957
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.9524	−1.19626	−0.598131	−	0.801398i	$-0.704090\pi$
$251$	−18.9524	−1.19626	−0.598131	+	0.801398i	$0.704090\pi$
$252$	0	0
$253$	29.7907	1.87292
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.1783	1.44582	0.722911	−	0.690941i	$-0.242804\pi$
$257$	23.1783	1.44582	0.722911	+	0.690941i	$0.242804\pi$
$258$	0	0
$259$	29.0467	1.80487
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.12223	−0.562501	−0.281250	−	0.959634i	$-0.590749\pi$
$263$	−9.12223	−0.562501	−0.281250	+	0.959634i	$0.590749\pi$
$264$	0	0
$265$	43.6451	2.68109
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.39138	−0.450661	−0.225330	−	0.974282i	$-0.572346\pi$
$269$	−7.39138	−0.450661	−0.225330	+	0.974282i	$0.572346\pi$
$270$	0	0
$271$	0.200638	0.0121879	0.00609393	−	0.999981i	$-0.498060\pi$
$271$	0.200638	0.0121879	0.00609393	+	0.999981i	$0.498060\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	41.7504	2.51765
$276$	0	0
$277$	−10.4787	−0.629603	−0.314801	−	0.949158i	$-0.601938\pi$
$277$	−10.4787	−0.629603	−0.314801	+	0.949158i	$0.601938\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−30.6913	−1.83089	−0.915444	−	0.402446i	$-0.868160\pi$
$281$	−30.6913	−1.83089	−0.915444	+	0.402446i	$0.868160\pi$
$282$	0	0
$283$	4.77320	0.283737	0.141869	−	0.989885i	$-0.454689\pi$
$283$	4.77320	0.283737	0.141869	+	0.989885i	$0.454689\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.71256	−0.396230
$288$	0	0
$289$	−13.9013	−0.817723
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.968987	0.0566088	0.0283044	−	0.999599i	$-0.490989\pi$
$293$	0.968987	0.0566088	0.0283044	+	0.999599i	$0.490989\pi$
$294$	0	0
$295$	−25.3793	−1.47764
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−47.6579	−2.75613
$300$	0	0
$301$	0.441336	0.0254382
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−43.6720	−2.50065
$306$	0	0
$307$	−4.02919	−0.229958	−0.114979	−	0.993368i	$-0.536680\pi$
$307$	−4.02919	−0.229958	−0.114979	+	0.993368i	$0.536680\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.01226	0.0574002	0.0287001	−	0.999588i	$-0.490863\pi$
$311$	1.01226	0.0574002	0.0287001	+	0.999588i	$0.490863\pi$
$312$	0	0
$313$	28.7177	1.62322	0.811611	−	0.584198i	$-0.198591\pi$
$313$	28.7177	1.62322	0.811611	+	0.584198i	$0.198591\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.4356	1.37244	0.686221	−	0.727393i	$-0.259268\pi$
$317$	24.4356	1.37244	0.686221	+	0.727393i	$0.259268\pi$
$318$	0	0
$319$	−14.6421	−0.819801
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.36492	0.354153
$324$	0	0
$325$	−66.7906	−3.70488
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−37.1226	−2.04664
$330$	0	0
$331$	30.2119	1.66060	0.830298	−	0.557320i	$-0.188170\pi$
$331$	30.2119	1.66060	0.830298	+	0.557320i	$0.188170\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.65831	0.145239
$336$	0	0
$337$	−11.9047	−0.648490	−0.324245	−	0.945973i	$-0.605110\pi$
$337$	−11.9047	−0.648490	−0.324245	+	0.945973i	$0.605110\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.793853	0.0429895
$342$	0	0
$343$	1.51083	0.0815772
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.7663	1.49057	0.745286	−	0.666745i	$-0.232313\pi$
$347$	27.7663	1.49057	0.745286	+	0.666745i	$0.232313\pi$
$348$	0	0
$349$	−6.68622	−0.357905	−0.178953	−	0.983858i	$-0.557271\pi$
$349$	−6.68622	−0.357905	−0.178953	+	0.983858i	$0.557271\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.8411	1.42861	0.714303	−	0.699836i	$-0.246744\pi$
$353$	26.8411	1.42861	0.714303	+	0.699836i	$0.246744\pi$
$354$	0	0
$355$	−33.0457	−1.75388
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.4973	0.554024	0.277012	−	0.960866i	$-0.410656\pi$
$359$	10.4973	0.554024	0.277012	+	0.960866i	$0.410656\pi$
$360$	0	0
$361$	−5.92612	−0.311901
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.2867	1.32357
$366$	0	0
$367$	−6.87562	−0.358905	−0.179452	−	0.983767i	$-0.557433\pi$
$367$	−6.87562	−0.358905	−0.179452	+	0.983767i	$0.557433\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	39.8495	2.06888
$372$	0	0
$373$	−17.9814	−0.931043	−0.465522	−	0.885037i	$-0.654133\pi$
$373$	−17.9814	−0.931043	−0.465522	+	0.885037i	$0.654133\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.4238	1.20639
$378$	0	0
$379$	−25.7458	−1.32247	−0.661237	−	0.750177i	$-0.729968\pi$
$379$	−25.7458	−1.32247	−0.661237	+	0.750177i	$0.729968\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.0322	−0.768108	−0.384054	−	0.923311i	$-0.625472\pi$
$383$	−15.0322	−0.768108	−0.384054	+	0.923311i	$0.625472\pi$
$384$	0	0
$385$	54.9833	2.80221
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.31272	0.117260	0.0586299	−	0.998280i	$-0.481327\pi$
$389$	2.31272	0.117260	0.0586299	+	0.998280i	$0.481327\pi$
$390$	0	0
$391$	14.1964	0.717941
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−38.6920	−1.94681
$396$	0	0
$397$	−30.3094	−1.52118	−0.760592	−	0.649230i	$-0.775091\pi$
$397$	−30.3094	−1.52118	−0.760592	+	0.649230i	$0.775091\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.1285	−0.755480	−0.377740	−	0.925912i	$-0.623299\pi$
$401$	−15.1285	−0.755480	−0.377740	+	0.925912i	$0.623299\pi$
$402$	0	0
$403$	−1.26997	−0.0632618
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−29.1057	−1.44271
$408$	0	0
$409$	−26.7870	−1.32453	−0.662265	−	0.749269i	$-0.730405\pi$
$409$	−26.7870	−1.32453	−0.662265	+	0.749269i	$0.730405\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−23.1722	−1.14023
$414$	0	0
$415$	26.7234	1.31180
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.6645	0.520993	0.260497	−	0.965475i	$-0.416114\pi$
$419$	10.6645	0.520993	0.260497	+	0.965475i	$0.416114\pi$
$420$	0	0
$421$	8.58075	0.418200	0.209100	−	0.977894i	$-0.432947\pi$
$421$	8.58075	0.418200	0.209100	+	0.977894i	$0.432947\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	19.8956	0.965080
$426$	0	0
$427$	−39.8741	−1.92964
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.481733	−0.0232043	−0.0116021	−	0.999933i	$-0.503693\pi$
$431$	−0.481733	−0.0232043	−0.0116021	+	0.999933i	$0.503693\pi$
$432$	0	0
$433$	23.4186	1.12543	0.562714	−	0.826652i	$-0.309757\pi$
$433$	23.4186	1.12543	0.562714	+	0.826652i	$0.309757\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.1601	1.39492
$438$	0	0
$439$	24.4298	1.16597	0.582985	−	0.812483i	$-0.301885\pi$
$439$	24.4298	1.16597	0.582985	+	0.812483i	$0.301885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.16664	0.435520	0.217760	−	0.976002i	$-0.430125\pi$
$443$	9.16664	0.435520	0.217760	+	0.976002i	$0.430125\pi$
$444$	0	0
$445$	−53.7882	−2.54981
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.859883	−0.0405804	−0.0202902	−	0.999794i	$-0.506459\pi$
$449$	−0.859883	−0.0405804	−0.0202902	+	0.999794i	$0.506459\pi$
$450$	0	0
$451$	6.72620	0.316724
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−87.9600	−4.12363
$456$	0	0
$457$	−29.1204	−1.36220	−0.681098	−	0.732193i	$-0.738497\pi$
$457$	−29.1204	−1.36220	−0.681098	+	0.732193i	$0.738497\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.40312	0.158499	0.0792495	−	0.996855i	$-0.474748\pi$
$461$	3.40312	0.158499	0.0792495	+	0.996855i	$0.474748\pi$
$462$	0	0
$463$	−11.0246	−0.512355	−0.256177	−	0.966630i	$-0.582463\pi$
$463$	−11.0246	−0.512355	−0.256177	+	0.966630i	$0.582463\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.0123	−0.509587	−0.254794	−	0.966995i	$-0.582008\pi$
$467$	−11.0123	−0.509587	−0.254794	+	0.966995i	$0.582008\pi$
$468$	0	0
$469$	2.42714	0.112075
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.442232	−0.0203339
$474$	0	0
$475$	40.8667	1.87509
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.19730	0.191780	0.0958899	−	0.995392i	$-0.469430\pi$
$479$	4.19730	0.191780	0.0958899	+	0.995392i	$0.469430\pi$
$480$	0	0
$481$	46.5621	2.12305
$482$	0	0
$483$	0	0
$484$	0	0
$485$	43.3081	1.96652
$486$	0	0
$487$	−9.18672	−0.416290	−0.208145	−	0.978098i	$-0.566743\pi$
$487$	−9.18672	−0.416290	−0.208145	+	0.978098i	$0.566743\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.7799	−1.34395	−0.671974	−	0.740575i	$-0.734553\pi$
$491$	−29.7799	−1.34395	−0.671974	+	0.740575i	$0.734553\pi$
$492$	0	0
$493$	−6.97750	−0.314251
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−30.1719	−1.35340
$498$	0	0
$499$	−24.9905	−1.11873	−0.559364	−	0.828922i	$-0.688954\pi$
$499$	−24.9905	−1.11873	−0.559364	+	0.828922i	$0.688954\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	35.1489	1.56721	0.783606	−	0.621258i	$-0.213378\pi$
$503$	35.1489	1.56721	0.783606	+	0.621258i	$0.213378\pi$
$504$	0	0
$505$	−3.58760	−0.159646
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.04340	0.400842	0.200421	−	0.979710i	$-0.435769\pi$
$509$	9.04340	0.400842	0.200421	+	0.979710i	$0.435769\pi$
$510$	0	0
$511$	23.0877	1.02134
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17.6949	−0.779729
$516$	0	0
$517$	37.1980	1.63597
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15.2988	−0.670251	−0.335126	−	0.942173i	$-0.608779\pi$
$521$	−15.2988	−0.670251	−0.335126	+	0.942173i	$0.608779\pi$
$522$	0	0
$523$	28.6061	1.25086	0.625429	−	0.780281i	$-0.284924\pi$
$523$	28.6061	1.25086	0.625429	+	0.780281i	$0.284924\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.378300	0.0164790
$528$	0	0
$529$	42.0389	1.82778
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.7603	−0.466080
$534$	0	0
$535$	34.5558	1.49398
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.3439	1.04857
$540$	0	0
$541$	13.6049	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$541$	13.6049	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	74.5495	3.19335
$546$	0	0
$547$	23.3364	0.997791	0.498896	−	0.866662i	$-0.333739\pi$
$547$	23.3364	0.997791	0.498896	+	0.866662i	$0.333739\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−14.3322	−0.610571
$552$	0	0
$553$	−35.3272	−1.50226
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.4516	1.92584	0.962922	−	0.269780i	$-0.0869509\pi$
$557$	45.4516	1.92584	0.962922	+	0.269780i	$0.0869509\pi$
$558$	0	0
$559$	0.707465	0.0299226
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.6927	1.41998	0.709990	−	0.704212i	$-0.248700\pi$
$563$	33.6927	1.41998	0.709990	+	0.704212i	$0.248700\pi$
$564$	0	0
$565$	49.2430	2.07167
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.12618	−0.298745	−0.149373	−	0.988781i	$-0.547725\pi$
$569$	−7.12618	−0.298745	−0.149373	+	0.988781i	$0.547725\pi$
$570$	0	0
$571$	8.76003	0.366596	0.183298	−	0.983057i	$-0.441323\pi$
$571$	8.76003	0.366596	0.183298	+	0.983057i	$0.441323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	91.1494	3.80119
$576$	0	0
$577$	29.7482	1.23843	0.619217	−	0.785220i	$-0.287450\pi$
$577$	29.7482	1.23843	0.619217	+	0.785220i	$0.287450\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.3994	1.01226
$582$	0	0
$583$	−39.9305	−1.65375
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.17314	0.0896949	0.0448475	−	0.998994i	$-0.485720\pi$
$587$	2.17314	0.0896949	0.0448475	+	0.998994i	$0.485720\pi$
$588$	0	0
$589$	0.777049	0.0320177
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.9329	−1.22920	−0.614599	−	0.788839i	$-0.710682\pi$
$593$	−29.9329	−1.22920	−0.614599	+	0.788839i	$0.710682\pi$
$594$	0	0
$595$	26.2016	1.07416
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.9309	1.67239	0.836195	−	0.548432i	$-0.184775\pi$
$599$	40.9309	1.67239	0.836195	+	0.548432i	$0.184775\pi$
$600$	0	0
$601$	−32.3803	−1.32082	−0.660411	−	0.750904i	$-0.729618\pi$
$601$	−32.3803	−1.32082	−0.660411	+	0.750904i	$0.729618\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.6813	−0.434256
$606$	0	0
$607$	−32.1400	−1.30452	−0.652261	−	0.757994i	$-0.726180\pi$
$607$	−32.1400	−1.30452	−0.652261	+	0.757994i	$0.726180\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−59.5079	−2.40743
$612$	0	0
$613$	37.5043	1.51479	0.757393	−	0.652959i	$-0.226473\pi$
$613$	37.5043	1.51479	0.757393	+	0.652959i	$0.226473\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	47.1330	1.89750	0.948752	−	0.316020i	$-0.102347\pi$
$617$	47.1330	1.89750	0.948752	+	0.316020i	$0.102347\pi$
$618$	0	0
$619$	31.0926	1.24972	0.624858	−	0.780739i	$-0.285157\pi$
$619$	31.0926	1.24972	0.624858	+	0.780739i	$0.285157\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−49.1106	−1.96757
$624$	0	0
$625$	46.2306	1.84922
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.8699	−0.553030
$630$	0	0
$631$	−35.8489	−1.42712	−0.713561	−	0.700593i	$-0.752919\pi$
$631$	−35.8489	−1.42712	−0.713561	+	0.700593i	$0.752919\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−43.6066	−1.73047
$636$	0	0
$637$	−38.9444	−1.54303
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.29887	0.0513024	0.0256512	−	0.999671i	$-0.491834\pi$
$641$	1.29887	0.0513024	0.0256512	+	0.999671i	$0.491834\pi$
$642$	0	0
$643$	−43.7887	−1.72686	−0.863428	−	0.504471i	$-0.831687\pi$
$643$	−43.7887	−1.72686	−0.863428	+	0.504471i	$0.831687\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.64330	−0.261175	−0.130588	−	0.991437i	$-0.541686\pi$
$647$	−6.64330	−0.261175	−0.130588	+	0.991437i	$0.541686\pi$
$648$	0	0
$649$	23.2193	0.911437
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.990949	0.0387788	0.0193894	−	0.999812i	$-0.493828\pi$
$653$	0.990949	0.0387788	0.0193894	+	0.999812i	$0.493828\pi$
$654$	0	0
$655$	−1.28075	−0.0500429
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.84897	−0.227844	−0.113922	−	0.993490i	$-0.536341\pi$
$659$	−5.84897	−0.227844	−0.113922	+	0.993490i	$0.536341\pi$
$660$	0	0
$661$	−19.4344	−0.755911	−0.377955	−	0.925824i	$-0.623373\pi$
$661$	−19.4344	−0.755911	−0.377955	+	0.925824i	$0.623373\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	53.8194	2.08703
$666$	0	0
$667$	−31.9666	−1.23775
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.9551	1.54245
$672$	0	0
$673$	48.2342	1.85929	0.929645	−	0.368456i	$-0.120113\pi$
$673$	48.2342	1.85929	0.929645	+	0.368456i	$0.120113\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.72178	−0.335205	−0.167603	−	0.985855i	$-0.553603\pi$
$677$	−8.72178	−0.335205	−0.167603	+	0.985855i	$0.553603\pi$
$678$	0	0
$679$	39.5419	1.51748
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17.4341	−0.667096	−0.333548	−	0.942733i	$-0.608246\pi$
$683$	−17.4341	−0.667096	−0.333548	+	0.942733i	$0.608246\pi$
$684$	0	0
$685$	22.9729	0.877748
$686$	0	0
$687$	0	0
$688$	0	0
$689$	63.8791	2.43360
$690$	0	0
$691$	−0.391097	−0.0148780	−0.00743902	−	0.999972i	$-0.502368\pi$
$691$	−0.391097	−0.0148780	−0.00743902	+	0.999972i	$0.502368\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−87.3657	−3.31397
$696$	0	0
$697$	3.20528	0.121409
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11.0894	0.418840	0.209420	−	0.977826i	$-0.432842\pi$
$701$	11.0894	0.418840	0.209420	+	0.977826i	$0.432842\pi$
$702$	0	0
$703$	−28.4896	−1.07450
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.27560	−0.123192
$708$	0	0
$709$	−42.6543	−1.60192	−0.800958	−	0.598721i	$-0.795676\pi$
$709$	−42.6543	−1.60192	−0.800958	+	0.598721i	$0.795676\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.73314	0.0649064
$714$	0	0
$715$	88.1387	3.29620
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.6096	−0.843197	−0.421598	−	0.906783i	$-0.638531\pi$
$719$	−22.6096	−0.843197	−0.421598	+	0.906783i	$0.638531\pi$
$720$	0	0
$721$	−16.1561	−0.601683
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−44.7999	−1.66383
$726$	0	0
$727$	−9.78043	−0.362736	−0.181368	−	0.983415i	$-0.558053\pi$
$727$	−9.78043	−0.362736	−0.181368	+	0.983415i	$0.558053\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.210740	−0.00779450
$732$	0	0
$733$	−52.9742	−1.95665	−0.978323	−	0.207086i	$-0.933602\pi$
$733$	−52.9742	−1.95665	−0.978323	+	0.207086i	$0.933602\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.43207	−0.0895863
$738$	0	0
$739$	−23.6615	−0.870404	−0.435202	−	0.900333i	$-0.643323\pi$
$739$	−23.6615	−0.870404	−0.435202	+	0.900333i	$0.643323\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.5325	−1.08344	−0.541721	−	0.840558i	$-0.682227\pi$
$743$	−29.5325	−1.08344	−0.541721	+	0.840558i	$0.682227\pi$
$744$	0	0
$745$	−9.32911	−0.341792
$746$	0	0
$747$	0	0
$748$	0	0
$749$	31.5506	1.15284
$750$	0	0
$751$	−34.8584	−1.27200	−0.636001	−	0.771688i	$-0.719413\pi$
$751$	−34.8584	−1.27200	−0.636001	+	0.771688i	$0.719413\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.9218	0.688635
$756$	0	0
$757$	−10.9987	−0.399753	−0.199876	−	0.979821i	$-0.564054\pi$
$757$	−10.9987	−0.399753	−0.199876	+	0.979821i	$0.564054\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−43.7554	−1.58613	−0.793066	−	0.609136i	$-0.791516\pi$
$761$	−43.7554	−1.58613	−0.793066	+	0.609136i	$0.791516\pi$
$762$	0	0
$763$	68.0664	2.46417
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−37.1453	−1.34124
$768$	0	0
$769$	35.6242	1.28464	0.642321	−	0.766436i	$-0.277972\pi$
$769$	35.6242	1.28464	0.642321	+	0.766436i	$0.277972\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−43.4788	−1.56382	−0.781912	−	0.623389i	$-0.785755\pi$
$773$	−43.4788	−1.56382	−0.781912	+	0.623389i	$0.785755\pi$
$774$	0	0
$775$	2.42892	0.0872494
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.58382	0.235890
$780$	0	0
$781$	30.2332	1.08183
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.2553	0.937091
$786$	0	0
$787$	28.3808	1.01166	0.505832	−	0.862632i	$-0.331185\pi$
$787$	28.3808	1.01166	0.505832	+	0.862632i	$0.331185\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	44.9607	1.59862
$792$	0	0
$793$	−63.9186	−2.26982
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.6765	−0.413603	−0.206802	−	0.978383i	$-0.566306\pi$
$797$	−11.6765	−0.413603	−0.206802	+	0.978383i	$0.566306\pi$
$798$	0	0
$799$	17.7262	0.627109
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.1346	−0.816402
$804$	0	0
$805$	120.039	4.23083
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.94305	0.208947	0.104473	−	0.994528i	$-0.466684\pi$
$809$	5.94305	0.208947	0.104473	+	0.994528i	$0.466684\pi$
$810$	0	0
$811$	−14.0429	−0.493114	−0.246557	−	0.969128i	$-0.579299\pi$
$811$	−14.0429	−0.493114	−0.246557	+	0.969128i	$0.579299\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	98.7650	3.45959
$816$	0	0
$817$	−0.432871	−0.0151442
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.65244	0.232172	0.116086	−	0.993239i	$-0.462965\pi$
$821$	6.65244	0.232172	0.116086	+	0.993239i	$0.462965\pi$
$822$	0	0
$823$	13.7336	0.478722	0.239361	−	0.970931i	$-0.423062\pi$
$823$	13.7336	0.478722	0.239361	+	0.970931i	$0.423062\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.7727	1.10485	0.552423	−	0.833564i	$-0.313703\pi$
$827$	31.7727	1.10485	0.552423	+	0.833564i	$0.313703\pi$
$828$	0	0
$829$	−47.2689	−1.64172	−0.820858	−	0.571132i	$-0.806504\pi$
$829$	−47.2689	−1.64172	−0.820858	+	0.571132i	$0.806504\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11.6008	0.401943
$834$	0	0
$835$	4.03761	0.139727
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.20519	−0.248751	−0.124375	−	0.992235i	$-0.539693\pi$
$839$	−7.20519	−0.248751	−0.124375	+	0.992235i	$0.539693\pi$
$840$	0	0
$841$	−13.2884	−0.458222
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−88.5118	−3.04490
$846$	0	0
$847$	−9.75240	−0.335096
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−63.5434	−2.17824
$852$	0	0
$853$	−2.52173	−0.0863425	−0.0431712	−	0.999068i	$-0.513746\pi$
$853$	−2.52173	−0.0863425	−0.0431712	+	0.999068i	$0.513746\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.80643	−0.334981	−0.167491	−	0.985874i	$-0.553566\pi$
$857$	−9.80643	−0.334981	−0.167491	+	0.985874i	$0.553566\pi$
$858$	0	0
$859$	−8.67890	−0.296120	−0.148060	−	0.988978i	$-0.547303\pi$
$859$	−8.67890	−0.296120	−0.148060	+	0.988978i	$0.547303\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.6430	0.736737	0.368368	−	0.929680i	$-0.379917\pi$
$863$	21.6430	0.736737	0.368368	+	0.929680i	$0.379917\pi$
$864$	0	0
$865$	−78.4661	−2.66793
$866$	0	0
$867$	0	0
$868$	0	0
$869$	35.3989	1.20083
$870$	0	0
$871$	3.89072	0.131832
$872$	0	0
$873$	0	0
$874$	0	0
$875$	93.8073	3.17127
$876$	0	0
$877$	45.7477	1.54479	0.772394	−	0.635143i	$-0.219059\pi$
$877$	45.7477	1.54479	0.772394	+	0.635143i	$0.219059\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34.5611	−1.16439	−0.582196	−	0.813049i	$-0.697806\pi$
$881$	−34.5611	−1.16439	−0.582196	+	0.813049i	$0.697806\pi$
$882$	0	0
$883$	33.4560	1.12589	0.562943	−	0.826496i	$-0.309669\pi$
$883$	33.4560	1.12589	0.562943	+	0.826496i	$0.309669\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.1106	−1.44751	−0.723756	−	0.690056i	$-0.757586\pi$
$887$	−43.1106	−1.44751	−0.723756	+	0.690056i	$0.757586\pi$
$888$	0	0
$889$	−39.8144	−1.33533
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.4106	1.21844
$894$	0	0
$895$	−37.4137	−1.25060
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.851835	−0.0284103
$900$	0	0
$901$	−19.0283	−0.633926
$902$	0	0
$903$	0	0
$904$	0	0
$905$	51.7247	1.71939
$906$	0	0
$907$	−42.2683	−1.40350	−0.701748	−	0.712425i	$-0.747597\pi$
$907$	−42.2683	−1.40350	−0.701748	+	0.712425i	$0.747597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.52503	−0.282447	−0.141223	−	0.989978i	$-0.545104\pi$
$911$	−8.52503	−0.282447	−0.141223	+	0.989978i	$0.545104\pi$
$912$	0	0
$913$	−24.4490	−0.809144
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.16937	−0.0386159
$918$	0	0
$919$	−15.2874	−0.504286	−0.252143	−	0.967690i	$-0.581135\pi$
$919$	−15.2874	−0.504286	−0.252143	+	0.967690i	$0.581135\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−48.3659	−1.59198
$924$	0	0
$925$	−89.0535	−2.92806
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.91452	0.128431	0.0642156	−	0.997936i	$-0.479545\pi$
$929$	3.91452	0.128431	0.0642156	+	0.997936i	$0.479545\pi$
$930$	0	0
$931$	23.8286	0.780951
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−26.2548	−0.858624
$936$	0	0
$937$	−26.6886	−0.871880	−0.435940	−	0.899976i	$-0.643584\pi$
$937$	−26.6886	−0.871880	−0.435940	+	0.899976i	$0.643584\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−45.0378	−1.46819	−0.734096	−	0.679046i	$-0.762394\pi$
$941$	−45.0378	−1.46819	−0.734096	+	0.679046i	$0.762394\pi$
$942$	0	0
$943$	14.6846	0.478197
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−50.6146	−1.64475	−0.822377	−	0.568942i	$-0.807353\pi$
$947$	−50.6146	−1.64475	−0.822377	+	0.568942i	$0.807353\pi$
$948$	0	0
$949$	37.0098	1.20139
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−59.8383	−1.93835	−0.969177	−	0.246367i	$-0.920763\pi$
$953$	−59.8383	−1.93835	−0.969177	+	0.246367i	$0.920763\pi$
$954$	0	0
$955$	94.3664	3.05363
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.9751	0.677320
$960$	0	0
$961$	−30.9538	−0.998510
$962$	0	0
$963$	0	0
$964$	0	0
$965$	42.3952	1.36475
$966$	0	0
$967$	−47.7477	−1.53546	−0.767732	−	0.640771i	$-0.778615\pi$
$967$	−47.7477	−1.53546	−0.767732	+	0.640771i	$0.778615\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.6740	0.856010	0.428005	−	0.903776i	$-0.359217\pi$
$971$	26.6740	0.856010	0.428005	+	0.903776i	$0.359217\pi$
$972$	0	0
$973$	−79.7681	−2.55725
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.0222	−0.928503	−0.464252	−	0.885703i	$-0.653677\pi$
$977$	−29.0222	−0.928503	−0.464252	+	0.885703i	$0.653677\pi$
$978$	0	0
$979$	49.2104	1.57277
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−30.5413	−0.974116	−0.487058	−	0.873370i	$-0.661930\pi$
$983$	−30.5413	−0.974116	−0.487058	+	0.873370i	$0.661930\pi$
$984$	0	0
$985$	54.5769	1.73896
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.965479	−0.0307005
$990$	0	0
$991$	−32.5262	−1.03323	−0.516614	−	0.856219i	$-0.672808\pi$
$991$	−32.5262	−1.03323	−0.516614	+	0.856219i	$0.672808\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	32.5240	1.03108
$996$	0	0
$997$	−17.3503	−0.549489	−0.274744	−	0.961517i	$-0.588593\pi$
$997$	−17.3503	−0.549489	−0.274744	+	0.961517i	$0.588593\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.g.1.1		7
3.2	odd	2		668.2.a.c.1.3	✓	7
12.11	even	2		2672.2.a.k.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.3	✓	7	3.2	odd	2
2672.2.a.k.1.5		7	12.11	even	2
6012.2.a.g.1.1		7	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-4.03761 q^{5} -3.68648 q^{7} +O(q^{10})\)	\(q-4.03761 q^{5} -3.68648 q^{7} +3.69397 q^{11} -5.90947 q^{13} +1.76032 q^{17} +3.61578 q^{19} +8.06467 q^{23} +11.3023 q^{25} -3.96378 q^{29} +0.214905 q^{31} +14.8846 q^{35} -7.87923 q^{37} +1.82086 q^{41} -0.119717 q^{43} +10.0699 q^{47} +6.59017 q^{49} -10.8096 q^{53} -14.9148 q^{55} +6.28572 q^{59} +10.8163 q^{61} +23.8601 q^{65} -0.658388 q^{67} +8.18447 q^{71} -6.26279 q^{73} -13.6178 q^{77} +9.58289 q^{79} -6.61862 q^{83} -7.10747 q^{85} +13.3218 q^{89} +21.7852 q^{91} -14.5991 q^{95} -10.7262 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) \) 7 * q + 2 * q^5 - 12 * q^7	\( 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) \) 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * q^97

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